LMFDB - Embedded newform 4410.2.b.c.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(881,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, 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("4410.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.b (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:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.c.881.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} -0.930151i q^{11} +0.331821i q^{13} +1.00000 q^{16} -3.26197 q^{17} -0.547123i q^{19} +1.00000 q^{20} -0.930151 q^{22} +0.801088i q^{23} +1.00000 q^{25} +0.331821 q^{26} +6.37170i q^{29} -0.101724i q^{31} -1.00000i q^{32} +3.26197i q^{34} +8.29385 q^{37} -0.547123 q^{38} -1.00000i q^{40} +6.57446 q^{41} +1.49501 q^{43} +0.930151i q^{44} +0.801088 q^{46} -6.57741 q^{47} -1.00000i q^{50} -0.331821i q^{52} -6.38009i q^{53} +0.930151i q^{55} +6.37170 q^{58} -6.11652 q^{59} -4.28130i q^{61} -0.101724 q^{62} -1.00000 q^{64} -0.331821i q^{65} -2.25102 q^{67} +3.26197 q^{68} +3.64725i q^{71} -2.28356i q^{73} -8.29385i q^{74} +0.547123i q^{76} +4.32957 q^{79} -1.00000 q^{80} -6.57446i q^{82} +15.4785 q^{83} +3.26197 q^{85} -1.49501i q^{86} +0.930151 q^{88} +0.219463 q^{89} -0.801088i q^{92} +6.57741i q^{94} +0.547123i q^{95} +3.65685i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^4 - 8 * q^5	$ 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} + 16 q^{22} + 8 q^{25} + 16 q^{37} + 16 q^{41} - 16 q^{43} + 16 q^{46} - 16 q^{47} - 16 q^{58} - 48 q^{59} - 8 q^{64} - 32 q^{67} - 8 q^{80} + 16 q^{83} - 16 q^{88} - 80 q^{89}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 + 16 * q^22 + 8 * q^25 + 16 * q^37 + 16 * q^41 - 16 * q^43 + 16 * q^46 - 16 * q^47 - 16 * q^58 - 48 * q^59 - 8 * q^64 - 32 * q^67 - 8 * q^80 + 16 * q^83 - 16 * q^88 - 80 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times$.

$n$	$1081$	$2647$	$3431$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	0	0
$10$	1.00000i	0.316228i
$11$	− 0.930151i	− 0.280451i	−0.990120	−	0.140226i	$-0.955217\pi$
$11$	− 0.930151i	− 0.280451i	0.990120	−	0.140226i	$-0.0447828\pi$
$12$	0	0
$13$	0.331821i	0.0920307i	0.998941	+	0.0460153i	$0.0146523\pi$
$13$	0.331821i	0.0920307i	−0.998941	+	0.0460153i	$0.985348\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.26197	−0.791145	−0.395572	−	0.918435i	$-0.629454\pi$
$17$	−3.26197	−0.791145	−0.395572	+	0.918435i	$0.629454\pi$
$18$	0	0
$19$	− 0.547123i	− 0.125519i	−0.998029	−	0.0627593i	$-0.980010\pi$
$19$	− 0.547123i	− 0.125519i	0.998029	−	0.0627593i	$-0.0199900\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−0.930151	−0.198309
$23$	0.801088i	0.167038i	0.996506	+	0.0835192i	$0.0266160\pi$
$23$	0.801088i	0.167038i	−0.996506	+	0.0835192i	$0.973384\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.331821	0.0650755
$27$	0	0
$28$	0	0
$29$	6.37170i	1.18320i	0.806233	+	0.591598i	$0.201503\pi$
$29$	6.37170i	1.18320i	−0.806233	+	0.591598i	$0.798497\pi$
$30$	0	0
$31$	− 0.101724i	− 0.0182702i	−0.999958	−	0.00913510i	$-0.997092\pi$
$31$	− 0.101724i	− 0.0182702i	0.999958	−	0.00913510i	$-0.00290783\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	3.26197i	0.559424i
$35$	0	0
$36$	0	0
$37$	8.29385	1.36350	0.681750	−	0.731585i	$-0.261219\pi$
$37$	8.29385	1.36350	0.681750	+	0.731585i	$0.261219\pi$
$38$	−0.547123	−0.0887550
$39$	0	0
$40$	− 1.00000i	− 0.158114i
$41$	6.57446	1.02676	0.513379	−	0.858162i	$-0.328393\pi$
$41$	6.57446	1.02676	0.513379	+	0.858162i	$0.328393\pi$
$42$	0	0
$43$	1.49501	0.227987	0.113994	−	0.993481i	$-0.463636\pi$
$43$	1.49501	0.227987	0.113994	+	0.993481i	$0.463636\pi$
$44$	0.930151i	0.140226i
$45$	0	0
$46$	0.801088	0.118114
$47$	−6.57741	−0.959413	−0.479707	−	0.877429i	$-0.659257\pi$
$47$	−6.57741	−0.959413	−0.479707	+	0.877429i	$0.659257\pi$
$48$	0	0
$49$	0	0
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	− 0.331821i	− 0.0460153i
$53$	− 6.38009i	− 0.876372i	−0.898884	−	0.438186i	$-0.855621\pi$
$53$	− 6.38009i	− 0.876372i	0.898884	−	0.438186i	$-0.144379\pi$
$54$	0	0
$55$	0.930151i	0.125422i
$56$	0	0
$57$	0	0
$58$	6.37170	0.836646
$59$	−6.11652	−0.796303	−0.398152	−	0.917320i	$-0.630348\pi$
$59$	−6.11652	−0.796303	−0.398152	+	0.917320i	$0.630348\pi$
$60$	0	0
$61$	− 4.28130i	− 0.548165i	−0.961706	−	0.274082i	$-0.911626\pi$
$61$	− 4.28130i	− 0.548165i	0.961706	−	0.274082i	$-0.0883741\pi$
$62$	−0.101724	−0.0129190
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 0.331821i	− 0.0411574i
$66$	0	0
$67$	−2.25102	−0.275006	−0.137503	−	0.990501i	$-0.543908\pi$
$67$	−2.25102	−0.275006	−0.137503	+	0.990501i	$0.543908\pi$
$68$	3.26197	0.395572
$69$	0	0
$70$	0	0
$71$	3.64725i	0.432849i	0.976299	+	0.216425i	$0.0694396\pi$
$71$	3.64725i	0.432849i	−0.976299	+	0.216425i	$0.930560\pi$
$72$	0	0
$73$	− 2.28356i	− 0.267270i	−0.991031	−	0.133635i	$-0.957335\pi$
$73$	− 2.28356i	− 0.267270i	0.991031	−	0.133635i	$-0.0426650\pi$
