LMFDB - Embedded newform 3960.2.a.bg.1.1

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:	$1$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.321637$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -4.21819 q^{7} -1.00000 q^{11} +5.57491 q^{13} +2.21819 q^{17} +0.643274 q^{19} +0.643274 q^{23} +1.00000 q^{25} -2.00000 q^{29} -1.35673 q^{31} +4.21819 q^{35} +2.00000 q^{37} -2.00000 q^{41} -4.86146 q^{43} -4.64327 q^{47} +10.7931 q^{49} +6.43637 q^{53} +1.00000 q^{55} -1.35673 q^{59} -0.0701770 q^{61} -5.57491 q^{65} -12.4364 q^{67} +5.14982 q^{71} +10.6546 q^{73} +4.21819 q^{77} +7.14982 q^{79} -13.9411 q^{83} -2.21819 q^{85} -12.3662 q^{89} -23.5160 q^{91} -0.643274 q^{95} -7.72292 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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
$7$	−4.21819	−1.59432	−0.797162	−	0.603765i	$-0.793667\pi$
$7$	−4.21819	−1.59432	−0.797162	+	0.603765i	$0.793667\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.57491	1.54620	0.773101	−	0.634283i	$-0.218704\pi$
$13$	5.57491	1.54620	0.773101	+	0.634283i	$0.218704\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.21819	0.537989	0.268995	−	0.963142i	$-0.413309\pi$
$17$	2.21819	0.537989	0.268995	+	0.963142i	$0.413309\pi$
$18$	0	0
$19$	0.643274	0.147577	0.0737886	−	0.997274i	$-0.476491\pi$
$19$	0.643274	0.147577	0.0737886	+	0.997274i	$0.476491\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.643274	0.134132	0.0670660	−	0.997749i	$-0.478636\pi$
$23$	0.643274	0.134132	0.0670660	+	0.997749i	$0.478636\pi$
$24$	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
$31$	−1.35673	−0.243675	−0.121838	−	0.992550i	$-0.538879\pi$
$31$	−1.35673	−0.243675	−0.121838	+	0.992550i	$0.538879\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.21819	0.713004
$36$	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
$43$	−4.86146	−0.741366	−0.370683	−	0.928759i	$-0.620876\pi$
$43$	−4.86146	−0.741366	−0.370683	+	0.928759i	$0.620876\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.64327	−0.677291	−0.338646	−	0.940914i	$-0.609969\pi$
$47$	−4.64327	−0.677291	−0.338646	+	0.940914i	$0.609969\pi$
$48$	0	0
$49$	10.7931	1.54187
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.43637	0.884104	0.442052	−	0.896989i	$-0.354251\pi$
$53$	6.43637	0.884104	0.442052	+	0.896989i	$0.354251\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.35673	−0.176631	−0.0883153	−	0.996093i	$-0.528148\pi$
$59$	−1.35673	−0.176631	−0.0883153	+	0.996093i	$0.528148\pi$
$60$	0	0
$61$	−0.0701770	−0.00898524	−0.00449262	−	0.999990i	$-0.501430\pi$
$61$	−0.0701770	−0.00898524	−0.00449262	+	0.999990i	$0.501430\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.57491	−0.691483
$66$	0	0
$67$	−12.4364	−1.51934	−0.759672	−	0.650306i	$-0.774641\pi$
$67$	−12.4364	−1.51934	−0.759672	+	0.650306i	$0.774641\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.14982	0.611172	0.305586	−	0.952165i	$-0.401148\pi$
$71$	5.14982	0.611172	0.305586	+	0.952165i	$0.401148\pi$
$72$	0	0
$73$	10.6546	1.24702	0.623511	−	0.781815i	$-0.285706\pi$
$73$	10.6546	1.24702	0.623511	+	0.781815i	$0.285706\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.21819	0.480707
$78$	0	0
$79$	7.14982	0.804418	0.402209	−	0.915548i	$-0.368242\pi$
$79$	7.14982	0.804418	0.402209	+	0.915548i	$0.368242\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.9411	−1.53024	−0.765118	−	0.643890i	$-0.777319\pi$
$83$	−13.9411	−1.53024	−0.765118	+	0.643890i	$0.777319\pi$
$84$	0	0
$85$	−2.21819	−0.240596
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.3662	−1.31081	−0.655407	−	0.755276i	$-0.727503\pi$
$89$	−12.3662	−1.31081	−0.655407	+	0.755276i	$0.727503\pi$
$90$	0	0
$91$	−23.5160	−2.46515
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.643274	−0.0659986
$96$	0	0
$97$	−7.72292	−0.784144	−0.392072	−	0.919935i	$-0.628242\pi$
$97$	−7.72292	−0.784144	−0.392072	+	0.919935i	$0.628242\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.4364	−1.43647	−0.718236	−	0.695799i	$-0.755050\pi$
$101$	−14.4364	−1.43647	−0.718236	+	0.695799i	$0.755050\pi$
$102$	0	0
$103$	16.8727	1.66252	0.831261	−	0.555883i	$-0.187620\pi$
