LMFDB - Embedded newform 1785.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1785,2,Mod(1,1785)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1785, 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("1785.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1785 = 3 \cdot 5 \cdot 7 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1785.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:	$14.2532967608$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1785.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} -1.00000 q^{5} +2.56155 q^{6} -1.00000 q^{7} +6.56155 q^{8} +1.00000 q^{9} -2.56155 q^{10} +3.12311 q^{11} +4.56155 q^{12} +3.56155 q^{13} -2.56155 q^{14} -1.00000 q^{15} +7.68466 q^{16} -1.00000 q^{17} +2.56155 q^{18} -7.12311 q^{19} -4.56155 q^{20} -1.00000 q^{21} +8.00000 q^{22} +8.68466 q^{23} +6.56155 q^{24} +1.00000 q^{25} +9.12311 q^{26} +1.00000 q^{27} -4.56155 q^{28} -1.12311 q^{29} -2.56155 q^{30} +2.43845 q^{31} +6.56155 q^{32} +3.12311 q^{33} -2.56155 q^{34} +1.00000 q^{35} +4.56155 q^{36} -3.56155 q^{37} -18.2462 q^{38} +3.56155 q^{39} -6.56155 q^{40} +3.56155 q^{41} -2.56155 q^{42} +14.2462 q^{44} -1.00000 q^{45} +22.2462 q^{46} -11.8078 q^{47} +7.68466 q^{48} +1.00000 q^{49} +2.56155 q^{50} -1.00000 q^{51} +16.2462 q^{52} -2.87689 q^{53} +2.56155 q^{54} -3.12311 q^{55} -6.56155 q^{56} -7.12311 q^{57} -2.87689 q^{58} +10.2462 q^{59} -4.56155 q^{60} -10.6847 q^{61} +6.24621 q^{62} -1.00000 q^{63} +1.43845 q^{64} -3.56155 q^{65} +8.00000 q^{66} -6.24621 q^{67} -4.56155 q^{68} +8.68466 q^{69} +2.56155 q^{70} -7.12311 q^{71} +6.56155 q^{72} +2.87689 q^{73} -9.12311 q^{74} +1.00000 q^{75} -32.4924 q^{76} -3.12311 q^{77} +9.12311 q^{78} +3.12311 q^{79} -7.68466 q^{80} +1.00000 q^{81} +9.12311 q^{82} -15.8078 q^{83} -4.56155 q^{84} +1.00000 q^{85} -1.12311 q^{87} +20.4924 q^{88} +2.00000 q^{89} -2.56155 q^{90} -3.56155 q^{91} +39.6155 q^{92} +2.43845 q^{93} -30.2462 q^{94} +7.12311 q^{95} +6.56155 q^{96} +6.00000 q^{97} +2.56155 q^{98} +3.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{2} + 2 q^{3} + 5 q^{4} - 2 q^{5} + q^{6} - 2 q^{7} + 9 q^{8} + 2 q^{9} - q^{10} - 2 q^{11} + 5 q^{12} + 3 q^{13} - q^{14} - 2 q^{15} + 3 q^{16} - 2 q^{17} + q^{18} - 6 q^{19} - 5 q^{20} - 2 q^{21}+ \cdots - 2 q^{99}+O(q^{100})$ 2 * q + q^2 + 2 * q^3 + 5 * q^4 - 2 * q^5 + q^6 - 2 * q^7 + 9 * q^8 + 2 * q^9 - q^10 - 2 * q^11 + 5 * q^12 + 3 * q^13 - q^14 - 2 * q^15 + 3 * q^16 - 2 * q^17 + q^18 - 6 * q^19 - 5 * q^20 - 2 * q^21 + 16 * q^22 + 5 * q^23 + 9 * q^24 + 2 * q^25 + 10 * q^26 + 2 * q^27 - 5 * q^28 + 6 * q^29 - q^30 + 9 * q^31 + 9 * q^32 - 2 * q^33 - q^34 + 2 * q^35 + 5 * q^36 - 3 * q^37 - 20 * q^38 + 3 * q^39 - 9 * q^40 + 3 * q^41 - q^42 + 12 * q^44 - 2 * q^45 + 28 * q^46 - 3 * q^47 + 3 * q^48 + 2 * q^49 + q^50 - 2 * q^51 + 16 * q^52 - 14 * q^53 + q^54 + 2 * q^55 - 9 * q^56 - 6 * q^57 - 14 * q^58 + 4 * q^59 - 5 * q^60 - 9 * q^61 - 4 * q^62 - 2 * q^63 + 7 * q^64 - 3 * q^65 + 16 * q^66 + 4 * q^67 - 5 * q^68 + 5 * q^69 + q^70 - 6 * q^71 + 9 * q^72 + 14 * q^73 - 10 * q^74 + 2 * q^75 - 32 * q^76 + 2 * q^77 + 10 * q^78 - 2 * q^79 - 3 * q^80 + 2 * q^81 + 10 * q^82 - 11 * q^83 - 5 * q^84 + 2 * q^85 + 6 * q^87 + 8 * q^88 + 4 * q^89 - q^90 - 3 * q^91 + 38 * q^92 + 9 * q^93 - 44 * q^94 + 6 * q^95 + 9 * q^96 + 12 * q^97 + q^98 - 2 * q^99

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

)

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

$2$	2.56155	1.81129	0.905646	−	0.424035i	$-0.139387\pi$
$2$	2.56155	1.81129	0.905646	+	0.424035i	$0.139387\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.56155	2.28078
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.56155	1.04575
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	6.56155	2.31986
$9$	1.00000	0.333333
$10$	−2.56155	−0.810034
$11$	3.12311	0.941652	0.470826	−	0.882226i	$-0.343956\pi$
$11$	3.12311	0.941652	0.470826	+	0.882226i	$0.343956\pi$
$12$	4.56155	1.31681
$13$	3.56155	0.987797	0.493899	−	0.869520i	$-0.335571\pi$
$13$	3.56155	0.987797	0.493899	+	0.869520i	$0.335571\pi$
$14$	−2.56155	−0.684604
$15$	−1.00000	−0.258199
$16$	7.68466	1.92116
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.56155	0.603764
$19$	−7.12311	−1.63415	−0.817076	−	0.576530i	$-0.804407\pi$
$19$	−7.12311	−1.63415	−0.817076	+	0.576530i	$0.804407\pi$
$20$	−4.56155	−1.01999
$21$	−1.00000	−0.218218
$22$	8.00000	1.70561
$23$	8.68466	1.81088	0.905438	−	0.424478i	$-0.139542\pi$
$23$	8.68466	1.81088	0.905438	+	0.424478i	$0.139542\pi$
$24$	6.56155	1.33937
$25$	1.00000	0.200000
$26$	9.12311	1.78919
$27$	1.00000	0.192450
$28$	−4.56155	−0.862052
$29$	−1.12311	−0.208555	−0.104278	−	0.994548i	$-0.533253\pi$
$29$	−1.12311	−0.208555	−0.104278	+	0.994548i	$0.533253\pi$
$30$	−2.56155	−0.467673
$31$	2.43845	0.437958	0.218979	−	0.975730i	$-0.429727\pi$
$31$	2.43845	0.437958	0.218979	+	0.975730i	$0.429727\pi$
$32$	6.56155	1.15993
$33$	3.12311	0.543663
$34$	−2.56155	−0.439303
$35$	1.00000	0.169031
$36$	4.56155	0.760259
$37$	−3.56155	−0.585516	−0.292758	−	0.956187i	$-0.594573\pi$
$37$	−3.56155	−0.585516	−0.292758	+	0.956187i	$0.594573\pi$
$38$	−18.2462	−2.95993
$39$	3.56155	0.570305
$40$	−6.56155	−1.03747
$41$	3.56155	0.556221	0.278111	−	0.960549i	$-0.410292\pi$
$41$	3.56155	0.556221	0.278111	+	0.960549i	$0.410292\pi$
