LMFDB - Embedded newform 4410.2.b.e.881.5

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			881.5
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.e.881.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000i q^{8} +1.00000i q^{10} -3.98174i q^{11} -0.0681483i q^{13} +1.00000 q^{16} +7.32780 q^{17} +2.03528i q^{19} -1.00000 q^{20} +3.98174 q^{22} +3.73205i q^{23} +1.00000 q^{25} +0.0681483 q^{26} +0.898979i q^{29} -4.82843i q^{31} +1.00000i q^{32} +7.32780i q^{34} -4.06815 q^{37} -2.03528 q^{38} -1.00000i q^{40} -1.68921 q^{41} -0.964724 q^{43} +3.98174i q^{44} -3.73205 q^{46} -1.66150 q^{47} +1.00000i q^{50} +0.0681483i q^{52} -13.2268i q^{53} -3.98174i q^{55} -0.898979 q^{58} -10.6422 q^{59} -7.52056i q^{61} +4.82843 q^{62} -1.00000 q^{64} -0.0681483i q^{65} +10.6629 q^{67} -7.32780 q^{68} +9.93426i q^{71} -11.6569i q^{73} -4.06815i q^{74} -2.03528i q^{76} +17.5498 q^{79} +1.00000 q^{80} -1.68921i q^{82} +14.3490 q^{83} +7.32780 q^{85} -0.964724i q^{86} -3.98174 q^{88} -1.82791 q^{89} -3.73205i q^{92} -1.66150i q^{94} +2.03528i q^{95} -17.1502i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$11$	− 3.98174i	− 1.20054i	−0.799798	−	0.600270i	$-0.795060\pi$
$11$	− 3.98174i	− 1.20054i	0.799798	−	0.600270i	$-0.204940\pi$
$12$	0	0
$13$	− 0.0681483i	− 0.0189010i	−0.999955	−	0.00945048i	$-0.996992\pi$
$13$	− 0.0681483i	− 0.0189010i	0.999955	−	0.00945048i	$-0.00300822\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.32780	1.77725	0.888627	−	0.458631i	$-0.151660\pi$
$17$	7.32780	1.77725	0.888627	+	0.458631i	$0.151660\pi$
$18$	0	0
$19$	2.03528i	0.466924i	0.972366	+	0.233462i	$0.0750055\pi$
$19$	2.03528i	0.466924i	−0.972366	+	0.233462i	$0.924994\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	3.98174	0.848910
$23$	3.73205i	0.778186i	0.921198	+	0.389093i	$0.127212\pi$
$23$	3.73205i	0.778186i	−0.921198	+	0.389093i	$0.872788\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.0681483	0.0133650
$27$	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
$31$	− 4.82843i	− 0.867211i	−0.901103	−	0.433606i	$-0.857241\pi$
$31$	− 4.82843i	− 0.867211i	0.901103	−	0.433606i	$-0.142759\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	7.32780i	1.25671i
$35$	0	0
$36$	0	0
$37$	−4.06815	−0.668799	−0.334400	−	0.942431i	$-0.608534\pi$
$37$	−4.06815	−0.668799	−0.334400	+	0.942431i	$0.608534\pi$
$38$	−2.03528	−0.330165
$39$	0	0
$40$	− 1.00000i	− 0.158114i
$41$	−1.68921	−0.263810	−0.131905	−	0.991262i	$-0.542109\pi$
$41$	−1.68921	−0.263810	−0.131905	+	0.991262i	$0.542109\pi$
$42$	0	0
$43$	−0.964724	−0.147119	−0.0735595	−	0.997291i	$-0.523436\pi$
$43$	−0.964724	−0.147119	−0.0735595	+	0.997291i	$0.523436\pi$
$44$	3.98174i	0.600270i
$45$	0	0
$46$	−3.73205	−0.550261
$47$	−1.66150	−0.242354	−0.121177	−	0.992631i	$-0.538667\pi$
$47$	−1.66150	−0.242354	−0.121177	+	0.992631i	$0.538667\pi$
$48$	0	0
$49$	0	0
$50$	1.00000i	0.141421i
$51$	0	0
$52$	0.0681483i	0.00945048i
$53$	− 13.2268i	− 1.81684i	−0.418060	−	0.908419i	$-0.637290\pi$
$53$	− 13.2268i	− 1.81684i	0.418060	−	0.908419i	$-0.362710\pi$
$54$	0	0
$55$	− 3.98174i	− 0.536898i
$56$	0	0
$57$	0	0
$58$	−0.898979	−0.118042
$59$	−10.6422	−1.38550	−0.692751	−	0.721177i	$-0.743601\pi$
$59$	−10.6422	−1.38550	−0.692751	+	0.721177i	$0.743601\pi$
$60$	0	0
$61$	− 7.52056i	− 0.962909i	−0.876471	−	0.481454i	$-0.840109\pi$
$61$	− 7.52056i	− 0.962909i	0.876471	−	0.481454i	$-0.159891\pi$
$62$	4.82843	0.613211
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 0.0681483i	− 0.00845276i
$66$	0	0
$67$	10.6629	1.30268	0.651341	−	0.758785i	$-0.274207\pi$
$67$	10.6629	1.30268	0.651341	+	0.758785i	$0.274207\pi$
$68$	−7.32780	−0.888627
$69$	0	0
$70$	0	0
$71$	9.93426i	1.17898i	0.807776	+	0.589490i	$0.200671\pi$
$71$	9.93426i	1.17898i	−0.807776	+	0.589490i	$0.799329\pi$
$72$	0	0
$73$	− 11.6569i	− 1.36433i	−0.731198	−	0.682166i	$-0.761038\pi$
$73$	− 11.6569i	− 1.36433i	0.731198	−	0.682166i	$-0.238962\pi$
$74$	− 4.06815i	− 0.472913i
$75$	0	0
$76$	− 2.03528i	− 0.233462i
$77$	0	0
$78$	0	0
$79$	17.5498	1.97450	0.987252	−	0.159163i	$-0.0508795\pi$
$79$	17.5498	1.97450	0.987252	+	0.159163i	$0.0508795\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	− 1.68921i	− 0.186542i
$83$	14.3490	1.57501	0.787503	−	0.616311i	$-0.211374\pi$
$83$	14.3490	1.57501	0.787503	+	0.616311i	$0.211374\pi$
$84$	0	0
$85$	7.32780	0.794812
$86$	− 0.964724i	− 0.104029i
$87$	0	0
$88$	−3.98174	−0.424455
$89$	−1.82791	−0.193758	−0.0968791	−	0.995296i	$-0.530886\pi$
$89$	−1.82791	−0.193758	−0.0968791	+	0.995296i	$0.530886\pi$
$90$	0	0
$91$	0	0
$92$	− 3.73205i	− 0.389093i
$93$	0	0
$94$	− 1.66150i	− 0.171370i
$95$	2.03528i	0.208815i
$96$	0	0
$97$	− 17.1502i	− 1.74134i	−0.491870	−	0.870668i	$-0.663687\pi$
$97$	− 17.1502i	− 1.74134i	0.491870	−	0.870668i	$-0.336313\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−6.73545	−0.670202	−0.335101	−	0.942182i	$-0.608771\pi$
