LMFDB - Embedded newform 7920.2.a.ck.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7920,2,Mod(1,7920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7920, 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("7920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7920.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:	$63.2415184009$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3960)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.17741$ of defining polynomial
Character	$\chi$	$=$	7920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.91852 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.91852 q^{7} +1.00000 q^{11} +3.43630 q^{13} -4.91852 q^{17} +4.35482 q^{19} +4.35482 q^{23} +1.00000 q^{25} -2.00000 q^{29} +6.35482 q^{31} +2.91852 q^{35} +2.00000 q^{37} -2.00000 q^{41} -7.27334 q^{43} -0.354819 q^{47} +1.51777 q^{49} -7.83705 q^{53} -1.00000 q^{55} +6.35482 q^{59} -15.0645 q^{61} -3.43630 q^{65} -1.83705 q^{67} -0.872594 q^{71} -10.7556 q^{73} -2.91852 q^{77} -2.87259 q^{79} -17.4652 q^{83} +4.91852 q^{85} +16.9015 q^{89} -10.0289 q^{91} -4.35482 q^{95} +16.5467 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5}+O(q^{10}) $ 3 * q - 3 * q^5	$ 3 q - 3 q^{5} + 3 q^{11} + 4 q^{13} - 6 q^{17} - 2 q^{19} - 2 q^{23} + 3 q^{25} - 6 q^{29} + 4 q^{31} + 6 q^{37} - 6 q^{41} + 2 q^{43} + 14 q^{47} + 7 q^{49} - 6 q^{53} - 3 q^{55} + 4 q^{59} - 4 q^{65} + 12 q^{67} + 10 q^{71} - 6 q^{73} + 4 q^{79} + 4 q^{83} + 6 q^{85} - 12 q^{89} + 20 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^11 + 4 * q^13 - 6 * q^17 - 2 * q^19 - 2 * q^23 + 3 * q^25 - 6 * q^29 + 4 * q^31 + 6 * q^37 - 6 * q^41 + 2 * q^43 + 14 * q^47 + 7 * q^49 - 6 * q^53 - 3 * q^55 + 4 * q^59 - 4 * q^65 + 12 * q^67 + 10 * q^71 - 6 * q^73 + 4 * q^79 + 4 * q^83 + 6 * q^85 - 12 * q^89 + 20 * q^91 + 2 * q^95 + 2 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.91852	−1.10310	−0.551549	−	0.834143i	$-0.685963\pi$
$7$	−2.91852	−1.10310	−0.551549	+	0.834143i	$0.685963\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.43630	0.953057	0.476529	−	0.879159i	$-0.341895\pi$
$13$	3.43630	0.953057	0.476529	+	0.879159i	$0.341895\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.91852	−1.19292	−0.596458	−	0.802644i	$-0.703426\pi$
$17$	−4.91852	−1.19292	−0.596458	+	0.802644i	$0.703426\pi$
$18$	0	0
$19$	4.35482	0.999064	0.499532	−	0.866295i	$-0.333505\pi$
$19$	4.35482	0.999064	0.499532	+	0.866295i	$0.333505\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.35482	0.908043	0.454021	−	0.890991i	$-0.349989\pi$
$23$	4.35482	0.908043	0.454021	+	0.890991i	$0.349989\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	6.35482	1.14136	0.570680	−	0.821173i	$-0.306680\pi$
$31$	6.35482	1.14136	0.570680	+	0.821173i	$0.306680\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.91852	0.493320
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−7.27334	−1.10917	−0.554587	−	0.832126i	$-0.687124\pi$
$43$	−7.27334	−1.10917	−0.554587	+	0.832126i	$0.687124\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.354819	−0.0517557	−0.0258779	−	0.999665i	$-0.508238\pi$
$47$	−0.354819	−0.0517557	−0.0258779	+	0.999665i	$0.508238\pi$
$48$	0	0
$49$	1.51777	0.216825
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.83705	−1.07650	−0.538250	−	0.842785i	$-0.680915\pi$
$53$	−7.83705	−1.07650	−0.538250	+	0.842785i	$0.680915\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.35482	0.827327	0.413664	−	0.910430i	$-0.364249\pi$
$59$	6.35482	0.827327	0.413664	+	0.910430i	$0.364249\pi$
$60$	0	0
$61$	−15.0645	−1.92881	−0.964403	−	0.264436i	$-0.914814\pi$
$61$	−15.0645	−1.92881	−0.964403	+	0.264436i	$0.914814\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.43630	−0.426220
$66$	0	0
$67$	−1.83705	−0.224431	−0.112215	−	0.993684i	$-0.535795\pi$
$67$	−1.83705	−0.224431	−0.112215	+	0.993684i	$0.535795\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.872594	−0.103558	−0.0517789	−	0.998659i	$-0.516489\pi$
$71$	−0.872594	−0.103558	−0.0517789	+	0.998659i	$0.516489\pi$
$72$	0	0
$73$	−10.7556	−1.25884	−0.629422	−	0.777064i	$-0.716708\pi$
$73$	−10.7556	−1.25884	−0.629422	+	0.777064i	$0.716708\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.91852	−0.332597
$78$	0	0
$79$	−2.87259	−0.323192	−0.161596	−	0.986857i	$-0.551664\pi$