$42$	−2.56155	−0.395256
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	14.2462	2.14770
$45$	−1.00000	−0.149071
$46$	22.2462	3.28002
$47$	−11.8078	−1.72234	−0.861170	−	0.508318i	$-0.830268\pi$
$47$	−11.8078	−1.72234	−0.861170	+	0.508318i	$0.830268\pi$
$48$	7.68466	1.10918
$49$	1.00000	0.142857
$50$	2.56155	0.362258
$51$	−1.00000	−0.140028
$52$	16.2462	2.25294
$53$	−2.87689	−0.395172	−0.197586	−	0.980286i	$-0.563310\pi$
$53$	−2.87689	−0.395172	−0.197586	+	0.980286i	$0.563310\pi$
$54$	2.56155	0.348583
$55$	−3.12311	−0.421119
$56$	−6.56155	−0.876824
$57$	−7.12311	−0.943478
$58$	−2.87689	−0.377755
$59$	10.2462	1.33394	0.666972	−	0.745083i	$-0.267590\pi$
$59$	10.2462	1.33394	0.666972	+	0.745083i	$0.267590\pi$
$60$	−4.56155	−0.588894
$61$	−10.6847	−1.36803	−0.684015	−	0.729468i	$-0.739768\pi$
$61$	−10.6847	−1.36803	−0.684015	+	0.729468i	$0.739768\pi$
$62$	6.24621	0.793270
$63$	−1.00000	−0.125988
$64$	1.43845	0.179806
$65$	−3.56155	−0.441756
$66$	8.00000	0.984732
$67$	−6.24621	−0.763096	−0.381548	−	0.924349i	$-0.624609\pi$
$67$	−6.24621	−0.763096	−0.381548	+	0.924349i	$0.624609\pi$
$68$	−4.56155	−0.553170
$69$	8.68466	1.04551
$70$	2.56155	0.306164
$71$	−7.12311	−0.845357	−0.422679	−	0.906280i	$-0.638910\pi$
$71$	−7.12311	−0.845357	−0.422679	+	0.906280i	$0.638910\pi$
$72$	6.56155	0.773286
$73$	2.87689	0.336715	0.168358	−	0.985726i	$-0.446154\pi$
$73$	2.87689	0.336715	0.168358	+	0.985726i	$0.446154\pi$
$74$	−9.12311	−1.06054
$75$	1.00000	0.115470
$76$	−32.4924	−3.72714
$77$	−3.12311	−0.355911
$78$	9.12311	1.03299
$79$	3.12311	0.351377	0.175688	−	0.984446i	$-0.443785\pi$
$79$	3.12311	0.351377	0.175688	+	0.984446i	$0.443785\pi$
$80$	−7.68466	−0.859171
$81$	1.00000	0.111111
$82$	9.12311	1.00748
$83$	−15.8078	−1.73513	−0.867564	−	0.497326i	$-0.834315\pi$
$83$	−15.8078	−1.73513	−0.867564	+	0.497326i	$0.834315\pi$
$84$	−4.56155	−0.497706
$85$	1.00000	0.108465
$86$	0	0
$87$	−1.12311	−0.120410
$88$	20.4924	2.18450
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	−2.56155	−0.270011
$91$	−3.56155	−0.373352
$92$	39.6155	4.13020
$93$	2.43845	0.252855
$94$	−30.2462	−3.11966
$95$	7.12311	0.730815
$96$	6.56155	0.669686
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	2.56155	0.258756
$99$	3.12311	0.313884
$100$	4.56155	0.456155
$101$	1.12311	0.111753	0.0558766	−	0.998438i	$-0.482205\pi$
$101$	1.12311	0.111753	0.0558766	+	0.998438i	$0.482205\pi$
$102$	−2.56155	−0.253632
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	23.3693	2.29155
$105$	1.00000	0.0975900
$106$	−7.36932	−0.715771
$107$	6.43845	0.622428	0.311214	−	0.950340i	$-0.399264\pi$
$107$	6.43845	0.622428	0.311214	+	0.950340i	$0.399264\pi$
$108$	4.56155	0.438936
$109$	12.2462	1.17297	0.586487	−	0.809959i	$-0.300510\pi$
$109$	12.2462	1.17297	0.586487	+	0.809959i	$0.300510\pi$
$110$	−8.00000	−0.762770
$111$	−3.56155	−0.338048
$112$	−7.68466	−0.726132
$113$	−8.24621	−0.775738	−0.387869	−	0.921714i	$-0.626789\pi$
$113$	−8.24621	−0.775738	−0.387869	+	0.921714i	$0.626789\pi$
$114$	−18.2462	−1.70891
$115$	−8.68466	−0.809849
$116$	−5.12311	−0.475668
$117$	3.56155	0.329266
$118$	26.2462	2.41616
$119$	1.00000	0.0916698
$120$	−6.56155	−0.598985
$121$	−1.24621	−0.113292
$122$	−27.3693	−2.47790
$123$	3.56155	0.321134
$124$	11.1231	0.998884
$125$	−1.00000	−0.0894427
$126$	−2.56155	−0.228201
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	−9.43845	−0.834249
$129$	0	0
$130$	−9.12311	−0.800149
$131$	−16.6847	−1.45775	−0.728873	−	0.684649i	$-0.759955\pi$
$131$	−16.6847	−1.45775	−0.728873	+	0.684649i	$0.759955\pi$
$132$	14.2462	1.23997
$133$	7.12311	0.617652
$134$	−16.0000	−1.38219
$135$	−1.00000	−0.0860663
$136$	−6.56155	−0.562649
$137$	−21.1231	−1.80467	−0.902334	−	0.431037i	$-0.858148\pi$
$137$	−21.1231	−1.80467	−0.902334	+	0.431037i	$0.858148\pi$
$138$	22.2462	1.89372
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	4.56155	0.385522
$141$	−11.8078	−0.994393
$142$	−18.2462	−1.53119
$143$	11.1231	0.930161
$144$	7.68466	0.640388
$145$	1.12311	0.0932688
$146$	7.36932	0.609889
$147$	1.00000	0.0824786
$148$	−16.2462	−1.33543
$149$	−18.6847	−1.53071	−0.765353	−	0.643610i	$-0.777436\pi$
$149$	−18.6847	−1.53071	−0.765353	+	0.643610i	$0.777436\pi$
$150$	2.56155	0.209150
$151$	−8.68466	−0.706747	−0.353374	−	0.935482i	$-0.614966\pi$
$151$	−8.68466	−0.706747	−0.353374	+	0.935482i	$0.614966\pi$
$152$	−46.7386	−3.79100
$153$	−1.00000	−0.0808452
$154$	−8.00000	−0.644658
$155$	−2.43845	−0.195861
$156$	16.2462	1.30074
$157$	−16.2462	−1.29659	−0.648294	−	0.761390i	$-0.724517\pi$
$157$	−16.2462	−1.29659	−0.648294	+	0.761390i	$0.724517\pi$
$158$	8.00000	0.636446
$159$	−2.87689	−0.228153
$160$	−6.56155	−0.518736
$161$	−8.68466	−0.684447
$162$	2.56155	0.201255
$163$	11.3153	0.886286	0.443143	−	0.896451i	$-0.353863\pi$
$163$	11.3153	0.886286	0.443143	+	0.896451i	$0.353863\pi$
$164$	16.2462	1.26862
$165$	−3.12311	−0.243133
$166$	−40.4924	−3.14282
$167$	4.87689	0.377385	0.188693	−	0.982036i	$-0.439575\pi$
$167$	4.87689	0.377385	0.188693	+	0.982036i	$0.439575\pi$
$168$	−6.56155	−0.506235
$169$	−0.315342	−0.0242570
$170$	2.56155	0.196462
$171$	−7.12311	−0.544718
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−2.87689	−0.218097
$175$	−1.00000	−0.0755929
$176$	24.0000	1.80907
$177$	10.2462	0.770152
$178$	5.12311	0.383993
