LMFDB - Embedded newform 387.2.y.a.316.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,2,Mod(10,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 10]))

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

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

chi := DirichletCharacter("387.10");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	387.y (of order $21$, degree $12$, 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:	$3.09021055822$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{21}]$

Embedding invariants

Embedding label			316.1
Root			$0.0747301 + 0.997204i$ of defining polynomial
Character	$\chi$	$=$	387.316
Dual form			387.2.y.a.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.445042 + 1.94986i) q^{4} +(-1.57775 + 2.73274i) q^{7} +O(q^{10})$	$q+(0.445042 + 1.94986i) q^{4} +(-1.57775 + 2.73274i) q^{7} +(-0.108219 + 0.0737826i) q^{13} +(-3.60388 + 1.73553i) q^{16} +(-5.64176 + 5.23479i) q^{19} +(4.94415 - 0.745211i) q^{25} +(-6.03061 - 1.86020i) q^{28} +(11.0109 + 1.65963i) q^{31} +(-1.30907 - 2.26738i) q^{37} +(-1.56856 + 6.36707i) q^{43} +(-1.47857 - 2.56097i) q^{49} +(-0.192028 - 0.178176i) q^{52} +(9.75955 - 1.47102i) q^{61} +(-4.98792 - 6.25465i) q^{64} +(8.26034 - 7.66447i) q^{67} +(6.46376 - 4.40692i) q^{73} +(-12.7179 - 8.67092i) q^{76} +(1.95561 - 3.38721i) q^{79} +(-0.0308860 - 0.412145i) q^{91} +(4.14459 - 18.1587i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{4} - q^{7}+O(q^{10}) $ 12 * q + 4 * q^4 - q^7	$ 12 q + 4 q^{4} - q^{7} - 7 q^{13} - 8 q^{16} + 8 q^{19} - 5 q^{25} + 2 q^{28} + 70 q^{31} + 11 q^{37} - 8 q^{43} - 55 q^{49} - 56 q^{52} + 14 q^{61} + 16 q^{64} + 5 q^{67} - 7 q^{73} - 16 q^{76} - 4 q^{79} + 7 q^{91} - 28 q^{97}+O(q^{100}) $ 12 * q + 4 * q^4 - q^7 - 7 * q^13 - 8 * q^16 + 8 * q^19 - 5 * q^25 + 2 * q^28 + 70 * q^31 + 11 * q^37 - 8 * q^43 - 55 * q^49 - 56 * q^52 + 14 * q^61 + 16 * q^64 + 5 * q^67 - 7 * q^73 - 16 * q^76 - 4 * q^79 + 7 * q^91 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/387\mathbb{Z}\right)^\times$.

$n$	$46$	$173$
$\chi(n)$	$e\left(\frac{13}{21}\right)$	$1$

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		−0.781831	−	0.623490i	$-0.785714\pi$
$2$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$3$		0			0
$3$		0			0
$4$	0.445042	+	1.94986i	0.222521	+	0.974928i
$5$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$5$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$6$		0			0
$7$	−1.57775	+	2.73274i	−0.596332	+	1.03288i	0.397025	+	0.917808i	$0.370043\pi$
$7$	−1.57775	+	2.73274i	−0.596332	+	1.03288i	−0.993357	+	0.115070i	$0.963291\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$11$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$12$		0			0
$13$	−0.108219	+	0.0737826i	−0.0300146	+	0.0204636i	−0.578234	−	0.815871i	$-0.696258\pi$
$13$	−0.108219	+	0.0737826i	−0.0300146	+	0.0204636i	0.548220	+	0.836334i	$0.315306\pi$
$14$		0			0
$15$		0			0
$16$	−3.60388	+	1.73553i	−0.900969	+	0.433884i
$17$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$17$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$18$		0			0
$19$	−5.64176	+	5.23479i	−1.29431	+	1.20094i	−0.327672	+	0.944792i	$0.606264\pi$
$19$	−5.64176	+	5.23479i	−1.29431	+	1.20094i	−0.966637	+	0.256151i	$0.917546\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$23$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$24$		0			0
$25$	4.94415	−	0.745211i	0.988831	−	0.149042i
$26$		0			0
$27$		0			0
$28$	−6.03061	−	1.86020i	−1.13968	−	0.351544i
$29$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$29$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$30$		0			0
$31$	11.0109	+	1.65963i	1.97762	+	0.298078i	0.989790	+	0.142535i	$0.0455255\pi$
$31$	11.0109	+	1.65963i	1.97762	+	0.298078i	0.987829	+	0.155543i	$0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.30907	−	2.26738i	−0.215210	−	0.372755i	0.738127	−	0.674661i	$-0.235710\pi$
$37$	−1.30907	−	2.26738i	−0.215210	−	0.372755i	−0.953338	+	0.301906i	$0.902377\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$41$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$42$		0			0
$43$	−1.56856	+	6.36707i	−0.239203	+	0.970970i
$43$	−1.56856	+	6.36707i	−0.239203	+	0.970970i
