LMFDB - Embedded newform 400.4.n.d.143.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.n (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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.1731891456.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.3
Root			$-1.35234 + 0.780776i$ of defining polynomial
Character	$\chi$	$=$	400.143
Dual form			400.4.n.d.207.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.12311 - 4.12311i) q^{3} +(4.12311 + 4.12311i) q^{7} -7.00000i q^{9} -13.8564i q^{11} +(-57.1314 - 57.1314i) q^{13} +(57.1314 - 57.1314i) q^{17} -96.9948 q^{19} +34.0000 q^{21} +(86.5852 - 86.5852i) q^{23} +(82.4621 + 82.4621i) q^{27} -174.000i q^{29} -193.990i q^{31} +(-57.1314 - 57.1314i) q^{33} -471.118 q^{39} +252.000 q^{41} +(-202.032 + 202.032i) q^{43} +(-284.494 - 284.494i) q^{47} -309.000i q^{49} -471.118i q^{51} +(399.920 + 399.920i) q^{53} +(-399.920 + 399.920i) q^{57} -872.954 q^{59} +56.0000 q^{61} +(28.8617 - 28.8617i) q^{63} +(317.479 + 317.479i) q^{67} -714.000i q^{69} -387.979i q^{71} +(399.920 + 399.920i) q^{73} +(57.1314 - 57.1314i) q^{77} -692.820 q^{79} +869.000 q^{81} +(-482.403 + 482.403i) q^{83} +(-717.420 - 717.420i) q^{87} +42.0000i q^{89} -471.118i q^{91} +(-799.840 - 799.840i) q^{93} +(742.709 - 742.709i) q^{97} -96.9948 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 272 q^{21} + 2016 q^{41} + 448 q^{61} + 6952 q^{81}+O(q^{100}) $ 8 * q + 272 * q^21 + 2016 * q^41 + 448 * q^61 + 6952 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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
$3$	4.12311	−	4.12311i	0.793492	−	0.793492i	−0.188568	−	0.982060i	$-0.560385\pi$
$3$	4.12311	−	4.12311i	0.793492	−	0.793492i	0.982060	+	0.188568i	$0.0603846\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	4.12311	+	4.12311i	0.222627	+	0.222627i	0.809604	−	0.586977i	$-0.199682\pi$
$7$	4.12311	+	4.12311i	0.222627	+	0.222627i	−0.586977	+	0.809604i	$0.699682\pi$
$8$		0			0
$9$		−	7.00000i		−	0.259259i
$10$		0			0
$11$		−	13.8564i		−	0.379806i	−0.981803	−	0.189903i	$-0.939183\pi$
$11$		−	13.8564i		−	0.379806i	0.981803	−	0.189903i	$-0.0608173\pi$
$12$		0			0
$13$	−57.1314	−	57.1314i	−1.21888	−	1.21888i	−0.968025	−	0.250852i	$-0.919289\pi$
$13$	−57.1314	−	57.1314i	−1.21888	−	1.21888i	−0.250852	−	0.968025i	$-0.580711\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	57.1314	−	57.1314i	0.815083	−	0.815083i	−0.170308	−	0.985391i	$-0.554476\pi$
$17$	57.1314	−	57.1314i	0.815083	−	0.815083i	0.985391	+	0.170308i	$0.0544763\pi$
$18$		0			0
$19$	−96.9948			−1.17117			−0.585583	−	0.810613i	$-0.699134\pi$
$19$	−96.9948			−1.17117			−0.585583	+	0.810613i	$0.699134\pi$
$20$		0			0
$21$	34.0000			0.353305
$22$		0			0
$23$	86.5852	−	86.5852i	0.784968	−	0.784968i	−0.195696	−	0.980665i	$-0.562697\pi$
$23$	86.5852	−	86.5852i	0.784968	−	0.784968i	0.980665	+	0.195696i	$0.0626967\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	82.4621	+	82.4621i	0.587772	+	0.587772i
$28$		0			0
$29$		−	174.000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	174.000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$		−	193.990i		−	1.12392i	−0.827164	−	0.561961i	$-0.810047\pi$
$31$		−	193.990i		−	1.12392i	0.827164	−	0.561961i	$-0.189953\pi$
$32$		0			0
$33$	−57.1314	−	57.1314i	−0.301373	−	0.301373i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$	−471.118			−1.93434
$40$		0			0
$41$	252.000			0.959897			0.479949	−	0.877297i	$-0.340655\pi$
$41$	252.000			0.959897			0.479949	+	0.877297i	$0.340655\pi$
$42$		0			0
$43$	−202.032	+	202.032i	−0.716503	+	0.716503i	−0.967887	−	0.251385i	$-0.919114\pi$
$43$	−202.032	+	202.032i	−0.716503	+	0.716503i	0.251385	+	0.967887i	$0.419114\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−284.494	−	284.494i	−0.882931	−	0.882931i	0.110901	−	0.993832i	$-0.464627\pi$
$47$	−284.494	−	284.494i	−0.882931	−	0.882931i	−0.993832	+	0.110901i	$0.964627\pi$
$48$		0			0
$49$		−	309.000i		−	0.900875i
$50$		0			0
$51$		−	471.118i		−	1.29352i
$52$		0			0
$53$	399.920	+	399.920i	1.03648	+	1.03648i	0.999309	+	0.0371671i	$0.0118334\pi$
$53$	399.920	+	399.920i	1.03648	+	1.03648i	0.0371671	+	0.999309i	$0.488167\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−399.920	+	399.920i	−0.929310	+	0.929310i
$58$		0			0
$59$	−872.954			−1.92625			−0.963126	−	0.269050i	$-0.913290\pi$
$59$	−872.954			−1.92625			−0.963126	+	0.269050i	$0.913290\pi$
$60$		0			0
$61$	56.0000			0.117542			0.0587710	−	0.998271i	$-0.481282\pi$
$61$	56.0000			0.117542			0.0587710	+	0.998271i	$0.481282\pi$
$62$		0			0
$63$	28.8617	−	28.8617i	0.0577181	−	0.0577181i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	317.479	+	317.479i	0.578899	+	0.578899i	0.934600	−	0.355701i	$-0.115758\pi$
$67$	317.479	+	317.479i	0.578899	+	0.578899i	−0.355701	+	0.934600i	$0.615758\pi$
$68$		0			0
$69$		−	714.000i		−	1.24573i
$70$		0			0
$71$		−	387.979i		−	0.648517i	−0.945969	−	0.324258i	$-0.894885\pi$
$71$		−	387.979i		−	0.648517i	0.945969	−	0.324258i	$-0.105115\pi$
$72$		0			0
$73$	399.920	+	399.920i	0.641193	+	0.641193i	0.950849	−	0.309656i	$-0.100214\pi$
$73$	399.920	+	399.920i	0.641193	+	0.641193i	−0.309656	+	0.950849i	$0.600214\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	57.1314	−	57.1314i	0.0845549	−	0.0845549i
$78$		0			0
$79$	−692.820			−0.986688			−0.493344	−	0.869834i	$-0.664226\pi$
$79$	−692.820			−0.986688			−0.493344	+	0.869834i	$0.664226\pi$
$80$		0			0
$81$	869.000			1.19204
$82$		0			0
$83$	−482.403	+	482.403i	−0.637960	+	0.637960i	−0.950052	−	0.312092i	$-0.898970\pi$
$83$	−482.403	+	482.403i	−0.637960	+	0.637960i	0.312092	+	0.950052i	$0.398970\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−717.420	−	717.420i	−0.884087	−	0.884087i
