LMFDB - Embedded newform 7623.2.a.cv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.6988960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1 $ x^8 - 9x^6 + 22x^4 - 11*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 693)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.25947$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.25947 q^{2} +3.10522 q^{4} +1.54336 q^{5} +1.00000 q^{7} -2.49721 q^{8} +O(q^{10})$	$q-2.25947 q^{2} +3.10522 q^{4} +1.54336 q^{5} +1.00000 q^{7} -2.49721 q^{8} -3.48718 q^{10} -2.53716 q^{13} -2.25947 q^{14} -0.568054 q^{16} +6.59570 q^{17} +7.56151 q^{19} +4.79248 q^{20} -5.61973 q^{23} -2.61803 q^{25} +5.73266 q^{26} +3.10522 q^{28} +5.45068 q^{29} -5.32545 q^{31} +6.27793 q^{32} -14.9028 q^{34} +1.54336 q^{35} -6.18609 q^{37} -17.0850 q^{38} -3.85410 q^{40} -9.78690 q^{41} -9.61149 q^{43} +12.6976 q^{46} -3.84418 q^{47} +1.00000 q^{49} +5.91538 q^{50} -7.87845 q^{52} -0.531600 q^{53} -2.49721 q^{56} -12.3157 q^{58} +1.76077 q^{59} +4.74760 q^{61} +12.0327 q^{62} -13.0487 q^{64} -3.91576 q^{65} -9.60824 q^{67} +20.4811 q^{68} -3.48718 q^{70} +9.94228 q^{71} -9.71346 q^{73} +13.9773 q^{74} +23.4802 q^{76} -2.07029 q^{79} -0.876713 q^{80} +22.1132 q^{82} -0.0442404 q^{83} +10.1795 q^{85} +21.7169 q^{86} -6.38297 q^{89} -2.53716 q^{91} -17.4505 q^{92} +8.68582 q^{94} +11.6702 q^{95} -13.2498 q^{97} -2.25947 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{4} + 8 q^{7}+O(q^{10}) $ 8 * q + 2 * q^4 + 8 * q^7	$ 8 q + 2 q^{4} + 8 q^{7} - 14 q^{10} - 2 q^{16} + 6 q^{19} - 12 q^{25} + 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} - 4 q^{40} - 16 q^{43} + 42 q^{46} + 8 q^{49} - 2 q^{52} - 30 q^{58} - 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} - 14 q^{73} + 34 q^{76} - 22 q^{79} + 36 q^{82} + 18 q^{85} - 32 q^{94} - 48 q^{97}+O(q^{100}) $ 8 * q + 2 * q^4 + 8 * q^7 - 14 * q^10 - 2 * q^16 + 6 * q^19 - 12 * q^25 + 2 * q^28 - 6 * q^31 - 24 * q^34 - 38 * q^37 - 4 * q^40 - 16 * q^43 + 42 * q^46 + 8 * q^49 - 2 * q^52 - 30 * q^58 - 28 * q^61 - 36 * q^64 - 36 * q^67 - 14 * q^70 - 14 * q^73 + 34 * q^76 - 22 * q^79 + 36 * q^82 + 18 * q^85 - 32 * q^94 - 48 * q^97

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 $

$2$	−2.25947	−1.59769	−0.798844	−	0.601538i	$-0.794555\pi$
$2$	−2.25947	−1.59769	−0.798844	+	0.601538i	$0.794555\pi$
$3$	0	0
$3$	0	0
$4$	3.10522	1.55261
$5$	1.54336	0.690212	0.345106	−	0.938564i	$-0.387843\pi$
$5$	1.54336	0.690212	0.345106	+	0.938564i	$0.387843\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.49721	−0.882898
$9$	0	0
$10$	−3.48718	−1.10274
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.53716	−0.703683	−0.351841	−	0.936060i	$-0.614444\pi$
$13$	−2.53716	−0.703683	−0.351841	+	0.936060i	$0.614444\pi$
$14$	−2.25947	−0.603870
$15$	0	0
$16$	−0.568054	−0.142014
$17$	6.59570	1.59969	0.799846	−	0.600206i	$-0.204915\pi$
$17$	6.59570	1.59969	0.799846	+	0.600206i	$0.204915\pi$
$18$	0	0
$19$	7.56151	1.73473	0.867365	−	0.497672i	$-0.165812\pi$
$19$	7.56151	1.73473	0.867365	+	0.497672i	$0.165812\pi$
$20$	4.79248	1.07163
$21$	0	0
$22$	0	0
$23$	−5.61973	−1.17179	−0.585897	−	0.810385i	$-0.699258\pi$
$23$	−5.61973	−1.17179	−0.585897	+	0.810385i	$0.699258\pi$
$24$	0	0
$25$	−2.61803	−0.523607
$26$	5.73266	1.12427
$27$	0	0
$28$	3.10522	0.586831
$29$	5.45068	1.01217	0.506083	−	0.862485i	$-0.331093\pi$
$29$	5.45068	1.01217	0.506083	+	0.862485i	$0.331093\pi$
$30$	0	0
$31$	−5.32545	−0.956478	−0.478239	−	0.878230i	$-0.658725\pi$
$31$	−5.32545	−0.956478	−0.478239	+	0.878230i	$0.658725\pi$
$32$	6.27793	1.10979
$33$	0	0
$34$	−14.9028	−2.55581
$35$	1.54336	0.260876
$36$	0	0
$37$	−6.18609	−1.01699	−0.508493	−	0.861066i	$-0.669797\pi$
$37$	−6.18609	−1.01699	−0.508493	+	0.861066i	$0.669797\pi$
$38$	−17.0850	−2.77156
$39$	0	0
$40$	−3.85410	−0.609387
$41$	−9.78690	−1.52846	−0.764229	−	0.644946i	$-0.776880\pi$
$41$	−9.78690	−1.52846	−0.764229	+	0.644946i	$0.776880\pi$
$42$	0	0
$43$	−9.61149	−1.46574	−0.732870	−	0.680369i	$-0.761819\pi$
$43$	−9.61149	−1.46574	−0.732870	+	0.680369i	$0.761819\pi$
$44$	0	0
$45$	0	0
$46$	12.6976	1.87216
$47$	−3.84418	−0.560731	−0.280366	−	0.959893i	$-0.590456\pi$
$47$	−3.84418	−0.560731	−0.280366	+	0.959893i	$0.590456\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.91538	0.836561
$51$	0	0
$52$	−7.87845	−1.09254
$53$	−0.531600	−0.0730209	−0.0365105	−	0.999333i	$-0.511624\pi$
$53$	−0.531600	−0.0730209	−0.0365105	+	0.999333i	$0.511624\pi$
$54$	0	0
$55$	0	0
$56$	−2.49721	−0.333704
$57$	0	0
$58$	−12.3157	−1.61712
$59$	1.76077	0.229233	0.114616	−	0.993410i	$-0.463436\pi$
$59$	1.76077	0.229233	0.114616	+	0.993410i	$0.463436\pi$
$60$	0	0
$61$	4.74760	0.607868	0.303934	−	0.952693i	$-0.401700\pi$
$61$	4.74760	0.607868	0.303934	+	0.952693i	$0.401700\pi$
$62$	12.0327	1.52815
$63$	0	0
$64$	−13.0487	−1.63109
$65$	−3.91576	−0.485691
$66$	0	0
$67$	−9.60824	−1.17383	−0.586917	−	0.809647i	$-0.699658\pi$
