LMFDB - Embedded newform 1520.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.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:	$12.1372611072$
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:	no (minimal twist has level 190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} +O(q^{10})$	$q+2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} -4.00000 q^{11} +5.68466 q^{13} +2.56155 q^{15} +3.43845 q^{17} +1.00000 q^{19} +6.56155 q^{21} -7.68466 q^{23} +1.00000 q^{25} +1.43845 q^{27} -5.68466 q^{29} +5.12311 q^{31} -10.2462 q^{33} +2.56155 q^{35} -6.00000 q^{37} +14.5616 q^{39} +12.2462 q^{41} +2.87689 q^{43} +3.56155 q^{45} -6.24621 q^{47} -0.438447 q^{49} +8.80776 q^{51} -4.56155 q^{53} -4.00000 q^{55} +2.56155 q^{57} -2.56155 q^{59} +11.1231 q^{61} +9.12311 q^{63} +5.68466 q^{65} +2.56155 q^{67} -19.6847 q^{69} -10.2462 q^{71} -1.68466 q^{73} +2.56155 q^{75} -10.2462 q^{77} +5.12311 q^{79} -7.00000 q^{81} -2.87689 q^{83} +3.43845 q^{85} -14.5616 q^{87} +2.00000 q^{89} +14.5616 q^{91} +13.1231 q^{93} +1.00000 q^{95} +6.00000 q^{97} -14.2462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9	$ 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 8 q^{11} - q^{13} + q^{15} + 11 q^{17} + 2 q^{19} + 9 q^{21} - 3 q^{23} + 2 q^{25} + 7 q^{27} + q^{29} + 2 q^{31} - 4 q^{33} + q^{35} - 12 q^{37} + 25 q^{39} + 8 q^{41} + 14 q^{43} + 3 q^{45} + 4 q^{47} - 5 q^{49} - 3 q^{51} - 5 q^{53} - 8 q^{55} + q^{57} - q^{59} + 14 q^{61} + 10 q^{63} - q^{65} + q^{67} - 27 q^{69} - 4 q^{71} + 9 q^{73} + q^{75} - 4 q^{77} + 2 q^{79} - 14 q^{81} - 14 q^{83} + 11 q^{85} - 25 q^{87} + 4 q^{89} + 25 q^{91} + 18 q^{93} + 2 q^{95} + 12 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 - 8 * q^11 - q^13 + q^15 + 11 * q^17 + 2 * q^19 + 9 * q^21 - 3 * q^23 + 2 * q^25 + 7 * q^27 + q^29 + 2 * q^31 - 4 * q^33 + q^35 - 12 * q^37 + 25 * q^39 + 8 * q^41 + 14 * q^43 + 3 * q^45 + 4 * q^47 - 5 * q^49 - 3 * q^51 - 5 * q^53 - 8 * q^55 + q^57 - q^59 + 14 * q^61 + 10 * q^63 - q^65 + q^67 - 27 * q^69 - 4 * q^71 + 9 * q^73 + q^75 - 4 * q^77 + 2 * q^79 - 14 * q^81 - 14 * q^83 + 11 * q^85 - 25 * q^87 + 4 * q^89 + 25 * q^91 + 18 * q^93 + 2 * q^95 + 12 * 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
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	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
$13$	5.68466	1.57664	0.788320	−	0.615265i	$-0.210951\pi$
$13$	5.68466	1.57664	0.788320	+	0.615265i	$0.210951\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	3.43845	0.833946	0.416973	−	0.908919i	$-0.363091\pi$
$17$	3.43845	0.833946	0.416973	+	0.908919i	$0.363091\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	6.56155	1.43185
$22$	0	0
$23$	−7.68466	−1.60236	−0.801181	−	0.598422i	$-0.795795\pi$
$23$	−7.68466	−1.60236	−0.801181	+	0.598422i	$0.795795\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	0	0
$31$	5.12311	0.920137	0.460068	−	0.887883i	$-0.347825\pi$
$31$	5.12311	0.920137	0.460068	+	0.887883i	$0.347825\pi$
$32$	0	0
$33$	−10.2462	−1.78364
$34$	0	0
$35$	2.56155	0.432981
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	14.5616	2.33171
$40$	0	0
$41$	12.2462	1.91254	0.956268	−	0.292490i	$-0.0944840\pi$
$41$	12.2462	1.91254	0.956268	+	0.292490i	$0.0944840\pi$
$42$	0	0
$43$	2.87689	0.438722	0.219361	−	0.975644i	$-0.429603\pi$
$43$	2.87689	0.438722	0.219361	+	0.975644i	$0.429603\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	−6.24621	−0.911104	−0.455552	−	0.890209i	$-0.650558\pi$
$47$	−6.24621	−0.911104	−0.455552	+	0.890209i	$0.650558\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	8.80776	1.23333
$52$	0	0
$53$	−4.56155	−0.626577	−0.313289	−	0.949658i	$-0.601431\pi$
$53$	−4.56155	−0.626577	−0.313289	+	0.949658i	$0.601431\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	2.56155	0.339286
$58$	0	0
$59$	−2.56155	−0.333486	−0.166743	−	0.986000i	$-0.553325\pi$
$59$	−2.56155	−0.333486	−0.166743	+	0.986000i	$0.553325\pi$
$60$	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
$63$	9.12311	1.14940
$64$	0	0
$65$	5.68466	0.705095
$66$	0	0
$67$	2.56155	0.312943	0.156472	−	0.987682i	$-0.449988\pi$
$67$	2.56155	0.312943	0.156472	+	0.987682i	$0.449988\pi$
$68$	0	0
$69$	−19.6847	−2.36975
$70$	0	0
$71$	−10.2462	−1.21600	−0.608001	−	0.793936i	$-0.708028\pi$
$71$	−10.2462	−1.21600	−0.608001	+	0.793936i	$0.708028\pi$
$72$	0	0
$73$	−1.68466	−0.197174	−0.0985872	−	0.995128i	$-0.531432\pi$
