LMFDB - Embedded newform 4410.2.b.b.881.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("4410.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.b.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} -1.00000i q^{10} +1.53225i q^{11} +1.48236i q^{13} +1.00000 q^{16} +2.42883 q^{17} -4.86370i q^{19} +1.00000 q^{20} -1.53225 q^{22} +0.267949i q^{23} +1.00000 q^{25} -1.48236 q^{26} +0.898979i q^{29} -0.828427i q^{31} +1.00000i q^{32} +2.42883i q^{34} -5.48236 q^{37} +4.86370 q^{38} +1.00000i q^{40} +8.76028 q^{41} +1.86370 q^{43} -1.53225i q^{44} -0.267949 q^{46} -7.45946 q^{47} +1.00000i q^{50} -1.48236i q^{52} -3.47015i q^{53} -1.53225i q^{55} -0.898979 q^{58} +6.25674 q^{59} -6.62158i q^{61} +0.828427 q^{62} -1.00000 q^{64} -1.48236i q^{65} +16.0340 q^{67} -2.42883 q^{68} +12.7627i q^{71} +0.343146i q^{73} -5.48236i q^{74} +4.86370i q^{76} -10.4488 q^{79} -1.00000 q^{80} +8.76028i q^{82} +5.45001 q^{83} -2.42883 q^{85} +1.86370i q^{86} +1.53225 q^{88} +15.9700 q^{89} -0.267949i q^{92} -7.45946i q^{94} +4.86370i q^{95} -14.9481i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^4 - 8 * q^5	$ 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} + 8 q^{25} - 16 q^{26} - 48 q^{37} + 8 q^{38} + 32 q^{41} - 16 q^{43} - 16 q^{46} + 16 q^{47} + 32 q^{58} + 48 q^{59} - 16 q^{62} - 8 q^{64} + 48 q^{67} + 48 q^{79} - 8 q^{80} - 16 q^{83} + 32 q^{89}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 + 8 * q^25 - 16 * q^26 - 48 * q^37 + 8 * q^38 + 32 * q^41 - 16 * q^43 - 16 * q^46 + 16 * q^47 + 32 * q^58 + 48 * q^59 - 16 * q^62 - 8 * q^64 + 48 * q^67 + 48 * q^79 - 8 * q^80 - 16 * q^83 + 32 * q^89

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	− 1.00000i	− 0.316228i
$11$	1.53225i	0.461991i	0.972955	+	0.230995i	$0.0741982\pi$
$11$	1.53225i	0.461991i	−0.972955	+	0.230995i	$0.925802\pi$
$12$	0	0
$13$	1.48236i	0.411133i	0.978643	+	0.205567i	$0.0659037\pi$
$13$	1.48236i	0.411133i	−0.978643	+	0.205567i	$0.934096\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.42883	0.589077	0.294538	−	0.955640i	$-0.404834\pi$
$17$	2.42883	0.589077	0.294538	+	0.955640i	$0.404834\pi$
$18$	0	0
$19$	− 4.86370i	− 1.11581i	−0.829905	−	0.557905i	$-0.811605\pi$
$19$	− 4.86370i	− 1.11581i	0.829905	−	0.557905i	$-0.188395\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.53225	−0.326677
$23$	0.267949i	0.0558713i	0.999610	+	0.0279356i	$0.00889335\pi$
$23$	0.267949i	0.0558713i	−0.999610	+	0.0279356i	$0.991107\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.48236	−0.290715
$27$	0	0
$28$	0	0
$29$	0.898979i	0.166936i	0.996510	+	0.0834681i	$0.0265997\pi$
$29$	0.898979i	0.166936i	−0.996510	+	0.0834681i	$0.973400\pi$
$30$	0	0
$31$	− 0.828427i	− 0.148790i	−0.997229	−	0.0743950i	$-0.976297\pi$
$31$	− 0.828427i	− 0.148790i	0.997229	−	0.0743950i	$-0.0237025\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	2.42883i	0.416540i
$35$	0	0
$36$	0	0
$37$	−5.48236	−0.901295	−0.450647	−	0.892702i	$-0.648807\pi$
$37$	−5.48236	−0.901295	−0.450647	+	0.892702i	$0.648807\pi$
$38$	4.86370	0.788997
$39$	0	0
$40$	1.00000i	0.158114i
$41$	8.76028	1.36813	0.684063	−	0.729423i	$-0.260211\pi$
$41$	8.76028	1.36813	0.684063	+	0.729423i	$0.260211\pi$
$42$	0	0
$43$	1.86370	0.284212	0.142106	−	0.989851i	$-0.454613\pi$
$43$	1.86370	0.284212	0.142106	+	0.989851i	$0.454613\pi$
$44$	− 1.53225i	− 0.230995i
$45$	0	0
$46$	−0.267949	−0.0395070
$47$	−7.45946	−1.08807	−0.544037	−	0.839061i	$-0.683105\pi$
$47$	−7.45946	−1.08807	−0.544037	+	0.839061i	$0.683105\pi$
$48$	0	0
$49$	0	0
$50$	1.00000i	0.141421i
$51$	0	0
$52$	− 1.48236i	− 0.205567i
$53$	− 3.47015i	− 0.476662i	−0.971184	−	0.238331i	$-0.923400\pi$
$53$	− 3.47015i	− 0.476662i	0.971184	−	0.238331i	$-0.0766004\pi$
$54$	0	0
$55$	− 1.53225i	− 0.206609i
$56$	0	0
$57$	0	0
$58$	−0.898979	−0.118042
$59$	6.25674	0.814558	0.407279	−	0.913304i	$-0.366478\pi$
$59$	6.25674	0.814558	0.407279	+	0.913304i	$0.366478\pi$
$60$	0	0
$61$	− 6.62158i	− 0.847806i	−0.905708	−	0.423903i	$-0.860660\pi$
$61$	− 6.62158i	− 0.847806i	0.905708	−	0.423903i	$-0.139340\pi$
$62$	0.828427	0.105210
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 1.48236i	− 0.183864i
$66$	0	0
$67$	16.0340	1.95887	0.979434	−	0.201763i	$-0.0646671\pi$
$67$	16.0340	1.95887	0.979434	+	0.201763i	$0.0646671\pi$
$68$	−2.42883	−0.294538
$69$	0	0
$70$	0	0
$71$	12.7627i	1.51465i	0.653037	+	0.757326i	$0.273495\pi$
$71$	12.7627i	1.51465i	−0.653037	+	0.757326i	$0.726505\pi$
$72$	0	0
$73$	0.343146i	0.0401622i	0.999798	+	0.0200811i	$0.00639244\pi$
$73$	0.343146i	0.0401622i	−0.999798	+	0.0200811i	$0.993608\pi$
$74$	− 5.48236i	− 0.637312i
$75$	0	0
$76$	4.86370i	0.557905i
$77$	0	0
$78$	0	0
$79$	−10.4488	−1.17558	−0.587789	−	0.809014i	$-0.700001\pi$
$79$	−10.4488	−1.17558	−0.587789	+	0.809014i	$0.700001\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	8.76028i	0.967411i
$83$	5.45001	0.598216	0.299108	−	0.954219i	$-0.403311\pi$
$83$	5.45001	0.598216	0.299108	+	0.954219i	$0.403311\pi$
$84$	0	0
