LMFDB - Embedded newform 3920.2.a.ce.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1960)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.18398$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.18398 q^{3} -1.00000 q^{5} -1.59819 q^{9} +O(q^{10})$	$q-1.18398 q^{3} -1.00000 q^{5} -1.59819 q^{9} -0.230234 q^{11} +2.27259 q^{13} +1.18398 q^{15} -6.53278 q^{17} -0.260186 q^{19} +8.87079 q^{23} +1.00000 q^{25} +5.44417 q^{27} +5.42662 q^{29} +4.87079 q^{31} +0.272593 q^{33} -1.15403 q^{37} -2.69070 q^{39} -4.43337 q^{41} -4.17723 q^{43} +1.59819 q^{45} +0.923793 q^{47} +7.73467 q^{51} -2.13487 q^{53} +0.230234 q^{55} +0.308055 q^{57} -5.07107 q^{59} -8.88833 q^{61} -2.27259 q^{65} -8.15968 q^{67} -10.5028 q^{69} +9.06556 q^{71} -6.00955 q^{73} -1.18398 q^{75} -0.112912 q^{79} -1.65120 q^{81} +8.72065 q^{83} +6.53278 q^{85} -6.42501 q^{87} -15.9170 q^{89} -5.76691 q^{93} +0.260186 q^{95} -4.29565 q^{97} +0.367959 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} + 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 4 * q^5 + 6 * q^9	$ 4 q + 2 q^{3} - 4 q^{5} + 6 q^{9} - 2 q^{11} - 10 q^{13} - 2 q^{15} - 6 q^{17} + 4 q^{23} + 4 q^{25} + 14 q^{27} - 2 q^{29} - 12 q^{31} - 18 q^{33} - 14 q^{39} - 12 q^{41} + 8 q^{43} - 6 q^{45} - 2 q^{47} - 2 q^{51} - 4 q^{53} + 2 q^{55} - 8 q^{57} + 8 q^{59} - 20 q^{61} + 10 q^{65} + 8 q^{67} - 24 q^{69} - 4 q^{71} - 16 q^{73} + 2 q^{75} - 22 q^{79} - 20 q^{81} + 36 q^{83} + 6 q^{85} - 18 q^{87} - 40 q^{89} - 32 q^{93} - 26 q^{97} - 12 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 + 6 * q^9 - 2 * q^11 - 10 * q^13 - 2 * q^15 - 6 * q^17 + 4 * q^23 + 4 * q^25 + 14 * q^27 - 2 * q^29 - 12 * q^31 - 18 * q^33 - 14 * q^39 - 12 * q^41 + 8 * q^43 - 6 * q^45 - 2 * q^47 - 2 * q^51 - 4 * q^53 + 2 * q^55 - 8 * q^57 + 8 * q^59 - 20 * q^61 + 10 * q^65 + 8 * q^67 - 24 * q^69 - 4 * q^71 - 16 * q^73 + 2 * q^75 - 22 * q^79 - 20 * q^81 + 36 * q^83 + 6 * q^85 - 18 * q^87 - 40 * q^89 - 32 * q^93 - 26 * q^97 - 12 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.18398	−0.683571	−0.341785	−	0.939778i	$-0.611032\pi$
$3$	−1.18398	−0.683571	−0.341785	+	0.939778i	$0.611032\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.59819	−0.532731
$10$	0	0
$11$	−0.230234	−0.0694182	−0.0347091	−	0.999397i	$-0.511050\pi$
$11$	−0.230234	−0.0694182	−0.0347091	+	0.999397i	$0.511050\pi$
$12$	0	0
$13$	2.27259	0.630304	0.315152	−	0.949041i	$-0.397945\pi$
$13$	2.27259	0.630304	0.315152	+	0.949041i	$0.397945\pi$
$14$	0	0
$15$	1.18398	0.305702
$16$	0	0
$17$	−6.53278	−1.58443	−0.792216	−	0.610241i	$-0.791073\pi$
$17$	−6.53278	−1.58443	−0.792216	+	0.610241i	$0.791073\pi$
$18$	0	0
$19$	−0.260186	−0.0596908	−0.0298454	−	0.999555i	$-0.509501\pi$
$19$	−0.260186	−0.0596908	−0.0298454	+	0.999555i	$0.509501\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.87079	1.84969	0.924843	−	0.380348i	$-0.124196\pi$
$23$	8.87079	1.84969	0.924843	+	0.380348i	$0.124196\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.44417	1.04773
$28$	0	0
$29$	5.42662	1.00770	0.503849	−	0.863792i	$-0.331917\pi$
$29$	5.42662	1.00770	0.503849	+	0.863792i	$0.331917\pi$
$30$	0	0
$31$	4.87079	0.874819	0.437409	−	0.899262i	$-0.355896\pi$
$31$	4.87079	0.874819	0.437409	+	0.899262i	$0.355896\pi$
$32$	0	0
$33$	0.272593	0.0474523
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.15403	−0.189721	−0.0948605	−	0.995491i	$-0.530240\pi$
$37$	−1.15403	−0.189721	−0.0948605	+	0.995491i	$0.530240\pi$
$38$	0	0
$39$	−2.69070	−0.430857
$40$	0	0
$41$	−4.43337	−0.692377	−0.346188	−	0.938165i	$-0.612524\pi$
$41$	−4.43337	−0.692377	−0.346188	+	0.938165i	$0.612524\pi$
$42$	0	0
$43$	−4.17723	−0.637021	−0.318511	−	0.947919i	$-0.603183\pi$
$43$	−4.17723	−0.637021	−0.318511	+	0.947919i	$0.603183\pi$
$44$	0	0
$45$	1.59819	0.238245
$46$	0	0
$47$	0.923793	0.134749	0.0673745	−	0.997728i	$-0.478538\pi$
$47$	0.923793	0.134749	0.0673745	+	0.997728i	$0.478538\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7.73467	1.08307
$52$	0	0
$53$	−2.13487	−0.293247	−0.146623	−	0.989192i	$-0.546841\pi$
$53$	−2.13487	−0.293247	−0.146623	+	0.989192i	$0.546841\pi$
$54$	0	0
$55$	0.230234	0.0310448
$56$	0	0
$57$	0.308055	0.0408029
$58$	0	0
$59$	−5.07107	−0.660197	−0.330098	−	0.943947i	$-0.607082\pi$
$59$	−5.07107	−0.660197	−0.330098	+	0.943947i	$0.607082\pi$
$60$	0	0
$61$	−8.88833	−1.13803	−0.569017	−	0.822326i	$-0.692676\pi$
$61$	−8.88833	−1.13803	−0.569017	+	0.822326i	$0.692676\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.27259	−0.281880
$66$	0	0
$67$	−8.15968	−0.996864	−0.498432	−	0.866929i	$-0.666091\pi$
$67$	−8.15968	−0.996864	−0.498432	+	0.866929i	$0.666091\pi$
$68$	0	0
$69$	−10.5028	−1.26439
$70$	0	0
