LMFDB - Embedded newform 2475.2.f.a.2276.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(2276,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.2276");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.f (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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 495)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2276.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	2475.2276
Dual form			2475.2.f.a.2276.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{4} +1.41421i q^{7} +3.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} -1.00000 q^{4} +1.41421i q^{7} +3.00000 q^{8} +(-3.00000 - 1.41421i) q^{11} -2.82843i q^{13} -1.41421i q^{14} -1.00000 q^{16} -2.00000 q^{17} +4.24264i q^{19} +(3.00000 + 1.41421i) q^{22} +2.82843i q^{26} -1.41421i q^{28} +6.00000 q^{29} -5.00000 q^{32} +2.00000 q^{34} -6.00000 q^{37} -4.24264i q^{38} +6.00000 q^{41} +1.41421i q^{43} +(3.00000 + 1.41421i) q^{44} -8.48528i q^{47} +5.00000 q^{49} +2.82843i q^{52} -4.24264i q^{53} +4.24264i q^{56} -6.00000 q^{58} -8.48528i q^{59} +8.48528i q^{61} +7.00000 q^{64} +2.00000 q^{68} -8.48528i q^{71} +11.3137i q^{73} +6.00000 q^{74} -4.24264i q^{76} +(2.00000 - 4.24264i) q^{77} +12.7279i q^{79} -6.00000 q^{82} -14.0000 q^{83} -1.41421i q^{86} +(-9.00000 - 4.24264i) q^{88} +7.07107i q^{89} +4.00000 q^{91} +8.48528i q^{94} -12.0000 q^{97} -5.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{4} + 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^4 + 6 * q^8	$ 2 q - 2 q^{2} - 2 q^{4} + 6 q^{8} - 6 q^{11} - 2 q^{16} - 4 q^{17} + 6 q^{22} + 12 q^{29} - 10 q^{32} + 4 q^{34} - 12 q^{37} + 12 q^{41} + 6 q^{44} + 10 q^{49} - 12 q^{58} + 14 q^{64} + 4 q^{68} + 12 q^{74} + 4 q^{77} - 12 q^{82} - 28 q^{83} - 18 q^{88} + 8 q^{91} - 24 q^{97} - 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^4 + 6 * q^8 - 6 * q^11 - 2 * q^16 - 4 * q^17 + 6 * q^22 + 12 * q^29 - 10 * q^32 + 4 * q^34 - 12 * q^37 + 12 * q^41 + 6 * q^44 + 10 * q^49 - 12 * q^58 + 14 * q^64 + 4 * q^68 + 12 * q^74 + 4 * q^77 - 12 * q^82 - 28 * q^83 - 18 * q^88 + 8 * q^91 - 24 * q^97 - 10 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000			−0.707107			−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000			−0.707107			−0.353553	+	0.935414i	$0.615027\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$			1.41421i			0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$			1.41421i			0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$	3.00000			1.06066
$9$		0			0
$10$		0			0
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$12$		0			0
$13$		−	2.82843i		−	0.784465i	−0.919866	−	0.392232i	$-0.871703\pi$
$13$		−	2.82843i		−	0.784465i	0.919866	−	0.392232i	$-0.128297\pi$
$14$		−	1.41421i		−	0.377964i
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$			4.24264i			0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$			4.24264i			0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$		0			0
$21$		0			0
$22$	3.00000	+	1.41421i	0.639602	+	0.301511i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$		0			0
$26$			2.82843i			0.554700i
$27$		0			0
$28$		−	1.41421i		−	0.267261i
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−5.00000			−0.883883
$33$		0			0
$34$	2.00000			0.342997
$35$		0			0
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		−	4.24264i		−	0.688247i
$39$		0			0
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$			1.41421i			0.215666i	0.994169	+	0.107833i	$0.0343911\pi$
$43$			1.41421i			0.215666i	−0.994169	+	0.107833i	$0.965609\pi$
$44$	3.00000	+	1.41421i	0.452267	+	0.213201i
$45$		0			0
$46$		0			0
$47$		−	8.48528i		−	1.23771i	−0.785507	−	0.618853i	$-0.787598\pi$
$47$		−	8.48528i		−	1.23771i	0.785507	−	0.618853i	$-0.212402\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$		0			0
$52$			2.82843i			0.392232i
$53$		−	4.24264i		−	0.582772i	−0.956606	−	0.291386i	$-0.905884\pi$
$53$		−	4.24264i		−	0.582772i	0.956606	−	0.291386i	$-0.0941163\pi$
$54$		0			0
$55$		0			0
$56$			4.24264i			0.566947i
$57$		0			0
$58$	−6.00000			−0.787839
$59$		−	8.48528i		−	1.10469i	−0.833616	−	0.552345i	$-0.813733\pi$
$59$		−	8.48528i		−	1.10469i	0.833616	−	0.552345i	$-0.186267\pi$
$60$		0			0
$61$			8.48528i			1.08643i	0.839594	+	0.543214i	$0.182793\pi$
$61$			8.48528i			1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$		0			0
$63$		0			0
$64$	7.00000			0.875000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	2.00000			0.242536
$69$		0			0
$70$		0			0
$71$		−	8.48528i		−	1.00702i	−0.863990	−	0.503509i	$-0.832042\pi$
$71$		−	8.48528i		−	1.00702i	0.863990	−	0.503509i	$-0.167958\pi$
$72$		0			0
$73$			11.3137i			1.32417i	0.749429	+	0.662085i	$0.230328\pi$
$73$			11.3137i			1.32417i	−0.749429	+	0.662085i	$0.769672\pi$
$74$	6.00000			0.697486
$75$		0			0
$76$		−	4.24264i		−	0.486664i
$77$	2.00000	−	4.24264i	0.227921	−	0.483494i
$78$		0			0
