LMFDB - Embedded newform 400.4.n.g.143.7

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:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.11007531417600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.7
Root			$-1.56290 + 0.418778i$ of defining polynomial
Character	$\chi$	$=$	400.143
Dual form			400.4.n.g.207.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.59346 - 6.59346i) q^{3} +(-17.4296 - 17.4296i) q^{7} -59.9473i q^{9} +O(q^{10})$	$q+(6.59346 - 6.59346i) q^{3} +(-17.4296 - 17.4296i) q^{7} -59.9473i q^{9} -1.16990i q^{11} +(32.6384 + 32.6384i) q^{13} +(90.9319 - 90.9319i) q^{17} -131.499 q^{19} -229.842 q^{21} +(-98.9577 + 98.9577i) q^{23} +(-217.237 - 217.237i) q^{27} -167.842i q^{29} -60.1965i q^{31} +(-7.71371 - 7.71371i) q^{33} +(-193.445 + 193.445i) q^{37} +430.399 q^{39} +28.7893 q^{41} +(76.6031 - 76.6031i) q^{43} +(-176.926 - 176.926i) q^{47} +264.579i q^{49} -1199.11i q^{51} +(327.372 + 327.372i) q^{53} +(-867.033 + 867.033i) q^{57} -141.846 q^{59} +369.315 q^{61} +(-1044.86 + 1044.86i) q^{63} +(67.8385 + 67.8385i) q^{67} +1304.95i q^{69} +46.7346i q^{71} +(-141.641 - 141.641i) q^{73} +(-20.3909 + 20.3909i) q^{77} +1032.70 q^{79} -1246.10 q^{81} +(498.417 - 498.417i) q^{83} +(-1106.66 - 1106.66i) q^{87} -1534.00i q^{89} -1137.74i q^{91} +(-396.903 - 396.903i) q^{93} +(1.20296 - 1.20296i) q^{97} -70.1327 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 1856 q^{21} - 1968 q^{41} - 1984 q^{61} - 9616 q^{81}+O(q^{100}) $ 16 * q - 1856 * q^21 - 1968 * q^41 - 1984 * q^61 - 9616 * q^81

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$	6.59346	−	6.59346i	1.26891	−	1.26891i	0.322260	−	0.946651i	$-0.395557\pi$
$3$	6.59346	−	6.59346i	1.26891	−	1.26891i	0.946651	−	0.322260i	$-0.104443\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−17.4296	−	17.4296i	−0.941107	−	0.941107i	0.0572523	−	0.998360i	$-0.481766\pi$
$7$	−17.4296	−	17.4296i	−0.941107	−	0.941107i	−0.998360	+	0.0572523i	$0.981766\pi$
$8$		0			0
$9$		−	59.9473i		−	2.22027i
$10$		0			0
$11$		−	1.16990i		−	0.0320672i	−0.999871	−	0.0160336i	$-0.994896\pi$
$11$		−	1.16990i		−	0.0320672i	0.999871	−	0.0160336i	$-0.00510388\pi$
$12$		0			0
$13$	32.6384	+	32.6384i	0.696327	+	0.696327i	0.963616	−	0.267289i	$-0.0861279\pi$
$13$	32.6384	+	32.6384i	0.696327	+	0.696327i	−0.267289	+	0.963616i	$0.586128\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	90.9319	−	90.9319i	1.29731	−	1.29731i	0.367142	−	0.930165i	$-0.380336\pi$
$17$	90.9319	−	90.9319i	1.29731	−	1.29731i	0.930165	−	0.367142i	$-0.119664\pi$
$18$		0			0
$19$	−131.499			−1.58779			−0.793893	−	0.608057i	$-0.791949\pi$
$19$	−131.499			−1.58779			−0.793893	+	0.608057i	$0.791949\pi$
$20$		0			0
$21$	−229.842			−2.38836
$22$		0			0
$23$	−98.9577	+	98.9577i	−0.897135	+	0.897135i	−0.995182	−	0.0980467i	$-0.968741\pi$
$23$	−98.9577	+	98.9577i	−0.897135	+	0.897135i	0.0980467	+	0.995182i	$0.468741\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−217.237	−	217.237i	−1.54842	−	1.54842i
$28$		0			0
$29$		−	167.842i		−	1.07474i	−0.843346	−	0.537370i	$-0.819418\pi$
$29$		−	167.842i		−	1.07474i	0.843346	−	0.537370i	$-0.180582\pi$
$30$		0			0
$31$		−	60.1965i		−	0.348761i	−0.984678	−	0.174381i	$-0.944208\pi$
$31$		−	60.1965i		−	0.348761i	0.984678	−	0.174381i	$-0.0557923\pi$
$32$		0			0
$33$	−7.71371	−	7.71371i	−0.0406905	−	0.0406905i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−193.445	+	193.445i	−0.859519	+	0.859519i	−0.991281	−	0.131763i	$-0.957936\pi$
$37$	−193.445	+	193.445i	−0.859519	+	0.859519i	0.131763	+	0.991281i	$0.457936\pi$
$38$		0			0
$39$	430.399			1.76715
$40$		0			0
$41$	28.7893			0.109662			0.0548309	−	0.998496i	$-0.482538\pi$
$41$	28.7893			0.109662			0.0548309	+	0.998496i	$0.482538\pi$
$42$		0			0
$43$	76.6031	−	76.6031i	0.271671	−	0.271671i	−0.558102	−	0.829773i	$-0.688470\pi$
$43$	76.6031	−	76.6031i	0.271671	−	0.271671i	0.829773	+	0.558102i	$0.188470\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−176.926	−	176.926i	−0.549091	−	0.549091i	0.377087	−	0.926178i	$-0.376926\pi$
$47$	−176.926	−	176.926i	−0.549091	−	0.549091i	−0.926178	+	0.377087i	$0.876926\pi$
$48$		0			0
$49$			264.579i			0.771366i
$50$		0			0
$51$		−	1199.11i		−	3.29233i
$52$		0			0
$53$	327.372	+	327.372i	0.848453	+	0.848453i	0.989940	−	0.141487i	$-0.0451883\pi$
$53$	327.372	+	327.372i	0.848453	+	0.848453i	−0.141487	+	0.989940i	$0.545188\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−867.033	+	867.033i	−2.01476	+	2.01476i
$58$		0			0
$59$	−141.846			−0.312996			−0.156498	−	0.987678i	$-0.550020\pi$
$59$	−141.846			−0.312996			−0.156498	+	0.987678i	$0.550020\pi$
$60$		0			0
$61$	369.315			0.775180			0.387590	−	0.921832i	$-0.373308\pi$
$61$	369.315			0.775180			0.387590	+	0.921832i	$0.373308\pi$
$62$		0			0
$63$	−1044.86	+	1044.86i	−2.08951	+	2.08951i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	67.8385	+	67.8385i	0.123698	+	0.123698i	0.766246	−	0.642548i	$-0.222122\pi$
$67$	67.8385	+	67.8385i	0.123698	+	0.123698i	−0.642548	+	0.766246i	$0.722122\pi$
$68$		0			0
$69$			1304.95i			2.27677i
$70$		0			0
$71$			46.7346i			0.0781179i	0.999237	+	0.0390590i	$0.0124360\pi$
$71$			46.7346i			0.0781179i	−0.999237	+	0.0390590i	$0.987564\pi$
$72$		0			0
$73$	−141.641	−	141.641i	−0.227093	−	0.227093i	0.584384	−	0.811477i	$-0.301336\pi$
$73$	−141.641	−	141.641i	−0.227093	−	0.227093i	−0.811477	+	0.584384i	$0.801336\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−20.3909	+	20.3909i	−0.0301787	+	0.0301787i
$78$		0			0
$79$	1032.70			1.47073			0.735364	−	0.677673i	$-0.237011\pi$
$79$	1032.70			1.47073			0.735364	+	0.677673i	$0.237011\pi$
$80$		0			0
$81$	−1246.10			−1.70933
$82$		0			0
