LMFDB - Embedded newform 1400.2.g.j.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(449,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1400.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	1400.449
Dual form			1400.2.g.j.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} +O(q^{10})$	$q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} -6.12311 q^{11} -2.00000i q^{13} +1.56155i q^{17} -3.56155 q^{19} -1.56155 q^{21} -1.43845i q^{23} +5.56155i q^{27} -3.43845 q^{29} -9.12311 q^{31} -9.56155i q^{33} +8.80776i q^{37} +3.12311 q^{39} -2.43845 q^{41} -6.56155i q^{43} -8.24621i q^{47} -1.00000 q^{49} -2.43845 q^{51} +1.12311i q^{53} -5.56155i q^{57} -11.3693 q^{59} +11.1231 q^{61} +0.561553i q^{63} +7.87689i q^{67} +2.24621 q^{69} +1.68466 q^{71} -6.43845i q^{73} -6.12311i q^{77} +5.68466 q^{79} -7.00000 q^{81} +1.31534i q^{83} -5.36932i q^{87} -9.80776 q^{89} +2.00000 q^{91} -14.2462i q^{93} +6.00000i q^{97} -3.43845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} - 8 q^{11} - 6 q^{19} + 2 q^{21} - 22 q^{29} - 20 q^{31} - 4 q^{39} - 18 q^{41} - 4 q^{49} - 18 q^{51} + 4 q^{59} + 28 q^{61} - 24 q^{69} - 18 q^{71} - 2 q^{79} - 28 q^{81} + 2 q^{89} + 8 q^{91} - 22 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 8 * q^11 - 6 * q^19 + 2 * q^21 - 22 * q^29 - 20 * q^31 - 4 * q^39 - 18 * q^41 - 4 * q^49 - 18 * q^51 + 4 * q^59 + 28 * q^61 - 24 * q^69 - 18 * q^71 - 2 * q^79 - 28 * q^81 + 2 * q^89 + 8 * q^91 - 22 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	−6.12311	−1.84619	−0.923093	−	0.384577i	$-0.874347\pi$
$11$	−6.12311	−1.84619	−0.923093	+	0.384577i	$0.874347\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.56155i	0.378732i	0.981907	+	0.189366i	$0.0606433\pi$
$17$	1.56155i	0.378732i	−0.981907	+	0.189366i	$0.939357\pi$
$18$	0	0
$19$	−3.56155	−0.817076	−0.408538	−	0.912741i	$-0.633961\pi$
$19$	−3.56155	−0.817076	−0.408538	+	0.912741i	$0.633961\pi$
$20$	0	0
$21$	−1.56155	−0.340759
$22$	0	0
$23$	− 1.43845i	− 0.299937i	−0.988691	−	0.149968i	$-0.952083\pi$
$23$	− 1.43845i	− 0.299937i	0.988691	−	0.149968i	$-0.0479172\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	−3.43845	−0.638504	−0.319252	−	0.947670i	$-0.603432\pi$
$29$	−3.43845	−0.638504	−0.319252	+	0.947670i	$0.603432\pi$
$30$	0	0
$31$	−9.12311	−1.63856	−0.819279	−	0.573395i	$-0.805626\pi$
$31$	−9.12311	−1.63856	−0.819279	+	0.573395i	$0.805626\pi$
$32$	0	0
$33$	− 9.56155i	− 1.66445i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.80776i	1.44799i	0.689807	+	0.723994i	$0.257696\pi$
$37$	8.80776i	1.44799i	−0.689807	+	0.723994i	$0.742304\pi$
$38$	0	0
$39$	3.12311	0.500097
$40$	0	0
$41$	−2.43845	−0.380821	−0.190411	−	0.981705i	$-0.560982\pi$
$41$	−2.43845	−0.380821	−0.190411	+	0.981705i	$0.560982\pi$
$42$	0	0
$43$	− 6.56155i	− 1.00063i	−0.865844	−	0.500314i	$-0.833218\pi$
$43$	− 6.56155i	− 1.00063i	0.865844	−	0.500314i	$-0.166782\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.24621i	− 1.20283i	−0.798935	−	0.601417i	$-0.794603\pi$
$47$	− 8.24621i	− 1.20283i	0.798935	−	0.601417i	$-0.205397\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−2.43845	−0.341451
$52$	0	0
$53$	1.12311i	0.154270i	0.997021	+	0.0771352i	$0.0245773\pi$
$53$	1.12311i	0.154270i	−0.997021	+	0.0771352i	$0.975423\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.56155i	− 0.736646i
$58$	0	0
$59$	−11.3693	−1.48016	−0.740079	−	0.672519i	$-0.765212\pi$
$59$	−11.3693	−1.48016	−0.740079	+	0.672519i	$0.765212\pi$
$60$	0	0
$61$	11.1231	1.42417	0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231	1.42417	0.712084	+	0.702094i	$0.247752\pi$
$62$	0	0
$63$	0.561553i	0.0707490i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.87689i	0.962316i	0.876634	+	0.481158i	$0.159784\pi$
$67$	7.87689i	0.962316i	−0.876634	+	0.481158i	$0.840216\pi$
$68$	0	0
$69$	2.24621	0.270412
$70$	0	0
$71$	1.68466	0.199932	0.0999661	−	0.994991i	$-0.468127\pi$
$71$	1.68466	0.199932	0.0999661	+	0.994991i	$0.468127\pi$
$72$	0	0
$73$	− 6.43845i	− 0.753563i	−0.926302	−	0.376782i	$-0.877031\pi$
$73$	− 6.43845i	− 0.753563i	0.926302	−	0.376782i	$-0.122969\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.12311i	− 0.697793i
$78$	0	0
$79$	5.68466	0.639574	0.319787	−	0.947489i	$-0.396389\pi$
$79$	5.68466	0.639574	0.319787	+	0.947489i	$0.396389\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	1.31534i	0.144377i	0.997391	+	0.0721887i	$0.0229984\pi$
$83$	1.31534i	0.144377i	−0.997391	+	0.0721887i	$0.977002\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.36932i	− 0.575651i
