LMFDB - Embedded newform 7623.2.a.dc.1.9

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 268x^{12} - 1395x^{10} + 3876x^{8} - 5635x^{6} + 4042x^{4} - 1272x^{2} + 121 $ x^16 - 26x^14 + 268x^12 - 1395x^10 + 3876x^8 - 5635x^6 + 4042x^4 - 1272*x^2 + 121
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			1.9
Root			$0.400970$ 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)

$f(q)$	$=$	$q+0.400970 q^{2} -1.83922 q^{4} -2.30562 q^{5} +1.00000 q^{7} -1.53941 q^{8} +O(q^{10})$	$q+0.400970 q^{2} -1.83922 q^{4} -2.30562 q^{5} +1.00000 q^{7} -1.53941 q^{8} -0.924485 q^{10} +4.49155 q^{13} +0.400970 q^{14} +3.06119 q^{16} -7.46771 q^{17} -1.34729 q^{19} +4.24055 q^{20} +4.27146 q^{23} +0.315883 q^{25} +1.80098 q^{26} -1.83922 q^{28} -1.12589 q^{29} -8.11240 q^{31} +4.30627 q^{32} -2.99433 q^{34} -2.30562 q^{35} +11.0657 q^{37} -0.540224 q^{38} +3.54930 q^{40} +1.16172 q^{41} +9.95451 q^{43} +1.71273 q^{46} -12.8441 q^{47} +1.00000 q^{49} +0.126660 q^{50} -8.26096 q^{52} -11.7297 q^{53} -1.53941 q^{56} -0.451447 q^{58} +1.04893 q^{59} -6.69637 q^{61} -3.25283 q^{62} -4.39569 q^{64} -10.3558 q^{65} -2.46445 q^{67} +13.7348 q^{68} -0.924485 q^{70} -8.42283 q^{71} -10.7770 q^{73} +4.43703 q^{74} +2.47797 q^{76} +2.71147 q^{79} -7.05793 q^{80} +0.465816 q^{82} +9.12912 q^{83} +17.2177 q^{85} +3.99146 q^{86} -5.28542 q^{89} +4.49155 q^{91} -7.85617 q^{92} -5.15009 q^{94} +3.10634 q^{95} +5.74601 q^{97} +0.400970 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 16 * q^7	$ 16 q + 20 q^{4} + 16 q^{7} - 6 q^{10} + 32 q^{16} + 10 q^{19} + 44 q^{25} + 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} - 68 q^{40} + 16 q^{43} + 80 q^{46} + 16 q^{49} + 2 q^{52} + 18 q^{58} - 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} - 14 q^{73} - 14 q^{76} + 54 q^{79} + 64 q^{82} + 30 q^{85} - 60 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 16 * q^7 - 6 * q^10 + 32 * q^16 + 10 * q^19 + 44 * q^25 + 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 - 68 * q^40 + 16 * q^43 + 80 * q^46 + 16 * q^49 + 2 * q^52 + 18 * q^58 - 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 - 14 * q^73 - 14 * q^76 + 54 * q^79 + 64 * q^82 + 30 * q^85 - 60 * q^94 + 108 * 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 $

$2$	0.400970	0.283529	0.141764	−	0.989900i	$-0.454722\pi$
$2$	0.400970	0.283529	0.141764	+	0.989900i	$0.454722\pi$
$3$	0	0
$3$	0	0
$4$	−1.83922	−0.919611
$5$	−2.30562	−1.03110	−0.515552	−	0.856858i	$-0.672413\pi$
$5$	−2.30562	−1.03110	−0.515552	+	0.856858i	$0.672413\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.53941	−0.544265
$9$	0	0
$10$	−0.924485	−0.292348
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.49155	1.24573	0.622866	−	0.782329i	$-0.285968\pi$
$13$	4.49155	1.24573	0.622866	+	0.782329i	$0.285968\pi$
$14$	0.400970	0.107164
$15$	0	0
$16$	3.06119	0.765297
$17$	−7.46771	−1.81119	−0.905593	−	0.424148i	$-0.860574\pi$
$17$	−7.46771	−1.81119	−0.905593	+	0.424148i	$0.860574\pi$
$18$	0	0
$19$	−1.34729	−0.309090	−0.154545	−	0.987986i	$-0.549391\pi$
$19$	−1.34729	−0.309090	−0.154545	+	0.987986i	$0.549391\pi$
$20$	4.24055	0.948216
$21$	0	0
$22$	0	0
$23$	4.27146	0.890661	0.445330	−	0.895366i	$-0.353086\pi$
$23$	4.27146	0.890661	0.445330	+	0.895366i	$0.353086\pi$
$24$	0	0
$25$	0.315883	0.0631766
$26$	1.80098	0.353201
$27$	0	0
$28$	−1.83922	−0.347580
$29$	−1.12589	−0.209072	−0.104536	−	0.994521i	$-0.533336\pi$
$29$	−1.12589	−0.209072	−0.104536	+	0.994521i	$0.533336\pi$
$30$	0	0
$31$	−8.11240	−1.45703	−0.728515	−	0.685030i	$-0.759789\pi$
$31$	−8.11240	−1.45703	−0.728515	+	0.685030i	$0.759789\pi$
$32$	4.30627	0.761249
$33$	0	0
$34$	−2.99433	−0.513523
$35$	−2.30562	−0.389721
$36$	0	0
$37$	11.0657	1.81920	0.909598	−	0.415490i	$-0.136390\pi$
$37$	11.0657	1.81920	0.909598	+	0.415490i	$0.136390\pi$
$38$	−0.540224	−0.0876359
$39$	0	0
$40$	3.54930	0.561194
$41$	1.16172	0.181430	0.0907152	−	0.995877i	$-0.471085\pi$
$41$	1.16172	0.181430	0.0907152	+	0.995877i	$0.471085\pi$
$42$	0	0
$43$	9.95451	1.51805	0.759024	−	0.651063i	$-0.225676\pi$
$43$	9.95451	1.51805	0.759024	+	0.651063i	$0.225676\pi$
$44$	0	0
$45$	0	0
$46$	1.71273	0.252528
$47$	−12.8441	−1.87350	−0.936751	−	0.349998i	$-0.886182\pi$
$47$	−12.8441	−1.87350	−0.936751	+	0.349998i	$0.886182\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.126660	0.0179124
$51$	0	0