$103$	16.8727	1.66252	0.831261	+	0.555883i	$0.187620\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.49526	−0.241226	−0.120613	−	0.992700i	$-0.538486\pi$
$107$	−2.49526	−0.241226	−0.120613	+	0.992700i	$0.538486\pi$
$108$	0	0
$109$	2.20690	0.211383	0.105691	−	0.994399i	$-0.466294\pi$
$109$	2.20690	0.211383	0.105691	+	0.994399i	$0.466294\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.7931	−1.67383	−0.836917	−	0.547330i	$-0.815644\pi$
$113$	−17.7931	−1.67383	−0.836917	+	0.547330i	$0.815644\pi$
$114$	0	0
$115$	−0.643274	−0.0599856
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.35673	−0.857730
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.9316	−0.970026	−0.485013	−	0.874507i	$-0.661185\pi$
$127$	−10.9316	−0.970026	−0.485013	+	0.874507i	$0.661185\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.14982	−0.624683	−0.312342	−	0.949970i	$-0.601113\pi$
$131$	−7.14982	−0.624683	−0.312342	+	0.949970i	$0.601113\pi$
$132$	0	0
$133$	−2.71345	−0.235286
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.7931	−1.17842	−0.589212	−	0.807978i	$-0.700562\pi$
$137$	−13.7931	−1.17842	−0.589212	+	0.807978i	$0.700562\pi$
$138$	0	0
$139$	9.07965	0.770126	0.385063	−	0.922890i	$-0.374180\pi$
$139$	9.07965	0.770126	0.385063	+	0.922890i	$0.374180\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.57491	−0.466198
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.1498	−1.40497	−0.702484	−	0.711699i	$-0.747926\pi$
$149$	−17.1498	−1.40497	−0.702484	+	0.711699i	$0.747926\pi$
$150$	0	0
$151$	4.43637	0.361027	0.180513	−	0.983573i	$-0.442224\pi$
$151$	4.43637	0.361027	0.180513	+	0.983573i	$0.442224\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.35673	0.108975
$156$	0	0
$157$	−22.0226	−1.75759	−0.878796	−	0.477197i	$-0.841653\pi$
$157$	−22.0226	−1.75759	−0.878796	+	0.477197i	$0.841653\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.71345	−0.213850
$162$	0	0
$163$	−9.28655	−0.727379	−0.363689	−	0.931520i	$-0.618483\pi$
$163$	−9.28655	−0.727379	−0.363689	+	0.931520i	$0.618483\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−25.0909	−1.94159	−0.970797	−	0.239901i	$-0.922885\pi$
$167$	−25.0909	−1.94159	−0.970797	+	0.239901i	$0.922885\pi$
$168$	0	0
$169$	18.0796	1.39074
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.06836	−0.233283	−0.116642	−	0.993174i	$-0.537213\pi$
$173$	−3.06836	−0.233283	−0.116642	+	0.993174i	$0.537213\pi$
$174$	0	0
$175$	−4.21819	−0.318865
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.8727	0.962154	0.481077	−	0.876678i	$-0.340246\pi$
$179$	12.8727	0.962154	0.481077	+	0.876678i	$0.340246\pi$
$180$	0	0
$181$	6.50655	0.483628	0.241814	−	0.970323i	$-0.422258\pi$
$181$	6.50655	0.483628	0.241814	+	0.970323i	$0.422258\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−2.21819	−0.162210
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.7931	−1.43218	−0.716089	−	0.698009i	$-0.754070\pi$
$191$	−19.7931	−1.43218	−0.716089	+	0.698009i	$0.754070\pi$
$192$	0	0
$193$	−19.9411	−1.43539	−0.717696	−	0.696356i	$-0.754803\pi$
$193$	−19.9411	−1.43539	−0.717696	+	0.696356i	$0.754803\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.64509	0.259702	0.129851	−	0.991534i	$-0.458550\pi$
$197$	3.64509	0.259702	0.129851	+	0.991534i	$0.458550\pi$
$198$	0	0
$199$	16.8727	1.19608	0.598039	−	0.801467i	$-0.295947\pi$
$199$	16.8727	1.19608	0.598039	+	0.801467i	$0.295947\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.43637	0.592117
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.643274	−0.0444962
$210$	0	0
$211$	8.22947	0.566540	0.283270	−	0.959040i	$-0.408581\pi$
$211$	8.22947	0.566540	0.283270	+	0.959040i	$0.408581\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.86146	0.331549
$216$	0	0
$217$	5.72292	0.388497
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.3662	0.831840
$222$	0	0
$223$	−21.8633	−1.46407	−0.732037	−	0.681265i	$-0.761430\pi$
$223$	−21.8633	−1.46407	−0.732037	+	0.681265i	$0.761430\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.3680	−1.28550	−0.642750	−	0.766076i	$-0.722207\pi$
$227$	−19.3680	−1.28550	−0.642750	+	0.766076i	$0.722207\pi$
