LMFDB - Embedded newform 400.5.b.d.351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,5,Mod(351,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("400.351");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	400.b (of order $2$, degree $1$, 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:	$41.3479852335$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	400.351
Dual form			400.5.b.d.351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.8564i q^{3} +27.7128i q^{7} -111.000 q^{9} +O(q^{10})$	$q-13.8564i q^{3} +27.7128i q^{7} -111.000 q^{9} -124.708i q^{11} -178.000 q^{13} +126.000 q^{17} +401.836i q^{19} +384.000 q^{21} +748.246i q^{23} +415.692i q^{27} -1422.00 q^{29} +332.554i q^{31} -1728.00 q^{33} -530.000 q^{37} +2466.44i q^{39} +162.000 q^{41} -1538.06i q^{43} +3491.81i q^{47} +1633.00 q^{49} -1745.91i q^{51} -594.000 q^{53} +5568.00 q^{57} +2369.45i q^{59} +626.000 q^{61} -3076.12i q^{63} +1094.66i q^{67} +10368.0 q^{69} +7731.87i q^{71} +6686.00 q^{73} +3456.00 q^{77} -1385.64i q^{79} -3231.00 q^{81} -4614.18i q^{83} +19703.8i q^{87} +8226.00 q^{89} -4932.88i q^{91} +4608.00 q^{93} +1598.00 q^{97} +13842.6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 222 q^{9}+O(q^{10}) $ 2 * q - 222 * q^9	$ 2 q - 222 q^{9} - 356 q^{13} + 252 q^{17} + 768 q^{21} - 2844 q^{29} - 3456 q^{33} - 1060 q^{37} + 324 q^{41} + 3266 q^{49} - 1188 q^{53} + 11136 q^{57} + 1252 q^{61} + 20736 q^{69} + 13372 q^{73} + 6912 q^{77} - 6462 q^{81} + 16452 q^{89} + 9216 q^{93} + 3196 q^{97}+O(q^{100}) $ 2 * q - 222 * q^9 - 356 * q^13 + 252 * q^17 + 768 * q^21 - 2844 * q^29 - 3456 * q^33 - 1060 * q^37 + 324 * q^41 + 3266 * q^49 - 1188 * q^53 + 11136 * q^57 + 1252 * q^61 + 20736 * q^69 + 13372 * q^73 + 6912 * q^77 - 6462 * q^81 + 16452 * q^89 + 9216 * q^93 + 3196 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 13.8564i	− 1.53960i	−0.638285	−	0.769800i	$-0.720356\pi$
$3$	− 13.8564i	− 1.53960i	0.638285	−	0.769800i	$-0.279644\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	27.7128i	0.565568i	0.959184	+	0.282784i	$0.0912579\pi$
$7$	27.7128i	0.565568i	−0.959184	+	0.282784i	$0.908742\pi$
$8$	0	0
$9$	−111.000	−1.37037
$10$	0	0
$11$	− 124.708i	− 1.03064i	−0.856997	−	0.515321i	$-0.827673\pi$
$11$	− 124.708i	− 1.03064i	0.856997	−	0.515321i	$-0.172327\pi$
$12$	0	0
$13$	−178.000	−1.05325	−0.526627	−	0.850096i	$-0.676544\pi$
$13$	−178.000	−1.05325	−0.526627	+	0.850096i	$0.676544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	126.000	0.435986	0.217993	−	0.975950i	$-0.430049\pi$
$17$	126.000	0.435986	0.217993	+	0.975950i	$0.430049\pi$
$18$	0	0
$19$	401.836i	1.11312i	0.830808	+	0.556559i	$0.187879\pi$
$19$	401.836i	1.11312i	−0.830808	+	0.556559i	$0.812121\pi$
$20$	0	0
$21$	384.000	0.870748
$22$	0	0
$23$	748.246i	1.41445i	0.706987	+	0.707227i	$0.250054\pi$
$23$	748.246i	1.41445i	−0.706987	+	0.707227i	$0.749946\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	415.692i	0.570222i
$28$	0	0
$29$	−1422.00	−1.69084	−0.845422	−	0.534099i	$-0.820651\pi$
$29$	−1422.00	−1.69084	−0.845422	+	0.534099i	$0.820651\pi$
$30$	0	0
$31$	332.554i	0.346050i	0.984917	+	0.173025i	$0.0553541\pi$
$31$	332.554i	0.346050i	−0.984917	+	0.173025i	$0.944646\pi$
$32$	0	0
$33$	−1728.00	−1.58678
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−530.000	−0.387144	−0.193572	−	0.981086i	$-0.562007\pi$
$37$	−530.000	−0.387144	−0.193572	+	0.981086i	$0.562007\pi$
$38$	0	0
$39$	2466.44i	1.62159i
$40$	0	0
$41$	162.000	0.0963712	0.0481856	−	0.998838i	$-0.484656\pi$
$41$	162.000	0.0963712	0.0481856	+	0.998838i	$0.484656\pi$
$42$	0	0
$43$	− 1538.06i	− 0.831834i	−0.909403	−	0.415917i	$-0.863461\pi$
$43$	− 1538.06i	− 0.831834i	0.909403	−	0.415917i	$-0.136539\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3491.81i	1.58072i	0.612642	+	0.790361i	$0.290107\pi$
$47$	3491.81i	1.58072i	−0.612642	+	0.790361i	$0.709893\pi$
$48$	0	0
$49$	1633.00	0.680133
$50$	0	0
$51$	− 1745.91i	− 0.671245i
$52$	0	0
$53$	−594.000	−0.211463	−0.105732	−	0.994395i	$-0.533718\pi$
$53$	−594.000	−0.211463	−0.105732	+	0.994395i	$0.533718\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5568.00	1.71376
$58$	0	0
$59$	2369.45i	0.680680i	0.940303	+	0.340340i	$0.110542\pi$
$59$	2369.45i	0.680680i	−0.940303	+	0.340340i	$0.889458\pi$
$60$	0	0
$61$	626.000	0.168234	0.0841172	−	0.996456i	$-0.473193\pi$
$61$	626.000	0.168234	0.0841172	+	0.996456i	$0.473193\pi$
$62$	0	0
$63$	− 3076.12i	− 0.775037i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1094.66i	0.243853i	0.992539	+	0.121926i	$0.0389072\pi$
$67$	1094.66i	0.243853i	−0.992539	+	0.121926i	$0.961093\pi$
$68$	0	0
$69$	10368.0	2.17769
$70$	0	0
$71$	7731.87i	1.53380i	0.641768	+	0.766899i	$0.278201\pi$
$71$	7731.87i	1.53380i	−0.641768	+	0.766899i	$0.721799\pi$
$72$	0	0
$73$	6686.00	1.25464	0.627322	−	0.778760i	$-0.284151\pi$
