LMFDB - Embedded newform 736.3.g.a.47.31

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [736,3,Mod(47,736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("736.47");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 736 = 2^{5} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	736.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.0545474569$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.31
Character	$\chi$	$=$	736.47
Dual form			736.3.g.a.47.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68155 q^{3} -5.82129i q^{5} -9.89299i q^{7} -1.80931 q^{9} +O(q^{10})$	$q+2.68155 q^{3} -5.82129i q^{5} -9.89299i q^{7} -1.80931 q^{9} +6.62174 q^{11} +3.58990i q^{13} -15.6101i q^{15} +30.4419 q^{17} -30.9283 q^{19} -26.5285i q^{21} +4.79583i q^{23} -8.88747 q^{25} -28.9857 q^{27} +12.1899i q^{29} -47.7450i q^{31} +17.7565 q^{33} -57.5900 q^{35} -40.7031i q^{37} +9.62650i q^{39} -31.4044 q^{41} -70.1419 q^{43} +10.5325i q^{45} -22.2026i q^{47} -48.8713 q^{49} +81.6314 q^{51} +53.9457i q^{53} -38.5471i q^{55} -82.9358 q^{57} -0.611750 q^{59} +10.5456i q^{61} +17.8995i q^{63} +20.8979 q^{65} +110.146 q^{67} +12.8602i q^{69} -120.786i q^{71} +89.2799 q^{73} -23.8322 q^{75} -65.5089i q^{77} +43.5141i q^{79} -61.4426 q^{81} +23.5417 q^{83} -177.211i q^{85} +32.6878i q^{87} +7.32067 q^{89} +35.5149 q^{91} -128.030i q^{93} +180.043i q^{95} +86.3316 q^{97} -11.9808 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 132 q^{9}+O(q^{10}) $ 44 * q + 132 * q^9	$ 44 q + 132 q^{9} - 8 q^{17} - 244 q^{25} + 48 q^{27} + 16 q^{33} - 96 q^{35} + 88 q^{41} + 128 q^{43} - 340 q^{49} - 160 q^{51} - 176 q^{57} - 16 q^{59} + 96 q^{65} + 288 q^{67} + 280 q^{73} - 160 q^{75} + 284 q^{81} + 480 q^{83} - 200 q^{89} - 192 q^{91} + 184 q^{97} - 256 q^{99}+O(q^{100}) $ 44 * q + 132 * q^9 - 8 * q^17 - 244 * q^25 + 48 * q^27 + 16 * q^33 - 96 * q^35 + 88 * q^41 + 128 * q^43 - 340 * q^49 - 160 * q^51 - 176 * q^57 - 16 * q^59 + 96 * q^65 + 288 * q^67 + 280 * q^73 - 160 * q^75 + 284 * q^81 + 480 * q^83 - 200 * q^89 - 192 * q^91 + 184 * q^97 - 256 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/736\mathbb{Z}\right)^\times$.

$n$	$97$	$415$	$645$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	2.68155	0.893849	0.446924	−	0.894572i	$-0.352519\pi$
$3$	2.68155	0.893849	0.446924	+	0.894572i	$0.352519\pi$
$4$	0	0
$5$	− 5.82129i	− 1.16426i	−0.813096	−	0.582129i	$-0.802220\pi$
$5$	− 5.82129i	− 1.16426i	0.813096	−	0.582129i	$-0.197780\pi$
$6$	0	0
$7$	− 9.89299i	− 1.41328i	−0.707571	−	0.706642i	$-0.750209\pi$
$7$	− 9.89299i	− 1.41328i	0.707571	−	0.706642i	$-0.249791\pi$
$8$	0	0
$9$	−1.80931	−0.201034
$10$	0	0
$11$	6.62174	0.601977	0.300988	−	0.953628i	$-0.402683\pi$
$11$	6.62174	0.601977	0.300988	+	0.953628i	$0.402683\pi$
$12$	0	0
$13$	3.58990i	0.276147i	0.990422	+	0.138073i	$0.0440909\pi$
$13$	3.58990i	0.276147i	−0.990422	+	0.138073i	$0.955909\pi$
$14$	0	0
$15$	− 15.6101i	− 1.04067i
$16$	0	0
$17$	30.4419	1.79070	0.895351	−	0.445362i	$-0.146925\pi$
$17$	30.4419	1.79070	0.895351	+	0.445362i	$0.146925\pi$
$18$	0	0
$19$	−30.9283	−1.62781	−0.813904	−	0.581000i	$-0.802662\pi$
$19$	−30.9283	−1.62781	−0.813904	+	0.581000i	$0.802662\pi$
$20$	0	0
$21$	− 26.5285i	− 1.26326i
$22$	0	0
$23$	4.79583i	0.208514i
$23$	4.79583i	0.208514i
$24$	0	0
$25$	−8.88747	−0.355499
$26$	0	0
$27$	−28.9857	−1.07354
$28$	0	0
$29$	12.1899i	0.420342i	0.977665	+	0.210171i	$0.0674021\pi$
$29$	12.1899i	0.420342i	−0.977665	+	0.210171i	$0.932598\pi$
$30$	0	0
$31$	− 47.7450i	− 1.54016i	−0.637947	−	0.770080i	$-0.720216\pi$
$31$	− 47.7450i	− 1.54016i	0.637947	−	0.770080i	$-0.279784\pi$
$32$	0	0
$33$	17.7565	0.538076
$34$	0	0
$35$	−57.5900	−1.64543
$36$	0	0
$37$	− 40.7031i	− 1.10008i	−0.835137	−	0.550041i	$-0.814612\pi$
$37$	− 40.7031i	− 1.10008i	0.835137	−	0.550041i	$-0.185388\pi$
$38$	0	0
$39$	9.62650i	0.246833i
$40$	0	0
$41$	−31.4044	−0.765962	−0.382981	−	0.923756i	$-0.625102\pi$
$41$	−31.4044	−0.765962	−0.382981	+	0.923756i	$0.625102\pi$
$42$	0	0
$43$	−70.1419	−1.63121	−0.815604	−	0.578611i	$-0.803595\pi$
$43$	−70.1419	−1.63121	−0.815604	+	0.578611i	$0.803595\pi$
$44$	0	0
$45$	10.5325i	0.234056i
$46$	0	0
$47$	− 22.2026i	− 0.472395i	−0.971705	−	0.236197i	$-0.924099\pi$
$47$	− 22.2026i	− 0.472395i	0.971705	−	0.236197i	$-0.0759012\pi$
$48$	0	0
$49$	−48.8713	−0.997373
$50$	0	0
$51$	81.6314	1.60062
$52$	0	0
$53$	53.9457i	1.01784i	0.860813	+	0.508922i	$0.169956\pi$
$53$	53.9457i	1.01784i	−0.860813	+	0.508922i	$0.830044\pi$
$54$	0	0
$55$	− 38.5471i	− 0.700857i
$56$	0	0
$57$	−82.9358	−1.45501
$58$	0	0
$59$	−0.611750	−0.0103686	−0.00518432	−	0.999987i	$-0.501650\pi$
$59$	−0.611750	−0.0103686	−0.00518432	+	0.999987i	$0.501650\pi$
$60$	0	0
$61$	10.5456i	0.172879i	0.996257	+	0.0864397i	$0.0275490\pi$
$61$	10.5456i	0.172879i	−0.996257	+	0.0864397i	$0.972451\pi$
