LMFDB - Embedded newform 6160.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1540)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +1.00000 q^{11} +6.24264 q^{13} +1.41421 q^{15} +0.585786 q^{17} -2.82843 q^{19} -1.41421 q^{21} -2.82843 q^{23} +1.00000 q^{25} +5.65685 q^{27} +8.82843 q^{29} -1.41421 q^{31} -1.41421 q^{33} -1.00000 q^{35} -5.65685 q^{37} -8.82843 q^{39} +9.07107 q^{41} +0.828427 q^{43} +1.00000 q^{45} -8.24264 q^{47} +1.00000 q^{49} -0.828427 q^{51} +4.00000 q^{53} -1.00000 q^{55} +4.00000 q^{57} -8.24264 q^{59} +0.585786 q^{61} -1.00000 q^{63} -6.24264 q^{65} -1.17157 q^{67} +4.00000 q^{69} +2.82843 q^{71} -7.89949 q^{73} -1.41421 q^{75} +1.00000 q^{77} +4.82843 q^{79} -5.00000 q^{81} -6.82843 q^{83} -0.585786 q^{85} -12.4853 q^{87} +11.6569 q^{89} +6.24264 q^{91} +2.00000 q^{93} +2.82843 q^{95} +4.82843 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7 - 2 * q^9	$ 2 q - 2 q^{5} + 2 q^{7} - 2 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{17} + 2 q^{25} + 12 q^{29} - 2 q^{35} - 12 q^{39} + 4 q^{41} - 4 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} + 4 q^{51} + 8 q^{53} - 2 q^{55} + 8 q^{57} - 8 q^{59} + 4 q^{61} - 2 q^{63} - 4 q^{65} - 8 q^{67} + 8 q^{69} + 4 q^{73} + 2 q^{77} + 4 q^{79} - 10 q^{81} - 8 q^{83} - 4 q^{85} - 8 q^{87} + 12 q^{89} + 4 q^{91} + 4 q^{93} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 - 2 * q^9 + 2 * q^11 + 4 * q^13 + 4 * q^17 + 2 * q^25 + 12 * q^29 - 2 * q^35 - 12 * q^39 + 4 * q^41 - 4 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 + 4 * q^51 + 8 * q^53 - 2 * q^55 + 8 * q^57 - 8 * q^59 + 4 * q^61 - 2 * q^63 - 4 * q^65 - 8 * q^67 + 8 * q^69 + 4 * q^73 + 2 * q^77 + 4 * q^79 - 10 * q^81 - 8 * q^83 - 4 * q^85 - 8 * q^87 + 12 * q^89 + 4 * q^91 + 4 * q^93 + 4 * q^97 - 2 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.24264	1.73140	0.865699	−	0.500566i	$-0.166875\pi$
$13$	6.24264	1.73140	0.865699	+	0.500566i	$0.166875\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	0.585786	0.142074	0.0710370	−	0.997474i	$-0.477369\pi$
$17$	0.585786	0.142074	0.0710370	+	0.997474i	$0.477369\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	−1.41421	−0.308607
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	8.82843	1.63940	0.819699	−	0.572795i	$-0.194141\pi$
$29$	8.82843	1.63940	0.819699	+	0.572795i	$0.194141\pi$
$30$	0	0
$31$	−1.41421	−0.254000	−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421	−0.254000	−0.127000	+	0.991903i	$0.540535\pi$
$32$	0	0
$33$	−1.41421	−0.246183
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−5.65685	−0.929981	−0.464991	−	0.885316i	$-0.653942\pi$
$37$	−5.65685	−0.929981	−0.464991	+	0.885316i	$0.653942\pi$
$38$	0	0
$39$	−8.82843	−1.41368
$40$	0	0
$41$	9.07107	1.41666	0.708331	−	0.705880i	$-0.249448\pi$
$41$	9.07107	1.41666	0.708331	+	0.705880i	$0.249448\pi$
$42$	0	0
$43$	0.828427	0.126334	0.0631670	−	0.998003i	$-0.479880\pi$
$43$	0.828427	0.126334	0.0631670	+	0.998003i	$0.479880\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−8.24264	−1.20231	−0.601156	−	0.799131i	$-0.705293\pi$
$47$	−8.24264	−1.20231	−0.601156	+	0.799131i	$0.705293\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.828427	−0.116003
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−8.24264	−1.07310	−0.536550	−	0.843868i	$-0.680273\pi$
$59$	−8.24264	−1.07310	−0.536550	+	0.843868i	$0.680273\pi$
$60$	0	0
$61$	0.585786	0.0750023	0.0375011	−	0.999297i	$-0.488060\pi$
$61$	0.585786	0.0750023	0.0375011	+	0.999297i	$0.488060\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.24264	−0.774304
$66$	0	0
$67$	−1.17157	−0.143130	−0.0715652	−	0.997436i	$-0.522799\pi$
$67$	−1.17157	−0.143130	−0.0715652	+	0.997436i	$0.522799\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	2.82843	0.335673	0.167836	−	0.985815i	$-0.446322\pi$
$71$	2.82843	0.335673	0.167836	+	0.985815i	$0.446322\pi$
$72$	0	0
$73$	−7.89949	−0.924566	−0.462283	−	0.886732i	$-0.652970\pi$
$73$	−7.89949	−0.924566	−0.462283	+	0.886732i	$0.652970\pi$
$74$	0	0
$75$	−1.41421	−0.163299
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	4.82843	0.543240	0.271620	−	0.962405i	$-0.412441\pi$
$79$	4.82843	0.543240	0.271620	+	0.962405i	$0.412441\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−6.82843	−0.749517	−0.374759	−	0.927122i	$-0.622274\pi$
$83$	−6.82843	−0.749517	−0.374759	+	0.927122i	$0.622274\pi$
$84$	0	0
