LMFDB - Embedded newform 2151.2.a.i.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 28 x^{15} - x^{14} + 319 x^{13} + 17 x^{12} - 1903 x^{11} - 91 x^{10} + 6377 x^{9} + 125 x^{8} + \cdots - 49 $ x^17 - 28x^15 - x^14 + 319x^13 + 17x^12 - 1903x^11 - 91x^10 + 6377x^9 + 125x^8 - 11967x^7 + 233x^6 + 11733x^5 - 503x^4 - 5015x^3 + 94x^2 + 609x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.61209$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61209 q^{2} +4.82304 q^{4} +3.37708 q^{5} +1.51525 q^{7} +7.37403 q^{8} +O(q^{10})$	\(q+2.61209 q^{2} +4.82304 q^{4} +3.37708 q^{5} +1.51525 q^{7} +7.37403 q^{8} +8.82124 q^{10} -1.81111 q^{11} -5.19479 q^{13} +3.95798 q^{14} +9.61560 q^{16} -4.20633 q^{17} +5.78551 q^{19} +16.2878 q^{20} -4.73079 q^{22} -1.61030 q^{23} +6.40464 q^{25} -13.5693 q^{26} +7.30811 q^{28} -4.27946 q^{29} +5.20254 q^{31} +10.3688 q^{32} -10.9873 q^{34} +5.11712 q^{35} +5.82542 q^{37} +15.1123 q^{38} +24.9027 q^{40} -9.18493 q^{41} -7.90741 q^{43} -8.73505 q^{44} -4.20625 q^{46} +5.88074 q^{47} -4.70401 q^{49} +16.7295 q^{50} -25.0546 q^{52} +2.70571 q^{53} -6.11626 q^{55} +11.1735 q^{56} -11.1783 q^{58} -11.3776 q^{59} -4.77350 q^{61} +13.5895 q^{62} +7.85303 q^{64} -17.5432 q^{65} -13.6469 q^{67} -20.2873 q^{68} +13.3664 q^{70} +8.95447 q^{71} +15.8938 q^{73} +15.2166 q^{74} +27.9037 q^{76} -2.74429 q^{77} +12.9357 q^{79} +32.4726 q^{80} -23.9919 q^{82} -11.4717 q^{83} -14.2051 q^{85} -20.6549 q^{86} -13.3552 q^{88} -12.2381 q^{89} -7.87141 q^{91} -7.76652 q^{92} +15.3611 q^{94} +19.5381 q^{95} +13.5363 q^{97} -12.2873 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * 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 $

$2$	2.61209	1.84703	0.923515	−	0.383563i	$-0.125303\pi$
$2$	2.61209	1.84703	0.923515	+	0.383563i	$0.125303\pi$
$3$	0	0
$3$	0	0
$4$	4.82304	2.41152
$5$	3.37708	1.51027	0.755137	−	0.655567i	$-0.227570\pi$
$5$	3.37708	1.51027	0.755137	+	0.655567i	$0.227570\pi$
$6$	0	0
$7$	1.51525	0.572711	0.286356	−	0.958123i	$-0.407556\pi$
$7$	1.51525	0.572711	0.286356	+	0.958123i	$0.407556\pi$
$8$	7.37403	2.60711
$9$	0	0
$10$	8.82124	2.78952
$11$	−1.81111	−0.546070	−0.273035	−	0.962004i	$-0.588028\pi$
$11$	−1.81111	−0.546070	−0.273035	+	0.962004i	$0.588028\pi$
$12$	0	0
$13$	−5.19479	−1.44077	−0.720387	−	0.693572i	$-0.756036\pi$
$13$	−5.19479	−1.44077	−0.720387	+	0.693572i	$0.756036\pi$
$14$	3.95798	1.05781
$15$	0	0
$16$	9.61560	2.40390
$17$	−4.20633	−1.02019	−0.510093	−	0.860119i	$-0.670389\pi$
$17$	−4.20633	−1.02019	−0.510093	+	0.860119i	$0.670389\pi$
$18$	0	0
$19$	5.78551	1.32729	0.663643	−	0.748049i	$-0.269009\pi$
$19$	5.78551	1.32729	0.663643	+	0.748049i	$0.269009\pi$
$20$	16.2878	3.64205
$21$	0	0
$22$	−4.73079	−1.00861
$23$	−1.61030	−0.335770	−0.167885	−	0.985807i	$-0.553694\pi$
$23$	−1.61030	−0.335770	−0.167885	+	0.985807i	$0.553694\pi$
$24$	0	0
$25$	6.40464	1.28093
$26$	−13.5693	−2.66115
$27$	0	0
$28$	7.30811	1.38110
$29$	−4.27946	−0.794675	−0.397338	−	0.917673i	$-0.630066\pi$
$29$	−4.27946	−0.794675	−0.397338	+	0.917673i	$0.630066\pi$
$30$	0	0
$31$	5.20254	0.934404	0.467202	−	0.884151i	$-0.345262\pi$
$31$	5.20254	0.934404	0.467202	+	0.884151i	$0.345262\pi$
$32$	10.3688	1.83296
$33$	0	0
$34$	−10.9873	−1.88431
$35$	5.11712	0.864951
$36$	0	0
$37$	5.82542	0.957694	0.478847	−	0.877898i	$-0.341055\pi$
$37$	5.82542	0.957694	0.478847	+	0.877898i	$0.341055\pi$
$38$	15.1123	2.45154
$39$	0	0
$40$	24.9027	3.93746
$41$	−9.18493	−1.43445	−0.717223	−	0.696844i	$-0.754587\pi$
$41$	−9.18493	−1.43445	−0.717223	+	0.696844i	$0.754587\pi$
$42$	0	0
$43$	−7.90741	−1.20587	−0.602934	−	0.797791i	$-0.706002\pi$
$43$	−7.90741	−1.20587	−0.602934	+	0.797791i	$0.706002\pi$
$44$	−8.73505	−1.31686
$45$	0	0
$46$	−4.20625	−0.620177
$47$	5.88074	0.857795	0.428897	−	0.903353i	$-0.358902\pi$
$47$	5.88074	0.857795	0.428897	+	0.903353i	$0.358902\pi$
$48$	0	0
$49$	−4.70401	−0.672002
$50$	16.7295	2.36591
$51$	0	0
$52$	−25.0546	−3.47445
$53$	2.70571	0.371658	0.185829	−	0.982582i	$-0.440503\pi$
$53$	2.70571	0.371658	0.185829	+	0.982582i	$0.440503\pi$
$54$	0	0
$55$	−6.11626	−0.824716
$56$	11.1735	1.49312
$57$	0	0
$58$	−11.1783	−1.46779
$59$	−11.3776	−1.48124	−0.740620	−	0.671924i	$-0.765468\pi$
$59$	−11.3776	−1.48124	−0.740620	+	0.671924i	$0.765468\pi$
$60$	0	0
$61$	−4.77350	−0.611184	−0.305592	−	0.952163i	$-0.598854\pi$
$61$	−4.77350	−0.611184	−0.305592	+	0.952163i	$0.598854\pi$
$62$	13.5895	1.72587
$63$	0	0
$64$	7.85303	0.981629
$65$	−17.5432	−2.17596
$66$	0	0
$67$	−13.6469	−1.66723	−0.833616	−	0.552344i	$-0.813733\pi$
$67$	−13.6469	−1.66723	−0.833616	+	0.552344i	$0.813733\pi$
$68$	−20.2873	−2.46020
$69$	0	0
$70$	13.3664	1.59759
$71$	8.95447	1.06270	0.531350	−	0.847153i	$-0.321685\pi$
$71$	8.95447	1.06270	0.531350	+	0.847153i	$0.321685\pi$
$72$	0	0
