LMFDB - Embedded newform 1444.3.c.d.721.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,3,Mod(721,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1444.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1444 = 2^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1444.c (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:	$39.3461501736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.1
Character	$\chi$	$=$	1444.721
Dual form			1444.3.c.d.721.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.75074i q^{3} +6.82813 q^{5} +2.75886 q^{7} -24.0710 q^{9} +O(q^{10})$	$q-5.75074i q^{3} +6.82813 q^{5} +2.75886 q^{7} -24.0710 q^{9} -17.6128 q^{11} -12.8526i q^{13} -39.2668i q^{15} -19.1912 q^{17} -15.8655i q^{21} -0.789449 q^{23} +21.6234 q^{25} +86.6696i q^{27} +7.93261i q^{29} +17.6050i q^{31} +101.287i q^{33} +18.8379 q^{35} -1.64247i q^{37} -73.9120 q^{39} +9.96021i q^{41} +21.3961 q^{43} -164.360 q^{45} +53.7969 q^{47} -41.3887 q^{49} +110.364i q^{51} -42.3224i q^{53} -120.263 q^{55} -93.5276i q^{59} -84.2074 q^{61} -66.4087 q^{63} -87.7593i q^{65} +12.1374i q^{67} +4.53992i q^{69} +20.8801i q^{71} -92.5299 q^{73} -124.350i q^{75} -48.5915 q^{77} +77.2834i q^{79} +281.775 q^{81} +35.3673 q^{83} -131.040 q^{85} +45.6184 q^{87} +2.37864i q^{89} -35.4586i q^{91} +101.242 q^{93} +149.491i q^{97} +423.959 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$.

$n$	$723$	$1085$
$\chi(n)$	$1$	$-1$

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
$3$	− 5.75074i	− 1.91691i	−0.285237	−	0.958457i	$-0.592072\pi$
$3$	− 5.75074i	− 1.91691i	0.285237	−	0.958457i	$-0.407928\pi$
$4$	0	0
$5$	6.82813	1.36563	0.682813	−	0.730593i	$-0.260756\pi$
$5$	6.82813	1.36563	0.682813	+	0.730593i	$0.260756\pi$
$6$	0	0
$7$	2.75886	0.394123	0.197062	−	0.980391i	$-0.436860\pi$
$7$	2.75886	0.394123	0.197062	+	0.980391i	$0.436860\pi$
$8$	0	0
$9$	−24.0710	−2.67456
$10$	0	0
$11$	−17.6128	−1.60117	−0.800584	−	0.599221i	$-0.795477\pi$
$11$	−17.6128	−1.60117	−0.800584	+	0.599221i	$0.795477\pi$
$12$	0	0
$13$	− 12.8526i	− 0.988662i	−0.869274	−	0.494331i	$-0.835413\pi$
$13$	− 12.8526i	− 0.988662i	0.869274	−	0.494331i	$-0.164587\pi$
$14$	0	0
$15$	− 39.2668i	− 2.61779i
$16$	0	0
$17$	−19.1912	−1.12889	−0.564447	−	0.825469i	$-0.690911\pi$
$17$	−19.1912	−1.12889	−0.564447	+	0.825469i	$0.690911\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	− 15.8655i	− 0.755501i
$22$	0	0
$23$	−0.789449	−0.0343239	−0.0171619	−	0.999853i	$-0.505463\pi$
$23$	−0.789449	−0.0343239	−0.0171619	+	0.999853i	$0.505463\pi$
$24$	0	0
$25$	21.6234	0.864935
$26$	0	0
$27$	86.6696i	3.20999i
$28$	0	0
$29$	7.93261i	0.273538i	0.990603	+	0.136769i	$0.0436719\pi$
$29$	7.93261i	0.273538i	−0.990603	+	0.136769i	$0.956328\pi$
$30$	0	0
$31$	17.6050i	0.567903i	0.958839	+	0.283951i	$0.0916454\pi$
$31$	17.6050i	0.567903i	−0.958839	+	0.283951i	$0.908355\pi$
$32$	0	0
$33$	101.287i	3.06930i
$34$	0	0
$35$	18.8379	0.538225
$36$	0	0
$37$	− 1.64247i	− 0.0443910i	−0.999754	−	0.0221955i	$-0.992934\pi$
$37$	− 1.64247i	− 0.0443910i	0.999754	−	0.0221955i	$-0.00706563\pi$
$38$	0	0
$39$	−73.9120	−1.89518
$40$	0	0
$41$	9.96021i	0.242932i	0.992596	+	0.121466i	$0.0387595\pi$
$41$	9.96021i	0.242932i	−0.992596	+	0.121466i	$0.961240\pi$
$42$	0	0
$43$	21.3961	0.497585	0.248792	−	0.968557i	$-0.419966\pi$
$43$	21.3961	0.497585	0.248792	+	0.968557i	$0.419966\pi$
$44$	0	0
$45$	−164.360	−3.65245
$46$	0	0
$47$	53.7969	1.14461	0.572307	−	0.820039i	$-0.306048\pi$
$47$	53.7969	1.14461	0.572307	+	0.820039i	$0.306048\pi$
$48$	0	0
$49$	−41.3887	−0.844667
$50$	0	0
$51$	110.364i	2.16399i
$52$	0	0
$53$	− 42.3224i	− 0.798535i	−0.916834	−	0.399268i	$-0.869264\pi$
$53$	− 42.3224i	− 0.798535i	0.916834	−	0.399268i	$-0.130736\pi$
$54$	0	0
$55$	−120.263	−2.18660
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 93.5276i	− 1.58521i	−0.609734	−	0.792606i	$-0.708724\pi$
$59$	− 93.5276i	− 1.58521i	0.609734	−	0.792606i	$-0.291276\pi$
$60$	0	0
$61$	−84.2074	−1.38045	−0.690224	−	0.723596i	$-0.742488\pi$
$61$	−84.2074	−1.38045	−0.690224	+	0.723596i	$0.742488\pi$
$62$	0	0
$63$	−66.4087	−1.05411
$64$	0	0
$65$	− 87.7593i	− 1.35014i
$66$	0	0
$67$	12.1374i	0.181155i	0.995889	+	0.0905777i	$0.0288713\pi$
$67$	12.1374i	0.181155i	−0.995889	+	0.0905777i	$0.971129\pi$
$68$	0	0
$69$	4.53992i	0.0657959i
$70$	0	0
$71$	20.8801i	0.294086i	0.989130	+	0.147043i	$0.0469756\pi$
$71$	20.8801i	0.294086i	−0.989130	+	0.147043i	$0.953024\pi$
$72$	0	0
$73$	−92.5299	−1.26753	−0.633767	−	0.773524i	$-0.718492\pi$
$73$	−92.5299	−1.26753	−0.633767	+	0.773524i	$0.718492\pi$
$74$	0	0
$75$	− 124.350i	− 1.65801i
$76$	0	0
