LMFDB - Embedded newform 1323.2.c.d.1322.9

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 $ x^12 - 9x^10 + 59x^8 - 180x^6 + 403x^4 - 198*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 189)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.9
Root			$1.90412 - 1.09935i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.d.1322.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.83424i q^{2} -1.36445 q^{4} -3.80824 q^{5} +1.16576i q^{8} +O(q^{10})$	$q+1.83424i q^{2} -1.36445 q^{4} -3.80824 q^{5} +1.16576i q^{8} -6.98525i q^{10} -1.16576i q^{11} -5.54030i q^{13} -4.86718 q^{16} +3.17700 q^{17} +0.631243i q^{19} +5.19615 q^{20} +2.13828 q^{22} -1.76183i q^{23} +9.50273 q^{25} +10.1622 q^{26} +4.83424i q^{29} -4.27781i q^{31} -6.59607i q^{32} +5.82739i q^{34} +4.23163 q^{37} -1.15785 q^{38} -4.43949i q^{40} +4.56491 q^{41} +7.23163 q^{43} +1.59061i q^{44} +3.23163 q^{46} -2.54576 q^{47} +17.4303i q^{50} +7.55945i q^{52} -0.0724126i q^{53} +4.43949i q^{55} -8.86718 q^{58} -6.98525 q^{59} -8.08606i q^{61} +7.84655 q^{62} +2.36445 q^{64} +21.0988i q^{65} +5.09880 q^{67} -4.33485 q^{68} -4.76183i q^{71} +5.59734i q^{73} +7.76183i q^{74} -0.861298i q^{76} +17.0988 q^{79} +18.5354 q^{80} +8.37315i q^{82} +10.1622 q^{83} -12.0988 q^{85} +13.2646i q^{86} +1.35899 q^{88} +11.5502 q^{89} +2.40393i q^{92} -4.66954i q^{94} -2.40393i q^{95} -0.688287i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{4}+O(q^{10}) $ 12 * q - 16 * q^4	$ 12 q - 16 q^{4} + 8 q^{16} - 40 q^{22} + 48 q^{25} - 16 q^{37} + 20 q^{43} - 28 q^{46} - 40 q^{58} + 28 q^{64} - 72 q^{67} + 72 q^{79} - 12 q^{85} + 148 q^{88}+O(q^{100}) $ 12 * q - 16 * q^4 + 8 * q^16 - 40 * q^22 + 48 * q^25 - 16 * q^37 + 20 * q^43 - 28 * q^46 - 40 * q^58 + 28 * q^64 - 72 * q^67 + 72 * q^79 - 12 * q^85 + 148 * 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 $

$2$	1.83424i	1.29701i	0.761212	+	0.648503i	$0.224605\pi$
$2$	1.83424i	1.29701i	−0.761212	+	0.648503i	$0.775395\pi$
$3$	0	0
$3$	0	0
$4$	−1.36445	−0.682224
$5$	−3.80824	−1.70310	−0.851549	−	0.524274i	$-0.824337\pi$
$5$	−3.80824	−1.70310	−0.851549	+	0.524274i	$0.824337\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.16576i	0.412157i
$9$	0	0
$10$	− 6.98525i	− 2.20893i
$11$	− 1.16576i	− 0.351489i	−0.984436	−	0.175744i	$-0.943767\pi$
$11$	− 1.16576i	− 0.351489i	0.984436	−	0.175744i	$-0.0562332\pi$
$12$	0	0
$13$	− 5.54030i	− 1.53660i	−0.640089	−	0.768301i	$-0.721103\pi$
$13$	− 5.54030i	− 1.53660i	0.640089	−	0.768301i	$-0.278897\pi$
$14$	0	0
$15$	0	0
$16$	−4.86718	−1.21679
$17$	3.17700	0.770536	0.385268	−	0.922805i	$-0.374109\pi$
$17$	3.17700	0.770536	0.385268	+	0.922805i	$0.374109\pi$
$18$	0	0
$19$	0.631243i	0.144817i	0.997375	+	0.0724085i	$0.0230685\pi$
$19$	0.631243i	0.144817i	−0.997375	+	0.0724085i	$0.976931\pi$
$20$	5.19615	1.16190
$21$	0	0
$22$	2.13828	0.455883
$23$	− 1.76183i	− 0.367367i	−0.982985	−	0.183684i	$-0.941198\pi$
$23$	− 1.76183i	− 0.367367i	0.982985	−	0.183684i	$-0.0588022\pi$
$24$	0	0
$25$	9.50273	1.90055
$26$	10.1622	1.99298
$27$	0	0
$28$	0	0
$29$	4.83424i	0.897696i	0.893608	+	0.448848i	$0.148166\pi$
$29$	4.83424i	0.897696i	−0.893608	+	0.448848i	$0.851834\pi$
$30$	0	0
$31$	− 4.27781i	− 0.768317i	−0.923267	−	0.384159i	$-0.874492\pi$
$31$	− 4.27781i	− 0.768317i	0.923267	−	0.384159i	$-0.125508\pi$
$32$	− 6.59607i	− 1.16603i
$33$	0	0
$34$	5.82739i	0.999390i
$35$	0	0
$36$	0	0
$37$	4.23163	0.695675	0.347837	−	0.937555i	$-0.386916\pi$
$37$	4.23163	0.695675	0.347837	+	0.937555i	$0.386916\pi$
$38$	−1.15785	−0.187828
$39$	0	0
$40$	− 4.43949i	− 0.701945i
$41$	4.56491	0.712919	0.356460	−	0.934311i	$-0.383984\pi$
$41$	4.56491	0.712919	0.356460	+	0.934311i	$0.383984\pi$
$42$	0	0
$43$	7.23163	1.10281	0.551406	−	0.834237i	$-0.314091\pi$
$43$	7.23163	1.10281	0.551406	+	0.834237i	$0.314091\pi$
$44$	1.59061i	0.239794i
$45$	0	0
$46$	3.23163	0.476477
$47$	−2.54576	−0.371337	−0.185669	−	0.982612i	$-0.559445\pi$
$47$	−2.54576	−0.371337	−0.185669	+	0.982612i	$0.559445\pi$
$48$	0	0
$49$	0	0
$50$	17.4303i	2.46502i
$51$	0	0
$52$	7.55945i	1.04831i
$53$	− 0.0724126i	− 0.00994664i	−0.999988	−	0.00497332i	$-0.998417\pi$
$53$	− 0.0724126i	− 0.00994664i	0.999988	−	0.00497332i	$-0.00158306\pi$
$54$	0	0
$55$	4.43949i	0.598620i
$56$	0	0
$57$	0	0
$58$	−8.86718	−1.16432
$59$	−6.98525	−0.909402	−0.454701	−	0.890644i	$-0.650254\pi$
$59$	−6.98525	−0.909402	−0.454701	+	0.890644i	$0.650254\pi$
$60$	0	0
$61$	− 8.08606i	− 1.03531i	−0.855588	−	0.517657i	$-0.826804\pi$
$61$	− 8.08606i	− 1.03531i	0.855588	−	0.517657i	$-0.173196\pi$
$62$	7.84655	0.996512
$63$	0	0
$64$	2.36445	0.295556
$65$	21.0988i	2.61698i
$66$	0	0
$67$	5.09880	0.622918	0.311459	−	0.950260i	$-0.399182\pi$
$67$	5.09880	0.622918	0.311459	+	0.950260i	$0.399182\pi$
$68$	−4.33485	−0.525678
$69$	0	0
$70$	0	0
$71$	− 4.76183i	− 0.565125i	−0.959249	−	0.282563i	$-0.908815\pi$
$71$	− 4.76183i	− 0.565125i	0.959249	−	0.282563i	$-0.0911845\pi$
$72$	0	0
$73$	5.59734i	0.655119i	0.944831	+	0.327560i	$0.106226\pi$
