LMFDB - Embedded newform 275.2.e.c.32.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,2,Mod(32,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.32"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	275.e (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.19588605559$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.1499238400.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 59x^4 - 66x^3 + 54x^2 - 26*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			32.3
Root			$0.500000 + 1.08454i$ of defining polynomial
Character	$\chi$	$=$	275.32
Dual form			275.2.e.c.43.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.584541 - 0.584541i) q^{2} +1.31662i q^{4} +(-3.69232 + 3.69232i) q^{7} +(1.93870 + 1.93870i) q^{8} +3.00000i q^{9} +3.31662 q^{11} +(-2.52324 - 2.52324i) q^{13} +4.31662i q^{14} -0.366750 q^{16} +(4.86140 - 4.86140i) q^{17} +(1.75362 + 1.75362i) q^{18} +(1.93870 - 1.93870i) q^{22} -2.94987 q^{26} +(-4.86140 - 4.86140i) q^{28} +6.63325 q^{31} +(-4.09178 + 4.09178i) q^{32} -5.68338i q^{34} -3.94987 q^{36} +(6.03049 + 6.03049i) q^{43} +4.36675i q^{44} -20.2665i q^{49} +(3.32216 - 3.32216i) q^{52} -14.3166 q^{56} -4.00000i q^{59} +(3.87740 - 3.87740i) q^{62} +(-11.0770 - 11.0770i) q^{63} +4.05013i q^{64} +(6.40065 + 6.40065i) q^{68} -8.00000 q^{71} +(-5.81610 + 5.81610i) q^{72} +(0.185080 + 0.185080i) q^{73} +(-12.2461 + 12.2461i) q^{77} -9.00000 q^{81} +(8.73881 + 8.73881i) q^{83} +7.05013 q^{86} +(6.42995 + 6.42995i) q^{88} -13.2665i q^{89} +18.6332 q^{91} +(-11.8466 - 11.8466i) q^{98} +9.94987i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 56 q^{16} + 56 q^{26} + 48 q^{36} - 88 q^{56} - 64 q^{71} - 72 q^{81} + 136 q^{86} + 96 q^{91}+O(q^{100}) $ 8 * q - 56 * q^16 + 56 * q^26 + 48 * q^36 - 88 * q^56 - 64 * q^71 - 72 * q^81 + 136 * q^86 + 96 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	0.584541	−	0.584541i	0.413333	−	0.413333i	−0.469565	−	0.882898i	$-0.655589\pi$
$2$	0.584541	−	0.584541i	0.413333	−	0.413333i	0.882898	+	0.469565i	$0.155589\pi$
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$			1.31662i			0.658312i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.69232	+	3.69232i	−1.39557	+	1.39557i	−0.583336	+	0.812231i	$0.698253\pi$
$7$	−3.69232	+	3.69232i	−1.39557	+	1.39557i	−0.812231	+	0.583336i	$0.801747\pi$
$8$	1.93870	+	1.93870i	0.685435	+	0.685435i
$9$			3.00000i			1.00000i
$10$		0			0
$11$	3.31662			1.00000
$11$	3.31662			1.00000
$12$		0			0
$13$	−2.52324	−	2.52324i	−0.699821	−	0.699821i	0.264550	−	0.964372i	$-0.414776\pi$
$13$	−2.52324	−	2.52324i	−0.699821	−	0.699821i	−0.964372	+	0.264550i	$0.914776\pi$
$14$			4.31662i			1.15367i
$15$		0			0
$16$	−0.366750			−0.0916876
$17$	4.86140	−	4.86140i	1.17906	−	1.17906i	0.199081	−	0.979983i	$-0.436204\pi$
$17$	4.86140	−	4.86140i	1.17906	−	1.17906i	0.979983	−	0.199081i	$-0.0637955\pi$
$18$	1.75362	+	1.75362i	0.413333	+	0.413333i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	1.93870	−	1.93870i	0.413333	−	0.413333i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$		0			0
$26$	−2.94987			−0.578518
$27$		0			0
$28$	−4.86140	−	4.86140i	−0.918719	−	0.918719i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	6.63325			1.19137			0.595683	−	0.803219i	$-0.296881\pi$
$31$	6.63325			1.19137			0.595683	+	0.803219i	$0.296881\pi$
$32$	−4.09178	+	4.09178i	−0.723332	+	0.723332i
$33$		0			0
$34$		−	5.68338i		−	0.974691i
$35$		0			0
$36$	−3.94987			−0.658312
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	6.03049	+	6.03049i	0.919640	+	0.919640i	0.997003	−	0.0773627i	$-0.0246499\pi$
$43$	6.03049	+	6.03049i	0.919640	+	0.919640i	−0.0773627	+	0.997003i	$0.524650\pi$
$44$			4.36675i			0.658312i
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	20.2665i		−	2.89521i
$50$		0			0
$51$		0			0
$52$	3.32216	−	3.32216i	0.460701	−	0.460701i
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$	−14.3166			−1.91314
$57$		0			0
