LMFDB - Embedded newform 2151.2.a.i.1.7

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			1.7
Root			$-0.685793$ 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)

$f(q)$	$=$	$q-0.685793 q^{2} -1.52969 q^{4} +0.233223 q^{5} +1.65270 q^{7} +2.42063 q^{8} +O(q^{10})$	\(q-0.685793 q^{2} -1.52969 q^{4} +0.233223 q^{5} +1.65270 q^{7} +2.42063 q^{8} -0.159942 q^{10} +5.18409 q^{11} +4.45942 q^{13} -1.13341 q^{14} +1.39932 q^{16} +1.66170 q^{17} -0.667306 q^{19} -0.356758 q^{20} -3.55521 q^{22} +6.83733 q^{23} -4.94561 q^{25} -3.05823 q^{26} -2.52812 q^{28} -0.562960 q^{29} -10.4012 q^{31} -5.80091 q^{32} -1.13958 q^{34} +0.385447 q^{35} +5.83904 q^{37} +0.457633 q^{38} +0.564547 q^{40} -8.92698 q^{41} +6.75631 q^{43} -7.93005 q^{44} -4.68899 q^{46} +1.55369 q^{47} -4.26858 q^{49} +3.39166 q^{50} -6.82152 q^{52} +6.92300 q^{53} +1.20905 q^{55} +4.00058 q^{56} +0.386074 q^{58} -3.29140 q^{59} +1.95221 q^{61} +7.13303 q^{62} +1.17958 q^{64} +1.04004 q^{65} +15.9049 q^{67} -2.54189 q^{68} -0.264337 q^{70} +0.604704 q^{71} -2.07421 q^{73} -4.00437 q^{74} +1.02077 q^{76} +8.56775 q^{77} -5.56785 q^{79} +0.326354 q^{80} +6.12206 q^{82} +2.10671 q^{83} +0.387547 q^{85} -4.63343 q^{86} +12.5488 q^{88} +2.79020 q^{89} +7.37008 q^{91} -10.4590 q^{92} -1.06551 q^{94} -0.155631 q^{95} +16.2122 q^{97} +2.92736 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.685793	−0.484929	−0.242464	−	0.970160i	$-0.577956\pi$
$2$	−0.685793	−0.484929	−0.242464	+	0.970160i	$0.577956\pi$
$3$	0	0
$3$	0	0
$4$	−1.52969	−0.764844
$5$	0.233223	0.104300	0.0521502	−	0.998639i	$-0.483393\pi$
$5$	0.233223	0.104300	0.0521502	+	0.998639i	$0.483393\pi$
$6$	0	0
$7$	1.65270	0.624662	0.312331	−	0.949973i	$-0.398890\pi$
$7$	1.65270	0.624662	0.312331	+	0.949973i	$0.398890\pi$
$8$	2.42063	0.855823
$9$	0	0
$10$	−0.159942	−0.0505782
$11$	5.18409	1.56306	0.781532	−	0.623866i	$-0.214439\pi$
$11$	5.18409	1.56306	0.781532	+	0.623866i	$0.214439\pi$
$12$	0	0
$13$	4.45942	1.23682	0.618410	−	0.785856i	$-0.287777\pi$
$13$	4.45942	1.23682	0.618410	+	0.785856i	$0.287777\pi$
$14$	−1.13341	−0.302916
$15$	0	0
$16$	1.39932	0.349831
$17$	1.66170	0.403023	0.201511	−	0.979486i	$-0.435415\pi$
$17$	1.66170	0.403023	0.201511	+	0.979486i	$0.435415\pi$
$18$	0	0
$19$	−0.667306	−0.153090	−0.0765452	−	0.997066i	$-0.524389\pi$
$19$	−0.667306	−0.153090	−0.0765452	+	0.997066i	$0.524389\pi$
$20$	−0.356758	−0.0797736
$21$	0	0
$22$	−3.55521	−0.757974
$23$	6.83733	1.42568	0.712841	−	0.701325i	$-0.247408\pi$
$23$	6.83733	1.42568	0.712841	+	0.701325i	$0.247408\pi$
$24$	0	0
$25$	−4.94561	−0.989121
$26$	−3.05823	−0.599769
$27$	0	0
$28$	−2.52812	−0.477769
$29$	−0.562960	−0.104539	−0.0522695	−	0.998633i	$-0.516645\pi$
$29$	−0.562960	−0.104539	−0.0522695	+	0.998633i	$0.516645\pi$
$30$	0	0
$31$	−10.4012	−1.86810	−0.934051	−	0.357140i	$-0.883752\pi$
$31$	−10.4012	−1.86810	−0.934051	+	0.357140i	$0.883752\pi$
$32$	−5.80091	−1.02547
$33$	0	0
$34$	−1.13958	−0.195437
$35$	0.385447	0.0651525
$36$	0	0
$37$	5.83904	0.959932	0.479966	−	0.877287i	$-0.340649\pi$
$37$	5.83904	0.959932	0.479966	+	0.877287i	$0.340649\pi$
$38$	0.457633	0.0742379
$39$	0	0
$40$	0.564547	0.0892627
$41$	−8.92698	−1.39416	−0.697080	−	0.716993i	$-0.745518\pi$
$41$	−8.92698	−1.39416	−0.697080	+	0.716993i	$0.745518\pi$
$42$	0	0
$43$	6.75631	1.03033	0.515164	−	0.857092i	$-0.327731\pi$
$43$	6.75631	1.03033	0.515164	+	0.857092i	$0.327731\pi$
$44$	−7.93005	−1.19550
$45$	0	0
$46$	−4.68899	−0.691354
$47$	1.55369	0.226629	0.113314	−	0.993559i	$-0.463853\pi$
$47$	1.55369	0.226629	0.113314	+	0.993559i	$0.463853\pi$
$48$	0	0
$49$	−4.26858	−0.609797
$50$	3.39166	0.479653
$51$	0	0
$52$	−6.82152	−0.945974
$53$	6.92300	0.950948	0.475474	−	0.879730i	$-0.342277\pi$
$53$	6.92300	0.950948	0.475474	+	0.879730i	$0.342277\pi$
$54$	0	0
$55$	1.20905	0.163028
$56$	4.00058	0.534600
$57$	0	0
$58$	0.386074	0.0506940
$59$	−3.29140	−0.428504	−0.214252	−	0.976778i	$-0.568731\pi$
$59$	−3.29140	−0.428504	−0.214252	+	0.976778i	$0.568731\pi$
$60$	0	0
$61$	1.95221	0.249954	0.124977	−	0.992160i	$-0.460114\pi$
$61$	1.95221	0.249954	0.124977	+	0.992160i	$0.460114\pi$
$62$	7.13303	0.905896
$63$	0	0
$64$	1.17958	0.147447
$65$	1.04004	0.129001
$66$	0	0
$67$	15.9049	1.94310	0.971548	−	0.236844i	$-0.0761132\pi$
$67$	15.9049	1.94310	0.971548	+	0.236844i	$0.0761132\pi$
$68$	−2.54189	−0.308249
$69$	0	0
$70$	−0.264337	−0.0315943
$71$	0.604704	0.0717652	0.0358826	−	0.999356i	$-0.488576\pi$
$71$	0.604704	0.0717652	0.0358826	+	0.999356i	$0.488576\pi$
$72$	0	0
$73$	−2.07421	−0.242768	−0.121384	−	0.992606i	$-0.538733\pi$
$73$	−2.07421	−0.242768	−0.121384	+	0.992606i	$0.538733\pi$
$74$	−4.00437	−0.465498
$75$	0	0
$76$	1.02077	0.117090
$77$	8.56775	0.976386
$78$	0	0
$79$	−5.56785	−0.626432	−0.313216	−	0.949682i	$-0.601406\pi$
$79$	−5.56785	−0.626432	−0.313216	+	0.949682i	$0.601406\pi$
