LMFDB - Embedded newform 2100.2.bo.c.1949.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1349,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2100.1349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.bo (of order $6$, degree $2$, 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:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			1949.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2100.1949
Dual form			2100.2.bo.c.1349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 1.50000i) q^{3} +(-2.59808 + 0.500000i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})$	$q+(0.866025 - 1.50000i) q^{3} +(-2.59808 + 0.500000i) q^{7} +(-1.50000 - 2.59808i) q^{9} -1.73205 q^{13} +(-4.50000 + 2.59808i) q^{19} +(-1.50000 + 4.33013i) q^{21} -5.19615 q^{27} +(9.00000 + 5.19615i) q^{31} +(-9.52628 + 5.50000i) q^{37} +(-1.50000 + 2.59808i) q^{39} +8.00000i q^{43} +(6.50000 - 2.59808i) q^{49} +9.00000i q^{57} +(-7.50000 + 4.33013i) q^{61} +(5.19615 + 6.00000i) q^{63} +(-9.52628 - 5.50000i) q^{67} +(-0.866025 + 1.50000i) q^{73} +(8.50000 + 14.7224i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(4.50000 - 0.866025i) q^{91} +(15.5885 - 9.00000i) q^{93} -19.0526 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} - 18 q^{19} - 6 q^{21} + 36 q^{31} - 6 q^{39} + 26 q^{49} - 30 q^{61} + 34 q^{79} - 18 q^{81} + 18 q^{91}+O(q^{100}) $ 4 * q - 6 * q^9 - 18 * q^19 - 6 * q^21 + 36 * q^31 - 6 * q^39 + 26 * q^49 - 30 * q^61 + 34 * q^79 - 18 * q^81 + 18 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$3$	0.866025	−	1.50000i	0.500000	−	0.866025i
$3$	0.866025	−	1.50000i	0.500000	−	0.866025i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.59808	+	0.500000i	−0.981981	+	0.188982i
$7$	−2.59808	+	0.500000i	−0.981981	+	0.188982i
$8$		0			0
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	−1.73205			−0.480384			−0.240192	−	0.970725i	$-0.577210\pi$
$13$	−1.73205			−0.480384			−0.240192	+	0.970725i	$0.577210\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	−4.50000	+	2.59808i	−1.03237	+	0.596040i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−4.50000	+	2.59808i	−1.03237	+	0.596040i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$	−1.50000	+	4.33013i	−0.327327	+	0.944911i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−5.19615			−1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	9.00000	+	5.19615i	1.61645	+	0.933257i	0.987829	+	0.155543i	$0.0497126\pi$
$31$	9.00000	+	5.19615i	1.61645	+	0.933257i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−9.52628	+	5.50000i	−1.56611	+	0.904194i	−0.569495	+	0.821995i	$0.692861\pi$
$37$	−9.52628	+	5.50000i	−1.56611	+	0.904194i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$		0			0
$39$	−1.50000	+	2.59808i	−0.240192	+	0.416025i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$			8.00000i			1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$			8.00000i			1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	6.50000	−	2.59808i	0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			9.00000i			1.19208i
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−7.50000	+	4.33013i	−0.960277	+	0.554416i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−7.50000	+	4.33013i	−0.960277	+	0.554416i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$	5.19615	+	6.00000i	0.654654	+	0.755929i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−9.52628	−	5.50000i	−1.16382	−	0.671932i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−9.52628	−	5.50000i	−1.16382	−	0.671932i	−0.952217	+	0.305424i	$0.901202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−0.866025	+	1.50000i	−0.101361	+	0.175562i	−0.912245	−	0.409644i	$-0.865653\pi$
$73$	−0.866025	+	1.50000i	−0.101361	+	0.175562i	0.810885	+	0.585206i	$0.198986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.50000	+	14.7224i	0.956325	+	1.65640i	0.731307	+	0.682048i	$0.238911\pi$
