LMFDB - Embedded newform 3120.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.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:	$24.9133254306$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 780)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.77200$ of defining polynomial
Character	$\chi$	$=$	3120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +4.77200 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +4.77200 q^{7} +1.00000 q^{9} -4.77200 q^{11} -1.00000 q^{13} -1.00000 q^{15} -4.77200 q^{17} -6.00000 q^{19} +4.77200 q^{21} -6.77200 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -6.00000 q^{31} -4.77200 q^{33} -4.77200 q^{35} +4.77200 q^{37} -1.00000 q^{39} -8.77200 q^{41} -8.00000 q^{43} -1.00000 q^{45} -6.00000 q^{47} +15.7720 q^{49} -4.77200 q^{51} -4.77200 q^{53} +4.77200 q^{55} -6.00000 q^{57} +3.54400 q^{59} -4.77200 q^{61} +4.77200 q^{63} +1.00000 q^{65} -7.54400 q^{67} -6.77200 q^{69} +10.3160 q^{71} +10.0000 q^{73} +1.00000 q^{75} -22.7720 q^{77} -1.22800 q^{79} +1.00000 q^{81} -7.54400 q^{83} +4.77200 q^{85} +2.00000 q^{87} +4.77200 q^{89} -4.77200 q^{91} -6.00000 q^{93} +6.00000 q^{95} +10.3160 q^{97} -4.77200 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} - q^{11} - 2 q^{13} - 2 q^{15} - q^{17} - 12 q^{19} + q^{21} - 5 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 12 q^{31} - q^{33} - q^{35} + q^{37} - 2 q^{39} - 9 q^{41} - 16 q^{43} - 2 q^{45} - 12 q^{47} + 23 q^{49} - q^{51} - q^{53} + q^{55} - 12 q^{57} - 10 q^{59} - q^{61} + q^{63} + 2 q^{65} + 2 q^{67} - 5 q^{69} - 5 q^{71} + 20 q^{73} + 2 q^{75} - 37 q^{77} - 11 q^{79} + 2 q^{81} + 2 q^{83} + q^{85} + 4 q^{87} + q^{89} - q^{91} - 12 q^{93} + 12 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 - q^11 - 2 * q^13 - 2 * q^15 - q^17 - 12 * q^19 + q^21 - 5 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 12 * q^31 - q^33 - q^35 + q^37 - 2 * q^39 - 9 * q^41 - 16 * q^43 - 2 * q^45 - 12 * q^47 + 23 * q^49 - q^51 - q^53 + q^55 - 12 * q^57 - 10 * q^59 - q^61 + q^63 + 2 * q^65 + 2 * q^67 - 5 * q^69 - 5 * q^71 + 20 * q^73 + 2 * q^75 - 37 * q^77 - 11 * q^79 + 2 * q^81 + 2 * q^83 + q^85 + 4 * q^87 + q^89 - q^91 - 12 * q^93 + 12 * q^95 - 5 * q^97 - 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
$7$	4.77200	1.80365	0.901824	−	0.432104i	$-0.142229\pi$
$7$	4.77200	1.80365	0.901824	+	0.432104i	$0.142229\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.77200	−1.43881	−0.719406	−	0.694589i	$-0.755586\pi$
$11$	−4.77200	−1.43881	−0.719406	+	0.694589i	$0.755586\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.77200	−1.15738	−0.578690	−	0.815547i	$-0.696436\pi$
$17$	−4.77200	−1.15738	−0.578690	+	0.815547i	$0.696436\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	4.77200	1.04134
$22$	0	0
$23$	−6.77200	−1.41206	−0.706030	−	0.708182i	$-0.749516\pi$
$23$	−6.77200	−1.41206	−0.706030	+	0.708182i	$0.749516\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	−4.77200	−0.830699
$34$	0	0
$35$	−4.77200	−0.806616
$36$	0	0
$37$	4.77200	0.784512	0.392256	−	0.919856i	$-0.371695\pi$
$37$	4.77200	0.784512	0.392256	+	0.919856i	$0.371695\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−8.77200	−1.36996	−0.684978	−	0.728564i	$-0.740188\pi$
$41$	−8.77200	−1.36996	−0.684978	+	0.728564i	$0.740188\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	15.7720	2.25314
$50$	0	0
$51$	−4.77200	−0.668214
$52$	0	0
$53$	−4.77200	−0.655485	−0.327742	−	0.944767i	$-0.606288\pi$
$53$	−4.77200	−0.655485	−0.327742	+	0.944767i	$0.606288\pi$
$54$	0	0
$55$	4.77200	0.643457
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	3.54400	0.461390	0.230695	−	0.973026i	$-0.425900\pi$
$59$	3.54400	0.461390	0.230695	+	0.973026i	$0.425900\pi$
$60$	0	0
$61$	−4.77200	−0.610992	−0.305496	−	0.952193i	$-0.598822\pi$
$61$	−4.77200	−0.610992	−0.305496	+	0.952193i	$0.598822\pi$
$62$	0	0
$63$	4.77200	0.601216
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−7.54400	−0.921647	−0.460823	−	0.887492i	$-0.652446\pi$
$67$	−7.54400	−0.921647	−0.460823	+	0.887492i	$0.652446\pi$
$68$	0	0
$69$	−6.77200	−0.815253
$70$	0	0
$71$	10.3160	1.22428	0.612142	−	0.790748i	$-0.290308\pi$
$71$	10.3160	1.22428	0.612142	+	0.790748i	$0.290308\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−22.7720	−2.59511