$58$		0			0
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	3.87740	−	3.87740i	0.492431	−	0.492431i
$63$	−11.0770	−	11.0770i	−1.39557	−	1.39557i
$64$			4.05013i			0.506266i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	6.40065	+	6.40065i	0.776192	+	0.776192i
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$	−5.81610	+	5.81610i	−0.685435	+	0.685435i
$73$	0.185080	+	0.185080i	0.0216620	+	0.0216620i	0.717855	−	0.696193i	$-0.245124\pi$
$73$	0.185080	+	0.185080i	0.0216620	+	0.0216620i	−0.696193	+	0.717855i	$0.745124\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.2461	+	12.2461i	−1.39557	+	1.39557i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	8.73881	+	8.73881i	0.959209	+	0.959209i	0.999200	−	0.0399913i	$-0.0127330\pi$
$83$	8.73881	+	8.73881i	0.959209	+	0.959209i	−0.0399913	+	0.999200i	$0.512733\pi$
$84$		0			0
$85$		0			0
$86$	7.05013			0.760235
$87$		0			0
$88$	6.42995	+	6.42995i	0.685435	+	0.685435i
$89$		−	13.2665i		−	1.40625i	−0.711068	−	0.703123i	$-0.751788\pi$
$89$		−	13.2665i		−	1.40625i	0.711068	−	0.703123i	$-0.248212\pi$
$90$		0			0
$91$	18.6332			1.95330
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	−11.8466	−	11.8466i	−1.19669	−	1.19669i
$99$			9.94987i			1.00000i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		−	9.78363i		−	0.959364i
$105$		0			0
$106$		0			0
$107$	13.4151	−	13.4151i	1.29689	−	1.29689i	0.366453	−	0.930436i	$-0.380572\pi$
$107$	13.4151	−	13.4151i	1.29689	−	1.29689i	0.930436	−	0.366453i	$-0.119428\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$	1.35416	−	1.35416i	0.127956	−	0.127956i
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.56973	−	7.56973i	0.699821	−	0.699821i
$118$	−2.33816	−	2.33816i	−0.215245	−	0.215245i
$119$			35.8997i			3.29092i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$			8.73350i			0.784292i
$125$		0			0
$126$	−12.9499			−1.15367
$127$	−0.984001	+	0.984001i	−0.0873160	+	0.0873160i	−0.749416	−	0.662100i	$-0.769666\pi$
$127$	−0.984001	+	0.984001i	−0.0873160	+	0.0873160i	0.662100	+	0.749416i	$0.269666\pi$
$128$	−5.81610	−	5.81610i	−0.514076	−	0.514076i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	18.8496			1.61634
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−4.67632	+	4.67632i	−0.392428	+	0.392428i
$143$	−8.36865	−	8.36865i	−0.699821	−	0.699821i
$144$		−	1.10025i		−	0.0916876i
$145$		0			0
$146$	0.216374			0.0179072
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	14.5842	+	14.5842i	1.17906	+	1.17906i
$154$			14.3166i			1.15367i
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−5.26086	+	5.26086i	−0.413333	+	0.413333i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	10.2164			0.792944
$167$	−6.40065	+	6.40065i	−0.495297	+	0.495297i	−0.909970	−	0.414673i	$-0.863896\pi$
$167$	−6.40065	+	6.40065i	−0.495297	+	0.495297i	0.414673	+	0.909970i	$0.363896\pi$
$168$		0			0
$169$		−	0.266499i		−	0.0204999i
$170$		0			0
$171$		0			0
$172$	−7.93989	+	7.93989i	−0.605411	+	0.605411i
$173$	17.2925	+	17.2925i	1.31473	+	1.31473i	0.917888	+	0.396839i	$0.129893\pi$
$173$	17.2925	+	17.2925i	1.31473	+	1.31473i	0.396839	+	0.917888i	$0.370107\pi$
$174$		0			0
$175$		0			0
$176$	−1.21637			−0.0916876
$177$		0			0
$178$	−7.75481	−	7.75481i	−0.581247	−	0.581247i
$179$			19.8997i			1.48738i	0.668526	+	0.743689i	$0.266925\pi$
$179$			19.8997i			1.48738i	−0.668526	+	0.743689i	$0.733075\pi$
$180$		0			0
$181$	−26.5330			−1.97218			−0.986091	−	0.166206i	$-0.946848\pi$
$181$	−26.5330			−1.97218			−0.986091	+	0.166206i	$0.946848\pi$
$182$	10.8919	−	10.8919i	0.807361	−	0.807361i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.1235	−	16.1235i	1.17906	−	1.17906i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.63325			0.479965			0.239983	−	0.970777i	$-0.422858\pi$
$191$	6.63325			0.479965			0.239983	+	0.970777i	$0.422858\pi$
$192$		0			0
$193$	−19.6307	−	19.6307i	−1.41305	−	1.41305i	−0.735221	−	0.677827i	$-0.762922\pi$