$88$	0	0
$89$	−9.80776	−1.03962	−0.519810	−	0.854282i	$-0.673997\pi$
$89$	−9.80776	−1.03962	−0.519810	+	0.854282i	$0.673997\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	− 14.2462i	− 1.47726i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	0	0
$99$	−3.43845	−0.345577
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	1.12311i	0.110663i	0.998468	+	0.0553314i	$0.0176215\pi$
$103$	1.12311i	0.110663i	−0.998468	+	0.0553314i	$0.982378\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.80776i	0.754805i	0.926049	+	0.377403i	$0.123183\pi$
$107$	7.80776i	0.754805i	−0.926049	+	0.377403i	$0.876817\pi$
$108$	0	0
$109$	−19.9309	−1.90903	−0.954516	−	0.298161i	$-0.903627\pi$
$109$	−19.9309	−1.90903	−0.954516	+	0.298161i	$0.903627\pi$
$110$	0	0
$111$	−13.7538	−1.30545
$112$	0	0
$113$	− 17.2462i	− 1.62239i	−0.584778	−	0.811194i	$-0.698818\pi$
$113$	− 17.2462i	− 1.62239i	0.584778	−	0.811194i	$-0.301182\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.12311i	− 0.103831i
$118$	0	0
$119$	−1.56155	−0.143147
$120$	0	0
$121$	26.4924	2.40840
$122$	0	0
$123$	− 3.80776i	− 0.343335i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.8078i	0.959034i	0.877533	+	0.479517i	$0.159188\pi$
$127$	10.8078i	0.959034i	−0.877533	+	0.479517i	$0.840812\pi$
$128$	0	0
$129$	10.2462	0.902129
$130$	0	0
$131$	−18.2462	−1.59418	−0.797089	−	0.603861i	$-0.793628\pi$
$131$	−18.2462	−1.59418	−0.797089	+	0.603861i	$0.793628\pi$
$132$	0	0
$133$	− 3.56155i	− 0.308826i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.6847i	− 1.25460i	−0.778780	−	0.627298i	$-0.784161\pi$
$137$	− 14.6847i	− 1.25460i	0.778780	−	0.627298i	$-0.215839\pi$
$138$	0	0
$139$	7.31534	0.620479	0.310240	−	0.950658i	$-0.399591\pi$
$139$	7.31534	0.620479	0.310240	+	0.950658i	$0.399591\pi$
$140$	0	0
$141$	12.8769	1.08443
$142$	0	0
$143$	12.2462i	1.02408i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.56155i	− 0.128795i
$148$	0	0
$149$	−13.4384	−1.10092	−0.550460	−	0.834861i	$-0.685548\pi$
$149$	−13.4384	−1.10092	−0.550460	+	0.834861i	$0.685548\pi$
$150$	0	0
$151$	−1.43845	−0.117059	−0.0585296	−	0.998286i	$-0.518641\pi$
$151$	−1.43845	−0.117059	−0.0585296	+	0.998286i	$0.518641\pi$
$152$	0	0
$153$	0.876894i	0.0708927i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.49242i	− 0.677769i	−0.940828	−	0.338885i	$-0.889950\pi$
$157$	− 8.49242i	− 0.677769i	0.940828	−	0.338885i	$-0.110050\pi$
$158$	0	0
$159$	−1.75379	−0.139084
$160$	0	0
$161$	1.43845	0.113366
$162$	0	0
$163$	− 14.9309i	− 1.16948i	−0.811222	−	0.584738i	$-0.801197\pi$
$163$	− 14.9309i	− 1.16948i	0.811222	−	0.584738i	$-0.198803\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.6155i	1.67266i	0.548227	+	0.836330i	$0.315303\pi$
$167$	21.6155i	1.67266i	−0.548227	+	0.836330i	$0.684697\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	18.4924i	1.40595i	0.711213	+	0.702976i	$0.248146\pi$
$173$	18.4924i	1.40595i	−0.711213	+	0.702976i	$0.751854\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 17.7538i	− 1.33446i
$178$	0	0
$179$	20.6847	1.54604	0.773022	−	0.634379i	$-0.218744\pi$
$179$	20.6847	1.54604	0.773022	+	0.634379i	$0.218744\pi$
$180$	0	0
$181$	−11.1231	−0.826774	−0.413387	−	0.910555i	$-0.635654\pi$
$181$	−11.1231	−0.826774	−0.413387	+	0.910555i	$0.635654\pi$
$182$	0	0
$183$	17.3693i	1.28398i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 9.56155i	− 0.699210i
$188$	0	0
$189$	−5.56155	−0.404543
$190$	0	0
$191$	25.3693	1.83566	0.917830	−	0.396974i	$-0.129940\pi$
$191$	25.3693	1.83566	0.917830	+	0.396974i	$0.129940\pi$
$192$	0	0
$193$	20.6155i	1.48394i	0.670434	+	0.741969i	$0.266108\pi$
$193$	20.6155i	1.48394i	−0.670434	+	0.741969i	$0.733892\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.561553i	0.0400090i	0.999800	+	0.0200045i	$0.00636805\pi$
$197$	0.561553i	0.0400090i	−0.999800	+	0.0200045i	$0.993632\pi$
$198$	0	0
$199$	4.87689	0.345714	0.172857	−	0.984947i	$-0.444700\pi$
$199$	4.87689	0.345714	0.172857	+	0.984947i	$0.444700\pi$
$200$	0	0
$201$	−12.3002	−0.867588
$202$	0	0
$203$	− 3.43845i	− 0.241332i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.807764i	− 0.0561435i
$208$	0	0
$209$	21.8078	1.50847
$210$	0	0
$211$	6.43845	0.443241	0.221620	−	0.975133i	$-0.428865\pi$
$211$	6.43845	0.443241	0.221620	+	0.975133i	$0.428865\pi$
$212$	0	0
$213$	2.63068i	0.180251i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 9.12311i	− 0.619317i
$218$	0	0
$219$	10.0540	0.679385
$220$	0	0
$221$	3.12311	0.210083