$83$	498.417	−	498.417i	0.659137	−	0.659137i	−0.296039	−	0.955176i	$-0.595666\pi$
$83$	498.417	−	498.417i	0.659137	−	0.659137i	0.955176	+	0.296039i	$0.0956659\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1106.66	−	1106.66i	−1.36375	−	1.36375i
$88$		0			0
$89$		−	1534.00i		−	1.82701i	−0.406831	−	0.913503i	$-0.633366\pi$
$89$		−	1534.00i		−	1.82701i	0.406831	−	0.913503i	$-0.366634\pi$
$90$		0			0
$91$		−	1137.74i		−	1.31064i
$92$		0			0
$93$	−396.903	−	396.903i	−0.442547	−	0.442547i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.20296	−	1.20296i	0.00125920	−	0.00125920i	−0.706477	−	0.707736i	$-0.749717\pi$
$97$	1.20296	−	1.20296i	0.00125920	−	0.00125920i	0.707736	+	0.706477i	$0.249717\pi$
$98$		0			0
$99$	−70.1327			−0.0711979
$100$		0			0
$101$	−473.842			−0.466822			−0.233411	−	0.972378i	$-0.574989\pi$
$101$	−473.842			−0.466822			−0.233411	+	0.972378i	$0.574989\pi$
$102$		0			0
$103$	104.230	−	104.230i	0.0997096	−	0.0997096i	−0.655492	−	0.755202i	$-0.727539\pi$
$103$	104.230	−	104.230i	0.0997096	−	0.0997096i	0.755202	+	0.655492i	$0.227539\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−170.692	−	170.692i	−0.154218	−	0.154218i	0.625781	−	0.779999i	$-0.284781\pi$
$107$	−170.692	−	170.692i	−0.154218	−	0.154218i	−0.779999	+	0.625781i	$0.784781\pi$
$108$		0			0
$109$		−	1587.84i		−	1.39530i	−0.716439	−	0.697649i	$-0.754229\pi$
$109$		−	1587.84i		−	1.39530i	0.716439	−	0.697649i	$-0.245771\pi$
$110$		0			0
$111$			2550.94i			2.18131i
$112$		0			0
$113$	844.278	+	844.278i	0.702858	+	0.702858i	0.965023	−	0.262165i	$-0.0844364\pi$
$113$	844.278	+	844.278i	0.702858	+	0.702858i	−0.262165	+	0.965023i	$0.584436\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1956.58	−	1956.58i	1.54604	−	1.54604i
$118$		0			0
$119$	−3169.80			−2.44181
$120$		0			0
$121$	1329.63			0.998972
$122$		0			0
$123$	189.821	−	189.821i	0.139151	−	0.139151i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	310.395	+	310.395i	0.216875	+	0.216875i	0.807180	−	0.590305i	$-0.200993\pi$
$127$	310.395	+	310.395i	0.216875	+	0.216875i	−0.590305	+	0.807180i	$0.700993\pi$
$128$		0			0
$129$		−	1010.16i		−	0.689453i
$130$		0			0
$131$		−	911.848i		−	0.608157i	−0.952647	−	0.304078i	$-0.901652\pi$
$131$		−	911.848i		−	0.608157i	0.952647	−	0.304078i	$-0.0983484\pi$
$132$		0			0
$133$	2291.97	+	2291.97i	1.49428	+	1.49428i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1101.13	−	1101.13i	0.686686	−	0.686686i	−0.274812	−	0.961498i	$-0.588616\pi$
$137$	1101.13	−	1101.13i	0.686686	−	0.686686i	0.961498	+	0.274812i	$0.0886156\pi$
$138$		0			0
$139$	−1380.67			−0.842497			−0.421248	−	0.906945i	$-0.638408\pi$
$139$	−1380.67			−0.842497			−0.421248	+	0.906945i	$0.638408\pi$
$140$		0			0
$141$	−2333.10			−1.39349
$142$		0			0
$143$	38.1838	−	38.1838i	0.0223293	−	0.0223293i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1744.49	+	1744.49i	0.978795	+	0.978795i
$148$		0			0
$149$			1992.16i			1.09533i	0.836698	+	0.547664i	$0.184483\pi$
$149$			1992.16i			1.09533i	−0.836698	+	0.547664i	$0.815517\pi$
$150$		0			0
$151$			895.775i			0.482762i	0.970430	+	0.241381i	$0.0776004\pi$
$151$			895.775i			0.482762i	−0.970430	+	0.241381i	$0.922400\pi$
$152$		0			0
$153$	−5451.12	−	5451.12i	−2.88037	−	2.88037i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−643.742	+	643.742i	−0.327237	+	0.327237i	−0.851535	−	0.524298i	$-0.824328\pi$
$157$	−643.742	+	643.742i	−0.327237	+	0.327237i	0.524298	+	0.851535i	$0.324328\pi$
$158$		0			0
$159$	4317.03			2.15322
$160$		0			0
$161$	3449.58			1.68860
$162$		0			0
$163$	1583.26	−	1583.26i	0.760798	−	0.760798i	−0.215668	−	0.976467i	$-0.569193\pi$
$163$	1583.26	−	1583.26i	0.760798	−	0.760798i	0.976467	+	0.215668i	$0.0691930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2151.02	+	2151.02i	0.996712	+	0.996712i	0.999995	−	0.00328303i	$-0.00104502\pi$
$167$	2151.02	+	2151.02i	0.996712	+	0.996712i	−0.00328303	+	0.999995i	$0.501045\pi$
$168$		0			0
$169$		−	66.4747i		−	0.0302570i
$170$		0			0
$171$			7883.02i			3.52532i
$172$		0			0
$173$	860.888	+	860.888i	0.378336	+	0.378336i	0.870502	−	0.492166i	$-0.163795\pi$
$173$	860.888	+	860.888i	0.378336	+	0.378336i	−0.492166	+	0.870502i	$0.663795\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−935.254	+	935.254i	−0.397164	+	0.397164i
$178$		0			0
$179$	2272.98			0.949109			0.474554	−	0.880226i	$-0.342609\pi$
$179$	2272.98			0.949109			0.474554	+	0.880226i	$0.342609\pi$
$180$		0			0
$181$	3155.16			1.29570			0.647848	−	0.761770i	$-0.275669\pi$
$181$	3155.16			1.29570			0.647848	+	0.761770i	$0.275669\pi$
$182$		0			0
$183$	2435.06	−	2435.06i	0.983635	−	0.983635i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−106.382	−	106.382i	−0.0416010	−	0.0416010i
$188$		0			0
$189$			7572.68i			2.91445i
$190$		0			0
$191$		−	5175.82i		−	1.96078i	−0.197065	−	0.980390i	$-0.563141\pi$
$191$		−	5175.82i		−	1.96078i	0.197065	−	0.980390i	$-0.436859\pi$
$192$		0			0
$193$	−1269.44	−	1269.44i	−0.473452	−	0.473452i	0.429578	−	0.903030i	$-0.358662\pi$
$193$	−1269.44	−	1269.44i	−0.473452	−	0.473452i	−0.903030	+	0.429578i	$0.858662\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1242.10	+	1242.10i	−0.449220	+	0.449220i	−0.895095	−	0.445875i	$-0.852892\pi$
$197$	−1242.10	+	1242.10i	−0.449220	+	0.449220i	0.445875	+	0.895095i	$0.352892\pi$
$198$		0			0
$199$	−2811.82			−1.00163			−0.500815	−	0.865554i	$-0.666966\pi$
$199$	−2811.82			−1.00163			−0.500815	+	0.865554i	$0.666966\pi$
$200$		0			0
$201$	894.580			0.313924
$202$		0			0
$203$	−2925.41	+	2925.41i	−1.01145	+	1.01145i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5932.25	+	5932.25i	1.99188	+	1.99188i
$208$		0			0
$209$			153.841i			0.0509159i
$210$		0			0
$211$			394.650i			0.128762i	0.997925	+	0.0643811i	$0.0205073\pi$