$78$	0	0
$79$	−1.22800	−0.138161	−0.0690803	−	0.997611i	$-0.522006\pi$
$79$	−1.22800	−0.138161	−0.0690803	+	0.997611i	$0.522006\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.54400	−0.828062	−0.414031	−	0.910263i	$-0.635879\pi$
$83$	−7.54400	−0.828062	−0.414031	+	0.910263i	$0.635879\pi$
$84$	0	0
$85$	4.77200	0.517596
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	4.77200	0.505831	0.252916	−	0.967488i	$-0.418611\pi$
$89$	4.77200	0.505831	0.252916	+	0.967488i	$0.418611\pi$
$90$	0	0
$91$	−4.77200	−0.500242
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	10.3160	1.04743	0.523716	−	0.851893i	$-0.324545\pi$
$97$	10.3160	1.04743	0.523716	+	0.851893i	$0.324545\pi$
$98$	0	0
$99$	−4.77200	−0.479604
$100$	0	0
$101$	3.54400	0.352642	0.176321	−	0.984333i	$-0.443580\pi$
$101$	3.54400	0.352642	0.176321	+	0.984333i	$0.443580\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	−4.77200	−0.465700
$106$	0	0
$107$	−6.77200	−0.654674	−0.327337	−	0.944908i	$-0.606151\pi$
$107$	−6.77200	−0.654674	−0.327337	+	0.944908i	$0.606151\pi$
$108$	0	0
$109$	7.54400	0.722585	0.361292	−	0.932453i	$-0.382336\pi$
$109$	7.54400	0.722585	0.361292	+	0.932453i	$0.382336\pi$
$110$	0	0
$111$	4.77200	0.452938
$112$	0	0
$113$	17.0880	1.60750	0.803752	−	0.594964i	$-0.202834\pi$
$113$	17.0880	1.60750	0.803752	+	0.594964i	$0.202834\pi$
$114$	0	0
$115$	6.77200	0.631492
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−22.7720	−2.08751
$120$	0	0
$121$	11.7720	1.07018
$122$	0	0
$123$	−8.77200	−0.790945
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.54400	0.846893	0.423447	−	0.905921i	$-0.360820\pi$
$127$	9.54400	0.846893	0.423447	+	0.905921i	$0.360820\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	21.5440	1.88231	0.941154	−	0.337978i	$-0.109743\pi$
$131$	21.5440	1.88231	0.941154	+	0.337978i	$0.109743\pi$
$132$	0	0
$133$	−28.6320	−2.48271
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	16.3160	1.38391	0.691953	−	0.721943i	$-0.256751\pi$
$139$	16.3160	1.38391	0.691953	+	0.721943i	$0.256751\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	4.77200	0.399055
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	15.7720	1.30085
$148$	0	0
$149$	−3.22800	−0.264448	−0.132224	−	0.991220i	$-0.542212\pi$
$149$	−3.22800	−0.264448	−0.132224	+	0.991220i	$0.542212\pi$
$150$	0	0
$151$	−4.45600	−0.362624	−0.181312	−	0.983426i	$-0.558034\pi$
$151$	−4.45600	−0.362624	−0.181312	+	0.983426i	$0.558034\pi$
$152$	0	0
$153$	−4.77200	−0.385793
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−4.77200	−0.378444
$160$	0	0
$161$	−32.3160	−2.54686
$162$	0	0
$163$	−22.3160	−1.74792	−0.873962	−	0.485994i	$-0.838458\pi$
$163$	−22.3160	−1.74792	−0.873962	+	0.485994i	$0.838458\pi$
$164$	0	0
$165$	4.77200	0.371500
$166$	0	0
$167$	−3.54400	−0.274243	−0.137122	−	0.990554i	$-0.543785\pi$
$167$	−3.54400	−0.274243	−0.137122	+	0.990554i	$0.543785\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	4.77200	0.360729
$176$	0	0
$177$	3.54400	0.266384
$178$	0	0
$179$	−6.45600	−0.482544	−0.241272	−	0.970458i	$-0.577565\pi$
$179$	−6.45600	−0.482544	−0.241272	+	0.970458i	$0.577565\pi$
$180$	0	0
$181$	−8.77200	−0.652018	−0.326009	−	0.945367i	$-0.605704\pi$
$181$	−8.77200	−0.652018	−0.326009	+	0.945367i	$0.605704\pi$
$182$	0	0
$183$	−4.77200	−0.352757
$184$	0	0
$185$	−4.77200	−0.350845
$186$	0	0
$187$	22.7720	1.66525
$188$	0	0
$189$	4.77200	0.347112
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−6.31601	−0.454636	−0.227318	−	0.973821i	$-0.572996\pi$
$193$	−6.31601	−0.454636	−0.227318	+	0.973821i	$0.572996\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−19.5440	−1.39245	−0.696226	−	0.717822i	$-0.745139\pi$
$197$	−19.5440	−1.39245	−0.696226	+	0.717822i	$0.745139\pi$
$198$	0	0
$199$	−27.0880	−1.92022	−0.960109	−	0.279626i	$-0.909789\pi$
$199$	−27.0880	−1.92022	−0.960109	+	0.279626i	$0.909789\pi$
$200$	0	0
$201$	−7.54400	−0.532113
$202$	0	0
$203$	9.54400	0.669858
$204$	0	0
$205$	8.77200	0.612663
$206$	0	0
$207$	−6.77200	−0.470687
$208$	0	0
$209$	28.6320	1.98052
$210$	0	0
$211$	−19.0880	−1.31407	−0.657036	−	0.753859i	$-0.728190\pi$
$211$	−19.0880	−1.31407	−0.657036	+	0.753859i	$0.728190\pi$
$212$	0	0
$213$	10.3160	0.706841
