LMFDB - Embedded newform 1764.4.f.b.881.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(881,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1764.881");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.f (of order $2$, degree $1$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.11
Character	$\chi$	$=$	1764.881
Dual form			1764.4.f.b.881.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.16450 q^{5} +O(q^{10})$	$q-1.16450 q^{5} +65.6114i q^{11} +57.5711i q^{13} -55.2913 q^{17} +122.831i q^{19} -54.4867i q^{23} -123.644 q^{25} +105.124i q^{29} -97.7959i q^{31} -130.009 q^{37} +69.2322 q^{41} -87.1042 q^{43} +236.694 q^{47} -412.811i q^{53} -76.4046i q^{55} +42.2102 q^{59} -844.401i q^{61} -67.0417i q^{65} +643.734 q^{67} -749.296i q^{71} +798.084i q^{73} +359.991 q^{79} -1304.90 q^{83} +64.3869 q^{85} -109.749 q^{89} -143.036i q^{95} +781.538i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 888 q^{25} + 864 q^{37} - 1248 q^{43} + 1056 q^{67} - 8064 q^{85}+O(q^{100}) $ 24 * q + 888 * q^25 + 864 * q^37 - 1248 * q^43 + 1056 * q^67 - 8064 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	−1.16450	−0.104156	−0.0520781	−	0.998643i	$-0.516584\pi$
$5$	−1.16450	−0.104156	−0.0520781	+	0.998643i	$0.516584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	65.6114i	1.79842i	0.437522	+	0.899208i	$0.355856\pi$
$11$	65.6114i	1.79842i	−0.437522	+	0.899208i	$0.644144\pi$
$12$	0	0
$13$	57.5711i	1.22826i	0.789206	+	0.614129i	$0.210493\pi$
$13$	57.5711i	1.22826i	−0.789206	+	0.614129i	$0.789507\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−55.2913	−0.788830	−0.394415	−	0.918932i	$-0.629053\pi$
$17$	−55.2913	−0.788830	−0.394415	+	0.918932i	$0.629053\pi$
$18$	0	0
$19$	122.831i	1.48312i	0.670887	+	0.741559i	$0.265913\pi$
$19$	122.831i	1.48312i	−0.670887	+	0.741559i	$0.734087\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 54.4867i	− 0.493968i	−0.969020	−	0.246984i	$-0.920560\pi$
$23$	− 54.4867i	− 0.493968i	0.969020	−	0.246984i	$-0.0794396\pi$
$24$	0	0
$25$	−123.644	−0.989151
$26$	0	0
$27$	0	0
$28$	0	0
$29$	105.124i	0.673138i	0.941659	+	0.336569i	$0.109267\pi$
$29$	105.124i	0.673138i	−0.941659	+	0.336569i	$0.890733\pi$
$30$	0	0
$31$	− 97.7959i	− 0.566602i	−0.959031	−	0.283301i	$-0.908570\pi$
$31$	− 97.7959i	− 0.566602i	0.959031	−	0.283301i	$-0.0914296\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−130.009	−0.577659	−0.288829	−	0.957381i	$-0.593266\pi$
$37$	−130.009	−0.577659	−0.288829	+	0.957381i	$0.593266\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	69.2322	0.263713	0.131857	−	0.991269i	$-0.457906\pi$
$41$	69.2322	0.263713	0.131857	+	0.991269i	$0.457906\pi$
$42$	0	0
$43$	−87.1042	−0.308913	−0.154457	−	0.988000i	$-0.549363\pi$
$43$	−87.1042	−0.308913	−0.154457	+	0.988000i	$0.549363\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	236.694	0.734581	0.367291	−	0.930106i	$-0.380285\pi$
$47$	236.694	0.734581	0.367291	+	0.930106i	$0.380285\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 412.811i	− 1.06989i	−0.844888	−	0.534943i	$-0.820333\pi$
$53$	− 412.811i	− 1.06989i	0.844888	−	0.534943i	$-0.179667\pi$
$54$	0	0
$55$	− 76.4046i	− 0.187316i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	42.2102	0.0931406	0.0465703	−	0.998915i	$-0.485171\pi$
$59$	42.2102	0.0931406	0.0465703	+	0.998915i	$0.485171\pi$
$60$	0	0
$61$	− 844.401i	− 1.77237i	−0.463332	−	0.886185i	$-0.653346\pi$
$61$	− 844.401i	− 1.77237i	0.463332	−	0.886185i	$-0.346654\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 67.0417i	− 0.127931i
$66$	0	0
$67$	643.734	1.17380	0.586900	−	0.809659i	$-0.300348\pi$
$67$	643.734	1.17380	0.586900	+	0.809659i	$0.300348\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 749.296i	− 1.25247i	−0.779636	−	0.626233i	$-0.784596\pi$
$71$	− 749.296i	− 1.25247i	0.779636	−	0.626233i	$-0.215404\pi$
$72$	0	0
$73$	798.084i	1.27957i	0.768553	+	0.639786i	$0.220977\pi$
$73$	798.084i	1.27957i	−0.768553	+	0.639786i	$0.779023\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	359.991	0.512686	0.256343	−	0.966586i	$-0.417482\pi$
$79$	359.991	0.512686	0.256343	+	0.966586i	$0.417482\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1304.90	−1.72568	−0.862840	−	0.505477i	$-0.831317\pi$
$83$	−1304.90	−1.72568	−0.862840	+	0.505477i	$0.831317\pi$
$84$	0	0
$85$	64.3869	0.0821616
