LMFDB - Embedded newform 3960.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,2,Mod(1,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, 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("3960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3960.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:	$31.6207592004$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -0.585786 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -0.585786 q^{7} -1.00000 q^{11} +1.41421 q^{13} +4.24264 q^{17} -1.17157 q^{19} +2.82843 q^{23} +1.00000 q^{25} +7.65685 q^{29} -4.82843 q^{31} +0.585786 q^{35} -3.65685 q^{37} -3.65685 q^{41} -9.07107 q^{43} -4.48528 q^{47} -6.65685 q^{49} -6.48528 q^{53} +1.00000 q^{55} +8.82843 q^{59} +8.82843 q^{61} -1.41421 q^{65} +8.48528 q^{67} +10.4853 q^{71} +7.07107 q^{73} +0.585786 q^{77} +0.485281 q^{79} +10.2426 q^{83} -4.24264 q^{85} +2.00000 q^{89} -0.828427 q^{91} +1.17157 q^{95} +8.82843 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 4 * q^7	$ 2 q - 2 q^{5} - 4 q^{7} - 2 q^{11} - 8 q^{19} + 2 q^{25} + 4 q^{29} - 4 q^{31} + 4 q^{35} + 4 q^{37} + 4 q^{41} - 4 q^{43} + 8 q^{47} - 2 q^{49} + 4 q^{53} + 2 q^{55} + 12 q^{59} + 12 q^{61} + 4 q^{71} + 4 q^{77} - 16 q^{79} + 12 q^{83} + 4 q^{89} + 4 q^{91} + 8 q^{95} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^7 - 2 * q^11 - 8 * q^19 + 2 * q^25 + 4 * q^29 - 4 * q^31 + 4 * q^35 + 4 * q^37 + 4 * q^41 - 4 * q^43 + 8 * q^47 - 2 * q^49 + 4 * q^53 + 2 * q^55 + 12 * q^59 + 12 * q^61 + 4 * q^71 + 4 * q^77 - 16 * q^79 + 12 * q^83 + 4 * q^89 + 4 * q^91 + 8 * q^95 + 12 * 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
$7$	−0.585786	−0.221406	−0.110703	−	0.993854i	$-0.535310\pi$
$7$	−0.585786	−0.221406	−0.110703	+	0.993854i	$0.535310\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.24264	1.02899	0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264	1.02899	0.514496	+	0.857493i	$0.327979\pi$
$18$	0	0
$19$	−1.17157	−0.268777	−0.134389	−	0.990929i	$-0.542907\pi$
$19$	−1.17157	−0.268777	−0.134389	+	0.990929i	$0.542907\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.65685	1.42184	0.710921	−	0.703272i	$-0.248278\pi$
$29$	7.65685	1.42184	0.710921	+	0.703272i	$0.248278\pi$
$30$	0	0
$31$	−4.82843	−0.867211	−0.433606	−	0.901103i	$-0.642759\pi$
$31$	−4.82843	−0.867211	−0.433606	+	0.901103i	$0.642759\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.585786	0.0990160
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	−9.07107	−1.38332	−0.691662	−	0.722221i	$-0.743121\pi$
$43$	−9.07107	−1.38332	−0.691662	+	0.722221i	$0.743121\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.48528	−0.654246	−0.327123	−	0.944982i	$-0.606079\pi$
$47$	−4.48528	−0.654246	−0.327123	+	0.944982i	$0.606079\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.48528	−0.890822	−0.445411	−	0.895326i	$-0.646942\pi$
$53$	−6.48528	−0.890822	−0.445411	+	0.895326i	$0.646942\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.82843	1.14936	0.574682	−	0.818377i	$-0.305126\pi$
$59$	8.82843	1.14936	0.574682	+	0.818377i	$0.305126\pi$
$60$	0	0
$61$	8.82843	1.13036	0.565182	−	0.824966i	$-0.308806\pi$
$61$	8.82843	1.13036	0.565182	+	0.824966i	$0.308806\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.41421	−0.175412
$66$	0	0
$67$	8.48528	1.03664	0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528	1.03664	0.518321	+	0.855186i	$0.326557\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.4853	1.24437	0.622187	−	0.782869i	$-0.286244\pi$
$71$	10.4853	1.24437	0.622187	+	0.782869i	$0.286244\pi$
$72$	0	0
$73$	7.07107	0.827606	0.413803	−	0.910366i	$-0.364200\pi$
$73$	7.07107	0.827606	0.413803	+	0.910366i	$0.364200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.585786	0.0667566
$78$	0	0
$79$	0.485281	0.0545984	0.0272992	−	0.999627i	$-0.491309\pi$
$79$	0.485281	0.0545984	0.0272992	+	0.999627i	$0.491309\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.2426	1.12428	0.562138	−	0.827043i	$-0.309979\pi$
$83$	10.2426	1.12428	0.562138	+	0.827043i	$0.309979\pi$
$84$	0	0
$85$	−4.24264	−0.460179
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.17157	0.120201
$96$	0	0