$67$	−9.60824	−1.17383	−0.586917	+	0.809647i	$0.699658\pi$
$68$	20.4811	2.48370
$69$	0	0
$70$	−3.48718	−0.416798
$71$	9.94228	1.17993	0.589966	−	0.807428i	$-0.299141\pi$
$71$	9.94228	1.17993	0.589966	+	0.807428i	$0.299141\pi$
$72$	0	0
$73$	−9.71346	−1.13687	−0.568437	−	0.822727i	$-0.692452\pi$
$73$	−9.71346	−1.13687	−0.568437	+	0.822727i	$0.692452\pi$
$74$	13.9773	1.62483
$75$	0	0
$76$	23.4802	2.69336
$77$	0	0
$78$	0	0
$79$	−2.07029	−0.232926	−0.116463	−	0.993195i	$-0.537156\pi$
$79$	−2.07029	−0.232926	−0.116463	+	0.993195i	$0.537156\pi$
$80$	−0.876713	−0.0980195
$81$	0	0
$82$	22.1132	2.44200
$83$	−0.0442404	−0.00485601	−0.00242801	−	0.999997i	$-0.500773\pi$
$83$	−0.0442404	−0.00485601	−0.00242801	+	0.999997i	$0.500773\pi$
$84$	0	0
$85$	10.1795	1.10413
$86$	21.7169	2.34179
$87$	0	0
$88$	0	0
$89$	−6.38297	−0.676594	−0.338297	−	0.941039i	$-0.609851\pi$
$89$	−6.38297	−0.676594	−0.338297	+	0.941039i	$0.609851\pi$
$90$	0	0
$91$	−2.53716	−0.265967
$92$	−17.4505	−1.81934
$93$	0	0
$94$	8.68582	0.895874
$95$	11.6702	1.19733
$96$	0	0
$97$	−13.2498	−1.34532	−0.672658	−	0.739953i	$-0.734848\pi$
$97$	−13.2498	−1.34532	−0.672658	+	0.739953i	$0.734848\pi$
$98$	−2.25947	−0.228241
$99$	0	0
$100$	−8.12957	−0.812957
$101$	5.53071	0.550326	0.275163	−	0.961398i	$-0.411268\pi$
$101$	5.53071	0.550326	0.275163	+	0.961398i	$0.411268\pi$
$102$	0	0
$103$	6.70087	0.660257	0.330128	−	0.943936i	$-0.392908\pi$
$103$	6.70087	0.660257	0.330128	+	0.943936i	$0.392908\pi$
$104$	6.33584	0.621280
$105$	0	0
$106$	1.20114	0.116665
$107$	−5.75478	−0.556335	−0.278168	−	0.960533i	$-0.589727\pi$
$107$	−5.75478	−0.556335	−0.278168	+	0.960533i	$0.589727\pi$
$108$	0	0
$109$	−13.7061	−1.31281	−0.656405	−	0.754409i	$-0.727924\pi$
$109$	−13.7061	−1.31281	−0.656405	+	0.754409i	$0.727924\pi$
$110$	0	0
$111$	0	0
$112$	−0.568054	−0.0536761
$113$	14.6044	1.37387	0.686933	−	0.726721i	$-0.258957\pi$
$113$	14.6044	1.37387	0.686933	+	0.726721i	$0.258957\pi$
$114$	0	0
$115$	−8.67327	−0.808787
$116$	16.9255	1.57150
$117$	0	0
$118$	−3.97841	−0.366242
$119$	6.59570	0.604627
$120$	0	0
$121$	0	0
$122$	−10.7271	−0.971184
$123$	0	0
$124$	−16.5367	−1.48504
$125$	−11.7574	−1.05161
$126$	0	0
$127$	−13.5063	−1.19849	−0.599244	−	0.800566i	$-0.704532\pi$
$127$	−13.5063	−1.19849	−0.599244	+	0.800566i	$0.704532\pi$
$128$	16.9273	1.49618
$129$	0	0
$130$	8.84756	0.775983
$131$	−4.30276	−0.375934	−0.187967	−	0.982175i	$-0.560190\pi$
$131$	−4.30276	−0.375934	−0.187967	+	0.982175i	$0.560190\pi$
$132$	0	0
$133$	7.56151	0.655666
$134$	21.7096	1.87542
$135$	0	0
$136$	−16.4709	−1.41236
$137$	−12.5285	−1.07038	−0.535191	−	0.844731i	$-0.679761\pi$
$137$	−12.5285	−1.07038	−0.535191	+	0.844731i	$0.679761\pi$
$138$	0	0
$139$	−11.2263	−0.952200	−0.476100	−	0.879391i	$-0.657950\pi$
$139$	−11.2263	−0.952200	−0.476100	+	0.879391i	$0.657950\pi$
$140$	4.79248	0.405038
$141$	0	0
$142$	−22.4643	−1.88516
$143$	0	0
$144$	0	0
$145$	8.41237	0.698609
$146$	21.9473	1.81637
$147$	0	0
$148$	−19.2092	−1.57898
$149$	−12.1151	−0.992505	−0.496252	−	0.868178i	$-0.665291\pi$
$149$	−12.1151	−0.992505	−0.496252	+	0.868178i	$0.665291\pi$
$150$	0	0
$151$	−10.7878	−0.877898	−0.438949	−	0.898512i	$-0.644649\pi$
$151$	−10.7878	−0.877898	−0.438949	+	0.898512i	$0.644649\pi$
$152$	−18.8827	−1.53159
$153$	0	0
$154$	0	0
$155$	−8.21909	−0.660173
$156$	0	0
$157$	−3.70295	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$157$	−3.70295	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$158$	4.67776	0.372143
$159$	0	0
$160$	9.68911	0.765992
$161$	−5.61973	−0.442897
$162$	0	0
$163$	22.5447	1.76584	0.882918	−	0.469526i	$-0.155575\pi$
$163$	22.5447	1.76584	0.882918	+	0.469526i	$0.155575\pi$
$164$	−30.3905	−2.37310
$165$	0	0
$166$	0.0999599	0.00775839
$167$	−14.2177	−1.10020	−0.550099	−	0.835099i	$-0.685410\pi$
$167$	−14.2177	−1.10020	−0.550099	+	0.835099i	$0.685410\pi$
$168$	0	0
$169$	−6.56280	−0.504830
$170$	−23.0004	−1.76405
$171$	0	0
$172$	−29.8458	−2.27572
$173$	−18.8789	−1.43534	−0.717670	−	0.696384i	$-0.754791\pi$
$173$	−18.8789	−1.43534	−0.717670	+	0.696384i	$0.754791\pi$
$174$	0	0
$175$	−2.61803	−0.197905
$176$	0	0
$177$	0	0
$178$	14.4222	1.08099
$179$	−4.39825	−0.328741	−0.164370	−	0.986399i	$-0.552559\pi$
$179$	−4.39825	−0.328741	−0.164370	+	0.986399i	$0.552559\pi$
$180$	0	0
$181$	4.87924	0.362671	0.181336	−	0.983421i	$-0.441958\pi$
$181$	4.87924	0.362671	0.181336	+	0.983421i	$0.441958\pi$
$182$	5.73266	0.424933
$183$	0	0
$184$	14.0337	1.03457
$185$	−9.54737	−0.701937
$186$	0	0
$187$	0	0
$188$	−11.9370	−0.870597
$189$	0	0
$190$	−26.3684	−1.91296
$191$	15.8326	1.14561	0.572805	−	0.819692i	$-0.305855\pi$
$191$	15.8326	1.14561	0.572805	+	0.819692i	$0.305855\pi$
$192$	0	0
$193$	8.33475	0.599948	0.299974	−	0.953947i	$-0.403022\pi$
$193$	8.33475	0.599948	0.299974	+	0.953947i	$0.403022\pi$
$194$	29.9376	2.14940
$195$	0	0
$196$	3.10522	0.221801
$197$	3.63957	0.259309	0.129654	−	0.991559i	$-0.458613\pi$
