LMFDB - Embedded newform 1176.2.k.a.881.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(881,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1176.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.k (of order $2$, degree $1$, 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:	$9.39040727770$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + \cdots + 6561 $ x^16 - 6x^15 + 19x^14 - 42x^13 + 65x^12 - 48x^11 - 94x^10 + 444x^9 - 962x^8 + 1332x^7 - 846x^6 - 1296x^5 + 5265x^4 - 10206x^3 + 13851x^2 - 13122*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.16
Root			$0.934861 - 1.45809i$ of defining polynomial
Character	$\chi$	$=$	1176.881
Dual form			1176.2.k.a.881.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.53866 + 0.795315i) q^{3} +3.80034 q^{5} +(1.73495 + 2.44744i) q^{9} +O(q^{10})$	$q+(1.53866 + 0.795315i) q^{3} +3.80034 q^{5} +(1.73495 + 2.44744i) q^{9} +0.357425i q^{11} -4.04570i q^{13} +(5.84742 + 3.02246i) q^{15} +0.103938 q^{17} -2.45507i q^{19} +1.33007i q^{23} +9.44255 q^{25} +(0.723015 + 5.14560i) q^{27} -4.97265i q^{29} +7.88669i q^{31} +(-0.284265 + 0.549955i) q^{33} -10.9124 q^{37} +(3.21760 - 6.22495i) q^{39} +6.15464 q^{41} +0.502751 q^{43} +(6.59339 + 9.30108i) q^{45} -11.4516 q^{47} +(0.159925 + 0.0826633i) q^{51} -5.86753i q^{53} +1.35833i q^{55} +(1.95255 - 3.77751i) q^{57} -7.54728 q^{59} +9.47414i q^{61} -15.3750i q^{65} +2.68750 q^{67} +(-1.05783 + 2.04653i) q^{69} -5.78975i q^{71} +0.235473i q^{73} +(14.5289 + 7.50980i) q^{75} +3.22495 q^{79} +(-2.97990 + 8.49236i) q^{81} -9.07747 q^{83} +0.394999 q^{85} +(3.95482 - 7.65122i) q^{87} -6.82427 q^{89} +(-6.27240 + 12.1349i) q^{93} -9.33007i q^{95} +5.14243i q^{97} +(-0.874774 + 0.620114i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^9	$ 16 q - 4 q^{9} + 8 q^{15} + 36 q^{25} + 4 q^{37} + 44 q^{39} + 20 q^{43} - 12 q^{51} - 8 q^{57} - 28 q^{67} - 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} + 20 q^{99}+O(q^{100}) $ 16 * q - 4 * q^9 + 8 * q^15 + 36 * q^25 + 4 * q^37 + 44 * q^39 + 20 * q^43 - 12 * q^51 - 8 * q^57 - 28 * q^67 - 56 * q^79 - 60 * q^81 + 16 * q^85 - 32 * q^93 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	1.53866	+	0.795315i	0.888346	+	0.459175i
$3$	1.53866	+	0.795315i	0.888346	+	0.459175i
$4$		0			0
$5$	3.80034			1.69956			0.849781	−	0.527136i	$-0.176734\pi$
$5$	3.80034			1.69956			0.849781	+	0.527136i	$0.176734\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.73495	+	2.44744i	0.578317	+	0.815812i
$10$		0			0
$11$			0.357425i			0.107768i	0.998547	+	0.0538838i	$0.0171600\pi$
$11$			0.357425i			0.107768i	−0.998547	+	0.0538838i	$0.982840\pi$
$12$		0			0
$13$		−	4.04570i		−	1.12207i	−0.827791	−	0.561037i	$-0.810403\pi$
$13$		−	4.04570i		−	1.12207i	0.827791	−	0.561037i	$-0.189597\pi$
$14$		0			0
$15$	5.84742	+	3.02246i	1.50980	+	0.780396i
$16$		0			0
$17$	0.103938			0.0252086			0.0126043	−	0.999921i	$-0.495988\pi$
$17$	0.103938			0.0252086			0.0126043	+	0.999921i	$0.495988\pi$
$18$		0			0
$19$		−	2.45507i		−	0.563231i	−0.959527	−	0.281615i	$-0.909130\pi$
$19$		−	2.45507i		−	0.563231i	0.959527	−	0.281615i	$-0.0908702\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.33007i			0.277340i	0.990339	+	0.138670i	$0.0442827\pi$
$23$			1.33007i			0.277340i	−0.990339	+	0.138670i	$0.955717\pi$
$24$		0			0
$25$	9.44255			1.88851
$26$		0			0
$27$	0.723015	+	5.14560i	0.139144	+	0.990272i
$28$		0			0
$29$		−	4.97265i		−	0.923398i	−0.887037	−	0.461699i	$-0.847240\pi$
$29$		−	4.97265i		−	0.923398i	0.887037	−	0.461699i	$-0.152760\pi$
$30$		0			0
$31$			7.88669i			1.41649i	0.705966	+	0.708246i	$0.250513\pi$
$31$			7.88669i			1.41649i	−0.705966	+	0.708246i	$0.749487\pi$
$32$		0			0
$33$	−0.284265	+	0.549955i	−0.0494842	+	0.0957348i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−10.9124			−1.79400			−0.896998	−	0.442035i	$-0.854257\pi$
$37$	−10.9124			−1.79400			−0.896998	+	0.442035i	$0.854257\pi$
$38$		0			0
$39$	3.21760	−	6.22495i	0.515228	−	0.996790i
$40$		0			0
$41$	6.15464			0.961193			0.480597	−	0.876942i	$-0.340420\pi$
$41$	6.15464			0.961193			0.480597	+	0.876942i	$0.340420\pi$
$42$		0			0
$43$	0.502751			0.0766688			0.0383344	−	0.999265i	$-0.487795\pi$
$43$	0.502751			0.0766688			0.0383344	+	0.999265i	$0.487795\pi$
$44$		0			0
$45$	6.59339	+	9.30108i	0.982885	+	1.38652i
$46$		0			0
$47$	−11.4516			−1.67038			−0.835190	−	0.549961i	$-0.814643\pi$
$47$	−11.4516			−1.67038			−0.835190	+	0.549961i	$0.814643\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	0.159925	+	0.0826633i	0.0223940	+	0.0115752i
$52$		0			0
$53$		−	5.86753i		−	0.805967i	−0.915207	−	0.402983i	$-0.867973\pi$
$53$		−	5.86753i		−	0.805967i	0.915207	−	0.402983i	$-0.132027\pi$
$54$		0			0
$55$			1.35833i			0.183158i
$56$		0			0
$57$	1.95255	−	3.77751i	0.258622	−	0.500344i
$58$		0			0
$59$	−7.54728			−0.982572			−0.491286	−	0.870998i	$-0.663473\pi$
$59$	−7.54728			−0.982572			−0.491286	+	0.870998i	$0.663473\pi$
$60$		0			0
$61$			9.47414i			1.21304i	0.795068	+	0.606520i	$0.207435\pi$
$61$			9.47414i			1.21304i	−0.795068	+	0.606520i	$0.792565\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	15.3750i		−	1.90703i
$66$		0			0
$67$	2.68750			0.328330			0.164165	−	0.986433i	$-0.447507\pi$
$67$	2.68750			0.328330			0.164165	+	0.986433i	$0.447507\pi$
$68$		0			0
$69$	−1.05783	+	2.04653i	−0.127348	+	0.246374i
$70$		0			0
$71$		−	5.78975i		−	0.687117i	−0.939131	−	0.343558i	$-0.888368\pi$
$71$		−	5.78975i		−	0.687117i	0.939131	−	0.343558i	$-0.111632\pi$
$72$		0			0
$73$			0.235473i			0.0275600i	0.999905	+	0.0137800i	$0.00438645\pi$
$73$			0.235473i			0.0275600i	−0.999905	+	0.0137800i	$0.995614\pi$
