LMFDB - Embedded newform 6040.2.a.o.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 $ x^15 - 5x^14 - 19x^13 + 119x^12 + 106x^11 - 1063x^10 - 48x^9 + 4510x^8 - 1130x^7 - 9512x^6 + 2839x^5 + 9383x^4 - 1781x^3 - 3692x^2 - 92x + 272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$3.03078$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.03078 q^{3} +1.00000 q^{5} +3.09463 q^{7} +6.18562 q^{9} +O(q^{10})$	$q+3.03078 q^{3} +1.00000 q^{5} +3.09463 q^{7} +6.18562 q^{9} +5.41918 q^{11} -1.41705 q^{13} +3.03078 q^{15} -1.72359 q^{17} -2.75038 q^{19} +9.37912 q^{21} +1.19937 q^{23} +1.00000 q^{25} +9.65489 q^{27} -4.80816 q^{29} +1.78515 q^{31} +16.4243 q^{33} +3.09463 q^{35} +0.781778 q^{37} -4.29476 q^{39} +3.31216 q^{41} -6.64280 q^{43} +6.18562 q^{45} +1.28217 q^{47} +2.57671 q^{49} -5.22382 q^{51} +2.00520 q^{53} +5.41918 q^{55} -8.33580 q^{57} -10.6911 q^{59} +15.2393 q^{61} +19.1422 q^{63} -1.41705 q^{65} +1.39864 q^{67} +3.63503 q^{69} -1.53640 q^{71} -11.0272 q^{73} +3.03078 q^{75} +16.7703 q^{77} -0.682313 q^{79} +10.7050 q^{81} +0.300775 q^{83} -1.72359 q^{85} -14.5725 q^{87} -2.89310 q^{89} -4.38524 q^{91} +5.41038 q^{93} -2.75038 q^{95} -5.13256 q^{97} +33.5210 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * 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
$3$	3.03078	1.74982	0.874910	−	0.484285i	$-0.160920\pi$
$3$	3.03078	1.74982	0.874910	+	0.484285i	$0.160920\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.09463	1.16966	0.584829	−	0.811156i	$-0.301161\pi$
$7$	3.09463	1.16966	0.584829	+	0.811156i	$0.301161\pi$
$8$	0	0
$9$	6.18562	2.06187
$10$	0	0
$11$	5.41918	1.63394	0.816972	−	0.576677i	$-0.195651\pi$
$11$	5.41918	1.63394	0.816972	+	0.576677i	$0.195651\pi$
$12$	0	0
$13$	−1.41705	−0.393019	−0.196509	−	0.980502i	$-0.562961\pi$
$13$	−1.41705	−0.393019	−0.196509	+	0.980502i	$0.562961\pi$
$14$	0	0
$15$	3.03078	0.782544
$16$	0	0
$17$	−1.72359	−0.418032	−0.209016	−	0.977912i	$-0.567026\pi$
$17$	−1.72359	−0.418032	−0.209016	+	0.977912i	$0.567026\pi$
$18$	0	0
$19$	−2.75038	−0.630981	−0.315491	−	0.948929i	$-0.602169\pi$
$19$	−2.75038	−0.630981	−0.315491	+	0.948929i	$0.602169\pi$
$20$	0	0
$21$	9.37912	2.04669
$22$	0	0
$23$	1.19937	0.250087	0.125043	−	0.992151i	$-0.460093\pi$
$23$	1.19937	0.250087	0.125043	+	0.992151i	$0.460093\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	9.65489	1.85808
$28$	0	0
$29$	−4.80816	−0.892853	−0.446426	−	0.894820i	$-0.647304\pi$
$29$	−4.80816	−0.892853	−0.446426	+	0.894820i	$0.647304\pi$
$30$	0	0
$31$	1.78515	0.320622	0.160311	−	0.987067i	$-0.448750\pi$
$31$	1.78515	0.320622	0.160311	+	0.987067i	$0.448750\pi$
$32$	0	0
$33$	16.4243	2.85911
$34$	0	0
$35$	3.09463	0.523087
$36$	0	0
$37$	0.781778	0.128523	0.0642617	−	0.997933i	$-0.479531\pi$
$37$	0.781778	0.128523	0.0642617	+	0.997933i	$0.479531\pi$
$38$	0	0
$39$	−4.29476	−0.687713
$40$	0	0
$41$	3.31216	0.517272	0.258636	−	0.965975i	$-0.416727\pi$
$41$	3.31216	0.517272	0.258636	+	0.965975i	$0.416727\pi$
$42$	0	0
$43$	−6.64280	−1.01302	−0.506509	−	0.862235i	$-0.669064\pi$
$43$	−6.64280	−1.01302	−0.506509	+	0.862235i	$0.669064\pi$
$44$	0	0
$45$	6.18562	0.922097
$46$	0	0
$47$	1.28217	0.187024	0.0935122	−	0.995618i	$-0.470191\pi$
$47$	1.28217	0.187024	0.0935122	+	0.995618i	$0.470191\pi$
$48$	0	0
$49$	2.57671	0.368102
$50$	0	0
$51$	−5.22382	−0.731481
$52$	0	0
$53$	2.00520	0.275435	0.137717	−	0.990472i	$-0.456023\pi$
$53$	2.00520	0.275435	0.137717	+	0.990472i	$0.456023\pi$
$54$	0	0
$55$	5.41918	0.730722
$56$	0	0
$57$	−8.33580	−1.10410
$58$	0	0
$59$	−10.6911	−1.39186	−0.695930	−	0.718109i	$-0.745008\pi$
$59$	−10.6911	−1.39186	−0.695930	+	0.718109i	$0.745008\pi$
$60$	0	0
$61$	15.2393	1.95119	0.975596	−	0.219571i	$-0.0704659\pi$
$61$	15.2393	1.95119	0.975596	+	0.219571i	$0.0704659\pi$
$62$	0	0
$63$	19.1422	2.41169
$64$	0	0
$65$	−1.41705	−0.175763
$66$	0	0
$67$	1.39864	0.170871	0.0854356	−	0.996344i	$-0.472772\pi$
$67$	1.39864	0.170871	0.0854356	+	0.996344i	$0.472772\pi$
$68$	0	0
$69$	3.63503	0.437607
$70$	0	0
$71$	−1.53640	−0.182337	−0.0911687	−	0.995835i	$-0.529060\pi$
$71$	−1.53640	−0.182337	−0.0911687	+	0.995835i	$0.529060\pi$
$72$	0	0
$73$	−11.0272	−1.29064	−0.645320	−	0.763912i	$-0.723276\pi$
$73$	−11.0272	−1.29064	−0.645320	+	0.763912i	$0.723276\pi$
$74$	0	0
$75$	3.03078	0.349964
$76$	0	0
$77$	16.7703	1.91116
$78$	0	0
$79$	−0.682313	−0.0767662	−0.0383831	−	0.999263i	$-0.512221\pi$
