LMFDB - Embedded newform 3744.2.g.f.1873.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1873,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("3744.1873");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1873.23
Character	$\chi$	$=$	3744.1873
Dual form			3744.2.g.f.1873.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.18566i q^{5} -2.70923 q^{7} +0.956799i q^{11} +1.00000i q^{13} -4.96511 q^{17} -5.59682i q^{19} +5.16007 q^{23} -12.5198 q^{25} +4.46122i q^{29} -6.37531 q^{31} +11.3399i q^{35} -1.66608i q^{37} +8.64689 q^{41} +7.67092i q^{43} -8.01021 q^{47} +0.339928 q^{49} -0.0797782i q^{53} +4.00484 q^{55} -4.70716i q^{59} +8.10132i q^{61} +4.18566 q^{65} -13.7670i q^{67} +1.15489 q^{71} +1.66608 q^{73} -2.59219i q^{77} -4.30121 q^{79} +14.0185i q^{83} +20.7823i q^{85} -11.6601 q^{89} -2.70923i q^{91} -23.4264 q^{95} -6.17018 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{7} - 24 q^{25} - 40 q^{31} + 24 q^{49} + 16 q^{55} - 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^7 - 24 * q^25 - 40 * q^31 + 24 * q^49 + 16 * q^55 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	− 4.18566i	− 1.87189i	−0.352152	−	0.935943i	$-0.614550\pi$
$5$	− 4.18566i	− 1.87189i	0.352152	−	0.935943i	$-0.385450\pi$
$6$	0	0
$7$	−2.70923	−1.02399	−0.511996	−	0.858988i	$-0.671094\pi$
$7$	−2.70923	−1.02399	−0.511996	+	0.858988i	$0.671094\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.956799i	0.288486i	0.989542	+	0.144243i	$0.0460747\pi$
$11$	0.956799i	0.288486i	−0.989542	+	0.144243i	$0.953925\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.96511	−1.20422	−0.602108	−	0.798415i	$-0.705672\pi$
$17$	−4.96511	−1.20422	−0.602108	+	0.798415i	$0.705672\pi$
$18$	0	0
$19$	− 5.59682i	− 1.28400i	−0.766705	−	0.641999i	$-0.778105\pi$
$19$	− 5.59682i	− 1.28400i	0.766705	−	0.641999i	$-0.221895\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.16007	1.07595	0.537974	−	0.842961i	$-0.319190\pi$
$23$	5.16007	1.07595	0.537974	+	0.842961i	$0.319190\pi$
$24$	0	0
$25$	−12.5198	−2.50396
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.46122i	0.828428i	0.910180	+	0.414214i	$0.135943\pi$
$29$	4.46122i	0.828428i	−0.910180	+	0.414214i	$0.864057\pi$
$30$	0	0
$31$	−6.37531	−1.14504	−0.572519	−	0.819891i	$-0.694034\pi$
$31$	−6.37531	−1.14504	−0.572519	+	0.819891i	$0.694034\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	11.3399i	1.91680i
$36$	0	0
$37$	− 1.66608i	− 0.273901i	−0.990578	−	0.136951i	$-0.956270\pi$
$37$	− 1.66608i	− 0.273901i	0.990578	−	0.136951i	$-0.0437301\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.64689	1.35042	0.675208	−	0.737627i	$-0.264054\pi$
$41$	8.64689	1.35042	0.675208	+	0.737627i	$0.264054\pi$
$42$	0	0
$43$	7.67092i	1.16980i	0.811104	+	0.584902i	$0.198867\pi$
$43$	7.67092i	1.16980i	−0.811104	+	0.584902i	$0.801133\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.01021	−1.16841	−0.584205	−	0.811606i	$-0.698593\pi$
$47$	−8.01021	−1.16841	−0.584205	+	0.811606i	$0.698593\pi$
$48$	0	0
$49$	0.339928	0.0485611
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.0797782i	− 0.0109584i	−0.999985	−	0.00547919i	$-0.998256\pi$
$53$	− 0.0797782i	− 0.0109584i	0.999985	−	0.00547919i	$-0.00174409\pi$
$54$	0	0
$55$	4.00484	0.540012
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.70716i	− 0.612820i	−0.951900	−	0.306410i	$-0.900872\pi$
$59$	− 4.70716i	− 0.612820i	0.951900	−	0.306410i	$-0.0991278\pi$
$60$	0	0
$61$	8.10132i	1.03727i	0.854996	+	0.518634i	$0.173559\pi$
$61$	8.10132i	1.03727i	−0.854996	+	0.518634i	$0.826441\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.18566	0.519168
$66$	0	0
$67$	− 13.7670i	− 1.68191i	−0.541108	−	0.840953i	$-0.681995\pi$
$67$	− 13.7670i	− 1.68191i	0.541108	−	0.840953i	$-0.318005\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.15489	0.137061	0.0685303	−	0.997649i	$-0.478169\pi$
$71$	1.15489	0.137061	0.0685303	+	0.997649i	$0.478169\pi$
$72$	0	0
$73$	1.66608	0.194999	0.0974997	−	0.995236i	$-0.468916\pi$
$73$	1.66608	0.194999	0.0974997	+	0.995236i	$0.468916\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.59219i	− 0.295407i
$78$	0	0
$79$	−4.30121	−0.483924	−0.241962	−	0.970286i	$-0.577791\pi$
$79$	−4.30121	−0.483924	−0.241962	+	0.970286i	$0.577791\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.0185i	1.53873i	0.638810	+	0.769365i	$0.279427\pi$
$83$	14.0185i	1.53873i	−0.638810	+	0.769365i	$0.720573\pi$
$84$	0	0
$85$	20.7823i	2.25415i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.6601	−1.23596	−0.617981	−	0.786193i	$-0.712049\pi$