$73$	6686.00	1.25464	0.627322	+	0.778760i	$0.284151\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3456.00	0.582898
$78$	0	0
$79$	− 1385.64i	− 0.222022i	−0.993819	−	0.111011i	$-0.964591\pi$
$79$	− 1385.64i	− 0.222022i	0.993819	−	0.111011i	$-0.0354089\pi$
$80$	0	0
$81$	−3231.00	−0.492455
$82$	0	0
$83$	− 4614.18i	− 0.669790i	−0.942255	−	0.334895i	$-0.891299\pi$
$83$	− 4614.18i	− 0.669790i	0.942255	−	0.334895i	$-0.108701\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19703.8i	2.60322i
$88$	0	0
$89$	8226.00	1.03851	0.519253	−	0.854621i	$-0.326210\pi$
$89$	8226.00	1.03851	0.519253	+	0.854621i	$0.326210\pi$
$90$	0	0
$91$	− 4932.88i	− 0.595687i
$92$	0	0
$93$	4608.00	0.532778
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1598.00	0.169837	0.0849187	−	0.996388i	$-0.472937\pi$
$97$	1598.00	0.169837	0.0849187	+	0.996388i	$0.472937\pi$
$98$	0	0
$99$	13842.6i	1.41236i
$100$	0	0
$101$	−4590.00	−0.449956	−0.224978	−	0.974364i	$-0.572231\pi$
$101$	−4590.00	−0.449956	−0.224978	+	0.974364i	$0.572231\pi$
$102$	0	0
$103$	− 8951.24i	− 0.843740i	−0.906656	−	0.421870i	$-0.861374\pi$
$103$	− 8951.24i	− 0.843740i	0.906656	−	0.421870i	$-0.138626\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3117.69i	0.272311i	0.990687	+	0.136156i	$0.0434747\pi$
$107$	3117.69i	0.272311i	−0.990687	+	0.136156i	$0.956525\pi$
$108$	0	0
$109$	9394.00	0.790674	0.395337	−	0.918536i	$-0.370628\pi$
$109$	9394.00	0.790674	0.395337	+	0.918536i	$0.370628\pi$
$110$	0	0
$111$	7343.90i	0.596047i
$112$	0	0
$113$	−18882.0	−1.47874	−0.739369	−	0.673301i	$-0.764876\pi$
$113$	−18882.0	−1.47874	−0.739369	+	0.673301i	$0.764876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	19758.0	1.44335
$118$	0	0
$119$	3491.81i	0.246580i
$120$	0	0
$121$	−911.000	−0.0622225
$122$	0	0
$123$	− 2244.74i	− 0.148373i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14632.4i	0.907208i	0.891203	+	0.453604i	$0.149862\pi$
$127$	14632.4i	0.907208i	−0.891203	+	0.453604i	$0.850138\pi$
$128$	0	0
$129$	−21312.0	−1.28069
$130$	0	0
$131$	− 20826.2i	− 1.21358i	−0.794864	−	0.606788i	$-0.792458\pi$
$131$	− 20826.2i	− 1.21358i	0.794864	−	0.606788i	$-0.207542\pi$
$132$	0	0
$133$	−11136.0	−0.629544
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8226.00	−0.438276	−0.219138	−	0.975694i	$-0.570324\pi$
$137$	−8226.00	−0.438276	−0.219138	+	0.975694i	$0.570324\pi$
$138$	0	0
$139$	28696.6i	1.48526i	0.669705	+	0.742628i	$0.266421\pi$
$139$	28696.6i	1.48526i	−0.669705	+	0.742628i	$0.733579\pi$
$140$	0	0
$141$	48384.0	2.43368
$142$	0	0
$143$	22198.0i	1.08553i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 22627.5i	− 1.04713i
$148$	0	0
$149$	−31662.0	−1.42615	−0.713076	−	0.701087i	$-0.752699\pi$
$149$	−31662.0	−1.42615	−0.713076	+	0.701087i	$0.752699\pi$
$150$	0	0
$151$	39601.6i	1.73684i	0.495832	+	0.868418i	$0.334863\pi$
$151$	39601.6i	1.73684i	−0.495832	+	0.868418i	$0.665137\pi$
$152$	0	0
$153$	−13986.0	−0.597463
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−29426.0	−1.19380	−0.596900	−	0.802315i	$-0.703601\pi$
$157$	−29426.0	−1.19380	−0.596900	+	0.802315i	$0.703601\pi$
$158$	0	0
$159$	8230.71i	0.325569i
$160$	0	0
$161$	−20736.0	−0.799969
$162$	0	0
$163$	18276.6i	0.687892i	0.938989	+	0.343946i	$0.111764\pi$
$163$	18276.6i	0.687892i	−0.938989	+	0.343946i	$0.888236\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15214.3i	0.545532i	0.962080	+	0.272766i	$0.0879384\pi$
$167$	15214.3i	0.545532i	−0.962080	+	0.272766i	$0.912062\pi$
$168$	0	0
$169$	3123.00	0.109345
$170$	0	0
$171$	− 44603.8i	− 1.52538i
$172$	0	0
$173$	20430.0	0.682616	0.341308	−	0.939952i	$-0.389130\pi$
$173$	20430.0	0.682616	0.341308	+	0.939952i	$0.389130\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	32832.0	1.04797
$178$	0	0
$179$	− 22322.7i	− 0.696691i	−0.937366	−	0.348345i	$-0.886744\pi$
$179$	− 22322.7i	− 0.696691i	0.937366	−	0.348345i	$-0.113256\pi$
$180$	0	0
$181$	−37934.0	−1.15790	−0.578951	−	0.815363i	$-0.696538\pi$
$181$	−37934.0	−1.15790	−0.578951	+	0.815363i	$0.696538\pi$
$182$	0	0
$183$	− 8674.11i	− 0.259014i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 15713.2i	− 0.449346i
$188$	0	0
$189$	−11520.0	−0.322499
$190$	0	0
$191$	− 13967.3i	− 0.382864i	−0.981506	−	0.191432i	$-0.938687\pi$
$191$	− 13967.3i	− 0.382864i	0.981506	−	0.191432i	$-0.0613131\pi$
$192$	0	0
$193$	−35138.0	−0.943327	−0.471664	−	0.881779i	$-0.656346\pi$
$193$	−35138.0	−0.943327	−0.471664	+	0.881779i	$0.656346\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−46098.0	−1.18782	−0.593909	−	0.804532i	$-0.702416\pi$
$197$	−46098.0	−1.18782	−0.593909	+	0.804532i	$0.702416\pi$
$198$	0	0
$199$	− 77845.3i	− 1.96574i	−0.184299	−	0.982870i	$-0.559001\pi$
$199$	− 77845.3i	− 1.96574i	0.184299	−	0.982870i	$-0.440999\pi$
$200$	0	0
$201$	15168.0	0.375436
$202$	0	0
$203$	− 39407.6i	− 0.956287i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 83055.3i	− 1.93833i