$79$	−2.87259	−0.323192	−0.161596	+	0.986857i	$0.551664\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−17.4652	−1.91706	−0.958528	−	0.284999i	$-0.908007\pi$
$83$	−17.4652	−1.91706	−0.958528	+	0.284999i	$0.908007\pi$
$84$	0	0
$85$	4.91852	0.533489
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.9015	1.79156	0.895778	−	0.444502i	$-0.146619\pi$
$89$	16.9015	1.79156	0.895778	+	0.444502i	$0.146619\pi$
$90$	0	0
$91$	−10.0289	−1.05132
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.35482	−0.446795
$96$	0	0
$97$	16.5467	1.68006	0.840031	−	0.542539i	$-0.182537\pi$
$97$	16.5467	1.68006	0.840031	+	0.542539i	$0.182537\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.162955	−0.0162146	−0.00810730	−	0.999967i	$-0.502581\pi$
$101$	−0.162955	−0.0162146	−0.00810730	+	0.999967i	$0.502581\pi$
$102$	0	0
$103$	11.6741	1.15028	0.575141	−	0.818054i	$-0.304947\pi$
$103$	11.6741	1.15028	0.575141	+	0.818054i	$0.304947\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.6282	1.89753	0.948763	−	0.315989i	$-0.102336\pi$
$107$	19.6282	1.89753	0.948763	+	0.315989i	$0.102336\pi$
$108$	0	0
$109$	11.4822	1.09980	0.549899	−	0.835231i	$-0.314666\pi$
$109$	11.4822	1.09980	0.549899	+	0.835231i	$0.314666\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.51777	−0.801285	−0.400642	−	0.916235i	$-0.631213\pi$
$113$	−8.51777	−0.801285	−0.400642	+	0.916235i	$0.631213\pi$
$114$	0	0
$115$	−4.35482	−0.406089
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.3548	1.31590
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.7911	1.22376	0.611882	−	0.790949i	$-0.290413\pi$
$127$	13.7911	1.22376	0.611882	+	0.790949i	$0.290413\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.87259	0.250980	0.125490	−	0.992095i	$-0.459950\pi$
$131$	2.87259	0.250980	0.125490	+	0.992095i	$0.459950\pi$
$132$	0	0
$133$	−12.7096	−1.10207
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.51777	−0.385979	−0.192990	−	0.981201i	$-0.561818\pi$
$137$	−4.51777	−0.385979	−0.192990	+	0.981201i	$0.561818\pi$
$138$	0	0
$139$	10.1919	0.864463	0.432231	−	0.901763i	$-0.357726\pi$
$139$	10.1919	0.864463	0.432231	+	0.901763i	$0.357726\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.43630	0.287358
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.8726	−1.05456	−0.527282	−	0.849690i	$-0.676789\pi$
$149$	−12.8726	−1.05456	−0.527282	+	0.849690i	$0.676789\pi$
$150$	0	0
$151$	9.83705	0.800527	0.400264	−	0.916400i	$-0.368919\pi$
$151$	9.83705	0.800527	0.400264	+	0.916400i	$0.368919\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.35482	−0.510431
$156$	0	0
$157$	10.8015	0.862053	0.431027	−	0.902339i	$-0.358151\pi$
$157$	10.8015	0.862053	0.431027	+	0.902339i	$0.358151\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.7096	−1.00166
$162$	0	0
$163$	−0.709639	−0.0555832	−0.0277916	−	0.999614i	$-0.508847\pi$
$163$	−0.709639	−0.0555832	−0.0277916	+	0.999614i	$0.508847\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.5926	−0.819681	−0.409841	−	0.912157i	$-0.634416\pi$
$167$	−10.5926	−0.819681	−0.409841	+	0.912157i	$0.634416\pi$
$168$	0	0
$169$	−1.19186	−0.0916819
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.208884	−0.0158811	−0.00794057	−	0.999968i	$-0.502528\pi$
$173$	−0.208884	−0.0158811	−0.00794057	+	0.999968i	$0.502528\pi$
$174$	0	0
$175$	−2.91852	−0.220620
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.6741	1.17154	0.585768	−	0.810479i	$-0.300793\pi$
$179$	15.6741	1.17154	0.585768	+	0.810479i	$0.300793\pi$
$180$	0	0
$181$	7.22741	0.537209	0.268605	−	0.963250i	$-0.413437\pi$
$181$	7.22741	0.537209	0.268605	+	0.963250i	$0.413437\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−4.91852	−0.359678
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.5178	0.761039	0.380520	−	0.924773i	$-0.375745\pi$
$191$	10.5178	0.761039	0.380520	+	0.924773i	$0.375745\pi$
$192$	0	0
$193$	11.4652	0.825284	0.412642	−	0.910893i	$-0.364606\pi$
$193$	11.4652	0.825284	0.412642	+	0.910893i	$0.364606\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.5008	1.17563	0.587815	−	0.808995i	$-0.299988\pi$
$197$	16.5008	1.17563	0.587815	+	0.808995i	$0.299988\pi$
$198$	0	0
$199$	11.6741	0.827554	0.413777	−	0.910378i	$-0.364209\pi$
$199$	11.6741	0.827554	0.413777	+	0.910378i	$0.364209\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.83705	0.409680
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.35482	0.301229
$210$	0	0
