LMFDB - Embedded newform 4410.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(1,4410)]

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([0, 0, 0]))

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

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

chi := DirichletCharacter("4410.1");

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.a (trivial)

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:	yes
Analytic conductor:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 630)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	4410.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} +0.645751 q^{11} +4.64575 q^{13} +1.00000 q^{16} -7.29150 q^{17} -6.29150 q^{19} -1.00000 q^{20} -0.645751 q^{22} -3.00000 q^{23} +1.00000 q^{25} -4.64575 q^{26} -2.00000 q^{31} -1.00000 q^{32} +7.29150 q^{34} +9.93725 q^{37} +6.29150 q^{38} +1.00000 q^{40} +6.64575 q^{41} +0.708497 q^{43} +0.645751 q^{44} +3.00000 q^{46} +11.5830 q^{47} -1.00000 q^{50} +4.64575 q^{52} -10.2915 q^{53} -0.645751 q^{55} +1.29150 q^{59} -0.708497 q^{61} +2.00000 q^{62} +1.00000 q^{64} -4.64575 q^{65} +15.2915 q^{67} -7.29150 q^{68} +13.2915 q^{71} -2.00000 q^{73} -9.93725 q^{74} -6.29150 q^{76} +0.708497 q^{79} -1.00000 q^{80} -6.64575 q^{82} +4.70850 q^{83} +7.29150 q^{85} -0.708497 q^{86} -0.645751 q^{88} -14.5830 q^{89} -3.00000 q^{92} -11.5830 q^{94} +6.29150 q^{95} +4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 4 q^{13} + 2 q^{16} - 4 q^{17} - 2 q^{19} - 2 q^{20} + 4 q^{22} - 6 q^{23} + 2 q^{25} - 4 q^{26} - 4 q^{31} - 2 q^{32} + 4 q^{34} + 4 q^{37} + 2 q^{38} + 2 q^{40} + 8 q^{41} + 12 q^{43} - 4 q^{44} + 6 q^{46} + 2 q^{47} - 2 q^{50} + 4 q^{52} - 10 q^{53} + 4 q^{55} - 8 q^{59} - 12 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{65} + 20 q^{67} - 4 q^{68} + 16 q^{71} - 4 q^{73} - 4 q^{74} - 2 q^{76} + 12 q^{79} - 2 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} - 12 q^{86} + 4 q^{88} - 8 q^{89} - 6 q^{92} - 2 q^{94} + 2 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 4 * q^13 + 2 * q^16 - 4 * q^17 - 2 * q^19 - 2 * q^20 + 4 * q^22 - 6 * q^23 + 2 * q^25 - 4 * q^26 - 4 * q^31 - 2 * q^32 + 4 * q^34 + 4 * q^37 + 2 * q^38 + 2 * q^40 + 8 * q^41 + 12 * q^43 - 4 * q^44 + 6 * q^46 + 2 * q^47 - 2 * q^50 + 4 * q^52 - 10 * q^53 + 4 * q^55 - 8 * q^59 - 12 * q^61 + 4 * q^62 + 2 * q^64 - 4 * q^65 + 20 * q^67 - 4 * q^68 + 16 * q^71 - 4 * q^73 - 4 * q^74 - 2 * q^76 + 12 * q^79 - 2 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 - 12 * q^86 + 4 * q^88 - 8 * q^89 - 6 * q^92 - 2 * q^94 + 2 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	0.645751	0.194701	0.0973507	−	0.995250i	$-0.468963\pi$
$11$	0.645751	0.194701	0.0973507	+	0.995250i	$0.468963\pi$
$12$	0	0
$13$	4.64575	1.28850	0.644250	−	0.764815i	$-0.277170\pi$
$13$	4.64575	1.28850	0.644250	+	0.764815i	$0.277170\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.29150	−1.76845	−0.884225	−	0.467062i	$-0.845312\pi$
$17$	−7.29150	−1.76845	−0.884225	+	0.467062i	$0.845312\pi$
$18$	0	0
$19$	−6.29150	−1.44337	−0.721685	−	0.692222i	$-0.756632\pi$
$19$	−6.29150	−1.44337	−0.721685	+	0.692222i	$0.756632\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−0.645751	−0.137675
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.64575	−0.911107
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.29150	1.25048
$35$	0	0
$36$	0	0
$37$	9.93725	1.63367	0.816837	−	0.576868i	$-0.195725\pi$
$37$	9.93725	1.63367	0.816837	+	0.576868i	$0.195725\pi$
$38$	6.29150	1.02062
$39$	0	0
$40$	1.00000	0.158114
$41$	6.64575	1.03789	0.518946	−	0.854807i	$-0.326325\pi$
$41$	6.64575	1.03789	0.518946	+	0.854807i	$0.326325\pi$
$42$	0	0
$43$	0.708497	0.108045	0.0540224	−	0.998540i	$-0.482796\pi$
$43$	0.708497	0.108045	0.0540224	+	0.998540i	$0.482796\pi$
$44$	0.645751	0.0973507
$45$	0	0
$46$	3.00000	0.442326
$47$	11.5830	1.68955	0.844777	−	0.535118i	$-0.179733\pi$
$47$	11.5830	1.68955	0.844777	+	0.535118i	$0.179733\pi$
$48$	0	0
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.64575	0.644250
$53$	−10.2915	−1.41365	−0.706823	−	0.707390i	$-0.749872\pi$
$53$	−10.2915	−1.41365	−0.706823	+	0.707390i	$0.749872\pi$
$54$	0	0
$55$	−0.645751	−0.0870731
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.29150	0.168139	0.0840697	−	0.996460i	$-0.473208\pi$
$59$	1.29150	0.168139	0.0840697	+	0.996460i	$0.473208\pi$
$60$	0	0
$61$	−0.708497	−0.0907138	−0.0453569	−	0.998971i	$-0.514443\pi$
$61$	−0.708497	−0.0907138	−0.0453569	+	0.998971i	$0.514443\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.64575	−0.576235
$66$	0	0
$67$	15.2915	1.86815	0.934077	−	0.357071i	$-0.116225\pi$