$228$	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
$233$	7.09093	0.464542	0.232271	−	0.972651i	$-0.425384\pi$
$233$	7.09093	0.464542	0.232271	+	0.972651i	$0.425384\pi$
$234$	0	0
$235$	4.64327	0.302894
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.43637	0.545704	0.272852	−	0.962056i	$-0.412033\pi$
$239$	8.43637	0.545704	0.272852	+	0.962056i	$0.412033\pi$
$240$	0	0
$241$	−3.72292	−0.239814	−0.119907	−	0.992785i	$-0.538260\pi$
$241$	−3.72292	−0.239814	−0.119907	+	0.992785i	$0.538260\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10.7931	−0.689546
$246$	0	0
$247$	3.58620	0.228184
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.21637	−0.329254	−0.164627	−	0.986356i	$-0.552642\pi$
$251$	−5.21637	−0.329254	−0.164627	+	0.986356i	$0.552642\pi$
$252$	0	0
$253$	−0.643274	−0.0404423
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.2295	0.887610	0.443805	−	0.896123i	$-0.353628\pi$
$257$	14.2295	0.887610	0.443805	+	0.896123i	$0.353628\pi$
$258$	0	0
$259$	−8.43637	−0.524211
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.93164	−0.427423	−0.213712	−	0.976897i	$-0.568555\pi$
$263$	−6.93164	−0.427423	−0.213712	+	0.976897i	$0.568555\pi$
$264$	0	0
$265$	−6.43637	−0.395383
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.50655	−0.396711	−0.198356	−	0.980130i	$-0.563560\pi$
$269$	−6.50655	−0.396711	−0.198356	+	0.980130i	$0.563560\pi$
$270$	0	0
$271$	4.43637	0.269490	0.134745	−	0.990880i	$-0.456978\pi$
$271$	4.43637	0.269490	0.134745	+	0.990880i	$0.456978\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	26.0113	1.56287	0.781433	−	0.623989i	$-0.214489\pi$
$277$	26.0113	1.56287	0.781433	+	0.623989i	$0.214489\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.8727	0.887234	0.443617	−	0.896217i	$-0.353695\pi$
$281$	14.8727	0.887234	0.443617	+	0.896217i	$0.353695\pi$
$282$	0	0
$283$	−18.7247	−1.11307	−0.556535	−	0.830824i	$-0.687869\pi$
$283$	−18.7247	−1.11307	−0.556535	+	0.830824i	$0.687869\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.43637	0.497983
$288$	0	0
$289$	−12.0796	−0.710568
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.5178	1.43235	0.716174	−	0.697922i	$-0.245892\pi$
$293$	24.5178	1.43235	0.716174	+	0.697922i	$0.245892\pi$
$294$	0	0
$295$	1.35673	0.0789916
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.58620	0.207395
$300$	0	0
$301$	20.5066	1.18198
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.0701770	0.00401832
$306$	0	0
$307$	−13.2978	−0.758947	−0.379474	−	0.925203i	$-0.623895\pi$
$307$	−13.2978	−0.758947	−0.379474	+	0.925203i	$0.623895\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.7360	1.17583	0.587916	−	0.808922i	$-0.299949\pi$
$311$	20.7360	1.17583	0.587916	+	0.808922i	$0.299949\pi$
$312$	0	0
$313$	−12.7135	−0.718607	−0.359303	−	0.933221i	$-0.616986\pi$
$313$	−12.7135	−0.718607	−0.359303	+	0.933221i	$0.616986\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.2865	0.633916	0.316958	−	0.948440i	$-0.397339\pi$
$317$	11.2865	0.633916	0.316958	+	0.948440i	$0.397339\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.42690	0.0793950
$324$	0	0
$325$	5.57491	0.309240
$326$	0	0
$327$	0	0
$328$	0	0
$329$	19.5862	1.07982
$330$	0	0
$331$	−17.3567	−0.954012	−0.477006	−	0.878900i	$-0.658278\pi$
$331$	−17.3567	−0.954012	−0.477006	+	0.878900i	$0.658278\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.4364	0.679472
$336$	0	0
$337$	15.5047	0.844597	0.422298	−	0.906457i	$-0.361223\pi$
$337$	15.5047	0.844597	0.422298	+	0.906457i	$0.361223\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.35673	0.0734708
$342$	0	0
$343$	−16.0000	−0.863919
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.3680	0.610267	0.305133	−	0.952310i	$-0.401299\pi$
$347$	11.3680	0.610267	0.305133	+	0.952310i	$0.401299\pi$
$348$	0	0
$349$	25.3567	1.35731	0.678657	−	0.734455i	$-0.262562\pi$
$349$	25.3567	1.35731	0.678657	+	0.734455i	$0.262562\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.9429	−1.32758	−0.663789	−	0.747920i	$-0.731053\pi$
$353$	−24.9429	−1.32758	−0.663789	+	0.747920i	$0.731053\pi$
$354$	0	0
$355$	−5.14982	−0.273324