$73$	−1.68466	−0.197174	−0.0985872	+	0.995128i	$0.531432\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	−10.2462	−1.16766
$78$	0	0
$79$	5.12311	0.576394	0.288197	−	0.957571i	$-0.406944\pi$
$79$	5.12311	0.576394	0.288197	+	0.957571i	$0.406944\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−2.87689	−0.315780	−0.157890	−	0.987457i	$-0.550469\pi$
$83$	−2.87689	−0.315780	−0.157890	+	0.987457i	$0.550469\pi$
$84$	0	0
$85$	3.43845	0.372952
$86$	0	0
$87$	−14.5616	−1.56116
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	14.5616	1.52647
$92$	0	0
$93$	13.1231	1.36080
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	−14.2462	−1.43180
$100$	0	0
$101$	−17.3693	−1.72831	−0.864156	−	0.503224i	$-0.832147\pi$
$101$	−17.3693	−1.72831	−0.864156	+	0.503224i	$0.832147\pi$
$102$	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
$105$	6.56155	0.640342
$106$	0	0
$107$	5.43845	0.525755	0.262877	−	0.964829i	$-0.415329\pi$
$107$	5.43845	0.525755	0.262877	+	0.964829i	$0.415329\pi$
$108$	0	0
$109$	−0.561553	−0.0537870	−0.0268935	−	0.999638i	$-0.508561\pi$
$109$	−0.561553	−0.0537870	−0.0268935	+	0.999638i	$0.508561\pi$
$110$	0	0
$111$	−15.3693	−1.45879
$112$	0	0
$113$	8.87689	0.835068	0.417534	−	0.908661i	$-0.362894\pi$
$113$	8.87689	0.835068	0.417534	+	0.908661i	$0.362894\pi$
$114$	0	0
$115$	−7.68466	−0.716598
$116$	0	0
$117$	20.2462	1.87176
$118$	0	0
$119$	8.80776	0.807406
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	31.3693	2.82848
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.1231	−1.16449	−0.582244	−	0.813014i	$-0.697825\pi$
$127$	−13.1231	−1.16449	−0.582244	+	0.813014i	$0.697825\pi$
$128$	0	0
$129$	7.36932	0.648832
$130$	0	0
$131$	16.4924	1.44095	0.720475	−	0.693481i	$-0.243924\pi$
$131$	16.4924	1.44095	0.720475	+	0.693481i	$0.243924\pi$
$132$	0	0
$133$	2.56155	0.222115
$134$	0	0
$135$	1.43845	0.123802
$136$	0	0
$137$	−14.8078	−1.26511	−0.632556	−	0.774514i	$-0.717994\pi$
$137$	−14.8078	−1.26511	−0.632556	+	0.774514i	$0.717994\pi$
$138$	0	0
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	0	0
$141$	−16.0000	−1.34744
$142$	0	0
$143$	−22.7386	−1.90150
$144$	0	0
$145$	−5.68466	−0.472085
$146$	0	0
$147$	−1.12311	−0.0926322
$148$	0	0
$149$	13.3693	1.09526	0.547629	−	0.836722i	$-0.315531\pi$
$149$	13.3693	1.09526	0.547629	+	0.836722i	$0.315531\pi$
$150$	0	0
$151$	−5.12311	−0.416912	−0.208456	−	0.978032i	$-0.566844\pi$
$151$	−5.12311	−0.416912	−0.208456	+	0.978032i	$0.566844\pi$
$152$	0	0
$153$	12.2462	0.990048
$154$	0	0
$155$	5.12311	0.411498
$156$	0	0
$157$	−20.2462	−1.61582	−0.807912	−	0.589303i	$-0.799402\pi$
$157$	−20.2462	−1.61582	−0.807912	+	0.589303i	$0.799402\pi$
$158$	0	0
$159$	−11.6847	−0.926654
$160$	0	0
$161$	−19.6847	−1.55137
$162$	0	0
$163$	15.3693	1.20382	0.601909	−	0.798565i	$-0.294407\pi$
$163$	15.3693	1.20382	0.601909	+	0.798565i	$0.294407\pi$
$164$	0	0
$165$	−10.2462	−0.797666
$166$	0	0
$167$	7.36932	0.570255	0.285127	−	0.958490i	$-0.407964\pi$
$167$	7.36932	0.570255	0.285127	+	0.958490i	$0.407964\pi$
$168$	0	0
$169$	19.3153	1.48580
$170$	0	0
$171$	3.56155	0.272359
$172$	0	0
$173$	20.2462	1.53929	0.769645	−	0.638471i	$-0.220433\pi$
$173$	20.2462	1.53929	0.769645	+	0.638471i	$0.220433\pi$
$174$	0	0
$175$	2.56155	0.193635
$176$	0	0
$177$	−6.56155	−0.493197
$178$	0	0
$179$	22.2462	1.66276	0.831380	−	0.555704i	$-0.187551\pi$
$179$	22.2462	1.66276	0.831380	+	0.555704i	$0.187551\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	28.4924	2.10622
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−13.7538	−1.00578
$188$	0	0
$189$	3.68466	0.268019
$190$	0	0
$191$	3.68466	0.266613	0.133306	−	0.991075i	$-0.457441\pi$
$191$	3.68466	0.266613	0.133306	+	0.991075i	$0.457441\pi$
$192$	0	0
$193$	−14.4924	−1.04319	−0.521594	−	0.853194i	$-0.674662\pi$
$193$	−14.4924	−1.04319	−0.521594	+	0.853194i	$0.674662\pi$
$194$	0	0
$195$	14.5616	1.04277
$196$	0	0
$197$	−20.2462	−1.44248	−0.721241	−	0.692684i	$-0.756428\pi$
$197$	−20.2462	−1.44248	−0.721241	+	0.692684i	$0.756428\pi$
$198$	0	0
$199$	−16.8078	−1.19147	−0.595735	−	0.803181i	$-0.703139\pi$
$199$	−16.8078	−1.19147	−0.595735	+	0.803181i	$0.703139\pi$
$200$	0	0
$201$	6.56155	0.462816
$202$	0	0
$203$	−14.5616	−1.02202
$204$	0	0
$205$	12.2462	0.855312
$206$	0	0
$207$	−27.3693	−1.90230
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−8.31534	−0.572452	−0.286226	−	0.958162i	$-0.592401\pi$
$211$	−8.31534	−0.572452	−0.286226	+	0.958162i	$0.592401\pi$
$212$	0	0
$213$	−26.2462	−1.79836
$214$	0	0
$215$	2.87689	0.196203
$216$	0	0