$179$	0.192236	0.0143684	0.00718419	−	0.999974i	$-0.497713\pi$
$179$	0.192236	0.0143684	0.00718419	+	0.999974i	$0.497713\pi$
$180$	−4.56155	−0.339998
$181$	19.5616	1.45400	0.726999	−	0.686638i	$-0.240914\pi$
$181$	19.5616	1.45400	0.726999	+	0.686638i	$0.240914\pi$
$182$	−9.12311	−0.676250
$183$	−10.6847	−0.789833
$184$	56.9848	4.20098
$185$	3.56155	0.261851
$186$	6.24621	0.457994
$187$	−3.12311	−0.228384
$188$	−53.8617	−3.92827
$189$	−1.00000	−0.0727393
$190$	18.2462	1.32372
$191$	14.9309	1.08036	0.540180	−	0.841550i	$-0.318356\pi$
$191$	14.9309	1.08036	0.540180	+	0.841550i	$0.318356\pi$
$192$	1.43845	0.103811
$193$	14.6847	1.05702	0.528512	−	0.848926i	$-0.322750\pi$
$193$	14.6847	1.05702	0.528512	+	0.848926i	$0.322750\pi$
$194$	15.3693	1.10345
$195$	−3.56155	−0.255048
$196$	4.56155	0.325825
$197$	3.75379	0.267446	0.133723	−	0.991019i	$-0.457307\pi$
$197$	3.75379	0.267446	0.133723	+	0.991019i	$0.457307\pi$
$198$	8.00000	0.568535
$199$	−6.24621	−0.442782	−0.221391	−	0.975185i	$-0.571060\pi$
$199$	−6.24621	−0.442782	−0.221391	+	0.975185i	$0.571060\pi$
$200$	6.56155	0.463972
$201$	−6.24621	−0.440574
$202$	2.87689	0.202418
$203$	1.12311	0.0788266
$204$	−4.56155	−0.319373
$205$	−3.56155	−0.248750
$206$	−5.75379	−0.400885
$207$	8.68466	0.603625
$208$	27.3693	1.89772
$209$	−22.2462	−1.53880
$210$	2.56155	0.176764
$211$	−7.12311	−0.490375	−0.245187	−	0.969476i	$-0.578849\pi$
$211$	−7.12311	−0.490375	−0.245187	+	0.969476i	$0.578849\pi$
$212$	−13.1231	−0.901299
$213$	−7.12311	−0.488067
$214$	16.4924	1.12740
$215$	0	0
$216$	6.56155	0.446457
$217$	−2.43845	−0.165533
$218$	31.3693	2.12460
$219$	2.87689	0.194403
$220$	−14.2462	−0.960479
$221$	−3.56155	−0.239576
$222$	−9.12311	−0.612303
$223$	3.31534	0.222012	0.111006	−	0.993820i	$-0.464593\pi$
$223$	3.31534	0.222012	0.111006	+	0.993820i	$0.464593\pi$
$224$	−6.56155	−0.438412
$225$	1.00000	0.0666667
$226$	−21.1231	−1.40509
$227$	−8.87689	−0.589180	−0.294590	−	0.955624i	$-0.595183\pi$
$227$	−8.87689	−0.589180	−0.294590	+	0.955624i	$0.595183\pi$
$228$	−32.4924	−2.15186
$229$	−3.75379	−0.248057	−0.124029	−	0.992279i	$-0.539581\pi$
$229$	−3.75379	−0.248057	−0.124029	+	0.992279i	$0.539581\pi$
$230$	−22.2462	−1.46687
$231$	−3.12311	−0.205485
$232$	−7.36932	−0.483819
$233$	2.19224	0.143618	0.0718091	−	0.997418i	$-0.477123\pi$
$233$	2.19224	0.143618	0.0718091	+	0.997418i	$0.477123\pi$
$234$	9.12311	0.596396
$235$	11.8078	0.770254
$236$	46.7386	3.04243
$237$	3.12311	0.202868
$238$	2.56155	0.166041
$239$	16.6847	1.07924	0.539620	−	0.841908i	$-0.318568\pi$
$239$	16.6847	1.07924	0.539620	+	0.841908i	$0.318568\pi$
$240$	−7.68466	−0.496043
$241$	−19.5616	−1.26007	−0.630035	−	0.776567i	$-0.716960\pi$
$241$	−19.5616	−1.26007	−0.630035	+	0.776567i	$0.716960\pi$
$242$	−3.19224	−0.205205
$243$	1.00000	0.0641500
$244$	−48.7386	−3.12017
$245$	−1.00000	−0.0638877
$246$	9.12311	0.581668
$247$	−25.3693	−1.61421
$248$	16.0000	1.01600
$249$	−15.8078	−1.00178
$250$	−2.56155	−0.162007
$251$	−16.4924	−1.04099	−0.520496	−	0.853864i	$-0.674253\pi$
$251$	−16.4924	−1.04099	−0.520496	+	0.853864i	$0.674253\pi$
$252$	−4.56155	−0.287351
$253$	27.1231	1.70522
$254$	51.2311	3.21452
$255$	1.00000	0.0626224
$256$	−27.0540	−1.69087
$257$	30.6847	1.91406	0.957028	−	0.289995i	$-0.0936536\pi$
$257$	30.6847	1.91406	0.957028	+	0.289995i	$0.0936536\pi$
$258$	0	0
$259$	3.56155	0.221304
$260$	−16.2462	−1.00755
$261$	−1.12311	−0.0695185
$262$	−42.7386	−2.64040
$263$	18.7386	1.15547	0.577737	−	0.816223i	$-0.303936\pi$
$263$	18.7386	1.15547	0.577737	+	0.816223i	$0.303936\pi$
$264$	20.4924	1.26122
$265$	2.87689	0.176726
$266$	18.2462	1.11875
$267$	2.00000	0.122398
$268$	−28.4924	−1.74045
$269$	1.31534	0.0801978	0.0400989	−	0.999196i	$-0.487233\pi$
$269$	1.31534	0.0801978	0.0400989	+	0.999196i	$0.487233\pi$
$270$	−2.56155	−0.155891
$271$	−11.1231	−0.675681	−0.337840	−	0.941203i	$-0.609696\pi$
$271$	−11.1231	−0.675681	−0.337840	+	0.941203i	$0.609696\pi$
$272$	−7.68466	−0.465951
$273$	−3.56155	−0.215555
$274$	−54.1080	−3.26878
$275$	3.12311	0.188330
$276$	39.6155	2.38457
$277$	32.9309	1.97862	0.989312	−	0.145813i	$-0.0465799\pi$
$277$	32.9309	1.97862	0.989312	+	0.145813i	$0.0465799\pi$
$278$	10.2462	0.614527
$279$	2.43845	0.145986
$280$	6.56155	0.392128
$281$	12.4384	0.742016	0.371008	−	0.928630i	$-0.379012\pi$
$281$	12.4384	0.742016	0.371008	+	0.928630i	$0.379012\pi$
$282$	−30.2462	−1.80114
$283$	26.2462	1.56018	0.780088	−	0.625670i	$-0.215174\pi$
$283$	26.2462	1.56018	0.780088	+	0.625670i	$0.215174\pi$
$284$	−32.4924	−1.92807
$285$	7.12311	0.421936
$286$	28.4924	1.68479
$287$	−3.56155	−0.210232
$288$	6.56155	0.386643
$289$	1.00000	0.0588235
$290$	2.87689	0.168937
$291$	6.00000	0.351726
$292$	13.1231	0.767972
$293$	14.4924	0.846656	0.423328	−	0.905976i	$-0.360862\pi$
$293$	14.4924	0.846656	0.423328	+	0.905976i	$0.360862\pi$
$294$	2.56155	0.149393
$295$	−10.2462	−0.596557
$296$	−23.3693	−1.35831
$297$	3.12311	0.181221
$298$	−47.8617	−2.77256
$299$	30.9309	1.78878
$300$	4.56155	0.263361
$301$	0	0
$302$	−22.2462	−1.28013
$303$	1.12311	0.0645207
$304$	−54.7386	−3.13948
$305$	10.6847	0.611802
$306$	−2.56155	−0.146434
$307$	−22.2462	−1.26966	−0.634829	−	0.772653i	$-0.718930\pi$
$307$	−22.2462	−1.26966	−0.634829	+	0.772653i	$0.718930\pi$
$308$	−14.2462	−0.811753
$309$	−2.24621	−0.127782
$310$	−6.24621	−0.354761