$79$	8.50000	+	14.7224i	0.956325	+	1.65640i	0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	4.50000	−	0.866025i	0.471728	−	0.0907841i
$92$		0			0
$93$	15.5885	−	9.00000i	1.61645	−	0.933257i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−19.0526			−1.93449			−0.967247	−	0.253837i	$-0.918307\pi$
$97$	−19.0526			−1.93449			−0.967247	+	0.253837i	$0.918307\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−7.79423	−	13.5000i	−0.767988	−	1.33019i	−0.938652	−	0.344865i	$-0.887925\pi$
$103$	−7.79423	−	13.5000i	−0.767988	−	1.33019i	0.170664	−	0.985329i	$-0.445409\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	−8.50000	+	14.7224i	−0.814152	+	1.41015i	0.0957826	+	0.995402i	$0.469465\pi$
$109$	−8.50000	+	14.7224i	−0.814152	+	1.41015i	−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$			19.0526i			1.80839i
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.59808	+	4.50000i	0.240192	+	0.416025i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	1.00000i		−	0.0887357i	−0.999015	−	0.0443678i	$-0.985873\pi$
$127$		−	1.00000i		−	0.0887357i	0.999015	−	0.0443678i	$-0.0141274\pi$
$128$		0			0
$129$	12.0000	+	6.92820i	1.05654	+	0.609994i
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	10.3923	−	9.00000i	0.901127	−	0.780399i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$			5.19615i			0.440732i	0.975417	+	0.220366i	$0.0707252\pi$
$139$			5.19615i			0.440732i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.73205	−	12.0000i	0.142857	−	0.989743i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	11.5000	−	19.9186i	0.935857	−	1.62095i	0.162758	−	0.986666i	$-0.447961\pi$
$151$	11.5000	−	19.9186i	0.935857	−	1.62095i	0.773099	−	0.634285i	$-0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.866025	−	1.50000i	0.0691164	−	0.119713i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	0.866025	−	1.50000i	0.0691164	−	0.119713i	0.898513	+	0.438948i	$0.144649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	21.6506	−	12.5000i	1.69581	−	0.979076i	0.746156	−	0.665771i	$-0.231897\pi$
$163$	21.6506	−	12.5000i	1.69581	−	0.979076i	0.949653	−	0.313304i	$-0.101436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−10.0000			−0.769231
$170$		0			0
$171$	13.5000	+	7.79423i	1.03237	+	0.596040i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$			6.92820i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$			6.92820i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$			15.0000i			1.10883i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	13.5000	−	2.59808i	0.981981	−	0.188982i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−1.73205	−	1.00000i	−0.124676	−	0.0719816i	0.436365	−	0.899770i	$-0.356266\pi$
$193$	−1.73205	−	1.00000i	−0.124676	−	0.0719816i	−0.561041	+	0.827788i	$0.689599\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	22.5000	+	12.9904i	1.59498	+	0.920864i	0.992434	+	0.122782i	$0.0391815\pi$
$199$	22.5000	+	12.9904i	1.59498	+	0.920864i	0.602549	+	0.798082i	$0.294152\pi$
$200$		0			0
$201$	−16.5000	+	9.52628i	−1.16382	+	0.671932i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−25.9808	−	9.00000i	−1.76369	−	0.610960i
$218$		0			0
$219$	1.50000	+	2.59808i	0.101361	+	0.175562i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−29.4449			−1.97177			−0.985887	−	0.167412i	$-0.946459\pi$
$223$	−29.4449			−1.97177			−0.985887	+	0.167412i	$0.946459\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	−25.5000	+	14.7224i	−1.68509	+	0.972886i	−0.726900	+	0.686743i	$0.759040\pi$
$229$	−25.5000	+	14.7224i	−1.68509	+	0.972886i	−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	29.4449			1.91265
