LMFDB - Embedded newform 450.6.f.a.107.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(107,450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("450.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	450.107
Dual form			450.6.f.a.143.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(-33.0000 - 33.0000i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})$	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(-33.0000 - 33.0000i) q^{7} +(-45.2548 - 45.2548i) q^{8} +258.801i q^{11} +(-213.000 + 213.000i) q^{13} -186.676 q^{14} -256.000 q^{16} +(462.448 - 462.448i) q^{17} +722.000i q^{19} +(732.000 + 732.000i) q^{22} +(810.344 + 810.344i) q^{23} +1204.91i q^{26} +(-528.000 + 528.000i) q^{28} +2537.10 q^{29} +2764.00 q^{31} +(-724.077 + 724.077i) q^{32} -2616.00i q^{34} +(4467.00 + 4467.00i) q^{37} +(2042.12 + 2042.12i) q^{38} +7632.51i q^{41} +(-3240.00 + 3240.00i) q^{43} +4140.82 q^{44} +4584.00 q^{46} +(9927.78 - 9927.78i) q^{47} -14629.0i q^{49} +(3408.00 + 3408.00i) q^{52} +(-7937.98 - 7937.98i) q^{53} +2986.82i q^{56} +(7176.00 - 7176.00i) q^{58} +11450.9 q^{59} +29750.0 q^{61} +(7817.77 - 7817.77i) q^{62} +4096.00i q^{64} +(19866.0 + 19866.0i) q^{67} +(-7399.17 - 7399.17i) q^{68} +71284.8i q^{71} +(45066.0 - 45066.0i) q^{73} +25269.2 q^{74} +11552.0 q^{76} +(8540.44 - 8540.44i) q^{77} +2992.00i q^{79} +(21588.0 + 21588.0i) q^{82} +(-28603.9 - 28603.9i) q^{83} +18328.2i q^{86} +(11712.0 - 11712.0i) q^{88} +11705.4 q^{89} +14058.0 q^{91} +(12965.5 - 12965.5i) q^{92} -56160.0i q^{94} +(111372. + 111372. i) q^{97} +(-41377.1 - 41377.1i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 132 q^{7}+O(q^{10}) $ 4 * q - 132 * q^7	$ 4 q - 132 q^{7} - 852 q^{13} - 1024 q^{16} + 2928 q^{22} - 2112 q^{28} + 11056 q^{31} + 17868 q^{37} - 12960 q^{43} + 18336 q^{46} + 13632 q^{52} + 28704 q^{58} + 119000 q^{61} + 79464 q^{67} + 180264 q^{73} + 46208 q^{76} + 86352 q^{82} + 46848 q^{88} + 56232 q^{91} + 445488 q^{97}+O(q^{100}) $ 4 * q - 132 * q^7 - 852 * q^13 - 1024 * q^16 + 2928 * q^22 - 2112 * q^28 + 11056 * q^31 + 17868 * q^37 - 12960 * q^43 + 18336 * q^46 + 13632 * q^52 + 28704 * q^58 + 119000 * q^61 + 79464 * q^67 + 180264 * q^73 + 46208 * q^76 + 86352 * q^82 + 46848 * q^88 + 56232 * q^91 + 445488 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.82843	−	2.82843i	0.500000	−	0.500000i
$2$	2.82843	−	2.82843i	0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$		−	16.0000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−33.0000	−	33.0000i	−0.254548	−	0.254548i	0.568285	−	0.822832i	$-0.307607\pi$
$7$	−33.0000	−	33.0000i	−0.254548	−	0.254548i	−0.822832	+	0.568285i	$0.807607\pi$
$8$	−45.2548	−	45.2548i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$			258.801i			0.644888i	0.946589	+	0.322444i	$0.104504\pi$
$11$			258.801i			0.644888i	−0.946589	+	0.322444i	$0.895496\pi$
$12$		0			0
$13$	−213.000	+	213.000i	−0.349560	+	0.349560i	−0.859945	−	0.510386i	$-0.829503\pi$
$13$	−213.000	+	213.000i	−0.349560	+	0.349560i	0.510386	+	0.859945i	$0.329503\pi$
$14$	−186.676			−0.254548
$15$		0			0
$16$	−256.000			−0.250000
$17$	462.448	−	462.448i	0.388097	−	0.388097i	−0.485911	−	0.874008i	$-0.661512\pi$
$17$	462.448	−	462.448i	0.388097	−	0.388097i	0.874008	+	0.485911i	$0.161512\pi$
$18$		0			0
$19$			722.000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			722.000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$		0			0
$22$	732.000	+	732.000i	0.322444	+	0.322444i
$23$	810.344	+	810.344i	0.319411	+	0.319411i	0.848541	−	0.529130i	$-0.177482\pi$
$23$	810.344	+	810.344i	0.319411	+	0.319411i	−0.529130	+	0.848541i	$0.677482\pi$
$24$		0			0
$25$		0			0
$26$			1204.91i			0.349560i
$27$		0			0
$28$	−528.000	+	528.000i	−0.127274	+	0.127274i
$29$	2537.10			0.560199			0.280100	−	0.959971i	$-0.409633\pi$
$29$	2537.10			0.560199			0.280100	+	0.959971i	$0.409633\pi$
$30$		0			0
$31$	2764.00			0.516575			0.258288	−	0.966068i	$-0.416842\pi$
$31$	2764.00			0.516575			0.258288	+	0.966068i	$0.416842\pi$
$32$	−724.077	+	724.077i	−0.125000	+	0.125000i
$33$		0			0
$34$		−	2616.00i		−	0.388097i
$35$		0			0
$36$		0			0
$37$	4467.00	+	4467.00i	0.536428	+	0.536428i	0.922478	−	0.386050i	$-0.126161\pi$
$37$	4467.00	+	4467.00i	0.536428	+	0.536428i	−0.386050	+	0.922478i	$0.626161\pi$
$38$	2042.12	+	2042.12i	0.229416	+	0.229416i
$39$		0			0
$40$		0			0
$41$			7632.51i			0.709100i	0.935037	+	0.354550i	$0.115366\pi$
$41$			7632.51i			0.709100i	−0.935037	+	0.354550i	$0.884634\pi$
$42$		0			0
$43$	−3240.00	+	3240.00i	−0.267223	+	0.267223i	−0.827980	−	0.560757i	$-0.810510\pi$
$43$	−3240.00	+	3240.00i	−0.267223	+	0.267223i	0.560757	+	0.827980i	$0.310510\pi$
$44$	4140.82			0.322444
$45$		0			0
$46$	4584.00			0.319411
$47$	9927.78	−	9927.78i	0.655552	−	0.655552i	−0.298772	−	0.954325i	$-0.596577\pi$
$47$	9927.78	−	9927.78i	0.655552	−	0.655552i	0.954325	+	0.298772i	$0.0965770\pi$
$48$		0			0
$49$		−	14629.0i		−	0.870411i
$50$		0			0
$51$		0			0
$52$	3408.00	+	3408.00i	0.174780	+	0.174780i
$53$	−7937.98	−	7937.98i	−0.388169	−	0.388169i	0.485865	−	0.874034i	$-0.338505\pi$
$53$	−7937.98	−	7937.98i	−0.388169	−	0.388169i	−0.874034	+	0.485865i	$0.838505\pi$
$54$		0			0
$55$		0			0
$56$			2986.82i			0.127274i
$57$		0			0
$58$	7176.00	−	7176.00i	0.280100	−	0.280100i
$59$	11450.9			0.428262			0.214131	−	0.976805i	$-0.431308\pi$
$59$	11450.9			0.428262			0.214131	+	0.976805i	$0.431308\pi$
$60$		0			0
$61$	29750.0			1.02368			0.511838	−	0.859082i	$-0.328965\pi$
$61$	29750.0			1.02368			0.511838	+	0.859082i	$0.328965\pi$
$62$	7817.77	−	7817.77i	0.258288	−	0.258288i
$63$		0			0
$64$			4096.00i			0.125000i
$65$		0			0
$66$		0			0
$67$	19866.0	+	19866.0i	0.540659	+	0.540659i	0.923722	−	0.383063i	$-0.125131\pi$
$67$	19866.0	+	19866.0i	0.540659	+	0.540659i	−0.383063	+	0.923722i	$0.625131\pi$
$68$	−7399.17	−	7399.17i	−0.194049	−	0.194049i
$69$		0			0
$70$		0			0
$71$			71284.8i			1.67823i	0.543954	+	0.839115i	$0.316926\pi$
$71$			71284.8i			1.67823i	−0.543954	+	0.839115i	$0.683074\pi$
$72$		0			0
$73$	45066.0	−	45066.0i	0.989787	−	0.989787i	−0.0101611	−	0.999948i	$-0.503234\pi$
$73$	45066.0	−	45066.0i	0.989787	−	0.989787i	0.999948	+	0.0101611i	$0.00323444\pi$
$74$	25269.2			0.536428
$75$		0			0
$76$	11552.0			0.229416
$77$	8540.44	−	8540.44i	0.164155	−	0.164155i
$78$		0			0
$79$			2992.00i			0.0539379i	0.999636	+	0.0269689i	$0.00858552\pi$
