LMFDB - Embedded newform 300.7.b.e.149.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,7,Mod(149,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	300.b (of order $2$, degree $1$, not 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:	$69.0162250860$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 406 x^{14} + 67561 x^{12} + 5921226 x^{10} + 291565644 x^{8} + 7924637994 x^{6} + \cdots + 276002078881 $ x^16 + 406x^14 + 67561x^12 + 5921226x^10 + 291565644x^8 + 7924637994x^6 + 107459649193x^4 + 554332708870*x^2 + 276002078881
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{18}\cdot 5^{26} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.8
Root			$-0.745166i$ of defining polynomial
Character	$\chi$	$=$	300.149
Dual form			300.7.b.e.149.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.99060 + 25.0836i) q^{3} -134.460i q^{7} +(-529.376 - 501.201i) q^{9} +O(q^{10})$	$q+(-9.99060 + 25.0836i) q^{3} -134.460i q^{7} +(-529.376 - 501.201i) q^{9} +1155.16i q^{11} -78.2272i q^{13} +54.6849 q^{17} -1984.42 q^{19} +(3372.73 + 1343.33i) q^{21} -16917.8 q^{23} +(17860.7 - 8271.35i) q^{27} -18486.2i q^{29} +38367.7 q^{31} +(-28975.6 - 11540.8i) q^{33} -21242.8i q^{37} +(1962.22 + 781.537i) q^{39} -88980.3i q^{41} -65565.7i q^{43} -163822. q^{47} +99569.6 q^{49} +(-546.335 + 1371.69i) q^{51} +97179.1 q^{53} +(19825.5 - 49776.4i) q^{57} +286230. i q^{59} +119504. q^{61} +(-67391.2 + 71179.6i) q^{63} +142653. i q^{67} +(169019. - 424360. i) q^{69} -363327. i q^{71} +576517. i q^{73} +155322. q^{77} +627119. q^{79} +(29036.1 + 530647. i) q^{81} +991316. q^{83} +(463701. + 184688. i) q^{87} -827592. i q^{89} -10518.4 q^{91} +(-383316. + 962400. i) q^{93} -848738. i q^{97} +(578968. - 611514. i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2984 q^{9}+O(q^{10}) $ 16 * q - 2984 * q^9	$ 16 q - 2984 q^{9} + 30544 q^{19} - 1736 q^{21} + 70064 q^{31} - 79216 q^{39} - 405120 q^{49} + 858240 q^{51} - 271216 q^{61} + 509880 q^{69} - 1415408 q^{79} - 2396224 q^{81} - 4008224 q^{91} - 5300160 q^{99}+O(q^{100}) $ 16 * q - 2984 * q^9 + 30544 * q^19 - 1736 * q^21 + 70064 * q^31 - 79216 * q^39 - 405120 * q^49 + 858240 * q^51 - 271216 * q^61 + 509880 * q^69 - 1415408 * q^79 - 2396224 * q^81 - 4008224 * q^91 - 5300160 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−9.99060	+	25.0836i	−0.370022	+	0.929023i
$3$	−9.99060	+	25.0836i	−0.370022	+	0.929023i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	134.460i		−	0.392010i	−0.980603	−	0.196005i	$-0.937203\pi$
$7$		−	134.460i		−	0.392010i	0.980603	−	0.196005i	$-0.0627969\pi$
$8$		0			0
$9$	−529.376	−	501.201i	−0.726167	−	0.687519i
$10$		0			0
$11$			1155.16i			0.867890i	0.900939	+	0.433945i	$0.142879\pi$
$11$			1155.16i			0.867890i	−0.900939	+	0.433945i	$0.857121\pi$
$12$		0			0
$13$		−	78.2272i		−	0.0356064i	−0.999842	−	0.0178032i	$-0.994333\pi$
$13$		−	78.2272i		−	0.0356064i	0.999842	−	0.0178032i	$-0.00566723\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	54.6849			0.0111306			0.00556532	−	0.999985i	$-0.498228\pi$
$17$	54.6849			0.0111306			0.00556532	+	0.999985i	$0.498228\pi$
$18$		0			0
$19$	−1984.42			−0.289316			−0.144658	−	0.989482i	$-0.546208\pi$
$19$	−1984.42			−0.289316			−0.144658	+	0.989482i	$0.546208\pi$
$20$		0			0
$21$	3372.73	+	1343.33i	0.364186	+	0.145053i
$22$		0			0
$23$	−16917.8			−1.39047			−0.695234	−	0.718784i	$-0.744699\pi$
$23$	−16917.8			−1.39047			−0.695234	+	0.718784i	$0.744699\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	17860.7	−	8271.35i	0.907418	−	0.420228i
$28$		0			0
$29$		−	18486.2i		−	0.757973i	−0.925402	−	0.378986i	$-0.876273\pi$
$29$		−	18486.2i		−	0.757973i	0.925402	−	0.378986i	$-0.123727\pi$
$30$		0			0
$31$	38367.7			1.28789			0.643947	−	0.765070i	$-0.277296\pi$
$31$	38367.7			1.28789			0.643947	+	0.765070i	$0.277296\pi$
$32$		0			0
$33$	−28975.6	−	11540.8i	−0.806290	−	0.321139i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	21242.8i		−	0.419378i	−0.977768	−	0.209689i	$-0.932755\pi$
$37$		−	21242.8i		−	0.419378i	0.977768	−	0.209689i	$-0.0672452\pi$
$38$		0			0
$39$	1962.22	+	781.537i	0.0330791	+	0.0131751i
$40$		0			0
$41$		−	88980.3i		−	1.29105i	−0.763740	−	0.645524i	$-0.776639\pi$
$41$		−	88980.3i		−	1.29105i	0.763740	−	0.645524i	$-0.223361\pi$
$42$		0			0
$43$		−	65565.7i		−	0.824654i	−0.911036	−	0.412327i	$-0.864716\pi$
$43$		−	65565.7i		−	0.824654i	0.911036	−	0.412327i	$-0.135284\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−163822.			−1.57790			−0.788950	−	0.614457i	$-0.789375\pi$
$47$	−163822.			−1.57790			−0.788950	+	0.614457i	$0.789375\pi$
$48$		0			0
$49$	99569.6			0.846328
$50$		0			0
$51$	−546.335	+	1371.69i	−0.00411859	+	0.0103406i
$52$		0			0
$53$	97179.1			0.652747			0.326374	−	0.945241i	$-0.394173\pi$
$53$	97179.1			0.652747			0.326374	+	0.945241i	$0.394173\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	19825.5	−	49776.4i	0.107053	−	0.268781i
$58$		0			0
$59$			286230.i			1.39367i	0.717233	+	0.696834i	$0.245408\pi$
$59$			286230.i			1.39367i	−0.717233	+	0.696834i	$0.754592\pi$
$60$		0			0
$61$	119504.			0.526496			0.263248	−	0.964728i	$-0.415206\pi$
$61$	119504.			0.526496			0.263248	+	0.964728i	$0.415206\pi$
$62$		0			0
$63$	−67391.2	+	71179.6i	−0.269514	+	0.284665i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			142653.i			0.474304i	0.971473	+	0.237152i	$0.0762139\pi$
$67$			142653.i			0.474304i	−0.971473	+	0.237152i	$0.923786\pi$
$68$		0			0
$69$	169019.	−	424360.i	0.514504	−	1.29178i
$70$		0			0
$71$		−	363327.i		−	1.01513i	−0.861613	−	0.507566i	$-0.830545\pi$
$71$		−	363327.i		−	1.01513i	0.861613	−	0.507566i	$-0.169455\pi$
$72$		0			0