$88$		0			0
$89$			42.0000i			0.0500224i	0.999687	+	0.0250112i	$0.00796214\pi$
$89$			42.0000i			0.0500224i	−0.999687	+	0.0250112i	$0.992038\pi$
$90$		0			0
$91$		−	471.118i		−	0.542710i
$92$		0			0
$93$	−799.840	−	799.840i	−0.891823	−	0.891823i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	742.709	−	742.709i	0.777429	−	0.777429i	−0.201964	−	0.979393i	$-0.564732\pi$
$97$	742.709	−	742.709i	0.777429	−	0.777429i	0.979393	+	0.201964i	$0.0647323\pi$
$98$		0			0
$99$	−96.9948			−0.0984682
$100$		0			0
$101$	−1050.00			−1.03444			−0.517222	−	0.855851i	$-0.673034\pi$
$101$	−1050.00			−1.03444			−0.517222	+	0.855851i	$0.673034\pi$
$102$		0			0
$103$	869.975	−	869.975i	0.832245	−	0.832245i	−0.155579	−	0.987823i	$-0.549724\pi$
$103$	869.975	−	869.975i	0.832245	−	0.832245i	0.987823	+	0.155579i	$0.0497242\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1298.78	+	1298.78i	1.17344	+	1.17344i	0.981385	+	0.192051i	$0.0615139\pi$
$107$	1298.78	+	1298.78i	1.17344	+	1.17344i	0.192051	+	0.981385i	$0.438486\pi$
$108$		0			0
$109$		−	272.000i		−	0.239017i	−0.992833	−	0.119509i	$-0.961868\pi$
$109$		−	272.000i		−	0.239017i	0.992833	−	0.119509i	$-0.0381319\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−799.840	−	799.840i	−0.665864	−	0.665864i	0.290892	−	0.956756i	$-0.406048\pi$
$113$	−799.840	−	799.840i	−0.665864	−	0.665864i	−0.956756	+	0.290892i	$0.906048\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−399.920	+	399.920i	−0.316005	+	0.316005i
$118$		0			0
$119$	471.118			0.362918
$120$		0			0
$121$	1139.00			0.855748
$122$		0			0
$123$	1039.02	−	1039.02i	0.761671	−	0.761671i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−28.8617	−	28.8617i	−0.0201659	−	0.0201659i	0.696952	−	0.717118i	$-0.254539\pi$
$127$	−28.8617	−	28.8617i	−0.0201659	−	0.0201659i	−0.717118	+	0.696952i	$0.754539\pi$
$128$		0			0
$129$			1666.00i			1.13708i
$130$		0			0
$131$			1842.90i			1.22912i	0.788869	+	0.614561i	$0.210667\pi$
$131$			1842.90i			1.22912i	−0.788869	+	0.614561i	$0.789333\pi$
$132$		0			0
$133$	−399.920	−	399.920i	−0.260733	−	0.260733i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	799.840	−	799.840i	0.498795	−	0.498795i	−0.412268	−	0.911063i	$-0.635263\pi$
$137$	799.840	−	799.840i	0.498795	−	0.498795i	0.911063	+	0.412268i	$0.135263\pi$
$138$		0			0
$139$	2036.89			1.24293			0.621464	−	0.783443i	$-0.286538\pi$
$139$	2036.89			1.24293			0.621464	+	0.783443i	$0.286538\pi$
$140$		0			0
$141$	−2346.00			−1.40120
$142$		0			0
$143$	−791.636	+	791.636i	−0.462937	+	0.462937i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1274.04	−	1274.04i	−0.714837	−	0.714837i
$148$		0			0
$149$		−	744.000i		−	0.409066i	−0.978860	−	0.204533i	$-0.934432\pi$
$149$		−	744.000i		−	0.409066i	0.978860	−	0.204533i	$-0.0655676\pi$
$150$		0			0
$151$			859.097i			0.462996i	0.972835	+	0.231498i	$0.0743626\pi$
$151$			859.097i			0.462996i	−0.972835	+	0.231498i	$0.925637\pi$
$152$		0			0
$153$	−399.920	−	399.920i	−0.211318	−	0.211318i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	799.840	−	799.840i	0.406587	−	0.406587i	−0.473960	−	0.880547i	$-0.657176\pi$
$157$	799.840	−	799.840i	0.406587	−	0.406587i	0.880547	+	0.473960i	$0.157176\pi$
$158$		0			0
$159$	3297.82			1.64487
$160$		0			0
$161$	714.000			0.349510
$162$		0			0
$163$	−1356.50	+	1356.50i	−0.651837	+	0.651837i	−0.953435	−	0.301598i	$-0.902480\pi$
$163$	−1356.50	+	1356.50i	−0.651837	+	0.651837i	0.301598	+	0.953435i	$0.402480\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1274.04	+	1274.04i	0.590348	+	0.590348i	0.937725	−	0.347377i	$-0.112928\pi$
$167$	1274.04	+	1274.04i	0.590348	+	0.590348i	−0.347377	+	0.937725i	$0.612928\pi$
$168$		0			0
$169$			4331.00i			1.97132i
$170$		0			0
$171$			678.964i			0.303635i
$172$		0			0
$173$	2856.57	+	2856.57i	1.25538	+	1.25538i	0.953274	+	0.302109i	$0.0976905\pi$
$173$	2856.57	+	2856.57i	1.25538	+	1.25538i	0.302109	+	0.953274i	$0.402309\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−3599.28	+	3599.28i	−1.52847	+	1.52847i
$178$		0			0
$179$	1842.90			0.769525			0.384762	−	0.923016i	$-0.374283\pi$
$179$	1842.90			0.769525			0.384762	+	0.923016i	$0.374283\pi$
$180$		0			0
$181$	−3094.00			−1.27058			−0.635291	−	0.772273i	$-0.719120\pi$
$181$	−3094.00			−1.27058			−0.635291	+	0.772273i	$0.719120\pi$
$182$		0			0
$183$	230.894	−	230.894i	0.0932687	−	0.0932687i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−791.636	−	791.636i	−0.309573	−	0.309573i
$188$		0			0
$189$			680.000i			0.261708i
$190$		0			0
$191$			3990.65i			1.51180i	0.654690	+	0.755898i	$0.272799\pi$
$191$			3990.65i			1.51180i	−0.654690	+	0.755898i	$0.727201\pi$
$192$		0			0
$193$	2799.44	+	2799.44i	1.04408	+	1.04408i	0.998982	+	0.0451010i	$0.0143610\pi$
$193$	2799.44	+	2799.44i	1.04408	+	1.04408i	0.0451010	+	0.998982i	$0.485639\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	399.920	−	399.920i	0.144635	−	0.144635i	−0.631081	−	0.775717i	$-0.717389\pi$
$197$	399.920	−	399.920i	0.144635	−	0.144635i	0.775717	+	0.631081i	$0.217389\pi$
$198$		0			0
$199$	3103.84			1.10565			0.552827	−	0.833296i	$-0.313549\pi$
$199$	3103.84			1.10565			0.552827	+	0.833296i	$0.313549\pi$
$200$		0			0
$201$	2618.00			0.918704
$202$		0			0
$203$	717.420	−	717.420i	0.248045	−	0.248045i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−606.097	−	606.097i	−0.203510	−	0.203510i
$208$		0			0
$209$			1344.00i			0.444815i
$210$		0			0
$211$			180.133i			0.0587720i	0.999568	+	0.0293860i	$0.00935520\pi$
$211$			180.133i			0.0587720i	−0.999568	+	0.0293860i	$0.990645\pi$
