LMFDB - Embedded newform 4620.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 3 $ x^3 - x^2 - 15*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.197906$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -0.802094 q^{13} +1.00000 q^{15} +5.28251 q^{17} +5.67832 q^{19} -1.00000 q^{21} -3.67832 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.87623 q^{29} +3.19791 q^{31} -1.00000 q^{33} -1.00000 q^{35} +3.19791 q^{37} -0.802094 q^{39} +1.19791 q^{41} +8.48042 q^{43} +1.00000 q^{45} -6.15874 q^{47} +1.00000 q^{49} +5.28251 q^{51} +6.07413 q^{53} -1.00000 q^{55} +5.67832 q^{57} +8.08461 q^{59} -14.6392 q^{61} -1.00000 q^{63} -0.802094 q^{65} +2.00000 q^{67} -3.67832 q^{69} +9.19791 q^{71} -6.96083 q^{73} +1.00000 q^{75} +1.00000 q^{77} +1.19791 q^{79} +1.00000 q^{81} -0.0741336 q^{83} +5.28251 q^{85} -4.87623 q^{87} +8.08461 q^{89} +0.802094 q^{91} +3.19791 q^{93} +5.67832 q^{95} +6.48042 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 2 q^{13} + 3 q^{15} + 2 q^{19} - 3 q^{21} + 4 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{31} - 3 q^{33} - 3 q^{35} + 10 q^{37} - 2 q^{39} + 4 q^{41} + 10 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} + 4 q^{53} - 3 q^{55} + 2 q^{57} + 8 q^{59} + 2 q^{61} - 3 q^{63} - 2 q^{65} + 6 q^{67} + 4 q^{69} + 28 q^{71} + 10 q^{73} + 3 q^{75} + 3 q^{77} + 4 q^{79} + 3 q^{81} + 14 q^{83} + 8 q^{89} + 2 q^{91} + 10 q^{93} + 2 q^{95} + 4 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 - 2 * q^13 + 3 * q^15 + 2 * q^19 - 3 * q^21 + 4 * q^23 + 3 * q^25 + 3 * q^27 + 10 * q^31 - 3 * q^33 - 3 * q^35 + 10 * q^37 - 2 * q^39 + 4 * q^41 + 10 * q^43 + 3 * q^45 + 12 * q^47 + 3 * q^49 + 4 * q^53 - 3 * q^55 + 2 * q^57 + 8 * q^59 + 2 * q^61 - 3 * q^63 - 2 * q^65 + 6 * q^67 + 4 * q^69 + 28 * q^71 + 10 * q^73 + 3 * q^75 + 3 * q^77 + 4 * q^79 + 3 * q^81 + 14 * q^83 + 8 * q^89 + 2 * q^91 + 10 * q^93 + 2 * q^95 + 4 * q^97 - 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.802094	−0.222461	−0.111230	−	0.993795i	$-0.535479\pi$
$13$	−0.802094	−0.222461	−0.111230	+	0.993795i	$0.535479\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	5.28251	1.28120	0.640599	−	0.767876i	$-0.278686\pi$
$17$	5.28251	1.28120	0.640599	+	0.767876i	$0.278686\pi$
$18$	0	0
$19$	5.67832	1.30270	0.651348	−	0.758779i	$-0.274204\pi$
$19$	5.67832	1.30270	0.651348	+	0.758779i	$0.274204\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−3.67832	−0.766983	−0.383492	−	0.923544i	$-0.625278\pi$
$23$	−3.67832	−0.766983	−0.383492	+	0.923544i	$0.625278\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.87623	−0.905493	−0.452746	−	0.891639i	$-0.649556\pi$
$29$	−4.87623	−0.905493	−0.452746	+	0.891639i	$0.649556\pi$
$30$	0	0
$31$	3.19791	0.574361	0.287180	−	0.957877i	$-0.407282\pi$
$31$	3.19791	0.574361	0.287180	+	0.957877i	$0.407282\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.19791	0.525732	0.262866	−	0.964832i	$-0.415332\pi$
$37$	3.19791	0.525732	0.262866	+	0.964832i	$0.415332\pi$
$38$	0	0
$39$	−0.802094	−0.128438
$40$	0	0
$41$	1.19791	0.187081	0.0935407	−	0.995615i	$-0.470181\pi$
$41$	1.19791	0.187081	0.0935407	+	0.995615i	$0.470181\pi$
$42$	0	0
$43$	8.48042	1.29325	0.646626	−	0.762807i	$-0.276180\pi$
$43$	8.48042	1.29325	0.646626	+	0.762807i	$0.276180\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−6.15874	−0.898344	−0.449172	−	0.893445i	$-0.648281\pi$
$47$	−6.15874	−0.898344	−0.449172	+	0.893445i	$0.648281\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.28251	0.739699
$52$	0	0
$53$	6.07413	0.834346	0.417173	−	0.908827i	$-0.363021\pi$
$53$	6.07413	0.834346	0.417173	+	0.908827i	$0.363021\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	5.67832	0.752112
$58$	0	0
$59$	8.08461	1.05253	0.526263	−	0.850322i	$-0.323593\pi$
$59$	8.08461	1.05253	0.526263	+	0.850322i	$0.323593\pi$
$60$	0	0
$61$	−14.6392	−1.87435	−0.937176	−	0.348857i	$-0.886570\pi$
$61$	−14.6392	−1.87435	−0.937176	+	0.348857i	$0.886570\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−0.802094	−0.0994876
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	−3.67832	−0.442818
$70$	0	0
$71$	9.19791	1.09159	0.545795	−	0.837919i	$-0.316228\pi$
$71$	9.19791	1.09159	0.545795	+	0.837919i	$0.316228\pi$
$72$	0	0
$73$	−6.96083	−0.814704	−0.407352	−	0.913271i	$-0.633548\pi$
$73$	−6.96083	−0.814704	−0.407352	+	0.913271i	$0.633548\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	1.19791	0.134775	0.0673875	−	0.997727i	$-0.478534\pi$