$217$	13.1231	0.890854
$218$	0	0
$219$	−4.31534	−0.291604
$220$	0	0
$221$	19.5464	1.31483
$222$	0	0
$223$	23.3693	1.56493	0.782463	−	0.622698i	$-0.213963\pi$
$223$	23.3693	1.56493	0.782463	+	0.622698i	$0.213963\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	−25.9309	−1.72109	−0.860546	−	0.509373i	$-0.829878\pi$
$227$	−25.9309	−1.72109	−0.860546	+	0.509373i	$0.829878\pi$
$228$	0	0
$229$	−14.4924	−0.957686	−0.478843	−	0.877900i	$-0.658944\pi$
$229$	−14.4924	−0.957686	−0.478843	+	0.877900i	$0.658944\pi$
$230$	0	0
$231$	−26.2462	−1.72687
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−6.24621	−0.407458
$236$	0	0
$237$	13.1231	0.852437
$238$	0	0
$239$	1.43845	0.0930454	0.0465227	−	0.998917i	$-0.485186\pi$
$239$	1.43845	0.0930454	0.0465227	+	0.998917i	$0.485186\pi$
$240$	0	0
$241$	23.1231	1.48949	0.744745	−	0.667349i	$-0.232571\pi$
$241$	23.1231	1.48949	0.744745	+	0.667349i	$0.232571\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	−0.438447	−0.0280114
$246$	0	0
$247$	5.68466	0.361706
$248$	0	0
$249$	−7.36932	−0.467011
$250$	0	0
$251$	−6.24621	−0.394257	−0.197129	−	0.980378i	$-0.563162\pi$
$251$	−6.24621	−0.394257	−0.197129	+	0.980378i	$0.563162\pi$
$252$	0	0
$253$	30.7386	1.93252
$254$	0	0
$255$	8.80776	0.551564
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	−15.3693	−0.955003
$260$	0	0
$261$	−20.2462	−1.25321
$262$	0	0
$263$	22.2462	1.37176	0.685880	−	0.727715i	$-0.259417\pi$
$263$	22.2462	1.37176	0.685880	+	0.727715i	$0.259417\pi$
$264$	0	0
$265$	−4.56155	−0.280214
$266$	0	0
$267$	5.12311	0.313529
$268$	0	0
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	21.9309	1.33221	0.666103	−	0.745860i	$-0.267961\pi$
$271$	21.9309	1.33221	0.666103	+	0.745860i	$0.267961\pi$
$272$	0	0
$273$	37.3002	2.25751
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	0.876894	0.0526875	0.0263437	−	0.999653i	$-0.491614\pi$
$277$	0.876894	0.0526875	0.0263437	+	0.999653i	$0.491614\pi$
$278$	0	0
$279$	18.2462	1.09237
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	21.1231	1.25564	0.627819	−	0.778359i	$-0.283948\pi$
$283$	21.1231	1.25564	0.627819	+	0.778359i	$0.283948\pi$
$284$	0	0
$285$	2.56155	0.151733
$286$	0	0
$287$	31.3693	1.85167
$288$	0	0
$289$	−5.17708	−0.304534
$290$	0	0
$291$	15.3693	0.900965
$292$	0	0
$293$	−22.1771	−1.29560	−0.647799	−	0.761811i	$-0.724311\pi$
$293$	−22.1771	−1.29560	−0.647799	+	0.761811i	$0.724311\pi$
$294$	0	0
$295$	−2.56155	−0.149139
$296$	0	0
$297$	−5.75379	−0.333869
$298$	0	0
$299$	−43.6847	−2.52635
$300$	0	0
$301$	7.36932	0.424760
$302$	0	0
$303$	−44.4924	−2.55602
$304$	0	0
$305$	11.1231	0.636907
$306$	0	0
$307$	−32.4924	−1.85444	−0.927220	−	0.374516i	$-0.877809\pi$
$307$	−32.4924	−1.85444	−0.927220	+	0.374516i	$0.877809\pi$
$308$	0	0
$309$	−5.75379	−0.327322
$310$	0	0
$311$	3.68466	0.208938	0.104469	−	0.994528i	$-0.466686\pi$
$311$	3.68466	0.208938	0.104469	+	0.994528i	$0.466686\pi$
$312$	0	0
$313$	5.05398	0.285668	0.142834	−	0.989747i	$-0.454379\pi$
$313$	5.05398	0.285668	0.142834	+	0.989747i	$0.454379\pi$
$314$	0	0
$315$	9.12311	0.514029
$316$	0	0
$317$	13.0540	0.733184	0.366592	−	0.930382i	$-0.380524\pi$
$317$	13.0540	0.733184	0.366592	+	0.930382i	$0.380524\pi$
$318$	0	0
$319$	22.7386	1.27312
$320$	0	0
$321$	13.9309	0.777545
$322$	0	0
$323$	3.43845	0.191320
$324$	0	0
$325$	5.68466	0.315328
$326$	0	0
$327$	−1.43845	−0.0795463
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	2.56155	0.140796	0.0703978	−	0.997519i	$-0.477573\pi$
$331$	2.56155	0.140796	0.0703978	+	0.997519i	$0.477573\pi$
$332$	0	0
$333$	−21.3693	−1.17103
$334$	0	0
$335$	2.56155	0.139953
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	22.7386	1.23499
$340$	0	0
$341$	−20.4924	−1.10973
$342$	0	0
$343$	−19.0540	−1.02882
$344$	0	0
$345$	−19.6847	−1.05979
$346$	0	0
$347$	8.63068	0.463319	0.231660	−	0.972797i	$-0.425584\pi$
$347$	8.63068	0.463319	0.231660	+	0.972797i	$0.425584\pi$
$348$	0	0
$349$	3.75379	0.200936	0.100468	−	0.994940i	$-0.467966\pi$
$349$	3.75379	0.200936	0.100468	+	0.994940i	$0.467966\pi$
$350$	0	0
$351$	8.17708	0.436460
$352$	0	0
$353$	−3.93087	−0.209219	−0.104610	−	0.994513i	$-0.533359\pi$
$353$	−3.93087	−0.209219	−0.104610	+	0.994513i	$0.533359\pi$
$354$	0	0
$355$	−10.2462	−0.543812
$356$	0	0
$357$	22.5616	1.19408
$358$	0	0
$359$	−1.43845	−0.0759183	−0.0379592	−	0.999279i	$-0.512086\pi$
$359$	−1.43845	−0.0759183	−0.0379592	+	0.999279i	$0.512086\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	12.8078	0.672233