$85$	−0.585786	−0.0635375
$86$	0	0
$87$	−12.4853	−1.33856
$88$	0	0
$89$	11.6569	1.23562	0.617812	−	0.786326i	$-0.288019\pi$
$89$	11.6569	1.23562	0.617812	+	0.786326i	$0.288019\pi$
$90$	0	0
$91$	6.24264	0.654407
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	4.82843	0.490252	0.245126	−	0.969491i	$-0.421171\pi$
$97$	4.82843	0.490252	0.245126	+	0.969491i	$0.421171\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−14.7279	−1.46548	−0.732742	−	0.680507i	$-0.761760\pi$
$101$	−14.7279	−1.46548	−0.732742	+	0.680507i	$0.761760\pi$
$102$	0	0
$103$	10.5858	1.04305	0.521524	−	0.853236i	$-0.325364\pi$
$103$	10.5858	1.04305	0.521524	+	0.853236i	$0.325364\pi$
$104$	0	0
$105$	1.41421	0.138013
$106$	0	0
$107$	7.31371	0.707043	0.353521	−	0.935426i	$-0.384984\pi$
$107$	7.31371	0.707043	0.353521	+	0.935426i	$0.384984\pi$
$108$	0	0
$109$	−14.4853	−1.38744	−0.693719	−	0.720246i	$-0.744029\pi$
$109$	−14.4853	−1.38744	−0.693719	+	0.720246i	$0.744029\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	2.82843	0.263752
$116$	0	0
$117$	−6.24264	−0.577132
$118$	0	0
$119$	0.585786	0.0536990
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.8284	−1.15670
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.6569	−1.21185	−0.605925	−	0.795522i	$-0.707197\pi$
$127$	−13.6569	−1.21185	−0.605925	+	0.795522i	$0.707197\pi$
$128$	0	0
$129$	−1.17157	−0.103151
$130$	0	0
$131$	2.34315	0.204722	0.102361	−	0.994747i	$-0.467360\pi$
$131$	2.34315	0.204722	0.102361	+	0.994747i	$0.467360\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0	0
$135$	−5.65685	−0.486864
$136$	0	0
$137$	0.686292	0.0586338	0.0293169	−	0.999570i	$-0.490667\pi$
$137$	0.686292	0.0586338	0.0293169	+	0.999570i	$0.490667\pi$
$138$	0	0
$139$	10.8284	0.918455	0.459228	−	0.888319i	$-0.348126\pi$
$139$	10.8284	0.918455	0.459228	+	0.888319i	$0.348126\pi$
$140$	0	0
$141$	11.6569	0.981684
$142$	0	0
$143$	6.24264	0.522036
$144$	0	0
$145$	−8.82843	−0.733161
$146$	0	0
$147$	−1.41421	−0.116642
$148$	0	0
$149$	−14.9706	−1.22644	−0.613218	−	0.789914i	$-0.710125\pi$
$149$	−14.9706	−1.22644	−0.613218	+	0.789914i	$0.710125\pi$
$150$	0	0
$151$	12.8284	1.04396	0.521981	−	0.852957i	$-0.325193\pi$
$151$	12.8284	1.04396	0.521981	+	0.852957i	$0.325193\pi$
$152$	0	0
$153$	−0.585786	−0.0473580
$154$	0	0
$155$	1.41421	0.113592
$156$	0	0
$157$	4.34315	0.346621	0.173310	−	0.984867i	$-0.444554\pi$
$157$	4.34315	0.346621	0.173310	+	0.984867i	$0.444554\pi$
$158$	0	0
$159$	−5.65685	−0.448618
$160$	0	0
$161$	−2.82843	−0.222911
$162$	0	0
$163$	−8.48528	−0.664619	−0.332309	−	0.943170i	$-0.607828\pi$
$163$	−8.48528	−0.664619	−0.332309	+	0.943170i	$0.607828\pi$
$164$	0	0
$165$	1.41421	0.110096
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	5.07107	0.385546	0.192773	−	0.981243i	$-0.438252\pi$
$173$	5.07107	0.385546	0.192773	+	0.981243i	$0.438252\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	11.6569	0.876183
$178$	0	0
$179$	−0.686292	−0.0512958	−0.0256479	−	0.999671i	$-0.508165\pi$
$179$	−0.686292	−0.0512958	−0.0256479	+	0.999671i	$0.508165\pi$
$180$	0	0
$181$	−10.4853	−0.779365	−0.389682	−	0.920949i	$-0.627415\pi$
$181$	−10.4853	−0.779365	−0.389682	+	0.920949i	$0.627415\pi$
$182$	0	0
$183$	−0.828427	−0.0612391
$184$	0	0
$185$	5.65685	0.415900
$186$	0	0
$187$	0.585786	0.0428369
$188$	0	0
$189$	5.65685	0.411476
$190$	0	0
$191$	10.3431	0.748404	0.374202	−	0.927347i	$-0.377917\pi$
$191$	10.3431	0.748404	0.374202	+	0.927347i	$0.377917\pi$
$192$	0	0
$193$	3.65685	0.263226	0.131613	−	0.991301i	$-0.457984\pi$
$193$	3.65685	0.263226	0.131613	+	0.991301i	$0.457984\pi$
$194$	0	0
$195$	8.82843	0.632217
$196$	0	0
$197$	0.343146	0.0244481	0.0122241	−	0.999925i	$-0.496109\pi$
$197$	0.343146	0.0244481	0.0122241	+	0.999925i	$0.496109\pi$
$198$	0	0
$199$	−7.07107	−0.501255	−0.250627	−	0.968084i	$-0.580637\pi$
$199$	−7.07107	−0.501255	−0.250627	+	0.968084i	$0.580637\pi$
$200$	0	0
$201$	1.65685	0.116865
$202$	0	0
$203$	8.82843	0.619634
$204$	0	0
$205$	−9.07107	−0.633551
$206$	0	0
$207$	2.82843	0.196589
$208$	0	0
$209$	−2.82843	−0.195646
$210$	0	0
$211$	−26.6274	−1.83311	−0.916553	−	0.399912i	$-0.869041\pi$
$211$	−26.6274	−1.83311	−0.916553	+	0.399912i	$0.869041\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	−0.828427	−0.0564983
$216$	0	0
$217$	−1.41421	−0.0960031
$218$	0	0
$219$	11.1716	0.754905
$220$	0	0
$221$	3.65685	0.245987
$222$	0	0
