LMFDB - Embedded newform 315.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.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:	$2.51528766367$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.43845 q^{8} +O(q^{10})$	$q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.43845 q^{8} +1.56155 q^{10} -2.56155 q^{11} +4.56155 q^{13} +1.56155 q^{14} -4.68466 q^{16} +4.56155 q^{17} +1.12311 q^{19} -0.438447 q^{20} +4.00000 q^{22} +5.12311 q^{23} +1.00000 q^{25} -7.12311 q^{26} -0.438447 q^{28} +5.68466 q^{29} +2.43845 q^{32} -7.12311 q^{34} +1.00000 q^{35} +6.00000 q^{37} -1.75379 q^{38} -2.43845 q^{40} +3.12311 q^{41} +9.12311 q^{43} -1.12311 q^{44} -8.00000 q^{46} -3.68466 q^{47} +1.00000 q^{49} -1.56155 q^{50} +2.00000 q^{52} -3.12311 q^{53} +2.56155 q^{55} -2.43845 q^{56} -8.87689 q^{58} +4.00000 q^{59} -9.36932 q^{61} +5.56155 q^{64} -4.56155 q^{65} -6.24621 q^{67} +2.00000 q^{68} -1.56155 q^{70} -8.00000 q^{71} +4.24621 q^{73} -9.36932 q^{74} +0.492423 q^{76} +2.56155 q^{77} -6.56155 q^{79} +4.68466 q^{80} -4.87689 q^{82} -4.00000 q^{83} -4.56155 q^{85} -14.2462 q^{86} -6.24621 q^{88} -7.12311 q^{89} -4.56155 q^{91} +2.24621 q^{92} +5.75379 q^{94} -1.12311 q^{95} -14.8078 q^{97} -1.56155 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 5 q^{4} - 2 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) $ 2 * q + q^2 + 5 * q^4 - 2 * q^5 - 2 * q^7 + 9 * q^8	$ 2 q + q^{2} + 5 q^{4} - 2 q^{5} - 2 q^{7} + 9 q^{8} - q^{10} - q^{11} + 5 q^{13} - q^{14} + 3 q^{16} + 5 q^{17} - 6 q^{19} - 5 q^{20} + 8 q^{22} + 2 q^{23} + 2 q^{25} - 6 q^{26} - 5 q^{28} - q^{29} + 9 q^{32} - 6 q^{34} + 2 q^{35} + 12 q^{37} - 20 q^{38} - 9 q^{40} - 2 q^{41} + 10 q^{43} + 6 q^{44} - 16 q^{46} + 5 q^{47} + 2 q^{49} + q^{50} + 4 q^{52} + 2 q^{53} + q^{55} - 9 q^{56} - 26 q^{58} + 8 q^{59} + 6 q^{61} + 7 q^{64} - 5 q^{65} + 4 q^{67} + 4 q^{68} + q^{70} - 16 q^{71} - 8 q^{73} + 6 q^{74} - 32 q^{76} + q^{77} - 9 q^{79} - 3 q^{80} - 18 q^{82} - 8 q^{83} - 5 q^{85} - 12 q^{86} + 4 q^{88} - 6 q^{89} - 5 q^{91} - 12 q^{92} + 28 q^{94} + 6 q^{95} - 9 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 + 5 * q^4 - 2 * q^5 - 2 * q^7 + 9 * q^8 - q^10 - q^11 + 5 * q^13 - q^14 + 3 * q^16 + 5 * q^17 - 6 * q^19 - 5 * q^20 + 8 * q^22 + 2 * q^23 + 2 * q^25 - 6 * q^26 - 5 * q^28 - q^29 + 9 * q^32 - 6 * q^34 + 2 * q^35 + 12 * q^37 - 20 * q^38 - 9 * q^40 - 2 * q^41 + 10 * q^43 + 6 * q^44 - 16 * q^46 + 5 * q^47 + 2 * q^49 + q^50 + 4 * q^52 + 2 * q^53 + q^55 - 9 * q^56 - 26 * q^58 + 8 * q^59 + 6 * q^61 + 7 * q^64 - 5 * q^65 + 4 * q^67 + 4 * q^68 + q^70 - 16 * q^71 - 8 * q^73 + 6 * q^74 - 32 * q^76 + q^77 - 9 * q^79 - 3 * q^80 - 18 * q^82 - 8 * q^83 - 5 * q^85 - 12 * q^86 + 4 * q^88 - 6 * q^89 - 5 * q^91 - 12 * q^92 + 28 * q^94 + 6 * q^95 - 9 * q^97 + q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.56155	−1.10418	−0.552092	−	0.833783i	$-0.686170\pi$
$2$	−1.56155	−1.10418	−0.552092	+	0.833783i	$0.686170\pi$
$3$	0	0
$3$	0	0
$4$	0.438447	0.219224
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.43845	0.862121
$9$	0	0
$10$	1.56155	0.493806
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	4.56155	1.26515	0.632574	−	0.774500i	$-0.281999\pi$
$13$	4.56155	1.26515	0.632574	+	0.774500i	$0.281999\pi$
$14$	1.56155	0.417343
$15$	0	0
$16$	−4.68466	−1.17116
$17$	4.56155	1.10634	0.553170	−	0.833069i	$-0.313418\pi$
$17$	4.56155	1.10634	0.553170	+	0.833069i	$0.313418\pi$
$18$	0	0
$19$	1.12311	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$19$	1.12311	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$20$	−0.438447	−0.0980398
$21$	0	0
$22$	4.00000	0.852803
$23$	5.12311	1.06824	0.534121	−	0.845408i	$-0.320643\pi$
$23$	5.12311	1.06824	0.534121	+	0.845408i	$0.320643\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−7.12311	−1.39696
$27$	0	0
$28$	−0.438447	−0.0828587
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	2.43845	0.431061
$33$	0	0
$34$	−7.12311	−1.22160
$35$	1.00000	0.169031
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−1.75379	−0.284502
$39$	0	0
$40$	−2.43845	−0.385552
$41$	3.12311	0.487747	0.243874	−	0.969807i	$-0.421582\pi$
$41$	3.12311	0.487747	0.243874	+	0.969807i	$0.421582\pi$
$42$	0	0
$43$	9.12311	1.39126	0.695630	−	0.718400i	$-0.255125\pi$
$43$	9.12311	1.39126	0.695630	+	0.718400i	$0.255125\pi$
$44$	−1.12311	−0.169315
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−3.68466	−0.537463	−0.268731	−	0.963215i	$-0.586604\pi$
$47$	−3.68466	−0.537463	−0.268731	+	0.963215i	$0.586604\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.56155	−0.220837
$51$	0	0
$52$	2.00000	0.277350
$53$	−3.12311	−0.428992	−0.214496	−	0.976725i	$-0.568811\pi$
$53$	−3.12311	−0.428992	−0.214496	+	0.976725i	$0.568811\pi$
$54$	0	0
$55$	2.56155	0.345400
$56$	−2.43845	−0.325851
$57$	0	0
$58$	−8.87689	−1.16559
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−9.36932	−1.19962	−0.599809	−	0.800143i	$-0.704757\pi$
$61$	−9.36932	−1.19962	−0.599809	+	0.800143i	$0.704757\pi$
$62$	0	0
$63$	0	0
$64$	5.56155	0.695194
$65$	−4.56155	−0.565791
$66$	0	0
$67$	−6.24621	−0.763096	−0.381548	−	0.924349i	$-0.624609\pi$