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$47$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$48$		0			0
$49$	−1.47857	−	2.56097i	−0.211225	−	0.365852i
$50$		0			0
$51$		0			0
$52$	−0.192028	−	0.178176i	−0.0266294	−	0.0247085i
$53$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$53$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$59$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$60$		0			0
$61$	9.75955	−	1.47102i	1.24958	−	0.188344i	0.509270	−	0.860607i	$-0.329915\pi$
$61$	9.75955	−	1.47102i	1.24958	−	0.188344i	0.740313	+	0.672262i	$0.234677\pi$
$62$		0			0
$63$		0			0
$64$	−4.98792	−	6.25465i	−0.623490	−	0.781831i
$65$		0			0
$66$		0			0
$67$	8.26034	−	7.66447i	1.00916	−	0.936365i	0.0111837	−	0.999937i	$-0.496440\pi$
$67$	8.26034	−	7.66447i	1.00916	−	0.936365i	0.997977	+	0.0635730i	$0.0202496\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$71$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$72$		0			0
$73$	6.46376	−	4.40692i	0.756526	−	0.515791i	−0.122556	−	0.992462i	$-0.539109\pi$
$73$	6.46376	−	4.40692i	0.756526	−	0.515791i	0.879082	+	0.476671i	$0.158157\pi$
$74$		0			0
$75$		0			0
$76$	−12.7179	−	8.67092i	−1.45884	−	0.994622i
$77$		0			0
$78$		0			0
$79$	1.95561	−	3.38721i	0.220023	−	0.381091i	−0.734792	−	0.678293i	$-0.762720\pi$
$79$	1.95561	−	3.38721i	0.220023	−	0.381091i	0.954815	+	0.297202i	$0.0960535\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$83$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$89$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$90$		0			0
$91$	−0.0308860	−	0.412145i	−0.00323773	−	0.0432046i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.14459	−	18.1587i	0.420820	−	1.84373i	−0.106841	−	0.994276i	$-0.534073\pi$
$97$	4.14459	−	18.1587i	0.420820	−	1.84373i	0.527660	−	0.849455i	$-0.323069\pi$
$98$		0			0
$99$		0			0
$100$	3.65341	+	9.30874i	0.365341	+	0.930874i
$101$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$101$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$102$		0			0
$103$	1.13426	−	15.1357i	0.111762	−	1.49136i	−0.606216	−	0.795300i	$-0.707313\pi$
$103$	1.13426	−	15.1357i	0.111762	−	1.49136i	0.717979	−	0.696065i	$-0.245067\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$107$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$108$		0			0
$109$	−19.7885	+	6.10395i	−1.89540	+	0.584652i	−0.909935	+	0.414751i	$0.863869\pi$
$109$	−19.7885	+	6.10395i	−1.89540	+	0.584652i	−0.985461	+	0.169901i	$0.945655\pi$
$110$		0			0
$111$		0			0
$112$	0.943242	−	12.5867i	0.0891280	−	1.18933i
$113$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$113$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.91066	+	4.77272i	0.900969	+	0.433884i
$122$		0			0
$123$		0			0
$124$	1.66428	+	22.2083i	0.149457	+	1.99436i
$125$		0			0
$126$		0			0
$127$	−11.3648	+	14.2510i	−1.00846	+	1.26457i	−0.0443678	+	0.999015i	$0.514127\pi$
$127$	−11.3648	+	14.2510i	−1.00846	+	1.26457i	−0.964095	+	0.265557i	$0.914444\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$131$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$132$		0			0
$133$	−5.40404	−	23.6766i	−0.468589	−	2.05302i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$137$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$138$		0			0
$139$	16.8938	+	11.5180i	1.43291	+	0.976941i	0.997036	+	0.0769336i	$0.0245129\pi$
$139$	16.8938	+	11.5180i	1.43291	+	0.976941i	0.435874	+	0.900008i	$0.356439\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	3.83847	−	3.56158i	0.315520	−	0.292760i
$149$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$149$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$150$		0			0
$151$	4.71945	+	5.91800i	0.384063	+	0.481600i	0.935857	−	0.352381i	$-0.114628\pi$
$151$	4.71945	+	5.91800i	0.384063	+	0.481600i	−0.551794	+	0.833981i	$0.686056\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	6.92948	+	2.13746i	0.553033	+	0.170588i	0.558661	−	0.829396i	$-0.311315\pi$
$157$	6.92948	+	2.13746i	0.553033	+	0.170588i	−0.00562830	+	0.999984i	$0.501792\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	16.3894	+	15.2071i	1.28371	+	1.19111i	0.970411	+	0.241460i	$0.0776264\pi$