$52$	−8.26096	−1.14559
$53$	−11.7297	−1.61120	−0.805602	−	0.592457i	$-0.798158\pi$
$53$	−11.7297	−1.61120	−0.805602	+	0.592457i	$0.798158\pi$
$54$	0	0
$55$	0	0
$56$	−1.53941	−0.205713
$57$	0	0
$58$	−0.451447	−0.0592779
$59$	1.04893	0.136559	0.0682793	−	0.997666i	$-0.478249\pi$
$59$	1.04893	0.136559	0.0682793	+	0.997666i	$0.478249\pi$
$60$	0	0
$61$	−6.69637	−0.857382	−0.428691	−	0.903451i	$-0.641025\pi$
$61$	−6.69637	−0.857382	−0.428691	+	0.903451i	$0.641025\pi$
$62$	−3.25283	−0.413110
$63$	0	0
$64$	−4.39569	−0.549461
$65$	−10.3558	−1.28448
$66$	0	0
$67$	−2.46445	−0.301080	−0.150540	−	0.988604i	$-0.548101\pi$
$67$	−2.46445	−0.301080	−0.150540	+	0.988604i	$0.548101\pi$
$68$	13.7348	1.66559
$69$	0	0
$70$	−0.924485	−0.110497
$71$	−8.42283	−0.999606	−0.499803	−	0.866139i	$-0.666594\pi$
$71$	−8.42283	−0.999606	−0.499803	+	0.866139i	$0.666594\pi$
$72$	0	0
$73$	−10.7770	−1.26135	−0.630673	−	0.776048i	$-0.717221\pi$
$73$	−10.7770	−1.26135	−0.630673	+	0.776048i	$0.717221\pi$
$74$	4.43703	0.515794
$75$	0	0
$76$	2.47797	0.284243
$77$	0	0
$78$	0	0
$79$	2.71147	0.305065	0.152532	−	0.988298i	$-0.451257\pi$
$79$	2.71147	0.305065	0.152532	+	0.988298i	$0.451257\pi$
$80$	−7.05793	−0.789101
$81$	0	0
$82$	0.465816	0.0514407
$83$	9.12912	1.00205	0.501025	−	0.865433i	$-0.332956\pi$
$83$	9.12912	1.00205	0.501025	+	0.865433i	$0.332956\pi$
$84$	0	0
$85$	17.2177	1.86752
$86$	3.99146	0.430410
$87$	0	0
$88$	0	0
$89$	−5.28542	−0.560253	−0.280126	−	0.959963i	$-0.590376\pi$
$89$	−5.28542	−0.560253	−0.280126	+	0.959963i	$0.590376\pi$
$90$	0	0
$91$	4.49155	0.470842
$92$	−7.85617	−0.819062
$93$	0	0
$94$	−5.15009	−0.531191
$95$	3.10634	0.318704
$96$	0	0
$97$	5.74601	0.583419	0.291709	−	0.956507i	$-0.405776\pi$
$97$	5.74601	0.583419	0.291709	+	0.956507i	$0.405776\pi$
$98$	0.400970	0.0405041
$99$	0	0
$100$	−0.580979	−0.0580979
$101$	−9.56357	−0.951611	−0.475805	−	0.879551i	$-0.657843\pi$
$101$	−9.56357	−0.951611	−0.475805	+	0.879551i	$0.657843\pi$
$102$	0	0
$103$	−0.807595	−0.0795747	−0.0397873	−	0.999208i	$-0.512668\pi$
$103$	−0.807595	−0.0795747	−0.0397873	+	0.999208i	$0.512668\pi$
$104$	−6.91435	−0.678008
$105$	0	0
$106$	−4.70328	−0.456823
$107$	16.8478	1.62874	0.814371	−	0.580345i	$-0.197082\pi$
$107$	16.8478	1.62874	0.814371	+	0.580345i	$0.197082\pi$
$108$	0	0
$109$	9.41870	0.902147	0.451074	−	0.892487i	$-0.351041\pi$
$109$	9.41870	0.902147	0.451074	+	0.892487i	$0.351041\pi$
$110$	0	0
$111$	0	0
$112$	3.06119	0.289255
$113$	4.35139	0.409344	0.204672	−	0.978831i	$-0.434387\pi$
$113$	4.35139	0.409344	0.204672	+	0.978831i	$0.434387\pi$
$114$	0	0
$115$	−9.84836	−0.918365
$116$	2.07076	0.192265
$117$	0	0
$118$	0.420588	0.0387183
$119$	−7.46771	−0.684564
$120$	0	0
$121$	0	0
$122$	−2.68504	−0.243093
$123$	0	0
$124$	14.9205	1.33990
$125$	10.7998	0.965963
$126$	0	0
$127$	−6.34739	−0.563240	−0.281620	−	0.959526i	$-0.590872\pi$
$127$	−6.34739	−0.563240	−0.281620	+	0.959526i	$0.590872\pi$
$128$	−10.3751	−0.917036
$129$	0	0
$130$	−4.15237	−0.364187
$131$	15.2292	1.33058	0.665290	−	0.746585i	$-0.268308\pi$
$131$	15.2292	1.33058	0.665290	+	0.746585i	$0.268308\pi$
$132$	0	0
$133$	−1.34729	−0.116825
$134$	−0.988170	−0.0853649
$135$	0	0
$136$	11.4959	0.985765
$137$	19.0993	1.63176	0.815880	−	0.578221i	$-0.196253\pi$
$137$	19.0993	1.63176	0.815880	+	0.578221i	$0.196253\pi$
$138$	0	0
$139$	−3.93905	−0.334106	−0.167053	−	0.985948i	$-0.553425\pi$
$139$	−3.93905	−0.334106	−0.167053	+	0.985948i	$0.553425\pi$
$140$	4.24055	0.358392
$141$	0	0
$142$	−3.37730	−0.283417
$143$	0	0
$144$	0	0
$145$	2.59587	0.215575
$146$	−4.32124	−0.357628
$147$	0	0
$148$	−20.3523	−1.67295
$149$	20.3815	1.66972	0.834859	−	0.550464i	$-0.185549\pi$
$149$	20.3815	1.66972	0.834859	+	0.550464i	$0.185549\pi$
$150$	0	0
$151$	−4.91718	−0.400154	−0.200077	−	0.979780i	$-0.564119\pi$
$151$	−4.91718	−0.400154	−0.200077	+	0.979780i	$0.564119\pi$
$152$	2.07404	0.168227
$153$	0	0
$154$	0	0
$155$	18.7041	1.50235
$156$	0	0
$157$	2.24200	0.178931	0.0894653	−	0.995990i	$-0.471484\pi$
$157$	2.24200	0.178931	0.0894653	+	0.995990i	$0.471484\pi$
$158$	1.08722	0.0864946
$159$	0	0
$160$	−9.92863	−0.784927
$161$	4.27146	0.336638
$162$	0	0
$163$	−9.83745	−0.770529	−0.385264	−	0.922806i	$-0.625890\pi$
$163$	−9.83745	−0.770529	−0.385264	+	0.922806i	$0.625890\pi$
$164$	−2.13666	−0.166845
$165$	0	0
$166$	3.66050	0.284110
$167$	−8.64481	−0.668955	−0.334478	−	0.942404i	$-0.608560\pi$