$67$	−6.24621	−0.763096	−0.381548	+	0.924349i	$0.624609\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	−1.56155	−0.186641
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	4.24621	0.496981	0.248491	−	0.968634i	$-0.420065\pi$
$73$	4.24621	0.496981	0.248491	+	0.968634i	$0.420065\pi$
$74$	−9.36932	−1.08916
$75$	0	0
$76$	0.492423	0.0564847
$77$	2.56155	0.291916
$78$	0	0
$79$	−6.56155	−0.738232	−0.369116	−	0.929383i	$-0.620340\pi$
$79$	−6.56155	−0.738232	−0.369116	+	0.929383i	$0.620340\pi$
$80$	4.68466	0.523761
$81$	0	0
$82$	−4.87689	−0.538563
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−4.56155	−0.494770
$86$	−14.2462	−1.53621
$87$	0	0
$88$	−6.24621	−0.665848
$89$	−7.12311	−0.755048	−0.377524	−	0.926000i	$-0.623224\pi$
$89$	−7.12311	−0.755048	−0.377524	+	0.926000i	$0.623224\pi$
$90$	0	0
$91$	−4.56155	−0.478181
$92$	2.24621	0.234184
$93$	0	0
$94$	5.75379	0.593458
$95$	−1.12311	−0.115228
$96$	0	0
$97$	−14.8078	−1.50350	−0.751750	−	0.659448i	$-0.770790\pi$
$97$	−14.8078	−1.50350	−0.751750	+	0.659448i	$0.770790\pi$
$98$	−1.56155	−0.157741
$99$	0	0
$100$	0.438447	0.0438447
$101$	−0.246211	−0.0244989	−0.0122495	−	0.999925i	$-0.503899\pi$
$101$	−0.246211	−0.0244989	−0.0122495	+	0.999925i	$0.503899\pi$
$102$	0	0
$103$	1.43845	0.141734	0.0708672	−	0.997486i	$-0.477423\pi$
$103$	1.43845	0.141734	0.0708672	+	0.997486i	$0.477423\pi$
$104$	11.1231	1.09071
$105$	0	0
$106$	4.87689	0.473686
$107$	11.3693	1.09911	0.549557	−	0.835456i	$-0.314797\pi$
$107$	11.3693	1.09911	0.549557	+	0.835456i	$0.314797\pi$
$108$	0	0
$109$	17.6847	1.69388	0.846942	−	0.531686i	$-0.178441\pi$
$109$	17.6847	1.69388	0.846942	+	0.531686i	$0.178441\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	4.68466	0.442659
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−5.12311	−0.477732
$116$	2.49242	0.231416
$117$	0	0
$118$	−6.24621	−0.575010
$119$	−4.56155	−0.418157
$120$	0	0
$121$	−4.43845	−0.403495
$122$	14.6307	1.32460
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.2462	0.909204	0.454602	−	0.890695i	$-0.349781\pi$
$127$	10.2462	0.909204	0.454602	+	0.890695i	$0.349781\pi$
$128$	−13.5616	−1.19868
$129$	0	0
$130$	7.12311	0.624738
$131$	9.12311	0.797089	0.398545	−	0.917149i	$-0.369515\pi$
$131$	9.12311	0.797089	0.398545	+	0.917149i	$0.369515\pi$
$132$	0	0
$133$	−1.12311	−0.0973856
$134$	9.75379	0.842599
$135$	0	0
$136$	11.1231	0.953798
$137$	8.87689	0.758404	0.379202	−	0.925314i	$-0.376199\pi$
$137$	8.87689	0.758404	0.379202	+	0.925314i	$0.376199\pi$
$138$	0	0
$139$	−6.87689	−0.583291	−0.291645	−	0.956527i	$-0.594203\pi$
$139$	−6.87689	−0.583291	−0.291645	+	0.956527i	$0.594203\pi$
$140$	0.438447	0.0370556
$141$	0	0
$142$	12.4924	1.04834
$143$	−11.6847	−0.977120
$144$	0	0
$145$	−5.68466	−0.472085
$146$	−6.63068	−0.548759
$147$	0	0
$148$	2.63068	0.216241
$149$	4.24621	0.347863	0.173932	−	0.984758i	$-0.444353\pi$
$149$	4.24621	0.347863	0.173932	+	0.984758i	$0.444353\pi$
$150$	0	0
$151$	21.9309	1.78471	0.892354	−	0.451335i	$-0.149052\pi$
$151$	21.9309	1.78471	0.892354	+	0.451335i	$0.149052\pi$
$152$	2.73863	0.222133
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	3.75379	0.299585	0.149792	−	0.988717i	$-0.452139\pi$
$157$	3.75379	0.299585	0.149792	+	0.988717i	$0.452139\pi$
$158$	10.2462	0.815145
$159$	0	0
$160$	−2.43845	−0.192776
$161$	−5.12311	−0.403757
$162$	0	0
$163$	1.12311	0.0879684	0.0439842	−	0.999032i	$-0.485995\pi$
$163$	1.12311	0.0879684	0.0439842	+	0.999032i	$0.485995\pi$
$164$	1.36932	0.106926
$165$	0	0
$166$	6.24621	0.484800
$167$	−21.9309	−1.69706	−0.848531	−	0.529146i	$-0.822512\pi$
$167$	−21.9309	−1.69706	−0.848531	+	0.529146i	$0.822512\pi$
$168$	0	0
$169$	7.80776	0.600597
$170$	7.12311	0.546317
$171$	0	0
$172$	4.00000	0.304997
$173$	8.56155	0.650923	0.325461	−	0.945555i	$-0.394480\pi$
$173$	8.56155	0.650923	0.325461	+	0.945555i	$0.394480\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	12.0000	0.904534
$177$	0	0
$178$	11.1231	0.833712
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	23.6155	1.75533	0.877664	−	0.479276i	$-0.159101\pi$
$181$	23.6155	1.75533	0.877664	+	0.479276i	$0.159101\pi$
$182$	7.12311	0.528000
$183$	0	0
$184$	12.4924	0.920954
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−11.6847	−0.854467
$188$	−1.61553	−0.117824
$189$	0	0
$190$	1.75379	0.127233
$191$	9.43845	0.682942	0.341471	−	0.939892i	$-0.389075\pi$
$191$	9.43845	0.682942	0.341471	+	0.939892i	$0.389075\pi$
$192$	0	0
$193$	−5.36932	−0.386492	−0.193246	−	0.981150i	$-0.561902\pi$
$193$	−5.36932	−0.386492	−0.193246	+	0.981150i	$0.561902\pi$
$194$	23.1231	1.66014
$195$	0	0
$196$	0.438447	0.0313177
$197$	7.12311	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$197$	7.12311	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$198$	0	0
$199$	−18.2462	−1.29344	−0.646720	−	0.762728i	$-0.723860\pi$
$199$	−18.2462	−1.29344	−0.646720	+	0.762728i	$0.723860\pi$
$200$	2.43845	0.172424
$201$	0	0
$202$	0.384472	0.0270513
$203$	−5.68466	−0.398985
$204$	0	0
$205$	−3.12311	−0.218127
$206$	−2.24621	−0.156501
$207$	0	0
$208$	−21.3693	−1.48170
$209$	−2.87689	−0.198999
$210$	0	0
$211$	−23.0540	−1.58710	−0.793551	−	0.608504i	$-0.791770\pi$