$79$			2992.00i			0.0539379i	−0.999636	+	0.0269689i	$0.991414\pi$
$80$		0			0
$81$		0			0
$82$	21588.0	+	21588.0i	0.354550	+	0.354550i
$83$	−28603.9	−	28603.9i	−0.455753	−	0.455753i	0.441505	−	0.897259i	$-0.354445\pi$
$83$	−28603.9	−	28603.9i	−0.455753	−	0.455753i	−0.897259	+	0.441505i	$0.854445\pi$
$84$		0			0
$85$		0			0
$86$			18328.2i			0.267223i
$87$		0			0
$88$	11712.0	−	11712.0i	0.161222	−	0.161222i
$89$	11705.4			0.156644			0.0783219	−	0.996928i	$-0.475044\pi$
$89$	11705.4			0.156644			0.0783219	+	0.996928i	$0.475044\pi$
$90$		0			0
$91$	14058.0			0.177959
$92$	12965.5	−	12965.5i	0.159706	−	0.159706i
$93$		0			0
$94$		−	56160.0i		−	0.655552i
$95$		0			0
$96$		0			0
$97$	111372.	+	111372.i	1.20184	+	1.20184i	0.973606	+	0.228234i	$0.0732950\pi$
$97$	111372.	+	111372.i	1.20184	+	1.20184i	0.228234	+	0.973606i	$0.426705\pi$
$98$	−41377.1	−	41377.1i	−0.435206	−	0.435206i
$99$		0			0
$100$		0			0
$101$			10173.9i			0.0992389i	0.998768	+	0.0496195i	$0.0158009\pi$
$101$			10173.9i			0.0992389i	−0.998768	+	0.0496195i	$0.984199\pi$
$102$		0			0
$103$	109833.	−	109833.i	1.02009	−	1.02009i	0.0202991	−	0.999794i	$-0.493538\pi$
$103$	109833.	−	109833.i	1.02009	−	1.02009i	0.999794	−	0.0202991i	$-0.00646184\pi$
$104$	19278.6			0.174780
$105$		0			0
$106$	−44904.0			−0.388169
$107$	131352.	−	131352.i	1.10912	−	1.10912i	0.115851	−	0.993267i	$-0.463040\pi$
$107$	131352.	−	131352.i	1.10912	−	1.10912i	0.993267	−	0.115851i	$-0.0369597\pi$
$108$		0			0
$109$			23758.0i			0.191533i	0.995404	+	0.0957665i	$0.0305302\pi$
$109$			23758.0i			0.191533i	−0.995404	+	0.0957665i	$0.969470\pi$
$110$		0			0
$111$		0			0
$112$	8448.00	+	8448.00i	0.0636369	+	0.0636369i
$113$	−82336.9	−	82336.9i	−0.606595	−	0.606595i	0.335460	−	0.942054i	$-0.391108\pi$
$113$	−82336.9	−	82336.9i	−0.606595	−	0.606595i	−0.942054	+	0.335460i	$0.891108\pi$
$114$		0			0
$115$		0			0
$116$		−	40593.6i		−	0.280100i
$117$		0			0
$118$	32388.0	−	32388.0i	0.214131	−	0.214131i
$119$	−30521.6			−0.197578
$120$		0			0
$121$	94073.0			0.584119
$122$	84145.7	−	84145.7i	0.511838	−	0.511838i
$123$		0			0
$124$		−	44224.0i		−	0.258288i
$125$		0			0
$126$		0			0
$127$	227979.	+	227979.i	1.25425	+	1.25425i	0.953795	+	0.300460i	$0.0971401\pi$
$127$	227979.	+	227979.i	1.25425	+	1.25425i	0.300460	+	0.953795i	$0.402860\pi$
$128$	11585.2	+	11585.2i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$		−	46563.0i		−	0.237062i	−0.992950	−	0.118531i	$-0.962181\pi$
$131$		−	46563.0i		−	0.237062i	0.992950	−	0.118531i	$-0.0378186\pi$
$132$		0			0
$133$	23826.0	−	23826.0i	0.116794	−	0.116794i
$134$	112379.			0.540659
$135$		0			0
$136$	−41856.0			−0.194049
$137$	150652.	−	150652.i	0.685762	−	0.685762i	−0.275531	−	0.961292i	$-0.588854\pi$
$137$	150652.	−	150652.i	0.685762	−	0.685762i	0.961292	+	0.275531i	$0.0888535\pi$
$138$		0			0
$139$			152756.i			0.670596i	0.942112	+	0.335298i	$0.108837\pi$
$139$			152756.i			0.670596i	−0.942112	+	0.335298i	$0.891163\pi$
$140$		0			0
$141$		0			0
$142$	201624.	+	201624.i	0.839115	+	0.839115i
$143$	−55124.6	−	55124.6i	−0.225427	−	0.225427i
$144$		0			0
$145$		0			0
$146$		−	254932.i		−	0.989787i
$147$		0			0
$148$	71472.0	−	71472.0i	0.268214	−	0.268214i
$149$	−462787.			−1.70772			−0.853858	−	0.520506i	$-0.825743\pi$
$149$	−462787.			−1.70772			−0.853858	+	0.520506i	$0.825743\pi$
$150$		0			0
$151$	−179168.			−0.639467			−0.319733	−	0.947508i	$-0.603593\pi$
$151$	−179168.			−0.639467			−0.319733	+	0.947508i	$0.603593\pi$
$152$	32674.0	−	32674.0i	0.114708	−	0.114708i
$153$		0			0
$154$		−	48312.0i		−	0.164155i
$155$		0			0
$156$		0			0
$157$	371961.	+	371961.i	1.20434	+	1.20434i	0.972834	+	0.231504i	$0.0743645\pi$
$157$	371961.	+	371961.i	1.20434	+	1.20434i	0.231504	+	0.972834i	$0.425635\pi$
$158$	8462.65	+	8462.65i	0.0269689	+	0.0269689i
$159$		0			0
$160$		0			0
$161$		−	53482.7i		−	0.162611i
$162$		0			0
$163$	189294.	−	189294.i	0.558043	−	0.558043i	−0.370707	−	0.928750i	$-0.620885\pi$
$163$	189294.	−	189294.i	0.558043	−	0.558043i	0.928750	+	0.370707i	$0.120885\pi$
$164$	122120.			0.354550
$165$		0			0
$166$	−161808.			−0.455753
$167$	258933.	−	258933.i	0.718448	−	0.718448i	−0.249839	−	0.968287i	$-0.580378\pi$
$167$	258933.	−	258933.i	0.718448	−	0.718448i	0.968287	+	0.249839i	$0.0803777\pi$
$168$		0			0
$169$			280555.i			0.755616i
$170$		0			0
$171$		0			0
$172$	51840.0	+	51840.0i	0.133612	+	0.133612i
$173$	−160164.	−	160164.i	−0.406864	−	0.406864i	0.473779	−	0.880644i	$-0.342889\pi$
$173$	−160164.	−	160164.i	−0.406864	−	0.406864i	−0.880644	+	0.473779i	$0.842889\pi$
$174$		0			0
$175$		0			0
$176$		−	66253.1i		−	0.161222i
$177$		0			0
$178$	33108.0	−	33108.0i	0.0783219	−	0.0783219i
$179$	−484420.			−1.13003			−0.565015	−	0.825081i	$-0.691130\pi$
$179$	−484420.			−1.13003			−0.565015	+	0.825081i	$0.691130\pi$
$180$		0			0
$181$	−266402.			−0.604423			−0.302212	−	0.953241i	$-0.597725\pi$
$181$	−266402.			−0.604423			−0.302212	+	0.953241i	$0.597725\pi$
$182$	39762.0	−	39762.0i	0.0889795	−	0.0889795i
$183$		0			0
$184$		−	73344.0i		−	0.159706i
$185$		0			0
$186$		0			0
$187$	119682.	+	119682.i	0.250279	+	0.250279i
$188$	−158844.	−	158844.i	−0.327776	−	0.327776i
$189$		0			0
$190$		0			0
$191$		−	43283.4i		−	0.0858496i	−0.999078	−	0.0429248i	$-0.986332\pi$
$191$		−	43283.4i		−	0.0858496i	0.999078	−	0.0429248i	$-0.0136676\pi$
$192$		0			0
$193$	160494.	−	160494.i	0.310146	−	0.310146i	−0.534820	−	0.844966i	$-0.679621\pi$
$193$	160494.	−	160494.i	0.310146	−	0.310146i	0.844966	+	0.534820i	$0.179621\pi$
$194$	630015.			1.20184
$195$		0			0
$196$	−234064.			−0.435206
$197$	387345.	−	387345.i	0.711102	−	0.711102i	−0.255664	−	0.966766i	$-0.582294\pi$
$197$	387345.	−	387345.i	0.711102	−	0.711102i	0.966766	+	0.255664i	$0.0822940\pi$
$198$		0			0
$199$			414340.i			0.741693i	0.928694	+	0.370846i	$0.120932\pi$
$199$			414340.i			0.741693i	−0.928694	+	0.370846i	$0.879068\pi$
$200$		0			0
$201$		0			0
$202$	28776.0	+	28776.0i	0.0496195	+	0.0496195i
$203$	−83724.3	−	83724.3i	−0.142597	−	0.142597i
$204$		0			0
$205$		0			0
$206$		−	621309.i		−	1.02009i
$207$		0			0
$208$	54528.0	−	54528.0i	0.0873899	−	0.0873899i
$209$	−186854.			−0.295895
$210$		0			0
$211$	589796.			0.912002			0.456001	−	0.889979i	$-0.349281\pi$
$211$	589796.			0.912002			0.456001	+	0.889979i	$0.349281\pi$
$212$	−127008.	+	127008.i	−0.194084	+	0.194084i
$213$		0			0
$214$		−	743040.i		−	1.10912i