$364$	0	0
$365$	−1.68466	−0.0881791
$366$	0	0
$367$	−6.24621	−0.326050	−0.163025	−	0.986622i	$-0.552125\pi$
$367$	−6.24621	−0.326050	−0.163025	+	0.986622i	$0.552125\pi$
$368$	0	0
$369$	43.6155	2.27053
$370$	0	0
$371$	−11.6847	−0.606637
$372$	0	0
$373$	−23.4384	−1.21360	−0.606798	−	0.794856i	$-0.707546\pi$
$373$	−23.4384	−1.21360	−0.606798	+	0.794856i	$0.707546\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	−32.3153	−1.66432
$378$	0	0
$379$	−10.5616	−0.542511	−0.271255	−	0.962507i	$-0.587439\pi$
$379$	−10.5616	−0.542511	−0.271255	+	0.962507i	$0.587439\pi$
$380$	0	0
$381$	−33.6155	−1.72218
$382$	0	0
$383$	−13.7538	−0.702786	−0.351393	−	0.936228i	$-0.614292\pi$
$383$	−13.7538	−0.702786	−0.351393	+	0.936228i	$0.614292\pi$
$384$	0	0
$385$	−10.2462	−0.522195
$386$	0	0
$387$	10.2462	0.520844
$388$	0	0
$389$	−7.12311	−0.361156	−0.180578	−	0.983561i	$-0.557797\pi$
$389$	−7.12311	−0.361156	−0.180578	+	0.983561i	$0.557797\pi$
$390$	0	0
$391$	−26.4233	−1.33628
$392$	0	0
$393$	42.2462	2.13104
$394$	0	0
$395$	5.12311	0.257771
$396$	0	0
$397$	−7.12311	−0.357498	−0.178749	−	0.983895i	$-0.557205\pi$
$397$	−7.12311	−0.357498	−0.178749	+	0.983895i	$0.557205\pi$
$398$	0	0
$399$	6.56155	0.328489
$400$	0	0
$401$	−3.75379	−0.187455	−0.0937276	−	0.995598i	$-0.529878\pi$
$401$	−3.75379	−0.187455	−0.0937276	+	0.995598i	$0.529878\pi$
$402$	0	0
$403$	29.1231	1.45073
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	24.7386	1.22325	0.611623	−	0.791149i	$-0.290517\pi$
$409$	24.7386	1.22325	0.611623	+	0.791149i	$0.290517\pi$
$410$	0	0
$411$	−37.9309	−1.87099
$412$	0	0
$413$	−6.56155	−0.322873
$414$	0	0
$415$	−2.87689	−0.141221
$416$	0	0
$417$	−42.2462	−2.06881
$418$	0	0
$419$	−23.8617	−1.16572	−0.582861	−	0.812572i	$-0.698067\pi$
$419$	−23.8617	−1.16572	−0.582861	+	0.812572i	$0.698067\pi$
$420$	0	0
$421$	−23.9309	−1.16632	−0.583160	−	0.812358i	$-0.698184\pi$
$421$	−23.9309	−1.16632	−0.583160	+	0.812358i	$0.698184\pi$
$422$	0	0
$423$	−22.2462	−1.08165
$424$	0	0
$425$	3.43845	0.166789
$426$	0	0
$427$	28.4924	1.37884
$428$	0	0
$429$	−58.2462	−2.81215
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	14.6307	0.703106	0.351553	−	0.936168i	$-0.385654\pi$
$433$	14.6307	0.703106	0.351553	+	0.936168i	$0.385654\pi$
$434$	0	0
$435$	−14.5616	−0.698173
$436$	0	0
$437$	−7.68466	−0.367607
$438$	0	0
$439$	13.1231	0.626332	0.313166	−	0.949698i	$-0.398610\pi$
$439$	13.1231	0.626332	0.313166	+	0.949698i	$0.398610\pi$
$440$	0	0
$441$	−1.56155	−0.0743597
$442$	0	0
$443$	−2.24621	−0.106721	−0.0533604	−	0.998575i	$-0.516993\pi$
$443$	−2.24621	−0.106721	−0.0533604	+	0.998575i	$0.516993\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	34.2462	1.61979
$448$	0	0
$449$	−28.7386	−1.35626	−0.678130	−	0.734942i	$-0.737209\pi$
$449$	−28.7386	−1.35626	−0.678130	+	0.734942i	$0.737209\pi$
$450$	0	0
$451$	−48.9848	−2.30661
$452$	0	0
$453$	−13.1231	−0.616577
$454$	0	0
$455$	14.5616	0.682656
$456$	0	0
$457$	6.31534	0.295419	0.147710	−	0.989031i	$-0.452810\pi$
$457$	6.31534	0.295419	0.147710	+	0.989031i	$0.452810\pi$
$458$	0	0
$459$	4.94602	0.230861
$460$	0	0
$461$	3.75379	0.174831	0.0874157	−	0.996172i	$-0.472139\pi$
$461$	3.75379	0.174831	0.0874157	+	0.996172i	$0.472139\pi$
$462$	0	0
$463$	30.2462	1.40566	0.702830	−	0.711358i	$-0.251919\pi$
$463$	30.2462	1.40566	0.702830	+	0.711358i	$0.251919\pi$
$464$	0	0
$465$	13.1231	0.608569
$466$	0	0
$467$	−18.2462	−0.844334	−0.422167	−	0.906518i	$-0.638730\pi$
$467$	−18.2462	−0.844334	−0.422167	+	0.906518i	$0.638730\pi$
$468$	0	0
$469$	6.56155	0.302984
$470$	0	0
$471$	−51.8617	−2.38966
$472$	0	0
$473$	−11.5076	−0.529119
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−16.2462	−0.743863
$478$	0	0
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	−34.1080	−1.55519
$482$	0	0
$483$	−50.4233	−2.29434
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	17.6155	0.798236	0.399118	−	0.916900i	$-0.369316\pi$
$487$	17.6155	0.798236	0.399118	+	0.916900i	$0.369316\pi$
$488$	0	0
$489$	39.3693	1.78034
$490$	0	0
$491$	1.12311	0.0506850	0.0253425	−	0.999679i	$-0.491932\pi$
$491$	1.12311	0.0506850	0.0253425	+	0.999679i	$0.491932\pi$
$492$	0	0
$493$	−19.5464	−0.880325
$494$	0	0
$495$	−14.2462	−0.640320
$496$	0	0
$497$	−26.2462	−1.17730
$498$	0	0
$499$	−42.1080	−1.88501	−0.942505	−	0.334191i	$-0.891537\pi$
$499$	−42.1080	−1.88501	−0.942505	+	0.334191i	$0.891537\pi$
$500$	0	0
$501$	18.8769	0.843357
$502$	0	0
$503$	7.05398	0.314521	0.157261	−	0.987557i	$-0.449734\pi$