$223$	21.8995	1.46650	0.733249	−	0.679960i	$-0.238003\pi$
$223$	21.8995	1.46650	0.733249	+	0.679960i	$0.238003\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	16.9706	1.12638	0.563188	−	0.826329i	$-0.309575\pi$
$227$	16.9706	1.12638	0.563188	+	0.826329i	$0.309575\pi$
$228$	0	0
$229$	0.828427	0.0547440	0.0273720	−	0.999625i	$-0.491286\pi$
$229$	0.828427	0.0547440	0.0273720	+	0.999625i	$0.491286\pi$
$230$	0	0
$231$	−1.41421	−0.0930484
$232$	0	0
$233$	14.4853	0.948962	0.474481	−	0.880266i	$-0.342636\pi$
$233$	14.4853	0.948962	0.474481	+	0.880266i	$0.342636\pi$
$234$	0	0
$235$	8.24264	0.537691
$236$	0	0
$237$	−6.82843	−0.443554
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	1.07107	0.0689935	0.0344968	−	0.999405i	$-0.489017\pi$
$241$	1.07107	0.0689935	0.0344968	+	0.999405i	$0.489017\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−17.6569	−1.12348
$248$	0	0
$249$	9.65685	0.611978
$250$	0	0
$251$	21.8995	1.38228	0.691142	−	0.722719i	$-0.257108\pi$
$251$	21.8995	1.38228	0.691142	+	0.722719i	$0.257108\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	0	0
$255$	0.828427	0.0518781
$256$	0	0
$257$	1.31371	0.0819469	0.0409734	−	0.999160i	$-0.486954\pi$
$257$	1.31371	0.0819469	0.0409734	+	0.999160i	$0.486954\pi$
$258$	0	0
$259$	−5.65685	−0.351500
$260$	0	0
$261$	−8.82843	−0.546466
$262$	0	0
$263$	27.3137	1.68424	0.842118	−	0.539294i	$-0.181309\pi$
$263$	27.3137	1.68424	0.842118	+	0.539294i	$0.181309\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	−16.4853	−1.00888
$268$	0	0
$269$	22.4853	1.37095	0.685476	−	0.728095i	$-0.259594\pi$
$269$	22.4853	1.37095	0.685476	+	0.728095i	$0.259594\pi$
$270$	0	0
$271$	32.4853	1.97334	0.986670	−	0.162733i	$-0.0520309\pi$
$271$	32.4853	1.97334	0.986670	+	0.162733i	$0.0520309\pi$
$272$	0	0
$273$	−8.82843	−0.534321
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	21.3137	1.28062	0.640308	−	0.768118i	$-0.278807\pi$
$277$	21.3137	1.28062	0.640308	+	0.768118i	$0.278807\pi$
$278$	0	0
$279$	1.41421	0.0846668
$280$	0	0
$281$	13.5147	0.806221	0.403110	−	0.915151i	$-0.367929\pi$
$281$	13.5147	0.806221	0.403110	+	0.915151i	$0.367929\pi$
$282$	0	0
$283$	11.3137	0.672530	0.336265	−	0.941767i	$-0.390836\pi$
$283$	11.3137	0.672530	0.336265	+	0.941767i	$0.390836\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	9.07107	0.535448
$288$	0	0
$289$	−16.6569	−0.979815
$290$	0	0
$291$	−6.82843	−0.400289
$292$	0	0
$293$	18.2426	1.06575	0.532873	−	0.846195i	$-0.321112\pi$
$293$	18.2426	1.06575	0.532873	+	0.846195i	$0.321112\pi$
$294$	0	0
$295$	8.24264	0.479905
$296$	0	0
$297$	5.65685	0.328244
$298$	0	0
$299$	−17.6569	−1.02112
$300$	0	0
$301$	0.828427	0.0477497
$302$	0	0
$303$	20.8284	1.19656
$304$	0	0
$305$	−0.585786	−0.0335420
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	−14.9706	−0.851646
$310$	0	0
$311$	20.2426	1.14785	0.573927	−	0.818906i	$-0.305419\pi$
$311$	20.2426	1.14785	0.573927	+	0.818906i	$0.305419\pi$
$312$	0	0
$313$	29.3137	1.65691	0.828454	−	0.560057i	$-0.189221\pi$
$313$	29.3137	1.65691	0.828454	+	0.560057i	$0.189221\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	17.3137	0.972435	0.486217	−	0.873838i	$-0.338376\pi$
$317$	17.3137	0.972435	0.486217	+	0.873838i	$0.338376\pi$
$318$	0	0
$319$	8.82843	0.494297
$320$	0	0
$321$	−10.3431	−0.577298
$322$	0	0
$323$	−1.65685	−0.0921898
$324$	0	0
$325$	6.24264	0.346279
$326$	0	0
$327$	20.4853	1.13284
$328$	0	0
$329$	−8.24264	−0.454431
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	5.65685	0.309994
$334$	0	0
$335$	1.17157	0.0640099
$336$	0	0
$337$	−25.7990	−1.40536	−0.702680	−	0.711506i	$-0.748014\pi$
$337$	−25.7990	−1.40536	−0.702680	+	0.711506i	$0.748014\pi$
$338$	0	0
$339$	−19.7990	−1.07533
$340$	0	0
$341$	−1.41421	−0.0765840
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−14.4853	−0.777611	−0.388805	−	0.921320i	$-0.627112\pi$
$347$	−14.4853	−0.777611	−0.388805	+	0.921320i	$0.627112\pi$
$348$	0	0
$349$	7.21320	0.386114	0.193057	−	0.981188i	$-0.438160\pi$
$349$	7.21320	0.386114	0.193057	+	0.981188i	$0.438160\pi$
$350$	0	0
$351$	35.3137	1.88491
$352$	0	0
$353$	25.3137	1.34731	0.673656	−	0.739045i	$-0.264723\pi$
$353$	25.3137	1.34731	0.673656	+	0.739045i	$0.264723\pi$
$354$	0	0
$355$	−2.82843	−0.150117
$356$	0	0
$357$	−0.828427	−0.0438450
$358$	0	0
$359$	6.48528	0.342280	0.171140	−	0.985247i	$-0.445255\pi$
$359$	6.48528	0.342280	0.171140	+	0.985247i	$0.445255\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−1.41421	−0.0742270