$73$	15.8938	1.86023	0.930116	−	0.367265i	$-0.119706\pi$
$73$	15.8938	1.86023	0.930116	+	0.367265i	$0.119706\pi$
$74$	15.2166	1.76889
$75$	0	0
$76$	27.9037	3.20078
$77$	−2.74429	−0.312741
$78$	0	0
$79$	12.9357	1.45538	0.727690	−	0.685907i	$-0.240594\pi$
$79$	12.9357	1.45538	0.727690	+	0.685907i	$0.240594\pi$
$80$	32.4726	3.63055
$81$	0	0
$82$	−23.9919	−2.64946
$83$	−11.4717	−1.25918	−0.629589	−	0.776928i	$-0.716777\pi$
$83$	−11.4717	−1.25918	−0.629589	+	0.776928i	$0.716777\pi$
$84$	0	0
$85$	−14.2051	−1.54076
$86$	−20.6549	−2.22728
$87$	0	0
$88$	−13.3552	−1.42367
$89$	−12.2381	−1.29723	−0.648617	−	0.761115i	$-0.724652\pi$
$89$	−12.2381	−1.29723	−0.648617	+	0.761115i	$0.724652\pi$
$90$	0	0
$91$	−7.87141	−0.825148
$92$	−7.76652	−0.809715
$93$	0	0
$94$	15.3611	1.58437
$95$	19.5381	2.00457
$96$	0	0
$97$	13.5363	1.37440	0.687200	−	0.726468i	$-0.258839\pi$
$97$	13.5363	1.37440	0.687200	+	0.726468i	$0.258839\pi$
$98$	−12.2873	−1.24121
$99$	0	0
$100$	30.8898	3.08898
$101$	16.7934	1.67101	0.835505	−	0.549483i	$-0.185175\pi$
$101$	16.7934	1.67101	0.835505	+	0.549483i	$0.185175\pi$
$102$	0	0
$103$	3.36206	0.331274	0.165637	−	0.986187i	$-0.447032\pi$
$103$	3.36206	0.331274	0.165637	+	0.986187i	$0.447032\pi$
$104$	−38.3065	−3.75626
$105$	0	0
$106$	7.06756	0.686463
$107$	−4.31621	−0.417264	−0.208632	−	0.977994i	$-0.566901\pi$
$107$	−4.31621	−0.417264	−0.208632	+	0.977994i	$0.566901\pi$
$108$	0	0
$109$	−4.07253	−0.390078	−0.195039	−	0.980796i	$-0.562483\pi$
$109$	−4.07253	−0.390078	−0.195039	+	0.980796i	$0.562483\pi$
$110$	−15.9762	−1.52328
$111$	0	0
$112$	14.5700	1.37674
$113$	4.09720	0.385432	0.192716	−	0.981255i	$-0.438270\pi$
$113$	4.09720	0.385432	0.192716	+	0.981255i	$0.438270\pi$
$114$	0	0
$115$	−5.43809	−0.507105
$116$	−20.6400	−1.91637
$117$	0	0
$118$	−29.7194	−2.73589
$119$	−6.37365	−0.584272
$120$	0	0
$121$	−7.71988	−0.701807
$122$	−12.4688	−1.12887
$123$	0	0
$124$	25.0920	2.25333
$125$	4.74359	0.424280
$126$	0	0
$127$	6.89364	0.611712	0.305856	−	0.952078i	$-0.401057\pi$
$127$	6.89364	0.611712	0.305856	+	0.952078i	$0.401057\pi$
$128$	−0.224702	−0.0198611
$129$	0	0
$130$	−45.8245	−4.01907
$131$	14.6134	1.27678	0.638390	−	0.769713i	$-0.279601\pi$
$131$	14.6134	1.27678	0.638390	+	0.769713i	$0.279601\pi$
$132$	0	0
$133$	8.76650	0.760152
$134$	−35.6469	−3.07943
$135$	0	0
$136$	−31.0176	−2.65974
$137$	3.00143	0.256429	0.128215	−	0.991746i	$-0.459075\pi$
$137$	3.00143	0.256429	0.128215	+	0.991746i	$0.459075\pi$
$138$	0	0
$139$	−10.8506	−0.920333	−0.460167	−	0.887833i	$-0.652210\pi$
$139$	−10.8506	−0.920333	−0.460167	+	0.887833i	$0.652210\pi$
$140$	24.6800	2.08584
$141$	0	0
$142$	23.3899	1.96284
$143$	9.40833	0.786764
$144$	0	0
$145$	−14.4521	−1.20018
$146$	41.5162	3.43590
$147$	0	0
$148$	28.0962	2.30950
$149$	12.7057	1.04089	0.520447	−	0.853894i	$-0.325765\pi$
$149$	12.7057	1.04089	0.520447	+	0.853894i	$0.325765\pi$
$150$	0	0
$151$	2.67730	0.217876	0.108938	−	0.994049i	$-0.465255\pi$
$151$	2.67730	0.217876	0.108938	+	0.994049i	$0.465255\pi$
$152$	42.6625	3.46039
$153$	0	0
$154$	−7.16834	−0.577641
$155$	17.5694	1.41121
$156$	0	0
$157$	21.5425	1.71928	0.859638	−	0.510904i	$-0.170689\pi$
$157$	21.5425	1.71928	0.859638	+	0.510904i	$0.170689\pi$
$158$	33.7893	2.68813
$159$	0	0
$160$	35.0162	2.76827
$161$	−2.44000	−0.192299
$162$	0	0
$163$	4.23695	0.331863	0.165932	−	0.986137i	$-0.446937\pi$
$163$	4.23695	0.331863	0.165932	+	0.986137i	$0.446937\pi$
$164$	−44.2992	−3.45919
$165$	0	0
$166$	−29.9651	−2.32574
$167$	1.29493	0.100205	0.0501025	−	0.998744i	$-0.484045\pi$
$167$	1.29493	0.100205	0.0501025	+	0.998744i	$0.484045\pi$
$168$	0	0
$169$	13.9858	1.07583
$170$	−37.1051	−2.84583
$171$	0	0
$172$	−38.1377	−2.90797
$173$	−6.34556	−0.482444	−0.241222	−	0.970470i	$-0.577548\pi$
$173$	−6.34556	−0.482444	−0.241222	+	0.970470i	$0.577548\pi$
$174$	0	0
$175$	9.70465	0.733602
$176$	−17.4149	−1.31270
$177$	0	0
$178$	−31.9670	−2.39603
$179$	−11.7553	−0.878631	−0.439316	−	0.898333i	$-0.644779\pi$
$179$	−11.7553	−0.878631	−0.439316	+	0.898333i	$0.644779\pi$
$180$	0	0
$181$	12.3657	0.919136	0.459568	−	0.888143i	$-0.348004\pi$
$181$	12.3657	0.919136	0.459568	+	0.888143i	$0.348004\pi$
$182$	−20.5609	−1.52407
$183$	0	0
$184$	−11.8744	−0.875391
$185$	19.6729	1.44638
$186$	0	0
$187$	7.61813	0.557093
$188$	28.3630	2.06859
$189$	0	0
$190$	51.0354	3.70249
$191$	−14.6901	−1.06293	−0.531467	−	0.847079i	$-0.678359\pi$
$191$	−14.6901	−1.06293	−0.531467	+	0.847079i	$0.678359\pi$
$192$	0	0
$193$	4.39441	0.316316	0.158158	−	0.987414i	$-0.449444\pi$
$193$	4.39441	0.316316	0.158158	+	0.987414i	$0.449444\pi$
$194$	35.3580	2.53856
$195$	0	0
$196$	−22.6876	−1.62054
$197$	−7.42773	−0.529204	−0.264602	−	0.964358i	$-0.585241\pi$
$197$	−7.42773	−0.529204	−0.264602	+	0.964358i	$0.585241\pi$
$198$	0	0
$199$	2.40583	0.170545	0.0852723	−	0.996358i	$-0.472824\pi$
