LMFDB - Embedded newform 117.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("117.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.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:	$6.90322347067$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.74166$ of defining polynomial
Character	$\chi$	$=$	117.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74166 q^{2} -0.483315 q^{4} -19.4833 q^{5} +7.48331 q^{7} -23.2583 q^{8} +O(q^{10})$	\(q+2.74166 q^{2} -0.483315 q^{4} -19.4833 q^{5} +7.48331 q^{7} -23.2583 q^{8} -53.4166 q^{10} -22.8999 q^{11} -13.0000 q^{13} +20.5167 q^{14} -59.8999 q^{16} -67.0334 q^{17} +16.5167 q^{19} +9.41657 q^{20} -62.7836 q^{22} +175.600 q^{23} +254.600 q^{25} -35.6415 q^{26} -3.61680 q^{28} -291.800 q^{29} +117.283 q^{31} +21.8418 q^{32} -183.783 q^{34} -145.800 q^{35} -154.766 q^{37} +45.2831 q^{38} +453.150 q^{40} +251.716 q^{41} -502.566 q^{43} +11.0679 q^{44} +481.434 q^{46} +281.733 q^{47} -287.000 q^{49} +698.025 q^{50} +6.28309 q^{52} -366.999 q^{53} +446.166 q^{55} -174.049 q^{56} -800.015 q^{58} +79.6663 q^{59} -194.865 q^{61} +321.550 q^{62} +539.082 q^{64} +253.283 q^{65} +400.082 q^{67} +32.3982 q^{68} -399.733 q^{70} -528.299 q^{71} -734.366 q^{73} -424.316 q^{74} -7.98276 q^{76} -171.367 q^{77} +113.266 q^{79} +1167.05 q^{80} +690.118 q^{82} +933.466 q^{83} +1306.03 q^{85} -1377.86 q^{86} +532.613 q^{88} -1190.91 q^{89} -97.2831 q^{91} -84.8699 q^{92} +772.415 q^{94} -321.800 q^{95} +557.165 q^{97} -786.856 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8	$ 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8} - 32 q^{10} + 44 q^{11} - 26 q^{13} + 56 q^{14} - 30 q^{16} - 164 q^{17} + 48 q^{19} - 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 26 q^{26} - 112 q^{28} - 404 q^{29} + 40 q^{31} + 126 q^{32} + 276 q^{34} - 112 q^{35} - 100 q^{37} - 104 q^{38} + 592 q^{40} - 200 q^{41} - 616 q^{43} + 980 q^{44} + 1352 q^{46} + 324 q^{47} - 574 q^{49} + 1194 q^{50} - 182 q^{52} + 164 q^{53} + 144 q^{55} + 56 q^{56} - 268 q^{58} - 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 312 q^{65} - 472 q^{67} - 1372 q^{68} - 560 q^{70} - 428 q^{71} - 900 q^{73} - 684 q^{74} + 448 q^{76} - 672 q^{77} - 432 q^{79} + 1032 q^{80} + 2832 q^{82} + 1388 q^{83} + 1744 q^{85} - 840 q^{86} - 1524 q^{88} - 960 q^{89} - 2744 q^{92} + 572 q^{94} - 464 q^{95} - 532 q^{97} + 574 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8 - 32 * q^10 + 44 * q^11 - 26 * q^13 + 56 * q^14 - 30 * q^16 - 164 * q^17 + 48 * q^19 - 56 * q^20 - 380 * q^22 - 8 * q^23 + 150 * q^25 + 26 * q^26 - 112 * q^28 - 404 * q^29 + 40 * q^31 + 126 * q^32 + 276 * q^34 - 112 * q^35 - 100 * q^37 - 104 * q^38 + 592 * q^40 - 200 * q^41 - 616 * q^43 + 980 * q^44 + 1352 * q^46 + 324 * q^47 - 574 * q^49 + 1194 * q^50 - 182 * q^52 + 164 * q^53 + 144 * q^55 + 56 * q^56 - 268 * q^58 - 140 * q^59 + 628 * q^61 + 688 * q^62 - 194 * q^64 + 312 * q^65 - 472 * q^67 - 1372 * q^68 - 560 * q^70 - 428 * q^71 - 900 * q^73 - 684 * q^74 + 448 * q^76 - 672 * q^77 - 432 * q^79 + 1032 * q^80 + 2832 * q^82 + 1388 * q^83 + 1744 * q^85 - 840 * q^86 - 1524 * q^88 - 960 * q^89 - 2744 * q^92 + 572 * q^94 - 464 * q^95 - 532 * q^97 + 574 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.74166	0.969322	0.484661	−	0.874702i	$-0.338943\pi$
$2$	2.74166	0.969322	0.484661	+	0.874702i	$0.338943\pi$
$3$	0	0
$3$	0	0
$4$	−0.483315	−0.0604143
$5$	−19.4833	−1.74264	−0.871320	−	0.490715i	$-0.836736\pi$
$5$	−19.4833	−1.74264	−0.871320	+	0.490715i	$0.836736\pi$
$6$	0	0
$7$	7.48331	0.404061	0.202031	−	0.979379i	$-0.435246\pi$
$7$	7.48331	0.404061	0.202031	+	0.979379i	$0.435246\pi$
$8$	−23.2583	−1.02788
$9$	0	0
$10$	−53.4166	−1.68918
$11$	−22.8999	−0.627689	−0.313844	−	0.949474i	$-0.601617\pi$
$11$	−22.8999	−0.627689	−0.313844	+	0.949474i	$0.601617\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	20.5167	0.391665
$15$	0	0
$16$	−59.8999	−0.935936
$17$	−67.0334	−0.956352	−0.478176	−	0.878264i	$-0.658702\pi$
$17$	−67.0334	−0.956352	−0.478176	+	0.878264i	$0.658702\pi$
$18$	0	0
$19$	16.5167	0.199431	0.0997155	−	0.995016i	$-0.468207\pi$
$19$	16.5167	0.199431	0.0997155	+	0.995016i	$0.468207\pi$
$20$	9.41657	0.105280
$21$	0	0
$22$	−62.7836	−0.608433
$23$	175.600	1.59196	0.795979	−	0.605324i	$-0.206956\pi$
$23$	175.600	1.59196	0.795979	+	0.605324i	$0.206956\pi$
$24$	0	0
$25$	254.600	2.03680
$26$	−35.6415	−0.268842
$27$	0	0
$28$	−3.61680	−0.0244111
$29$	−291.800	−1.86848	−0.934239	−	0.356648i	$-0.883920\pi$
$29$	−291.800	−1.86848	−0.934239	+	0.356648i	$0.883920\pi$
$30$	0	0
$31$	117.283	0.679505	0.339753	−	0.940515i	$-0.389657\pi$
$31$	117.283	0.679505	0.339753	+	0.940515i	$0.389657\pi$
$32$	21.8418	0.120660
$33$	0	0
$34$	−183.783	−0.927013
$35$	−145.800	−0.704133
$36$	0	0
$37$	−154.766	−0.687661	−0.343830	−	0.939032i	$-0.611724\pi$
$37$	−154.766	−0.687661	−0.343830	+	0.939032i	$0.611724\pi$
$38$	45.2831	0.193313
$39$	0	0
$40$	453.150	1.79123
$41$	251.716	0.958815	0.479407	−	0.877592i	$-0.340852\pi$
$41$	251.716	0.958815	0.479407	+	0.877592i	$0.340852\pi$
$42$	0	0
$43$	−502.566	−1.78234	−0.891170	−	0.453669i	$-0.850115\pi$
$43$	−502.566	−1.78234	−0.891170	+	0.453669i	$0.850115\pi$
$44$	11.0679	0.0379214
$45$	0	0
$46$	481.434	1.54312
$47$	281.733	0.874361	0.437181	−	0.899374i	$-0.355977\pi$