$74$	− 8.29385i	− 0.964140i
$75$	0	0
$76$	0.547123i	0.0627593i
$77$	0	0
$78$	0	0
$79$	4.32957	0.487115	0.243557	−	0.969887i	$-0.421686\pi$
$79$	4.32957	0.487115	0.243557	+	0.969887i	$0.421686\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	− 6.57446i	− 0.726028i
$83$	15.4785	1.69899	0.849493	−	0.527600i	$-0.176908\pi$
$83$	15.4785	1.69899	0.849493	+	0.527600i	$0.176908\pi$
$84$	0	0
$85$	3.26197	0.353811
$86$	− 1.49501i	− 0.161211i
$87$	0	0
$88$	0.930151	0.0991545
$89$	0.219463	0.0232631	0.0116315	−	0.999932i	$-0.496297\pi$
$89$	0.219463	0.0232631	0.0116315	+	0.999932i	$0.496297\pi$
$90$	0	0
$91$	0	0
$92$	− 0.801088i	− 0.0835192i
$93$	0	0
$94$	6.57741i	0.678408i
$95$	0.547123i	0.0561336i
$96$	0	0
$97$	3.65685i	0.371297i	0.982616	+	0.185649i	$0.0594386\pi$
$97$	3.65685i	0.371297i	−0.982616	+	0.185649i	$0.940561\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−10.7119	−1.06587	−0.532937	−	0.846155i	$-0.678912\pi$
$101$	−10.7119	−1.06587	−0.532937	+	0.846155i	$0.678912\pi$
$102$	0	0
$103$	− 1.91082i	− 0.188279i	−0.995559	−	0.0941393i	$-0.969990\pi$
$103$	− 1.91082i	− 0.188279i	0.995559	−	0.0941393i	$-0.0300099\pi$
$104$	−0.331821	−0.0325378
$105$	0	0
$106$	−6.38009	−0.619689
$107$	− 0.972287i	− 0.0939945i	−0.998895	−	0.0469973i	$-0.985035\pi$
$107$	− 0.972287i	− 0.0939945i	0.998895	−	0.0469973i	$-0.0149652\pi$
$108$	0	0
$109$	18.4020	1.76259	0.881295	−	0.472566i	$-0.156672\pi$
$109$	18.4020	1.76259	0.881295	+	0.472566i	$0.156672\pi$
$110$	0.930151	0.0886864
$111$	0	0
$112$	0	0
$113$	4.16160i	0.391490i	0.980655	+	0.195745i	$0.0627125\pi$
$113$	4.16160i	0.391490i	−0.980655	+	0.195745i	$0.937287\pi$
$114$	0	0
$115$	− 0.801088i	− 0.0747018i
$116$	− 6.37170i	− 0.591598i
$117$	0	0
$118$	6.11652i	0.563071i
$119$	0	0
$120$	0	0
$121$	10.1348	0.921347
$122$	−4.28130	−0.387611
$123$	0	0
$124$	0.101724i	0.00913510i
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.9832	1.41828	0.709141	−	0.705066i	$-0.249083\pi$
$127$	15.9832	1.41828	0.709141	+	0.705066i	$0.249083\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−0.331821	−0.0291027
$131$	5.82805	0.509199	0.254600	−	0.967047i	$-0.418056\pi$
$131$	5.82805	0.509199	0.254600	+	0.967047i	$0.418056\pi$
$132$	0	0
$133$	0	0
$134$	2.25102i	0.194459i
$135$	0	0
$136$	− 3.26197i	− 0.279712i
$137$	− 5.58316i	− 0.477002i	−0.971142	−	0.238501i	$-0.923344\pi$
$137$	− 5.58316i	− 0.477002i	0.971142	−	0.238501i	$-0.0766560\pi$
$138$	0	0
$139$	12.8599i	1.09076i	0.838187	+	0.545382i	$0.183616\pi$
$139$	12.8599i	1.09076i	−0.838187	+	0.545382i	$0.816384\pi$
$140$	0	0
$141$	0	0
$142$	3.64725	0.306071
$143$	0.308644	0.0258101
$144$	0	0
$145$	− 6.37170i	− 0.529141i
$146$	−2.28356	−0.188989
$147$	0	0
$148$	−8.29385	−0.681750
$149$	− 7.93694i	− 0.650219i	−0.945676	−	0.325110i	$-0.894599\pi$
$149$	− 7.93694i	− 0.650219i	0.945676	−	0.325110i	$-0.105401\pi$
$150$	0	0
$151$	13.5236	1.10053	0.550266	−	0.834989i	$-0.314526\pi$
$151$	13.5236	1.10053	0.550266	+	0.834989i	$0.314526\pi$
$152$	0.547123	0.0443775
$153$	0	0
$154$	0	0
$155$	0.101724i	0.00817068i
$156$	0	0
$157$	− 15.3196i	− 1.22264i	−0.791385	−	0.611318i	$-0.790639\pi$
$157$	− 15.3196i	− 1.22264i	0.791385	−	0.611318i	$-0.209361\pi$
$158$	− 4.32957i	− 0.344442i
$159$	0	0
$160$	1.00000i	0.0790569i
$161$	0	0
$162$	0	0
$163$	13.2661	1.03908	0.519542	−	0.854445i	$-0.326103\pi$
$163$	13.2661	1.03908	0.519542	+	0.854445i	$0.326103\pi$
$164$	−6.57446	−0.513379
$165$	0	0
$166$	− 15.4785i	− 1.20136i
$167$	15.3462	1.18753	0.593764	−	0.804639i	$-0.297641\pi$
$167$	15.3462	1.18753	0.593764	+	0.804639i	$0.297641\pi$
$168$	0	0
$169$	12.8899	0.991530
$170$	− 3.26197i	− 0.250182i
$171$	0	0
$172$	−1.49501	−0.113994
$173$	−7.23532	−0.550091	−0.275046	−	0.961431i	$-0.588693\pi$
$173$	−7.23532	−0.550091	−0.275046	+	0.961431i	$0.588693\pi$
$174$	0	0
$175$	0	0
$176$	− 0.930151i	− 0.0701128i
$177$	0	0
$178$	− 0.219463i	− 0.0164495i
$179$	− 14.8082i	− 1.10682i	−0.832910	−	0.553408i	$-0.813327\pi$
$179$	− 14.8082i	− 1.10682i	0.832910	−	0.553408i	$-0.186673\pi$
$180$	0	0
$181$	− 15.0930i	− 1.12185i	−0.827866	−	0.560926i	$-0.810445\pi$
$181$	− 15.0930i	− 1.12185i	0.827866	−	0.560926i	$-0.189555\pi$
$182$	0	0
$183$	0	0
$184$	−0.801088	−0.0590570
$185$	−8.29385	−0.609776
$186$	0	0
$187$	3.03413i	0.221877i
$188$	6.57741	0.479707
$189$	0	0
$190$	0.547123	0.0396924
$191$	− 20.9314i	− 1.51454i	−0.653101	−	0.757270i	$-0.726533\pi$
$191$	− 20.9314i	− 1.51454i	0.653101	−	0.757270i	$-0.273467\pi$
$192$	0	0
$193$	15.6014	1.12302	0.561508	−	0.827472i	$-0.310222\pi$
$193$	15.6014	1.12302	0.561508	+	0.827472i	$0.310222\pi$
$194$	3.65685	0.262547
$195$	0	0
$196$	0	0
$197$	− 2.65498i	− 0.189159i	−0.995517	−	0.0945796i	$-0.969849\pi$
$197$	− 2.65498i	− 0.189159i	0.995517	−	0.0945796i	$-0.0301507\pi$
$198$	0	0
$199$	11.6840i	0.828253i	0.910219	+	0.414127i	$0.135913\pi$