$89$	−11.6601	−1.23596	−0.617981	+	0.786193i	$0.712049\pi$
$90$	0	0
$91$	− 2.70923i	− 0.284004i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−23.4264	−2.40350
$96$	0	0
$97$	−6.17018	−0.626487	−0.313243	−	0.949673i	$-0.601416\pi$
$97$	−6.17018	−0.626487	−0.313243	+	0.949673i	$0.601416\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.2218i	1.81313i	0.422063	+	0.906567i	$0.361306\pi$
$101$	18.2218i	1.81313i	−0.422063	+	0.906567i	$0.638694\pi$
$102$	0	0
$103$	1.61651	0.159280	0.0796399	−	0.996824i	$-0.474623\pi$
$103$	1.61651	0.159280	0.0796399	+	0.996824i	$0.474623\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.6645i	1.03097i	0.856898	+	0.515487i	$0.172389\pi$
$107$	10.6645i	1.03097i	−0.856898	+	0.515487i	$0.827611\pi$
$108$	0	0
$109$	− 15.0942i	− 1.44576i	−0.690972	−	0.722882i	$-0.742817\pi$
$109$	− 15.0942i	− 1.44576i	0.690972	−	0.722882i	$-0.257183\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.5746	1.37106	0.685530	−	0.728044i	$-0.259570\pi$
$113$	14.5746	1.37106	0.685530	+	0.728044i	$0.259570\pi$
$114$	0	0
$115$	− 21.5983i	− 2.01405i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.4516	1.23311
$120$	0	0
$121$	10.0845	0.916776
$122$	0	0
$123$	0	0
$124$	0	0
$125$	31.4753i	2.81524i
$126$	0	0
$127$	16.1471	1.43283	0.716414	−	0.697676i	$-0.245782\pi$
$127$	16.1471	1.43283	0.716414	+	0.697676i	$0.245782\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.42633i	0.823582i	0.911278	+	0.411791i	$0.135097\pi$
$131$	9.42633i	0.823582i	−0.911278	+	0.411791i	$0.864903\pi$
$132$	0	0
$133$	15.1631i	1.31480i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.68642	−0.400388	−0.200194	−	0.979756i	$-0.564157\pi$
$137$	−4.68642	−0.400388	−0.200194	+	0.979756i	$0.564157\pi$
$138$	0	0
$139$	− 9.95019i	− 0.843964i	−0.906604	−	0.421982i	$-0.861334\pi$
$139$	− 9.95019i	− 0.843964i	0.906604	−	0.421982i	$-0.138666\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.956799	−0.0800116
$144$	0	0
$145$	18.6732	1.55072
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.19106i	0.507191i	0.967310	+	0.253596i	$0.0816132\pi$
$149$	6.19106i	0.507191i	−0.967310	+	0.253596i	$0.918387\pi$
$150$	0	0
$151$	8.48441	0.690451	0.345226	−	0.938520i	$-0.387802\pi$
$151$	8.48441	0.690451	0.345226	+	0.938520i	$0.387802\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	26.6849i	2.14338i
$156$	0	0
$157$	16.0239i	1.27885i	0.768855	+	0.639423i	$0.220827\pi$
$157$	16.0239i	1.27885i	−0.768855	+	0.639423i	$0.779173\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.9798	−1.10176
$162$	0	0
$163$	16.6039i	1.30052i	0.759712	+	0.650260i	$0.225340\pi$
$163$	16.6039i	1.30052i	−0.759712	+	0.650260i	$0.774660\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.35953	0.492115	0.246058	−	0.969255i	$-0.420865\pi$
$167$	6.35953	0.492115	0.246058	+	0.969255i	$0.420865\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.02859i	− 0.154231i	−0.997022	−	0.0771154i	$-0.975429\pi$
$173$	− 2.02859i	− 0.154231i	0.997022	−	0.0771154i	$-0.0245710\pi$
$174$	0	0
$175$	33.9190	2.56403
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 17.9220i	− 1.33955i	−0.742563	−	0.669777i	$-0.766390\pi$
$179$	− 17.9220i	− 1.33955i	0.742563	−	0.669777i	$-0.233610\pi$
$180$	0	0
$181$	− 3.41179i	− 0.253596i	−0.991929	−	0.126798i	$-0.959530\pi$
$181$	− 3.41179i	− 0.253596i	0.991929	−	0.126798i	$-0.0404701\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.97363	−0.512712
$186$	0	0
$187$	− 4.75061i	− 0.347399i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.0171	0.941886	0.470943	−	0.882164i	$-0.343914\pi$
$191$	13.0171	0.941886	0.470943	+	0.882164i	$0.343914\pi$
$192$	0	0
$193$	2.02198	0.145546	0.0727728	−	0.997349i	$-0.476815\pi$
$193$	2.02198	0.145546	0.0727728	+	0.997349i	$0.476815\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.5454i	1.17881i	0.807838	+	0.589405i	$0.200638\pi$
$197$	16.5454i	1.17881i	−0.807838	+	0.589405i	$0.799362\pi$
$198$	0	0
$199$	1.11725	0.0791999	0.0395999	−	0.999216i	$-0.487392\pi$
$199$	1.11725	0.0791999	0.0395999	+	0.999216i	$0.487392\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 12.0865i	− 0.848304i
$204$	0	0
$205$	− 36.1930i	− 2.52783i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.35503	0.370415
$210$	0	0
$211$	22.3717i	1.54013i	0.637964	+	0.770066i	$0.279777\pi$
$211$	22.3717i	1.54013i	−0.637964	+	0.770066i	$0.720223\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	32.1079	2.18974
$216$	0	0
$217$	17.2722	1.17251