$74$		0			0
$75$	14.5289	+	7.50980i	1.67765	+	0.867157i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.22495			0.362835			0.181418	−	0.983406i	$-0.441931\pi$
$79$	3.22495			0.362835			0.181418	+	0.983406i	$0.441931\pi$
$80$		0			0
$81$	−2.97990	+	8.49236i	−0.331100	+	0.943596i
$82$		0			0
$83$	−9.07747			−0.996382			−0.498191	−	0.867067i	$-0.666002\pi$
$83$	−9.07747			−0.996382			−0.498191	+	0.867067i	$0.666002\pi$
$84$		0			0
$85$	0.394999			0.0428436
$86$		0			0
$87$	3.95482	−	7.65122i	0.424001	−	0.820297i
$88$		0			0
$89$	−6.82427			−0.723371			−0.361685	−	0.932300i	$-0.617799\pi$
$89$	−6.82427			−0.723371			−0.361685	+	0.932300i	$0.617799\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−6.27240	+	12.1349i	−0.650418	+	1.25833i
$94$		0			0
$95$		−	9.33007i		−	0.957245i
$96$		0			0
$97$			5.14243i			0.522134i	0.965321	+	0.261067i	$0.0840744\pi$
$97$			5.14243i			0.522134i	−0.965321	+	0.261067i	$0.915926\pi$
$98$		0			0
$99$	−0.874774	+	0.620114i	−0.0879181	+	0.0623238i
$100$		0			0
$101$	12.8778			1.28139			0.640695	−	0.767795i	$-0.278646\pi$
$101$	12.8778			1.28139			0.640695	+	0.767795i	$0.278646\pi$
$102$		0			0
$103$			5.63632i			0.555364i	0.960673	+	0.277682i	$0.0895661\pi$
$103$			5.63632i			0.555364i	−0.960673	+	0.277682i	$0.910434\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.80732i			0.851436i	0.904856	+	0.425718i	$0.139978\pi$
$107$			8.80732i			0.851436i	−0.904856	+	0.425718i	$0.860022\pi$
$108$		0			0
$109$	−4.47725			−0.428843			−0.214421	−	0.976741i	$-0.568787\pi$
$109$	−4.47725			−0.428843			−0.214421	+	0.976741i	$0.568787\pi$
$110$		0			0
$111$	−16.7905	−	8.67883i	−1.59369	−	0.823758i
$112$		0			0
$113$		−	4.00000i		−	0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$		−	4.00000i		−	0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$		0			0
$115$			5.05473i			0.471356i
$116$		0			0
$117$	9.90159	−	7.01908i	0.915402	−	0.648914i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.8722			0.988386
$122$		0			0
$123$	9.46990	+	4.89487i	0.853872	+	0.441356i
$124$		0			0
$125$	16.8832			1.51008
$126$		0			0
$127$	12.9198			1.14645			0.573223	−	0.819399i	$-0.305693\pi$
$127$	12.9198			1.14645			0.573223	+	0.819399i	$0.305693\pi$
$128$		0			0
$129$	0.773563	+	0.399845i	0.0681084	+	0.0352044i
$130$		0			0
$131$	−5.32769			−0.465482			−0.232741	−	0.972539i	$-0.574769\pi$
$131$	−5.32769			−0.465482			−0.232741	+	0.972539i	$0.574769\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.74770	+	19.5550i	0.236484	+	1.68303i
$136$		0			0
$137$			5.05043i			0.431487i	0.976450	+	0.215744i	$0.0692175\pi$
$137$			5.05043i			0.431487i	−0.976450	+	0.215744i	$0.930782\pi$
$138$		0			0
$139$			21.2651i			1.80368i	0.432067	+	0.901841i	$0.357784\pi$
$139$			21.2651i			1.80368i	−0.432067	+	0.901841i	$0.642216\pi$
$140$		0			0
$141$	−17.6200	−	9.10759i	−1.48388	−	0.766997i
$142$		0			0
$143$	1.44603			0.120923
$144$		0			0
$145$		−	18.8977i		−	1.56937i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	12.1800i		−	0.997827i	−0.866652	−	0.498913i	$-0.833733\pi$
$149$		−	12.1800i		−	0.997827i	0.866652	−	0.498913i	$-0.166267\pi$
$150$		0			0
$151$	−8.21760			−0.668739			−0.334369	−	0.942442i	$-0.608523\pi$
$151$	−8.21760			−0.668739			−0.334369	+	0.942442i	$0.608523\pi$
$152$		0			0
$153$	0.180327	+	0.254381i	0.0145786	+	0.0205655i
$154$		0			0
$155$			29.9721i			2.40741i
$156$		0			0
$157$		−	12.9446i		−	1.03309i	−0.856260	−	0.516546i	$-0.827218\pi$
$157$		−	12.9446i		−	1.03309i	0.856260	−	0.516546i	$-0.172782\pi$
$158$		0			0
$159$	4.66653	−	9.02813i	0.370080	−	0.715977i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.18475			−0.171123			−0.0855613	−	0.996333i	$-0.527268\pi$
$163$	−2.18475			−0.171123			−0.0855613	+	0.996333i	$0.527268\pi$
$164$		0			0
$165$	−1.08030	+	2.09001i	−0.0841014	+	0.162707i
$166$		0			0
$167$	0.464592			0.0359512			0.0179756	−	0.999838i	$-0.494278\pi$
$167$	0.464592			0.0359512			0.0179756	+	0.999838i	$0.494278\pi$
$168$		0			0
$169$	−3.36765			−0.259050
$170$		0			0
$171$	6.00862	−	4.25942i	0.459491	−	0.325726i
$172$		0			0
$173$	−9.25173			−0.703396			−0.351698	−	0.936113i	$-0.614396\pi$
$173$	−9.25173			−0.703396			−0.351698	+	0.936113i	$0.614396\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−11.6127	−	6.00246i	−0.872864	−	0.451173i
$178$		0			0
$179$		−	2.04492i		−	0.152845i	−0.997076	−	0.0764224i	$-0.975650\pi$
$179$		−	2.04492i		−	0.152845i	0.997076	−	0.0764224i	$-0.0243497\pi$
$180$		0			0
$181$		−	17.6193i		−	1.30963i	−0.755790	−	0.654815i	$-0.772747\pi$
$181$		−	17.6193i		−	1.30963i	0.755790	−	0.654815i	$-0.227253\pi$
$182$		0			0
$183$	−7.53492	+	14.5775i	−0.556998	+	1.07760i
$184$		0			0
$185$	−41.4710			−3.04901
$186$		0			0
$187$			0.0371500i			0.00271667i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	22.4948i		−	1.62767i	−0.581097	−	0.813834i	$-0.697376\pi$
$191$		−	22.4948i		−	1.62767i	0.581097	−	0.813834i	$-0.302624\pi$
$192$		0			0
$193$	9.63970			0.693881			0.346940	−	0.937887i	$-0.387221\pi$
$193$	9.63970			0.693881			0.346940	+	0.937887i	$0.387221\pi$
$194$		0			0
$195$	12.2280	−	23.6569i	0.875662	−	1.69411i
$196$		0			0
$197$		−	15.3750i		−	1.09542i	−0.836667	−	0.547712i	$-0.815499\pi$
$197$		−	15.3750i		−	1.09542i	0.836667	−	0.547712i	$-0.184501\pi$
$198$		0			0
$199$			4.58379i			0.324936i	0.986714	+	0.162468i	$0.0519454\pi$
$199$			4.58379i			0.324936i	−0.986714	+	0.162468i	$0.948055\pi$
$200$		0			0
$201$	4.13515	+	2.13741i	0.291671	+	0.150761i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	23.3897			1.63361
$206$		0			0