$71$	9.06556	1.07588	0.537942	−	0.842982i	$-0.319202\pi$
$71$	9.06556	1.07588	0.537942	+	0.842982i	$0.319202\pi$
$72$	0	0
$73$	−6.00955	−0.703365	−0.351682	−	0.936119i	$-0.614390\pi$
$73$	−6.00955	−0.703365	−0.351682	+	0.936119i	$0.614390\pi$
$74$	0	0
$75$	−1.18398	−0.136714
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.112912	−0.0127035	−0.00635177	−	0.999980i	$-0.502022\pi$
$79$	−0.112912	−0.0127035	−0.00635177	+	0.999980i	$0.502022\pi$
$80$	0	0
$81$	−1.65120	−0.183467
$82$	0	0
$83$	8.72065	0.957216	0.478608	−	0.878029i	$-0.341141\pi$
$83$	8.72065	0.957216	0.478608	+	0.878029i	$0.341141\pi$
$84$	0	0
$85$	6.53278	0.708579
$86$	0	0
$87$	−6.42501	−0.688833
$88$	0	0
$89$	−15.9170	−1.68720	−0.843601	−	0.536970i	$-0.819569\pi$
$89$	−15.9170	−1.68720	−0.843601	+	0.536970i	$0.819569\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.76691	−0.598001
$94$	0	0
$95$	0.260186	0.0266945
$96$	0	0
$97$	−4.29565	−0.436157	−0.218079	−	0.975931i	$-0.569979\pi$
$97$	−4.29565	−0.436157	−0.218079	+	0.975931i	$0.569979\pi$
$98$	0	0
$99$	0.367959	0.0369812
$100$	0	0
$101$	−7.38712	−0.735046	−0.367523	−	0.930014i	$-0.619794\pi$
$101$	−7.38712	−0.735046	−0.367523	+	0.930014i	$0.619794\pi$
$102$	0	0
$103$	−10.5806	−1.04254	−0.521271	−	0.853391i	$-0.674542\pi$
$103$	−10.5806	−1.04254	−0.521271	+	0.853391i	$0.674542\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.4661	1.30182	0.650910	−	0.759155i	$-0.274388\pi$
$107$	13.4661	1.30182	0.650910	+	0.759155i	$0.274388\pi$
$108$	0	0
$109$	8.50568	0.814697	0.407348	−	0.913273i	$-0.366453\pi$
$109$	8.50568	0.814697	0.407348	+	0.913273i	$0.366453\pi$
$110$	0	0
$111$	1.36634	0.129688
$112$	0	0
$113$	18.3968	1.73063	0.865313	−	0.501232i	$-0.167120\pi$
$113$	18.3968	1.73063	0.865313	+	0.501232i	$0.167120\pi$
$114$	0	0
$115$	−8.87079	−0.827205
$116$	0	0
$117$	−3.63204	−0.335782
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9470	−0.995181
$122$	0	0
$123$	5.24902	0.473288
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.41032	−0.746295	−0.373147	−	0.927772i	$-0.621721\pi$
$127$	−8.41032	−0.746295	−0.373147	+	0.927772i	$0.621721\pi$
$128$	0	0
$129$	4.94575	0.435449
$130$	0	0
$131$	−1.78217	−0.155709	−0.0778546	−	0.996965i	$-0.524807\pi$
$131$	−1.78217	−0.155709	−0.0778546	+	0.996965i	$0.524807\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.44417	−0.468559
$136$	0	0
$137$	−11.3137	−0.966595	−0.483298	−	0.875456i	$-0.660561\pi$
$137$	−11.3137	−0.966595	−0.483298	+	0.875456i	$0.660561\pi$
$138$	0	0
$139$	15.4390	1.30952	0.654761	−	0.755836i	$-0.272769\pi$
$139$	15.4390	1.30952	0.654761	+	0.755836i	$0.272769\pi$
$140$	0	0
$141$	−1.09375	−0.0921105
$142$	0	0
$143$	−0.523229	−0.0437546
$144$	0	0
$145$	−5.42662	−0.450656
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.2811	1.90726	0.953631	−	0.300978i	$-0.0973130\pi$
$149$	23.2811	1.90726	0.953631	+	0.300978i	$0.0973130\pi$
$150$	0	0
$151$	−22.3398	−1.81798	−0.908992	−	0.416813i	$-0.863147\pi$
$151$	−22.3398	−1.81798	−0.908992	+	0.416813i	$0.863147\pi$
$152$	0	0
$153$	10.4406	0.844076
$154$	0	0
$155$	−4.87079	−0.391231
$156$	0	0
$157$	3.52603	0.281407	0.140704	−	0.990052i	$-0.455063\pi$
$157$	3.52603	0.281407	0.140704	+	0.990052i	$0.455063\pi$
$158$	0	0
$159$	2.52764	0.200455
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.2522	1.27297	0.636485	−	0.771289i	$-0.280388\pi$
$163$	16.2522	1.27297	0.636485	+	0.771289i	$0.280388\pi$
$164$	0	0
$165$	−0.272593	−0.0212213
$166$	0	0
$167$	−3.99714	−0.309308	−0.154654	−	0.987969i	$-0.549426\pi$
$167$	−3.99714	−0.309308	−0.154654	+	0.987969i	$0.549426\pi$
$168$	0	0
$169$	−7.83532	−0.602717
$170$	0	0
$171$	0.415828	0.0317991
$172$	0	0
$173$	−17.6214	−1.33973	−0.669865	−	0.742483i	$-0.733648\pi$
$173$	−17.6214	−1.33973	−0.669865	+	0.742483i	$0.733648\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00404	0.451291
$178$	0	0
$179$	−25.2811	−1.88960	−0.944799	−	0.327650i	$-0.893743\pi$
$179$	−25.2811	−1.88960	−0.944799	+	0.327650i	$0.893743\pi$
$180$	0	0
$181$	−18.4950	−1.37473	−0.687363	−	0.726315i	$-0.741232\pi$
$181$	−18.4950	−1.37473	−0.687363	+	0.726315i	$0.741232\pi$
$182$	0	0
$183$	10.5236	0.777927
$184$	0	0
$185$	1.15403	0.0848458
$186$	0	0
$187$	1.50407	0.109988
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.8658	−1.43744	−0.718719	−	0.695301i	$-0.755271\pi$
$191$	−19.8658	−1.43744	−0.718719	+	0.695301i	$0.755271\pi$
$192$	0	0
$193$	−11.4335	−0.823002	−0.411501	−	0.911409i	$-0.634995\pi$
$193$	−11.4335	−0.823002	−0.411501	+	0.911409i	$0.634995\pi$
$194$	0	0
$195$	2.69070	0.192685
$196$	0	0
$197$	−4.56273	−0.325081	−0.162541	−	0.986702i	$-0.551969\pi$
$197$	−4.56273	−0.325081	−0.162541	+	0.986702i	$0.551969\pi$
$198$	0	0