$215$		0			0
$216$		0			0
$217$	−91212.0	−	91212.0i	−0.131493	−	0.131493i
$218$	67197.8	+	67197.8i	0.0957665	+	0.0957665i
$219$		0			0
$220$		0			0
$221$			197003.i			0.271326i
$222$		0			0
$223$	330381.	−	330381.i	0.444890	−	0.444890i	−0.448761	−	0.893652i	$-0.648135\pi$
$223$	330381.	−	330381.i	0.444890	−	0.444890i	0.893652	+	0.448761i	$0.148135\pi$
$224$	47789.1			0.0636369
$225$		0			0
$226$	−465768.			−0.606595
$227$	−549430.	+	549430.i	−0.707698	+	0.707698i	−0.966051	−	0.258353i	$-0.916820\pi$
$227$	−549430.	+	549430.i	−0.707698	+	0.707698i	0.258353	+	0.966051i	$0.416820\pi$
$228$		0			0
$229$			630494.i			0.794497i	0.917711	+	0.397248i	$0.130035\pi$
$229$			630494.i			0.794497i	−0.917711	+	0.397248i	$0.869965\pi$
$230$		0			0
$231$		0			0
$232$	−114816.	−	114816.i	−0.140050	−	0.140050i
$233$	−244516.	−	244516.i	−0.295065	−	0.295065i	0.544012	−	0.839077i	$-0.316904\pi$
$233$	−244516.	−	244516.i	−0.295065	−	0.295065i	−0.839077	+	0.544012i	$0.816904\pi$
$234$		0			0
$235$		0			0
$236$		−	183214.i		−	0.214131i
$237$		0			0
$238$	−86328.0	+	86328.0i	−0.0987892	+	0.0987892i
$239$	−792670.			−0.897630			−0.448815	−	0.893625i	$-0.648154\pi$
$239$	−792670.			−0.897630			−0.448815	+	0.893625i	$0.648154\pi$
$240$		0			0
$241$	356612.			0.395506			0.197753	−	0.980252i	$-0.436636\pi$
$241$	356612.			0.395506			0.197753	+	0.980252i	$0.436636\pi$
$242$	266079.	−	266079.i	0.292060	−	0.292060i
$243$		0			0
$244$		−	476000.i		−	0.511838i
$245$		0			0
$246$		0			0
$247$	−153786.	−	153786.i	−0.160389	−	0.160389i
$248$	−125084.	−	125084.i	−0.129144	−	0.129144i
$249$		0			0
$250$		0			0
$251$			1.15393e6i			1.15610i	0.816003	+	0.578048i	$0.196185\pi$
$251$			1.15393e6i			1.15610i	−0.816003	+	0.578048i	$0.803815\pi$
$252$		0			0
$253$	−209718.	+	209718.i	−0.205984	+	0.205984i
$254$	1.28964e6			1.25425
$255$		0			0
$256$	65536.0			0.0625000
$257$	−859945.	+	859945.i	−0.812153	+	0.812153i	−0.984956	−	0.172803i	$-0.944718\pi$
$257$	−859945.	+	859945.i	−0.812153	+	0.812153i	0.172803	+	0.984956i	$0.444718\pi$
$258$		0			0
$259$		−	294822.i		−	0.273093i
$260$		0			0
$261$		0			0
$262$	−131700.	−	131700.i	−0.118531	−	0.118531i
$263$	679047.	+	679047.i	0.605356	+	0.605356i	0.941729	−	0.336373i	$-0.109200\pi$
$263$	679047.	+	679047.i	0.605356	+	0.605356i	−0.336373	+	0.941729i	$0.609200\pi$
$264$		0			0
$265$		0			0
$266$		−	134780.i		−	0.116794i
$267$		0			0
$268$	317856.	−	317856.i	0.270329	−	0.270329i
$269$	−672535.			−0.566675			−0.283338	−	0.959020i	$-0.591442\pi$
$269$	−672535.			−0.566675			−0.283338	+	0.959020i	$0.591442\pi$
$270$		0			0
$271$	1.57142e6			1.29978			0.649889	−	0.760029i	$-0.274815\pi$
$271$	1.57142e6			1.29978			0.649889	+	0.760029i	$0.274815\pi$
$272$	−118387.	+	118387.i	−0.0970243	+	0.0970243i
$273$		0			0
$274$		−	852216.i		−	0.685762i
$275$		0			0
$276$		0			0
$277$	−1.04804e6	−	1.04804e6i	−0.820689	−	0.820689i	0.165517	−	0.986207i	$-0.447071\pi$
$277$	−1.04804e6	−	1.04804e6i	−0.820689	−	0.820689i	−0.986207	+	0.165517i	$0.947071\pi$
$278$	432059.	+	432059.i	0.335298	+	0.335298i
$279$		0			0
$280$		0			0
$281$			1.74425e6i			1.31778i	0.752240	+	0.658889i	$0.228973\pi$
$281$			1.74425e6i			1.31778i	−0.752240	+	0.658889i	$0.771027\pi$
$282$		0			0
$283$	−1.43080e6	+	1.43080e6i	−1.06197	+	1.06197i	−0.0640246	+	0.997948i	$0.520394\pi$
$283$	−1.43080e6	+	1.43080e6i	−1.06197	+	1.06197i	−0.997948	+	0.0640246i	$0.979606\pi$
$284$	1.14056e6			0.839115
$285$		0			0
$286$	−311832.			−0.225427
$287$	251873.	−	251873.i	0.180500	−	0.180500i
$288$		0			0
$289$			992141.i			0.698761i
$290$		0			0
$291$		0			0
$292$	−721056.	−	721056.i	−0.494894	−	0.494894i
$293$	666018.	+	666018.i	0.453228	+	0.453228i	0.896425	−	0.443196i	$-0.146156\pi$
$293$	666018.	+	666018.i	0.453228	+	0.453228i	−0.443196	+	0.896425i	$0.646156\pi$
$294$		0			0
$295$		0			0
$296$		−	404307.i		−	0.268214i
$297$		0			0
$298$	−1.30896e6	+	1.30896e6i	−0.853858	+	0.853858i
$299$	−345207.			−0.223306
$300$		0			0
$301$	213840.			0.136042
$302$	−506764.	+	506764.i	−0.319733	+	0.319733i
$303$		0			0
$304$		−	184832.i		−	0.114708i
$305$		0			0
$306$		0			0
$307$	−1.08216e6	−	1.08216e6i	−0.655308	−	0.655308i	0.298958	−	0.954266i	$-0.403361\pi$
$307$	−1.08216e6	−	1.08216e6i	−0.655308	−	0.655308i	−0.954266	+	0.298958i	$0.903361\pi$
$308$	−136647.	−	136647.i	−0.0820773	−	0.0820773i
$309$		0			0
$310$		0			0
$311$			1.79768e6i			1.05393i	0.849887	+	0.526966i	$0.176670\pi$
$311$			1.79768e6i			1.05393i	−0.849887	+	0.526966i	$0.823330\pi$
$312$		0			0
$313$	−799266.	+	799266.i	−0.461138	+	0.461138i	−0.899028	−	0.437891i	$-0.855726\pi$
$313$	−799266.	+	799266.i	−0.461138	+	0.461138i	0.437891	+	0.899028i	$0.355726\pi$
$314$	2.10413e6			1.20434
$315$		0			0
$316$	47872.0			0.0269689
$317$	−906831.	+	906831.i	−0.506848	+	0.506848i	−0.913558	−	0.406709i	$-0.866676\pi$
$317$	−906831.	+	906831.i	−0.506848	+	0.506848i	0.406709	+	0.913558i	$0.366676\pi$
$318$		0			0
$319$			656604.i			0.361266i
$320$		0			0
$321$		0			0
$322$	−151272.	−	151272.i	−0.0813053	−	0.0813053i
$323$	333887.	+	333887.i	0.178071	+	0.178071i
$324$		0			0
$325$		0			0
$326$		−	1.07081e6i		−	0.558043i
$327$		0			0
$328$	345408.	−	345408.i	0.177275	−	0.177275i
$329$	−655233.			−0.333738
$330$		0			0
$331$	107546.			0.0539541			0.0269770	−	0.999636i	$-0.491412\pi$
$331$	107546.			0.0539541			0.0269770	+	0.999636i	$0.491412\pi$
$332$	−457662.	+	457662.i	−0.227877	+	0.227877i
$333$		0			0
$334$		−	1.46474e6i		−	0.718448i
$335$		0			0
$336$		0			0
$337$	−1.31587e6	−	1.31587e6i	−0.631156	−	0.631156i	0.317202	−	0.948358i	$-0.397257\pi$
$337$	−1.31587e6	−	1.31587e6i	−0.631156	−	0.631156i	−0.948358	+	0.317202i	$0.897257\pi$
$338$	793529.	+	793529.i	0.377808	+	0.377808i
$339$		0			0
$340$		0			0
$341$			715326.i			0.333133i
$342$		0			0
$343$	−1.03739e6	+	1.03739e6i	−0.476108	+	0.476108i
$344$	293251.			0.133612
$345$		0			0
$346$	−906024.			−0.406864
$347$	−1.78519e6	+	1.78519e6i	−0.795905	+	0.795905i	−0.982447	−	0.186542i	$-0.940272\pi$
$347$	−1.78519e6	+	1.78519e6i	−0.795905	+	0.795905i	0.186542	+	0.982447i	$0.440272\pi$
$348$		0			0
$349$		−	1.47933e6i		−	0.650134i	−0.945691	−	0.325067i	$-0.894613\pi$
$349$		−	1.47933e6i		−	0.650134i	0.945691	−	0.325067i	$-0.105387\pi$
$350$		0			0
$351$		0			0
$352$	−187392.	−	187392.i	−0.0806110	−	0.0806110i
$353$	2.22216e6	+	2.22216e6i	0.949156	+	0.949156i	0.998769	−	0.0496125i	$-0.0157986\pi$