$211$			394.650i			0.128762i	−0.997925	+	0.0643811i	$0.979493\pi$
$212$		0			0
$213$	308.142	+	308.142i	0.0991247	+	0.0991247i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1049.20	+	1049.20i	−0.328222	+	0.328222i
$218$		0			0
$219$	−1867.80			−0.576322
$220$		0			0
$221$	5935.73			1.80670
$222$		0			0
$223$	2751.79	−	2751.79i	0.826337	−	0.826337i	−0.160671	−	0.987008i	$-0.551366\pi$
$223$	2751.79	−	2751.79i	0.826337	−	0.826337i	0.987008	+	0.160671i	$0.0513657\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3015.51	+	3015.51i	0.881703	+	0.881703i	0.993708	−	0.112005i	$-0.0357272\pi$
$227$	3015.51	+	3015.51i	0.881703	+	0.881703i	−0.112005	+	0.993708i	$0.535727\pi$
$228$		0			0
$229$		−	1700.95i		−	0.490837i	−0.969417	−	0.245418i	$-0.921075\pi$
$229$		−	1700.95i		−	0.490837i	0.969417	−	0.245418i	$-0.0789253\pi$
$230$		0			0
$231$			268.893i			0.0765882i
$232$		0			0
$233$	3464.22	+	3464.22i	0.974028	+	0.974028i	0.999671	−	0.0256428i	$-0.00816325\pi$
$233$	3464.22	+	3464.22i	0.974028	+	0.974028i	−0.0256428	+	0.999671i	$0.508163\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	6809.04	−	6809.04i	1.86622	−	1.86622i
$238$		0			0
$239$	−2487.89			−0.673341			−0.336670	−	0.941623i	$-0.609301\pi$
$239$	−2487.89			−0.673341			−0.336670	+	0.941623i	$0.609301\pi$
$240$		0			0
$241$	−1219.79			−0.326031			−0.163016	−	0.986624i	$-0.552122\pi$
$241$	−1219.79			−0.326031			−0.163016	+	0.986624i	$0.552122\pi$
$242$		0			0
$243$	−2350.74	+	2350.74i	−0.620577	+	0.620577i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4291.91	−	4291.91i	−1.10562	−	1.10562i
$248$		0			0
$249$		−	6572.58i		−	1.67277i
$250$		0			0
$251$			5887.04i			1.48042i	0.672373	+	0.740212i	$0.265275\pi$
$251$			5887.04i			1.48042i	−0.672373	+	0.740212i	$0.734725\pi$
$252$		0			0
$253$	115.771	+	115.771i	0.0287686	+	0.0287686i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4191.85	+	4191.85i	−1.01743	+	1.01743i	−0.0175881	+	0.999845i	$0.505599\pi$
$257$	−4191.85	+	4191.85i	−1.01743	+	1.01743i	−0.999845	+	0.0175881i	$0.994401\pi$
$258$		0			0
$259$	6743.33			1.61780
$260$		0			0
$261$	−10061.7			−2.38622
$262$		0			0
$263$	2969.43	−	2969.43i	0.696210	−	0.696210i	−0.267381	−	0.963591i	$-0.586158\pi$
$263$	2969.43	−	2969.43i	0.696210	−	0.696210i	0.963591	+	0.267381i	$0.0861582\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−10114.4	−	10114.4i	−2.31831	−	2.31831i
$268$		0			0
$269$			2180.00i			0.494116i	0.969001	+	0.247058i	$0.0794638\pi$
$269$			2180.00i			0.494116i	−0.969001	+	0.247058i	$0.920536\pi$
$270$		0			0
$271$		−	2376.34i		−	0.532666i	−0.963881	−	0.266333i	$-0.914188\pi$
$271$		−	2376.34i		−	0.532666i	0.963881	−	0.266333i	$-0.0858121\pi$
$272$		0			0
$273$	−7501.67	−	7501.67i	−1.66308	−	1.66308i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4845.71	−	4845.71i	1.05108	−	1.05108i	0.0524618	−	0.998623i	$-0.483293\pi$
$277$	4845.71	−	4845.71i	1.05108	−	1.05108i	0.998623	−	0.0524618i	$-0.0167068\pi$
$278$		0			0
$279$	−3608.62			−0.774345
$280$		0			0
$281$	−4898.52			−1.03993			−0.519967	−	0.854186i	$-0.674056\pi$
$281$	−4898.52			−1.03993			−0.519967	+	0.854186i	$0.674056\pi$
$282$		0			0
$283$	1790.83	−	1790.83i	0.376163	−	0.376163i	−0.493553	−	0.869716i	$-0.664302\pi$
$283$	1790.83	−	1790.83i	0.376163	−	0.376163i	0.869716	+	0.493553i	$0.164302\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−501.785	−	501.785i	−0.103204	−	0.103204i
$288$		0			0
$289$		−	11624.2i		−	2.36601i
$290$		0			0
$291$		−	15.8633i		−	0.00319562i
$292$		0			0
$293$	3456.25	+	3456.25i	0.689135	+	0.689135i	0.962041	−	0.272906i	$-0.0879848\pi$
$293$	3456.25	+	3456.25i	0.689135	+	0.689135i	−0.272906	+	0.962041i	$0.587985\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−254.146	+	254.146i	−0.0496534	+	0.0496534i
$298$		0			0
$299$	−6459.63			−1.24940
$300$		0			0
$301$	−2670.31			−0.511343
$302$		0			0
$303$	−3124.26	+	3124.26i	−0.592356	+	0.592356i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	3755.56	+	3755.56i	0.698179	+	0.698179i	0.964018	−	0.265839i	$-0.0856488\pi$
$307$	3755.56	+	3755.56i	0.698179	+	0.698179i	−0.265839	+	0.964018i	$0.585649\pi$
$308$		0			0
$309$		−	1374.47i		−	0.253045i
$310$		0			0
$311$		−	2866.12i		−	0.522581i	−0.965260	−	0.261291i	$-0.915852\pi$
$311$		−	2866.12i		−	0.522581i	0.965260	−	0.261291i	$-0.0841481\pi$
$312$		0			0
$313$	3070.75	+	3070.75i	0.554534	+	0.554534i	0.927746	−	0.373212i	$-0.121744\pi$
$313$	3070.75	+	3070.75i	0.554534	+	0.554534i	−0.373212	+	0.927746i	$0.621744\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3355.28	−	3355.28i	0.594484	−	0.594484i	−0.344355	−	0.938840i	$-0.611902\pi$
$317$	3355.28	−	3355.28i	0.594484	−	0.594484i	0.938840	+	0.344355i	$0.111902\pi$
$318$		0			0
$319$	−196.359			−0.0344639
$320$		0			0
$321$	−2250.89			−0.391379
$322$		0			0
$323$	−11957.5	+	11957.5i	−2.05985	+	2.05985i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−10469.4	−	10469.4i	−1.77051	−	1.77051i
$328$		0			0
$329$			6167.47i			1.03351i
$330$		0			0
$331$		−	7909.93i		−	1.31350i	−0.754108	−	0.656751i	$-0.771930\pi$
$331$		−	7909.93i		−	1.31350i	0.754108	−	0.656751i	$-0.228070\pi$
$332$		0			0
$333$	11596.5	+	11596.5i	1.90836	+	1.90836i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2979.68	+	2979.68i	−0.481642	+	0.481642i	−0.905656	−	0.424014i	$-0.860621\pi$
$337$	−2979.68	+	2979.68i	−0.481642	+	0.481642i	0.424014	+	0.905656i	$0.360621\pi$
$338$		0			0
$339$	11133.4			1.78373
$340$		0			0
$341$	−70.4241			−0.0111838
$342$		0			0
$343$	−1366.85	+	1366.85i	−0.215169	+	0.215169i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2971.02	+	2971.02i	0.459633	+	0.459633i	0.898535	−	0.438902i	$-0.144632\pi$