$97$	8.82843	0.896391	0.448195	−	0.893936i	$-0.352067\pi$
$97$	8.82843	0.896391	0.448195	+	0.893936i	$0.352067\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.4853	1.83935	0.919677	−	0.392675i	$-0.128450\pi$
$101$	18.4853	1.83935	0.919677	+	0.392675i	$0.128450\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.58579	0.443325	0.221662	−	0.975123i	$-0.428852\pi$
$107$	4.58579	0.443325	0.221662	+	0.975123i	$0.428852\pi$
$108$	0	0
$109$	9.31371	0.892091	0.446046	−	0.895010i	$-0.352832\pi$
$109$	9.31371	0.892091	0.446046	+	0.895010i	$0.352832\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.65685	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$113$	7.65685	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$114$	0	0
$115$	−2.82843	−0.263752
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.48528	−0.227825
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.55635	−0.847989	−0.423994	−	0.905665i	$-0.639372\pi$
$127$	−9.55635	−0.847989	−0.423994	+	0.905665i	$0.639372\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.17157	0.102361	0.0511804	−	0.998689i	$-0.483702\pi$
$131$	1.17157	0.102361	0.0511804	+	0.998689i	$0.483702\pi$
$132$	0	0
$133$	0.686292	0.0595090
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.31371	−0.453981	−0.226990	−	0.973897i	$-0.572889\pi$
$137$	−5.31371	−0.453981	−0.226990	+	0.973897i	$0.572889\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.41421	−0.118262
$144$	0	0
$145$	−7.65685	−0.635867
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.48528	−0.203602	−0.101801	−	0.994805i	$-0.532461\pi$
$149$	−2.48528	−0.203602	−0.101801	+	0.994805i	$0.532461\pi$
$150$	0	0
$151$	10.1421	0.825355	0.412678	−	0.910877i	$-0.364594\pi$
$151$	10.1421	0.825355	0.412678	+	0.910877i	$0.364594\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.82843	0.387829
$156$	0	0
$157$	5.51472	0.440122	0.220061	−	0.975486i	$-0.429374\pi$
$157$	5.51472	0.440122	0.220061	+	0.975486i	$0.429374\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.65685	−0.130578
$162$	0	0
$163$	11.3137	0.886158	0.443079	−	0.896483i	$-0.353886\pi$
$163$	11.3137	0.886158	0.443079	+	0.896483i	$0.353886\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.7279	1.44921	0.724605	−	0.689164i	$-0.242022\pi$
$167$	18.7279	1.44921	0.724605	+	0.689164i	$0.242022\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.8995	1.66499	0.832494	−	0.554034i	$-0.186912\pi$
$173$	21.8995	1.66499	0.832494	+	0.554034i	$0.186912\pi$
$174$	0	0
$175$	−0.585786	−0.0442813
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.6569	1.31974	0.659868	−	0.751382i	$-0.270612\pi$
$179$	17.6569	1.31974	0.659868	+	0.751382i	$0.270612\pi$
$180$	0	0
$181$	23.3137	1.73289	0.866447	−	0.499269i	$-0.166398\pi$
$181$	23.3137	1.73289	0.866447	+	0.499269i	$0.166398\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.65685	0.268857
$186$	0	0
$187$	−4.24264	−0.310253
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	17.8995	1.28843	0.644217	−	0.764843i	$-0.277183\pi$
$193$	17.8995	1.28843	0.644217	+	0.764843i	$0.277183\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.24264	0.587264	0.293632	−	0.955919i	$-0.405136\pi$
$197$	8.24264	0.587264	0.293632	+	0.955919i	$0.405136\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.48528	−0.314805
$204$	0	0
$205$	3.65685	0.255406
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.17157	0.0810394
$210$	0	0
$211$	−20.4853	−1.41026	−0.705132	−	0.709076i	$-0.749112\pi$
$211$	−20.4853	−1.41026	−0.705132	+	0.709076i	$0.749112\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.07107	0.618642
$216$	0	0
$217$	2.82843	0.192006
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	14.8284	0.992985	0.496492	−	0.868041i	$-0.334621\pi$
$223$	14.8284	0.992985	0.496492	+	0.868041i	$0.334621\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.07107	−0.0710893	−0.0355446	−	0.999368i	$-0.511317\pi$
$227$	−1.07107	−0.0710893	−0.0355446	+	0.999368i	$0.511317\pi$
$228$	0	0
$229$	−9.31371	−0.615467	−0.307734	−	0.951473i	$-0.599571\pi$
$229$	−9.31371	−0.615467	−0.307734	+	0.951473i	$0.599571\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.72792	−0.571785	−0.285893	−	0.958262i	$-0.592290\pi$
