LMFDB - Embedded newform 1792.3.d.j.1023.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(1023,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1792.1023");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.d (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:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 5x^{14} + 48x^{12} + 180x^{10} + 1056x^{8} + 2880x^{6} + 12288x^{4} + 20480x^{2} + 65536 $ x^16 + 5x^14 + 48x^12 + 180x^10 + 1056x^8 + 2880x^6 + 12288x^4 + 20480*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{38} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1023.14
Root			$-1.09337 - 1.67467i$ of defining polynomial
Character	$\chi$	$=$	1792.1023
Dual form			1792.3.d.j.1023.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.56747i q^{3} +5.73252 q^{5} +2.64575i q^{7} -11.8618 q^{9} +O(q^{10})$	$q+4.56747i q^{3} +5.73252 q^{5} +2.64575i q^{7} -11.8618 q^{9} +1.40065i q^{11} +19.0821 q^{13} +26.1831i q^{15} -32.2699 q^{17} +12.5675i q^{19} -12.0844 q^{21} -15.8893i q^{23} +7.86180 q^{25} -13.0712i q^{27} +3.29194 q^{29} +22.6705i q^{31} -6.39741 q^{33} +15.1668i q^{35} -54.1537 q^{37} +87.1569i q^{39} +7.59607 q^{41} +20.8478i q^{43} -67.9980 q^{45} +21.6384i q^{47} -7.00000 q^{49} -147.392i q^{51} +0.356667 q^{53} +8.02924i q^{55} -57.4016 q^{57} -26.8583i q^{59} -86.2287 q^{61} -31.3834i q^{63} +109.389 q^{65} +114.523i q^{67} +72.5739 q^{69} +104.792i q^{71} +24.3974 q^{73} +35.9085i q^{75} -3.70576 q^{77} +117.128i q^{79} -47.0539 q^{81} +79.2706i q^{83} -184.988 q^{85} +15.0359i q^{87} -2.66078 q^{89} +50.4865i q^{91} -103.547 q^{93} +72.0433i q^{95} -52.0930 q^{97} -16.6142i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 96 q^{9}+O(q^{10}) $ 16 * q - 96 * q^9	$ 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) $ 16 * q - 96 * q^9 - 160 * q^17 + 32 * q^25 + 64 * q^33 - 256 * q^41 - 112 * q^49 - 112 * q^57 - 144 * q^65 + 224 * q^73 + 96 * q^81 + 1024 * q^89 + 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	4.56747i	1.52249i	0.648464	+	0.761245i	$0.275412\pi$
$3$	4.56747i	1.52249i	−0.648464	+	0.761245i	$0.724588\pi$
$4$	0	0
$5$	5.73252	1.14650	0.573252	−	0.819379i	$-0.305682\pi$
$5$	5.73252	1.14650	0.573252	+	0.819379i	$0.305682\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−11.8618	−1.31798
$10$	0	0
$11$	1.40065i	0.127332i	0.997971	+	0.0636658i	$0.0202792\pi$
$11$	1.40065i	0.127332i	−0.997971	+	0.0636658i	$0.979721\pi$
$12$	0	0
$13$	19.0821	1.46785	0.733927	−	0.679228i	$-0.237685\pi$
$13$	19.0821	1.46785	0.733927	+	0.679228i	$0.237685\pi$
$14$	0	0
$15$	26.1831i	1.74554i
$16$	0	0
$17$	−32.2699	−1.89823	−0.949114	−	0.314932i	$-0.898018\pi$
$17$	−32.2699	−1.89823	−0.949114	+	0.314932i	$0.898018\pi$
$18$	0	0
$19$	12.5675i	0.661446i	0.943728	+	0.330723i	$0.107293\pi$
$19$	12.5675i	0.661446i	−0.943728	+	0.330723i	$0.892707\pi$
$20$	0	0
$21$	−12.0844	−0.575447
$22$	0	0
$23$	− 15.8893i	− 0.690839i	−0.938448	−	0.345419i	$-0.887737\pi$
$23$	− 15.8893i	− 0.690839i	0.938448	−	0.345419i	$-0.112263\pi$
$24$	0	0
$25$	7.86180	0.314472
$26$	0	0
$27$	− 13.0712i	− 0.484118i
$28$	0	0
$29$	3.29194	0.113515	0.0567576	−	0.998388i	$-0.481924\pi$
$29$	3.29194	0.113515	0.0567576	+	0.998388i	$0.481924\pi$
$30$	0	0
$31$	22.6705i	0.731306i	0.930751	+	0.365653i	$0.119154\pi$
$31$	22.6705i	0.731306i	−0.930751	+	0.365653i	$0.880846\pi$
$32$	0	0
$33$	−6.39741	−0.193861
$34$	0	0
$35$	15.1668i	0.433338i
$36$	0	0
$37$	−54.1537	−1.46361	−0.731807	−	0.681512i	$-0.761323\pi$
$37$	−54.1537	−1.46361	−0.731807	+	0.681512i	$0.761323\pi$
$38$	0	0
$39$	87.1569i	2.23479i
$40$	0	0
$41$	7.59607	0.185270	0.0926350	−	0.995700i	$-0.470471\pi$
$41$	7.59607	0.185270	0.0926350	+	0.995700i	$0.470471\pi$
$42$	0	0
$43$	20.8478i	0.484833i	0.970172	+	0.242417i	$0.0779401\pi$
$43$	20.8478i	0.484833i	−0.970172	+	0.242417i	$0.922060\pi$
$44$	0	0
$45$	−67.9980	−1.51107
$46$	0	0
$47$	21.6384i	0.460392i	0.973144	+	0.230196i	$0.0739367\pi$
$47$	21.6384i	0.460392i	−0.973144	+	0.230196i	$0.926063\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 147.392i	− 2.89004i
$52$	0	0
$53$	0.356667	0.00672957	0.00336479	−	0.999994i	$-0.498929\pi$
$53$	0.356667	0.00672957	0.00336479	+	0.999994i	$0.498929\pi$
$54$	0	0
$55$	8.02924i	0.145986i
$56$	0	0
$57$	−57.4016	−1.00705
$58$	0	0
$59$	− 26.8583i	− 0.455226i	−0.973752	−	0.227613i	$-0.926908\pi$
$59$	− 26.8583i	− 0.455226i	0.973752	−	0.227613i	$-0.0730921\pi$
$60$	0	0
$61$	−86.2287	−1.41359	−0.706793	−	0.707420i	$-0.749859\pi$
$61$	−86.2287	−1.41359	−0.706793	+	0.707420i	$0.749859\pi$
$62$	0	0
$63$	− 31.3834i	− 0.498149i
$64$	0	0
$65$	109.389	1.68290
$66$	0	0
$67$	114.523i	1.70929i	0.519211	+	0.854646i	$0.326226\pi$
$67$	114.523i	1.70929i	−0.519211	+	0.854646i	$0.673774\pi$
$68$	0	0
$69$	72.5739	1.05180
$70$	0	0
$71$	104.792i	1.47594i	0.674834	+	0.737969i	$0.264215\pi$
$71$	104.792i	1.47594i	−0.674834	+	0.737969i	$0.735785\pi$