$101$	−6.73545	−0.670202	−0.335101	+	0.942182i	$0.608771\pi$
$102$	0	0
$103$	− 0.520042i	− 0.0512413i	−0.999672	−	0.0256206i	$-0.991844\pi$
$103$	− 0.520042i	− 0.0512413i	0.999672	−	0.0256206i	$-0.00815619\pi$
$104$	−0.0681483	−0.00668250
$105$	0	0
$106$	13.2268	1.28470
$107$	6.12701i	0.592320i	0.955138	+	0.296160i	$0.0957062\pi$
$107$	6.12701i	0.592320i	−0.955138	+	0.296160i	$0.904294\pi$
$108$	0	0
$109$	13.5546	1.29829	0.649147	−	0.760663i	$-0.275126\pi$
$109$	13.5546	1.29829	0.649147	+	0.760663i	$0.275126\pi$
$110$	3.98174	0.379644
$111$	0	0
$112$	0	0
$113$	11.6982i	1.10047i	0.835009	+	0.550236i	$0.185462\pi$
$113$	11.6982i	1.10047i	−0.835009	+	0.550236i	$0.814538\pi$
$114$	0	0
$115$	3.73205i	0.348016i
$116$	− 0.898979i	− 0.0834681i
$117$	0	0
$118$	− 10.6422i	− 0.979698i
$119$	0	0
$120$	0	0
$121$	−4.85425	−0.441296
$122$	7.52056	0.680879
$123$	0	0
$124$	4.82843i	0.433606i
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.7530	0.954173	0.477086	−	0.878856i	$-0.341693\pi$
$127$	10.7530	0.954173	0.477086	+	0.878856i	$0.341693\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0.0681483	0.00597701
$131$	3.70915	0.324070	0.162035	−	0.986785i	$-0.448194\pi$
$131$	3.70915	0.324070	0.162035	+	0.986785i	$0.448194\pi$
$132$	0	0
$133$	0	0
$134$	10.6629i	0.921135i
$135$	0	0
$136$	− 7.32780i	− 0.628354i
$137$	− 13.5206i	− 1.15514i	−0.816341	−	0.577570i	$-0.804001\pi$
$137$	− 13.5206i	− 1.15514i	0.816341	−	0.577570i	$-0.195999\pi$
$138$	0	0
$139$	− 9.06251i	− 0.768672i	−0.923193	−	0.384336i	$-0.874430\pi$
$139$	− 9.06251i	− 0.768672i	0.923193	−	0.384336i	$-0.125570\pi$
$140$	0	0
$141$	0	0
$142$	−9.93426	−0.833664
$143$	−0.271349	−0.0226913
$144$	0	0
$145$	0.898979i	0.0746562i
$146$	11.6569	0.964728
$147$	0	0
$148$	4.06815	0.334400
$149$	− 15.2865i	− 1.25232i	−0.779696	−	0.626158i	$-0.784626\pi$
$149$	− 15.2865i	− 1.25232i	0.779696	−	0.626158i	$-0.215374\pi$
$150$	0	0
$151$	−9.87883	−0.803928	−0.401964	−	0.915656i	$-0.631672\pi$
$151$	−9.87883	−0.803928	−0.401964	+	0.915656i	$0.631672\pi$
$152$	2.03528	0.165083
$153$	0	0
$154$	0	0
$155$	− 4.82843i	− 0.387829i
$156$	0	0
$157$	14.2884i	1.14034i	0.821528	+	0.570168i	$0.193122\pi$
$157$	14.2884i	1.14034i	−0.821528	+	0.570168i	$0.806878\pi$
$158$	17.5498i	1.39619i
$159$	0	0
$160$	1.00000i	0.0790569i
$161$	0	0
$162$	0	0
$163$	9.97758	0.781504	0.390752	−	0.920496i	$-0.372215\pi$
$163$	9.97758	0.781504	0.390752	+	0.920496i	$0.372215\pi$
$164$	1.68921	0.131905
$165$	0	0
$166$	14.3490i	1.11370i
$167$	13.7778	1.06616	0.533079	−	0.846065i	$-0.321035\pi$
$167$	13.7778	1.06616	0.533079	+	0.846065i	$0.321035\pi$
$168$	0	0
$169$	12.9954	0.999643
$170$	7.32780i	0.562017i
$171$	0	0
$172$	0.964724	0.0735595
$173$	−8.27276	−0.628966	−0.314483	−	0.949263i	$-0.601831\pi$
$173$	−8.27276	−0.628966	−0.314483	+	0.949263i	$0.601831\pi$
$174$	0	0
$175$	0	0
$176$	− 3.98174i	− 0.300135i
$177$	0	0
$178$	− 1.82791i	− 0.137008i
$179$	11.5182i	0.860907i	0.902613	+	0.430454i	$0.141646\pi$
$179$	11.5182i	0.860907i	−0.902613	+	0.430454i	$0.858354\pi$
$180$	0	0
$181$	− 16.5924i	− 1.23330i	−0.787237	−	0.616650i	$-0.788489\pi$
$181$	− 16.5924i	− 1.23330i	0.787237	−	0.616650i	$-0.211511\pi$
$182$	0	0
$183$	0	0
$184$	3.73205	0.275130
$185$	−4.06815	−0.299096
$186$	0	0
$187$	− 29.1774i	− 2.13366i
$188$	1.66150	0.121177
$189$	0	0
$190$	−2.03528	−0.147654
$191$	18.4336i	1.33381i	0.745142	+	0.666905i	$0.232382\pi$
$191$	18.4336i	1.33381i	−0.745142	+	0.666905i	$0.767618\pi$
$192$	0	0
$193$	−6.35827	−0.457679	−0.228839	−	0.973464i	$-0.573493\pi$
$193$	−6.35827	−0.457679	−0.228839	+	0.973464i	$0.573493\pi$
$194$	17.1502	1.23131
$195$	0	0
$196$	0	0
$197$	− 12.8389i	− 0.914734i	−0.889278	−	0.457367i	$-0.848792\pi$
$197$	− 12.8389i	− 0.914734i	0.889278	−	0.457367i	$-0.151208\pi$
$198$	0	0
$199$	− 9.62962i	− 0.682626i	−0.939950	−	0.341313i	$-0.889128\pi$
$199$	− 9.62962i	− 0.682626i	0.939950	−	0.341313i	$-0.110872\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	− 6.73545i	− 0.473905i
$203$	0	0
$204$	0	0
$205$	−1.68921	−0.117980
$206$	0.520042	0.0362331
$207$	0	0
$208$	− 0.0681483i	− 0.00472524i
$209$	8.10394	0.560561
$210$	0	0
$211$	−14.5619	−1.00248	−0.501241	−	0.865308i	$-0.667123\pi$
$211$	−14.5619	−1.00248	−0.501241	+	0.865308i	$0.667123\pi$
$212$	13.2268i	0.908419i
$213$	0	0
$214$	−6.12701	−0.418834
$215$	−0.964724	−0.0657936
$216$	0	0
$217$	0	0
$218$	13.5546i	0.918033i
$219$	0	0
$220$	3.98174i	0.268449i
$221$	− 0.499378i	− 0.0335918i
$222$	0	0
$223$	23.6609i	1.58445i	0.610227	+	0.792227i	$0.291078\pi$
$223$	23.6609i	1.58445i	−0.610227	+	0.792227i	$0.708922\pi$
$224$	0	0
$225$	0	0
$226$	−11.6982	−0.778152
$227$	−8.64048	−0.573489	−0.286744	−	0.958007i	$-0.592573\pi$
$227$	−8.64048	−0.573489	−0.286744	+	0.958007i	$0.592573\pi$
$228$	0	0
$229$	14.0754i	0.930126i	0.885278	+	0.465063i	$0.153968\pi$
$229$	14.0754i	0.930126i	−0.885278	+	0.465063i	$0.846032\pi$