$163$	16.3894	+	15.2071i	1.28371	+	1.19111i	0.313304	+	0.949653i	$0.398564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$167$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$168$		0			0
$169$	−4.74317	+	12.0854i	−0.364859	+	0.929645i
$170$		0			0
$171$		0			0
$172$	−13.1129	−	0.224844i	−0.999853	−	0.0171442i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−5.76416	+	14.6868i	−0.435729	+	1.11022i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−2.23550	−	2.07424i	−0.166164	−	0.154177i	0.592708	−	0.805417i	$-0.298059\pi$
$181$	−2.23550	−	2.07424i	−0.166164	−	0.154177i	−0.758872	+	0.651240i	$0.774249\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$191$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$192$		0			0
$193$	−12.6666	−	15.8834i	−0.911760	−	1.14331i	−0.989238	−	0.146317i	$-0.953258\pi$
$193$	−12.6666	−	15.8834i	−0.911760	−	1.14331i	0.0774780	−	0.996994i	$-0.475313\pi$
$194$		0			0
$195$		0			0
$196$	4.33549	−	4.02274i	0.309678	−	0.287339i
$197$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$197$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$198$		0			0
$199$	−21.3747	+	10.2935i	−1.51521	+	0.729687i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−21.3747	+	10.2935i	−1.51521	+	0.729687i	−0.522778	+	0.852469i	$0.675104\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.261956	−	0.453722i	0.0181634	−	0.0314599i
$209$		0			0
$210$		0			0
$211$	5.98127	+	26.2057i	0.411768	+	1.80407i	0.575768	+	0.817613i	$0.304703\pi$
$211$	5.98127	+	26.2057i	0.411768	+	1.80407i	−0.164000	+	0.986460i	$0.552440\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−21.9078	+	27.4715i	−1.48720	+	1.86489i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−24.8613	−	11.9726i	−1.66484	−	0.801744i	−0.998421	−	0.0561655i	$-0.982113\pi$
$223$	−24.8613	−	11.9726i	−1.66484	−	0.801744i	−0.666418	−	0.745579i	$-0.732173\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$227$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$228$		0			0
$229$	3.94686	+	10.0564i	0.260816	+	0.664547i	0.999959	−	0.00900024i	$-0.00286491\pi$
$229$	3.94686	+	10.0564i	0.260816	+	0.664547i	−0.739144	+	0.673548i	$0.764770\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$233$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$239$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$240$		0			0
$241$	2.31005	−	30.8255i	0.148803	−	1.98564i	−0.0189841	−	0.999820i	$-0.506043\pi$
$241$	2.31005	−	30.8255i	0.148803	−	1.98564i	0.167787	−	0.985823i	$-0.446338\pi$
$242$		0			0
$243$		0			0
$244$	7.21168	+	18.3751i	0.461680	+	1.17634i
$245$		0			0
$246$		0			0
$247$	0.224310	−	0.982768i	0.0142725	−	0.0625321i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$251$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	9.97584	−	12.5093i	0.623490	−	0.781831i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	8.26154			0.513347
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$263$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	18.6208	+	12.6955i	1.13745	+	0.775498i
$269$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$269$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$270$		0			0
$271$	12.5420	−	8.55099i	0.761872	−	0.519436i	−0.118940	−	0.992901i	$-0.537950\pi$
$271$	12.5420	−	8.55099i	0.761872	−	0.519436i	0.880812	+	0.473466i	$0.156997\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−24.3304	+	22.5753i	−1.46187	+	1.35642i	−0.679306	+	0.733855i	$0.737719\pi$
$277$	−24.3304	+	22.5753i	−1.46187	+	1.35642i	−0.782568	+	0.622566i	$0.786090\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$281$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$282$		0			0
$283$	32.9677	−	4.96908i	1.95972	−	0.295381i	0.959835	−	0.280566i	$-0.0905223\pi$
$283$	32.9677	−	4.96908i	1.95972	−	0.295381i	0.999890	−	0.0148145i	$-0.00471579\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.8101	+	2.53372i	0.988831	+	0.149042i
$290$		0			0
$291$		0			0
$292$	11.4695	+	10.6421i	0.671202	+	0.622784i
$293$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$293$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−14.9248	−	14.3321i	−0.860249	−	0.826088i
$302$		0			0
$303$		0			0
$304$	11.2470	−	28.6570i	0.645062	−	1.64359i
$305$		0			0
$306$		0			0
