LMFDB - Embedded newform 2100.3.e.a.449.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2100,3,Mod(449,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.449"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2100.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$57.2208555157$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.13463425024.14
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 6x^{6} + 6x^{5} - 10x^{4} + 18x^{3} + 338x^{2} + 618x + 333 $ x^8 - 2x^7 - 6x^6 + 6x^5 - 10x^4 + 18x^3 + 338x^2 + 618*x + 333
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$0.555499 + 2.40882i$ of defining polynomial
Character	$\chi$	$=$	2100.449
Dual form			2100.3.e.a.449.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.99477 + 0.177124i) q^{3} +2.64575i q^{7} +(8.93725 - 1.06089i) q^{9} +14.1009i q^{11} +20.2288i q^{13} -21.8363 q^{17} -6.93725 q^{19} +(-0.468627 - 7.92341i) q^{21} -7.73550 q^{23} +(-26.5771 + 4.76013i) q^{27} +37.3073i q^{29} +15.2915 q^{31} +(-2.49760 - 42.2288i) q^{33} +12.1255i q^{37} +(-3.58301 - 60.5804i) q^{39} -42.3026i q^{41} +16.1255i q^{43} +71.8744 q^{47} -7.00000 q^{49} +(65.3948 - 3.86775i) q^{51} -101.446 q^{53} +(20.7755 - 1.22876i) q^{57} +25.7041i q^{59} +78.1033 q^{61} +(2.80686 + 23.6458i) q^{63} +123.166i q^{67} +(23.1660 - 1.37014i) q^{69} -34.5671i q^{71} -100.539i q^{73} -37.3073 q^{77} +42.5830 q^{79} +(78.7490 - 18.9629i) q^{81} +82.1075 q^{83} +(-6.60804 - 111.727i) q^{87} +42.3026i q^{89} -53.5203 q^{91} +(-45.7945 + 2.70850i) q^{93} -48.3320i q^{97} +(14.9595 + 126.023i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9} + 8 q^{19} + 28 q^{21} + 80 q^{31} + 56 q^{39} - 56 q^{49} + 248 q^{51} + 392 q^{61} + 16 q^{69} + 256 q^{79} + 376 q^{81} - 280 q^{91} + 416 q^{99}+O(q^{100}) $ 8 * q + 8 * q^9 + 8 * q^19 + 28 * q^21 + 80 * q^31 + 56 * q^39 - 56 * q^49 + 248 * q^51 + 392 * q^61 + 16 * q^69 + 256 * q^79 + 376 * q^81 - 280 * q^91 + 416 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.99477	+	0.177124i	−0.998256	+	0.0590414i
$3$	−2.99477	+	0.177124i	−0.998256	+	0.0590414i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			2.64575i			0.377964i
$7$			2.64575i			0.377964i
$8$		0			0
$9$	8.93725	−	1.06089i	0.993028	−	0.117877i
$10$		0			0
$11$			14.1009i			1.28190i	0.767585	+	0.640948i	$0.221458\pi$
$11$			14.1009i			1.28190i	−0.767585	+	0.640948i	$0.778542\pi$
$12$		0			0
$13$			20.2288i			1.55606i	0.628228	+	0.778029i	$0.283780\pi$
$13$			20.2288i			1.55606i	−0.628228	+	0.778029i	$0.716220\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−21.8363			−1.28449			−0.642246	−	0.766499i	$-0.721997\pi$
$17$	−21.8363			−1.28449			−0.642246	+	0.766499i	$0.721997\pi$
$18$		0			0
$19$	−6.93725			−0.365119			−0.182559	−	0.983195i	$-0.558438\pi$
$19$	−6.93725			−0.365119			−0.182559	+	0.983195i	$0.558438\pi$
$20$		0			0
$21$	−0.468627	−	7.92341i	−0.0223156	−	0.377305i
$22$		0			0
$23$	−7.73550			−0.336326			−0.168163	−	0.985759i	$-0.553783\pi$
$23$	−7.73550			−0.336326			−0.168163	+	0.985759i	$0.553783\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−26.5771	+	4.76013i	−0.984336	+	0.176301i
$28$		0			0
$29$			37.3073i			1.28646i	0.765673	+	0.643230i	$0.222406\pi$
$29$			37.3073i			1.28646i	−0.765673	+	0.643230i	$0.777594\pi$
$30$		0			0
$31$	15.2915			0.493274			0.246637	−	0.969108i	$-0.420674\pi$
$31$	15.2915			0.493274			0.246637	+	0.969108i	$0.420674\pi$
$32$		0			0
$33$	−2.49760	−	42.2288i	−0.0756850	−	1.27966i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			12.1255i			0.327716i	0.986484	+	0.163858i	$0.0523939\pi$
$37$			12.1255i			0.327716i	−0.986484	+	0.163858i	$0.947606\pi$
$38$		0			0
$39$	−3.58301	−	60.5804i	−0.0918719	−	1.55334i
$40$		0			0
$41$		−	42.3026i		−	1.03177i	−0.856658	−	0.515885i	$-0.827463\pi$
$41$		−	42.3026i		−	1.03177i	0.856658	−	0.515885i	$-0.172537\pi$
$42$		0			0
$43$			16.1255i			0.375011i	0.982264	+	0.187506i	$0.0600403\pi$
$43$			16.1255i			0.375011i	−0.982264	+	0.187506i	$0.939960\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	71.8744			1.52924			0.764621	−	0.644480i	$-0.222926\pi$
$47$	71.8744			1.52924			0.764621	+	0.644480i	$0.222926\pi$
$48$		0			0
$49$	−7.00000			−0.142857
$50$		0			0
$51$	65.3948	−	3.86775i	1.28225	−	0.0758382i
$52$		0			0
$53$	−101.446			−1.91408			−0.957040	−	0.289956i	$-0.906359\pi$
$53$	−101.446			−1.91408			−0.957040	+	0.289956i	$0.906359\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	20.7755	−	1.22876i	0.364482	−	0.0215571i
$58$		0			0
$59$			25.7041i			0.435663i	0.975986	+	0.217831i	$0.0698983\pi$
$59$			25.7041i			0.435663i	−0.975986	+	0.217831i	$0.930102\pi$
$60$		0			0
$61$	78.1033			1.28038			0.640191	−	0.768216i	$-0.278855\pi$
$61$	78.1033			1.28038			0.640191	+	0.768216i	$0.278855\pi$
$62$		0			0
$63$	2.80686	+	23.6458i	0.0445533	+	0.375329i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			123.166i			1.83830i	0.393909	+	0.919149i	$0.371122\pi$
$67$			123.166i			1.83830i	−0.393909	+	0.919149i	$0.628878\pi$
$68$		0			0
$69$	23.1660	−	1.37014i	0.335739	−	0.0198572i
$70$		0			0
$71$		−	34.5671i		−	0.486860i	−0.969918	−	0.243430i	$-0.921727\pi$
$71$		−	34.5671i		−	0.486860i	0.969918	−	0.243430i	$-0.0782726\pi$
$72$		0			0
$73$		−	100.539i		−	1.37724i	−0.725122	−	0.688620i	$-0.758217\pi$
$73$		−	100.539i		−	1.37724i	0.725122	−	0.688620i	$-0.241783\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−37.3073			−0.484511
$78$		0			0
$79$	42.5830			0.539025			0.269513	−	0.962997i	$-0.413137\pi$
$79$	42.5830			0.539025			0.269513	+	0.962997i	$0.413137\pi$
