LMFDB - Embedded newform 920.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, 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("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.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:	$7.34623698596$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +1.00000 q^{5} +5.12311 q^{7} +3.56155 q^{9} +O(q^{10})$	$q+2.56155 q^{3} +1.00000 q^{5} +5.12311 q^{7} +3.56155 q^{9} -4.00000 q^{11} -0.561553 q^{13} +2.56155 q^{15} -3.12311 q^{17} +4.00000 q^{19} +13.1231 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.43845 q^{27} -8.56155 q^{29} +1.43845 q^{31} -10.2462 q^{33} +5.12311 q^{35} -7.12311 q^{37} -1.43845 q^{39} +0.561553 q^{41} -9.12311 q^{43} +3.56155 q^{45} -3.68466 q^{47} +19.2462 q^{49} -8.00000 q^{51} -4.24621 q^{53} -4.00000 q^{55} +10.2462 q^{57} -6.24621 q^{59} +11.1231 q^{61} +18.2462 q^{63} -0.561553 q^{65} +6.24621 q^{67} +2.56155 q^{69} +3.68466 q^{71} +16.5616 q^{73} +2.56155 q^{75} -20.4924 q^{77} -10.2462 q^{79} -7.00000 q^{81} +12.0000 q^{83} -3.12311 q^{85} -21.9309 q^{87} +10.0000 q^{89} -2.87689 q^{91} +3.68466 q^{93} +4.00000 q^{95} -16.2462 q^{97} -14.2462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9	$ 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} - 8 q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} + 18 q^{21} + 2 q^{23} + 2 q^{25} + 7 q^{27} - 13 q^{29} + 7 q^{31} - 4 q^{33} + 2 q^{35} - 6 q^{37} - 7 q^{39} - 3 q^{41} - 10 q^{43} + 3 q^{45} + 5 q^{47} + 22 q^{49} - 16 q^{51} + 8 q^{53} - 8 q^{55} + 4 q^{57} + 4 q^{59} + 14 q^{61} + 20 q^{63} + 3 q^{65} - 4 q^{67} + q^{69} - 5 q^{71} + 29 q^{73} + q^{75} - 8 q^{77} - 4 q^{79} - 14 q^{81} + 24 q^{83} + 2 q^{85} - 15 q^{87} + 20 q^{89} - 14 q^{91} - 5 q^{93} + 8 q^{95} - 16 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 - 8 * q^11 + 3 * q^13 + q^15 + 2 * q^17 + 8 * q^19 + 18 * q^21 + 2 * q^23 + 2 * q^25 + 7 * q^27 - 13 * q^29 + 7 * q^31 - 4 * q^33 + 2 * q^35 - 6 * q^37 - 7 * q^39 - 3 * q^41 - 10 * q^43 + 3 * q^45 + 5 * q^47 + 22 * q^49 - 16 * q^51 + 8 * q^53 - 8 * q^55 + 4 * q^57 + 4 * q^59 + 14 * q^61 + 20 * q^63 + 3 * q^65 - 4 * q^67 + q^69 - 5 * q^71 + 29 * q^73 + q^75 - 8 * q^77 - 4 * q^79 - 14 * q^81 + 24 * q^83 + 2 * q^85 - 15 * q^87 + 20 * q^89 - 14 * q^91 - 5 * q^93 + 8 * q^95 - 16 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	5.12311	1.93635	0.968176	−	0.250270i	$-0.0805195\pi$
$7$	5.12311	1.93635	0.968176	+	0.250270i	$0.0805195\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−0.561553	−0.155747	−0.0778734	−	0.996963i	$-0.524813\pi$
$13$	−0.561553	−0.155747	−0.0778734	+	0.996963i	$0.524813\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	13.1231	2.86370
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	−8.56155	−1.58984	−0.794920	−	0.606714i	$-0.792487\pi$
$29$	−8.56155	−1.58984	−0.794920	+	0.606714i	$0.792487\pi$
$30$	0	0
$31$	1.43845	0.258353	0.129176	−	0.991622i	$-0.458767\pi$
$31$	1.43845	0.258353	0.129176	+	0.991622i	$0.458767\pi$
$32$	0	0
$33$	−10.2462	−1.78364
$34$	0	0
$35$	5.12311	0.865963
$36$	0	0
$37$	−7.12311	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$37$	−7.12311	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$38$	0	0
$39$	−1.43845	−0.230336
$40$	0	0
$41$	0.561553	0.0876998	0.0438499	−	0.999038i	$-0.486038\pi$
$41$	0.561553	0.0876998	0.0438499	+	0.999038i	$0.486038\pi$
$42$	0	0
$43$	−9.12311	−1.39126	−0.695630	−	0.718400i	$-0.744875\pi$
$43$	−9.12311	−1.39126	−0.695630	+	0.718400i	$0.744875\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	−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
$49$	19.2462	2.74946
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	−4.24621	−0.583262	−0.291631	−	0.956531i	$-0.594198\pi$
$53$	−4.24621	−0.583262	−0.291631	+	0.956531i	$0.594198\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	10.2462	1.35714
$58$	0	0
$59$	−6.24621	−0.813187	−0.406594	−	0.913609i	$-0.633284\pi$
$59$	−6.24621	−0.813187	−0.406594	+	0.913609i	$0.633284\pi$
$60$	0	0
$61$	11.1231	1.42417	0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231	1.42417	0.712084	+	0.702094i	$0.247752\pi$
$62$	0	0
$63$	18.2462	2.29881
$64$	0	0
$65$	−0.561553	−0.0696521
$66$	0	0
$67$	6.24621	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$68$	0	0
$69$	2.56155	0.308375
$70$	0	0
$71$	3.68466	0.437289	0.218644	−	0.975805i	$-0.429837\pi$
$71$	3.68466	0.437289	0.218644	+	0.975805i	$0.429837\pi$
$72$	0	0