$230$	−3.73205	−0.246084
$231$	0	0
$232$	0.898979	0.0590209
$233$	20.1976i	1.32319i	0.749863	+	0.661593i	$0.230119\pi$
$233$	20.1976i	1.32319i	−0.749863	+	0.661593i	$0.769881\pi$
$234$	0	0
$235$	−1.66150	−0.108384
$236$	10.6422	0.692751
$237$	0	0
$238$	0	0
$239$	23.5040i	1.52035i	0.649721	+	0.760173i	$0.274886\pi$
$239$	23.5040i	1.52035i	−0.649721	+	0.760173i	$0.725114\pi$
$240$	0	0
$241$	− 14.8685i	− 0.957765i	−0.877879	−	0.478883i	$-0.841042\pi$
$241$	− 14.8685i	− 0.957765i	0.877879	−	0.478883i	$-0.158958\pi$
$242$	− 4.85425i	− 0.312043i
$243$	0	0
$244$	7.52056i	0.481454i
$245$	0	0
$246$	0	0
$247$	0.138701	0.00882531
$248$	−4.82843	−0.306605
$249$	0	0
$250$	1.00000i	0.0632456i
$251$	4.31736	0.272509	0.136255	−	0.990674i	$-0.456493\pi$
$251$	4.31736	0.272509	0.136255	+	0.990674i	$0.456493\pi$
$252$	0	0
$253$	14.8601	0.934244
$254$	10.7530i	0.674702i
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.83324	0.301489	0.150744	−	0.988573i	$-0.451833\pi$
$257$	4.83324	0.301489	0.150744	+	0.988573i	$0.451833\pi$
$258$	0	0
$259$	0	0
$260$	0.0681483i	0.00422638i
$261$	0	0
$262$	3.70915i	0.229152i
$263$	− 6.66490i	− 0.410975i	−0.978660	−	0.205488i	$-0.934122\pi$
$263$	− 6.66490i	− 0.410975i	0.978660	−	0.205488i	$-0.0658780\pi$
$264$	0	0
$265$	− 13.2268i	− 0.812515i
$266$	0	0
$267$	0	0
$268$	−10.6629	−0.651341
$269$	31.7400	1.93522	0.967612	−	0.252441i	$-0.0812333\pi$
$269$	31.7400	1.93522	0.967612	+	0.252441i	$0.0812333\pi$
$270$	0	0
$271$	− 17.4641i	− 1.06087i	−0.847726	−	0.530434i	$-0.822029\pi$
$271$	− 17.4641i	− 1.06087i	0.847726	−	0.530434i	$-0.177971\pi$
$272$	7.32780	0.444313
$273$	0	0
$274$	13.5206	0.816807
$275$	− 3.98174i	− 0.240108i
$276$	0	0
$277$	−23.5047	−1.41226	−0.706130	−	0.708082i	$-0.749561\pi$
$277$	−23.5047	−1.41226	−0.706130	+	0.708082i	$0.749561\pi$
$278$	9.06251	0.543533
$279$	0	0
$280$	0	0
$281$	15.4159i	0.919635i	0.888013	+	0.459817i	$0.152085\pi$
$281$	15.4159i	0.919635i	−0.888013	+	0.459817i	$0.847915\pi$
$282$	0	0
$283$	− 27.7108i	− 1.64724i	−0.567144	−	0.823619i	$-0.691952\pi$
$283$	− 27.7108i	− 1.64724i	0.567144	−	0.823619i	$-0.308048\pi$
$284$	− 9.93426i	− 0.589490i
$285$	0	0
$286$	− 0.271349i	− 0.0160452i
$287$	0	0
$288$	0	0
$289$	36.6967	2.15863
$290$	−0.898979	−0.0527899
$291$	0	0
$292$	11.6569i	0.682166i
$293$	−2.84377	−0.166135	−0.0830673	−	0.996544i	$-0.526472\pi$
$293$	−2.84377	−0.166135	−0.0830673	+	0.996544i	$0.526472\pi$
$294$	0	0
$295$	−10.6422	−0.619615
$296$	4.06815i	0.236456i
$297$	0	0
$298$	15.2865	0.885522
$299$	0.254333	0.0147085
$300$	0	0
$301$	0	0
$302$	− 9.87883i	− 0.568463i
$303$	0	0
$304$	2.03528i	0.116731i
$305$	− 7.52056i	− 0.430626i
$306$	0	0
$307$	2.52180i	0.143927i	0.997407	+	0.0719634i	$0.0229265\pi$
$307$	2.52180i	0.143927i	−0.997407	+	0.0719634i	$0.977074\pi$
$308$	0	0
$309$	0	0
$310$	4.82843	0.274236
$311$	10.9939	0.623410	0.311705	−	0.950179i	$-0.399100\pi$
$311$	10.9939	0.623410	0.311705	+	0.950179i	$0.399100\pi$
$312$	0	0
$313$	− 24.9461i	− 1.41004i	−0.709188	−	0.705020i	$-0.750938\pi$
$313$	− 24.9461i	− 1.41004i	0.709188	−	0.705020i	$-0.249062\pi$
$314$	−14.2884	−0.806339
$315$	0	0
$316$	−17.5498	−0.987252
$317$	15.0693i	0.846377i	0.906042	+	0.423188i	$0.139089\pi$
$317$	15.0693i	0.846377i	−0.906042	+	0.423188i	$0.860911\pi$
$318$	0	0
$319$	3.57950	0.200414
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	14.9141i	0.829843i
$324$	0	0
$325$	− 0.0681483i	− 0.00378019i
$326$	9.97758i	0.552607i
$327$	0	0
$328$	1.68921i	0.0932711i
$329$	0	0
$330$	0	0
$331$	30.1808	1.65889	0.829444	−	0.558590i	$-0.188658\pi$
$331$	30.1808	1.65889	0.829444	+	0.558590i	$0.188658\pi$
$332$	−14.3490	−0.787503
$333$	0	0
$334$	13.7778i	0.753888i
$335$	10.6629	0.582577
$336$	0	0
$337$	−21.5911	−1.17614	−0.588071	−	0.808809i	$-0.700113\pi$
$337$	−21.5911	−1.17614	−0.588071	+	0.808809i	$0.700113\pi$
$338$	12.9954i	0.706854i
$339$	0	0
$340$	−7.32780	−0.397406
$341$	−19.2255	−1.04112
$342$	0	0
$343$	0	0
$344$	0.964724i	0.0520144i
$345$	0	0
$346$	− 8.27276i	− 0.444746i
$347$	− 23.1842i	− 1.24459i	−0.782781	−	0.622297i	$-0.786200\pi$
$347$	− 23.1842i	− 1.24459i	0.782781	−	0.622297i	$-0.213800\pi$
$348$	0	0
$349$	− 10.7287i	− 0.574292i	−0.957887	−	0.287146i	$-0.907294\pi$
$349$	− 10.7287i	− 0.574292i	0.957887	−	0.287146i	$-0.0927064\pi$
$350$	0	0
$351$	0	0
$352$	3.98174	0.212227
$353$	8.74756	0.465585	0.232793	−	0.972526i	$-0.425214\pi$
$353$	8.74756	0.465585	0.232793	+	0.972526i	$0.425214\pi$
$354$	0	0
$355$	9.93426i	0.527256i
$356$	1.82791	0.0968791
$357$	0	0
$358$	−11.5182	−0.608753
$359$	− 25.7566i	− 1.35938i	−0.733498	−	0.679691i	$-0.762114\pi$
$359$	− 25.7566i	− 1.35938i	0.733498	−	0.679691i	$-0.237886\pi$
$360$	0	0
$361$	14.8577	0.781982
$362$	16.5924	0.872075
$363$	0	0
$364$	0	0
$365$	− 11.6569i	− 0.610148i
$366$	0	0
$367$	19.9377i	1.04074i	0.853941	+	0.520369i	$0.174206\pi$