$80$		0			0
$81$	78.7490	−	18.9629i	0.972210	−	0.234110i
$82$		0			0
$83$	82.1075			0.989247			0.494623	−	0.869107i	$-0.335306\pi$
$83$	82.1075			0.989247			0.494623	+	0.869107i	$0.335306\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.60804	−	111.727i	−0.0759545	−	1.28422i
$88$		0			0
$89$			42.3026i			0.475310i	0.971350	+	0.237655i	$0.0763787\pi$
$89$			42.3026i			0.475310i	−0.971350	+	0.237655i	$0.923621\pi$
$90$		0			0
$91$	−53.5203			−0.588135
$92$		0			0
$93$	−45.7945	+	2.70850i	−0.492414	+	0.0291236i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	48.3320i		−	0.498268i	−0.968469	−	0.249134i	$-0.919854\pi$
$97$		−	48.3320i		−	0.498268i	0.968469	−	0.249134i	$-0.0801460\pi$
$98$		0			0
$99$	14.9595	+	126.023i	0.151106	+	1.27296i
$100$		0			0
$101$		−	80.7374i		−	0.799380i	−0.916650	−	0.399690i	$-0.869118\pi$
$101$		−	80.7374i		−	0.799380i	0.916650	−	0.399690i	$-0.130882\pi$
$102$		0			0
$103$			48.2510i			0.468456i	0.972182	+	0.234228i	$0.0752563\pi$
$103$			48.2510i			0.468456i	−0.972182	+	0.234228i	$0.924744\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−196.527			−1.83670			−0.918351	−	0.395767i	$-0.870479\pi$
$107$	−196.527			−1.83670			−0.918351	+	0.395767i	$0.870479\pi$
$108$		0			0
$109$	53.2915			0.488913			0.244456	−	0.969660i	$-0.421390\pi$
$109$	53.2915			0.488913			0.244456	+	0.969660i	$0.421390\pi$
$110$		0			0
$111$	−2.14772	−	36.3130i	−0.0193488	−	0.327144i
$112$		0			0
$113$	−140.124			−1.24003			−0.620017	−	0.784589i	$-0.712874\pi$
$113$	−140.124			−1.24003			−0.620017	+	0.784589i	$0.712874\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	21.4605	+	180.790i	0.183423	+	1.54521i
$118$		0			0
$119$		−	57.7735i		−	0.485492i
$120$		0			0
$121$	−77.8340			−0.643256
$122$		0			0
$123$	7.49281	+	126.686i	0.0609172	+	1.02997i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			115.579i			0.910071i	0.890473	+	0.455036i	$0.150374\pi$
$127$			115.579i			0.910071i	−0.890473	+	0.455036i	$0.849626\pi$
$128$		0			0
$129$	−2.85622	−	48.2921i	−0.0221412	−	0.374357i
$130$		0			0
$131$		−	82.5929i		−	0.630480i	−0.949012	−	0.315240i	$-0.897915\pi$
$131$		−	82.5929i		−	0.630480i	0.949012	−	0.315240i	$-0.102085\pi$
$132$		0			0
$133$		−	18.3542i		−	0.138002i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−153.739			−1.12218			−0.561092	−	0.827753i	$-0.689619\pi$
$137$	−153.739			−1.12218			−0.561092	+	0.827753i	$0.689619\pi$
$138$		0			0
$139$	−220.516			−1.58645			−0.793224	−	0.608930i	$-0.791599\pi$
$139$	−220.516			−1.58645			−0.793224	+	0.608930i	$0.791599\pi$
$140$		0			0
$141$	−215.247	+	12.7307i	−1.52657	+	0.0902887i
$142$		0			0
$143$	−285.243			−1.99470
$144$		0			0
$145$		0			0
$146$		0			0
$147$	20.9634	−	1.23987i	0.142608	−	0.00843449i
$148$		0			0
$149$		−	194.272i		−	1.30384i	−0.758288	−	0.651920i	$-0.773964\pi$
$149$		−	194.272i		−	1.30384i	0.758288	−	0.651920i	$-0.226036\pi$
$150$		0			0
$151$	128.081			0.848219			0.424109	−	0.905611i	$-0.360587\pi$
$151$	128.081			0.848219			0.424109	+	0.905611i	$0.360587\pi$
$152$		0			0
$153$	−195.157	+	23.1660i	−1.27554	+	0.151412i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	147.682i		−	0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$157$		−	147.682i		−	0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$158$		0			0
$159$	303.808	−	17.9686i	1.91074	−	0.113010i
$160$		0			0
$161$		−	20.4662i		−	0.127119i
$162$		0			0
$163$		−	210.369i		−	1.29060i	−0.763927	−	0.645302i	$-0.776731\pi$
$163$		−	210.369i		−	1.29060i	0.763927	−	0.645302i	$-0.223269\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	85.0905			0.509524			0.254762	−	0.967004i	$-0.418003\pi$
$167$	85.0905			0.509524			0.254762	+	0.967004i	$0.418003\pi$
$168$		0			0
$169$	−240.203			−1.42132
$170$		0			0
$171$	−62.0000	+	7.35968i	−0.362573	+	0.0430391i
$172$		0			0
$173$	93.4681			0.540278			0.270139	−	0.962821i	$-0.412930\pi$
$173$	93.4681			0.540278			0.270139	+	0.962821i	$0.412930\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−4.55282	−	76.9778i	−0.0257222	−	0.434903i
$178$		0			0
$179$			196.527i			1.09792i	0.835850	+	0.548958i	$0.184975\pi$
$179$			196.527i			1.09792i	−0.835850	+	0.548958i	$0.815025\pi$
$180$		0			0
$181$	−164.022			−0.906200			−0.453100	−	0.891460i	$-0.649682\pi$
$181$	−164.022			−0.906200			−0.453100	+	0.891460i	$0.649682\pi$
$182$		0			0
$183$	−233.901	+	13.8340i	−1.27815	+	0.0755956i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	307.911i		−	1.64658i
$188$		0			0
$189$	−12.5941	−	70.3163i	−0.0666356	−	0.372044i
$190$		0			0
$191$		−	230.209i		−	1.20528i	−0.798011	−	0.602642i	$-0.794115\pi$
$191$		−	230.209i		−	1.20528i	0.798011	−	0.602642i	$-0.205885\pi$
$192$		0			0
$193$		−	184.620i		−	0.956578i	−0.878202	−	0.478289i	$-0.841257\pi$
$193$		−	184.620i		−	0.956578i	0.878202	−	0.478289i	$-0.158743\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−91.4558			−0.464243			−0.232121	−	0.972687i	$-0.574567\pi$
$197$	−91.4558			−0.464243			−0.232121	+	0.972687i	$0.574567\pi$
$198$		0			0
$199$	203.749			1.02386			0.511932	−	0.859026i	$-0.328930\pi$
$199$	203.749			1.02386			0.511932	+	0.859026i	$0.328930\pi$
$200$		0			0
$201$	−21.8157	−	368.853i	−0.108536	−	1.83509i
$202$		0			0
$203$	−98.7060			−0.486236
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−69.1341	+	8.20653i	−0.333981	+	0.0396451i
$208$		0			0
$209$		−	97.8212i		−	0.468044i
$210$		0			0
$211$	188.243			0.892147			0.446074	−	0.894996i	$-0.352822\pi$
$211$	188.243			0.892147			0.446074	+	0.894996i	$0.352822\pi$
$212$		0			0