$77$	−48.5915	−0.631058
$78$	0	0
$79$	77.2834i	0.978271i	0.872208	+	0.489135i	$0.162688\pi$
$79$	77.2834i	0.978271i	−0.872208	+	0.489135i	$0.837312\pi$
$80$	0	0
$81$	281.775	3.47871
$82$	0	0
$83$	35.3673	0.426112	0.213056	−	0.977040i	$-0.431658\pi$
$83$	35.3673	0.426112	0.213056	+	0.977040i	$0.431658\pi$
$84$	0	0
$85$	−131.040	−1.54165
$86$	0	0
$87$	45.6184	0.524350
$88$	0	0
$89$	2.37864i	0.0267263i	0.999911	+	0.0133631i	$0.00425374\pi$
$89$	2.37864i	0.0267263i	−0.999911	+	0.0133631i	$0.995746\pi$
$90$	0	0
$91$	− 35.4586i	− 0.389655i
$92$	0	0
$93$	101.242	1.08862
$94$	0	0
$95$	0	0
$96$	0	0
$97$	149.491i	1.54114i	0.637352	+	0.770572i	$0.280030\pi$
$97$	149.491i	1.54114i	−0.637352	+	0.770572i	$0.719970\pi$
$98$	0	0
$99$	423.959	4.28242
$100$	0	0
$101$	−133.086	−1.31769	−0.658843	−	0.752280i	$-0.728954\pi$
$101$	−133.086	−1.31769	−0.658843	+	0.752280i	$0.728954\pi$
$102$	0	0
$103$	137.082i	1.33089i	0.746446	+	0.665446i	$0.231759\pi$
$103$	137.082i	1.33089i	−0.746446	+	0.665446i	$0.768241\pi$
$104$	0	0
$105$	− 108.332i	− 1.03173i
$106$	0	0
$107$	− 30.0392i	− 0.280740i	−0.990099	−	0.140370i	$-0.955171\pi$
$107$	− 30.0392i	− 0.280740i	0.990099	−	0.140370i	$-0.0448292\pi$
$108$	0	0
$109$	− 209.654i	− 1.92344i	−0.274043	−	0.961718i	$-0.588361\pi$
$109$	− 209.654i	− 1.92344i	0.274043	−	0.961718i	$-0.411639\pi$
$110$	0	0
$111$	−9.44541	−0.0850938
$112$	0	0
$113$	− 129.448i	− 1.14556i	−0.819710	−	0.572778i	$-0.805866\pi$
$113$	− 129.448i	− 1.14556i	0.819710	−	0.572778i	$-0.194134\pi$
$114$	0	0
$115$	−5.39046	−0.0468736
$116$	0	0
$117$	309.375i	2.64423i
$118$	0	0
$119$	−52.9459	−0.444924
$120$	0	0
$121$	189.212	1.56374
$122$	0	0
$123$	57.2786	0.465680
$124$	0	0
$125$	−23.0560	−0.184448
$126$	0	0
$127$	− 181.551i	− 1.42954i	−0.699361	−	0.714769i	$-0.746532\pi$
$127$	− 181.551i	− 1.42954i	0.699361	−	0.714769i	$-0.253468\pi$
$128$	0	0
$129$	− 123.044i	− 0.953827i
$130$	0	0
$131$	88.5960	0.676305	0.338153	−	0.941091i	$-0.390198\pi$
$131$	88.5960	0.676305	0.338153	+	0.941091i	$0.390198\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	591.791i	4.38364i
$136$	0	0
$137$	−98.2621	−0.717241	−0.358621	−	0.933483i	$-0.616753\pi$
$137$	−98.2621	−0.717241	−0.358621	+	0.933483i	$0.616753\pi$
$138$	0	0
$139$	−197.214	−1.41881	−0.709403	−	0.704804i	$-0.751035\pi$
$139$	−197.214	−1.41881	−0.709403	+	0.704804i	$0.751035\pi$
$140$	0	0
$141$	− 309.372i	− 2.19413i
$142$	0	0
$143$	226.371i	1.58301i
$144$	0	0
$145$	54.1649i	0.373551i
$146$	0	0
$147$	238.016i	1.61915i
$148$	0	0
$149$	62.8936	0.422105	0.211052	−	0.977475i	$-0.432311\pi$
$149$	62.8936	0.422105	0.211052	+	0.977475i	$0.432311\pi$
$150$	0	0
$151$	− 8.91053i	− 0.0590101i	−0.999565	−	0.0295051i	$-0.990607\pi$
$151$	− 8.91053i	− 0.0590101i	0.999565	−	0.0295051i	$-0.00939312\pi$
$152$	0	0
$153$	461.952	3.01929
$154$	0	0
$155$	120.209i	0.775543i
$156$	0	0
$157$	36.0961	0.229911	0.114956	−	0.993371i	$-0.463327\pi$
$157$	36.0961	0.229911	0.114956	+	0.993371i	$0.463327\pi$
$158$	0	0
$159$	−243.385	−1.53072
$160$	0	0
$161$	−2.17798	−0.0135278
$162$	0	0
$163$	103.257	0.633478	0.316739	−	0.948513i	$-0.397412\pi$
$163$	103.257	0.633478	0.316739	+	0.948513i	$0.397412\pi$
$164$	0	0
$165$	691.601i	4.19152i
$166$	0	0
$167$	313.989i	1.88018i	0.340933	+	0.940088i	$0.389257\pi$
$167$	313.989i	1.88018i	−0.340933	+	0.940088i	$0.610743\pi$
$168$	0	0
$169$	3.81056	0.0225477
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 19.7204i	− 0.113991i	−0.998374	−	0.0569953i	$-0.981848\pi$
$173$	− 19.7204i	− 0.113991i	0.998374	−	0.0569953i	$-0.0181520\pi$
$174$	0	0
$175$	59.6560	0.340891
$176$	0	0
$177$	−537.853	−3.03872
$178$	0	0
$179$	− 202.249i	− 1.12988i	−0.825131	−	0.564942i	$-0.808899\pi$
$179$	− 202.249i	− 1.12988i	0.825131	−	0.564942i	$-0.191101\pi$
$180$	0	0
$181$	− 128.567i	− 0.710317i	−0.934806	−	0.355159i	$-0.884427\pi$
$181$	− 128.567i	− 0.710317i	0.934806	−	0.355159i	$-0.115573\pi$
$182$	0	0
$183$	484.255i	2.64620i
$184$	0	0
$185$	− 11.2150i	− 0.0606216i
$186$	0	0
$187$	338.012	1.80755
$188$	0	0
$189$	239.110i	1.26513i
$190$	0	0
$191$	−14.8420	−0.0777066	−0.0388533	−	0.999245i	$-0.512370\pi$
$191$	−14.8420	−0.0777066	−0.0388533	+	0.999245i	$0.512370\pi$
$192$	0	0
$193$	− 152.793i	− 0.791675i	−0.918321	−	0.395838i	$-0.870454\pi$
$193$	− 152.793i	− 0.791675i	0.918321	−	0.395838i	$-0.129546\pi$
$194$	0	0
$195$	−504.681	−2.58811
$196$	0	0
$197$	−171.910	−0.872642	−0.436321	−	0.899791i	$-0.643719\pi$
$197$	−171.910	−0.872642	−0.436321	+	0.899791i	$0.643719\pi$
$198$	0	0
$199$	−282.805	−1.42113	−0.710565	−	0.703632i	$-0.751561\pi$
$199$	−282.805	−1.42113	−0.710565	+	0.703632i	$0.751561\pi$