$67$	15.2915	1.86815	0.934077	+	0.357071i	$0.116225\pi$
$68$	−7.29150	−0.884225
$69$	0	0
$70$	0	0
$71$	13.2915	1.57741	0.788706	−	0.614771i	$-0.210752\pi$
$71$	13.2915	1.57741	0.788706	+	0.614771i	$0.210752\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−9.93725	−1.15518
$75$	0	0
$76$	−6.29150	−0.721685
$77$	0	0
$78$	0	0
$79$	0.708497	0.0797122	0.0398561	−	0.999205i	$-0.487310\pi$
$79$	0.708497	0.0797122	0.0398561	+	0.999205i	$0.487310\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−6.64575	−0.733900
$83$	4.70850	0.516825	0.258412	−	0.966035i	$-0.416801\pi$
$83$	4.70850	0.516825	0.258412	+	0.966035i	$0.416801\pi$
$84$	0	0
$85$	7.29150	0.790875
$86$	−0.708497	−0.0763992
$87$	0	0
$88$	−0.645751	−0.0688373
$89$	−14.5830	−1.54580	−0.772898	−	0.634531i	$-0.781193\pi$
$89$	−14.5830	−1.54580	−0.772898	+	0.634531i	$0.781193\pi$
$90$	0	0
$91$	0	0
$92$	−3.00000	−0.312772
$93$	0	0
$94$	−11.5830	−1.19470
$95$	6.29150	0.645495
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−6.70850	−0.661008	−0.330504	−	0.943805i	$-0.607219\pi$
$103$	−6.70850	−0.661008	−0.330504	+	0.943805i	$0.607219\pi$
$104$	−4.64575	−0.455553
$105$	0	0
$106$	10.2915	0.999599
$107$	−15.8745	−1.53465	−0.767323	−	0.641260i	$-0.778412\pi$
$107$	−15.8745	−1.53465	−0.767323	+	0.641260i	$0.778412\pi$
$108$	0	0
$109$	−0.583005	−0.0558418	−0.0279209	−	0.999610i	$-0.508889\pi$
$109$	−0.583005	−0.0558418	−0.0279209	+	0.999610i	$0.508889\pi$
$110$	0.645751	0.0615700
$111$	0	0
$112$	0	0
$113$	−8.58301	−0.807421	−0.403711	−	0.914887i	$-0.632280\pi$
$113$	−8.58301	−0.807421	−0.403711	+	0.914887i	$0.632280\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	0	0
$118$	−1.29150	−0.118892
$119$	0	0
$120$	0	0
$121$	−10.5830	−0.962091
$122$	0.708497	0.0641443
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.6458	1.29960	0.649800	−	0.760105i	$-0.274853\pi$
$127$	14.6458	1.29960	0.649800	+	0.760105i	$0.274853\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.64575	0.407459
$131$	0.645751	0.0564196	0.0282098	−	0.999602i	$-0.491019\pi$
$131$	0.645751	0.0564196	0.0282098	+	0.999602i	$0.491019\pi$
$132$	0	0
$133$	0	0
$134$	−15.2915	−1.32098
$135$	0	0
$136$	7.29150	0.625241
$137$	15.8745	1.35625	0.678125	−	0.734946i	$-0.262793\pi$
$137$	15.8745	1.35625	0.678125	+	0.734946i	$0.262793\pi$
$138$	0	0
$139$	−10.5830	−0.897639	−0.448819	−	0.893622i	$-0.648155\pi$
$139$	−10.5830	−0.897639	−0.448819	+	0.893622i	$0.648155\pi$
$140$	0	0
$141$	0	0
$142$	−13.2915	−1.11540
$143$	3.00000	0.250873
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	9.93725	0.816837
$149$	9.87451	0.808951	0.404476	−	0.914549i	$-0.367454\pi$
$149$	9.87451	0.808951	0.404476	+	0.914549i	$0.367454\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	6.29150	0.510308
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	15.3542	1.22540	0.612701	−	0.790315i	$-0.290083\pi$
$157$	15.3542	1.22540	0.612701	+	0.790315i	$0.290083\pi$
$158$	−0.708497	−0.0563650
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	6.64575	0.518946
$165$	0	0
$166$	−4.70850	−0.365450
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	8.58301	0.660231
$170$	−7.29150	−0.559233
$171$	0	0
$172$	0.708497	0.0540224
$173$	7.70850	0.586066	0.293033	−	0.956102i	$-0.405335\pi$
$173$	7.70850	0.586066	0.293033	+	0.956102i	$0.405335\pi$
$174$	0	0
$175$	0	0
$176$	0.645751	0.0486753
$177$	0	0
$178$	14.5830	1.09304
$179$	10.0627	0.752125	0.376062	−	0.926594i	$-0.377278\pi$
$179$	10.0627	0.752125	0.376062	+	0.926594i	$0.377278\pi$
$180$	0	0
$181$	11.2915	0.839291	0.419645	−	0.907688i	$-0.362154\pi$
$181$	11.2915	0.839291	0.419645	+	0.907688i	$0.362154\pi$
$182$	0	0
$183$	0	0
$184$	3.00000	0.221163
$185$	−9.93725	−0.730601
$186$	0	0
$187$	−4.70850	−0.344319
$188$	11.5830	0.844777
$189$	0	0
$190$	−6.29150	−0.456434
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	23.8745	1.71852	0.859262	−	0.511535i	$-0.170923\pi$
$193$	23.8745	1.71852	0.859262	+	0.511535i	$0.170923\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	0	0
$197$	4.29150	0.305757	0.152878	−	0.988245i	$-0.451146\pi$
$197$	4.29150	0.305757	0.152878	+	0.988245i	$0.451146\pi$
$198$	0	0
$199$	13.8745	0.983538	0.491769	−	0.870726i	$-0.336350\pi$
$199$	13.8745	0.983538	0.491769	+	0.870726i	$0.336350\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	12.0000	0.844317
$203$	0	0
$204$	0	0
$205$	−6.64575	−0.464159
$206$	6.70850	0.467403
$207$	0	0
$208$	4.64575	0.322125
$209$	−4.06275	−0.281026
$210$	0	0
$211$	−2.29150	−0.157754	−0.0788768	−	0.996884i	$-0.525133\pi$
$211$	−2.29150	−0.157754	−0.0788768	+	0.996884i	$0.525133\pi$
$212$	−10.2915	−0.706823
