LMFDB - Embedded newform 1344.4.c.h.673.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 386x^{10} + 54793x^{8} + 3447408x^{6} + 90154296x^{4} + 707138208x^{2} + 525876624 $ x^12 + 386x^10 + 54793x^8 + 3447408x^6 + 90154296x^4 + 707138208*x^2 + 525876624
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.11
Root			$-9.31944i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.h.673.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +8.40898i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +8.40898i q^{5} +7.00000 q^{7} -9.00000 q^{9} +42.3001i q^{11} -9.41890i q^{13} -25.2269 q^{15} +107.603 q^{17} -124.796i q^{19} +21.0000i q^{21} +55.2983 q^{23} +54.2890 q^{25} -27.0000i q^{27} -47.1557i q^{29} +318.539 q^{31} -126.900 q^{33} +58.8629i q^{35} +88.8183i q^{37} +28.2567 q^{39} +105.791 q^{41} -202.714i q^{43} -75.6808i q^{45} +176.014 q^{47} +49.0000 q^{49} +322.810i q^{51} +393.735i q^{53} -355.701 q^{55} +374.388 q^{57} +898.392i q^{59} -904.556i q^{61} -63.0000 q^{63} +79.2034 q^{65} +67.0601i q^{67} +165.895i q^{69} -620.245 q^{71} +481.892 q^{73} +162.867i q^{75} +296.101i q^{77} -375.551 q^{79} +81.0000 q^{81} -976.706i q^{83} +904.835i q^{85} +141.467 q^{87} +1187.44 q^{89} -65.9323i q^{91} +955.617i q^{93} +1049.41 q^{95} +599.694 q^{97} -380.701i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 84 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q + 84 * q^7 - 108 * q^9	$ 12 q + 84 q^{7} - 108 q^{9} + 24 q^{15} + 24 q^{17} - 80 q^{23} - 564 q^{25} + 640 q^{31} - 408 q^{33} - 120 q^{39} + 1416 q^{41} + 1536 q^{47} + 588 q^{49} - 1392 q^{55} - 336 q^{57} - 756 q^{63} - 2880 q^{65} - 1392 q^{71} + 2472 q^{73} + 544 q^{79} + 972 q^{81} - 720 q^{87} + 888 q^{89} - 2368 q^{95} - 2712 q^{97}+O(q^{100}) $ 12 * q + 84 * q^7 - 108 * q^9 + 24 * q^15 + 24 * q^17 - 80 * q^23 - 564 * q^25 + 640 * q^31 - 408 * q^33 - 120 * q^39 + 1416 * q^41 + 1536 * q^47 + 588 * q^49 - 1392 * q^55 - 336 * q^57 - 756 * q^63 - 2880 * q^65 - 1392 * q^71 + 2472 * q^73 + 544 * q^79 + 972 * q^81 - 720 * q^87 + 888 * q^89 - 2368 * q^95 - 2712 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	8.40898i	0.752122i	0.926595	+	0.376061i	$0.122722\pi$
$5$	8.40898i	0.752122i	−0.926595	+	0.376061i	$0.877278\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	42.3001i	1.15945i	0.814811	+	0.579726i	$0.196840\pi$
$11$	42.3001i	1.15945i	−0.814811	+	0.579726i	$0.803160\pi$
$12$	0	0
$13$	− 9.41890i	− 0.200949i	−0.994940	−	0.100474i	$-0.967964\pi$
$13$	− 9.41890i	− 0.200949i	0.994940	−	0.100474i	$-0.0320360\pi$
$14$	0	0
$15$	−25.2269	−0.434238
$16$	0	0
$17$	107.603	1.53516	0.767578	−	0.640956i	$-0.221462\pi$
$17$	107.603	1.53516	0.767578	+	0.640956i	$0.221462\pi$
$18$	0	0
$19$	− 124.796i	− 1.50685i	−0.657534	−	0.753425i	$-0.728400\pi$
$19$	− 124.796i	− 1.50685i	0.657534	−	0.753425i	$-0.271600\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	55.2983	0.501326	0.250663	−	0.968074i	$-0.419351\pi$
$23$	55.2983	0.501326	0.250663	+	0.968074i	$0.419351\pi$
$24$	0	0
$25$	54.2890	0.434312
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 47.1557i	− 0.301951i	−0.988537	−	0.150976i	$-0.951758\pi$
$29$	− 47.1557i	− 0.301951i	0.988537	−	0.150976i	$-0.0482415\pi$
$30$	0	0
$31$	318.539	1.84553	0.922763	−	0.385368i	$-0.125926\pi$
$31$	318.539	1.84553	0.922763	+	0.385368i	$0.125926\pi$
$32$	0	0
$33$	−126.900	−0.669410
$34$	0	0
$35$	58.8629i	0.284275i
$36$	0	0
$37$	88.8183i	0.394639i	0.980339	+	0.197319i	$0.0632236\pi$
$37$	88.8183i	0.394639i	−0.980339	+	0.197319i	$0.936776\pi$
$38$	0	0
$39$	28.2567	0.116018
$40$	0	0
$41$	105.791	0.402969	0.201484	−	0.979492i	$-0.435423\pi$
$41$	105.791	0.402969	0.201484	+	0.979492i	$0.435423\pi$
$42$	0	0
$43$	− 202.714i	− 0.718922i	−0.933160	−	0.359461i	$-0.882961\pi$
$43$	− 202.714i	− 0.718922i	0.933160	−	0.359461i	$-0.117039\pi$
$44$	0	0
$45$	− 75.6808i	− 0.250707i
$46$	0	0
$47$	176.014	0.546261	0.273131	−	0.961977i	$-0.411941\pi$
$47$	176.014	0.546261	0.273131	+	0.961977i	$0.411941\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	322.810i	0.886322i
$52$	0	0
$53$	393.735i	1.02045i	0.860042	+	0.510223i	$0.170437\pi$
$53$	393.735i	1.02045i	−0.860042	+	0.510223i	$0.829563\pi$
$54$	0	0
$55$	−355.701	−0.872050
$56$	0	0
$57$	374.388	0.869980
$58$	0	0
$59$	898.392i	1.98238i	0.132434	+	0.991192i	$0.457721\pi$
$59$	898.392i	1.98238i	−0.132434	+	0.991192i	$0.542279\pi$
$60$	0	0
$61$	− 904.556i	− 1.89863i	−0.314324	−	0.949316i	$-0.601778\pi$
$61$	− 904.556i	− 1.89863i	0.314324	−	0.949316i	$-0.398222\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	79.2034	0.151138
$66$	0	0
$67$	67.0601i	0.122279i	0.998129	+	0.0611395i	$0.0194735\pi$
$67$	67.0601i	0.122279i	−0.998129	+	0.0611395i	$0.980527\pi$
$68$	0	0
$69$	165.895i	0.289441i