$233$	−8.72792	−0.571785	−0.285893	+	0.958262i	$0.592290\pi$
$234$	0	0
$235$	4.48528	0.292587
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.1421	−1.17352	−0.586759	−	0.809762i	$-0.699596\pi$
$239$	−18.1421	−1.17352	−0.586759	+	0.809762i	$0.699596\pi$
$240$	0	0
$241$	−12.1421	−0.782144	−0.391072	−	0.920360i	$-0.627896\pi$
$241$	−12.1421	−0.782144	−0.391072	+	0.920360i	$0.627896\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.65685	0.425291
$246$	0	0
$247$	−1.65685	−0.105423
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.51472	−0.600564	−0.300282	−	0.953851i	$-0.597081\pi$
$251$	−9.51472	−0.600564	−0.300282	+	0.953851i	$0.597081\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.79899	0.611244	0.305622	−	0.952153i	$-0.401136\pi$
$257$	9.79899	0.611244	0.305622	+	0.952153i	$0.401136\pi$
$258$	0	0
$259$	2.14214	0.133106
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.7279	1.15481	0.577407	−	0.816457i	$-0.304065\pi$
$263$	18.7279	1.15481	0.577407	+	0.816457i	$0.304065\pi$
$264$	0	0
$265$	6.48528	0.398388
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.65685	0.101020	0.0505101	−	0.998724i	$-0.483915\pi$
$269$	1.65685	0.101020	0.0505101	+	0.998724i	$0.483915\pi$
$270$	0	0
$271$	−12.4853	−0.758427	−0.379213	−	0.925309i	$-0.623805\pi$
$271$	−12.4853	−0.758427	−0.379213	+	0.925309i	$0.623805\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	14.5858	0.876375	0.438187	−	0.898884i	$-0.355621\pi$
$277$	14.5858	0.876375	0.438187	+	0.898884i	$0.355621\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.68629	0.398871	0.199435	−	0.979911i	$-0.436089\pi$
$281$	6.68629	0.398871	0.199435	+	0.979911i	$0.436089\pi$
$282$	0	0
$283$	−10.2426	−0.608862	−0.304431	−	0.952534i	$-0.598466\pi$
$283$	−10.2426	−0.608862	−0.304431	+	0.952534i	$0.598466\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.14214	0.126446
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.41421	−0.549984	−0.274992	−	0.961446i	$-0.588675\pi$
$293$	−9.41421	−0.549984	−0.274992	+	0.961446i	$0.588675\pi$
$294$	0	0
$295$	−8.82843	−0.514011
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	5.31371	0.306277
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.82843	−0.505514
$306$	0	0
$307$	−10.2426	−0.584578	−0.292289	−	0.956330i	$-0.594417\pi$
$307$	−10.2426	−0.584578	−0.292289	+	0.956330i	$0.594417\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.828427	−0.0469758	−0.0234879	−	0.999724i	$-0.507477\pi$
$311$	−0.828427	−0.0469758	−0.0234879	+	0.999724i	$0.507477\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	0	0
$319$	−7.65685	−0.428702
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.97056	−0.276570
$324$	0	0
$325$	1.41421	0.0784465
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.62742	0.144854
$330$	0	0
$331$	14.4853	0.796183	0.398092	−	0.917346i	$-0.369673\pi$
$331$	14.4853	0.796183	0.398092	+	0.917346i	$0.369673\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.48528	−0.463600
$336$	0	0
$337$	18.8701	1.02792	0.513959	−	0.857815i	$-0.328178\pi$
$337$	18.8701	1.02792	0.513959	+	0.857815i	$0.328178\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.82843	0.261474
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.2426	−0.979316	−0.489658	−	0.871915i	$-0.662878\pi$
$347$	−18.2426	−0.979316	−0.489658	+	0.871915i	$0.662878\pi$
$348$	0	0
$349$	−11.1716	−0.598001	−0.299000	−	0.954253i	$-0.596653\pi$
$349$	−11.1716	−0.598001	−0.299000	+	0.954253i	$0.596653\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.4853	−0.558075	−0.279038	−	0.960280i	$-0.590015\pi$
$353$	−10.4853	−0.558075	−0.279038	+	0.960280i	$0.590015\pi$
$354$	0	0
$355$	−10.4853	−0.556501
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.48528	0.447836	0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528	0.447836	0.223918	+	0.974608i	$0.428115\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.07107	−0.370117
$366$	0	0
$367$	−21.1716	−1.10515	−0.552574	−	0.833464i	$-0.686354\pi$
$367$	−21.1716	−1.10515	−0.552574	+	0.833464i	$0.686354\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.79899	0.197234
$372$	0	0