$85$	−2.42883	−0.263443
$86$	1.86370i	0.200968i
$87$	0	0
$88$	1.53225	0.163338
$89$	15.9700	1.69282	0.846411	−	0.532531i	$-0.178759\pi$
$89$	15.9700	1.69282	0.846411	+	0.532531i	$0.178759\pi$
$90$	0	0
$91$	0	0
$92$	− 0.267949i	− 0.0279356i
$93$	0	0
$94$	− 7.45946i	− 0.769384i
$95$	4.86370i	0.499005i
$96$	0	0
$97$	− 14.9481i	− 1.51775i	−0.651234	−	0.758877i	$-0.725748\pi$
$97$	− 14.9481i	− 1.51775i	0.651234	−	0.758877i	$-0.274252\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−2.73545	−0.272188	−0.136094	−	0.990696i	$-0.543455\pi$
$101$	−2.73545	−0.272188	−0.136094	+	0.990696i	$0.543455\pi$
$102$	0	0
$103$	6.17690i	0.608628i	0.952572	+	0.304314i	$0.0984272\pi$
$103$	6.17690i	0.608628i	−0.952572	+	0.304314i	$0.901573\pi$
$104$	1.48236	0.145358
$105$	0	0
$106$	3.47015	0.337051
$107$	4.56993i	0.441792i	0.975297	+	0.220896i	$0.0708981\pi$
$107$	4.56993i	0.441792i	−0.975297	+	0.220896i	$0.929102\pi$
$108$	0	0
$109$	−5.95867	−0.570737	−0.285369	−	0.958418i	$-0.592116\pi$
$109$	−5.95867	−0.570737	−0.285369	+	0.958418i	$0.592116\pi$
$110$	1.53225	0.146094
$111$	0	0
$112$	0	0
$113$	19.8977i	1.87182i	0.352237	+	0.935911i	$0.385421\pi$
$113$	19.8977i	1.87182i	−0.352237	+	0.935911i	$0.614579\pi$
$114$	0	0
$115$	− 0.267949i	− 0.0249864i
$116$	− 0.898979i	− 0.0834681i
$117$	0	0
$118$	6.25674i	0.575979i
$119$	0	0
$120$	0	0
$121$	8.65221	0.786565
$122$	6.62158	0.599490
$123$	0	0
$124$	0.828427i	0.0743950i
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−21.2025	−1.88142	−0.940708	−	0.339219i	$-0.889837\pi$
$127$	−21.2025	−1.88142	−0.940708	+	0.339219i	$0.889837\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	1.48236	0.130012
$131$	7.46170	0.651932	0.325966	−	0.945382i	$-0.394311\pi$
$131$	7.46170	0.651932	0.325966	+	0.945382i	$0.394311\pi$
$132$	0	0
$133$	0	0
$134$	16.0340i	1.38513i
$135$	0	0
$136$	− 2.42883i	− 0.208270i
$137$	0.621578i	0.0531050i	0.999647	+	0.0265525i	$0.00845292\pi$
$137$	0.621578i	0.0531050i	−0.999647	+	0.0265525i	$0.991547\pi$
$138$	0	0
$139$	18.5334i	1.57198i	0.618237	+	0.785992i	$0.287847\pi$
$139$	18.5334i	1.57198i	−0.618237	+	0.785992i	$0.712153\pi$
$140$	0	0
$141$	0	0
$142$	−12.7627	−1.07102
$143$	−2.27135	−0.189940
$144$	0	0
$145$	− 0.898979i	− 0.0746562i
$146$	−0.343146	−0.0283989
$147$	0	0
$148$	5.48236	0.450647
$149$	13.9834i	1.14557i	0.819707	+	0.572783i	$0.194136\pi$
$149$	13.9834i	1.14557i	−0.819707	+	0.572783i	$0.805864\pi$
$150$	0	0
$151$	19.6768	1.60127	0.800637	−	0.599149i	$-0.204494\pi$
$151$	19.6768	1.60127	0.800637	+	0.599149i	$0.204494\pi$
$152$	−4.86370	−0.394498
$153$	0	0
$154$	0	0
$155$	0.828427i	0.0665409i
$156$	0	0
$157$	− 9.06010i	− 0.723075i	−0.932358	−	0.361538i	$-0.882252\pi$
$157$	− 9.06010i	− 0.723075i	0.932358	−	0.361538i	$-0.117748\pi$
$158$	− 10.4488i	− 0.831259i
$159$	0	0
$160$	− 1.00000i	− 0.0790569i
$161$	0	0
$162$	0	0
$163$	11.8204	0.925844	0.462922	−	0.886399i	$-0.346801\pi$
$163$	11.8204	0.925844	0.462922	+	0.886399i	$0.346801\pi$
$164$	−8.76028	−0.684063
$165$	0	0
$166$	5.45001i	0.423002i
$167$	15.7778	1.22092	0.610462	−	0.792046i	$-0.290984\pi$
$167$	15.7778	1.22092	0.610462	+	0.792046i	$0.290984\pi$
$168$	0	0
$169$	10.8026	0.830969
$170$	− 2.42883i	− 0.186282i
$171$	0	0
$172$	−1.86370	−0.142106
$173$	20.2221	1.53746	0.768730	−	0.639573i	$-0.220889\pi$
$173$	20.2221	1.53746	0.768730	+	0.639573i	$0.220889\pi$
$174$	0	0
$175$	0	0
$176$	1.53225i	0.115498i
$177$	0	0
$178$	15.9700i	1.19701i
$179$	− 6.86662i	− 0.513235i	−0.966513	−	0.256618i	$-0.917392\pi$
$179$	− 6.86662i	− 0.513235i	0.966513	−	0.256618i	$-0.0826081\pi$
$180$	0	0
$181$	16.3066i	1.21206i	0.795441	+	0.606031i	$0.207239\pi$
$181$	16.3066i	1.21206i	−0.795441	+	0.606031i	$0.792761\pi$
$182$	0	0
$183$	0	0
$184$	0.267949	0.0197535
$185$	5.48236	0.403071
$186$	0	0
$187$	3.72157i	0.272148i
$188$	7.45946	0.544037
$189$	0	0
$190$	−4.86370	−0.352850
$191$	17.1623i	1.24182i	0.783882	+	0.620910i	$0.213237\pi$
$191$	17.1623i	1.24182i	−0.783882	+	0.620910i	$0.786763\pi$
$192$	0	0
$193$	9.05521	0.651808	0.325904	−	0.945403i	$-0.394331\pi$
$193$	9.05521	0.651808	0.325904	+	0.945403i	$0.394331\pi$
$194$	14.9481	1.07321
$195$	0	0
$196$	0	0
$197$	21.7379i	1.54876i	0.632720	+	0.774380i	$0.281938\pi$
$197$	21.7379i	1.54876i	−0.632720	+	0.774380i	$0.718062\pi$
$198$	0	0
$199$	− 8.32656i	− 0.590254i	−0.955458	−	0.295127i	$-0.904638\pi$
$199$	− 8.32656i	− 0.590254i	0.955458	−	0.295127i	$-0.0953620\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	− 2.73545i	− 0.192466i
$203$	0	0
$204$	0	0
$205$	−8.76028	−0.611844
$206$	−6.17690	−0.430365
$207$	0	0
$208$	1.48236i	0.102783i
$209$	7.45241	0.515494
$210$	0	0
$211$	−19.9330	−1.37225	−0.686123	−	0.727486i	$-0.740689\pi$
$211$	−19.9330	−1.37225	−0.686123	+	0.727486i	$0.740689\pi$
$212$	3.47015i	0.238331i
$213$	0	0
$214$	−4.56993	−0.312394
$215$	−1.86370	−0.127104
$216$	0	0
$217$	0	0
$218$	− 5.95867i	− 0.403572i
$219$	0	0
$220$	1.53225i	0.103304i
$221$	3.60040i	0.242189i
$222$	0	0