$200$	0	0
$201$	69.7991	0.347259
$202$	0	0
$203$	21.8850i	0.107808i
$204$	0	0
$205$	68.0096i	0.331754i
$206$	0	0
$207$	19.0029	0.0918013
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 50.4914i	− 0.239296i	−0.992816	−	0.119648i	$-0.961823\pi$
$211$	− 50.4914i	− 0.239296i	0.992816	−	0.119648i	$-0.0381766\pi$
$212$	0	0
$213$	120.076	0.563738
$214$	0	0
$215$	146.096	0.679515
$216$	0	0
$217$	48.5698i	0.223824i
$218$	0	0
$219$	532.116i	2.42975i
$220$	0	0
$221$	246.657i	1.11609i
$222$	0	0
$223$	118.674i	0.532172i	0.963949	+	0.266086i	$0.0857305\pi$
$223$	118.674i	0.532172i	−0.963949	+	0.266086i	$0.914270\pi$
$224$	0	0
$225$	−520.497	−2.31332
$226$	0	0
$227$	− 302.726i	− 1.33359i	−0.745240	−	0.666797i	$-0.767665\pi$
$227$	− 302.726i	− 1.33359i	0.745240	−	0.666797i	$-0.232335\pi$
$228$	0	0
$229$	−44.9515	−0.196295	−0.0981474	−	0.995172i	$-0.531292\pi$
$229$	−44.9515	−0.196295	−0.0981474	+	0.995172i	$0.531292\pi$
$230$	0	0
$231$	279.437i	1.20968i
$232$	0	0
$233$	102.188	0.438574	0.219287	−	0.975660i	$-0.429627\pi$
$233$	102.188	0.438574	0.219287	+	0.975660i	$0.429627\pi$
$234$	0	0
$235$	367.332	1.56312
$236$	0	0
$237$	444.437	1.87526
$238$	0	0
$239$	213.634	0.893865	0.446932	−	0.894568i	$-0.352516\pi$
$239$	213.634	0.893865	0.446932	+	0.894568i	$0.352516\pi$
$240$	0	0
$241$	− 309.435i	− 1.28396i	−0.766720	−	0.641981i	$-0.778113\pi$
$241$	− 309.435i	− 1.28396i	0.766720	−	0.641981i	$-0.221887\pi$
$242$	0	0
$243$	− 840.390i	− 3.45840i
$244$	0	0
$245$	−282.607	−1.15350
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 203.388i	− 0.816819i
$250$	0	0
$251$	230.358	0.917761	0.458880	−	0.888498i	$-0.348251\pi$
$251$	230.358	0.917761	0.458880	+	0.888498i	$0.348251\pi$
$252$	0	0
$253$	13.9045	0.0549583
$254$	0	0
$255$	753.578i	2.95521i
$256$	0	0
$257$	− 282.699i	− 1.10000i	−0.835166	−	0.549999i	$-0.814628\pi$
$257$	− 282.699i	− 1.10000i	0.835166	−	0.549999i	$-0.185372\pi$
$258$	0	0
$259$	− 4.53135i	− 0.0174956i
$260$	0	0
$261$	− 190.946i	− 0.731595i
$262$	0	0
$263$	366.504	1.39355	0.696776	−	0.717288i	$-0.254617\pi$
$263$	366.504	1.39355	0.696776	+	0.717288i	$0.254617\pi$
$264$	0	0
$265$	− 288.983i	− 1.09050i
$266$	0	0
$267$	13.6789	0.0512320
$268$	0	0
$269$	− 432.551i	− 1.60799i	−0.594633	−	0.803997i	$-0.702703\pi$
$269$	− 432.551i	− 1.60799i	0.594633	−	0.803997i	$-0.297297\pi$
$270$	0	0
$271$	396.249	1.46217	0.731087	−	0.682285i	$-0.239014\pi$
$271$	396.249	1.46217	0.731087	+	0.682285i	$0.239014\pi$
$272$	0	0
$273$	−203.913	−0.746935
$274$	0	0
$275$	−380.849	−1.38491
$276$	0	0
$277$	−534.131	−1.92827	−0.964136	−	0.265408i	$-0.914493\pi$
$277$	−534.131	−1.92827	−0.964136	+	0.265408i	$0.914493\pi$
$278$	0	0
$279$	− 423.770i	− 1.51889i
$280$	0	0
$281$	282.009i	1.00359i	0.864986	+	0.501796i	$0.167327\pi$
$281$	282.009i	1.00359i	−0.864986	+	0.501796i	$0.832673\pi$
$282$	0	0
$283$	132.139	0.466922	0.233461	−	0.972366i	$-0.424995\pi$
$283$	132.139	0.466922	0.233461	+	0.972366i	$0.424995\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	27.4789i	0.0957452i
$288$	0	0
$289$	79.3023	0.274402
$290$	0	0
$291$	859.684	2.95424
$292$	0	0
$293$	− 131.914i	− 0.450218i	−0.974334	−	0.225109i	$-0.927726\pi$
$293$	− 131.914i	− 0.450218i	0.974334	−	0.225109i	$-0.0722739\pi$
$294$	0	0
$295$	− 638.618i	− 2.16481i
$296$	0	0
$297$	− 1526.50i	− 5.13973i
$298$	0	0
$299$	10.1465i	0.0339347i
$300$	0	0
$301$	59.0291	0.196110
$302$	0	0
$303$	765.345i	2.52589i
$304$	0	0
$305$	−574.979	−1.88518
$306$	0	0
$307$	− 135.470i	− 0.441271i	−0.975356	−	0.220635i	$-0.929187\pi$
$307$	− 135.470i	− 0.441271i	0.975356	−	0.220635i	$-0.0708131\pi$
$308$	0	0
$309$	788.322	2.55120
$310$	0	0
$311$	−186.467	−0.599571	−0.299786	−	0.954007i	$-0.596915\pi$
$311$	−186.467	−0.599571	−0.299786	+	0.954007i	$0.596915\pi$
$312$	0	0
$313$	−475.815	−1.52017	−0.760087	−	0.649821i	$-0.774844\pi$
$313$	−475.815	−1.52017	−0.760087	+	0.649821i	$0.774844\pi$
$314$	0	0
$315$	−453.447	−1.43952
$316$	0	0
$317$	95.1263i	0.300083i	0.988680	+	0.150042i	$0.0479407\pi$
$317$	95.1263i	0.300083i	−0.988680	+	0.150042i	$0.952059\pi$
$318$	0	0
$319$	− 139.716i	− 0.437981i
$320$	0	0
$321$	−172.748	−0.538154
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 277.917i	− 0.855128i
$326$	0	0
$327$	−1205.67	−3.68706
$328$	0	0
$329$	148.418	0.451119
$330$	0	0
$331$	153.982i	0.465203i	0.972572	+	0.232602i	$0.0747238\pi$
$331$	153.982i	0.465203i	−0.972572	+	0.232602i	$0.925276\pi$
$332$	0	0
$333$	39.5359i	0.118726i
$334$	0	0
$335$	82.8758i	0.247391i
$336$	0	0
$337$	307.465i	0.912360i	0.889887	+	0.456180i	$0.150783\pi$
$337$	307.465i	0.912360i	−0.889887	+	0.456180i	$0.849217\pi$