$212$		0			0
$213$	−1599.68	−	1599.68i	−0.514593	−	0.514593i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	799.840	−	799.840i	0.250215	−	0.250215i
$218$		0			0
$219$	3297.82			1.01756
$220$		0			0
$221$	−6528.00			−1.98697
$222$		0			0
$223$	−3756.15	+	3756.15i	−1.12794	+	1.12794i	−0.137427	+	0.990512i	$0.543883\pi$
$223$	−3756.15	+	3756.15i	−1.12794	+	1.12794i	−0.990512	+	0.137427i	$0.956117\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3401.56	−	3401.56i	−0.994580	−	0.994580i	0.00540561	−	0.999985i	$-0.498279\pi$
$227$	−3401.56	−	3401.56i	−0.994580	−	0.994580i	−0.999985	+	0.00540561i	$0.998279\pi$
$228$		0			0
$229$		−	2170.00i		−	0.626191i	−0.949722	−	0.313095i	$-0.898634\pi$
$229$		−	2170.00i		−	0.626191i	0.949722	−	0.313095i	$-0.101366\pi$
$230$		0			0
$231$		−	471.118i		−	0.134187i
$232$		0			0
$233$	−1199.76	−	1199.76i	−0.337334	−	0.337334i	0.518029	−	0.855363i	$-0.326666\pi$
$233$	−1199.76	−	1199.76i	−0.337334	−	0.337334i	−0.855363	+	0.518029i	$0.826666\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2856.57	+	2856.57i	−0.782929	+	0.782929i
$238$		0			0
$239$	−2798.99			−0.757539			−0.378770	−	0.925491i	$-0.623653\pi$
$239$	−2798.99			−0.757539			−0.378770	+	0.925491i	$0.623653\pi$
$240$		0			0
$241$	5908.00			1.57912			0.789560	−	0.613674i	$-0.210309\pi$
$241$	5908.00			1.57912			0.789560	+	0.613674i	$0.210309\pi$
$242$		0			0
$243$	1356.50	−	1356.50i	0.358105	−	0.358105i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5541.45	+	5541.45i	1.42751	+	1.42751i
$248$		0			0
$249$			3978.00i			1.01243i
$250$		0			0
$251$			3588.81i			0.902485i	0.892401	+	0.451242i	$0.149019\pi$
$251$			3588.81i			0.902485i	−0.892401	+	0.451242i	$0.850981\pi$
$252$		0			0
$253$	−1199.76	−	1199.76i	−0.298135	−	0.298135i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4799.04	−	4799.04i	1.16481	−	1.16481i	0.181399	−	0.983410i	$-0.441937\pi$
$257$	4799.04	−	4799.04i	1.16481	−	1.16481i	0.983410	−	0.181399i	$-0.0580627\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1218.00			−0.288859
$262$		0			0
$263$	3723.16	−	3723.16i	0.872928	−	0.872928i	−0.119862	−	0.992791i	$-0.538245\pi$
$263$	3723.16	−	3723.16i	0.872928	−	0.872928i	0.992791	+	0.119862i	$0.0382453\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	173.170	+	173.170i	0.0396924	+	0.0396924i
$268$		0			0
$269$		−	2856.00i		−	0.647336i	−0.946171	−	0.323668i	$-0.895084\pi$
$269$		−	2856.00i		−	0.647336i	0.946171	−	0.323668i	$-0.104916\pi$
$270$		0			0
$271$		−	6013.68i		−	1.34799i	−0.738736	−	0.673995i	$-0.764577\pi$
$271$		−	6013.68i		−	1.34799i	0.738736	−	0.673995i	$-0.235423\pi$
$272$		0			0
$273$	−1942.47	−	1942.47i	−0.430636	−	0.430636i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1599.68	−	1599.68i	0.346987	−	0.346987i	−0.511999	−	0.858986i	$-0.671095\pi$
$277$	1599.68	−	1599.68i	0.346987	−	0.346987i	0.858986	+	0.511999i	$0.171095\pi$
$278$		0			0
$279$	−1357.93			−0.291387
$280$		0			0
$281$	636.000			0.135020			0.0675099	−	0.997719i	$-0.478495\pi$
$281$	636.000			0.135020			0.0675099	+	0.997719i	$0.478495\pi$
$282$		0			0
$283$	5661.02	−	5661.02i	1.18909	−	1.18909i	0.211773	−	0.977319i	$-0.432076\pi$
$283$	5661.02	−	5661.02i	1.18909	−	1.18909i	0.977319	−	0.211773i	$-0.0679237\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1039.02	+	1039.02i	0.213699	+	0.213699i
$288$		0			0
$289$		−	1615.00i		−	0.328720i
$290$		0			0
$291$		−	6124.53i		−	1.23377i
$292$		0			0
$293$	−1942.47	−	1942.47i	−0.387305	−	0.387305i	0.486420	−	0.873725i	$-0.338302\pi$
$293$	−1942.47	−	1942.47i	−0.387305	−	0.387305i	−0.873725	+	0.486420i	$0.838302\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	1142.63	−	1142.63i	0.223239	−	0.223239i
$298$		0			0
$299$	−9893.47			−1.91356
$300$		0			0
$301$	−1666.00			−0.319025
$302$		0			0
$303$	−4329.26	+	4329.26i	−0.820824	+	0.820824i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1331.76	−	1331.76i	−0.247582	−	0.247582i	0.572395	−	0.819978i	$-0.306014\pi$
$307$	−1331.76	−	1331.76i	−0.247582	−	0.247582i	−0.819978	+	0.572395i	$0.806014\pi$
$308$		0			0
$309$		−	7174.00i		−	1.32076i
$310$		0			0
$311$			4849.74i			0.884256i	0.896952	+	0.442128i	$0.145776\pi$
$311$			4849.74i			0.884256i	−0.896952	+	0.442128i	$0.854224\pi$
$312$		0			0
$313$	−2742.31	−	2742.31i	−0.495222	−	0.495222i	0.414725	−	0.909947i	$-0.363878\pi$
$313$	−2742.31	−	2742.31i	−0.495222	−	0.495222i	−0.909947	+	0.414725i	$0.863878\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1199.76	−	1199.76i	0.212572	−	0.212572i	−0.592787	−	0.805359i	$-0.701973\pi$
$317$	1199.76	−	1199.76i	0.212572	−	0.212572i	0.805359	+	0.592787i	$0.201973\pi$
$318$		0			0
$319$	−2411.01			−0.423169
$320$		0			0
$321$	10710.0			1.86222
$322$		0			0
$323$	−5541.45	+	5541.45i	−0.954597	+	0.954597i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1121.48	−	1121.48i	−0.189658	−	0.189658i
$328$		0			0
$329$		−	2346.00i		−	0.393128i
$330$		0			0
$331$		−	5722.70i		−	0.950296i	−0.879906	−	0.475148i	$-0.842395\pi$
$331$		−	5722.70i		−	0.950296i	0.879906	−	0.475148i	$-0.157605\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4799.04	+	4799.04i	−0.775728	+	0.775728i	−0.979101	−	0.203373i	$-0.934810\pi$
$337$	−4799.04	+	4799.04i	−0.775728	+	0.775728i	0.203373	+	0.979101i	$0.434810\pi$
$338$		0			0
$339$	−6595.65			−1.05672
$340$		0			0
$341$	−2688.00			−0.426872
$342$		0			0
$343$	2688.26	−	2688.26i	0.423186	−	0.423186i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3203.65	+	3203.65i	0.495623	+	0.495623i	0.910072	−	0.414450i	$-0.136026\pi$
$347$	3203.65	+	3203.65i	0.495623	+	0.495623i	−0.414450	+	0.910072i	$0.636026\pi$
