LMFDB - Embedded newform 4410.2.d.d.4409.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(4409,4410)]

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, 1, 1]))

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

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

chi := DirichletCharacter("4410.4409");

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.d (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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4409.2
Character	$\chi$	$=$	4410.4409
Dual form			4410.2.d.d.4409.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +(-2.12190 + 0.705358i) q^{5} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +(-2.12190 + 0.705358i) q^{5} +1.00000 q^{8} +(-2.12190 + 0.705358i) q^{10} -3.38091i q^{11} +0.920905 q^{13} +1.00000 q^{16} +7.01247i q^{17} +0.709774i q^{19} +(-2.12190 + 0.705358i) q^{20} -3.38091i q^{22} -2.11904 q^{23} +(4.00494 - 2.99340i) q^{25} +0.920905 q^{26} -0.0694157i q^{29} -4.09651i q^{31} +1.00000 q^{32} +7.01247i q^{34} +2.81463i q^{37} +0.709774i q^{38} +(-2.12190 + 0.705358i) q^{40} +10.8635 q^{41} -7.35468i q^{43} -3.38091i q^{44} -2.11904 q^{46} +10.9853i q^{47} +(4.00494 - 2.99340i) q^{50} +0.920905 q^{52} +2.79453 q^{53} +(2.38475 + 7.17396i) q^{55} -0.0694157i q^{58} -5.80717 q^{59} +5.18196i q^{61} -4.09651i q^{62} +1.00000 q^{64} +(-1.95407 + 0.649567i) q^{65} +6.80832i q^{67} +7.01247i q^{68} +13.9389i q^{71} +6.18863 q^{73} +2.81463i q^{74} +0.709774i q^{76} +4.57479 q^{79} +(-2.12190 + 0.705358i) q^{80} +10.8635 q^{82} +8.39481i q^{83} +(-4.94630 - 14.8798i) q^{85} -7.35468i q^{86} -3.38091i q^{88} +0.636877 q^{89} -2.11904 q^{92} +10.9853i q^{94} +(-0.500645 - 1.50607i) q^{95} +3.00409 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{2} + 24 q^{4} + 24 q^{8}+O(q^{10}) $ 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^8	$ 24 q + 24 q^{2} + 24 q^{4} + 24 q^{8} + 24 q^{16} - 32 q^{23} - 16 q^{25} + 24 q^{32} - 32 q^{46} - 16 q^{50} + 32 q^{53} + 24 q^{64} - 32 q^{85} - 32 q^{92} - 32 q^{95}+O(q^{100}) $ 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^8 + 24 * q^16 - 32 * q^23 - 16 * q^25 + 24 * q^32 - 32 * q^46 - 16 * q^50 + 32 * q^53 + 24 * q^64 - 32 * q^85 - 32 * q^92 - 32 * q^95

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−2.12190	+	0.705358i	−0.948944	+	0.315445i
$5$	−2.12190	+	0.705358i	−0.948944	+	0.315445i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	1.00000			0.353553
$9$		0			0
$10$	−2.12190	+	0.705358i	−0.671005	+	0.223054i
$11$		−	3.38091i		−	1.01938i	−0.860358	−	0.509691i	$-0.829760\pi$
$11$		−	3.38091i		−	1.01938i	0.860358	−	0.509691i	$-0.170240\pi$
$12$		0			0
$13$	0.920905			0.255413			0.127707	−	0.991812i	$-0.459238\pi$
$13$	0.920905			0.255413			0.127707	+	0.991812i	$0.459238\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$			7.01247i			1.70077i	0.526158	+	0.850387i	$0.323632\pi$
$17$			7.01247i			1.70077i	−0.526158	+	0.850387i	$0.676368\pi$
$18$		0			0
$19$			0.709774i			0.162833i	0.996680	+	0.0814167i	$0.0259445\pi$
$19$			0.709774i			0.162833i	−0.996680	+	0.0814167i	$0.974056\pi$
$20$	−2.12190	+	0.705358i	−0.474472	+	0.157723i
$21$		0			0
$22$		−	3.38091i		−	0.720812i
$23$	−2.11904			−0.441851			−0.220926	−	0.975291i	$-0.570908\pi$
$23$	−2.11904			−0.441851			−0.220926	+	0.975291i	$0.570908\pi$
$24$		0			0
$25$	4.00494	−	2.99340i	0.800988	−	0.598680i
$26$	0.920905			0.180604
$27$		0			0
$28$		0			0
$29$		−	0.0694157i		−	0.0128902i	−0.999979	−	0.00644509i	$-0.997948\pi$
$29$		−	0.0694157i		−	0.0128902i	0.999979	−	0.00644509i	$-0.00205155\pi$
$30$		0			0
$31$		−	4.09651i		−	0.735755i	−0.929874	−	0.367878i	$-0.880085\pi$
$31$		−	4.09651i		−	0.735755i	0.929874	−	0.367878i	$-0.119915\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			7.01247i			1.20263i
$35$		0			0
$36$		0			0
$37$			2.81463i			0.462723i	0.972868	+	0.231361i	$0.0743179\pi$
$37$			2.81463i			0.462723i	−0.972868	+	0.231361i	$0.925682\pi$
$38$			0.709774i			0.115141i
$39$		0			0
$40$	−2.12190	+	0.705358i	−0.335502	+	0.111527i
$41$	10.8635			1.69660			0.848299	−	0.529518i	$-0.177627\pi$
$41$	10.8635			1.69660			0.848299	+	0.529518i	$0.177627\pi$
$42$		0			0
$43$		−	7.35468i		−	1.12158i	−0.827959	−	0.560789i	$-0.810498\pi$
$43$		−	7.35468i		−	1.12158i	0.827959	−	0.560789i	$-0.189502\pi$
$44$		−	3.38091i		−	0.509691i
$45$		0			0
$46$	−2.11904			−0.312436
$47$			10.9853i			1.60237i	0.598414	+	0.801187i	$0.295798\pi$
$47$			10.9853i			1.60237i	−0.598414	+	0.801187i	$0.704202\pi$
$48$		0			0
$49$		0			0
$50$	4.00494	−	2.99340i	0.566384	−	0.423331i
$51$		0			0
$52$	0.920905			0.127707
$53$	2.79453			0.383858			0.191929	−	0.981409i	$-0.438526\pi$
$53$	2.79453			0.383858			0.191929	+	0.981409i	$0.438526\pi$
$54$		0			0
$55$	2.38475	+	7.17396i	0.321559	+	0.967336i
$56$		0			0
$57$		0			0
$58$		−	0.0694157i		−	0.00911473i
$59$	−5.80717			−0.756030			−0.378015	−	0.925800i	$-0.623393\pi$
$59$	−5.80717			−0.756030			−0.378015	+	0.925800i	$0.623393\pi$
$60$		0			0
$61$			5.18196i			0.663482i	0.943371	+	0.331741i	$0.107636\pi$
$61$			5.18196i			0.663482i	−0.943371	+	0.331741i	$0.892364\pi$
$62$		−	4.09651i		−	0.520258i
$63$		0			0
$64$	1.00000			0.125000
$65$	−1.95407	+	0.649567i	−0.242373	+	0.0805689i
$66$		0			0
$67$			6.80832i			0.831769i	0.909417	+	0.415884i	$0.136528\pi$
$67$			6.80832i			0.831769i	−0.909417	+	0.415884i	$0.863472\pi$
$68$			7.01247i			0.850387i
$69$		0			0
$70$		0			0
$71$			13.9389i			1.65424i	0.562023	+	0.827122i	$0.310023\pi$
$71$			13.9389i			1.65424i	−0.562023	+	0.827122i	$0.689977\pi$
$72$		0			0
$73$	6.18863			0.724324			0.362162	−	0.932115i	$-0.382039\pi$
$73$	6.18863			0.724324			0.362162	+	0.932115i	$0.382039\pi$
