LMFDB - Embedded newform 1323.2.c.f.1322.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1322,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1322");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.c (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:	$10.5642081874$
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} + 24x^{14} + 212x^{12} + 872x^{10} + 1815x^{8} + 1928x^{6} + 996x^{4} + 200x^{2} + 1 $ x^16 + 24x^14 + 212x^12 + 872x^10 + 1815x^8 + 1928x^6 + 996x^4 + 200*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.15
Root			$1.54124i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.f.1322.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.67093i q^{2} -5.13386 q^{4} -3.20318 q^{5} -8.37032i q^{8} +O(q^{10})$	$q+2.67093i q^{2} -5.13386 q^{4} -3.20318 q^{5} -8.37032i q^{8} -8.55547i q^{10} -4.06357i q^{11} +2.99862i q^{13} +12.0888 q^{16} -2.54255 q^{17} -0.165299i q^{19} +16.4447 q^{20} +10.8535 q^{22} -4.04395i q^{23} +5.26038 q^{25} -8.00910 q^{26} +2.12182i q^{29} +7.70685i q^{31} +15.5477i q^{32} -6.79097i q^{34} +11.8011 q^{37} +0.441503 q^{38} +26.8117i q^{40} +0.629916 q^{41} +3.94455 q^{43} +20.8618i q^{44} +10.8011 q^{46} +5.45154 q^{47} +14.0501i q^{50} -15.3945i q^{52} +2.48535i q^{53} +13.0164i q^{55} -5.66724 q^{58} -8.84313 q^{59} -10.8324i q^{61} -20.5844 q^{62} -17.3492 q^{64} -9.60513i q^{65} -1.14425 q^{67} +13.0531 q^{68} -9.33455i q^{71} -6.10764i q^{73} +31.5199i q^{74} +0.848624i q^{76} -12.8712 q^{79} -38.7226 q^{80} +1.68246i q^{82} -3.76846 q^{83} +8.14425 q^{85} +10.5356i q^{86} -34.0134 q^{88} +18.1382 q^{89} +20.7611i q^{92} +14.5607i q^{94} +0.529484i q^{95} +12.1949i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 48 q^{16} + 64 q^{22} - 16 q^{25} + 32 q^{37} - 16 q^{43} + 16 q^{46} - 32 q^{64} + 48 q^{67} - 64 q^{79} + 64 q^{85} - 176 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 48 * q^16 + 64 * q^22 - 16 * q^25 + 32 * q^37 - 16 * q^43 + 16 * q^46 - 32 * q^64 + 48 * q^67 - 64 * q^79 + 64 * q^85 - 176 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

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

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 $

$2$	2.67093i	1.88863i	0.329040	+	0.944316i	$0.393275\pi$
$2$	2.67093i	1.88863i	−0.329040	+	0.944316i	$0.606725\pi$
$3$	0	0
$3$	0	0
$4$	−5.13386	−2.56693
$5$	−3.20318	−1.43251	−0.716253	−	0.697840i	$-0.754144\pi$
$5$	−3.20318	−1.43251	−0.716253	+	0.697840i	$0.754144\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 8.37032i	− 2.95935i
$9$	0	0
$10$	− 8.55547i	− 2.70548i
$11$	− 4.06357i	− 1.22521i	−0.790388	−	0.612606i	$-0.790121\pi$
$11$	− 4.06357i	− 1.22521i	0.790388	−	0.612606i	$-0.209879\pi$
$12$	0	0
$13$	2.99862i	0.831668i	0.909441	+	0.415834i	$0.136510\pi$
$13$	2.99862i	0.831668i	−0.909441	+	0.415834i	$0.863490\pi$
$14$	0	0
$15$	0	0
$16$	12.0888	3.02220
$17$	−2.54255	−0.616659	−0.308329	−	0.951280i	$-0.599770\pi$
$17$	−2.54255	−0.616659	−0.308329	+	0.951280i	$0.599770\pi$
$18$	0	0
$19$	− 0.165299i	− 0.0379223i	−0.999820	−	0.0189611i	$-0.993964\pi$
$19$	− 0.165299i	− 0.0379223i	0.999820	−	0.0189611i	$-0.00603588\pi$
$20$	16.4447	3.67714
$21$	0	0
$22$	10.8535	2.31398
$23$	− 4.04395i	− 0.843222i	−0.906777	−	0.421611i	$-0.861465\pi$
$23$	− 4.04395i	− 0.843222i	0.906777	−	0.421611i	$-0.138535\pi$
$24$	0	0
$25$	5.26038	1.05208
$26$	−8.00910	−1.57071
$27$	0	0
$28$	0	0
$29$	2.12182i	0.394013i	0.980402	+	0.197006i	$0.0631220\pi$
$29$	2.12182i	0.394013i	−0.980402	+	0.197006i	$0.936878\pi$
$30$	0	0
$31$	7.70685i	1.38419i	0.721806	+	0.692095i	$0.243312\pi$
$31$	7.70685i	1.38419i	−0.721806	+	0.692095i	$0.756688\pi$
$32$	15.5477i	2.74847i
$33$	0	0
$34$	− 6.79097i	− 1.16464i
$35$	0	0
$36$	0	0
$37$	11.8011	1.94009	0.970045	−	0.242926i	$-0.0781074\pi$
$37$	11.8011	1.94009	0.970045	+	0.242926i	$0.0781074\pi$
$38$	0.441503	0.0716212
$39$	0	0
$40$	26.8117i	4.23929i
$41$	0.629916	0.0983763	0.0491881	−	0.998790i	$-0.484337\pi$
$41$	0.629916	0.0983763	0.0491881	+	0.998790i	$0.484337\pi$
$42$	0	0
$43$	3.94455	0.601539	0.300769	−	0.953697i	$-0.402757\pi$
$43$	3.94455	0.601539	0.300769	+	0.953697i	$0.402757\pi$
$44$	20.8618i	3.14504i
$45$	0	0
$46$	10.8011	1.59254
$47$	5.45154	0.795188	0.397594	−	0.917561i	$-0.369845\pi$
$47$	5.45154	0.795188	0.397594	+	0.917561i	$0.369845\pi$
$48$	0	0
$49$	0	0
$50$	14.0501i	1.98698i
$51$	0	0
$52$	− 15.3945i	− 2.13483i
$53$	2.48535i	0.341389i	0.985324	+	0.170694i	$0.0546011\pi$
$53$	2.48535i	0.341389i	−0.985324	+	0.170694i	$0.945399\pi$
$54$	0	0
$55$	13.0164i	1.75513i
$56$	0	0
$57$	0	0
$58$	−5.66724	−0.744145
$59$	−8.84313	−1.15128	−0.575639	−	0.817704i	$-0.695247\pi$
$59$	−8.84313	−1.15128	−0.575639	+	0.817704i	$0.695247\pi$
$60$	0	0
$61$	− 10.8324i	− 1.38694i	−0.720484	−	0.693472i	$-0.756080\pi$
$61$	− 10.8324i	− 1.38694i	0.720484	−	0.693472i	$-0.243920\pi$
$62$	−20.5844	−2.61423
$63$	0	0