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−28.6320	−1.94367
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	4.77200	0.321000
$222$	0	0
$223$	−26.6320	−1.78341	−0.891706	−	0.452616i	$-0.850491\pi$
$223$	−26.6320	−1.78341	−0.891706	+	0.452616i	$0.850491\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	0	0
$229$	−17.0880	−1.12921	−0.564604	−	0.825362i	$-0.690971\pi$
$229$	−17.0880	−1.12921	−0.564604	+	0.825362i	$0.690971\pi$
$230$	0	0
$231$	−22.7720	−1.49829
$232$	0	0
$233$	−14.3160	−0.937873	−0.468936	−	0.883232i	$-0.655363\pi$
$233$	−14.3160	−0.937873	−0.468936	+	0.883232i	$0.655363\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	−1.22800	−0.0797671
$238$	0	0
$239$	10.3160	0.667287	0.333643	−	0.942699i	$-0.391722\pi$
$239$	10.3160	0.667287	0.333643	+	0.942699i	$0.391722\pi$
$240$	0	0
$241$	29.0880	1.87372	0.936862	−	0.349700i	$-0.113717\pi$
$241$	29.0880	1.87372	0.936862	+	0.349700i	$0.113717\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−15.7720	−1.00764
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	−7.54400	−0.478082
$250$	0	0
$251$	−15.0880	−0.952347	−0.476173	−	0.879351i	$-0.657977\pi$
$251$	−15.0880	−0.952347	−0.476173	+	0.879351i	$0.657977\pi$
$252$	0	0
$253$	32.3160	2.03169
$254$	0	0
$255$	4.77200	0.298834
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	22.7720	1.41498
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	4.77200	0.293142
$266$	0	0
$267$	4.77200	0.292042
$268$	0	0
$269$	−19.5440	−1.19162	−0.595809	−	0.803126i	$-0.703169\pi$
$269$	−19.5440	−1.19162	−0.595809	+	0.803126i	$0.703169\pi$
$270$	0	0
$271$	−13.0880	−0.795040	−0.397520	−	0.917594i	$-0.630129\pi$
$271$	−13.0880	−0.795040	−0.397520	+	0.917594i	$0.630129\pi$
$272$	0	0
$273$	−4.77200	−0.288815
$274$	0	0
$275$	−4.77200	−0.287763
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	19.0880	1.13466	0.567332	−	0.823489i	$-0.307976\pi$
$283$	19.0880	1.13466	0.567332	+	0.823489i	$0.307976\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	−41.8600	−2.47092
$288$	0	0
$289$	5.77200	0.339530
$290$	0	0
$291$	10.3160	0.604735
$292$	0	0
$293$	23.5440	1.37546	0.687728	−	0.725969i	$-0.258608\pi$
$293$	23.5440	1.37546	0.687728	+	0.725969i	$0.258608\pi$
$294$	0	0
$295$	−3.54400	−0.206340
$296$	0	0
$297$	−4.77200	−0.276900
$298$	0	0
$299$	6.77200	0.391635
$300$	0	0
$301$	−38.1760	−2.20043
$302$	0	0
$303$	3.54400	0.203598
$304$	0	0
$305$	4.77200	0.273244
$306$	0	0
$307$	15.8600	0.905179	0.452589	−	0.891719i	$-0.350500\pi$
$307$	15.8600	0.905179	0.452589	+	0.891719i	$0.350500\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	−3.08801	−0.175105	−0.0875524	−	0.996160i	$-0.527905\pi$
$311$	−3.08801	−0.175105	−0.0875524	+	0.996160i	$0.527905\pi$
$312$	0	0
$313$	9.08801	0.513685	0.256842	−	0.966453i	$-0.417318\pi$
$313$	9.08801	0.513685	0.256842	+	0.966453i	$0.417318\pi$
$314$	0	0
$315$	−4.77200	−0.268872
$316$	0	0
$317$	−19.5440	−1.09770	−0.548850	−	0.835921i	$-0.684934\pi$
$317$	−19.5440	−1.09770	−0.548850	+	0.835921i	$0.684934\pi$
$318$	0	0
$319$	−9.54400	−0.534362
$320$	0	0
$321$	−6.77200	−0.377976
$322$	0	0
$323$	28.6320	1.59313
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	7.54400	0.417184
$328$	0	0
$329$	−28.6320	−1.57853
$330$	0	0
$331$	−35.5440	−1.95368	−0.976838	−	0.213982i	$-0.931357\pi$
$331$	−35.5440	−1.95368	−0.976838	+	0.213982i	$0.931357\pi$
$332$	0	0
$333$	4.77200	0.261504
$334$	0	0
$335$	7.54400	0.412173
$336$	0	0
$337$	3.54400	0.193054	0.0965271	−	0.995330i	$-0.469227\pi$
$337$	3.54400	0.193054	0.0965271	+	0.995330i	$0.469227\pi$
$338$	0	0
$339$	17.0880	0.928093
$340$	0	0
$341$	28.6320	1.55051
$342$	0	0
$343$	41.8600	2.26023
$344$	0	0
$345$	6.77200	0.364592
$346$	0	0
$347$	−3.68399	−0.197767	−0.0988836	−	0.995099i	$-0.531527\pi$
$347$	−3.68399	−0.197767	−0.0988836	+	0.995099i	$0.531527\pi$
$348$	0	0
$349$	12.4560	0.666754	0.333377	−	0.942794i	$-0.391812\pi$
$349$	12.4560	0.666754	0.333377	+	0.942794i	$0.391812\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	−10.3160	−0.547517
$356$	0	0
$357$	−22.7720	−1.20522
$358$	0	0
$359$	−34.6320	−1.82781	−0.913904	−	0.405931i	$-0.866947\pi$
$359$	−34.6320	−1.82781	−0.913904	+	0.405931i	$0.866947\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	11.7720	0.617870