$208$	0	0
$209$	50112.0	1.14723
$210$	0	0
$211$	52058.5i	1.16930i	0.811285	+	0.584651i	$0.198769\pi$
$211$	52058.5i	1.16930i	−0.811285	+	0.584651i	$0.801231\pi$
$212$	0	0
$213$	107136.	2.36144
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9216.00	−0.195714
$218$	0	0
$219$	− 92643.9i	− 1.93165i
$220$	0	0
$221$	−22428.0	−0.459204
$222$	0	0
$223$	29597.3i	0.595172i	0.954695	+	0.297586i	$0.0961814\pi$
$223$	29597.3i	0.595172i	−0.954695	+	0.297586i	$0.903819\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8106.00i	− 0.157309i	−0.996902	−	0.0786547i	$-0.974938\pi$
$227$	− 8106.00i	− 0.157309i	0.996902	−	0.0786547i	$-0.0250625\pi$
$228$	0	0
$229$	26770.0	0.510478	0.255239	−	0.966878i	$-0.417846\pi$
$229$	26770.0	0.510478	0.255239	+	0.966878i	$0.417846\pi$
$230$	0	0
$231$	− 47887.7i	− 0.897430i
$232$	0	0
$233$	28062.0	0.516900	0.258450	−	0.966025i	$-0.416788\pi$
$233$	28062.0	0.516900	0.258450	+	0.966025i	$0.416788\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−19200.0	−0.341826
$238$	0	0
$239$	14466.1i	0.253253i	0.991950	+	0.126627i	$0.0404150\pi$
$239$	14466.1i	0.253253i	−0.991950	+	0.126627i	$0.959585\pi$
$240$	0	0
$241$	−58622.0	−1.00931	−0.504657	−	0.863320i	$-0.668381\pi$
$241$	−58622.0	−1.00931	−0.504657	+	0.863320i	$0.668381\pi$
$242$	0	0
$243$	78441.1i	1.32841i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 71526.8i	− 1.17240i
$248$	0	0
$249$	−63936.0	−1.03121
$250$	0	0
$251$	92159.0i	1.46282i	0.681939	+	0.731409i	$0.261137\pi$
$251$	92159.0i	1.46282i	−0.681939	+	0.731409i	$0.738863\pi$
$252$	0	0
$253$	93312.0	1.45779
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−47106.0	−0.713198	−0.356599	−	0.934258i	$-0.616064\pi$
$257$	−47106.0	−0.713198	−0.356599	+	0.934258i	$0.616064\pi$
$258$	0	0
$259$	− 14687.8i	− 0.218956i
$260$	0	0
$261$	157842.	2.31708
$262$	0	0
$263$	− 57614.9i	− 0.832959i	−0.909145	−	0.416479i	$-0.863264\pi$
$263$	− 57614.9i	− 0.832959i	0.909145	−	0.416479i	$-0.136736\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 113983.i	− 1.59888i
$268$	0	0
$269$	44082.0	0.609196	0.304598	−	0.952481i	$-0.401478\pi$
$269$	44082.0	0.609196	0.304598	+	0.952481i	$0.401478\pi$
$270$	0	0
$271$	− 79203.2i	− 1.07846i	−0.842158	−	0.539230i	$-0.818715\pi$
$271$	− 79203.2i	− 1.07846i	0.842158	−	0.539230i	$-0.181285\pi$
$272$	0	0
$273$	−68352.0	−0.917120
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−40402.0	−0.526554	−0.263277	−	0.964720i	$-0.584803\pi$
$277$	−40402.0	−0.526554	−0.263277	+	0.964720i	$0.584803\pi$
$278$	0	0
$279$	− 36913.5i	− 0.474216i
$280$	0	0
$281$	−153054.	−1.93835	−0.969175	−	0.246375i	$-0.920761\pi$
$281$	−153054.	−1.93835	−0.969175	+	0.246375i	$0.920761\pi$
$282$	0	0
$283$	− 29971.4i	− 0.374226i	−0.982338	−	0.187113i	$-0.940087\pi$
$283$	− 29971.4i	− 0.374226i	0.982338	−	0.187113i	$-0.0599131\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4489.48i	0.0545044i
$288$	0	0
$289$	−67645.0	−0.809916
$290$	0	0
$291$	− 22142.5i	− 0.261482i
$292$	0	0
$293$	−100242.	−1.16765	−0.583827	−	0.811878i	$-0.698446\pi$
$293$	−100242.	−1.16765	−0.583827	+	0.811878i	$0.698446\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	51840.0	0.587695
$298$	0	0
$299$	− 133188.i	− 1.48978i
$300$	0	0
$301$	42624.0	0.470458
$302$	0	0
$303$	63600.9i	0.692752i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	146033.i	1.54943i	0.632308	+	0.774717i	$0.282108\pi$
$307$	146033.i	1.54943i	−0.632308	+	0.774717i	$0.717892\pi$
$308$	0	0
$309$	−124032.	−1.29902
$310$	0	0
$311$	− 53624.3i	− 0.554422i	−0.960809	−	0.277211i	$-0.910590\pi$
$311$	− 53624.3i	− 0.554422i	0.960809	−	0.277211i	$-0.0894102\pi$
$312$	0	0
$313$	−145634.	−1.48653	−0.743266	−	0.668996i	$-0.766724\pi$
$313$	−145634.	−1.48653	−0.743266	+	0.668996i	$0.766724\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9486.00	0.0943984	0.0471992	−	0.998885i	$-0.484970\pi$
$317$	9486.00	0.0943984	0.0471992	+	0.998885i	$0.484970\pi$
$318$	0	0
$319$	177334.i	1.74265i
$320$	0	0
$321$	43200.0	0.419251
$322$	0	0
$323$	50631.3i	0.485304i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 130167.i	− 1.21732i
$328$	0	0
$329$	−96768.0	−0.894005
$330$	0	0
$331$	94542.3i	0.862919i	0.902132	+	0.431459i	$0.142001\pi$
$331$	94542.3i	0.862919i	−0.902132	+	0.431459i	$0.857999\pi$
$332$	0	0
$333$	58830.0	0.530531
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−145474.	−1.28093	−0.640465	−	0.767987i	$-0.721258\pi$
$337$	−145474.	−1.28093	−0.640465	+	0.767987i	$0.721258\pi$
$338$	0	0
$339$	261637.i	2.27667i
$340$	0	0
$341$	41472.0	0.356653
$342$	0	0
$343$	111793.i	0.950229i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	182198.i	1.51316i	0.653902	+	0.756579i	$0.273131\pi$
$347$	182198.i	1.51316i	−0.653902	+	0.756579i	$0.726869\pi$
$348$	0	0
$349$	626.000	0.00513953	0.00256977	−	0.999997i	$-0.499182\pi$