$72$	0	0
$73$	24.3974	0.334211	0.167106	−	0.985939i	$-0.446558\pi$
$73$	24.3974	0.334211	0.167106	+	0.985939i	$0.446558\pi$
$74$	0	0
$75$	35.9085i	0.478781i
$76$	0	0
$77$	−3.70576	−0.0481268
$78$	0	0
$79$	117.128i	1.48263i	0.671157	+	0.741315i	$0.265798\pi$
$79$	117.128i	1.48263i	−0.671157	+	0.741315i	$0.734202\pi$
$80$	0	0
$81$	−47.0539	−0.580913
$82$	0	0
$83$	79.2706i	0.955067i	0.878614	+	0.477534i	$0.158469\pi$
$83$	79.2706i	0.955067i	−0.878614	+	0.477534i	$0.841531\pi$
$84$	0	0
$85$	−184.988	−2.17633
$86$	0	0
$87$	15.0359i	0.172826i
$88$	0	0
$89$	−2.66078	−0.0298964	−0.0149482	−	0.999888i	$-0.504758\pi$
$89$	−2.66078	−0.0298964	−0.0149482	+	0.999888i	$0.504758\pi$
$90$	0	0
$91$	50.4865i	0.554797i
$92$	0	0
$93$	−103.547	−1.11341
$94$	0	0
$95$	72.0433i	0.758350i
$96$	0	0
$97$	−52.0930	−0.537042	−0.268521	−	0.963274i	$-0.586535\pi$
$97$	−52.0930	−0.537042	−0.268521	+	0.963274i	$0.586535\pi$
$98$	0	0
$99$	− 16.6142i	− 0.167820i
$100$	0	0
$101$	91.4742	0.905685	0.452842	−	0.891591i	$-0.350410\pi$
$101$	91.4742	0.905685	0.452842	+	0.891591i	$0.350410\pi$
$102$	0	0
$103$	− 39.7891i	− 0.386302i	−0.981169	−	0.193151i	$-0.938129\pi$
$103$	− 39.7891i	− 0.386302i	0.981169	−	0.193151i	$-0.0618708\pi$
$104$	0	0
$105$	−69.2740	−0.659753
$106$	0	0
$107$	− 82.6631i	− 0.772552i	−0.922383	−	0.386276i	$-0.873761\pi$
$107$	− 82.6631i	− 0.772552i	0.922383	−	0.386276i	$-0.126239\pi$
$108$	0	0
$109$	29.4719	0.270384	0.135192	−	0.990819i	$-0.456835\pi$
$109$	29.4719	0.270384	0.135192	+	0.990819i	$0.456835\pi$
$110$	0	0
$111$	− 247.346i	− 2.22834i
$112$	0	0
$113$	159.133	1.40826	0.704130	−	0.710071i	$-0.251337\pi$
$113$	159.133	1.40826	0.704130	+	0.710071i	$0.251337\pi$
$114$	0	0
$115$	− 91.0857i	− 0.792049i
$116$	0	0
$117$	−226.348	−1.93460
$118$	0	0
$119$	− 85.3781i	− 0.717463i
$120$	0	0
$121$	119.038	0.983787
$122$	0	0
$123$	34.6948i	0.282072i
$124$	0	0
$125$	−98.2451	−0.785961
$126$	0	0
$127$	16.0834i	0.126641i	0.997993	+	0.0633205i	$0.0201690\pi$
$127$	16.0834i	0.126641i	−0.997993	+	0.0633205i	$0.979831\pi$
$128$	0	0
$129$	−95.2219	−0.738154
$130$	0	0
$131$	− 118.136i	− 0.901799i	−0.892575	−	0.450899i	$-0.851103\pi$
$131$	− 118.136i	− 0.901799i	0.892575	−	0.450899i	$-0.148897\pi$
$132$	0	0
$133$	−33.2504	−0.250003
$134$	0	0
$135$	− 74.9308i	− 0.555043i
$136$	0	0
$137$	19.1708	0.139933	0.0699664	−	0.997549i	$-0.477711\pi$
$137$	19.1708	0.139933	0.0699664	+	0.997549i	$0.477711\pi$
$138$	0	0
$139$	− 104.954i	− 0.755062i	−0.925997	−	0.377531i	$-0.876773\pi$
$139$	− 104.954i	− 0.755062i	0.925997	−	0.377531i	$-0.123227\pi$
$140$	0	0
$141$	−98.8329	−0.700942
$142$	0	0
$143$	26.7273i	0.186904i
$144$	0	0
$145$	18.8711	0.130146
$146$	0	0
$147$	− 31.9723i	− 0.217499i
$148$	0	0
$149$	−82.3906	−0.552957	−0.276478	−	0.961020i	$-0.589167\pi$
$149$	−82.3906	−0.552957	−0.276478	+	0.961020i	$0.589167\pi$
$150$	0	0
$151$	57.7395i	0.382381i	0.981553	+	0.191190i	$0.0612347\pi$
$151$	57.7395i	0.382381i	−0.981553	+	0.191190i	$0.938765\pi$
$152$	0	0
$153$	382.779	2.50182
$154$	0	0
$155$	129.959i	0.838445i
$156$	0	0
$157$	−3.72975	−0.0237564	−0.0118782	−	0.999929i	$-0.503781\pi$
$157$	−3.72975	−0.0237564	−0.0118782	+	0.999929i	$0.503781\pi$
$158$	0	0
$159$	1.62907i	0.0102457i
$160$	0	0
$161$	42.0391	0.261112
$162$	0	0
$163$	77.7069i	0.476729i	0.971176	+	0.238365i	$0.0766113\pi$
$163$	77.7069i	0.476729i	−0.971176	+	0.238365i	$0.923389\pi$
$164$	0	0
$165$	−36.6733	−0.222262
$166$	0	0
$167$	− 62.0837i	− 0.371759i	−0.982573	−	0.185879i	$-0.940487\pi$
$167$	− 62.0837i	− 0.371759i	0.982573	−	0.185879i	$-0.0595133\pi$
$168$	0	0
$169$	195.127	1.15459
$170$	0	0
$171$	− 149.073i	− 0.871771i
$172$	0	0
$173$	195.614	1.13072	0.565358	−	0.824846i	$-0.308738\pi$
$173$	195.614	1.13072	0.565358	+	0.824846i	$0.308738\pi$
$174$	0	0
$175$	20.8004i	0.118859i
$176$	0	0
$177$	122.675	0.693077
$178$	0	0
$179$	72.2099i	0.403407i	0.979447	+	0.201704i	$0.0646477\pi$
$179$	72.2099i	0.403407i	−0.979447	+	0.201704i	$0.935352\pi$
$180$	0	0
$181$	−140.980	−0.778895	−0.389448	−	0.921049i	$-0.627334\pi$
$181$	−140.980	−0.778895	−0.389448	+	0.921049i	$0.627334\pi$
$182$	0	0
$183$	− 393.847i	− 2.15217i
$184$	0	0
$185$	−310.437	−1.67804
$186$	0	0
$187$	− 45.1987i	− 0.241704i
$188$	0	0
$189$	34.5831	0.182979
$190$	0	0
$191$	− 284.473i	− 1.48939i	−0.667407	−	0.744693i	$-0.732596\pi$
$191$	− 284.473i	− 1.48939i	0.667407	−	0.744693i	$-0.267404\pi$
$192$	0	0
$193$	−123.850	−0.641710	−0.320855	−	0.947128i	$-0.603970\pi$
$193$	−123.850	−0.641710	−0.320855	+	0.947128i	$0.603970\pi$
$194$	0	0
$195$	499.629i	2.56220i
$196$	0	0
$197$	108.098	0.548721	0.274361	−	0.961627i	$-0.411534\pi$
$197$	108.098	0.548721	0.274361	+	0.961627i	$0.411534\pi$
$198$	0	0
$199$	− 331.854i	− 1.66761i	−0.552060	−	0.833804i	$-0.686158\pi$
$199$	− 331.854i	− 1.66761i	0.552060	−	0.833804i	$-0.313842\pi$
$200$	0	0
$201$	−523.079	−2.60238