$211$	−23.0540	−1.58710	−0.793551	+	0.608504i	$0.791770\pi$
$212$	−1.36932	−0.0940451
$213$	0	0
$214$	−17.7538	−1.21362
$215$	−9.12311	−0.622191
$216$	0	0
$217$	0	0
$218$	−27.6155	−1.87036
$219$	0	0
$220$	1.12311	0.0757198
$221$	20.8078	1.39968
$222$	0	0
$223$	−6.56155	−0.439394	−0.219697	−	0.975568i	$-0.570507\pi$
$223$	−6.56155	−0.439394	−0.219697	+	0.975568i	$0.570507\pi$
$224$	−2.43845	−0.162926
$225$	0	0
$226$	−21.8617	−1.45422
$227$	−23.6847	−1.57201	−0.786003	−	0.618223i	$-0.787853\pi$
$227$	−23.6847	−1.57201	−0.786003	+	0.618223i	$0.787853\pi$
$228$	0	0
$229$	19.1231	1.26369	0.631845	−	0.775095i	$-0.282298\pi$
$229$	19.1231	1.26369	0.631845	+	0.775095i	$0.282298\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	13.8617	0.910068
$233$	3.12311	0.204601	0.102301	−	0.994754i	$-0.467380\pi$
$233$	3.12311	0.204601	0.102301	+	0.994754i	$0.467380\pi$
$234$	0	0
$235$	3.68466	0.240361
$236$	1.75379	0.114162
$237$	0	0
$238$	7.12311	0.461722
$239$	0.807764	0.0522499	0.0261250	−	0.999659i	$-0.491683\pi$
$239$	0.807764	0.0522499	0.0261250	+	0.999659i	$0.491683\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	6.93087	0.445533
$243$	0	0
$244$	−4.10795	−0.262985
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	5.12311	0.325975
$248$	0	0
$249$	0	0
$250$	1.56155	0.0987613
$251$	17.1231	1.08080	0.540400	−	0.841408i	$-0.318273\pi$
$251$	17.1231	1.08080	0.540400	+	0.841408i	$0.318273\pi$
$252$	0	0
$253$	−13.1231	−0.825043
$254$	−16.0000	−1.00393
$255$	0	0
$256$	10.0540	0.628373
$257$	−22.4924	−1.40304	−0.701519	−	0.712650i	$-0.747495\pi$
$257$	−22.4924	−1.40304	−0.701519	+	0.712650i	$0.747495\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	−2.00000	−0.124035
$261$	0	0
$262$	−14.2462	−0.880134
$263$	21.1231	1.30251	0.651253	−	0.758860i	$-0.274244\pi$
$263$	21.1231	1.30251	0.651253	+	0.758860i	$0.274244\pi$
$264$	0	0
$265$	3.12311	0.191851
$266$	1.75379	0.107532
$267$	0	0
$268$	−2.73863	−0.167289
$269$	−28.7386	−1.75223	−0.876113	−	0.482106i	$-0.839872\pi$
$269$	−28.7386	−1.75223	−0.876113	+	0.482106i	$0.839872\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−21.3693	−1.29571
$273$	0	0
$274$	−13.8617	−0.837418
$275$	−2.56155	−0.154467
$276$	0	0
$277$	16.2462	0.976140	0.488070	−	0.872804i	$-0.337701\pi$
$277$	16.2462	0.976140	0.488070	+	0.872804i	$0.337701\pi$
$278$	10.7386	0.644060
$279$	0	0
$280$	2.43845	0.145725
$281$	−16.5616	−0.987979	−0.493990	−	0.869468i	$-0.664462\pi$
$281$	−16.5616	−0.987979	−0.493990	+	0.869468i	$0.664462\pi$
$282$	0	0
$283$	−23.6847	−1.40791	−0.703953	−	0.710246i	$-0.748584\pi$
$283$	−23.6847	−1.40791	−0.703953	+	0.710246i	$0.748584\pi$
$284$	−3.50758	−0.208136
$285$	0	0
$286$	18.2462	1.07892
$287$	−3.12311	−0.184351
$288$	0	0
$289$	3.80776	0.223986
$290$	8.87689	0.521269
$291$	0	0
$292$	1.86174	0.108950
$293$	−9.68466	−0.565784	−0.282892	−	0.959152i	$-0.591294\pi$
$293$	−9.68466	−0.565784	−0.282892	+	0.959152i	$0.591294\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	14.6307	0.850391
$297$	0	0
$298$	−6.63068	−0.384105
$299$	23.3693	1.35148
$300$	0	0
$301$	−9.12311	−0.525847
$302$	−34.2462	−1.97065
$303$	0	0
$304$	−5.26137	−0.301760
$305$	9.36932	0.536486
$306$	0	0
$307$	−31.6847	−1.80834	−0.904169	−	0.427174i	$-0.859509\pi$
$307$	−31.6847	−1.80834	−0.904169	+	0.427174i	$0.859509\pi$
$308$	1.12311	0.0639949
$309$	0	0
$310$	0	0
$311$	9.61553	0.545247	0.272623	−	0.962121i	$-0.412109\pi$
$311$	9.61553	0.545247	0.272623	+	0.962121i	$0.412109\pi$
$312$	0	0
$313$	31.3002	1.76919	0.884596	−	0.466359i	$-0.154434\pi$
$313$	31.3002	1.76919	0.884596	+	0.466359i	$0.154434\pi$
$314$	−5.86174	−0.330797
$315$	0	0
$316$	−2.87689	−0.161838
$317$	22.4924	1.26330	0.631650	−	0.775254i	$-0.282378\pi$
$317$	22.4924	1.26330	0.631650	+	0.775254i	$0.282378\pi$
$318$	0	0
$319$	−14.5616	−0.815290
$320$	−5.56155	−0.310900
$321$	0	0
$322$	8.00000	0.445823
$323$	5.12311	0.285057
$324$	0	0
$325$	4.56155	0.253029
$326$	−1.75379	−0.0971334
$327$	0	0
$328$	7.61553	0.420497
$329$	3.68466	0.203142
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−1.75379	−0.0962517
$333$	0	0
$334$	34.2462	1.87387
$335$	6.24621	0.341267
$336$	0	0
$337$	−34.4924	−1.87892	−0.939461	−	0.342656i	$-0.888674\pi$
$337$	−34.4924	−1.87892	−0.939461	+	0.342656i	$0.888674\pi$
$338$	−12.1922	−0.663170
$339$	0	0
$340$	−2.00000	−0.108465
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	22.2462	1.19944
$345$	0	0
$346$	−13.3693	−0.718739
$347$	1.12311	0.0602915	0.0301457	−	0.999546i	$-0.490403\pi$
$347$	1.12311	0.0602915	0.0301457	+	0.999546i	$0.490403\pi$
$348$	0	0
$349$	−22.4924	−1.20399	−0.601996	−	0.798499i	$-0.705628\pi$
$349$	−22.4924	−1.20399	−0.601996	+	0.798499i	$0.705628\pi$
$350$	1.56155	0.0834685
$351$	0	0
$352$	−6.24621	−0.332924
$353$	14.8078	0.788138	0.394069	−	0.919081i	$-0.371067\pi$
$353$	14.8078	0.788138	0.394069	+	0.919081i	$0.371067\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	−3.12311	−0.165524
$357$	0	0
$358$	31.2311	1.65061
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	−36.8769	−1.93821
$363$	0	0
$364$	−2.00000	−0.104828
$365$	−4.24621	−0.222257