$364$	0	0
$365$	7.89949	0.413478
$366$	0	0
$367$	−4.92893	−0.257288	−0.128644	−	0.991691i	$-0.541062\pi$
$367$	−4.92893	−0.257288	−0.128644	+	0.991691i	$0.541062\pi$
$368$	0	0
$369$	−9.07107	−0.472221
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	−2.68629	−0.139091	−0.0695455	−	0.997579i	$-0.522155\pi$
$373$	−2.68629	−0.139091	−0.0695455	+	0.997579i	$0.522155\pi$
$374$	0	0
$375$	1.41421	0.0730297
$376$	0	0
$377$	55.1127	2.83845
$378$	0	0
$379$	−7.51472	−0.386005	−0.193003	−	0.981198i	$-0.561823\pi$
$379$	−7.51472	−0.386005	−0.193003	+	0.981198i	$0.561823\pi$
$380$	0	0
$381$	19.3137	0.989471
$382$	0	0
$383$	19.0711	0.974486	0.487243	−	0.873266i	$-0.338003\pi$
$383$	19.0711	0.974486	0.487243	+	0.873266i	$0.338003\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−0.828427	−0.0421113
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−1.65685	−0.0837907
$392$	0	0
$393$	−3.31371	−0.167154
$394$	0	0
$395$	−4.82843	−0.242945
$396$	0	0
$397$	−22.9706	−1.15286	−0.576430	−	0.817147i	$-0.695555\pi$
$397$	−22.9706	−1.15286	−0.576430	+	0.817147i	$0.695555\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−18.9706	−0.947345	−0.473672	−	0.880701i	$-0.657072\pi$
$401$	−18.9706	−0.947345	−0.473672	+	0.880701i	$0.657072\pi$
$402$	0	0
$403$	−8.82843	−0.439775
$404$	0	0
$405$	5.00000	0.248452
$406$	0	0
$407$	−5.65685	−0.280400
$408$	0	0
$409$	−25.5563	−1.26368	−0.631840	−	0.775099i	$-0.717700\pi$
$409$	−25.5563	−1.26368	−0.631840	+	0.775099i	$0.717700\pi$
$410$	0	0
$411$	−0.970563	−0.0478743
$412$	0	0
$413$	−8.24264	−0.405594
$414$	0	0
$415$	6.82843	0.335194
$416$	0	0
$417$	−15.3137	−0.749916
$418$	0	0
$419$	−22.3848	−1.09357	−0.546784	−	0.837274i	$-0.684148\pi$
$419$	−22.3848	−1.09357	−0.546784	+	0.837274i	$0.684148\pi$
$420$	0	0
$421$	−5.65685	−0.275698	−0.137849	−	0.990453i	$-0.544019\pi$
$421$	−5.65685	−0.275698	−0.137849	+	0.990453i	$0.544019\pi$
$422$	0	0
$423$	8.24264	0.400771
$424$	0	0
$425$	0.585786	0.0284148
$426$	0	0
$427$	0.585786	0.0283482
$428$	0	0
$429$	−8.82843	−0.426240
$430$	0	0
$431$	−8.82843	−0.425250	−0.212625	−	0.977134i	$-0.568201\pi$
$431$	−8.82843	−0.425250	−0.212625	+	0.977134i	$0.568201\pi$
$432$	0	0
$433$	−8.82843	−0.424267	−0.212134	−	0.977241i	$-0.568041\pi$
$433$	−8.82843	−0.424267	−0.212134	+	0.977241i	$0.568041\pi$
$434$	0	0
$435$	12.4853	0.598623
$436$	0	0
$437$	8.00000	0.382692
$438$	0	0
$439$	29.4558	1.40585	0.702925	−	0.711264i	$-0.251877\pi$
$439$	29.4558	1.40585	0.702925	+	0.711264i	$0.251877\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	24.2843	1.15378	0.576890	−	0.816822i	$-0.304266\pi$
$443$	24.2843	1.15378	0.576890	+	0.816822i	$0.304266\pi$
$444$	0	0
$445$	−11.6569	−0.552588
$446$	0	0
$447$	21.1716	1.00138
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	9.07107	0.427140
$452$	0	0
$453$	−18.1421	−0.852392
$454$	0	0
$455$	−6.24264	−0.292660
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	3.31371	0.154671
$460$	0	0
$461$	−33.5563	−1.56287	−0.781437	−	0.623984i	$-0.785513\pi$
$461$	−33.5563	−1.56287	−0.781437	+	0.623984i	$0.785513\pi$
$462$	0	0
$463$	−7.51472	−0.349239	−0.174619	−	0.984636i	$-0.555869\pi$
$463$	−7.51472	−0.349239	−0.174619	+	0.984636i	$0.555869\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	15.0711	0.697406	0.348703	−	0.937233i	$-0.386622\pi$
$467$	15.0711	0.697406	0.348703	+	0.937233i	$0.386622\pi$
$468$	0	0
$469$	−1.17157	−0.0540982
$470$	0	0
$471$	−6.14214	−0.283015
$472$	0	0
$473$	0.828427	0.0380911
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	−4.00000	−0.183147
$478$	0	0
$479$	−9.17157	−0.419060	−0.209530	−	0.977802i	$-0.567193\pi$
$479$	−9.17157	−0.419060	−0.209530	+	0.977802i	$0.567193\pi$
$480$	0	0
$481$	−35.3137	−1.61017
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	−4.82843	−0.219248
$486$	0	0
$487$	−31.7990	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$487$	−31.7990	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	4.82843	0.217904	0.108952	−	0.994047i	$-0.465251\pi$
$491$	4.82843	0.217904	0.108952	+	0.994047i	$0.465251\pi$
$492$	0	0
$493$	5.17157	0.232916
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	2.82843	0.126872
$498$	0	0
$499$	20.4853	0.917047	0.458524	−	0.888682i	$-0.348378\pi$
$499$	20.4853	0.917047	0.458524	+	0.888682i	$0.348378\pi$
$500$	0	0
$501$	−5.65685	−0.252730
$502$	0	0