$373$	15.7574	0.815885	0.407943	−	0.913008i	$-0.366246\pi$
$373$	15.7574	0.815885	0.407943	+	0.913008i	$0.366246\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.8284	0.557692
$378$	0	0
$379$	7.31371	0.375680	0.187840	−	0.982200i	$-0.439851\pi$
$379$	7.31371	0.375680	0.187840	+	0.982200i	$0.439851\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.970563	0.0495934	0.0247967	−	0.999693i	$-0.492106\pi$
$383$	0.970563	0.0495934	0.0247967	+	0.999693i	$0.492106\pi$
$384$	0	0
$385$	−0.585786	−0.0298544
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.31371	−0.168012	−0.0840058	−	0.996465i	$-0.526771\pi$
$389$	−3.31371	−0.168012	−0.0840058	+	0.996465i	$0.526771\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.485281	−0.0244172
$396$	0	0
$397$	18.4853	0.927750	0.463875	−	0.885901i	$-0.346459\pi$
$397$	18.4853	0.927750	0.463875	+	0.885901i	$0.346459\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.31371	0.465104	0.232552	−	0.972584i	$-0.425292\pi$
$401$	9.31371	0.465104	0.232552	+	0.972584i	$0.425292\pi$
$402$	0	0
$403$	−6.82843	−0.340148
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.65685	0.181264
$408$	0	0
$409$	6.97056	0.344672	0.172336	−	0.985038i	$-0.444868\pi$
$409$	6.97056	0.344672	0.172336	+	0.985038i	$0.444868\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.17157	−0.254476
$414$	0	0
$415$	−10.2426	−0.502791
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.6274	−1.30083	−0.650417	−	0.759577i	$-0.725406\pi$
$419$	−26.6274	−1.30083	−0.650417	+	0.759577i	$0.725406\pi$
$420$	0	0
$421$	33.6569	1.64033	0.820167	−	0.572124i	$-0.193880\pi$
$421$	33.6569	1.64033	0.820167	+	0.572124i	$0.193880\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.24264	0.205798
$426$	0	0
$427$	−5.17157	−0.250270
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.97056	−0.432097	−0.216048	−	0.976383i	$-0.569317\pi$
$431$	−8.97056	−0.432097	−0.216048	+	0.976383i	$0.569317\pi$
$432$	0	0
$433$	−15.1716	−0.729099	−0.364550	−	0.931184i	$-0.618777\pi$
$433$	−15.1716	−0.729099	−0.364550	+	0.931184i	$0.618777\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.31371	−0.158516
$438$	0	0
$439$	−39.3137	−1.87634	−0.938170	−	0.346174i	$-0.887481\pi$
$439$	−39.3137	−1.87634	−0.938170	+	0.346174i	$0.887481\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.6569	0.738893	0.369446	−	0.929252i	$-0.379547\pi$
$449$	15.6569	0.738893	0.369446	+	0.929252i	$0.379547\pi$
$450$	0	0
$451$	3.65685	0.172195
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.828427	0.0388373
$456$	0	0
$457$	−33.6985	−1.57635	−0.788174	−	0.615452i	$-0.788973\pi$
$457$	−33.6985	−1.57635	−0.788174	+	0.615452i	$0.788973\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.4558	−0.533552	−0.266776	−	0.963759i	$-0.585958\pi$
$461$	−11.4558	−0.533552	−0.266776	+	0.963759i	$0.585958\pi$
$462$	0	0
$463$	3.51472	0.163343	0.0816714	−	0.996659i	$-0.473974\pi$
$463$	3.51472	0.163343	0.0816714	+	0.996659i	$0.473974\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.82843	0.130884	0.0654420	−	0.997856i	$-0.479154\pi$
$467$	2.82843	0.130884	0.0654420	+	0.997856i	$0.479154\pi$
$468$	0	0
$469$	−4.97056	−0.229519
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.07107	0.417088
$474$	0	0
$475$	−1.17157	−0.0537555
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.1421	1.19446	0.597232	−	0.802068i	$-0.296267\pi$
$479$	26.1421	1.19446	0.597232	+	0.802068i	$0.296267\pi$
$480$	0	0
$481$	−5.17157	−0.235803
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.82843	−0.400878
$486$	0	0
$487$	27.7990	1.25969	0.629846	−	0.776720i	$-0.283118\pi$
$487$	27.7990	1.25969	0.629846	+	0.776720i	$0.283118\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.02944	−0.317234	−0.158617	−	0.987340i	$-0.550704\pi$
$491$	−7.02944	−0.317234	−0.158617	+	0.987340i	$0.550704\pi$
$492$	0	0
$493$	32.4853	1.46306
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.14214	−0.275512
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.92893	0.130595	0.0652973	−	0.997866i	$-0.479200\pi$
$503$	2.92893	0.130595	0.0652973	+	0.997866i	$0.479200\pi$
$504$	0	0
$505$	−18.4853	−0.822584