$73$	5.59734i	0.655119i	−0.944831	+	0.327560i	$0.893774\pi$
$74$	7.76183i	0.902294i
$75$	0	0
$76$	− 0.861298i	− 0.0987976i
$77$	0	0
$78$	0	0
$79$	17.0988	1.92377	0.961883	−	0.273462i	$-0.0881687\pi$
$79$	17.0988	1.92377	0.961883	+	0.273462i	$0.0881687\pi$
$80$	18.5354	2.07232
$81$	0	0
$82$	8.37315i	0.924660i
$83$	10.1622	1.11545	0.557726	−	0.830025i	$-0.311674\pi$
$83$	10.1622	1.11545	0.557726	+	0.830025i	$0.311674\pi$
$84$	0	0
$85$	−12.0988	−1.31230
$86$	13.2646i	1.43035i
$87$	0	0
$88$	1.35899	0.144869
$89$	11.5502	1.22431	0.612157	−	0.790736i	$-0.290302\pi$
$89$	11.5502	1.22431	0.612157	+	0.790736i	$0.290302\pi$
$90$	0	0
$91$	0	0
$92$	2.40393i	0.250627i
$93$	0	0
$94$	− 4.66954i	− 0.481627i
$95$	− 2.40393i	− 0.246638i
$96$	0	0
$97$	− 0.688287i	− 0.0698849i	−0.999389	−	0.0349425i	$-0.988875\pi$
$97$	− 0.688287i	− 0.0698849i	0.999389	−	0.0349425i	$-0.0111248\pi$
$98$	0	0
$99$	0	0
$100$	−12.9660	−1.29660
$101$	−6.58406	−0.655138	−0.327569	−	0.944827i	$-0.606229\pi$
$101$	−6.58406	−0.655138	−0.327569	+	0.944827i	$0.606229\pi$
$102$	0	0
$103$	9.34854i	0.921139i	0.887624	+	0.460570i	$0.152355\pi$
$103$	9.34854i	0.921139i	−0.887624	+	0.460570i	$0.847645\pi$
$104$	6.45864	0.633322
$105$	0	0
$106$	0.132822	0.0129009
$107$	− 15.7673i	− 1.52428i	−0.647411	−	0.762141i	$-0.724148\pi$
$107$	− 15.7673i	− 1.52428i	0.647411	−	0.762141i	$-0.275852\pi$
$108$	0	0
$109$	−14.4633	−1.38533	−0.692664	−	0.721260i	$-0.743563\pi$
$109$	−14.4633	−1.38533	−0.692664	+	0.721260i	$0.743563\pi$
$110$	−8.14310	−0.776414
$111$	0	0
$112$	0	0
$113$	− 19.8606i	− 1.86833i	−0.356840	−	0.934166i	$-0.616146\pi$
$113$	− 19.8606i	− 1.86833i	0.356840	−	0.934166i	$-0.383854\pi$
$114$	0	0
$115$	6.70948i	0.625662i
$116$	− 6.59607i	− 0.612430i
$117$	0	0
$118$	− 12.8126i	− 1.17950i
$119$	0	0
$120$	0	0
$121$	9.64101	0.876456
$122$	14.8318	1.34281
$123$	0	0
$124$	5.83685i	0.524165i
$125$	−17.1475	−1.53372
$126$	0	0
$127$	−8.90666	−0.790338	−0.395169	−	0.918608i	$-0.629314\pi$
$127$	−8.90666	−0.790338	−0.395169	+	0.918608i	$0.629314\pi$
$128$	− 8.85517i	− 0.782694i
$129$	0	0
$130$	−38.7003	−3.39424
$131$	6.75519	0.590204	0.295102	−	0.955466i	$-0.404646\pi$
$131$	6.75519	0.590204	0.295102	+	0.955466i	$0.404646\pi$
$132$	0	0
$133$	0	0
$134$	9.35245i	0.807928i
$135$	0	0
$136$	3.70361i	0.317582i
$137$	− 16.8606i	− 1.44050i	−0.693714	−	0.720251i	$-0.744027\pi$
$137$	− 16.8606i	− 1.44050i	0.693714	−	0.720251i	$-0.255973\pi$
$138$	0	0
$139$	− 0.813709i	− 0.0690179i	−0.999404	−	0.0345089i	$-0.989013\pi$
$139$	− 0.813709i	− 0.0690179i	0.999404	−	0.0345089i	$-0.0109867\pi$
$140$	0	0
$141$	0	0
$142$	8.73436	0.732971
$143$	−6.45864	−0.540098
$144$	0	0
$145$	− 18.4100i	− 1.52887i
$146$	−10.2669	−0.849693
$147$	0	0
$148$	−5.77383	−0.474606
$149$	8.43032i	0.690638i	0.938485	+	0.345319i	$0.112229\pi$
$149$	8.43032i	0.690638i	−0.938485	+	0.345319i	$0.887771\pi$
$150$	0	0
$151$	3.03948	0.247349	0.123675	−	0.992323i	$-0.460532\pi$
$151$	3.03948	0.247349	0.123675	+	0.992323i	$0.460532\pi$
$152$	−0.735875	−0.0596874
$153$	0	0
$154$	0	0
$155$	16.2910i	1.30852i
$156$	0	0
$157$	− 3.80824i	− 0.303931i	−0.988386	−	0.151966i	$-0.951440\pi$
$157$	− 3.80824i	− 0.303931i	0.988386	−	0.151966i	$-0.0485603\pi$
$158$	31.3634i	2.49514i
$159$	0	0
$160$	25.1195i	1.98587i
$161$	0	0
$162$	0	0
$163$	−11.0933	−0.868898	−0.434449	−	0.900696i	$-0.643057\pi$
$163$	−11.0933	−0.868898	−0.434449	+	0.900696i	$0.643057\pi$
$164$	−6.22858	−0.486371
$165$	0	0
$166$	18.6400i	1.44675i
$167$	3.70361	0.286594	0.143297	−	0.989680i	$-0.454230\pi$
$167$	3.70361	0.286594	0.143297	+	0.989680i	$0.454230\pi$
$168$	0	0
$169$	−17.6949	−1.36114
$170$	− 22.1921i	− 1.70206i
$171$	0	0
$172$	−9.86718	−0.752365
$173$	−9.63564	−0.732584	−0.366292	−	0.930500i	$-0.619373\pi$
$173$	−9.63564	−0.732584	−0.366292	+	0.930500i	$0.619373\pi$
$174$	0	0
$175$	0	0
$176$	5.67395i	0.427690i
$177$	0	0
$178$	21.1858i	1.58794i
$179$	− 18.4512i	− 1.37911i	−0.724233	−	0.689556i	$-0.757806\pi$
$179$	− 18.4512i	− 1.37911i	0.724233	−	0.689556i	$-0.242194\pi$
$180$	0	0
$181$	− 16.6209i	− 1.23542i	−0.786406	−	0.617710i	$-0.788060\pi$
$181$	− 16.6209i	− 1.23542i	0.786406	−	0.617710i	$-0.211940\pi$
$182$	0	0
$183$	0	0
$184$	2.05387	0.151413
$185$	−16.1151	−1.18480
$186$	0	0
$187$	− 3.70361i	− 0.270835i
$188$	3.47356	0.253335
$189$	0	0
$190$	4.40939	0.319890
$191$	− 7.16576i	− 0.518496i	−0.965811	−	0.259248i	$-0.916525\pi$
$191$	− 7.16576i	− 0.518496i	0.965811	−	0.259248i	$-0.0834747\pi$
$192$	0	0
$193$	−3.17776	−0.228740	−0.114370	−	0.993438i	$-0.536485\pi$
$193$	−3.17776	−0.228740	−0.114370	+	0.993438i	$0.536485\pi$
$194$	1.26249	0.0906411
$195$	0	0
$196$	0	0
$197$	1.93305i	0.137724i	0.997626	+	0.0688619i	$0.0219368\pi$
$197$	1.93305i	0.137724i	−0.997626	+	0.0688619i	$0.978063\pi$
$198$	0	0
$199$	− 8.66025i	− 0.613909i	−0.951724	−	0.306955i	$-0.900690\pi$
$199$	− 8.66025i	− 0.613909i	0.951724	−	0.306955i	$-0.0993100\pi$
$200$	11.0779i	0.783324i
$201$	0	0