$211$	15.3193	1.05462	0.527311	−	0.849672i	$-0.323200\pi$
$211$	15.3193	1.05462	0.527311	+	0.849672i	$0.323200\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.27334	0.496038
$216$	0	0
$217$	−18.5467	−1.25903
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−16.9015	−1.13692
$222$	0	0
$223$	27.5822	1.84704	0.923521	−	0.383547i	$-0.125298\pi$
$223$	27.5822	1.84704	0.923521	+	0.383547i	$0.125298\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.95407	0.527930	0.263965	−	0.964532i	$-0.414970\pi$
$227$	7.95407	0.527930	0.263965	+	0.964532i	$0.414970\pi$
$228$	0	0
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−28.5926	−1.87316	−0.936582	−	0.350448i	$-0.886029\pi$
$233$	−28.5926	−1.87316	−0.936582	+	0.350448i	$0.886029\pi$
$234$	0	0
$235$	0.354819	0.0231459
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.83705	0.377567	0.188784	−	0.982019i	$-0.439546\pi$
$239$	5.83705	0.377567	0.188784	+	0.982019i	$0.439546\pi$
$240$	0	0
$241$	20.5467	1.32353	0.661764	−	0.749712i	$-0.269808\pi$
$241$	20.5467	1.32353	0.661764	+	0.749712i	$0.269808\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.51777	−0.0969670
$246$	0	0
$247$	14.9645	0.952165
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.7741	−1.24813	−0.624065	−	0.781372i	$-0.714520\pi$
$251$	−19.7741	−1.24813	−0.624065	+	0.781372i	$0.714520\pi$
$252$	0	0
$253$	4.35482	0.273785
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.31927	−0.581320	−0.290660	−	0.956826i	$-0.593875\pi$
$257$	−9.31927	−0.581320	−0.290660	+	0.956826i	$0.593875\pi$
$258$	0	0
$259$	−5.83705	−0.362696
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.79112	0.603746	0.301873	−	0.953348i	$-0.402388\pi$
$263$	9.79112	0.603746	0.301873	+	0.953348i	$0.402388\pi$
$264$	0	0
$265$	7.83705	0.481426
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.22741	−0.440663	−0.220332	−	0.975425i	$-0.570714\pi$
$269$	−7.22741	−0.440663	−0.220332	+	0.975425i	$0.570714\pi$
$270$	0	0
$271$	9.83705	0.597558	0.298779	−	0.954322i	$-0.403421\pi$
$271$	9.83705	0.597558	0.298779	+	0.954322i	$0.403421\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	9.59925	0.576763	0.288382	−	0.957516i	$-0.406883\pi$
$277$	9.59925	0.576763	0.288382	+	0.957516i	$0.406883\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.6741	−0.815728	−0.407864	−	0.913043i	$-0.633726\pi$
$281$	−13.6741	−0.815728	−0.407864	+	0.913043i	$0.633726\pi$
$282$	0	0
$283$	12.3089	0.731688	0.365844	−	0.930676i	$-0.380780\pi$
$283$	12.3089	0.731688	0.365844	+	0.930676i	$0.380780\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.83705	0.344550
$288$	0	0
$289$	7.19186	0.423051
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.82666	0.515659	0.257830	−	0.966190i	$-0.416993\pi$
$293$	8.82666	0.515659	0.257830	+	0.966190i	$0.416993\pi$
$294$	0	0
$295$	−6.35482	−0.369992
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.9645	0.865417
$300$	0	0
$301$	21.2274	1.22353
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.0645	0.862588
$306$	0	0
$307$	−13.1104	−0.748249	−0.374125	−	0.927378i	$-0.622057\pi$
$307$	−13.1104	−0.748249	−0.374125	+	0.927378i	$0.622057\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.09186	0.118618	0.0593092	−	0.998240i	$-0.481110\pi$
$311$	2.09186	0.118618	0.0593092	+	0.998240i	$0.481110\pi$
$312$	0	0
$313$	−22.7096	−1.28362	−0.641812	−	0.766862i	$-0.721817\pi$
$313$	−22.7096	−1.28362	−0.641812	+	0.766862i	$0.721817\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.29036	0.0724739	0.0362370	−	0.999343i	$-0.488463\pi$
$317$	1.29036	0.0724739	0.0362370	+	0.999343i	$0.488463\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−21.4193	−1.19180
$324$	0	0
$325$	3.43630	0.190611
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.03555	0.0570916
$330$	0	0
$331$	22.3548	1.22873	0.614366	−	0.789021i	$-0.289412\pi$
$331$	22.3548	1.22873	0.614366	+	0.789021i	$0.289412\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.83705	0.100369
$336$	0	0
$337$	−1.62816	−0.0886916	−0.0443458	−	0.999016i	$-0.514120\pi$
$337$	−1.62816	−0.0886916	−0.0443458	+	0.999016i	$0.514120\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.35482	0.344133
$342$	0	0
$343$	16.0000	0.863919
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.0459291	0.00246560	0.00123280	−	0.999999i	$-0.499608\pi$
$347$	0.0459291	0.00246560	0.00123280	+	0.999999i	$0.499608\pi$