$199$	11.6840i	0.828253i	−0.910219	+	0.414127i	$0.864087\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	10.7119i	0.753687i
$203$	0	0
$204$	0	0
$205$	−6.57446	−0.459180
$206$	−1.91082	−0.133133
$207$	0	0
$208$	0.331821i	0.0230077i
$209$	−0.508907	−0.0352018
$210$	0	0
$211$	−4.36407	−0.300435	−0.150218	−	0.988653i	$-0.547997\pi$
$211$	−4.36407	−0.300435	−0.150218	+	0.988653i	$0.547997\pi$
$212$	6.38009i	0.438186i
$213$	0	0
$214$	−0.972287	−0.0664642
$215$	−1.49501	−0.101959
$216$	0	0
$217$	0	0
$218$	− 18.4020i	− 1.24634i
$219$	0	0
$220$	− 0.930151i	− 0.0627108i
$221$	− 1.08239i	− 0.0728096i
$222$	0	0
$223$	− 24.2623i	− 1.62472i	−0.583154	−	0.812362i	$-0.698182\pi$
$223$	− 24.2623i	− 1.62472i	0.583154	−	0.812362i	$-0.301818\pi$
$224$	0	0
$225$	0	0
$226$	4.16160	0.276825
$227$	−29.2809	−1.94344	−0.971722	−	0.236129i	$-0.924121\pi$
$227$	−29.2809	−1.94344	−0.971722	+	0.236129i	$0.924121\pi$
$228$	0	0
$229$	4.84710i	0.320305i	0.987092	+	0.160153i	$0.0511987\pi$
$229$	4.84710i	0.320305i	−0.987092	+	0.160153i	$0.948801\pi$
$230$	−0.801088	−0.0528222
$231$	0	0
$232$	−6.37170	−0.418323
$233$	2.66683i	0.174710i	0.996177	+	0.0873549i	$0.0278414\pi$
$233$	2.66683i	0.174710i	−0.996177	+	0.0873549i	$0.972159\pi$
$234$	0	0
$235$	6.57741	0.429063
$236$	6.11652	0.398152
$237$	0	0
$238$	0	0
$239$	18.9906i	1.22840i	0.789152	+	0.614198i	$0.210521\pi$
$239$	18.9906i	1.22840i	−0.789152	+	0.614198i	$0.789479\pi$
$240$	0	0
$241$	− 14.4045i	− 0.927874i	−0.885868	−	0.463937i	$-0.846436\pi$
$241$	− 14.4045i	− 0.927874i	0.885868	−	0.463937i	$-0.153564\pi$
$242$	− 10.1348i	− 0.651491i
$243$	0	0
$244$	4.28130i	0.274082i
$245$	0	0
$246$	0	0
$247$	0.181547	0.0115516
$248$	0.101724	0.00645949
$249$	0	0
$250$	1.00000i	0.0632456i
$251$	20.5072	1.29440	0.647201	−	0.762319i	$-0.275939\pi$
$251$	20.5072	1.29440	0.647201	+	0.762319i	$0.275939\pi$
$252$	0	0
$253$	0.745133	0.0468461
$254$	− 15.9832i	− 1.00288i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.5842	−0.847359	−0.423679	−	0.905812i	$-0.639262\pi$
$257$	−13.5842	−0.847359	−0.423679	+	0.905812i	$0.639262\pi$
$258$	0	0
$259$	0	0
$260$	0.331821i	0.0205787i
$261$	0	0
$262$	− 5.82805i	− 0.360058i
$263$	− 27.6507i	− 1.70501i	−0.522716	−	0.852507i	$-0.675081\pi$
$263$	− 27.6507i	− 1.70501i	0.522716	−	0.852507i	$-0.324919\pi$
$264$	0	0
$265$	6.38009i	0.391926i
$266$	0	0
$267$	0	0
$268$	2.25102	0.137503
$269$	13.1708	0.803039	0.401520	−	0.915850i	$-0.368482\pi$
$269$	13.1708	0.803039	0.401520	+	0.915850i	$0.368482\pi$
$270$	0	0
$271$	13.1590i	0.799354i	0.916656	+	0.399677i	$0.130878\pi$
$271$	13.1590i	0.799354i	−0.916656	+	0.399677i	$0.869122\pi$
$272$	−3.26197	−0.197786
$273$	0	0
$274$	−5.58316	−0.337291
$275$	− 0.930151i	− 0.0560902i
$276$	0	0
$277$	10.0071	0.601269	0.300634	−	0.953739i	$-0.402802\pi$
$277$	10.0071	0.601269	0.300634	+	0.953739i	$0.402802\pi$
$278$	12.8599	0.771287
$279$	0	0
$280$	0	0
$281$	− 3.54033i	− 0.211199i	−0.994409	−	0.105599i	$-0.966324\pi$
$281$	− 3.54033i	− 0.211199i	0.994409	−	0.105599i	$-0.0336761\pi$
$282$	0	0
$283$	− 15.5581i	− 0.924831i	−0.886663	−	0.462416i	$-0.846983\pi$
$283$	− 15.5581i	− 0.924831i	0.886663	−	0.462416i	$-0.153017\pi$
$284$	− 3.64725i	− 0.216425i
$285$	0	0
$286$	− 0.308644i	− 0.0182505i
$287$	0	0
$288$	0	0
$289$	−6.35953	−0.374090
$290$	−6.37170	−0.374159
$291$	0	0
$292$	2.28356i	0.133635i
$293$	4.25813	0.248762	0.124381	−	0.992235i	$-0.460305\pi$
$293$	4.25813	0.248762	0.124381	+	0.992235i	$0.460305\pi$
$294$	0	0
$295$	6.11652	0.356118
$296$	8.29385i	0.482070i
$297$	0	0
$298$	−7.93694	−0.459775
$299$	−0.265818	−0.0153727
$300$	0	0
$301$	0	0
$302$	− 13.5236i	− 0.778194i
$303$	0	0
$304$	− 0.547123i	− 0.0313796i
$305$	4.28130i	0.245147i
$306$	0	0
$307$	− 3.65097i	− 0.208372i	−0.994558	−	0.104186i	$-0.966776\pi$
$307$	− 3.65097i	− 0.208372i	0.994558	−	0.104186i	$-0.0332237\pi$
$308$	0	0
$309$	0	0
$310$	0.101724	0.00577754
$311$	8.15121	0.462213	0.231106	−	0.972929i	$-0.425765\pi$
$311$	8.15121	0.462213	0.231106	+	0.972929i	$0.425765\pi$
$312$	0	0
$313$	7.07921i	0.400140i	0.979782	+	0.200070i	$0.0641170\pi$
$313$	7.07921i	0.400140i	−0.979782	+	0.200070i	$0.935883\pi$
$314$	−15.3196	−0.864535
$315$	0	0
$316$	−4.32957	−0.243557
$317$	0.0850185i	0.00477511i	0.999997	+	0.00238756i	$0.000759984\pi$
$317$	0.0850185i	0.00477511i	−0.999997	+	0.00238756i	$0.999240\pi$
$318$	0	0
$319$	5.92665	0.331829
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	1.78470i	0.0993033i
$324$	0	0
$325$	0.331821i	0.0184061i
$326$	− 13.2661i	− 0.734743i
$327$	0	0
$328$	6.57446i	0.363014i
$329$	0	0
$330$	0	0
$331$	−6.64860	−0.365440	−0.182720	−	0.983165i	$-0.558490\pi$
$331$	−6.64860	−0.365440	−0.182720	+	0.983165i	$0.558490\pi$
$332$	−15.4785	−0.849493
$333$	0	0
$334$	− 15.3462i	− 0.839709i
$335$	2.25102	0.122986
$336$	0	0
$337$	−5.65367	−0.307975	−0.153987	−	0.988073i	$-0.549212\pi$
$337$	−5.65367	−0.307975	−0.153987	+	0.988073i	$0.549212\pi$