$73$			576517.i			1.48199i	0.671513	+	0.740993i	$0.265645\pi$
$73$			576517.i			1.48199i	−0.671513	+	0.740993i	$0.734355\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	155322.			0.340222
$78$		0			0
$79$	627119.			1.27195			0.635973	−	0.771711i	$-0.280599\pi$
$79$	627119.			1.27195			0.635973	+	0.771711i	$0.280599\pi$
$80$		0			0
$81$	29036.1	+	530647.i	0.0546366	+	0.998506i
$82$		0			0
$83$	991316.			1.73372			0.866858	−	0.498556i	$-0.166136\pi$
$83$	991316.			1.73372			0.866858	+	0.498556i	$0.166136\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	463701.	+	184688.i	0.704174	+	0.280467i
$88$		0			0
$89$		−	827592.i		−	1.17394i	−0.809608	−	0.586970i	$-0.800321\pi$
$89$		−	827592.i		−	1.17394i	0.809608	−	0.586970i	$-0.199679\pi$
$90$		0			0
$91$	−10518.4			−0.0139581
$92$		0			0
$93$	−383316.	+	962400.i	−0.476550	+	1.19648i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	848738.i		−	0.929948i	−0.885324	−	0.464974i	$-0.846064\pi$
$97$		−	848738.i		−	0.929948i	0.885324	−	0.464974i	$-0.153936\pi$
$98$		0			0
$99$	578968.	−	611514.i	0.596690	−	0.630233i
$100$		0			0
$101$			1.11358e6i			1.08083i	0.841398	+	0.540417i	$0.181733\pi$
$101$			1.11358e6i			1.08083i	−0.841398	+	0.540417i	$0.818267\pi$
$102$		0			0
$103$			2.09668e6i			1.91876i	0.282118	+	0.959380i	$0.408963\pi$
$103$			2.09668e6i			1.91876i	−0.282118	+	0.959380i	$0.591037\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−242367.			−0.197844			−0.0989218	−	0.995095i	$-0.531539\pi$
$107$	−242367.			−0.197844			−0.0989218	+	0.995095i	$0.531539\pi$
$108$		0			0
$109$	2.23474e6			1.72563			0.862816	−	0.505519i	$-0.168699\pi$
$109$	2.23474e6			1.72563			0.862816	+	0.505519i	$0.168699\pi$
$110$		0			0
$111$	532846.	+	212228.i	0.389612	+	0.155179i
$112$		0			0
$113$	245486.			0.170134			0.0850672	−	0.996375i	$-0.472890\pi$
$113$	245486.			0.170134			0.0850672	+	0.996375i	$0.472890\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−39207.5	+	41411.6i	−0.0244800	+	0.0258562i
$118$		0			0
$119$		−	7352.90i		−	0.00436333i
$120$		0			0
$121$	437163.			0.246767
$122$		0			0
$123$	2.23195e6	+	888967.i	1.19941	+	0.477717i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			484140.i			0.236352i	0.992993	+	0.118176i	$0.0377047\pi$
$127$			484140.i			0.236352i	−0.992993	+	0.118176i	$0.962295\pi$
$128$		0			0
$129$	1.64463e6	+	655041.i	0.766122	+	0.305140i
$130$		0			0
$131$		−	3.46823e6i		−	1.54274i	−0.636385	−	0.771372i	$-0.719571\pi$
$131$		−	3.46823e6i		−	1.54274i	0.636385	−	0.771372i	$-0.280429\pi$
$132$		0			0
$133$			266824.i			0.113415i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.74492e6			−0.678600			−0.339300	−	0.940678i	$-0.610190\pi$
$137$	−1.74492e6			−0.678600			−0.339300	+	0.940678i	$0.610190\pi$
$138$		0			0
$139$	3.54561e6			1.32022			0.660111	−	0.751168i	$-0.270509\pi$
$139$	3.54561e6			1.32022			0.660111	+	0.751168i	$0.270509\pi$
$140$		0			0
$141$	1.63668e6	−	4.10926e6i	0.583859	−	1.46591i
$142$		0			0
$143$	90365.0			0.0309024
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−994761.	+	2.49757e6i	−0.313160	+	0.786258i
$148$		0			0
$149$		−	4.13539e6i		−	1.25014i	−0.780570	−	0.625069i	$-0.785071\pi$
$149$		−	4.13539e6i		−	1.25014i	0.780570	−	0.625069i	$-0.214929\pi$
$150$		0			0
$151$	3.42874e6			0.995873			0.497937	−	0.867213i	$-0.334091\pi$
$151$	3.42874e6			0.995873			0.497937	+	0.867213i	$0.334091\pi$
$152$		0			0
$153$	−28948.8	−	27408.1i	−0.00808270	−	0.00765252i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	5.35519e6i		−	1.38381i	−0.721990	−	0.691904i	$-0.756772\pi$
$157$		−	5.35519e6i		−	1.38381i	0.721990	−	0.691904i	$-0.243228\pi$
$158$		0			0
$159$	−970878.	+	2.43760e6i	−0.241531	+	0.606417i
$160$		0			0
$161$			2.27476e6i			0.545078i
$162$		0			0
$163$			590537.i			0.136359i	0.997673	+	0.0681796i	$0.0217191\pi$
$163$			590537.i			0.136359i	−0.997673	+	0.0681796i	$0.978281\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.06892e6			0.873634			0.436817	−	0.899550i	$-0.356106\pi$
$167$	4.06892e6			0.873634			0.436817	+	0.899550i	$0.356106\pi$
$168$		0			0
$169$	4.82069e6			0.998732
$170$		0			0
$171$	1.05050e6	+	994593.i	0.210092	+	0.198910i
$172$		0			0
$173$	−4.80964e6			−0.928912			−0.464456	−	0.885596i	$-0.653750\pi$
$173$	−4.80964e6			−0.928912			−0.464456	+	0.885596i	$0.653750\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−7.17968e6	−	2.85961e6i	−1.29475	−	0.515688i
$178$		0			0
$179$		−	616529.i		−	0.107496i	−0.998555	−	0.0537482i	$-0.982883\pi$
$179$		−	616529.i		−	0.107496i	0.998555	−	0.0537482i	$-0.0171168\pi$
$180$		0			0
$181$	169854.			0.0286444			0.0143222	−	0.999897i	$-0.495441\pi$
$181$	169854.			0.0286444			0.0143222	+	0.999897i	$0.495441\pi$
$182$		0			0
$183$	−1.19392e6	+	2.99760e6i	−0.194815	+	0.489126i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			63169.8i			0.00966017i
$188$		0			0
$189$	−1.11216e6	−	2.40154e6i	−0.164734	−	0.355717i
$190$		0			0
$191$			5.63748e6i			0.809068i	0.914523	+	0.404534i	$0.132566\pi$
$191$			5.63748e6i			0.809068i	−0.914523	+	0.404534i	$0.867434\pi$
$192$		0			0
$193$		−	1.93534e6i		−	0.269207i	−0.990900	−	0.134603i	$-0.957024\pi$
$193$		−	1.93534e6i		−	0.269207i	0.990900	−	0.134603i	$-0.0429760\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.63667e6			0.737265			0.368632	−	0.929575i	$-0.379826\pi$
$197$	5.63667e6			0.737265			0.368632	+	0.929575i	$0.379826\pi$
$198$		0			0
$199$	−661790.			−0.0839772			−0.0419886	−	0.999118i	$-0.513369\pi$
$199$	−661790.			−0.0839772			−0.0419886	+	0.999118i	$0.513369\pi$
$200$		0			0
$201$	−3.57825e6	−	1.42519e6i	−0.440639	−	0.175503i
$202$		0			0
$203$	−2.48565e6			−0.297133