$338$	0	0
$339$	−744.421	−2.19593
$340$	0	0
$341$	− 310.074i	− 0.909307i
$342$	0	0
$343$	−249.370	−0.727026
$344$	0	0
$345$	30.9992i	0.0898527i
$346$	0	0
$347$	−91.8160	−0.264599	−0.132300	−	0.991210i	$-0.542236\pi$
$347$	−91.8160	−0.264599	−0.132300	+	0.991210i	$0.542236\pi$
$348$	0	0
$349$	385.100	1.10344	0.551720	−	0.834029i	$-0.313972\pi$
$349$	385.100	1.10344	0.551720	+	0.834029i	$0.313972\pi$
$350$	0	0
$351$	1113.93	3.17359
$352$	0	0
$353$	58.5192	0.165777	0.0828884	−	0.996559i	$-0.473586\pi$
$353$	58.5192	0.165777	0.0828884	+	0.996559i	$0.473586\pi$
$354$	0	0
$355$	142.572i	0.401612i
$356$	0	0
$357$	304.478i	0.852880i
$358$	0	0
$359$	−531.008	−1.47913	−0.739565	−	0.673085i	$-0.764969\pi$
$359$	−531.008	−1.47913	−0.739565	+	0.673085i	$0.764969\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	− 1088.11i	− 2.99755i
$364$	0	0
$365$	−631.807	−1.73098
$366$	0	0
$367$	−449.087	−1.22367	−0.611835	−	0.790986i	$-0.709568\pi$
$367$	−449.087	−1.22367	−0.611835	+	0.790986i	$0.709568\pi$
$368$	0	0
$369$	− 239.753i	− 0.649736i
$370$	0	0
$371$	− 116.762i	− 0.314721i
$372$	0	0
$373$	− 19.5131i	− 0.0523140i	−0.999658	−	0.0261570i	$-0.991673\pi$
$373$	− 19.5131i	− 0.0523140i	0.999658	−	0.0261570i	$-0.00832698\pi$
$374$	0	0
$375$	132.589i	0.353571i
$376$	0	0
$377$	101.955	0.270437
$378$	0	0
$379$	− 386.321i	− 1.01932i	−0.860377	−	0.509658i	$-0.829772\pi$
$379$	− 386.321i	− 1.01932i	0.860377	−	0.509658i	$-0.170228\pi$
$380$	0	0
$381$	−1044.05	−2.74030
$382$	0	0
$383$	− 267.488i	− 0.698401i	−0.937048	−	0.349201i	$-0.886453\pi$
$383$	− 267.488i	− 0.698401i	0.937048	−	0.349201i	$-0.113547\pi$
$384$	0	0
$385$	−331.789	−0.861789
$386$	0	0
$387$	−515.027	−1.33082
$388$	0	0
$389$	116.526	0.299554	0.149777	−	0.988720i	$-0.452144\pi$
$389$	116.526	0.299554	0.149777	+	0.988720i	$0.452144\pi$
$390$	0	0
$391$	15.1505	0.0387480
$392$	0	0
$393$	− 509.492i	− 1.29642i
$394$	0	0
$395$	527.701i	1.33595i
$396$	0	0
$397$	671.719	1.69199	0.845993	−	0.533194i	$-0.179008\pi$
$397$	671.719	1.69199	0.845993	+	0.533194i	$0.179008\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 364.432i	− 0.908809i	−0.890795	−	0.454404i	$-0.849852\pi$
$401$	− 364.432i	− 0.908809i	0.890795	−	0.454404i	$-0.150148\pi$
$402$	0	0
$403$	226.270	0.561464
$404$	0	0
$405$	1924.00	4.75061
$406$	0	0
$407$	28.9285i	0.0710775i
$408$	0	0
$409$	589.056i	1.44023i	0.693852	+	0.720117i	$0.255912\pi$
$409$	589.056i	1.44023i	−0.693852	+	0.720117i	$0.744088\pi$
$410$	0	0
$411$	565.080i	1.37489i
$412$	0	0
$413$	− 258.030i	− 0.624770i
$414$	0	0
$415$	241.492	0.581909
$416$	0	0
$417$	1134.13i	2.71973i
$418$	0	0
$419$	409.167	0.976532	0.488266	−	0.872695i	$-0.337630\pi$
$419$	409.167	0.976532	0.488266	+	0.872695i	$0.337630\pi$
$420$	0	0
$421$	278.311i	0.661072i	0.943793	+	0.330536i	$0.107230\pi$
$421$	278.311i	0.661072i	−0.943793	+	0.330536i	$0.892770\pi$
$422$	0	0
$423$	−1294.95	−3.06134
$424$	0	0
$425$	−414.979	−0.976420
$426$	0	0
$427$	−232.317	−0.544067
$428$	0	0
$429$	1301.80	3.03450
$430$	0	0
$431$	− 52.1950i	− 0.121102i	−0.998165	−	0.0605510i	$-0.980714\pi$
$431$	− 52.1950i	− 0.121102i	0.998165	−	0.0605510i	$-0.0192858\pi$
$432$	0	0
$433$	527.042i	1.21719i	0.793482	+	0.608594i	$0.208266\pi$
$433$	527.042i	1.21719i	−0.793482	+	0.608594i	$0.791734\pi$
$434$	0	0
$435$	311.489	0.716066
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 846.293i	− 1.92777i	−0.266311	−	0.963887i	$-0.585805\pi$
$439$	− 846.293i	− 1.92777i	0.266311	−	0.963887i	$-0.414195\pi$
$440$	0	0
$441$	996.268	2.25911
$442$	0	0
$443$	−299.040	−0.675034	−0.337517	−	0.941319i	$-0.609587\pi$
$443$	−299.040	−0.675034	−0.337517	+	0.941319i	$0.609587\pi$
$444$	0	0
$445$	16.2417i	0.0364981i
$446$	0	0
$447$	− 361.685i	− 0.809139i
$448$	0	0
$449$	134.257i	0.299014i	0.988761	+	0.149507i	$0.0477686\pi$
$449$	134.257i	0.299014i	−0.988761	+	0.149507i	$0.952231\pi$
$450$	0	0
$451$	− 175.428i	− 0.388975i
$452$	0	0
$453$	−51.2422	−0.113117
$454$	0	0
$455$	− 242.116i	− 0.532123i
$456$	0	0
$457$	−310.728	−0.679931	−0.339965	−	0.940438i	$-0.610415\pi$
$457$	−310.728	−0.679931	−0.339965	+	0.940438i	$0.610415\pi$
$458$	0	0
$459$	− 1663.29i	− 3.62373i
$460$	0	0
$461$	114.926	0.249296	0.124648	−	0.992201i	$-0.460220\pi$
$461$	114.926	0.249296	0.124648	+	0.992201i	$0.460220\pi$
$462$	0	0
$463$	365.036	0.788415	0.394208	−	0.919021i	$-0.371019\pi$
$463$	365.036	0.788415	0.394208	+	0.919021i	$0.371019\pi$
$464$	0	0
$465$	691.292	1.48665
$466$	0	0
$467$	194.349	0.416164	0.208082	−	0.978111i	$-0.433278\pi$
$467$	194.349	0.416164	0.208082	+	0.978111i	$0.433278\pi$
$468$	0	0
$469$	33.4855i	0.0713976i
