LMFDB - Embedded newform 4410.2.a.bo.1.1

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			1.1
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} -4.64575 q^{11} -0.645751 q^{13} +1.00000 q^{16} +3.29150 q^{17} +4.29150 q^{19} -1.00000 q^{20} +4.64575 q^{22} -3.00000 q^{23} +1.00000 q^{25} +0.645751 q^{26} -2.00000 q^{31} -1.00000 q^{32} -3.29150 q^{34} -5.93725 q^{37} -4.29150 q^{38} +1.00000 q^{40} +1.35425 q^{41} +11.2915 q^{43} -4.64575 q^{44} +3.00000 q^{46} -9.58301 q^{47} -1.00000 q^{50} -0.645751 q^{52} +0.291503 q^{53} +4.64575 q^{55} -9.29150 q^{59} -11.2915 q^{61} +2.00000 q^{62} +1.00000 q^{64} +0.645751 q^{65} +4.70850 q^{67} +3.29150 q^{68} +2.70850 q^{71} -2.00000 q^{73} +5.93725 q^{74} +4.29150 q^{76} +11.2915 q^{79} -1.00000 q^{80} -1.35425 q^{82} +15.2915 q^{83} -3.29150 q^{85} -11.2915 q^{86} +4.64575 q^{88} +6.58301 q^{89} -3.00000 q^{92} +9.58301 q^{94} -4.29150 q^{95} +4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	−4.64575	−1.40075	−0.700373	−	0.713777i	$-0.746983\pi$
$11$	−4.64575	−1.40075	−0.700373	+	0.713777i	$0.746983\pi$
$12$	0	0
$13$	−0.645751	−0.179099	−0.0895496	−	0.995982i	$-0.528543\pi$
$13$	−0.645751	−0.179099	−0.0895496	+	0.995982i	$0.528543\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	3.29150	0.798307	0.399153	−	0.916884i	$-0.369304\pi$
$17$	3.29150	0.798307	0.399153	+	0.916884i	$0.369304\pi$
$18$	0	0
$19$	4.29150	0.984538	0.492269	−	0.870443i	$-0.336168\pi$
$19$	4.29150	0.984538	0.492269	+	0.870443i	$0.336168\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	4.64575	0.990478
$23$	−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
$26$	0.645751	0.126642
$27$	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
$34$	−3.29150	−0.564488
$35$	0	0
$36$	0	0
$37$	−5.93725	−0.976079	−0.488039	−	0.872822i	$-0.662288\pi$
$37$	−5.93725	−0.976079	−0.488039	+	0.872822i	$0.662288\pi$
$38$	−4.29150	−0.696174
$39$	0	0
$40$	1.00000	0.158114
$41$	1.35425	0.211498	0.105749	−	0.994393i	$-0.466276\pi$
$41$	1.35425	0.211498	0.105749	+	0.994393i	$0.466276\pi$
$42$	0	0
$43$	11.2915	1.72194	0.860969	−	0.508657i	$-0.169858\pi$
$43$	11.2915	1.72194	0.860969	+	0.508657i	$0.169858\pi$
$44$	−4.64575	−0.700373
$45$	0	0
$46$	3.00000	0.442326
$47$	−9.58301	−1.39782	−0.698912	−	0.715207i	$-0.746332\pi$
$47$	−9.58301	−1.39782	−0.698912	+	0.715207i	$0.746332\pi$
$48$	0	0
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−0.645751	−0.0895496
$53$	0.291503	0.0400410	0.0200205	−	0.999800i	$-0.493627\pi$
$53$	0.291503	0.0400410	0.0200205	+	0.999800i	$0.493627\pi$
$54$	0	0
$55$	4.64575	0.626433
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.29150	−1.20965	−0.604825	−	0.796358i	$-0.706757\pi$
$59$	−9.29150	−1.20965	−0.604825	+	0.796358i	$0.706757\pi$
$60$	0	0
$61$	−11.2915	−1.44573	−0.722864	−	0.690990i	$-0.757175\pi$
$61$	−11.2915	−1.44573	−0.722864	+	0.690990i	$0.757175\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0.645751	0.0800956
$66$	0	0
$67$	4.70850	0.575235	0.287617	−	0.957745i	$-0.407137\pi$
$67$	4.70850	0.575235	0.287617	+	0.957745i	$0.407137\pi$
$68$	3.29150	0.399153
$69$	0	0
$70$	0	0
$71$	2.70850	0.321440	0.160720	−	0.987000i	$-0.448618\pi$
$71$	2.70850	0.321440	0.160720	+	0.987000i	$0.448618\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	5.93725	0.690192
$75$	0	0
$76$	4.29150	0.492269
$77$	0	0
$78$	0	0
$79$	11.2915	1.27039	0.635197	−	0.772350i	$-0.280919\pi$
$79$	11.2915	1.27039	0.635197	+	0.772350i	$0.280919\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−1.35425	−0.149552
$83$	15.2915	1.67846	0.839230	−	0.543776i	$-0.183006\pi$
$83$	15.2915	1.67846	0.839230	+	0.543776i	$0.183006\pi$
$84$	0	0
$85$	−3.29150	−0.357014
$86$	−11.2915	−1.21759
$87$	0	0
$88$	4.64575	0.495239
$89$	6.58301	0.697797	0.348899	−	0.937160i	$-0.386556\pi$
$89$	6.58301	0.697797	0.348899	+	0.937160i	$0.386556\pi$
$90$	0	0
$91$	0	0
$92$	−3.00000	−0.312772
$93$	0	0
$94$	9.58301	0.988412
$95$	−4.29150	−0.440299
$96$	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
$103$	−17.2915	−1.70378	−0.851891	−	0.523719i	$-0.824544\pi$
$103$	−17.2915	−1.70378	−0.851891	+	0.523719i	$0.824544\pi$
$104$	0.645751	0.0633211
$105$	0	0
$106$	−0.291503	−0.0283132
$107$	15.8745	1.53465	0.767323	−	0.641260i	$-0.221588\pi$