$347$	2971.02	+	2971.02i	0.459633	+	0.459633i	−0.438902	+	0.898535i	$0.644632\pi$
$348$		0			0
$349$			5351.31i			0.820771i	0.911912	+	0.410386i	$0.134606\pi$
$349$			5351.31i			0.820771i	−0.911912	+	0.410386i	$0.865394\pi$
$350$		0			0
$351$		−	14180.5i		−	2.15641i
$352$		0			0
$353$	−1506.52	−	1506.52i	−0.227150	−	0.227150i	0.584351	−	0.811501i	$-0.301349\pi$
$353$	−1506.52	−	1506.52i	−0.227150	−	0.227150i	−0.811501	+	0.584351i	$0.801349\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−20900.0	+	20900.0i	−3.09844	+	3.09844i
$358$		0			0
$359$	−10529.1			−1.54792			−0.773960	−	0.633234i	$-0.781727\pi$
$359$	−10529.1			−1.54792			−0.773960	+	0.633234i	$0.781727\pi$
$360$		0			0
$361$	10433.0			1.52107
$362$		0			0
$363$	8766.87	−	8766.87i	1.26761	−	1.26761i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−438.136	−	438.136i	−0.0623175	−	0.0623175i	0.675261	−	0.737579i	$-0.264031\pi$
$367$	−438.136	−	438.136i	−0.0623175	−	0.0623175i	−0.737579	+	0.675261i	$0.764031\pi$
$368$		0			0
$369$		−	1725.84i		−	0.243479i
$370$		0			0
$371$		−	11411.9i		−	1.59697i
$372$		0			0
$373$	−7090.32	−	7090.32i	−0.984244	−	0.984244i	0.0156341	−	0.999878i	$-0.495023\pi$
$373$	−7090.32	−	7090.32i	−0.984244	−	0.984244i	−0.999878	+	0.0156341i	$0.995023\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5478.09	−	5478.09i	0.748371	−	0.748371i
$378$		0			0
$379$	5073.01			0.687554			0.343777	−	0.939051i	$-0.388294\pi$
$379$	5073.01			0.687554			0.343777	+	0.939051i	$0.388294\pi$
$380$		0			0
$381$	4093.15			0.550390
$382$		0			0
$383$	−3284.88	+	3284.88i	−0.438250	+	0.438250i	−0.891423	−	0.453173i	$-0.850292\pi$
$383$	−3284.88	+	3284.88i	−0.438250	+	0.438250i	0.453173	+	0.891423i	$0.350292\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4592.15	−	4592.15i	−0.603184	−	0.603184i
$388$		0			0
$389$			5470.95i			0.713080i	0.934280	+	0.356540i	$0.116044\pi$
$389$			5470.95i			0.713080i	−0.934280	+	0.356540i	$0.883956\pi$
$390$		0			0
$391$			17996.8i			2.32772i
$392$		0			0
$393$	−6012.23	−	6012.23i	−0.771697	−	0.771697i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−9163.71	+	9163.71i	−1.15847	+	1.15847i	−0.173667	+	0.984804i	$0.555562\pi$
$397$	−9163.71	+	9163.71i	−1.15847	+	1.15847i	−0.984804	+	0.173667i	$0.944438\pi$
$398$		0			0
$399$	30224.0			3.79221
$400$		0			0
$401$	−7823.26			−0.974252			−0.487126	−	0.873332i	$-0.661955\pi$
$401$	−7823.26			−0.974252			−0.487126	+	0.873332i	$0.661955\pi$
$402$		0			0
$403$	1964.71	−	1964.71i	0.242852	−	0.242852i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	226.312	+	226.312i	0.0275624	+	0.0275624i
$408$		0			0
$409$			13142.9i			1.58893i	0.607307	+	0.794467i	$0.292250\pi$
$409$			13142.9i			1.58893i	−0.607307	+	0.794467i	$0.707750\pi$
$410$		0			0
$411$		−	14520.5i		−	1.74269i
$412$		0			0
$413$	2472.31	+	2472.31i	0.294562	+	0.294562i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−9103.40	+	9103.40i	−1.06905	+	1.06905i
$418$		0			0
$419$	−2754.53			−0.321164			−0.160582	−	0.987023i	$-0.551337\pi$
$419$	−2754.53			−0.321164			−0.160582	+	0.987023i	$0.551337\pi$
$420$		0			0
$421$	−11368.2			−1.31604			−0.658019	−	0.753001i	$-0.728606\pi$
$421$	−11368.2			−1.31604			−0.658019	+	0.753001i	$0.728606\pi$
$422$		0			0
$423$	−10606.2	+	10606.2i	−1.21913	+	1.21913i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−6437.00	−	6437.00i	−0.729528	−	0.729528i
$428$		0			0
$429$		−	503.526i		−	0.0566677i
$430$		0			0
$431$			924.819i			0.103357i	0.998664	+	0.0516786i	$0.0164571\pi$
$431$			924.819i			0.103357i	−0.998664	+	0.0516786i	$0.983543\pi$
$432$		0			0
$433$	586.803	+	586.803i	0.0651269	+	0.0651269i	0.738920	−	0.673793i	$-0.235336\pi$
$433$	586.803	+	586.803i	0.0651269	+	0.0651269i	−0.673793	+	0.738920i	$0.735336\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	13012.8	−	13012.8i	1.42446	−	1.42446i
$438$		0			0
$439$	7929.82			0.862117			0.431059	−	0.902324i	$-0.358140\pi$
$439$	7929.82			0.862117			0.431059	+	0.902324i	$0.358140\pi$
$440$		0			0
$441$	15860.8			1.71264
$442$		0			0
$443$	−5109.87	+	5109.87i	−0.548030	+	0.548030i	−0.925871	−	0.377841i	$-0.876667\pi$
$443$	−5109.87	+	5109.87i	−0.548030	+	0.548030i	0.377841	+	0.925871i	$0.376667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	13135.2	+	13135.2i	1.38987	+	1.38987i
$448$		0			0
$449$		−	201.051i		−	0.0211318i	−0.999944	−	0.0105659i	$-0.996637\pi$
$449$		−	201.051i		−	0.0211318i	0.999944	−	0.0105659i	$-0.00336330\pi$
$450$		0			0
$451$		−	33.6808i		−	0.00351655i
$452$		0			0
$453$	5906.25	+	5906.25i	0.612583	+	0.612583i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−767.997	+	767.997i	−0.0786114	+	0.0786114i	−0.745319	−	0.666708i	$-0.767703\pi$
$457$	−767.997	+	767.997i	−0.0786114	+	0.0786114i	0.666708	+	0.745319i	$0.267703\pi$
$458$		0			0
$459$	−39507.5			−4.01754
$460$		0			0
$461$	−10233.1			−1.03385			−0.516925	−	0.856030i	$-0.672924\pi$
$461$	−10233.1			−1.03385			−0.516925	+	0.856030i	$0.672924\pi$
$462$		0			0
$463$	−1507.13	+	1507.13i	−0.151279	+	0.151279i	−0.778689	−	0.627410i	$-0.784115\pi$
$463$	−1507.13	+	1507.13i	−0.151279	+	0.151279i	0.627410	+	0.778689i	$0.284115\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7687.25	+	7687.25i	0.761721	+	0.761721i	0.976633	−	0.214912i	$-0.0689466\pi$
$467$	7687.25	+	7687.25i	0.761721	+	0.761721i	−0.214912	+	0.976633i	$0.568947\pi$
$468$		0			0
$469$		−	2364.79i		−	0.232827i
$470$		0			0
$471$			8488.96i			0.830469i
$472$		0			0
$473$	−89.6183	−	89.6183i	−0.00871174	−	0.00871174i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	19625.1	−	19625.1i	1.88380	−	1.88380i
$478$		0			0
$479$	−17640.5			−1.68270			−0.841352	−	0.540487i	$-0.818240\pi$