$64$	−17.3492	−2.16865
$65$	− 9.60513i	− 1.19137i
$66$	0	0
$67$	−1.14425	−0.139792	−0.0698961	−	0.997554i	$-0.522267\pi$
$67$	−1.14425	−0.139792	−0.0698961	+	0.997554i	$0.522267\pi$
$68$	13.0531	1.58292
$69$	0	0
$70$	0	0
$71$	− 9.33455i	− 1.10781i	−0.832581	−	0.553904i	$-0.813138\pi$
$71$	− 9.33455i	− 1.10781i	0.832581	−	0.553904i	$-0.186862\pi$
$72$	0	0
$73$	− 6.10764i	− 0.714846i	−0.933943	−	0.357423i	$-0.883656\pi$
$73$	− 6.10764i	− 0.714846i	0.933943	−	0.357423i	$-0.116344\pi$
$74$	31.5199i	3.66411i
$75$	0	0
$76$	0.848624i	0.0973438i
$77$	0	0
$78$	0	0
$79$	−12.8712	−1.44813	−0.724064	−	0.689733i	$-0.757728\pi$
$79$	−12.8712	−1.44813	−0.724064	+	0.689733i	$0.757728\pi$
$80$	−38.7226	−4.32932
$81$	0	0
$82$	1.68246i	0.185797i
$83$	−3.76846	−0.413642	−0.206821	−	0.978379i	$-0.566312\pi$
$83$	−3.76846	−0.413642	−0.206821	+	0.978379i	$0.566312\pi$
$84$	0	0
$85$	8.14425	0.883368
$86$	10.5356i	1.13609i
$87$	0	0
$88$	−34.0134	−3.62584
$89$	18.1382	1.92264	0.961322	−	0.275426i	$-0.0888188\pi$
$89$	18.1382	1.92264	0.961322	+	0.275426i	$0.0888188\pi$
$90$	0	0
$91$	0	0
$92$	20.7611i	2.16449i
$93$	0	0
$94$	14.5607i	1.50182i
$95$	0.529484i	0.0543239i
$96$	0	0
$97$	12.1949i	1.23820i	0.785311	+	0.619101i	$0.212503\pi$
$97$	12.1949i	1.23820i	−0.785311	+	0.619101i	$0.787497\pi$
$98$	0	0
$99$	0	0
$100$	−27.0060	−2.70060
$101$	14.9295	1.48554	0.742769	−	0.669548i	$-0.233512\pi$
$101$	14.9295	1.48554	0.742769	+	0.669548i	$0.233512\pi$
$102$	0	0
$103$	− 11.3531i	− 1.11865i	−0.828947	−	0.559327i	$-0.811060\pi$
$103$	− 11.3531i	− 1.11865i	0.828947	−	0.559327i	$-0.188940\pi$
$104$	25.0994	2.46120
$105$	0	0
$106$	−6.63819	−0.644758
$107$	4.32637i	0.418246i	0.977889	+	0.209123i	$0.0670609\pi$
$107$	4.32637i	0.418246i	−0.977889	+	0.209123i	$0.932939\pi$
$108$	0	0
$109$	7.88083	0.754847	0.377423	−	0.926041i	$-0.376810\pi$
$109$	7.88083	0.754847	0.377423	+	0.926041i	$0.376810\pi$
$110$	−34.7658	−3.31479
$111$	0	0
$112$	0	0
$113$	5.01207i	0.471496i	0.971814	+	0.235748i	$0.0757540\pi$
$113$	5.01207i	0.471496i	−0.971814	+	0.235748i	$0.924246\pi$
$114$	0	0
$115$	12.9535i	1.20792i
$116$	− 10.8932i	− 1.01140i
$117$	0	0
$118$	− 23.6194i	− 2.17434i
$119$	0	0
$120$	0	0
$121$	−5.51261	−0.501146
$122$	28.9325	2.61943
$123$	0	0
$124$	− 39.5659i	− 3.55312i
$125$	−0.834029	−0.0745978
$126$	0	0
$127$	17.5311	1.55564	0.777819	−	0.628489i	$-0.216326\pi$
$127$	17.5311	1.55564	0.777819	+	0.628489i	$0.216326\pi$
$128$	− 15.2430i	− 1.34731i
$129$	0	0
$130$	25.6546	2.25006
$131$	−4.83988	−0.422862	−0.211431	−	0.977393i	$-0.567812\pi$
$131$	−4.83988	−0.422862	−0.211431	+	0.977393i	$0.567812\pi$
$132$	0	0
$133$	0	0
$134$	− 3.05621i	− 0.264016i
$135$	0	0
$136$	21.2819i	1.82491i
$137$	4.04006i	0.345166i	0.984995	+	0.172583i	$0.0552113\pi$
$137$	4.04006i	0.345166i	−0.984995	+	0.172583i	$0.944789\pi$
$138$	0	0
$139$	− 4.14265i	− 0.351375i	−0.984446	−	0.175687i	$-0.943785\pi$
$139$	− 4.14265i	− 0.351375i	0.984446	−	0.175687i	$-0.0562148\pi$
$140$	0	0
$141$	0	0
$142$	24.9319	2.09224
$143$	12.1851	1.01897
$144$	0	0
$145$	− 6.79659i	− 0.564426i
$146$	16.3131	1.35008
$147$	0	0
$148$	−60.5852	−4.98007
$149$	− 5.17025i	− 0.423564i	−0.977317	−	0.211782i	$-0.932073\pi$
$149$	− 5.17025i	− 0.423564i	0.977317	−	0.211782i	$-0.0679266\pi$
$150$	0	0
$151$	9.86820	0.803063	0.401531	−	0.915845i	$-0.368478\pi$
$151$	9.86820	0.803063	0.401531	+	0.915845i	$0.368478\pi$
$152$	−1.38361	−0.112225
$153$	0	0
$154$	0	0
$155$	− 24.6864i	− 1.98286i
$156$	0	0
$157$	− 11.3871i	− 0.908788i	−0.890801	−	0.454394i	$-0.849856\pi$
$157$	− 11.3871i	− 0.908788i	0.890801	−	0.454394i	$-0.150144\pi$
$158$	− 34.3782i	− 2.73498i
$159$	0	0
$160$	− 49.8021i	− 3.93720i
$161$	0	0
$162$	0	0
$163$	−2.94760	−0.230874	−0.115437	−	0.993315i	$-0.536827\pi$
$163$	−2.94760	−0.230874	−0.115437	+	0.993315i	$0.536827\pi$
$164$	−3.23390	−0.252525
$165$	0	0
$166$	− 10.0653i	− 0.781218i
$167$	17.0818	1.32183	0.660914	−	0.750461i	$-0.270169\pi$
$167$	17.0818	1.32183	0.660914	+	0.750461i	$0.270169\pi$
$168$	0	0
$169$	4.00828	0.308329
$170$	21.7527i	1.66836i
$171$	0	0
$172$	−20.2508	−1.54411
$173$	13.3937	1.01830	0.509151	−	0.860677i	$-0.329960\pi$
$173$	13.3937	1.01830	0.509151	+	0.860677i	$0.329960\pi$
$174$	0	0
$175$	0	0
$176$	− 49.1237i	− 3.70284i
$177$	0	0
$178$	48.4458i	3.63117i
$179$	8.04603i	0.601389i	0.953721	+	0.300694i	$0.0972184\pi$
$179$	8.04603i	0.601389i	−0.953721	+	0.300694i	$0.902782\pi$
$180$	0	0
$181$	3.34572i	0.248686i	0.992239	+	0.124343i	$0.0396823\pi$
$181$	3.34572i	0.248686i	−0.992239	+	0.124343i	$0.960318\pi$
$182$	0	0
$183$	0	0
$184$	−33.8492	−2.49539
$185$	−37.8011	−2.77919
$186$	0	0
$187$	10.3318i	0.755538i
$188$	−27.9874	−2.04119
$189$	0	0
$190$	−1.41421	−0.102598
$191$	− 12.9344i	− 0.935898i	−0.883756	−	0.467949i	$-0.844993\pi$
$191$	− 12.9344i	− 0.935898i	0.883756	−	0.467949i	$-0.155007\pi$
$192$	0	0
$193$	17.2153	1.23919	0.619593	−	0.784923i	$-0.287298\pi$
