LMFDB - Embedded newform 2601.2.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, 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("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.4848615424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 16x^{4} - 8x^{2} + 1 $ x^8 - 8x^6 + 16x^4 - 8*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 153)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.688039$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14144 q^{2} +2.58579 q^{4} -2.79793 q^{5} +2.93015 q^{7} -1.25443 q^{8} +O(q^{10})$	$q-2.14144 q^{2} +2.58579 q^{4} -2.79793 q^{5} +2.93015 q^{7} -1.25443 q^{8} +5.99162 q^{10} +1.15894 q^{11} +4.41421 q^{13} -6.27476 q^{14} -2.48528 q^{16} +6.41421 q^{19} -7.23486 q^{20} -2.48181 q^{22} +2.79793 q^{23} +2.82843 q^{25} -9.45280 q^{26} +7.57675 q^{28} +9.55274 q^{29} -7.83938 q^{31} +7.83095 q^{32} -8.19837 q^{35} +1.39942 q^{37} -13.7357 q^{38} +3.50981 q^{40} -0.480049 q^{41} +8.07107 q^{43} +2.99678 q^{44} -5.99162 q^{46} +6.05692 q^{47} +1.58579 q^{49} -6.05692 q^{50} +11.4142 q^{52} -10.3398 q^{53} -3.24264 q^{55} -3.67567 q^{56} -20.4567 q^{58} +6.42433 q^{59} -0.765367 q^{61} +16.7876 q^{62} -11.7990 q^{64} -12.3507 q^{65} +0.585786 q^{67} +17.5563 q^{70} -12.8307 q^{71} -11.9832 q^{73} -2.99678 q^{74} +16.5858 q^{76} +3.39587 q^{77} -2.48181 q^{79} +6.95365 q^{80} +1.02800 q^{82} -10.3398 q^{83} -17.2837 q^{86} -1.45381 q^{88} +11.2268 q^{89} +12.9343 q^{91} +7.23486 q^{92} -12.9706 q^{94} -17.9465 q^{95} +6.30864 q^{97} -3.39587 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{4}+O(q^{10}) $ 8 * q + 32 * q^4	$ 8 q + 32 q^{4} + 24 q^{13} + 48 q^{16} + 40 q^{19} + 8 q^{43} + 24 q^{49} + 80 q^{52} + 8 q^{55} + 64 q^{64} + 16 q^{67} + 16 q^{70} + 144 q^{76} + 32 q^{94}+O(q^{100}) $ 8 * q + 32 * q^4 + 24 * q^13 + 48 * q^16 + 40 * q^19 + 8 * q^43 + 24 * q^49 + 80 * q^52 + 8 * q^55 + 64 * q^64 + 16 * q^67 + 16 * q^70 + 144 * q^76 + 32 * q^94

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 $

$2$	−2.14144	−1.51423	−0.757115	−	0.653281i	$-0.773392\pi$
$2$	−2.14144	−1.51423	−0.757115	+	0.653281i	$0.773392\pi$
$3$	0	0
$3$	0	0
$4$	2.58579	1.29289
$5$	−2.79793	−1.25127	−0.625637	−	0.780115i	$-0.715161\pi$
$5$	−2.79793	−1.25127	−0.625637	+	0.780115i	$0.715161\pi$
$6$	0	0
$7$	2.93015	1.10749	0.553747	−	0.832685i	$-0.313198\pi$
$7$	2.93015	1.10749	0.553747	+	0.832685i	$0.313198\pi$
$8$	−1.25443	−0.443508
$9$	0	0
$10$	5.99162	1.89472
$11$	1.15894	0.349434	0.174717	−	0.984619i	$-0.444099\pi$
$11$	1.15894	0.349434	0.174717	+	0.984619i	$0.444099\pi$
$12$	0	0
$13$	4.41421	1.22428	0.612141	−	0.790748i	$-0.290308\pi$
$13$	4.41421	1.22428	0.612141	+	0.790748i	$0.290308\pi$
$14$	−6.27476	−1.67700
$15$	0	0
$16$	−2.48528	−0.621320
$17$	0	0
$17$	0	0
$18$	0	0
$19$	6.41421	1.47152	0.735761	−	0.677242i	$-0.236825\pi$
$19$	6.41421	1.47152	0.735761	+	0.677242i	$0.236825\pi$
$20$	−7.23486	−1.61776
$21$	0	0
$22$	−2.48181	−0.529124
$23$	2.79793	0.583409	0.291705	−	0.956508i	$-0.405778\pi$
$23$	2.79793	0.583409	0.291705	+	0.956508i	$0.405778\pi$
$24$	0	0
$25$	2.82843	0.565685
$26$	−9.45280	−1.85385
$27$	0	0
$28$	7.57675	1.43187
$29$	9.55274	1.77390	0.886950	−	0.461866i	$-0.152820\pi$
$29$	9.55274	1.77390	0.886950	+	0.461866i	$0.152820\pi$
$30$	0	0
$31$	−7.83938	−1.40799	−0.703997	−	0.710203i	$-0.748603\pi$
$31$	−7.83938	−1.40799	−0.703997	+	0.710203i	$0.748603\pi$
$32$	7.83095	1.38433
$33$	0	0
$34$	0	0
$35$	−8.19837	−1.38578
$36$	0	0
$37$	1.39942	0.230063	0.115031	−	0.993362i	$-0.463303\pi$
$37$	1.39942	0.230063	0.115031	+	0.993362i	$0.463303\pi$
$38$	−13.7357	−2.22822
$39$	0	0
$40$	3.50981	0.554950
$41$	−0.480049	−0.0749711	−0.0374856	−	0.999297i	$-0.511935\pi$
$41$	−0.480049	−0.0749711	−0.0374856	+	0.999297i	$0.511935\pi$
$42$	0	0
$43$	8.07107	1.23083	0.615413	−	0.788205i	$-0.288989\pi$
$43$	8.07107	1.23083	0.615413	+	0.788205i	$0.288989\pi$
$44$	2.99678	0.451781
$45$	0	0
$46$	−5.99162	−0.883416
$47$	6.05692	0.883493	0.441746	−	0.897140i	$-0.354359\pi$
$47$	6.05692	0.883493	0.441746	+	0.897140i	$0.354359\pi$
$48$	0	0
$49$	1.58579	0.226541
$50$	−6.05692	−0.856578
$51$	0	0
$52$	11.4142	1.58287
$53$	−10.3398	−1.42028	−0.710141	−	0.704059i	$-0.751369\pi$
$53$	−10.3398	−1.42028	−0.710141	+	0.704059i	$0.751369\pi$
$54$	0	0
$55$	−3.24264	−0.437238
$56$	−3.67567	−0.491182
$57$	0	0
$58$	−20.4567	−2.68609
$59$	6.42433	0.836377	0.418189	−	0.908360i	$-0.362665\pi$
$59$	6.42433	0.836377	0.418189	+	0.908360i	$0.362665\pi$
$60$	0	0
$61$	−0.765367	−0.0979952	−0.0489976	−	0.998799i	$-0.515603\pi$
$61$	−0.765367	−0.0979952	−0.0489976	+	0.998799i	$0.515603\pi$
$62$	16.7876	2.13203
$63$	0	0
$64$	−11.7990	−1.47487
$65$	−12.3507	−1.53191
$66$	0	0
$67$	0.585786	0.0715652	0.0357826	−	0.999360i	$-0.488608\pi$
$67$	0.585786	0.0715652	0.0357826	+	0.999360i	$0.488608\pi$
$68$	0	0
$69$	0	0
$70$	17.5563	2.09839
