LMFDB - Embedded newform 784.3.c.d.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,3,Mod(97,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("784.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	784.c (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:	$21.3624527258$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	no (minimal twist has level 196)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$-0.765367i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.3.c.d.97.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.46088i q^{3} +8.47343i q^{5} -10.8995 q^{9} +O(q^{10})$	$q-4.46088i q^{3} +8.47343i q^{5} -10.8995 q^{9} -3.89949 q^{11} -19.1886i q^{13} +37.7990 q^{15} +13.3827i q^{17} +6.70259i q^{19} -26.0000 q^{23} -46.7990 q^{25} +8.47343i q^{27} -11.7990 q^{29} -36.5838i q^{31} +17.3952i q^{33} -32.0000 q^{37} -85.5980 q^{39} +20.9594i q^{41} -79.2965 q^{43} -92.3561i q^{45} -14.3019i q^{47} +59.6985 q^{51} -13.7990 q^{53} -33.0421i q^{55} +29.8995 q^{57} +8.45090i q^{59} -31.6520i q^{61} +162.593 q^{65} +31.3970 q^{67} +115.983i q^{69} -95.5980 q^{71} -18.3144i q^{73} +208.765i q^{75} -79.7990 q^{79} -60.2965 q^{81} +141.381i q^{83} -113.397 q^{85} +52.6339i q^{87} +116.454i q^{89} -163.196 q^{93} -56.7939 q^{95} +137.346i q^{97} +42.5025 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 24 q^{11} + 72 q^{15} - 104 q^{23} - 108 q^{25} + 32 q^{29} - 128 q^{37} - 184 q^{39} - 40 q^{43} + 120 q^{51} + 24 q^{53} + 80 q^{57} + 96 q^{65} - 112 q^{67} - 224 q^{71} - 240 q^{79} + 36 q^{81} - 216 q^{85} - 336 q^{93} + 248 q^{95} + 368 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 24 * q^11 + 72 * q^15 - 104 * q^23 - 108 * q^25 + 32 * q^29 - 128 * q^37 - 184 * q^39 - 40 * q^43 + 120 * q^51 + 24 * q^53 + 80 * q^57 + 96 * q^65 - 112 * q^67 - 224 * q^71 - 240 * q^79 + 36 * q^81 - 216 * q^85 - 336 * q^93 + 248 * q^95 + 368 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 4.46088i	− 1.48696i	−0.668757	−	0.743481i	$-0.733173\pi$
$3$	− 4.46088i	− 1.48696i	0.668757	−	0.743481i	$-0.266827\pi$
$4$	0	0
$5$	8.47343i	1.69469i	0.531046	+	0.847343i	$0.321799\pi$
$5$	8.47343i	1.69469i	−0.531046	+	0.847343i	$0.678201\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−10.8995	−1.21105
$10$	0	0
$11$	−3.89949	−0.354500	−0.177250	−	0.984166i	$-0.556720\pi$
$11$	−3.89949	−0.354500	−0.177250	+	0.984166i	$0.556720\pi$
$12$	0	0
$13$	− 19.1886i	− 1.47604i	−0.674777	−	0.738022i	$-0.735760\pi$
$13$	− 19.1886i	− 1.47604i	0.674777	−	0.738022i	$-0.264240\pi$
$14$	0	0
$15$	37.7990	2.51993
$16$	0	0
$17$	13.3827i	0.787215i	0.919279	+	0.393607i	$0.128773\pi$
$17$	13.3827i	0.787215i	−0.919279	+	0.393607i	$0.871227\pi$
$18$	0	0
$19$	6.70259i	0.352768i	0.984321	+	0.176384i	$0.0564401\pi$
$19$	6.70259i	0.352768i	−0.984321	+	0.176384i	$0.943560\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−26.0000	−1.13043	−0.565217	−	0.824942i	$-0.691208\pi$
$23$	−26.0000	−1.13043	−0.565217	+	0.824942i	$0.691208\pi$
$24$	0	0
$25$	−46.7990	−1.87196
$26$	0	0
$27$	8.47343i	0.313831i
$28$	0	0
$29$	−11.7990	−0.406862	−0.203431	−	0.979089i	$-0.565209\pi$
$29$	−11.7990	−0.406862	−0.203431	+	0.979089i	$0.565209\pi$
$30$	0	0
$31$	− 36.5838i	− 1.18012i	−0.807359	−	0.590061i	$-0.799104\pi$
$31$	− 36.5838i	− 1.18012i	0.807359	−	0.590061i	$-0.200896\pi$
$32$	0	0
$33$	17.3952i	0.527127i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−32.0000	−0.864865	−0.432432	−	0.901666i	$-0.642345\pi$
$37$	−32.0000	−0.864865	−0.432432	+	0.901666i	$0.642345\pi$
$38$	0	0
$39$	−85.5980	−2.19482
$40$	0	0
$41$	20.9594i	0.511205i	0.966782	+	0.255602i	$0.0822738\pi$
$41$	20.9594i	0.511205i	−0.966782	+	0.255602i	$0.917726\pi$
$42$	0	0
$43$	−79.2965	−1.84410	−0.922052	−	0.387066i	$-0.873488\pi$
$43$	−79.2965	−1.84410	−0.922052	+	0.387066i	$0.873488\pi$
$44$	0	0
$45$	− 92.3561i	− 2.05236i
$46$	0	0
$47$	− 14.3019i	− 0.304295i	−0.988358	−	0.152148i	$-0.951381\pi$
$47$	− 14.3019i	− 0.304295i	0.988358	−	0.152148i	$-0.0486189\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	59.6985	1.17056
$52$	0	0
$53$	−13.7990	−0.260358	−0.130179	−	0.991490i	$-0.541555\pi$
$53$	−13.7990	−0.260358	−0.130179	+	0.991490i	$0.541555\pi$
$54$	0	0
$55$	− 33.0421i	− 0.600765i
$56$	0	0
$57$	29.8995	0.524553
$58$	0	0
$59$	8.45090i	0.143236i	0.997432	+	0.0716178i	$0.0228162\pi$
$59$	8.45090i	0.143236i	−0.997432	+	0.0716178i	$0.977184\pi$
$60$	0	0
$61$	− 31.6520i	− 0.518885i	−0.965759	−	0.259443i	$-0.916461\pi$
$61$	− 31.6520i	− 0.518885i	0.965759	−	0.259443i	$-0.0835389\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	162.593	2.50143
$66$	0	0
$67$	31.3970	0.468611	0.234306	−	0.972163i	$-0.424718\pi$
$67$	31.3970	0.468611	0.234306	+	0.972163i	$0.424718\pi$
$68$	0	0
$69$	115.983i	1.68091i
$70$	0	0
$71$	−95.5980	−1.34645	−0.673225	−	0.739438i	$-0.735092\pi$