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−22.5000	−	12.9904i	−1.44935	−	0.836784i	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−22.5000	−	12.9904i	−1.44935	−	0.836784i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$	7.79423	+	13.5000i	0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	7.79423	−	4.50000i	0.495935	−	0.286328i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	22.0000	−	19.0526i	1.36701	−	1.18387i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	15.0000	−	8.66025i	0.911185	−	0.526073i	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	15.0000	−	8.66025i	0.911185	−	0.526073i	0.880812	+	0.473466i	$0.156997\pi$
$272$		0			0
$273$	2.59808	−	7.50000i	0.157243	−	0.453921i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	26.8468	+	15.5000i	1.61307	+	0.931305i	0.988654	+	0.150210i	$0.0479951\pi$
$277$	26.8468	+	15.5000i	1.61307	+	0.931305i	0.624413	+	0.781094i	$0.285338\pi$
$278$		0			0
$279$		−	31.1769i		−	1.86651i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−16.4545	+	28.5000i	−0.978117	+	1.69415i	−0.308879	+	0.951101i	$0.599954\pi$
$283$	−16.4545	+	28.5000i	−0.978117	+	1.69415i	−0.669238	+	0.743048i	$0.733379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.50000	−	14.7224i	−0.500000	−	0.866025i
$290$		0			0
$291$	−16.5000	+	28.5788i	−0.967247	+	1.67532i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−4.00000	−	20.7846i	−0.230556	−	1.19800i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−31.1769			−1.77936			−0.889680	−	0.456584i	$-0.849073\pi$
$307$	−31.1769			−1.77936			−0.889680	+	0.456584i	$0.849073\pi$
$308$		0			0
$309$	−27.0000			−1.53598
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−13.8564	−	24.0000i	−0.783210	−	1.35656i	−0.930062	−	0.367402i	$-0.880247\pi$
$313$	−13.8564	−	24.0000i	−0.783210	−	1.35656i	0.146852	−	0.989158i	$-0.453086\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	14.7224	+	25.5000i	0.814152	+	1.41015i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.5000	−	26.8468i	−0.851957	−	1.47563i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−15.5000	−	26.8468i	−0.851957	−	1.47563i	0.0274825	−	0.999622i	$-0.491251\pi$
$332$		0			0
$333$	28.5788	+	16.5000i	1.56611	+	0.904194i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			34.0000i			1.85210i	0.377403	+	0.926049i	$0.376817\pi$
$337$			34.0000i			1.85210i	−0.377403	+	0.926049i	$0.623183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−15.5885	+	10.0000i	−0.841698	+	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$		−	34.6410i		−	1.85429i	−0.374701	−	0.927146i	$-0.622255\pi$
$349$		−	34.6410i		−	1.85429i	0.374701	−	0.927146i	$-0.377745\pi$
$350$		0			0
$351$	9.00000			0.480384
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	4.00000	−	6.92820i	0.210526	−	0.364642i
$362$		0			0
$363$	−19.0526			−1.00000
$364$		0			0
$365$		0			0
$366$		0			0
$367$	19.0526	−	33.0000i	0.994535	−	1.72259i	0.406855	−	0.913493i	$-0.366625\pi$
$367$	19.0526	−	33.0000i	0.994535	−	1.72259i	0.587680	−	0.809093i	$-0.300041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.2583	−	6.50000i	0.582934	−	0.336557i	−0.179364	−	0.983783i	$-0.557404\pi$
$373$	11.2583	−	6.50000i	0.582934	−	0.336557i	0.762299	+	0.647225i	$0.224071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−37.0000			−1.90056			−0.950281	−	0.311393i	$-0.899204\pi$
$379$	−37.0000			−1.90056			−0.950281	+	0.311393i	$0.899204\pi$
$380$		0			0
$381$	−1.50000	−	0.866025i	−0.0768473	−	0.0443678i
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	20.7846	−	12.0000i	1.05654	−	0.609994i
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−10.3923	−	18.0000i	−0.521575	−	0.903394i	−0.999685	−	0.0250943i	$-0.992011\pi$
$397$	−10.3923	−	18.0000i	−0.521575	−	0.903394i	0.478110	−	0.878300i	$-0.341322\pi$
$398$		0			0
$399$	−4.50000	−	23.3827i	−0.225282	−	1.17060i