$207$	−3.25527	+	2.30761i	−0.226257	+	0.160390i
$208$		0			0
$209$	0.877501			0.0606980
$210$		0			0
$211$	−0.870400			−0.0599208			−0.0299604	−	0.999551i	$-0.509538\pi$
$211$	−0.870400			−0.0599208			−0.0299604	+	0.999551i	$0.509538\pi$
$212$		0			0
$213$	4.60467	−	8.90846i	0.315507	−	0.610397i
$214$		0			0
$215$	1.91062			0.130303
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−0.187275	+	0.362312i	−0.0126549	+	0.0244828i
$220$		0			0
$221$		−	0.420501i		−	0.0282860i
$222$		0			0
$223$		−	1.21373i		−	0.0812777i	−0.999174	−	0.0406388i	$-0.987061\pi$
$223$		−	1.21373i		−	0.0812777i	0.999174	−	0.0406388i	$-0.0129393\pi$
$224$		0			0
$225$	16.3823	+	23.1100i	1.09216	+	1.54067i
$226$		0			0
$227$	−13.3441			−0.885680			−0.442840	−	0.896601i	$-0.646029\pi$
$227$	−13.3441			−0.885680			−0.442840	+	0.896601i	$0.646029\pi$
$228$		0			0
$229$			11.0924i			0.733004i	0.930417	+	0.366502i	$0.119445\pi$
$229$			11.0924i			0.733004i	−0.930417	+	0.366502i	$0.880555\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.18003i		−	0.535891i	−0.963434	−	0.267946i	$-0.913655\pi$
$233$		−	8.18003i		−	0.535891i	0.963434	−	0.267946i	$-0.0863448\pi$
$234$		0			0
$235$	−43.5197			−2.83892
$236$		0			0
$237$	4.96210	+	2.56485i	0.322323	+	0.166605i
$238$		0			0
$239$		−	22.5944i		−	1.46151i	−0.682638	−	0.730757i	$-0.739167\pi$
$239$		−	22.5944i		−	1.46151i	0.682638	−	0.730757i	$-0.260833\pi$
$240$		0			0
$241$		−	4.89740i		−	0.315469i	−0.987482	−	0.157735i	$-0.949581\pi$
$241$		−	4.89740i		−	0.315469i	0.987482	−	0.157735i	$-0.0504191\pi$
$242$		0			0
$243$	−11.3392	+	10.6969i	−0.727407	+	0.686206i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−9.93245			−0.631987
$248$		0			0
$249$	−13.9671	−	7.21945i	−0.885132	−	0.457514i
$250$		0			0
$251$	9.17857			0.579346			0.289673	−	0.957126i	$-0.406453\pi$
$251$	9.17857			0.579346			0.289673	+	0.957126i	$0.406453\pi$
$252$		0			0
$253$	−0.475401			−0.0298882
$254$		0			0
$255$	0.607769	+	0.314148i	0.0380600	+	0.0196727i
$256$		0			0
$257$	−12.6211			−0.787282			−0.393641	−	0.919264i	$-0.628785\pi$
$257$	−12.6211			−0.787282			−0.393641	+	0.919264i	$0.628785\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	12.1703	−	8.62730i	0.753320	−	0.534016i
$262$		0			0
$263$			29.1550i			1.79777i	0.438182	+	0.898886i	$0.355623\pi$
$263$			29.1550i			1.79777i	−0.438182	+	0.898886i	$0.644377\pi$
$264$		0			0
$265$		−	22.2986i		−	1.36979i
$266$		0			0
$267$	−10.5002	−	5.42744i	−0.642603	−	0.332154i
$268$		0			0
$269$	−4.47280			−0.272712			−0.136356	−	0.990660i	$-0.543539\pi$
$269$	−4.47280			−0.272712			−0.136356	+	0.990660i	$0.543539\pi$
$270$		0			0
$271$		−	16.7414i		−	1.01697i	−0.861071	−	0.508485i	$-0.830206\pi$
$271$		−	16.7414i		−	1.01697i	0.861071	−	0.508485i	$-0.169794\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			3.37500i			0.203520i
$276$		0			0
$277$	1.02185			0.0613969			0.0306984	−	0.999529i	$-0.490227\pi$
$277$	1.02185			0.0613969			0.0306984	+	0.999529i	$0.490227\pi$
$278$		0			0
$279$	−19.3022	+	13.6830i	−1.15559	+	0.819181i
$280$		0			0
$281$			13.9453i			0.831907i	0.909386	+	0.415953i	$0.136552\pi$
$281$			13.9453i			0.831907i	−0.909386	+	0.415953i	$0.863448\pi$
$282$		0			0
$283$		−	16.2104i		−	0.963607i	−0.876279	−	0.481803i	$-0.839982\pi$
$283$		−	16.2104i		−	0.963607i	0.876279	−	0.481803i	$-0.160018\pi$
$284$		0			0
$285$	7.42034	−	14.3558i	0.439543	−	0.850365i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.9892			−0.999365
$290$		0			0
$291$	−4.08985	+	7.91245i	−0.239751	+	0.463836i
$292$		0			0
$293$	−19.2067			−1.12207			−0.561034	−	0.827793i	$-0.689596\pi$
$293$	−19.2067			−1.12207			−0.561034	+	0.827793i	$0.689596\pi$
$294$		0			0
$295$	−28.6822			−1.66994
$296$		0			0
$297$	−1.83917	+	0.258423i	−0.106719	+	0.0149952i
$298$		0			0
$299$	5.38108			0.311196
$300$		0			0
$301$		0			0
$302$		0			0
$303$	19.8146	+	10.2419i	1.13832	+	0.588382i
$304$		0			0
$305$			36.0049i			2.06164i
$306$		0			0
$307$		−	0.480498i		−	0.0274235i	−0.999906	−	0.0137117i	$-0.995635\pi$
$307$		−	0.480498i		−	0.0274235i	0.999906	−	0.0137117i	$-0.00436472\pi$
$308$		0			0
$309$	−4.48265	+	8.67239i	−0.255009	+	0.493355i
$310$		0			0
$311$	−9.33306			−0.529229			−0.264615	−	0.964354i	$-0.585245\pi$
$311$	−9.33306			−0.529229			−0.264615	+	0.964354i	$0.585245\pi$
$312$		0			0
$313$			17.9148i			1.01261i	0.862356	+	0.506303i	$0.168988\pi$
$313$			17.9148i			1.01261i	−0.862356	+	0.506303i	$0.831012\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			22.3174i			1.25347i	0.779232	+	0.626736i	$0.215609\pi$
$317$			22.3174i			1.25347i	−0.779232	+	0.626736i	$0.784391\pi$
$318$		0			0
$319$	1.77735			0.0995123
$320$		0			0
$321$	−7.00459	+	13.5515i	−0.390958	+	0.756370i
$322$		0			0
$323$		−	0.255174i		−	0.0141983i
$324$		0			0
$325$		−	38.2017i		−	2.11905i
$326$		0			0
$327$	−6.88896	−	3.56082i	−0.380961	−	0.196914i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	14.1172			0.775951			0.387976	−	0.921670i	$-0.373174\pi$
$331$	14.1172			0.775951			0.387976	+	0.921670i	$0.373174\pi$
$332$		0			0
$333$	−18.9325	−	26.7075i	−1.03750	−	1.46356i
$334$		0			0
$335$	10.2134			0.558018
$336$		0			0
$337$	−18.4042			−1.00254			−0.501270	−	0.865291i	$-0.667134\pi$
$337$	−18.4042			−1.00254			−0.501270	+	0.865291i	$0.667134\pi$
$338$		0			0
$339$	3.18126	−	6.15464i	0.172782	−	0.334274i
$340$		0			0
$341$	−2.81890			−0.152652
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−4.02010	+	7.77751i	−0.216435	+	0.418727i
$346$		0			0
$347$		−	31.9245i		−	1.71380i	−0.515484	−	0.856899i	$-0.672388\pi$
$347$		−	31.9245i		−	1.71380i	0.515484	−	0.856899i	$-0.327612\pi$