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−35.3137	−1.56525	−0.782626	−	0.622492i	$-0.786120\pi$
$509$	−35.3137	−1.56525	−0.782626	+	0.622492i	$0.786120\pi$
$510$	0	0
$511$	−4.14214	−0.183237
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.48528	0.197262
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.6274	0.903704	0.451852	−	0.892093i	$-0.350764\pi$
$521$	20.6274	0.903704	0.451852	+	0.892093i	$0.350764\pi$
$522$	0	0
$523$	23.8995	1.04505	0.522526	−	0.852623i	$-0.324990\pi$
$523$	23.8995	1.04505	0.522526	+	0.852623i	$0.324990\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.4853	−0.892353
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.17157	−0.224006
$534$	0	0
$535$	−4.58579	−0.198261
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.65685	0.286731
$540$	0	0
$541$	20.3431	0.874620	0.437310	−	0.899311i	$-0.355931\pi$
$541$	20.3431	0.874620	0.437310	+	0.899311i	$0.355931\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.31371	−0.398955
$546$	0	0
$547$	−38.7279	−1.65589	−0.827943	−	0.560812i	$-0.810489\pi$
$547$	−38.7279	−1.65589	−0.827943	+	0.560812i	$0.810489\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.97056	−0.382159
$552$	0	0
$553$	−0.284271	−0.0120884
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.21320	−0.0514051	−0.0257025	−	0.999670i	$-0.508182\pi$
$557$	−1.21320	−0.0514051	−0.0257025	+	0.999670i	$0.508182\pi$
$558$	0	0
$559$	−12.8284	−0.542585
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.07107	0.382300	0.191150	−	0.981561i	$-0.438778\pi$
$563$	9.07107	0.382300	0.191150	+	0.981561i	$0.438778\pi$
$564$	0	0
$565$	−7.65685	−0.322126
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.1421	1.01209	0.506045	−	0.862507i	$-0.331107\pi$
$569$	24.1421	1.01209	0.506045	+	0.862507i	$0.331107\pi$
$570$	0	0
$571$	−6.34315	−0.265452	−0.132726	−	0.991153i	$-0.542373\pi$
$571$	−6.34315	−0.265452	−0.132726	+	0.991153i	$0.542373\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.82843	0.117954
$576$	0	0
$577$	3.17157	0.132034	0.0660172	−	0.997818i	$-0.478971\pi$
$577$	3.17157	0.132034	0.0660172	+	0.997818i	$0.478971\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	6.48528	0.268593
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.51472	−0.310166	−0.155083	−	0.987901i	$-0.549564\pi$
$587$	−7.51472	−0.310166	−0.155083	+	0.987901i	$0.549564\pi$
$588$	0	0
$589$	5.65685	0.233087
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.5858	−1.25601	−0.628004	−	0.778210i	$-0.716128\pi$
$593$	−30.5858	−1.25601	−0.628004	+	0.778210i	$0.716128\pi$
$594$	0	0
$595$	2.48528	0.101887
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.6569	−0.558004	−0.279002	−	0.960291i	$-0.590004\pi$
$599$	−13.6569	−0.558004	−0.279002	+	0.960291i	$0.590004\pi$
$600$	0	0
$601$	−8.14214	−0.332125	−0.166062	−	0.986115i	$-0.553105\pi$
$601$	−8.14214	−0.332125	−0.166062	+	0.986115i	$0.553105\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	0.384776	0.0156176	0.00780879	−	0.999970i	$-0.497514\pi$
$607$	0.384776	0.0156176	0.00780879	+	0.999970i	$0.497514\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.34315	−0.256616
$612$	0	0
$613$	35.5563	1.43611	0.718054	−	0.695988i	$-0.245033\pi$
$613$	35.5563	1.43611	0.718054	+	0.695988i	$0.245033\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−39.6569	−1.59652	−0.798262	−	0.602310i	$-0.794247\pi$
$617$	−39.6569	−1.59652	−0.798262	+	0.602310i	$0.794247\pi$
$618$	0	0
$619$	−35.4558	−1.42509	−0.712545	−	0.701626i	$-0.752458\pi$
$619$	−35.4558	−1.42509	−0.712545	+	0.701626i	$0.752458\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.17157	−0.0469381
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−15.5147	−0.618612
$630$	0	0
$631$	22.6274	0.900783	0.450392	−	0.892831i	$-0.351284\pi$
$631$	22.6274	0.900783	0.450392	+	0.892831i	$0.351284\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.55635	0.379232
$636$	0	0
$637$	−9.41421	−0.373005
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.68629	0.106102	0.0530511	−	0.998592i	$-0.483105\pi$
$641$	2.68629	0.106102	0.0530511	+	0.998592i	$0.483105\pi$
$642$	0	0
$643$	29.6569	1.16955	0.584776	−	0.811195i	$-0.301182\pi$