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.43637	0.234143	0.117071	−	0.993123i	$-0.462649\pi$
$359$	4.43637	0.234143	0.117071	+	0.993123i	$0.462649\pi$
$360$	0	0
$361$	−18.5862	−0.978221
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.6546	−0.557685
$366$	0	0
$367$	11.1498	0.582016	0.291008	−	0.956721i	$-0.406009\pi$
$367$	11.1498	0.582016	0.291008	+	0.956721i	$0.406009\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−27.1498	−1.40955
$372$	0	0
$373$	3.27526	0.169587	0.0847933	−	0.996399i	$-0.472977\pi$
$373$	3.27526	0.169587	0.0847933	+	0.996399i	$0.472977\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11.1498	−0.574245
$378$	0	0
$379$	−11.5862	−0.595143	−0.297572	−	0.954700i	$-0.596177\pi$
$379$	−11.5862	−0.595143	−0.297572	+	0.954700i	$0.596177\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.3793	−0.990236	−0.495118	−	0.868826i	$-0.664875\pi$
$383$	−19.3793	−0.990236	−0.495118	+	0.868826i	$0.664875\pi$
$384$	0	0
$385$	−4.21819	−0.214979
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−29.9524	−1.51865	−0.759323	−	0.650714i	$-0.774470\pi$
$389$	−29.9524	−1.51865	−0.759323	+	0.650714i	$0.774470\pi$
$390$	0	0
$391$	1.42690	0.0721616
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.14982	−0.359747
$396$	0	0
$397$	−5.00947	−0.251418	−0.125709	−	0.992067i	$-0.540121\pi$
$397$	−5.00947	−0.251418	−0.125709	+	0.992067i	$0.540121\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.9524	−0.796625	−0.398312	−	0.917250i	$-0.630404\pi$
$401$	−15.9524	−0.796625	−0.398312	+	0.917250i	$0.630404\pi$
$402$	0	0
$403$	−7.56363	−0.376771
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	15.2865	0.755871	0.377936	−	0.925832i	$-0.376634\pi$
$409$	15.2865	0.755871	0.377936	+	0.925832i	$0.376634\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.72292	0.281607
$414$	0	0
$415$	13.9411	0.684342
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	7.79310	0.379812	0.189906	−	0.981802i	$-0.439182\pi$
$421$	7.79310	0.379812	0.189906	+	0.981802i	$0.439182\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.21819	0.107598
$426$	0	0
$427$	0.296019	0.0143254
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.14035	−0.199434	−0.0997169	−	0.995016i	$-0.531794\pi$
$431$	−4.14035	−0.199434	−0.0997169	+	0.995016i	$0.531794\pi$
$432$	0	0
$433$	−13.1498	−0.631940	−0.315970	−	0.948769i	$-0.602330\pi$
$433$	−13.1498	−0.631940	−0.315970	+	0.948769i	$0.602330\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.413802	0.0197948
$438$	0	0
$439$	7.12725	0.340165	0.170083	−	0.985430i	$-0.445597\pi$
$439$	7.12725	0.340165	0.170083	+	0.985430i	$0.445597\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−22.2996	−1.05949	−0.529744	−	0.848157i	$-0.677712\pi$
$443$	−22.2996	−1.05949	−0.529744	+	0.848157i	$0.677712\pi$
$444$	0	0
$445$	12.3662	0.586214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.2389	1.56864	0.784321	−	0.620355i	$-0.213011\pi$
$449$	33.2389	1.56864	0.784321	+	0.620355i	$0.213011\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	0	0
$453$	0	0
$454$	0	0
$455$	23.5160	1.10245
$456$	0	0
$457$	26.6771	1.24790	0.623952	−	0.781463i	$-0.285526\pi$
$457$	26.6771	1.24790	0.623952	+	0.781463i	$0.285526\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.7360	1.33837	0.669185	−	0.743096i	$-0.266643\pi$
$461$	28.7360	1.33837	0.669185	+	0.743096i	$0.266643\pi$
$462$	0	0
$463$	−27.0095	−1.25524	−0.627618	−	0.778521i	$-0.715970\pi$
$463$	−27.0095	−1.25524	−0.627618	+	0.778521i	$0.715970\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.43637	0.205291	0.102645	−	0.994718i	$-0.467269\pi$
$467$	4.43637	0.205291	0.102645	+	0.994718i	$0.467269\pi$
$468$	0	0
$469$	52.4589	2.42233
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.86146	0.223530
$474$	0	0
$475$	0.643274	0.0295155
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−34.5957	−1.58072	−0.790358	−	0.612645i	$-0.790106\pi$
$479$	−34.5957	−1.58072	−0.790358	+	0.612645i	$0.790106\pi$
$480$	0	0
$481$	11.1498	0.508388
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.72292	0.350680
$486$	0	0
$487$	0.576727	0.0261340	0.0130670	−	0.999915i	$-0.495841\pi$