$204$		0			0
$205$		0			0
$206$		0			0
$207$	8.95588e6	+	8.47923e6i	1.00971	+	0.955972i
$208$		0			0
$209$		−	2.29232e6i		−	0.251095i
$210$		0			0
$211$	726505.			0.0773376			0.0386688	−	0.999252i	$-0.487688\pi$
$211$	726505.			0.0773376			0.0386688	+	0.999252i	$0.487688\pi$
$212$		0			0
$213$	9.11355e6	+	3.62985e6i	0.943080	+	0.375621i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	5.15890e6i		−	0.504868i
$218$		0			0
$219$	−1.44611e7	−	5.75976e6i	−1.37680	−	0.548368i
$220$		0			0
$221$		−	4277.84i		−	0.000396322i
$222$		0			0
$223$			19464.1i			0.00175517i	1.00000		0.000877585i	$0.000279344\pi$
$223$			19464.1i			0.00175517i	−1.00000		0.000877585i	$0.999721\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.74686e6			0.149342			0.0746708	−	0.997208i	$-0.476209\pi$
$227$	1.74686e6			0.149342			0.0746708	+	0.997208i	$0.476209\pi$
$228$		0			0
$229$	−803494.			−0.0669077			−0.0334539	−	0.999440i	$-0.510651\pi$
$229$	−803494.			−0.0669077			−0.0334539	+	0.999440i	$0.510651\pi$
$230$		0			0
$231$	−1.55177e6	+	3.89605e6i	−0.125890	+	0.316074i
$232$		0			0
$233$	1.41647e7			1.11980			0.559900	−	0.828561i	$-0.310840\pi$
$233$	1.41647e7			1.11980			0.559900	+	0.828561i	$0.310840\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−6.26530e6	+	1.57304e7i	−0.470648	+	1.18167i
$238$		0			0
$239$			1.51609e7i			1.11054i	0.831672	+	0.555268i	$0.187384\pi$
$239$			1.51609e7i			1.11054i	−0.831672	+	0.555268i	$0.812616\pi$
$240$		0			0
$241$	−2.44786e7			−1.74878			−0.874391	−	0.485222i	$-0.838739\pi$
$241$	−2.44786e7			−1.74878			−0.874391	+	0.485222i	$0.838739\pi$
$242$		0			0
$243$	−1.36006e7	−	4.57316e6i	−0.947852	−	0.318711i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			155235.i			0.0103015i
$248$		0			0
$249$	−9.90384e6	+	2.48658e7i	−0.641513	+	1.61066i
$250$		0			0
$251$		−	5.38840e6i		−	0.340752i	−0.985379	−	0.170376i	$-0.945502\pi$
$251$		−	5.38840e6i		−	0.340752i	0.985379	−	0.170376i	$-0.0544982\pi$
$252$		0			0
$253$		−	1.95428e7i		−	1.20677i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.81725e7			−1.65969			−0.829843	−	0.557998i	$-0.811570\pi$
$257$	−2.81725e7			−1.65969			−0.829843	+	0.557998i	$0.811570\pi$
$258$		0			0
$259$	−2.85629e6			−0.164401
$260$		0			0
$261$	−9.26530e6	+	9.78614e6i	−0.521120	+	0.550415i
$262$		0			0
$263$	1.14892e7			0.631570			0.315785	−	0.948831i	$-0.397732\pi$
$263$	1.14892e7			0.631570			0.315785	+	0.948831i	$0.397732\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.07590e7	+	8.26814e6i	1.09062	+	0.434384i
$268$		0			0
$269$			1.46873e7i			0.754546i	0.926102	+	0.377273i	$0.123138\pi$
$269$			1.46873e7i			0.754546i	−0.926102	+	0.377273i	$0.876862\pi$
$270$		0			0
$271$	−1.77493e6			−0.0891812			−0.0445906	−	0.999005i	$-0.514198\pi$
$271$	−1.77493e6			−0.0891812			−0.0445906	+	0.999005i	$0.514198\pi$
$272$		0			0
$273$	105085.	−	263839.i	0.00516479	−	0.0129674i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	2.74584e7i		−	1.29192i	−0.763371	−	0.645960i	$-0.776457\pi$
$277$		−	2.74584e7i		−	1.29192i	0.763371	−	0.645960i	$-0.223543\pi$
$278$		0			0
$279$	−2.03109e7	−	1.92299e7i	−0.935226	−	0.885452i
$280$		0			0
$281$			2.73409e6i			0.123224i	0.998100	+	0.0616118i	$0.0196241\pi$
$281$			2.73409e6i			0.123224i	−0.998100	+	0.0616118i	$0.980376\pi$
$282$		0			0
$283$		−	3.51523e7i		−	1.55094i	−0.631386	−	0.775469i	$-0.717513\pi$
$283$		−	3.51523e7i		−	1.55094i	0.631386	−	0.775469i	$-0.282487\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.19642e7			−0.506104
$288$		0			0
$289$	−2.41346e7			−0.999876
$290$		0			0
$291$	2.12894e7	+	8.47941e6i	0.863943	+	0.344102i
$292$		0			0
$293$	−4.66113e7			−1.85306			−0.926528	−	0.376227i	$-0.877221\pi$
$293$	−4.66113e7			−1.85306			−0.926528	+	0.376227i	$0.877221\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	9.55475e6	+	2.06320e7i	0.364712	+	0.787539i
$298$		0			0
$299$			1.32343e6i			0.0495095i
$300$		0			0
$301$	−8.81594e6			−0.323273
$302$		0			0
$303$	−2.79327e7	−	1.11254e7i	−1.00412	−	0.399932i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	3.11960e7i		−	1.07816i	−0.842254	−	0.539081i	$-0.818772\pi$
$307$		−	3.11960e7i		−	1.07816i	0.842254	−	0.539081i	$-0.181228\pi$
$308$		0			0
$309$	−5.25923e7	−	2.09471e7i	−1.78257	−	0.709984i
$310$		0			0
$311$		−	3.51987e7i		−	1.17016i	−0.810975	−	0.585081i	$-0.801063\pi$
$311$		−	3.51987e7i		−	1.17016i	0.810975	−	0.585081i	$-0.198937\pi$
$312$		0			0
$313$		−	3.60591e7i		−	1.17593i	−0.808886	−	0.587965i	$-0.799929\pi$
$313$		−	3.60591e7i		−	1.17593i	0.808886	−	0.587965i	$-0.200071\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.74502e7			1.80349			0.901745	−	0.432269i	$-0.142287\pi$
$317$	5.74502e7			1.80349			0.901745	+	0.432269i	$0.142287\pi$
$318$		0			0
$319$	2.13545e7			0.657837
$320$		0			0
$321$	2.42139e6	−	6.07944e6i	0.0732065	−	0.183801i
$322$		0			0
$323$	−108518.			−0.00322027
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−2.23264e7	+	5.60554e7i	−0.638522	+	1.60315i
$328$		0			0
$329$			2.20275e7i			0.618553i
$330$		0			0
$331$	3.26327e7			0.899849			0.449924	−	0.893067i	$-0.351451\pi$
$331$	3.26327e7			0.899849			0.449924	+	0.893067i	$0.351451\pi$
$332$		0			0
$333$	−1.06469e7	+	1.12454e7i	−0.288330	+	0.304539i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	4.34020e7i		−	1.13402i	−0.823712	−	0.567009i	$-0.808100\pi$
$337$		−	4.34020e7i		−	1.13402i	0.823712	−	0.567009i	$-0.191900\pi$
$338$		0			0
$339$	−2.45256e6	+	6.15769e6i	−0.0629535	+	0.158059i
$340$		0			0
$341$			4.43209e7i			1.11775i
$342$		0			0
$343$		−	2.92071e7i		−	0.723779i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.50726e7			−0.839421			−0.419710	−	0.907658i	$-0.637868\pi$