$213$	0	0
$214$	15.8745	1.08516
$215$	−0.708497	−0.0483191
$216$	0	0
$217$	0	0
$218$	0.583005	0.0394861
$219$	0	0
$220$	−0.645751	−0.0435365
$221$	−33.8745	−2.27865
$222$	0	0
$223$	5.29150	0.354345	0.177173	−	0.984180i	$-0.443305\pi$
$223$	5.29150	0.354345	0.177173	+	0.984180i	$0.443305\pi$
$224$	0	0
$225$	0	0
$226$	8.58301	0.570933
$227$	−7.29150	−0.483954	−0.241977	−	0.970282i	$-0.577796\pi$
$227$	−7.29150	−0.483954	−0.241977	+	0.970282i	$0.577796\pi$
$228$	0	0
$229$	−6.70850	−0.443310	−0.221655	−	0.975125i	$-0.571146\pi$
$229$	−6.70850	−0.443310	−0.221655	+	0.975125i	$0.571146\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	0	0
$233$	27.8745	1.82612	0.913060	−	0.407826i	$-0.133713\pi$
$233$	27.8745	1.82612	0.913060	+	0.407826i	$0.133713\pi$
$234$	0	0
$235$	−11.5830	−0.755592
$236$	1.29150	0.0840697
$237$	0	0
$238$	0	0
$239$	8.58301	0.555188	0.277594	−	0.960698i	$-0.410463\pi$
$239$	8.58301	0.555188	0.277594	+	0.960698i	$0.410463\pi$
$240$	0	0
$241$	4.41699	0.284524	0.142262	−	0.989829i	$-0.454563\pi$
$241$	4.41699	0.284524	0.142262	+	0.989829i	$0.454563\pi$
$242$	10.5830	0.680301
$243$	0	0
$244$	−0.708497	−0.0453569
$245$	0	0
$246$	0	0
$247$	−29.2288	−1.85978
$248$	2.00000	0.127000
$249$	0	0
$250$	1.00000	0.0632456
$251$	−1.93725	−0.122278	−0.0611392	−	0.998129i	$-0.519473\pi$
$251$	−1.93725	−0.122278	−0.0611392	+	0.998129i	$0.519473\pi$
$252$	0	0
$253$	−1.93725	−0.121794
$254$	−14.6458	−0.918956
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.7085	1.04225	0.521124	−	0.853481i	$-0.325513\pi$
$257$	16.7085	1.04225	0.521124	+	0.853481i	$0.325513\pi$
$258$	0	0
$259$	0	0
$260$	−4.64575	−0.288117
$261$	0	0
$262$	−0.645751	−0.0398946
$263$	−14.5830	−0.899227	−0.449613	−	0.893223i	$-0.648438\pi$
$263$	−14.5830	−0.899227	−0.449613	+	0.893223i	$0.648438\pi$
$264$	0	0
$265$	10.2915	0.632202
$266$	0	0
$267$	0	0
$268$	15.2915	0.934077
$269$	9.87451	0.602059	0.301030	−	0.953615i	$-0.402670\pi$
$269$	9.87451	0.602059	0.301030	+	0.953615i	$0.402670\pi$
$270$	0	0
$271$	−25.1660	−1.52873	−0.764363	−	0.644786i	$-0.776946\pi$
$271$	−25.1660	−1.52873	−0.764363	+	0.644786i	$0.776946\pi$
$272$	−7.29150	−0.442112
$273$	0	0
$274$	−15.8745	−0.959014
$275$	0.645751	0.0389403
$276$	0	0
$277$	5.41699	0.325476	0.162738	−	0.986669i	$-0.447968\pi$
$277$	5.41699	0.325476	0.162738	+	0.986669i	$0.447968\pi$
$278$	10.5830	0.634726
$279$	0	0
$280$	0	0
$281$	−10.5203	−0.627586	−0.313793	−	0.949491i	$-0.601600\pi$
$281$	−10.5203	−0.627586	−0.313793	+	0.949491i	$0.601600\pi$
$282$	0	0
$283$	−23.8745	−1.41919	−0.709596	−	0.704609i	$-0.751123\pi$
$283$	−23.8745	−1.41919	−0.709596	+	0.704609i	$0.751123\pi$
$284$	13.2915	0.788706
$285$	0	0
$286$	−3.00000	−0.177394
$287$	0	0
$288$	0	0
$289$	36.1660	2.12741
$290$	0	0
$291$	0	0
$292$	−2.00000	−0.117041
$293$	22.2915	1.30228	0.651142	−	0.758956i	$-0.274290\pi$
$293$	22.2915	1.30228	0.651142	+	0.758956i	$0.274290\pi$
$294$	0	0
$295$	−1.29150	−0.0751942
$296$	−9.93725	−0.577591
$297$	0	0
$298$	−9.87451	−0.572015
$299$	−13.9373	−0.806012
$300$	0	0
$301$	0	0
$302$	−14.0000	−0.805609
$303$	0	0
$304$	−6.29150	−0.360842
$305$	0.708497	0.0405684
$306$	0	0
$307$	23.2915	1.32932	0.664658	−	0.747148i	$-0.268577\pi$
$307$	23.2915	1.32932	0.664658	+	0.747148i	$0.268577\pi$
$308$	0	0
$309$	0	0
$310$	−2.00000	−0.113592
$311$	9.87451	0.559932	0.279966	−	0.960010i	$-0.409677\pi$
$311$	9.87451	0.559932	0.279966	+	0.960010i	$0.409677\pi$
$312$	0	0
$313$	−29.8745	−1.68861	−0.844304	−	0.535865i	$-0.819985\pi$
$313$	−29.8745	−1.68861	−0.844304	+	0.535865i	$0.819985\pi$
$314$	−15.3542	−0.866490
$315$	0	0
$316$	0.708497	0.0398561
$317$	32.5830	1.83004	0.915022	−	0.403404i	$-0.132173\pi$
$317$	32.5830	1.83004	0.915022	+	0.403404i	$0.132173\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	45.8745	2.55253
$324$	0	0
$325$	4.64575	0.257700
$326$	−8.00000	−0.443079
$327$	0	0
$328$	−6.64575	−0.366950
$329$	0	0
$330$	0	0
$331$	29.4575	1.61913	0.809566	−	0.587029i	$-0.199703\pi$
$331$	29.4575	1.61913	0.809566	+	0.587029i	$0.199703\pi$
$332$	4.70850	0.258412
$333$	0	0
$334$	−9.00000	−0.492458
$335$	−15.2915	−0.835464
$336$	0	0
$337$	0.708497	0.0385943	0.0192972	−	0.999814i	$-0.493857\pi$
$337$	0.708497	0.0385943	0.0192972	+	0.999814i	$0.493857\pi$
$338$	−8.58301	−0.466854
$339$	0	0
$340$	7.29150	0.395437
$341$	−1.29150	−0.0699388
$342$	0	0
$343$	0	0
$344$	−0.708497	−0.0381996
$345$	0	0
$346$	−7.70850	−0.414411
$347$	−1.29150	−0.0693315	−0.0346657	−	0.999399i	$-0.511037\pi$
$347$	−1.29150	−0.0693315	−0.0346657	+	0.999399i	$0.511037\pi$
$348$	0	0
$349$	−22.5830	−1.20884	−0.604420	−	0.796666i	$-0.706595\pi$