$487$	0.576727	0.0261340	0.0130670	+	0.999915i	$0.495841\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−40.4589	−1.82589	−0.912943	−	0.408086i	$-0.866196\pi$
$491$	−40.4589	−1.82589	−0.912943	+	0.408086i	$0.866196\pi$
$492$	0	0
$493$	−4.43637	−0.199804
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−21.7229	−0.974406
$498$	0	0
$499$	17.4269	0.780135	0.390068	−	0.920786i	$-0.372452\pi$
$499$	17.4269	0.780135	0.390068	+	0.920786i	$0.372452\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.21819	0.366431	0.183215	−	0.983073i	$-0.441349\pi$
$503$	8.21819	0.366431	0.183215	+	0.983073i	$0.441349\pi$
$504$	0	0
$505$	14.4364	0.642410
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−29.5386	−1.30928	−0.654638	−	0.755943i	$-0.727179\pi$
$509$	−29.5386	−1.30928	−0.654638	+	0.755943i	$0.727179\pi$
$510$	0	0
$511$	−44.9429	−1.98816
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.8727	−0.743502
$516$	0	0
$517$	4.64327	0.204211
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.3793	−0.586158	−0.293079	−	0.956088i	$-0.594680\pi$
$521$	−13.3793	−0.586158	−0.293079	+	0.956088i	$0.594680\pi$
$522$	0	0
$523$	−11.9887	−0.524230	−0.262115	−	0.965037i	$-0.584420\pi$
$523$	−11.9887	−0.524230	−0.262115	+	0.965037i	$0.584420\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.00947	−0.131095
$528$	0	0
$529$	−22.5862	−0.982009
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.1498	−0.482953
$534$	0	0
$535$	2.49526	0.107880
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10.7931	−0.464892
$540$	0	0
$541$	−8.50655	−0.365725	−0.182863	−	0.983138i	$-0.558536\pi$
$541$	−8.50655	−0.365725	−0.182863	+	0.983138i	$0.558536\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.20690	−0.0945333
$546$	0	0
$547$	33.0244	1.41202	0.706010	−	0.708201i	$-0.250493\pi$
$547$	33.0244	1.41202	0.706010	+	0.708201i	$0.250493\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.28655	−0.0548088
$552$	0	0
$553$	−30.1593	−1.28250
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23.0909	−0.978394	−0.489197	−	0.872173i	$-0.662710\pi$
$557$	−23.0909	−0.978394	−0.489197	+	0.872173i	$0.662710\pi$
$558$	0	0
$559$	−27.1022	−1.14630
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.3680	1.82774	0.913872	−	0.406003i	$-0.133078\pi$
$563$	43.3680	1.82774	0.913872	+	0.406003i	$0.133078\pi$
$564$	0	0
$565$	17.7931	0.748561
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.30912	−0.306414	−0.153207	−	0.988194i	$-0.548960\pi$
$569$	−7.30912	−0.306414	−0.153207	+	0.988194i	$0.548960\pi$
$570$	0	0
$571$	−1.22000	−0.0510555	−0.0255277	−	0.999674i	$-0.508127\pi$
$571$	−1.22000	−0.0510555	−0.0255277	+	0.999674i	$0.508127\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.643274	0.0268264
$576$	0	0
$577$	−37.1498	−1.54657	−0.773284	−	0.634060i	$-0.781387\pi$
$577$	−37.1498	−1.54657	−0.773284	+	0.634060i	$0.781387\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	58.8062	2.43969
$582$	0	0
$583$	−6.43637	−0.266567
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.4684	1.62904	0.814518	−	0.580138i	$-0.197002\pi$
$587$	39.4684	1.62904	0.814518	+	0.580138i	$0.197002\pi$
$588$	0	0
$589$	−0.872747	−0.0359609
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−40.5178	−1.66387	−0.831934	−	0.554875i	$-0.812766\pi$
$593$	−40.5178	−1.66387	−0.831934	+	0.554875i	$0.812766\pi$
$594$	0	0
$595$	9.35673	0.383588
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.5066	1.08303	0.541514	−	0.840692i	$-0.317851\pi$
$599$	26.5066	1.08303	0.541514	+	0.840692i	$0.317851\pi$
$600$	0	0
$601$	5.00947	0.204341	0.102170	−	0.994767i	$-0.467421\pi$
$601$	5.00947	0.204341	0.102170	+	0.994767i	$0.467421\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−16.3585	−0.663973	−0.331986	−	0.943284i	$-0.607719\pi$
$607$	−16.3585	−0.663973	−0.331986	+	0.943284i	$0.607719\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−25.8858	−1.04723
$612$	0	0
$613$	38.8840	1.57051	0.785256	−	0.619172i	$-0.212532\pi$
$613$	38.8840	1.57051	0.785256	+	0.619172i	$0.212532\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.2200	0.612734	0.306367	−	0.951913i	$-0.400886\pi$
$617$	15.2200	0.612734	0.306367	+	0.951913i	$0.400886\pi$