$349$	626.000	0.00513953	0.00256977	+	0.999997i	$0.499182\pi$
$350$	0	0
$351$	− 73993.2i	− 0.600589i
$352$	0	0
$353$	−25218.0	−0.202377	−0.101189	−	0.994867i	$-0.532265\pi$
$353$	−25218.0	−0.202377	−0.101189	+	0.994867i	$0.532265\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	48384.0	0.379634
$358$	0	0
$359$	− 45144.2i	− 0.350278i	−0.984544	−	0.175139i	$-0.943963\pi$
$359$	− 45144.2i	− 0.350278i	0.984544	−	0.175139i	$-0.0560375\pi$
$360$	0	0
$361$	−31151.0	−0.239033
$362$	0	0
$363$	12623.2i	0.0957978i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 14022.7i	− 0.104112i	−0.998644	−	0.0520558i	$-0.983423\pi$
$367$	− 14022.7i	− 0.104112i	0.998644	−	0.0520558i	$-0.0165774\pi$
$368$	0	0
$369$	−17982.0	−0.132064
$370$	0	0
$371$	− 16461.4i	− 0.119597i
$372$	0	0
$373$	203182.	1.46039	0.730193	−	0.683241i	$-0.239430\pi$
$373$	203182.	1.46039	0.730193	+	0.683241i	$0.239430\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	253116.	1.78089
$378$	0	0
$379$	− 122892.i	− 0.855553i	−0.903885	−	0.427776i	$-0.859297\pi$
$379$	− 122892.i	− 0.855553i	0.903885	−	0.427776i	$-0.140703\pi$
$380$	0	0
$381$	202752.	1.39674
$382$	0	0
$383$	− 131691.i	− 0.897758i	−0.893592	−	0.448879i	$-0.851823\pi$
$383$	− 131691.i	− 0.897758i	0.893592	−	0.448879i	$-0.148177\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	170725.i	1.13992i
$388$	0	0
$389$	150930.	0.997416	0.498708	−	0.866770i	$-0.333808\pi$
$389$	150930.	0.997416	0.498708	+	0.866770i	$0.333808\pi$
$390$	0	0
$391$	94279.0i	0.616682i
$392$	0	0
$393$	−288576.	−1.86842
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−36146.0	−0.229340	−0.114670	−	0.993404i	$-0.536581\pi$
$397$	−36146.0	−0.229340	−0.114670	+	0.993404i	$0.536581\pi$
$398$	0	0
$399$	154305.i	0.969246i
$400$	0	0
$401$	−156798.	−0.975106	−0.487553	−	0.873093i	$-0.662110\pi$
$401$	−156798.	−0.975106	−0.487553	+	0.873093i	$0.662110\pi$
$402$	0	0
$403$	− 59194.6i	− 0.364478i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	66095.1i	0.399007i
$408$	0	0
$409$	254050.	1.51870	0.759351	−	0.650681i	$-0.225517\pi$
$409$	254050.	1.51870	0.759351	+	0.650681i	$0.225517\pi$
$410$	0	0
$411$	113983.i	0.674770i
$412$	0	0
$413$	−65664.0	−0.384970
$414$	0	0
$415$	0	0
$416$	0	0
$417$	397632.	2.28670
$418$	0	0
$419$	169727.i	0.966770i	0.875408	+	0.483385i	$0.160593\pi$
$419$	169727.i	0.966770i	−0.875408	+	0.483385i	$0.839407\pi$
$420$	0	0
$421$	−37742.0	−0.212942	−0.106471	−	0.994316i	$-0.533955\pi$
$421$	−37742.0	−0.212942	−0.106471	+	0.994316i	$0.533955\pi$
$422$	0	0
$423$	− 387591.i	− 2.16617i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	17348.2i	0.0951479i
$428$	0	0
$429$	307584.	1.67128
$430$	0	0
$431$	− 175090.i	− 0.942553i	−0.881986	−	0.471276i	$-0.843793\pi$
$431$	− 175090.i	− 0.942553i	0.881986	−	0.471276i	$-0.156207\pi$
$432$	0	0
$433$	165886.	0.884777	0.442389	−	0.896823i	$-0.354131\pi$
$433$	165886.	0.884777	0.442389	+	0.896823i	$0.354131\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−300672.	−1.57445
$438$	0	0
$439$	− 169353.i	− 0.878747i	−0.898304	−	0.439373i	$-0.855200\pi$
$439$	− 169353.i	− 0.878747i	0.898304	−	0.439373i	$-0.144800\pi$
$440$	0	0
$441$	−181263.	−0.932034
$442$	0	0
$443$	− 317630.i	− 1.61851i	−0.587460	−	0.809254i	$-0.699872\pi$
$443$	− 317630.i	− 1.61851i	0.587460	−	0.809254i	$-0.300128\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	438722.i	2.19570i
$448$	0	0
$449$	192834.	0.956513	0.478257	−	0.878220i	$-0.341269\pi$
$449$	192834.	0.956513	0.478257	+	0.878220i	$0.341269\pi$
$450$	0	0
$451$	− 20202.6i	− 0.0993242i
$452$	0	0
$453$	548736.	2.67403
$454$	0	0
$455$	0	0
$456$	0	0
$457$	151262.	0.724265	0.362132	−	0.932127i	$-0.382049\pi$
$457$	151262.	0.724265	0.362132	+	0.932127i	$0.382049\pi$
$458$	0	0
$459$	52377.2i	0.248609i
$460$	0	0
$461$	−119502.	−0.562307	−0.281153	−	0.959663i	$-0.590717\pi$
$461$	−119502.	−0.562307	−0.281153	+	0.959663i	$0.590717\pi$
$462$	0	0
$463$	288269.i	1.34473i	0.740219	+	0.672366i	$0.234722\pi$
$463$	288269.i	1.34473i	−0.740219	+	0.672366i	$0.765278\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 96399.0i	− 0.442017i	−0.975272	−	0.221008i	$-0.929065\pi$
$467$	− 96399.0i	− 0.442017i	0.975272	−	0.221008i	$-0.0709348\pi$
$468$	0	0
$469$	−30336.0	−0.137915
$470$	0	0
$471$	407739.i	1.83798i
$472$	0	0
$473$	−191808.	−0.857323
$474$	0	0
$475$	0	0
$476$	0	0
$477$	65934.0	0.289783
$478$	0	0
$479$	292315.i	1.27403i	0.770851	+	0.637015i	$0.219831\pi$
$479$	292315.i	1.27403i	−0.770851	+	0.637015i	$0.780169\pi$
$480$	0	0
$481$	94340.0	0.407761
$482$	0	0
$483$	287326.i	1.23163i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	194572.i	0.820392i	0.911997	+	0.410196i	$0.134540\pi$
$487$	194572.i	0.820392i	−0.911997	+	0.410196i	$0.865460\pi$
$488$	0	0
$489$	253248.	1.05908
$490$	0	0
$491$	− 39033.5i	− 0.161910i	−0.996718	−	0.0809551i	$-0.974203\pi$