$71$	−12.8307	−1.52273	−0.761363	−	0.648326i	$-0.775470\pi$
$71$	−12.8307	−1.52273	−0.761363	+	0.648326i	$0.775470\pi$
$72$	0	0
$73$	−11.9832	−1.40253	−0.701266	−	0.712900i	$-0.747381\pi$
$73$	−11.9832	−1.40253	−0.701266	+	0.712900i	$0.747381\pi$
$74$	−2.99678	−0.348368
$75$	0	0
$76$	16.5858	1.90252
$77$	3.39587	0.386996
$78$	0	0
$79$	−2.48181	−0.279225	−0.139613	−	0.990206i	$-0.544586\pi$
$79$	−2.48181	−0.279225	−0.139613	+	0.990206i	$0.544586\pi$
$80$	6.95365	0.777442
$81$	0	0
$82$	1.02800	0.113524
$83$	−10.3398	−1.13494	−0.567471	−	0.823393i	$-0.692078\pi$
$83$	−10.3398	−1.13494	−0.567471	+	0.823393i	$0.692078\pi$
$84$	0	0
$85$	0	0
$86$	−17.2837	−1.86375
$87$	0	0
$88$	−1.45381	−0.154977
$89$	11.2268	1.19004	0.595021	−	0.803710i	$-0.297144\pi$
$89$	11.2268	1.19004	0.595021	+	0.803710i	$0.297144\pi$
$90$	0	0
$91$	12.9343	1.35588
$92$	7.23486	0.754286
$93$	0	0
$94$	−12.9706	−1.33781
$95$	−17.9465	−1.84128
$96$	0	0
$97$	6.30864	0.640546	0.320273	−	0.947325i	$-0.396225\pi$
$97$	6.30864	0.640546	0.320273	+	0.947325i	$0.396225\pi$
$98$	−3.39587	−0.343035
$99$	0	0
$100$	7.31371	0.731371
$101$	1.25443	0.124820	0.0624102	−	0.998051i	$-0.480121\pi$
$101$	1.25443	0.124820	0.0624102	+	0.998051i	$0.480121\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	−5.53732	−0.542979
$105$	0	0
$106$	22.1421	2.15063
$107$	5.11582	0.494565	0.247282	−	0.968943i	$-0.420462\pi$
$107$	5.11582	0.494565	0.247282	+	0.968943i	$0.420462\pi$
$108$	0	0
$109$	3.50981	0.336179	0.168089	−	0.985772i	$-0.446240\pi$
$109$	3.50981	0.336179	0.168089	+	0.985772i	$0.446240\pi$
$110$	6.94394	0.662078
$111$	0	0
$112$	−7.28225	−0.688108
$113$	2.11904	0.199343	0.0996713	−	0.995020i	$-0.468221\pi$
$113$	2.11904	0.199343	0.0996713	+	0.995020i	$0.468221\pi$
$114$	0	0
$115$	−7.82843	−0.730005
$116$	24.7013	2.29346
$117$	0	0
$118$	−13.7574	−1.26647
$119$	0	0
$120$	0	0
$121$	−9.65685	−0.877896
$122$	1.63899	0.148387
$123$	0	0
$124$	−20.2710	−1.82039
$125$	6.07591	0.543446
$126$	0	0
$127$	6.89949	0.612231	0.306116	−	0.951994i	$-0.400971\pi$
$127$	6.89949	0.612231	0.306116	+	0.951994i	$0.400971\pi$
$128$	9.60498	0.848969
$129$	0	0
$130$	26.4483	2.31967
$131$	−0.198843	−0.0173730	−0.00868649	−	0.999962i	$-0.502765\pi$
$131$	−0.198843	−0.0173730	−0.00868649	+	0.999962i	$0.502765\pi$
$132$	0	0
$133$	18.7946	1.62970
$134$	−1.25443	−0.108366
$135$	0	0
$136$	0	0
$137$	0.887016	0.0757829	0.0378914	−	0.999282i	$-0.487936\pi$
$137$	0.887016	0.0757829	0.0378914	+	0.999282i	$0.487936\pi$
$138$	0	0
$139$	−9.87285	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$139$	−9.87285	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$140$	−21.1992	−1.79166
$141$	0	0
$142$	27.4763	2.30576
$143$	5.11582	0.427806
$144$	0	0
$145$	−26.7279	−2.21963
$146$	25.6614	2.12376
$147$	0	0
$148$	3.61859	0.297447
$149$	−2.66105	−0.218001	−0.109001	−	0.994042i	$-0.534765\pi$
$149$	−2.66105	−0.218001	−0.109001	+	0.994042i	$0.534765\pi$
$150$	0	0
$151$	5.41421	0.440602	0.220301	−	0.975432i	$-0.429296\pi$
$151$	5.41421	0.440602	0.220301	+	0.975432i	$0.429296\pi$
$152$	−8.04618	−0.652631
$153$	0	0
$154$	−7.27208	−0.586001
$155$	21.9341	1.76179
$156$	0	0
$157$	18.7990	1.50032	0.750161	−	0.661255i	$-0.229976\pi$
$157$	18.7990	1.50032	0.750161	+	0.661255i	$0.229976\pi$
$158$	5.31466	0.422812
$159$	0	0
$160$	−21.9105	−1.73218
$161$	8.19837	0.646122
$162$	0	0
$163$	8.47343	0.663690	0.331845	−	0.943334i	$-0.392329\pi$
$163$	8.47343	0.663690	0.331845	+	0.943334i	$0.392329\pi$
$164$	−1.24131	−0.0969296
$165$	0	0
$166$	22.1421	1.71856
$167$	−5.79471	−0.448408	−0.224204	−	0.974542i	$-0.571978\pi$
$167$	−5.79471	−0.448408	−0.224204	+	0.974542i	$0.571978\pi$
$168$	0	0
$169$	6.48528	0.498868
$170$	0	0
$171$	0	0
$172$	20.8701	1.59133
$173$	17.6653	1.34307	0.671535	−	0.740973i	$-0.265635\pi$
$173$	17.6653	1.34307	0.671535	+	0.740973i	$0.265635\pi$
$174$	0	0
$175$	8.28772	0.626493
$176$	−2.88030	−0.217110
$177$	0	0
$178$	−24.0416	−1.80200
$179$	−18.1708	−1.35815	−0.679073	−	0.734070i	$-0.737618\pi$
$179$	−18.1708	−1.35815	−0.679073	+	0.734070i	$0.737618\pi$
$180$	0	0
$181$	−8.60474	−0.639586	−0.319793	−	0.947487i	$-0.603613\pi$
$181$	−8.60474	−0.639586	−0.319793	+	0.947487i	$0.603613\pi$
$182$	−27.6981	−2.05312
$183$	0	0
$184$	−3.50981	−0.258747
$185$	−3.91548	−0.287872
$186$	0	0
$187$	0	0
$188$	15.6619	1.14226
$189$	0	0
$190$	38.4315	2.78812
$191$	22.8211	1.65127	0.825637	−	0.564201i	$-0.190816\pi$
$191$	22.8211	1.65127	0.825637	+	0.564201i	$0.190816\pi$
$192$	0	0
$193$	6.49435	0.467474	0.233737	−	0.972300i	$-0.424905\pi$
$193$	6.49435	0.467474	0.233737	+	0.972300i	$0.424905\pi$
$194$	−13.5096	−0.969934
$195$	0	0
$196$	4.10051	0.292893
$197$	−18.6254	−1.32701	−0.663503	−	0.748173i	$-0.730931\pi$
$197$	−18.6254	−1.32701	−0.663503	+	0.748173i	$0.730931\pi$
$198$	0	0
$199$	−13.4370	−0.952527	−0.476264	−	0.879303i	$-0.658009\pi$