$167$	−8.64481	−0.668955	−0.334478	+	0.942404i	$0.608560\pi$
$168$	0	0
$169$	7.17400	0.551847
$170$	6.90378	0.529496
$171$	0	0
$172$	−18.3086	−1.39601
$173$	−15.9672	−1.21397	−0.606984	−	0.794714i	$-0.707621\pi$
$173$	−15.9672	−1.21397	−0.606984	+	0.794714i	$0.707621\pi$
$174$	0	0
$175$	0.315883	0.0238785
$176$	0	0
$177$	0	0
$178$	−2.11929	−0.158848
$179$	17.5759	1.31368	0.656841	−	0.754029i	$-0.271892\pi$
$179$	17.5759	1.31368	0.656841	+	0.754029i	$0.271892\pi$
$180$	0	0
$181$	20.7645	1.54341	0.771707	−	0.635979i	$-0.219403\pi$
$181$	20.7645	1.54341	0.771707	+	0.635979i	$0.219403\pi$
$182$	1.80098	0.133497
$183$	0	0
$184$	−6.57554	−0.484756
$185$	−25.5134	−1.87578
$186$	0	0
$187$	0	0
$188$	23.6231	1.72289
$189$	0	0
$190$	1.24555	0.0903618
$191$	−10.5001	−0.759762	−0.379881	−	0.925035i	$-0.624035\pi$
$191$	−10.5001	−0.759762	−0.379881	+	0.925035i	$0.624035\pi$
$192$	0	0
$193$	−12.9500	−0.932164	−0.466082	−	0.884742i	$-0.654335\pi$
$193$	−12.9500	−0.932164	−0.466082	+	0.884742i	$0.654335\pi$
$194$	2.30398	0.165416
$195$	0	0
$196$	−1.83922	−0.131373
$197$	−0.664214	−0.0473233	−0.0236617	−	0.999720i	$-0.507532\pi$
$197$	−0.664214	−0.0473233	−0.0236617	+	0.999720i	$0.507532\pi$
$198$	0	0
$199$	13.3550	0.946710	0.473355	−	0.880872i	$-0.343043\pi$
$199$	13.3550	0.946710	0.473355	+	0.880872i	$0.343043\pi$
$200$	−0.486274	−0.0343848
$201$	0	0
$202$	−3.83470	−0.269809
$203$	−1.12589	−0.0790218
$204$	0	0
$205$	−2.67849	−0.187074
$206$	−0.323821	−0.0225617
$207$	0	0
$208$	13.7495	0.953354
$209$	0	0
$210$	0	0
$211$	13.0795	0.900428	0.450214	−	0.892921i	$-0.351348\pi$
$211$	13.0795	0.900428	0.450214	+	0.892921i	$0.351348\pi$
$212$	21.5736	1.48168
$213$	0	0
$214$	6.75548	0.461795
$215$	−22.9513	−1.56527
$216$	0	0
$217$	−8.11240	−0.550706
$218$	3.77662	0.255785
$219$	0	0
$220$	0	0
$221$	−33.5416	−2.25625
$222$	0	0
$223$	10.2293	0.685004	0.342502	−	0.939517i	$-0.388726\pi$
$223$	10.2293	0.685004	0.342502	+	0.939517i	$0.388726\pi$
$224$	4.30627	0.287725
$225$	0	0
$226$	1.74478	0.116061
$227$	2.98867	0.198365	0.0991824	−	0.995069i	$-0.468377\pi$
$227$	2.98867	0.198365	0.0991824	+	0.995069i	$0.468377\pi$
$228$	0	0
$229$	20.1017	1.32836	0.664179	−	0.747574i	$-0.268781\pi$
$229$	20.1017	1.32836	0.664179	+	0.747574i	$0.268781\pi$
$230$	−3.94890	−0.260383
$231$	0	0
$232$	1.73321	0.113791
$233$	4.67052	0.305976	0.152988	−	0.988228i	$-0.451110\pi$
$233$	4.67052	0.305976	0.152988	+	0.988228i	$0.451110\pi$
$234$	0	0
$235$	29.6136	1.93178
$236$	−1.92921	−0.125581
$237$	0	0
$238$	−2.99433	−0.194094
$239$	−22.7673	−1.47269	−0.736347	−	0.676604i	$-0.763451\pi$
$239$	−22.7673	−1.47269	−0.736347	+	0.676604i	$0.763451\pi$
$240$	0	0
$241$	−4.83714	−0.311587	−0.155794	−	0.987790i	$-0.549793\pi$
$241$	−4.83714	−0.311587	−0.155794	+	0.987790i	$0.549793\pi$
$242$	0	0
$243$	0	0
$244$	12.3161	0.788459
$245$	−2.30562	−0.147301
$246$	0	0
$247$	−6.05143	−0.385043
$248$	12.4883	0.793011
$249$	0	0
$250$	4.33039	0.273878
$251$	16.0556	1.01342	0.506710	−	0.862117i	$-0.330862\pi$
$251$	16.0556	1.01342	0.506710	+	0.862117i	$0.330862\pi$
$252$	0	0
$253$	0	0
$254$	−2.54512	−0.159695
$255$	0	0
$256$	4.63128	0.289455
$257$	−12.1743	−0.759411	−0.379706	−	0.925107i	$-0.623975\pi$
$257$	−12.1743	−0.759411	−0.379706	+	0.925107i	$0.623975\pi$
$258$	0	0
$259$	11.0657	0.687591
$260$	19.0466	1.18122
$261$	0	0
$262$	6.10645	0.377258
$263$	18.8786	1.16411	0.582053	−	0.813151i	$-0.302250\pi$
$263$	18.8786	1.16411	0.582053	+	0.813151i	$0.302250\pi$
$264$	0	0
$265$	27.0443	1.66132
$266$	−0.540224	−0.0331233
$267$	0	0
$268$	4.53267	0.276877
$269$	0.711037	0.0433527	0.0216763	−	0.999765i	$-0.493100\pi$
$269$	0.711037	0.0433527	0.0216763	+	0.999765i	$0.493100\pi$
$270$	0	0
$271$	14.7238	0.894404	0.447202	−	0.894433i	$-0.352420\pi$
$271$	14.7238	0.894404	0.447202	+	0.894433i	$0.352420\pi$
$272$	−22.8601	−1.38609
$273$	0	0
$274$	7.65823	0.462651
$275$	0	0
$276$	0	0
$277$	17.5829	1.05645	0.528227	−	0.849103i	$-0.322857\pi$
$277$	17.5829	1.05645	0.528227	+	0.849103i	$0.322857\pi$
$278$	−1.57944	−0.0947286
$279$	0	0
$280$	3.54930	0.212111
$281$	−14.9728	−0.893200	−0.446600	−	0.894734i	$-0.647365\pi$
$281$	−14.9728	−0.893200	−0.446600	+	0.894734i	$0.647365\pi$
$282$	0	0
$283$	17.4418	1.03681	0.518404	−	0.855136i	$-0.326526\pi$
$283$	17.4418	1.03681	0.518404	+	0.855136i	$0.326526\pi$