$307$	−14.9470	−	25.8890i	−0.853072	−	1.47756i	−0.878422	−	0.477885i	$-0.841403\pi$
$307$	−14.9470	−	25.8890i	−0.853072	−	1.47756i	0.0253503	−	0.999679i	$-0.491930\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$311$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$312$		0			0
$313$	−7.86118	−	1.18488i	−0.444340	−	0.0669735i	−0.0769384	−	0.997036i	$-0.524514\pi$
$313$	−7.86118	−	1.18488i	−0.444340	−	0.0669735i	−0.367402	+	0.930062i	$0.619753\pi$
$314$		0			0
$315$		0			0
$316$	7.47489	+	2.30570i	0.420496	+	0.129706i
$317$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$317$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.480069	+	0.445439i	−0.0266294	+	0.0247085i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−30.0305	+	20.4745i	−1.65063	+	1.12538i	−0.798671	+	0.601768i	$0.794463\pi$
$331$	−30.0305	+	20.4745i	−1.65063	+	1.12538i	−0.851957	+	0.523612i	$0.824584\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	12.6601	−	21.9279i	0.689638	−	1.19449i	−0.282316	−	0.959321i	$-0.591103\pi$
$337$	12.6601	−	21.9279i	0.689638	−	1.19449i	0.971955	−	0.235168i	$-0.0755639\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−12.7572			−0.688824
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$347$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$348$		0			0
$349$	−0.593853	−	7.92442i	−0.0317882	−	0.424184i	−0.990282	−	0.139072i	$-0.955588\pi$
$349$	−0.593853	−	7.92442i	−0.0317882	−	0.424184i	0.958494	−	0.285112i	$-0.0920309\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$353$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$359$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$360$		0			0
$361$	3.00657	−	40.1200i	0.158241	−	2.11158i
$362$		0			0
$363$		0			0
$364$	0.789878	−	0.243645i	0.0414009	−	0.0127705i
$365$		0			0
$366$		0			0
$367$	17.3536	−	5.35288i	0.905851	−	0.279418i	0.193383	−	0.981123i	$-0.438054\pi$
$367$	17.3536	−	5.35288i	0.905851	−	0.279418i	0.712467	+	0.701705i	$0.247578\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−14.1051	−	35.9393i	−0.730337	−	1.86087i	−0.393780	−	0.919205i	$-0.628833\pi$
$373$	−14.1051	−	35.9393i	−0.730337	−	1.86087i	−0.336557	−	0.941663i	$-0.609263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−18.0677	−	8.70095i	−0.928076	−	0.446938i	−0.0921284	−	0.995747i	$-0.529367\pi$
$379$	−18.0677	−	8.70095i	−0.928076	−	0.446938i	−0.835948	+	0.548809i	$0.815081\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$383$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	37.2513			1.89115
$389$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$389$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−30.9416	−	21.0956i	−1.55291	−	1.05876i	−0.969037	−	0.246916i	$-0.920583\pi$
$397$	−30.9416	−	21.0956i	−1.55291	−	1.05876i	−0.583877	−	0.811842i	$-0.698465\pi$
$398$		0			0
$399$		0			0
$400$	−16.5248	+	11.2664i	−0.826239	+	0.563320i
$401$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$401$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$402$		0			0
$403$	−1.31404	+	0.632810i	−0.0654572	+	0.0315225i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	20.0935	+	25.1965i	0.993561	+	1.24589i	0.969224	+	0.246182i	$0.0791762\pi$
$409$	20.0935	+	25.1965i	0.993561	+	1.24589i	0.0243376	+	0.999704i	$0.492252\pi$
$410$		0			0
$411$		0			0
$412$	30.0172	−	4.52437i	1.47884	−	0.222900i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$419$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$420$		0			0
$421$	−3.51216	−	3.25880i	−0.171172	−	0.158824i	0.589935	−	0.807451i	$-0.299153\pi$
$421$	−3.51216	−	3.25880i	−0.171172	−	0.158824i	−0.761107	+	0.648626i	$0.775344\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−11.3782	+	28.9912i	−0.550630	+	1.40298i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	14.6436	−	37.3112i	0.703726	−	1.79306i	0.101350	−	0.994851i	$-0.467684\pi$
$433$	14.6436	−	37.3112i	0.703726	−	1.79306i	0.602376	−	0.798213i	$-0.294221\pi$
$434$		0			0
$435$		0			0
$436$	−20.7085	−	35.8682i	−0.991759	−	1.71778i
$437$		0			0
$438$		0			0
$439$	29.8332	+	27.6812i	1.42386	+	1.32115i	0.874028	+	0.485876i	$0.161499\pi$