$199$	−13.4370	−0.952527	−0.476264	+	0.879303i	$0.658009\pi$
$200$	−3.54806	−0.250886
$201$	0	0
$202$	−2.68629	−0.189007
$203$	27.9910	1.96458
$204$	0	0
$205$	1.34315	0.0938094
$206$	−2.14144	−0.149202
$207$	0	0
$208$	−10.9706	−0.760672
$209$	7.43370	0.514200
$210$	0	0
$211$	−8.92177	−0.614200	−0.307100	−	0.951677i	$-0.599359\pi$
$211$	−8.92177	−0.614200	−0.307100	+	0.951677i	$0.599359\pi$
$212$	−26.7365	−1.83627
$213$	0	0
$214$	−10.9552	−0.748885
$215$	−22.5823	−1.54010
$216$	0	0
$217$	−22.9706	−1.55934
$218$	−7.51606	−0.509052
$219$	0	0
$220$	−8.38478	−0.565302
$221$	0	0
$222$	0	0
$223$	−6.07107	−0.406549	−0.203274	−	0.979122i	$-0.565158\pi$
$223$	−6.07107	−0.406549	−0.203274	+	0.979122i	$0.565158\pi$
$224$	22.9459	1.53314
$225$	0	0
$226$	−4.53781	−0.301851
$227$	−20.9433	−1.39006	−0.695028	−	0.718982i	$-0.744608\pi$
$227$	−20.9433	−1.39006	−0.695028	+	0.718982i	$0.744608\pi$
$228$	0	0
$229$	14.5858	0.963856	0.481928	−	0.876211i	$-0.339937\pi$
$229$	14.5858	0.963856	0.481928	+	0.876211i	$0.339937\pi$
$230$	16.7641	1.10540
$231$	0	0
$232$	−11.9832	−0.786738
$233$	13.3108	0.872018	0.436009	−	0.899942i	$-0.356392\pi$
$233$	13.3108	0.872018	0.436009	+	0.899942i	$0.356392\pi$
$234$	0	0
$235$	−16.9469	−1.10549
$236$	16.6120	1.08135
$237$	0	0
$238$	0	0
$239$	23.1885	1.49994	0.749969	−	0.661473i	$-0.230068\pi$
$239$	23.1885	1.49994	0.749969	+	0.661473i	$0.230068\pi$
$240$	0	0
$241$	−9.50143	−0.612041	−0.306020	−	0.952025i	$-0.598998\pi$
$241$	−9.50143	−0.612041	−0.306020	+	0.952025i	$0.598998\pi$
$242$	20.6796	1.32934
$243$	0	0
$244$	−1.97908	−0.126697
$245$	−4.43692	−0.283465
$246$	0	0
$247$	28.3137	1.80156
$248$	9.83395	0.624456
$249$	0	0
$250$	−13.0112	−0.822903
$251$	2.66105	0.167964	0.0839819	−	0.996467i	$-0.473236\pi$
$251$	2.66105	0.167964	0.0839819	+	0.996467i	$0.473236\pi$
$252$	0	0
$253$	3.24264	0.203863
$254$	−14.7749	−0.927059
$255$	0	0
$256$	3.02944	0.189340
$257$	10.7072	0.667898	0.333949	−	0.942591i	$-0.391619\pi$
$257$	10.7072	0.667898	0.333949	+	0.942591i	$0.391619\pi$
$258$	0	0
$259$	4.10051	0.254793
$260$	−31.9362	−1.98060
$261$	0	0
$262$	0.425811	0.0263067
$263$	−9.97240	−0.614924	−0.307462	−	0.951560i	$-0.599480\pi$
$263$	−9.97240	−0.614924	−0.307462	+	0.951560i	$0.599480\pi$
$264$	0	0
$265$	28.9301	1.77716
$266$	−40.2476	−2.46774
$267$	0	0
$268$	1.51472	0.0925262
$269$	13.0296	0.794427	0.397213	−	0.917726i	$-0.369977\pi$
$269$	13.0296	0.794427	0.397213	+	0.917726i	$0.369977\pi$
$270$	0	0
$271$	32.0711	1.94818	0.974089	−	0.226164i	$-0.0726184\pi$
$271$	32.0711	1.94818	0.974089	+	0.226164i	$0.0726184\pi$
$272$	0	0
$273$	0	0
$274$	−1.89949	−0.114753
$275$	3.27798	0.197670
$276$	0	0
$277$	11.7975	0.708845	0.354422	−	0.935085i	$-0.384678\pi$
$277$	11.7975	0.708845	0.354422	+	0.935085i	$0.384678\pi$
$278$	21.1422	1.26802
$279$	0	0
$280$	10.2843	0.614603
$281$	32.6413	1.94722	0.973608	−	0.228228i	$-0.0732933\pi$
$281$	32.6413	1.94722	0.973608	+	0.228228i	$0.0732933\pi$
$282$	0	0
$283$	3.37849	0.200831	0.100415	−	0.994946i	$-0.467983\pi$
$283$	3.37849	0.200831	0.100415	+	0.994946i	$0.467983\pi$
$284$	−33.1775	−1.96872
$285$	0	0
$286$	−10.9552	−0.647797
$287$	−1.40662	−0.0830300
$288$	0	0
$289$	0	0
$290$	57.2364	3.36104
$291$	0	0
$292$	−30.9861	−1.81332
$293$	14.2553	0.832803	0.416401	−	0.909181i	$-0.363291\pi$
$293$	14.2553	0.832803	0.416401	+	0.909181i	$0.363291\pi$
$294$	0	0
$295$	−17.9749	−1.04654
$296$	−1.75547	−0.102035
$297$	0	0
$298$	5.69848	0.330104
$299$	12.3507	0.714258
$300$	0	0
$301$	23.6494	1.36313
$302$	−11.5942	−0.667174
$303$	0	0
$304$	−15.9411	−0.914286
$305$	2.14144	0.122619
$306$	0	0
$307$	−3.89949	−0.222556	−0.111278	−	0.993789i	$-0.535494\pi$
$307$	−3.89949	−0.222556	−0.111278	+	0.993789i	$0.535494\pi$
$308$	8.78101	0.500344
$309$	0	0
$310$	−46.9706	−2.66775
$311$	19.7844	1.12187	0.560934	−	0.827860i	$-0.310442\pi$
$311$	19.7844	1.12187	0.560934	+	0.827860i	$0.310442\pi$
$312$	0	0
$313$	27.0279	1.52771	0.763855	−	0.645388i	$-0.223304\pi$
$313$	27.0279	1.52771	0.763855	+	0.645388i	$0.223304\pi$
$314$	−40.2570	−2.27183
$315$	0	0
$316$	−6.41743	−0.361009
$317$	−9.55274	−0.536535	−0.268268	−	0.963344i	$-0.586451\pi$
$317$	−9.55274	−0.536535	−0.268268	+	0.963344i	$0.586451\pi$
$318$	0	0
$319$	11.0711	0.619861
$320$	33.0128	1.84547
$321$	0	0
$322$	−17.5563	−0.978377
$323$	0	0
$324$	0	0
$325$	12.4853	0.692559
$326$	−18.1454	−1.00498
$327$	0	0
$328$	0.602188	0.0332503
$329$	17.7477	0.978462
$330$	0	0
$331$	27.1421	1.49187	0.745933	−	0.666021i	$-0.232004\pi$
$331$	27.1421	1.49187	0.745933	+	0.666021i	$0.232004\pi$
$332$	−26.7365	−1.46736
$333$	0	0
$334$	12.4090	0.678993
$335$	−1.63899	−0.0895476
$336$	0	0
$337$	−9.50143	−0.517576	−0.258788	−	0.965934i	$-0.583323\pi$
$337$	−9.50143	−0.517576	−0.258788	+	0.965934i	$0.583323\pi$
$338$	−13.8879	−0.755401
$339$	0	0
$340$	0	0
$341$	−9.08538	−0.492001