$284$	15.4915	0.919249
$285$	0	0
$286$	0	0
$287$	1.16172	0.0685742
$288$	0	0
$289$	38.7667	2.28039
$290$	1.04087	0.0611217
$291$	0	0
$292$	19.8212	1.15995
$293$	26.0123	1.51966	0.759828	−	0.650124i	$-0.225283\pi$
$293$	26.0123	1.51966	0.759828	+	0.650124i	$0.225283\pi$
$294$	0	0
$295$	−2.41843	−0.140806
$296$	−17.0347	−0.990124
$297$	0	0
$298$	8.17237	0.473413
$299$	19.1855	1.10952
$300$	0	0
$301$	9.95451	0.573768
$302$	−1.97164	−0.113455
$303$	0	0
$304$	−4.12431	−0.236546
$305$	15.4393	0.884051
$306$	0	0
$307$	−23.5921	−1.34647	−0.673236	−	0.739428i	$-0.735096\pi$
$307$	−23.5921	−1.34647	−0.673236	+	0.739428i	$0.735096\pi$
$308$	0	0
$309$	0	0
$310$	7.49979	0.425960
$311$	4.55191	0.258115	0.129058	−	0.991637i	$-0.458805\pi$
$311$	4.55191	0.258115	0.129058	+	0.991637i	$0.458805\pi$
$312$	0	0
$313$	−13.1273	−0.742000	−0.371000	−	0.928633i	$-0.620985\pi$
$313$	−13.1273	−0.742000	−0.371000	+	0.928633i	$0.620985\pi$
$314$	0.898973	0.0507320
$315$	0	0
$316$	−4.98700	−0.280541
$317$	−8.94352	−0.502318	−0.251159	−	0.967946i	$-0.580812\pi$
$317$	−8.94352	−0.502318	−0.251159	+	0.967946i	$0.580812\pi$
$318$	0	0
$319$	0	0
$320$	10.1348	0.566552
$321$	0	0
$322$	1.71273	0.0954466
$323$	10.0612	0.559820
$324$	0	0
$325$	1.41880	0.0787010
$326$	−3.94453	−0.218467
$327$	0	0
$328$	−1.78837	−0.0987462
$329$	−12.8441	−0.708117
$330$	0	0
$331$	25.9302	1.42525	0.712627	−	0.701543i	$-0.247505\pi$
$331$	25.9302	1.42525	0.712627	+	0.701543i	$0.247505\pi$
$332$	−16.7905	−0.921497
$333$	0	0
$334$	−3.46631	−0.189668
$335$	5.68208	0.310445
$336$	0	0
$337$	−15.7560	−0.858285	−0.429142	−	0.903237i	$-0.641184\pi$
$337$	−15.7560	−0.858285	−0.429142	+	0.903237i	$0.641184\pi$
$338$	2.87656	0.156464
$339$	0	0
$340$	−31.6672	−1.71739
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−15.3241	−0.826220
$345$	0	0
$346$	−6.40239	−0.344195
$347$	−2.22968	−0.119696	−0.0598478	−	0.998208i	$-0.519062\pi$
$347$	−2.22968	−0.119696	−0.0598478	+	0.998208i	$0.519062\pi$
$348$	0	0
$349$	9.61826	0.514854	0.257427	−	0.966298i	$-0.417125\pi$
$349$	9.61826	0.514854	0.257427	+	0.966298i	$0.417125\pi$
$350$	0.126660	0.00677024
$351$	0	0
$352$	0	0
$353$	9.98576	0.531488	0.265744	−	0.964044i	$-0.414382\pi$
$353$	9.98576	0.531488	0.265744	+	0.964044i	$0.414382\pi$
$354$	0	0
$355$	19.4198	1.03070
$356$	9.72106	0.515215
$357$	0	0
$358$	7.04740	0.372466
$359$	10.2469	0.540808	0.270404	−	0.962747i	$-0.412843\pi$
$359$	10.2469	0.540808	0.270404	+	0.962747i	$0.412843\pi$
$360$	0	0
$361$	−17.1848	−0.904463
$362$	8.32595	0.437602
$363$	0	0
$364$	−8.26096	−0.432992
$365$	24.8475	1.30058
$366$	0	0
$367$	3.69263	0.192754	0.0963769	−	0.995345i	$-0.469275\pi$
$367$	3.69263	0.192754	0.0963769	+	0.995345i	$0.469275\pi$
$368$	13.0757	0.681620
$369$	0	0
$370$	−10.2301	−0.531838
$371$	−11.7297	−0.608978
$372$	0	0
$373$	17.2657	0.893986	0.446993	−	0.894538i	$-0.352495\pi$
$373$	17.2657	0.893986	0.446993	+	0.894538i	$0.352495\pi$
$374$	0	0
$375$	0	0
$376$	19.7723	1.01968
$377$	−5.05697	−0.260447
$378$	0	0
$379$	−5.18952	−0.266568	−0.133284	−	0.991078i	$-0.542552\pi$
$379$	−5.18952	−0.266568	−0.133284	+	0.991078i	$0.542552\pi$
$380$	−5.71326	−0.293084
$381$	0	0
$382$	−4.21023	−0.215414
$383$	17.6212	0.900400	0.450200	−	0.892928i	$-0.351353\pi$
$383$	17.6212	0.900400	0.450200	+	0.892928i	$0.351353\pi$
$384$	0	0
$385$	0	0
$386$	−5.19258	−0.264295
$387$	0	0
$388$	−10.5682	−0.536519
$389$	−10.6188	−0.538396	−0.269198	−	0.963085i	$-0.586759\pi$
$389$	−10.6188	−0.538396	−0.269198	+	0.963085i	$0.586759\pi$
$390$	0	0
$391$	−31.8980	−1.61315
$392$	−1.53941	−0.0777521
$393$	0	0
$394$	−0.266330	−0.0134175
$395$	−6.25163	−0.314554
$396$	0	0
$397$	22.6262	1.13557	0.567787	−	0.823175i	$-0.307800\pi$
$397$	22.6262	1.13557	0.567787	+	0.823175i	$0.307800\pi$
$398$	5.35495	0.268419
$399$	0	0
$400$	0.966976	0.0483488
$401$	−4.60950	−0.230187	−0.115094	−	0.993355i	$-0.536717\pi$
$401$	−4.60950	−0.230187	−0.115094	+	0.993355i	$0.536717\pi$
$402$	0	0
$403$	−36.4372	−1.81507
$404$	17.5895	0.875112
$405$	0	0
$406$	−0.451447	−0.0224049
$407$	0	0
$408$	0	0
$409$	21.0486	1.04079	0.520394	−	0.853926i	$-0.325785\pi$
$409$	21.0486	1.04079	0.520394	+	0.853926i	$0.325785\pi$
$410$	−1.07399	−0.0530408
$411$	0	0
$412$	1.48535	0.0731778
$413$	1.04893	0.0516143
$414$	0	0
$415$	−21.0483	−1.03322
$416$	19.3418	0.948311
$417$	0	0
$418$	0	0
