LMFDB - Embedded newform 225.6.f.c.107.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,Mod(107,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.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:	$36.0863594579$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.3
Character	$\chi$	$=$	225.107
Dual form			225.6.f.c.143.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.41913 + 5.41913i) q^{2} -26.7338i q^{4} +(-10.9755 - 10.9755i) q^{7} +(-28.5379 - 28.5379i) q^{8} +O(q^{10})$	\(q+(-5.41913 + 5.41913i) q^{2} -26.7338i q^{4} +(-10.9755 - 10.9755i) q^{7} +(-28.5379 - 28.5379i) q^{8} -194.798i q^{11} +(194.671 - 194.671i) q^{13} +118.955 q^{14} +1164.78 q^{16} +(-1427.43 + 1427.43i) q^{17} -2273.37i q^{19} +(1055.63 + 1055.63i) q^{22} +(354.348 + 354.348i) q^{23} +2109.89i q^{26} +(-293.417 + 293.417i) q^{28} +3373.99 q^{29} +4493.52 q^{31} +(-5398.90 + 5398.90i) q^{32} -15470.9i q^{34} +(9731.76 + 9731.76i) q^{37} +(12319.7 + 12319.7i) q^{38} +16191.0i q^{41} +(-6182.12 + 6182.12i) q^{43} -5207.69 q^{44} -3840.52 q^{46} +(-5123.99 + 5123.99i) q^{47} -16566.1i q^{49} +(-5204.30 - 5204.30i) q^{52} +(11304.4 + 11304.4i) q^{53} +626.435i q^{56} +(-18284.1 + 18284.1i) q^{58} -32365.9 q^{59} +5599.54 q^{61} +(-24351.0 + 24351.0i) q^{62} -21241.5i q^{64} +(-40504.9 - 40504.9i) q^{67} +(38160.8 + 38160.8i) q^{68} -22026.4i q^{71} +(-26237.5 + 26237.5i) q^{73} -105475. q^{74} -60775.9 q^{76} +(-2138.00 + 2138.00i) q^{77} +70839.5i q^{79} +(-87741.0 - 87741.0i) q^{82} +(-12450.6 - 12450.6i) q^{83} -67003.4i q^{86} +(-5559.12 + 5559.12i) q^{88} -91155.9 q^{89} -4273.21 q^{91} +(9473.10 - 9473.10i) q^{92} -55535.1i q^{94} +(1795.86 + 1795.86i) q^{97} +(89773.7 + 89773.7i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 5664 q^{16} - 49176 q^{31} - 116976 q^{46} - 87048 q^{61} + 241680 q^{76} - 489096 q^{91}+O(q^{100}) $ 24 * q + 5664 * q^16 - 49176 * q^31 - 116976 * q^46 - 87048 * q^61 + 241680 * q^76 - 489096 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

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

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 $

$2$	−5.41913	+	5.41913i	−0.957975	+	0.957975i	−0.999152	−	0.0411767i	$-0.986889\pi$
$2$	−5.41913	+	5.41913i	−0.957975	+	0.957975i	0.0411767	+	0.999152i	$0.486889\pi$
$3$		0			0
$3$		0			0
$4$		−	26.7338i		−	0.835433i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−10.9755	−	10.9755i	−0.0846601	−	0.0846601i	0.663509	−	0.748169i	$-0.269067\pi$
$7$	−10.9755	−	10.9755i	−0.0846601	−	0.0846601i	−0.748169	+	0.663509i	$0.769067\pi$
$8$	−28.5379	−	28.5379i	−0.157651	−	0.157651i
$9$		0			0
$10$		0			0
$11$		−	194.798i		−	0.485402i	−0.970101	−	0.242701i	$-0.921967\pi$
$11$		−	194.798i		−	0.485402i	0.970101	−	0.242701i	$-0.0780334\pi$
$12$		0			0
$13$	194.671	−	194.671i	0.319479	−	0.319479i	−0.529088	−	0.848567i	$-0.677466\pi$
$13$	194.671	−	194.671i	0.319479	−	0.319479i	0.848567	+	0.529088i	$0.177466\pi$
$14$	118.955			0.162205
$15$		0			0
$16$	1164.78			1.13748
$17$	−1427.43	+	1427.43i	−1.19794	+	1.19794i	−0.223154	+	0.974783i	$0.571635\pi$
$17$	−1427.43	+	1427.43i	−1.19794	+	1.19794i	−0.974783	+	0.223154i	$0.928365\pi$
$18$		0			0
$19$		−	2273.37i		−	1.44473i	−0.691513	−	0.722364i	$-0.743056\pi$
$19$		−	2273.37i		−	1.44473i	0.691513	−	0.722364i	$-0.256944\pi$
$20$		0			0
$21$		0			0
$22$	1055.63	+	1055.63i	0.465003	+	0.465003i
$23$	354.348	+	354.348i	0.139672	+	0.139672i	0.773486	−	0.633813i	$-0.218511\pi$
$23$	354.348	+	354.348i	0.139672	+	0.139672i	−0.633813	+	0.773486i	$0.718511\pi$
$24$		0			0
$25$		0			0
$26$			2109.89i			0.612106i
$27$		0			0
$28$	−293.417	+	293.417i	−0.0707278	+	0.0707278i
$29$	3373.99			0.744987			0.372494	−	0.928035i	$-0.378503\pi$
$29$	3373.99			0.744987			0.372494	+	0.928035i	$0.378503\pi$
$30$		0			0
$31$	4493.52			0.839813			0.419906	−	0.907567i	$-0.362063\pi$
$31$	4493.52			0.839813			0.419906	+	0.907567i	$0.362063\pi$
$32$	−5398.90	+	5398.90i	−0.932031	+	0.932031i
$33$		0			0
$34$		−	15470.9i		−	2.29519i
$35$		0			0
$36$		0			0
$37$	9731.76	+	9731.76i	1.16866	+	1.16866i	0.982525	+	0.186132i	$0.0595953\pi$
$37$	9731.76	+	9731.76i	1.16866	+	1.16866i	0.186132	+	0.982525i	$0.440405\pi$
$38$	12319.7	+	12319.7i	1.38401	+	1.38401i
$39$		0			0
$40$		0			0
$41$			16191.0i			1.50423i	0.659033	+	0.752114i	$0.270966\pi$
$41$			16191.0i			1.50423i	−0.659033	+	0.752114i	$0.729034\pi$
$42$		0			0
$43$	−6182.12	+	6182.12i	−0.509878	+	0.509878i	−0.914489	−	0.404611i	$-0.867407\pi$
$43$	−6182.12	+	6182.12i	−0.509878	+	0.509878i	0.404611	+	0.914489i	$0.367407\pi$
$44$	−5207.69			−0.405521
$45$		0			0
$46$	−3840.52			−0.267606
$47$	−5123.99	+	5123.99i	−0.338348	+	0.338348i	−0.855745	−	0.517397i	$-0.826901\pi$
$47$	−5123.99	+	5123.99i	−0.338348	+	0.338348i	0.517397	+	0.855745i	$0.326901\pi$
$48$		0			0
$49$		−	16566.1i		−	0.985665i
$50$		0			0
$51$		0			0
$52$	−5204.30	−	5204.30i	−0.266903	−	0.266903i
$53$	11304.4	+	11304.4i	0.552787	+	0.552787i	0.927244	−	0.374457i	$-0.122171\pi$
$53$	11304.4	+	11304.4i	0.552787	+	0.552787i	−0.374457	+	0.927244i	$0.622171\pi$
$54$		0			0
$55$		0			0
$56$			626.435i			0.0266935i
$57$		0			0
$58$	−18284.1	+	18284.1i	−0.713679	+	0.713679i
$59$	−32365.9			−1.21048			−0.605240	−	0.796043i	$-0.706923\pi$
$59$	−32365.9			−1.21048			−0.605240	+	0.796043i	$0.706923\pi$
$60$		0			0
$61$	5599.54			0.192676			0.0963379	−	0.995349i	$-0.469287\pi$
$61$	5599.54			0.192676			0.0963379	+	0.995349i	$0.469287\pi$
$62$	−24351.0	+	24351.0i	−0.804520	+	0.804520i
$63$		0			0
$64$		−	21241.5i		−	0.648240i
$65$		0			0
$66$		0			0
$67$	−40504.9	−	40504.9i	−1.10235	−	1.10235i	−0.994126	−	0.108226i	$-0.965483\pi$
$67$	−40504.9	−	40504.9i	−1.10235	−	1.10235i	−0.108226	−	0.994126i	$-0.534517\pi$
$68$	38160.8	+	38160.8i	1.00080	+	1.00080i
$69$		0			0
$70$		0			0
$71$		−	22026.4i		−	0.518557i	−0.965803	−	0.259279i	$-0.916515\pi$
$71$		−	22026.4i		−	0.518557i	0.965803	−	0.259279i	$-0.0834848\pi$
$72$		0			0
$73$	−26237.5	+	26237.5i	−0.576255	+	0.576255i	−0.933869	−	0.357614i	$-0.883590\pi$
$73$	−26237.5	+	26237.5i	−0.576255	+	0.576255i	0.357614	+	0.933869i	$0.383590\pi$
$74$	−105475.			−2.23909
$75$		0			0
$76$	−60775.9			−1.20697
$77$	−2138.00	+	2138.00i	−0.0410942	+	0.0410942i
$78$		0			0
$79$			70839.5i			1.27705i	0.769601	+	0.638525i	$0.220455\pi$