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4.96511i	− 0.333989i
$222$	0	0
$223$	−25.3166	−1.69532	−0.847661	−	0.530538i	$-0.821990\pi$
$223$	−25.3166	−1.69532	−0.847661	+	0.530538i	$0.821990\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 0.936542i	− 0.0621604i	−0.999517	−	0.0310802i	$-0.990105\pi$
$227$	− 0.936542i	− 0.0621604i	0.999517	−	0.0310802i	$-0.00989474\pi$
$228$	0	0
$229$	3.40878i	0.225259i	0.993637	+	0.112629i	$0.0359272\pi$
$229$	3.40878i	0.225259i	−0.993637	+	0.112629i	$0.964073\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.0000	−0.851656	−0.425828	−	0.904804i	$-0.640017\pi$
$233$	−13.0000	−0.851656	−0.425828	+	0.904804i	$0.640017\pi$
$234$	0	0
$235$	33.5280i	2.18713i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.91302	−0.382482	−0.191241	−	0.981543i	$-0.561251\pi$
$239$	−5.91302	−0.382482	−0.191241	+	0.981543i	$0.561251\pi$
$240$	0	0
$241$	−12.0209	−0.774333	−0.387166	−	0.922010i	$-0.626546\pi$
$241$	−12.0209	−0.774333	−0.387166	+	0.922010i	$0.626546\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.42282i	− 0.0909008i
$246$	0	0
$247$	5.59682	0.356117
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 7.48790i	− 0.472632i	−0.971676	−	0.236316i	$-0.924060\pi$
$251$	− 7.48790i	− 0.472632i	0.971676	−	0.236316i	$-0.0759400\pi$
$252$	0	0
$253$	4.93715i	0.310396i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.5397	−1.21885	−0.609426	−	0.792843i	$-0.708600\pi$
$257$	−19.5397	−1.21885	−0.609426	+	0.792843i	$0.708600\pi$
$258$	0	0
$259$	4.51378i	0.280473i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.2733	−0.880130	−0.440065	−	0.897966i	$-0.645045\pi$
$263$	−14.2733	−0.880130	−0.440065	+	0.897966i	$0.645045\pi$
$264$	0	0
$265$	−0.333925	−0.0205128
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.47978i	0.577992i	0.957330	+	0.288996i	$0.0933215\pi$
$269$	9.47978i	0.577992i	−0.957330	+	0.288996i	$0.906679\pi$
$270$	0	0
$271$	−29.1684	−1.77185	−0.885926	−	0.463827i	$-0.846476\pi$
$271$	−29.1684	−1.77185	−0.885926	+	0.463827i	$0.846476\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 11.9789i	− 0.722356i
$276$	0	0
$277$	5.56665i	0.334468i	0.985917	+	0.167234i	$0.0534835\pi$
$277$	5.56665i	0.334468i	−0.985917	+	0.167234i	$0.946516\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.67325	−0.0998179	−0.0499090	−	0.998754i	$-0.515893\pi$
$281$	−1.67325	−0.0998179	−0.0499090	+	0.998754i	$0.515893\pi$
$282$	0	0
$283$	− 20.4580i	− 1.21610i	−0.793897	−	0.608052i	$-0.791951\pi$
$283$	− 20.4580i	− 1.21610i	0.793897	−	0.608052i	$-0.208049\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−23.4264	−1.38282
$288$	0	0
$289$	7.65230	0.450135
$290$	0	0
$291$	0	0
$292$	0	0
$293$	29.8221i	1.74223i	0.491083	+	0.871113i	$0.336601\pi$
$293$	29.8221i	1.74223i	−0.491083	+	0.871113i	$0.663399\pi$
$294$	0	0
$295$	−19.7026	−1.14713
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.16007i	0.298415i
$300$	0	0
$301$	− 20.7823i	− 1.19787i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	33.9094	1.94165
$306$	0	0
$307$	− 9.81086i	− 0.559935i	−0.960009	−	0.279968i	$-0.909676\pi$
$307$	− 9.81086i	− 0.559935i	0.960009	−	0.279968i	$-0.0903237\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.24920	0.297655	0.148827	−	0.988863i	$-0.452450\pi$
$311$	5.24920	0.297655	0.148827	+	0.988863i	$0.452450\pi$
$312$	0	0
$313$	3.78596	0.213995	0.106998	−	0.994259i	$-0.465876\pi$
$313$	3.78596	0.213995	0.106998	+	0.994259i	$0.465876\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.9601i	− 0.896407i	−0.893931	−	0.448204i	$-0.852064\pi$
$317$	− 15.9601i	− 0.896407i	0.893931	−	0.448204i	$-0.147936\pi$
$318$	0	0
$319$	−4.26849	−0.238990
$320$	0	0
$321$	0	0
$322$	0	0
$323$	27.7888i	1.54621i
$324$	0	0
$325$	− 12.5198i	− 0.694473i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	21.7015	1.19644
$330$	0	0
$331$	30.6147i	1.68274i	0.540463	+	0.841368i	$0.318249\pi$
$331$	30.6147i	1.68274i	−0.540463	+	0.841368i	$0.681751\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−57.6240	−3.14834
$336$	0	0
$337$	−19.8679	−1.08227	−0.541137	−	0.840934i	$-0.682006\pi$
$337$	−19.8679	−1.08227	−0.541137	+	0.840934i	$0.682006\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 6.09989i	− 0.330327i
$342$	0	0
$343$	18.0437	0.974267
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.59683i	0.193088i	0.995329	+	0.0965440i	$0.0307788\pi$
$347$	3.59683i	0.193088i	−0.995329	+	0.0965440i	$0.969221\pi$
$348$	0	0
$349$	− 31.9651i	− 1.71105i	−0.517758	−	0.855527i	$-0.673233\pi$