$202$	0	0
$203$	8.70966i	0.0429047i
$204$	0	0
$205$	43.5446	0.212413
$206$	0	0
$207$	188.476i	0.910510i
$208$	0	0
$209$	−17.6026	−0.0842229
$210$	0	0
$211$	26.3950i	0.125095i	0.998042	+	0.0625475i	$0.0199225\pi$
$211$	26.3950i	0.125095i	−0.998042	+	0.0625475i	$0.980078\pi$
$212$	0	0
$213$	−478.633	−2.24710
$214$	0	0
$215$	119.511i	0.555864i
$216$	0	0
$217$	−59.9804	−0.276408
$218$	0	0
$219$	111.434i	0.508833i
$220$	0	0
$221$	−615.777	−2.78632
$222$	0	0
$223$	− 161.183i	− 0.722796i	−0.932412	−	0.361398i	$-0.882300\pi$
$223$	− 161.183i	− 0.722796i	0.932412	−	0.361398i	$-0.117700\pi$
$224$	0	0
$225$	−93.2551	−0.414467
$226$	0	0
$227$	− 171.279i	− 0.754533i	−0.926105	−	0.377266i	$-0.876864\pi$
$227$	− 171.279i	− 0.754533i	0.926105	−	0.377266i	$-0.123136\pi$
$228$	0	0
$229$	229.251	1.00110	0.500548	−	0.865709i	$-0.333132\pi$
$229$	229.251	1.00110	0.500548	+	0.865709i	$0.333132\pi$
$230$	0	0
$231$	− 16.9260i	− 0.0732726i
$232$	0	0
$233$	270.154	1.15946	0.579730	−	0.814808i	$-0.303158\pi$
$233$	270.154	1.15946	0.579730	+	0.814808i	$0.303158\pi$
$234$	0	0
$235$	124.043i	0.527841i
$236$	0	0
$237$	−534.978	−2.25729
$238$	0	0
$239$	− 157.155i	− 0.657551i	−0.944408	−	0.328776i	$-0.893364\pi$
$239$	− 157.155i	− 0.657551i	0.944408	−	0.328776i	$-0.106636\pi$
$240$	0	0
$241$	−97.7124	−0.405445	−0.202723	−	0.979236i	$-0.564979\pi$
$241$	−97.7124	−0.405445	−0.202723	+	0.979236i	$0.564979\pi$
$242$	0	0
$243$	− 332.558i	− 1.36855i
$244$	0	0
$245$	−40.1276	−0.163786
$246$	0	0
$247$	239.814i	0.970906i
$248$	0	0
$249$	−362.066	−1.45408
$250$	0	0
$251$	313.145i	1.24759i	0.781587	+	0.623796i	$0.214410\pi$
$251$	313.145i	1.24759i	−0.781587	+	0.623796i	$0.785590\pi$
$252$	0	0
$253$	22.2553	0.0879655
$254$	0	0
$255$	− 844.927i	− 3.31344i
$256$	0	0
$257$	−348.855	−1.35741	−0.678707	−	0.734409i	$-0.737459\pi$
$257$	−348.855	−1.35741	−0.678707	+	0.734409i	$0.737459\pi$
$258$	0	0
$259$	− 143.277i	− 0.553194i
$260$	0	0
$261$	−39.0484	−0.149611
$262$	0	0
$263$	384.364i	1.46146i	0.682667	+	0.730729i	$0.260820\pi$
$263$	384.364i	1.46146i	−0.682667	+	0.730729i	$0.739180\pi$
$264$	0	0
$265$	2.04460	0.00771548
$266$	0	0
$267$	− 12.1530i	− 0.0455170i
$268$	0	0
$269$	37.7613	0.140376	0.0701882	−	0.997534i	$-0.477640\pi$
$269$	37.7613	0.140376	0.0701882	+	0.997534i	$0.477640\pi$
$270$	0	0
$271$	308.730i	1.13922i	0.821914	+	0.569612i	$0.192907\pi$
$271$	308.730i	1.13922i	−0.821914	+	0.569612i	$0.807093\pi$
$272$	0	0
$273$	−230.596	−0.844673
$274$	0	0
$275$	11.0116i	0.0400422i
$276$	0	0
$277$	−244.210	−0.881623	−0.440812	−	0.897600i	$-0.645309\pi$
$277$	−244.210	−0.881623	−0.440812	+	0.897600i	$0.645309\pi$
$278$	0	0
$279$	− 268.913i	− 0.963845i
$280$	0	0
$281$	−266.569	−0.948646	−0.474323	−	0.880351i	$-0.657307\pi$
$281$	−266.569	−0.948646	−0.474323	+	0.880351i	$0.657307\pi$
$282$	0	0
$283$	− 165.605i	− 0.585177i	−0.956238	−	0.292589i	$-0.905483\pi$
$283$	− 165.605i	− 0.585177i	0.956238	−	0.292589i	$-0.0945166\pi$
$284$	0	0
$285$	−329.056	−1.15458
$286$	0	0
$287$	20.0973i	0.0700255i
$288$	0	0
$289$	752.346	2.60327
$290$	0	0
$291$	− 237.933i	− 0.817641i
$292$	0	0
$293$	−34.3652	−0.117288	−0.0586438	−	0.998279i	$-0.518678\pi$
$293$	−34.3652	−0.117288	−0.0586438	+	0.998279i	$0.518678\pi$
$294$	0	0
$295$	− 153.966i	− 0.521918i
$296$	0	0
$297$	18.3081	0.0616434
$298$	0	0
$299$	− 303.201i	− 1.01405i
$300$	0	0
$301$	−55.1582	−0.183250
$302$	0	0
$303$	417.806i	1.37890i
$304$	0	0
$305$	−494.308	−1.62068
$306$	0	0
$307$	− 222.934i	− 0.726170i	−0.931756	−	0.363085i	$-0.881724\pi$
$307$	− 222.934i	− 0.726170i	0.931756	−	0.363085i	$-0.118276\pi$
$308$	0	0
$309$	181.736	0.588141
$310$	0	0
$311$	419.934i	1.35027i	0.737694	+	0.675135i	$0.235915\pi$
$311$	419.934i	1.35027i	−0.737694	+	0.675135i	$0.764085\pi$
$312$	0	0
$313$	293.869	0.938878	0.469439	−	0.882965i	$-0.344456\pi$
$313$	293.869	0.938878	0.469439	+	0.882965i	$0.344456\pi$
$314$	0	0
$315$	− 179.906i	− 0.571130i
$316$	0	0
$317$	423.461	1.33584	0.667919	−	0.744234i	$-0.267185\pi$
$317$	423.461	1.33584	0.667919	+	0.744234i	$0.267185\pi$
$318$	0	0
$319$	4.61085i	0.0144541i
$320$	0	0
$321$	377.561	1.17620
$322$	0	0
$323$	− 405.551i	− 1.25558i
$324$	0	0
$325$	150.020	0.461599
$326$	0	0
$327$	134.612i	0.411658i
$328$	0	0
$329$	−57.2499	−0.174012
$330$	0	0
$331$	126.666i	0.382678i	0.981524	+	0.191339i	$0.0612830\pi$
$331$	126.666i	0.382678i	−0.981524	+	0.191339i	$0.938717\pi$
$332$	0	0
$333$	642.361	1.92901
$334$	0	0
$335$	656.503i	1.95971i
$336$	0	0
$337$	302.404	0.897341	0.448671	−	0.893697i	$-0.351898\pi$
$337$	302.404	0.897341	0.448671	+	0.893697i	$0.351898\pi$
$338$	0	0
$339$	726.838i	2.14406i
$340$	0	0
$341$	−31.7533	−0.0931183
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	416.031	1.20589
$346$	0	0
$347$	− 320.532i	− 0.923724i	−0.886952	−	0.461862i	$-0.847182\pi$
$347$	− 320.532i	− 0.923724i	0.886952	−	0.461862i	$-0.152818\pi$
$348$	0	0