$643$	29.6569	1.16955	0.584776	+	0.811195i	$0.301182\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.6274	−1.51860	−0.759300	−	0.650740i	$-0.774459\pi$
$647$	−38.6274	−1.51860	−0.759300	+	0.650740i	$0.774459\pi$
$648$	0	0
$649$	−8.82843	−0.346546
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−33.7990	−1.32266	−0.661328	−	0.750097i	$-0.730007\pi$
$653$	−33.7990	−1.32266	−0.661328	+	0.750097i	$0.730007\pi$
$654$	0	0
$655$	−1.17157	−0.0457771
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22.6274	−0.881439	−0.440720	−	0.897645i	$-0.645277\pi$
$659$	−22.6274	−0.881439	−0.440720	+	0.897645i	$0.645277\pi$
$660$	0	0
$661$	28.3431	1.10242	0.551210	−	0.834366i	$-0.314166\pi$
$661$	28.3431	1.10242	0.551210	+	0.834366i	$0.314166\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.686292	−0.0266132
$666$	0	0
$667$	21.6569	0.838557
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.82843	−0.340818
$672$	0	0
$673$	6.10051	0.235157	0.117579	−	0.993064i	$-0.462487\pi$
$673$	6.10051	0.235157	0.117579	+	0.993064i	$0.462487\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.10051	−0.234461	−0.117231	−	0.993105i	$-0.537402\pi$
$677$	−6.10051	−0.234461	−0.117231	+	0.993105i	$0.537402\pi$
$678$	0	0
$679$	−5.17157	−0.198467
$680$	0	0
$681$	0	0
$682$	0	0
$683$	35.7990	1.36981	0.684905	−	0.728632i	$-0.259844\pi$
$683$	35.7990	1.36981	0.684905	+	0.728632i	$0.259844\pi$
$684$	0	0
$685$	5.31371	0.203026
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.17157	−0.349409
$690$	0	0
$691$	17.6569	0.671698	0.335849	−	0.941916i	$-0.390977\pi$
$691$	17.6569	0.671698	0.335849	+	0.941916i	$0.390977\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16.0000	0.606915
$696$	0	0
$697$	−15.5147	−0.587662
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.2843	0.992743	0.496372	−	0.868110i	$-0.334665\pi$
$701$	26.2843	0.992743	0.496372	+	0.868110i	$0.334665\pi$
$702$	0	0
$703$	4.28427	0.161584
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.8284	−0.407245
$708$	0	0
$709$	35.3137	1.32623	0.663117	−	0.748516i	$-0.269233\pi$
$709$	35.3137	1.32623	0.663117	+	0.748516i	$0.269233\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−13.6569	−0.511453
$714$	0	0
$715$	1.41421	0.0528886
$716$	0	0
$717$	0	0
$718$	0	0
$719$	5.65685	0.210965	0.105483	−	0.994421i	$-0.466361\pi$
$719$	5.65685	0.210965	0.105483	+	0.994421i	$0.466361\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.65685	0.284368
$726$	0	0
$727$	28.2843	1.04901	0.524503	−	0.851409i	$-0.324251\pi$
$727$	28.2843	1.04901	0.524503	+	0.851409i	$0.324251\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−38.4853	−1.42343
$732$	0	0
$733$	4.44365	0.164130	0.0820650	−	0.996627i	$-0.473848\pi$
$733$	4.44365	0.164130	0.0820650	+	0.996627i	$0.473848\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.48528	−0.312559
$738$	0	0
$739$	−20.2843	−0.746169	−0.373084	−	0.927797i	$-0.621700\pi$
$739$	−20.2843	−0.746169	−0.373084	+	0.927797i	$0.621700\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.5269	−1.56016	−0.780081	−	0.625679i	$-0.784822\pi$
$743$	−42.5269	−1.56016	−0.780081	+	0.625679i	$0.784822\pi$
$744$	0	0
$745$	2.48528	0.0910537
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.68629	−0.0981550
$750$	0	0
$751$	2.34315	0.0855026	0.0427513	−	0.999086i	$-0.486388\pi$
$751$	2.34315	0.0855026	0.0427513	+	0.999086i	$0.486388\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.1421	−0.369110
$756$	0	0
$757$	5.51472	0.200436	0.100218	−	0.994966i	$-0.468046\pi$
$757$	5.51472	0.200436	0.100218	+	0.994966i	$0.468046\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.828427	0.0300305	0.0150152	−	0.999887i	$-0.495220\pi$
$761$	0.828427	0.0300305	0.0150152	+	0.999887i	$0.495220\pi$
$762$	0	0
$763$	−5.45584	−0.197515
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.4853	0.450817
$768$	0	0
$769$	−27.1716	−0.979832	−0.489916	−	0.871770i	$-0.662973\pi$
$769$	−27.1716	−0.979832	−0.489916	+	0.871770i	$0.662973\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−11.8579	−0.426498	−0.213249	−	0.976998i	$-0.568405\pi$
$773$	−11.8579	−0.426498	−0.213249	+	0.976998i	$0.568405\pi$
$774$	0	0
$775$	−4.82843	−0.173442