$439$	29.8332	+	27.6812i	1.42386	+	1.32115i	0.549834	+	0.835274i	$0.314691\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$443$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	24.9620	−	3.76242i	1.17934	−	0.177757i
$449$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$449$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	34.8990	−	16.8065i	1.63251	−	0.786175i	0.632577	−	0.774498i	$-0.281997\pi$
$457$	34.8990	−	16.8065i	1.63251	−	0.786175i	0.999932	−	0.0116771i	$-0.00371703\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$461$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$462$		0			0
$463$	−35.5283	−	24.2228i	−1.65114	−	1.12573i	−0.848242	−	0.529609i	$-0.822338\pi$
$463$	−35.5283	−	24.2228i	−1.65114	−	1.12573i	−0.802897	−	0.596118i	$-0.796709\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	7.91228	+	34.6660i	0.365355	+	1.60072i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−23.9927	+	30.0859i	−1.10086	+	1.38044i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	0.308960	+	0.148787i	0.0140874	+	0.00678411i
$482$		0			0
$483$		0			0
$484$	−4.89546	+	21.4484i	−0.222521	+	0.974928i
$485$		0			0
$486$		0			0
$487$	14.9878	+	38.1883i	0.679162	+	1.73048i	0.682930	+	0.730484i	$0.260705\pi$
$487$	14.9878	+	38.1883i	0.679162	+	1.73048i	−0.00376769	+	0.999993i	$0.501199\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$491$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−42.5623	+	13.1287i	−1.91110	+	0.589498i
$497$		0			0
$498$		0			0
$499$	−3.21339	+	42.8798i	−0.143851	+	1.91956i	0.198696	+	0.980061i	$0.436329\pi$
$499$	−3.21339	+	42.8798i	−0.143851	+	1.91956i	−0.342547	+	0.939501i	$0.611290\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$503$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−32.8452	−	15.8174i	−1.45727	−	0.701785i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	1.84477	+	24.6168i	0.0816079	+	1.08898i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$521$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$522$		0			0
$523$	21.5000	−	37.2391i	0.940129	−	1.62835i	0.174908	−	0.984585i	$-0.444037\pi$
$523$	21.5000	−	37.2391i	0.940129	−	1.62835i	0.765222	−	0.643767i	$-0.222629\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−19.0035	+	12.9564i	−0.826239	+	0.563320i
$530$		0			0
$531$		0			0
$532$	43.7610	−	21.0742i	1.89728	−	0.913682i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	23.0040	−	3.46730i	0.989019	−	0.149071i	0.365444	−	0.930834i	$-0.380917\pi$
$541$	23.0040	−	3.46730i	0.989019	−	0.149071i	0.623576	+	0.781763i	$0.285679\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−25.2326	−	3.80321i	−1.07887	−	0.162613i	−0.414523	−	0.910039i	$-0.636052\pi$
$547$	−25.2326	−	3.80321i	−1.07887	−	0.162613i	−0.664346	+	0.747425i	$0.731290\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	6.17090	+	10.6883i	0.262414	+	0.454514i
$554$		0			0
$555$		0			0
$556$	−14.9400	+	38.0664i	−0.633595	+	1.61437i
$557$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$557$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$558$		0			0
$559$	−0.300031	−	0.804772i	−0.0126900	−	0.0340382i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$563$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$569$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$570$		0			0
$571$	1.99285	+	0.300374i	0.0833982	+	0.0125703i	0.190609	−	0.981666i	$-0.438954\pi$
$571$	1.99285	+	0.300374i	0.0833982	+	0.0125703i	−0.107211	+	0.994236i	$0.534192\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	45.3009	−	6.82801i	1.88590	−	0.284254i	0.898078	−	0.439837i	$-0.144964\pi$
$577$	45.3009	−	6.82801i	1.88590	−	0.284254i	0.987823	+	0.155583i	$0.0497256\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$587$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$588$		0			0
$589$	−70.8087	+	48.2766i	−2.91762	+	1.98920i
$590$		0			0
$591$		0			0
$592$	8.65285	+	5.89941i	0.355630	+	0.242464i
$593$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$593$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$599$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$600$		0			0
$601$	−47.3336			−1.93078			−0.965389	−	0.260815i	$-0.916009\pi$
$601$	−47.3336			−1.93078			−0.965389	+	0.260815i	$0.916009\pi$
$602$		0			0
$603$		0			0