$470$	0	0
$471$	− 207.579i	− 0.440720i
$472$	0	0
$473$	−376.847	−0.796717
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1018.74i	2.13573i
$478$	0	0
$479$	−461.286	−0.963019	−0.481510	−	0.876441i	$-0.659911\pi$
$479$	−461.286	−0.963019	−0.481510	+	0.876441i	$0.659911\pi$
$480$	0	0
$481$	−21.1100	−0.0438877
$482$	0	0
$483$	12.5250i	0.0259317i
$484$	0	0
$485$	1020.74i	2.10463i
$486$	0	0
$487$	− 309.851i	− 0.636245i	−0.948050	−	0.318123i	$-0.896948\pi$
$487$	− 309.851i	− 0.636245i	0.948050	−	0.318123i	$-0.103052\pi$
$488$	0	0
$489$	− 593.804i	− 1.21432i
$490$	0	0
$491$	568.890	1.15864	0.579318	−	0.815102i	$-0.303319\pi$
$491$	568.890	1.15864	0.579318	+	0.815102i	$0.303319\pi$
$492$	0	0
$493$	− 152.236i	− 0.308796i
$494$	0	0
$495$	2894.85	5.84818
$496$	0	0
$497$	57.6054i	0.115906i
$498$	0	0
$499$	−428.122	−0.857960	−0.428980	−	0.903314i	$-0.641127\pi$
$499$	−428.122	−0.857960	−0.428980	+	0.903314i	$0.641127\pi$
$500$	0	0
$501$	1805.67	3.60413
$502$	0	0
$503$	−937.256	−1.86333	−0.931666	−	0.363316i	$-0.881645\pi$
$503$	−937.256	−1.86333	−0.931666	+	0.363316i	$0.881645\pi$
$504$	0	0
$505$	−908.731	−1.79947
$506$	0	0
$507$	− 21.9136i	− 0.0432220i
$508$	0	0
$509$	− 770.738i	− 1.51422i	−0.653288	−	0.757110i	$-0.726611\pi$
$509$	− 770.738i	− 1.51422i	0.653288	−	0.757110i	$-0.273389\pi$
$510$	0	0
$511$	−255.278	−0.499565
$512$	0	0
$513$	0	0
$514$	0	0
$515$	936.013i	1.81750i
$516$	0	0
$517$	−947.516	−1.83272
$518$	0	0
$519$	−113.407	−0.218510
$520$	0	0
$521$	− 815.494i	− 1.56525i	−0.622495	−	0.782624i	$-0.713881\pi$
$521$	− 815.494i	− 1.56525i	0.622495	−	0.782624i	$-0.286119\pi$
$522$	0	0
$523$	− 331.709i	− 0.634244i	−0.948385	−	0.317122i	$-0.897284\pi$
$523$	− 331.709i	− 0.634244i	0.948385	−	0.317122i	$-0.102716\pi$
$524$	0	0
$525$	− 343.066i	− 0.653459i
$526$	0	0
$527$	− 337.861i	− 0.641102i
$528$	0	0
$529$	−528.377	−0.998822
$530$	0	0
$531$	2251.30i	4.23975i
$532$	0	0
$533$	128.015	0.240178
$534$	0	0
$535$	− 205.111i	− 0.383386i
$536$	0	0
$537$	−1163.08	−2.16589
$538$	0	0
$539$	728.972	1.35245
$540$	0	0
$541$	33.7990	0.0624750	0.0312375	−	0.999512i	$-0.490055\pi$
$541$	33.7990	0.0624750	0.0312375	+	0.999512i	$0.490055\pi$
$542$	0	0
$543$	−739.358	−1.36162
$544$	0	0
$545$	− 1431.55i	− 2.62669i
$546$	0	0
$547$	65.2038i	0.119203i	0.998222	+	0.0596013i	$0.0189829\pi$
$547$	65.2038i	0.119203i	−0.998222	+	0.0596013i	$0.981017\pi$
$548$	0	0
$549$	2026.96	3.69209
$550$	0	0
$551$	0	0
$552$	0	0
$553$	213.214i	0.385559i
$554$	0	0
$555$	−64.4945	−0.116206
$556$	0	0
$557$	−575.655	−1.03349	−0.516746	−	0.856139i	$-0.672857\pi$
$557$	−575.655	−1.03349	−0.516746	+	0.856139i	$0.672857\pi$
$558$	0	0
$559$	− 274.996i	− 0.491943i
$560$	0	0
$561$	− 1943.82i	− 3.46492i
$562$	0	0
$563$	− 177.893i	− 0.315973i	−0.987441	−	0.157987i	$-0.949500\pi$
$563$	− 177.893i	− 0.315973i	0.987441	−	0.157987i	$-0.0505003\pi$
$564$	0	0
$565$	− 883.887i	− 1.56440i
$566$	0	0
$567$	777.380	1.37104
$568$	0	0
$569$	− 680.533i	− 1.19602i	−0.801490	−	0.598008i	$-0.795959\pi$
$569$	− 680.533i	− 1.19602i	0.801490	−	0.598008i	$-0.204041\pi$
$570$	0	0
$571$	447.590	0.783870	0.391935	−	0.919993i	$-0.371806\pi$
$571$	447.590	0.783870	0.391935	+	0.919993i	$0.371806\pi$
$572$	0	0
$573$	85.3522i	0.148957i
$574$	0	0
$575$	−17.0706	−0.0296879
$576$	0	0
$577$	308.500	0.534662	0.267331	−	0.963605i	$-0.413858\pi$
$577$	308.500	0.534662	0.267331	+	0.963605i	$0.413858\pi$
$578$	0	0
$579$	−878.675	−1.51757
$580$	0	0
$581$	97.5735	0.167941
$582$	0	0
$583$	745.417i	1.27859i
$584$	0	0
$585$	2112.46i	3.61104i
$586$	0	0
$587$	−158.149	−0.269418	−0.134709	−	0.990885i	$-0.543010\pi$
$587$	−158.149	−0.269418	−0.134709	+	0.990885i	$0.543010\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	988.613i	1.67278i
$592$	0	0
$593$	839.401	1.41552	0.707758	−	0.706455i	$-0.249707\pi$
$593$	839.401	1.41552	0.707758	+	0.706455i	$0.249707\pi$
$594$	0	0
$595$	−361.522	−0.607600
$596$	0	0
$597$	1626.34i	2.72418i
$598$	0	0
$599$	397.756i	0.664034i	0.943273	+	0.332017i	$0.107729\pi$
$599$	397.756i	0.664034i	−0.943273	+	0.332017i	$0.892271\pi$
$600$	0	0
$601$	999.928i	1.66377i	0.554946	+	0.831887i	$0.312739\pi$
$601$	999.928i	1.66377i	−0.554946	+	0.831887i	$0.687261\pi$
$602$	0	0
$603$	− 292.160i	− 0.484511i
$604$	0	0
$605$	1291.97	2.13548
$606$	0	0
$607$	− 390.793i	− 0.643810i	−0.946772	−	0.321905i	$-0.895677\pi$
$607$	− 390.793i	− 0.643810i	0.946772	−	0.321905i	$-0.104323\pi$
$608$	0	0
$609$	125.855	0.206658
$610$	0	0
$611$	− 691.430i	− 1.13164i
$612$	0	0
$613$	−644.013	−1.05059	−0.525296	−	0.850919i	$-0.676045\pi$
$613$	−644.013	−1.05059	−0.525296	+	0.850919i	$0.676045\pi$