$79$	1.19791	0.134775	0.0673875	+	0.997727i	$0.478534\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.0741336	−0.00813722	−0.00406861	−	0.999992i	$-0.501295\pi$
$83$	−0.0741336	−0.00813722	−0.00406861	+	0.999992i	$0.501295\pi$
$84$	0	0
$85$	5.28251	0.572969
$86$	0	0
$87$	−4.87623	−0.522787
$88$	0	0
$89$	8.08461	0.856966	0.428483	−	0.903550i	$-0.359048\pi$
$89$	8.08461	0.856966	0.428483	+	0.903550i	$0.359048\pi$
$90$	0	0
$91$	0.802094	0.0840823
$92$	0	0
$93$	3.19791	0.331607
$94$	0	0
$95$	5.67832	0.582584
$96$	0	0
$97$	6.48042	0.657987	0.328993	−	0.944332i	$-0.393291\pi$
$97$	6.48042	0.657987	0.328993	+	0.944332i	$0.393291\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−7.35664	−0.732014	−0.366007	−	0.930612i	$-0.619275\pi$
$101$	−7.35664	−0.732014	−0.366007	+	0.930612i	$0.619275\pi$
$102$	0	0
$103$	−7.83706	−0.772209	−0.386104	−	0.922455i	$-0.626180\pi$
$103$	−7.83706	−0.772209	−0.386104	+	0.922455i	$0.626180\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	12.5546	1.21369	0.606847	−	0.794819i	$-0.292434\pi$
$107$	12.5546	1.21369	0.606847	+	0.794819i	$0.292434\pi$
$108$	0	0
$109$	−7.19791	−0.689434	−0.344717	−	0.938707i	$-0.612025\pi$
$109$	−7.19791	−0.689434	−0.344717	+	0.938707i	$0.612025\pi$
$110$	0	0
$111$	3.19791	0.303532
$112$	0	0
$113$	16.6392	1.56528	0.782640	−	0.622475i	$-0.213873\pi$
$113$	16.6392	1.56528	0.782640	+	0.622475i	$0.213873\pi$
$114$	0	0
$115$	−3.67832	−0.343005
$116$	0	0
$117$	−0.802094	−0.0741536
$118$	0	0
$119$	−5.28251	−0.484247
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.19791	0.108012
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	21.4413	1.90260	0.951302	−	0.308262i	$-0.0997472\pi$
$127$	21.4413	1.90260	0.951302	+	0.308262i	$0.0997472\pi$
$128$	0	0
$129$	8.48042	0.746659
$130$	0	0
$131$	10.9504	0.956738	0.478369	−	0.878159i	$-0.341228\pi$
$131$	10.9504	0.956738	0.478369	+	0.878159i	$0.341228\pi$
$132$	0	0
$133$	−5.67832	−0.492373
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−13.3566	−1.14114	−0.570568	−	0.821251i	$-0.693277\pi$
$137$	−13.3566	−1.14114	−0.570568	+	0.821251i	$0.693277\pi$
$138$	0	0
$139$	6.56502	0.556838	0.278419	−	0.960460i	$-0.410190\pi$
$139$	6.56502	0.556838	0.278419	+	0.960460i	$0.410190\pi$
$140$	0	0
$141$	−6.15874	−0.518659
$142$	0	0
$143$	0.802094	0.0670745
$144$	0	0
$145$	−4.87623	−0.404949
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	18.3175	1.50063	0.750313	−	0.661083i	$-0.229903\pi$
$149$	18.3175	1.50063	0.750313	+	0.661083i	$0.229903\pi$
$150$	0	0
$151$	14.5650	1.18528	0.592642	−	0.805466i	$-0.298085\pi$
$151$	14.5650	1.18528	0.592642	+	0.805466i	$0.298085\pi$
$152$	0	0
$153$	5.28251	0.427066
$154$	0	0
$155$	3.19791	0.256862
$156$	0	0
$157$	19.4413	1.55158	0.775790	−	0.630991i	$-0.217351\pi$
$157$	19.4413	1.55158	0.775790	+	0.630991i	$0.217351\pi$
$158$	0	0
$159$	6.07413	0.481710
$160$	0	0
$161$	3.67832	0.289892
$162$	0	0
$163$	19.1979	1.50370	0.751848	−	0.659336i	$-0.229163\pi$
$163$	19.1979	1.50370	0.751848	+	0.659336i	$0.229163\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−14.5650	−1.12708	−0.563538	−	0.826090i	$-0.690560\pi$
$167$	−14.5650	−1.12708	−0.563538	+	0.826090i	$0.690560\pi$
$168$	0	0
$169$	−12.3566	−0.950511
$170$	0	0
$171$	5.67832	0.434232
$172$	0	0
$173$	5.35664	0.407258	0.203629	−	0.979048i	$-0.434726\pi$
$173$	5.35664	0.407258	0.203629	+	0.979048i	$0.434726\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	8.08461	0.607676
$178$	0	0
$179$	−16.7238	−1.24999	−0.624996	−	0.780628i	$-0.714900\pi$
$179$	−16.7238	−1.24999	−0.624996	+	0.780628i	$0.714900\pi$
$180$	0	0
$181$	−1.20838	−0.0898180	−0.0449090	−	0.998991i	$-0.514300\pi$
$181$	−1.20838	−0.0898180	−0.0449090	+	0.998991i	$0.514300\pi$
$182$	0	0
$183$	−14.6392	−1.08216
$184$	0	0
$185$	3.19791	0.235115
$186$	0	0
$187$	−5.28251	−0.386295
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	12.9608	0.932941	0.466471	−	0.884537i	$-0.345525\pi$
$193$	12.9608	0.932941	0.466471	+	0.884537i	$0.345525\pi$
$194$	0	0
$195$	−0.802094	−0.0574392
$196$	0	0
$197$	−24.3175	−1.73255	−0.866274	−	0.499569i	$-0.833492\pi$
$197$	−24.3175	−1.73255	−0.866274	+	0.499569i	$0.833492\pi$
$198$	0	0
$199$	−16.5650	−1.17426	−0.587132	−	0.809491i	$-0.699743\pi$
$199$	−16.5650	−1.17426	−0.587132	+	0.809491i	$0.699743\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	4.87623	0.342244
$204$	0	0
$205$	1.19791	0.0836654
$206$	0	0
$207$	−3.67832	−0.255661
$208$	0	0
$209$	−5.67832	−0.392778
$210$	0	0
$211$	17.7525	1.22213	0.611065	−	0.791581i	$-0.290741\pi$