$366$	0	0
$367$	3.68466	0.192338	0.0961688	−	0.995365i	$-0.469341\pi$
$367$	3.68466	0.192338	0.0961688	+	0.995365i	$0.469341\pi$
$368$	−24.0000	−1.25109
$369$	0	0
$370$	9.36932	0.487088
$371$	3.12311	0.162144
$372$	0	0
$373$	29.3693	1.52069	0.760343	−	0.649522i	$-0.225031\pi$
$373$	29.3693	1.52069	0.760343	+	0.649522i	$0.225031\pi$
$374$	18.2462	0.943489
$375$	0	0
$376$	−8.98485	−0.463358
$377$	25.9309	1.33551
$378$	0	0
$379$	16.4924	0.847159	0.423579	−	0.905859i	$-0.360773\pi$
$379$	16.4924	0.847159	0.423579	+	0.905859i	$0.360773\pi$
$380$	−0.492423	−0.0252607
$381$	0	0
$382$	−14.7386	−0.754094
$383$	10.2462	0.523557	0.261778	−	0.965128i	$-0.415691\pi$
$383$	10.2462	0.523557	0.261778	+	0.965128i	$0.415691\pi$
$384$	0	0
$385$	−2.56155	−0.130549
$386$	8.38447	0.426758
$387$	0	0
$388$	−6.49242	−0.329603
$389$	−3.93087	−0.199303	−0.0996515	−	0.995022i	$-0.531773\pi$
$389$	−3.93087	−0.199303	−0.0996515	+	0.995022i	$0.531773\pi$
$390$	0	0
$391$	23.3693	1.18184
$392$	2.43845	0.123160
$393$	0	0
$394$	−11.1231	−0.560374
$395$	6.56155	0.330148
$396$	0	0
$397$	23.4384	1.17634	0.588171	−	0.808737i	$-0.299848\pi$
$397$	23.4384	1.17634	0.588171	+	0.808737i	$0.299848\pi$
$398$	28.4924	1.42820
$399$	0	0
$400$	−4.68466	−0.234233
$401$	−27.4384	−1.37021	−0.685105	−	0.728444i	$-0.740244\pi$
$401$	−27.4384	−1.37021	−0.685105	+	0.728444i	$0.740244\pi$
$402$	0	0
$403$	0	0
$404$	−0.107951	−0.00537074
$405$	0	0
$406$	8.87689	0.440553
$407$	−15.3693	−0.761829
$408$	0	0
$409$	−26.4924	−1.30997	−0.654983	−	0.755644i	$-0.727324\pi$
$409$	−26.4924	−1.30997	−0.654983	+	0.755644i	$0.727324\pi$
$410$	4.87689	0.240853
$411$	0	0
$412$	0.630683	0.0310715
$413$	−4.00000	−0.196827
$414$	0	0
$415$	4.00000	0.196352
$416$	11.1231	0.545355
$417$	0	0
$418$	4.49242	0.219732
$419$	−9.75379	−0.476504	−0.238252	−	0.971203i	$-0.576574\pi$
$419$	−9.75379	−0.476504	−0.238252	+	0.971203i	$0.576574\pi$
$420$	0	0
$421$	9.68466	0.472001	0.236001	−	0.971753i	$-0.424163\pi$
$421$	9.68466	0.472001	0.236001	+	0.971753i	$0.424163\pi$
$422$	36.0000	1.75245
$423$	0	0
$424$	−7.61553	−0.369843
$425$	4.56155	0.221268
$426$	0	0
$427$	9.36932	0.453413
$428$	4.98485	0.240952
$429$	0	0
$430$	14.2462	0.687013
$431$	−0.807764	−0.0389086	−0.0194543	−	0.999811i	$-0.506193\pi$
$431$	−0.807764	−0.0389086	−0.0194543	+	0.999811i	$0.506193\pi$
$432$	0	0
$433$	−8.24621	−0.396288	−0.198144	−	0.980173i	$-0.563491\pi$
$433$	−8.24621	−0.396288	−0.198144	+	0.980173i	$0.563491\pi$
$434$	0	0
$435$	0	0
$436$	7.75379	0.371339
$437$	5.75379	0.275241
$438$	0	0
$439$	−15.3693	−0.733537	−0.366769	−	0.930312i	$-0.619536\pi$
$439$	−15.3693	−0.733537	−0.366769	+	0.930312i	$0.619536\pi$
$440$	6.24621	0.297776
$441$	0	0
$442$	−32.4924	−1.54551
$443$	27.3693	1.30036	0.650178	−	0.759782i	$-0.274694\pi$
$443$	27.3693	1.30036	0.650178	+	0.759782i	$0.274694\pi$
$444$	0	0
$445$	7.12311	0.337668
$446$	10.2462	0.485172
$447$	0	0
$448$	−5.56155	−0.262759
$449$	−18.8078	−0.887593	−0.443797	−	0.896128i	$-0.646369\pi$
$449$	−18.8078	−0.887593	−0.443797	+	0.896128i	$0.646369\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	6.13826	0.288719
$453$	0	0
$454$	36.9848	1.73578
$455$	4.56155	0.213849
$456$	0	0
$457$	−8.87689	−0.415244	−0.207622	−	0.978209i	$-0.566572\pi$
$457$	−8.87689	−0.415244	−0.207622	+	0.978209i	$0.566572\pi$
$458$	−29.8617	−1.39535
$459$	0	0
$460$	−2.24621	−0.104730
$461$	4.87689	0.227140	0.113570	−	0.993530i	$-0.463771\pi$
$461$	4.87689	0.227140	0.113570	+	0.993530i	$0.463771\pi$
$462$	0	0
$463$	−20.4924	−0.952364	−0.476182	−	0.879347i	$-0.657980\pi$
$463$	−20.4924	−0.952364	−0.476182	+	0.879347i	$0.657980\pi$
$464$	−26.6307	−1.23630
$465$	0	0
$466$	−4.87689	−0.225918
$467$	−26.5616	−1.22912	−0.614561	−	0.788869i	$-0.710667\pi$
$467$	−26.5616	−1.22912	−0.614561	+	0.788869i	$0.710667\pi$
$468$	0	0
$469$	6.24621	0.288423
$470$	−5.75379	−0.265402
$471$	0	0
$472$	9.75379	0.448955
$473$	−23.3693	−1.07452
$474$	0	0
$475$	1.12311	0.0515316
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	−1.26137	−0.0576935
$479$	−13.1231	−0.599610	−0.299805	−	0.954001i	$-0.596922\pi$
$479$	−13.1231	−0.599610	−0.299805	+	0.954001i	$0.596922\pi$
$480$	0	0
$481$	27.3693	1.24793
$482$	−19.1231	−0.871034
$483$	0	0
$484$	−1.94602	−0.0884557
$485$	14.8078	0.672386
$486$	0	0
$487$	5.12311	0.232150	0.116075	−	0.993240i	$-0.462969\pi$
$487$	5.12311	0.232150	0.116075	+	0.993240i	$0.462969\pi$
$488$	−22.8466	−1.03422
$489$	0	0
$490$	1.56155	0.0705438
$491$	−4.17708	−0.188509	−0.0942545	−	0.995548i	$-0.530047\pi$
$491$	−4.17708	−0.188509	−0.0942545	+	0.995548i	$0.530047\pi$
$492$	0	0
$493$	25.9309	1.16787
$494$	−8.00000	−0.359937
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	−4.17708	−0.186992	−0.0934959	−	0.995620i	$-0.529804\pi$
$499$	−4.17708	−0.186992	−0.0934959	+	0.995620i	$0.529804\pi$
$500$	−0.438447	−0.0196080
$501$	0	0
$502$	−26.7386	−1.19340
$503$	−10.0691	−0.448960	−0.224480	−	0.974479i	$-0.572068\pi$
$503$	−10.0691	−0.448960	−0.224480	+	0.974479i	$0.572068\pi$
$504$	0	0
$505$	0.246211	0.0109563
$506$	20.4924	0.910999
$507$	0	0
$508$	4.49242	0.199319