$342$	0	0
$343$	−15.8645	−0.856601
$344$	−10.1246	−0.545881
$345$	0	0
$346$	−37.8293	−2.03372
$347$	−29.6183	−1.58999	−0.794997	−	0.606613i	$-0.792528\pi$
$347$	−29.6183	−1.58999	−0.794997	+	0.606613i	$0.792528\pi$
$348$	0	0
$349$	15.7279	0.841896	0.420948	−	0.907085i	$-0.361697\pi$
$349$	15.7279	0.841896	0.420948	+	0.907085i	$0.361697\pi$
$350$	−17.7477	−0.948654
$351$	0	0
$352$	9.07562	0.483732
$353$	17.2837	0.919921	0.459961	−	0.887939i	$-0.347864\pi$
$353$	17.2837	0.919921	0.459961	+	0.887939i	$0.347864\pi$
$354$	0	0
$355$	35.8995	1.90535
$356$	29.0302	1.53860
$357$	0	0
$358$	38.9117	2.05655
$359$	11.2268	0.592529	0.296265	−	0.955106i	$-0.404259\pi$
$359$	11.2268	0.592529	0.296265	+	0.955106i	$0.404259\pi$
$360$	0	0
$361$	22.1421	1.16538
$362$	18.4266	0.968480
$363$	0	0
$364$	33.4454	1.75301
$365$	33.5283	1.75495
$366$	0	0
$367$	0.765367	0.0399518	0.0199759	−	0.999800i	$-0.493641\pi$
$367$	0.765367	0.0399518	0.0199759	+	0.999800i	$0.493641\pi$
$368$	−6.95365	−0.362484
$369$	0	0
$370$	8.38478	0.435904
$371$	−30.2972	−1.57295
$372$	0	0
$373$	−32.0416	−1.65905	−0.829526	−	0.558468i	$-0.811390\pi$
$373$	−32.0416	−1.65905	−0.829526	+	0.558468i	$0.811390\pi$
$374$	0	0
$375$	0	0
$376$	−7.59798	−0.391836
$377$	42.1678	2.17175
$378$	0	0
$379$	−19.2430	−0.988444	−0.494222	−	0.869336i	$-0.664547\pi$
$379$	−19.2430	−0.988444	−0.494222	+	0.869336i	$0.664547\pi$
$380$	−46.4059	−2.38057
$381$	0	0
$382$	−48.8701	−2.50041
$383$	−14.6227	−0.747185	−0.373593	−	0.927593i	$-0.621874\pi$
$383$	−14.6227	−0.747185	−0.373593	+	0.927593i	$0.621874\pi$
$384$	0	0
$385$	−9.50143	−0.484238
$386$	−13.9073	−0.707863
$387$	0	0
$388$	16.3128	0.828157
$389$	2.14144	0.108576	0.0542878	−	0.998525i	$-0.482711\pi$
$389$	2.14144	0.108576	0.0542878	+	0.998525i	$0.482711\pi$
$390$	0	0
$391$	0	0
$392$	−1.98926	−0.100473
$393$	0	0
$394$	39.8853	2.00939
$395$	6.94394	0.349387
$396$	0	0
$397$	−20.4567	−1.02669	−0.513345	−	0.858182i	$-0.671594\pi$
$397$	−20.4567	−1.02669	−0.513345	+	0.858182i	$0.671594\pi$
$398$	28.7747	1.44235
$399$	0	0
$400$	−7.02944	−0.351472
$401$	20.9433	1.04586	0.522930	−	0.852376i	$-0.324839\pi$
$401$	20.9433	1.04586	0.522930	+	0.852376i	$0.324839\pi$
$402$	0	0
$403$	−34.6047	−1.72378
$404$	3.24369	0.161379
$405$	0	0
$406$	−59.9411	−2.97483
$407$	1.62184	0.0803918
$408$	0	0
$409$	−2.79899	−0.138401	−0.0692006	−	0.997603i	$-0.522045\pi$
$409$	−2.79899	−0.138401	−0.0692006	+	0.997603i	$0.522045\pi$
$410$	−2.87627	−0.142049
$411$	0	0
$412$	2.58579	0.127393
$413$	18.8243	0.926282
$414$	0	0
$415$	28.9301	1.42012
$416$	34.5675	1.69481
$417$	0	0
$418$	−15.9189	−0.778617
$419$	29.0559	1.41947	0.709737	−	0.704467i	$-0.248814\pi$
$419$	29.0559	1.41947	0.709737	+	0.704467i	$0.248814\pi$
$420$	0	0
$421$	15.8284	0.771430	0.385715	−	0.922618i	$-0.373955\pi$
$421$	15.8284	0.771430	0.385715	+	0.922618i	$0.373955\pi$
$422$	19.1055	0.930040
$423$	0	0
$424$	12.9706	0.629906
$425$	0	0
$426$	0	0
$427$	−2.24264	−0.108529
$428$	13.2284	0.639419
$429$	0	0
$430$	48.3588	2.33207
$431$	2.99678	0.144350	0.0721748	−	0.997392i	$-0.477006\pi$
$431$	2.99678	0.144350	0.0721748	+	0.997392i	$0.477006\pi$
$432$	0	0
$433$	−1.20101	−0.0577169	−0.0288584	−	0.999584i	$-0.509187\pi$
$433$	−1.20101	−0.0577169	−0.0288584	+	0.999584i	$0.509187\pi$
$434$	49.1902	2.36120
$435$	0	0
$436$	9.07562	0.434643
$437$	17.9465	0.858499
$438$	0	0
$439$	19.8226	0.946082	0.473041	−	0.881040i	$-0.343156\pi$
$439$	19.8226	0.946082	0.473041	+	0.881040i	$0.343156\pi$
$440$	4.06766	0.193918
$441$	0	0
$442$	0	0
$443$	−12.1138	−0.575546	−0.287773	−	0.957699i	$-0.592915\pi$
$443$	−12.1138	−0.575546	−0.287773	+	0.957699i	$0.592915\pi$
$444$	0	0
$445$	−31.4119	−1.48907
$446$	13.0009	0.615608
$447$	0	0
$448$	−34.5728	−1.63341
$449$	−4.63577	−0.218775	−0.109388	−	0.993999i	$-0.534889\pi$
$449$	−4.63577	−0.218775	−0.109388	+	0.993999i	$0.534889\pi$
$450$	0	0
$451$	−0.556349	−0.0261975
$452$	5.47939	0.257729
$453$	0	0
$454$	44.8490	2.10487
$455$	−36.1893	−1.69658
$456$	0	0
$457$	−14.5563	−0.680917	−0.340459	−	0.940259i	$-0.610582\pi$
$457$	−14.5563	−0.680917	−0.340459	+	0.940259i	$0.610582\pi$
$458$	−31.2347	−1.45950
$459$	0	0
$460$	−20.2426	−0.943818
$461$	14.2553	0.663935	0.331967	−	0.943291i	$-0.392288\pi$
$461$	14.2553	0.663935	0.331967	+	0.943291i	$0.392288\pi$
$462$	0	0
$463$	−28.5269	−1.32576	−0.662879	−	0.748727i	$-0.730666\pi$
$463$	−28.5269	−1.32576	−0.662879	+	0.748727i	$0.730666\pi$
$464$	−23.7412	−1.10216
$465$	0	0
$466$	−28.5043	−1.32044
$467$	−14.9901	−0.693660	−0.346830	−	0.937928i	$-0.612742\pi$
$467$	−14.9901	−0.693660	−0.346830	+	0.937928i	$0.612742\pi$
$468$	0	0
$469$	1.71644	0.0792580
$470$	36.2908	1.67397
$471$	0	0
$472$	−8.05887	−0.370940
$473$	9.35390	0.430093
$474$	0	0
$475$	18.1421	0.832418
$476$	0	0
$477$	0	0
$478$	−49.6569	−2.27125
$479$	3.47682	0.158860	0.0794301	−	0.996840i	$-0.474690\pi$
