LMFDB - Embedded newform 6615.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6615,2,Mod(1,6615)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6615, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6615.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6615 = 3^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6615.a (trivial)

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:	yes
Analytic conductor:	$52.8210409371$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1496809.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - x^{2} + 7x - 1 $ x^5 - 7x^3 - x^2 + 7x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 945)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.31063$ of defining polynomial
Character	$\chi$	$=$	6615.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31063 q^{2} -0.282239 q^{4} +1.00000 q^{5} +2.99118 q^{8} +O(q^{10})$	$q-1.31063 q^{2} -0.282239 q^{4} +1.00000 q^{5} +2.99118 q^{8} -1.31063 q^{10} -5.03641 q^{11} +3.53882 q^{13} -3.35586 q^{16} +1.51043 q^{17} +4.76299 q^{19} -0.282239 q^{20} +6.60089 q^{22} +4.44233 q^{23} +1.00000 q^{25} -4.63810 q^{26} +6.72577 q^{29} -6.17693 q^{31} -1.58405 q^{32} -1.97962 q^{34} +8.15127 q^{37} -6.24253 q^{38} +2.99118 q^{40} -4.64966 q^{41} -3.64084 q^{43} +1.42147 q^{44} -5.82227 q^{46} -1.87632 q^{47} -1.31063 q^{50} -0.998795 q^{52} +9.38953 q^{53} -5.03641 q^{55} -8.81503 q^{58} -6.99799 q^{59} +4.93311 q^{61} +8.09569 q^{62} +8.78784 q^{64} +3.53882 q^{65} +7.04716 q^{67} -0.426302 q^{68} -9.98514 q^{71} +8.14326 q^{73} -10.6833 q^{74} -1.34430 q^{76} -15.4611 q^{79} -3.35586 q^{80} +6.09400 q^{82} -8.26057 q^{83} +1.51043 q^{85} +4.77181 q^{86} -15.0648 q^{88} -1.80423 q^{89} -1.25380 q^{92} +2.45916 q^{94} +4.76299 q^{95} +11.5788 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{4} + 5 q^{5} + 3 q^{8}+O(q^{10}) $ 5 * q + 4 * q^4 + 5 * q^5 + 3 * q^8	$ 5 q + 4 q^{4} + 5 q^{5} + 3 q^{8} + 3 q^{11} + 10 q^{13} + 6 q^{16} + q^{17} + 13 q^{19} + 4 q^{20} - 6 q^{22} - 4 q^{23} + 5 q^{25} + 5 q^{26} + 12 q^{29} + 5 q^{31} + 16 q^{32} + 16 q^{34} + 8 q^{37} - 5 q^{38} + 3 q^{40} - 9 q^{41} + 8 q^{43} - 6 q^{44} - 26 q^{46} - 2 q^{47} + 5 q^{52} - 6 q^{53} + 3 q^{55} + 20 q^{58} + 14 q^{59} + 15 q^{61} + 20 q^{62} - 9 q^{64} + 10 q^{65} - 18 q^{67} - 30 q^{68} + 13 q^{71} + 35 q^{73} - 9 q^{74} + 30 q^{76} + q^{79} + 6 q^{80} + 39 q^{82} - 10 q^{83} + q^{85} + 25 q^{86} - 46 q^{88} - 11 q^{89} - 43 q^{92} - 29 q^{94} + 13 q^{95} + 22 q^{97}+O(q^{100}) $ 5 * q + 4 * q^4 + 5 * q^5 + 3 * q^8 + 3 * q^11 + 10 * q^13 + 6 * q^16 + q^17 + 13 * q^19 + 4 * q^20 - 6 * q^22 - 4 * q^23 + 5 * q^25 + 5 * q^26 + 12 * q^29 + 5 * q^31 + 16 * q^32 + 16 * q^34 + 8 * q^37 - 5 * q^38 + 3 * q^40 - 9 * q^41 + 8 * q^43 - 6 * q^44 - 26 * q^46 - 2 * q^47 + 5 * q^52 - 6 * q^53 + 3 * q^55 + 20 * q^58 + 14 * q^59 + 15 * q^61 + 20 * q^62 - 9 * q^64 + 10 * q^65 - 18 * q^67 - 30 * q^68 + 13 * q^71 + 35 * q^73 - 9 * q^74 + 30 * q^76 + q^79 + 6 * q^80 + 39 * q^82 - 10 * q^83 + q^85 + 25 * q^86 - 46 * q^88 - 11 * q^89 - 43 * q^92 - 29 * q^94 + 13 * q^95 + 22 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−1.31063	−0.926758	−0.463379	−	0.886160i	$-0.653363\pi$
$2$	−1.31063	−0.926758	−0.463379	+	0.886160i	$0.653363\pi$
$3$	0	0
$3$	0	0
$4$	−0.282239	−0.141120
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.99118	1.05754
$9$	0	0
$10$	−1.31063	−0.414459
$11$	−5.03641	−1.51853	−0.759267	−	0.650779i	$-0.774442\pi$
$11$	−5.03641	−1.51853	−0.759267	+	0.650779i	$0.774442\pi$
$12$	0	0
$13$	3.53882	0.981493	0.490747	−	0.871302i	$-0.336724\pi$
$13$	3.53882	0.981493	0.490747	+	0.871302i	$0.336724\pi$
$14$	0	0
$15$	0	0
$16$	−3.35586	−0.838966
$17$	1.51043	0.366333	0.183166	−	0.983082i	$-0.441365\pi$
$17$	1.51043	0.366333	0.183166	+	0.983082i	$0.441365\pi$
$18$	0	0
$19$	4.76299	1.09270	0.546352	−	0.837555i	$-0.316016\pi$
$19$	4.76299	1.09270	0.546352	+	0.837555i	$0.316016\pi$
$20$	−0.282239	−0.0631106
$21$	0	0
$22$	6.60089	1.40731
$23$	4.44233	0.926290	0.463145	−	0.886283i	$-0.346721\pi$
$23$	4.44233	0.926290	0.463145	+	0.886283i	$0.346721\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.63810	−0.909607
$27$	0	0
$28$	0	0
$29$	6.72577	1.24895	0.624473	−	0.781047i	$-0.285314\pi$
$29$	6.72577	1.24895	0.624473	+	0.781047i	$0.285314\pi$
$30$	0	0
$31$	−6.17693	−1.10941	−0.554704	−	0.832048i	$-0.687169\pi$
$31$	−6.17693	−1.10941	−0.554704	+	0.832048i	$0.687169\pi$
$32$	−1.58405	−0.280024
$33$	0	0
$34$	−1.97962	−0.339502
$35$	0	0
$36$	0	0
$37$	8.15127	1.34006	0.670030	−	0.742334i	$-0.266281\pi$
$37$	8.15127	1.34006	0.670030	+	0.742334i	$0.266281\pi$
$38$	−6.24253	−1.01267
$39$	0	0
$40$	2.99118	0.472947
$41$	−4.64966	−0.726155	−0.363078	−	0.931759i	$-0.618274\pi$
$41$	−4.64966	−0.726155	−0.363078	+	0.931759i	$0.618274\pi$
$42$	0	0
$43$	−3.64084	−0.555223	−0.277612	−	0.960693i	$-0.589543\pi$
$43$	−3.64084	−0.555223	−0.277612	+	0.960693i	$0.589543\pi$
$44$	1.42147	0.214295
$45$	0	0
$46$	−5.82227	−0.858447