$419$	−14.0827	−0.687984	−0.343992	−	0.938973i	$-0.611779\pi$
$419$	−14.0827	−0.687984	−0.343992	+	0.938973i	$0.611779\pi$
$420$	0	0
$421$	27.5676	1.34356	0.671781	−	0.740750i	$-0.265530\pi$
$421$	27.5676	1.34356	0.671781	+	0.740750i	$0.265530\pi$
$422$	5.24448	0.255297
$423$	0	0
$424$	18.0569	0.876922
$425$	−2.35892	−0.114424
$426$	0	0
$427$	−6.69637	−0.324060
$428$	−30.9869	−1.49781
$429$	0	0
$430$	−9.20279	−0.443798
$431$	34.4396	1.65890	0.829448	−	0.558584i	$-0.188655\pi$
$431$	34.4396	1.65890	0.829448	+	0.558584i	$0.188655\pi$
$432$	0	0
$433$	16.1385	0.775564	0.387782	−	0.921751i	$-0.373241\pi$
$433$	16.1385	0.775564	0.387782	+	0.921751i	$0.373241\pi$
$434$	−3.25283	−0.156141
$435$	0	0
$436$	−17.3231	−0.829625
$437$	−5.75491	−0.275294
$438$	0	0
$439$	19.3036	0.921309	0.460655	−	0.887580i	$-0.347615\pi$
$439$	19.3036	0.921309	0.460655	+	0.887580i	$0.347615\pi$
$440$	0	0
$441$	0	0
$442$	−13.4492	−0.639712
$443$	−13.1312	−0.623880	−0.311940	−	0.950102i	$-0.600979\pi$
$443$	−13.1312	−0.623880	−0.311940	+	0.950102i	$0.600979\pi$
$444$	0	0
$445$	12.1862	0.577679
$446$	4.10164	0.194218
$447$	0	0
$448$	−4.39569	−0.207677
$449$	12.7511	0.601763	0.300882	−	0.953662i	$-0.402719\pi$
$449$	12.7511	0.601763	0.300882	+	0.953662i	$0.402719\pi$
$450$	0	0
$451$	0	0
$452$	−8.00317	−0.376437
$453$	0	0
$454$	1.19837	0.0562421
$455$	−10.3558	−0.485488
$456$	0	0
$457$	−9.19436	−0.430094	−0.215047	−	0.976604i	$-0.568991\pi$
$457$	−9.19436	−0.430094	−0.215047	+	0.976604i	$0.568991\pi$
$458$	8.06018	0.376628
$459$	0	0
$460$	18.1133	0.844539
$461$	−16.0714	−0.748521	−0.374260	−	0.927324i	$-0.622103\pi$
$461$	−16.0714	−0.748521	−0.374260	+	0.927324i	$0.622103\pi$
$462$	0	0
$463$	32.9203	1.52994	0.764968	−	0.644068i	$-0.222755\pi$
$463$	32.9203	1.52994	0.764968	+	0.644068i	$0.222755\pi$
$464$	−3.44655	−0.160002
$465$	0	0
$466$	1.87274	0.0867530
$467$	8.50986	0.393790	0.196895	−	0.980425i	$-0.436914\pi$
$467$	8.50986	0.393790	0.196895	+	0.980425i	$0.436914\pi$
$468$	0	0
$469$	−2.46445	−0.113798
$470$	11.8742	0.547714
$471$	0	0
$472$	−1.61473	−0.0743240
$473$	0	0
$474$	0	0
$475$	−0.425586	−0.0195272
$476$	13.7348	0.629533
$477$	0	0
$478$	−9.12901	−0.417551
$479$	9.52240	0.435089	0.217545	−	0.976050i	$-0.430195\pi$
$479$	9.52240	0.435089	0.217545	+	0.976050i	$0.430195\pi$
$480$	0	0
$481$	49.7023	2.26623
$482$	−1.93955	−0.0883439
$483$	0	0
$484$	0	0
$485$	−13.2481	−0.601566
$486$	0	0
$487$	−1.09438	−0.0495910	−0.0247955	−	0.999693i	$-0.507893\pi$
$487$	−1.09438	−0.0495910	−0.0247955	+	0.999693i	$0.507893\pi$
$488$	10.3085	0.466643
$489$	0	0
$490$	−0.924485	−0.0417640
$491$	−20.6684	−0.932751	−0.466375	−	0.884587i	$-0.654440\pi$
$491$	−20.6684	−0.932751	−0.466375	+	0.884587i	$0.654440\pi$
$492$	0	0
$493$	8.40780	0.378668
$494$	−2.42644	−0.109171
$495$	0	0
$496$	−24.8336	−1.11506
$497$	−8.42283	−0.377816
$498$	0	0
$499$	23.8350	1.06700	0.533500	−	0.845800i	$-0.320876\pi$
$499$	23.8350	1.06700	0.533500	+	0.845800i	$0.320876\pi$
$500$	−19.8632	−0.888311
$501$	0	0
$502$	6.43781	0.287333
$503$	−15.1096	−0.673706	−0.336853	−	0.941557i	$-0.609362\pi$
$503$	−15.1096	−0.673706	−0.336853	+	0.941557i	$0.609362\pi$
$504$	0	0
$505$	22.0500	0.981210
$506$	0	0
$507$	0	0
$508$	11.6743	0.517962
$509$	−1.67629	−0.0743004	−0.0371502	−	0.999310i	$-0.511828\pi$
$509$	−1.67629	−0.0743004	−0.0371502	+	0.999310i	$0.511828\pi$
$510$	0	0
$511$	−10.7770	−0.476744
$512$	22.6072	0.999105
$513$	0	0
$514$	−4.88152	−0.215315
$515$	1.86201	0.0820498
$516$	0	0
$517$	0	0
$518$	4.43703	0.194952
$519$	0	0
$520$	15.9419	0.699097
$521$	35.9717	1.57595	0.787974	−	0.615709i	$-0.211130\pi$
$521$	35.9717	1.57595	0.787974	+	0.615709i	$0.211130\pi$
$522$	0	0
$523$	−15.9612	−0.697933	−0.348967	−	0.937135i	$-0.613467\pi$
$523$	−15.9612	−0.697933	−0.348967	+	0.937135i	$0.613467\pi$
$524$	−28.0099	−1.22362
$525$	0	0
$526$	7.56976	0.330057
$527$	60.5811	2.63895
$528$	0	0
$529$	−4.75463	−0.206723
$530$	10.8440	0.471032
$531$	0	0
$532$	2.47797	0.107434
$533$	5.21793	0.226014
$534$	0	0
$535$	−38.8447	−1.67940
$536$	3.79380	0.163867
$537$	0	0
$538$	0.285104	0.0122917
$539$	0	0
$540$	0	0
$541$	−7.68655	−0.330471	−0.165235	−	0.986254i	$-0.552838\pi$
$541$	−7.68655	−0.330471	−0.165235	+	0.986254i	$0.552838\pi$
$542$	5.90379	0.253589
$543$	0	0
$544$	−32.1580	−1.37876
$545$	−21.7159	−0.930208
$546$	0	0
