LMFDB - Embedded newform 2240.2.h.b.671.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(671,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2240.671");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.h (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2240.671
Dual form			2240.2.h.b.671.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000 q^{5} +(1.73205 - 2.00000i) q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000 q^{5} +(1.73205 - 2.00000i) q^{7} +2.00000 q^{9} -5.19615 q^{11} -1.00000 q^{13} +1.00000i q^{15} -1.73205i q^{17} -2.00000i q^{19} +(-2.00000 - 1.73205i) q^{21} +1.00000 q^{25} -5.00000i q^{27} +1.73205i q^{29} -3.46410 q^{31} +5.19615i q^{33} +(-1.73205 + 2.00000i) q^{35} +6.92820i q^{37} +1.00000i q^{39} -10.3923i q^{41} -3.46410 q^{43} -2.00000 q^{45} -12.1244 q^{47} +(-1.00000 - 6.92820i) q^{49} -1.73205 q^{51} +3.46410i q^{53} +5.19615 q^{55} -2.00000 q^{57} -6.00000i q^{59} -8.00000 q^{61} +(3.46410 - 4.00000i) q^{63} +1.00000 q^{65} +10.3923 q^{67} +6.92820i q^{73} -1.00000i q^{75} +(-9.00000 + 10.3923i) q^{77} -1.00000i q^{79} +1.00000 q^{81} -12.0000i q^{83} +1.73205i q^{85} +1.73205 q^{87} +6.92820i q^{89} +(-1.73205 + 2.00000i) q^{91} +3.46410i q^{93} +2.00000i q^{95} +8.66025i q^{97} -10.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^5 + 8 * q^9	$ 4 q - 4 q^{5} + 8 q^{9} - 4 q^{13} - 8 q^{21} + 4 q^{25} - 8 q^{45} - 4 q^{49} - 8 q^{57} - 32 q^{61} + 4 q^{65} - 36 q^{77} + 4 q^{81}+O(q^{100}) $ 4 * q - 4 * q^5 + 8 * q^9 - 4 * q^13 - 8 * q^21 + 4 * q^25 - 8 * q^45 - 4 * q^49 - 8 * q^57 - 32 * q^61 + 4 * q^65 - 36 * q^77 + 4 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$		−	1.00000i		−	0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$	−5.19615			−1.56670			−0.783349	−	0.621582i	$-0.786490\pi$
$11$	−5.19615			−1.56670			−0.783349	+	0.621582i	$0.786490\pi$
$12$		0			0
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$		0			0
$15$			1.00000i			0.258199i
$16$		0			0
$17$		−	1.73205i		−	0.420084i	−0.977692	−	0.210042i	$-0.932640\pi$
$17$		−	1.73205i		−	0.420084i	0.977692	−	0.210042i	$-0.0673601\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$	−2.00000	−	1.73205i	−0.436436	−	0.377964i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	5.00000i		−	0.962250i
$28$		0			0
$29$			1.73205i			0.321634i	0.986984	+	0.160817i	$0.0514129\pi$
$29$			1.73205i			0.321634i	−0.986984	+	0.160817i	$0.948587\pi$
$30$		0			0
$31$	−3.46410			−0.622171			−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410			−0.622171			−0.311086	+	0.950382i	$0.600693\pi$
$32$		0			0
$33$			5.19615i			0.904534i
$34$		0			0
$35$	−1.73205	+	2.00000i	−0.292770	+	0.338062i
$36$		0			0
$37$			6.92820i			1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$			6.92820i			1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$			1.00000i			0.160128i
$40$		0			0
$41$		−	10.3923i		−	1.62301i	−0.584349	−	0.811503i	$-0.698650\pi$
$41$		−	10.3923i		−	1.62301i	0.584349	−	0.811503i	$-0.301350\pi$
$42$		0			0
$43$	−3.46410			−0.528271			−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410			−0.528271			−0.264135	+	0.964486i	$0.585087\pi$
$44$		0			0
$45$	−2.00000			−0.298142
$46$		0			0
$47$	−12.1244			−1.76852			−0.884260	−	0.466996i	$-0.845336\pi$
$47$	−12.1244			−1.76852			−0.884260	+	0.466996i	$0.845336\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$	−1.73205			−0.242536
$52$		0			0
$53$			3.46410i			0.475831i	0.971286	+	0.237915i	$0.0764641\pi$
$53$			3.46410i			0.475831i	−0.971286	+	0.237915i	$0.923536\pi$
$54$		0			0
$55$	5.19615			0.700649
$56$		0			0
$57$	−2.00000			−0.264906
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	3.46410	−	4.00000i	0.436436	−	0.503953i
$64$		0			0
$65$	1.00000			0.124035
$66$		0			0
$67$	10.3923			1.26962			0.634811	−	0.772667i	$-0.281078\pi$
$67$	10.3923			1.26962			0.634811	+	0.772667i	$0.281078\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	−9.00000	+	10.3923i	−1.02565	+	1.18431i
$78$		0			0
$79$		−	1.00000i		−	0.112509i	−0.998416	−	0.0562544i	$-0.982084\pi$
$79$		−	1.00000i		−	0.112509i	0.998416	−	0.0562544i	$-0.0179158\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$			1.73205i			0.187867i
$86$		0			0
$87$	1.73205			0.185695
$88$		0			0
$89$			6.92820i			0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$			6.92820i			0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$		0			0
$91$	−1.73205	+	2.00000i	−0.181568	+	0.209657i
$92$		0			0
$93$			3.46410i			0.359211i
$94$		0			0
$95$			2.00000i			0.205196i
$96$		0			0
$97$			8.66025i			0.879316i	0.898165	+	0.439658i	$0.144900\pi$
$97$			8.66025i			0.879316i	−0.898165	+	0.439658i	$0.855100\pi$
$98$		0			0
$99$	−10.3923			−1.04447
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	1.73205			0.170664			0.0853320	−	0.996353i	$-0.472805\pi$
$103$	1.73205			0.170664			0.0853320	+	0.996353i	$0.472805\pi$