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.28427	0.153500
$780$	0	0
$781$	−10.4853	−0.375193
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.51472	−0.196829
$786$	0	0
$787$	23.8995	0.851925	0.425962	−	0.904741i	$-0.359936\pi$
$787$	23.8995	0.851925	0.425962	+	0.904741i	$0.359936\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.48528	−0.159478
$792$	0	0
$793$	12.4853	0.443365
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.6274	0.588973	0.294487	−	0.955656i	$-0.404851\pi$
$797$	16.6274	0.588973	0.294487	+	0.955656i	$0.404851\pi$
$798$	0	0
$799$	−19.0294	−0.673213
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.07107	−0.249533
$804$	0	0
$805$	1.65685	0.0583964
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.0294	0.739356	0.369678	−	0.929160i	$-0.379468\pi$
$809$	21.0294	0.739356	0.369678	+	0.929160i	$0.379468\pi$
$810$	0	0
$811$	−14.6274	−0.513638	−0.256819	−	0.966460i	$-0.582674\pi$
$811$	−14.6274	−0.513638	−0.256819	+	0.966460i	$0.582674\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.3137	−0.396302
$816$	0	0
$817$	10.6274	0.371806
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.3137	0.743854	0.371927	−	0.928262i	$-0.378697\pi$
$821$	21.3137	0.743854	0.371927	+	0.928262i	$0.378697\pi$
$822$	0	0
$823$	−2.62742	−0.0915860	−0.0457930	−	0.998951i	$-0.514581\pi$
$823$	−2.62742	−0.0915860	−0.0457930	+	0.998951i	$0.514581\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.8701	−0.586629	−0.293315	−	0.956016i	$-0.594758\pi$
$827$	−16.8701	−0.586629	−0.293315	+	0.956016i	$0.594758\pi$
$828$	0	0
$829$	−38.6274	−1.34159	−0.670793	−	0.741645i	$-0.734046\pi$
$829$	−38.6274	−1.34159	−0.670793	+	0.741645i	$0.734046\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−28.2426	−0.978550
$834$	0	0
$835$	−18.7279	−0.648106
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−15.8579	−0.547474	−0.273737	−	0.961805i	$-0.588260\pi$
$839$	−15.8579	−0.547474	−0.273737	+	0.961805i	$0.588260\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.0000	0.378412
$846$	0	0
$847$	−0.585786	−0.0201279
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.3431	−0.354558
$852$	0	0
$853$	37.4142	1.28104	0.640519	−	0.767942i	$-0.278719\pi$
$853$	37.4142	1.28104	0.640519	+	0.767942i	$0.278719\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.8995	−1.43126	−0.715630	−	0.698480i	$-0.753860\pi$
$857$	−41.8995	−1.43126	−0.715630	+	0.698480i	$0.753860\pi$
$858$	0	0
$859$	7.85786	0.268107	0.134053	−	0.990974i	$-0.457201\pi$
$859$	7.85786	0.268107	0.134053	+	0.990974i	$0.457201\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.3137	0.929769	0.464885	−	0.885371i	$-0.346096\pi$
$863$	27.3137	0.929769	0.464885	+	0.885371i	$0.346096\pi$
$864$	0	0
$865$	−21.8995	−0.744605
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.485281	−0.0164620
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.585786	0.0198032
$876$	0	0
$877$	−14.3848	−0.485739	−0.242870	−	0.970059i	$-0.578089\pi$
$877$	−14.3848	−0.485739	−0.242870	+	0.970059i	$0.578089\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.34315	−0.0789426	−0.0394713	−	0.999221i	$-0.512567\pi$
$881$	−2.34315	−0.0789426	−0.0394713	+	0.999221i	$0.512567\pi$
$882$	0	0
$883$	−16.2843	−0.548009	−0.274005	−	0.961728i	$-0.588348\pi$
$883$	−16.2843	−0.548009	−0.274005	+	0.961728i	$0.588348\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.10051	−0.137681	−0.0688407	−	0.997628i	$-0.521930\pi$
$887$	−4.10051	−0.137681	−0.0688407	+	0.997628i	$0.521930\pi$
$888$	0	0
$889$	5.59798	0.187750
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.25483	0.175846
$894$	0	0
$895$	−17.6569	−0.590204
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−36.9706	−1.23304
$900$	0	0
$901$	−27.5147	−0.916648
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.3137	−0.774974
$906$	0	0
$907$	39.5147	1.31206	0.656032	−	0.754733i	$-0.272233\pi$
$907$	39.5147	1.31206	0.656032	+	0.754733i	$0.272233\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.6863	−1.21547	−0.607736	−	0.794139i	$-0.707922\pi$
$911$	−36.6863	−1.21547	−0.607736	+	0.794139i	$0.707922\pi$
$912$	0	0
$913$	−10.2426	−0.338982
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.686292	−0.0226633