$503$	13.1716	0.587291	0.293646	−	0.955914i	$-0.405131\pi$
$503$	13.1716	0.587291	0.293646	+	0.955914i	$0.405131\pi$
$504$	0	0
$505$	14.7279	0.655384
$506$	0	0
$507$	−36.7279	−1.63114
$508$	0	0
$509$	−5.79899	−0.257036	−0.128518	−	0.991707i	$-0.541022\pi$
$509$	−5.79899	−0.257036	−0.128518	+	0.991707i	$0.541022\pi$
$510$	0	0
$511$	−7.89949	−0.349453
$512$	0	0
$513$	−16.0000	−0.706417
$514$	0	0
$515$	−10.5858	−0.466465
$516$	0	0
$517$	−8.24264	−0.362511
$518$	0	0
$519$	−7.17157	−0.314797
$520$	0	0
$521$	−12.1421	−0.531957	−0.265978	−	0.963979i	$-0.585695\pi$
$521$	−12.1421	−0.531957	−0.265978	+	0.963979i	$0.585695\pi$
$522$	0	0
$523$	26.6274	1.16434	0.582168	−	0.813069i	$-0.302205\pi$
$523$	26.6274	1.16434	0.582168	+	0.813069i	$0.302205\pi$
$524$	0	0
$525$	−1.41421	−0.0617213
$526$	0	0
$527$	−0.828427	−0.0360869
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	8.24264	0.357700
$532$	0	0
$533$	56.6274	2.45281
$534$	0	0
$535$	−7.31371	−0.316199
$536$	0	0
$537$	0.970563	0.0418829
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−25.3137	−1.08832	−0.544161	−	0.838981i	$-0.683152\pi$
$541$	−25.3137	−1.08832	−0.544161	+	0.838981i	$0.683152\pi$
$542$	0	0
$543$	14.8284	0.636349
$544$	0	0
$545$	14.4853	0.620481
$546$	0	0
$547$	−34.6274	−1.48056	−0.740281	−	0.672298i	$-0.765307\pi$
$547$	−34.6274	−1.48056	−0.740281	+	0.672298i	$0.765307\pi$
$548$	0	0
$549$	−0.585786	−0.0250008
$550$	0	0
$551$	−24.9706	−1.06378
$552$	0	0
$553$	4.82843	0.205326
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	0	0
$559$	5.17157	0.218734
$560$	0	0
$561$	−0.828427	−0.0349762
$562$	0	0
$563$	19.7990	0.834428	0.417214	−	0.908808i	$-0.363007\pi$
$563$	19.7990	0.834428	0.417214	+	0.908808i	$0.363007\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	0	0
$567$	−5.00000	−0.209980
$568$	0	0
$569$	8.82843	0.370107	0.185053	−	0.982728i	$-0.440754\pi$
$569$	8.82843	0.370107	0.185053	+	0.982728i	$0.440754\pi$
$570$	0	0
$571$	−9.65685	−0.404127	−0.202063	−	0.979372i	$-0.564765\pi$
$571$	−9.65685	−0.404127	−0.202063	+	0.979372i	$0.564765\pi$
$572$	0	0
$573$	−14.6274	−0.611069
$574$	0	0
$575$	−2.82843	−0.117954
$576$	0	0
$577$	44.8284	1.86623	0.933116	−	0.359576i	$-0.117079\pi$
$577$	44.8284	1.86623	0.933116	+	0.359576i	$0.117079\pi$
$578$	0	0
$579$	−5.17157	−0.214923
$580$	0	0
$581$	−6.82843	−0.283291
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	6.24264	0.258101
$586$	0	0
$587$	24.0416	0.992304	0.496152	−	0.868236i	$-0.334746\pi$
$587$	24.0416	0.992304	0.496152	+	0.868236i	$0.334746\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−0.485281	−0.0199618
$592$	0	0
$593$	40.8701	1.67833	0.839166	−	0.543875i	$-0.183044\pi$
$593$	40.8701	1.67833	0.839166	+	0.543875i	$0.183044\pi$
$594$	0	0
$595$	−0.585786	−0.0240149
$596$	0	0
$597$	10.0000	0.409273
$598$	0	0
$599$	27.1127	1.10779	0.553897	−	0.832585i	$-0.313140\pi$
$599$	27.1127	1.10779	0.553897	+	0.832585i	$0.313140\pi$
$600$	0	0
$601$	−29.5563	−1.20563	−0.602814	−	0.797882i	$-0.705954\pi$
$601$	−29.5563	−1.20563	−0.602814	+	0.797882i	$0.705954\pi$
$602$	0	0
$603$	1.17157	0.0477101
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−16.2843	−0.660958	−0.330479	−	0.943813i	$-0.607210\pi$
$607$	−16.2843	−0.660958	−0.330479	+	0.943813i	$0.607210\pi$
$608$	0	0
$609$	−12.4853	−0.505929
$610$	0	0
$611$	−51.4558	−2.08168
$612$	0	0
$613$	36.4264	1.47125	0.735624	−	0.677390i	$-0.236889\pi$
$613$	36.4264	1.47125	0.735624	+	0.677390i	$0.236889\pi$
$614$	0	0
$615$	12.8284	0.517292
$616$	0	0
$617$	39.3137	1.58271	0.791355	−	0.611357i	$-0.209376\pi$
$617$	39.3137	1.58271	0.791355	+	0.611357i	$0.209376\pi$
$618$	0	0
$619$	25.6985	1.03291	0.516455	−	0.856315i	$-0.327251\pi$
$619$	25.6985	1.03291	0.516455	+	0.856315i	$0.327251\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	0	0
$623$	11.6569	0.467022
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.00000	0.159745
$628$	0	0
$629$	−3.31371	−0.132126
$630$	0	0
$631$	40.2843	1.60369	0.801846	−	0.597531i	$-0.203852\pi$
$631$	40.2843	1.60369	0.801846	+	0.597531i	$0.203852\pi$
$632$	0	0
$633$	37.6569	1.49673
$634$	0	0
$635$	13.6569	0.541956
$636$	0	0
$637$	6.24264	0.247342
$638$	0	0
$639$	−2.82843	−0.111891
$640$	0	0
$641$	33.9411	1.34059	0.670297	−	0.742093i	$-0.266167\pi$
$641$	33.9411	1.34059	0.670297	+	0.742093i	$0.266167\pi$
$642$	0	0
$643$	43.8406	1.72891	0.864453	−	0.502714i	$-0.167665\pi$