$104$		0			0
$105$	2.00000	+	1.73205i	0.195180	+	0.169031i
$106$		0			0
$107$	−10.3923			−1.00466			−0.502331	−	0.864675i	$-0.667524\pi$
$107$	−10.3923			−1.00466			−0.502331	+	0.864675i	$0.667524\pi$
$108$		0			0
$109$			12.1244i			1.16130i	0.814152	+	0.580651i	$0.197202\pi$
$109$			12.1244i			1.16130i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		0			0
$111$	6.92820			0.657596
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.00000			−0.184900
$118$		0			0
$119$	−3.46410	−	3.00000i	−0.317554	−	0.275010i
$120$		0			0
$121$	16.0000			1.45455
$122$		0			0
$123$	−10.3923			−0.937043
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	14.0000i		−	1.24230i	−0.783692	−	0.621150i	$-0.786666\pi$
$127$		−	14.0000i		−	1.24230i	0.783692	−	0.621150i	$-0.213334\pi$
$128$		0			0
$129$			3.46410i			0.304997i
$130$		0			0
$131$		−	12.0000i		−	1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$		−	12.0000i		−	1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$		0			0
$133$	−4.00000	−	3.46410i	−0.346844	−	0.300376i
$134$		0			0
$135$			5.00000i			0.430331i
$136$		0			0
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$		0			0
$139$		−	16.0000i		−	1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$		−	16.0000i		−	1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$		0			0
$141$			12.1244i			1.02105i
$142$		0			0
$143$	5.19615			0.434524
$144$		0			0
$145$		−	1.73205i		−	0.143839i
$146$		0			0
$147$	−6.92820	+	1.00000i	−0.571429	+	0.0824786i
$148$		0			0
$149$		−	13.8564i		−	1.13516i	−0.823318	−	0.567581i	$-0.807880\pi$
$149$		−	13.8564i		−	1.13516i	0.823318	−	0.567581i	$-0.192120\pi$
$150$		0			0
$151$			17.0000i			1.38344i	0.722166	+	0.691720i	$0.243147\pi$
$151$			17.0000i			1.38344i	−0.722166	+	0.691720i	$0.756853\pi$
$152$		0			0
$153$		−	3.46410i		−	0.280056i
$154$		0			0
$155$	3.46410			0.278243
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$	3.46410			0.274721
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		−	5.19615i		−	0.404520i
$166$		0			0
$167$	−5.19615			−0.402090			−0.201045	−	0.979582i	$-0.564434\pi$
$167$	−5.19615			−0.402090			−0.201045	+	0.979582i	$0.564434\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$		−	4.00000i		−	0.305888i
$172$		0			0
$173$	−9.00000			−0.684257			−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000			−0.684257			−0.342129	+	0.939653i	$0.611148\pi$
$174$		0			0
$175$	1.73205	−	2.00000i	0.130931	−	0.151186i
$176$		0			0
$177$	−6.00000			−0.450988
$178$		0			0
$179$	−24.2487			−1.81243			−0.906217	−	0.422813i	$-0.861043\pi$
$179$	−24.2487			−1.81243			−0.906217	+	0.422813i	$0.861043\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$			8.00000i			0.591377i
$184$		0			0
$185$		−	6.92820i		−	0.509372i
$186$		0			0
$187$			9.00000i			0.658145i
$188$		0			0
$189$	−10.0000	−	8.66025i	−0.727393	−	0.629941i
$190$		0			0
$191$			15.0000i			1.08536i	0.839939	+	0.542681i	$0.182591\pi$
$191$			15.0000i			1.08536i	−0.839939	+	0.542681i	$0.817409\pi$
$192$		0			0
$193$	4.00000			0.287926			0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000			0.287926			0.143963	+	0.989583i	$0.454015\pi$
$194$		0			0
$195$		−	1.00000i		−	0.0716115i
$196$		0			0
$197$		−	3.46410i		−	0.246807i	−0.992357	−	0.123404i	$-0.960619\pi$
$197$		−	3.46410i		−	0.246807i	0.992357	−	0.123404i	$-0.0393809\pi$
$198$		0			0
$199$	−10.3923			−0.736691			−0.368345	−	0.929689i	$-0.620076\pi$
$199$	−10.3923			−0.736691			−0.368345	+	0.929689i	$0.620076\pi$
$200$		0			0
$201$		−	10.3923i		−	0.733017i
$202$		0			0
$203$	3.46410	+	3.00000i	0.243132	+	0.210559i
$204$		0			0
$205$			10.3923i			0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			10.3923i			0.718851i
$210$		0			0
$211$	5.19615			0.357718			0.178859	−	0.983875i	$-0.442759\pi$
$211$	5.19615			0.357718			0.178859	+	0.983875i	$0.442759\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.46410			0.236250
$216$		0			0
$217$	−6.00000	+	6.92820i	−0.407307	+	0.470317i
$218$		0			0
$219$	6.92820			0.468165
$220$		0			0
$221$			1.73205i			0.116510i
$222$		0			0
$223$	−8.66025			−0.579934			−0.289967	−	0.957037i	$-0.593644\pi$
$223$	−8.66025			−0.579934			−0.289967	+	0.957037i	$0.593644\pi$
$224$		0			0
$225$	2.00000			0.133333
$226$		0			0
$227$			15.0000i			0.995585i	0.867296	+	0.497792i	$0.165856\pi$
$227$			15.0000i			0.995585i	−0.867296	+	0.497792i	$0.834144\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$	10.3923	+	9.00000i	0.683763	+	0.592157i
$232$		0			0
$233$	12.0000			0.786146			0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000			0.786146			0.393073	+	0.919507i	$0.371412\pi$
$234$		0			0
$235$	12.1244			0.790906
$236$		0			0
$237$	−1.00000			−0.0649570
$238$		0			0
$239$		−	9.00000i		−	0.582162i	−0.956698	−	0.291081i	$-0.905985\pi$
$239$		−	9.00000i		−	0.582162i	0.956698	−	0.291081i	$-0.0940149\pi$
$240$		0			0