$353$	2.22216e6	+	2.22216e6i	0.949156	+	0.949156i	−0.0496125	+	0.998769i	$0.515799\pi$
$354$		0			0
$355$		0			0
$356$		−	187287.i		−	0.0783219i
$357$		0			0
$358$	−1.37015e6	+	1.37015e6i	−0.565015	+	0.565015i
$359$	−33075.6			−0.0135448			−0.00677239	−	0.999977i	$-0.502156\pi$
$359$	−33075.6			−0.0135448			−0.00677239	+	0.999977i	$0.502156\pi$
$360$		0			0
$361$	1.95482e6			0.789474
$362$	−753499.	+	753499.i	−0.302212	+	0.302212i
$363$		0			0
$364$		−	224928.i		−	0.0889795i
$365$		0			0
$366$		0			0
$367$	−2.24097e6	−	2.24097e6i	−0.868501	−	0.868501i	0.123806	−	0.992306i	$-0.460490\pi$
$367$	−2.24097e6	−	2.24097e6i	−0.868501	−	0.868501i	−0.992306	+	0.123806i	$0.960490\pi$
$368$	−207448.	−	207448.i	−0.0798528	−	0.0798528i
$369$		0			0
$370$		0			0
$371$			523907.i			0.197615i
$372$		0			0
$373$	−1.01738e6	+	1.01738e6i	−0.378625	+	0.378625i	−0.870606	−	0.491981i	$-0.836273\pi$
$373$	−1.01738e6	+	1.01738e6i	−0.378625	+	0.378625i	0.491981	+	0.870606i	$0.336273\pi$
$374$	677024.			0.250279
$375$		0			0
$376$	−898560.			−0.327776
$377$	−540402.	+	540402.i	−0.195823	+	0.195823i
$378$		0			0
$379$			2.12654e6i			0.760459i	0.924892	+	0.380229i	$0.124155\pi$
$379$			2.12654e6i			0.760459i	−0.924892	+	0.380229i	$0.875845\pi$
$380$		0			0
$381$		0			0
$382$	−122424.	−	122424.i	−0.0429248	−	0.0429248i
$383$	−1.07621e6	−	1.07621e6i	−0.374885	−	0.374885i	0.494368	−	0.869253i	$-0.335400\pi$
$383$	−1.07621e6	−	1.07621e6i	−0.374885	−	0.374885i	−0.869253	+	0.494368i	$0.835400\pi$
$384$		0			0
$385$		0			0
$386$		−	907891.i		−	0.310146i
$387$		0			0
$388$	1.78195e6	−	1.78195e6i	0.600920	−	0.600920i
$389$	−4.19716e6			−1.40631			−0.703156	−	0.711036i	$-0.748226\pi$
$389$	−4.19716e6			−1.40631			−0.703156	+	0.711036i	$0.748226\pi$
$390$		0			0
$391$	749484.			0.247925
$392$	−662033.	+	662033.i	−0.217603	+	0.217603i
$393$		0			0
$394$		−	2.19115e6i		−	0.711102i
$395$		0			0
$396$		0			0
$397$	−2.02632e6	−	2.02632e6i	−0.645257	−	0.645257i	0.306586	−	0.951843i	$-0.400813\pi$
$397$	−2.02632e6	−	2.02632e6i	−0.645257	−	0.645257i	−0.951843	+	0.306586i	$0.900813\pi$
$398$	1.17193e6	+	1.17193e6i	0.370846	+	0.370846i
$399$		0			0
$400$		0			0
$401$		−	4.02761e6i		−	1.25080i	−0.780306	−	0.625398i	$-0.784937\pi$
$401$		−	4.02761e6i		−	1.25080i	0.780306	−	0.625398i	$-0.215063\pi$
$402$		0			0
$403$	−588732.	+	588732.i	−0.180574	+	0.180574i
$404$	162782.			0.0496195
$405$		0			0
$406$	−473616.			−0.142597
$407$	−1.15606e6	+	1.15606e6i	−0.345936	+	0.345936i
$408$		0			0
$409$			5.65537e6i			1.67168i	0.548973	+	0.835840i	$0.315019\pi$
$409$			5.65537e6i			1.67168i	−0.548973	+	0.835840i	$0.684981\pi$
$410$		0			0
$411$		0			0
$412$	−1.75733e6	−	1.75733e6i	−0.510047	−	0.510047i
$413$	−377879.	−	377879.i	−0.109013	−	0.109013i
$414$		0			0
$415$		0			0
$416$		−	308457.i		−	0.0873899i
$417$		0			0
$418$	−528504.	+	528504.i	−0.147947	+	0.147947i
$419$	−43567.7			−0.0121235			−0.00606177	−	0.999982i	$-0.501930\pi$
$419$	−43567.7			−0.0121235			−0.00606177	+	0.999982i	$0.501930\pi$
$420$		0			0
$421$	−3.65783e6			−1.00582			−0.502908	−	0.864340i	$-0.667736\pi$
$421$	−3.65783e6			−1.00582			−0.502908	+	0.864340i	$0.667736\pi$
$422$	1.66820e6	−	1.66820e6i	0.456001	−	0.456001i
$423$		0			0
$424$			718464.i			0.194084i
$425$		0			0
$426$		0			0
$427$	−981750.	−	981750.i	−0.260574	−	0.260574i
$428$	−2.10163e6	−	2.10163e6i	−0.554559	−	0.554559i
$429$		0			0
$430$		0			0
$431$			249959.i			0.0648151i	0.999475	+	0.0324076i	$0.0103175\pi$
$431$			249959.i			0.0648151i	−0.999475	+	0.0324076i	$0.989683\pi$
$432$		0			0
$433$	1.11368e6	−	1.11368e6i	0.285456	−	0.285456i	−0.549824	−	0.835281i	$-0.685305\pi$
$433$	1.11368e6	−	1.11368e6i	0.285456	−	0.285456i	0.835281	+	0.549824i	$0.185305\pi$
$434$	−515973.			−0.131493
$435$		0			0
$436$	380128.			0.0957665
$437$	−585069.	+	585069.i	−0.146556	+	0.146556i
$438$		0			0
$439$		−	2.89281e6i		−	0.716405i	−0.933644	−	0.358203i	$-0.883390\pi$
$439$		−	2.89281e6i		−	0.716405i	0.933644	−	0.358203i	$-0.116610\pi$
$440$		0			0
$441$		0			0
$442$	557208.	+	557208.i	0.135663	+	0.135663i
$443$	−739934.	−	739934.i	−0.179136	−	0.179136i	0.611843	−	0.790979i	$-0.290428\pi$
$443$	−739934.	−	739934.i	−0.179136	−	0.179136i	−0.790979	+	0.611843i	$0.790428\pi$
$444$		0			0
$445$		0			0
$446$		−	1.86892e6i		−	0.444890i
$447$		0			0
$448$	135168.	−	135168.i	0.0318184	−	0.0318184i
$449$	−1.73408e6			−0.405932			−0.202966	−	0.979186i	$-0.565058\pi$
$449$	−1.73408e6			−0.405932			−0.202966	+	0.979186i	$0.565058\pi$
$450$		0			0
$451$	−1.97530e6			−0.457290
$452$	−1.31739e6	+	1.31739e6i	−0.303297	+	0.303297i
$453$		0			0
$454$			3.10805e6i			0.707698i
$455$		0			0
$456$		0			0
$457$	2.99052e6	+	2.99052e6i	0.669817	+	0.669817i	0.957674	−	0.287856i	$-0.0929426\pi$
$457$	2.99052e6	+	2.99052e6i	0.669817	+	0.669817i	−0.287856	+	0.957674i	$0.592943\pi$
$458$	1.78331e6	+	1.78331e6i	0.397248	+	0.397248i
$459$		0			0
$460$		0			0
$461$		−	3.77512e6i		−	0.827329i	−0.910429	−	0.413665i	$-0.864249\pi$
$461$		−	3.77512e6i		−	0.827329i	0.910429	−	0.413665i	$-0.135751\pi$
$462$		0			0
$463$	1.77997e6	−	1.77997e6i	0.385887	−	0.385887i	−0.487331	−	0.873218i	$-0.662029\pi$
$463$	1.77997e6	−	1.77997e6i	0.385887	−	0.385887i	0.873218	+	0.487331i	$0.162029\pi$
$464$	−649497.			−0.140050
$465$		0			0
$466$	−1.38319e6			−0.295065
$467$	1.10978e6	−	1.10978e6i	0.235475	−	0.235475i	−0.579498	−	0.814973i	$-0.696751\pi$
$467$	1.10978e6	−	1.10978e6i	0.235475	−	0.235475i	0.814973	+	0.579498i	$0.196751\pi$
$468$		0			0
$469$		−	1.31116e6i		−	0.275247i
$470$		0			0
$471$		0			0
$472$	−518208.	−	518208.i	−0.107065	−	0.107065i
$473$	−838516.	−	838516.i	−0.172329	−	0.172329i
$474$		0			0
$475$		0			0
$476$			488345.i			0.0987892i
$477$		0			0
$478$	−2.24201e6	+	2.24201e6i	−0.448815	+	0.448815i
$479$	521344.			0.103821			0.0519106	−	0.998652i	$-0.483469\pi$
$479$	521344.			0.103821			0.0519106	+	0.998652i	$0.483469\pi$
$480$		0			0
$481$	−1.90294e6			−0.375027
$482$	1.00865e6	−	1.00865e6i	0.197753	−	0.197753i
$483$		0			0
$484$		−	1.50517e6i		−	0.292060i
$485$		0			0
$486$		0			0
$487$	2.98473e6	+	2.98473e6i	0.570272	+	0.570272i	0.932204	−	0.361932i	$-0.117883\pi$
$487$	2.98473e6	+	2.98473e6i	0.570272	+	0.570272i	−0.361932	+	0.932204i	$0.617883\pi$
$488$	−1.34633e6	−	1.34633e6i	−0.255919	−	0.255919i
$489$		0			0
$490$		0			0