$349$	−22.5830	−1.20884	−0.604420	+	0.796666i	$0.706595\pi$
$350$	0	0
$351$	0	0
$352$	−0.645751	−0.0344187
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	0	0
$355$	−13.2915	−0.705440
$356$	−14.5830	−0.772898
$357$	0	0
$358$	−10.0627	−0.531833
$359$	−2.58301	−0.136326	−0.0681629	−	0.997674i	$-0.521714\pi$
$359$	−2.58301	−0.136326	−0.0681629	+	0.997674i	$0.521714\pi$
$360$	0	0
$361$	20.5830	1.08332
$362$	−11.2915	−0.593468
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	21.3542	1.11468	0.557341	−	0.830283i	$-0.311821\pi$
$367$	21.3542	1.11468	0.557341	+	0.830283i	$0.311821\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	9.93725	0.516613
$371$	0	0
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	4.70850	0.243471
$375$	0	0
$376$	−11.5830	−0.597348
$377$	0	0
$378$	0	0
$379$	12.2915	0.631372	0.315686	−	0.948864i	$-0.397765\pi$
$379$	12.2915	0.631372	0.315686	+	0.948864i	$0.397765\pi$
$380$	6.29150	0.322747
$381$	0	0
$382$	−6.00000	−0.306987
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	0	0
$385$	0	0
$386$	−23.8745	−1.21518
$387$	0	0
$388$	4.00000	0.203069
$389$	25.2915	1.28233	0.641165	−	0.767403i	$-0.278451\pi$
$389$	25.2915	1.28233	0.641165	+	0.767403i	$0.278451\pi$
$390$	0	0
$391$	21.8745	1.10624
$392$	0	0
$393$	0	0
$394$	−4.29150	−0.216203
$395$	−0.708497	−0.0356484
$396$	0	0
$397$	1.41699	0.0711169	0.0355585	−	0.999368i	$-0.488679\pi$
$397$	1.41699	0.0711169	0.0355585	+	0.999368i	$0.488679\pi$
$398$	−13.8745	−0.695466
$399$	0	0
$400$	1.00000	0.0500000
$401$	−9.22876	−0.460862	−0.230431	−	0.973089i	$-0.574014\pi$
$401$	−9.22876	−0.460862	−0.230431	+	0.973089i	$0.574014\pi$
$402$	0	0
$403$	−9.29150	−0.462843
$404$	−12.0000	−0.597022
$405$	0	0
$406$	0	0
$407$	6.41699	0.318079
$408$	0	0
$409$	−11.4170	−0.564534	−0.282267	−	0.959336i	$-0.591086\pi$
$409$	−11.4170	−0.564534	−0.282267	+	0.959336i	$0.591086\pi$
$410$	6.64575	0.328210
$411$	0	0
$412$	−6.70850	−0.330504
$413$	0	0
$414$	0	0
$415$	−4.70850	−0.231131
$416$	−4.64575	−0.227777
$417$	0	0
$418$	4.06275	0.198715
$419$	−37.9373	−1.85336	−0.926678	−	0.375856i	$-0.877349\pi$
$419$	−37.9373	−1.85336	−0.926678	+	0.375856i	$0.877349\pi$
$420$	0	0
$421$	−31.8745	−1.55347	−0.776734	−	0.629828i	$-0.783125\pi$
$421$	−31.8745	−1.55347	−0.776734	+	0.629828i	$0.783125\pi$
$422$	2.29150	0.111549
$423$	0	0
$424$	10.2915	0.499800
$425$	−7.29150	−0.353690
$426$	0	0
$427$	0	0
$428$	−15.8745	−0.767323
$429$	0	0
$430$	0.708497	0.0341668
$431$	33.8745	1.63168	0.815839	−	0.578279i	$-0.196276\pi$
$431$	33.8745	1.63168	0.815839	+	0.578279i	$0.196276\pi$
$432$	0	0
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	0	0
$436$	−0.583005	−0.0279209
$437$	18.8745	0.902890
$438$	0	0
$439$	9.16601	0.437470	0.218735	−	0.975784i	$-0.429807\pi$
$439$	9.16601	0.437470	0.218735	+	0.975784i	$0.429807\pi$
$440$	0.645751	0.0307850
$441$	0	0
$442$	33.8745	1.61125
$443$	28.7085	1.36398	0.681991	−	0.731361i	$-0.261114\pi$
$443$	28.7085	1.36398	0.681991	+	0.731361i	$0.261114\pi$
$444$	0	0
$445$	14.5830	0.691301
$446$	−5.29150	−0.250560
$447$	0	0
$448$	0	0
$449$	−23.8118	−1.12375	−0.561873	−	0.827223i	$-0.689919\pi$
$449$	−23.8118	−1.12375	−0.561873	+	0.827223i	$0.689919\pi$
$450$	0	0
$451$	4.29150	0.202079
$452$	−8.58301	−0.403711
$453$	0	0
$454$	7.29150	0.342207
$455$	0	0
$456$	0	0
$457$	−17.2915	−0.808862	−0.404431	−	0.914568i	$-0.632530\pi$
$457$	−17.2915	−0.808862	−0.404431	+	0.914568i	$0.632530\pi$
$458$	6.70850	0.313467
$459$	0	0
$460$	3.00000	0.139876
$461$	17.1660	0.799501	0.399750	−	0.916624i	$-0.369097\pi$
$461$	17.1660	0.799501	0.399750	+	0.916624i	$0.369097\pi$
$462$	0	0
$463$	15.9373	0.740667	0.370334	−	0.928899i	$-0.379243\pi$
$463$	15.9373	0.740667	0.370334	+	0.928899i	$0.379243\pi$
$464$	0	0
$465$	0	0
$466$	−27.8745	−1.29126
$467$	−19.2915	−0.892704	−0.446352	−	0.894857i	$-0.647277\pi$
$467$	−19.2915	−0.892704	−0.446352	+	0.894857i	$0.647277\pi$
$468$	0	0
$469$	0	0
$470$	11.5830	0.534284
$471$	0	0
$472$	−1.29150	−0.0594462
$473$	0.457513	0.0210365
$474$	0	0
$475$	−6.29150	−0.288674
$476$	0	0
$477$	0	0
$478$	−8.58301	−0.392578
$479$	−37.7490	−1.72480	−0.862398	−	0.506230i	$-0.831039\pi$
$479$	−37.7490	−1.72480	−0.862398	+	0.506230i	$0.831039\pi$
$480$	0	0
$481$	46.1660	2.10499
$482$	−4.41699	−0.201189
$483$	0	0
$484$	−10.5830	−0.481046
$485$	−4.00000	−0.181631
$486$	0	0
$487$	23.8745	1.08186	0.540929	−	0.841069i	$-0.318073\pi$
$487$	23.8745	1.08186	0.540929	+	0.841069i	$0.318073\pi$
$488$	0.708497	0.0320722
$489$	0	0
$490$	0	0
$491$	39.8745	1.79951	0.899756	−	0.436394i	$-0.143745\pi$
$491$	39.8745	1.79951	0.899756	+	0.436394i	$0.143745\pi$