$367$	19.9377i	1.04074i	−0.853941	+	0.520369i	$0.825794\pi$
$368$	3.73205i	0.194547i
$369$	0	0
$370$	− 4.06815i	− 0.211493i
$371$	0	0
$372$	0	0
$373$	1.03152	0.0534103	0.0267052	−	0.999643i	$-0.491498\pi$
$373$	1.03152	0.0534103	0.0267052	+	0.999643i	$0.491498\pi$
$374$	29.1774	1.50873
$375$	0	0
$376$	1.66150i	0.0856852i
$377$	0.0612640	0.00315525
$378$	0	0
$379$	5.09497	0.261711	0.130855	−	0.991401i	$-0.458228\pi$
$379$	5.09497	0.261711	0.130855	+	0.991401i	$0.458228\pi$
$380$	− 2.03528i	− 0.104407i
$381$	0	0
$382$	−18.4336	−0.943147
$383$	−8.46750	−0.432669	−0.216335	−	0.976319i	$-0.569410\pi$
$383$	−8.46750	−0.432669	−0.216335	+	0.976319i	$0.569410\pi$
$384$	0	0
$385$	0	0
$386$	− 6.35827i	− 0.323628i
$387$	0	0
$388$	17.1502i	0.870668i
$389$	− 5.74150i	− 0.291106i	−0.989350	−	0.145553i	$-0.953504\pi$
$389$	− 5.74150i	− 0.291106i	0.989350	−	0.145553i	$-0.0464961\pi$
$390$	0	0
$391$	27.3477i	1.38303i
$392$	0	0
$393$	0	0
$394$	12.8389	0.646815
$395$	17.5498	0.883025
$396$	0	0
$397$	− 16.8758i	− 0.846973i	−0.905902	−	0.423487i	$-0.860806\pi$
$397$	− 16.8758i	− 0.846973i	0.905902	−	0.423487i	$-0.139194\pi$
$398$	9.62962	0.482689
$399$	0	0
$400$	1.00000	0.0500000
$401$	− 1.88876i	− 0.0943203i	−0.998887	−	0.0471602i	$-0.984983\pi$
$401$	− 1.88876i	− 0.0943203i	0.998887	−	0.0471602i	$-0.0150171\pi$
$402$	0	0
$403$	−0.329049	−0.0163911
$404$	6.73545	0.335101
$405$	0	0
$406$	0	0
$407$	16.1983i	0.802920i
$408$	0	0
$409$	32.9802i	1.63076i	0.578923	+	0.815382i	$0.303473\pi$
$409$	32.9802i	1.63076i	−0.578923	+	0.815382i	$0.696527\pi$
$410$	− 1.68921i	− 0.0834242i
$411$	0	0
$412$	0.520042i	0.0256206i
$413$	0	0
$414$	0	0
$415$	14.3490	0.704364
$416$	0.0681483	0.00334125
$417$	0	0
$418$	8.10394i	0.396377i
$419$	−0.300470	−0.0146789	−0.00733945	−	0.999973i	$-0.502336\pi$
$419$	−0.300470	−0.0146789	−0.00733945	+	0.999973i	$0.502336\pi$
$420$	0	0
$421$	28.8625	1.40667	0.703335	−	0.710858i	$-0.251693\pi$
$421$	28.8625	1.40667	0.703335	+	0.710858i	$0.251693\pi$
$422$	− 14.5619i	− 0.708862i
$423$	0	0
$424$	−13.2268	−0.642349
$425$	7.32780	0.355451
$426$	0	0
$427$	0	0
$428$	− 6.12701i	− 0.296160i
$429$	0	0
$430$	− 0.964724i	− 0.0465231i
$431$	33.5897i	1.61796i	0.587839	+	0.808978i	$0.299979\pi$
$431$	33.5897i	1.61796i	−0.587839	+	0.808978i	$0.700021\pi$
$432$	0	0
$433$	− 11.2207i	− 0.539234i	−0.962968	−	0.269617i	$-0.913103\pi$
$433$	− 11.2207i	− 0.539234i	0.962968	−	0.269617i	$-0.0868971\pi$
$434$	0	0
$435$	0	0
$436$	−13.5546	−0.649147
$437$	−7.59575	−0.363354
$438$	0	0
$439$	16.8673i	0.805031i	0.915413	+	0.402515i	$0.131864\pi$
$439$	16.8673i	0.805031i	−0.915413	+	0.402515i	$0.868136\pi$
$440$	−3.98174	−0.189822
$441$	0	0
$442$	0.499378	0.0237530
$443$	− 2.05646i	− 0.0977052i	−0.998806	−	0.0488526i	$-0.984444\pi$
$443$	− 2.05646i	− 0.0977052i	0.998806	−	0.0488526i	$-0.0155565\pi$
$444$	0	0
$445$	−1.82791	−0.0866513
$446$	−23.6609	−1.12038
$447$	0	0
$448$	0	0
$449$	12.5892i	0.594122i	0.954858	+	0.297061i	$0.0960065\pi$
$449$	12.5892i	0.594122i	−0.954858	+	0.297061i	$0.903994\pi$
$450$	0	0
$451$	6.72600i	0.316715i
$452$	− 11.6982i	− 0.550236i
$453$	0	0
$454$	− 8.64048i	− 0.405518i
$455$	0	0
$456$	0	0
$457$	34.9529	1.63503	0.817515	−	0.575907i	$-0.195351\pi$
$457$	34.9529	1.63503	0.817515	+	0.575907i	$0.195351\pi$
$458$	−14.0754	−0.657698
$459$	0	0
$460$	− 3.73205i	− 0.174008i
$461$	−23.3750	−1.08868	−0.544341	−	0.838864i	$-0.683220\pi$
$461$	−23.3750	−1.08868	−0.544341	+	0.838864i	$0.683220\pi$
$462$	0	0
$463$	−6.35693	−0.295431	−0.147716	−	0.989030i	$-0.547192\pi$
$463$	−6.35693	−0.295431	−0.147716	+	0.989030i	$0.547192\pi$
$464$	0.898979i	0.0417341i
$465$	0	0
$466$	−20.1976	−0.935634
$467$	7.79315	0.360624	0.180312	−	0.983609i	$-0.442289\pi$
$467$	7.79315	0.360624	0.180312	+	0.983609i	$0.442289\pi$
$468$	0	0
$469$	0	0
$470$	− 1.66150i	− 0.0766392i
$471$	0	0
$472$	10.6422i	0.489849i
$473$	3.84128i	0.176622i
$474$	0	0
$475$	2.03528i	0.0933849i
$476$	0	0
$477$	0	0
$478$	−23.5040	−1.07505
$479$	27.5040	1.25669	0.628344	−	0.777935i	$-0.283733\pi$
$479$	27.5040	1.25669	0.628344	+	0.777935i	$0.283733\pi$
$480$	0	0
$481$	0.277238i	0.0126409i
$482$	14.8685	0.677242
$483$	0	0
$484$	4.85425	0.220648
$485$	− 17.1502i	− 0.778750i
$486$	0	0
$487$	35.4468	1.60625	0.803124	−	0.595812i	$-0.203170\pi$
$487$	35.4468	1.60625	0.803124	+	0.595812i	$0.203170\pi$
$488$	−7.52056	−0.340440
$489$	0	0
$490$	0	0
$491$	1.64349i	0.0741695i	0.999312	+	0.0370848i	$0.0118072\pi$
$491$	1.64349i	0.0741695i	−0.999312	+	0.0370848i	$0.988193\pi$
$492$	0	0
$493$	6.58755i	0.296688i
$494$	0.138701i	0.00624044i
$495$	0	0
$496$	− 4.82843i	− 0.216803i
$497$	0	0
$498$	0	0
$499$	−34.7330	−1.55486	−0.777430	−	0.628969i	$-0.783477\pi$
$499$	−34.7330	−1.55486	−0.777430	+	0.628969i	$0.783477\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	4.31736i	0.192693i
$503$	−28.9613	−1.29132	−0.645660	−	0.763625i	$-0.723418\pi$
