LMFDB - Embedded newform 4410.2.d.d.4409.15

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			4409.15
Character	$\chi$	$=$	4410.4409
Dual form			4410.2.d.d.4409.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +(0.613159 - 2.15036i) q^{5} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +(0.613159 - 2.15036i) q^{5} +1.00000 q^{8} +(0.613159 - 2.15036i) q^{10} -2.77878i q^{11} -4.99059 q^{13} +1.00000 q^{16} +4.36915i q^{17} -1.15115i q^{19} +(0.613159 - 2.15036i) q^{20} -2.77878i q^{22} +2.40542 q^{23} +(-4.24807 - 2.63702i) q^{25} -4.99059 q^{26} -6.90583i q^{29} -5.46069i q^{31} +1.00000 q^{32} +4.36915i q^{34} -0.263205i q^{37} -1.15115i q^{38} +(0.613159 - 2.15036i) q^{40} -6.42959 q^{41} -2.17610i q^{43} -2.77878i q^{44} +2.40542 q^{46} +6.93514i q^{47} +(-4.24807 - 2.63702i) q^{50} -4.99059 q^{52} +5.34425 q^{53} +(-5.97537 - 1.70383i) q^{55} -6.90583i q^{58} -11.1458 q^{59} -8.51333i q^{61} -5.46069i q^{62} +1.00000 q^{64} +(-3.06003 + 10.7316i) q^{65} -11.9095i q^{67} +4.36915i q^{68} -4.98480i q^{71} +9.07862 q^{73} -0.263205i q^{74} -1.15115i q^{76} -17.2713 q^{79} +(0.613159 - 2.15036i) q^{80} -6.42959 q^{82} -9.01901i q^{83} +(9.39524 + 2.67898i) q^{85} -2.17610i q^{86} -2.77878i q^{88} -11.4960 q^{89} +2.40542 q^{92} +6.93514i q^{94} +(-2.47539 - 0.705838i) q^{95} +1.18281 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.613159	−	2.15036i	0.274213	−	0.961669i
$5$	0.613159	−	2.15036i	0.274213	−	0.961669i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	1.00000			0.353553
$9$		0			0
$10$	0.613159	−	2.15036i	0.193898	−	0.680003i
$11$		−	2.77878i		−	0.837834i	−0.908024	−	0.418917i	$-0.862410\pi$
$11$		−	2.77878i		−	0.837834i	0.908024	−	0.418917i	$-0.137590\pi$
$12$		0			0
$13$	−4.99059			−1.38414			−0.692071	−	0.721830i	$-0.743301\pi$
$13$	−4.99059			−1.38414			−0.692071	+	0.721830i	$0.743301\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$			4.36915i			1.05968i	0.848099	+	0.529838i	$0.177747\pi$
$17$			4.36915i			1.05968i	−0.848099	+	0.529838i	$0.822253\pi$
$18$		0			0
$19$		−	1.15115i		−	0.264092i	−0.991244	−	0.132046i	$-0.957845\pi$
$19$		−	1.15115i		−	0.264092i	0.991244	−	0.132046i	$-0.0421547\pi$
$20$	0.613159	−	2.15036i	0.137106	−	0.480835i
$21$		0			0
$22$		−	2.77878i		−	0.592438i
$23$	2.40542			0.501565			0.250783	−	0.968043i	$-0.419312\pi$
$23$	2.40542			0.501565			0.250783	+	0.968043i	$0.419312\pi$
$24$		0			0
$25$	−4.24807	−	2.63702i	−0.849615	−	0.527404i
$26$	−4.99059			−0.978736
$27$		0			0
$28$		0			0
$29$		−	6.90583i		−	1.28238i	−0.767382	−	0.641190i	$-0.778441\pi$
$29$		−	6.90583i		−	1.28238i	0.767382	−	0.641190i	$-0.221559\pi$
$30$		0			0
$31$		−	5.46069i		−	0.980768i	−0.871506	−	0.490384i	$-0.836856\pi$
$31$		−	5.46069i		−	0.980768i	0.871506	−	0.490384i	$-0.163144\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			4.36915i			0.749303i
$35$		0			0
$36$		0			0
$37$		−	0.263205i		−	0.0432706i	−0.999766	−	0.0216353i	$-0.993113\pi$
$37$		−	0.263205i		−	0.0432706i	0.999766	−	0.0216353i	$-0.00688726\pi$
$38$		−	1.15115i		−	0.186741i
$39$		0			0
$40$	0.613159	−	2.15036i	0.0969489	−	0.340001i
$41$	−6.42959			−1.00413			−0.502067	−	0.864829i	$-0.667427\pi$
$41$	−6.42959			−1.00413			−0.502067	+	0.864829i	$0.667427\pi$
$42$		0			0
$43$		−	2.17610i		−	0.331852i	−0.986138	−	0.165926i	$-0.946939\pi$
$43$		−	2.17610i		−	0.331852i	0.986138	−	0.165926i	$-0.0530613\pi$
$44$		−	2.77878i		−	0.418917i
$45$		0			0
$46$	2.40542			0.354660
$47$			6.93514i			1.01159i	0.862653	+	0.505797i	$0.168801\pi$
$47$			6.93514i			1.01159i	−0.862653	+	0.505797i	$0.831199\pi$
$48$		0			0
$49$		0			0
$50$	−4.24807	−	2.63702i	−0.600768	−	0.372931i
$51$		0			0
$52$	−4.99059			−0.692071
$53$	5.34425			0.734089			0.367045	−	0.930203i	$-0.380370\pi$
$53$	5.34425			0.734089			0.367045	+	0.930203i	$0.380370\pi$
$54$		0			0
$55$	−5.97537	−	1.70383i	−0.805719	−	0.229745i
$56$		0			0
$57$		0			0
$58$		−	6.90583i		−	0.906780i
$59$	−11.1458			−1.45106			−0.725530	−	0.688191i	$-0.758405\pi$
$59$	−11.1458			−1.45106			−0.725530	+	0.688191i	$0.758405\pi$
$60$		0			0
$61$		−	8.51333i		−	1.09002i	−0.838429	−	0.545010i	$-0.816526\pi$
$61$		−	8.51333i		−	1.09002i	0.838429	−	0.545010i	$-0.183474\pi$
$62$		−	5.46069i		−	0.693508i
$63$		0			0
$64$	1.00000			0.125000
$65$	−3.06003	+	10.7316i	−0.379549	+	1.33109i
$66$		0			0
$67$		−	11.9095i		−	1.45497i	−0.686122	−	0.727487i	$-0.740688\pi$
$67$		−	11.9095i		−	1.45497i	0.686122	−	0.727487i	$-0.259312\pi$
$68$			4.36915i			0.529838i
$69$		0			0
$70$		0			0
$71$		−	4.98480i		−	0.591587i	−0.955252	−	0.295794i	$-0.904416\pi$
$71$		−	4.98480i		−	0.591587i	0.955252	−	0.295794i	$-0.0955841\pi$
$72$		0			0
$73$	9.07862			1.06257			0.531286	−	0.847192i	$-0.321709\pi$
$73$	9.07862			1.06257			0.531286	+	0.847192i	$0.321709\pi$
$74$		−	0.263205i		−	0.0305969i
$75$		0			0
$76$		−	1.15115i		−	0.132046i
$77$		0			0
$78$		0			0
$79$	−17.2713			−1.94317			−0.971584	−	0.236694i	$-0.923936\pi$
$79$	−17.2713			−1.94317			−0.971584	+	0.236694i	$0.923936\pi$
$80$	0.613159	−	2.15036i	0.0685532	−	0.240417i
$81$		0			0
$82$	−6.42959			−0.710030
$83$		−	9.01901i		−	0.989965i	−0.868903	−	0.494982i	$-0.835174\pi$
$83$		−	9.01901i		−	0.989965i	0.868903	−	0.494982i	$-0.164826\pi$
$84$		0			0
$85$	9.39524	+	2.67898i	1.01906	+	0.290577i
$86$		−	2.17610i		−	0.234655i
$87$		0			0
$88$		−	2.77878i		−	0.296219i
$89$	−11.4960			−1.21857			−0.609287	−	0.792950i	$-0.708544\pi$
$89$	−11.4960			−1.21857			−0.609287	+	0.792950i	$0.708544\pi$
$90$		0			0