$47$	−1.87632	−0.273689	−0.136845	−	0.990593i	$-0.543696\pi$
$47$	−1.87632	−0.273689	−0.136845	+	0.990593i	$0.543696\pi$
$48$	0	0
$49$	0	0
$50$	−1.31063	−0.185352
$51$	0	0
$52$	−0.998795	−0.138508
$53$	9.38953	1.28975	0.644876	−	0.764288i	$-0.276909\pi$
$53$	9.38953	1.28975	0.644876	+	0.764288i	$0.276909\pi$
$54$	0	0
$55$	−5.03641	−0.679109
$56$	0	0
$57$	0	0
$58$	−8.81503	−1.15747
$59$	−6.99799	−0.911061	−0.455530	−	0.890220i	$-0.650550\pi$
$59$	−6.99799	−0.911061	−0.455530	+	0.890220i	$0.650550\pi$
$60$	0	0
$61$	4.93311	0.631620	0.315810	−	0.948823i	$-0.397724\pi$
$61$	4.93311	0.631620	0.315810	+	0.948823i	$0.397724\pi$
$62$	8.09569	1.02815
$63$	0	0
$64$	8.78784	1.09848
$65$	3.53882	0.438937
$66$	0	0
$67$	7.04716	0.860948	0.430474	−	0.902603i	$-0.358346\pi$
$67$	7.04716	0.860948	0.430474	+	0.902603i	$0.358346\pi$
$68$	−0.426302	−0.0516968
$69$	0	0
$70$	0	0
$71$	−9.98514	−1.18502	−0.592509	−	0.805564i	$-0.701863\pi$
$71$	−9.98514	−1.18502	−0.592509	+	0.805564i	$0.701863\pi$
$72$	0	0
$73$	8.14326	0.953096	0.476548	−	0.879148i	$-0.341888\pi$
$73$	8.14326	0.953096	0.476548	+	0.879148i	$0.341888\pi$
$74$	−10.6833	−1.24191
$75$	0	0
$76$	−1.34430	−0.154202
$77$	0	0
$78$	0	0
$79$	−15.4611	−1.73951	−0.869755	−	0.493484i	$-0.835723\pi$
$79$	−15.4611	−1.73951	−0.869755	+	0.493484i	$0.835723\pi$
$80$	−3.35586	−0.375197
$81$	0	0
$82$	6.09400	0.672970
$83$	−8.26057	−0.906716	−0.453358	−	0.891329i	$-0.649774\pi$
$83$	−8.26057	−0.906716	−0.453358	+	0.891329i	$0.649774\pi$
$84$	0	0
$85$	1.51043	0.163829
$86$	4.77181	0.514558
$87$	0	0
$88$	−15.0648	−1.60591
$89$	−1.80423	−0.191248	−0.0956239	−	0.995418i	$-0.530485\pi$
$89$	−1.80423	−0.191248	−0.0956239	+	0.995418i	$0.530485\pi$
$90$	0	0
$91$	0	0
$92$	−1.25380	−0.130718
$93$	0	0
$94$	2.45916	0.253643
$95$	4.76299	0.488672
$96$	0	0
$97$	11.5788	1.17565	0.587826	−	0.808988i	$-0.299984\pi$
$97$	11.5788	1.17565	0.587826	+	0.808988i	$0.299984\pi$
$98$	0	0
$99$	0	0
$100$	−0.282239	−0.0282239
$101$	6.11732	0.608696	0.304348	−	0.952561i	$-0.401561\pi$
$101$	6.11732	0.608696	0.304348	+	0.952561i	$0.401561\pi$
$102$	0	0
$103$	−8.11406	−0.799502	−0.399751	−	0.916624i	$-0.630903\pi$
$103$	−8.11406	−0.799502	−0.399751	+	0.916624i	$0.630903\pi$
$104$	10.5853	1.03797
$105$	0	0
$106$	−12.3062	−1.19529
$107$	13.0703	1.26356	0.631778	−	0.775149i	$-0.282325\pi$
$107$	13.0703	1.26356	0.631778	+	0.775149i	$0.282325\pi$
$108$	0	0
$109$	3.56319	0.341292	0.170646	−	0.985332i	$-0.445415\pi$
$109$	3.56319	0.341292	0.170646	+	0.985332i	$0.445415\pi$
$110$	6.60089	0.629370
$111$	0	0
$112$	0	0
$113$	−13.0731	−1.22981	−0.614905	−	0.788601i	$-0.710806\pi$
$113$	−13.0731	−1.22981	−0.614905	+	0.788601i	$0.710806\pi$
$114$	0	0
$115$	4.44233	0.414249
$116$	−1.89828	−0.176251
$117$	0	0
$118$	9.17180	0.844333
$119$	0	0
$120$	0	0
$121$	14.3654	1.30595
$122$	−6.46550	−0.585358
$123$	0	0
$124$	1.74337	0.156559
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.84697	0.518835	0.259417	−	0.965765i	$-0.416470\pi$
$127$	5.84697	0.518835	0.259417	+	0.965765i	$0.416470\pi$
$128$	−8.34953	−0.738001
$129$	0	0
$130$	−4.63810	−0.406788
$131$	−11.6473	−1.01763	−0.508816	−	0.860876i	$-0.669916\pi$
$131$	−11.6473	−1.01763	−0.508816	+	0.860876i	$0.669916\pi$
$132$	0	0
$133$	0	0
$134$	−9.23625	−0.797890
$135$	0	0
$136$	4.51797	0.387412
$137$	−0.702983	−0.0600599	−0.0300299	−	0.999549i	$-0.509560\pi$
$137$	−0.702983	−0.0600599	−0.0300299	+	0.999549i	$0.509560\pi$
$138$	0	0
$139$	22.9376	1.94554	0.972770	−	0.231773i	$-0.0744528\pi$
$139$	22.9376	1.94554	0.972770	+	0.231773i	$0.0744528\pi$
$140$	0	0
$141$	0	0
$142$	13.0869	1.09823
$143$	−17.8230	−1.49043
$144$	0	0
$145$	6.72577	0.558545
$146$	−10.6728	−0.883289
$147$	0	0
$148$	−2.30061	−0.189109
$149$	6.76299	0.554046	0.277023	−	0.960863i	$-0.410652\pi$
$149$	6.76299	0.554046	0.277023	+	0.960863i	$0.410652\pi$
$150$	0	0
$151$	−23.5504	−1.91650	−0.958250	−	0.285931i	$-0.907697\pi$
$151$	−23.5504	−1.91650	−0.958250	+	0.285931i	$0.907697\pi$
$152$	14.2470	1.15558
$153$	0	0
$154$	0	0
$155$	−6.17693	−0.496143
$156$	0	0
$157$	8.23093	0.656900	0.328450	−	0.944521i	$-0.393474\pi$
$157$	8.23093	0.656900	0.328450	+	0.944521i	$0.393474\pi$
$158$	20.2638	1.61210
$159$	0	0
$160$	−1.58405	−0.125230
$161$	0	0
$162$	0	0
$163$	21.8926	1.71476	0.857382	−	0.514681i	$-0.172089\pi$
$163$	21.8926	1.71476	0.857382	+	0.514681i	$0.172089\pi$
$164$	1.31232	0.102475
$165$	0	0
$166$	10.8266	0.840306
$167$	−14.9250	−1.15493	−0.577466	−	0.816415i	$-0.695958\pi$
$167$	−14.9250	−1.15493	−0.577466	+	0.816415i	$0.695958\pi$
$168$	0	0
$169$	−0.476726	−0.0366713
$170$	−1.97962	−0.151830
$171$	0	0
$172$	1.02759	0.0783529
$173$	15.5672	1.18355	0.591776	−	0.806103i	$-0.298427\pi$
$173$	15.5672	1.18355	0.591776	+	0.806103i	$0.298427\pi$
$174$	0	0
$175$	0	0
$176$	16.9015	1.27400
$177$	0	0
$178$	2.36468	0.177240
$179$	−7.03649	−0.525932	−0.262966	−	0.964805i	$-0.584701\pi$
$179$	−7.03649	−0.525932	−0.262966	+	0.964805i	$0.584701\pi$
$180$	0	0
$181$	13.1249	0.975569	0.487784	−	0.872964i	$-0.337805\pi$