$80$	0.326354	0.0364875
$81$	0	0
$82$	6.12206	0.676068
$83$	2.10671	0.231242	0.115621	−	0.993293i	$-0.463114\pi$
$83$	2.10671	0.231242	0.115621	+	0.993293i	$0.463114\pi$
$84$	0	0
$85$	0.387547	0.0420354
$86$	−4.63343	−0.499636
$87$	0	0
$88$	12.5488	1.33771
$89$	2.79020	0.295760	0.147880	−	0.989005i	$-0.452755\pi$
$89$	2.79020	0.295760	0.147880	+	0.989005i	$0.452755\pi$
$90$	0	0
$91$	7.37008	0.772594
$92$	−10.4590	−1.09043
$93$	0	0
$94$	−1.06551	−0.109899
$95$	−0.155631	−0.0159674
$96$	0	0
$97$	16.2122	1.64610	0.823051	−	0.567967i	$-0.192270\pi$
$97$	16.2122	1.64610	0.823051	+	0.567967i	$0.192270\pi$
$98$	2.92736	0.295708
$99$	0	0
$100$	7.56524	0.756524
$101$	6.89676	0.686253	0.343127	−	0.939289i	$-0.388514\pi$
$101$	6.89676	0.686253	0.343127	+	0.939289i	$0.388514\pi$
$102$	0	0
$103$	9.14974	0.901551	0.450776	−	0.892637i	$-0.351147\pi$
$103$	9.14974	0.901551	0.450776	+	0.892637i	$0.351147\pi$
$104$	10.7946	1.05850
$105$	0	0
$106$	−4.74774	−0.461142
$107$	−12.7104	−1.22876	−0.614380	−	0.789010i	$-0.710594\pi$
$107$	−12.7104	−1.22876	−0.614380	+	0.789010i	$0.710594\pi$
$108$	0	0
$109$	−6.31670	−0.605031	−0.302515	−	0.953145i	$-0.597826\pi$
$109$	−6.31670	−0.605031	−0.302515	+	0.953145i	$0.597826\pi$
$110$	−0.829157	−0.0790570
$111$	0	0
$112$	2.31266	0.218526
$113$	−14.2804	−1.34338	−0.671692	−	0.740831i	$-0.734432\pi$
$113$	−14.2804	−1.34338	−0.671692	+	0.740831i	$0.734432\pi$
$114$	0	0
$115$	1.59462	0.148699
$116$	0.861154	0.0799561
$117$	0	0
$118$	2.25722	0.207794
$119$	2.74630	0.251753
$120$	0	0
$121$	15.8748	1.44317
$122$	−1.33881	−0.121210
$123$	0	0
$124$	15.9105	1.42881
$125$	−2.31954	−0.207466
$126$	0	0
$127$	−4.49934	−0.399252	−0.199626	−	0.979872i	$-0.563973\pi$
$127$	−4.49934	−0.399252	−0.199626	+	0.979872i	$0.563973\pi$
$128$	10.7929	0.953965
$129$	0	0
$130$	−0.713250	−0.0625562
$131$	18.9702	1.65743	0.828715	−	0.559671i	$-0.189072\pi$
$131$	18.9702	1.65743	0.828715	+	0.559671i	$0.189072\pi$
$132$	0	0
$133$	−1.10286	−0.0956298
$134$	−10.9075	−0.942262
$135$	0	0
$136$	4.02238	0.344916
$137$	−17.4891	−1.49420	−0.747099	−	0.664713i	$-0.768554\pi$
$137$	−17.4891	−1.49420	−0.747099	+	0.664713i	$0.768554\pi$
$138$	0	0
$139$	−12.7730	−1.08339	−0.541697	−	0.840574i	$-0.682218\pi$
$139$	−12.7730	−1.08339	−0.541697	+	0.840574i	$0.682218\pi$
$140$	−0.589614	−0.0498315
$141$	0	0
$142$	−0.414702	−0.0348010
$143$	23.1180	1.93323
$144$	0	0
$145$	−0.131295	−0.0109035
$146$	1.42248	0.117725
$147$	0	0
$148$	−8.93191	−0.734198
$149$	−5.81414	−0.476313	−0.238156	−	0.971227i	$-0.576543\pi$
$149$	−5.81414	−0.476313	−0.238156	+	0.971227i	$0.576543\pi$
$150$	0	0
$151$	6.01295	0.489327	0.244663	−	0.969608i	$-0.421323\pi$
$151$	6.01295	0.489327	0.244663	+	0.969608i	$0.421323\pi$
$152$	−1.61530	−0.131018
$153$	0	0
$154$	−5.87570	−0.473477
$155$	−2.42579	−0.194844
$156$	0	0
$157$	16.1084	1.28559	0.642797	−	0.766037i	$-0.277774\pi$
$157$	16.1084	1.28559	0.642797	+	0.766037i	$0.277774\pi$
$158$	3.81839	0.303775
$159$	0	0
$160$	−1.35291	−0.106957
$161$	11.3001	0.890570
$162$	0	0
$163$	14.8630	1.16416	0.582081	−	0.813131i	$-0.302239\pi$
$163$	14.8630	1.16416	0.582081	+	0.813131i	$0.302239\pi$
$164$	13.6555	1.06632
$165$	0	0
$166$	−1.44477	−0.112136
$167$	−6.87803	−0.532238	−0.266119	−	0.963940i	$-0.585741\pi$
$167$	−6.87803	−0.532238	−0.266119	+	0.963940i	$0.585741\pi$
$168$	0	0
$169$	6.88639	0.529723
$170$	−0.265777	−0.0203842
$171$	0	0
$172$	−10.3351	−0.788041
$173$	24.7535	1.88197	0.940987	−	0.338444i	$-0.109901\pi$
$173$	24.7535	1.88197	0.940987	+	0.338444i	$0.109901\pi$
$174$	0	0
$175$	−8.17361	−0.617866
$176$	7.25423	0.546808
$177$	0	0
$178$	−1.91350	−0.143423
$179$	−2.32161	−0.173526	−0.0867628	−	0.996229i	$-0.527652\pi$
$179$	−2.32161	−0.173526	−0.0867628	+	0.996229i	$0.527652\pi$
$180$	0	0
$181$	−4.64163	−0.345009	−0.172505	−	0.985009i	$-0.555186\pi$
$181$	−4.64163	−0.345009	−0.172505	+	0.985009i	$0.555186\pi$
$182$	−5.05434	−0.374653
$183$	0	0
$184$	16.5507	1.22013
$185$	1.36180	0.100121
$186$	0	0
$187$	8.61443	0.629950
$188$	−2.37666	−0.173336
$189$	0	0
$190$	0.106731	0.00774305
$191$	−4.24207	−0.306946	−0.153473	−	0.988153i	$-0.549046\pi$
$191$	−4.24207	−0.306946	−0.153473	+	0.988153i	$0.549046\pi$
$192$	0	0
$193$	−3.08489	−0.222055	−0.111028	−	0.993817i	$-0.535414\pi$
$193$	−3.08489	−0.222055	−0.111028	+	0.993817i	$0.535414\pi$
$194$	−11.1182	−0.798242
$195$	0	0
$196$	6.52960	0.466400
$197$	−11.5228	−0.820966	−0.410483	−	0.911868i	$-0.634640\pi$
$197$	−11.5228	−0.820966	−0.410483	+	0.911868i	$0.634640\pi$
$198$	0	0
$199$	−5.80470	−0.411485	−0.205742	−	0.978606i	$-0.565961\pi$
$199$	−5.80470	−0.411485	−0.205742	+	0.978606i	$0.565961\pi$
$200$	−11.9715	−0.846513
$201$	0	0
$202$	−4.72975	−0.332784
$203$	−0.930404	−0.0653016
$204$	0	0
$205$	−2.08198	−0.145412
$206$	−6.27483	−0.437188