$614$	0	0
$615$	391.106	0.635944
$616$	0	0
$617$	933.427	1.51285	0.756424	−	0.654082i	$-0.226945\pi$
$617$	933.427	1.51285	0.756424	+	0.654082i	$0.226945\pi$
$618$	0	0
$619$	790.626	1.27726	0.638632	−	0.769513i	$-0.279501\pi$
$619$	790.626	1.27726	0.638632	+	0.769513i	$0.279501\pi$
$620$	0	0
$621$	− 68.4213i	− 0.110179i
$622$	0	0
$623$	6.56234i	0.0105334i
$624$	0	0
$625$	−698.014	−1.11682
$626$	0	0
$627$	0	0
$628$	0	0
$629$	31.5209i	0.0501128i
$630$	0	0
$631$	−270.291	−0.428353	−0.214177	−	0.976795i	$-0.568707\pi$
$631$	−270.291	−0.428353	−0.214177	+	0.976795i	$0.568707\pi$
$632$	0	0
$633$	−290.363	−0.458709
$634$	0	0
$635$	− 1239.66i	− 1.95221i
$636$	0	0
$637$	531.952i	0.835090i
$638$	0	0
$639$	− 502.606i	− 0.786551i
$640$	0	0
$641$	697.561i	1.08824i	0.839008	+	0.544119i	$0.183136\pi$
$641$	697.561i	1.08824i	−0.839008	+	0.544119i	$0.816864\pi$
$642$	0	0
$643$	−477.897	−0.743230	−0.371615	−	0.928387i	$-0.621196\pi$
$643$	−477.897	−0.743230	−0.371615	+	0.928387i	$0.621196\pi$
$644$	0	0
$645$	− 840.159i	− 1.30257i
$646$	0	0
$647$	−468.082	−0.723465	−0.361732	−	0.932282i	$-0.617815\pi$
$647$	−468.082	−0.723465	−0.361732	+	0.932282i	$0.617815\pi$
$648$	0	0
$649$	1647.29i	2.53819i
$650$	0	0
$651$	279.312	0.429051
$652$	0	0
$653$	142.918	0.218864	0.109432	−	0.993994i	$-0.465097\pi$
$653$	142.918	0.218864	0.109432	+	0.993994i	$0.465097\pi$
$654$	0	0
$655$	604.945	0.923580
$656$	0	0
$657$	2227.29	3.39009
$658$	0	0
$659$	− 290.460i	− 0.440759i	−0.975414	−	0.220380i	$-0.929270\pi$
$659$	− 290.460i	− 0.440759i	0.975414	−	0.220380i	$-0.0707296\pi$
$660$	0	0
$661$	599.707i	0.907272i	0.891187	+	0.453636i	$0.149873\pi$
$661$	599.707i	0.907272i	−0.891187	+	0.453636i	$0.850127\pi$
$662$	0	0
$663$	1418.46	2.13946
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6.26240i	− 0.00938890i
$668$	0	0
$669$	682.465	1.02013
$670$	0	0
$671$	1483.13	2.21033
$672$	0	0
$673$	− 591.892i	− 0.879484i	−0.898124	−	0.439742i	$-0.855070\pi$
$673$	− 591.892i	− 0.879484i	0.898124	−	0.439742i	$-0.144930\pi$
$674$	0	0
$675$	1874.09i	2.77643i
$676$	0	0
$677$	− 401.259i	− 0.592701i	−0.955079	−	0.296350i	$-0.904230\pi$
$677$	− 401.259i	− 0.592701i	0.955079	−	0.296350i	$-0.0957696\pi$
$678$	0	0
$679$	412.426i	0.607401i
$680$	0	0
$681$	−1740.90	−2.55638
$682$	0	0
$683$	205.532i	0.300925i	0.988616	+	0.150463i	$0.0480763\pi$
$683$	205.532i	0.300925i	−0.988616	+	0.150463i	$0.951924\pi$
$684$	0	0
$685$	−670.946	−0.979484
$686$	0	0
$687$	258.505i	0.376280i
$688$	0	0
$689$	−543.953	−0.789481
$690$	0	0
$691$	−543.541	−0.786601	−0.393300	−	0.919410i	$-0.628667\pi$
$691$	−543.541	−0.786601	−0.393300	+	0.919410i	$0.628667\pi$
$692$	0	0
$693$	1169.65	1.68780
$694$	0	0
$695$	−1346.60	−1.93756
$696$	0	0
$697$	− 191.148i	− 0.274245i
$698$	0	0
$699$	− 587.655i	− 0.840709i
$700$	0	0
$701$	−883.595	−1.26048	−0.630239	−	0.776401i	$-0.717043\pi$
$701$	−883.595	−1.26048	−0.630239	+	0.776401i	$0.717043\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 2112.43i	− 2.99636i
$706$	0	0
$707$	−367.167	−0.519331
$708$	0	0
$709$	196.040	0.276502	0.138251	−	0.990397i	$-0.455852\pi$
$709$	196.040	0.276502	0.138251	+	0.990397i	$0.455852\pi$
$710$	0	0
$711$	− 1860.29i	− 2.61644i
$712$	0	0
$713$	− 13.8982i	− 0.0194926i
$714$	0	0
$715$	1545.69i	2.16181i
$716$	0	0
$717$	− 1228.55i	− 1.71346i
$718$	0	0
$719$	448.293	0.623495	0.311747	−	0.950165i	$-0.399086\pi$
$719$	448.293	0.623495	0.311747	+	0.950165i	$0.399086\pi$
$720$	0	0
$721$	378.190i	0.524536i
$722$	0	0
$723$	−1779.48	−2.46125
$724$	0	0
$725$	171.530i	0.236593i
$726$	0	0
$727$	0.338250	0.000465269 0	0.000232634	−	1.00000i	$-0.499926\pi$
$727$	0.338250	0.000465269 0	0.000232634		1.00000i	$0.499926\pi$
$728$	0	0
$729$	−2296.89	−3.15074
$730$	0	0
$731$	−410.618	−0.561721
$732$	0	0
$733$	1370.46	1.86966	0.934831	−	0.355092i	$-0.115551\pi$
$733$	1370.46	1.86966	0.934831	+	0.355092i	$0.115551\pi$
$734$	0	0
$735$	1625.20i	2.21116i
$736$	0	0
$737$	− 213.774i	− 0.290060i
$738$	0	0
$739$	−679.036	−0.918858	−0.459429	−	0.888214i	$-0.651946\pi$
$739$	−679.036	−0.918858	−0.459429	+	0.888214i	$0.651946\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 290.098i	− 0.390442i	−0.980759	−	0.195221i	$-0.937458\pi$
$743$	− 290.098i	− 0.390442i	0.980759	−	0.195221i	$-0.0625424\pi$
$744$	0	0
$745$	429.446	0.576437
$746$	0	0
$747$	−851.327	−1.13966
$748$	0	0
$749$	− 82.8740i	− 0.110646i
$750$	0	0
$751$	− 826.661i	− 1.10075i	−0.834918	−	0.550374i	$-0.814485\pi$
$751$	− 826.661i	− 1.10075i	0.834918	−	0.550374i	$-0.185515\pi$
$752$	0	0
$753$	− 1324.73i	− 1.75927i
$754$	0	0
$755$	− 60.8423i	− 0.0805858i