$181$	13.1249	0.975569	0.487784	+	0.872964i	$0.337805\pi$
$182$	0	0
$183$	0	0
$184$	13.2878	0.979590
$185$	8.15127	0.599293
$186$	0	0
$187$	−7.60714	−0.556289
$188$	0.529570	0.0386229
$189$	0	0
$190$	−6.24253	−0.452881
$191$	19.0020	1.37493	0.687467	−	0.726215i	$-0.258722\pi$
$191$	19.0020	1.37493	0.687467	+	0.726215i	$0.258722\pi$
$192$	0	0
$193$	5.23017	0.376476	0.188238	−	0.982123i	$-0.439722\pi$
$193$	5.23017	0.376476	0.188238	+	0.982123i	$0.439722\pi$
$194$	−15.1756	−1.08954
$195$	0	0
$196$	0	0
$197$	−16.1141	−1.14808	−0.574041	−	0.818827i	$-0.694625\pi$
$197$	−16.1141	−1.14808	−0.574041	+	0.818827i	$0.694625\pi$
$198$	0	0
$199$	−11.4159	−0.809250	−0.404625	−	0.914483i	$-0.632598\pi$
$199$	−11.4159	−0.809250	−0.404625	+	0.914483i	$0.632598\pi$
$200$	2.99118	0.211508
$201$	0	0
$202$	−8.01756	−0.564114
$203$	0	0
$204$	0	0
$205$	−4.64966	−0.324746
$206$	10.6346	0.740945
$207$	0	0
$208$	−11.8758	−0.823439
$209$	−23.9884	−1.65931
$210$	0	0
$211$	21.2493	1.46286	0.731431	−	0.681915i	$-0.238853\pi$
$211$	21.2493	1.46286	0.731431	+	0.681915i	$0.238853\pi$
$212$	−2.65009	−0.182009
$213$	0	0
$214$	−17.1304	−1.17101
$215$	−3.64084	−0.248303
$216$	0	0
$217$	0	0
$218$	−4.67004	−0.316295
$219$	0	0
$220$	1.42147	0.0958356
$221$	5.34514	0.359553
$222$	0	0
$223$	19.4592	1.30308	0.651541	−	0.758613i	$-0.274123\pi$
$223$	19.4592	1.30308	0.651541	+	0.758613i	$0.274123\pi$
$224$	0	0
$225$	0	0
$226$	17.1340	1.13974
$227$	−4.13693	−0.274577	−0.137289	−	0.990531i	$-0.543839\pi$
$227$	−4.13693	−0.274577	−0.137289	+	0.990531i	$0.543839\pi$
$228$	0	0
$229$	−10.1592	−0.671339	−0.335670	−	0.941980i	$-0.608963\pi$
$229$	−10.1592	−0.671339	−0.335670	+	0.941980i	$0.608963\pi$
$230$	−5.82227	−0.383909
$231$	0	0
$232$	20.1180	1.32081
$233$	−17.9788	−1.17783	−0.588914	−	0.808195i	$-0.700445\pi$
$233$	−17.9788	−1.17783	−0.588914	+	0.808195i	$0.700445\pi$
$234$	0	0
$235$	−1.87632	−0.122397
$236$	1.97511	0.128568
$237$	0	0
$238$	0	0
$239$	2.64692	0.171215	0.0856076	−	0.996329i	$-0.472717\pi$
$239$	2.64692	0.171215	0.0856076	+	0.996329i	$0.472717\pi$
$240$	0	0
$241$	−12.9158	−0.831977	−0.415989	−	0.909370i	$-0.636564\pi$
$241$	−12.9158	−0.831977	−0.415989	+	0.909370i	$0.636564\pi$
$242$	−18.8278	−1.21030
$243$	0	0
$244$	−1.39232	−0.0891339
$245$	0	0
$246$	0	0
$247$	16.8554	1.07248
$248$	−18.4763	−1.17325
$249$	0	0
$250$	−1.31063	−0.0828918
$251$	−21.8811	−1.38112	−0.690560	−	0.723275i	$-0.742636\pi$
$251$	−21.8811	−1.38112	−0.690560	+	0.723275i	$0.742636\pi$
$252$	0	0
$253$	−22.3734	−1.40660
$254$	−7.66323	−0.480834
$255$	0	0
$256$	−6.63250	−0.414531
$257$	−21.7986	−1.35976	−0.679881	−	0.733323i	$-0.737968\pi$
$257$	−21.7986	−1.35976	−0.679881	+	0.733323i	$0.737968\pi$
$258$	0	0
$259$	0	0
$260$	−0.998795	−0.0619426
$261$	0	0
$262$	15.2654	0.943098
$263$	−17.2970	−1.06658	−0.533290	−	0.845933i	$-0.679045\pi$
$263$	−17.2970	−1.06658	−0.533290	+	0.845933i	$0.679045\pi$
$264$	0	0
$265$	9.38953	0.576794
$266$	0	0
$267$	0	0
$268$	−1.98899	−0.121497
$269$	−24.0691	−1.46752	−0.733760	−	0.679409i	$-0.762236\pi$
$269$	−24.0691	−1.46752	−0.733760	+	0.679409i	$0.762236\pi$
$270$	0	0
$271$	24.7793	1.50524	0.752619	−	0.658456i	$-0.228790\pi$
$271$	24.7793	1.50524	0.752619	+	0.658456i	$0.228790\pi$
$272$	−5.06879	−0.307341
$273$	0	0
$274$	0.921353	0.0556610
$275$	−5.03641	−0.303707
$276$	0	0
$277$	14.3347	0.861287	0.430644	−	0.902522i	$-0.358287\pi$
$277$	14.3347	0.861287	0.430644	+	0.902522i	$0.358287\pi$
$278$	−30.0628	−1.80304
$279$	0	0
$280$	0	0
$281$	15.0012	0.894895	0.447447	−	0.894310i	$-0.352333\pi$
$281$	15.0012	0.894895	0.447447	+	0.894310i	$0.352333\pi$
$282$	0	0
$283$	21.3013	1.26623	0.633116	−	0.774057i	$-0.281776\pi$
$283$	21.3013	1.26623	0.633116	+	0.774057i	$0.281776\pi$
$284$	2.81820	0.167229
$285$	0	0
$286$	23.3594	1.38127
$287$	0	0
$288$	0	0
$289$	−14.7186	−0.865800
$290$	−8.81503	−0.517636
$291$	0	0
$292$	−2.29835	−0.134501
$293$	25.8451	1.50989	0.754944	−	0.655790i	$-0.227664\pi$
$293$	25.8451	1.50989	0.754944	+	0.655790i	$0.227664\pi$
$294$	0	0
$295$	−6.99799	−0.407439
$296$	24.3819	1.41717
$297$	0	0
$298$	−8.86380	−0.513466
$299$	15.7206	0.909147
$300$	0	0
$301$	0	0
$302$	30.8659	1.77613
$303$	0	0
$304$	−15.9839	−0.916742
$305$	4.93311	0.282469
$306$	0	0
$307$	11.4637	0.654266	0.327133	−	0.944978i	$-0.393917\pi$
$307$	11.4637	0.654266	0.327133	+	0.944978i	$0.393917\pi$
$308$	0	0
$309$	0	0
$310$	8.09569	0.459804
$311$	17.0156	0.964868	0.482434	−	0.875932i	$-0.339753\pi$
$311$	17.0156	0.964868	0.482434	+	0.875932i	$0.339753\pi$
$312$	0	0
$313$	18.2063	1.02908	0.514540	−	0.857467i	$-0.327963\pi$
$313$	18.2063	1.02908	0.514540	+	0.857467i	$0.327963\pi$
$314$	−10.7877	−0.608787
$315$	0	0
$316$	4.36373	0.245479
$317$	−7.60894	−0.427360	−0.213680	−	0.976904i	$-0.568545\pi$
$317$	−7.60894	−0.427360	−0.213680	+	0.976904i	$0.568545\pi$
$318$	0	0
$319$	−33.8738	−1.89657
$320$	8.78784	0.491255
$321$	0	0
$322$	0	0
$323$	7.19416	0.400294
$324$	0	0
$325$	3.53882	0.196299
$326$	−28.6932	−1.58917
$327$	0	0
$328$	−13.9080	−0.767939
$329$	0	0