$197$	3.63957	0.259309	0.129654	+	0.991559i	$0.458613\pi$
$198$	0	0
$199$	−4.58189	−0.324801	−0.162401	−	0.986725i	$-0.551924\pi$
$199$	−4.58189	−0.324801	−0.162401	+	0.986725i	$0.551924\pi$
$200$	6.53779	0.462291
$201$	0	0
$202$	−12.4965	−0.879250
$203$	5.45068	0.382562
$204$	0	0
$205$	−15.1047	−1.05496
$206$	−15.1404	−1.05488
$207$	0	0
$208$	1.44125	0.0999325
$209$	0	0
$210$	0	0
$211$	−1.16946	−0.0805087	−0.0402543	−	0.999189i	$-0.512817\pi$
$211$	−1.16946	−0.0805087	−0.0402543	+	0.999189i	$0.512817\pi$
$212$	−1.65074	−0.113373
$213$	0	0
$214$	13.0028	0.888850
$215$	−14.8340	−1.01167
$216$	0	0
$217$	−5.32545	−0.361515
$218$	30.9686	2.09746
$219$	0	0
$220$	0	0
$221$	−16.7344	−1.12568
$222$	0	0
$223$	−3.93300	−0.263373	−0.131687	−	0.991291i	$-0.542039\pi$
$223$	−3.93300	−0.263373	−0.131687	+	0.991291i	$0.542039\pi$
$224$	6.27793	0.419462
$225$	0	0
$226$	−32.9982	−2.19501
$227$	24.3591	1.61677	0.808384	−	0.588655i	$-0.200342\pi$
$227$	24.3591	1.61677	0.808384	+	0.588655i	$0.200342\pi$
$228$	0	0
$229$	−24.2275	−1.60100	−0.800498	−	0.599336i	$-0.795431\pi$
$229$	−24.2275	−1.60100	−0.800498	+	0.599336i	$0.795431\pi$
$230$	19.5970	1.29219
$231$	0	0
$232$	−13.6115	−0.893638
$233$	28.1961	1.84719	0.923594	−	0.383372i	$-0.125237\pi$
$233$	28.1961	1.84719	0.923594	+	0.383372i	$0.125237\pi$
$234$	0	0
$235$	−5.93296	−0.387024
$236$	5.46757	0.355909
$237$	0	0
$238$	−14.9028	−0.966005
$239$	18.6392	1.20567	0.602834	−	0.797867i	$-0.294038\pi$
$239$	18.6392	1.20567	0.602834	+	0.797867i	$0.294038\pi$
$240$	0	0
$241$	17.4814	1.12608	0.563039	−	0.826430i	$-0.309632\pi$
$241$	17.4814	1.12608	0.563039	+	0.826430i	$0.309632\pi$
$242$	0	0
$243$	0	0
$244$	14.7423	0.943782
$245$	1.54336	0.0986018
$246$	0	0
$247$	−19.1848	−1.22070
$248$	13.2988	0.844473
$249$	0	0
$250$	26.5655	1.68015
$251$	4.04114	0.255075	0.127537	−	0.991834i	$-0.459293\pi$
$251$	4.04114	0.255075	0.127537	+	0.991834i	$0.459293\pi$
$252$	0	0
$253$	0	0
$254$	30.5171	1.91481
$255$	0	0
$256$	−12.1495	−0.759341
$257$	−9.68475	−0.604118	−0.302059	−	0.953289i	$-0.597674\pi$
$257$	−9.68475	−0.604118	−0.302059	+	0.953289i	$0.597674\pi$
$258$	0	0
$259$	−6.18609	−0.384385
$260$	−12.1593	−0.754088
$261$	0	0
$262$	9.72197	0.600625
$263$	5.08831	0.313759	0.156879	−	0.987618i	$-0.449857\pi$
$263$	5.08831	0.313759	0.156879	+	0.987618i	$0.449857\pi$
$264$	0	0
$265$	−0.820452	−0.0504000
$266$	−17.0850	−1.04755
$267$	0	0
$268$	−29.8357	−1.82251
$269$	0.224607	0.0136945	0.00684725	−	0.999977i	$-0.497820\pi$
$269$	0.224607	0.0136945	0.00684725	+	0.999977i	$0.497820\pi$
$270$	0	0
$271$	24.3590	1.47971	0.739853	−	0.672769i	$-0.234895\pi$
$271$	24.3590	1.47971	0.739853	+	0.672769i	$0.234895\pi$
$272$	−3.74671	−0.227178
$273$	0	0
$274$	28.3078	1.71014
$275$	0	0
$276$	0	0
$277$	17.4997	1.05146	0.525729	−	0.850652i	$-0.323793\pi$
$277$	17.4997	1.05146	0.525729	+	0.850652i	$0.323793\pi$
$278$	25.3655	1.52132
$279$	0	0
$280$	−3.85410	−0.230327
$281$	−14.3550	−0.856345	−0.428173	−	0.903697i	$-0.640842\pi$
$281$	−14.3550	−0.856345	−0.428173	+	0.903697i	$0.640842\pi$
$282$	0	0
$283$	−28.0055	−1.66475	−0.832376	−	0.554212i	$-0.813020\pi$
$283$	−28.0055	−1.66475	−0.832376	+	0.554212i	$0.813020\pi$
$284$	30.8730	1.83197
$285$	0	0
$286$	0	0
$287$	−9.78690	−0.577702
$288$	0	0
$289$	26.5032	1.55901
$290$	−19.0075	−1.11616
$291$	0	0
$292$	−30.1624	−1.76512
$293$	−13.3587	−0.780421	−0.390211	−	0.920726i	$-0.627598\pi$
$293$	−13.3587	−0.780421	−0.390211	+	0.920726i	$0.627598\pi$
$294$	0	0
$295$	2.71750	0.158219
$296$	15.4480	0.897895
$297$	0	0
$298$	27.3737	1.58571
$299$	14.2582	0.824571
$300$	0	0
$301$	−9.61149	−0.553997
$302$	24.3747	1.40261
$303$	0	0
$304$	−4.29535	−0.246355
$305$	7.32727	0.419558
$306$	0	0
$307$	3.14669	0.179591	0.0897955	−	0.995960i	$-0.471379\pi$
$307$	3.14669	0.179591	0.0897955	+	0.995960i	$0.471379\pi$
$308$	0	0
$309$	0	0
$310$	18.5708	1.05475
$311$	−24.5598	−1.39266	−0.696329	−	0.717722i	$-0.745185\pi$
$311$	−24.5598	−1.39266	−0.696329	+	0.717722i	$0.745185\pi$
$312$	0	0
$313$	−21.5445	−1.21776	−0.608882	−	0.793261i	$-0.708382\pi$
$313$	−21.5445	−1.21776	−0.608882	+	0.793261i	$0.708382\pi$
$314$	8.36670	0.472160
$315$	0	0
$316$	−6.42870	−0.361642
$317$	7.38572	0.414823	0.207412	−	0.978254i	$-0.433496\pi$
$317$	7.38572	0.414823	0.207412	+	0.978254i	$0.433496\pi$
$318$	0	0
$319$	0	0
$320$	−20.1389	−1.12580
$321$	0	0
$322$	12.6976	0.707611
$323$	49.8735	2.77503
$324$	0	0
$325$	6.64238	0.368453
$326$	−50.9391	−2.82126
$327$	0	0
$328$	24.4400	1.34947
$329$	−3.84418	−0.211937
$330$	0	0
$331$	−21.0897	−1.15919	−0.579597	−	0.814903i	$-0.696790\pi$
$331$	−21.0897	−1.15919	−0.579597	+	0.814903i	$0.696790\pi$
$332$	−0.137376	−0.00753949
$333$	0	0
$334$	32.1245	1.75777
$335$	−14.8290	−0.810195
$336$	0	0
$337$	20.2973	1.10566	0.552831	−	0.833293i	$-0.313548\pi$
$337$	20.2973	1.10566	0.552831	+	0.833293i	$0.313548\pi$
$338$	14.8285	0.806562
$339$	0	0