$211$	17.7525	1.22213	0.611065	+	0.791581i	$0.290741\pi$
$212$	0	0
$213$	9.19791	0.630230
$214$	0	0
$215$	8.48042	0.578360
$216$	0	0
$217$	−3.19791	−0.217088
$218$	0	0
$219$	−6.96083	−0.470370
$220$	0	0
$221$	−4.23707	−0.285016
$222$	0	0
$223$	20.6496	1.38280	0.691401	−	0.722472i	$-0.256994\pi$
$223$	20.6496	1.38280	0.691401	+	0.722472i	$0.256994\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	2.46994	0.163936	0.0819680	−	0.996635i	$-0.473879\pi$
$227$	2.46994	0.163936	0.0819680	+	0.996635i	$0.473879\pi$
$228$	0	0
$229$	−4.80209	−0.317331	−0.158666	−	0.987332i	$-0.550719\pi$
$229$	−4.80209	−0.317331	−0.158666	+	0.987332i	$0.550719\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−25.5154	−1.67157	−0.835784	−	0.549058i	$-0.814987\pi$
$233$	−25.5154	−1.67157	−0.835784	+	0.549058i	$0.814987\pi$
$234$	0	0
$235$	−6.15874	−0.401752
$236$	0	0
$237$	1.19791	0.0778123
$238$	0	0
$239$	−13.2720	−0.858497	−0.429248	−	0.903186i	$-0.641222\pi$
$239$	−13.2720	−0.858497	−0.429248	+	0.903186i	$0.641222\pi$
$240$	0	0
$241$	2.39581	0.154328	0.0771639	−	0.997018i	$-0.475414\pi$
$241$	2.39581	0.154328	0.0771639	+	0.997018i	$0.475414\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−4.55455	−0.289799
$248$	0	0
$249$	−0.0741336	−0.00469803
$250$	0	0
$251$	−14.1483	−0.893031	−0.446515	−	0.894776i	$-0.647335\pi$
$251$	−14.1483	−0.893031	−0.446515	+	0.894776i	$0.647335\pi$
$252$	0	0
$253$	3.67832	0.231254
$254$	0	0
$255$	5.28251	0.330804
$256$	0	0
$257$	−16.9504	−1.05733	−0.528667	−	0.848829i	$-0.677308\pi$
$257$	−16.9504	−1.05733	−0.528667	+	0.848829i	$0.677308\pi$
$258$	0	0
$259$	−3.19791	−0.198708
$260$	0	0
$261$	−4.87623	−0.301831
$262$	0	0
$263$	−1.60419	−0.0989185	−0.0494593	−	0.998776i	$-0.515750\pi$
$263$	−1.60419	−0.0989185	−0.0494593	+	0.998776i	$0.515750\pi$
$264$	0	0
$265$	6.07413	0.373131
$266$	0	0
$267$	8.08461	0.494770
$268$	0	0
$269$	3.91539	0.238726	0.119363	−	0.992851i	$-0.461915\pi$
$269$	3.91539	0.238726	0.119363	+	0.992851i	$0.461915\pi$
$270$	0	0
$271$	5.67832	0.344934	0.172467	−	0.985015i	$-0.444826\pi$
$271$	5.67832	0.344934	0.172467	+	0.985015i	$0.444826\pi$
$272$	0	0
$273$	0.802094	0.0485450
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−13.6042	−0.817396	−0.408698	−	0.912670i	$-0.634017\pi$
$277$	−13.6042	−0.817396	−0.408698	+	0.912670i	$0.634017\pi$
$278$	0	0
$279$	3.19791	0.191454
$280$	0	0
$281$	2.96083	0.176629	0.0883143	−	0.996093i	$-0.471852\pi$
$281$	2.96083	0.176629	0.0883143	+	0.996093i	$0.471852\pi$
$282$	0	0
$283$	7.44545	0.442586	0.221293	−	0.975207i	$-0.428972\pi$
$283$	7.44545	0.442586	0.221293	+	0.975207i	$0.428972\pi$
$284$	0	0
$285$	5.67832	0.336355
$286$	0	0
$287$	−1.19791	−0.0707101
$288$	0	0
$289$	10.9049	0.641466
$290$	0	0
$291$	6.48042	0.379889
$292$	0	0
$293$	−5.28251	−0.308608	−0.154304	−	0.988023i	$-0.549313\pi$
$293$	−5.28251	−0.308608	−0.154304	+	0.988023i	$0.549313\pi$
$294$	0	0
$295$	8.08461	0.470704
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	2.95036	0.170624
$300$	0	0
$301$	−8.48042	−0.488803
$302$	0	0
$303$	−7.35664	−0.422628
$304$	0	0
$305$	−14.6392	−0.838236
$306$	0	0
$307$	−12.9608	−0.739714	−0.369857	−	0.929089i	$-0.620593\pi$
$307$	−12.9608	−0.739714	−0.369857	+	0.929089i	$0.620593\pi$
$308$	0	0
$309$	−7.83706	−0.445835
$310$	0	0
$311$	22.1692	1.25710	0.628550	−	0.777769i	$-0.283649\pi$
$311$	22.1692	1.25710	0.628550	+	0.777769i	$0.283649\pi$
$312$	0	0
$313$	34.0063	1.92215	0.961074	−	0.276291i	$-0.0891053\pi$
$313$	34.0063	1.92215	0.961074	+	0.276291i	$0.0891053\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−18.9608	−1.06495	−0.532473	−	0.846447i	$-0.678737\pi$
$317$	−18.9608	−1.06495	−0.532473	+	0.846447i	$0.678737\pi$
$318$	0	0
$319$	4.87623	0.273016
$320$	0	0
$321$	12.5546	0.700727
$322$	0	0
$323$	29.9958	1.66901
$324$	0	0
$325$	−0.802094	−0.0444922
$326$	0	0
$327$	−7.19791	−0.398045
$328$	0	0
$329$	6.15874	0.339542
$330$	0	0
$331$	−19.4308	−1.06801	−0.534006	−	0.845481i	$-0.679314\pi$
$331$	−19.4308	−1.06801	−0.534006	+	0.845481i	$0.679314\pi$
$332$	0	0
$333$	3.19791	0.175244
$334$	0	0
$335$	2.00000	0.109272
$336$	0	0
$337$	−20.7979	−1.13293	−0.566467	−	0.824085i	$-0.691690\pi$
$337$	−20.7979	−1.13293	−0.566467	+	0.824085i	$0.691690\pi$
$338$	0	0
$339$	16.6392	0.903715
$340$	0	0
$341$	−3.19791	−0.173176
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.67832	−0.198034
$346$	0	0
$347$	−32.3175	−1.73489	−0.867447	−	0.497530i	$-0.834240\pi$
$347$	−32.3175	−1.73489	−0.867447	+	0.497530i	$0.834240\pi$