$338$	− 12.8899i	− 0.701118i
$339$	0	0
$340$	−3.26197	−0.176905
$341$	−0.0946188	−0.00512390
$342$	0	0
$343$	0	0
$344$	1.49501i	0.0806057i
$345$	0	0
$346$	7.23532i	0.388973i
$347$	13.0479i	0.700447i	0.936666	+	0.350224i	$0.113894\pi$
$347$	13.0479i	0.700447i	−0.936666	+	0.350224i	$0.886106\pi$
$348$	0	0
$349$	32.2182i	1.72460i	0.506398	+	0.862300i	$0.330977\pi$
$349$	32.2182i	1.72460i	−0.506398	+	0.862300i	$0.669023\pi$
$350$	0	0
$351$	0	0
$352$	−0.930151	−0.0495772
$353$	18.2963	0.973813	0.486907	−	0.873454i	$-0.338125\pi$
$353$	18.2963	0.973813	0.486907	+	0.873454i	$0.338125\pi$
$354$	0	0
$355$	− 3.64725i	− 0.193576i
$356$	−0.219463	−0.0116315
$357$	0	0
$358$	−14.8082	−0.782637
$359$	− 36.9146i	− 1.94828i	−0.225949	−	0.974139i	$-0.572548\pi$
$359$	− 36.9146i	− 1.94828i	0.225949	−	0.974139i	$-0.427452\pi$
$360$	0	0
$361$	18.7007	0.984245
$362$	−15.0930	−0.793269
$363$	0	0
$364$	0	0
$365$	2.28356i	0.119527i
$366$	0	0
$367$	− 16.2459i	− 0.848028i	−0.905656	−	0.424014i	$-0.860621\pi$
$367$	− 16.2459i	− 0.848028i	0.905656	−	0.424014i	$-0.139379\pi$
$368$	0.801088i	0.0417596i
$369$	0	0
$370$	8.29385i	0.431177i
$371$	0	0
$372$	0	0
$373$	12.2333	0.633415	0.316708	−	0.948523i	$-0.397423\pi$
$373$	12.2333	0.633415	0.316708	+	0.948523i	$0.397423\pi$
$374$	3.03413	0.156891
$375$	0	0
$376$	− 6.57741i	− 0.339204i
$377$	−2.11427	−0.108890
$378$	0	0
$379$	24.8754	1.27776	0.638882	−	0.769305i	$-0.279397\pi$
$379$	24.8754	1.27776	0.638882	+	0.769305i	$0.279397\pi$
$380$	− 0.547123i	− 0.0280668i
$381$	0	0
$382$	−20.9314	−1.07094
$383$	−18.0549	−0.922563	−0.461282	−	0.887254i	$-0.652610\pi$
$383$	−18.0549	−0.922563	−0.461282	+	0.887254i	$0.652610\pi$
$384$	0	0
$385$	0	0
$386$	− 15.6014i	− 0.794092i
$387$	0	0
$388$	− 3.65685i	− 0.185649i
$389$	− 9.78892i	− 0.496318i	−0.968719	−	0.248159i	$-0.920174\pi$
$389$	− 9.78892i	− 0.496318i	0.968719	−	0.248159i	$-0.0798255\pi$
$390$	0	0
$391$	− 2.61313i	− 0.132151i
$392$	0	0
$393$	0	0
$394$	−2.65498	−0.133756
$395$	−4.32957	−0.217844
$396$	0	0
$397$	14.0160i	0.703444i	0.936105	+	0.351722i	$0.114404\pi$
$397$	14.0160i	0.703444i	−0.936105	+	0.351722i	$0.885596\pi$
$398$	11.6840	0.585664
$399$	0	0
$400$	1.00000	0.0500000
$401$	9.34596i	0.466715i	0.972391	+	0.233357i	$0.0749712\pi$
$401$	9.34596i	0.466715i	−0.972391	+	0.233357i	$0.925029\pi$
$402$	0	0
$403$	0.0337542	0.00168142
$404$	10.7119	0.532937
$405$	0	0
$406$	0	0
$407$	− 7.71453i	− 0.382395i
$408$	0	0
$409$	17.7620i	0.878274i	0.898420	+	0.439137i	$0.144716\pi$
$409$	17.7620i	0.878274i	−0.898420	+	0.439137i	$0.855284\pi$
$410$	6.57446i	0.324690i
$411$	0	0
$412$	1.91082i	0.0941393i
$413$	0	0
$414$	0	0
$415$	−15.4785	−0.759809
$416$	0.331821	0.0162689
$417$	0	0
$418$	0.508907i	0.0248914i
$419$	−17.3073	−0.845519	−0.422759	−	0.906242i	$-0.638938\pi$
$419$	−17.3073	−0.845519	−0.422759	+	0.906242i	$0.638938\pi$
$420$	0	0
$421$	3.79621	0.185016	0.0925079	−	0.995712i	$-0.470512\pi$
$421$	3.79621	0.185016	0.0925079	+	0.995712i	$0.470512\pi$
$422$	4.36407i	0.212440i
$423$	0	0
$424$	6.38009	0.309844
$425$	−3.26197	−0.158229
$426$	0	0
$427$	0	0
$428$	0.972287i	0.0469973i
$429$	0	0
$430$	1.49501i	0.0720959i
$431$	1.92214i	0.0925864i	0.998928	+	0.0462932i	$0.0147409\pi$
$431$	1.92214i	0.0925864i	−0.998928	+	0.0462932i	$0.985259\pi$
$432$	0	0
$433$	− 17.5127i	− 0.841604i	−0.907152	−	0.420802i	$-0.861749\pi$
$433$	− 17.5127i	− 0.841604i	0.907152	−	0.420802i	$-0.138251\pi$
$434$	0	0
$435$	0	0
$436$	−18.4020	−0.881295
$437$	0.438293	0.0209664
$438$	0	0
$439$	40.3613i	1.92634i	0.268893	+	0.963170i	$0.413342\pi$
$439$	40.3613i	1.92634i	−0.268893	+	0.963170i	$0.586658\pi$
$440$	−0.930151	−0.0443432
$441$	0	0
$442$	−1.08239	−0.0514841
$443$	8.00694i	0.380421i	0.981743	+	0.190211i	$0.0609171\pi$
$443$	8.00694i	0.380421i	−0.981743	+	0.190211i	$0.939083\pi$
$444$	0	0
$445$	−0.219463	−0.0104036
$446$	−24.2623	−1.14885
$447$	0	0
$448$	0	0
$449$	27.4212i	1.29408i	0.762454	+	0.647042i	$0.223994\pi$
$449$	27.4212i	1.29408i	−0.762454	+	0.647042i	$0.776006\pi$
$450$	0	0
$451$	− 6.11524i	− 0.287956i
$452$	− 4.16160i	− 0.195745i
$453$	0	0
$454$	29.2809i	1.37422i
$455$	0	0
$456$	0	0
$457$	−3.49526	−0.163501	−0.0817506	−	0.996653i	$-0.526051\pi$
$457$	−3.49526	−0.163501	−0.0817506	+	0.996653i	$0.526051\pi$
$458$	4.84710	0.226490
$459$	0	0
$460$	0.801088i	0.0373509i
$461$	28.4061	1.32301	0.661503	−	0.749942i	$-0.269919\pi$
$461$	28.4061	1.32301	0.661503	+	0.749942i	$0.269919\pi$
$462$	0	0
$463$	−42.4813	−1.97428	−0.987138	−	0.159871i	$-0.948892\pi$
$463$	−42.4813	−1.97428	−0.987138	+	0.159871i	$0.948892\pi$
$464$	6.37170i	0.295799i
$465$	0	0
$466$	2.66683	0.123538
$467$	−11.8480	−0.548262	−0.274131	−	0.961692i	$-0.588390\pi$
$467$	−11.8480	−0.548262	−0.274131	+	0.961692i	$0.588390\pi$
$468$	0	0
$469$	0	0
$470$	− 6.57741i	− 0.303393i
$471$	0	0
$472$	− 6.11652i	− 0.281536i
$473$	− 1.39059i	− 0.0639393i
$474$	0	0
$475$	− 0.547123i	− 0.0251037i