$223$	7.16604i	0.479873i	0.970789	+	0.239937i	$0.0771267\pi$
$223$	7.16604i	0.479873i	−0.970789	+	0.239937i	$0.922873\pi$
$224$	0	0
$225$	0	0
$226$	−19.8977	−1.32358
$227$	15.8544	1.05229	0.526147	−	0.850393i	$-0.323636\pi$
$227$	15.8544	1.05229	0.526147	+	0.850393i	$0.323636\pi$
$228$	0	0
$229$	− 28.2175i	− 1.86466i	−0.361603	−	0.932332i	$-0.617770\pi$
$229$	− 28.2175i	− 1.86466i	0.361603	−	0.932332i	$-0.382230\pi$
$230$	0.267949	0.0176680
$231$	0	0
$232$	0.898979	0.0590209
$233$	24.2973i	1.59177i	0.605447	+	0.795886i	$0.292994\pi$
$233$	24.2973i	1.59177i	−0.605447	+	0.795886i	$0.707006\pi$
$234$	0	0
$235$	7.45946	0.486601
$236$	−6.25674	−0.407279
$237$	0	0
$238$	0	0
$239$	− 19.9081i	− 1.28774i	−0.765133	−	0.643872i	$-0.777327\pi$
$239$	− 19.9081i	− 1.28774i	0.765133	−	0.643872i	$-0.222673\pi$
$240$	0	0
$241$	20.5254i	1.32216i	0.750318	+	0.661078i	$0.229901\pi$
$241$	20.5254i	1.32216i	−0.750318	+	0.661078i	$0.770099\pi$
$242$	8.65221i	0.556185i
$243$	0	0
$244$	6.62158i	0.423903i
$245$	0	0
$246$	0	0
$247$	7.20977	0.458747
$248$	−0.828427	−0.0526052
$249$	0	0
$250$	− 1.00000i	− 0.0632456i
$251$	5.86787	0.370376	0.185188	−	0.982703i	$-0.440711\pi$
$251$	5.86787	0.370376	0.185188	+	0.982703i	$0.440711\pi$
$252$	0	0
$253$	−0.410565	−0.0258120
$254$	− 21.2025i	− 1.33036i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.66166	−0.477921	−0.238961	−	0.971029i	$-0.576807\pi$
$257$	−7.66166	−0.477921	−0.238961	+	0.971029i	$0.576807\pi$
$258$	0	0
$259$	0	0
$260$	1.48236i	0.0919322i
$261$	0	0
$262$	7.46170i	0.460985i
$263$	8.46286i	0.521842i	0.965360	+	0.260921i	$0.0840262\pi$
$263$	8.46286i	0.521842i	−0.965360	+	0.260921i	$0.915974\pi$
$264$	0	0
$265$	3.47015i	0.213170i
$266$	0	0
$267$	0	0
$268$	−16.0340	−0.979434
$269$	17.0431	1.03914	0.519568	−	0.854429i	$-0.326093\pi$
$269$	17.0431	1.03914	0.519568	+	0.854429i	$0.326093\pi$
$270$	0	0
$271$	10.5359i	0.640010i	0.947416	+	0.320005i	$0.103685\pi$
$271$	10.5359i	0.640010i	−0.947416	+	0.320005i	$0.896315\pi$
$272$	2.42883	0.147269
$273$	0	0
$274$	−0.621578	−0.0375509
$275$	1.53225i	0.0923981i
$276$	0	0
$277$	−8.09122	−0.486154	−0.243077	−	0.970007i	$-0.578157\pi$
$277$	−8.09122	−0.486154	−0.243077	+	0.970007i	$0.578157\pi$
$278$	−18.5334	−1.11156
$279$	0	0
$280$	0	0
$281$	− 11.1684i	− 0.666253i	−0.942882	−	0.333127i	$-0.891896\pi$
$281$	− 11.1684i	− 0.666253i	0.942882	−	0.333127i	$-0.108104\pi$
$282$	0	0
$283$	− 7.21592i	− 0.428942i	−0.976730	−	0.214471i	$-0.931197\pi$
$283$	− 7.21592i	− 0.428942i	0.976730	−	0.214471i	$-0.0688028\pi$
$284$	− 12.7627i	− 0.757326i
$285$	0	0
$286$	− 2.27135i	− 0.134308i
$287$	0	0
$288$	0	0
$289$	−11.1008	−0.652989
$290$	0.898979	0.0527899
$291$	0	0
$292$	− 0.343146i	− 0.0200811i
$293$	18.2573	1.06660	0.533300	−	0.845926i	$-0.320952\pi$
$293$	18.2573	1.06660	0.533300	+	0.845926i	$0.320952\pi$
$294$	0	0
$295$	−6.25674	−0.364281
$296$	5.48236i	0.318656i
$297$	0	0
$298$	−13.9834	−0.810037
$299$	−0.397198	−0.0229705
$300$	0	0
$301$	0	0
$302$	19.6768i	1.13227i
$303$	0	0
$304$	− 4.86370i	− 0.278953i
$305$	6.62158i	0.379150i
$306$	0	0
$307$	3.42078i	0.195234i	0.995224	+	0.0976172i	$0.0311221\pi$
$307$	3.42078i	0.195234i	−0.995224	+	0.0976172i	$0.968878\pi$
$308$	0	0
$309$	0	0
$310$	−0.828427	−0.0470515
$311$	5.69089	0.322701	0.161350	−	0.986897i	$-0.448415\pi$
$311$	5.69089	0.322701	0.161350	+	0.986897i	$0.448415\pi$
$312$	0	0
$313$	13.3467i	0.754401i	0.926132	+	0.377200i	$0.123113\pi$
$313$	13.3467i	0.754401i	−0.926132	+	0.377200i	$0.876887\pi$
$314$	9.06010	0.511291
$315$	0	0
$316$	10.4488	0.587789
$317$	12.5266i	0.703565i	0.936082	+	0.351782i	$0.114424\pi$
$317$	12.5266i	0.703565i	−0.936082	+	0.351782i	$0.885576\pi$
$318$	0	0
$319$	−1.37746	−0.0771230
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	− 11.8131i	− 0.657298i
$324$	0	0
$325$	1.48236i	0.0822266i
$326$	11.8204i	0.654670i
$327$	0	0
$328$	− 8.76028i	− 0.483705i
$329$	0	0
$330$	0	0
$331$	−1.28183	−0.0704559	−0.0352279	−	0.999379i	$-0.511216\pi$
$331$	−1.28183	−0.0704559	−0.0352279	+	0.999379i	$0.511216\pi$
$332$	−5.45001	−0.299108
$333$	0	0
$334$	15.7778i	0.863323i
$335$	−16.0340	−0.876033
$336$	0	0
$337$	−13.1058	−0.713920	−0.356960	−	0.934120i	$-0.616187\pi$
$337$	−13.1058	−0.713920	−0.356960	+	0.934120i	$0.616187\pi$
$338$	10.8026i	0.587584i
$339$	0	0
$340$	2.42883	0.131722
$341$	1.26936	0.0687396
$342$	0	0
$343$	0	0
$344$	− 1.86370i	− 0.100484i
$345$	0	0
$346$	20.2221i	1.08715i
$347$	14.2852i	0.766871i	0.923568	+	0.383436i	$0.125259\pi$
$347$	14.2852i	0.766871i	−0.923568	+	0.383436i	$0.874741\pi$
$348$	0	0
$349$	13.2713i	0.710399i	0.934791	+	0.355200i	$0.115587\pi$
$349$	13.2713i	0.710399i	−0.934791	+	0.355200i	$0.884413\pi$
$350$	0	0
$351$	0	0
$352$	−1.53225	−0.0816692
$353$	−32.6463	−1.73759	−0.868794	−	0.495173i	$-0.835105\pi$
$353$	−32.6463	−1.73759	−0.868794	+	0.495173i	$0.835105\pi$
$354$	0	0
$355$	− 12.7627i	− 0.677373i
$356$	−15.9700	−0.846411
$357$	0	0
$358$	6.86662	0.362912