$348$	0	0
$349$	30.3548	1.62486	0.812428	−	0.583061i	$-0.198145\pi$
$349$	30.3548	1.62486	0.812428	+	0.583061i	$0.198145\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11.3904	−0.606248	−0.303124	−	0.952951i	$-0.598030\pi$
$353$	−11.3904	−0.606248	−0.303124	+	0.952951i	$0.598030\pi$
$354$	0	0
$355$	0.872594	0.0463125
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.83705	0.519179	0.259590	−	0.965719i	$-0.416413\pi$
$359$	9.83705	0.519179	0.259590	+	0.965719i	$0.416413\pi$
$360$	0	0
$361$	−0.0355483	−0.00187096
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.7556	0.562972
$366$	0	0
$367$	−6.87259	−0.358746	−0.179373	−	0.983781i	$-0.557407\pi$
$367$	−6.87259	−0.358746	−0.179373	+	0.983781i	$0.557407\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	22.8726	1.18749
$372$	0	0
$373$	9.69111	0.501787	0.250893	−	0.968015i	$-0.419276\pi$
$373$	9.69111	0.501787	0.250893	+	0.968015i	$0.419276\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.87259	−0.353957
$378$	0	0
$379$	−6.96445	−0.357740	−0.178870	−	0.983873i	$-0.557244\pi$
$379$	−6.96445	−0.357740	−0.178870	+	0.983873i	$0.557244\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.44668	−0.431605	−0.215803	−	0.976437i	$-0.569237\pi$
$383$	−8.44668	−0.431605	−0.215803	+	0.976437i	$0.569237\pi$
$384$	0	0
$385$	2.91852	0.148742
$386$	0	0
$387$	0	0
$388$	0	0
$389$	17.8660	0.905840	0.452920	−	0.891551i	$-0.350382\pi$
$389$	17.8660	0.905840	0.452920	+	0.891551i	$0.350382\pi$
$390$	0	0
$391$	−21.4193	−1.08322
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.87259	0.144536
$396$	0	0
$397$	29.2563	1.46833	0.734166	−	0.678970i	$-0.237573\pi$
$397$	29.2563	1.46833	0.734166	+	0.678970i	$0.237573\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.8660	1.59131	0.795655	−	0.605750i	$-0.207127\pi$
$401$	31.8660	1.59131	0.795655	+	0.605750i	$0.207127\pi$
$402$	0	0
$403$	21.8370	1.08778
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	5.29036	0.261592	0.130796	−	0.991409i	$-0.458247\pi$
$409$	5.29036	0.261592	0.130796	+	0.991409i	$0.458247\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−18.5467	−0.912623
$414$	0	0
$415$	17.4652	0.857333
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−1.48223	−0.0722393	−0.0361196	−	0.999347i	$-0.511500\pi$
$421$	−1.48223	−0.0722393	−0.0361196	+	0.999347i	$0.511500\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.91852	−0.238583
$426$	0	0
$427$	43.9660	2.12766
$428$	0	0
$429$	0	0
$430$	0	0
$431$	34.1289	1.64393	0.821966	−	0.569537i	$-0.192877\pi$
$431$	34.1289	1.64393	0.821966	+	0.569537i	$0.192877\pi$
$432$	0	0
$433$	−8.87259	−0.426390	−0.213195	−	0.977010i	$-0.568387\pi$
$433$	−8.87259	−0.426390	−0.213195	+	0.977010i	$0.568387\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.9645	0.907193
$438$	0	0
$439$	−35.6741	−1.70263	−0.851316	−	0.524654i	$-0.824195\pi$
$439$	−35.6741	−1.70263	−0.851316	+	0.524654i	$0.824195\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.7452	0.653054	0.326527	−	0.945188i	$-0.394122\pi$
$443$	13.7452	0.653054	0.326527	+	0.945188i	$0.394122\pi$
$444$	0	0
$445$	−16.9015	−0.801208
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.5756	−1.15979	−0.579897	−	0.814690i	$-0.696907\pi$
$449$	−24.5756	−1.15979	−0.579897	+	0.814690i	$0.696907\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.0289	0.470163
$456$	0	0
$457$	−27.5571	−1.28907	−0.644533	−	0.764577i	$-0.722948\pi$
$457$	−27.5571	−1.28907	−0.644533	+	0.764577i	$0.722948\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.90814	0.275170	0.137585	−	0.990490i	$-0.456066\pi$
$461$	5.90814	0.275170	0.137585	+	0.990490i	$0.456066\pi$
$462$	0	0
$463$	−7.25632	−0.337230	−0.168615	−	0.985682i	$-0.553929\pi$
$463$	−7.25632	−0.337230	−0.168615	+	0.985682i	$0.553929\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.83705	0.455204	0.227602	−	0.973754i	$-0.426911\pi$
$467$	9.83705	0.455204	0.227602	+	0.973754i	$0.426911\pi$
$468$	0	0
$469$	5.36146	0.247569
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.27334	−0.334429
$474$	0	0
$475$	4.35482	0.199813
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.2208	−0.832528	−0.416264	−	0.909244i	$-0.636661\pi$
$479$	−18.2208	−0.832528	−0.416264	+	0.909244i	$0.636661\pi$
$480$	0	0
$481$	6.87259	0.313363
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.5467	−0.751346
$486$	0	0