$918$	0	0
$919$	12.6863	0.418482	0.209241	−	0.977864i	$-0.432901\pi$
$919$	12.6863	0.418482	0.209241	+	0.977864i	$0.432901\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.8284	0.488084
$924$	0	0
$925$	−3.65685	−0.120237
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.34315	0.208112	0.104056	−	0.994571i	$-0.466818\pi$
$929$	6.34315	0.208112	0.104056	+	0.994571i	$0.466818\pi$
$930$	0	0
$931$	7.79899	0.255602
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.24264	0.138749
$936$	0	0
$937$	11.0711	0.361676	0.180838	−	0.983513i	$-0.442119\pi$
$937$	11.0711	0.361676	0.180838	+	0.983513i	$0.442119\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−34.4853	−1.12419	−0.562094	−	0.827073i	$-0.690004\pi$
$941$	−34.4853	−1.12419	−0.562094	+	0.827073i	$0.690004\pi$
$942$	0	0
$943$	−10.3431	−0.336819
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.4558	1.21715	0.608576	−	0.793496i	$-0.291741\pi$
$947$	37.4558	1.21715	0.608576	+	0.793496i	$0.291741\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21.8995	−0.709394	−0.354697	−	0.934981i	$-0.615416\pi$
$953$	−21.8995	−0.709394	−0.354697	+	0.934981i	$0.615416\pi$
$954$	0	0
$955$	3.31371	0.107229
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.11270	0.100514
$960$	0	0
$961$	−7.68629	−0.247945
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−17.8995	−0.576205
$966$	0	0
$967$	−44.8701	−1.44292	−0.721462	−	0.692454i	$-0.756529\pi$
$967$	−44.8701	−1.44292	−0.721462	+	0.692454i	$0.756529\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.9411	−1.21759	−0.608794	−	0.793328i	$-0.708347\pi$
$971$	−37.9411	−1.21759	−0.608794	+	0.793328i	$0.708347\pi$
$972$	0	0
$973$	9.37258	0.300471
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.28427	−0.201052	−0.100526	−	0.994934i	$-0.532053\pi$
$977$	−6.28427	−0.201052	−0.100526	+	0.994934i	$0.532053\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	56.9706	1.81708	0.908539	−	0.417799i	$-0.137198\pi$
$983$	56.9706	1.81708	0.908539	+	0.417799i	$0.137198\pi$
$984$	0	0
$985$	−8.24264	−0.262632
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25.6569	−0.815841
$990$	0	0
$991$	−15.4558	−0.490971	−0.245486	−	0.969400i	$-0.578947\pi$
$991$	−15.4558	−0.490971	−0.245486	+	0.969400i	$0.578947\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.9706	0.538003
$996$	0	0
$997$	0.242641	0.00768451	0.00384225	−	0.999993i	$-0.498777\pi$
$997$	0.242641	0.00768451	0.00384225	+	0.999993i	$0.498777\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3960.2.a.u.1.2	✓	2
3.2	odd	2		3960.2.a.bb.1.2	yes	2
4.3	odd	2		7920.2.a.bx.1.1		2
12.11	even	2		7920.2.a.cf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.2.a.u.1.2	✓	2	1.1	even	1	trivial
3960.2.a.bb.1.2	yes	2	3.2	odd	2
7920.2.a.bx.1.1		2	4.3	odd	2
7920.2.a.cf.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3960.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -0.585786 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -0.585786 q^{7} -1.00000 q^{11} +1.41421 q^{13} +4.24264 q^{17} -1.17157 q^{19} +2.82843 q^{23} +1.00000 q^{25} +7.65685 q^{29} -4.82843 q^{31} +0.585786 q^{35} -3.65685 q^{37} -3.65685 q^{41} -9.07107 q^{43} -4.48528 q^{47} -6.65685 q^{49} -6.48528 q^{53} +1.00000 q^{55} +8.82843 q^{59} +8.82843 q^{61} -1.41421 q^{65} +8.48528 q^{67} +10.4853 q^{71} +7.07107 q^{73} +0.585786 q^{77} +0.485281 q^{79} +10.2426 q^{83} -4.24264 q^{85} +2.00000 q^{89} -0.828427 q^{91} +1.17157 q^{95} +8.82843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 4 * q^7	\( 2 q - 2 q^{5} - 4 q^{7} - 2 q^{11} - 8 q^{19} + 2 q^{25} + 4 q^{29} - 4 q^{31} + 4 q^{35} + 4 q^{37} + 4 q^{41} - 4 q^{43} + 8 q^{47} - 2 q^{49} + 4 q^{53} + 2 q^{55} + 12 q^{59} + 12 q^{61} + 4 q^{71} + 4 q^{77} - 16 q^{79} + 12 q^{83} + 4 q^{89} + 4 q^{91} + 8 q^{95} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^7 - 2 * q^11 - 8 * q^19 + 2 * q^25 + 4 * q^29 - 4 * q^31 + 4 * q^35 + 4 * q^37 + 4 * q^41 - 4 * q^43 + 8 * q^47 - 2 * q^49 + 4 * q^53 + 2 * q^55 + 12 * q^59 + 12 * q^61 + 4 * q^71 + 4 * q^77 - 16 * q^79 + 12 * q^83 + 4 * q^89 + 4 * q^91 + 8 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more