$491$			2.65071e6i			0.496202i	0.968734	+	0.248101i	$0.0798066\pi$
$491$			2.65071e6i			0.496202i	−0.968734	+	0.248101i	$0.920193\pi$
$492$		0			0
$493$	1.17328e6	−	1.17328e6i	0.217412	−	0.217412i
$494$	−869945.			−0.160389
$495$		0			0
$496$	−707584.			−0.129144
$497$	2.35240e6	−	2.35240e6i	0.427189	−	0.427189i
$498$		0			0
$499$			1.33298e6i			0.239648i	0.992795	+	0.119824i	$0.0382330\pi$
$499$			1.33298e6i			0.239648i	−0.992795	+	0.119824i	$0.961767\pi$
$500$		0			0
$501$		0			0
$502$	3.26380e6	+	3.26380e6i	0.578048	+	0.578048i
$503$	3.60041e6	+	3.60041e6i	0.634500	+	0.634500i	0.949193	−	0.314693i	$-0.101902\pi$
$503$	3.60041e6	+	3.60041e6i	0.634500	+	0.634500i	−0.314693	+	0.949193i	$0.601902\pi$
$504$		0			0
$505$		0			0
$506$			1.18634e6i			0.205984i
$507$		0			0
$508$	3.64766e6	−	3.64766e6i	0.627127	−	0.627127i
$509$	1.32877e6			0.227329			0.113665	−	0.993519i	$-0.463741\pi$
$509$	1.32877e6			0.227329			0.113665	+	0.993519i	$0.463741\pi$
$510$		0			0
$511$	−2.97436e6			−0.503896
$512$	185364.	−	185364.i	0.0312500	−	0.0312500i
$513$		0			0
$514$			4.86458e6i			0.812153i
$515$		0			0
$516$		0			0
$517$	2.56932e6	+	2.56932e6i	0.422758	+	0.422758i
$518$	−833883.	−	833883.i	−0.136546	−	0.136546i
$519$		0			0
$520$		0			0
$521$		−	7.53594e6i		−	1.21631i	−0.793820	−	0.608153i	$-0.791911\pi$
$521$		−	7.53594e6i		−	1.21631i	0.793820	−	0.608153i	$-0.208089\pi$
$522$		0			0
$523$	1.14775e6	−	1.14775e6i	0.183481	−	0.183481i	−0.609390	−	0.792871i	$-0.708585\pi$
$523$	1.14775e6	−	1.14775e6i	0.183481	−	0.183481i	0.792871	+	0.609390i	$0.208585\pi$
$524$	−745008.			−0.118531
$525$		0			0
$526$	3.84127e6			0.605356
$527$	1.27821e6	−	1.27821e6i	0.200481	−	0.200481i
$528$		0			0
$529$		−	5.12303e6i		−	0.795953i
$530$		0			0
$531$		0			0
$532$	−381216.	−	381216.i	−0.0583972	−	0.0583972i
$533$	−1.62572e6	−	1.62572e6i	−0.247873	−	0.247873i
$534$		0			0
$535$		0			0
$536$		−	1.79807e6i		−	0.270329i
$537$		0			0
$538$	−1.90222e6	+	1.90222e6i	−0.283338	+	0.283338i
$539$	3.78600e6			0.561318
$540$		0			0
$541$	8642.00			0.00126947			0.000634733	−	1.00000i	$-0.499798\pi$
$541$	8642.00			0.00126947			0.000634733		1.00000i	$0.499798\pi$
$542$	4.44465e6	−	4.44465e6i	0.649889	−	0.649889i
$543$		0			0
$544$			669696.i			0.0970243i
$545$		0			0
$546$		0			0
$547$	7.56553e6	+	7.56553e6i	1.08111	+	1.08111i	0.996406	+	0.0847073i	$0.0269955\pi$
$547$	7.56553e6	+	7.56553e6i	1.08111	+	1.08111i	0.0847073	+	0.996406i	$0.473004\pi$
$548$	−2.41043e6	−	2.41043e6i	−0.342881	−	0.342881i
$549$		0			0
$550$		0			0
$551$			1.83179e6i			0.257037i
$552$		0			0
$553$	98736.0	−	98736.0i	0.0137298	−	0.0137298i
$554$	−5.92862e6			−0.820689
$555$		0			0
$556$	2.44410e6			0.335298
$557$	7.92605e6	−	7.92605e6i	1.08248	−	1.08248i	0.0861989	−	0.996278i	$-0.472528\pi$
$557$	7.92605e6	−	7.92605e6i	1.08248	−	1.08248i	0.996278	−	0.0861989i	$-0.0274720\pi$
$558$		0			0
$559$		−	1.38024e6i		−	0.186821i
$560$		0			0
$561$		0			0
$562$	4.93348e6	+	4.93348e6i	0.658889	+	0.658889i
$563$	−7.04595e6	−	7.04595e6i	−0.936847	−	0.936847i	0.0612744	−	0.998121i	$-0.480484\pi$
$563$	−7.04595e6	−	7.04595e6i	−0.936847	−	0.936847i	−0.998121	+	0.0612744i	$0.980484\pi$
$564$		0			0
$565$		0			0
$566$			8.09384e6i			1.06197i
$567$		0			0
$568$	3.22598e6	−	3.22598e6i	0.419557	−	0.419557i
$569$	4.20382e6			0.544332			0.272166	−	0.962250i	$-0.412260\pi$
$569$	4.20382e6			0.544332			0.272166	+	0.962250i	$0.412260\pi$
$570$		0			0
$571$	1.26883e7			1.62860			0.814298	−	0.580448i	$-0.197122\pi$
$571$	1.26883e7			1.62860			0.814298	+	0.580448i	$0.197122\pi$
$572$	−881994.	+	881994.i	−0.112713	+	0.112713i
$573$		0			0
$574$		−	1.42481e6i		−	0.180500i
$575$		0			0
$576$		0			0
$577$	3.48634e6	+	3.48634e6i	0.435943	+	0.435943i	0.890644	−	0.454701i	$-0.150254\pi$
$577$	3.48634e6	+	3.48634e6i	0.435943	+	0.435943i	−0.454701	+	0.890644i	$0.650254\pi$
$578$	2.80620e6	+	2.80620e6i	0.349381	+	0.349381i
$579$		0			0
$580$		0			0
$581$			1.88786e6i			0.232022i
$582$		0			0
$583$	2.05436e6	−	2.05436e6i	0.250325	−	0.250325i
$584$	−4.07891e6			−0.494894
$585$		0			0
$586$	3.76757e6			0.453228
$587$	7.17632e6	−	7.17632e6i	0.859621	−	0.859621i	−0.131673	−	0.991293i	$-0.542035\pi$
$587$	7.17632e6	−	7.17632e6i	0.859621	−	0.859621i	0.991293	+	0.131673i	$0.0420348\pi$
$588$		0			0
$589$			1.99561e6i			0.237021i
$590$		0			0
$591$		0			0
$592$	−1.14355e6	−	1.14355e6i	−0.134107	−	0.134107i
$593$	−993240.	−	993240.i	−0.115989	−	0.115989i	0.646730	−	0.762719i	$-0.276136\pi$
$593$	−993240.	−	993240.i	−0.115989	−	0.115989i	−0.762719	+	0.646730i	$0.776136\pi$
$594$		0			0
$595$		0			0
$596$			7.40460e6i			0.853858i
$597$		0			0
$598$	−976392.	+	976392.i	−0.111653	+	0.111653i
$599$	1.15789e7			1.31856			0.659278	−	0.751899i	$-0.270862\pi$
$599$	1.15789e7			1.31856			0.659278	+	0.751899i	$0.270862\pi$
$600$		0			0
$601$	−1.10155e7			−1.24399			−0.621996	−	0.783020i	$-0.713678\pi$
$601$	−1.10155e7			−1.24399			−0.621996	+	0.783020i	$0.713678\pi$
$602$	604831.	−	604831.i	0.0680210	−	0.0680210i
$603$		0			0
$604$			2.86669e6i			0.319733i
$605$		0			0
$606$		0			0
$607$	6.27601e6	+	6.27601e6i	0.691372	+	0.691372i	0.962534	−	0.271162i	$-0.0874078\pi$
$607$	6.27601e6	+	6.27601e6i	0.691372	+	0.691372i	−0.271162	+	0.962534i	$0.587408\pi$
$608$	−522784.	−	522784.i	−0.0573539	−	0.0573539i
$609$		0			0
$610$		0			0
$611$			4.22923e6i			0.458309i
$612$		0			0
$613$	5.31846e6	−	5.31846e6i	0.571656	−	0.571656i	−0.360935	−	0.932591i	$-0.617542\pi$
$613$	5.31846e6	−	5.31846e6i	0.571656	−	0.571656i	0.932591	+	0.360935i	$0.117542\pi$
$614$	−6.12162e6			−0.655308
$615$		0			0
$616$	−772992.			−0.0820773
$617$	−3.55725e6	+	3.55725e6i	−0.376185	+	0.376185i	−0.869724	−	0.493539i	$-0.835703\pi$
$617$	−3.55725e6	+	3.55725e6i	−0.376185	+	0.376185i	0.493539	+	0.869724i	$0.335703\pi$
$618$		0			0
$619$		−	1.89359e7i		−	1.98636i	−0.116586	−	0.993181i	$-0.537195\pi$
$619$		−	1.89359e7i		−	1.98636i	0.116586	−	0.993181i	$-0.462805\pi$
$620$		0			0
$621$		0			0
$622$	5.08462e6	+	5.08462e6i	0.526966	+	0.526966i
$623$	−386280.	−	386280.i	−0.0398733	−	0.0398733i
$624$		0			0
$625$		0			0
$626$			4.52133e6i			0.461138i
$627$		0			0
$628$	5.95138e6	−	5.95138e6i	0.602169	−	0.602169i
$629$	4.13151e6			0.416373
$630$		0			0
$631$	8.62094e6			0.861948			0.430974	−	0.902364i	$-0.358170\pi$
$631$	8.62094e6			0.861948			0.430974	+	0.902364i	$0.358170\pi$