$330$	0	0
$331$	−8.37499	−0.460331	−0.230166	−	0.973151i	$-0.573927\pi$
$331$	−8.37499	−0.460331	−0.230166	+	0.973151i	$0.573927\pi$
$332$	2.33146	0.127955
$333$	0	0
$334$	19.5612	1.07034
$335$	7.04716	0.385028
$336$	0	0
$337$	−13.8554	−0.754753	−0.377376	−	0.926060i	$-0.623174\pi$
$337$	−13.8554	−0.754753	−0.377376	+	0.926060i	$0.623174\pi$
$338$	0.624814	0.0339854
$339$	0	0
$340$	−0.426302	−0.0231195
$341$	31.1095	1.68468
$342$	0	0
$343$	0	0
$344$	−10.8904	−0.587172
$345$	0	0
$346$	−20.4029	−1.09687
$347$	−18.2012	−0.977093	−0.488546	−	0.872538i	$-0.662473\pi$
$347$	−18.2012	−0.977093	−0.488546	+	0.872538i	$0.662473\pi$
$348$	0	0
$349$	36.3329	1.94486	0.972428	−	0.233203i	$-0.0749205\pi$
$349$	36.3329	1.94486	0.972428	+	0.233203i	$0.0749205\pi$
$350$	0	0
$351$	0	0
$352$	7.97794	0.425225
$353$	−8.52324	−0.453646	−0.226823	−	0.973936i	$-0.572834\pi$
$353$	−8.52324	−0.453646	−0.226823	+	0.973936i	$0.572834\pi$
$354$	0	0
$355$	−9.98514	−0.529956
$356$	0.509224	0.0269888
$357$	0	0
$358$	9.22226	0.487411
$359$	−1.40293	−0.0740439	−0.0370219	−	0.999314i	$-0.511787\pi$
$359$	−1.40293	−0.0740439	−0.0370219	+	0.999314i	$0.511787\pi$
$360$	0	0
$361$	3.68607	0.194004
$362$	−17.2020	−0.904116
$363$	0	0
$364$	0	0
$365$	8.14326	0.426238
$366$	0	0
$367$	−2.59707	−0.135566	−0.0677829	−	0.997700i	$-0.521593\pi$
$367$	−2.59707	−0.135566	−0.0677829	+	0.997700i	$0.521593\pi$
$368$	−14.9079	−0.777125
$369$	0	0
$370$	−10.6833	−0.555400
$371$	0	0
$372$	0	0
$373$	−37.8792	−1.96131	−0.980656	−	0.195737i	$-0.937290\pi$
$373$	−37.8792	−1.96131	−0.980656	+	0.195737i	$0.937290\pi$
$374$	9.97017	0.515545
$375$	0	0
$376$	−5.61240	−0.289438
$377$	23.8013	1.22583
$378$	0	0
$379$	12.7422	0.654522	0.327261	−	0.944934i	$-0.393874\pi$
$379$	12.7422	0.654522	0.327261	+	0.944934i	$0.393874\pi$
$380$	−1.34430	−0.0689613
$381$	0	0
$382$	−24.9046	−1.27423
$383$	29.3761	1.50105	0.750523	−	0.660844i	$-0.229802\pi$
$383$	29.3761	1.50105	0.750523	+	0.660844i	$0.229802\pi$
$384$	0	0
$385$	0	0
$386$	−6.85484	−0.348902
$387$	0	0
$388$	−3.26800	−0.165907
$389$	10.3953	0.527061	0.263530	−	0.964651i	$-0.415113\pi$
$389$	10.3953	0.527061	0.263530	+	0.964651i	$0.415113\pi$
$390$	0	0
$391$	6.70983	0.339330
$392$	0	0
$393$	0	0
$394$	21.1197	1.06399
$395$	−15.4611	−0.777932
$396$	0	0
$397$	26.7290	1.34149	0.670746	−	0.741687i	$-0.265974\pi$
$397$	26.7290	1.34149	0.670746	+	0.741687i	$0.265974\pi$
$398$	14.9620	0.749978
$399$	0	0
$400$	−3.35586	−0.167793
$401$	21.4408	1.07070	0.535352	−	0.844629i	$-0.320179\pi$
$401$	21.4408	1.07070	0.535352	+	0.844629i	$0.320179\pi$
$402$	0	0
$403$	−21.8591	−1.08888
$404$	−1.72655	−0.0858989
$405$	0	0
$406$	0	0
$407$	−41.0531	−2.03493
$408$	0	0
$409$	19.2899	0.953822	0.476911	−	0.878952i	$-0.341756\pi$
$409$	19.2899	0.953822	0.476911	+	0.878952i	$0.341756\pi$
$410$	6.09400	0.300961
$411$	0	0
$412$	2.29010	0.112825
$413$	0	0
$414$	0	0
$415$	−8.26057	−0.405496
$416$	−5.60568	−0.274841
$417$	0	0
$418$	31.4400	1.53778
$419$	31.2412	1.52623	0.763116	−	0.646261i	$-0.223668\pi$
$419$	31.2412	1.52623	0.763116	+	0.646261i	$0.223668\pi$
$420$	0	0
$421$	34.1516	1.66445	0.832223	−	0.554442i	$-0.187068\pi$
$421$	34.1516	1.66445	0.832223	+	0.554442i	$0.187068\pi$
$422$	−27.8500	−1.35572
$423$	0	0
$424$	28.0858	1.36397
$425$	1.51043	0.0732666
$426$	0	0
$427$	0	0
$428$	−3.68896	−0.178313
$429$	0	0
$430$	4.77181	0.230117
$431$	34.5936	1.66631	0.833157	−	0.553037i	$-0.186531\pi$
$431$	34.5936	1.66631	0.833157	+	0.553037i	$0.186531\pi$
$432$	0	0
$433$	−32.0346	−1.53948	−0.769742	−	0.638356i	$-0.779615\pi$
$433$	−32.0346	−1.53948	−0.769742	+	0.638356i	$0.779615\pi$
$434$	0	0
$435$	0	0
$436$	−1.00567	−0.0481630
$437$	21.1588	1.01216
$438$	0	0
$439$	−5.92959	−0.283004	−0.141502	−	0.989938i	$-0.545193\pi$
$439$	−5.92959	−0.283004	−0.141502	+	0.989938i	$0.545193\pi$
$440$	−15.0648	−0.718186
$441$	0	0
$442$	−7.00552	−0.333219
$443$	−0.328274	−0.0155968	−0.00779839	−	0.999970i	$-0.502482\pi$
$443$	−0.328274	−0.0155968	−0.00779839	+	0.999970i	$0.502482\pi$
$444$	0	0
$445$	−1.80423	−0.0855286
$446$	−25.5038	−1.20764
$447$	0	0
$448$	0	0
$449$	3.28231	0.154902	0.0774509	−	0.996996i	$-0.475322\pi$
$449$	3.28231	0.154902	0.0774509	+	0.996996i	$0.475322\pi$
$450$	0	0
$451$	23.4176	1.10269
$452$	3.68973	0.173550
$453$	0	0
$454$	5.42199	0.254467
$455$	0	0
$456$	0	0
$457$	−21.8041	−1.01995	−0.509977	−	0.860188i	$-0.670346\pi$
$457$	−21.8041	−1.01995	−0.509977	+	0.860188i	$0.670346\pi$
$458$	13.3150	0.622169
$459$	0	0
$460$	−1.25380	−0.0584587
$461$	6.55824	0.305448	0.152724	−	0.988269i	$-0.451195\pi$
$461$	6.55824	0.305448	0.152724	+	0.988269i	$0.451195\pi$
$462$	0	0
$463$	4.40172	0.204565	0.102283	−	0.994755i	$-0.467385\pi$
$463$	4.40172	0.204565	0.102283	+	0.994755i	$0.467385\pi$
$464$	−22.5708	−1.04782
$465$	0	0
$466$	23.5636	1.09156
$467$	−7.67108	−0.354975	−0.177488	−	0.984123i	$-0.556797\pi$
$467$	−7.67108	−0.354975	−0.177488	+	0.984123i	$0.556797\pi$
$468$	0	0
$469$	0	0
$470$	2.45916	0.113433
$471$	0	0
$472$	−20.9322	−0.963485