$348$	0	0
$349$	−33.9958	−1.81975	−0.909877	−	0.414879i	$-0.863824\pi$
$349$	−33.9958	−1.81975	−0.909877	+	0.414879i	$0.863824\pi$
$350$	0	0
$351$	−0.802094	−0.0428126
$352$	0	0
$353$	4.39581	0.233965	0.116983	−	0.993134i	$-0.462678\pi$
$353$	4.39581	0.233965	0.116983	+	0.993134i	$0.462678\pi$
$354$	0	0
$355$	9.19791	0.488174
$356$	0	0
$357$	−5.28251	−0.279580
$358$	0	0
$359$	8.48042	0.447579	0.223790	−	0.974637i	$-0.428157\pi$
$359$	8.48042	0.447579	0.223790	+	0.974637i	$0.428157\pi$
$360$	0	0
$361$	13.2433	0.697018
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−6.96083	−0.364347
$366$	0	0
$367$	12.6287	0.659212	0.329606	−	0.944119i	$-0.393084\pi$
$367$	12.6287	0.659212	0.329606	+	0.944119i	$0.393084\pi$
$368$	0	0
$369$	1.19791	0.0623605
$370$	0	0
$371$	−6.07413	−0.315353
$372$	0	0
$373$	32.7979	1.69821	0.849105	−	0.528224i	$-0.177142\pi$
$373$	32.7979	1.69821	0.849105	+	0.528224i	$0.177142\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	3.91119	0.201437
$378$	0	0
$379$	−15.2825	−0.785010	−0.392505	−	0.919750i	$-0.628391\pi$
$379$	−15.2825	−0.785010	−0.392505	+	0.919750i	$0.628391\pi$
$380$	0	0
$381$	21.4413	1.09847
$382$	0	0
$383$	19.9321	1.01848	0.509242	−	0.860623i	$-0.329926\pi$
$383$	19.9321	1.01848	0.509242	+	0.860623i	$0.329926\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	8.48042	0.431084
$388$	0	0
$389$	39.0412	1.97947	0.989735	−	0.142917i	$-0.0456482\pi$
$389$	39.0412	1.97947	0.989735	+	0.142917i	$0.0456482\pi$
$390$	0	0
$391$	−19.4308	−0.982657
$392$	0	0
$393$	10.9504	0.552373
$394$	0	0
$395$	1.19791	0.0602732
$396$	0	0
$397$	25.6741	1.28855	0.644274	−	0.764795i	$-0.277160\pi$
$397$	25.6741	1.28855	0.644274	+	0.764795i	$0.277160\pi$
$398$	0	0
$399$	−5.67832	−0.284272
$400$	0	0
$401$	3.36712	0.168146	0.0840729	−	0.996460i	$-0.473207\pi$
$401$	3.36712	0.168146	0.0840729	+	0.996460i	$0.473207\pi$
$402$	0	0
$403$	−2.56502	−0.127773
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−3.19791	−0.158514
$408$	0	0
$409$	31.5259	1.55885	0.779427	−	0.626494i	$-0.215511\pi$
$409$	31.5259	1.55885	0.779427	+	0.626494i	$0.215511\pi$
$410$	0	0
$411$	−13.3566	−0.658835
$412$	0	0
$413$	−8.08461	−0.397817
$414$	0	0
$415$	−0.0741336	−0.00363908
$416$	0	0
$417$	6.56502	0.321491
$418$	0	0
$419$	−1.51958	−0.0742365	−0.0371183	−	0.999311i	$-0.511818\pi$
$419$	−1.51958	−0.0742365	−0.0371183	+	0.999311i	$0.511818\pi$
$420$	0	0
$421$	−6.24334	−0.304282	−0.152141	−	0.988359i	$-0.548617\pi$
$421$	−6.24334	−0.304282	−0.152141	+	0.988359i	$0.548617\pi$
$422$	0	0
$423$	−6.15874	−0.299448
$424$	0	0
$425$	5.28251	0.256239
$426$	0	0
$427$	14.6392	0.708438
$428$	0	0
$429$	0.802094	0.0387255
$430$	0	0
$431$	−23.5259	−1.13320	−0.566600	−	0.823993i	$-0.691742\pi$
$431$	−23.5259	−1.13320	−0.566600	+	0.823993i	$0.691742\pi$
$432$	0	0
$433$	15.6042	0.749889	0.374945	−	0.927047i	$-0.377662\pi$
$433$	15.6042	0.749889	0.374945	+	0.927047i	$0.377662\pi$
$434$	0	0
$435$	−4.87623	−0.233797
$436$	0	0
$437$	−20.8867	−0.999146
$438$	0	0
$439$	−15.9259	−0.760100	−0.380050	−	0.924966i	$-0.624093\pi$
$439$	−15.9259	−0.760100	−0.380050	+	0.924966i	$0.624093\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−11.9217	−0.566415	−0.283208	−	0.959059i	$-0.591399\pi$
$443$	−11.9217	−0.566415	−0.283208	+	0.959059i	$0.591399\pi$
$444$	0	0
$445$	8.08461	0.383247
$446$	0	0
$447$	18.3175	0.866387
$448$	0	0
$449$	−10.5755	−0.499088	−0.249544	−	0.968363i	$-0.580281\pi$
$449$	−10.5755	−0.499088	−0.249544	+	0.968363i	$0.580281\pi$
$450$	0	0
$451$	−1.19791	−0.0564072
$452$	0	0
$453$	14.5650	0.684324
$454$	0	0
$455$	0.802094	0.0376028
$456$	0	0
$457$	−17.5895	−0.822803	−0.411401	−	0.911454i	$-0.634961\pi$
$457$	−17.5895	−0.822803	−0.411401	+	0.911454i	$0.634961\pi$
$458$	0	0
$459$	5.28251	0.246566
$460$	0	0
$461$	−37.7525	−1.75831	−0.879154	−	0.476539i	$-0.841891\pi$
$461$	−37.7525	−1.75831	−0.879154	+	0.476539i	$0.841891\pi$
$462$	0	0
$463$	8.39581	0.390186	0.195093	−	0.980785i	$-0.437499\pi$
$463$	8.39581	0.390186	0.195093	+	0.980785i	$0.437499\pi$
$464$	0	0
$465$	3.19791	0.148299
$466$	0	0
$467$	8.81257	0.407797	0.203898	−	0.978992i	$-0.434639\pi$
$467$	8.81257	0.407797	0.203898	+	0.978992i	$0.434639\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	19.4413	0.895806
$472$	0	0
$473$	−8.48042	−0.389930
$474$	0	0
$475$	5.67832	0.260539
$476$	0	0
$477$	6.07413	0.278115
$478$	0	0
$479$	−7.76293	−0.354697	−0.177349	−	0.984148i	$-0.556752\pi$
$479$	−7.76293	−0.354697	−0.177349	+	0.984148i	$0.556752\pi$