$107$	15.8745	1.53465	0.767323	+	0.641260i	$0.221588\pi$
$108$	0	0
$109$	20.5830	1.97149	0.985747	−	0.168234i	$-0.0538063\pi$
$109$	20.5830	1.97149	0.985747	+	0.168234i	$0.0538063\pi$
$110$	−4.64575	−0.442955
$111$	0	0
$112$	0	0
$113$	12.5830	1.18371	0.591855	−	0.806045i	$-0.298396\pi$
$113$	12.5830	1.18371	0.591855	+	0.806045i	$0.298396\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	0	0
$118$	9.29150	0.855352
$119$	0	0
$120$	0	0
$121$	10.5830	0.962091
$122$	11.2915	1.02228
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.35425	0.830055	0.415028	−	0.909809i	$-0.363772\pi$
$127$	9.35425	0.830055	0.415028	+	0.909809i	$0.363772\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.645751	−0.0566361
$131$	−4.64575	−0.405901	−0.202951	−	0.979189i	$-0.565053\pi$
$131$	−4.64575	−0.405901	−0.202951	+	0.979189i	$0.565053\pi$
$132$	0	0
$133$	0	0
$134$	−4.70850	−0.406752
$135$	0	0
$136$	−3.29150	−0.282244
$137$	−15.8745	−1.35625	−0.678125	−	0.734946i	$-0.737207\pi$
$137$	−15.8745	−1.35625	−0.678125	+	0.734946i	$0.737207\pi$
$138$	0	0
$139$	10.5830	0.897639	0.448819	−	0.893622i	$-0.351845\pi$
$139$	10.5830	0.897639	0.448819	+	0.893622i	$0.351845\pi$
$140$	0	0
$141$	0	0
$142$	−2.70850	−0.227292
$143$	3.00000	0.250873
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	−5.93725	−0.488039
$149$	−21.8745	−1.79203	−0.896015	−	0.444024i	$-0.853550\pi$
$149$	−21.8745	−1.79203	−0.896015	+	0.444024i	$0.853550\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−4.29150	−0.348087
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	20.6458	1.64771	0.823855	−	0.566800i	$-0.191819\pi$
$157$	20.6458	1.64771	0.823855	+	0.566800i	$0.191819\pi$
$158$	−11.2915	−0.898304
$159$	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	$0.398564\pi$
$164$	1.35425	0.105749
$165$	0	0
$166$	−15.2915	−1.18685
$167$	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
$169$	−12.5830	−0.967923
$170$	3.29150	0.252447
$171$	0	0
$172$	11.2915	0.860969
$173$	18.2915	1.39068	0.695339	−	0.718682i	$-0.255254\pi$
$173$	18.2915	1.39068	0.695339	+	0.718682i	$0.255254\pi$
$174$	0	0
$175$	0	0
$176$	−4.64575	−0.350187
$177$	0	0
$178$	−6.58301	−0.493417
$179$	25.9373	1.93864	0.969321	−	0.245800i	$-0.0790505\pi$
$179$	25.9373	1.93864	0.969321	+	0.245800i	$0.0790505\pi$
$180$	0	0
$181$	0.708497	0.0526622	0.0263311	−	0.999653i	$-0.491618\pi$
$181$	0.708497	0.0526622	0.0263311	+	0.999653i	$0.491618\pi$
$182$	0	0
$183$	0	0
$184$	3.00000	0.221163
$185$	5.93725	0.436516
$186$	0	0
$187$	−15.2915	−1.11823
$188$	−9.58301	−0.698912
$189$	0	0
$190$	4.29150	0.311338
$191$	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
$193$	−7.87451	−0.566819	−0.283410	−	0.958999i	$-0.591466\pi$
$193$	−7.87451	−0.566819	−0.283410	+	0.958999i	$0.591466\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	0	0
$197$	−6.29150	−0.448251	−0.224126	−	0.974560i	$-0.571953\pi$
$197$	−6.29150	−0.448251	−0.224126	+	0.974560i	$0.571953\pi$
$198$	0	0
$199$	−17.8745	−1.26709	−0.633545	−	0.773706i	$-0.718401\pi$
$199$	−17.8745	−1.26709	−0.633545	+	0.773706i	$0.718401\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	12.0000	0.844317
$203$	0	0
$204$	0	0
$205$	−1.35425	−0.0945848
$206$	17.2915	1.20476
$207$	0	0
$208$	−0.645751	−0.0447748
$209$	−19.9373	−1.37909
$210$	0	0
$211$	8.29150	0.570811	0.285405	−	0.958407i	$-0.407872\pi$
$211$	8.29150	0.570811	0.285405	+	0.958407i	$0.407872\pi$
$212$	0.291503	0.0200205
$213$	0	0
$214$	−15.8745	−1.08516
$215$	−11.2915	−0.770074
$216$	0	0
$217$	0	0
$218$	−20.5830	−1.39406
$219$	0	0
$220$	4.64575	0.313216
$221$	−2.12549	−0.142976
$222$	0	0
$223$	−5.29150	−0.354345	−0.177173	−	0.984180i	$-0.556695\pi$
$223$	−5.29150	−0.354345	−0.177173	+	0.984180i	$0.556695\pi$
$224$	0	0
$225$	0	0
$226$	−12.5830	−0.837009
$227$	3.29150	0.218465	0.109232	−	0.994016i	$-0.465161\pi$
$227$	3.29150	0.218465	0.109232	+	0.994016i	$0.465161\pi$
$228$	0	0
$229$	−17.2915	−1.14265	−0.571327	−	0.820722i	$-0.693571\pi$
$229$	−17.2915	−1.14265	−0.571327	+	0.820722i	$0.693571\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	0	0
$233$	−3.87451	−0.253827	−0.126914	−	0.991914i	$-0.540507\pi$
$233$	−3.87451	−0.253827	−0.126914	+	0.991914i	$0.540507\pi$
$234$	0	0
$235$	9.58301	0.625126
$236$	−9.29150	−0.604825
$237$	0	0
$238$	0	0