$359$	− 6.24337i	− 0.329512i	−0.986334	−	0.164756i	$-0.947316\pi$
$359$	− 6.24337i	− 0.329512i	0.986334	−	0.164756i	$-0.0526837\pi$
$360$	0	0
$361$	−4.65561	−0.245032
$362$	−16.3066	−0.857057
$363$	0	0
$364$	0	0
$365$	− 0.343146i	− 0.0179611i
$366$	0	0
$367$	12.5892i	0.657152i	0.944478	+	0.328576i	$0.106569\pi$
$367$	12.5892i	0.657152i	−0.944478	+	0.328576i	$0.893431\pi$
$368$	0.267949i	0.0139678i
$369$	0	0
$370$	5.48236i	0.285014i
$371$	0	0
$372$	0	0
$373$	35.9583	1.86185	0.930924	−	0.365212i	$-0.119004\pi$
$373$	35.9583	1.86185	0.930924	+	0.365212i	$0.119004\pi$
$374$	−3.72157	−0.192438
$375$	0	0
$376$	7.45946i	0.384692i
$377$	−1.33261	−0.0686331
$378$	0	0
$379$	−11.5899	−0.595331	−0.297666	−	0.954670i	$-0.596208\pi$
$379$	−11.5899	−0.595331	−0.297666	+	0.954670i	$0.596208\pi$
$380$	− 4.86370i	− 0.249503i
$381$	0	0
$382$	−17.1623	−0.878099
$383$	−4.46750	−0.228279	−0.114139	−	0.993465i	$-0.536411\pi$
$383$	−4.46750	−0.228279	−0.114139	+	0.993465i	$0.536411\pi$
$384$	0	0
$385$	0	0
$386$	9.05521i	0.460898i
$387$	0	0
$388$	14.9481i	0.758877i
$389$	− 12.9554i	− 0.656867i	−0.944527	−	0.328433i	$-0.893479\pi$
$389$	− 12.9554i	− 0.656867i	0.944527	−	0.328433i	$-0.106521\pi$
$390$	0	0
$391$	0.650802i	0.0329125i
$392$	0	0
$393$	0	0
$394$	−21.7379	−1.09514
$395$	10.4488	0.525734
$396$	0	0
$397$	− 9.27991i	− 0.465745i	−0.972507	−	0.232873i	$-0.925187\pi$
$397$	− 9.27991i	− 0.465745i	0.972507	−	0.232873i	$-0.0748125\pi$
$398$	8.32656	0.417373
$399$	0	0
$400$	1.00000	0.0500000
$401$	7.43927i	0.371500i	0.982597	+	0.185750i	$0.0594714\pi$
$401$	7.43927i	0.371500i	−0.982597	+	0.185750i	$0.940529\pi$
$402$	0	0
$403$	1.22803	0.0611725
$404$	2.73545	0.136094
$405$	0	0
$406$	0	0
$407$	− 8.40035i	− 0.416390i
$408$	0	0
$409$	− 16.0096i	− 0.791625i	−0.918331	−	0.395812i	$-0.870463\pi$
$409$	− 16.0096i	− 0.791625i	0.918331	−	0.395812i	$-0.129537\pi$
$410$	− 8.76028i	− 0.432639i
$411$	0	0
$412$	− 6.17690i	− 0.304314i
$413$	0	0
$414$	0	0
$415$	−5.45001	−0.267530
$416$	−1.48236	−0.0726788
$417$	0	0
$418$	7.45241i	0.364509i
$419$	28.4419	1.38948	0.694738	−	0.719263i	$-0.255520\pi$
$419$	28.4419	1.38948	0.694738	+	0.719263i	$0.255520\pi$
$420$	0	0
$421$	17.8345	0.869199	0.434600	−	0.900624i	$-0.356890\pi$
$421$	17.8345	0.869199	0.434600	+	0.900624i	$0.356890\pi$
$422$	− 19.9330i	− 0.970324i
$423$	0	0
$424$	−3.47015	−0.168526
$425$	2.42883	0.117815
$426$	0	0
$427$	0	0
$428$	− 4.56993i	− 0.220896i
$429$	0	0
$430$	− 1.86370i	− 0.0898758i
$431$	− 30.8927i	− 1.48805i	−0.668152	−	0.744025i	$-0.732914\pi$
$431$	− 30.8927i	− 1.48805i	0.668152	−	0.744025i	$-0.267086\pi$
$432$	0	0
$433$	− 15.2207i	− 0.731462i	−0.930721	−	0.365731i	$-0.880819\pi$
$433$	− 15.2207i	− 0.731462i	0.930721	−	0.365731i	$-0.119181\pi$
$434$	0	0
$435$	0	0
$436$	5.95867	0.285369
$437$	1.30323	0.0623417
$438$	0	0
$439$	− 14.3246i	− 0.683674i	−0.939759	−	0.341837i	$-0.888951\pi$
$439$	− 14.3246i	− 0.683674i	0.939759	−	0.341837i	$-0.111049\pi$
$440$	−1.53225	−0.0730472
$441$	0	0
$442$	−3.60040	−0.171253
$443$	5.15748i	0.245039i	0.992466	+	0.122520i	$0.0390974\pi$
$443$	5.15748i	0.245039i	−0.992466	+	0.122520i	$0.960903\pi$
$444$	0	0
$445$	−15.9700	−0.757053
$446$	−7.16604	−0.339322
$447$	0	0
$448$	0	0
$449$	− 19.9377i	− 0.940918i	−0.882422	−	0.470459i	$-0.844088\pi$
$449$	− 19.9377i	− 0.940918i	0.882422	−	0.470459i	$-0.155912\pi$
$450$	0	0
$451$	13.4229i	0.632061i
$452$	− 19.8977i	− 0.935911i
$453$	0	0
$454$	15.8544i	0.744085i
$455$	0	0
$456$	0	0
$457$	11.3399	0.530459	0.265229	−	0.964185i	$-0.414552\pi$
$457$	11.3399	0.530459	0.265229	+	0.964185i	$0.414552\pi$
$458$	28.2175	1.31852
$459$	0	0
$460$	0.267949i	0.0124932i
$461$	2.01890	0.0940298	0.0470149	−	0.998894i	$-0.485029\pi$
$461$	2.01890	0.0940298	0.0470149	+	0.998894i	$0.485029\pi$
$462$	0	0
$463$	−27.2844	−1.26801	−0.634007	−	0.773327i	$-0.718591\pi$
$463$	−27.2844	−1.26801	−0.634007	+	0.773327i	$0.718591\pi$
$464$	0.898979i	0.0417341i
$465$	0	0
$466$	−24.2973	−1.12555
$467$	0.692130	0.0320280	0.0160140	−	0.999872i	$-0.494902\pi$
$467$	0.692130	0.0320280	0.0160140	+	0.999872i	$0.494902\pi$
$468$	0	0
$469$	0	0
$470$	7.45946i	0.344079i
$471$	0	0
$472$	− 6.25674i	− 0.287990i
$473$	2.85566i	0.131303i
$474$	0	0
$475$	− 4.86370i	− 0.223162i
$476$	0	0
$477$	0	0
$478$	19.9081	0.910573
$479$	15.9081	0.726858	0.363429	−	0.931622i	$-0.381606\pi$
$479$	15.9081	0.726858	0.363429	+	0.931622i	$0.381606\pi$
$480$	0	0
$481$	− 8.12684i	− 0.370552i
$482$	−20.5254	−0.934905
$483$	0	0
$484$	−8.65221	−0.393282
$485$	14.9481i	0.678760i
$486$	0	0
$487$	−28.7498	−1.30278	−0.651390	−	0.758743i	$-0.725814\pi$
$487$	−28.7498	−1.30278	−0.651390	+	0.758743i	$0.725814\pi$
$488$	−6.62158	−0.299745
$489$	0	0
$490$	0	0
$491$	5.45753i	0.246295i	0.992388	+	0.123148i	$0.0392988\pi$
$491$	5.45753i	0.246295i	−0.992388	+	0.123148i	$0.960701\pi$
$492$	0	0
$493$	2.18346i	0.0983383i
$494$	7.20977i	0.324383i
$495$	0	0
$496$	− 0.828427i	− 0.0371975i
$497$	0	0
$498$	0	0
$499$	−8.86296	−0.396760	−0.198380	−	0.980125i	$-0.563568\pi$