$348$		0			0
$349$		−	1778.00i		−	0.272705i	−0.990660	−	0.136353i	$-0.956462\pi$
$349$		−	1778.00i		−	0.272705i	0.990660	−	0.136353i	$-0.0435380\pi$
$350$		0			0
$351$		−	9422.36i		−	1.43284i
$352$		0			0
$353$	3656.41	+	3656.41i	0.551306	+	0.551306i	0.926818	−	0.375511i	$-0.122533\pi$
$353$	3656.41	+	3656.41i	0.551306	+	0.551306i	−0.375511	+	0.926818i	$0.622533\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1942.47	−	1942.47i	0.287973	−	0.287973i
$358$		0			0
$359$	−11445.4			−1.68263			−0.841316	−	0.540544i	$-0.818218\pi$
$359$	−11445.4			−1.68263			−0.841316	+	0.540544i	$0.818218\pi$
$360$		0			0
$361$	2549.00			0.371629
$362$		0			0
$363$	4696.22	−	4696.22i	0.679029	−	0.679029i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−6930.94	−	6930.94i	−0.985810	−	0.985810i	0.0140910	−	0.999901i	$-0.495515\pi$
$367$	−6930.94	−	6930.94i	−0.985810	−	0.985810i	−0.999901	+	0.0140910i	$0.995515\pi$
$368$		0			0
$369$		−	1764.00i		−	0.248862i
$370$		0			0
$371$			3297.82i			0.461495i
$372$		0			0
$373$	8798.24	+	8798.24i	1.22133	+	1.22133i	0.967160	+	0.254169i	$0.0818019\pi$
$373$	8798.24	+	8798.24i	1.22133	+	1.22133i	0.254169	+	0.967160i	$0.418198\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9940.87	+	9940.87i	−1.35804	+	1.35804i
$378$		0			0
$379$	−872.954			−0.118313			−0.0591565	−	0.998249i	$-0.518841\pi$
$379$	−872.954			−0.118313			−0.0591565	+	0.998249i	$0.518841\pi$
$380$		0			0
$381$	−238.000			−0.0320029
$382$		0			0
$383$	−1941.98	+	1941.98i	−0.259088	+	0.259088i	−0.824683	−	0.565595i	$-0.808647\pi$
$383$	−1941.98	+	1941.98i	−0.259088	+	0.259088i	0.565595	+	0.824683i	$0.308647\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1414.23	+	1414.23i	0.185760	+	0.185760i
$388$		0			0
$389$		−	11880.0i		−	1.54843i	−0.632922	−	0.774216i	$-0.718144\pi$
$389$		−	11880.0i		−	1.54843i	0.632922	−	0.774216i	$-0.281856\pi$
$390$		0			0
$391$		−	9893.47i		−	1.27963i
$392$		0			0
$393$	7598.48	+	7598.48i	0.975299	+	0.975299i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−57.1314	+	57.1314i	−0.00722253	+	0.00722253i	−0.710709	−	0.703486i	$-0.751626\pi$
$397$	−57.1314	+	57.1314i	−0.00722253	+	0.00722253i	0.703486	+	0.710709i	$0.251626\pi$
$398$		0			0
$399$	−3297.82			−0.413779
$400$		0			0
$401$	−498.000			−0.0620173			−0.0310086	−	0.999519i	$-0.509872\pi$
$401$	−498.000			−0.0620173			−0.0310086	+	0.999519i	$0.509872\pi$
$402$		0			0
$403$	−11082.9	+	11082.9i	−1.36992	+	1.36992i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			7700.00i			0.930906i	0.885073	+	0.465453i	$0.154109\pi$
$409$			7700.00i			0.930906i	−0.885073	+	0.465453i	$0.845891\pi$
$410$		0			0
$411$		−	6595.65i		−	0.791580i
$412$		0			0
$413$	−3599.28	−	3599.28i	−0.428835	−	0.428835i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	8398.32	−	8398.32i	0.986253	−	0.986253i
$418$		0			0
$419$	−3976.79			−0.463673			−0.231836	−	0.972755i	$-0.574473\pi$
$419$	−3976.79			−0.463673			−0.231836	+	0.972755i	$0.574473\pi$
$420$		0			0
$421$	11768.0			1.36232			0.681161	−	0.732134i	$-0.261476\pi$
$421$	11768.0			1.36232			0.681161	+	0.732134i	$0.261476\pi$
$422$		0			0
$423$	−1991.46	+	1991.46i	−0.228908	+	0.228908i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	230.894	+	230.894i	0.0261680	+	0.0261680i
$428$		0			0
$429$			6528.00i			0.734673i
$430$		0			0
$431$			17542.2i			1.96051i	0.197745	+	0.980254i	$0.436638\pi$
$431$			17542.2i			1.96051i	−0.197745	+	0.980254i	$0.563362\pi$
$432$		0			0
$433$	−7598.48	−	7598.48i	−0.843325	−	0.843325i	0.145965	−	0.989290i	$-0.453371\pi$
$433$	−7598.48	−	7598.48i	−0.843325	−	0.843325i	−0.989290	+	0.145965i	$0.953371\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8398.32	+	8398.32i	−0.919328	+	0.919328i
$438$		0			0
$439$	1745.91			0.189812			0.0949062	−	0.995486i	$-0.469745\pi$
$439$	1745.91			0.189812			0.0949062	+	0.995486i	$0.469745\pi$
$440$		0			0
$441$	−2163.00			−0.233560
$442$		0			0
$443$	−1471.95	+	1471.95i	−0.157865	+	0.157865i	−0.781620	−	0.623755i	$-0.785606\pi$
$443$	−1471.95	+	1471.95i	−0.157865	+	0.157865i	0.623755	+	0.781620i	$0.285606\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−3067.59	−	3067.59i	−0.324591	−	0.324591i
$448$		0			0
$449$		−	15252.0i		−	1.60309i	−0.597936	−	0.801544i	$-0.704012\pi$
$449$		−	15252.0i		−	1.60309i	0.597936	−	0.801544i	$-0.295988\pi$
$450$		0			0
$451$		−	3491.81i		−	0.364575i
$452$		0			0
$453$	3542.15	+	3542.15i	0.367383	+	0.367383i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6398.72	−	6398.72i	0.654966	−	0.654966i	−0.299219	−	0.954185i	$-0.596726\pi$
$457$	6398.72	−	6398.72i	0.654966	−	0.654966i	0.954185	+	0.299219i	$0.0967260\pi$
$458$		0			0
$459$	9422.36			0.958165
$460$		0			0
$461$	−15582.0			−1.57424			−0.787122	−	0.616798i	$-0.788430\pi$
$461$	−15582.0			−1.57424			−0.787122	+	0.616798i	$0.788430\pi$
$462$		0			0
$463$	11919.9	−	11919.9i	1.19647	−	1.19647i	0.221251	−	0.975217i	$-0.428986\pi$
$463$	11919.9	−	11919.9i	1.19647	−	1.19647i	0.975217	−	0.221251i	$-0.0710140\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3005.74	+	3005.74i	0.297836	+	0.297836i	0.840166	−	0.542330i	$-0.182458\pi$
$467$	3005.74	+	3005.74i	0.297836	+	0.297836i	−0.542330	+	0.840166i	$0.682458\pi$
$468$		0			0
$469$			2618.00i			0.257757i
$470$		0			0
$471$		−	6595.65i		−	0.645247i
$472$		0			0
$473$	2799.44	+	2799.44i	0.272132	+	0.272132i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	2799.44	−	2799.44i	0.268716	−	0.268716i
$478$		0			0
$479$	−387.979			−0.0370088			−0.0185044	−	0.999829i	$-0.505890\pi$