$479$	−17640.5			−1.68270			−0.841352	+	0.540487i	$0.818240\pi$
$480$		0			0
$481$	−12627.5			−1.19701
$482$		0			0
$483$	22744.6	−	22744.6i	2.14268	−	2.14268i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−10374.7	−	10374.7i	−0.965349	−	0.965349i	0.0340709	−	0.999419i	$-0.489153\pi$
$487$	−10374.7	−	10374.7i	−0.965349	−	0.965349i	−0.999419	+	0.0340709i	$0.989153\pi$
$488$		0			0
$489$		−	20878.3i		−	1.93077i
$490$		0			0
$491$			11766.0i			1.08145i	0.841200	+	0.540724i	$0.181850\pi$
$491$			11766.0i			1.08145i	−0.841200	+	0.540724i	$0.818150\pi$
$492$		0			0
$493$	−15262.2	−	15262.2i	−1.39427	−	1.39427i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	814.563	−	814.563i	0.0735174	−	0.0735174i
$498$		0			0
$499$	7654.08			0.686661			0.343331	−	0.939215i	$-0.388445\pi$
$499$	7654.08			0.686661			0.343331	+	0.939215i	$0.388445\pi$
$500$		0			0
$501$	28365.3			2.52948
$502$		0			0
$503$	−8907.38	+	8907.38i	−0.789584	+	0.789584i	−0.981426	−	0.191842i	$-0.938554\pi$
$503$	−8907.38	+	8907.38i	−0.789584	+	0.789584i	0.191842	+	0.981426i	$0.438554\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−438.298	−	438.298i	−0.0383935	−	0.0383935i
$508$		0			0
$509$		−	12672.2i		−	1.10351i	−0.834007	−	0.551754i	$-0.813959\pi$
$509$		−	12672.2i		−	1.10351i	0.834007	−	0.551754i	$-0.186041\pi$
$510$		0			0
$511$			4937.47i			0.427438i
$512$		0			0
$513$	28566.4	+	28566.4i	2.45855	+	2.45855i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−206.986	+	206.986i	−0.0176078	+	0.0176078i
$518$		0			0
$519$	11352.5			0.960150
$520$		0			0
$521$	−13205.5			−1.11045			−0.555225	−	0.831701i	$-0.687368\pi$
$521$	−13205.5			−1.11045			−0.555225	+	0.831701i	$0.687368\pi$
$522$		0			0
$523$	11937.7	−	11937.7i	0.998088	−	0.998088i	−0.00191034	−	0.999998i	$-0.500608\pi$
$523$	11937.7	−	11937.7i	0.998088	−	0.998088i	0.999998	+	0.00191034i	$0.000608082\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5473.78	−	5473.78i	−0.452451	−	0.452451i
$528$		0			0
$529$		−	7418.26i		−	0.609703i
$530$		0			0
$531$			8503.27i			0.694935i
$532$		0			0
$533$	939.636	+	939.636i	0.0763606	+	0.0763606i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	14986.8	−	14986.8i	1.20433	−	1.20433i
$538$		0			0
$539$	309.532			0.0247356
$540$		0			0
$541$	12291.6			0.976814			0.488407	−	0.872616i	$-0.337578\pi$
$541$	12291.6			0.976814			0.488407	+	0.872616i	$0.337578\pi$
$542$		0			0
$543$	20803.4	−	20803.4i	1.64412	−	1.64412i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4788.11	+	4788.11i	0.374268	+	0.374268i	0.869029	−	0.494761i	$-0.164744\pi$
$547$	4788.11	+	4788.11i	0.374268	+	0.374268i	−0.494761	+	0.869029i	$0.664744\pi$
$548$		0			0
$549$		−	22139.5i		−	1.72111i
$550$		0			0
$551$			22071.1i			1.70646i
$552$		0			0
$553$	−17999.4	−	17999.4i	−1.38411	−	1.38411i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3849.08	−	3849.08i	0.292802	−	0.292802i	−0.545384	−	0.838186i	$-0.683616\pi$
$557$	3849.08	−	3849.08i	0.292802	−	0.292802i	0.838186	+	0.545384i	$0.183616\pi$
$558$		0			0
$559$	5000.40			0.378344
$560$		0			0
$561$	−1402.84			−0.105576
$562$		0			0
$563$	−8314.72	+	8314.72i	−0.622422	+	0.622422i	−0.946150	−	0.323728i	$-0.895064\pi$
$563$	−8314.72	+	8314.72i	−0.622422	+	0.622422i	0.323728	+	0.946150i	$0.395064\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	21719.0	+	21719.0i	1.60867	+	1.60867i
$568$		0			0
$569$			22679.9i			1.67099i	0.549499	+	0.835495i	$0.314819\pi$
$569$			22679.9i			1.67099i	−0.549499	+	0.835495i	$0.685181\pi$
$570$		0			0
$571$			4725.35i			0.346322i	0.984894	+	0.173161i	$0.0553980\pi$
$571$			4725.35i			0.346322i	−0.984894	+	0.173161i	$0.944602\pi$
$572$		0			0
$573$	−34126.5	−	34126.5i	−2.48806	−	2.48806i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	15787.1	−	15787.1i	1.13904	−	1.13904i	0.150413	−	0.988623i	$-0.451940\pi$
$577$	15787.1	−	15787.1i	1.13904	−	1.13904i	0.988623	−	0.150413i	$-0.0480603\pi$
$578$		0			0
$579$	−16740.0			−1.20154
$580$		0			0
$581$	−17374.4			−1.24064
$582$		0			0
$583$	382.994	−	382.994i	0.0272075	−	0.0272075i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	11559.0	+	11559.0i	0.812760	+	0.812760i	0.985047	−	0.172287i	$-0.0551156\pi$
$587$	11559.0	+	11559.0i	0.812760	+	0.812760i	−0.172287	+	0.985047i	$0.555116\pi$
$588$		0			0
$589$			7915.78i			0.553759i
$590$		0			0
$591$			16379.5i			1.14004i
$592$		0			0
$593$	8435.85	+	8435.85i	0.584180	+	0.584180i	0.936049	−	0.351869i	$-0.114454\pi$
$593$	8435.85	+	8435.85i	0.584180	+	0.584180i	−0.351869	+	0.936049i	$0.614454\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−18539.6	+	18539.6i	−1.27098	+	1.27098i
$598$		0			0
$599$	−17552.9			−1.19732			−0.598659	−	0.801004i	$-0.704299\pi$
$599$	−17552.9			−1.19732			−0.598659	+	0.801004i	$0.704299\pi$
$600$		0			0
$601$	−592.834			−0.0402366			−0.0201183	−	0.999798i	$-0.506404\pi$
$601$	−592.834			−0.0402366			−0.0201183	+	0.999798i	$0.506404\pi$
$602$		0			0
$603$	4066.74	−	4066.74i	0.274644	−	0.274644i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	4939.94	+	4939.94i	0.330323	+	0.330323i	0.852709	−	0.522386i	$-0.174958\pi$
$607$	4939.94	+	4939.94i	0.330323	+	0.330323i	−0.522386	+	0.852709i	$0.674958\pi$
$608$		0			0
$609$			38577.1i			2.56687i
$610$		0			0
$611$		−	11549.1i		−	0.764693i
$612$		0			0
$613$	20209.0	+	20209.0i	1.33154	+	1.33154i	0.903996	+	0.427541i	$0.140620\pi$
$613$	20209.0	+	20209.0i	1.33154	+	1.33154i	0.427541	+	0.903996i	$0.359380\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1312.19	+	1312.19i	−0.0856186	+	0.0856186i	−0.748619	−	0.663000i	$-0.769283\pi$
$617$	−1312.19	+	1312.19i	−0.0856186	+	0.0856186i	0.663000	+	0.748619i	$0.269283\pi$
$618$		0			0