$79$			70839.5i			1.27705i	−0.769601	+	0.638525i	$0.779545\pi$
$80$		0			0
$81$		0			0
$82$	−87741.0	−	87741.0i	−1.44101	−	1.44101i
$83$	−12450.6	−	12450.6i	−0.198378	−	0.198378i	0.600926	−	0.799304i	$-0.294799\pi$
$83$	−12450.6	−	12450.6i	−0.198378	−	0.198378i	−0.799304	+	0.600926i	$0.794799\pi$
$84$		0			0
$85$		0			0
$86$		−	67003.4i		−	0.976901i
$87$		0			0
$88$	−5559.12	+	5559.12i	−0.0765243	+	0.0765243i
$89$	−91155.9			−1.21986			−0.609930	−	0.792456i	$-0.708802\pi$
$89$	−91155.9			−1.21986			−0.609930	+	0.792456i	$0.708802\pi$
$90$		0			0
$91$	−4273.21			−0.0540942
$92$	9473.10	−	9473.10i	0.116687	−	0.116687i
$93$		0			0
$94$		−	55535.1i		−	0.648258i
$95$		0			0
$96$		0			0
$97$	1795.86	+	1795.86i	0.0193796	+	0.0193796i	0.716730	−	0.697351i	$-0.245638\pi$
$97$	1795.86	+	1795.86i	0.0193796	+	0.0193796i	−0.697351	+	0.716730i	$0.745638\pi$
$98$	89773.7	+	89773.7i	0.944243	+	0.944243i
$99$		0			0
$100$		0			0
$101$			129300.i			1.26123i	0.776094	+	0.630617i	$0.217198\pi$
$101$			129300.i			1.26123i	−0.776094	+	0.630617i	$0.782802\pi$
$102$		0			0
$103$	−138191.	+	138191.i	−1.28347	+	1.28347i	−0.344789	+	0.938680i	$0.612050\pi$
$103$	−138191.	+	138191.i	−1.28347	+	1.28347i	−0.938680	+	0.344789i	$0.887950\pi$
$104$	−11111.0			−0.100732
$105$		0			0
$106$	−122520.			−1.05911
$107$	69822.5	−	69822.5i	0.589571	−	0.589571i	−0.347944	−	0.937515i	$-0.613120\pi$
$107$	69822.5	−	69822.5i	0.589571	−	0.589571i	0.937515	+	0.347944i	$0.113120\pi$
$108$		0			0
$109$			131350.i			1.05893i	0.848333	+	0.529463i	$0.177606\pi$
$109$			131350.i			1.05893i	−0.848333	+	0.529463i	$0.822394\pi$
$110$		0			0
$111$		0			0
$112$	−12784.1	−	12784.1i	−0.0962996	−	0.0962996i
$113$	−26187.1	−	26187.1i	−0.192926	−	0.192926i	0.604033	−	0.796959i	$-0.293559\pi$
$113$	−26187.1	−	26187.1i	−0.192926	−	0.192926i	−0.796959	+	0.604033i	$0.793559\pi$
$114$		0			0
$115$		0			0
$116$		−	90199.8i		−	0.622387i
$117$		0			0
$118$	175395.	−	175395.i	1.15961	−	1.15961i
$119$	31333.6			0.202835
$120$		0			0
$121$	123105.			0.764385
$122$	−30344.6	+	30344.6i	−0.184579	+	0.184579i
$123$		0			0
$124$		−	120129.i		−	0.701607i
$125$		0			0
$126$		0			0
$127$	218514.	+	218514.i	1.20218	+	1.20218i	0.973501	+	0.228683i	$0.0734419\pi$
$127$	218514.	+	218514.i	1.20218	+	1.20218i	0.228683	+	0.973501i	$0.426558\pi$
$128$	−57654.3	−	57654.3i	−0.311033	−	0.311033i
$129$		0			0
$130$		0			0
$131$			273444.i			1.39216i	0.717962	+	0.696082i	$0.245075\pi$
$131$			273444.i			1.39216i	−0.717962	+	0.696082i	$0.754925\pi$
$132$		0			0
$133$	−24951.3	+	24951.3i	−0.122311	+	0.122311i
$134$	439002.			2.11205
$135$		0			0
$136$	81472.1			0.377713
$137$	−74124.9	+	74124.9i	−0.337414	+	0.337414i	−0.855393	−	0.517979i	$-0.826684\pi$
$137$	−74124.9	+	74124.9i	−0.337414	+	0.337414i	0.517979	+	0.855393i	$0.326684\pi$
$138$		0			0
$139$		−	224787.i		−	0.986811i	−0.869799	−	0.493406i	$-0.835752\pi$
$139$		−	224787.i		−	0.986811i	0.869799	−	0.493406i	$-0.164248\pi$
$140$		0			0
$141$		0			0
$142$	119364.	+	119364.i	0.496765	+	0.496765i
$143$	−37921.4	−	37921.4i	−0.155076	−	0.155076i
$144$		0			0
$145$		0			0
$146$		−	284368.i		−	1.10408i
$147$		0			0
$148$	260167.	−	260167.i	0.976334	−	0.976334i
$149$	164888.			0.608449			0.304224	−	0.952600i	$-0.401603\pi$
$149$	164888.			0.608449			0.304224	+	0.952600i	$0.401603\pi$
$150$		0			0
$151$	392244.			1.39996			0.699978	−	0.714165i	$-0.253193\pi$
$151$	392244.			1.39996			0.699978	+	0.714165i	$0.253193\pi$
$152$	−64877.2	+	64877.2i	−0.227763	+	0.227763i
$153$		0			0
$154$		−	23172.2i		−	0.0787344i
$155$		0			0
$156$		0			0
$157$	107211.	+	107211.i	0.347129	+	0.347129i	0.859039	−	0.511910i	$-0.171062\pi$
$157$	107211.	+	107211.i	0.347129	+	0.347129i	−0.511910	+	0.859039i	$0.671062\pi$
$158$	−383888.	−	383888.i	−1.22338	−	1.22338i
$159$		0			0
$160$		0			0
$161$		−	7778.29i		−	0.0236494i
$162$		0			0
$163$	72934.4	−	72934.4i	0.215012	−	0.215012i	−0.591380	−	0.806393i	$-0.701417\pi$
$163$	72934.4	−	72934.4i	0.215012	−	0.215012i	0.806393	+	0.591380i	$0.201417\pi$
$164$	432847.			1.25668
$165$		0			0
$166$	134942.			0.380083
$167$	39616.8	−	39616.8i	0.109923	−	0.109923i	−0.650006	−	0.759929i	$-0.725234\pi$
$167$	39616.8	−	39616.8i	0.109923	−	0.109923i	0.759929	+	0.650006i	$0.225234\pi$
$168$		0			0
$169$			295500.i			0.795867i
$170$		0			0
$171$		0			0
$172$	165272.	+	165272.i	0.425969	+	0.425969i
$173$	−303757.	−	303757.i	−0.771633	−	0.771633i	0.206759	−	0.978392i	$-0.433708\pi$
$173$	−303757.	−	303757.i	−0.771633	−	0.771633i	−0.978392	+	0.206759i	$0.933708\pi$
$174$		0			0
$175$		0			0
$176$		−	226897.i		−	0.552138i
$177$		0			0
$178$	493985.	−	493985.i	1.16859	−	1.16859i
$179$	337098.			0.786363			0.393181	−	0.919461i	$-0.371374\pi$
$179$	337098.			0.786363			0.393181	+	0.919461i	$0.371374\pi$
$180$		0			0
$181$	45494.0			0.103219			0.0516093	−	0.998667i	$-0.483565\pi$
$181$	45494.0			0.103219			0.0516093	+	0.998667i	$0.483565\pi$
$182$	23157.1	−	23157.1i	0.0518209	−	0.0518209i
$183$		0			0
$184$		−	20224.7i		−	0.0440391i
$185$		0			0
$186$		0			0
$187$	278061.	+	278061.i	0.581481	+	0.581481i
$188$	136984.	+	136984.i	0.282667	+	0.282667i
$189$		0			0
$190$		0			0
$191$			843927.i			1.67387i	0.547303	+	0.836935i	$0.315655\pi$
$191$			843927.i			1.67387i	−0.547303	+	0.836935i	$0.684345\pi$
$192$		0			0
$193$	−454404.	+	454404.i	−0.878111	+	0.878111i	−0.993339	−	0.115228i	$-0.963240\pi$
$193$	−454404.	+	454404.i	−0.878111	+	0.878111i	0.115228	+	0.993339i	$0.463240\pi$
$194$	−19464.0			−0.0371303
$195$		0			0
$196$	−442875.			−0.823457
$197$	725036.	−	725036.i	1.33105	−	1.33105i	0.426617	−	0.904433i	$-0.359705\pi$
$197$	725036.	−	725036.i	1.33105	−	1.33105i	0.904433	−	0.426617i	$-0.140295\pi$
$198$		0			0
$199$			838663.i			1.50126i	0.660725	+	0.750628i	$0.270249\pi$
$199$			838663.i			1.50126i	−0.660725	+	0.750628i	$0.729751\pi$
$200$		0			0
$201$		0			0
$202$	−700694.	−	700694.i	−1.20823	−	1.20823i
$203$	−37031.2	−	37031.2i	−0.0630707	−	0.0630707i
$204$		0			0
$205$		0			0
$206$		−	1.49774e6i		−	2.45906i
$207$		0			0
$208$	226749.	−	226749.i	0.363402	−	0.363402i
$209$	−442846.			−0.701274
$210$		0			0
$211$	−781145.			−1.20788			−0.603942	−	0.797028i	$-0.706404\pi$
$211$	−781145.			−1.20788			−0.603942	+	0.797028i	$0.706404\pi$
$212$	302210.	−	302210.i	0.461817	−	0.461817i
$213$		0			0