$473$	18.3368	0.843125
$474$	0	0
$475$	4.76299	0.218541
$476$	0	0
$477$	0	0
$478$	−3.46915	−0.158675
$479$	14.6947	0.671416	0.335708	−	0.941966i	$-0.391024\pi$
$479$	14.6947	0.671416	0.335708	+	0.941966i	$0.391024\pi$
$480$	0	0
$481$	28.8459	1.31526
$482$	16.9278	0.771042
$483$	0	0
$484$	−4.05448	−0.184295
$485$	11.5788	0.525767
$486$	0	0
$487$	−6.65182	−0.301423	−0.150711	−	0.988578i	$-0.548156\pi$
$487$	−6.65182	−0.301423	−0.150711	+	0.988578i	$0.548156\pi$
$488$	14.7558	0.667964
$489$	0	0
$490$	0	0
$491$	8.53854	0.385339	0.192669	−	0.981264i	$-0.438285\pi$
$491$	8.53854	0.385339	0.192669	+	0.981264i	$0.438285\pi$
$492$	0	0
$493$	10.1588	0.457530
$494$	−22.0912	−0.993931
$495$	0	0
$496$	20.7289	0.930756
$497$	0	0
$498$	0	0
$499$	17.0753	0.764394	0.382197	−	0.924081i	$-0.375168\pi$
$499$	17.0753	0.764394	0.382197	+	0.924081i	$0.375168\pi$
$500$	−0.282239	−0.0126221
$501$	0	0
$502$	28.6781	1.27996
$503$	18.4481	0.822562	0.411281	−	0.911509i	$-0.365082\pi$
$503$	18.4481	0.822562	0.411281	+	0.911509i	$0.365082\pi$
$504$	0	0
$505$	6.11732	0.272217
$506$	29.3233	1.30358
$507$	0	0
$508$	−1.65024	−0.0732177
$509$	2.61998	0.116129	0.0580644	−	0.998313i	$-0.481507\pi$
$509$	2.61998	0.116129	0.0580644	+	0.998313i	$0.481507\pi$
$510$	0	0
$511$	0	0
$512$	25.3918	1.12217
$513$	0	0
$514$	28.5700	1.26017
$515$	−8.11406	−0.357548
$516$	0	0
$517$	9.44990	0.415606
$518$	0	0
$519$	0	0
$520$	10.5853	0.464194
$521$	37.0839	1.62467	0.812337	−	0.583189i	$-0.198195\pi$
$521$	37.0839	1.62467	0.812337	+	0.583189i	$0.198195\pi$
$522$	0	0
$523$	−29.1153	−1.27312	−0.636562	−	0.771226i	$-0.719644\pi$
$523$	−29.1153	−1.27312	−0.636562	+	0.771226i	$0.719644\pi$
$524$	3.28733	0.143608
$525$	0	0
$526$	22.6701	0.988461
$527$	−9.32981	−0.406413
$528$	0	0
$529$	−3.26570	−0.141987
$530$	−12.3062	−0.534549
$531$	0	0
$532$	0	0
$533$	−16.4543	−0.712716
$534$	0	0
$535$	13.0703	0.565080
$536$	21.0793	0.910488
$537$	0	0
$538$	31.5458	1.36004
$539$	0	0
$540$	0	0
$541$	22.5277	0.968543	0.484271	−	0.874918i	$-0.339085\pi$
$541$	22.5277	0.968543	0.484271	+	0.874918i	$0.339085\pi$
$542$	−32.4766	−1.39499
$543$	0	0
$544$	−2.39260	−0.102582
$545$	3.56319	0.152630
$546$	0	0
$547$	18.6039	0.795444	0.397722	−	0.917506i	$-0.369801\pi$
$547$	18.6039	0.795444	0.397722	+	0.917506i	$0.369801\pi$
$548$	0.198409	0.00847562
$549$	0	0
$550$	6.60089	0.281463
$551$	32.0348	1.36473
$552$	0	0
$553$	0	0
$554$	−18.7875	−0.798205
$555$	0	0
$556$	−6.47388	−0.274554
$557$	−12.7155	−0.538771	−0.269386	−	0.963032i	$-0.586821\pi$
$557$	−12.7155	−0.538771	−0.269386	+	0.963032i	$0.586821\pi$
$558$	0	0
$559$	−12.8843	−0.544948
$560$	0	0
$561$	0	0
$562$	−19.6610	−0.829351
$563$	33.6385	1.41770	0.708848	−	0.705361i	$-0.249215\pi$
$563$	33.6385	1.41770	0.708848	+	0.705361i	$0.249215\pi$
$564$	0	0
$565$	−13.0731	−0.549988
$566$	−27.9182	−1.17349
$567$	0	0
$568$	−29.8674	−1.25321
$569$	−24.2693	−1.01742	−0.508711	−	0.860938i	$-0.669878\pi$
$569$	−24.2693	−1.01742	−0.508711	+	0.860938i	$0.669878\pi$
$570$	0	0
$571$	21.3012	0.891428	0.445714	−	0.895175i	$-0.352950\pi$
$571$	21.3012	0.891428	0.445714	+	0.895175i	$0.352950\pi$
$572$	5.03034	0.210329
$573$	0	0
$574$	0	0
$575$	4.44233	0.185258
$576$	0	0
$577$	−21.0371	−0.875786	−0.437893	−	0.899027i	$-0.644275\pi$
$577$	−21.0371	−0.875786	−0.437893	+	0.899027i	$0.644275\pi$
$578$	19.2907	0.802387
$579$	0	0
$580$	−1.89828	−0.0788217
$581$	0	0
$582$	0	0
$583$	−47.2895	−1.95853
$584$	24.3579	1.00794
$585$	0	0
$586$	−33.8735	−1.39930
$587$	29.7134	1.22640	0.613202	−	0.789926i	$-0.289881\pi$
$587$	29.7134	1.22640	0.613202	+	0.789926i	$0.289881\pi$
$588$	0	0
$589$	−29.4206	−1.21226
$590$	9.17180	0.377597
$591$	0	0
$592$	−27.3545	−1.12426
$593$	29.2413	1.20079	0.600397	−	0.799702i	$-0.295009\pi$
$593$	29.2413	1.20079	0.600397	+	0.799702i	$0.295009\pi$
$594$	0	0
$595$	0	0
$596$	−1.90878	−0.0781867
$597$	0	0
$598$	−20.6040	−0.842559
$599$	43.9553	1.79597	0.897983	−	0.440031i	$-0.145032\pi$
$599$	43.9553	1.79597	0.897983	+	0.440031i	$0.145032\pi$
$600$	0	0
$601$	−32.1919	−1.31313	−0.656567	−	0.754267i	$-0.727992\pi$
$601$	−32.1919	−1.31313	−0.656567	+	0.754267i	$0.727992\pi$
$602$	0	0
$603$	0	0
$604$	6.64683	0.270456
$605$	14.3654	0.584037
$606$	0	0
$607$	25.5814	1.03832	0.519159	−	0.854678i	$-0.326245\pi$
$607$	25.5814	1.03832	0.519159	+	0.854678i	$0.326245\pi$
$608$	−7.54483	−0.305983
$609$	0	0
$610$	−6.46550	−0.261780
$611$	−6.63996	−0.268624
$612$	0	0
$613$	9.60644	0.388001	0.194000	−	0.981001i	$-0.437854\pi$
$613$	9.60644	0.388001	0.194000	+	0.981001i	$0.437854\pi$
$614$	−15.0247	−0.606346
$615$	0	0
$616$	0	0
$617$	13.4891	0.543052	0.271526	−	0.962431i	$-0.412472\pi$
$617$	13.4891	0.543052	0.271526	+	0.962431i	$0.412472\pi$
$618$	0	0
$619$	−7.39154	−0.297091	−0.148546	−	0.988906i	$-0.547459\pi$
$619$	−7.39154	−0.297091	−0.148546	+	0.988906i	$0.547459\pi$
$620$	1.74337	0.0700154
$621$	0	0
$622$	−22.3013	−0.894199
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−23.8618	−0.953708
$627$	0	0
$628$	−2.32309	−0.0927014