$222$	0	0
$223$	25.3693i	1.69886i	0.527705	+	0.849428i	$0.323053\pi$
$223$	25.3693i	1.69886i	−0.527705	+	0.849428i	$0.676947\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.3693i	0.754608i	0.926089	+	0.377304i	$0.123149\pi$
$227$	11.3693i	0.754608i	−0.926089	+	0.377304i	$0.876851\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	9.56155	0.629104
$232$	0	0
$233$	0.561553i	0.0367885i	0.999831	+	0.0183943i	$0.00585541\pi$
$233$	0.561553i	0.0367885i	−0.999831	+	0.0183943i	$0.994145\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.87689i	0.576616i
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−12.0540	−0.776465	−0.388232	−	0.921562i	$-0.626914\pi$
$241$	−12.0540	−0.776465	−0.388232	+	0.921562i	$0.626914\pi$
$242$	0	0
$243$	5.75379i	0.369106i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.12311i	0.453232i
$248$	0	0
$249$	−2.05398	−0.130165
$250$	0	0
$251$	−24.6847	−1.55808	−0.779041	−	0.626973i	$-0.784294\pi$
$251$	−24.6847	−1.55808	−0.779041	+	0.626973i	$0.784294\pi$
$252$	0	0
$253$	8.80776i	0.553739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.2462i	0.763898i	0.924183	+	0.381949i	$0.124747\pi$
$257$	12.2462i	0.763898i	−0.924183	+	0.381949i	$0.875253\pi$
$258$	0	0
$259$	−8.80776	−0.547288
$260$	0	0
$261$	−1.93087	−0.119518
$262$	0	0
$263$	− 1.68466i	− 0.103880i	−0.998650	−	0.0519402i	$-0.983459\pi$
$263$	− 1.68466i	− 0.103880i	0.998650	−	0.0519402i	$-0.0165405\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 15.3153i	− 0.937284i
$268$	0	0
$269$	−28.2462	−1.72220	−0.861101	−	0.508434i	$-0.830225\pi$
$269$	−28.2462	−1.72220	−0.861101	+	0.508434i	$0.830225\pi$
$270$	0	0
$271$	−0.246211	−0.0149563	−0.00747813	−	0.999972i	$-0.502380\pi$
$271$	−0.246211	−0.0149563	−0.00747813	+	0.999972i	$0.502380\pi$
$272$	0	0
$273$	3.12311i	0.189019i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.4924i	1.35144i	0.737159	+	0.675719i	$0.236167\pi$
$277$	22.4924i	1.35144i	−0.737159	+	0.675719i	$0.763833\pi$
$278$	0	0
$279$	−5.12311	−0.306712
$280$	0	0
$281$	7.43845	0.443741	0.221870	−	0.975076i	$-0.428784\pi$
$281$	7.43845	0.443741	0.221870	+	0.975076i	$0.428784\pi$
$282$	0	0
$283$	1.31534i	0.0781889i	0.999236	+	0.0390945i	$0.0124473\pi$
$283$	1.31534i	0.0781889i	−0.999236	+	0.0390945i	$0.987553\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.43845i	− 0.143937i
$288$	0	0
$289$	14.5616	0.856562
$290$	0	0
$291$	−9.36932	−0.549239
$292$	0	0
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 34.0540i	− 1.97601i
$298$	0	0
$299$	−2.87689	−0.166375
$300$	0	0
$301$	6.56155	0.378202
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.6847i	0.952244i	0.879379	+	0.476122i	$0.157958\pi$
$307$	16.6847i	0.952244i	−0.879379	+	0.476122i	$0.842042\pi$
$308$	0	0
$309$	−1.75379	−0.0997696
$310$	0	0
$311$	11.1231	0.630733	0.315367	−	0.948970i	$-0.397872\pi$
$311$	11.1231	0.630733	0.315367	+	0.948970i	$0.397872\pi$
$312$	0	0
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.31534i	− 0.467036i	−0.972352	−	0.233518i	$-0.924976\pi$
$317$	− 8.31534i	− 0.467036i	0.972352	−	0.233518i	$-0.0750238\pi$
$318$	0	0
$319$	21.0540	1.17880
$320$	0	0
$321$	−12.1922	−0.680504
$322$	0	0
$323$	− 5.56155i	− 0.309453i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 31.1231i	− 1.72111i
$328$	0	0
$329$	8.24621	0.454628
$330$	0	0
$331$	27.0000	1.48405	0.742027	−	0.670370i	$-0.233865\pi$
$331$	27.0000	1.48405	0.742027	+	0.670370i	$0.233865\pi$
$332$	0	0
$333$	4.94602i	0.271040i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.68466i	0.364137i	0.983286	+	0.182068i	$0.0582792\pi$
$337$	6.68466i	0.364137i	−0.983286	+	0.182068i	$0.941721\pi$
$338$	0	0
$339$	26.9309	1.46268
$340$	0	0
$341$	55.8617	3.02508
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.6155i	− 0.999334i	−0.866218	−	0.499667i	$-0.833456\pi$
$347$	− 18.6155i	− 0.999334i	0.866218	−	0.499667i	$-0.166544\pi$
$348$	0	0
$349$	21.1231	1.13069	0.565347	−	0.824853i	$-0.308742\pi$
$349$	21.1231	1.13069	0.565347	+	0.824853i	$0.308742\pi$
$350$	0	0
$351$	11.1231	0.593707
$352$	0	0
$353$	28.2462i	1.50339i	0.659509	+	0.751697i	$0.270764\pi$
$353$	28.2462i	1.50339i	−0.659509	+	0.751697i	$0.729236\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.43845i	− 0.129056i
$358$	0	0
$359$	2.31534	0.122199	0.0610995	−	0.998132i	$-0.480539\pi$
$359$	2.31534	0.122199	0.0610995	+	0.998132i	$0.480539\pi$
$360$	0	0
$361$	−6.31534	−0.332386
$362$	0	0
$363$	41.3693i	2.17133i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	34.2462i	1.78764i	0.448428	+	0.893819i	$0.351984\pi$