$604$	−9.43890	+	11.8360i	−0.384063	+	0.481600i
$605$		0			0
$606$		0			0
$607$	0.113578	+	1.51560i	0.00461001	+	0.0615162i	0.999034	−	0.0439364i	$-0.0139899\pi$
$607$	0.113578	+	1.51560i	0.00461001	+	0.0615162i	−0.994424	+	0.105453i	$0.966371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−9.23274	+	40.4513i	−0.372907	+	1.63381i	0.345664	+	0.938358i	$0.387654\pi$
$613$	−9.23274	+	40.4513i	−0.372907	+	1.63381i	−0.718571	+	0.695454i	$0.755203\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$617$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$618$		0			0
$619$	3.12447	−	41.6931i	0.125583	−	1.67579i	−0.477902	−	0.878413i	$-0.658603\pi$
$619$	3.12447	−	41.6931i	0.125583	−	1.67579i	0.603485	−	0.797374i	$-0.293778\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	23.8893	−	7.36888i	0.955573	−	0.294755i
$626$		0			0
$627$		0			0
$628$	−1.08383	+	14.4627i	−0.0432496	+	0.577126i
$629$		0			0
$630$		0			0
$631$	−15.7097	−	40.0276i	−0.625392	−	1.59347i	−0.794087	−	0.607804i	$-0.792051\pi$
$631$	−15.7097	−	40.0276i	−0.625392	−	1.59347i	0.168695	−	0.985668i	$-0.446045\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.348965	+	0.168053i	0.0138265	+	0.00665849i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$641$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$642$		0			0
$643$	−13.4015	+	16.8050i	−0.528505	+	0.662725i	−0.972391	−	0.233359i	$-0.925028\pi$
$643$	−13.4015	+	16.8050i	−0.528505	+	0.662725i	0.443885	+	0.896084i	$0.353600\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$647$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−22.3577	+	38.7247i	−0.875596	+	1.51658i
$653$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$653$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$659$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$660$		0			0
$661$	29.2027	−	14.0633i	1.13585	−	0.546999i	0.231099	−	0.972930i	$-0.425768\pi$
$661$	29.2027	−	14.0633i	1.13585	−	0.546999i	0.904756	+	0.425931i	$0.140053\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	29.7810	+	9.18623i	1.14797	+	0.354103i	0.809642	−	0.586924i	$-0.199661\pi$
$673$	29.7810	+	9.18623i	1.14797	+	0.354103i	0.338332	+	0.941027i	$0.390137\pi$
$674$		0			0
$675$		0			0
$676$	−25.6757	−	3.86999i	−0.987526	−	0.148846i
$677$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$677$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$678$		0			0
$679$	43.0837	+	39.9759i	1.65340	+	1.53413i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$683$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−5.39740	−	25.6684i	−0.205774	−	0.978600i
$689$		0			0
$690$		0			0
$691$	8.38842	−	21.3733i	0.319110	−	0.813080i	−0.677852	−	0.735198i	$-0.737089\pi$
$691$	8.38842	−	21.3733i	0.319110	−	0.813080i	0.996963	−	0.0778818i	$-0.0248157\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−31.2025	−	4.70302i	−1.17934	−	0.177757i
$701$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$701$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$702$		0			0
$703$	19.2547	+	5.93929i	0.726205	+	0.224005i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−32.1930	−	40.3688i	−1.20903	−	1.51608i	−0.795920	−	0.605401i	$-0.793013\pi$
$709$	−32.1930	−	40.3688i	−1.20903	−	1.51608i	−0.413114	−	0.910679i	$-0.635559\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$719$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$720$		0			0
$721$	39.5723	+	26.9800i	1.47375	+	1.00479i
$722$		0			0
$723$		0			0
$724$	3.04958	−	5.28203i	0.113337	−	0.196305i
$725$		0			0
$726$		0			0
$727$	8.64758	+	37.8875i	0.320721	+	1.40517i	0.836273	+	0.548313i	$0.184730\pi$
$727$	8.64758	+	37.8875i	0.320721	+	1.40517i	−0.515552	+	0.856858i	$0.672413\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−26.8101	+	33.6188i	−0.990252	+	1.24174i	−0.0199601	+	0.999801i	$0.506354\pi$
$733$	−26.8101	+	33.6188i	−0.990252	+	1.24174i	−0.970292	+	0.241936i	$0.922218\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−7.27697	−	3.50441i	−0.267688	−	0.128912i	0.295226	−	0.955428i	$-0.404605\pi$
$739$	−7.27697	−	3.50441i	−0.267688	−	0.128912i	−0.562913	+	0.826516i	$0.690320\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$743$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	51.6016	−	15.9170i	1.88297	−	0.580819i	0.891057	−	0.453892i	$-0.149965\pi$