$503$	−28.9613	−1.29132	−0.645660	+	0.763625i	$0.723418\pi$
$504$	0	0
$505$	−6.73545	−0.299724
$506$	14.8601i	0.660610i
$507$	0	0
$508$	−10.7530	−0.477086
$509$	−21.4621	−0.951291	−0.475646	−	0.879637i	$-0.657786\pi$
$509$	−21.4621	−0.951291	−0.475646	+	0.879637i	$0.657786\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	4.83324i	0.213185i
$515$	− 0.520042i	− 0.0229158i
$516$	0	0
$517$	6.61565i	0.290956i
$518$	0	0
$519$	0	0
$520$	−0.0681483	−0.00298850
$521$	−40.2436	−1.76310	−0.881552	−	0.472087i	$-0.843501\pi$
$521$	−40.2436	−1.76310	−0.881552	+	0.472087i	$0.843501\pi$
$522$	0	0
$523$	− 29.6889i	− 1.29820i	−0.760701	−	0.649102i	$-0.775145\pi$
$523$	− 29.6889i	− 1.29820i	0.760701	−	0.649102i	$-0.224855\pi$
$524$	−3.70915	−0.162035
$525$	0	0
$526$	6.66490	0.290603
$527$	− 35.3818i	− 1.54125i
$528$	0	0
$529$	9.07180	0.394426
$530$	13.2268	0.574535
$531$	0	0
$532$	0	0
$533$	0.115117i	0.00498627i
$534$	0	0
$535$	6.12701i	0.264894i
$536$	− 10.6629i	− 0.460567i
$537$	0	0
$538$	31.7400i	1.36841i
$539$	0	0
$540$	0	0
$541$	−17.2583	−0.741992	−0.370996	−	0.928635i	$-0.620984\pi$
$541$	−17.2583	−0.741992	−0.370996	+	0.928635i	$0.620984\pi$
$542$	17.4641	0.750147
$543$	0	0
$544$	7.32780i	0.314177i
$545$	13.5546	0.580615
$546$	0	0
$547$	−17.9703	−0.768354	−0.384177	−	0.923260i	$-0.625515\pi$
$547$	−17.9703	−0.768354	−0.384177	+	0.923260i	$0.625515\pi$
$548$	13.5206i	0.577570i
$549$	0	0
$550$	3.98174	0.169782
$551$	−1.82967	−0.0779466
$552$	0	0
$553$	0	0
$554$	− 23.5047i	− 0.998619i
$555$	0	0
$556$	9.06251i	0.384336i
$557$	45.2546i	1.91750i	0.284256	+	0.958748i	$0.408253\pi$
$557$	45.2546i	1.91750i	−0.284256	+	0.958748i	$0.591747\pi$
$558$	0	0
$559$	0.0657443i	0.00278069i
$560$	0	0
$561$	0	0
$562$	−15.4159	−0.650280
$563$	−15.8667	−0.668703	−0.334351	−	0.942448i	$-0.608517\pi$
$563$	−15.8667	−0.668703	−0.334351	+	0.942448i	$0.608517\pi$
$564$	0	0
$565$	11.6982i	0.492146i
$566$	27.7108	1.16477
$567$	0	0
$568$	9.93426	0.416832
$569$	− 0.0605496i	− 0.00253837i	−0.999999	−	0.00126918i	$-0.999596\pi$
$569$	− 0.0605496i	− 0.00253837i	0.999999	−	0.00126918i	$-0.000403994\pi$
$570$	0	0
$571$	−13.5649	−0.567674	−0.283837	−	0.958873i	$-0.591607\pi$
$571$	−13.5649	−0.567674	−0.283837	+	0.958873i	$0.591607\pi$
$572$	0.271349	0.0113457
$573$	0	0
$574$	0	0
$575$	3.73205i	0.155637i
$576$	0	0
$577$	− 43.6768i	− 1.81829i	−0.416481	−	0.909144i	$-0.636737\pi$
$577$	− 43.6768i	− 1.81829i	0.416481	−	0.909144i	$-0.363263\pi$
$578$	36.6967i	1.52638i
$579$	0	0
$580$	− 0.898979i	− 0.0373281i
$581$	0	0
$582$	0	0
$583$	−52.6656	−2.18119
$584$	−11.6569	−0.482364
$585$	0	0
$586$	− 2.84377i	− 0.117475i
$587$	45.4100	1.87427	0.937135	−	0.348967i	$-0.113468\pi$
$587$	45.4100	1.87427	0.937135	+	0.348967i	$0.113468\pi$
$588$	0	0
$589$	9.82718	0.404922
$590$	− 10.6422i	− 0.438134i
$591$	0	0
$592$	−4.06815	−0.167200
$593$	−25.7012	−1.05542	−0.527711	−	0.849424i	$-0.676950\pi$
$593$	−25.7012	−1.05542	−0.527711	+	0.849424i	$0.676950\pi$
$594$	0	0
$595$	0	0
$596$	15.2865i	0.626158i
$597$	0	0
$598$	0.254333i	0.0104005i
$599$	− 26.5982i	− 1.08677i	−0.839483	−	0.543386i	$-0.817142\pi$
$599$	− 26.5982i	− 1.08677i	0.839483	−	0.543386i	$-0.182858\pi$
$600$	0	0
$601$	− 12.9681i	− 0.528979i	−0.964389	−	0.264489i	$-0.914797\pi$
$601$	− 12.9681i	− 0.528979i	0.964389	−	0.264489i	$-0.0852034\pi$
$602$	0	0
$603$	0	0
$604$	9.87883	0.401964
$605$	−4.85425	−0.197353
$606$	0	0
$607$	16.3115i	0.662064i	0.943620	+	0.331032i	$0.107397\pi$
$607$	16.3115i	0.662064i	−0.943620	+	0.331032i	$0.892603\pi$
$608$	−2.03528	−0.0825413
$609$	0	0
$610$	7.52056	0.304498
$611$	0.113228i	0.00458073i
$612$	0	0
$613$	−14.3879	−0.581122	−0.290561	−	0.956856i	$-0.593842\pi$
$613$	−14.3879	−0.581122	−0.290561	+	0.956856i	$0.593842\pi$
$614$	−2.52180	−0.101772
$615$	0	0
$616$	0	0
$617$	− 8.65760i	− 0.348542i	−0.984698	−	0.174271i	$-0.944243\pi$
$617$	− 8.65760i	− 0.348542i	0.984698	−	0.174271i	$-0.0557569\pi$
$618$	0	0
$619$	44.8553i	1.80289i	0.432896	+	0.901444i	$0.357492\pi$
$619$	44.8553i	1.80289i	−0.432896	+	0.901444i	$0.642508\pi$
$620$	4.82843i	0.193914i
$621$	0	0
$622$	10.9939i	0.440817i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	24.9461	0.997049
$627$	0	0
$628$	− 14.2884i	− 0.570168i
$629$	−29.8106	−1.18863
$630$	0	0
$631$	−38.1878	−1.52023	−0.760116	−	0.649788i	$-0.774858\pi$
$631$	−38.1878	−1.52023	−0.760116	+	0.649788i	$0.774858\pi$
$632$	− 17.5498i	− 0.698093i
$633$	0	0
$634$	−15.0693	−0.598479
$635$	10.7530	0.426719
$636$	0	0
$637$	0	0
$638$	3.57950i	0.141714i
$639$	0	0
$640$	− 1.00000i	− 0.0395285i
$641$	− 8.55094i	− 0.337742i	−0.985638	−	0.168871i	$-0.945988\pi$
$641$	− 8.55094i	− 0.337742i	0.985638	−	0.168871i	$-0.0540121\pi$
$642$	0	0
$643$	− 2.75058i	− 0.108472i	−0.998528	−	0.0542361i	$-0.982728\pi$
$643$	− 2.75058i	− 0.108472i	0.998528	−	0.0542361i	$-0.0172724\pi$
$644$	0	0
$645$	0	0