$479$	3.47682	0.158860	0.0794301	+	0.996840i	$0.474690\pi$
$480$	0	0
$481$	6.17733	0.281662
$482$	20.3468	0.926771
$483$	0	0
$484$	−24.9706	−1.13503
$485$	−17.6512	−0.801498
$486$	0	0
$487$	−30.5152	−1.38278	−0.691388	−	0.722483i	$-0.743001\pi$
$487$	−30.5152	−1.38278	−0.691388	+	0.722483i	$0.743001\pi$
$488$	0.960099	0.0434616
$489$	0	0
$490$	9.50143	0.429231
$491$	11.5942	0.523241	0.261620	−	0.965171i	$-0.415743\pi$
$491$	11.5942	0.523241	0.261620	+	0.965171i	$0.415743\pi$
$492$	0	0
$493$	0	0
$494$	−60.6322	−2.72797
$495$	0	0
$496$	19.4831	0.874815
$497$	−37.5960	−1.68641
$498$	0	0
$499$	35.8728	1.60589	0.802943	−	0.596055i	$-0.203266\pi$
$499$	35.8728	1.60589	0.802943	+	0.596055i	$0.203266\pi$
$500$	15.7110	0.702618
$501$	0	0
$502$	−5.69848	−0.254336
$503$	−35.1318	−1.56645	−0.783225	−	0.621738i	$-0.786427\pi$
$503$	−35.1318	−1.56645	−0.783225	+	0.621738i	$0.786427\pi$
$504$	0	0
$505$	−3.50981	−0.156184
$506$	−6.94394	−0.308696
$507$	0	0
$508$	17.8406	0.791550
$509$	−23.7081	−1.05084	−0.525421	−	0.850842i	$-0.676092\pi$
$509$	−23.7081	−1.05084	−0.525421	+	0.850842i	$0.676092\pi$
$510$	0	0
$511$	−35.1127	−1.55329
$512$	−25.6973	−1.13567
$513$	0	0
$514$	−22.9289	−1.01135
$515$	−2.79793	−0.123292
$516$	0	0
$517$	7.01962	0.308722
$518$	−8.78101	−0.385815
$519$	0	0
$520$	15.4930	0.679415
$521$	−17.6653	−0.773932	−0.386966	−	0.922094i	$-0.626477\pi$
$521$	−17.6653	−0.773932	−0.386966	+	0.922094i	$0.626477\pi$
$522$	0	0
$523$	0.443651	0.0193995	0.00969975	−	0.999953i	$-0.496912\pi$
$523$	0.443651	0.0193995	0.00969975	+	0.999953i	$0.496912\pi$
$524$	−0.514165	−0.0224614
$525$	0	0
$526$	21.3553	0.931137
$527$	0	0
$528$	0	0
$529$	−15.1716	−0.659634
$530$	−61.9522	−2.69103
$531$	0	0
$532$	48.5989	2.10703
$533$	−2.11904	−0.0917858
$534$	0	0
$535$	−14.3137	−0.618836
$536$	−0.734828	−0.0317397
$537$	0	0
$538$	−27.9021	−1.20294
$539$	1.83783	0.0791611
$540$	0	0
$541$	−29.9581	−1.28800	−0.644000	−	0.765026i	$-0.722726\pi$
$541$	−29.9581	−1.28800	−0.644000	+	0.765026i	$0.722726\pi$
$542$	−68.6784	−2.94999
$543$	0	0
$544$	0	0
$545$	−9.82021	−0.420652
$546$	0	0
$547$	16.8925	0.722270	0.361135	−	0.932514i	$-0.382389\pi$
$547$	16.8925	0.722270	0.361135	+	0.932514i	$0.382389\pi$
$548$	2.29363	0.0979791
$549$	0	0
$550$	−7.01962	−0.299318
$551$	61.2733	2.61033
$552$	0	0
$553$	−7.27208	−0.309240
$554$	−25.2638	−1.07335
$555$	0	0
$556$	−25.5291	−1.08267
$557$	11.7464	0.497712	0.248856	−	0.968540i	$-0.419945\pi$
$557$	11.7464	0.497712	0.248856	+	0.968540i	$0.419945\pi$
$558$	0	0
$559$	35.6274	1.50688
$560$	20.3752	0.861011
$561$	0	0
$562$	−69.8995	−2.94853
$563$	−15.8771	−0.669141	−0.334571	−	0.942371i	$-0.608591\pi$
$563$	−15.8771	−0.669141	−0.334571	+	0.942371i	$0.608591\pi$
$564$	0	0
$565$	−5.92893	−0.249432
$566$	−7.23486	−0.304104
$567$	0	0
$568$	16.0952	0.675341
$569$	19.4252	0.814346	0.407173	−	0.913351i	$-0.366515\pi$
$569$	19.4252	0.814346	0.407173	+	0.913351i	$0.366515\pi$
$570$	0	0
$571$	−44.6858	−1.87004	−0.935021	−	0.354593i	$-0.884619\pi$
$571$	−44.6858	−1.87004	−0.935021	+	0.354593i	$0.884619\pi$
$572$	13.2284	0.553108
$573$	0	0
$574$	3.01219	0.125727
$575$	7.91375	0.330026
$576$	0	0
$577$	10.3137	0.429365	0.214683	−	0.976684i	$-0.431128\pi$
$577$	10.3137	0.429365	0.214683	+	0.976684i	$0.431128\pi$
$578$	0	0
$579$	0	0
$580$	−69.1127	−2.86975
$581$	−30.2972	−1.25694
$582$	0	0
$583$	−11.9832	−0.496295
$584$	15.0321	0.622034
$585$	0	0
$586$	−30.5269	−1.26106
$587$	−3.02846	−0.124998	−0.0624990	−	0.998045i	$-0.519907\pi$
$587$	−3.02846	−0.124998	−0.0624990	+	0.998045i	$0.519907\pi$
$588$	0	0
$589$	−50.2834	−2.07189
$590$	38.4922	1.58470
$591$	0	0
$592$	−3.47795	−0.142943
$593$	15.1423	0.621820	0.310910	−	0.950439i	$-0.399366\pi$
$593$	15.1423	0.621820	0.310910	+	0.950439i	$0.399366\pi$
$594$	0	0
$595$	0	0
$596$	−6.88090	−0.281853
$597$	0	0
$598$	−26.4483	−1.08155
$599$	−8.93319	−0.365000	−0.182500	−	0.983206i	$-0.558419\pi$
$599$	−8.93319	−0.365000	−0.182500	+	0.983206i	$0.558419\pi$
$600$	0	0
$601$	−3.43289	−0.140030	−0.0700152	−	0.997546i	$-0.522305\pi$
$601$	−3.43289	−0.140030	−0.0700152	+	0.997546i	$0.522305\pi$
$602$	−50.6440	−2.06410
$603$	0	0
$604$	14.0000	0.569652
$605$	27.0192	1.09849
$606$	0	0
$607$	25.1033	1.01891	0.509455	−	0.860497i	$-0.329847\pi$
$607$	25.1033	1.01891	0.509455	+	0.860497i	$0.329847\pi$
$608$	50.2294	2.03707
$609$	0	0
$610$	−4.58579	−0.185673
$611$	26.7365	1.08164
$612$	0	0
$613$	−3.68629	−0.148888	−0.0744440	−	0.997225i	$-0.523718\pi$
$613$	−3.68629	−0.148888	−0.0744440	+	0.997225i	$0.523718\pi$
$614$	8.35055	0.337001
$615$	0	0
$616$	−4.25988	−0.171636
$617$	−4.35456	−0.175308	−0.0876540	−	0.996151i	$-0.527937\pi$
$617$	−4.35456	−0.175308	−0.0876540	+	0.996151i	$0.527937\pi$
$618$	0	0
$619$	−2.48181	−0.0997524	−0.0498762	−	0.998755i	$-0.515883\pi$
$619$	−2.48181	−0.0997524	−0.0498762	+	0.998755i	$0.515883\pi$