$347$	−3.50726e7			−0.839421			−0.419710	+	0.907658i	$0.637868\pi$
$348$		0			0
$349$	−1.41882e7			−0.333772			−0.166886	−	0.985976i	$-0.553371\pi$
$349$	−1.41882e7			−0.333772			−0.166886	+	0.985976i	$0.553371\pi$
$350$		0			0
$351$	−647045.	−	1.39719e6i	−0.0149628	−	0.0323099i
$352$		0			0
$353$	−1.86131e7			−0.423149			−0.211575	−	0.977362i	$-0.567859\pi$
$353$	−1.86131e7			−0.423149			−0.211575	+	0.977362i	$0.567859\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	184437.	+	73459.9i	0.00405363	+	0.00161453i
$358$		0			0
$359$			5.28748e6i			0.114279i	0.998366	+	0.0571394i	$0.0181979\pi$
$359$			5.28748e6i			0.114279i	−0.998366	+	0.0571394i	$0.981802\pi$
$360$		0			0
$361$	−4.31080e7			−0.916296
$362$		0			0
$363$	−4.36752e6	+	1.09656e7i	−0.0913094	+	0.229252i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	1.00133e6i		−	0.0202571i	−0.999949	−	0.0101286i	$-0.996776\pi$
$367$		−	1.00133e6i		−	0.0202571i	0.999949	−	0.0101286i	$-0.00322408\pi$
$368$		0			0
$369$	−4.45970e7	+	4.71040e7i	−0.887619	+	0.937516i
$370$		0			0
$371$		−	1.30666e7i		−	0.255884i
$372$		0			0
$373$			4.27333e7i			0.823455i	0.911307	+	0.411728i	$0.135074\pi$
$373$			4.27333e7i			0.823455i	−0.911307	+	0.411728i	$0.864926\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.44612e6			−0.0269887
$378$		0			0
$379$	1.05217e8			1.93272			0.966358	−	0.257202i	$-0.0828006\pi$
$379$	1.05217e8			1.93272			0.966358	+	0.257202i	$0.0828006\pi$
$380$		0			0
$381$	−1.21440e7	−	4.83685e6i	−0.219577	−	0.0874556i
$382$		0			0
$383$	6.60163e7			1.17505			0.587523	−	0.809208i	$-0.300103\pi$
$383$	6.60163e7			1.17505			0.587523	+	0.809208i	$0.300103\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.28616e7	+	3.47089e7i	−0.566965	+	0.598836i
$388$		0			0
$389$		−	6.54648e7i		−	1.11214i	−0.831136	−	0.556070i	$-0.812309\pi$
$389$		−	6.54648e7i		−	1.11214i	0.831136	−	0.556070i	$-0.187691\pi$
$390$		0			0
$391$	−925149.			−0.0154768
$392$		0			0
$393$	8.69957e7	+	3.46497e7i	1.43324	+	0.570850i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	5.36739e7i		−	0.857811i	−0.903349	−	0.428905i	$-0.858899\pi$
$397$		−	5.36739e7i		−	0.857811i	0.903349	−	0.428905i	$-0.141101\pi$
$398$		0			0
$399$	−6.69291e6	−	2.66573e6i	−0.105365	−	0.0419660i
$400$		0			0
$401$			8.82767e6i			0.136903i	0.997654	+	0.0684515i	$0.0218058\pi$
$401$			8.82767e6i			0.136903i	−0.997654	+	0.0684515i	$0.978194\pi$
$402$		0			0
$403$		−	3.00139e6i		−	0.0458572i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.45388e7			0.363974
$408$		0			0
$409$	1.11796e8			1.63402			0.817011	−	0.576622i	$-0.195629\pi$
$409$	1.11796e8			1.63402			0.817011	+	0.576622i	$0.195629\pi$
$410$		0			0
$411$	1.74328e7	−	4.37689e7i	0.251097	−	0.630435i
$412$		0			0
$413$	3.84863e7			0.546332
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.54228e7	+	8.89368e7i	−0.488512	+	1.22652i
$418$		0			0
$419$		−	2.09574e7i		−	0.284902i	−0.989802	−	0.142451i	$-0.954502\pi$
$419$		−	2.09574e7i		−	0.284902i	0.989802	−	0.142451i	$-0.0454983\pi$
$420$		0			0
$421$	−1.87381e7			−0.251118			−0.125559	−	0.992086i	$-0.540073\pi$
$421$	−1.87381e7			−0.251118			−0.125559	+	0.992086i	$0.540073\pi$
$422$		0			0
$423$	8.67236e7	+	8.21080e7i	1.14582	+	1.08484i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	1.60685e7i		−	0.206392i
$428$		0			0
$429$	−902801.	+	2.26668e6i	−0.0114346	+	0.0287090i
$430$		0			0
$431$			7.51443e6i			0.0938565i	0.998898	+	0.0469283i	$0.0149432\pi$
$431$			7.51443e6i			0.0938565i	−0.998898	+	0.0469283i	$0.985057\pi$
$432$		0			0
$433$			6.88652e7i			0.848274i	0.905598	+	0.424137i	$0.139422\pi$
$433$			6.88652e7i			0.848274i	−0.905598	+	0.424137i	$0.860578\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.35721e7			0.402285
$438$		0			0
$439$	1.30096e8			1.53769			0.768846	−	0.639434i	$-0.220831\pi$
$439$	1.30096e8			1.53769			0.768846	+	0.639434i	$0.220831\pi$
$440$		0			0
$441$	−5.27097e7	−	4.99044e7i	−0.614575	−	0.581866i
$442$		0			0
$443$	−1.19372e8			−1.37306			−0.686531	−	0.727100i	$-0.740868\pi$
$443$	−1.19372e8			−1.37306			−0.686531	+	0.727100i	$0.740868\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.03731e8	+	4.13151e7i	1.16141	+	0.462579i
$448$		0			0
$449$		−	5.58450e7i		−	0.616943i	−0.951234	−	0.308471i	$-0.900183\pi$
$449$		−	5.58450e7i		−	0.616943i	0.951234	−	0.308471i	$-0.0998174\pi$
$450$		0			0
$451$	1.02787e8			1.12049
$452$		0			0
$453$	−3.42552e7	+	8.60053e7i	−0.368495	+	0.925189i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	1.17246e8i		−	1.22843i	−0.789140	−	0.614213i	$-0.789473\pi$
$457$		−	1.17246e8i		−	1.22843i	0.789140	−	0.614213i	$-0.210527\pi$
$458$		0			0
$459$	976711.	−	452318.i	0.0101002	−	0.00467741i
$460$		0			0
$461$			1.77582e8i			1.81258i	0.422657	+	0.906290i	$0.361097\pi$
$461$			1.77582e8i			1.81258i	−0.422657	+	0.906290i	$0.638903\pi$
$462$		0			0
$463$			1.25613e8i			1.26558i	0.774323	+	0.632791i	$0.218091\pi$
$463$			1.25613e8i			1.26558i	−0.774323	+	0.632791i	$0.781909\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.66195e7			0.261366			0.130683	−	0.991424i	$-0.458283\pi$
$467$	2.66195e7			0.261366			0.130683	+	0.991424i	$0.458283\pi$
$468$		0			0
$469$	1.91810e7			0.185932
$470$		0			0
$471$	1.34327e8	+	5.35016e7i	1.28559	+	0.512040i
$472$		0			0
$473$	7.57390e7			0.715709
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−5.14442e7	−	4.87062e7i	−0.474003	−	0.448776i
$478$		0			0
$479$		−	1.63522e8i		−	1.48789i	−0.668242	−	0.743944i	$-0.732953\pi$
$479$		−	1.63522e8i		−	1.48789i	0.668242	−	0.743944i	$-0.267047\pi$
$480$		0			0
$481$	−1.66176e6			−0.0149325