$91$		0			0
$92$	2.40542			0.250783
$93$		0			0
$94$			6.93514i			0.715305i
$95$	−2.47539	−	0.705838i	−0.253969	−	0.0724175i
$96$		0			0
$97$	1.18281			0.120097			0.0600483	−	0.998195i	$-0.480875\pi$
$97$	1.18281			0.120097			0.0600483	+	0.998195i	$0.480875\pi$
$98$		0			0
$99$		0			0
$100$	−4.24807	−	2.63702i	−0.424807	−	0.263702i
$101$	2.06376			0.205352			0.102676	−	0.994715i	$-0.467260\pi$
$101$	2.06376			0.205352			0.102676	+	0.994715i	$0.467260\pi$
$102$		0			0
$103$	−0.730836			−0.0720114			−0.0360057	−	0.999352i	$-0.511463\pi$
$103$	−0.730836			−0.0720114			−0.0360057	+	0.999352i	$0.511463\pi$
$104$	−4.99059			−0.489368
$105$		0			0
$106$	5.34425			0.519079
$107$	9.98628			0.965410			0.482705	−	0.875783i	$-0.339654\pi$
$107$	9.98628			0.965410			0.482705	+	0.875783i	$0.339654\pi$
$108$		0			0
$109$	9.34460			0.895050			0.447525	−	0.894271i	$-0.352306\pi$
$109$	9.34460			0.895050			0.447525	+	0.894271i	$0.352306\pi$
$110$	−5.97537	−	1.70383i	−0.569729	−	0.162454i
$111$		0			0
$112$		0			0
$113$	−13.8803			−1.30575			−0.652876	−	0.757464i	$-0.726438\pi$
$113$	−13.8803			−1.30575			−0.652876	+	0.757464i	$0.726438\pi$
$114$		0			0
$115$	1.47491	−	5.17252i	0.137536	−	0.482340i
$116$		−	6.90583i		−	0.641190i
$117$		0			0
$118$	−11.1458			−1.02605
$119$		0			0
$120$		0			0
$121$	3.27837			0.298034
$122$		−	8.51333i		−	0.770761i
$123$		0			0
$124$		−	5.46069i		−	0.490384i
$125$	−8.27528	+	7.51796i	−0.740163	+	0.672427i
$126$		0			0
$127$			20.2794i			1.79950i	0.436404	+	0.899751i	$0.356252\pi$
$127$			20.2794i			1.79950i	−0.436404	+	0.899751i	$0.643748\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	−3.06003	+	10.7316i	−0.268382	+	0.941220i
$131$	−22.4671			−1.96296			−0.981480	−	0.191564i	$-0.938644\pi$
$131$	−22.4671			−1.96296			−0.981480	+	0.191564i	$0.938644\pi$
$132$		0			0
$133$		0			0
$134$		−	11.9095i		−	1.02882i
$135$		0			0
$136$			4.36915i			0.374652i
$137$	13.8659			1.18464			0.592320	−	0.805703i	$-0.298212\pi$
$137$	13.8659			1.18464			0.592320	+	0.805703i	$0.298212\pi$
$138$		0			0
$139$		−	3.75476i		−	0.318475i	−0.987240	−	0.159237i	$-0.949097\pi$
$139$		−	3.75476i		−	0.318475i	0.987240	−	0.159237i	$-0.0509035\pi$
$140$		0			0
$141$		0			0
$142$		−	4.98480i		−	0.418315i
$143$			13.8678i			1.15968i
$144$		0			0
$145$	−14.8500	−	4.23437i	−1.23323	−	0.351645i
$146$	9.07862			0.751352
$147$		0			0
$148$		−	0.263205i		−	0.0216353i
$149$			17.2247i			1.41111i	0.708657	+	0.705553i	$0.249301\pi$
$149$			17.2247i			1.41111i	−0.708657	+	0.705553i	$0.750699\pi$
$150$		0			0
$151$	2.47598			0.201492			0.100746	−	0.994912i	$-0.467877\pi$
$151$	2.47598			0.201492			0.100746	+	0.994912i	$0.467877\pi$
$152$		−	1.15115i		−	0.0933707i
$153$		0			0
$154$		0			0
$155$	−11.7424	−	3.34827i	−0.943175	−	0.268939i
$156$		0			0
$157$	−12.4192			−0.991158			−0.495579	−	0.868563i	$-0.665044\pi$
$157$	−12.4192			−0.991158			−0.495579	+	0.868563i	$0.665044\pi$
$158$	−17.2713			−1.37403
$159$		0			0
$160$	0.613159	−	2.15036i	0.0484744	−	0.170001i
$161$		0			0
$162$		0			0
$163$			13.2214i			1.03558i	0.855509	+	0.517789i	$0.173245\pi$
$163$			13.2214i			1.03558i	−0.855509	+	0.517789i	$0.826755\pi$
$164$	−6.42959			−0.502067
$165$		0			0
$166$		−	9.01901i		−	0.700011i
$167$			5.89817i			0.456414i	0.973613	+	0.228207i	$0.0732863\pi$
$167$			5.89817i			0.456414i	−0.973613	+	0.228207i	$0.926714\pi$
$168$		0			0
$169$	11.9060			0.915848
$170$	9.39524	+	2.67898i	0.720582	+	0.205469i
$171$		0			0
$172$		−	2.17610i		−	0.165926i
$173$		−	6.32052i		−	0.480540i	−0.970706	−	0.240270i	$-0.922764\pi$
$173$		−	6.32052i		−	0.480540i	0.970706	−	0.240270i	$-0.0772360\pi$
$174$		0			0
$175$		0			0
$176$		−	2.77878i		−	0.209459i
$177$		0			0
$178$	−11.4960			−0.861661
$179$		−	17.3662i		−	1.29801i	−0.760782	−	0.649007i	$-0.775185\pi$
$179$		−	17.3662i		−	1.29801i	0.760782	−	0.649007i	$-0.224815\pi$
$180$		0			0
$181$			4.98718i			0.370694i	0.982673	+	0.185347i	$0.0593409\pi$
$181$			4.98718i			0.370694i	−0.982673	+	0.185347i	$0.940659\pi$
$182$		0			0
$183$		0			0
$184$	2.40542			0.177330
$185$	−0.565984	−	0.161386i	−0.0416120	−	0.0118653i
$186$		0			0
$187$	12.1409			0.887832
$188$			6.93514i			0.505797i
$189$		0			0
$190$	−2.47539	−	0.705838i	−0.179583	−	0.0512069i
$191$			10.4489i			0.756055i	0.925794	+	0.378028i	$0.123398\pi$
$191$			10.4489i			0.756055i	−0.925794	+	0.378028i	$0.876602\pi$
$192$		0			0
$193$		−	11.3765i		−	0.818899i	−0.912333	−	0.409449i	$-0.865721\pi$
$193$		−	11.3765i		−	0.818899i	0.912333	−	0.409449i	$-0.134279\pi$
$194$	1.18281			0.0849211
$195$		0			0
$196$		0			0
$197$	−9.34972			−0.666140			−0.333070	−	0.942902i	$-0.608085\pi$
$197$	−9.34972			−0.666140			−0.333070	+	0.942902i	$0.608085\pi$
$198$		0			0
$199$		−	16.1828i		−	1.14717i	−0.819148	−	0.573583i	$-0.805553\pi$
$199$		−	16.1828i		−	1.14717i	0.819148	−	0.573583i	$-0.194447\pi$
$200$	−4.24807	−	2.63702i	−0.300384	−	0.186465i
$201$		0			0
$202$	2.06376			0.145206
$203$		0			0
$204$		0			0
$205$	−3.94236	+	13.8259i	−0.275346	+	0.965645i
$206$	−0.730836			−0.0509198
$207$		0			0
$208$	−4.99059			−0.346035
$209$	−3.19880			−0.221265
$210$		0			0
$211$	25.6010			1.76245			0.881224	−	0.472699i	$-0.156720\pi$
$211$	25.6010			1.76245			0.881224	+	0.472699i	$0.156720\pi$
$212$	5.34425			0.367045
$213$		0			0
$214$	9.98628			0.682648
$215$	−4.67940	−	1.33430i	−0.319132	−	0.0909982i
$216$		0			0
$217$		0			0
$218$	9.34460			0.632896
$219$		0			0
$220$	−5.97537	−	1.70383i	−0.402860	−	0.114872i
$221$		−	21.8047i		−	1.46674i