$79$	−0.682313	−0.0767662	−0.0383831	+	0.999263i	$0.512221\pi$
$80$	0	0
$81$	10.7050	1.18944
$82$	0	0
$83$	0.300775	0.0330143	0.0165072	−	0.999864i	$-0.494745\pi$
$83$	0.300775	0.0330143	0.0165072	+	0.999864i	$0.494745\pi$
$84$	0	0
$85$	−1.72359	−0.186949
$86$	0	0
$87$	−14.5725	−1.56233
$88$	0	0
$89$	−2.89310	−0.306668	−0.153334	−	0.988174i	$-0.549001\pi$
$89$	−2.89310	−0.306668	−0.153334	+	0.988174i	$0.549001\pi$
$90$	0	0
$91$	−4.38524	−0.459698
$92$	0	0
$93$	5.41038	0.561031
$94$	0	0
$95$	−2.75038	−0.282183
$96$	0	0
$97$	−5.13256	−0.521133	−0.260566	−	0.965456i	$-0.583909\pi$
$97$	−5.13256	−0.521133	−0.260566	+	0.965456i	$0.583909\pi$
$98$	0	0
$99$	33.5210	3.36898
$100$	0	0
$101$	−2.36866	−0.235690	−0.117845	−	0.993032i	$-0.537599\pi$
$101$	−2.36866	−0.235690	−0.117845	+	0.993032i	$0.537599\pi$
$102$	0	0
$103$	−2.82072	−0.277933	−0.138967	−	0.990297i	$-0.544378\pi$
$103$	−2.82072	−0.277933	−0.138967	+	0.990297i	$0.544378\pi$
$104$	0	0
$105$	9.37912	0.915309
$106$	0	0
$107$	−1.01058	−0.0976960	−0.0488480	−	0.998806i	$-0.515555\pi$
$107$	−1.01058	−0.0976960	−0.0488480	+	0.998806i	$0.515555\pi$
$108$	0	0
$109$	8.44639	0.809018	0.404509	−	0.914534i	$-0.367442\pi$
$109$	8.44639	0.809018	0.404509	+	0.914534i	$0.367442\pi$
$110$	0	0
$111$	2.36940	0.224893
$112$	0	0
$113$	−6.22601	−0.585693	−0.292847	−	0.956159i	$-0.594603\pi$
$113$	−6.22601	−0.585693	−0.292847	+	0.956159i	$0.594603\pi$
$114$	0	0
$115$	1.19937	0.111842
$116$	0	0
$117$	−8.76533	−0.810355
$118$	0	0
$119$	−5.33386	−0.488955
$120$	0	0
$121$	18.3675	1.66977
$122$	0	0
$123$	10.0384	0.905134
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.26611	−0.112349	−0.0561746	−	0.998421i	$-0.517890\pi$
$127$	−1.26611	−0.112349	−0.0561746	+	0.998421i	$0.517890\pi$
$128$	0	0
$129$	−20.1328	−1.77260
$130$	0	0
$131$	−20.0427	−1.75114	−0.875569	−	0.483093i	$-0.839513\pi$
$131$	−20.0427	−1.75114	−0.875569	+	0.483093i	$0.839513\pi$
$132$	0	0
$133$	−8.51141	−0.738033
$134$	0	0
$135$	9.65489	0.830961
$136$	0	0
$137$	18.7135	1.59881	0.799403	−	0.600795i	$-0.205149\pi$
$137$	18.7135	1.59881	0.799403	+	0.600795i	$0.205149\pi$
$138$	0	0
$139$	15.0588	1.27727	0.638637	−	0.769508i	$-0.279499\pi$
$139$	15.0588	1.27727	0.638637	+	0.769508i	$0.279499\pi$
$140$	0	0
$141$	3.88599	0.327259
$142$	0	0
$143$	−7.67925	−0.642171
$144$	0	0
$145$	−4.80816	−0.399296
$146$	0	0
$147$	7.80944	0.644112
$148$	0	0
$149$	11.8276	0.968957	0.484478	−	0.874803i	$-0.339009\pi$
$149$	11.8276	0.968957	0.484478	+	0.874803i	$0.339009\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−10.6615	−0.861928
$154$	0	0
$155$	1.78515	0.143386
$156$	0	0
$157$	−12.3132	−0.982701	−0.491351	−	0.870962i	$-0.663497\pi$
$157$	−12.3132	−0.982701	−0.491351	+	0.870962i	$0.663497\pi$
$158$	0	0
$159$	6.07730	0.481962
$160$	0	0
$161$	3.71161	0.292516
$162$	0	0
$163$	−18.9737	−1.48613	−0.743067	−	0.669217i	$-0.766630\pi$
$163$	−18.9737	−1.48613	−0.743067	+	0.669217i	$0.766630\pi$
$164$	0	0
$165$	16.4243	1.27863
$166$	0	0
$167$	1.90713	0.147578	0.0737890	−	0.997274i	$-0.476491\pi$
$167$	1.90713	0.147578	0.0737890	+	0.997274i	$0.476491\pi$
$168$	0	0
$169$	−10.9920	−0.845536
$170$	0	0
$171$	−17.0128	−1.30100
$172$	0	0
$173$	−13.5173	−1.02770	−0.513851	−	0.857880i	$-0.671782\pi$
$173$	−13.5173	−1.02770	−0.513851	+	0.857880i	$0.671782\pi$
$174$	0	0
$175$	3.09463	0.233932
$176$	0	0
$177$	−32.4023	−2.43551
$178$	0	0
$179$	−12.0824	−0.903084	−0.451542	−	0.892250i	$-0.649126\pi$
$179$	−12.0824	−0.903084	−0.451542	+	0.892250i	$0.649126\pi$
$180$	0	0
$181$	−2.49892	−0.185743	−0.0928716	−	0.995678i	$-0.529605\pi$
$181$	−2.49892	−0.185743	−0.0928716	+	0.995678i	$0.529605\pi$
$182$	0	0
$183$	46.1869	3.41424
$184$	0	0
$185$	0.781778	0.0574775
$186$	0	0
$187$	−9.34044	−0.683041
$188$	0	0
$189$	29.8783	2.17333
$190$	0	0
$191$	−16.6392	−1.20397	−0.601987	−	0.798506i	$-0.705624\pi$
$191$	−16.6392	−1.20397	−0.601987	+	0.798506i	$0.705624\pi$
$192$	0	0
$193$	−17.6495	−1.27044	−0.635219	−	0.772332i	$-0.719090\pi$
$193$	−17.6495	−1.27044	−0.635219	+	0.772332i	$0.719090\pi$
$194$	0	0
$195$	−4.29476	−0.307554
$196$	0	0
$197$	−1.29775	−0.0924608	−0.0462304	−	0.998931i	$-0.514721\pi$
$197$	−1.29775	−0.0924608	−0.0462304	+	0.998931i	$0.514721\pi$
$198$	0	0
$199$	8.39272	0.594944	0.297472	−	0.954730i	$-0.403856\pi$
$199$	8.39272	0.594944	0.297472	+	0.954730i	$0.403856\pi$
$200$	0	0
$201$	4.23897	0.298994
$202$	0	0
$203$	−14.8795	−1.04433
$204$	0	0