$47$	281.733	0.874361	0.437181	+	0.899374i	$0.355977\pi$
$48$	0	0
$49$	−287.000	−0.836735
$50$	698.025	1.97431
$51$	0	0
$52$	6.28309	0.0167559
$53$	−366.999	−0.951154	−0.475577	−	0.879674i	$-0.657761\pi$
$53$	−366.999	−0.951154	−0.475577	+	0.879674i	$0.657761\pi$
$54$	0	0
$55$	446.166	1.09384
$56$	−174.049	−0.415328
$57$	0	0
$58$	−800.015	−1.81116
$59$	79.6663	0.175791	0.0878955	−	0.996130i	$-0.471986\pi$
$59$	79.6663	0.175791	0.0878955	+	0.996130i	$0.471986\pi$
$60$	0	0
$61$	−194.865	−0.409016	−0.204508	−	0.978865i	$-0.565559\pi$
$61$	−194.865	−0.409016	−0.204508	+	0.978865i	$0.565559\pi$
$62$	321.550	0.658660
$63$	0	0
$64$	539.082	1.05289
$65$	253.283	0.483322
$66$	0	0
$67$	400.082	0.729519	0.364759	−	0.931102i	$-0.381151\pi$
$67$	400.082	0.729519	0.364759	+	0.931102i	$0.381151\pi$
$68$	32.3982	0.0577774
$69$	0	0
$70$	−399.733	−0.682532
$71$	−528.299	−0.883065	−0.441532	−	0.897245i	$-0.645565\pi$
$71$	−528.299	−0.883065	−0.441532	+	0.897245i	$0.645565\pi$
$72$	0	0
$73$	−734.366	−1.17741	−0.588706	−	0.808347i	$-0.700362\pi$
$73$	−734.366	−1.17741	−0.588706	+	0.808347i	$0.700362\pi$
$74$	−424.316	−0.666565
$75$	0	0
$76$	−7.98276	−0.0120485
$77$	−171.367	−0.253625
$78$	0	0
$79$	113.266	0.161309	0.0806545	−	0.996742i	$-0.474299\pi$
$79$	113.266	0.161309	0.0806545	+	0.996742i	$0.474299\pi$
$80$	1167.05	1.63100
$81$	0	0
$82$	690.118	0.929400
$83$	933.466	1.23447	0.617236	−	0.786778i	$-0.288252\pi$
$83$	933.466	1.23447	0.617236	+	0.786778i	$0.288252\pi$
$84$	0	0
$85$	1306.03	1.66658
$86$	−1377.86	−1.72766
$87$	0	0
$88$	532.613	0.645191
$89$	−1190.91	−1.41839	−0.709195	−	0.705012i	$-0.750941\pi$
$89$	−1190.91	−1.41839	−0.709195	+	0.705012i	$0.750941\pi$
$90$	0	0
$91$	−97.2831	−0.112066
$92$	−84.8699	−0.0961771
$93$	0	0
$94$	772.415	0.847538
$95$	−321.800	−0.347536
$96$	0	0
$97$	557.165	0.583211	0.291606	−	0.956539i	$-0.405811\pi$
$97$	557.165	0.583211	0.291606	+	0.956539i	$0.405811\pi$
$98$	−786.856	−0.811066
$99$	0	0
$100$	−123.052	−0.123052
$101$	286.766	0.282518	0.141259	−	0.989973i	$-0.454885\pi$
$101$	286.766	0.282518	0.141259	+	0.989973i	$0.454885\pi$
$102$	0	0
$103$	−1911.36	−1.82847	−0.914234	−	0.405187i	$-0.867206\pi$
$103$	−1911.36	−1.82847	−0.914234	+	0.405187i	$0.867206\pi$
$104$	302.358	0.285084
$105$	0	0
$106$	−1006.19	−0.921975
$107$	−834.334	−0.753814	−0.376907	−	0.926251i	$-0.623012\pi$
$107$	−834.334	−0.753814	−0.376907	+	0.926251i	$0.623012\pi$
$108$	0	0
$109$	−1077.66	−0.946986	−0.473493	−	0.880798i	$-0.657007\pi$
$109$	−1077.66	−0.946986	−0.473493	+	0.880798i	$0.657007\pi$
$110$	1223.23	1.06028
$111$	0	0
$112$	−448.250	−0.378175
$113$	166.065	0.138248	0.0691241	−	0.997608i	$-0.477980\pi$
$113$	166.065	0.138248	0.0691241	+	0.997608i	$0.477980\pi$
$114$	0	0
$115$	−3421.26	−2.77421
$116$	141.031	0.112883
$117$	0	0
$118$	218.418	0.170398
$119$	−501.632	−0.386424
$120$	0	0
$121$	−806.595	−0.606007
$122$	−534.254	−0.396468
$123$	0	0
$124$	−56.6847	−0.0410519
$125$	−2525.03	−1.80676
$126$	0	0
$127$	1296.16	0.905637	0.452819	−	0.891603i	$-0.350419\pi$
$127$	1296.16	0.905637	0.452819	+	0.891603i	$0.350419\pi$
$128$	1303.24	0.899934
$129$	0	0
$130$	694.415	0.468494
$131$	197.201	0.131523	0.0657617	−	0.997835i	$-0.479052\pi$
$131$	197.201	0.131523	0.0657617	+	0.997835i	$0.479052\pi$
$132$	0	0
$133$	123.600	0.0805823
$134$	1096.89	0.707139
$135$	0	0
$136$	1559.09	0.983018
$137$	546.915	0.341066	0.170533	−	0.985352i	$-0.445451\pi$
$137$	546.915	0.341066	0.170533	+	0.985352i	$0.445451\pi$
$138$	0	0
$139$	609.666	0.372023	0.186012	−	0.982548i	$-0.440444\pi$
$139$	609.666	0.372023	0.186012	+	0.982548i	$0.440444\pi$
$140$	70.4672	0.0425397
$141$	0	0
$142$	−1448.42	−0.855974
$143$	297.699	0.174090
$144$	0	0
$145$	5685.23	3.25609
$146$	−2013.38	−1.14129
$147$	0	0
$148$	74.8009	0.0415446
$149$	2165.08	1.19040	0.595202	−	0.803576i	$-0.297072\pi$
$149$	2165.08	1.19040	0.595202	+	0.803576i	$0.297072\pi$
$150$	0	0
$151$	−846.549	−0.456233	−0.228116	−	0.973634i	$-0.573257\pi$
$151$	−846.549	−0.456233	−0.228116	+	0.973634i	$0.573257\pi$
$152$	−384.151	−0.204992
$153$	0	0
$154$	−469.830	−0.245844
$155$	−2285.06	−1.18413
$156$	0	0
$157$	1653.60	0.840581	0.420291	−	0.907390i	$-0.361928\pi$
$157$	1653.60	0.840581	0.420291	+	0.907390i	$0.361928\pi$
$158$	310.536	0.156360
$159$	0	0
$160$	−425.550	−0.210267
$161$	1314.07	0.643248
$162$	0	0
$163$	−2866.51	−1.37744	−0.688720	−	0.725027i	$-0.741827\pi$
$163$	−2866.51	−1.37744	−0.688720	+	0.725027i	$0.741827\pi$
$164$	−121.658	−0.0579262
$165$	0	0
$166$	2559.24	1.19660
$167$	−729.066	−0.337825	−0.168913	−	0.985631i	$-0.554026\pi$
$167$	−729.066	−0.337825	−0.168913	+	0.985631i	$0.554026\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	3580.69	1.61545
$171$	0	0
$172$	242.898	0.107679
$173$	3834.83	1.68530	0.842650	−	0.538462i	$-0.180995\pi$
$173$	3834.83	1.68530	0.842650	+	0.538462i	$0.180995\pi$
$174$	0	0
$175$	1905.25	0.822990
$176$	1371.70	0.587476
$177$	0	0
$178$	−3265.08	−1.37488
$179$	283.862	0.118530	0.0592649	−	0.998242i	$-0.481124\pi$
$179$	283.862	0.118530	0.0592649	+	0.998242i	$0.481124\pi$
$180$	0	0
$181$	2363.60	0.970634	0.485317	−	0.874338i	$-0.338704\pi$
$181$	2363.60	0.970634	0.485317	+	0.874338i	$0.338704\pi$
$182$	−266.717	−0.108628
$183$	0	0
$184$	−4084.15	−1.63635
$185$	3015.36	1.19835
$186$	0	0