$340$	31.6097	1.71428
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	24.0019	1.29410
$345$	0	0
$346$	42.6565	2.29323
$347$	16.4180	0.881366	0.440683	−	0.897663i	$-0.354736\pi$
$347$	16.4180	0.881366	0.440683	+	0.897663i	$0.354736\pi$
$348$	0	0
$349$	28.3977	1.52009	0.760047	−	0.649868i	$-0.225176\pi$
$349$	28.3977	1.52009	0.760047	+	0.649868i	$0.225176\pi$
$350$	5.91538	0.316190
$351$	0	0
$352$	0	0
$353$	−6.54122	−0.348154	−0.174077	−	0.984732i	$-0.555694\pi$
$353$	−6.54122	−0.348154	−0.174077	+	0.984732i	$0.555694\pi$
$354$	0	0
$355$	15.3445	0.814403
$356$	−19.8205	−1.05049
$357$	0	0
$358$	9.93773	0.525225
$359$	3.56688	0.188253	0.0941265	−	0.995560i	$-0.469994\pi$
$359$	3.56688	0.188253	0.0941265	+	0.995560i	$0.469994\pi$
$360$	0	0
$361$	38.1765	2.00929
$362$	−11.0245	−0.579436
$363$	0	0
$364$	−7.87845	−0.412943
$365$	−14.9914	−0.784685
$366$	0	0
$367$	6.38020	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$367$	6.38020	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$368$	3.19231	0.166411
$369$	0	0
$370$	21.5720	1.12148
$371$	−0.531600	−0.0275993
$372$	0	0
$373$	24.6135	1.27444	0.637218	−	0.770683i	$-0.280085\pi$
$373$	24.6135	1.27444	0.637218	+	0.770683i	$0.280085\pi$
$374$	0	0
$375$	0	0
$376$	9.59974	0.495069
$377$	−13.8293	−0.712243
$378$	0	0
$379$	31.5946	1.62291	0.811453	−	0.584418i	$-0.198677\pi$
$379$	31.5946	1.62291	0.811453	+	0.584418i	$0.198677\pi$
$380$	36.2384	1.85899
$381$	0	0
$382$	−35.7734	−1.83033
$383$	17.0187	0.869614	0.434807	−	0.900524i	$-0.356817\pi$
$383$	17.0187	0.869614	0.434807	+	0.900524i	$0.356817\pi$
$384$	0	0
$385$	0	0
$386$	−18.8321	−0.958531
$387$	0	0
$388$	−41.1436	−2.08875
$389$	−2.70796	−0.137299	−0.0686495	−	0.997641i	$-0.521869\pi$
$389$	−2.70796	−0.137299	−0.0686495	+	0.997641i	$0.521869\pi$
$390$	0	0
$391$	−37.0660	−1.87451
$392$	−2.49721	−0.126128
$393$	0	0
$394$	−8.22352	−0.414295
$395$	−3.19520	−0.160768
$396$	0	0
$397$	−20.9423	−1.05106	−0.525531	−	0.850774i	$-0.676133\pi$
$397$	−20.9423	−1.05106	−0.525531	+	0.850774i	$0.676133\pi$
$398$	10.3526	0.518931
$399$	0	0
$400$	1.48718	0.0743592
$401$	−12.7290	−0.635658	−0.317829	−	0.948148i	$-0.602954\pi$
$401$	−12.7290	−0.635658	−0.317829	+	0.948148i	$0.602954\pi$
$402$	0	0
$403$	13.5115	0.673057
$404$	17.1741	0.854441
$405$	0	0
$406$	−12.3157	−0.611216
$407$	0	0
$408$	0	0
$409$	−8.25063	−0.407967	−0.203984	−	0.978974i	$-0.565389\pi$
$409$	−8.25063	−0.407967	−0.203984	+	0.978974i	$0.565389\pi$
$410$	34.1287	1.68550
$411$	0	0
$412$	20.8077	1.02512
$413$	1.76077	0.0866418
$414$	0	0
$415$	−0.0682789	−0.00335168
$416$	−15.9281	−0.780941
$417$	0	0
$418$	0	0
$419$	−5.81767	−0.284212	−0.142106	−	0.989851i	$-0.545387\pi$
$419$	−5.81767	−0.284212	−0.142106	+	0.989851i	$0.545387\pi$
$420$	0	0
$421$	7.60495	0.370643	0.185321	−	0.982678i	$-0.440667\pi$
$421$	7.60495	0.370643	0.185321	+	0.982678i	$0.440667\pi$
$422$	2.64235	0.128628
$423$	0	0
$424$	1.32752	0.0644700
$425$	−17.2678	−0.837609
$426$	0	0
$427$	4.74760	0.229753
$428$	−17.8698	−0.863771
$429$	0	0
$430$	33.5171	1.61634
$431$	−23.3450	−1.12449	−0.562244	−	0.826971i	$-0.690062\pi$
$431$	−23.3450	−1.12449	−0.562244	+	0.826971i	$0.690062\pi$
$432$	0	0
$433$	−13.5912	−0.653153	−0.326576	−	0.945171i	$-0.605895\pi$
$433$	−13.5912	−0.653153	−0.326576	+	0.945171i	$0.605895\pi$
$434$	12.0327	0.577588
$435$	0	0
$436$	−42.5605	−2.03828
$437$	−42.4936	−2.03275
$438$	0	0
$439$	37.4203	1.78597	0.892986	−	0.450084i	$-0.148606\pi$
$439$	37.4203	1.78597	0.892986	+	0.450084i	$0.148606\pi$
$440$	0	0
$441$	0	0
$442$	37.8109	1.79848
$443$	−29.2812	−1.39119	−0.695597	−	0.718432i	$-0.744860\pi$
$443$	−29.2812	−1.39119	−0.695597	+	0.718432i	$0.744860\pi$
$444$	0	0
$445$	−9.85124	−0.466993
$446$	8.88651	0.420789
$447$	0	0
$448$	−13.0487	−0.616493
$449$	−4.90362	−0.231416	−0.115708	−	0.993283i	$-0.536914\pi$
$449$	−4.90362	−0.231416	−0.115708	+	0.993283i	$0.536914\pi$
$450$	0	0
$451$	0	0
$452$	45.3498	2.13308
$453$	0	0
$454$	−55.0387	−2.58309
$455$	−3.91576	−0.183574
$456$	0	0
$457$	−1.55754	−0.0728585	−0.0364292	−	0.999336i	$-0.511598\pi$
$457$	−1.55754	−0.0728585	−0.0364292	+	0.999336i	$0.511598\pi$
$458$	54.7413	2.55789
$459$	0	0
$460$	−26.9324	−1.25573
$461$	17.6580	0.822414	0.411207	−	0.911542i	$-0.365107\pi$
$461$	17.6580	0.822414	0.411207	+	0.911542i	$0.365107\pi$
$462$	0	0
$463$	−4.54726	−0.211329	−0.105664	−	0.994402i	$-0.533697\pi$
$463$	−4.54726	−0.211329	−0.105664	+	0.994402i	$0.533697\pi$
$464$	−3.09628	−0.143741
$465$	0	0
$466$	−63.7083	−2.95123
$467$	−2.04275	−0.0945271	−0.0472636	−	0.998882i	$-0.515050\pi$
$467$	−2.04275	−0.0945271	−0.0472636	+	0.998882i	$0.515050\pi$
$468$	0	0
$469$	−9.60824	−0.443667
$470$	13.4054	0.618344
$471$	0	0
$472$	−4.39701	−0.202389
$473$	0	0
$474$	0	0
$475$	−19.7963	−0.908317
$476$	20.4811	0.938749
$477$	0	0
$478$	−42.1147	−1.92628
$479$	8.13148	0.371537	0.185768	−	0.982594i	$-0.440523\pi$