$349$	380.678	1.09077	0.545385	−	0.838186i	$-0.316384\pi$
$349$	380.678	1.09077	0.545385	+	0.838186i	$0.316384\pi$
$350$	0	0
$351$	− 249.426i	− 0.710614i
$352$	0	0
$353$	−364.369	−1.03221	−0.516104	−	0.856526i	$-0.672618\pi$
$353$	−364.369	−1.03221	−0.516104	+	0.856526i	$0.672618\pi$
$354$	0	0
$355$	600.720i	1.69217i
$356$	0	0
$357$	389.962	1.09233
$358$	0	0
$359$	111.995i	0.311965i	0.987760	+	0.155982i	$0.0498543\pi$
$359$	111.995i	0.311965i	−0.987760	+	0.155982i	$0.950146\pi$
$360$	0	0
$361$	203.059	0.562489
$362$	0	0
$363$	543.704i	1.49781i
$364$	0	0
$365$	139.859	0.383174
$366$	0	0
$367$	439.042i	1.19630i	0.801384	+	0.598150i	$0.204097\pi$
$367$	439.042i	1.19630i	−0.801384	+	0.598150i	$0.795903\pi$
$368$	0	0
$369$	−90.1030	−0.244182
$370$	0	0
$371$	0.943653i	0.00254354i
$372$	0	0
$373$	254.996	0.683637	0.341818	−	0.939766i	$-0.388957\pi$
$373$	254.996	0.683637	0.341818	+	0.939766i	$0.388957\pi$
$374$	0	0
$375$	− 448.732i	− 1.19662i
$376$	0	0
$377$	62.8172	0.166624
$378$	0	0
$379$	− 603.048i	− 1.59116i	−0.605852	−	0.795578i	$-0.707167\pi$
$379$	− 603.048i	− 1.59116i	0.605852	−	0.795578i	$-0.292833\pi$
$380$	0	0
$381$	−73.4605	−0.192810
$382$	0	0
$383$	73.3855i	0.191607i	0.995400	+	0.0958035i	$0.0305420\pi$
$383$	73.3855i	0.191607i	−0.995400	+	0.0958035i	$0.969458\pi$
$384$	0	0
$385$	−21.2434	−0.0551776
$386$	0	0
$387$	− 247.293i	− 0.639000i
$388$	0	0
$389$	340.800	0.876092	0.438046	−	0.898953i	$-0.355671\pi$
$389$	340.800	0.876092	0.438046	+	0.898953i	$0.355671\pi$
$390$	0	0
$391$	512.745i	1.31137i
$392$	0	0
$393$	539.581	1.37298
$394$	0	0
$395$	671.438i	1.69984i
$396$	0	0
$397$	111.540	0.280957	0.140478	−	0.990084i	$-0.455136\pi$
$397$	111.540	0.280957	0.140478	+	0.990084i	$0.455136\pi$
$398$	0	0
$399$	− 151.870i	− 0.380627i
$400$	0	0
$401$	340.535	0.849215	0.424607	−	0.905378i	$-0.360412\pi$
$401$	340.535	0.849215	0.424607	+	0.905378i	$0.360412\pi$
$402$	0	0
$403$	432.600i	1.07345i
$404$	0	0
$405$	−269.738	−0.666019
$406$	0	0
$407$	− 75.8502i	− 0.186364i
$408$	0	0
$409$	−666.959	−1.63071	−0.815354	−	0.578963i	$-0.803457\pi$
$409$	−666.959	−1.63071	−0.815354	+	0.578963i	$0.803457\pi$
$410$	0	0
$411$	87.5620i	0.213046i
$412$	0	0
$413$	71.0604	0.172059
$414$	0	0
$415$	454.420i	1.09499i
$416$	0	0
$417$	479.373	1.14958
$418$	0	0
$419$	− 200.191i	− 0.477783i	−0.971046	−	0.238891i	$-0.923216\pi$
$419$	− 200.191i	− 0.477783i	0.971046	−	0.238891i	$-0.0767840\pi$
$420$	0	0
$421$	−15.9136	−0.0377996	−0.0188998	−	0.999821i	$-0.506016\pi$
$421$	−15.9136	−0.0377996	−0.0188998	+	0.999821i	$0.506016\pi$
$422$	0	0
$423$	− 256.671i	− 0.606786i
$424$	0	0
$425$	−253.699	−0.596940
$426$	0	0
$427$	− 228.140i	− 0.534285i
$428$	0	0
$429$	−122.076	−0.284560
$430$	0	0
$431$	628.013i	1.45711i	0.684989	+	0.728553i	$0.259807\pi$
$431$	628.013i	1.45711i	−0.684989	+	0.728553i	$0.740193\pi$
$432$	0	0
$433$	789.232	1.82271	0.911353	−	0.411625i	$-0.135039\pi$
$433$	789.232	1.82271	0.911353	+	0.411625i	$0.135039\pi$
$434$	0	0
$435$	86.1933i	0.198146i
$436$	0	0
$437$	199.688	0.456952
$438$	0	0
$439$	665.570i	1.51610i	0.652194	+	0.758052i	$0.273849\pi$
$439$	665.570i	1.51610i	−0.652194	+	0.758052i	$0.726151\pi$
$440$	0	0
$441$	83.0326	0.188283
$442$	0	0
$443$	− 507.152i	− 1.14481i	−0.819970	−	0.572406i	$-0.806010\pi$
$443$	− 507.152i	− 1.14481i	0.819970	−	0.572406i	$-0.193990\pi$
$444$	0	0
$445$	−15.2530	−0.0342764
$446$	0	0
$447$	− 376.317i	− 0.841872i
$448$	0	0
$449$	−279.029	−0.621446	−0.310723	−	0.950501i	$-0.600571\pi$
$449$	−279.029	−0.621446	−0.310723	+	0.950501i	$0.600571\pi$
$450$	0	0
$451$	10.6394i	0.0235907i
$452$	0	0
$453$	−263.723	−0.582171
$454$	0	0
$455$	289.415i	0.636077i
$456$	0	0
$457$	720.881	1.57742	0.788710	−	0.614765i	$-0.210749\pi$
$457$	720.881	1.57742	0.788710	+	0.614765i	$0.210749\pi$
$458$	0	0
$459$	421.805i	0.918966i
$460$	0	0
$461$	483.262	1.04829	0.524145	−	0.851629i	$-0.324385\pi$
$461$	483.262	1.04829	0.524145	+	0.851629i	$0.324385\pi$
$462$	0	0
$463$	39.6326i	0.0855995i	0.999084	+	0.0427997i	$0.0136277\pi$
$463$	39.6326i	0.0855995i	−0.999084	+	0.0427997i	$0.986372\pi$
$464$	0	0
$465$	−593.584	−1.27652
$466$	0	0
$467$	17.7868i	0.0380874i	0.999819	+	0.0190437i	$0.00606216\pi$
$467$	17.7868i	0.0380874i	−0.999819	+	0.0190437i	$0.993938\pi$
$468$	0	0
$469$	−302.998	−0.646052
$470$	0	0
$471$	− 17.0355i	− 0.0361689i
$472$	0	0
$473$	−29.2005	−0.0617346
$474$	0	0
$475$	98.8029i	0.208006i
$476$	0	0
$477$	−4.23072	−0.00886943
$478$	0	0
$479$	668.616i	1.39586i	0.716166	+	0.697930i	$0.245895\pi$
$479$	668.616i	1.39586i	−0.716166	+	0.697930i	$0.754105\pi$
$480$	0	0
$481$	−1033.37	−2.14837
$482$	0	0
$483$	192.012i	0.397541i
$484$	0	0
$485$	−298.624	−0.615720
$486$	0	0
$487$	418.484i	0.859311i	0.902993	+	0.429656i	$0.141365\pi$
$487$	418.484i	0.859311i	−0.902993	+	0.429656i	$0.858635\pi$
$488$	0	0
$489$	−354.924	−0.725816
$490$	0	0