$479$	−387.979			−0.0370088			−0.0185044	+	0.999829i	$0.505890\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	2943.90	−	2943.90i	0.277333	−	0.277333i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−1356.50	−	1356.50i	−0.126220	−	0.126220i	0.641175	−	0.767395i	$-0.278447\pi$
$487$	−1356.50	−	1356.50i	−0.126220	−	0.126220i	−0.767395	+	0.641175i	$0.778447\pi$
$488$		0			0
$489$			11186.0i			1.03445i
$490$		0			0
$491$			14369.1i			1.32071i	0.750954	+	0.660354i	$0.229594\pi$
$491$			14369.1i			1.32071i	−0.750954	+	0.660354i	$0.770406\pi$
$492$		0			0
$493$	−9940.87	−	9940.87i	−0.908142	−	0.908142i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1599.68	−	1599.68i	0.144377	−	0.144377i
$498$		0			0
$499$	7357.75			0.660077			0.330038	−	0.943968i	$-0.392938\pi$
$499$	7357.75			0.660077			0.330038	+	0.943968i	$0.392938\pi$
$500$		0			0
$501$	10506.0			0.936873
$502$		0			0
$503$	12579.6	−	12579.6i	1.11510	−	1.11510i	0.122653	−	0.992450i	$-0.460860\pi$
$503$	12579.6	−	12579.6i	1.11510	−	1.11510i	0.992450	−	0.122653i	$-0.0391401\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	17857.2	+	17857.2i	1.56423	+	1.56423i
$508$		0			0
$509$			20370.0i			1.77384i	0.461923	+	0.886920i	$0.347159\pi$
$509$			20370.0i			1.77384i	−0.461923	+	0.886920i	$0.652841\pi$
$510$		0			0
$511$			3297.82i			0.285493i
$512$		0			0
$513$	−7998.40	−	7998.40i	−0.688378	−	0.688378i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−3942.07	+	3942.07i	−0.335342	+	0.335342i
$518$		0			0
$519$	23555.9			1.99227
$520$		0			0
$521$	−18438.0			−1.55045			−0.775224	−	0.631686i	$-0.782363\pi$
$521$	−18438.0			−1.55045			−0.775224	+	0.631686i	$0.782363\pi$
$522$		0			0
$523$	−3904.58	+	3904.58i	−0.326454	+	0.326454i	−0.851236	−	0.524782i	$-0.824147\pi$
$523$	−3904.58	+	3904.58i	−0.326454	+	0.326454i	0.524782	+	0.851236i	$0.324147\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11082.9	−	11082.9i	−0.916089	−	0.916089i
$528$		0			0
$529$		−	2827.00i		−	0.232350i
$530$		0			0
$531$			6110.68i			0.499399i
$532$		0			0
$533$	−14397.1	−	14397.1i	−1.17000	−	1.17000i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	7598.48	−	7598.48i	0.610612	−	0.610612i
$538$		0			0
$539$	−4281.63			−0.342157
$540$		0			0
$541$	6322.00			0.502410			0.251205	−	0.967934i	$-0.419173\pi$
$541$	6322.00			0.502410			0.251205	+	0.967934i	$0.419173\pi$
$542$		0			0
$543$	−12756.9	+	12756.9i	−1.00820	+	1.00820i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1587.40	−	1587.40i	−0.124081	−	0.124081i	0.642340	−	0.766420i	$-0.277964\pi$
$547$	−1587.40	−	1587.40i	−0.124081	−	0.124081i	−0.766420	+	0.642340i	$0.777964\pi$
$548$		0			0
$549$		−	392.000i		−	0.0304739i
$550$		0			0
$551$			16877.1i			1.30488i
$552$		0			0
$553$	−2856.57	−	2856.57i	−0.219663	−	0.219663i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12797.4	−	12797.4i	0.973510	−	0.973510i	−0.0261483	−	0.999658i	$-0.508324\pi$
$557$	12797.4	−	12797.4i	0.973510	−	0.973510i	0.999658	+	0.0261483i	$0.00832422\pi$
$558$		0			0
$559$	23084.8			1.74666
$560$		0			0
$561$	−6528.00			−0.491288
$562$		0			0
$563$	729.790	−	729.790i	0.0546305	−	0.0546305i	−0.679264	−	0.733894i	$-0.737701\pi$
$563$	729.790	−	729.790i	0.0546305	−	0.0546305i	0.733894	+	0.679264i	$0.237701\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	3582.98	+	3582.98i	0.265381	+	0.265381i
$568$		0			0
$569$			5196.00i			0.382825i	0.981510	+	0.191413i	$0.0613069\pi$
$569$			5196.00i			0.382825i	−0.981510	+	0.191413i	$0.938693\pi$
$570$		0			0
$571$		−	5528.71i		−	0.405200i	−0.979262	−	0.202600i	$-0.935061\pi$
$571$		−	5528.71i		−	0.405200i	0.979262	−	0.202600i	$-0.0649391\pi$
$572$		0			0
$573$	16453.9	+	16453.9i	1.19960	+	1.19960i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−17253.7	+	17253.7i	−1.24485	+	1.24485i	−0.286890	+	0.957963i	$0.592621\pi$
$577$	−17253.7	+	17253.7i	−1.24485	+	1.24485i	−0.957963	+	0.286890i	$0.907379\pi$
$578$		0			0
$579$	23084.8			1.65694
$580$		0			0
$581$	−3978.00			−0.284054
$582$		0			0
$583$	5541.45	−	5541.45i	0.393660	−	0.393660i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13989.7	−	13989.7i	−0.983674	−	0.983674i	0.0161949	−	0.999869i	$-0.494845\pi$
$587$	−13989.7	−	13989.7i	−0.983674	−	0.983674i	−0.999869	+	0.0161949i	$0.994845\pi$
$588$		0			0
$589$			18816.0i			1.31630i
$590$		0			0
$591$		−	3297.82i		−	0.229534i
$592$		0			0
$593$	9255.29	+	9255.29i	0.640926	+	0.640926i	0.950783	−	0.309857i	$-0.100281\pi$
$593$	9255.29	+	9255.29i	0.640926	+	0.640926i	−0.309857	+	0.950783i	$0.600281\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	12797.4	−	12797.4i	0.877327	−	0.877327i
$598$		0			0
$599$	−10586.3			−0.722111			−0.361055	−	0.932544i	$-0.617583\pi$
$599$	−10586.3			−0.722111			−0.361055	+	0.932544i	$0.617583\pi$
$600$		0			0
$601$	−17612.0			−1.19536			−0.597678	−	0.801737i	$-0.703910\pi$
$601$	−17612.0			−1.19536			−0.597678	+	0.801737i	$0.703910\pi$
$602$		0			0
$603$	2222.35	−	2222.35i	0.150085	−	0.150085i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	13338.2	+	13338.2i	0.891899	+	0.891899i	0.994702	−	0.102803i	$-0.0327810\pi$
$607$	13338.2	+	13338.2i	0.891899	+	0.891899i	−0.102803	+	0.994702i	$0.532781\pi$
$608$		0			0
$609$		−	5916.00i		−	0.393643i
$610$		0			0
$611$			32507.1i			2.15237i
$612$		0			0
$613$	5198.96	+	5198.96i	0.342551	+	0.342551i	0.857326	−	0.514774i	$-0.172124\pi$
$613$	5198.96	+	5198.96i	0.342551	+	0.342551i	−0.514774	+	0.857326i	$0.672124\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−12397.5	+	12397.5i	−0.808923	+	0.808923i	−0.984471	−	0.175548i	$-0.943830\pi$
$617$	−12397.5	+	12397.5i	−0.808923	+	0.808923i	0.175548	+	0.984471i	$0.443830\pi$
$618$		0			0