$79$			12.7279i			1.43200i	0.698099	+	0.716002i	$0.254030\pi$
$79$			12.7279i			1.43200i	−0.698099	+	0.716002i	$0.745970\pi$
$80$		0			0
$81$		0			0
$82$	−6.00000			−0.662589
$83$	−14.0000			−1.53670			−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000			−1.53670			−0.768350	+	0.640030i	$0.778922\pi$
$84$		0			0
$85$		0			0
$86$		−	1.41421i		−	0.152499i
$87$		0			0
$88$	−9.00000	−	4.24264i	−0.959403	−	0.452267i
$89$			7.07107i			0.749532i	0.927119	+	0.374766i	$0.122277\pi$
$89$			7.07107i			0.749532i	−0.927119	+	0.374766i	$0.877723\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$			8.48528i			0.875190i
$95$		0			0
$96$		0			0
$97$	−12.0000			−1.21842			−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000			−1.21842			−0.609208	+	0.793011i	$0.708512\pi$
$98$	−5.00000			−0.505076
$99$		0			0
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	12.0000			1.18240			0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000			1.18240			0.591198	+	0.806527i	$0.298655\pi$
$104$		−	8.48528i		−	0.832050i
$105$		0			0
$106$			4.24264i			0.412082i
$107$	14.0000			1.35343			0.676716	−	0.736245i	$-0.263403\pi$
$107$	14.0000			1.35343			0.676716	+	0.736245i	$0.263403\pi$
$108$		0			0
$109$			16.9706i			1.62549i	0.582623	+	0.812743i	$0.302026\pi$
$109$			16.9706i			1.62549i	−0.582623	+	0.812743i	$0.697974\pi$
$110$		0			0
$111$		0			0
$112$		−	1.41421i		−	0.133631i
$113$			12.7279i			1.19734i	0.800995	+	0.598671i	$0.204304\pi$
$113$			12.7279i			1.19734i	−0.800995	+	0.598671i	$0.795696\pi$
$114$		0			0
$115$		0			0
$116$	−6.00000			−0.557086
$117$		0			0
$118$			8.48528i			0.781133i
$119$		−	2.82843i		−	0.259281i
$120$		0			0
$121$	7.00000	+	8.48528i	0.636364	+	0.771389i
$122$		−	8.48528i		−	0.768221i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	18.3848i		−	1.63139i	−0.578486	−	0.815693i	$-0.696356\pi$
$127$		−	18.3848i		−	1.63139i	0.578486	−	0.815693i	$-0.303644\pi$
$128$	3.00000			0.265165
$129$		0			0
$130$		0			0
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$	−6.00000			−0.520266
$134$		0			0
$135$		0			0
$136$	−6.00000			−0.514496
$137$			4.24264i			0.362473i	0.983440	+	0.181237i	$0.0580100\pi$
$137$			4.24264i			0.362473i	−0.983440	+	0.181237i	$0.941990\pi$
$138$		0			0
$139$			4.24264i			0.359856i	0.983680	+	0.179928i	$0.0575865\pi$
$139$			4.24264i			0.359856i	−0.983680	+	0.179928i	$0.942414\pi$
$140$		0			0
$141$		0			0
$142$			8.48528i			0.712069i
$143$	−4.00000	+	8.48528i	−0.334497	+	0.709575i
$144$		0			0
$145$		0			0
$146$		−	11.3137i		−	0.936329i
$147$		0			0
$148$	6.00000			0.493197
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		0			0
$151$			4.24264i			0.345261i	0.984987	+	0.172631i	$0.0552267\pi$
$151$			4.24264i			0.345261i	−0.984987	+	0.172631i	$0.944773\pi$
$152$			12.7279i			1.03237i
$153$		0			0
$154$	−2.00000	+	4.24264i	−0.161165	+	0.341882i
$155$		0			0
$156$		0			0
$157$	24.0000			1.91541			0.957704	−	0.287754i	$-0.0929087\pi$
$157$	24.0000			1.91541			0.957704	+	0.287754i	$0.0929087\pi$
$158$		−	12.7279i		−	1.01258i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	24.0000			1.87983			0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000			1.87983			0.939913	+	0.341415i	$0.110906\pi$
$164$	−6.00000			−0.468521
$165$		0			0
$166$	14.0000			1.08661
$167$	8.00000			0.619059			0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000			0.619059			0.309529	+	0.950890i	$0.399829\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$		0			0
$172$		−	1.41421i		−	0.107833i
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$		0			0
$176$	3.00000	+	1.41421i	0.226134	+	0.106600i
$177$		0			0
$178$		−	7.07107i		−	0.529999i
$179$			5.65685i			0.422813i	0.977398	+	0.211407i	$0.0678044\pi$
$179$			5.65685i			0.422813i	−0.977398	+	0.211407i	$0.932196\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	−4.00000			−0.296500
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	6.00000	+	2.82843i	0.438763	+	0.206835i
$188$			8.48528i			0.618853i
$189$		0			0
$190$		0			0
$191$		−	2.82843i		−	0.204658i	−0.994751	−	0.102329i	$-0.967371\pi$
$191$		−	2.82843i		−	0.204658i	0.994751	−	0.102329i	$-0.0326294\pi$
$192$		0			0
$193$		−	19.7990i		−	1.42516i	−0.701590	−	0.712581i	$-0.747526\pi$
$193$		−	19.7990i		−	1.42516i	0.701590	−	0.712581i	$-0.252474\pi$
$194$	12.0000			0.861550
$195$		0			0
$196$	−5.00000			−0.357143
$197$	−14.0000			−0.997459			−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000			−0.997459			−0.498729	+	0.866758i	$0.666200\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		0			0
$201$		0			0
$202$	−6.00000			−0.422159
$203$			8.48528i			0.595550i
$204$		0			0
$205$		0			0
$206$	−12.0000			−0.836080
$207$		0			0
$208$			2.82843i			0.196116i
$209$	6.00000	−	12.7279i	0.415029	−	0.880409i
$210$		0			0
$211$			4.24264i			0.292075i	0.989279	+	0.146038i	$0.0466521\pi$
$211$			4.24264i			0.292075i	−0.989279	+	0.146038i	$0.953348\pi$