$311$	22.4384	1.27237	0.636184	−	0.771538i	$-0.280512\pi$
$311$	22.4384	1.27237	0.636184	+	0.771538i	$0.280512\pi$
$312$	23.3693	1.32303
$313$	24.7386	1.39831	0.699155	−	0.714970i	$-0.253560\pi$
$313$	24.7386	1.39831	0.699155	+	0.714970i	$0.253560\pi$
$314$	−41.6155	−2.34850
$315$	1.00000	0.0563436
$316$	14.2462	0.801412
$317$	−24.0540	−1.35101	−0.675503	−	0.737357i	$-0.736073\pi$
$317$	−24.0540	−1.35101	−0.675503	+	0.737357i	$0.736073\pi$
$318$	−7.36932	−0.413251
$319$	−3.50758	−0.196387
$320$	−1.43845	−0.0804116
$321$	6.43845	0.359359
$322$	−22.2462	−1.23973
$323$	7.12311	0.396340
$324$	4.56155	0.253420
$325$	3.56155	0.197559
$326$	28.9848	1.60532
$327$	12.2462	0.677217
$328$	23.3693	1.29035
$329$	11.8078	0.650983
$330$	−8.00000	−0.440386
$331$	−0.192236	−0.0105662	−0.00528312	−	0.999986i	$-0.501682\pi$
$331$	−0.192236	−0.0105662	−0.00528312	+	0.999986i	$0.501682\pi$
$332$	−72.1080	−3.95744
$333$	−3.56155	−0.195172
$334$	12.4924	0.683555
$335$	6.24621	0.341267
$336$	−7.68466	−0.419232
$337$	7.75379	0.422376	0.211188	−	0.977445i	$-0.432267\pi$
$337$	7.75379	0.422376	0.211188	+	0.977445i	$0.432267\pi$
$338$	−0.807764	−0.0439366
$339$	−8.24621	−0.447873
$340$	4.56155	0.247385
$341$	7.61553	0.412404
$342$	−18.2462	−0.986642
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−8.68466	−0.467566
$346$	−35.8617	−1.92794
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−5.12311	−0.274627
$349$	25.1231	1.34481	0.672405	−	0.740184i	$-0.265262\pi$
$349$	25.1231	1.34481	0.672405	+	0.740184i	$0.265262\pi$
$350$	−2.56155	−0.136921
$351$	3.56155	0.190102
$352$	20.4924	1.09225
$353$	−15.5616	−0.828258	−0.414129	−	0.910218i	$-0.635914\pi$
$353$	−15.5616	−0.828258	−0.414129	+	0.910218i	$0.635914\pi$
$354$	26.2462	1.39497
$355$	7.12311	0.378055
$356$	9.12311	0.483524
$357$	1.00000	0.0529256
$358$	0.492423	0.0260253
$359$	−2.43845	−0.128696	−0.0643482	−	0.997928i	$-0.520497\pi$
$359$	−2.43845	−0.128696	−0.0643482	+	0.997928i	$0.520497\pi$
$360$	−6.56155	−0.345824
$361$	31.7386	1.67045
$362$	50.1080	2.63362
$363$	−1.24621	−0.0654091
$364$	−16.2462	−0.851533
$365$	−2.87689	−0.150584
$366$	−27.3693	−1.43062
$367$	−9.36932	−0.489074	−0.244537	−	0.969640i	$-0.578636\pi$
$367$	−9.36932	−0.489074	−0.244537	+	0.969640i	$0.578636\pi$
$368$	66.7386	3.47899
$369$	3.56155	0.185407
$370$	9.12311	0.474288
$371$	2.87689	0.149361
$372$	11.1231	0.576706
$373$	23.3693	1.21002	0.605009	−	0.796219i	$-0.293170\pi$
$373$	23.3693	1.21002	0.605009	+	0.796219i	$0.293170\pi$
$374$	−8.00000	−0.413670
$375$	−1.00000	−0.0516398
$376$	−77.4773	−3.99558
$377$	−4.00000	−0.206010
$378$	−2.56155	−0.131752
$379$	0.492423	0.0252940	0.0126470	−	0.999920i	$-0.495974\pi$
$379$	0.492423	0.0252940	0.0126470	+	0.999920i	$0.495974\pi$
$380$	32.4924	1.66683
$381$	20.0000	1.02463
$382$	38.2462	1.95685
$383$	34.7386	1.77506	0.887531	−	0.460749i	$-0.152419\pi$
$383$	34.7386	1.77506	0.887531	+	0.460749i	$0.152419\pi$
$384$	−9.43845	−0.481654
$385$	3.12311	0.159168
$386$	37.6155	1.91458
$387$	0	0
$388$	27.3693	1.38947
$389$	−36.0540	−1.82801	−0.914005	−	0.405704i	$-0.867026\pi$
$389$	−36.0540	−1.82801	−0.914005	+	0.405704i	$0.867026\pi$
$390$	−9.12311	−0.461966
$391$	−8.68466	−0.439202
$392$	6.56155	0.331408
$393$	−16.6847	−0.841630
$394$	9.61553	0.484423
$395$	−3.12311	−0.157140
$396$	14.2462	0.715899
$397$	−24.7386	−1.24160	−0.620798	−	0.783970i	$-0.713191\pi$
$397$	−24.7386	−1.24160	−0.620798	+	0.783970i	$0.713191\pi$
$398$	−16.0000	−0.802008
$399$	7.12311	0.356601
$400$	7.68466	0.384233
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	−16.0000	−0.798007
$403$	8.68466	0.432614
$404$	5.12311	0.254884
$405$	−1.00000	−0.0496904
$406$	2.87689	0.142778
$407$	−11.1231	−0.551352
$408$	−6.56155	−0.324845
$409$	27.3693	1.35333	0.676663	−	0.736293i	$-0.263425\pi$
$409$	27.3693	1.35333	0.676663	+	0.736293i	$0.263425\pi$
$410$	−9.12311	−0.450558
$411$	−21.1231	−1.04193
$412$	−10.2462	−0.504795
$413$	−10.2462	−0.504183
$414$	22.2462	1.09334
$415$	15.8078	0.775973
$416$	23.3693	1.14578
$417$	4.00000	0.195881
$418$	−56.9848	−2.78722
$419$	−21.5616	−1.05335	−0.526675	−	0.850066i	$-0.676562\pi$
$419$	−21.5616	−1.05335	−0.526675	+	0.850066i	$0.676562\pi$
$420$	4.56155	0.222581
$421$	−17.3153	−0.843898	−0.421949	−	0.906620i	$-0.638654\pi$
$421$	−17.3153	−0.843898	−0.421949	+	0.906620i	$0.638654\pi$
$422$	−18.2462	−0.888212
$423$	−11.8078	−0.574113
$424$	−18.8769	−0.916743
$425$	−1.00000	−0.0485071
$426$	−18.2462	−0.884032
$427$	10.6847	0.517067
$428$	29.3693	1.41962
$429$	11.1231	0.537029
$430$	0	0
$431$	24.4924	1.17976	0.589879	−	0.807491i	$-0.299175\pi$
$431$	24.4924	1.17976	0.589879	+	0.807491i	$0.299175\pi$
$432$	7.68466	0.369728
$433$	7.17708	0.344908	0.172454	−	0.985018i	$-0.444830\pi$
$433$	7.17708	0.344908	0.172454	+	0.985018i	$0.444830\pi$
$434$	−6.24621	−0.299828
$435$	1.12311	0.0538488
$436$	55.8617	2.67529
$437$	−61.8617	−2.95925
$438$	7.36932	0.352120
$439$	20.4924	0.978050	0.489025	−	0.872270i	$-0.337353\pi$
$439$	20.4924	0.978050	0.489025	+	0.872270i	$0.337353\pi$
$440$	−20.4924	−0.976938
$441$	1.00000	0.0476190
$442$	−9.12311	−0.433942
$443$	−24.4924	−1.16367	−0.581835	−	0.813307i	$-0.697665\pi$
$443$	−24.4924	−1.16367	−0.581835	+	0.813307i	$0.697665\pi$
$444$	−16.2462	−0.771011
$445$	−2.00000	−0.0948091
$446$	8.49242	0.402128