$482$		0			0
$483$	−5.70593e7	−	2.27262e7i	−0.506390	−	0.201691i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			6.56192e6i			0.0568125i	0.999596	+	0.0284062i	$0.00904320\pi$
$487$			6.56192e6i			0.0568125i	−0.999596	+	0.0284062i	$0.990957\pi$
$488$		0			0
$489$	−1.48128e7	−	5.89982e6i	−0.126681	−	0.0504559i
$490$		0			0
$491$		−	1.29952e8i		−	1.09784i	−0.835874	−	0.548921i	$-0.815039\pi$
$491$		−	1.29952e8i		−	1.09784i	0.835874	−	0.548921i	$-0.184961\pi$
$492$		0			0
$493$		−	1.01092e6i		−	0.00843673i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.88527e7			−0.397942
$498$		0			0
$499$	1.09191e8			0.878788			0.439394	−	0.898295i	$-0.355193\pi$
$499$	1.09191e8			0.878788			0.439394	+	0.898295i	$0.355193\pi$
$500$		0			0
$501$	−4.06509e7	+	1.02063e8i	−0.323264	+	0.811626i
$502$		0			0
$503$	3.22774e7			0.253626			0.126813	−	0.991927i	$-0.459525\pi$
$503$	3.22774e7			0.253626			0.126813	+	0.991927i	$0.459525\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−4.81616e7	+	1.20920e8i	−0.369553	+	0.927845i
$508$		0			0
$509$			1.11482e8i			0.845382i	0.906274	+	0.422691i	$0.138914\pi$
$509$			1.11482e8i			0.845382i	−0.906274	+	0.422691i	$0.861086\pi$
$510$		0			0
$511$	7.75183e7			0.580953
$512$		0			0
$513$	−3.54432e7	+	1.64138e7i	−0.262531	+	0.121579i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	1.89241e8i		−	1.36944i
$518$		0			0
$519$	4.80513e7	−	1.20643e8i	0.343718	−	0.862981i
$520$		0			0
$521$		−	2.70153e8i		−	1.91028i	−0.296160	−	0.955138i	$-0.595706\pi$
$521$		−	2.70153e8i		−	1.91028i	0.296160	−	0.955138i	$-0.404294\pi$
$522$		0			0
$523$		−	1.29085e8i		−	0.902341i	−0.892438	−	0.451171i	$-0.851007\pi$
$523$		−	1.29085e8i		−	0.902341i	0.892438	−	0.451171i	$-0.148993\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.09813e6			0.0143351
$528$		0			0
$529$	1.38177e8			0.933401
$530$		0			0
$531$	1.43459e8	−	1.51523e8i	0.958172	−	1.01203i
$532$		0			0
$533$	−6.96068e6			−0.0459695
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.54648e7	+	6.15949e6i	0.0998667	+	0.0397761i
$538$		0			0
$539$			1.15019e8i			0.734519i
$540$		0			0
$541$	−1.19668e8			−0.755765			−0.377883	−	0.925853i	$-0.623348\pi$
$541$	−1.19668e8			−0.755765			−0.377883	+	0.925853i	$0.623348\pi$
$542$		0			0
$543$	−1.69694e6	+	4.26055e6i	−0.0105991	+	0.0266113i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			7.12552e7i			0.435366i	0.976020	+	0.217683i	$0.0698499\pi$
$547$			7.12552e7i			0.435366i	−0.976020	+	0.217683i	$0.930150\pi$
$548$		0			0
$549$	−6.32628e7	−	5.98958e7i	−0.382324	−	0.361975i
$550$		0			0
$551$			3.66844e7i			0.219294i
$552$		0			0
$553$		−	8.43221e7i		−	0.498616i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.53675e8			−1.46795			−0.733976	−	0.679176i	$-0.762337\pi$
$557$	−2.53675e8			−1.46795			−0.733976	+	0.679176i	$0.762337\pi$
$558$		0			0
$559$	−5.12902e6			−0.0293629
$560$		0			0
$561$	−1.58453e6	−	631105.i	−0.00897452	−	0.00357448i
$562$		0			0
$563$	4.15634e7			0.232909			0.116454	−	0.993196i	$-0.462847\pi$
$563$	4.15634e7			0.232909			0.116454	+	0.993196i	$0.462847\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7.13506e7	−	3.90418e6i	0.391425	−	0.0214181i
$568$		0			0
$569$			1.69480e8i			0.919986i	0.887923	+	0.459993i	$0.152148\pi$
$569$			1.69480e8i			0.919986i	−0.887923	+	0.459993i	$0.847852\pi$
$570$		0			0
$571$	2.30661e8			1.23898			0.619492	−	0.785003i	$-0.287338\pi$
$571$	2.30661e8			1.23898			0.619492	+	0.785003i	$0.287338\pi$
$572$		0			0
$573$	−1.41408e8	−	5.63218e7i	−0.751642	−	0.299373i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	9.38558e7i		−	0.488578i	−0.969703	−	0.244289i	$-0.921446\pi$
$577$		−	9.38558e7i		−	0.488578i	0.969703	−	0.244289i	$-0.0785545\pi$
$578$		0			0
$579$	4.85454e7	+	1.93352e7i	0.250099	+	0.0996125i
$580$		0			0
$581$		−	1.33292e8i		−	0.679634i
$582$		0			0
$583$			1.12258e8i			0.566513i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.28514e8			1.12979			0.564897	−	0.825162i	$-0.308916\pi$
$587$	2.28514e8			1.12979			0.564897	+	0.825162i	$0.308916\pi$
$588$		0			0
$589$	−7.61376e7			−0.372609
$590$		0			0
$591$	−5.63137e7	+	1.41388e8i	−0.272805	+	0.684936i
$592$		0			0
$593$	1.47213e8			0.705966			0.352983	−	0.935630i	$-0.385167\pi$
$593$	1.47213e8			0.705966			0.352983	+	0.935630i	$0.385167\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.61169e6	−	1.66001e7i	0.0310734	−	0.0780167i
$598$		0			0
$599$			1.04762e8i			0.487440i	0.969846	+	0.243720i	$0.0783678\pi$
$599$			1.04762e8i			0.487440i	−0.969846	+	0.243720i	$0.921632\pi$
$600$		0			0
$601$	−2.92674e8			−1.34822			−0.674110	−	0.738631i	$-0.735473\pi$
$601$	−2.92674e8			−1.34822			−0.674110	+	0.738631i	$0.735473\pi$
$602$		0			0
$603$	7.14978e7	−	7.55170e7i	0.326092	−	0.344424i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			2.94250e8i			1.31568i	0.753158	+	0.657840i	$0.228530\pi$
$607$			2.94250e8i			1.31568i	−0.753158	+	0.657840i	$0.771470\pi$
$608$		0			0
$609$	2.48331e7	−	6.23490e7i	0.109946	−	0.276043i
$610$		0			0
$611$			1.28154e7i			0.0561833i
$612$		0			0
$613$		−	1.10142e8i		−	0.478158i	−0.971000	−	0.239079i	$-0.923155\pi$
$613$		−	1.10142e8i		−	0.478158i	0.971000	−	0.239079i	$-0.0768454\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.60731e8			−1.53578			−0.767888	−	0.640584i	$-0.778692\pi$
$617$	−3.60731e8			−1.53578			−0.767888	+	0.640584i	$0.778692\pi$
$618$		0			0
$619$	−4.02744e8			−1.69808			−0.849038	−	0.528332i	$-0.822818\pi$
$619$	−4.02744e8			−1.69808			−0.849038	+	0.528332i	$0.822818\pi$
$620$		0			0
$621$	−3.02164e8	+	1.39933e8i	−1.26174	+	0.584314i
$622$		0			0
$623$	−1.11278e8			−0.460197