$503$	7.05398	0.314521	0.157261	+	0.987557i	$0.449734\pi$
$504$	0	0
$505$	−17.3693	−0.772924
$506$	0	0
$507$	49.4773	2.19736
$508$	0	0
$509$	2.49242	0.110475	0.0552373	−	0.998473i	$-0.482408\pi$
$509$	2.49242	0.110475	0.0552373	+	0.998473i	$0.482408\pi$
$510$	0	0
$511$	−4.31534	−0.190899
$512$	0	0
$513$	1.43845	0.0635090
$514$	0	0
$515$	−2.24621	−0.0989799
$516$	0	0
$517$	24.9848	1.09883
$518$	0	0
$519$	51.8617	2.27648
$520$	0	0
$521$	−3.12311	−0.136826	−0.0684129	−	0.997657i	$-0.521794\pi$
$521$	−3.12311	−0.136826	−0.0684129	+	0.997657i	$0.521794\pi$
$522$	0	0
$523$	31.6847	1.38547	0.692737	−	0.721191i	$-0.256405\pi$
$523$	31.6847	1.38547	0.692737	+	0.721191i	$0.256405\pi$
$524$	0	0
$525$	6.56155	0.286370
$526$	0	0
$527$	17.6155	0.767344
$528$	0	0
$529$	36.0540	1.56756
$530$	0	0
$531$	−9.12311	−0.395909
$532$	0	0
$533$	69.6155	3.01538
$534$	0	0
$535$	5.43845	0.235125
$536$	0	0
$537$	56.9848	2.45908
$538$	0	0
$539$	1.75379	0.0755410
$540$	0	0
$541$	−0.384472	−0.0165297	−0.00826487	−	0.999966i	$-0.502631\pi$
$541$	−0.384472	−0.0165297	−0.00826487	+	0.999966i	$0.502631\pi$
$542$	0	0
$543$	−46.1080	−1.97868
$544$	0	0
$545$	−0.561553	−0.0240543
$546$	0	0
$547$	−16.4924	−0.705165	−0.352583	−	0.935781i	$-0.614696\pi$
$547$	−16.4924	−0.705165	−0.352583	+	0.935781i	$0.614696\pi$
$548$	0	0
$549$	39.6155	1.69075
$550$	0	0
$551$	−5.68466	−0.242175
$552$	0	0
$553$	13.1231	0.558051
$554$	0	0
$555$	−15.3693	−0.652391
$556$	0	0
$557$	39.6155	1.67856	0.839282	−	0.543697i	$-0.182976\pi$
$557$	39.6155	1.67856	0.839282	+	0.543697i	$0.182976\pi$
$558$	0	0
$559$	16.3542	0.691707
$560$	0	0
$561$	−35.2311	−1.48746
$562$	0	0
$563$	8.49242	0.357913	0.178956	−	0.983857i	$-0.442728\pi$
$563$	8.49242	0.357913	0.178956	+	0.983857i	$0.442728\pi$
$564$	0	0
$565$	8.87689	0.373454
$566$	0	0
$567$	−17.9309	−0.753026
$568$	0	0
$569$	−3.12311	−0.130927	−0.0654637	−	0.997855i	$-0.520853\pi$
$569$	−3.12311	−0.130927	−0.0654637	+	0.997855i	$0.520853\pi$
$570$	0	0
$571$	−21.6155	−0.904582	−0.452291	−	0.891870i	$-0.649393\pi$
$571$	−21.6155	−0.904582	−0.452291	+	0.891870i	$0.649393\pi$
$572$	0	0
$573$	9.43845	0.394297
$574$	0	0
$575$	−7.68466	−0.320472
$576$	0	0
$577$	−10.3153	−0.429433	−0.214717	−	0.976676i	$-0.568883\pi$
$577$	−10.3153	−0.429433	−0.214717	+	0.976676i	$0.568883\pi$
$578$	0	0
$579$	−37.1231	−1.54278
$580$	0	0
$581$	−7.36932	−0.305731
$582$	0	0
$583$	18.2462	0.755681
$584$	0	0
$585$	20.2462	0.837078
$586$	0	0
$587$	7.36932	0.304164	0.152082	−	0.988368i	$-0.451402\pi$
$587$	7.36932	0.304164	0.152082	+	0.988368i	$0.451402\pi$
$588$	0	0
$589$	5.12311	0.211094
$590$	0	0
$591$	−51.8617	−2.13331
$592$	0	0
$593$	7.75379	0.318410	0.159205	−	0.987246i	$-0.449107\pi$
$593$	7.75379	0.318410	0.159205	+	0.987246i	$0.449107\pi$
$594$	0	0
$595$	8.80776	0.361083
$596$	0	0
$597$	−43.0540	−1.76208
$598$	0	0
$599$	11.8617	0.484658	0.242329	−	0.970194i	$-0.422089\pi$
$599$	11.8617	0.484658	0.242329	+	0.970194i	$0.422089\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	0	0
$603$	9.12311	0.371522
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	21.1231	0.857360	0.428680	−	0.903456i	$-0.358979\pi$
$607$	21.1231	0.857360	0.428680	+	0.903456i	$0.358979\pi$
$608$	0	0
$609$	−37.3002	−1.51148
$610$	0	0
$611$	−35.5076	−1.43648
$612$	0	0
$613$	5.36932	0.216865	0.108432	−	0.994104i	$-0.465417\pi$
$613$	5.36932	0.216865	0.108432	+	0.994104i	$0.465417\pi$
$614$	0	0
$615$	31.3693	1.26493
$616$	0	0
$617$	12.2462	0.493014	0.246507	−	0.969141i	$-0.420717\pi$
$617$	12.2462	0.493014	0.246507	+	0.969141i	$0.420717\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	−11.0540	−0.443581
$622$	0	0
$623$	5.12311	0.205253
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.2462	−0.409194
$628$	0	0
$629$	−20.6307	−0.822599
$630$	0	0
$631$	20.4924	0.815790	0.407895	−	0.913029i	$-0.366263\pi$
$631$	20.4924	0.815790	0.407895	+	0.913029i	$0.366263\pi$
$632$	0	0
$633$	−21.3002	−0.846606
$634$	0	0
$635$	−13.1231	−0.520775
$636$	0	0
$637$	−2.49242	−0.0987534
$638$	0	0
$639$	−36.4924	−1.44362
$640$	0	0
$641$	21.8617	0.863487	0.431743	−	0.901996i	$-0.357899\pi$
$641$	21.8617	0.863487	0.431743	+	0.901996i	$0.357899\pi$
$642$	0	0
$643$	33.6155	1.32567	0.662834	−	0.748767i	$-0.269354\pi$
$643$	33.6155	1.32567	0.662834	+	0.748767i	$0.269354\pi$
$644$	0	0
$645$	7.36932	0.290167
$646$	0	0
$647$	−5.43845	−0.213807	−0.106904	−	0.994269i	$-0.534094\pi$
$647$	−5.43845	−0.213807	−0.106904	+	0.994269i	$0.534094\pi$