$447$	−18.6847	−0.883754
$448$	−1.43845	−0.0679602
$449$	−40.2462	−1.89934	−0.949668	−	0.313258i	$-0.898580\pi$
$449$	−40.2462	−1.89934	−0.949668	+	0.313258i	$0.898580\pi$
$450$	2.56155	0.120753
$451$	11.1231	0.523767
$452$	−37.6155	−1.76929
$453$	−8.68466	−0.408041
$454$	−22.7386	−1.06718
$455$	3.56155	0.166968
$456$	−46.7386	−2.18874
$457$	−25.1231	−1.17521	−0.587605	−	0.809148i	$-0.699929\pi$
$457$	−25.1231	−1.17521	−0.587605	+	0.809148i	$0.699929\pi$
$458$	−9.61553	−0.449304
$459$	−1.00000	−0.0466760
$460$	−39.6155	−1.84708
$461$	4.24621	0.197766	0.0988829	−	0.995099i	$-0.468473\pi$
$461$	4.24621	0.197766	0.0988829	+	0.995099i	$0.468473\pi$
$462$	−8.00000	−0.372194
$463$	−8.87689	−0.412544	−0.206272	−	0.978495i	$-0.566133\pi$
$463$	−8.87689	−0.412544	−0.206272	+	0.978495i	$0.566133\pi$
$464$	−8.63068	−0.400669
$465$	−2.43845	−0.113080
$466$	5.61553	0.260134
$467$	38.0540	1.76093	0.880464	−	0.474113i	$-0.157231\pi$
$467$	38.0540	1.76093	0.880464	+	0.474113i	$0.157231\pi$
$468$	16.2462	0.750981
$469$	6.24621	0.288423
$470$	30.2462	1.39515
$471$	−16.2462	−0.748586
$472$	67.2311	3.09456
$473$	0	0
$474$	8.00000	0.367452
$475$	−7.12311	−0.326831
$476$	4.56155	0.209078
$477$	−2.87689	−0.131724
$478$	42.7386	1.95482
$479$	1.56155	0.0713492	0.0356746	−	0.999363i	$-0.488642\pi$
$479$	1.56155	0.0713492	0.0356746	+	0.999363i	$0.488642\pi$
$480$	−6.56155	−0.299493
$481$	−12.6847	−0.578371
$482$	−50.1080	−2.28235
$483$	−8.68466	−0.395166
$484$	−5.68466	−0.258394
$485$	−6.00000	−0.272446
$486$	2.56155	0.116194
$487$	21.1771	0.959625	0.479813	−	0.877371i	$-0.340705\pi$
$487$	21.1771	0.959625	0.479813	+	0.877371i	$0.340705\pi$
$488$	−70.1080	−3.17364
$489$	11.3153	0.511697
$490$	−2.56155	−0.115719
$491$	−38.0540	−1.71735	−0.858676	−	0.512519i	$-0.828712\pi$
$491$	−38.0540	−1.71735	−0.858676	+	0.512519i	$0.828712\pi$
$492$	16.2462	0.732436
$493$	1.12311	0.0505821
$494$	−64.9848	−2.92381
$495$	−3.12311	−0.140373
$496$	18.7386	0.841389
$497$	7.12311	0.319515
$498$	−40.4924	−1.81451
$499$	−13.7538	−0.615704	−0.307852	−	0.951434i	$-0.599610\pi$
$499$	−13.7538	−0.615704	−0.307852	+	0.951434i	$0.599610\pi$
$500$	−4.56155	−0.203999
$501$	4.87689	0.217884
$502$	−42.2462	−1.88554
$503$	−40.9848	−1.82742	−0.913712	−	0.406362i	$-0.866797\pi$
$503$	−40.9848	−1.82742	−0.913712	+	0.406362i	$0.866797\pi$
$504$	−6.56155	−0.292275
$505$	−1.12311	−0.0499775
$506$	69.4773	3.08864
$507$	−0.315342	−0.0140048
$508$	91.2311	4.04772
$509$	37.6155	1.66728	0.833639	−	0.552309i	$-0.186253\pi$
$509$	37.6155	1.66728	0.833639	+	0.552309i	$0.186253\pi$
$510$	2.56155	0.113427
$511$	−2.87689	−0.127266
$512$	−50.4233	−2.22842
$513$	−7.12311	−0.314493
$514$	78.6004	3.46691
$515$	2.24621	0.0989799
$516$	0	0
$517$	−36.8769	−1.62184
$518$	9.12311	0.400846
$519$	−14.0000	−0.614532
$520$	−23.3693	−1.02481
$521$	8.73863	0.382846	0.191423	−	0.981508i	$-0.438690\pi$
$521$	8.73863	0.382846	0.191423	+	0.981508i	$0.438690\pi$
$522$	−2.87689	−0.125918
$523$	11.8078	0.516317	0.258159	−	0.966103i	$-0.416884\pi$
$523$	11.8078	0.516317	0.258159	+	0.966103i	$0.416884\pi$
$524$	−76.1080	−3.32479
$525$	−1.00000	−0.0436436
$526$	48.0000	2.09290
$527$	−2.43845	−0.106220
$528$	24.0000	1.04447
$529$	52.4233	2.27927
$530$	7.36932	0.320103
$531$	10.2462	0.444648
$532$	32.4924	1.40873
$533$	12.6847	0.549434
$534$	5.12311	0.221698
$535$	−6.43845	−0.278358
$536$	−40.9848	−1.77028
$537$	0.192236	0.00829559
$538$	3.36932	0.145262
$539$	3.12311	0.134522
$540$	−4.56155	−0.196298
$541$	1.12311	0.0482861	0.0241430	−	0.999709i	$-0.492314\pi$
$541$	1.12311	0.0482861	0.0241430	+	0.999709i	$0.492314\pi$
$542$	−28.4924	−1.22385
$543$	19.5616	0.839467
$544$	−6.56155	−0.281324
$545$	−12.2462	−0.524570
$546$	−9.12311	−0.390433
$547$	28.6847	1.22647	0.613234	−	0.789902i	$-0.289868\pi$
$547$	28.6847	1.22647	0.613234	+	0.789902i	$0.289868\pi$
$548$	−96.3542	−4.11605
$549$	−10.6847	−0.456010
$550$	8.00000	0.341121
$551$	8.00000	0.340811
$552$	56.9848	2.42544
$553$	−3.12311	−0.132808
$554$	84.3542	3.58386
$555$	3.56155	0.151179
$556$	18.2462	0.773812
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	6.24621	0.264423
$559$	0	0
$560$	7.68466	0.324736
$561$	−3.12311	−0.131858
$562$	31.8617	1.34401
$563$	26.9309	1.13500	0.567500	−	0.823373i	$-0.307911\pi$
$563$	26.9309	1.13500	0.567500	+	0.823373i	$0.307911\pi$
$564$	−53.8617	−2.26799
$565$	8.24621	0.346921
$566$	67.2311	2.82593
$567$	−1.00000	−0.0419961
$568$	−46.7386	−1.96111
$569$	−19.1771	−0.803945	−0.401973	−	0.915652i	$-0.631675\pi$
$569$	−19.1771	−0.803945	−0.401973	+	0.915652i	$0.631675\pi$
$570$	18.2462	0.764250
$571$	13.3693	0.559488	0.279744	−	0.960075i	$-0.409750\pi$
$571$	13.3693	0.559488	0.279744	+	0.960075i	$0.409750\pi$
$572$	50.7386	2.12149
$573$	14.9309	0.623746
$574$	−9.12311	−0.380791
$575$	8.68466	0.362175
$576$	1.43845	0.0599353
$577$	12.7386	0.530316	0.265158	−	0.964205i	$-0.414576\pi$
$577$	12.7386	0.530316	0.265158	+	0.964205i	$0.414576\pi$
$578$	2.56155	0.106547
$579$	14.6847	0.610274
$580$	5.12311	0.212725
$581$	15.8078	0.655817
$582$	15.3693	0.637079
$583$	−8.98485	−0.372114
$584$	18.8769	0.781131
$585$	−3.56155	−0.147252
$586$	37.1231	1.53354
$587$	40.4924	1.67130	0.835651	−	0.549261i	$-0.185091\pi$
$587$	40.4924	1.67130	0.835651	+	0.549261i	$0.185091\pi$
$588$	4.56155	0.188115