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−109.749	−0.130712	−0.0653560	−	0.997862i	$-0.520818\pi$
$89$	−109.749	−0.130712	−0.0653560	+	0.997862i	$0.520818\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 143.036i	− 0.154476i
$96$	0	0
$97$	781.538i	0.818074i	0.912518	+	0.409037i	$0.134135\pi$
$97$	781.538i	0.818074i	−0.912518	+	0.409037i	$0.865865\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−124.539	−0.122694	−0.0613472	−	0.998116i	$-0.519540\pi$
$101$	−124.539	−0.122694	−0.0613472	+	0.998116i	$0.519540\pi$
$102$	0	0
$103$	426.866i	0.408353i	0.978934	+	0.204176i	$0.0654516\pi$
$103$	426.866i	0.408353i	−0.978934	+	0.204176i	$0.934548\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2012.89i	− 1.81863i	−0.416107	−	0.909316i	$-0.636606\pi$
$107$	− 2012.89i	− 1.81863i	0.416107	−	0.909316i	$-0.363394\pi$
$108$	0	0
$109$	−1005.16	−0.883275	−0.441637	−	0.897194i	$-0.645602\pi$
$109$	−1005.16	−0.883275	−0.441637	+	0.897194i	$0.645602\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1151.55i	0.958660i	0.877635	+	0.479330i	$0.159120\pi$
$113$	1151.55i	0.958660i	−0.877635	+	0.479330i	$0.840880\pi$
$114$	0	0
$115$	63.4499i	0.0514499i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2973.85	−2.23430
$122$	0	0
$123$	0	0
$124$	0	0
$125$	289.546	0.207183
$126$	0	0
$127$	66.4170	0.0464060	0.0232030	−	0.999731i	$-0.492614\pi$
$127$	66.4170	0.0464060	0.0232030	+	0.999731i	$0.492614\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1856.24	1.23802	0.619010	−	0.785383i	$-0.287534\pi$
$131$	1856.24	1.23802	0.619010	+	0.785383i	$0.287534\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	795.248i	0.495932i	0.968769	+	0.247966i	$0.0797621\pi$
$137$	795.248i	0.495932i	−0.968769	+	0.247966i	$0.920238\pi$
$138$	0	0
$139$	1076.39i	0.656820i	0.944535	+	0.328410i	$0.106513\pi$
$139$	1076.39i	0.656820i	−0.944535	+	0.328410i	$0.893487\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3777.32	−2.20892
$144$	0	0
$145$	− 122.417i	− 0.0701115i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1598.74i	− 0.879022i	−0.898237	−	0.439511i	$-0.855152\pi$
$149$	− 1598.74i	− 0.879022i	0.898237	−	0.439511i	$-0.144848\pi$
$150$	0	0
$151$	−2260.06	−1.21802	−0.609011	−	0.793162i	$-0.708433\pi$
$151$	−2260.06	−1.21802	−0.609011	+	0.793162i	$0.708433\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	113.884i	0.0590152i
$156$	0	0
$157$	− 1342.50i	− 0.682441i	−0.939983	−	0.341221i	$-0.889160\pi$
$157$	− 1342.50i	− 0.682441i	0.939983	−	0.341221i	$-0.110840\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1044.42	−0.501872	−0.250936	−	0.968004i	$-0.580738\pi$
$163$	−1044.42	−0.501872	−0.250936	+	0.968004i	$0.580738\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2538.50	−1.17626	−0.588129	−	0.808767i	$-0.700135\pi$
$167$	−2538.50	−1.17626	−0.588129	+	0.808767i	$0.700135\pi$
$168$	0	0
$169$	−1117.43	−0.508618
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3924.69	1.72479	0.862396	−	0.506234i	$-0.168963\pi$
$173$	3924.69	1.72479	0.862396	+	0.506234i	$0.168963\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1960.28i	− 0.818538i	−0.912414	−	0.409269i	$-0.865784\pi$
$179$	− 1960.28i	− 0.818538i	0.912414	−	0.409269i	$-0.134216\pi$
$180$	0	0
$181$	2865.74i	1.17685i	0.808553	+	0.588423i	$0.200251\pi$
$181$	2865.74i	1.17685i	−0.808553	+	0.588423i	$0.799749\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	151.396	0.0601668
$186$	0	0
$187$	− 3627.74i	− 1.41864i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 2018.35i	− 0.764621i	−0.924034	−	0.382311i	$-0.875128\pi$
$191$	− 2018.35i	− 0.764621i	0.924034	−	0.382311i	$-0.124872\pi$
$192$	0	0
$193$	1471.88	0.548955	0.274478	−	0.961594i	$-0.411495\pi$
$193$	1471.88	0.548955	0.274478	+	0.961594i	$0.411495\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5503.83i	− 1.99051i	−0.0972758	−	0.995257i	$-0.531013\pi$
$197$	− 5503.83i	− 1.99051i	0.0972758	−	0.995257i	$-0.468987\pi$
$198$	0	0
$199$	4688.80i	1.67025i	0.550058	+	0.835126i	$0.314605\pi$
$199$	4688.80i	1.67025i	−0.550058	+	0.835126i	$0.685395\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−80.6210	−0.0274674
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8059.08	−2.66726
$210$	0	0
$211$	−1542.29	−0.503202	−0.251601	−	0.967831i	$-0.580957\pi$
$211$	−1542.29	−0.503202	−0.251601	+	0.967831i	$0.580957\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	101.433	0.0321752