$241$		−	20.7846i		−	1.33885i	−0.742878	−	0.669427i	$-0.766540\pi$
$241$		−	20.7846i		−	1.33885i	0.742878	−	0.669427i	$-0.233460\pi$
$242$		0			0
$243$		−	16.0000i		−	1.02640i
$244$		0			0
$245$	1.00000	+	6.92820i	0.0638877	+	0.442627i
$246$		0			0
$247$			2.00000i			0.127257i
$248$		0			0
$249$	−12.0000			−0.760469
$250$		0			0
$251$		−	12.0000i		−	0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$		−	12.0000i		−	0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	1.73205			0.108465
$256$		0			0
$257$			6.92820i			0.432169i	0.976375	+	0.216085i	$0.0693287\pi$
$257$			6.92820i			0.432169i	−0.976375	+	0.216085i	$0.930671\pi$
$258$		0			0
$259$	13.8564	+	12.0000i	0.860995	+	0.745644i
$260$		0			0
$261$			3.46410i			0.214423i
$262$		0			0
$263$		−	6.00000i		−	0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$		−	6.00000i		−	0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$		0			0
$265$		−	3.46410i		−	0.212798i
$266$		0			0
$267$	6.92820			0.423999
$268$		0			0
$269$	−24.0000			−1.46331			−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000			−1.46331			−0.731653	+	0.681677i	$0.761251\pi$
$270$		0			0
$271$	27.7128			1.68343			0.841717	−	0.539919i	$-0.181545\pi$
$271$	27.7128			1.68343			0.841717	+	0.539919i	$0.181545\pi$
$272$		0			0
$273$	2.00000	+	1.73205i	0.121046	+	0.104828i
$274$		0			0
$275$	−5.19615			−0.313340
$276$		0			0
$277$		−	27.7128i		−	1.66510i	−0.553949	−	0.832551i	$-0.686880\pi$
$277$		−	27.7128i		−	1.66510i	0.553949	−	0.832551i	$-0.313120\pi$
$278$		0			0
$279$	−6.92820			−0.414781
$280$		0			0
$281$	27.0000			1.61068			0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000			1.61068			0.805342	+	0.592810i	$0.201981\pi$
$282$		0			0
$283$			31.0000i			1.84276i	0.388664	+	0.921379i	$0.372937\pi$
$283$			31.0000i			1.84276i	−0.388664	+	0.921379i	$0.627063\pi$
$284$		0			0
$285$	2.00000			0.118470
$286$		0			0
$287$	−20.7846	−	18.0000i	−1.22688	−	1.06251i
$288$		0			0
$289$	14.0000			0.823529
$290$		0			0
$291$	8.66025			0.507673
$292$		0			0
$293$	27.0000			1.57736			0.788678	−	0.614806i	$-0.210766\pi$
$293$	27.0000			1.57736			0.788678	+	0.614806i	$0.210766\pi$
$294$		0			0
$295$			6.00000i			0.349334i
$296$		0			0
$297$			25.9808i			1.50756i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−6.00000	+	6.92820i	−0.345834	+	0.399335i
$302$		0			0
$303$			6.00000i			0.344691i
$304$		0			0
$305$	8.00000			0.458079
$306$		0			0
$307$		−	1.00000i		−	0.0570730i	−0.999593	−	0.0285365i	$-0.990915\pi$
$307$		−	1.00000i		−	0.0570730i	0.999593	−	0.0285365i	$-0.00908469\pi$
$308$		0			0
$309$		−	1.73205i		−	0.0985329i
$310$		0			0
$311$	31.1769			1.76788			0.883940	−	0.467600i	$-0.154881\pi$
$311$	31.1769			1.76788			0.883940	+	0.467600i	$0.154881\pi$
$312$		0			0
$313$		−	25.9808i		−	1.46852i	−0.678869	−	0.734260i	$-0.737529\pi$
$313$		−	25.9808i		−	1.46852i	0.678869	−	0.734260i	$-0.262471\pi$
$314$		0			0
$315$	−3.46410	+	4.00000i	−0.195180	+	0.225374i
$316$		0			0
$317$			17.3205i			0.972817i	0.873732	+	0.486408i	$0.161693\pi$
$317$			17.3205i			0.972817i	−0.873732	+	0.486408i	$0.838307\pi$
$318$		0			0
$319$		−	9.00000i		−	0.503903i
$320$		0			0
$321$			10.3923i			0.580042i
$322$		0			0
$323$	−3.46410			−0.192748
$324$		0			0
$325$	−1.00000			−0.0554700
$326$		0			0
$327$	12.1244			0.670478
$328$		0			0
$329$	−21.0000	+	24.2487i	−1.15777	+	1.33687i
$330$		0			0
$331$	3.46410			0.190404			0.0952021	−	0.995458i	$-0.469650\pi$
$331$	3.46410			0.190404			0.0952021	+	0.995458i	$0.469650\pi$
$332$		0			0
$333$			13.8564i			0.759326i
$334$		0			0
$335$	−10.3923			−0.567792
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$		0			0
$339$			12.0000i			0.651751i
$340$		0			0
$341$	18.0000			0.974755
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	27.7128			1.48770			0.743851	−	0.668346i	$-0.232997\pi$
$347$	27.7128			1.48770			0.743851	+	0.668346i	$0.232997\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$			5.00000i			0.266880i
$352$		0			0
$353$		−	15.5885i		−	0.829690i	−0.909892	−	0.414845i	$-0.863836\pi$
$353$		−	15.5885i		−	0.829690i	0.909892	−	0.414845i	$-0.136164\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−3.00000	+	3.46410i	−0.158777	+	0.183340i
$358$		0			0
$359$		−	24.0000i		−	1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$		−	24.0000i		−	1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		−	16.0000i		−	0.839782i
$364$		0			0
$365$		−	6.92820i		−	0.362639i
$366$		0			0
$367$	−15.5885			−0.813711			−0.406855	−	0.913493i	$-0.633375\pi$
$367$	−15.5885			−0.813711			−0.406855	+	0.913493i	$0.633375\pi$
$368$		0			0
$369$		−	20.7846i		−	1.08200i
$370$		0			0
$371$	6.92820	+	6.00000i	0.359694	+	0.311504i
$372$		0			0
$373$		−	34.6410i		−	1.79364i	−0.442392	−	0.896822i	$-0.645870\pi$
$373$		−	34.6410i		−	1.79364i	0.442392	−	0.896822i	$-0.354130\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$		−	1.73205i		−	0.0892052i