$476$	0	0
$477$	0	0
$478$	18.9906	0.868608
$479$	−11.4231	−0.521933	−0.260967	−	0.965348i	$-0.584041\pi$
$479$	−11.4231	−0.521933	−0.260967	+	0.965348i	$0.584041\pi$
$480$	0	0
$481$	2.75208i	0.125484i
$482$	−14.4045	−0.656106
$483$	0	0
$484$	−10.1348	−0.460674
$485$	− 3.65685i	− 0.166049i
$486$	0	0
$487$	−19.0897	−0.865039	−0.432519	−	0.901625i	$-0.642375\pi$
$487$	−19.0897	−0.865039	−0.432519	+	0.901625i	$0.642375\pi$
$488$	4.28130	0.193806
$489$	0	0
$490$	0	0
$491$	− 4.04288i	− 0.182453i	−0.995830	−	0.0912264i	$-0.970921\pi$
$491$	− 4.04288i	− 0.182453i	0.995830	−	0.0912264i	$-0.0290787\pi$
$492$	0	0
$493$	− 20.7843i	− 0.936079i
$494$	− 0.181547i	− 0.00816818i
$495$	0	0
$496$	− 0.101724i	− 0.00456755i
$497$	0	0
$498$	0	0
$499$	−30.4594	−1.36355	−0.681776	−	0.731561i	$-0.738792\pi$
$499$	−30.4594	−1.36355	−0.681776	+	0.731561i	$0.738792\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	− 20.5072i	− 0.915280i
$503$	1.83400	0.0817739	0.0408869	−	0.999164i	$-0.486982\pi$
$503$	1.83400	0.0817739	0.0408869	+	0.999164i	$0.486982\pi$
$504$	0	0
$505$	10.7119	0.476674
$506$	− 0.745133i	− 0.0331252i
$507$	0	0
$508$	−15.9832	−0.709141
$509$	5.75147	0.254930	0.127465	−	0.991843i	$-0.459316\pi$
$509$	5.75147	0.254930	0.127465	+	0.991843i	$0.459316\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	13.5842i	0.599173i
$515$	1.91082i	0.0842008i
$516$	0	0
$517$	6.11798i	0.269069i
$518$	0	0
$519$	0	0
$520$	0.331821	0.0145513
$521$	30.8399	1.35112	0.675561	−	0.737304i	$-0.263901\pi$
$521$	30.8399	1.35112	0.675561	+	0.737304i	$0.263901\pi$
$522$	0	0
$523$	16.2217i	0.709326i	0.934994	+	0.354663i	$0.115404\pi$
$523$	16.2217i	0.709326i	−0.934994	+	0.354663i	$0.884596\pi$
$524$	−5.82805	−0.254600
$525$	0	0
$526$	−27.6507	−1.20563
$527$	0.331821i	0.0144544i
$528$	0	0
$529$	22.3583	0.972098
$530$	6.38009	0.277133
$531$	0	0
$532$	0	0
$533$	2.18155i	0.0944933i
$534$	0	0
$535$	0.972287i	0.0420356i
$536$	− 2.25102i	− 0.0972293i
$537$	0	0
$538$	− 13.1708i	− 0.567835i
$539$	0	0
$540$	0	0
$541$	21.6951	0.932746	0.466373	−	0.884588i	$-0.345561\pi$
$541$	21.6951	0.932746	0.466373	+	0.884588i	$0.345561\pi$
$542$	13.1590	0.565229
$543$	0	0
$544$	3.26197i	0.139856i
$545$	−18.4020	−0.788255
$546$	0	0
$547$	25.2142	1.07808	0.539041	−	0.842279i	$-0.318787\pi$
$547$	25.2142	1.07808	0.539041	+	0.842279i	$0.318787\pi$
$548$	5.58316i	0.238501i
$549$	0	0
$550$	−0.930151	−0.0396618
$551$	3.48610	0.148513
$552$	0	0
$553$	0	0
$554$	− 10.0071i	− 0.425161i
$555$	0	0
$556$	− 12.8599i	− 0.545382i
$557$	36.5741i	1.54970i	0.632148	+	0.774848i	$0.282173\pi$
$557$	36.5741i	1.54970i	−0.632148	+	0.774848i	$0.717827\pi$
$558$	0	0
$559$	0.496077i	0.0209818i
$560$	0	0
$561$	0	0
$562$	−3.54033	−0.149340
$563$	32.4995	1.36969	0.684845	−	0.728689i	$-0.259870\pi$
$563$	32.4995	1.36969	0.684845	+	0.728689i	$0.259870\pi$
$564$	0	0
$565$	− 4.16160i	− 0.175080i
$566$	−15.5581	−0.653955
$567$	0	0
$568$	−3.64725	−0.153035
$569$	− 11.4915i	− 0.481750i	−0.970556	−	0.240875i	$-0.922566\pi$
$569$	− 11.4915i	− 0.481750i	0.970556	−	0.240875i	$-0.0774345\pi$
$570$	0	0
$571$	−38.0006	−1.59028	−0.795139	−	0.606428i	$-0.792602\pi$
$571$	−38.0006	−1.59028	−0.795139	+	0.606428i	$0.792602\pi$
$572$	−0.308644	−0.0129051
$573$	0	0
$574$	0	0
$575$	0.801088i	0.0334077i
$576$	0	0
$577$	− 33.6416i	− 1.40052i	−0.713890	−	0.700258i	$-0.753068\pi$
$577$	− 33.6416i	− 1.40052i	0.713890	−	0.700258i	$-0.246932\pi$
$578$	6.35953i	0.264522i
$579$	0	0
$580$	6.37170i	0.264571i
$581$	0	0
$582$	0	0
$583$	−5.93444	−0.245780
$584$	2.28356	0.0944943
$585$	0	0
$586$	− 4.25813i	− 0.175902i
$587$	26.2459	1.08328	0.541641	−	0.840610i	$-0.317803\pi$
$587$	26.2459	1.08328	0.541641	+	0.840610i	$0.317803\pi$
$588$	0	0
$589$	−0.0556556	−0.00229325
$590$	− 6.11652i	− 0.251813i
$591$	0	0
$592$	8.29385	0.340875
$593$	−19.2070	−0.788735	−0.394367	−	0.918953i	$-0.629036\pi$
$593$	−19.2070	−0.788735	−0.394367	+	0.918953i	$0.629036\pi$
$594$	0	0
$595$	0	0
$596$	7.93694i	0.325110i
$597$	0	0
$598$	0.265818i	0.0108701i
$599$	26.6429i	1.08860i	0.838891	+	0.544300i	$0.183205\pi$
$599$	26.6429i	1.08860i	−0.838891	+	0.544300i	$0.816795\pi$
$600$	0	0
$601$	2.45770i	0.100252i	0.998743	+	0.0501258i	$0.0159622\pi$
$601$	2.45770i	0.100252i	−0.998743	+	0.0501258i	$0.984038\pi$
$602$	0	0
$603$	0	0
$604$	−13.5236	−0.550266
$605$	−10.1348	−0.412039
$606$	0	0
$607$	− 8.93681i	− 0.362734i	−0.983415	−	0.181367i	$-0.941948\pi$
$607$	− 8.93681i	− 0.362734i	0.983415	−	0.181367i	$-0.0580522\pi$
$608$	−0.547123	−0.0221887
$609$	0	0
$610$	4.28130	0.173345
$611$	− 2.18252i	− 0.0882955i
$612$	0	0
$613$	27.9482	1.12882	0.564409	−	0.825495i	$-0.309104\pi$
$613$	27.9482	1.12882	0.564409	+	0.825495i	$0.309104\pi$
$614$	−3.65097	−0.147341
$615$	0	0
$616$	0	0
$617$	− 12.4353i	− 0.500626i	−0.968165	−	0.250313i	$-0.919466\pi$
$617$	− 12.4353i	− 0.500626i	0.968165	−	0.250313i	$-0.0805335\pi$