$624$		0			0
$625$		0			0
$626$		0			0
$627$	5.74998e7	+	2.29017e7i	0.233273	+	0.0929106i
$628$		0			0
$629$		−	1.16166e6i		−	0.00466795i
$630$		0			0
$631$	−7.78200e7			−0.309744			−0.154872	−	0.987935i	$-0.549497\pi$
$631$	−7.78200e7			−0.309744			−0.154872	+	0.987935i	$0.549497\pi$
$632$		0			0
$633$	−7.25822e6	+	1.82234e7i	−0.0286167	+	0.0718484i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	7.78905e6i		−	0.0301347i
$638$		0			0
$639$	−1.82100e8	+	1.92336e8i	−0.697921	+	0.737155i
$640$		0			0
$641$			4.82128e8i			1.83058i	0.402797	+	0.915290i	$0.368038\pi$
$641$			4.82128e8i			1.83058i	−0.402797	+	0.915290i	$0.631962\pi$
$642$		0			0
$643$			1.57608e7i			0.0592850i	0.999561	+	0.0296425i	$0.00943688\pi$
$643$			1.57608e7i			0.0592850i	−0.999561	+	0.0296425i	$0.990563\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.10892e8			0.778660			0.389330	−	0.921098i	$-0.372707\pi$
$647$	2.10892e8			0.778660			0.389330	+	0.921098i	$0.372707\pi$
$648$		0			0
$649$	−3.30642e8			−1.20955
$650$		0			0
$651$	1.29404e8	+	5.15405e7i	0.469034	+	0.186812i
$652$		0			0
$653$	2.03936e8			0.732410			0.366205	−	0.930534i	$-0.380657\pi$
$653$	2.03936e8			0.732410			0.366205	+	0.930534i	$0.380657\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	2.88951e8	−	3.05194e8i	1.01889	−	1.07617i
$658$		0			0
$659$		−	4.80405e8i		−	1.67862i	−0.543656	−	0.839308i	$-0.682960\pi$
$659$		−	4.80405e8i		−	1.67862i	0.543656	−	0.839308i	$-0.317040\pi$
$660$		0			0
$661$	−1.13313e8			−0.392351			−0.196176	−	0.980569i	$-0.562852\pi$
$661$	−1.13313e8			−0.392351			−0.196176	+	0.980569i	$0.562852\pi$
$662$		0			0
$663$	107304.	+	42738.2i	0.000368192	+	0.000146648i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			3.12746e8i			1.05394i
$668$		0			0
$669$	−488229.	−	194458.i	−0.00163059	−	0.000649452i
$670$		0			0
$671$			1.38047e8i			0.456940i
$672$		0			0
$673$		−	6.78723e7i		−	0.222663i	−0.993783	−	0.111331i	$-0.964489\pi$
$673$		−	6.78723e7i		−	0.222663i	0.993783	−	0.111331i	$-0.0355115\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.22505e8			1.03937			0.519685	−	0.854358i	$-0.326049\pi$
$677$	3.22505e8			1.03937			0.519685	+	0.854358i	$0.326049\pi$
$678$		0			0
$679$	−1.14121e8			−0.364549
$680$		0			0
$681$	−1.74522e7	+	4.38176e7i	−0.0552598	+	0.138742i
$682$		0			0
$683$	−4.17199e8			−1.30943			−0.654714	−	0.755877i	$-0.727211\pi$
$683$	−4.17199e8			−1.30943			−0.654714	+	0.755877i	$0.727211\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	8.02739e6	−	2.01545e7i	0.0247574	−	0.0621588i
$688$		0			0
$689$		−	7.60204e6i		−	0.0232420i
$690$		0			0
$691$	−3.88809e7			−0.117842			−0.0589212	−	0.998263i	$-0.518766\pi$
$691$	−3.88809e7			−0.117842			−0.0589212	+	0.998263i	$0.518766\pi$
$692$		0			0
$693$	−8.22239e7	−	7.78478e7i	−0.247058	−	0.233909i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	4.86587e6i		−	0.0143702i
$698$		0			0
$699$	−1.41514e8	+	3.55302e8i	−0.414351	+	1.04032i
$700$		0			0
$701$		−	7.02379e7i		−	0.203900i	−0.994790	−	0.101950i	$-0.967492\pi$
$701$		−	7.02379e7i		−	0.203900i	0.994790	−	0.101950i	$-0.0325082\pi$
$702$		0			0
$703$			4.21546e7i			0.121333i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.49732e8			0.423698
$708$		0			0
$709$	−3.27456e8			−0.918785			−0.459393	−	0.888233i	$-0.651933\pi$
$709$	−3.27456e8			−0.918785			−0.459393	+	0.888233i	$0.651933\pi$
$710$		0			0
$711$	−3.31981e8	−	3.14313e8i	−0.923645	−	0.874486i
$712$		0			0
$713$	−6.49097e8			−1.79078
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−3.80291e8	−	1.51467e8i	−1.03171	−	0.410923i
$718$		0			0
$719$			6.46847e8i			1.74026i	0.492819	+	0.870132i	$0.335966\pi$
$719$			6.46847e8i			1.74026i	−0.492819	+	0.870132i	$0.664034\pi$
$720$		0			0
$721$	2.81919e8			0.752173
$722$		0			0
$723$	2.44556e8	−	6.14012e8i	0.647089	−	1.62466i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			4.36086e8i			1.13493i	0.823398	+	0.567465i	$0.192076\pi$
$727$			4.36086e8i			1.13493i	−0.823398	+	0.567465i	$0.807924\pi$
$728$		0			0
$729$	2.50590e8	−	2.95465e8i	0.646816	−	0.762646i
$730$		0			0
$731$		−	3.58545e6i		−	0.00917893i
$732$		0			0
$733$		−	4.27585e8i		−	1.08570i	−0.839829	−	0.542851i	$-0.817345\pi$
$733$		−	4.27585e8i		−	1.08570i	0.839829	−	0.542851i	$-0.182655\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.64787e8			−0.411643
$738$		0			0
$739$	−3.65022e8			−0.904452			−0.452226	−	0.891903i	$-0.649370\pi$
$739$	−3.65022e8			−0.904452			−0.452226	+	0.891903i	$0.649370\pi$
$740$		0			0
$741$	−3.89387e6	−	1.55090e6i	−0.00957032	−	0.00381178i
$742$		0			0
$743$	6.75024e8			1.64571			0.822854	−	0.568253i	$-0.192380\pi$
$743$	6.75024e8			1.64571			0.822854	+	0.568253i	$0.192380\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−5.24778e8	−	4.96848e8i	−1.25897	−	1.19196i
$748$		0			0
$749$			3.25885e7i			0.0775567i
$750$		0			0
$751$	−5.14564e8			−1.21484			−0.607421	−	0.794380i	$-0.707796\pi$
$751$	−5.14564e8			−1.21484			−0.607421	+	0.794380i	$0.707796\pi$
$752$		0			0
$753$	1.35161e8	+	5.38334e7i	0.316567	+	0.126086i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			2.32695e8i			0.536413i	0.963361	+	0.268207i	$0.0864310\pi$
$757$			2.32695e8i			0.536413i	−0.963361	+	0.268207i	$0.913569\pi$
$758$		0			0
$759$	4.90204e8	+	1.95245e8i	1.12112	+	0.446533i
$760$		0			0
$761$			3.18044e7i			0.0721661i	0.999349	+	0.0360831i	$0.0114881\pi$
$761$			3.18044e7i			0.0721661i	−0.999349	+	0.0360831i	$0.988512\pi$
$762$		0			0
$763$		−	3.00482e8i		−	0.676465i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.23910e7			0.0496234
$768$		0			0
$769$	−6.46145e8			−1.42086			−0.710430	−	0.703768i	$-0.751499\pi$