$222$		0			0
$223$	4.98924			0.334104			0.167052	−	0.985948i	$-0.446575\pi$
$223$	4.98924			0.334104			0.167052	+	0.985948i	$0.446575\pi$
$224$		0			0
$225$		0			0
$226$	−13.8803			−0.923307
$227$		−	10.2580i		−	0.680849i	−0.940272	−	0.340425i	$-0.889429\pi$
$227$		−	10.2580i		−	0.680849i	0.940272	−	0.340425i	$-0.110571\pi$
$228$		0			0
$229$		−	0.506318i		−	0.0334585i	−0.999860	−	0.0167292i	$-0.994675\pi$
$229$		−	0.506318i		−	0.0334585i	0.999860	−	0.0167292i	$-0.00532533\pi$
$230$	1.47491	−	5.17252i	0.0972524	−	0.341066i
$231$		0			0
$232$		−	6.90583i		−	0.453390i
$233$	4.50799			0.295328			0.147664	−	0.989038i	$-0.452825\pi$
$233$	4.50799			0.295328			0.147664	+	0.989038i	$0.452825\pi$
$234$		0			0
$235$	14.9130	+	4.25234i	0.972818	+	0.277392i
$236$	−11.1458			−0.725530
$237$		0			0
$238$		0			0
$239$			6.22961i			0.402960i	0.979493	+	0.201480i	$0.0645751\pi$
$239$			6.22961i			0.402960i	−0.979493	+	0.201480i	$0.935425\pi$
$240$		0			0
$241$			15.8611i			1.02171i	0.859668	+	0.510853i	$0.170670\pi$
$241$			15.8611i			1.02171i	−0.859668	+	0.510853i	$0.829330\pi$
$242$	3.27837			0.210742
$243$		0			0
$244$		−	8.51333i		−	0.545010i
$245$		0			0
$246$		0			0
$247$			5.74493i			0.365541i
$248$		−	5.46069i		−	0.346754i
$249$		0			0
$250$	−8.27528	+	7.51796i	−0.523375	+	0.475478i
$251$	29.1750			1.84151			0.920753	−	0.390145i	$-0.127575\pi$
$251$	29.1750			1.84151			0.920753	+	0.390145i	$0.127575\pi$
$252$		0			0
$253$		−	6.68414i		−	0.420228i
$254$			20.2794i			1.27244i
$255$		0			0
$256$	1.00000			0.0625000
$257$			15.4333i			0.962700i	0.876528	+	0.481350i	$0.159853\pi$
$257$			15.4333i			0.962700i	−0.876528	+	0.481350i	$0.840147\pi$
$258$		0			0
$259$		0			0
$260$	−3.06003	+	10.7316i	−0.189775	+	0.665543i
$261$		0			0
$262$	−22.4671			−1.38802
$263$	1.14308			0.0704856			0.0352428	−	0.999379i	$-0.488780\pi$
$263$	1.14308			0.0704856			0.0352428	+	0.999379i	$0.488780\pi$
$264$		0			0
$265$	3.27687	−	11.4920i	0.201297	−	0.705951i
$266$		0			0
$267$		0			0
$268$		−	11.9095i		−	0.727487i
$269$	4.82268			0.294044			0.147022	−	0.989133i	$-0.453031\pi$
$269$	4.82268			0.294044			0.147022	+	0.989133i	$0.453031\pi$
$270$		0			0
$271$		−	23.8234i		−	1.44717i	−0.690235	−	0.723585i	$-0.742493\pi$
$271$		−	23.8234i		−	1.44717i	0.690235	−	0.723585i	$-0.257507\pi$
$272$			4.36915i			0.264919i
$273$		0			0
$274$	13.8659			0.837667
$275$	−7.32770	+	11.8045i	−0.441877	+	0.711836i
$276$		0			0
$277$		−	28.3130i		−	1.70117i	−0.525841	−	0.850583i	$-0.676249\pi$
$277$		−	28.3130i		−	1.70117i	0.525841	−	0.850583i	$-0.323751\pi$
$278$		−	3.75476i		−	0.225196i
$279$		0			0
$280$		0			0
$281$		−	32.3121i		−	1.92758i	−0.266670	−	0.963788i	$-0.585923\pi$
$281$		−	32.3121i		−	1.92758i	0.266670	−	0.963788i	$-0.414077\pi$
$282$		0			0
$283$	23.7635			1.41260			0.706298	−	0.707915i	$-0.250364\pi$
$283$	23.7635			1.41260			0.706298	+	0.707915i	$0.250364\pi$
$284$		−	4.98480i		−	0.295794i
$285$		0			0
$286$			13.8678i			0.820018i
$287$		0			0
$288$		0			0
$289$	−2.08949			−0.122911
$290$	−14.8500	−	4.23437i	−0.872022	−	0.248651i
$291$		0			0
$292$	9.07862			0.531286
$293$		−	10.9857i		−	0.641790i	−0.947115	−	0.320895i	$-0.896016\pi$
$293$		−	10.9857i		−	0.641790i	0.947115	−	0.320895i	$-0.103984\pi$
$294$		0			0
$295$	−6.83414	+	23.9674i	−0.397899	+	1.39544i
$296$		−	0.263205i		−	0.0152985i
$297$		0			0
$298$			17.2247i			0.997803i
$299$	−12.0045			−0.694237
$300$		0			0
$301$		0			0
$302$	2.47598			0.142477
$303$		0			0
$304$		−	1.15115i		−	0.0660230i
$305$	−18.3067	−	5.22002i	−1.04824	−	0.298898i
$306$		0			0
$307$	28.4731			1.62505			0.812523	−	0.582930i	$-0.198094\pi$
$307$	28.4731			1.62505			0.812523	+	0.582930i	$0.198094\pi$
$308$		0			0
$309$		0			0
$310$	−11.7424	−	3.34827i	−0.666925	−	0.190169i
$311$	8.53207			0.483809			0.241905	−	0.970300i	$-0.422228\pi$
$311$	8.53207			0.483809			0.241905	+	0.970300i	$0.422228\pi$
$312$		0			0
$313$	−14.0371			−0.793424			−0.396712	−	0.917943i	$-0.629849\pi$
$313$	−14.0371			−0.793424			−0.396712	+	0.917943i	$0.629849\pi$
$314$	−12.4192			−0.700854
$315$		0			0
$316$	−17.2713			−0.971584
$317$	−34.4661			−1.93581			−0.967906	−	0.251313i	$-0.919138\pi$
$317$	−34.4661			−1.93581			−0.967906	+	0.251313i	$0.919138\pi$
$318$		0			0
$319$	−19.1898			−1.07442
$320$	0.613159	−	2.15036i	0.0342766	−	0.120209i
$321$		0			0
$322$		0			0
$323$	5.02955			0.279852
$324$		0			0
$325$	21.2004	+	13.1603i	1.17599	+	0.730002i
$326$			13.2214i			0.732264i
$327$		0			0
$328$	−6.42959			−0.355015
$329$		0			0
$330$		0			0
$331$	5.22730			0.287318			0.143659	−	0.989627i	$-0.454113\pi$
$331$	5.22730			0.287318			0.143659	+	0.989627i	$0.454113\pi$
$332$		−	9.01901i		−	0.494982i
$333$		0			0
$334$			5.89817i			0.322733i
$335$	−25.6096	−	7.30240i	−1.39920	−	0.398973i
$336$		0			0
$337$		−	34.0655i		−	1.85567i	−0.372997	−	0.927833i	$-0.621670\pi$
$337$		−	34.0655i		−	1.85567i	0.372997	−	0.927833i	$-0.378330\pi$
$338$	11.9060			0.647603
$339$		0			0
$340$	9.39524	+	2.67898i	0.509528	+	0.145288i
$341$	−15.1741			−0.821721
$342$		0			0
$343$		0			0
$344$		−	2.17610i		−	0.117328i
$345$		0			0
$346$		−	6.32052i		−	0.339793i
$347$	2.14401			0.115096			0.0575482	−	0.998343i	$-0.481672\pi$
$347$	2.14401			0.115096			0.0575482	+	0.998343i	$0.481672\pi$
$348$		0			0
$349$		−	8.45360i		−	0.452511i	−0.974068	−	0.226255i	$-0.927352\pi$
$349$		−	8.45360i		−	0.452511i	0.974068	−	0.226255i	$-0.0726484\pi$
$350$		0			0
$351$		0			0
$352$		−	2.77878i		−	0.148110i