$499$	−8.86296	−0.396760	−0.198380	+	0.980125i	$0.563568\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	5.86787i	0.261896i
$503$	−9.36536	−0.417581	−0.208790	−	0.977960i	$-0.566953\pi$
$503$	−9.36536	−0.417581	−0.208790	+	0.977960i	$0.566953\pi$
$504$	0	0
$505$	2.73545	0.121726
$506$	− 0.410565i	− 0.0182518i
$507$	0	0
$508$	21.2025	0.940708
$509$	35.0328	1.55280	0.776400	−	0.630240i	$-0.217043\pi$
$509$	35.0328	1.55280	0.776400	+	0.630240i	$0.217043\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 7.66166i	− 0.337941i
$515$	− 6.17690i	− 0.272187i
$516$	0	0
$517$	− 11.4298i	− 0.502680i
$518$	0	0
$519$	0	0
$520$	−1.48236	−0.0650059
$521$	−19.9961	−0.876047	−0.438023	−	0.898964i	$-0.644321\pi$
$521$	−19.9961	−0.876047	−0.438023	+	0.898964i	$0.644321\pi$
$522$	0	0
$523$	33.5029i	1.46498i	0.680777	+	0.732491i	$0.261642\pi$
$523$	33.5029i	1.46498i	−0.680777	+	0.732491i	$0.738358\pi$
$524$	−7.46170	−0.325966
$525$	0	0
$526$	−8.46286	−0.368998
$527$	− 2.01210i	− 0.0876487i
$528$	0	0
$529$	22.9282	0.996878
$530$	−3.47015	−0.150734
$531$	0	0
$532$	0	0
$533$	12.9859i	0.562482i
$534$	0	0
$535$	− 4.56993i	− 0.197575i
$536$	− 16.0340i	− 0.692565i
$537$	0	0
$538$	17.0431i	0.734781i
$539$	0	0
$540$	0	0
$541$	37.7532	1.62314	0.811568	−	0.584258i	$-0.198614\pi$
$541$	37.7532	1.62314	0.811568	+	0.584258i	$0.198614\pi$
$542$	−10.5359	−0.452556
$543$	0	0
$544$	2.42883i	0.104135i
$545$	5.95867	0.255241
$546$	0	0
$547$	5.07130	0.216833	0.108417	−	0.994106i	$-0.465422\pi$
$547$	5.07130	0.216833	0.108417	+	0.994106i	$0.465422\pi$
$548$	− 0.621578i	− 0.0265525i
$549$	0	0
$550$	−1.53225	−0.0653354
$551$	4.37237	0.186269
$552$	0	0
$553$	0	0
$554$	− 8.09122i	− 0.343763i
$555$	0	0
$556$	− 18.5334i	− 0.785992i
$557$	− 34.3556i	− 1.45569i	−0.685740	−	0.727846i	$-0.740522\pi$
$557$	− 34.3556i	− 1.45569i	0.685740	−	0.727846i	$-0.259478\pi$
$558$	0	0
$559$	2.76268i	0.116849i
$560$	0	0
$561$	0	0
$562$	11.1684	0.471112
$563$	−47.0586	−1.98328	−0.991641	−	0.129028i	$-0.958814\pi$
$563$	−47.0586	−1.98328	−0.991641	+	0.129028i	$0.958814\pi$
$564$	0	0
$565$	− 19.8977i	− 0.837104i
$566$	7.21592	0.303308
$567$	0	0
$568$	12.7627	0.535510
$569$	− 44.1869i	− 1.85241i	−0.377019	−	0.926206i	$-0.623051\pi$
$569$	− 44.1869i	− 1.85241i	0.377019	−	0.926206i	$-0.376949\pi$
$570$	0	0
$571$	41.1608	1.72253	0.861263	−	0.508159i	$-0.169674\pi$
$571$	41.1608	1.72253	0.861263	+	0.508159i	$0.169674\pi$
$572$	2.27135	0.0949699
$573$	0	0
$574$	0	0
$575$	0.267949i	0.0111743i
$576$	0	0
$577$	14.1212i	0.587872i	0.955825	+	0.293936i	$0.0949653\pi$
$577$	14.1212i	0.587872i	−0.955825	+	0.293936i	$0.905035\pi$
$578$	− 11.1008i	− 0.461733i
$579$	0	0
$580$	0.898979i	0.0373281i
$581$	0	0
$582$	0	0
$583$	5.31714	0.220214
$584$	0.343146	0.0141995
$585$	0	0
$586$	18.2573i	0.754200i
$587$	−37.7819	−1.55942	−0.779712	−	0.626138i	$-0.784635\pi$
$587$	−37.7819	−1.55942	−0.779712	+	0.626138i	$0.784635\pi$
$588$	0	0
$589$	−4.02922	−0.166021
$590$	− 6.25674i	− 0.257586i
$591$	0	0
$592$	−5.48236	−0.225324
$593$	−26.1961	−1.07575	−0.537873	−	0.843026i	$-0.680772\pi$
$593$	−26.1961	−1.07575	−0.537873	+	0.843026i	$0.680772\pi$
$594$	0	0
$595$	0	0
$596$	− 13.9834i	− 0.572783i
$597$	0	0
$598$	− 0.397198i	− 0.0162426i
$599$	4.80023i	0.196132i	0.995180	+	0.0980661i	$0.0312657\pi$
$599$	4.80023i	0.196132i	−0.995180	+	0.0980661i	$0.968734\pi$
$600$	0	0
$601$	− 37.3722i	− 1.52444i	−0.647317	−	0.762221i	$-0.724109\pi$
$601$	− 37.3722i	− 1.52444i	0.647317	−	0.762221i	$-0.275891\pi$
$602$	0	0
$603$	0	0
$604$	−19.6768	−0.800637
$605$	−8.65221	−0.351762
$606$	0	0
$607$	− 37.2390i	− 1.51148i	−0.654869	−	0.755742i	$-0.727276\pi$
$607$	− 37.2390i	− 1.51148i	0.654869	−	0.755742i	$-0.272724\pi$
$608$	4.86370	0.197249
$609$	0	0
$610$	−6.62158	−0.268100
$611$	− 11.0576i	− 0.447343i
$612$	0	0
$613$	−9.85954	−0.398223	−0.199112	−	0.979977i	$-0.563806\pi$
$613$	−9.85954	−0.398223	−0.199112	+	0.979977i	$0.563806\pi$
$614$	−3.42078	−0.138052
$615$	0	0
$616$	0	0
$617$	31.3545i	1.26229i	0.775666	+	0.631143i	$0.217414\pi$
$617$	31.3545i	1.26229i	−0.775666	+	0.631143i	$0.782586\pi$
$618$	0	0
$619$	− 15.2355i	− 0.612366i	−0.951973	−	0.306183i	$-0.900948\pi$
$619$	− 15.2355i	− 0.612366i	0.951973	−	0.306183i	$-0.0990519\pi$
$620$	− 0.828427i	− 0.0332704i
$621$	0	0
$622$	5.69089i	0.228184i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.3467	−0.533442
$627$	0	0
$628$	9.06010i	0.361538i
$629$	−13.3157	−0.530932
$630$	0	0
$631$	−1.00406	−0.0399710	−0.0199855	−	0.999800i	$-0.506362\pi$
$631$	−1.00406	−0.0399710	−0.0199855	+	0.999800i	$0.506362\pi$
$632$	10.4488i	0.415629i
$633$	0	0
$634$	−12.5266	−0.497495
$635$	21.2025	0.841394
$636$	0	0
$637$	0	0
$638$	− 1.37746i	− 0.0545342i
$639$	0	0
$640$	1.00000i	0.0395285i
$641$	23.4045i	0.924423i	0.886770	+	0.462211i	$0.152944\pi$
$641$	23.4045i	0.924423i	−0.886770	+	0.462211i	$0.847056\pi$
$642$	0	0
$643$	− 33.4475i	− 1.31904i	−0.751686	−	0.659521i	$-0.770759\pi$