$74$			2.81463i			0.327194i
$75$		0			0
$76$			0.709774i			0.0814167i
$77$		0			0
$78$		0			0
$79$	4.57479			0.514705			0.257352	−	0.966318i	$-0.417150\pi$
$79$	4.57479			0.514705			0.257352	+	0.966318i	$0.417150\pi$
$80$	−2.12190	+	0.705358i	−0.237236	+	0.0788614i
$81$		0			0
$82$	10.8635			1.19968
$83$			8.39481i			0.921450i	0.887543	+	0.460725i	$0.152411\pi$
$83$			8.39481i			0.921450i	−0.887543	+	0.460725i	$0.847589\pi$
$84$		0			0
$85$	−4.94630	−	14.8798i	−0.536501	−	1.61394i
$86$		−	7.35468i		−	0.793076i
$87$		0			0
$88$		−	3.38091i		−	0.360406i
$89$	0.636877			0.0675089			0.0337544	−	0.999430i	$-0.489254\pi$
$89$	0.636877			0.0675089			0.0337544	+	0.999430i	$0.489254\pi$
$90$		0			0
$91$		0			0
$92$	−2.11904			−0.220926
$93$		0			0
$94$			10.9853i			1.13305i
$95$	−0.500645	−	1.50607i	−0.0513651	−	0.154520i
$96$		0			0
$97$	3.00409			0.305019			0.152510	−	0.988302i	$-0.451264\pi$
$97$	3.00409			0.305019			0.152510	+	0.988302i	$0.451264\pi$
$98$		0			0
$99$		0			0
$100$	4.00494	−	2.99340i	0.400494	−	0.299340i
$101$	1.93226			0.192267			0.0961334	−	0.995368i	$-0.469352\pi$
$101$	1.93226			0.192267			0.0961334	+	0.995368i	$0.469352\pi$
$102$		0			0
$103$	11.2781			1.11127			0.555634	−	0.831427i	$-0.312476\pi$
$103$	11.2781			1.11127			0.555634	+	0.831427i	$0.312476\pi$
$104$	0.920905			0.0903022
$105$		0			0
$106$	2.79453			0.271428
$107$	−16.9650			−1.64007			−0.820036	−	0.572312i	$-0.806046\pi$
$107$	−16.9650			−1.64007			−0.820036	+	0.572312i	$0.806046\pi$
$108$		0			0
$109$	15.5084			1.48543			0.742717	−	0.669605i	$-0.233537\pi$
$109$	15.5084			1.48543			0.742717	+	0.669605i	$0.233537\pi$
$110$	2.38475	+	7.17396i	0.227377	+	0.684010i
$111$		0			0
$112$		0			0
$113$	18.0732			1.70018			0.850091	−	0.526636i	$-0.176547\pi$
$113$	18.0732			1.70018			0.850091	+	0.526636i	$0.176547\pi$
$114$		0			0
$115$	4.49640	−	1.49468i	0.419292	−	0.139380i
$116$		−	0.0694157i		−	0.00644509i
$117$		0			0
$118$	−5.80717			−0.534594
$119$		0			0
$120$		0			0
$121$	−0.430536			−0.0391396
$122$			5.18196i			0.469152i
$123$		0			0
$124$		−	4.09651i		−	0.367878i
$125$	−6.38668	+	9.17662i	−0.571242	+	0.820782i
$126$		0			0
$127$			2.72006i			0.241367i	0.992691	+	0.120683i	$0.0385086\pi$
$127$			2.72006i			0.241367i	−0.992691	+	0.120683i	$0.961491\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	−1.95407	+	0.649567i	−0.171383	+	0.0569708i
$131$	−15.6215			−1.36486			−0.682430	−	0.730951i	$-0.739077\pi$
$131$	−15.6215			−1.36486			−0.682430	+	0.730951i	$0.739077\pi$
$132$		0			0
$133$		0			0
$134$			6.80832i			0.588149i
$135$		0			0
$136$			7.01247i			0.601314i
$137$	22.4821			1.92078			0.960389	−	0.278662i	$-0.0898908\pi$
$137$	22.4821			1.92078			0.960389	+	0.278662i	$0.0898908\pi$
$138$		0			0
$139$		−	1.58386i		−	0.134341i	−0.997741	−	0.0671707i	$-0.978603\pi$
$139$		−	1.58386i		−	0.134341i	0.997741	−	0.0671707i	$-0.0213972\pi$
$140$		0			0
$141$		0			0
$142$			13.9389i			1.16973i
$143$		−	3.11349i		−	0.260364i
$144$		0			0
$145$	0.0489629	+	0.147293i	0.00406615	+	0.0122320i
$146$	6.18863			0.512174
$147$		0			0
$148$			2.81463i			0.231361i
$149$		−	8.51774i		−	0.697800i	−0.937160	−	0.348900i	$-0.886555\pi$
$149$		−	8.51774i		−	0.697800i	0.937160	−	0.348900i	$-0.113445\pi$
$150$		0			0
$151$	15.3762			1.25129			0.625647	−	0.780106i	$-0.284835\pi$
$151$	15.3762			1.25129			0.625647	+	0.780106i	$0.284835\pi$
$152$			0.709774i			0.0575703i
$153$		0			0
$154$		0			0
$155$	2.88951	+	8.69240i	0.232091	+	0.698190i
$156$		0			0
$157$	0.162670			0.0129825			0.00649125	−	0.999979i	$-0.497934\pi$
$157$	0.162670			0.0129825			0.00649125	+	0.999979i	$0.497934\pi$
$158$	4.57479			0.363951
$159$		0			0
$160$	−2.12190	+	0.705358i	−0.167751	+	0.0557634i
$161$		0			0
$162$		0			0
$163$			20.9458i			1.64060i	0.571934	+	0.820299i	$0.306193\pi$
$163$			20.9458i			1.64060i	−0.571934	+	0.820299i	$0.693807\pi$
$164$	10.8635			0.848299
$165$		0			0
$166$			8.39481i			0.651564i
$167$		−	7.27807i		−	0.563194i	−0.959533	−	0.281597i	$-0.909136\pi$
$167$		−	7.27807i		−	0.563194i	0.959533	−	0.281597i	$-0.0908641\pi$
$168$		0			0
$169$	−12.1519			−0.934764
$170$	−4.94630	−	14.8798i	−0.379364	−	1.14123i
$171$		0			0
$172$		−	7.35468i		−	0.560789i
$173$			14.5567i			1.10673i	0.832939	+	0.553364i	$0.186656\pi$
$173$			14.5567i			1.10673i	−0.832939	+	0.553364i	$0.813344\pi$
$174$		0			0
$175$		0			0
$176$		−	3.38091i		−	0.254845i
$177$		0			0
$178$	0.636877			0.0477360
$179$		−	2.40612i		−	0.179842i	−0.995949	−	0.0899210i	$-0.971339\pi$
$179$		−	2.40612i		−	0.179842i	0.995949	−	0.0899210i	$-0.0286615\pi$
$180$		0			0
$181$		−	1.39280i		−	0.103526i	−0.998659	−	0.0517632i	$-0.983516\pi$
$181$		−	1.39280i		−	0.103526i	0.998659	−	0.0517632i	$-0.0164841\pi$
$182$		0			0
$183$		0			0
$184$	−2.11904			−0.156218
$185$	−1.98532	−	5.97237i	−0.145964	−	0.439098i
$186$		0			0
$187$	23.7085			1.73374
$188$			10.9853i			0.801187i
$189$		0			0
$190$	−0.500645	−	1.50607i	−0.0363206	−	0.109262i
$191$			21.8400i			1.58029i	0.612920	+	0.790145i	$0.289995\pi$
$191$			21.8400i			1.58029i	−0.612920	+	0.790145i	$0.710005\pi$
$192$		0			0
$193$		−	11.1452i		−	0.802247i	−0.916024	−	0.401124i	$-0.868620\pi$
$193$		−	11.1452i		−	0.802247i	0.916024	−	0.401124i	$-0.131380\pi$
$194$	3.00409			0.215681
$195$		0			0
$196$		0			0
$197$	18.5178			1.31934			0.659669	−	0.751556i	$-0.270696\pi$
$197$	18.5178			1.31934			0.659669	+	0.751556i	$0.270696\pi$
$198$		0			0
$199$		−	7.96353i		−	0.564520i	−0.959338	−	0.282260i	$-0.908916\pi$
$199$		−	7.96353i		−	0.564520i	0.959338	−	0.282260i	$-0.0910841\pi$
$200$	4.00494	−	2.99340i	0.283192	−	0.211665i
$201$		0			0
$202$	1.93226			0.135953
$203$		0			0
$204$		0			0