$646$	−14.9141	−0.586788
$647$	−27.7626	−1.09146	−0.545731	−	0.837961i	$-0.683748\pi$
$647$	−27.7626	−1.09146	−0.545731	+	0.837961i	$0.683748\pi$
$648$	0	0
$649$	42.3746i	1.66335i
$650$	0.0681483	0.00267300
$651$	0	0
$652$	−9.97758	−0.390752
$653$	− 14.1199i	− 0.552553i	−0.961078	−	0.276277i	$-0.910899\pi$
$653$	− 14.1199i	− 0.552553i	0.961078	−	0.276277i	$-0.0891006\pi$
$654$	0	0
$655$	3.70915	0.144928
$656$	−1.68921	−0.0659526
$657$	0	0
$658$	0	0
$659$	− 9.92570i	− 0.386650i	−0.981135	−	0.193325i	$-0.938073\pi$
$659$	− 9.92570i	− 0.386650i	0.981135	−	0.193325i	$-0.0619272\pi$
$660$	0	0
$661$	18.3631i	0.714241i	0.934058	+	0.357121i	$0.116241\pi$
$661$	18.3631i	0.714241i	−0.934058	+	0.357121i	$0.883759\pi$
$662$	30.1808i	1.17301i
$663$	0	0
$664$	− 14.3490i	− 0.556849i
$665$	0	0
$666$	0	0
$667$	−3.35504	−0.129908
$668$	−13.7778	−0.533079
$669$	0	0
$670$	10.6629i	0.411944i
$671$	−29.9449	−1.15601
$672$	0	0
$673$	−32.6050	−1.25683	−0.628415	−	0.777878i	$-0.716296\pi$
$673$	−32.6050	−1.25683	−0.628415	+	0.777878i	$0.716296\pi$
$674$	− 21.5911i	− 0.831658i
$675$	0	0
$676$	−12.9954	−0.499821
$677$	18.3385	0.704804	0.352402	−	0.935849i	$-0.385365\pi$
$677$	18.3385	0.704804	0.352402	+	0.935849i	$0.385365\pi$
$678$	0	0
$679$	0	0
$680$	− 7.32780i	− 0.281008i
$681$	0	0
$682$	− 19.2255i	− 0.736184i
$683$	− 24.4770i	− 0.936587i	−0.883573	−	0.468294i	$-0.844869\pi$
$683$	− 24.4770i	− 0.936587i	0.883573	−	0.468294i	$-0.155131\pi$
$684$	0	0
$685$	− 13.5206i	− 0.516594i
$686$	0	0
$687$	0	0
$688$	−0.964724	−0.0367798
$689$	−0.901383	−0.0343400
$690$	0	0
$691$	18.3734i	0.698957i	0.936944	+	0.349479i	$0.113641\pi$
$691$	18.3734i	0.698957i	−0.936944	+	0.349479i	$0.886359\pi$
$692$	8.27276	0.314483
$693$	0	0
$694$	23.1842	0.880061
$695$	− 9.06251i	− 0.343761i
$696$	0	0
$697$	−12.3782	−0.468858
$698$	10.7287	0.406085
$699$	0	0
$700$	0	0
$701$	− 14.2399i	− 0.537834i	−0.963163	−	0.268917i	$-0.913334\pi$
$701$	− 14.2399i	− 0.537834i	0.963163	−	0.268917i	$-0.0866658\pi$
$702$	0	0
$703$	− 8.27981i	− 0.312279i
$704$	3.98174i	0.150067i
$705$	0	0
$706$	8.74756i	0.329219i
$707$	0	0
$708$	0	0
$709$	37.5171	1.40898	0.704492	−	0.709712i	$-0.251175\pi$
$709$	37.5171	1.40898	0.704492	+	0.709712i	$0.251175\pi$
$710$	−9.93426	−0.372826
$711$	0	0
$712$	1.82791i	0.0685039i
$713$	18.0199	0.674852
$714$	0	0
$715$	−0.271349	−0.0101479
$716$	− 11.5182i	− 0.430454i
$717$	0	0
$718$	25.7566	0.961229
$719$	5.31467	0.198204	0.0991019	−	0.995077i	$-0.468403\pi$
$719$	5.31467	0.198204	0.0991019	+	0.995077i	$0.468403\pi$
$720$	0	0
$721$	0	0
$722$	14.8577i	0.552945i
$723$	0	0
$724$	16.5924i	0.616650i
$725$	0.898979i	0.0333873i
$726$	0	0
$727$	2.93413i	0.108821i	0.998519	+	0.0544104i	$0.0173279\pi$
$727$	2.93413i	0.108821i	−0.998519	+	0.0544104i	$0.982672\pi$
$728$	0	0
$729$	0	0
$730$	11.6569	0.431440
$731$	−7.06931	−0.261468
$732$	0	0
$733$	− 13.6482i	− 0.504108i	−0.967713	−	0.252054i	$-0.918894\pi$
$733$	− 13.6482i	− 0.504108i	0.967713	−	0.252054i	$-0.0811061\pi$
$734$	−19.9377	−0.735914
$735$	0	0
$736$	−3.73205	−0.137565
$737$	− 42.4569i	− 1.56392i
$738$	0	0
$739$	31.3300	1.15249	0.576246	−	0.817277i	$-0.304517\pi$
$739$	31.3300	1.15249	0.576246	+	0.817277i	$0.304517\pi$
$740$	4.06815	0.149548
$741$	0	0
$742$	0	0
$743$	− 4.72061i	− 0.173182i	−0.996244	−	0.0865911i	$-0.972403\pi$
$743$	− 4.72061i	− 0.173182i	0.996244	−	0.0865911i	$-0.0275974\pi$
$744$	0	0
$745$	− 15.2865i	− 0.560053i
$746$	1.03152i	0.0377668i
$747$	0	0
$748$	29.1774i	1.06683i
$749$	0	0
$750$	0	0
$751$	−25.6788	−0.937032	−0.468516	−	0.883455i	$-0.655211\pi$
$751$	−25.6788	−0.937032	−0.468516	+	0.883455i	$0.655211\pi$
$752$	−1.66150	−0.0605886
$753$	0	0
$754$	0.0612640i	0.00223110i
$755$	−9.87883	−0.359527
$756$	0	0
$757$	−37.8781	−1.37670	−0.688352	−	0.725377i	$-0.741665\pi$
$757$	−37.8781	−1.37670	−0.688352	+	0.725377i	$0.741665\pi$
$758$	5.09497i	0.185058i
$759$	0	0
$760$	2.03528	0.0738272
$761$	20.7066	0.750615	0.375308	−	0.926900i	$-0.377537\pi$
$761$	20.7066	0.750615	0.375308	+	0.926900i	$0.377537\pi$
$762$	0	0
$763$	0	0
$764$	− 18.4336i	− 0.666905i
$765$	0	0
$766$	− 8.46750i	− 0.305943i
$767$	0.725251i	0.0261873i
$768$	0	0
$769$	3.34563i	0.120647i	0.998179	+	0.0603233i	$0.0192132\pi$
$769$	3.34563i	0.120647i	−0.998179	+	0.0603233i	$0.980787\pi$
$770$	0	0
$771$	0	0
$772$	6.35827	0.228839
$773$	28.7264	1.03322	0.516609	−	0.856221i	$-0.327194\pi$
$773$	28.7264	1.03322	0.516609	+	0.856221i	$0.327194\pi$
$774$	0	0
$775$	− 4.82843i	− 0.173442i
$776$	−17.1502	−0.615656
$777$	0	0
$778$	5.74150	0.205843
$779$	− 3.43801i	− 0.123180i
$780$	0	0
$781$	39.5556	1.41541
$782$	−27.3477	−0.977953
$783$	0	0
$784$	0	0
$785$	14.2884i	0.509974i
$786$	0	0
$787$	− 5.49030i	− 0.195708i	−0.995201	−	0.0978541i	$-0.968802\pi$
$787$	− 5.49030i	− 0.195708i	0.995201	−	0.0978541i	$-0.0311978\pi$
$788$	12.8389i	0.457367i
$789$	0	0
$790$	17.5498i	0.624393i
$791$	0	0