$632$	135402.	−	135402.i	0.0134845	−	0.0134845i
$633$		0			0
$634$			5.12981e6i			0.506848i
$635$		0			0
$636$		0			0
$637$	3.11598e6	+	3.11598e6i	0.304261	+	0.304261i
$638$	1.85716e6	+	1.85716e6i	0.180633	+	0.180633i
$639$		0			0
$640$		0			0
$641$		−	2.95492e6i		−	0.284054i	−0.989863	−	0.142027i	$-0.954638\pi$
$641$		−	2.95492e6i		−	0.284054i	0.989863	−	0.142027i	$-0.0453619\pi$
$642$		0			0
$643$	8.32876e6	−	8.32876e6i	0.794425	−	0.794425i	−0.187785	−	0.982210i	$-0.560131\pi$
$643$	8.32876e6	−	8.32876e6i	0.794425	−	0.794425i	0.982210	+	0.187785i	$0.0601308\pi$
$644$	−855724.			−0.0813053
$645$		0			0
$646$	1.88875e6			0.178071
$647$	2.45184e6	−	2.45184e6i	0.230267	−	0.230267i	−0.582537	−	0.812804i	$-0.697940\pi$
$647$	2.45184e6	−	2.45184e6i	0.230267	−	0.230267i	0.812804	+	0.582537i	$0.197940\pi$
$648$		0			0
$649$			2.96350e6i			0.276181i
$650$		0			0
$651$		0			0
$652$	−3.02870e6	−	3.02870e6i	−0.279022	−	0.279022i
$653$	−1.32744e7	−	1.32744e7i	−1.21824	−	1.21824i	−0.968249	−	0.249988i	$-0.919573\pi$
$653$	−1.32744e7	−	1.32744e7i	−1.21824	−	1.21824i	−0.249988	−	0.968249i	$-0.580427\pi$
$654$		0			0
$655$		0			0
$656$		−	1.95392e6i		−	0.177275i
$657$		0			0
$658$	−1.85328e6	+	1.85328e6i	−0.166869	+	0.166869i
$659$	1.75882e7			1.57764			0.788818	−	0.614626i	$-0.210693\pi$
$659$	1.75882e7			1.57764			0.788818	+	0.614626i	$0.210693\pi$
$660$		0			0
$661$	−6.27670e6			−0.558763			−0.279381	−	0.960180i	$-0.590129\pi$
$661$	−6.27670e6			−0.558763			−0.279381	+	0.960180i	$0.590129\pi$
$662$	304186.	−	304186.i	0.0269770	−	0.0269770i
$663$		0			0
$664$			2.58893e6i			0.227877i
$665$		0			0
$666$		0			0
$667$	2.05592e6	+	2.05592e6i	0.178934	+	0.178934i
$668$	−4.14292e6	−	4.14292e6i	−0.359224	−	0.359224i
$669$		0			0
$670$		0			0
$671$			7.69933e6i			0.660156i
$672$		0			0
$673$	5.06163e6	−	5.06163e6i	0.430777	−	0.430777i	−0.458115	−	0.888893i	$-0.651475\pi$
$673$	5.06163e6	−	5.06163e6i	0.430777	−	0.430777i	0.888893	+	0.458115i	$0.151475\pi$
$674$	−7.44366e6			−0.631156
$675$		0			0
$676$	4.48888e6			0.377808
$677$	4.86767e6	−	4.86767e6i	0.408178	−	0.408178i	−0.472925	−	0.881103i	$-0.656802\pi$
$677$	4.86767e6	−	4.86767e6i	0.408178	−	0.408178i	0.881103	+	0.472925i	$0.156802\pi$
$678$		0			0
$679$		−	7.35055e6i		−	0.611851i
$680$		0			0
$681$		0			0
$682$	2.02325e6	+	2.02325e6i	0.166567	+	0.166567i
$683$	9.89201e6	+	9.89201e6i	0.811395	+	0.811395i	0.984843	−	0.173448i	$-0.0554907\pi$
$683$	9.89201e6	+	9.89201e6i	0.811395	+	0.811395i	−0.173448	+	0.984843i	$0.555491\pi$
$684$		0			0
$685$		0			0
$686$			5.86835e6i			0.476108i
$687$		0			0
$688$	829440.	−	829440.i	0.0668058	−	0.0668058i
$689$	3.38158e6			0.271376
$690$		0			0
$691$	−3.09191e6			−0.246338			−0.123169	−	0.992386i	$-0.539306\pi$
$691$	−3.09191e6			−0.246338			−0.123169	+	0.992386i	$0.539306\pi$
$692$	−2.56262e6	+	2.56262e6i	−0.203432	+	0.203432i
$693$		0			0
$694$			1.00986e7i			0.795905i
$695$		0			0
$696$		0			0
$697$	3.52964e6	+	3.52964e6i	0.275200	+	0.275200i
$698$	−4.18419e6	−	4.18419e6i	−0.325067	−	0.325067i
$699$		0			0
$700$		0			0
$701$		−	1.23298e7i		−	0.947675i	−0.880612	−	0.473837i	$-0.842868\pi$
$701$		−	1.23298e7i		−	0.947675i	0.880612	−	0.473837i	$-0.157132\pi$
$702$		0			0
$703$	−3.22517e6	+	3.22517e6i	−0.246130	+	0.246130i
$704$	−1.06005e6			−0.0806110
$705$		0			0
$706$	1.25704e7			0.949156
$707$	335737.	−	335737.i	0.0252610	−	0.0252610i
$708$		0			0
$709$		−	6.51293e6i		−	0.486587i	−0.969953	−	0.243294i	$-0.921772\pi$
$709$		−	6.51293e6i		−	0.486587i	0.969953	−	0.243294i	$-0.0782278\pi$
$710$		0			0
$711$		0			0
$712$	−529728.	−	529728.i	−0.0391609	−	0.0391609i
$713$	2.23979e6	+	2.23979e6i	0.165000	+	0.165000i
$714$		0			0
$715$		0			0
$716$			7.75073e6i			0.565015i
$717$		0			0
$718$	−93552.0	+	93552.0i	−0.00677239	+	0.00677239i
$719$	1.02328e7			0.738195			0.369098	−	0.929391i	$-0.379667\pi$
$719$	1.02328e7			0.738195			0.369098	+	0.929391i	$0.379667\pi$
$720$		0			0
$721$	−7.24898e6			−0.519324
$722$	5.52905e6	−	5.52905e6i	0.394737	−	0.394737i
$723$		0			0
$724$			4.26243e6i			0.302212i
$725$		0			0
$726$		0			0
$727$	−1.39935e7	−	1.39935e7i	−0.981949	−	0.981949i	0.0178905	−	0.999840i	$-0.494305\pi$
$727$	−1.39935e7	−	1.39935e7i	−0.981949	−	0.981949i	−0.999840	+	0.0178905i	$0.994305\pi$
$728$	−636192.	−	636192.i	−0.0444898	−	0.0444898i
$729$		0			0
$730$		0			0
$731$			2.99666e6i			0.207417i
$732$		0			0
$733$	−6.89732e6	+	6.89732e6i	−0.474155	+	0.474155i	−0.903256	−	0.429101i	$-0.858830\pi$
$733$	−6.89732e6	+	6.89732e6i	−0.474155	+	0.474155i	0.429101	+	0.903256i	$0.358830\pi$
$734$	−1.26768e7			−0.868501
$735$		0			0
$736$	−1.17350e6			−0.0798528
$737$	−5.14134e6	+	5.14134e6i	−0.348665	+	0.348665i
$738$		0			0
$739$		−	9.89996e6i		−	0.666841i	−0.942778	−	0.333421i	$-0.891797\pi$
$739$		−	9.89996e6i		−	0.666841i	0.942778	−	0.333421i	$-0.108203\pi$
$740$		0			0
$741$		0			0
$742$	1.48183e6	+	1.48183e6i	0.0988073	+	0.0988073i
$743$	−1.42800e7	−	1.42800e7i	−0.948976	−	0.948976i	0.0497843	−	0.998760i	$-0.484147\pi$
$743$	−1.42800e7	−	1.42800e7i	−0.948976	−	0.948976i	−0.998760	+	0.0497843i	$0.984147\pi$
$744$		0			0
$745$		0			0
$746$			5.75514e6i			0.378625i
$747$		0			0
$748$	1.91491e6	−	1.91491e6i	0.125140	−	0.125140i
$749$	−8.66924e6			−0.564646
$750$		0			0
$751$	5.78649e6			0.374382			0.187191	−	0.982324i	$-0.440062\pi$
$751$	5.78649e6			0.374382			0.187191	+	0.982324i	$0.440062\pi$
$752$	−2.54151e6	+	2.54151e6i	−0.163888	+	0.163888i
$753$		0			0
$754$			3.05698e6i			0.195823i
$755$		0			0
$756$		0			0
$757$	986385.	+	986385.i	0.0625614	+	0.0625614i	0.737695	−	0.675134i	$-0.235914\pi$
$757$	986385.	+	986385.i	0.0625614	+	0.0625614i	−0.675134	+	0.737695i	$0.735914\pi$
$758$	6.01476e6	+	6.01476e6i	0.380229	+	0.380229i
$759$		0			0
$760$		0			0
$761$		−	2.59161e7i		−	1.62221i	−0.584900	−	0.811105i	$-0.698866\pi$
$761$		−	2.59161e7i		−	1.62221i	0.584900	−	0.811105i	$-0.301134\pi$
$762$		0			0
$763$	784014.	−	784014.i	0.0487542	−	0.0487542i
$764$	−692535.			−0.0429248
$765$		0			0
$766$	−6.08794e6			−0.374885
$767$	−2.43904e6	+	2.43904e6i	−0.149703	+	0.149703i
$768$		0			0
$769$			7.56460e6i			0.461286i	0.973038	+	0.230643i	$0.0740829\pi$
$769$			7.56460e6i			0.461286i	−0.973038	+	0.230643i	$0.925917\pi$
$770$		0			0
$771$		0			0
$772$	−2.56790e6	−	2.56790e6i	−0.155073	−	0.155073i
$773$	2.11501e7	+	2.11501e7i	1.27311	+	1.27311i	0.944449	+	0.328657i	$0.106596\pi$