$378$		0			0
$379$	10.3923			0.533817			0.266908	−	0.963722i	$-0.413998\pi$
$379$	10.3923			0.533817			0.266908	+	0.963722i	$0.413998\pi$
$380$		0			0
$381$	−14.0000			−0.717242
$382$		0			0
$383$	−10.3923			−0.531022			−0.265511	−	0.964108i	$-0.585541\pi$
$383$	−10.3923			−0.531022			−0.265511	+	0.964108i	$0.585541\pi$
$384$		0			0
$385$	9.00000	−	10.3923i	0.458682	−	0.529641i
$386$		0			0
$387$	−6.92820			−0.352180
$388$		0			0
$389$			8.66025i			0.439092i	0.975602	+	0.219546i	$0.0704577\pi$
$389$			8.66025i			0.439092i	−0.975602	+	0.219546i	$0.929542\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−12.0000			−0.605320
$394$		0			0
$395$			1.00000i			0.0503155i
$396$		0			0
$397$	11.0000			0.552074			0.276037	−	0.961147i	$-0.410979\pi$
$397$	11.0000			0.552074			0.276037	+	0.961147i	$0.410979\pi$
$398$		0			0
$399$	−3.46410	+	4.00000i	−0.173422	+	0.200250i
$400$		0			0
$401$	21.0000			1.04869			0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000			1.04869			0.524345	+	0.851506i	$0.324310\pi$
$402$		0			0
$403$	3.46410			0.172559
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$		−	36.0000i		−	1.78445i
$408$		0			0
$409$			10.3923i			0.513866i	0.966429	+	0.256933i	$0.0827120\pi$
$409$			10.3923i			0.513866i	−0.966429	+	0.256933i	$0.917288\pi$
$410$		0			0
$411$			12.0000i			0.591916i
$412$		0			0
$413$	−12.0000	−	10.3923i	−0.590481	−	0.511372i
$414$		0			0
$415$			12.0000i			0.589057i
$416$		0			0
$417$	−16.0000			−0.783523
$418$		0			0
$419$		−	6.00000i		−	0.293119i	−0.989202	−	0.146560i	$-0.953180\pi$
$419$		−	6.00000i		−	0.293119i	0.989202	−	0.146560i	$-0.0468200\pi$
$420$		0			0
$421$		−	39.8372i		−	1.94154i	−0.240002	−	0.970772i	$-0.577148\pi$
$421$		−	39.8372i		−	1.94154i	0.240002	−	0.970772i	$-0.422852\pi$
$422$		0			0
$423$	−24.2487			−1.17901
$424$		0			0
$425$		−	1.73205i		−	0.0840168i
$426$		0			0
$427$	−13.8564	+	16.0000i	−0.670559	+	0.774294i
$428$		0			0
$429$		−	5.19615i		−	0.250873i
$430$		0			0
$431$			27.0000i			1.30054i	0.759701	+	0.650272i	$0.225345\pi$
$431$			27.0000i			1.30054i	−0.759701	+	0.650272i	$0.774655\pi$
$432$		0			0
$433$			13.8564i			0.665896i	0.942945	+	0.332948i	$0.108043\pi$
$433$			13.8564i			0.665896i	−0.942945	+	0.332948i	$0.891957\pi$
$434$		0			0
$435$	−1.73205			−0.0830455
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−27.7128			−1.32266			−0.661330	−	0.750095i	$-0.730008\pi$
$439$	−27.7128			−1.32266			−0.661330	+	0.750095i	$0.730008\pi$
$440$		0			0
$441$	−2.00000	−	13.8564i	−0.0952381	−	0.659829i
$442$		0			0
$443$	24.2487			1.15209			0.576046	−	0.817418i	$-0.304595\pi$
$443$	24.2487			1.15209			0.576046	+	0.817418i	$0.304595\pi$
$444$		0			0
$445$		−	6.92820i		−	0.328428i
$446$		0			0
$447$	−13.8564			−0.655386
$448$		0			0
$449$	−15.0000			−0.707894			−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000			−0.707894			−0.353947	+	0.935266i	$0.615161\pi$
$450$		0			0
$451$			54.0000i			2.54276i
$452$		0			0
$453$	17.0000			0.798730
$454$		0			0
$455$	1.73205	−	2.00000i	0.0811998	−	0.0937614i
$456$		0			0
$457$	−34.0000			−1.59045			−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000			−1.59045			−0.795226	+	0.606313i	$0.792648\pi$
$458$		0			0
$459$	−8.66025			−0.404226
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$			4.00000i			0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$			4.00000i			0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$		0			0
$465$		−	3.46410i		−	0.160644i
$466$		0			0
$467$		−	21.0000i		−	0.971764i	−0.874024	−	0.485882i	$-0.838498\pi$
$467$		−	21.0000i		−	0.971764i	0.874024	−	0.485882i	$-0.161502\pi$
$468$		0			0
$469$	18.0000	−	20.7846i	0.831163	−	0.959744i
$470$		0			0
$471$			14.0000i			0.645086i
$472$		0			0
$473$	18.0000			0.827641
$474$		0			0
$475$		−	2.00000i		−	0.0917663i
$476$		0			0
$477$			6.92820i			0.317221i
$478$		0			0
$479$	13.8564			0.633115			0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564			0.633115			0.316558	+	0.948573i	$0.397473\pi$
$480$		0			0
$481$		−	6.92820i		−	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	8.66025i		−	0.393242i
$486$		0			0
$487$		−	16.0000i		−	0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$		−	16.0000i		−	0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−22.5167			−1.01616			−0.508081	−	0.861309i	$-0.669645\pi$
$491$	−22.5167			−1.01616			−0.508081	+	0.861309i	$0.669645\pi$
$492$		0			0
$493$	3.00000			0.135113
$494$		0			0
$495$	10.3923			0.467099
$496$		0			0
$497$		0			0
$498$		0			0
$499$	29.4449			1.31813			0.659067	−	0.752085i	$-0.270952\pi$
$499$	29.4449			1.31813			0.659067	+	0.752085i	$0.270952\pi$
$500$		0			0
$501$			5.19615i			0.232147i
$502$		0			0
$503$	15.5885			0.695055			0.347527	−	0.937670i	$-0.387021\pi$
$503$	15.5885			0.695055			0.347527	+	0.937670i	$0.387021\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$			12.0000i			0.532939i