$643$	43.8406	1.72891	0.864453	+	0.502714i	$0.167665\pi$
$644$	0	0
$645$	1.17157	0.0461306
$646$	0	0
$647$	2.58579	0.101658	0.0508289	−	0.998707i	$-0.483814\pi$
$647$	2.58579	0.101658	0.0508289	+	0.998707i	$0.483814\pi$
$648$	0	0
$649$	−8.24264	−0.323552
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	10.9706	0.429311	0.214656	−	0.976690i	$-0.431137\pi$
$653$	10.9706	0.429311	0.214656	+	0.976690i	$0.431137\pi$
$654$	0	0
$655$	−2.34315	−0.0915543
$656$	0	0
$657$	7.89949	0.308189
$658$	0	0
$659$	−45.1127	−1.75734	−0.878671	−	0.477428i	$-0.841569\pi$
$659$	−45.1127	−1.75734	−0.878671	+	0.477428i	$0.841569\pi$
$660$	0	0
$661$	46.9706	1.82694	0.913472	−	0.406903i	$-0.133391\pi$
$661$	46.9706	1.82694	0.913472	+	0.406903i	$0.133391\pi$
$662$	0	0
$663$	−5.17157	−0.200847
$664$	0	0
$665$	2.82843	0.109682
$666$	0	0
$667$	−24.9706	−0.966864
$668$	0	0
$669$	−30.9706	−1.19739
$670$	0	0
$671$	0.585786	0.0226140
$672$	0	0
$673$	7.17157	0.276444	0.138222	−	0.990401i	$-0.455861\pi$
$673$	7.17157	0.276444	0.138222	+	0.990401i	$0.455861\pi$
$674$	0	0
$675$	5.65685	0.217732
$676$	0	0
$677$	−26.7279	−1.02724	−0.513619	−	0.858019i	$-0.671695\pi$
$677$	−26.7279	−1.02724	−0.513619	+	0.858019i	$0.671695\pi$
$678$	0	0
$679$	4.82843	0.185298
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	−38.6274	−1.47804	−0.739019	−	0.673685i	$-0.764710\pi$
$683$	−38.6274	−1.47804	−0.739019	+	0.673685i	$0.764710\pi$
$684$	0	0
$685$	−0.686292	−0.0262219
$686$	0	0
$687$	−1.17157	−0.0446983
$688$	0	0
$689$	24.9706	0.951303
$690$	0	0
$691$	−39.7574	−1.51244	−0.756221	−	0.654317i	$-0.772956\pi$
$691$	−39.7574	−1.51244	−0.756221	+	0.654317i	$0.772956\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−10.8284	−0.410746
$696$	0	0
$697$	5.31371	0.201271
$698$	0	0
$699$	−20.4853	−0.774825
$700$	0	0
$701$	−17.7990	−0.672259	−0.336129	−	0.941816i	$-0.609118\pi$
$701$	−17.7990	−0.672259	−0.336129	+	0.941816i	$0.609118\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	0	0
$705$	−11.6569	−0.439023
$706$	0	0
$707$	−14.7279	−0.553901
$708$	0	0
$709$	49.5980	1.86269	0.931346	−	0.364136i	$-0.118636\pi$
$709$	49.5980	1.86269	0.931346	+	0.364136i	$0.118636\pi$
$710$	0	0
$711$	−4.82843	−0.181080
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	−6.24264	−0.233462
$716$	0	0
$717$	11.3137	0.422518
$718$	0	0
$719$	37.6985	1.40592	0.702958	−	0.711231i	$-0.251862\pi$
$719$	37.6985	1.40592	0.702958	+	0.711231i	$0.251862\pi$
$720$	0	0
$721$	10.5858	0.394235
$722$	0	0
$723$	−1.51472	−0.0563330
$724$	0	0
$725$	8.82843	0.327880
$726$	0	0
$727$	22.8701	0.848203	0.424102	−	0.905615i	$-0.360590\pi$
$727$	22.8701	0.848203	0.424102	+	0.905615i	$0.360590\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	0.485281	0.0179488
$732$	0	0
$733$	18.2426	0.673807	0.336904	−	0.941539i	$-0.390620\pi$
$733$	18.2426	0.673807	0.336904	+	0.941539i	$0.390620\pi$
$734$	0	0
$735$	1.41421	0.0521641
$736$	0	0
$737$	−1.17157	−0.0431554
$738$	0	0
$739$	8.28427	0.304742	0.152371	−	0.988323i	$-0.451309\pi$
$739$	8.28427	0.304742	0.152371	+	0.988323i	$0.451309\pi$
$740$	0	0
$741$	24.9706	0.917317
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	14.9706	0.548479
$746$	0	0
$747$	6.82843	0.249839
$748$	0	0
$749$	7.31371	0.267237
$750$	0	0
$751$	38.1421	1.39183	0.695913	−	0.718126i	$-0.255000\pi$
$751$	38.1421	1.39183	0.695913	+	0.718126i	$0.255000\pi$
$752$	0	0
$753$	−30.9706	−1.12863
$754$	0	0
$755$	−12.8284	−0.466874
$756$	0	0
$757$	−51.2548	−1.86289	−0.931444	−	0.363884i	$-0.881450\pi$
$757$	−51.2548	−1.86289	−0.931444	+	0.363884i	$0.881450\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	−15.2132	−0.551478	−0.275739	−	0.961233i	$-0.588923\pi$
$761$	−15.2132	−0.551478	−0.275739	+	0.961233i	$0.588923\pi$
$762$	0	0
$763$	−14.4853	−0.524402
$764$	0	0
$765$	0.585786	0.0211792
$766$	0	0
$767$	−51.4558	−1.85796
$768$	0	0
$769$	−8.78680	−0.316860	−0.158430	−	0.987370i	$-0.550643\pi$
$769$	−8.78680	−0.316860	−0.158430	+	0.987370i	$0.550643\pi$
$770$	0	0
$771$	−1.85786	−0.0669094
$772$	0	0
$773$	−0.828427	−0.0297965	−0.0148982	−	0.999889i	$-0.504742\pi$
$773$	−0.828427	−0.0297965	−0.0148982	+	0.999889i	$0.504742\pi$
$774$	0	0
$775$	−1.41421	−0.0508001
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	−25.6569	−0.919252
$780$	0	0
$781$	2.82843	0.101209
$782$	0	0
$783$	49.9411	1.78475
$784$	0	0
$785$	−4.34315	−0.155014
$786$	0	0