$479$	8.13148	0.371537	0.185768	+	0.982594i	$0.440523\pi$
$480$	0	0
$481$	15.6951	0.715636
$482$	−39.4988	−1.79912
$483$	0	0
$484$	0	0
$485$	−20.4493	−0.928555
$486$	0	0
$487$	−28.7773	−1.30402	−0.652011	−	0.758209i	$-0.726075\pi$
$487$	−28.7773	−1.30402	−0.652011	+	0.758209i	$0.726075\pi$
$488$	−11.8558	−0.536685
$489$	0	0
$490$	−3.48718	−0.157535
$491$	5.29273	0.238858	0.119429	−	0.992843i	$-0.461894\pi$
$491$	5.29273	0.238858	0.119429	+	0.992843i	$0.461894\pi$
$492$	0	0
$493$	35.9510	1.61915
$494$	43.3476	1.95030
$495$	0	0
$496$	3.02514	0.135833
$497$	9.94228	0.445972
$498$	0	0
$499$	−25.4668	−1.14005	−0.570025	−	0.821628i	$-0.693066\pi$
$499$	−25.4668	−1.14005	−0.570025	+	0.821628i	$0.693066\pi$
$500$	−36.5092	−1.63274
$501$	0	0
$502$	−9.13085	−0.407530
$503$	42.0133	1.87328	0.936640	−	0.350293i	$-0.113918\pi$
$503$	42.0133	1.87328	0.936640	+	0.350293i	$0.113918\pi$
$504$	0	0
$505$	8.53588	0.379842
$506$	0	0
$507$	0	0
$508$	−41.9399	−1.86078
$509$	−31.9838	−1.41766	−0.708829	−	0.705380i	$-0.750776\pi$
$509$	−31.9838	−1.41766	−0.708829	+	0.705380i	$0.750776\pi$
$510$	0	0
$511$	−9.71346	−0.429698
$512$	−6.40330	−0.282989
$513$	0	0
$514$	21.8824	0.965193
$515$	10.3419	0.455717
$516$	0	0
$517$	0	0
$518$	13.9773	0.614127
$519$	0	0
$520$	9.77849	0.428815
$521$	33.9429	1.48707	0.743533	−	0.668699i	$-0.233148\pi$
$521$	33.9429	1.48707	0.743533	+	0.668699i	$0.233148\pi$
$522$	0	0
$523$	−17.1926	−0.751779	−0.375890	−	0.926664i	$-0.622663\pi$
$523$	−17.1926	−0.751779	−0.375890	+	0.926664i	$0.622663\pi$
$524$	−13.3610	−0.583679
$525$	0	0
$526$	−11.4969	−0.501289
$527$	−35.1250	−1.53007
$528$	0	0
$529$	8.58133	0.373101
$530$	1.85379	0.0805234
$531$	0	0
$532$	23.4802	1.01799
$533$	24.8310	1.07555
$534$	0	0
$535$	−8.88170	−0.383989
$536$	23.9938	1.03638
$537$	0	0
$538$	−0.507492	−0.0218796
$539$	0	0
$540$	0	0
$541$	−7.65418	−0.329079	−0.164539	−	0.986370i	$-0.552614\pi$
$541$	−7.65418	−0.329079	−0.164539	+	0.986370i	$0.552614\pi$
$542$	−55.0386	−2.36411
$543$	0	0
$544$	41.4073	1.77532
$545$	−21.1535	−0.906117
$546$	0	0
$547$	39.9128	1.70655	0.853275	−	0.521462i	$-0.174613\pi$
$547$	39.9128	1.70655	0.853275	+	0.521462i	$0.174613\pi$
$548$	−38.9038	−1.66189
$549$	0	0
$550$	0	0
$551$	41.2154	1.75583
$552$	0	0
$553$	−2.07029	−0.0880376
$554$	−39.5402	−1.67990
$555$	0	0
$556$	−34.8600	−1.47840
$557$	−17.3280	−0.734211	−0.367105	−	0.930179i	$-0.619651\pi$
$557$	−17.3280	−0.734211	−0.367105	+	0.930179i	$0.619651\pi$
$558$	0	0
$559$	24.3859	1.03142
$560$	−0.876713	−0.0370479
$561$	0	0
$562$	32.4346	1.36817
$563$	−36.7312	−1.54803	−0.774017	−	0.633164i	$-0.781756\pi$
$563$	−36.7312	−1.54803	−0.774017	+	0.633164i	$0.781756\pi$
$564$	0	0
$565$	22.5399	0.948259
$566$	63.2776	2.65976
$567$	0	0
$568$	−24.8280	−1.04176
$569$	−30.9746	−1.29852	−0.649261	−	0.760566i	$-0.724922\pi$
$569$	−30.9746	−1.29852	−0.649261	+	0.760566i	$0.724922\pi$
$570$	0	0
$571$	−26.7450	−1.11925	−0.559623	−	0.828748i	$-0.689054\pi$
$571$	−26.7450	−1.11925	−0.559623	+	0.828748i	$0.689054\pi$
$572$	0	0
$573$	0	0
$574$	22.1132	0.922989
$575$	14.7126	0.613559
$576$	0	0
$577$	−30.6529	−1.27610	−0.638048	−	0.769996i	$-0.720258\pi$
$577$	−30.6529	−1.27610	−0.638048	+	0.769996i	$0.720258\pi$
$578$	−59.8833	−2.49082
$579$	0	0
$580$	26.1222	1.08467
$581$	−0.0442404	−0.00183540
$582$	0	0
$583$	0	0
$584$	24.2566	1.00374
$585$	0	0
$586$	30.1835	1.24687
$587$	−16.6367	−0.686670	−0.343335	−	0.939213i	$-0.611557\pi$
$587$	−16.6367	−0.686670	−0.343335	+	0.939213i	$0.611557\pi$
$588$	0	0
$589$	−40.2684	−1.65923
$590$	−6.14013	−0.252785
$591$	0	0
$592$	3.51403	0.144426
$593$	37.7740	1.55119	0.775596	−	0.631230i	$-0.217450\pi$
$593$	37.7740	1.55119	0.775596	+	0.631230i	$0.217450\pi$
$594$	0	0
$595$	10.1795	0.417321
$596$	−37.6199	−1.54097
$597$	0	0
$598$	−32.2160	−1.31741
$599$	25.0536	1.02366	0.511832	−	0.859086i	$-0.328967\pi$
$599$	25.0536	1.02366	0.511832	+	0.859086i	$0.328967\pi$
$600$	0	0
$601$	42.9247	1.75093	0.875467	−	0.483277i	$-0.160554\pi$
$601$	42.9247	1.75093	0.875467	+	0.483277i	$0.160554\pi$
$602$	21.7169	0.885115
$603$	0	0
$604$	−33.4985	−1.36303
$605$	0	0
$606$	0	0
$607$	−8.72896	−0.354298	−0.177149	−	0.984184i	$-0.556687\pi$
$607$	−8.72896	−0.354298	−0.177149	+	0.984184i	$0.556687\pi$
$608$	47.4706	1.92519
$609$	0	0
$610$	−16.5558	−0.670323
$611$	9.75332	0.394577
$612$	0	0
$613$	−8.79301	−0.355146	−0.177573	−	0.984108i	$-0.556825\pi$
$613$	−8.79301	−0.355146	−0.177573	+	0.984108i	$0.556825\pi$
$614$	−7.10986	−0.286931
$615$	0	0
$616$	0	0
$617$	8.31423	0.334718	0.167359	−	0.985896i	$-0.446476\pi$
$617$	8.31423	0.334718	0.167359	+	0.985896i	$0.446476\pi$
$618$	0	0
$619$	2.35834	0.0947899	0.0473949	−	0.998876i	$-0.484908\pi$
$619$	2.35834	0.0947899	0.0473949	+	0.998876i	$0.484908\pi$
$620$	−25.5221	−1.02499
$621$	0	0
$622$	55.4922	2.22504
$623$	−6.38297	−0.255728
$624$	0	0
$625$	−5.05573	−0.202229
$626$	48.6791	1.94561
$627$	0	0