$643$	− 33.4475i	− 1.31904i	0.751686	−	0.659521i	$-0.229241\pi$
$644$	0	0
$645$	0	0
$646$	11.8131	0.464780
$647$	−16.5708	−0.651465	−0.325733	−	0.945462i	$-0.605611\pi$
$647$	−16.5708	−0.651465	−0.325733	+	0.945462i	$0.605611\pi$
$648$	0	0
$649$	9.58689i	0.376318i
$650$	−1.48236	−0.0581430
$651$	0	0
$652$	−11.8204	−0.462922
$653$	6.02906i	0.235935i	0.993017	+	0.117968i	$0.0376379\pi$
$653$	6.02906i	0.235935i	−0.993017	+	0.117968i	$0.962362\pi$
$654$	0	0
$655$	−7.46170	−0.291553
$656$	8.76028	0.342031
$657$	0	0
$658$	0	0
$659$	− 36.3672i	− 1.41666i	−0.705880	−	0.708332i	$-0.749448\pi$
$659$	− 36.3672i	− 1.41666i	0.705880	−	0.708332i	$-0.250552\pi$
$660$	0	0
$661$	− 11.4349i	− 0.444765i	−0.974959	−	0.222383i	$-0.928617\pi$
$661$	− 11.4349i	− 0.444765i	0.974959	−	0.222383i	$-0.0713834\pi$
$662$	− 1.28183i	− 0.0498198i
$663$	0	0
$664$	− 5.45001i	− 0.211501i
$665$	0	0
$666$	0	0
$667$	−0.240881	−0.00932694
$668$	−15.7778	−0.610462
$669$	0	0
$670$	− 16.0340i	− 0.619449i
$671$	10.1459	0.391679
$672$	0	0
$673$	10.8070	0.416581	0.208290	−	0.978067i	$-0.433210\pi$
$673$	10.8070	0.416581	0.208290	+	0.978067i	$0.433210\pi$
$674$	− 13.1058i	− 0.504818i
$675$	0	0
$676$	−10.8026	−0.415485
$677$	14.0456	0.539816	0.269908	−	0.962886i	$-0.413007\pi$
$677$	14.0456	0.539816	0.269908	+	0.962886i	$0.413007\pi$
$678$	0	0
$679$	0	0
$680$	2.42883i	0.0931412i
$681$	0	0
$682$	1.26936i	0.0486062i
$683$	36.4770i	1.39575i	0.716217	+	0.697877i	$0.245872\pi$
$683$	36.4770i	1.39575i	−0.716217	+	0.697877i	$0.754128\pi$
$684$	0	0
$685$	− 0.621578i	− 0.0237493i
$686$	0	0
$687$	0	0
$688$	1.86370	0.0710530
$689$	5.14402	0.195972
$690$	0	0
$691$	23.7673i	0.904150i	0.891980	+	0.452075i	$0.149316\pi$
$691$	23.7673i	0.904150i	−0.891980	+	0.452075i	$0.850684\pi$
$692$	−20.2221	−0.768730
$693$	0	0
$694$	−14.2852	−0.542260
$695$	− 18.5334i	− 0.703012i
$696$	0	0
$697$	21.2772	0.805931
$698$	−13.2713	−0.502328
$699$	0	0
$700$	0	0
$701$	1.74502i	0.0659086i	0.999457	+	0.0329543i	$0.0104916\pi$
$701$	1.74502i	0.0659086i	−0.999457	+	0.0329543i	$0.989508\pi$
$702$	0	0
$703$	26.6646i	1.00567i
$704$	− 1.53225i	− 0.0577488i
$705$	0	0
$706$	− 32.6463i	− 1.22866i
$707$	0	0
$708$	0	0
$709$	−12.1232	−0.455298	−0.227649	−	0.973743i	$-0.573104\pi$
$709$	−12.1232	−0.455298	−0.227649	+	0.973743i	$0.573104\pi$
$710$	12.7627	0.478975
$711$	0	0
$712$	− 15.9700i	− 0.598503i
$713$	0.221976	0.00831308
$714$	0	0
$715$	2.27135	0.0849436
$716$	6.86662i	0.256618i
$717$	0	0
$718$	6.24337	0.233000
$719$	−1.78635	−0.0666197	−0.0333098	−	0.999445i	$-0.510605\pi$
$719$	−1.78635	−0.0666197	−0.0333098	+	0.999445i	$0.510605\pi$
$720$	0	0
$721$	0	0
$722$	− 4.65561i	− 0.173264i
$723$	0	0
$724$	− 16.3066i	− 0.606031i
$725$	0.898979i	0.0333873i
$726$	0	0
$727$	29.8785i	1.10813i	0.832472	+	0.554066i	$0.186925\pi$
$727$	29.8785i	1.10813i	−0.832472	+	0.554066i	$0.813075\pi$
$728$	0	0
$729$	0	0
$730$	0.343146	0.0127004
$731$	4.52661	0.167423
$732$	0	0
$733$	13.7911i	0.509385i	0.967022	+	0.254692i	$0.0819742\pi$
$733$	13.7911i	0.509385i	−0.967022	+	0.254692i	$0.918026\pi$
$734$	−12.5892	−0.464677
$735$	0	0
$736$	−0.267949	−0.00987674
$737$	24.5681i	0.904979i
$738$	0	0
$739$	7.36698	0.270999	0.135499	−	0.990777i	$-0.456736\pi$
$739$	7.36698	0.270999	0.135499	+	0.990777i	$0.456736\pi$
$740$	−5.48236	−0.201536
$741$	0	0
$742$	0	0
$743$	− 11.0774i	− 0.406389i	−0.979138	−	0.203194i	$-0.934868\pi$
$743$	− 11.0774i	− 0.406389i	0.979138	−	0.203194i	$-0.0651323\pi$
$744$	0	0
$745$	− 13.9834i	− 0.512313i
$746$	35.9583i	1.31653i
$747$	0	0
$748$	− 3.72157i	− 0.136074i
$749$	0	0
$750$	0	0
$751$	24.3757	0.889483	0.444741	−	0.895659i	$-0.353296\pi$
$751$	24.3757	0.889483	0.444741	+	0.895659i	$0.353296\pi$
$752$	−7.45946	−0.272018
$753$	0	0
$754$	− 1.33261i	− 0.0485309i
$755$	−19.6768	−0.716112
$756$	0	0
$757$	19.6761	0.715139	0.357569	−	0.933887i	$-0.383606\pi$
$757$	19.6761	0.715139	0.357569	+	0.933887i	$0.383606\pi$
$758$	− 11.5899i	− 0.420963i
$759$	0	0
$760$	4.86370	0.176425
$761$	−49.8337	−1.80647	−0.903234	−	0.429147i	$-0.858814\pi$
$761$	−49.8337	−1.80647	−0.903234	+	0.429147i	$0.858814\pi$
$762$	0	0
$763$	0	0
$764$	− 17.1623i	− 0.620910i
$765$	0	0
$766$	− 4.46750i	− 0.161417i
$767$	9.27475i	0.334892i
$768$	0	0
$769$	− 31.0584i	− 1.12000i	−0.828494	−	0.559998i	$-0.810802\pi$
$769$	− 31.0584i	− 1.12000i	0.828494	−	0.559998i	$-0.189198\pi$
$770$	0	0
$771$	0	0
$772$	−9.05521	−0.325904
$773$	−3.97049	−0.142809	−0.0714043	−	0.997447i	$-0.522748\pi$
$773$	−3.97049	−0.142809	−0.0714043	+	0.997447i	$0.522748\pi$
$774$	0	0
$775$	− 0.828427i	− 0.0297580i
$776$	−14.9481	−0.536607
$777$	0	0
$778$	12.9554	0.464475
$779$	− 42.6074i	− 1.52657i
$780$	0	0
$781$	−19.5556	−0.699755
$782$	−0.650802	−0.0232726
$783$	0	0
$784$	0	0
$785$	9.06010i	0.323369i
$786$	0	0
$787$	44.8026i	1.59704i	0.601969	+	0.798519i	$0.294383\pi$
$787$	44.8026i	1.59704i	−0.601969	+	0.798519i	$0.705617\pi$
$788$	− 21.7379i	− 0.774380i
$789$	0	0
$790$	10.4488i	0.371750i
$791$	0	0
$792$	0	0