$349$	− 31.9651i	− 1.71105i	0.517758	−	0.855527i	$-0.326767\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.7537	−0.891707	−0.445854	−	0.895106i	$-0.647100\pi$
$353$	−16.7537	−0.891707	−0.445854	+	0.895106i	$0.647100\pi$
$354$	0	0
$355$	− 4.83399i	− 0.256562i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.7820	1.67739	0.838694	−	0.544603i	$-0.183320\pi$
$359$	31.7820	1.67739	0.838694	+	0.544603i	$0.183320\pi$
$360$	0	0
$361$	−12.3244	−0.648651
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 6.97363i	− 0.365017i
$366$	0	0
$367$	−27.8497	−1.45374	−0.726872	−	0.686773i	$-0.759027\pi$
$367$	−27.8497	−1.45374	−0.726872	+	0.686773i	$0.759027\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.216137i	0.0112213i
$372$	0	0
$373$	− 29.9442i	− 1.55045i	−0.631683	−	0.775227i	$-0.717635\pi$
$373$	− 29.9442i	− 1.55045i	0.631683	−	0.775227i	$-0.282365\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.46122	−0.229765
$378$	0	0
$379$	− 5.41028i	− 0.277907i	−0.990299	−	0.138954i	$-0.955626\pi$
$379$	− 5.41028i	− 0.277907i	0.990299	−	0.138954i	$-0.0443739\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.35907	0.120543	0.0602714	−	0.998182i	$-0.480803\pi$
$383$	2.35907	0.120543	0.0602714	+	0.998182i	$0.480803\pi$
$384$	0	0
$385$	−10.8500	−0.552969
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 34.2038i	− 1.73420i	−0.498131	−	0.867102i	$-0.665980\pi$
$389$	− 34.2038i	− 1.73420i	0.498131	−	0.867102i	$-0.334020\pi$
$390$	0	0
$391$	−25.6203	−1.29567
$392$	0	0
$393$	0	0
$394$	0	0
$395$	18.0034i	0.905850i
$396$	0	0
$397$	16.8214i	0.844240i	0.906540	+	0.422120i	$0.138714\pi$
$397$	16.8214i	0.844240i	−0.906540	+	0.422120i	$0.861286\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.03043	0.151333	0.0756663	−	0.997133i	$-0.475892\pi$
$401$	3.03043	0.151333	0.0756663	+	0.997133i	$0.475892\pi$
$402$	0	0
$403$	− 6.37531i	− 0.317577i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.59410	0.0790166
$408$	0	0
$409$	−36.7145	−1.81542	−0.907708	−	0.419602i	$-0.862169\pi$
$409$	−36.7145	−1.81542	−0.907708	+	0.419602i	$0.862169\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.7528i	0.627523i
$414$	0	0
$415$	58.6767	2.88033
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 18.2783i	− 0.892956i	−0.894795	−	0.446478i	$-0.852678\pi$
$419$	− 18.2783i	− 0.892956i	0.894795	−	0.446478i	$-0.147322\pi$
$420$	0	0
$421$	0.991796i	0.0483372i	0.999708	+	0.0241686i	$0.00769385\pi$
$421$	0.991796i	0.0483372i	−0.999708	+	0.0241686i	$0.992306\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	62.1621	3.01530
$426$	0	0
$427$	− 21.9484i	− 1.06216i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.05847	0.0509847	0.0254923	−	0.999675i	$-0.491885\pi$
$431$	1.05847	0.0509847	0.0254923	+	0.999675i	$0.491885\pi$
$432$	0	0
$433$	−26.5935	−1.27800	−0.639002	−	0.769205i	$-0.720652\pi$
$433$	−26.5935	−1.27800	−0.639002	+	0.769205i	$0.720652\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 28.8800i	− 1.38152i
$438$	0	0
$439$	−28.5184	−1.36111	−0.680554	−	0.732698i	$-0.738261\pi$
$439$	−28.5184	−1.36111	−0.680554	+	0.732698i	$0.738261\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.16454i	0.387909i	0.981011	+	0.193955i	$0.0621315\pi$
$443$	8.16454i	0.387909i	−0.981011	+	0.193955i	$0.937869\pi$
$444$	0	0
$445$	48.8051i	2.31358i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.19420	0.0563577	0.0281788	−	0.999603i	$-0.491029\pi$
$449$	1.19420	0.0563577	0.0281788	+	0.999603i	$0.491029\pi$
$450$	0	0
$451$	8.27333i	0.389576i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−11.3399	−0.531624
$456$	0	0
$457$	−15.9989	−0.748397	−0.374198	−	0.927349i	$-0.622082\pi$
$457$	−15.9989	−0.748397	−0.374198	+	0.927349i	$0.622082\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.35688i	0.296069i	0.988982	+	0.148035i	$0.0472947\pi$
$461$	6.35688i	0.296069i	−0.988982	+	0.148035i	$0.952705\pi$
$462$	0	0
$463$	−17.3832	−0.807866	−0.403933	−	0.914789i	$-0.632357\pi$
$463$	−17.3832	−0.807866	−0.403933	+	0.914789i	$0.632357\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.59410i	− 0.0737661i	−0.999320	−	0.0368831i	$-0.988257\pi$
$467$	− 1.59410i	− 0.0737661i	0.999320	−	0.0368831i	$-0.0117429\pi$
$468$	0	0
$469$	37.2980i	1.72226i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.33952	−0.337472
$474$	0	0
$475$	70.0710i	3.21508i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17.9730	0.821206	0.410603	−	0.911814i	$-0.365318\pi$
$479$	17.9730	0.821206	0.410603	+	0.911814i	$0.365318\pi$
$480$	0	0
$481$	1.66608	0.0759665
$482$	0	0