$213$	6.12267	+	103.520i	0.0287449	+	0.486011i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			40.4575i			0.186440i
$218$		0			0
$219$	17.8078	+	301.089i	0.0813143	+	1.37484i
$220$		0			0
$221$		−	441.722i		−	1.99874i
$222$		0			0
$223$			154.125i			0.691146i	0.938392	+	0.345573i	$0.112315\pi$
$223$			154.125i			0.691146i	−0.938392	+	0.345573i	$0.887685\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	233.592			1.02904			0.514519	−	0.857479i	$-0.327970\pi$
$227$	233.592			1.02904			0.514519	+	0.857479i	$0.327970\pi$
$228$		0			0
$229$	−275.269			−1.20205			−0.601025	−	0.799231i	$-0.705241\pi$
$229$	−275.269			−1.20205			−0.601025	+	0.799231i	$0.705241\pi$
$230$		0			0
$231$	111.727	−	6.60804i	0.483666	−	0.0286062i
$232$		0			0
$233$	−35.4518			−0.152154			−0.0760769	−	0.997102i	$-0.524239\pi$
$233$	−35.4518			−0.152154			−0.0760769	+	0.997102i	$0.524239\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−127.526	+	7.54249i	−0.538085	+	0.0318248i
$238$		0			0
$239$			56.8888i			0.238028i	0.992893	+	0.119014i	$0.0379734\pi$
$239$			56.8888i			0.238028i	−0.992893	+	0.119014i	$0.962027\pi$
$240$		0			0
$241$	−209.749			−0.870328			−0.435164	−	0.900351i	$-0.643310\pi$
$241$	−209.749			−0.870328			−0.435164	+	0.900351i	$0.643310\pi$
$242$		0			0
$243$	−232.476	+	70.7379i	−0.956692	+	0.291102i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	140.332i		−	0.568146i
$248$		0			0
$249$	−245.893	+	14.5432i	−0.987521	+	0.0584066i
$250$		0			0
$251$			77.1123i			0.307220i	0.988132	+	0.153610i	$0.0490900\pi$
$251$			77.1123i			0.307220i	−0.988132	+	0.153610i	$0.950910\pi$
$252$		0			0
$253$		−	109.077i		−	0.431135i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−158.335			−0.616090			−0.308045	−	0.951372i	$-0.599675\pi$
$257$	−158.335			−0.616090			−0.308045	+	0.951372i	$0.599675\pi$
$258$		0			0
$259$	−32.0810			−0.123865
$260$		0			0
$261$	39.5791	+	333.425i	0.151644	+	1.27749i
$262$		0			0
$263$	−150.114			−0.570776			−0.285388	−	0.958412i	$-0.592122\pi$
$263$	−150.114			−0.570776			−0.285388	+	0.958412i	$0.592122\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−7.49281	−	126.686i	−0.0280630	−	0.474480i
$268$		0			0
$269$			190.804i			0.709308i	0.934998	+	0.354654i	$0.115401\pi$
$269$			190.804i			0.709308i	−0.934998	+	0.354654i	$0.884599\pi$
$270$		0			0
$271$	−342.450			−1.26365			−0.631826	−	0.775110i	$-0.717694\pi$
$271$	−342.450			−1.26365			−0.631826	+	0.775110i	$0.717694\pi$
$272$		0			0
$273$	160.281	−	9.47974i	0.587109	−	0.0347243i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			93.2470i			0.336632i	0.985733	+	0.168316i	$0.0538329\pi$
$277$			93.2470i			0.336632i	−0.985733	+	0.168316i	$0.946167\pi$
$278$		0			0
$279$	136.664	−	16.2226i	0.489835	−	0.0581456i
$280$		0			0
$281$			251.075i			0.893505i	0.894657	+	0.446753i	$0.147420\pi$
$281$			251.075i			0.893505i	−0.894657	+	0.446753i	$0.852580\pi$
$282$		0			0
$283$			16.6497i			0.0588328i	0.999567	+	0.0294164i	$0.00936489\pi$
$283$			16.6497i			0.0588328i	−0.999567	+	0.0294164i	$0.990635\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	111.922			0.389972
$288$		0			0
$289$	187.826			0.649917
$290$		0			0
$291$	8.56078	+	144.743i	0.0294185	+	0.497399i
$292$		0			0
$293$	−307.322			−1.04888			−0.524440	−	0.851448i	$-0.675725\pi$
$293$	−307.322			−1.04888			−0.524440	+	0.851448i	$0.675725\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−67.1219	−	374.759i	−0.226000	−	1.26182i
$298$		0			0
$299$		−	156.480i		−	0.523343i
$300$		0			0
$301$	−42.6640			−0.141741
$302$		0			0
$303$	14.3006	+	241.790i	0.0471965	+	0.797985i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	126.775i		−	0.412948i	−0.978452	−	0.206474i	$-0.933801\pi$
$307$		−	126.775i		−	0.412948i	0.978452	−	0.206474i	$-0.0661990\pi$
$308$		0			0
$309$	−8.54642	−	144.500i	−0.0276583	−	0.467639i
$310$		0			0
$311$		−	328.915i		−	1.05761i	−0.848745	−	0.528803i	$-0.822641\pi$
$311$		−	328.915i		−	1.05761i	0.848745	−	0.528803i	$-0.177359\pi$
$312$		0			0
$313$		−	259.284i		−	0.828382i	−0.910190	−	0.414191i	$-0.864065\pi$
$313$		−	259.284i		−	0.828382i	0.910190	−	0.414191i	$-0.135935\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.62027			0.0271933			0.0135966	−	0.999908i	$-0.495672\pi$
$317$	8.62027			0.0271933			0.0135966	+	0.999908i	$0.495672\pi$
$318$		0			0
$319$	−526.065			−1.64911
$320$		0			0
$321$	588.553	−	34.8097i	1.83350	−	0.108442i
$322$		0			0
$323$	151.484			0.468992
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−159.596	+	9.43922i	−0.488060	+	0.0288661i
$328$		0			0
$329$			190.162i			0.577999i
$330$		0			0
$331$	−474.369			−1.43314			−0.716569	−	0.697516i	$-0.754288\pi$
$331$	−474.369			−1.43314			−0.716569	+	0.697516i	$0.754288\pi$
$332$		0			0
$333$	12.8638	+	108.369i	0.0386301	+	0.325431i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			609.195i			1.80770i	0.427850	+	0.903850i	$0.359271\pi$
$337$			609.195i			1.80770i	−0.427850	+	0.903850i	$0.640729\pi$
$338$		0			0
$339$	419.638	−	24.8193i	1.23787	−	0.0732133i
$340$		0			0
$341$			215.623i			0.632326i
$342$		0			0
$343$		−	18.5203i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	427.136			1.23094			0.615470	−	0.788161i	$-0.288966\pi$
$347$	427.136			1.23094			0.615470	+	0.788161i	$0.288966\pi$
$348$		0			0
$349$	−316.759			−0.907620			−0.453810	−	0.891098i	$-0.649936\pi$
$349$	−316.759			−0.907620			−0.453810	+	0.891098i	$0.649936\pi$
$350$		0			0
$351$	−96.2915	−	537.621i	−0.274335	−	1.53168i
$352$		0			0
$353$	−86.4606			−0.244931			−0.122465	−	0.992473i	$-0.539080\pi$