$620$	56.7168	2.27780
$621$	0	0
$622$	−42.3671	−1.69877
$623$	32.8963	1.31796
$624$	0	0
$625$	−31.1421	−1.24569
$626$	−57.8789	−2.31330
$627$	0	0
$628$	48.6102	1.93976
$629$	0	0
$630$	0	0
$631$	−11.1421	−0.443561	−0.221781	−	0.975097i	$-0.571187\pi$
$631$	−11.1421	−0.443561	−0.221781	+	0.975097i	$0.571187\pi$
$632$	3.11326	0.123839
$633$	0	0
$634$	20.4567	0.812438
$635$	−19.3043	−0.766069
$636$	0	0
$637$	7.00000	0.277350
$638$	−23.7081	−0.938612
$639$	0	0
$640$	−26.8741	−1.06229
$641$	1.15894	0.0457754	0.0228877	−	0.999738i	$-0.492714\pi$
$641$	1.15894	0.0457754	0.0228877	+	0.999738i	$0.492714\pi$
$642$	0	0
$643$	33.0740	1.30431	0.652155	−	0.758086i	$-0.273865\pi$
$643$	33.0740	1.30431	0.652155	+	0.758086i	$0.273865\pi$
$644$	21.1992	0.835366
$645$	0	0
$646$	0	0
$647$	−20.5274	−0.807017	−0.403508	−	0.914976i	$-0.632209\pi$
$647$	−20.5274	−0.807017	−0.403508	+	0.914976i	$0.632209\pi$
$648$	0	0
$649$	7.44543	0.292259
$650$	−26.7365	−1.04869
$651$	0	0
$652$	21.9105	0.858080
$653$	−12.7484	−0.498882	−0.249441	−	0.968390i	$-0.580247\pi$
$653$	−12.7484	−0.498882	−0.249441	+	0.968390i	$0.580247\pi$
$654$	0	0
$655$	0.556349	0.0217384
$656$	1.19306	0.0465811
$657$	0	0
$658$	−38.0057	−1.48162
$659$	41.2071	1.60520	0.802599	−	0.596518i	$-0.203450\pi$
$659$	41.2071	1.60520	0.802599	+	0.596518i	$0.203450\pi$
$660$	0	0
$661$	−26.2132	−1.01958	−0.509788	−	0.860300i	$-0.670276\pi$
$661$	−26.2132	−1.01958	−0.509788	+	0.860300i	$0.670276\pi$
$662$	−58.1234	−2.25903
$663$	0	0
$664$	12.9706	0.503355
$665$	−52.5861	−2.03920
$666$	0	0
$667$	26.7279	1.03491
$668$	−14.9839	−0.579744
$669$	0	0
$670$	3.50981	0.135596
$671$	−0.887016	−0.0342429
$672$	0	0
$673$	44.0836	1.69930	0.849649	−	0.527349i	$-0.176814\pi$
$673$	44.0836	1.69930	0.849649	+	0.527349i	$0.176814\pi$
$674$	20.3468	0.783729
$675$	0	0
$676$	16.7696	0.644983
$677$	−29.8172	−1.14597	−0.572983	−	0.819567i	$-0.694214\pi$
$677$	−29.8172	−1.14597	−0.572983	+	0.819567i	$0.694214\pi$
$678$	0	0
$679$	18.4853	0.709400
$680$	0	0
$681$	0	0
$682$	19.4558	0.745003
$683$	−33.0951	−1.26635	−0.633175	−	0.774008i	$-0.718249\pi$
$683$	−33.0951	−1.26635	−0.633175	+	0.774008i	$0.718249\pi$
$684$	0	0
$685$	−2.48181	−0.0948251
$686$	33.9729	1.29709
$687$	0	0
$688$	−20.0589	−0.764737
$689$	−45.6421	−1.73883
$690$	0	0
$691$	9.63274	0.366447	0.183223	−	0.983071i	$-0.441347\pi$
$691$	9.63274	0.366447	0.183223	+	0.983071i	$0.441347\pi$
$692$	45.6788	1.73645
$693$	0	0
$694$	63.4260	2.40762
$695$	27.6236	1.04782
$696$	0	0
$697$	0	0
$698$	−33.6805	−1.27482
$699$	0	0
$700$	21.4303	0.809988
$701$	−23.7081	−0.895442	−0.447721	−	0.894173i	$-0.647764\pi$
$701$	−23.7081	−0.895442	−0.447721	+	0.894173i	$0.647764\pi$
$702$	0	0
$703$	8.97616	0.338542
$704$	−13.6743	−0.515371
$705$	0	0
$706$	−37.0122	−1.39297
$707$	3.67567	0.138238
$708$	0	0
$709$	29.6411	1.11319	0.556597	−	0.830783i	$-0.312107\pi$
$709$	29.6411	1.11319	0.556597	+	0.830783i	$0.312107\pi$
$710$	−76.8768	−2.88513
$711$	0	0
$712$	−14.0833	−0.527793
$713$	−21.9341	−0.821437
$714$	0	0
$715$	−14.3137	−0.535302
$716$	−46.9857	−1.75594
$717$	0	0
$718$	−24.0416	−0.897226
$719$	3.75803	0.140151	0.0700755	−	0.997542i	$-0.477676\pi$
$719$	3.75803	0.140151	0.0700755	+	0.997542i	$0.477676\pi$
$720$	0	0
$721$	2.93015	0.109125
$722$	−47.4162	−1.76465
$723$	0	0
$724$	−22.2500	−0.826916
$725$	27.0192	1.00347
$726$	0	0
$727$	3.85786	0.143080	0.0715401	−	0.997438i	$-0.477209\pi$
$727$	3.85786	0.143080	0.0715401	+	0.997438i	$0.477209\pi$
$728$	−16.2252	−0.601345
$729$	0	0
$730$	−71.7990	−2.65740
$731$	0	0
$732$	0	0
$733$	−20.8701	−0.770853	−0.385427	−	0.922739i	$-0.625946\pi$
$733$	−20.8701	−0.770853	−0.385427	+	0.922739i	$0.625946\pi$
$734$	−1.63899	−0.0604963
$735$	0	0
$736$	21.9105	0.807631
$737$	0.678892	0.0250073
$738$	0	0
$739$	−6.89949	−0.253802	−0.126901	−	0.991915i	$-0.540503\pi$
$739$	−6.89949	−0.253802	−0.126901	+	0.991915i	$0.540503\pi$
$740$	−10.1246	−0.372187
$741$	0	0
$742$	64.8798	2.38181
$743$	1.35778	0.0498123	0.0249061	−	0.999690i	$-0.492071\pi$
$743$	1.35778	0.0498123	0.0249061	+	0.999690i	$0.492071\pi$
$744$	0	0
$745$	7.44543	0.272779
$746$	68.6154	2.51219
$747$	0	0
$748$	0	0
$749$	14.9901	0.547727
$750$	0	0
$751$	−4.59220	−0.167572	−0.0837859	−	0.996484i	$-0.526701\pi$
$751$	−4.59220	−0.167572	−0.0837859	+	0.996484i	$0.526701\pi$
$752$	−15.0532	−0.548932
$753$	0	0
$754$	−90.3001	−3.28854
$755$	−15.1486	−0.551314
$756$	0	0
$757$	−9.97056	−0.362386	−0.181193	−	0.983448i	$-0.557996\pi$
$757$	−9.97056	−0.362386	−0.181193	+	0.983448i	$0.557996\pi$
$758$	41.2077	1.49673
$759$	0	0
$760$	22.5127	0.816620
$761$	−19.9448	−0.722998	−0.361499	−	0.932372i	$-0.617735\pi$
$761$	−19.9448	−0.722998	−0.361499	+	0.932372i	$0.617735\pi$
$762$	0	0
$763$	10.2843	0.372316
$764$	59.0104	2.13492
$765$	0	0
$766$	31.3137	1.13141
$767$	28.3584	1.02396
$768$	0	0