$212$			4.24264i			0.291386i
$213$		0			0
$214$	−14.0000			−0.957020
$215$		0			0
$216$		0			0
$217$		0			0
$218$		−	16.9706i		−	1.14939i
$219$		0			0
$220$		0			0
$221$			5.65685i			0.380521i
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$		−	7.07107i		−	0.472456i
$225$		0			0
$226$		−	12.7279i		−	0.846649i
$227$	−2.00000			−0.132745			−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000			−0.132745			−0.0663723	+	0.997795i	$0.521143\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$		0			0
$231$		0			0
$232$	18.0000			1.18176
$233$	10.0000			0.655122			0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000			0.655122			0.327561	+	0.944830i	$0.393773\pi$
$234$		0			0
$235$		0			0
$236$			8.48528i			0.552345i
$237$		0			0
$238$			2.82843i			0.183340i
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$			8.48528i			0.546585i	0.961931	+	0.273293i	$0.0881127\pi$
$241$			8.48528i			0.546585i	−0.961931	+	0.273293i	$0.911887\pi$
$242$	−7.00000	−	8.48528i	−0.449977	−	0.545455i
$243$		0			0
$244$		−	8.48528i		−	0.543214i
$245$		0			0
$246$		0			0
$247$	12.0000			0.763542
$248$		0			0
$249$		0			0
$250$		0			0
$251$			16.9706i			1.07117i	0.844481	+	0.535586i	$0.179909\pi$
$251$			16.9706i			1.07117i	−0.844481	+	0.535586i	$0.820091\pi$
$252$		0			0
$253$		0			0
$254$			18.3848i			1.15356i
$255$		0			0
$256$	−17.0000			−1.06250
$257$			29.6985i			1.85254i	0.376860	+	0.926270i	$0.377004\pi$
$257$			29.6985i			1.85254i	−0.376860	+	0.926270i	$0.622996\pi$
$258$		0			0
$259$		−	8.48528i		−	0.527250i
$260$		0			0
$261$		0			0
$262$	−18.0000			−1.11204
$263$	8.00000			0.493301			0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000			0.493301			0.246651	+	0.969104i	$0.420670\pi$
$264$		0			0
$265$		0			0
$266$	6.00000			0.367884
$267$		0			0
$268$		0			0
$269$		−	12.7279i		−	0.776035i	−0.921652	−	0.388018i	$-0.873160\pi$
$269$		−	12.7279i		−	0.776035i	0.921652	−	0.388018i	$-0.126840\pi$
$270$		0			0
$271$			12.7279i			0.773166i	0.922255	+	0.386583i	$0.126345\pi$
$271$			12.7279i			0.773166i	−0.922255	+	0.386583i	$0.873655\pi$
$272$	2.00000			0.121268
$273$		0			0
$274$		−	4.24264i		−	0.256307i
$275$		0			0
$276$		0			0
$277$			31.1127i			1.86938i	0.355463	+	0.934690i	$0.384323\pi$
$277$			31.1127i			1.86938i	−0.355463	+	0.934690i	$0.615677\pi$
$278$		−	4.24264i		−	0.254457i
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$		−	1.41421i		−	0.0840663i	−0.999116	−	0.0420331i	$-0.986616\pi$
$283$		−	1.41421i		−	0.0840663i	0.999116	−	0.0420331i	$-0.0133835\pi$
$284$			8.48528i			0.503509i
$285$		0			0
$286$	4.00000	−	8.48528i	0.236525	−	0.501745i
$287$			8.48528i			0.500870i
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		−	11.3137i		−	0.662085i
$293$	10.0000			0.584206			0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000			0.584206			0.292103	+	0.956387i	$0.405645\pi$
$294$		0			0
$295$		0			0
$296$	−18.0000			−1.04623
$297$		0			0
$298$	−18.0000			−1.04271
$299$		0			0
$300$		0			0
$301$	−2.00000			−0.115278
$302$		−	4.24264i		−	0.244137i
$303$		0			0
$304$		−	4.24264i		−	0.243332i
$305$		0			0
$306$		0			0
$307$		−	24.0416i		−	1.37213i	−0.727541	−	0.686064i	$-0.759337\pi$
$307$		−	24.0416i		−	1.37213i	0.727541	−	0.686064i	$-0.240663\pi$
$308$	−2.00000	+	4.24264i	−0.113961	+	0.241747i
$309$		0			0
$310$		0			0
$311$		−	16.9706i		−	0.962312i	−0.876635	−	0.481156i	$-0.840217\pi$
$311$		−	16.9706i		−	0.962312i	0.876635	−	0.481156i	$-0.159783\pi$
$312$		0			0
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$	−24.0000			−1.35440
$315$		0			0
$316$		−	12.7279i		−	0.716002i
$317$			21.2132i			1.19145i	0.803188	+	0.595726i	$0.203136\pi$
$317$			21.2132i			1.19145i	−0.803188	+	0.595726i	$0.796864\pi$
$318$		0			0
$319$	−18.0000	−	8.48528i	−1.00781	−	0.475085i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	8.48528i		−	0.472134i
$324$		0			0
$325$		0			0
$326$	−24.0000			−1.32924
$327$		0			0
$328$	18.0000			0.993884
$329$	12.0000			0.661581
$330$		0			0
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$	14.0000			0.768350
$333$		0			0
$334$	−8.00000			−0.437741
$335$		0			0
$336$		0			0
$337$			11.3137i			0.616297i	0.951338	+	0.308148i	$0.0997094\pi$
$337$			11.3137i			0.616297i	−0.951338	+	0.308148i	$0.900291\pi$
$338$	−5.00000			−0.271964
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			16.9706i			0.916324i
$344$			4.24264i			0.228748i
$345$		0			0
$346$	−14.0000			−0.752645
$347$	−14.0000			−0.751559			−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000			−0.751559			−0.375780	+	0.926709i	$0.622625\pi$
$348$		0			0
$349$			8.48528i			0.454207i	0.973871	+	0.227103i	$0.0729255\pi$
$349$			8.48528i			0.454207i	−0.973871	+	0.227103i	$0.927074\pi$
$350$		0			0
$351$		0			0
$352$	15.0000	+	7.07107i	0.799503	+	0.376889i
$353$			4.24264i			0.225813i	0.993606	+	0.112906i	$0.0360161\pi$
$353$			4.24264i			0.225813i	−0.993606	+	0.112906i	$0.963984\pi$