$628$	−11.4985	−0.458838
$629$	−40.8016	−1.62686
$630$	0	0
$631$	40.3297	1.60550	0.802750	−	0.596316i	$-0.203369\pi$
$631$	40.3297	1.60550	0.802750	+	0.596316i	$0.203369\pi$
$632$	5.16995	0.205649
$633$	0	0
$634$	−16.6878	−0.662758
$635$	−20.8451	−0.827211
$636$	0	0
$637$	−2.53716	−0.100526
$638$	0	0
$639$	0	0
$640$	26.1250	1.03268
$641$	−12.6982	−0.501551	−0.250775	−	0.968045i	$-0.580686\pi$
$641$	−12.6982	−0.501551	−0.250775	+	0.968045i	$0.580686\pi$
$642$	0	0
$643$	14.7309	0.580929	0.290465	−	0.956886i	$-0.406190\pi$
$643$	14.7309	0.580929	0.290465	+	0.956886i	$0.406190\pi$
$644$	−17.4505	−0.687645
$645$	0	0
$646$	−112.688	−4.43364
$647$	4.64105	0.182459	0.0912293	−	0.995830i	$-0.470920\pi$
$647$	4.64105	0.182459	0.0912293	+	0.995830i	$0.470920\pi$
$648$	0	0
$649$	0	0
$650$	−15.0083	−0.588673
$651$	0	0
$652$	70.0062	2.74165
$653$	22.8707	0.895001	0.447500	−	0.894284i	$-0.352314\pi$
$653$	22.8707	0.895001	0.447500	+	0.894284i	$0.352314\pi$
$654$	0	0
$655$	−6.64072	−0.259474
$656$	5.55949	0.217062
$657$	0	0
$658$	8.68582	0.338609
$659$	17.8597	0.695717	0.347858	−	0.937547i	$-0.386909\pi$
$659$	17.8597	0.695717	0.347858	+	0.937547i	$0.386909\pi$
$660$	0	0
$661$	−18.7110	−0.727774	−0.363887	−	0.931443i	$-0.618551\pi$
$661$	−18.7110	−0.727774	−0.363887	+	0.931443i	$0.618551\pi$
$662$	47.6516	1.85203
$663$	0	0
$664$	0.110478	0.00428736
$665$	11.6702	0.452549
$666$	0	0
$667$	−30.6313	−1.18605
$668$	−44.1491	−1.70818
$669$	0	0
$670$	33.5057	1.29444
$671$	0	0
$672$	0	0
$673$	−7.15894	−0.275957	−0.137978	−	0.990435i	$-0.544060\pi$
$673$	−7.15894	−0.275957	−0.137978	+	0.990435i	$0.544060\pi$
$674$	−45.8611	−1.76650
$675$	0	0
$676$	−20.3789	−0.783804
$677$	−23.0053	−0.884166	−0.442083	−	0.896974i	$-0.645760\pi$
$677$	−23.0053	−0.884166	−0.442083	+	0.896974i	$0.645760\pi$
$678$	0	0
$679$	−13.2498	−0.508482
$680$	−25.4205	−0.974831
$681$	0	0
$682$	0	0
$683$	18.0889	0.692152	0.346076	−	0.938206i	$-0.387514\pi$
$683$	18.0889	0.692152	0.346076	+	0.938206i	$0.387514\pi$
$684$	0	0
$685$	−19.3360	−0.738792
$686$	−2.25947	−0.0862671
$687$	0	0
$688$	5.45985	0.208155
$689$	1.34876	0.0513836
$690$	0	0
$691$	−26.6850	−1.01514	−0.507572	−	0.861609i	$-0.669457\pi$
$691$	−26.6850	−1.01514	−0.507572	+	0.861609i	$0.669457\pi$
$692$	−58.6232	−2.22852
$693$	0	0
$694$	−37.0961	−1.40815
$695$	−17.3262	−0.657221
$696$	0	0
$697$	−64.5514	−2.44506
$698$	−64.1638	−2.42864
$699$	0	0
$700$	−8.12957	−0.307269
$701$	−14.2569	−0.538477	−0.269238	−	0.963074i	$-0.586772\pi$
$701$	−14.2569	−0.538477	−0.269238	+	0.963074i	$0.586772\pi$
$702$	0	0
$703$	−46.7762	−1.76420
$704$	0	0
$705$	0	0
$706$	14.7797	0.556242
$707$	5.53071	0.208004
$708$	0	0
$709$	16.0584	0.603085	0.301543	−	0.953453i	$-0.402498\pi$
$709$	16.0584	0.603085	0.301543	+	0.953453i	$0.402498\pi$
$710$	−34.6706	−1.30116
$711$	0	0
$712$	15.9396	0.597363
$713$	29.9276	1.12080
$714$	0	0
$715$	0	0
$716$	−13.6575	−0.510406
$717$	0	0
$718$	−8.05928	−0.300770
$719$	37.0621	1.38218	0.691091	−	0.722768i	$-0.257131\pi$
$719$	37.0621	1.38218	0.691091	+	0.722768i	$0.257131\pi$
$720$	0	0
$721$	6.70087	0.249554
$722$	−86.2588	−3.21022
$723$	0	0
$724$	15.1511	0.563087
$725$	−14.2701	−0.529977
$726$	0	0
$727$	14.6597	0.543698	0.271849	−	0.962340i	$-0.412365\pi$
$727$	14.6597	0.543698	0.271849	+	0.962340i	$0.412365\pi$
$728$	6.33584	0.234822
$729$	0	0
$730$	33.8726	1.25368
$731$	−63.3945	−2.34473
$732$	0	0
$733$	28.9555	1.06949	0.534747	−	0.845012i	$-0.320407\pi$
$733$	28.9555	1.06949	0.534747	+	0.845012i	$0.320407\pi$
$734$	−14.4159	−0.532100
$735$	0	0
$736$	−35.2802	−1.30045
$737$	0	0
$738$	0	0
$739$	−8.43921	−0.310441	−0.155221	−	0.987880i	$-0.549609\pi$
$739$	−8.43921	−0.310441	−0.155221	+	0.987880i	$0.549609\pi$
$740$	−29.6467	−1.08983
$741$	0	0
$742$	1.20114	0.0440951
$743$	−47.7420	−1.75149	−0.875743	−	0.482778i	$-0.839628\pi$
$743$	−47.7420	−1.75149	−0.875743	+	0.482778i	$0.839628\pi$
$744$	0	0
$745$	−18.6979	−0.685039
$746$	−55.6135	−2.03615
$747$	0	0
$748$	0	0
$749$	−5.75478	−0.210275
$750$	0	0
$751$	−0.931618	−0.0339952	−0.0169976	−	0.999856i	$-0.505411\pi$
$751$	−0.931618	−0.0339952	−0.0169976	+	0.999856i	$0.505411\pi$
$752$	2.18370	0.0796314
$753$	0	0
$754$	31.2468	1.13794
$755$	−16.6495	−0.605936
$756$	0	0
$757$	−15.9127	−0.578355	−0.289178	−	0.957275i	$-0.593382\pi$
$757$	−15.9127	−0.578355	−0.289178	+	0.957275i	$0.593382\pi$
$758$	−71.3872	−2.59290
$759$	0	0
$760$	−29.1428	−1.05712
$761$	−0.221975	−0.00804657	−0.00402328	−	0.999992i	$-0.501281\pi$
$761$	−0.221975	−0.00804657	−0.00402328	+	0.999992i	$0.501281\pi$
$762$	0	0
$763$	−13.7061	−0.496195
$764$	49.1638	1.77868
$765$	0	0
$766$	−38.4532	−1.38937
$767$	−4.46736	−0.161307
$768$	0	0
$769$	44.0307	1.58779	0.793894	−	0.608056i	$-0.208051\pi$
$769$	44.0307	1.58779	0.793894	+	0.608056i	$0.208051\pi$
$770$	0	0
$771$	0	0
$772$	25.8812	0.931485
$773$	−44.3759	−1.59609	−0.798045	−	0.602598i	$-0.794132\pi$