$629$	12.3119	0.490908
$630$	0	0
$631$	41.7055	1.66027	0.830134	−	0.557564i	$-0.188264\pi$
$631$	41.7055	1.66027	0.830134	+	0.557564i	$0.188264\pi$
$632$	−46.2469	−1.83960
$633$	0	0
$634$	9.97253	0.396060
$635$	5.84697	0.232030
$636$	0	0
$637$	0	0
$638$	44.3961	1.75766
$639$	0	0
$640$	−8.34953	−0.330044
$641$	11.3295	0.447488	0.223744	−	0.974648i	$-0.428172\pi$
$641$	11.3295	0.447488	0.223744	+	0.974648i	$0.428172\pi$
$642$	0	0
$643$	−29.5254	−1.16437	−0.582184	−	0.813057i	$-0.697802\pi$
$643$	−29.5254	−1.16437	−0.582184	+	0.813057i	$0.697802\pi$
$644$	0	0
$645$	0	0
$646$	−9.42891	−0.370975
$647$	8.04017	0.316092	0.158046	−	0.987432i	$-0.449481\pi$
$647$	8.04017	0.316092	0.158046	+	0.987432i	$0.449481\pi$
$648$	0	0
$649$	35.2447	1.38348
$650$	−4.63810	−0.181921
$651$	0	0
$652$	−6.17896	−0.241987
$653$	27.6792	1.08317	0.541586	−	0.840645i	$-0.317824\pi$
$653$	27.6792	1.08317	0.541586	+	0.840645i	$0.317824\pi$
$654$	0	0
$655$	−11.6473	−0.455098
$656$	15.6036	0.609219
$657$	0	0
$658$	0	0
$659$	−16.8807	−0.657579	−0.328790	−	0.944403i	$-0.606641\pi$
$659$	−16.8807	−0.657579	−0.328790	+	0.944403i	$0.606641\pi$
$660$	0	0
$661$	−10.8542	−0.422179	−0.211089	−	0.977467i	$-0.567701\pi$
$661$	−10.8542	−0.422179	−0.211089	+	0.977467i	$0.567701\pi$
$662$	10.9765	0.426616
$663$	0	0
$664$	−24.7089	−0.958890
$665$	0	0
$666$	0	0
$667$	29.8781	1.15689
$668$	4.21242	0.162984
$669$	0	0
$670$	−9.23625	−0.356827
$671$	−24.8451	−0.959136
$672$	0	0
$673$	−28.3291	−1.09201	−0.546003	−	0.837783i	$-0.683851\pi$
$673$	−28.3291	−1.09201	−0.546003	+	0.837783i	$0.683851\pi$
$674$	18.1594	0.699473
$675$	0	0
$676$	0.134551	0.00517503
$677$	−0.241433	−0.00927904	−0.00463952	−	0.999989i	$-0.501477\pi$
$677$	−0.241433	−0.00927904	−0.00463952	+	0.999989i	$0.501477\pi$
$678$	0	0
$679$	0	0
$680$	4.51797	0.173256
$681$	0	0
$682$	−40.7732	−1.56129
$683$	−15.4885	−0.592650	−0.296325	−	0.955087i	$-0.595761\pi$
$683$	−15.4885	−0.592650	−0.296325	+	0.955087i	$0.595761\pi$
$684$	0	0
$685$	−0.702983	−0.0268596
$686$	0	0
$687$	0	0
$688$	12.2182	0.465813
$689$	33.2279	1.26588
$690$	0	0
$691$	−17.1871	−0.653830	−0.326915	−	0.945054i	$-0.606009\pi$
$691$	−17.1871	−0.653830	−0.326915	+	0.945054i	$0.606009\pi$
$692$	−4.39367	−0.167022
$693$	0	0
$694$	23.8551	0.905528
$695$	22.9376	0.870072
$696$	0	0
$697$	−7.02299	−0.266015
$698$	−47.6191	−1.80241
$699$	0	0
$700$	0	0
$701$	9.16592	0.346192	0.173096	−	0.984905i	$-0.444623\pi$
$701$	9.16592	0.346192	0.173096	+	0.984905i	$0.444623\pi$
$702$	0	0
$703$	38.8244	1.46429
$704$	−44.2591	−1.66808
$705$	0	0
$706$	11.1708	0.420420
$707$	0	0
$708$	0	0
$709$	15.6945	0.589420	0.294710	−	0.955587i	$-0.404777\pi$
$709$	15.6945	0.589420	0.294710	+	0.955587i	$0.404777\pi$
$710$	13.0869	0.491141
$711$	0	0
$712$	−5.39677	−0.202253
$713$	−27.4399	−1.02763
$714$	0	0
$715$	−17.8230	−0.666541
$716$	1.98597	0.0742193
$717$	0	0
$718$	1.83873	0.0686208
$719$	36.1133	1.34680	0.673399	−	0.739280i	$-0.264834\pi$
$719$	36.1133	1.34680	0.673399	+	0.739280i	$0.264834\pi$
$720$	0	0
$721$	0	0
$722$	−4.83109	−0.179794
$723$	0	0
$724$	−3.70437	−0.137672
$725$	6.72577	0.249789
$726$	0	0
$727$	26.8410	0.995476	0.497738	−	0.867327i	$-0.334164\pi$
$727$	26.8410	0.995476	0.497738	+	0.867327i	$0.334164\pi$
$728$	0	0
$729$	0	0
$730$	−10.6728	−0.395019
$731$	−5.49923	−0.203397
$732$	0	0
$733$	29.5977	1.09322	0.546609	−	0.837388i	$-0.315919\pi$
$733$	29.5977	1.09322	0.546609	+	0.837388i	$0.315919\pi$
$734$	3.40381	0.125637
$735$	0	0
$736$	−7.03689	−0.259383
$737$	−35.4924	−1.30738
$738$	0	0
$739$	−32.8130	−1.20705	−0.603523	−	0.797346i	$-0.706237\pi$
$739$	−32.8130	−1.20705	−0.603523	+	0.797346i	$0.706237\pi$
$740$	−2.30061	−0.0845720
$741$	0	0
$742$	0	0
$743$	16.2198	0.595047	0.297524	−	0.954714i	$-0.403839\pi$
$743$	16.2198	0.595047	0.297524	+	0.954714i	$0.403839\pi$
$744$	0	0
$745$	6.76299	0.247777
$746$	49.6458	1.81766
$747$	0	0
$748$	2.14703	0.0785033
$749$	0	0
$750$	0	0
$751$	−0.0884793	−0.00322865	−0.00161433	−	0.999999i	$-0.500514\pi$
$751$	−0.0884793	−0.00322865	−0.00161433	+	0.999999i	$0.500514\pi$
$752$	6.29666	0.229616
$753$	0	0
$754$	−31.1948	−1.13605
$755$	−23.5504	−0.857085
$756$	0	0
$757$	38.0943	1.38456	0.692280	−	0.721629i	$-0.256606\pi$
$757$	38.0943	1.38456	0.692280	+	0.721629i	$0.256606\pi$
$758$	−16.7003	−0.606583
$759$	0	0
$760$	14.2470	0.516791
$761$	−11.2223	−0.406809	−0.203405	−	0.979095i	$-0.565201\pi$
$761$	−11.2223	−0.406809	−0.203405	+	0.979095i	$0.565201\pi$
$762$	0	0
$763$	0	0
$764$	−5.36310	−0.194030
$765$	0	0
$766$	−38.5012	−1.39111
$767$	−24.7646	−0.894200
$768$	0	0
$769$	−10.2232	−0.368658	−0.184329	−	0.982865i	$-0.559011\pi$
$769$	−10.2232	−0.368658	−0.184329	+	0.982865i	$0.559011\pi$
$770$	0	0
$771$	0	0
$772$	−1.47616	−0.0531281
$773$	38.8202	1.39627	0.698134	−	0.715967i	$-0.254014\pi$
$773$	38.8202	1.39627	0.698134	+	0.715967i	$0.254014\pi$
$774$	0	0
$775$	−6.17693	−0.221882
$776$	34.6343	1.24330
$777$	0	0
$778$	−13.6244	−0.488458
$779$	−22.1463	−0.793473
$780$	0	0
$781$	50.2893	1.79949