$348$		0			0
$349$			14.7367i			0.788840i	0.918930	+	0.394420i	$0.129055\pi$
$349$			14.7367i			0.788840i	−0.918930	+	0.394420i	$0.870945\pi$
$350$		0			0
$351$	20.8175	−	2.92510i	1.11116	−	0.156130i
$352$		0			0
$353$	27.1373			1.44437			0.722185	−	0.691700i	$-0.243138\pi$
$353$	27.1373			1.44437			0.722185	+	0.691700i	$0.243138\pi$
$354$		0			0
$355$		−	22.0030i		−	1.16780i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	19.5697i		−	1.03285i	−0.856332	−	0.516425i	$-0.827262\pi$
$359$		−	19.5697i		−	1.03285i	0.856332	−	0.516425i	$-0.172738\pi$
$360$		0			0
$361$	12.9727			0.682771
$362$		0			0
$363$	16.7287	+	8.64686i	0.878029	+	0.453842i
$364$		0			0
$365$			0.894875i			0.0468399i
$366$		0			0
$367$			1.34243i			0.0700741i	0.999386	+	0.0350371i	$0.0111549\pi$
$367$			1.34243i			0.0700741i	−0.999386	+	0.0350371i	$0.988845\pi$
$368$		0			0
$369$	10.6780	+	15.0631i	0.555874	+	0.784153i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	13.0476			0.675577			0.337788	−	0.941222i	$-0.390321\pi$
$373$	13.0476			0.675577			0.337788	+	0.941222i	$0.390321\pi$
$374$		0			0
$375$	25.9775	+	13.4274i	1.34147	+	0.693390i
$376$		0			0
$377$	−20.1178			−1.03612
$378$		0			0
$379$	−20.0822			−1.03156			−0.515778	−	0.856722i	$-0.672497\pi$
$379$	−20.0822			−1.03156			−0.515778	+	0.856722i	$0.672497\pi$
$380$		0			0
$381$	19.8792	+	10.2753i	1.01844	+	0.526420i
$382$		0			0
$383$	22.5227			1.15086			0.575428	−	0.817853i	$-0.304836\pi$
$383$	22.5227			1.15086			0.575428	+	0.817853i	$0.304836\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.872248	+	1.23045i	0.0443388	+	0.0625474i
$388$		0			0
$389$		−	37.1697i		−	1.88458i	−0.334802	−	0.942289i	$-0.608669\pi$
$389$		−	37.1697i		−	1.88458i	0.334802	−	0.942289i	$-0.391331\pi$
$390$		0			0
$391$			0.138245i			0.00699136i
$392$		0			0
$393$	−8.19750	−	4.23719i	−0.413509	−	0.213738i
$394$		0			0
$395$	12.2559			0.616661
$396$		0			0
$397$		−	27.7461i		−	1.39254i	−0.717782	−	0.696268i	$-0.754842\pi$
$397$		−	27.7461i		−	1.39254i	0.717782	−	0.696268i	$-0.245158\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			22.7745i			1.13730i	0.822579	+	0.568651i	$0.192535\pi$
$401$			22.7745i			1.13730i	−0.822579	+	0.568651i	$0.807465\pi$
$402$		0			0
$403$	31.9071			1.58941
$404$		0			0
$405$	−11.3246	+	32.2738i	−0.562725	+	1.60370i
$406$		0			0
$407$		−	3.90038i		−	0.193334i
$408$		0			0
$409$			26.1943i			1.29523i	0.761969	+	0.647613i	$0.224233\pi$
$409$			26.1943i			1.29523i	−0.761969	+	0.647613i	$0.775767\pi$
$410$		0			0
$411$	−4.01668	+	7.77089i	−0.198128	+	0.383310i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−34.4974			−1.69341
$416$		0			0
$417$	−16.9124	+	32.7198i	−0.828206	+	1.60229i
$418$		0			0
$419$	−8.93992			−0.436744			−0.218372	−	0.975866i	$-0.570075\pi$
$419$	−8.93992			−0.436744			−0.218372	+	0.975866i	$0.570075\pi$
$420$		0			0
$421$	−5.00735			−0.244043			−0.122022	−	0.992527i	$-0.538938\pi$
$421$	−5.00735			−0.244043			−0.122022	+	0.992527i	$0.538938\pi$
$422$		0			0
$423$	−19.8679	−	28.0270i	−0.966009	−	1.36272i
$424$		0			0
$425$	0.981439			0.0476068
$426$		0			0
$427$		0			0
$428$		0			0
$429$	2.22495	+	1.15005i	0.107422	+	0.0555249i
$430$		0			0
$431$		−	6.49482i		−	0.312845i	−0.987690	−	0.156422i	$-0.950004\pi$
$431$		−	6.49482i		−	0.312845i	0.987690	−	0.156422i	$-0.0499961\pi$
$432$		0			0
$433$			1.05254i			0.0505818i	0.999680	+	0.0252909i	$0.00805120\pi$
$433$			1.05254i			0.0505818i	−0.999680	+	0.0252909i	$0.991949\pi$
$434$		0			0
$435$	15.0296	−	29.0772i	0.720616	−	1.39414i
$436$		0			0
$437$	3.26542			0.156206
$438$		0			0
$439$			29.9056i			1.42732i	0.700494	+	0.713658i	$0.252963\pi$
$439$			29.9056i			1.42732i	−0.700494	+	0.713658i	$0.747037\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	30.8425i		−	1.46537i	−0.680568	−	0.732685i	$-0.738267\pi$
$443$		−	30.8425i		−	1.46537i	0.680568	−	0.732685i	$-0.261733\pi$
$444$		0			0
$445$	−25.9345			−1.22941
$446$		0			0
$447$	9.68695	−	18.7409i	0.458177	−	0.886415i
$448$		0			0
$449$			36.6953i			1.73176i	0.500253	+	0.865879i	$0.333240\pi$
$449$			36.6953i			1.73176i	−0.500253	+	0.865879i	$0.666760\pi$
$450$		0			0
$451$			2.19982i			0.103585i
$452$		0			0
$453$	−12.6441	−	6.53558i	−0.594071	−	0.307068i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−23.7500			−1.11098			−0.555489	−	0.831524i	$-0.687469\pi$
$457$	−23.7500			−1.11098			−0.555489	+	0.831524i	$0.687469\pi$
$458$		0			0
$459$	0.0751487	+	0.534823i	0.00350764	+	0.0249634i
$460$		0			0
$461$	10.5938			0.493404			0.246702	−	0.969091i	$-0.420653\pi$
$461$	10.5938			0.493404			0.246702	+	0.969091i	$0.420653\pi$
$462$		0			0
$463$	−0.367649			−0.0170861			−0.00854305	−	0.999964i	$-0.502719\pi$
$463$	−0.367649			−0.0170861			−0.00854305	+	0.999964i	$0.502719\pi$
$464$		0			0
$465$	−23.8372	+	46.1168i	−1.10542	+	2.13862i
$466$		0			0
$467$	−31.5694			−1.46086			−0.730428	−	0.682989i	$-0.760680\pi$
$467$	−31.5694			−1.46086			−0.730428	+	0.682989i	$0.760680\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	10.2950	−	19.9173i	0.474370	−	0.917742i
$472$		0			0
$473$			0.179696i			0.00826241i
$474$		0			0
$475$		−	23.1821i		−	1.06367i
$476$		0			0
$477$	14.3604	−	10.1799i	0.657518	−	0.466104i
$478$		0			0
$479$	12.0299			0.549662			0.274831	−	0.961493i	$-0.411378\pi$
$479$	12.0299			0.549662			0.274831	+	0.961493i	$0.411378\pi$
$480$		0			0
$481$			44.1484i			2.01300i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			19.5430i			0.887400i
$486$		0			0
$487$	−18.9547			−0.858922			−0.429461	−	0.903086i	$-0.641296\pi$
$487$	−18.9547			−0.858922			−0.429461	+	0.903086i	$0.641296\pi$