$70$	0	0
$71$	−620.245	−1.03675	−0.518377	−	0.855152i	$-0.673464\pi$
$71$	−620.245	−1.03675	−0.518377	+	0.855152i	$0.673464\pi$
$72$	0	0
$73$	481.892	0.772620	0.386310	−	0.922369i	$-0.373750\pi$
$73$	481.892	0.772620	0.386310	+	0.922369i	$0.373750\pi$
$74$	0	0
$75$	162.867i	0.250750i
$76$	0	0
$77$	296.101i	0.438232i
$78$	0	0
$79$	−375.551	−0.534845	−0.267422	−	0.963579i	$-0.586172\pi$
$79$	−375.551	−0.534845	−0.267422	+	0.963579i	$0.586172\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 976.706i	− 1.29166i	−0.763483	−	0.645828i	$-0.776512\pi$
$83$	− 976.706i	− 1.29166i	0.763483	−	0.645828i	$-0.223488\pi$
$84$	0	0
$85$	904.835i	1.15462i
$86$	0	0
$87$	141.467	0.174332
$88$	0	0
$89$	1187.44	1.41426	0.707128	−	0.707086i	$-0.249991\pi$
$89$	1187.44	1.41426	0.707128	+	0.707086i	$0.249991\pi$
$90$	0	0
$91$	− 65.9323i	− 0.0759515i
$92$	0	0
$93$	955.617i	1.06551i
$94$	0	0
$95$	1049.41	1.13334
$96$	0	0
$97$	599.694	0.627729	0.313864	−	0.949468i	$-0.398376\pi$
$97$	599.694	0.627729	0.313864	+	0.949468i	$0.398376\pi$
$98$	0	0
$99$	− 380.701i	− 0.386484i
$100$	0	0
$101$	− 727.366i	− 0.716591i	−0.933608	−	0.358295i	$-0.883358\pi$
$101$	− 727.366i	− 0.716591i	0.933608	−	0.358295i	$-0.116642\pi$
$102$	0	0
$103$	−890.145	−0.851539	−0.425770	−	0.904832i	$-0.639997\pi$
$103$	−890.145	−0.851539	−0.425770	+	0.904832i	$0.639997\pi$
$104$	0	0
$105$	−176.589	−0.164127
$106$	0	0
$107$	− 729.014i	− 0.658658i	−0.944215	−	0.329329i	$-0.893177\pi$
$107$	− 729.014i	− 0.658658i	0.944215	−	0.329329i	$-0.106823\pi$
$108$	0	0
$109$	1123.45i	0.987216i	0.869685	+	0.493608i	$0.164322\pi$
$109$	1123.45i	0.987216i	−0.869685	+	0.493608i	$0.835678\pi$
$110$	0	0
$111$	−266.455	−0.227845
$112$	0	0
$113$	397.884	0.331237	0.165619	−	0.986190i	$-0.447038\pi$
$113$	397.884	0.331237	0.165619	+	0.986190i	$0.447038\pi$
$114$	0	0
$115$	465.003i	0.377058i
$116$	0	0
$117$	84.7701i	0.0669829i
$118$	0	0
$119$	753.223	0.580234
$120$	0	0
$121$	−458.303	−0.344330
$122$	0	0
$123$	317.372i	0.232654i
$124$	0	0
$125$	1507.64i	1.07878i
$126$	0	0
$127$	−2718.70	−1.89957	−0.949784	−	0.312905i	$-0.898698\pi$
$127$	−2718.70	−1.89957	−0.949784	+	0.312905i	$0.898698\pi$
$128$	0	0
$129$	608.143	0.415070
$130$	0	0
$131$	305.953i	0.204055i	0.994782	+	0.102028i	$0.0325330\pi$
$131$	305.953i	0.204055i	−0.994782	+	0.102028i	$0.967467\pi$
$132$	0	0
$133$	− 873.571i	− 0.569536i
$134$	0	0
$135$	227.043	0.144746
$136$	0	0
$137$	422.151	0.263261	0.131631	−	0.991299i	$-0.457979\pi$
$137$	422.151	0.263261	0.131631	+	0.991299i	$0.457979\pi$
$138$	0	0
$139$	1681.97i	1.02635i	0.858284	+	0.513176i	$0.171531\pi$
$139$	1681.97i	1.02635i	−0.858284	+	0.513176i	$0.828469\pi$
$140$	0	0
$141$	528.042i	0.315384i
$142$	0	0
$143$	398.421	0.232990
$144$	0	0
$145$	396.531	0.227104
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	536.114i	0.294766i	0.989079	+	0.147383i	$0.0470850\pi$
$149$	536.114i	0.294766i	−0.989079	+	0.147383i	$0.952915\pi$
$150$	0	0
$151$	148.402	0.0799789	0.0399894	−	0.999200i	$-0.487268\pi$
$151$	148.402	0.0799789	0.0399894	+	0.999200i	$0.487268\pi$
$152$	0	0
$153$	−968.430	−0.511718
$154$	0	0
$155$	2678.59i	1.38806i
$156$	0	0
$157$	− 1115.22i	− 0.566906i	−0.958986	−	0.283453i	$-0.908520\pi$
$157$	− 1115.22i	− 0.566906i	0.958986	−	0.283453i	$-0.0914800\pi$
$158$	0	0
$159$	−1181.20	−0.589154
$160$	0	0
$161$	387.088	0.189483
$162$	0	0
$163$	2519.03i	1.21046i	0.796050	+	0.605231i	$0.206919\pi$
$163$	2519.03i	1.21046i	−0.796050	+	0.605231i	$0.793081\pi$
$164$	0	0
$165$	− 1067.10i	− 0.503478i
$166$	0	0
$167$	−3077.29	−1.42592	−0.712958	−	0.701206i	$-0.752645\pi$
$167$	−3077.29	−1.42592	−0.712958	+	0.701206i	$0.752645\pi$
$168$	0	0
$169$	2108.28	0.959620
$170$	0	0
$171$	1123.16i	0.502283i
$172$	0	0
$173$	4020.80i	1.76703i	0.468406	+	0.883513i	$0.344828\pi$
$173$	4020.80i	1.76703i	−0.468406	+	0.883513i	$0.655172\pi$
$174$	0	0
$175$	380.023	0.164155
$176$	0	0
$177$	−2695.18	−1.14453
$178$	0	0
$179$	1912.07i	0.798408i	0.916862	+	0.399204i	$0.130713\pi$
$179$	1912.07i	0.798408i	−0.916862	+	0.399204i	$0.869287\pi$
$180$	0	0
$181$	1165.84i	0.478765i	0.970925	+	0.239383i	$0.0769450\pi$
$181$	1165.84i	0.478765i	−0.970925	+	0.239383i	$0.923055\pi$
$182$	0	0
$183$	2713.67	1.09618
$184$	0	0
$185$	−746.872	−0.296817
$186$	0	0
$187$	4551.64i	1.77994i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	−4062.94	−1.53918	−0.769592	−	0.638536i	$-0.779540\pi$
$191$	−4062.94	−1.53918	−0.769592	+	0.638536i	$0.779540\pi$
$192$	0	0
$193$	3782.90	1.41088	0.705438	−	0.708772i	$-0.250750\pi$
$193$	3782.90	1.41088	0.705438	+	0.708772i	$0.250750\pi$
$194$	0	0
$195$	237.610i	0.0872596i
$196$	0	0
$197$	− 1109.48i	− 0.401254i	−0.979668	−	0.200627i	$-0.935702\pi$
$197$	− 1109.48i	− 0.401254i	0.979668	−	0.200627i	$-0.0642980\pi$