$62$	0	0
$63$	17.8995i	0.284119i
$64$	0	0
$65$	20.8979	0.321506
$66$	0	0
$67$	110.146	1.64397	0.821984	−	0.569510i	$-0.192867\pi$
$67$	110.146	1.64397	0.821984	+	0.569510i	$0.192867\pi$
$68$	0	0
$69$	12.8602i	0.186380i
$70$	0	0
$71$	− 120.786i	− 1.70122i	−0.525801	−	0.850608i	$-0.676234\pi$
$71$	− 120.786i	− 1.70122i	0.525801	−	0.850608i	$-0.323766\pi$
$72$	0	0
$73$	89.2799	1.22301	0.611506	−	0.791240i	$-0.290564\pi$
$73$	89.2799	1.22301	0.611506	+	0.791240i	$0.290564\pi$
$74$	0	0
$75$	−23.8322	−0.317762
$76$	0	0
$77$	− 65.5089i	− 0.850764i
$78$	0	0
$79$	43.5141i	0.550811i	0.961328	+	0.275406i	$0.0888122\pi$
$79$	43.5141i	0.550811i	−0.961328	+	0.275406i	$0.911188\pi$
$80$	0	0
$81$	−61.4426	−0.758551
$82$	0	0
$83$	23.5417	0.283635	0.141818	−	0.989893i	$-0.454705\pi$
$83$	23.5417	0.283635	0.141818	+	0.989893i	$0.454705\pi$
$84$	0	0
$85$	− 177.211i	− 2.08484i
$86$	0	0
$87$	32.6878i	0.375722i
$88$	0	0
$89$	7.32067	0.0822547	0.0411273	−	0.999154i	$-0.486905\pi$
$89$	7.32067	0.0822547	0.0411273	+	0.999154i	$0.486905\pi$
$90$	0	0
$91$	35.5149	0.390274
$92$	0	0
$93$	− 128.030i	− 1.37667i
$94$	0	0
$95$	180.043i	1.89519i
$96$	0	0
$97$	86.3316	0.890016	0.445008	−	0.895527i	$-0.353201\pi$
$97$	86.3316	0.890016	0.445008	+	0.895527i	$0.353201\pi$
$98$	0	0
$99$	−11.9808	−0.121018
$100$	0	0
$101$	− 10.0607i	− 0.0996104i	−0.998759	−	0.0498052i	$-0.984140\pi$
$101$	− 10.0607i	− 0.0996104i	0.998759	−	0.0498052i	$-0.0158601\pi$
$102$	0	0
$103$	77.6553i	0.753935i	0.926226	+	0.376967i	$0.123033\pi$
$103$	77.6553i	0.753935i	−0.926226	+	0.376967i	$0.876967\pi$
$104$	0	0
$105$	−154.430	−1.47077
$106$	0	0
$107$	−33.3659	−0.311831	−0.155915	−	0.987770i	$-0.549833\pi$
$107$	−33.3659	−0.311831	−0.155915	+	0.987770i	$0.549833\pi$
$108$	0	0
$109$	156.517i	1.43594i	0.696076	+	0.717968i	$0.254928\pi$
$109$	156.517i	1.43594i	−0.696076	+	0.717968i	$0.745072\pi$
$110$	0	0
$111$	− 109.147i	− 0.983308i
$112$	0	0
$113$	163.295	1.44509	0.722544	−	0.691325i	$-0.242972\pi$
$113$	163.295	1.44509	0.722544	+	0.691325i	$0.242972\pi$
$114$	0	0
$115$	27.9179	0.242765
$116$	0	0
$117$	− 6.49524i	− 0.0555149i
$118$	0	0
$119$	− 301.162i	− 2.53077i
$120$	0	0
$121$	−77.1525	−0.637624
$122$	0	0
$123$	−84.2124	−0.684654
$124$	0	0
$125$	− 93.7958i	− 0.750366i
$126$	0	0
$127$	41.1850i	0.324292i	0.986767	+	0.162146i	$0.0518415\pi$
$127$	41.1850i	0.324292i	−0.986767	+	0.162146i	$0.948159\pi$
$128$	0	0
$129$	−188.089	−1.45805
$130$	0	0
$131$	42.5817	0.325051	0.162526	−	0.986704i	$-0.448036\pi$
$131$	42.5817	0.325051	0.162526	+	0.986704i	$0.448036\pi$
$132$	0	0
$133$	305.974i	2.30056i
$134$	0	0
$135$	168.734i	1.24988i
$136$	0	0
$137$	−23.2365	−0.169609	−0.0848046	−	0.996398i	$-0.527027\pi$
$137$	−23.2365	−0.169609	−0.0848046	+	0.996398i	$0.527027\pi$
$138$	0	0
$139$	111.827	0.804512	0.402256	−	0.915527i	$-0.368226\pi$
$139$	111.827	0.804512	0.402256	+	0.915527i	$0.368226\pi$
$140$	0	0
$141$	− 59.5372i	− 0.422250i
$142$	0	0
$143$	23.7714i	0.166234i
$144$	0	0
$145$	70.9611	0.489387
$146$	0	0
$147$	−131.051	−0.891501
$148$	0	0
$149$	− 219.579i	− 1.47369i	−0.676064	−	0.736843i	$-0.736316\pi$
$149$	− 219.579i	− 1.47369i	0.676064	−	0.736843i	$-0.263684\pi$
$150$	0	0
$151$	− 133.093i	− 0.881412i	−0.897652	−	0.440706i	$-0.854728\pi$
$151$	− 133.093i	− 0.881412i	0.897652	−	0.440706i	$-0.145272\pi$
$152$	0	0
$153$	−55.0788	−0.359992
$154$	0	0
$155$	−277.937	−1.79314
$156$	0	0
$157$	122.190i	0.778283i	0.921178	+	0.389141i	$0.127228\pi$
$157$	122.190i	0.778283i	−0.921178	+	0.389141i	$0.872772\pi$
$158$	0	0
$159$	144.658i	0.909798i
$160$	0	0
$161$	47.4451	0.294690
$162$	0	0
$163$	−8.99765	−0.0552003	−0.0276002	−	0.999619i	$-0.508787\pi$
$163$	−8.99765	−0.0552003	−0.0276002	+	0.999619i	$0.508787\pi$
$164$	0	0
$165$	− 103.366i	− 0.626460i
$166$	0	0
$167$	− 76.3383i	− 0.457115i	−0.973530	−	0.228558i	$-0.926599\pi$
$167$	− 76.3383i	− 0.457115i	0.973530	−	0.228558i	$-0.0734010\pi$
$168$	0	0
$169$	156.113	0.923743
$170$	0	0
$171$	55.9589	0.327245
$172$	0	0
$173$	− 17.1179i	− 0.0989474i	−0.998775	−	0.0494737i	$-0.984246\pi$
$173$	− 17.1179i	− 0.0989474i	0.998775	−	0.0494737i	$-0.0157544\pi$
$174$	0	0
$175$	87.9237i	0.502421i
$176$	0	0
$177$	−1.64044	−0.00926800
$178$	0	0
$179$	134.176	0.749587	0.374794	−	0.927108i	$-0.377714\pi$
$179$	134.176	0.749587	0.374794	+	0.927108i	$0.377714\pi$
$180$	0	0
$181$	− 148.616i	− 0.821084i	−0.911842	−	0.410542i	$-0.865340\pi$
$181$	− 148.616i	− 0.821084i	0.911842	−	0.410542i	$-0.134660\pi$
$182$	0	0
$183$	28.2786i	0.154528i
$184$	0	0
$185$	−236.945	−1.28078
$186$	0	0
$187$	201.579	1.07796
$188$	0	0
$189$	286.755i	1.51722i
$190$	0	0
$191$	158.705i	0.830915i	0.909613	+	0.415458i	$0.136379\pi$
$191$	158.705i	0.830915i	−0.909613	+	0.415458i	$0.863621\pi$
$192$	0	0