$214$			756754.i			1.12959i
$215$		0			0
$216$		0			0
$217$	−49318.6	−	49318.6i	−0.0710986	−	0.0710986i
$218$	−711804.	−	711804.i	−1.01442	−	1.01442i
$219$		0			0
$220$		0			0
$221$			555759.i			0.765431i
$222$		0			0
$223$	−323966.	+	323966.i	−0.436252	+	0.436252i	−0.890749	−	0.454496i	$-0.849819\pi$
$223$	−323966.	+	323966.i	−0.436252	+	0.436252i	0.454496	+	0.890749i	$0.349819\pi$
$224$	118511.			0.157812
$225$		0			0
$226$	283822.			0.369637
$227$	531318.	−	531318.i	0.684368	−	0.684368i	−0.276613	−	0.960981i	$-0.589212\pi$
$227$	531318.	−	531318.i	0.684368	−	0.684368i	0.960981	+	0.276613i	$0.0892121\pi$
$228$		0			0
$229$			242575.i			0.305673i	0.988251	+	0.152837i	$0.0488408\pi$
$229$			242575.i			0.305673i	−0.988251	+	0.152837i	$0.951159\pi$
$230$		0			0
$231$		0			0
$232$	−96286.7	−	96286.7i	−0.117448	−	0.117448i
$233$	1.07761e6	+	1.07761e6i	1.30038	+	1.30038i	0.928135	+	0.372244i	$0.121411\pi$
$233$	1.07761e6	+	1.07761e6i	1.30038	+	1.30038i	0.372244	+	0.928135i	$0.378589\pi$
$234$		0			0
$235$		0			0
$236$			865265.i			1.01127i
$237$		0			0
$238$	−169801.	+	169801.i	−0.194311	+	0.194311i
$239$	−876668.			−0.992752			−0.496376	−	0.868108i	$-0.665336\pi$
$239$	−876668.			−0.992752			−0.496376	+	0.868108i	$0.665336\pi$
$240$		0			0
$241$	−1.17411e6			−1.30217			−0.651085	−	0.759005i	$-0.725686\pi$
$241$	−1.17411e6			−1.30217			−0.651085	+	0.759005i	$0.725686\pi$
$242$	−667121.	+	667121.i	−0.732262	+	0.732262i
$243$		0			0
$244$		−	149697.i		−	0.160968i
$245$		0			0
$246$		0			0
$247$	−442558.	−	442558.i	−0.461560	−	0.461560i
$248$	−128236.	−	128236.i	−0.132398	−	0.132398i
$249$		0			0
$250$		0			0
$251$		−	1.72668e6i		−	1.72993i	−0.501835	−	0.864963i	$-0.667342\pi$
$251$		−	1.72668e6i		−	1.72993i	0.501835	−	0.864963i	$-0.332658\pi$
$252$		0			0
$253$	69026.2	−	69026.2i	0.0677973	−	0.0677973i
$254$	−2.36832e6			−2.30332
$255$		0			0
$256$	1.30460e6			1.24416
$257$	−54001.3	+	54001.3i	−0.0510001	+	0.0510001i	−0.732147	−	0.681147i	$-0.761482\pi$
$257$	−54001.3	+	54001.3i	−0.0510001	+	0.0510001i	0.681147	+	0.732147i	$0.261482\pi$
$258$		0			0
$259$		−	213622.i		−	0.197877i
$260$		0			0
$261$		0			0
$262$	−1.48183e6	−	1.48183e6i	−1.33366	−	1.33366i
$263$	−131461.	−	131461.i	−0.117195	−	0.117195i	0.646077	−	0.763272i	$-0.276408\pi$
$263$	−131461.	−	131461.i	−0.117195	−	0.117195i	−0.763272	+	0.646077i	$0.776408\pi$
$264$		0			0
$265$		0			0
$266$		−	270429.i		−	0.234341i
$267$		0			0
$268$	−1.08285e6	+	1.08285e6i	−0.920941	+	0.920941i
$269$	1.47111e6			1.23955			0.619774	−	0.784781i	$-0.287224\pi$
$269$	1.47111e6			1.23955			0.619774	+	0.784781i	$0.287224\pi$
$270$		0			0
$271$	−206507.			−0.170809			−0.0854047	−	0.996346i	$-0.527218\pi$
$271$	−206507.			−0.170809			−0.0854047	+	0.996346i	$0.527218\pi$
$272$	−1.66265e6	+	1.66265e6i	−1.36264	+	1.36264i
$273$		0			0
$274$		−	803384.i		−	0.646468i
$275$		0			0
$276$		0			0
$277$	872040.	+	872040.i	0.682869	+	0.682869i	0.960646	−	0.277777i	$-0.0895976\pi$
$277$	872040.	+	872040.i	0.682869	+	0.682869i	−0.277777	+	0.960646i	$0.589598\pi$
$278$	1.21815e6	+	1.21815e6i	0.945341	+	0.945341i
$279$		0			0
$280$		0			0
$281$			1.61973e6i			1.22371i	0.790971	+	0.611854i	$0.209576\pi$
$281$			1.61973e6i			1.22371i	−0.790971	+	0.611854i	$0.790424\pi$
$282$		0			0
$283$	786552.	−	786552.i	0.583796	−	0.583796i	−0.352148	−	0.935944i	$-0.614549\pi$
$283$	786552.	−	786552.i	0.583796	−	0.583796i	0.935944	+	0.352148i	$0.114549\pi$
$284$	−588849.			−0.433220
$285$		0			0
$286$	411001.			0.297117
$287$	177704.	−	177704.i	0.127348	−	0.127348i
$288$		0			0
$289$		−	2.65528e6i		−	1.87011i
$290$		0			0
$291$		0			0
$292$	701429.	+	701429.i	0.481423	+	0.481423i
$293$	852510.	+	852510.i	0.580137	+	0.580137i	0.934941	−	0.354804i	$-0.115452\pi$
$293$	852510.	+	852510.i	0.580137	+	0.580137i	−0.354804	+	0.934941i	$0.615452\pi$
$294$		0			0
$295$		0			0
$296$		−	555449.i		−	0.368481i
$297$		0			0
$298$	−893550.	+	893550.i	−0.582879	+	0.582879i
$299$	137962.			0.0892448
$300$		0			0
$301$	135704.			0.0863327
$302$	−2.12562e6	+	2.12562e6i	−1.34112	+	1.34112i
$303$		0			0
$304$		−	2.64798e6i		−	1.64336i
$305$		0			0
$306$		0			0
$307$	642741.	+	642741.i	0.389216	+	0.389216i	0.874408	−	0.485192i	$-0.161250\pi$
$307$	642741.	+	642741.i	0.389216	+	0.389216i	−0.485192	+	0.874408i	$0.661250\pi$
$308$	57156.9	+	57156.9i	0.0343314	+	0.0343314i
$309$		0			0
$310$		0			0
$311$			2.78173e6i			1.63085i	0.578863	+	0.815425i	$0.303497\pi$
$311$			2.78173e6i			1.63085i	−0.578863	+	0.815425i	$0.696503\pi$
$312$		0			0
$313$	−751913.	+	751913.i	−0.433817	+	0.433817i	−0.889925	−	0.456107i	$-0.849243\pi$
$313$	−751913.	+	751913.i	−0.433817	+	0.433817i	0.456107	+	0.889925i	$0.349243\pi$
$314$	−1.16198e6			−0.665082
$315$		0			0
$316$	1.89381e6			1.06689
$317$	4557.50	−	4557.50i	0.00254729	−	0.00254729i	−0.705832	−	0.708379i	$-0.749427\pi$
$317$	4557.50	−	4557.50i	0.00254729	−	0.00254729i	0.708379	+	0.705832i	$0.249427\pi$
$318$		0			0
$319$		−	657245.i		−	0.361618i
$320$		0			0
$321$		0			0
$322$	42151.5	+	42151.5i	0.0226555	+	0.0226555i
$323$	3.24509e6	+	3.24509e6i	1.73069	+	1.73069i
$324$		0			0
$325$		0			0
$326$			790481.i			0.411953i
$327$		0			0
$328$	462057.	−	462057.i	0.237143	−	0.237143i
$329$	112477.			0.0572891
$330$		0			0
$331$	−2.75226e6			−1.38076			−0.690382	−	0.723445i	$-0.742557\pi$
$331$	−2.75226e6			−1.38076			−0.690382	+	0.723445i	$0.742557\pi$
$332$	−332852.	+	332852.i	−0.165732	+	0.165732i
$333$		0			0
$334$			429377.i			0.210607i
$335$		0			0
$336$		0			0
$337$	949071.	+	949071.i	0.455223	+	0.455223i	0.897084	−	0.441861i	$-0.145682\pi$
$337$	949071.	+	949071.i	0.455223	+	0.455223i	−0.441861	+	0.897084i	$0.645682\pi$
$338$	−1.60135e6	−	1.60135e6i	−0.762420	−	0.762420i
$339$		0			0
$340$		0			0
$341$		−	875327.i		−	0.407647i
$342$		0			0
$343$	−366286.	+	366286.i	−0.168107	+	0.168107i
$344$	352850.			0.160766
$345$		0			0
$346$	3.29219e6			1.47841
$347$	1.15265e6	−	1.15265e6i	0.513895	−	0.513895i	−0.401823	−	0.915717i	$-0.631623\pi$
$347$	1.15265e6	−	1.15265e6i	0.513895	−	0.513895i	0.915717	+	0.401823i	$0.131623\pi$
$348$		0			0
$349$		−	1.38647e6i		−	0.609320i	−0.952461	−	0.304660i	$-0.901457\pi$
$349$		−	1.38647e6i		−	0.609320i	0.952461	−	0.304660i	$-0.0985429\pi$
$350$		0			0
$351$		0			0
$352$	1.05169e6	+	1.05169e6i	0.452410	+	0.452410i