$769$	−6.46145e8			−1.42086			−0.710430	+	0.703768i	$0.751499\pi$
$770$		0			0
$771$	2.81460e8	−	7.06668e8i	0.614121	−	1.54189i
$772$		0			0
$773$	−1.82196e8			−0.394457			−0.197228	−	0.980358i	$-0.563194\pi$
$773$	−1.82196e8			−0.394457			−0.197228	+	0.980358i	$0.563194\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2.85361e7	−	7.16462e7i	0.0608319	−	0.152732i
$778$		0			0
$779$			1.76574e8i			0.373521i
$780$		0			0
$781$	4.19701e8			0.881022
$782$		0			0
$783$	−1.52906e8	−	3.30177e8i	−0.318522	−	0.687799i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1.57411e8i			0.322932i	0.986878	+	0.161466i	$0.0516222\pi$
$787$			1.57411e8i			0.322932i	−0.986878	+	0.161466i	$0.948378\pi$
$788$		0			0
$789$	−1.14784e8	+	2.88190e8i	−0.233695	+	0.586743i
$790$		0			0
$791$		−	3.30080e7i		−	0.0666944i
$792$		0			0
$793$		−	9.34850e6i		−	0.0187466i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−5.11553e7			−0.101045			−0.0505225	−	0.998723i	$-0.516089\pi$
$797$	−5.11553e7			−0.101045			−0.0505225	+	0.998723i	$0.516089\pi$
$798$		0			0
$799$	−8.95860e6			−0.0175631
$800$		0			0
$801$	−4.14790e8	+	4.38107e8i	−0.807106	+	0.852477i
$802$		0			0
$803$	−6.65971e8			−1.28620
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−3.68411e8	−	1.46735e8i	−0.700990	−	0.279199i
$808$		0			0
$809$		−	4.30177e8i		−	0.812459i	−0.913771	−	0.406229i	$-0.866843\pi$
$809$		−	4.30177e8i		−	0.812459i	0.913771	−	0.406229i	$-0.133157\pi$
$810$		0			0
$811$	1.35268e8			0.253590			0.126795	−	0.991929i	$-0.459531\pi$
$811$	1.35268e8			0.253590			0.126795	+	0.991929i	$0.459531\pi$
$812$		0			0
$813$	1.77326e7	−	4.45216e7i	0.0329990	−	0.0828513i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			1.30110e8i			0.238586i
$818$		0			0
$819$	5.56818e6	+	5.27183e6i	0.0101359	+	0.00959642i
$820$		0			0
$821$		−	6.01637e8i		−	1.08719i	−0.839348	−	0.543595i	$-0.817063\pi$
$821$		−	6.01637e8i		−	1.08719i	0.839348	−	0.543595i	$-0.182937\pi$
$822$		0			0
$823$			5.45245e8i			0.978119i	0.872250	+	0.489060i	$0.162660\pi$
$823$			5.45245e8i			0.978119i	−0.872250	+	0.489060i	$0.837340\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.83751e8			−0.501673			−0.250836	−	0.968030i	$-0.580706\pi$
$827$	−2.83751e8			−0.501673			−0.250836	+	0.968030i	$0.580706\pi$
$828$		0			0
$829$	−9.17319e7			−0.161011			−0.0805057	−	0.996754i	$-0.525654\pi$
$829$	−9.17319e7			−0.161011			−0.0805057	+	0.996754i	$0.525654\pi$
$830$		0			0
$831$	6.88755e8	+	2.74326e8i	1.20022	+	0.478039i
$832$		0			0
$833$	5.44495e6			0.00942018
$834$		0			0
$835$		0			0
$836$		0			0
$837$	6.85274e8	−	3.17353e8i	1.16866	−	0.541210i
$838$		0			0
$839$		−	7.01084e8i		−	1.18709i	−0.804800	−	0.593545i	$-0.797728\pi$
$839$		−	7.01084e8i		−	1.18709i	0.804800	−	0.593545i	$-0.202272\pi$
$840$		0			0
$841$	2.53084e8			0.425477
$842$		0			0
$843$	−6.85808e7	−	2.73152e7i	−0.114477	−	0.0455955i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	5.87807e7i		−	0.0967352i
$848$		0			0
$849$	8.81747e8	+	3.51193e8i	1.44086	+	0.573882i
$850$		0			0
$851$			3.59382e8i			0.583132i
$852$		0			0
$853$			1.36003e8i			0.219130i	0.993980	+	0.109565i	$0.0349457\pi$
$853$			1.36003e8i			0.219130i	−0.993980	+	0.109565i	$0.965054\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.07337e9			1.70532			0.852661	−	0.522465i	$-0.174987\pi$
$857$	1.07337e9			1.70532			0.852661	+	0.522465i	$0.174987\pi$
$858$		0			0
$859$	4.00149e8			0.631309			0.315654	−	0.948874i	$-0.397776\pi$
$859$	4.00149e8			0.631309			0.315654	+	0.948874i	$0.397776\pi$
$860$		0			0
$861$	1.19530e8	−	3.00107e8i	0.187270	−	0.470182i
$862$		0			0
$863$	4.98364e8			0.775379			0.387690	−	0.921790i	$-0.373273\pi$
$863$	4.98364e8			0.775379			0.387690	+	0.921790i	$0.373273\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	2.41119e8	−	6.05383e8i	0.369977	−	0.928908i
$868$		0			0
$869$			7.24424e8i			1.10391i
$870$		0			0
$871$	1.11593e7			0.0168882
$872$		0			0
$873$	−4.25389e8	+	4.49301e8i	−0.639356	+	0.675297i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			1.06775e9i			1.58296i	0.611194	+	0.791481i	$0.290689\pi$
$877$			1.06775e9i			1.58296i	−0.611194	+	0.791481i	$0.709311\pi$
$878$		0			0
$879$	4.65675e8	−	1.16918e9i	0.685672	−	1.72153i
$880$		0			0
$881$			7.83809e7i			0.114626i	0.998356	+	0.0573129i	$0.0182533\pi$
$881$			7.83809e7i			0.114626i	−0.998356	+	0.0573129i	$0.981747\pi$
$882$		0			0
$883$			8.34186e8i			1.21166i	0.795594	+	0.605830i	$0.207159\pi$
$883$			8.34186e8i			1.21166i	−0.795594	+	0.605830i	$0.792841\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.77272e8			−0.970493			−0.485247	−	0.874377i	$-0.661270\pi$
$887$	−6.77272e8			−0.970493			−0.485247	+	0.874377i	$0.661270\pi$
$888$		0			0
$889$	6.50972e7			0.0926525
$890$		0			0
$891$	−6.12983e8	+	3.35414e7i	−0.866594	+	0.0474185i
$892$		0			0
$893$	3.25092e8			0.456512
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−3.31965e7	−	1.32219e7i	−0.0459954	−	0.0183196i
$898$		0			0
$899$		−	7.09273e8i		−	0.976189i
$900$		0			0
$901$	5.31422e6			0.00726550
$902$		0			0
$903$	8.80765e7	−	2.21136e8i	0.119618	−	0.300328i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	3.70732e8i		−	0.496865i	−0.968649	−	0.248432i	$-0.920085\pi$
$907$		−	3.70732e8i		−	0.496865i	0.968649	−	0.248432i	$-0.0799154\pi$
$908$		0			0
$909$	5.58129e8	−	5.89504e8i	0.743093	−	0.784865i
$910$		0			0
$911$			5.03944e8i			0.666541i	0.942831	+	0.333271i	$0.108152\pi$
$911$			5.03944e8i			0.666541i	−0.942831	+	0.333271i	$0.891848\pi$
$912$		0			0
$913$			1.14513e9i			1.50467i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.66336e8			−0.604771
$918$		0			0