$193$	144.529	0.748857	0.374428	−	0.927256i	$-0.377839\pi$
$193$	144.529	0.748857	0.374428	+	0.927256i	$0.377839\pi$
$194$	0	0
$195$	56.0387	0.287378
$196$	0	0
$197$	69.8122i	0.354376i	0.984177	+	0.177188i	$0.0567001\pi$
$197$	69.8122i	0.354376i	−0.984177	+	0.177188i	$0.943300\pi$
$198$	0	0
$199$	− 221.104i	− 1.11108i	−0.831491	−	0.555538i	$-0.812512\pi$
$199$	− 221.104i	− 1.11108i	0.831491	−	0.555538i	$-0.187488\pi$
$200$	0	0
$201$	295.361	1.46946
$202$	0	0
$203$	120.595	0.594063
$204$	0	0
$205$	182.814i	0.891778i
$206$	0	0
$207$	− 8.67714i	− 0.0419185i
$208$	0	0
$209$	−204.800	−0.979902
$210$	0	0
$211$	−61.1501	−0.289811	−0.144905	−	0.989446i	$-0.546288\pi$
$211$	−61.1501	−0.289811	−0.144905	+	0.989446i	$0.546288\pi$
$212$	0	0
$213$	− 323.894i	− 1.52063i
$214$	0	0
$215$	408.317i	1.89915i
$216$	0	0
$217$	−472.340	−2.17668
$218$	0	0
$219$	239.408	1.09319
$220$	0	0
$221$	109.284i	0.494496i
$222$	0	0
$223$	77.1161i	0.345812i	0.984938	+	0.172906i	$0.0553157\pi$
$223$	77.1161i	0.345812i	−0.984938	+	0.172906i	$0.944684\pi$
$224$	0	0
$225$	16.0802	0.0714675
$226$	0	0
$227$	−154.098	−0.678845	−0.339422	−	0.940634i	$-0.610232\pi$
$227$	−154.098	−0.678845	−0.339422	+	0.940634i	$0.610232\pi$
$228$	0	0
$229$	246.803i	1.07774i	0.842388	+	0.538871i	$0.181149\pi$
$229$	246.803i	1.07774i	−0.842388	+	0.538871i	$0.818851\pi$
$230$	0	0
$231$	− 175.665i	− 0.760455i
$232$	0	0
$233$	378.862	1.62602	0.813009	−	0.582252i	$-0.197828\pi$
$233$	378.862	1.62602	0.813009	+	0.582252i	$0.197828\pi$
$234$	0	0
$235$	−129.248	−0.549990
$236$	0	0
$237$	116.685i	0.492342i
$238$	0	0
$239$	118.684i	0.496584i	0.968685	+	0.248292i	$0.0798692\pi$
$239$	118.684i	0.496584i	−0.968685	+	0.248292i	$0.920131\pi$
$240$	0	0
$241$	301.069	1.24925	0.624623	−	0.780926i	$-0.285252\pi$
$241$	301.069	1.24925	0.624623	+	0.780926i	$0.285252\pi$
$242$	0	0
$243$	96.1097	0.395513
$244$	0	0
$245$	284.494i	1.16120i
$246$	0	0
$247$	− 111.030i	− 0.449513i
$248$	0	0
$249$	63.1282	0.253527
$250$	0	0
$251$	195.686	0.779626	0.389813	−	0.920894i	$-0.372540\pi$
$251$	195.686	0.779626	0.389813	+	0.920894i	$0.372540\pi$
$252$	0	0
$253$	31.7568i	0.125521i
$254$	0	0
$255$	− 475.201i	− 1.86353i
$256$	0	0
$257$	−269.964	−1.05044	−0.525221	−	0.850966i	$-0.676018\pi$
$257$	−269.964	−1.05044	−0.525221	+	0.850966i	$0.676018\pi$
$258$	0	0
$259$	−402.675	−1.55473
$260$	0	0
$261$	− 22.0553i	− 0.0845031i
$262$	0	0
$263$	255.115i	0.970020i	0.874509	+	0.485010i	$0.161184\pi$
$263$	255.115i	0.970020i	−0.874509	+	0.485010i	$0.838816\pi$
$264$	0	0
$265$	314.034	1.18503
$266$	0	0
$267$	19.6307	0.0735232
$268$	0	0
$269$	− 23.4786i	− 0.0872811i	−0.999047	−	0.0436405i	$-0.986104\pi$
$269$	− 23.4786i	− 0.0872811i	0.999047	−	0.0436405i	$-0.0138956\pi$
$270$	0	0
$271$	319.924i	1.18053i	0.807210	+	0.590265i	$0.200977\pi$
$271$	319.924i	1.18053i	−0.807210	+	0.590265i	$0.799023\pi$
$272$	0	0
$273$	95.2349	0.348846
$274$	0	0
$275$	−58.8506	−0.214002
$276$	0	0
$277$	− 399.173i	− 1.44106i	−0.693424	−	0.720529i	$-0.743899\pi$
$277$	− 399.173i	− 1.44106i	0.693424	−	0.720529i	$-0.256101\pi$
$278$	0	0
$279$	86.3853i	0.309625i
$280$	0	0
$281$	247.717	0.881557	0.440778	−	0.897616i	$-0.354703\pi$
$281$	247.717	0.881557	0.440778	+	0.897616i	$0.354703\pi$
$282$	0	0
$283$	189.081	0.668130	0.334065	−	0.942550i	$-0.391580\pi$
$283$	189.081	0.668130	0.334065	+	0.942550i	$0.391580\pi$
$284$	0	0
$285$	482.794i	1.69401i
$286$	0	0
$287$	310.684i	1.08252i
$288$	0	0
$289$	637.711	2.20661
$290$	0	0
$291$	231.502	0.795540
$292$	0	0
$293$	41.3099i	0.140989i	0.997512	+	0.0704947i	$0.0224578\pi$
$293$	41.3099i	0.140989i	−0.997512	+	0.0704947i	$0.977542\pi$
$294$	0	0
$295$	3.56118i	0.0120718i
$296$	0	0
$297$	−191.936	−0.646248
$298$	0	0
$299$	−17.2166	−0.0575805
$300$	0	0
$301$	693.914i	2.30536i
$302$	0	0
$303$	− 26.9781i	− 0.0890367i
$304$	0	0
$305$	61.3893	0.201276
$306$	0	0
$307$	14.2003	0.0462550	0.0231275	−	0.999733i	$-0.492638\pi$
$307$	14.2003	0.0462550	0.0231275	+	0.999733i	$0.492638\pi$
$308$	0	0
$309$	208.236i	0.673904i
$310$	0	0
$311$	143.262i	0.460651i	0.973114	+	0.230325i	$0.0739791\pi$
$311$	143.262i	0.460651i	−0.973114	+	0.230325i	$0.926021\pi$
$312$	0	0
$313$	506.132	1.61704	0.808518	−	0.588472i	$-0.200270\pi$
$313$	506.132	1.61704	0.808518	+	0.588472i	$0.200270\pi$
$314$	0	0
$315$	104.198	0.330788
$316$	0	0
$317$	397.899i	1.25520i	0.778536	+	0.627601i	$0.215963\pi$
$317$	397.899i	1.25520i	−0.778536	+	0.627601i	$0.784037\pi$
$318$	0	0
$319$	80.7185i	0.253036i
$320$	0	0
$321$	−89.4722	−0.278730
$322$	0	0
$323$	−941.518	−2.91492
$324$	0	0
$325$	− 31.9052i	− 0.0981698i
$326$	0	0
$327$	419.708i	1.28351i
$328$	0	0
$329$	−219.650	−0.667628
$330$	0	0
$331$	124.468	0.376037	0.188018	−	0.982166i	$-0.439794\pi$
$331$	124.468	0.376037	0.188018	+	0.982166i	$0.439794\pi$
$332$	0	0
$333$	73.6444i	0.221154i
$334$	0	0