$751$	51.6016	−	15.9170i	1.88297	−	0.580819i	0.991912	−	0.126927i	$-0.0405114\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	4.04496	−	53.9762i	0.147016	−	1.96180i	−0.106081	−	0.994357i	$-0.533830\pi$
$757$	4.04496	−	53.9762i	0.147016	−	1.96180i	0.253098	−	0.967441i	$-0.418551\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$761$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$762$		0			0
$763$	14.5408	−	63.7073i	0.526412	−	2.30636i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.45674	+	32.7829i	0.0885923	+	1.18218i	0.847432	+	0.530904i	$0.178148\pi$
$769$	2.45674	+	32.7829i	0.0885923	+	1.18218i	−0.758839	+	0.651278i	$0.774233\pi$
$770$		0			0
$771$		0			0
$772$	25.3331	−	31.7668i	0.911760	−	1.14331i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	55.6764			1.99996
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	9.77324	+	6.66328i	0.349044	+	0.237974i
$785$		0			0
$786$		0			0
$787$	14.2398	−	9.70851i	0.507593	−	0.346071i	−0.282289	−	0.959329i	$-0.591094\pi$
$787$	14.2398	−	9.70851i	0.507593	−	0.346071i	0.789883	+	0.613258i	$0.210141\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.947636	+	0.879278i	−0.0336515	+	0.0312241i
$794$		0			0
$795$		0			0
$796$	−29.5835	−	37.0965i	−1.04856	−	1.31485i
$797$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$797$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$809$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$810$		0			0
$811$	−26.0618	−	45.1403i	−0.915153	−	1.58509i	−0.806677	−	0.590992i	$-0.798736\pi$
$811$	−26.0618	−	45.1403i	−0.915153	−	1.58509i	−0.108476	−	0.994099i	$-0.534597\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−24.4809	−	44.1326i	−0.856477	−	1.54400i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$821$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$822$		0			0
$823$	−10.8127	−	18.7281i	−0.376907	−	0.652822i	0.613704	−	0.789536i	$-0.289679\pi$
$823$	−10.8127	−	18.7281i	−0.376907	−	0.652822i	−0.990610	+	0.136715i	$0.956346\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$827$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$828$		0			0
$829$	37.5575	+	5.66088i	1.30443	+	0.196611i	0.764238	−	0.644934i	$-0.223115\pi$
$829$	37.5575	+	5.66088i	1.30443	+	0.196611i	0.540188	+	0.841544i	$0.318353\pi$
$830$		0			0
$831$		0			0
$832$	1.00127	+	0.308852i	0.0347129	+	0.0107075i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$839$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$840$		0			0
$841$	21.2585	−	19.7250i	0.733052	−	0.680173i
$842$		0			0
$843$		0			0
$844$	−48.4354	+	23.3252i	−1.66721	+	0.802888i
$845$		0			0
$846$		0			0
$847$	−28.6791	+	19.5531i	−0.985426	+	0.671852i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−9.83036	+	17.0267i	−0.336585	+	0.582982i	−0.983788	−	0.179335i	$-0.942605\pi$
$853$	−9.83036	+	17.0267i	−0.336585	+	0.582982i	0.647203	+	0.762318i	$0.275939\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$857$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$858$		0			0
$859$	−43.0000			−1.46714			−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−43.0000			−1.46714			−0.733571	+	0.679613i	$0.762148\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$863$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	−63.3153	−	30.4910i	−2.14906	−	1.03493i
$869$		0			0
$870$		0			0
$871$	−0.328422	+	1.43891i	−0.0111282	+	0.0487557i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−4.14092	+	55.2567i	−0.139829	+	1.86589i	0.282266	+	0.959336i	$0.408914\pi$
$877$	−4.14092	+	55.2567i	−0.139829	+	1.86589i	−0.422095	+	0.906552i	$0.638705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$881$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$882$		0			0
$883$	56.6874	−	17.4857i	1.90768	−	0.588442i	0.936101	−	0.351731i	$-0.114407\pi$
$883$	56.6874	−	17.4857i	1.90768	−	0.588442i	0.971580	−	0.236710i	$-0.0760692\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$887$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$888$		0			0
$889$	−21.0135	−	53.5415i	−0.704770	−	1.79572i
$890$		0			0
$891$		0			0