$619$	18868.7			1.22520			0.612599	−	0.790394i	$-0.290124\pi$
$619$	18868.7			1.22520			0.612599	+	0.790394i	$0.290124\pi$
$620$		0			0
$621$	42994.5			2.77828
$622$		0			0
$623$	−26736.9	+	26736.9i	−1.71941	+	1.71941i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1014.35	+	1014.35i	0.0646078	+	0.0646078i
$628$		0			0
$629$			35180.7i			2.23012i
$630$		0			0
$631$		−	11365.7i		−	0.717057i	−0.933519	−	0.358529i	$-0.883279\pi$
$631$		−	11365.7i		−	0.717057i	0.933519	−	0.358529i	$-0.116721\pi$
$632$		0			0
$633$	2602.11	+	2602.11i	0.163388	+	0.163388i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−8635.41	+	8635.41i	−0.537123	+	0.537123i
$638$		0			0
$639$	2801.61			0.173443
$640$		0			0
$641$	19324.2			1.19073			0.595366	−	0.803454i	$-0.297007\pi$
$641$	19324.2			1.19073			0.595366	+	0.803454i	$0.297007\pi$
$642$		0			0
$643$	−10220.8	+	10220.8i	−0.626858	+	0.626858i	−0.947276	−	0.320418i	$-0.896177\pi$
$643$	−10220.8	+	10220.8i	−0.626858	+	0.626858i	0.320418	+	0.947276i	$0.396177\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12395.5	+	12395.5i	0.753196	+	0.753196i	0.975074	−	0.221879i	$-0.0712189\pi$
$647$	12395.5	+	12395.5i	0.753196	+	0.753196i	−0.221879	+	0.975074i	$0.571219\pi$
$648$		0			0
$649$			165.946i			0.0100369i
$650$		0			0
$651$			13835.7i			0.832969i
$652$		0			0
$653$	−6708.87	−	6708.87i	−0.402050	−	0.402050i	0.476905	−	0.878955i	$-0.341759\pi$
$653$	−6708.87	−	6708.87i	−0.402050	−	0.402050i	−0.878955	+	0.476905i	$0.841759\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−8490.98	+	8490.98i	−0.504208	+	0.504208i
$658$		0			0
$659$	15423.1			0.911682			0.455841	−	0.890061i	$-0.349339\pi$
$659$	15423.1			0.911682			0.455841	+	0.890061i	$0.349339\pi$
$660$		0			0
$661$	−26793.1			−1.57660			−0.788299	−	0.615293i	$-0.789038\pi$
$661$	−26793.1			−1.57660			−0.788299	+	0.615293i	$0.789038\pi$
$662$		0			0
$663$	39137.0	−	39137.0i	2.29254	−	2.29254i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	16609.3	+	16609.3i	0.964188	+	0.964188i
$668$		0			0
$669$		−	36287.6i		−	2.09710i
$670$		0			0
$671$		−	432.064i		−	0.0248579i
$672$		0			0
$673$	−15270.8	−	15270.8i	−0.874662	−	0.874662i	0.118314	−	0.992976i	$-0.462251\pi$
$673$	−15270.8	−	15270.8i	−0.874662	−	0.874662i	−0.992976	+	0.118314i	$0.962251\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9246.91	+	9246.91i	−0.524945	+	0.524945i	−0.919061	−	0.394116i	$-0.871051\pi$
$677$	−9246.91	+	9246.91i	−0.524945	+	0.524945i	0.394116	+	0.919061i	$0.371051\pi$
$678$		0			0
$679$	−41.9341			−0.00237008
$680$		0			0
$681$	39765.3			2.23761
$682$		0			0
$683$	−6073.88	+	6073.88i	−0.340279	+	0.340279i	−0.856472	−	0.516193i	$-0.827349\pi$
$683$	−6073.88	+	6073.88i	−0.340279	+	0.340279i	0.516193	+	0.856472i	$0.327349\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−11215.1	−	11215.1i	−0.622828	−	0.622828i
$688$		0			0
$689$			21369.8i			1.18160i
$690$		0			0
$691$			7454.55i			0.410397i	0.978720	+	0.205198i	$0.0657840\pi$
$691$			7454.55i			0.410397i	−0.978720	+	0.205198i	$0.934216\pi$
$692$		0			0
$693$	1222.38	+	1222.38i	0.0670049	+	0.0670049i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2617.87	−	2617.87i	0.142265	−	0.142265i
$698$		0			0
$699$	45682.4			2.47191
$700$		0			0
$701$	15565.3			0.838651			0.419325	−	0.907836i	$-0.362267\pi$
$701$	15565.3			0.838651			0.419325	+	0.907836i	$0.362267\pi$
$702$		0			0
$703$	25437.9	−	25437.9i	1.36473	−	1.36473i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8258.85	+	8258.85i	0.439330	+	0.439330i
$708$		0			0
$709$		−	25227.6i		−	1.33631i	−0.744024	−	0.668153i	$-0.767085\pi$
$709$		−	25227.6i		−	1.33631i	0.744024	−	0.668153i	$-0.232915\pi$
$710$		0			0
$711$		−	61907.4i		−	3.26541i
$712$		0			0
$713$	5956.90	+	5956.90i	0.312886	+	0.312886i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−16403.8	+	16403.8i	−0.854410	+	0.854410i
$718$		0			0
$719$	−35468.2			−1.83970			−0.919848	−	0.392275i	$-0.871688\pi$
$719$	−35468.2			−1.83970			−0.919848	+	0.392275i	$0.871688\pi$
$720$		0			0
$721$	−3633.37			−0.187675
$722$		0			0
$723$	−8042.62	+	8042.62i	−0.413705	+	0.413705i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−24949.4	−	24949.4i	−1.27280	−	1.27280i	−0.944615	−	0.328182i	$-0.893564\pi$
$727$	−24949.4	−	24949.4i	−1.27280	−	1.27280i	−0.328182	−	0.944615i	$-0.606436\pi$
$728$		0			0
$729$		−	2645.78i		−	0.134420i
$730$		0			0
$731$		−	13931.3i		−	0.704882i
$732$		0			0
$733$	−6598.27	−	6598.27i	−0.332486	−	0.332486i	0.521044	−	0.853530i	$-0.325543\pi$
$733$	−6598.27	−	6598.27i	−0.332486	−	0.332486i	−0.853530	+	0.521044i	$0.825543\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	79.3645	−	79.3645i	0.00396666	−	0.00396666i
$738$		0			0
$739$	−978.265			−0.0486956			−0.0243478	−	0.999704i	$-0.507751\pi$
$739$	−978.265			−0.0486956			−0.0243478	+	0.999704i	$0.507751\pi$
$740$		0			0
$741$	−56597.1			−2.80586
$742$		0			0
$743$	−5941.82	+	5941.82i	−0.293384	+	0.293384i	−0.838415	−	0.545032i	$-0.816518\pi$
$743$	−5941.82	+	5941.82i	−0.293384	+	0.293384i	0.545032	+	0.838415i	$0.316518\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−29878.8	−	29878.8i	−1.46346	−	1.46346i
$748$		0			0
$749$			5950.16i			0.290272i
$750$		0			0
$751$			21538.9i			1.04656i	0.852162	+	0.523278i	$0.175291\pi$
$751$			21538.9i			1.04656i	−0.852162	+	0.523278i	$0.824709\pi$
$752$		0			0
$753$	38815.9	+	38815.9i	1.87853	+	1.87853i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−4265.56	+	4265.56i	−0.204801	+	0.204801i	−0.802053	−	0.597252i	$-0.796259\pi$
$757$	−4265.56	+	4265.56i	−0.204801	+	0.204801i	0.597252	+	0.802053i	$0.296259\pi$
$758$		0			0
$759$	1526.66			0.0730097
$760$		0			0
$761$	−6743.11			−0.321206			−0.160603	−	0.987019i	$-0.551344\pi$