$239$	−12.5830	−0.813927	−0.406963	−	0.913444i	$-0.633412\pi$
$239$	−12.5830	−0.813927	−0.406963	+	0.913444i	$0.633412\pi$
$240$	0	0
$241$	25.5830	1.64795	0.823973	−	0.566629i	$-0.191753\pi$
$241$	25.5830	1.64795	0.823973	+	0.566629i	$0.191753\pi$
$242$	−10.5830	−0.680301
$243$	0	0
$244$	−11.2915	−0.722864
$245$	0	0
$246$	0	0
$247$	−2.77124	−0.176330
$248$	2.00000	0.127000
$249$	0	0
$250$	1.00000	0.0632456
$251$	13.9373	0.879712	0.439856	−	0.898068i	$-0.355030\pi$
$251$	13.9373	0.879712	0.439856	+	0.898068i	$0.355030\pi$
$252$	0	0
$253$	13.9373	0.876228
$254$	−9.35425	−0.586938
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.2915	1.70240	0.851199	−	0.524844i	$-0.175876\pi$
$257$	27.2915	1.70240	0.851199	+	0.524844i	$0.175876\pi$
$258$	0	0
$259$	0	0
$260$	0.645751	0.0400478
$261$	0	0
$262$	4.64575	0.287015
$263$	6.58301	0.405925	0.202963	−	0.979186i	$-0.434943\pi$
$263$	6.58301	0.405925	0.202963	+	0.979186i	$0.434943\pi$
$264$	0	0
$265$	−0.291503	−0.0179069
$266$	0	0
$267$	0	0
$268$	4.70850	0.287617
$269$	−21.8745	−1.33371	−0.666856	−	0.745186i	$-0.732360\pi$
$269$	−21.8745	−1.33371	−0.666856	+	0.745186i	$0.732360\pi$
$270$	0	0
$271$	17.1660	1.04276	0.521380	−	0.853324i	$-0.325417\pi$
$271$	17.1660	1.04276	0.521380	+	0.853324i	$0.325417\pi$
$272$	3.29150	0.199577
$273$	0	0
$274$	15.8745	0.959014
$275$	−4.64575	−0.280149
$276$	0	0
$277$	26.5830	1.59722	0.798609	−	0.601850i	$-0.205570\pi$
$277$	26.5830	1.59722	0.798609	+	0.601850i	$0.205570\pi$
$278$	−10.5830	−0.634726
$279$	0	0
$280$	0	0
$281$	26.5203	1.58207	0.791033	−	0.611773i	$-0.209544\pi$
$281$	26.5203	1.58207	0.791033	+	0.611773i	$0.209544\pi$
$282$	0	0
$283$	7.87451	0.468091	0.234045	−	0.972226i	$-0.424804\pi$
$283$	7.87451	0.468091	0.234045	+	0.972226i	$0.424804\pi$
$284$	2.70850	0.160720
$285$	0	0
$286$	−3.00000	−0.177394
$287$	0	0
$288$	0	0
$289$	−6.16601	−0.362706
$290$	0	0
$291$	0	0
$292$	−2.00000	−0.117041
$293$	11.7085	0.684018	0.342009	−	0.939697i	$-0.388893\pi$
$293$	11.7085	0.684018	0.342009	+	0.939697i	$0.388893\pi$
$294$	0	0
$295$	9.29150	0.540972
$296$	5.93725	0.345096
$297$	0	0
$298$	21.8745	1.26716
$299$	1.93725	0.112034
$300$	0	0
$301$	0	0
$302$	−14.0000	−0.805609
$303$	0	0
$304$	4.29150	0.246135
$305$	11.2915	0.646550
$306$	0	0
$307$	12.7085	0.725312	0.362656	−	0.931923i	$-0.381870\pi$
$307$	12.7085	0.725312	0.362656	+	0.931923i	$0.381870\pi$
$308$	0	0
$309$	0	0
$310$	−2.00000	−0.113592
$311$	−21.8745	−1.24039	−0.620195	−	0.784448i	$-0.712946\pi$
$311$	−21.8745	−1.24039	−0.620195	+	0.784448i	$0.712946\pi$
$312$	0	0
$313$	1.87451	0.105953	0.0529767	−	0.998596i	$-0.483129\pi$
$313$	1.87451	0.105953	0.0529767	+	0.998596i	$0.483129\pi$
$314$	−20.6458	−1.16511
$315$	0	0
$316$	11.2915	0.635197
$317$	11.4170	0.641242	0.320621	−	0.947208i	$-0.396108\pi$
$317$	11.4170	0.641242	0.320621	+	0.947208i	$0.396108\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	14.1255	0.785963
$324$	0	0
$325$	−0.645751	−0.0358198
$326$	−8.00000	−0.443079
$327$	0	0
$328$	−1.35425	−0.0747759
$329$	0	0
$330$	0	0
$331$	−23.4575	−1.28934	−0.644671	−	0.764460i	$-0.723006\pi$
$331$	−23.4575	−1.28934	−0.644671	+	0.764460i	$0.723006\pi$
$332$	15.2915	0.839230
$333$	0	0
$334$	−9.00000	−0.492458
$335$	−4.70850	−0.257253
$336$	0	0
$337$	11.2915	0.615087	0.307544	−	0.951534i	$-0.400493\pi$
$337$	11.2915	0.615087	0.307544	+	0.951534i	$0.400493\pi$
$338$	12.5830	0.684425
$339$	0	0
$340$	−3.29150	−0.178507
$341$	9.29150	0.503163
$342$	0	0
$343$	0	0
$344$	−11.2915	−0.608797
$345$	0	0
$346$	−18.2915	−0.983357
$347$	9.29150	0.498794	0.249397	−	0.968401i	$-0.419768\pi$
$347$	9.29150	0.498794	0.249397	+	0.968401i	$0.419768\pi$
$348$	0	0
$349$	−1.41699	−0.0758500	−0.0379250	−	0.999281i	$-0.512075\pi$
$349$	−1.41699	−0.0758500	−0.0379250	+	0.999281i	$0.512075\pi$
$350$	0	0
$351$	0	0
$352$	4.64575	0.247619
$353$	−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
$355$	−2.70850	−0.143752
$356$	6.58301	0.348899
$357$	0	0
$358$	−25.9373	−1.37083
$359$	18.5830	0.980774	0.490387	−	0.871505i	$-0.336856\pi$
$359$	18.5830	0.980774	0.490387	+	0.871505i	$0.336856\pi$
$360$	0	0
$361$	−0.583005	−0.0306845
$362$	−0.708497	−0.0372378
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	26.6458	1.39090	0.695448	−	0.718576i	$-0.255206\pi$