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 3183.18i	− 0.968887i
$222$	0	0
$223$	544.576i	0.163531i	0.996652	+	0.0817657i	$0.0260559\pi$
$223$	544.576i	0.163531i	−0.996652	+	0.0817657i	$0.973944\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1674.34	0.489558	0.244779	−	0.969579i	$-0.421285\pi$
$227$	1674.34	0.489558	0.244779	+	0.969579i	$0.421285\pi$
$228$	0	0
$229$	− 4660.45i	− 1.34485i	−0.740165	−	0.672426i	$-0.765252\pi$
$229$	− 4660.45i	− 1.34485i	0.740165	−	0.672426i	$-0.234748\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6484.23i	− 1.82316i	−0.411124	−	0.911580i	$-0.634864\pi$
$233$	− 6484.23i	− 1.82316i	0.411124	−	0.911580i	$-0.365136\pi$
$234$	0	0
$235$	−275.630	−0.0765112
$236$	0	0
$237$	0	0
$238$	0	0
$239$	890.490i	0.241008i	0.992713	+	0.120504i	$0.0384511\pi$
$239$	890.490i	0.241008i	−0.992713	+	0.120504i	$0.961549\pi$
$240$	0	0
$241$	4970.68i	1.32859i	0.747471	+	0.664294i	$0.231268\pi$
$241$	4970.68i	1.32859i	−0.747471	+	0.664294i	$0.768732\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7071.49	−1.82165
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2469.89	0.621107	0.310554	−	0.950556i	$-0.399486\pi$
$251$	2469.89	0.621107	0.310554	+	0.950556i	$0.399486\pi$
$252$	0	0
$253$	3574.95	0.888360
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3513.82	0.852864	0.426432	−	0.904520i	$-0.359770\pi$
$257$	3513.82	0.852864	0.426432	+	0.904520i	$0.359770\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2826.41i	− 0.662676i	−0.943512	−	0.331338i	$-0.892500\pi$
$263$	− 2826.41i	− 0.662676i	0.943512	−	0.331338i	$-0.107500\pi$
$264$	0	0
$265$	480.720i	0.111435i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8008.18	−1.81512	−0.907560	−	0.419923i	$-0.862057\pi$
$269$	−8008.18	−1.81512	−0.907560	+	0.419923i	$0.862057\pi$
$270$	0	0
$271$	− 3157.34i	− 0.707730i	−0.935297	−	0.353865i	$-0.884867\pi$
$271$	− 3157.34i	− 0.707730i	0.935297	−	0.353865i	$-0.115133\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 8112.45i	− 1.77891i
$276$	0	0
$277$	2255.61	0.489266	0.244633	−	0.969616i	$-0.421332\pi$
$277$	2255.61	0.489266	0.244633	+	0.969616i	$0.421332\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7277.95i	1.54507i	0.634970	+	0.772537i	$0.281013\pi$
$281$	7277.95i	1.54507i	−0.634970	+	0.772537i	$0.718987\pi$
$282$	0	0
$283$	− 14.9603i	− 0.00314239i	−0.999999	−	0.00157119i	$-0.999500\pi$
$283$	− 14.9603i	− 0.00314239i	0.999999	−	0.00157119i	$-0.000500127\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1855.87	−0.377747
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8446.59	−1.68415	−0.842074	−	0.539362i	$-0.818665\pi$
$293$	−8446.59	−1.68415	−0.842074	+	0.539362i	$0.818665\pi$
$294$	0	0
$295$	−49.1539	−0.00970118
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3136.86	0.606721
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	983.307i	0.184603i
$306$	0	0
$307$	7870.82i	1.46323i	0.681719	+	0.731615i	$0.261233\pi$
$307$	7870.82i	1.46323i	−0.681719	+	0.731615i	$0.738767\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6061.34	−1.10517	−0.552584	−	0.833457i	$-0.686358\pi$
$311$	−6061.34	−1.10517	−0.552584	+	0.833457i	$0.686358\pi$
$312$	0	0
$313$	5641.75i	1.01882i	0.860524	+	0.509410i	$0.170136\pi$
$313$	5641.75i	1.01882i	−0.860524	+	0.509410i	$0.829864\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4490.28i	− 0.795581i	−0.917476	−	0.397791i	$-0.869777\pi$
$317$	− 4490.28i	− 0.795581i	0.917476	−	0.397791i	$-0.130223\pi$
$318$	0	0
$319$	−6897.32	−1.21058
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6791.46i	− 1.16993i
$324$	0	0
$325$	− 7118.32i	− 1.21493i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−10914.4	−1.81242	−0.906212	−	0.422824i	$-0.861039\pi$
$331$	−10914.4	−1.81242	−0.906212	+	0.422824i	$0.861039\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−749.630	−0.122259
$336$	0	0
$337$	−6590.94	−1.06537	−0.532687	−	0.846312i	$-0.678818\pi$
$337$	−6590.94	−1.06537	−0.532687	+	0.846312i	$0.678818\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6416.52	1.01899
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5326.11i	0.823979i	0.911189	+	0.411989i	$0.135166\pi$
$347$	5326.11i	0.823979i	−0.911189	+	0.411989i	$0.864834\pi$
$348$	0	0
$349$	8102.39i	1.24272i	0.783523	+	0.621362i	$0.213420\pi$