$205$	−23.0513	+	7.66267i	−1.60998	+	0.535184i
$206$	11.2781			0.785785
$207$		0			0
$208$	0.920905			0.0638533
$209$	2.39968			0.165989
$210$		0			0
$211$	−14.4828			−0.997038			−0.498519	−	0.866879i	$-0.666123\pi$
$211$	−14.4828			−0.997038			−0.498519	+	0.866879i	$0.666123\pi$
$212$	2.79453			0.191929
$213$		0			0
$214$	−16.9650			−1.15971
$215$	5.18768	+	15.6059i	0.353797	+	1.06431i
$216$		0			0
$217$		0			0
$218$	15.5084			1.05036
$219$		0			0
$220$	2.38475	+	7.17396i	0.160780	+	0.483668i
$221$			6.45782i			0.434400i
$222$		0			0
$223$	−19.7255			−1.32092			−0.660458	−	0.750863i	$-0.729638\pi$
$223$	−19.7255			−1.32092			−0.660458	+	0.750863i	$0.729638\pi$
$224$		0			0
$225$		0			0
$226$	18.0732			1.20221
$227$			11.5064i			0.763708i	0.924223	+	0.381854i	$0.124714\pi$
$227$			11.5064i			0.763708i	−0.924223	+	0.381854i	$0.875286\pi$
$228$		0			0
$229$			19.3140i			1.27630i	0.769910	+	0.638152i	$0.220301\pi$
$229$			19.3140i			1.27630i	−0.769910	+	0.638152i	$0.779699\pi$
$230$	4.49640	−	1.49468i	0.296484	−	0.0985565i
$231$		0			0
$232$		−	0.0694157i		−	0.00455736i
$233$	−20.4744			−1.34132			−0.670661	−	0.741764i	$-0.733990\pi$
$233$	−20.4744			−1.34132			−0.670661	+	0.741764i	$0.733990\pi$
$234$		0			0
$235$	−7.74858	−	23.3098i	−0.505462	−	1.52056i
$236$	−5.80717			−0.378015
$237$		0			0
$238$		0			0
$239$			15.0439i			0.973106i	0.873651	+	0.486553i	$0.161746\pi$
$239$			15.0439i			0.973106i	−0.873651	+	0.486553i	$0.838254\pi$
$240$		0			0
$241$		−	10.0812i		−	0.649384i	−0.945820	−	0.324692i	$-0.894739\pi$
$241$		−	10.0812i		−	0.649384i	0.945820	−	0.324692i	$-0.105261\pi$
$242$	−0.430536			−0.0276759
$243$		0			0
$244$			5.18196i			0.331741i
$245$		0			0
$246$		0			0
$247$			0.653635i			0.0415898i
$248$		−	4.09651i		−	0.260129i
$249$		0			0
$250$	−6.38668	+	9.17662i	−0.403929	+	0.580380i
$251$	−19.1132			−1.20641			−0.603206	−	0.797585i	$-0.706110\pi$
$251$	−19.1132			−1.20641			−0.603206	+	0.797585i	$0.706110\pi$
$252$		0			0
$253$			7.16429i			0.450415i
$254$			2.72006i			0.170672i
$255$		0			0
$256$	1.00000			0.0625000
$257$			3.42573i			0.213691i	0.994276	+	0.106846i	$0.0340751\pi$
$257$			3.42573i			0.213691i	−0.994276	+	0.106846i	$0.965925\pi$
$258$		0			0
$259$		0			0
$260$	−1.95407	+	0.649567i	−0.121186	+	0.0402845i
$261$		0			0
$262$	−15.6215			−0.965101
$263$	12.4097			0.765217			0.382609	−	0.923911i	$-0.375026\pi$
$263$	12.4097			0.765217			0.382609	+	0.923911i	$0.375026\pi$
$264$		0			0
$265$	−5.92971	+	1.97114i	−0.364259	+	0.121086i
$266$		0			0
$267$		0			0
$268$			6.80832i			0.415884i
$269$	−26.6150			−1.62275			−0.811373	−	0.584529i	$-0.801279\pi$
$269$	−26.6150			−1.62275			−0.811373	+	0.584529i	$0.801279\pi$
$270$		0			0
$271$			4.13615i			0.251253i	0.992078	+	0.125626i	$0.0400941\pi$
$271$			4.13615i			0.251253i	−0.992078	+	0.125626i	$0.959906\pi$
$272$			7.01247i			0.425193i
$273$		0			0
$274$	22.4821			1.35820
$275$	−10.1204	−	13.5403i	−0.610284	−	0.816513i
$276$		0			0
$277$			19.9414i			1.19817i	0.800687	+	0.599083i	$0.204468\pi$
$277$			19.9414i			1.19817i	−0.800687	+	0.599083i	$0.795532\pi$
$278$		−	1.58386i		−	0.0949937i
$279$		0			0
$280$		0			0
$281$		−	11.1429i		−	0.664728i	−0.943151	−	0.332364i	$-0.892154\pi$
$281$		−	11.1429i		−	0.664728i	0.943151	−	0.332364i	$-0.107846\pi$
$282$		0			0
$283$	29.8097			1.77200			0.886001	−	0.463684i	$-0.153473\pi$
$283$	29.8097			1.77200			0.886001	+	0.463684i	$0.153473\pi$
$284$			13.9389i			0.827122i
$285$		0			0
$286$		−	3.11349i		−	0.184105i
$287$		0			0
$288$		0			0
$289$	−32.1747			−1.89263
$290$	0.0489629	+	0.147293i	0.00287520	+	0.00864936i
$291$		0			0
$292$	6.18863			0.362162
$293$		−	2.22037i		−	0.129715i	−0.997895	−	0.0648576i	$-0.979341\pi$
$293$		−	2.22037i		−	0.129715i	0.997895	−	0.0648576i	$-0.0206593\pi$
$294$		0			0
$295$	12.3223	−	4.09613i	0.717430	−	0.238486i
$296$			2.81463i			0.163597i
$297$		0			0
$298$		−	8.51774i		−	0.493419i
$299$	−1.95144			−0.112855
$300$		0			0
$301$		0			0
$302$	15.3762			0.884798
$303$		0			0
$304$			0.709774i			0.0407083i
$305$	−3.65513	−	10.9956i	−0.209292	−	0.629607i
$306$		0			0
$307$	21.1861			1.20916			0.604578	−	0.796546i	$-0.293342\pi$
$307$	21.1861			1.20916			0.604578	+	0.796546i	$0.293342\pi$
$308$		0			0
$309$		0			0
$310$	2.88951	+	8.69240i	0.164113	+	0.493695i
$311$	4.88322			0.276902			0.138451	−	0.990369i	$-0.455788\pi$
$311$	4.88322			0.276902			0.138451	+	0.990369i	$0.455788\pi$
$312$		0			0
$313$	8.41182			0.475464			0.237732	−	0.971331i	$-0.423596\pi$
$313$	8.41182			0.475464			0.237732	+	0.971331i	$0.423596\pi$
$314$	0.162670			0.00918001
$315$		0			0
$316$	4.57479			0.257352
$317$	−1.70288			−0.0956433			−0.0478216	−	0.998856i	$-0.515228\pi$
$317$	−1.70288			−0.0956433			−0.0478216	+	0.998856i	$0.515228\pi$
$318$		0			0
$319$	−0.234688			−0.0131400
$320$	−2.12190	+	0.705358i	−0.118618	+	0.0394307i
$321$		0			0
$322$		0			0
$323$	−4.97727			−0.276943
$324$		0			0
$325$	3.68817	−	2.75664i	0.204583	−	0.152911i
$326$			20.9458i			1.16008i
$327$		0			0
$328$	10.8635			0.599838
$329$		0			0
$330$		0			0
$331$	−1.17708			−0.0646979			−0.0323490	−	0.999477i	$-0.510299\pi$
$331$	−1.17708			−0.0646979			−0.0323490	+	0.999477i	$0.510299\pi$
$332$			8.39481i			0.460725i
$333$		0			0
$334$		−	7.27807i		−	0.398238i
$335$	−4.80230	−	14.4466i	−0.262378	−	0.789302i
$336$		0			0
$337$		−	24.3592i		−	1.32693i	−0.748207	−	0.663465i	$-0.769085\pi$
$337$		−	24.3592i		−	1.32693i	0.748207	−	0.663465i	$-0.230915\pi$
$338$	−12.1519			−0.660978
$339$		0			0
$340$	−4.94630	−	14.8798i	−0.268251	−	0.806969i
$341$	−13.8499			−0.750016
$342$		0			0
$343$		0			0
$344$		−	7.35468i		−	0.396538i
$345$		0			0
$346$			14.5567i			0.782575i