$919$	3.59392e7			0.0463043			0.0231522	−	0.999732i	$-0.492630\pi$
$919$	3.59392e7			0.0463043			0.0231522	+	0.999732i	$0.492630\pi$
$920$		0			0
$921$	7.82509e8	+	3.11667e8i	1.00164	+	0.398944i
$922$		0			0
$923$	−2.84220e7			−0.0361451
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.05086e9	−	1.10993e9i	1.31918	−	1.39334i
$928$		0			0
$929$		−	4.60827e8i		−	0.574766i	−0.957816	−	0.287383i	$-0.907215\pi$
$929$		−	4.60827e8i		−	0.574766i	0.957816	−	0.287383i	$-0.0927853\pi$
$930$		0			0
$931$	−1.97588e8			−0.244856
$932$		0			0
$933$	8.82911e8	+	3.51656e8i	1.08711	+	0.432986i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			5.28388e8i			0.642295i	0.947029	+	0.321147i	$0.104068\pi$
$937$			5.28388e8i			0.642295i	−0.947029	+	0.321147i	$0.895932\pi$
$938$		0			0
$939$	9.04492e8	+	3.60252e8i	1.09247	+	0.435120i
$940$		0			0
$941$		−	2.96965e8i		−	0.356399i	−0.983994	−	0.178200i	$-0.942973\pi$
$941$		−	2.96965e8i		−	0.356399i	0.983994	−	0.178200i	$-0.0570273\pi$
$942$		0			0
$943$			1.50535e9i			1.79516i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.43742e8			−0.640240			−0.320120	−	0.947377i	$-0.603723\pi$
$947$	−5.43742e8			−0.640240			−0.320120	+	0.947377i	$0.603723\pi$
$948$		0			0
$949$	4.50993e7			0.0527681
$950$		0			0
$951$	−5.73962e8	+	1.44106e9i	−0.667332	+	1.67548i
$952$		0			0
$953$	1.43357e9			1.65630			0.828150	−	0.560506i	$-0.189393\pi$
$953$	1.43357e9			1.65630			0.828150	+	0.560506i	$0.189393\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−2.13345e8	+	5.35649e8i	−0.243414	+	0.611146i
$958$		0			0
$959$			2.34621e8i			0.266018i
$960$		0			0
$961$	5.84575e8			0.658673
$962$		0			0
$963$	1.28303e8	+	1.21475e8i	0.143667	+	0.136021i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1.86723e8i		−	0.206500i	−0.994655	−	0.103250i	$-0.967076\pi$
$967$		−	1.86723e8i		−	0.206500i	0.994655	−	0.103250i	$-0.0329241\pi$
$968$		0			0
$969$	1.08416e6	−	2.72202e6i	0.00119157	−	0.00299171i
$970$		0			0
$971$		−	7.68364e8i		−	0.839284i	−0.907690	−	0.419642i	$-0.862156\pi$
$971$		−	7.68364e8i		−	0.839284i	0.907690	−	0.419642i	$-0.137844\pi$
$972$		0			0
$973$		−	4.76741e8i		−	0.517540i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.60128e7			−0.0922316			−0.0461158	−	0.998936i	$-0.514684\pi$
$977$	−8.60128e7			−0.0922316			−0.0461158	+	0.998936i	$0.514684\pi$
$978$		0			0
$979$	9.56002e8			1.01885
$980$		0			0
$981$	−1.18302e9	−	1.12006e9i	−1.25310	−	1.18640i
$982$		0			0
$983$	9.61067e8			1.01180			0.505898	−	0.862593i	$-0.331161\pi$
$983$	9.61067e8			1.01180			0.505898	+	0.862593i	$0.331161\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−5.52529e8	−	2.20068e8i	−0.574650	−	0.228879i
$988$		0			0
$989$			1.10923e9i			1.14665i
$990$		0			0
$991$	−6.24596e8			−0.641768			−0.320884	−	0.947118i	$-0.603980\pi$
$991$	−6.24596e8			−0.641768			−0.320884	+	0.947118i	$0.603980\pi$
$992$		0			0
$993$	−3.26021e8	+	8.18547e8i	−0.332964	+	0.835980i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1.53088e9i			1.54474i	0.635171	+	0.772371i	$0.280929\pi$
$997$			1.53088e9i			1.54474i	−0.635171	+	0.772371i	$0.719071\pi$
$998$		0			0
$999$	−1.75707e8	−	3.79411e8i	−0.176235	−	0.380552i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.7.b.e.149.8		16
3.2	odd	2	inner	300.7.b.e.149.10		16
5.2	odd	4		60.7.g.a.41.8	yes	8
5.3	odd	4		300.7.g.h.101.1		8
5.4	even	2	inner	300.7.b.e.149.9		16
15.2	even	4		60.7.g.a.41.7	✓	8
15.8	even	4		300.7.g.h.101.2		8
15.14	odd	2	inner	300.7.b.e.149.7		16
20.7	even	4		240.7.l.c.161.1		8
60.47	odd	4		240.7.l.c.161.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.7.g.a.41.7	✓	8	15.2	even	4
60.7.g.a.41.8	yes	8	5.2	odd	4
240.7.l.c.161.1		8	20.7	even	4
240.7.l.c.161.2		8	60.47	odd	4
300.7.b.e.149.7		16	15.14	odd	2	inner
300.7.b.e.149.8		16	1.1	even	1	trivial
300.7.b.e.149.9		16	5.4	even	2	inner
300.7.b.e.149.10		16	3.2	odd	2	inner
300.7.g.h.101.1		8	5.3	odd	4
300.7.g.h.101.2		8	15.8	even	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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	300.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-9.99060 + 25.0836i) q^{3} -134.460i q^{7} +(-529.376 - 501.201i) q^{9} +O(q^{10})\)	\(q+(-9.99060 + 25.0836i) q^{3} -134.460i q^{7} +(-529.376 - 501.201i) q^{9} +1155.16i q^{11} -78.2272i q^{13} +54.6849 q^{17} -1984.42 q^{19} +(3372.73 + 1343.33i) q^{21} -16917.8 q^{23} +(17860.7 - 8271.35i) q^{27} -18486.2i q^{29} +38367.7 q^{31} +(-28975.6 - 11540.8i) q^{33} -21242.8i q^{37} +(1962.22 + 781.537i) q^{39} -88980.3i q^{41} -65565.7i q^{43} -163822. q^{47} +99569.6 q^{49} +(-546.335 + 1371.69i) q^{51} +97179.1 q^{53} +(19825.5 - 49776.4i) q^{57} +286230. i q^{59} +119504. q^{61} +(-67391.2 + 71179.6i) q^{63} +142653. i q^{67} +(169019. - 424360. i) q^{69} -363327. i q^{71} +576517. i q^{73} +155322. q^{77} +627119. q^{79} +(29036.1 + 530647. i) q^{81} +991316. q^{83} +(463701. + 184688. i) q^{87} -827592. i q^{89} -10518.4 q^{91} +(-383316. + 962400. i) q^{93} -848738. i q^{97} +(578968. - 611514. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2984 q^{9}+O(q^{10}) \) 16 * q - 2984 * q^9	\( 16 q - 2984 q^{9} + 30544 q^{19} - 1736 q^{21} + 70064 q^{31} - 79216 q^{39} - 405120 q^{49} + 858240 q^{51} - 271216 q^{61} + 509880 q^{69} - 1415408 q^{79} - 2396224 q^{81} - 4008224 q^{91} - 5300160 q^{99}+O(q^{100}) \) 16 * q - 2984 * q^9 + 30544 * q^19 - 1736 * q^21 + 70064 * q^31 - 79216 * q^39 - 405120 * q^49 + 858240 * q^51 - 271216 * q^61 + 509880 * q^69 - 1415408 * q^79 - 2396224 * q^81 - 4008224 * q^91 - 5300160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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