$349$	8102.39i	1.24272i	−0.783523	+	0.621362i	$0.786580\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−554.855	−0.0836599	−0.0418300	−	0.999125i	$-0.513319\pi$
$353$	−554.855	−0.0836599	−0.0418300	+	0.999125i	$0.513319\pi$
$354$	0	0
$355$	872.557i	0.130452i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	403.480i	0.0593172i	0.999560	+	0.0296586i	$0.00944201\pi$
$359$	403.480i	0.0593172i	−0.999560	+	0.0296586i	$0.990558\pi$
$360$	0	0
$361$	−8228.34	−1.19964
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 929.371i	− 0.133275i
$366$	0	0
$367$	6486.91i	0.922653i	0.887230	+	0.461327i	$0.152626\pi$
$367$	6486.91i	0.922653i	−0.887230	+	0.461327i	$0.847374\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7245.81	1.00583	0.502914	−	0.864337i	$-0.332261\pi$
$373$	7245.81	1.00583	0.502914	+	0.864337i	$0.332261\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6052.09	−0.826787
$378$	0	0
$379$	−8439.30	−1.14379	−0.571897	−	0.820326i	$-0.693792\pi$
$379$	−8439.30	−1.14379	−0.571897	+	0.820326i	$0.693792\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6309.13	0.841728	0.420864	−	0.907124i	$-0.361727\pi$
$383$	6309.13	0.841728	0.420864	+	0.907124i	$0.361727\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 3442.94i	− 0.448751i	−0.974503	−	0.224375i	$-0.927966\pi$
$389$	− 3442.94i	− 0.448751i	0.974503	−	0.224375i	$-0.0720342\pi$
$390$	0	0
$391$	3012.64i	0.389657i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−419.211	−0.0533994
$396$	0	0
$397$	− 10299.2i	− 1.30202i	−0.759071	−	0.651008i	$-0.774346\pi$
$397$	− 10299.2i	− 1.30202i	0.759071	−	0.651008i	$-0.225654\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 139.553i	− 0.0173789i	−0.999962	−	0.00868944i	$-0.997234\pi$
$401$	− 139.553i	− 0.0173789i	0.999962	−	0.00868944i	$-0.00276597\pi$
$402$	0	0
$403$	5630.22	0.695934
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8530.08i	− 1.03887i
$408$	0	0
$409$	5064.38i	0.612268i	0.951988	+	0.306134i	$0.0990355\pi$
$409$	5064.38i	0.612268i	−0.951988	+	0.306134i	$0.900964\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1519.56	0.179740
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1888.80	−0.220224	−0.110112	−	0.993919i	$-0.535121\pi$
$419$	−1888.80	−0.220224	−0.110112	+	0.993919i	$0.535121\pi$
$420$	0	0
$421$	12973.6	1.50188	0.750941	−	0.660369i	$-0.229600\pi$
$421$	12973.6	1.50188	0.750941	+	0.660369i	$0.229600\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6836.44	0.780273
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 9230.76i	− 1.03162i	−0.856702	−	0.515812i	$-0.827490\pi$
$431$	− 9230.76i	− 1.03162i	0.856702	−	0.515812i	$-0.172510\pi$
$432$	0	0
$433$	− 9931.42i	− 1.10225i	−0.834423	−	0.551124i	$-0.814199\pi$
$433$	− 9931.42i	− 1.10225i	0.834423	−	0.551124i	$-0.185801\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6692.64	0.732614
$438$	0	0
$439$	− 6563.42i	− 0.713565i	−0.934187	−	0.356783i	$-0.883874\pi$
$439$	− 6563.42i	− 0.713565i	0.934187	−	0.356783i	$-0.116126\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8705.74i	− 0.933685i	−0.884341	−	0.466842i	$-0.845392\pi$
$443$	− 8705.74i	− 0.933685i	0.884341	−	0.466842i	$-0.154608\pi$
$444$	0	0
$445$	127.803	0.0136145
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11408.9i	1.19915i	0.800317	+	0.599577i	$0.204665\pi$
$449$	11408.9i	1.19915i	−0.800317	+	0.599577i	$0.795335\pi$
$450$	0	0
$451$	4542.42i	0.474266i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8681.67	−0.888647	−0.444323	−	0.895866i	$-0.646556\pi$
$457$	−8681.67	−0.888647	−0.444323	+	0.895866i	$0.646556\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10427.6	1.05349	0.526746	−	0.850023i	$-0.323412\pi$
$461$	10427.6	1.05349	0.526746	+	0.850023i	$0.323412\pi$
$462$	0	0
$463$	4918.91	0.493738	0.246869	−	0.969049i	$-0.420598\pi$
$463$	4918.91	0.493738	0.246869	+	0.969049i	$0.420598\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11313.2	−1.12102	−0.560508	−	0.828149i	$-0.689394\pi$
$467$	−11313.2	−1.12102	−0.560508	+	0.828149i	$0.689394\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 5715.02i	− 0.555554i
$474$	0	0
$475$	− 15187.3i	− 1.46703i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11148.6	−1.06345	−0.531727	−	0.846916i	$-0.678457\pi$
$479$	−11148.6	−1.06345	−0.531727	+	0.846916i	$0.678457\pi$
$480$	0	0