$71$	−95.5980	−1.34645	−0.673225	+	0.739438i	$0.735092\pi$
$72$	0	0
$73$	− 18.3144i	− 0.250882i	−0.992101	−	0.125441i	$-0.959965\pi$
$73$	− 18.3144i	− 0.250882i	0.992101	−	0.125441i	$-0.0400346\pi$
$74$	0	0
$75$	208.765i	2.78353i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−79.7990	−1.01011	−0.505057	−	0.863086i	$-0.668528\pi$
$79$	−79.7990	−1.01011	−0.505057	+	0.863086i	$0.668528\pi$
$80$	0	0
$81$	−60.2965	−0.744401
$82$	0	0
$83$	141.381i	1.70338i	0.524044	+	0.851691i	$0.324423\pi$
$83$	141.381i	1.70338i	−0.524044	+	0.851691i	$0.675577\pi$
$84$	0	0
$85$	−113.397	−1.33408
$86$	0	0
$87$	52.6339i	0.604988i
$88$	0	0
$89$	116.454i	1.30847i	0.756291	+	0.654235i	$0.227009\pi$
$89$	116.454i	1.30847i	−0.756291	+	0.654235i	$0.772991\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−163.196	−1.75480
$94$	0	0
$95$	−56.7939	−0.597831
$96$	0	0
$97$	137.346i	1.41593i	0.706245	+	0.707967i	$0.250388\pi$
$97$	137.346i	1.41593i	−0.706245	+	0.707967i	$0.749612\pi$
$98$	0	0
$99$	42.5025	0.429318
$100$	0	0
$101$	− 90.5177i	− 0.896214i	−0.893980	−	0.448107i	$-0.852098\pi$
$101$	− 90.5177i	− 0.896214i	0.893980	−	0.448107i	$-0.147902\pi$
$102$	0	0
$103$	− 14.2568i	− 0.138416i	−0.997602	−	0.0692078i	$-0.977953\pi$
$103$	− 14.2568i	− 0.138416i	0.997602	−	0.0692078i	$-0.0220472\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	49.5980	0.463533	0.231766	−	0.972772i	$-0.425550\pi$
$107$	49.5980	0.463533	0.231766	+	0.972772i	$0.425550\pi$
$108$	0	0
$109$	147.196	1.35042	0.675211	−	0.737625i	$-0.264053\pi$
$109$	147.196	1.35042	0.675211	+	0.737625i	$0.264053\pi$
$110$	0	0
$111$	142.748i	1.28602i
$112$	0	0
$113$	74.3015	0.657536	0.328768	−	0.944411i	$-0.393367\pi$
$113$	74.3015	0.657536	0.328768	+	0.944411i	$0.393367\pi$
$114$	0	0
$115$	− 220.309i	− 1.91573i
$116$	0	0
$117$	209.146i	1.78757i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−105.794	−0.874330
$122$	0	0
$123$	93.4975	0.760142
$124$	0	0
$125$	− 184.712i	− 1.47770i
$126$	0	0
$127$	76.9949	0.606259	0.303130	−	0.952949i	$-0.401968\pi$
$127$	76.9949	0.606259	0.303130	+	0.952949i	$0.401968\pi$
$128$	0	0
$129$	353.732i	2.74211i
$130$	0	0
$131$	− 155.189i	− 1.18465i	−0.805699	−	0.592325i	$-0.798210\pi$
$131$	− 155.189i	− 1.18465i	0.805699	−	0.592325i	$-0.201790\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−71.7990	−0.531844
$136$	0	0
$137$	−130.101	−0.949639	−0.474819	−	0.880083i	$-0.657487\pi$
$137$	−130.101	−0.949639	−0.474819	+	0.880083i	$0.657487\pi$
$138$	0	0
$139$	− 200.740i	− 1.44417i	−0.691804	−	0.722086i	$-0.743184\pi$
$139$	− 200.740i	− 1.44417i	0.691804	−	0.722086i	$-0.256816\pi$
$140$	0	0
$141$	−63.7990	−0.452475
$142$	0	0
$143$	74.8257i	0.523257i
$144$	0	0
$145$	− 99.9779i	− 0.689503i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	169.397	1.13689	0.568446	−	0.822720i	$-0.307545\pi$
$149$	169.397	1.13689	0.568446	+	0.822720i	$0.307545\pi$
$150$	0	0
$151$	2.20101	0.0145762	0.00728811	−	0.999973i	$-0.497680\pi$
$151$	2.20101	0.0145762	0.00728811	+	0.999973i	$0.497680\pi$
$152$	0	0
$153$	− 145.864i	− 0.953361i
$154$	0	0
$155$	309.990	1.99993
$156$	0	0
$157$	− 161.085i	− 1.02602i	−0.858382	−	0.513010i	$-0.828530\pi$
$157$	− 161.085i	− 1.02602i	0.858382	−	0.513010i	$-0.171470\pi$
$158$	0	0
$159$	61.5557i	0.387143i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	222.291	1.36375	0.681876	−	0.731468i	$-0.261165\pi$
$163$	222.291	1.36375	0.681876	+	0.731468i	$0.261165\pi$
$164$	0	0
$165$	−147.397	−0.893315
$166$	0	0
$167$	207.084i	1.24003i	0.784592	+	0.620013i	$0.212873\pi$
$167$	207.084i	1.24003i	−0.784592	+	0.620013i	$0.787127\pi$
$168$	0	0
$169$	−199.201	−1.17870
$170$	0	0
$171$	− 73.0549i	− 0.427221i
$172$	0	0
$173$	− 277.830i	− 1.60595i	−0.596011	−	0.802976i	$-0.703249\pi$
$173$	− 277.830i	− 1.60595i	0.596011	−	0.802976i	$-0.296751\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	37.6985	0.212986
$178$	0	0
$179$	−340.392	−1.90163	−0.950815	−	0.309758i	$-0.899752\pi$
$179$	−340.392	−1.90163	−0.950815	+	0.309758i	$0.899752\pi$
$180$	0	0
$181$	− 184.174i	− 1.01753i	−0.860904	−	0.508767i	$-0.830101\pi$
$181$	− 184.174i	− 1.01753i	0.860904	−	0.508767i	$-0.169899\pi$
$182$	0	0
$183$	−141.196	−0.771563
$184$	0	0
$185$	− 271.150i	− 1.46567i
$186$	0	0
$187$	− 52.1856i	− 0.279067i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−229.588	−1.20203	−0.601015	−	0.799237i	$-0.705237\pi$
$191$	−229.588	−1.20203	−0.601015	+	0.799237i	$0.705237\pi$
$192$	0	0
$193$	−41.4975	−0.215013	−0.107506	−	0.994204i	$-0.534287\pi$
$193$	−41.4975	−0.215013	−0.107506	+	0.994204i	$0.534287\pi$
$194$	0	0
$195$	− 725.308i	− 3.71953i
$196$	0	0
$197$	253.598	1.28730	0.643650	−	0.765320i	$-0.277419\pi$
$197$	253.598	1.28730	0.643650	+	0.765320i	$0.277419\pi$
$198$	0	0
$199$	112.351i	0.564579i	0.959329	+	0.282289i	$0.0910939\pi$