$347$	25.8225			1.38622			0.693111	−	0.720831i	$-0.256239\pi$
$347$	25.8225			1.38622			0.693111	+	0.720831i	$0.256239\pi$
$348$		0			0
$349$		−	18.7142i		−	1.00175i	−0.865520	−	0.500874i	$-0.833012\pi$
$349$		−	18.7142i		−	1.00175i	0.865520	−	0.500874i	$-0.166988\pi$
$350$		0			0
$351$		0			0
$352$		−	3.38091i		−	0.180203i
$353$		−	20.5673i		−	1.09469i	−0.836908	−	0.547344i	$-0.815639\pi$
$353$		−	20.5673i		−	1.09469i	0.836908	−	0.547344i	$-0.184361\pi$
$354$		0			0
$355$	−9.83191	−	29.5770i	−0.521824	−	1.56978i
$356$	0.636877			0.0337544
$357$		0			0
$358$		−	2.40612i		−	0.127168i
$359$		−	17.8940i		−	0.944411i	−0.881489	−	0.472205i	$-0.843458\pi$
$359$		−	17.8940i		−	0.944411i	0.881489	−	0.472205i	$-0.156542\pi$
$360$		0			0
$361$	18.4962			0.973485
$362$		−	1.39280i		−	0.0732042i
$363$		0			0
$364$		0			0
$365$	−13.1317	+	4.36519i	−0.687343	+	0.228485i
$366$		0			0
$367$	−31.5003			−1.64430			−0.822152	−	0.569269i	$-0.807226\pi$
$367$	−31.5003			−1.64430			−0.822152	+	0.569269i	$0.807226\pi$
$368$	−2.11904			−0.110463
$369$		0			0
$370$	−1.98532	−	5.97237i	−0.103212	−	0.310489i
$371$		0			0
$372$		0			0
$373$		−	28.6159i		−	1.48168i	−0.671683	−	0.740838i	$-0.734428\pi$
$373$		−	28.6159i		−	1.48168i	0.671683	−	0.740838i	$-0.265572\pi$
$374$	23.7085			1.22594
$375$		0			0
$376$			10.9853i			0.566525i
$377$		−	0.0639253i		−	0.00329232i
$378$		0			0
$379$	−1.43606			−0.0737655			−0.0368827	−	0.999320i	$-0.511743\pi$
$379$	−1.43606			−0.0737655			−0.0368827	+	0.999320i	$0.511743\pi$
$380$	−0.500645	−	1.50607i	−0.0256825	−	0.0772599i
$381$		0			0
$382$			21.8400i			1.11743i
$383$			22.5484i			1.15217i	0.817390	+	0.576084i	$0.195420\pi$
$383$			22.5484i			1.15217i	−0.817390	+	0.576084i	$0.804580\pi$
$384$		0			0
$385$		0			0
$386$		−	11.1452i		−	0.567274i
$387$		0			0
$388$	3.00409			0.152510
$389$		−	22.8059i		−	1.15631i	−0.815928	−	0.578153i	$-0.803774\pi$
$389$		−	22.8059i		−	1.15631i	0.815928	−	0.578153i	$-0.196226\pi$
$390$		0			0
$391$		−	14.8597i		−	0.751489i
$392$		0			0
$393$		0			0
$394$	18.5178			0.932913
$395$	−9.70727	+	3.22687i	−0.488426	+	0.162361i
$396$		0			0
$397$	24.2195			1.21554			0.607772	−	0.794112i	$-0.292064\pi$
$397$	24.2195			1.21554			0.607772	+	0.794112i	$0.292064\pi$
$398$		−	7.96353i		−	0.399176i
$399$		0			0
$400$	4.00494	−	2.99340i	0.200247	−	0.149670i
$401$			25.4197i			1.26940i	0.772759	+	0.634699i	$0.218876\pi$
$401$			25.4197i			1.26940i	−0.772759	+	0.634699i	$0.781124\pi$
$402$		0			0
$403$		−	3.77250i		−	0.187922i
$404$	1.93226			0.0961334
$405$		0			0
$406$		0			0
$407$	9.51601			0.471691
$408$		0			0
$409$			0.367496i			0.0181715i	0.999959	+	0.00908574i	$0.00289212\pi$
$409$			0.367496i			0.0181715i	−0.999959	+	0.00908574i	$0.997108\pi$
$410$	−23.0513	+	7.66267i	−1.13842	+	0.378432i
$411$		0			0
$412$	11.2781			0.555634
$413$		0			0
$414$		0			0
$415$	−5.92134	−	17.8130i	−0.290667	−	0.874404i
$416$	0.920905			0.0451511
$417$		0			0
$418$	2.39968			0.117372
$419$	−18.4189			−0.899823			−0.449912	−	0.893073i	$-0.648545\pi$
$419$	−18.4189			−0.899823			−0.449912	+	0.893073i	$0.648545\pi$
$420$		0			0
$421$	34.8600			1.69897			0.849486	−	0.527612i	$-0.176912\pi$
$421$	34.8600			1.69897			0.849486	+	0.527612i	$0.176912\pi$
$422$	−14.4828			−0.705012
$423$		0			0
$424$	2.79453			0.135714
$425$	20.9911	+	28.0845i	1.01822	+	1.36230i
$426$		0			0
$427$		0			0
$428$	−16.9650			−0.820036
$429$		0			0
$430$	5.18768	+	15.6059i	0.250172	+	0.752584i
$431$			21.1923i			1.02079i	0.859939	+	0.510397i	$0.170502\pi$
$431$			21.1923i			1.02079i	−0.859939	+	0.510397i	$0.829498\pi$
$432$		0			0
$433$	15.9350			0.765785			0.382893	−	0.923793i	$-0.374928\pi$
$433$	15.9350			0.765785			0.382893	+	0.923793i	$0.374928\pi$
$434$		0			0
$435$		0			0
$436$	15.5084			0.742717
$437$		−	1.50404i		−	0.0719481i
$438$		0			0
$439$		−	39.1511i		−	1.86858i	−0.356516	−	0.934289i	$-0.616036\pi$
$439$		−	39.1511i		−	1.86858i	0.356516	−	0.934289i	$-0.383964\pi$
$440$	2.38475	+	7.17396i	0.113688	+	0.342005i
$441$		0			0
$442$			6.45782i			0.307167i
$443$	7.35834			0.349605			0.174803	−	0.984604i	$-0.444071\pi$
$443$	7.35834			0.349605			0.174803	+	0.984604i	$0.444071\pi$
$444$		0			0
$445$	−1.35139	+	0.449226i	−0.0640621	+	0.0212954i
$446$	−19.7255			−0.934029
$447$		0			0
$448$		0			0
$449$		−	32.8021i		−	1.54803i	−0.633170	−	0.774013i	$-0.718247\pi$
$449$		−	32.8021i		−	1.54803i	0.633170	−	0.774013i	$-0.281753\pi$
$450$		0			0
$451$		−	36.7286i		−	1.72948i
$452$	18.0732			0.850091
$453$		0			0
$454$			11.5064i			0.540023i
$455$		0			0
$456$		0			0
$457$			8.59771i			0.402184i	0.979572	+	0.201092i	$0.0644490\pi$
$457$			8.59771i			0.402184i	−0.979572	+	0.201092i	$0.935551\pi$
$458$			19.3140i			0.902483i
$459$		0			0
$460$	4.49640	−	1.49468i	0.209646	−	0.0696900i
$461$	24.8133			1.15567			0.577836	−	0.816153i	$-0.303897\pi$
$461$	24.8133			1.15567			0.577836	+	0.816153i	$0.303897\pi$
$462$		0			0
$463$		−	18.1377i		−	0.842930i	−0.906845	−	0.421465i	$-0.861516\pi$
$463$		−	18.1377i		−	0.842930i	0.906845	−	0.421465i	$-0.138484\pi$
$464$		−	0.0694157i		−	0.00322254i
$465$		0			0
$466$	−20.4744			−0.948458
$467$		−	42.4895i		−	1.96618i	−0.183130	−	0.983089i	$-0.558623\pi$
$467$		−	42.4895i		−	1.96618i	0.183130	−	0.983089i	$-0.441377\pi$
$468$		0			0
$469$		0			0
$470$	−7.74858	−	23.3098i	−0.357416	−	1.07520i
$471$		0			0
$472$	−5.80717			−0.267297
$473$	−24.8655			−1.14332
$474$		0			0
$475$	2.12464	+	2.84260i	0.0974851	+	0.130428i
$476$		0			0
$477$		0			0
$478$			15.0439i			0.688090i
$479$	−28.8898			−1.32001			−0.660005	−	0.751262i	$-0.729446\pi$
$479$	−28.8898			−1.32001			−0.660005	+	0.751262i	$0.729446\pi$
$480$		0			0
$481$			2.59201i			0.118185i