$202$	− 12.0768i	− 0.849718i
$203$	0	0
$204$	0	0
$205$	−17.3843	−1.21417
$206$	−17.1475	−1.19472
$207$	0	0
$208$	26.9656i	1.86973i
$209$	0.735875	0.0509016
$210$	0	0
$211$	12.4183	0.854912	0.427456	−	0.904036i	$-0.359410\pi$
$211$	12.4183	0.854912	0.427456	+	0.904036i	$0.359410\pi$
$212$	0.0988033i	0.00678584i
$213$	0	0
$214$	28.9210	1.97700
$215$	−27.5398	−1.87820
$216$	0	0
$217$	0	0
$218$	− 26.5291i	− 1.79678i
$219$	0	0
$220$	− 6.05745i	− 0.408393i
$221$	− 17.6015i	− 1.18401i
$222$	0	0
$223$	18.9366i	1.26809i	0.773297	+	0.634044i	$0.218606\pi$
$223$	18.9366i	1.26809i	−0.773297	+	0.634044i	$0.781394\pi$
$224$	0	0
$225$	0	0
$226$	36.4292	2.42324
$227$	−5.72276	−0.379833	−0.189917	−	0.981800i	$-0.560822\pi$
$227$	−5.72276	−0.379833	−0.189917	+	0.981800i	$0.560822\pi$
$228$	0	0
$229$	− 11.1376i	− 0.735996i	−0.929827	−	0.367998i	$-0.880043\pi$
$229$	− 11.1376i	− 0.735996i	0.929827	−	0.367998i	$-0.119957\pi$
$230$	−12.3068	−0.811488
$231$	0	0
$232$	−5.63555	−0.369992
$233$	10.3370i	0.677198i	0.940931	+	0.338599i	$0.109953\pi$
$233$	10.3370i	0.677198i	−0.940931	+	0.338599i	$0.890047\pi$
$234$	0	0
$235$	9.69488	0.632424
$236$	9.53101	0.620416
$237$	0	0
$238$	0	0
$239$	− 16.5806i	− 1.07251i	−0.844056	−	0.536255i	$-0.819839\pi$
$239$	− 16.5806i	− 1.07251i	0.844056	−	0.536255i	$-0.180161\pi$
$240$	0	0
$241$	23.8932i	1.53910i	0.638587	+	0.769549i	$0.279519\pi$
$241$	23.8932i	1.53910i	−0.638587	+	0.769549i	$0.720481\pi$
$242$	17.6840i	1.13677i
$243$	0	0
$244$	11.0330i	0.706316i
$245$	0	0
$246$	0	0
$247$	3.49727	0.222526
$248$	4.98689	0.316668
$249$	0	0
$250$	− 31.4527i	− 1.98924i
$251$	−19.9233	−1.25755	−0.628774	−	0.777588i	$-0.716443\pi$
$251$	−19.9233	−1.25755	−0.628774	+	0.777588i	$0.716443\pi$
$252$	0	0
$253$	−2.05387	−0.129125
$254$	− 16.3370i	− 1.02507i
$255$	0	0
$256$	20.9714	1.31072
$257$	23.2049	1.44748	0.723742	−	0.690070i	$-0.242420\pi$
$257$	23.2049	1.44748	0.723742	+	0.690070i	$0.242420\pi$
$258$	0	0
$259$	0	0
$260$	− 28.7882i	− 1.78537i
$261$	0	0
$262$	12.3907i	0.765498i
$263$	9.62246i	0.593347i	0.954979	+	0.296673i	$0.0958772\pi$
$263$	9.62246i	0.593347i	−0.954979	+	0.296673i	$0.904123\pi$
$264$	0	0
$265$	0.275765i	0.0169401i
$266$	0	0
$267$	0	0
$268$	−6.95705	−0.424970
$269$	20.8303	1.27005	0.635024	−	0.772493i	$-0.280990\pi$
$269$	20.8303	1.27005	0.635024	+	0.772493i	$0.280990\pi$
$270$	0	0
$271$	− 19.6838i	− 1.19571i	−0.801606	−	0.597853i	$-0.796021\pi$
$271$	− 19.6838i	− 1.19571i	0.801606	−	0.597853i	$-0.203979\pi$
$272$	−15.4630	−0.937584
$273$	0	0
$274$	30.9265	1.86834
$275$	− 11.0779i	− 0.668021i
$276$	0	0
$277$	8.31604	0.499662	0.249831	−	0.968289i	$-0.419625\pi$
$277$	8.31604	0.499662	0.249831	+	0.968289i	$0.419625\pi$
$278$	1.49254	0.0895166
$279$	0	0
$280$	0	0
$281$	− 22.7618i	− 1.35786i	−0.734204	−	0.678928i	$-0.762445\pi$
$281$	− 22.7618i	− 1.35786i	0.734204	−	0.678928i	$-0.237555\pi$
$282$	0	0
$283$	− 17.0334i	− 1.01253i	−0.862378	−	0.506266i	$-0.831026\pi$
$283$	− 17.0334i	− 1.01253i	0.862378	−	0.506266i	$-0.168974\pi$
$284$	6.49727i	0.385542i
$285$	0	0
$286$	− 11.8467i	− 0.700511i
$287$	0	0
$288$	0	0
$289$	−6.90666	−0.406274
$290$	33.7684	1.98295
$291$	0	0
$292$	− 7.63728i	− 0.446938i
$293$	−21.4159	−1.25113	−0.625564	−	0.780173i	$-0.715131\pi$
$293$	−21.4159	−1.25113	−0.625564	+	0.780173i	$0.715131\pi$
$294$	0	0
$295$	26.6015	1.54880
$296$	4.93305i	0.286728i
$297$	0	0
$298$	−15.4633	−0.895762
$299$	−9.76106	−0.564497
$300$	0	0
$301$	0	0
$302$	5.57514i	0.320813i
$303$	0	0
$304$	− 3.07237i	− 0.176212i
$305$	30.7937i	1.76324i
$306$	0	0
$307$	19.4537i	1.11028i	0.831756	+	0.555142i	$0.187336\pi$
$307$	19.4537i	1.11028i	−0.831756	+	0.555142i	$0.812664\pi$
$308$	0	0
$309$	0	0
$310$	−29.8816	−1.69716
$311$	−6.81412	−0.386393	−0.193197	−	0.981160i	$-0.561885\pi$
$311$	−6.81412	−0.386393	−0.193197	+	0.981160i	$0.561885\pi$
$312$	0	0
$313$	16.6417i	0.940643i	0.882495	+	0.470322i	$0.155862\pi$
$313$	16.6417i	0.940643i	−0.882495	+	0.470322i	$0.844138\pi$
$314$	6.98525	0.394200
$315$	0	0
$316$	−23.3304	−1.31244
$317$	− 5.07787i	− 0.285202i	−0.989780	−	0.142601i	$-0.954453\pi$
$317$	− 5.07787i	− 0.285202i	0.989780	−	0.142601i	$-0.0455465\pi$
$318$	0	0
$319$	5.63555	0.315530
$320$	−9.00440	−0.503361
$321$	0	0
$322$	0	0
$323$	2.00546i	0.111587i
$324$	0	0
$325$	− 52.6479i	− 2.92038i
$326$	− 20.3479i	− 1.12697i
$327$	0	0
$328$	5.32157i	0.293835i
$329$	0	0
$330$	0	0
$331$	16.3304	0.897602	0.448801	−	0.893632i	$-0.351851\pi$
$331$	16.3304	0.897602	0.448801	+	0.893632i	$0.351851\pi$
$332$	−13.8659	−0.760988
$333$	0	0
$334$	6.79333i	0.371714i
$335$	−19.4175	−1.06089
$336$	0	0
$337$	−3.95506	−0.215446	−0.107723	−	0.994181i	$-0.534356\pi$
$337$	−3.95506	−0.215446	−0.107723	+	0.994181i	$0.534356\pi$
$338$	− 32.4567i	− 1.76541i
$339$	0	0
$340$	16.5082	0.895282
$341$	−4.98689	−0.270055
$342$	0	0
$343$	0	0
$344$	8.43032i	0.454532i
$345$	0	0
$346$	− 17.6741i	− 0.950166i
$347$	7.33697i	0.393869i	0.980417	+	0.196935i	$0.0630987\pi$