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−15.5885	−	9.00000i	−0.776516	−	0.448322i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−34.5000	−	19.9186i	−1.70592	−	0.984911i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−34.5000	−	19.9186i	−1.70592	−	0.984911i	−0.766426	−	0.642333i	$-0.777967\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	7.79423	+	4.50000i	0.381685	+	0.220366i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	41.0000			1.99822			0.999109	−	0.0422075i	$-0.0134391\pi$
$421$	41.0000			1.99822			0.999109	+	0.0422075i	$0.0134391\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	17.3205	−	15.0000i	0.838198	−	0.725901i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	−41.5692			−1.99769			−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	−41.5692			−1.99769			−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	7.50000	−	4.33013i	0.357955	−	0.206666i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	7.50000	−	4.33013i	0.357955	−	0.206666i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	−16.5000	−	12.9904i	−0.785714	−	0.618590i
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−19.9186	−	34.5000i	−0.935857	−	1.62095i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	26.8468	−	15.5000i	1.25584	−	0.725059i	0.283577	−	0.958950i	$-0.408479\pi$
$457$	26.8468	−	15.5000i	1.25584	−	0.725059i	0.972263	+	0.233890i	$0.0751456\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$			43.0000i			1.99838i	0.0402476	+	0.999190i	$0.487185\pi$
$463$			43.0000i			1.99838i	−0.0402476	+	0.999190i	$0.512815\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$	27.5000	+	9.52628i	1.26983	+	0.439883i
$470$		0			0
$471$	−1.50000	−	2.59808i	−0.0691164	−	0.119713i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	16.5000	−	9.52628i	0.752335	−	0.434361i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	38.1051	+	22.0000i	1.72671	+	0.996915i	0.902597	+	0.430486i	$0.141658\pi$
$487$	38.1051	+	22.0000i	1.72671	+	0.996915i	0.824110	+	0.566429i	$0.191675\pi$
$488$		0			0
$489$		−	43.3013i		−	1.95815i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	21.5000	+	37.2391i	0.962472	+	1.66705i	0.716258	+	0.697835i	$0.245853\pi$
$499$	21.5000	+	37.2391i	0.962472	+	1.66705i	0.246214	+	0.969216i	$0.420813\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−8.66025	+	15.0000i	−0.384615	+	0.666173i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	1.50000	−	4.33013i	0.0663561	−	0.191554i
$512$		0			0
$513$	23.3827	−	13.5000i	1.03237	−	0.596040i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	22.5167	+	39.0000i	0.984585	+	1.70535i	0.643767	+	0.765222i	$0.277371\pi$
$523$	22.5167	+	39.0000i	0.984585	+	1.70535i	0.340818	+	0.940129i	$0.389296\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	0.365444	−	0.930834i	$-0.380917\pi$
$542$		0			0
$543$	10.3923	+	6.00000i	0.445976	+	0.257485i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	40.0000i		−	1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$		−	40.0000i		−	1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$		0			0
$549$	22.5000	+	12.9904i	0.960277	+	0.554416i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−29.4449	−	34.0000i	−1.25212	−	1.44583i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$		−	13.8564i		−	0.586064i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7.79423	−	22.5000i	0.327327	−	0.944911i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	15.5000	−	26.8468i	0.648655	−	1.12350i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	15.5000	−	26.8468i	0.648655	−	1.12350i	0.983444	−	0.181210i	$-0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	6.92820	−	12.0000i	0.288425	−	0.499567i	−0.685009	−	0.728535i	$-0.740202\pi$
$577$	6.92820	−	12.0000i	0.288425	−	0.499567i	0.973434	+	0.228968i	$0.0735351\pi$
$578$		0			0