$199$	−14.4429	−1.02383	−0.511916	−	0.859036i	$-0.671064\pi$
$199$	−14.4429	−1.02383	−0.511916	+	0.859036i	$0.671064\pi$
$200$	0	0
$201$	9.66089	0.681427
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.43337	0.309640
$206$	0	0
$207$	−14.1772	−0.985385
$208$	0	0
$209$	0.0599037	0.00414363
$210$	0	0
$211$	−6.63651	−0.456876	−0.228438	−	0.973558i	$-0.573362\pi$
$211$	−6.63651	−0.456876	−0.228438	+	0.973558i	$0.573362\pi$
$212$	0	0
$213$	−10.7334	−0.735443
$214$	0	0
$215$	4.17723	0.284884
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.11518	0.480800
$220$	0	0
$221$	−14.8463	−0.998673
$222$	0	0
$223$	−20.3325	−1.36156	−0.680782	−	0.732486i	$-0.738360\pi$
$223$	−20.3325	−1.36156	−0.680782	+	0.732486i	$0.738360\pi$
$224$	0	0
$225$	−1.59819	−0.106546
$226$	0	0
$227$	23.9312	1.58837	0.794185	−	0.607676i	$-0.207898\pi$
$227$	23.9312	1.58837	0.794185	+	0.607676i	$0.207898\pi$
$228$	0	0
$229$	−2.59534	−0.171505	−0.0857523	−	0.996316i	$-0.527329\pi$
$229$	−2.59534	−0.171505	−0.0857523	+	0.996316i	$0.527329\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.1638	−0.993412	−0.496706	−	0.867919i	$-0.665457\pi$
$233$	−15.1638	−0.993412	−0.496706	+	0.867919i	$0.665457\pi$
$234$	0	0
$235$	−0.923793	−0.0602616
$236$	0	0
$237$	0.133685	0.00868377
$238$	0	0
$239$	−10.5656	−0.683431	−0.341716	−	0.939803i	$-0.611008\pi$
$239$	−10.5656	−0.683431	−0.341716	+	0.939803i	$0.611008\pi$
$240$	0	0
$241$	−19.6762	−1.26745	−0.633726	−	0.773557i	$-0.718475\pi$
$241$	−19.6762	−1.26745	−0.633726	+	0.773557i	$0.718475\pi$
$242$	0	0
$243$	−14.3775	−0.922318
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.591297	−0.0376233
$248$	0	0
$249$	−10.3251	−0.654325
$250$	0	0
$251$	−25.0498	−1.58113	−0.790564	−	0.612380i	$-0.790212\pi$
$251$	−25.0498	−1.58113	−0.790564	+	0.612380i	$0.790212\pi$
$252$	0	0
$253$	−2.04236	−0.128402
$254$	0	0
$255$	−7.73467	−0.484364
$256$	0	0
$257$	−1.71500	−0.106979	−0.0534894	−	0.998568i	$-0.517034\pi$
$257$	−1.71500	−0.106979	−0.0534894	+	0.998568i	$0.517034\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−8.67279	−0.536832
$262$	0	0
$263$	22.1173	1.36381	0.681906	−	0.731440i	$-0.261151\pi$
$263$	22.1173	1.36381	0.681906	+	0.731440i	$0.261151\pi$
$264$	0	0
$265$	2.13487	0.131144
$266$	0	0
$267$	18.8454	1.15332
$268$	0	0
$269$	31.9261	1.94657	0.973283	−	0.229608i	$-0.0737443\pi$
$269$	31.9261	1.94657	0.973283	+	0.229608i	$0.0737443\pi$
$270$	0	0
$271$	−25.7767	−1.56582	−0.782910	−	0.622135i	$-0.786266\pi$
$271$	−25.7767	−1.56582	−0.782910	+	0.622135i	$0.786266\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.230234	−0.0138836
$276$	0	0
$277$	−8.01755	−0.481728	−0.240864	−	0.970559i	$-0.577431\pi$
$277$	−8.01755	−0.481728	−0.240864	+	0.970559i	$0.577431\pi$
$278$	0	0
$279$	−7.78445	−0.466043
$280$	0	0
$281$	−3.13207	−0.186844	−0.0934218	−	0.995627i	$-0.529781\pi$
$281$	−3.13207	−0.186844	−0.0934218	+	0.995627i	$0.529781\pi$
$282$	0	0
$283$	3.17047	0.188465	0.0942325	−	0.995550i	$-0.469960\pi$
$283$	3.17047	0.188465	0.0942325	+	0.995550i	$0.469960\pi$
$284$	0	0
$285$	−0.308055	−0.0182476
$286$	0	0
$287$	0	0
$288$	0	0
$289$	25.6772	1.51042
$290$	0	0
$291$	5.08596	0.298144
$292$	0	0
$293$	17.5264	1.02390	0.511952	−	0.859014i	$-0.328923\pi$
$293$	17.5264	1.02390	0.511952	+	0.859014i	$0.328923\pi$
$294$	0	0
$295$	5.07107	0.295249
$296$	0	0
$297$	−1.25343	−0.0727316
$298$	0	0
$299$	20.1597	1.16586
$300$	0	0
$301$	0	0
$302$	0	0
$303$	8.74620	0.502456
$304$	0	0
$305$	8.88833	0.508944
$306$	0	0
$307$	2.11063	0.120460	0.0602300	−	0.998185i	$-0.480817\pi$
$307$	2.11063	0.120460	0.0602300	+	0.998185i	$0.480817\pi$
$308$	0	0
$309$	12.5273	0.712651
$310$	0	0
$311$	20.8149	1.18031	0.590153	−	0.807291i	$-0.299067\pi$
$311$	20.8149	1.18031	0.590153	+	0.807291i	$0.299067\pi$
$312$	0	0
$313$	21.3860	1.20881	0.604405	−	0.796678i	$-0.293411\pi$
$313$	21.3860	1.20881	0.604405	+	0.796678i	$0.293411\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.4909	0.870058	0.435029	−	0.900417i	$-0.356738\pi$
$317$	15.4909	0.870058	0.435029	+	0.900417i	$0.356738\pi$
$318$	0	0
$319$	−1.24939	−0.0699526
$320$	0	0
$321$	−15.9436	−0.889886
$322$	0	0
$323$	1.69974	0.0945760
$324$	0	0
$325$	2.27259	0.126061
$326$	0	0
$327$	−10.0706	−0.556903
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.52360	0.468499	0.234250	−	0.972176i	$-0.424737\pi$
$331$	8.52360	0.468499	0.234250	+	0.972176i	$0.424737\pi$
$332$	0	0
$333$	1.84436	0.101070
$334$	0	0
$335$	8.15968	0.445811
$336$	0	0
$337$	−18.1246	−0.987309	−0.493655	−	0.869658i	$-0.664339\pi$
$337$	−18.1246	−0.987309	−0.493655	+	0.869658i	$0.664339\pi$
$338$	0	0
$339$	−21.7814	−1.18301
$340$	0	0
$341$	−1.12142	−0.0607284
$342$	0	0
$343$	0	0
$344$	0	0
$345$	10.5028	0.565453