$367$	26.6458	1.39090	0.695448	+	0.718576i	$0.255206\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	−5.93725	−0.308663
$371$	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	$0.533022\pi$
$374$	15.2915	0.790705
$375$	0	0
$376$	9.58301	0.494206
$377$	0	0
$378$	0	0
$379$	1.70850	0.0877596	0.0438798	−	0.999037i	$-0.486028\pi$
$379$	1.70850	0.0877596	0.0438798	+	0.999037i	$0.486028\pi$
$380$	−4.29150	−0.220149
$381$	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
$386$	7.87451	0.400802
$387$	0	0
$388$	4.00000	0.203069
$389$	14.7085	0.745750	0.372875	−	0.927881i	$-0.378372\pi$
$389$	14.7085	0.745750	0.372875	+	0.927881i	$0.378372\pi$
$390$	0	0
$391$	−9.87451	−0.499375
$392$	0	0
$393$	0	0
$394$	6.29150	0.316961
$395$	−11.2915	−0.568137
$396$	0	0
$397$	22.5830	1.13341	0.566704	−	0.823921i	$-0.308218\pi$
$397$	22.5830	1.13341	0.566704	+	0.823921i	$0.308218\pi$
$398$	17.8745	0.895968
$399$	0	0
$400$	1.00000	0.0500000
$401$	17.2288	0.860363	0.430182	−	0.902742i	$-0.358450\pi$
$401$	17.2288	0.860363	0.430182	+	0.902742i	$0.358450\pi$
$402$	0	0
$403$	1.29150	0.0643343
$404$	−12.0000	−0.597022
$405$	0	0
$406$	0	0
$407$	27.5830	1.36724
$408$	0	0
$409$	−32.5830	−1.61113	−0.805563	−	0.592510i	$-0.798137\pi$
$409$	−32.5830	−1.61113	−0.805563	+	0.592510i	$0.798137\pi$
$410$	1.35425	0.0668816
$411$	0	0
$412$	−17.2915	−0.851891
$413$	0	0
$414$	0	0
$415$	−15.2915	−0.750630
$416$	0.645751	0.0316606
$417$	0	0
$418$	19.9373	0.975163
$419$	−22.0627	−1.07784	−0.538918	−	0.842358i	$-0.681167\pi$
$419$	−22.0627	−1.07784	−0.538918	+	0.842358i	$0.681167\pi$
$420$	0	0
$421$	−0.125492	−0.00611611	−0.00305806	−	0.999995i	$-0.500973\pi$
$421$	−0.125492	−0.00611611	−0.00305806	+	0.999995i	$0.500973\pi$
$422$	−8.29150	−0.403624
$423$	0	0
$424$	−0.291503	−0.0141566
$425$	3.29150	0.159661
$426$	0	0
$427$	0	0
$428$	15.8745	0.767323
$429$	0	0
$430$	11.2915	0.544525
$431$	2.12549	0.102381	0.0511907	−	0.998689i	$-0.483698\pi$
$431$	2.12549	0.102381	0.0511907	+	0.998689i	$0.483698\pi$
$432$	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
$436$	20.5830	0.985747
$437$	−12.8745	−0.615871
$438$	0	0
$439$	−33.1660	−1.58293	−0.791464	−	0.611216i	$-0.790681\pi$
$439$	−33.1660	−1.58293	−0.791464	+	0.611216i	$0.790681\pi$
$440$	−4.64575	−0.221478
$441$	0	0
$442$	2.12549	0.101099
$443$	39.2915	1.86680	0.933398	−	0.358843i	$-0.116829\pi$
$443$	39.2915	1.86680	0.933398	+	0.358843i	$0.116829\pi$
$444$	0	0
$445$	−6.58301	−0.312064
$446$	5.29150	0.250560
$447$	0	0
$448$	0	0
$449$	23.8118	1.12375	0.561873	−	0.827223i	$-0.310081\pi$
$449$	23.8118	1.12375	0.561873	+	0.827223i	$0.310081\pi$
$450$	0	0
$451$	−6.29150	−0.296255
$452$	12.5830	0.591855
$453$	0	0
$454$	−3.29150	−0.154478
$455$	0	0
$456$	0	0
$457$	−6.70850	−0.313810	−0.156905	−	0.987614i	$-0.550152\pi$
$457$	−6.70850	−0.313810	−0.156905	+	0.987614i	$0.550152\pi$
$458$	17.2915	0.807979
$459$	0	0
$460$	3.00000	0.139876
$461$	−25.1660	−1.17210	−0.586049	−	0.810276i	$-0.699317\pi$
$461$	−25.1660	−1.17210	−0.586049	+	0.810276i	$0.699317\pi$
$462$	0	0
$463$	0.0627461	0.00291606	0.00145803	−	0.999999i	$-0.499536\pi$
$463$	0.0627461	0.00291606	0.00145803	+	0.999999i	$0.499536\pi$
$464$	0	0
$465$	0	0
$466$	3.87451	0.179483
$467$	−8.70850	−0.402981	−0.201491	−	0.979490i	$-0.564579\pi$
$467$	−8.70850	−0.402981	−0.201491	+	0.979490i	$0.564579\pi$
$468$	0	0
$469$	0	0
$470$	−9.58301	−0.442031
$471$	0	0
$472$	9.29150	0.427676
$473$	−52.4575	−2.41200
$474$	0	0
$475$	4.29150	0.196908
$476$	0	0
$477$	0	0
$478$	12.5830	0.575533
$479$	25.7490	1.17650	0.588251	−	0.808678i	$-0.299817\pi$
$479$	25.7490	1.17650	0.588251	+	0.808678i	$0.299817\pi$
$480$	0	0
$481$	3.83399	0.174815
$482$	−25.5830	−1.16527
$483$	0	0
$484$	10.5830	0.481046
$485$	−4.00000	−0.181631
$486$	0	0
$487$	−7.87451	−0.356828	−0.178414	−	0.983956i	$-0.557097\pi$
$487$	−7.87451	−0.356828	−0.178414	+	0.983956i	$0.557097\pi$
$488$	11.2915	0.511142
$489$	0	0
$490$	0	0
$491$	8.12549	0.366698	0.183349	−	0.983048i	$-0.441306\pi$
$491$	8.12549	0.366698	0.183349	+	0.983048i	$0.441306\pi$
$492$	0	0
$493$	0	0
$494$	2.77124	0.124684
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	0	0
$499$	14.5830	0.652825	0.326412	−	0.945227i	$-0.394160\pi$
$499$	14.5830	0.652825	0.326412	+	0.945227i	$0.394160\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−13.9373	−0.622050