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−3.08801	−0.161193	−0.0805963	−	0.996747i	$-0.525682\pi$
$367$	−3.08801	−0.161193	−0.0805963	+	0.996747i	$0.525682\pi$
$368$	0	0
$369$	−8.77200	−0.456652
$370$	0	0
$371$	−22.7720	−1.18226
$372$	0	0
$373$	26.6320	1.37895	0.689477	−	0.724308i	$-0.257841\pi$
$373$	26.6320	1.37895	0.689477	+	0.724308i	$0.257841\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	9.54400	0.488954
$382$	0	0
$383$	33.0880	1.69072	0.845359	−	0.534198i	$-0.179386\pi$
$383$	33.0880	1.69072	0.845359	+	0.534198i	$0.179386\pi$
$384$	0	0
$385$	22.7720	1.16057
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	15.5440	0.788112	0.394056	−	0.919086i	$-0.371072\pi$
$389$	15.5440	0.788112	0.394056	+	0.919086i	$0.371072\pi$
$390$	0	0
$391$	32.3160	1.63429
$392$	0	0
$393$	21.5440	1.08675
$394$	0	0
$395$	1.22800	0.0617873
$396$	0	0
$397$	−3.22800	−0.162009	−0.0810043	−	0.996714i	$-0.525813\pi$
$397$	−3.22800	−0.162009	−0.0810043	+	0.996714i	$0.525813\pi$
$398$	0	0
$399$	−28.6320	−1.43339
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−22.7720	−1.12877
$408$	0	0
$409$	−17.0880	−0.844948	−0.422474	−	0.906375i	$-0.638838\pi$
$409$	−17.0880	−0.844948	−0.422474	+	0.906375i	$0.638838\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	16.9120	0.832185
$414$	0	0
$415$	7.54400	0.370321
$416$	0	0
$417$	16.3160	0.798998
$418$	0	0
$419$	21.5440	1.05249	0.526247	−	0.850332i	$-0.323599\pi$
$419$	21.5440	1.05249	0.526247	+	0.850332i	$0.323599\pi$
$420$	0	0
$421$	−0.455996	−0.0222239	−0.0111119	−	0.999938i	$-0.503537\pi$
$421$	−0.455996	−0.0222239	−0.0111119	+	0.999938i	$0.503537\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−4.77200	−0.231476
$426$	0	0
$427$	−22.7720	−1.10201
$428$	0	0
$429$	4.77200	0.230394
$430$	0	0
$431$	10.6320	0.512126	0.256063	−	0.966660i	$-0.417575\pi$
$431$	10.6320	0.512126	0.256063	+	0.966660i	$0.417575\pi$
$432$	0	0
$433$	23.5440	1.13145	0.565726	−	0.824593i	$-0.308596\pi$
$433$	23.5440	1.13145	0.565726	+	0.824593i	$0.308596\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	40.6320	1.94369
$438$	0	0
$439$	13.2280	0.631338	0.315669	−	0.948869i	$-0.397771\pi$
$439$	13.2280	0.631338	0.315669	+	0.948869i	$0.397771\pi$
$440$	0	0
$441$	15.7720	0.751048
$442$	0	0
$443$	31.4040	1.49205	0.746025	−	0.665918i	$-0.231960\pi$
$443$	31.4040	1.49205	0.746025	+	0.665918i	$0.231960\pi$
$444$	0	0
$445$	−4.77200	−0.226215
$446$	0	0
$447$	−3.22800	−0.152679
$448$	0	0
$449$	23.2280	1.09620	0.548099	−	0.836414i	$-0.315352\pi$
$449$	23.2280	1.09620	0.548099	+	0.836414i	$0.315352\pi$
$450$	0	0
$451$	41.8600	1.97111
$452$	0	0
$453$	−4.45600	−0.209361
$454$	0	0
$455$	4.77200	0.223715
$456$	0	0
$457$	−16.7720	−0.784561	−0.392281	−	0.919846i	$-0.628314\pi$
$457$	−16.7720	−0.784561	−0.392281	+	0.919846i	$0.628314\pi$
$458$	0	0
$459$	−4.77200	−0.222738
$460$	0	0
$461$	−24.7720	−1.15375	−0.576874	−	0.816833i	$-0.695727\pi$
$461$	−24.7720	−1.15375	−0.576874	+	0.816833i	$0.695727\pi$
$462$	0	0
$463$	−33.4040	−1.55242	−0.776208	−	0.630477i	$-0.782859\pi$
$463$	−33.4040	−1.55242	−0.776208	+	0.630477i	$0.782859\pi$
$464$	0	0
$465$	6.00000	0.278243
$466$	0	0
$467$	−13.8600	−0.641365	−0.320682	−	0.947187i	$-0.603912\pi$
$467$	−13.8600	−0.641365	−0.320682	+	0.947187i	$0.603912\pi$
$468$	0	0
$469$	−36.0000	−1.66233
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	38.1760	1.75534
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−4.77200	−0.218495
$478$	0	0
$479$	−31.2280	−1.42684	−0.713422	−	0.700735i	$-0.752856\pi$
$479$	−31.2280	−1.42684	−0.713422	+	0.700735i	$0.752856\pi$
$480$	0	0
$481$	−4.77200	−0.217585
$482$	0	0
$483$	−32.3160	−1.47043
$484$	0	0
$485$	−10.3160	−0.468426
$486$	0	0
$487$	30.3160	1.37375	0.686875	−	0.726776i	$-0.258982\pi$
$487$	30.3160	1.37375	0.686875	+	0.726776i	$0.258982\pi$
$488$	0	0
$489$	−22.3160	−1.00916
$490$	0	0
$491$	−15.0880	−0.680912	−0.340456	−	0.940260i	$-0.610581\pi$
$491$	−15.0880	−0.680912	−0.340456	+	0.940260i	$0.610581\pi$
$492$	0	0
$493$	−9.54400	−0.429840
$494$	0	0
$495$	4.77200	0.214486
$496$	0	0
$497$	49.2280	2.20818
$498$	0	0
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	0	0
$501$	−3.54400	−0.158334
$502$	0	0