$491$	− 381.031i	− 0.776030i	−0.921653	−	0.388015i	$-0.873161\pi$
$491$	− 381.031i	− 0.776030i	0.921653	−	0.388015i	$-0.126839\pi$
$492$	0	0
$493$	−106.231	−0.215478
$494$	0	0
$495$	− 95.2412i	− 0.192406i
$496$	0	0
$497$	−277.253	−0.557852
$498$	0	0
$499$	− 438.392i	− 0.878541i	−0.898355	−	0.439271i	$-0.855237\pi$
$499$	− 438.392i	− 0.878541i	0.898355	−	0.439271i	$-0.144763\pi$
$500$	0	0
$501$	283.565	0.565999
$502$	0	0
$503$	− 754.754i	− 1.50050i	−0.661151	−	0.750252i	$-0.729932\pi$
$503$	− 754.754i	− 1.50050i	0.661151	−	0.750252i	$-0.270068\pi$
$504$	0	0
$505$	524.378	1.03837
$506$	0	0
$507$	891.235i	1.75786i
$508$	0	0
$509$	494.029	0.970588	0.485294	−	0.874351i	$-0.338713\pi$
$509$	494.029	0.970588	0.485294	+	0.874351i	$0.338713\pi$
$510$	0	0
$511$	64.5495i	0.126320i
$512$	0	0
$513$	164.272	0.320218
$514$	0	0
$515$	− 228.092i	− 0.442897i
$516$	0	0
$517$	−30.3078	−0.0586224
$518$	0	0
$519$	893.460i	1.72150i
$520$	0	0
$521$	32.8747	0.0630993	0.0315496	−	0.999502i	$-0.489956\pi$
$521$	32.8747	0.0630993	0.0315496	+	0.999502i	$0.489956\pi$
$522$	0	0
$523$	28.2755i	0.0540640i	0.999635	+	0.0270320i	$0.00860560\pi$
$523$	28.2755i	0.0540640i	−0.999635	+	0.0270320i	$0.991394\pi$
$524$	0	0
$525$	−95.0051	−0.180962
$526$	0	0
$527$	− 731.574i	− 1.38819i
$528$	0	0
$529$	276.531	0.522742
$530$	0	0
$531$	318.588i	0.599977i
$532$	0	0
$533$	144.949	0.271949
$534$	0	0
$535$	− 473.868i	− 0.885735i
$536$	0	0
$537$	−329.817	−0.614183
$538$	0	0
$539$	− 9.80453i	− 0.0181902i
$540$	0	0
$541$	−1071.59	−1.98077	−0.990383	−	0.138352i	$-0.955820\pi$
$541$	−1071.59	−1.98077	−0.990383	+	0.138352i	$0.955820\pi$
$542$	0	0
$543$	− 643.922i	− 1.18586i
$544$	0	0
$545$	168.948	0.309997
$546$	0	0
$547$	− 986.888i	− 1.80418i	−0.431545	−	0.902091i	$-0.642032\pi$
$547$	− 986.888i	− 1.80418i	0.431545	−	0.902091i	$-0.357968\pi$
$548$	0	0
$549$	1022.83	1.86307
$550$	0	0
$551$	41.3714i	0.0750842i
$552$	0	0
$553$	−309.891	−0.560382
$554$	0	0
$555$	− 1417.91i	− 2.55480i
$556$	0	0
$557$	483.550	0.868133	0.434067	−	0.900881i	$-0.357078\pi$
$557$	483.550	0.868133	0.434067	+	0.900881i	$0.357078\pi$
$558$	0	0
$559$	397.821i	0.711665i
$560$	0	0
$561$	206.444	0.367993
$562$	0	0
$563$	520.893i	0.925210i	0.886564	+	0.462605i	$0.153085\pi$
$563$	520.893i	0.925210i	−0.886564	+	0.462605i	$0.846915\pi$
$564$	0	0
$565$	912.236	1.61458
$566$	0	0
$567$	− 124.493i	− 0.219564i
$568$	0	0
$569$	732.959	1.28815	0.644077	−	0.764961i	$-0.277242\pi$
$569$	732.959	1.28815	0.644077	+	0.764961i	$0.277242\pi$
$570$	0	0
$571$	999.584i	1.75058i	0.483595	+	0.875292i	$0.339331\pi$
$571$	999.584i	1.75058i	−0.483595	+	0.875292i	$0.660669\pi$
$572$	0	0
$573$	1299.32	2.26758
$574$	0	0
$575$	− 124.918i	− 0.217249i
$576$	0	0
$577$	465.859	0.807381	0.403690	−	0.914896i	$-0.367727\pi$
$577$	465.859	0.807381	0.403690	+	0.914896i	$0.367727\pi$
$578$	0	0
$579$	− 565.682i	− 0.976998i
$580$	0	0
$581$	−209.730	−0.360981
$582$	0	0
$583$	0.499565i	0 0.000856887i
$584$	0	0
$585$	−1297.54	−2.21803
$586$	0	0
$587$	574.851i	0.979303i	0.871918	+	0.489651i	$0.162876\pi$
$587$	574.851i	0.979303i	−0.871918	+	0.489651i	$0.837124\pi$
$588$	0	0
$589$	−284.911	−0.483719
$590$	0	0
$591$	493.735i	0.835423i
$592$	0	0
$593$	943.055	1.59031	0.795156	−	0.606405i	$-0.207389\pi$
$593$	943.055	1.59031	0.795156	+	0.606405i	$0.207389\pi$
$594$	0	0
$595$	− 489.432i	− 0.822574i
$596$	0	0
$597$	1515.73	2.53892
$598$	0	0
$599$	− 9.26699i	− 0.0154708i	−0.999970	−	0.00773538i	$-0.997538\pi$
$599$	− 9.26699i	− 0.0154708i	0.999970	−	0.00773538i	$-0.00246227\pi$
$600$	0	0
$601$	−57.7003	−0.0960072	−0.0480036	−	0.998847i	$-0.515286\pi$
$601$	−57.7003	−0.0960072	−0.0480036	+	0.998847i	$0.515286\pi$
$602$	0	0
$603$	− 1358.44i	− 2.25281i
$604$	0	0
$605$	682.389	1.12792
$606$	0	0
$607$	− 1024.68i	− 1.68810i	−0.536264	−	0.844050i	$-0.680165\pi$
$607$	− 1024.68i	− 1.68810i	0.536264	−	0.844050i	$-0.319835\pi$
$608$	0	0
$609$	−39.7811	−0.0653221
$610$	0	0
$611$	412.906i	0.675788i
$612$	0	0
$613$	−404.818	−0.660389	−0.330195	−	0.943913i	$-0.607114\pi$
$613$	−404.818	−0.660389	−0.330195	+	0.943913i	$0.607114\pi$
$614$	0	0
$615$	198.889i	0.323396i
$616$	0	0
$617$	−894.209	−1.44928	−0.724642	−	0.689125i	$-0.757995\pi$
$617$	−894.209	−1.44928	−0.724642	+	0.689125i	$0.757995\pi$
$618$	0	0
$619$	779.388i	1.25911i	0.776957	+	0.629554i	$0.216762\pi$
$619$	779.388i	1.25911i	−0.776957	+	0.629554i	$0.783238\pi$
$620$	0	0
$621$	−207.692	−0.334447
$622$	0	0
$623$	− 7.03977i	− 0.0112998i
$624$	0	0
$625$	−759.737	−1.21558
$626$	0	0
$627$	− 80.3993i	− 0.128229i
$628$	0	0
$629$	1747.53	2.77827
$630$	0	0
$631$	780.191i	1.23644i	0.786007	+	0.618218i	$0.212145\pi$
$631$	780.191i	1.23644i	−0.786007	+	0.618218i	$0.787855\pi$
$632$	0	0
$633$	−120.559	−0.190456
$634$	0	0
$635$	92.1985i	0.145194i
$636$	0	0
$637$	−133.575	−0.209693
$638$	0	0
$639$	− 1243.02i	− 1.94525i
$640$	0	0
$641$	−23.3139	−0.0363712	−0.0181856	−	0.999835i	$-0.505789\pi$