$503$	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
$506$	−13.9373	−0.619587
$507$	0	0
$508$	9.35425	0.415028
$509$	−41.0405	−1.81909	−0.909544	−	0.415607i	$-0.863569\pi$
$509$	−41.0405	−1.81909	−0.909544	+	0.415607i	$0.863569\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−27.2915	−1.20378
$515$	17.2915	0.761955
$516$	0	0
$517$	44.5203	1.95800
$518$	0	0
$519$	0	0
$520$	−0.645751	−0.0283181
$521$	17.2288	0.754806	0.377403	−	0.926049i	$-0.376817\pi$
$521$	17.2288	0.754806	0.377403	+	0.926049i	$0.376817\pi$
$522$	0	0
$523$	−15.1660	−0.663163	−0.331582	−	0.943427i	$-0.607582\pi$
$523$	−15.1660	−0.663163	−0.331582	+	0.943427i	$0.607582\pi$
$524$	−4.64575	−0.202951
$525$	0	0
$526$	−6.58301	−0.287033
$527$	−6.58301	−0.286760
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0.291503	0.0126621
$531$	0	0
$532$	0	0
$533$	−0.874508	−0.0378791
$534$	0	0
$535$	−15.8745	−0.686315
$536$	−4.70850	−0.203376
$537$	0	0
$538$	21.8745	0.943077
$539$	0	0
$540$	0	0
$541$	42.4575	1.82539	0.912696	−	0.408640i	$-0.133997\pi$
$541$	42.4575	1.82539	0.912696	+	0.408640i	$0.133997\pi$
$542$	−17.1660	−0.737343
$543$	0	0
$544$	−3.29150	−0.141122
$545$	−20.5830	−0.881679
$546$	0	0
$547$	−19.2915	−0.824845	−0.412423	−	0.910993i	$-0.635317\pi$
$547$	−19.2915	−0.824845	−0.412423	+	0.910993i	$0.635317\pi$
$548$	−15.8745	−0.678125
$549$	0	0
$550$	4.64575	0.198096
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−26.5830	−1.12940
$555$	0	0
$556$	10.5830	0.448819
$557$	38.0405	1.61183	0.805914	−	0.592032i	$-0.201674\pi$
$557$	38.0405	1.61183	0.805914	+	0.592032i	$0.201674\pi$
$558$	0	0
$559$	−7.29150	−0.308398
$560$	0	0
$561$	0	0
$562$	−26.5203	−1.11869
$563$	31.1660	1.31349	0.656745	−	0.754112i	$-0.271933\pi$
$563$	31.1660	1.31349	0.656745	+	0.754112i	$0.271933\pi$
$564$	0	0
$565$	−12.5830	−0.529371
$566$	−7.87451	−0.330990
$567$	0	0
$568$	−2.70850	−0.113646
$569$	11.8118	0.495175	0.247587	−	0.968866i	$-0.420362\pi$
$569$	11.8118	0.495175	0.247587	+	0.968866i	$0.420362\pi$
$570$	0	0
$571$	−9.41699	−0.394089	−0.197044	−	0.980395i	$-0.563134\pi$
$571$	−9.41699	−0.394089	−0.197044	+	0.980395i	$0.563134\pi$
$572$	3.00000	0.125436
$573$	0	0
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−15.1660	−0.631369	−0.315685	−	0.948864i	$-0.602234\pi$
$577$	−15.1660	−0.631369	−0.315685	+	0.948864i	$0.602234\pi$
$578$	6.16601	0.256472
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.35425	−0.0560872
$584$	2.00000	0.0827606
$585$	0	0
$586$	−11.7085	−0.483674
$587$	−12.5830	−0.519356	−0.259678	−	0.965695i	$-0.583616\pi$
$587$	−12.5830	−0.519356	−0.259678	+	0.965695i	$0.583616\pi$
$588$	0	0
$589$	−8.58301	−0.353657
$590$	−9.29150	−0.382525
$591$	0	0
$592$	−5.93725	−0.244020
$593$	2.12549	0.0872835	0.0436418	−	0.999047i	$-0.486104\pi$
$593$	2.12549	0.0872835	0.0436418	+	0.999047i	$0.486104\pi$
$594$	0	0
$595$	0	0
$596$	−21.8745	−0.896015
$597$	0	0
$598$	−1.93725	−0.0792202
$599$	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
$601$	11.1660	0.455471	0.227736	−	0.973723i	$-0.426868\pi$
$601$	11.1660	0.455471	0.227736	+	0.973723i	$0.426868\pi$
$602$	0	0
$603$	0	0
$604$	14.0000	0.569652
$605$	−10.5830	−0.430260
$606$	0	0
$607$	10.7712	0.437191	0.218596	−	0.975816i	$-0.429852\pi$
$607$	10.7712	0.437191	0.218596	+	0.975816i	$0.429852\pi$
$608$	−4.29150	−0.174043
$609$	0	0
$610$	−11.2915	−0.457180
$611$	6.18824	0.250349
$612$	0	0
$613$	−11.3542	−0.458594	−0.229297	−	0.973357i	$-0.573643\pi$
$613$	−11.3542	−0.458594	−0.229297	+	0.973357i	$0.573643\pi$
$614$	−12.7085	−0.512873
$615$	0	0
$616$	0	0
$617$	−15.2915	−0.615613	−0.307806	−	0.951449i	$-0.599595\pi$
$617$	−15.2915	−0.615613	−0.307806	+	0.951449i	$0.599595\pi$
$618$	0	0
$619$	4.87451	0.195923	0.0979615	−	0.995190i	$-0.468768\pi$
$619$	4.87451	0.195923	0.0979615	+	0.995190i	$0.468768\pi$
$620$	2.00000	0.0803219
$621$	0	0
$622$	21.8745	0.877088
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−1.87451	−0.0749204
$627$	0	0
$628$	20.6458	0.823855
$629$	−19.5425	−0.779210
$630$	0	0
$631$	−29.7490	−1.18429	−0.592145	−	0.805832i	$-0.701719\pi$
$631$	−29.7490	−1.18429	−0.592145	+	0.805832i	$0.701719\pi$
$632$	−11.2915	−0.449152
$633$	0	0
$634$	−11.4170	−0.453427
$635$	−9.35425	−0.371212