$353$	−86.4606			−0.244931			−0.122465	+	0.992473i	$0.539080\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	10.2331	+	173.018i	0.0286642	+	0.484645i
$358$		0			0
$359$			286.213i			0.797252i	0.917114	+	0.398626i	$0.130513\pi$
$359$			286.213i			0.797252i	−0.917114	+	0.398626i	$0.869487\pi$
$360$		0			0
$361$	−312.875			−0.866688
$362$		0			0
$363$	233.095	−	13.7863i	0.642134	−	0.0379788i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	111.616i		−	0.304130i	−0.988371	−	0.152065i	$-0.951408\pi$
$367$		−	111.616i		−	0.304130i	0.988371	−	0.152065i	$-0.0485923\pi$
$368$		0			0
$369$	−44.8784	−	378.069i	−0.121622	−	1.02458i
$370$		0			0
$371$		−	268.402i		−	0.723454i
$372$		0			0
$373$			85.0850i			0.228110i	0.993474	+	0.114055i	$0.0363840\pi$
$373$			85.0850i			0.228110i	−0.993474	+	0.114055i	$0.963616\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−754.681			−2.00181
$378$		0			0
$379$	26.5464			0.0700433			0.0350217	−	0.999387i	$-0.488850\pi$
$379$	26.5464			0.0700433			0.0350217	+	0.999387i	$0.488850\pi$
$380$		0			0
$381$	−20.4719	−	346.132i	−0.0537319	−	0.908484i
$382$		0			0
$383$	−115.547			−0.301690			−0.150845	−	0.988557i	$-0.548199\pi$
$383$	−115.547			−0.301690			−0.150845	+	0.988557i	$0.548199\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	17.1074	+	144.118i	0.0442052	+	0.372397i
$388$		0			0
$389$		−	385.718i		−	0.991563i	−0.868447	−	0.495782i	$-0.834882\pi$
$389$		−	385.718i		−	0.991563i	0.868447	−	0.495782i	$-0.165118\pi$
$390$		0			0
$391$	168.915			0.432008
$392$		0			0
$393$	14.6292	+	247.346i	0.0372244	+	0.629380i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	47.5934i		−	0.119883i	−0.998202	−	0.0599413i	$-0.980909\pi$
$397$		−	47.5934i		−	0.119883i	0.998202	−	0.0599413i	$-0.0190914\pi$
$398$		0			0
$399$	3.25098	+	54.9667i	0.00814783	+	0.137761i
$400$		0			0
$401$			458.563i			1.14355i	0.820411	+	0.571775i	$0.193745\pi$
$401$			458.563i			1.14355i	−0.820411	+	0.571775i	$0.806255\pi$
$402$		0			0
$403$			309.328i			0.767563i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−170.980			−0.420098
$408$		0			0
$409$	57.8666			0.141483			0.0707416	−	0.997495i	$-0.477463\pi$
$409$	57.8666			0.141483			0.0707416	+	0.997495i	$0.477463\pi$
$410$		0			0
$411$	460.413	−	27.2310i	1.12023	−	0.0662554i
$412$		0			0
$413$	−68.0066			−0.164665
$414$		0			0
$415$		0			0
$416$		0			0
$417$	660.395	−	39.0588i	1.58368	−	0.0936662i
$418$		0			0
$419$			115.304i			0.275190i	0.990489	+	0.137595i	$0.0439372\pi$
$419$			115.304i			0.275190i	−0.990489	+	0.137595i	$0.956063\pi$
$420$		0			0
$421$	622.664			1.47901			0.739506	−	0.673150i	$-0.235059\pi$
$421$	622.664			1.47901			0.739506	+	0.673150i	$0.235059\pi$
$422$		0			0
$423$	642.360	−	76.2510i	1.51858	−	0.180262i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			206.642i			0.483939i
$428$		0			0
$429$	854.235	−	50.5234i	1.99122	−	0.117770i
$430$		0			0
$431$			170.666i			0.395978i	0.980204	+	0.197989i	$0.0634410\pi$
$431$			170.666i			0.395978i	−0.980204	+	0.197989i	$0.936559\pi$
$432$		0			0
$433$		−	251.992i		−	0.581968i	−0.956728	−	0.290984i	$-0.906017\pi$
$433$		−	251.992i		−	0.581968i	0.956728	−	0.290984i	$-0.0939826\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	53.6631			0.122799
$438$		0			0
$439$	452.089			1.02982			0.514908	−	0.857246i	$-0.327826\pi$
$439$	452.089			1.02982			0.514908	+	0.857246i	$0.327826\pi$
$440$		0			0
$441$	−62.5608	+	7.42624i	−0.141861	+	0.0168396i
$442$		0			0
$443$	−109.182			−0.246460			−0.123230	−	0.992378i	$-0.539325\pi$
$443$	−109.182			−0.246460			−0.123230	+	0.992378i	$0.539325\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	34.4103	+	581.800i	0.0769806	+	1.30157i
$448$		0			0
$449$		−	565.005i		−	1.25836i	−0.777259	−	0.629181i	$-0.783390\pi$
$449$		−	565.005i		−	1.25836i	0.777259	−	0.629181i	$-0.216610\pi$
$450$		0			0
$451$	596.502			1.32262
$452$		0			0
$453$	−383.573	+	22.6863i	−0.846739	+	0.0500801i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	595.705i		−	1.30351i	−0.758429	−	0.651756i	$-0.774033\pi$
$457$		−	595.705i		−	1.30351i	0.758429	−	0.651756i	$-0.225967\pi$
$458$		0			0
$459$	580.346	−	103.944i	1.26437	−	0.226457i
$460$		0			0
$461$			445.590i			0.966573i	0.875462	+	0.483286i	$0.160557\pi$
$461$			445.590i			0.966573i	−0.875462	+	0.483286i	$0.839443\pi$
$462$		0			0
$463$		−	333.919i		−	0.721207i	−0.932719	−	0.360604i	$-0.882571\pi$
$463$		−	333.919i		−	0.721207i	0.932719	−	0.360604i	$-0.117429\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	448.244			0.959838			0.479919	−	0.877313i	$-0.340666\pi$
$467$	448.244			0.959838			0.479919	+	0.877313i	$0.340666\pi$
$468$		0			0
$469$	−325.867			−0.694812
$470$		0			0
$471$	26.1581	+	442.274i	0.0555374	+	0.939011i
$472$		0			0
$473$	−227.383			−0.480725
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−906.651	+	107.624i	−1.90074	+	0.225626i
$478$		0			0
$479$		−	13.7014i		−	0.0286043i	−0.999898	−	0.0143021i	$-0.995447\pi$
$479$		−	13.7014i		−	0.0286043i	0.999898	−	0.0143021i	$-0.00455267\pi$
$480$		0			0
$481$	−245.284			−0.509945
$482$		0			0
$483$	3.62506	+	61.2915i	0.00750531	+	0.126898i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	11.2549i		−	0.0231107i	−0.999933	−	0.0115554i	$-0.996322\pi$
$487$		−	11.2549i		−	0.0231107i	0.999933	−	0.0115554i	$-0.00367827\pi$
$488$		0			0
$489$	37.2614	+	630.005i	0.0761992	+	1.28835i
$490$		0			0
$491$			285.642i			0.581756i	0.956760	+	0.290878i	$0.0939473\pi$
$491$			285.642i			0.581756i	−0.956760	+	0.290878i	$0.906053\pi$
$492$		0			0
$493$		−	814.656i		−	1.65245i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	91.4558			0.184016