$187$	1535.06	0.600291
$188$	−136.166	−0.0528240
$189$	0	0
$190$	−882.265	−0.336875
$191$	−2514.26	−0.952491	−0.476246	−	0.879312i	$-0.658003\pi$
$191$	−2514.26	−0.952491	−0.476246	+	0.879312i	$0.658003\pi$
$192$	0	0
$193$	2420.73	0.902839	0.451420	−	0.892312i	$-0.350918\pi$
$193$	2420.73	0.902839	0.451420	+	0.892312i	$0.350918\pi$
$194$	1527.55	0.565320
$195$	0	0
$196$	138.711	0.0505508
$197$	4633.65	1.67581	0.837903	−	0.545819i	$-0.183781\pi$
$197$	4633.65	1.67581	0.837903	+	0.545819i	$0.183781\pi$
$198$	0	0
$199$	3054.17	1.08796	0.543980	−	0.839098i	$-0.316917\pi$
$199$	3054.17	1.08796	0.543980	+	0.839098i	$0.316917\pi$
$200$	−5921.56	−2.09359
$201$	0	0
$202$	786.215	0.273851
$203$	−2183.63	−0.754979
$204$	0	0
$205$	−4904.26	−1.67087
$206$	−5240.30	−1.77237
$207$	0	0
$208$	778.699	0.259582
$209$	−378.230	−0.125181
$210$	0	0
$211$	−4031.60	−1.31539	−0.657694	−	0.753285i	$-0.728468\pi$
$211$	−4031.60	−1.31539	−0.657694	+	0.753285i	$0.728468\pi$
$212$	177.376	0.0574634
$213$	0	0
$214$	−2287.46	−0.730689
$215$	9791.66	3.10598
$216$	0	0
$217$	877.666	0.274562
$218$	−2954.59	−0.917935
$219$	0	0
$220$	−215.638	−0.0660834
$221$	871.434	0.265244
$222$	0	0
$223$	3784.95	1.13659	0.568294	−	0.822826i	$-0.307604\pi$
$223$	3784.95	1.13659	0.568294	+	0.822826i	$0.307604\pi$
$224$	163.449	0.0487539
$225$	0	0
$226$	455.292	0.134007
$227$	−2013.83	−0.588821	−0.294411	−	0.955679i	$-0.595123\pi$
$227$	−2013.83	−0.588821	−0.294411	+	0.955679i	$0.595123\pi$
$228$	0	0
$229$	−3050.73	−0.880340	−0.440170	−	0.897915i	$-0.645082\pi$
$229$	−3050.73	−0.880340	−0.440170	+	0.897915i	$0.645082\pi$
$230$	−9379.93	−2.68910
$231$	0	0
$232$	6786.78	1.92058
$233$	−5587.49	−1.57103	−0.785513	−	0.618846i	$-0.787601\pi$
$233$	−5587.49	−1.57103	−0.785513	+	0.618846i	$0.787601\pi$
$234$	0	0
$235$	−5489.09	−1.52370
$236$	−38.5039	−0.0106203
$237$	0	0
$238$	−1375.30	−0.374570
$239$	1335.69	0.361501	0.180750	−	0.983529i	$-0.442147\pi$
$239$	1335.69	0.361501	0.180750	+	0.983529i	$0.442147\pi$
$240$	0	0
$241$	−571.558	−0.152769	−0.0763845	−	0.997078i	$-0.524338\pi$
$241$	−571.558	−0.152769	−0.0763845	+	0.997078i	$0.524338\pi$
$242$	−2211.41	−0.587416
$243$	0	0
$244$	94.1813	0.0247104
$245$	5591.71	1.45813
$246$	0	0
$247$	−214.717	−0.0553122
$248$	−2727.81	−0.698452
$249$	0	0
$250$	−6922.76	−1.75134
$251$	−4088.60	−1.02817	−0.514084	−	0.857740i	$-0.671868\pi$
$251$	−4088.60	−1.02817	−0.514084	+	0.857740i	$0.671868\pi$
$252$	0	0
$253$	−4021.21	−0.999254
$254$	3553.64	0.877854
$255$	0	0
$256$	−739.607	−0.180568
$257$	−3050.23	−0.740342	−0.370171	−	0.928964i	$-0.620701\pi$
$257$	−3050.23	−0.740342	−0.370171	+	0.928964i	$0.620701\pi$
$258$	0	0
$259$	−1158.17	−0.277857
$260$	−122.415	−0.0291996
$261$	0	0
$262$	540.659	0.127489
$263$	−5770.99	−1.35306	−0.676530	−	0.736415i	$-0.736517\pi$
$263$	−5770.99	−1.35306	−0.676530	+	0.736415i	$0.736517\pi$
$264$	0	0
$265$	7150.35	1.65752
$266$	338.868	0.0781102
$267$	0	0
$268$	−193.365	−0.0440734
$269$	2079.40	0.471314	0.235657	−	0.971836i	$-0.424276\pi$
$269$	2079.40	0.471314	0.235657	+	0.971836i	$0.424276\pi$
$270$	0	0
$271$	6012.00	1.34761	0.673807	−	0.738908i	$-0.264658\pi$
$271$	6012.00	1.34761	0.673807	+	0.738908i	$0.264658\pi$
$272$	4015.29	0.895084
$273$	0	0
$274$	1499.45	0.330603
$275$	−5830.30	−1.27847
$276$	0	0
$277$	−735.201	−0.159473	−0.0797364	−	0.996816i	$-0.525408\pi$
$277$	−735.201	−0.159473	−0.0797364	+	0.996816i	$0.525408\pi$
$278$	1671.50	0.360610
$279$	0	0
$280$	3391.06	0.723767
$281$	1902.92	0.403981	0.201990	−	0.979387i	$-0.435259\pi$
$281$	1902.92	0.403981	0.201990	+	0.979387i	$0.435259\pi$
$282$	0	0
$283$	2125.71	0.446502	0.223251	−	0.974761i	$-0.428333\pi$
$283$	2125.71	0.446502	0.223251	+	0.974761i	$0.428333\pi$
$284$	255.335	0.0533498
$285$	0	0
$286$	816.187	0.168749
$287$	1883.67	0.387420
$288$	0	0
$289$	−419.527	−0.0853913
$290$	15586.9	3.15620
$291$	0	0
$292$	354.930	0.0711325
$293$	1641.03	0.327200	0.163600	−	0.986527i	$-0.447689\pi$
$293$	1641.03	0.327200	0.163600	+	0.986527i	$0.447689\pi$
$294$	0	0
$295$	−1552.16	−0.306341
$296$	3599.61	0.706835
$297$	0	0
$298$	5935.91	1.15389
$299$	−2282.79	−0.441530
$300$	0	0
$301$	−3760.86	−0.720174
$302$	−2320.95	−0.442237
$303$	0	0
$304$	−989.348	−0.186655
$305$	3796.62	0.712767
$306$	0	0
$307$	−3373.27	−0.627111	−0.313555	−	0.949570i	$-0.601520\pi$
$307$	−3373.27	−0.627111	−0.313555	+	0.949570i	$0.601520\pi$
$308$	82.8242	0.0153226
$309$	0	0
$310$	−6264.86	−1.14781
$311$	868.525	0.158359	0.0791793	−	0.996860i	$-0.474770\pi$
$311$	868.525	0.158359	0.0791793	+	0.996860i	$0.474770\pi$
$312$	0	0
$313$	−4343.19	−0.784319	−0.392159	−	0.919897i	$-0.628272\pi$
$313$	−4343.19	−0.784319	−0.392159	+	0.919897i	$0.628272\pi$
$314$	4533.59	0.814794
$315$	0	0
$316$	−54.7431	−0.00974537
$317$	3277.65	0.580730	0.290365	−	0.956916i	$-0.406223\pi$
$317$	3277.65	0.580730	0.290365	+	0.956916i	$0.406223\pi$
$318$	0	0
$319$	6682.18	1.17282
$320$	−10503.1	−1.83482
$321$	0	0
$322$	3602.72	0.623515
$323$	−1107.17	−0.190726
$324$	0	0
$325$	−3309.79	−0.564906
$326$	−7859.00	−1.33518
$327$	0	0
$328$	−5854.49	−0.985550
$329$	2108.30	0.353295
$330$	0	0
$331$	5589.62	0.928197	0.464099	−	0.885784i	$-0.346378\pi$
$331$	5589.62	0.928197	0.464099	+	0.885784i	$0.346378\pi$
$332$	−451.158	−0.0745798
$333$	0	0
$334$	−1998.85	−0.327461