$756$	0	0
$757$	−822.012	−1.08588	−0.542941	−	0.839771i	$-0.682689\pi$
$757$	−822.012	−1.08588	−0.542941	+	0.839771i	$0.682689\pi$
$758$	0	0
$759$	− 79.9609i	− 0.105350i
$760$	0	0
$761$	1476.85	1.94067	0.970334	−	0.241767i	$-0.0777270\pi$
$761$	1476.85	1.94067	0.970334	+	0.241767i	$0.0777270\pi$
$762$	0	0
$763$	− 578.408i	− 0.758071i
$764$	0	0
$765$	3154.27	4.12323
$766$	0	0
$767$	−1202.07	−1.56724
$768$	0	0
$769$	470.359	0.611651	0.305825	−	0.952088i	$-0.401068\pi$
$769$	470.359	0.611651	0.305825	+	0.952088i	$0.401068\pi$
$770$	0	0
$771$	−1625.73	−2.10860
$772$	0	0
$773$	569.944i	0.737315i	0.929565	+	0.368657i	$0.120182\pi$
$773$	569.944i	0.737315i	−0.929565	+	0.368657i	$0.879818\pi$
$774$	0	0
$775$	380.679i	0.491199i
$776$	0	0
$777$	−26.0586	−0.0335375
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 367.758i	− 0.470881i
$782$	0	0
$783$	−687.516	−0.878054
$784$	0	0
$785$	246.469	0.313973
$786$	0	0
$787$	1090.30i	1.38539i	0.721232	+	0.692694i	$0.243576\pi$
$787$	1090.30i	1.38539i	−0.721232	+	0.692694i	$0.756424\pi$
$788$	0	0
$789$	− 2107.67i	− 2.67132i
$790$	0	0
$791$	− 357.129i	− 0.451491i
$792$	0	0
$793$	1082.28i	1.36480i
$794$	0	0
$795$	−1661.86	−2.09040
$796$	0	0
$797$	− 1040.58i	− 1.30562i	−0.757523	−	0.652809i	$-0.773591\pi$
$797$	− 1040.58i	− 1.30562i	0.757523	−	0.652809i	$-0.226409\pi$
$798$	0	0
$799$	−1032.43	−1.29215
$800$	0	0
$801$	− 57.2563i	− 0.0714810i
$802$	0	0
$803$	1629.72	2.02953
$804$	0	0
$805$	−14.8716	−0.0184740
$806$	0	0
$807$	−2487.49	−3.08239
$808$	0	0
$809$	−855.949	−1.05803	−0.529017	−	0.848611i	$-0.677439\pi$
$809$	−855.949	−1.05803	−0.529017	+	0.848611i	$0.677439\pi$
$810$	0	0
$811$	394.372i	0.486279i	0.969991	+	0.243139i	$0.0781772\pi$
$811$	394.372i	0.486279i	−0.969991	+	0.243139i	$0.921823\pi$
$812$	0	0
$813$	− 2278.73i	− 2.80286i
$814$	0	0
$815$	705.052	0.865094
$816$	0	0
$817$	0	0
$818$	0	0
$819$	853.525i	1.04215i
$820$	0	0
$821$	205.319	0.250084	0.125042	−	0.992151i	$-0.460093\pi$
$821$	205.319	0.250084	0.125042	+	0.992151i	$0.460093\pi$
$822$	0	0
$823$	−495.057	−0.601528	−0.300764	−	0.953699i	$-0.597242\pi$
$823$	−495.057	−0.601528	−0.300764	+	0.953699i	$0.597242\pi$
$824$	0	0
$825$	2190.17i	2.65475i
$826$	0	0
$827$	1243.35i	1.50344i	0.659482	+	0.751721i	$0.270776\pi$
$827$	1243.35i	1.50344i	−0.659482	+	0.751721i	$0.729224\pi$
$828$	0	0
$829$	− 122.213i	− 0.147423i	−0.997280	−	0.0737114i	$-0.976516\pi$
$829$	− 122.213i	− 0.147423i	0.997280	−	0.0737114i	$-0.0234844\pi$
$830$	0	0
$831$	3071.65i	3.69633i
$832$	0	0
$833$	794.298	0.953539
$834$	0	0
$835$	2143.96i	2.56762i
$836$	0	0
$837$	−1525.82	−1.82296
$838$	0	0
$839$	12.3002i	0.0146606i	0.999973	+	0.00733030i	$0.00233333\pi$
$839$	12.3002i	0.0146606i	−0.999973	+	0.00733030i	$0.997667\pi$
$840$	0	0
$841$	778.074	0.925177
$842$	0	0
$843$	1621.76	1.92380
$844$	0	0
$845$	26.0190	0.0307917
$846$	0	0
$847$	522.011	0.616306
$848$	0	0
$849$	− 759.897i	− 0.895049i
$850$	0	0
$851$	1.29665i	0.00152367i
$852$	0	0
$853$	143.963	0.168773	0.0843864	−	0.996433i	$-0.473107\pi$
$853$	143.963	0.168773	0.0843864	+	0.996433i	$0.473107\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	384.698i	0.448889i	0.974487	+	0.224445i	$0.0720568\pi$
$857$	384.698i	0.448889i	−0.974487	+	0.224445i	$0.927943\pi$
$858$	0	0
$859$	−946.146	−1.10145	−0.550725	−	0.834686i	$-0.685649\pi$
$859$	−946.146	−1.10145	−0.550725	+	0.834686i	$0.685649\pi$
$860$	0	0
$861$	158.024	0.183535
$862$	0	0
$863$	1148.72i	1.33108i	0.746363	+	0.665539i	$0.231798\pi$
$863$	1148.72i	1.33108i	−0.746363	+	0.665539i	$0.768202\pi$
$864$	0	0
$865$	− 134.653i	− 0.155668i
$866$	0	0
$867$	− 456.047i	− 0.526006i
$868$	0	0
$869$	− 1361.18i	− 1.56638i
$870$	0	0
$871$	155.997	0.179101
$872$	0	0
$873$	− 3598.40i	− 4.12188i
$874$	0	0
$875$	−63.6084	−0.0726953
$876$	0	0
$877$	341.612i	0.389523i	0.980851	+	0.194762i	$0.0623933\pi$
$877$	341.612i	0.389523i	−0.980851	+	0.194762i	$0.937607\pi$
$878$	0	0
$879$	−758.603	−0.863030
$880$	0	0
$881$	299.341	0.339774	0.169887	−	0.985464i	$-0.445660\pi$
$881$	299.341	0.339774	0.169887	+	0.985464i	$0.445660\pi$
$882$	0	0
$883$	652.623	0.739097	0.369549	−	0.929211i	$-0.379512\pi$
$883$	652.623	0.739097	0.369549	+	0.929211i	$0.379512\pi$
$884$	0	0
$885$	−3672.53	−4.14975
$886$	0	0
$887$	− 1571.53i	− 1.77174i	−0.463932	−	0.885871i	$-0.653562\pi$
$887$	− 1571.53i	− 1.77174i	0.463932	−	0.885871i	$-0.346438\pi$
$888$	0	0
$889$	− 500.875i	− 0.563414i
$890$	0	0
$891$	−4962.86	−5.56999
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 1380.98i	− 1.54300i
$896$	0	0
$897$	58.3498	0.0650499
$898$	0	0
$899$	−139.654	−0.155343
$900$	0	0