$509$	28.2462	1.25199	0.625996	−	0.779827i	$-0.284693\pi$
$509$	28.2462	1.25199	0.625996	+	0.779827i	$0.284693\pi$
$510$	0	0
$511$	−4.24621	−0.187841
$512$	11.4233	0.504843
$513$	0	0
$514$	35.1231	1.54921
$515$	−1.43845	−0.0633856
$516$	0	0
$517$	9.43845	0.415102
$518$	9.36932	0.411664
$519$	0	0
$520$	−11.1231	−0.487780
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	7.50758	0.328283	0.164142	−	0.986437i	$-0.447515\pi$
$523$	7.50758	0.328283	0.164142	+	0.986437i	$0.447515\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−32.9848	−1.43821
$527$	0	0
$528$	0	0
$529$	3.24621	0.141140
$530$	−4.87689	−0.211839
$531$	0	0
$532$	−0.492423	−0.0213492
$533$	14.2462	0.617072
$534$	0	0
$535$	−11.3693	−0.491538
$536$	−15.2311	−0.657881
$537$	0	0
$538$	44.8769	1.93478
$539$	−2.56155	−0.110334
$540$	0	0
$541$	−17.1922	−0.739152	−0.369576	−	0.929201i	$-0.620497\pi$
$541$	−17.1922	−0.739152	−0.369576	+	0.929201i	$0.620497\pi$
$542$	24.9848	1.07319
$543$	0	0
$544$	11.1231	0.476899
$545$	−17.6847	−0.757528
$546$	0	0
$547$	14.2462	0.609124	0.304562	−	0.952493i	$-0.401490\pi$
$547$	14.2462	0.609124	0.304562	+	0.952493i	$0.401490\pi$
$548$	3.89205	0.166260
$549$	0	0
$550$	4.00000	0.170561
$551$	6.38447	0.271988
$552$	0	0
$553$	6.56155	0.279026
$554$	−25.3693	−1.07784
$555$	0	0
$556$	−3.01515	−0.127871
$557$	4.87689	0.206641	0.103320	−	0.994648i	$-0.467053\pi$
$557$	4.87689	0.206641	0.103320	+	0.994648i	$0.467053\pi$
$558$	0	0
$559$	41.6155	1.76015
$560$	−4.68466	−0.197963
$561$	0	0
$562$	25.8617	1.09091
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	36.9848	1.55459
$567$	0	0
$568$	−19.5076	−0.818520
$569$	−34.9848	−1.46664	−0.733320	−	0.679883i	$-0.762031\pi$
$569$	−34.9848	−1.46664	−0.733320	+	0.679883i	$0.762031\pi$
$570$	0	0
$571$	7.50758	0.314182	0.157091	−	0.987584i	$-0.449788\pi$
$571$	7.50758	0.314182	0.157091	+	0.987584i	$0.449788\pi$
$572$	−5.12311	−0.214208
$573$	0	0
$574$	4.87689	0.203558
$575$	5.12311	0.213648
$576$	0	0
$577$	13.0540	0.543444	0.271722	−	0.962376i	$-0.412407\pi$
$577$	13.0540	0.543444	0.271722	+	0.962376i	$0.412407\pi$
$578$	−5.94602	−0.247322
$579$	0	0
$580$	−2.49242	−0.103492
$581$	4.00000	0.165948
$582$	0	0
$583$	8.00000	0.331326
$584$	10.3542	0.428458
$585$	0	0
$586$	15.1231	0.624730
$587$	9.75379	0.402582	0.201291	−	0.979531i	$-0.435486\pi$
$587$	9.75379	0.402582	0.201291	+	0.979531i	$0.435486\pi$
$588$	0	0
$589$	0	0
$590$	6.24621	0.257152
$591$	0	0
$592$	−28.1080	−1.15523
$593$	23.4384	0.962502	0.481251	−	0.876583i	$-0.340183\pi$
$593$	23.4384	0.962502	0.481251	+	0.876583i	$0.340183\pi$
$594$	0	0
$595$	4.56155	0.187005
$596$	1.86174	0.0762598
$597$	0	0
$598$	−36.4924	−1.49229
$599$	−8.80776	−0.359875	−0.179938	−	0.983678i	$-0.557590\pi$
$599$	−8.80776	−0.359875	−0.179938	+	0.983678i	$0.557590\pi$
$600$	0	0
$601$	−26.4924	−1.08065	−0.540324	−	0.841457i	$-0.681698\pi$
$601$	−26.4924	−1.08065	−0.540324	+	0.841457i	$0.681698\pi$
$602$	14.2462	0.580632
$603$	0	0
$604$	9.61553	0.391250
$605$	4.43845	0.180449
$606$	0	0
$607$	−4.94602	−0.200753	−0.100376	−	0.994950i	$-0.532005\pi$
$607$	−4.94602	−0.200753	−0.100376	+	0.994950i	$0.532005\pi$
$608$	2.73863	0.111066
$609$	0	0
$610$	−14.6307	−0.592379
$611$	−16.8078	−0.679969
$612$	0	0
$613$	−8.73863	−0.352950	−0.176475	−	0.984305i	$-0.556469\pi$
$613$	−8.73863	−0.352950	−0.176475	+	0.984305i	$0.556469\pi$
$614$	49.4773	1.99674
$615$	0	0
$616$	6.24621	0.251667
$617$	−15.7538	−0.634224	−0.317112	−	0.948388i	$-0.602713\pi$
$617$	−15.7538	−0.634224	−0.317112	+	0.948388i	$0.602713\pi$
$618$	0	0
$619$	−42.1080	−1.69246	−0.846231	−	0.532817i	$-0.821134\pi$
$619$	−42.1080	−1.69246	−0.846231	+	0.532817i	$0.821134\pi$
$620$	0	0
$621$	0	0
$622$	−15.0152	−0.602053
$623$	7.12311	0.285381
$624$	0	0
$625$	1.00000	0.0400000
$626$	−48.8769	−1.95351
$627$	0	0
$628$	1.64584	0.0656761
$629$	27.3693	1.09129
$630$	0	0
$631$	8.80776	0.350632	0.175316	−	0.984512i	$-0.443905\pi$
$631$	8.80776	0.350632	0.175316	+	0.984512i	$0.443905\pi$
$632$	−16.0000	−0.636446
$633$	0	0
$634$	−35.1231	−1.39492
$635$	−10.2462	−0.406608
$636$	0	0
$637$	4.56155	0.180735
$638$	22.7386	0.900231
$639$	0	0
$640$	13.5616	0.536067
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−2.56155	−0.101018	−0.0505089	−	0.998724i	$-0.516084\pi$
$643$	−2.56155	−0.101018	−0.0505089	+	0.998724i	$0.516084\pi$
$644$	−2.24621	−0.0885131
$645$	0	0
$646$	−8.00000	−0.314756
$647$	−3.50758	−0.137897	−0.0689486	−	0.997620i	$-0.521964\pi$
$647$	−3.50758	−0.137897	−0.0689486	+	0.997620i	$0.521964\pi$
$648$	0	0
$649$	−10.2462	−0.402199
$650$	−7.12311	−0.279391
$651$	0	0
$652$	0.492423	0.0192848
$653$	−49.2311	−1.92656	−0.963280	−	0.268499i	$-0.913473\pi$
$653$	−49.2311	−1.92656	−0.963280	+	0.268499i	$0.913473\pi$
$654$	0	0
$655$	−9.12311	−0.356469
$656$	−14.6307	−0.571232
$657$	0	0
$658$	−5.75379	−0.224306
$659$	36.1771	1.40926	0.704629	−	0.709575i	$-0.251113\pi$
$659$	36.1771	1.40926	0.704629	+	0.709575i	$0.251113\pi$
$660$	0	0
$661$	3.12311	0.121475	0.0607374	−	0.998154i	$-0.480655\pi$
$661$	3.12311	0.121475	0.0607374	+	0.998154i	$0.480655\pi$
$662$	−18.7386	−0.728298
$663$	0	0
$664$	−9.75379	−0.378520