$618$	0	0
$619$	−11.5160	−0.462868	−0.231434	−	0.972851i	$-0.574342\pi$
$619$	−11.5160	−0.462868	−0.231434	+	0.972851i	$0.574342\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	52.1629	2.08986
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.43637	0.176890
$630$	0	0
$631$	10.5731	0.420908	0.210454	−	0.977604i	$-0.432506\pi$
$631$	10.5731	0.420908	0.210454	+	0.977604i	$0.432506\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.9316	0.433809
$636$	0	0
$637$	60.1706	2.38405
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.8062	1.21677	0.608386	−	0.793641i	$-0.291817\pi$
$641$	30.8062	1.21677	0.608386	+	0.793641i	$0.291817\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−44.2520	−1.73973	−0.869864	−	0.493292i	$-0.835793\pi$
$647$	−44.2520	−1.73973	−0.869864	+	0.493292i	$0.835793\pi$
$648$	0	0
$649$	1.35673	0.0532561
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.57673	0.100835	0.0504176	−	0.998728i	$-0.483945\pi$
$653$	2.57673	0.100835	0.0504176	+	0.998728i	$0.483945\pi$
$654$	0	0
$655$	7.14982	0.279367
$656$	0	0
$657$	0	0
$658$	0	0
$659$	31.5862	1.23042	0.615212	−	0.788362i	$-0.289070\pi$
$659$	31.5862	1.23042	0.615212	+	0.788362i	$0.289070\pi$
$660$	0	0
$661$	10.8727	0.422901	0.211450	−	0.977389i	$-0.432181\pi$
$661$	10.8727	0.422901	0.211450	+	0.977389i	$0.432181\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.71345	0.105223
$666$	0	0
$667$	−1.28655	−0.0498154
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.0701770	0.00270915
$672$	0	0
$673$	−19.6451	−0.757263	−0.378631	−	0.925548i	$-0.623605\pi$
$673$	−19.6451	−0.757263	−0.378631	+	0.925548i	$0.623605\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.2502	1.43164	0.715821	−	0.698284i	$-0.246053\pi$
$677$	37.2502	1.43164	0.715821	+	0.698284i	$0.246053\pi$
$678$	0	0
$679$	32.5767	1.25018
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25.8633	0.989631	0.494815	−	0.868998i	$-0.335236\pi$
$683$	25.8633	0.989631	0.494815	+	0.868998i	$0.335236\pi$
$684$	0	0
$685$	13.7931	0.527007
$686$	0	0
$687$	0	0
$688$	0	0
$689$	35.8822	1.36700
$690$	0	0
$691$	2.29965	0.0874828	0.0437414	−	0.999043i	$-0.486072\pi$
$691$	2.29965	0.0874828	0.0437414	+	0.999043i	$0.486072\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.07965	−0.344411
$696$	0	0
$697$	−4.43637	−0.168040
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.159296	−0.00601651	−0.00300825	−	0.999995i	$-0.500958\pi$
$701$	−0.159296	−0.00601651	−0.00300825	+	0.999995i	$0.500958\pi$
$702$	0	0
$703$	1.28655	0.0485231
$704$	0	0
$705$	0	0
$706$	0	0
$707$	60.8953	2.29020
$708$	0	0
$709$	−38.8251	−1.45811	−0.729054	−	0.684456i	$-0.760040\pi$
$709$	−38.8251	−1.45811	−0.729054	+	0.684456i	$0.760040\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.872747	−0.0326846
$714$	0	0
$715$	5.57491	0.208490
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28.2520	−1.05362	−0.526812	−	0.849982i	$-0.676613\pi$
$719$	−28.2520	−1.05362	−0.526812	+	0.849982i	$0.676613\pi$
$720$	0	0
$721$	−71.1724	−2.65060
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−19.7266	−0.731617	−0.365809	−	0.930690i	$-0.619208\pi$
$727$	−19.7266	−0.731617	−0.365809	+	0.930690i	$0.619208\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−10.7836	−0.398847
$732$	0	0
$733$	−36.0302	−1.33081	−0.665403	−	0.746484i	$-0.731740\pi$
$733$	−36.0302	−1.33081	−0.665403	+	0.746484i	$0.731740\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.4364	0.458100
$738$	0	0
$739$	−5.22000	−0.192021	−0.0960104	−	0.995380i	$-0.530608\pi$
$739$	−5.22000	−0.192021	−0.0960104	+	0.995380i	$0.530608\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.0684	1.21316	0.606580	−	0.795022i	$-0.292541\pi$
$743$	33.0684	1.21316	0.606580	+	0.795022i	$0.292541\pi$
$744$	0	0
$745$	17.1498	0.628321
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.5255	0.384593
$750$	0	0
$751$	−7.12725	−0.260077	−0.130039	−	0.991509i	$-0.541510\pi$
$751$	−7.12725	−0.260077	−0.130039	+	0.991509i	$0.541510\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.43637	−0.161456
$756$	0	0
$757$	35.3091	1.28333	0.641666	−	0.766984i	$-0.278244\pi$