$354$		0			0
$355$		0			0
$356$		−	7.07107i		−	0.374766i
$357$		0			0
$358$		−	5.65685i		−	0.298974i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		0			0
$364$	−4.00000			−0.209657
$365$		0			0
$366$		0			0
$367$	−12.0000			−0.626395			−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000			−0.626395			−0.313197	+	0.949688i	$0.601400\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.00000			0.311504
$372$		0			0
$373$		−	22.6274i		−	1.17160i	−0.810454	−	0.585802i	$-0.800780\pi$
$373$		−	22.6274i		−	1.17160i	0.810454	−	0.585802i	$-0.199220\pi$
$374$	−6.00000	−	2.82843i	−0.310253	−	0.146254i
$375$		0			0
$376$		−	25.4558i		−	1.31278i
$377$		−	16.9706i		−	0.874028i
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$			2.82843i			0.144715i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$			19.7990i			1.00774i
$387$		0			0
$388$	12.0000			0.609208
$389$		−	21.2132i		−	1.07555i	−0.843088	−	0.537776i	$-0.819265\pi$
$389$		−	21.2132i		−	1.07555i	0.843088	−	0.537776i	$-0.180735\pi$
$390$		0			0
$391$		0			0
$392$	15.0000			0.757614
$393$		0			0
$394$	14.0000			0.705310
$395$		0			0
$396$		0			0
$397$	6.00000			0.301131			0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000			0.301131			0.150566	+	0.988600i	$0.451890\pi$
$398$	−20.0000			−1.00251
$399$		0			0
$400$		0			0
$401$		−	26.8701i		−	1.34183i	−0.741536	−	0.670913i	$-0.765902\pi$
$401$		−	26.8701i		−	1.34183i	0.741536	−	0.670913i	$-0.234098\pi$
$402$		0			0
$403$		0			0
$404$	−6.00000			−0.298511
$405$		0			0
$406$		−	8.48528i		−	0.421117i
$407$	18.0000	+	8.48528i	0.892227	+	0.420600i
$408$		0			0
$409$		−	8.48528i		−	0.419570i	−0.977748	−	0.209785i	$-0.932724\pi$
$409$		−	8.48528i		−	0.419570i	0.977748	−	0.209785i	$-0.0672764\pi$
$410$		0			0
$411$		0			0
$412$	−12.0000			−0.591198
$413$	12.0000			0.590481
$414$		0			0
$415$		0			0
$416$			14.1421i			0.693375i
$417$		0			0
$418$	−6.00000	+	12.7279i	−0.293470	+	0.622543i
$419$			39.5980i			1.93449i	0.253849	+	0.967244i	$0.418303\pi$
$419$			39.5980i			1.93449i	−0.253849	+	0.967244i	$0.581697\pi$
$420$		0			0
$421$	24.0000			1.16969			0.584844	−	0.811146i	$-0.301156\pi$
$421$	24.0000			1.16969			0.584844	+	0.811146i	$0.301156\pi$
$422$		−	4.24264i		−	0.206529i
$423$		0			0
$424$		−	12.7279i		−	0.618123i
$425$		0			0
$426$		0			0
$427$	−12.0000			−0.580721
$428$	−14.0000			−0.676716
$429$		0			0
$430$		0			0
$431$	−18.0000			−0.867029			−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000			−0.867029			−0.433515	+	0.901146i	$0.642727\pi$
$432$		0			0
$433$	18.0000			0.865025			0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000			0.865025			0.432512	+	0.901628i	$0.357627\pi$
$434$		0			0
$435$		0			0
$436$		−	16.9706i		−	0.812743i
$437$		0			0
$438$		0			0
$439$		−	38.1838i		−	1.82241i	−0.411951	−	0.911206i	$-0.635153\pi$
$439$		−	38.1838i		−	1.82241i	0.411951	−	0.911206i	$-0.364847\pi$
$440$		0			0
$441$		0			0
$442$		−	5.65685i		−	0.269069i
$443$			25.4558i			1.20944i	0.796437	+	0.604722i	$0.206716\pi$
$443$			25.4558i			1.20944i	−0.796437	+	0.604722i	$0.793284\pi$
$444$		0			0
$445$		0			0
$446$	24.0000			1.13643
$447$		0			0
$448$			9.89949i			0.467707i
$449$			32.5269i			1.53504i	0.641025	+	0.767520i	$0.278509\pi$
$449$			32.5269i			1.53504i	−0.641025	+	0.767520i	$0.721491\pi$
$450$		0			0
$451$	−18.0000	−	8.48528i	−0.847587	−	0.399556i
$452$		−	12.7279i		−	0.598671i
$453$		0			0
$454$	2.00000			0.0938647
$455$		0			0
$456$		0			0
$457$		−	28.2843i		−	1.32308i	−0.749909	−	0.661541i	$-0.769903\pi$
$457$		−	28.2843i		−	1.32308i	0.749909	−	0.661541i	$-0.230097\pi$
$458$	−14.0000			−0.654177
$459$		0			0
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$	−6.00000			−0.278543
$465$		0			0
$466$	−10.0000			−0.463241
$467$		−	8.48528i		−	0.392652i	−0.980539	−	0.196326i	$-0.937099\pi$
$467$		−	8.48528i		−	0.392652i	0.980539	−	0.196326i	$-0.0629011\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		−	25.4558i		−	1.17170i
$473$	2.00000	−	4.24264i	0.0919601	−	0.195077i
$474$		0			0
$475$		0			0
$476$			2.82843i			0.129641i
$477$		0			0
$478$	6.00000			0.274434
$479$	−6.00000			−0.274147			−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000			−0.274147			−0.137073	+	0.990561i	$0.543770\pi$
$480$		0			0
$481$			16.9706i			0.773791i
$482$		−	8.48528i		−	0.386494i
$483$		0			0
$484$	−7.00000	−	8.48528i	−0.318182	−	0.385695i
$485$		0			0
$486$		0			0
$487$	12.0000			0.543772			0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000			0.543772			0.271886	+	0.962329i	$0.412353\pi$
$488$			25.4558i			1.15233i
$489$		0			0
$490$		0			0
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$		0			0
$493$	−12.0000			−0.540453
$494$	−12.0000			−0.539906
$495$		0			0
$496$		0			0
$497$	12.0000			0.538274
$498$		0			0
$499$	−32.0000			−1.43252			−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000			−1.43252			−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		−	16.9706i		−	0.757433i