$487$	−16.2919	−0.738255	−0.369128	−	0.929379i	$-0.620343\pi$
$487$	−16.2919	−0.738255	−0.369128	+	0.929379i	$0.620343\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.63854	−0.299593	−0.149797	−	0.988717i	$-0.547862\pi$
$491$	−6.63854	−0.299593	−0.149797	+	0.988717i	$0.547862\pi$
$492$	0	0
$493$	9.83705	0.443038
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.54668	0.114234
$498$	0	0
$499$	−37.4193	−1.67512	−0.837558	−	0.546348i	$-0.816018\pi$
$499$	−37.4193	−1.67512	−0.837558	+	0.546348i	$0.816018\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.08148	−0.0482207	−0.0241103	−	0.999709i	$-0.507675\pi$
$503$	−1.08148	−0.0482207	−0.0241103	+	0.999709i	$0.507675\pi$
$504$	0	0
$505$	0.162955	0.00725139
$506$	0	0
$507$	0	0
$508$	0	0
$509$	36.8304	1.63248	0.816240	−	0.577714i	$-0.196055\pi$
$509$	36.8304	1.63248	0.816240	+	0.577714i	$0.196055\pi$
$510$	0	0
$511$	31.3904	1.38863
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.6741	−0.514422
$516$	0	0
$517$	−0.354819	−0.0156049
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.4467	0.632920	0.316460	−	0.948606i	$-0.397506\pi$
$521$	14.4467	0.632920	0.316460	+	0.948606i	$0.397506\pi$
$522$	0	0
$523$	28.4007	1.24188	0.620939	−	0.783859i	$-0.286751\pi$
$523$	28.4007	1.24188	0.620939	+	0.783859i	$0.286751\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31.2563	−1.36155
$528$	0	0
$529$	−4.03555	−0.175459
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.87259	−0.297685
$534$	0	0
$535$	−19.6282	−0.848599
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.51777	0.0653752
$540$	0	0
$541$	−9.22741	−0.396717	−0.198359	−	0.980129i	$-0.563561\pi$
$541$	−9.22741	−0.396717	−0.198359	+	0.980129i	$0.563561\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.4822	−0.491845
$546$	0	0
$547$	−18.0541	−0.771937	−0.385968	−	0.922512i	$-0.626133\pi$
$547$	−18.0541	−0.771937	−0.385968	+	0.922512i	$0.626133\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.70964	−0.371043
$552$	0	0
$553$	8.38373	0.356512
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.5926	0.533566	0.266783	−	0.963757i	$-0.414039\pi$
$557$	12.5926	0.533566	0.266783	+	0.963757i	$0.414039\pi$
$558$	0	0
$559$	−24.9934	−1.05711
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.9541	−1.34670	−0.673352	−	0.739322i	$-0.735146\pi$
$563$	−31.9541	−1.34670	−0.673352	+	0.739322i	$0.735146\pi$
$564$	0	0
$565$	8.51777	0.358345
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.5111	1.48870	0.744352	−	0.667787i	$-0.232758\pi$
$569$	35.5111	1.48870	0.744352	+	0.667787i	$0.232758\pi$
$570$	0	0
$571$	11.9371	0.499550	0.249775	−	0.968304i	$-0.419643\pi$
$571$	11.9371	0.499550	0.249775	+	0.968304i	$0.419643\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.35482	0.181609
$576$	0	0
$577$	−32.8726	−1.36850	−0.684252	−	0.729246i	$-0.739871\pi$
$577$	−32.8726	−1.36850	−0.684252	+	0.729246i	$0.739871\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	50.9726	2.11470
$582$	0	0
$583$	−7.83705	−0.324577
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.8949	1.72919	0.864593	−	0.502473i	$-0.167576\pi$
$587$	41.8949	1.72919	0.864593	+	0.502473i	$0.167576\pi$
$588$	0	0
$589$	27.6741	1.14029
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−24.8267	−1.01951	−0.509754	−	0.860320i	$-0.670264\pi$
$593$	−24.8267	−1.01951	−0.509754	+	0.860320i	$0.670264\pi$
$594$	0	0
$595$	−14.3548	−0.588490
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.2274	−1.11248	−0.556241	−	0.831021i	$-0.687757\pi$
$599$	−27.2274	−1.11248	−0.556241	+	0.831021i	$0.687757\pi$
$600$	0	0
$601$	−29.2563	−1.19339	−0.596695	−	0.802468i	$-0.703520\pi$
$601$	−29.2563	−1.19339	−0.596695	+	0.802468i	$0.703520\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	39.2104	1.59150	0.795750	−	0.605625i	$-0.207077\pi$
$607$	39.2104	1.59150	0.795750	+	0.605625i	$0.207077\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.21926	−0.0493262
$612$	0	0
$613$	−6.07484	−0.245360	−0.122680	−	0.992446i	$-0.539149\pi$
$613$	−6.07484	−0.245360	−0.122680	+	0.992446i	$0.539149\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.9371	1.04419	0.522093	−	0.852888i	$-0.325151\pi$
$617$	25.9371	1.04419	0.522093	+	0.852888i	$0.325151\pi$
$618$	0	0
$619$	−22.0289	−0.885417	−0.442708	−	0.896666i	$-0.645982\pi$
$619$	−22.0289	−0.885417	−0.442708	+	0.896666i	$0.645982\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−49.3274	−1.97626