$503$	4.91199	0.219015	0.109507	−	0.993986i	$-0.465073\pi$
$503$	4.91199	0.219015	0.109507	+	0.993986i	$0.465073\pi$
$504$	0	0
$505$	−3.54400	−0.157706
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	15.8600	0.702983	0.351491	−	0.936191i	$-0.385675\pi$
$509$	15.8600	0.702983	0.351491	+	0.936191i	$0.385675\pi$
$510$	0	0
$511$	47.7200	2.11101
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	28.6320	1.25923
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	0	0
$523$	39.0880	1.70920	0.854600	−	0.519287i	$-0.173803\pi$
$523$	39.0880	1.70920	0.854600	+	0.519287i	$0.173803\pi$
$524$	0	0
$525$	4.77200	0.208267
$526$	0	0
$527$	28.6320	1.24723
$528$	0	0
$529$	22.8600	0.993913
$530$	0	0
$531$	3.54400	0.153797
$532$	0	0
$533$	8.77200	0.379958
$534$	0	0
$535$	6.77200	0.292779
$536$	0	0
$537$	−6.45600	−0.278597
$538$	0	0
$539$	−75.2640	−3.24185
$540$	0	0
$541$	6.91199	0.297170	0.148585	−	0.988900i	$-0.452528\pi$
$541$	6.91199	0.297170	0.148585	+	0.988900i	$0.452528\pi$
$542$	0	0
$543$	−8.77200	−0.376443
$544$	0	0
$545$	−7.54400	−0.323150
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	0	0
$549$	−4.77200	−0.203664
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−5.86001	−0.249193
$554$	0	0
$555$	−4.77200	−0.202560
$556$	0	0
$557$	−1.08801	−0.0461004	−0.0230502	−	0.999734i	$-0.507338\pi$
$557$	−1.08801	−0.0461004	−0.0230502	+	0.999734i	$0.507338\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	22.7720	0.961435
$562$	0	0
$563$	29.2280	1.23181	0.615907	−	0.787819i	$-0.288790\pi$
$563$	29.2280	1.23181	0.615907	+	0.787819i	$0.288790\pi$
$564$	0	0
$565$	−17.0880	−0.718898
$566$	0	0
$567$	4.77200	0.200405
$568$	0	0
$569$	−22.6320	−0.948783	−0.474392	−	0.880314i	$-0.657332\pi$
$569$	−22.6320	−0.948783	−0.474392	+	0.880314i	$0.657332\pi$
$570$	0	0
$571$	−9.22800	−0.386180	−0.193090	−	0.981181i	$-0.561851\pi$
$571$	−9.22800	−0.386180	−0.193090	+	0.981181i	$0.561851\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	−6.77200	−0.282412
$576$	0	0
$577$	−0.772002	−0.0321389	−0.0160694	−	0.999871i	$-0.505115\pi$
$577$	−0.772002	−0.0321389	−0.0160694	+	0.999871i	$0.505115\pi$
$578$	0	0
$579$	−6.31601	−0.262484
$580$	0	0
$581$	−36.0000	−1.49353
$582$	0	0
$583$	22.7720	0.943120
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	0	0
$589$	36.0000	1.48335
$590$	0	0
$591$	−19.5440	−0.803933
$592$	0	0
$593$	−3.54400	−0.145535	−0.0727674	−	0.997349i	$-0.523183\pi$
$593$	−3.54400	−0.145535	−0.0727674	+	0.997349i	$0.523183\pi$
$594$	0	0
$595$	22.7720	0.933561
$596$	0	0
$597$	−27.0880	−1.10864
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−7.22800	−0.294836	−0.147418	−	0.989074i	$-0.547096\pi$
$601$	−7.22800	−0.294836	−0.147418	+	0.989074i	$0.547096\pi$
$602$	0	0
$603$	−7.54400	−0.307216
$604$	0	0
$605$	−11.7720	−0.478600
$606$	0	0
$607$	−38.1760	−1.54952	−0.774758	−	0.632257i	$-0.782128\pi$
$607$	−38.1760	−1.54952	−0.774758	+	0.632257i	$0.782128\pi$
$608$	0	0
$609$	9.54400	0.386743
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	−13.6840	−0.552691	−0.276346	−	0.961058i	$-0.589123\pi$
$613$	−13.6840	−0.552691	−0.276346	+	0.961058i	$0.589123\pi$
$614$	0	0
$615$	8.77200	0.353721
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−29.0880	−1.16915	−0.584573	−	0.811341i	$-0.698738\pi$
$619$	−29.0880	−1.16915	−0.584573	+	0.811341i	$0.698738\pi$
$620$	0	0
$621$	−6.77200	−0.271751
$622$	0	0
$623$	22.7720	0.912341
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	28.6320	1.14345
$628$	0	0
$629$	−22.7720	−0.907979
$630$	0	0
$631$	22.6320	0.900966	0.450483	−	0.892785i	$-0.351252\pi$
$631$	22.6320	0.900966	0.450483	+	0.892785i	$0.351252\pi$
$632$	0	0
$633$	−19.0880	−0.758680
$634$	0	0
$635$	−9.54400	−0.378742
$636$	0	0
$637$	−15.7720	−0.624909
$638$	0	0
$639$	10.3160	0.408095
$640$	0	0
$641$	−31.5440	−1.24591	−0.622957	−	0.782256i	$-0.714069\pi$
$641$	−31.5440	−1.24591	−0.622957	+	0.782256i	$0.714069\pi$
$642$	0	0
$643$	43.8600	1.72967	0.864835	−	0.502056i	$-0.167423\pi$
$643$	43.8600	1.72967	0.864835	+	0.502056i	$0.167423\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−44.3160	−1.74224	−0.871121	−	0.491068i	$-0.836606\pi$
$647$	−44.3160	−1.74224	−0.871121	+	0.491068i	$0.836606\pi$
$648$	0	0
$649$	−16.9120	−0.663854
$650$	0	0
$651$	−28.6320	−1.12218