$353$			9.99587i			0.532026i	0.963969	+	0.266013i	$0.0857065\pi$
$353$			9.99587i			0.532026i	−0.963969	+	0.266013i	$0.914294\pi$
$354$		0			0
$355$	−10.7191	−	3.05648i	−0.568911	−	0.162221i
$356$	−11.4960			−0.609287
$357$		0			0
$358$		−	17.3662i		−	0.917835i
$359$			13.2296i			0.698233i	0.937079	+	0.349117i	$0.113518\pi$
$359$			13.2296i			0.698233i	−0.937079	+	0.349117i	$0.886482\pi$
$360$		0			0
$361$	17.6749			0.930255
$362$			4.98718i			0.262120i
$363$		0			0
$364$		0			0
$365$	5.56664	−	19.5223i	0.291371	−	1.02184i
$366$		0			0
$367$	14.6459			0.764511			0.382256	−	0.924057i	$-0.375147\pi$
$367$	14.6459			0.764511			0.382256	+	0.924057i	$0.375147\pi$
$368$	2.40542			0.125391
$369$		0			0
$370$	−0.565984	−	0.161386i	−0.0294241	−	0.00839006i
$371$		0			0
$372$		0			0
$373$			12.3430i			0.639095i	0.947570	+	0.319547i	$0.103531\pi$
$373$			12.3430i			0.639095i	−0.947570	+	0.319547i	$0.896469\pi$
$374$	12.1409			0.627792
$375$		0			0
$376$			6.93514i			0.357652i
$377$			34.4642i			1.77500i
$378$		0			0
$379$	−14.9088			−0.765814			−0.382907	−	0.923787i	$-0.625077\pi$
$379$	−14.9088			−0.765814			−0.382907	+	0.923787i	$0.625077\pi$
$380$	−2.47539	−	0.705838i	−0.126985	−	0.0362087i
$381$		0			0
$382$			10.4489i			0.534612i
$383$			29.5094i			1.50786i	0.656956	+	0.753929i	$0.271844\pi$
$383$			29.5094i			1.50786i	−0.656956	+	0.753929i	$0.728156\pi$
$384$		0			0
$385$		0			0
$386$		−	11.3765i		−	0.579049i
$387$		0			0
$388$	1.18281			0.0600483
$389$		−	4.24434i		−	0.215196i	−0.994194	−	0.107598i	$-0.965684\pi$
$389$		−	4.24434i		−	0.215196i	0.994194	−	0.107598i	$-0.0343160\pi$
$390$		0			0
$391$			10.5097i			0.531496i
$392$		0			0
$393$		0			0
$394$	−9.34972			−0.471032
$395$	−10.5900	+	37.1394i	−0.532842	+	1.86868i
$396$		0			0
$397$	20.0023			1.00388			0.501942	−	0.864901i	$-0.332619\pi$
$397$	20.0023			1.00388			0.501942	+	0.864901i	$0.332619\pi$
$398$		−	16.1828i		−	0.811168i
$399$		0			0
$400$	−4.24807	−	2.63702i	−0.212404	−	0.131851i
$401$		−	30.0841i		−	1.50233i	−0.660116	−	0.751163i	$-0.729493\pi$
$401$		−	30.0841i		−	1.50233i	0.660116	−	0.751163i	$-0.270507\pi$
$402$		0			0
$403$			27.2521i			1.35752i
$404$	2.06376			0.102676
$405$		0			0
$406$		0			0
$407$	−0.731388			−0.0362536
$408$		0			0
$409$		−	33.8806i		−	1.67529i	−0.546218	−	0.837643i	$-0.683933\pi$
$409$		−	33.8806i		−	1.67529i	0.546218	−	0.837643i	$-0.316067\pi$
$410$	−3.94236	+	13.8259i	−0.194699	+	0.682814i
$411$		0			0
$412$	−0.730836			−0.0360057
$413$		0			0
$414$		0			0
$415$	−19.3941	−	5.53008i	−0.952019	−	0.271461i
$416$	−4.99059			−0.244684
$417$		0			0
$418$	−3.19880			−0.156458
$419$	19.5576			0.955452			0.477726	−	0.878509i	$-0.341461\pi$
$419$	19.5576			0.955452			0.477726	+	0.878509i	$0.341461\pi$
$420$		0			0
$421$	−1.14027			−0.0555735			−0.0277867	−	0.999614i	$-0.508846\pi$
$421$	−1.14027			−0.0555735			−0.0277867	+	0.999614i	$0.508846\pi$
$422$	25.6010			1.24624
$423$		0			0
$424$	5.34425			0.259540
$425$	11.5215	−	18.5605i	0.558877	−	0.900315i
$426$		0			0
$427$		0			0
$428$	9.98628			0.482705
$429$		0			0
$430$	−4.67940	−	1.33430i	−0.225661	−	0.0643454i
$431$			16.2043i			0.780536i	0.920701	+	0.390268i	$0.127618\pi$
$431$			16.2043i			0.780536i	−0.920701	+	0.390268i	$0.872382\pi$
$432$		0			0
$433$	−12.5614			−0.603663			−0.301831	−	0.953361i	$-0.597598\pi$
$433$	−12.5614			−0.603663			−0.301831	+	0.953361i	$0.597598\pi$
$434$		0			0
$435$		0			0
$436$	9.34460			0.447525
$437$		−	2.76900i		−	0.132459i
$438$		0			0
$439$		−	36.7462i		−	1.75380i	−0.480671	−	0.876901i	$-0.659607\pi$
$439$		−	36.7462i		−	1.75380i	0.480671	−	0.876901i	$-0.340393\pi$
$440$	−5.97537	−	1.70383i	−0.284865	−	0.0812271i
$441$		0			0
$442$		−	21.8047i		−	1.03714i
$443$	−24.0144			−1.14096			−0.570479	−	0.821312i	$-0.693242\pi$
$443$	−24.0144			−1.14096			−0.570479	+	0.821312i	$0.693242\pi$
$444$		0			0
$445$	−7.04887	+	24.7205i	−0.334148	+	1.17186i
$446$	4.98924			0.236247
$447$		0			0
$448$		0			0
$449$			8.98128i			0.423853i	0.977286	+	0.211926i	$0.0679737\pi$
$449$			8.98128i			0.423853i	−0.977286	+	0.211926i	$0.932026\pi$
$450$		0			0
$451$			17.8664i			0.841298i
$452$	−13.8803			−0.652876
$453$		0			0
$454$		−	10.2580i		−	0.481433i
$455$		0			0
$456$		0			0
$457$			27.8745i			1.30391i	0.758257	+	0.651956i	$0.226051\pi$
$457$			27.8745i			1.30391i	−0.758257	+	0.651956i	$0.773949\pi$
$458$		−	0.506318i		−	0.0236587i
$459$		0			0
$460$	1.47491	−	5.17252i	0.0687678	−	0.241170i
$461$	10.3589			0.482462			0.241231	−	0.970468i	$-0.422449\pi$
$461$	10.3589			0.482462			0.241231	+	0.970468i	$0.422449\pi$
$462$		0			0
$463$			5.95243i			0.276633i	0.990388	+	0.138316i	$0.0441691\pi$
$463$			5.95243i			0.276633i	−0.990388	+	0.138316i	$0.955831\pi$
$464$		−	6.90583i		−	0.320595i
$465$		0			0
$466$	4.50799			0.208828
$467$			26.0215i			1.20413i	0.798447	+	0.602065i	$0.205655\pi$
$467$			26.0215i			1.20413i	−0.798447	+	0.602065i	$0.794345\pi$
$468$		0			0
$469$		0			0
$470$	14.9130	+	4.25234i	0.687887	+	0.196146i
$471$		0			0
$472$	−11.1458			−0.513027
$473$	−6.04691			−0.278037
$474$		0			0
$475$	−3.03561	+	4.89017i	−0.139283	+	0.224376i
$476$		0			0
$477$		0			0
$478$			6.22961i			0.284936i
$479$	−1.59497			−0.0728759			−0.0364380	−	0.999336i	$-0.511601\pi$
$479$	−1.59497			−0.0728759			−0.0364380	+	0.999336i	$0.511601\pi$
$480$		0			0
$481$			1.31355i			0.0598926i
$482$			15.8611i			0.722456i
$483$		0			0
$484$	3.27837			0.149017
$485$	0.725253	−	2.54347i	0.0329320	−	0.115493i
$486$		0			0
$487$		−	41.4338i		−	1.87755i	−0.344536	−	0.938773i	$-0.611964\pi$