$199$	2.40583	0.170545	0.0852723	+	0.996358i	$0.472824\pi$
$200$	47.2281	3.33953
$201$	0	0
$202$	43.8660	3.08640
$203$	−6.48445	−0.455119
$204$	0	0
$205$	−31.0182	−2.16641
$206$	8.78203	0.611873
$207$	0	0
$208$	−49.9510	−3.46348
$209$	−10.4782	−0.724792
$210$	0	0
$211$	−2.66000	−0.183122	−0.0915610	−	0.995799i	$-0.529186\pi$
$211$	−2.66000	−0.183122	−0.0915610	+	0.995799i	$0.529186\pi$
$212$	13.0497	0.896259
$213$	0	0
$214$	−11.2743	−0.770698
$215$	−26.7039	−1.82119
$216$	0	0
$217$	7.88316	0.535144
$218$	−10.6378	−0.720485
$219$	0	0
$220$	−29.4989	−1.98882
$221$	21.8510	1.46986
$222$	0	0
$223$	−3.95756	−0.265018	−0.132509	−	0.991182i	$-0.542303\pi$
$223$	−3.95756	−0.265018	−0.132509	+	0.991182i	$0.542303\pi$
$224$	15.7113	1.04976
$225$	0	0
$226$	10.7023	0.711905
$227$	−7.72720	−0.512872	−0.256436	−	0.966561i	$-0.582548\pi$
$227$	−7.72720	−0.512872	−0.256436	+	0.966561i	$0.582548\pi$
$228$	0	0
$229$	−2.94024	−0.194297	−0.0971483	−	0.995270i	$-0.530972\pi$
$229$	−2.94024	−0.194297	−0.0971483	+	0.995270i	$0.530972\pi$
$230$	−14.2048	−0.936638
$231$	0	0
$232$	−31.5569	−2.07181
$233$	−11.8488	−0.776243	−0.388121	−	0.921608i	$-0.626876\pi$
$233$	−11.8488	−0.776243	−0.388121	+	0.921608i	$0.626876\pi$
$234$	0	0
$235$	19.8597	1.29551
$236$	−54.8747	−3.57204
$237$	0	0
$238$	−16.6486	−1.07917
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	19.4625	1.25369	0.626845	−	0.779144i	$-0.284346\pi$
$241$	19.4625	1.25369	0.626845	+	0.779144i	$0.284346\pi$
$242$	−20.1650	−1.29626
$243$	0	0
$244$	−23.0228	−1.47388
$245$	−15.8858	−1.01491
$246$	0	0
$247$	−30.0545	−1.91232
$248$	38.3637	2.43610
$249$	0	0
$250$	12.3907	0.783657
$251$	13.4635	0.849807	0.424903	−	0.905239i	$-0.360308\pi$
$251$	13.4635	0.849807	0.424903	+	0.905239i	$0.360308\pi$
$252$	0	0
$253$	2.91642	0.183354
$254$	18.0068	1.12985
$255$	0	0
$256$	−16.2930	−1.01831
$257$	−6.36188	−0.396843	−0.198422	−	0.980117i	$-0.563582\pi$
$257$	−6.36188	−0.396843	−0.198422	+	0.980117i	$0.563582\pi$
$258$	0	0
$259$	8.82698	0.548482
$260$	−84.6114	−5.24738
$261$	0	0
$262$	38.1716	2.35825
$263$	−10.8435	−0.668641	−0.334320	−	0.942459i	$-0.608507\pi$
$263$	−10.8435	−0.668641	−0.334320	+	0.942459i	$0.608507\pi$
$264$	0	0
$265$	9.13738	0.561305
$266$	22.8989	1.40402
$267$	0	0
$268$	−65.8194	−4.02056
$269$	−26.6051	−1.62214	−0.811070	−	0.584949i	$-0.801115\pi$
$269$	−26.6051	−1.62214	−0.811070	+	0.584949i	$0.801115\pi$
$270$	0	0
$271$	−7.15319	−0.434525	−0.217263	−	0.976113i	$-0.569713\pi$
$271$	−7.15319	−0.434525	−0.217263	+	0.976113i	$0.569713\pi$
$272$	−40.4464	−2.45242
$273$	0	0
$274$	7.84002	0.473633
$275$	−11.5995	−0.699477
$276$	0	0
$277$	−4.53801	−0.272663	−0.136331	−	0.990663i	$-0.543531\pi$
$277$	−4.53801	−0.272663	−0.136331	+	0.990663i	$0.543531\pi$
$278$	−28.3427	−1.69988
$279$	0	0
$280$	37.7338	2.25503
$281$	19.3559	1.15468	0.577339	−	0.816504i	$-0.304091\pi$
$281$	19.3559	1.15468	0.577339	+	0.816504i	$0.304091\pi$
$282$	0	0
$283$	−12.6799	−0.753744	−0.376872	−	0.926265i	$-0.623000\pi$
$283$	−12.6799	−0.753744	−0.376872	+	0.926265i	$0.623000\pi$
$284$	43.1877	2.56272
$285$	0	0
$286$	24.5754	1.45318
$287$	−13.9175	−0.821523
$288$	0	0
$289$	0.693231	0.0407783
$290$	−37.7501	−2.21676
$291$	0	0
$292$	76.6565	4.48598
$293$	22.0234	1.28662	0.643310	−	0.765606i	$-0.277561\pi$
$293$	22.0234	1.28662	0.643310	+	0.765606i	$0.277561\pi$
$294$	0	0
$295$	−38.4231	−2.23708
$296$	42.9569	2.49682
$297$	0	0
$298$	33.1885	1.92256
$299$	8.36514	0.483769
$300$	0	0
$301$	−11.9817	−0.690614
$302$	6.99336	0.402423
$303$	0	0
$304$	55.6311	3.19066
$305$	−16.1205	−0.923056
$306$	0	0
$307$	−12.2717	−0.700384	−0.350192	−	0.936678i	$-0.613884\pi$
$307$	−12.2717	−0.700384	−0.350192	+	0.936678i	$0.613884\pi$
$308$	−13.2358	−0.754179
$309$	0	0
$310$	45.8929	2.60654
$311$	0.538105	0.0305131	0.0152566	−	0.999884i	$-0.495143\pi$
$311$	0.538105	0.0305131	0.0152566	+	0.999884i	$0.495143\pi$
$312$	0	0
$313$	13.2842	0.750866	0.375433	−	0.926849i	$-0.377494\pi$
$313$	13.2842	0.750866	0.375433	+	0.926849i	$0.377494\pi$
$314$	56.2709	3.17555
$315$	0	0
$316$	62.3893	3.50967
$317$	13.1115	0.736412	0.368206	−	0.929744i	$-0.379972\pi$
$317$	13.1115	0.736412	0.368206	+	0.929744i	$0.379972\pi$
$318$	0	0
$319$	7.75057	0.433949
$320$	26.5203	1.48253
$321$	0	0
$322$	−6.37352	−0.355182
$323$	−24.3358	−1.35408
$324$	0	0
$325$	−33.2708	−1.84553
$326$	11.0673	0.612961
$327$	0	0
$328$	−67.7300	−3.73976
$329$	8.91080	0.491269
$330$	0	0
$331$	34.9597	1.92156	0.960781	−	0.277309i	$-0.0894425\pi$
$331$	34.9597	1.92156	0.960781	+	0.277309i	$0.0894425\pi$
$332$	−55.3282	−3.03653
$333$	0	0
$334$	3.38249	0.185081
$335$	−46.0866	−2.51798
$336$	0	0
$337$	−8.10105	−0.441292	−0.220646	−	0.975354i	$-0.570817\pi$
$337$	−8.10105	−0.441292	−0.220646	+	0.975354i	$0.570817\pi$