$353$	336957.	+	336957.i	0.143925	+	0.143925i	0.775398	−	0.631473i	$-0.217549\pi$
$353$	336957.	+	336957.i	0.143925	+	0.143925i	−0.631473	+	0.775398i	$0.717549\pi$
$354$		0			0
$355$		0			0
$356$			2.43695e6i			1.01911i
$357$		0			0
$358$	−1.82677e6	+	1.82677e6i	−0.753316	+	0.753316i
$359$	3.85085e6			1.57696			0.788480	−	0.615061i	$-0.210869\pi$
$359$	3.85085e6			1.57696			0.788480	+	0.615061i	$0.210869\pi$
$360$		0			0
$361$	−2.69210e6			−1.08724
$362$	−246538.	+	246538.i	−0.0988808	+	0.0988808i
$363$		0			0
$364$			114239.i			0.0451921i
$365$		0			0
$366$		0			0
$367$	1.08759e6	+	1.08759e6i	0.421501	+	0.421501i	0.885720	−	0.464219i	$-0.153665\pi$
$367$	1.08759e6	+	1.08759e6i	0.421501	+	0.421501i	−0.464219	+	0.885720i	$0.653665\pi$
$368$	412740.	+	412740.i	0.158875	+	0.158875i
$369$		0			0
$370$		0			0
$371$		−	248143.i		−	0.0935981i
$372$		0			0
$373$	2.46018e6	−	2.46018e6i	0.915575	−	0.915575i	−0.0811283	−	0.996704i	$-0.525852\pi$
$373$	2.46018e6	−	2.46018e6i	0.915575	−	0.915575i	0.996704	+	0.0811283i	$0.0258524\pi$
$374$	−3.01369e6			−1.11409
$375$		0			0
$376$	292456.			0.106682
$377$	656817.	−	656817.i	0.238008	−	0.238008i
$378$		0			0
$379$		−	236094.i		−	0.0844282i	−0.999109	−	0.0422141i	$-0.986559\pi$
$379$		−	236094.i		−	0.0844282i	0.999109	−	0.0422141i	$-0.0134412\pi$
$380$		0			0
$381$		0			0
$382$	−4.57335e6	−	4.57335e6i	−1.60352	−	1.60352i
$383$	−1.14707e6	−	1.14707e6i	−0.399570	−	0.399570i	0.478511	−	0.878081i	$-0.341177\pi$
$383$	−1.14707e6	−	1.14707e6i	−0.399570	−	0.399570i	−0.878081	+	0.478511i	$0.841177\pi$
$384$		0			0
$385$		0			0
$386$		−	4.92495e6i		−	1.68242i
$387$		0			0
$388$	48010.3	−	48010.3i	0.0161903	−	0.0161903i
$389$	−4.13785e6			−1.38644			−0.693219	−	0.720727i	$-0.743808\pi$
$389$	−4.13785e6			−1.38644			−0.693219	+	0.720727i	$0.743808\pi$
$390$		0			0
$391$	−1.01162e6			−0.334638
$392$	−472762.	+	472762.i	−0.155391	+	0.155391i
$393$		0			0
$394$			7.85812e6i			2.55022i
$395$		0			0
$396$		0			0
$397$	−3.87558e6	−	3.87558e6i	−1.23413	−	1.23413i	−0.962363	−	0.271766i	$-0.912392\pi$
$397$	−3.87558e6	−	3.87558e6i	−1.23413	−	1.23413i	−0.271766	−	0.962363i	$-0.587608\pi$
$398$	−4.54482e6	−	4.54482e6i	−1.43817	−	1.43817i
$399$		0			0
$400$		0			0
$401$		−	3.46163e6i		−	1.07503i	−0.843255	−	0.537514i	$-0.819363\pi$
$401$		−	3.46163e6i		−	1.07503i	0.843255	−	0.537514i	$-0.180637\pi$
$402$		0			0
$403$	874756.	−	874756.i	0.268302	−	0.268302i
$404$	3.45669e6			1.05368
$405$		0			0
$406$	401353.			0.120840
$407$	1.89572e6	−	1.89572e6i	0.567269	−	0.567269i
$408$		0			0
$409$			5.21009e6i			1.54006i	0.638009	+	0.770029i	$0.279758\pi$
$409$			5.21009e6i			1.54006i	−0.638009	+	0.770029i	$0.720242\pi$
$410$		0			0
$411$		0			0
$412$	3.69437e6	+	3.69437e6i	1.07225	+	1.07225i
$413$	355231.	+	355231.i	0.102479	+	0.102479i
$414$		0			0
$415$		0			0
$416$			2.10201e6i			0.595528i
$417$		0			0
$418$	2.39984e6	−	2.39984e6i	0.671803	−	0.671803i
$419$	1.98439e6			0.552195			0.276097	−	0.961130i	$-0.410959\pi$
$419$	1.98439e6			0.552195			0.276097	+	0.961130i	$0.410959\pi$
$420$		0			0
$421$	−3.18296e6			−0.875236			−0.437618	−	0.899161i	$-0.644178\pi$
$421$	−3.18296e6			−0.875236			−0.437618	+	0.899161i	$0.644178\pi$
$422$	4.23312e6	−	4.23312e6i	1.15712	−	1.15712i
$423$		0			0
$424$		−	645209.i		−	0.174295i
$425$		0			0
$426$		0			0
$427$	−61457.6	−	61457.6i	−0.0163120	−	0.0163120i
$428$	−1.86662e6	−	1.86662e6i	−0.492547	−	0.492547i
$429$		0			0
$430$		0			0
$431$		−	3.76453e6i		−	0.976153i	−0.872801	−	0.488077i	$-0.837699\pi$
$431$		−	3.76453e6i		−	0.976153i	0.872801	−	0.488077i	$-0.162301\pi$
$432$		0			0
$433$	−116515.	+	116515.i	−0.0298651	+	0.0298651i	−0.721882	−	0.692017i	$-0.756723\pi$
$433$	−116515.	+	116515.i	−0.0298651	+	0.0298651i	0.692017	+	0.721882i	$0.256723\pi$
$434$	534527.			0.136221
$435$		0			0
$436$	3.51150e6			0.884661
$437$	805565.	−	805565.i	0.201789	−	0.201789i
$438$		0			0
$439$			2.35575e6i			0.583403i	0.956509	+	0.291701i	$0.0942213\pi$
$439$			2.35575e6i			0.583403i	−0.956509	+	0.291701i	$0.905779\pi$
$440$		0			0
$441$		0			0
$442$	−3.01173e6	−	3.01173e6i	−0.733264	−	0.733264i
$443$	−4.26536e6	−	4.26536e6i	−1.03263	−	1.03263i	−0.999449	−	0.0331846i	$-0.989435\pi$
$443$	−4.26536e6	−	4.26536e6i	−1.03263	−	1.03263i	−0.0331846	−	0.999449i	$-0.510565\pi$
$444$		0			0
$445$		0			0
$446$		−	3.51123e6i		−	0.835837i
$447$		0			0
$448$	−233136.	+	233136.i	−0.0548801	+	0.0548801i
$449$	−1.38214e6			−0.323546			−0.161773	−	0.986828i	$-0.551721\pi$
$449$	−1.38214e6			−0.323546			−0.161773	+	0.986828i	$0.551721\pi$
$450$		0			0
$451$	3.15396e6			0.730155
$452$	−700082.	+	700082.i	−0.161177	+	0.161177i
$453$		0			0
$454$			5.75856e6i			1.31122i
$455$		0			0
$456$		0			0
$457$	1.05756e6	+	1.05756e6i	0.236872	+	0.236872i	0.815554	−	0.578682i	$-0.196433\pi$
$457$	1.05756e6	+	1.05756e6i	0.236872	+	0.236872i	−0.578682	+	0.815554i	$0.696433\pi$
$458$	−1.31454e6	−	1.31454e6i	−0.292827	−	0.292827i
$459$		0			0
$460$		0			0
$461$			4.58869e6i			1.00563i	0.864395	+	0.502813i	$0.167702\pi$
$461$			4.58869e6i			1.00563i	−0.864395	+	0.502813i	$0.832298\pi$
$462$		0			0
$463$	65050.0	−	65050.0i	0.0141025	−	0.0141025i	−0.700020	−	0.714123i	$-0.746826\pi$
$463$	65050.0	−	65050.0i	0.0141025	−	0.0141025i	0.714123	+	0.700020i	$0.246826\pi$
$464$	3.92997e6			0.847412
$465$		0			0
$466$	−1.16794e7			−2.49146
$467$	1.59074e6	−	1.59074e6i	0.337525	−	0.337525i	−0.517910	−	0.855435i	$-0.673290\pi$
$467$	1.59074e6	−	1.59074e6i	0.337525	−	0.337525i	0.855435	+	0.517910i	$0.173290\pi$
$468$		0			0
$469$			889122.i			0.186650i
$470$		0			0
$471$		0			0
$472$	923656.	+	923656.i	0.190834	+	0.190834i
$473$	1.20426e6	+	1.20426e6i	0.247496	+	0.247496i
$474$		0			0
$475$		0			0
$476$		−	837667.i		−	0.169455i
$477$		0			0
$478$	4.75078e6	−	4.75078e6i	0.951031	−	0.951031i
$479$	2.60690e6			0.519141			0.259570	−	0.965724i	$-0.416419\pi$
$479$	2.60690e6			0.519141			0.259570	+	0.965724i	$0.416419\pi$
$480$		0			0
$481$	3.78898e6			0.746722
$482$	6.36268e6	−	6.36268e6i	1.24745	−	1.24745i
$483$		0			0
$484$		−	3.29107e6i		−	0.638592i
$485$		0			0
$486$		0			0
$487$	2.21227e6	+	2.21227e6i	0.422683	+	0.422683i	0.886127	−	0.463443i	$-0.153386\pi$
$487$	2.21227e6	+	2.21227e6i	0.422683	+	0.422683i	−0.463443	+	0.886127i	$0.653386\pi$