$335$	− 641.192i	− 1.91400i
$336$	0	0
$337$	−480.788	−1.42667	−0.713335	−	0.700823i	$-0.752816\pi$
$337$	−480.788	−1.42667	−0.713335	+	0.700823i	$0.752816\pi$
$338$	0	0
$339$	437.883	1.29169
$340$	0	0
$341$	− 316.155i	− 0.927140i
$342$	0	0
$343$	− 1.27323i	− 0.00371203i
$344$	0	0
$345$	74.8633	0.216995
$346$	0	0
$347$	246.836	0.711343	0.355672	−	0.934611i	$-0.384252\pi$
$347$	246.836	0.711343	0.355672	+	0.934611i	$0.384252\pi$
$348$	0	0
$349$	− 412.101i	− 1.18081i	−0.807109	−	0.590403i	$-0.798969\pi$
$349$	− 412.101i	− 1.18081i	0.807109	−	0.590403i	$-0.201031\pi$
$350$	0	0
$351$	− 104.056i	− 0.296455i
$352$	0	0
$353$	−614.680	−1.74130	−0.870652	−	0.491900i	$-0.836303\pi$
$353$	−614.680	−1.74130	−0.870652	+	0.491900i	$0.836303\pi$
$354$	0	0
$355$	−703.133	−1.98066
$356$	0	0
$357$	− 807.579i	− 2.26213i
$358$	0	0
$359$	− 631.559i	− 1.75922i	−0.475698	−	0.879609i	$-0.657804\pi$
$359$	− 631.559i	− 1.75922i	0.475698	−	0.879609i	$-0.342196\pi$
$360$	0	0
$361$	595.562	1.64976
$362$	0	0
$363$	−206.888	−0.569940
$364$	0	0
$365$	− 519.725i	− 1.42390i
$366$	0	0
$367$	674.284i	1.83729i	0.395087	+	0.918644i	$0.370714\pi$
$367$	674.284i	1.83729i	−0.395087	+	0.918644i	$0.629286\pi$
$368$	0	0
$369$	56.8203	0.153985
$370$	0	0
$371$	533.684	1.43850
$372$	0	0
$373$	660.303i	1.77025i	0.465355	+	0.885124i	$0.345927\pi$
$373$	660.303i	1.77025i	−0.465355	+	0.885124i	$0.654073\pi$
$374$	0	0
$375$	− 251.518i	− 0.670714i
$376$	0	0
$377$	−43.7606	−0.116076
$378$	0	0
$379$	−333.991	−0.881243	−0.440621	−	0.897693i	$-0.645242\pi$
$379$	−333.991	−0.881243	−0.440621	+	0.897693i	$0.645242\pi$
$380$	0	0
$381$	110.440i	0.289868i
$382$	0	0
$383$	− 606.133i	− 1.58259i	−0.611433	−	0.791296i	$-0.709406\pi$
$383$	− 606.133i	− 1.58259i	0.611433	−	0.791296i	$-0.290594\pi$
$384$	0	0
$385$	−381.346	−0.990510
$386$	0	0
$387$	126.908	0.327929
$388$	0	0
$389$	− 703.583i	− 1.80870i	−0.426796	−	0.904348i	$-0.640358\pi$
$389$	− 703.583i	− 1.80870i	0.426796	−	0.904348i	$-0.359642\pi$
$390$	0	0
$391$	145.994i	0.373387i
$392$	0	0
$393$	114.185	0.290547
$394$	0	0
$395$	253.308	0.641287
$396$	0	0
$397$	− 331.481i	− 0.834965i	−0.908685	−	0.417483i	$-0.862912\pi$
$397$	− 331.481i	− 0.834965i	0.908685	−	0.417483i	$-0.137088\pi$
$398$	0	0
$399$	820.483i	2.05635i
$400$	0	0
$401$	−171.314	−0.427218	−0.213609	−	0.976919i	$-0.568522\pi$
$401$	−171.314	−0.427218	−0.213609	+	0.976919i	$0.568522\pi$
$402$	0	0
$403$	171.400	0.425310
$404$	0	0
$405$	357.676i	0.883150i
$406$	0	0
$407$	− 269.525i	− 0.662224i
$408$	0	0
$409$	−593.101	−1.45013	−0.725063	−	0.688683i	$-0.758189\pi$
$409$	−593.101	−1.45013	−0.725063	+	0.688683i	$0.758189\pi$
$410$	0	0
$411$	−62.3097	−0.151605
$412$	0	0
$413$	6.05204i	0.0146538i
$414$	0	0
$415$	− 137.043i	− 0.330225i
$416$	0	0
$417$	299.870	0.719112
$418$	0	0
$419$	51.3017	0.122438	0.0612192	−	0.998124i	$-0.480501\pi$
$419$	51.3017	0.122438	0.0612192	+	0.998124i	$0.480501\pi$
$420$	0	0
$421$	256.352i	0.608911i	0.952527	+	0.304456i	$0.0984745\pi$
$421$	256.352i	0.608911i	−0.952527	+	0.304456i	$0.901526\pi$
$422$	0	0
$423$	40.1713i	0.0949675i
$424$	0	0
$425$	−270.552	−0.636593
$426$	0	0
$427$	104.328	0.244328
$428$	0	0
$429$	63.7442i	0.148588i
$430$	0	0
$431$	540.299i	1.25360i	0.779182	+	0.626798i	$0.215635\pi$
$431$	540.299i	1.25360i	−0.779182	+	0.626798i	$0.784365\pi$
$432$	0	0
$433$	−193.753	−0.447466	−0.223733	−	0.974650i	$-0.571824\pi$
$433$	−193.753	−0.447466	−0.223733	+	0.974650i	$0.571824\pi$
$434$	0	0
$435$	190.285	0.437438
$436$	0	0
$437$	− 148.327i	− 0.339421i
$438$	0	0
$439$	− 292.776i	− 0.666916i	−0.942765	−	0.333458i	$-0.891784\pi$
$439$	− 292.776i	− 0.666916i	0.942765	−	0.333458i	$-0.108216\pi$
$440$	0	0
$441$	88.4233	0.200506
$442$	0	0
$443$	−63.9694	−0.144400	−0.0722002	−	0.997390i	$-0.523002\pi$
$443$	−63.9694	−0.144400	−0.0722002	+	0.997390i	$0.523002\pi$
$444$	0	0
$445$	− 42.6158i	− 0.0957657i
$446$	0	0
$447$	− 588.812i	− 1.31725i
$448$	0	0
$449$	−420.907	−0.937433	−0.468716	−	0.883349i	$-0.655283\pi$
$449$	−420.907	−0.937433	−0.468716	+	0.883349i	$0.655283\pi$
$450$	0	0
$451$	−207.952	−0.461091
$452$	0	0
$453$	− 356.895i	− 0.787849i
$454$	0	0
$455$	− 206.743i	− 0.454380i
$456$	0	0
$457$	−355.239	−0.777329	−0.388664	−	0.921379i	$-0.627063\pi$
$457$	−355.239	−0.777329	−0.388664	+	0.921379i	$0.627063\pi$
$458$	0	0
$459$	−882.379	−1.92239
$460$	0	0
$461$	− 159.661i	− 0.346335i	−0.984892	−	0.173168i	$-0.944600\pi$
$461$	− 159.661i	− 0.346335i	0.984892	−	0.173168i	$-0.0554002\pi$
$462$	0	0
$463$	213.425i	0.460961i	0.973077	+	0.230481i	$0.0740298\pi$
$463$	213.425i	0.460961i	−0.973077	+	0.230481i	$0.925970\pi$
$464$	0	0
$465$	−745.302	−1.60280
$466$	0	0
$467$	774.059	1.65751	0.828757	−	0.559608i	$-0.189048\pi$
$467$	774.059	1.65751	0.828757	+	0.559608i	$0.189048\pi$
$468$	0	0
$469$	− 1089.67i	− 2.32339i