$193$	−19.6307	−	19.6307i	−1.41305	−	1.41305i	−0.677827	−	0.735221i	$-0.737078\pi$
$194$		0			0
$195$		0			0
$196$	26.6834			1.90596
$197$	2.15308	−	2.15308i	0.153401	−	0.153401i	−0.626234	−	0.779635i	$-0.715405\pi$
$197$	2.15308	−	2.15308i	0.153401	−	0.153401i	0.779635	+	0.626234i	$0.215405\pi$
$198$	5.81610	+	5.81610i	0.413333	+	0.413333i
$199$		−	24.0000i		−	1.70131i	−0.525720	−	0.850657i	$-0.676204\pi$
$199$		−	24.0000i		−	1.70131i	0.525720	−	0.850657i	$-0.323796\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.925400	+	0.925400i	0.0641650	+	0.0641650i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		−	15.6834i		−	1.07209i
$215$		0			0
$216$		0			0
$217$	−24.4921	+	24.4921i	−1.66263	+	1.66263i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−24.5330			−1.65027
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		−	30.2164i		−	2.01892i
$225$		0			0
$226$		0			0
$227$	−20.7998	+	20.7998i	−1.38053	+	1.38053i	−0.536855	+	0.843674i	$0.680388\pi$
$227$	−20.7998	+	20.7998i	−1.38053	+	1.38053i	−0.843674	+	0.536855i	$0.819612\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.23156	−	5.23156i	−0.342731	−	0.342731i	0.514662	−	0.857393i	$-0.327917\pi$
$233$	−5.23156	−	5.23156i	−0.342731	−	0.342731i	−0.857393	+	0.514662i	$0.827917\pi$
$234$		−	8.84962i		−	0.578518i
$235$		0			0
$236$	5.26650			0.342820
$237$		0			0
$238$	20.9849	+	20.9849i	1.36025	+	1.36025i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	6.42995	−	6.42995i	0.413333	−	0.413333i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	12.8599	+	12.8599i	0.816604	+	0.816604i
$249$		0			0
$250$		0			0
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$	14.5842	−	14.5842i	0.918719	−	0.918719i
$253$		0			0
$254$			1.15038i			0.0721811i
$255$		0			0
$256$	−14.8997			−0.931234
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.7853	−	13.7853i	−0.850037	−	0.850037i	0.140100	−	0.990137i	$-0.455258\pi$
$263$	−13.7853	−	13.7853i	−0.850037	−	0.850037i	−0.990137	+	0.140100i	$0.955258\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	14.0000i		−	0.853595i	−0.904347	−	0.426798i	$-0.859642\pi$
$269$		−	14.0000i		−	0.853595i	0.904347	−	0.426798i	$-0.140358\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−1.78292	+	1.78292i	−0.108106	+	0.108106i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	21.9689	−	21.9689i	1.31998	−	1.31998i	0.406194	−	0.913787i	$-0.366856\pi$
$277$	21.9689	−	21.9689i	1.31998	−	1.31998i	0.913787	−	0.406194i	$-0.133144\pi$
$278$		0			0
$279$			19.8997i			1.19137i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	3.32216	+	3.32216i	0.197482	+	0.197482i	0.798920	−	0.601438i	$-0.205405\pi$
$283$	3.32216	+	3.32216i	0.197482	+	0.197482i	−0.601438	+	0.798920i	$0.705405\pi$
$284$		−	10.5330i		−	0.625018i
$285$		0			0
$286$	−9.78363			−0.578518
$287$		0			0
$288$	−12.2754	−	12.2754i	−0.723332	−	0.723332i
$289$		−	30.2665i		−	1.78038i
$290$		0			0
$291$		0			0
$292$	−0.243681	+	0.243681i	−0.0142603	+	0.0142603i
$293$	−16.9224	−	16.9224i	−0.988616	−	0.988616i	0.0113203	−	0.999936i	$-0.496397\pi$
$293$	−16.9224	−	16.9224i	−0.988616	−	0.988616i	−0.999936	+	0.0113203i	$0.996397\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−44.5330			−2.56684
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	17.0501			0.974691
$307$	10.7068	−	10.7068i	0.611070	−	0.611070i	−0.332155	−	0.943225i	$-0.607776\pi$
$307$	10.7068	−	10.7068i	0.611070	−	0.611070i	0.943225	+	0.332155i	$0.107776\pi$
$308$	−16.1235	−	16.1235i	−0.918719	−	0.918719i
$309$		0			0
$310$		0			0
$311$	32.0000			1.81455			0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000			1.81455			0.907277	+	0.420534i	$0.138157\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	11.8496i		−	0.658312i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.63325			0.364596			0.182298	−	0.983243i	$-0.441646\pi$
$331$	6.63325			0.364596			0.182298	+	0.983243i	$0.441646\pi$