$619$	19108.0			1.24073			0.620367	−	0.784311i	$-0.286984\pi$
$619$	19108.0			1.24073			0.620367	+	0.784311i	$0.286984\pi$
$620$		0			0
$621$	14280.0			0.922764
$622$		0			0
$623$	−173.170	+	173.170i	−0.0111363	+	0.0111363i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	5541.45	+	5541.45i	0.352958	+	0.352958i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			4572.61i			0.288483i	0.989543	+	0.144242i	$0.0460742\pi$
$631$			4572.61i			0.288483i	−0.989543	+	0.144242i	$0.953926\pi$
$632$		0			0
$633$	742.709	+	742.709i	0.0466351	+	0.0466351i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−17653.6	+	17653.6i	−1.09806	+	1.09806i
$638$		0			0
$639$	−2715.86			−0.168134
$640$		0			0
$641$	−10404.0			−0.641082			−0.320541	−	0.947235i	$-0.603865\pi$
$641$	−10404.0			−0.641082			−0.320541	+	0.947235i	$0.603865\pi$
$642$		0			0
$643$	−1034.90	+	1034.90i	−0.0634719	+	0.0634719i	−0.738130	−	0.674658i	$-0.764291\pi$
$643$	−1034.90	+	1034.90i	−0.0634719	+	0.0634719i	0.674658	+	0.738130i	$0.264291\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3500.52	−	3500.52i	−0.212704	−	0.212704i	0.592711	−	0.805415i	$-0.298058\pi$
$647$	−3500.52	−	3500.52i	−0.212704	−	0.212704i	−0.805415	+	0.592711i	$0.798058\pi$
$648$		0			0
$649$			12096.0i			0.731602i
$650$		0			0
$651$		−	6595.65i		−	0.397087i
$652$		0			0
$653$	11597.7	+	11597.7i	0.695026	+	0.695026i	0.963333	−	0.268307i	$-0.0864642\pi$
$653$	11597.7	+	11597.7i	0.695026	+	0.695026i	−0.268307	+	0.963333i	$0.586464\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	2799.44	−	2799.44i	0.166235	−	0.166235i
$658$		0			0
$659$	−762.102			−0.0450490			−0.0225245	−	0.999746i	$-0.507170\pi$
$659$	−762.102			−0.0450490			−0.0225245	+	0.999746i	$0.507170\pi$
$660$		0			0
$661$	18592.0			1.09402			0.547008	−	0.837127i	$-0.315767\pi$
$661$	18592.0			1.09402			0.547008	+	0.837127i	$0.315767\pi$
$662$		0			0
$663$	−26915.6	+	26915.6i	−1.57665	+	1.57665i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−15065.8	−	15065.8i	−0.874589	−	0.874589i
$668$		0			0
$669$			30974.0i			1.79002i
$670$		0			0
$671$		−	775.959i		−	0.0446432i
$672$		0			0
$673$	9998.00	+	9998.00i	0.572652	+	0.572652i	0.932869	−	0.360217i	$-0.117297\pi$
$673$	9998.00	+	9998.00i	0.572652	+	0.572652i	−0.360217	+	0.932869i	$0.617297\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7141.43	+	7141.43i	−0.405417	+	0.405417i	−0.880137	−	0.474720i	$-0.842549\pi$
$677$	−7141.43	+	7141.43i	−0.405417	+	0.405417i	0.474720	+	0.880137i	$0.342549\pi$
$678$		0			0
$679$	6124.53			0.346153
$680$		0			0
$681$	−28050.0			−1.57838
$682$		0			0
$683$	−5974.38	+	5974.38i	−0.334705	+	0.334705i	−0.854370	−	0.519665i	$-0.826057\pi$
$683$	−5974.38	+	5974.38i	−0.334705	+	0.334705i	0.519665	+	0.854370i	$0.326057\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−8947.14	−	8947.14i	−0.496877	−	0.496877i
$688$		0			0
$689$		−	45696.0i		−	2.52667i
$690$		0			0
$691$		−	29001.5i		−	1.59662i	−0.602244	−	0.798312i	$-0.705726\pi$
$691$		−	29001.5i		−	1.59662i	0.602244	−	0.798312i	$-0.294274\pi$
$692$		0			0
$693$	−399.920	−	399.920i	−0.0219217	−	0.0219217i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	14397.1	−	14397.1i	0.782396	−	0.782396i
$698$		0			0
$699$	−9893.47			−0.535344
$700$		0			0
$701$	3408.00			0.183621			0.0918105	−	0.995776i	$-0.470735\pi$
$701$	3408.00			0.183621			0.0918105	+	0.995776i	$0.470735\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4329.26	−	4329.26i	−0.230295	−	0.230295i
$708$		0			0
$709$		−	16486.0i		−	0.873265i	−0.899640	−	0.436632i	$-0.856171\pi$
$709$		−	16486.0i		−	0.873265i	0.899640	−	0.436632i	$-0.143829\pi$
$710$		0			0
$711$			4849.74i			0.255808i
$712$		0			0
$713$	−16796.6	−	16796.6i	−0.882243	−	0.882243i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−11540.5	+	11540.5i	−0.601101	+	0.601101i
$718$		0			0
$719$	2521.87			0.130806			0.0654032	−	0.997859i	$-0.479167\pi$
$719$	2521.87			0.130806			0.0654032	+	0.997859i	$0.479167\pi$
$720$		0			0
$721$	7174.00			0.370560
$722$		0			0
$723$	24359.3	−	24359.3i	1.25302	−	1.25302i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1653.37	+	1653.37i	0.0843465	+	0.0843465i	0.748021	−	0.663675i	$-0.231004\pi$
$727$	1653.37	+	1653.37i	0.0843465	+	0.0843465i	−0.663675	+	0.748021i	$0.731004\pi$
$728$		0			0
$729$			12277.0i			0.623736i
$730$		0			0
$731$			23084.8i			1.16802i
$732$		0			0
$733$	−6055.93	−	6055.93i	−0.305158	−	0.305158i	0.537870	−	0.843028i	$-0.319229\pi$
$733$	−6055.93	−	6055.93i	−0.305158	−	0.305158i	−0.843028	+	0.537870i	$0.819229\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4399.12	−	4399.12i	0.219869	−	0.219869i
$738$		0			0
$739$	−25204.8			−1.25463			−0.627316	−	0.778764i	$-0.715847\pi$
$739$	−25204.8			−1.25463			−0.627316	+	0.778764i	$0.715847\pi$
$740$		0			0
$741$	45696.0			2.26543
$742$		0			0
$743$	1125.61	−	1125.61i	0.0555781	−	0.0555781i	−0.678771	−	0.734350i	$-0.737487\pi$
$743$	1125.61	−	1125.61i	0.0555781	−	0.0555781i	0.734350	+	0.678771i	$0.237487\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	3376.82	+	3376.82i	0.165397	+	0.165397i
$748$		0			0
$749$			10710.0i			0.522476i
$750$		0			0
$751$		−	11140.6i		−	0.541311i	−0.962676	−	0.270655i	$-0.912760\pi$
$751$		−	11140.6i		−	0.541311i	0.962676	−	0.270655i	$-0.0872404\pi$
$752$		0			0
$753$	14797.0	+	14797.0i	0.716114	+	0.716114i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	8798.24	−	8798.24i	0.422427	−	0.422427i	−0.463611	−	0.886039i	$-0.653447\pi$
$757$	8798.24	−	8798.24i	0.422427	−	0.422427i	0.886039	+	0.463611i	$0.153447\pi$
$758$		0			0
$759$	−9893.47			−0.473136
$760$		0			0