$508$		0			0
$509$	12.0000			0.531891			0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000			0.531891			0.265945	+	0.963988i	$0.414316\pi$
$510$		0			0
$511$	13.8564	+	12.0000i	0.612971	+	0.530849i
$512$		0			0
$513$	−10.0000			−0.441511
$514$		0			0
$515$	−1.73205			−0.0763233
$516$		0			0
$517$	63.0000			2.77074
$518$		0			0
$519$			9.00000i			0.395056i
$520$		0			0
$521$		−	17.3205i		−	0.758825i	−0.925228	−	0.379413i	$-0.876126\pi$
$521$		−	17.3205i		−	0.758825i	0.925228	−	0.379413i	$-0.123874\pi$
$522$		0			0
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$		0			0
$525$	−2.00000	−	1.73205i	−0.0872872	−	0.0755929i
$526$		0			0
$527$			6.00000i			0.261364i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		−	12.0000i		−	0.520756i
$532$		0			0
$533$			10.3923i			0.450141i
$534$		0			0
$535$	10.3923			0.449299
$536$		0			0
$537$			24.2487i			1.04641i
$538$		0			0
$539$	5.19615	+	36.0000i	0.223814	+	1.55063i
$540$		0			0
$541$		−	32.9090i		−	1.41487i	−0.706780	−	0.707433i	$-0.749853\pi$
$541$		−	32.9090i		−	1.41487i	0.706780	−	0.707433i	$-0.250147\pi$
$542$		0			0
$543$		−	22.0000i		−	0.944110i
$544$		0			0
$545$		−	12.1244i		−	0.519350i
$546$		0			0
$547$	27.7128			1.18491			0.592457	−	0.805602i	$-0.298158\pi$
$547$	27.7128			1.18491			0.592457	+	0.805602i	$0.298158\pi$
$548$		0			0
$549$	−16.0000			−0.682863
$550$		0			0
$551$	3.46410			0.147576
$552$		0			0
$553$	−2.00000	−	1.73205i	−0.0850487	−	0.0736543i
$554$		0			0
$555$	−6.92820			−0.294086
$556$		0			0
$557$		−	27.7128i		−	1.17423i	−0.809504	−	0.587115i	$-0.800264\pi$
$557$		−	27.7128i		−	1.17423i	0.809504	−	0.587115i	$-0.199736\pi$
$558$		0			0
$559$	3.46410			0.146516
$560$		0			0
$561$	9.00000			0.379980
$562$		0			0
$563$			36.0000i			1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$			36.0000i			1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$		0			0
$565$	12.0000			0.504844
$566$		0			0
$567$	1.73205	−	2.00000i	0.0727393	−	0.0839921i
$568$		0			0
$569$	−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
$571$	−3.46410			−0.144968			−0.0724841	−	0.997370i	$-0.523093\pi$
$571$	−3.46410			−0.144968			−0.0724841	+	0.997370i	$0.523093\pi$
$572$		0			0
$573$	15.0000			0.626634
$574$		0			0
$575$		0			0
$576$		0			0
$577$			19.0526i			0.793168i	0.917998	+	0.396584i	$0.129805\pi$
$577$			19.0526i			0.793168i	−0.917998	+	0.396584i	$0.870195\pi$
$578$		0			0
$579$		−	4.00000i		−	0.166234i
$580$		0			0
$581$	−24.0000	−	20.7846i	−0.995688	−	0.862291i
$582$		0			0
$583$		−	18.0000i		−	0.745484i
$584$		0			0
$585$	2.00000			0.0826898
$586$		0			0
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$			6.92820i			0.285472i
$590$		0			0
$591$	−3.46410			−0.142494
$592$		0			0
$593$			39.8372i			1.63592i	0.575278	+	0.817958i	$0.304894\pi$
$593$			39.8372i			1.63592i	−0.575278	+	0.817958i	$0.695106\pi$
$594$		0			0
$595$	3.46410	+	3.00000i	0.142014	+	0.122988i
$596$		0			0
$597$			10.3923i			0.425329i
$598$		0			0
$599$		−	9.00000i		−	0.367730i	−0.982952	−	0.183865i	$-0.941139\pi$
$599$		−	9.00000i		−	0.367730i	0.982952	−	0.183865i	$-0.0588609\pi$
$600$		0			0
$601$		−	24.2487i		−	0.989126i	−0.869142	−	0.494563i	$-0.835328\pi$
$601$		−	24.2487i		−	0.989126i	0.869142	−	0.494563i	$-0.164672\pi$
$602$		0			0
$603$	20.7846			0.846415
$604$		0			0
$605$	−16.0000			−0.650493
$606$		0			0
$607$	−8.66025			−0.351509			−0.175754	−	0.984434i	$-0.556236\pi$
$607$	−8.66025			−0.351509			−0.175754	+	0.984434i	$0.556236\pi$
$608$		0			0
$609$	3.00000	−	3.46410i	0.121566	−	0.140372i
$610$		0			0
$611$	12.1244			0.490499
$612$		0			0
$613$		−	24.2487i		−	0.979396i	−0.871892	−	0.489698i	$-0.837107\pi$
$613$		−	24.2487i		−	0.979396i	0.871892	−	0.489698i	$-0.162893\pi$
$614$		0			0
$615$	10.3923			0.419058
$616$		0			0
$617$	24.0000			0.966204			0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000			0.966204			0.483102	+	0.875564i	$0.339510\pi$
$618$		0			0
$619$		−	4.00000i		−	0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$		−	4.00000i		−	0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.8564	+	12.0000i	0.555145	+	0.480770i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	10.3923			0.415029
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$			13.0000i			0.517522i	0.965941	+	0.258761i	$0.0833142\pi$
$631$			13.0000i			0.517522i	−0.965941	+	0.258761i	$0.916686\pi$
$632$		0			0
$633$		−	5.19615i		−	0.206529i
$634$		0			0
$635$			14.0000i			0.555573i
$636$		0			0
$637$	1.00000	+	6.92820i	0.0396214	+	0.274505i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$			1.00000i			0.0394362i	0.999806	+	0.0197181i	$0.00627687\pi$
$643$			1.00000i			0.0394362i	−0.999806	+	0.0197181i	$0.993723\pi$
$644$		0			0
$645$		−	3.46410i		−	0.136399i
$646$		0			0
$647$	−24.2487			−0.953315			−0.476658	−	0.879089i	$-0.658152\pi$