$347$	7.33697i	0.393869i	−0.980417	+	0.196935i	$0.936901\pi$
$348$	0	0
$349$	− 25.5093i	− 1.36548i	−0.730660	−	0.682741i	$-0.760788\pi$
$349$	− 25.5093i	− 1.36548i	0.730660	−	0.682741i	$-0.239212\pi$
$350$	0	0
$351$	0	0
$352$	−7.68942	−0.409847
$353$	0.0207896	0.00110652	0.000553260	−	1.00000i	$-0.499824\pi$
$353$	0.0207896	0.00110652	0.000553260		1.00000i	$0.499824\pi$
$354$	0	0
$355$	18.1342i	0.962464i
$356$	−15.7596	−0.835257
$357$	0	0
$358$	33.8441	1.78872
$359$	36.2240i	1.91183i	0.293644	+	0.955915i	$0.405132\pi$
$359$	36.2240i	1.91183i	−0.293644	+	0.955915i	$0.594868\pi$
$360$	0	0
$361$	18.6015	0.979028
$362$	30.4867	1.60235
$363$	0	0
$364$	0	0
$365$	− 21.3160i	− 1.11573i
$366$	0	0
$367$	− 8.87897i	− 0.463479i	−0.972778	−	0.231739i	$-0.925558\pi$
$367$	− 8.87897i	− 0.463479i	0.972778	−	0.231739i	$-0.0744416\pi$
$368$	8.57514i	0.447010i
$369$	0	0
$370$	− 29.5590i	− 1.53670i
$371$	0	0
$372$	0	0
$373$	−12.8188	−0.663731	−0.331865	−	0.943327i	$-0.607678\pi$
$373$	−12.8188	−0.663731	−0.331865	+	0.943327i	$0.607678\pi$
$374$	6.79333	0.351274
$375$	0	0
$376$	− 2.96774i	− 0.153049i
$377$	26.7831	1.37940
$378$	0	0
$379$	3.13828	0.161203	0.0806013	−	0.996746i	$-0.474316\pi$
$379$	3.13828	0.161203	0.0806013	+	0.996746i	$0.474316\pi$
$380$	3.28003i	0.168262i
$381$	0	0
$382$	13.1437	0.672492
$383$	−27.5190	−1.40616	−0.703078	−	0.711113i	$-0.748192\pi$
$383$	−27.5190	−1.40616	−0.703078	+	0.711113i	$0.748192\pi$
$384$	0	0
$385$	0	0
$386$	− 5.82878i	− 0.296677i
$387$	0	0
$388$	0.939131i	0.0476772i
$389$	18.2436i	0.924989i	0.886622	+	0.462494i	$0.153045\pi$
$389$	18.2436i	0.924989i	−0.886622	+	0.462494i	$0.846955\pi$
$390$	0	0
$391$	− 5.59734i	− 0.283070i
$392$	0	0
$393$	0	0
$394$	−3.54568	−0.178629
$395$	−65.1164	−3.27636
$396$	0	0
$397$	− 1.16919i	− 0.0586798i	−0.999569	−	0.0293399i	$-0.990659\pi$
$397$	− 1.16919i	− 0.0586798i	0.999569	−	0.0293399i	$-0.00934052\pi$
$398$	15.8850	0.796244
$399$	0	0
$400$	−46.2515	−2.31257
$401$	− 13.9594i	− 0.697101i	−0.937290	−	0.348551i	$-0.886674\pi$
$401$	− 13.9594i	− 0.697101i	0.937290	−	0.348551i	$-0.113326\pi$
$402$	0	0
$403$	−23.7003	−1.18060
$404$	8.98361	0.446951
$405$	0	0
$406$	0	0
$407$	− 4.93305i	− 0.244522i
$408$	0	0
$409$	20.3721i	1.00733i	0.863898	+	0.503667i	$0.168016\pi$
$409$	20.3721i	1.00733i	−0.863898	+	0.503667i	$0.831984\pi$
$410$	− 31.8870i	− 1.57479i
$411$	0	0
$412$	− 12.7556i	− 0.628423i
$413$	0	0
$414$	0	0
$415$	−38.7003	−1.89972
$416$	−36.5442	−1.79173
$417$	0	0
$418$	1.34977i	0.0660196i
$419$	25.5207	1.24677	0.623383	−	0.781917i	$-0.285758\pi$
$419$	25.5207	1.24677	0.623383	+	0.781917i	$0.285758\pi$
$420$	0	0
$421$	0.0933442	0.00454932	0.00227466	−	0.999997i	$-0.499276\pi$
$421$	0.0933442	0.00454932	0.00227466	+	0.999997i	$0.499276\pi$
$422$	22.7782i	1.10883i
$423$	0	0
$424$	0.0844155	0.00409958
$425$	30.1902	1.46444
$426$	0	0
$427$	0	0
$428$	21.5136i	1.03990i
$429$	0	0
$430$	− 50.5147i	− 2.43603i
$431$	− 33.1712i	− 1.59780i	−0.601463	−	0.798901i	$-0.705415\pi$
$431$	− 33.1712i	− 1.59780i	0.601463	−	0.798901i	$-0.294585\pi$
$432$	0	0
$433$	17.9518i	0.862706i	0.902183	+	0.431353i	$0.141964\pi$
$433$	17.9518i	0.862706i	−0.902183	+	0.431353i	$0.858036\pi$
$434$	0	0
$435$	0	0
$436$	19.7344	0.945104
$437$	1.11214	0.0532010
$438$	0	0
$439$	10.0368i	0.479032i	0.970892	+	0.239516i	$0.0769887\pi$
$439$	10.0368i	0.479032i	−0.970892	+	0.239516i	$0.923011\pi$
$440$	−5.17536	−0.246726
$441$	0	0
$442$	32.2855	1.53566
$443$	− 26.3579i	− 1.25230i	−0.779702	−	0.626151i	$-0.784629\pi$
$443$	− 26.3579i	− 1.25230i	0.779702	−	0.626151i	$-0.215371\pi$
$444$	0	0
$445$	−43.9858	−2.08513
$446$	−34.7343	−1.64472
$447$	0	0
$448$	0	0
$449$	24.0264i	1.13388i	0.823761	+	0.566938i	$0.191872\pi$
$449$	24.0264i	1.13388i	−0.823761	+	0.566938i	$0.808128\pi$
$450$	0	0
$451$	− 5.32157i	− 0.250583i
$452$	27.0988i	1.27462i
$453$	0	0
$454$	− 10.4969i	− 0.492646i
$455$	0	0
$456$	0	0
$457$	20.8222	0.974023	0.487012	−	0.873395i	$-0.338087\pi$
$457$	20.8222	0.974023	0.487012	+	0.873395i	$0.338087\pi$
$458$	20.4291	0.954591
$459$	0	0
$460$	− 9.15474i	− 0.426842i
$461$	−8.35237	−0.389008	−0.194504	−	0.980902i	$-0.562310\pi$
$461$	−8.35237	−0.389008	−0.194504	+	0.980902i	$0.562310\pi$
$462$	0	0
$463$	11.8133	0.549011	0.274506	−	0.961586i	$-0.411486\pi$
$463$	11.8133	0.549011	0.274506	+	0.961586i	$0.411486\pi$
$464$	− 23.5291i	− 1.09231i
$465$	0	0
$466$	−18.9605	−0.878329
$467$	−29.1578	−1.34926	−0.674630	−	0.738156i	$-0.735697\pi$
$467$	−29.1578	−1.34926	−0.674630	+	0.738156i	$0.735697\pi$
$468$	0	0
$469$	0	0
$470$	17.7828i	0.820258i
$471$	0	0
$472$	− 8.14310i	− 0.374817i
$473$	− 8.43032i	− 0.387626i
$474$	0	0
$475$	5.99853i	0.275231i
$476$	0	0
$477$	0	0
$478$	30.4129	1.39105
$479$	40.5236	1.85157	0.925785	−	0.378051i	$-0.123406\pi$
$479$	40.5236	1.85157	0.925785	+	0.378051i	$0.123406\pi$
$480$	0	0
$481$	− 23.4445i	− 1.06898i
$482$	−43.8260	−1.99622
$483$	0	0
$484$	−13.1547	−0.597939
$485$	2.62116i	0.119021i
$486$	0	0