$367$	34.2462i	1.78764i	−0.448428	+	0.893819i	$0.648016\pi$
$368$	0	0
$369$	−1.36932	−0.0712838
$370$	0	0
$371$	−1.12311	−0.0583087
$372$	0	0
$373$	23.0540i	1.19369i	0.802357	+	0.596845i	$0.203579\pi$
$373$	23.0540i	1.19369i	−0.802357	+	0.596845i	$0.796421\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.87689i	0.354178i
$378$	0	0
$379$	−32.8617	−1.68799	−0.843997	−	0.536348i	$-0.819804\pi$
$379$	−32.8617	−1.68799	−0.843997	+	0.536348i	$0.819804\pi$
$380$	0	0
$381$	−16.8769	−0.864629
$382$	0	0
$383$	3.75379i	0.191810i	0.995391	+	0.0959048i	$0.0305744\pi$
$383$	3.75379i	0.191810i	−0.995391	+	0.0959048i	$0.969426\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3.68466i	− 0.187302i
$388$	0	0
$389$	−6.56155	−0.332684	−0.166342	−	0.986068i	$-0.553196\pi$
$389$	−6.56155	−0.332684	−0.166342	+	0.986068i	$0.553196\pi$
$390$	0	0
$391$	2.24621	0.113596
$392$	0	0
$393$	− 28.4924i	− 1.43725i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.50758i	0.276417i	0.990403	+	0.138209i	$0.0441345\pi$
$397$	5.50758i	0.276417i	−0.990403	+	0.138209i	$0.955866\pi$
$398$	0	0
$399$	5.56155	0.278426
$400$	0	0
$401$	1.49242	0.0745280	0.0372640	−	0.999305i	$-0.488136\pi$
$401$	1.49242	0.0745280	0.0372640	+	0.999305i	$0.488136\pi$
$402$	0	0
$403$	18.2462i	0.908909i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 53.9309i	− 2.67325i
$408$	0	0
$409$	25.8078	1.27611	0.638056	−	0.769990i	$-0.279739\pi$
$409$	25.8078	1.27611	0.638056	+	0.769990i	$0.279739\pi$
$410$	0	0
$411$	22.9309	1.13110
$412$	0	0
$413$	− 11.3693i	− 0.559448i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.4233i	0.559401i
$418$	0	0
$419$	−1.80776	−0.0883151	−0.0441575	−	0.999025i	$-0.514060\pi$
$419$	−1.80776	−0.0883151	−0.0441575	+	0.999025i	$0.514060\pi$
$420$	0	0
$421$	33.9309	1.65369	0.826845	−	0.562430i	$-0.190134\pi$
$421$	33.9309	1.65369	0.826845	+	0.562430i	$0.190134\pi$
$422$	0	0
$423$	− 4.63068i	− 0.225152i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.1231i	0.538285i
$428$	0	0
$429$	−19.1231	−0.923272
$430$	0	0
$431$	−30.2462	−1.45691	−0.728454	−	0.685094i	$-0.759761\pi$
$431$	−30.2462	−1.45691	−0.728454	+	0.685094i	$0.759761\pi$
$432$	0	0
$433$	− 34.5464i	− 1.66019i	−0.557619	−	0.830097i	$-0.688285\pi$
$433$	− 34.5464i	− 1.66019i	0.557619	−	0.830097i	$-0.311715\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.12311i	0.245071i
$438$	0	0
$439$	−1.12311	−0.0536029	−0.0268015	−	0.999641i	$-0.508532\pi$
$439$	−1.12311	−0.0536029	−0.0268015	+	0.999641i	$0.508532\pi$
$440$	0	0
$441$	−0.561553	−0.0267406
$442$	0	0
$443$	− 18.4384i	− 0.876037i	−0.898966	−	0.438019i	$-0.855680\pi$
$443$	− 18.4384i	− 0.876037i	0.898966	−	0.438019i	$-0.144320\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 20.9848i	− 0.992549i
$448$	0	0
$449$	14.3693	0.678130	0.339065	−	0.940763i	$-0.389889\pi$
$449$	14.3693	0.678130	0.339065	+	0.940763i	$0.389889\pi$
$450$	0	0
$451$	14.9309	0.703067
$452$	0	0
$453$	− 2.24621i	− 0.105536i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 37.1080i	− 1.73584i	−0.496707	−	0.867918i	$-0.665458\pi$
$457$	− 37.1080i	− 1.73584i	0.496707	−	0.867918i	$-0.334542\pi$
$458$	0	0
$459$	−8.68466	−0.405365
$460$	0	0
$461$	−26.4924	−1.23388	−0.616938	−	0.787012i	$-0.711627\pi$
$461$	−26.4924	−1.23388	−0.616938	+	0.787012i	$0.711627\pi$
$462$	0	0
$463$	− 12.4924i	− 0.580572i	−0.956940	−	0.290286i	$-0.906250\pi$
$463$	− 12.4924i	− 0.580572i	0.956940	−	0.290286i	$-0.0937505\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 33.1231i	− 1.53275i	−0.642391	−	0.766377i	$-0.722057\pi$
$467$	− 33.1231i	− 1.53275i	0.642391	−	0.766377i	$-0.277943\pi$
$468$	0	0
$469$	−7.87689	−0.363721
$470$	0	0
$471$	13.2614	0.611052
$472$	0	0
$473$	40.1771i	1.84734i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.630683i	0.0288770i
$478$	0	0
$479$	−8.73863	−0.399278	−0.199639	−	0.979869i	$-0.563977\pi$
$479$	−8.73863	−0.399278	−0.199639	+	0.979869i	$0.563977\pi$
$480$	0	0
$481$	17.6155	0.803199
$482$	0	0
$483$	2.24621i	0.102206i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.4384i	1.42461i	0.701869	+	0.712306i	$0.252349\pi$
$487$	31.4384i	1.42461i	−0.701869	+	0.712306i	$0.747651\pi$
$488$	0	0
$489$	23.3153	1.05436
$490$	0	0
$491$	8.31534	0.375266	0.187633	−	0.982239i	$-0.439918\pi$
$491$	8.31534	0.375266	0.187633	+	0.982239i	$0.439918\pi$
$492$	0	0
$493$	− 5.36932i	− 0.241822i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.68466i	0.0755673i
$498$	0	0