$647$	−24.2487			−0.953315			−0.476658	+	0.879089i	$0.658152\pi$
$648$		0			0
$649$			31.1769i			1.22380i
$650$		0			0
$651$	6.92820	+	6.00000i	0.271538	+	0.235159i
$652$		0			0
$653$			38.1051i			1.49117i	0.666411	+	0.745584i	$0.267829\pi$
$653$			38.1051i			1.49117i	−0.666411	+	0.745584i	$0.732171\pi$
$654$		0			0
$655$			12.0000i			0.468879i
$656$		0			0
$657$			13.8564i			0.540590i
$658$		0			0
$659$	12.1244			0.472298			0.236149	−	0.971717i	$-0.424115\pi$
$659$	12.1244			0.472298			0.236149	+	0.971717i	$0.424115\pi$
$660$		0			0
$661$	20.0000			0.777910			0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000			0.777910			0.388955	+	0.921257i	$0.372836\pi$
$662$		0			0
$663$	1.73205			0.0672673
$664$		0			0
$665$	4.00000	+	3.46410i	0.155113	+	0.134332i
$666$		0			0
$667$		0			0
$668$		0			0
$669$			8.66025i			0.334825i
$670$		0			0
$671$	41.5692			1.60476
$672$		0			0
$673$	−40.0000			−1.54189			−0.770943	−	0.636904i	$-0.780215\pi$
$673$	−40.0000			−1.54189			−0.770943	+	0.636904i	$0.780215\pi$
$674$		0			0
$675$		−	5.00000i		−	0.192450i
$676$		0			0
$677$	3.00000			0.115299			0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000			0.115299			0.0576497	+	0.998337i	$0.481639\pi$
$678$		0			0
$679$	17.3205	+	15.0000i	0.664700	+	0.575647i
$680$		0			0
$681$	15.0000			0.574801
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$		−	22.0000i		−	0.839352i
$688$		0			0
$689$		−	3.46410i		−	0.131972i
$690$		0			0
$691$			8.00000i			0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$			8.00000i			0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$		0			0
$693$	−18.0000	+	20.7846i	−0.683763	+	0.789542i
$694$		0			0
$695$			16.0000i			0.606915i
$696$		0			0
$697$	−18.0000			−0.681799
$698$		0			0
$699$		−	12.0000i		−	0.453882i
$700$		0			0
$701$			25.9808i			0.981280i	0.871362	+	0.490640i	$0.163237\pi$
$701$			25.9808i			0.981280i	−0.871362	+	0.490640i	$0.836763\pi$
$702$		0			0
$703$	13.8564			0.522604
$704$		0			0
$705$		−	12.1244i		−	0.456630i
$706$		0			0
$707$	−10.3923	+	12.0000i	−0.390843	+	0.451306i
$708$		0			0
$709$			1.73205i			0.0650485i	0.999471	+	0.0325243i	$0.0103546\pi$
$709$			1.73205i			0.0650485i	−0.999471	+	0.0325243i	$0.989645\pi$
$710$		0			0
$711$		−	2.00000i		−	0.0750059i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−5.19615			−0.194325
$716$		0			0
$717$	−9.00000			−0.336111
$718$		0			0
$719$	−6.92820			−0.258378			−0.129189	−	0.991620i	$-0.541237\pi$
$719$	−6.92820			−0.258378			−0.129189	+	0.991620i	$0.541237\pi$
$720$		0			0
$721$	3.00000	−	3.46410i	0.111726	−	0.129010i
$722$		0			0
$723$	−20.7846			−0.772988
$724$		0			0
$725$			1.73205i			0.0643268i
$726$		0			0
$727$	−17.3205			−0.642382			−0.321191	−	0.947014i	$-0.604083\pi$
$727$	−17.3205			−0.642382			−0.321191	+	0.947014i	$0.604083\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$			6.00000i			0.221918i
$732$		0			0
$733$	−19.0000			−0.701781			−0.350891	−	0.936416i	$-0.614121\pi$
$733$	−19.0000			−0.701781			−0.350891	+	0.936416i	$0.614121\pi$
$734$		0			0
$735$	6.92820	−	1.00000i	0.255551	−	0.0368856i
$736$		0			0
$737$	−54.0000			−1.98912
$738$		0			0
$739$	25.9808			0.955718			0.477859	−	0.878437i	$-0.341413\pi$
$739$	25.9808			0.955718			0.477859	+	0.878437i	$0.341413\pi$
$740$		0			0
$741$	2.00000			0.0734718
$742$		0			0
$743$			30.0000i			1.10059i	0.834969	+	0.550297i	$0.185485\pi$
$743$			30.0000i			1.10059i	−0.834969	+	0.550297i	$0.814515\pi$
$744$		0			0
$745$			13.8564i			0.507659i
$746$		0			0
$747$		−	24.0000i		−	0.878114i
$748$		0			0
$749$	−18.0000	+	20.7846i	−0.657706	+	0.759453i
$750$		0			0
$751$			7.00000i			0.255434i	0.991811	+	0.127717i	$0.0407649\pi$
$751$			7.00000i			0.255434i	−0.991811	+	0.127717i	$0.959235\pi$
$752$		0			0
$753$	−12.0000			−0.437304
$754$		0			0
$755$		−	17.0000i		−	0.618693i
$756$		0			0
$757$		−	34.6410i		−	1.25905i	−0.776981	−	0.629525i	$-0.783250\pi$
$757$		−	34.6410i		−	1.25905i	0.776981	−	0.629525i	$-0.216750\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	13.8564i		−	0.502294i	−0.967949	−	0.251147i	$-0.919192\pi$
$761$		−	13.8564i		−	0.502294i	0.967949	−	0.251147i	$-0.0808078\pi$
$762$		0			0
$763$	24.2487	+	21.0000i	0.877862	+	0.760251i
$764$		0			0
$765$			3.46410i			0.125245i
$766$		0			0
$767$			6.00000i			0.216647i
$768$		0			0
$769$		−	31.1769i		−	1.12427i	−0.827046	−	0.562134i	$-0.809980\pi$
$769$		−	31.1769i		−	1.12427i	0.827046	−	0.562134i	$-0.190020\pi$
$770$		0			0
$771$	6.92820			0.249513
$772$		0			0
$773$	−15.0000			−0.539513			−0.269756	−	0.962929i	$-0.586943\pi$
$773$	−15.0000			−0.539513			−0.269756	+	0.962929i	$0.586943\pi$
$774$		0			0
$775$	−3.46410			−0.124434
$776$		0			0
$777$	12.0000	−	13.8564i	0.430498	−	0.497096i
$778$		0			0
$779$	−20.7846			−0.744686
$780$		0			0
$781$		0			0
$782$		0			0
$783$	8.66025			0.309492
$784$		0			0