$487$		−	41.4338i		−	1.87755i	0.344536	−	0.938773i	$-0.388036\pi$
$488$		−	8.51333i		−	0.385380i
$489$		0			0
$490$		0			0
$491$			20.3769i			0.919595i	0.888024	+	0.459797i	$0.152078\pi$
$491$			20.3769i			0.919595i	−0.888024	+	0.459797i	$0.847922\pi$
$492$		0			0
$493$	30.1726			1.35891
$494$			5.74493i			0.258476i
$495$		0			0
$496$		−	5.46069i		−	0.245192i
$497$		0			0
$498$		0			0
$499$	32.1885			1.44096			0.720478	−	0.693478i	$-0.243923\pi$
$499$	32.1885			1.44096			0.720478	+	0.693478i	$0.243923\pi$
$500$	−8.27528	+	7.51796i	−0.370082	+	0.336214i
$501$		0			0
$502$	29.1750			1.30214
$503$			29.2316i			1.30337i	0.758489	+	0.651685i	$0.225938\pi$
$503$			29.2316i			1.30337i	−0.758489	+	0.651685i	$0.774062\pi$
$504$		0			0
$505$	1.26541	−	4.43782i	0.0563101	−	0.197480i
$506$		−	6.68414i		−	0.297146i
$507$		0			0
$508$			20.2794i			0.899751i
$509$	−1.78126			−0.0789531			−0.0394766	−	0.999220i	$-0.512569\pi$
$509$	−1.78126			−0.0789531			−0.0394766	+	0.999220i	$0.512569\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$		0			0
$514$			15.4333i			0.680732i
$515$	−0.448118	+	1.57156i	−0.0197465	+	0.0692511i
$516$		0			0
$517$	19.2712			0.847548
$518$		0			0
$519$		0			0
$520$	−3.06003	+	10.7316i	−0.134191	+	0.470610i
$521$	26.5076			1.16132			0.580659	−	0.814147i	$-0.302795\pi$
$521$	26.5076			1.16132			0.580659	+	0.814147i	$0.302795\pi$
$522$		0			0
$523$	29.8765			1.30641			0.653204	−	0.757182i	$-0.273424\pi$
$523$	29.8765			1.30641			0.653204	+	0.757182i	$0.273424\pi$
$524$	−22.4671			−0.981480
$525$		0			0
$526$	1.14308			0.0498408
$527$	23.8586			1.03930
$528$		0			0
$529$	−17.2139			−0.748432
$530$	3.27687	−	11.4920i	0.142338	−	0.499183i
$531$		0			0
$532$		0			0
$533$	32.0875			1.38986
$534$		0			0
$535$	6.12318	−	21.4741i	0.264728	−	0.928405i
$536$		−	11.9095i		−	0.514411i
$537$		0			0
$538$	4.82268			0.207920
$539$		0			0
$540$		0			0
$541$	25.9957			1.11764			0.558821	−	0.829288i	$-0.311254\pi$
$541$	25.9957			1.11764			0.558821	+	0.829288i	$0.311254\pi$
$542$		−	23.8234i		−	1.02330i
$543$		0			0
$544$			4.36915i			0.187326i
$545$	5.72972	−	20.0942i	0.245434	−	0.860742i
$546$		0			0
$547$		−	14.0745i		−	0.601784i	−0.953658	−	0.300892i	$-0.902716\pi$
$547$		−	14.0745i		−	0.601784i	0.953658	−	0.300892i	$-0.0972844\pi$
$548$	13.8659			0.592320
$549$		0			0
$550$	−7.32770	+	11.8045i	−0.312454	+	0.503344i
$551$	−7.94965			−0.338667
$552$		0			0
$553$		0			0
$554$		−	28.3130i		−	1.20291i
$555$		0			0
$556$		−	3.75476i		−	0.159237i
$557$	40.4162			1.71249			0.856244	−	0.516572i	$-0.172792\pi$
$557$	40.4162			1.71249			0.856244	+	0.516572i	$0.172792\pi$
$558$		0			0
$559$			10.8600i			0.459331i
$560$		0			0
$561$		0			0
$562$		−	32.3121i		−	1.36300i
$563$			37.1252i			1.56464i	0.622877	+	0.782320i	$0.285964\pi$
$563$			37.1252i			1.56464i	−0.622877	+	0.782320i	$0.714036\pi$
$564$		0			0
$565$	−8.51085	+	29.8477i	−0.358054	+	1.25570i
$566$	23.7635			0.998856
$567$		0			0
$568$		−	4.98480i		−	0.209158i
$569$			4.27025i			0.179018i	0.995986	+	0.0895091i	$0.0285298\pi$
$569$			4.27025i			0.179018i	−0.995986	+	0.0895091i	$0.971470\pi$
$570$		0			0
$571$	−0.494299			−0.0206858			−0.0103429	−	0.999947i	$-0.503292\pi$
$571$	−0.494299			−0.0206858			−0.0103429	+	0.999947i	$0.503292\pi$
$572$			13.8678i			0.579841i
$573$		0			0
$574$		0			0
$575$	−10.2184	−	6.34315i	−0.426137	−	0.264527i
$576$		0			0
$577$	−19.1893			−0.798862			−0.399431	−	0.916763i	$-0.630792\pi$
$577$	−19.1893			−0.798862			−0.399431	+	0.916763i	$0.630792\pi$
$578$	−2.08949			−0.0869113
$579$		0			0
$580$	−14.8500	−	4.23437i	−0.616613	−	0.175823i
$581$		0			0
$582$		0			0
$583$		−	14.8505i		−	0.615045i
$584$	9.07862			0.375676
$585$		0			0
$586$		−	10.9857i		−	0.453814i
$587$			5.48894i			0.226553i	0.993564	+	0.113276i	$0.0361345\pi$
$587$			5.48894i			0.226553i	−0.993564	+	0.113276i	$0.963865\pi$
$588$		0			0
$589$	−6.28607			−0.259013
$590$	−6.83414	+	23.9674i	−0.281357	+	0.986724i
$591$		0			0
$592$		−	0.263205i		−	0.0108176i
$593$		−	19.5359i		−	0.802242i	−0.916025	−	0.401121i	$-0.868621\pi$
$593$		−	19.5359i		−	0.802242i	0.916025	−	0.401121i	$-0.131379\pi$
$594$		0			0
$595$		0			0
$596$			17.2247i			0.705553i
$597$		0			0
$598$	−12.0045			−0.490900
$599$			36.6241i			1.49642i	0.663461	+	0.748210i	$0.269087\pi$
$599$			36.6241i			1.49642i	−0.663461	+	0.748210i	$0.730913\pi$
$600$		0			0
$601$			7.98824i			0.325847i	0.986639	+	0.162924i	$0.0520924\pi$
$601$			7.98824i			0.325847i	−0.986639	+	0.162924i	$0.947908\pi$
$602$		0			0
$603$		0			0
$604$	2.47598			0.100746
$605$	2.01016	−	7.04967i	0.0817247	−	0.286610i
$606$		0			0
$607$	43.1621			1.75190			0.875948	−	0.482406i	$-0.160237\pi$
$607$	43.1621			1.75190			0.875948	+	0.482406i	$0.160237\pi$
$608$		−	1.15115i		−	0.0466853i
$609$		0			0
$610$	−18.3067	−	5.22002i	−0.741217	−	0.211353i
$611$		−	34.6105i		−	1.40019i
$612$		0			0
$613$		−	18.0127i		−	0.727528i	−0.931491	−	0.363764i	$-0.881492\pi$
$613$		−	18.0127i		−	0.727528i	0.931491	−	0.363764i	$-0.118508\pi$
$614$	28.4731			1.14908
$615$		0			0
$616$		0			0
$617$	9.20267			0.370486			0.185243	−	0.982693i	$-0.440693\pi$
$617$	9.20267			0.370486			0.185243	+	0.982693i	$0.440693\pi$
$618$		0			0
$619$			2.50956i			0.100868i	0.998727	+	0.0504338i	$0.0160604\pi$
$619$			2.50956i			0.100868i	−0.998727	+	0.0504338i	$0.983940\pi$
$620$	−11.7424	−	3.34827i	−0.471587	−	0.134470i
$621$		0			0
$622$	8.53207			0.342105
$623$		0			0
$624$		0			0
$625$	11.0922	+	22.4045i	0.443690	+	0.896180i