$338$	36.5322	1.98709
$339$	0	0
$340$	−68.5117	−3.71557
$341$	−9.42238	−0.510250
$342$	0	0
$343$	−17.7345	−0.957574
$344$	−58.3095	−3.14384
$345$	0	0
$346$	−16.5752	−0.891088
$347$	−16.2093	−0.870161	−0.435081	−	0.900392i	$-0.643280\pi$
$347$	−16.2093	−0.870161	−0.435081	+	0.900392i	$0.643280\pi$
$348$	0	0
$349$	18.7650	1.00447	0.502233	−	0.864732i	$-0.332512\pi$
$349$	18.7650	1.00447	0.502233	+	0.864732i	$0.332512\pi$
$350$	25.3494	1.35498
$351$	0	0
$352$	−18.7790	−1.00092
$353$	−12.1856	−0.648576	−0.324288	−	0.945958i	$-0.605125\pi$
$353$	−12.1856	−0.648576	−0.324288	+	0.945958i	$0.605125\pi$
$354$	0	0
$355$	30.2399	1.60497
$356$	−59.0247	−3.12830
$357$	0	0
$358$	−30.7059	−1.62286
$359$	13.8706	0.732064	0.366032	−	0.930602i	$-0.380716\pi$
$359$	13.8706	0.732064	0.366032	+	0.930602i	$0.380716\pi$
$360$	0	0
$361$	14.4721	0.761690
$362$	32.3004	1.69767
$363$	0	0
$364$	−37.9641	−1.98986
$365$	53.6747	2.80946
$366$	0	0
$367$	15.5803	0.813285	0.406643	−	0.913587i	$-0.366699\pi$
$367$	15.5803	0.813285	0.406643	+	0.913587i	$0.366699\pi$
$368$	−15.4840	−0.807157
$369$	0	0
$370$	51.3875	2.67151
$371$	4.09983	0.212852
$372$	0	0
$373$	−21.0641	−1.09066	−0.545329	−	0.838222i	$-0.683595\pi$
$373$	−21.0641	−1.09066	−0.545329	+	0.838222i	$0.683595\pi$
$374$	19.8993	1.02897
$375$	0	0
$376$	43.3648	2.23637
$377$	22.2309	1.14495
$378$	0	0
$379$	−20.9514	−1.07620	−0.538101	−	0.842880i	$-0.680858\pi$
$379$	−20.9514	−1.07620	−0.538101	+	0.842880i	$0.680858\pi$
$380$	94.2330	4.83405
$381$	0	0
$382$	−38.3718	−1.96327
$383$	−15.5647	−0.795319	−0.397660	−	0.917533i	$-0.630178\pi$
$383$	−15.5647	−0.795319	−0.397660	+	0.917533i	$0.630178\pi$
$384$	0	0
$385$	−9.26767	−0.472324
$386$	11.4786	0.584245
$387$	0	0
$388$	65.2860	3.31439
$389$	−0.448236	−0.0227265	−0.0113632	−	0.999935i	$-0.503617\pi$
$389$	−0.448236	−0.0227265	−0.0113632	+	0.999935i	$0.503617\pi$
$390$	0	0
$391$	6.77344	0.342548
$392$	−34.6876	−1.75199
$393$	0	0
$394$	−19.4019	−0.977455
$395$	43.6848	2.19802
$396$	0	0
$397$	26.4977	1.32988	0.664941	−	0.746896i	$-0.268457\pi$
$397$	26.4977	1.32988	0.664941	+	0.746896i	$0.268457\pi$
$398$	6.28425	0.315001
$399$	0	0
$400$	61.5845	3.07922
$401$	−9.64358	−0.481578	−0.240789	−	0.970578i	$-0.577406\pi$
$401$	−9.64358	−0.481578	−0.240789	+	0.970578i	$0.577406\pi$
$402$	0	0
$403$	−27.0261	−1.34627
$404$	80.9954	4.02967
$405$	0	0
$406$	−16.9380	−0.840619
$407$	−10.5505	−0.522968
$408$	0	0
$409$	19.5094	0.964677	0.482339	−	0.875985i	$-0.339787\pi$
$409$	19.5094	0.964677	0.482339	+	0.875985i	$0.339787\pi$
$410$	−81.0225	−4.00142
$411$	0	0
$412$	16.2154	0.798873
$413$	−17.2400	−0.848323
$414$	0	0
$415$	−38.7407	−1.90171
$416$	−53.8636	−2.64088
$417$	0	0
$418$	−27.3700	−1.33871
$419$	−32.7841	−1.60161	−0.800803	−	0.598927i	$-0.795594\pi$
$419$	−32.7841	−1.60161	−0.800803	+	0.598927i	$0.795594\pi$
$420$	0	0
$421$	30.1382	1.46885	0.734423	−	0.678691i	$-0.237453\pi$
$421$	30.1382	1.46885	0.734423	+	0.678691i	$0.237453\pi$
$422$	−6.94817	−0.338232
$423$	0	0
$424$	19.9520	0.968954
$425$	−26.9401	−1.30679
$426$	0	0
$427$	−7.23305	−0.350032
$428$	−20.8172	−1.00624
$429$	0	0
$430$	−69.7532	−3.36380
$431$	26.1324	1.25875	0.629376	−	0.777101i	$-0.283311\pi$
$431$	26.1324	1.25875	0.629376	+	0.777101i	$0.283311\pi$
$432$	0	0
$433$	−8.44070	−0.405634	−0.202817	−	0.979217i	$-0.565010\pi$
$433$	−8.44070	−0.405634	−0.202817	+	0.979217i	$0.565010\pi$
$434$	20.5915	0.988426
$435$	0	0
$436$	−19.6420	−0.940680
$437$	−9.31638	−0.445663
$438$	0	0
$439$	−36.0272	−1.71948	−0.859742	−	0.510729i	$-0.829375\pi$
$439$	−36.0272	−1.71948	−0.859742	+	0.510729i	$0.829375\pi$
$440$	−45.1015	−2.15013
$441$	0	0
$442$	57.0769	2.71487
$443$	−18.8845	−0.897229	−0.448614	−	0.893725i	$-0.648082\pi$
$443$	−18.8845	−0.897229	−0.448614	+	0.893725i	$0.648082\pi$
$444$	0	0
$445$	−41.3289	−1.95918
$446$	−10.3375	−0.489496
$447$	0	0
$448$	11.8993	0.562190
$449$	17.5504	0.828254	0.414127	−	0.910219i	$-0.364087\pi$
$449$	17.5504	0.828254	0.414127	+	0.910219i	$0.364087\pi$
$450$	0	0
$451$	16.6349	0.783308
$452$	19.7609	0.929476
$453$	0	0
$454$	−20.1842	−0.947291
$455$	−26.5823	−1.24620
$456$	0	0
$457$	−16.0884	−0.752584	−0.376292	−	0.926501i	$-0.622801\pi$
$457$	−16.0884	−0.752584	−0.376292	+	0.926501i	$0.622801\pi$
$458$	−7.68019	−0.358872
$459$	0	0
$460$	−26.2281	−1.22289
$461$	16.4470	0.766014	0.383007	−	0.923745i	$-0.374888\pi$
$461$	16.4470	0.766014	0.383007	+	0.923745i	$0.374888\pi$
$462$	0	0
$463$	16.4060	0.762454	0.381227	−	0.924482i	$-0.375502\pi$
$463$	16.4060	0.762454	0.381227	+	0.924482i	$0.375502\pi$
$464$	−41.1495	−1.91032
$465$	0	0
$466$	−30.9503	−1.43374
$467$	−8.49665	−0.393178	−0.196589	−	0.980486i	$-0.562986\pi$
$467$	−8.49665	−0.393178	−0.196589	+	0.980486i	$0.562986\pi$
$468$	0	0
$469$	−20.6785	−0.954842
$470$	51.8755	2.39284