$488$	−159799.	−	159799.i	−0.0303756	−	0.0303756i
$489$		0			0
$490$		0			0
$491$			5.58021e6i			1.04459i	0.852764	+	0.522297i	$0.174925\pi$
$491$			5.58021e6i			1.04459i	−0.852764	+	0.522297i	$0.825075\pi$
$492$		0			0
$493$	−4.81615e6	+	4.81615e6i	−0.892448	+	0.892448i
$494$	4.79656e6			0.884325
$495$		0			0
$496$	5.23398e6			0.955274
$497$	−241750.	+	241750.i	−0.0439011	+	0.0439011i
$498$		0			0
$499$		−	1.39810e6i		−	0.251354i	−0.992071	−	0.125677i	$-0.959890\pi$
$499$		−	1.39810e6i		−	0.251354i	0.992071	−	0.125677i	$-0.0401104\pi$
$500$		0			0
$501$		0			0
$502$	9.35710e6	+	9.35710e6i	1.65723	+	1.65723i
$503$	−471587.	−	471587.i	−0.0831078	−	0.0831078i	0.664331	−	0.747439i	$-0.268717\pi$
$503$	−471587.	−	471587.i	−0.0831078	−	0.0831078i	−0.747439	+	0.664331i	$0.768717\pi$
$504$		0			0
$505$		0			0
$506$			748123.i			0.129896i
$507$		0			0
$508$	5.84173e6	−	5.84173e6i	1.00434	−	1.00434i
$509$	3.97703e6			0.680399			0.340200	−	0.940353i	$-0.389505\pi$
$509$	3.97703e6			0.680399			0.340200	+	0.940353i	$0.389505\pi$
$510$		0			0
$511$	575938.			0.0975717
$512$	−5.22486e6	+	5.22486e6i	−0.880845	+	0.880845i
$513$		0			0
$514$		−	585280.i		−	0.0977137i
$515$		0			0
$516$		0			0
$517$	998140.	+	998140.i	0.164235	+	0.164235i
$518$	1.15764e6	+	1.15764e6i	0.189561	+	0.189561i
$519$		0			0
$520$		0			0
$521$			1.70745e6i			0.275585i	0.990461	+	0.137792i	$0.0440006\pi$
$521$			1.70745e6i			0.275585i	−0.990461	+	0.137792i	$0.955999\pi$
$522$		0			0
$523$	3.03074e6	−	3.03074e6i	0.484501	−	0.484501i	−0.422065	−	0.906566i	$-0.638695\pi$
$523$	3.03074e6	−	3.03074e6i	0.484501	−	0.484501i	0.906566	+	0.422065i	$0.138695\pi$
$524$	7.31022e6			1.16306
$525$		0			0
$526$	1.42481e6			0.224539
$527$	−6.41421e6	+	6.41421e6i	−1.00604	+	1.00604i
$528$		0			0
$529$		−	6.18522e6i		−	0.960983i
$530$		0			0
$531$		0			0
$532$	667045.	+	667045.i	0.102182	+	0.102182i
$533$	3.15191e6	+	3.15191e6i	0.480569	+	0.480569i
$534$		0			0
$535$		0			0
$536$			2.31185e6i			0.347574i
$537$		0			0
$538$	−7.97211e6	+	7.97211e6i	−1.18746	+	1.18746i
$539$	−3.22703e6			−0.478444
$540$		0			0
$541$	7.93011e6			1.16489			0.582446	−	0.812869i	$-0.302096\pi$
$541$	7.93011e6			1.16489			0.582446	+	0.812869i	$0.302096\pi$
$542$	1.11909e6	−	1.11909e6i	0.163631	−	0.163631i
$543$		0			0
$544$		−	1.54132e7i		−	2.23303i
$545$		0			0
$546$		0			0
$547$	347229.	+	347229.i	0.0496189	+	0.0496189i	0.731481	−	0.681862i	$-0.238830\pi$
$547$	347229.	+	347229.i	0.0496189	+	0.0496189i	−0.681862	+	0.731481i	$0.738830\pi$
$548$	1.98164e6	+	1.98164e6i	0.281886	+	0.281886i
$549$		0			0
$550$		0			0
$551$		−	7.67032e6i		−	1.07630i
$552$		0			0
$553$	777498.	−	777498.i	0.108115	−	0.108115i
$554$	−9.45139e6			−1.30834
$555$		0			0
$556$	−6.00942e6			−0.824414
$557$	−5.98254e6	+	5.98254e6i	−0.817048	+	0.817048i	−0.985679	−	0.168631i	$-0.946065\pi$
$557$	−5.98254e6	+	5.98254e6i	−0.817048	+	0.817048i	0.168631	+	0.985679i	$0.446065\pi$
$558$		0			0
$559$			2.40696e6i			0.325791i
$560$		0			0
$561$		0			0
$562$	−8.77754e6	−	8.77754e6i	−1.17228	−	1.17228i
$563$	−8.86463e6	−	8.86463e6i	−1.17866	−	1.17866i	−0.980085	−	0.198577i	$-0.936368\pi$
$563$	−8.86463e6	−	8.86463e6i	−1.17866	−	1.17866i	−0.198577	−	0.980085i	$-0.563632\pi$
$564$		0			0
$565$		0			0
$566$			8.52485e6i			1.11852i
$567$		0			0
$568$	−628586.	+	628586.i	−0.0817512	+	0.0817512i
$569$	−263589.			−0.0341308			−0.0170654	−	0.999854i	$-0.505432\pi$
$569$	−263589.			−0.0341308			−0.0170654	+	0.999854i	$0.505432\pi$
$570$		0			0
$571$	1.24926e7			1.60347			0.801737	−	0.597677i	$-0.203910\pi$
$571$	1.24926e7			1.60347			0.801737	+	0.597677i	$0.203910\pi$
$572$	−1.01378e6	+	1.01378e6i	−0.129555	+	0.129555i
$573$		0			0
$574$			1.92600e6i			0.243992i
$575$		0			0
$576$		0			0
$577$	3.28115e6	+	3.28115e6i	0.410286	+	0.410286i	0.881838	−	0.471552i	$-0.156306\pi$
$577$	3.28115e6	+	3.28115e6i	0.410286	+	0.410286i	−0.471552	+	0.881838i	$0.656306\pi$
$578$	1.43893e7	+	1.43893e7i	1.79152	+	1.79152i
$579$		0			0
$580$		0			0
$581$			273302.i			0.0335894i
$582$		0			0
$583$	2.20207e6	−	2.20207e6i	0.268324	−	0.268324i
$584$	1.49753e6			0.181695
$585$		0			0
$586$	−9.23972e6			−1.11151
$587$	−3.85645e6	+	3.85645e6i	−0.461948	+	0.461948i	−0.899293	−	0.437346i	$-0.855919\pi$
$587$	−3.85645e6	+	3.85645e6i	−0.461948	+	0.461948i	0.437346	+	0.899293i	$0.355919\pi$
$588$		0			0
$589$		−	1.02154e7i		−	1.21330i
$590$		0			0
$591$		0			0
$592$	1.13354e7	+	1.13354e7i	1.32933	+	1.32933i
$593$	−7.34203e6	−	7.34203e6i	−0.857392	−	0.857392i	0.133638	−	0.991030i	$-0.457334\pi$
$593$	−7.34203e6	−	7.34203e6i	−0.857392	−	0.857392i	−0.991030	+	0.133638i	$0.957334\pi$
$594$		0			0
$595$		0			0
$596$		−	4.40810e6i		−	0.508318i
$597$		0			0
$598$	−747636.	+	747636.i	−0.0854943	+	0.0854943i
$599$	−2.16225e6			−0.246228			−0.123114	−	0.992393i	$-0.539288\pi$
$599$	−2.16225e6			−0.246228			−0.123114	+	0.992393i	$0.539288\pi$
$600$		0			0
$601$	4.31939e6			0.487793			0.243897	−	0.969801i	$-0.421574\pi$
$601$	4.31939e6			0.487793			0.243897	+	0.969801i	$0.421574\pi$
$602$	−735395.	+	735395.i	−0.0827046	+	0.0827046i
$603$		0			0
$604$		−	1.04862e7i		−	1.16957i
$605$		0			0
$606$		0			0
$607$	−1.11920e7	−	1.11920e7i	−1.23292	−	1.23292i	−0.962837	−	0.270082i	$-0.912949\pi$
$607$	−1.11920e7	−	1.11920e7i	−1.23292	−	1.23292i	−0.270082	−	0.962837i	$-0.587051\pi$
$608$	1.22737e7	+	1.22737e7i	1.34653	+	1.34653i
$609$		0			0
$610$		0			0
$611$			1.99498e6i			0.216190i
$612$		0			0
$613$	−9.40080e6	+	9.40080e6i	−1.01045	+	1.01045i	−0.0105019	+	0.999945i	$0.503343\pi$
$613$	−9.40080e6	+	9.40080e6i	−1.01045	+	1.01045i	−0.999945	+	0.0105019i	$0.996657\pi$
$614$	−6.96619e6			−0.745718
$615$		0			0
$616$	122028.			0.0129571
$617$	−9.23092e6	+	9.23092e6i	−0.976185	+	0.976185i	−0.999723	−	0.0235377i	$-0.992507\pi$
$617$	−9.23092e6	+	9.23092e6i	−0.976185	+	0.976185i	0.0235377	+	0.999723i	$0.492507\pi$
$618$		0			0
$619$			1.66306e7i			1.74454i	0.489027	+	0.872269i	$0.337352\pi$
$619$			1.66306e7i			1.74454i	−0.489027	+	0.872269i	$0.662648\pi$
$620$		0			0
$621$		0			0
$622$	−1.50745e7	−	1.50745e7i	−1.56231	−	1.56231i
$623$	1.00048e6	+	1.00048e6i	0.103273	+	0.103273i
$624$		0			0
$625$		0			0
$626$		−	8.14942e6i		−	0.831172i
$627$		0			0
$628$	2.86617e6	−	2.86617e6i	0.290003	−	0.290003i
$629$	−2.77829e7			−2.79995
$630$		0			0