$491$	− 39033.5i	− 0.161910i	0.996718	−	0.0809551i	$-0.0257971\pi$
$492$	0	0
$493$	−179172.	−0.737185
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−214272.	−0.867466
$498$	0	0
$499$	159252.i	0.639562i	0.947491	+	0.319781i	$0.103609\pi$
$499$	159252.i	0.639562i	−0.947491	+	0.319781i	$0.896391\pi$
$500$	0	0
$501$	210816.	0.839901
$502$	0	0
$503$	− 363398.i	− 1.43631i	−0.695886	−	0.718153i	$-0.744988\pi$
$503$	− 363398.i	− 1.43631i	0.695886	−	0.718153i	$-0.255012\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 43273.6i	− 0.168348i
$508$	0	0
$509$	−259470.	−1.00150	−0.500751	−	0.865592i	$-0.666943\pi$
$509$	−259470.	−1.00150	−0.500751	+	0.865592i	$0.666943\pi$
$510$	0	0
$511$	185288.i	0.709586i
$512$	0	0
$513$	−167040.	−0.634725
$514$	0	0
$515$	0	0
$516$	0	0
$517$	435456.	1.62916
$518$	0	0
$519$	− 283086.i	− 1.05096i
$520$	0	0
$521$	17442.0	0.0642571	0.0321285	−	0.999484i	$-0.489771\pi$
$521$	17442.0	0.0642571	0.0321285	+	0.999484i	$0.489771\pi$
$522$	0	0
$523$	− 6969.77i	− 0.0254809i	−0.999919	−	0.0127405i	$-0.995944\pi$
$523$	− 6969.77i	− 0.0254809i	0.999919	−	0.0127405i	$-0.00405553\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	41901.8i	0.150873i
$528$	0	0
$529$	−280031.	−1.00068
$530$	0	0
$531$	− 263008.i	− 0.932783i
$532$	0	0
$533$	−28836.0	−0.101503
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−309312.	−1.07263
$538$	0	0
$539$	− 203648.i	− 0.700974i
$540$	0	0
$541$	231922.	0.792405	0.396203	−	0.918163i	$-0.370328\pi$
$541$	231922.	0.792405	0.396203	+	0.918163i	$0.370328\pi$
$542$	0	0
$543$	525629.i	1.78271i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 233134.i	− 0.779168i	−0.920991	−	0.389584i	$-0.872619\pi$
$547$	− 233134.i	− 0.779168i	0.920991	−	0.389584i	$-0.127381\pi$
$548$	0	0
$549$	−69486.0	−0.230543
$550$	0	0
$551$	− 571410.i	− 1.88211i
$552$	0	0
$553$	38400.0	0.125569
$554$	0	0
$555$	0	0
$556$	0	0
$557$	552654.	1.78132	0.890662	−	0.454666i	$-0.150241\pi$
$557$	552654.	1.78132	0.890662	+	0.454666i	$0.150241\pi$
$558$	0	0
$559$	273775.i	0.876133i
$560$	0	0
$561$	−217728.	−0.691813
$562$	0	0
$563$	− 101387.i	− 0.319865i	−0.987128	−	0.159933i	$-0.948872\pi$
$563$	− 101387.i	− 0.319865i	0.987128	−	0.159933i	$-0.0511277\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 89540.1i	− 0.278517i
$568$	0	0
$569$	−483102.	−1.49216	−0.746078	−	0.665858i	$-0.768065\pi$
$569$	−483102.	−1.49216	−0.746078	+	0.665858i	$0.768065\pi$
$570$	0	0
$571$	− 221993.i	− 0.680876i	−0.940267	−	0.340438i	$-0.889425\pi$
$571$	− 221993.i	− 0.680876i	0.940267	−	0.340438i	$-0.110575\pi$
$572$	0	0
$573$	−193536.	−0.589458
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−176642.	−0.530570	−0.265285	−	0.964170i	$-0.585466\pi$
$577$	−176642.	−0.530570	−0.265285	+	0.964170i	$0.585466\pi$
$578$	0	0
$579$	486886.i	1.45235i
$580$	0	0
$581$	127872.	0.378812
$582$	0	0
$583$	74076.3i	0.217943i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	209634.i	0.608394i	0.952609	+	0.304197i	$0.0983880\pi$
$587$	209634.i	0.608394i	−0.952609	+	0.304197i	$0.901612\pi$
$588$	0	0
$589$	−133632.	−0.385194
$590$	0	0
$591$	638753.i	1.82876i
$592$	0	0
$593$	−653634.	−1.85877	−0.929384	−	0.369114i	$-0.879661\pi$
$593$	−653634.	−1.85877	−0.929384	+	0.369114i	$0.879661\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.07866e6	−3.02646
$598$	0	0
$599$	5237.72i	0.0145978i	0.999973	+	0.00729892i	$0.00232334\pi$
$599$	5237.72i	0.0145978i	−0.999973	+	0.00729892i	$0.997677\pi$
$600$	0	0
$601$	77858.0	0.215553	0.107777	−	0.994175i	$-0.465627\pi$
$601$	77858.0	0.215553	0.107777	+	0.994175i	$0.465627\pi$
$602$	0	0
$603$	− 121507.i	− 0.334169i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 153529.i	− 0.416690i	−0.978055	−	0.208345i	$-0.933192\pi$
$607$	− 153529.i	− 0.416690i	0.978055	−	0.208345i	$-0.0668077\pi$
$608$	0	0
$609$	−546048.	−1.47230
$610$	0	0
$611$	− 621543.i	− 1.66490i
$612$	0	0
$613$	−197522.	−0.525647	−0.262824	−	0.964844i	$-0.584654\pi$
$613$	−197522.	−0.525647	−0.262824	+	0.964844i	$0.584654\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	394398.	1.03601	0.518006	−	0.855377i	$-0.326675\pi$
$617$	394398.	1.03601	0.518006	+	0.855377i	$0.326675\pi$
$618$	0	0
$619$	− 262066.i	− 0.683958i	−0.939708	−	0.341979i	$-0.888903\pi$
$619$	− 262066.i	− 0.683958i	0.939708	−	0.341979i	$-0.111097\pi$
$620$	0	0
$621$	−311040.	−0.806553
$622$	0	0
$623$	227966.i	0.587345i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 694372.i	− 1.76627i
$628$	0	0
$629$	−66780.0	−0.168789
$630$	0	0
$631$	659149.i	1.65548i	0.561109	+	0.827742i	$0.310375\pi$
$631$	659149.i	1.65548i	−0.561109	+	0.827742i	$0.689625\pi$
$632$	0	0
$633$	721344.	1.80026
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−290674.	−0.716353
$638$	0	0
$639$	− 858238.i	− 2.10187i
$640$	0	0
$641$	314946.	0.766514	0.383257	−	0.923642i	$-0.374802\pi$