$499$	−16.4924	−0.738302	−0.369151	−	0.929369i	$-0.620352\pi$
$499$	−16.4924	−0.738302	−0.369151	+	0.929369i	$0.620352\pi$
$500$	0	0
$501$	−33.7538	−1.50801
$502$	0	0
$503$	− 36.2462i	− 1.61614i	−0.589087	−	0.808069i	$-0.700513\pi$
$503$	− 36.2462i	− 1.61614i	0.589087	−	0.808069i	$-0.299487\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.0540i	0.624159i
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	6.43845	0.284820
$512$	0	0
$513$	− 19.8078i	− 0.874534i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	50.4924i	2.22065i
$518$	0	0
$519$	−28.8769	−1.26755
$520$	0	0
$521$	−31.5616	−1.38274	−0.691368	−	0.722502i	$-0.742992\pi$
$521$	−31.5616	−1.38274	−0.691368	+	0.722502i	$0.742992\pi$
$522$	0	0
$523$	− 40.6847i	− 1.77902i	−0.456920	−	0.889508i	$-0.651047\pi$
$523$	− 40.6847i	− 1.77902i	0.456920	−	0.889508i	$-0.348953\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 14.2462i	− 0.620575i
$528$	0	0
$529$	20.9309	0.910038
$530$	0	0
$531$	−6.38447	−0.277062
$532$	0	0
$533$	4.87689i	0.211242i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	32.3002i	1.39386i
$538$	0	0
$539$	6.12311	0.263741
$540$	0	0
$541$	−20.4233	−0.878066	−0.439033	−	0.898471i	$-0.644679\pi$
$541$	−20.4233	−0.878066	−0.439033	+	0.898471i	$0.644679\pi$
$542$	0	0
$543$	− 17.3693i	− 0.745389i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15.7386i	− 0.672935i	−0.941695	−	0.336468i	$-0.890768\pi$
$547$	− 15.7386i	− 0.672935i	0.941695	−	0.336468i	$-0.109232\pi$
$548$	0	0
$549$	6.24621	0.266582
$550$	0	0
$551$	12.2462	0.521706
$552$	0	0
$553$	5.68466i	0.241736i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.43845i	0.230434i	0.993340	+	0.115217i	$0.0367564\pi$
$557$	5.43845i	0.230434i	−0.993340	+	0.115217i	$0.963244\pi$
$558$	0	0
$559$	−13.1231	−0.555048
$560$	0	0
$561$	14.9309	0.630382
$562$	0	0
$563$	− 1.12311i	− 0.0473333i	−0.999720	−	0.0236666i	$-0.992466\pi$
$563$	− 1.12311i	− 0.0473333i	0.999720	−	0.0236666i	$-0.00753403\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 7.00000i	− 0.293972i
$568$	0	0
$569$	−45.9848	−1.92778	−0.963892	−	0.266292i	$-0.914202\pi$
$569$	−45.9848	−1.92778	−0.963892	+	0.266292i	$0.914202\pi$
$570$	0	0
$571$	39.6847	1.66075	0.830376	−	0.557204i	$-0.188126\pi$
$571$	39.6847	1.66075	0.830376	+	0.557204i	$0.188126\pi$
$572$	0	0
$573$	39.6155i	1.65496i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.5464i	1.02188i	0.859616	+	0.510940i	$0.170703\pi$
$577$	24.5464i	1.02188i	−0.859616	+	0.510940i	$0.829297\pi$
$578$	0	0
$579$	−32.1922	−1.33786
$580$	0	0
$581$	−1.31534	−0.0545696
$582$	0	0
$583$	− 6.87689i	− 0.284812i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 37.1771i	− 1.53446i	−0.641371	−	0.767231i	$-0.721634\pi$
$587$	− 37.1771i	− 1.53446i	0.641371	−	0.767231i	$-0.278366\pi$
$588$	0	0
$589$	32.4924	1.33883
$590$	0	0
$591$	−0.876894	−0.0360706
$592$	0	0
$593$	− 24.6847i	− 1.01368i	−0.862041	−	0.506839i	$-0.830814\pi$
$593$	− 24.6847i	− 1.01368i	0.862041	−	0.506839i	$-0.169186\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.61553i	0.311683i
$598$	0	0
$599$	29.3002	1.19717	0.598587	−	0.801058i	$-0.295729\pi$
$599$	29.3002	1.19717	0.598587	+	0.801058i	$0.295729\pi$
$600$	0	0
$601$	−42.0540	−1.71542	−0.857709	−	0.514136i	$-0.828113\pi$
$601$	−42.0540	−1.71542	−0.857709	+	0.514136i	$0.828113\pi$
$602$	0	0
$603$	4.42329i	0.180130i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 12.6307i	− 0.512664i	−0.966589	−	0.256332i	$-0.917486\pi$
$607$	− 12.6307i	− 0.512664i	0.966589	−	0.256332i	$-0.0825140\pi$
$608$	0	0
$609$	5.36932	0.217576
$610$	0	0
$611$	−16.4924	−0.667212
$612$	0	0
$613$	− 5.68466i	− 0.229601i	−0.993389	−	0.114801i	$-0.963377\pi$
$613$	− 5.68466i	− 0.229601i	0.993389	−	0.114801i	$-0.0366229\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.80776i	− 0.113036i	−0.998402	−	0.0565182i	$-0.982000\pi$
$617$	− 2.80776i	− 0.113036i	0.998402	−	0.0565182i	$-0.0179999\pi$
$618$	0	0
$619$	−4.63068	−0.186123	−0.0930614	−	0.995660i	$-0.529665\pi$
$619$	−4.63068	−0.186123	−0.0930614	+	0.995660i	$0.529665\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	− 9.80776i	− 0.392940i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	34.0540i	1.35998i
$628$	0	0
$629$	−13.7538	−0.548399
$630$	0	0
$631$	−3.05398	−0.121577	−0.0607884	−	0.998151i	$-0.519361\pi$
$631$	−3.05398	−0.121577	−0.0607884	+	0.998151i	$0.519361\pi$
$632$	0	0
$633$	10.0540i	0.399610i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	0.946025	0.0374242