$492$	0	0
$493$	0	0
$494$	29.2288	1.31506
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	0	0
$499$	−6.58301	−0.294696	−0.147348	−	0.989085i	$-0.547074\pi$
$499$	−6.58301	−0.294696	−0.147348	+	0.989085i	$0.547074\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	1.93725	0.0864639
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	1.93725	0.0861214
$507$	0	0
$508$	14.6458	0.649800
$509$	33.0405	1.46450	0.732248	−	0.681038i	$-0.238471\pi$
$509$	33.0405	1.46450	0.732248	+	0.681038i	$0.238471\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.7085	−0.736980
$515$	6.70850	0.295612
$516$	0	0
$517$	7.47974	0.328959
$518$	0	0
$519$	0	0
$520$	4.64575	0.203730
$521$	−9.22876	−0.404319	−0.202160	−	0.979353i	$-0.564796\pi$
$521$	−9.22876	−0.404319	−0.202160	+	0.979353i	$0.564796\pi$
$522$	0	0
$523$	27.1660	1.18789	0.593943	−	0.804507i	$-0.297570\pi$
$523$	27.1660	1.18789	0.593943	+	0.804507i	$0.297570\pi$
$524$	0.645751	0.0282098
$525$	0	0
$526$	14.5830	0.635849
$527$	14.5830	0.635246
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−10.2915	−0.447034
$531$	0	0
$532$	0	0
$533$	30.8745	1.33732
$534$	0	0
$535$	15.8745	0.686315
$536$	−15.2915	−0.660492
$537$	0	0
$538$	−9.87451	−0.425720
$539$	0	0
$540$	0	0
$541$	−10.4575	−0.449604	−0.224802	−	0.974405i	$-0.572173\pi$
$541$	−10.4575	−0.449604	−0.224802	+	0.974405i	$0.572173\pi$
$542$	25.1660	1.08097
$543$	0	0
$544$	7.29150	0.312621
$545$	0.583005	0.0249732
$546$	0	0
$547$	−8.70850	−0.372348	−0.186174	−	0.982517i	$-0.559609\pi$
$547$	−8.70850	−0.372348	−0.186174	+	0.982517i	$0.559609\pi$
$548$	15.8745	0.678125
$549$	0	0
$550$	−0.645751	−0.0275349
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−5.41699	−0.230146
$555$	0	0
$556$	−10.5830	−0.448819
$557$	−36.0405	−1.52709	−0.763543	−	0.645757i	$-0.776542\pi$
$557$	−36.0405	−1.52709	−0.763543	+	0.645757i	$0.776542\pi$
$558$	0	0
$559$	3.29150	0.139216
$560$	0	0
$561$	0	0
$562$	10.5203	0.443770
$563$	−11.1660	−0.470591	−0.235296	−	0.971924i	$-0.575606\pi$
$563$	−11.1660	−0.470591	−0.235296	+	0.971924i	$0.575606\pi$
$564$	0	0
$565$	8.58301	0.361090
$566$	23.8745	1.00352
$567$	0	0
$568$	−13.2915	−0.557699
$569$	−35.8118	−1.50131	−0.750654	−	0.660696i	$-0.770261\pi$
$569$	−35.8118	−1.50131	−0.750654	+	0.660696i	$0.770261\pi$
$570$	0	0
$571$	−30.5830	−1.27986	−0.639929	−	0.768434i	$-0.721036\pi$
$571$	−30.5830	−1.27986	−0.639929	+	0.768434i	$0.721036\pi$
$572$	3.00000	0.125436
$573$	0	0
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	27.1660	1.13094	0.565468	−	0.824770i	$-0.308696\pi$
$577$	27.1660	1.13094	0.565468	+	0.824770i	$0.308696\pi$
$578$	−36.1660	−1.50431
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.64575	−0.275239
$584$	2.00000	0.0827606
$585$	0	0
$586$	−22.2915	−0.920853
$587$	8.58301	0.354259	0.177129	−	0.984188i	$-0.443319\pi$
$587$	8.58301	0.354259	0.177129	+	0.984188i	$0.443319\pi$
$588$	0	0
$589$	12.5830	0.518474
$590$	1.29150	0.0531703
$591$	0	0
$592$	9.93725	0.408419
$593$	33.8745	1.39106	0.695530	−	0.718497i	$-0.255170\pi$
$593$	33.8745	1.39106	0.695530	+	0.718497i	$0.255170\pi$
$594$	0	0
$595$	0	0
$596$	9.87451	0.404476
$597$	0	0
$598$	13.9373	0.569937
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−31.1660	−1.27129	−0.635644	−	0.771982i	$-0.719265\pi$
$601$	−31.1660	−1.27129	−0.635644	+	0.771982i	$0.719265\pi$
$602$	0	0
$603$	0	0
$604$	14.0000	0.569652
$605$	10.5830	0.430260
$606$	0	0
$607$	37.2288	1.51107	0.755534	−	0.655109i	$-0.227377\pi$
$607$	37.2288	1.51107	0.755534	+	0.655109i	$0.227377\pi$
$608$	6.29150	0.255154
$609$	0	0
$610$	−0.708497	−0.0286862
$611$	53.8118	2.17699
$612$	0	0
$613$	−16.6458	−0.672316	−0.336158	−	0.941806i	$-0.609128\pi$
$613$	−16.6458	−0.672316	−0.336158	+	0.941806i	$0.609128\pi$
$614$	−23.2915	−0.939969
$615$	0	0
$616$	0	0
$617$	−4.70850	−0.189557	−0.0947785	−	0.995498i	$-0.530214\pi$
$617$	−4.70850	−0.189557	−0.0947785	+	0.995498i	$0.530214\pi$
$618$	0	0
$619$	−26.8745	−1.08018	−0.540089	−	0.841608i	$-0.681609\pi$
$619$	−26.8745	−1.08018	−0.540089	+	0.841608i	$0.681609\pi$
$620$	2.00000	0.0803219
$621$	0	0
$622$	−9.87451	−0.395932
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	29.8745	1.19403
$627$	0	0
$628$	15.3542	0.612701
$629$	−72.4575	−2.88907
$630$	0	0
$631$	33.7490	1.34353	0.671764	−	0.740766i	$-0.265537\pi$
$631$	33.7490	1.34353	0.671764	+	0.740766i	$0.265537\pi$
$632$	−0.708497	−0.0281825
$633$	0	0
$634$	−32.5830	−1.29404
$635$	−14.6458	−0.581199
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	13.1033	0.517548	0.258774	−	0.965938i	$-0.416682\pi$
$641$	13.1033	0.517548	0.258774	+	0.965938i	$0.416682\pi$