$498$		0			0
$499$	−780.818			−1.56477			−0.782383	−	0.622798i	$-0.785996\pi$
$499$	−780.818			−1.56477			−0.782383	+	0.622798i	$0.785996\pi$
$500$		0			0
$501$	−254.826	+	15.0716i	−0.508635	+	0.0300830i
$502$		0			0
$503$	420.771			0.836522			0.418261	−	0.908327i	$-0.362640\pi$
$503$	420.771			0.836522			0.418261	+	0.908327i	$0.362640\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	719.351	−	42.5457i	1.41884	−	0.0839166i
$508$		0			0
$509$		−	102.973i		−	0.202305i	−0.994871	−	0.101152i	$-0.967747\pi$
$509$		−	102.973i		−	0.202305i	0.994871	−	0.101152i	$-0.0322530\pi$
$510$		0			0
$511$	266.000			0.520548
$512$		0			0
$513$	184.372	−	33.0222i	0.359400	−	0.0643708i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			1013.49i			1.96033i
$518$		0			0
$519$	−279.915	+	16.5555i	−0.539335	+	0.0318988i
$520$		0			0
$521$		−	76.7836i		−	0.147377i	−0.997281	−	0.0736887i	$-0.976523\pi$
$521$		−	76.7836i		−	0.147377i	0.997281	−	0.0736887i	$-0.0234771\pi$
$522$		0			0
$523$		−	404.561i		−	0.773539i	−0.922176	−	0.386769i	$-0.873591\pi$
$523$		−	404.561i		−	0.773539i	0.922176	−	0.386769i	$-0.126409\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−333.911			−0.633606
$528$		0			0
$529$	−469.162			−0.886885
$530$		0			0
$531$	27.2693	+	229.724i	0.0513546	+	0.432625i
$532$		0			0
$533$	855.728			1.60549
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−34.8097	−	588.553i	−0.0648226	−	1.09600i
$538$		0			0
$539$		−	98.7060i		−	0.183128i
$540$		0			0
$541$	592.907			1.09595			0.547973	−	0.836496i	$-0.315400\pi$
$541$	592.907			1.09595			0.547973	+	0.836496i	$0.315400\pi$
$542$		0			0
$543$	491.208	−	29.0523i	0.904619	−	0.0535034i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			793.689i			1.45099i	0.688230	+	0.725493i	$0.258388\pi$
$547$			793.689i			1.45099i	−0.688230	+	0.725493i	$0.741612\pi$
$548$		0			0
$549$	698.029	−	82.8591i	1.27145	−	0.150927i
$550$		0			0
$551$		−	258.811i		−	0.469711i
$552$		0			0
$553$			112.664i			0.203732i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−368.963			−0.662411			−0.331206	−	0.943559i	$-0.607455\pi$
$557$	−368.963			−0.662411			−0.331206	+	0.943559i	$0.607455\pi$
$558$		0			0
$559$	−326.199			−0.583540
$560$		0			0
$561$	54.5385	+	922.122i	0.0972167	+	1.64371i
$562$		0			0
$563$	529.796			0.941022			0.470511	−	0.882394i	$-0.344070\pi$
$563$	529.796			0.941022			0.470511	+	0.882394i	$0.344070\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	50.1712	+	208.350i	0.0884853	+	0.367461i
$568$		0			0
$569$			1068.04i			1.87705i	0.345216	+	0.938523i	$0.387806\pi$
$569$			1068.04i			1.87705i	−0.345216	+	0.938523i	$0.612194\pi$
$570$		0			0
$571$	−118.191			−0.206989			−0.103495	−	0.994630i	$-0.533002\pi$
$571$	−118.191			−0.206989			−0.103495	+	0.994630i	$0.533002\pi$
$572$		0			0
$573$	40.7757	+	689.423i	0.0711618	+	1.20318i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	586.559i		−	1.01657i	−0.861190	−	0.508284i	$-0.830280\pi$
$577$		−	586.559i		−	1.01657i	0.861190	−	0.508284i	$-0.169720\pi$
$578$		0			0
$579$	32.7006	+	552.893i	0.0564778	+	0.954909i
$580$		0			0
$581$			217.236i			0.373900i
$582$		0			0
$583$		−	1430.48i		−	2.45365i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	809.558			1.37914			0.689572	−	0.724217i	$-0.257799\pi$
$587$	809.558			1.37914			0.689572	+	0.724217i	$0.257799\pi$
$588$		0			0
$589$	−106.081			−0.180104
$590$		0			0
$591$	273.889	−	16.1991i	0.463433	−	0.0274096i
$592$		0			0
$593$	480.314			0.809973			0.404986	−	0.914323i	$-0.367276\pi$
$593$	480.314			0.809973			0.404986	+	0.914323i	$0.367276\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−610.181	+	36.0889i	−1.02208	+	0.0604504i
$598$		0			0
$599$			386.689i			0.645557i	0.946474	+	0.322779i	$0.104617\pi$
$599$			386.689i			0.645557i	−0.946474	+	0.322779i	$0.895383\pi$
$600$		0			0
$601$	−474.369			−0.789299			−0.394649	−	0.918832i	$-0.629134\pi$
$601$	−474.369			−0.789299			−0.394649	+	0.918832i	$0.629134\pi$
$602$		0			0
$603$	130.666	+	1100.77i	0.216693	+	1.82548i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			434.199i			0.715319i	0.933852	+	0.357660i	$0.116425\pi$
$607$			434.199i			0.715319i	−0.933852	+	0.357660i	$0.883575\pi$
$608$		0			0
$609$	295.601	−	17.4832i	0.485388	−	0.0287081i
$610$		0			0
$611$			1453.93i			2.37959i
$612$		0			0
$613$			614.782i			1.00291i	0.865185	+	0.501453i	$0.167201\pi$
$613$			614.782i			1.00291i	−0.865185	+	0.501453i	$0.832799\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	527.697			0.855263			0.427632	−	0.903953i	$-0.359348\pi$
$617$	527.697			0.855263			0.427632	+	0.903953i	$0.359348\pi$
$618$		0			0
$619$	679.970			1.09850			0.549249	−	0.835659i	$-0.314914\pi$
$619$	679.970			1.09850			0.549249	+	0.835659i	$0.314914\pi$
$620$		0			0
$621$	205.587	−	36.8220i	0.331058	−	0.0592946i
$622$		0			0
$623$	−111.922			−0.179650
$624$		0			0
$625$		0			0
$626$		0			0
$627$	17.3265	+	292.952i	0.0276340	+	0.467227i
$628$		0			0
$629$		−	264.776i		−	0.420948i
$630$		0			0
$631$	−929.239			−1.47265			−0.736323	−	0.676631i	$-0.763439\pi$
$631$	−929.239			−1.47265			−0.736323	+	0.676631i	$0.763439\pi$
$632$		0			0
$633$	−563.744	+	33.3424i	−0.890591	+	0.0526737i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	141.601i		−	0.222294i
$638$		0			0
$639$	−36.6719	−	308.935i	−0.0573895	−	0.483466i
$640$		0			0
$641$		−	798.925i		−	1.24637i	−0.782073	−	0.623187i	$-0.785838\pi$
$641$		−	798.925i		−	1.24637i	0.782073	−	0.623187i	$-0.214162\pi$
$642$		0			0
$643$			306.051i			0.475973i	0.971268	+	0.237987i	$0.0764875\pi$