$648$	0	0
$649$	10.2462	0.402199
$650$	0	0
$651$	33.6155	1.31750
$652$	0	0
$653$	−7.12311	−0.278749	−0.139374	−	0.990240i	$-0.544509\pi$
$653$	−7.12311	−0.278749	−0.139374	+	0.990240i	$0.544509\pi$
$654$	0	0
$655$	16.4924	0.644412
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	14.0691	0.548056	0.274028	−	0.961722i	$-0.411644\pi$
$659$	14.0691	0.548056	0.274028	+	0.961722i	$0.411644\pi$
$660$	0	0
$661$	−10.8078	−0.420373	−0.210187	−	0.977661i	$-0.567407\pi$
$661$	−10.8078	−0.420373	−0.210187	+	0.977661i	$0.567407\pi$
$662$	0	0
$663$	50.0691	1.94452
$664$	0	0
$665$	2.56155	0.0993328
$666$	0	0
$667$	43.6847	1.69148
$668$	0	0
$669$	59.8617	2.31439
$670$	0	0
$671$	−44.4924	−1.71761
$672$	0	0
$673$	21.3693	0.823727	0.411863	−	0.911246i	$-0.364878\pi$
$673$	21.3693	0.823727	0.411863	+	0.911246i	$0.364878\pi$
$674$	0	0
$675$	1.43845	0.0553659
$676$	0	0
$677$	40.5616	1.55891	0.779454	−	0.626460i	$-0.215497\pi$
$677$	40.5616	1.55891	0.779454	+	0.626460i	$0.215497\pi$
$678$	0	0
$679$	15.3693	0.589820
$680$	0	0
$681$	−66.4233	−2.54535
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−14.8078	−0.565776
$686$	0	0
$687$	−37.1231	−1.41633
$688$	0	0
$689$	−25.9309	−0.987887
$690$	0	0
$691$	17.1231	0.651394	0.325697	−	0.945474i	$-0.394401\pi$
$691$	17.1231	0.651394	0.325697	+	0.945474i	$0.394401\pi$
$692$	0	0
$693$	−36.4924	−1.38623
$694$	0	0
$695$	−16.4924	−0.625593
$696$	0	0
$697$	42.1080	1.59495
$698$	0	0
$699$	25.6155	0.968868
$700$	0	0
$701$	−20.8769	−0.788509	−0.394255	−	0.919001i	$-0.628997\pi$
$701$	−20.8769	−0.788509	−0.394255	+	0.919001i	$0.628997\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	−44.4924	−1.67331
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	18.2462	0.684286
$712$	0	0
$713$	−39.3693	−1.47439
$714$	0	0
$715$	−22.7386	−0.850377
$716$	0	0
$717$	3.68466	0.137606
$718$	0	0
$719$	25.4384	0.948694	0.474347	−	0.880338i	$-0.342684\pi$
$719$	25.4384	0.948694	0.474347	+	0.880338i	$0.342684\pi$
$720$	0	0
$721$	−5.75379	−0.214282
$722$	0	0
$723$	59.2311	2.20283
$724$	0	0
$725$	−5.68466	−0.211123
$726$	0	0
$727$	24.3153	0.901806	0.450903	−	0.892573i	$-0.351102\pi$
$727$	24.3153	0.901806	0.450903	+	0.892573i	$0.351102\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	9.89205	0.365871
$732$	0	0
$733$	−36.8769	−1.36208	−0.681040	−	0.732247i	$-0.738472\pi$
$733$	−36.8769	−1.36208	−0.681040	+	0.732247i	$0.738472\pi$
$734$	0	0
$735$	−1.12311	−0.0414264
$736$	0	0
$737$	−10.2462	−0.377424
$738$	0	0
$739$	−25.1231	−0.924168	−0.462084	−	0.886836i	$-0.652898\pi$
$739$	−25.1231	−0.924168	−0.462084	+	0.886836i	$0.652898\pi$
$740$	0	0
$741$	14.5616	0.534932
$742$	0	0
$743$	−18.8769	−0.692526	−0.346263	−	0.938137i	$-0.612550\pi$
$743$	−18.8769	−0.692526	−0.346263	+	0.938137i	$0.612550\pi$
$744$	0	0
$745$	13.3693	0.489814
$746$	0	0
$747$	−10.2462	−0.374889
$748$	0	0
$749$	13.9309	0.509023
$750$	0	0
$751$	34.8769	1.27268	0.636338	−	0.771410i	$-0.280448\pi$
$751$	34.8769	1.27268	0.636338	+	0.771410i	$0.280448\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	0	0
$755$	−5.12311	−0.186449
$756$	0	0
$757$	42.4924	1.54441	0.772207	−	0.635371i	$-0.219153\pi$
$757$	42.4924	1.54441	0.772207	+	0.635371i	$0.219153\pi$
$758$	0	0
$759$	78.7386	2.85803
$760$	0	0
$761$	41.5464	1.50606	0.753028	−	0.657989i	$-0.228593\pi$
$761$	41.5464	1.50606	0.753028	+	0.657989i	$0.228593\pi$
$762$	0	0
$763$	−1.43845	−0.0520753
$764$	0	0
$765$	12.2462	0.442763
$766$	0	0
$767$	−14.5616	−0.525787
$768$	0	0
$769$	27.4384	0.989456	0.494728	−	0.869048i	$-0.335268\pi$
$769$	27.4384	0.989456	0.494728	+	0.869048i	$0.335268\pi$
$770$	0	0
$771$	35.8617	1.29153
$772$	0	0
$773$	10.8078	0.388728	0.194364	−	0.980929i	$-0.437736\pi$
$773$	10.8078	0.388728	0.194364	+	0.980929i	$0.437736\pi$
$774$	0	0
$775$	5.12311	0.184027
$776$	0	0
$777$	−39.3693	−1.41237
$778$	0	0
$779$	12.2462	0.438766
$780$	0	0
$781$	40.9848	1.46655
$782$	0	0
$783$	−8.17708	−0.292225
$784$	0	0
$785$	−20.2462	−0.722618
$786$	0	0
$787$	−11.1922	−0.398960	−0.199480	−	0.979902i	$-0.563925\pi$
$787$	−11.1922	−0.398960	−0.199480	+	0.979902i	$0.563925\pi$
$788$	0	0
$789$	56.9848	2.02871
$790$	0	0
$791$	22.7386	0.808493
$792$	0	0
$793$	63.2311	2.24540
$794$	0	0
$795$	−11.6847	−0.414412
$796$	0	0
$797$	−11.3002	−0.400273	−0.200137	−	0.979768i	$-0.564139\pi$
$797$	−11.3002	−0.400273	−0.200137	+	0.979768i	$0.564139\pi$
$798$	0	0
$799$	−21.4773	−0.759811
$800$	0	0
$801$	7.12311	0.251683
$802$	0	0
$803$	6.73863	0.237801