$626$	−14.0371			−0.561035
$627$		0			0
$628$	−12.4192			−0.495579
$629$	1.14998			0.0458527
$630$		0			0
$631$	−11.9933			−0.477444			−0.238722	−	0.971088i	$-0.576728\pi$
$631$	−11.9933			−0.477444			−0.238722	+	0.971088i	$0.576728\pi$
$632$	−17.2713			−0.687014
$633$		0			0
$634$	−34.4661			−1.36883
$635$	43.6079	+	12.4345i	1.73053	+	0.493447i
$636$		0			0
$637$		0			0
$638$	−19.1898			−0.759731
$639$		0			0
$640$	0.613159	−	2.15036i	0.0242372	−	0.0850003i
$641$		−	24.8898i		−	0.983088i	−0.870853	−	0.491544i	$-0.836433\pi$
$641$		−	24.8898i		−	0.983088i	0.870853	−	0.491544i	$-0.163567\pi$
$642$		0			0
$643$	21.9094			0.864021			0.432011	−	0.901868i	$-0.357804\pi$
$643$	21.9094			0.864021			0.432011	+	0.901868i	$0.357804\pi$
$644$		0			0
$645$		0			0
$646$	5.02955			0.197885
$647$		−	23.1287i		−	0.909284i	−0.890674	−	0.454642i	$-0.849767\pi$
$647$		−	23.1287i		−	0.909284i	0.890674	−	0.454642i	$-0.150233\pi$
$648$		0			0
$649$			30.9717i			1.21575i
$650$	21.2004	+	13.1603i	0.831548	+	0.516189i
$651$		0			0
$652$			13.2214i			0.517789i
$653$	−23.9838			−0.938558			−0.469279	−	0.883050i	$-0.655486\pi$
$653$	−23.9838			−0.938558			−0.469279	+	0.883050i	$0.655486\pi$
$654$		0			0
$655$	−13.7759	+	48.3123i	−0.538269	+	1.88772i
$656$	−6.42959			−0.251034
$657$		0			0
$658$		0			0
$659$			30.5511i			1.19010i	0.803688	+	0.595052i	$0.202868\pi$
$659$			30.5511i			1.19010i	−0.803688	+	0.595052i	$0.797132\pi$
$660$		0			0
$661$		−	49.9771i		−	1.94389i	−0.235218	−	0.971943i	$-0.575580\pi$
$661$		−	49.9771i		−	1.94389i	0.235218	−	0.971943i	$-0.424420\pi$
$662$	5.22730			0.203165
$663$		0			0
$664$		−	9.01901i		−	0.350005i
$665$		0			0
$666$		0			0
$667$		−	16.6114i		−	0.643197i
$668$			5.89817i			0.228207i
$669$		0			0
$670$	−25.6096	−	7.30240i	−0.989386	−	0.282116i
$671$	−23.6567			−0.913256
$672$		0			0
$673$			20.0023i			0.771033i	0.922701	+	0.385517i	$0.125977\pi$
$673$			20.0023i			0.771033i	−0.922701	+	0.385517i	$0.874023\pi$
$674$		−	34.0655i		−	1.31215i
$675$		0			0
$676$	11.9060			0.457924
$677$		−	20.6494i		−	0.793622i	−0.917900	−	0.396811i	$-0.870117\pi$
$677$		−	20.6494i		−	0.793622i	0.917900	−	0.396811i	$-0.129883\pi$
$678$		0			0
$679$		0			0
$680$	9.39524	+	2.67898i	0.360291	+	0.102734i
$681$		0			0
$682$	−15.1741			−0.581045
$683$	−42.4127			−1.62288			−0.811439	−	0.584437i	$-0.801315\pi$
$683$	−42.4127			−1.62288			−0.811439	+	0.584437i	$0.801315\pi$
$684$		0			0
$685$	8.50197	−	29.8165i	0.324844	−	1.13923i
$686$		0			0
$687$		0			0
$688$		−	2.17610i		−	0.0829631i
$689$	−26.6710			−1.01608
$690$		0			0
$691$			20.9488i			0.796929i	0.917184	+	0.398464i	$0.130457\pi$
$691$			20.9488i			0.796929i	−0.917184	+	0.398464i	$0.869543\pi$
$692$		−	6.32052i		−	0.240270i
$693$		0			0
$694$	2.14401			0.0813854
$695$	−8.07408	−	2.30226i	−0.306267	−	0.0873298i
$696$		0			0
$697$		−	28.0919i		−	1.06406i
$698$		−	8.45360i		−	0.319973i
$699$		0			0
$700$		0			0
$701$		−	6.47725i		−	0.244642i	−0.992491	−	0.122321i	$-0.960966\pi$
$701$		−	6.47725i		−	0.244642i	0.992491	−	0.122321i	$-0.0390338\pi$
$702$		0			0
$703$	−0.302988			−0.0114274
$704$		−	2.77878i		−	0.104729i
$705$		0			0
$706$			9.99587i			0.376199i
$707$		0			0
$708$		0			0
$709$	−19.2508			−0.722981			−0.361490	−	0.932376i	$-0.617732\pi$
$709$	−19.2508			−0.722981			−0.361490	+	0.932376i	$0.617732\pi$
$710$	−10.7191	−	3.05648i	−0.402281	−	0.114707i
$711$		0			0
$712$	−11.4960			−0.430831
$713$		−	13.1353i		−	0.491919i
$714$		0			0
$715$	29.8207	+	8.50314i	1.11523	+	0.318000i
$716$		−	17.3662i		−	0.649007i
$717$		0			0
$718$			13.2296i			0.493726i
$719$	30.4513			1.13564			0.567821	−	0.823152i	$-0.307787\pi$
$719$	30.4513			1.13564			0.567821	+	0.823152i	$0.307787\pi$
$720$		0			0
$721$		0			0
$722$	17.6749			0.657790
$723$		0			0
$724$			4.98718i			0.185347i
$725$	−18.2108	+	29.3365i	−0.676333	+	1.08953i
$726$		0			0
$727$	26.6783			0.989444			0.494722	−	0.869051i	$-0.335270\pi$
$727$	26.6783			0.989444			0.494722	+	0.869051i	$0.335270\pi$
$728$		0			0
$729$		0			0
$730$	5.56664	−	19.5223i	0.206030	−	0.722552i
$731$	9.50772			0.351656
$732$		0			0
$733$	−36.8010			−1.35928			−0.679638	−	0.733548i	$-0.737863\pi$
$733$	−36.8010			−1.35928			−0.679638	+	0.733548i	$0.737863\pi$
$734$	14.6459			0.540591
$735$		0			0
$736$	2.40542			0.0886650
$737$	−33.0938			−1.21903
$738$		0			0
$739$	37.2928			1.37184			0.685919	−	0.727678i	$-0.259400\pi$
$739$	37.2928			1.37184			0.685919	+	0.727678i	$0.259400\pi$
$740$	−0.565984	−	0.161386i	−0.0208060	−	0.00593267i
$741$		0			0
$742$		0			0
$743$	−52.6525			−1.93163			−0.965817	−	0.259224i	$-0.916533\pi$
$743$	−52.6525			−1.93163			−0.965817	+	0.259224i	$0.916533\pi$
$744$		0			0
$745$	37.0394	+	10.5615i	1.35702	+	0.386943i
$746$			12.3430i			0.451908i
$747$		0			0
$748$	12.1409			0.443916
$749$		0			0
$750$		0			0
$751$	41.2261			1.50436			0.752182	−	0.658956i	$-0.229002\pi$
$751$	41.2261			1.50436			0.752182	+	0.658956i	$0.229002\pi$
$752$			6.93514i			0.252898i
$753$		0			0
$754$			34.4642i			1.25511i
$755$	1.51817	−	5.32424i	0.0552518	−	0.193769i
$756$		0			0
$757$		−	10.9683i		−	0.398648i	−0.979934	−	0.199324i	$-0.936125\pi$
$757$		−	10.9683i		−	0.398648i	0.979934	−	0.199324i	$-0.0638747\pi$
$758$	−14.9088			−0.541512
$759$		0			0
$760$	−2.47539	−	0.705838i	−0.0897917	−	0.0256034i
$761$	34.3263			1.24433			0.622163	−	0.782888i	$-0.286254\pi$
$761$	34.3263			1.24433			0.622163	+	0.782888i	$0.286254\pi$
$762$		0			0
$763$		0			0
$764$			10.4489i			0.378028i
$765$		0			0