$207$	0	0
$208$	6.24017	0.432678
$209$	−3.45938	−0.239290
$210$	0	0
$211$	−26.4316	−1.81963	−0.909815	−	0.415015i	$-0.863776\pi$
$211$	−26.4316	−1.81963	−0.909815	+	0.415015i	$0.863776\pi$
$212$	−10.5900	−0.727327
$213$	0	0
$214$	8.71669	0.595861
$215$	1.57573	0.107464
$216$	0	0
$217$	−17.1900	−1.16693
$218$	4.33195	0.293397
$219$	0	0
$220$	−1.84947	−0.124691
$221$	7.41023	0.498466
$222$	0	0
$223$	−9.77559	−0.654622	−0.327311	−	0.944917i	$-0.606142\pi$
$223$	−9.77559	−0.654622	−0.327311	+	0.944917i	$0.606142\pi$
$224$	−9.58717	−0.640570
$225$	0	0
$226$	9.79336	0.651445
$227$	19.0132	1.26195	0.630975	−	0.775804i	$-0.282655\pi$
$227$	19.0132	1.26195	0.630975	+	0.775804i	$0.282655\pi$
$228$	0	0
$229$	15.6028	1.03106	0.515532	−	0.856870i	$-0.327594\pi$
$229$	15.6028	1.03106	0.515532	+	0.856870i	$0.327594\pi$
$230$	−1.09358	−0.0721085
$231$	0	0
$232$	−1.36272	−0.0894670
$233$	−7.21397	−0.472603	−0.236301	−	0.971680i	$-0.575935\pi$
$233$	−7.21397	−0.472603	−0.236301	+	0.971680i	$0.575935\pi$
$234$	0	0
$235$	0.362355	0.0236375
$236$	5.03482	0.327739
$237$	0	0
$238$	−1.88339	−0.122082
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	10.1261	0.652277	0.326138	−	0.945322i	$-0.394252\pi$
$241$	10.1261	0.652277	0.326138	+	0.945322i	$0.394252\pi$
$242$	−10.8868	−0.699833
$243$	0	0
$244$	−2.98627	−0.191176
$245$	−0.995531	−0.0636021
$246$	0	0
$247$	−2.97579	−0.189345
$248$	−25.1774	−1.59877
$249$	0	0
$250$	1.59072	0.100606
$251$	14.9369	0.942811	0.471405	−	0.881917i	$-0.343747\pi$
$251$	14.9369	0.942811	0.471405	+	0.881917i	$0.343747\pi$
$252$	0	0
$253$	35.4454	2.22843
$254$	3.08561	0.193609
$255$	0	0
$256$	−9.76083	−0.610052
$257$	−22.2050	−1.38511	−0.692556	−	0.721364i	$-0.743516\pi$
$257$	−22.2050	−1.38511	−0.692556	+	0.721364i	$0.743516\pi$
$258$	0	0
$259$	9.65018	0.599633
$260$	−1.59093	−0.0986655
$261$	0	0
$262$	−13.0096	−0.803735
$263$	24.5780	1.51555	0.757774	−	0.652517i	$-0.226287\pi$
$263$	24.5780	1.51555	0.757774	+	0.652517i	$0.226287\pi$
$264$	0	0
$265$	1.61460	0.0991842
$266$	0.756331	0.0463736
$267$	0	0
$268$	−24.3296	−1.48617
$269$	26.8012	1.63410	0.817050	−	0.576567i	$-0.195608\pi$
$269$	26.8012	1.63410	0.817050	+	0.576567i	$0.195608\pi$
$270$	0	0
$271$	−21.1228	−1.28312	−0.641559	−	0.767074i	$-0.721712\pi$
$271$	−21.1228	−1.28312	−0.641559	+	0.767074i	$0.721712\pi$
$272$	2.32526	0.140990
$273$	0	0
$274$	11.9939	0.724579
$275$	−25.6385	−1.54606
$276$	0	0
$277$	−23.6773	−1.42263	−0.711317	−	0.702871i	$-0.751901\pi$
$277$	−23.6773	−1.42263	−0.711317	+	0.702871i	$0.751901\pi$
$278$	8.75965	0.525369
$279$	0	0
$280$	0.933027	0.0557590
$281$	15.6713	0.934874	0.467437	−	0.884027i	$-0.345178\pi$
$281$	15.6713	0.934874	0.467437	+	0.884027i	$0.345178\pi$
$282$	0	0
$283$	16.9417	1.00708	0.503539	−	0.863972i	$-0.332031\pi$
$283$	16.9417	1.00708	0.503539	+	0.863972i	$0.332031\pi$
$284$	−0.925009	−0.0548892
$285$	0	0
$286$	−15.8542	−0.937477
$287$	−14.7536	−0.870879
$288$	0	0
$289$	−14.2387	−0.837573
$290$	0.0900412	0.00528740
$291$	0	0
$292$	3.17290	0.185680
$293$	18.8194	1.09944	0.549720	−	0.835349i	$-0.314734\pi$
$293$	18.8194	1.09944	0.549720	+	0.835349i	$0.314734\pi$
$294$	0	0
$295$	−0.767631	−0.0446932
$296$	14.1342	0.821532
$297$	0	0
$298$	3.98729	0.230978
$299$	30.4905	1.76331
$300$	0	0
$301$	11.1662	0.643607
$302$	−4.12363	−0.237289
$303$	0	0
$304$	−0.933777	−0.0535558
$305$	0.455299	0.0260703
$306$	0	0
$307$	3.96841	0.226489	0.113245	−	0.993567i	$-0.463876\pi$
$307$	3.96841	0.226489	0.113245	+	0.993567i	$0.463876\pi$
$308$	−13.1060	−0.746783
$309$	0	0
$310$	1.66359	0.0944853
$311$	−8.91507	−0.505527	−0.252764	−	0.967528i	$-0.581340\pi$
$311$	−8.91507	−0.505527	−0.252764	+	0.967528i	$0.581340\pi$
$312$	0	0
$313$	−12.5115	−0.707194	−0.353597	−	0.935398i	$-0.615042\pi$
$313$	−12.5115	−0.707194	−0.353597	+	0.935398i	$0.615042\pi$
$314$	−11.0470	−0.623421
$315$	0	0
$316$	8.51708	0.479123
$317$	22.2463	1.24948	0.624738	−	0.780835i	$-0.285206\pi$
$317$	22.2463	1.24948	0.624738	+	0.780835i	$0.285206\pi$
$318$	0	0
$319$	−2.91844	−0.163401
$320$	0.275104	0.0153788
$321$	0	0
$322$	−7.74950	−0.431863
$323$	−1.10887	−0.0616989
$324$	0	0
$325$	−22.0545	−1.22336
$326$	−10.1930	−0.564535
$327$	0	0
$328$	−21.6090	−1.19316
$329$	2.56778	0.141566
$330$	0	0
$331$	11.1006	0.610146	0.305073	−	0.952329i	$-0.401319\pi$
$331$	11.1006	0.610146	0.305073	+	0.952329i	$0.401319\pi$
$332$	−3.22261	−0.176864
$333$	0	0
$334$	4.71690	0.258098
$335$	3.70939	0.202666
$336$	0	0
$337$	4.08075	0.222293	0.111146	−	0.993804i	$-0.464548\pi$
$337$	4.08075	0.222293	0.111146	+	0.993804i	$0.464548\pi$
$338$	−4.72264	−0.256878
$339$	0	0
$340$	−0.592827	−0.0321505
$341$	−53.9206	−2.91996
$342$	0	0
$343$	−18.6236	−1.00558
$344$	16.3546	0.881779
$345$	0	0
$346$	−16.9758	−0.912623
$347$	−11.8649	−0.636940	−0.318470	−	0.947933i	$-0.603169\pi$