$480$	0	0
$481$	−2.56502	−0.116955
$482$	0	0
$483$	3.67832	0.167369
$484$	0	0
$485$	6.48042	0.294261
$486$	0	0
$487$	9.18743	0.416322	0.208161	−	0.978095i	$-0.433252\pi$
$487$	9.18743	0.416322	0.208161	+	0.978095i	$0.433252\pi$
$488$	0	0
$489$	19.1979	0.868159
$490$	0	0
$491$	−23.8371	−1.07575	−0.537876	−	0.843024i	$-0.680773\pi$
$491$	−23.8371	−1.07575	−0.537876	+	0.843024i	$0.680773\pi$
$492$	0	0
$493$	−25.7587	−1.16011
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	−9.19791	−0.412582
$498$	0	0
$499$	17.6783	0.791390	0.395695	−	0.918382i	$-0.370504\pi$
$499$	17.6783	0.791390	0.395695	+	0.918382i	$0.370504\pi$
$500$	0	0
$501$	−14.5650	−0.650717
$502$	0	0
$503$	−7.28251	−0.324711	−0.162356	−	0.986732i	$-0.551909\pi$
$503$	−7.28251	−0.324711	−0.162356	+	0.986732i	$0.551909\pi$
$504$	0	0
$505$	−7.35664	−0.327366
$506$	0	0
$507$	−12.3566	−0.548778
$508$	0	0
$509$	−21.1937	−0.939395	−0.469697	−	0.882828i	$-0.655637\pi$
$509$	−21.1937	−0.939395	−0.469697	+	0.882828i	$0.655637\pi$
$510$	0	0
$511$	6.96083	0.307929
$512$	0	0
$513$	5.67832	0.250704
$514$	0	0
$515$	−7.83706	−0.345342
$516$	0	0
$517$	6.15874	0.270861
$518$	0	0
$519$	5.35664	0.235131
$520$	0	0
$521$	−8.23287	−0.360689	−0.180344	−	0.983604i	$-0.557721\pi$
$521$	−8.23287	−0.360689	−0.180344	+	0.983604i	$0.557721\pi$
$522$	0	0
$523$	19.1874	0.839008	0.419504	−	0.907754i	$-0.362204\pi$
$523$	19.1874	0.839008	0.419504	+	0.907754i	$0.362204\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	16.8930	0.735869
$528$	0	0
$529$	−9.46994	−0.411737
$530$	0	0
$531$	8.08461	0.350842
$532$	0	0
$533$	−0.960833	−0.0416183
$534$	0	0
$535$	12.5546	0.542781
$536$	0	0
$537$	−16.7238	−0.721684
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−7.98953	−0.343497	−0.171748	−	0.985141i	$-0.554942\pi$
$541$	−7.98953	−0.343497	−0.171748	+	0.985141i	$0.554942\pi$
$542$	0	0
$543$	−1.20838	−0.0518564
$544$	0	0
$545$	−7.19791	−0.308324
$546$	0	0
$547$	−4.33215	−0.185229	−0.0926147	−	0.995702i	$-0.529522\pi$
$547$	−4.33215	−0.185229	−0.0926147	+	0.995702i	$0.529522\pi$
$548$	0	0
$549$	−14.6392	−0.624784
$550$	0	0
$551$	−27.6888	−1.17958
$552$	0	0
$553$	−1.19791	−0.0509401
$554$	0	0
$555$	3.19791	0.135744
$556$	0	0
$557$	27.6741	1.17259	0.586295	−	0.810098i	$-0.300586\pi$
$557$	27.6741	1.17259	0.586295	+	0.810098i	$0.300586\pi$
$558$	0	0
$559$	−6.80209	−0.287698
$560$	0	0
$561$	−5.28251	−0.223028
$562$	0	0
$563$	20.9608	0.883394	0.441697	−	0.897164i	$-0.354377\pi$
$563$	20.9608	0.883394	0.441697	+	0.897164i	$0.354377\pi$
$564$	0	0
$565$	16.6392	0.700015
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−17.5196	−0.734459	−0.367230	−	0.930130i	$-0.619694\pi$
$569$	−17.5196	−0.734459	−0.367230	+	0.930130i	$0.619694\pi$
$570$	0	0
$571$	6.01047	0.251530	0.125765	−	0.992060i	$-0.459861\pi$
$571$	6.01047	0.251530	0.125765	+	0.992060i	$0.459861\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.67832	−0.153397
$576$	0	0
$577$	−31.4266	−1.30831	−0.654153	−	0.756362i	$-0.726975\pi$
$577$	−31.4266	−1.30831	−0.654153	+	0.756362i	$0.726975\pi$
$578$	0	0
$579$	12.9608	0.538634
$580$	0	0
$581$	0.0741336	0.00307558
$582$	0	0
$583$	−6.07413	−0.251565
$584$	0	0
$585$	−0.802094	−0.0331625
$586$	0	0
$587$	−6.80209	−0.280752	−0.140376	−	0.990098i	$-0.544831\pi$
$587$	−6.80209	−0.280752	−0.140376	+	0.990098i	$0.544831\pi$
$588$	0	0
$589$	18.1587	0.748218
$590$	0	0
$591$	−24.3175	−1.00029
$592$	0	0
$593$	−42.4657	−1.74386	−0.871930	−	0.489631i	$-0.837131\pi$
$593$	−42.4657	−1.74386	−0.871930	+	0.489631i	$0.837131\pi$
$594$	0	0
$595$	−5.28251	−0.216562
$596$	0	0
$597$	−16.5650	−0.677961
$598$	0	0
$599$	4.16921	0.170349	0.0851747	−	0.996366i	$-0.472855\pi$
$599$	4.16921	0.170349	0.0851747	+	0.996366i	$0.472855\pi$
$600$	0	0
$601$	14.6182	0.596289	0.298145	−	0.954521i	$-0.403632\pi$
$601$	14.6182	0.596289	0.298145	+	0.954521i	$0.403632\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−23.6741	−0.960903	−0.480451	−	0.877021i	$-0.659527\pi$
$607$	−23.6741	−0.960903	−0.480451	+	0.877021i	$0.659527\pi$
$608$	0	0
$609$	4.87623	0.197595
$610$	0	0
$611$	4.93989	0.199847
$612$	0	0
$613$	−47.2000	−1.90639	−0.953195	−	0.302358i	$-0.902226\pi$
$613$	−47.2000	−1.90639	−0.953195	+	0.302358i	$0.902226\pi$
$614$	0	0
$615$	1.19791	0.0483042
$616$	0	0
$617$	5.67412	0.228432	0.114216	−	0.993456i	$-0.463564\pi$
$617$	5.67412	0.228432	0.114216	+	0.993456i	$0.463564\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−3.67832	−0.147606
$622$	0	0