$642$	0	0
$643$	−6.70850	−0.264557	−0.132279	−	0.991213i	$-0.542229\pi$
$643$	−6.70850	−0.264557	−0.132279	+	0.991213i	$0.542229\pi$
$644$	0	0
$645$	0	0
$646$	−45.8745	−1.80491
$647$	14.1660	0.556923	0.278462	−	0.960447i	$-0.410175\pi$
$647$	14.1660	0.556923	0.278462	+	0.960447i	$0.410175\pi$
$648$	0	0
$649$	0.833990	0.0327370
$650$	−4.64575	−0.182221
$651$	0	0
$652$	8.00000	0.313304
$653$	−15.4575	−0.604899	−0.302450	−	0.953165i	$-0.597804\pi$
$653$	−15.4575	−0.604899	−0.302450	+	0.953165i	$0.597804\pi$
$654$	0	0
$655$	−0.645751	−0.0252316
$656$	6.64575	0.259473
$657$	0	0
$658$	0	0
$659$	−13.2915	−0.517763	−0.258882	−	0.965909i	$-0.583354\pi$
$659$	−13.2915	−0.517763	−0.258882	+	0.965909i	$0.583354\pi$
$660$	0	0
$661$	−22.5830	−0.878377	−0.439189	−	0.898395i	$-0.644734\pi$
$661$	−22.5830	−0.878377	−0.439189	+	0.898395i	$0.644734\pi$
$662$	−29.4575	−1.14490
$663$	0	0
$664$	−4.70850	−0.182725
$665$	0	0
$666$	0	0
$667$	0	0
$668$	9.00000	0.348220
$669$	0	0
$670$	15.2915	0.590762
$671$	−0.457513	−0.0176621
$672$	0	0
$673$	−1.41699	−0.0546211	−0.0273106	−	0.999627i	$-0.508694\pi$
$673$	−1.41699	−0.0546211	−0.0273106	+	0.999627i	$0.508694\pi$
$674$	−0.708497	−0.0272903
$675$	0	0
$676$	8.58301	0.330116
$677$	17.1255	0.658186	0.329093	−	0.944297i	$-0.393257\pi$
$677$	17.1255	0.658186	0.329093	+	0.944297i	$0.393257\pi$
$678$	0	0
$679$	0	0
$680$	−7.29150	−0.279616
$681$	0	0
$682$	1.29150	0.0494542
$683$	29.1660	1.11601	0.558003	−	0.829839i	$-0.311568\pi$
$683$	29.1660	1.11601	0.558003	+	0.829839i	$0.311568\pi$
$684$	0	0
$685$	−15.8745	−0.606534
$686$	0	0
$687$	0	0
$688$	0.708497	0.0270112
$689$	−47.8118	−1.82148
$690$	0	0
$691$	13.4170	0.510407	0.255203	−	0.966887i	$-0.417858\pi$
$691$	13.4170	0.510407	0.255203	+	0.966887i	$0.417858\pi$
$692$	7.70850	0.293033
$693$	0	0
$694$	1.29150	0.0490248
$695$	10.5830	0.401436
$696$	0	0
$697$	−48.4575	−1.83546
$698$	22.5830	0.854779
$699$	0	0
$700$	0	0
$701$	30.4575	1.15036	0.575182	−	0.818025i	$-0.304931\pi$
$701$	30.4575	1.15036	0.575182	+	0.818025i	$0.304931\pi$
$702$	0	0
$703$	−62.5203	−2.35800
$704$	0.645751	0.0243377
$705$	0	0
$706$	12.0000	0.451626
$707$	0	0
$708$	0	0
$709$	−21.1660	−0.794906	−0.397453	−	0.917622i	$-0.630106\pi$
$709$	−21.1660	−0.794906	−0.397453	+	0.917622i	$0.630106\pi$
$710$	13.2915	0.498821
$711$	0	0
$712$	14.5830	0.546521
$713$	6.00000	0.224702
$714$	0	0
$715$	−3.00000	−0.112194
$716$	10.0627	0.376062
$717$	0	0
$718$	2.58301	0.0963969
$719$	−20.1255	−0.750554	−0.375277	−	0.926913i	$-0.622452\pi$
$719$	−20.1255	−0.750554	−0.375277	+	0.926913i	$0.622452\pi$
$720$	0	0
$721$	0	0
$722$	−20.5830	−0.766020
$723$	0	0
$724$	11.2915	0.419645
$725$	0	0
$726$	0	0
$727$	−25.3542	−0.940337	−0.470169	−	0.882577i	$-0.655807\pi$
$727$	−25.3542	−0.940337	−0.470169	+	0.882577i	$0.655807\pi$
$728$	0	0
$729$	0	0
$730$	−2.00000	−0.0740233
$731$	−5.16601	−0.191072
$732$	0	0
$733$	−48.5203	−1.79214	−0.896068	−	0.443916i	$-0.853589\pi$
$733$	−48.5203	−1.79214	−0.896068	+	0.443916i	$0.853589\pi$
$734$	−21.3542	−0.788200
$735$	0	0
$736$	3.00000	0.110581
$737$	9.87451	0.363732
$738$	0	0
$739$	−32.2915	−1.18786	−0.593931	−	0.804516i	$-0.702425\pi$
$739$	−32.2915	−1.18786	−0.593931	+	0.804516i	$0.702425\pi$
$740$	−9.93725	−0.365301
$741$	0	0
$742$	0	0
$743$	−34.7490	−1.27482	−0.637409	−	0.770526i	$-0.719994\pi$
$743$	−34.7490	−1.27482	−0.637409	+	0.770526i	$0.719994\pi$
$744$	0	0
$745$	−9.87451	−0.361774
$746$	4.00000	0.146450
$747$	0	0
$748$	−4.70850	−0.172160
$749$	0	0
$750$	0	0
$751$	35.8745	1.30908	0.654540	−	0.756028i	$-0.272862\pi$
$751$	35.8745	1.30908	0.654540	+	0.756028i	$0.272862\pi$
$752$	11.5830	0.422389
$753$	0	0
$754$	0	0
$755$	−14.0000	−0.509512
$756$	0	0
$757$	−9.16601	−0.333144	−0.166572	−	0.986029i	$-0.553270\pi$
$757$	−9.16601	−0.333144	−0.166572	+	0.986029i	$0.553270\pi$
$758$	−12.2915	−0.446447
$759$	0	0
$760$	−6.29150	−0.228217
$761$	28.0627	1.01727	0.508637	−	0.860981i	$-0.330150\pi$
$761$	28.0627	1.01727	0.508637	+	0.860981i	$0.330150\pi$
$762$	0	0
$763$	0	0
$764$	6.00000	0.217072
$765$	0	0
$766$	−21.0000	−0.758761
$767$	6.00000	0.216647
$768$	0	0
$769$	12.1660	0.438718	0.219359	−	0.975644i	$-0.429603\pi$
$769$	12.1660	0.438718	0.219359	+	0.975644i	$0.429603\pi$
$770$	0	0
$771$	0	0
$772$	23.8745	0.859262
$773$	18.8745	0.678869	0.339434	−	0.940630i	$-0.389764\pi$
$773$	18.8745	0.678869	0.339434	+	0.940630i	$0.389764\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	−4.00000	−0.143592
$777$	0	0
$778$	−25.2915	−0.906744
$779$	−41.8118	−1.49806
$780$	0	0
$781$	8.58301	0.307124
$782$	−21.8745	−0.782231
$783$	0	0
$784$	0	0
$785$	−15.3542	−0.548017