$193$	17.2153	1.23919	0.619593	+	0.784923i	$0.287298\pi$
$194$	−32.5717	−2.33851
$195$	0	0
$196$	0	0
$197$	− 19.3681i	− 1.37992i	−0.723848	−	0.689960i	$-0.757628\pi$
$197$	− 19.3681i	− 1.37992i	0.723848	−	0.689960i	$-0.242372\pi$
$198$	0	0
$199$	26.0519i	1.84677i	0.383874	+	0.923385i	$0.374590\pi$
$199$	26.0519i	1.84677i	−0.383874	+	0.923385i	$0.625410\pi$
$200$	− 44.0310i	− 3.11346i
$201$	0	0
$202$	39.8756i	2.80563i
$203$	0	0
$204$	0	0
$205$	−2.01773	−0.140925
$206$	30.3233	2.11272
$207$	0	0
$208$	36.2497i	2.51347i
$209$	−0.671705	−0.0464628
$210$	0	0
$211$	13.6672	0.940892	0.470446	−	0.882429i	$-0.344093\pi$
$211$	13.6672	0.940892	0.470446	+	0.882429i	$0.344093\pi$
$212$	− 12.7594i	− 0.876322i
$213$	0	0
$214$	−11.5554	−0.789912
$215$	−12.6351	−0.861708
$216$	0	0
$217$	0	0
$218$	21.0491i	1.42563i
$219$	0	0
$220$	− 66.8242i	− 4.50528i
$221$	− 7.62414i	− 0.512855i
$222$	0	0
$223$	− 14.7000i	− 0.984387i	−0.870486	−	0.492193i	$-0.836195\pi$
$223$	− 14.7000i	− 0.984387i	0.870486	−	0.492193i	$-0.163805\pi$
$224$	0	0
$225$	0	0
$226$	−13.3869	−0.890483
$227$	−9.40612	−0.624306	−0.312153	−	0.950032i	$-0.601050\pi$
$227$	−9.40612	−0.624306	−0.312153	+	0.950032i	$0.601050\pi$
$228$	0	0
$229$	− 17.4687i	− 1.15436i	−0.816616	−	0.577182i	$-0.804152\pi$
$229$	− 17.4687i	− 1.15436i	0.816616	−	0.577182i	$-0.195848\pi$
$230$	−34.5979	−2.28132
$231$	0	0
$232$	17.7603	1.16602
$233$	− 14.8245i	− 0.971185i	−0.874185	−	0.485593i	$-0.838604\pi$
$233$	− 14.8245i	− 0.971185i	0.874185	−	0.485593i	$-0.161396\pi$
$234$	0	0
$235$	−17.4623	−1.13911
$236$	45.3994	2.95525
$237$	0	0
$238$	0	0
$239$	13.1924i	0.853347i	0.904406	+	0.426674i	$0.140315\pi$
$239$	13.1924i	0.853347i	−0.904406	+	0.426674i	$0.859685\pi$
$240$	0	0
$241$	− 26.9521i	− 1.73614i	−0.496442	−	0.868070i	$-0.665361\pi$
$241$	− 26.9521i	− 1.73614i	0.496442	−	0.868070i	$-0.334639\pi$
$242$	− 14.7238i	− 0.946480i
$243$	0	0
$244$	55.6119i	3.56019i
$245$	0	0
$246$	0	0
$247$	0.495670	0.0315387
$248$	64.5088	4.09631
$249$	0	0
$250$	− 2.22763i	− 0.140888i
$251$	8.08092	0.510063	0.255032	−	0.966933i	$-0.417914\pi$
$251$	8.08092	0.510063	0.255032	+	0.966933i	$0.417914\pi$
$252$	0	0
$253$	−16.4329	−1.03313
$254$	46.8244i	2.93803i
$255$	0	0
$256$	6.01469	0.375918
$257$	−10.5161	−0.655975	−0.327988	−	0.944682i	$-0.606370\pi$
$257$	−10.5161	−0.655975	−0.327988	+	0.944682i	$0.606370\pi$
$258$	0	0
$259$	0	0
$260$	49.3114i	3.05816i
$261$	0	0
$262$	− 12.9270i	− 0.798631i
$263$	− 14.4891i	− 0.893435i	−0.894675	−	0.446717i	$-0.852593\pi$
$263$	− 14.4891i	− 0.893435i	0.894675	−	0.446717i	$-0.147407\pi$
$264$	0	0
$265$	− 7.96103i	− 0.489042i
$266$	0	0
$267$	0	0
$268$	5.87441	0.358837
$269$	−16.7458	−1.02101	−0.510505	−	0.859875i	$-0.670541\pi$
$269$	−16.7458	−1.02101	−0.510505	+	0.859875i	$0.670541\pi$
$270$	0	0
$271$	− 2.57864i	− 0.156641i	−0.996928	−	0.0783206i	$-0.975044\pi$
$271$	− 2.57864i	− 0.156641i	0.996928	−	0.0783206i	$-0.0249558\pi$
$272$	−30.7364	−1.86367
$273$	0	0
$274$	−10.7907	−0.651891
$275$	− 21.3759i	− 1.28902i
$276$	0	0
$277$	3.39424	0.203940	0.101970	−	0.994787i	$-0.467485\pi$
$277$	3.39424	0.203940	0.101970	+	0.994787i	$0.467485\pi$
$278$	11.0647	0.663617
$279$	0	0
$280$	0	0
$281$	23.8408i	1.42222i	0.703079	+	0.711111i	$0.251808\pi$
$281$	23.8408i	1.42222i	−0.703079	+	0.711111i	$0.748192\pi$
$282$	0	0
$283$	− 16.0699i	− 0.955255i	−0.878563	−	0.477627i	$-0.841497\pi$
$283$	− 16.0699i	− 0.955255i	0.878563	−	0.477627i	$-0.158503\pi$
$284$	47.9223i	2.84366i
$285$	0	0
$286$	32.5455i	1.92446i
$287$	0	0
$288$	0	0
$289$	−10.5354	−0.619732
$290$	18.1532	1.06599
$291$	0	0
$292$	31.3558i	1.83496i
$293$	−10.6137	−0.620061	−0.310030	−	0.950727i	$-0.600339\pi$
$293$	−10.6137	−0.620061	−0.310030	+	0.950727i	$0.600339\pi$
$294$	0	0
$295$	28.3262	1.64921
$296$	− 98.7790i	− 5.74141i
$297$	0	0
$298$	13.8094	0.799956
$299$	12.1263	0.701280
$300$	0	0
$301$	0	0
$302$	26.3573i	1.51669i
$303$	0	0
$304$	− 1.99827i	− 0.114609i
$305$	34.6981i	1.98681i
$306$	0	0
$307$	− 3.38473i	− 0.193177i	−0.995324	−	0.0965883i	$-0.969207\pi$
$307$	− 3.38473i	− 0.193177i	0.995324	−	0.0965883i	$-0.0307930\pi$
$308$	0	0
$309$	0	0
$310$	65.9357	3.74490
$311$	−30.4634	−1.72742	−0.863711	−	0.503987i	$-0.831866\pi$
$311$	−30.4634	−1.72742	−0.863711	+	0.503987i	$0.831866\pi$
$312$	0	0
$313$	21.1291i	1.19429i	0.802134	+	0.597144i	$0.203698\pi$
$313$	21.1291i	1.19429i	−0.802134	+	0.597144i	$0.796302\pi$
$314$	30.4141	1.71637
$315$	0	0
$316$	66.0792	3.71724
$317$	2.94574i	0.165449i	0.996572	+	0.0827247i	$0.0263622\pi$
$317$	2.94574i	0.165449i	−0.996572	+	0.0827247i	$0.973638\pi$
$318$	0	0
$319$	8.62218	0.482750
$320$	55.5726	3.10660
$321$	0	0
$322$	0	0
$323$	0.420282i	0.0233851i
$324$	0	0
$325$	15.7739i	0.874977i
$326$	− 7.87282i	− 0.436035i
$327$	0	0
$328$	− 5.27259i	− 0.291130i
$329$	0	0
$330$	0	0
$331$	14.5134	0.797729	0.398864	−	0.917010i	$-0.369404\pi$