$482$		−	10.0812i		−	0.459184i
$483$		0			0
$484$	−0.430536			−0.0195698
$485$	−6.37439	+	2.11896i	−0.289446	+	0.0962170i
$486$		0			0
$487$			5.09237i			0.230757i	0.993322	+	0.115379i	$0.0368081\pi$
$487$			5.09237i			0.230757i	−0.993322	+	0.115379i	$0.963192\pi$
$488$			5.18196i			0.234576i
$489$		0			0
$490$		0			0
$491$			32.8067i			1.48055i	0.672305	+	0.740274i	$0.265304\pi$
$491$			32.8067i			1.48055i	−0.672305	+	0.740274i	$0.734696\pi$
$492$		0			0
$493$	0.486775			0.0219233
$494$			0.653635i			0.0294084i
$495$		0			0
$496$		−	4.09651i		−	0.183939i
$497$		0			0
$498$		0			0
$499$	−21.5080			−0.962831			−0.481416	−	0.876492i	$-0.659877\pi$
$499$	−21.5080			−0.962831			−0.481416	+	0.876492i	$0.659877\pi$
$500$	−6.38668	+	9.17662i	−0.285621	+	0.410391i
$501$		0			0
$502$	−19.1132			−0.853063
$503$			2.64306i			0.117848i	0.998262	+	0.0589241i	$0.0187670\pi$
$503$			2.64306i			0.117848i	−0.998262	+	0.0589241i	$0.981233\pi$
$504$		0			0
$505$	−4.10006	+	1.36293i	−0.182450	+	0.0606497i
$506$			7.16429i			0.318492i
$507$		0			0
$508$			2.72006i			0.120683i
$509$	−13.6587			−0.605413			−0.302706	−	0.953084i	$-0.597890\pi$
$509$	−13.6587			−0.605413			−0.302706	+	0.953084i	$0.597890\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$		0			0
$514$			3.42573i			0.151102i
$515$	−23.9311	+	7.95511i	−1.05453	+	0.350544i
$516$		0			0
$517$	37.1404			1.63343
$518$		0			0
$519$		0			0
$520$	−1.95407	+	0.649567i	−0.0856917	+	0.0284854i
$521$	−18.4460			−0.808136			−0.404068	−	0.914729i	$-0.632404\pi$
$521$	−18.4460			−0.808136			−0.404068	+	0.914729i	$0.632404\pi$
$522$		0			0
$523$	−29.3443			−1.28314			−0.641568	−	0.767066i	$-0.721716\pi$
$523$	−29.3443			−1.28314			−0.641568	+	0.767066i	$0.721716\pi$
$524$	−15.6215			−0.682430
$525$		0			0
$526$	12.4097			0.541090
$527$	28.7267			1.25135
$528$		0			0
$529$	−18.5097			−0.804768
$530$	−5.92971	+	1.97114i	−0.257570	+	0.0856208i
$531$		0			0
$532$		0			0
$533$	10.0043			0.433333
$534$		0			0
$535$	35.9981	−	11.9664i	1.55634	−	0.517353i
$536$			6.80832i			0.294075i
$537$		0			0
$538$	−26.6150			−1.14745
$539$		0			0
$540$		0			0
$541$	−11.3037			−0.485983			−0.242991	−	0.970028i	$-0.578129\pi$
$541$	−11.3037			−0.485983			−0.242991	+	0.970028i	$0.578129\pi$
$542$			4.13615i			0.177663i
$543$		0			0
$544$			7.01247i			0.300657i
$545$	−32.9073	+	10.9390i	−1.40959	+	0.468574i
$546$		0			0
$547$			2.71136i			0.115929i	0.998319	+	0.0579646i	$0.0184611\pi$
$547$			2.71136i			0.115929i	−0.998319	+	0.0579646i	$0.981539\pi$
$548$	22.4821			0.960389
$549$		0			0
$550$	−10.1204	−	13.5403i	−0.431536	−	0.577362i
$551$	0.0492695			0.00209895
$552$		0			0
$553$		0			0
$554$			19.9414i			0.847231i
$555$		0			0
$556$		−	1.58386i		−	0.0671707i
$557$	3.21745			0.136328			0.0681638	−	0.997674i	$-0.478286\pi$
$557$	3.21745			0.136328			0.0681638	+	0.997674i	$0.478286\pi$
$558$		0			0
$559$		−	6.77296i		−	0.286466i
$560$		0			0
$561$		0			0
$562$		−	11.1429i		−	0.470034i
$563$			16.3125i			0.687489i	0.939063	+	0.343744i	$0.111695\pi$
$563$			16.3125i			0.687489i	−0.939063	+	0.343744i	$0.888305\pi$
$564$		0			0
$565$	−38.3495	+	12.7481i	−1.61338	+	0.536315i
$566$	29.8097			1.25299
$567$		0			0
$568$			13.9389i			0.584863i
$569$			24.2232i			1.01549i	0.861508	+	0.507744i	$0.169520\pi$
$569$			24.2232i			1.01549i	−0.861508	+	0.507744i	$0.830480\pi$
$570$		0			0
$571$	−40.9276			−1.71277			−0.856383	−	0.516341i	$-0.827294\pi$
$571$	−40.9276			−1.71277			−0.856383	+	0.516341i	$0.827294\pi$
$572$		−	3.11349i		−	0.130182i
$573$		0			0
$574$		0			0
$575$	−8.48665	+	6.34315i	−0.353918	+	0.264527i
$576$		0			0
$577$	−39.0626			−1.62620			−0.813098	−	0.582127i	$-0.802221\pi$
$577$	−39.0626			−1.62620			−0.813098	+	0.582127i	$0.802221\pi$
$578$	−32.1747			−1.33829
$579$		0			0
$580$	0.0489629	+	0.147293i	0.00203307	+	0.00611602i
$581$		0			0
$582$		0			0
$583$		−	9.44803i		−	0.391297i
$584$	6.18863			0.256087
$585$		0			0
$586$		−	2.22037i		−	0.0917224i
$587$		−	30.6482i		−	1.26499i	−0.774566	−	0.632493i	$-0.782032\pi$
$587$		−	30.6482i		−	1.26499i	0.774566	−	0.632493i	$-0.217968\pi$
$588$		0			0
$589$	2.90760			0.119806
$590$	12.3223	−	4.09613i	0.507299	−	0.168635i
$591$		0			0
$592$			2.81463i			0.115681i
$593$		−	14.7950i		−	0.607557i	−0.952743	−	0.303778i	$-0.901752\pi$
$593$		−	14.7950i		−	0.607557i	0.952743	−	0.303778i	$-0.0982482\pi$
$594$		0			0
$595$		0			0
$596$		−	8.51774i		−	0.348900i
$597$		0			0
$598$	−1.95144			−0.0798002
$599$			6.34967i			0.259440i	0.991551	+	0.129720i	$0.0414079\pi$
$599$			6.34967i			0.259440i	−0.991551	+	0.129720i	$0.958592\pi$
$600$		0			0
$601$		−	25.7817i		−	1.05166i	−0.850591	−	0.525828i	$-0.823756\pi$
$601$		−	25.7817i		−	1.05166i	0.850591	−	0.525828i	$-0.176244\pi$
$602$		0			0
$603$		0			0
$604$	15.3762			0.625647
$605$	0.913555	−	0.303682i	0.0371413	−	0.0123464i
$606$		0			0
$607$	40.3279			1.63686			0.818429	−	0.574608i	$-0.194846\pi$
$607$	40.3279			1.63686			0.818429	+	0.574608i	$0.194846\pi$
$608$			0.709774i			0.0287851i
$609$		0			0
$610$	−3.65513	−	10.9956i	−0.147992	−	0.445199i
$611$			10.1164i			0.409268i
$612$		0			0
$613$		−	29.1887i		−	1.17892i	−0.807798	−	0.589460i	$-0.799341\pi$
$613$		−	29.1887i		−	1.17892i	0.807798	−	0.589460i	$-0.200659\pi$
$614$	21.1861			0.855003
$615$		0			0
$616$		0			0
$617$	−3.83743			−0.154489			−0.0772445	−	0.997012i	$-0.524612\pi$
$617$	−3.83743			−0.154489			−0.0772445	+	0.997012i	$0.524612\pi$
$618$		0			0
$619$		−	16.0331i		−	0.644424i	−0.946668	−	0.322212i	$-0.895574\pi$
$619$		−	16.0331i		−	0.644424i	0.946668	−	0.322212i	$-0.104426\pi$
$620$	2.88951	+	8.69240i	0.116045	+	0.349095i
$621$		0			0
$622$	4.88322			0.195799
$623$		0			0
$624$		0			0