$623$	−8.08461	−0.323903
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.67832	−0.226770
$628$	0	0
$629$	16.8930	0.673567
$630$	0	0
$631$	16.2224	0.645804	0.322902	−	0.946432i	$-0.395342\pi$
$631$	16.2224	0.645804	0.322902	+	0.946432i	$0.395342\pi$
$632$	0	0
$633$	17.7525	0.705597
$634$	0	0
$635$	21.4413	0.850870
$636$	0	0
$637$	−0.802094	−0.0317801
$638$	0	0
$639$	9.19791	0.363864
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	−0.480417	−0.0189458	−0.00947289	−	0.999955i	$-0.503015\pi$
$643$	−0.480417	−0.0189458	−0.00947289	+	0.999955i	$0.503015\pi$
$644$	0	0
$645$	8.48042	0.333916
$646$	0	0
$647$	25.9007	1.01826	0.509131	−	0.860689i	$-0.329967\pi$
$647$	25.9007	1.01826	0.509131	+	0.860689i	$0.329967\pi$
$648$	0	0
$649$	−8.08461	−0.317349
$650$	0	0
$651$	−3.19791	−0.125336
$652$	0	0
$653$	−16.4909	−0.645338	−0.322669	−	0.946512i	$-0.604580\pi$
$653$	−16.4909	−0.645338	−0.322669	+	0.946512i	$0.604580\pi$
$654$	0	0
$655$	10.9504	0.427866
$656$	0	0
$657$	−6.96083	−0.271568
$658$	0	0
$659$	−27.8580	−1.08519	−0.542597	−	0.839993i	$-0.682559\pi$
$659$	−27.8580	−1.08519	−0.542597	+	0.839993i	$0.682559\pi$
$660$	0	0
$661$	11.9895	0.466339	0.233169	−	0.972436i	$-0.425090\pi$
$661$	11.9895	0.466339	0.233169	+	0.972436i	$0.425090\pi$
$662$	0	0
$663$	−4.23707	−0.164554
$664$	0	0
$665$	−5.67832	−0.220196
$666$	0	0
$667$	17.9363	0.694498
$668$	0	0
$669$	20.6496	0.798361
$670$	0	0
$671$	14.6392	0.565138
$672$	0	0
$673$	20.6496	0.795985	0.397992	−	0.917389i	$-0.369707\pi$
$673$	20.6496	0.795985	0.397992	+	0.917389i	$0.369707\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	11.9958	0.461036	0.230518	−	0.973068i	$-0.425958\pi$
$677$	11.9958	0.461036	0.230518	+	0.973068i	$0.425958\pi$
$678$	0	0
$679$	−6.48042	−0.248696
$680$	0	0
$681$	2.46994	0.0946485
$682$	0	0
$683$	37.6951	1.44236	0.721181	−	0.692747i	$-0.243600\pi$
$683$	37.6951	1.44236	0.721181	+	0.692747i	$0.243600\pi$
$684$	0	0
$685$	−13.3566	−0.510331
$686$	0	0
$687$	−4.80209	−0.183211
$688$	0	0
$689$	−4.87203	−0.185609
$690$	0	0
$691$	−39.4266	−1.49986	−0.749929	−	0.661518i	$-0.769912\pi$
$691$	−39.4266	−1.49986	−0.749929	+	0.661518i	$0.769912\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	6.56502	0.249025
$696$	0	0
$697$	6.32795	0.239688
$698$	0	0
$699$	−25.5154	−0.965080
$700$	0	0
$701$	−23.1238	−0.873373	−0.436686	−	0.899614i	$-0.643848\pi$
$701$	−23.1238	−0.873373	−0.436686	+	0.899614i	$0.643848\pi$
$702$	0	0
$703$	18.1587	0.684870
$704$	0	0
$705$	−6.15874	−0.231952
$706$	0	0
$707$	7.35664	0.276675
$708$	0	0
$709$	32.9566	1.23771	0.618856	−	0.785504i	$-0.287596\pi$
$709$	32.9566	1.23771	0.618856	+	0.785504i	$0.287596\pi$
$710$	0	0
$711$	1.19791	0.0449250
$712$	0	0
$713$	−11.7629	−0.440525
$714$	0	0
$715$	0.802094	0.0299966
$716$	0	0
$717$	−13.2720	−0.495653
$718$	0	0
$719$	−33.3420	−1.24345	−0.621723	−	0.783237i	$-0.713567\pi$
$719$	−33.3420	−1.24345	−0.621723	+	0.783237i	$0.713567\pi$
$720$	0	0
$721$	7.83706	0.291867
$722$	0	0
$723$	2.39581	0.0891012
$724$	0	0
$725$	−4.87623	−0.181099
$726$	0	0
$727$	−38.2538	−1.41876	−0.709378	−	0.704828i	$-0.751024\pi$
$727$	−38.2538	−1.41876	−0.709378	+	0.704828i	$0.751024\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	44.7979	1.65691
$732$	0	0
$733$	−4.39581	−0.162363	−0.0811815	−	0.996699i	$-0.525869\pi$
$733$	−4.39581	−0.162363	−0.0811815	+	0.996699i	$0.525869\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	1.45592	0.0535569	0.0267785	−	0.999641i	$-0.491475\pi$
$739$	1.45592	0.0535569	0.0267785	+	0.999641i	$0.491475\pi$
$740$	0	0
$741$	−4.55455	−0.167316
$742$	0	0
$743$	−40.4657	−1.48454	−0.742272	−	0.670099i	$-0.766252\pi$
$743$	−40.4657	−1.48454	−0.742272	+	0.670099i	$0.766252\pi$
$744$	0	0
$745$	18.3175	0.671100
$746$	0	0
$747$	−0.0741336	−0.00271241
$748$	0	0
$749$	−12.5546	−0.458733
$750$	0	0
$751$	45.9958	1.67841	0.839205	−	0.543815i	$-0.183021\pi$
$751$	45.9958	1.67841	0.839205	+	0.543815i	$0.183021\pi$
$752$	0	0
$753$	−14.1483	−0.515592
$754$	0	0
$755$	14.5650	0.530075
$756$	0	0
$757$	10.3175	0.374995	0.187498	−	0.982265i	$-0.439962\pi$
$757$	10.3175	0.374995	0.187498	+	0.982265i	$0.439962\pi$
$758$	0	0
$759$	3.67832	0.133515
$760$	0	0
$761$	−30.5650	−1.10798	−0.553991	−	0.832523i	$-0.686896\pi$
$761$	−30.5650	−1.10798	−0.553991	+	0.832523i	$0.686896\pi$
$762$	0	0
$763$	7.19791	0.260582
$764$	0	0
$765$	5.28251	0.190990
$766$	0	0
$767$	−6.48462	−0.234146
$768$	0	0
$769$	−24.0741	−0.868135	−0.434068	−	0.900880i	$-0.642922\pi$