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.83705	−0.392229
$630$	0	0
$631$	9.41928	0.374976	0.187488	−	0.982267i	$-0.439966\pi$
$631$	9.41928	0.374976	0.187488	+	0.982267i	$0.439966\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.7911	−0.547284
$636$	0	0
$637$	5.21552	0.206647
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22.9726	0.907363	0.453682	−	0.891164i	$-0.350110\pi$
$641$	22.9726	0.907363	0.453682	+	0.891164i	$0.350110\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.1208	−0.476517	−0.238258	−	0.971202i	$-0.576576\pi$
$647$	−12.1208	−0.476517	−0.238258	+	0.971202i	$0.576576\pi$
$648$	0	0
$649$	6.35482	0.249448
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.2919	0.715816	0.357908	−	0.933757i	$-0.383490\pi$
$653$	18.2919	0.715816	0.357908	+	0.933757i	$0.383490\pi$
$654$	0	0
$655$	−2.87259	−0.112242
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13.0355	−0.507793	−0.253896	−	0.967231i	$-0.581712\pi$
$659$	−13.0355	−0.507793	−0.253896	+	0.967231i	$0.581712\pi$
$660$	0	0
$661$	−17.6741	−0.687442	−0.343721	−	0.939072i	$-0.611688\pi$
$661$	−17.6741	−0.687442	−0.343721	+	0.939072i	$0.611688\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.7096	0.492859
$666$	0	0
$667$	−8.70964	−0.337239
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15.0645	−0.581557
$672$	0	0
$673$	−32.5008	−1.25281	−0.626406	−	0.779497i	$-0.715475\pi$
$673$	−32.5008	−1.25281	−0.626406	+	0.779497i	$0.715475\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−36.9763	−1.42112	−0.710558	−	0.703638i	$-0.751557\pi$
$677$	−36.9763	−1.42112	−0.710558	+	0.703638i	$0.751557\pi$
$678$	0	0
$679$	−48.2919	−1.85327
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−31.5822	−1.20846	−0.604230	−	0.796810i	$-0.706519\pi$
$683$	−31.5822	−1.20846	−0.604230	+	0.796810i	$0.706519\pi$
$684$	0	0
$685$	4.51777	0.172615
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−26.9304	−1.02597
$690$	0	0
$691$	6.25481	0.237944	0.118972	−	0.992898i	$-0.462040\pi$
$691$	6.25481	0.237944	0.118972	+	0.992898i	$0.462040\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.1919	−0.386599
$696$	0	0
$697$	9.83705	0.372605
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.3837	1.44973	0.724867	−	0.688889i	$-0.241901\pi$
$701$	38.3837	1.44973	0.724867	+	0.688889i	$0.241901\pi$
$702$	0	0
$703$	8.70964	0.328490
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.475587	0.0178863
$708$	0	0
$709$	37.5400	1.40985	0.704923	−	0.709284i	$-0.250982\pi$
$709$	37.5400	1.40985	0.704923	+	0.709284i	$0.250982\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.6741	1.03640
$714$	0	0
$715$	−3.43630	−0.128510
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28.1208	−1.04873	−0.524364	−	0.851494i	$-0.675697\pi$
$719$	−28.1208	−1.04873	−0.524364	+	0.851494i	$0.675697\pi$
$720$	0	0
$721$	−34.0711	−1.26887
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	31.1645	1.15583	0.577913	−	0.816098i	$-0.303867\pi$
$727$	31.1645	1.15583	0.577913	+	0.816098i	$0.303867\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	35.7741	1.32315
$732$	0	0
$733$	48.9134	1.80666	0.903329	−	0.428949i	$-0.141116\pi$
$733$	48.9134	1.80666	0.903329	+	0.428949i	$0.141116\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.83705	−0.0676684
$738$	0	0
$739$	15.9371	0.586254	0.293127	−	0.956074i	$-0.405304\pi$
$739$	15.9371	0.586254	0.293127	+	0.956074i	$0.405304\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30.2089	−1.10826	−0.554128	−	0.832431i	$-0.686948\pi$
$743$	−30.2089	−1.10826	−0.554128	+	0.832431i	$0.686948\pi$
$744$	0	0
$745$	12.8726	0.471615
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−57.2852	−2.09316
$750$	0	0
$751$	35.6741	1.30177	0.650883	−	0.759178i	$-0.274399\pi$
$751$	35.6741	1.30177	0.650883	+	0.759178i	$0.274399\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.83705	−0.358007
$756$	0	0
$757$	−7.51114	−0.272997	−0.136498	−	0.990640i	$-0.543585\pi$
$757$	−7.51114	−0.272997	−0.136498	+	0.990640i	$0.543585\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37.6401	−1.36445	−0.682225	−	0.731142i	$-0.738988\pi$
$761$	−37.6401	−1.36445	−0.682225	+	0.731142i	$0.738988\pi$
$762$	0	0
$763$	−33.5111	−1.21318
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.8370	0.788490
$768$	0	0