$579$	−3.00000	+	1.73205i	−0.124676	+	0.0719816i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	−54.0000			−2.22503
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	38.9711	−	22.5000i	1.59498	−	0.920864i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$		−	1.73205i		−	0.0706518i	−0.999376	−	0.0353259i	$-0.988753\pi$
$601$		−	1.73205i		−	0.0706518i	0.999376	−	0.0353259i	$-0.0112469\pi$
$602$		0			0
$603$			33.0000i			1.34386i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	19.9186	+	34.5000i	0.808470	+	1.40031i	0.913923	+	0.405887i	$0.133038\pi$
$607$	19.9186	+	34.5000i	0.808470	+	1.40031i	−0.105453	+	0.994424i	$0.533629\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8.66025	−	5.00000i	−0.349784	−	0.201948i	0.314806	−	0.949156i	$-0.398061\pi$
$613$	−8.66025	−	5.00000i	−0.349784	−	0.201948i	−0.664590	+	0.747208i	$0.731394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	33.0000	+	19.0526i	1.32638	+	0.765787i	0.984738	−	0.174042i	$-0.0556830\pi$
$619$	33.0000	+	19.0526i	1.32638	+	0.765787i	0.341644	+	0.939829i	$0.389016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−43.0000			−1.71180			−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000			−1.71180			−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$	11.2583	−	19.5000i	0.447478	−	0.775055i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−11.2583	+	4.50000i	−0.446071	+	0.178296i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	19.0526			0.751360			0.375680	−	0.926750i	$-0.377409\pi$
$643$	19.0526			0.751360			0.375680	+	0.926750i	$0.377409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−36.0000	+	31.1769i	−1.41095	+	1.22192i
$652$		0			0
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	5.19615			0.202721
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−43.5000	−	25.1147i	−1.69195	−	0.976850i	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−43.5000	−	25.1147i	−1.69195	−	0.976850i	−0.739014	−	0.673690i	$-0.764708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−25.5000	+	44.1673i	−0.985887	+	1.70761i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	13.0000i		−	0.501113i	−0.968102	−	0.250557i	$-0.919386\pi$
$673$		−	13.0000i		−	0.501113i	0.968102	−	0.250557i	$-0.0806136\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	49.5000	−	9.52628i	1.89964	−	0.365585i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			51.0000i			1.94577i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−28.5000	+	16.4545i	−1.08419	+	0.625958i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−28.5000	+	16.4545i	−1.08419	+	0.625958i	−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	28.5788	−	49.5000i	1.07787	−	1.86693i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	15.5000	+	26.8468i	0.582115	+	1.00825i	0.995228	+	0.0975728i	$0.0311079\pi$
$709$	15.5000	+	26.8468i	0.582115	+	1.00825i	−0.413114	+	0.910679i	$0.635559\pi$
$710$		0			0
$711$	25.5000	−	44.1673i	0.956325	−	1.65640i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	27.0000	+	31.1769i	1.00553	+	1.16109i
$722$		0			0
$723$	−38.9711	+	22.5000i	−1.44935	+	0.836784i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−22.5167			−0.835097			−0.417548	−	0.908655i	$-0.637111\pi$
$727$	−22.5167			−0.835097			−0.417548	+	0.908655i	$0.637111\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	16.4545	+	28.5000i	0.607760	+	1.05267i	0.991609	+	0.129275i	$0.0412651\pi$
$733$	16.4545	+	28.5000i	0.607760	+	1.05267i	−0.383849	+	0.923396i	$0.625402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−18.5000	+	32.0429i	−0.680534	+	1.17872i	0.294285	+	0.955718i	$0.404919\pi$
$739$	−18.5000	+	32.0429i	−0.680534	+	1.17872i	−0.974818	+	0.223001i	$0.928415\pi$
$740$		0			0
$741$		−	15.5885i		−	0.572656i
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	5.50000	+	9.52628i	0.200698	+	0.347619i	0.948753	−	0.316017i	$-0.102346\pi$