$335$	−7794.92	−1.27129
$336$	0	0
$337$	901.544	0.145728	0.0728638	−	0.997342i	$-0.476786\pi$
$337$	901.544	0.145728	0.0728638	+	0.997342i	$0.476786\pi$
$338$	463.340	0.0745633
$339$	0	0
$340$	−631.225	−0.100685
$341$	−2685.77	−0.426518
$342$	0	0
$343$	−4714.49	−0.742153
$344$	11688.9	1.83204
$345$	0	0
$346$	10513.8	1.63360
$347$	812.318	0.125670	0.0628350	−	0.998024i	$-0.479986\pi$
$347$	812.318	0.125670	0.0628350	+	0.998024i	$0.479986\pi$
$348$	0	0
$349$	4437.96	0.680683	0.340342	−	0.940302i	$-0.389457\pi$
$349$	4437.96	0.680683	0.340342	+	0.940302i	$0.389457\pi$
$350$	5223.54	0.797743
$351$	0	0
$352$	−500.174	−0.0757368
$353$	−7115.35	−1.07284	−0.536419	−	0.843952i	$-0.680223\pi$
$353$	−7115.35	−1.07284	−0.536419	+	0.843952i	$0.680223\pi$
$354$	0	0
$355$	10293.0	1.53886
$356$	575.587	0.0856911
$357$	0	0
$358$	778.253	0.114894
$359$	−4693.98	−0.690081	−0.345040	−	0.938588i	$-0.612135\pi$
$359$	−4693.98	−0.690081	−0.345040	+	0.938588i	$0.612135\pi$
$360$	0	0
$361$	−6586.20	−0.960227
$362$	6480.17	0.940857
$363$	0	0
$364$	47.0184	0.00677042
$365$	14307.9	2.05181
$366$	0	0
$367$	9243.98	1.31480	0.657400	−	0.753542i	$-0.271656\pi$
$367$	9243.98	1.31480	0.657400	+	0.753542i	$0.271656\pi$
$368$	−10518.4	−1.48997
$369$	0	0
$370$	8267.09	1.16158
$371$	−2746.37	−0.384324
$372$	0	0
$373$	−4311.99	−0.598569	−0.299285	−	0.954164i	$-0.596748\pi$
$373$	−4311.99	−0.598569	−0.299285	+	0.954164i	$0.596748\pi$
$374$	4208.60	0.581876
$375$	0	0
$376$	−6552.64	−0.898741
$377$	3793.40	0.518223
$378$	0	0
$379$	−2382.73	−0.322936	−0.161468	−	0.986878i	$-0.551623\pi$
$379$	−2382.73	−0.322936	−0.161468	+	0.986878i	$0.551623\pi$
$380$	155.531	0.0209962
$381$	0	0
$382$	−6893.25	−0.923271
$383$	−4845.81	−0.646499	−0.323250	−	0.946314i	$-0.604775\pi$
$383$	−4845.81	−0.646499	−0.323250	+	0.946314i	$0.604775\pi$
$384$	0	0
$385$	3338.80	0.441976
$386$	6636.81	0.875142
$387$	0	0
$388$	−269.286	−0.0352343
$389$	−9561.50	−1.24624	−0.623120	−	0.782127i	$-0.714135\pi$
$389$	−9561.50	−1.24624	−0.623120	+	0.782127i	$0.714135\pi$
$390$	0	0
$391$	−11771.0	−1.52247
$392$	6675.14	0.860066
$393$	0	0
$394$	12703.9	1.62440
$395$	−2206.79	−0.281103
$396$	0	0
$397$	−7440.11	−0.940575	−0.470287	−	0.882513i	$-0.655850\pi$
$397$	−7440.11	−0.940575	−0.470287	+	0.882513i	$0.655850\pi$
$398$	8373.48	1.05458
$399$	0	0
$400$	−15250.5	−1.90631
$401$	8687.80	1.08192	0.540958	−	0.841050i	$-0.318062\pi$
$401$	8687.80	1.08192	0.540958	+	0.841050i	$0.318062\pi$
$402$	0	0
$403$	−1524.68	−0.188461
$404$	−138.598	−0.0170681
$405$	0	0
$406$	−5986.76	−0.731818
$407$	3544.13	0.431637
$408$	0	0
$409$	2556.10	0.309024	0.154512	−	0.987991i	$-0.450619\pi$
$409$	2556.10	0.309024	0.154512	+	0.987991i	$0.450619\pi$
$410$	−13445.8	−1.61961
$411$	0	0
$412$	923.790	0.110466
$413$	596.168	0.0710303
$414$	0	0
$415$	−18187.0	−2.15124
$416$	−283.943	−0.0334650
$417$	0	0
$418$	−1036.98	−0.121340
$419$	3347.46	0.390296	0.195148	−	0.980774i	$-0.437481\pi$
$419$	3347.46	0.390296	0.195148	+	0.980774i	$0.437481\pi$
$420$	0	0
$421$	−1854.48	−0.214684	−0.107342	−	0.994222i	$-0.534234\pi$
$421$	−1854.48	−0.214684	−0.107342	+	0.994222i	$0.534234\pi$
$422$	−11053.3	−1.27503
$423$	0	0
$424$	8535.79	0.977675
$425$	−17066.7	−1.94789
$426$	0	0
$427$	−1458.24	−0.165267
$428$	403.246	0.0455412
$429$	0	0
$430$	26845.4	3.01069
$431$	14043.1	1.56945	0.784725	−	0.619844i	$-0.212804\pi$
$431$	14043.1	1.56945	0.784725	+	0.619844i	$0.212804\pi$
$432$	0	0
$433$	3086.47	0.342555	0.171278	−	0.985223i	$-0.445210\pi$
$433$	3086.47	0.342555	0.171278	+	0.985223i	$0.445210\pi$
$434$	2406.26	0.266139
$435$	0	0
$436$	520.851	0.0572116
$437$	2900.32	0.317486
$438$	0	0
$439$	2837.68	0.308508	0.154254	−	0.988031i	$-0.450703\pi$
$439$	2837.68	0.308508	0.154254	+	0.988031i	$0.450703\pi$
$440$	−10377.1	−1.12434
$441$	0	0
$442$	2389.17	0.257107
$443$	−18309.4	−1.96367	−0.981834	−	0.189744i	$-0.939234\pi$
$443$	−18309.4	−1.96367	−0.981834	+	0.189744i	$0.939234\pi$
$444$	0	0
$445$	23203.0	2.47174
$446$	10377.0	1.10172
$447$	0	0
$448$	4034.12	0.425433
$449$	−13861.2	−1.45690	−0.728451	−	0.685098i	$-0.759759\pi$
$449$	−13861.2	−1.45690	−0.728451	+	0.685098i	$0.759759\pi$
$450$	0	0
$451$	−5764.26	−0.601837
$452$	−80.2614	−0.00835217
$453$	0	0
$454$	−5521.23	−0.570758
$455$	1895.40	0.195291
$456$	0	0
$457$	−8990.36	−0.920243	−0.460122	−	0.887856i	$-0.652194\pi$
$457$	−8990.36	−0.920243	−0.460122	+	0.887856i	$0.652194\pi$
$458$	−8364.05	−0.853333
$459$	0	0
$460$	1653.55	0.167602
$461$	3406.90	0.344198	0.172099	−	0.985080i	$-0.444945\pi$
$461$	3406.90	0.344198	0.172099	+	0.985080i	$0.444945\pi$
$462$	0	0
$463$	−7498.45	−0.752662	−0.376331	−	0.926485i	$-0.622814\pi$
$463$	−7498.45	−0.752662	−0.376331	+	0.926485i	$0.622814\pi$
$464$	17478.8	1.74878
$465$	0	0
$466$	−15319.0	−1.52283
$467$	−7711.38	−0.764112	−0.382056	−	0.924139i	$-0.624784\pi$
$467$	−7711.38	−0.764112	−0.382056	+	0.924139i	$0.624784\pi$
$468$	0	0
$469$	2993.94	0.294770
$470$	−15049.2	−1.47695
$471$	0	0
$472$	−1852.91	−0.180693
$473$	11508.7	1.11875
$474$	0	0
$475$	4205.14	0.406200
$476$	242.446	0.0233456
$477$	0	0
$478$	3662.01	0.350411
$479$	9439.82	0.900451	0.450226	−	0.892915i	$-0.351344\pi$
$479$	9439.82	0.900451	0.450226	+	0.892915i	$0.351344\pi$
$480$	0	0
$481$	2011.96	0.190723
$482$	−1567.02	−0.148082
$483$	0	0