$471$	0	0
$472$	−83.8990	−3.86176
$473$	14.3212	0.658489
$474$	0	0
$475$	37.0541	1.70016
$476$	−30.7403	−1.40898
$477$	0	0
$478$	−2.61209	−0.119474
$479$	12.2941	0.561733	0.280866	−	0.959747i	$-0.409378\pi$
$479$	12.2941	0.561733	0.280866	+	0.959747i	$0.409378\pi$
$480$	0	0
$481$	−30.2618	−1.37982
$482$	50.8379	2.31560
$483$	0	0
$484$	−37.2332	−1.69242
$485$	45.7130	2.07572
$486$	0	0
$487$	−6.54124	−0.296412	−0.148206	−	0.988957i	$-0.547350\pi$
$487$	−6.54124	−0.296412	−0.148206	+	0.988957i	$0.547350\pi$
$488$	−35.1999	−1.59343
$489$	0	0
$490$	−41.4952	−1.87456
$491$	6.04335	0.272733	0.136366	−	0.990658i	$-0.456458\pi$
$491$	6.04335	0.272733	0.136366	+	0.990658i	$0.456458\pi$
$492$	0	0
$493$	18.0008	0.810716
$494$	−78.5051	−3.53211
$495$	0	0
$496$	50.0255	2.24621
$497$	13.5683	0.608620
$498$	0	0
$499$	3.33402	0.149251	0.0746257	−	0.997212i	$-0.476224\pi$
$499$	3.33402	0.149251	0.0746257	+	0.997212i	$0.476224\pi$
$500$	22.8785	1.02316
$501$	0	0
$502$	35.1679	1.56962
$503$	21.2393	0.947016	0.473508	−	0.880790i	$-0.342988\pi$
$503$	21.2393	0.947016	0.473508	+	0.880790i	$0.342988\pi$
$504$	0	0
$505$	56.7127	2.52368
$506$	7.61798	0.338660
$507$	0	0
$508$	33.2483	1.47515
$509$	36.5559	1.62031	0.810155	−	0.586216i	$-0.199383\pi$
$509$	36.5559	1.62031	0.810155	+	0.586216i	$0.199383\pi$
$510$	0	0
$511$	24.0832	1.06538
$512$	−42.1095	−1.86099
$513$	0	0
$514$	−16.6178	−0.732981
$515$	11.3539	0.500315
$516$	0	0
$517$	−10.6507	−0.468416
$518$	23.0569	1.01306
$519$	0	0
$520$	−129.364	−5.67299
$521$	37.0788	1.62445	0.812225	−	0.583344i	$-0.198256\pi$
$521$	37.0788	1.62445	0.812225	+	0.583344i	$0.198256\pi$
$522$	0	0
$523$	−39.4260	−1.72398	−0.861989	−	0.506927i	$-0.830781\pi$
$523$	−39.4260	−1.72398	−0.861989	+	0.506927i	$0.830781\pi$
$524$	70.4810	3.07898
$525$	0	0
$526$	−28.3243	−1.23500
$527$	−21.8836	−0.953265
$528$	0	0
$529$	−20.4069	−0.887259
$530$	23.8677	1.03675
$531$	0	0
$532$	42.2811	1.83312
$533$	47.7138	2.06671
$534$	0	0
$535$	−14.5762	−0.630183
$536$	−100.633	−4.34667
$537$	0	0
$538$	−69.4950	−2.99614
$539$	8.51949	0.366960
$540$	0	0
$541$	10.1924	0.438206	0.219103	−	0.975702i	$-0.429687\pi$
$541$	10.1924	0.438206	0.219103	+	0.975702i	$0.429687\pi$
$542$	−18.6848	−0.802581
$543$	0	0
$544$	−43.6145	−1.86996
$545$	−13.7533	−0.589125
$546$	0	0
$547$	−2.30286	−0.0984631	−0.0492315	−	0.998787i	$-0.515677\pi$
$547$	−2.30286	−0.0984631	−0.0492315	+	0.998787i	$0.515677\pi$
$548$	14.4760	0.618384
$549$	0	0
$550$	−30.2990	−1.29196
$551$	−24.7588	−1.05476
$552$	0	0
$553$	19.6008	0.833512
$554$	−11.8537	−0.503616
$555$	0	0
$556$	−52.3327	−2.21940
$557$	−44.3467	−1.87903	−0.939514	−	0.342510i	$-0.888723\pi$
$557$	−44.3467	−1.87903	−0.939514	+	0.342510i	$0.888723\pi$
$558$	0	0
$559$	41.0773	1.73738
$560$	49.2042	2.07926
$561$	0	0
$562$	50.5595	2.13272
$563$	34.9527	1.47308	0.736540	−	0.676394i	$-0.236458\pi$
$563$	34.9527	1.47308	0.736540	+	0.676394i	$0.236458\pi$
$564$	0	0
$565$	13.8366	0.582108
$566$	−33.1212	−1.39219
$567$	0	0
$568$	66.0305	2.77058
$569$	−23.8350	−0.999213	−0.499607	−	0.866252i	$-0.666522\pi$
$569$	−23.8350	−0.999213	−0.499607	+	0.866252i	$0.666522\pi$
$570$	0	0
$571$	−11.4127	−0.477609	−0.238804	−	0.971068i	$-0.576755\pi$
$571$	−11.4127	−0.477609	−0.238804	+	0.971068i	$0.576755\pi$
$572$	45.3767	1.89730
$573$	0	0
$574$	−36.3538	−1.51738
$575$	−10.3134	−0.430097
$576$	0	0
$577$	27.7090	1.15354	0.576770	−	0.816907i	$-0.304313\pi$
$577$	27.7090	1.15354	0.576770	+	0.816907i	$0.304313\pi$
$578$	1.81079	0.0753187
$579$	0	0
$580$	−69.7028	−2.89425
$581$	−17.3825	−0.721146
$582$	0	0
$583$	−4.90034	−0.202951
$584$	117.202	4.84984
$585$	0	0
$586$	57.5272	2.37643
$587$	−5.40269	−0.222993	−0.111496	−	0.993765i	$-0.535564\pi$
$587$	−5.40269	−0.222993	−0.111496	+	0.993765i	$0.535564\pi$
$588$	0	0
$589$	30.0993	1.24022
$590$	−100.365	−4.13195
$591$	0	0
$592$	56.0149	2.30220
$593$	−4.73054	−0.194260	−0.0971301	−	0.995272i	$-0.530966\pi$
$593$	−4.73054	−0.194260	−0.0971301	+	0.995272i	$0.530966\pi$
$594$	0	0
$595$	−21.5243	−0.882410
$596$	61.2801	2.51013
$597$	0	0
$598$	21.8505	0.893535
$599$	7.45633	0.304657	0.152329	−	0.988330i	$-0.451323\pi$
$599$	7.45633	0.304657	0.152329	+	0.988330i	$0.451323\pi$
$600$	0	0
$601$	−9.54063	−0.389170	−0.194585	−	0.980886i	$-0.562336\pi$
$601$	−9.54063	−0.389170	−0.194585	+	0.980886i	$0.562336\pi$
$602$	−31.2974	−1.27559
$603$	0	0
$604$	12.9127	0.525411
$605$	−26.0706	−1.05992
$606$	0	0
$607$	18.1299	0.735870	0.367935	−	0.929851i	$-0.380065\pi$
$607$	18.1299	0.735870	0.367935	+	0.929851i	$0.380065\pi$
$608$	59.9887	2.43286
$609$	0	0
$610$	−42.1082	−1.70491
$611$	−30.5492	−1.23589
$612$	0	0
$613$	−9.92145	−0.400724	−0.200362	−	0.979722i	$-0.564212\pi$
$613$	−9.92145	−0.400724	−0.200362	+	0.979722i	$0.564212\pi$
$614$	−32.0549	−1.29363