$331$	14.5134	0.797729	0.398864	+	0.917010i	$0.369404\pi$
$332$	19.3468	1.06179
$333$	0	0
$334$	45.6242i	2.49645i
$335$	3.66524	0.200253
$336$	0	0
$337$	−3.40462	−0.185462	−0.0927308	−	0.995691i	$-0.529560\pi$
$337$	−3.40462	−0.185462	−0.0927308	+	0.995691i	$0.529560\pi$
$338$	10.7058i	0.582320i
$339$	0	0
$340$	−41.8114	−2.26754
$341$	31.3173	1.69593
$342$	0	0
$343$	0	0
$344$	− 33.0172i	− 1.78017i
$345$	0	0
$346$	35.7735i	1.92320i
$347$	− 33.0097i	− 1.77206i	−0.463632	−	0.886028i	$-0.653454\pi$
$347$	− 33.0097i	− 1.77206i	0.463632	−	0.886028i	$-0.346546\pi$
$348$	0	0
$349$	− 22.4821i	− 1.20344i	−0.798708	−	0.601719i	$-0.794483\pi$
$349$	− 22.4821i	− 1.20344i	0.798708	−	0.601719i	$-0.205517\pi$
$350$	0	0
$351$	0	0
$352$	63.1791	3.36746
$353$	1.31897	0.0702018	0.0351009	−	0.999384i	$-0.488825\pi$
$353$	1.31897	0.0702018	0.0351009	+	0.999384i	$0.488825\pi$
$354$	0	0
$355$	29.9003i	1.58694i
$356$	−93.1190	−4.93529
$357$	0	0
$358$	−21.4904	−1.13580
$359$	15.5097i	0.818572i	0.912406	+	0.409286i	$0.134222\pi$
$359$	15.5097i	0.818572i	−0.912406	+	0.409286i	$0.865778\pi$
$360$	0	0
$361$	18.9727	0.998562
$362$	−8.93619	−0.469676
$363$	0	0
$364$	0	0
$365$	19.5639i	1.02402i
$366$	0	0
$367$	− 20.6751i	− 1.07923i	−0.841911	−	0.539617i	$-0.818569\pi$
$367$	− 20.6751i	− 1.07923i	0.841911	−	0.539617i	$-0.181431\pi$
$368$	− 48.8865i	− 2.54839i
$369$	0	0
$370$	− 100.964i	− 5.24887i
$371$	0	0
$372$	0	0
$373$	12.6559	0.655299	0.327649	−	0.944799i	$-0.393743\pi$
$373$	12.6559	0.655299	0.327649	+	0.944799i	$0.393743\pi$
$374$	−27.5956	−1.42693
$375$	0	0
$376$	− 45.6311i	− 2.35324i
$377$	−6.36255	−0.327688
$378$	0	0
$379$	−35.1126	−1.80361	−0.901806	−	0.432142i	$-0.857758\pi$
$379$	−35.1126	−1.80361	−0.901806	+	0.432142i	$0.857758\pi$
$380$	− 2.71830i	− 0.139446i
$381$	0	0
$382$	34.5468	1.76757
$383$	−5.18897	−0.265144	−0.132572	−	0.991173i	$-0.542324\pi$
$383$	−5.18897	−0.265144	−0.132572	+	0.991173i	$0.542324\pi$
$384$	0	0
$385$	0	0
$386$	45.9809i	2.34037i
$387$	0	0
$388$	− 62.6068i	− 3.17838i
$389$	23.3663i	1.18472i	0.805674	+	0.592359i	$0.201803\pi$
$389$	23.3663i	1.18472i	−0.805674	+	0.592359i	$0.798197\pi$
$390$	0	0
$391$	10.2819i	0.519980i
$392$	0	0
$393$	0	0
$394$	51.7308	2.60616
$395$	41.2289	2.07445
$396$	0	0
$397$	− 24.5539i	− 1.23233i	−0.787618	−	0.616164i	$-0.788686\pi$
$397$	− 24.5539i	− 1.23233i	0.787618	−	0.616164i	$-0.211314\pi$
$398$	−69.5828	−3.48787
$399$	0	0
$400$	63.5916	3.17958
$401$	21.4007i	1.06870i	0.845264	+	0.534349i	$0.179443\pi$
$401$	21.4007i	1.06870i	−0.845264	+	0.534349i	$0.820557\pi$
$402$	0	0
$403$	−23.1099	−1.15119
$404$	−76.6458	−3.81327
$405$	0	0
$406$	0	0
$407$	− 47.9546i	− 2.37702i
$408$	0	0
$409$	16.8267i	0.832026i	0.909359	+	0.416013i	$0.136573\pi$
$409$	16.8267i	0.832026i	−0.909359	+	0.416013i	$0.863427\pi$
$410$	− 5.38922i	− 0.266155i
$411$	0	0
$412$	58.2852i	2.87151i
$413$	0	0
$414$	0	0
$415$	12.0711	0.592545
$416$	−46.6216	−2.28581
$417$	0	0
$418$	− 1.79408i	− 0.0877512i
$419$	32.3652	1.58115	0.790573	−	0.612368i	$-0.209783\pi$
$419$	32.3652	1.58115	0.790573	+	0.612368i	$0.209783\pi$
$420$	0	0
$421$	−10.5354	−0.513466	−0.256733	−	0.966482i	$-0.582646\pi$
$421$	−10.5354	−0.513466	−0.256733	+	0.966482i	$0.582646\pi$
$422$	36.5042i	1.77700i
$423$	0	0
$424$	20.8032	1.01029
$425$	−13.3748	−0.648771
$426$	0	0
$427$	0	0
$428$	− 22.2110i	− 1.07361i
$429$	0	0
$430$	− 33.7475i	− 1.62745i
$431$	1.20142i	0.0578702i	0.999581	+	0.0289351i	$0.00921162\pi$
$431$	1.20142i	0.0578702i	−0.999581	+	0.0289351i	$0.990788\pi$
$432$	0	0
$433$	29.1553i	1.40111i	0.713596	+	0.700557i	$0.247065\pi$
$433$	29.1553i	1.40111i	−0.713596	+	0.700557i	$0.752935\pi$
$434$	0	0
$435$	0	0
$436$	−40.4591	−1.93764
$437$	−0.668462	−0.0319769
$438$	0	0
$439$	10.9225i	0.521301i	0.965433	+	0.260651i	$0.0839370\pi$
$439$	10.9225i	0.521301i	−0.965433	+	0.260651i	$0.916063\pi$
$440$	108.951	5.19404
$441$	0	0
$442$	20.3635	0.968595
$443$	− 4.49622i	− 0.213622i	−0.994279	−	0.106811i	$-0.965936\pi$
$443$	− 4.49622i	− 0.213622i	0.994279	−	0.106811i	$-0.0340639\pi$
$444$	0	0
$445$	−58.0999	−2.75420
$446$	39.2627	1.85914
$447$	0	0
$448$	0	0
$449$	− 13.3794i	− 0.631414i	−0.948857	−	0.315707i	$-0.897758\pi$
$449$	− 13.3794i	− 0.631414i	0.948857	−	0.315707i	$-0.102242\pi$
$450$	0	0
$451$	− 2.55971i	− 0.120532i
$452$	− 25.7313i	− 1.21030i
$453$	0	0
$454$	− 25.1231i	− 1.17908i
$455$	0	0
$456$	0	0
$457$	−28.1243	−1.31560	−0.657799	−	0.753194i	$-0.728512\pi$
$457$	−28.1243	−1.31560	−0.657799	+	0.753194i	$0.728512\pi$
$458$	46.6576	2.18017
$459$	0	0
$460$	− 66.5015i	− 3.10065i
$461$	−19.1329	−0.891108	−0.445554	−	0.895255i	$-0.646993\pi$
$461$	−19.1329	−0.891108	−0.445554	+	0.895255i	$0.646993\pi$
$462$	0	0
$463$	−14.3116	−0.665119	−0.332559	−	0.943082i	$-0.607912\pi$
$463$	−14.3116	−0.665119	−0.332559	+	0.943082i	$0.607912\pi$
$464$	25.6503i	1.19079i
$465$	0	0
$466$	39.5952	1.83421