$73$	16.5616	1.93838	0.969192	−	0.246308i	$-0.0792175\pi$
$73$	16.5616	1.93838	0.969192	+	0.246308i	$0.0792175\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	−20.4924	−2.33533
$78$	0	0
$79$	−10.2462	−1.15279	−0.576394	−	0.817172i	$-0.695541\pi$
$79$	−10.2462	−1.15279	−0.576394	+	0.817172i	$0.695541\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−3.12311	−0.338748
$86$	0	0
$87$	−21.9309	−2.35124
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−2.87689	−0.301580
$92$	0	0
$93$	3.68466	0.382081
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−16.2462	−1.64955	−0.824776	−	0.565459i	$-0.808699\pi$
$97$	−16.2462	−1.64955	−0.824776	+	0.565459i	$0.808699\pi$
$98$	0	0
$99$	−14.2462	−1.43180
$100$	0	0
$101$	0.246211	0.0244989	0.0122495	−	0.999925i	$-0.496101\pi$
$101$	0.246211	0.0244989	0.0122495	+	0.999925i	$0.496101\pi$
$102$	0	0
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	0	0
$105$	13.1231	1.28068
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	8.24621	0.789844	0.394922	−	0.918715i	$-0.370772\pi$
$109$	8.24621	0.789844	0.394922	+	0.918715i	$0.370772\pi$
$110$	0	0
$111$	−18.2462	−1.73185
$112$	0	0
$113$	20.2462	1.90460	0.952302	−	0.305158i	$-0.0987093\pi$
$113$	20.2462	1.90460	0.952302	+	0.305158i	$0.0987093\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	1.43845	0.129700
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.8078	−1.49145	−0.745724	−	0.666255i	$-0.767896\pi$
$127$	−16.8078	−1.49145	−0.745724	+	0.666255i	$0.767896\pi$
$128$	0	0
$129$	−23.3693	−2.05755
$130$	0	0
$131$	9.93087	0.867664	0.433832	−	0.900994i	$-0.357161\pi$
$131$	9.93087	0.867664	0.433832	+	0.900994i	$0.357161\pi$
$132$	0	0
$133$	20.4924	1.77692
$134$	0	0
$135$	1.43845	0.123802
$136$	0	0
$137$	15.1231	1.29205	0.646027	−	0.763315i	$-0.276429\pi$
$137$	15.1231	1.29205	0.646027	+	0.763315i	$0.276429\pi$
$138$	0	0
$139$	0.315342	0.0267469	0.0133735	−	0.999911i	$-0.495743\pi$
$139$	0.315342	0.0267469	0.0133735	+	0.999911i	$0.495743\pi$
$140$	0	0
$141$	−9.43845	−0.794861
$142$	0	0
$143$	2.24621	0.187838
$144$	0	0
$145$	−8.56155	−0.710998
$146$	0	0
$147$	49.3002	4.06621
$148$	0	0
$149$	−4.24621	−0.347863	−0.173932	−	0.984758i	$-0.555647\pi$
$149$	−4.24621	−0.347863	−0.173932	+	0.984758i	$0.555647\pi$
$150$	0	0
$151$	−16.8078	−1.36780	−0.683898	−	0.729577i	$-0.739717\pi$
$151$	−16.8078	−1.36780	−0.683898	+	0.729577i	$0.739717\pi$
$152$	0	0
$153$	−11.1231	−0.899250
$154$	0	0
$155$	1.43845	0.115539
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−10.8769	−0.862594
$160$	0	0
$161$	5.12311	0.403757
$162$	0	0
$163$	−0.315342	−0.0246995	−0.0123497	−	0.999924i	$-0.503931\pi$
$163$	−0.315342	−0.0246995	−0.0123497	+	0.999924i	$0.503931\pi$
$164$	0	0
$165$	−10.2462	−0.797666
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	14.2462	1.08944
$172$	0	0
$173$	10.4924	0.797724	0.398862	−	0.917011i	$-0.369405\pi$
$173$	10.4924	0.797724	0.398862	+	0.917011i	$0.369405\pi$
$174$	0	0
$175$	5.12311	0.387270
$176$	0	0
$177$	−16.0000	−1.20263
$178$	0	0
$179$	−7.68466	−0.574378	−0.287189	−	0.957874i	$-0.592721\pi$
$179$	−7.68466	−0.574378	−0.287189	+	0.957874i	$0.592721\pi$
$180$	0	0
$181$	3.12311	0.232139	0.116069	−	0.993241i	$-0.462971\pi$
$181$	3.12311	0.232139	0.116069	+	0.993241i	$0.462971\pi$
$182$	0	0
$183$	28.4924	2.10622
$184$	0	0
$185$	−7.12311	−0.523701
$186$	0	0
$187$	12.4924	0.913536
$188$	0	0
$189$	7.36932	0.536039
$190$	0	0
$191$	−2.87689	−0.208165	−0.104082	−	0.994569i	$-0.533191\pi$
$191$	−2.87689	−0.208165	−0.104082	+	0.994569i	$0.533191\pi$
$192$	0	0
$193$	24.5616	1.76798	0.883990	−	0.467507i	$-0.154848\pi$
$193$	24.5616	1.76798	0.883990	+	0.467507i	$0.154848\pi$
$194$	0	0
$195$	−1.43845	−0.103009
$196$	0	0
$197$	−11.4384	−0.814956	−0.407478	−	0.913215i	$-0.633592\pi$
$197$	−11.4384	−0.814956	−0.407478	+	0.913215i	$0.633592\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	−43.8617	−3.07849
$204$	0	0
$205$	0.561553	0.0392205
$206$	0	0
$207$	3.56155	0.247545
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−14.2462	−0.980750	−0.490375	−	0.871512i	$-0.663140\pi$
$211$	−14.2462	−0.980750	−0.490375	+	0.871512i	$0.663140\pi$
$212$	0	0
$213$	9.43845	0.646712
$214$	0	0
$215$	−9.12311	−0.622191
$216$	0	0
$217$	7.36932	0.500262
$218$	0	0
$219$	42.4233	2.86670
$220$	0	0
$221$	1.75379	0.117973
$222$	0	0
$223$	−20.4924	−1.37227	−0.686137	−	0.727472i	$-0.740695\pi$
$223$	−20.4924	−1.37227	−0.686137	+	0.727472i	$0.740695\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	9.12311	0.605522	0.302761	−	0.953067i	$-0.402092\pi$