$332$	−11.5057	+	11.5057i	−0.631459	+	0.631459i
$333$		0			0
$334$			7.48287i			0.409445i
$335$		0			0
$336$		0			0
$337$	24.6772	−	24.6772i	1.34425	−	1.34425i	0.452474	−	0.891778i	$-0.350541\pi$
$337$	24.6772	−	24.6772i	1.34425	−	1.34425i	0.891778	−	0.452474i	$-0.149459\pi$
$338$	−0.155780	−	0.155780i	−0.00847329	−	0.00847329i
$339$		0			0
$340$		0			0
$341$	22.0000			1.19137
$342$		0			0
$343$	48.9842	+	48.9842i	2.64490	+	2.64490i
$344$			23.3826i			1.26071i
$345$		0			0
$346$	20.2164			1.08684
$347$	−23.5081	+	23.5081i	−1.26198	+	1.26198i	−0.311849	+	0.950132i	$0.600948\pi$
$347$	−23.5081	+	23.5081i	−1.26198	+	1.26198i	−0.950132	+	0.311849i	$0.899052\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−13.5709	+	13.5709i	−0.723332	+	0.723332i
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$	17.4670			0.925749
$357$		0			0
$358$	11.6322	+	11.6322i	0.614781	+	0.614781i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$	−15.5096	+	15.5096i	−0.815167	+	0.815167i
$363$		0			0
$364$			24.5330i			1.28588i
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−22.3390	−	22.3390i	−1.15667	−	1.15667i	−0.985187	−	0.171484i	$-0.945144\pi$
$373$	−22.3390	−	22.3390i	−1.15667	−	1.15667i	−0.171484	−	0.985187i	$-0.554856\pi$
$374$		−	18.8496i		−	0.974691i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			36.0000i			1.84920i	0.380945	+	0.924598i	$0.375599\pi$
$379$			36.0000i			1.84920i	−0.380945	+	0.924598i	$0.624401\pi$
$380$		0			0
$381$		0			0
$382$	3.87740	−	3.87740i	0.198385	−	0.198385i
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$	−22.9499			−1.16812
$387$	−18.0915	+	18.0915i	−0.919640	+	0.919640i
$388$		0			0
$389$			26.0000i			1.31825i	0.752032	+	0.659126i	$0.229074\pi$
$389$			26.0000i			1.31825i	−0.752032	+	0.659126i	$0.770926\pi$
$390$		0			0
$391$		0			0
$392$	39.2907	−	39.2907i	1.98448	−	1.98448i
$393$		0			0
$394$		−	2.51713i		−	0.126811i
$395$		0			0
$396$	−13.1003			−0.658312
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	−14.0290	−	14.0290i	−0.703209	−	0.703209i
$399$		0			0
$400$		0			0
$401$	39.7995			1.98749			0.993746	−	0.111664i	$-0.0356180\pi$
$401$	39.7995			1.98749			0.993746	+	0.111664i	$0.0356180\pi$
$402$		0			0
$403$	−16.7373	−	16.7373i	−0.833744	−	0.833744i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	14.7693	+	14.7693i	0.726749	+	0.726749i
$414$		0			0
$415$		0			0
$416$	20.6491			1.01241
$417$		0			0
$418$		0			0
$419$			19.8997i			0.972166i	0.873913	+	0.486083i	$0.161575\pi$
$419$			19.8997i			0.972166i	−0.873913	+	0.486083i	$0.838425\pi$
$420$		0			0
$421$	−26.5330			−1.29314			−0.646570	−	0.762855i	$-0.723797\pi$
$421$	−26.5330			−1.29314			−0.646570	+	0.762855i	$0.723797\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	17.6627	+	17.6627i	0.853759	+	0.853759i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$			28.6332i			1.37444i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	60.7995			2.89521
$442$	−14.3405	+	14.3405i	−0.682110	+	0.682110i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−14.9544	−	14.9544i	−0.706528	−	0.706528i
$449$		−	13.2665i		−	0.626085i	−0.949739	−	0.313042i	$-0.898652\pi$
$449$		−	13.2665i		−	0.626085i	0.949739	−	0.313042i	$-0.101348\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$			24.3166i			1.14124i
$455$		0			0
$456$		0			0
$457$	−29.3535	+	29.3535i	−1.37310	+	1.37310i	−0.517287	+	0.855812i	$0.673058\pi$
$457$	−29.3535	+	29.3535i	−1.37310	+	1.37310i	−0.855812	+	0.517287i	$0.826942\pi$
$458$	3.50724	+	3.50724i	0.163883	+	0.163883i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	−6.11612			−0.283324
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	9.96649	+	9.96649i	0.460701	+	0.460701i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	7.75481	−	7.75481i	0.356944	−	0.356944i
$473$	20.0009	+	20.0009i	0.919640	+	0.919640i
$474$		0			0
$475$		0			0
$476$	−47.2665			−2.16646
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$			14.4829i			0.658312i