$643$			306.051i			0.475973i	−0.971268	+	0.237987i	$0.923513\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	856.527			1.32384			0.661922	−	0.749573i	$-0.269741\pi$
$647$	856.527			1.32384			0.661922	+	0.749573i	$0.269741\pi$
$648$		0			0
$649$	−362.450			−0.558474
$650$		0			0
$651$	−7.16601	−	121.161i	−0.0110077	−	0.186115i
$652$		0			0
$653$	610.533			0.934966			0.467483	−	0.884002i	$-0.345161\pi$
$653$	610.533			0.934966			0.467483	+	0.884002i	$0.345161\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−106.661	−	898.539i	−0.162345	−	1.36764i
$658$		0			0
$659$			41.3318i			0.0627190i	0.999508	+	0.0313595i	$0.00998367\pi$
$659$			41.3318i			0.0627190i	−0.999508	+	0.0313595i	$0.990016\pi$
$660$		0			0
$661$	259.285			0.392262			0.196131	−	0.980578i	$-0.437162\pi$
$661$	259.285			0.392262			0.196131	+	0.980578i	$0.437162\pi$
$662$		0			0
$663$	78.2398	+	1322.85i	0.118009	+	1.99526i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	288.591i		−	0.432670i
$668$		0			0
$669$	−27.2994	−	461.570i	−0.0408062	−	0.689940i
$670$		0			0
$671$			1101.32i			1.64132i
$672$		0			0
$673$			113.498i			0.168645i	0.996439	+	0.0843225i	$0.0268726\pi$
$673$			113.498i			0.168645i	−0.996439	+	0.0843225i	$0.973127\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1239.43			1.83077			0.915387	−	0.402575i	$-0.131885\pi$
$677$	1239.43			1.83077			0.915387	+	0.402575i	$0.131885\pi$
$678$		0			0
$679$	127.875			0.188328
$680$		0			0
$681$	−699.553	+	41.3748i	−1.02724	+	0.0607559i
$682$		0			0
$683$	1286.58			1.88371			0.941856	−	0.336017i	$-0.109080\pi$
$683$	1286.58			1.88371			0.941856	+	0.336017i	$0.109080\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	824.367	−	48.7569i	1.19995	−	0.0709707i
$688$		0			0
$689$		−	2052.13i		−	2.97842i
$690$		0			0
$691$	−888.278			−1.28550			−0.642748	−	0.766077i	$-0.722206\pi$
$691$	−888.278			−1.28550			−0.642748	+	0.766077i	$0.722206\pi$
$692$		0			0
$693$	−333.425	+	39.5791i	−0.481133	+	0.0571127i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			923.733i			1.32530i
$698$		0			0
$699$	106.170	−	6.27938i	0.151888	−	0.00898338i
$700$		0			0
$701$		−	270.171i		−	0.385408i	−0.981257	−	0.192704i	$-0.938274\pi$
$701$		−	270.171i		−	0.385408i	0.981257	−	0.192704i	$-0.0617257\pi$
$702$		0			0
$703$		−	84.1176i		−	0.119655i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	213.611			0.302137
$708$		0			0
$709$	−81.8824			−0.115490			−0.0577450	−	0.998331i	$-0.518391\pi$
$709$	−81.8824			−0.115490			−0.0577450	+	0.998331i	$0.518391\pi$
$710$		0			0
$711$	380.575	−	45.1760i	0.535267	−	0.0635386i
$712$		0			0
$713$	−118.287			−0.165901
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−10.0764	−	170.369i	−0.0140535	−	0.237613i
$718$		0			0
$719$			680.552i			0.946526i	0.880921	+	0.473263i	$0.156924\pi$
$719$			680.552i			0.946526i	−0.880921	+	0.473263i	$0.843076\pi$
$720$		0			0
$721$	−127.660			−0.177060
$722$		0			0
$723$	628.149	−	37.1517i	0.868810	−	0.0513854i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	333.344i		−	0.458520i	−0.973365	−	0.229260i	$-0.926369\pi$
$727$		−	333.344i		−	0.458520i	0.973365	−	0.229260i	$-0.0736306\pi$
$728$		0			0
$729$	683.682	−	253.021i	0.937836	−	0.347079i
$730$		0			0
$731$		−	352.122i		−	0.481699i
$732$		0			0
$733$		−	825.261i		−	1.12587i	−0.826502	−	0.562934i	$-0.809672\pi$
$733$		−	825.261i		−	1.12587i	0.826502	−	0.562934i	$-0.190328\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1736.75			−2.35651
$738$		0			0
$739$	−1074.77			−1.45435			−0.727176	−	0.686451i	$-0.759168\pi$
$739$	−1074.77			−1.45435			−0.727176	+	0.686451i	$0.759168\pi$
$740$		0			0
$741$	24.8562	+	420.262i	0.0335442	+	0.567155i
$742$		0			0
$743$	−171.465			−0.230774			−0.115387	−	0.993321i	$-0.536811\pi$
$743$	−171.465			−0.230774			−0.115387	+	0.993321i	$0.536811\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	733.816	−	87.1072i	0.982350	−	0.116609i
$748$		0			0
$749$		−	519.962i		−	0.694208i
$750$		0			0
$751$	352.640			0.469561			0.234781	−	0.972048i	$-0.424563\pi$
$751$	352.640			0.469561			0.234781	+	0.972048i	$0.424563\pi$
$752$		0			0
$753$	−13.6585	−	230.933i	−0.0181387	−	0.306684i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	624.790i		−	0.825349i	−0.910878	−	0.412675i	$-0.864595\pi$
$757$		−	624.790i		−	0.825349i	0.910878	−	0.412675i	$-0.135405\pi$
$758$		0			0
$759$	19.3202	+	326.660i	0.0254548	+	0.430383i
$760$		0			0
$761$		−	539.144i		−	0.708468i	−0.935157	−	0.354234i	$-0.884742\pi$
$761$		−	539.144i		−	0.708468i	0.935157	−	0.354234i	$-0.115258\pi$
$762$		0			0
$763$			140.996i			0.184792i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−519.962			−0.677916
$768$		0			0
$769$	357.409			0.464771			0.232386	−	0.972624i	$-0.425347\pi$
$769$	357.409			0.464771			0.232386	+	0.972624i	$0.425347\pi$
$770$		0			0
$771$	474.176	−	28.0450i	0.615015	−	0.0363748i
$772$		0			0
$773$	232.707			0.301044			0.150522	−	0.988607i	$-0.451905\pi$
$773$	232.707			0.301044			0.150522	+	0.988607i	$0.451905\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	96.0752	−	5.68233i	0.123649	−	0.00731317i
$778$		0			0
$779$			293.464i			0.376718i
$780$		0			0
$781$	487.425			0.624104
$782$		0			0
$783$	−177.588	−	991.520i	−0.226804	−	1.26631i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	1202.71i		−	1.52823i	−0.645082	−	0.764114i	$-0.723177\pi$
$787$		−	1202.71i		−	1.52823i	0.645082	−	0.764114i	$-0.276823\pi$
$788$		0			0
$789$	449.557	−	26.5889i	0.569781	−	0.0336995i
$790$		0			0
$791$		−	370.733i		−	0.468688i
$792$		0			0