$199$	112.351i	0.564579i	−0.959329	+	0.282289i	$0.908906\pi$
$200$	0	0
$201$	− 140.058i	− 0.696807i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−177.598	−0.866332
$206$	0	0
$207$	283.387	1.36902
$208$	0	0
$209$	− 26.1367i	− 0.125056i
$210$	0	0
$211$	−353.789	−1.67672	−0.838362	−	0.545113i	$-0.816487\pi$
$211$	−353.789	−1.67672	−0.838362	+	0.545113i	$0.816487\pi$
$212$	0	0
$213$	426.452i	2.00212i
$214$	0	0
$215$	− 671.913i	− 3.12518i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−81.6985	−0.373052
$220$	0	0
$221$	256.794	1.16196
$222$	0	0
$223$	126.653i	0.567951i	0.958832	+	0.283976i	$0.0916534\pi$
$223$	126.653i	0.567951i	−0.958832	+	0.283976i	$0.908347\pi$
$224$	0	0
$225$	510.085	2.26705
$226$	0	0
$227$	225.847i	0.994920i	0.867487	+	0.497460i	$0.165734\pi$
$227$	225.847i	0.994920i	−0.867487	+	0.497460i	$0.834266\pi$
$228$	0	0
$229$	− 258.283i	− 1.12787i	−0.825818	−	0.563937i	$-0.809286\pi$
$229$	− 258.283i	− 1.12787i	0.825818	−	0.563937i	$-0.190714\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	57.6985	0.247633	0.123816	−	0.992305i	$-0.460487\pi$
$233$	57.6985	0.247633	0.123816	+	0.992305i	$0.460487\pi$
$234$	0	0
$235$	121.186	0.515685
$236$	0	0
$237$	355.974i	1.50200i
$238$	0	0
$239$	−82.6030	−0.345619	−0.172810	−	0.984955i	$-0.555285\pi$
$239$	−82.6030	−0.345619	−0.172810	+	0.984955i	$0.555285\pi$
$240$	0	0
$241$	− 130.666i	− 0.542181i	−0.962554	−	0.271091i	$-0.912616\pi$
$241$	− 130.666i	− 0.542181i	0.962554	−	0.271091i	$-0.0873843\pi$
$242$	0	0
$243$	345.236i	1.42073i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	128.613	0.520701
$248$	0	0
$249$	630.683	2.53286
$250$	0	0
$251$	− 149.854i	− 0.597029i	−0.954405	−	0.298514i	$-0.903509\pi$
$251$	− 149.854i	− 0.597029i	0.954405	−	0.298514i	$-0.0964911\pi$
$252$	0	0
$253$	101.387	0.400739
$254$	0	0
$255$	505.851i	1.98373i
$256$	0	0
$257$	− 5.22239i	− 0.0203206i	−0.999948	−	0.0101603i	$-0.996766\pi$
$257$	− 5.22239i	− 0.0203206i	0.999948	−	0.0101603i	$-0.00323417\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	128.603	0.492732
$262$	0	0
$263$	133.407	0.507251	0.253626	−	0.967302i	$-0.418377\pi$
$263$	133.407	0.507251	0.253626	+	0.967302i	$0.418377\pi$
$264$	0	0
$265$	− 116.925i	− 0.441225i
$266$	0	0
$267$	519.487	1.94565
$268$	0	0
$269$	4.12519i	0.0153353i	0.999971	+	0.00766765i	$0.00244071\pi$
$269$	4.12519i	0.0153353i	−0.999971	+	0.00766765i	$0.997559\pi$
$270$	0	0
$271$	315.759i	1.16516i	0.812773	+	0.582580i	$0.197957\pi$
$271$	315.759i	1.16516i	−0.812773	+	0.582580i	$0.802043\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	182.492	0.663609
$276$	0	0
$277$	−260.191	−0.939317	−0.469659	−	0.882848i	$-0.655623\pi$
$277$	−260.191	−0.939317	−0.469659	+	0.882848i	$0.655623\pi$
$278$	0	0
$279$	398.745i	1.42919i
$280$	0	0
$281$	−372.894	−1.32703	−0.663513	−	0.748165i	$-0.730935\pi$
$281$	−372.894	−1.32703	−0.663513	+	0.748165i	$0.730935\pi$
$282$	0	0
$283$	79.9602i	0.282545i	0.989971	+	0.141273i	$0.0451194\pi$
$283$	79.9602i	0.282545i	−0.989971	+	0.141273i	$0.954881\pi$
$284$	0	0
$285$	253.351i	0.888952i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	109.905	0.380293
$290$	0	0
$291$	612.683	2.10544
$292$	0	0
$293$	112.038i	0.382382i	0.981553	+	0.191191i	$0.0612351\pi$
$293$	112.038i	0.382382i	−0.981553	+	0.191191i	$0.938765\pi$
$294$	0	0
$295$	−71.6081	−0.242739
$296$	0	0
$297$	− 33.0421i	− 0.111253i
$298$	0	0
$299$	498.903i	1.66857i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−403.789	−1.33264
$304$	0	0
$305$	268.201	0.879348
$306$	0	0
$307$	− 146.177i	− 0.476148i	−0.971247	−	0.238074i	$-0.923484\pi$
$307$	− 146.177i	− 0.476148i	0.971247	−	0.238074i	$-0.0765160\pi$
$308$	0	0
$309$	−63.5980	−0.205819
$310$	0	0
$311$	− 422.955i	− 1.35998i	−0.733220	−	0.679992i	$-0.761983\pi$
$311$	− 422.955i	− 1.35998i	0.733220	−	0.679992i	$-0.238017\pi$
$312$	0	0
$313$	87.7150i	0.280239i	0.990135	+	0.140120i	$0.0447488\pi$
$313$	87.7150i	0.280239i	−0.990135	+	0.140120i	$0.955251\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	431.387	1.36084	0.680421	−	0.732822i	$-0.261797\pi$
$317$	431.387	1.36084	0.680421	+	0.732822i	$0.261797\pi$
$318$	0	0
$319$	46.0101	0.144232
$320$	0	0
$321$	− 221.251i	− 0.689255i
$322$	0	0
$323$	−89.6985	−0.277704
$324$	0	0
$325$	898.005i	2.76309i
$326$	0	0
$327$	− 656.624i	− 2.00803i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	341.487	1.03168	0.515842	−	0.856684i	$-0.327479\pi$
$331$	341.487	1.03168	0.515842	+	0.856684i	$0.327479\pi$
$332$	0	0
$333$	348.784	1.04740
$334$	0	0
$335$	266.040i	0.794149i
$336$	0	0
$337$	−391.377	−1.16136	−0.580678	−	0.814134i	$-0.697212\pi$
$337$	−391.377	−1.16136	−0.580678	+	0.814134i	$0.697212\pi$
$338$	0	0
$339$	− 331.451i	− 0.977730i
$340$	0	0
$341$	142.658i	0.418352i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−982.774	−2.84862
$346$	0	0
$347$	608.482	1.75355	0.876776	−	0.480900i	$-0.159690\pi$