$485$		0			0
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−2.43275			−0.109234
$497$	29.5386	−	29.5386i	1.32499	−	1.32499i
$498$		0			0
$499$		−	4.00000i		−	0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$		−	4.00000i		−	0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$		0			0
$501$		0			0
$502$	−16.3671	+	16.3671i	−0.730501	+	0.730501i
$503$	−5.66033	−	5.66033i	−0.252381	−	0.252381i	0.569565	−	0.821946i	$-0.307112\pi$
$503$	−5.66033	−	5.66033i	−0.252381	−	0.252381i	−0.821946	+	0.569565i	$0.807112\pi$
$504$		−	42.9499i		−	1.91314i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1.29556	−	1.29556i	−0.0574812	−	0.0574812i
$509$		−	34.0000i		−	1.50702i	−0.657434	−	0.753512i	$-0.728358\pi$
$509$		−	34.0000i		−	1.50702i	0.657434	−	0.753512i	$-0.271642\pi$
$510$		0			0
$511$	−1.36675			−0.0604615
$512$	2.92270	−	2.92270i	0.129166	−	0.129166i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	39.7995			1.74365			0.871824	−	0.489820i	$-0.162937\pi$
$521$	39.7995			1.74365			0.871824	+	0.489820i	$0.162937\pi$
$522$		0			0
$523$	−25.4761	−	25.4761i	−1.11399	−	1.11399i	−0.992605	−	0.121387i	$-0.961266\pi$
$523$	−25.4761	−	25.4761i	−1.11399	−	1.11399i	−0.121387	−	0.992605i	$-0.538734\pi$
$524$		0			0
$525$		0			0
$526$	−16.1161			−0.702696
$527$	32.2469	−	32.2469i	1.40470	−	1.40470i
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−8.18357	−	8.18357i	−0.352819	−	0.352819i
$539$		−	67.2164i		−	2.89521i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$			39.7836i			1.70571i
$545$		0			0
$546$		0			0
$547$	18.8318	−	18.8318i	0.805189	−	0.805189i	−0.178713	−	0.983901i	$-0.557193\pi$
$547$	18.8318	−	18.8318i	0.805189	−	0.805189i	0.983901	+	0.178713i	$0.0571933\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	25.6834i		−	1.09118i
$555$		0			0
$556$		0			0
$557$	−32.0618	+	32.0618i	−1.35850	+	1.35850i	−0.482739	+	0.875764i	$0.660358\pi$
$557$	−32.0618	+	32.0618i	−1.35850	+	1.35850i	−0.875764	+	0.482739i	$0.839642\pi$
$558$	11.6322	+	11.6322i	0.492431	+	0.492431i
$559$		−	30.4327i		−	1.28717i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−30.8927	−	30.8927i	−1.30197	−	1.30197i	−0.927059	−	0.374915i	$-0.877672\pi$
$563$	−30.8927	−	30.8927i	−1.30197	−	1.30197i	−0.374915	−	0.927059i	$-0.622328\pi$
$564$		0			0
$565$		0			0
$566$	3.88388			0.163252
$567$	33.2309	−	33.2309i	1.39557	−	1.39557i
$568$	−15.5096	−	15.5096i	−0.650769	−	0.650769i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	11.0184	−	11.0184i	0.460701	−	0.460701i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−12.1504			−0.506266
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−17.6920	−	17.6920i	−0.735890	−	0.735890i
$579$		0			0
$580$		0			0
$581$	−64.5330			−2.67728
$582$		0			0
$583$		0			0
$584$			0.717630i			0.0296957i
$585$		0			0
$586$	−19.7836			−0.817254
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	34.4000	+	34.4000i	1.41264	+	1.41264i	0.739686	+	0.672953i	$0.234974\pi$
$593$	34.4000	+	34.4000i	1.41264	+	1.41264i	0.672953	+	0.739686i	$0.265026\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			19.8997i			0.813082i	0.913633	+	0.406541i	$0.133265\pi$
$599$			19.8997i			0.813082i	−0.913633	+	0.406541i	$0.866735\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−26.0313	+	26.0313i	−1.06096	+	1.06096i
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−9.10897	+	9.10897i	−0.369722	+	0.369722i	−0.867376	−	0.497654i	$-0.834195\pi$
$607$	−9.10897	+	9.10897i	−0.369722	+	0.369722i	0.497654	+	0.867376i	$0.334195\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−19.2019	+	19.2019i	−0.776192	+	0.776192i
$613$	31.6917	+	31.6917i	1.28001	+	1.28001i	0.940658	+	0.339357i	$0.110209\pi$
$613$	31.6917	+	31.6917i	1.28001	+	1.28001i	0.339357	+	0.940658i	$0.389791\pi$
$614$		−	12.5171i		−	0.505150i
$615$		0			0
$616$	−47.4829			−1.91314
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		−	46.4327i		−	1.86629i	−0.359501	−	0.933145i	$-0.617053\pi$