$198$	0	0
$199$	−5309.30	−1.89129	−0.945644	−	0.325204i	$-0.894567\pi$
$199$	−5309.30	−1.89129	−0.945644	+	0.325204i	$0.894567\pi$
$200$	0	0
$201$	−201.180	−0.0705978
$202$	0	0
$203$	− 330.090i	− 0.114127i
$204$	0	0
$205$	889.592i	0.303082i
$206$	0	0
$207$	−497.685	−0.167109
$208$	0	0
$209$	5278.89	1.74712
$210$	0	0
$211$	− 3106.33i	− 1.01350i	−0.862093	−	0.506751i	$-0.830846\pi$
$211$	− 3106.33i	− 1.01350i	0.862093	−	0.506751i	$-0.169154\pi$
$212$	0	0
$213$	− 1860.73i	− 0.598570i
$214$	0	0
$215$	1704.62	0.540717
$216$	0	0
$217$	2229.77	0.697543
$218$	0	0
$219$	1445.68i	0.446072i
$220$	0	0
$221$	− 1013.51i	− 0.308488i
$222$	0	0
$223$	5998.73	1.80137	0.900683	−	0.434476i	$-0.143067\pi$
$223$	5998.73	1.80137	0.900683	+	0.434476i	$0.143067\pi$
$224$	0	0
$225$	−488.601	−0.144771
$226$	0	0
$227$	3581.68i	1.04725i	0.851950	+	0.523623i	$0.175420\pi$
$227$	3581.68i	1.04725i	−0.851950	+	0.523623i	$0.824580\pi$
$228$	0	0
$229$	− 272.597i	− 0.0786626i	−0.999226	−	0.0393313i	$-0.987477\pi$
$229$	− 272.597i	− 0.0786626i	0.999226	−	0.0393313i	$-0.0125228\pi$
$230$	0	0
$231$	−888.303	−0.253013
$232$	0	0
$233$	3776.00	1.06169	0.530845	−	0.847469i	$-0.321875\pi$
$233$	3776.00	1.06169	0.530845	+	0.847469i	$0.321875\pi$
$234$	0	0
$235$	1480.10i	0.410855i
$236$	0	0
$237$	− 1126.65i	− 0.308793i
$238$	0	0
$239$	3897.69	1.05490	0.527449	−	0.849587i	$-0.323148\pi$
$239$	3897.69	1.05490	0.527449	+	0.849587i	$0.323148\pi$
$240$	0	0
$241$	−3216.40	−0.859695	−0.429847	−	0.902902i	$-0.641433\pi$
$241$	−3216.40	−0.859695	−0.429847	+	0.902902i	$0.641433\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	412.040i	0.107446i
$246$	0	0
$247$	−1175.44	−0.302800
$248$	0	0
$249$	2930.12	0.745738
$250$	0	0
$251$	− 2656.75i	− 0.668097i	−0.942556	−	0.334049i	$-0.891585\pi$
$251$	− 2656.75i	− 0.668097i	0.942556	−	0.334049i	$-0.108415\pi$
$252$	0	0
$253$	2339.13i	0.581264i
$254$	0	0
$255$	−2714.50	−0.666623
$256$	0	0
$257$	−5167.55	−1.25425	−0.627126	−	0.778918i	$-0.715769\pi$
$257$	−5167.55	−1.25425	−0.627126	+	0.778918i	$0.715769\pi$
$258$	0	0
$259$	621.728i	0.149159i
$260$	0	0
$261$	424.401i	0.100650i
$262$	0	0
$263$	−64.3415	−0.0150854	−0.00754271	−	0.999972i	$-0.502401\pi$
$263$	−64.3415	−0.0150854	−0.00754271	+	0.999972i	$0.502401\pi$
$264$	0	0
$265$	−3310.91	−0.767500
$266$	0	0
$267$	3562.33i	0.816521i
$268$	0	0
$269$	4208.58i	0.953908i	0.878928	+	0.476954i	$0.158259\pi$
$269$	4208.58i	0.953908i	−0.878928	+	0.476954i	$0.841741\pi$
$270$	0	0
$271$	767.619	0.172065	0.0860324	−	0.996292i	$-0.472581\pi$
$271$	767.619	0.172065	0.0860324	+	0.996292i	$0.472581\pi$
$272$	0	0
$273$	197.797	0.0438506
$274$	0	0
$275$	2296.43i	0.503564i
$276$	0	0
$277$	− 2572.37i	− 0.557974i	−0.960295	−	0.278987i	$-0.910001\pi$
$277$	− 2572.37i	− 0.557974i	0.960295	−	0.278987i	$-0.0899987\pi$
$278$	0	0
$279$	−2866.85	−0.615175
$280$	0	0
$281$	6947.17	1.47485	0.737426	−	0.675428i	$-0.236041\pi$
$281$	6947.17	1.47485	0.737426	+	0.675428i	$0.236041\pi$
$282$	0	0
$283$	1351.64i	0.283910i	0.989873	+	0.141955i	$0.0453388\pi$
$283$	1351.64i	0.283910i	−0.989873	+	0.141955i	$0.954661\pi$
$284$	0	0
$285$	3148.22i	0.654331i
$286$	0	0
$287$	740.535	0.152308
$288$	0	0
$289$	6665.48	1.35670
$290$	0	0
$291$	1799.08i	0.362419i
$292$	0	0
$293$	− 1261.75i	− 0.251578i	−0.992057	−	0.125789i	$-0.959854\pi$
$293$	− 1261.75i	− 0.251578i	0.992057	−	0.125789i	$-0.0401463\pi$
$294$	0	0
$295$	−7554.56	−1.49099
$296$	0	0
$297$	1142.10	0.223137
$298$	0	0
$299$	− 520.849i	− 0.100741i
$300$	0	0
$301$	− 1419.00i	− 0.271727i
$302$	0	0
$303$	2182.10	0.413724
$304$	0	0
$305$	7606.39	1.42800
$306$	0	0
$307$	9167.14i	1.70422i	0.523361	+	0.852111i	$0.324678\pi$
$307$	9167.14i	1.70422i	−0.523361	+	0.852111i	$0.675322\pi$
$308$	0	0
$309$	− 2670.43i	− 0.491637i
$310$	0	0
$311$	−2425.05	−0.442161	−0.221081	−	0.975256i	$-0.570958\pi$
$311$	−2425.05	−0.442161	−0.221081	+	0.975256i	$0.570958\pi$
$312$	0	0
$313$	−3837.73	−0.693039	−0.346520	−	0.938043i	$-0.612637\pi$
$313$	−3837.73	−0.693039	−0.346520	+	0.938043i	$0.612637\pi$
$314$	0	0
$315$	− 529.766i	− 0.0947585i
$316$	0	0
$317$	− 5810.18i	− 1.02944i	−0.857359	−	0.514719i	$-0.827896\pi$
$317$	− 5810.18i	− 1.02944i	0.857359	−	0.514719i	$-0.172104\pi$
$318$	0	0
$319$	1994.69	0.350098
$320$	0	0
$321$	2187.04	0.380277
$322$	0	0
$323$	− 13428.5i	− 2.31325i
$324$	0	0
$325$	− 511.343i	− 0.0872745i
$326$	0	0
$327$	−3370.34	−0.569969
$328$	0	0
$329$	1232.10	0.206467
$330$	0	0
$331$	5253.39i	0.872364i	0.899858	+	0.436182i	$0.143670\pi$
$331$	5253.39i	0.872364i	−0.899858	+	0.436182i	$0.856330\pi$
$332$	0	0
$333$	− 799.365i	− 0.131546i
$334$	0	0
$335$	−563.907	−0.0919688
$336$	0	0
$337$	−5689.30	−0.919632	−0.459816	−	0.888014i	$-0.652085\pi$
$337$	−5689.30	−0.919632	−0.459816	+	0.888014i	$0.652085\pi$