$205$	3.31216	0.231331
$206$	0	0
$207$	7.41886	0.515646
$208$	0	0
$209$	−14.9048	−1.03099
$210$	0	0
$211$	9.09943	0.626431	0.313215	−	0.949682i	$-0.398594\pi$
$211$	9.09943	0.626431	0.313215	+	0.949682i	$0.398594\pi$
$212$	0	0
$213$	−4.65650	−0.319058
$214$	0	0
$215$	−6.64280	−0.453035
$216$	0	0
$217$	5.52436	0.375018
$218$	0	0
$219$	−33.4211	−2.25839
$220$	0	0
$221$	2.44241	0.164294
$222$	0	0
$223$	−26.0186	−1.74233	−0.871167	−	0.490987i	$-0.836636\pi$
$223$	−26.0186	−1.74233	−0.871167	+	0.490987i	$0.836636\pi$
$224$	0	0
$225$	6.18562	0.412374
$226$	0	0
$227$	7.20219	0.478026	0.239013	−	0.971016i	$-0.423176\pi$
$227$	7.20219	0.478026	0.239013	+	0.971016i	$0.423176\pi$
$228$	0	0
$229$	3.43748	0.227155	0.113578	−	0.993529i	$-0.463769\pi$
$229$	3.43748	0.227155	0.113578	+	0.993529i	$0.463769\pi$
$230$	0	0
$231$	50.8272	3.34418
$232$	0	0
$233$	−12.0992	−0.792643	−0.396321	−	0.918112i	$-0.629713\pi$
$233$	−12.0992	−0.792643	−0.396321	+	0.918112i	$0.629713\pi$
$234$	0	0
$235$	1.28217	0.0836399
$236$	0	0
$237$	−2.06794	−0.134327
$238$	0	0
$239$	11.4121	0.738190	0.369095	−	0.929392i	$-0.379668\pi$
$239$	11.4121	0.738190	0.369095	+	0.929392i	$0.379668\pi$
$240$	0	0
$241$	13.6491	0.879215	0.439607	−	0.898190i	$-0.355118\pi$
$241$	13.6491	0.879215	0.439607	+	0.898190i	$0.355118\pi$
$242$	0	0
$243$	3.47977	0.223227
$244$	0	0
$245$	2.57671	0.164620
$246$	0	0
$247$	3.89743	0.247988
$248$	0	0
$249$	0.911581	0.0577691
$250$	0	0
$251$	−21.6637	−1.36740	−0.683701	−	0.729762i	$-0.739631\pi$
$251$	−21.6637	−1.36740	−0.683701	+	0.729762i	$0.739631\pi$
$252$	0	0
$253$	6.49962	0.408628
$254$	0	0
$255$	−5.22382	−0.327128
$256$	0	0
$257$	3.42757	0.213806	0.106903	−	0.994269i	$-0.465907\pi$
$257$	3.42757	0.213806	0.106903	+	0.994269i	$0.465907\pi$
$258$	0	0
$259$	2.41931	0.150329
$260$	0	0
$261$	−29.7414	−1.84095
$262$	0	0
$263$	7.92507	0.488681	0.244340	−	0.969690i	$-0.421429\pi$
$263$	7.92507	0.488681	0.244340	+	0.969690i	$0.421429\pi$
$264$	0	0
$265$	2.00520	0.123178
$266$	0	0
$267$	−8.76833	−0.536613
$268$	0	0
$269$	32.4896	1.98093	0.990463	−	0.137781i	$-0.0439971\pi$
$269$	32.4896	1.98093	0.990463	+	0.137781i	$0.0439971\pi$
$270$	0	0
$271$	8.08602	0.491190	0.245595	−	0.969372i	$-0.421017\pi$
$271$	8.08602	0.491190	0.245595	+	0.969372i	$0.421017\pi$
$272$	0	0
$273$	−13.2907	−0.804389
$274$	0	0
$275$	5.41918	0.326789
$276$	0	0
$277$	31.3114	1.88132	0.940660	−	0.339351i	$-0.110207\pi$
$277$	31.3114	1.88132	0.940660	+	0.339351i	$0.110207\pi$
$278$	0	0
$279$	11.0422	0.661081
$280$	0	0
$281$	9.91669	0.591580	0.295790	−	0.955253i	$-0.404417\pi$
$281$	9.91669	0.591580	0.295790	+	0.955253i	$0.404417\pi$
$282$	0	0
$283$	28.6758	1.70460	0.852298	−	0.523056i	$-0.175208\pi$
$283$	28.6758	1.70460	0.852298	+	0.523056i	$0.175208\pi$
$284$	0	0
$285$	−8.33580	−0.493770
$286$	0	0
$287$	10.2499	0.605032
$288$	0	0
$289$	−14.0292	−0.825249
$290$	0	0
$291$	−15.5557	−0.911889
$292$	0	0
$293$	25.4493	1.48676	0.743381	−	0.668868i	$-0.233221\pi$
$293$	25.4493	1.48676	0.743381	+	0.668868i	$0.233221\pi$
$294$	0	0
$295$	−10.6911	−0.622459
$296$	0	0
$297$	52.3216	3.03601
$298$	0	0
$299$	−1.69957	−0.0982888
$300$	0	0
$301$	−20.5570	−1.18488
$302$	0	0
$303$	−7.17888	−0.412416
$304$	0	0
$305$	15.2393	0.872600
$306$	0	0
$307$	20.7837	1.18619	0.593095	−	0.805132i	$-0.297906\pi$
$307$	20.7837	1.18619	0.593095	+	0.805132i	$0.297906\pi$
$308$	0	0
$309$	−8.54896	−0.486333
$310$	0	0
$311$	9.65365	0.547408	0.273704	−	0.961814i	$-0.411751\pi$
$311$	9.65365	0.547408	0.273704	+	0.961814i	$0.411751\pi$
$312$	0	0
$313$	8.53952	0.482682	0.241341	−	0.970440i	$-0.422413\pi$
$313$	8.53952	0.482682	0.241341	+	0.970440i	$0.422413\pi$
$314$	0	0
$315$	19.1422	1.07854
$316$	0	0
$317$	−24.9613	−1.40196	−0.700982	−	0.713179i	$-0.747255\pi$
$317$	−24.9613	−1.40196	−0.700982	+	0.713179i	$0.747255\pi$
$318$	0	0
$319$	−26.0563	−1.45887
$320$	0	0
$321$	−3.06283	−0.170950
$322$	0	0
$323$	4.74053	0.263770
$324$	0	0
$325$	−1.41705	−0.0786038
$326$	0	0
$327$	25.5991	1.41564
$328$	0	0
$329$	3.96785	0.218755
$330$	0	0
$331$	9.73208	0.534923	0.267462	−	0.963569i	$-0.413815\pi$
$331$	9.73208	0.534923	0.267462	+	0.963569i	$0.413815\pi$
$332$	0	0
$333$	4.83578	0.264999
$334$	0	0
$335$	1.39864	0.0764159
$336$	0	0
$337$	16.2054	0.882766	0.441383	−	0.897319i	$-0.354488\pi$
$337$	16.2054	0.882766	0.441383	+	0.897319i	$0.354488\pi$
$338$	0	0
$339$	−18.8696	−1.02486
$340$	0	0
$341$	9.67403	0.523878