$641$	314946.	0.766514	0.383257	+	0.923642i	$0.374802\pi$
$642$	0	0
$643$	563554.i	1.36306i	0.731792	+	0.681528i	$0.238684\pi$
$643$	563554.i	1.36306i	−0.731792	+	0.681528i	$0.761316\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	161871.i	0.386687i	0.981131	+	0.193343i	$0.0619331\pi$
$647$	161871.i	0.386687i	−0.981131	+	0.193343i	$0.938067\pi$
$648$	0	0
$649$	295488.	0.701537
$650$	0	0
$651$	127701.i	0.301322i
$652$	0	0
$653$	472014.	1.10695	0.553476	−	0.832865i	$-0.313301\pi$
$653$	472014.	1.10695	0.553476	+	0.832865i	$0.313301\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−742146.	−1.71933
$658$	0	0
$659$	− 508184.i	− 1.17017i	−0.810971	−	0.585086i	$-0.801061\pi$
$659$	− 508184.i	− 1.17017i	0.810971	−	0.585086i	$-0.198939\pi$
$660$	0	0
$661$	−16558.0	−0.0378970	−0.0189485	−	0.999820i	$-0.506032\pi$
$661$	−16558.0	−0.0378970	−0.0189485	+	0.999820i	$0.506032\pi$
$662$	0	0
$663$	310771.i	0.706991i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.06401e6i	− 2.39162i
$668$	0	0
$669$	410112.	0.916326
$670$	0	0
$671$	− 78067.0i	− 0.173389i
$672$	0	0
$673$	543550.	1.20008	0.600039	−	0.799971i	$-0.295152\pi$
$673$	543550.	1.20008	0.600039	+	0.799971i	$0.295152\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	419310.	0.914867	0.457433	−	0.889244i	$-0.348769\pi$
$677$	419310.	0.914867	0.457433	+	0.889244i	$0.348769\pi$
$678$	0	0
$679$	44285.1i	0.0960545i
$680$	0	0
$681$	−112320.	−0.242194
$682$	0	0
$683$	− 912735.i	− 1.95661i	−0.207178	−	0.978303i	$-0.566428\pi$
$683$	− 912735.i	− 1.95661i	0.207178	−	0.978303i	$-0.433572\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 370936.i	− 0.785933i
$688$	0	0
$689$	105732.	0.222725
$690$	0	0
$691$	− 17001.8i	− 0.0356073i	−0.999842	−	0.0178037i	$-0.994333\pi$
$691$	− 17001.8i	− 0.0356073i	0.999842	−	0.0178037i	$-0.00566738\pi$
$692$	0	0
$693$	−383616.	−0.798786
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20412.0	0.0420165
$698$	0	0
$699$	− 388838.i	− 0.795820i
$700$	0	0
$701$	−462222.	−0.940621	−0.470310	−	0.882501i	$-0.655858\pi$
$701$	−462222.	−0.940621	−0.470310	+	0.882501i	$0.655858\pi$
$702$	0	0
$703$	− 212973.i	− 0.430937i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 127202.i	− 0.254480i
$708$	0	0
$709$	−162158.	−0.322586	−0.161293	−	0.986907i	$-0.551566\pi$
$709$	−162158.	−0.322586	−0.161293	+	0.986907i	$0.551566\pi$
$710$	0	0
$711$	153806.i	0.304253i
$712$	0	0
$713$	−248832.	−0.489471
$714$	0	0
$715$	0	0
$716$	0	0
$717$	200448.	0.389909
$718$	0	0
$719$	− 204022.i	− 0.394656i	−0.980338	−	0.197328i	$-0.936774\pi$
$719$	− 204022.i	− 0.394656i	0.980338	−	0.197328i	$-0.0632264\pi$
$720$	0	0
$721$	248064.	0.477192
$722$	0	0
$723$	812290.i	1.55394i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 144910.i	− 0.274177i	−0.990559	−	0.137088i	$-0.956226\pi$
$727$	− 144910.i	− 0.274177i	0.990559	−	0.137088i	$-0.0437744\pi$
$728$	0	0
$729$	825201.	1.55276
$730$	0	0
$731$	− 193796.i	− 0.362668i
$732$	0	0
$733$	885902.	1.64884	0.824419	−	0.565981i	$-0.191502\pi$
$733$	885902.	1.64884	0.824419	+	0.565981i	$0.191502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	136512.	0.251325
$738$	0	0
$739$	− 114274.i	− 0.209246i	−0.994512	−	0.104623i	$-0.966636\pi$
$739$	− 114274.i	− 0.209246i	0.994512	−	0.104623i	$-0.0333636\pi$
$740$	0	0
$741$	−991104.	−1.80502
$742$	0	0
$743$	− 581387.i	− 1.05314i	−0.850131	−	0.526572i	$-0.823477\pi$
$743$	− 581387.i	− 1.05314i	0.850131	−	0.526572i	$-0.176523\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	512174.i	0.917860i
$748$	0	0
$749$	−86400.0	−0.154010
$750$	0	0
$751$	− 1.11178e6i	− 1.97124i	−0.168970	−	0.985621i	$-0.554044\pi$
$751$	− 1.11178e6i	− 1.97124i	0.168970	−	0.985621i	$-0.445956\pi$
$752$	0	0
$753$	1.27699e6	2.25215
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−376786.	−0.657511	−0.328755	−	0.944415i	$-0.606629\pi$
$757$	−376786.	−0.657511	−0.328755	+	0.944415i	$0.606629\pi$
$758$	0	0
$759$	− 1.29297e6i	− 2.24442i
$760$	0	0
$761$	307170.	0.530407	0.265204	−	0.964192i	$-0.414561\pi$
$761$	307170.	0.530407	0.265204	+	0.964192i	$0.414561\pi$
$762$	0	0
$763$	260334.i	0.447180i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 421761.i	− 0.716929i
$768$	0	0
$769$	−110014.	−0.186035	−0.0930176	−	0.995664i	$-0.529651\pi$
$769$	−110014.	−0.186035	−0.0930176	+	0.995664i	$0.529651\pi$
$770$	0	0
$771$	652720.i	1.09804i
$772$	0	0
$773$	−685458.	−1.14715	−0.573577	−	0.819152i	$-0.694445\pi$
$773$	−685458.	−1.14715	−0.573577	+	0.819152i	$0.694445\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−203520.	−0.337105
$778$	0	0
$779$	65097.4i	0.107273i
$780$	0	0
$781$	964224.	1.58080
$782$	0	0
$783$	− 591114.i	− 0.964157i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	382700.i	0.617887i	0.951080	+	0.308944i	$0.0999754\pi$
$787$	382700.i	0.617887i	−0.951080	+	0.308944i	$0.900025\pi$