$338$	0	0
$339$	1193.65i	0.191240i
$340$	0	0
$341$	13474.2i	2.13980i
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−1395.01	−0.217695
$346$	0	0
$347$	− 9445.38i	− 1.46125i	−0.682777	−	0.730626i	$-0.739228\pi$
$347$	− 9445.38i	− 1.46125i	0.682777	−	0.730626i	$-0.260772\pi$
$348$	0	0
$349$	9480.35i	1.45407i	0.686599	+	0.727037i	$0.259103\pi$
$349$	9480.35i	1.45407i	−0.686599	+	0.727037i	$0.740897\pi$
$350$	0	0
$351$	−254.310	−0.0386726
$352$	0	0
$353$	6172.70	0.930708	0.465354	−	0.885125i	$-0.345927\pi$
$353$	6172.70	0.930708	0.465354	+	0.885125i	$0.345927\pi$
$354$	0	0
$355$	− 5215.63i	− 0.779766i
$356$	0	0
$357$	2259.67i	0.334998i
$358$	0	0
$359$	7209.01	1.05982	0.529912	−	0.848052i	$-0.322225\pi$
$359$	7209.01	1.05982	0.529912	+	0.848052i	$0.322225\pi$
$360$	0	0
$361$	−8715.02	−1.27060
$362$	0	0
$363$	− 1374.91i	− 0.198799i
$364$	0	0
$365$	4052.22i	0.581104i
$366$	0	0
$367$	3196.42	0.454638	0.227319	−	0.973820i	$-0.427004\pi$
$367$	3196.42	0.454638	0.227319	+	0.973820i	$0.427004\pi$
$368$	0	0
$369$	−952.116	−0.134323
$370$	0	0
$371$	2756.14i	0.385692i
$372$	0	0
$373$	− 11311.7i	− 1.57023i	−0.619347	−	0.785117i	$-0.712603\pi$
$373$	− 11311.7i	− 1.57023i	0.619347	−	0.785117i	$-0.287397\pi$
$374$	0	0
$375$	−4522.91	−0.622833
$376$	0	0
$377$	−444.155	−0.0606767
$378$	0	0
$379$	1601.01i	0.216987i	0.994097	+	0.108494i	$0.0346027\pi$
$379$	1601.01i	0.216987i	−0.994097	+	0.108494i	$0.965397\pi$
$380$	0	0
$381$	− 8156.09i	− 1.09672i
$382$	0	0
$383$	−5392.30	−0.719408	−0.359704	−	0.933066i	$-0.617122\pi$
$383$	−5392.30	−0.719408	−0.359704	+	0.933066i	$0.617122\pi$
$384$	0	0
$385$	−2489.91	−0.329604
$386$	0	0
$387$	1824.43i	0.239641i
$388$	0	0
$389$	3958.04i	0.515888i	0.966160	+	0.257944i	$0.0830450\pi$
$389$	3958.04i	0.515888i	−0.966160	+	0.257944i	$0.916955\pi$
$390$	0	0
$391$	5950.28	0.769613
$392$	0	0
$393$	−917.858	−0.117811
$394$	0	0
$395$	− 3158.00i	− 0.402269i
$396$	0	0
$397$	924.331i	0.116854i	0.998292	+	0.0584268i	$0.0186084\pi$
$397$	924.331i	0.116854i	−0.998292	+	0.0584268i	$0.981392\pi$
$398$	0	0
$399$	2620.71	0.328822
$400$	0	0
$401$	−5168.91	−0.643698	−0.321849	−	0.946791i	$-0.604304\pi$
$401$	−5168.91	−0.643698	−0.321849	+	0.946791i	$0.604304\pi$
$402$	0	0
$403$	− 3000.29i	− 0.370856i
$404$	0	0
$405$	681.128i	0.0835691i
$406$	0	0
$407$	−3757.03	−0.457565
$408$	0	0
$409$	−10204.9	−1.23374	−0.616868	−	0.787067i	$-0.711599\pi$
$409$	−10204.9	−1.23374	−0.616868	+	0.787067i	$0.711599\pi$
$410$	0	0
$411$	1266.45i	0.151994i
$412$	0	0
$413$	6288.74i	0.749271i
$414$	0	0
$415$	8213.10	0.971483
$416$	0	0
$417$	−5045.91	−0.592564
$418$	0	0
$419$	− 1698.81i	− 0.198072i	−0.995084	−	0.0990361i	$-0.968424\pi$
$419$	− 1698.81i	− 0.198072i	0.995084	−	0.0990361i	$-0.0315759\pi$
$420$	0	0
$421$	978.378i	0.113262i	0.998395	+	0.0566309i	$0.0180358\pi$
$421$	978.378i	0.113262i	−0.998395	+	0.0566309i	$0.981964\pi$
$422$	0	0
$423$	−1584.13	−0.182087
$424$	0	0
$425$	5841.68	0.666737
$426$	0	0
$427$	− 6331.89i	− 0.717615i
$428$	0	0
$429$	1195.26i	0.134517i
$430$	0	0
$431$	5647.32	0.631141	0.315571	−	0.948902i	$-0.397804\pi$
$431$	5647.32	0.631141	0.315571	+	0.948902i	$0.397804\pi$
$432$	0	0
$433$	4698.41	0.521458	0.260729	−	0.965412i	$-0.416037\pi$
$433$	4698.41	0.521458	0.260729	+	0.965412i	$0.416037\pi$
$434$	0	0
$435$	1189.59i	0.131119i
$436$	0	0
$437$	− 6901.00i	− 0.755423i
$438$	0	0
$439$	−11114.9	−1.20840	−0.604199	−	0.796833i	$-0.706507\pi$
$439$	−11114.9	−1.20840	−0.604199	+	0.796833i	$0.706507\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	− 8622.15i	− 0.924720i	−0.886692	−	0.462360i	$-0.847003\pi$
$443$	− 8622.15i	− 0.924720i	0.886692	−	0.462360i	$-0.152997\pi$
$444$	0	0
$445$	9985.19i	1.06369i
$446$	0	0
$447$	−1608.34	−0.170183
$448$	0	0
$449$	14647.9	1.53959	0.769797	−	0.638288i	$-0.220357\pi$
$449$	14647.9	1.53959	0.769797	+	0.638288i	$0.220357\pi$
$450$	0	0
$451$	4474.96i	0.467223i
$452$	0	0
$453$	445.207i	0.0461758i
$454$	0	0
$455$	554.424	0.0571248
$456$	0	0
$457$	1697.97	0.173802	0.0869011	−	0.996217i	$-0.472304\pi$
$457$	1697.97	0.173802	0.0869011	+	0.996217i	$0.472304\pi$
$458$	0	0
$459$	− 2905.29i	− 0.295441i
$460$	0	0
$461$	− 1444.97i	− 0.145985i	−0.997333	−	0.0729923i	$-0.976745\pi$
$461$	− 1444.97i	− 0.145985i	0.997333	−	0.0729923i	$-0.0232549\pi$
$462$	0	0
$463$	−5328.42	−0.534844	−0.267422	−	0.963580i	$-0.586172\pi$
$463$	−5328.42	−0.534844	−0.267422	+	0.963580i	$0.586172\pi$
$464$	0	0
$465$	−8035.77	−0.801397
$466$	0	0
$467$	− 3489.64i	− 0.345785i	−0.984941	−	0.172892i	$-0.944689\pi$
$467$	− 3489.64i	− 0.345785i	0.984941	−	0.172892i	$-0.0553113\pi$
$468$	0	0
$469$	469.421i	0.0462171i
$470$	0	0
$471$	3345.66	0.327303
$472$	0	0
$473$	8574.85	0.833556
$474$	0	0
$475$	− 6775.05i	− 0.654443i