$227$	9.12311	0.605522	0.302761	+	0.953067i	$0.402092\pi$
$228$	0	0
$229$	0.246211	0.0162701	0.00813505	−	0.999967i	$-0.497411\pi$
$229$	0.246211	0.0162701	0.00813505	+	0.999967i	$0.497411\pi$
$230$	0	0
$231$	−52.4924	−3.45375
$232$	0	0
$233$	−12.5616	−0.822935	−0.411467	−	0.911425i	$-0.634984\pi$
$233$	−12.5616	−0.822935	−0.411467	+	0.911425i	$0.634984\pi$
$234$	0	0
$235$	−3.68466	−0.240361
$236$	0	0
$237$	−26.2462	−1.70487
$238$	0	0
$239$	−8.80776	−0.569727	−0.284863	−	0.958568i	$-0.591948\pi$
$239$	−8.80776	−0.569727	−0.284863	+	0.958568i	$0.591948\pi$
$240$	0	0
$241$	−18.4924	−1.19120	−0.595601	−	0.803281i	$-0.703086\pi$
$241$	−18.4924	−1.19120	−0.595601	+	0.803281i	$0.703086\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	19.2462	1.22960
$246$	0	0
$247$	−2.24621	−0.142923
$248$	0	0
$249$	30.7386	1.94798
$250$	0	0
$251$	6.24621	0.394257	0.197129	−	0.980378i	$-0.436838\pi$
$251$	6.24621	0.394257	0.197129	+	0.980378i	$0.436838\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	−9.05398	−0.564771	−0.282386	−	0.959301i	$-0.591126\pi$
$257$	−9.05398	−0.564771	−0.282386	+	0.959301i	$0.591126\pi$
$258$	0	0
$259$	−36.4924	−2.26753
$260$	0	0
$261$	−30.4924	−1.88743
$262$	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
$265$	−4.24621	−0.260843
$266$	0	0
$267$	25.6155	1.56764
$268$	0	0
$269$	−23.3002	−1.42064	−0.710319	−	0.703880i	$-0.751449\pi$
$269$	−23.3002	−1.42064	−0.710319	+	0.703880i	$0.751449\pi$
$270$	0	0
$271$	−2.24621	−0.136448	−0.0682238	−	0.997670i	$-0.521733\pi$
$271$	−2.24621	−0.136448	−0.0682238	+	0.997670i	$0.521733\pi$
$272$	0	0
$273$	−7.36932	−0.446011
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−11.4384	−0.687270	−0.343635	−	0.939103i	$-0.611658\pi$
$277$	−11.4384	−0.687270	−0.343635	+	0.939103i	$0.611658\pi$
$278$	0	0
$279$	5.12311	0.306712
$280$	0	0
$281$	30.4924	1.81903	0.909513	−	0.415676i	$-0.136455\pi$
$281$	30.4924	1.81903	0.909513	+	0.415676i	$0.136455\pi$
$282$	0	0
$283$	22.8769	1.35989	0.679945	−	0.733263i	$-0.262004\pi$
$283$	22.8769	1.35989	0.679945	+	0.733263i	$0.262004\pi$
$284$	0	0
$285$	10.2462	0.606933
$286$	0	0
$287$	2.87689	0.169818
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	−41.6155	−2.43955
$292$	0	0
$293$	−12.8769	−0.752276	−0.376138	−	0.926564i	$-0.622748\pi$
$293$	−12.8769	−0.752276	−0.376138	+	0.926564i	$0.622748\pi$
$294$	0	0
$295$	−6.24621	−0.363668
$296$	0	0
$297$	−5.75379	−0.333869
$298$	0	0
$299$	−0.561553	−0.0324754
$300$	0	0
$301$	−46.7386	−2.69397
$302$	0	0
$303$	0.630683	0.0362318
$304$	0	0
$305$	11.1231	0.636907
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−5.75379	−0.327322
$310$	0	0
$311$	13.9309	0.789947	0.394974	−	0.918692i	$-0.370754\pi$
$311$	13.9309	0.789947	0.394974	+	0.918692i	$0.370754\pi$
$312$	0	0
$313$	15.1231	0.854808	0.427404	−	0.904061i	$-0.359428\pi$
$313$	15.1231	0.854808	0.427404	+	0.904061i	$0.359428\pi$
$314$	0	0
$315$	18.2462	1.02806
$316$	0	0
$317$	20.7386	1.16480	0.582399	−	0.812903i	$-0.302114\pi$
$317$	20.7386	1.16480	0.582399	+	0.812903i	$0.302114\pi$
$318$	0	0
$319$	34.2462	1.91742
$320$	0	0
$321$	−30.7386	−1.71566
$322$	0	0
$323$	−12.4924	−0.695097
$324$	0	0
$325$	−0.561553	−0.0311493
$326$	0	0
$327$	21.1231	1.16811
$328$	0	0
$329$	−18.8769	−1.04072
$330$	0	0
$331$	10.5616	0.580515	0.290258	−	0.956949i	$-0.406259\pi$
$331$	10.5616	0.580515	0.290258	+	0.956949i	$0.406259\pi$
$332$	0	0
$333$	−25.3693	−1.39023
$334$	0	0
$335$	6.24621	0.341267
$336$	0	0
$337$	15.7538	0.858164	0.429082	−	0.903266i	$-0.358837\pi$
$337$	15.7538	0.858164	0.429082	+	0.903266i	$0.358837\pi$
$338$	0	0
$339$	51.8617	2.81674
$340$	0	0
$341$	−5.75379	−0.311585
$342$	0	0
$343$	62.7386	3.38757
$344$	0	0
$345$	2.56155	0.137909
$346$	0	0
$347$	−16.4924	−0.885360	−0.442680	−	0.896680i	$-0.645972\pi$
$347$	−16.4924	−0.885360	−0.442680	+	0.896680i	$0.645972\pi$
$348$	0	0
$349$	−2.80776	−0.150296	−0.0751481	−	0.997172i	$-0.523943\pi$
$349$	−2.80776	−0.150296	−0.0751481	+	0.997172i	$0.523943\pi$
$350$	0	0
$351$	−0.807764	−0.0431153
$352$	0	0
$353$	34.8078	1.85263	0.926315	−	0.376750i	$-0.122958\pi$
$353$	34.8078	1.85263	0.926315	+	0.376750i	$0.122958\pi$
$354$	0	0
$355$	3.68466	0.195561
$356$	0	0
$357$	−40.9848	−2.16915
$358$	0	0
$359$	−13.7538	−0.725897	−0.362949	−	0.931809i	$-0.618230\pi$
$359$	−13.7538	−0.725897	−0.362949	+	0.931809i	$0.618230\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	12.8078	0.672233