$631$	−872532.			−0.0872385			−0.0436192	−	0.999048i	$-0.513889\pi$
$631$	−872532.			−0.0872385			−0.0436192	+	0.999048i	$0.513889\pi$
$632$	2.02161e6	−	2.02161e6i	0.201328	−	0.201328i
$633$		0			0
$634$			49395.4i			0.00488049i
$635$		0			0
$636$		0			0
$637$	−3.22493e6	−	3.22493e6i	−0.314899	−	0.314899i
$638$	3.56169e6	+	3.56169e6i	0.346422	+	0.346422i
$639$		0			0
$640$		0			0
$641$		−	2.87418e6i		−	0.276292i	−0.990412	−	0.138146i	$-0.955886\pi$
$641$		−	2.87418e6i		−	0.276292i	0.990412	−	0.138146i	$-0.0441143\pi$
$642$		0			0
$643$	−1.25870e7	+	1.25870e7i	−1.20059	+	1.20059i	−0.226599	+	0.973988i	$0.572761\pi$
$643$	−1.25870e7	+	1.25870e7i	−1.20059	+	1.20059i	−0.973988	+	0.226599i	$0.927239\pi$
$644$	−207944.			−0.0197575
$645$		0			0
$646$	−3.51711e7			−3.31592
$647$	1.15062e7	−	1.15062e7i	1.08062	−	1.08062i	0.0841657	−	0.996452i	$-0.473178\pi$
$647$	1.15062e7	−	1.15062e7i	1.08062	−	1.08062i	0.996452	−	0.0841657i	$-0.0268225\pi$
$648$		0			0
$649$			6.30479e6i			0.587569i
$650$		0			0
$651$		0			0
$652$	−1.94982e6	−	1.94982e6i	−0.179628	−	0.179628i
$653$	1.10167e7	+	1.10167e7i	1.01104	+	1.01104i	0.999938	+	0.0111059i	$0.00353519\pi$
$653$	1.10167e7	+	1.10167e7i	1.01104	+	1.01104i	0.0111059	+	0.999938i	$0.496465\pi$
$654$		0			0
$655$		0			0
$656$			1.88590e7i			1.71104i
$657$		0			0
$658$	−609525.	+	609525.i	−0.0548816	+	0.0548816i
$659$	−3.56502e6			−0.319778			−0.159889	−	0.987135i	$-0.551114\pi$
$659$	−3.56502e6			−0.319778			−0.159889	+	0.987135i	$0.551114\pi$
$660$		0			0
$661$	609051.			0.0542188			0.0271094	−	0.999632i	$-0.491370\pi$
$661$	609051.			0.0542188			0.0271094	+	0.999632i	$0.491370\pi$
$662$	1.49148e7	−	1.49148e7i	1.32274	−	1.32274i
$663$		0			0
$664$			710627.i			0.0625492i
$665$		0			0
$666$		0			0
$667$	1.19557e6	+	1.19557e6i	0.104054	+	0.104054i
$668$	−1.05911e6	−	1.05911e6i	−0.0918332	−	0.0918332i
$669$		0			0
$670$		0			0
$671$		−	1.09078e6i		−	0.0935253i
$672$		0			0
$673$	−6.32688e6	+	6.32688e6i	−0.538458	+	0.538458i	−0.923076	−	0.384618i	$-0.874333\pi$
$673$	−6.32688e6	+	6.32688e6i	−0.538458	+	0.538458i	0.384618	+	0.923076i	$0.374333\pi$
$674$	−1.02863e7			−0.872184
$675$		0			0
$676$	7.89984e6			0.664893
$677$	−4.90614e6	+	4.90614e6i	−0.411404	+	0.411404i	−0.882227	−	0.470824i	$-0.843957\pi$
$677$	−4.90614e6	+	4.90614e6i	−0.411404	+	0.411404i	0.470824	+	0.882227i	$0.343957\pi$
$678$		0			0
$679$		−	39420.9i		−	0.00328135i
$680$		0			0
$681$		0			0
$682$	4.74350e6	+	4.74350e6i	0.390516	+	0.390516i
$683$	−1.03107e7	−	1.03107e7i	−0.845741	−	0.845741i	0.143857	−	0.989598i	$-0.454049\pi$
$683$	−1.03107e7	−	1.03107e7i	−0.845741	−	0.845741i	−0.989598	+	0.143857i	$0.954049\pi$
$684$		0			0
$685$		0			0
$686$		−	3.96990e6i		−	0.322084i
$687$		0			0
$688$	−7.20084e6	+	7.20084e6i	−0.579979	+	0.579979i
$689$	4.40127e6			0.353208
$690$		0			0
$691$	−4.75927e6			−0.379180			−0.189590	−	0.981863i	$-0.560716\pi$
$691$	−4.75927e6			−0.379180			−0.189590	+	0.981863i	$0.560716\pi$
$692$	−8.12059e6	+	8.12059e6i	−0.644648	+	0.644648i
$693$		0			0
$694$			1.24927e7i			0.984597i
$695$		0			0
$696$		0			0
$697$	−2.31116e7	−	2.31116e7i	−1.80197	−	1.80197i
$698$	7.51343e6	+	7.51343e6i	0.583714	+	0.583714i
$699$		0			0
$700$		0			0
$701$			1.82391e6i			0.140187i	0.997540	+	0.0700937i	$0.0223298\pi$
$701$			1.82391e6i			0.140187i	−0.997540	+	0.0700937i	$0.977670\pi$
$702$		0			0
$703$	2.21239e7	−	2.21239e7i	1.68839	−	1.68839i
$704$	−4.13780e6			−0.314657
$705$		0			0
$706$	−3.65202e6			−0.275754
$707$	1.41913e6	−	1.41913e6i	0.106776	−	0.106776i
$708$		0			0
$709$		−	2.24106e6i		−	0.167432i	−0.996490	−	0.0837158i	$-0.973321\pi$
$709$		−	2.24106e6i		−	0.167432i	0.996490	−	0.0837158i	$-0.0266788\pi$
$710$		0			0
$711$		0			0
$712$	2.60140e6	+	2.60140e6i	0.192312	+	0.192312i
$713$	1.59227e6	+	1.59227e6i	0.117299	+	0.117299i
$714$		0			0
$715$		0			0
$716$		−	9.01191e6i		−	0.656953i
$717$		0			0
$718$	−2.08682e7	+	2.08682e7i	−1.51069	+	1.51069i
$719$	1.26499e6			0.0912565			0.0456283	−	0.998958i	$-0.485471\pi$
$719$	1.26499e6			0.0912565			0.0456283	+	0.998958i	$0.485471\pi$
$720$		0			0
$721$	3.03342e6			0.217317
$722$	1.45889e7	−	1.45889e7i	1.04155	−	1.04155i
$723$		0			0
$724$		−	1.21623e6i		−	0.0862322i
$725$		0			0
$726$		0			0
$727$	1.55230e7	+	1.55230e7i	1.08928	+	1.08928i	0.995603	+	0.0936761i	$0.0298618\pi$
$727$	1.55230e7	+	1.55230e7i	1.08928	+	1.08928i	0.0936761	+	0.995603i	$0.470138\pi$
$728$	121949.	+	121949.i	0.00852802	+	0.00852802i
$729$		0			0
$730$		0			0
$731$		−	1.76492e7i		−	1.22160i
$732$		0			0
$733$	6.63098e6	−	6.63098e6i	0.455846	−	0.455846i	−0.441443	−	0.897289i	$-0.645533\pi$
$733$	6.63098e6	−	6.63098e6i	0.455846	−	0.455846i	0.897289	+	0.441443i	$0.145533\pi$
$734$	−1.17875e7			−0.807576
$735$		0			0
$736$	−3.82618e6			−0.260358
$737$	−7.89025e6	+	7.89025e6i	−0.535084	+	0.535084i
$738$		0			0
$739$			6.71487e6i			0.452300i	0.974092	+	0.226150i	$0.0726140\pi$
$739$			6.71487e6i			0.452300i	−0.974092	+	0.226150i	$0.927386\pi$
$740$		0			0
$741$		0			0
$742$	1.34472e6	+	1.34472e6i	0.0896646	+	0.0896646i
$743$	−7.01273e6	−	7.01273e6i	−0.466031	−	0.466031i	0.434595	−	0.900626i	$-0.356892\pi$
$743$	−7.01273e6	−	7.01273e6i	−0.466031	−	0.466031i	−0.900626	+	0.434595i	$0.856892\pi$
$744$		0			0
$745$		0			0
$746$			2.66640e7i			1.75420i
$747$		0			0
$748$	7.43363e6	−	7.43363e6i	0.485788	−	0.485788i
$749$	−1.53267e6			−0.0998262
$750$		0			0
$751$	1.08177e7			0.699896			0.349948	−	0.936769i	$-0.386199\pi$
$751$	1.08177e7			0.699896			0.349948	+	0.936769i	$0.386199\pi$
$752$	−5.96834e6	+	5.96834e6i	−0.384866	+	0.384866i
$753$		0			0
$754$			7.11875e6i			0.456011i
$755$		0			0
$756$		0			0
$757$	−1.46756e7	−	1.46756e7i	−0.930798	−	0.930798i	0.0669581	−	0.997756i	$-0.478671\pi$
$757$	−1.46756e7	−	1.46756e7i	−0.930798	−	0.930798i	−0.997756	+	0.0669581i	$0.978671\pi$
$758$	1.27943e6	+	1.27943e6i	0.0808802	+	0.0808802i
$759$		0			0
$760$		0			0
$761$		−	1.75265e7i		−	1.09707i	−0.836127	−	0.548535i	$-0.815186\pi$
$761$		−	1.75265e7i		−	1.09707i	0.836127	−	0.548535i	$-0.184814\pi$
$762$		0			0
$763$	1.44163e6	−	1.44163e6i	0.0896487	−	0.0896487i
$764$	2.25614e7			1.39841
$765$		0			0
$766$	1.24322e7			0.765556
$767$	−6.30069e6	+	6.30069e6i	−0.386723	+	0.386723i
$768$		0			0
$769$		−	1.48329e7i		−	0.904501i	−0.891891	−	0.452251i	$-0.850621\pi$