$761$	−6743.11			−0.321206			−0.160603	+	0.987019i	$0.551344\pi$
$762$		0			0
$763$	−27675.4	+	27675.4i	−1.31313	+	1.31313i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4629.61	−	4629.61i	−0.217947	−	0.217947i
$768$		0			0
$769$			12774.9i			0.599059i	0.954087	+	0.299530i	$0.0968297\pi$
$769$			12774.9i			0.599059i	−0.954087	+	0.299530i	$0.903170\pi$
$770$		0			0
$771$			55277.6i			2.58207i
$772$		0			0
$773$	−18865.0	−	18865.0i	−0.877785	−	0.877785i	0.115520	−	0.993305i	$-0.463147\pi$
$773$	−18865.0	−	18865.0i	−0.877785	−	0.877785i	−0.993305	+	0.115520i	$0.963147\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	44461.8	−	44461.8i	2.05284	−	2.05284i
$778$		0			0
$779$	−3785.77			−0.174120
$780$		0			0
$781$	54.6750			0.00250502
$782$		0			0
$783$	−36461.5	+	36461.5i	−1.66415	+	1.66415i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−24315.1	−	24315.1i	−1.10132	−	1.10132i	−0.994252	−	0.107068i	$-0.965854\pi$
$787$	−24315.1	−	24315.1i	−1.10132	−	1.10132i	−0.107068	−	0.994252i	$-0.534146\pi$
$788$		0			0
$789$		−	39157.7i		−	1.76686i
$790$		0			0
$791$		−	29430.8i		−	1.32293i
$792$		0			0
$793$	12053.8	+	12053.8i	0.539779	+	0.539779i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19005.7	−	19005.7i	0.844689	−	0.844689i	−0.144776	−	0.989464i	$-0.546246\pi$
$797$	19005.7	−	19005.7i	0.844689	−	0.844689i	0.989464	+	0.144776i	$0.0462461\pi$
$798$		0			0
$799$	−32176.4			−1.42468
$800$		0			0
$801$	−91959.2			−4.05645
$802$		0			0
$803$	−165.706	+	165.706i	−0.00728224	+	0.00728224i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	14373.8	+	14373.8i	0.626989	+	0.626989i
$808$		0			0
$809$		−	13371.3i		−	0.581102i	−0.956860	−	0.290551i	$-0.906161\pi$
$809$		−	13371.3i		−	0.581102i	0.956860	−	0.290551i	$-0.0938386\pi$
$810$		0			0
$811$			6759.66i			0.292680i	0.989234	+	0.146340i	$0.0467494\pi$
$811$			6759.66i			0.292680i	−0.989234	+	0.146340i	$0.953251\pi$
$812$		0			0
$813$	−15668.3	−	15668.3i	−0.675906	−	0.675906i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10073.2	+	10073.2i	−0.431356	+	0.431356i
$818$		0			0
$819$	−68204.7			−2.90997
$820$		0			0
$821$	18576.0			0.789655			0.394827	−	0.918755i	$-0.370804\pi$
$821$	18576.0			0.789655			0.394827	+	0.918755i	$0.370804\pi$
$822$		0			0
$823$	15133.5	−	15133.5i	0.640974	−	0.640974i	−0.309821	−	0.950795i	$-0.600269\pi$
$823$	15133.5	−	15133.5i	0.640974	−	0.640974i	0.950795	+	0.309821i	$0.100269\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5338.40	+	5338.40i	0.224467	+	0.224467i	0.810377	−	0.585909i	$-0.199263\pi$
$827$	5338.40	+	5338.40i	0.224467	+	0.224467i	−0.585909	+	0.810377i	$0.699263\pi$
$828$		0			0
$829$		−	19209.5i		−	0.804794i	−0.915465	−	0.402397i	$-0.868177\pi$
$829$		−	19209.5i		−	0.804794i	0.915465	−	0.402397i	$-0.131823\pi$
$830$		0			0
$831$		−	63900.0i		−	2.66747i
$832$		0			0
$833$	24058.6	+	24058.6i	1.00070	+	1.00070i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−13076.9	+	13076.9i	−0.540028	+	0.540028i
$838$		0			0
$839$	39469.1			1.62411			0.812053	−	0.583584i	$-0.198350\pi$
$839$	39469.1			1.62411			0.812053	+	0.583584i	$0.198350\pi$
$840$		0			0
$841$	−3781.94			−0.155067
$842$		0			0
$843$	−32298.2	+	32298.2i	−1.31958	+	1.31958i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−23174.9	−	23174.9i	−0.940140	−	0.940140i
$848$		0			0
$849$		−	23615.6i		−	0.954634i
$850$		0			0
$851$		−	38285.8i		−	1.54221i
$852$		0			0
$853$	−380.996	−	380.996i	−0.0152932	−	0.0152932i	0.699419	−	0.714712i	$-0.253442\pi$
$853$	−380.996	−	380.996i	−0.0152932	−	0.0152932i	−0.714712	+	0.699419i	$0.753442\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6593.35	+	6593.35i	−0.262806	+	0.262806i	−0.826193	−	0.563387i	$-0.809498\pi$
$857$	−6593.35	+	6593.35i	−0.262806	+	0.262806i	0.563387	+	0.826193i	$0.309498\pi$
$858$		0			0
$859$	46458.5			1.84534			0.922669	−	0.385593i	$-0.126003\pi$
$859$	46458.5			1.84534			0.922669	+	0.385593i	$0.126003\pi$
$860$		0			0
$861$	−6617.00			−0.261912
$862$		0			0
$863$	28617.9	−	28617.9i	1.12881	−	1.12881i	0.138442	−	0.990371i	$-0.455790\pi$
$863$	28617.9	−	28617.9i	1.12881	−	1.12881i	0.990371	−	0.138442i	$-0.0442095\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−76643.7	−	76643.7i	−3.00226	−	3.00226i
$868$		0			0
$869$		−	1208.16i		−	0.0471621i
$870$		0			0
$871$			4428.27i			0.172269i
$872$		0			0
$873$	−72.1142	−	72.1142i	−0.00279576	−	0.00279576i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	15237.5	−	15237.5i	0.586697	−	0.586697i	−0.350038	−	0.936735i	$-0.613831\pi$
$877$	15237.5	−	15237.5i	0.586697	−	0.586697i	0.936735	+	0.350038i	$0.113831\pi$
$878$		0			0
$879$	45577.3			1.74890
$880$		0			0
$881$	−48427.8			−1.85196			−0.925978	−	0.377577i	$-0.876757\pi$
$881$	−48427.8			−1.85196			−0.925978	+	0.377577i	$0.876757\pi$
$882$		0			0
$883$	30054.7	−	30054.7i	1.14544	−	1.14544i	0.157998	−	0.987439i	$-0.449496\pi$
$883$	30054.7	−	30054.7i	1.14544	−	1.14544i	0.987439	−	0.157998i	$-0.0505039\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9648.05	+	9648.05i	0.365220	+	0.365220i	0.865730	−	0.500511i	$-0.166854\pi$
$887$	9648.05	+	9648.05i	0.365220	+	0.365220i	−0.500511	+	0.865730i	$0.666854\pi$
$888$		0			0
$889$		−	10820.1i		−	0.408205i
$890$		0			0
$891$			1457.82i			0.0548136i
$892$		0			0
$893$	23265.6	+	23265.6i	0.871839	+	0.871839i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−42591.3	+	42591.3i	−1.58538	+	1.58538i
$898$		0			0
$899$	−10103.5			−0.374828
$900$		0			0
$901$	59537.1			2.20141
$902$		0			0
$903$	−17606.6	+	17606.6i	−0.648849	+	0.648849i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	15205.8	+	15205.8i	0.556672	+	0.556672i	0.928358	−	0.371686i	$-0.121220\pi$