$640$	0	0
$641$	−20.5616	−0.812133	−0.406066	−	0.913844i	$-0.633100\pi$
$641$	−20.5616	−0.812133	−0.406066	+	0.913844i	$0.633100\pi$
$642$	0	0
$643$	46.7386i	1.84319i	0.388151	+	0.921596i	$0.373114\pi$
$643$	46.7386i	1.84319i	−0.388151	+	0.921596i	$0.626886\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 0.630683i	− 0.0247947i	−0.999923	−	0.0123974i	$-0.996054\pi$
$647$	− 0.630683i	− 0.0247947i	0.999923	−	0.0123974i	$-0.00394630\pi$
$648$	0	0
$649$	69.6155	2.73265
$650$	0	0
$651$	14.2462	0.558353
$652$	0	0
$653$	23.8617i	0.933782i	0.884315	+	0.466891i	$0.154626\pi$
$653$	23.8617i	0.933782i	−0.884315	+	0.466891i	$0.845374\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.61553i	− 0.141055i
$658$	0	0
$659$	−14.4384	−0.562442	−0.281221	−	0.959643i	$-0.590739\pi$
$659$	−14.4384	−0.562442	−0.281221	+	0.959643i	$0.590739\pi$
$660$	0	0
$661$	10.4924	0.408108	0.204054	−	0.978960i	$-0.434588\pi$
$661$	10.4924	0.408108	0.204054	+	0.978960i	$0.434588\pi$
$662$	0	0
$663$	4.87689i	0.189403i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.94602i	0.191511i
$668$	0	0
$669$	−39.6155	−1.53162
$670$	0	0
$671$	−68.1080	−2.62928
$672$	0	0
$673$	− 13.5076i	− 0.520679i	−0.965517	−	0.260339i	$-0.916166\pi$
$673$	− 13.5076i	− 0.520679i	0.965517	−	0.260339i	$-0.0838345\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.3693i	1.51309i	0.653944	+	0.756543i	$0.273113\pi$
$677$	39.3693i	1.51309i	−0.653944	+	0.756543i	$0.726887\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−17.7538	−0.680327
$682$	0	0
$683$	− 25.8769i	− 0.990152i	−0.868850	−	0.495076i	$-0.835140\pi$
$683$	− 25.8769i	− 0.990152i	0.868850	−	0.495076i	$-0.164860\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.8617i	0.834077i
$688$	0	0
$689$	2.24621	0.0855738
$690$	0	0
$691$	−21.5616	−0.820240	−0.410120	−	0.912032i	$-0.634513\pi$
$691$	−21.5616	−0.820240	−0.410120	+	0.912032i	$0.634513\pi$
$692$	0	0
$693$	− 3.43845i	− 0.130616i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3.80776i	− 0.144229i
$698$	0	0
$699$	−0.876894	−0.0331672
$700$	0	0
$701$	−30.1080	−1.13716	−0.568581	−	0.822627i	$-0.692507\pi$
$701$	−30.1080	−1.13716	−0.568581	+	0.822627i	$0.692507\pi$
$702$	0	0
$703$	− 31.3693i	− 1.18312i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−12.6307	−0.474355	−0.237178	−	0.971466i	$-0.576222\pi$
$709$	−12.6307	−0.474355	−0.237178	+	0.971466i	$0.576222\pi$
$710$	0	0
$711$	3.19224	0.119718
$712$	0	0
$713$	13.1231i	0.491464i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.24621i	0.233269i
$718$	0	0
$719$	−21.7538	−0.811279	−0.405640	−	0.914033i	$-0.632951\pi$
$719$	−21.7538	−0.811279	−0.405640	+	0.914033i	$0.632951\pi$
$720$	0	0
$721$	−1.12311	−0.0418266
$722$	0	0
$723$	− 18.8229i	− 0.700032i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	39.6155i	1.46926i	0.678469	+	0.734629i	$0.262644\pi$
$727$	39.6155i	1.46926i	−0.678469	+	0.734629i	$0.737356\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	10.2462	0.378970
$732$	0	0
$733$	− 5.75379i	− 0.212521i	−0.994338	−	0.106261i	$-0.966112\pi$
$733$	− 5.75379i	− 0.212521i	0.994338	−	0.106261i	$-0.0338878\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 48.2311i	− 1.77661i
$738$	0	0
$739$	28.1771	1.03651	0.518255	−	0.855226i	$-0.326582\pi$
$739$	28.1771	1.03651	0.518255	+	0.855226i	$0.326582\pi$
$740$	0	0
$741$	−11.1231	−0.408617
$742$	0	0
$743$	− 30.7386i	− 1.12769i	−0.825880	−	0.563846i	$-0.809321\pi$
$743$	− 30.7386i	− 1.12769i	0.825880	−	0.563846i	$-0.190679\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.738634i	0.0270252i
$748$	0	0
$749$	−7.80776	−0.285289
$750$	0	0
$751$	2.63068	0.0959950	0.0479975	−	0.998847i	$-0.484716\pi$
$751$	2.63068	0.0959950	0.0479975	+	0.998847i	$0.484716\pi$
$752$	0	0
$753$	− 38.5464i	− 1.40471i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.6847i	0.933525i	0.884383	+	0.466762i	$0.154580\pi$
$757$	25.6847i	0.933525i	−0.884383	+	0.466762i	$0.845420\pi$
$758$	0	0
$759$	−13.7538	−0.499231
$760$	0	0
$761$	−27.4233	−0.994094	−0.497047	−	0.867724i	$-0.665582\pi$
$761$	−27.4233	−0.994094	−0.497047	+	0.867724i	$0.665582\pi$
$762$	0	0
$763$	− 19.9309i	− 0.721546i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.7386i	0.821044i
$768$	0	0
$769$	−20.3002	−0.732043	−0.366022	−	0.930606i	$-0.619280\pi$
$769$	−20.3002	−0.732043	−0.366022	+	0.930606i	$0.619280\pi$
$770$	0	0
$771$	−19.1231	−0.688702
$772$	0	0
$773$	− 29.6155i	− 1.06520i	−0.846368	−	0.532598i	$-0.821216\pi$