$769$		−	1.48329e7i		−	0.904501i	0.891891	−	0.452251i	$-0.149379\pi$
$770$		0			0
$771$		0			0
$772$	1.21480e7	+	1.21480e7i	0.733602	+	0.733602i
$773$	−1.30854e7	−	1.30854e7i	−0.787658	−	0.787658i	0.193451	−	0.981110i	$-0.438032\pi$
$773$	−1.30854e7	−	1.30854e7i	−0.787658	−	0.787658i	−0.981110	+	0.193451i	$0.938032\pi$
$774$		0			0
$775$		0			0
$776$		−	102500.i		−	0.00611042i
$777$		0			0
$778$	2.24235e7	−	2.24235e7i	1.32817	−	1.32817i
$779$	3.68081e7			2.17320
$780$		0			0
$781$	−4.29068e6			−0.251709
$782$	5.48209e6	−	5.48209e6i	0.320575	−	0.320575i
$783$		0			0
$784$		−	1.92959e7i		−	1.12118i
$785$		0			0
$786$		0			0
$787$	1.07428e7	+	1.07428e7i	0.618273	+	0.618273i	0.945088	−	0.326815i	$-0.105975\pi$
$787$	1.07428e7	+	1.07428e7i	0.618273	+	0.618273i	−0.326815	+	0.945088i	$0.605975\pi$
$788$	−1.93830e7	−	1.93830e7i	−1.11200	−	1.11200i
$789$		0			0
$790$		0			0
$791$			574832.i			0.0326663i
$792$		0			0
$793$	1.09007e6	−	1.09007e6i	0.0615559	−	0.0615559i
$794$	4.20045e7			2.36453
$795$		0			0
$796$	2.24207e7			1.25420
$797$	−1.18496e7	+	1.18496e7i	−0.660780	+	0.660780i	−0.955564	−	0.294784i	$-0.904752\pi$
$797$	−1.18496e7	+	1.18496e7i	−0.660780	+	0.660780i	0.294784	+	0.955564i	$0.404752\pi$
$798$		0			0
$799$		−	1.46283e7i		−	0.810639i
$800$		0			0
$801$		0			0
$802$	1.87590e7	+	1.87590e7i	1.02985	+	1.02985i
$803$	5.11100e6	+	5.11100e6i	0.279716	+	0.279716i
$804$		0			0
$805$		0			0
$806$			9.48083e6i			0.514054i
$807$		0			0
$808$	3.68996e6	−	3.68996e6i	0.198835	−	0.198835i
$809$	−2.70325e7			−1.45216			−0.726081	−	0.687610i	$-0.758660\pi$
$809$	−2.70325e7			−1.45216			−0.726081	+	0.687610i	$0.758660\pi$
$810$		0			0
$811$	2.53514e7			1.35347			0.676737	−	0.736225i	$-0.263394\pi$
$811$	2.53514e7			1.35347			0.676737	+	0.736225i	$0.263394\pi$
$812$	−989986.	+	989986.i	−0.0526913	+	0.0526913i
$813$		0			0
$814$			2.05463e7i			1.08686i
$815$		0			0
$816$		0			0
$817$	1.40542e7	+	1.40542e7i	0.736635	+	0.736635i
$818$	−2.82341e7	−	2.82341e7i	−1.47534	−	1.47534i
$819$		0			0
$820$		0			0
$821$		−	1.15371e7i		−	0.597366i	−0.954352	−	0.298683i	$-0.903453\pi$
$821$		−	1.15371e7i		−	0.597366i	0.954352	−	0.298683i	$-0.0965472\pi$
$822$		0			0
$823$	1.30562e7	−	1.30562e7i	0.671918	−	0.671918i	−0.286240	−	0.958158i	$-0.592405\pi$
$823$	1.30562e7	−	1.30562e7i	0.671918	−	0.671918i	0.958158	+	0.286240i	$0.0924055\pi$
$824$	7.88735e6			0.404681
$825$		0			0
$826$	−3.85009e6			−0.196345
$827$	−6.81499e6	+	6.81499e6i	−0.346499	+	0.346499i	−0.858804	−	0.512305i	$-0.828792\pi$
$827$	−6.81499e6	+	6.81499e6i	−0.346499	+	0.346499i	0.512305	+	0.858804i	$0.328792\pi$
$828$		0			0
$829$		−	1.23963e7i		−	0.626476i	−0.949675	−	0.313238i	$-0.898586\pi$
$829$		−	1.23963e7i		−	0.626476i	0.949675	−	0.313238i	$-0.101414\pi$
$830$		0			0
$831$		0			0
$832$	−4.13510e6	−	4.13510e6i	−0.207099	−	0.207099i
$833$	2.36470e7	+	2.36470e7i	1.18077	+	1.18077i
$834$		0			0
$835$		0			0
$836$			1.18390e7i			0.585867i
$837$		0			0
$838$	−1.07537e7	+	1.07537e7i	−0.528989	+	0.528989i
$839$	3.23818e7			1.58817			0.794083	−	0.607809i	$-0.207951\pi$
$839$	3.23818e7			1.58817			0.794083	+	0.607809i	$0.207951\pi$
$840$		0			0
$841$	−9.12733e6			−0.444994
$842$	1.72488e7	−	1.72488e7i	0.838455	−	0.838455i
$843$		0			0
$844$			2.08830e7i			1.00911i
$845$		0			0
$846$		0			0
$847$	−1.35114e6	−	1.35114e6i	−0.0647129	−	0.0647129i
$848$	1.31672e7	+	1.31672e7i	0.628787	+	0.628787i
$849$		0			0
$850$		0			0
$851$			6.89687e6i			0.326458i
$852$		0			0
$853$	3.39744e6	−	3.39744e6i	0.159875	−	0.159875i	−0.622636	−	0.782511i	$-0.713938\pi$
$853$	3.39744e6	−	3.39744e6i	0.159875	−	0.159875i	0.782511	+	0.622636i	$0.213938\pi$
$854$	666093.			0.0312529
$855$		0			0
$856$	−3.98518e6			−0.185893
$857$	1.93484e7	−	1.93484e7i	0.899897	−	0.899897i	−0.0955295	−	0.995427i	$-0.530454\pi$
$857$	1.93484e7	−	1.93484e7i	0.899897	−	0.899897i	0.995427	+	0.0955295i	$0.0304544\pi$
$858$		0			0
$859$		−	2.39307e6i		−	0.110656i	−0.998468	−	0.0553278i	$-0.982380\pi$
$859$		−	2.39307e6i		−	0.110656i	0.998468	−	0.0553278i	$-0.0176204\pi$
$860$		0			0
$861$		0			0
$862$	2.04005e7	+	2.04005e7i	0.935130	+	0.935130i
$863$	−2.53145e7	−	2.53145e7i	−1.15702	−	1.15702i	−0.985113	−	0.171909i	$-0.945006\pi$
$863$	−2.53145e7	−	2.53145e7i	−1.15702	−	1.15702i	−0.171909	−	0.985113i	$-0.554994\pi$
$864$		0			0
$865$		0			0
$866$		−	1.26282e6i		−	0.0572200i
$867$		0			0
$868$	−1.31848e6	+	1.31848e6i	−0.0593981	+	0.0593981i
$869$	1.37994e7			0.619882
$870$		0			0
$871$	−1.57702e7			−0.704356
$872$	3.74847e6	−	3.74847e6i	0.166941	−	0.166941i
$873$		0			0
$874$			8.73091e6i			0.386617i
$875$		0			0
$876$		0			0
$877$	2.14638e7	+	2.14638e7i	0.942339	+	0.942339i	0.998426	−	0.0560871i	$-0.0178625\pi$
$877$	2.14638e7	+	2.14638e7i	0.942339	+	0.942339i	−0.0560871	+	0.998426i	$0.517862\pi$
$878$	−1.27661e7	−	1.27661e7i	−0.558885	−	0.558885i
$879$		0			0
$880$		0			0
$881$			3.01600e7i			1.30916i	0.755994	+	0.654578i	$0.227154\pi$
$881$			3.01600e7i			1.30916i	−0.755994	+	0.654578i	$0.772846\pi$
$882$		0			0
$883$	2.03899e7	−	2.03899e7i	0.880061	−	0.880061i	−0.113480	−	0.993540i	$-0.536200\pi$
$883$	2.03899e7	−	2.03899e7i	0.880061	−	0.880061i	0.993540	+	0.113480i	$0.0361996\pi$
$884$	1.48576e7			0.639466
$885$		0			0
$886$	4.62290e7			1.97848
$887$	6.74195e6	−	6.74195e6i	0.287724	−	0.287724i	−0.548455	−	0.836180i	$-0.684784\pi$
$887$	6.74195e6	−	6.74195e6i	0.287724	−	0.287724i	0.836180	+	0.548455i	$0.184784\pi$
$888$		0			0
$889$		−	4.79661e6i		−	0.203554i
$890$		0			0
$891$		0			0
$892$	8.66086e6	+	8.66086e6i	0.364459	+	0.364459i
$893$	1.16487e7	+	1.16487e7i	0.488820	+	0.488820i
$894$		0			0
$895$		0			0
$896$			1.26557e6i			0.0526642i
$897$		0			0
$898$	7.48998e6	−	7.48998e6i	0.309949	−	0.309949i
$899$	1.51611e7			0.625650
$900$		0			0
$901$	−3.22726e7			−1.32441
$902$	−1.70917e7	+	1.70917e7i	−0.699471	+	0.699471i
$903$		0			0
$904$			1.49465e6i			0.0608301i
$905$		0			0
$906$		0			0
$907$	−2.01745e7	−	2.01745e7i	−0.814301	−	0.814301i	0.170974	−	0.985276i	$-0.445309\pi$
$907$	−2.01745e7	−	2.01745e7i	−0.814301	−	0.814301i	−0.985276	+	0.170974i	$0.945309\pi$
$908$	−1.42042e7	−	1.42042e7i	−0.571744	−	0.571744i
$909$		0			0
$910$		0			0
$911$		−	4.76912e7i		−	1.90389i	−0.306266	−	0.951946i	$-0.599080\pi$
$911$		−	4.76912e7i		−	1.90389i	0.306266	−	0.951946i	$-0.400920\pi$