$346$	0	0
$347$	−4.10226	−0.220221	−0.110110	−	0.993919i	$-0.535120\pi$
$347$	−4.10226	−0.220221	−0.110110	+	0.993919i	$0.535120\pi$
$348$	0	0
$349$	8.67850	0.464549	0.232275	−	0.972650i	$-0.425383\pi$
$349$	8.67850	0.464549	0.232275	+	0.972650i	$0.425383\pi$
$350$	0	0
$351$	12.3724	0.660388
$352$	0	0
$353$	−30.1162	−1.60292	−0.801462	−	0.598045i	$-0.795944\pi$
$353$	−30.1162	−1.60292	−0.801462	+	0.598045i	$0.795944\pi$
$354$	0	0
$355$	−9.06556	−0.481150
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.2843	−1.91501	−0.957505	−	0.288416i	$-0.906872\pi$
$359$	−36.2843	−1.91501	−0.957505	+	0.288416i	$0.906872\pi$
$360$	0	0
$361$	−18.9323	−0.996437
$362$	0	0
$363$	12.9610	0.680277
$364$	0	0
$365$	6.00955	0.314554
$366$	0	0
$367$	−10.7115	−0.559134	−0.279567	−	0.960126i	$-0.590191\pi$
$367$	−10.7115	−0.559134	−0.279567	+	0.960126i	$0.590191\pi$
$368$	0	0
$369$	7.08539	0.368850
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5.64335	0.292202	0.146101	−	0.989270i	$-0.453328\pi$
$373$	5.64335	0.292202	0.146101	+	0.989270i	$0.453328\pi$
$374$	0	0
$375$	1.18398	0.0611404
$376$	0	0
$377$	12.3325	0.635156
$378$	0	0
$379$	1.59944	0.0821575	0.0410787	−	0.999156i	$-0.486921\pi$
$379$	1.59944	0.0821575	0.0410787	+	0.999156i	$0.486921\pi$
$380$	0	0
$381$	9.95764	0.510145
$382$	0	0
$383$	−28.8894	−1.47618	−0.738089	−	0.674704i	$-0.764271\pi$
$383$	−28.8894	−1.47618	−0.738089	+	0.674704i	$0.764271\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.67601	0.339361
$388$	0	0
$389$	−23.5226	−1.19265	−0.596323	−	0.802745i	$-0.703372\pi$
$389$	−23.5226	−1.19265	−0.596323	+	0.802745i	$0.703372\pi$
$390$	0	0
$391$	−57.9509	−2.93070
$392$	0	0
$393$	2.11006	0.106438
$394$	0	0
$395$	0.112912	0.00568120
$396$	0	0
$397$	16.5806	0.832159	0.416079	−	0.909328i	$-0.363404\pi$
$397$	16.5806	0.832159	0.416079	+	0.909328i	$0.363404\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.52161	−0.0759858	−0.0379929	−	0.999278i	$-0.512096\pi$
$401$	−1.52161	−0.0759858	−0.0379929	+	0.999278i	$0.512096\pi$
$402$	0	0
$403$	11.0693	0.551402
$404$	0	0
$405$	1.65120	0.0820488
$406$	0	0
$407$	0.265697	0.0131701
$408$	0	0
$409$	5.42927	0.268460	0.134230	−	0.990950i	$-0.457144\pi$
$409$	5.42927	0.268460	0.134230	+	0.990950i	$0.457144\pi$
$410$	0	0
$411$	13.3952	0.660736
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−8.72065	−0.428080
$416$	0	0
$417$	−18.2795	−0.895150
$418$	0	0
$419$	12.4199	0.606750	0.303375	−	0.952871i	$-0.401886\pi$
$419$	12.4199	0.606750	0.303375	+	0.952871i	$0.401886\pi$
$420$	0	0
$421$	−20.6804	−1.00790	−0.503951	−	0.863732i	$-0.668121\pi$
$421$	−20.6804	−1.00790	−0.503951	+	0.863732i	$0.668121\pi$
$422$	0	0
$423$	−1.47640	−0.0717850
$424$	0	0
$425$	−6.53278	−0.316886
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.619492	0.0299093
$430$	0	0
$431$	31.6152	1.52285	0.761424	−	0.648254i	$-0.224500\pi$
$431$	31.6152	1.52285	0.761424	+	0.648254i	$0.224500\pi$
$432$	0	0
$433$	31.1247	1.49576	0.747880	−	0.663834i	$-0.231072\pi$
$433$	31.1247	1.49576	0.747880	+	0.663834i	$0.231072\pi$
$434$	0	0
$435$	6.42501	0.308055
$436$	0	0
$437$	−2.30805	−0.110409
$438$	0	0
$439$	0.244398	0.0116645	0.00583223	−	0.999983i	$-0.498144\pi$
$439$	0.244398	0.0116645	0.00583223	+	0.999983i	$0.498144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.45481	0.164143	0.0820716	−	0.996626i	$-0.473846\pi$
$443$	3.45481	0.164143	0.0820716	+	0.996626i	$0.473846\pi$
$444$	0	0
$445$	15.9170	0.754540
$446$	0	0
$447$	−27.5643	−1.30375
$448$	0	0
$449$	−15.5329	−0.733044	−0.366522	−	0.930409i	$-0.619452\pi$
$449$	−15.5329	−0.733044	−0.366522	+	0.930409i	$0.619452\pi$
$450$	0	0
$451$	1.02071	0.0480636
$452$	0	0
$453$	26.4498	1.24272
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.5379	−0.492943	−0.246471	−	0.969150i	$-0.579271\pi$
$457$	−10.5379	−0.492943	−0.246471	+	0.969150i	$0.579271\pi$
$458$	0	0
$459$	−35.5655	−1.66006
$460$	0	0
$461$	−5.98972	−0.278969	−0.139485	−	0.990224i	$-0.544545\pi$
$461$	−5.98972	−0.278969	−0.139485	+	0.990224i	$0.544545\pi$
$462$	0	0
$463$	33.3601	1.55038	0.775188	−	0.631731i	$-0.217655\pi$
$463$	33.3601	1.55038	0.775188	+	0.631731i	$0.217655\pi$
$464$	0	0
$465$	5.76691	0.267434
$466$	0	0
$467$	19.7076	0.911958	0.455979	−	0.889991i	$-0.349289\pi$
$467$	19.7076	0.911958	0.455979	+	0.889991i	$0.349289\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4.17474	−0.192362
$472$	0	0
$473$	0.961740	0.0442209
$474$	0	0
$475$	−0.260186	−0.0119382
$476$	0	0
$477$	3.41193	0.156222
$478$	0	0
$479$	0.325600	0.0148771	0.00743853	−	0.999972i	$-0.497632\pi$
$479$	0.325600	0.0148771	0.00743853	+	0.999972i	$0.497632\pi$
$480$	0	0
$481$	−2.62263	−0.119582