$793$			1579.93i			1.99235i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1108.42			−1.39074			−0.695368	−	0.718654i	$-0.744758\pi$
$797$	−1108.42			−1.39074			−0.695368	+	0.718654i	$0.744758\pi$
$798$		0			0
$799$	−1569.47			−1.96430
$800$		0			0
$801$	44.8784	+	378.069i	0.0560280	+	0.471996i
$802$		0			0
$803$	1417.68			1.76548
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−33.7960	−	571.413i	−0.0418786	−	0.708071i
$808$		0			0
$809$		−	867.089i		−	1.07180i	−0.844281	−	0.535901i	$-0.819972\pi$
$809$		−	867.089i		−	1.07180i	0.844281	−	0.535901i	$-0.180028\pi$
$810$		0			0
$811$	−1008.35			−1.24335			−0.621673	−	0.783277i	$-0.713547\pi$
$811$	−1008.35			−1.24335			−0.621673	+	0.783277i	$0.713547\pi$
$812$		0			0
$813$	1025.56	−	60.6562i	1.26145	−	0.0746078i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	111.867i		−	0.136924i
$818$		0			0
$819$	−478.324	+	56.7792i	−0.584034	+	0.0693275i
$820$		0			0
$821$			1316.06i			1.60300i	0.597996	+	0.801499i	$0.295964\pi$
$821$			1316.06i			1.60300i	−0.597996	+	0.801499i	$0.704036\pi$
$822$		0			0
$823$		−	992.146i		−	1.20552i	−0.797921	−	0.602762i	$-0.794067\pi$
$823$		−	992.146i		−	1.20552i	0.797921	−	0.602762i	$-0.205933\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	39.2488			0.0474593			0.0237296	−	0.999718i	$-0.492446\pi$
$827$	39.2488			0.0474593			0.0237296	+	0.999718i	$0.492446\pi$
$828$		0			0
$829$	−1083.74			−1.30729			−0.653643	−	0.756803i	$-0.726760\pi$
$829$	−1083.74			−1.30729			−0.653643	+	0.756803i	$0.726760\pi$
$830$		0			0
$831$	−16.5163	−	279.253i	−0.0198752	−	0.336045i
$832$		0			0
$833$	152.854			0.183499
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−406.403	+	72.7895i	−0.485548	+	0.0869648i
$838$		0			0
$839$			570.971i			0.680537i	0.940328	+	0.340269i	$0.110518\pi$
$839$			570.971i			0.680537i	−0.940328	+	0.340269i	$0.889482\pi$
$840$		0			0
$841$	−550.838			−0.654980
$842$		0			0
$843$	−44.4715	−	751.911i	−0.0527539	−	0.891947i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	205.929i		−	0.243128i
$848$		0			0
$849$	−2.94907	−	49.8619i	−0.00347357	−	0.0587302i
$850$		0			0
$851$		−	93.7967i		−	0.110219i
$852$		0			0
$853$		−	381.290i		−	0.446999i	−0.974704	−	0.223499i	$-0.928252\pi$
$853$		−	381.290i		−	0.446999i	0.974704	−	0.223499i	$-0.0717481\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	342.531			0.399686			0.199843	−	0.979828i	$-0.435957\pi$
$857$	342.531			0.399686			0.199843	+	0.979828i	$0.435957\pi$
$858$		0			0
$859$	−522.051			−0.607743			−0.303871	−	0.952713i	$-0.598279\pi$
$859$	−522.051			−0.607743			−0.303871	+	0.952713i	$0.598279\pi$
$860$		0			0
$861$	−335.180	+	19.8241i	−0.389292	+	0.0230245i
$862$		0			0
$863$	−118.373			−0.137165			−0.0685825	−	0.997645i	$-0.521848\pi$
$863$	−118.373			−0.137165			−0.0685825	+	0.997645i	$0.521848\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−562.495	+	33.2686i	−0.648784	+	0.0383721i
$868$		0			0
$869$			600.457i			0.690974i
$870$		0			0
$871$	−2491.50			−2.86050
$872$		0			0
$873$	−51.2751	−	431.956i	−0.0587343	−	0.494794i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	60.7321i		−	0.0692499i	−0.999400	−	0.0346249i	$-0.988976\pi$
$877$		−	60.7321i		−	0.0692499i	0.999400	−	0.0346249i	$-0.0110237\pi$
$878$		0			0
$879$	920.357	−	54.4342i	1.04705	−	0.0619274i
$880$		0			0
$881$			817.935i			0.928417i	0.885726	+	0.464208i	$0.153661\pi$
$881$			817.935i			0.928417i	−0.885726	+	0.464208i	$0.846339\pi$
$882$		0			0
$883$			19.9764i			0.0226233i	0.999936	+	0.0113117i	$0.00360069\pi$
$883$			19.9764i			0.0226233i	−0.999936	+	0.0113117i	$0.996399\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	682.008			0.768893			0.384446	−	0.923147i	$-0.374392\pi$
$887$	682.008			0.768893			0.384446	+	0.923147i	$0.374392\pi$
$888$		0			0
$889$	−305.793			−0.343975
$890$		0			0
$891$	267.393	+	1110.43i	0.300105	+	1.24627i
$892$		0			0
$893$	−498.611			−0.558355
$894$		0			0
$895$		0			0
$896$		0			0
$897$	27.7163	+	468.620i	0.0308989	+	0.522430i
$898$		0			0
$899$			570.485i			0.634578i
$900$		0			0
$901$	2215.22			2.45862
$902$		0			0
$903$	127.769	−	7.55684i	0.141494	−	0.00836859i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	649.895i		−	0.716533i	−0.933619	−	0.358266i	$-0.883368\pi$
$907$		−	649.895i		−	0.716533i	0.933619	−	0.358266i	$-0.116632\pi$
$908$		0			0
$909$	−85.6536	−	721.570i	−0.0942284	−	0.793807i
$910$		0			0
$911$			1522.75i			1.67152i	0.549099	+	0.835758i	$0.314971\pi$
$911$			1522.75i			1.67152i	−0.549099	+	0.835758i	$0.685029\pi$
$912$		0			0
$913$			1157.79i			1.26811i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	218.520			0.238299
$918$		0			0
$919$	753.652			0.820079			0.410039	−	0.912068i	$-0.365515\pi$
$919$	753.652			0.820079			0.410039	+	0.912068i	$0.365515\pi$
$920$		0			0
$921$	22.4550	+	379.662i	0.0243811	+	0.412228i
$922$		0			0
$923$	699.249			0.757582
$924$		0			0
$925$		0			0
$926$		0			0
$927$	51.1891	+	431.231i	0.0552202	+	0.465190i
$928$		0			0
$929$		−	555.100i		−	0.597525i	−0.954328	−	0.298762i	$-0.903426\pi$
$929$		−	555.100i		−	0.597525i	0.954328	−	0.298762i	$-0.0965738\pi$
$930$		0			0
$931$	48.5608			0.0521598
$932$		0			0
$933$	58.2589	+	985.025i	0.0624426	+	1.05576i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			231.357i			0.246912i	0.992350	+	0.123456i	$0.0393978\pi$
$937$			231.357i			0.246912i	−0.992350	+	0.123456i	$0.960602\pi$
$938$		0			0
$939$	45.9254	+	776.494i	0.0489089	+	0.826937i
$940$		0			0
$941$		−	525.513i		−	0.558463i	−0.960224	−	0.279231i	$-0.909920\pi$