$589$	−17.3693	−0.715690
$590$	−26.2462	−1.08054
$591$	3.75379	0.154410
$592$	−27.3693	−1.12487
$593$	−39.5616	−1.62460	−0.812299	−	0.583241i	$-0.801784\pi$
$593$	−39.5616	−1.62460	−0.812299	+	0.583241i	$0.801784\pi$
$594$	8.00000	0.328244
$595$	−1.00000	−0.0409960
$596$	−85.2311	−3.49120
$597$	−6.24621	−0.255640
$598$	79.2311	3.24000
$599$	−24.9848	−1.02085	−0.510427	−	0.859921i	$-0.670512\pi$
$599$	−24.9848	−1.02085	−0.510427	+	0.859921i	$0.670512\pi$
$600$	6.56155	0.267874
$601$	13.5076	0.550986	0.275493	−	0.961303i	$-0.411159\pi$
$601$	13.5076	0.550986	0.275493	+	0.961303i	$0.411159\pi$
$602$	0	0
$603$	−6.24621	−0.254365
$604$	−39.6155	−1.61193
$605$	1.24621	0.0506657
$606$	2.87689	0.116866
$607$	17.7538	0.720604	0.360302	−	0.932836i	$-0.382674\pi$
$607$	17.7538	0.720604	0.360302	+	0.932836i	$0.382674\pi$
$608$	−46.7386	−1.89550
$609$	1.12311	0.0455105
$610$	27.3693	1.10815
$611$	−42.0540	−1.70132
$612$	−4.56155	−0.184390
$613$	20.2462	0.817737	0.408868	−	0.912593i	$-0.365924\pi$
$613$	20.2462	0.817737	0.408868	+	0.912593i	$0.365924\pi$
$614$	−56.9848	−2.29972
$615$	−3.56155	−0.143616
$616$	−20.4924	−0.825663
$617$	4.93087	0.198509	0.0992547	−	0.995062i	$-0.468354\pi$
$617$	4.93087	0.198509	0.0992547	+	0.995062i	$0.468354\pi$
$618$	−5.75379	−0.231451
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−11.1231	−0.446715
$621$	8.68466	0.348503
$622$	57.4773	2.30463
$623$	−2.00000	−0.0801283
$624$	27.3693	1.09565
$625$	1.00000	0.0400000
$626$	63.3693	2.53275
$627$	−22.2462	−0.888428
$628$	−74.1080	−2.95723
$629$	3.56155	0.142008
$630$	2.56155	0.102055
$631$	−40.9848	−1.63158	−0.815790	−	0.578348i	$-0.803698\pi$
$631$	−40.9848	−1.63158	−0.815790	+	0.578348i	$0.803698\pi$
$632$	20.4924	0.815145
$633$	−7.12311	−0.283118
$634$	−61.6155	−2.44707
$635$	−20.0000	−0.793676
$636$	−13.1231	−0.520365
$637$	3.56155	0.141114
$638$	−8.98485	−0.355713
$639$	−7.12311	−0.281786
$640$	9.43845	0.373087
$641$	−34.9848	−1.38182	−0.690909	−	0.722942i	$-0.742790\pi$
$641$	−34.9848	−1.38182	−0.690909	+	0.722942i	$0.742790\pi$
$642$	16.4924	0.650904
$643$	17.8617	0.704398	0.352199	−	0.935925i	$-0.385434\pi$
$643$	17.8617	0.704398	0.352199	+	0.935925i	$0.385434\pi$
$644$	−39.6155	−1.56107
$645$	0	0
$646$	18.2462	0.717888
$647$	−39.2311	−1.54233	−0.771166	−	0.636634i	$-0.780326\pi$
$647$	−39.2311	−1.54233	−0.771166	+	0.636634i	$0.780326\pi$
$648$	6.56155	0.257762
$649$	32.0000	1.25611
$650$	9.12311	0.357838
$651$	−2.43845	−0.0955703
$652$	51.6155	2.02142
$653$	9.31534	0.364537	0.182269	−	0.983249i	$-0.441656\pi$
$653$	9.31534	0.364537	0.182269	+	0.983249i	$0.441656\pi$
$654$	31.3693	1.22664
$655$	16.6847	0.651924
$656$	27.3693	1.06859
$657$	2.87689	0.112238
$658$	30.2462	1.17912
$659$	24.4924	0.954089	0.477045	−	0.878879i	$-0.341708\pi$
$659$	24.4924	0.954089	0.477045	+	0.878879i	$0.341708\pi$
$660$	−14.2462	−0.554533
$661$	44.2462	1.72098	0.860489	−	0.509469i	$-0.170158\pi$
$661$	44.2462	1.72098	0.860489	+	0.509469i	$0.170158\pi$
$662$	−0.492423	−0.0191385
$663$	−3.56155	−0.138319
$664$	−103.723	−4.02525
$665$	−7.12311	−0.276222
$666$	−9.12311	−0.353513
$667$	−9.75379	−0.377668
$668$	22.2462	0.860732
$669$	3.31534	0.128179
$670$	16.0000	0.618134
$671$	−33.3693	−1.28821
$672$	−6.56155	−0.253117
$673$	−15.1771	−0.585033	−0.292517	−	0.956260i	$-0.594493\pi$
$673$	−15.1771	−0.585033	−0.292517	+	0.956260i	$0.594493\pi$
$674$	19.8617	0.765046
$675$	1.00000	0.0384900
$676$	−1.43845	−0.0553249
$677$	−18.4924	−0.710722	−0.355361	−	0.934729i	$-0.615642\pi$
$677$	−18.4924	−0.710722	−0.355361	+	0.934729i	$0.615642\pi$
$678$	−21.1231	−0.811228
$679$	−6.00000	−0.230259
$680$	6.56155	0.251624
$681$	−8.87689	−0.340163
$682$	19.5076	0.746984
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	−32.4924	−1.24238
$685$	21.1231	0.807072
$686$	−2.56155	−0.0978005
$687$	−3.75379	−0.143216
$688$	0	0
$689$	−10.2462	−0.390350
$690$	−22.2462	−0.846899
$691$	−1.56155	−0.0594043	−0.0297021	−	0.999559i	$-0.509456\pi$
$691$	−1.56155	−0.0594043	−0.0297021	+	0.999559i	$0.509456\pi$
$692$	−63.8617	−2.42766
$693$	−3.12311	−0.118637
$694$	−10.2462	−0.388941
$695$	−4.00000	−0.151729
$696$	−7.36932	−0.279333
$697$	−3.56155	−0.134903
$698$	64.3542	2.43584
$699$	2.19224	0.0829180
$700$	−4.56155	−0.172410
$701$	3.17708	0.119997	0.0599983	−	0.998198i	$-0.480890\pi$
$701$	3.17708	0.119997	0.0599983	+	0.998198i	$0.480890\pi$
$702$	9.12311	0.344329
$703$	25.3693	0.956822
$704$	4.49242	0.169315
$705$	11.8078	0.444706
$706$	−39.8617	−1.50022
$707$	−1.12311	−0.0422387
$708$	46.7386	1.75655
$709$	−27.3693	−1.02788	−0.513938	−	0.857827i	$-0.671814\pi$
$709$	−27.3693	−1.02788	−0.513938	+	0.857827i	$0.671814\pi$
$710$	18.2462	0.684768
$711$	3.12311	0.117126
$712$	13.1231	0.491809
$713$	21.1771	0.793088
$714$	2.56155	0.0958637
$715$	−11.1231	−0.415981
$716$	0.876894	0.0327711
$717$	16.6847	0.623100
$718$	−6.24621	−0.233107
$719$	−1.56155	−0.0582361	−0.0291180	−	0.999576i	$-0.509270\pi$
$719$	−1.56155	−0.0582361	−0.0291180	+	0.999576i	$0.509270\pi$
$720$	−7.68466	−0.286390
$721$	2.24621	0.0836533
$722$	81.3002	3.02568
$723$	−19.5616	−0.727502
$724$	89.2311	3.31625
$725$	−1.12311	−0.0417111
$726$	−3.19224	−0.118475
$727$	−1.56155	−0.0579148	−0.0289574	−	0.999581i	$-0.509219\pi$
$727$	−1.56155	−0.0579148	−0.0289574	+	0.999581i	$0.509219\pi$