$769$	53.2563	1.92047	0.960236	−	0.279189i	$-0.0900657\pi$
$769$	53.2563	1.92047	0.960236	+	0.279189i	$0.0900657\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−38.6178	−1.38899	−0.694493	−	0.719500i	$-0.744371\pi$
$773$	−38.6178	−1.38899	−0.694493	+	0.719500i	$0.744371\pi$
$774$	0	0
$775$	6.35482	0.228272
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.70964	−0.312055
$780$	0	0
$781$	−0.872594	−0.0312239
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.8015	−0.385522
$786$	0	0
$787$	45.8200	1.63331	0.816654	−	0.577128i	$-0.195827\pi$
$787$	45.8200	1.63331	0.816654	+	0.577128i	$0.195827\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24.8593	0.883896
$792$	0	0
$793$	−51.7659	−1.83826
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.90663	0.173802	0.0869009	−	0.996217i	$-0.472304\pi$
$797$	4.90663	0.173802	0.0869009	+	0.996217i	$0.472304\pi$
$798$	0	0
$799$	1.74519	0.0617403
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.7556	−0.379556
$804$	0	0
$805$	12.7096	0.447956
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36.8386	1.29517	0.647587	−	0.761991i	$-0.275778\pi$
$809$	36.8386	1.29517	0.647587	+	0.761991i	$0.275778\pi$
$810$	0	0
$811$	−49.5400	−1.73959	−0.869793	−	0.493417i	$-0.835748\pi$
$811$	−49.5400	−1.73959	−0.869793	+	0.493417i	$0.835748\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.709639	0.0248576
$816$	0	0
$817$	−31.6741	−1.10814
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12.1289	−0.423302	−0.211651	−	0.977345i	$-0.567884\pi$
$821$	−12.1289	−0.423302	−0.211651	+	0.977345i	$0.567884\pi$
$822$	0	0
$823$	11.2904	0.393557	0.196779	−	0.980448i	$-0.436952\pi$
$823$	11.2904	0.393557	0.196779	+	0.980448i	$0.436952\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31.3023	−1.08849	−0.544243	−	0.838928i	$-0.683183\pi$
$827$	−31.3023	−1.08849	−0.544243	+	0.838928i	$0.683183\pi$
$828$	0	0
$829$	9.80814	0.340651	0.170325	−	0.985388i	$-0.445518\pi$
$829$	9.80814	0.340651	0.170325	+	0.985388i	$0.445518\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.46521	−0.258654
$834$	0	0
$835$	10.5926	0.366572
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28.2208	0.974289	0.487145	−	0.873321i	$-0.338038\pi$
$839$	28.2208	0.974289	0.487145	+	0.873321i	$0.338038\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.19186	0.0410014
$846$	0	0
$847$	−2.91852	−0.100282
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.70964	0.298563
$852$	0	0
$853$	13.8911	0.475623	0.237811	−	0.971311i	$-0.423570\pi$
$853$	13.8911	0.475623	0.237811	+	0.971311i	$0.423570\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.86221	−0.200249	−0.100125	−	0.994975i	$-0.531924\pi$
$857$	−5.86221	−0.200249	−0.100125	+	0.994975i	$0.531924\pi$
$858$	0	0
$859$	12.2837	0.419115	0.209558	−	0.977796i	$-0.432798\pi$
$859$	12.2837	0.419115	0.209558	+	0.977796i	$0.432798\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.2274	−0.382186	−0.191093	−	0.981572i	$-0.561203\pi$
$863$	−11.2274	−0.382186	−0.191093	+	0.981572i	$0.561203\pi$
$864$	0	0
$865$	0.208884	0.00710226
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.87259	−0.0974461
$870$	0	0
$871$	−6.31263	−0.213895
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.91852	0.0986641
$876$	0	0
$877$	12.2378	0.413241	0.206620	−	0.978421i	$-0.433753\pi$
$877$	12.2378	0.413241	0.206620	+	0.978421i	$0.433753\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.6385	1.03224	0.516119	−	0.856517i	$-0.327376\pi$
$881$	30.6385	1.03224	0.516119	+	0.856517i	$0.327376\pi$
$882$	0	0
$883$	29.6030	0.996220	0.498110	−	0.867114i	$-0.334028\pi$
$883$	29.6030	0.996220	0.498110	+	0.867114i	$0.334028\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.7200	−1.06505	−0.532527	−	0.846413i	$-0.678758\pi$
$887$	−31.7200	−1.06505	−0.532527	+	0.846413i	$0.678758\pi$
$888$	0	0
$889$	−40.2497	−1.34993
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.54517	−0.0517073
$894$	0	0
$895$	−15.6741	−0.523927
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.7096	−0.423890
$900$	0	0
$901$	38.5467	1.28418
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.22741	−0.240247
$906$	0	0
$907$	1.12741	0.0374349	0.0187175	−	0.999825i	$-0.494042\pi$
$907$	1.12741	0.0374349	0.0187175	+	0.999825i	$0.494042\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46.9015	1.55392	0.776958	−	0.629552i	$-0.216762\pi$