$619$		−	46.4327i		−	1.86629i	0.359501	−	0.933145i	$-0.382947\pi$
$620$		0			0
$621$		0			0
$622$	18.7053	−	18.7053i	0.750014	−	0.750014i
$623$	48.9842	+	48.9842i	1.96251	+	1.96251i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−48.0000			−1.91085			−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000			−1.91085			−0.955425	+	0.295234i	$0.904602\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−51.1373	+	51.1373i	−2.02613	+	2.02613i
$638$		0			0
$639$		−	24.0000i		−	0.949425i
$640$		0			0
$641$	39.7995			1.57199			0.785993	−	0.618236i	$-0.212152\pi$
$641$	39.7995			1.57199			0.785993	+	0.618236i	$0.212152\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	−17.4483	−	17.4483i	−0.685435	−	0.685435i
$649$		−	13.2665i		−	0.520756i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.555240	+	0.555240i	−0.0216620	+	0.0216620i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	42.0000			1.63361			0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000			1.63361			0.816805	+	0.576913i	$0.195743\pi$
$662$	3.87740	−	3.87740i	0.150700	−	0.150700i
$663$		0			0
$664$			33.8839i			1.31495i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−8.42725	−	8.42725i	−0.326060	−	0.326060i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−14.2141	−	14.2141i	−0.547911	−	0.547911i	0.377925	−	0.925836i	$-0.376638\pi$
$673$	−14.2141	−	14.2141i	−0.547911	−	0.547911i	−0.925836	+	0.377925i	$0.876638\pi$
$674$		−	28.8496i		−	1.11125i
$675$		0			0
$676$	0.350879			0.0134954
$677$	27.3855	−	27.3855i	1.05251	−	1.05251i	0.0539677	−	0.998543i	$-0.482813\pi$
$677$	27.3855	−	27.3855i	1.05251	−	1.05251i	0.998543	−	0.0539677i	$-0.0171868\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	12.8599	−	12.8599i	0.492431	−	0.492431i
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$	57.2665			2.18645
$687$		0			0
$688$	−2.21168	−	2.21168i	−0.0843196	−	0.0843196i
$689$		0			0
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$	−22.7678	+	22.7678i	−0.865501	+	0.865501i
$693$	−36.7382	−	36.7382i	−1.39557	−	1.39557i
$694$			27.4829i			1.04324i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$			13.4327i			0.506266i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			53.0660i			1.99294i	0.0839773	+	0.996468i	$0.473238\pi$
$709$			53.0660i			1.99294i	−0.0839773	+	0.996468i	$0.526762\pi$
$710$		0			0
$711$		0			0
$712$	25.7198	−	25.7198i	0.963890	−	0.963890i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−26.2005			−0.979159
$717$		0			0
$718$		0			0
$719$		−	46.4327i		−	1.73165i	−0.500348	−	0.865825i	$-0.666794\pi$
$719$		−	46.4327i		−	1.73165i	0.500348	−	0.865825i	$-0.333206\pi$
$720$		0			0
$721$		0			0
$722$	−11.1063	+	11.1063i	−0.413333	+	0.413333i
$723$		0			0
$724$		−	34.9340i		−	1.29831i
$725$		0			0
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$	36.1243	+	36.1243i	1.33886	+	1.33886i
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	58.6332			2.16863
$732$		0			0
$733$	−7.93989	−	7.93989i	−0.293266	−	0.293266i	0.545103	−	0.838369i	$-0.316491\pi$
$733$	−7.93989	−	7.93989i	−0.293266	−	0.293266i	−0.838369	+	0.545103i	$0.816491\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.4936	−	16.4936i	−0.605092	−	0.605092i	0.336567	−	0.941659i	$-0.390734\pi$
$743$	−16.4936	−	16.4936i	−0.605092	−	0.605092i	−0.941659	+	0.336567i	$0.890734\pi$
$744$		0			0
$745$		0			0
$746$	−26.1161			−0.956179
$747$	−26.2164	+	26.2164i	−0.959209	+	0.959209i
$748$	21.2285	+	21.2285i	0.776192	+	0.776192i
$749$			99.0660i			3.61979i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	21.0435	+	21.0435i	0.764333	+	0.764333i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$			8.73350i			0.315967i
$765$		0			0
$766$		0			0
$767$	−10.0930	+	10.0930i	−0.364436	+	0.364436i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	25.8463	−	25.8463i	0.930227	−	0.930227i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$			21.1504i			0.760235i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	15.1981	+	15.1981i	0.544877	+	0.544877i