$484$	389.839	0.0366115
$485$	−10855.4	−1.01633
$486$	0	0
$487$	−6156.20	−0.572821	−0.286411	−	0.958107i	$-0.592462\pi$
$487$	−6156.20	−0.572821	−0.286411	+	0.958107i	$0.592462\pi$
$488$	4532.25	0.420420
$489$	0	0
$490$	15330.6	1.41340
$491$	−3842.74	−0.353198	−0.176599	−	0.984283i	$-0.556510\pi$
$491$	−3842.74	−0.353198	−0.176599	+	0.984283i	$0.556510\pi$
$492$	0	0
$493$	19560.3	1.78692
$494$	−588.680	−0.0536153
$495$	0	0
$496$	−7025.24	−0.635973
$497$	−3953.43	−0.356812
$498$	0	0
$499$	−12842.4	−1.15211	−0.576056	−	0.817410i	$-0.695409\pi$
$499$	−12842.4	−1.15211	−0.576056	+	0.817410i	$0.695409\pi$
$500$	1220.38	0.109154
$501$	0	0
$502$	−11209.5	−0.996627
$503$	−8580.11	−0.760573	−0.380287	−	0.924869i	$-0.624175\pi$
$503$	−8580.11	−0.760573	−0.380287	+	0.924869i	$0.624175\pi$
$504$	0	0
$505$	−5587.16	−0.492327
$506$	−11024.8	−0.968599
$507$	0	0
$508$	−626.455	−0.0547135
$509$	43.5957	0.00379635	0.00189818	−	0.999998i	$-0.499396\pi$
$509$	43.5957	0.00379635	0.00189818	+	0.999998i	$0.499396\pi$
$510$	0	0
$511$	−5495.49	−0.475746
$512$	−12453.7	−1.07496
$513$	0	0
$514$	−8362.68	−0.717630
$515$	37239.7	3.18636
$516$	0	0
$517$	−6451.66	−0.548827
$518$	−3175.29	−0.269333
$519$	0	0
$520$	−5890.94	−0.496798
$521$	−11368.1	−0.955939	−0.477969	−	0.878377i	$-0.658627\pi$
$521$	−11368.1	−0.955939	−0.477969	+	0.878377i	$0.658627\pi$
$522$	0	0
$523$	−5229.53	−0.437230	−0.218615	−	0.975811i	$-0.570154\pi$
$523$	−5229.53	−0.437230	−0.218615	+	0.975811i	$0.570154\pi$
$524$	−95.3103	−0.00794590
$525$	0	0
$526$	−15822.1	−1.31155
$527$	−7861.88	−0.649846
$528$	0	0
$529$	18668.2	1.53433
$530$	19603.8	1.60667
$531$	0	0
$532$	−59.7375	−0.00486832
$533$	−3272.31	−0.265927
$534$	0	0
$535$	16255.6	1.31363
$536$	−9305.24	−0.749860
$537$	0	0
$538$	5701.01	0.456855
$539$	6572.27	0.525209
$540$	0	0
$541$	−6567.99	−0.521959	−0.260980	−	0.965344i	$-0.584046\pi$
$541$	−6567.99	−0.521959	−0.260980	+	0.965344i	$0.584046\pi$
$542$	16482.9	1.30627
$543$	0	0
$544$	−1464.13	−0.115393
$545$	20996.5	1.65026
$546$	0	0
$547$	−13675.7	−1.06897	−0.534487	−	0.845177i	$-0.679495\pi$
$547$	−13675.7	−1.06897	−0.534487	+	0.845177i	$0.679495\pi$
$548$	−264.332	−0.0206053
$549$	0	0
$550$	−15984.7	−1.23925
$551$	−4819.57	−0.372632
$552$	0	0
$553$	847.604	0.0651786
$554$	−2015.67	−0.154581
$555$	0	0
$556$	−294.661	−0.0224755
$557$	−4527.96	−0.344445	−0.172222	−	0.985058i	$-0.555095\pi$
$557$	−4527.96	−0.344445	−0.172222	+	0.985058i	$0.555095\pi$
$558$	0	0
$559$	6533.36	0.494332
$560$	8733.39	0.659023
$561$	0	0
$562$	5217.15	0.391588
$563$	−18441.8	−1.38051	−0.690256	−	0.723566i	$-0.742502\pi$
$563$	−18441.8	−1.38051	−0.690256	+	0.723566i	$0.742502\pi$
$564$	0	0
$565$	−3235.49	−0.240917
$566$	5827.96	0.432804
$567$	0	0
$568$	12287.4	0.907687
$569$	13553.5	0.998578	0.499289	−	0.866436i	$-0.333595\pi$
$569$	13553.5	0.998578	0.499289	+	0.866436i	$0.333595\pi$
$570$	0	0
$571$	14815.5	1.08583	0.542915	−	0.839788i	$-0.317321\pi$
$571$	14815.5	1.08583	0.542915	+	0.839788i	$0.317321\pi$
$572$	−143.882	−0.0105175
$573$	0	0
$574$	5164.37	0.375534
$575$	44707.6	3.24249
$576$	0	0
$577$	21596.2	1.55816	0.779081	−	0.626923i	$-0.215686\pi$
$577$	21596.2	1.55816	0.779081	+	0.626923i	$0.215686\pi$
$578$	−1150.20	−0.0827716
$579$	0	0
$580$	−2747.75	−0.196714
$581$	6985.42	0.498802
$582$	0	0
$583$	8404.23	0.597029
$584$	17080.1	1.21024
$585$	0	0
$586$	4499.13	0.317163
$587$	918.801	0.0646047	0.0323024	−	0.999478i	$-0.489716\pi$
$587$	918.801	0.0646047	0.0323024	+	0.999478i	$0.489716\pi$
$588$	0	0
$589$	1937.13	0.135514
$590$	−4255.50	−0.296943
$591$	0	0
$592$	9270.49	0.643606
$593$	−19816.0	−1.37226	−0.686128	−	0.727481i	$-0.740691\pi$
$593$	−19816.0	−1.37226	−0.686128	+	0.727481i	$0.740691\pi$
$594$	0	0
$595$	9773.45	0.673399
$596$	−1046.42	−0.0719175
$597$	0	0
$598$	−6258.64	−0.427985
$599$	5141.86	0.350736	0.175368	−	0.984503i	$-0.443889\pi$
$599$	5141.86	0.350736	0.175368	+	0.984503i	$0.443889\pi$
$600$	0	0
$601$	12380.9	0.840312	0.420156	−	0.907452i	$-0.361975\pi$
$601$	12380.9	0.840312	0.420156	+	0.907452i	$0.361975\pi$
$602$	−10311.0	−0.698081
$603$	0	0
$604$	409.150	0.0275630
$605$	15715.1	1.05605
$606$	0	0
$607$	−23717.0	−1.58590	−0.792951	−	0.609286i	$-0.791456\pi$
$607$	−23717.0	−1.58590	−0.792951	+	0.609286i	$0.791456\pi$
$608$	360.754	0.0240633
$609$	0	0
$610$	10409.0	0.690901
$611$	−3662.53	−0.242504
$612$	0	0
$613$	−26157.1	−1.72345	−0.861726	−	0.507373i	$-0.830617\pi$
$613$	−26157.1	−1.72345	−0.861726	+	0.507373i	$0.830617\pi$
$614$	−9248.36	−0.607872
$615$	0	0
$616$	3985.71	0.260696
$617$	−23613.9	−1.54077	−0.770387	−	0.637576i	$-0.779937\pi$
$617$	−23613.9	−1.54077	−0.770387	+	0.637576i	$0.779937\pi$
$618$	0	0
$619$	23345.4	1.51588	0.757940	−	0.652324i	$-0.226206\pi$
$619$	23345.4	1.51588	0.757940	+	0.652324i	$0.226206\pi$
$620$	1104.40	0.0715387
$621$	0	0
$622$	2381.20	0.153501
$623$	−8911.99	−0.573116
$624$	0	0
$625$	17371.0	1.11174
$626$	−11907.5	−0.760258
$627$	0	0
$628$	−799.207	−0.0507832
$629$	10374.5	0.657645
$630$	0	0
$631$	15245.7	0.961841	0.480921	−	0.876764i	$-0.340302\pi$
$631$	15245.7	0.961841	0.480921	+	0.876764i	$0.340302\pi$
$632$	−2634.38	−0.165807
$633$	0	0
$634$	8986.20	0.562914
$635$	−25253.6	−1.57820
$636$	0	0
$637$	3731.00	0.232068
$638$	18320.3	1.13684
$639$	0	0
$640$	−25391.5	−1.56826