$625$	7.07911	−	23.9768i	0.283164	−	0.959071i
$626$	8.41182			0.336204
$627$		0			0
$628$	0.162670			0.00649125
$629$	−19.7375			−0.786986
$630$		0			0
$631$	−28.5075			−1.13487			−0.567433	−	0.823420i	$-0.692063\pi$
$631$	−28.5075			−1.13487			−0.567433	+	0.823420i	$0.692063\pi$
$632$	4.57479			0.181976
$633$		0			0
$634$	−1.70288			−0.0676300
$635$	−1.91862	−	5.77171i	−0.0761380	−	0.229043i
$636$		0			0
$637$		0			0
$638$	−0.234688			−0.00929139
$639$		0			0
$640$	−2.12190	+	0.705358i	−0.0838756	+	0.0278817i
$641$		−	33.5431i		−	1.32487i	−0.749118	−	0.662437i	$-0.769522\pi$
$641$		−	33.5431i		−	1.32487i	0.749118	−	0.662437i	$-0.230478\pi$
$642$		0			0
$643$	−24.8624			−0.980476			−0.490238	−	0.871589i	$-0.663090\pi$
$643$	−24.8624			−0.980476			−0.490238	+	0.871589i	$0.663090\pi$
$644$		0			0
$645$		0			0
$646$	−4.97727			−0.195828
$647$			9.73984i			0.382913i	0.981501	+	0.191456i	$0.0613211\pi$
$647$			9.73984i			0.382913i	−0.981501	+	0.191456i	$0.938679\pi$
$648$		0			0
$649$			19.6335i			0.770683i
$650$	3.68817	−	2.75664i	0.144662	−	0.108124i
$651$		0			0
$652$			20.9458i			0.820299i
$653$	11.2009			0.438325			0.219163	−	0.975688i	$-0.429668\pi$
$653$	11.2009			0.438325			0.219163	+	0.975688i	$0.429668\pi$
$654$		0			0
$655$	33.1474	−	11.0188i	1.29517	−	0.430539i
$656$	10.8635			0.424149
$657$		0			0
$658$		0			0
$659$		−	34.4041i		−	1.34019i	−0.742274	−	0.670096i	$-0.766253\pi$
$659$		−	34.4041i		−	1.34019i	0.742274	−	0.670096i	$-0.233747\pi$
$660$		0			0
$661$		−	13.2794i		−	0.516509i	−0.966077	−	0.258254i	$-0.916853\pi$
$661$		−	13.2794i		−	0.516509i	0.966077	−	0.258254i	$-0.0831473\pi$
$662$	−1.17708			−0.0457483
$663$		0			0
$664$			8.39481i			0.325782i
$665$		0			0
$666$		0			0
$667$			0.147095i			0.00569554i
$668$		−	7.27807i		−	0.281597i
$669$		0			0
$670$	−4.80230	−	14.4466i	−0.185529	−	0.558121i
$671$	17.5197			0.676341
$672$		0			0
$673$			24.6142i			0.948807i	0.880307	+	0.474404i	$0.157336\pi$
$673$			24.6142i			0.948807i	−0.880307	+	0.474404i	$0.842664\pi$
$674$		−	24.3592i		−	0.938281i
$675$		0			0
$676$	−12.1519			−0.467382
$677$		−	40.0283i		−	1.53841i	−0.639001	−	0.769206i	$-0.720652\pi$
$677$		−	40.0283i		−	1.53841i	0.639001	−	0.769206i	$-0.279348\pi$
$678$		0			0
$679$		0			0
$680$	−4.94630	−	14.8798i	−0.189682	−	0.570613i
$681$		0			0
$682$	−13.8499			−0.530341
$683$	−14.3162			−0.547796			−0.273898	−	0.961759i	$-0.588313\pi$
$683$	−14.3162			−0.547796			−0.273898	+	0.961759i	$0.588313\pi$
$684$		0			0
$685$	−47.7049	+	15.8579i	−1.82271	+	0.605901i
$686$		0			0
$687$		0			0
$688$		−	7.35468i		−	0.280395i
$689$	2.57349			0.0980423
$690$		0			0
$691$		−	12.3377i		−	0.469348i	−0.972074	−	0.234674i	$-0.924598\pi$
$691$		−	12.3377i		−	0.469348i	0.972074	−	0.234674i	$-0.0754023\pi$
$692$			14.5567i			0.553364i
$693$		0			0
$694$	25.8225			0.980207
$695$	1.11719	+	3.36080i	0.0423774	+	0.127482i
$696$		0			0
$697$			76.1801i			2.88553i
$698$		−	18.7142i		−	0.708343i
$699$		0			0
$700$		0			0
$701$			1.12731i			0.0425777i	0.999773	+	0.0212889i	$0.00677697\pi$
$701$			1.12731i			0.0425777i	−0.999773	+	0.0212889i	$0.993223\pi$
$702$		0			0
$703$	−1.99775			−0.0753467
$704$		−	3.38091i		−	0.127423i
$705$		0			0
$706$		−	20.5673i		−	0.774061i
$707$		0			0
$708$		0			0
$709$	−20.6496			−0.775512			−0.387756	−	0.921762i	$-0.626750\pi$
$709$	−20.6496			−0.775512			−0.387756	+	0.921762i	$0.626750\pi$
$710$	−9.83191	−	29.5770i	−0.368985	−	1.11000i
$711$		0			0
$712$	0.636877			0.0238680
$713$			8.68069i			0.325094i
$714$		0			0
$715$	2.19613	+	6.60653i	0.0821305	+	0.247070i
$716$		−	2.40612i		−	0.0899210i
$717$		0			0
$718$		−	17.8940i		−	0.667799i
$719$	−23.4309			−0.873824			−0.436912	−	0.899504i	$-0.643928\pi$
$719$	−23.4309			−0.873824			−0.436912	+	0.899504i	$0.643928\pi$
$720$		0			0
$721$		0			0
$722$	18.4962			0.688358
$723$		0			0
$724$		−	1.39280i		−	0.0517632i
$725$	−0.207789	−	0.278006i	−0.00771709	−	0.0103249i
$726$		0			0
$727$	18.0552			0.669631			0.334815	−	0.942284i	$-0.391326\pi$
$727$	18.0552			0.669631			0.334815	+	0.942284i	$0.391326\pi$
$728$		0			0
$729$		0			0
$730$	−13.1317	+	4.36519i	−0.486025	+	0.161563i
$731$	51.5745			1.90755
$732$		0			0
$733$	−31.2282			−1.15344			−0.576720	−	0.816942i	$-0.695668\pi$
$733$	−31.2282			−1.15344			−0.576720	+	0.816942i	$0.695668\pi$
$734$	−31.5003			−1.16270
$735$		0			0
$736$	−2.11904			−0.0781090
$737$	23.0183			0.847890
$738$		0			0
$739$	−3.41350			−0.125568			−0.0627838	−	0.998027i	$-0.519998\pi$
$739$	−3.41350			−0.125568			−0.0627838	+	0.998027i	$0.519998\pi$
$740$	−1.98532	−	5.97237i	−0.0729819	−	0.219549i
$741$		0			0
$742$		0			0
$743$	−44.2909			−1.62487			−0.812437	−	0.583049i	$-0.801860\pi$
$743$	−44.2909			−1.62487			−0.812437	+	0.583049i	$0.801860\pi$
$744$		0			0
$745$	6.00805	+	18.0738i	0.220118	+	0.662173i
$746$		−	28.6159i		−	1.04770i
$747$		0			0
$748$	23.7085			0.866869
$749$		0			0
$750$		0			0
$751$	32.6273			1.19059			0.595293	−	0.803508i	$-0.297036\pi$
$751$	32.6273			1.19059			0.595293	+	0.803508i	$0.297036\pi$
$752$			10.9853i			0.400594i
$753$		0			0
$754$		−	0.0639253i		−	0.00232802i
$755$	−32.6267	+	10.8457i	−1.18741	+	0.394715i
$756$		0			0
$757$			52.3185i			1.90155i	0.309887	+	0.950773i	$0.399709\pi$
$757$			52.3185i			1.90155i	−0.309887	+	0.950773i	$0.600291\pi$
$758$	−1.43606			−0.0521601
$759$		0			0
$760$	−0.500645	−	1.50607i	−0.0181603	−	0.0546310i
$761$	22.5653			0.817993			0.408996	−	0.912536i	$-0.365879\pi$
$761$	22.5653			0.817993			0.408996	+	0.912536i	$0.365879\pi$
$762$		0			0
$763$		0			0
$764$			21.8400i			0.790145i
$765$		0			0
$766$			22.5484i			0.814706i
$767$	−5.34786			−0.193100
$768$		0			0