$779$		0			0
$780$		0			0
$781$	−26.5330			−0.949425
$782$		0			0
$783$		0			0
$784$			7.43275i			0.265455i
$785$		0			0
$786$		0			0
$787$	7.99849	−	7.99849i	0.285115	−	0.285115i	−0.550030	−	0.835145i	$-0.685384\pi$
$787$	7.99849	−	7.99849i	0.285115	−	0.285115i	0.835145	+	0.550030i	$0.185384\pi$
$788$	2.83480	+	2.83480i	0.100986	+	0.100986i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−19.2898	+	19.2898i	−0.685435	+	0.685435i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	31.5990			1.12000
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	39.7995			1.40625
$802$	23.2644	−	23.2644i	0.821495	−	0.821495i
$803$	0.613841	+	0.613841i	0.0216620	+	0.0216620i
$804$		0			0
$805$		0			0
$806$	−19.5673			−0.689227
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$			55.8997i			1.95330i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$	17.2665			0.600778
$827$	−15.3831	+	15.3831i	−0.534924	+	0.534924i	−0.922034	−	0.387110i	$-0.873473\pi$
$827$	−15.3831	+	15.3831i	−0.534924	+	0.534924i	0.387110	+	0.922034i	$0.373473\pi$
$828$		0			0
$829$			53.0660i			1.84306i	0.388309	+	0.921529i	$0.373059\pi$
$829$			53.0660i			1.84306i	−0.388309	+	0.921529i	$0.626941\pi$
$830$		0			0
$831$		0			0
$832$	10.2194	−	10.2194i	0.354296	−	0.354296i
$833$	−98.5236	−	98.5236i	−3.41364	−	3.41364i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	11.6322	+	11.6322i	0.401828	+	0.401828i
$839$			56.0000i			1.93333i	0.256036	+	0.966667i	$0.417584\pi$
$839$			56.0000i			1.93333i	−0.256036	+	0.966667i	$0.582416\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	−15.5096	+	15.5096i	−0.534497	+	0.534497i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−40.6156	+	40.6156i	−1.39557	+	1.39557i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	37.1083	+	37.1083i	1.27056	+	1.27056i	0.945792	+	0.324772i	$0.105288\pi$
$853$	37.1083	+	37.1083i	1.27056	+	1.27056i	0.324772	+	0.945792i	$0.394712\pi$
$854$		0			0
$855$		0			0
$856$	52.0159			1.77787
$857$	39.0763	−	39.0763i	1.33482	−	1.33482i	0.433824	−	0.900998i	$-0.357164\pi$
$857$	39.0763	−	39.0763i	1.33482	−	1.33482i	0.900998	−	0.433824i	$-0.142836\pi$
$858$		0			0
$859$		−	46.4327i		−	1.58426i	−0.610349	−	0.792132i	$-0.708971\pi$
$859$		−	46.4327i		−	1.58426i	0.610349	−	0.792132i	$-0.291029\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	−32.2469	−	32.2469i	−1.09453	−	1.09453i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	41.7846	−	41.7846i	1.41097	−	1.41097i	0.657597	−	0.753370i	$-0.271573\pi$
$877$	41.7846	−	41.7846i	1.41097	−	1.41097i	0.753370	−	0.657597i	$-0.228427\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$	35.5398	−	35.5398i	1.19669	−	1.19669i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		−	32.3008i		−	1.08639i
$885$		0			0
$886$		0			0
$887$	35.9392	−	35.9392i	1.20672	−	1.20672i	0.234639	−	0.972083i	$-0.424609\pi$
$887$	35.9392	−	35.9392i	1.20672	−	1.20672i	0.972083	−	0.234639i	$-0.0753906\pi$
$888$		0			0
$889$		−	7.26650i		−	0.243711i
$890$		0			0
$891$	−29.8496			−1.00000
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	42.9499			1.43485
$897$		0			0
$898$	−7.75481	−	7.75481i	−0.258781	−	0.258781i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	−27.3855	−	27.3855i	−0.908820	−	0.908820i
$909$		0			0
$910$		0			0
$911$	−59.6992			−1.97792			−0.988962	−	0.148168i	$-0.952662\pi$
$911$	−59.6992			−1.97792			−0.988962	+	0.148168i	$0.952662\pi$
$912$		0			0
$913$	28.9833	+	28.9833i	0.959209	+	0.959209i
$914$			34.3166i			1.13509i
$915$		0			0
$916$	−7.89975			−0.261015
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	20.1859	+	20.1859i	0.664428	+	0.664428i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	14.0000i		−	0.459325i	−0.973270	−	0.229663i	$-0.926238\pi$