$347$	608.482	1.75355	0.876776	+	0.480900i	$0.159690\pi$
$348$	0	0
$349$	− 93.1627i	− 0.266942i	−0.991053	−	0.133471i	$-0.957388\pi$
$349$	− 93.1627i	− 0.266942i	0.991053	−	0.133471i	$-0.0426123\pi$
$350$	0	0
$351$	162.593	0.463228
$352$	0	0
$353$	− 92.3786i	− 0.261696i	−0.991402	−	0.130848i	$-0.958230\pi$
$353$	− 92.3786i	− 0.261696i	0.991402	−	0.130848i	$-0.0417700\pi$
$354$	0	0
$355$	− 810.043i	− 2.28181i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	289.176	0.805504	0.402752	−	0.915309i	$-0.368054\pi$
$359$	289.176	0.805504	0.402752	+	0.915309i	$0.368054\pi$
$360$	0	0
$361$	316.075	0.875555
$362$	0	0
$363$	471.935i	1.30010i
$364$	0	0
$365$	155.186	0.425167
$366$	0	0
$367$	529.971i	1.44406i	0.691860	+	0.722032i	$0.256791\pi$
$367$	529.971i	1.44406i	−0.691860	+	0.722032i	$0.743209\pi$
$368$	0	0
$369$	− 228.447i	− 0.619097i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	465.799	1.24879	0.624395	−	0.781108i	$-0.285345\pi$
$373$	465.799	1.24879	0.624395	+	0.781108i	$0.285345\pi$
$374$	0	0
$375$	−823.980	−2.19728
$376$	0	0
$377$	226.406i	0.600546i
$378$	0	0
$379$	−91.8793	−0.242426	−0.121213	−	0.992627i	$-0.538678\pi$
$379$	−91.8793	−0.242426	−0.121213	+	0.992627i	$0.538678\pi$
$380$	0	0
$381$	− 343.466i	− 0.901485i
$382$	0	0
$383$	− 311.275i	− 0.812729i	−0.913711	−	0.406364i	$-0.866796\pi$
$383$	− 311.275i	− 0.812729i	0.913711	−	0.406364i	$-0.133204\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	864.291	2.23331
$388$	0	0
$389$	−542.593	−1.39484	−0.697420	−	0.716662i	$-0.745669\pi$
$389$	−542.593	−1.39484	−0.697420	+	0.716662i	$0.745669\pi$
$390$	0	0
$391$	− 347.949i	− 0.889895i
$392$	0	0
$393$	−692.281	−1.76153
$394$	0	0
$395$	− 676.171i	− 1.71183i
$396$	0	0
$397$	163.775i	0.412532i	0.978496	+	0.206266i	$0.0661313\pi$
$397$	163.775i	0.412532i	−0.978496	+	0.206266i	$0.933869\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−143.407	−0.357624	−0.178812	−	0.983883i	$-0.557225\pi$
$401$	−143.407	−0.357624	−0.178812	+	0.983883i	$0.557225\pi$
$402$	0	0
$403$	−701.990	−1.74191
$404$	0	0
$405$	− 510.918i	− 1.26153i
$406$	0	0
$407$	124.784	0.306594
$408$	0	0
$409$	290.003i	0.709053i	0.935046	+	0.354526i	$0.115358\pi$
$409$	290.003i	0.709053i	−0.935046	+	0.354526i	$0.884642\pi$
$410$	0	0
$411$	580.363i	1.41208i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1197.98	−2.88670
$416$	0	0
$417$	−895.477	−2.14743
$418$	0	0
$419$	340.170i	0.811860i	0.913904	+	0.405930i	$0.133052\pi$
$419$	340.170i	0.811860i	−0.913904	+	0.405930i	$0.866948\pi$
$420$	0	0
$421$	−6.18081	−0.0146813	−0.00734063	−	0.999973i	$-0.502337\pi$
$421$	−6.18081	−0.0146813	−0.00734063	+	0.999973i	$0.502337\pi$
$422$	0	0
$423$	155.883i	0.368518i
$424$	0	0
$425$	− 626.295i	− 1.47363i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	333.789	0.778063
$430$	0	0
$431$	28.5929	0.0663409	0.0331705	−	0.999450i	$-0.489440\pi$
$431$	28.5929	0.0663409	0.0331705	+	0.999450i	$0.489440\pi$
$432$	0	0
$433$	243.736i	0.562900i	0.959576	+	0.281450i	$0.0908153\pi$
$433$	243.736i	0.562900i	−0.959576	+	0.281450i	$0.909185\pi$
$434$	0	0
$435$	−445.990	−1.02526
$436$	0	0
$437$	− 174.267i	− 0.398781i
$438$	0	0
$439$	494.910i	1.12736i	0.825994	+	0.563679i	$0.190615\pi$
$439$	494.910i	1.12736i	−0.825994	+	0.563679i	$0.809385\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−79.3970	−0.179226	−0.0896128	−	0.995977i	$-0.528563\pi$
$443$	−79.3970	−0.179226	−0.0896128	+	0.995977i	$0.528563\pi$
$444$	0	0
$445$	−986.764	−2.21745
$446$	0	0
$447$	− 755.660i	− 1.69052i
$448$	0	0
$449$	804.362	1.79145	0.895726	−	0.444607i	$-0.146657\pi$
$449$	804.362	1.79145	0.895726	+	0.444607i	$0.146657\pi$
$450$	0	0
$451$	− 81.7311i	− 0.181222i
$452$	0	0
$453$	− 9.81845i	− 0.0216743i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	537.477	1.17610	0.588050	−	0.808825i	$-0.299896\pi$
$457$	537.477	1.17610	0.588050	+	0.808825i	$0.299896\pi$
$458$	0	0
$459$	−113.397	−0.247052
$460$	0	0
$461$	524.278i	1.13726i	0.822593	+	0.568631i	$0.192527\pi$
$461$	524.278i	1.13726i	−0.822593	+	0.568631i	$0.807473\pi$
$462$	0	0
$463$	506.995	1.09502	0.547511	−	0.836799i	$-0.315576\pi$
$463$	506.995	1.09502	0.547511	+	0.836799i	$0.315576\pi$
$464$	0	0
$465$	− 1382.83i	− 2.97383i
$466$	0	0
$467$	205.807i	0.440700i	0.975421	+	0.220350i	$0.0707199\pi$
$467$	205.807i	0.440700i	−0.975421	+	0.220350i	$0.929280\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−718.583	−1.52565
$472$	0	0
$473$	309.216	0.653734
$474$	0	0
$475$	− 313.675i	− 0.660368i
$476$	0	0
$477$	150.402	0.315308
$478$	0	0
$479$	− 496.032i	− 1.03556i	−0.855514	−	0.517779i	$-0.826759\pi$
$479$	− 496.032i	− 1.03556i	0.855514	−	0.517779i	$-0.173241\pi$
$480$	0	0
$481$	614.034i	1.27658i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1163.79	−2.39956
$486$	0	0