$788$	0	0
$789$	−798336.	−1.28242
$790$	0	0
$791$	− 523273.i	− 0.836326i
$792$	0	0
$793$	−111428.	−0.177194
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−672498.	−1.05870	−0.529352	−	0.848402i	$-0.677565\pi$
$797$	−672498.	−1.05870	−0.529352	+	0.848402i	$0.677565\pi$
$798$	0	0
$799$	439969.i	0.689173i
$800$	0	0
$801$	−913086.	−1.42314
$802$	0	0
$803$	− 833795.i	− 1.29309i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 610818.i	− 0.937918i
$808$	0	0
$809$	18594.0	0.0284103	0.0142051	−	0.999899i	$-0.495478\pi$
$809$	18594.0	0.0284103	0.0142051	+	0.999899i	$0.495478\pi$
$810$	0	0
$811$	1.05507e6i	1.60413i	0.597238	+	0.802064i	$0.296265\pi$
$811$	1.05507e6i	1.60413i	−0.597238	+	0.802064i	$0.703735\pi$
$812$	0	0
$813$	−1.09747e6	−1.66040
$814$	0	0
$815$	0	0
$816$	0	0
$817$	618048.	0.925930
$818$	0	0
$819$	547550.i	0.816311i
$820$	0	0
$821$	−250542.	−0.371701	−0.185851	−	0.982578i	$-0.559504\pi$
$821$	−250542.	−0.371701	−0.185851	+	0.982578i	$0.559504\pi$
$822$	0	0
$823$	807413.i	1.19205i	0.802964	+	0.596027i	$0.203255\pi$
$823$	807413.i	1.19205i	−0.802964	+	0.596027i	$0.796745\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	316882.i	0.463326i	0.972796	+	0.231663i	$0.0744167\pi$
$827$	316882.i	0.463326i	−0.972796	+	0.231663i	$0.925583\pi$
$828$	0	0
$829$	428914.	0.624110	0.312055	−	0.950064i	$-0.398983\pi$
$829$	428914.	0.624110	0.312055	+	0.950064i	$0.398983\pi$
$830$	0	0
$831$	559827.i	0.810684i
$832$	0	0
$833$	205758.	0.296529
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−138240.	−0.197325
$838$	0	0
$839$	829805.i	1.17883i	0.807830	+	0.589416i	$0.200642\pi$
$839$	829805.i	1.17883i	−0.807830	+	0.589416i	$0.799358\pi$
$840$	0	0
$841$	1.31480e6	1.85895
$842$	0	0
$843$	2.12078e6i	2.98428i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 25246.4i	− 0.0351910i
$848$	0	0
$849$	−415296.	−0.576159
$850$	0	0
$851$	− 396570.i	− 0.547597i
$852$	0	0
$853$	51502.0	0.0707825	0.0353913	−	0.999374i	$-0.488732\pi$
$853$	51502.0	0.0707825	0.0353913	+	0.999374i	$0.488732\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	574110.	0.781688	0.390844	−	0.920457i	$-0.372183\pi$
$857$	574110.	0.781688	0.390844	+	0.920457i	$0.372183\pi$
$858$	0	0
$859$	− 173330.i	− 0.234902i	−0.993079	−	0.117451i	$-0.962528\pi$
$859$	− 173330.i	− 0.234902i	0.993079	−	0.117451i	$-0.0374723\pi$
$860$	0	0
$861$	62208.0	0.0839151
$862$	0	0
$863$	246422.i	0.330871i	0.986221	+	0.165435i	$0.0529029\pi$
$863$	246422.i	0.330871i	−0.986221	+	0.165435i	$0.947097\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	937317.i	1.24695i
$868$	0	0
$869$	−172800.	−0.228825
$870$	0	0
$871$	− 194849.i	− 0.256839i
$872$	0	0
$873$	−177378.	−0.232740
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.39579e6	1.81477	0.907384	−	0.420304i	$-0.138076\pi$
$877$	1.39579e6	1.81477	0.907384	+	0.420304i	$0.138076\pi$
$878$	0	0
$879$	1.38899e6i	1.79772i
$880$	0	0
$881$	−702270.	−0.904799	−0.452400	−	0.891815i	$-0.649432\pi$
$881$	−702270.	−0.904799	−0.452400	+	0.891815i	$0.649432\pi$
$882$	0	0
$883$	− 776028.i	− 0.995305i	−0.867377	−	0.497652i	$-0.834196\pi$
$883$	− 776028.i	− 0.995305i	0.867377	−	0.497652i	$-0.165804\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	382853.i	0.486614i	0.969949	+	0.243307i	$0.0782322\pi$
$887$	382853.i	0.486614i	−0.969949	+	0.243307i	$0.921768\pi$
$888$	0	0
$889$	−405504.	−0.513088
$890$	0	0
$891$	402930.i	0.507545i
$892$	0	0
$893$	−1.40314e6	−1.75953
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.84550e6	−2.29367
$898$	0	0
$899$	− 472891.i	− 0.585116i
$900$	0	0
$901$	−74844.0	−0.0921950
$902$	0	0
$903$	− 590615.i	− 0.724318i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 66829.4i	− 0.0812369i	−0.999175	−	0.0406184i	$-0.987067\pi$
$907$	− 66829.4i	− 0.0812369i	0.999175	−	0.0406184i	$-0.0129328\pi$
$908$	0	0
$909$	509490.	0.616606
$910$	0	0
$911$	− 1.43115e6i	− 1.72444i	−0.506538	−	0.862218i	$-0.669075\pi$
$911$	− 1.43115e6i	− 1.72444i	0.506538	−	0.862218i	$-0.330925\pi$
$912$	0	0
$913$	−575424.	−0.690314
$914$	0	0
$915$	0	0
$916$	0	0
$917$	577152.	0.686359
$918$	0	0
$919$	1.45584e6i	1.72378i	0.507095	+	0.861890i	$0.330719\pi$
$919$	1.45584e6i	1.72378i	−0.507095	+	0.861890i	$0.669281\pi$
$920$	0	0
$921$	2.02349e6	2.38551
$922$	0	0
$923$	− 1.37627e6i	− 1.61548i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	993587.i	1.15624i
$928$	0	0
$929$	−582462.	−0.674895	−0.337447	−	0.941344i	$-0.609564\pi$
$929$	−582462.	−0.674895	−0.337447	+	0.941344i	$0.609564\pi$
$930$	0	0
$931$	656198.i	0.757069i
$932$	0	0
$933$	−743040.	−0.853589
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−887138.	−1.01044	−0.505222	−	0.862990i	$-0.668589\pi$
$937$	−887138.	−1.01044	−0.505222	+	0.862990i	$0.668589\pi$
$938$	0	0
$939$	2.01796e6i	2.28866i
$940$	0	0
$941$	−777294.	−0.877821	−0.438911	−	0.898531i	$-0.644636\pi$