$929$		−	14.0000i		−	0.459325i	0.973270	−	0.229663i	$-0.0737623\pi$
$930$		0			0
$931$		0			0
$932$	6.88801	−	6.88801i	0.225624	−	0.225624i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	29.3509			0.959364
$937$	−34.7701	+	34.7701i	−1.13589	+	1.13589i	−0.146712	+	0.989179i	$0.546869\pi$
$937$	−34.7701	+	34.7701i	−1.13589	+	1.13589i	−0.989179	+	0.146712i	$0.953131\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$			1.46700i			0.0477468i
$945$		0			0
$946$	23.3826			0.760235
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		−	0.934003i		−	0.0303190i
$950$		0			0
$951$		0			0
$952$	−69.5989	+	69.5989i	−2.25571	+	2.25571i
$953$	9.16757	+	9.16757i	0.296967	+	0.296967i	0.839825	−	0.542858i	$-0.182658\pi$
$953$	9.16757	+	9.16757i	0.296967	+	0.296967i	−0.542858	+	0.839825i	$0.682658\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$	40.2454	+	40.2454i	1.29689	+	1.29689i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−35.1989	+	35.1989i	−1.13192	+	1.13192i	−0.142063	+	0.989858i	$0.545374\pi$
$967$	−35.1989	+	35.1989i	−1.13192	+	1.13192i	−0.989858	+	0.142063i	$0.954626\pi$
$968$	21.3257	+	21.3257i	0.685435	+	0.685435i
$969$		0			0
$970$		0			0
$971$	−59.6992			−1.91584			−0.957920	−	0.287035i	$-0.907330\pi$
$971$	−59.6992			−1.91584			−0.957920	+	0.287035i	$0.907330\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		−	44.0000i		−	1.40625i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	6.63325			0.210712			0.105356	−	0.994435i	$-0.466402\pi$
$991$	6.63325			0.210712			0.105356	+	0.994435i	$0.466402\pi$
$992$	−27.1418	+	27.1418i	−0.861754	+	0.861754i
$993$		0			0
$994$		−	34.5330i		−	1.09532i
$995$		0			0
$996$		0			0
$997$	16.5522	−	16.5522i	0.524214	−	0.524214i	−0.394627	−	0.918841i	$-0.629126\pi$
$997$	16.5522	−	16.5522i	0.524214	−	0.524214i	0.918841	+	0.394627i	$0.129126\pi$
$998$	−2.33816	−	2.33816i	−0.0740132	−	0.0740132i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.2.e.c.32.3	yes	8
5.2	odd	4	inner	275.2.e.c.43.3	yes	8
5.3	odd	4	inner	275.2.e.c.43.2	yes	8
5.4	even	2	inner	275.2.e.c.32.2	✓	8
11.10	odd	2	inner	275.2.e.c.32.2	✓	8
55.32	even	4	inner	275.2.e.c.43.2	yes	8
55.43	even	4	inner	275.2.e.c.43.3	yes	8
55.54	odd	2	CM	275.2.e.c.32.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.e.c.32.2	✓	8	5.4	even	2	inner
275.2.e.c.32.2	✓	8	11.10	odd	2	inner
275.2.e.c.32.3	yes	8	1.1	even	1	trivial
275.2.e.c.32.3	yes	8	55.54	odd	2	CM
275.2.e.c.43.2	yes	8	5.3	odd	4	inner
275.2.e.c.43.2	yes	8	55.32	even	4	inner
275.2.e.c.43.3	yes	8	5.2	odd	4	inner
275.2.e.c.43.3	yes	8	55.43	even	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	275.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.584541 - 0.584541i) q^{2} +1.31662i q^{4} +(-3.69232 + 3.69232i) q^{7} +(1.93870 + 1.93870i) q^{8} +3.00000i q^{9} +3.31662 q^{11} +(-2.52324 - 2.52324i) q^{13} +4.31662i q^{14} -0.366750 q^{16} +(4.86140 - 4.86140i) q^{17} +(1.75362 + 1.75362i) q^{18} +(1.93870 - 1.93870i) q^{22} -2.94987 q^{26} +(-4.86140 - 4.86140i) q^{28} +6.63325 q^{31} +(-4.09178 + 4.09178i) q^{32} -5.68338i q^{34} -3.94987 q^{36} +(6.03049 + 6.03049i) q^{43} +4.36675i q^{44} -20.2665i q^{49} +(3.32216 - 3.32216i) q^{52} -14.3166 q^{56} -4.00000i q^{59} +(3.87740 - 3.87740i) q^{62} +(-11.0770 - 11.0770i) q^{63} +4.05013i q^{64} +(6.40065 + 6.40065i) q^{68} -8.00000 q^{71} +(-5.81610 + 5.81610i) q^{72} +(0.185080 + 0.185080i) q^{73} +(-12.2461 + 12.2461i) q^{77} -9.00000 q^{81} +(8.73881 + 8.73881i) q^{83} +7.05013 q^{86} +(6.42995 + 6.42995i) q^{88} -13.2665i q^{89} +18.6332 q^{91} +(-11.8466 - 11.8466i) q^{98} +9.94987i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 56 q^{16} + 56 q^{26} + 48 q^{36} - 88 q^{56} - 64 q^{71} - 72 q^{81} + 136 q^{86} + 96 q^{91}+O(q^{100}) \) 8 * q - 56 * q^16 + 56 * q^26 + 48 * q^36 - 88 * q^56 - 64 * q^71 - 72 * q^81 + 136 * q^86 + 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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