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	−45.1033	−1.78147	−0.890736	−	0.454521i	$-0.849810\pi$
$641$	−45.1033	−1.78147	−0.890736	+	0.454521i	$0.849810\pi$
$642$	0	0
$643$	−17.2915	−0.681910	−0.340955	−	0.940080i	$-0.610750\pi$
$643$	−17.2915	−0.681910	−0.340955	+	0.940080i	$0.610750\pi$
$644$	0	0
$645$	0	0
$646$	−14.1255	−0.555760
$647$	−28.1660	−1.10732	−0.553660	−	0.832743i	$-0.686769\pi$
$647$	−28.1660	−1.10732	−0.553660	+	0.832743i	$0.686769\pi$
$648$	0	0
$649$	43.1660	1.69441
$650$	0.645751	0.0253285
$651$	0	0
$652$	8.00000	0.313304
$653$	37.4575	1.46583	0.732913	−	0.680323i	$-0.238160\pi$
$653$	37.4575	1.46583	0.732913	+	0.680323i	$0.238160\pi$
$654$	0	0
$655$	4.64575	0.181525
$656$	1.35425	0.0528745
$657$	0	0
$658$	0	0
$659$	−2.70850	−0.105508	−0.0527540	−	0.998608i	$-0.516800\pi$
$659$	−2.70850	−0.105508	−0.0527540	+	0.998608i	$0.516800\pi$
$660$	0	0
$661$	−1.41699	−0.0551147	−0.0275574	−	0.999620i	$-0.508773\pi$
$661$	−1.41699	−0.0551147	−0.0275574	+	0.999620i	$0.508773\pi$
$662$	23.4575	0.911702
$663$	0	0
$664$	−15.2915	−0.593425
$665$	0	0
$666$	0	0
$667$	0	0
$668$	9.00000	0.348220
$669$	0	0
$670$	4.70850	0.181905
$671$	52.4575	2.02510
$672$	0	0
$673$	−22.5830	−0.870511	−0.435255	−	0.900307i	$-0.643342\pi$
$673$	−22.5830	−0.870511	−0.435255	+	0.900307i	$0.643342\pi$
$674$	−11.2915	−0.434932
$675$	0	0
$676$	−12.5830	−0.483962
$677$	48.8745	1.87840	0.939200	−	0.343371i	$-0.111569\pi$
$677$	48.8745	1.87840	0.939200	+	0.343371i	$0.111569\pi$
$678$	0	0
$679$	0	0
$680$	3.29150	0.126223
$681$	0	0
$682$	−9.29150	−0.355790
$683$	−13.1660	−0.503783	−0.251892	−	0.967755i	$-0.581053\pi$
$683$	−13.1660	−0.503783	−0.251892	+	0.967755i	$0.581053\pi$
$684$	0	0
$685$	15.8745	0.606534
$686$	0	0
$687$	0	0
$688$	11.2915	0.430485
$689$	−0.188238	−0.00717130
$690$	0	0
$691$	34.5830	1.31560	0.657800	−	0.753193i	$-0.271487\pi$
$691$	34.5830	1.31560	0.657800	+	0.753193i	$0.271487\pi$
$692$	18.2915	0.695339
$693$	0	0
$694$	−9.29150	−0.352701
$695$	−10.5830	−0.401436
$696$	0	0
$697$	4.45751	0.168840
$698$	1.41699	0.0536340
$699$	0	0
$700$	0	0
$701$	−22.4575	−0.848209	−0.424104	−	0.905613i	$-0.639411\pi$
$701$	−22.4575	−0.848209	−0.424104	+	0.905613i	$0.639411\pi$
$702$	0	0
$703$	−25.4797	−0.960987
$704$	−4.64575	−0.175093
$705$	0	0
$706$	12.0000	0.451626
$707$	0	0
$708$	0	0
$709$	21.1660	0.794906	0.397453	−	0.917622i	$-0.369894\pi$
$709$	21.1660	0.794906	0.397453	+	0.917622i	$0.369894\pi$
$710$	2.70850	0.101648
$711$	0	0
$712$	−6.58301	−0.246709
$713$	6.00000	0.224702
$714$	0	0
$715$	−3.00000	−0.112194
$716$	25.9373	0.969321
$717$	0	0
$718$	−18.5830	−0.693512
$719$	−51.8745	−1.93459	−0.967296	−	0.253649i	$-0.918369\pi$
$719$	−51.8745	−1.93459	−0.967296	+	0.253649i	$0.918369\pi$
$720$	0	0
$721$	0	0
$722$	0.583005	0.0216972
$723$	0	0
$724$	0.708497	0.0263311
$725$	0	0
$726$	0	0
$727$	−30.6458	−1.13659	−0.568294	−	0.822826i	$-0.692396\pi$
$727$	−30.6458	−1.13659	−0.568294	+	0.822826i	$0.692396\pi$
$728$	0	0
$729$	0	0
$730$	−2.00000	−0.0740233
$731$	37.1660	1.37463
$732$	0	0
$733$	−11.4797	−0.424014	−0.212007	−	0.977268i	$-0.568000\pi$
$733$	−11.4797	−0.424014	−0.212007	+	0.977268i	$0.568000\pi$
$734$	−26.6458	−0.983513
$735$	0	0
$736$	3.00000	0.110581
$737$	−21.8745	−0.805758
$738$	0	0
$739$	−21.7085	−0.798560	−0.399280	−	0.916829i	$-0.630740\pi$
$739$	−21.7085	−0.798560	−0.399280	+	0.916829i	$0.630740\pi$
$740$	5.93725	0.218258
$741$	0	0
$742$	0	0
$743$	28.7490	1.05470	0.527350	−	0.849648i	$-0.323186\pi$
$743$	28.7490	1.05470	0.527350	+	0.849648i	$0.323186\pi$
$744$	0	0
$745$	21.8745	0.801420
$746$	4.00000	0.146450
$747$	0	0
$748$	−15.2915	−0.559113
$749$	0	0
$750$	0	0
$751$	4.12549	0.150541	0.0752707	−	0.997163i	$-0.476018\pi$
$751$	4.12549	0.150541	0.0752707	+	0.997163i	$0.476018\pi$
$752$	−9.58301	−0.349456
$753$	0	0
$754$	0	0
$755$	−14.0000	−0.509512
$756$	0	0
$757$	33.1660	1.20544	0.602720	−	0.797953i	$-0.294084\pi$
$757$	33.1660	1.20544	0.602720	+	0.797953i	$0.294084\pi$
$758$	−1.70850	−0.0620554
$759$	0	0
$760$	4.29150	0.155669
$761$	43.9373	1.59272	0.796362	−	0.604820i	$-0.206755\pi$
$761$	43.9373	1.59272	0.796362	+	0.604820i	$0.206755\pi$
$762$	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
$769$	−30.1660	−1.08781	−0.543907	−	0.839145i	$-0.683056\pi$
$769$	−30.1660	−1.08781	−0.543907	+	0.839145i	$0.683056\pi$