$470$	0	0
$471$	327.659i	0.695667i
$472$	0	0
$473$	−464.462	−0.981949
$474$	0	0
$475$	274.875	0.578684
$476$	0	0
$477$	− 97.6044i	− 0.204621i
$478$	0	0
$479$	− 455.636i	− 0.951224i	−0.879655	−	0.475612i	$-0.842227\pi$
$479$	− 455.636i	− 0.951224i	0.879655	−	0.475612i	$-0.157773\pi$
$480$	0	0
$481$	146.120	0.303784
$482$	0	0
$483$	127.226	0.263409
$484$	0	0
$485$	− 502.562i	− 1.03621i
$486$	0	0
$487$	118.418i	0.243157i	0.992582	+	0.121579i	$0.0387957\pi$
$487$	118.418i	0.243157i	−0.992582	+	0.121579i	$0.961204\pi$
$488$	0	0
$489$	−24.1276	−0.0493408
$490$	0	0
$491$	−621.373	−1.26553	−0.632763	−	0.774346i	$-0.718079\pi$
$491$	−621.373	−1.26553	−0.632763	+	0.774346i	$0.718079\pi$
$492$	0	0
$493$	371.084i	0.752707i
$494$	0	0
$495$	69.7436i	0.140896i
$496$	0	0
$497$	−1194.94	−2.40430
$498$	0	0
$499$	26.2624	0.0526300	0.0263150	−	0.999654i	$-0.491623\pi$
$499$	26.2624	0.0526300	0.0263150	+	0.999654i	$0.491623\pi$
$500$	0	0
$501$	− 204.705i	− 0.408592i
$502$	0	0
$503$	272.529i	0.541808i	0.962606	+	0.270904i	$0.0873225\pi$
$503$	272.529i	0.541808i	−0.962606	+	0.270904i	$0.912677\pi$
$504$	0	0
$505$	−58.5660	−0.115972
$506$	0	0
$507$	418.623	0.825687
$508$	0	0
$509$	− 406.140i	− 0.797918i	−0.916969	−	0.398959i	$-0.869372\pi$
$509$	− 406.140i	− 0.797918i	0.916969	−	0.398959i	$-0.130628\pi$
$510$	0	0
$511$	− 883.246i	− 1.72847i
$512$	0	0
$513$	896.479	1.74752
$514$	0	0
$515$	452.054	0.877776
$516$	0	0
$517$	− 147.020i	− 0.284371i
$518$	0	0
$519$	− 45.9024i	− 0.0884440i
$520$	0	0
$521$	−25.1954	−0.0483596	−0.0241798	−	0.999708i	$-0.507697\pi$
$521$	−25.1954	−0.0483596	−0.0241798	+	0.999708i	$0.507697\pi$
$522$	0	0
$523$	229.774	0.439338	0.219669	−	0.975574i	$-0.429502\pi$
$523$	229.774	0.439338	0.219669	+	0.975574i	$0.429502\pi$
$524$	0	0
$525$	235.772i	0.449089i
$526$	0	0
$527$	− 1453.45i	− 2.75797i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	1.10684	0.00208445
$532$	0	0
$533$	− 112.739i	− 0.211518i
$534$	0	0
$535$	194.233i	0.363052i
$536$	0	0
$537$	359.800	0.670018
$538$	0	0
$539$	−323.613	−0.600396
$540$	0	0
$541$	765.739i	1.41541i	0.706506	+	0.707707i	$0.250270\pi$
$541$	765.739i	1.41541i	−0.706506	+	0.707707i	$0.749730\pi$
$542$	0	0
$543$	− 398.521i	− 0.733925i
$544$	0	0
$545$	911.132	1.67180
$546$	0	0
$547$	346.375	0.633226	0.316613	−	0.948555i	$-0.397454\pi$
$547$	346.375	0.633226	0.316613	+	0.948555i	$0.397454\pi$
$548$	0	0
$549$	− 19.0803i	− 0.0347547i
$550$	0	0
$551$	− 377.014i	− 0.684236i
$552$	0	0
$553$	430.485	0.778453
$554$	0	0
$555$	−635.378	−1.14482
$556$	0	0
$557$	− 137.182i	− 0.246287i	−0.992389	−	0.123143i	$-0.960702\pi$
$557$	− 137.182i	− 0.246287i	0.992389	−	0.123143i	$-0.0392975\pi$
$558$	0	0
$559$	− 251.803i	− 0.450452i
$560$	0	0
$561$	540.542	0.963534
$562$	0	0
$563$	487.016	0.865037	0.432518	−	0.901625i	$-0.357625\pi$
$563$	487.016	0.865037	0.432518	+	0.901625i	$0.357625\pi$
$564$	0	0
$565$	− 950.589i	− 1.68246i
$566$	0	0
$567$	607.851i	1.07205i
$568$	0	0
$569$	−132.078	−0.232123	−0.116062	−	0.993242i	$-0.537027\pi$
$569$	−132.078	−0.232123	−0.116062	+	0.993242i	$0.537027\pi$
$570$	0	0
$571$	−524.715	−0.918941	−0.459470	−	0.888193i	$-0.651961\pi$
$571$	−524.715	−0.918941	−0.459470	+	0.888193i	$0.651961\pi$
$572$	0	0
$573$	425.574i	0.742713i
$574$	0	0
$575$	− 42.6228i	− 0.0741267i
$576$	0	0
$577$	580.590	1.00622	0.503111	−	0.864222i	$-0.332189\pi$
$577$	580.590	1.00622	0.503111	+	0.864222i	$0.332189\pi$
$578$	0	0
$579$	387.562	0.669365
$580$	0	0
$581$	− 232.898i	− 0.400857i
$582$	0	0
$583$	357.215i	0.612718i
$584$	0	0
$585$	−37.8107	−0.0646337
$586$	0	0
$587$	−273.163	−0.465354	−0.232677	−	0.972554i	$-0.574748\pi$
$587$	−273.163	−0.465354	−0.232677	+	0.972554i	$0.574748\pi$
$588$	0	0
$589$	1476.67i	2.50708i
$590$	0	0
$591$	187.205i	0.316759i
$592$	0	0
$593$	482.084	0.812958	0.406479	−	0.913660i	$-0.366756\pi$
$593$	482.084	0.812958	0.406479	+	0.913660i	$0.366756\pi$
$594$	0	0
$595$	−1753.15	−2.94647
$596$	0	0
$597$	− 592.901i	− 0.993134i
$598$	0	0
$599$	861.737i	1.43863i	0.694686	+	0.719313i	$0.255543\pi$
$599$	861.737i	1.43863i	−0.694686	+	0.719313i	$0.744457\pi$
$600$	0	0
$601$	−102.961	−0.171317	−0.0856585	−	0.996325i	$-0.527299\pi$
$601$	−102.961	−0.171317	−0.0856585	+	0.996325i	$0.527299\pi$
$602$	0	0
$603$	−199.288	−0.330494
$604$	0	0
$605$	449.128i	0.742360i
$606$	0	0
$607$	− 598.408i	− 0.985846i	−0.870073	−	0.492923i	$-0.835928\pi$
$607$	− 598.408i	− 0.985846i	0.870073	−	0.492923i	$-0.164072\pi$
$608$	0	0
$609$	323.380	0.531002
$610$	0	0
$611$	79.7051	0.130450
$612$	0	0
$613$	1109.83i	1.81048i	0.424896	+	0.905242i	$0.360311\pi$
$613$	1109.83i	1.81048i	−0.424896	+	0.905242i	$0.639689\pi$
$614$	0	0
$615$	490.225i	0.797115i
$616$	0	0
$617$	334.282	0.541786	0.270893	−	0.962609i	$-0.412681\pi$
$617$	334.282	0.541786	0.270893	+	0.962609i	$0.412681\pi$