$488$		0			0
$489$	−3.36159	−	1.73756i	−0.152016	−	0.0785753i
$490$		0			0
$491$		−	15.8373i		−	0.714727i	−0.933965	−	0.357364i	$-0.883676\pi$
$491$		−	15.8373i		−	0.714727i	0.933965	−	0.357364i	$-0.116324\pi$
$492$		0			0
$493$		−	0.516847i		−	0.0232776i
$494$		0			0
$495$	−3.32444	+	2.35664i	−0.149422	+	0.105923i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	20.1976			0.904169			0.452084	−	0.891975i	$-0.350681\pi$
$499$	20.1976			0.904169			0.452084	+	0.891975i	$0.350681\pi$
$500$		0			0
$501$	0.714849	+	0.369497i	0.0319371	+	0.0165079i
$502$		0			0
$503$	36.8663			1.64379			0.821893	−	0.569641i	$-0.192918\pi$
$503$	36.8663			1.64379			0.821893	+	0.569641i	$0.192918\pi$
$504$		0			0
$505$	48.9400			2.17780
$506$		0			0
$507$	−5.18167	−	2.67834i	−0.230126	−	0.118949i
$508$		0			0
$509$	10.2639			0.454941			0.227471	−	0.973785i	$-0.426954\pi$
$509$	10.2639			0.454941			0.227471	+	0.973785i	$0.426954\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	12.6328	−	1.77505i	0.557752	−	0.0783704i
$514$		0			0
$515$			21.4199i			0.943875i
$516$		0			0
$517$		−	4.09307i		−	0.180013i
$518$		0			0
$519$	−14.2353	−	7.35804i	−0.624859	−	0.322982i
$520$		0			0
$521$	−15.9777			−0.699998			−0.349999	−	0.936750i	$-0.613818\pi$
$521$	−15.9777			−0.699998			−0.349999	+	0.936750i	$0.613818\pi$
$522$		0			0
$523$			0.781385i			0.0341676i	0.999854	+	0.0170838i	$0.00543821\pi$
$523$			0.781385i			0.0341676i	−0.999854	+	0.0170838i	$0.994562\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			0.819726i			0.0357078i
$528$		0			0
$529$	21.2309			0.923083
$530$		0			0
$531$	−13.0942	−	18.4715i	−0.568238	−	0.801595i
$532$		0			0
$533$		−	24.8998i		−	1.07853i
$534$		0			0
$535$			33.4708i			1.44707i
$536$		0			0
$537$	1.62636	−	3.14644i	0.0701825	−	0.135779i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7.26725			−0.312443			−0.156222	−	0.987722i	$-0.549931\pi$
$541$	−7.26725			−0.312443			−0.156222	+	0.987722i	$0.549931\pi$
$542$		0			0
$543$	14.0129	−	27.1100i	0.601349	−	1.16340i
$544$		0			0
$545$	−17.0150			−0.728845
$546$		0			0
$547$	41.2546			1.76392			0.881960	−	0.471325i	$-0.156224\pi$
$547$	41.2546			1.76392			0.881960	+	0.471325i	$0.156224\pi$
$548$		0			0
$549$	−23.1874	+	16.4372i	−0.989613	+	0.701521i
$550$		0			0
$551$	−12.2082			−0.520086
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−63.8097	−	32.9825i	−2.70857	−	1.40003i
$556$		0			0
$557$			6.33698i			0.268506i	0.990947	+	0.134253i	$0.0428635\pi$
$557$			6.33698i			0.268506i	−0.990947	+	0.134253i	$0.957136\pi$
$558$		0			0
$559$		−	2.03398i		−	0.0860281i
$560$		0			0
$561$	−0.0295459	+	0.0571611i	−0.00124743	+	0.00241335i
$562$		0			0
$563$	15.4626			0.651671			0.325836	−	0.945426i	$-0.394354\pi$
$563$	15.4626			0.651671			0.325836	+	0.945426i	$0.394354\pi$
$564$		0			0
$565$		−	15.2013i		−	0.639525i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			11.8200i			0.495519i	0.968822	+	0.247760i	$0.0796943\pi$
$569$			11.8200i			0.495519i	−0.968822	+	0.247760i	$0.920306\pi$
$570$		0			0
$571$	36.0772			1.50978			0.754892	−	0.655849i	$-0.227689\pi$
$571$	36.0772			1.50978			0.754892	+	0.655849i	$0.227689\pi$
$572$		0			0
$573$	17.8905	−	34.6119i	0.747385	−	1.44593i
$574$		0			0
$575$			12.5593i			0.523759i
$576$		0			0
$577$			23.5851i			0.981861i	0.871199	+	0.490930i	$0.163343\pi$
$577$			23.5851i			0.981861i	−0.871199	+	0.490930i	$0.836657\pi$
$578$		0			0
$579$	14.8322	+	7.66659i	0.616406	+	0.318613i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.09720			0.0868570
$584$		0			0
$585$	37.6293	−	26.6748i	1.55578	−	1.10287i
$586$		0			0
$587$	0.287490			0.0118660			0.00593298	−	0.999982i	$-0.498111\pi$
$587$	0.287490			0.0118660			0.00593298	+	0.999982i	$0.498111\pi$
$588$		0			0
$589$	19.3623			0.797812
$590$		0			0
$591$	12.2280	−	23.6569i	0.502991	−	0.973115i
$592$		0			0
$593$	11.4318			0.469447			0.234723	−	0.972062i	$-0.424582\pi$
$593$	11.4318			0.469447			0.234723	+	0.972062i	$0.424582\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.64555	+	7.05289i	−0.149202	+	0.288655i
$598$		0			0
$599$		−	21.6901i		−	0.886235i	−0.896464	−	0.443117i	$-0.853873\pi$
$599$		−	21.6901i		−	0.886235i	0.896464	−	0.443117i	$-0.146127\pi$
$600$		0			0
$601$			23.7036i			0.966889i	0.875375	+	0.483445i	$0.160615\pi$
$601$			23.7036i			0.966889i	−0.875375	+	0.483445i	$0.839385\pi$
$602$		0			0
$603$	4.66268	+	6.57749i	0.189879	+	0.267856i
$604$		0			0
$605$	41.3182			1.67982
$606$		0			0
$607$		−	21.3655i		−	0.867199i	−0.901106	−	0.433600i	$-0.857243\pi$
$607$		−	21.3655i		−	0.867199i	0.901106	−	0.433600i	$-0.142757\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			46.3295i			1.87429i
$612$		0			0
$613$	39.6496			1.60143			0.800716	−	0.599044i	$-0.204453\pi$
$613$	39.6496			1.60143			0.800716	+	0.599044i	$0.204453\pi$
$614$		0			0
$615$	35.9888	+	18.6022i	1.45121	+	0.750112i
$616$		0			0
$617$			28.6296i			1.15258i	0.817244	+	0.576292i	$0.195501\pi$
$617$			28.6296i			1.15258i	−0.817244	+	0.576292i	$0.804499\pi$
$618$		0			0
$619$			38.0507i			1.52939i	0.644394	+	0.764694i	$0.277110\pi$
$619$			38.0507i			1.52939i	−0.644394	+	0.764694i	$0.722890\pi$
$620$		0			0
$621$	−6.84404	+	0.961664i	−0.274642	+	0.0385903i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	16.9490			0.677960
$626$		0			0
$627$	1.35018	+	0.697889i	0.0539208	+	0.0278710i
$628$		0			0
$629$	−1.13422			−0.0452242
$630$		0			0
$631$	−3.65235			−0.145398			−0.0726989	−	0.997354i	$-0.523161\pi$
$631$	−3.65235			−0.145398			−0.0726989	+	0.997354i	$0.523161\pi$
$632$		0			0
$633$	−1.33925	−	0.692242i	−0.0532304	−	0.0275141i