$761$	−19110.0			−0.910298			−0.455149	−	0.890415i	$-0.650414\pi$
$761$	−19110.0			−0.910298			−0.455149	+	0.890415i	$0.650414\pi$
$762$		0			0
$763$	1121.48	−	1121.48i	0.0532116	−	0.0532116i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	49873.1	+	49873.1i	2.34787	+	2.34787i
$768$		0			0
$769$		−	17458.0i		−	0.818663i	−0.912386	−	0.409331i	$-0.865762\pi$
$769$		−	17458.0i		−	0.818663i	0.912386	−	0.409331i	$-0.134238\pi$
$770$		0			0
$771$		−	39573.9i		−	1.84853i
$772$		0			0
$773$	−12054.7	−	12054.7i	−0.560904	−	0.560904i	0.368660	−	0.929564i	$-0.379817\pi$
$773$	−12054.7	−	12054.7i	−0.560904	−	0.560904i	−0.929564	+	0.368660i	$0.879817\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24442.7			−1.12420
$780$		0			0
$781$	−5376.00			−0.246310
$782$		0			0
$783$	14348.4	−	14348.4i	0.654879	−	0.654879i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−10443.8	−	10443.8i	−0.473040	−	0.473040i	0.429857	−	0.902897i	$-0.358564\pi$
$787$	−10443.8	−	10443.8i	−0.473040	−	0.473040i	−0.902897	+	0.429857i	$0.858564\pi$
$788$		0			0
$789$		−	30702.0i		−	1.38532i
$790$		0			0
$791$		−	6595.65i		−	0.296478i
$792$		0			0
$793$	−3199.36	−	3199.36i	−0.143269	−	0.143269i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3256.49	−	3256.49i	0.144731	−	0.144731i	−0.631028	−	0.775760i	$-0.717367\pi$
$797$	3256.49	−	3256.49i	0.144731	−	0.144731i	0.775760	+	0.631028i	$0.217367\pi$
$798$		0			0
$799$	−32507.1			−1.43932
$800$		0			0
$801$	294.000			0.0129688
$802$		0			0
$803$	5541.45	−	5541.45i	0.243529	−	0.243529i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−11775.6	−	11775.6i	−0.513656	−	0.513656i
$808$		0			0
$809$		−	12762.0i		−	0.554621i	−0.960780	−	0.277310i	$-0.910557\pi$
$809$		−	12762.0i		−	0.554621i	0.960780	−	0.277310i	$-0.0894430\pi$
$810$		0			0
$811$		−	1842.90i		−	0.0797941i	−0.999204	−	0.0398971i	$-0.987297\pi$
$811$		−	1842.90i		−	0.0797941i	0.999204	−	0.0398971i	$-0.0127030\pi$
$812$		0			0
$813$	−24795.0	−	24795.0i	−1.06962	−	1.06962i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	19596.1	−	19596.1i	0.839143	−	0.839143i
$818$		0			0
$819$	−3297.82			−0.140702
$820$		0			0
$821$	18768.0			0.797817			0.398908	−	0.916991i	$-0.369389\pi$
$821$	18768.0			0.797817			0.398908	+	0.916991i	$0.369389\pi$
$822$		0			0
$823$	−16306.9	+	16306.9i	−0.690671	+	0.690671i	−0.962380	−	0.271709i	$-0.912411\pi$
$823$	−16306.9	+	16306.9i	−0.690671	+	0.690671i	0.271709	+	0.962380i	$0.412411\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−14113.4	−	14113.4i	−0.593435	−	0.593435i	0.345123	−	0.938558i	$-0.387837\pi$
$827$	−14113.4	−	14113.4i	−0.593435	−	0.593435i	−0.938558	+	0.345123i	$0.887837\pi$
$828$		0			0
$829$			19208.0i			0.804730i	0.915479	+	0.402365i	$0.131812\pi$
$829$			19208.0i			0.804730i	−0.915479	+	0.402365i	$0.868188\pi$
$830$		0			0
$831$		−	13191.3i		−	0.550663i
$832$		0			0
$833$	−17653.6	−	17653.6i	−0.734287	−	0.734287i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	15996.8	−	15996.8i	0.660610	−	0.660610i
$838$		0			0
$839$	−38216.0			−1.57254			−0.786270	−	0.617883i	$-0.787991\pi$
$839$	−38216.0			−1.57254			−0.786270	+	0.617883i	$0.787991\pi$
$840$		0			0
$841$	−5887.00			−0.241379
$842$		0			0
$843$	2622.30	−	2622.30i	0.107137	−	0.107137i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	4696.22	+	4696.22i	0.190512	+	0.190512i
$848$		0			0
$849$		−	46682.0i		−	1.88707i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−5198.96	−	5198.96i	−0.208686	−	0.208686i	0.595023	−	0.803709i	$-0.297143\pi$
$853$	−5198.96	−	5198.96i	−0.208686	−	0.208686i	−0.803709	+	0.595023i	$0.797143\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3999.20	+	3999.20i	−0.159405	+	0.159405i	−0.782303	−	0.622898i	$-0.785955\pi$
$857$	−3999.20	+	3999.20i	−0.159405	+	0.159405i	0.622898	+	0.782303i	$0.285955\pi$
$858$		0			0
$859$	4946.74			0.196485			0.0982424	−	0.995163i	$-0.468678\pi$
$859$	4946.74			0.196485			0.0982424	+	0.995163i	$0.468678\pi$
$860$		0			0
$861$	8568.00			0.339137
$862$		0			0
$863$	7013.40	−	7013.40i	0.276638	−	0.276638i	−0.555127	−	0.831766i	$-0.687330\pi$
$863$	7013.40	−	7013.40i	0.276638	−	0.276638i	0.831766	+	0.555127i	$0.187330\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−6658.82	−	6658.82i	−0.260836	−	0.260836i
$868$		0			0
$869$			9600.00i			0.374750i
$870$		0			0
$871$		−	36276.1i		−	1.41121i
$872$		0			0
$873$	−5198.96	−	5198.96i	−0.201556	−	0.201556i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	11997.6	−	11997.6i	0.461950	−	0.461950i	−0.437344	−	0.899294i	$-0.644081\pi$
$877$	11997.6	−	11997.6i	0.461950	−	0.461950i	0.899294	+	0.437344i	$0.144081\pi$
$878$		0			0
$879$	−16018.0			−0.614646
$880$		0			0
$881$	13020.0			0.497906			0.248953	−	0.968516i	$-0.419914\pi$
$881$	13020.0			0.497906			0.248953	+	0.968516i	$0.419914\pi$
$882$		0			0
$883$	9668.68	−	9668.68i	0.368490	−	0.368490i	−0.498436	−	0.866926i	$-0.666092\pi$
$883$	9668.68	−	9668.68i	0.368490	−	0.368490i	0.866926	+	0.498436i	$0.166092\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13297.0	+	13297.0i	0.503348	+	0.503348i	0.912477	−	0.409128i	$-0.134167\pi$
$887$	13297.0	+	13297.0i	0.503348	+	0.503348i	−0.409128	+	0.912477i	$0.634167\pi$
$888$		0			0
$889$		−	238.000i		−	0.00897892i
$890$		0			0
$891$		−	12041.2i		−	0.452745i
$892$		0			0
$893$	27594.5	+	27594.5i	1.03406	+	1.03406i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−40791.8	+	40791.8i	−1.51839	+	1.51839i
$898$		0			0
$899$	−33754.2			−1.25224
$900$		0			0
$901$	45696.0			1.68963
$902$		0			0
$903$	−6869.09	+	6869.09i	−0.253144	+	0.253144i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	9322.34	+	9322.34i	0.341283	+	0.341283i	0.856849	−	0.515567i	$-0.172419\pi$