$618$	0	0
$619$	846.483	1.36750	0.683750	−	0.729716i	$-0.260348\pi$
$619$	846.483	1.36750	0.683750	+	0.729716i	$0.260348\pi$
$620$	0	0
$621$	− 139.010i	− 0.223849i
$622$	0	0
$623$	− 72.4233i	− 0.116249i
$624$	0	0
$625$	−768.200	−1.22912
$626$	0	0
$627$	−549.180	−0.875884
$628$	0	0
$629$	− 1239.08i	− 1.96992i
$630$	0	0
$631$	209.381i	0.331824i	0.986141	+	0.165912i	$0.0530567\pi$
$631$	209.381i	0.331824i	−0.986141	+	0.165912i	$0.946943\pi$
$632$	0	0
$633$	−163.977	−0.259047
$634$	0	0
$635$	239.750	0.377559
$636$	0	0
$637$	− 175.443i	− 0.275421i
$638$	0	0
$639$	218.540i	0.342003i
$640$	0	0
$641$	−898.820	−1.40222	−0.701108	−	0.713055i	$-0.747311\pi$
$641$	−898.820	−1.40222	−0.701108	+	0.713055i	$0.747311\pi$
$642$	0	0
$643$	−1045.93	−1.62663	−0.813317	−	0.581821i	$-0.802340\pi$
$643$	−1045.93	−1.62663	−0.813317	+	0.581821i	$0.802340\pi$
$644$	0	0
$645$	1094.92i	1.69755i
$646$	0	0
$647$	− 708.110i	− 1.09445i	−0.836985	−	0.547225i	$-0.815684\pi$
$647$	− 708.110i	− 1.09445i	0.836985	−	0.547225i	$-0.184316\pi$
$648$	0	0
$649$	−4.05085	−0.00624168
$650$	0	0
$651$	−1266.60	−1.94563
$652$	0	0
$653$	− 551.482i	− 0.844537i	−0.906471	−	0.422268i	$-0.861234\pi$
$653$	− 551.482i	− 0.844537i	0.906471	−	0.422268i	$-0.138766\pi$
$654$	0	0
$655$	− 247.881i	− 0.378444i
$656$	0	0
$657$	−161.535	−0.245867
$658$	0	0
$659$	−226.444	−0.343618	−0.171809	−	0.985130i	$-0.554961\pi$
$659$	−226.444	−0.343618	−0.171809	+	0.985130i	$0.554961\pi$
$660$	0	0
$661$	− 148.152i	− 0.224133i	−0.993701	−	0.112067i	$-0.964253\pi$
$661$	− 148.152i	− 0.224133i	0.993701	−	0.112067i	$-0.0357470\pi$
$662$	0	0
$663$	293.049i	0.442005i
$664$	0	0
$665$	1781.16	2.67844
$666$	0	0
$667$	−58.4608	−0.0876473
$668$	0	0
$669$	206.790i	0.309104i
$670$	0	0
$671$	69.8305i	0.104069i
$672$	0	0
$673$	165.615	0.246085	0.123043	−	0.992401i	$-0.460735\pi$
$673$	165.615	0.246085	0.123043	+	0.992401i	$0.460735\pi$
$674$	0	0
$675$	257.609	0.381643
$676$	0	0
$677$	188.215i	0.278013i	0.990291	+	0.139006i	$0.0443909\pi$
$677$	188.215i	0.278013i	−0.990291	+	0.139006i	$0.955609\pi$
$678$	0	0
$679$	− 854.078i	− 1.25785i
$680$	0	0
$681$	−413.220	−0.606785
$682$	0	0
$683$	−30.3901	−0.0444950	−0.0222475	−	0.999752i	$-0.507082\pi$
$683$	−30.3901	−0.0444950	−0.0222475	+	0.999752i	$0.507082\pi$
$684$	0	0
$685$	135.266i	0.197469i
$686$	0	0
$687$	661.813i	0.963338i
$688$	0	0
$689$	−193.660	−0.281074
$690$	0	0
$691$	1200.18	1.73687	0.868435	−	0.495804i	$-0.165126\pi$
$691$	1200.18	1.73687	0.868435	+	0.495804i	$0.165126\pi$
$692$	0	0
$693$	118.526i	0.171033i
$694$	0	0
$695$	− 650.979i	− 0.936660i
$696$	0	0
$697$	−956.011	−1.37161
$698$	0	0
$699$	1015.94	1.45341
$700$	0	0
$701$	934.154i	1.33260i	0.745683	+	0.666301i	$0.232123\pi$
$701$	934.154i	1.33260i	−0.745683	+	0.666301i	$0.767877\pi$
$702$	0	0
$703$	1258.88i	1.79072i
$704$	0	0
$705$	−346.584	−0.491608
$706$	0	0
$707$	−99.5300	−0.140778
$708$	0	0
$709$	812.723i	1.14629i	0.819452	+	0.573147i	$0.194278\pi$
$709$	812.723i	1.14629i	−0.819452	+	0.573147i	$0.805722\pi$
$710$	0	0
$711$	− 78.7304i	− 0.110732i
$712$	0	0
$713$	228.977	0.321146
$714$	0	0
$715$	138.380	0.193539
$716$	0	0
$717$	318.255i	0.443871i
$718$	0	0
$719$	1138.42i	1.58334i	0.610948	+	0.791671i	$0.290789\pi$
$719$	1138.42i	1.58334i	−0.610948	+	0.791671i	$0.709211\pi$
$720$	0	0
$721$	768.243	1.06552
$722$	0	0
$723$	807.329	1.11664
$724$	0	0
$725$	− 108.338i	− 0.149431i
$726$	0	0
$727$	514.392i	0.707555i	0.935330	+	0.353777i	$0.115103\pi$
$727$	514.392i	0.707555i	−0.935330	+	0.353777i	$0.884897\pi$
$728$	0	0
$729$	810.706	1.11208
$730$	0	0
$731$	−2135.26	−2.92101
$732$	0	0
$733$	349.612i	0.476960i	0.971147	+	0.238480i	$0.0766492\pi$
$733$	349.612i	0.476960i	−0.971147	+	0.238480i	$0.923351\pi$
$734$	0	0
$735$	762.885i	1.03794i
$736$	0	0
$737$	729.358	0.989630
$738$	0	0
$739$	−192.564	−0.260574	−0.130287	−	0.991476i	$-0.541590\pi$
$739$	−192.564	−0.260574	−0.130287	+	0.991476i	$0.541590\pi$
$740$	0	0
$741$	− 297.732i	− 0.401797i
$742$	0	0
$743$	− 238.623i	− 0.321162i	−0.987023	−	0.160581i	$-0.948663\pi$
$743$	− 238.623i	− 0.321162i	0.987023	−	0.160581i	$-0.0513367\pi$
$744$	0	0
$745$	−1278.24	−1.71575
$746$	0	0
$747$	−42.5942	−0.0570204
$748$	0	0
$749$	330.089i	0.440706i
$750$	0	0
$751$	− 823.600i	− 1.09667i	−0.836259	−	0.548335i	$-0.815262\pi$
$751$	− 823.600i	− 1.09667i	0.836259	−	0.548335i	$-0.184738\pi$
$752$	0	0
$753$	524.742	0.696868
$754$	0	0
$755$	−774.774	−1.02619
$756$	0	0
$757$	− 937.789i	− 1.23882i	−0.785067	−	0.619411i	$-0.787371\pi$
$757$	− 937.789i	− 1.23882i	0.785067	−	0.619411i	$-0.212629\pi$
$758$	0	0
$759$	85.1572i	0.112197i
$760$	0	0
$761$	−867.888	−1.14046	−0.570229	−	0.821486i	$-0.693145\pi$
$761$	−867.888	−1.14046	−0.570229	+	0.821486i	$0.693145\pi$
$762$	0	0
$763$	1548.42	2.02939
$764$	0	0