$634$		0			0
$635$	49.0996			1.94846
$636$		0			0
$637$		0			0
$638$		0			0
$639$	14.1700	−	10.0449i	0.560558	−	0.397371i
$640$		0			0
$641$		−	24.5447i		−	0.969456i	−0.874665	−	0.484728i	$-0.838919\pi$
$641$		−	24.5447i		−	0.969456i	0.874665	−	0.484728i	$-0.161081\pi$
$642$		0			0
$643$			27.3936i			1.08030i	0.841569	+	0.540149i	$0.181632\pi$
$643$			27.3936i			1.08030i	−0.841569	+	0.540149i	$0.818368\pi$
$644$		0			0
$645$	2.93980	+	1.51955i	0.115754	+	0.0598321i
$646$		0			0
$647$	32.2362			1.26733			0.633667	−	0.773606i	$-0.281549\pi$
$647$	32.2362			1.26733			0.633667	+	0.773606i	$0.281549\pi$
$648$		0			0
$649$		−	2.69758i		−	0.105889i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	15.5916i		−	0.610145i	−0.952329	−	0.305073i	$-0.901319\pi$
$653$		−	15.5916i		−	0.610145i	0.952329	−	0.305073i	$-0.0986808\pi$
$654$		0			0
$655$	−20.2470			−0.791116
$656$		0			0
$657$	−0.576304	+	0.408533i	−0.0224838	+	0.0159384i
$658$		0			0
$659$		−	35.1100i		−	1.36769i	−0.729626	−	0.683847i	$-0.760306\pi$
$659$		−	35.1100i		−	1.36769i	0.729626	−	0.683847i	$-0.239694\pi$
$660$		0			0
$661$		−	8.03767i		−	0.312629i	−0.987707	−	0.156314i	$-0.950039\pi$
$661$		−	8.03767i		−	0.312629i	0.987707	−	0.156314i	$-0.0499613\pi$
$662$		0			0
$663$	0.334431	−	0.647008i	0.0129882	−	0.0251277i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	6.61400			0.256095
$668$		0			0
$669$	0.965301	−	1.86753i	0.0373207	−	0.0722027i
$670$		0			0
$671$	−3.38629			−0.130726
$672$		0			0
$673$	28.1744			1.08604			0.543022	−	0.839719i	$-0.317280\pi$
$673$	28.1744			1.08604			0.543022	+	0.839719i	$0.317280\pi$
$674$		0			0
$675$	6.82711	+	48.5876i	0.262776	+	1.87014i
$676$		0			0
$677$	34.7687			1.33627			0.668135	−	0.744040i	$-0.267093\pi$
$677$	34.7687			1.33627			0.668135	+	0.744040i	$0.267093\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−20.5320	−	10.6128i	−0.786790	−	0.406682i
$682$		0			0
$683$			47.0417i			1.80000i	0.435889	+	0.900001i	$0.356434\pi$
$683$			47.0417i			1.80000i	−0.435889	+	0.900001i	$0.643566\pi$
$684$		0			0
$685$			19.1933i			0.733339i
$686$		0			0
$687$	−8.82192	+	17.0674i	−0.336577	+	0.651161i
$688$		0			0
$689$	−23.7382			−0.904354
$690$		0			0
$691$		−	31.3799i		−	1.19375i	−0.802335	−	0.596874i	$-0.796409\pi$
$691$		−	31.3799i		−	1.19375i	0.802335	−	0.596874i	$-0.203591\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			80.8145i			3.06547i
$696$		0			0
$697$	0.639700			0.0242304
$698$		0			0
$699$	6.50569	−	12.5863i	0.246068	−	0.476057i
$700$		0			0
$701$			29.9818i			1.13240i	0.824268	+	0.566199i	$0.191587\pi$
$701$			29.9818i			1.13240i	−0.824268	+	0.566199i	$0.808413\pi$
$702$		0			0
$703$			26.7908i			1.01043i
$704$		0			0
$705$	−66.9621	−	34.6119i	−2.52194	−	1.30356i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	23.0903			0.867174			0.433587	−	0.901112i	$-0.357248\pi$
$709$	23.0903			0.867174			0.433587	+	0.901112i	$0.357248\pi$
$710$		0			0
$711$	5.59513	+	7.89286i	0.209834	+	0.296005i
$712$		0			0
$713$	−10.4899			−0.392849
$714$		0			0
$715$	5.49540			0.205516
$716$		0			0
$717$	17.9697	−	34.7652i	0.671090	−	1.29833i
$718$		0			0
$719$	45.0680			1.68075			0.840376	−	0.542004i	$-0.182334\pi$
$719$	45.0680			1.68075			0.840376	+	0.542004i	$0.182334\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	3.89498	−	7.53544i	0.144856	−	0.280246i
$724$		0			0
$725$		−	46.9545i		−	1.74385i
$726$		0			0
$727$			3.14662i			0.116702i	0.998296	+	0.0583508i	$0.0185842\pi$
$727$			3.14662i			0.116702i	−0.998296	+	0.0583508i	$0.981416\pi$
$728$		0			0
$729$	−25.9545	+	7.44070i	−0.961278	+	0.275582i
$730$		0			0
$731$	0.0522549			0.00193272
$732$		0			0
$733$			17.2731i			0.637998i	0.947755	+	0.318999i	$0.103347\pi$
$733$			17.2731i			0.637998i	−0.947755	+	0.318999i	$0.896653\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			0.960578i			0.0353834i
$738$		0			0
$739$	1.99310			0.0733174			0.0366587	−	0.999328i	$-0.488329\pi$
$739$	1.99310			0.0733174			0.0366587	+	0.999328i	$0.488329\pi$
$740$		0			0
$741$	−15.2827	−	7.89942i	−0.561423	−	0.290192i
$742$		0			0
$743$			5.54435i			0.203402i	0.994815	+	0.101701i	$0.0324286\pi$
$743$			5.54435i			0.203402i	−0.994815	+	0.101701i	$0.967571\pi$
$744$		0			0
$745$		−	46.2882i		−	1.69587i
$746$		0			0
$747$	−15.7490	−	22.2166i	−0.576224	−	0.812861i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−44.1794			−1.61213			−0.806065	−	0.591826i	$-0.798407\pi$
$751$	−44.1794			−1.61213			−0.806065	+	0.591826i	$0.798407\pi$
$752$		0			0
$753$	14.1227	+	7.29985i	0.514660	+	0.266021i
$754$		0			0
$755$	−31.2296			−1.13656
$756$		0			0
$757$	10.6250			0.386172			0.193086	−	0.981182i	$-0.438150\pi$
$757$	10.6250			0.386172			0.193086	+	0.981182i	$0.438150\pi$
$758$		0			0
$759$	−0.731481	−	0.378094i	−0.0265511	−	0.0137239i
$760$		0			0
$761$	27.8168			1.00836			0.504178	−	0.863600i	$-0.331795\pi$
$761$	27.8168			1.00836			0.504178	+	0.863600i	$0.331795\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.685303	+	0.966735i	0.0247772	+	0.0349524i
$766$		0			0
$767$			30.5340i			1.10252i
$768$		0			0
$769$			10.2707i			0.370369i	0.982704	+	0.185185i	$0.0592883\pi$
$769$			10.2707i			0.370369i	−0.982704	+	0.185185i	$0.940712\pi$
$770$		0			0
$771$	−19.4196	−	10.0377i	−0.699379	−	0.361500i
$772$		0			0
$773$	40.5906			1.45994			0.729972	−	0.683477i	$-0.239533\pi$
$773$	40.5906			1.45994			0.729972	+	0.683477i	$0.239533\pi$
$774$		0			0
$775$			74.4705i			2.67506i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	15.1100i		−	0.541374i
$780$		0			0
$781$	2.06940			0.0740489
$782$		0			0