$342$	0	0
$343$	−13.6884	−0.739106
$344$	0	0
$345$	3.63503	0.195704
$346$	0	0
$347$	−16.4764	−0.884499	−0.442250	−	0.896892i	$-0.645819\pi$
$347$	−16.4764	−0.884499	−0.442250	+	0.896892i	$0.645819\pi$
$348$	0	0
$349$	−5.52688	−0.295847	−0.147924	−	0.988999i	$-0.547259\pi$
$349$	−5.52688	−0.295847	−0.147924	+	0.988999i	$0.547259\pi$
$350$	0	0
$351$	−13.6815	−0.730263
$352$	0	0
$353$	−15.6064	−0.830645	−0.415323	−	0.909674i	$-0.636331\pi$
$353$	−15.6064	−0.830645	−0.415323	+	0.909674i	$0.636331\pi$
$354$	0	0
$355$	−1.53640	−0.0815438
$356$	0	0
$357$	−16.1658	−0.855583
$358$	0	0
$359$	−3.74173	−0.197481	−0.0987406	−	0.995113i	$-0.531481\pi$
$359$	−3.74173	−0.197481	−0.0987406	+	0.995113i	$0.531481\pi$
$360$	0	0
$361$	−11.4354	−0.601863
$362$	0	0
$363$	55.6678	2.92180
$364$	0	0
$365$	−11.0272	−0.577192
$366$	0	0
$367$	1.72539	0.0900647	0.0450323	−	0.998986i	$-0.485661\pi$
$367$	1.72539	0.0900647	0.0450323	+	0.998986i	$0.485661\pi$
$368$	0	0
$369$	20.4877	1.06655
$370$	0	0
$371$	6.20533	0.322165
$372$	0	0
$373$	−3.12202	−0.161652	−0.0808261	−	0.996728i	$-0.525756\pi$
$373$	−3.12202	−0.161652	−0.0808261	+	0.996728i	$0.525756\pi$
$374$	0	0
$375$	3.03078	0.156509
$376$	0	0
$377$	6.81340	0.350908
$378$	0	0
$379$	−1.14664	−0.0588989	−0.0294495	−	0.999566i	$-0.509375\pi$
$379$	−1.14664	−0.0588989	−0.0294495	+	0.999566i	$0.509375\pi$
$380$	0	0
$381$	−3.83730	−0.196591
$382$	0	0
$383$	16.0580	0.820524	0.410262	−	0.911968i	$-0.365437\pi$
$383$	16.0580	0.820524	0.410262	+	0.911968i	$0.365437\pi$
$384$	0	0
$385$	16.7703	0.854695
$386$	0	0
$387$	−41.0898	−2.08871
$388$	0	0
$389$	28.5795	1.44904	0.724518	−	0.689256i	$-0.242062\pi$
$389$	28.5795	1.44904	0.724518	+	0.689256i	$0.242062\pi$
$390$	0	0
$391$	−2.06723	−0.104544
$392$	0	0
$393$	−60.7449	−3.06418
$394$	0	0
$395$	−0.682313	−0.0343309
$396$	0	0
$397$	−5.41796	−0.271919	−0.135960	−	0.990714i	$-0.543412\pi$
$397$	−5.41796	−0.271919	−0.135960	+	0.990714i	$0.543412\pi$
$398$	0	0
$399$	−25.7962	−1.29142
$400$	0	0
$401$	−7.95934	−0.397471	−0.198735	−	0.980053i	$-0.563683\pi$
$401$	−7.95934	−0.397471	−0.198735	+	0.980053i	$0.563683\pi$
$402$	0	0
$403$	−2.52964	−0.126010
$404$	0	0
$405$	10.7050	0.531935
$406$	0	0
$407$	4.23660	0.210000
$408$	0	0
$409$	−10.2725	−0.507944	−0.253972	−	0.967212i	$-0.581737\pi$
$409$	−10.2725	−0.507944	−0.253972	+	0.967212i	$0.581737\pi$
$410$	0	0
$411$	56.7166	2.79762
$412$	0	0
$413$	−33.0849	−1.62800
$414$	0	0
$415$	0.300775	0.0147644
$416$	0	0
$417$	45.6400	2.23500
$418$	0	0
$419$	19.9851	0.976336	0.488168	−	0.872750i	$-0.337665\pi$
$419$	19.9851	0.976336	0.488168	+	0.872750i	$0.337665\pi$
$420$	0	0
$421$	−26.5754	−1.29521	−0.647603	−	0.761978i	$-0.724229\pi$
$421$	−26.5754	−1.29521	−0.647603	+	0.761978i	$0.724229\pi$
$422$	0	0
$423$	7.93104	0.385620
$424$	0	0
$425$	−1.72359	−0.0836064
$426$	0	0
$427$	47.1599	2.28223
$428$	0	0
$429$	−23.2741	−1.12368
$430$	0	0
$431$	−4.70177	−0.226476	−0.113238	−	0.993568i	$-0.536122\pi$
$431$	−4.70177	−0.226476	−0.113238	+	0.993568i	$0.536122\pi$
$432$	0	0
$433$	−3.09422	−0.148699	−0.0743494	−	0.997232i	$-0.523688\pi$
$433$	−3.09422	−0.148699	−0.0743494	+	0.997232i	$0.523688\pi$
$434$	0	0
$435$	−14.5725	−0.698696
$436$	0	0
$437$	−3.29874	−0.157800
$438$	0	0
$439$	−26.7957	−1.27889	−0.639444	−	0.768838i	$-0.720835\pi$
$439$	−26.7957	−1.27889	−0.639444	+	0.768838i	$0.720835\pi$
$440$	0	0
$441$	15.9385	0.758978
$442$	0	0
$443$	33.8176	1.60672	0.803362	−	0.595491i	$-0.203043\pi$
$443$	33.8176	1.60672	0.803362	+	0.595491i	$0.203043\pi$
$444$	0	0
$445$	−2.89310	−0.137146
$446$	0	0
$447$	35.8469	1.69550
$448$	0	0
$449$	14.2958	0.674662	0.337331	−	0.941386i	$-0.390476\pi$
$449$	14.2958	0.674662	0.337331	+	0.941386i	$0.390476\pi$
$450$	0	0
$451$	17.9492	0.845194
$452$	0	0
$453$	−3.03078	−0.142398
$454$	0	0
$455$	−4.38524	−0.205583
$456$	0	0
$457$	5.99090	0.280243	0.140121	−	0.990134i	$-0.455251\pi$
$457$	5.99090	0.280243	0.140121	+	0.990134i	$0.455251\pi$
$458$	0	0
$459$	−16.6411	−0.776739
$460$	0	0
$461$	−4.80230	−0.223666	−0.111833	−	0.993727i	$-0.535672\pi$
$461$	−4.80230	−0.223666	−0.111833	+	0.993727i	$0.535672\pi$
$462$	0	0
$463$	4.83136	0.224532	0.112266	−	0.993678i	$-0.464189\pi$
$463$	4.83136	0.224532	0.112266	+	0.993678i	$0.464189\pi$
$464$	0	0
$465$	5.41038	0.250901
$466$	0	0
$467$	−1.56083	−0.0722266	−0.0361133	−	0.999348i	$-0.511498\pi$
$467$	−1.56083	−0.0722266	−0.0361133	+	0.999348i	$0.511498\pi$