$907$	9322.34	+	9322.34i	0.341283	+	0.341283i	−0.515567	+	0.856849i	$0.672419\pi$
$908$		0			0
$909$			7350.00i			0.268189i
$910$		0			0
$911$			2133.89i			0.0776057i	0.999247	+	0.0388029i	$0.0123544\pi$
$911$			2133.89i			0.0776057i	−0.999247	+	0.0388029i	$0.987646\pi$
$912$		0			0
$913$	6684.38	+	6684.38i	0.242301	+	0.242301i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7598.48	+	7598.48i	−0.273636	+	0.273636i
$918$		0			0
$919$	22613.7			0.811704			0.405852	−	0.913939i	$-0.366975\pi$
$919$	22613.7			0.811704			0.405852	+	0.913939i	$0.366975\pi$
$920$		0			0
$921$	−10982.0			−0.392909
$922$		0			0
$923$	−22165.8	+	22165.8i	−0.790462	+	0.790462i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−6089.83	−	6089.83i	−0.215767	−	0.215767i
$928$		0			0
$929$		−	33684.0i		−	1.18960i	−0.803875	−	0.594799i	$-0.797232\pi$
$929$		−	33684.0i		−	1.18960i	0.803875	−	0.594799i	$-0.202768\pi$
$930$		0			0
$931$			29971.4i			1.05507i
$932$		0			0
$933$	19996.0	+	19996.0i	0.701650	+	0.701650i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−20738.7	+	20738.7i	−0.723057	+	0.723057i	−0.969227	−	0.246170i	$-0.920828\pi$
$937$	−20738.7	+	20738.7i	−0.723057	+	0.723057i	0.246170	+	0.969227i	$0.420828\pi$
$938$		0			0
$939$	−22613.7			−0.785909
$940$		0			0
$941$	−33054.0			−1.14509			−0.572545	−	0.819873i	$-0.694044\pi$
$941$	−33054.0			−1.14509			−0.572545	+	0.819873i	$0.694044\pi$
$942$		0			0
$943$	21819.5	−	21819.5i	0.753489	−	0.753489i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8398.77	+	8398.77i	0.288198	+	0.288198i	0.836367	−	0.548170i	$-0.184675\pi$
$947$	8398.77	+	8398.77i	0.288198	+	0.288198i	−0.548170	+	0.836367i	$0.684675\pi$
$948$		0			0
$949$		−	45696.0i		−	1.56307i
$950$		0			0
$951$		−	9893.47i		−	0.337348i
$952$		0			0
$953$	15197.0	+	15197.0i	0.516556	+	0.516556i	0.916528	−	0.399972i	$-0.130980\pi$
$953$	15197.0	+	15197.0i	0.516556	+	0.516556i	−0.399972	+	0.916528i	$0.630980\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−9940.87	+	9940.87i	−0.335781	+	0.335781i
$958$		0			0
$959$	6595.65			0.222090
$960$		0			0
$961$	−7841.00			−0.263200
$962$		0			0
$963$	9091.45	−	9091.45i	0.304224	−	0.304224i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−25484.9	−	25484.9i	−0.847507	−	0.847507i	0.142314	−	0.989822i	$-0.454546\pi$
$967$	−25484.9	−	25484.9i	−0.847507	−	0.847507i	−0.989822	+	0.142314i	$0.954546\pi$
$968$		0			0
$969$			45696.0i			1.51493i
$970$		0			0
$971$		−	52668.2i		−	1.74068i	−0.492449	−	0.870341i	$-0.663898\pi$
$971$		−	52668.2i		−	1.74068i	0.492449	−	0.870341i	$-0.336102\pi$
$972$		0			0
$973$	8398.32	+	8398.32i	0.276709	+	0.276709i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	40391.9	−	40391.9i	1.32267	−	1.32267i	0.411069	−	0.911604i	$-0.365155\pi$
$977$	40391.9	−	40391.9i	1.32267	−	1.32267i	0.911604	−	0.411069i	$-0.134845\pi$
$978$		0			0
$979$	581.969			0.0189988
$980$		0			0
$981$	−1904.00			−0.0619674
$982$		0			0
$983$	−22104.0	+	22104.0i	−0.717200	+	0.717200i	−0.968031	−	0.250831i	$-0.919296\pi$
$983$	−22104.0	+	22104.0i	−0.717200	+	0.717200i	0.250831	+	0.968031i	$0.419296\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−9672.81	−	9672.81i	−0.311944	−	0.311944i
$988$		0			0
$989$			34986.0i			1.12486i
$990$		0			0
$991$			18235.0i			0.584515i	0.956340	+	0.292258i	$0.0944065\pi$
$991$			18235.0i			0.584515i	−0.956340	+	0.292258i	$0.905593\pi$
$992$		0			0
$993$	−23595.3	−	23595.3i	−0.754052	−	0.754052i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−12854.6	+	12854.6i	−0.408333	+	0.408333i	−0.881157	−	0.472824i	$-0.843235\pi$
$997$	−12854.6	+	12854.6i	−0.408333	+	0.408333i	0.472824	+	0.881157i	$0.343235\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.n.d.143.3	yes	8
4.3	odd	2	inner	400.4.n.d.143.2	yes	8
5.2	odd	4	inner	400.4.n.d.207.1	yes	8
5.3	odd	4	inner	400.4.n.d.207.3	yes	8
5.4	even	2	inner	400.4.n.d.143.1	✓	8
20.3	even	4	inner	400.4.n.d.207.2	yes	8
20.7	even	4	inner	400.4.n.d.207.4	yes	8
20.19	odd	2	inner	400.4.n.d.143.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.4.n.d.143.1	✓	8	5.4	even	2	inner
400.4.n.d.143.2	yes	8	4.3	odd	2	inner
400.4.n.d.143.3	yes	8	1.1	even	1	trivial
400.4.n.d.143.4	yes	8	20.19	odd	2	inner
400.4.n.d.207.1	yes	8	5.2	odd	4	inner
400.4.n.d.207.2	yes	8	20.3	even	4	inner
400.4.n.d.207.3	yes	8	5.3	odd	4	inner
400.4.n.d.207.4	yes	8	20.7	even	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.12311 - 4.12311i) q^{3} +(4.12311 + 4.12311i) q^{7} -7.00000i q^{9} -13.8564i q^{11} +(-57.1314 - 57.1314i) q^{13} +(57.1314 - 57.1314i) q^{17} -96.9948 q^{19} +34.0000 q^{21} +(86.5852 - 86.5852i) q^{23} +(82.4621 + 82.4621i) q^{27} -174.000i q^{29} -193.990i q^{31} +(-57.1314 - 57.1314i) q^{33} -471.118 q^{39} +252.000 q^{41} +(-202.032 + 202.032i) q^{43} +(-284.494 - 284.494i) q^{47} -309.000i q^{49} -471.118i q^{51} +(399.920 + 399.920i) q^{53} +(-399.920 + 399.920i) q^{57} -872.954 q^{59} +56.0000 q^{61} +(28.8617 - 28.8617i) q^{63} +(317.479 + 317.479i) q^{67} -714.000i q^{69} -387.979i q^{71} +(399.920 + 399.920i) q^{73} +(57.1314 - 57.1314i) q^{77} -692.820 q^{79} +869.000 q^{81} +(-482.403 + 482.403i) q^{83} +(-717.420 - 717.420i) q^{87} +42.0000i q^{89} -471.118i q^{91} +(-799.840 - 799.840i) q^{93} +(742.709 - 742.709i) q^{97} -96.9948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 272 q^{21} + 2016 q^{41} + 448 q^{61} + 6952 q^{81}+O(q^{100}) \) 8 * q + 272 * q^21 + 2016 * q^41 + 448 * q^61 + 6952 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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