$641$	−23.3139	−0.0363712	−0.0181856	+	0.999835i	$0.505789\pi$
$642$	0	0
$643$	530.706i	0.825360i	0.910876	+	0.412680i	$0.135407\pi$
$643$	530.706i	0.825360i	−0.910876	+	0.412680i	$0.864593\pi$
$644$	0	0
$645$	−545.862	−0.846297
$646$	0	0
$647$	213.435i	0.329883i	0.986303	+	0.164942i	$0.0527436\pi$
$647$	213.435i	0.329883i	−0.986303	+	0.164942i	$0.947256\pi$
$648$	0	0
$649$	37.6190	0.0579646
$650$	0	0
$651$	− 273.959i	− 0.420828i
$652$	0	0
$653$	−274.874	−0.420941	−0.210470	−	0.977600i	$-0.567500\pi$
$653$	−274.874	−0.420941	−0.210470	+	0.977600i	$0.567500\pi$
$654$	0	0
$655$	− 677.215i	− 1.03392i
$656$	0	0
$657$	−289.397	−0.440483
$658$	0	0
$659$	1234.48i	1.87327i	0.350313	+	0.936633i	$0.386075\pi$
$659$	1234.48i	1.87327i	−0.350313	+	0.936633i	$0.613925\pi$
$660$	0	0
$661$	−582.733	−0.881593	−0.440797	−	0.897607i	$-0.645304\pi$
$661$	−582.733	−0.881593	−0.440797	+	0.897607i	$0.645304\pi$
$662$	0	0
$663$	− 2812.54i	− 4.24215i
$664$	0	0
$665$	−190.609	−0.286630
$666$	0	0
$667$	− 52.3066i	− 0.0784207i
$668$	0	0
$669$	736.201	1.10045
$670$	0	0
$671$	− 120.776i	− 0.179994i
$672$	0	0
$673$	−399.145	−0.593083	−0.296542	−	0.955020i	$-0.595833\pi$
$673$	−399.145	−0.593083	−0.296542	+	0.955020i	$0.595833\pi$
$674$	0	0
$675$	− 102.763i	− 0.152241i
$676$	0	0
$677$	754.467	1.11443	0.557214	−	0.830369i	$-0.311870\pi$
$677$	754.467	1.11443	0.557214	+	0.830369i	$0.311870\pi$
$678$	0	0
$679$	− 137.825i	− 0.202983i
$680$	0	0
$681$	782.312	1.14877
$682$	0	0
$683$	− 288.264i	− 0.422055i	−0.977480	−	0.211028i	$-0.932319\pi$
$683$	− 288.264i	− 0.422055i	0.977480	−	0.211028i	$-0.0676810\pi$
$684$	0	0
$685$	109.897	0.160433
$686$	0	0
$687$	1047.10i	1.52416i
$688$	0	0
$689$	6.80596	0.00987803
$690$	0	0
$691$	− 156.692i	− 0.226761i	−0.993552	−	0.113380i	$-0.963832\pi$
$691$	− 156.692i	− 0.226761i	0.993552	−	0.113380i	$-0.0361678\pi$
$692$	0	0
$693$	43.9570	0.0634300
$694$	0	0
$695$	− 601.649i	− 0.865682i
$696$	0	0
$697$	−245.124	−0.351685
$698$	0	0
$699$	1233.92i	1.76527i
$700$	0	0
$701$	1126.50	1.60700	0.803498	−	0.595307i	$-0.202970\pi$
$701$	1126.50	1.60700	0.803498	+	0.595307i	$0.202970\pi$
$702$	0	0
$703$	− 680.575i	− 0.968102i
$704$	0	0
$705$	−566.562	−0.803633
$706$	0	0
$707$	242.018i	0.342317i
$708$	0	0
$709$	−1096.17	−1.54608	−0.773041	−	0.634356i	$-0.781266\pi$
$709$	−1096.17	−1.54608	−0.773041	+	0.634356i	$0.781266\pi$
$710$	0	0
$711$	− 1389.35i	− 1.95407i
$712$	0	0
$713$	360.218	0.505214
$714$	0	0
$715$	153.215i	0.214286i
$716$	0	0
$717$	717.800	1.00112
$718$	0	0
$719$	605.362i	0.841949i	0.907072	+	0.420975i	$0.138312\pi$
$719$	605.362i	0.841949i	−0.907072	+	0.420975i	$0.861688\pi$
$720$	0	0
$721$	105.272	0.146009
$722$	0	0
$723$	− 446.298i	− 0.617287i
$724$	0	0
$725$	25.8806	0.0356974
$726$	0	0
$727$	443.659i	0.610260i	0.952311	+	0.305130i	$0.0986999\pi$
$727$	443.659i	0.610260i	−0.952311	+	0.305130i	$0.901300\pi$
$728$	0	0
$729$	1095.46	1.50269
$730$	0	0
$731$	− 672.757i	− 0.920325i
$732$	0	0
$733$	−750.026	−1.02323	−0.511614	−	0.859216i	$-0.670952\pi$
$733$	−750.026	−1.02323	−0.511614	+	0.859216i	$0.670952\pi$
$734$	0	0
$735$	− 183.282i	− 0.249363i
$736$	0	0
$737$	−160.406	−0.217647
$738$	0	0
$739$	619.293i	0.838015i	0.907983	+	0.419007i	$0.137622\pi$
$739$	619.293i	0.838015i	−0.907983	+	0.419007i	$0.862378\pi$
$740$	0	0
$741$	−1095.34	−1.47819
$742$	0	0
$743$	30.5255i	0.0410842i	0.999789	+	0.0205421i	$0.00653921\pi$
$743$	30.5255i	0.0410842i	−0.999789	+	0.0205421i	$0.993461\pi$
$744$	0	0
$745$	−472.306	−0.633967
$746$	0	0
$747$	− 940.291i	− 1.25876i
$748$	0	0
$749$	218.706	0.291997
$750$	0	0
$751$	968.214i	1.28923i	0.764506	+	0.644616i	$0.222983\pi$
$751$	968.214i	1.28923i	−0.764506	+	0.644616i	$0.777017\pi$
$752$	0	0
$753$	−1430.28	−1.89945
$754$	0	0
$755$	330.993i	0.438401i
$756$	0	0
$757$	1171.15	1.54710	0.773550	−	0.633736i	$-0.218479\pi$
$757$	1171.15	1.54710	0.773550	+	0.633736i	$0.218479\pi$
$758$	0	0
$759$	101.650i	0.133927i
$760$	0	0
$761$	−235.996	−0.310113	−0.155057	−	0.987906i	$-0.549556\pi$
$761$	−235.996	−0.310113	−0.155057	+	0.987906i	$0.549556\pi$
$762$	0	0
$763$	77.9753i	0.102196i
$764$	0	0
$765$	2194.29	2.86835
$766$	0	0
$767$	− 512.513i	− 0.668205i
$768$	0	0
$769$	−124.257	−0.161582	−0.0807912	−	0.996731i	$-0.525745\pi$
$769$	−124.257	−0.161582	−0.0807912	+	0.996731i	$0.525745\pi$
$770$	0	0
$771$	− 1593.39i	− 2.06665i
$772$	0	0
$773$	−178.223	−0.230560	−0.115280	−	0.993333i	$-0.536776\pi$
$773$	−178.223	−0.230560	−0.115280	+	0.993333i	$0.536776\pi$
$774$	0	0
$775$	178.231i	0.229975i
$776$	0	0
$777$	654.415	0.842233
$778$	0	0
$779$	95.4634i	0.122546i
$780$	0	0
$781$	−146.776	−0.187933
$782$	0	0
$783$	− 43.0296i	− 0.0549548i
$784$	0	0
$785$	−21.3809	−0.0272368
$786$	0	0
$787$	− 1107.90i	− 1.40775i	−0.710326	−	0.703873i	$-0.751453\pi$
$787$	− 1107.90i	− 1.40775i	0.710326	−	0.703873i	$-0.248547\pi$
$788$	0	0
$789$	−1755.57	−2.22506