$468$	0	0
$469$	4.32827	0.199861
$470$	0	0
$471$	−37.3186	−1.71955
$472$	0	0
$473$	−35.9985	−1.65521
$474$	0	0
$475$	−2.75038	−0.126196
$476$	0	0
$477$	12.4034	0.567911
$478$	0	0
$479$	−21.6163	−0.987674	−0.493837	−	0.869554i	$-0.664406\pi$
$479$	−21.6163	−0.987674	−0.493837	+	0.869554i	$0.664406\pi$
$480$	0	0
$481$	−1.10782	−0.0505122
$482$	0	0
$483$	11.2491	0.511850
$484$	0	0
$485$	−5.13256	−0.233058
$486$	0	0
$487$	20.1968	0.915206	0.457603	−	0.889157i	$-0.348708\pi$
$487$	20.1968	0.915206	0.457603	+	0.889157i	$0.348708\pi$
$488$	0	0
$489$	−57.5050	−2.60047
$490$	0	0
$491$	−8.73029	−0.393992	−0.196996	−	0.980404i	$-0.563119\pi$
$491$	−8.73029	−0.393992	−0.196996	+	0.980404i	$0.563119\pi$
$492$	0	0
$493$	8.28729	0.373241
$494$	0	0
$495$	33.5210	1.50666
$496$	0	0
$497$	−4.75459	−0.213273
$498$	0	0
$499$	−35.0505	−1.56908	−0.784538	−	0.620080i	$-0.787100\pi$
$499$	−35.0505	−1.56908	−0.784538	+	0.620080i	$0.787100\pi$
$500$	0	0
$501$	5.78008	0.258235
$502$	0	0
$503$	31.1432	1.38861	0.694304	−	0.719681i	$-0.255712\pi$
$503$	31.1432	1.38861	0.694304	+	0.719681i	$0.255712\pi$
$504$	0	0
$505$	−2.36866	−0.105404
$506$	0	0
$507$	−33.3142	−1.47954
$508$	0	0
$509$	−4.22002	−0.187049	−0.0935246	−	0.995617i	$-0.529813\pi$
$509$	−4.22002	−0.187049	−0.0935246	+	0.995617i	$0.529813\pi$
$510$	0	0
$511$	−34.1252	−1.50961
$512$	0	0
$513$	−26.5547	−1.17242
$514$	0	0
$515$	−2.82072	−0.124296
$516$	0	0
$517$	6.94834	0.305587
$518$	0	0
$519$	−40.9679	−1.79829
$520$	0	0
$521$	18.0776	0.791995	0.395997	−	0.918252i	$-0.370399\pi$
$521$	18.0776	0.791995	0.395997	+	0.918252i	$0.370399\pi$
$522$	0	0
$523$	−1.86352	−0.0814860	−0.0407430	−	0.999170i	$-0.512972\pi$
$523$	−1.86352	−0.0814860	−0.0407430	+	0.999170i	$0.512972\pi$
$524$	0	0
$525$	9.37912	0.409339
$526$	0	0
$527$	−3.07686	−0.134030
$528$	0	0
$529$	−21.5615	−0.937457
$530$	0	0
$531$	−66.1309	−2.86984
$532$	0	0
$533$	−4.69349	−0.203298
$534$	0	0
$535$	−1.01058	−0.0436910
$536$	0	0
$537$	−36.6192	−1.58024
$538$	0	0
$539$	13.9637	0.601457
$540$	0	0
$541$	1.40127	0.0602454	0.0301227	−	0.999546i	$-0.490410\pi$
$541$	1.40127	0.0602454	0.0301227	+	0.999546i	$0.490410\pi$
$542$	0	0
$543$	−7.57367	−0.325017
$544$	0	0
$545$	8.44639	0.361804
$546$	0	0
$547$	19.8559	0.848976	0.424488	−	0.905433i	$-0.360454\pi$
$547$	19.8559	0.848976	0.424488	+	0.905433i	$0.360454\pi$
$548$	0	0
$549$	94.2645	4.02311
$550$	0	0
$551$	13.2243	0.563373
$552$	0	0
$553$	−2.11150	−0.0897903
$554$	0	0
$555$	2.36940	0.100575
$556$	0	0
$557$	5.64088	0.239012	0.119506	−	0.992833i	$-0.461869\pi$
$557$	5.64088	0.239012	0.119506	+	0.992833i	$0.461869\pi$
$558$	0	0
$559$	9.41318	0.398135
$560$	0	0
$561$	−28.3088	−1.19520
$562$	0	0
$563$	9.71512	0.409443	0.204722	−	0.978820i	$-0.434371\pi$
$563$	9.71512	0.409443	0.204722	+	0.978820i	$0.434371\pi$
$564$	0	0
$565$	−6.22601	−0.261930
$566$	0	0
$567$	33.1279	1.39124
$568$	0	0
$569$	17.7644	0.744722	0.372361	−	0.928088i	$-0.378548\pi$
$569$	17.7644	0.744722	0.372361	+	0.928088i	$0.378548\pi$
$570$	0	0
$571$	24.4883	1.02480	0.512401	−	0.858746i	$-0.328756\pi$
$571$	24.4883	1.02480	0.512401	+	0.858746i	$0.328756\pi$
$572$	0	0
$573$	−50.4299	−2.10674
$574$	0	0
$575$	1.19937	0.0500173
$576$	0	0
$577$	−10.1062	−0.420725	−0.210362	−	0.977623i	$-0.567464\pi$
$577$	−10.1062	−0.420725	−0.210362	+	0.977623i	$0.567464\pi$
$578$	0	0
$579$	−53.4917	−2.22304
$580$	0	0
$581$	0.930785	0.0386155
$582$	0	0
$583$	10.8665	0.450045
$584$	0	0
$585$	−8.76533	−0.362402
$586$	0	0
$587$	9.54277	0.393872	0.196936	−	0.980416i	$-0.436901\pi$
$587$	9.54277	0.393872	0.196936	+	0.980416i	$0.436901\pi$
$588$	0	0
$589$	−4.90984	−0.202306
$590$	0	0
$591$	−3.93319	−0.161790
$592$	0	0
$593$	10.5554	0.433458	0.216729	−	0.976232i	$-0.430461\pi$
$593$	10.5554	0.433458	0.216729	+	0.976232i	$0.430461\pi$
$594$	0	0
$595$	−5.33386	−0.218667
$596$	0	0
$597$	25.4365	1.04105
$598$	0	0
$599$	−37.3924	−1.52781	−0.763906	−	0.645328i	$-0.776721\pi$
$599$	−37.3924	−1.52781	−0.763906	+	0.645328i	$0.776721\pi$
$600$	0	0
$601$	17.9190	0.730932	0.365466	−	0.930825i	$-0.380910\pi$
$601$	17.9190	0.730932	0.365466	+	0.930825i	$0.380910\pi$
$602$	0	0
$603$	8.65146	0.352315
$604$	0	0
$605$	18.3675	0.746746
$606$	0	0
$607$	−5.29570	−0.214946	−0.107473	−	0.994208i	$-0.534276\pi$
$607$	−5.29570	−0.214946	−0.107473	+	0.994208i	$0.534276\pi$
$608$	0	0
$609$	−45.0963	−1.82740
$610$	0	0
$611$	−1.81691	−0.0735041