$364$	0	0
$365$	16.5616	0.866871
$366$	0	0
$367$	20.4924	1.06970	0.534848	−	0.844948i	$-0.320369\pi$
$367$	20.4924	1.06970	0.534848	+	0.844948i	$0.320369\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−21.7538	−1.12940
$372$	0	0
$373$	−24.7386	−1.28092	−0.640459	−	0.767992i	$-0.721256\pi$
$373$	−24.7386	−1.28092	−0.640459	+	0.767992i	$0.721256\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	4.80776	0.247612
$378$	0	0
$379$	−14.2462	−0.731779	−0.365889	−	0.930658i	$-0.619235\pi$
$379$	−14.2462	−0.731779	−0.365889	+	0.930658i	$0.619235\pi$
$380$	0	0
$381$	−43.0540	−2.20572
$382$	0	0
$383$	33.6155	1.71767	0.858837	−	0.512250i	$-0.171188\pi$
$383$	33.6155	1.71767	0.858837	+	0.512250i	$0.171188\pi$
$384$	0	0
$385$	−20.4924	−1.04439
$386$	0	0
$387$	−32.4924	−1.65168
$388$	0	0
$389$	7.61553	0.386123	0.193061	−	0.981187i	$-0.438158\pi$
$389$	7.61553	0.386123	0.193061	+	0.981187i	$0.438158\pi$
$390$	0	0
$391$	−3.12311	−0.157942
$392$	0	0
$393$	25.4384	1.28320
$394$	0	0
$395$	−10.2462	−0.515543
$396$	0	0
$397$	3.93087	0.197285	0.0986423	−	0.995123i	$-0.468550\pi$
$397$	3.93087	0.197285	0.0986423	+	0.995123i	$0.468550\pi$
$398$	0	0
$399$	52.4924	2.62791
$400$	0	0
$401$	−28.7386	−1.43514	−0.717569	−	0.696487i	$-0.754745\pi$
$401$	−28.7386	−1.43514	−0.717569	+	0.696487i	$0.754745\pi$
$402$	0	0
$403$	−0.807764	−0.0402376
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	28.4924	1.41232
$408$	0	0
$409$	−2.31534	−0.114486	−0.0572431	−	0.998360i	$-0.518231\pi$
$409$	−2.31534	−0.114486	−0.0572431	+	0.998360i	$0.518231\pi$
$410$	0	0
$411$	38.7386	1.91084
$412$	0	0
$413$	−32.0000	−1.57462
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	0.807764	0.0395564
$418$	0	0
$419$	−9.12311	−0.445693	−0.222846	−	0.974854i	$-0.571535\pi$
$419$	−9.12311	−0.445693	−0.222846	+	0.974854i	$0.571535\pi$
$420$	0	0
$421$	26.4924	1.29116	0.645581	−	0.763692i	$-0.276615\pi$
$421$	26.4924	1.29116	0.645581	+	0.763692i	$0.276615\pi$
$422$	0	0
$423$	−13.1231	−0.638067
$424$	0	0
$425$	−3.12311	−0.151493
$426$	0	0
$427$	56.9848	2.75769
$428$	0	0
$429$	5.75379	0.277796
$430$	0	0
$431$	−33.6155	−1.61920	−0.809602	−	0.586980i	$-0.800317\pi$
$431$	−33.6155	−1.61920	−0.809602	+	0.586980i	$0.800317\pi$
$432$	0	0
$433$	24.7386	1.18886	0.594431	−	0.804146i	$-0.297377\pi$
$433$	24.7386	1.18886	0.594431	+	0.804146i	$0.297377\pi$
$434$	0	0
$435$	−21.9309	−1.05150
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	21.3002	1.01660	0.508301	−	0.861179i	$-0.330274\pi$
$439$	21.3002	1.01660	0.508301	+	0.861179i	$0.330274\pi$
$440$	0	0
$441$	68.5464	3.26411
$442$	0	0
$443$	−15.6847	−0.745201	−0.372600	−	0.927992i	$-0.621534\pi$
$443$	−15.6847	−0.745201	−0.372600	+	0.927992i	$0.621534\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	0	0
$447$	−10.8769	−0.514459
$448$	0	0
$449$	16.7386	0.789945	0.394972	−	0.918693i	$-0.370754\pi$
$449$	16.7386	0.789945	0.394972	+	0.918693i	$0.370754\pi$
$450$	0	0
$451$	−2.24621	−0.105770
$452$	0	0
$453$	−43.0540	−2.02285
$454$	0	0
$455$	−2.87689	−0.134871
$456$	0	0
$457$	−0.876894	−0.0410194	−0.0205097	−	0.999790i	$-0.506529\pi$
$457$	−0.876894	−0.0410194	−0.0205097	+	0.999790i	$0.506529\pi$
$458$	0	0
$459$	−4.49242	−0.209688
$460$	0	0
$461$	23.4384	1.09164	0.545819	−	0.837903i	$-0.316219\pi$
$461$	23.4384	1.09164	0.545819	+	0.837903i	$0.316219\pi$
$462$	0	0
$463$	10.2462	0.476182	0.238091	−	0.971243i	$-0.423478\pi$
$463$	10.2462	0.476182	0.238091	+	0.971243i	$0.423478\pi$
$464$	0	0
$465$	3.68466	0.170872
$466$	0	0
$467$	35.3693	1.63670	0.818348	−	0.574722i	$-0.194890\pi$
$467$	35.3693	1.63670	0.818348	+	0.574722i	$0.194890\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	−5.12311	−0.236060
$472$	0	0
$473$	36.4924	1.67792
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−15.1231	−0.692439
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	13.1231	0.597122
$484$	0	0
$485$	−16.2462	−0.737702
$486$	0	0
$487$	−3.05398	−0.138389	−0.0691944	−	0.997603i	$-0.522043\pi$
$487$	−3.05398	−0.138389	−0.0691944	+	0.997603i	$0.522043\pi$
$488$	0	0
$489$	−0.807764	−0.0365284
$490$	0	0
$491$	10.5616	0.476636	0.238318	−	0.971187i	$-0.423404\pi$
$491$	10.5616	0.476636	0.238318	+	0.971187i	$0.423404\pi$
$492$	0	0
$493$	26.7386	1.20425
$494$	0	0
$495$	−14.2462	−0.640320
$496$	0	0
$497$	18.8769	0.846744
$498$	0	0
$499$	0.946025	0.0423499	0.0211749	−	0.999776i	$-0.493259\pi$
$499$	0.946025	0.0423499	0.0211749	+	0.999776i	$0.493259\pi$