$618$	0	0
$619$	41.9175i	1.68481i	0.538849	+	0.842403i	$0.318859\pi$
$619$	41.9175i	1.68481i	−0.538849	+	0.842403i	$0.681141\pi$
$620$	− 0.101724i	− 0.00408534i
$621$	0	0
$622$	− 8.15121i	− 0.326834i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	7.07921	0.282942
$627$	0	0
$628$	15.3196i	0.611318i
$629$	−27.0543	−1.07873
$630$	0	0
$631$	−5.24943	−0.208976	−0.104488	−	0.994526i	$-0.533320\pi$
$631$	−5.24943	−0.208976	−0.104488	+	0.994526i	$0.533320\pi$
$632$	4.32957i	0.172221i
$633$	0	0
$634$	0.0850185	0.00337652
$635$	−15.9832	−0.634275
$636$	0	0
$637$	0	0
$638$	− 5.92665i	− 0.234638i
$639$	0	0
$640$	− 1.00000i	− 0.0395285i
$641$	30.4266i	1.20178i	0.799332	+	0.600890i	$0.205187\pi$
$641$	30.4266i	1.20178i	−0.799332	+	0.600890i	$0.794813\pi$
$642$	0	0
$643$	− 42.6780i	− 1.68306i	−0.540212	−	0.841529i	$-0.681656\pi$
$643$	− 42.6780i	− 1.68306i	0.540212	−	0.841529i	$-0.318344\pi$
$644$	0	0
$645$	0	0
$646$	1.78470	0.0702180
$647$	18.2966	0.719315	0.359657	−	0.933084i	$-0.382894\pi$
$647$	18.2966	0.719315	0.359657	+	0.933084i	$0.382894\pi$
$648$	0	0
$649$	5.68929i	0.223324i
$650$	0.331821	0.0130151
$651$	0	0
$652$	−13.2661	−0.519542
$653$	44.1790i	1.72886i	0.502755	+	0.864429i	$0.332320\pi$
$653$	44.1790i	1.72886i	−0.502755	+	0.864429i	$0.667680\pi$
$654$	0	0
$655$	−5.82805	−0.227721
$656$	6.57446	0.256690
$657$	0	0
$658$	0	0
$659$	40.0063i	1.55842i	0.626760	+	0.779212i	$0.284381\pi$
$659$	40.0063i	1.55842i	−0.626760	+	0.779212i	$0.715619\pi$
$660$	0	0
$661$	− 5.14596i	− 0.200155i	−0.994980	−	0.100077i	$-0.968091\pi$
$661$	− 5.14596i	− 0.200155i	0.994980	−	0.100077i	$-0.0319090\pi$
$662$	6.64860i	0.258405i
$663$	0	0
$664$	15.4785i	0.600682i
$665$	0	0
$666$	0	0
$667$	−5.10429	−0.197639
$668$	−15.3462	−0.593764
$669$	0	0
$670$	− 2.25102i	− 0.0869646i
$671$	−3.98226	−0.153733
$672$	0	0
$673$	−24.7124	−0.952594	−0.476297	−	0.879284i	$-0.658021\pi$
$673$	−24.7124	−0.952594	−0.476297	+	0.879284i	$0.658021\pi$
$674$	5.65367i	0.217771i
$675$	0	0
$676$	−12.8899	−0.495765
$677$	−22.5424	−0.866375	−0.433187	−	0.901304i	$-0.642611\pi$
$677$	−22.5424	−0.866375	−0.433187	+	0.901304i	$0.642611\pi$
$678$	0	0
$679$	0	0
$680$	3.26197i	0.125091i
$681$	0	0
$682$	0.0946188i	0.00362314i
$683$	4.15882i	0.159133i	0.996830	+	0.0795664i	$0.0253536\pi$
$683$	4.15882i	0.159133i	−0.996830	+	0.0795664i	$0.974646\pi$
$684$	0	0
$685$	5.58316i	0.213322i
$686$	0	0
$687$	0	0
$688$	1.49501	0.0569968
$689$	2.11705	0.0806531
$690$	0	0
$691$	27.1058i	1.03115i	0.856843	+	0.515577i	$0.172422\pi$
$691$	27.1058i	1.03115i	−0.856843	+	0.515577i	$0.827578\pi$
$692$	7.23532	0.275046
$693$	0	0
$694$	13.0479	0.495291
$695$	− 12.8599i	− 0.487805i
$696$	0	0
$697$	−21.4457	−0.812314
$698$	32.2182	1.21948
$699$	0	0
$700$	0	0
$701$	− 6.04705i	− 0.228394i	−0.993458	−	0.114197i	$-0.963571\pi$
$701$	− 6.04705i	− 0.228394i	0.993458	−	0.114197i	$-0.0364295\pi$
$702$	0	0
$703$	− 4.53775i	− 0.171145i
$704$	0.930151i	0.0350564i
$705$	0	0
$706$	− 18.2963i	− 0.688590i
$707$	0	0
$708$	0	0
$709$	−42.3571	−1.59075	−0.795377	−	0.606115i	$-0.792727\pi$
$709$	−42.3571	−1.59075	−0.795377	+	0.606115i	$0.792727\pi$
$710$	−3.64725	−0.136879
$711$	0	0
$712$	0.219463i	0.00822474i
$713$	0.0814900	0.00305182
$714$	0	0
$715$	−0.308644	−0.0115426
$716$	14.8082i	0.553408i
$717$	0	0
$718$	−36.9146	−1.37764
$719$	23.8704	0.890214	0.445107	−	0.895477i	$-0.353166\pi$
$719$	23.8704	0.890214	0.445107	+	0.895477i	$0.353166\pi$
$720$	0	0
$721$	0	0
$722$	− 18.7007i	− 0.695966i
$723$	0	0
$724$	15.0930i	0.560926i
$725$	6.37170i	0.236639i
$726$	0	0
$727$	− 10.6041i	− 0.393283i	−0.980475	−	0.196641i	$-0.936997\pi$
$727$	− 10.6041i	− 0.393283i	0.980475	−	0.196641i	$-0.0630034\pi$
$728$	0	0
$729$	0	0
$730$	2.28356	0.0845182
$731$	−4.87669	−0.180371
$732$	0	0
$733$	− 28.8949i	− 1.06726i	−0.845719	−	0.533628i	$-0.820828\pi$
$733$	− 28.8949i	− 1.06726i	0.845719	−	0.533628i	$-0.179172\pi$
$734$	−16.2459	−0.599646
$735$	0	0
$736$	0.801088	0.0295285
$737$	2.09379i	0.0771258i
$738$	0	0
$739$	−4.21872	−0.155188	−0.0775940	−	0.996985i	$-0.524724\pi$
$739$	−4.21872	−0.155188	−0.0775940	+	0.996985i	$0.524724\pi$
$740$	8.29385	0.304888
$741$	0	0
$742$	0	0
$743$	1.75683i	0.0644519i	0.999481	+	0.0322259i	$0.0102596\pi$
$743$	1.75683i	0.0644519i	−0.999481	+	0.0322259i	$0.989740\pi$
$744$	0	0
$745$	7.93694i	0.290787i
$746$	− 12.2333i	− 0.447892i
$747$	0	0
$748$	− 3.03413i	− 0.110939i
$749$	0	0
$750$	0	0
$751$	−8.16801	−0.298055	−0.149028	−	0.988833i	$-0.547614\pi$
$751$	−8.16801	−0.298055	−0.149028	+	0.988833i	$0.547614\pi$
$752$	−6.57741	−0.239853
$753$	0	0
$754$	2.11427i	0.0769971i
$755$	−13.5236	−0.492173
$756$	0	0
$757$	23.7897	0.864653	0.432326	−	0.901717i	$-0.357693\pi$
$757$	23.7897	0.864653	0.432326	+	0.901717i	$0.357693\pi$
$758$	− 24.8754i	− 0.903516i
$759$	0	0
$760$	−0.547123	−0.0198462
$761$	29.0511	1.05310	0.526550	−	0.850144i	$-0.323485\pi$