$787$	−15.7990	−0.563173	−0.281587	−	0.959536i	$-0.590861\pi$
$787$	−15.7990	−0.563173	−0.281587	+	0.959536i	$0.590861\pi$
$788$	0	0
$789$	−38.6274	−1.37517
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	3.65685	0.129859
$794$	0	0
$795$	5.65685	0.200628
$796$	0	0
$797$	14.6863	0.520215	0.260108	−	0.965580i	$-0.416242\pi$
$797$	14.6863	0.520215	0.260108	+	0.965580i	$0.416242\pi$
$798$	0	0
$799$	−4.82843	−0.170817
$800$	0	0
$801$	−11.6569	−0.411875
$802$	0	0
$803$	−7.89949	−0.278767
$804$	0	0
$805$	2.82843	0.0996890
$806$	0	0
$807$	−31.7990	−1.11938
$808$	0	0
$809$	−41.3137	−1.45251	−0.726256	−	0.687424i	$-0.758741\pi$
$809$	−41.3137	−1.45251	−0.726256	+	0.687424i	$0.758741\pi$
$810$	0	0
$811$	−24.2843	−0.852736	−0.426368	−	0.904550i	$-0.640207\pi$
$811$	−24.2843	−0.852736	−0.426368	+	0.904550i	$0.640207\pi$
$812$	0	0
$813$	−45.9411	−1.61123
$814$	0	0
$815$	8.48528	0.297226
$816$	0	0
$817$	−2.34315	−0.0819763
$818$	0	0
$819$	−6.24264	−0.218136
$820$	0	0
$821$	44.3431	1.54759	0.773793	−	0.633438i	$-0.218357\pi$
$821$	44.3431	1.54759	0.773793	+	0.633438i	$0.218357\pi$
$822$	0	0
$823$	−52.2843	−1.82252	−0.911258	−	0.411837i	$-0.864887\pi$
$823$	−52.2843	−1.82252	−0.911258	+	0.411837i	$0.864887\pi$
$824$	0	0
$825$	−1.41421	−0.0492366
$826$	0	0
$827$	−4.82843	−0.167901	−0.0839504	−	0.996470i	$-0.526754\pi$
$827$	−4.82843	−0.167901	−0.0839504	+	0.996470i	$0.526754\pi$
$828$	0	0
$829$	16.6274	0.577494	0.288747	−	0.957405i	$-0.406761\pi$
$829$	16.6274	0.577494	0.288747	+	0.957405i	$0.406761\pi$
$830$	0	0
$831$	−30.1421	−1.04562
$832$	0	0
$833$	0.585786	0.0202963
$834$	0	0
$835$	−4.00000	−0.138426
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−47.5563	−1.64183	−0.820914	−	0.571052i	$-0.806535\pi$
$839$	−47.5563	−1.64183	−0.820914	+	0.571052i	$0.806535\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	0	0
$843$	−19.1127	−0.658276
$844$	0	0
$845$	−25.9706	−0.893415
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	16.0000	0.548473
$852$	0	0
$853$	49.5563	1.69678	0.848389	−	0.529374i	$-0.177573\pi$
$853$	49.5563	1.69678	0.848389	+	0.529374i	$0.177573\pi$
$854$	0	0
$855$	−2.82843	−0.0967302
$856$	0	0
$857$	−35.2132	−1.20286	−0.601430	−	0.798925i	$-0.705402\pi$
$857$	−35.2132	−1.20286	−0.601430	+	0.798925i	$0.705402\pi$
$858$	0	0
$859$	−35.0711	−1.19661	−0.598305	−	0.801269i	$-0.704159\pi$
$859$	−35.0711	−1.19661	−0.598305	+	0.801269i	$0.704159\pi$
$860$	0	0
$861$	−12.8284	−0.437192
$862$	0	0
$863$	−40.2843	−1.37129	−0.685646	−	0.727935i	$-0.740480\pi$
$863$	−40.2843	−1.37129	−0.685646	+	0.727935i	$0.740480\pi$
$864$	0	0
$865$	−5.07107	−0.172421
$866$	0	0
$867$	23.5563	0.800016
$868$	0	0
$869$	4.82843	0.163793
$870$	0	0
$871$	−7.31371	−0.247816
$872$	0	0
$873$	−4.82843	−0.163417
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−6.48528	−0.218992	−0.109496	−	0.993987i	$-0.534924\pi$
$877$	−6.48528	−0.218992	−0.109496	+	0.993987i	$0.534924\pi$
$878$	0	0
$879$	−25.7990	−0.870178
$880$	0	0
$881$	7.17157	0.241616	0.120808	−	0.992676i	$-0.461451\pi$
$881$	7.17157	0.241616	0.120808	+	0.992676i	$0.461451\pi$
$882$	0	0
$883$	−35.3137	−1.18840	−0.594200	−	0.804317i	$-0.702531\pi$
$883$	−35.3137	−1.18840	−0.594200	+	0.804317i	$0.702531\pi$
$884$	0	0
$885$	−11.6569	−0.391841
$886$	0	0
$887$	21.8579	0.733915	0.366958	−	0.930238i	$-0.380399\pi$
$887$	21.8579	0.733915	0.366958	+	0.930238i	$0.380399\pi$
$888$	0	0
$889$	−13.6569	−0.458036
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	23.3137	0.780164
$894$	0	0
$895$	0.686292	0.0229402
$896$	0	0
$897$	24.9706	0.833743
$898$	0	0
$899$	−12.4853	−0.416407
$900$	0	0
$901$	2.34315	0.0780615
$902$	0	0
$903$	−1.17157	−0.0389875
$904$	0	0
$905$	10.4853	0.348543
$906$	0	0
$907$	−38.8284	−1.28928	−0.644638	−	0.764488i	$-0.722992\pi$
$907$	−38.8284	−1.28928	−0.644638	+	0.764488i	$0.722992\pi$
$908$	0	0
$909$	14.7279	0.488494
$910$	0	0
$911$	−34.1421	−1.13118	−0.565590	−	0.824687i	$-0.691351\pi$
$911$	−34.1421	−1.13118	−0.565590	+	0.824687i	$0.691351\pi$
$912$	0	0
$913$	−6.82843	−0.225988
$914$	0	0
$915$	0.828427	0.0273870
$916$	0	0
$917$	2.34315	0.0773775
$918$	0	0
$919$	−6.62742	−0.218618	−0.109309	−	0.994008i	$-0.534864\pi$
$919$	−6.62742	−0.218618	−0.109309	+	0.994008i	$0.534864\pi$
$920$	0	0
$921$	−22.6274	−0.745599
$922$	0	0
$923$	17.6569	0.581182
$924$	0	0
$925$	−5.65685	−0.185996
$926$	0	0
$927$	−10.5858	−0.347683
$928$	0	0