$487$	−473.377	−0.972026	−0.486013	−	0.873952i	$-0.661549\pi$
$487$	−473.377	−0.972026	−0.486013	+	0.873952i	$0.661549\pi$
$488$	0	0
$489$	− 991.616i	− 2.02785i
$490$	0	0
$491$	143.417	0.292092	0.146046	−	0.989278i	$-0.453345\pi$
$491$	143.417	0.292092	0.146046	+	0.989278i	$0.453345\pi$
$492$	0	0
$493$	− 157.902i	− 0.320288i
$494$	0	0
$495$	360.142i	0.727560i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	446.995	0.895781	0.447891	−	0.894088i	$-0.352175\pi$
$499$	446.995	0.895781	0.447891	+	0.894088i	$0.352175\pi$
$500$	0	0
$501$	923.779	1.84387
$502$	0	0
$503$	− 660.346i	− 1.31282i	−0.754406	−	0.656408i	$-0.772075\pi$
$503$	− 660.346i	− 1.31282i	0.754406	−	0.656408i	$-0.227925\pi$
$504$	0	0
$505$	766.995	1.51880
$506$	0	0
$507$	888.613i	1.75269i
$508$	0	0
$509$	− 486.392i	− 0.955583i	−0.878473	−	0.477792i	$-0.841437\pi$
$509$	− 486.392i	− 0.955583i	0.878473	−	0.477792i	$-0.158563\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−56.7939	−0.110709
$514$	0	0
$515$	120.804	0.234571
$516$	0	0
$517$	55.7701i	0.107872i
$518$	0	0
$519$	−1239.37	−2.38799
$520$	0	0
$521$	− 102.781i	− 0.197276i	−0.995123	−	0.0986378i	$-0.968551\pi$
$521$	− 102.781i	− 0.197276i	0.995123	−	0.0986378i	$-0.0314485\pi$
$522$	0	0
$523$	517.260i	0.989025i	0.869171	+	0.494512i	$0.164653\pi$
$523$	517.260i	0.989025i	−0.869171	+	0.494512i	$0.835347\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	489.588	0.929009
$528$	0	0
$529$	147.000	0.277883
$530$	0	0
$531$	− 92.1105i	− 0.173466i
$532$	0	0
$533$	402.181	0.754561
$534$	0	0
$535$	420.265i	0.785542i
$536$	0	0
$537$	1518.45i	2.82765i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−450.402	−0.832536	−0.416268	−	0.909242i	$-0.636662\pi$
$541$	−450.402	−0.832536	−0.416268	+	0.909242i	$0.636662\pi$
$542$	0	0
$543$	−821.578	−1.51303
$544$	0	0
$545$	1247.25i	2.28854i
$546$	0	0
$547$	−195.095	−0.356664	−0.178332	−	0.983970i	$-0.557070\pi$
$547$	−195.095	−0.356664	−0.178332	+	0.983970i	$0.557070\pi$
$548$	0	0
$549$	344.991i	0.628399i
$550$	0	0
$551$	− 79.0838i	− 0.143528i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1209.57	−2.17940
$556$	0	0
$557$	−689.960	−1.23871	−0.619353	−	0.785112i	$-0.712605\pi$
$557$	−689.960	−1.23871	−0.619353	+	0.785112i	$0.712605\pi$
$558$	0	0
$559$	1521.59i	2.72198i
$560$	0	0
$561$	−232.794	−0.414962
$562$	0	0
$563$	− 836.876i	− 1.48646i	−0.669037	−	0.743229i	$-0.733293\pi$
$563$	− 836.876i	− 1.48646i	0.669037	−	0.743229i	$-0.266707\pi$
$564$	0	0
$565$	629.589i	1.11432i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.81919	0.0137420	0.00687100	−	0.999976i	$-0.497813\pi$
$569$	7.81919	0.0137420	0.00687100	+	0.999976i	$0.497813\pi$
$570$	0	0
$571$	−1.70859	−0.00299227	−0.00149613	−	0.999999i	$-0.500476\pi$
$571$	−1.70859	−0.00299227	−0.00149613	+	0.999999i	$0.500476\pi$
$572$	0	0
$573$	1024.17i	1.78737i
$574$	0	0
$575$	1216.77	2.11613
$576$	0	0
$577$	216.880i	0.375875i	0.982181	+	0.187938i	$0.0601803\pi$
$577$	216.880i	0.375875i	−0.982181	+	0.187938i	$0.939820\pi$
$578$	0	0
$579$	185.115i	0.319716i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	53.8091	0.0922969
$584$	0	0
$585$	−1772.18	−3.02937
$586$	0	0
$587$	50.9983i	0.0868795i	0.999056	+	0.0434398i	$0.0138317\pi$
$587$	50.9983i	0.0868795i	−0.999056	+	0.0434398i	$0.986168\pi$
$588$	0	0
$589$	245.206	0.416309
$590$	0	0
$591$	− 1131.27i	− 1.91416i
$592$	0	0
$593$	− 650.773i	− 1.09743i	−0.836011	−	0.548713i	$-0.815118\pi$
$593$	− 650.773i	− 1.09743i	0.836011	−	0.548713i	$-0.184882\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	501.186	0.839507
$598$	0	0
$599$	33.7889	0.0564088	0.0282044	−	0.999602i	$-0.491021\pi$
$599$	33.7889	0.0564088	0.0282044	+	0.999602i	$0.491021\pi$
$600$	0	0
$601$	− 415.133i	− 0.690737i	−0.938467	−	0.345368i	$-0.887754\pi$
$601$	− 415.133i	− 0.690737i	0.938467	−	0.345368i	$-0.112246\pi$
$602$	0	0
$603$	−342.211	−0.567514
$604$	0	0
$605$	− 896.437i	− 1.48171i
$606$	0	0
$607$	435.418i	0.717329i	0.933467	+	0.358664i	$0.116768\pi$
$607$	435.418i	0.717329i	−0.933467	+	0.358664i	$0.883232\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−274.432	−0.449153
$612$	0	0
$613$	424.764	0.692926	0.346463	−	0.938064i	$-0.387383\pi$
$613$	424.764	0.692926	0.346463	+	0.938064i	$0.387383\pi$
$614$	0	0
$615$	792.244i	1.28820i
$616$	0	0
$617$	−726.563	−1.17757	−0.588787	−	0.808289i	$-0.700394\pi$
$617$	−726.563	−1.17757	−0.588787	+	0.808289i	$0.700394\pi$
$618$	0	0
$619$	708.920i	1.14527i	0.819812	+	0.572633i	$0.194078\pi$
$619$	708.920i	1.14527i	−0.819812	+	0.572633i	$0.805922\pi$
$620$	0	0
$621$	− 220.309i	− 0.354765i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	395.171	0.632273
$626$	0	0
$627$	−116.593	−0.185954
$628$	0	0
$629$	− 428.245i	− 0.680835i
$630$	0	0
$631$	−207.176	−0.328329	−0.164165	−	0.986433i	$-0.552493\pi$
$631$	−207.176	−0.328329	−0.164165	+	0.986433i	$0.552493\pi$