$500$	0	0
$501$	−20.4924	−0.915534
$502$	0	0
$503$	25.6155	1.14214	0.571070	−	0.820901i	$-0.306528\pi$
$503$	25.6155	1.14214	0.571070	+	0.820901i	$0.306528\pi$
$504$	0	0
$505$	0.246211	0.0109563
$506$	0	0
$507$	−32.4924	−1.44304
$508$	0	0
$509$	−8.56155	−0.379484	−0.189742	−	0.981834i	$-0.560765\pi$
$509$	−8.56155	−0.379484	−0.189742	+	0.981834i	$0.560765\pi$
$510$	0	0
$511$	84.8466	3.75339
$512$	0	0
$513$	5.75379	0.254036
$514$	0	0
$515$	−2.24621	−0.0989799
$516$	0	0
$517$	14.7386	0.648204
$518$	0	0
$519$	26.8769	1.17976
$520$	0	0
$521$	−17.8617	−0.782537	−0.391269	−	0.920277i	$-0.627964\pi$
$521$	−17.8617	−0.782537	−0.391269	+	0.920277i	$0.627964\pi$
$522$	0	0
$523$	−23.8617	−1.04340	−0.521701	−	0.853129i	$-0.674702\pi$
$523$	−23.8617	−1.04340	−0.521701	+	0.853129i	$0.674702\pi$
$524$	0	0
$525$	13.1231	0.572739
$526$	0	0
$527$	−4.49242	−0.195693
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−22.2462	−0.965403
$532$	0	0
$533$	−0.315342	−0.0136590
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−19.6847	−0.849456
$538$	0	0
$539$	−76.9848	−3.31597
$540$	0	0
$541$	33.6847	1.44822	0.724108	−	0.689686i	$-0.242252\pi$
$541$	33.6847	1.44822	0.724108	+	0.689686i	$0.242252\pi$
$542$	0	0
$543$	8.00000	0.343313
$544$	0	0
$545$	8.24621	0.353229
$546$	0	0
$547$	0.946025	0.0404491	0.0202245	−	0.999795i	$-0.493562\pi$
$547$	0.946025	0.0404491	0.0202245	+	0.999795i	$0.493562\pi$
$548$	0	0
$549$	39.6155	1.69075
$550$	0	0
$551$	−34.2462	−1.45894
$552$	0	0
$553$	−52.4924	−2.23220
$554$	0	0
$555$	−18.2462	−0.774509
$556$	0	0
$557$	−7.75379	−0.328539	−0.164269	−	0.986416i	$-0.552527\pi$
$557$	−7.75379	−0.328539	−0.164269	+	0.986416i	$0.552527\pi$
$558$	0	0
$559$	5.12311	0.216684
$560$	0	0
$561$	32.0000	1.35104
$562$	0	0
$563$	−39.2311	−1.65339	−0.826696	−	0.562649i	$-0.809782\pi$
$563$	−39.2311	−1.65339	−0.826696	+	0.562649i	$0.809782\pi$
$564$	0	0
$565$	20.2462	0.851765
$566$	0	0
$567$	−35.8617	−1.50605
$568$	0	0
$569$	−20.7386	−0.869409	−0.434704	−	0.900573i	$-0.643147\pi$
$569$	−20.7386	−0.869409	−0.434704	+	0.900573i	$0.643147\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	−7.36932	−0.307858
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−32.4233	−1.34980	−0.674900	−	0.737910i	$-0.735813\pi$
$577$	−32.4233	−1.34980	−0.674900	+	0.737910i	$0.735813\pi$
$578$	0	0
$579$	62.9157	2.61469
$580$	0	0
$581$	61.4773	2.55051
$582$	0	0
$583$	16.9848	0.703440
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−36.1771	−1.49319	−0.746594	−	0.665280i	$-0.768312\pi$
$587$	−36.1771	−1.49319	−0.746594	+	0.665280i	$0.768312\pi$
$588$	0	0
$589$	5.75379	0.237081
$590$	0	0
$591$	−29.3002	−1.20525
$592$	0	0
$593$	26.9848	1.10813	0.554067	−	0.832472i	$-0.313075\pi$
$593$	26.9848	1.10813	0.554067	+	0.832472i	$0.313075\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	61.4773	2.51610
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	18.1771	0.741459	0.370729	−	0.928741i	$-0.379108\pi$
$601$	18.1771	0.741459	0.370729	+	0.928741i	$0.379108\pi$
$602$	0	0
$603$	22.2462	0.905936
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	−112.354	−4.55282
$610$	0	0
$611$	2.06913	0.0837081
$612$	0	0
$613$	3.12311	0.126141	0.0630705	−	0.998009i	$-0.479911\pi$
$613$	3.12311	0.126141	0.0630705	+	0.998009i	$0.479911\pi$
$614$	0	0
$615$	1.43845	0.0580038
$616$	0	0
$617$	35.6155	1.43383	0.716914	−	0.697162i	$-0.245554\pi$
$617$	35.6155	1.43383	0.716914	+	0.697162i	$0.245554\pi$
$618$	0	0
$619$	−28.9848	−1.16500	−0.582500	−	0.812831i	$-0.697925\pi$
$619$	−28.9848	−1.16500	−0.582500	+	0.812831i	$0.697925\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	51.2311	2.05253
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−40.9848	−1.63678
$628$	0	0
$629$	22.2462	0.887015
$630$	0	0
$631$	−31.3693	−1.24879	−0.624396	−	0.781108i	$-0.714655\pi$
$631$	−31.3693	−1.24879	−0.624396	+	0.781108i	$0.714655\pi$
$632$	0	0
$633$	−36.4924	−1.45044
$634$	0	0
$635$	−16.8078	−0.666996
$636$	0	0
$637$	−10.8078	−0.428219
$638$	0	0
$639$	13.1231	0.519142
$640$	0	0
$641$	23.7538	0.938218	0.469109	−	0.883140i	$-0.344575\pi$
$641$	23.7538	0.938218	0.469109	+	0.883140i	$0.344575\pi$
$642$	0	0
$643$	44.3542	1.74916	0.874579	−	0.484884i	$-0.161138\pi$
$643$	44.3542	1.74916	0.874579	+	0.484884i	$0.161138\pi$
$644$	0	0
$645$	−23.3693	−0.920166
$646$	0	0
$647$	−1.43845	−0.0565512	−0.0282756	−	0.999600i	$-0.509002\pi$
$647$	−1.43845	−0.0565512	−0.0282756	+	0.999600i	$0.509002\pi$
$648$	0	0