$770$	0	0
$771$	0	0
$772$	−7.87451	−0.283410
$773$	−12.8745	−0.463064	−0.231532	−	0.972827i	$-0.574374\pi$
$773$	−12.8745	−0.463064	−0.231532	+	0.972827i	$0.574374\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	−4.00000	−0.143592
$777$	0	0
$778$	−14.7085	−0.527325
$779$	5.81176	0.208228
$780$	0	0
$781$	−12.5830	−0.450255
$782$	9.87451	0.353112
$783$	0	0
$784$	0	0
$785$	−20.6458	−0.736878
$786$	0	0
$787$	38.4575	1.37086	0.685431	−	0.728137i	$-0.259614\pi$
$787$	38.4575	1.37086	0.685431	+	0.728137i	$0.259614\pi$
$788$	−6.29150	−0.224126
$789$	0	0
$790$	11.2915	0.401734
$791$	0	0
$792$	0	0
$793$	7.29150	0.258929
$794$	−22.5830	−0.801441
$795$	0	0
$796$	−17.8745	−0.633545
$797$	37.7490	1.33714	0.668569	−	0.743650i	$-0.266907\pi$
$797$	37.7490	1.33714	0.668569	+	0.743650i	$0.266907\pi$
$798$	0	0
$799$	−31.5425	−1.11589
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−17.2288	−0.608369
$803$	9.29150	0.327890
$804$	0	0
$805$	0	0
$806$	−1.29150	−0.0454912
$807$	0	0
$808$	12.0000	0.422159
$809$	−11.8118	−0.415279	−0.207640	−	0.978205i	$-0.566578\pi$
$809$	−11.8118	−0.415279	−0.207640	+	0.978205i	$0.566578\pi$
$810$	0	0
$811$	16.8745	0.592544	0.296272	−	0.955104i	$-0.404256\pi$
$811$	16.8745	0.592544	0.296272	+	0.955104i	$0.404256\pi$
$812$	0	0
$813$	0	0
$814$	−27.5830	−0.966784
$815$	−8.00000	−0.280228
$816$	0	0
$817$	48.4575	1.69531
$818$	32.5830	1.13924
$819$	0	0
$820$	−1.35425	−0.0472924
$821$	15.8745	0.554024	0.277012	−	0.960866i	$-0.410656\pi$
$821$	15.8745	0.554024	0.277012	+	0.960866i	$0.410656\pi$
$822$	0	0
$823$	−18.7085	−0.652137	−0.326069	−	0.945346i	$-0.605724\pi$
$823$	−18.7085	−0.652137	−0.326069	+	0.945346i	$0.605724\pi$
$824$	17.2915	0.602378
$825$	0	0
$826$	0	0
$827$	48.5830	1.68940	0.844698	−	0.535243i	$-0.179780\pi$
$827$	48.5830	1.68940	0.844698	+	0.535243i	$0.179780\pi$
$828$	0	0
$829$	−10.1255	−0.351673	−0.175836	−	0.984419i	$-0.556263\pi$
$829$	−10.1255	−0.351673	−0.175836	+	0.984419i	$0.556263\pi$
$830$	15.2915	0.530776
$831$	0	0
$832$	−0.645751	−0.0223874
$833$	0	0
$834$	0	0
$835$	−9.00000	−0.311458
$836$	−19.9373	−0.689544
$837$	0	0
$838$	22.0627	0.762145
$839$	−40.4575	−1.39675	−0.698374	−	0.715733i	$-0.746093\pi$
$839$	−40.4575	−1.39675	−0.698374	+	0.715733i	$0.746093\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0.125492	0.00432474
$843$	0	0
$844$	8.29150	0.285405
$845$	12.5830	0.432869
$846$	0	0
$847$	0	0
$848$	0.291503	0.0100102
$849$	0	0
$850$	−3.29150	−0.112898
$851$	17.8118	0.610579
$852$	0	0
$853$	−49.8118	−1.70552	−0.852761	−	0.522301i	$-0.825074\pi$
$853$	−49.8118	−1.70552	−0.852761	+	0.522301i	$0.825074\pi$
$854$	0	0
$855$	0	0
$856$	−15.8745	−0.542580
$857$	15.8745	0.542263	0.271131	−	0.962542i	$-0.412602\pi$
$857$	15.8745	0.542263	0.271131	+	0.962542i	$0.412602\pi$
$858$	0	0
$859$	41.1660	1.40457	0.702283	−	0.711898i	$-0.252164\pi$
$859$	41.1660	1.40457	0.702283	+	0.711898i	$0.252164\pi$
$860$	−11.2915	−0.385037
$861$	0	0
$862$	−2.12549	−0.0723945
$863$	22.1660	0.754540	0.377270	−	0.926103i	$-0.376863\pi$
$863$	22.1660	0.754540	0.377270	+	0.926103i	$0.376863\pi$
$864$	0	0
$865$	−18.2915	−0.621930
$866$	8.00000	0.271851
$867$	0	0
$868$	0	0
$869$	−52.4575	−1.77950
$870$	0	0
$871$	−3.04052	−0.103024
$872$	−20.5830	−0.697029
$873$	0	0
$874$	12.8745	0.435487
$875$	0	0
$876$	0	0
$877$	−49.6863	−1.67779	−0.838893	−	0.544296i	$-0.816797\pi$
$877$	−49.6863	−1.67779	−0.838893	+	0.544296i	$0.816797\pi$
$878$	33.1660	1.11930
$879$	0	0
$880$	4.64575	0.156608
$881$	37.3542	1.25850	0.629248	−	0.777204i	$-0.283363\pi$
$881$	37.3542	1.25850	0.629248	+	0.777204i	$0.283363\pi$
$882$	0	0
$883$	32.5830	1.09651	0.548253	−	0.836313i	$-0.315293\pi$
$883$	32.5830	1.09651	0.548253	+	0.836313i	$0.315293\pi$
$884$	−2.12549	−0.0714880
$885$	0	0
$886$	−39.2915	−1.32002
$887$	1.16601	0.0391508	0.0195754	−	0.999808i	$-0.493769\pi$
$887$	1.16601	0.0391508	0.0195754	+	0.999808i	$0.493769\pi$
$888$	0	0
$889$	0	0
$890$	6.58301	0.220663
$891$	0	0
$892$	−5.29150	−0.177173
$893$	−41.1255	−1.37621
$894$	0	0
$895$	−25.9373	−0.866987
$896$	0	0
$897$	0	0
$898$	−23.8118	−0.794609
$899$	0	0
$900$	0	0
$901$	0.959482	0.0319650
$902$	6.29150	0.209484
$903$	0	0
$904$	−12.5830	−0.418505
$905$	−0.708497	−0.0235512
$906$	0	0