$347$	−11.8649	−0.636940	−0.318470	+	0.947933i	$0.603169\pi$
$348$	0	0
$349$	2.67544	0.143213	0.0716066	−	0.997433i	$-0.477187\pi$
$349$	2.67544	0.143213	0.0716066	+	0.997433i	$0.477187\pi$
$350$	5.60540	0.299621
$351$	0	0
$352$	−30.0725	−1.60287
$353$	5.19771	0.276646	0.138323	−	0.990387i	$-0.455829\pi$
$353$	5.19771	0.276646	0.138323	+	0.990387i	$0.455829\pi$
$354$	0	0
$355$	0.141031	0.00748514
$356$	−4.26813	−0.226211
$357$	0	0
$358$	1.59215	0.0841475
$359$	−21.2739	−1.12280	−0.561398	−	0.827546i	$-0.689736\pi$
$359$	−21.2739	−1.12280	−0.561398	+	0.827546i	$0.689736\pi$
$360$	0	0
$361$	−18.5547	−0.976563
$362$	3.18319	0.167305
$363$	0	0
$364$	−11.2739	−0.590914
$365$	−0.483754	−0.0253208
$366$	0	0
$367$	4.58942	0.239566	0.119783	−	0.992800i	$-0.461780\pi$
$367$	4.58942	0.239566	0.119783	+	0.992800i	$0.461780\pi$
$368$	9.56765	0.498748
$369$	0	0
$370$	−0.933910	−0.0485517
$371$	11.4416	0.594021
$372$	0	0
$373$	29.7025	1.53794	0.768970	−	0.639285i	$-0.220770\pi$
$373$	29.7025	1.53794	0.768970	+	0.639285i	$0.220770\pi$
$374$	−5.90771	−0.305481
$375$	0	0
$376$	3.76091	0.193954
$377$	−2.51047	−0.129296
$378$	0	0
$379$	−3.23657	−0.166251	−0.0831257	−	0.996539i	$-0.526490\pi$
$379$	−3.23657	−0.166251	−0.0831257	+	0.996539i	$0.526490\pi$
$380$	0.238067	0.0122126
$381$	0	0
$382$	2.90918	0.148847
$383$	29.2957	1.49694	0.748469	−	0.663169i	$-0.230789\pi$
$383$	29.2957	1.49694	0.748469	+	0.663169i	$0.230789\pi$
$384$	0	0
$385$	1.99820	0.101837
$386$	2.11560	0.107681
$387$	0	0
$388$	−24.7997	−1.25901
$389$	0.233036	0.0118154	0.00590770	−	0.999983i	$-0.498120\pi$
$389$	0.233036	0.0118154	0.00590770	+	0.999983i	$0.498120\pi$
$390$	0	0
$391$	11.3616	0.574582
$392$	−10.3327	−0.521879
$393$	0	0
$394$	7.90226	0.398110
$395$	−1.29855	−0.0653371
$396$	0	0
$397$	−5.36962	−0.269494	−0.134747	−	0.990880i	$-0.543022\pi$
$397$	−5.36962	−0.269494	−0.134747	+	0.990880i	$0.543022\pi$
$398$	3.98082	0.199541
$399$	0	0
$400$	−6.92051	−0.346025
$401$	−16.5917	−0.828552	−0.414276	−	0.910151i	$-0.635965\pi$
$401$	−16.5917	−0.828552	−0.414276	+	0.910151i	$0.635965\pi$
$402$	0	0
$403$	−46.3831	−2.31051
$404$	−10.5499	−0.524877
$405$	0	0
$406$	0.638064	0.0316666
$407$	30.2701	1.50043
$408$	0	0
$409$	37.1563	1.83726	0.918630	−	0.395120i	$-0.129297\pi$
$409$	37.1563	1.83726	0.918630	+	0.395120i	$0.129297\pi$
$410$	1.42780	0.0705142
$411$	0	0
$412$	−13.9963	−0.689546
$413$	−5.43971	−0.267670
$414$	0	0
$415$	0.491333	0.0241186
$416$	−25.8687	−1.26832
$417$	0	0
$418$	2.37241	0.116039
$419$	−24.8231	−1.21269	−0.606343	−	0.795203i	$-0.707364\pi$
$419$	−24.8231	−1.21269	−0.606343	+	0.795203i	$0.707364\pi$
$420$	0	0
$421$	−15.9595	−0.777820	−0.388910	−	0.921276i	$-0.627148\pi$
$421$	−15.9595	−0.777820	−0.388910	+	0.921276i	$0.627148\pi$
$422$	18.1266	0.882390
$423$	0	0
$424$	16.7581	0.813843
$425$	−8.21814	−0.398638
$426$	0	0
$427$	3.22641	0.156137
$428$	19.4429	0.939810
$429$	0	0
$430$	−1.08062	−0.0521122
$431$	−11.5670	−0.557164	−0.278582	−	0.960412i	$-0.589864\pi$
$431$	−11.5670	−0.557164	−0.278582	+	0.960412i	$0.589864\pi$
$432$	0	0
$433$	13.7395	0.660277	0.330138	−	0.943933i	$-0.392905\pi$
$433$	13.7395	0.660277	0.330138	+	0.943933i	$0.392905\pi$
$434$	11.7888	0.565879
$435$	0	0
$436$	9.66259	0.462754
$437$	−4.56259	−0.218258
$438$	0	0
$439$	19.2056	0.916632	0.458316	−	0.888789i	$-0.348453\pi$
$439$	19.2056	0.916632	0.458316	+	0.888789i	$0.348453\pi$
$440$	2.92667	0.139523
$441$	0	0
$442$	−5.08188	−0.241720
$443$	18.3235	0.870574	0.435287	−	0.900292i	$-0.356647\pi$
$443$	18.3235	0.870574	0.435287	+	0.900292i	$0.356647\pi$
$444$	0	0
$445$	0.650738	0.0308479
$446$	6.70403	0.317445
$447$	0	0
$448$	1.94948	0.0921045
$449$	−6.13214	−0.289394	−0.144697	−	0.989476i	$-0.546221\pi$
$449$	−6.13214	−0.289394	−0.144697	+	0.989476i	$0.546221\pi$
$450$	0	0
$451$	−46.2783	−2.17916
$452$	21.8445	1.02748
$453$	0	0
$454$	−13.0391	−0.611955
$455$	1.71887	0.0805819
$456$	0	0
$457$	−13.3238	−0.623261	−0.311631	−	0.950203i	$-0.600875\pi$
$457$	−13.3238	−0.623261	−0.311631	+	0.950203i	$0.600875\pi$
$458$	−10.7003	−0.499993
$459$	0	0
$460$	−2.43928	−0.113732
$461$	−28.4495	−1.32503	−0.662514	−	0.749050i	$-0.730510\pi$
$461$	−28.4495	−1.32503	−0.662514	+	0.749050i	$0.730510\pi$
$462$	0	0
$463$	36.3822	1.69082	0.845411	−	0.534116i	$-0.179355\pi$
$463$	36.3822	1.69082	0.845411	+	0.534116i	$0.179355\pi$
$464$	−0.787764	−0.0365710
$465$	0	0
$466$	4.94729	0.229179
$467$	−13.7343	−0.635546	−0.317773	−	0.948167i	$-0.602935\pi$
$467$	−13.7343	−0.635546	−0.317773	+	0.948167i	$0.602935\pi$
$468$	0	0
$469$	26.2861	1.21378
$470$	−0.248501	−0.0114625
$471$	0	0
$472$	−7.96729	−0.366724
$473$	35.0254	1.61047
$474$	0	0
$475$	3.30023	0.151425
$476$	−4.20098	−0.192552
$477$	0	0
$478$	0.685793	0.0313674
$479$	11.2032	0.511888	0.255944	−	0.966692i	$-0.417614\pi$