$783$	25.5873	−	3.59530i	0.914415	−	0.128486i
$784$		0			0
$785$		−	49.1938i		−	1.75580i
$786$		0			0
$787$		−	26.1222i		−	0.931155i	−0.885007	−	0.465578i	$-0.845847\pi$
$787$		−	26.1222i		−	0.931155i	0.885007	−	0.465578i	$-0.154153\pi$
$788$		0			0
$789$	−23.1874	+	44.8596i	−0.825492	+	1.59704i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	38.3295			1.36112
$794$		0			0
$795$	17.7344	−	34.3099i	0.628973	−	1.21685i
$796$		0			0
$797$	−38.0284			−1.34704			−0.673518	−	0.739171i	$-0.735218\pi$
$797$	−38.0284			−1.34704			−0.673518	+	0.739171i	$0.735218\pi$
$798$		0			0
$799$	−1.19025			−0.0421080
$800$		0			0
$801$	−11.8398	−	16.7020i	−0.418337	−	0.590135i
$802$		0			0
$803$	−0.0841637			−0.00297007
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−6.88212	−	3.55729i	−0.242262	−	0.125222i
$808$		0			0
$809$			29.8551i			1.04965i	0.851210	+	0.524825i	$0.175869\pi$
$809$			29.8551i			1.04965i	−0.851210	+	0.524825i	$0.824131\pi$
$810$		0			0
$811$			15.4099i			0.541114i	0.962704	+	0.270557i	$0.0872079\pi$
$811$			15.4099i			0.541114i	−0.962704	+	0.270557i	$0.912792\pi$
$812$		0			0
$813$	13.3147	−	25.7594i	0.466967	−	0.903420i
$814$		0			0
$815$	−8.30278			−0.290834
$816$		0			0
$817$		−	1.23429i		−	0.0431822i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			28.9723i			1.01114i	0.862785	+	0.505570i	$0.168718\pi$
$821$			28.9723i			1.01114i	−0.862785	+	0.505570i	$0.831282\pi$
$822$		0			0
$823$	−7.17925			−0.250253			−0.125126	−	0.992141i	$-0.539934\pi$
$823$	−7.17925			−0.250253			−0.125126	+	0.992141i	$0.539934\pi$
$824$		0			0
$825$	−2.68419	+	5.19298i	−0.0934513	+	0.180796i
$826$		0			0
$827$		−	37.6512i		−	1.30926i	−0.755949	−	0.654630i	$-0.772824\pi$
$827$		−	37.6512i		−	1.30926i	0.755949	−	0.654630i	$-0.227176\pi$
$828$		0			0
$829$			50.1585i			1.74208i	0.491216	+	0.871038i	$0.336553\pi$
$829$			50.1585i			1.74208i	−0.491216	+	0.871038i	$0.663447\pi$
$830$		0			0
$831$	1.57228	+	0.812690i	0.0545417	+	0.0281919i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	1.76561			0.0611013
$836$		0			0
$837$	−40.5818	+	5.70220i	−1.40271	+	0.197097i
$838$		0			0
$839$	−10.2849			−0.355073			−0.177536	−	0.984114i	$-0.556813\pi$
$839$	−10.2849			−0.355073			−0.177536	+	0.984114i	$0.556813\pi$
$840$		0			0
$841$	4.27275			0.147336
$842$		0			0
$843$	−11.0909	+	21.4571i	−0.381991	+	0.739021i
$844$		0			0
$845$	−12.7982			−0.440271
$846$		0			0
$847$		0			0
$848$		0			0
$849$	12.8923	−	24.9423i	0.442464	−	0.856016i
$850$		0			0
$851$		−	14.5144i		−	0.497546i
$852$		0			0
$853$		−	29.0278i		−	0.993891i	−0.867782	−	0.496946i	$-0.834455\pi$
$853$		−	29.0278i		−	0.993891i	0.867782	−	0.496946i	$-0.165545\pi$
$854$		0			0
$855$	22.8348	−	16.1872i	0.780933	−	0.553591i
$856$		0			0
$857$	−15.9777			−0.545789			−0.272895	−	0.962044i	$-0.587981\pi$
$857$	−15.9777			−0.545789			−0.272895	+	0.962044i	$0.587981\pi$
$858$		0			0
$859$		−	5.75333i		−	0.196301i	−0.995172	−	0.0981505i	$-0.968707\pi$
$859$		−	5.75333i		−	0.196301i	0.995172	−	0.0981505i	$-0.0312926\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.4401i		−	0.423467i	−0.977327	−	0.211733i	$-0.932089\pi$
$863$		−	12.4401i		−	0.423467i	0.977327	−	0.211733i	$-0.0679108\pi$
$864$		0			0
$865$	−35.1597			−1.19547
$866$		0			0
$867$	−26.1406	−	13.5118i	−0.887781	−	0.458883i
$868$		0			0
$869$			1.15268i			0.0391019i
$870$		0			0
$871$		−	10.8728i		−	0.368411i
$872$		0			0
$873$	−12.5858	+	8.92185i	−0.425964	+	0.301959i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−43.9345			−1.48356			−0.741781	−	0.670642i	$-0.766019\pi$
$877$	−43.9345			−1.48356			−0.741781	+	0.670642i	$0.766019\pi$
$878$		0			0
$879$	−29.5526	−	15.2754i	−0.996784	−	0.515226i
$880$		0			0
$881$	51.0805			1.72095			0.860473	−	0.509496i	$-0.170168\pi$
$881$	51.0805			1.72095			0.860473	+	0.509496i	$0.170168\pi$
$882$		0			0
$883$	−34.3823			−1.15706			−0.578529	−	0.815662i	$-0.696373\pi$
$883$	−34.3823			−1.15706			−0.578529	+	0.815662i	$0.696373\pi$
$884$		0			0
$885$	−44.1322	−	22.8114i	−1.48349	−	0.766796i
$886$		0			0
$887$	−41.7658			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$887$	−41.7658			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−3.03538	−	1.06509i	−0.101689	−	0.0356818i
$892$		0			0
$893$			28.1143i			0.940810i
$894$		0			0
$895$		−	7.77140i		−	0.259769i
$896$		0			0
$897$	8.27965	+	4.27965i	0.276449	+	0.142893i
$898$		0			0
$899$	39.2178			1.30799
$900$		0			0
$901$		−	0.609858i		−	0.0203173i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	66.9591i		−	2.22580i
$906$		0			0
$907$	−37.9678			−1.26070			−0.630350	−	0.776311i	$-0.717089\pi$
$907$	−37.9678			−1.26070			−0.630350	+	0.776311i	$0.717089\pi$
$908$		0			0
$909$	22.3424	+	31.5176i	0.741049	+	1.04537i
$910$		0			0
$911$			55.0007i			1.82225i	0.412127	+	0.911126i	$0.364786\pi$
$911$			55.0007i			1.82225i	−0.412127	+	0.911126i	$0.635214\pi$
$912$		0			0
$913$		−	3.24451i		−	0.107378i
$914$		0			0
$915$	−28.6352	+	55.3993i	−0.946652	+	1.83145i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	27.1778			0.896513			0.448256	−	0.893905i	$-0.352045\pi$
$919$	27.1778			0.896513			0.448256	+	0.893905i	$0.352045\pi$
$920$		0			0
$921$	0.382147	−	0.739324i	0.0125922	−	0.0243615i
$922$		0			0
$923$	−23.4236			−0.770996
$924$		0			0
$925$	−103.041			−3.38798
$926$		0			0
$927$	−13.7946	+	9.77874i	−0.453072	+	0.321176i
$928$		0			0
$929$	1.96572			0.0644932			0.0322466	−	0.999480i	$-0.489734\pi$
$929$	1.96572			0.0644932			0.0322466	+	0.999480i	$0.489734\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−14.3604	−	7.42271i	−0.470138	−	0.243009i
$934$		0			0