$757$	35.3091	1.28333	0.641666	+	0.766984i	$0.278244\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.1688	1.27487	0.637433	−	0.770505i	$-0.279996\pi$
$761$	35.1688	1.27487	0.637433	+	0.770505i	$0.279996\pi$
$762$	0	0
$763$	−9.30912	−0.337013
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.56363	−0.273107
$768$	0	0
$769$	18.9905	0.684816	0.342408	−	0.939551i	$-0.388757\pi$
$769$	18.9905	0.684816	0.342408	+	0.939551i	$0.388757\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−51.4495	−1.85051	−0.925254	−	0.379347i	$-0.876149\pi$
$773$	−51.4495	−1.85051	−0.925254	+	0.379347i	$0.876149\pi$
$774$	0	0
$775$	−1.35673	−0.0487350
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.28655	−0.0460954
$780$	0	0
$781$	−5.14982	−0.184275
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.0226	0.786019
$786$	0	0
$787$	−9.41562	−0.335630	−0.167815	−	0.985818i	$-0.553671\pi$
$787$	−9.41562	−0.335630	−0.167815	+	0.985818i	$0.553671\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	75.0546	2.66863
$792$	0	0
$793$	−0.391230	−0.0138930
$794$	0	0
$795$	0	0
$796$	0	0
$797$	53.4458	1.89315	0.946574	−	0.322485i	$-0.104518\pi$
$797$	53.4458	1.89315	0.946574	+	0.322485i	$0.104518\pi$
$798$	0	0
$799$	−10.2996	−0.364375
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.6546	−0.375991
$804$	0	0
$805$	2.71345	0.0956366
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.14619	−0.110614	−0.0553072	−	0.998469i	$-0.517614\pi$
$809$	−3.14619	−0.110614	−0.0553072	+	0.998469i	$0.517614\pi$
$810$	0	0
$811$	−26.8251	−0.941958	−0.470979	−	0.882144i	$-0.656099\pi$
$811$	−26.8251	−0.941958	−0.470979	+	0.882144i	$0.656099\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.28655	0.325294
$816$	0	0
$817$	−3.12725	−0.109409
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.8596	0.623306	0.311653	−	0.950196i	$-0.399117\pi$
$821$	17.8596	0.623306	0.311653	+	0.950196i	$0.399117\pi$
$822$	0	0
$823$	−21.2865	−0.742002	−0.371001	−	0.928632i	$-0.620985\pi$
$823$	−21.2865	−0.742002	−0.371001	+	0.928632i	$0.620985\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.3775	−0.499954	−0.249977	−	0.968252i	$-0.580423\pi$
$827$	−14.3775	−0.499954	−0.249977	+	0.968252i	$0.580423\pi$
$828$	0	0
$829$	29.0796	1.00998	0.504989	−	0.863126i	$-0.331497\pi$
$829$	29.0796	1.00998	0.504989	+	0.863126i	$0.331497\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23.9411	0.829510
$834$	0	0
$835$	25.0909	0.868308
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.5957	0.849137	0.424568	−	0.905396i	$-0.360426\pi$
$839$	24.5957	0.849137	0.424568	+	0.905396i	$0.360426\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−18.0796	−0.621959
$846$	0	0
$847$	−4.21819	−0.144939
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.28655	0.0441023
$852$	0	0
$853$	14.5880	0.499484	0.249742	−	0.968312i	$-0.419654\pi$
$853$	14.5880	0.499484	0.249742	+	0.968312i	$0.419654\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40.1040	−1.36993	−0.684964	−	0.728577i	$-0.740182\pi$
$857$	−40.1040	−1.36993	−0.684964	+	0.728577i	$0.740182\pi$
$858$	0	0
$859$	29.8157	1.01730	0.508649	−	0.860974i	$-0.330145\pi$
$859$	29.8157	1.01730	0.508649	+	0.860974i	$0.330145\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.5066	0.357647	0.178824	−	0.983881i	$-0.442771\pi$
$863$	10.5066	0.357647	0.178824	+	0.983881i	$0.442771\pi$
$864$	0	0
$865$	3.06836	0.104327
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.14982	−0.242541
$870$	0	0
$871$	−69.3317	−2.34921
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.21819	0.142601
$876$	0	0
$877$	−18.4477	−0.622933	−0.311467	−	0.950257i	$-0.600820\pi$
$877$	−18.4477	−0.622933	−0.311467	+	0.950257i	$0.600820\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.4589	−0.554516	−0.277258	−	0.960796i	$-0.589426\pi$
$881$	−16.4589	−0.554516	−0.277258	+	0.960796i	$0.589426\pi$
$882$	0	0
$883$	36.0451	1.21302	0.606508	−	0.795078i	$-0.292570\pi$
$883$	36.0451	1.21302	0.606508	+	0.795078i	$0.292570\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.24076	−0.276698	−0.138349	−	0.990384i	$-0.544179\pi$