$479$	11.2032	0.511888	0.255944	+	0.966692i	$0.417614\pi$
$480$	0	0
$481$	26.0387	1.18726
$482$	−6.94438	−0.316308
$483$	0	0
$484$	−24.2836	−1.10380
$485$	3.78106	0.171689
$486$	0	0
$487$	−20.3055	−0.920130	−0.460065	−	0.887885i	$-0.652174\pi$
$487$	−20.3055	−0.920130	−0.460065	+	0.887885i	$0.652174\pi$
$488$	4.72558	0.213917
$489$	0	0
$490$	0.682728	0.0308425
$491$	34.9339	1.57654	0.788271	−	0.615328i	$-0.210976\pi$
$491$	34.9339	1.57654	0.788271	+	0.615328i	$0.210976\pi$
$492$	0	0
$493$	−0.935473	−0.0421316
$494$	2.04078	0.0918189
$495$	0	0
$496$	−14.5546	−0.653520
$497$	0.999395	0.0448290
$498$	0	0
$499$	−29.6343	−1.32661	−0.663306	−	0.748348i	$-0.730847\pi$
$499$	−29.6343	−1.32661	−0.663306	+	0.748348i	$0.730847\pi$
$500$	3.54818	0.158679
$501$	0	0
$502$	−10.2436	−0.457196
$503$	−20.7122	−0.923509	−0.461755	−	0.887008i	$-0.652780\pi$
$503$	−20.7122	−0.923509	−0.461755	+	0.887008i	$0.652780\pi$
$504$	0	0
$505$	1.60848	0.0715765
$506$	−24.3082	−1.08063
$507$	0	0
$508$	6.88259	0.305366
$509$	−10.5093	−0.465816	−0.232908	−	0.972499i	$-0.574824\pi$
$509$	−10.5093	−0.465816	−0.232908	+	0.972499i	$0.574824\pi$
$510$	0	0
$511$	−3.42805	−0.151648
$512$	−14.8919	−0.658134
$513$	0	0
$514$	15.2281	0.671681
$515$	2.13393	0.0940322
$516$	0	0
$517$	8.05446	0.354235
$518$	−6.61802	−0.290779
$519$	0	0
$520$	2.51755	0.110402
$521$	−9.62742	−0.421785	−0.210892	−	0.977509i	$-0.567637\pi$
$521$	−9.62742	−0.421785	−0.210892	+	0.977509i	$0.567637\pi$
$522$	0	0
$523$	−10.6331	−0.464952	−0.232476	−	0.972602i	$-0.574683\pi$
$523$	−10.6331	−0.464952	−0.232476	+	0.972602i	$0.574683\pi$
$524$	−29.0184	−1.26768
$525$	0	0
$526$	−16.8554	−0.734932
$527$	−17.2836	−0.752887
$528$	0	0
$529$	23.7491	1.03257
$530$	−1.10728	−0.0480973
$531$	0	0
$532$	1.68703	0.0731419
$533$	−39.8091	−1.72433
$534$	0	0
$535$	−2.96435	−0.128160
$536$	38.5000	1.66295
$537$	0	0
$538$	−18.3801	−0.792422
$539$	−22.1287	−0.953152
$540$	0	0
$541$	−39.0776	−1.68008	−0.840039	−	0.542525i	$-0.817468\pi$
$541$	−39.0776	−1.68008	−0.840039	+	0.542525i	$0.817468\pi$
$542$	14.4859	0.622221
$543$	0	0
$544$	−9.63940	−0.413286
$545$	−1.47320	−0.0631049
$546$	0	0
$547$	−43.4081	−1.85600	−0.927998	−	0.372585i	$-0.878472\pi$
$547$	−43.4081	−1.85600	−0.927998	+	0.372585i	$0.878472\pi$
$548$	26.7529	1.14283
$549$	0	0
$550$	17.5827	0.749728
$551$	0.375667	0.0160039
$552$	0	0
$553$	−9.20199	−0.391308
$554$	16.2377	0.689876
$555$	0	0
$556$	19.5388	0.828628
$557$	41.0837	1.74077	0.870385	−	0.492371i	$-0.163870\pi$
$557$	41.0837	1.74077	0.870385	+	0.492371i	$0.163870\pi$
$558$	0	0
$559$	30.1292	1.27433
$560$	0.539366	0.0227924
$561$	0	0
$562$	−10.7473	−0.453347
$563$	5.65138	0.238177	0.119089	−	0.992884i	$-0.462003\pi$
$563$	5.65138	0.238177	0.119089	+	0.992884i	$0.462003\pi$
$564$	0	0
$565$	−3.33051	−0.140115
$566$	−11.6185	−0.488361
$567$	0	0
$568$	1.46377	0.0614183
$569$	20.2009	0.846866	0.423433	−	0.905927i	$-0.360825\pi$
$569$	20.2009	0.846866	0.423433	+	0.905927i	$0.360825\pi$
$570$	0	0
$571$	12.0127	0.502716	0.251358	−	0.967894i	$-0.419123\pi$
$571$	12.0127	0.502716	0.251358	+	0.967894i	$0.419123\pi$
$572$	−35.3634	−1.47862
$573$	0	0
$574$	10.1179	0.422314
$575$	−33.8148	−1.41017
$576$	0	0
$577$	34.2894	1.42748	0.713742	−	0.700409i	$-0.246999\pi$
$577$	34.2894	1.42748	0.713742	+	0.700409i	$0.246999\pi$
$578$	9.76482	0.406163
$579$	0	0
$580$	0.200841	0.00833946
$581$	3.48176	0.144448
$582$	0	0
$583$	35.8895	1.48639
$584$	−5.02091	−0.207767
$585$	0	0
$586$	−12.9062	−0.533150
$587$	12.5725	0.518921	0.259461	−	0.965754i	$-0.416455\pi$
$587$	12.5725	0.518921	0.259461	+	0.965754i	$0.416455\pi$
$588$	0	0
$589$	6.94075	0.285989
$590$	0.526435	0.0216730
$591$	0	0
$592$	8.17071	0.335814
$593$	−4.65387	−0.191112	−0.0955559	−	0.995424i	$-0.530463\pi$
$593$	−4.65387	−0.191112	−0.0955559	+	0.995424i	$0.530463\pi$
$594$	0	0
$595$	0.640500	0.0262579
$596$	8.89382	0.364305
$597$	0	0
$598$	−20.9102	−0.855080
$599$	−30.3155	−1.23866	−0.619328	−	0.785132i	$-0.712595\pi$
$599$	−30.3155	−1.23866	−0.619328	+	0.785132i	$0.712595\pi$
$600$	0	0
$601$	28.8598	1.17722	0.588608	−	0.808419i	$-0.299676\pi$
$601$	28.8598	1.17722	0.588608	+	0.808419i	$0.299676\pi$
$602$	−7.65767	−0.312103
$603$	0	0
$604$	−9.19794	−0.374259
$605$	3.70237	0.150523
$606$	0	0
$607$	−13.6954	−0.555879	−0.277939	−	0.960599i	$-0.589651\pi$
$607$	−13.6954	−0.555879	−0.277939	+	0.960599i	$0.589651\pi$
$608$	3.87098	0.156989
$609$	0	0
$610$	−0.312241	−0.0126423
$611$	6.92854	0.280299
$612$	0	0
$613$	39.2347	1.58468	0.792338	−	0.610082i	$-0.208864\pi$
$613$	39.2347	1.58468	0.792338	+	0.610082i	$0.208864\pi$
$614$	−2.72151	−0.109831
$615$	0	0
$616$	20.7394	0.835614
$617$	−29.1159	−1.17216	−0.586081	−	0.810252i	$-0.699330\pi$
$617$	−29.1159	−1.17216	−0.586081	+	0.810252i	$0.699330\pi$
$618$	0	0