$782$	−8.79412	−0.314477
$783$	0	0
$784$	0	0
$785$	8.23093	0.293774
$786$	0	0
$787$	18.2164	0.649345	0.324672	−	0.945827i	$-0.394746\pi$
$787$	18.2164	0.649345	0.324672	+	0.945827i	$0.394746\pi$
$788$	4.54803	0.162017
$789$	0	0
$790$	20.2638	0.720955
$791$	0	0
$792$	0	0
$793$	17.4574	0.619930
$794$	−35.0320	−1.24324
$795$	0	0
$796$	3.22201	0.114201
$797$	39.3577	1.39412	0.697060	−	0.717013i	$-0.254491\pi$
$797$	39.3577	1.39412	0.697060	+	0.717013i	$0.254491\pi$
$798$	0	0
$799$	−2.83404	−0.100261
$800$	−1.58405	−0.0560047
$801$	0	0
$802$	−28.1011	−0.992284
$803$	−41.0128	−1.44731
$804$	0	0
$805$	0	0
$806$	28.6492	1.00913
$807$	0	0
$808$	18.2980	0.643721
$809$	−27.7918	−0.977107	−0.488554	−	0.872534i	$-0.662475\pi$
$809$	−27.7918	−0.977107	−0.488554	+	0.872534i	$0.662475\pi$
$810$	0	0
$811$	20.0045	0.702452	0.351226	−	0.936291i	$-0.385765\pi$
$811$	20.0045	0.702452	0.351226	+	0.936291i	$0.385765\pi$
$812$	0	0
$813$	0	0
$814$	53.8056	1.88589
$815$	21.8926	0.766866
$816$	0	0
$817$	−17.3413	−0.606695
$818$	−25.2819	−0.883962
$819$	0	0
$820$	1.31232	0.0458281
$821$	−33.9141	−1.18361	−0.591805	−	0.806081i	$-0.701585\pi$
$821$	−33.9141	−1.18361	−0.591805	+	0.806081i	$0.701585\pi$
$822$	0	0
$823$	−1.25765	−0.0438390	−0.0219195	−	0.999760i	$-0.506978\pi$
$823$	−1.25765	−0.0438390	−0.0219195	+	0.999760i	$0.506978\pi$
$824$	−24.2706	−0.845506
$825$	0	0
$826$	0	0
$827$	−12.2053	−0.424421	−0.212210	−	0.977224i	$-0.568066\pi$
$827$	−12.2053	−0.424421	−0.212210	+	0.977224i	$0.568066\pi$
$828$	0	0
$829$	−14.2893	−0.496289	−0.248145	−	0.968723i	$-0.579821\pi$
$829$	−14.2893	−0.496289	−0.248145	+	0.968723i	$0.579821\pi$
$830$	10.8266	0.375796
$831$	0	0
$832$	31.0986	1.07815
$833$	0	0
$834$	0	0
$835$	−14.9250	−0.516501
$836$	6.77046	0.234161
$837$	0	0
$838$	−40.9458	−1.41445
$839$	20.9146	0.722052	0.361026	−	0.932556i	$-0.382427\pi$
$839$	20.9146	0.722052	0.361026	+	0.932556i	$0.382427\pi$
$840$	0	0
$841$	16.2360	0.559864
$842$	−44.7602	−1.54254
$843$	0	0
$844$	−5.99739	−0.206439
$845$	−0.476726	−0.0163999
$846$	0	0
$847$	0	0
$848$	−31.5100	−1.08206
$849$	0	0
$850$	−1.97962	−0.0679004
$851$	36.2106	1.24128
$852$	0	0
$853$	−14.7825	−0.506142	−0.253071	−	0.967448i	$-0.581441\pi$
$853$	−14.7825	−0.506142	−0.253071	+	0.967448i	$0.581441\pi$
$854$	0	0
$855$	0	0
$856$	39.0957	1.33626
$857$	−21.0414	−0.718762	−0.359381	−	0.933191i	$-0.617012\pi$
$857$	−21.0414	−0.718762	−0.359381	+	0.933191i	$0.617012\pi$
$858$	0	0
$859$	−11.9756	−0.408601	−0.204300	−	0.978908i	$-0.565492\pi$
$859$	−11.9756	−0.408601	−0.204300	+	0.978908i	$0.565492\pi$
$860$	1.02759	0.0350405
$861$	0	0
$862$	−45.3395	−1.54427
$863$	−54.7502	−1.86372	−0.931859	−	0.362821i	$-0.881814\pi$
$863$	−54.7502	−1.86372	−0.931859	+	0.362821i	$0.881814\pi$
$864$	0	0
$865$	15.5672	0.529300
$866$	41.9856	1.42673
$867$	0	0
$868$	0	0
$869$	77.8684	2.64150
$870$	0	0
$871$	24.9387	0.845014
$872$	10.6582	0.360931
$873$	0	0
$874$	−27.7314	−0.938029
$875$	0	0
$876$	0	0
$877$	−26.9431	−0.909804	−0.454902	−	0.890542i	$-0.650326\pi$
$877$	−26.9431	−0.909804	−0.454902	+	0.890542i	$0.650326\pi$
$878$	7.77153	0.262276
$879$	0	0
$880$	16.9015	0.569749
$881$	5.55657	0.187206	0.0936028	−	0.995610i	$-0.470162\pi$
$881$	5.55657	0.187206	0.0936028	+	0.995610i	$0.470162\pi$
$882$	0	0
$883$	0.333833	0.0112344	0.00561719	−	0.999984i	$-0.498212\pi$
$883$	0.333833	0.0112344	0.00561719	+	0.999984i	$0.498212\pi$
$884$	−1.50861	−0.0507400
$885$	0	0
$886$	0.430247	0.0144544
$887$	−50.7137	−1.70280	−0.851400	−	0.524517i	$-0.824246\pi$
$887$	−50.7137	−1.70280	−0.851400	+	0.524517i	$0.824246\pi$
$888$	0	0
$889$	0	0
$890$	2.36468	0.0792644
$891$	0	0
$892$	−5.49214	−0.183890
$893$	−8.93688	−0.299061
$894$	0	0
$895$	−7.03649	−0.235204
$896$	0	0
$897$	0	0
$898$	−4.30190	−0.143556
$899$	−41.5446	−1.38559
$900$	0	0
$901$	14.1822	0.472478
$902$	−30.6919	−1.02193
$903$	0	0
$904$	−39.1039	−1.30058
$905$	13.1249	0.436288
$906$	0	0
$907$	−3.62385	−0.120328	−0.0601640	−	0.998189i	$-0.519162\pi$
$907$	−3.62385	−0.120328	−0.0601640	+	0.998189i	$0.519162\pi$
$908$	1.16760	0.0387483
$909$	0	0
$910$	0	0
$911$	−35.5729	−1.17858	−0.589291	−	0.807921i	$-0.700593\pi$
$911$	−35.5729	−1.17858	−0.589291	+	0.807921i	$0.700593\pi$
$912$	0	0
$913$	41.6036	1.37688
$914$	28.5772	0.945250
$915$	0	0
$916$	2.86733	0.0947391
$917$	0	0
$918$	0	0
$919$	−28.6085	−0.943707	−0.471854	−	0.881677i	$-0.656415\pi$
$919$	−28.6085	−0.943707	−0.471854	+	0.881677i	$0.656415\pi$
$920$	13.2878	0.438086
$921$	0	0
$922$	−8.59545	−0.283076
$923$	−35.3357	−1.16309
$924$	0	0
$925$	8.15127	0.268012
$926$	−5.76904	−0.189582
$927$	0	0
$928$	−10.6540	−0.349734
$929$	23.8330	0.781934	0.390967	−	0.920405i	$-0.372141\pi$
$929$	23.8330	0.781934	0.390967	+	0.920405i	$0.372141\pi$
$930$	0	0
$931$	0	0
$932$	5.07432	0.166215
$933$	0	0
$934$	10.0540	0.328976
$935$	−7.60714	−0.248780
$936$	0	0
$937$	47.1193	1.53932	0.769661	−	0.638453i	$-0.220425\pi$
$937$	47.1193	1.53932	0.769661	+	0.638453i	$0.220425\pi$
$938$	0	0
$939$	0	0
$940$	0.529570	0.0172727