$481$	− 7484.77i	− 0.709514i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 910.102i	− 0.0852075i
$486$	0	0
$487$	13467.0	1.25307	0.626536	−	0.779392i	$-0.284472\pi$
$487$	13467.0	1.25307	0.626536	+	0.779392i	$0.284472\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 5130.89i	− 0.471597i	−0.971802	−	0.235798i	$-0.924229\pi$
$491$	− 5130.89i	− 0.471597i	0.971802	−	0.235798i	$-0.0757705\pi$
$492$	0	0
$493$	− 5812.43i	− 0.530992i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7937.42	0.712079	0.356040	−	0.934471i	$-0.384127\pi$
$499$	7937.42	0.712079	0.356040	+	0.934471i	$0.384127\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14153.2	−1.25459	−0.627294	−	0.778782i	$-0.715838\pi$
$503$	−14153.2	−1.25459	−0.627294	+	0.778782i	$0.715838\pi$
$504$	0	0
$505$	145.027	0.0127794
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17700.6	−1.54139	−0.770694	−	0.637205i	$-0.780090\pi$
$509$	−17700.6	−1.54139	−0.770694	+	0.637205i	$0.780090\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 497.086i	− 0.0425325i
$516$	0	0
$517$	15529.8i	1.32108i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22840.8	1.92068	0.960339	−	0.278836i	$-0.0899485\pi$
$521$	22840.8	1.92068	0.960339	+	0.278836i	$0.0899485\pi$
$522$	0	0
$523$	20678.5i	1.72889i	0.502731	+	0.864443i	$0.332329\pi$
$523$	20678.5i	1.72889i	−0.502731	+	0.864443i	$0.667671\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5407.27i	0.446953i
$528$	0	0
$529$	9198.20	0.755995
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3985.77i	0.323908i
$534$	0	0
$535$	2344.02i	0.189422i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17993.8	−1.42997	−0.714985	−	0.699140i	$-0.753566\pi$
$541$	−17993.8	−1.42997	−0.714985	+	0.699140i	$0.753566\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1170.51	0.0919986
$546$	0	0
$547$	−11216.0	−0.876712	−0.438356	−	0.898801i	$-0.644439\pi$
$547$	−11216.0	−0.876712	−0.438356	+	0.898801i	$0.644439\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12912.4	−0.998344
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21659.7i	1.64767i	0.566831	+	0.823834i	$0.308169\pi$
$557$	21659.7i	1.64767i	−0.566831	+	0.823834i	$0.691831\pi$
$558$	0	0
$559$	− 5014.68i	− 0.379425i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18477.7	−1.38320	−0.691600	−	0.722281i	$-0.743094\pi$
$563$	−18477.7	−1.38320	−0.691600	+	0.722281i	$0.743094\pi$
$564$	0	0
$565$	− 1340.98i	− 0.0998505i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16756.2i	1.23455i	0.786748	+	0.617274i	$0.211763\pi$
$569$	16756.2i	1.23455i	−0.786748	+	0.617274i	$0.788237\pi$
$570$	0	0
$571$	−20090.6	−1.47245	−0.736224	−	0.676738i	$-0.763393\pi$
$571$	−20090.6	−1.47245	−0.736224	+	0.676738i	$0.763393\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6736.95i	0.488609i
$576$	0	0
$577$	− 9051.76i	− 0.653084i	−0.945183	−	0.326542i	$-0.894117\pi$
$577$	− 9051.76i	− 0.653084i	0.945183	−	0.326542i	$-0.105883\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	27085.1	1.92410
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15886.2	1.11703	0.558513	−	0.829496i	$-0.311372\pi$
$587$	15886.2	1.11703	0.558513	+	0.829496i	$0.311372\pi$
$588$	0	0
$589$	12012.3	0.840338
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25951.6	1.79714	0.898570	−	0.438831i	$-0.144607\pi$
$593$	25951.6	1.79714	0.898570	+	0.438831i	$0.144607\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15920.2i	1.08594i	0.839751	+	0.542972i	$0.182701\pi$
$599$	15920.2i	1.08594i	−0.839751	+	0.542972i	$0.817299\pi$
$600$	0	0
$601$	5872.84i	0.398599i	0.979939	+	0.199300i	$0.0638667\pi$
$601$	5872.84i	0.398599i	−0.979939	+	0.199300i	$0.936133\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3463.06	0.232716
$606$	0	0
$607$	27690.4i	1.85160i	0.378016	+	0.925799i	$0.376606\pi$
$607$	27690.4i	1.85160i	−0.378016	+	0.925799i	$0.623394\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13626.7i	0.902255i
$612$	0	0
$613$	−8012.72	−0.527945	−0.263973	−	0.964530i	$-0.585033\pi$
$613$	−8012.72	−0.527945	−0.263973	+	0.964530i	$0.585033\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1546.21i	0.100888i	0.998727	+	0.0504442i	$0.0160637\pi$
$617$	1546.21i	0.100888i	−0.998727	+	0.0504442i	$0.983936\pi$
$618$	0	0
$619$	− 11480.3i	− 0.745450i	−0.927942	−	0.372725i	$-0.878424\pi$