$483$	0	0
$484$	0	0
$485$	25.8263i	1.17271i
$486$	0	0
$487$	−27.5020	−1.24623	−0.623117	−	0.782129i	$-0.714134\pi$
$487$	−27.5020	−1.24623	−0.623117	+	0.782129i	$0.714134\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 6.45310i	− 0.291224i	−0.989342	−	0.145612i	$-0.953485\pi$
$491$	− 6.45310i	− 0.291224i	0.989342	−	0.145612i	$-0.0465152\pi$
$492$	0	0
$493$	− 22.1504i	− 0.997606i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.12887	−0.140349
$498$	0	0
$499$	− 37.8522i	− 1.69450i	−0.531197	−	0.847249i	$-0.678257\pi$
$499$	− 37.8522i	− 1.69450i	0.531197	−	0.847249i	$-0.321743\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.57973	0.204200	0.102100	−	0.994774i	$-0.467444\pi$
$503$	4.57973	0.204200	0.102100	+	0.994774i	$0.467444\pi$
$504$	0	0
$505$	76.2702	3.39398
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 13.7626i	− 0.610016i	−0.952350	−	0.305008i	$-0.901341\pi$
$509$	− 13.7626i	− 0.610016i	0.952350	−	0.305008i	$-0.0986591\pi$
$510$	0	0
$511$	−4.51378	−0.199678
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 6.76618i	− 0.298154i
$516$	0	0
$517$	− 7.66416i	− 0.337069i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.5790	−1.42731	−0.713656	−	0.700496i	$-0.752962\pi$
$521$	−32.5790	−1.42731	−0.713656	+	0.700496i	$0.752962\pi$
$522$	0	0
$523$	40.2919i	1.76184i	0.473264	+	0.880921i	$0.343076\pi$
$523$	40.2919i	1.76184i	−0.473264	+	0.880921i	$0.656924\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.6541	1.37887
$528$	0	0
$529$	3.62631	0.157666
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.64689i	0.374538i
$534$	0	0
$535$	44.6379	1.92986
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.325242i	0.0140092i
$540$	0	0
$541$	− 28.6390i	− 1.23129i	−0.788025	−	0.615643i	$-0.788896\pi$
$541$	− 28.6390i	− 1.23129i	0.788025	−	0.615643i	$-0.211104\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−63.1793	−2.70630
$546$	0	0
$547$	− 6.51086i	− 0.278384i	−0.990265	−	0.139192i	$-0.955549\pi$
$547$	− 6.51086i	− 0.278384i	0.990265	−	0.139192i	$-0.0444505\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	24.9686	1.06370
$552$	0	0
$553$	11.6530	0.495534
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6.44138i	− 0.272930i	−0.990645	−	0.136465i	$-0.956426\pi$
$557$	− 6.44138i	− 0.272930i	0.990645	−	0.136465i	$-0.0435741\pi$
$558$	0	0
$559$	−7.67092	−0.324445
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.3919i	1.44945i	0.689040	+	0.724724i	$0.258033\pi$
$563$	34.3919i	1.44945i	−0.689040	+	0.724724i	$0.741967\pi$
$564$	0	0
$565$	− 61.0043i	− 2.56647i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−43.8071	−1.83649	−0.918244	−	0.396014i	$-0.870393\pi$
$569$	−43.8071	−1.83649	−0.918244	+	0.396014i	$0.870393\pi$
$570$	0	0
$571$	29.1661i	1.22056i	0.792185	+	0.610281i	$0.208944\pi$
$571$	29.1661i	1.22056i	−0.792185	+	0.610281i	$0.791056\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−64.6030	−2.69413
$576$	0	0
$577$	−10.5041	−0.437291	−0.218646	−	0.975804i	$-0.570164\pi$
$577$	−10.5041	−0.437291	−0.218646	+	0.975804i	$0.570164\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 37.9793i	− 1.57565i
$582$	0	0
$583$	0.0763317	0.00316134
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.6896i	1.63817i	0.573675	+	0.819083i	$0.305517\pi$
$587$	39.6896i	1.63817i	−0.573675	+	0.819083i	$0.694483\pi$
$588$	0	0
$589$	35.6814i	1.47023i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.9670	0.778882	0.389441	−	0.921051i	$-0.372668\pi$
$593$	18.9670	0.778882	0.389441	+	0.921051i	$0.372668\pi$
$594$	0	0
$595$	− 56.3040i	− 2.30824i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.68323	0.0687750	0.0343875	−	0.999409i	$-0.489052\pi$
$599$	1.68323	0.0687750	0.0343875	+	0.999409i	$0.489052\pi$
$600$	0	0
$601$	−7.76439	−0.316716	−0.158358	−	0.987382i	$-0.550620\pi$
$601$	−7.76439	−0.316716	−0.158358	+	0.987382i	$0.550620\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 42.2105i	− 1.71610i
$606$	0	0
$607$	4.45424	0.180792	0.0903961	−	0.995906i	$-0.471187\pi$
$607$	4.45424	0.180792	0.0903961	+	0.995906i	$0.471187\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 8.01021i	− 0.324058i
$612$	0	0
$613$	− 20.7056i	− 0.836293i	−0.908380	−	0.418147i	$-0.862680\pi$
$613$	− 20.7056i	− 0.836293i	0.908380	−	0.418147i	$-0.137320\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.2889	1.25964	0.629821	−	0.776740i	$-0.283128\pi$
$617$	31.2889	1.25964	0.629821	+	0.776740i	$0.283128\pi$
$618$	0	0
$619$	13.1646i	0.529129i	0.964368	+	0.264565i	$0.0852282\pi$