$728$	−23.3693	−0.866125
$729$	1.00000	0.0370370
$730$	−7.36932	−0.272751
$731$	0	0
$732$	−48.7386	−1.80143
$733$	7.06913	0.261104	0.130552	−	0.991441i	$-0.458325\pi$
$733$	7.06913	0.261104	0.130552	+	0.991441i	$0.458325\pi$
$734$	−24.0000	−0.885856
$735$	−1.00000	−0.0368856
$736$	56.9848	2.10049
$737$	−19.5076	−0.718571
$738$	9.12311	0.335826
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	16.2462	0.597223
$741$	−25.3693	−0.931965
$742$	7.36932	0.270536
$743$	28.1922	1.03427	0.517136	−	0.855903i	$-0.326998\pi$
$743$	28.1922	1.03427	0.517136	+	0.855903i	$0.326998\pi$
$744$	16.0000	0.586588
$745$	18.6847	0.684553
$746$	59.8617	2.19169
$747$	−15.8078	−0.578376
$748$	−14.2462	−0.520893
$749$	−6.43845	−0.235256
$750$	−2.56155	−0.0935347
$751$	47.6155	1.73752	0.868758	−	0.495237i	$-0.164919\pi$
$751$	47.6155	1.73752	0.868758	+	0.495237i	$0.164919\pi$
$752$	−90.7386	−3.30890
$753$	−16.4924	−0.601017
$754$	−10.2462	−0.373145
$755$	8.68466	0.316067
$756$	−4.56155	−0.165902
$757$	15.7538	0.572581	0.286291	−	0.958143i	$-0.407578\pi$
$757$	15.7538	0.572581	0.286291	+	0.958143i	$0.407578\pi$
$758$	1.26137	0.0458149
$759$	27.1231	0.984506
$760$	46.7386	1.69539
$761$	0.246211	0.00892515	0.00446258	−	0.999990i	$-0.498580\pi$
$761$	0.246211	0.00892515	0.00446258	+	0.999990i	$0.498580\pi$
$762$	51.2311	1.85591
$763$	−12.2462	−0.443343
$764$	68.1080	2.46406
$765$	1.00000	0.0361551
$766$	88.9848	3.21515
$767$	36.4924	1.31767
$768$	−27.0540	−0.976226
$769$	−25.1231	−0.905962	−0.452981	−	0.891520i	$-0.649640\pi$
$769$	−25.1231	−0.905962	−0.452981	+	0.891520i	$0.649640\pi$
$770$	8.00000	0.288300
$771$	30.6847	1.10508
$772$	66.9848	2.41084
$773$	26.3002	0.945952	0.472976	−	0.881075i	$-0.343180\pi$
$773$	26.3002	0.945952	0.472976	+	0.881075i	$0.343180\pi$
$774$	0	0
$775$	2.43845	0.0875916
$776$	39.3693	1.41328
$777$	3.56155	0.127770
$778$	−92.3542	−3.31106
$779$	−25.3693	−0.908950
$780$	−16.2462	−0.581708
$781$	−22.2462	−0.796032
$782$	−22.2462	−0.795523
$783$	−1.12311	−0.0401365
$784$	7.68466	0.274452
$785$	16.2462	0.579852
$786$	−42.7386	−1.52444
$787$	26.2462	0.935576	0.467788	−	0.883841i	$-0.345051\pi$
$787$	26.2462	0.935576	0.467788	+	0.883841i	$0.345051\pi$
$788$	17.1231	0.609985
$789$	18.7386	0.667113
$790$	−8.00000	−0.284627
$791$	8.24621	0.293202
$792$	20.4924	0.728167
$793$	−38.0540	−1.35134
$794$	−63.3693	−2.24889
$795$	2.87689	0.102033
$796$	−28.4924	−1.00989
$797$	−33.4233	−1.18391	−0.591957	−	0.805970i	$-0.701644\pi$
$797$	−33.4233	−1.18391	−0.591957	+	0.805970i	$0.701644\pi$
$798$	18.2462	0.645909
$799$	11.8078	0.417729
$800$	6.56155	0.231986
$801$	2.00000	0.0706665
$802$	35.8617	1.26632
$803$	8.98485	0.317068
$804$	−28.4924	−1.00485
$805$	8.68466	0.306094
$806$	22.2462	0.783589
$807$	1.31534	0.0463022
$808$	7.36932	0.259252
$809$	−47.4773	−1.66921	−0.834606	−	0.550847i	$-0.814305\pi$
$809$	−47.4773	−1.66921	−0.834606	+	0.550847i	$0.814305\pi$
$810$	−2.56155	−0.0900038
$811$	15.4233	0.541585	0.270793	−	0.962638i	$-0.412714\pi$
$811$	15.4233	0.541585	0.270793	+	0.962638i	$0.412714\pi$
$812$	5.12311	0.179786
$813$	−11.1231	−0.390104
$814$	−28.4924	−0.998659
$815$	−11.3153	−0.396359
$816$	−7.68466	−0.269017
$817$	0	0
$818$	70.1080	2.45127
$819$	−3.56155	−0.124451
$820$	−16.2462	−0.567342
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	−54.1080	−1.88723
$823$	−0.684658	−0.0238657	−0.0119328	−	0.999929i	$-0.503798\pi$
$823$	−0.684658	−0.0238657	−0.0119328	+	0.999929i	$0.503798\pi$
$824$	−14.7386	−0.513445
$825$	3.12311	0.108733
$826$	−26.2462	−0.913222
$827$	6.43845	0.223887	0.111943	−	0.993715i	$-0.464292\pi$
$827$	6.43845	0.223887	0.111943	+	0.993715i	$0.464292\pi$
$828$	39.6155	1.37673
$829$	−26.9848	−0.937222	−0.468611	−	0.883405i	$-0.655245\pi$
$829$	−26.9848	−0.937222	−0.468611	+	0.883405i	$0.655245\pi$
$830$	40.4924	1.40551
$831$	32.9309	1.14236
$832$	5.12311	0.177612
$833$	−1.00000	−0.0346479
$834$	10.2462	0.354797
$835$	−4.87689	−0.168772
$836$	−101.477	−3.50966
$837$	2.43845	0.0842851
$838$	−55.2311	−1.90793
$839$	5.75379	0.198643	0.0993214	−	0.995055i	$-0.468333\pi$
$839$	5.75379	0.198643	0.0993214	+	0.995055i	$0.468333\pi$
$840$	6.56155	0.226395
$841$	−27.7386	−0.956505
$842$	−44.3542	−1.52855
$843$	12.4384	0.428403
$844$	−32.4924	−1.11844
$845$	0.315342	0.0108481
$846$	−30.2462	−1.03989
$847$	1.24621	0.0428203
$848$	−22.1080	−0.759190
$849$	26.2462	0.900768
$850$	−2.56155	−0.0878605
$851$	−30.9309	−1.06030
$852$	−32.4924	−1.11317
$853$	34.9848	1.19786	0.598929	−	0.800802i	$-0.295593\pi$
$853$	34.9848	1.19786	0.598929	+	0.800802i	$0.295593\pi$
$854$	27.3693	0.936559
$855$	7.12311	0.243605
$856$	42.2462	1.44395
$857$	12.2462	0.418323	0.209161	−	0.977881i	$-0.432927\pi$
$857$	12.2462	0.418323	0.209161	+	0.977881i	$0.432927\pi$
$858$	28.4924	0.972715
$859$	14.7386	0.502876	0.251438	−	0.967873i	$-0.419097\pi$
$859$	14.7386	0.502876	0.251438	+	0.967873i	$0.419097\pi$
$860$	0	0
$861$	−3.56155	−0.121377
$862$	62.7386	2.13689
$863$	−23.6155	−0.803882	−0.401941	−	0.915666i	$-0.631664\pi$
$863$	−23.6155	−0.803882	−0.401941	+	0.915666i	$0.631664\pi$
$864$	6.56155	0.223229
$865$	14.0000	0.476014
$866$	18.3845	0.624730
$867$	1.00000	0.0339618
$868$	−11.1231	−0.377543
$869$	9.75379	0.330875
$870$	2.87689	0.0975359
$871$	−22.2462	−0.753784
$872$	80.3542	2.72114