$619$	− 11480.3i	− 0.745450i	0.927942	−	0.372725i	$-0.121576\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15118.3	0.967572
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7188.38	0.455675
$630$	0	0
$631$	−12482.7	−0.787524	−0.393762	−	0.919212i	$-0.628827\pi$
$631$	−12482.7	−0.787524	−0.393762	+	0.919212i	$0.628827\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−77.3428	−0.00483347
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4170.78i	0.256998i	0.991710	+	0.128499i	$0.0410160\pi$
$641$	4170.78i	0.256998i	−0.991710	+	0.128499i	$0.958984\pi$
$642$	0	0
$643$	− 18330.0i	− 1.12421i	−0.827066	−	0.562105i	$-0.809992\pi$
$643$	− 18330.0i	− 1.12421i	0.827066	−	0.562105i	$-0.190008\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10099.3	0.613671	0.306835	−	0.951763i	$-0.400730\pi$
$647$	10099.3	0.613671	0.306835	+	0.951763i	$0.400730\pi$
$648$	0	0
$649$	2769.47i	0.167506i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14974.3i	0.897381i	0.893687	+	0.448691i	$0.148110\pi$
$653$	14974.3i	0.897381i	−0.893687	+	0.448691i	$0.851890\pi$
$654$	0	0
$655$	−2161.60	−0.128948
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 25422.5i	− 1.50276i	−0.659870	−	0.751380i	$-0.729389\pi$
$659$	− 25422.5i	− 1.50276i	0.659870	−	0.751380i	$-0.270611\pi$
$660$	0	0
$661$	− 6684.89i	− 0.393362i	−0.980468	−	0.196681i	$-0.936984\pi$
$661$	− 6684.89i	− 0.393362i	0.980468	−	0.196681i	$-0.0630163\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5727.85	0.332509
$668$	0	0
$669$	0	0
$670$	0	0
$671$	55402.3	3.18746
$672$	0	0
$673$	4182.44	0.239556	0.119778	−	0.992801i	$-0.461782\pi$
$673$	4182.44	0.239556	0.119778	+	0.992801i	$0.461782\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12231.7	−0.694390	−0.347195	−	0.937793i	$-0.612866\pi$
$677$	−12231.7	−0.694390	−0.347195	+	0.937793i	$0.612866\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 16172.7i	− 0.906046i	−0.891499	−	0.453023i	$-0.850345\pi$
$683$	− 16172.7i	− 0.906046i	0.891499	−	0.453023i	$-0.149655\pi$
$684$	0	0
$685$	− 926.068i	− 0.0516544i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23766.0	1.31410
$690$	0	0
$691$	− 20821.9i	− 1.14631i	−0.819447	−	0.573155i	$-0.805719\pi$
$691$	− 20821.9i	− 1.14631i	0.819447	−	0.573155i	$-0.194281\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1253.45i	− 0.0684119i
$696$	0	0
$697$	−3827.94	−0.208025
$698$	0	0
$699$	0	0
$700$	0	0
$701$	35212.3i	1.89722i	0.316451	+	0.948609i	$0.397509\pi$
$701$	35212.3i	1.89722i	−0.316451	+	0.948609i	$0.602491\pi$
$702$	0	0
$703$	− 15969.1i	− 0.856737i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20847.2	1.10428	0.552138	−	0.833753i	$-0.313812\pi$
$709$	20847.2	1.10428	0.552138	+	0.833753i	$0.313812\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5328.58	−0.279883
$714$	0	0
$715$	4398.70	0.230073
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10894.0	−0.565057	−0.282528	−	0.959259i	$-0.591173\pi$
$719$	−10894.0	−0.565057	−0.282528	+	0.959259i	$0.591173\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 12997.9i	− 0.665835i
$726$	0	0
$727$	− 31196.0i	− 1.59147i	−0.605646	−	0.795734i	$-0.707085\pi$
$727$	− 31196.0i	− 1.59147i	0.605646	−	0.795734i	$-0.292915\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4816.10	0.243680
$732$	0	0
$733$	17927.0i	0.903343i	0.892184	+	0.451672i	$0.149172\pi$
$733$	17927.0i	0.903343i	−0.892184	+	0.451672i	$0.850828\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42236.3i	2.11098i
$738$	0	0
$739$	15060.7	0.749683	0.374841	−	0.927089i	$-0.377697\pi$
$739$	15060.7	0.749683	0.374841	+	0.927089i	$0.377697\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 8750.44i	− 0.432062i	−0.976386	−	0.216031i	$-0.930689\pi$
$743$	− 8750.44i	− 0.432062i	0.976386	−	0.216031i	$-0.0693113\pi$
$744$	0	0
$745$	1861.74i	0.0915556i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−22831.3	−1.10936	−0.554678	−	0.832065i	$-0.687159\pi$
$751$	−22831.3	−1.10936	−0.554678	+	0.832065i	$0.687159\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2631.85	0.126865
$756$	0	0
$757$	12087.5	0.580353	0.290176	−	0.956973i	$-0.406286\pi$
$757$	12087.5	0.580353	0.290176	+	0.956973i	$0.406286\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−36718.9	−1.74909	−0.874547	−	0.484941i	$-0.838841\pi$
$761$	−36718.9	−1.74909	−0.874547	+	0.484941i	$0.838841\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2430.09i	0.114401i