$773$	2.11501e7	+	2.11501e7i	1.27311	+	1.27311i	0.328657	+	0.944449i	$0.393404\pi$
$774$		0			0
$775$		0			0
$776$		−	1.00802e7i		−	0.600920i
$777$		0			0
$778$	−1.18714e7	+	1.18714e7i	−0.703156	+	0.703156i
$779$	−5.51067e6			−0.325358
$780$		0			0
$781$	−1.84486e7			−1.08227
$782$	2.11986e6	−	2.11986e6i	0.123963	−	0.123963i
$783$		0			0
$784$			3.74502e6i			0.217603i
$785$		0			0
$786$		0			0
$787$	−5.24337e6	−	5.24337e6i	−0.301768	−	0.301768i	0.539937	−	0.841705i	$-0.318448\pi$
$787$	−5.24337e6	−	5.24337e6i	−0.301768	−	0.301768i	−0.841705	+	0.539937i	$0.818448\pi$
$788$	−6.19751e6	−	6.19751e6i	−0.355551	−	0.355551i
$789$		0			0
$790$		0			0
$791$			5.43424e6i			0.308814i
$792$		0			0
$793$	−6.33675e6	+	6.33675e6i	−0.357836	+	0.357836i
$794$	−1.14626e7			−0.645257
$795$		0			0
$796$	6.62944e6			0.370846
$797$	−1.41641e7	+	1.41641e7i	−0.789848	+	0.789848i	−0.981469	−	0.191621i	$-0.938626\pi$
$797$	−1.41641e7	+	1.41641e7i	−0.789848	+	0.789848i	0.191621	+	0.981469i	$0.438626\pi$
$798$		0			0
$799$		−	9.18216e6i		−	0.508836i
$800$		0			0
$801$		0			0
$802$	−1.13918e7	−	1.13918e7i	−0.625398	−	0.625398i
$803$	1.16631e7	+	1.16631e7i	0.638302	+	0.638302i
$804$		0			0
$805$		0			0
$806$			3.33037e6i			0.180574i
$807$		0			0
$808$	460416.	−	460416.i	0.0248097	−	0.0248097i
$809$	−1.10071e7			−0.591290			−0.295645	−	0.955298i	$-0.595535\pi$
$809$	−1.10071e7			−0.591290			−0.295645	+	0.955298i	$0.595535\pi$
$810$		0			0
$811$	5.99290e6			0.319952			0.159976	−	0.987121i	$-0.448858\pi$
$811$	5.99290e6			0.319952			0.159976	+	0.987121i	$0.448858\pi$
$812$	−1.33959e6	+	1.33959e6i	−0.0712986	+	0.0712986i
$813$		0			0
$814$			6.53969e6i			0.345936i
$815$		0			0
$816$		0			0
$817$	−2.33928e6	−	2.33928e6i	−0.122610	−	0.122610i
$818$	1.59958e7	+	1.59958e7i	0.835840	+	0.835840i
$819$		0			0
$820$		0			0
$821$		−	3.27886e7i		−	1.69772i	−0.528619	−	0.848859i	$-0.677290\pi$
$821$		−	3.27886e7i		−	1.69772i	0.528619	−	0.848859i	$-0.322710\pi$
$822$		0			0
$823$	−9.94173e6	+	9.94173e6i	−0.511637	+	0.511637i	−0.915028	−	0.403391i	$-0.867832\pi$
$823$	−9.94173e6	+	9.94173e6i	−0.511637	+	0.511637i	0.403391	+	0.915028i	$0.367832\pi$
$824$	−9.94095e6			−0.510047
$825$		0			0
$826$	−2.13761e6			−0.109013
$827$	−7.37928e6	+	7.37928e6i	−0.375189	+	0.375189i	−0.869363	−	0.494174i	$-0.835471\pi$
$827$	−7.37928e6	+	7.37928e6i	−0.375189	+	0.375189i	0.494174	+	0.869363i	$0.335471\pi$
$828$		0			0
$829$			1.82419e7i			0.921899i	0.887426	+	0.460950i	$0.152491\pi$
$829$			1.82419e7i			0.921899i	−0.887426	+	0.460950i	$0.847509\pi$
$830$		0			0
$831$		0			0
$832$	−872448.	−	872448.i	−0.0436949	−	0.0436949i
$833$	−6.76515e6	−	6.76515e6i	−0.337804	−	0.337804i
$834$		0			0
$835$		0			0
$836$			2.98967e6i			0.147947i
$837$		0			0
$838$	−123228.	+	123228.i	−0.00606177	+	0.00606177i
$839$	2.70804e7			1.32816			0.664080	−	0.747661i	$-0.268823\pi$
$839$	2.70804e7			1.32816			0.664080	+	0.747661i	$0.268823\pi$
$840$		0			0
$841$	−1.40743e7			−0.686177
$842$	−1.03459e7	+	1.03459e7i	−0.502908	+	0.502908i
$843$		0			0
$844$		−	9.43674e6i		−	0.456001i
$845$		0			0
$846$		0			0
$847$	−3.10441e6	−	3.10441e6i	−0.148686	−	0.148686i
$848$	2.03212e6	+	2.03212e6i	0.0970421	+	0.0970421i
$849$		0			0
$850$		0			0
$851$			7.23962e6i			0.342682i
$852$		0			0
$853$	−1.86524e7	+	1.86524e7i	−0.877731	+	0.877731i	−0.993300	−	0.115568i	$-0.963131\pi$
$853$	−1.86524e7	+	1.86524e7i	−0.877731	+	0.877731i	0.115568	+	0.993300i	$0.463131\pi$
$854$	−5.55362e6			−0.260574
$855$		0			0
$856$	−1.18886e7			−0.554559
$857$	2.73159e6	−	2.73159e6i	0.127047	−	0.127047i	−0.640724	−	0.767771i	$-0.721366\pi$
$857$	2.73159e6	−	2.73159e6i	0.127047	−	0.127047i	0.767771	+	0.640724i	$0.221366\pi$
$858$		0			0
$859$			1.85975e7i			0.859946i	0.902842	+	0.429973i	$0.141477\pi$
$859$			1.85975e7i			0.859946i	−0.902842	+	0.429973i	$0.858523\pi$
$860$		0			0
$861$		0			0
$862$	706992.	+	706992.i	0.0324076	+	0.0324076i
$863$	1.12074e7	+	1.12074e7i	0.512245	+	0.512245i	0.915214	−	0.402969i	$-0.132022\pi$
$863$	1.12074e7	+	1.12074e7i	0.512245	+	0.512245i	−0.402969	+	0.915214i	$0.632022\pi$
$864$		0			0
$865$		0			0
$866$		−	6.29991e6i		−	0.285456i
$867$		0			0
$868$	−1.45939e6	+	1.45939e6i	−0.0657465	+	0.0657465i
$869$	−774333.			−0.0347839
$870$		0			0
$871$	−8.46292e6			−0.377985
$872$	1.07516e6	−	1.07516e6i	0.0478833	−	0.0478833i
$873$		0			0
$874$			3.30965e6i			0.146556i
$875$		0			0
$876$		0			0
$877$	−2.76523e7	−	2.76523e7i	−1.21404	−	1.21404i	−0.969686	−	0.244353i	$-0.921424\pi$
$877$	−2.76523e7	−	2.76523e7i	−1.21404	−	1.21404i	−0.244353	−	0.969686i	$-0.578576\pi$
$878$	−8.18211e6	−	8.18211e6i	−0.358203	−	0.358203i
$879$		0			0
$880$		0			0
$881$		−	1.14119e7i		−	0.495357i	−0.968842	−	0.247678i	$-0.920332\pi$
$881$		−	1.14119e7i		−	0.495357i	0.968842	−	0.247678i	$-0.0796677\pi$
$882$		0			0
$883$	−1.53482e7	+	1.53482e7i	−0.662453	+	0.662453i	−0.955958	−	0.293505i	$-0.905178\pi$
$883$	−1.53482e7	+	1.53482e7i	−0.662453	+	0.662453i	0.293505	+	0.955958i	$0.405178\pi$
$884$	3.15204e6			0.135663
$885$		0			0
$886$	−4.18570e6			−0.179136
$887$	−1.86085e7	+	1.86085e7i	−0.794151	+	0.794151i	−0.982166	−	0.188015i	$-0.939794\pi$
$887$	−1.86085e7	+	1.86085e7i	−0.794151	+	0.794151i	0.188015	+	0.982166i	$0.439794\pi$
$888$		0			0
$889$		−	1.50466e7i		−	0.638535i
$890$		0			0
$891$		0			0
$892$	−5.28610e6	−	5.28610e6i	−0.222445	−	0.222445i
$893$	7.16786e6	+	7.16786e6i	0.300788	+	0.300788i
$894$		0			0
$895$		0			0
$896$		−	764626.i		−	0.0318184i
$897$		0			0
$898$	−4.90472e6	+	4.90472e6i	−0.202966	+	0.202966i
$899$	7.01254e6			0.289385
$900$		0			0
$901$	−7.34180e6			−0.301294
$902$	−5.58700e6	+	5.58700e6i	−0.228645	+	0.228645i
$903$		0			0
$904$			7.45229e6i			0.303297i
$905$		0			0
$906$		0			0
$907$	−2.18773e7	−	2.18773e7i	−0.883030	−	0.883030i	0.110812	−	0.993841i	$-0.464655\pi$
$907$	−2.18773e7	−	2.18773e7i	−0.883030	−	0.883030i	−0.993841	+	0.110812i	$0.964655\pi$
$908$	8.79089e6	+	8.79089e6i	0.353849	+	0.353849i
$909$		0			0
$910$		0			0
$911$			1.41972e7i			0.566771i	0.959006	+	0.283385i	$0.0914575\pi$
$911$			1.41972e7i			0.566771i	−0.959006	+	0.283385i	$0.908542\pi$
$912$		0			0
$913$	7.40272e6	−	7.40272e6i	0.293910	−	0.293910i
$914$	1.69169e7			0.669817
$915$		0			0
$916$	1.00879e7			0.397248
$917$	−1.53658e6	+	1.53658e6i	−0.0603436	+	0.0603436i
$918$		0			0