$476$	0	0
$477$	− 3543.61i	− 0.340149i
$478$	0	0
$479$	7492.93	0.714740	0.357370	−	0.933963i	$-0.383673\pi$
$479$	7492.93	0.714740	0.357370	+	0.933963i	$0.383673\pi$
$480$	0	0
$481$	836.571	0.0793022
$482$	0	0
$483$	1161.26i	0.109398i
$484$	0	0
$485$	5042.81i	0.472129i
$486$	0	0
$487$	4445.02	0.413600	0.206800	−	0.978383i	$-0.433695\pi$
$487$	4445.02	0.413600	0.206800	+	0.978383i	$0.433695\pi$
$488$	0	0
$489$	−7557.08	−0.698861
$490$	0	0
$491$	− 7213.35i	− 0.663001i	−0.943455	−	0.331501i	$-0.892445\pi$
$491$	− 7213.35i	− 0.663001i	0.943455	−	0.331501i	$-0.107555\pi$
$492$	0	0
$493$	− 5074.11i	− 0.463542i
$494$	0	0
$495$	3201.31	0.290683
$496$	0	0
$497$	−4341.71	−0.391856
$498$	0	0
$499$	13727.6i	1.23153i	0.787931	+	0.615763i	$0.211152\pi$
$499$	13727.6i	1.23153i	−0.787931	+	0.615763i	$0.788848\pi$
$500$	0	0
$501$	− 9231.88i	− 0.823253i
$502$	0	0
$503$	8699.38	0.771146	0.385573	−	0.922677i	$-0.374004\pi$
$503$	8699.38	0.771146	0.385573	+	0.922677i	$0.374004\pi$
$504$	0	0
$505$	6116.41	0.538964
$506$	0	0
$507$	6324.85i	0.554037i
$508$	0	0
$509$	− 12917.9i	− 1.12491i	−0.826829	−	0.562453i	$-0.809858\pi$
$509$	− 12917.9i	− 1.12491i	0.826829	−	0.562453i	$-0.190142\pi$
$510$	0	0
$511$	3373.25	0.292023
$512$	0	0
$513$	−3369.49	−0.289993
$514$	0	0
$515$	− 7485.21i	− 0.640462i
$516$	0	0
$517$	7445.42i	0.633364i
$518$	0	0
$519$	−12062.4	−1.02019
$520$	0	0
$521$	9852.45	0.828490	0.414245	−	0.910165i	$-0.364046\pi$
$521$	9852.45	0.828490	0.414245	+	0.910165i	$0.364046\pi$
$522$	0	0
$523$	− 14109.6i	− 1.17967i	−0.807523	−	0.589835i	$-0.799193\pi$
$523$	− 14109.6i	− 1.17967i	0.807523	−	0.589835i	$-0.200807\pi$
$524$	0	0
$525$	1140.07i	0.0947747i
$526$	0	0
$527$	34275.9	2.83317
$528$	0	0
$529$	−9109.10	−0.748672
$530$	0	0
$531$	− 8085.53i	− 0.660795i
$532$	0	0
$533$	− 996.432i	− 0.0809761i
$534$	0	0
$535$	6130.27	0.495392
$536$	0	0
$537$	−5736.22	−0.460961
$538$	0	0
$539$	2072.71i	0.165636i
$540$	0	0
$541$	− 1338.43i	− 0.106366i	−0.998585	−	0.0531828i	$-0.983063\pi$
$541$	− 1338.43i	− 0.106366i	0.998585	−	0.0531828i	$-0.0169366\pi$
$542$	0	0
$543$	−3497.53	−0.276415
$544$	0	0
$545$	−9447.03	−0.742507
$546$	0	0
$547$	− 12462.6i	− 0.974151i	−0.873360	−	0.487076i	$-0.838064\pi$
$547$	− 12462.6i	− 0.974151i	0.873360	−	0.487076i	$-0.161936\pi$
$548$	0	0
$549$	8141.00i	0.632877i
$550$	0	0
$551$	−5884.84	−0.454995
$552$	0	0
$553$	−2628.85	−0.202152
$554$	0	0
$555$	− 2240.61i	− 0.171367i
$556$	0	0
$557$	− 17426.2i	− 1.32562i	−0.748787	−	0.662811i	$-0.769363\pi$
$557$	− 17426.2i	− 1.32562i	0.748787	−	0.662811i	$-0.230637\pi$
$558$	0	0
$559$	−1909.35	−0.144466
$560$	0	0
$561$	−13654.9	−1.02765
$562$	0	0
$563$	− 20585.9i	− 1.54102i	−0.637429	−	0.770509i	$-0.720002\pi$
$563$	− 20585.9i	− 1.54102i	0.637429	−	0.770509i	$-0.279998\pi$
$564$	0	0
$565$	3345.80i	0.249131i
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−4334.40	−0.319345	−0.159672	−	0.987170i	$-0.551044\pi$
$569$	−4334.40	−0.319345	−0.159672	+	0.987170i	$0.551044\pi$
$570$	0	0
$571$	14014.8i	1.02715i	0.858046	+	0.513573i	$0.171679\pi$
$571$	14014.8i	1.02715i	−0.858046	+	0.513573i	$0.828321\pi$
$572$	0	0
$573$	− 12188.8i	− 0.888648i
$574$	0	0
$575$	3002.09	0.217732
$576$	0	0
$577$	782.119	0.0564299	0.0282150	−	0.999602i	$-0.491018\pi$
$577$	782.119	0.0564299	0.0282150	+	0.999602i	$0.491018\pi$
$578$	0	0
$579$	11348.7i	0.814569i
$580$	0	0
$581$	− 6836.94i	− 0.488200i
$582$	0	0
$583$	−16655.0	−1.18316
$584$	0	0
$585$	−712.830	−0.0503793
$586$	0	0
$587$	− 11299.3i	− 0.794500i	−0.917710	−	0.397250i	$-0.869965\pi$
$587$	− 11299.3i	− 0.794500i	0.917710	−	0.397250i	$-0.130035\pi$
$588$	0	0
$589$	− 39752.4i	− 2.78093i
$590$	0	0
$591$	3328.44	0.231664
$592$	0	0
$593$	−16620.0	−1.15093	−0.575464	−	0.817827i	$-0.695179\pi$
$593$	−16620.0	−1.15093	−0.575464	+	0.817827i	$0.695179\pi$
$594$	0	0
$595$	6333.84i	0.436407i
$596$	0	0
$597$	− 15927.9i	− 1.09194i
$598$	0	0
$599$	2792.53	0.190484	0.0952419	−	0.995454i	$-0.469638\pi$
$599$	2792.53	0.190484	0.0952419	+	0.995454i	$0.469638\pi$
$600$	0	0
$601$	−25010.4	−1.69750	−0.848748	−	0.528798i	$-0.822643\pi$
$601$	−25010.4	−1.69750	−0.848748	+	0.528798i	$0.822643\pi$
$602$	0	0
$603$	− 603.541i	− 0.0407597i
$604$	0	0
$605$	− 3853.86i	− 0.258978i
$606$	0	0
$607$	13307.9	0.889868	0.444934	−	0.895563i	$-0.353227\pi$
$607$	13307.9	0.889868	0.444934	+	0.895563i	$0.353227\pi$
$608$	0	0
$609$	990.269	0.0658912
$610$	0	0
$611$	− 1657.86i	− 0.109770i
$612$	0	0
$613$	− 28351.9i	− 1.86806i	−0.357192	−	0.934031i	$-0.616266\pi$
$613$	− 28351.9i	− 1.86806i	0.357192	−	0.934031i	$-0.383734\pi$
$614$	0	0
$615$	−2668.78	−0.174984
$616$	0	0
$617$	−23107.5	−1.50774	−0.753868	−	0.657026i	$-0.771814\pi$
$617$	−23107.5	−1.50774	−0.753868	+	0.657026i	$0.771814\pi$
$618$	0	0