$761$	29.0511	1.05310	0.526550	+	0.850144i	$0.323485\pi$
$762$	0	0
$763$	0	0
$764$	20.9314i	0.757270i
$765$	0	0
$766$	18.0549i	0.652351i
$767$	− 2.02959i	− 0.0732843i
$768$	0	0
$769$	37.9013i	1.36676i	0.730064	+	0.683379i	$0.239490\pi$
$769$	37.9013i	1.36676i	−0.730064	+	0.683379i	$0.760510\pi$
$770$	0	0
$771$	0	0
$772$	−15.6014	−0.561508
$773$	34.8149	1.25220	0.626102	−	0.779742i	$-0.284650\pi$
$773$	34.8149	1.25220	0.626102	+	0.779742i	$0.284650\pi$
$774$	0	0
$775$	− 0.101724i	− 0.00365404i
$776$	−3.65685	−0.131273
$777$	0	0
$778$	−9.78892	−0.350950
$779$	− 3.59704i	− 0.128877i
$780$	0	0
$781$	3.39250	0.121393
$782$	−2.61313	−0.0934452
$783$	0	0
$784$	0	0
$785$	15.3196i	0.546780i
$786$	0	0
$787$	44.7309i	1.59448i	0.603660	+	0.797242i	$0.293709\pi$
$787$	44.7309i	1.59448i	−0.603660	+	0.797242i	$0.706291\pi$
$788$	2.65498i	0.0945796i
$789$	0	0
$790$	4.32957i	0.154039i
$791$	0	0
$792$	0	0
$793$	1.42063	0.0504480
$794$	14.0160	0.497410
$795$	0	0
$796$	− 11.6840i	− 0.414127i
$797$	−15.9485	−0.564925	−0.282463	−	0.959278i	$-0.591151\pi$
$797$	−15.9485	−0.564925	−0.282463	+	0.959278i	$0.591151\pi$
$798$	0	0
$799$	21.4553	0.759035
$800$	− 1.00000i	− 0.0353553i
$801$	0	0
$802$	9.34596	0.330017
$803$	−2.12405	−0.0749562
$804$	0	0
$805$	0	0
$806$	− 0.0337542i	− 0.00118894i
$807$	0	0
$808$	− 10.7119i	− 0.376844i
$809$	31.5898i	1.11064i	0.831638	+	0.555318i	$0.187404\pi$
$809$	31.5898i	1.11064i	−0.831638	+	0.555318i	$0.812596\pi$
$810$	0	0
$811$	− 50.5850i	− 1.77628i	−0.459572	−	0.888140i	$-0.651997\pi$
$811$	− 50.5850i	− 1.77628i	0.459572	−	0.888140i	$-0.348003\pi$
$812$	0	0
$813$	0	0
$814$	−7.71453	−0.270394
$815$	−13.2661	−0.464692
$816$	0	0
$817$	− 0.817955i	− 0.0286166i
$818$	17.7620	0.621034
$819$	0	0
$820$	6.57446	0.229590
$821$	10.9221i	0.381182i	0.981669	+	0.190591i	$0.0610405\pi$
$821$	10.9221i	0.381182i	−0.981669	+	0.190591i	$0.938960\pi$
$822$	0	0
$823$	−33.5641	−1.16997	−0.584986	−	0.811043i	$-0.698900\pi$
$823$	−33.5641	−1.16997	−0.584986	+	0.811043i	$0.698900\pi$
$824$	1.91082	0.0665665
$825$	0	0
$826$	0	0
$827$	− 33.7958i	− 1.17520i	−0.809153	−	0.587598i	$-0.800074\pi$
$827$	− 33.7958i	− 1.17520i	0.809153	−	0.587598i	$-0.199926\pi$
$828$	0	0
$829$	23.2218i	0.806526i	0.915084	+	0.403263i	$0.132124\pi$
$829$	23.2218i	0.806526i	−0.915084	+	0.403263i	$0.867876\pi$
$830$	15.4785i	0.537266i
$831$	0	0
$832$	− 0.331821i	− 0.0115038i
$833$	0	0
$834$	0	0
$835$	−15.3462	−0.531079
$836$	0.508907	0.0176009
$837$	0	0
$838$	17.3073i	0.597872i
$839$	−22.7535	−0.785537	−0.392768	−	0.919637i	$-0.628483\pi$
$839$	−22.7535	−0.785537	−0.392768	+	0.919637i	$0.628483\pi$
$840$	0	0
$841$	−11.5986	−0.399952
$842$	− 3.79621i	− 0.130826i
$843$	0	0
$844$	4.36407	0.150218
$845$	−12.8899	−0.443426
$846$	0	0
$847$	0	0
$848$	− 6.38009i	− 0.219093i
$849$	0	0
$850$	3.26197i	0.111885i
$851$	6.64410i	0.227757i
$852$	0	0
$853$	− 58.0358i	− 1.98711i	−0.113353	−	0.993555i	$-0.536159\pi$
$853$	− 58.0358i	− 1.98711i	0.113353	−	0.993555i	$-0.463841\pi$
$854$	0	0
$855$	0	0
$856$	0.972287	0.0332321
$857$	−14.2966	−0.488364	−0.244182	−	0.969729i	$-0.578519\pi$
$857$	−14.2966	−0.488364	−0.244182	+	0.969729i	$0.578519\pi$
$858$	0	0
$859$	− 43.3580i	− 1.47936i	−0.672960	−	0.739679i	$-0.734978\pi$
$859$	− 43.3580i	− 1.47936i	0.672960	−	0.739679i	$-0.265022\pi$
$860$	1.49501	0.0509795
$861$	0	0
$862$	1.92214	0.0654685
$863$	22.8329i	0.777242i	0.921398	+	0.388621i	$0.127048\pi$
$863$	22.8329i	0.777242i	−0.921398	+	0.388621i	$0.872952\pi$
$864$	0	0
$865$	7.23532	0.246008
$866$	−17.5127	−0.595104
$867$	0	0
$868$	0	0
$869$	− 4.02715i	− 0.136612i
$870$	0	0
$871$	− 0.746937i	− 0.0253090i
$872$	18.4020i	0.623170i
$873$	0	0
$874$	− 0.438293i	− 0.0148255i
$875$	0	0
$876$	0	0
$877$	11.7859	0.397982	0.198991	−	0.980001i	$-0.436234\pi$
$877$	11.7859	0.397982	0.198991	+	0.980001i	$0.436234\pi$
$878$	40.3613	1.36213
$879$	0	0
$880$	0.930151i	0.0313554i
$881$	−14.1084	−0.475323	−0.237662	−	0.971348i	$-0.576381\pi$
$881$	−14.1084	−0.475323	−0.237662	+	0.971348i	$0.576381\pi$
$882$	0	0
$883$	25.9258	0.872474	0.436237	−	0.899832i	$-0.356311\pi$
$883$	25.9258	0.872474	0.436237	+	0.899832i	$0.356311\pi$
$884$	1.08239i	0.0364048i
$885$	0	0
$886$	8.00694	0.268999
$887$	33.2065	1.11497	0.557483	−	0.830188i	$-0.311767\pi$
$887$	33.2065	1.11497	0.557483	+	0.830188i	$0.311767\pi$
$888$	0	0
$889$	0	0
$890$	0.219463i	0.00735643i
$891$	0	0
$892$	24.2623i	0.812362i
$893$	3.59865i	0.120424i
$894$	0	0
$895$	14.8082i	0.494983i
$896$	0	0
$897$	0	0
$898$	27.4212	0.915056
$899$	0.648156	0.0216172
$900$	0	0
$901$	20.8117i	0.693337i
$902$	−6.11524	−0.203615
$903$	0	0
$904$	−4.16160	−0.138413
$905$	15.0930i	0.501707i
$906$	0	0
$907$	−11.9492	−0.396768	−0.198384	−	0.980124i	$-0.563569\pi$
$907$	−11.9492	−0.396768	−0.198384	+	0.980124i	$0.563569\pi$
$908$	29.2809	0.971722
$909$	0	0
$910$	0	0
$911$	− 20.2523i	− 0.670988i	−0.942042	−	0.335494i	$-0.891097\pi$