$887$	−8.24076	−0.276698	−0.138349	+	0.990384i	$0.544179\pi$
$888$	0	0
$889$	46.1117	1.54654
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.98690	−0.0999528
$894$	0	0
$895$	−12.8727	−0.430288
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.71345	0.0904987
$900$	0	0
$901$	14.2771	0.475638
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.50655	−0.216285
$906$	0	0
$907$	3.14982	0.104588	0.0522941	−	0.998632i	$-0.483347\pi$
$907$	3.14982	0.104588	0.0522941	+	0.998632i	$0.483347\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.6338	−0.584234	−0.292117	−	0.956383i	$-0.594360\pi$
$911$	−17.6338	−0.584234	−0.292117	+	0.956383i	$0.594360\pi$
$912$	0	0
$913$	13.9411	0.461383
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.1593	0.995948
$918$	0	0
$919$	5.70035	0.188037	0.0940186	−	0.995570i	$-0.470029\pi$
$919$	5.70035	0.188037	0.0940186	+	0.995570i	$0.470029\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.7098	0.944995
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.1404	0.529548	0.264774	−	0.964311i	$-0.414703\pi$
$929$	16.1404	0.529548	0.264774	+	0.964311i	$0.414703\pi$
$930$	0	0
$931$	6.94292	0.227545
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.21819	0.0725425
$936$	0	0
$937$	−53.6640	−1.75313	−0.876564	−	0.481286i	$-0.840170\pi$
$937$	−53.6640	−1.75313	−0.876564	+	0.481286i	$0.840170\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12.2771	0.400221	0.200111	−	0.979773i	$-0.435870\pi$
$941$	12.2771	0.400221	0.200111	+	0.979773i	$0.435870\pi$
$942$	0	0
$943$	−1.28655	−0.0418958
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.3091	1.08240	0.541200	−	0.840894i	$-0.317970\pi$
$947$	33.3091	1.08240	0.541200	+	0.840894i	$0.317970\pi$
$948$	0	0
$949$	59.3982	1.92815
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.0815	0.520930	0.260465	−	0.965483i	$-0.416124\pi$
$953$	16.0815	0.520930	0.260465	+	0.965483i	$0.416124\pi$
$954$	0	0
$955$	19.7931	0.640490
$956$	0	0
$957$	0	0
$958$	0	0
$959$	58.1819	1.87879
$960$	0	0
$961$	−29.1593	−0.940622
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.9411	0.641927
$966$	0	0
$967$	44.2408	1.42269	0.711343	−	0.702845i	$-0.248087\pi$
$967$	44.2408	1.42269	0.711343	+	0.702845i	$0.248087\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16.7324	−0.536968	−0.268484	−	0.963284i	$-0.586523\pi$
$971$	−16.7324	−0.536968	−0.268484	+	0.963284i	$0.586523\pi$
$972$	0	0
$973$	−38.2996	−1.22783
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.66585	−0.213259	−0.106630	−	0.994299i	$-0.534006\pi$
$977$	−6.66585	−0.213259	−0.106630	+	0.994299i	$0.534006\pi$
$978$	0	0
$979$	12.3662	0.395225
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.3662	0.713371	0.356685	−	0.934225i	$-0.383907\pi$
$983$	22.3662	0.713371	0.356685	+	0.934225i	$0.383907\pi$
$984$	0	0
$985$	−3.64509	−0.116142
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.12725	−0.0994409
$990$	0	0
$991$	−18.0891	−0.574620	−0.287310	−	0.957838i	$-0.592761\pi$
$991$	−18.0891	−0.574620	−0.287310	+	0.957838i	$0.592761\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.8727	−0.534902
$996$	0	0
$997$	5.55234	0.175844	0.0879222	−	0.996127i	$-0.471977\pi$
$997$	5.55234	0.175844	0.0879222	+	0.996127i	$0.471977\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -4.21819 q^{7} -1.00000 q^{11} +5.57491 q^{13} +2.21819 q^{17} +0.643274 q^{19} +0.643274 q^{23} +1.00000 q^{25} -2.00000 q^{29} -1.35673 q^{31} +4.21819 q^{35} +2.00000 q^{37} -2.00000 q^{41} -4.86146 q^{43} -4.64327 q^{47} +10.7931 q^{49} +6.43637 q^{53} +1.00000 q^{55} -1.35673 q^{59} -0.0701770 q^{61} -5.57491 q^{65} -12.4364 q^{67} +5.14982 q^{71} +10.6546 q^{73} +4.21819 q^{77} +7.14982 q^{79} -13.9411 q^{83} -2.21819 q^{85} -12.3662 q^{89} -23.5160 q^{91} -0.643274 q^{95} -7.72292 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{11} + 4 q^{13} - 6 q^{17} + 2 q^{19} + 2 q^{23} + 3 q^{25} - 6 q^{29} - 4 q^{31} + 6 q^{37} - 6 q^{41} - 2 q^{43} - 14 q^{47} + 7 q^{49} - 6 q^{53} + 3 q^{55} - 4 q^{59} - 4 q^{65}+ \cdots + 2 q^{97}+O(q^{100}) \) 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

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