$792$	0	0
$793$	−0.512514	−0.0181999
$794$	16.8758	0.598901
$795$	0	0
$796$	9.62962i	0.341313i
$797$	−50.8854	−1.80245	−0.901227	−	0.433347i	$-0.857332\pi$
$797$	−50.8854	−1.80245	−0.901227	+	0.433347i	$0.857332\pi$
$798$	0	0
$799$	−12.1751	−0.430725
$800$	1.00000i	0.0353553i
$801$	0	0
$802$	1.88876	0.0666945
$803$	−46.4146	−1.63793
$804$	0	0
$805$	0	0
$806$	− 0.329049i	− 0.0115903i
$807$	0	0
$808$	6.73545i	0.236952i
$809$	2.08433i	0.0732812i	0.999329	+	0.0366406i	$0.0116657\pi$
$809$	2.08433i	0.0732812i	−0.999329	+	0.0366406i	$0.988334\pi$
$810$	0	0
$811$	27.0199i	0.948797i	0.880310	+	0.474398i	$0.157334\pi$
$811$	27.0199i	0.948797i	−0.880310	+	0.474398i	$0.842666\pi$
$812$	0	0
$813$	0	0
$814$	−16.1983	−0.567750
$815$	9.97758	0.349499
$816$	0	0
$817$	− 1.96348i	− 0.0686934i
$818$	−32.9802	−1.15312
$819$	0	0
$820$	1.68921	0.0589898
$821$	− 9.30787i	− 0.324847i	−0.986721	−	0.162423i	$-0.948069\pi$
$821$	− 9.30787i	− 0.324847i	0.986721	−	0.162423i	$-0.0519311\pi$
$822$	0	0
$823$	−40.5186	−1.41239	−0.706195	−	0.708018i	$-0.749590\pi$
$823$	−40.5186	−1.41239	−0.706195	+	0.708018i	$0.749590\pi$
$824$	−0.520042	−0.0181165
$825$	0	0
$826$	0	0
$827$	− 14.0131i	− 0.487284i	−0.969865	−	0.243642i	$-0.921658\pi$
$827$	− 14.0131i	− 0.487284i	0.969865	−	0.243642i	$-0.0783423\pi$
$828$	0	0
$829$	21.7707i	0.756129i	0.925779	+	0.378064i	$0.123410\pi$
$829$	21.7707i	0.756129i	−0.925779	+	0.378064i	$0.876590\pi$
$830$	14.3490i	0.498061i
$831$	0	0
$832$	0.0681483i	0.00236262i
$833$	0	0
$834$	0	0
$835$	13.7778	0.476801
$836$	−8.10394	−0.280281
$837$	0	0
$838$	− 0.300470i	− 0.0103796i
$839$	−53.7026	−1.85402	−0.927009	−	0.375039i	$-0.877629\pi$
$839$	−53.7026	−1.85402	−0.927009	+	0.375039i	$0.877629\pi$
$840$	0	0
$841$	28.1918	0.972132
$842$	28.8625i	0.994666i
$843$	0	0
$844$	14.5619	0.501241
$845$	12.9954	0.447054
$846$	0	0
$847$	0	0
$848$	− 13.2268i	− 0.454210i
$849$	0	0
$850$	7.32780i	0.251342i
$851$	− 15.1825i	− 0.520451i
$852$	0	0
$853$	− 7.95355i	− 0.272324i	−0.990687	−	0.136162i	$-0.956523\pi$
$853$	− 7.95355i	− 0.272324i	0.990687	−	0.136162i	$-0.0434768\pi$
$854$	0	0
$855$	0	0
$856$	6.12701	0.209417
$857$	−2.29303	−0.0783283	−0.0391641	−	0.999233i	$-0.512470\pi$
$857$	−2.29303	−0.0783283	−0.0391641	+	0.999233i	$0.512470\pi$
$858$	0	0
$859$	44.0003i	1.50127i	0.660716	+	0.750636i	$0.270253\pi$
$859$	44.0003i	1.50127i	−0.660716	+	0.750636i	$0.729747\pi$
$860$	0.964724	0.0328968
$861$	0	0
$862$	−33.5897	−1.14407
$863$	33.1352i	1.12794i	0.825797	+	0.563968i	$0.190726\pi$
$863$	33.1352i	1.12794i	−0.825797	+	0.563968i	$0.809274\pi$
$864$	0	0
$865$	−8.27276	−0.281282
$866$	11.2207	0.381296
$867$	0	0
$868$	0	0
$869$	− 69.8787i	− 2.37047i
$870$	0	0
$871$	− 0.726659i	− 0.0246219i
$872$	− 13.5546i	− 0.459016i
$873$	0	0
$874$	− 7.59575i	− 0.256930i
$875$	0	0
$876$	0	0
$877$	−5.68871	−0.192094	−0.0960471	−	0.995377i	$-0.530620\pi$
$877$	−5.68871	−0.192094	−0.0960471	+	0.995377i	$0.530620\pi$
$878$	−16.8673	−0.569243
$879$	0	0
$880$	− 3.98174i	− 0.134224i
$881$	22.4073	0.754923	0.377461	−	0.926025i	$-0.376797\pi$
$881$	22.4073	0.754923	0.377461	+	0.926025i	$0.376797\pi$
$882$	0	0
$883$	−18.7564	−0.631204	−0.315602	−	0.948892i	$-0.602206\pi$
$883$	−18.7564	−0.631204	−0.315602	+	0.948892i	$0.602206\pi$
$884$	0.499378i	0.0167959i
$885$	0	0
$886$	2.05646	0.0690880
$887$	−1.47090	−0.0493880	−0.0246940	−	0.999695i	$-0.507861\pi$
$887$	−1.47090	−0.0493880	−0.0246940	+	0.999695i	$0.507861\pi$
$888$	0	0
$889$	0	0
$890$	− 1.82791i	− 0.0612717i
$891$	0	0
$892$	− 23.6609i	− 0.792227i
$893$	− 3.38161i	− 0.113161i
$894$	0	0
$895$	11.5182i	0.385009i
$896$	0	0
$897$	0	0
$898$	−12.5892	−0.420108
$899$	4.34066	0.144769
$900$	0	0
$901$	− 96.9233i	− 3.22898i
$902$	−6.72600	−0.223951
$903$	0	0
$904$	11.6982	0.389076
$905$	− 16.5924i	− 0.551549i
$906$	0	0
$907$	18.8459	0.625767	0.312883	−	0.949792i	$-0.398705\pi$
$907$	18.8459	0.625767	0.312883	+	0.949792i	$0.398705\pi$
$908$	8.64048	0.286744
$909$	0	0
$910$	0	0
$911$	− 5.45859i	− 0.180851i	−0.995903	−	0.0904256i	$-0.971177\pi$
$911$	− 5.45859i	− 0.180851i	0.995903	−	0.0904256i	$-0.0288227\pi$
$912$	0	0
$913$	− 57.1339i	− 1.89086i
$914$	34.9529i	1.15614i
$915$	0	0
$916$	− 14.0754i	− 0.465063i
$917$	0	0
$918$	0	0
$919$	29.9611	0.988324	0.494162	−	0.869370i	$-0.335475\pi$
$919$	29.9611	0.988324	0.494162	+	0.869370i	$0.335475\pi$
$920$	3.73205	0.123042
$921$	0	0
$922$	− 23.3750i	− 0.769814i
$923$	0.677003	0.0222838
$924$	0	0
$925$	−4.06815	−0.133760
$926$	− 6.35693i	− 0.208902i
$927$	0	0
$928$	−0.898979	−0.0295104
$929$	46.7863	1.53501	0.767504	−	0.641044i	$-0.221498\pi$
$929$	46.7863	1.53501	0.767504	+	0.641044i	$0.221498\pi$
$930$	0	0
$931$	0	0
$932$	− 20.1976i	− 0.661593i
$933$	0	0
$934$	7.79315i	0.255000i
$935$	− 29.1774i	− 0.954203i
$936$	0	0
$937$	19.1632i	0.626036i	0.949747	+	0.313018i	$0.101340\pi$
$937$	19.1632i	0.626036i	−0.949747	+	0.313018i	$0.898660\pi$