$487$	12.4183	0.562728	0.281364	−	0.959601i	$-0.409213\pi$
$487$	12.4183	0.562728	0.281364	+	0.959601i	$0.409213\pi$
$488$	9.42637	0.426712
$489$	0	0
$490$	0	0
$491$	− 6.17122i	− 0.278503i	−0.990257	−	0.139252i	$-0.955530\pi$
$491$	− 6.17122i	− 0.278503i	0.990257	−	0.139252i	$-0.0444697\pi$
$492$	0	0
$493$	15.3584i	0.691708i
$494$	6.41484i	0.288617i
$495$	0	0
$496$	20.8209i	0.934884i
$497$	0	0
$498$	0	0
$499$	−2.96052	−0.132531	−0.0662656	−	0.997802i	$-0.521108\pi$
$499$	−2.96052	−0.132531	−0.0662656	+	0.997802i	$0.521108\pi$
$500$	23.3969	1.04634
$501$	0	0
$502$	− 36.5442i	− 1.63105i
$503$	−4.33485	−0.193282	−0.0966408	−	0.995319i	$-0.530810\pi$
$503$	−4.33485	−0.193282	−0.0966408	+	0.995319i	$0.530810\pi$
$504$	0	0
$505$	25.0737	1.11577
$506$	− 3.76729i	− 0.167476i
$507$	0	0
$508$	12.1527	0.539188
$509$	28.9485	1.28312	0.641560	−	0.767073i	$-0.278288\pi$
$509$	28.9485	1.28312	0.641560	+	0.767073i	$0.278288\pi$
$510$	0	0
$511$	0	0
$512$	20.7564i	0.917311i
$513$	0	0
$514$	42.5635i	1.87740i
$515$	− 35.6015i	− 1.56879i
$516$	0	0
$517$	2.96774i	0.130521i
$518$	0	0
$519$	0	0
$520$	−24.5961	−1.07861
$521$	12.1606	0.532766	0.266383	−	0.963867i	$-0.414171\pi$
$521$	12.1606	0.532766	0.266383	+	0.963867i	$0.414171\pi$
$522$	0	0
$523$	38.2857i	1.67412i	0.547113	+	0.837059i	$0.315727\pi$
$523$	38.2857i	1.67412i	−0.547113	+	0.837059i	$0.684273\pi$
$524$	−9.21711	−0.402651
$525$	0	0
$526$	−17.6499	−0.769574
$527$	− 13.5906i	− 0.592016i
$528$	0	0
$529$	19.8960	0.865041
$530$	−0.505820	−0.0219714
$531$	0	0
$532$	0	0
$533$	− 25.2910i	− 1.09547i
$534$	0	0
$535$	60.0457i	2.59600i
$536$	5.94396i	0.256740i
$537$	0	0
$538$	38.2079i	1.64726i
$539$	0	0
$540$	0	0
$541$	0.508189	0.0218487	0.0109244	−	0.999940i	$-0.496523\pi$
$541$	0.508189	0.0218487	0.0109244	+	0.999940i	$0.496523\pi$
$542$	36.1049	1.55084
$543$	0	0
$544$	− 20.9557i	− 0.898470i
$545$	55.0796	2.35935
$546$	0	0
$547$	−40.2964	−1.72295	−0.861475	−	0.507800i	$-0.830459\pi$
$547$	−40.2964	−1.72295	−0.861475	+	0.507800i	$0.830459\pi$
$548$	23.0055i	0.982745i
$549$	0	0
$550$	20.3195	0.866427
$551$	−3.05158	−0.130002
$552$	0	0
$553$	0	0
$554$	15.2536i	0.648065i
$555$	0	0
$556$	1.11026i	0.0470857i
$557$	43.4358i	1.84043i	0.391410	+	0.920216i	$0.371987\pi$
$557$	43.4358i	1.84043i	−0.391410	+	0.920216i	$0.628013\pi$
$558$	0	0
$559$	− 40.0653i	− 1.69458i
$560$	0	0
$561$	0	0
$562$	41.7507	1.76115
$563$	6.33322	0.266913	0.133457	−	0.991055i	$-0.457392\pi$
$563$	6.33322	0.266913	0.133457	+	0.991055i	$0.457392\pi$
$564$	0	0
$565$	75.6342i	3.18195i
$566$	31.2434	1.31326
$567$	0	0
$568$	5.55114	0.232921
$569$	8.23271i	0.345133i	0.984998	+	0.172567i	$0.0552060\pi$
$569$	8.23271i	0.345133i	−0.984998	+	0.172567i	$0.944794\pi$
$570$	0	0
$571$	8.49727	0.355600	0.177800	−	0.984067i	$-0.443102\pi$
$571$	8.49727	0.355600	0.177800	+	0.984067i	$0.443102\pi$
$572$	8.81248	0.368468
$573$	0	0
$574$	0	0
$575$	− 16.7422i	− 0.698198i
$576$	0	0
$577$	39.4798i	1.64357i	0.569801	+	0.821783i	$0.307020\pi$
$577$	39.4798i	1.64357i	−0.569801	+	0.821783i	$0.692980\pi$
$578$	− 12.6685i	− 0.526940i
$579$	0	0
$580$	25.1195i	1.04303i
$581$	0	0
$582$	0	0
$583$	−0.0844155	−0.00349613
$584$	−6.52514	−0.270012
$585$	0	0
$586$	− 39.2819i	− 1.62272i
$587$	−24.1120	−0.995207	−0.497603	−	0.867405i	$-0.665787\pi$
$587$	−24.1120	−0.995207	−0.497603	+	0.867405i	$0.665787\pi$
$588$	0	0
$589$	2.70034	0.111265
$590$	48.7937i	2.00880i
$591$	0	0
$592$	−20.5961	−0.846493
$593$	−18.9366	−0.777633	−0.388816	−	0.921315i	$-0.627116\pi$
$593$	−18.9366	−0.777633	−0.388816	+	0.921315i	$0.627116\pi$
$594$	0	0
$595$	0	0
$596$	− 11.5027i	− 0.471170i
$597$	0	0
$598$	− 17.9042i	− 0.732156i
$599$	− 34.6794i	− 1.41696i	−0.705730	−	0.708481i	$-0.749381\pi$
$599$	− 34.6794i	− 1.41696i	0.705730	−	0.708481i	$-0.250619\pi$
$600$	0	0
$601$	13.2709i	0.541330i	0.962674	+	0.270665i	$0.0872436\pi$
$601$	13.2709i	0.541330i	−0.962674	+	0.270665i	$0.912756\pi$
$602$	0	0
$603$	0	0
$604$	−4.14721	−0.168748
$605$	−36.7153	−1.49269
$606$	0	0
$607$	25.0148i	1.01532i	0.861557	+	0.507660i	$0.169490\pi$
$607$	25.0148i	1.01532i	−0.861557	+	0.507660i	$0.830510\pi$
$608$	4.16372	0.168861
$609$	0	0
$610$	−56.4831	−2.28693
$611$	14.1043i	0.570597i
$612$	0	0
$613$	−44.1097	−1.78157	−0.890787	−	0.454420i	$-0.849846\pi$
$613$	−44.1097	−1.78157	−0.890787	+	0.454420i	$0.849846\pi$
$614$	−35.6829	−1.44004
$615$	0	0
$616$	0	0
$617$	19.3370i	0.778477i	0.921137	+	0.389239i	$0.127262\pi$
$617$	19.3370i	0.778477i	−0.921137	+	0.389239i	$0.872738\pi$
$618$	0	0
$619$	11.0598i	0.444531i	0.974986	+	0.222265i	$0.0713452\pi$
$619$	11.0598i	0.444531i	−0.974986	+	0.222265i	$0.928655\pi$
$620$	− 22.2282i	− 0.892704i
$621$	0	0
$622$	− 12.4987i	− 0.501154i
$623$	0	0
$624$	0	0
$625$	17.7882	0.711529
$626$	−30.5249	−1.22002
$627$	0	0
$628$	5.19615i	0.207349i
$629$	13.4439	0.536043
$630$	0	0
$631$	−27.6015	−1.09880	−0.549400	−	0.835560i	$-0.685144\pi$
$631$	−27.6015	−1.09880	−0.549400	+	0.835560i	$0.685144\pi$
$632$	19.9330i	0.792894i