$773$	− 29.6155i	− 1.06520i	0.846368	−	0.532598i	$-0.178784\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 13.7538i	− 0.493414i
$778$	0	0
$779$	8.68466	0.311160
$780$	0	0
$781$	−10.3153	−0.369112
$782$	0	0
$783$	− 19.1231i	− 0.683404i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9.75379i	0.347685i	0.984773	+	0.173843i	$0.0556184\pi$
$787$	9.75379i	0.347685i	−0.984773	+	0.173843i	$0.944382\pi$
$788$	0	0
$789$	2.63068	0.0936548
$790$	0	0
$791$	17.2462	0.613205
$792$	0	0
$793$	− 22.2462i	− 0.789986i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.384472i	0.0136187i	0.999977	+	0.00680935i	$0.00216750\pi$
$797$	0.384472i	0.0136187i	−0.999977	+	0.00680935i	$0.997833\pi$
$798$	0	0
$799$	12.8769	0.455552
$800$	0	0
$801$	−5.50758	−0.194601
$802$	0	0
$803$	39.4233i	1.39122i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 44.1080i	− 1.55267i
$808$	0	0
$809$	12.4233	0.436780	0.218390	−	0.975862i	$-0.429920\pi$
$809$	12.4233	0.436780	0.218390	+	0.975862i	$0.429920\pi$
$810$	0	0
$811$	2.38447	0.0837301	0.0418651	−	0.999123i	$-0.486670\pi$
$811$	2.38447	0.0837301	0.0418651	+	0.999123i	$0.486670\pi$
$812$	0	0
$813$	− 0.384472i	− 0.0134840i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	23.3693i	0.817589i
$818$	0	0
$819$	1.12311	0.0392445
$820$	0	0
$821$	−11.8617	−0.413978	−0.206989	−	0.978343i	$-0.566366\pi$
$821$	−11.8617	−0.413978	−0.206989	+	0.978343i	$0.566366\pi$
$822$	0	0
$823$	− 16.5616i	− 0.577299i	−0.957435	−	0.288650i	$-0.906794\pi$
$823$	− 16.5616i	− 0.577299i	0.957435	−	0.288650i	$-0.0932063\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.24621i	0.251975i	0.992032	+	0.125988i	$0.0402100\pi$
$827$	7.24621i	0.251975i	−0.992032	+	0.125988i	$0.959790\pi$
$828$	0	0
$829$	−14.2462	−0.494791	−0.247396	−	0.968915i	$-0.579575\pi$
$829$	−14.2462	−0.494791	−0.247396	+	0.968915i	$0.579575\pi$
$830$	0	0
$831$	−35.1231	−1.21841
$832$	0	0
$833$	− 1.56155i	− 0.0541046i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 50.7386i	− 1.75378i
$838$	0	0
$839$	−7.12311	−0.245917	−0.122958	−	0.992412i	$-0.539238\pi$
$839$	−7.12311	−0.245917	−0.122958	+	0.992412i	$0.539238\pi$
$840$	0	0
$841$	−17.1771	−0.592313
$842$	0	0
$843$	11.6155i	0.400060i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.4924i	0.910290i
$848$	0	0
$849$	−2.05398	−0.0704923
$850$	0	0
$851$	12.6695	0.434305
$852$	0	0
$853$	− 19.1231i	− 0.654763i	−0.944892	−	0.327381i	$-0.893834\pi$
$853$	− 19.1231i	− 0.654763i	0.944892	−	0.327381i	$-0.106166\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.0540i	1.43654i	0.695766	+	0.718268i	$0.255065\pi$
$857$	42.0540i	1.43654i	−0.695766	+	0.718268i	$0.744935\pi$
$858$	0	0
$859$	11.1771	0.381357	0.190679	−	0.981653i	$-0.438931\pi$
$859$	11.1771	0.381357	0.190679	+	0.981653i	$0.438931\pi$
$860$	0	0
$861$	3.80776	0.129768
$862$	0	0
$863$	3.19224i	0.108665i	0.998523	+	0.0543325i	$0.0173031\pi$
$863$	3.19224i	0.108665i	−0.998523	+	0.0543325i	$0.982697\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	22.7386i	0.772244i
$868$	0	0
$869$	−34.8078	−1.18077
$870$	0	0
$871$	15.7538	0.533797
$872$	0	0
$873$	3.36932i	0.114034i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 17.6155i	− 0.594834i	−0.954748	−	0.297417i	$-0.903875\pi$
$877$	− 17.6155i	− 0.594834i	0.954748	−	0.297417i	$-0.0961252\pi$
$878$	0	0
$879$	−46.8466	−1.58010
$880$	0	0
$881$	8.73863	0.294412	0.147206	−	0.989106i	$-0.452972\pi$
$881$	8.73863	0.294412	0.147206	+	0.989106i	$0.452972\pi$
$882$	0	0
$883$	− 31.4924i	− 1.05980i	−0.848059	−	0.529902i	$-0.822229\pi$
$883$	− 31.4924i	− 1.05980i	0.848059	−	0.529902i	$-0.177771\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 8.38447i	− 0.281523i	−0.990044	−	0.140762i	$-0.955045\pi$
$887$	− 8.38447i	− 0.281523i	0.990044	−	0.140762i	$-0.0449551\pi$
$888$	0	0
$889$	−10.8078	−0.362481
$890$	0	0
$891$	42.8617	1.43592
$892$	0	0
$893$	29.3693i	0.982807i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.49242i	− 0.149998i
$898$	0	0
$899$	31.3693	1.04623
$900$	0	0
$901$	−1.75379	−0.0584272
$902$	0	0
$903$	10.2462i	0.340973i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22.4233	0.742917	0.371458	−	0.928450i	$-0.378858\pi$
$911$	22.4233	0.742917	0.371458	+	0.928450i	$0.378858\pi$
$912$	0	0
$913$	− 8.05398i	− 0.266548i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 18.2462i	− 0.602543i
$918$	0	0
$919$	−38.8078	−1.28015	−0.640075	−	0.768312i	$-0.721097\pi$
$919$	−38.8078	−1.28015	−0.640075	+	0.768312i	$0.721097\pi$
$920$	0	0
$921$	−26.0540	−0.858508
$922$	0	0