$619$	− 792.761i	− 0.0514762i	−0.999669	−	0.0257381i	$-0.991806\pi$
$619$	− 792.761i	− 0.0514762i	0.999669	−	0.0257381i	$-0.00819359\pi$
$620$	0	0
$621$	− 1493.05i	− 0.0964802i
$622$	0	0
$623$	8312.10	0.534538
$624$	0	0
$625$	−5891.58	−0.377061
$626$	0	0
$627$	15836.7i	1.00870i
$628$	0	0
$629$	9557.15i	0.605832i
$630$	0	0
$631$	−24342.8	−1.53577	−0.767884	−	0.640589i	$-0.778690\pi$
$631$	−24342.8	−1.53577	−0.767884	+	0.640589i	$0.778690\pi$
$632$	0	0
$633$	9319.00	0.585145
$634$	0	0
$635$	− 22861.5i	− 1.42871i
$636$	0	0
$637$	− 461.526i	− 0.0287070i
$638$	0	0
$639$	5582.20	0.345585
$640$	0	0
$641$	8587.39	0.529144	0.264572	−	0.964366i	$-0.414769\pi$
$641$	8587.39	0.529144	0.264572	+	0.964366i	$0.414769\pi$
$642$	0	0
$643$	21410.4i	1.31313i	0.754269	+	0.656565i	$0.227991\pi$
$643$	21410.4i	1.31313i	−0.754269	+	0.656565i	$0.772009\pi$
$644$	0	0
$645$	5113.86i	0.312183i
$646$	0	0
$647$	−17879.1	−1.08640	−0.543200	−	0.839603i	$-0.682788\pi$
$647$	−17879.1	−1.08640	−0.543200	+	0.839603i	$0.682788\pi$
$648$	0	0
$649$	−38002.1	−2.29848
$650$	0	0
$651$	6689.32i	0.402727i
$652$	0	0
$653$	27765.7i	1.66394i	0.554819	+	0.831971i	$0.312788\pi$
$653$	27765.7i	1.66394i	−0.554819	+	0.831971i	$0.687212\pi$
$654$	0	0
$655$	−2572.75	−0.153474
$656$	0	0
$657$	−4337.03	−0.257540
$658$	0	0
$659$	− 13334.8i	− 0.788240i	−0.919059	−	0.394120i	$-0.871049\pi$
$659$	− 13334.8i	− 0.788240i	0.919059	−	0.394120i	$-0.128951\pi$
$660$	0	0
$661$	− 21256.8i	− 1.25082i	−0.780295	−	0.625412i	$-0.784931\pi$
$661$	− 21256.8i	− 1.25082i	0.780295	−	0.625412i	$-0.215069\pi$
$662$	0	0
$663$	3040.52	0.178105
$664$	0	0
$665$	7345.85	0.428360
$666$	0	0
$667$	− 2607.63i	− 0.151376i
$668$	0	0
$669$	17996.2i	1.04002i
$670$	0	0
$671$	38262.8	2.20137
$672$	0	0
$673$	8963.29	0.513387	0.256693	−	0.966493i	$-0.417367\pi$
$673$	8963.29	0.513387	0.256693	+	0.966493i	$0.417367\pi$
$674$	0	0
$675$	− 1465.80i	− 0.0835834i
$676$	0	0
$677$	25166.0i	1.42867i	0.699805	+	0.714334i	$0.253270\pi$
$677$	25166.0i	1.42867i	−0.699805	+	0.714334i	$0.746730\pi$
$678$	0	0
$679$	4197.86	0.237259
$680$	0	0
$681$	−10745.1	−0.604627
$682$	0	0
$683$	791.146i	0.0443226i	0.999754	+	0.0221613i	$0.00705475\pi$
$683$	791.146i	0.0443226i	−0.999754	+	0.0221613i	$0.992945\pi$
$684$	0	0
$685$	3549.86i	0.198005i
$686$	0	0
$687$	817.792	0.0454159
$688$	0	0
$689$	3708.55	0.205057
$690$	0	0
$691$	2389.97i	0.131576i	0.997834	+	0.0657878i	$0.0209561\pi$
$691$	2389.97i	0.131576i	−0.997834	+	0.0657878i	$0.979044\pi$
$692$	0	0
$693$	− 2664.91i	− 0.146077i
$694$	0	0
$695$	−14143.7	−0.771941
$696$	0	0
$697$	11383.4	0.618620
$698$	0	0
$699$	11328.0i	0.612967i
$700$	0	0
$701$	− 6711.69i	− 0.361622i	−0.983518	−	0.180811i	$-0.942128\pi$
$701$	− 6711.69i	− 0.361622i	0.983518	−	0.180811i	$-0.0578722\pi$
$702$	0	0
$703$	11084.2	0.594662
$704$	0	0
$705$	−4440.29	−0.237207
$706$	0	0
$707$	− 5091.56i	− 0.270846i
$708$	0	0
$709$	15903.4i	0.842405i	0.906967	+	0.421202i	$0.138392\pi$
$709$	15903.4i	0.842405i	−0.906967	+	0.421202i	$0.861608\pi$
$710$	0	0
$711$	3379.96	0.178282
$712$	0	0
$713$	17614.7	0.925210
$714$	0	0
$715$	3350.31i	0.175237i
$716$	0	0
$717$	11693.1i	0.609046i
$718$	0	0
$719$	35162.7	1.82385	0.911924	−	0.410360i	$-0.134597\pi$
$719$	35162.7	1.82385	0.911924	+	0.410360i	$0.134597\pi$
$720$	0	0
$721$	−6231.01	−0.321852
$722$	0	0
$723$	− 9649.19i	− 0.496345i
$724$	0	0
$725$	− 2560.04i	− 0.131141i
$726$	0	0
$727$	−8519.06	−0.434600	−0.217300	−	0.976105i	$-0.569725\pi$
$727$	−8519.06	−0.434600	−0.217300	+	0.976105i	$0.569725\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 21812.7i	− 1.10366i
$732$	0	0
$733$	16085.0i	0.810524i	0.914201	+	0.405262i	$0.132820\pi$
$733$	16085.0i	0.810524i	−0.914201	+	0.405262i	$0.867180\pi$
$734$	0	0
$735$	−1236.12	−0.0620340
$736$	0	0
$737$	−2836.65	−0.141777
$738$	0	0
$739$	− 24279.1i	− 1.20855i	−0.796775	−	0.604276i	$-0.793462\pi$
$739$	− 24279.1i	− 1.20855i	0.796775	−	0.604276i	$-0.206538\pi$
$740$	0	0
$741$	− 3526.32i	− 0.174821i
$742$	0	0
$743$	−16834.0	−0.831195	−0.415598	−	0.909549i	$-0.636428\pi$
$743$	−16834.0	−0.831195	−0.415598	+	0.909549i	$0.636428\pi$
$744$	0	0
$745$	−4508.18	−0.221700
$746$	0	0
$747$	8790.35i	0.430552i
$748$	0	0
$749$	− 5103.10i	− 0.248949i
$750$	0	0
$751$	−30640.0	−1.48877	−0.744386	−	0.667749i	$-0.767258\pi$
$751$	−30640.0	−1.48877	−0.744386	+	0.667749i	$0.767258\pi$
$752$	0	0
$753$	7970.24	0.385726
$754$	0	0
$755$	1247.91i	0.0601539i
$756$	0	0
$757$	− 25016.9i	− 1.20113i	−0.799575	−	0.600566i	$-0.794942\pi$
$757$	− 25016.9i	− 1.20113i	0.799575	−	0.600566i	$-0.205058\pi$
$758$	0	0
$759$	−7017.38	−0.335593
$760$	0	0
$761$	31113.8	1.48210	0.741048	−	0.671452i	$-0.234329\pi$
$761$	31113.8	1.48210	0.741048	+	0.671452i	$0.234329\pi$
$762$	0	0
$763$	7864.12i	0.373133i