$619$	−42.2518	−1.69824	−0.849121	−	0.528198i	$-0.822868\pi$
$619$	−42.2518	−1.69824	−0.849121	+	0.528198i	$0.822868\pi$
$620$	3.71070	0.149025
$621$	0	0
$622$	6.11389	0.245145
$623$	4.61136	0.184750
$624$	0	0
$625$	24.1871	0.967483
$626$	8.58032	0.342938
$627$	0	0
$628$	−24.6409	−0.983279
$629$	9.70275	0.386874
$630$	0	0
$631$	−42.6549	−1.69807	−0.849033	−	0.528339i	$-0.822815\pi$
$631$	−42.6549	−1.69807	−0.849033	+	0.528339i	$0.822815\pi$
$632$	−13.4777	−0.536115
$633$	0	0
$634$	−15.2563	−0.605906
$635$	−1.04935	−0.0416421
$636$	0	0
$637$	−19.0354	−0.754209
$638$	2.00144	0.0792379
$639$	0	0
$640$	2.51715	0.0994990
$641$	14.5072	0.573001	0.286501	−	0.958080i	$-0.407508\pi$
$641$	14.5072	0.573001	0.286501	+	0.958080i	$0.407508\pi$
$642$	0	0
$643$	3.88358	0.153153	0.0765767	−	0.997064i	$-0.475601\pi$
$643$	3.88358	0.153153	0.0765767	+	0.997064i	$0.475601\pi$
$644$	−17.2856	−0.681147
$645$	0	0
$646$	0.760451	0.0299196
$647$	−35.8971	−1.41126	−0.705630	−	0.708580i	$-0.749336\pi$
$647$	−35.8971	−1.41126	−0.705630	+	0.708580i	$0.749336\pi$
$648$	0	0
$649$	−17.0630	−0.669779
$650$	15.1248	0.593244
$651$	0	0
$652$	−22.7358	−0.890403
$653$	−15.4474	−0.604503	−0.302251	−	0.953228i	$-0.597738\pi$
$653$	−15.4474	−0.604503	−0.302251	+	0.953228i	$0.597738\pi$
$654$	0	0
$655$	4.42427	0.172871
$656$	−12.4917	−0.487721
$657$	0	0
$658$	−1.76096	−0.0686495
$659$	−8.53827	−0.332604	−0.166302	−	0.986075i	$-0.553183\pi$
$659$	−8.53827	−0.332604	−0.166302	+	0.986075i	$0.553183\pi$
$660$	0	0
$661$	13.6374	0.530434	0.265217	−	0.964189i	$-0.414556\pi$
$661$	13.6374	0.530434	0.265217	+	0.964189i	$0.414556\pi$
$662$	−7.61273	−0.295877
$663$	0	0
$664$	5.09958	0.197902
$665$	−0.257211	−0.00997423
$666$	0	0
$667$	−3.84915	−0.149040
$668$	10.5213	0.407079
$669$	0	0
$670$	−2.54387	−0.0982784
$671$	10.1204	0.390694
$672$	0	0
$673$	−9.05683	−0.349115	−0.174558	−	0.984647i	$-0.555850\pi$
$673$	−9.05683	−0.349115	−0.174558	+	0.984647i	$0.555850\pi$
$674$	−2.79855	−0.107796
$675$	0	0
$676$	−10.5340	−0.405155
$677$	−19.5993	−0.753264	−0.376632	−	0.926363i	$-0.622918\pi$
$677$	−19.5993	−0.753264	−0.376632	+	0.926363i	$0.622918\pi$
$678$	0	0
$679$	26.7940	1.02826
$680$	0.938110	0.0359749
$681$	0	0
$682$	36.9783	1.41597
$683$	−13.8092	−0.528395	−0.264198	−	0.964469i	$-0.585107\pi$
$683$	−13.8092	−0.528395	−0.264198	+	0.964469i	$0.585107\pi$
$684$	0	0
$685$	−4.07886	−0.155845
$686$	12.7719	0.487634
$687$	0	0
$688$	9.45427	0.360441
$689$	30.8726	1.17615
$690$	0	0
$691$	22.2557	0.846648	0.423324	−	0.905978i	$-0.360863\pi$
$691$	22.2557	0.846648	0.423324	+	0.905978i	$0.360863\pi$
$692$	−37.8651	−1.43942
$693$	0	0
$694$	8.13684	0.308870
$695$	−2.97896	−0.112999
$696$	0	0
$697$	−14.8340	−0.561878
$698$	−1.83480	−0.0694482
$699$	0	0
$700$	12.5031	0.472572
$701$	−41.6957	−1.57482	−0.787412	−	0.616427i	$-0.788579\pi$
$701$	−41.6957	−1.57482	−0.787412	+	0.616427i	$0.788579\pi$
$702$	0	0
$703$	−3.89642	−0.146956
$704$	6.11503	0.230469
$705$	0	0
$706$	−3.56455	−0.134154
$707$	11.3983	0.428676
$708$	0	0
$709$	−19.5858	−0.735562	−0.367781	−	0.929912i	$-0.619882\pi$
$709$	−19.5858	−0.735562	−0.367781	+	0.929912i	$0.619882\pi$
$710$	−0.0967179	−0.00362976
$711$	0	0
$712$	6.75405	0.253119
$713$	−71.1161	−2.66332
$714$	0	0
$715$	5.39165	0.201636
$716$	3.55135	0.132720
$717$	0	0
$718$	14.5895	0.544476
$719$	−10.4987	−0.391537	−0.195769	−	0.980650i	$-0.562720\pi$
$719$	−10.4987	−0.391537	−0.195769	+	0.980650i	$0.562720\pi$
$720$	0	0
$721$	15.1218	0.563165
$722$	12.7247	0.473563
$723$	0	0
$724$	7.10024	0.263878
$725$	2.78418	0.103402
$726$	0	0
$727$	11.6805	0.433206	0.216603	−	0.976260i	$-0.430502\pi$
$727$	11.6805	0.433206	0.216603	+	0.976260i	$0.430502\pi$
$728$	17.8403	0.661204
$729$	0	0
$730$	0.331755	0.0122788
$731$	11.2270	0.415245
$732$	0	0
$733$	−41.8585	−1.54608	−0.773040	−	0.634358i	$-0.781265\pi$
$733$	−41.8585	−1.54608	−0.773040	+	0.634358i	$0.781265\pi$
$734$	−3.14739	−0.116172
$735$	0	0
$736$	−39.6628	−1.46199
$737$	82.4526	3.03718
$738$	0	0
$739$	−1.14025	−0.0419447	−0.0209723	−	0.999780i	$-0.506676\pi$
$739$	−1.14025	−0.0419447	−0.0209723	+	0.999780i	$0.506676\pi$
$740$	−2.08312	−0.0765772
$741$	0	0
$742$	−7.84660	−0.288058
$743$	−44.0060	−1.61442	−0.807211	−	0.590262i	$-0.799024\pi$
$743$	−44.0060	−1.61442	−0.807211	+	0.590262i	$0.799024\pi$
$744$	0	0
$745$	−1.35599	−0.0496796
$746$	−20.3698	−0.745791
$747$	0	0
$748$	−13.1774	−0.481813
$749$	−21.0065	−0.767560
$750$	0	0
$751$	−32.2458	−1.17667	−0.588333	−	0.808618i	$-0.700216\pi$
$751$	−32.2458	−1.17667	−0.588333	+	0.808618i	$0.700216\pi$
$752$	2.17411	0.0792818
$753$	0	0
$754$	1.72166	0.0626993
$755$	1.40236	0.0510370
$756$	0	0
$757$	2.39218	0.0869451	0.0434725	−	0.999055i	$-0.486158\pi$
$757$	2.39218	0.0869451	0.0434725	+	0.999055i	$0.486158\pi$
$758$	2.21961	0.0806200
$759$	0	0
$760$	−0.376726	−0.0136653