$641$	−10192.7	−0.628063	−0.314032	−	0.949413i	$-0.601680\pi$
$641$	−10192.7	−0.628063	−0.314032	+	0.949413i	$0.601680\pi$
$642$	0	0
$643$	−5506.31	−0.337710	−0.168855	−	0.985641i	$-0.554007\pi$
$643$	−5506.31	−0.337710	−0.168855	+	0.985641i	$0.554007\pi$
$644$	−635.108	−0.0388614
$645$	0	0
$646$	−3035.48	−0.184875
$647$	13297.5	0.808005	0.404003	−	0.914758i	$-0.367619\pi$
$647$	13297.5	0.808005	0.404003	+	0.914758i	$0.367619\pi$
$648$	0	0
$649$	−1824.35	−0.110342
$650$	−9074.32	−0.547576
$651$	0	0
$652$	1385.43	0.0832171
$653$	12440.2	0.745519	0.372760	−	0.927928i	$-0.378412\pi$
$653$	12440.2	0.745519	0.372760	+	0.927928i	$0.378412\pi$
$654$	0	0
$655$	−3842.14	−0.229198
$656$	−15077.7	−0.897389
$657$	0	0
$658$	5780.23	0.342457
$659$	9562.87	0.565276	0.282638	−	0.959227i	$-0.408791\pi$
$659$	9562.87	0.565276	0.282638	+	0.959227i	$0.408791\pi$
$660$	0	0
$661$	2409.69	0.141795	0.0708973	−	0.997484i	$-0.477414\pi$
$661$	2409.69	0.141795	0.0708973	+	0.997484i	$0.477414\pi$
$662$	15324.8	0.899722
$663$	0	0
$664$	−21710.9	−1.26889
$665$	−2408.13	−0.140426
$666$	0	0
$667$	−51239.9	−2.97454
$668$	352.368	0.0204095
$669$	0	0
$670$	−21371.0	−1.23229
$671$	4462.40	0.256735
$672$	0	0
$673$	7929.02	0.454147	0.227074	−	0.973878i	$-0.427084\pi$
$673$	7929.02	0.454147	0.227074	+	0.973878i	$0.427084\pi$
$674$	2471.72	0.141257
$675$	0	0
$676$	−81.6802	−0.00464726
$677$	2628.26	0.149206	0.0746030	−	0.997213i	$-0.476231\pi$
$677$	2628.26	0.149206	0.0746030	+	0.997213i	$0.476231\pi$
$678$	0	0
$679$	4169.44	0.235653
$680$	−30376.1	−1.71305
$681$	0	0
$682$	−7363.46	−0.413433
$683$	−10021.5	−0.561437	−0.280719	−	0.959790i	$-0.590573\pi$
$683$	−10021.5	−0.561437	−0.280719	+	0.959790i	$0.590573\pi$
$684$	0	0
$685$	−10655.7	−0.594356
$686$	−12925.5	−0.719385
$687$	0	0
$688$	30103.7	1.66816
$689$	4770.99	0.263803
$690$	0	0
$691$	23987.2	1.32057	0.660286	−	0.751014i	$-0.270435\pi$
$691$	23987.2	1.32057	0.660286	+	0.751014i	$0.270435\pi$
$692$	−1853.43	−0.101816
$693$	0	0
$694$	2227.10	0.121815
$695$	−11878.3	−0.648303
$696$	0	0
$697$	−16873.4	−0.916964
$698$	12167.4	0.659801
$699$	0	0
$700$	−920.835	−0.0497204
$701$	3763.71	0.202787	0.101393	−	0.994846i	$-0.467670\pi$
$701$	3763.71	0.202787	0.101393	+	0.994846i	$0.467670\pi$
$702$	0	0
$703$	−2556.23	−0.137141
$704$	−12344.9	−0.660890
$705$	0	0
$706$	−19507.8	−1.03993
$707$	2145.96	0.114155
$708$	0	0
$709$	−36047.8	−1.90946	−0.954728	−	0.297479i	$-0.903854\pi$
$709$	−36047.8	−1.90946	−0.954728	+	0.297479i	$0.903854\pi$
$710$	28219.9	1.49166
$711$	0	0
$712$	27698.7	1.45794
$713$	20594.9	1.08174
$714$	0	0
$715$	−5800.15	−0.303376
$716$	−137.195	−0.00716090
$717$	0	0
$718$	−12869.3	−0.668910
$719$	3944.18	0.204580	0.102290	−	0.994755i	$-0.467383\pi$
$719$	3944.18	0.204580	0.102290	+	0.994755i	$0.467383\pi$
$720$	0	0
$721$	−14303.3	−0.738812
$722$	−18057.1	−0.930770
$723$	0	0
$724$	−1142.36	−0.0586402
$725$	−74292.1	−3.80571
$726$	0	0
$727$	−20447.8	−1.04315	−0.521573	−	0.853206i	$-0.674655\pi$
$727$	−20447.8	−1.04315	−0.521573	+	0.853206i	$0.674655\pi$
$728$	2262.64	0.115191
$729$	0	0
$730$	39227.3	1.98886
$731$	33688.7	1.70454
$732$	0	0
$733$	−13536.2	−0.682089	−0.341045	−	0.940047i	$-0.610781\pi$
$733$	−13536.2	−0.682089	−0.341045	+	0.940047i	$0.610781\pi$
$734$	25343.8	1.27447
$735$	0	0
$736$	3835.40	0.192085
$737$	−9161.83	−0.457911
$738$	0	0
$739$	15839.1	0.788433	0.394217	−	0.919018i	$-0.371016\pi$
$739$	15839.1	0.788433	0.394217	+	0.919018i	$0.371016\pi$
$740$	−1457.37	−0.0723972
$741$	0	0
$742$	−7529.60	−0.372534
$743$	1664.92	0.0822075	0.0411037	−	0.999155i	$-0.486913\pi$
$743$	1664.92	0.0822075	0.0411037	+	0.999155i	$0.486913\pi$
$744$	0	0
$745$	−42182.9	−2.07445
$746$	−11822.0	−0.580206
$747$	0	0
$748$	−741.916	−0.0362662
$749$	−6243.58	−0.304587
$750$	0	0
$751$	22399.1	1.08835	0.544177	−	0.838970i	$-0.316842\pi$
$751$	22399.1	1.08835	0.544177	+	0.838970i	$0.316842\pi$
$752$	−16875.8	−0.818346
$753$	0	0
$754$	10400.2	0.502325
$755$	16493.6	0.795050
$756$	0	0
$757$	23798.9	1.14265	0.571326	−	0.820723i	$-0.306429\pi$
$757$	23798.9	1.14265	0.571326	+	0.820723i	$0.306429\pi$
$758$	−6532.64	−0.313029
$759$	0	0
$760$	7484.53	0.357227
$761$	−13693.5	−0.652285	−0.326142	−	0.945321i	$-0.605749\pi$
$761$	−13693.5	−0.652285	−0.326142	+	0.945321i	$0.605749\pi$
$762$	0	0
$763$	−8064.50	−0.382640
$764$	1215.18	0.0575441
$765$	0	0
$766$	−13285.5	−0.626666
$767$	−1035.66	−0.0487556
$768$	0	0
$769$	16299.9	0.764358	0.382179	−	0.924088i	$-0.375174\pi$
$769$	16299.9	0.764358	0.382179	+	0.924088i	$0.375174\pi$
$770$	9153.84	0.428418
$771$	0	0
$772$	−1169.97	−0.0545444
$773$	−33532.2	−1.56024	−0.780122	−	0.625628i	$-0.784843\pi$
$773$	−33532.2	−1.56024	−0.780122	+	0.625628i	$0.784843\pi$
$774$	0	0
$775$	29860.2	1.38401
$776$	−12958.7	−0.599473
$777$	0	0
$778$	−26214.3	−1.20801
$779$	4157.51	0.191217
$780$	0	0
$781$	12098.0	0.554290
$782$	−32272.1	−1.47577
$783$	0	0
$784$	17191.3	0.783130
$785$	−32217.5	−1.46483
$786$	0	0
$787$	16163.3	0.732097	0.366049	−	0.930596i	$-0.380710\pi$
$787$	16163.3	0.732097	0.366049	+	0.930596i	$0.380710\pi$
$788$	−2239.51	−0.101243
$789$	0	0
$790$	−6050.27	−0.272480
$791$	1242.71	0.0558607
$792$	0	0
$793$	2533.25	0.113441
$794$	−20398.2	−0.911720
$795$	0	0
$796$	−1476.12	−0.0657284
$797$	39636.4	1.76160	0.880798	−	0.473492i	$-0.157007\pi$