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.29565	0.195055
$486$	0	0
$487$	16.9204	0.766737	0.383369	−	0.923595i	$-0.374764\pi$
$487$	16.9204	0.766737	0.383369	+	0.923595i	$0.374764\pi$
$488$	0	0
$489$	−19.2423	−0.870165
$490$	0	0
$491$	−15.0203	−0.677859	−0.338929	−	0.940812i	$-0.610065\pi$
$491$	−15.0203	−0.677859	−0.338929	+	0.940812i	$0.610065\pi$
$492$	0	0
$493$	−35.4509	−1.59663
$494$	0	0
$495$	−0.367959	−0.0165385
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−41.0431	−1.83734	−0.918670	−	0.395025i	$-0.870736\pi$
$499$	−41.0431	−1.83734	−0.918670	+	0.395025i	$0.870736\pi$
$500$	0	0
$501$	4.73254	0.211434
$502$	0	0
$503$	8.29741	0.369963	0.184982	−	0.982742i	$-0.440777\pi$
$503$	8.29741	0.369963	0.184982	+	0.982742i	$0.440777\pi$
$504$	0	0
$505$	7.38712	0.328722
$506$	0	0
$507$	9.27686	0.412000
$508$	0	0
$509$	−8.71266	−0.386182	−0.193091	−	0.981181i	$-0.561851\pi$
$509$	−8.71266	−0.386182	−0.193091	+	0.981181i	$0.561851\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.41650	−0.0625398
$514$	0	0
$515$	10.5806	0.466239
$516$	0	0
$517$	−0.212689	−0.00935404
$518$	0	0
$519$	20.8634	0.915800
$520$	0	0
$521$	27.1056	1.18752	0.593760	−	0.804642i	$-0.297643\pi$
$521$	27.1056	1.18752	0.593760	+	0.804642i	$0.297643\pi$
$522$	0	0
$523$	−1.37186	−0.0599870	−0.0299935	−	0.999550i	$-0.509549\pi$
$523$	−1.37186	−0.0599870	−0.0299935	+	0.999550i	$0.509549\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31.8198	−1.38609
$528$	0	0
$529$	55.6908	2.42134
$530$	0	0
$531$	8.10454	0.351707
$532$	0	0
$533$	−10.0753	−0.436408
$534$	0	0
$535$	−13.4661	−0.582191
$536$	0	0
$537$	29.9323	1.29167
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.56559	0.368263	0.184132	−	0.982902i	$-0.441053\pi$
$541$	8.56559	0.368263	0.184132	+	0.982902i	$0.441053\pi$
$542$	0	0
$543$	21.8977	0.939722
$544$	0	0
$545$	−8.50568	−0.364343
$546$	0	0
$547$	−44.2056	−1.89009	−0.945046	−	0.326936i	$-0.893984\pi$
$547$	−44.2056	−1.89009	−0.945046	+	0.326936i	$0.893984\pi$
$548$	0	0
$549$	14.2053	0.606266
$550$	0	0
$551$	−1.41193	−0.0601503
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.36634	−0.0579981
$556$	0	0
$557$	−13.3385	−0.565171	−0.282586	−	0.959242i	$-0.591192\pi$
$557$	−13.3385	−0.565171	−0.282586	+	0.959242i	$0.591192\pi$
$558$	0	0
$559$	−9.49313	−0.401517
$560$	0	0
$561$	−1.78079	−0.0751849
$562$	0	0
$563$	29.7320	1.25306	0.626528	−	0.779399i	$-0.284476\pi$
$563$	29.7320	1.25306	0.626528	+	0.779399i	$0.284476\pi$
$564$	0	0
$565$	−18.3968	−0.773960
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.5859	0.988774	0.494387	−	0.869242i	$-0.335393\pi$
$569$	23.5859	0.988774	0.494387	+	0.869242i	$0.335393\pi$
$570$	0	0
$571$	−32.6434	−1.36608	−0.683042	−	0.730379i	$-0.739343\pi$
$571$	−32.6434	−1.36608	−0.683042	+	0.730379i	$0.739343\pi$
$572$	0	0
$573$	23.5207	0.982591
$574$	0	0
$575$	8.87079	0.369937
$576$	0	0
$577$	28.8195	1.19977	0.599886	−	0.800085i	$-0.295212\pi$
$577$	28.8195	1.19977	0.599886	+	0.800085i	$0.295212\pi$
$578$	0	0
$579$	13.5370	0.562580
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.491520	0.0203567
$584$	0	0
$585$	3.63204	0.150166
$586$	0	0
$587$	−1.82205	−0.0752039	−0.0376019	−	0.999293i	$-0.511972\pi$
$587$	−1.82205	−0.0752039	−0.0376019	+	0.999293i	$0.511972\pi$
$588$	0	0
$589$	−1.26731	−0.0522186
$590$	0	0
$591$	5.40218	0.222216
$592$	0	0
$593$	−30.9369	−1.27042	−0.635212	−	0.772338i	$-0.719087\pi$
$593$	−30.9369	−1.27042	−0.635212	+	0.772338i	$0.719087\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.1001	0.699861
$598$	0	0
$599$	32.2912	1.31938	0.659691	−	0.751537i	$-0.270687\pi$
$599$	32.2912	1.31938	0.659691	+	0.751537i	$0.270687\pi$
$600$	0	0
$601$	25.2829	1.03131	0.515655	−	0.856797i	$-0.327549\pi$
$601$	25.2829	1.03131	0.515655	+	0.856797i	$0.327549\pi$
$602$	0	0
$603$	13.0407	0.531060
$604$	0	0
$605$	10.9470	0.445059
$606$	0	0
$607$	−9.28852	−0.377010	−0.188505	−	0.982072i	$-0.560364\pi$
$607$	−9.28852	−0.377010	−0.188505	+	0.982072i	$0.560364\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.09941	0.0849329
$612$	0	0
$613$	40.0609	1.61805	0.809023	−	0.587777i	$-0.199997\pi$
$613$	40.0609	1.61805	0.809023	+	0.587777i	$0.199997\pi$
$614$	0	0
$615$	−5.24902	−0.211661
$616$	0	0
$617$	27.9860	1.12667	0.563336	−	0.826228i	$-0.309518\pi$
$617$	27.9860	1.12667	0.563336	+	0.826228i	$0.309518\pi$
$618$	0	0
$619$	−45.8048	−1.84105	−0.920525	−	0.390684i	$-0.872239\pi$
$619$	−45.8048	−1.84105	−0.920525	+	0.390684i	$0.872239\pi$
$620$	0	0
$621$	48.2940	1.93797
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.0709248	−0.00283246
$628$	0	0
$629$	7.53901	0.300600
$630$	0	0
$631$	−22.5847	−0.899082	−0.449541	−	0.893260i	$-0.648412\pi$
$631$	−22.5847	−0.899082	−0.449541	+	0.893260i	$0.648412\pi$