$773$	−44.3759	−1.59609	−0.798045	+	0.602598i	$0.794132\pi$
$774$	0	0
$775$	13.9422	0.500819
$776$	33.0876	1.18778
$777$	0	0
$778$	6.11856	0.219361
$779$	−74.0038	−2.65146
$780$	0	0
$781$	0	0
$782$	83.7497	2.99488
$783$	0	0
$784$	−0.568054	−0.0202876
$785$	−5.71498	−0.203977
$786$	0	0
$787$	17.2307	0.614210	0.307105	−	0.951676i	$-0.400640\pi$
$787$	17.2307	0.614210	0.307105	+	0.951676i	$0.400640\pi$
$788$	11.3017	0.402605
$789$	0	0
$790$	7.21948	0.256857
$791$	14.6044	0.519272
$792$	0	0
$793$	−12.0454	−0.427746
$794$	47.3185	1.67927
$795$	0	0
$796$	−14.2278	−0.504290
$797$	32.7314	1.15941	0.579703	−	0.814828i	$-0.303168\pi$
$797$	32.7314	1.15941	0.579703	+	0.814828i	$0.303168\pi$
$798$	0	0
$799$	−25.3551	−0.896997
$800$	−16.4358	−0.581094
$801$	0	0
$802$	28.7609	1.01558
$803$	0	0
$804$	0	0
$805$	−8.67327	−0.305693
$806$	−30.5289	−1.07534
$807$	0	0
$808$	−13.8113	−0.485882
$809$	−36.9725	−1.29988	−0.649942	−	0.759984i	$-0.725207\pi$
$809$	−36.9725	−1.29988	−0.649942	+	0.759984i	$0.725207\pi$
$810$	0	0
$811$	16.4600	0.577990	0.288995	−	0.957331i	$-0.406679\pi$
$811$	16.4600	0.577990	0.288995	+	0.957331i	$0.406679\pi$
$812$	16.9255	0.593970
$813$	0	0
$814$	0	0
$815$	34.7946	1.21880
$816$	0	0
$817$	−72.6774	−2.54266
$818$	18.6421	0.651805
$819$	0	0
$820$	−46.9035	−1.63794
$821$	7.48406	0.261195	0.130598	−	0.991435i	$-0.458310\pi$
$821$	7.48406	0.261195	0.130598	+	0.991435i	$0.458310\pi$
$822$	0	0
$823$	−47.6217	−1.65999	−0.829994	−	0.557772i	$-0.811656\pi$
$823$	−47.6217	−1.65999	−0.829994	+	0.557772i	$0.811656\pi$
$824$	−16.7335	−0.582939
$825$	0	0
$826$	−3.97841	−0.138427
$827$	−37.8258	−1.31533	−0.657666	−	0.753310i	$-0.728456\pi$
$827$	−37.8258	−1.31533	−0.657666	+	0.753310i	$0.728456\pi$
$828$	0	0
$829$	−6.27021	−0.217773	−0.108887	−	0.994054i	$-0.534729\pi$
$829$	−6.27021	−0.217773	−0.108887	+	0.994054i	$0.534729\pi$
$830$	0.154274	0.00535494
$831$	0	0
$832$	33.1067	1.14777
$833$	6.59570	0.228527
$834$	0	0
$835$	−21.9431	−0.759371
$836$	0	0
$837$	0	0
$838$	13.1449	0.454082
$839$	−13.8793	−0.479166	−0.239583	−	0.970876i	$-0.577011\pi$
$839$	−13.8793	−0.479166	−0.239583	+	0.970876i	$0.577011\pi$
$840$	0	0
$841$	0.709871	0.0244783
$842$	−17.1832	−0.592172
$843$	0	0
$844$	−3.63142	−0.124998
$845$	−10.1288	−0.348440
$846$	0	0
$847$	0	0
$848$	0.301978	0.0103700
$849$	0	0
$850$	39.0160	1.33824
$851$	34.7641	1.19170
$852$	0	0
$853$	25.4608	0.871762	0.435881	−	0.900004i	$-0.356437\pi$
$853$	25.4608	0.871762	0.435881	+	0.900004i	$0.356437\pi$
$854$	−10.7271	−0.367073
$855$	0	0
$856$	14.3709	0.491187
$857$	−50.8757	−1.73788	−0.868941	−	0.494916i	$-0.835199\pi$
$857$	−50.8757	−1.73788	−0.868941	+	0.494916i	$0.835199\pi$
$858$	0	0
$859$	−37.2976	−1.27258	−0.636288	−	0.771452i	$-0.719531\pi$
$859$	−37.2976	−1.27258	−0.636288	+	0.771452i	$0.719531\pi$
$860$	−46.0629	−1.57073
$861$	0	0
$862$	52.7474	1.79658
$863$	12.8773	0.438348	0.219174	−	0.975686i	$-0.429664\pi$
$863$	12.8773	0.438348	0.219174	+	0.975686i	$0.429664\pi$
$864$	0	0
$865$	−29.1370	−0.990689
$866$	30.7090	1.04353
$867$	0	0
$868$	−16.5367	−0.561291
$869$	0	0
$870$	0	0
$871$	24.3777	0.826007
$872$	34.2271	1.15908
$873$	0	0
$874$	96.0133	3.24770
$875$	−11.7574	−0.397472
$876$	0	0
$877$	10.8105	0.365046	0.182523	−	0.983202i	$-0.441574\pi$
$877$	10.8105	0.365046	0.182523	+	0.983202i	$0.441574\pi$
$878$	−84.5501	−2.85343
$879$	0	0
$880$	0	0
$881$	19.9303	0.671467	0.335734	−	0.941957i	$-0.391016\pi$
$881$	19.9303	0.671467	0.335734	+	0.941957i	$0.391016\pi$
$882$	0	0
$883$	−27.1591	−0.913976	−0.456988	−	0.889473i	$-0.651072\pi$
$883$	−27.1591	−0.913976	−0.456988	+	0.889473i	$0.651072\pi$
$884$	−51.9639	−1.74773
$885$	0	0
$886$	66.1602	2.22269
$887$	−6.37071	−0.213908	−0.106954	−	0.994264i	$-0.534110\pi$
$887$	−6.37071	−0.213908	−0.106954	+	0.994264i	$0.534110\pi$
$888$	0	0
$889$	−13.5063	−0.452986
$890$	22.2586	0.746110
$891$	0	0
$892$	−12.2128	−0.408916
$893$	−29.0678	−0.972718
$894$	0	0
$895$	−6.78809	−0.226901
$896$	16.9273	0.565502
$897$	0	0
$898$	11.0796	0.369731
$899$	−29.0273	−0.968114
$900$	0	0
$901$	−3.50628	−0.116811
$902$	0	0
$903$	0	0
$904$	−36.4703	−1.21298
$905$	7.53044	0.250320
$906$	0	0
$907$	−11.2413	−0.373262	−0.186631	−	0.982430i	$-0.559757\pi$
$907$	−11.2413	−0.373262	−0.186631	+	0.982430i	$0.559757\pi$
$908$	75.6403	2.51021
$909$	0	0
$910$	8.84756	0.293294
$911$	32.3301	1.07114	0.535572	−	0.844490i	$-0.320096\pi$
$911$	32.3301	1.07114	0.535572	+	0.844490i	$0.320096\pi$
$912$	0	0
$913$	0	0
$914$	3.51921	0.116405
$915$	0	0
$916$	−75.2315	−2.48572
$917$	−4.30276	−0.142090
$918$	0	0
$919$	16.9783	0.560063	0.280031	−	0.959991i	$-0.409655\pi$
$919$	16.9783	0.560063	0.280031	+	0.959991i	$0.409655\pi$
$920$	21.6590	0.714076
$921$	0	0
$922$	−39.8977	−1.31396
$923$	−25.2252	−0.830298
$924$	0	0
$925$	16.1954	0.532501
$926$	10.2744	0.337638
$927$	0	0
$928$	34.2189	1.12329