$633$	0	0
$634$	9.31405	0.369908
$635$	33.9187	1.34602
$636$	0	0
$637$	0	0
$638$	10.3370i	0.409245i
$639$	0	0
$640$	33.7227i	1.33301i
$641$	− 6.07241i	− 0.239846i	−0.992783	−	0.119923i	$-0.961735\pi$
$641$	− 6.07241i	− 0.239846i	0.992783	−	0.119923i	$-0.0382648\pi$
$642$	0	0
$643$	− 13.2709i	− 0.523352i	−0.965156	−	0.261676i	$-0.915725\pi$
$643$	− 13.2709i	− 0.523352i	0.965156	−	0.261676i	$-0.0842752\pi$
$644$	0	0
$645$	0	0
$646$	−3.67850	−0.144729
$647$	30.7376	1.20842	0.604210	−	0.796825i	$-0.293489\pi$
$647$	30.7376	1.20842	0.604210	+	0.796825i	$0.293489\pi$
$648$	0	0
$649$	8.14310i	0.319645i
$650$	96.5691	3.78775
$651$	0	0
$652$	15.1363	0.592783
$653$	37.2185i	1.45647i	0.685325	+	0.728237i	$0.259660\pi$
$653$	37.2185i	1.45647i	−0.685325	+	0.728237i	$0.740340\pi$
$654$	0	0
$655$	−25.7254	−1.00518
$656$	−22.2182	−0.867476
$657$	0	0
$658$	0	0
$659$	46.2545i	1.80182i	0.434005	+	0.900911i	$0.357100\pi$
$659$	46.2545i	1.80182i	−0.434005	+	0.900911i	$0.642900\pi$
$660$	0	0
$661$	13.1889i	0.512989i	0.966546	+	0.256495i	$0.0825676\pi$
$661$	13.1889i	0.512989i	−0.966546	+	0.256495i	$0.917432\pi$
$662$	29.9540i	1.16419i
$663$	0	0
$664$	11.8467i	0.459742i
$665$	0	0
$666$	0	0
$667$	8.51712	0.329784
$668$	−5.05339	−0.195521
$669$	0	0
$670$	− 35.6164i	− 1.37598i
$671$	−9.42637	−0.363901
$672$	0	0
$673$	40.9265	1.57760	0.788800	−	0.614649i	$-0.210703\pi$
$673$	40.9265	1.57760	0.788800	+	0.614649i	$0.210703\pi$
$674$	− 7.25455i	− 0.279435i
$675$	0	0
$676$	24.1437	0.928605
$677$	1.61796	0.0621834	0.0310917	−	0.999517i	$-0.490102\pi$
$677$	1.61796	0.0621834	0.0310917	+	0.999517i	$0.490102\pi$
$678$	0	0
$679$	0	0
$680$	− 14.1043i	− 0.540874i
$681$	0	0
$682$	− 9.14716i	− 0.350263i
$683$	− 26.5555i	− 1.01612i	−0.861322	−	0.508059i	$-0.830363\pi$
$683$	− 26.5555i	− 1.01612i	0.861322	−	0.508059i	$-0.169637\pi$
$684$	0	0
$685$	64.2094i	2.45332i
$686$	0	0
$687$	0	0
$688$	−35.1976	−1.34190
$689$	−0.401187	−0.0152840
$690$	0	0
$691$	− 37.6337i	− 1.43165i	−0.698278	−	0.715826i	$-0.746050\pi$
$691$	− 37.6337i	− 1.43165i	0.698278	−	0.715826i	$-0.253950\pi$
$692$	13.1473	0.499787
$693$	0	0
$694$	−13.4578	−0.510851
$695$	3.09880i	0.117544i
$696$	0	0
$697$	14.5027	0.549330
$698$	46.7903	1.77104
$699$	0	0
$700$	0	0
$701$	24.3228i	0.918659i	0.888266	+	0.459330i	$0.151910\pi$
$701$	24.3228i	0.918659i	−0.888266	+	0.459330i	$0.848090\pi$
$702$	0	0
$703$	2.67118i	0.100746i
$704$	− 2.75637i	− 0.103885i
$705$	0	0
$706$	0.0381332i	0.00143516i
$707$	0	0
$708$	0	0
$709$	1.03402	0.0388334	0.0194167	−	0.999811i	$-0.493819\pi$
$709$	1.03402	0.0388334	0.0194167	+	0.999811i	$0.493819\pi$
$710$	−33.2626	−1.24832
$711$	0	0
$712$	13.4647i	0.504610i
$713$	−7.53678	−0.282255
$714$	0	0
$715$	24.5961	0.919841
$716$	25.1758i	0.940863i
$717$	0	0
$718$	−66.4436	−2.47965
$719$	16.7463	0.624532	0.312266	−	0.949995i	$-0.398912\pi$
$719$	16.7463	0.624532	0.312266	+	0.949995i	$0.398912\pi$
$720$	0	0
$721$	0	0
$722$	34.1197i	1.26981i
$723$	0	0
$724$	22.6783i	0.842834i
$725$	45.9385i	1.70611i
$726$	0	0
$727$	− 32.8976i	− 1.22011i	−0.792361	−	0.610053i	$-0.791148\pi$
$727$	− 32.8976i	− 1.22011i	0.792361	−	0.610053i	$-0.208852\pi$
$728$	0	0
$729$	0	0
$730$	39.0988	1.44711
$731$	22.9749	0.849757
$732$	0	0
$733$	19.9120i	0.735466i	0.929931	+	0.367733i	$0.119866\pi$
$733$	19.9120i	0.735466i	−0.929931	+	0.367733i	$0.880134\pi$
$734$	16.2862	0.601135
$735$	0	0
$736$	−11.6212	−0.428362
$737$	− 5.94396i	− 0.218949i
$738$	0	0
$739$	−50.6698	−1.86392	−0.931959	−	0.362563i	$-0.881902\pi$
$739$	−50.6698	−1.86392	−0.931959	+	0.362563i	$0.881902\pi$
$740$	21.9882	0.808301
$741$	0	0
$742$	0	0
$743$	44.2491i	1.62334i	0.584115	+	0.811671i	$0.301442\pi$
$743$	44.2491i	1.62334i	−0.584115	+	0.811671i	$0.698558\pi$
$744$	0	0
$745$	− 32.1047i	− 1.17623i
$746$	− 23.5127i	− 0.860863i
$747$	0	0
$748$	5.05339i	0.184770i
$749$	0	0
$750$	0	0
$751$	2.34460	0.0855557	0.0427779	−	0.999085i	$-0.486379\pi$
$751$	2.34460	0.0855557	0.0427779	+	0.999085i	$0.486379\pi$
$752$	12.3907	0.451841
$753$	0	0
$754$	49.1268i	1.78909i
$755$	−11.5751	−0.421260
$756$	0	0
$757$	−24.8530	−0.903298	−0.451649	−	0.892196i	$-0.649164\pi$
$757$	−24.8530	−0.903298	−0.451649	+	0.892196i	$0.649164\pi$
$758$	5.75637i	0.209081i
$759$	0	0
$760$	2.80239	0.101653
$761$	−0.882087	−0.0319756	−0.0159878	−	0.999872i	$-0.505089\pi$
$761$	−0.882087	−0.0319756	−0.0159878	+	0.999872i	$0.505089\pi$
$762$	0	0
$763$	0	0
$764$	9.77730i	0.353730i
$765$	0	0
$766$	− 50.4766i	− 1.82379i
$767$	38.7003i	1.39739i
$768$	0	0
$769$	20.7049i	0.746638i	0.927703	+	0.373319i	$0.121780\pi$
$769$	20.7049i	0.746638i	−0.927703	+	0.373319i	$0.878220\pi$
$770$	0	0
$771$	0	0
$772$	4.33589	0.156052
$773$	9.40558	0.338295	0.169148	−	0.985591i	$-0.445899\pi$
$773$	9.40558	0.338295	0.169148	+	0.985591i	$0.445899\pi$
$774$	0	0
$775$	− 40.6509i	− 1.46022i
$776$	0.802375	0.0288036
$777$	0	0
$778$	−33.4633	−1.19972
$779$	2.88157i	0.103243i
$780$	0	0
$781$	−5.55114	−0.198635
$782$	10.2669	0.367143
$783$	0	0