$665$	1.12311	0.0435522
$666$	0	0
$667$	29.1231	1.12765
$668$	−9.61553	−0.372036
$669$	0	0
$670$	−9.75379	−0.376822
$671$	24.0000	0.926510
$672$	0	0
$673$	−25.8617	−0.996897	−0.498448	−	0.866919i	$-0.666097\pi$
$673$	−25.8617	−0.996897	−0.498448	+	0.866919i	$0.666097\pi$
$674$	53.8617	2.07468
$675$	0	0
$676$	3.42329	0.131665
$677$	23.9309	0.919738	0.459869	−	0.887987i	$-0.347896\pi$
$677$	23.9309	0.919738	0.459869	+	0.887987i	$0.347896\pi$
$678$	0	0
$679$	14.8078	0.568270
$680$	−11.1231	−0.426552
$681$	0	0
$682$	0	0
$683$	−42.7386	−1.63535	−0.817674	−	0.575681i	$-0.804737\pi$
$683$	−42.7386	−1.63535	−0.817674	+	0.575681i	$0.804737\pi$
$684$	0	0
$685$	−8.87689	−0.339169
$686$	1.56155	0.0596204
$687$	0	0
$688$	−42.7386	−1.62940
$689$	−14.2462	−0.542737
$690$	0	0
$691$	8.49242	0.323067	0.161533	−	0.986867i	$-0.448356\pi$
$691$	8.49242	0.323067	0.161533	+	0.986867i	$0.448356\pi$
$692$	3.75379	0.142698
$693$	0	0
$694$	−1.75379	−0.0665729
$695$	6.87689	0.260855
$696$	0	0
$697$	14.2462	0.539614
$698$	35.1231	1.32943
$699$	0	0
$700$	−0.438447	−0.0165717
$701$	−0.0691303	−0.00261102	−0.00130551	−	0.999999i	$-0.500416\pi$
$701$	−0.0691303	−0.00261102	−0.00130551	+	0.999999i	$0.500416\pi$
$702$	0	0
$703$	6.73863	0.254152
$704$	−14.2462	−0.536924
$705$	0	0
$706$	−23.1231	−0.870250
$707$	0.246211	0.00925973
$708$	0	0
$709$	−18.1771	−0.682655	−0.341327	−	0.939945i	$-0.610876\pi$
$709$	−18.1771	−0.682655	−0.341327	+	0.939945i	$0.610876\pi$
$710$	−12.4924	−0.468832
$711$	0	0
$712$	−17.3693	−0.650943
$713$	0	0
$714$	0	0
$715$	11.6847	0.436981
$716$	−8.76894	−0.327711
$717$	0	0
$718$	12.4924	0.466213
$719$	−49.6155	−1.85035	−0.925173	−	0.379544i	$-0.876081\pi$
$719$	−49.6155	−1.85035	−0.925173	+	0.379544i	$0.876081\pi$
$720$	0	0
$721$	−1.43845	−0.0535706
$722$	27.6998	1.03088
$723$	0	0
$724$	10.3542	0.384809
$725$	5.68466	0.211123
$726$	0	0
$727$	19.5076	0.723496	0.361748	−	0.932276i	$-0.382180\pi$
$727$	19.5076	0.723496	0.361748	+	0.932276i	$0.382180\pi$
$728$	−11.1231	−0.412250
$729$	0	0
$730$	6.63068	0.245413
$731$	41.6155	1.53921
$732$	0	0
$733$	−5.68466	−0.209968	−0.104984	−	0.994474i	$-0.533479\pi$
$733$	−5.68466	−0.209968	−0.104984	+	0.994474i	$0.533479\pi$
$734$	−5.75379	−0.212376
$735$	0	0
$736$	12.4924	0.460477
$737$	16.0000	0.589368
$738$	0	0
$739$	6.06913	0.223257	0.111628	−	0.993750i	$-0.464393\pi$
$739$	6.06913	0.223257	0.111628	+	0.993750i	$0.464393\pi$
$740$	−2.63068	−0.0967058
$741$	0	0
$742$	−4.87689	−0.179036
$743$	−32.9848	−1.21010	−0.605048	−	0.796189i	$-0.706846\pi$
$743$	−32.9848	−1.21010	−0.605048	+	0.796189i	$0.706846\pi$
$744$	0	0
$745$	−4.24621	−0.155569
$746$	−45.8617	−1.67912
$747$	0	0
$748$	−5.12311	−0.187319
$749$	−11.3693	−0.415426
$750$	0	0
$751$	45.9309	1.67604	0.838021	−	0.545639i	$-0.183713\pi$
$751$	45.9309	1.67604	0.838021	+	0.545639i	$0.183713\pi$
$752$	17.2614	0.629457
$753$	0	0
$754$	−40.4924	−1.47465
$755$	−21.9309	−0.798146
$756$	0	0
$757$	14.6307	0.531761	0.265881	−	0.964006i	$-0.414337\pi$
$757$	14.6307	0.531761	0.265881	+	0.964006i	$0.414337\pi$
$758$	−25.7538	−0.935420
$759$	0	0
$760$	−2.73863	−0.0993407
$761$	−31.7538	−1.15107	−0.575537	−	0.817776i	$-0.695207\pi$
$761$	−31.7538	−1.15107	−0.575537	+	0.817776i	$0.695207\pi$
$762$	0	0
$763$	−17.6847	−0.640228
$764$	4.13826	0.149717
$765$	0	0
$766$	−16.0000	−0.578103
$767$	18.2462	0.658833
$768$	0	0
$769$	−9.50758	−0.342852	−0.171426	−	0.985197i	$-0.554837\pi$
$769$	−9.50758	−0.342852	−0.171426	+	0.985197i	$0.554837\pi$
$770$	4.00000	0.144150
$771$	0	0
$772$	−2.35416	−0.0847281
$773$	−8.06913	−0.290226	−0.145113	−	0.989415i	$-0.546355\pi$
$773$	−8.06913	−0.290226	−0.145113	+	0.989415i	$0.546355\pi$
$774$	0	0
$775$	0	0
$776$	−36.1080	−1.29620
$777$	0	0
$778$	6.13826	0.220067
$779$	3.50758	0.125672
$780$	0	0
$781$	20.4924	0.733277
$782$	−36.4924	−1.30497
$783$	0	0
$784$	−4.68466	−0.167309
$785$	−3.75379	−0.133978
$786$	0	0
$787$	−3.82292	−0.136272	−0.0681362	−	0.997676i	$-0.521705\pi$
$787$	−3.82292	−0.136272	−0.0681362	+	0.997676i	$0.521705\pi$
$788$	3.12311	0.111256
$789$	0	0
$790$	−10.2462	−0.364544
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−42.7386	−1.51769
$794$	−36.6004	−1.29890
$795$	0	0
$796$	−8.00000	−0.283552
$797$	13.0540	0.462396	0.231198	−	0.972907i	$-0.425736\pi$
$797$	13.0540	0.462396	0.231198	+	0.972907i	$0.425736\pi$
$798$	0	0
$799$	−16.8078	−0.594616
$800$	2.43845	0.0862121
$801$	0	0
$802$	42.8466	1.51297
$803$	−10.8769	−0.383837
$804$	0	0
$805$	5.12311	0.180566
$806$	0	0
$807$	0	0
$808$	−0.600373	−0.0211211
$809$	53.5464	1.88259	0.941296	−	0.337584i	$-0.109610\pi$
$809$	53.5464	1.88259	0.941296	+	0.337584i	$0.109610\pi$
$810$	0	0
$811$	−21.6155	−0.759024	−0.379512	−	0.925187i	$-0.623908\pi$
$811$	−21.6155	−0.759024	−0.379512	+	0.925187i	$0.623908\pi$
$812$	−2.49242	−0.0874669
$813$	0	0
$814$	24.0000	0.841200
$815$	−1.12311	−0.0393407
$816$	0	0
$817$	10.2462	0.358470
$818$	41.3693	1.44644
$819$	0	0
$820$	−1.36932	−0.0478186
$821$	−40.4233	−1.41078	−0.705391	−	0.708818i	$-0.749229\pi$
$821$	−40.4233	−1.41078	−0.705391	+	0.708818i	$0.749229\pi$
$822$	0	0