$615$	0	0
$616$	−20.2365	−0.815351
$617$	−35.3477	−1.42305	−0.711523	−	0.702663i	$-0.751994\pi$
$617$	−35.3477	−1.42305	−0.711523	+	0.702663i	$0.751994\pi$
$618$	0	0
$619$	9.08076	0.364987	0.182493	−	0.983207i	$-0.441583\pi$
$619$	9.08076	0.364987	0.182493	+	0.983207i	$0.441583\pi$
$620$	84.7377	3.40315
$621$	0	0
$622$	1.40558	0.0563587
$623$	−18.5438	−0.742940
$624$	0	0
$625$	−16.0038	−0.640150
$626$	34.6995	1.38687
$627$	0	0
$628$	103.900	4.14606
$629$	−24.5037	−0.977025
$630$	0	0
$631$	−39.7709	−1.58325	−0.791626	−	0.611005i	$-0.790765\pi$
$631$	−39.7709	−1.58325	−0.791626	+	0.611005i	$0.790765\pi$
$632$	95.3882	3.79434
$633$	0	0
$634$	34.2483	1.36018
$635$	23.2804	0.923853
$636$	0	0
$637$	24.4363	0.968203
$638$	20.2452	0.801516
$639$	0	0
$640$	−0.758837	−0.0299957
$641$	6.15905	0.243268	0.121634	−	0.992575i	$-0.461187\pi$
$641$	6.15905	0.243268	0.121634	+	0.992575i	$0.461187\pi$
$642$	0	0
$643$	−25.0556	−0.988098	−0.494049	−	0.869434i	$-0.664484\pi$
$643$	−25.0556	−0.988098	−0.494049	+	0.869434i	$0.664484\pi$
$644$	−11.7682	−0.463733
$645$	0	0
$646$	−63.5673	−2.50102
$647$	40.7329	1.60138	0.800688	−	0.599081i	$-0.204467\pi$
$647$	40.7329	1.60138	0.800688	+	0.599081i	$0.204467\pi$
$648$	0	0
$649$	20.6061	0.808862
$650$	−86.9064	−3.40875
$651$	0	0
$652$	20.4349	0.800294
$653$	23.1924	0.907590	0.453795	−	0.891106i	$-0.350070\pi$
$653$	23.1924	0.907590	0.453795	+	0.891106i	$0.350070\pi$
$654$	0	0
$655$	49.3506	1.92829
$656$	−88.3186	−3.44826
$657$	0	0
$658$	23.2759	0.907387
$659$	−7.57308	−0.295005	−0.147503	−	0.989062i	$-0.547124\pi$
$659$	−7.57308	−0.295005	−0.147503	+	0.989062i	$0.547124\pi$
$660$	0	0
$661$	41.4061	1.61051	0.805256	−	0.592928i	$-0.202028\pi$
$661$	41.4061	1.61051	0.805256	+	0.592928i	$0.202028\pi$
$662$	91.3182	3.54918
$663$	0	0
$664$	−84.5924	−3.28282
$665$	29.6051	1.14804
$666$	0	0
$667$	6.89119	0.266828
$668$	6.24551	0.241646
$669$	0	0
$670$	−120.382	−4.65078
$671$	8.64534	0.333749
$672$	0	0
$673$	44.6166	1.71984	0.859922	−	0.510425i	$-0.170512\pi$
$673$	44.6166	1.71984	0.859922	+	0.510425i	$0.170512\pi$
$674$	−21.1607	−0.815080
$675$	0	0
$676$	67.4540	2.59439
$677$	17.3065	0.665143	0.332572	−	0.943078i	$-0.392084\pi$
$677$	17.3065	0.665143	0.332572	+	0.943078i	$0.392084\pi$
$678$	0	0
$679$	20.5109	0.787135
$680$	−104.749	−4.01694
$681$	0	0
$682$	−24.6121	−0.942447
$683$	−24.0336	−0.919620	−0.459810	−	0.888017i	$-0.652082\pi$
$683$	−24.0336	−0.919620	−0.459810	+	0.888017i	$0.652082\pi$
$684$	0	0
$685$	10.1361	0.387279
$686$	−46.3242	−1.76867
$687$	0	0
$688$	−76.0345	−2.89879
$689$	−14.0556	−0.535475
$690$	0	0
$691$	5.88092	0.223721	0.111860	−	0.993724i	$-0.464319\pi$
$691$	5.88092	0.223721	0.111860	+	0.993724i	$0.464319\pi$
$692$	−30.6048	−1.16342
$693$	0	0
$694$	−42.3402	−1.60721
$695$	−36.6432	−1.38996
$696$	0	0
$697$	38.6349	1.46340
$698$	49.0159	1.85528
$699$	0	0
$700$	46.8058	1.76909
$701$	23.2511	0.878183	0.439091	−	0.898442i	$-0.355300\pi$
$701$	23.2511	0.878183	0.439091	+	0.898442i	$0.355300\pi$
$702$	0	0
$703$	33.7030	1.27113
$704$	−14.2227	−0.536039
$705$	0	0
$706$	−31.8300	−1.19794
$707$	25.4463	0.957006
$708$	0	0
$709$	4.12147	0.154785	0.0773925	−	0.997001i	$-0.475341\pi$
$709$	4.12147	0.154785	0.0773925	+	0.997001i	$0.475341\pi$
$710$	78.9895	2.96442
$711$	0	0
$712$	−90.2440	−3.38204
$713$	−8.37763	−0.313745
$714$	0	0
$715$	31.7727	1.18823
$716$	−56.6962	−2.11884
$717$	0	0
$718$	36.2314	1.35214
$719$	31.9220	1.19049	0.595244	−	0.803545i	$-0.297055\pi$
$719$	31.9220	1.19049	0.595244	+	0.803545i	$0.297055\pi$
$720$	0	0
$721$	5.09437	0.189724
$722$	37.8025	1.40686
$723$	0	0
$724$	59.6402	2.21651
$725$	−27.4084	−1.01792
$726$	0	0
$727$	−46.2291	−1.71454	−0.857271	−	0.514865i	$-0.827842\pi$
$727$	−46.2291	−1.71454	−0.857271	+	0.514865i	$0.827842\pi$
$728$	−58.0440	−2.15125
$729$	0	0
$730$	140.203	5.18916
$731$	33.2612	1.23021
$732$	0	0
$733$	−47.6780	−1.76103	−0.880513	−	0.474022i	$-0.842802\pi$
$733$	−47.6780	−1.76103	−0.880513	+	0.474022i	$0.842802\pi$
$734$	40.6972	1.50216
$735$	0	0
$736$	−16.6968	−0.615453
$737$	24.7160	0.910426
$738$	0	0
$739$	32.2779	1.18736	0.593681	−	0.804701i	$-0.297674\pi$
$739$	32.2779	1.18736	0.593681	+	0.804701i	$0.297674\pi$
$740$	94.8831	3.48797
$741$	0	0
$742$	10.7091	0.393145
$743$	23.4925	0.861855	0.430928	−	0.902387i	$-0.358186\pi$
$743$	23.4925	0.861855	0.430928	+	0.902387i	$0.358186\pi$
$744$	0	0
$745$	42.9082	1.57203
$746$	−55.0214	−2.01448
$747$	0	0
$748$	36.7425	1.34344
$749$	−6.54014	−0.238972
$750$	0	0
$751$	−15.2842	−0.557727	−0.278863	−	0.960331i	$-0.589958\pi$
$751$	−15.2842	−0.557727	−0.278863	+	0.960331i	$0.589958\pi$
$752$	56.5469	2.06205
$753$	0	0
$754$	58.0691	2.11475
$755$	9.04145	0.329052
$756$	0	0
$757$	12.6201	0.458687	0.229343	−	0.973346i	$-0.426342\pi$