$941$	−9.07950	−0.295983	−0.147992	−	0.988989i	$-0.547281\pi$
$941$	−9.07950	−0.295983	−0.147992	+	0.988989i	$0.547281\pi$
$942$	0	0
$943$	−20.6553	−0.672630
$944$	23.4843	0.764348
$945$	0	0
$946$	−24.0328	−0.781373
$947$	20.9866	0.681973	0.340986	−	0.940068i	$-0.389239\pi$
$947$	20.9866	0.681973	0.340986	+	0.940068i	$0.389239\pi$
$948$	0	0
$949$	28.8176	0.935457
$950$	−6.24253	−0.202535
$951$	0	0
$952$	0	0
$953$	−8.81353	−0.285498	−0.142749	−	0.989759i	$-0.545594\pi$
$953$	−8.81353	−0.285498	−0.142749	+	0.989759i	$0.545594\pi$
$954$	0	0
$955$	19.0020	0.614890
$956$	−0.747065	−0.0241618
$957$	0	0
$958$	−19.2593	−0.622240
$959$	0	0
$960$	0	0
$961$	7.15441	0.230787
$962$	−37.8064	−1.21893
$963$	0	0
$964$	3.64533	0.117408
$965$	5.23017	0.168365
$966$	0	0
$967$	53.4961	1.72032	0.860159	−	0.510026i	$-0.170364\pi$
$967$	53.4961	1.72032	0.860159	+	0.510026i	$0.170364\pi$
$968$	42.9695	1.38109
$969$	0	0
$970$	−15.1756	−0.487259
$971$	−39.1951	−1.25783	−0.628915	−	0.777474i	$-0.716501\pi$
$971$	−39.1951	−1.25783	−0.628915	+	0.777474i	$0.716501\pi$
$972$	0	0
$973$	0	0
$974$	8.71810	0.279346
$975$	0	0
$976$	−16.5548	−0.529907
$977$	22.7558	0.728022	0.364011	−	0.931395i	$-0.381407\pi$
$977$	22.7558	0.728022	0.364011	+	0.931395i	$0.381407\pi$
$978$	0	0
$979$	9.08683	0.290416
$980$	0	0
$981$	0	0
$982$	−11.1909	−0.357116
$983$	45.1621	1.44045	0.720224	−	0.693741i	$-0.244039\pi$
$983$	45.1621	1.44045	0.720224	+	0.693741i	$0.244039\pi$
$984$	0	0
$985$	−16.1141	−0.513438
$986$	−13.3145	−0.424019
$987$	0	0
$988$	−4.75725	−0.151348
$989$	−16.1738	−0.514298
$990$	0	0
$991$	−3.97132	−0.126153	−0.0630766	−	0.998009i	$-0.520091\pi$
$991$	−3.97132	−0.126153	−0.0630766	+	0.998009i	$0.520091\pi$
$992$	9.78458	0.310661
$993$	0	0
$994$	0	0
$995$	−11.4159	−0.361907
$996$	0	0
$997$	17.3270	0.548751	0.274376	−	0.961623i	$-0.411529\pi$
$997$	17.3270	0.548751	0.274376	+	0.961623i	$0.411529\pi$
$998$	−22.3794	−0.708408
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6615.2.a.bq.1.2		5
3.2	odd	2		6615.2.a.bl.1.4		5
7.3	odd	6		945.2.j.h.541.4	yes	10
7.5	odd	6		945.2.j.h.676.4	yes	10
7.6	odd	2		6615.2.a.bm.1.2		5
21.5	even	6		945.2.j.f.676.2	yes	10
21.17	even	6		945.2.j.f.541.2	✓	10
21.20	even	2		6615.2.a.bp.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.j.f.541.2	✓	10	21.17	even	6
945.2.j.f.676.2	yes	10	21.5	even	6
945.2.j.h.541.4	yes	10	7.3	odd	6
945.2.j.h.676.4	yes	10	7.5	odd	6
6615.2.a.bl.1.4		5	3.2	odd	2
6615.2.a.bm.1.2		5	7.6	odd	2
6615.2.a.bp.1.4		5	21.20	even	2
6615.2.a.bq.1.2		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6615 = 3^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6615.a (trivial)

\(f(q)\)	\(=\)	\(q-1.31063 q^{2} -0.282239 q^{4} +1.00000 q^{5} +2.99118 q^{8} +O(q^{10})\)	\(q-1.31063 q^{2} -0.282239 q^{4} +1.00000 q^{5} +2.99118 q^{8} -1.31063 q^{10} -5.03641 q^{11} +3.53882 q^{13} -3.35586 q^{16} +1.51043 q^{17} +4.76299 q^{19} -0.282239 q^{20} +6.60089 q^{22} +4.44233 q^{23} +1.00000 q^{25} -4.63810 q^{26} +6.72577 q^{29} -6.17693 q^{31} -1.58405 q^{32} -1.97962 q^{34} +8.15127 q^{37} -6.24253 q^{38} +2.99118 q^{40} -4.64966 q^{41} -3.64084 q^{43} +1.42147 q^{44} -5.82227 q^{46} -1.87632 q^{47} -1.31063 q^{50} -0.998795 q^{52} +9.38953 q^{53} -5.03641 q^{55} -8.81503 q^{58} -6.99799 q^{59} +4.93311 q^{61} +8.09569 q^{62} +8.78784 q^{64} +3.53882 q^{65} +7.04716 q^{67} -0.426302 q^{68} -9.98514 q^{71} +8.14326 q^{73} -10.6833 q^{74} -1.34430 q^{76} -15.4611 q^{79} -3.35586 q^{80} +6.09400 q^{82} -8.26057 q^{83} +1.51043 q^{85} +4.77181 q^{86} -15.0648 q^{88} -1.80423 q^{89} -1.25380 q^{92} +2.45916 q^{94} +4.76299 q^{95} +11.5788 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{4} + 5 q^{5} + 3 q^{8}+O(q^{10}) \) 5 * q + 4 * q^4 + 5 * q^5 + 3 * q^8	\( 5 q + 4 q^{4} + 5 q^{5} + 3 q^{8} + 3 q^{11} + 10 q^{13} + 6 q^{16} + q^{17} + 13 q^{19} + 4 q^{20} - 6 q^{22} - 4 q^{23} + 5 q^{25} + 5 q^{26} + 12 q^{29} + 5 q^{31} + 16 q^{32} + 16 q^{34} + 8 q^{37} - 5 q^{38} + 3 q^{40} - 9 q^{41} + 8 q^{43} - 6 q^{44} - 26 q^{46} - 2 q^{47} + 5 q^{52} - 6 q^{53} + 3 q^{55} + 20 q^{58} + 14 q^{59} + 15 q^{61} + 20 q^{62} - 9 q^{64} + 10 q^{65} - 18 q^{67} - 30 q^{68} + 13 q^{71} + 35 q^{73} - 9 q^{74} + 30 q^{76} + q^{79} + 6 q^{80} + 39 q^{82} - 10 q^{83} + q^{85} + 25 q^{86} - 46 q^{88} - 11 q^{89} - 43 q^{92} - 29 q^{94} + 13 q^{95} + 22 q^{97}+O(q^{100}) \) 5 * q + 4 * q^4 + 5 * q^5 + 3 * q^8 + 3 * q^11 + 10 * q^13 + 6 * q^16 + q^17 + 13 * q^19 + 4 * q^20 - 6 * q^22 - 4 * q^23 + 5 * q^25 + 5 * q^26 + 12 * q^29 + 5 * q^31 + 16 * q^32 + 16 * q^34 + 8 * q^37 - 5 * q^38 + 3 * q^40 - 9 * q^41 + 8 * q^43 - 6 * q^44 - 26 * q^46 - 2 * q^47 + 5 * q^52 - 6 * q^53 + 3 * q^55 + 20 * q^58 + 14 * q^59 + 15 * q^61 + 20 * q^62 - 9 * q^64 + 10 * q^65 - 18 * q^67 - 30 * q^68 + 13 * q^71 + 35 * q^73 - 9 * q^74 + 30 * q^76 + q^79 + 6 * q^80 + 39 * q^82 - 10 * q^83 + q^85 + 25 * q^86 - 46 * q^88 - 11 * q^89 - 43 * q^92 - 29 * q^94 + 13 * q^95 + 22 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more