$784$	0	0
$785$	14.5027i	0.517625i
$786$	0	0
$787$	− 25.1557i	− 0.896705i	−0.893857	−	0.448352i	$-0.852011\pi$
$787$	− 25.1557i	− 0.896705i	0.893857	−	0.448352i	$-0.147989\pi$
$788$	− 2.63754i	− 0.0939585i
$789$	0	0
$790$	− 119.439i	− 4.24946i
$791$	0	0
$792$	0	0
$793$	−44.7991	−1.59086
$794$	2.14457	0.0761080
$795$	0	0
$796$	11.8165i	0.418824i
$797$	30.7168	1.08805	0.544023	−	0.839071i	$-0.316901\pi$
$797$	30.7168	1.08805	0.544023	+	0.839071i	$0.316901\pi$
$798$	0	0
$799$	−8.08789	−0.286129
$800$	− 62.6807i	− 2.21610i
$801$	0	0
$802$	25.6050	0.904144
$803$	6.52514	0.230267
$804$	0	0
$805$	0	0
$806$	− 43.4722i	− 1.53124i
$807$	0	0
$808$	− 7.67541i	− 0.270020i
$809$	− 14.2755i	− 0.501899i	−0.968000	−	0.250950i	$-0.919257\pi$
$809$	− 14.2755i	− 0.501899i	0.968000	−	0.250950i	$-0.0807428\pi$
$810$	0	0
$811$	− 10.0160i	− 0.351711i	−0.984416	−	0.175855i	$-0.943731\pi$
$811$	− 10.0160i	− 0.351711i	0.984416	−	0.175855i	$-0.0562691\pi$
$812$	0	0
$813$	0	0
$814$	9.04841	0.317146
$815$	42.2462	1.47982
$816$	0	0
$817$	4.56491i	0.159706i
$818$	−37.3674	−1.30652
$819$	0	0
$820$	23.7200	0.828337
$821$	12.4609i	0.434887i	0.976073	+	0.217444i	$0.0697718\pi$
$821$	12.4609i	0.434887i	−0.976073	+	0.217444i	$0.930228\pi$
$822$	0	0
$823$	−4.59607	−0.160209	−0.0801045	−	0.996786i	$-0.525525\pi$
$823$	−4.59607	−0.160209	−0.0801045	+	0.996786i	$0.525525\pi$
$824$	−10.8981	−0.379654
$825$	0	0
$826$	0	0
$827$	37.9330i	1.31906i	0.751678	+	0.659531i	$0.229245\pi$
$827$	37.9330i	1.31906i	−0.751678	+	0.659531i	$0.770755\pi$
$828$	0	0
$829$	− 36.7992i	− 1.27809i	−0.769170	−	0.639044i	$-0.779330\pi$
$829$	− 36.7992i	− 1.27809i	0.769170	−	0.639044i	$-0.220670\pi$
$830$	− 70.9858i	− 2.46395i
$831$	0	0
$832$	− 13.0997i	− 0.454152i
$833$	0	0
$834$	0	0
$835$	−14.1043	−0.488098
$836$	−1.00406	−0.0347263
$837$	0	0
$838$	46.8111i	1.61706i
$839$	−21.3569	−0.737323	−0.368662	−	0.929564i	$-0.620184\pi$
$839$	−21.3569	−0.737323	−0.368662	+	0.929564i	$0.620184\pi$
$840$	0	0
$841$	5.63009	0.194141
$842$	0.171216i	0.00590049i
$843$	0	0
$844$	−16.9441	−0.583242
$845$	67.3864	2.31816
$846$	0	0
$847$	0	0
$848$	0.352445i	0.0121030i
$849$	0	0
$850$	55.3762i	1.89939i
$851$	− 7.45541i	− 0.255568i
$852$	0	0
$853$	− 49.5031i	− 1.69495i	−0.530833	−	0.847476i	$-0.678121\pi$
$853$	− 49.5031i	− 1.69495i	0.530833	−	0.847476i	$-0.321879\pi$
$854$	0	0
$855$	0	0
$856$	18.3808	0.628244
$857$	−14.0162	−0.478784	−0.239392	−	0.970923i	$-0.576948\pi$
$857$	−14.0162	−0.478784	−0.239392	+	0.970923i	$0.576948\pi$
$858$	0	0
$859$	− 1.21490i	− 0.0414517i	−0.999785	−	0.0207259i	$-0.993402\pi$
$859$	− 1.21490i	− 0.0414517i	0.999785	−	0.0207259i	$-0.00659772\pi$
$860$	37.5766	1.28135
$861$	0	0
$862$	60.8441	2.07236
$863$	− 36.0724i	− 1.22792i	−0.789337	−	0.613960i	$-0.789576\pi$
$863$	− 36.0724i	− 1.22792i	0.789337	−	0.613960i	$-0.210424\pi$
$864$	0	0
$865$	36.6949	1.24766
$866$	−32.9279	−1.11893
$867$	0	0
$868$	0	0
$869$	− 19.9330i	− 0.676182i
$870$	0	0
$871$	− 28.2489i	− 0.957177i
$872$	− 16.8606i	− 0.570973i
$873$	0	0
$874$	2.03994i	0.0690020i
$875$	0	0
$876$	0	0
$877$	−11.1921	−0.377932	−0.188966	−	0.981984i	$-0.560514\pi$
$877$	−11.1921	−0.377932	−0.188966	+	0.981984i	$0.560514\pi$
$878$	−18.4100	−0.621307
$879$	0	0
$880$	− 21.6078i	− 0.728398i
$881$	−4.16372	−0.140279	−0.0701397	−	0.997537i	$-0.522345\pi$
$881$	−4.16372	−0.140279	−0.0701397	+	0.997537i	$0.522345\pi$
$882$	0	0
$883$	−38.7433	−1.30382	−0.651908	−	0.758298i	$-0.726031\pi$
$883$	−38.7433	−1.30382	−0.651908	+	0.758298i	$0.726031\pi$
$884$	24.0164i	0.807758i
$885$	0	0
$886$	48.3468	1.62424
$887$	−38.0824	−1.27868	−0.639342	−	0.768923i	$-0.720793\pi$
$887$	−38.0824	−1.27868	−0.639342	+	0.768923i	$0.720793\pi$
$888$	0	0
$889$	0	0
$890$	− 80.6807i	− 2.70442i
$891$	0	0
$892$	− 25.8380i	− 0.865120i
$893$	− 1.60699i	− 0.0537759i
$894$	0	0
$895$	70.2669i	2.34876i
$896$	0	0
$897$	0	0
$898$	−44.0702	−1.47064
$899$	20.6800	0.689716
$900$	0	0
$901$	− 0.230055i	− 0.00766425i
$902$	9.76106	0.325008
$903$	0	0
$904$	23.1527	0.770046
$905$	63.2964i	2.10404i
$906$	0	0
$907$	1.54767	0.0513894	0.0256947	−	0.999670i	$-0.491820\pi$
$907$	1.54767	0.0513894	0.0256947	+	0.999670i	$0.491820\pi$
$908$	7.80841	0.259131
$909$	0	0
$910$	0	0
$911$	− 38.9704i	− 1.29115i	−0.763699	−	0.645573i	$-0.776619\pi$
$911$	− 38.9704i	− 1.29115i	0.763699	−	0.645573i	$-0.223381\pi$
$912$	0	0
$913$	− 11.8467i	− 0.392069i
$914$	38.1931i	1.26331i
$915$	0	0
$916$	15.1967i	0.502114i
$917$	0	0
$918$	0	0
$919$	11.7398	0.387261	0.193630	−	0.981075i	$-0.437974\pi$
$919$	11.7398	0.387261	0.193630	+	0.981075i	$0.437974\pi$
$920$	−7.82162	−0.257871
$921$	0	0
$922$	− 15.3203i	− 0.504546i
$923$	−26.3819	−0.868372
$924$	0	0
$925$	40.2120	1.32216
$926$	21.6685i	0.712071i
$927$	0	0
$928$	31.8870	1.04674
$929$	−44.6284	−1.46421	−0.732105	−	0.681192i	$-0.761462\pi$
$929$	−44.6284	−1.46421	−0.732105	+	0.681192i	$0.761462\pi$
$930$	0	0
$931$	0	0
$932$	− 14.1043i	− 0.462000i
$933$	0	0
$934$	− 53.4824i	− 1.75000i