$467$	1.43785	0.0665360	0.0332680	−	0.999446i	$-0.489409\pi$
$467$	1.43785	0.0665360	0.0332680	+	0.999446i	$0.489409\pi$
$468$	0	0
$469$	0	0
$470$	− 46.6405i	− 2.15136i
$471$	0	0
$472$	74.0198i	3.40704i
$473$	− 16.0290i	− 0.737013i
$474$	0	0
$475$	− 0.869536i	− 0.0398971i
$476$	0	0
$477$	0	0
$478$	−35.2360	−1.61166
$479$	−4.31575	−0.197192	−0.0985958	−	0.995128i	$-0.531435\pi$
$479$	−4.31575	−0.197192	−0.0985958	+	0.995128i	$0.531435\pi$
$480$	0	0
$481$	35.3870i	1.61351i
$482$	71.9872	3.27893
$483$	0	0
$484$	28.3009	1.28641
$485$	− 39.0624i	− 1.77373i
$486$	0	0
$487$	−37.7443	−1.71036	−0.855180	−	0.518332i	$-0.826553\pi$
$487$	−37.7443	−1.71036	−0.855180	+	0.518332i	$0.826553\pi$
$488$	−90.6704	−4.10446
$489$	0	0
$490$	0	0
$491$	− 25.3545i	− 1.14423i	−0.820173	−	0.572116i	$-0.806123\pi$
$491$	− 25.3545i	− 1.14423i	0.820173	−	0.572116i	$-0.193877\pi$
$492$	0	0
$493$	− 5.39484i	− 0.242972i
$494$	1.32390i	0.0595650i
$495$	0	0
$496$	93.1666i	4.18330i
$497$	0	0
$498$	0	0
$499$	0.0116489	0.000521477 0	0.000260738	−	1.00000i	$-0.499917\pi$
$499$	0.0116489	0.000521477 0	0.000260738		1.00000i	$0.499917\pi$
$500$	4.28179	0.191487
$501$	0	0
$502$	21.5836i	0.963322i
$503$	24.9237	1.11129	0.555646	−	0.831419i	$-0.312471\pi$
$503$	24.9237	1.11129	0.555646	+	0.831419i	$0.312471\pi$
$504$	0	0
$505$	−47.8218	−2.12804
$506$	− 43.8910i	− 1.95119i
$507$	0	0
$508$	−90.0024	−3.99321
$509$	21.3672	0.947084	0.473542	−	0.880771i	$-0.342975\pi$
$509$	21.3672	0.947084	0.473542	+	0.880771i	$0.342975\pi$
$510$	0	0
$511$	0	0
$512$	− 14.4212i	− 0.637335i
$513$	0	0
$514$	− 28.0877i	− 1.23890i
$515$	36.3660i	1.60248i
$516$	0	0
$517$	− 22.1527i	− 0.974275i
$518$	0	0
$519$	0	0
$520$	−80.3980	−3.52568
$521$	10.0430	0.439990	0.219995	−	0.975501i	$-0.429396\pi$
$521$	10.0430	0.439990	0.219995	+	0.975501i	$0.429396\pi$
$522$	0	0
$523$	15.5654i	0.680628i	0.940312	+	0.340314i	$0.110533\pi$
$523$	15.5654i	0.680628i	−0.940312	+	0.340314i	$0.889467\pi$
$524$	24.8473	1.08546
$525$	0	0
$526$	38.6993	1.68737
$527$	− 19.5950i	− 0.853573i
$528$	0	0
$529$	6.64647	0.288977
$530$	21.2633	0.923620
$531$	0	0
$532$	0	0
$533$	1.88888i	0.0818164i
$534$	0	0
$535$	− 13.8581i	− 0.599140i
$536$	9.57773i	0.413695i
$537$	0	0
$538$	− 44.7269i	− 1.92831i
$539$	0	0
$540$	0	0
$541$	28.8241	1.23924	0.619622	−	0.784901i	$-0.287286\pi$
$541$	28.8241	1.23924	0.619622	+	0.784901i	$0.287286\pi$
$542$	6.88736	0.295838
$543$	0	0
$544$	− 39.5308i	− 1.69487i
$545$	−25.2437	−1.08132
$546$	0	0
$547$	−24.3469	−1.04100	−0.520500	−	0.853862i	$-0.674254\pi$
$547$	−24.3469	−1.04100	−0.520500	+	0.853862i	$0.674254\pi$
$548$	− 20.7411i	− 0.886016i
$549$	0	0
$550$	57.0935	2.43448
$551$	0.350736	0.0149419
$552$	0	0
$553$	0	0
$554$	9.06576i	0.385167i
$555$	0	0
$556$	21.2678i	0.901954i
$557$	− 8.26112i	− 0.350035i	−0.984565	−	0.175017i	$-0.944002\pi$
$557$	− 8.26112i	− 0.350035i	0.984565	−	0.175017i	$-0.0559982\pi$
$558$	0	0
$559$	11.8282i	0.500280i
$560$	0	0
$561$	0	0
$562$	−63.6771	−2.68605
$563$	−0.276886	−0.0116694	−0.00583468	−	0.999983i	$-0.501857\pi$
$563$	−0.276886	−0.0116694	−0.00583468	+	0.999983i	$0.501857\pi$
$564$	0	0
$565$	− 16.0546i	− 0.675421i
$566$	42.9215	1.80412
$567$	0	0
$568$	−78.1332	−3.27839
$569$	36.4807i	1.52935i	0.644415	+	0.764676i	$0.277101\pi$
$569$	36.4807i	1.52935i	−0.644415	+	0.764676i	$0.722899\pi$
$570$	0	0
$571$	10.4091	0.435605	0.217803	−	0.975993i	$-0.430111\pi$
$571$	10.4091	0.435605	0.217803	+	0.975993i	$0.430111\pi$
$572$	−62.5566	−2.61562
$573$	0	0
$574$	0	0
$575$	− 21.2727i	− 0.887133i
$576$	0	0
$577$	11.0368i	0.459467i	0.973254	+	0.229734i	$0.0737855\pi$
$577$	11.0368i	0.459467i	−0.973254	+	0.229734i	$0.926215\pi$
$578$	− 28.1394i	− 1.17045i
$579$	0	0
$580$	34.8927i	1.44884i
$581$	0	0
$582$	0	0
$583$	10.0994	0.418274
$584$	−51.1229	−2.11548
$585$	0	0
$586$	− 28.3485i	− 1.17107i
$587$	19.3497	0.798649	0.399324	−	0.916810i	$-0.369245\pi$
$587$	19.3497	0.798649	0.399324	+	0.916810i	$0.369245\pi$
$588$	0	0
$589$	1.27394	0.0524916
$590$	75.6571i	3.11476i
$591$	0	0
$592$	142.661	5.86334
$593$	33.4553	1.37385	0.686923	−	0.726730i	$-0.258961\pi$
$593$	33.4553	1.37385	0.686923	+	0.726730i	$0.258961\pi$
$594$	0	0
$595$	0	0
$596$	26.5434i	1.08726i
$597$	0	0
$598$	32.3884i	1.32446i
$599$	− 7.34670i	− 0.300178i	−0.988672	−	0.150089i	$-0.952044\pi$
$599$	− 7.34670i	− 0.300178i	0.988672	−	0.150089i	$-0.0479560\pi$
$600$	0	0
$601$	− 26.5631i	− 1.08353i	−0.840530	−	0.541766i	$-0.817756\pi$
$601$	− 26.5631i	− 1.08353i	0.840530	−	0.541766i	$-0.182244\pi$
$602$	0	0
$603$	0	0
$604$	−50.6620	−2.06141
$605$	17.6579	0.717895
$606$	0	0
$607$	− 21.3489i	− 0.866527i	−0.901267	−	0.433263i	$-0.857362\pi$
$607$	− 21.3489i	− 0.866527i	0.901267	−	0.433263i	$-0.142638\pi$
$608$	2.57002	0.104228
$609$	0	0
$610$	−92.6761	−3.75234
$611$	16.3471i	0.661332i
$612$	0	0
$613$	−17.4572	−0.705088	−0.352544	−	0.935795i	$-0.614683\pi$