$793$	9.81558	0.348561
$794$	9.27991	0.329332
$795$	0	0
$796$	8.32656i	0.295127i
$797$	−38.4813	−1.36308	−0.681539	−	0.731781i	$-0.738689\pi$
$797$	−38.4813	−1.36308	−0.681539	+	0.731781i	$0.738689\pi$
$798$	0	0
$799$	−18.1177	−0.640959
$800$	1.00000i	0.0353553i
$801$	0	0
$802$	−7.43927	−0.262690
$803$	−0.525785	−0.0185546
$804$	0	0
$805$	0	0
$806$	1.22803i	0.0432555i
$807$	0	0
$808$	2.73545i	0.0962328i
$809$	17.3550i	0.610168i	0.952326	+	0.305084i	$0.0986845\pi$
$809$	17.3550i	0.610168i	−0.952326	+	0.305084i	$0.901316\pi$
$810$	0	0
$811$	32.7270i	1.14920i	0.818434	+	0.574601i	$0.194843\pi$
$811$	32.7270i	1.14920i	−0.818434	+	0.574601i	$0.805157\pi$
$812$	0	0
$813$	0	0
$814$	8.40035	0.294432
$815$	−11.8204	−0.414050
$816$	0	0
$817$	− 9.06450i	− 0.317127i
$818$	16.0096	0.559763
$819$	0	0
$820$	8.76028	0.305922
$821$	− 17.7932i	− 0.620985i	−0.950576	−	0.310493i	$-0.899506\pi$
$821$	− 17.7932i	− 0.620985i	0.950576	−	0.310493i	$-0.100494\pi$
$822$	0	0
$823$	9.82165	0.342361	0.171181	−	0.985240i	$-0.445242\pi$
$823$	9.82165	0.342361	0.171181	+	0.985240i	$0.445242\pi$
$824$	6.17690	0.215182
$825$	0	0
$826$	0	0
$827$	− 7.78482i	− 0.270705i	−0.990798	−	0.135352i	$-0.956783\pi$
$827$	− 7.78482i	− 0.270705i	0.990798	−	0.135352i	$-0.0432167\pi$
$828$	0	0
$829$	− 15.1283i	− 0.525426i	−0.964874	−	0.262713i	$-0.915383\pi$
$829$	− 15.1283i	− 0.525426i	0.964874	−	0.262713i	$-0.0846172\pi$
$830$	− 5.45001i	− 0.189172i
$831$	0	0
$832$	− 1.48236i	− 0.0513917i
$833$	0	0
$834$	0	0
$835$	−15.7778	−0.546014
$836$	−7.45241	−0.257747
$837$	0	0
$838$	28.4419i	0.982508i
$839$	−25.2077	−0.870265	−0.435133	−	0.900366i	$-0.643299\pi$
$839$	−25.2077	−0.870265	−0.435133	+	0.900366i	$0.643299\pi$
$840$	0	0
$841$	28.1918	0.972132
$842$	17.8345i	0.614617i
$843$	0	0
$844$	19.9330	0.686123
$845$	−10.8026	−0.371621
$846$	0	0
$847$	0	0
$848$	− 3.47015i	− 0.119166i
$849$	0	0
$850$	2.42883i	0.0833080i
$851$	− 1.46899i	− 0.0503565i
$852$	0	0
$853$	− 23.3020i	− 0.797846i	−0.916984	−	0.398923i	$-0.869384\pi$
$853$	− 23.3020i	− 0.797846i	0.916984	−	0.398923i	$-0.130616\pi$
$854$	0	0
$855$	0	0
$856$	4.56993	0.156197
$857$	26.6060	0.908842	0.454421	−	0.890787i	$-0.349846\pi$
$857$	26.6060	0.908842	0.454421	+	0.890787i	$0.349846\pi$
$858$	0	0
$859$	− 56.5854i	− 1.93067i	−0.261017	−	0.965334i	$-0.584058\pi$
$859$	− 56.5854i	− 1.93067i	0.261017	−	0.965334i	$-0.415942\pi$
$860$	1.86370	0.0635518
$861$	0	0
$862$	30.8927	1.05221
$863$	34.0566i	1.15930i	0.814865	+	0.579650i	$0.196811\pi$
$863$	34.0566i	1.15930i	−0.814865	+	0.579650i	$0.803189\pi$
$864$	0	0
$865$	−20.2221	−0.687573
$866$	15.2207	0.517222
$867$	0	0
$868$	0	0
$869$	− 16.0101i	− 0.543106i
$870$	0	0
$871$	23.7682i	0.805356i
$872$	5.95867i	0.201786i
$873$	0	0
$874$	1.30323i	0.0440823i
$875$	0	0
$876$	0	0
$877$	24.1382	0.815089	0.407545	−	0.913185i	$-0.366385\pi$
$877$	24.1382	0.815089	0.407545	+	0.913185i	$0.366385\pi$
$878$	14.3246	0.483431
$879$	0	0
$880$	− 1.53225i	− 0.0516521i
$881$	37.2609	1.25535	0.627676	−	0.778475i	$-0.284006\pi$
$881$	37.2609	1.25535	0.627676	+	0.778475i	$0.284006\pi$
$882$	0	0
$883$	48.5544	1.63398	0.816992	−	0.576648i	$-0.195640\pi$
$883$	48.5544	1.63398	0.816992	+	0.576648i	$0.195640\pi$
$884$	− 3.60040i	− 0.121094i
$885$	0	0
$886$	−5.15748	−0.173269
$887$	−17.4709	−0.586615	−0.293308	−	0.956018i	$-0.594756\pi$
$887$	−17.4709	−0.586615	−0.293308	+	0.956018i	$0.594756\pi$
$888$	0	0
$889$	0	0
$890$	− 15.9700i	− 0.535317i
$891$	0	0
$892$	− 7.16604i	− 0.239937i
$893$	36.2806i	1.21408i
$894$	0	0
$895$	6.86662i	0.229526i
$896$	0	0
$897$	0	0
$898$	19.9377	0.665329
$899$	0.744739	0.0248384
$900$	0	0
$901$	− 8.42840i	− 0.280791i
$902$	−13.4229	−0.446935
$903$	0	0
$904$	19.8977	0.661789
$905$	− 16.3066i	− 0.542051i
$906$	0	0
$907$	−21.4520	−0.712302	−0.356151	−	0.934428i	$-0.615911\pi$
$907$	−21.4520	−0.712302	−0.356151	+	0.934428i	$0.615911\pi$
$908$	−15.8544	−0.526147
$909$	0	0
$910$	0	0
$911$	− 42.2281i	− 1.39908i	−0.714593	−	0.699540i	$-0.753388\pi$
$911$	− 42.2281i	− 1.39908i	0.714593	−	0.699540i	$-0.246612\pi$
$912$	0	0
$913$	8.35077i	0.276370i
$914$	11.3399i	0.375091i
$915$	0	0
$916$	28.2175i	0.932332i
$917$	0	0
$918$	0	0
$919$	−56.1631	−1.85265	−0.926325	−	0.376726i	$-0.877050\pi$
$919$	−56.1631	−1.85265	−0.926325	+	0.376726i	$0.877050\pi$
$920$	−0.267949	−0.00883402
$921$	0	0
$922$	2.01890i	0.0664891i
$923$	−18.9189	−0.622724
$924$	0	0
$925$	−5.48236	−0.180259
$926$	− 27.2844i	− 0.896621i
$927$	0	0
$928$	−0.898979	−0.0295104
$929$	18.5388	0.608239	0.304120	−	0.952634i	$-0.401638\pi$
$929$	18.5388	0.608239	0.304120	+	0.952634i	$0.401638\pi$
$930$	0	0
$931$	0	0
$932$	− 24.2973i	− 0.795886i
$933$	0	0
$934$	0.692130i	0.0226472i
$935$	− 3.72157i	− 0.121708i
$936$	0	0
$937$	− 39.5337i	− 1.29151i	−0.763545	−	0.645755i	$-0.776543\pi$
$937$	− 39.5337i	− 1.29151i	0.763545	−	0.645755i	$-0.223457\pi$
$938$	0	0
$939$	0	0
$940$	−7.45946	−0.243301
$941$	50.4949	1.64609	0.823043	−	0.567979i	$-0.192274\pi$