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	−21.5440	−0.841794
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	46.1760	1.79876	0.899381	−	0.437165i	$-0.144018\pi$
$659$	46.1760	1.79876	0.899381	+	0.437165i	$0.144018\pi$
$660$	0	0
$661$	34.6320	1.34703	0.673515	−	0.739174i	$-0.264784\pi$
$661$	34.6320	1.34703	0.673515	+	0.739174i	$0.264784\pi$
$662$	0	0
$663$	4.77200	0.185329
$664$	0	0
$665$	28.6320	1.11030
$666$	0	0
$667$	−13.5440	−0.524426
$668$	0	0
$669$	−26.6320	−1.02965
$670$	0	0
$671$	22.7720	0.879103
$672$	0	0
$673$	−13.0880	−0.504506	−0.252253	−	0.967661i	$-0.581171\pi$
$673$	−13.0880	−0.504506	−0.252253	+	0.967661i	$0.581171\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−42.9480	−1.65063	−0.825313	−	0.564675i	$-0.809001\pi$
$677$	−42.9480	−1.65063	−0.825313	+	0.564675i	$0.809001\pi$
$678$	0	0
$679$	49.2280	1.88920
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	0	0
$683$	−41.7200	−1.59637	−0.798186	−	0.602411i	$-0.794207\pi$
$683$	−41.7200	−1.59637	−0.798186	+	0.602411i	$0.794207\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	−17.0880	−0.651948
$688$	0	0
$689$	4.77200	0.181799
$690$	0	0
$691$	−28.1760	−1.07187	−0.535933	−	0.844260i	$-0.680040\pi$
$691$	−28.1760	−1.07187	−0.535933	+	0.844260i	$0.680040\pi$
$692$	0	0
$693$	−22.7720	−0.865037
$694$	0	0
$695$	−16.3160	−0.618901
$696$	0	0
$697$	41.8600	1.58556
$698$	0	0
$699$	−14.3160	−0.541481
$700$	0	0
$701$	39.5440	1.49356	0.746778	−	0.665073i	$-0.231600\pi$
$701$	39.5440	1.49356	0.746778	+	0.665073i	$0.231600\pi$
$702$	0	0
$703$	−28.6320	−1.07988
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	16.9120	0.636041
$708$	0	0
$709$	−14.6320	−0.549517	−0.274758	−	0.961513i	$-0.588598\pi$
$709$	−14.6320	−0.549517	−0.274758	+	0.961513i	$0.588598\pi$
$710$	0	0
$711$	−1.22800	−0.0460535
$712$	0	0
$713$	40.6320	1.52168
$714$	0	0
$715$	−4.77200	−0.178463
$716$	0	0
$717$	10.3160	0.385258
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	57.2640	2.13262
$722$	0	0
$723$	29.0880	1.08179
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	33.5440	1.24408	0.622039	−	0.782986i	$-0.286304\pi$
$727$	33.5440	1.24408	0.622039	+	0.782986i	$0.286304\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	38.1760	1.41199
$732$	0	0
$733$	−30.3160	−1.11975	−0.559874	−	0.828578i	$-0.689150\pi$
$733$	−30.3160	−1.11975	−0.559874	+	0.828578i	$0.689150\pi$
$734$	0	0
$735$	−15.7720	−0.581759
$736$	0	0
$737$	36.0000	1.32608
$738$	0	0
$739$	−21.0880	−0.775735	−0.387867	−	0.921715i	$-0.626788\pi$
$739$	−21.0880	−0.775735	−0.387867	+	0.921715i	$0.626788\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	3.22800	0.118265
$746$	0	0
$747$	−7.54400	−0.276021
$748$	0	0
$749$	−32.3160	−1.18080
$750$	0	0
$751$	−5.86001	−0.213835	−0.106917	−	0.994268i	$-0.534098\pi$
$751$	−5.86001	−0.213835	−0.106917	+	0.994268i	$0.534098\pi$
$752$	0	0
$753$	−15.0880	−0.549838
$754$	0	0
$755$	4.45600	0.162170
$756$	0	0
$757$	−7.54400	−0.274191	−0.137096	−	0.990558i	$-0.543777\pi$
$757$	−7.54400	−0.274191	−0.137096	+	0.990558i	$0.543777\pi$
$758$	0	0
$759$	32.3160	1.17300
$760$	0	0
$761$	−17.0880	−0.619440	−0.309720	−	0.950828i	$-0.600235\pi$
$761$	−17.0880	−0.619440	−0.309720	+	0.950828i	$0.600235\pi$
$762$	0	0
$763$	36.0000	1.30329
$764$	0	0
$765$	4.77200	0.172532
$766$	0	0
$767$	−3.54400	−0.127967
$768$	0	0
$769$	−54.6320	−1.97008	−0.985040	−	0.172324i	$-0.944872\pi$
$769$	−54.6320	−1.97008	−0.985040	+	0.172324i	$0.944872\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	−14.6320	−0.526277	−0.263138	−	0.964758i	$-0.584758\pi$
$773$	−14.6320	−0.526277	−0.263138	+	0.964758i	$0.584758\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	22.7720	0.816941
$778$	0	0
$779$	52.6320	1.88574
$780$	0	0
$781$	−49.2280	−1.76152
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	15.5440	0.554084	0.277042	−	0.960858i	$-0.410646\pi$
$787$	15.5440	0.554084	0.277042	+	0.960858i	$0.410646\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	81.5440	2.89937
$792$	0	0
$793$	4.77200	0.169459
$794$	0	0
$795$	4.77200	0.169245
$796$	0	0
$797$	−0.139991	−0.00495872	−0.00247936	−	0.999997i	$-0.500789\pi$
$797$	−0.139991	−0.00495872	−0.00247936	+	0.999997i	$0.500789\pi$
$798$	0	0
$799$	28.6320	1.01293