$768$	0	0
$769$	− 2652.68i	− 0.124393i	−0.998064	−	0.0621964i	$-0.980189\pi$
$769$	− 2652.68i	− 0.124393i	0.998064	−	0.0621964i	$-0.0198105\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24670.0	1.14789	0.573944	−	0.818895i	$-0.305413\pi$
$773$	24670.0	1.14789	0.573944	+	0.818895i	$0.305413\pi$
$774$	0	0
$775$	12091.9i	0.560455i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8503.82i	0.391118i
$780$	0	0
$781$	49162.4	2.25245
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1563.35i	0.0710805i
$786$	0	0
$787$	17103.9i	0.774699i	0.921933	+	0.387349i	$0.126609\pi$
$787$	17103.9i	0.774699i	−0.921933	+	0.387349i	$0.873391\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	48613.1	2.17693
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26143.0	−1.16190	−0.580948	−	0.813940i	$-0.697318\pi$
$797$	−26143.0	−1.16190	−0.580948	+	0.813940i	$0.697318\pi$
$798$	0	0
$799$	−13087.1	−0.579460
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−52363.4	−2.30120
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24531.7i	1.06612i	0.846079	+	0.533058i	$0.178957\pi$
$809$	24531.7i	1.06612i	−0.846079	+	0.533058i	$0.821043\pi$
$810$	0	0
$811$	− 8145.19i	− 0.352671i	−0.984330	−	0.176336i	$-0.943576\pi$
$811$	− 8145.19i	− 0.352671i	0.984330	−	0.176336i	$-0.0564244\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1216.23	0.0522731
$816$	0	0
$817$	− 10699.1i	− 0.458155i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24100.1i	1.02448i	0.858843	+	0.512240i	$0.171184\pi$
$821$	24100.1i	1.02448i	−0.858843	+	0.512240i	$0.828816\pi$
$822$	0	0
$823$	32871.6	1.39226	0.696131	−	0.717915i	$-0.254903\pi$
$823$	32871.6	1.39226	0.696131	+	0.717915i	$0.254903\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5891.21i	0.247712i	0.992300	+	0.123856i	$0.0395260\pi$
$827$	5891.21i	0.247712i	−0.992300	+	0.123856i	$0.960474\pi$
$828$	0	0
$829$	− 10904.2i	− 0.456836i	−0.973563	−	0.228418i	$-0.926645\pi$
$829$	− 10904.2i	− 0.456836i	0.973563	−	0.228418i	$-0.0733553\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2956.09	0.122515
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38021.0	1.56452	0.782259	−	0.622953i	$-0.214067\pi$
$839$	38021.0	1.56452	0.782259	+	0.622953i	$0.214067\pi$
$840$	0	0
$841$	13338.0	0.546885
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1301.25	0.0529757
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7083.78i	0.285345i
$852$	0	0
$853$	− 30803.2i	− 1.23644i	−0.786006	−	0.618218i	$-0.787855\pi$
$853$	− 30803.2i	− 1.23644i	0.786006	−	0.618218i	$-0.212145\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5246.09	−0.209105	−0.104553	−	0.994519i	$-0.533341\pi$
$857$	−5246.09	−0.209105	−0.104553	+	0.994519i	$0.533341\pi$
$858$	0	0
$859$	30597.9i	1.21535i	0.794185	+	0.607677i	$0.207898\pi$
$859$	30597.9i	1.21535i	−0.794185	+	0.607677i	$0.792102\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 2320.00i	− 0.0915108i	−0.998953	−	0.0457554i	$-0.985431\pi$
$863$	− 2320.00i	− 0.0915108i	0.998953	−	0.0457554i	$-0.0145695\pi$
$864$	0	0
$865$	−4570.32	−0.179648
$866$	0	0
$867$	0	0
$868$	0	0
$869$	23619.5i	0.922022i
$870$	0	0
$871$	37060.5i	1.44173i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28230.0	1.08695	0.543477	−	0.839424i	$-0.317108\pi$
$877$	28230.0	1.08695	0.543477	+	0.839424i	$0.317108\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16404.3	0.627325	0.313663	−	0.949535i	$-0.398444\pi$
$881$	16404.3	0.627325	0.313663	+	0.949535i	$0.398444\pi$
$882$	0	0
$883$	12816.6	0.488463	0.244232	−	0.969717i	$-0.421464\pi$
$883$	12816.6	0.488463	0.244232	+	0.969717i	$0.421464\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4848.20	−0.183525	−0.0917625	−	0.995781i	$-0.529250\pi$
$887$	−4848.20	−0.183525	−0.0917625	+	0.995781i	$0.529250\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	29073.2i	1.08947i
$894$	0	0
$895$	2282.75i	0.0852558i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10280.7	0.381401
$900$	0	0
$901$	22824.9i	0.843959i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 3337.16i	− 0.122576i
$906$	0	0
$907$	−35931.3	−1.31541	−0.657706	−	0.753275i	$-0.728473\pi$
$907$	−35931.3	−1.31541	−0.657706	+	0.753275i	$0.728473\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 26942.0i	− 0.979834i	−0.871769	−	0.489917i	$-0.837027\pi$
$911$	− 26942.0i	− 0.979834i	0.871769	−	0.489917i	$-0.162973\pi$