$923$	− 3.36932i	− 0.110902i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.630683i	0.0207144i
$928$	0	0
$929$	34.3542	1.12712	0.563562	−	0.826074i	$-0.309431\pi$
$929$	34.3542	1.12712	0.563562	+	0.826074i	$0.309431\pi$
$930$	0	0
$931$	3.56155	0.116725
$932$	0	0
$933$	17.3693i	0.568646i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 32.3002i	− 1.05520i	−0.849493	−	0.527601i	$-0.823092\pi$
$937$	− 32.3002i	− 1.05520i	0.849493	−	0.527601i	$-0.176908\pi$
$938$	0	0
$939$	34.3542	1.12111
$940$	0	0
$941$	23.3693	0.761818	0.380909	−	0.924613i	$-0.375611\pi$
$941$	23.3693	0.761818	0.380909	+	0.924613i	$0.375611\pi$
$942$	0	0
$943$	3.50758i	0.114222i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.4924i	− 0.535932i	−0.963428	−	0.267966i	$-0.913649\pi$
$947$	− 16.4924i	− 0.535932i	0.963428	−	0.267966i	$-0.0863514\pi$
$948$	0	0
$949$	−12.8769	−0.418002
$950$	0	0
$951$	12.9848	0.421062
$952$	0	0
$953$	36.1231i	1.17014i	0.810982	+	0.585071i	$0.198933\pi$
$953$	36.1231i	1.17014i	−0.810982	+	0.585071i	$0.801067\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.8769i	1.06276i
$958$	0	0
$959$	14.6847	0.474192
$960$	0	0
$961$	52.2311	1.68487
$962$	0	0
$963$	4.38447i	0.141288i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 41.3693i	− 1.33035i	−0.746689	−	0.665174i	$-0.768357\pi$
$967$	− 41.3693i	− 1.33035i	0.746689	−	0.665174i	$-0.231643\pi$
$968$	0	0
$969$	8.68466	0.278991
$970$	0	0
$971$	−14.6847	−0.471253	−0.235627	−	0.971844i	$-0.575714\pi$
$971$	−14.6847	−0.471253	−0.235627	+	0.971844i	$0.575714\pi$
$972$	0	0
$973$	7.31534i	0.234519i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 16.7538i	− 0.536001i	−0.963419	−	0.268001i	$-0.913637\pi$
$977$	− 16.7538i	− 0.536001i	0.963419	−	0.268001i	$-0.0863629\pi$
$978$	0	0
$979$	60.0540	1.91933
$980$	0	0
$981$	−11.1922	−0.357341
$982$	0	0
$983$	− 28.3542i	− 0.904357i	−0.891927	−	0.452179i	$-0.850647\pi$
$983$	− 28.3542i	− 0.904357i	0.891927	−	0.452179i	$-0.149353\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.8769i	0.409876i
$988$	0	0
$989$	−9.43845	−0.300125
$990$	0	0
$991$	48.6695	1.54604	0.773019	−	0.634383i	$-0.218746\pi$
$991$	48.6695	1.54604	0.773019	+	0.634383i	$0.218746\pi$
$992$	0	0
$993$	42.1619i	1.33797i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 4.24621i	− 0.134479i	−0.997737	−	0.0672394i	$-0.978581\pi$
$997$	− 4.24621i	− 0.134479i	0.997737	−	0.0672394i	$-0.0214191\pi$
$998$	0	0
$999$	−48.9848	−1.54981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.g.j.449.3		4
4.3	odd	2		2800.2.g.v.449.2		4
5.2	odd	4		1400.2.a.o.1.2	✓	2
5.3	odd	4		1400.2.a.q.1.1	yes	2
5.4	even	2	inner	1400.2.g.j.449.2		4
20.3	even	4		2800.2.a.bj.1.2		2
20.7	even	4		2800.2.a.bo.1.1		2
20.19	odd	2		2800.2.g.v.449.3		4
35.13	even	4		9800.2.a.bt.1.2		2
35.27	even	4		9800.2.a.bx.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.a.o.1.2	✓	2	5.2	odd	4
1400.2.a.q.1.1	yes	2	5.3	odd	4
1400.2.g.j.449.2		4	5.4	even	2	inner
1400.2.g.j.449.3		4	1.1	even	1	trivial
2800.2.a.bj.1.2		2	20.3	even	4
2800.2.a.bo.1.1		2	20.7	even	4
2800.2.g.v.449.2		4	4.3	odd	2
2800.2.g.v.449.3		4	20.19	odd	2
9800.2.a.bt.1.2		2	35.13	even	4
9800.2.a.bx.1.1		2	35.27	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} +O(q^{10})\)	\(q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} -6.12311 q^{11} -2.00000i q^{13} +1.56155i q^{17} -3.56155 q^{19} -1.56155 q^{21} -1.43845i q^{23} +5.56155i q^{27} -3.43845 q^{29} -9.12311 q^{31} -9.56155i q^{33} +8.80776i q^{37} +3.12311 q^{39} -2.43845 q^{41} -6.56155i q^{43} -8.24621i q^{47} -1.00000 q^{49} -2.43845 q^{51} +1.12311i q^{53} -5.56155i q^{57} -11.3693 q^{59} +11.1231 q^{61} +0.561553i q^{63} +7.87689i q^{67} +2.24621 q^{69} +1.68466 q^{71} -6.43845i q^{73} -6.12311i q^{77} +5.68466 q^{79} -7.00000 q^{81} +1.31534i q^{83} -5.36932i q^{87} -9.80776 q^{89} +2.00000 q^{91} -14.2462i q^{93} +6.00000i q^{97} -3.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} - 8 q^{11} - 6 q^{19} + 2 q^{21} - 22 q^{29} - 20 q^{31} - 4 q^{39} - 18 q^{41} - 4 q^{49} - 18 q^{51} + 4 q^{59} + 28 q^{61} - 24 q^{69} - 18 q^{71} - 2 q^{79} - 28 q^{81} + 2 q^{89} + 8 q^{91} - 22 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 8 * q^11 - 6 * q^19 + 2 * q^21 - 22 * q^29 - 20 * q^31 - 4 * q^39 - 18 * q^41 - 4 * q^49 - 18 * q^51 + 4 * q^59 + 28 * q^61 - 24 * q^69 - 18 * q^71 - 2 * q^79 - 28 * q^81 + 2 * q^89 + 8 * q^91 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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