$800$	0	0
$801$	4.77200	0.168610
$802$	0	0
$803$	−47.7200	−1.68400
$804$	0	0
$805$	32.3160	1.13899
$806$	0	0
$807$	−19.5440	−0.687982
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	−35.5440	−1.24812	−0.624059	−	0.781377i	$-0.714518\pi$
$811$	−35.5440	−1.24812	−0.624059	+	0.781377i	$0.714518\pi$
$812$	0	0
$813$	−13.0880	−0.459016
$814$	0	0
$815$	22.3160	0.781696
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	−4.77200	−0.166747
$820$	0	0
$821$	−13.6840	−0.477575	−0.238787	−	0.971072i	$-0.576750\pi$
$821$	−13.6840	−0.477575	−0.238787	+	0.971072i	$0.576750\pi$
$822$	0	0
$823$	34.1760	1.19130	0.595650	−	0.803244i	$-0.296894\pi$
$823$	34.1760	1.19130	0.595650	+	0.803244i	$0.296894\pi$
$824$	0	0
$825$	−4.77200	−0.166140
$826$	0	0
$827$	51.2640	1.78262	0.891312	−	0.453390i	$-0.149785\pi$
$827$	51.2640	1.78262	0.891312	+	0.453390i	$0.149785\pi$
$828$	0	0
$829$	44.1760	1.53430	0.767148	−	0.641470i	$-0.221675\pi$
$829$	44.1760	1.53430	0.767148	+	0.641470i	$0.221675\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−75.2640	−2.60774
$834$	0	0
$835$	3.54400	0.122645
$836$	0	0
$837$	−6.00000	−0.207390
$838$	0	0
$839$	9.40401	0.324663	0.162331	−	0.986736i	$-0.448099\pi$
$839$	9.40401	0.324663	0.162331	+	0.986736i	$0.448099\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	18.0000	0.619953
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	56.1760	1.93023
$848$	0	0
$849$	19.0880	0.655099
$850$	0	0
$851$	−32.3160	−1.10778
$852$	0	0
$853$	−5.68399	−0.194616	−0.0973081	−	0.995254i	$-0.531023\pi$
$853$	−5.68399	−0.194616	−0.0973081	+	0.995254i	$0.531023\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	1.68399	0.0575242	0.0287621	−	0.999586i	$-0.490843\pi$
$857$	1.68399	0.0575242	0.0287621	+	0.999586i	$0.490843\pi$
$858$	0	0
$859$	35.4040	1.20797	0.603985	−	0.796996i	$-0.293579\pi$
$859$	35.4040	1.20797	0.603985	+	0.796996i	$0.293579\pi$
$860$	0	0
$861$	−41.8600	−1.42659
$862$	0	0
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	5.77200	0.196027
$868$	0	0
$869$	5.86001	0.198787
$870$	0	0
$871$	7.54400	0.255619
$872$	0	0
$873$	10.3160	0.349144
$874$	0	0
$875$	−4.77200	−0.161323
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	23.5440	0.794120
$880$	0	0
$881$	−9.08801	−0.306183	−0.153091	−	0.988212i	$-0.548923\pi$
$881$	−9.08801	−0.306183	−0.153091	+	0.988212i	$0.548923\pi$
$882$	0	0
$883$	−11.0880	−0.373141	−0.186571	−	0.982442i	$-0.559737\pi$
$883$	−11.0880	−0.373141	−0.186571	+	0.982442i	$0.559737\pi$
$884$	0	0
$885$	−3.54400	−0.119130
$886$	0	0
$887$	29.8600	1.00260	0.501300	−	0.865273i	$-0.332855\pi$
$887$	29.8600	1.00260	0.501300	+	0.865273i	$0.332855\pi$
$888$	0	0
$889$	45.5440	1.52750
$890$	0	0
$891$	−4.77200	−0.159868
$892$	0	0
$893$	36.0000	1.20469
$894$	0	0
$895$	6.45600	0.215800
$896$	0	0
$897$	6.77200	0.226111
$898$	0	0
$899$	−12.0000	−0.400222
$900$	0	0
$901$	22.7720	0.758645
$902$	0	0
$903$	−38.1760	−1.27042
$904$	0	0
$905$	8.77200	0.291591
$906$	0	0
$907$	0.632011	0.0209856	0.0104928	−	0.999945i	$-0.496660\pi$
$907$	0.632011	0.0209856	0.0104928	+	0.999945i	$0.496660\pi$
$908$	0	0
$909$	3.54400	0.117547
$910$	0	0
$911$	−28.0000	−0.927681	−0.463841	−	0.885919i	$-0.653529\pi$
$911$	−28.0000	−0.927681	−0.463841	+	0.885919i	$0.653529\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	0	0
$915$	4.77200	0.157758
$916$	0	0
$917$	102.808	3.39502
$918$	0	0
$919$	38.7720	1.27897	0.639485	−	0.768803i	$-0.279148\pi$
$919$	38.7720	1.27897	0.639485	+	0.768803i	$0.279148\pi$
$920$	0	0
$921$	15.8600	0.522605
$922$	0	0
$923$	−10.3160	−0.339555
$924$	0	0
$925$	4.77200	0.156902
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	−35.2280	−1.15579	−0.577897	−	0.816110i	$-0.696126\pi$
$929$	−35.2280	−1.15579	−0.577897	+	0.816110i	$0.696126\pi$
$930$	0	0
$931$	−94.6320	−3.10144
$932$	0	0
$933$	−3.08801	−0.101097
$934$	0	0
$935$	−22.7720	−0.744724
$936$	0	0
$937$	−12.1760	−0.397773	−0.198887	−	0.980023i	$-0.563733\pi$
$937$	−12.1760	−0.397773	−0.198887	+	0.980023i	$0.563733\pi$
$938$	0	0
$939$	9.08801	0.296576
$940$	0	0
$941$	−3.22800	−0.105230	−0.0526149	−	0.998615i	$-0.516756\pi$
$941$	−3.22800	−0.105230	−0.0526149	+	0.998615i	$0.516756\pi$
$942$	0	0
$943$	59.4040	1.93446
$944$	0	0
$945$	−4.77200	−0.155233