$938$	0	0
$939$	0	0
$940$	1.66150	0.0541921
$941$	−50.4949	−1.64609	−0.823043	−	0.567979i	$-0.807726\pi$
$941$	−50.4949	−1.64609	−0.823043	+	0.567979i	$0.807726\pi$
$942$	0	0
$943$	− 6.30422i	− 0.205294i
$944$	−10.6422	−0.346375
$945$	0	0
$946$	−3.84128	−0.124891
$947$	36.0618i	1.17185i	0.810365	+	0.585925i	$0.199269\pi$
$947$	36.0618i	1.17185i	−0.810365	+	0.585925i	$0.800731\pi$
$948$	0	0
$949$	−0.794395	−0.0257872
$950$	−2.03528	−0.0660331
$951$	0	0
$952$	0	0
$953$	− 29.7944i	− 0.965137i	−0.875858	−	0.482568i	$-0.839704\pi$
$953$	− 29.7944i	− 0.965137i	0.875858	−	0.482568i	$-0.160296\pi$
$954$	0	0
$955$	18.4336i	0.596498i
$956$	− 23.5040i	− 0.760173i
$957$	0	0
$958$	27.5040i	0.888613i
$959$	0	0
$960$	0	0
$961$	7.68629	0.247945
$962$	−0.277238	−0.00893850
$963$	0	0
$964$	14.8685i	0.478883i
$965$	−6.35827	−0.204680
$966$	0	0
$967$	−11.5198	−0.370453	−0.185226	−	0.982696i	$-0.559302\pi$
$967$	−11.5198	−0.370453	−0.185226	+	0.982696i	$0.559302\pi$
$968$	4.85425i	0.156022i
$969$	0	0
$970$	17.1502	0.550659
$971$	−42.4693	−1.36291	−0.681453	−	0.731862i	$-0.738652\pi$
$971$	−42.4693	−1.36291	−0.681453	+	0.731862i	$0.738652\pi$
$972$	0	0
$973$	0	0
$974$	35.4468i	1.13579i
$975$	0	0
$976$	− 7.52056i	− 0.240727i
$977$	18.1610i	0.581023i	0.956871	+	0.290512i	$0.0938255\pi$
$977$	18.1610i	0.581023i	−0.956871	+	0.290512i	$0.906175\pi$
$978$	0	0
$979$	7.27827i	0.232614i
$980$	0	0
$981$	0	0
$982$	−1.64349	−0.0524458
$983$	12.3998	0.395491	0.197746	−	0.980253i	$-0.436638\pi$
$983$	12.3998	0.395491	0.197746	+	0.980253i	$0.436638\pi$
$984$	0	0
$985$	− 12.8389i	− 0.409082i
$986$	−6.58755	−0.209790
$987$	0	0
$988$	−0.138701	−0.00441266
$989$	− 3.60040i	− 0.114486i
$990$	0	0
$991$	21.7201	0.689962	0.344981	−	0.938610i	$-0.387885\pi$
$991$	21.7201	0.689962	0.344981	+	0.938610i	$0.387885\pi$
$992$	4.82843	0.153303
$993$	0	0
$994$	0	0
$995$	− 9.62962i	− 0.305280i
$996$	0	0
$997$	− 42.4763i	− 1.34524i	−0.739989	−	0.672619i	$-0.765169\pi$
$997$	− 42.4763i	− 1.34524i	0.739989	−	0.672619i	$-0.234831\pi$
$998$	− 34.7330i	− 1.09945i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.e.881.5		8
3.2	odd	2		4410.2.b.b.881.4		8
7.4	even	3		630.2.be.a.341.1	✓	8
7.5	odd	6		630.2.be.b.521.3	yes	8
7.6	odd	2		4410.2.b.b.881.5		8
21.5	even	6		630.2.be.a.521.1	yes	8
21.11	odd	6		630.2.be.b.341.3	yes	8
21.20	even	2	inner	4410.2.b.e.881.4		8
35.4	even	6		3150.2.bf.b.1601.4		8
35.12	even	12		3150.2.bp.f.899.4		8
35.18	odd	12		3150.2.bp.a.1349.4		8
35.19	odd	6		3150.2.bf.c.1151.2		8
35.32	odd	12		3150.2.bp.d.1349.1		8
35.33	even	12		3150.2.bp.c.899.1		8
105.32	even	12		3150.2.bp.c.1349.1		8
105.47	odd	12		3150.2.bp.a.899.4		8
105.53	even	12		3150.2.bp.f.1349.4		8
105.68	odd	12		3150.2.bp.d.899.1		8
105.74	odd	6		3150.2.bf.c.1601.2		8
105.89	even	6		3150.2.bf.b.1151.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.be.a.341.1	✓	8	7.4	even	3
630.2.be.a.521.1	yes	8	21.5	even	6
630.2.be.b.341.3	yes	8	21.11	odd	6
630.2.be.b.521.3	yes	8	7.5	odd	6
3150.2.bf.b.1151.4		8	105.89	even	6
3150.2.bf.b.1601.4		8	35.4	even	6
3150.2.bf.c.1151.2		8	35.19	odd	6
3150.2.bf.c.1601.2		8	105.74	odd	6
3150.2.bp.a.899.4		8	105.47	odd	12
3150.2.bp.a.1349.4		8	35.18	odd	12
3150.2.bp.c.899.1		8	35.33	even	12
3150.2.bp.c.1349.1		8	105.32	even	12
3150.2.bp.d.899.1		8	105.68	odd	12
3150.2.bp.d.1349.1		8	35.32	odd	12
3150.2.bp.f.899.4		8	35.12	even	12
3150.2.bp.f.1349.4		8	105.53	even	12
4410.2.b.b.881.4		8	3.2	odd	2
4410.2.b.b.881.5		8	7.6	odd	2
4410.2.b.e.881.4		8	21.20	even	2	inner
4410.2.b.e.881.5		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000i q^{8} +1.00000i q^{10} -3.98174i q^{11} -0.0681483i q^{13} +1.00000 q^{16} +7.32780 q^{17} +2.03528i q^{19} -1.00000 q^{20} +3.98174 q^{22} +3.73205i q^{23} +1.00000 q^{25} +0.0681483 q^{26} +0.898979i q^{29} -4.82843i q^{31} +1.00000i q^{32} +7.32780i q^{34} -4.06815 q^{37} -2.03528 q^{38} -1.00000i q^{40} -1.68921 q^{41} -0.964724 q^{43} +3.98174i q^{44} -3.73205 q^{46} -1.66150 q^{47} +1.00000i q^{50} +0.0681483i q^{52} -13.2268i q^{53} -3.98174i q^{55} -0.898979 q^{58} -10.6422 q^{59} -7.52056i q^{61} +4.82843 q^{62} -1.00000 q^{64} -0.0681483i q^{65} +10.6629 q^{67} -7.32780 q^{68} +9.93426i q^{71} -11.6569i q^{73} -4.06815i q^{74} -2.03528i q^{76} +17.5498 q^{79} +1.00000 q^{80} -1.68921i q^{82} +14.3490 q^{83} +7.32780 q^{85} -0.964724i q^{86} -3.98174 q^{88} -1.82791 q^{89} -3.73205i q^{92} -1.66150i q^{94} +2.03528i q^{95} -17.1502i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 8 q^{5} + 8 q^{16} - 8 q^{20} + 8 q^{25} + 16 q^{26} - 48 q^{37} - 8 q^{38} - 32 q^{41} - 16 q^{43} - 16 q^{46} - 16 q^{47} + 32 q^{58} - 48 q^{59} + 16 q^{62} - 8 q^{64} + 48 q^{67} + 48 q^{79}+ \cdots - 32 q^{89}+O(q^{100}) \) 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

L-functions

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