$503$	−16.0000			−0.713405			−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000			−0.713405			−0.356702	+	0.934218i	$0.616099\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$			18.3848i			0.815693i
$509$		−	4.24264i		−	0.188052i	−0.995570	−	0.0940259i	$-0.970026\pi$
$509$		−	4.24264i		−	0.188052i	0.995570	−	0.0940259i	$-0.0299736\pi$
$510$		0			0
$511$	−16.0000			−0.707798
$512$	11.0000			0.486136
$513$		0			0
$514$		−	29.6985i		−	1.30994i
$515$		0			0
$516$		0			0
$517$	−12.0000	+	25.4558i	−0.527759	+	1.11955i
$518$			8.48528i			0.372822i
$519$		0			0
$520$		0			0
$521$			7.07107i			0.309789i	0.987931	+	0.154895i	$0.0495038\pi$
$521$			7.07107i			0.309789i	−0.987931	+	0.154895i	$0.950496\pi$
$522$		0			0
$523$			7.07107i			0.309196i	0.987977	+	0.154598i	$0.0494083\pi$
$523$			7.07107i			0.309196i	−0.987977	+	0.154598i	$0.950592\pi$
$524$	−18.0000			−0.786334
$525$		0			0
$526$	−8.00000			−0.348817
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$	6.00000			0.260133
$533$		−	16.9706i		−	0.735077i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$			12.7279i			0.548740i
$539$	−15.0000	−	7.07107i	−0.646096	−	0.304572i
$540$		0			0
$541$			33.9411i			1.45924i	0.683851	+	0.729621i	$0.260304\pi$
$541$			33.9411i			1.45924i	−0.683851	+	0.729621i	$0.739696\pi$
$542$		−	12.7279i		−	0.546711i
$543$		0			0
$544$	10.0000			0.428746
$545$		0			0
$546$		0			0
$547$			9.89949i			0.423272i	0.977349	+	0.211636i	$0.0678791\pi$
$547$			9.89949i			0.423272i	−0.977349	+	0.211636i	$0.932121\pi$
$548$		−	4.24264i		−	0.181237i
$549$		0			0
$550$		0			0
$551$			25.4558i			1.08446i
$552$		0			0
$553$	−18.0000			−0.765438
$554$		−	31.1127i		−	1.32185i
$555$		0			0
$556$		−	4.24264i		−	0.179928i
$557$	−34.0000			−1.44063			−0.720313	−	0.693649i	$-0.756002\pi$
$557$	−34.0000			−1.44063			−0.720313	+	0.693649i	$0.756002\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$		0			0
$561$		0			0
$562$	−6.00000			−0.253095
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		0			0
$565$		0			0
$566$			1.41421i			0.0594438i
$567$		0			0
$568$		−	25.4558i		−	1.06810i
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$			4.24264i			0.177549i	0.996052	+	0.0887745i	$0.0282950\pi$
$571$			4.24264i			0.177549i	−0.996052	+	0.0887745i	$0.971705\pi$
$572$	4.00000	−	8.48528i	0.167248	−	0.354787i
$573$		0			0
$574$		−	8.48528i		−	0.354169i
$575$		0			0
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	13.0000			0.540729
$579$		0			0
$580$		0			0
$581$		−	19.7990i		−	0.821401i
$582$		0			0
$583$	−6.00000	+	12.7279i	−0.248495	+	0.527137i
$584$			33.9411i			1.40449i
$585$		0			0
$586$	−10.0000			−0.413096
$587$		−	42.4264i		−	1.75113i	−0.483105	−	0.875563i	$-0.660491\pi$
$587$		−	42.4264i		−	1.75113i	0.483105	−	0.875563i	$-0.339509\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	6.00000			0.246598
$593$	−2.00000			−0.0821302			−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000			−0.0821302			−0.0410651	+	0.999156i	$0.513075\pi$
$594$		0			0
$595$		0			0
$596$	−18.0000			−0.737309
$597$		0			0
$598$		0			0
$599$		−	11.3137i		−	0.462266i	−0.972922	−	0.231133i	$-0.925757\pi$
$599$		−	11.3137i		−	0.462266i	0.972922	−	0.231133i	$-0.0742432\pi$
$600$		0			0
$601$		−	33.9411i		−	1.38449i	−0.721664	−	0.692244i	$-0.756622\pi$
$601$		−	33.9411i		−	1.38449i	0.721664	−	0.692244i	$-0.243378\pi$
$602$	2.00000			0.0815139
$603$		0			0
$604$		−	4.24264i		−	0.172631i
$605$		0			0
$606$		0			0
$607$			32.5269i			1.32023i	0.751166	+	0.660113i	$0.229492\pi$
$607$			32.5269i			1.32023i	−0.751166	+	0.660113i	$0.770508\pi$
$608$		−	21.2132i		−	0.860309i
$609$		0			0
$610$		0			0
$611$	−24.0000			−0.970936
$612$		0			0
$613$		−	28.2843i		−	1.14239i	−0.820814	−	0.571195i	$-0.806480\pi$
$613$		−	28.2843i		−	1.14239i	0.820814	−	0.571195i	$-0.193520\pi$
$614$			24.0416i			0.970241i
$615$		0			0
$616$	6.00000	−	12.7279i	0.241747	−	0.512823i
$617$		−	4.24264i		−	0.170802i	−0.996347	−	0.0854011i	$-0.972783\pi$
$617$		−	4.24264i		−	0.170802i	0.996347	−	0.0854011i	$-0.0272172\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$			16.9706i			0.680458i
$623$	−10.0000			−0.400642
$624$		0			0
$625$		0			0
$626$	6.00000			0.239808
$627$		0			0
$628$	−24.0000			−0.957704
$629$	12.0000			0.478471
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$			38.1838i			1.51887i
$633$		0			0
$634$		−	21.2132i		−	0.842484i
$635$		0			0
$636$		0			0
$637$		−	14.1421i		−	0.560332i
$638$	18.0000	+	8.48528i	0.712627	+	0.335936i
$639$		0			0
$640$		0			0
$641$			15.5563i			0.614439i	0.951639	+	0.307219i	$0.0993986\pi$
$641$			15.5563i			0.614439i	−0.951639	+	0.307219i	$0.900601\pi$
$642$		0			0
$643$	−12.0000			−0.473234			−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000			−0.473234			−0.236617	+	0.971603i	$0.576039\pi$
$644$		0			0
$645$		0			0
$646$			8.48528i			0.333849i