$757$	12.6201	0.458687	0.229343	+	0.973346i	$0.426342\pi$
$758$	−54.7271	−1.98778
$759$	0	0
$760$	144.075	5.22614
$761$	−10.8503	−0.393322	−0.196661	−	0.980472i	$-0.563010\pi$
$761$	−10.8503	−0.393322	−0.196661	+	0.980472i	$0.563010\pi$
$762$	0	0
$763$	−6.17091	−0.223402
$764$	−70.8506	−2.56329
$765$	0	0
$766$	−40.6565	−1.46898
$767$	59.1043	2.13413
$768$	0	0
$769$	10.8756	0.392183	0.196091	−	0.980586i	$-0.437175\pi$
$769$	10.8756	0.392183	0.196091	+	0.980586i	$0.437175\pi$
$770$	−24.2080	−0.872397
$771$	0	0
$772$	21.1944	0.762802
$773$	−3.92153	−0.141048	−0.0705238	−	0.997510i	$-0.522467\pi$
$773$	−3.92153	−0.141048	−0.0705238	+	0.997510i	$0.522467\pi$
$774$	0	0
$775$	33.3204	1.19691
$776$	99.8170	3.58322
$777$	0	0
$778$	−1.17084	−0.0419765
$779$	−53.1395	−1.90392
$780$	0	0
$781$	−16.2175	−0.580309
$782$	17.6929	0.632696
$783$	0	0
$784$	−45.2319	−1.61543
$785$	72.7505	2.59658
$786$	0	0
$787$	19.7133	0.702703	0.351351	−	0.936244i	$-0.385722\pi$
$787$	19.7133	0.702703	0.351351	+	0.936244i	$0.385722\pi$
$788$	−35.8242	−1.27618
$789$	0	0
$790$	114.109	4.05981
$791$	6.20829	0.220741
$792$	0	0
$793$	24.7973	0.880578
$794$	69.2145	2.45633
$795$	0	0
$796$	11.6034	0.411272
$797$	15.3888	0.545101	0.272550	−	0.962142i	$-0.412133\pi$
$797$	15.3888	0.545101	0.272550	+	0.962142i	$0.412133\pi$
$798$	0	0
$799$	−24.7364	−0.875110
$800$	66.4084	2.34789
$801$	0	0
$802$	−25.1899	−0.889488
$803$	−28.7855	−1.01582
$804$	0	0
$805$	−8.24008	−0.290425
$806$	−70.5947	−2.48659
$807$	0	0
$808$	123.835	4.35651
$809$	−28.9637	−1.01831	−0.509156	−	0.860675i	$-0.670042\pi$
$809$	−28.9637	−1.01831	−0.509156	+	0.860675i	$0.670042\pi$
$810$	0	0
$811$	−2.56669	−0.0901287	−0.0450643	−	0.998984i	$-0.514349\pi$
$811$	−2.56669	−0.0901287	−0.0450643	+	0.998984i	$0.514349\pi$
$812$	−31.2747	−1.09753
$813$	0	0
$814$	−27.5589	−0.965938
$815$	14.3085	0.501205
$816$	0	0
$817$	−45.7484	−1.60053
$818$	50.9604	1.78179
$819$	0	0
$820$	−149.602	−5.22433
$821$	33.8121	1.18005	0.590026	−	0.807384i	$-0.299117\pi$
$821$	33.8121	1.18005	0.590026	+	0.807384i	$0.299117\pi$
$822$	0	0
$823$	−1.22448	−0.0426825	−0.0213413	−	0.999772i	$-0.506794\pi$
$823$	−1.22448	−0.0426825	−0.0213413	+	0.999772i	$0.506794\pi$
$824$	24.7920	0.863669
$825$	0	0
$826$	−45.0324	−1.56688
$827$	−39.3468	−1.36822	−0.684111	−	0.729378i	$-0.739810\pi$
$827$	−39.3468	−1.36822	−0.684111	+	0.729378i	$0.739810\pi$
$828$	0	0
$829$	33.8463	1.17553	0.587765	−	0.809032i	$-0.300008\pi$
$829$	33.8463	1.17553	0.587765	+	0.809032i	$0.300008\pi$
$830$	−101.194	−3.51251
$831$	0	0
$832$	−40.7948	−1.41431
$833$	19.7866	0.685567
$834$	0	0
$835$	4.37309	0.151337
$836$	−50.5367	−1.74785
$837$	0	0
$838$	−85.6351	−2.95822
$839$	10.5766	0.365146	0.182573	−	0.983192i	$-0.441557\pi$
$839$	10.5766	0.365146	0.182573	+	0.983192i	$0.441557\pi$
$840$	0	0
$841$	−10.6862	−0.368491
$842$	78.7239	2.71300
$843$	0	0
$844$	−12.8293	−0.441602
$845$	47.2311	1.62480
$846$	0	0
$847$	−11.6976	−0.401933
$848$	26.0170	0.893428
$849$	0	0
$850$	−70.3700	−2.41367
$851$	−9.38066	−0.321565
$852$	0	0
$853$	−23.1942	−0.794153	−0.397076	−	0.917786i	$-0.629975\pi$
$853$	−23.1942	−0.794153	−0.397076	+	0.917786i	$0.629975\pi$
$854$	−18.8934	−0.646519
$855$	0	0
$856$	−31.8279	−1.08785
$857$	16.4028	0.560309	0.280154	−	0.959955i	$-0.409614\pi$
$857$	16.4028	0.560309	0.280154	+	0.959955i	$0.409614\pi$
$858$	0	0
$859$	23.0997	0.788151	0.394075	−	0.919078i	$-0.371065\pi$
$859$	23.0997	0.788151	0.394075	+	0.919078i	$0.371065\pi$
$860$	−128.794	−4.39184
$861$	0	0
$862$	68.2602	2.32495
$863$	−6.21210	−0.211462	−0.105731	−	0.994395i	$-0.533718\pi$
$863$	−6.21210	−0.211462	−0.105731	+	0.994395i	$0.533718\pi$
$864$	0	0
$865$	−21.4294	−0.728622
$866$	−22.0479	−0.749218
$867$	0	0
$868$	38.0207	1.29051
$869$	−23.4280	−0.794740
$870$	0	0
$871$	70.8926	2.40211
$872$	−30.0310	−1.01698
$873$	0	0
$874$	−24.3353	−0.823153
$875$	7.18773	0.242990
$876$	0	0
$877$	−2.74715	−0.0927646	−0.0463823	−	0.998924i	$-0.514769\pi$
$877$	−2.74715	−0.0927646	−0.0463823	+	0.998924i	$0.514769\pi$
$878$	−94.1064	−3.17594
$879$	0	0
$880$	−58.8115	−1.98253
$881$	28.6124	0.963976	0.481988	−	0.876178i	$-0.339915\pi$
$881$	28.6124	0.963976	0.481988	+	0.876178i	$0.339915\pi$
$882$	0	0
$883$	3.52285	0.118553	0.0592767	−	0.998242i	$-0.481121\pi$
$883$	3.52285	0.118553	0.0592767	+	0.998242i	$0.481121\pi$
$884$	105.388	3.54459
$885$	0	0
$886$	−49.3280	−1.65721
$887$	14.5892	0.489859	0.244929	−	0.969541i	$-0.421235\pi$
$887$	14.5892	0.489859	0.244929	+	0.969541i	$0.421235\pi$
$888$	0	0
$889$	10.4456	0.350334
$890$	−107.955	−3.61866
$891$	0	0
$892$	−19.0875	−0.639096
$893$	34.0231	1.13854
$894$	0	0
$895$	−39.6985	−1.32697
$896$	−0.340480	−0.0113747
$897$	0	0
$898$	45.8433	1.52981
$899$	−22.2641	−0.742548
$900$	0	0
$901$	−11.3811	−0.379160
$902$	43.4520	1.44679
$903$	0	0