$911$	− 20.2523i	− 0.670988i	0.942042	−	0.335494i	$-0.108903\pi$
$912$	0	0
$913$	− 14.3973i	− 0.476482i
$914$	3.49526i	0.115613i
$915$	0	0
$916$	− 4.84710i	− 0.160153i
$917$	0	0
$918$	0	0
$919$	−31.6974	−1.04560	−0.522801	−	0.852455i	$-0.675113\pi$
$919$	−31.6974	−1.04560	−0.522801	+	0.852455i	$0.675113\pi$
$920$	0.801088	0.0264111
$921$	0	0
$922$	− 28.4061i	− 0.935507i
$923$	−1.21024	−0.0398354
$924$	0	0
$925$	8.29385	0.272700
$926$	42.4813i	1.39602i
$927$	0	0
$928$	6.37170	0.209161
$929$	20.5149	0.673073	0.336537	−	0.941670i	$-0.390744\pi$
$929$	20.5149	0.673073	0.336537	+	0.941670i	$0.390744\pi$
$930$	0	0
$931$	0	0
$932$	− 2.66683i	− 0.0873549i
$933$	0	0
$934$	11.8480i	0.387680i
$935$	− 3.03413i	− 0.0992266i
$936$	0	0
$937$	− 29.3157i	− 0.957701i	−0.877896	−	0.478850i	$-0.841054\pi$
$937$	− 29.3157i	− 0.957701i	0.877896	−	0.478850i	$-0.158946\pi$
$938$	0	0
$939$	0	0
$940$	−6.57741	−0.214531
$941$	32.6165	1.06327	0.531634	−	0.846974i	$-0.321578\pi$
$941$	32.6165	1.06327	0.531634	+	0.846974i	$0.321578\pi$
$942$	0	0
$943$	5.26672i	0.171508i
$944$	−6.11652	−0.199076
$945$	0	0
$946$	−1.39059	−0.0452119
$947$	32.0156i	1.04037i	0.854054	+	0.520184i	$0.174137\pi$
$947$	32.0156i	1.04037i	−0.854054	+	0.520184i	$0.825863\pi$
$948$	0	0
$949$	0.757733	0.0245971
$950$	−0.547123	−0.0177510
$951$	0	0
$952$	0	0
$953$	− 35.3686i	− 1.14570i	−0.819660	−	0.572850i	$-0.805838\pi$
$953$	− 35.3686i	− 1.14570i	0.819660	−	0.572850i	$-0.194162\pi$
$954$	0	0
$955$	20.9314i	0.677323i
$956$	− 18.9906i	− 0.614198i
$957$	0	0
$958$	11.4231i	0.369063i
$959$	0	0
$960$	0	0
$961$	30.9897	0.999666
$962$	2.75208	0.0887305
$963$	0	0
$964$	14.4045i	0.463937i
$965$	−15.6014	−0.502228
$966$	0	0
$967$	−30.3515	−0.976037	−0.488019	−	0.872833i	$-0.662280\pi$
$967$	−30.3515	−0.976037	−0.488019	+	0.872833i	$0.662280\pi$
$968$	10.1348i	0.325745i
$969$	0	0
$970$	−3.65685	−0.117415
$971$	−8.15046	−0.261561	−0.130780	−	0.991411i	$-0.541748\pi$
$971$	−8.15046	−0.261561	−0.130780	+	0.991411i	$0.541748\pi$
$972$	0	0
$973$	0	0
$974$	19.0897i	0.611675i
$975$	0	0
$976$	− 4.28130i	− 0.137041i
$977$	62.0995i	1.98674i	0.114960	+	0.993370i	$0.463326\pi$
$977$	62.0995i	1.98674i	−0.114960	+	0.993370i	$0.536674\pi$
$978$	0	0
$979$	− 0.204134i	− 0.00652416i
$980$	0	0
$981$	0	0
$982$	−4.04288	−0.129014
$983$	−20.2029	−0.644374	−0.322187	−	0.946676i	$-0.604418\pi$
$983$	−20.2029	−0.644374	−0.322187	+	0.946676i	$0.604418\pi$
$984$	0	0
$985$	2.65498i	0.0845946i
$986$	−20.7843	−0.661908
$987$	0	0
$988$	−0.181547	−0.00577578
$989$	1.19764i	0.0380826i
$990$	0	0
$991$	−50.7136	−1.61097	−0.805485	−	0.592616i	$-0.798095\pi$
$991$	−50.7136	−1.61097	−0.805485	+	0.592616i	$0.798095\pi$
$992$	−0.101724	−0.00322974
$993$	0	0
$994$	0	0
$995$	− 11.6840i	− 0.370406i
$996$	0	0
$997$	− 14.7493i	− 0.467115i	−0.972343	−	0.233557i	$-0.924963\pi$
$997$	− 14.7493i	− 0.467115i	0.972343	−	0.233557i	$-0.0750367\pi$
$998$	30.4594i	0.964177i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.c.881.2	✓	8
3.2	odd	2		4410.2.b.f.881.7	yes	8
7.6	odd	2		4410.2.b.f.881.2	yes	8
21.20	even	2	inner	4410.2.b.c.881.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.b.c.881.2	✓	8	1.1	even	1	trivial
4410.2.b.c.881.7	yes	8	21.20	even	2	inner
4410.2.b.f.881.2	yes	8	7.6	odd	2
4410.2.b.f.881.7	yes	8	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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} -0.930151i q^{11} +0.331821i q^{13} +1.00000 q^{16} -3.26197 q^{17} -0.547123i q^{19} +1.00000 q^{20} -0.930151 q^{22} +0.801088i q^{23} +1.00000 q^{25} +0.331821 q^{26} +6.37170i q^{29} -0.101724i q^{31} -1.00000i q^{32} +3.26197i q^{34} +8.29385 q^{37} -0.547123 q^{38} -1.00000i q^{40} +6.57446 q^{41} +1.49501 q^{43} +0.930151i q^{44} +0.801088 q^{46} -6.57741 q^{47} -1.00000i q^{50} -0.331821i q^{52} -6.38009i q^{53} +0.930151i q^{55} +6.37170 q^{58} -6.11652 q^{59} -4.28130i q^{61} -0.101724 q^{62} -1.00000 q^{64} -0.331821i q^{65} -2.25102 q^{67} +3.26197 q^{68} +3.64725i q^{71} -2.28356i q^{73} -8.29385i q^{74} +0.547123i q^{76} +4.32957 q^{79} -1.00000 q^{80} -6.57446i q^{82} +15.4785 q^{83} +3.26197 q^{85} -1.49501i q^{86} +0.930151 q^{88} +0.219463 q^{89} -0.801088i q^{92} +6.57741i q^{94} +0.547123i q^{95} +3.65685i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^4 - 8 * q^5	\( 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} + 16 q^{22} + 8 q^{25} + 16 q^{37} + 16 q^{41} - 16 q^{43} + 16 q^{46} - 16 q^{47} - 16 q^{58} - 48 q^{59} - 8 q^{64} - 32 q^{67} - 8 q^{80} + 16 q^{83} - 16 q^{88} - 80 q^{89}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 + 16 * q^22 + 8 * q^25 + 16 * q^37 + 16 * q^41 - 16 * q^43 + 16 * q^46 - 16 * q^47 - 16 * q^58 - 48 * q^59 - 8 * q^64 - 32 * q^67 - 8 * q^80 + 16 * q^83 - 16 * q^88 - 80 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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