$632$	0	0
$633$	1578.21i	2.49323i
$634$	0	0
$635$	652.411i	1.02742i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1041.97	1.63063
$640$	0	0
$641$	912.583	1.42369	0.711843	−	0.702339i	$-0.247861\pi$
$641$	912.583	1.42369	0.711843	+	0.702339i	$0.247861\pi$
$642$	0	0
$643$	656.917i	1.02164i	0.859686	+	0.510822i	$0.170659\pi$
$643$	656.917i	1.02164i	−0.859686	+	0.510822i	$0.829341\pi$
$644$	0	0
$645$	−2997.33	−4.64702
$646$	0	0
$647$	− 38.1068i	− 0.0588976i	−0.999566	−	0.0294488i	$-0.990625\pi$
$647$	− 38.1068i	− 0.0588976i	0.999566	−	0.0294488i	$-0.00937520\pi$
$648$	0	0
$649$	− 32.9542i	− 0.0507769i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−376.583	−0.576697	−0.288348	−	0.957526i	$-0.593106\pi$
$653$	−376.583	−0.576697	−0.288348	+	0.957526i	$0.593106\pi$
$654$	0	0
$655$	1314.98	2.00761
$656$	0	0
$657$	199.618i	0.303832i
$658$	0	0
$659$	954.291	1.44809	0.724045	−	0.689753i	$-0.242281\pi$
$659$	954.291	1.44809	0.724045	+	0.689753i	$0.242281\pi$
$660$	0	0
$661$	− 1166.37i	− 1.76456i	−0.470724	−	0.882281i	$-0.656007\pi$
$661$	− 1166.37i	− 1.76456i	0.470724	−	0.882281i	$-0.343993\pi$
$662$	0	0
$663$	− 1145.53i	− 1.72780i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	306.774	0.459931
$668$	0	0
$669$	564.985	0.844521
$670$	0	0
$671$	123.427i	0.183945i
$672$	0	0
$673$	−774.101	−1.15022	−0.575112	−	0.818075i	$-0.695041\pi$
$673$	−774.101	−1.15022	−0.575112	+	0.818075i	$0.695041\pi$
$674$	0	0
$675$	− 396.548i	− 0.587478i
$676$	0	0
$677$	561.488i	0.829377i	0.909964	+	0.414688i	$0.136109\pi$
$677$	561.488i	0.829377i	−0.909964	+	0.414688i	$0.863891\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1007.48	1.47941
$682$	0	0
$683$	−352.161	−0.515608	−0.257804	−	0.966197i	$-0.582999\pi$
$683$	−352.161	−0.515608	−0.257804	+	0.966197i	$0.582999\pi$
$684$	0	0
$685$	− 1102.40i	− 1.60934i
$686$	0	0
$687$	−1152.17	−1.67710
$688$	0	0
$689$	264.783i	0.384300i
$690$	0	0
$691$	− 1047.50i	− 1.51592i	−0.652300	−	0.757961i	$-0.726196\pi$
$691$	− 1047.50i	− 1.51592i	0.652300	−	0.757961i	$-0.273804\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1700.95	2.44742
$696$	0	0
$697$	−280.492	−0.402428
$698$	0	0
$699$	− 257.386i	− 0.368221i
$700$	0	0
$701$	809.990	1.15548	0.577739	−	0.816222i	$-0.303935\pi$
$701$	809.990	1.15548	0.577739	+	0.816222i	$0.303935\pi$
$702$	0	0
$703$	− 214.483i	− 0.305097i
$704$	0	0
$705$	− 540.596i	− 0.766803i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−459.960	−0.648744	−0.324372	−	0.945930i	$-0.605153\pi$
$709$	−459.960	−0.648744	−0.324372	+	0.945930i	$0.605153\pi$
$710$	0	0
$711$	869.769	1.22330
$712$	0	0
$713$	951.178i	1.33405i
$714$	0	0
$715$	−634.030	−0.886756
$716$	0	0
$717$	368.483i	0.513923i
$718$	0	0
$719$	86.0365i	0.119661i	0.998209	+	0.0598307i	$0.0190561\pi$
$719$	86.0365i	0.119661i	−0.998209	+	0.0598307i	$0.980944\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−582.884	−0.806202
$724$	0	0
$725$	552.181	0.761629
$726$	0	0
$727$	− 78.4124i	− 0.107858i	−0.998545	−	0.0539288i	$-0.982826\pi$
$727$	− 78.4124i	− 0.107858i	0.998545	−	0.0539288i	$-0.0171744\pi$
$728$	0	0
$729$	997.392	1.36816
$730$	0	0
$731$	− 1061.20i	− 1.45171i
$732$	0	0
$733$	519.569i	0.708826i	0.935089	+	0.354413i	$0.115319\pi$
$733$	519.569i	0.708826i	−0.935089	+	0.354413i	$0.884681\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−122.432	−0.166123
$738$	0	0
$739$	−1163.48	−1.57439	−0.787197	−	0.616702i	$-0.788469\pi$
$739$	−1163.48	−1.57439	−0.787197	+	0.616702i	$0.788469\pi$
$740$	0	0
$741$	− 573.728i	− 0.774262i
$742$	0	0
$743$	66.8040	0.0899112	0.0449556	−	0.998989i	$-0.485685\pi$
$743$	66.8040	0.0899112	0.0449556	+	0.998989i	$0.485685\pi$
$744$	0	0
$745$	1435.37i	1.92668i
$746$	0	0
$747$	− 1540.98i	− 2.06289i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−3.38687	−0.00450981	−0.00225491	−	0.999997i	$-0.500718\pi$
$751$	−3.38687	−0.00450981	−0.00225491	+	0.999997i	$0.500718\pi$
$752$	0	0
$753$	−668.482	−0.887759
$754$	0	0
$755$	18.6501i	0.0247021i
$756$	0	0
$757$	−700.402	−0.925234	−0.462617	−	0.886558i	$-0.653089\pi$
$757$	−700.402	−0.925234	−0.462617	+	0.886558i	$0.653089\pi$
$758$	0	0
$759$	− 452.275i	− 0.595883i
$760$	0	0
$761$	611.948i	0.804136i	0.915610	+	0.402068i	$0.131709\pi$
$761$	611.948i	0.804136i	−0.915610	+	0.402068i	$0.868291\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1235.97	1.61565
$766$	0	0
$767$	162.161	0.211422
$768$	0	0
$769$	638.310i	0.830052i	0.909810	+	0.415026i	$0.136227\pi$
$769$	638.310i	0.830052i	−0.909810	+	0.415026i	$0.863773\pi$
$770$	0	0
$771$	−23.2965	−0.0302159
$772$	0	0
$773$	− 475.005i	− 0.614496i	−0.951629	−	0.307248i	$-0.900592\pi$
$773$	− 475.005i	− 0.614496i	0.951629	−	0.307248i	$-0.0994081\pi$
$774$	0	0
$775$	1712.08i	2.20914i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−140.482	−0.180337
$780$	0	0
$781$	372.784	0.477316