$901$	812.217i	0.901462i
$902$	0	0
$903$	− 339.461i	− 0.375926i
$904$	0	0
$905$	− 877.876i	− 0.970028i
$906$	0	0
$907$	− 205.632i	− 0.226717i	−0.993554	−	0.113358i	$-0.963839\pi$
$907$	− 205.632i	− 0.226717i	0.993554	−	0.113358i	$-0.0361608\pi$
$908$	0	0
$909$	3203.53	3.52423
$910$	0	0
$911$	− 1109.37i	− 1.21775i	−0.793264	−	0.608877i	$-0.791620\pi$
$911$	− 1109.37i	− 1.21775i	0.793264	−	0.608877i	$-0.208380\pi$
$912$	0	0
$913$	−622.918	−0.682276
$914$	0	0
$915$	3306.56i	3.61372i
$916$	0	0
$917$	244.424	0.266548
$918$	0	0
$919$	−1218.25	−1.32562	−0.662811	−	0.748787i	$-0.730637\pi$
$919$	−1218.25	−1.32562	−0.662811	+	0.748787i	$0.730637\pi$
$920$	0	0
$921$	−779.054	−0.845878
$922$	0	0
$923$	268.364	0.290752
$924$	0	0
$925$	− 35.5157i	− 0.0383954i
$926$	0	0
$927$	− 3299.70i	− 3.55955i
$928$	0	0
$929$	−733.241	−0.789280	−0.394640	−	0.918836i	$-0.629131\pi$
$929$	−733.241	−0.789280	−0.394640	+	0.918836i	$0.629131\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1072.32i	1.14933i
$934$	0	0
$935$	2307.99	2.46844
$936$	0	0
$937$	−469.706	−0.501287	−0.250643	−	0.968079i	$-0.580642\pi$
$937$	−469.706	−0.501287	−0.250643	+	0.968079i	$0.580642\pi$
$938$	0	0
$939$	2736.29i	2.91404i
$940$	0	0
$941$	− 539.224i	− 0.573033i	−0.958075	−	0.286516i	$-0.907503\pi$
$941$	− 539.224i	− 0.573033i	0.958075	−	0.286516i	$-0.0924973\pi$
$942$	0	0
$943$	− 7.86308i	− 0.00833837i
$944$	0	0
$945$	1632.67i	1.72770i
$946$	0	0
$947$	757.758	0.800167	0.400083	−	0.916479i	$-0.368981\pi$
$947$	757.758	0.800167	0.400083	+	0.916479i	$0.368981\pi$
$948$	0	0
$949$	1189.25i	1.25316i
$950$	0	0
$951$	547.047	0.575233
$952$	0	0
$953$	− 1663.75i	− 1.74581i	−0.487893	−	0.872904i	$-0.662234\pi$
$953$	− 1663.75i	− 1.74581i	0.487893	−	0.872904i	$-0.337766\pi$
$954$	0	0
$955$	−101.343	−0.106118
$956$	0	0
$957$	−803.470	−0.839572
$958$	0	0
$959$	−271.092	−0.282682
$960$	0	0
$961$	651.065	0.677487
$962$	0	0
$963$	723.074i	0.750855i
$964$	0	0
$965$	− 1043.29i	− 1.08113i
$966$	0	0
$967$	1859.03	1.92247	0.961234	−	0.275736i	$-0.0889214\pi$
$967$	1859.03	1.92247	0.961234	+	0.275736i	$0.0889214\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1467.24i	1.51106i	0.655112	+	0.755532i	$0.272622\pi$
$971$	1467.24i	1.51106i	−0.655112	+	0.755532i	$0.727378\pi$
$972$	0	0
$973$	−544.086	−0.559184
$974$	0	0
$975$	−1598.23	−1.63921
$976$	0	0
$977$	− 556.493i	− 0.569594i	−0.958588	−	0.284797i	$-0.908074\pi$
$977$	− 556.493i	− 0.569594i	0.958588	−	0.284797i	$-0.0919261\pi$
$978$	0	0
$979$	− 41.8946i	− 0.0427932i
$980$	0	0
$981$	5046.60i	5.14434i
$982$	0	0
$983$	1023.28i	1.04098i	0.853868	+	0.520490i	$0.174251\pi$
$983$	1023.28i	1.04098i	−0.853868	+	0.520490i	$0.825749\pi$
$984$	0	0
$985$	−1173.83	−1.19170
$986$	0	0
$987$	− 853.515i	− 0.864757i
$988$	0	0
$989$	−16.8912	−0.0170790
$990$	0	0
$991$	− 1227.70i	− 1.23885i	−0.785054	−	0.619427i	$-0.787365\pi$
$991$	− 1227.70i	− 1.23885i	0.785054	−	0.619427i	$-0.212635\pi$
$992$	0	0
$993$	885.512	0.891754
$994$	0	0
$995$	−1931.03	−1.94073
$996$	0	0
$997$	−1309.55	−1.31349	−0.656745	−	0.754113i	$-0.728067\pi$
$997$	−1309.55	−1.31349	−0.656745	+	0.754113i	$0.728067\pi$
$998$	0	0
$999$	142.352	0.142495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.3.c.d.721.1	✓	24
19.18	odd	2	inner	1444.3.c.d.721.24	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1444.3.c.d.721.1	✓	24	1.1	even	1	trivial
1444.3.c.d.721.24	yes	24	19.18	odd	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:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1444.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.75074i q^{3} +6.82813 q^{5} +2.75886 q^{7} -24.0710 q^{9} +O(q^{10})\)	\(q-5.75074i q^{3} +6.82813 q^{5} +2.75886 q^{7} -24.0710 q^{9} -17.6128 q^{11} -12.8526i q^{13} -39.2668i q^{15} -19.1912 q^{17} -15.8655i q^{21} -0.789449 q^{23} +21.6234 q^{25} +86.6696i q^{27} +7.93261i q^{29} +17.6050i q^{31} +101.287i q^{33} +18.8379 q^{35} -1.64247i q^{37} -73.9120 q^{39} +9.96021i q^{41} +21.3961 q^{43} -164.360 q^{45} +53.7969 q^{47} -41.3887 q^{49} +110.364i q^{51} -42.3224i q^{53} -120.263 q^{55} -93.5276i q^{59} -84.2074 q^{61} -66.4087 q^{63} -87.7593i q^{65} +12.1374i q^{67} +4.53992i q^{69} +20.8801i q^{71} -92.5299 q^{73} -124.350i q^{75} -48.5915 q^{77} +77.2834i q^{79} +281.775 q^{81} +35.3673 q^{83} -131.040 q^{85} +45.6184 q^{87} +2.37864i q^{89} -35.4586i q^{91} +101.242 q^{93} +149.491i q^{97} +423.959 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Introduction

L-functions

Modular forms

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Database

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