$766$			29.5094i			1.06622i
$767$	55.6241			2.00847
$768$		0			0
$769$			5.17853i			0.186742i	0.995631	+	0.0933712i	$0.0297643\pi$
$769$			5.17853i			0.186742i	−0.995631	+	0.0933712i	$0.970236\pi$
$770$		0			0
$771$		0			0
$772$		−	11.3765i		−	0.409449i
$773$		−	16.5304i		−	0.594556i	−0.954791	−	0.297278i	$-0.903921\pi$
$773$		−	16.5304i		−	0.594556i	0.954791	−	0.297278i	$-0.0960787\pi$
$774$		0			0
$775$	−14.3999	+	23.1974i	−0.517261	+	0.833275i
$776$	1.18281			0.0424605
$777$		0			0
$778$		−	4.24434i		−	0.152167i
$779$			7.40143i			0.265184i
$780$		0			0
$781$	−13.8517			−0.495652
$782$			10.5097i			0.375824i
$783$		0			0
$784$		0			0
$785$	−7.61492	+	26.7057i	−0.271788	+	0.953166i
$786$		0			0
$787$	−31.8222			−1.13434			−0.567169	−	0.823601i	$-0.691961\pi$
$787$	−31.8222			−1.13434			−0.567169	+	0.823601i	$0.691961\pi$
$788$	−9.34972			−0.333070
$789$		0			0
$790$	−10.5900	+	37.1394i	−0.376776	+	1.32136i
$791$		0			0
$792$		0			0
$793$			42.4866i			1.50874i
$794$	20.0023			0.709854
$795$		0			0
$796$		−	16.1828i		−	0.573583i
$797$			6.73189i			0.238456i	0.992867	+	0.119228i	$0.0380419\pi$
$797$			6.73189i			0.238456i	−0.992867	+	0.119228i	$0.961958\pi$
$798$		0			0
$799$	−30.3007			−1.07196
$800$	−4.24807	−	2.63702i	−0.150192	−	0.0932327i
$801$		0			0
$802$		−	30.0841i		−	1.06231i
$803$		−	25.2275i		−	0.890260i
$804$		0			0
$805$		0			0
$806$			27.2521i			0.959913i
$807$		0			0
$808$	2.06376			0.0726028
$809$			42.1400i			1.48156i	0.671746	+	0.740782i	$0.265545\pi$
$809$			42.1400i			1.48156i	−0.671746	+	0.740782i	$0.734455\pi$
$810$		0			0
$811$		−	23.8570i		−	0.837732i	−0.908048	−	0.418866i	$-0.862428\pi$
$811$		−	23.8570i		−	0.837732i	0.908048	−	0.418866i	$-0.137572\pi$
$812$		0			0
$813$		0			0
$814$	−0.731388			−0.0256351
$815$	28.4307	+	8.10679i	0.995883	+	0.283969i
$816$		0			0
$817$	−2.50502			−0.0876396
$818$		−	33.8806i		−	1.18461i
$819$		0			0
$820$	−3.94236	+	13.8259i	−0.137673	+	0.482822i
$821$		−	19.0413i		−	0.664546i	−0.943183	−	0.332273i	$-0.892185\pi$
$821$		−	19.0413i		−	0.664546i	0.943183	−	0.332273i	$-0.107815\pi$
$822$		0			0
$823$			34.5413i			1.20404i	0.798483	+	0.602018i	$0.205636\pi$
$823$			34.5413i			1.20404i	−0.798483	+	0.602018i	$0.794364\pi$
$824$	−0.730836			−0.0254599
$825$		0			0
$826$		0			0
$827$	22.7399			0.790744			0.395372	−	0.918521i	$-0.370616\pi$
$827$	22.7399			0.790744			0.395372	+	0.918521i	$0.370616\pi$
$828$		0			0
$829$			56.9916i			1.97940i	0.143151	+	0.989701i	$0.454277\pi$
$829$			56.9916i			1.97940i	−0.143151	+	0.989701i	$0.545723\pi$
$830$	−19.3941	−	5.53008i	−0.673179	−	0.191952i
$831$		0			0
$832$	−4.99059			−0.173018
$833$		0			0
$834$		0			0
$835$	12.6832	+	3.61651i	0.438919	+	0.125155i
$836$	−3.19880			−0.110633
$837$		0			0
$838$	19.5576			0.675606
$839$	−7.83951			−0.270650			−0.135325	−	0.990801i	$-0.543208\pi$
$839$	−7.83951			−0.270650			−0.135325	+	0.990801i	$0.543208\pi$
$840$		0			0
$841$	−18.6905			−0.644500
$842$	−1.14027			−0.0392964
$843$		0			0
$844$	25.6010			0.881224
$845$	7.30028	−	25.6022i	0.251137	−	0.880743i
$846$		0			0
$847$		0			0
$848$	5.34425			0.183522
$849$		0			0
$850$	11.5215	−	18.5605i	0.395186	−	0.636619i
$851$		−	0.633118i		−	0.0217030i
$852$		0			0
$853$	−21.5981			−0.739504			−0.369752	−	0.929131i	$-0.620557\pi$
$853$	−21.5981			−0.739504			−0.369752	+	0.929131i	$0.620557\pi$
$854$		0			0
$855$		0			0
$856$	9.98628			0.341324
$857$		−	21.7490i		−	0.742931i	−0.928447	−	0.371466i	$-0.878855\pi$
$857$		−	21.7490i		−	0.742931i	0.928447	−	0.371466i	$-0.121145\pi$
$858$		0			0
$859$			20.0985i			0.685751i	0.939381	+	0.342875i	$0.111401\pi$
$859$			20.0985i			0.685751i	−0.939381	+	0.342875i	$0.888599\pi$
$860$	−4.67940	−	1.33430i	−0.159566	−	0.0454991i
$861$		0			0
$862$			16.2043i			0.551922i
$863$	16.1578			0.550017			0.275008	−	0.961442i	$-0.411319\pi$
$863$	16.1578			0.550017			0.275008	+	0.961442i	$0.411319\pi$
$864$		0			0
$865$	−13.5914	−	3.87548i	−0.462121	−	0.131770i
$866$	−12.5614			−0.426854
$867$		0			0
$868$		0			0
$869$			47.9931i			1.62805i
$870$		0			0
$871$			59.4353i			2.01389i
$872$	9.34460			0.316448
$873$		0			0
$874$		−	2.76900i		−	0.0936629i
$875$		0			0
$876$		0			0
$877$		−	8.88215i		−	0.299929i	−0.988691	−	0.149964i	$-0.952084\pi$
$877$		−	8.88215i		−	0.299929i	0.988691	−	0.149964i	$-0.0479159\pi$
$878$		−	36.7462i		−	1.24013i
$879$		0			0
$880$	−5.97537	−	1.70383i	−0.201430	−	0.0574362i
$881$	26.2658			0.884918			0.442459	−	0.896789i	$-0.354106\pi$
$881$	26.2658			0.884918			0.442459	+	0.896789i	$0.354106\pi$
$882$		0			0
$883$			55.5073i			1.86797i	0.357312	+	0.933985i	$0.383693\pi$
$883$			55.5073i			1.86797i	−0.357312	+	0.933985i	$0.616307\pi$
$884$		−	21.8047i		−	0.733370i
$885$		0			0
$886$	−24.0144			−0.806779
$887$			20.4941i			0.688126i	0.938947	+	0.344063i	$0.111803\pi$
$887$			20.4941i			0.688126i	−0.938947	+	0.344063i	$0.888197\pi$
$888$		0			0
$889$		0			0
$890$	−7.04887	+	24.7205i	−0.236279	+	0.828633i
$891$		0			0
$892$	4.98924			0.167052
$893$	7.98339			0.267154
$894$		0			0
$895$	−37.3436	−	10.6483i	−1.24826	−	0.355932i
$896$		0			0
$897$		0			0
$898$			8.98128i			0.299709i
$899$	−37.7106			−1.25772
$900$		0			0
$901$			23.3498i			0.777896i
$902$			17.8664i			0.594887i
$903$		0			0
$904$	−13.8803			−0.461653
$905$	10.7242	+	3.05793i	0.356485	+	0.101649i
$906$		0			0
$907$		−	22.8967i		−	0.760274i	−0.924930	−	0.380137i	$-0.875877\pi$
$907$		−	22.8967i		−	0.760274i	0.924930	−	0.380137i	$-0.124123\pi$
$908$		−	10.2580i		−	0.340425i
$909$		0			0