$769$	−7.24264	−0.261176	−0.130588	−	0.991437i	$-0.541687\pi$
$769$	−7.24264	−0.261176	−0.130588	+	0.991437i	$0.541687\pi$
$770$	20.3468	0.733247
$771$	0	0
$772$	16.7930	0.604394
$773$	−46.8966	−1.68675	−0.843376	−	0.537324i	$-0.819435\pi$
$773$	−46.8966	−1.68675	−0.843376	+	0.537324i	$0.819435\pi$
$774$	0	0
$775$	−22.1731	−0.796482
$776$	−7.91375	−0.284087
$777$	0	0
$778$	−4.58579	−0.164408
$779$	−3.07914	−0.110322
$780$	0	0
$781$	−14.8701	−0.532092
$782$	0	0
$783$	0	0
$784$	−3.94113	−0.140754
$785$	−52.5983	−1.87731
$786$	0	0
$787$	2.72191	0.0970257	0.0485128	−	0.998823i	$-0.484552\pi$
$787$	2.72191	0.0970257	0.0485128	+	0.998823i	$0.484552\pi$
$788$	−48.1614	−1.71568
$789$	0	0
$790$	−14.8701	−0.529053
$791$	6.20911	0.220770
$792$	0	0
$793$	−3.37849	−0.119974
$794$	43.8068	1.55465
$795$	0	0
$796$	−34.7453	−1.23152
$797$	−9.30061	−0.329444	−0.164722	−	0.986340i	$-0.552673\pi$
$797$	−9.30061	−0.329444	−0.164722	+	0.986340i	$0.552673\pi$
$798$	0	0
$799$	0	0
$800$	22.1493	0.783095
$801$	0	0
$802$	−44.8490	−1.58367
$803$	−13.8879	−0.490092
$804$	0	0
$805$	−22.9385	−0.808475
$806$	74.1040	2.61020
$807$	0	0
$808$	−1.57359	−0.0553588
$809$	−11.3906	−0.400471	−0.200236	−	0.979748i	$-0.564171\pi$
$809$	−11.3906	−0.400471	−0.200236	+	0.979748i	$0.564171\pi$
$810$	0	0
$811$	30.2751	1.06310	0.531552	−	0.847026i	$-0.321609\pi$
$811$	30.2751	1.06310	0.531552	+	0.847026i	$0.321609\pi$
$812$	72.3787	2.53999
$813$	0	0
$814$	−3.47309	−0.121732
$815$	−23.7081	−0.830458
$816$	0	0
$817$	51.7696	1.81119
$818$	5.99388	0.209571
$819$	0	0
$820$	3.47309	0.121285
$821$	−50.2804	−1.75480	−0.877399	−	0.479761i	$-0.840724\pi$
$821$	−50.2804	−1.75480	−0.877399	+	0.479761i	$0.840724\pi$
$822$	0	0
$823$	45.2973	1.57896	0.789482	−	0.613773i	$-0.210349\pi$
$823$	45.2973	1.57896	0.789482	+	0.613773i	$0.210349\pi$
$824$	−1.25443	−0.0437001
$825$	0	0
$826$	−40.3111	−1.40260
$827$	7.71491	0.268274	0.134137	−	0.990963i	$-0.457174\pi$
$827$	7.71491	0.268274	0.134137	+	0.990963i	$0.457174\pi$
$828$	0	0
$829$	12.2426	0.425204	0.212602	−	0.977139i	$-0.431806\pi$
$829$	12.2426	0.425204	0.212602	+	0.977139i	$0.431806\pi$
$830$	−61.9522	−2.15039
$831$	0	0
$832$	−52.0833	−1.80566
$833$	0	0
$834$	0	0
$835$	16.2132	0.561081
$836$	19.2220	0.664805
$837$	0	0
$838$	−62.2216	−2.14941
$839$	4.43692	0.153180	0.0765898	−	0.997063i	$-0.475597\pi$
$839$	4.43692	0.153180	0.0765898	+	0.997063i	$0.475597\pi$
$840$	0	0
$841$	62.2548	2.14672
$842$	−33.8957	−1.16812
$843$	0	0
$844$	−23.0698	−0.794095
$845$	−18.1454	−0.624220
$846$	0	0
$847$	−28.2960	−0.972264
$848$	25.6973	0.882450
$849$	0	0
$850$	0	0
$851$	3.91548	0.134221
$852$	0	0
$853$	14.0936	0.482557	0.241278	−	0.970456i	$-0.422433\pi$
$853$	14.0936	0.482557	0.241278	+	0.970456i	$0.422433\pi$
$854$	4.80249	0.164338
$855$	0	0
$856$	−6.41743	−0.219343
$857$	−18.4266	−0.629440	−0.314720	−	0.949185i	$-0.601911\pi$
$857$	−18.4266	−0.629440	−0.314720	+	0.949185i	$0.601911\pi$
$858$	0	0
$859$	−12.7279	−0.434271	−0.217136	−	0.976141i	$-0.569671\pi$
$859$	−12.7279	−0.434271	−0.217136	+	0.976141i	$0.569671\pi$
$860$	−58.3930	−1.99119
$861$	0	0
$862$	−6.41743	−0.218579
$863$	−27.1040	−0.922629	−0.461315	−	0.887237i	$-0.652622\pi$
$863$	−27.1040	−0.922629	−0.461315	+	0.887237i	$0.652622\pi$
$864$	0	0
$865$	−49.4264	−1.68055
$866$	2.57190	0.0873966
$867$	0	0
$868$	−59.3970	−2.01606
$869$	−2.87627	−0.0975709
$870$	0	0
$871$	2.58579	0.0876160
$872$	−4.40281	−0.149098
$873$	0	0
$874$	−38.4315	−1.29997
$875$	17.8033	0.601863
$876$	0	0
$877$	41.9413	1.41626	0.708129	−	0.706083i	$-0.249539\pi$
$877$	41.9413	1.41626	0.708129	+	0.706083i	$0.249539\pi$
$878$	−42.4490	−1.43259
$879$	0	0
$880$	8.05887	0.271665
$881$	16.7876	0.565588	0.282794	−	0.959181i	$-0.408739\pi$
$881$	16.7876	0.565588	0.282794	+	0.959181i	$0.408739\pi$
$882$	0	0
$883$	25.0000	0.841317	0.420658	−	0.907219i	$-0.361799\pi$
$883$	25.0000	0.841317	0.420658	+	0.907219i	$0.361799\pi$
$884$	0	0
$885$	0	0
$886$	25.9411	0.871509
$887$	7.31722	0.245688	0.122844	−	0.992426i	$-0.460798\pi$
$887$	7.31722	0.245688	0.122844	+	0.992426i	$0.460798\pi$
$888$	0	0
$889$	20.2166	0.678042
$890$	67.2669	2.25479
$891$	0	0
$892$	−15.6985	−0.525624
$893$	38.8504	1.30008
$894$	0	0
$895$	50.8406	1.69941
$896$	28.1441	0.940227
$897$	0	0
$898$	9.92724	0.331276
$899$	−74.8875	−2.49764
$900$	0	0
$901$	0	0
$902$	1.19139	0.0396690
$903$	0	0
$904$	−2.65819	−0.0884100
$905$	24.0755	0.800297
$906$	0	0
$907$	50.3378	1.67144	0.835720	−	0.549155i	$-0.185051\pi$
$907$	50.3378	1.67144	0.835720	+	0.549155i	$0.185051\pi$
$908$	−54.1549	−1.79719
$909$	0	0
$910$	77.4975	2.56902
$911$	28.1782	0.933584	0.466792	−	0.884367i	$-0.345410\pi$
$911$	28.1782	0.933584	0.466792	+	0.884367i	$0.345410\pi$
$912$	0	0
$913$	−11.9832	−0.396587
$914$	31.1716	1.03107
$915$	0	0
$916$	37.7157	1.24616
$917$	−0.582640	−0.0192405
$918$	0	0
$919$	43.8284	1.44577	0.722883	−	0.690970i	$-0.242816\pi$