$912$	0	0
$913$	− 85616.3i	− 3.10349i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−18097.9	−0.649615	−0.324807	−	0.945780i	$-0.605299\pi$
$919$	−18097.9	−0.649615	−0.324807	+	0.945780i	$0.605299\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	43137.8	1.53835
$924$	0	0
$925$	16074.8	0.571392
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27036.0	0.954815	0.477407	−	0.878682i	$-0.341577\pi$
$929$	27036.0	0.954815	0.477407	+	0.878682i	$0.341577\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4224.51i	0.147761i
$936$	0	0
$937$	22915.3i	0.798942i	0.916746	+	0.399471i	$0.130806\pi$
$937$	22915.3i	0.798942i	−0.916746	+	0.399471i	$0.869194\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	29976.8	1.03849	0.519244	−	0.854626i	$-0.326214\pi$
$941$	29976.8	1.03849	0.519244	+	0.854626i	$0.326214\pi$
$942$	0	0
$943$	− 3772.23i	− 0.130266i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48949.6i	1.67967i	0.542841	+	0.839835i	$0.317349\pi$
$947$	48949.6i	1.67967i	−0.542841	+	0.839835i	$0.682651\pi$
$948$	0	0
$949$	−45946.6	−1.57164
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10983.8i	0.373349i	0.982422	+	0.186674i	$0.0597709\pi$
$953$	10983.8i	0.373349i	−0.982422	+	0.186674i	$0.940229\pi$
$954$	0	0
$955$	2350.37i	0.0796401i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	20227.0	0.678962
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1714.01	−0.0571771
$966$	0	0
$967$	−45637.1	−1.51767	−0.758836	−	0.651281i	$-0.774232\pi$
$967$	−45637.1	−1.51767	−0.758836	+	0.651281i	$0.774232\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29554.0	−0.976760	−0.488380	−	0.872631i	$-0.662412\pi$
$971$	−29554.0	−0.976760	−0.488380	+	0.872631i	$0.662412\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 20249.3i	− 0.663083i	−0.943441	−	0.331541i	$-0.892431\pi$
$977$	− 20249.3i	− 0.663083i	0.943441	−	0.331541i	$-0.107569\pi$
$978$	0	0
$979$	− 7200.78i	− 0.235074i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	221.639	0.00719143	0.00359571	−	0.999994i	$-0.498855\pi$
$983$	221.639	0.00719143	0.00359571	+	0.999994i	$0.498855\pi$
$984$	0	0
$985$	6409.22i	0.207325i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4746.02i	0.152593i
$990$	0	0
$991$	−46233.1	−1.48198	−0.740991	−	0.671515i	$-0.765644\pi$
$991$	−46233.1	−1.48198	−0.740991	+	0.671515i	$0.765644\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 5460.12i	− 0.173967i
$996$	0	0
$997$	− 873.051i	− 0.0277330i	−0.999904	−	0.0138665i	$-0.995586\pi$
$997$	− 873.051i	− 0.0277330i	0.999904	−	0.0138665i	$-0.00441399\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.f.b.881.11	✓	24
3.2	odd	2	inner	1764.4.f.b.881.14	yes	24
7.2	even	3		1764.4.t.c.521.13		48
7.3	odd	6		1764.4.t.c.1097.12		48
7.4	even	3		1764.4.t.c.1097.14		48
7.5	odd	6		1764.4.t.c.521.11		48
7.6	odd	2	inner	1764.4.f.b.881.13	yes	24
21.2	odd	6		1764.4.t.c.521.12		48
21.5	even	6		1764.4.t.c.521.14		48
21.11	odd	6		1764.4.t.c.1097.11		48
21.17	even	6		1764.4.t.c.1097.13		48
21.20	even	2	inner	1764.4.f.b.881.12	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.4.f.b.881.11	✓	24	1.1	even	1	trivial
1764.4.f.b.881.12	yes	24	21.20	even	2	inner
1764.4.f.b.881.13	yes	24	7.6	odd	2	inner
1764.4.f.b.881.14	yes	24	3.2	odd	2	inner
1764.4.t.c.521.11		48	7.5	odd	6
1764.4.t.c.521.12		48	21.2	odd	6
1764.4.t.c.521.13		48	7.2	even	3
1764.4.t.c.521.14		48	21.5	even	6
1764.4.t.c.1097.11		48	21.11	odd	6
1764.4.t.c.1097.12		48	7.3	odd	6
1764.4.t.c.1097.13		48	21.17	even	6
1764.4.t.c.1097.14		48	7.4	even	3

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.16450 q^{5} +O(q^{10})\)	\(q-1.16450 q^{5} +65.6114i q^{11} +57.5711i q^{13} -55.2913 q^{17} +122.831i q^{19} -54.4867i q^{23} -123.644 q^{25} +105.124i q^{29} -97.7959i q^{31} -130.009 q^{37} +69.2322 q^{41} -87.1042 q^{43} +236.694 q^{47} -412.811i q^{53} -76.4046i q^{55} +42.2102 q^{59} -844.401i q^{61} -67.0417i q^{65} +643.734 q^{67} -749.296i q^{71} +798.084i q^{73} +359.991 q^{79} -1304.90 q^{83} +64.3869 q^{85} -109.749 q^{89} -143.036i q^{95} +781.538i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 888 q^{25} + 864 q^{37} - 1248 q^{43} + 1056 q^{67} - 8064 q^{85}+O(q^{100}) \) 24 * q + 888 * q^25 + 864 * q^37 - 1248 * q^43 + 1056 * q^67 - 8064 * q^85

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