$873$	6.00000	0.203069
$874$	−158.462	−5.36006
$875$	1.00000	0.0338062
$876$	13.1231	0.443389
$877$	−16.7386	−0.565224	−0.282612	−	0.959234i	$-0.591201\pi$
$877$	−16.7386	−0.565224	−0.282612	+	0.959234i	$0.591201\pi$
$878$	52.4924	1.77153
$879$	14.4924	0.488817
$880$	−24.0000	−0.809040
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	2.56155	0.0862520
$883$	−17.7538	−0.597463	−0.298731	−	0.954337i	$-0.596563\pi$
$883$	−17.7538	−0.597463	−0.298731	+	0.954337i	$0.596563\pi$
$884$	−16.2462	−0.546419
$885$	−10.2462	−0.344423
$886$	−62.7386	−2.10775
$887$	−28.8769	−0.969591	−0.484796	−	0.874627i	$-0.661106\pi$
$887$	−28.8769	−0.969591	−0.484796	+	0.874627i	$0.661106\pi$
$888$	−23.3693	−0.784223
$889$	−20.0000	−0.670778
$890$	−5.12311	−0.171727
$891$	3.12311	0.104628
$892$	15.1231	0.506359
$893$	84.1080	2.81457
$894$	−47.8617	−1.60074
$895$	−0.192236	−0.00642574
$896$	9.43845	0.315316
$897$	30.9309	1.03275
$898$	−103.093	−3.44025
$899$	−2.73863	−0.0913385
$900$	4.56155	0.152052
$901$	2.87689	0.0958432
$902$	28.4924	0.948694
$903$	0	0
$904$	−54.1080	−1.79960
$905$	−19.5616	−0.650248
$906$	−22.2462	−0.739081
$907$	10.2462	0.340220	0.170110	−	0.985425i	$-0.445588\pi$
$907$	10.2462	0.340220	0.170110	+	0.985425i	$0.445588\pi$
$908$	−40.4924	−1.34379
$909$	1.12311	0.0372511
$910$	9.12311	0.302428
$911$	7.50758	0.248737	0.124369	−	0.992236i	$-0.460309\pi$
$911$	7.50758	0.248737	0.124369	+	0.992236i	$0.460309\pi$
$912$	−54.7386	−1.81258
$913$	−49.3693	−1.63389
$914$	−64.3542	−2.12865
$915$	10.6847	0.353224
$916$	−17.1231	−0.565763
$917$	16.6847	0.550976
$918$	−2.56155	−0.0845438
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	−56.9848	−1.87873
$921$	−22.2462	−0.733038
$922$	10.8769	0.358211
$923$	−25.3693	−0.835041
$924$	−14.2462	−0.468666
$925$	−3.56155	−0.117103
$926$	−22.7386	−0.747238
$927$	−2.24621	−0.0737753
$928$	−7.36932	−0.241910
$929$	−50.6847	−1.66291	−0.831455	−	0.555592i	$-0.812492\pi$
$929$	−50.6847	−1.66291	−0.831455	+	0.555592i	$0.812492\pi$
$930$	−6.24621	−0.204821
$931$	−7.12311	−0.233450
$932$	10.0000	0.327561
$933$	22.4384	0.734602
$934$	97.4773	3.18955
$935$	3.12311	0.102136
$936$	23.3693	0.763850
$937$	−21.3153	−0.696342	−0.348171	−	0.937431i	$-0.613197\pi$
$937$	−21.3153	−0.696342	−0.348171	+	0.937431i	$0.613197\pi$
$938$	16.0000	0.522419
$939$	24.7386	0.807315
$940$	53.8617	1.75678
$941$	38.7926	1.26460	0.632301	−	0.774722i	$-0.282110\pi$
$941$	38.7926	1.26460	0.632301	+	0.774722i	$0.282110\pi$
$942$	−41.6155	−1.35591
$943$	30.9309	1.00725
$944$	78.7386	2.56272
$945$	1.00000	0.0325300
$946$	0	0
$947$	12.6847	0.412196	0.206098	−	0.978531i	$-0.433923\pi$
$947$	12.6847	0.412196	0.206098	+	0.978531i	$0.433923\pi$
$948$	14.2462	0.462695
$949$	10.2462	0.332606
$950$	−18.2462	−0.591985
$951$	−24.0540	−0.780004
$952$	6.56155	0.212661
$953$	−0.246211	−0.00797556	−0.00398778	−	0.999992i	$-0.501269\pi$
$953$	−0.246211	−0.00797556	−0.00398778	+	0.999992i	$0.501269\pi$
$954$	−7.36932	−0.238590
$955$	−14.9309	−0.483152
$956$	76.1080	2.46151
$957$	−3.50758	−0.113384
$958$	4.00000	0.129234
$959$	21.1231	0.682101
$960$	−1.43845	−0.0464257
$961$	−25.0540	−0.808193
$962$	−32.4924	−1.04760
$963$	6.43845	0.207476
$964$	−89.2311	−2.87394
$965$	−14.6847	−0.472716
$966$	−22.2462	−0.715760
$967$	−10.6307	−0.341860	−0.170930	−	0.985283i	$-0.554677\pi$
$967$	−10.6307	−0.341860	−0.170930	+	0.985283i	$0.554677\pi$
$968$	−8.17708	−0.262821
$969$	7.12311	0.228827
$970$	−15.3693	−0.493479
$971$	−52.9848	−1.70036	−0.850182	−	0.526488i	$-0.823508\pi$
$971$	−52.9848	−1.70036	−0.850182	+	0.526488i	$0.823508\pi$
$972$	4.56155	0.146312
$973$	−4.00000	−0.128234
$974$	54.2462	1.73816
$975$	3.56155	0.114061
$976$	−82.1080	−2.62821
$977$	9.50758	0.304174	0.152087	−	0.988367i	$-0.451401\pi$
$977$	9.50758	0.304174	0.152087	+	0.988367i	$0.451401\pi$
$978$	28.9848	0.926833
$979$	6.24621	0.199630
$980$	−4.56155	−0.145713
$981$	12.2462	0.390991
$982$	−97.4773	−3.11062
$983$	53.8617	1.71792	0.858961	−	0.512040i	$-0.171110\pi$
$983$	53.8617	1.71792	0.858961	+	0.512040i	$0.171110\pi$
$984$	23.3693	0.744987
$985$	−3.75379	−0.119606
$986$	2.87689	0.0916190
$987$	11.8078	0.375845
$988$	−115.723	−3.68165
$989$	0	0
$990$	−8.00000	−0.254257
$991$	47.6155	1.51256	0.756279	−	0.654250i	$-0.227015\pi$
$991$	47.6155	1.51256	0.756279	+	0.654250i	$0.227015\pi$
$992$	16.0000	0.508001
$993$	−0.192236	−0.00610042
$994$	18.2462	0.578735
$995$	6.24621	0.198018
$996$	−72.1080	−2.28483
$997$	50.9848	1.61471	0.807353	−	0.590069i	$-0.200899\pi$
$997$	50.9848	1.61471	0.807353	+	0.590069i	$0.200899\pi$
$998$	−35.2311	−1.11522
$999$	−3.56155	−0.112683

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1785.2.a.s.1.2	✓	2
3.2	odd	2		5355.2.a.v.1.1		2
5.4	even	2		8925.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1785.2.a.s.1.2	✓	2	1.1	even	1	trivial
5355.2.a.v.1.1		2	3.2	odd	2
8925.2.a.be.1.1		2	5.4	even	2

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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 $\F_{q}$

Introduction

L-functions

Modular forms

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Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1785.2.a.s.1.2

Level	$1785$
Weight	$2$
Character	1785.1
Self dual	yes
Analytic conductor	$14.253$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$