$612$	0	0
$613$	13.7632	0.555891	0.277946	−	0.960597i	$-0.410347\pi$
$613$	13.7632	0.555891	0.277946	+	0.960597i	$0.410347\pi$
$614$	0	0
$615$	10.0384	0.404788
$616$	0	0
$617$	−26.0647	−1.04932	−0.524662	−	0.851310i	$-0.675808\pi$
$617$	−26.0647	−1.04932	−0.524662	+	0.851310i	$0.675808\pi$
$618$	0	0
$619$	−8.44129	−0.339284	−0.169642	−	0.985506i	$-0.554261\pi$
$619$	−8.44129	−0.339284	−0.169642	+	0.985506i	$0.554261\pi$
$620$	0	0
$621$	11.5798	0.464682
$622$	0	0
$623$	−8.95305	−0.358697
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−45.1732	−1.80404
$628$	0	0
$629$	−1.34746	−0.0537269
$630$	0	0
$631$	20.7769	0.827114	0.413557	−	0.910478i	$-0.364286\pi$
$631$	20.7769	0.827114	0.413557	+	0.910478i	$0.364286\pi$
$632$	0	0
$633$	27.5784	1.09614
$634$	0	0
$635$	−1.26611	−0.0502441
$636$	0	0
$637$	−3.65133	−0.144671
$638$	0	0
$639$	−9.50360	−0.375956
$640$	0	0
$641$	1.07389	0.0424162	0.0212081	−	0.999775i	$-0.493249\pi$
$641$	1.07389	0.0424162	0.0212081	+	0.999775i	$0.493249\pi$
$642$	0	0
$643$	−31.9492	−1.25995	−0.629977	−	0.776614i	$-0.716936\pi$
$643$	−31.9492	−1.25995	−0.629977	+	0.776614i	$0.716936\pi$
$644$	0	0
$645$	−20.1328	−0.792730
$646$	0	0
$647$	32.2620	1.26835	0.634175	−	0.773190i	$-0.281340\pi$
$647$	32.2620	1.26835	0.634175	+	0.773190i	$0.281340\pi$
$648$	0	0
$649$	−57.9369	−2.27422
$650$	0	0
$651$	16.7431	0.656215
$652$	0	0
$653$	32.5599	1.27417	0.637084	−	0.770794i	$-0.280140\pi$
$653$	32.5599	1.27417	0.637084	+	0.770794i	$0.280140\pi$
$654$	0	0
$655$	−20.0427	−0.783133
$656$	0	0
$657$	−68.2102	−2.66113
$658$	0	0
$659$	−37.7144	−1.46915	−0.734573	−	0.678530i	$-0.762617\pi$
$659$	−37.7144	−1.46915	−0.734573	+	0.678530i	$0.762617\pi$
$660$	0	0
$661$	13.6592	0.531282	0.265641	−	0.964072i	$-0.414416\pi$
$661$	13.6592	0.531282	0.265641	+	0.964072i	$0.414416\pi$
$662$	0	0
$663$	7.40241	0.287486
$664$	0	0
$665$	−8.51141	−0.330058
$666$	0	0
$667$	−5.76678	−0.223291
$668$	0	0
$669$	−78.8566	−3.04877
$670$	0	0
$671$	82.5845	3.18814
$672$	0	0
$673$	−7.38718	−0.284755	−0.142377	−	0.989812i	$-0.545475\pi$
$673$	−7.38718	−0.284755	−0.142377	+	0.989812i	$0.545475\pi$
$674$	0	0
$675$	9.65489	0.371617
$676$	0	0
$677$	−13.1513	−0.505444	−0.252722	−	0.967539i	$-0.581326\pi$
$677$	−13.1513	−0.505444	−0.252722	+	0.967539i	$0.581326\pi$
$678$	0	0
$679$	−15.8834	−0.609548
$680$	0	0
$681$	21.8282	0.836460
$682$	0	0
$683$	6.08907	0.232992	0.116496	−	0.993191i	$-0.462834\pi$
$683$	6.08907	0.232992	0.116496	+	0.993191i	$0.462834\pi$
$684$	0	0
$685$	18.7135	0.715008
$686$	0	0
$687$	10.4182	0.397481
$688$	0	0
$689$	−2.84146	−0.108251
$690$	0	0
$691$	−43.0682	−1.63839	−0.819195	−	0.573515i	$-0.805579\pi$
$691$	−43.0682	−1.63839	−0.819195	+	0.573515i	$0.805579\pi$
$692$	0	0
$693$	103.735	3.94056
$694$	0	0
$695$	15.0588	0.571214
$696$	0	0
$697$	−5.70880	−0.216236
$698$	0	0
$699$	−36.6699	−1.38698
$700$	0	0
$701$	−31.2841	−1.18159	−0.590793	−	0.806823i	$-0.701185\pi$
$701$	−31.2841	−1.18159	−0.590793	+	0.806823i	$0.701185\pi$
$702$	0	0
$703$	−2.15019	−0.0810959
$704$	0	0
$705$	3.88599	0.146355
$706$	0	0
$707$	−7.33011	−0.275677
$708$	0	0
$709$	−29.6724	−1.11437	−0.557186	−	0.830388i	$-0.688119\pi$
$709$	−29.6724	−1.11437	−0.557186	+	0.830388i	$0.688119\pi$
$710$	0	0
$711$	−4.22053	−0.158282
$712$	0	0
$713$	2.14106	0.0801832
$714$	0	0
$715$	−7.67925	−0.287188
$716$	0	0
$717$	34.5877	1.29170
$718$	0	0
$719$	34.8316	1.29900	0.649500	−	0.760362i	$-0.274978\pi$
$719$	34.8316	1.29900	0.649500	+	0.760362i	$0.274978\pi$
$720$	0	0
$721$	−8.72906	−0.325087
$722$	0	0
$723$	41.3673	1.53847
$724$	0	0
$725$	−4.80816	−0.178571
$726$	0	0
$727$	−31.3443	−1.16250	−0.581248	−	0.813726i	$-0.697435\pi$
$727$	−31.3443	−1.16250	−0.581248	+	0.813726i	$0.697435\pi$
$728$	0	0
$729$	−21.5686	−0.798835
$730$	0	0
$731$	11.4495	0.423473
$732$	0	0
$733$	−42.8777	−1.58372	−0.791862	−	0.610700i	$-0.790888\pi$
$733$	−42.8777	−1.58372	−0.791862	+	0.610700i	$0.790888\pi$
$734$	0	0
$735$	7.80944	0.288056
$736$	0	0
$737$	7.57949	0.279194
$738$	0	0
$739$	46.0530	1.69409	0.847044	−	0.531523i	$-0.178380\pi$
$739$	46.0530	1.69409	0.847044	+	0.531523i	$0.178380\pi$
$740$	0	0
$741$	11.8122	0.433934
$742$	0	0
$743$	37.7175	1.38372	0.691860	−	0.722031i	$-0.256791\pi$
$743$	37.7175	1.38372	0.691860	+	0.722031i	$0.256791\pi$
$744$	0	0
$745$	11.8276	0.433331
$746$	0	0
$747$	1.86048	0.0680713
$748$	0	0
$749$	−3.12735	−0.114271
$750$	0	0