$761$	2.14906	0.0779035	0.0389517	−	0.999241i	$-0.487598\pi$
$761$	2.14906	0.0779035	0.0389517	+	0.999241i	$0.487598\pi$
$762$	0	0
$763$	−10.4396	−0.377940
$764$	6.48905	0.234766
$765$	0	0
$766$	−20.0907	−0.725908
$767$	−14.6777	−0.529983
$768$	0	0
$769$	−3.11069	−0.112174	−0.0560872	−	0.998426i	$-0.517862\pi$
$769$	−3.11069	−0.112174	−0.0560872	+	0.998426i	$0.517862\pi$
$770$	−1.37035	−0.0493839
$771$	0	0
$772$	4.71892	0.169838
$773$	7.46796	0.268604	0.134302	−	0.990940i	$-0.457121\pi$
$773$	7.46796	0.268604	0.134302	+	0.990940i	$0.457121\pi$
$774$	0	0
$775$	51.4400	1.84778
$776$	39.2439	1.40877
$777$	0	0
$778$	−0.159815	−0.00572963
$779$	5.95703	0.213433
$780$	0	0
$781$	3.13484	0.112174
$782$	−7.79172	−0.278631
$783$	0	0
$784$	−5.97313	−0.213326
$785$	3.75686	0.134088
$786$	0	0
$787$	−15.3410	−0.546847	−0.273423	−	0.961894i	$-0.588156\pi$
$787$	−15.3410	−0.546847	−0.273423	+	0.961894i	$0.588156\pi$
$788$	17.6263	0.627912
$789$	0	0
$790$	0.890536	0.0316838
$791$	−23.6012	−0.839160
$792$	0	0
$793$	8.70570	0.309148
$794$	3.68245	0.130685
$795$	0	0
$796$	8.87939	0.314722
$797$	26.0812	0.923844	0.461922	−	0.886921i	$-0.347160\pi$
$797$	26.0812	0.923844	0.461922	+	0.886921i	$0.347160\pi$
$798$	0	0
$799$	2.58177	0.0913365
$800$	28.6890	1.01431
$801$	0	0
$802$	11.3785	0.401788
$803$	−10.7529	−0.379462
$804$	0	0
$805$	2.63543	0.0928868
$806$	31.8092	1.12043
$807$	0	0
$808$	16.6945	0.587311
$809$	−27.2744	−0.958917	−0.479459	−	0.877564i	$-0.659167\pi$
$809$	−27.2744	−0.958917	−0.479459	+	0.877564i	$0.659167\pi$
$810$	0	0
$811$	32.1959	1.13055	0.565276	−	0.824902i	$-0.308770\pi$
$811$	32.1959	1.13055	0.565276	+	0.824902i	$0.308770\pi$
$812$	1.42323	0.0499455
$813$	0	0
$814$	−20.7590	−0.727603
$815$	3.46640	0.121423
$816$	0	0
$817$	−4.50853	−0.157733
$818$	−25.4815	−0.890939
$819$	0	0
$820$	3.18478	0.111217
$821$	5.90650	0.206138	0.103069	−	0.994674i	$-0.467134\pi$
$821$	5.90650	0.206138	0.103069	+	0.994674i	$0.467134\pi$
$822$	0	0
$823$	−19.5416	−0.681179	−0.340589	−	0.940212i	$-0.610627\pi$
$823$	−19.5416	−0.681179	−0.340589	+	0.940212i	$0.610627\pi$
$824$	22.1482	0.771569
$825$	0	0
$826$	3.73051	0.129801
$827$	−43.9949	−1.52985	−0.764926	−	0.644118i	$-0.777224\pi$
$827$	−43.9949	−1.52985	−0.764926	+	0.644118i	$0.777224\pi$
$828$	0	0
$829$	28.9351	1.00496	0.502479	−	0.864590i	$-0.332421\pi$
$829$	28.9351	1.00496	0.502479	+	0.864590i	$0.332421\pi$
$830$	−0.336953	−0.0116958
$831$	0	0
$832$	5.26022	0.182365
$833$	−7.09312	−0.245762
$834$	0	0
$835$	−1.60411	−0.0555127
$836$	5.29177	0.183020
$837$	0	0
$838$	17.0235	0.588067
$839$	−11.4135	−0.394038	−0.197019	−	0.980400i	$-0.563126\pi$
$839$	−11.4135	−0.394038	−0.197019	+	0.980400i	$0.563126\pi$
$840$	0	0
$841$	−28.6831	−0.989072
$842$	10.9449	0.377187
$843$	0	0
$844$	40.4322	1.39173
$845$	1.60606	0.0552503
$846$	0	0
$847$	26.2363	0.901491
$848$	9.68753	0.332671
$849$	0	0
$850$	5.63594	0.193311
$851$	39.9234	1.36856
$852$	0	0
$853$	15.7678	0.539878	0.269939	−	0.962877i	$-0.412996\pi$
$853$	15.7678	0.539878	0.269939	+	0.962877i	$0.412996\pi$
$854$	−2.21265	−0.0757153
$855$	0	0
$856$	−30.7672	−1.05160
$857$	−14.0055	−0.478419	−0.239209	−	0.970968i	$-0.576888\pi$
$857$	−14.0055	−0.478419	−0.239209	+	0.970968i	$0.576888\pi$
$858$	0	0
$859$	−34.9741	−1.19330	−0.596651	−	0.802501i	$-0.703502\pi$
$859$	−34.9741	−1.19330	−0.596651	+	0.802501i	$0.703502\pi$
$860$	−2.41037	−0.0821930
$861$	0	0
$862$	7.93258	0.270185
$863$	−18.7791	−0.639246	−0.319623	−	0.947545i	$-0.603556\pi$
$863$	−18.7791	−0.639246	−0.319623	+	0.947545i	$0.603556\pi$
$864$	0	0
$865$	5.77308	0.196291
$866$	−9.42243	−0.320187
$867$	0	0
$868$	26.2953	0.892521
$869$	−28.8643	−0.979153
$870$	0	0
$871$	70.9267	2.40326
$872$	−15.2904	−0.517799
$873$	0	0
$874$	3.12899	0.105840
$875$	−3.83351	−0.129596
$876$	0	0
$877$	39.0477	1.31855	0.659274	−	0.751903i	$-0.270864\pi$
$877$	39.0477	1.31855	0.659274	+	0.751903i	$0.270864\pi$
$878$	−13.1710	−0.444501
$879$	0	0
$880$	1.69185	0.0570323
$881$	2.35959	0.0794967	0.0397484	−	0.999210i	$-0.487344\pi$
$881$	2.35959	0.0794967	0.0397484	+	0.999210i	$0.487344\pi$
$882$	0	0
$883$	34.4766	1.16023	0.580115	−	0.814535i	$-0.303008\pi$
$883$	34.4766	1.16023	0.580115	+	0.814535i	$0.303008\pi$
$884$	−11.3353	−0.381249
$885$	0	0
$886$	−12.5661	−0.422166
$887$	23.7203	0.796450	0.398225	−	0.917288i	$-0.369626\pi$
$887$	23.7203	0.796450	0.398225	+	0.917288i	$0.369626\pi$
$888$	0	0
$889$	−7.43606	−0.249397
$890$	−0.446271	−0.0149590
$891$	0	0
$892$	14.9536	0.500684
$893$	−1.03679	−0.0346947
$894$	0	0
$895$	−0.541453	−0.0180988
$896$	17.8374	0.595906
$897$	0	0
$898$	4.20538	0.140335
$899$	5.85543	0.195290
$900$	0	0
$901$	11.5040	0.383253
$902$	31.7373	1.05674
$903$	0	0
$904$	−34.5675	−1.14970
$905$	−1.08253	−0.0359846
$906$	0	0
$907$	4.80693	0.159612	0.0798058	−	0.996810i	$-0.474570\pi$