$649$	24.9848	0.980741
$650$	0	0
$651$	18.8769	0.739844
$652$	0	0
$653$	−35.7926	−1.40067	−0.700337	−	0.713813i	$-0.746967\pi$
$653$	−35.7926	−1.40067	−0.700337	+	0.713813i	$0.746967\pi$
$654$	0	0
$655$	9.93087	0.388031
$656$	0	0
$657$	58.9848	2.30122
$658$	0	0
$659$	14.2462	0.554954	0.277477	−	0.960732i	$-0.410502\pi$
$659$	14.2462	0.554954	0.277477	+	0.960732i	$0.410502\pi$
$660$	0	0
$661$	10.4924	0.408108	0.204054	−	0.978960i	$-0.434588\pi$
$661$	10.4924	0.408108	0.204054	+	0.978960i	$0.434588\pi$
$662$	0	0
$663$	4.49242	0.174471
$664$	0	0
$665$	20.4924	0.794662
$666$	0	0
$667$	−8.56155	−0.331505
$668$	0	0
$669$	−52.4924	−2.02947
$670$	0	0
$671$	−44.4924	−1.71761
$672$	0	0
$673$	−4.56155	−0.175835	−0.0879175	−	0.996128i	$-0.528021\pi$
$673$	−4.56155	−0.175835	−0.0879175	+	0.996128i	$0.528021\pi$
$674$	0	0
$675$	1.43845	0.0553659
$676$	0	0
$677$	20.7386	0.797050	0.398525	−	0.917157i	$-0.369522\pi$
$677$	20.7386	0.797050	0.398525	+	0.917157i	$0.369522\pi$
$678$	0	0
$679$	−83.2311	−3.19411
$680$	0	0
$681$	23.3693	0.895514
$682$	0	0
$683$	−18.5616	−0.710238	−0.355119	−	0.934821i	$-0.615560\pi$
$683$	−18.5616	−0.710238	−0.355119	+	0.934821i	$0.615560\pi$
$684$	0	0
$685$	15.1231	0.577824
$686$	0	0
$687$	0.630683	0.0240621
$688$	0	0
$689$	2.38447	0.0908411
$690$	0	0
$691$	6.24621	0.237617	0.118809	−	0.992917i	$-0.462093\pi$
$691$	6.24621	0.237617	0.118809	+	0.992917i	$0.462093\pi$
$692$	0	0
$693$	−72.9848	−2.77247
$694$	0	0
$695$	0.315342	0.0119616
$696$	0	0
$697$	−1.75379	−0.0664295
$698$	0	0
$699$	−32.1771	−1.21705
$700$	0	0
$701$	38.9848	1.47244	0.736219	−	0.676744i	$-0.236610\pi$
$701$	38.9848	1.47244	0.736219	+	0.676744i	$0.236610\pi$
$702$	0	0
$703$	−28.4924	−1.07461
$704$	0	0
$705$	−9.43845	−0.355472
$706$	0	0
$707$	1.26137	0.0474386
$708$	0	0
$709$	−40.7386	−1.52997	−0.764986	−	0.644047i	$-0.777254\pi$
$709$	−40.7386	−1.52997	−0.764986	+	0.644047i	$0.777254\pi$
$710$	0	0
$711$	−36.4924	−1.36857
$712$	0	0
$713$	1.43845	0.0538703
$714$	0	0
$715$	2.24621	0.0840035
$716$	0	0
$717$	−22.5616	−0.842577
$718$	0	0
$719$	38.7386	1.44471	0.722354	−	0.691524i	$-0.243060\pi$
$719$	38.7386	1.44471	0.722354	+	0.691524i	$0.243060\pi$
$720$	0	0
$721$	−11.5076	−0.428565
$722$	0	0
$723$	−47.3693	−1.76168
$724$	0	0
$725$	−8.56155	−0.317968
$726$	0	0
$727$	37.1231	1.37682	0.688410	−	0.725322i	$-0.258309\pi$
$727$	37.1231	1.37682	0.688410	+	0.725322i	$0.258309\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	28.4924	1.05383
$732$	0	0
$733$	11.1231	0.410841	0.205421	−	0.978674i	$-0.434144\pi$
$733$	11.1231	0.410841	0.205421	+	0.978674i	$0.434144\pi$
$734$	0	0
$735$	49.3002	1.81846
$736$	0	0
$737$	−24.9848	−0.920329
$738$	0	0
$739$	27.5464	1.01331	0.506655	−	0.862149i	$-0.330882\pi$
$739$	27.5464	1.01331	0.506655	+	0.862149i	$0.330882\pi$
$740$	0	0
$741$	−5.75379	−0.211371
$742$	0	0
$743$	−13.7538	−0.504578	−0.252289	−	0.967652i	$-0.581183\pi$
$743$	−13.7538	−0.504578	−0.252289	+	0.967652i	$0.581183\pi$
$744$	0	0
$745$	−4.24621	−0.155569
$746$	0	0
$747$	42.7386	1.56372
$748$	0	0
$749$	−61.4773	−2.24633
$750$	0	0
$751$	49.6155	1.81050	0.905248	−	0.424883i	$-0.139685\pi$
$751$	49.6155	1.81050	0.905248	+	0.424883i	$0.139685\pi$
$752$	0	0
$753$	16.0000	0.583072
$754$	0	0
$755$	−16.8078	−0.611697
$756$	0	0
$757$	24.8769	0.904166	0.452083	−	0.891976i	$-0.350681\pi$
$757$	24.8769	0.904166	0.452083	+	0.891976i	$0.350681\pi$
$758$	0	0
$759$	−10.2462	−0.371914
$760$	0	0
$761$	−11.3002	−0.409631	−0.204816	−	0.978801i	$-0.565660\pi$
$761$	−11.3002	−0.409631	−0.204816	+	0.978801i	$0.565660\pi$
$762$	0	0
$763$	42.2462	1.52942
$764$	0	0
$765$	−11.1231	−0.402157
$766$	0	0
$767$	3.50758	0.126651
$768$	0	0
$769$	38.4924	1.38807	0.694036	−	0.719940i	$-0.255831\pi$
$769$	38.4924	1.38807	0.694036	+	0.719940i	$0.255831\pi$
$770$	0	0
$771$	−23.1922	−0.835248
$772$	0	0
$773$	−4.24621	−0.152726	−0.0763628	−	0.997080i	$-0.524331\pi$
$773$	−4.24621	−0.152726	−0.0763628	+	0.997080i	$0.524331\pi$
$774$	0	0
$775$	1.43845	0.0516705
$776$	0	0
$777$	−93.4773	−3.35348
$778$	0	0
$779$	2.24621	0.0804789
$780$	0	0
$781$	−14.7386	−0.527390
$782$	0	0
$783$	−12.3153	−0.440114
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−46.2462	−1.64850	−0.824250	−	0.566226i	$-0.808403\pi$
$787$	−46.2462	−1.64850	−0.824250	+	0.566226i	$0.808403\pi$
$788$	0	0
$789$	−20.4924	−0.729550
$790$	0	0
$791$	103.723	3.68798
$792$	0	0
$793$	−6.24621	−0.221809
$794$	0	0
$795$	−10.8769	−0.385764
$796$	0	0