$632$	0	0
$633$	7.85749	0.312307
$634$	0	0
$635$	8.41032	0.333753
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−14.4885	−0.573157
$640$	0	0
$641$	−33.5125	−1.32366	−0.661832	−	0.749652i	$-0.730221\pi$
$641$	−33.5125	−1.32366	−0.661832	+	0.749652i	$0.730221\pi$
$642$	0	0
$643$	46.4980	1.83370	0.916851	−	0.399231i	$-0.130723\pi$
$643$	46.4980	1.83370	0.916851	+	0.399231i	$0.130723\pi$
$644$	0	0
$645$	−4.94575	−0.194739
$646$	0	0
$647$	−1.57565	−0.0619453	−0.0309726	−	0.999520i	$-0.509860\pi$
$647$	−1.57565	−0.0619453	−0.0309726	+	0.999520i	$0.509860\pi$
$648$	0	0
$649$	1.16753	0.0458297
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.1311	0.474727	0.237364	−	0.971421i	$-0.423717\pi$
$653$	12.1311	0.474727	0.237364	+	0.971421i	$0.423717\pi$
$654$	0	0
$655$	1.78217	0.0696352
$656$	0	0
$657$	9.60442	0.374704
$658$	0	0
$659$	−35.1682	−1.36996	−0.684979	−	0.728563i	$-0.740189\pi$
$659$	−35.1682	−1.36996	−0.684979	+	0.728563i	$0.740189\pi$
$660$	0	0
$661$	1.62425	0.0631759	0.0315880	−	0.999501i	$-0.489944\pi$
$661$	1.62425	0.0631759	0.0315880	+	0.999501i	$0.489944\pi$
$662$	0	0
$663$	17.5778	0.682664
$664$	0	0
$665$	0	0
$666$	0	0
$667$	48.1384	1.86393
$668$	0	0
$669$	24.0733	0.930726
$670$	0	0
$671$	2.04640	0.0790003
$672$	0	0
$673$	−24.3194	−0.937443	−0.468721	−	0.883346i	$-0.655285\pi$
$673$	−24.3194	−0.937443	−0.468721	+	0.883346i	$0.655285\pi$
$674$	0	0
$675$	5.44417	0.209546
$676$	0	0
$677$	−12.1553	−0.467165	−0.233582	−	0.972337i	$-0.575045\pi$
$677$	−12.1553	−0.467165	−0.233582	+	0.972337i	$0.575045\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−28.3341	−1.08576
$682$	0	0
$683$	32.5158	1.24418	0.622091	−	0.782945i	$-0.286283\pi$
$683$	32.5158	1.24418	0.622091	+	0.782945i	$0.286283\pi$
$684$	0	0
$685$	11.3137	0.432275
$686$	0	0
$687$	3.07282	0.117236
$688$	0	0
$689$	−4.85169	−0.184834
$690$	0	0
$691$	20.3311	0.773432	0.386716	−	0.922199i	$-0.373609\pi$
$691$	20.3311	0.773432	0.386716	+	0.922199i	$0.373609\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.4390	−0.585636
$696$	0	0
$697$	28.9622	1.09702
$698$	0	0
$699$	17.9536	0.679068
$700$	0	0
$701$	7.35321	0.277727	0.138863	−	0.990312i	$-0.455655\pi$
$701$	7.35321	0.277727	0.138863	+	0.990312i	$0.455655\pi$
$702$	0	0
$703$	0.300262	0.0113246
$704$	0	0
$705$	1.09375	0.0411931
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−11.6638	−0.438041	−0.219021	−	0.975720i	$-0.570286\pi$
$709$	−11.6638	−0.438041	−0.219021	+	0.975720i	$0.570286\pi$
$710$	0	0
$711$	0.180454	0.00676757
$712$	0	0
$713$	43.2077	1.61814
$714$	0	0
$715$	0.523229	0.0195676
$716$	0	0
$717$	12.5094	0.467173
$718$	0	0
$719$	−5.68571	−0.212041	−0.106021	−	0.994364i	$-0.533811\pi$
$719$	−5.68571	−0.212041	−0.106021	+	0.994364i	$0.533811\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	23.2962	0.866394
$724$	0	0
$725$	5.42662	0.201540
$726$	0	0
$727$	18.6873	0.693074	0.346537	−	0.938036i	$-0.387357\pi$
$727$	18.6873	0.693074	0.346537	+	0.938036i	$0.387357\pi$
$728$	0	0
$729$	21.9763	0.813936
$730$	0	0
$731$	27.2889	1.00932
$732$	0	0
$733$	−1.83694	−0.0678488	−0.0339244	−	0.999424i	$-0.510801\pi$
$733$	−1.83694	−0.0678488	−0.0339244	+	0.999424i	$0.510801\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.87864	0.0692005
$738$	0	0
$739$	24.5634	0.903579	0.451789	−	0.892125i	$-0.350786\pi$
$739$	24.5634	0.903579	0.451789	+	0.892125i	$0.350786\pi$
$740$	0	0
$741$	0.700083	0.0257182
$742$	0	0
$743$	27.1926	0.997601	0.498800	−	0.866717i	$-0.333774\pi$
$743$	27.1926	0.997601	0.498800	+	0.866717i	$0.333774\pi$
$744$	0	0
$745$	−23.2811	−0.852954
$746$	0	0
$747$	−13.9373	−0.509939
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−0.328457	−0.0119856	−0.00599278	−	0.999982i	$-0.501908\pi$
$751$	−0.328457	−0.0119856	−0.00599278	+	0.999982i	$0.501908\pi$
$752$	0	0
$753$	29.6584	1.08081
$754$	0	0
$755$	22.3398	0.813027
$756$	0	0
$757$	−48.6331	−1.76760	−0.883801	−	0.467864i	$-0.845024\pi$
$757$	−48.6331	−1.76760	−0.883801	+	0.467864i	$0.845024\pi$
$758$	0	0
$759$	2.41811	0.0877718
$760$	0	0
$761$	8.02253	0.290817	0.145408	−	0.989372i	$-0.453550\pi$
$761$	8.02253	0.290817	0.145408	+	0.989372i	$0.453550\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−10.4406	−0.377482
$766$	0	0
$767$	−11.5245	−0.416125
$768$	0	0
$769$	−35.6370	−1.28510	−0.642552	−	0.766242i	$-0.722124\pi$
$769$	−35.6370	−1.28510	−0.642552	+	0.766242i	$0.722124\pi$
$770$	0	0
$771$	2.03053	0.0731276
$772$	0	0
$773$	−49.5955	−1.78383	−0.891914	−	0.452206i	$-0.850637\pi$
$773$	−49.5955	−1.78383	−0.891914	+	0.452206i	$0.850637\pi$
$774$	0	0
$775$	4.87079	0.174964
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.15350	0.0413285
$780$	0	0