$785$	14.0000			0.499681
$786$		0			0
$787$			29.0000i			1.03374i	0.856064	+	0.516869i	$0.172903\pi$
$787$			29.0000i			1.03374i	−0.856064	+	0.516869i	$0.827097\pi$
$788$		0			0
$789$	−6.00000			−0.213606
$790$		0			0
$791$	−20.7846	+	24.0000i	−0.739016	+	0.853342i
$792$		0			0
$793$	8.00000			0.284088
$794$		0			0
$795$	−3.46410			−0.122859
$796$		0			0
$797$	15.0000			0.531327			0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000			0.531327			0.265664	+	0.964066i	$0.414409\pi$
$798$		0			0
$799$			21.0000i			0.742927i
$800$		0			0
$801$			13.8564i			0.489592i
$802$		0			0
$803$		−	36.0000i		−	1.27041i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			24.0000i			0.844840i
$808$		0			0
$809$	−27.0000			−0.949269			−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000			−0.949269			−0.474635	+	0.880183i	$0.657420\pi$
$810$		0			0
$811$		−	44.0000i		−	1.54505i	−0.634985	−	0.772524i	$-0.718994\pi$
$811$		−	44.0000i		−	1.54505i	0.634985	−	0.772524i	$-0.281006\pi$
$812$		0			0
$813$		−	27.7128i		−	0.971931i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			6.92820i			0.242387i
$818$		0			0
$819$	−3.46410	+	4.00000i	−0.121046	+	0.139771i
$820$		0			0
$821$			39.8372i			1.39033i	0.718852	+	0.695163i	$0.244668\pi$
$821$			39.8372i			1.39033i	−0.718852	+	0.695163i	$0.755332\pi$
$822$		0			0
$823$		−	14.0000i		−	0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$		−	14.0000i		−	0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$		0			0
$825$			5.19615i			0.180907i
$826$		0			0
$827$	−34.6410			−1.20459			−0.602293	−	0.798275i	$-0.705746\pi$
$827$	−34.6410			−1.20459			−0.602293	+	0.798275i	$0.705746\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$		0			0
$831$	−27.7128			−0.961347
$832$		0			0
$833$	−12.0000	+	1.73205i	−0.415775	+	0.0600120i
$834$		0			0
$835$	5.19615			0.179820
$836$		0			0
$837$			17.3205i			0.598684i
$838$		0			0
$839$	−13.8564			−0.478376			−0.239188	−	0.970973i	$-0.576881\pi$
$839$	−13.8564			−0.478376			−0.239188	+	0.970973i	$0.576881\pi$
$840$		0			0
$841$	26.0000			0.896552
$842$		0			0
$843$		−	27.0000i		−	0.929929i
$844$		0			0
$845$	12.0000			0.412813
$846$		0			0
$847$	27.7128	−	32.0000i	0.952224	−	1.09953i
$848$		0			0
$849$	31.0000			1.06392
$850$		0			0
$851$		0			0
$852$		0			0
$853$	46.0000			1.57501			0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000			1.57501			0.787505	+	0.616308i	$0.211372\pi$
$854$		0			0
$855$			4.00000i			0.136797i
$856$		0			0
$857$		−	34.6410i		−	1.18331i	−0.806190	−	0.591657i	$-0.798474\pi$
$857$		−	34.6410i		−	1.18331i	0.806190	−	0.591657i	$-0.201526\pi$
$858$		0			0
$859$			56.0000i			1.91070i	0.295484	+	0.955348i	$0.404519\pi$
$859$			56.0000i			1.91070i	−0.295484	+	0.955348i	$0.595481\pi$
$860$		0			0
$861$	−18.0000	+	20.7846i	−0.613438	+	0.708338i
$862$		0			0
$863$		−	6.00000i		−	0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$		−	6.00000i		−	0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$		0			0
$865$	9.00000			0.306009
$866$		0			0
$867$		−	14.0000i		−	0.475465i
$868$		0			0
$869$			5.19615i			0.176267i
$870$		0			0
$871$	−10.3923			−0.352130
$872$		0			0
$873$			17.3205i			0.586210i
$874$		0			0
$875$	−1.73205	+	2.00000i	−0.0585540	+	0.0676123i
$876$		0			0
$877$		−	45.0333i		−	1.52067i	−0.649533	−	0.760334i	$-0.725035\pi$
$877$		−	45.0333i		−	1.52067i	0.649533	−	0.760334i	$-0.274965\pi$
$878$		0			0
$879$		−	27.0000i		−	0.910687i
$880$		0			0
$881$		−	48.4974i		−	1.63392i	−0.576695	−	0.816960i	$-0.695658\pi$
$881$		−	48.4974i		−	1.63392i	0.576695	−	0.816960i	$-0.304342\pi$
$882$		0			0
$883$	−13.8564			−0.466305			−0.233153	−	0.972440i	$-0.574904\pi$
$883$	−13.8564			−0.466305			−0.233153	+	0.972440i	$0.574904\pi$
$884$		0			0
$885$	6.00000			0.201688
$886$		0			0
$887$	−51.9615			−1.74470			−0.872349	−	0.488884i	$-0.837404\pi$
$887$	−51.9615			−1.74470			−0.872349	+	0.488884i	$0.837404\pi$
$888$		0			0
$889$	−28.0000	−	24.2487i	−0.939090	−	0.813276i
$890$		0			0
$891$	−5.19615			−0.174078
$892$		0			0
$893$			24.2487i			0.811452i
$894$		0			0
$895$	24.2487			0.810545
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	6.00000i		−	0.200111i
$900$		0			0
$901$	6.00000			0.199889
$902$		0			0
$903$	6.92820	+	6.00000i	0.230556	+	0.199667i
$904$		0			0
$905$	−22.0000			−0.731305
$906$		0			0
$907$	−48.4974			−1.61033			−0.805165	−	0.593051i	$-0.797923\pi$
$907$	−48.4974			−1.61033			−0.805165	+	0.593051i	$0.797923\pi$
$908$		0			0
$909$	−12.0000			−0.398015
$910$		0			0
$911$			24.0000i			0.795155i	0.917568	+	0.397578i	$0.130149\pi$
$911$			24.0000i			0.795155i	−0.917568	+	0.397578i	$0.869851\pi$
$912$		0			0
$913$			62.3538i			2.06361i
$914$		0			0
$915$		−	8.00000i		−	0.264472i
$916$		0			0
$917$	−24.0000	−	20.7846i	−0.792550	−	0.686368i
$918$		0			0
$919$		−	55.0000i		−	1.81428i	−0.420826	−	0.907141i	$-0.638260\pi$