$619$	13.1646i	0.529129i	−0.964368	+	0.264565i	$0.914772\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	31.5898	1.26562
$624$	0	0
$625$	69.1461	2.76584
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.27224i	0.329836i
$630$	0	0
$631$	6.38498	0.254182	0.127091	−	0.991891i	$-0.459436\pi$
$631$	6.38498	0.254182	0.127091	+	0.991891i	$0.459436\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 67.5865i	− 2.68209i
$636$	0	0
$637$	0.339928i	0.0134684i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.671299	0.0265147	0.0132574	−	0.999912i	$-0.495780\pi$
$641$	0.671299	0.0265147	0.0132574	+	0.999912i	$0.495780\pi$
$642$	0	0
$643$	− 2.22333i	− 0.0876794i	−0.999039	−	0.0438397i	$-0.986041\pi$
$643$	− 2.22333i	− 0.0876794i	0.999039	−	0.0438397i	$-0.0139591\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.6075	0.849480	0.424740	−	0.905315i	$-0.360366\pi$
$647$	21.6075	0.849480	0.424740	+	0.905315i	$0.360366\pi$
$648$	0	0
$649$	4.50380	0.176790
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.85741i	0.190085i	0.995473	+	0.0950426i	$0.0302987\pi$
$653$	4.85741i	0.190085i	−0.995473	+	0.0950426i	$0.969701\pi$
$654$	0	0
$655$	39.4554	1.54165
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 28.1414i	− 1.09623i	−0.836402	−	0.548116i	$-0.815345\pi$
$659$	− 28.1414i	− 1.09623i	0.836402	−	0.548116i	$-0.184655\pi$
$660$	0	0
$661$	− 25.8016i	− 1.00356i	−0.864994	−	0.501782i	$-0.832678\pi$
$661$	− 25.8016i	− 1.00356i	0.864994	−	0.501782i	$-0.167322\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	63.4675	2.46116
$666$	0	0
$667$	23.0202i	0.891346i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.75134	−0.299237
$672$	0	0
$673$	13.6803	0.527336	0.263668	−	0.964613i	$-0.415068\pi$
$673$	13.6803	0.527336	0.263668	+	0.964613i	$0.415068\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 18.7430i	− 0.720351i	−0.932885	−	0.360175i	$-0.882717\pi$
$677$	− 18.7430i	− 0.720351i	0.932885	−	0.360175i	$-0.117283\pi$
$678$	0	0
$679$	16.7164	0.641518
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.0389i	1.14941i	0.818362	+	0.574704i	$0.194883\pi$
$683$	30.0389i	1.14941i	−0.818362	+	0.574704i	$0.805117\pi$
$684$	0	0
$685$	19.6158i	0.749480i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.0797782	0.00303931
$690$	0	0
$691$	− 33.8302i	− 1.28696i	−0.765462	−	0.643481i	$-0.777490\pi$
$691$	− 33.8302i	− 1.28696i	0.765462	−	0.643481i	$-0.222510\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−41.6482	−1.57981
$696$	0	0
$697$	−42.9327	−1.62619
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.20542i	0.347684i	0.984774	+	0.173842i	$0.0556182\pi$
$701$	9.20542i	0.347684i	−0.984774	+	0.173842i	$0.944382\pi$
$702$	0	0
$703$	−9.32472	−0.351689
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 49.3670i	− 1.85664i
$708$	0	0
$709$	42.5563i	1.59824i	0.601174	+	0.799119i	$0.294700\pi$
$709$	42.5563i	1.59824i	−0.601174	+	0.799119i	$0.705300\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−32.8970	−1.23200
$714$	0	0
$715$	4.00484i	0.149773i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25.0205	−0.933107	−0.466554	−	0.884493i	$-0.654505\pi$
$719$	−25.0205	−0.933107	−0.466554	+	0.884493i	$0.654505\pi$
$720$	0	0
$721$	−4.37951	−0.163101
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 55.8535i	− 2.07435i
$726$	0	0
$727$	−17.7041	−0.656609	−0.328305	−	0.944572i	$-0.606477\pi$
$727$	−17.7041	−0.656609	−0.328305	+	0.944572i	$0.606477\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 38.0869i	− 1.40870i
$732$	0	0
$733$	− 20.1480i	− 0.744183i	−0.928196	−	0.372092i	$-0.878641\pi$
$733$	− 20.1480i	− 0.744183i	0.928196	−	0.372092i	$-0.121359\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.1722	0.485206
$738$	0	0
$739$	− 12.9725i	− 0.477202i	−0.971118	−	0.238601i	$-0.923311\pi$
$739$	− 12.9725i	− 0.477202i	0.971118	−	0.238601i	$-0.0766888\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.3522	1.15020	0.575100	−	0.818083i	$-0.304963\pi$
$743$	31.3522	1.15020	0.575100	+	0.818083i	$0.304963\pi$
$744$	0	0
$745$	25.9137	0.949404
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 28.8925i	− 1.05571i
$750$	0	0
$751$	13.4231	0.489814	0.244907	−	0.969547i	$-0.421243\pi$
$751$	13.4231	0.489814	0.244907	+	0.969547i	$0.421243\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 35.5129i	− 1.29245i
$756$	0	0
$757$	− 9.49056i	− 0.344940i	−0.985015	−	0.172470i	$-0.944825\pi$
$757$	− 9.49056i	− 0.344940i	0.985015	−	0.172470i	$-0.0551748\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.4197	−0.486466	−0.243233	−	0.969968i	$-0.578208\pi$