$647$		−	16.9706i		−	0.667182i	−0.942718	−	0.333591i	$-0.891740\pi$
$647$		−	16.9706i		−	0.667182i	0.942718	−	0.333591i	$-0.108260\pi$
$648$		0			0
$649$	−12.0000	+	25.4558i	−0.471041	+	0.999229i
$650$		0			0
$651$		0			0
$652$	−24.0000			−0.939913
$653$		−	29.6985i		−	1.16219i	−0.813835	−	0.581096i	$-0.802624\pi$
$653$		−	29.6985i		−	1.16219i	0.813835	−	0.581096i	$-0.197376\pi$
$654$		0			0
$655$		0			0
$656$	−6.00000			−0.234261
$657$		0			0
$658$	−12.0000			−0.467809
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$	−12.0000			−0.466393
$663$		0			0
$664$	−42.0000			−1.62992
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−8.00000			−0.309529
$669$		0			0
$670$		0			0
$671$	12.0000	−	25.4558i	0.463255	−	0.982712i
$672$		0			0
$673$			36.7696i			1.41736i	0.705529	+	0.708681i	$0.250709\pi$
$673$			36.7696i			1.41736i	−0.705529	+	0.708681i	$0.749291\pi$
$674$		−	11.3137i		−	0.435788i
$675$		0			0
$676$	−5.00000			−0.192308
$677$	−46.0000			−1.76792			−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000			−1.76792			−0.883962	+	0.467559i	$0.845134\pi$
$678$		0			0
$679$		−	16.9706i		−	0.651270i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	33.9411i		−	1.29872i	−0.760481	−	0.649361i	$-0.775037\pi$
$683$		−	33.9411i		−	1.29872i	0.760481	−	0.649361i	$-0.224963\pi$
$684$		0			0
$685$		0			0
$686$		−	16.9706i		−	0.647939i
$687$		0			0
$688$		−	1.41421i		−	0.0539164i
$689$	−12.0000			−0.457164
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$	−14.0000			−0.532200
$693$		0			0
$694$	14.0000			0.531433
$695$		0			0
$696$		0			0
$697$	−12.0000			−0.454532
$698$		−	8.48528i		−	0.321173i
$699$		0			0
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$		−	25.4558i		−	0.960085i
$704$	−21.0000	−	9.89949i	−0.791467	−	0.373101i
$705$		0			0
$706$		−	4.24264i		−	0.159674i
$707$			8.48528i			0.319122i
$708$		0			0
$709$	−30.0000			−1.12667			−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000			−1.12667			−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$			21.2132i			0.794998i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	5.65685i		−	0.211407i
$717$		0			0
$718$	24.0000			0.895672
$719$		−	31.1127i		−	1.16031i	−0.814507	−	0.580154i	$-0.802992\pi$
$719$		−	31.1127i		−	1.16031i	0.814507	−	0.580154i	$-0.197008\pi$
$720$		0			0
$721$			16.9706i			0.632017i
$722$	−1.00000			−0.0372161
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−48.0000			−1.78022			−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000			−1.78022			−0.890111	+	0.455744i	$0.849373\pi$
$728$	12.0000			0.444750
$729$		0			0
$730$		0			0
$731$		−	2.82843i		−	0.104613i
$732$		0			0
$733$		−	31.1127i		−	1.14917i	−0.818444	−	0.574587i	$-0.805163\pi$
$733$		−	31.1127i		−	1.14917i	0.818444	−	0.574587i	$-0.194837\pi$
$734$	12.0000			0.442928
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	46.6690i		−	1.71675i	−0.513024	−	0.858374i	$-0.671475\pi$
$739$		−	46.6690i		−	1.71675i	0.513024	−	0.858374i	$-0.328525\pi$
$740$		0			0
$741$		0			0
$742$	−6.00000			−0.220267
$743$	−10.0000			−0.366864			−0.183432	−	0.983032i	$-0.558721\pi$
$743$	−10.0000			−0.366864			−0.183432	+	0.983032i	$0.558721\pi$
$744$		0			0
$745$		0			0
$746$			22.6274i			0.828449i
$747$		0			0
$748$	−6.00000	−	2.82843i	−0.219382	−	0.103418i
$749$			19.7990i			0.723439i
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$			8.48528i			0.309426i
$753$		0			0
$754$			16.9706i			0.618031i
$755$		0			0
$756$		0			0
$757$	42.0000			1.52652			0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000			1.52652			0.763258	+	0.646094i	$0.223599\pi$
$758$	16.0000			0.581146
$759$		0			0
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$	−24.0000			−0.868858
$764$			2.82843i			0.102329i
$765$		0			0
$766$		0			0
$767$	−24.0000			−0.866590
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$			19.7990i			0.712581i
$773$		−	4.24264i		−	0.152597i	−0.997085	−	0.0762986i	$-0.975690\pi$
$773$		−	4.24264i		−	0.152597i	0.997085	−	0.0762986i	$-0.0243102\pi$
$774$		0			0
$775$		0			0
$776$	−36.0000			−1.29232
$777$		0			0
$778$			21.2132i			0.760530i
$779$			25.4558i			0.912050i
$780$		0			0
$781$	−12.0000	+	25.4558i	−0.429394	+	0.910882i
$782$		0			0
$783$		0			0
$784$	−5.00000			−0.178571
$785$		0			0
$786$		0			0
$787$		−	9.89949i		−	0.352879i	−0.984311	−	0.176439i	$-0.943542\pi$
$787$		−	9.89949i		−	0.352879i	0.984311	−	0.176439i	$-0.0564580\pi$
$788$	14.0000			0.498729
$789$		0			0
$790$		0			0
$791$	−18.0000			−0.640006
$792$		0			0
$793$	24.0000			0.852265
$794$	−6.00000			−0.212932
$795$		0			0
$796$	−20.0000			−0.708881
$797$		−	29.6985i		−	1.05197i	−0.850493	−	0.525987i	$-0.823696\pi$
$797$		−	29.6985i		−	1.05197i	0.850493	−	0.525987i	$-0.176304\pi$
$798$		0			0
$799$			16.9706i			0.600375i
$800$		0			0
$801$		0			0
$802$			26.8701i			0.948815i
$803$	16.0000	−	33.9411i	0.564628	−	1.19776i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	18.0000			0.633238