$941$		−	525.513i		−	0.558463i	0.960224	−	0.279231i	$-0.0900796\pi$
$942$		0			0
$943$			327.231i			0.347011i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−571.855			−0.603860			−0.301930	−	0.953330i	$-0.597631\pi$
$947$	−571.855			−0.603860			−0.301930	+	0.953330i	$0.597631\pi$
$948$		0			0
$949$	2033.77			2.14307
$950$		0			0
$951$	−25.8157	+	1.52686i	−0.0271458	+	0.00160553i
$952$		0			0
$953$	−596.119			−0.625518			−0.312759	−	0.949833i	$-0.601253\pi$
$953$	−596.119			−0.625518			−0.312759	+	0.949833i	$0.601253\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	1575.44	−	93.1790i	1.64623	−	0.0973657i
$958$		0			0
$959$		−	406.756i		−	0.424146i
$960$		0			0
$961$	−727.170			−0.756680
$962$		0			0
$963$	−1756.41	+	208.494i	−1.82390	+	0.216505i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1519.98i		−	1.57185i	−0.618324	−	0.785924i	$-0.712188\pi$
$967$		−	1519.98i		−	1.57185i	0.618324	−	0.785924i	$-0.287812\pi$
$968$		0			0
$969$	−453.660	+	26.8316i	−0.468173	+	0.0276899i
$970$		0			0
$971$		−	1077.39i		−	1.10957i	−0.831995	−	0.554783i	$-0.812801\pi$
$971$		−	1077.39i		−	1.10957i	0.831995	−	0.554783i	$-0.187199\pi$
$972$		0			0
$973$		−	583.431i		−	0.599621i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1800.57			1.84296			0.921480	−	0.388427i	$-0.126981\pi$
$977$	1800.57			1.84296			0.921480	+	0.388427i	$0.126981\pi$
$978$		0			0
$979$	−596.502			−0.609297
$980$		0			0
$981$	476.280	−	56.5365i	0.485504	−	0.0576315i
$982$		0			0
$983$	−512.798			−0.521666			−0.260833	−	0.965384i	$-0.583997\pi$
$983$	−512.798			−0.521666			−0.260833	+	0.965384i	$0.583997\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−33.6823	−	569.490i	−0.0341259	−	0.576991i
$988$		0			0
$989$		−	124.739i		−	0.126126i
$990$		0			0
$991$	−191.749			−0.193490			−0.0967452	−	0.995309i	$-0.530843\pi$
$991$	−191.749			−0.193490			−0.0967452	+	0.995309i	$0.530843\pi$
$992$		0			0
$993$	1420.62	−	84.0222i	1.43064	−	0.0846145i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1677.56i			1.68260i	0.540565	+	0.841302i	$0.318211\pi$
$997$			1677.56i			1.68260i	−0.540565	+	0.841302i	$0.681789\pi$
$998$		0			0
$999$	−57.7189	−	322.260i	−0.0577767	−	0.322583i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.3.e.a.449.2		8
3.2	odd	2	inner	2100.3.e.a.449.8		8
5.2	odd	4		84.3.c.a.29.2	yes	4
5.3	odd	4		2100.3.g.a.701.3		4
5.4	even	2	inner	2100.3.e.a.449.7		8
15.2	even	4		84.3.c.a.29.1	✓	4
15.8	even	4		2100.3.g.a.701.4		4
15.14	odd	2	inner	2100.3.e.a.449.1		8
20.7	even	4		336.3.d.a.113.3		4
35.2	odd	12		588.3.p.f.557.1		8
35.12	even	12		588.3.p.h.557.4		8
35.17	even	12		588.3.p.h.569.1		8
35.27	even	4		588.3.c.h.197.3		4
35.32	odd	12		588.3.p.f.569.4		8
40.27	even	4		1344.3.d.g.449.2		4
40.37	odd	4		1344.3.d.a.449.3		4
45.2	even	12		2268.3.bg.a.701.3		8
45.7	odd	12		2268.3.bg.a.701.2		8
45.22	odd	12		2268.3.bg.a.2213.3		8
45.32	even	12		2268.3.bg.a.2213.2		8
60.47	odd	4		336.3.d.a.113.4		4
105.2	even	12		588.3.p.f.557.4		8
105.17	odd	12		588.3.p.h.569.4		8
105.32	even	12		588.3.p.f.569.1		8
105.47	odd	12		588.3.p.h.557.1		8
105.62	odd	4		588.3.c.h.197.4		4
120.77	even	4		1344.3.d.a.449.4		4
120.107	odd	4		1344.3.d.g.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.c.a.29.1	✓	4	15.2	even	4
84.3.c.a.29.2	yes	4	5.2	odd	4
336.3.d.a.113.3		4	20.7	even	4
336.3.d.a.113.4		4	60.47	odd	4
588.3.c.h.197.3		4	35.27	even	4
588.3.c.h.197.4		4	105.62	odd	4
588.3.p.f.557.1		8	35.2	odd	12
588.3.p.f.557.4		8	105.2	even	12
588.3.p.f.569.1		8	105.32	even	12
588.3.p.f.569.4		8	35.32	odd	12
588.3.p.h.557.1		8	105.47	odd	12
588.3.p.h.557.4		8	35.12	even	12
588.3.p.h.569.1		8	35.17	even	12
588.3.p.h.569.4		8	105.17	odd	12
1344.3.d.a.449.3		4	40.37	odd	4
1344.3.d.a.449.4		4	120.77	even	4
1344.3.d.g.449.1		4	120.107	odd	4
1344.3.d.g.449.2		4	40.27	even	4
2100.3.e.a.449.1		8	15.14	odd	2	inner
2100.3.e.a.449.2		8	1.1	even	1	trivial
2100.3.e.a.449.7		8	5.4	even	2	inner
2100.3.e.a.449.8		8	3.2	odd	2	inner
2100.3.g.a.701.3		4	5.3	odd	4
2100.3.g.a.701.4		4	15.8	even	4
2268.3.bg.a.701.2		8	45.7	odd	12
2268.3.bg.a.701.3		8	45.2	even	12
2268.3.bg.a.2213.2		8	45.32	even	12
2268.3.bg.a.2213.3		8	45.22	odd	12

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2100.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.99477 + 0.177124i) q^{3} +2.64575i q^{7} +(8.93725 - 1.06089i) q^{9} +14.1009i q^{11} +20.2288i q^{13} -21.8363 q^{17} -6.93725 q^{19} +(-0.468627 - 7.92341i) q^{21} -7.73550 q^{23} +(-26.5771 + 4.76013i) q^{27} +37.3073i q^{29} +15.2915 q^{31} +(-2.49760 - 42.2288i) q^{33} +12.1255i q^{37} +(-3.58301 - 60.5804i) q^{39} -42.3026i q^{41} +16.1255i q^{43} +71.8744 q^{47} -7.00000 q^{49} +(65.3948 - 3.86775i) q^{51} -101.446 q^{53} +(20.7755 - 1.22876i) q^{57} +25.7041i q^{59} +78.1033 q^{61} +(2.80686 + 23.6458i) q^{63} +123.166i q^{67} +(23.1660 - 1.37014i) q^{69} -34.5671i q^{71} -100.539i q^{73} -37.3073 q^{77} +42.5830 q^{79} +(78.7490 - 18.9629i) q^{81} +82.1075 q^{83} +(-6.60804 - 111.727i) q^{87} +42.3026i q^{89} -53.5203 q^{91} +(-45.7945 + 2.70850i) q^{93} -48.3320i q^{97} +(14.9595 + 126.023i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9} + 8 q^{19} + 28 q^{21} + 80 q^{31} + 56 q^{39} - 56 q^{49} + 248 q^{51} + 392 q^{61} + 16 q^{69} + 256 q^{79} + 376 q^{81} - 280 q^{91} + 416 q^{99}+O(q^{100}) \) 8 * q + 8 * q^9 + 8 * q^19 + 28 * q^21 + 80 * q^31 + 56 * q^39 - 56 * q^49 + 248 * q^51 + 392 * q^61 + 16 * q^69 + 256 * q^79 + 376 * q^81 - 280 * q^91 + 416 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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