$919$	43.8284	1.44577	0.722883	+	0.690970i	$0.242816\pi$
$920$	9.82021	0.323763
$921$	0	0
$922$	−30.5269	−1.00535
$923$	−56.6375	−1.86425
$924$	0	0
$925$	3.95815	0.130143
$926$	61.0888	2.00750
$927$	0	0
$928$	74.8070	2.45566
$929$	−20.8268	−0.683306	−0.341653	−	0.939826i	$-0.610987\pi$
$929$	−20.8268	−0.683306	−0.341653	+	0.939826i	$0.610987\pi$
$930$	0	0
$931$	10.1716	0.333360
$932$	34.4188	1.12743
$933$	0	0
$934$	32.1005	1.05036
$935$	0	0
$936$	0	0
$937$	−4.28427	−0.139961	−0.0699805	−	0.997548i	$-0.522294\pi$
$937$	−4.28427	−0.139961	−0.0699805	+	0.997548i	$0.522294\pi$
$938$	−3.67567	−0.120015
$939$	0	0
$940$	−43.8210	−1.42928
$941$	26.7380	0.871635	0.435817	−	0.900035i	$-0.356459\pi$
$941$	26.7380	0.871635	0.435817	+	0.900035i	$0.356459\pi$
$942$	0	0
$943$	−1.34315	−0.0437388
$944$	−15.9663	−0.519658
$945$	0	0
$946$	−20.0309	−0.651259
$947$	−47.7637	−1.55211	−0.776056	−	0.630664i	$-0.782783\pi$
$947$	−47.7637	−1.55211	−0.776056	+	0.630664i	$0.782783\pi$
$948$	0	0
$949$	−52.8966	−1.71710
$950$	−38.8504	−1.26047
$951$	0	0
$952$	0	0
$953$	−23.7081	−0.767980	−0.383990	−	0.923337i	$-0.625450\pi$
$953$	−23.7081	−0.767980	−0.383990	+	0.923337i	$0.625450\pi$
$954$	0	0
$955$	−63.8518	−2.06620
$956$	59.9605	1.93926
$957$	0	0
$958$	−7.44543	−0.240551
$959$	2.59909	0.0839290
$960$	0	0
$961$	30.4558	0.982447
$962$	−13.2284	−0.426501
$963$	0	0
$964$	−24.5687	−0.791303
$965$	−18.1708	−0.584938
$966$	0	0
$967$	−5.38478	−0.173163	−0.0865814	−	0.996245i	$-0.527594\pi$
$967$	−5.38478	−0.173163	−0.0865814	+	0.996245i	$0.527594\pi$
$968$	12.1138	0.389354
$969$	0	0
$970$	37.7990	1.21365
$971$	−59.0104	−1.89373	−0.946867	−	0.321625i	$-0.895771\pi$
$971$	−59.0104	−1.89373	−0.946867	+	0.321625i	$0.895771\pi$
$972$	0	0
$973$	−28.9289	−0.927419
$974$	65.3467	2.09384
$975$	0	0
$976$	1.90215	0.0608864
$977$	−43.5007	−1.39171	−0.695855	−	0.718182i	$-0.744974\pi$
$977$	−43.5007	−1.39171	−0.695855	+	0.718182i	$0.744974\pi$
$978$	0	0
$979$	13.0112	0.415841
$980$	−11.4729	−0.366490
$981$	0	0
$982$	−24.8284	−0.792307
$983$	41.9690	1.33860	0.669301	−	0.742991i	$-0.266593\pi$
$983$	41.9690	1.33860	0.669301	+	0.742991i	$0.266593\pi$
$984$	0	0
$985$	52.1127	1.66045
$986$	0	0
$987$	0	0
$988$	73.2132	2.32922
$989$	22.5823	0.718075
$990$	0	0
$991$	57.8602	1.83799	0.918995	−	0.394270i	$-0.129002\pi$
$991$	57.8602	1.83799	0.918995	+	0.394270i	$0.129002\pi$
$992$	−61.3898	−1.94913
$993$	0	0
$994$	80.5097	2.55361
$995$	37.5960	1.19187
$996$	0	0
$997$	−34.3646	−1.08834	−0.544169	−	0.838976i	$-0.683155\pi$
$997$	−34.3646	−1.08834	−0.544169	+	0.838976i	$0.683155\pi$
$998$	−76.8196	−2.43168
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bl.1.3		8
3.2	odd	2	inner	2601.2.a.bl.1.6		8
17.10	odd	16		153.2.l.d.100.2	yes	8
17.12	odd	16		153.2.l.d.127.2	yes	8
17.16	even	2	inner	2601.2.a.bl.1.4		8
51.29	even	16		153.2.l.d.127.1	yes	8
51.44	even	16		153.2.l.d.100.1	✓	8
51.50	odd	2	inner	2601.2.a.bl.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.2.l.d.100.1	✓	8	51.44	even	16
153.2.l.d.100.2	yes	8	17.10	odd	16
153.2.l.d.127.1	yes	8	51.29	even	16
153.2.l.d.127.2	yes	8	17.12	odd	16
2601.2.a.bl.1.3		8	1.1	even	1	trivial
2601.2.a.bl.1.4		8	17.16	even	2	inner
2601.2.a.bl.1.5		8	51.50	odd	2	inner
2601.2.a.bl.1.6		8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14144 q^{2} +2.58579 q^{4} -2.79793 q^{5} +2.93015 q^{7} -1.25443 q^{8} +O(q^{10})\)	\(q-2.14144 q^{2} +2.58579 q^{4} -2.79793 q^{5} +2.93015 q^{7} -1.25443 q^{8} +5.99162 q^{10} +1.15894 q^{11} +4.41421 q^{13} -6.27476 q^{14} -2.48528 q^{16} +6.41421 q^{19} -7.23486 q^{20} -2.48181 q^{22} +2.79793 q^{23} +2.82843 q^{25} -9.45280 q^{26} +7.57675 q^{28} +9.55274 q^{29} -7.83938 q^{31} +7.83095 q^{32} -8.19837 q^{35} +1.39942 q^{37} -13.7357 q^{38} +3.50981 q^{40} -0.480049 q^{41} +8.07107 q^{43} +2.99678 q^{44} -5.99162 q^{46} +6.05692 q^{47} +1.58579 q^{49} -6.05692 q^{50} +11.4142 q^{52} -10.3398 q^{53} -3.24264 q^{55} -3.67567 q^{56} -20.4567 q^{58} +6.42433 q^{59} -0.765367 q^{61} +16.7876 q^{62} -11.7990 q^{64} -12.3507 q^{65} +0.585786 q^{67} +17.5563 q^{70} -12.8307 q^{71} -11.9832 q^{73} -2.99678 q^{74} +16.5858 q^{76} +3.39587 q^{77} -2.48181 q^{79} +6.95365 q^{80} +1.02800 q^{82} -10.3398 q^{83} -17.2837 q^{86} -1.45381 q^{88} +11.2268 q^{89} +12.9343 q^{91} +7.23486 q^{92} -12.9706 q^{94} -17.9465 q^{95} +6.30864 q^{97} -3.39587 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{4}+O(q^{10}) \) 8 * q + 32 * q^4	\( 8 q + 32 q^{4} + 24 q^{13} + 48 q^{16} + 40 q^{19} + 8 q^{43} + 24 q^{49} + 80 q^{52} + 8 q^{55} + 64 q^{64} + 16 q^{67} + 16 q^{70} + 144 q^{76} + 32 q^{94}+O(q^{100}) \) 8 * q + 32 * q^4 + 24 * q^13 + 48 * q^16 + 40 * q^19 + 8 * q^43 + 24 * q^49 + 80 * q^52 + 8 * q^55 + 64 * q^64 + 16 * q^67 + 16 * q^70 + 144 * q^76 + 32 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more