$786$	0	0
$787$	−14.4575	−0.515355	−0.257677	−	0.966231i	$-0.582957\pi$
$787$	−14.4575	−0.515355	−0.257677	+	0.966231i	$0.582957\pi$
$788$	4.29150	0.152878
$789$	0	0
$790$	0.708497	0.0252072
$791$	0	0
$792$	0	0
$793$	−3.29150	−0.116885
$794$	−1.41699	−0.0502873
$795$	0	0
$796$	13.8745	0.491769
$797$	−25.7490	−0.912077	−0.456038	−	0.889960i	$-0.650732\pi$
$797$	−25.7490	−0.912077	−0.456038	+	0.889960i	$0.650732\pi$
$798$	0	0
$799$	−84.4575	−2.98789
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	9.22876	0.325879
$803$	−1.29150	−0.0455761
$804$	0	0
$805$	0	0
$806$	9.29150	0.327279
$807$	0	0
$808$	12.0000	0.422159
$809$	35.8118	1.25907	0.629537	−	0.776970i	$-0.283245\pi$
$809$	35.8118	1.25907	0.629537	+	0.776970i	$0.283245\pi$
$810$	0	0
$811$	−14.8745	−0.522315	−0.261157	−	0.965296i	$-0.584104\pi$
$811$	−14.8745	−0.522315	−0.261157	+	0.965296i	$0.584104\pi$
$812$	0	0
$813$	0	0
$814$	−6.41699	−0.224916
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−4.45751	−0.155949
$818$	11.4170	0.399186
$819$	0	0
$820$	−6.64575	−0.232080
$821$	−15.8745	−0.554024	−0.277012	−	0.960866i	$-0.589344\pi$
$821$	−15.8745	−0.554024	−0.277012	+	0.960866i	$0.589344\pi$
$822$	0	0
$823$	−29.2915	−1.02104	−0.510519	−	0.859867i	$-0.670547\pi$
$823$	−29.2915	−1.02104	−0.510519	+	0.859867i	$0.670547\pi$
$824$	6.70850	0.233702
$825$	0	0
$826$	0	0
$827$	27.4170	0.953382	0.476691	−	0.879071i	$-0.341836\pi$
$827$	27.4170	0.953382	0.476691	+	0.879071i	$0.341836\pi$
$828$	0	0
$829$	−41.8745	−1.45436	−0.727181	−	0.686446i	$-0.759170\pi$
$829$	−41.8745	−1.45436	−0.727181	+	0.686446i	$0.759170\pi$
$830$	4.70850	0.163434
$831$	0	0
$832$	4.64575	0.161062
$833$	0	0
$834$	0	0
$835$	−9.00000	−0.311458
$836$	−4.06275	−0.140513
$837$	0	0
$838$	37.9373	1.31052
$839$	12.4575	0.430081	0.215041	−	0.976605i	$-0.431012\pi$
$839$	12.4575	0.430081	0.215041	+	0.976605i	$0.431012\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	31.8745	1.09847
$843$	0	0
$844$	−2.29150	−0.0788768
$845$	−8.58301	−0.295264
$846$	0	0
$847$	0	0
$848$	−10.2915	−0.353412
$849$	0	0
$850$	7.29150	0.250096
$851$	−29.8118	−1.02193
$852$	0	0
$853$	−2.18824	−0.0749238	−0.0374619	−	0.999298i	$-0.511927\pi$
$853$	−2.18824	−0.0749238	−0.0374619	+	0.999298i	$0.511927\pi$
$854$	0	0
$855$	0	0
$856$	15.8745	0.542580
$857$	−15.8745	−0.542263	−0.271131	−	0.962542i	$-0.587398\pi$
$857$	−15.8745	−0.542263	−0.271131	+	0.962542i	$0.587398\pi$
$858$	0	0
$859$	−1.16601	−0.0397838	−0.0198919	−	0.999802i	$-0.506332\pi$
$859$	−1.16601	−0.0397838	−0.0198919	+	0.999802i	$0.506332\pi$
$860$	−0.708497	−0.0241596
$861$	0	0
$862$	−33.8745	−1.15377
$863$	−20.1660	−0.686459	−0.343229	−	0.939252i	$-0.611521\pi$
$863$	−20.1660	−0.686459	−0.343229	+	0.939252i	$0.611521\pi$
$864$	0	0
$865$	−7.70850	−0.262097
$866$	8.00000	0.271851
$867$	0	0
$868$	0	0
$869$	0.457513	0.0155201
$870$	0	0
$871$	71.0405	2.40712
$872$	0.583005	0.0197430
$873$	0	0
$874$	−18.8745	−0.638440
$875$	0	0
$876$	0	0
$877$	29.6863	1.00243	0.501217	−	0.865322i	$-0.332886\pi$
$877$	29.6863	1.00243	0.501217	+	0.865322i	$0.332886\pi$
$878$	−9.16601	−0.309338
$879$	0	0
$880$	−0.645751	−0.0217683
$881$	42.6458	1.43677	0.718386	−	0.695645i	$-0.244881\pi$
$881$	42.6458	1.43677	0.718386	+	0.695645i	$0.244881\pi$
$882$	0	0
$883$	11.4170	0.384212	0.192106	−	0.981374i	$-0.438468\pi$
$883$	11.4170	0.384212	0.192106	+	0.981374i	$0.438468\pi$
$884$	−33.8745	−1.13932
$885$	0	0
$886$	−28.7085	−0.964481
$887$	−41.1660	−1.38222	−0.691110	−	0.722750i	$-0.742878\pi$
$887$	−41.1660	−1.38222	−0.691110	+	0.722750i	$0.742878\pi$
$888$	0	0
$889$	0	0
$890$	−14.5830	−0.488823
$891$	0	0
$892$	5.29150	0.177173
$893$	−72.8745	−2.43865
$894$	0	0
$895$	−10.0627	−0.336361
$896$	0	0
$897$	0	0
$898$	23.8118	0.794609
$899$	0	0
$900$	0	0
$901$	75.0405	2.49996
$902$	−4.29150	−0.142891
$903$	0	0
$904$	8.58301	0.285467
$905$	−11.2915	−0.375342
$906$	0	0
$907$	21.7490	0.722164	0.361082	−	0.932534i	$-0.382407\pi$
$907$	21.7490	0.722164	0.361082	+	0.932534i	$0.382407\pi$
$908$	−7.29150	−0.241977
$909$	0	0
$910$	0	0
$911$	−47.6235	−1.57784	−0.788919	−	0.614497i	$-0.789359\pi$
$911$	−47.6235	−1.57784	−0.788919	+	0.614497i	$0.789359\pi$
$912$	0	0
$913$	3.04052	0.100626
$914$	17.2915	0.571952
$915$	0	0
$916$	−6.70850	−0.221655
$917$	0	0
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	−3.00000	−0.0989071
$921$	0	0
$922$	−17.1660	−0.565332
$923$	61.7490	2.03249
$924$	0	0
$925$	9.93725	0.326735
$926$	−15.9373	−0.523731
$927$	0	0
$928$	0	0
$929$	13.1033	0.429904	0.214952	−	0.976625i	$-0.431040\pi$
$929$	13.1033	0.429904	0.214952	+	0.976625i	$0.431040\pi$
$930$	0	0
$931$	0	0