$764$	0	0
$765$	− 8143.51i	− 0.384875i
$766$	0	0
$767$	8461.86	0.398357
$768$	0	0
$769$	−25683.5	−1.20438	−0.602192	−	0.798351i	$-0.705706\pi$
$769$	−25683.5	−1.20438	−0.602192	+	0.798351i	$0.705706\pi$
$770$	0	0
$771$	− 15502.6i	− 0.724143i
$772$	0	0
$773$	10195.1i	0.474375i	0.971464	+	0.237187i	$0.0762255\pi$
$773$	10195.1i	0.474375i	−0.971464	+	0.237187i	$0.923774\pi$
$774$	0	0
$775$	17293.2	0.801534
$776$	0	0
$777$	−1865.18	−0.0861173
$778$	0	0
$779$	− 13202.2i	− 0.607214i
$780$	0	0
$781$	− 26236.5i	− 1.20207i
$782$	0	0
$783$	−1273.20	−0.0581106
$784$	0	0
$785$	9377.87	0.426383
$786$	0	0
$787$	− 34242.3i	− 1.55096i	−0.631372	−	0.775480i	$-0.717508\pi$
$787$	− 34242.3i	− 1.55096i	0.631372	−	0.775480i	$-0.282492\pi$
$788$	0	0
$789$	− 193.024i	− 0.00870957i
$790$	0	0
$791$	2785.19	0.125196
$792$	0	0
$793$	−8519.92	−0.381528
$794$	0	0
$795$	− 9932.72i	− 0.443116i
$796$	0	0
$797$	20801.8i	0.924512i	0.886746	+	0.462256i	$0.152960\pi$
$797$	20801.8i	0.924512i	−0.886746	+	0.462256i	$0.847040\pi$
$798$	0	0
$799$	18939.7	0.838596
$800$	0	0
$801$	−10687.0	−0.471418
$802$	0	0
$803$	20384.1i	0.895815i
$804$	0	0
$805$	3255.02i	0.142515i
$806$	0	0
$807$	−12625.7	−0.550739
$808$	0	0
$809$	−15209.7	−0.660993	−0.330496	−	0.943807i	$-0.607216\pi$
$809$	−15209.7	−0.660993	−0.330496	+	0.943807i	$0.607216\pi$
$810$	0	0
$811$	− 25337.6i	− 1.09707i	−0.836128	−	0.548534i	$-0.815186\pi$
$811$	− 25337.6i	− 1.09707i	0.836128	−	0.548534i	$-0.184814\pi$
$812$	0	0
$813$	2302.86i	0.0993416i
$814$	0	0
$815$	−21182.4	−0.910416
$816$	0	0
$817$	−25297.9	−1.08331
$818$	0	0
$819$	593.391i	0.0253172i
$820$	0	0
$821$	17766.3i	0.755237i	0.925961	+	0.377618i	$0.123257\pi$
$821$	17766.3i	0.755237i	−0.925961	+	0.377618i	$0.876743\pi$
$822$	0	0
$823$	−6605.85	−0.279788	−0.139894	−	0.990167i	$-0.544676\pi$
$823$	−6605.85	−0.279788	−0.139894	+	0.990167i	$0.544676\pi$
$824$	0	0
$825$	−6889.30	−0.290733
$826$	0	0
$827$	11302.2i	0.475229i	0.971360	+	0.237614i	$0.0763655\pi$
$827$	11302.2i	0.475229i	−0.971360	+	0.237614i	$0.923635\pi$
$828$	0	0
$829$	− 30888.8i	− 1.29410i	−0.762446	−	0.647051i	$-0.776002\pi$
$829$	− 30888.8i	− 1.29410i	0.762446	−	0.647051i	$-0.223998\pi$
$830$	0	0
$831$	7717.12	0.322147
$832$	0	0
$833$	5272.56	0.219308
$834$	0	0
$835$	− 25876.9i	− 1.07246i
$836$	0	0
$837$	− 8600.55i	− 0.355172i
$838$	0	0
$839$	−11316.7	−0.465671	−0.232835	−	0.972516i	$-0.574800\pi$
$839$	−11316.7	−0.465671	−0.232835	+	0.972516i	$0.574800\pi$
$840$	0	0
$841$	22165.3	0.908825
$842$	0	0
$843$	20841.5i	0.851506i
$844$	0	0
$845$	17728.5i	0.721751i
$846$	0	0
$847$	−3208.12	−0.130144
$848$	0	0
$849$	−4054.91	−0.163915
$850$	0	0
$851$	4911.50i	0.197843i
$852$	0	0
$853$	13992.3i	0.561650i	0.959759	+	0.280825i	$0.0906081\pi$
$853$	13992.3i	0.561650i	−0.959759	+	0.280825i	$0.909392\pi$
$854$	0	0
$855$	−9444.66	−0.377778
$856$	0	0
$857$	30193.9	1.20350	0.601752	−	0.798683i	$-0.294470\pi$
$857$	30193.9	1.20350	0.601752	+	0.798683i	$0.294470\pi$
$858$	0	0
$859$	19915.5i	0.791047i	0.918456	+	0.395523i	$0.129437\pi$
$859$	19915.5i	0.791047i	−0.918456	+	0.395523i	$0.870563\pi$
$860$	0	0
$861$	2221.60i	0.0879350i
$862$	0	0
$863$	34621.8	1.36563	0.682815	−	0.730591i	$-0.260756\pi$
$863$	34621.8	1.36563	0.682815	+	0.730591i	$0.260756\pi$
$864$	0	0
$865$	−33810.8	−1.32902
$866$	0	0
$867$	19996.4i	0.783292i
$868$	0	0
$869$	− 15885.8i	− 0.620127i
$870$	0	0
$871$	631.632	0.0245718
$872$	0	0
$873$	−5397.24	−0.209243
$874$	0	0
$875$	10553.5i	0.407740i
$876$	0	0
$877$	16375.7i	0.630521i	0.949005	+	0.315260i	$0.102092\pi$
$877$	16375.7i	0.630521i	−0.949005	+	0.315260i	$0.897908\pi$
$878$	0	0
$879$	3785.26	0.145249
$880$	0	0
$881$	−25301.9	−0.967586	−0.483793	−	0.875183i	$-0.660741\pi$
$881$	−25301.9	−0.967586	−0.483793	+	0.875183i	$0.660741\pi$
$882$	0	0
$883$	20642.4i	0.786718i	0.919385	+	0.393359i	$0.128687\pi$
$883$	20642.4i	0.786718i	−0.919385	+	0.393359i	$0.871313\pi$
$884$	0	0
$885$	− 22663.7i	− 0.860826i
$886$	0	0
$887$	13123.7	0.496788	0.248394	−	0.968659i	$-0.420097\pi$
$887$	13123.7	0.496788	0.248394	+	0.968659i	$0.420097\pi$
$888$	0	0
$889$	−19030.9	−0.717969
$890$	0	0
$891$	3426.31i	0.128828i
$892$	0	0
$893$	− 21965.8i	− 0.823133i
$894$	0	0
$895$	−16078.6	−0.600500
$896$	0	0
$897$	1562.55	0.0581627
$898$	0	0
$899$	− 15020.9i	− 0.557259i
$900$	0	0
$901$	42367.2i	1.56654i
$902$	0	0
$903$	4257.00	0.156882
$904$	0	0
$905$	−9803.56	−0.360090
$906$	0	0
$907$	13064.4i	0.478277i	0.970985	+	0.239138i	$0.0768650\pi$
$907$	13064.4i	0.478277i	−0.970985	+	0.239138i	$0.923135\pi$
$908$	0	0
$909$	6546.30i	0.238864i
$910$	0	0
$911$	−52055.4	−1.89316	−0.946582	−	0.322462i	$-0.895490\pi$
$911$	−52055.4	−1.89316	−0.946582	+	0.322462i	$0.895490\pi$
$912$	0	0
$913$	41314.8	1.49761
$914$	0	0