$797$	39636.4	1.76160	0.880798	+	0.473492i	$0.157007\pi$
$798$	0	0
$799$	−18885.5	−0.836197
$800$	5560.90	0.245760
$801$	0	0
$802$	23819.0	1.04872
$803$	16816.9	0.739048
$804$	0	0
$805$	−25602.4	−1.12095
$806$	−4180.15	−0.182679
$807$	0	0
$808$	−6669.71	−0.290396
$809$	23811.2	1.03481	0.517403	−	0.855742i	$-0.326899\pi$
$809$	23811.2	1.03481	0.517403	+	0.855742i	$0.326899\pi$
$810$	0	0
$811$	27218.6	1.17851	0.589256	−	0.807946i	$-0.299421\pi$
$811$	27218.6	1.17851	0.589256	+	0.807946i	$0.299421\pi$
$812$	1055.38	0.0456116
$813$	0	0
$814$	9716.80	0.418395
$815$	55849.2	2.40038
$816$	0	0
$817$	−8300.73	−0.355454
$818$	7007.94	0.299544
$819$	0	0
$820$	2370.30	0.100944
$821$	43094.8	1.83193	0.915967	−	0.401253i	$-0.131425\pi$
$821$	43094.8	1.83193	0.915967	+	0.401253i	$0.131425\pi$
$822$	0	0
$823$	26541.1	1.12414	0.562068	−	0.827091i	$-0.310006\pi$
$823$	26541.1	1.12414	0.562068	+	0.827091i	$0.310006\pi$
$824$	44455.1	1.87945
$825$	0	0
$826$	1634.49	0.0688512
$827$	44898.7	1.88788	0.943942	−	0.330112i	$-0.107087\pi$
$827$	44898.7	1.88788	0.943942	+	0.330112i	$0.107087\pi$
$828$	0	0
$829$	−7137.48	−0.299029	−0.149514	−	0.988760i	$-0.547771\pi$
$829$	−7137.48	−0.299029	−0.149514	+	0.988760i	$0.547771\pi$
$830$	−49862.6	−2.08525
$831$	0	0
$832$	−7008.06	−0.292020
$833$	19238.6	0.800213
$834$	0	0
$835$	14204.6	0.588708
$836$	182.804	0.00756270
$837$	0	0
$838$	9177.59	0.378323
$839$	4387.17	0.180527	0.0902634	−	0.995918i	$-0.471229\pi$
$839$	4387.17	0.180527	0.0902634	+	0.995918i	$0.471229\pi$
$840$	0	0
$841$	60758.1	2.49121
$842$	−5084.36	−0.208098
$843$	0	0
$844$	1948.53	0.0794683
$845$	−3292.68	−0.134049
$846$	0	0
$847$	−6036.01	−0.244864
$848$	21983.2	0.890219
$849$	0	0
$850$	−46791.0	−1.88814
$851$	−27176.9	−1.09473
$852$	0	0
$853$	−9328.85	−0.374459	−0.187230	−	0.982316i	$-0.559951\pi$
$853$	−9328.85	−0.374459	−0.187230	+	0.982316i	$0.559951\pi$
$854$	−3997.99	−0.160197
$855$	0	0
$856$	19405.2	0.774833
$857$	5010.39	0.199710	0.0998552	−	0.995002i	$-0.468162\pi$
$857$	5010.39	0.199710	0.0998552	+	0.995002i	$0.468162\pi$
$858$	0	0
$859$	30233.4	1.20088	0.600438	−	0.799672i	$-0.294993\pi$
$859$	30233.4	1.20088	0.600438	+	0.799672i	$0.294993\pi$
$860$	−4732.45	−0.187646
$861$	0	0
$862$	38501.4	1.52130
$863$	−4334.93	−0.170988	−0.0854940	−	0.996339i	$-0.527247\pi$
$863$	−4334.93	−0.170988	−0.0854940	+	0.996339i	$0.527247\pi$
$864$	0	0
$865$	−74715.2	−2.93687
$866$	8462.05	0.332047
$867$	0	0
$868$	−424.189	−0.0165875
$869$	−2593.78	−0.101252
$870$	0	0
$871$	−5201.06	−0.202332
$872$	25064.7	0.973391
$873$	0	0
$874$	7951.69	0.307746
$875$	−18895.6	−0.730043
$876$	0	0
$877$	34683.3	1.33543	0.667716	−	0.744416i	$-0.267272\pi$
$877$	34683.3	1.33543	0.667716	+	0.744416i	$0.267272\pi$
$878$	7779.95	0.299044
$879$	0	0
$880$	−26725.3	−1.02376
$881$	18269.2	0.698642	0.349321	−	0.937003i	$-0.386412\pi$
$881$	18269.2	0.698642	0.349321	+	0.937003i	$0.386412\pi$
$882$	0	0
$883$	−14592.0	−0.556128	−0.278064	−	0.960563i	$-0.589693\pi$
$883$	−14592.0	−0.556128	−0.278064	+	0.960563i	$0.589693\pi$
$884$	−421.177	−0.0160246
$885$	0	0
$886$	−50198.0	−1.90343
$887$	−30459.3	−1.15301	−0.576507	−	0.817092i	$-0.695585\pi$
$887$	−30459.3	−1.15301	−0.576507	+	0.817092i	$0.695585\pi$
$888$	0	0
$889$	9699.60	0.365933
$890$	63614.6	2.39592
$891$	0	0
$892$	−1829.32	−0.0686662
$893$	4653.30	0.174375
$894$	0	0
$895$	−5530.57	−0.206555
$896$	9752.58	0.363628
$897$	0	0
$898$	−38002.6	−1.41221
$899$	−34223.2	−1.26964
$900$	0	0
$901$	24601.2	0.909638
$902$	−15803.6	−0.583374
$903$	0	0
$904$	−3862.39	−0.142103
$905$	−46050.7	−1.69147
$906$	0	0
$907$	−9364.89	−0.342840	−0.171420	−	0.985198i	$-0.554836\pi$
$907$	−9364.89	−0.342840	−0.171420	+	0.985198i	$0.554836\pi$
$908$	973.313	0.0355733
$909$	0	0
$910$	5196.53	0.189300
$911$	−32479.8	−1.18123	−0.590616	−	0.806952i	$-0.701115\pi$
$911$	−32479.8	−1.18123	−0.590616	+	0.806952i	$0.701115\pi$
$912$	0	0
$913$	−21376.3	−0.774864
$914$	−24648.5	−0.892012
$915$	0	0
$916$	1474.46	0.0531851
$917$	1475.72	0.0531435
$918$	0	0
$919$	295.958	0.0106232	0.00531161	−	0.999986i	$-0.498309\pi$
$919$	295.958	0.0106232	0.00531161	+	0.999986i	$0.498309\pi$
$920$	79572.9	2.85157
$921$	0	0
$922$	9340.56	0.333639
$923$	6867.89	0.244918
$924$	0	0
$925$	−39403.5	−1.40062
$926$	−20558.2	−0.729572
$927$	0	0
$928$	−6373.42	−0.225450
$929$	5620.38	0.198492	0.0992458	−	0.995063i	$-0.468357\pi$
$929$	5620.38	0.198492	0.0992458	+	0.995063i	$0.468357\pi$
$930$	0	0
$931$	−4740.29	−0.166871
$932$	2700.52	0.0949125
$933$	0	0
$934$	−21142.0	−0.740670
$935$	−29908.0	−1.04609
$936$	0	0
$937$	−32583.1	−1.13601	−0.568006	−	0.823024i	$-0.692285\pi$
$937$	−32583.1	−1.13601	−0.568006	+	0.823024i	$0.692285\pi$
$938$	8208.35	0.285727
$939$	0	0
$940$	2652.96	0.0920532
$941$	−8812.99	−0.305308	−0.152654	−	0.988280i	$-0.548782\pi$
$941$	−8812.99	−0.305308	−0.152654	+	0.988280i	$0.548782\pi$
$942$	0	0
$943$	44201.2	1.52639
$944$	−4772.00	−0.164529
$945$	0	0
$946$	31552.9	1.08443
$947$	−13426.8	−0.460732	−0.230366	−	0.973104i	$-0.573992\pi$
$947$	−13426.8	−0.460732	−0.230366	+	0.973104i	$0.573992\pi$
$948$	0	0
$949$	9546.76	0.326555
$950$	11529.1	0.393739
$951$	0	0
$952$	11667.1	0.397199
$953$	13394.6	0.455293	0.227647	−	0.973744i	$-0.426897\pi$
$953$	13394.6	0.455293	0.227647	+	0.973744i	$0.426897\pi$