$932$	27.8745	0.913060
$933$	0	0
$934$	19.2915	0.631237
$935$	4.70850	0.153984
$936$	0	0
$937$	57.6235	1.88248	0.941239	−	0.337741i	$-0.109663\pi$
$937$	57.6235	1.88248	0.941239	+	0.337741i	$0.109663\pi$
$938$	0	0
$939$	0	0
$940$	−11.5830	−0.377796
$941$	−3.41699	−0.111391	−0.0556954	−	0.998448i	$-0.517738\pi$
$941$	−3.41699	−0.111391	−0.0556954	+	0.998448i	$0.517738\pi$
$942$	0	0
$943$	−19.9373	−0.649246
$944$	1.29150	0.0420348
$945$	0	0
$946$	−0.457513	−0.0148750
$947$	45.0405	1.46362	0.731810	−	0.681509i	$-0.238676\pi$
$947$	45.0405	1.46362	0.731810	+	0.681509i	$0.238676\pi$
$948$	0	0
$949$	−9.29150	−0.301615
$950$	6.29150	0.204123
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	8.58301	0.277594
$957$	0	0
$958$	37.7490	1.21962
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−46.1660	−1.48845
$963$	0	0
$964$	4.41699	0.142262
$965$	−23.8745	−0.768548
$966$	0	0
$967$	−29.2915	−0.941951	−0.470976	−	0.882146i	$-0.656098\pi$
$967$	−29.2915	−0.941951	−0.470976	+	0.882146i	$0.656098\pi$
$968$	10.5830	0.340151
$969$	0	0
$970$	4.00000	0.128432
$971$	53.8118	1.72690	0.863451	−	0.504433i	$-0.168298\pi$
$971$	53.8118	1.72690	0.863451	+	0.504433i	$0.168298\pi$
$972$	0	0
$973$	0	0
$974$	−23.8745	−0.764989
$975$	0	0
$976$	−0.708497	−0.0226784
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	−9.41699	−0.300968
$980$	0	0
$981$	0	0
$982$	−39.8745	−1.27245
$983$	49.3320	1.57345	0.786724	−	0.617305i	$-0.211776\pi$
$983$	49.3320	1.57345	0.786724	+	0.617305i	$0.211776\pi$
$984$	0	0
$985$	−4.29150	−0.136739
$986$	0	0
$987$	0	0
$988$	−29.2288	−0.929891
$989$	−2.12549	−0.0675867
$990$	0	0
$991$	−5.29150	−0.168090	−0.0840451	−	0.996462i	$-0.526784\pi$
$991$	−5.29150	−0.168090	−0.0840451	+	0.996462i	$0.526784\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	0	0
$995$	−13.8745	−0.439851
$996$	0	0
$997$	47.7490	1.51223	0.756113	−	0.654441i	$-0.227096\pi$
$997$	47.7490	1.51223	0.756113	+	0.654441i	$0.227096\pi$
$998$	6.58301	0.208381
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.bo.1.2		2
3.2	odd	2		4410.2.a.ca.1.1		2
7.3	odd	6		630.2.k.j.541.1	yes	4
7.5	odd	6		630.2.k.j.361.1	yes	4
7.6	odd	2		4410.2.a.bq.1.2		2
21.5	even	6		630.2.k.i.361.1	✓	4
21.17	even	6		630.2.k.i.541.1	yes	4
21.20	even	2		4410.2.a.bv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.k.i.361.1	✓	4	21.5	even	6
630.2.k.i.541.1	yes	4	21.17	even	6
630.2.k.j.361.1	yes	4	7.5	odd	6
630.2.k.j.541.1	yes	4	7.3	odd	6
4410.2.a.bo.1.2		2	1.1	even	1	trivial
4410.2.a.bq.1.2		2	7.6	odd	2
4410.2.a.bv.1.1		2	21.20	even	2
4410.2.a.ca.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} +0.645751 q^{11} +4.64575 q^{13} +1.00000 q^{16} -7.29150 q^{17} -6.29150 q^{19} -1.00000 q^{20} -0.645751 q^{22} -3.00000 q^{23} +1.00000 q^{25} -4.64575 q^{26} -2.00000 q^{31} -1.00000 q^{32} +7.29150 q^{34} +9.93725 q^{37} +6.29150 q^{38} +1.00000 q^{40} +6.64575 q^{41} +0.708497 q^{43} +0.645751 q^{44} +3.00000 q^{46} +11.5830 q^{47} -1.00000 q^{50} +4.64575 q^{52} -10.2915 q^{53} -0.645751 q^{55} +1.29150 q^{59} -0.708497 q^{61} +2.00000 q^{62} +1.00000 q^{64} -4.64575 q^{65} +15.2915 q^{67} -7.29150 q^{68} +13.2915 q^{71} -2.00000 q^{73} -9.93725 q^{74} -6.29150 q^{76} +0.708497 q^{79} -1.00000 q^{80} -6.64575 q^{82} +4.70850 q^{83} +7.29150 q^{85} -0.708497 q^{86} -0.645751 q^{88} -14.5830 q^{89} -3.00000 q^{92} -11.5830 q^{94} +6.29150 q^{95} +4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 4 q^{13} + 2 q^{16} - 4 q^{17} - 2 q^{19} - 2 q^{20} + 4 q^{22} - 6 q^{23} + 2 q^{25} - 4 q^{26} - 4 q^{31} - 2 q^{32} + 4 q^{34} + 4 q^{37} + 2 q^{38} + 2 q^{40} + 8 q^{41} + 12 q^{43} - 4 q^{44} + 6 q^{46} + 2 q^{47} - 2 q^{50} + 4 q^{52} - 10 q^{53} + 4 q^{55} - 8 q^{59} - 12 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{65} + 20 q^{67} - 4 q^{68} + 16 q^{71} - 4 q^{73} - 4 q^{74} - 2 q^{76} + 12 q^{79} - 2 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} - 12 q^{86} + 4 q^{88} - 8 q^{89} - 6 q^{92} - 2 q^{94} + 2 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 4 * q^13 + 2 * q^16 - 4 * q^17 - 2 * q^19 - 2 * q^20 + 4 * q^22 - 6 * q^23 + 2 * q^25 - 4 * q^26 - 4 * q^31 - 2 * q^32 + 4 * q^34 + 4 * q^37 + 2 * q^38 + 2 * q^40 + 8 * q^41 + 12 * q^43 - 4 * q^44 + 6 * q^46 + 2 * q^47 - 2 * q^50 + 4 * q^52 - 10 * q^53 + 4 * q^55 - 8 * q^59 - 12 * q^61 + 4 * q^62 + 2 * q^64 - 4 * q^65 + 20 * q^67 - 4 * q^68 + 16 * q^71 - 4 * q^73 - 4 * q^74 - 2 * q^76 + 12 * q^79 - 2 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 - 12 * q^86 + 4 * q^88 - 8 * q^89 - 6 * q^92 - 2 * q^94 + 2 * q^95 + 8 * q^97

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