$761$	−13.4197	−0.486466	−0.243233	+	0.969968i	$0.578208\pi$
$762$	0	0
$763$	40.8937i	1.48045i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.70716	0.169966
$768$	0	0
$769$	6.84250	0.246747	0.123373	−	0.992360i	$-0.460629\pi$
$769$	6.84250	0.246747	0.123373	+	0.992360i	$0.460629\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.6355i	0.742207i	0.928591	+	0.371104i	$0.121021\pi$
$773$	20.6355i	0.742207i	−0.928591	+	0.371104i	$0.878979\pi$
$774$	0	0
$775$	79.8174	2.86713
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 48.3950i	− 1.73393i
$780$	0	0
$781$	1.10500i	0.0395400i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	67.0706	2.39385
$786$	0	0
$787$	− 4.33162i	− 0.154406i	−0.997015	−	0.0772028i	$-0.975401\pi$
$787$	− 4.33162i	− 0.154406i	0.997015	−	0.0772028i	$-0.0245989\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−39.4859	−1.40396
$792$	0	0
$793$	−8.10132	−0.287686
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 26.9591i	− 0.954941i	−0.878648	−	0.477471i	$-0.841554\pi$
$797$	− 26.9591i	− 0.954941i	0.878648	−	0.477471i	$-0.158446\pi$
$798$	0	0
$799$	39.7716	1.40702
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.59410i	0.0562545i
$804$	0	0
$805$	58.5148i	2.06238i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.217636	0.00765166	0.00382583	−	0.999993i	$-0.498782\pi$
$809$	0.217636	0.00765166	0.00382583	+	0.999993i	$0.498782\pi$
$810$	0	0
$811$	− 10.9498i	− 0.384501i	−0.981346	−	0.192251i	$-0.938421\pi$
$811$	− 10.9498i	− 0.384501i	0.981346	−	0.192251i	$-0.0615786\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	69.4984	2.43442
$816$	0	0
$817$	42.9327	1.50203
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 10.6224i	− 0.370725i	−0.982670	−	0.185362i	$-0.940654\pi$
$821$	− 10.6224i	− 0.370725i	0.982670	−	0.185362i	$-0.0593459\pi$
$822$	0	0
$823$	−28.0573	−0.978017	−0.489008	−	0.872279i	$-0.662641\pi$
$823$	−28.0573	−0.978017	−0.489008	+	0.872279i	$0.662641\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.91954i	0.101522i	0.998711	+	0.0507611i	$0.0161647\pi$
$827$	2.91954i	0.101522i	−0.998711	+	0.0507611i	$0.983835\pi$
$828$	0	0
$829$	33.7216i	1.17120i	0.810600	+	0.585600i	$0.199141\pi$
$829$	33.7216i	1.17120i	−0.810600	+	0.585600i	$0.800859\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.68778	−0.0584780
$834$	0	0
$835$	− 26.6189i	− 0.921183i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.6980	−0.680050	−0.340025	−	0.940416i	$-0.610436\pi$
$839$	−19.6980	−0.680050	−0.340025	+	0.940416i	$0.610436\pi$
$840$	0	0
$841$	9.09751	0.313707
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.18566i	0.143991i
$846$	0	0
$847$	−27.3213	−0.938772
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 8.59706i	− 0.294704i
$852$	0	0
$853$	− 11.3421i	− 0.388347i	−0.980967	−	0.194174i	$-0.937798\pi$
$853$	− 11.3421i	− 0.388347i	0.980967	−	0.194174i	$-0.0622025\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.8724	−0.542192	−0.271096	−	0.962552i	$-0.587386\pi$
$857$	−15.8724	−0.542192	−0.271096	+	0.962552i	$0.587386\pi$
$858$	0	0
$859$	− 23.4990i	− 0.801776i	−0.916127	−	0.400888i	$-0.868702\pi$
$859$	− 23.4990i	− 0.801776i	0.916127	−	0.400888i	$-0.131298\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.45060	0.219581	0.109790	−	0.993955i	$-0.464982\pi$
$863$	6.45060	0.219581	0.109790	+	0.993955i	$0.464982\pi$
$864$	0	0
$865$	−8.49099	−0.288702
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4.11539i	− 0.139605i
$870$	0	0
$871$	13.7670	0.466477
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 85.2738i	− 2.88278i
$876$	0	0
$877$	43.8038i	1.47915i	0.673074	+	0.739575i	$0.264973\pi$
$877$	43.8038i	1.47915i	−0.673074	+	0.739575i	$0.735027\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	54.1923	1.82579	0.912893	−	0.408199i	$-0.133843\pi$
$881$	54.1923	1.82579	0.912893	+	0.408199i	$0.133843\pi$
$882$	0	0
$883$	− 32.4922i	− 1.09345i	−0.837312	−	0.546725i	$-0.815874\pi$
$883$	− 32.4922i	− 1.09345i	0.837312	−	0.546725i	$-0.184126\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.3801	−1.02006	−0.510032	−	0.860156i	$-0.670366\pi$
$887$	−30.3801	−1.02006	−0.510032	+	0.860156i	$0.670366\pi$
$888$	0	0
$889$	−43.7463	−1.46720
$890$	0	0
$891$	0	0
$892$	0	0
$893$	44.8317i	1.50024i
$894$	0	0
$895$	−75.0155	−2.50749
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 28.4416i	− 0.948582i
$900$	0	0
$901$	0.396107i	0.0131962i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.2806	−0.474703