$907$	−41.7490	−1.38625	−0.693127	−	0.720816i	$-0.743767\pi$
$907$	−41.7490	−1.38625	−0.693127	+	0.720816i	$0.743767\pi$
$908$	3.29150	0.109232
$909$	0	0
$910$	0	0
$911$	47.6235	1.57784	0.788919	−	0.614497i	$-0.210641\pi$
$911$	47.6235	1.57784	0.788919	+	0.614497i	$0.210641\pi$
$912$	0	0
$913$	−71.0405	−2.35110
$914$	6.70850	0.221897
$915$	0	0
$916$	−17.2915	−0.571327
$917$	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
$922$	25.1660	0.828798
$923$	−1.74902	−0.0575696
$924$	0	0
$925$	−5.93725	−0.195216
$926$	−0.0627461	−0.00206196
$927$	0	0
$928$	0	0
$929$	−45.1033	−1.47979	−0.739895	−	0.672722i	$-0.765125\pi$
$929$	−45.1033	−1.47979	−0.739895	+	0.672722i	$0.765125\pi$
$930$	0	0
$931$	0	0
$932$	−3.87451	−0.126914
$933$	0	0
$934$	8.70850	0.284951
$935$	15.2915	0.500086
$936$	0	0
$937$	−37.6235	−1.22911	−0.614553	−	0.788875i	$-0.710664\pi$
$937$	−37.6235	−1.22911	−0.614553	+	0.788875i	$0.710664\pi$
$938$	0	0
$939$	0	0
$940$	9.58301	0.312563
$941$	−24.5830	−0.801383	−0.400692	−	0.916213i	$-0.631230\pi$
$941$	−24.5830	−0.801383	−0.400692	+	0.916213i	$0.631230\pi$
$942$	0	0
$943$	−4.06275	−0.132301
$944$	−9.29150	−0.302413
$945$	0	0
$946$	52.4575	1.70554
$947$	−29.0405	−0.943690	−0.471845	−	0.881682i	$-0.656412\pi$
$947$	−29.0405	−0.943690	−0.471845	+	0.881682i	$0.656412\pi$
$948$	0	0
$949$	1.29150	0.0419239
$950$	−4.29150	−0.139235
$951$	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
$956$	−12.5830	−0.406963
$957$	0	0
$958$	−25.7490	−0.831913
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−3.83399	−0.123613
$963$	0	0
$964$	25.5830	0.823973
$965$	7.87451	0.253489
$966$	0	0
$967$	−18.7085	−0.601625	−0.300812	−	0.953683i	$-0.597258\pi$
$967$	−18.7085	−0.601625	−0.300812	+	0.953683i	$0.597258\pi$
$968$	−10.5830	−0.340151
$969$	0	0
$970$	4.00000	0.128432
$971$	6.18824	0.198590	0.0992950	−	0.995058i	$-0.468341\pi$
$971$	6.18824	0.198590	0.0992950	+	0.995058i	$0.468341\pi$
$972$	0	0
$973$	0	0
$974$	7.87451	0.252316
$975$	0	0
$976$	−11.2915	−0.361432
$977$	−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
$979$	−30.5830	−0.977437
$980$	0	0
$981$	0	0
$982$	−8.12549	−0.259295
$983$	−35.3320	−1.12692	−0.563458	−	0.826145i	$-0.690529\pi$
$983$	−35.3320	−1.12692	−0.563458	+	0.826145i	$0.690529\pi$
$984$	0	0
$985$	6.29150	0.200464
$986$	0	0
$987$	0	0
$988$	−2.77124	−0.0881650
$989$	−33.8745	−1.07715
$990$	0	0
$991$	5.29150	0.168090	0.0840451	−	0.996462i	$-0.473216\pi$
$991$	5.29150	0.168090	0.0840451	+	0.996462i	$0.473216\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	0	0
$995$	17.8745	0.566660
$996$	0	0
$997$	−15.7490	−0.498776	−0.249388	−	0.968404i	$-0.580229\pi$
$997$	−15.7490	−0.498776	−0.249388	+	0.968404i	$0.580229\pi$
$998$	−14.5830	−0.461617
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.bo.1.1		2
3.2	odd	2		4410.2.a.ca.1.2		2
7.3	odd	6		630.2.k.j.541.2	yes	4
7.5	odd	6		630.2.k.j.361.2	yes	4
7.6	odd	2		4410.2.a.bq.1.1		2
21.5	even	6		630.2.k.i.361.2	✓	4
21.17	even	6		630.2.k.i.541.2	yes	4
21.20	even	2		4410.2.a.bv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.k.i.361.2	✓	4	21.5	even	6
630.2.k.i.541.2	yes	4	21.17	even	6
630.2.k.j.361.2	yes	4	7.5	odd	6
630.2.k.j.541.2	yes	4	7.3	odd	6
4410.2.a.bo.1.1		2	1.1	even	1	trivial
4410.2.a.bq.1.1		2	7.6	odd	2
4410.2.a.bv.1.2		2	21.20	even	2
4410.2.a.ca.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} -4.64575 q^{11} -0.645751 q^{13} +1.00000 q^{16} +3.29150 q^{17} +4.29150 q^{19} -1.00000 q^{20} +4.64575 q^{22} -3.00000 q^{23} +1.00000 q^{25} +0.645751 q^{26} -2.00000 q^{31} -1.00000 q^{32} -3.29150 q^{34} -5.93725 q^{37} -4.29150 q^{38} +1.00000 q^{40} +1.35425 q^{41} +11.2915 q^{43} -4.64575 q^{44} +3.00000 q^{46} -9.58301 q^{47} -1.00000 q^{50} -0.645751 q^{52} +0.291503 q^{53} +4.64575 q^{55} -9.29150 q^{59} -11.2915 q^{61} +2.00000 q^{62} +1.00000 q^{64} +0.645751 q^{65} +4.70850 q^{67} +3.29150 q^{68} +2.70850 q^{71} -2.00000 q^{73} +5.93725 q^{74} +4.29150 q^{76} +11.2915 q^{79} -1.00000 q^{80} -1.35425 q^{82} +15.2915 q^{83} -3.29150 q^{85} -11.2915 q^{86} +4.64575 q^{88} +6.58301 q^{89} -3.00000 q^{92} +9.58301 q^{94} -4.29150 q^{95} +4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) \) 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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