$941$	−777294.	−0.877821	−0.438911	+	0.898531i	$0.644636\pi$
$942$	0	0
$943$	121216.i	0.136313i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.44299e6i	1.60903i	0.593933	+	0.804515i	$0.297575\pi$
$947$	1.44299e6i	1.60903i	−0.593933	+	0.804515i	$0.702425\pi$
$948$	0	0
$949$	−1.19011e6	−1.32146
$950$	0	0
$951$	− 131442.i	− 0.145336i
$952$	0	0
$953$	584478.	0.643550	0.321775	−	0.946816i	$-0.395721\pi$
$953$	584478.	0.643550	0.321775	+	0.946816i	$0.395721\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.45722e6	2.68299
$958$	0	0
$959$	− 227966.i	− 0.247875i
$960$	0	0
$961$	812929.	0.880250
$962$	0	0
$963$	− 346064.i	− 0.373167i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 316785.i	− 0.338775i	−0.985549	−	0.169388i	$-0.945821\pi$
$967$	− 316785.i	− 0.338775i	0.985549	−	0.169388i	$-0.0541790\pi$
$968$	0	0
$969$	701568.	0.747175
$970$	0	0
$971$	− 1.18734e6i	− 1.25932i	−0.776870	−	0.629662i	$-0.783194\pi$
$971$	− 1.18734e6i	− 1.25932i	0.776870	−	0.629662i	$-0.216806\pi$
$972$	0	0
$973$	−795264.	−0.840012
$974$	0	0
$975$	0	0
$976$	0	0
$977$	665982.	0.697707	0.348854	−	0.937177i	$-0.386571\pi$
$977$	665982.	0.697707	0.348854	+	0.937177i	$0.386571\pi$
$978$	0	0
$979$	− 1.02585e6i	− 1.07033i
$980$	0	0
$981$	−1.04273e6	−1.08352
$982$	0	0
$983$	− 73078.7i	− 0.0756282i	−0.999285	−	0.0378141i	$-0.987961\pi$
$983$	− 73078.7i	− 0.0756282i	0.999285	−	0.0378141i	$-0.0120395\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.34086e6i	1.37641i
$988$	0	0
$989$	1.15085e6	1.17659
$990$	0	0
$991$	764098.i	0.778039i	0.921229	+	0.389020i	$0.127186\pi$
$991$	764098.i	0.778039i	−0.921229	+	0.389020i	$0.872814\pi$
$992$	0	0
$993$	1.31002e6	1.32855
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−979730.	−0.985635	−0.492817	−	0.870133i	$-0.664033\pi$
$997$	−979730.	−0.985635	−0.492817	+	0.870133i	$0.664033\pi$
$998$	0	0
$999$	− 220317.i	− 0.220758i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.5.b.d.351.1		2
4.3	odd	2	inner	400.5.b.d.351.2		2
5.2	odd	4		400.5.h.b.399.1		4
5.3	odd	4		400.5.h.b.399.3		4
5.4	even	2		16.5.c.a.15.2	yes	2
15.14	odd	2		144.5.g.c.127.1		2
20.3	even	4		400.5.h.b.399.2		4
20.7	even	4		400.5.h.b.399.4		4
20.19	odd	2		16.5.c.a.15.1	✓	2
35.34	odd	2		784.5.d.a.687.1		2
40.19	odd	2		64.5.c.c.63.2		2
40.29	even	2		64.5.c.c.63.1		2
60.59	even	2		144.5.g.c.127.2		2
80.19	odd	4		256.5.d.f.127.3		4
80.29	even	4		256.5.d.f.127.1		4
80.59	odd	4		256.5.d.f.127.2		4
80.69	even	4		256.5.d.f.127.4		4
120.29	odd	2		576.5.g.h.127.1		2
120.59	even	2		576.5.g.h.127.2		2
140.139	even	2		784.5.d.a.687.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.5.c.a.15.1	✓	2	20.19	odd	2
16.5.c.a.15.2	yes	2	5.4	even	2
64.5.c.c.63.1		2	40.29	even	2
64.5.c.c.63.2		2	40.19	odd	2
144.5.g.c.127.1		2	15.14	odd	2
144.5.g.c.127.2		2	60.59	even	2
256.5.d.f.127.1		4	80.29	even	4
256.5.d.f.127.2		4	80.59	odd	4
256.5.d.f.127.3		4	80.19	odd	4
256.5.d.f.127.4		4	80.69	even	4
400.5.b.d.351.1		2	1.1	even	1	trivial
400.5.b.d.351.2		2	4.3	odd	2	inner
400.5.h.b.399.1		4	5.2	odd	4
400.5.h.b.399.2		4	20.3	even	4
400.5.h.b.399.3		4	5.3	odd	4
400.5.h.b.399.4		4	20.7	even	4
576.5.g.h.127.1		2	120.29	odd	2
576.5.g.h.127.2		2	120.59	even	2
784.5.d.a.687.1		2	35.34	odd	2
784.5.d.a.687.2		2	140.139	even	2

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	400.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-13.8564i q^{3} +27.7128i q^{7} -111.000 q^{9} +O(q^{10})\)	\(q-13.8564i q^{3} +27.7128i q^{7} -111.000 q^{9} -124.708i q^{11} -178.000 q^{13} +126.000 q^{17} +401.836i q^{19} +384.000 q^{21} +748.246i q^{23} +415.692i q^{27} -1422.00 q^{29} +332.554i q^{31} -1728.00 q^{33} -530.000 q^{37} +2466.44i q^{39} +162.000 q^{41} -1538.06i q^{43} +3491.81i q^{47} +1633.00 q^{49} -1745.91i q^{51} -594.000 q^{53} +5568.00 q^{57} +2369.45i q^{59} +626.000 q^{61} -3076.12i q^{63} +1094.66i q^{67} +10368.0 q^{69} +7731.87i q^{71} +6686.00 q^{73} +3456.00 q^{77} -1385.64i q^{79} -3231.00 q^{81} -4614.18i q^{83} +19703.8i q^{87} +8226.00 q^{89} -4932.88i q^{91} +4608.00 q^{93} +1598.00 q^{97} +13842.6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 222 q^{9}+O(q^{10}) \) 2 * q - 222 * q^9	\( 2 q - 222 q^{9} - 356 q^{13} + 252 q^{17} + 768 q^{21} - 2844 q^{29} - 3456 q^{33} - 1060 q^{37} + 324 q^{41} + 3266 q^{49} - 1188 q^{53} + 11136 q^{57} + 1252 q^{61} + 20736 q^{69} + 13372 q^{73} + 6912 q^{77} - 6462 q^{81} + 16452 q^{89} + 9216 q^{93} + 3196 q^{97}+O(q^{100}) \) 2 * q - 222 * q^9 - 356 * q^13 + 252 * q^17 + 768 * q^21 - 2844 * q^29 - 3456 * q^33 - 1060 * q^37 + 324 * q^41 + 3266 * q^49 - 1188 * q^53 + 11136 * q^57 + 1252 * q^61 + 20736 * q^69 + 13372 * q^73 + 6912 * q^77 - 6462 * q^81 + 16452 * q^89 + 9216 * q^93 + 3196 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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