$935$			0.141182i			0.00461715i
$936$		0			0
$937$		−	31.0157i		−	1.01324i	−0.862170	−	0.506620i	$-0.830895\pi$
$937$		−	31.0157i		−	1.01324i	0.862170	−	0.506620i	$-0.169105\pi$
$938$		0			0
$939$	−14.2479	+	27.5648i	−0.464963	+	0.899544i
$940$		0			0
$941$	−33.5828			−1.09477			−0.547384	−	0.836882i	$-0.684376\pi$
$941$	−33.5828			−1.09477			−0.547384	+	0.836882i	$0.684376\pi$
$942$		0			0
$943$			8.18613i			0.266577i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			10.2662i			0.333606i	0.985990	+	0.166803i	$0.0533444\pi$
$947$			10.2662i			0.333606i	−0.985990	+	0.166803i	$0.946656\pi$
$948$		0			0
$949$	0.952650			0.0309243
$950$		0			0
$951$	−17.7494	+	34.3389i	−0.575563	+	1.11352i
$952$		0			0
$953$			1.00920i			0.0326911i	0.999866	+	0.0163455i	$0.00520318\pi$
$953$			1.00920i			0.0326911i	−0.999866	+	0.0163455i	$0.994797\pi$
$954$		0			0
$955$		−	85.4879i		−	2.76632i
$956$		0			0
$957$	2.73473	+	1.41355i	0.0884014	+	0.0456936i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.1999			−1.00645
$962$		0			0
$963$	−21.5554	+	15.2803i	−0.694612	+	0.492400i
$964$		0			0
$965$	36.6341			1.17929
$966$		0			0
$967$	1.83020			0.0588552			0.0294276	−	0.999567i	$-0.490632\pi$
$967$	1.83020			0.0588552			0.0294276	+	0.999567i	$0.490632\pi$
$968$		0			0
$969$	0.202944	−	0.392627i	0.00651950	−	0.0126130i
$970$		0			0
$971$	−14.5696			−0.467559			−0.233780	−	0.972290i	$-0.575109\pi$
$971$	−14.5696			−0.467559			−0.233780	+	0.972290i	$0.575109\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	30.3823	−	58.7794i	0.973014	−	1.88245i
$976$		0			0
$977$		−	6.85447i		−	0.219294i	−0.993971	−	0.109647i	$-0.965028\pi$
$977$		−	6.85447i		−	0.219294i	0.993971	−	0.109647i	$-0.0349720\pi$
$978$		0			0
$979$		−	2.43916i		−	0.0779559i
$980$		0			0
$981$	−7.76780	−	10.9578i	−0.248007	−	0.349855i
$982$		0			0
$983$	−59.1868			−1.88777			−0.943883	−	0.330281i	$-0.892856\pi$
$983$	−59.1868			−1.88777			−0.943883	+	0.330281i	$0.892856\pi$
$984$		0			0
$985$		−	58.4302i		−	1.86174i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			0.668696i			0.0212633i
$990$		0			0
$991$	56.4285			1.79251			0.896256	−	0.443537i	$-0.146277\pi$
$991$	56.4285			1.79251			0.896256	+	0.443537i	$0.146277\pi$
$992$		0			0
$993$	21.7216	+	11.2276i	0.689313	+	0.356297i
$994$		0			0
$995$			17.4199i			0.552249i
$996$		0			0
$997$			52.6348i			1.66696i	0.552549	+	0.833480i	$0.313655\pi$
$997$			52.6348i			1.66696i	−0.552549	+	0.833480i	$0.686345\pi$
$998$		0			0
$999$	−7.88987	−	56.1511i	−0.249624	−	1.77654i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.2.k.a.881.16		16
3.2	odd	2	inner	1176.2.k.a.881.2		16
4.3	odd	2		2352.2.k.i.881.1		16
7.2	even	3		168.2.u.a.17.2	✓	16
7.3	odd	6		168.2.u.a.89.4	yes	16
7.4	even	3		1176.2.u.b.1097.5		16
7.5	odd	6		1176.2.u.b.521.7		16
7.6	odd	2	inner	1176.2.k.a.881.1		16
12.11	even	2		2352.2.k.i.881.15		16
21.2	odd	6		168.2.u.a.17.4	yes	16
21.5	even	6		1176.2.u.b.521.5		16
21.11	odd	6		1176.2.u.b.1097.7		16
21.17	even	6		168.2.u.a.89.2	yes	16
21.20	even	2	inner	1176.2.k.a.881.15		16
28.3	even	6		336.2.bc.f.257.5		16
28.23	odd	6		336.2.bc.f.17.7		16
28.27	even	2		2352.2.k.i.881.16		16
84.23	even	6		336.2.bc.f.17.5		16
84.59	odd	6		336.2.bc.f.257.7		16
84.83	odd	2		2352.2.k.i.881.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.u.a.17.2	✓	16	7.2	even	3
168.2.u.a.17.4	yes	16	21.2	odd	6
168.2.u.a.89.2	yes	16	21.17	even	6
168.2.u.a.89.4	yes	16	7.3	odd	6
336.2.bc.f.17.5		16	84.23	even	6
336.2.bc.f.17.7		16	28.23	odd	6
336.2.bc.f.257.5		16	28.3	even	6
336.2.bc.f.257.7		16	84.59	odd	6
1176.2.k.a.881.1		16	7.6	odd	2	inner
1176.2.k.a.881.2		16	3.2	odd	2	inner
1176.2.k.a.881.15		16	21.20	even	2	inner
1176.2.k.a.881.16		16	1.1	even	1	trivial
1176.2.u.b.521.5		16	21.5	even	6
1176.2.u.b.521.7		16	7.5	odd	6
1176.2.u.b.1097.5		16	7.4	even	3
1176.2.u.b.1097.7		16	21.11	odd	6
2352.2.k.i.881.1		16	4.3	odd	2
2352.2.k.i.881.2		16	84.83	odd	2
2352.2.k.i.881.15		16	12.11	even	2
2352.2.k.i.881.16		16	28.27	even	2

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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.53866 + 0.795315i) q^{3} +3.80034 q^{5} +(1.73495 + 2.44744i) q^{9} +O(q^{10})\)	\(q+(1.53866 + 0.795315i) q^{3} +3.80034 q^{5} +(1.73495 + 2.44744i) q^{9} +0.357425i q^{11} -4.04570i q^{13} +(5.84742 + 3.02246i) q^{15} +0.103938 q^{17} -2.45507i q^{19} +1.33007i q^{23} +9.44255 q^{25} +(0.723015 + 5.14560i) q^{27} -4.97265i q^{29} +7.88669i q^{31} +(-0.284265 + 0.549955i) q^{33} -10.9124 q^{37} +(3.21760 - 6.22495i) q^{39} +6.15464 q^{41} +0.502751 q^{43} +(6.59339 + 9.30108i) q^{45} -11.4516 q^{47} +(0.159925 + 0.0826633i) q^{51} -5.86753i q^{53} +1.35833i q^{55} +(1.95255 - 3.77751i) q^{57} -7.54728 q^{59} +9.47414i q^{61} -15.3750i q^{65} +2.68750 q^{67} +(-1.05783 + 2.04653i) q^{69} -5.78975i q^{71} +0.235473i q^{73} +(14.5289 + 7.50980i) q^{75} +3.22495 q^{79} +(-2.97990 + 8.49236i) q^{81} -9.07747 q^{83} +0.394999 q^{85} +(3.95482 - 7.65122i) q^{87} -6.82427 q^{89} +(-6.27240 + 12.1349i) q^{93} -9.33007i q^{95} +5.14243i q^{97} +(-0.874774 + 0.620114i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^9	\( 16 q - 4 q^{9} + 8 q^{15} + 36 q^{25} + 4 q^{37} + 44 q^{39} + 20 q^{43} - 12 q^{51} - 8 q^{57} - 28 q^{67} - 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} + 20 q^{99}+O(q^{100}) \) 16 * q - 4 * q^9 + 8 * q^15 + 36 * q^25 + 4 * q^37 + 44 * q^39 + 20 * q^43 - 12 * q^51 - 8 * q^57 - 28 * q^67 - 56 * q^79 - 60 * q^81 + 16 * q^85 - 32 * q^93 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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