$907$	15205.8	+	15205.8i	0.556672	+	0.556672i	−0.371686	+	0.928358i	$0.621220\pi$
$908$		0			0
$909$			28405.6i			1.03647i
$910$		0			0
$911$			47786.3i			1.73790i	0.494897	+	0.868951i	$0.335206\pi$
$911$			47786.3i			1.73790i	−0.494897	+	0.868951i	$0.664794\pi$
$912$		0			0
$913$	−583.100	−	583.100i	−0.0211367	−	0.0211367i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15893.1	+	15893.1i	−0.572341	+	0.572341i
$918$		0			0
$919$	−34276.9			−1.23035			−0.615174	−	0.788391i	$-0.710914\pi$
$919$	−34276.9			−1.23035			−0.615174	+	0.788391i	$0.710914\pi$
$920$		0			0
$921$	49524.2			1.77185
$922$		0			0
$923$	−1525.34	+	1525.34i	−0.0543956	+	0.0543956i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−6248.31	−	6248.31i	−0.221382	−	0.221382i
$928$		0			0
$929$		−	30514.6i		−	1.07767i	−0.842412	−	0.538833i	$-0.818865\pi$
$929$		−	30514.6i		−	1.07767i	0.842412	−	0.538833i	$-0.181135\pi$
$930$		0			0
$931$		−	34791.8i		−	1.22477i
$932$		0			0
$933$	−18897.6	−	18897.6i	−0.663109	−	0.663109i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−6725.38	+	6725.38i	−0.234481	+	0.234481i	−0.814560	−	0.580079i	$-0.803022\pi$
$937$	−6725.38	+	6725.38i	−0.234481	+	0.234481i	0.580079	+	0.814560i	$0.303022\pi$
$938$		0			0
$939$	40493.8			1.40731
$940$		0			0
$941$	32999.8			1.14321			0.571607	−	0.820528i	$-0.306320\pi$
$941$	32999.8			1.14321			0.571607	+	0.820528i	$0.306320\pi$
$942$		0			0
$943$	−2848.93	+	2848.93i	−0.0983815	+	0.0983815i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15569.2	+	15569.2i	0.534245	+	0.534245i	0.921833	−	0.387588i	$-0.126692\pi$
$947$	15569.2	+	15569.2i	0.534245	+	0.534245i	−0.387588	+	0.921833i	$0.626692\pi$
$948$		0			0
$949$		−	9245.84i		−	0.316262i
$950$		0			0
$951$		−	44245.8i		−	1.50870i
$952$		0			0
$953$	1494.47	+	1494.47i	0.0507980	+	0.0507980i	0.732049	−	0.681251i	$-0.238564\pi$
$953$	1494.47	+	1494.47i	0.0507980	+	0.0507980i	−0.681251	+	0.732049i	$0.738564\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1294.69	+	1294.69i	−0.0437317	+	0.0437317i
$958$		0			0
$959$	−38384.4			−1.29249
$960$		0			0
$961$	26167.4			0.878365
$962$		0			0
$963$	−10232.5	+	10232.5i	−0.342407	+	0.342407i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	35907.1	+	35907.1i	1.19410	+	1.19410i	0.975905	+	0.218196i	$0.0700172\pi$
$967$	35907.1	+	35907.1i	1.19410	+	1.19410i	0.218196	+	0.975905i	$0.429983\pi$
$968$		0			0
$969$			157682.i			5.22752i
$970$		0			0
$971$		−	8479.77i		−	0.280256i	−0.990133	−	0.140128i	$-0.955249\pi$
$971$		−	8479.77i		−	0.280256i	0.990133	−	0.140128i	$-0.0447514\pi$
$972$		0			0
$973$	24064.5	+	24064.5i	0.792880	+	0.792880i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−39251.4	+	39251.4i	−1.28533	+	1.28533i	−0.347734	+	0.937593i	$0.613049\pi$
$977$	−39251.4	+	39251.4i	−1.28533	+	1.28533i	−0.937593	+	0.347734i	$0.886951\pi$
$978$		0			0
$979$	−1794.63			−0.0585870
$980$		0			0
$981$	−95186.8			−3.09794
$982$		0			0
$983$	−39743.4	+	39743.4i	−1.28954	+	1.28954i	−0.354476	+	0.935065i	$0.615341\pi$
$983$	−39743.4	+	39743.4i	−1.28954	+	1.28954i	−0.935065	+	0.354476i	$0.884659\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	40664.9	+	40664.9i	1.31143	+	1.31143i
$988$		0			0
$989$			15160.9i			0.487451i
$990$		0			0
$991$		−	42627.2i		−	1.36639i	−0.730234	−	0.683197i	$-0.760589\pi$
$991$		−	42627.2i		−	1.36639i	0.730234	−	0.683197i	$-0.239411\pi$
$992$		0			0
$993$	−52153.8	−	52153.8i	−1.66672	−	1.66672i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−28755.5	+	28755.5i	−0.913435	+	0.913435i	−0.996541	−	0.0831060i	$-0.973516\pi$
$997$	−28755.5	+	28755.5i	−0.913435	+	0.913435i	0.0831060	+	0.996541i	$0.473516\pi$
$998$		0			0
$999$	84046.8			2.66179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.n.g.143.7	yes	16
4.3	odd	2	inner	400.4.n.g.143.2	yes	16
5.2	odd	4	inner	400.4.n.g.207.1	yes	16
5.3	odd	4	inner	400.4.n.g.207.7	yes	16
5.4	even	2	inner	400.4.n.g.143.1	✓	16
20.3	even	4	inner	400.4.n.g.207.2	yes	16
20.7	even	4	inner	400.4.n.g.207.8	yes	16
20.19	odd	2	inner	400.4.n.g.143.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.4.n.g.143.1	✓	16	5.4	even	2	inner
400.4.n.g.143.2	yes	16	4.3	odd	2	inner
400.4.n.g.143.7	yes	16	1.1	even	1	trivial
400.4.n.g.143.8	yes	16	20.19	odd	2	inner
400.4.n.g.207.1	yes	16	5.2	odd	4	inner
400.4.n.g.207.2	yes	16	20.3	even	4	inner
400.4.n.g.207.7	yes	16	5.3	odd	4	inner
400.4.n.g.207.8	yes	16	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)

\(f(q)\)	\(=\)	\(q+(6.59346 - 6.59346i) q^{3} +(-17.4296 - 17.4296i) q^{7} -59.9473i q^{9} +O(q^{10})\)	\(q+(6.59346 - 6.59346i) q^{3} +(-17.4296 - 17.4296i) q^{7} -59.9473i q^{9} -1.16990i q^{11} +(32.6384 + 32.6384i) q^{13} +(90.9319 - 90.9319i) q^{17} -131.499 q^{19} -229.842 q^{21} +(-98.9577 + 98.9577i) q^{23} +(-217.237 - 217.237i) q^{27} -167.842i q^{29} -60.1965i q^{31} +(-7.71371 - 7.71371i) q^{33} +(-193.445 + 193.445i) q^{37} +430.399 q^{39} +28.7893 q^{41} +(76.6031 - 76.6031i) q^{43} +(-176.926 - 176.926i) q^{47} +264.579i q^{49} -1199.11i q^{51} +(327.372 + 327.372i) q^{53} +(-867.033 + 867.033i) q^{57} -141.846 q^{59} +369.315 q^{61} +(-1044.86 + 1044.86i) q^{63} +(67.8385 + 67.8385i) q^{67} +1304.95i q^{69} +46.7346i q^{71} +(-141.641 - 141.641i) q^{73} +(-20.3909 + 20.3909i) q^{77} +1032.70 q^{79} -1246.10 q^{81} +(498.417 - 498.417i) q^{83} +(-1106.66 - 1106.66i) q^{87} -1534.00i q^{89} -1137.74i q^{91} +(-396.903 - 396.903i) q^{93} +(1.20296 - 1.20296i) q^{97} -70.1327 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 1856 q^{21} - 1968 q^{41} - 1984 q^{61} - 9616 q^{81}+O(q^{100}) \) 16 * q - 1856 * q^21 - 1968 * q^41 - 1984 * q^61 - 9616 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more