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$		−	46.6690i		−	1.63877i	−0.573242	−	0.819386i	$-0.694315\pi$
$811$		−	46.6690i		−	1.63877i	0.573242	−	0.819386i	$-0.305685\pi$
$812$		−	8.48528i		−	0.297775i
$813$		0			0
$814$	−18.0000	−	8.48528i	−0.630900	−	0.297409i
$815$		0			0
$816$		0			0
$817$	−6.00000			−0.209913
$818$			8.48528i			0.296681i
$819$		0			0
$820$		0			0
$821$	30.0000			1.04701			0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000			1.04701			0.523504	+	0.852023i	$0.324625\pi$
$822$		0			0
$823$	−24.0000			−0.836587			−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000			−0.836587			−0.418294	+	0.908312i	$0.637372\pi$
$824$	36.0000			1.25412
$825$		0			0
$826$	−12.0000			−0.417533
$827$	34.0000			1.18230			0.591148	−	0.806563i	$-0.298675\pi$
$827$	34.0000			1.18230			0.591148	+	0.806563i	$0.298675\pi$
$828$		0			0
$829$	−6.00000			−0.208389			−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000			−0.208389			−0.104194	+	0.994557i	$0.533226\pi$
$830$		0			0
$831$		0			0
$832$		−	19.7990i		−	0.686406i
$833$	−10.0000			−0.346479
$834$		0			0
$835$		0			0
$836$	−6.00000	+	12.7279i	−0.207514	+	0.440204i
$837$		0			0
$838$		−	39.5980i		−	1.36789i
$839$			50.9117i			1.75767i	0.477129	+	0.878833i	$0.341677\pi$
$839$			50.9117i			1.75767i	−0.477129	+	0.878833i	$0.658323\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$	−24.0000			−0.827095
$843$		0			0
$844$		−	4.24264i		−	0.146038i
$845$		0			0
$846$		0			0
$847$	−12.0000	+	9.89949i	−0.412325	+	0.340151i
$848$			4.24264i			0.145693i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			28.2843i			0.968435i	0.874948	+	0.484218i	$0.160896\pi$
$853$			28.2843i			0.968435i	−0.874948	+	0.484218i	$0.839104\pi$
$854$	12.0000			0.410632
$855$		0			0
$856$	42.0000			1.43553
$857$	26.0000			0.888143			0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000			0.888143			0.444072	+	0.895991i	$0.353534\pi$
$858$		0			0
$859$	48.0000			1.63774			0.818869	−	0.573980i	$-0.194601\pi$
$859$	48.0000			1.63774			0.818869	+	0.573980i	$0.194601\pi$
$860$		0			0
$861$		0			0
$862$	18.0000			0.613082
$863$			33.9411i			1.15537i	0.816260	+	0.577685i	$0.196044\pi$
$863$			33.9411i			1.15537i	−0.816260	+	0.577685i	$0.803956\pi$
$864$		0			0
$865$		0			0
$866$	−18.0000			−0.611665
$867$		0			0
$868$		0			0
$869$	18.0000	−	38.1838i	0.610608	−	1.29530i
$870$		0			0
$871$		0			0
$872$			50.9117i			1.72409i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			11.3137i			0.382037i	0.981586	+	0.191018i	$0.0611790\pi$
$877$			11.3137i			0.382037i	−0.981586	+	0.191018i	$0.938821\pi$
$878$			38.1838i			1.28864i
$879$		0			0
$880$		0			0
$881$			21.2132i			0.714691i	0.933972	+	0.357345i	$0.116318\pi$
$881$			21.2132i			0.714691i	−0.933972	+	0.357345i	$0.883682\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		−	5.65685i		−	0.190261i
$885$		0			0
$886$		−	25.4558i		−	0.855206i
$887$	−2.00000			−0.0671534			−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000			−0.0671534			−0.0335767	+	0.999436i	$0.510690\pi$
$888$		0			0
$889$	26.0000			0.872012
$890$		0			0
$891$		0			0
$892$	24.0000			0.803579
$893$	36.0000			1.20469
$894$		0			0
$895$		0			0
$896$			4.24264i			0.141737i
$897$		0			0
$898$		−	32.5269i		−	1.08544i
$899$		0			0
$900$		0			0
$901$			8.48528i			0.282686i
$902$	18.0000	+	8.48528i	0.599334	+	0.282529i
$903$		0			0
$904$			38.1838i			1.26997i
$905$		0			0
$906$		0			0
$907$	12.0000			0.398453			0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000			0.398453			0.199227	+	0.979953i	$0.436157\pi$
$908$	2.00000			0.0663723
$909$		0			0
$910$		0			0
$911$			2.82843i			0.0937100i	0.998902	+	0.0468550i	$0.0149199\pi$
$911$			2.82843i			0.0937100i	−0.998902	+	0.0468550i	$0.985080\pi$
$912$		0			0
$913$	42.0000	+	19.7990i	1.39000	+	0.655251i
$914$			28.2843i			0.935561i
$915$		0			0
$916$	−14.0000			−0.462573
$917$			25.4558i			0.840626i
$918$		0			0
$919$		−	12.7279i		−	0.419855i	−0.977717	−	0.209928i	$-0.932677\pi$
$919$		−	12.7279i		−	0.419855i	0.977717	−	0.209928i	$-0.0673229\pi$
$920$		0			0
$921$		0			0
$922$	18.0000			0.592798
$923$	−24.0000			−0.789970
$924$		0			0
$925$		0			0
$926$	−24.0000			−0.788689
$927$		0			0
$928$	−30.0000			−0.984798
$929$		−	55.1543i		−	1.80955i	−0.425885	−	0.904777i	$-0.640037\pi$
$929$		−	55.1543i		−	1.80955i	0.425885	−	0.904777i	$-0.359963\pi$
$930$		0			0
$931$			21.2132i			0.695235i
$932$	−10.0000			−0.327561
$933$		0			0
$934$			8.48528i			0.277647i
$935$		0			0
$936$		0			0
$937$			2.82843i			0.0924007i	0.998932	+	0.0462003i	$0.0147113\pi$
$937$			2.82843i			0.0924007i	−0.998932	+	0.0462003i	$0.985289\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	30.0000			0.977972			0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000			0.977972			0.488986	+	0.872292i	$0.337367\pi$
$942$		0			0
$943$		0			0
$944$			8.48528i			0.276172i