$769$	−24.0741	−0.868135	−0.434068	+	0.900880i	$0.642922\pi$
$770$	0	0
$771$	−16.9504	−0.610452
$772$	0	0
$773$	18.8126	0.676641	0.338320	−	0.941031i	$-0.390141\pi$
$773$	18.8126	0.676641	0.338320	+	0.941031i	$0.390141\pi$
$774$	0	0
$775$	3.19791	0.114872
$776$	0	0
$777$	−3.19791	−0.114724
$778$	0	0
$779$	6.80209	0.243710
$780$	0	0
$781$	−9.19791	−0.329127
$782$	0	0
$783$	−4.87623	−0.174262
$784$	0	0
$785$	19.4413	0.693888
$786$	0	0
$787$	5.28671	0.188451	0.0942254	−	0.995551i	$-0.469963\pi$
$787$	5.28671	0.188451	0.0942254	+	0.995551i	$0.469963\pi$
$788$	0	0
$789$	−1.60419	−0.0571106
$790$	0	0
$791$	−16.6392	−0.591620
$792$	0	0
$793$	11.7420	0.416970
$794$	0	0
$795$	6.07413	0.215427
$796$	0	0
$797$	−6.81257	−0.241313	−0.120657	−	0.992694i	$-0.538500\pi$
$797$	−6.81257	−0.241313	−0.120657	+	0.992694i	$0.538500\pi$
$798$	0	0
$799$	−32.5336	−1.15096
$800$	0	0
$801$	8.08461	0.285655
$802$	0	0
$803$	6.96083	0.245642
$804$	0	0
$805$	3.67832	0.129644
$806$	0	0
$807$	3.91539	0.137828
$808$	0	0
$809$	−35.4475	−1.24627	−0.623134	−	0.782115i	$-0.714141\pi$
$809$	−35.4475	−1.24627	−0.623134	+	0.782115i	$0.714141\pi$
$810$	0	0
$811$	1.92167	0.0674788	0.0337394	−	0.999431i	$-0.489258\pi$
$811$	1.92167	0.0674788	0.0337394	+	0.999431i	$0.489258\pi$
$812$	0	0
$813$	5.67832	0.199147
$814$	0	0
$815$	19.1979	0.672473
$816$	0	0
$817$	48.1545	1.68471
$818$	0	0
$819$	0.802094	0.0280274
$820$	0	0
$821$	−40.4021	−1.41004	−0.705021	−	0.709186i	$-0.749063\pi$
$821$	−40.4021	−1.41004	−0.705021	+	0.709186i	$0.749063\pi$
$822$	0	0
$823$	53.4371	1.86270	0.931349	−	0.364127i	$-0.118633\pi$
$823$	53.4371	1.86270	0.931349	+	0.364127i	$0.118633\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	6.73423	0.234172	0.117086	−	0.993122i	$-0.462645\pi$
$827$	6.73423	0.234172	0.117086	+	0.993122i	$0.462645\pi$
$828$	0	0
$829$	47.4475	1.64792	0.823960	−	0.566648i	$-0.191760\pi$
$829$	47.4475	1.64792	0.823960	+	0.566648i	$0.191760\pi$
$830$	0	0
$831$	−13.6042	−0.471924
$832$	0	0
$833$	5.28251	0.183028
$834$	0	0
$835$	−14.5650	−0.504043
$836$	0	0
$837$	3.19791	0.110536
$838$	0	0
$839$	19.4413	0.671186	0.335593	−	0.942007i	$-0.391063\pi$
$839$	19.4413	0.671186	0.335593	+	0.942007i	$0.391063\pi$
$840$	0	0
$841$	−5.22240	−0.180083
$842$	0	0
$843$	2.96083	0.101977
$844$	0	0
$845$	−12.3566	−0.425081
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	7.44545	0.255527
$850$	0	0
$851$	−11.7629	−0.403228
$852$	0	0
$853$	51.6846	1.76965	0.884823	−	0.465927i	$-0.154279\pi$
$853$	51.6846	1.76965	0.884823	+	0.465927i	$0.154279\pi$
$854$	0	0
$855$	5.67832	0.194195
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	−23.1979	−0.791502	−0.395751	−	0.918358i	$-0.629516\pi$
$859$	−23.1979	−0.791502	−0.395751	+	0.918358i	$0.629516\pi$
$860$	0	0
$861$	−1.19791	−0.0408245
$862$	0	0
$863$	3.65738	0.124499	0.0622493	−	0.998061i	$-0.480173\pi$
$863$	3.65738	0.124499	0.0622493	+	0.998061i	$0.480173\pi$
$864$	0	0
$865$	5.35664	0.182131
$866$	0	0
$867$	10.9049	0.370351
$868$	0	0
$869$	−1.19791	−0.0406362
$870$	0	0
$871$	−1.60419	−0.0543559
$872$	0	0
$873$	6.48042	0.219329
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−35.8371	−1.21013	−0.605066	−	0.796175i	$-0.706853\pi$
$877$	−35.8371	−1.21013	−0.605066	+	0.796175i	$0.706853\pi$
$878$	0	0
$879$	−5.28251	−0.178175
$880$	0	0
$881$	−54.6496	−1.84119	−0.920596	−	0.390515i	$-0.872297\pi$
$881$	−54.6496	−1.84119	−0.920596	+	0.390515i	$0.872297\pi$
$882$	0	0
$883$	13.0392	0.438803	0.219401	−	0.975635i	$-0.429590\pi$
$883$	13.0392	0.438803	0.219401	+	0.975635i	$0.429590\pi$
$884$	0	0
$885$	8.08461	0.271761
$886$	0	0
$887$	−10.1650	−0.341308	−0.170654	−	0.985331i	$-0.554588\pi$
$887$	−10.1650	−0.341308	−0.170654	+	0.985331i	$0.554588\pi$
$888$	0	0
$889$	−21.4413	−0.719116
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−34.9713	−1.17027
$894$	0	0
$895$	−16.7238	−0.559014
$896$	0	0
$897$	2.95036	0.0985097
$898$	0	0
$899$	−15.5937	−0.520080
$900$	0	0
$901$	32.0867	1.06896
$902$	0	0
$903$	−8.48042	−0.282211
$904$	0	0
$905$	−1.20838	−0.0401678
$906$	0	0
$907$	−28.8021	−0.956358	−0.478179	−	0.878262i	$-0.658703\pi$
$907$	−28.8021	−0.956358	−0.478179	+	0.878262i	$0.658703\pi$
$908$	0	0
$909$	−7.35664	−0.244005
$910$	0	0
$911$	6.73423	0.223115	0.111558	−	0.993758i	$-0.464416\pi$
$911$	6.73423	0.223115	0.111558	+	0.993758i	$0.464416\pi$
$912$	0	0
$913$	0.0741336	0.00245346
$914$	0	0
$915$	−14.6392	−0.483956
$916$	0	0
$917$	−10.9504	−0.361613
$918$	0	0
$919$	−12.9399	−0.426848	−0.213424	−	0.976960i	$-0.568462\pi$