$797$	27.1231	0.960750	0.480375	−	0.877063i	$-0.340501\pi$
$797$	27.1231	0.960750	0.480375	+	0.877063i	$0.340501\pi$
$798$	0	0
$799$	11.5076	0.407109
$800$	0	0
$801$	35.6155	1.25841
$802$	0	0
$803$	−66.2462	−2.33778
$804$	0	0
$805$	5.12311	0.180566
$806$	0	0
$807$	−59.6847	−2.10100
$808$	0	0
$809$	−10.4924	−0.368894	−0.184447	−	0.982842i	$-0.559049\pi$
$809$	−10.4924	−0.368894	−0.184447	+	0.982842i	$0.559049\pi$
$810$	0	0
$811$	−8.31534	−0.291991	−0.145996	−	0.989285i	$-0.546639\pi$
$811$	−8.31534	−0.291991	−0.145996	+	0.989285i	$0.546639\pi$
$812$	0	0
$813$	−5.75379	−0.201794
$814$	0	0
$815$	−0.315342	−0.0110459
$816$	0	0
$817$	−36.4924	−1.27671
$818$	0	0
$819$	−10.2462	−0.358032
$820$	0	0
$821$	−31.7538	−1.10821	−0.554107	−	0.832445i	$-0.686940\pi$
$821$	−31.7538	−1.10821	−0.554107	+	0.832445i	$0.686940\pi$
$822$	0	0
$823$	−17.4384	−0.607866	−0.303933	−	0.952693i	$-0.598300\pi$
$823$	−17.4384	−0.607866	−0.303933	+	0.952693i	$0.598300\pi$
$824$	0	0
$825$	−10.2462	−0.356727
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	50.4924	1.75367	0.876837	−	0.480787i	$-0.159649\pi$
$829$	50.4924	1.75367	0.876837	+	0.480787i	$0.159649\pi$
$830$	0	0
$831$	−29.3002	−1.01641
$832$	0	0
$833$	−60.1080	−2.08262
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	2.06913	0.0715196
$838$	0	0
$839$	9.61553	0.331965	0.165982	−	0.986129i	$-0.446920\pi$
$839$	9.61553	0.331965	0.165982	+	0.986129i	$0.446920\pi$
$840$	0	0
$841$	44.3002	1.52759
$842$	0	0
$843$	78.1080	2.69018
$844$	0	0
$845$	−12.6847	−0.436366
$846$	0	0
$847$	25.6155	0.880160
$848$	0	0
$849$	58.6004	2.01116
$850$	0	0
$851$	−7.12311	−0.244177
$852$	0	0
$853$	22.9848	0.786986	0.393493	−	0.919328i	$-0.371267\pi$
$853$	22.9848	0.786986	0.393493	+	0.919328i	$0.371267\pi$
$854$	0	0
$855$	14.2462	0.487210
$856$	0	0
$857$	34.1771	1.16747	0.583733	−	0.811945i	$-0.301591\pi$
$857$	34.1771	1.16747	0.583733	+	0.811945i	$0.301591\pi$
$858$	0	0
$859$	38.4233	1.31099	0.655493	−	0.755201i	$-0.272461\pi$
$859$	38.4233	1.31099	0.655493	+	0.755201i	$0.272461\pi$
$860$	0	0
$861$	7.36932	0.251146
$862$	0	0
$863$	18.4233	0.627136	0.313568	−	0.949566i	$-0.398476\pi$
$863$	18.4233	0.627136	0.313568	+	0.949566i	$0.398476\pi$
$864$	0	0
$865$	10.4924	0.356753
$866$	0	0
$867$	−18.5616	−0.630383
$868$	0	0
$869$	40.9848	1.39032
$870$	0	0
$871$	−3.50758	−0.118850
$872$	0	0
$873$	−57.8617	−1.95832
$874$	0	0
$875$	5.12311	0.173193
$876$	0	0
$877$	−40.7386	−1.37565	−0.687823	−	0.725878i	$-0.741433\pi$
$877$	−40.7386	−1.37565	−0.687823	+	0.725878i	$0.741433\pi$
$878$	0	0
$879$	−32.9848	−1.11255
$880$	0	0
$881$	24.1080	0.812217	0.406109	−	0.913825i	$-0.366885\pi$
$881$	24.1080	0.812217	0.406109	+	0.913825i	$0.366885\pi$
$882$	0	0
$883$	−40.4924	−1.36268	−0.681339	−	0.731968i	$-0.738602\pi$
$883$	−40.4924	−1.36268	−0.681339	+	0.731968i	$0.738602\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	0	0
$887$	−33.4384	−1.12275	−0.561377	−	0.827560i	$-0.689728\pi$
$887$	−33.4384	−1.12275	−0.561377	+	0.827560i	$0.689728\pi$
$888$	0	0
$889$	−86.1080	−2.88797
$890$	0	0
$891$	28.0000	0.938035
$892$	0	0
$893$	−14.7386	−0.493210
$894$	0	0
$895$	−7.68466	−0.256870
$896$	0	0
$897$	−1.43845	−0.0480284
$898$	0	0
$899$	−12.3153	−0.410740
$900$	0	0
$901$	13.2614	0.441800
$902$	0	0
$903$	−119.723	−3.98415
$904$	0	0
$905$	3.12311	0.103816
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	0.876894	0.0290848
$910$	0	0
$911$	−20.4924	−0.678944	−0.339472	−	0.940616i	$-0.610248\pi$
$911$	−20.4924	−0.678944	−0.339472	+	0.940616i	$0.610248\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	0	0
$915$	28.4924	0.941930
$916$	0	0
$917$	50.8769	1.68010
$918$	0	0
$919$	−35.8617	−1.18297	−0.591485	−	0.806316i	$-0.701458\pi$
$919$	−35.8617	−1.18297	−0.591485	+	0.806316i	$0.701458\pi$
$920$	0	0
$921$	30.7386	1.01287
$922$	0	0
$923$	−2.06913	−0.0681063
$924$	0	0
$925$	−7.12311	−0.234206
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	13.0540	0.428287	0.214144	−	0.976802i	$-0.431304\pi$
$929$	13.0540	0.428287	0.214144	+	0.976802i	$0.431304\pi$
$930$	0	0
$931$	76.9848	2.52308
$932$	0	0
$933$	35.6847	1.16826
$934$	0	0
$935$	12.4924	0.408546
$936$	0	0
$937$	41.3693	1.35148	0.675738	−	0.737142i	$-0.263825\pi$
$937$	41.3693	1.35148	0.675738	+	0.737142i	$0.263825\pi$
$938$	0	0
$939$	38.7386	1.26419
$940$	0	0
$941$	−60.6004	−1.97552	−0.987758	−	0.155995i	$-0.950142\pi$
$941$	−60.6004	−1.97552	−0.987758	+	0.155995i	$0.950142\pi$
$942$	0	0
$943$	0.561553	0.0182867
$944$	0	0