$769$			16.0796i			0.579845i	0.957050	+	0.289923i	$0.0936296\pi$
$769$			16.0796i			0.579845i	−0.957050	+	0.289923i	$0.906370\pi$
$770$		0			0
$771$		0			0
$772$		−	11.1452i		−	0.401124i
$773$		−	17.4875i		−	0.628980i	−0.949261	−	0.314490i	$-0.898166\pi$
$773$		−	17.4875i		−	0.628980i	0.949261	−	0.314490i	$-0.101834\pi$
$774$		0			0
$775$	−12.2625	−	16.4063i	−0.440482	−	0.589331i
$776$	3.00409			0.107841
$777$		0			0
$778$		−	22.8059i		−	0.817632i
$779$			7.71065i			0.276263i
$780$		0			0
$781$	47.1261			1.68631
$782$		−	14.8597i		−	0.531383i
$783$		0			0
$784$		0			0
$785$	−0.345170	+	0.114741i	−0.0123197	+	0.00409527i
$786$		0			0
$787$	38.6094			1.37628			0.688139	−	0.725579i	$-0.258428\pi$
$787$	38.6094			1.37628			0.688139	+	0.725579i	$0.258428\pi$
$788$	18.5178			0.659669
$789$		0			0
$790$	−9.70727	+	3.22687i	−0.345369	+	0.114807i
$791$		0			0
$792$		0			0
$793$			4.77209i			0.169462i
$794$	24.2195			0.859519
$795$		0			0
$796$		−	7.96353i		−	0.282260i
$797$		−	21.4449i		−	0.759617i	−0.925065	−	0.379808i	$-0.875990\pi$
$797$		−	21.4449i		−	0.759617i	0.925065	−	0.379808i	$-0.124010\pi$
$798$		0			0
$799$	−77.0343			−2.72528
$800$	4.00494	−	2.99340i	0.141596	−	0.105833i
$801$		0			0
$802$			25.4197i			0.897600i
$803$		−	20.9232i		−	0.738363i
$804$		0			0
$805$		0			0
$806$		−	3.77250i		−	0.132881i
$807$		0			0
$808$	1.93226			0.0679766
$809$		−	3.55074i		−	0.124837i	−0.998050	−	0.0624187i	$-0.980119\pi$
$809$		−	3.55074i		−	0.124837i	0.998050	−	0.0624187i	$-0.0198814\pi$
$810$		0			0
$811$		−	35.8128i		−	1.25756i	−0.777585	−	0.628778i	$-0.783555\pi$
$811$		−	35.8128i		−	1.25756i	0.777585	−	0.628778i	$-0.216445\pi$
$812$		0			0
$813$		0			0
$814$	9.51601			0.333536
$815$	−14.7743	−	44.4449i	−0.517520	−	1.55684i
$816$		0			0
$817$	5.22016			0.182630
$818$			0.367496i			0.0128492i
$819$		0			0
$820$	−23.0513	+	7.66267i	−0.804988	+	0.267592i
$821$			38.3868i			1.33971i	0.742492	+	0.669855i	$0.233644\pi$
$821$			38.3868i			1.33971i	−0.742492	+	0.669855i	$0.766356\pi$
$822$		0			0
$823$			38.9006i			1.35599i	0.735067	+	0.677995i	$0.237151\pi$
$823$			38.9006i			1.35599i	−0.735067	+	0.677995i	$0.762849\pi$
$824$	11.2781			0.392892
$825$		0			0
$826$		0			0
$827$	−26.0633			−0.906309			−0.453154	−	0.891432i	$-0.649701\pi$
$827$	−26.0633			−0.906309			−0.453154	+	0.891432i	$0.649701\pi$
$828$		0			0
$829$			49.6495i			1.72440i	0.506570	+	0.862199i	$0.330913\pi$
$829$			49.6495i			1.72440i	−0.506570	+	0.862199i	$0.669087\pi$
$830$	−5.92134	−	17.8130i	−0.205533	−	0.618297i
$831$		0			0
$832$	0.920905			0.0319266
$833$		0			0
$834$		0			0
$835$	5.13364	+	15.4434i	0.177657	+	0.534439i
$836$	2.39968			0.0829947
$837$		0			0
$838$	−18.4189			−0.636271
$839$	55.2237			1.90653			0.953267	−	0.302131i	$-0.0976979\pi$
$839$	55.2237			1.90653			0.953267	+	0.302131i	$0.0976979\pi$
$840$		0			0
$841$	28.9952			0.999834
$842$	34.8600			1.20135
$843$		0			0
$844$	−14.4828			−0.498519
$845$	25.7852	−	8.57146i	0.887039	−	0.294867i
$846$		0			0
$847$		0			0
$848$	2.79453			0.0959644
$849$		0			0
$850$	20.9911	+	28.0845i	0.719990	+	0.963291i
$851$		−	5.96433i		−	0.204454i
$852$		0			0
$853$	−1.73556			−0.0594244			−0.0297122	−	0.999558i	$-0.509459\pi$
$853$	−1.73556			−0.0594244			−0.0297122	+	0.999558i	$0.509459\pi$
$854$		0			0
$855$		0			0
$856$	−16.9650			−0.579853
$857$			25.8348i			0.882499i	0.897385	+	0.441249i	$0.145465\pi$
$857$			25.8348i			0.882499i	−0.897385	+	0.441249i	$0.854535\pi$
$858$		0			0
$859$			1.84701i			0.0630193i	0.999503	+	0.0315097i	$0.0100315\pi$
$859$			1.84701i			0.0630193i	−0.999503	+	0.0315097i	$0.989969\pi$
$860$	5.18768	+	15.6059i	0.176898	+	0.532157i
$861$		0			0
$862$			21.1923i			0.721811i
$863$	9.14735			0.311379			0.155690	−	0.987806i	$-0.450240\pi$
$863$	9.14735			0.311379			0.155690	+	0.987806i	$0.450240\pi$
$864$		0			0
$865$	−10.2677	−	30.8880i	−0.349112	−	1.05022i
$866$	15.9350			0.541492
$867$		0			0
$868$		0			0
$869$		−	15.4670i		−	0.524681i
$870$		0			0
$871$			6.26982i			0.212445i
$872$	15.5084			0.525181
$873$		0			0
$874$		−	1.50404i		−	0.0508750i
$875$		0			0
$876$		0			0
$877$		−	22.0644i		−	0.745063i	−0.928020	−	0.372532i	$-0.878490\pi$
$877$		−	22.0644i		−	0.745063i	0.928020	−	0.372532i	$-0.121510\pi$
$878$		−	39.1511i		−	1.32128i
$879$		0			0
$880$	2.38475	+	7.17396i	0.0803899	+	0.241834i
$881$	4.44738			0.149836			0.0749180	−	0.997190i	$-0.476130\pi$
$881$	4.44738			0.149836			0.0749180	+	0.997190i	$0.476130\pi$
$882$		0			0
$883$			15.3381i			0.516168i	0.966122	+	0.258084i	$0.0830911\pi$
$883$			15.3381i			0.516168i	−0.966122	+	0.258084i	$0.916909\pi$
$884$			6.45782i			0.217200i
$885$		0			0
$886$	7.35834			0.247208
$887$		−	57.3582i		−	1.92590i	−0.269679	−	0.962950i	$-0.586918\pi$
$887$		−	57.3582i		−	1.92590i	0.269679	−	0.962950i	$-0.413082\pi$
$888$		0			0
$889$		0			0
$890$	−1.35139	+	0.449226i	−0.0452987	+	0.0150581i
$891$		0			0
$892$	−19.7255			−0.660458
$893$	−7.79710			−0.260920
$894$		0			0
$895$	1.69718	+	5.10556i	0.0567304	+	0.170660i
$896$		0			0
$897$		0			0
$898$		−	32.8021i		−	1.09462i
$899$	−0.284362			−0.00948401
$900$		0			0
$901$			19.5965i			0.652855i
$902$		−	36.7286i		−	1.22293i
$903$		0			0
$904$	18.0732			0.601105
$905$	0.982425	+	2.95540i	0.0326569	+	0.0982407i
$906$		0			0
$907$			48.6872i			1.61663i	0.588750	+	0.808315i	$0.299620\pi$
$907$			48.6872i			1.61663i	−0.588750	+	0.808315i	$0.700380\pi$
$908$			11.5064i			0.381854i
$909$		0			0
$910$		0			0
$911$			12.0295i			0.398557i	0.979943	+	0.199278i	$0.0638598\pi$
$911$			12.0295i			0.398557i	−0.979943	+	0.199278i	$0.936140\pi$
$912$		0			0
$913$	28.3821			0.939309
$914$			8.59771i			0.284387i
$915$		0			0
$916$			19.3140i			0.638152i
$917$		0			0