$790$	0	0
$791$	421.028i	0.532273i
$792$	0	0
$793$	−1645.43	−2.07494
$794$	0	0
$795$	9.33867i	0.0117468i
$796$	0	0
$797$	−1094.69	−1.37351	−0.686755	−	0.726889i	$-0.740966\pi$
$797$	−1094.69	−1.37351	−0.686755	+	0.726889i	$0.740966\pi$
$798$	0	0
$799$	− 698.269i	− 0.873929i
$800$	0	0
$801$	31.5617	0.0394028
$802$	0	0
$803$	34.1722i	0.0425556i
$804$	0	0
$805$	240.990	0.299367
$806$	0	0
$807$	172.473i	0.213722i
$808$	0	0
$809$	1386.75	1.71416	0.857079	−	0.515185i	$-0.172277\pi$
$809$	1386.75	1.71416	0.857079	+	0.515185i	$0.172277\pi$
$810$	0	0
$811$	312.204i	0.384962i	0.981301	+	0.192481i	$0.0616534\pi$
$811$	312.204i	0.384962i	−0.981301	+	0.192481i	$0.938347\pi$
$812$	0	0
$813$	−1410.11	−1.73446
$814$	0	0
$815$	445.456i	0.546572i
$816$	0	0
$817$	−262.005	−0.320691
$818$	0	0
$819$	− 598.861i	− 0.731209i
$820$	0	0
$821$	1092.89	1.33117	0.665583	−	0.746324i	$-0.268183\pi$
$821$	1092.89	1.33117	0.665583	+	0.746324i	$0.268183\pi$
$822$	0	0
$823$	− 907.162i	− 1.10226i	−0.834419	−	0.551131i	$-0.814196\pi$
$823$	− 907.162i	− 1.10226i	0.834419	−	0.551131i	$-0.185804\pi$
$824$	0	0
$825$	−50.2952	−0.0609638
$826$	0	0
$827$	607.144i	0.734152i	0.930191	+	0.367076i	$0.119641\pi$
$827$	607.144i	0.734152i	−0.930191	+	0.367076i	$0.880359\pi$
$828$	0	0
$829$	−427.969	−0.516247	−0.258124	−	0.966112i	$-0.583104\pi$
$829$	−427.969	−0.516247	−0.258124	+	0.966112i	$0.583104\pi$
$830$	0	0
$831$	− 1115.42i	− 1.34226i
$832$	0	0
$833$	225.889	0.271176
$834$	0	0
$835$	− 355.896i	− 0.426223i
$836$	0	0
$837$	296.330	0.354038
$838$	0	0
$839$	− 1133.09i	− 1.35053i	−0.737575	−	0.675265i	$-0.764029\pi$
$839$	− 1133.09i	− 1.35053i	0.737575	−	0.675265i	$-0.235971\pi$
$840$	0	0
$841$	−830.163	−0.987114
$842$	0	0
$843$	− 1217.55i	− 1.44430i
$844$	0	0
$845$	1118.57	1.32375
$846$	0	0
$847$	314.945i	0.371836i
$848$	0	0
$849$	756.397	0.890927
$850$	0	0
$851$	860.464i	1.01112i
$852$	0	0
$853$	169.502	0.198712	0.0993562	−	0.995052i	$-0.468322\pi$
$853$	169.502	0.198712	0.0993562	+	0.995052i	$0.468322\pi$
$854$	0	0
$855$	− 854.563i	− 0.999489i
$856$	0	0
$857$	234.079	0.273138	0.136569	−	0.990631i	$-0.456393\pi$
$857$	234.079	0.273138	0.136569	+	0.990631i	$0.456393\pi$
$858$	0	0
$859$	− 894.342i	− 1.04114i	−0.853818	−	0.520571i	$-0.825719\pi$
$859$	− 894.342i	− 1.04114i	0.853818	−	0.520571i	$-0.174281\pi$
$860$	0	0
$861$	−91.7939	−0.106613
$862$	0	0
$863$	− 778.580i	− 0.902178i	−0.892479	−	0.451089i	$-0.851036\pi$
$863$	− 778.580i	− 0.902178i	0.892479	−	0.451089i	$-0.148964\pi$
$864$	0	0
$865$	1121.36	1.29637
$866$	0	0
$867$	3436.32i	3.96346i
$868$	0	0
$869$	−164.055	−0.188786
$870$	0	0
$871$	2185.33i	2.50899i
$872$	0	0
$873$	617.917	0.707809
$874$	0	0
$875$	− 259.932i	− 0.297065i
$876$	0	0
$877$	17.2780	0.0197013	0.00985064	−	0.999951i	$-0.496864\pi$
$877$	17.2780	0.0197013	0.00985064	+	0.999951i	$0.496864\pi$
$878$	0	0
$879$	− 156.962i	− 0.178569i
$880$	0	0
$881$	770.918	0.875049	0.437524	−	0.899207i	$-0.355855\pi$
$881$	770.918	0.875049	0.437524	+	0.899207i	$0.355855\pi$
$882$	0	0
$883$	776.362i	0.879232i	0.898186	+	0.439616i	$0.144886\pi$
$883$	776.362i	0.879232i	−0.898186	+	0.439616i	$0.855114\pi$
$884$	0	0
$885$	703.235	0.794616
$886$	0	0
$887$	1630.80i	1.83856i	0.393603	+	0.919280i	$0.371228\pi$
$887$	1630.80i	1.83856i	−0.393603	+	0.919280i	$0.628772\pi$
$888$	0	0
$889$	−42.5527	−0.0478658
$890$	0	0
$891$	− 65.9059i	− 0.0739685i
$892$	0	0
$893$	−271.940	−0.304524
$894$	0	0
$895$	413.945i	0.462508i
$896$	0	0
$897$	1384.86	1.54388
$898$	0	0
$899$	74.6299i	0.0830144i
$900$	0	0
$901$	−11.5096	−0.0127743
$902$	0	0
$903$	− 251.933i	− 0.278996i
$904$	0	0
$905$	−808.171	−0.893007
$906$	0	0
$907$	953.863i	1.05167i	0.850587	+	0.525834i	$0.176247\pi$
$907$	953.863i	1.05167i	−0.850587	+	0.525834i	$0.823753\pi$
$908$	0	0
$909$	−1085.05	−1.19367
$910$	0	0
$911$	1681.15i	1.84539i	0.385534	+	0.922694i	$0.374017\pi$
$911$	1681.15i	1.84539i	−0.385534	+	0.922694i	$0.625983\pi$
$912$	0	0
$913$	−111.030	−0.121610
$914$	0	0
$915$	− 2257.74i	− 2.46747i
$916$	0	0
$917$	312.557	0.340848
$918$	0	0
$919$	− 504.991i	− 0.549500i	−0.961516	−	0.274750i	$-0.911405\pi$
$919$	− 504.991i	− 0.549500i	0.961516	−	0.274750i	$-0.0885952\pi$
$920$	0	0
$921$	1018.24	1.10559
$922$	0	0
$923$	1999.64i	2.16646i
$924$	0	0
$925$	−425.746	−0.460266
$926$	0	0
$927$	471.971i	0.509138i
$928$	0	0
$929$	983.851	1.05904	0.529521	−	0.848297i	$-0.322372\pi$
$929$	983.851	1.05904	0.529521	+	0.848297i	$0.322372\pi$
$930$	0	0
$931$	− 87.9723i	− 0.0944923i
$932$	0	0
$933$	−1918.04	−2.05577
$934$	0	0
$935$	− 259.103i	− 0.277115i
$936$	0	0
$937$	389.648	0.415846	0.207923	−	0.978145i	$-0.433330\pi$
$937$	389.648	0.415846	0.207923	+	0.978145i	$0.433330\pi$
$938$	0	0
$939$	1342.24i	1.42943i
$940$	0	0
$941$	−875.465	−0.930356	−0.465178	−	0.885217i	$-0.654010\pi$
$941$	−875.465	−0.930356	−0.465178	+	0.885217i	$0.654010\pi$
$942$	0	0