$919$		−	1.37905e7i		−	0.538631i	−0.963052	−	0.269315i	$-0.913203\pi$
$919$		−	1.37905e7i		−	0.538631i	0.963052	−	0.269315i	$-0.0867974\pi$
$920$		0			0
$921$		0			0
$922$	−1.06776e7	−	1.06776e7i	−0.413665	−	0.413665i
$923$	−1.51837e7	−	1.51837e7i	−0.586641	−	0.586641i
$924$		0			0
$925$		0			0
$926$		−	1.00690e7i		−	0.385887i
$927$		0			0
$928$	−1.83706e6	+	1.83706e6i	−0.0700249	+	0.0700249i
$929$	−4.48053e7			−1.70330			−0.851649	−	0.524113i	$-0.824397\pi$
$929$	−4.48053e7			−1.70330			−0.851649	+	0.524113i	$0.824397\pi$
$930$		0			0
$931$	1.05621e7			0.399372
$932$	−3.91226e6	+	3.91226e6i	−0.147532	+	0.147532i
$933$		0			0
$934$		−	6.27787e6i		−	0.235475i
$935$		0			0
$936$		0			0
$937$	−2.64785e7	−	2.64785e7i	−0.985244	−	0.985244i	0.0146490	−	0.999893i	$-0.495337\pi$
$937$	−2.64785e7	−	2.64785e7i	−0.985244	−	0.985244i	−0.999893	+	0.0146490i	$0.995337\pi$
$938$	−3.70851e6	−	3.70851e6i	−0.137623	−	0.137623i
$939$		0			0
$940$		0			0
$941$		−	3.46220e7i		−	1.27461i	−0.770610	−	0.637307i	$-0.780048\pi$
$941$		−	3.46220e7i		−	1.27461i	0.770610	−	0.637307i	$-0.219952\pi$
$942$		0			0
$943$	−6.18496e6	+	6.18496e6i	−0.226495	+	0.226495i
$944$	−2.93143e6			−0.107065
$945$		0			0
$946$	−4.74336e6			−0.172329
$947$	−505689.	+	505689.i	−0.0183235	+	0.0183235i	−0.716209	−	0.697886i	$-0.754124\pi$
$947$	−505689.	+	505689.i	−0.0183235	+	0.0183235i	0.697886	+	0.716209i	$0.254124\pi$
$948$		0			0
$949$			1.91981e7i			0.691979i
$950$		0			0
$951$		0			0
$952$	1.38125e6	+	1.38125e6i	0.0493946	+	0.0493946i
$953$	2.13467e7	+	2.13467e7i	0.761374	+	0.761374i	0.976571	−	0.215197i	$-0.0690393\pi$
$953$	2.13467e7	+	2.13467e7i	0.761374	+	0.761374i	−0.215197	+	0.976571i	$0.569039\pi$
$954$		0			0
$955$		0			0
$956$			1.26827e7i			0.448815i
$957$		0			0
$958$	1.47458e6	−	1.47458e6i	0.0519106	−	0.0519106i
$959$	−9.94303e6			−0.349118
$960$		0			0
$961$	−2.09895e7			−0.733150
$962$	−5.38233e6	+	5.38233e6i	−0.187514	+	0.187514i
$963$		0			0
$964$		−	5.70579e6i		−	0.197753i
$965$		0			0
$966$		0			0
$967$	−4.35296e6	−	4.35296e6i	−0.149699	−	0.149699i	0.628285	−	0.777983i	$-0.283757\pi$
$967$	−4.35296e6	−	4.35296e6i	−0.149699	−	0.149699i	−0.777983	+	0.628285i	$0.783757\pi$
$968$	−4.25726e6	−	4.25726e6i	−0.146030	−	0.146030i
$969$		0			0
$970$		0			0
$971$			5.77443e7i			1.96545i	0.185081	+	0.982723i	$0.440745\pi$
$971$			5.77443e7i			1.96545i	−0.185081	+	0.982723i	$0.559255\pi$
$972$		0			0
$973$	5.04095e6	−	5.04095e6i	0.170699	−	0.170699i
$974$	1.68842e7			0.570272
$975$		0			0
$976$	−7.61600e6			−0.255919
$977$	−3.22881e7	+	3.22881e7i	−1.08219	+	1.08219i	−0.0858902	+	0.996305i	$0.527373\pi$
$977$	−3.22881e7	+	3.22881e7i	−1.08219	+	1.08219i	−0.996305	+	0.0858902i	$0.972627\pi$
$978$		0			0
$979$			3.02938e6i			0.101018i
$980$		0			0
$981$		0			0
$982$	7.49735e6	+	7.49735e6i	0.248101	+	0.248101i
$983$	1.46650e7	+	1.46650e7i	0.484059	+	0.484059i	0.906425	−	0.422366i	$-0.138800\pi$
$983$	1.46650e7	+	1.46650e7i	0.484059	+	0.484059i	−0.422366	+	0.906425i	$0.638800\pi$
$984$		0			0
$985$		0			0
$986$		−	6.63705e6i		−	0.217412i
$987$		0			0
$988$	−2.46058e6	+	2.46058e6i	−0.0801945	+	0.0801945i
$989$	−5.25103e6			−0.170708
$990$		0			0
$991$	5.45331e7			1.76391			0.881954	−	0.471335i	$-0.156228\pi$
$991$	5.45331e7			1.76391			0.881954	+	0.471335i	$0.156228\pi$
$992$	−2.00135e6	+	2.00135e6i	−0.0645719	+	0.0645719i
$993$		0			0
$994$		−	1.33072e7i		−	0.427189i
$995$		0			0
$996$		0			0
$997$	289683.	+	289683.i	0.00922965	+	0.00922965i	0.711707	−	0.702477i	$-0.247923\pi$
$997$	289683.	+	289683.i	0.00922965	+	0.00922965i	−0.702477	+	0.711707i	$0.747923\pi$
$998$	3.77024e6	+	3.77024e6i	0.119824	+	0.119824i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.f.a.107.2	yes	4
3.2	odd	2	inner	450.6.f.a.107.1	✓	4
5.2	odd	4		450.6.f.c.143.2	yes	4
5.3	odd	4	inner	450.6.f.a.143.1	yes	4
5.4	even	2		450.6.f.c.107.1	yes	4
15.2	even	4		450.6.f.c.143.1	yes	4
15.8	even	4	inner	450.6.f.a.143.2	yes	4
15.14	odd	2		450.6.f.c.107.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.6.f.a.107.1	✓	4	3.2	odd	2	inner
450.6.f.a.107.2	yes	4	1.1	even	1	trivial
450.6.f.a.143.1	yes	4	5.3	odd	4	inner
450.6.f.a.143.2	yes	4	15.8	even	4	inner
450.6.f.c.107.1	yes	4	5.4	even	2
450.6.f.c.107.2	yes	4	15.14	odd	2
450.6.f.c.143.1	yes	4	15.2	even	4
450.6.f.c.143.2	yes	4	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(-33.0000 - 33.0000i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(-33.0000 - 33.0000i) q^{7} +(-45.2548 - 45.2548i) q^{8} +258.801i q^{11} +(-213.000 + 213.000i) q^{13} -186.676 q^{14} -256.000 q^{16} +(462.448 - 462.448i) q^{17} +722.000i q^{19} +(732.000 + 732.000i) q^{22} +(810.344 + 810.344i) q^{23} +1204.91i q^{26} +(-528.000 + 528.000i) q^{28} +2537.10 q^{29} +2764.00 q^{31} +(-724.077 + 724.077i) q^{32} -2616.00i q^{34} +(4467.00 + 4467.00i) q^{37} +(2042.12 + 2042.12i) q^{38} +7632.51i q^{41} +(-3240.00 + 3240.00i) q^{43} +4140.82 q^{44} +4584.00 q^{46} +(9927.78 - 9927.78i) q^{47} -14629.0i q^{49} +(3408.00 + 3408.00i) q^{52} +(-7937.98 - 7937.98i) q^{53} +2986.82i q^{56} +(7176.00 - 7176.00i) q^{58} +11450.9 q^{59} +29750.0 q^{61} +(7817.77 - 7817.77i) q^{62} +4096.00i q^{64} +(19866.0 + 19866.0i) q^{67} +(-7399.17 - 7399.17i) q^{68} +71284.8i q^{71} +(45066.0 - 45066.0i) q^{73} +25269.2 q^{74} +11552.0 q^{76} +(8540.44 - 8540.44i) q^{77} +2992.00i q^{79} +(21588.0 + 21588.0i) q^{82} +(-28603.9 - 28603.9i) q^{83} +18328.2i q^{86} +(11712.0 - 11712.0i) q^{88} +11705.4 q^{89} +14058.0 q^{91} +(12965.5 - 12965.5i) q^{92} -56160.0i q^{94} +(111372. + 111372. i) q^{97} +(-41377.1 - 41377.1i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 132 q^{7}+O(q^{10}) \) 4 * q - 132 * q^7	\( 4 q - 132 q^{7} - 852 q^{13} - 1024 q^{16} + 2928 q^{22} - 2112 q^{28} + 11056 q^{31} + 17868 q^{37} - 12960 q^{43} + 18336 q^{46} + 13632 q^{52} + 28704 q^{58} + 119000 q^{61} + 79464 q^{67} + 180264 q^{73} + 46208 q^{76} + 86352 q^{82} + 46848 q^{88} + 56232 q^{91} + 445488 q^{97}+O(q^{100}) \) 4 * q - 132 * q^7 - 852 * q^13 - 1024 * q^16 + 2928 * q^22 - 2112 * q^28 + 11056 * q^31 + 17868 * q^37 - 12960 * q^43 + 18336 * q^46 + 13632 * q^52 + 28704 * q^58 + 119000 * q^61 + 79464 * q^67 + 180264 * q^73 + 46208 * q^76 + 86352 * q^82 + 46848 * q^88 + 56232 * q^91 + 445488 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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