$912$		0			0
$913$	−2.42534e6	+	2.42534e6i	−0.0962932	+	0.0962932i
$914$	−1.14621e7			−0.453835
$915$		0			0
$916$	6.48496e6			0.255369
$917$	3.00118e6	−	3.00118e6i	0.117861	−	0.117861i
$918$		0			0
$919$			1.43195e7i			0.559293i	0.960103	+	0.279646i	$0.0902172\pi$
$919$			1.43195e7i			0.559293i	−0.960103	+	0.279646i	$0.909783\pi$
$920$		0			0
$921$		0			0
$922$	−2.48667e7	−	2.48667e7i	−0.963365	−	0.963365i
$923$	−4.28788e6	−	4.28788e6i	−0.165668	−	0.165668i
$924$		0			0
$925$		0			0
$926$			705028.i			0.0270196i
$927$		0			0
$928$	−1.82158e7	+	1.82158e7i	−0.694351	+	0.694351i
$929$	−2.60483e7			−0.990241			−0.495120	−	0.868824i	$-0.664876\pi$
$929$	−2.60483e7			−0.990241			−0.495120	+	0.868824i	$0.664876\pi$
$930$		0			0
$931$	−3.76608e7			−1.42402
$932$	2.88085e7	−	2.88085e7i	1.08638	−	1.08638i
$933$		0			0
$934$			1.72408e7i			0.646681i
$935$		0			0
$936$		0			0
$937$	−1.81854e7	−	1.81854e7i	−0.676665	−	0.676665i	0.282579	−	0.959244i	$-0.408810\pi$
$937$	−1.81854e7	−	1.81854e7i	−0.676665	−	0.676665i	−0.959244	+	0.282579i	$0.908810\pi$
$938$	−4.81826e6	−	4.81826e6i	−0.178806	−	0.178806i
$939$		0			0
$940$		0			0
$941$		−	1.70102e6i		−	0.0626231i	−0.999510	−	0.0313115i	$-0.990032\pi$
$941$		−	1.70102e6i		−	0.0626231i	0.999510	−	0.0313115i	$-0.00996840\pi$
$942$		0			0
$943$	−5.73725e6	+	5.73725e6i	−0.210099	+	0.210099i
$944$	−3.76993e7			−1.37690
$945$		0			0
$946$	−1.30521e7			−0.474190
$947$	3.59453e7	−	3.59453e7i	1.30247	−	1.30247i	0.375744	−	0.926724i	$-0.377387\pi$
$947$	3.59453e7	−	3.59453e7i	1.30247	−	1.30247i	0.926724	−	0.375744i	$-0.122613\pi$
$948$		0			0
$949$			1.02153e7i			0.368203i
$950$		0			0
$951$		0			0
$952$	−894196.	−	894196.i	−0.0319772	−	0.0319772i
$953$	3.75538e7	+	3.75538e7i	1.33943	+	1.33943i	0.896607	+	0.442827i	$0.146024\pi$
$953$	3.75538e7	+	3.75538e7i	1.33943	+	1.33943i	0.442827	+	0.896607i	$0.353976\pi$
$954$		0			0
$955$		0			0
$956$			2.34367e7i			0.829377i
$957$		0			0
$958$	−1.41271e7	+	1.41271e7i	−0.497324	+	0.497324i
$959$	1.62711e6			0.0571309
$960$		0			0
$961$	−8.43743e6			−0.294715
$962$	−2.05329e7	+	2.05329e7i	−0.715341	+	0.715341i
$963$		0			0
$964$			3.13886e7i			1.08788i
$965$		0			0
$966$		0			0
$967$	−7.16001e6	−	7.16001e6i	−0.246234	−	0.246234i	0.573189	−	0.819423i	$-0.305706\pi$
$967$	−7.16001e6	−	7.16001e6i	−0.246234	−	0.246234i	−0.819423	+	0.573189i	$0.805706\pi$
$968$	−3.51316e6	−	3.51316e6i	−0.120506	−	0.120506i
$969$		0			0
$970$		0			0
$971$		−	2.40544e7i		−	0.818740i	−0.912368	−	0.409370i	$-0.865748\pi$
$971$		−	2.40544e7i		−	0.818740i	0.912368	−	0.409370i	$-0.134252\pi$
$972$		0			0
$973$	−2.46715e6	+	2.46715e6i	−0.0835435	+	0.0835435i
$974$	−2.39771e7			−0.809840
$975$		0			0
$976$	6.52225e6			0.219166
$977$	−3.64967e7	+	3.64967e7i	−1.22326	+	1.22326i	−0.256789	+	0.966467i	$0.582665\pi$
$977$	−3.64967e7	+	3.64967e7i	−1.22326	+	1.22326i	−0.966467	+	0.256789i	$0.917335\pi$
$978$		0			0
$979$			1.77569e7i			0.592122i
$980$		0			0
$981$		0			0
$982$	−3.02399e7	−	3.02399e7i	−1.00069	−	1.00069i
$983$	−2.25567e7	−	2.25567e7i	−0.744546	−	0.744546i	0.228903	−	0.973449i	$-0.426486\pi$
$983$	−2.25567e7	−	2.25567e7i	−0.744546	−	0.744546i	−0.973449	+	0.228903i	$0.926486\pi$
$984$		0			0
$985$		0			0
$986$		−	5.21987e7i		−	1.70989i
$987$		0			0
$988$	−1.18313e7	+	1.18313e7i	−0.385602	+	0.385602i
$989$	−4.38125e6			−0.142432
$990$		0			0
$991$	3.40802e6			0.110235			0.0551173	−	0.998480i	$-0.482447\pi$
$991$	3.40802e6			0.110235			0.0551173	+	0.998480i	$0.482447\pi$
$992$	−2.42601e7	+	2.42601e7i	−0.782731	+	0.782731i
$993$		0			0
$994$		−	2.62015e6i		−	0.0841123i
$995$		0			0
$996$		0			0
$997$	−2.74529e7	−	2.74529e7i	−0.874684	−	0.874684i	0.118295	−	0.992979i	$-0.462257\pi$
$997$	−2.74529e7	−	2.74529e7i	−0.874684	−	0.874684i	−0.992979	+	0.118295i	$0.962257\pi$
$998$	7.57647e6	+	7.57647e6i	0.240791	+	0.240791i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.f.c.107.3	✓	24
3.2	odd	2	inner	225.6.f.c.107.10	yes	24
5.2	odd	4	inner	225.6.f.c.143.4	yes	24
5.3	odd	4	inner	225.6.f.c.143.10	yes	24
5.4	even	2	inner	225.6.f.c.107.9	yes	24
15.2	even	4	inner	225.6.f.c.143.9	yes	24
15.8	even	4	inner	225.6.f.c.143.3	yes	24
15.14	odd	2	inner	225.6.f.c.107.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.6.f.c.107.3	✓	24	1.1	even	1	trivial
225.6.f.c.107.4	yes	24	15.14	odd	2	inner
225.6.f.c.107.9	yes	24	5.4	even	2	inner
225.6.f.c.107.10	yes	24	3.2	odd	2	inner
225.6.f.c.143.3	yes	24	15.8	even	4	inner
225.6.f.c.143.4	yes	24	5.2	odd	4	inner
225.6.f.c.143.9	yes	24	15.2	even	4	inner
225.6.f.c.143.10	yes	24	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.41913 + 5.41913i) q^{2} -26.7338i q^{4} +(-10.9755 - 10.9755i) q^{7} +(-28.5379 - 28.5379i) q^{8} +O(q^{10})\)	\(q+(-5.41913 + 5.41913i) q^{2} -26.7338i q^{4} +(-10.9755 - 10.9755i) q^{7} +(-28.5379 - 28.5379i) q^{8} -194.798i q^{11} +(194.671 - 194.671i) q^{13} +118.955 q^{14} +1164.78 q^{16} +(-1427.43 + 1427.43i) q^{17} -2273.37i q^{19} +(1055.63 + 1055.63i) q^{22} +(354.348 + 354.348i) q^{23} +2109.89i q^{26} +(-293.417 + 293.417i) q^{28} +3373.99 q^{29} +4493.52 q^{31} +(-5398.90 + 5398.90i) q^{32} -15470.9i q^{34} +(9731.76 + 9731.76i) q^{37} +(12319.7 + 12319.7i) q^{38} +16191.0i q^{41} +(-6182.12 + 6182.12i) q^{43} -5207.69 q^{44} -3840.52 q^{46} +(-5123.99 + 5123.99i) q^{47} -16566.1i q^{49} +(-5204.30 - 5204.30i) q^{52} +(11304.4 + 11304.4i) q^{53} +626.435i q^{56} +(-18284.1 + 18284.1i) q^{58} -32365.9 q^{59} +5599.54 q^{61} +(-24351.0 + 24351.0i) q^{62} -21241.5i q^{64} +(-40504.9 - 40504.9i) q^{67} +(38160.8 + 38160.8i) q^{68} -22026.4i q^{71} +(-26237.5 + 26237.5i) q^{73} -105475. q^{74} -60775.9 q^{76} +(-2138.00 + 2138.00i) q^{77} +70839.5i q^{79} +(-87741.0 - 87741.0i) q^{82} +(-12450.6 - 12450.6i) q^{83} -67003.4i q^{86} +(-5559.12 + 5559.12i) q^{88} -91155.9 q^{89} -4273.21 q^{91} +(9473.10 - 9473.10i) q^{92} -55535.1i q^{94} +(1795.86 + 1795.86i) q^{97} +(89773.7 + 89773.7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 5664 q^{16} - 49176 q^{31} - 116976 q^{46} - 87048 q^{61} + 241680 q^{76} - 489096 q^{91}+O(q^{100}) \) 24 * q + 5664 * q^16 - 49176 * q^31 - 116976 * q^46 - 87048 * q^61 + 241680 * q^76 - 489096 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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