$751$	48.8223	1.78155	0.890776	−	0.454443i	$-0.150162\pi$
$751$	48.8223	1.78155	0.890776	+	0.454443i	$0.150162\pi$
$752$	0	0
$753$	−65.6580	−2.39271
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−28.9942	−1.05381	−0.526906	−	0.849923i	$-0.676648\pi$
$757$	−28.9942	−1.05381	−0.526906	+	0.849923i	$0.676648\pi$
$758$	0	0
$759$	19.6989	0.715025
$760$	0	0
$761$	21.5898	0.782631	0.391315	−	0.920257i	$-0.372020\pi$
$761$	21.5898	0.782631	0.391315	+	0.920257i	$0.372020\pi$
$762$	0	0
$763$	26.1384	0.946275
$764$	0	0
$765$	−10.6615	−0.385466
$766$	0	0
$767$	15.1498	0.547028
$768$	0	0
$769$	−42.9020	−1.54709	−0.773543	−	0.633744i	$-0.781517\pi$
$769$	−42.9020	−1.54709	−0.773543	+	0.633744i	$0.781517\pi$
$770$	0	0
$771$	10.3882	0.374122
$772$	0	0
$773$	−36.6170	−1.31702	−0.658512	−	0.752571i	$-0.728814\pi$
$773$	−36.6170	−1.31702	−0.658512	+	0.752571i	$0.728814\pi$
$774$	0	0
$775$	1.78515	0.0641244
$776$	0	0
$777$	7.33239	0.263048
$778$	0	0
$779$	−9.10971	−0.326389
$780$	0	0
$781$	−8.32604	−0.297929
$782$	0	0
$783$	−46.4223	−1.65900
$784$	0	0
$785$	−12.3132	−0.439477
$786$	0	0
$787$	−19.6285	−0.699679	−0.349840	−	0.936810i	$-0.613764\pi$
$787$	−19.6285	−0.699679	−0.349840	+	0.936810i	$0.613764\pi$
$788$	0	0
$789$	24.0191	0.855104
$790$	0	0
$791$	−19.2672	−0.685061
$792$	0	0
$793$	−21.5949	−0.766856
$794$	0	0
$795$	6.07730	0.215540
$796$	0	0
$797$	−5.55050	−0.196609	−0.0983044	−	0.995156i	$-0.531342\pi$
$797$	−5.55050	−0.196609	−0.0983044	+	0.995156i	$0.531342\pi$
$798$	0	0
$799$	−2.20994	−0.0781821
$800$	0	0
$801$	−17.8956	−0.632309
$802$	0	0
$803$	−59.7586	−2.10883
$804$	0	0
$805$	3.71161	0.130817
$806$	0	0
$807$	98.4687	3.46626
$808$	0	0
$809$	−26.9535	−0.947633	−0.473817	−	0.880624i	$-0.657124\pi$
$809$	−26.9535	−0.947633	−0.473817	+	0.880624i	$0.657124\pi$
$810$	0	0
$811$	25.9811	0.912318	0.456159	−	0.889898i	$-0.349225\pi$
$811$	25.9811	0.912318	0.456159	+	0.889898i	$0.349225\pi$
$812$	0	0
$813$	24.5069	0.859495
$814$	0	0
$815$	−18.9737	−0.664619
$816$	0	0
$817$	18.2702	0.639195
$818$	0	0
$819$	−27.1254	−0.947839
$820$	0	0
$821$	30.2413	1.05543	0.527714	−	0.849422i	$-0.323049\pi$
$821$	30.2413	1.05543	0.527714	+	0.849422i	$0.323049\pi$
$822$	0	0
$823$	15.9863	0.557249	0.278624	−	0.960400i	$-0.410122\pi$
$823$	15.9863	0.557249	0.278624	+	0.960400i	$0.410122\pi$
$824$	0	0
$825$	16.4243	0.571822
$826$	0	0
$827$	46.8328	1.62854	0.814268	−	0.580489i	$-0.197139\pi$
$827$	46.8328	1.62854	0.814268	+	0.580489i	$0.197139\pi$
$828$	0	0
$829$	−25.0377	−0.869594	−0.434797	−	0.900528i	$-0.643180\pi$
$829$	−25.0377	−0.869594	−0.434797	+	0.900528i	$0.643180\pi$
$830$	0	0
$831$	94.8979	3.29197
$832$	0	0
$833$	−4.44119	−0.153878
$834$	0	0
$835$	1.90713	0.0659989
$836$	0	0
$837$	17.2354	0.595743
$838$	0	0
$839$	56.6948	1.95732	0.978660	−	0.205486i	$-0.0658774\pi$
$839$	56.6948	1.95732	0.978660	+	0.205486i	$0.0658774\pi$
$840$	0	0
$841$	−5.88160	−0.202814
$842$	0	0
$843$	30.0553	1.03516
$844$	0	0
$845$	−10.9920	−0.378135
$846$	0	0
$847$	56.8406	1.95307
$848$	0	0
$849$	86.9098	2.98274
$850$	0	0
$851$	0.937643	0.0321420
$852$	0	0
$853$	−14.6457	−0.501460	−0.250730	−	0.968057i	$-0.580671\pi$
$853$	−14.6457	−0.501460	−0.250730	+	0.968057i	$0.580671\pi$
$854$	0	0
$855$	−17.0128	−0.581826
$856$	0	0
$857$	−33.0398	−1.12862	−0.564309	−	0.825564i	$-0.690857\pi$
$857$	−33.0398	−1.12862	−0.564309	+	0.825564i	$0.690857\pi$
$858$	0	0
$859$	−34.8998	−1.19076	−0.595382	−	0.803443i	$-0.702999\pi$
$859$	−34.8998	−1.19076	−0.595382	+	0.803443i	$0.702999\pi$
$860$	0	0
$861$	31.0651	1.05870
$862$	0	0
$863$	−41.6558	−1.41798	−0.708989	−	0.705219i	$-0.750849\pi$
$863$	−41.6558	−1.41798	−0.708989	+	0.705219i	$0.750849\pi$
$864$	0	0
$865$	−13.5173	−0.459602
$866$	0	0
$867$	−42.5195	−1.44404
$868$	0	0
$869$	−3.69758	−0.125432
$870$	0	0
$871$	−1.98194	−0.0671556
$872$	0	0
$873$	−31.7481	−1.07451
$874$	0	0
$875$	3.09463	0.104617
$876$	0	0
$877$	5.25198	0.177347	0.0886734	−	0.996061i	$-0.471737\pi$
$877$	5.25198	0.177347	0.0886734	+	0.996061i	$0.471737\pi$
$878$	0	0
$879$	77.1311	2.60157
$880$	0	0
$881$	−8.21617	−0.276810	−0.138405	−	0.990376i	$-0.544198\pi$
$881$	−8.21617	−0.276810	−0.138405	+	0.990376i	$0.544198\pi$
$882$	0	0
$883$	−51.8528	−1.74499	−0.872493	−	0.488627i	$-0.837498\pi$
$883$	−51.8528	−1.74499	−0.872493	+	0.488627i	$0.837498\pi$
$884$	0	0
$885$	−32.4023	−1.08919
$886$	0	0
$887$	17.4136	0.584690	0.292345	−	0.956313i	$-0.405564\pi$