$804$	0	0
$805$	−19.6847	−0.693793
$806$	0	0
$807$	−66.6004	−2.34444
$808$	0	0
$809$	−12.5616	−0.441641	−0.220820	−	0.975315i	$-0.570873\pi$
$809$	−12.5616	−0.441641	−0.220820	+	0.975315i	$0.570873\pi$
$810$	0	0
$811$	20.8078	0.730659	0.365330	−	0.930878i	$-0.380956\pi$
$811$	20.8078	0.730659	0.365330	+	0.930878i	$0.380956\pi$
$812$	0	0
$813$	56.1771	1.97022
$814$	0	0
$815$	15.3693	0.538364
$816$	0	0
$817$	2.87689	0.100650
$818$	0	0
$819$	51.8617	1.81220
$820$	0	0
$821$	−17.3693	−0.606193	−0.303097	−	0.952960i	$-0.598021\pi$
$821$	−17.3693	−0.606193	−0.303097	+	0.952960i	$0.598021\pi$
$822$	0	0
$823$	29.4384	1.02616	0.513080	−	0.858341i	$-0.328504\pi$
$823$	29.4384	1.02616	0.513080	+	0.858341i	$0.328504\pi$
$824$	0	0
$825$	−10.2462	−0.356727
$826$	0	0
$827$	10.5616	0.367261	0.183631	−	0.982995i	$-0.441215\pi$
$827$	10.5616	0.367261	0.183631	+	0.982995i	$0.441215\pi$
$828$	0	0
$829$	−21.0540	−0.731235	−0.365617	−	0.930765i	$-0.619142\pi$
$829$	−21.0540	−0.731235	−0.365617	+	0.930765i	$0.619142\pi$
$830$	0	0
$831$	2.24621	0.0779202
$832$	0	0
$833$	−1.50758	−0.0522345
$834$	0	0
$835$	7.36932	0.255026
$836$	0	0
$837$	7.36932	0.254721
$838$	0	0
$839$	−20.4924	−0.707477	−0.353738	−	0.935344i	$-0.615090\pi$
$839$	−20.4924	−0.707477	−0.353738	+	0.935344i	$0.615090\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	0	0
$843$	5.12311	0.176449
$844$	0	0
$845$	19.3153	0.664468
$846$	0	0
$847$	12.8078	0.440080
$848$	0	0
$849$	54.1080	1.85698
$850$	0	0
$851$	46.1080	1.58056
$852$	0	0
$853$	−24.7386	−0.847035	−0.423517	−	0.905888i	$-0.639205\pi$
$853$	−24.7386	−0.847035	−0.423517	+	0.905888i	$0.639205\pi$
$854$	0	0
$855$	3.56155	0.121803
$856$	0	0
$857$	14.6307	0.499775	0.249887	−	0.968275i	$-0.419606\pi$
$857$	14.6307	0.499775	0.249887	+	0.968275i	$0.419606\pi$
$858$	0	0
$859$	52.9848	1.80782	0.903910	−	0.427723i	$-0.140684\pi$
$859$	52.9848	1.80782	0.903910	+	0.427723i	$0.140684\pi$
$860$	0	0
$861$	80.3542	2.73846
$862$	0	0
$863$	−2.24621	−0.0764619	−0.0382310	−	0.999269i	$-0.512172\pi$
$863$	−2.24621	−0.0764619	−0.0382310	+	0.999269i	$0.512172\pi$
$864$	0	0
$865$	20.2462	0.688392
$866$	0	0
$867$	−13.2614	−0.450380
$868$	0	0
$869$	−20.4924	−0.695158
$870$	0	0
$871$	14.5616	0.493399
$872$	0	0
$873$	21.3693	0.723242
$874$	0	0
$875$	2.56155	0.0865963
$876$	0	0
$877$	−3.93087	−0.132736	−0.0663680	−	0.997795i	$-0.521141\pi$
$877$	−3.93087	−0.132736	−0.0663680	+	0.997795i	$0.521141\pi$
$878$	0	0
$879$	−56.8078	−1.91608
$880$	0	0
$881$	42.9848	1.44820	0.724098	−	0.689697i	$-0.242256\pi$
$881$	42.9848	1.44820	0.724098	+	0.689697i	$0.242256\pi$
$882$	0	0
$883$	6.38447	0.214855	0.107427	−	0.994213i	$-0.465739\pi$
$883$	6.38447	0.214855	0.107427	+	0.994213i	$0.465739\pi$
$884$	0	0
$885$	−6.56155	−0.220564
$886$	0	0
$887$	−28.4924	−0.956682	−0.478341	−	0.878174i	$-0.658762\pi$
$887$	−28.4924	−0.956682	−0.478341	+	0.878174i	$0.658762\pi$
$888$	0	0
$889$	−33.6155	−1.12743
$890$	0	0
$891$	28.0000	0.938035
$892$	0	0
$893$	−6.24621	−0.209021
$894$	0	0
$895$	22.2462	0.743609
$896$	0	0
$897$	−111.901	−3.73625
$898$	0	0
$899$	−29.1231	−0.971310
$900$	0	0
$901$	−15.6847	−0.522532
$902$	0	0
$903$	18.8769	0.628184
$904$	0	0
$905$	−18.0000	−0.598340
$906$	0	0
$907$	20.1771	0.669969	0.334984	−	0.942224i	$-0.391269\pi$
$907$	20.1771	0.669969	0.334984	+	0.942224i	$0.391269\pi$
$908$	0	0
$909$	−61.8617	−2.05182
$910$	0	0
$911$	−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
$913$	11.5076	0.380845
$914$	0	0
$915$	28.4924	0.941930
$916$	0	0
$917$	42.2462	1.39509
$918$	0	0
$919$	2.06913	0.0682543	0.0341272	−	0.999417i	$-0.489135\pi$
$919$	2.06913	0.0682543	0.0341272	+	0.999417i	$0.489135\pi$
$920$	0	0
$921$	−83.2311	−2.74256
$922$	0	0
$923$	−58.2462	−1.91720
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−19.3002	−0.633219	−0.316609	−	0.948556i	$-0.602544\pi$
$929$	−19.3002	−0.633219	−0.316609	+	0.948556i	$0.602544\pi$
$930$	0	0
$931$	−0.438447	−0.0143695
$932$	0	0
$933$	9.43845	0.309001
$934$	0	0
$935$	−13.7538	−0.449797
$936$	0	0
$937$	40.5616	1.32509	0.662544	−	0.749023i	$-0.269477\pi$
$937$	40.5616	1.32509	0.662544	+	0.749023i	$0.269477\pi$
$938$	0	0
$939$	12.9460	0.422478
$940$	0	0
$941$	54.8078	1.78668	0.893341	−	0.449379i	$-0.148355\pi$
$941$	54.8078	1.78668	0.893341	+	0.449379i	$0.148355\pi$
$942$	0	0
$943$	−94.1080	−3.06458
$944$	0	0
$945$	3.68466	0.119862
$946$	0	0
$947$	34.2462	1.11285	0.556426	−	0.830897i	$-0.312172\pi$
$947$	34.2462	1.11285	0.556426	+	0.830897i	$0.312172\pi$