$929$	46.4853	1.52513	0.762566	−	0.646910i	$-0.223939\pi$
$929$	46.4853	1.52513	0.762566	+	0.646910i	$0.223939\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	0	0
$933$	−28.6274	−0.937220
$934$	0	0
$935$	−0.585786	−0.0191573
$936$	0	0
$937$	−35.4142	−1.15693	−0.578466	−	0.815707i	$-0.696348\pi$
$937$	−35.4142	−1.15693	−0.578466	+	0.815707i	$0.696348\pi$
$938$	0	0
$939$	−41.4558	−1.35286
$940$	0	0
$941$	18.9289	0.617066	0.308533	−	0.951214i	$-0.400162\pi$
$941$	18.9289	0.617066	0.308533	+	0.951214i	$0.400162\pi$
$942$	0	0
$943$	−25.6569	−0.835502
$944$	0	0
$945$	−5.65685	−0.184017
$946$	0	0
$947$	31.5980	1.02680	0.513398	−	0.858151i	$-0.328386\pi$
$947$	31.5980	1.02680	0.513398	+	0.858151i	$0.328386\pi$
$948$	0	0
$949$	−49.3137	−1.60079
$950$	0	0
$951$	−24.4853	−0.793990
$952$	0	0
$953$	−20.6274	−0.668188	−0.334094	−	0.942540i	$-0.608430\pi$
$953$	−20.6274	−0.668188	−0.334094	+	0.942540i	$0.608430\pi$
$954$	0	0
$955$	−10.3431	−0.334696
$956$	0	0
$957$	−12.4853	−0.403592
$958$	0	0
$959$	0.686292	0.0221615
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	−7.31371	−0.235681
$964$	0	0
$965$	−3.65685	−0.117718
$966$	0	0
$967$	49.7990	1.60143	0.800714	−	0.599047i	$-0.204454\pi$
$967$	49.7990	1.60143	0.800714	+	0.599047i	$0.204454\pi$
$968$	0	0
$969$	2.34315	0.0752727
$970$	0	0
$971$	25.6985	0.824704	0.412352	−	0.911025i	$-0.364707\pi$
$971$	25.6985	0.824704	0.412352	+	0.911025i	$0.364707\pi$
$972$	0	0
$973$	10.8284	0.347143
$974$	0	0
$975$	−8.82843	−0.282736
$976$	0	0
$977$	−38.2843	−1.22482	−0.612411	−	0.790539i	$-0.709800\pi$
$977$	−38.2843	−1.22482	−0.612411	+	0.790539i	$0.709800\pi$
$978$	0	0
$979$	11.6569	0.372555
$980$	0	0
$981$	14.4853	0.462479
$982$	0	0
$983$	13.4142	0.427847	0.213923	−	0.976850i	$-0.431376\pi$
$983$	13.4142	0.427847	0.213923	+	0.976850i	$0.431376\pi$
$984$	0	0
$985$	−0.343146	−0.0109335
$986$	0	0
$987$	11.6569	0.371042
$988$	0	0
$989$	−2.34315	−0.0745077
$990$	0	0
$991$	34.4264	1.09359	0.546795	−	0.837266i	$-0.315848\pi$
$991$	34.4264	1.09359	0.546795	+	0.837266i	$0.315848\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.07107	0.224168
$996$	0	0
$997$	1.75736	0.0556561	0.0278281	−	0.999613i	$-0.491141\pi$
$997$	1.75736	0.0556561	0.0278281	+	0.999613i	$0.491141\pi$
$998$	0	0
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.z.1.1		2
4.3	odd	2		1540.2.a.d.1.2	✓	2
20.3	even	4		7700.2.e.m.1849.4		4
20.7	even	4		7700.2.e.m.1849.1		4
20.19	odd	2		7700.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1540.2.a.d.1.2	✓	2	4.3	odd	2
6160.2.a.z.1.1		2	1.1	even	1	trivial
7700.2.a.q.1.1		2	20.19	odd	2
7700.2.e.m.1849.1		4	20.7	even	4
7700.2.e.m.1849.4		4	20.3	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +1.00000 q^{11} +6.24264 q^{13} +1.41421 q^{15} +0.585786 q^{17} -2.82843 q^{19} -1.41421 q^{21} -2.82843 q^{23} +1.00000 q^{25} +5.65685 q^{27} +8.82843 q^{29} -1.41421 q^{31} -1.41421 q^{33} -1.00000 q^{35} -5.65685 q^{37} -8.82843 q^{39} +9.07107 q^{41} +0.828427 q^{43} +1.00000 q^{45} -8.24264 q^{47} +1.00000 q^{49} -0.828427 q^{51} +4.00000 q^{53} -1.00000 q^{55} +4.00000 q^{57} -8.24264 q^{59} +0.585786 q^{61} -1.00000 q^{63} -6.24264 q^{65} -1.17157 q^{67} +4.00000 q^{69} +2.82843 q^{71} -7.89949 q^{73} -1.41421 q^{75} +1.00000 q^{77} +4.82843 q^{79} -5.00000 q^{81} -6.82843 q^{83} -0.585786 q^{85} -12.4853 q^{87} +11.6569 q^{89} +6.24264 q^{91} +2.00000 q^{93} +2.82843 q^{95} +4.82843 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7 - 2 * q^9	\( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{17} + 2 q^{25} + 12 q^{29} - 2 q^{35} - 12 q^{39} + 4 q^{41} - 4 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} + 4 q^{51} + 8 q^{53} - 2 q^{55} + 8 q^{57} - 8 q^{59} + 4 q^{61} - 2 q^{63} - 4 q^{65} - 8 q^{67} + 8 q^{69} + 4 q^{73} + 2 q^{77} + 4 q^{79} - 10 q^{81} - 8 q^{83} - 4 q^{85} - 8 q^{87} + 12 q^{89} + 4 q^{91} + 4 q^{93} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 - 2 * q^9 + 2 * q^11 + 4 * q^13 + 4 * q^17 + 2 * q^25 + 12 * q^29 - 2 * q^35 - 12 * q^39 + 4 * q^41 - 4 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 + 4 * q^51 + 8 * q^53 - 2 * q^55 + 8 * q^57 - 8 * q^59 + 4 * q^61 - 2 * q^63 - 4 * q^65 - 8 * q^67 + 8 * q^69 + 4 * q^73 + 2 * q^77 + 4 * q^79 - 10 * q^81 - 8 * q^83 - 4 * q^85 - 8 * q^87 + 12 * q^89 + 4 * q^91 + 4 * q^93 + 4 * q^97 - 2 * q^99

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