$943$	− 120.696i	− 0.127992i
$944$	0	0
$945$	198.248	0.209787
$946$	0	0
$947$	− 1251.29i	− 1.32132i	−0.750685	−	0.660661i	$-0.770276\pi$
$947$	− 1251.29i	− 1.32132i	0.750685	−	0.660661i	$-0.229724\pi$
$948$	0	0
$949$	465.554	0.490573
$950$	0	0
$951$	1934.14i	2.03380i
$952$	0	0
$953$	882.129	0.925633	0.462817	−	0.886454i	$-0.346839\pi$
$953$	882.129	0.925633	0.462817	+	0.886454i	$0.346839\pi$
$954$	0	0
$955$	− 1630.75i	− 1.70759i
$956$	0	0
$957$	−21.0599	−0.0220062
$958$	0	0
$959$	50.7211i	0.0528896i
$960$	0	0
$961$	447.049	0.465192
$962$	0	0
$963$	980.533i	1.01821i
$964$	0	0
$965$	−709.973	−0.735724
$966$	0	0
$967$	− 1410.24i	− 1.45836i	−0.684320	−	0.729182i	$-0.739901\pi$
$967$	− 1410.24i	− 1.45836i	0.684320	−	0.729182i	$-0.260099\pi$
$968$	0	0
$969$	1852.34	1.91160
$970$	0	0
$971$	− 678.550i	− 0.698815i	−0.936971	−	0.349408i	$-0.886383\pi$
$971$	− 678.550i	− 0.698815i	0.936971	−	0.349408i	$-0.113617\pi$
$972$	0	0
$973$	277.681	0.285387
$974$	0	0
$975$	685.210i	0.702780i
$976$	0	0
$977$	111.815	0.114447	0.0572235	−	0.998361i	$-0.481775\pi$
$977$	111.815	0.114447	0.0572235	+	0.998361i	$0.481775\pi$
$978$	0	0
$979$	− 3.72682i	− 0.00380676i
$980$	0	0
$981$	−349.590	−0.356360
$982$	0	0
$983$	202.226i	0.205723i	0.994696	+	0.102862i	$0.0327999\pi$
$983$	202.226i	0.205723i	−0.994696	+	0.102862i	$0.967200\pi$
$984$	0	0
$985$	619.675	0.629111
$986$	0	0
$987$	− 261.487i	− 0.264931i
$988$	0	0
$989$	331.257	0.334942
$990$	0	0
$991$	189.064i	0.190781i	0.995440	+	0.0953907i	$0.0304100\pi$
$991$	189.064i	0.190781i	−0.995440	+	0.0953907i	$0.969590\pi$
$992$	0	0
$993$	−578.546	−0.582624
$994$	0	0
$995$	− 1902.36i	− 1.91192i
$996$	0	0
$997$	−1632.91	−1.63783	−0.818914	−	0.573917i	$-0.805423\pi$
$997$	−1632.91	−1.63783	−0.818914	+	0.573917i	$0.805423\pi$
$998$	0	0
$999$	707.853i	0.708562i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.d.j.1023.14		16
4.3	odd	2	inner	1792.3.d.j.1023.4		16
8.3	odd	2	inner	1792.3.d.j.1023.13		16
8.5	even	2	inner	1792.3.d.j.1023.3		16
16.3	odd	4		56.3.g.b.43.2	yes	8
16.5	even	4		56.3.g.b.43.1	✓	8
16.11	odd	4		224.3.g.b.15.2		8
16.13	even	4		224.3.g.b.15.1		8
48.5	odd	4		504.3.g.b.379.8		8
48.11	even	4		2016.3.g.b.1135.2		8
48.29	odd	4		2016.3.g.b.1135.7		8
48.35	even	4		504.3.g.b.379.7		8
112.3	even	12		392.3.k.n.275.5		16
112.5	odd	12		392.3.k.n.67.5		16
112.13	odd	4		1568.3.g.m.687.8		8
112.19	even	12		392.3.k.n.67.7		16
112.27	even	4		1568.3.g.m.687.7		8
112.37	even	12		392.3.k.o.67.5		16
112.51	odd	12		392.3.k.o.67.7		16
112.53	even	12		392.3.k.o.275.7		16
112.67	odd	12		392.3.k.o.275.5		16
112.69	odd	4		392.3.g.m.99.1		8
112.83	even	4		392.3.g.m.99.2		8
112.101	odd	12		392.3.k.n.275.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.b.43.1	✓	8	16.5	even	4
56.3.g.b.43.2	yes	8	16.3	odd	4
224.3.g.b.15.1		8	16.13	even	4
224.3.g.b.15.2		8	16.11	odd	4
392.3.g.m.99.1		8	112.69	odd	4
392.3.g.m.99.2		8	112.83	even	4
392.3.k.n.67.5		16	112.5	odd	12
392.3.k.n.67.7		16	112.19	even	12
392.3.k.n.275.5		16	112.3	even	12
392.3.k.n.275.7		16	112.101	odd	12
392.3.k.o.67.5		16	112.37	even	12
392.3.k.o.67.7		16	112.51	odd	12
392.3.k.o.275.5		16	112.67	odd	12
392.3.k.o.275.7		16	112.53	even	12
504.3.g.b.379.7		8	48.35	even	4
504.3.g.b.379.8		8	48.5	odd	4
1568.3.g.m.687.7		8	112.27	even	4
1568.3.g.m.687.8		8	112.13	odd	4
1792.3.d.j.1023.3		16	8.5	even	2	inner
1792.3.d.j.1023.4		16	4.3	odd	2	inner
1792.3.d.j.1023.13		16	8.3	odd	2	inner
1792.3.d.j.1023.14		16	1.1	even	1	trivial
2016.3.g.b.1135.2		8	48.11	even	4
2016.3.g.b.1135.7		8	48.29	odd	4

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.56747i q^{3} +5.73252 q^{5} +2.64575i q^{7} -11.8618 q^{9} +O(q^{10})\)	\(q+4.56747i q^{3} +5.73252 q^{5} +2.64575i q^{7} -11.8618 q^{9} +1.40065i q^{11} +19.0821 q^{13} +26.1831i q^{15} -32.2699 q^{17} +12.5675i q^{19} -12.0844 q^{21} -15.8893i q^{23} +7.86180 q^{25} -13.0712i q^{27} +3.29194 q^{29} +22.6705i q^{31} -6.39741 q^{33} +15.1668i q^{35} -54.1537 q^{37} +87.1569i q^{39} +7.59607 q^{41} +20.8478i q^{43} -67.9980 q^{45} +21.6384i q^{47} -7.00000 q^{49} -147.392i q^{51} +0.356667 q^{53} +8.02924i q^{55} -57.4016 q^{57} -26.8583i q^{59} -86.2287 q^{61} -31.3834i q^{63} +109.389 q^{65} +114.523i q^{67} +72.5739 q^{69} +104.792i q^{71} +24.3974 q^{73} +35.9085i q^{75} -3.70576 q^{77} +117.128i q^{79} -47.0539 q^{81} +79.2706i q^{83} -184.988 q^{85} +15.0359i q^{87} -2.66078 q^{89} +50.4865i q^{91} -103.547 q^{93} +72.0433i q^{95} -52.0930 q^{97} -16.6142i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 96 q^{9}+O(q^{10}) \) 16 * q - 96 * q^9	\( 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) \) 16 * q - 96 * q^9 - 160 * q^17 + 32 * q^25 + 64 * q^33 - 256 * q^41 - 112 * q^49 - 112 * q^57 - 144 * q^65 + 224 * q^73 + 96 * q^81 + 1024 * q^89 + 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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