$954$	0	0
$955$	48986.2	1.65985
$956$	−645.560	−0.0218398
$957$	0	0
$958$	25880.7	0.872828
$959$	4092.74	0.137812
$960$	0	0
$961$	−16035.7	−0.538273
$962$	5516.11	0.184872
$963$	0	0
$964$	276.243	0.00922944
$965$	−47163.8	−1.57332
$966$	0	0
$967$	45590.8	1.51613	0.758066	−	0.652178i	$-0.226144\pi$
$967$	45590.8	1.51613	0.758066	+	0.652178i	$0.226144\pi$
$968$	18760.1	0.622904
$969$	0	0
$970$	−29761.8	−0.985149
$971$	−264.763	−0.00875041	−0.00437521	−	0.999990i	$-0.501393\pi$
$971$	−264.763	−0.00875041	−0.00437521	+	0.999990i	$0.501393\pi$
$972$	0	0
$973$	4562.32	0.150320
$974$	−16878.2	−0.555248
$975$	0	0
$976$	11672.4	0.382812
$977$	−610.521	−0.0199921	−0.00999606	−	0.999950i	$-0.503182\pi$
$977$	−610.521	−0.0199921	−0.00999606	+	0.999950i	$0.503182\pi$
$978$	0	0
$979$	27271.8	0.890308
$980$	−2702.56	−0.0880918
$981$	0	0
$982$	−10535.5	−0.342363
$983$	57829.7	1.87638	0.938190	−	0.346121i	$-0.112501\pi$
$983$	57829.7	1.87638	0.938190	+	0.346121i	$0.112501\pi$
$984$	0	0
$985$	−90278.8	−2.92033
$986$	53627.7	1.73210
$987$	0	0
$988$	103.776	0.00334165
$989$	−88250.4	−2.83741
$990$	0	0
$991$	−56780.7	−1.82008	−0.910039	−	0.414522i	$-0.863949\pi$
$991$	−56780.7	−1.82008	−0.910039	+	0.414522i	$0.863949\pi$
$992$	2561.67	0.0819890
$993$	0	0
$994$	−10838.9	−0.345866
$995$	−59505.3	−1.89592
$996$	0	0
$997$	18616.6	0.591369	0.295684	−	0.955286i	$-0.404452\pi$
$997$	18616.6	0.591369	0.295684	+	0.955286i	$0.404452\pi$
$998$	−35209.4	−1.11677
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.a.c.1.2		2
3.2	odd	2		39.4.a.b.1.1	✓	2
4.3	odd	2		1872.4.a.t.1.1		2
12.11	even	2		624.4.a.r.1.2		2
13.12	even	2		1521.4.a.s.1.1		2
15.14	odd	2		975.4.a.j.1.2		2
21.20	even	2		1911.4.a.h.1.1		2
24.5	odd	2		2496.4.a.bc.1.1		2
24.11	even	2		2496.4.a.s.1.1		2
39.5	even	4		507.4.b.f.337.3		4
39.8	even	4		507.4.b.f.337.2		4
39.38	odd	2		507.4.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.1	✓	2	3.2	odd	2
117.4.a.c.1.2		2	1.1	even	1	trivial
507.4.a.f.1.2		2	39.38	odd	2
507.4.b.f.337.2		4	39.8	even	4
507.4.b.f.337.3		4	39.5	even	4
624.4.a.r.1.2		2	12.11	even	2
975.4.a.j.1.2		2	15.14	odd	2
1521.4.a.s.1.1		2	13.12	even	2
1872.4.a.t.1.1		2	4.3	odd	2
1911.4.a.h.1.1		2	21.20	even	2
2496.4.a.s.1.1		2	24.11	even	2
2496.4.a.bc.1.1		2	24.5	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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q+2.74166 q^{2} -0.483315 q^{4} -19.4833 q^{5} +7.48331 q^{7} -23.2583 q^{8} +O(q^{10})\)	\(q+2.74166 q^{2} -0.483315 q^{4} -19.4833 q^{5} +7.48331 q^{7} -23.2583 q^{8} -53.4166 q^{10} -22.8999 q^{11} -13.0000 q^{13} +20.5167 q^{14} -59.8999 q^{16} -67.0334 q^{17} +16.5167 q^{19} +9.41657 q^{20} -62.7836 q^{22} +175.600 q^{23} +254.600 q^{25} -35.6415 q^{26} -3.61680 q^{28} -291.800 q^{29} +117.283 q^{31} +21.8418 q^{32} -183.783 q^{34} -145.800 q^{35} -154.766 q^{37} +45.2831 q^{38} +453.150 q^{40} +251.716 q^{41} -502.566 q^{43} +11.0679 q^{44} +481.434 q^{46} +281.733 q^{47} -287.000 q^{49} +698.025 q^{50} +6.28309 q^{52} -366.999 q^{53} +446.166 q^{55} -174.049 q^{56} -800.015 q^{58} +79.6663 q^{59} -194.865 q^{61} +321.550 q^{62} +539.082 q^{64} +253.283 q^{65} +400.082 q^{67} +32.3982 q^{68} -399.733 q^{70} -528.299 q^{71} -734.366 q^{73} -424.316 q^{74} -7.98276 q^{76} -171.367 q^{77} +113.266 q^{79} +1167.05 q^{80} +690.118 q^{82} +933.466 q^{83} +1306.03 q^{85} -1377.86 q^{86} +532.613 q^{88} -1190.91 q^{89} -97.2831 q^{91} -84.8699 q^{92} +772.415 q^{94} -321.800 q^{95} +557.165 q^{97} -786.856 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8	\( 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8} - 32 q^{10} + 44 q^{11} - 26 q^{13} + 56 q^{14} - 30 q^{16} - 164 q^{17} + 48 q^{19} - 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 26 q^{26} - 112 q^{28} - 404 q^{29} + 40 q^{31} + 126 q^{32} + 276 q^{34} - 112 q^{35} - 100 q^{37} - 104 q^{38} + 592 q^{40} - 200 q^{41} - 616 q^{43} + 980 q^{44} + 1352 q^{46} + 324 q^{47} - 574 q^{49} + 1194 q^{50} - 182 q^{52} + 164 q^{53} + 144 q^{55} + 56 q^{56} - 268 q^{58} - 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 312 q^{65} - 472 q^{67} - 1372 q^{68} - 560 q^{70} - 428 q^{71} - 900 q^{73} - 684 q^{74} + 448 q^{76} - 672 q^{77} - 432 q^{79} + 1032 q^{80} + 2832 q^{82} + 1388 q^{83} + 1744 q^{85} - 840 q^{86} - 1524 q^{88} - 960 q^{89} - 2744 q^{92} + 572 q^{94} - 464 q^{95} - 532 q^{97} + 574 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8 - 32 * q^10 + 44 * q^11 - 26 * q^13 + 56 * q^14 - 30 * q^16 - 164 * q^17 + 48 * q^19 - 56 * q^20 - 380 * q^22 - 8 * q^23 + 150 * q^25 + 26 * q^26 - 112 * q^28 - 404 * q^29 + 40 * q^31 + 126 * q^32 + 276 * q^34 - 112 * q^35 - 100 * q^37 - 104 * q^38 + 592 * q^40 - 200 * q^41 - 616 * q^43 + 980 * q^44 + 1352 * q^46 + 324 * q^47 - 574 * q^49 + 1194 * q^50 - 182 * q^52 + 164 * q^53 + 144 * q^55 + 56 * q^56 - 268 * q^58 - 140 * q^59 + 628 * q^61 + 688 * q^62 - 194 * q^64 + 312 * q^65 - 472 * q^67 - 1372 * q^68 - 560 * q^70 - 428 * q^71 - 900 * q^73 - 684 * q^74 + 448 * q^76 - 672 * q^77 - 432 * q^79 + 1032 * q^80 + 2832 * q^82 + 1388 * q^83 + 1744 * q^85 - 840 * q^86 - 1524 * q^88 - 960 * q^89 - 2744 * q^92 + 572 * q^94 - 464 * q^95 - 532 * q^97 + 574 * q^98

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