$946$	0	0
$947$	−8.45600	−0.274783	−0.137391	−	0.990517i	$-0.543872\pi$
$947$	−8.45600	−0.274783	−0.137391	+	0.990517i	$0.543872\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	−19.5440	−0.633758
$952$	0	0
$953$	56.4920	1.82996	0.914978	−	0.403504i	$-0.132208\pi$
$953$	56.4920	1.82996	0.914978	+	0.403504i	$0.132208\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	−9.54400	−0.308514
$958$	0	0
$959$	−28.6320	−0.924576
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−6.77200	−0.218225
$964$	0	0
$965$	6.31601	0.203319
$966$	0	0
$967$	−26.6320	−0.856428	−0.428214	−	0.903677i	$-0.640857\pi$
$967$	−26.6320	−0.856428	−0.428214	+	0.903677i	$0.640857\pi$
$968$	0	0
$969$	28.6320	0.919793
$970$	0	0
$971$	43.0880	1.38276	0.691380	−	0.722491i	$-0.257003\pi$
$971$	43.0880	1.38276	0.691380	+	0.722491i	$0.257003\pi$
$972$	0	0
$973$	77.8600	2.49608
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	7.54400	0.241354	0.120677	−	0.992692i	$-0.461493\pi$
$977$	7.54400	0.241354	0.120677	+	0.992692i	$0.461493\pi$
$978$	0	0
$979$	−22.7720	−0.727796
$980$	0	0
$981$	7.54400	0.240862
$982$	0	0
$983$	23.5440	0.750937	0.375469	−	0.926835i	$-0.377482\pi$
$983$	23.5440	0.750937	0.375469	+	0.926835i	$0.377482\pi$
$984$	0	0
$985$	19.5440	0.622724
$986$	0	0
$987$	−28.6320	−0.911367
$988$	0	0
$989$	54.1760	1.72270
$990$	0	0
$991$	−9.86001	−0.313214	−0.156607	−	0.987661i	$-0.550056\pi$
$991$	−9.86001	−0.313214	−0.156607	+	0.987661i	$0.550056\pi$
$992$	0	0
$993$	−35.5440	−1.12795
$994$	0	0
$995$	27.0880	0.858748
$996$	0	0
$997$	35.5440	1.12569	0.562845	−	0.826562i	$-0.309707\pi$
$997$	35.5440	1.12569	0.562845	+	0.826562i	$0.309707\pi$
$998$	0	0
$999$	4.77200	0.150979

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.be.1.2		2
3.2	odd	2		9360.2.a.cu.1.2		2
4.3	odd	2		780.2.a.e.1.1	✓	2
12.11	even	2		2340.2.a.l.1.1		2
20.3	even	4		3900.2.h.i.1249.2		4
20.7	even	4		3900.2.h.i.1249.3		4
20.19	odd	2		3900.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
780.2.a.e.1.1	✓	2	4.3	odd	2
2340.2.a.l.1.1		2	12.11	even	2
3120.2.a.be.1.2		2	1.1	even	1	trivial
3900.2.a.s.1.2		2	20.19	odd	2
3900.2.h.i.1249.2		4	20.3	even	4
3900.2.h.i.1249.3		4	20.7	even	4
9360.2.a.cu.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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.77200 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.77200 q^{7} +1.00000 q^{9} -4.77200 q^{11} -1.00000 q^{13} -1.00000 q^{15} -4.77200 q^{17} -6.00000 q^{19} +4.77200 q^{21} -6.77200 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -6.00000 q^{31} -4.77200 q^{33} -4.77200 q^{35} +4.77200 q^{37} -1.00000 q^{39} -8.77200 q^{41} -8.00000 q^{43} -1.00000 q^{45} -6.00000 q^{47} +15.7720 q^{49} -4.77200 q^{51} -4.77200 q^{53} +4.77200 q^{55} -6.00000 q^{57} +3.54400 q^{59} -4.77200 q^{61} +4.77200 q^{63} +1.00000 q^{65} -7.54400 q^{67} -6.77200 q^{69} +10.3160 q^{71} +10.0000 q^{73} +1.00000 q^{75} -22.7720 q^{77} -1.22800 q^{79} +1.00000 q^{81} -7.54400 q^{83} +4.77200 q^{85} +2.00000 q^{87} +4.77200 q^{89} -4.77200 q^{91} -6.00000 q^{93} +6.00000 q^{95} +10.3160 q^{97} -4.77200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} - q^{11} - 2 q^{13} - 2 q^{15} - q^{17} - 12 q^{19} + q^{21} - 5 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 12 q^{31} - q^{33} - q^{35} + q^{37} - 2 q^{39} - 9 q^{41} - 16 q^{43} - 2 q^{45} - 12 q^{47} + 23 q^{49} - q^{51} - q^{53} + q^{55} - 12 q^{57} - 10 q^{59} - q^{61} + q^{63} + 2 q^{65} + 2 q^{67} - 5 q^{69} - 5 q^{71} + 20 q^{73} + 2 q^{75} - 37 q^{77} - 11 q^{79} + 2 q^{81} + 2 q^{83} + q^{85} + 4 q^{87} + q^{89} - q^{91} - 12 q^{93} + 12 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 - q^11 - 2 * q^13 - 2 * q^15 - q^17 - 12 * q^19 + q^21 - 5 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 12 * q^31 - q^33 - q^35 + q^37 - 2 * q^39 - 9 * q^41 - 16 * q^43 - 2 * q^45 - 12 * q^47 + 23 * q^49 - q^51 - q^53 + q^55 - 12 * q^57 - 10 * q^59 - q^61 + q^63 + 2 * q^65 + 2 * q^67 - 5 * q^69 - 5 * q^71 + 20 * q^73 + 2 * q^75 - 37 * q^77 - 11 * q^79 + 2 * q^81 + 2 * q^83 + q^85 + 4 * q^87 + q^89 - q^91 - 12 * q^93 + 12 * q^95 - 5 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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