$919$	−12.9399	−0.426848	−0.213424	+	0.976960i	$0.568462\pi$
$920$	0	0
$921$	−12.9608	−0.427074
$922$	0	0
$923$	−7.37759	−0.242836
$924$	0	0
$925$	3.19791	0.105146
$926$	0	0
$927$	−7.83706	−0.257403
$928$	0	0
$929$	42.4657	1.39326	0.696628	−	0.717433i	$-0.254683\pi$
$929$	42.4657	1.39326	0.696628	+	0.717433i	$0.254683\pi$
$930$	0	0
$931$	5.67832	0.186099
$932$	0	0
$933$	22.1692	0.725787
$934$	0	0
$935$	−5.28251	−0.172757
$936$	0	0
$937$	−16.1378	−0.527199	−0.263599	−	0.964632i	$-0.584910\pi$
$937$	−16.1378	−0.527199	−0.263599	+	0.964632i	$0.584910\pi$
$938$	0	0
$939$	34.0063	1.10975
$940$	0	0
$941$	6.47622	0.211119	0.105559	−	0.994413i	$-0.466337\pi$
$941$	6.47622	0.211119	0.105559	+	0.994413i	$0.466337\pi$
$942$	0	0
$943$	−4.40628	−0.143488
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−23.9958	−0.779759	−0.389879	−	0.920866i	$-0.627483\pi$
$947$	−23.9958	−0.779759	−0.389879	+	0.920866i	$0.627483\pi$
$948$	0	0
$949$	5.58325	0.181240
$950$	0	0
$951$	−18.9608	−0.614847
$952$	0	0
$953$	−6.22660	−0.201699	−0.100850	−	0.994902i	$-0.532156\pi$
$953$	−6.22660	−0.201699	−0.100850	+	0.994902i	$0.532156\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.87623	0.157626
$958$	0	0
$959$	13.3566	0.431309
$960$	0	0
$961$	−20.7734	−0.670110
$962$	0	0
$963$	12.5546	0.404565
$964$	0	0
$965$	12.9608	0.417224
$966$	0	0
$967$	−41.7378	−1.34220	−0.671098	−	0.741368i	$-0.734177\pi$
$967$	−41.7378	−1.34220	−0.671098	+	0.741368i	$0.734177\pi$
$968$	0	0
$969$	29.9958	0.963604
$970$	0	0
$971$	−3.29298	−0.105677	−0.0528384	−	0.998603i	$-0.516827\pi$
$971$	−3.29298	−0.105677	−0.0528384	+	0.998603i	$0.516827\pi$
$972$	0	0
$973$	−6.56502	−0.210465
$974$	0	0
$975$	−0.802094	−0.0256876
$976$	0	0
$977$	−52.6392	−1.68408	−0.842038	−	0.539418i	$-0.818644\pi$
$977$	−52.6392	−1.68408	−0.842038	+	0.539418i	$0.818644\pi$
$978$	0	0
$979$	−8.08461	−0.258385
$980$	0	0
$981$	−7.19791	−0.229811
$982$	0	0
$983$	27.0392	0.862415	0.431208	−	0.902253i	$-0.358088\pi$
$983$	27.0392	0.862415	0.431208	+	0.902253i	$0.358088\pi$
$984$	0	0
$985$	−24.3175	−0.774819
$986$	0	0
$987$	6.15874	0.196035
$988$	0	0
$989$	−31.1937	−0.991902
$990$	0	0
$991$	−11.7567	−0.373462	−0.186731	−	0.982411i	$-0.559789\pi$
$991$	−11.7567	−0.373462	−0.186731	+	0.982411i	$0.559789\pi$
$992$	0	0
$993$	−19.4308	−0.616617
$994$	0	0
$995$	−16.5650	−0.525147
$996$	0	0
$997$	−1.20838	−0.0382697	−0.0191348	−	0.999817i	$-0.506091\pi$
$997$	−1.20838	−0.0382697	−0.0191348	+	0.999817i	$0.506091\pi$
$998$	0	0
$999$	3.19791	0.101177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.w.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.w.1.2	✓	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -0.802094 q^{13} +1.00000 q^{15} +5.28251 q^{17} +5.67832 q^{19} -1.00000 q^{21} -3.67832 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.87623 q^{29} +3.19791 q^{31} -1.00000 q^{33} -1.00000 q^{35} +3.19791 q^{37} -0.802094 q^{39} +1.19791 q^{41} +8.48042 q^{43} +1.00000 q^{45} -6.15874 q^{47} +1.00000 q^{49} +5.28251 q^{51} +6.07413 q^{53} -1.00000 q^{55} +5.67832 q^{57} +8.08461 q^{59} -14.6392 q^{61} -1.00000 q^{63} -0.802094 q^{65} +2.00000 q^{67} -3.67832 q^{69} +9.19791 q^{71} -6.96083 q^{73} +1.00000 q^{75} +1.00000 q^{77} +1.19791 q^{79} +1.00000 q^{81} -0.0741336 q^{83} +5.28251 q^{85} -4.87623 q^{87} +8.08461 q^{89} +0.802094 q^{91} +3.19791 q^{93} +5.67832 q^{95} +6.48042 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 2 q^{13} + 3 q^{15} + 2 q^{19} - 3 q^{21} + 4 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{31} - 3 q^{33} - 3 q^{35} + 10 q^{37} - 2 q^{39} + 4 q^{41} + 10 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} + 4 q^{53} - 3 q^{55} + 2 q^{57} + 8 q^{59} + 2 q^{61} - 3 q^{63} - 2 q^{65} + 6 q^{67} + 4 q^{69} + 28 q^{71} + 10 q^{73} + 3 q^{75} + 3 q^{77} + 4 q^{79} + 3 q^{81} + 14 q^{83} + 8 q^{89} + 2 q^{91} + 10 q^{93} + 2 q^{95} + 4 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 - 2 * q^13 + 3 * q^15 + 2 * q^19 - 3 * q^21 + 4 * q^23 + 3 * q^25 + 3 * q^27 + 10 * q^31 - 3 * q^33 - 3 * q^35 + 10 * q^37 - 2 * q^39 + 4 * q^41 + 10 * q^43 + 3 * q^45 + 12 * q^47 + 3 * q^49 + 4 * q^53 - 3 * q^55 + 2 * q^57 + 8 * q^59 + 2 * q^61 - 3 * q^63 - 2 * q^65 + 6 * q^67 + 4 * q^69 + 28 * q^71 + 10 * q^73 + 3 * q^75 + 3 * q^77 + 4 * q^79 + 3 * q^81 + 14 * q^83 + 8 * q^89 + 2 * q^91 + 10 * q^93 + 2 * q^95 + 4 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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