$547$	15.2442	0.651794	0.325897	−	0.945405i	$-0.394334\pi$
$547$	15.2442	0.651794	0.325897	+	0.945405i	$0.394334\pi$
$548$	−35.1278	−1.50058
$549$	0	0
$550$	0	0
$551$	1.51690	0.0646221
$552$	0	0
$553$	2.71147	0.115304
$554$	7.05022	0.299535
$555$	0	0
$556$	7.24479	0.307248
$557$	2.09493	0.0887649	0.0443825	−	0.999015i	$-0.485868\pi$
$557$	2.09493	0.0887649	0.0443825	+	0.999015i	$0.485868\pi$
$558$	0	0
$559$	44.7112	1.89108
$560$	−7.05793	−0.298252
$561$	0	0
$562$	−6.00363	−0.253248
$563$	−9.86821	−0.415895	−0.207948	−	0.978140i	$-0.566678\pi$
$563$	−9.86821	−0.415895	−0.207948	+	0.978140i	$0.566678\pi$
$564$	0	0
$565$	−10.0326	−0.422077
$566$	6.99365	0.293965
$567$	0	0
$568$	12.9662	0.544051
$569$	−15.2708	−0.640186	−0.320093	−	0.947386i	$-0.603714\pi$
$569$	−15.2708	−0.640186	−0.320093	+	0.947386i	$0.603714\pi$
$570$	0	0
$571$	−20.5400	−0.859573	−0.429787	−	0.902930i	$-0.641411\pi$
$571$	−20.5400	−0.859573	−0.429787	+	0.902930i	$0.641411\pi$
$572$	0	0
$573$	0	0
$574$	0.465816	0.0194428
$575$	1.34928	0.0562689
$576$	0	0
$577$	44.2582	1.84249	0.921246	−	0.388980i	$-0.127172\pi$
$577$	44.2582	1.84249	0.921246	+	0.388980i	$0.127172\pi$
$578$	15.5443	0.646557
$579$	0	0
$580$	−4.77438	−0.198245
$581$	9.12912	0.378740
$582$	0	0
$583$	0	0
$584$	16.5902	0.686507
$585$	0	0
$586$	10.4302	0.430866
$587$	−3.56942	−0.147326	−0.0736628	−	0.997283i	$-0.523469\pi$
$587$	−3.56942	−0.147326	−0.0736628	+	0.997283i	$0.523469\pi$
$588$	0	0
$589$	10.9298	0.450354
$590$	−0.969716	−0.0399226
$591$	0	0
$592$	33.8743	1.39222
$593$	−11.2563	−0.462240	−0.231120	−	0.972925i	$-0.574239\pi$
$593$	−11.2563	−0.462240	−0.231120	+	0.972925i	$0.574239\pi$
$594$	0	0
$595$	17.2177	0.705857
$596$	−37.4861	−1.53549
$597$	0	0
$598$	7.69280	0.314582
$599$	43.6263	1.78252	0.891262	−	0.453488i	$-0.149821\pi$
$599$	43.6263	1.78252	0.891262	+	0.453488i	$0.149821\pi$
$600$	0	0
$601$	−17.7286	−0.723166	−0.361583	−	0.932340i	$-0.617764\pi$
$601$	−17.7286	−0.723166	−0.361583	+	0.932340i	$0.617764\pi$
$602$	3.99146	0.162680
$603$	0	0
$604$	9.04378	0.367986
$605$	0	0
$606$	0	0
$607$	−7.16734	−0.290913	−0.145457	−	0.989365i	$-0.546465\pi$
$607$	−7.16734	−0.290913	−0.145457	+	0.989365i	$0.546465\pi$
$608$	−5.80181	−0.235294
$609$	0	0
$610$	6.19069	0.250654
$611$	−57.6898	−2.33388
$612$	0	0
$613$	−12.3784	−0.499957	−0.249979	−	0.968251i	$-0.580424\pi$
$613$	−12.3784	−0.499957	−0.249979	+	0.968251i	$0.580424\pi$
$614$	−9.45972	−0.381763
$615$	0	0
$616$	0	0
$617$	−6.11777	−0.246292	−0.123146	−	0.992389i	$-0.539298\pi$
$617$	−6.11777	−0.246292	−0.123146	+	0.992389i	$0.539298\pi$
$618$	0	0
$619$	38.8234	1.56045	0.780223	−	0.625501i	$-0.215106\pi$
$619$	38.8234	1.56045	0.780223	+	0.625501i	$0.215106\pi$
$620$	−34.4010	−1.38158
$621$	0	0
$622$	1.82518	0.0731831
$623$	−5.28542	−0.211756
$624$	0	0
$625$	−26.4796	−1.05919
$626$	−5.26366	−0.210378
$627$	0	0
$628$	−4.12353	−0.164547
$629$	−82.6357	−3.29490
$630$	0	0
$631$	−9.70799	−0.386469	−0.193234	−	0.981153i	$-0.561898\pi$
$631$	−9.70799	−0.386469	−0.193234	+	0.981153i	$0.561898\pi$
$632$	−4.17408	−0.166036
$633$	0	0
$634$	−3.58609	−0.142422
$635$	14.6347	0.580759
$636$	0	0
$637$	4.49155	0.177962
$638$	0	0
$639$	0	0
$640$	23.9210	0.945560
$641$	14.8483	0.586473	0.293236	−	0.956040i	$-0.405268\pi$
$641$	14.8483	0.586473	0.293236	+	0.956040i	$0.405268\pi$
$642$	0	0
$643$	−1.05956	−0.0417851	−0.0208926	−	0.999782i	$-0.506651\pi$
$643$	−1.05956	−0.0417851	−0.0208926	+	0.999782i	$0.506651\pi$
$644$	−7.85617	−0.309576
$645$	0	0
$646$	4.03424	0.158725
$647$	12.5622	0.493872	0.246936	−	0.969032i	$-0.420576\pi$
$647$	12.5622	0.493872	0.246936	+	0.969032i	$0.420576\pi$
$648$	0	0
$649$	0	0
$650$	0.568898	0.0223140
$651$	0	0
$652$	18.0933	0.708587
$653$	−29.6143	−1.15890	−0.579449	−	0.815009i	$-0.696732\pi$
$653$	−29.6143	−1.15890	−0.579449	+	0.815009i	$0.696732\pi$
$654$	0	0
$655$	−35.1127	−1.37197
$656$	3.55625	0.138848
$657$	0	0
$658$	−5.15009	−0.200771
$659$	9.45208	0.368201	0.184100	−	0.982907i	$-0.441063\pi$
$659$	9.45208	0.368201	0.184100	+	0.982907i	$0.441063\pi$
$660$	0	0
$661$	−8.94284	−0.347836	−0.173918	−	0.984760i	$-0.555643\pi$
$661$	−8.94284	−0.347836	−0.173918	+	0.984760i	$0.555643\pi$
$662$	10.3972	0.404100
$663$	0	0
$664$	−14.0535	−0.545381
$665$	3.10634	0.120459
$666$	0	0
$667$	−4.80918	−0.186212
$668$	15.8997	0.615179
$669$	0	0
$670$	2.27834	0.0880201
$671$	0	0
$672$	0	0