$911$	46.9015	1.55392	0.776958	+	0.629552i	$0.216762\pi$
$912$	0	0
$913$	−17.4652	−0.578014
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.38373	−0.276855
$918$	0	0
$919$	−14.2548	−0.470223	−0.235111	−	0.971968i	$-0.575546\pi$
$919$	−14.2548	−0.470223	−0.235111	+	0.971968i	$0.575546\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.99849	−0.0986965
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.1289	1.51344	0.756720	−	0.653739i	$-0.226800\pi$
$929$	46.1289	1.51344	0.756720	+	0.653739i	$0.226800\pi$
$930$	0	0
$931$	6.60963	0.216622
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.91852	0.160853
$936$	0	0
$937$	2.01189	0.0657256	0.0328628	−	0.999460i	$-0.489538\pi$
$937$	2.01189	0.0657256	0.0328628	+	0.999460i	$0.489538\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	36.5467	1.19139	0.595694	−	0.803212i	$-0.296877\pi$
$941$	36.5467	1.19139	0.595694	+	0.803212i	$0.296877\pi$
$942$	0	0
$943$	−8.70964	−0.283625
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.51114	0.309070	0.154535	−	0.987987i	$-0.450612\pi$
$947$	9.51114	0.309070	0.154535	+	0.987987i	$0.450612\pi$
$948$	0	0
$949$	−36.9593	−1.19975
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14.6637	0.475004	0.237502	−	0.971387i	$-0.423671\pi$
$953$	14.6637	0.475004	0.237502	+	0.971387i	$0.423671\pi$
$954$	0	0
$955$	−10.5178	−0.340347
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.1852	0.425773
$960$	0	0
$961$	9.38373	0.302701
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−11.4652	−0.369078
$966$	0	0
$967$	−4.27998	−0.137635	−0.0688174	−	0.997629i	$-0.521923\pi$
$967$	−4.27998	−0.137635	−0.0688174	+	0.997629i	$0.521923\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.8030	−1.34152	−0.670761	−	0.741673i	$-0.734032\pi$
$971$	−41.8030	−1.34152	−0.670761	+	0.741673i	$0.734032\pi$
$972$	0	0
$973$	−29.7452	−0.953587
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.1563	0.996779	0.498389	−	0.866953i	$-0.333925\pi$
$977$	31.1563	0.996779	0.498389	+	0.866953i	$0.333925\pi$
$978$	0	0
$979$	16.9015	0.540174
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.90150	0.220124	0.110062	−	0.993925i	$-0.464895\pi$
$983$	6.90150	0.220124	0.110062	+	0.993925i	$0.464895\pi$
$984$	0	0
$985$	−16.5008	−0.525758
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−31.6741	−1.00718
$990$	0	0
$991$	−35.4482	−1.12605	−0.563024	−	0.826440i	$-0.690362\pi$
$991$	−35.4482	−1.12605	−0.563024	+	0.826440i	$0.690362\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.6741	−0.370094
$996$	0	0
$997$	36.2378	1.14766	0.573831	−	0.818974i	$-0.305457\pi$
$997$	36.2378	1.14766	0.573831	+	0.818974i	$0.305457\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.ck.1.1		3
3.2	odd	2		7920.2.a.cl.1.1		3
4.3	odd	2		3960.2.a.bg.1.3	✓	3
12.11	even	2		3960.2.a.bh.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.2.a.bg.1.3	✓	3	4.3	odd	2
3960.2.a.bh.1.3	yes	3	12.11	even	2
7920.2.a.ck.1.1		3	1.1	even	1	trivial
7920.2.a.cl.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.91852 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.91852 q^{7} +1.00000 q^{11} +3.43630 q^{13} -4.91852 q^{17} +4.35482 q^{19} +4.35482 q^{23} +1.00000 q^{25} -2.00000 q^{29} +6.35482 q^{31} +2.91852 q^{35} +2.00000 q^{37} -2.00000 q^{41} -7.27334 q^{43} -0.354819 q^{47} +1.51777 q^{49} -7.83705 q^{53} -1.00000 q^{55} +6.35482 q^{59} -15.0645 q^{61} -3.43630 q^{65} -1.83705 q^{67} -0.872594 q^{71} -10.7556 q^{73} -2.91852 q^{77} -2.87259 q^{79} -17.4652 q^{83} +4.91852 q^{85} +16.9015 q^{89} -10.0289 q^{91} -4.35482 q^{95} +16.5467 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5}+O(q^{10}) \) 3 * q - 3 * q^5	\( 3 q - 3 q^{5} + 3 q^{11} + 4 q^{13} - 6 q^{17} - 2 q^{19} - 2 q^{23} + 3 q^{25} - 6 q^{29} + 4 q^{31} + 6 q^{37} - 6 q^{41} + 2 q^{43} + 14 q^{47} + 7 q^{49} - 6 q^{53} - 3 q^{55} + 4 q^{59} - 4 q^{65} + 12 q^{67} + 10 q^{71} - 6 q^{73} + 4 q^{79} + 4 q^{83} + 6 q^{85} - 12 q^{89} + 20 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^11 + 4 * q^13 - 6 * q^17 - 2 * q^19 - 2 * q^23 + 3 * q^25 - 6 * q^29 + 4 * q^31 + 6 * q^37 - 6 * q^41 + 2 * q^43 + 14 * q^47 + 7 * q^49 - 6 * q^53 - 3 * q^55 + 4 * q^59 - 4 * q^65 + 12 * q^67 + 10 * q^71 - 6 * q^73 + 4 * q^79 + 4 * q^83 + 6 * q^85 - 12 * q^89 + 20 * q^91 + 2 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more