$945$	7.36932	0.239724
$946$	0	0
$947$	−35.1922	−1.14359	−0.571797	−	0.820395i	$-0.693754\pi$
$947$	−35.1922	−1.14359	−0.571797	+	0.820395i	$0.693754\pi$
$948$	0	0
$949$	−9.30019	−0.301897
$950$	0	0
$951$	53.1231	1.72263
$952$	0	0
$953$	13.5076	0.437553	0.218777	−	0.975775i	$-0.429793\pi$
$953$	13.5076	0.437553	0.218777	+	0.975775i	$0.429793\pi$
$954$	0	0
$955$	−2.87689	−0.0930941
$956$	0	0
$957$	87.7235	2.83570
$958$	0	0
$959$	77.4773	2.50187
$960$	0	0
$961$	−28.9309	−0.933254
$962$	0	0
$963$	−42.7386	−1.37723
$964$	0	0
$965$	24.5616	0.790664
$966$	0	0
$967$	0.177081	0.00569454	0.00284727	−	0.999996i	$-0.499094\pi$
$967$	0.177081	0.00569454	0.00284727	+	0.999996i	$0.499094\pi$
$968$	0	0
$969$	−32.0000	−1.02799
$970$	0	0
$971$	3.36932	0.108127	0.0540633	−	0.998538i	$-0.482783\pi$
$971$	3.36932	0.108127	0.0540633	+	0.998538i	$0.482783\pi$
$972$	0	0
$973$	1.61553	0.0517915
$974$	0	0
$975$	−1.43845	−0.0460672
$976$	0	0
$977$	48.1080	1.53911	0.769555	−	0.638581i	$-0.220478\pi$
$977$	48.1080	1.53911	0.769555	+	0.638581i	$0.220478\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	29.3693	0.937690
$982$	0	0
$983$	37.1231	1.18404	0.592022	−	0.805922i	$-0.298330\pi$
$983$	37.1231	1.18404	0.592022	+	0.805922i	$0.298330\pi$
$984$	0	0
$985$	−11.4384	−0.364459
$986$	0	0
$987$	−48.3542	−1.53913
$988$	0	0
$989$	−9.12311	−0.290098
$990$	0	0
$991$	−12.4924	−0.396835	−0.198417	−	0.980118i	$-0.563580\pi$
$991$	−12.4924	−0.396835	−0.198417	+	0.980118i	$0.563580\pi$
$992$	0	0
$993$	27.0540	0.858532
$994$	0	0
$995$	24.0000	0.760851
$996$	0	0
$997$	−48.7386	−1.54357	−0.771784	−	0.635885i	$-0.780635\pi$
$997$	−48.7386	−1.54357	−0.771784	+	0.635885i	$0.780635\pi$
$998$	0	0
$999$	−10.2462	−0.324176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.f.1.2	✓	2
3.2	odd	2		8280.2.a.bb.1.2		2
4.3	odd	2		1840.2.a.k.1.1		2
5.2	odd	4		4600.2.e.m.4049.1		4
5.3	odd	4		4600.2.e.m.4049.4		4
5.4	even	2		4600.2.a.r.1.1		2
8.3	odd	2		7360.2.a.bm.1.2		2
8.5	even	2		7360.2.a.bj.1.1		2
20.19	odd	2		9200.2.a.bx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.f.1.2	✓	2	1.1	even	1	trivial
1840.2.a.k.1.1		2	4.3	odd	2
4600.2.a.r.1.1		2	5.4	even	2
4600.2.e.m.4049.1		4	5.2	odd	4
4600.2.e.m.4049.4		4	5.3	odd	4
7360.2.a.bj.1.1		2	8.5	even	2
7360.2.a.bm.1.2		2	8.3	odd	2
8280.2.a.bb.1.2		2	3.2	odd	2
9200.2.a.bx.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +1.00000 q^{5} +5.12311 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q+2.56155 q^{3} +1.00000 q^{5} +5.12311 q^{7} +3.56155 q^{9} -4.00000 q^{11} -0.561553 q^{13} +2.56155 q^{15} -3.12311 q^{17} +4.00000 q^{19} +13.1231 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.43845 q^{27} -8.56155 q^{29} +1.43845 q^{31} -10.2462 q^{33} +5.12311 q^{35} -7.12311 q^{37} -1.43845 q^{39} +0.561553 q^{41} -9.12311 q^{43} +3.56155 q^{45} -3.68466 q^{47} +19.2462 q^{49} -8.00000 q^{51} -4.24621 q^{53} -4.00000 q^{55} +10.2462 q^{57} -6.24621 q^{59} +11.1231 q^{61} +18.2462 q^{63} -0.561553 q^{65} +6.24621 q^{67} +2.56155 q^{69} +3.68466 q^{71} +16.5616 q^{73} +2.56155 q^{75} -20.4924 q^{77} -10.2462 q^{79} -7.00000 q^{81} +12.0000 q^{83} -3.12311 q^{85} -21.9309 q^{87} +10.0000 q^{89} -2.87689 q^{91} +3.68466 q^{93} +4.00000 q^{95} -16.2462 q^{97} -14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9	\( 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} - 8 q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} + 18 q^{21} + 2 q^{23} + 2 q^{25} + 7 q^{27} - 13 q^{29} + 7 q^{31} - 4 q^{33} + 2 q^{35} - 6 q^{37} - 7 q^{39} - 3 q^{41} - 10 q^{43} + 3 q^{45} + 5 q^{47} + 22 q^{49} - 16 q^{51} + 8 q^{53} - 8 q^{55} + 4 q^{57} + 4 q^{59} + 14 q^{61} + 20 q^{63} + 3 q^{65} - 4 q^{67} + q^{69} - 5 q^{71} + 29 q^{73} + q^{75} - 8 q^{77} - 4 q^{79} - 14 q^{81} + 24 q^{83} + 2 q^{85} - 15 q^{87} + 20 q^{89} - 14 q^{91} - 5 q^{93} + 8 q^{95} - 16 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 - 8 * q^11 + 3 * q^13 + q^15 + 2 * q^17 + 8 * q^19 + 18 * q^21 + 2 * q^23 + 2 * q^25 + 7 * q^27 - 13 * q^29 + 7 * q^31 - 4 * q^33 + 2 * q^35 - 6 * q^37 - 7 * q^39 - 3 * q^41 - 10 * q^43 + 3 * q^45 + 5 * q^47 + 22 * q^49 - 16 * q^51 + 8 * q^53 - 8 * q^55 + 4 * q^57 + 4 * q^59 + 14 * q^61 + 20 * q^63 + 3 * q^65 - 4 * q^67 + q^69 - 5 * q^71 + 29 * q^73 + q^75 - 8 * q^77 - 4 * q^79 - 14 * q^81 + 24 * q^83 + 2 * q^85 - 15 * q^87 + 20 * q^89 - 14 * q^91 - 5 * q^93 + 8 * q^95 - 16 * q^97 - 12 * q^99

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