$941$	50.4949	1.64609	0.823043	+	0.567979i	$0.192274\pi$
$942$	0	0
$943$	2.34731i	0.0764389i
$944$	6.25674	0.203639
$945$	0	0
$946$	−2.85566	−0.0928455
$947$	− 39.1628i	− 1.27262i	−0.771433	−	0.636310i	$-0.780460\pi$
$947$	− 39.1628i	− 1.27262i	0.771433	−	0.636310i	$-0.219540\pi$
$948$	0	0
$949$	−0.508666	−0.0165120
$950$	4.86370	0.157799
$951$	0	0
$952$	0	0
$953$	− 52.9933i	− 1.71662i	−0.513130	−	0.858311i	$-0.671514\pi$
$953$	− 52.9933i	− 1.71662i	0.513130	−	0.858311i	$-0.328486\pi$
$954$	0	0
$955$	− 17.1623i	− 0.555358i
$956$	19.9081i	0.643872i
$957$	0	0
$958$	15.9081i	0.513966i
$959$	0	0
$960$	0	0
$961$	30.3137	0.977862
$962$	8.12684	0.262020
$963$	0	0
$964$	− 20.5254i	− 0.661078i
$965$	−9.05521	−0.291498
$966$	0	0
$967$	30.6208	0.984700	0.492350	−	0.870397i	$-0.336138\pi$
$967$	30.6208	0.984700	0.492350	+	0.870397i	$0.336138\pi$
$968$	− 8.65221i	− 0.278093i
$969$	0	0
$970$	−14.9481	−0.479956
$971$	12.0710	0.387376	0.193688	−	0.981063i	$-0.437955\pi$
$971$	12.0710	0.387376	0.193688	+	0.981063i	$0.437955\pi$
$972$	0	0
$973$	0	0
$974$	− 28.7498i	− 0.921205i
$975$	0	0
$976$	− 6.62158i	− 0.211952i
$977$	11.2328i	0.359370i	0.983724	+	0.179685i	$0.0575079\pi$
$977$	11.2328i	0.359370i	−0.983724	+	0.179685i	$0.942492\pi$
$978$	0	0
$979$	24.4701i	0.782068i
$980$	0	0
$981$	0	0
$982$	−5.45753	−0.174157
$983$	−22.7921	−0.726954	−0.363477	−	0.931603i	$-0.618410\pi$
$983$	−22.7921	−0.726954	−0.363477	+	0.931603i	$0.618410\pi$
$984$	0	0
$985$	− 21.7379i	− 0.692627i
$986$	−2.18346	−0.0695357
$987$	0	0
$988$	−7.20977	−0.229373
$989$	0.499378i	0.0158793i
$990$	0	0
$991$	−8.82113	−0.280212	−0.140106	−	0.990136i	$-0.544744\pi$
$991$	−8.82113	−0.280212	−0.140106	+	0.990136i	$0.544744\pi$
$992$	0.828427	0.0263026
$993$	0	0
$994$	0	0
$995$	8.32656i	0.263970i
$996$	0	0
$997$	− 46.4763i	− 1.47192i	−0.677026	−	0.735960i	$-0.736731\pi$
$997$	− 46.4763i	− 1.47192i	0.677026	−	0.735960i	$-0.263269\pi$
$998$	− 8.86296i	− 0.280552i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.b.881.7		8
3.2	odd	2		4410.2.b.e.881.2		8
7.4	even	3		630.2.be.b.341.2	yes	8
7.5	odd	6		630.2.be.a.521.4	yes	8
7.6	odd	2		4410.2.b.e.881.7		8
21.5	even	6		630.2.be.b.521.2	yes	8
21.11	odd	6		630.2.be.a.341.4	✓	8
21.20	even	2	inner	4410.2.b.b.881.2		8
35.4	even	6		3150.2.bf.c.1601.3		8
35.12	even	12		3150.2.bp.d.899.3		8
35.18	odd	12		3150.2.bp.c.1349.3		8
35.19	odd	6		3150.2.bf.b.1151.1		8
35.32	odd	12		3150.2.bp.f.1349.2		8
35.33	even	12		3150.2.bp.a.899.2		8
105.32	even	12		3150.2.bp.a.1349.2		8
105.47	odd	12		3150.2.bp.c.899.3		8
105.53	even	12		3150.2.bp.d.1349.3		8
105.68	odd	12		3150.2.bp.f.899.2		8
105.74	odd	6		3150.2.bf.b.1601.1		8
105.89	even	6		3150.2.bf.c.1151.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.be.a.341.4	✓	8	21.11	odd	6
630.2.be.a.521.4	yes	8	7.5	odd	6
630.2.be.b.341.2	yes	8	7.4	even	3
630.2.be.b.521.2	yes	8	21.5	even	6
3150.2.bf.b.1151.1		8	35.19	odd	6
3150.2.bf.b.1601.1		8	105.74	odd	6
3150.2.bf.c.1151.3		8	105.89	even	6
3150.2.bf.c.1601.3		8	35.4	even	6
3150.2.bp.a.899.2		8	35.33	even	12
3150.2.bp.a.1349.2		8	105.32	even	12
3150.2.bp.c.899.3		8	105.47	odd	12
3150.2.bp.c.1349.3		8	35.18	odd	12
3150.2.bp.d.899.3		8	35.12	even	12
3150.2.bp.d.1349.3		8	105.53	even	12
3150.2.bp.f.899.2		8	105.68	odd	12
3150.2.bp.f.1349.2		8	35.32	odd	12
4410.2.b.b.881.2		8	21.20	even	2	inner
4410.2.b.b.881.7		8	1.1	even	1	trivial
4410.2.b.e.881.2		8	3.2	odd	2
4410.2.b.e.881.7		8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} -1.00000i q^{10} +1.53225i q^{11} +1.48236i q^{13} +1.00000 q^{16} +2.42883 q^{17} -4.86370i q^{19} +1.00000 q^{20} -1.53225 q^{22} +0.267949i q^{23} +1.00000 q^{25} -1.48236 q^{26} +0.898979i q^{29} -0.828427i q^{31} +1.00000i q^{32} +2.42883i q^{34} -5.48236 q^{37} +4.86370 q^{38} +1.00000i q^{40} +8.76028 q^{41} +1.86370 q^{43} -1.53225i q^{44} -0.267949 q^{46} -7.45946 q^{47} +1.00000i q^{50} -1.48236i q^{52} -3.47015i q^{53} -1.53225i q^{55} -0.898979 q^{58} +6.25674 q^{59} -6.62158i q^{61} +0.828427 q^{62} -1.00000 q^{64} -1.48236i q^{65} +16.0340 q^{67} -2.42883 q^{68} +12.7627i q^{71} +0.343146i q^{73} -5.48236i q^{74} +4.86370i q^{76} -10.4488 q^{79} -1.00000 q^{80} +8.76028i q^{82} +5.45001 q^{83} -2.42883 q^{85} +1.86370i q^{86} +1.53225 q^{88} +15.9700 q^{89} -0.267949i q^{92} -7.45946i q^{94} +4.86370i q^{95} -14.9481i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^4 - 8 * q^5	\( 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} + 8 q^{25} - 16 q^{26} - 48 q^{37} + 8 q^{38} + 32 q^{41} - 16 q^{43} - 16 q^{46} + 16 q^{47} + 32 q^{58} + 48 q^{59} - 16 q^{62} - 8 q^{64} + 48 q^{67} + 48 q^{79} - 8 q^{80} - 16 q^{83} + 32 q^{89}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 + 8 * q^25 - 16 * q^26 - 48 * q^37 + 8 * q^38 + 32 * q^41 - 16 * q^43 - 16 * q^46 + 16 * q^47 + 32 * q^58 + 48 * q^59 - 16 * q^62 - 8 * q^64 + 48 * q^67 + 48 * q^79 - 8 * q^80 - 16 * q^83 + 32 * q^89

Introduction

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