$906$	0	0
$907$	29.0224i	0.963671i	0.876262	+	0.481836i	$0.160030\pi$
$907$	29.0224i	0.963671i	−0.876262	+	0.481836i	$0.839970\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	48.1729	1.59604	0.798019	−	0.602632i	$-0.205881\pi$
$911$	48.1729	1.59604	0.798019	+	0.602632i	$0.205881\pi$
$912$	0	0
$913$	−13.4129	−0.443902
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 25.5381i	− 0.843342i
$918$	0	0
$919$	−25.1063	−0.828180	−0.414090	−	0.910236i	$-0.635900\pi$
$919$	−25.1063	−0.828180	−0.414090	+	0.910236i	$0.635900\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.15489i	0.0380138i
$924$	0	0
$925$	20.8589i	0.685837i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45.7495	−1.50099	−0.750497	−	0.660874i	$-0.770186\pi$
$929$	−45.7495	−1.50099	−0.750497	+	0.660874i	$0.770186\pi$
$930$	0	0
$931$	− 1.90251i	− 0.0623523i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.8845	−0.650291
$936$	0	0
$937$	−41.4498	−1.35411	−0.677053	−	0.735934i	$-0.736743\pi$
$937$	−41.4498	−1.35411	−0.677053	+	0.735934i	$0.736743\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.6008i	0.899760i	0.893089	+	0.449880i	$0.148533\pi$
$941$	27.6008i	0.899760i	−0.893089	+	0.449880i	$0.851467\pi$
$942$	0	0
$943$	44.6185	1.45298
$944$	0	0
$945$	0	0
$946$	0	0
$947$	60.4406i	1.96405i	0.188740	+	0.982027i	$0.439560\pi$
$947$	60.4406i	1.96405i	−0.188740	+	0.982027i	$0.560440\pi$
$948$	0	0
$949$	1.66608i	0.0540831i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−23.3201	−0.755412	−0.377706	−	0.925926i	$-0.623287\pi$
$953$	−23.3201	−0.755412	−0.377706	+	0.925926i	$0.623287\pi$
$954$	0	0
$955$	− 54.4853i	− 1.76310i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.6966	0.409994
$960$	0	0
$961$	9.64452	0.311114
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 8.46334i	− 0.272445i
$966$	0	0
$967$	28.4046	0.913432	0.456716	−	0.889613i	$-0.349026\pi$
$967$	28.4046	0.913432	0.456716	+	0.889613i	$0.349026\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 32.3884i	− 1.03939i	−0.854351	−	0.519697i	$-0.826045\pi$
$971$	− 32.3884i	− 1.03939i	0.854351	−	0.519697i	$-0.173955\pi$
$972$	0	0
$973$	26.9574i	0.864213i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.7641	1.30416	0.652079	−	0.758151i	$-0.273897\pi$
$977$	40.7641	1.30416	0.652079	+	0.758151i	$0.273897\pi$
$978$	0	0
$979$	− 11.1563i	− 0.356558i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−44.1753	−1.40897	−0.704486	−	0.709718i	$-0.748822\pi$
$983$	−44.1753	−1.40897	−0.704486	+	0.709718i	$0.748822\pi$
$984$	0	0
$985$	69.2534	2.20660
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39.5825i	1.25865i
$990$	0	0
$991$	−27.7306	−0.880893	−0.440446	−	0.897779i	$-0.645180\pi$
$991$	−27.7306	−0.880893	−0.440446	+	0.897779i	$0.645180\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 4.67644i	− 0.148253i
$996$	0	0
$997$	15.5011i	0.490925i	0.969406	+	0.245463i	$0.0789398\pi$
$997$	15.5011i	0.490925i	−0.969406	+	0.245463i	$0.921060\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.g.f.1873.23		24
3.2	odd	2	inner	3744.2.g.f.1873.12		24
4.3	odd	2		936.2.g.f.469.23	yes	24
8.3	odd	2		936.2.g.f.469.24	yes	24
8.5	even	2	inner	3744.2.g.f.1873.24		24
12.11	even	2		936.2.g.f.469.2	yes	24
24.5	odd	2	inner	3744.2.g.f.1873.11		24
24.11	even	2		936.2.g.f.469.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.g.f.469.1	✓	24	24.11	even	2
936.2.g.f.469.2	yes	24	12.11	even	2
936.2.g.f.469.23	yes	24	4.3	odd	2
936.2.g.f.469.24	yes	24	8.3	odd	2
3744.2.g.f.1873.11		24	24.5	odd	2	inner
3744.2.g.f.1873.12		24	3.2	odd	2	inner
3744.2.g.f.1873.23		24	1.1	even	1	trivial
3744.2.g.f.1873.24		24	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.18566i q^{5} -2.70923 q^{7} +0.956799i q^{11} +1.00000i q^{13} -4.96511 q^{17} -5.59682i q^{19} +5.16007 q^{23} -12.5198 q^{25} +4.46122i q^{29} -6.37531 q^{31} +11.3399i q^{35} -1.66608i q^{37} +8.64689 q^{41} +7.67092i q^{43} -8.01021 q^{47} +0.339928 q^{49} -0.0797782i q^{53} +4.00484 q^{55} -4.70716i q^{59} +8.10132i q^{61} +4.18566 q^{65} -13.7670i q^{67} +1.15489 q^{71} +1.66608 q^{73} -2.59219i q^{77} -4.30121 q^{79} +14.0185i q^{83} +20.7823i q^{85} -11.6601 q^{89} -2.70923i q^{91} -23.4264 q^{95} -6.17018 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{7} - 24 q^{25} - 40 q^{31} + 24 q^{49} + 16 q^{55} - 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^7 - 24 * q^25 - 40 * q^31 + 24 * q^49 + 16 * q^55 - 16 * q^97

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