$781$	−2.08720	−0.0746859
$782$	0	0
$783$	29.5434	1.05580
$784$	0	0
$785$	−3.52603	−0.125849
$786$	0	0
$787$	21.4434	0.764376	0.382188	−	0.924085i	$-0.375171\pi$
$787$	21.4434	0.764376	0.382188	+	0.924085i	$0.375171\pi$
$788$	0	0
$789$	−26.1865	−0.932262
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−20.1996	−0.717307
$794$	0	0
$795$	−2.52764	−0.0896461
$796$	0	0
$797$	−35.1429	−1.24482	−0.622412	−	0.782690i	$-0.713847\pi$
$797$	−35.1429	−1.24482	−0.622412	+	0.782690i	$0.713847\pi$
$798$	0	0
$799$	−6.03494	−0.213501
$800$	0	0
$801$	25.4385	0.898825
$802$	0	0
$803$	1.38360	0.0488263
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−37.7998	−1.33062
$808$	0	0
$809$	−2.32597	−0.0817768	−0.0408884	−	0.999164i	$-0.513019\pi$
$809$	−2.32597	−0.0817768	−0.0408884	+	0.999164i	$0.513019\pi$
$810$	0	0
$811$	−11.8045	−0.414512	−0.207256	−	0.978287i	$-0.566453\pi$
$811$	−11.8045	−0.414512	−0.207256	+	0.978287i	$0.566453\pi$
$812$	0	0
$813$	30.5190	1.07035
$814$	0	0
$815$	−16.2522	−0.569289
$816$	0	0
$817$	1.08686	0.0380243
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−51.7790	−1.80710	−0.903550	−	0.428483i	$-0.859048\pi$
$821$	−51.7790	−1.80710	−0.903550	+	0.428483i	$0.859048\pi$
$822$	0	0
$823$	40.6259	1.41613	0.708064	−	0.706148i	$-0.249569\pi$
$823$	40.6259	1.41613	0.708064	+	0.706148i	$0.249569\pi$
$824$	0	0
$825$	0.272593	0.00949045
$826$	0	0
$827$	−12.6201	−0.438846	−0.219423	−	0.975630i	$-0.570417\pi$
$827$	−12.6201	−0.438846	−0.219423	+	0.975630i	$0.570417\pi$
$828$	0	0
$829$	50.4905	1.75361	0.876803	−	0.480849i	$-0.159672\pi$
$829$	50.4905	1.75361	0.876803	+	0.480849i	$0.159672\pi$
$830$	0	0
$831$	9.49261	0.329295
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.99714	0.138327
$836$	0	0
$837$	26.5174	0.916574
$838$	0	0
$839$	−4.41032	−0.152261	−0.0761305	−	0.997098i	$-0.524257\pi$
$839$	−4.41032	−0.152261	−0.0761305	+	0.997098i	$0.524257\pi$
$840$	0	0
$841$	0.448205	0.0154553
$842$	0	0
$843$	3.70831	0.127721
$844$	0	0
$845$	7.83532	0.269543
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.75378	−0.128829
$850$	0	0
$851$	−10.2371	−0.350924
$852$	0	0
$853$	−20.6785	−0.708018	−0.354009	−	0.935242i	$-0.615182\pi$
$853$	−20.6785	−0.708018	−0.354009	+	0.935242i	$0.615182\pi$
$854$	0	0
$855$	−0.415828	−0.0142210
$856$	0	0
$857$	5.93055	0.202584	0.101292	−	0.994857i	$-0.467702\pi$
$857$	5.93055	0.202584	0.101292	+	0.994857i	$0.467702\pi$
$858$	0	0
$859$	31.3832	1.07078	0.535390	−	0.844605i	$-0.320165\pi$
$859$	31.3832	1.07078	0.535390	+	0.844605i	$0.320165\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.1710	−0.890871	−0.445435	−	0.895314i	$-0.646951\pi$
$863$	−26.1710	−0.890871	−0.445435	+	0.895314i	$0.646951\pi$
$864$	0	0
$865$	17.6214	0.599145
$866$	0	0
$867$	−30.4013	−1.03248
$868$	0	0
$869$	0.0259961	0.000881857 0
$870$	0	0
$871$	−18.5436	−0.628327
$872$	0	0
$873$	6.86527	0.232354
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−52.2304	−1.76369	−0.881847	−	0.471536i	$-0.843700\pi$
$877$	−52.2304	−1.76369	−0.881847	+	0.471536i	$0.843700\pi$
$878$	0	0
$879$	−20.7509	−0.699910
$880$	0	0
$881$	−33.5930	−1.13178	−0.565888	−	0.824482i	$-0.691467\pi$
$881$	−33.5930	−1.13178	−0.565888	+	0.824482i	$0.691467\pi$
$882$	0	0
$883$	−30.5923	−1.02951	−0.514757	−	0.857336i	$-0.672118\pi$
$883$	−30.5923	−1.02951	−0.514757	+	0.857336i	$0.672118\pi$
$884$	0	0
$885$	−6.00404	−0.201824
$886$	0	0
$887$	49.0305	1.64628	0.823141	−	0.567837i	$-0.192220\pi$
$887$	49.0305	1.64628	0.823141	+	0.567837i	$0.192220\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0.380163	0.0127359
$892$	0	0
$893$	−0.240358	−0.00804328
$894$	0	0
$895$	25.2811	0.845054
$896$	0	0
$897$	−23.8686	−0.796951
$898$	0	0
$899$	26.4319	0.881553
$900$	0	0
$901$	13.9466	0.464629
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.4950	0.614796
$906$	0	0
$907$	36.9013	1.22529	0.612644	−	0.790359i	$-0.290106\pi$
$907$	36.9013	1.22529	0.612644	+	0.790359i	$0.290106\pi$
$908$	0	0
$909$	11.8060	0.391582
$910$	0	0
$911$	−34.6118	−1.14674	−0.573371	−	0.819296i	$-0.694364\pi$
$911$	−34.6118	−1.14674	−0.573371	+	0.819296i	$0.694364\pi$
$912$	0	0
$913$	−2.00779	−0.0664483
$914$	0	0
$915$	−10.5236	−0.347899
$916$	0	0
$917$	0	0
$918$	0	0
$919$	30.8170	1.01656	0.508279	−	0.861192i	$-0.330282\pi$
$919$	30.8170	1.01656	0.508279	+	0.861192i	$0.330282\pi$
$920$	0	0
$921$	−2.49894	−0.0823429
$922$	0	0
$923$	20.6023	0.678134
$924$	0	0
$925$	−1.15403	−0.0379442
$926$	0	0
$927$	16.9099	0.555395
$928$	0	0
$929$	17.3387	0.568865	0.284433	−	0.958696i	$-0.408195\pi$
$929$	17.3387	0.568865	0.284433	+	0.958696i	$0.408195\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.6444	−0.806823
$934$	0	0
$935$	−1.50407	−0.0491883