$823$	−3.50758	−0.122266	−0.0611332	−	0.998130i	$-0.519471\pi$
$823$	−3.50758	−0.122266	−0.0611332	+	0.998130i	$0.519471\pi$
$824$	3.50758	0.122192
$825$	0	0
$826$	6.24621	0.217333
$827$	−19.3693	−0.673537	−0.336769	−	0.941587i	$-0.609334\pi$
$827$	−19.3693	−0.673537	−0.336769	+	0.941587i	$0.609334\pi$
$828$	0	0
$829$	43.1231	1.49773	0.748864	−	0.662724i	$-0.230600\pi$
$829$	43.1231	1.49773	0.748864	+	0.662724i	$0.230600\pi$
$830$	−6.24621	−0.216809
$831$	0	0
$832$	25.3693	0.879523
$833$	4.56155	0.158048
$834$	0	0
$835$	21.9309	0.758949
$836$	−1.26137	−0.0436253
$837$	0	0
$838$	15.2311	0.526148
$839$	37.1231	1.28163	0.640816	−	0.767695i	$-0.278596\pi$
$839$	37.1231	1.28163	0.640816	+	0.767695i	$0.278596\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	−15.1231	−0.521177
$843$	0	0
$844$	−10.1080	−0.347930
$845$	−7.80776	−0.268595
$846$	0	0
$847$	4.43845	0.152507
$848$	14.6307	0.502420
$849$	0	0
$850$	−7.12311	−0.244321
$851$	30.7386	1.05371
$852$	0	0
$853$	−56.7386	−1.94269	−0.971347	−	0.237666i	$-0.923618\pi$
$853$	−56.7386	−1.94269	−0.971347	+	0.237666i	$0.923618\pi$
$854$	−14.6307	−0.500652
$855$	0	0
$856$	27.7235	0.947569
$857$	32.2462	1.10151	0.550755	−	0.834667i	$-0.314340\pi$
$857$	32.2462	1.10151	0.550755	+	0.834667i	$0.314340\pi$
$858$	0	0
$859$	16.4924	0.562714	0.281357	−	0.959603i	$-0.409215\pi$
$859$	16.4924	0.562714	0.281357	+	0.959603i	$0.409215\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	1.26137	0.0429623
$863$	42.2462	1.43808	0.719039	−	0.694970i	$-0.244582\pi$
$863$	42.2462	1.43808	0.719039	+	0.694970i	$0.244582\pi$
$864$	0	0
$865$	−8.56155	−0.291102
$866$	12.8769	0.437575
$867$	0	0
$868$	0	0
$869$	16.8078	0.570164
$870$	0	0
$871$	−28.4924	−0.965429
$872$	43.1231	1.46033
$873$	0	0
$874$	−8.98485	−0.303917
$875$	1.00000	0.0338062
$876$	0	0
$877$	−23.7538	−0.802108	−0.401054	−	0.916054i	$-0.631356\pi$
$877$	−23.7538	−0.802108	−0.401054	+	0.916054i	$0.631356\pi$
$878$	24.0000	0.809961
$879$	0	0
$880$	−12.0000	−0.404520
$881$	−45.8617	−1.54512	−0.772561	−	0.634941i	$-0.781024\pi$
$881$	−45.8617	−1.54512	−0.772561	+	0.634941i	$0.781024\pi$
$882$	0	0
$883$	24.4924	0.824236	0.412118	−	0.911131i	$-0.364789\pi$
$883$	24.4924	0.824236	0.412118	+	0.911131i	$0.364789\pi$
$884$	9.12311	0.306843
$885$	0	0
$886$	−42.7386	−1.43583
$887$	12.4924	0.419454	0.209727	−	0.977760i	$-0.432742\pi$
$887$	12.4924	0.419454	0.209727	+	0.977760i	$0.432742\pi$
$888$	0	0
$889$	−10.2462	−0.343647
$890$	−11.1231	−0.372847
$891$	0	0
$892$	−2.87689	−0.0963255
$893$	−4.13826	−0.138482
$894$	0	0
$895$	20.0000	0.668526
$896$	13.5616	0.453060
$897$	0	0
$898$	29.3693	0.980067
$899$	0	0
$900$	0	0
$901$	−14.2462	−0.474610
$902$	12.4924	0.415952
$903$	0	0
$904$	34.1383	1.13542
$905$	−23.6155	−0.785007
$906$	0	0
$907$	50.1080	1.66381	0.831904	−	0.554920i	$-0.187251\pi$
$907$	50.1080	1.66381	0.831904	+	0.554920i	$0.187251\pi$
$908$	−10.3845	−0.344621
$909$	0	0
$910$	−7.12311	−0.236129
$911$	−4.49242	−0.148841	−0.0744203	−	0.997227i	$-0.523711\pi$
$911$	−4.49242	−0.148841	−0.0744203	+	0.997227i	$0.523711\pi$
$912$	0	0
$913$	10.2462	0.339100
$914$	13.8617	0.458506
$915$	0	0
$916$	8.38447	0.277031
$917$	−9.12311	−0.301271
$918$	0	0
$919$	−13.3002	−0.438733	−0.219366	−	0.975643i	$-0.570399\pi$
$919$	−13.3002	−0.438733	−0.219366	+	0.975643i	$0.570399\pi$
$920$	−12.4924	−0.411863
$921$	0	0
$922$	−7.61553	−0.250804
$923$	−36.4924	−1.20116
$924$	0	0
$925$	6.00000	0.197279
$926$	32.0000	1.05159
$927$	0	0
$928$	13.8617	0.455034
$929$	52.1080	1.70961	0.854803	−	0.518952i	$-0.173678\pi$
$929$	52.1080	1.70961	0.854803	+	0.518952i	$0.173678\pi$
$930$	0	0
$931$	1.12311	0.0368083
$932$	1.36932	0.0448535
$933$	0	0
$934$	41.4773	1.35718
$935$	11.6847	0.382129
$936$	0	0
$937$	22.6695	0.740580	0.370290	−	0.928916i	$-0.379258\pi$
$937$	22.6695	0.740580	0.370290	+	0.928916i	$0.379258\pi$
$938$	−9.75379	−0.318472
$939$	0	0
$940$	1.61553	0.0526927
$941$	13.8617	0.451880	0.225940	−	0.974141i	$-0.427455\pi$
$941$	13.8617	0.451880	0.225940	+	0.974141i	$0.427455\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	−18.7386	−0.609891
$945$	0	0
$946$	36.4924	1.18647
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	19.3693	0.628755
$950$	−1.75379	−0.0569004
$951$	0	0
$952$	−11.1231	−0.360502
$953$	24.8769	0.805842	0.402921	−	0.915235i	$-0.367995\pi$
$953$	24.8769	0.805842	0.402921	+	0.915235i	$0.367995\pi$
$954$	0	0
$955$	−9.43845	−0.305421
$956$	0.354162	0.0114544
$957$	0	0
$958$	20.4924	0.662080
$959$	−8.87689	−0.286650
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−42.7386	−1.37795
$963$	0	0
$964$	5.36932	0.172934
$965$	5.36932	0.172844
$966$	0	0
$967$	−26.8769	−0.864303	−0.432151	−	0.901801i	$-0.642245\pi$
$967$	−26.8769	−0.864303	−0.432151	+	0.901801i	$0.642245\pi$
$968$	−10.8229	−0.347862
$969$	0	0
$970$	−23.1231	−0.742438
$971$	49.4773	1.58780	0.793901	−	0.608048i	$-0.208047\pi$
$971$	49.4773	1.58780	0.793901	+	0.608048i	$0.208047\pi$
$972$	0	0
$973$	6.87689	0.220463
$974$	−8.00000	−0.256337
$975$	0	0
$976$	43.8920	1.40495
$977$	49.2311	1.57504	0.787521	−	0.616288i	$-0.211364\pi$
$977$	49.2311	1.57504	0.787521	+	0.616288i	$0.211364\pi$