$887$	17.4136	0.584690	0.292345	+	0.956313i	$0.405564\pi$
$888$	0	0
$889$	−3.91814	−0.131410
$890$	0	0
$891$	58.0123	1.94348
$892$	0	0
$893$	−3.52647	−0.118009
$894$	0	0
$895$	−12.0824	−0.403871
$896$	0	0
$897$	−5.15103	−0.171988
$898$	0	0
$899$	−8.58327	−0.286268
$900$	0	0
$901$	−3.45613	−0.115141
$902$	0	0
$903$	−62.3036	−2.07333
$904$	0	0
$905$	−2.49892	−0.0830669
$906$	0	0
$907$	16.5532	0.549639	0.274820	−	0.961496i	$-0.411382\pi$
$907$	16.5532	0.549639	0.274820	+	0.961496i	$0.411382\pi$
$908$	0	0
$909$	−14.6516	−0.485963
$910$	0	0
$911$	47.5793	1.57637	0.788186	−	0.615436i	$-0.211020\pi$
$911$	47.5793	1.57637	0.788186	+	0.615436i	$0.211020\pi$
$912$	0	0
$913$	1.62995	0.0539435
$914$	0	0
$915$	46.1869	1.52689
$916$	0	0
$917$	−62.0246	−2.04823
$918$	0	0
$919$	24.7970	0.817978	0.408989	−	0.912539i	$-0.365881\pi$
$919$	24.7970	0.817978	0.408989	+	0.912539i	$0.365881\pi$
$920$	0	0
$921$	62.9909	2.07562
$922$	0	0
$923$	2.17716	0.0716621
$924$	0	0
$925$	0.781778	0.0257047
$926$	0	0
$927$	−17.4479	−0.573063
$928$	0	0
$929$	−18.9492	−0.621702	−0.310851	−	0.950459i	$-0.600614\pi$
$929$	−18.9492	−0.621702	−0.310851	+	0.950459i	$0.600614\pi$
$930$	0	0
$931$	−7.08694	−0.232265
$932$	0	0
$933$	29.2581	0.957866
$934$	0	0
$935$	−9.34044	−0.305465
$936$	0	0
$937$	24.3373	0.795064	0.397532	−	0.917588i	$-0.369867\pi$
$937$	24.3373	0.795064	0.397532	+	0.917588i	$0.369867\pi$
$938$	0	0
$939$	25.8814	0.844608
$940$	0	0
$941$	−2.90793	−0.0947959	−0.0473980	−	0.998876i	$-0.515093\pi$
$941$	−2.90793	−0.0947959	−0.0473980	+	0.998876i	$0.515093\pi$
$942$	0	0
$943$	3.97251	0.129363
$944$	0	0
$945$	29.8783	0.971941
$946$	0	0
$947$	−1.00615	−0.0326956	−0.0163478	−	0.999866i	$-0.505204\pi$
$947$	−1.00615	−0.0326956	−0.0163478	+	0.999866i	$0.505204\pi$
$948$	0	0
$949$	15.6261	0.507246
$950$	0	0
$951$	−75.6521	−2.45319
$952$	0	0
$953$	30.2354	0.979421	0.489711	−	0.871885i	$-0.337102\pi$
$953$	30.2354	0.979421	0.489711	+	0.871885i	$0.337102\pi$
$954$	0	0
$955$	−16.6392	−0.538433
$956$	0	0
$957$	−78.9708	−2.55276
$958$	0	0
$959$	57.9114	1.87006
$960$	0	0
$961$	−27.8132	−0.897202
$962$	0	0
$963$	−6.25103	−0.201437
$964$	0	0
$965$	−17.6495	−0.568158
$966$	0	0
$967$	−5.32528	−0.171249	−0.0856247	−	0.996327i	$-0.527289\pi$
$967$	−5.32528	−0.171249	−0.0856247	+	0.996327i	$0.527289\pi$
$968$	0	0
$969$	14.3675	0.461550
$970$	0	0
$971$	32.1270	1.03100	0.515502	−	0.856888i	$-0.327605\pi$
$971$	32.1270	1.03100	0.515502	+	0.856888i	$0.327605\pi$
$972$	0	0
$973$	46.6015	1.49397
$974$	0	0
$975$	−4.29476	−0.137543
$976$	0	0
$977$	−10.4786	−0.335239	−0.167619	−	0.985852i	$-0.553608\pi$
$977$	−10.4786	−0.335239	−0.167619	+	0.985852i	$0.553608\pi$
$978$	0	0
$979$	−15.6782	−0.501078
$980$	0	0
$981$	52.2461	1.66809
$982$	0	0
$983$	−8.94051	−0.285158	−0.142579	−	0.989783i	$-0.545539\pi$
$983$	−8.94051	−0.285158	−0.142579	+	0.989783i	$0.545539\pi$
$984$	0	0
$985$	−1.29775	−0.0413497
$986$	0	0
$987$	12.0257	0.382782
$988$	0	0
$989$	−7.96719	−0.253342
$990$	0	0
$991$	42.0696	1.33639	0.668193	−	0.743988i	$-0.267068\pi$
$991$	42.0696	1.33639	0.668193	+	0.743988i	$0.267068\pi$
$992$	0	0
$993$	29.4958	0.936020
$994$	0	0
$995$	8.39272	0.266067
$996$	0	0
$997$	−38.2332	−1.21086	−0.605429	−	0.795899i	$-0.706998\pi$
$997$	−38.2332	−1.21086	−0.605429	+	0.795899i	$0.706998\pi$
$998$	0	0
$999$	7.54798	0.238808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.14	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.14	✓	15	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+3.03078 q^{3} +1.00000 q^{5} +3.09463 q^{7} +6.18562 q^{9} +O(q^{10})\)	\(q+3.03078 q^{3} +1.00000 q^{5} +3.09463 q^{7} +6.18562 q^{9} +5.41918 q^{11} -1.41705 q^{13} +3.03078 q^{15} -1.72359 q^{17} -2.75038 q^{19} +9.37912 q^{21} +1.19937 q^{23} +1.00000 q^{25} +9.65489 q^{27} -4.80816 q^{29} +1.78515 q^{31} +16.4243 q^{33} +3.09463 q^{35} +0.781778 q^{37} -4.29476 q^{39} +3.31216 q^{41} -6.64280 q^{43} +6.18562 q^{45} +1.28217 q^{47} +2.57671 q^{49} -5.22382 q^{51} +2.00520 q^{53} +5.41918 q^{55} -8.33580 q^{57} -10.6911 q^{59} +15.2393 q^{61} +19.1422 q^{63} -1.41705 q^{65} +1.39864 q^{67} +3.63503 q^{69} -1.53640 q^{71} -11.0272 q^{73} +3.03078 q^{75} +16.7703 q^{77} -0.682313 q^{79} +10.7050 q^{81} +0.300775 q^{83} -1.72359 q^{85} -14.5725 q^{87} -2.89310 q^{89} -4.38524 q^{91} +5.41038 q^{93} -2.75038 q^{95} -5.13256 q^{97} +33.5210 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more