$613$	−17.4572	−0.705088	−0.352544	+	0.935795i	$0.614683\pi$
$614$	9.04036	0.364839
$615$	0	0
$616$	0	0
$617$	39.7469i	1.60015i	0.599901	+	0.800074i	$0.295207\pi$
$617$	39.7469i	1.60015i	−0.599901	+	0.800074i	$0.704793\pi$
$618$	0	0
$619$	9.35594i	0.376047i	0.982165	+	0.188023i	$0.0602081\pi$
$619$	9.35594i	0.376047i	−0.982165	+	0.188023i	$0.939792\pi$
$620$	126.737i	5.08987i
$621$	0	0
$622$	− 81.3656i	− 3.26246i
$623$	0	0
$624$	0	0
$625$	−23.6303	−0.945213
$626$	−56.4343	−2.25557
$627$	0	0
$628$	58.4597i	2.33279i
$629$	−30.0049	−1.19637
$630$	0	0
$631$	−17.1161	−0.681382	−0.340691	−	0.940175i	$-0.610661\pi$
$631$	−17.1161	−0.681382	−0.340691	+	0.940175i	$0.610661\pi$
$632$	107.736i	4.28552i
$633$	0	0
$634$	−7.86787	−0.312473
$635$	−56.1554	−2.22846
$636$	0	0
$637$	0	0
$638$	23.0292i	0.911736i
$639$	0	0
$640$	48.8262i	1.93002i
$641$	6.26895i	0.247609i	0.992307	+	0.123804i	$0.0395095\pi$
$641$	6.26895i	0.247609i	−0.992307	+	0.123804i	$0.960490\pi$
$642$	0	0
$643$	− 37.8182i	− 1.49141i	−0.666279	−	0.745703i	$-0.732114\pi$
$643$	− 37.8182i	− 1.49141i	0.666279	−	0.745703i	$-0.267886\pi$
$644$	0	0
$645$	0	0
$646$	−1.12254	−0.0441658
$647$	26.0490	1.02409	0.512045	−	0.858958i	$-0.328888\pi$
$647$	26.0490	1.02409	0.512045	+	0.858958i	$0.328888\pi$
$648$	0	0
$649$	35.9347i	1.41056i
$650$	−42.1309	−1.65251
$651$	0	0
$652$	15.1325	0.592636
$653$	− 26.8906i	− 1.05231i	−0.850388	−	0.526156i	$-0.823633\pi$
$653$	− 26.8906i	− 1.05231i	0.850388	−	0.526156i	$-0.176367\pi$
$654$	0	0
$655$	15.5030	0.605753
$656$	7.61493	0.297313
$657$	0	0
$658$	0	0
$659$	8.10738i	0.315819i	0.987454	+	0.157909i	$0.0504754\pi$
$659$	8.10738i	0.315819i	−0.987454	+	0.157909i	$0.949525\pi$
$660$	0	0
$661$	29.4100i	1.14392i	0.820283	+	0.571958i	$0.193816\pi$
$661$	29.4100i	1.14392i	−0.820283	+	0.571958i	$0.806184\pi$
$662$	38.7643i	1.50662i
$663$	0	0
$664$	31.5432i	1.22411i
$665$	0	0
$666$	0	0
$667$	8.58055	0.332240
$668$	−87.6955	−3.39304
$669$	0	0
$670$	9.78959i	0.378205i
$671$	−44.0181	−1.69930
$672$	0	0
$673$	−15.3797	−0.592843	−0.296421	−	0.955057i	$-0.595793\pi$
$673$	−15.3797	−0.592843	−0.296421	+	0.955057i	$0.595793\pi$
$674$	− 9.09351i	− 0.350269i
$675$	0	0
$676$	−20.5779	−0.791459
$677$	24.7993	0.953114	0.476557	−	0.879143i	$-0.341885\pi$
$677$	24.7993	0.953114	0.476557	+	0.879143i	$0.341885\pi$
$678$	0	0
$679$	0	0
$680$	− 68.1700i	− 2.61420i
$681$	0	0
$682$	83.6463i	3.20298i
$683$	− 39.8402i	− 1.52444i	−0.647316	−	0.762222i	$-0.724108\pi$
$683$	− 39.8402i	− 1.52444i	0.647316	−	0.762222i	$-0.275892\pi$
$684$	0	0
$685$	− 12.9411i	− 0.494452i
$686$	0	0
$687$	0	0
$688$	47.6849	1.81797
$689$	−7.45262	−0.283922
$690$	0	0
$691$	− 12.7920i	− 0.486631i	−0.969947	−	0.243316i	$-0.921765\pi$
$691$	− 12.7920i	− 0.486631i	0.969947	−	0.243316i	$-0.0782351\pi$
$692$	−68.7612	−2.61391
$693$	0	0
$694$	88.1666	3.34676
$695$	13.2696i	0.503346i
$696$	0	0
$697$	−1.60159	−0.0606646
$698$	60.0480	2.27285
$699$	0	0
$700$	0	0
$701$	32.7936i	1.23860i	0.785155	+	0.619299i	$0.212583\pi$
$701$	32.7936i	1.23860i	−0.785155	+	0.619299i	$0.787417\pi$
$702$	0	0
$703$	− 1.95071i	− 0.0735726i
$704$	70.4996i	2.65705i
$705$	0	0
$706$	3.52288i	0.132585i
$707$	0	0
$708$	0	0
$709$	26.3711	0.990387	0.495193	−	0.868783i	$-0.335097\pi$
$709$	26.3711	0.990387	0.495193	+	0.868783i	$0.335097\pi$
$710$	−79.8615	−2.99715
$711$	0	0
$712$	− 151.822i	− 5.68979i
$713$	31.1661	1.16718
$714$	0	0
$715$	−39.0311	−1.45968
$716$	− 41.3072i	− 1.54372i
$717$	0	0
$718$	−41.4253	−1.54598
$719$	−33.9578	−1.26641	−0.633207	−	0.773983i	$-0.718262\pi$
$719$	−33.9578	−1.26641	−0.633207	+	0.773983i	$0.718262\pi$
$720$	0	0
$721$	0	0
$722$	50.6747i	1.88592i
$723$	0	0
$724$	− 17.1765i	− 0.638359i
$725$	11.1616i	0.414531i
$726$	0	0
$727$	14.3061i	0.530583i	0.964168	+	0.265292i	$0.0854682\pi$
$727$	14.3061i	0.530583i	−0.964168	+	0.265292i	$0.914532\pi$
$728$	0	0
$729$	0	0
$730$	−52.2538	−1.93400
$731$	−10.0292	−0.370944
$732$	0	0
$733$	21.1450i	0.781009i	0.920601	+	0.390504i	$0.127699\pi$
$733$	21.1450i	0.781009i	−0.920601	+	0.390504i	$0.872301\pi$
$734$	55.2218	2.03828
$735$	0	0
$736$	62.8741	2.31757
$737$	4.64974i	0.171275i
$738$	0	0
$739$	28.5709	1.05100	0.525499	−	0.850794i	$-0.323879\pi$
$739$	28.5709	1.05100	0.525499	+	0.850794i	$0.323879\pi$
$740$	194.065	7.13399
$741$	0	0
$742$	0	0
$743$	0.925908i	0.0339683i	0.999856	+	0.0169841i	$0.00540648\pi$
$743$	0.925908i	0.0339683i	−0.999856	+	0.0169841i	$0.994594\pi$
$744$	0	0
$745$	16.5613i	0.606758i
$746$	33.8031i	1.23762i
$747$	0	0
$748$	− 53.0422i	− 1.93941i
$749$	0	0
$750$	0	0
$751$	−7.07444	−0.258150	−0.129075	−	0.991635i	$-0.541201\pi$
$751$	−7.07444	−0.258150	−0.129075	+	0.991635i	$0.541201\pi$
$752$	65.9025	2.40322
$753$	0	0
$754$	− 16.9939i	− 0.618882i
$755$	−31.6096	−1.15039
$756$	0	0
$757$	14.3950	0.523196	0.261598	−	0.965177i	$-0.415750\pi$
$757$	14.3950	0.523196	0.261598	+	0.965177i	$0.415750\pi$