$978$	0	0
$979$	18.2462	0.583151
$980$	−0.438447	−0.0140057
$981$	0	0
$982$	6.52273	0.208149
$983$	10.4233	0.332451	0.166226	−	0.986088i	$-0.446842\pi$
$983$	10.4233	0.332451	0.166226	+	0.986088i	$0.446842\pi$
$984$	0	0
$985$	−7.12311	−0.226961
$986$	−40.4924	−1.28954
$987$	0	0
$988$	2.24621	0.0714615
$989$	46.7386	1.48620
$990$	0	0
$991$	−20.4924	−0.650963	−0.325482	−	0.945548i	$-0.605526\pi$
$991$	−20.4924	−0.650963	−0.325482	+	0.945548i	$0.605526\pi$
$992$	0	0
$993$	0	0
$994$	−12.4924	−0.396236
$995$	18.2462	0.578444
$996$	0	0
$997$	9.68466	0.306716	0.153358	−	0.988171i	$-0.450991\pi$
$997$	9.68466	0.306716	0.153358	+	0.988171i	$0.450991\pi$
$998$	6.52273	0.206473
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.a.e.1.1		2
3.2	odd	2		35.2.a.b.1.2	✓	2
4.3	odd	2		5040.2.a.bt.1.2		2
5.2	odd	4		1575.2.d.e.1324.2		4
5.3	odd	4		1575.2.d.e.1324.3		4
5.4	even	2		1575.2.a.p.1.2		2
7.6	odd	2		2205.2.a.x.1.1		2
12.11	even	2		560.2.a.i.1.2		2
15.2	even	4		175.2.b.b.99.3		4
15.8	even	4		175.2.b.b.99.2		4
15.14	odd	2		175.2.a.f.1.1		2
21.2	odd	6		245.2.e.i.116.1		4
21.5	even	6		245.2.e.h.116.1		4
21.11	odd	6		245.2.e.i.226.1		4
21.17	even	6		245.2.e.h.226.1		4
21.20	even	2		245.2.a.d.1.2		2
24.5	odd	2		2240.2.a.bh.1.2		2
24.11	even	2		2240.2.a.bd.1.1		2
33.32	even	2		4235.2.a.m.1.1		2
39.38	odd	2		5915.2.a.l.1.1		2
60.23	odd	4		2800.2.g.t.449.4		4
60.47	odd	4		2800.2.g.t.449.1		4
60.59	even	2		2800.2.a.bi.1.1		2
84.83	odd	2		3920.2.a.bs.1.1		2
105.62	odd	4		1225.2.b.f.99.3		4
105.83	odd	4		1225.2.b.f.99.2		4
105.104	even	2		1225.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.b.1.2	✓	2	3.2	odd	2
175.2.a.f.1.1		2	15.14	odd	2
175.2.b.b.99.2		4	15.8	even	4
175.2.b.b.99.3		4	15.2	even	4
245.2.a.d.1.2		2	21.20	even	2
245.2.e.h.116.1		4	21.5	even	6
245.2.e.h.226.1		4	21.17	even	6
245.2.e.i.116.1		4	21.2	odd	6
245.2.e.i.226.1		4	21.11	odd	6
315.2.a.e.1.1		2	1.1	even	1	trivial
560.2.a.i.1.2		2	12.11	even	2
1225.2.a.s.1.1		2	105.104	even	2
1225.2.b.f.99.2		4	105.83	odd	4
1225.2.b.f.99.3		4	105.62	odd	4
1575.2.a.p.1.2		2	5.4	even	2
1575.2.d.e.1324.2		4	5.2	odd	4
1575.2.d.e.1324.3		4	5.3	odd	4
2205.2.a.x.1.1		2	7.6	odd	2
2240.2.a.bd.1.1		2	24.11	even	2
2240.2.a.bh.1.2		2	24.5	odd	2
2800.2.a.bi.1.1		2	60.59	even	2
2800.2.g.t.449.1		4	60.47	odd	4
2800.2.g.t.449.4		4	60.23	odd	4
3920.2.a.bs.1.1		2	84.83	odd	2
4235.2.a.m.1.1		2	33.32	even	2
5040.2.a.bt.1.2		2	4.3	odd	2
5915.2.a.l.1.1		2	39.38	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.43845 q^{8} +O(q^{10})\)	\(q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.43845 q^{8} +1.56155 q^{10} -2.56155 q^{11} +4.56155 q^{13} +1.56155 q^{14} -4.68466 q^{16} +4.56155 q^{17} +1.12311 q^{19} -0.438447 q^{20} +4.00000 q^{22} +5.12311 q^{23} +1.00000 q^{25} -7.12311 q^{26} -0.438447 q^{28} +5.68466 q^{29} +2.43845 q^{32} -7.12311 q^{34} +1.00000 q^{35} +6.00000 q^{37} -1.75379 q^{38} -2.43845 q^{40} +3.12311 q^{41} +9.12311 q^{43} -1.12311 q^{44} -8.00000 q^{46} -3.68466 q^{47} +1.00000 q^{49} -1.56155 q^{50} +2.00000 q^{52} -3.12311 q^{53} +2.56155 q^{55} -2.43845 q^{56} -8.87689 q^{58} +4.00000 q^{59} -9.36932 q^{61} +5.56155 q^{64} -4.56155 q^{65} -6.24621 q^{67} +2.00000 q^{68} -1.56155 q^{70} -8.00000 q^{71} +4.24621 q^{73} -9.36932 q^{74} +0.492423 q^{76} +2.56155 q^{77} -6.56155 q^{79} +4.68466 q^{80} -4.87689 q^{82} -4.00000 q^{83} -4.56155 q^{85} -14.2462 q^{86} -6.24621 q^{88} -7.12311 q^{89} -4.56155 q^{91} +2.24621 q^{92} +5.75379 q^{94} -1.12311 q^{95} -14.8078 q^{97} -1.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 5 q^{4} - 2 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) 2 * q + q^2 + 5 * q^4 - 2 * q^5 - 2 * q^7 + 9 * q^8	\( 2 q + q^{2} + 5 q^{4} - 2 q^{5} - 2 q^{7} + 9 q^{8} - q^{10} - q^{11} + 5 q^{13} - q^{14} + 3 q^{16} + 5 q^{17} - 6 q^{19} - 5 q^{20} + 8 q^{22} + 2 q^{23} + 2 q^{25} - 6 q^{26} - 5 q^{28} - q^{29} + 9 q^{32} - 6 q^{34} + 2 q^{35} + 12 q^{37} - 20 q^{38} - 9 q^{40} - 2 q^{41} + 10 q^{43} + 6 q^{44} - 16 q^{46} + 5 q^{47} + 2 q^{49} + q^{50} + 4 q^{52} + 2 q^{53} + q^{55} - 9 q^{56} - 26 q^{58} + 8 q^{59} + 6 q^{61} + 7 q^{64} - 5 q^{65} + 4 q^{67} + 4 q^{68} + q^{70} - 16 q^{71} - 8 q^{73} + 6 q^{74} - 32 q^{76} + q^{77} - 9 q^{79} - 3 q^{80} - 18 q^{82} - 8 q^{83} - 5 q^{85} - 12 q^{86} + 4 q^{88} - 6 q^{89} - 5 q^{91} - 12 q^{92} + 28 q^{94} + 6 q^{95} - 9 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 + 5 * q^4 - 2 * q^5 - 2 * q^7 + 9 * q^8 - q^10 - q^11 + 5 * q^13 - q^14 + 3 * q^16 + 5 * q^17 - 6 * q^19 - 5 * q^20 + 8 * q^22 + 2 * q^23 + 2 * q^25 - 6 * q^26 - 5 * q^28 - q^29 + 9 * q^32 - 6 * q^34 + 2 * q^35 + 12 * q^37 - 20 * q^38 - 9 * q^40 - 2 * q^41 + 10 * q^43 + 6 * q^44 - 16 * q^46 + 5 * q^47 + 2 * q^49 + q^50 + 4 * q^52 + 2 * q^53 + q^55 - 9 * q^56 - 26 * q^58 + 8 * q^59 + 6 * q^61 + 7 * q^64 - 5 * q^65 + 4 * q^67 + 4 * q^68 + q^70 - 16 * q^71 - 8 * q^73 + 6 * q^74 - 32 * q^76 + q^77 - 9 * q^79 - 3 * q^80 - 18 * q^82 - 8 * q^83 - 5 * q^85 - 12 * q^86 + 4 * q^88 - 6 * q^89 - 5 * q^91 - 12 * q^92 + 28 * q^94 + 6 * q^95 - 9 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more