$915$	22819.2i	0.824458i
$916$	0	0
$917$	2141.67i	0.0771256i
$918$	0	0
$919$	24860.3	0.892347	0.446173	−	0.894947i	$-0.352786\pi$
$919$	24860.3	0.892347	0.446173	+	0.894947i	$0.352786\pi$
$920$	0	0
$921$	−27501.4	−0.983933
$922$	0	0
$923$	5842.03i	0.208334i
$924$	0	0
$925$	4821.86i	0.171396i
$926$	0	0
$927$	8011.30	0.283846
$928$	0	0
$929$	32081.0	1.13298	0.566492	−	0.824067i	$-0.308300\pi$
$929$	32081.0	1.13298	0.566492	+	0.824067i	$0.308300\pi$
$930$	0	0
$931$	− 6115.00i	− 0.215264i
$932$	0	0
$933$	− 7275.16i	− 0.255282i
$934$	0	0
$935$	−38274.6	−1.33873
$936$	0	0
$937$	50156.5	1.74871	0.874355	−	0.485287i	$-0.161285\pi$
$937$	50156.5	1.74871	0.874355	+	0.485287i	$0.161285\pi$
$938$	0	0
$939$	− 11513.2i	− 0.400126i
$940$	0	0
$941$	755.090i	0.0261586i	0.999914	+	0.0130793i	$0.00416339\pi$
$941$	755.090i	0.0261586i	−0.999914	+	0.0130793i	$0.995837\pi$
$942$	0	0
$943$	5850.05	0.202019
$944$	0	0
$945$	1589.30	0.0547088
$946$	0	0
$947$	− 49944.3i	− 1.71380i	−0.515480	−	0.856902i	$-0.672386\pi$
$947$	− 49944.3i	− 1.71380i	0.515480	−	0.856902i	$-0.327614\pi$
$948$	0	0
$949$	− 4538.90i	− 0.155257i
$950$	0	0
$951$	17430.5	0.594347
$952$	0	0
$953$	−14174.2	−0.481791	−0.240895	−	0.970551i	$-0.577441\pi$
$953$	−14174.2	−0.481791	−0.240895	+	0.970551i	$0.577441\pi$
$954$	0	0
$955$	− 34165.2i	− 1.15765i
$956$	0	0
$957$	5984.08i	0.202129i
$958$	0	0
$959$	2955.06	0.0995034
$960$	0	0
$961$	71676.1	2.40597
$962$	0	0
$963$	6561.13i	0.219553i
$964$	0	0
$965$	31810.3i	1.06115i
$966$	0	0
$967$	−39816.7	−1.32411	−0.662057	−	0.749453i	$-0.730317\pi$
$967$	−39816.7	−1.32411	−0.662057	+	0.749453i	$0.730317\pi$
$968$	0	0
$969$	40285.4	1.33555
$970$	0	0
$971$	10300.1i	0.340416i	0.985408	+	0.170208i	$0.0544440\pi$
$971$	10300.1i	0.340416i	−0.985408	+	0.170208i	$0.945556\pi$
$972$	0	0
$973$	11773.8i	0.387924i
$974$	0	0
$975$	1534.03	0.0503879
$976$	0	0
$977$	39564.3	1.29557	0.647786	−	0.761822i	$-0.275695\pi$
$977$	39564.3	1.29557	0.647786	+	0.761822i	$0.275695\pi$
$978$	0	0
$979$	50229.0i	1.63976i
$980$	0	0
$981$	− 10111.0i	− 0.329072i
$982$	0	0
$983$	4441.00	0.144095	0.0720477	−	0.997401i	$-0.477047\pi$
$983$	4441.00	0.144095	0.0720477	+	0.997401i	$0.477047\pi$
$984$	0	0
$985$	9329.59	0.301792
$986$	0	0
$987$	3696.29i	0.119204i
$988$	0	0
$989$	− 11209.8i	− 0.360414i
$990$	0	0
$991$	31663.9	1.01497	0.507485	−	0.861661i	$-0.330575\pi$
$991$	31663.9	1.01497	0.507485	+	0.861661i	$0.330575\pi$
$992$	0	0
$993$	−15760.2	−0.503660
$994$	0	0
$995$	− 44645.8i	− 1.42248i
$996$	0	0
$997$	− 18057.4i	− 0.573605i	−0.957990	−	0.286803i	$-0.907408\pi$
$997$	− 18057.4i	− 0.573605i	0.957990	−	0.286803i	$-0.0925924\pi$
$998$	0	0
$999$	2398.09	0.0759483

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.h.673.11	yes	12
4.3	odd	2		1344.4.c.e.673.5	✓	12
8.3	odd	2		1344.4.c.e.673.8	yes	12
8.5	even	2	inner	1344.4.c.h.673.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.e.673.5	✓	12	4.3	odd	2
1344.4.c.e.673.8	yes	12	8.3	odd	2
1344.4.c.h.673.2	yes	12	8.5	even	2	inner
1344.4.c.h.673.11	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +8.40898i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +8.40898i q^{5} +7.00000 q^{7} -9.00000 q^{9} +42.3001i q^{11} -9.41890i q^{13} -25.2269 q^{15} +107.603 q^{17} -124.796i q^{19} +21.0000i q^{21} +55.2983 q^{23} +54.2890 q^{25} -27.0000i q^{27} -47.1557i q^{29} +318.539 q^{31} -126.900 q^{33} +58.8629i q^{35} +88.8183i q^{37} +28.2567 q^{39} +105.791 q^{41} -202.714i q^{43} -75.6808i q^{45} +176.014 q^{47} +49.0000 q^{49} +322.810i q^{51} +393.735i q^{53} -355.701 q^{55} +374.388 q^{57} +898.392i q^{59} -904.556i q^{61} -63.0000 q^{63} +79.2034 q^{65} +67.0601i q^{67} +165.895i q^{69} -620.245 q^{71} +481.892 q^{73} +162.867i q^{75} +296.101i q^{77} -375.551 q^{79} +81.0000 q^{81} -976.706i q^{83} +904.835i q^{85} +141.467 q^{87} +1187.44 q^{89} -65.9323i q^{91} +955.617i q^{93} +1049.41 q^{95} +599.694 q^{97} -380.701i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q + 84 * q^7 - 108 * q^9	\( 12 q + 84 q^{7} - 108 q^{9} + 24 q^{15} + 24 q^{17} - 80 q^{23} - 564 q^{25} + 640 q^{31} - 408 q^{33} - 120 q^{39} + 1416 q^{41} + 1536 q^{47} + 588 q^{49} - 1392 q^{55} - 336 q^{57} - 756 q^{63} - 2880 q^{65} - 1392 q^{71} + 2472 q^{73} + 544 q^{79} + 972 q^{81} - 720 q^{87} + 888 q^{89} - 2368 q^{95} - 2712 q^{97}+O(q^{100}) \) 12 * q + 84 * q^7 - 108 * q^9 + 24 * q^15 + 24 * q^17 - 80 * q^23 - 564 * q^25 + 640 * q^31 - 408 * q^33 - 120 * q^39 + 1416 * q^41 + 1536 * q^47 + 588 * q^49 - 1392 * q^55 - 336 * q^57 - 756 * q^63 - 2880 * q^65 - 1392 * q^71 + 2472 * q^73 + 544 * q^79 + 972 * q^81 - 720 * q^87 + 888 * q^89 - 2368 * q^95 - 2712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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