$935$	14.1043i	0.461259i
$936$	0	0
$937$	33.3351i	1.08901i	0.838758	+	0.544505i	$0.183282\pi$
$937$	33.3351i	1.08901i	−0.838758	+	0.544505i	$0.816718\pi$
$938$	0	0
$939$	0	0
$940$	−13.2282	−0.431455
$941$	−33.9984	−1.10832	−0.554159	−	0.832411i	$-0.686960\pi$
$941$	−33.9984	−1.10832	−0.554159	+	0.832411i	$0.686960\pi$
$942$	0	0
$943$	− 8.04260i	− 0.261903i
$944$	33.9984	1.10655
$945$	0	0
$946$	15.4633	0.502754
$947$	− 46.1461i	− 1.49955i	−0.661694	−	0.749774i	$-0.730162\pi$
$947$	− 46.1461i	− 1.49955i	0.661694	−	0.749774i	$-0.269838\pi$
$948$	0	0
$949$	31.0109	1.00666
$950$	−11.0028	−0.356977
$951$	0	0
$952$	0	0
$953$	35.1143i	1.13746i	0.822523	+	0.568731i	$0.192566\pi$
$953$	35.1143i	1.13746i	−0.822523	+	0.568731i	$0.807434\pi$
$954$	0	0
$955$	27.2890i	0.883050i
$956$	22.6234i	0.731692i
$957$	0	0
$958$	74.3301i	2.40150i
$959$	0	0
$960$	0	0
$961$	12.7003	0.409688
$962$	43.0028	1.38647
$963$	0	0
$964$	− 32.6011i	− 1.05001i
$965$	12.1017	0.389567
$966$	0	0
$967$	−13.4148	−0.431392	−0.215696	−	0.976461i	$-0.569202\pi$
$967$	−13.4148	−0.431392	−0.215696	+	0.976461i	$0.569202\pi$
$968$	11.2391i	0.361238i
$969$	0	0
$970$	−4.80785	−0.154371
$971$	−18.3892	−0.590137	−0.295069	−	0.955476i	$-0.595343\pi$
$971$	−18.3892	−0.590137	−0.295069	+	0.955476i	$0.595343\pi$
$972$	0	0
$973$	0	0
$974$	22.7782i	0.729861i
$975$	0	0
$976$	39.3563i	1.25976i
$977$	22.6530i	0.724734i	0.932035	+	0.362367i	$0.118031\pi$
$977$	22.6530i	0.724734i	−0.932035	+	0.362367i	$0.881969\pi$
$978$	0	0
$979$	− 13.4647i	− 0.430333i
$980$	0	0
$981$	0	0
$982$	11.3195	0.361220
$983$	−39.3408	−1.25478	−0.627388	−	0.778707i	$-0.715876\pi$
$983$	−39.3408	−1.25478	−0.627388	+	0.778707i	$0.715876\pi$
$984$	0	0
$985$	− 7.36151i	− 0.234557i
$986$	−28.1710	−0.897149
$987$	0	0
$988$	−4.77184	−0.151813
$989$	− 12.7409i	− 0.405137i
$990$	0	0
$991$	0.951593	0.0302284	0.0151142	−	0.999886i	$-0.495189\pi$
$991$	0.951593	0.0302284	0.0151142	+	0.999886i	$0.495189\pi$
$992$	−28.2168	−0.895883
$993$	0	0
$994$	0	0
$995$	32.9804i	1.04555i
$996$	0	0
$997$	− 31.4640i	− 0.996475i	−0.867041	−	0.498238i	$-0.833981\pi$
$997$	− 31.4640i	− 0.996475i	0.867041	−	0.498238i	$-0.166019\pi$
$998$	− 5.43032i	− 0.171894i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.d.1322.9		12
3.2	odd	2	inner	1323.2.c.d.1322.4		12
7.4	even	3		189.2.p.d.26.2	✓	12
7.5	odd	6		189.2.p.d.80.5	yes	12
7.6	odd	2	inner	1323.2.c.d.1322.10		12
21.5	even	6		189.2.p.d.80.2	yes	12
21.11	odd	6		189.2.p.d.26.5	yes	12
21.20	even	2	inner	1323.2.c.d.1322.3		12
63.4	even	3		567.2.s.f.26.5		12
63.5	even	6		567.2.i.f.269.2		12
63.11	odd	6		567.2.i.f.215.2		12
63.25	even	3		567.2.i.f.215.5		12
63.32	odd	6		567.2.s.f.26.2		12
63.40	odd	6		567.2.i.f.269.5		12
63.47	even	6		567.2.s.f.458.5		12
63.61	odd	6		567.2.s.f.458.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.p.d.26.2	✓	12	7.4	even	3
189.2.p.d.26.5	yes	12	21.11	odd	6
189.2.p.d.80.2	yes	12	21.5	even	6
189.2.p.d.80.5	yes	12	7.5	odd	6
567.2.i.f.215.2		12	63.11	odd	6
567.2.i.f.215.5		12	63.25	even	3
567.2.i.f.269.2		12	63.5	even	6
567.2.i.f.269.5		12	63.40	odd	6
567.2.s.f.26.2		12	63.32	odd	6
567.2.s.f.26.5		12	63.4	even	3
567.2.s.f.458.2		12	63.61	odd	6
567.2.s.f.458.5		12	63.47	even	6
1323.2.c.d.1322.3		12	21.20	even	2	inner
1323.2.c.d.1322.4		12	3.2	odd	2	inner
1323.2.c.d.1322.9		12	1.1	even	1	trivial
1323.2.c.d.1322.10		12	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)

\(f(q)\)	\(=\)	\(q+1.83424i q^{2} -1.36445 q^{4} -3.80824 q^{5} +1.16576i q^{8} +O(q^{10})\)	\(q+1.83424i q^{2} -1.36445 q^{4} -3.80824 q^{5} +1.16576i q^{8} -6.98525i q^{10} -1.16576i q^{11} -5.54030i q^{13} -4.86718 q^{16} +3.17700 q^{17} +0.631243i q^{19} +5.19615 q^{20} +2.13828 q^{22} -1.76183i q^{23} +9.50273 q^{25} +10.1622 q^{26} +4.83424i q^{29} -4.27781i q^{31} -6.59607i q^{32} +5.82739i q^{34} +4.23163 q^{37} -1.15785 q^{38} -4.43949i q^{40} +4.56491 q^{41} +7.23163 q^{43} +1.59061i q^{44} +3.23163 q^{46} -2.54576 q^{47} +17.4303i q^{50} +7.55945i q^{52} -0.0724126i q^{53} +4.43949i q^{55} -8.86718 q^{58} -6.98525 q^{59} -8.08606i q^{61} +7.84655 q^{62} +2.36445 q^{64} +21.0988i q^{65} +5.09880 q^{67} -4.33485 q^{68} -4.76183i q^{71} +5.59734i q^{73} +7.76183i q^{74} -0.861298i q^{76} +17.0988 q^{79} +18.5354 q^{80} +8.37315i q^{82} +10.1622 q^{83} -12.0988 q^{85} +13.2646i q^{86} +1.35899 q^{88} +11.5502 q^{89} +2.40393i q^{92} -4.66954i q^{94} -2.40393i q^{95} -0.688287i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{4}+O(q^{10}) \) 12 * q - 16 * q^4	\( 12 q - 16 q^{4} + 8 q^{16} - 40 q^{22} + 48 q^{25} - 16 q^{37} + 20 q^{43} - 28 q^{46} - 40 q^{58} + 28 q^{64} - 72 q^{67} + 72 q^{79} - 12 q^{85} + 148 q^{88}+O(q^{100}) \) 12 * q - 16 * q^4 + 8 * q^16 - 40 * q^22 + 48 * q^25 - 16 * q^37 + 20 * q^43 - 28 * q^46 - 40 * q^58 + 28 * q^64 - 72 * q^67 + 72 * q^79 - 12 * q^85 + 148 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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