$673$	7.68207	0.296122	0.148061	−	0.988978i	$-0.452697\pi$
$673$	7.68207	0.296122	0.148061	+	0.988978i	$0.452697\pi$
$674$	−6.31769	−0.243348
$675$	0	0
$676$	−13.1946	−0.507484
$677$	35.0474	1.34698	0.673490	−	0.739197i	$-0.264795\pi$
$677$	35.0474	1.34698	0.673490	+	0.739197i	$0.264795\pi$
$678$	0	0
$679$	5.74601	0.220512
$680$	−26.5052	−1.01643
$681$	0	0
$682$	0	0
$683$	23.5098	0.899577	0.449789	−	0.893135i	$-0.351499\pi$
$683$	23.5098	0.899577	0.449789	+	0.893135i	$0.351499\pi$
$684$	0	0
$685$	−44.0356	−1.68251
$686$	0.400970	0.0153091
$687$	0	0
$688$	30.4726	1.16176
$689$	−52.6847	−2.00713
$690$	0	0
$691$	−16.2445	−0.617968	−0.308984	−	0.951067i	$-0.599989\pi$
$691$	−16.2445	−0.617968	−0.308984	+	0.951067i	$0.599989\pi$
$692$	29.3673	1.11638
$693$	0	0
$694$	−0.894036	−0.0339371
$695$	9.08195	0.344498
$696$	0	0
$697$	−8.67540	−0.328604
$698$	3.85664	0.145976
$699$	0	0
$700$	−0.580979	−0.0219589
$701$	22.5424	0.851413	0.425706	−	0.904861i	$-0.360026\pi$
$701$	22.5424	0.851413	0.425706	+	0.904861i	$0.360026\pi$
$702$	0	0
$703$	−14.9088	−0.562295
$704$	0	0
$705$	0	0
$706$	4.00399	0.150692
$707$	−9.56357	−0.359675
$708$	0	0
$709$	24.1185	0.905791	0.452896	−	0.891564i	$-0.350391\pi$
$709$	24.1185	0.905791	0.452896	+	0.891564i	$0.350391\pi$
$710$	7.78678	0.292233
$711$	0	0
$712$	8.13644	0.304926
$713$	−34.6518	−1.29772
$714$	0	0
$715$	0	0
$716$	−32.3259	−1.20808
$717$	0	0
$718$	4.10868	0.153335
$719$	−32.2087	−1.20118	−0.600591	−	0.799557i	$-0.705068\pi$
$719$	−32.2087	−1.20118	−0.600591	+	0.799557i	$0.705068\pi$
$720$	0	0
$721$	−0.807595	−0.0300764
$722$	−6.89059	−0.256441
$723$	0	0
$724$	−38.1906	−1.41934
$725$	−0.355648	−0.0132084
$726$	0	0
$727$	7.29214	0.270450	0.135225	−	0.990815i	$-0.456824\pi$
$727$	7.29214	0.270450	0.135225	+	0.990815i	$0.456824\pi$
$728$	−6.91435	−0.256263
$729$	0	0
$730$	9.96313	0.368752
$731$	−74.3374	−2.74947
$732$	0	0
$733$	5.81868	0.214918	0.107459	−	0.994210i	$-0.465729\pi$
$733$	5.81868	0.214918	0.107459	+	0.994210i	$0.465729\pi$
$734$	1.48063	0.0546512
$735$	0	0
$736$	18.3941	0.678014
$737$	0	0
$738$	0	0
$739$	−14.1113	−0.519093	−0.259546	−	0.965731i	$-0.583573\pi$
$739$	−14.1113	−0.519093	−0.259546	+	0.965731i	$0.583573\pi$
$740$	46.9248	1.72499
$741$	0	0
$742$	−4.70328	−0.172663
$743$	−2.28107	−0.0836843	−0.0418422	−	0.999124i	$-0.513323\pi$
$743$	−2.28107	−0.0836843	−0.0418422	+	0.999124i	$0.513323\pi$
$744$	0	0
$745$	−46.9920	−1.72165
$746$	6.92304	0.253471
$747$	0	0
$748$	0	0
$749$	16.8478	0.615607
$750$	0	0
$751$	−25.4870	−0.930033	−0.465016	−	0.885302i	$-0.653952\pi$
$751$	−25.4870	−0.930033	−0.465016	+	0.885302i	$0.653952\pi$
$752$	−39.3181	−1.43378
$753$	0	0
$754$	−2.02770	−0.0738443
$755$	11.3371	0.412601
$756$	0	0
$757$	25.1462	0.913953	0.456976	−	0.889479i	$-0.348932\pi$
$757$	25.1462	0.913953	0.456976	+	0.889479i	$0.348932\pi$
$758$	−2.08084	−0.0755796
$759$	0	0
$760$	−4.78195	−0.173460
$761$	−28.4523	−1.03139	−0.515697	−	0.856771i	$-0.672467\pi$
$761$	−28.4523	−1.03139	−0.515697	+	0.856771i	$0.672467\pi$
$762$	0	0
$763$	9.41870	0.340980
$764$	19.3120	0.698685
$765$	0	0
$766$	7.06556	0.255289
$767$	4.71130	0.170115
$768$	0	0
$769$	−45.0119	−1.62317	−0.811586	−	0.584233i	$-0.801395\pi$
$769$	−45.0119	−1.62317	−0.811586	+	0.584233i	$0.801395\pi$
$770$	0	0
$771$	0	0
$772$	23.8180	0.857229
$773$	−30.8653	−1.11015	−0.555074	−	0.831801i	$-0.687310\pi$
$773$	−30.8653	−1.11015	−0.555074	+	0.831801i	$0.687310\pi$
$774$	0	0
$775$	−2.56257	−0.0920502
$776$	−8.84548	−0.317534
$777$	0	0
$778$	−4.25783	−0.152651
$779$	−1.56518	−0.0560783
$780$	0	0
$781$	0	0
$782$	−12.7902	−0.457375
$783$	0	0
$784$	3.06119	0.109328
$785$	−5.16919	−0.184496
$786$	0	0
$787$	−35.6535	−1.27091	−0.635456	−	0.772137i	$-0.719188\pi$
$787$	−35.6535	−1.27091	−0.635456	+	0.772137i	$0.719188\pi$
$788$	1.22164	0.0435191
$789$	0	0
$790$	−2.50672	−0.0891850
$791$	4.35139	0.154718
$792$	0	0
$793$	−30.0771	−1.06807
$794$	9.07241	0.321968
$795$	0	0
$796$	−24.5628	−0.870605
$797$	−44.3444	−1.57076	−0.785379	−	0.619015i	$-0.787532\pi$
$797$	−44.3444	−1.57076	−0.785379	+	0.619015i	$0.787532\pi$
$798$	0	0
$799$	95.9158	3.39326
$800$	1.36028	0.0480931
$801$	0	0
$802$	−1.84827	−0.0652647
$803$	0	0
$804$	0	0
$805$	−9.84836	−0.347109
$806$	−14.6102	−0.514624
$807$	0	0
$808$	14.7223	0.517928
$809$	−6.41321	−0.225476	−0.112738	−	0.993625i	$-0.535962\pi$