$758$	− 93.7832i	− 3.40636i
$759$	0	0
$760$	4.43195	0.160764
$761$	−19.7726	−0.716757	−0.358378	−	0.933576i	$-0.616670\pi$
$761$	−19.7726	−0.716757	−0.358378	+	0.933576i	$0.616670\pi$
$762$	0	0
$763$	0	0
$764$	66.4032i	2.40238i
$765$	0	0
$766$	− 13.8594i	− 0.500759i
$767$	− 26.5172i	− 0.957480i
$768$	0	0
$769$	14.3584i	0.517779i	0.965907	+	0.258889i	$0.0833565\pi$
$769$	14.3584i	0.517779i	−0.965907	+	0.258889i	$0.916643\pi$
$770$	0	0
$771$	0	0
$772$	−88.3810	−3.18090
$773$	−16.2164	−0.583263	−0.291632	−	0.956531i	$-0.594198\pi$
$773$	−16.2164	−0.583263	−0.291632	+	0.956531i	$0.594198\pi$
$774$	0	0
$775$	40.5409i	1.45627i
$776$	102.075	3.66428
$777$	0	0
$778$	−62.4097	−2.23750
$779$	− 0.104125i	− 0.00373065i
$780$	0	0
$781$	−37.9316	−1.35730
$782$	−27.4623	−0.982051
$783$	0	0
$784$	0	0
$785$	36.4749i	1.30184i
$786$	0	0
$787$	− 49.9936i	− 1.78208i	−0.453927	−	0.891039i	$-0.649977\pi$
$787$	− 49.9936i	− 1.78208i	0.453927	−	0.891039i	$-0.350023\pi$
$788$	99.4331i	3.54216i
$789$	0	0
$790$	110.120i	3.91788i
$791$	0	0
$792$	0	0
$793$	32.4822	1.15348
$794$	65.5818	2.32741
$795$	0	0
$796$	− 133.747i	− 4.74053i
$797$	−10.9957	−0.389488	−0.194744	−	0.980854i	$-0.562388\pi$
$797$	−10.9957	−0.389488	−0.194744	+	0.980854i	$0.562388\pi$
$798$	0	0
$799$	−13.8608	−0.490360
$800$	81.7867i	2.89160i
$801$	0	0
$802$	−57.1597	−2.01838
$803$	−24.8188	−0.875838
$804$	0	0
$805$	0	0
$806$	− 61.7249i	− 2.17417i
$807$	0	0
$808$	− 124.964i	− 4.39623i
$809$	35.3075i	1.24135i	0.784069	+	0.620674i	$0.213141\pi$
$809$	35.3075i	1.24135i	−0.784069	+	0.620674i	$0.786859\pi$
$810$	0	0
$811$	20.7045i	0.727034i	0.931588	+	0.363517i	$0.118424\pi$
$811$	20.7045i	0.727034i	−0.931588	+	0.363517i	$0.881576\pi$
$812$	0	0
$813$	0	0
$814$	128.083	4.48932
$815$	9.44169	0.330728
$816$	0	0
$817$	− 0.652032i	− 0.0228117i
$818$	−44.9429	−1.57139
$819$	0	0
$820$	10.3588	0.361744
$821$	11.6830i	0.407739i	0.978998	+	0.203869i	$0.0653518\pi$
$821$	11.6830i	0.407739i	−0.978998	+	0.203869i	$0.934648\pi$
$822$	0	0
$823$	26.2902	0.916420	0.458210	−	0.888844i	$-0.348491\pi$
$823$	26.2902	0.916420	0.458210	+	0.888844i	$0.348491\pi$
$824$	−95.0290	−3.31049
$825$	0	0
$826$	0	0
$827$	30.4447i	1.05867i	0.848414	+	0.529333i	$0.177558\pi$
$827$	30.4447i	1.05867i	−0.848414	+	0.529333i	$0.822442\pi$
$828$	0	0
$829$	− 29.8109i	− 1.03537i	−0.855570	−	0.517687i	$-0.826793\pi$
$829$	− 29.8109i	− 1.03537i	0.855570	−	0.517687i	$-0.173207\pi$
$830$	32.2410i	1.11910i
$831$	0	0
$832$	− 52.0236i	− 1.80359i
$833$	0	0
$834$	0	0
$835$	−54.7161	−1.89353
$836$	3.44844	0.119267
$837$	0	0
$838$	86.4453i	2.98620i
$839$	20.1662	0.696215	0.348107	−	0.937455i	$-0.386824\pi$
$839$	20.1662	0.696215	0.348107	+	0.937455i	$0.386824\pi$
$840$	0	0
$841$	24.4979	0.844754
$842$	− 28.1394i	− 0.969748i
$843$	0	0
$844$	−70.1657	−2.41520
$845$	−12.8392	−0.441683
$846$	0	0
$847$	0	0
$848$	30.0449i	1.03175i
$849$	0	0
$850$	− 35.7230i	− 1.22529i
$851$	− 47.7231i	− 1.63593i
$852$	0	0
$853$	− 35.2714i	− 1.20767i	−0.797109	−	0.603835i	$-0.793639\pi$
$853$	− 35.2714i	− 1.20767i	0.797109	−	0.603835i	$-0.206361\pi$
$854$	0	0
$855$	0	0
$856$	36.2131	1.23774
$857$	−42.9336	−1.46658	−0.733291	−	0.679915i	$-0.762017\pi$
$857$	−42.9336	−1.46658	−0.733291	+	0.679915i	$0.762017\pi$
$858$	0	0
$859$	− 39.9956i	− 1.36463i	−0.731057	−	0.682317i	$-0.760973\pi$
$859$	− 39.9956i	− 1.36463i	0.731057	−	0.682317i	$-0.239027\pi$
$860$	64.8670	2.21194
$861$	0	0
$862$	−3.20890	−0.109296
$863$	48.2349i	1.64194i	0.570975	+	0.820968i	$0.306565\pi$
$863$	48.2349i	1.64194i	−0.570975	+	0.820968i	$0.693435\pi$
$864$	0	0
$865$	−42.9024	−1.45872
$866$	−77.8718	−2.64619
$867$	0	0
$868$	0	0
$869$	52.3032i	1.77426i
$870$	0	0
$871$	− 3.43117i	− 0.116261i
$872$	− 65.9651i	− 2.23386i
$873$	0	0
$874$	− 1.78542i	− 0.0603926i
$875$	0	0
$876$	0	0
$877$	−41.3378	−1.39588	−0.697939	−	0.716157i	$-0.745899\pi$
$877$	−41.3378	−1.39588	−0.697939	+	0.716157i	$0.745899\pi$
$878$	−29.1731	−0.984546
$879$	0	0
$880$	157.352i	5.30434i
$881$	43.3796	1.46150	0.730749	−	0.682647i	$-0.239171\pi$
$881$	43.3796	1.46150	0.730749	+	0.682647i	$0.239171\pi$
$882$	0	0
$883$	12.4836	0.420105	0.210053	−	0.977690i	$-0.432637\pi$
$883$	12.4836	0.420105	0.210053	+	0.977690i	$0.432637\pi$
$884$	39.1413i	1.31646i
$885$	0	0
$886$	12.0091	0.403453
$887$	26.8236	0.900649	0.450325	−	0.892865i	$-0.351308\pi$
$887$	26.8236	0.900649	0.450325	+	0.892865i	$0.351308\pi$
$888$	0	0
$889$	0	0
$890$	− 155.181i	− 5.20167i
$891$	0	0
$892$	75.4679i	2.52685i
$893$	− 0.901135i	− 0.0301553i
$894$	0	0
$895$	− 25.7729i	− 0.861493i
$896$	0	0
$897$	0	0
$898$	35.7355	1.19251
$899$	−16.3526	−0.545389
$900$	0	0
$901$	− 6.31912i	− 0.210521i
$902$	6.83679	0.227640
$903$	0	0
$904$	41.9526	1.39532
$905$	− 10.7170i	− 0.356244i
$906$	0	0
$907$	−14.7707	−0.490454	−0.245227	−	0.969466i	$-0.578863\pi$
$907$	−14.7707	−0.490454	−0.245227	+	0.969466i	$0.578863\pi$
$908$	48.2897	1.60255