$910$		0			0
$911$			34.3522i			1.13814i	0.822289	+	0.569070i	$0.192697\pi$
$911$			34.3522i			1.13814i	−0.822289	+	0.569070i	$0.807303\pi$
$912$		0			0
$913$	−25.0619			−0.829426
$914$			27.8745i			0.922005i
$915$		0			0
$916$		−	0.506318i		−	0.0167292i
$917$		0			0
$918$		0			0
$919$	−44.1328			−1.45581			−0.727903	−	0.685680i	$-0.759505\pi$
$919$	−44.1328			−1.45581			−0.727903	+	0.685680i	$0.759505\pi$
$920$	1.47491	−	5.17252i	0.0486262	−	0.170533i
$921$		0			0
$922$	10.3589			0.341152
$923$			24.8771i			0.818841i
$924$		0			0
$925$	−0.694076	+	1.11811i	−0.0228211	+	0.0367633i
$926$			5.95243i			0.195609i
$927$		0			0
$928$		−	6.90583i		−	0.226695i
$929$	−6.70019			−0.219826			−0.109913	−	0.993941i	$-0.535057\pi$
$929$	−6.70019			−0.219826			−0.109913	+	0.993941i	$0.535057\pi$
$930$		0			0
$931$		0			0
$932$	4.50799			0.147664
$933$		0			0
$934$			26.0215i			0.851449i
$935$	7.44431	−	26.1073i	0.243455	−	0.853801i
$936$		0			0
$937$	26.7610			0.874243			0.437122	−	0.899402i	$-0.355998\pi$
$937$	26.7610			0.874243			0.437122	+	0.899402i	$0.355998\pi$
$938$		0			0
$939$		0			0
$940$	14.9130	+	4.25234i	0.486409	+	0.138696i
$941$	24.3001			0.792160			0.396080	−	0.918216i	$-0.370370\pi$
$941$	24.3001			0.792160			0.396080	+	0.918216i	$0.370370\pi$
$942$		0			0
$943$	−15.4659			−0.503639
$944$	−11.1458			−0.362765
$945$		0			0
$946$	−6.04691			−0.196602
$947$	28.7965			0.935760			0.467880	−	0.883792i	$-0.345018\pi$
$947$	28.7965			0.935760			0.467880	+	0.883792i	$0.345018\pi$
$948$		0			0
$949$	−45.3077			−1.47075
$950$	−3.03561	+	4.89017i	−0.0984881	+	0.158658i
$951$		0			0
$952$		0			0
$953$	11.1030			0.359661			0.179830	−	0.983698i	$-0.442445\pi$
$953$	11.1030			0.359661			0.179830	+	0.983698i	$0.442445\pi$
$954$		0			0
$955$	22.4689	+	6.40683i	0.727075	+	0.207320i
$956$			6.22961i			0.201480i
$957$		0			0
$958$	−1.59497			−0.0515311
$959$		0			0
$960$		0			0
$961$	1.18090			0.0380934
$962$			1.31355i			0.0423505i
$963$		0			0
$964$			15.8611i			0.510853i
$965$	−24.4636	−	6.97560i	−0.787510	−	0.224553i
$966$		0			0
$967$			45.1712i			1.45261i	0.687374	+	0.726304i	$0.258763\pi$
$967$			45.1712i			1.45261i	−0.687374	+	0.726304i	$0.741237\pi$
$968$	3.27837			0.105371
$969$		0			0
$970$	0.725253	−	2.54347i	0.0232865	−	0.0816660i
$971$	47.0341			1.50940			0.754698	−	0.656073i	$-0.227784\pi$
$971$	47.0341			1.50940			0.754698	+	0.656073i	$0.227784\pi$
$972$		0			0
$973$		0			0
$974$		−	41.4338i		−	1.32763i
$975$		0			0
$976$		−	8.51333i		−	0.272505i
$977$	51.1929			1.63781			0.818904	−	0.573931i	$-0.194582\pi$
$977$	51.1929			1.63781			0.818904	+	0.573931i	$0.194582\pi$
$978$		0			0
$979$			31.9449i			1.02096i
$980$		0			0
$981$		0			0
$982$			20.3769i			0.650252i
$983$		−	1.89193i		−	0.0603432i	−0.999545	−	0.0301716i	$-0.990395\pi$
$983$		−	1.89193i		−	0.0603432i	0.999545	−	0.0301716i	$-0.00960538\pi$
$984$		0			0
$985$	−5.73286	+	20.1052i	−0.182664	+	0.640606i
$986$	30.1726			0.960892
$987$		0			0
$988$			5.74493i			0.182770i
$989$		−	5.23444i		−	0.166446i
$990$		0			0
$991$	−20.8992			−0.663886			−0.331943	−	0.943299i	$-0.607704\pi$
$991$	−20.8992			−0.663886			−0.331943	+	0.943299i	$0.607704\pi$
$992$		−	5.46069i		−	0.173377i
$993$		0			0
$994$		0			0
$995$	−34.7987	−	9.92260i	−1.10319	−	0.314567i
$996$		0			0
$997$	−60.9512			−1.93034			−0.965172	−	0.261617i	$-0.915744\pi$
$997$	−60.9512			−1.93034			−0.965172	+	0.261617i	$0.915744\pi$
$998$	32.1885			1.01891
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.d.d.4409.15	yes	24
3.2	odd	2		4410.2.d.c.4409.10	yes	24
5.4	even	2		4410.2.d.c.4409.16	yes	24
7.6	odd	2	inner	4410.2.d.d.4409.10	yes	24
15.14	odd	2	inner	4410.2.d.d.4409.9	yes	24
21.20	even	2		4410.2.d.c.4409.15	yes	24
35.34	odd	2		4410.2.d.c.4409.9	✓	24
105.104	even	2	inner	4410.2.d.d.4409.16	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.d.c.4409.9	✓	24	35.34	odd	2
4410.2.d.c.4409.10	yes	24	3.2	odd	2
4410.2.d.c.4409.15	yes	24	21.20	even	2
4410.2.d.c.4409.16	yes	24	5.4	even	2
4410.2.d.d.4409.9	yes	24	15.14	odd	2	inner
4410.2.d.d.4409.10	yes	24	7.6	odd	2	inner
4410.2.d.d.4409.15	yes	24	1.1	even	1	trivial
4410.2.d.d.4409.16	yes	24	105.104	even	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(0.613159 - 2.15036i) q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(0.613159 - 2.15036i) q^{5} +1.00000 q^{8} +(0.613159 - 2.15036i) q^{10} -2.77878i q^{11} -4.99059 q^{13} +1.00000 q^{16} +4.36915i q^{17} -1.15115i q^{19} +(0.613159 - 2.15036i) q^{20} -2.77878i q^{22} +2.40542 q^{23} +(-4.24807 - 2.63702i) q^{25} -4.99059 q^{26} -6.90583i q^{29} -5.46069i q^{31} +1.00000 q^{32} +4.36915i q^{34} -0.263205i q^{37} -1.15115i q^{38} +(0.613159 - 2.15036i) q^{40} -6.42959 q^{41} -2.17610i q^{43} -2.77878i q^{44} +2.40542 q^{46} +6.93514i q^{47} +(-4.24807 - 2.63702i) q^{50} -4.99059 q^{52} +5.34425 q^{53} +(-5.97537 - 1.70383i) q^{55} -6.90583i q^{58} -11.1458 q^{59} -8.51333i q^{61} -5.46069i q^{62} +1.00000 q^{64} +(-3.06003 + 10.7316i) q^{65} -11.9095i q^{67} +4.36915i q^{68} -4.98480i q^{71} +9.07862 q^{73} -0.263205i q^{74} -1.15115i q^{76} -17.2713 q^{79} +(0.613159 - 2.15036i) q^{80} -6.42959 q^{82} -9.01901i q^{83} +(9.39524 + 2.67898i) q^{85} -2.17610i q^{86} -2.77878i q^{88} -11.4960 q^{89} +2.40542 q^{92} +6.93514i q^{94} +(-2.47539 - 0.705838i) q^{95} +1.18281 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{8}+O(q^{10}) \) 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

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