$765$	320.630i	0.419124i
$766$	0	0
$767$	− 2.19612i	− 0.00286326i
$768$	0	0
$769$	−457.768	−0.595277	−0.297638	−	0.954679i	$-0.596199\pi$
$769$	−457.768	−0.595277	−0.297638	+	0.954679i	$0.596199\pi$
$770$	0	0
$771$	−723.921	−0.938937
$772$	0	0
$773$	− 1102.59i	− 1.42638i	−0.700973	−	0.713188i	$-0.747251\pi$
$773$	− 1102.59i	− 1.42638i	0.700973	−	0.713188i	$-0.252749\pi$
$774$	0	0
$775$	424.332i	0.547525i
$776$	0	0
$777$	−1079.79	−1.38969
$778$	0	0
$779$	971.287	1.24684
$780$	0	0
$781$	− 799.816i	− 1.02409i
$782$	0	0
$783$	− 353.333i	− 0.451255i
$784$	0	0
$785$	711.306	0.906123
$786$	0	0
$787$	326.075	0.414326	0.207163	−	0.978306i	$-0.433577\pi$
$787$	326.075	0.414326	0.207163	+	0.978306i	$0.433577\pi$
$788$	0	0
$789$	684.103i	0.867051i
$790$	0	0
$791$	− 1615.48i	− 2.04232i
$792$	0	0
$793$	−37.8578	−0.0477400
$794$	0	0
$795$	842.096	1.05924
$796$	0	0
$797$	202.946i	0.254638i	0.991862	+	0.127319i	$0.0406372\pi$
$797$	202.946i	0.254638i	−0.991862	+	0.127319i	$0.959363\pi$
$798$	0	0
$799$	− 675.888i	− 0.845918i
$800$	0	0
$801$	−13.2453	−0.0165360
$802$	0	0
$803$	591.189	0.736225
$804$	0	0
$805$	− 276.192i	− 0.343096i
$806$	0	0
$807$	− 62.9590i	− 0.0780161i
$808$	0	0
$809$	506.745	0.626384	0.313192	−	0.949690i	$-0.398602\pi$
$809$	506.745	0.626384	0.313192	+	0.949690i	$0.398602\pi$
$810$	0	0
$811$	1137.44	1.40251	0.701257	−	0.712909i	$-0.252623\pi$
$811$	1137.44	1.40251	0.701257	+	0.712909i	$0.252623\pi$
$812$	0	0
$813$	857.890i	1.05522i
$814$	0	0
$815$	52.3780i	0.0642675i
$816$	0	0
$817$	2169.37	2.65529
$818$	0	0
$819$	−64.2574	−0.0784584
$820$	0	0
$821$	− 317.244i	− 0.386412i	−0.981158	−	0.193206i	$-0.938111\pi$
$821$	− 317.244i	− 0.386412i	0.981158	−	0.193206i	$-0.0618886\pi$
$822$	0	0
$823$	785.795i	0.954793i	0.878688	+	0.477397i	$0.158420\pi$
$823$	785.795i	0.954793i	−0.878688	+	0.477397i	$0.841580\pi$
$824$	0	0
$825$	−157.811	−0.191286
$826$	0	0
$827$	214.732	0.259652	0.129826	−	0.991537i	$-0.458558\pi$
$827$	214.732	0.259652	0.129826	+	0.991537i	$0.458558\pi$
$828$	0	0
$829$	− 730.294i	− 0.880934i	−0.897769	−	0.440467i	$-0.854813\pi$
$829$	− 730.294i	− 0.880934i	0.897769	−	0.440467i	$-0.145187\pi$
$830$	0	0
$831$	− 1070.40i	− 1.28809i
$832$	0	0
$833$	−1487.74	−1.78600
$834$	0	0
$835$	−444.388	−0.532201
$836$	0	0
$837$	1383.92i	1.65343i
$838$	0	0
$839$	− 250.308i	− 0.298341i	−0.988811	−	0.149171i	$-0.952340\pi$
$839$	− 250.308i	− 0.298341i	0.988811	−	0.149171i	$-0.0476603\pi$
$840$	0	0
$841$	692.406	0.823313
$842$	0	0
$843$	664.266	0.787978
$844$	0	0
$845$	− 908.777i	− 1.07548i
$846$	0	0
$847$	763.269i	0.901144i
$848$	0	0
$849$	507.029	0.597207
$850$	0	0
$851$	195.205	0.229383
$852$	0	0
$853$	− 764.203i	− 0.895900i	−0.894059	−	0.447950i	$-0.852154\pi$
$853$	− 764.203i	− 0.895900i	0.894059	−	0.447950i	$-0.147846\pi$
$854$	0	0
$855$	− 325.753i	− 0.380998i
$856$	0	0
$857$	−8.70183	−0.0101538	−0.00507691	−	0.999987i	$-0.501616\pi$
$857$	−8.70183	−0.0101538	−0.00507691	+	0.999987i	$0.501616\pi$
$858$	0	0
$859$	−1550.95	−1.80553	−0.902764	−	0.430137i	$-0.858465\pi$
$859$	−1550.95	−1.80553	−0.902764	+	0.430137i	$0.858465\pi$
$860$	0	0
$861$	833.113i	0.967611i
$862$	0	0
$863$	43.2061i	0.0500651i	0.999687	+	0.0250325i	$0.00796893\pi$
$863$	43.2061i	0.0500651i	−0.999687	+	0.0250325i	$0.992031\pi$
$864$	0	0
$865$	−99.6483	−0.115200
$866$	0	0
$867$	1710.05	1.97238
$868$	0	0
$869$	288.139i	0.331576i
$870$	0	0
$871$	395.413i	0.453976i
$872$	0	0
$873$	−156.200	−0.178924
$874$	0	0
$875$	−927.921	−1.06048
$876$	0	0
$877$	− 1234.87i	− 1.40806i	−0.710169	−	0.704031i	$-0.751382\pi$
$877$	− 1234.87i	− 1.40806i	0.710169	−	0.704031i	$-0.248618\pi$
$878$	0	0
$879$	110.774i	0.126023i
$880$	0	0
$881$	−558.653	−0.634112	−0.317056	−	0.948407i	$-0.602694\pi$
$881$	−558.653	−0.634112	−0.317056	+	0.948407i	$0.602694\pi$
$882$	0	0
$883$	−124.038	−0.140473	−0.0702366	−	0.997530i	$-0.522375\pi$
$883$	−124.038	−0.140473	−0.0702366	+	0.997530i	$0.522375\pi$
$884$	0	0
$885$	9.54946i	0.0107903i
$886$	0	0
$887$	1108.66i	1.24990i	0.780664	+	0.624951i	$0.214881\pi$
$887$	1108.66i	1.24990i	−0.780664	+	0.624951i	$0.785119\pi$
$888$	0	0
$889$	407.443	0.458316
$890$	0	0
$891$	−406.857	−0.456630
$892$	0	0
$893$	686.688i	0.768968i
$894$	0	0
$895$	− 781.079i	− 0.872714i
$896$	0	0
$897$	−46.1671	−0.0514683
$898$	0	0
$899$	582.007	0.647394
$900$	0	0
$901$	1642.21i	1.82265i
$902$	0	0
$903$	1860.76i	2.06064i
$904$	0	0
$905$	−865.138	−0.955954
$906$	0	0
$907$	430.673	0.474833	0.237416	−	0.971408i	$-0.423699\pi$
$907$	430.673	0.474833	0.237416	+	0.971408i	$0.423699\pi$
$908$	0	0
$909$	18.2028i	0.0200251i
$910$	0	0
$911$	978.784i	1.07441i	0.843453	+	0.537203i	$0.180519\pi$
$911$	978.784i	1.07441i	−0.843453	+	0.537203i	$0.819481\pi$
$912$	0	0
$913$	155.887	0.170742
$914$	0	0
$915$	164.618	0.179911
$916$	0	0
$917$	− 421.261i	− 0.459390i