$909$	0	0
$910$	0	0
$911$	− 25.1293i	− 0.832571i	−0.909234	−	0.416286i	$-0.863332\pi$
$911$	− 25.1293i	− 0.832571i	0.909234	−	0.416286i	$-0.136668\pi$
$912$	0	0
$913$	15.3134i	0.506800i
$914$	− 75.1179i	− 2.48468i
$915$	0	0
$916$	89.6818i	2.96317i
$917$	0	0
$918$	0	0
$919$	41.2856	1.36189	0.680943	−	0.732337i	$-0.261570\pi$
$919$	41.2856	1.36189	0.680943	+	0.732337i	$0.261570\pi$
$920$	108.425	3.57467
$921$	0	0
$922$	− 51.1026i	− 1.68297i
$923$	27.9908	0.921328
$924$	0	0
$925$	62.0782	2.04112
$926$	− 38.2254i	− 1.25616i
$927$	0	0
$928$	−32.9895	−1.08293
$929$	−1.85221	−0.0607690	−0.0303845	−	0.999538i	$-0.509673\pi$
$929$	−1.85221	−0.0607690	−0.0303845	+	0.999538i	$0.509673\pi$
$930$	0	0
$931$	0	0
$932$	76.1069i	2.49297i
$933$	0	0
$934$	3.84041i	0.125662i
$935$	− 33.0947i	− 1.08231i
$936$	0	0
$937$	47.2417i	1.54332i	0.636036	+	0.771659i	$0.280573\pi$
$937$	47.2417i	1.54332i	−0.636036	+	0.771659i	$0.719427\pi$
$938$	0	0
$939$	0	0
$940$	89.6488	2.92402
$941$	40.7764	1.32927	0.664636	−	0.747168i	$-0.268587\pi$
$941$	40.7764	1.32927	0.664636	+	0.747168i	$0.268587\pi$
$942$	0	0
$943$	− 2.54735i	− 0.0829530i
$944$	−106.903	−3.47939
$945$	0	0
$946$	42.8122	1.39195
$947$	− 12.8681i	− 0.418159i	−0.977899	−	0.209079i	$-0.932953\pi$
$947$	− 12.8681i	− 0.418159i	0.977899	−	0.209079i	$-0.0670467\pi$
$948$	0	0
$949$	18.3145	0.594514
$950$	2.32247	0.0753509
$951$	0	0
$952$	0	0
$953$	− 35.7507i	− 1.15808i	−0.815300	−	0.579039i	$-0.803428\pi$
$953$	− 35.7507i	− 1.15808i	0.815300	−	0.579039i	$-0.196572\pi$
$954$	0	0
$955$	41.4311i	1.34068i
$956$	− 67.7281i	− 2.19048i
$957$	0	0
$958$	− 11.5271i	− 0.372422i
$959$	0	0
$960$	0	0
$961$	−28.3955	−0.915984
$962$	−94.5162	−3.04733
$963$	0	0
$964$	138.368i	4.45655i
$965$	−55.1438	−1.77514
$966$	0	0
$967$	−34.2307	−1.10078	−0.550392	−	0.834906i	$-0.685522\pi$
$967$	−34.2307	−1.10078	−0.550392	+	0.834906i	$0.685522\pi$
$968$	46.1423i	1.48307i
$969$	0	0
$970$	104.333	3.34993
$971$	−3.99904	−0.128335	−0.0641677	−	0.997939i	$-0.520439\pi$
$971$	−3.99904	−0.128335	−0.0641677	+	0.997939i	$0.520439\pi$
$972$	0	0
$973$	0	0
$974$	− 100.812i	− 3.23024i
$975$	0	0
$976$	− 130.950i	− 4.19162i
$977$	43.4806i	1.39107i	0.718493	+	0.695535i	$0.244832\pi$
$977$	43.4806i	1.39107i	−0.718493	+	0.695535i	$0.755168\pi$
$978$	0	0
$979$	− 73.7058i	− 2.35565i
$980$	0	0
$981$	0	0
$982$	67.7200	2.16103
$983$	31.2017	0.995179	0.497589	−	0.867413i	$-0.334219\pi$
$983$	31.2017	0.995179	0.497589	+	0.867413i	$0.334219\pi$
$984$	0	0
$985$	62.0395i	1.97674i
$986$	14.4092	0.458884
$987$	0	0
$988$	−2.54470	−0.0809577
$989$	− 15.9516i	− 0.507231i
$990$	0	0
$991$	26.4989	0.841766	0.420883	−	0.907115i	$-0.361720\pi$
$991$	26.4989	0.841766	0.420883	+	0.907115i	$0.361720\pi$
$992$	−119.824	−3.80441
$993$	0	0
$994$	0	0
$995$	− 83.4490i	− 2.64551i
$996$	0	0
$997$	− 18.8775i	− 0.597857i	−0.954276	−	0.298928i	$-0.903371\pi$
$997$	− 18.8775i	− 0.597857i	0.954276	−	0.298928i	$-0.0966292\pi$
$998$	0.0311134i	0 0.000984877i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.f.1322.15	yes	16
3.2	odd	2	inner	1323.2.c.f.1322.2	yes	16
7.6	odd	2	inner	1323.2.c.f.1322.16	yes	16
21.20	even	2	inner	1323.2.c.f.1322.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.c.f.1322.1	✓	16	21.20	even	2	inner
1323.2.c.f.1322.2	yes	16	3.2	odd	2	inner
1323.2.c.f.1322.15	yes	16	1.1	even	1	trivial
1323.2.c.f.1322.16	yes	16	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.67093i q^{2} -5.13386 q^{4} -3.20318 q^{5} -8.37032i q^{8} +O(q^{10})\)	\(q+2.67093i q^{2} -5.13386 q^{4} -3.20318 q^{5} -8.37032i q^{8} -8.55547i q^{10} -4.06357i q^{11} +2.99862i q^{13} +12.0888 q^{16} -2.54255 q^{17} -0.165299i q^{19} +16.4447 q^{20} +10.8535 q^{22} -4.04395i q^{23} +5.26038 q^{25} -8.00910 q^{26} +2.12182i q^{29} +7.70685i q^{31} +15.5477i q^{32} -6.79097i q^{34} +11.8011 q^{37} +0.441503 q^{38} +26.8117i q^{40} +0.629916 q^{41} +3.94455 q^{43} +20.8618i q^{44} +10.8011 q^{46} +5.45154 q^{47} +14.0501i q^{50} -15.3945i q^{52} +2.48535i q^{53} +13.0164i q^{55} -5.66724 q^{58} -8.84313 q^{59} -10.8324i q^{61} -20.5844 q^{62} -17.3492 q^{64} -9.60513i q^{65} -1.14425 q^{67} +13.0531 q^{68} -9.33455i q^{71} -6.10764i q^{73} +31.5199i q^{74} +0.848624i q^{76} -12.8712 q^{79} -38.7226 q^{80} +1.68246i q^{82} -3.76846 q^{83} +8.14425 q^{85} +10.5356i q^{86} -34.0134 q^{88} +18.1382 q^{89} +20.7611i q^{92} +14.5607i q^{94} +0.529484i q^{95} +12.1949i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 48 q^{16} + 64 q^{22} - 16 q^{25} + 32 q^{37} - 16 q^{43} + 16 q^{46} - 32 q^{64} + 48 q^{67} - 64 q^{79} + 64 q^{85} - 176 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 48 * q^16 + 64 * q^22 - 16 * q^25 + 32 * q^37 - 16 * q^43 + 16 * q^46 - 32 * q^64 + 48 * q^67 - 64 * q^79 + 64 * q^85 - 176 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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