$904$	30.2129	1.00487
$905$	41.7599	1.38815
$906$	0	0
$907$	−4.88786	−0.162299	−0.0811494	−	0.996702i	$-0.525859\pi$
$907$	−4.88786	−0.162299	−0.0811494	+	0.996702i	$0.525859\pi$
$908$	−37.2686	−1.23680
$909$	0	0
$910$	−69.4356	−2.30177
$911$	5.15994	0.170956	0.0854782	−	0.996340i	$-0.472758\pi$
$911$	5.15994	0.170956	0.0854782	+	0.996340i	$0.472758\pi$
$912$	0	0
$913$	20.7765	0.687600
$914$	−42.0244	−1.39004
$915$	0	0
$916$	−14.1809	−0.468550
$917$	22.1430	0.731226
$918$	0	0
$919$	−42.1796	−1.39138	−0.695688	−	0.718344i	$-0.744901\pi$
$919$	−42.1796	−1.39138	−0.695688	+	0.718344i	$0.744901\pi$
$920$	−40.1007	−1.32208
$921$	0	0
$922$	42.9612	1.41485
$923$	−46.5165	−1.53111
$924$	0	0
$925$	37.3098	1.22674
$926$	42.8541	1.40827
$927$	0	0
$928$	−44.3727	−1.45661
$929$	21.6446	0.710135	0.355068	−	0.934841i	$-0.384458\pi$
$929$	21.6446	0.710135	0.355068	+	0.934841i	$0.384458\pi$
$930$	0	0
$931$	−27.2151	−0.891939
$932$	−57.1473	−1.87192
$933$	0	0
$934$	−22.1940	−0.726211
$935$	25.7270	0.841363
$936$	0	0
$937$	29.9841	0.979537	0.489768	−	0.871853i	$-0.337081\pi$
$937$	29.9841	0.979537	0.489768	+	0.871853i	$0.337081\pi$
$938$	−54.0141	−1.76362
$939$	0	0
$940$	95.7841	3.12413
$941$	−17.9745	−0.585953	−0.292976	−	0.956120i	$-0.594646\pi$
$941$	−17.9745	−0.585953	−0.292976	+	0.956120i	$0.594646\pi$
$942$	0	0
$943$	14.7905	0.481644
$944$	−109.403	−3.56075
$945$	0	0
$946$	37.4083	1.21625
$947$	46.7980	1.52073	0.760365	−	0.649496i	$-0.225020\pi$
$947$	46.7980	1.52073	0.760365	+	0.649496i	$0.225020\pi$
$948$	0	0
$949$	−82.5651	−2.68018
$950$	96.7889	3.14025
$951$	0	0
$952$	−46.9995	−1.52326
$953$	18.8544	0.610755	0.305378	−	0.952231i	$-0.401217\pi$
$953$	18.8544	0.610755	0.305378	+	0.952231i	$0.401217\pi$
$954$	0	0
$955$	−49.6094	−1.60532
$956$	−4.82304	−0.155988
$957$	0	0
$958$	32.1134	1.03754
$959$	4.54792	0.146860
$960$	0	0
$961$	−3.93357	−0.126889
$962$	−79.0467	−2.54857
$963$	0	0
$964$	93.8684	3.02330
$965$	14.8402	0.477724
$966$	0	0
$967$	−42.6277	−1.37081	−0.685407	−	0.728160i	$-0.740376\pi$
$967$	−42.6277	−1.37081	−0.685407	+	0.728160i	$0.740376\pi$
$968$	−56.9266	−1.82969
$969$	0	0
$970$	119.407	3.83392
$971$	−59.8886	−1.92192	−0.960958	−	0.276694i	$-0.910761\pi$
$971$	−59.8886	−1.92192	−0.960958	+	0.276694i	$0.910761\pi$
$972$	0	0
$973$	−16.4413	−0.527085
$974$	−17.0863	−0.547482
$975$	0	0
$976$	−45.9001	−1.46922
$977$	14.3310	0.458490	0.229245	−	0.973369i	$-0.426374\pi$
$977$	14.3310	0.458490	0.229245	+	0.973369i	$0.426374\pi$
$978$	0	0
$979$	22.1645	0.708381
$980$	−76.6178	−2.44747
$981$	0	0
$982$	15.7858	0.503745
$983$	13.2676	0.423170	0.211585	−	0.977360i	$-0.432137\pi$
$983$	13.2676	0.423170	0.211585	+	0.977360i	$0.432137\pi$
$984$	0	0
$985$	−25.0840	−0.799243
$986$	47.0198	1.49742
$987$	0	0
$988$	−144.954	−4.61160
$989$	12.7333	0.404895
$990$	0	0
$991$	4.13056	0.131211	0.0656057	−	0.997846i	$-0.479102\pi$
$991$	4.13056	0.131211	0.0656057	+	0.997846i	$0.479102\pi$
$992$	53.9440	1.71272
$993$	0	0
$994$	35.4416	1.12414
$995$	8.12467	0.257569
$996$	0	0
$997$	−29.5640	−0.936300	−0.468150	−	0.883649i	$-0.655079\pi$
$997$	−29.5640	−0.936300	−0.468150	+	0.883649i	$0.655079\pi$
$998$	8.70878	0.275672
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.i.1.16		17
3.2	odd	2		239.2.a.b.1.2	✓	17
12.11	even	2		3824.2.a.p.1.12		17
15.14	odd	2		5975.2.a.g.1.16		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.2.a.b.1.2	✓	17	3.2	odd	2
2151.2.a.i.1.16		17	1.1	even	1	trivial
3824.2.a.p.1.12		17	12.11	even	2
5975.2.a.g.1.16		17	15.14	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61209 q^{2} +4.82304 q^{4} +3.37708 q^{5} +1.51525 q^{7} +7.37403 q^{8} +O(q^{10})\)	\(q+2.61209 q^{2} +4.82304 q^{4} +3.37708 q^{5} +1.51525 q^{7} +7.37403 q^{8} +8.82124 q^{10} -1.81111 q^{11} -5.19479 q^{13} +3.95798 q^{14} +9.61560 q^{16} -4.20633 q^{17} +5.78551 q^{19} +16.2878 q^{20} -4.73079 q^{22} -1.61030 q^{23} +6.40464 q^{25} -13.5693 q^{26} +7.30811 q^{28} -4.27946 q^{29} +5.20254 q^{31} +10.3688 q^{32} -10.9873 q^{34} +5.11712 q^{35} +5.82542 q^{37} +15.1123 q^{38} +24.9027 q^{40} -9.18493 q^{41} -7.90741 q^{43} -8.73505 q^{44} -4.20625 q^{46} +5.88074 q^{47} -4.70401 q^{49} +16.7295 q^{50} -25.0546 q^{52} +2.70571 q^{53} -6.11626 q^{55} +11.1735 q^{56} -11.1783 q^{58} -11.3776 q^{59} -4.77350 q^{61} +13.5895 q^{62} +7.85303 q^{64} -17.5432 q^{65} -13.6469 q^{67} -20.2873 q^{68} +13.3664 q^{70} +8.95447 q^{71} +15.8938 q^{73} +15.2166 q^{74} +27.9037 q^{76} -2.74429 q^{77} +12.9357 q^{79} +32.4726 q^{80} -23.9919 q^{82} -11.4717 q^{83} -14.2051 q^{85} -20.6549 q^{86} -13.3552 q^{88} -12.2381 q^{89} -7.87141 q^{91} -7.76652 q^{92} +15.3611 q^{94} +19.5381 q^{95} +13.5363 q^{97} -12.2873 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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