$892$	12.2805	−	53.8044i	0.411181	−	1.80150i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−6.07631	−	26.6221i	−0.201761	−	0.883971i	−0.969864	−	0.243646i	$-0.921657\pi$
$907$	−6.07631	−	26.6221i	−0.201761	−	0.883971i	0.768104	−	0.640325i	$-0.221200\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$911$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−17.8521	+	12.1713i	−0.589849	+	0.402152i
$917$		0			0
$918$		0			0
$919$	36.4894	−	17.5724i	1.20368	−	0.579660i	0.278954	−	0.960304i	$-0.410012\pi$
$919$	36.4894	−	17.5724i	1.20368	−	0.579660i	0.924721	+	0.380645i	$0.124298\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−8.16193	−	10.2347i	−0.268363	−	0.336516i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$929$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$930$		0			0
$931$	21.7479	+	6.70833i	0.712758	+	0.219857i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	43.8523	+	40.6890i	1.43259	+	1.32925i	0.860892	+	0.508788i	$0.169906\pi$
$937$	43.8523	+	40.6890i	1.43259	+	1.32925i	0.571700	+	0.820463i	$0.306284\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$941$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	−0.374349	+	0.953827i	−0.0121519	+	0.0309625i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$953$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	88.8631	+	27.4107i	2.86655	+	0.884215i
$962$		0			0
$963$		0			0
$964$	61.1333	−	9.21436i	1.96897	−	0.296775i
$965$		0			0
$966$		0			0
$967$	−38.5714	−	48.3670i	−1.24037	−	1.55538i	−0.700569	−	0.713584i	$-0.747070\pi$
$967$	−38.5714	−	48.3670i	−1.24037	−	1.55538i	−0.539802	−	0.841792i	$-0.681501\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$971$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$972$		0			0
$973$	−58.1297	+	27.9938i	−1.86355	+	0.897439i
$974$		0			0
$975$		0			0
$976$	−32.6192	+	22.2394i	−1.04412	+	0.711866i
$977$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$977$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	2.01608			0.0641402
$989$		0			0
$990$		0			0
$991$	6.84987	−	8.58946i	0.217593	−	0.272853i	−0.661040	−	0.750351i	$-0.729885\pi$
$991$	6.84987	−	8.58946i	0.217593	−	0.272853i	0.878633	+	0.477498i	$0.158456\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	55.5870	+	26.7693i	1.76046	+	0.847792i	0.972788	+	0.231696i	$0.0744275\pi$
$997$	55.5870	+	26.7693i	1.76046	+	0.847792i	0.787671	+	0.616096i	$0.211287\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.2.y.a.316.1	yes	12
3.2	odd	2	CM	387.2.y.a.316.1	yes	12
43.23	even	21	inner	387.2.y.a.109.1	✓	12
129.23	odd	42	inner	387.2.y.a.109.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
387.2.y.a.109.1	✓	12	43.23	even	21	inner
387.2.y.a.109.1	✓	12	129.23	odd	42	inner
387.2.y.a.316.1	yes	12	1.1	even	1	trivial
387.2.y.a.316.1	yes	12	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	387.y (of order \(21\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(0.445042 + 1.94986i) q^{4} +(-1.57775 + 2.73274i) q^{7} +O(q^{10})\)	\(q+(0.445042 + 1.94986i) q^{4} +(-1.57775 + 2.73274i) q^{7} +(-0.108219 + 0.0737826i) q^{13} +(-3.60388 + 1.73553i) q^{16} +(-5.64176 + 5.23479i) q^{19} +(4.94415 - 0.745211i) q^{25} +(-6.03061 - 1.86020i) q^{28} +(11.0109 + 1.65963i) q^{31} +(-1.30907 - 2.26738i) q^{37} +(-1.56856 + 6.36707i) q^{43} +(-1.47857 - 2.56097i) q^{49} +(-0.192028 - 0.178176i) q^{52} +(9.75955 - 1.47102i) q^{61} +(-4.98792 - 6.25465i) q^{64} +(8.26034 - 7.66447i) q^{67} +(6.46376 - 4.40692i) q^{73} +(-12.7179 - 8.67092i) q^{76} +(1.95561 - 3.38721i) q^{79} +(-0.0308860 - 0.412145i) q^{91} +(4.14459 - 18.1587i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{4} - q^{7}+O(q^{10}) \) 12 * q + 4 * q^4 - q^7	\( 12 q + 4 q^{4} - q^{7} - 7 q^{13} - 8 q^{16} + 8 q^{19} - 5 q^{25} + 2 q^{28} + 70 q^{31} + 11 q^{37} - 8 q^{43} - 55 q^{49} - 56 q^{52} + 14 q^{61} + 16 q^{64} + 5 q^{67} - 7 q^{73} - 16 q^{76} - 4 q^{79} + 7 q^{91} - 28 q^{97}+O(q^{100}) \) 12 * q + 4 * q^4 - q^7 - 7 * q^13 - 8 * q^16 + 8 * q^19 - 5 * q^25 + 2 * q^28 + 70 * q^31 + 11 * q^37 - 8 * q^43 - 55 * q^49 - 56 * q^52 + 14 * q^61 + 16 * q^64 + 5 * q^67 - 7 * q^73 - 16 * q^76 - 4 * q^79 + 7 * q^91 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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