$948$	0	0
$949$	−9.57671	−0.310873
$950$	0	0
$951$	33.4384	1.08432
$952$	0	0
$953$	44.1080	1.42880	0.714398	−	0.699739i	$-0.246700\pi$
$953$	44.1080	1.42880	0.714398	+	0.699739i	$0.246700\pi$
$954$	0	0
$955$	3.68466	0.119233
$956$	0	0
$957$	58.2462	1.88283
$958$	0	0
$959$	−37.9309	−1.22485
$960$	0	0
$961$	−4.75379	−0.153348
$962$	0	0
$963$	19.3693	0.624168
$964$	0	0
$965$	−14.4924	−0.466528
$966$	0	0
$967$	15.5076	0.498690	0.249345	−	0.968415i	$-0.419785\pi$
$967$	15.5076	0.498690	0.249345	+	0.968415i	$0.419785\pi$
$968$	0	0
$969$	8.80776	0.282946
$970$	0	0
$971$	56.4924	1.81293	0.906464	−	0.422283i	$-0.138771\pi$
$971$	56.4924	1.81293	0.906464	+	0.422283i	$0.138771\pi$
$972$	0	0
$973$	−42.2462	−1.35435
$974$	0	0
$975$	14.5616	0.466343
$976$	0	0
$977$	28.7386	0.919430	0.459715	−	0.888066i	$-0.347952\pi$
$977$	28.7386	0.919430	0.459715	+	0.888066i	$0.347952\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−18.8769	−0.602079	−0.301040	−	0.953612i	$-0.597334\pi$
$983$	−18.8769	−0.602079	−0.301040	+	0.953612i	$0.597334\pi$
$984$	0	0
$985$	−20.2462	−0.645098
$986$	0	0
$987$	−40.9848	−1.30456
$988$	0	0
$989$	−22.1080	−0.702992
$990$	0	0
$991$	−2.87689	−0.0913876	−0.0456938	−	0.998955i	$-0.514550\pi$
$991$	−2.87689	−0.0913876	−0.0456938	+	0.998955i	$0.514550\pi$
$992$	0	0
$993$	6.56155	0.208225
$994$	0	0
$995$	−16.8078	−0.532842
$996$	0	0
$997$	−16.7386	−0.530118	−0.265059	−	0.964232i	$-0.585391\pi$
$997$	−16.7386	−0.530118	−0.265059	+	0.964232i	$0.585391\pi$
$998$	0	0
$999$	−8.63068	−0.273063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.n.1.2		2
4.3	odd	2		190.2.a.d.1.1	✓	2
5.4	even	2		7600.2.a.y.1.1		2
8.3	odd	2		6080.2.a.bh.1.2		2
8.5	even	2		6080.2.a.bb.1.1		2
12.11	even	2		1710.2.a.w.1.1		2
20.3	even	4		950.2.b.f.799.3		4
20.7	even	4		950.2.b.f.799.2		4
20.19	odd	2		950.2.a.h.1.2		2
28.27	even	2		9310.2.a.bc.1.2		2
60.59	even	2		8550.2.a.br.1.2		2
76.75	even	2		3610.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.d.1.1	✓	2	4.3	odd	2
950.2.a.h.1.2		2	20.19	odd	2
950.2.b.f.799.2		4	20.7	even	4
950.2.b.f.799.3		4	20.3	even	4
1520.2.a.n.1.2		2	1.1	even	1	trivial
1710.2.a.w.1.1		2	12.11	even	2
3610.2.a.t.1.2		2	76.75	even	2
6080.2.a.bb.1.1		2	8.5	even	2
6080.2.a.bh.1.2		2	8.3	odd	2
7600.2.a.y.1.1		2	5.4	even	2
8550.2.a.br.1.2		2	60.59	even	2
9310.2.a.bc.1.2		2	28.27	even	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q+2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} -4.00000 q^{11} +5.68466 q^{13} +2.56155 q^{15} +3.43845 q^{17} +1.00000 q^{19} +6.56155 q^{21} -7.68466 q^{23} +1.00000 q^{25} +1.43845 q^{27} -5.68466 q^{29} +5.12311 q^{31} -10.2462 q^{33} +2.56155 q^{35} -6.00000 q^{37} +14.5616 q^{39} +12.2462 q^{41} +2.87689 q^{43} +3.56155 q^{45} -6.24621 q^{47} -0.438447 q^{49} +8.80776 q^{51} -4.56155 q^{53} -4.00000 q^{55} +2.56155 q^{57} -2.56155 q^{59} +11.1231 q^{61} +9.12311 q^{63} +5.68466 q^{65} +2.56155 q^{67} -19.6847 q^{69} -10.2462 q^{71} -1.68466 q^{73} +2.56155 q^{75} -10.2462 q^{77} +5.12311 q^{79} -7.00000 q^{81} -2.87689 q^{83} +3.43845 q^{85} -14.5616 q^{87} +2.00000 q^{89} +14.5616 q^{91} +13.1231 q^{93} +1.00000 q^{95} +6.00000 q^{97} -14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9	\( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 8 q^{11} - q^{13} + q^{15} + 11 q^{17} + 2 q^{19} + 9 q^{21} - 3 q^{23} + 2 q^{25} + 7 q^{27} + q^{29} + 2 q^{31} - 4 q^{33} + q^{35} - 12 q^{37} + 25 q^{39} + 8 q^{41} + 14 q^{43} + 3 q^{45} + 4 q^{47} - 5 q^{49} - 3 q^{51} - 5 q^{53} - 8 q^{55} + q^{57} - q^{59} + 14 q^{61} + 10 q^{63} - q^{65} + q^{67} - 27 q^{69} - 4 q^{71} + 9 q^{73} + q^{75} - 4 q^{77} + 2 q^{79} - 14 q^{81} - 14 q^{83} + 11 q^{85} - 25 q^{87} + 4 q^{89} + 25 q^{91} + 18 q^{93} + 2 q^{95} + 12 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 - 8 * q^11 - q^13 + q^15 + 11 * q^17 + 2 * q^19 + 9 * q^21 - 3 * q^23 + 2 * q^25 + 7 * q^27 + q^29 + 2 * q^31 - 4 * q^33 + q^35 - 12 * q^37 + 25 * q^39 + 8 * q^41 + 14 * q^43 + 3 * q^45 + 4 * q^47 - 5 * q^49 - 3 * q^51 - 5 * q^53 - 8 * q^55 + q^57 - q^59 + 14 * q^61 + 10 * q^63 - q^65 + q^67 - 27 * q^69 - 4 * q^71 + 9 * q^73 + q^75 - 4 * q^77 + 2 * q^79 - 14 * q^81 - 14 * q^83 + 11 * q^85 - 25 * q^87 + 4 * q^89 + 25 * q^91 + 18 * q^93 + 2 * q^95 + 12 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

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Database

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