$751$	5.50000	+	9.52628i	0.200698	+	0.347619i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			55.0000i			1.99901i	0.0314762	+	0.999505i	$0.489979\pi$
$757$			55.0000i			1.99901i	−0.0314762	+	0.999505i	$0.510021\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	14.7224	−	42.5000i	0.532988	−	1.53860i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	55.4256i		−	1.99870i	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$		−	55.4256i		−	1.99870i	0.0360609	−	0.999350i	$-0.488519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−9.52628	−	49.5000i	−0.341753	−	1.77580i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	25.1147	−	43.5000i	0.895244	−	1.55061i	0.0617409	−	0.998092i	$-0.480335\pi$
$787$	25.1147	−	43.5000i	0.895244	−	1.55061i	0.833503	−	0.552515i	$-0.186332\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	12.9904	−	7.50000i	0.461302	−	0.266333i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$			43.3013i			1.52051i	0.649623	+	0.760257i	$0.274927\pi$
$811$			43.3013i			1.52051i	−0.649623	+	0.760257i	$0.725073\pi$
$812$		0			0
$813$		−	30.0000i		−	1.05215i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−20.7846	−	36.0000i	−0.727161	−	1.25948i
$818$		0			0
$819$	−9.00000	−	10.3923i	−0.314485	−	0.363137i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	4.33013	+	2.50000i	0.150939	+	0.0871445i	0.573567	−	0.819159i	$-0.305559\pi$
$823$	4.33013	+	2.50000i	0.150939	+	0.0871445i	−0.422628	+	0.906303i	$0.638892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−19.5000	−	11.2583i	−0.677263	−	0.391018i	0.121560	−	0.992584i	$-0.461210\pi$
$829$	−19.5000	−	11.2583i	−0.677263	−	0.391018i	−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$	46.5000	−	26.8468i	1.61307	−	0.931305i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−46.7654	−	27.0000i	−1.61645	−	0.933257i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	19.0526	+	22.0000i	0.654654	+	0.755929i
$848$		0			0
$849$	28.5000	+	49.3634i	0.978117	+	1.69415i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	6.92820			0.237217			0.118609	−	0.992941i	$-0.462157\pi$
$853$	6.92820			0.237217			0.118609	+	0.992941i	$0.462157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	−15.0000	+	8.66025i	−0.511793	+	0.295484i	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−15.0000	+	8.66025i	−0.511793	+	0.295484i	0.221777	+	0.975097i	$0.428814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−29.4449			−1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$	16.5000	+	9.52628i	0.559081	+	0.322786i
$872$		0			0
$873$	28.5788	+	49.5000i	0.967247	+	1.67532i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−21.6506	+	12.5000i	−0.731090	+	0.422095i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$	−21.6506	+	12.5000i	−0.731090	+	0.422095i	0.0877308	+	0.996144i	$0.472038\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$			47.0000i			1.58168i	0.612026	+	0.790838i	$0.290355\pi$
$883$			47.0000i			1.58168i	−0.612026	+	0.790838i	$0.709645\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	0.500000	+	2.59808i	0.0167695	+	0.0871367i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−34.6410	−	12.0000i	−1.15278	−	0.399335i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−16.4545	−	9.50000i	−0.546362	−	0.315442i	0.201291	−	0.979531i	$-0.435486\pi$
$907$	−16.4545	−	9.50000i	−0.546362	−	0.315442i	−0.747653	+	0.664089i	$0.768820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	26.0000	+	45.0333i	0.857661	+	1.48551i	0.874154	+	0.485648i	$0.161416\pi$
$919$	26.0000	+	45.0333i	0.857661	+	1.48551i	−0.0164935	+	0.999864i	$0.505250\pi$
$920$		0			0
$921$	−27.0000	+	46.7654i	−0.889680	+	1.54097i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−23.3827	+	40.5000i	−0.767988	+	1.33019i