$945$		0			0
$946$	−2.00000	+	4.24264i	−0.0650256	+	0.137940i
$947$			33.9411i			1.10294i	0.834195	+	0.551469i	$0.185933\pi$
$947$			33.9411i			1.10294i	−0.834195	+	0.551469i	$0.814067\pi$
$948$		0			0
$949$	32.0000			1.03876
$950$		0			0
$951$		0			0
$952$		−	8.48528i		−	0.275010i
$953$	−46.0000			−1.49009			−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000			−1.49009			−0.745043	+	0.667016i	$0.767571\pi$
$954$		0			0
$955$		0			0
$956$	6.00000			0.194054
$957$		0			0
$958$	6.00000			0.193851
$959$	−6.00000			−0.193750
$960$		0			0
$961$	−31.0000			−1.00000
$962$		−	16.9706i		−	0.547153i
$963$		0			0
$964$		−	8.48528i		−	0.273293i
$965$		0			0
$966$		0			0
$967$		−	52.3259i		−	1.68269i	−0.540500	−	0.841344i	$-0.681765\pi$
$967$		−	52.3259i		−	1.68269i	0.540500	−	0.841344i	$-0.318235\pi$
$968$	21.0000	+	25.4558i	0.674966	+	0.818182i
$969$		0			0
$970$		0			0
$971$			19.7990i			0.635380i	0.948195	+	0.317690i	$0.102907\pi$
$971$			19.7990i			0.635380i	−0.948195	+	0.317690i	$0.897093\pi$
$972$		0			0
$973$	−6.00000			−0.192351
$974$	−12.0000			−0.384505
$975$		0			0
$976$		−	8.48528i		−	0.271607i
$977$		−	38.1838i		−	1.22161i	−0.791782	−	0.610803i	$-0.790847\pi$
$977$		−	38.1838i		−	1.22161i	0.791782	−	0.610803i	$-0.209153\pi$
$978$		0			0
$979$	10.0000	−	21.2132i	0.319601	−	0.677977i
$980$		0			0
$981$		0			0
$982$	−36.0000			−1.14881
$983$			50.9117i			1.62383i	0.583775	+	0.811915i	$0.301575\pi$
$983$			50.9117i			1.62383i	−0.583775	+	0.811915i	$0.698425\pi$
$984$		0			0
$985$		0			0
$986$	12.0000			0.382158
$987$		0			0
$988$	−12.0000			−0.381771
$989$		0			0
$990$		0			0
$991$	−12.0000			−0.381193			−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000			−0.381193			−0.190596	+	0.981669i	$0.561042\pi$
$992$		0			0
$993$		0			0
$994$	−12.0000			−0.380617
$995$		0			0
$996$		0			0
$997$		−	39.5980i		−	1.25408i	−0.778987	−	0.627040i	$-0.784266\pi$
$997$		−	39.5980i		−	1.25408i	0.778987	−	0.627040i	$-0.215734\pi$
$998$	32.0000			1.01294
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.f.a.2276.2		2
3.2	odd	2		2475.2.f.d.2276.2		2
5.2	odd	4		495.2.d.a.494.1	✓	4
5.3	odd	4		495.2.d.a.494.4	yes	4
5.4	even	2		2475.2.f.c.2276.1		2
11.10	odd	2		2475.2.f.d.2276.1		2
15.2	even	4		495.2.d.b.494.4	yes	4
15.8	even	4		495.2.d.b.494.1	yes	4
15.14	odd	2		2475.2.f.b.2276.1		2
33.32	even	2	inner	2475.2.f.a.2276.1		2
55.32	even	4		495.2.d.b.494.3	yes	4
55.43	even	4		495.2.d.b.494.2	yes	4
55.54	odd	2		2475.2.f.b.2276.2		2
165.32	odd	4		495.2.d.a.494.2	yes	4
165.98	odd	4		495.2.d.a.494.3	yes	4
165.164	even	2		2475.2.f.c.2276.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.d.a.494.1	✓	4	5.2	odd	4
495.2.d.a.494.2	yes	4	165.32	odd	4
495.2.d.a.494.3	yes	4	165.98	odd	4
495.2.d.a.494.4	yes	4	5.3	odd	4
495.2.d.b.494.1	yes	4	15.8	even	4
495.2.d.b.494.2	yes	4	55.43	even	4
495.2.d.b.494.3	yes	4	55.32	even	4
495.2.d.b.494.4	yes	4	15.2	even	4
2475.2.f.a.2276.1		2	33.32	even	2	inner
2475.2.f.a.2276.2		2	1.1	even	1	trivial
2475.2.f.b.2276.1		2	15.14	odd	2
2475.2.f.b.2276.2		2	55.54	odd	2
2475.2.f.c.2276.1		2	5.4	even	2
2475.2.f.c.2276.2		2	165.164	even	2
2475.2.f.d.2276.1		2	11.10	odd	2
2475.2.f.d.2276.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{4} +1.41421i q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{4} +1.41421i q^{7} +3.00000 q^{8} +(-3.00000 - 1.41421i) q^{11} -2.82843i q^{13} -1.41421i q^{14} -1.00000 q^{16} -2.00000 q^{17} +4.24264i q^{19} +(3.00000 + 1.41421i) q^{22} +2.82843i q^{26} -1.41421i q^{28} +6.00000 q^{29} -5.00000 q^{32} +2.00000 q^{34} -6.00000 q^{37} -4.24264i q^{38} +6.00000 q^{41} +1.41421i q^{43} +(3.00000 + 1.41421i) q^{44} -8.48528i q^{47} +5.00000 q^{49} +2.82843i q^{52} -4.24264i q^{53} +4.24264i q^{56} -6.00000 q^{58} -8.48528i q^{59} +8.48528i q^{61} +7.00000 q^{64} +2.00000 q^{68} -8.48528i q^{71} +11.3137i q^{73} +6.00000 q^{74} -4.24264i q^{76} +(2.00000 - 4.24264i) q^{77} +12.7279i q^{79} -6.00000 q^{82} -14.0000 q^{83} -1.41421i q^{86} +(-9.00000 - 4.24264i) q^{88} +7.07107i q^{89} +4.00000 q^{91} +8.48528i q^{94} -12.0000 q^{97} -5.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{4} + 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^4 + 6 * q^8	\( 2 q - 2 q^{2} - 2 q^{4} + 6 q^{8} - 6 q^{11} - 2 q^{16} - 4 q^{17} + 6 q^{22} + 12 q^{29} - 10 q^{32} + 4 q^{34} - 12 q^{37} + 12 q^{41} + 6 q^{44} + 10 q^{49} - 12 q^{58} + 14 q^{64} + 4 q^{68} + 12 q^{74} + 4 q^{77} - 12 q^{82} - 28 q^{83} - 18 q^{88} + 8 q^{91} - 24 q^{97} - 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^4 + 6 * q^8 - 6 * q^11 - 2 * q^16 - 4 * q^17 + 6 * q^22 + 12 * q^29 - 10 * q^32 + 4 * q^34 - 12 * q^37 + 12 * q^41 + 6 * q^44 + 10 * q^49 - 12 * q^58 + 14 * q^64 + 4 * q^68 + 12 * q^74 + 4 * q^77 - 12 * q^82 - 28 * q^83 - 18 * q^88 + 8 * q^91 - 24 * q^97 - 10 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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