$918$	0	0
$919$	1052.91i	1.14572i	0.819654	+	0.572858i	$0.194165\pi$
$919$	1052.91i	1.14572i	−0.819654	+	0.572858i	$0.805835\pi$
$920$	0	0
$921$	38.0787	0.0413450
$922$	0	0
$923$	433.611	0.469785
$924$	0	0
$925$	361.747i	0.391078i
$926$	0	0
$927$	− 140.502i	− 0.151567i
$928$	0	0
$929$	975.050	1.04957	0.524785	−	0.851235i	$-0.324146\pi$
$929$	975.050	1.04957	0.524785	+	0.851235i	$0.324146\pi$
$930$	0	0
$931$	1511.51	1.62353
$932$	0	0
$933$	384.165i	0.411752i
$934$	0	0
$935$	− 1173.45i	− 1.25503i
$936$	0	0
$937$	1295.34	1.38244	0.691219	−	0.722645i	$-0.257074\pi$
$937$	1295.34	1.38244	0.691219	+	0.722645i	$0.257074\pi$
$938$	0	0
$939$	1357.22	1.44539
$940$	0	0
$941$	213.605i	0.226998i	0.993538	+	0.113499i	$0.0362059\pi$
$941$	213.605i	0.226998i	−0.993538	+	0.113499i	$0.963794\pi$
$942$	0	0
$943$	− 150.610i	− 0.159714i
$944$	0	0
$945$	1669.29	1.76644
$946$	0	0
$947$	−1439.62	−1.52019	−0.760097	−	0.649810i	$-0.774848\pi$
$947$	−1439.62	−1.52019	−0.760097	+	0.649810i	$0.774848\pi$
$948$	0	0
$949$	320.506i	0.337731i
$950$	0	0
$951$	1066.98i	1.12196i
$952$	0	0
$953$	−1256.44	−1.31841	−0.659204	−	0.751964i	$-0.729107\pi$
$953$	−1256.44	−1.31841	−0.659204	+	0.751964i	$0.729107\pi$
$954$	0	0
$955$	923.868	0.967401
$956$	0	0
$957$	216.450i	0.226176i
$958$	0	0
$959$	229.878i	0.239706i
$960$	0	0
$961$	−1318.58	−1.37209
$962$	0	0
$963$	60.3692	0.0626887
$964$	0	0
$965$	− 841.348i	− 0.871863i
$966$	0	0
$967$	1408.92i	1.45700i	0.685047	+	0.728499i	$0.259782\pi$
$967$	1408.92i	1.45700i	−0.685047	+	0.728499i	$0.740218\pi$
$968$	0	0
$969$	−2524.72	−2.60550
$970$	0	0
$971$	−527.196	−0.542941	−0.271470	−	0.962447i	$-0.587510\pi$
$971$	−527.196	−0.542941	−0.271470	+	0.962447i	$0.587510\pi$
$972$	0	0
$973$	− 1106.30i	− 1.13700i
$974$	0	0
$975$	− 85.5552i	− 0.0877490i
$976$	0	0
$977$	−1728.15	−1.76883	−0.884417	−	0.466698i	$-0.845444\pi$
$977$	−1728.15	−1.76883	−0.884417	+	0.466698i	$0.845444\pi$
$978$	0	0
$979$	48.4756	0.0495154
$980$	0	0
$981$	− 283.188i	− 0.288672i
$982$	0	0
$983$	431.863i	0.439332i	0.975575	+	0.219666i	$0.0704967\pi$
$983$	431.863i	0.439332i	−0.975575	+	0.219666i	$0.929503\pi$
$984$	0	0
$985$	406.397	0.412586
$986$	0	0
$987$	−589.001	−0.596759
$988$	0	0
$989$	− 336.389i	− 0.340130i
$990$	0	0
$991$	1658.76i	1.67382i	0.547340	+	0.836910i	$0.315640\pi$
$991$	1658.76i	1.67382i	−0.547340	+	0.836910i	$0.684360\pi$
$992$	0	0
$993$	333.767	0.336120
$994$	0	0
$995$	−1287.11	−1.29358
$996$	0	0
$997$	− 677.009i	− 0.679046i	−0.940598	−	0.339523i	$-0.889734\pi$
$997$	− 677.009i	− 0.679046i	0.940598	−	0.339523i	$-0.110266\pi$
$998$	0	0
$999$	1179.81i	1.18099i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	736.3.g.a.47.31		44
4.3	odd	2		184.3.g.a.139.7	✓	44
8.3	odd	2	inner	736.3.g.a.47.32		44
8.5	even	2		184.3.g.a.139.8	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.3.g.a.139.7	✓	44	4.3	odd	2
184.3.g.a.139.8	yes	44	8.5	even	2
736.3.g.a.47.31		44	1.1	even	1	trivial
736.3.g.a.47.32		44	8.3	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:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	736.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.68155 q^{3} -5.82129i q^{5} -9.89299i q^{7} -1.80931 q^{9} +O(q^{10})\)	\(q+2.68155 q^{3} -5.82129i q^{5} -9.89299i q^{7} -1.80931 q^{9} +6.62174 q^{11} +3.58990i q^{13} -15.6101i q^{15} +30.4419 q^{17} -30.9283 q^{19} -26.5285i q^{21} +4.79583i q^{23} -8.88747 q^{25} -28.9857 q^{27} +12.1899i q^{29} -47.7450i q^{31} +17.7565 q^{33} -57.5900 q^{35} -40.7031i q^{37} +9.62650i q^{39} -31.4044 q^{41} -70.1419 q^{43} +10.5325i q^{45} -22.2026i q^{47} -48.8713 q^{49} +81.6314 q^{51} +53.9457i q^{53} -38.5471i q^{55} -82.9358 q^{57} -0.611750 q^{59} +10.5456i q^{61} +17.8995i q^{63} +20.8979 q^{65} +110.146 q^{67} +12.8602i q^{69} -120.786i q^{71} +89.2799 q^{73} -23.8322 q^{75} -65.5089i q^{77} +43.5141i q^{79} -61.4426 q^{81} +23.5417 q^{83} -177.211i q^{85} +32.6878i q^{87} +7.32067 q^{89} +35.5149 q^{91} -128.030i q^{93} +180.043i q^{95} +86.3316 q^{97} -11.9808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 132 q^{9}+O(q^{10}) \) 44 * q + 132 * q^9	\( 44 q + 132 q^{9} - 8 q^{17} - 244 q^{25} + 48 q^{27} + 16 q^{33} - 96 q^{35} + 88 q^{41} + 128 q^{43} - 340 q^{49} - 160 q^{51} - 176 q^{57} - 16 q^{59} + 96 q^{65} + 288 q^{67} + 280 q^{73} - 160 q^{75} + 284 q^{81} + 480 q^{83} - 200 q^{89} - 192 q^{91} + 184 q^{97} - 256 q^{99}+O(q^{100}) \) 44 * q + 132 * q^9 - 8 * q^17 - 244 * q^25 + 48 * q^27 + 16 * q^33 - 96 * q^35 + 88 * q^41 + 128 * q^43 - 340 * q^49 - 160 * q^51 - 176 * q^57 - 16 * q^59 + 96 * q^65 + 288 * q^67 + 280 * q^73 - 160 * q^75 + 284 * q^81 + 480 * q^83 - 200 * q^89 - 192 * q^91 + 184 * q^97 - 256 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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