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−22.5000	+	28.5788i	−0.737408	+	0.936634i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−55.4256			−1.81068			−0.905338	−	0.424691i	$-0.860383\pi$
$937$	−55.4256			−1.81068			−0.905338	+	0.424691i	$0.860383\pi$
$938$		0			0
$939$	−48.0000			−1.56642
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$	1.50000	−	2.59808i	0.0486921	−	0.0843371i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	38.5000	+	66.6840i	1.24194	+	2.15110i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			41.0000i			1.31847i	0.751936	+	0.659236i	$0.229120\pi$
$967$			41.0000i			1.31847i	−0.751936	+	0.659236i	$0.770880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$	−2.59808	−	13.5000i	−0.0832905	−	0.432790i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	51.0000			1.62830
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	22.0000	−	38.1051i	0.698853	−	1.21045i	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	22.0000	−	38.1051i	0.698853	−	1.21045i	0.968864	−	0.247592i	$-0.0796392\pi$
$992$		0			0
$993$	−53.6936			−1.70391
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−11.2583	+	19.5000i	−0.356555	+	0.617571i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	−11.2583	+	19.5000i	−0.356555	+	0.617571i	0.630828	+	0.775923i	$0.282715\pi$
$998$		0			0
$999$	49.5000	−	28.5788i	1.56611	−	0.904194i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.bo.c.1949.2		4
3.2	odd	2	CM	2100.2.bo.c.1949.2		4
5.2	odd	4		2100.2.bi.b.101.1	✓	2
5.3	odd	4		2100.2.bi.h.101.1	yes	2
5.4	even	2	inner	2100.2.bo.c.1949.1		4
7.5	odd	6	inner	2100.2.bo.c.1349.1		4
15.2	even	4		2100.2.bi.b.101.1	✓	2
15.8	even	4		2100.2.bi.h.101.1	yes	2
15.14	odd	2	inner	2100.2.bo.c.1949.1		4
21.5	even	6	inner	2100.2.bo.c.1349.1		4
35.12	even	12		2100.2.bi.b.1601.1	yes	2
35.19	odd	6	inner	2100.2.bo.c.1349.2		4
35.33	even	12		2100.2.bi.h.1601.1	yes	2
105.47	odd	12		2100.2.bi.b.1601.1	yes	2
105.68	odd	12		2100.2.bi.h.1601.1	yes	2
105.89	even	6	inner	2100.2.bo.c.1349.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.bi.b.101.1	✓	2	5.2	odd	4
2100.2.bi.b.101.1	✓	2	15.2	even	4
2100.2.bi.b.1601.1	yes	2	35.12	even	12
2100.2.bi.b.1601.1	yes	2	105.47	odd	12
2100.2.bi.h.101.1	yes	2	5.3	odd	4
2100.2.bi.h.101.1	yes	2	15.8	even	4
2100.2.bi.h.1601.1	yes	2	35.33	even	12
2100.2.bi.h.1601.1	yes	2	105.68	odd	12
2100.2.bo.c.1349.1		4	7.5	odd	6	inner
2100.2.bo.c.1349.1		4	21.5	even	6	inner
2100.2.bo.c.1349.2		4	35.19	odd	6	inner
2100.2.bo.c.1349.2		4	105.89	even	6	inner
2100.2.bo.c.1949.1		4	5.4	even	2	inner
2100.2.bo.c.1949.1		4	15.14	odd	2	inner
2100.2.bo.c.1949.2		4	1.1	even	1	trivial
2100.2.bo.c.1949.2		4	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.bo (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 1.50000i) q^{3} +(-2.59808 + 0.500000i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(0.866025 - 1.50000i) q^{3} +(-2.59808 + 0.500000i) q^{7} +(-1.50000 - 2.59808i) q^{9} -1.73205 q^{13} +(-4.50000 + 2.59808i) q^{19} +(-1.50000 + 4.33013i) q^{21} -5.19615 q^{27} +(9.00000 + 5.19615i) q^{31} +(-9.52628 + 5.50000i) q^{37} +(-1.50000 + 2.59808i) q^{39} +8.00000i q^{43} +(6.50000 - 2.59808i) q^{49} +9.00000i q^{57} +(-7.50000 + 4.33013i) q^{61} +(5.19615 + 6.00000i) q^{63} +(-9.52628 - 5.50000i) q^{67} +(-0.866025 + 1.50000i) q^{73} +(8.50000 + 14.7224i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(4.50000 - 0.866025i) q^{91} +(15.5885 - 9.00000i) q^{93} -19.0526 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} - 18 q^{19} - 6 q^{21} + 36 q^{31} - 6 q^{39} + 26 q^{49} - 30 q^{61} + 34 q^{79} - 18 q^{81} + 18 q^{91}+O(q^{100}) \) 4 * q - 6 * q^9 - 18 * q^19 - 6 * q^21 + 36 * q^31 - 6 * q^39 + 26 * q^49 - 30 * q^61 + 34 * q^79 - 18 * q^81 + 18 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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