$936$	0	0
$937$	−44.3736	−1.44962	−0.724811	−	0.688948i	$-0.758073\pi$
$937$	−44.3736	−1.44962	−0.724811	+	0.688948i	$0.758073\pi$
$938$	0	0
$939$	−25.3206	−0.826307
$940$	0	0
$941$	−1.65934	−0.0540929	−0.0270465	−	0.999634i	$-0.508610\pi$
$941$	−1.65934	−0.0540929	−0.0270465	+	0.999634i	$0.508610\pi$
$942$	0	0
$943$	−39.3275	−1.28068
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.4486	1.21692	0.608458	−	0.793586i	$-0.291789\pi$
$947$	37.4486	1.21692	0.608458	+	0.793586i	$0.291789\pi$
$948$	0	0
$949$	−13.6573	−0.443333
$950$	0	0
$951$	−18.3409	−0.594746
$952$	0	0
$953$	37.5875	1.21758	0.608789	−	0.793332i	$-0.291656\pi$
$953$	37.5875	1.21758	0.608789	+	0.793332i	$0.291656\pi$
$954$	0	0
$955$	19.8658	0.642842
$956$	0	0
$957$	1.47926	0.0478176
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−7.27545	−0.234692
$962$	0	0
$963$	−21.5215	−0.693519
$964$	0	0
$965$	11.4335	0.368058
$966$	0	0
$967$	24.5757	0.790302	0.395151	−	0.918616i	$-0.370692\pi$
$967$	24.5757	0.790302	0.395151	+	0.918616i	$0.370692\pi$
$968$	0	0
$969$	−2.01245	−0.0646494
$970$	0	0
$971$	48.6354	1.56078	0.780392	−	0.625290i	$-0.215019\pi$
$971$	48.6354	1.56078	0.780392	+	0.625290i	$0.215019\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.69070	−0.0861714
$976$	0	0
$977$	2.33743	0.0747811	0.0373906	−	0.999301i	$-0.488095\pi$
$977$	2.33743	0.0747811	0.0373906	+	0.999301i	$0.488095\pi$
$978$	0	0
$979$	3.66465	0.117123
$980$	0	0
$981$	−13.5937	−0.434014
$982$	0	0
$983$	28.7331	0.916442	0.458221	−	0.888838i	$-0.348487\pi$
$983$	28.7331	0.916442	0.458221	+	0.888838i	$0.348487\pi$
$984$	0	0
$985$	4.56273	0.145381
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−37.0553	−1.17829
$990$	0	0
$991$	7.84787	0.249296	0.124648	−	0.992201i	$-0.460220\pi$
$991$	7.84787	0.249296	0.124648	+	0.992201i	$0.460220\pi$
$992$	0	0
$993$	−10.0918	−0.320253
$994$	0	0
$995$	14.4429	0.457871
$996$	0	0
$997$	−34.4523	−1.09112	−0.545558	−	0.838073i	$-0.683682\pi$
$997$	−34.4523	−1.09112	−0.545558	+	0.838073i	$0.683682\pi$
$998$	0	0
$999$	−6.28272	−0.198776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.ce.1.2		4
4.3	odd	2		1960.2.a.x.1.3	✓	4
7.6	odd	2		3920.2.a.cd.1.3		4
20.19	odd	2		9800.2.a.cs.1.2		4
28.3	even	6		1960.2.q.x.961.3		8
28.11	odd	6		1960.2.q.y.961.2		8
28.19	even	6		1960.2.q.x.361.3		8
28.23	odd	6		1960.2.q.y.361.2		8
28.27	even	2		1960.2.a.y.1.2	yes	4
140.139	even	2		9800.2.a.cl.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.a.x.1.3	✓	4	4.3	odd	2
1960.2.a.y.1.2	yes	4	28.27	even	2
1960.2.q.x.361.3		8	28.19	even	6
1960.2.q.x.961.3		8	28.3	even	6
1960.2.q.y.361.2		8	28.23	odd	6
1960.2.q.y.961.2		8	28.11	odd	6
3920.2.a.cd.1.3		4	7.6	odd	2
3920.2.a.ce.1.2		4	1.1	even	1	trivial
9800.2.a.cl.1.3		4	140.139	even	2
9800.2.a.cs.1.2		4	20.19	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q-1.18398 q^{3} -1.00000 q^{5} -1.59819 q^{9} +O(q^{10})\)	\(q-1.18398 q^{3} -1.00000 q^{5} -1.59819 q^{9} -0.230234 q^{11} +2.27259 q^{13} +1.18398 q^{15} -6.53278 q^{17} -0.260186 q^{19} +8.87079 q^{23} +1.00000 q^{25} +5.44417 q^{27} +5.42662 q^{29} +4.87079 q^{31} +0.272593 q^{33} -1.15403 q^{37} -2.69070 q^{39} -4.43337 q^{41} -4.17723 q^{43} +1.59819 q^{45} +0.923793 q^{47} +7.73467 q^{51} -2.13487 q^{53} +0.230234 q^{55} +0.308055 q^{57} -5.07107 q^{59} -8.88833 q^{61} -2.27259 q^{65} -8.15968 q^{67} -10.5028 q^{69} +9.06556 q^{71} -6.00955 q^{73} -1.18398 q^{75} -0.112912 q^{79} -1.65120 q^{81} +8.72065 q^{83} +6.53278 q^{85} -6.42501 q^{87} -15.9170 q^{89} -5.76691 q^{93} +0.260186 q^{95} -4.29565 q^{97} +0.367959 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} + 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 4 * q^5 + 6 * q^9	\( 4 q + 2 q^{3} - 4 q^{5} + 6 q^{9} - 2 q^{11} - 10 q^{13} - 2 q^{15} - 6 q^{17} + 4 q^{23} + 4 q^{25} + 14 q^{27} - 2 q^{29} - 12 q^{31} - 18 q^{33} - 14 q^{39} - 12 q^{41} + 8 q^{43} - 6 q^{45} - 2 q^{47} - 2 q^{51} - 4 q^{53} + 2 q^{55} - 8 q^{57} + 8 q^{59} - 20 q^{61} + 10 q^{65} + 8 q^{67} - 24 q^{69} - 4 q^{71} - 16 q^{73} + 2 q^{75} - 22 q^{79} - 20 q^{81} + 36 q^{83} + 6 q^{85} - 18 q^{87} - 40 q^{89} - 32 q^{93} - 26 q^{97} - 12 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 + 6 * q^9 - 2 * q^11 - 10 * q^13 - 2 * q^15 - 6 * q^17 + 4 * q^23 + 4 * q^25 + 14 * q^27 - 2 * q^29 - 12 * q^31 - 18 * q^33 - 14 * q^39 - 12 * q^41 + 8 * q^43 - 6 * q^45 - 2 * q^47 - 2 * q^51 - 4 * q^53 + 2 * q^55 - 8 * q^57 + 8 * q^59 - 20 * q^61 + 10 * q^65 + 8 * q^67 - 24 * q^69 - 4 * q^71 - 16 * q^73 + 2 * q^75 - 22 * q^79 - 20 * q^81 + 36 * q^83 + 6 * q^85 - 18 * q^87 - 40 * q^89 - 32 * q^93 - 26 * q^97 - 12 * q^99

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