$782$	0	0
$783$	− 99.9779i	− 0.127686i
$784$	0	0
$785$	1364.94	1.73878
$786$	0	0
$787$	1475.43i	1.87476i	0.348313	+	0.937378i	$0.386755\pi$
$787$	1475.43i	1.87476i	−0.348313	+	0.937378i	$0.613245\pi$
$788$	0	0
$789$	− 595.114i	− 0.754263i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−607.357	−0.765897
$794$	0	0
$795$	−521.588	−0.656085
$796$	0	0
$797$	330.869i	0.415143i	0.978220	+	0.207572i	$0.0665560\pi$
$797$	330.869i	0.415143i	−0.978220	+	0.207572i	$0.933444\pi$
$798$	0	0
$799$	191.397	0.239546
$800$	0	0
$801$	− 1269.29i	− 1.58463i
$802$	0	0
$803$	71.4170i	0.0889377i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.4020	0.0228030
$808$	0	0
$809$	−735.889	−0.909628	−0.454814	−	0.890586i	$-0.650294\pi$
$809$	−735.889	−0.909628	−0.454814	+	0.890586i	$0.650294\pi$
$810$	0	0
$811$	− 1129.32i	− 1.39251i	−0.717796	−	0.696253i	$-0.754849\pi$
$811$	− 1129.32i	− 1.39251i	0.717796	−	0.696253i	$-0.245151\pi$
$812$	0	0
$813$	1408.56	1.73255
$814$	0	0
$815$	1883.57i	2.31113i
$816$	0	0
$817$	− 531.492i	− 0.650541i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1453.96	−1.77096	−0.885481	−	0.464676i	$-0.846171\pi$
$821$	−1453.96	−1.77096	−0.885481	+	0.464676i	$0.846171\pi$
$822$	0	0
$823$	584.965	0.710771	0.355386	−	0.934720i	$-0.384350\pi$
$823$	584.965	0.710771	0.355386	+	0.934720i	$0.384350\pi$
$824$	0	0
$825$	− 814.078i	− 0.986761i
$826$	0	0
$827$	1453.16	1.75714	0.878570	−	0.477613i	$-0.158498\pi$
$827$	1453.16	1.75714	0.878570	+	0.477613i	$0.158498\pi$
$828$	0	0
$829$	− 403.586i	− 0.486835i	−0.969922	−	0.243417i	$-0.921731\pi$
$829$	− 403.586i	− 0.486835i	0.969922	−	0.243417i	$-0.0782685\pi$
$830$	0	0
$831$	1160.68i	1.39673i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1754.71	−2.10145
$836$	0	0
$837$	309.990	0.370358
$838$	0	0
$839$	− 191.525i	− 0.228278i	−0.993465	−	0.114139i	$-0.963589\pi$
$839$	− 191.525i	− 0.228278i	0.993465	−	0.114139i	$-0.0364109\pi$
$840$	0	0
$841$	−701.784	−0.834464
$842$	0	0
$843$	1663.44i	1.97324i
$844$	0	0
$845$	− 1687.92i	− 1.99753i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	356.693	0.420134
$850$	0	0
$851$	832.000	0.977673
$852$	0	0
$853$	− 220.893i	− 0.258960i	−0.991582	−	0.129480i	$-0.958669\pi$
$853$	− 220.893i	− 0.258960i	0.991582	−	0.129480i	$-0.0413308\pi$
$854$	0	0
$855$	619.025	0.724006
$856$	0	0
$857$	939.501i	1.09627i	0.836391	+	0.548133i	$0.184661\pi$
$857$	939.501i	1.09627i	−0.836391	+	0.548133i	$0.815339\pi$
$858$	0	0
$859$	− 930.534i	− 1.08328i	−0.840612	−	0.541638i	$-0.817804\pi$
$859$	− 930.534i	− 1.08328i	0.840612	−	0.541638i	$-0.182196\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1172.16	−1.35824	−0.679120	−	0.734028i	$-0.737638\pi$
$863$	−1172.16	−1.35824	−0.679120	+	0.734028i	$0.737638\pi$
$864$	0	0
$865$	2354.17	2.72158
$866$	0	0
$867$	− 490.272i	− 0.565480i
$868$	0	0
$869$	311.176	0.358085
$870$	0	0
$871$	− 602.463i	− 0.691691i
$872$	0	0
$873$	− 1497.00i	− 1.71478i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−501.407	−0.571730	−0.285865	−	0.958270i	$-0.592281\pi$
$877$	−501.407	−0.571730	−0.285865	+	0.958270i	$0.592281\pi$
$878$	0	0
$879$	499.789	0.568588
$880$	0	0
$881$	− 1037.14i	− 1.17723i	−0.808412	−	0.588616i	$-0.799673\pi$
$881$	− 1037.14i	− 1.17723i	0.808412	−	0.588616i	$-0.200327\pi$
$882$	0	0
$883$	−103.226	−0.116904	−0.0584520	−	0.998290i	$-0.518616\pi$
$883$	−103.226	−0.116904	−0.0584520	+	0.998290i	$0.518616\pi$
$884$	0	0
$885$	319.435i	0.360944i
$886$	0	0
$887$	540.015i	0.608811i	0.952543	+	0.304405i	$0.0984577\pi$
$887$	540.015i	0.608811i	−0.952543	+	0.304405i	$0.901542\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	235.126	0.263890
$892$	0	0
$893$	95.8596	0.107346
$894$	0	0
$895$	− 2884.29i	− 3.22267i
$896$	0	0
$897$	2225.55	2.48110
$898$	0	0
$899$	431.651i	0.480146i
$900$	0	0
$901$	− 184.667i	− 0.204958i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1560.58	1.72440
$906$	0	0
$907$	722.382	0.796452	0.398226	−	0.917287i	$-0.369626\pi$
$907$	722.382	0.796452	0.398226	+	0.917287i	$0.369626\pi$
$908$	0	0
$909$	986.597i	1.08537i
$910$	0	0
$911$	−1292.59	−1.41887	−0.709436	−	0.704770i	$-0.751050\pi$
$911$	−1292.59	−1.41887	−0.709436	+	0.704770i	$0.751050\pi$
$912$	0	0
$913$	− 551.314i	− 0.603848i
$914$	0	0
$915$	− 1196.41i	− 1.30756i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1076.98	−1.17191	−0.585955	−	0.810344i	$-0.699280\pi$
$919$	−1076.98	−1.17191	−0.585955	+	0.810344i	$0.699280\pi$
$920$	0	0
$921$	−652.080	−0.708013
$922$	0	0
$923$	1834.39i	1.98742i
$924$	0	0
$925$	1497.57	1.61899
$926$	0	0
$927$	155.392i	0.167629i
$928$	0	0
$929$	1252.03i	1.34772i	0.738860	+	0.673859i	$0.235365\pi$
$929$	1252.03i	1.34772i	−0.738860	+	0.673859i	$0.764635\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1886.75	−2.02224
$934$	0	0
$935$	442.191	0.472931
$936$	0	0