$929$	−7.11309	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$929$	−7.11309	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$930$	0	0
$931$	7.56151	0.247819
$932$	87.5551	2.86796
$933$	0	0
$934$	4.61554	0.151025
$935$	0	0
$936$	0	0
$937$	−15.6317	−0.510665	−0.255332	−	0.966853i	$-0.582185\pi$
$937$	−15.6317	−0.510665	−0.255332	+	0.966853i	$0.582185\pi$
$938$	21.7096	0.708843
$939$	0	0
$940$	−18.4231	−0.600897
$941$	−5.05065	−0.164646	−0.0823232	−	0.996606i	$-0.526234\pi$
$941$	−5.05065	−0.164646	−0.0823232	+	0.996606i	$0.526234\pi$
$942$	0	0
$943$	54.9997	1.79104
$944$	−1.00021	−0.0325541
$945$	0	0
$946$	0	0
$947$	−20.3081	−0.659925	−0.329963	−	0.943994i	$-0.607036\pi$
$947$	−20.3081	−0.659925	−0.329963	+	0.943994i	$0.607036\pi$
$948$	0	0
$949$	24.6447	0.799999
$950$	44.7292	1.45121
$951$	0	0
$952$	−16.4709	−0.533823
$953$	−29.3431	−0.950515	−0.475257	−	0.879847i	$-0.657645\pi$
$953$	−29.3431	−0.950515	−0.475257	+	0.879847i	$0.657645\pi$
$954$	0	0
$955$	24.4355	0.790714
$956$	57.8787	1.87193
$957$	0	0
$958$	−18.3729	−0.593600
$959$	−12.5285	−0.404567
$960$	0	0
$961$	−2.63962	−0.0851491
$962$	−35.4627	−1.14336
$963$	0	0
$964$	54.2837	1.74836
$965$	12.8635	0.414092
$966$	0	0
$967$	−35.2155	−1.13245	−0.566227	−	0.824249i	$-0.691598\pi$
$967$	−35.2155	−1.13245	−0.566227	+	0.824249i	$0.691598\pi$
$968$	0	0
$969$	0	0
$970$	46.2046	1.48354
$971$	4.93270	0.158298	0.0791490	−	0.996863i	$-0.474780\pi$
$971$	4.93270	0.158298	0.0791490	+	0.996863i	$0.474780\pi$
$972$	0	0
$973$	−11.2263	−0.359898
$974$	65.0215	2.08342
$975$	0	0
$976$	−2.69689	−0.0863255
$977$	30.3307	0.970366	0.485183	−	0.874413i	$-0.338753\pi$
$977$	30.3307	0.970366	0.485183	+	0.874413i	$0.338753\pi$
$978$	0	0
$979$	0	0
$980$	4.79248	0.153090
$981$	0	0
$982$	−11.9588	−0.381620
$983$	−40.7718	−1.30042	−0.650209	−	0.759756i	$-0.725319\pi$
$983$	−40.7718	−1.30042	−0.650209	+	0.759756i	$0.725319\pi$
$984$	0	0
$985$	5.61718	0.178978
$986$	−81.2303	−2.58690
$987$	0	0
$988$	−59.5730	−1.89527
$989$	54.0140	1.71754
$990$	0	0
$991$	11.0927	0.352370	0.176185	−	0.984357i	$-0.443624\pi$
$991$	11.0927	0.352370	0.176185	+	0.984357i	$0.443624\pi$
$992$	−33.4328	−1.06149
$993$	0	0
$994$	−22.4643	−0.712525
$995$	−7.07151	−0.224182
$996$	0	0
$997$	26.3185	0.833515	0.416757	−	0.909018i	$-0.363166\pi$
$997$	26.3185	0.833515	0.416757	+	0.909018i	$0.363166\pi$
$998$	57.5415	1.82144
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cv.1.1		8
3.2	odd	2	inner	7623.2.a.cv.1.8		8
11.7	odd	10		693.2.m.h.379.4	yes	16
11.8	odd	10		693.2.m.h.64.4	yes	16
11.10	odd	2		7623.2.a.cu.1.8		8
33.8	even	10		693.2.m.h.64.1	✓	16
33.29	even	10		693.2.m.h.379.1	yes	16
33.32	even	2		7623.2.a.cu.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.h.64.1	✓	16	33.8	even	10
693.2.m.h.64.4	yes	16	11.8	odd	10
693.2.m.h.379.1	yes	16	33.29	even	10
693.2.m.h.379.4	yes	16	11.7	odd	10
7623.2.a.cu.1.1		8	33.32	even	2
7623.2.a.cu.1.8		8	11.10	odd	2
7623.2.a.cv.1.1		8	1.1	even	1	trivial
7623.2.a.cv.1.8		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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-2.25947 q^{2} +3.10522 q^{4} +1.54336 q^{5} +1.00000 q^{7} -2.49721 q^{8} +O(q^{10})\)	\(q-2.25947 q^{2} +3.10522 q^{4} +1.54336 q^{5} +1.00000 q^{7} -2.49721 q^{8} -3.48718 q^{10} -2.53716 q^{13} -2.25947 q^{14} -0.568054 q^{16} +6.59570 q^{17} +7.56151 q^{19} +4.79248 q^{20} -5.61973 q^{23} -2.61803 q^{25} +5.73266 q^{26} +3.10522 q^{28} +5.45068 q^{29} -5.32545 q^{31} +6.27793 q^{32} -14.9028 q^{34} +1.54336 q^{35} -6.18609 q^{37} -17.0850 q^{38} -3.85410 q^{40} -9.78690 q^{41} -9.61149 q^{43} +12.6976 q^{46} -3.84418 q^{47} +1.00000 q^{49} +5.91538 q^{50} -7.87845 q^{52} -0.531600 q^{53} -2.49721 q^{56} -12.3157 q^{58} +1.76077 q^{59} +4.74760 q^{61} +12.0327 q^{62} -13.0487 q^{64} -3.91576 q^{65} -9.60824 q^{67} +20.4811 q^{68} -3.48718 q^{70} +9.94228 q^{71} -9.71346 q^{73} +13.9773 q^{74} +23.4802 q^{76} -2.07029 q^{79} -0.876713 q^{80} +22.1132 q^{82} -0.0442404 q^{83} +10.1795 q^{85} +21.7169 q^{86} -6.38297 q^{89} -2.53716 q^{91} -17.4505 q^{92} +8.68582 q^{94} +11.6702 q^{95} -13.2498 q^{97} -2.25947 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} + 8 q^{7}+O(q^{10}) \) 8 * q + 2 * q^4 + 8 * q^7	\( 8 q + 2 q^{4} + 8 q^{7} - 14 q^{10} - 2 q^{16} + 6 q^{19} - 12 q^{25} + 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} - 4 q^{40} - 16 q^{43} + 42 q^{46} + 8 q^{49} - 2 q^{52} - 30 q^{58} - 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} - 14 q^{73} + 34 q^{76} - 22 q^{79} + 36 q^{82} + 18 q^{85} - 32 q^{94} - 48 q^{97}+O(q^{100}) \) 8 * q + 2 * q^4 + 8 * q^7 - 14 * q^10 - 2 * q^16 + 6 * q^19 - 12 * q^25 + 2 * q^28 - 6 * q^31 - 24 * q^34 - 38 * q^37 - 4 * q^40 - 16 * q^43 + 42 * q^46 + 8 * q^49 - 2 * q^52 - 30 * q^58 - 28 * q^61 - 36 * q^64 - 36 * q^67 - 14 * q^70 - 14 * q^73 + 34 * q^76 - 22 * q^79 + 36 * q^82 + 18 * q^85 - 32 * q^94 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more