$919$		−	55.0000i		−	1.81428i	0.420826	−	0.907141i	$-0.361740\pi$
$920$		0			0
$921$	−1.00000			−0.0329511
$922$		0			0
$923$		0			0
$924$		0			0
$925$			6.92820i			0.227798i
$926$		0			0
$927$	3.46410			0.113776
$928$		0			0
$929$			24.2487i			0.795574i	0.917478	+	0.397787i	$0.130222\pi$
$929$			24.2487i			0.795574i	−0.917478	+	0.397787i	$0.869778\pi$
$930$		0			0
$931$	−13.8564	+	2.00000i	−0.454125	+	0.0655474i
$932$		0			0
$933$		−	31.1769i		−	1.02069i
$934$		0			0
$935$		−	9.00000i		−	0.294331i
$936$		0			0
$937$			29.4449i			0.961922i	0.876742	+	0.480961i	$0.159712\pi$
$937$			29.4449i			0.961922i	−0.876742	+	0.480961i	$0.840288\pi$
$938$		0			0
$939$	−25.9808			−0.847850
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	10.0000	+	8.66025i	0.325300	+	0.281718i
$946$		0			0
$947$	−10.3923			−0.337705			−0.168852	−	0.985641i	$-0.554006\pi$
$947$	−10.3923			−0.337705			−0.168852	+	0.985641i	$0.554006\pi$
$948$		0			0
$949$		−	6.92820i		−	0.224899i
$950$		0			0
$951$	17.3205			0.561656
$952$		0			0
$953$	54.0000			1.74923			0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000			1.74923			0.874616	+	0.484817i	$0.161114\pi$
$954$		0			0
$955$		−	15.0000i		−	0.485389i
$956$		0			0
$957$	−9.00000			−0.290929
$958$		0			0
$959$	−20.7846	+	24.0000i	−0.671170	+	0.775000i
$960$		0			0
$961$	−19.0000			−0.612903
$962$		0			0
$963$	−20.7846			−0.669775
$964$		0			0
$965$	−4.00000			−0.128765
$966$		0			0
$967$		−	22.0000i		−	0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$		−	22.0000i		−	0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$		0			0
$969$			3.46410i			0.111283i
$970$		0			0
$971$		−	42.0000i		−	1.34784i	−0.738802	−	0.673922i	$-0.764608\pi$
$971$		−	42.0000i		−	1.34784i	0.738802	−	0.673922i	$-0.235392\pi$
$972$		0			0
$973$	−32.0000	−	27.7128i	−1.02587	−	0.888432i
$974$		0			0
$975$			1.00000i			0.0320256i
$976$		0			0
$977$	−6.00000			−0.191957			−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000			−0.191957			−0.0959785	+	0.995383i	$0.530598\pi$
$978$		0			0
$979$		−	36.0000i		−	1.15056i
$980$		0			0
$981$			24.2487i			0.774202i
$982$		0			0
$983$	32.9090			1.04963			0.524816	−	0.851215i	$-0.324134\pi$
$983$	32.9090			1.04963			0.524816	+	0.851215i	$0.324134\pi$
$984$		0			0
$985$			3.46410i			0.110375i
$986$		0			0
$987$	24.2487	+	21.0000i	0.771845	+	0.668437i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	20.0000i		−	0.635321i	−0.948205	−	0.317660i	$-0.897103\pi$
$991$		−	20.0000i		−	0.635321i	0.948205	−	0.317660i	$-0.102897\pi$
$992$		0			0
$993$		−	3.46410i		−	0.109930i
$994$		0			0
$995$	10.3923			0.329458
$996$		0			0
$997$	1.00000			0.0316703			0.0158352	−	0.999875i	$-0.494959\pi$
$997$	1.00000			0.0316703			0.0158352	+	0.999875i	$0.494959\pi$
$998$		0			0
$999$	34.6410			1.09599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.h.b.671.2	yes	4
4.3	odd	2	inner	2240.2.h.b.671.3	yes	4
7.6	odd	2		2240.2.h.d.671.3	yes	4
8.3	odd	2		2240.2.h.d.671.1	yes	4
8.5	even	2		2240.2.h.d.671.4	yes	4
28.27	even	2		2240.2.h.d.671.2	yes	4
56.13	odd	2	inner	2240.2.h.b.671.1	✓	4
56.27	even	2	inner	2240.2.h.b.671.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.h.b.671.1	✓	4	56.13	odd	2	inner
2240.2.h.b.671.2	yes	4	1.1	even	1	trivial
2240.2.h.b.671.3	yes	4	4.3	odd	2	inner
2240.2.h.b.671.4	yes	4	56.27	even	2	inner
2240.2.h.d.671.1	yes	4	8.3	odd	2
2240.2.h.d.671.2	yes	4	28.27	even	2
2240.2.h.d.671.3	yes	4	7.6	odd	2
2240.2.h.d.671.4	yes	4	8.5	even	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000 q^{5} +(1.73205 - 2.00000i) q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000 q^{5} +(1.73205 - 2.00000i) q^{7} +2.00000 q^{9} -5.19615 q^{11} -1.00000 q^{13} +1.00000i q^{15} -1.73205i q^{17} -2.00000i q^{19} +(-2.00000 - 1.73205i) q^{21} +1.00000 q^{25} -5.00000i q^{27} +1.73205i q^{29} -3.46410 q^{31} +5.19615i q^{33} +(-1.73205 + 2.00000i) q^{35} +6.92820i q^{37} +1.00000i q^{39} -10.3923i q^{41} -3.46410 q^{43} -2.00000 q^{45} -12.1244 q^{47} +(-1.00000 - 6.92820i) q^{49} -1.73205 q^{51} +3.46410i q^{53} +5.19615 q^{55} -2.00000 q^{57} -6.00000i q^{59} -8.00000 q^{61} +(3.46410 - 4.00000i) q^{63} +1.00000 q^{65} +10.3923 q^{67} +6.92820i q^{73} -1.00000i q^{75} +(-9.00000 + 10.3923i) q^{77} -1.00000i q^{79} +1.00000 q^{81} -12.0000i q^{83} +1.73205i q^{85} +1.73205 q^{87} +6.92820i q^{89} +(-1.73205 + 2.00000i) q^{91} +3.46410i q^{93} +2.00000i q^{95} +8.66025i q^{97} -10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^5 + 8 * q^9	\( 4 q - 4 q^{5} + 8 q^{9} - 4 q^{13} - 8 q^{21} + 4 q^{25} - 8 q^{45} - 4 q^{49} - 8 q^{57} - 32 q^{61} + 4 q^{65} - 36 q^{77} + 4 q^{81}+O(q^{100}) \) 4 * q - 4 * q^5 + 8 * q^9 - 4 * q^13 - 8 * q^21 + 4 * q^25 - 8 * q^45 - 4 * q^49 - 8 * q^57 - 32 * q^61 + 4 * q^65 - 36 * q^77 + 4 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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