$918$		0			0
$919$	32.3342			1.06661			0.533304	−	0.845924i	$-0.320950\pi$
$919$	32.3342			1.06661			0.533304	+	0.845924i	$0.320950\pi$
$920$	4.49640	−	1.49468i	0.148242	−	0.0492783i
$921$		0			0
$922$	24.8133			0.817184
$923$			12.8364i			0.422515i
$924$		0			0
$925$	8.42532	+	11.2724i	0.277023	+	0.370635i
$926$		−	18.1377i		−	0.596041i
$927$		0			0
$928$		−	0.0694157i		−	0.00227868i
$929$	44.5657			1.46215			0.731077	−	0.682295i	$-0.239018\pi$
$929$	44.5657			1.46215			0.731077	+	0.682295i	$0.239018\pi$
$930$		0			0
$931$		0			0
$932$	−20.4744			−0.670661
$933$		0			0
$934$		−	42.4895i		−	1.39030i
$935$	−50.3072	+	16.7230i	−1.64522	+	0.546900i
$936$		0			0
$937$	−28.3290			−0.925468			−0.462734	−	0.886497i	$-0.653131\pi$
$937$	−28.3290			−0.925468			−0.462734	+	0.886497i	$0.653131\pi$
$938$		0			0
$939$		0			0
$940$	−7.74858	−	23.3098i	−0.252731	−	0.760282i
$941$	−10.3362			−0.336950			−0.168475	−	0.985706i	$-0.553884\pi$
$941$	−10.3362			−0.336950			−0.168475	+	0.985706i	$0.553884\pi$
$942$		0			0
$943$	−23.0203			−0.749644
$944$	−5.80717			−0.189007
$945$		0			0
$946$	−24.8655			−0.808447
$947$	−6.51990			−0.211868			−0.105934	−	0.994373i	$-0.533783\pi$
$947$	−6.51990			−0.211868			−0.105934	+	0.994373i	$0.533783\pi$
$948$		0			0
$949$	5.69914			0.185002
$950$	2.12464	+	2.84260i	0.0689324	+	0.0922263i
$951$		0			0
$952$		0			0
$953$	−44.0930			−1.42831			−0.714155	−	0.699987i	$-0.753189\pi$
$953$	−44.0930			−1.42831			−0.714155	+	0.699987i	$0.753189\pi$
$954$		0			0
$955$	−15.4050	−	46.3424i	−0.498495	−	1.49961i
$956$			15.0439i			0.486553i
$957$		0			0
$958$	−28.8898			−0.933387
$959$		0			0
$960$		0			0
$961$	14.2186			0.458664
$962$			2.59201i			0.0835697i
$963$		0			0
$964$		−	10.0812i		−	0.324692i
$965$	7.86133	+	23.6490i	0.253065	+	0.761287i
$966$		0			0
$967$			29.4111i			0.945797i	0.881117	+	0.472898i	$0.156792\pi$
$967$			29.4111i			0.945797i	−0.881117	+	0.472898i	$0.843208\pi$
$968$	−0.430536			−0.0138379
$969$		0			0
$970$	−6.37439	+	2.11896i	−0.204669	+	0.0680357i
$971$	−35.2117			−1.13000			−0.564998	−	0.825092i	$-0.691123\pi$
$971$	−35.2117			−1.13000			−0.564998	+	0.825092i	$0.691123\pi$
$972$		0			0
$973$		0			0
$974$			5.09237i			0.163170i
$975$		0			0
$976$			5.18196i			0.165870i
$977$	34.1109			1.09131			0.545653	−	0.838011i	$-0.316282\pi$
$977$	34.1109			1.09131			0.545653	+	0.838011i	$0.316282\pi$
$978$		0			0
$979$		−	2.15322i		−	0.0688173i
$980$		0			0
$981$		0			0
$982$			32.8067i			1.04691i
$983$		−	48.7659i		−	1.55539i	−0.628640	−	0.777696i	$-0.716388\pi$
$983$		−	48.7659i		−	1.55539i	0.628640	−	0.777696i	$-0.283612\pi$
$984$		0			0
$985$	−39.2930	+	13.0617i	−1.25198	+	0.416179i
$986$	0.486775			0.0155021
$987$		0			0
$988$			0.653635i			0.0207949i
$989$			15.5849i			0.495571i
$990$		0			0
$991$	−3.99403			−0.126875			−0.0634373	−	0.997986i	$-0.520206\pi$
$991$	−3.99403			−0.126875			−0.0634373	+	0.997986i	$0.520206\pi$
$992$		−	4.09651i		−	0.130064i
$993$		0			0
$994$		0			0
$995$	5.61714	+	16.8978i	0.178075	+	0.535698i
$996$		0			0
$997$	−35.8258			−1.13461			−0.567307	−	0.823506i	$-0.692015\pi$
$997$	−35.8258			−1.13461			−0.567307	+	0.823506i	$0.692015\pi$
$998$	−21.5080			−0.680825
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.d.d.4409.2	yes	24
3.2	odd	2		4410.2.d.c.4409.23	yes	24
5.4	even	2		4410.2.d.c.4409.1	✓	24
7.6	odd	2	inner	4410.2.d.d.4409.23	yes	24
15.14	odd	2	inner	4410.2.d.d.4409.24	yes	24
21.20	even	2		4410.2.d.c.4409.2	yes	24
35.34	odd	2		4410.2.d.c.4409.24	yes	24
105.104	even	2	inner	4410.2.d.d.4409.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.d.c.4409.1	✓	24	5.4	even	2
4410.2.d.c.4409.2	yes	24	21.20	even	2
4410.2.d.c.4409.23	yes	24	3.2	odd	2
4410.2.d.c.4409.24	yes	24	35.34	odd	2
4410.2.d.d.4409.1	yes	24	105.104	even	2	inner
4410.2.d.d.4409.2	yes	24	1.1	even	1	trivial
4410.2.d.d.4409.23	yes	24	7.6	odd	2	inner
4410.2.d.d.4409.24	yes	24	15.14	odd	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(-2.12190 + 0.705358i) q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(-2.12190 + 0.705358i) q^{5} +1.00000 q^{8} +(-2.12190 + 0.705358i) q^{10} -3.38091i q^{11} +0.920905 q^{13} +1.00000 q^{16} +7.01247i q^{17} +0.709774i q^{19} +(-2.12190 + 0.705358i) q^{20} -3.38091i q^{22} -2.11904 q^{23} +(4.00494 - 2.99340i) q^{25} +0.920905 q^{26} -0.0694157i q^{29} -4.09651i q^{31} +1.00000 q^{32} +7.01247i q^{34} +2.81463i q^{37} +0.709774i q^{38} +(-2.12190 + 0.705358i) q^{40} +10.8635 q^{41} -7.35468i q^{43} -3.38091i q^{44} -2.11904 q^{46} +10.9853i q^{47} +(4.00494 - 2.99340i) q^{50} +0.920905 q^{52} +2.79453 q^{53} +(2.38475 + 7.17396i) q^{55} -0.0694157i q^{58} -5.80717 q^{59} +5.18196i q^{61} -4.09651i q^{62} +1.00000 q^{64} +(-1.95407 + 0.649567i) q^{65} +6.80832i q^{67} +7.01247i q^{68} +13.9389i q^{71} +6.18863 q^{73} +2.81463i q^{74} +0.709774i q^{76} +4.57479 q^{79} +(-2.12190 + 0.705358i) q^{80} +10.8635 q^{82} +8.39481i q^{83} +(-4.94630 - 14.8798i) q^{85} -7.35468i q^{86} -3.38091i q^{88} +0.636877 q^{89} -2.11904 q^{92} +10.9853i q^{94} +(-0.500645 - 1.50607i) q^{95} +3.00409 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{8}+O(q^{10}) \) 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^8	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{8} + 24 q^{16} - 32 q^{23} - 16 q^{25} + 24 q^{32} - 32 q^{46} - 16 q^{50} + 32 q^{53} + 24 q^{64} - 32 q^{85} - 32 q^{92} - 32 q^{95}+O(q^{100}) \) 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^8 + 24 * q^16 - 32 * q^23 - 16 * q^25 + 24 * q^32 - 32 * q^46 - 16 * q^50 + 32 * q^53 + 24 * q^64 - 32 * q^85 - 32 * q^92 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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