$907$	4.80693	0.159612	0.0798058	+	0.996810i	$0.474570\pi$
$908$	−29.0842	−0.965195
$909$	0	0
$910$	−1.17879	−0.0390765
$911$	16.9817	0.562630	0.281315	−	0.959615i	$-0.409229\pi$
$911$	16.9817	0.562630	0.281315	+	0.959615i	$0.409229\pi$
$912$	0	0
$913$	10.9214	0.361445
$914$	9.13737	0.302237
$915$	0	0
$916$	−23.8675	−0.788604
$917$	31.3520	1.03533
$918$	0	0
$919$	21.8814	0.721801	0.360900	−	0.932604i	$-0.382469\pi$
$919$	21.8814	0.721801	0.360900	+	0.932604i	$0.382469\pi$
$920$	3.86000	0.127260
$921$	0	0
$922$	19.5105	0.642544
$923$	2.69663	0.0887606
$924$	0	0
$925$	−28.8776	−0.949489
$926$	−24.9506	−0.819928
$927$	0	0
$928$	3.26568	0.107201
$929$	29.3245	0.962105	0.481053	−	0.876692i	$-0.340255\pi$
$929$	29.3245	0.962105	0.481053	+	0.876692i	$0.340255\pi$
$930$	0	0
$931$	2.84845	0.0933542
$932$	11.0351	0.361468
$933$	0	0
$934$	9.41885	0.308194
$935$	2.00908	0.0657040
$936$	0	0
$937$	7.37774	0.241020	0.120510	−	0.992712i	$-0.461547\pi$
$937$	7.37774	0.241020	0.120510	+	0.992712i	$0.461547\pi$
$938$	−18.0268	−0.588595
$939$	0	0
$940$	−0.554291	−0.0180790
$941$	44.8956	1.46355	0.731777	−	0.681544i	$-0.238691\pi$
$941$	44.8956	1.46355	0.731777	+	0.681544i	$0.238691\pi$
$942$	0	0
$943$	−61.0368	−1.98763
$944$	−4.60574	−0.149904
$945$	0	0
$946$	−24.0201	−0.780962
$947$	−35.8074	−1.16358	−0.581792	−	0.813338i	$-0.697648\pi$
$947$	−35.8074	−1.16358	−0.581792	+	0.813338i	$0.697648\pi$
$948$	0	0
$949$	−9.24977	−0.300260
$950$	−2.26327	−0.0734303
$951$	0	0
$952$	6.64779	0.215456
$953$	55.0689	1.78386	0.891928	−	0.452177i	$-0.149352\pi$
$953$	55.0689	1.78386	0.891928	+	0.452177i	$0.149352\pi$
$954$	0	0
$955$	−0.989348	−0.0320146
$956$	1.52969	0.0494737
$957$	0	0
$958$	−7.68308	−0.248229
$959$	−28.9043	−0.933368
$960$	0	0
$961$	77.1840	2.48981
$962$	−17.8571	−0.575737
$963$	0	0
$964$	−15.4897	−0.498890
$965$	−0.719467	−0.0231605
$966$	0	0
$967$	−13.8418	−0.445121	−0.222561	−	0.974919i	$-0.571442\pi$
$967$	−13.8418	−0.445121	−0.222561	+	0.974919i	$0.571442\pi$
$968$	38.4272	1.23510
$969$	0	0
$970$	−2.59302	−0.0832570
$971$	−9.70128	−0.311329	−0.155664	−	0.987810i	$-0.549752\pi$
$971$	−9.70128	−0.311329	−0.155664	+	0.987810i	$0.549752\pi$
$972$	0	0
$973$	−21.1100	−0.676756
$974$	13.9254	0.446197
$975$	0	0
$976$	2.73177	0.0874418
$977$	41.1282	1.31581	0.657904	−	0.753102i	$-0.271443\pi$
$977$	41.1282	1.31581	0.657904	+	0.753102i	$0.271443\pi$
$978$	0	0
$979$	14.4647	0.462292
$980$	1.52285	0.0486457
$981$	0	0
$982$	−23.9574	−0.764511
$983$	56.1687	1.79150	0.895751	−	0.444556i	$-0.146639\pi$
$983$	56.1687	1.79150	0.895751	+	0.444556i	$0.146639\pi$
$984$	0	0
$985$	−2.68738	−0.0856271
$986$	0.641541	0.0204308
$987$	0	0
$988$	4.55204	0.144820
$989$	46.1952	1.46892
$990$	0	0
$991$	−46.0357	−1.46237	−0.731186	−	0.682178i	$-0.761033\pi$
$991$	−46.0357	−1.46237	−0.731186	+	0.682178i	$0.761033\pi$
$992$	60.3362	1.91568
$993$	0	0
$994$	−0.685377	−0.0217388
$995$	−1.35379	−0.0429180
$996$	0	0
$997$	20.7597	0.657465	0.328732	−	0.944423i	$-0.393379\pi$
$997$	20.7597	0.657465	0.328732	+	0.944423i	$0.393379\pi$
$998$	20.3230	0.643312
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.i.1.7		17
3.2	odd	2		239.2.a.b.1.11	✓	17
12.11	even	2		3824.2.a.p.1.1		17
15.14	odd	2		5975.2.a.g.1.7		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.2.a.b.1.11	✓	17	3.2	odd	2
2151.2.a.i.1.7		17	1.1	even	1	trivial
3824.2.a.p.1.1		17	12.11	even	2
5975.2.a.g.1.7		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)

\(f(q)\)	\(=\)	\(q-0.685793 q^{2} -1.52969 q^{4} +0.233223 q^{5} +1.65270 q^{7} +2.42063 q^{8} +O(q^{10})\)	\(q-0.685793 q^{2} -1.52969 q^{4} +0.233223 q^{5} +1.65270 q^{7} +2.42063 q^{8} -0.159942 q^{10} +5.18409 q^{11} +4.45942 q^{13} -1.13341 q^{14} +1.39932 q^{16} +1.66170 q^{17} -0.667306 q^{19} -0.356758 q^{20} -3.55521 q^{22} +6.83733 q^{23} -4.94561 q^{25} -3.05823 q^{26} -2.52812 q^{28} -0.562960 q^{29} -10.4012 q^{31} -5.80091 q^{32} -1.13958 q^{34} +0.385447 q^{35} +5.83904 q^{37} +0.457633 q^{38} +0.564547 q^{40} -8.92698 q^{41} +6.75631 q^{43} -7.93005 q^{44} -4.68899 q^{46} +1.55369 q^{47} -4.26858 q^{49} +3.39166 q^{50} -6.82152 q^{52} +6.92300 q^{53} +1.20905 q^{55} +4.00058 q^{56} +0.386074 q^{58} -3.29140 q^{59} +1.95221 q^{61} +7.13303 q^{62} +1.17958 q^{64} +1.04004 q^{65} +15.9049 q^{67} -2.54189 q^{68} -0.264337 q^{70} +0.604704 q^{71} -2.07421 q^{73} -4.00437 q^{74} +1.02077 q^{76} +8.56775 q^{77} -5.56785 q^{79} +0.326354 q^{80} +6.12206 q^{82} +2.10671 q^{83} +0.387547 q^{85} -4.63343 q^{86} +12.5488 q^{88} +2.79020 q^{89} +7.37008 q^{91} -10.4590 q^{92} -1.06551 q^{94} -0.155631 q^{95} +16.2122 q^{97} +2.92736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 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

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