$937$	− 857.272i	− 0.914911i	−0.889232	−	0.457456i	$-0.848761\pi$
$937$	− 857.272i	− 0.914911i	0.889232	−	0.457456i	$-0.151239\pi$
$938$	0	0
$939$	391.286	0.416705
$940$	0	0
$941$	708.319i	0.752730i	0.926472	+	0.376365i	$0.122826\pi$
$941$	708.319i	0.752730i	−0.926472	+	0.376365i	$0.877174\pi$
$942$	0	0
$943$	− 544.944i	− 0.577884i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−483.095	−0.510132	−0.255066	−	0.966924i	$-0.582097\pi$
$947$	−483.095	−0.510132	−0.255066	+	0.966924i	$0.582097\pi$
$948$	0	0
$949$	−351.427	−0.370313
$950$	0	0
$951$	− 1924.37i	− 2.02352i
$952$	0	0
$953$	67.5778	0.0709106	0.0354553	−	0.999371i	$-0.488712\pi$
$953$	67.5778	0.0709106	0.0354553	+	0.999371i	$0.488712\pi$
$954$	0	0
$955$	− 1945.40i	− 2.03706i
$956$	0	0
$957$	− 205.246i	− 0.214468i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−377.372	−0.392686
$962$	0	0
$963$	−540.593	−0.561363
$964$	0	0
$965$	− 351.626i	− 0.364379i
$966$	0	0
$967$	806.382	0.833901	0.416950	−	0.908929i	$-0.363099\pi$
$967$	806.382	0.833901	0.416950	+	0.908929i	$0.363099\pi$
$968$	0	0
$969$	400.135i	0.412936i
$970$	0	0
$971$	− 691.124i	− 0.711765i	−0.934531	−	0.355883i	$-0.884180\pi$
$971$	− 691.124i	− 0.711765i	0.934531	−	0.355883i	$-0.115820\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4005.90	4.10861
$976$	0	0
$977$	359.648	0.368115	0.184057	−	0.982916i	$-0.441077\pi$
$977$	359.648	0.368115	0.184057	+	0.982916i	$0.441077\pi$
$978$	0	0
$979$	− 454.111i	− 0.463852i
$980$	0	0
$981$	−1604.36	−1.63543
$982$	0	0
$983$	391.751i	0.398526i	0.979946	+	0.199263i	$0.0638548\pi$
$983$	391.751i	0.398526i	−0.979946	+	0.199263i	$0.936145\pi$
$984$	0	0
$985$	2148.84i	2.18157i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2061.71	2.08464
$990$	0	0
$991$	−1409.39	−1.42219	−0.711093	−	0.703098i	$-0.751800\pi$
$991$	−1409.39	−1.42219	−0.711093	+	0.703098i	$0.751800\pi$
$992$	0	0
$993$	− 1523.34i	− 1.53407i
$994$	0	0
$995$	−952.000	−0.956784
$996$	0	0
$997$	460.568i	0.461954i	0.972959	+	0.230977i	$0.0741922\pi$
$997$	460.568i	0.461954i	−0.972959	+	0.230977i	$0.925808\pi$
$998$	0	0
$999$	− 271.150i	− 0.271421i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.3.c.d.97.1		4
4.3	odd	2		196.3.b.b.97.4	yes	4
7.2	even	3		784.3.s.g.129.4		8
7.3	odd	6		784.3.s.g.705.4		8
7.4	even	3		784.3.s.g.705.1		8
7.5	odd	6		784.3.s.g.129.1		8
7.6	odd	2	inner	784.3.c.d.97.4		4
12.11	even	2		1764.3.d.e.685.1		4
28.3	even	6		196.3.h.c.117.1		8
28.11	odd	6		196.3.h.c.117.4		8
28.19	even	6		196.3.h.c.129.4		8
28.23	odd	6		196.3.h.c.129.1		8
28.27	even	2		196.3.b.b.97.1	✓	4
84.11	even	6		1764.3.z.k.901.4		8
84.23	even	6		1764.3.z.k.325.1		8
84.47	odd	6		1764.3.z.k.325.4		8
84.59	odd	6		1764.3.z.k.901.1		8
84.83	odd	2		1764.3.d.e.685.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.3.b.b.97.1	✓	4	28.27	even	2
196.3.b.b.97.4	yes	4	4.3	odd	2
196.3.h.c.117.1		8	28.3	even	6
196.3.h.c.117.4		8	28.11	odd	6
196.3.h.c.129.1		8	28.23	odd	6
196.3.h.c.129.4		8	28.19	even	6
784.3.c.d.97.1		4	1.1	even	1	trivial
784.3.c.d.97.4		4	7.6	odd	2	inner
784.3.s.g.129.1		8	7.5	odd	6
784.3.s.g.129.4		8	7.2	even	3
784.3.s.g.705.1		8	7.4	even	3
784.3.s.g.705.4		8	7.3	odd	6
1764.3.d.e.685.1		4	12.11	even	2
1764.3.d.e.685.4		4	84.83	odd	2
1764.3.z.k.325.1		8	84.23	even	6
1764.3.z.k.325.4		8	84.47	odd	6
1764.3.z.k.901.1		8	84.59	odd	6
1764.3.z.k.901.4		8	84.11	even	6

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.46088i q^{3} +8.47343i q^{5} -10.8995 q^{9} +O(q^{10})\)	\(q-4.46088i q^{3} +8.47343i q^{5} -10.8995 q^{9} -3.89949 q^{11} -19.1886i q^{13} +37.7990 q^{15} +13.3827i q^{17} +6.70259i q^{19} -26.0000 q^{23} -46.7990 q^{25} +8.47343i q^{27} -11.7990 q^{29} -36.5838i q^{31} +17.3952i q^{33} -32.0000 q^{37} -85.5980 q^{39} +20.9594i q^{41} -79.2965 q^{43} -92.3561i q^{45} -14.3019i q^{47} +59.6985 q^{51} -13.7990 q^{53} -33.0421i q^{55} +29.8995 q^{57} +8.45090i q^{59} -31.6520i q^{61} +162.593 q^{65} +31.3970 q^{67} +115.983i q^{69} -95.5980 q^{71} -18.3144i q^{73} +208.765i q^{75} -79.7990 q^{79} -60.2965 q^{81} +141.381i q^{83} -113.397 q^{85} +52.6339i q^{87} +116.454i q^{89} -163.196 q^{93} -56.7939 q^{95} +137.346i q^{97} +42.5025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 24 q^{11} + 72 q^{15} - 104 q^{23} - 108 q^{25} + 32 q^{29} - 128 q^{37} - 184 q^{39} - 40 q^{43} + 120 q^{51} + 24 q^{53} + 80 q^{57} + 96 q^{65} - 112 q^{67} - 224 q^{71} - 240 q^{79} + 36 q^{81} - 216 q^{85} - 336 q^{93} + 248 q^{95} + 368 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 24 * q^11 + 72 * q^15 - 104 * q^23 - 108 * q^25 + 32 * q^29 - 128 * q^37 - 184 * q^39 - 40 * q^43 + 120 * q^51 + 24 * q^53 + 80 * q^57 + 96 * q^65 - 112 * q^67 - 224 * q^71 - 240 * q^79 + 36 * q^81 - 216 * q^85 - 336 * q^93 + 248 * q^95 + 368 * q^99

Introduction

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