$809$	−6.41321	−0.225476	−0.112738	+	0.993625i	$0.535962\pi$
$810$	0	0
$811$	34.5913	1.21466	0.607332	−	0.794448i	$-0.292240\pi$
$811$	34.5913	1.21466	0.607332	+	0.794448i	$0.292240\pi$
$812$	2.07076	0.0726693
$813$	0	0
$814$	0	0
$815$	22.6814	0.794496
$816$	0	0
$817$	−13.4116	−0.469214
$818$	8.43987	0.295093
$819$	0	0
$820$	4.92634	0.172035
$821$	13.8043	0.481772	0.240886	−	0.970553i	$-0.422562\pi$
$821$	13.8043	0.481772	0.240886	+	0.970553i	$0.422562\pi$
$822$	0	0
$823$	−56.7973	−1.97983	−0.989915	−	0.141666i	$-0.954754\pi$
$823$	−56.7973	−1.97983	−0.989915	+	0.141666i	$0.954754\pi$
$824$	1.24322	0.0433097
$825$	0	0
$826$	0.420588	0.0146341
$827$	−5.29390	−0.184087	−0.0920434	−	0.995755i	$-0.529340\pi$
$827$	−5.29390	−0.184087	−0.0920434	+	0.995755i	$0.529340\pi$
$828$	0	0
$829$	30.6618	1.06493	0.532463	−	0.846453i	$-0.321266\pi$
$829$	30.6618	1.06493	0.532463	+	0.846453i	$0.321266\pi$
$830$	−8.43973	−0.292947
$831$	0	0
$832$	−19.7434	−0.684481
$833$	−7.46771	−0.258741
$834$	0	0
$835$	19.9316	0.689763
$836$	0	0
$837$	0	0
$838$	−5.64673	−0.195063
$839$	15.4714	0.534133	0.267067	−	0.963678i	$-0.413946\pi$
$839$	15.4714	0.534133	0.267067	+	0.963678i	$0.413946\pi$
$840$	0	0
$841$	−27.7324	−0.956289
$842$	11.0538	0.380938
$843$	0	0
$844$	−24.0561	−0.828044
$845$	−16.5405	−0.569011
$846$	0	0
$847$	0	0
$848$	−35.9069	−1.23305
$849$	0	0
$850$	−0.945857	−0.0324426
$851$	47.2668	1.62029
$852$	0	0
$853$	−36.3631	−1.24505	−0.622524	−	0.782600i	$-0.713893\pi$
$853$	−36.3631	−1.24505	−0.622524	+	0.782600i	$0.713893\pi$
$854$	−2.68504	−0.0918803
$855$	0	0
$856$	−25.9358	−0.886467
$857$	−17.0833	−0.583554	−0.291777	−	0.956486i	$-0.594246\pi$
$857$	−17.0833	−0.583554	−0.291777	+	0.956486i	$0.594246\pi$
$858$	0	0
$859$	21.3315	0.727822	0.363911	−	0.931434i	$-0.381441\pi$
$859$	21.3315	0.727822	0.363911	+	0.931434i	$0.381441\pi$
$860$	42.2126	1.43944
$861$	0	0
$862$	13.8092	0.470345
$863$	−14.9495	−0.508887	−0.254443	−	0.967088i	$-0.581892\pi$
$863$	−14.9495	−0.508887	−0.254443	+	0.967088i	$0.581892\pi$
$864$	0	0
$865$	36.8144	1.25173
$866$	6.47104	0.219895
$867$	0	0
$868$	14.9205	0.506435
$869$	0	0
$870$	0	0
$871$	−11.0692	−0.375065
$872$	−14.4993	−0.491007
$873$	0	0
$874$	−2.30755	−0.0780539
$875$	10.7998	0.365100
$876$	0	0
$877$	17.2499	0.582488	0.291244	−	0.956649i	$-0.405931\pi$
$877$	17.2499	0.582488	0.291244	+	0.956649i	$0.405931\pi$
$878$	7.74015	0.261218
$879$	0	0
$880$	0	0
$881$	2.58251	0.0870069	0.0435035	−	0.999053i	$-0.486148\pi$
$881$	2.58251	0.0870069	0.0435035	+	0.999053i	$0.486148\pi$
$882$	0	0
$883$	36.2072	1.21847	0.609234	−	0.792990i	$-0.291477\pi$
$883$	36.2072	1.21847	0.609234	+	0.792990i	$0.291477\pi$
$884$	61.6904	2.07487
$885$	0	0
$886$	−5.26520	−0.176888
$887$	−53.1757	−1.78546	−0.892732	−	0.450587i	$-0.851215\pi$
$887$	−53.1757	−1.78546	−0.892732	+	0.450587i	$0.851215\pi$
$888$	0	0
$889$	−6.34739	−0.212885
$890$	4.88629	0.163789
$891$	0	0
$892$	−18.8140	−0.629938
$893$	17.3047	0.579081
$894$	0	0
$895$	−40.5233	−1.35454
$896$	−10.3751	−0.346607
$897$	0	0
$898$	5.11282	0.170617
$899$	9.13365	0.304624
$900$	0	0
$901$	87.5943	2.91819
$902$	0	0
$903$	0	0
$904$	−6.69858	−0.222792
$905$	−47.8751	−1.59142
$906$	0	0
$907$	48.9616	1.62574	0.812872	−	0.582443i	$-0.197903\pi$
$907$	48.9616	1.62574	0.812872	+	0.582443i	$0.197903\pi$
$908$	−5.49682	−0.182418
$909$	0	0
$910$	−4.15237	−0.137650
$911$	−0.998096	−0.0330684	−0.0165342	−	0.999863i	$-0.505263\pi$
$911$	−0.998096	−0.0330684	−0.0165342	+	0.999863i	$0.505263\pi$
$912$	0	0
$913$	0	0
$914$	−3.68667	−0.121944
$915$	0	0
$916$	−36.9715	−1.22157
$917$	15.2292	0.502912
$918$	0	0
$919$	53.3533	1.75996	0.879982	−	0.475008i	$-0.157555\pi$
$919$	53.3533	1.75996	0.879982	+	0.475008i	$0.157555\pi$
$920$	15.1607	0.499834
$921$	0	0
$922$	−6.44416	−0.212227
$923$	−37.8315	−1.24524
$924$	0	0
$925$	3.49547	0.114930
$926$	13.2001	0.433781
$927$	0	0
$928$	−4.84837	−0.159156
$929$	−2.72724	−0.0894778	−0.0447389	−	0.998999i	$-0.514246\pi$
$929$	−2.72724	−0.0894778	−0.0447389	+	0.998999i	$0.514246\pi$
$930$	0	0
$931$	−1.34729	−0.0441557
$932$	−8.59013	−0.281379
$933$	0	0
$934$	3.41220	0.111651
$935$	0	0
$936$	0	0
$937$	12.6735	0.414024	0.207012	−	0.978338i	$-0.433626\pi$
$937$	12.6735	0.414024	0.207012	+	0.978338i	$0.433626\pi$
$938$	−0.988170	−0.0322649
$939$	0	0
$940$	−54.4659	−1.77648
$941$	6.00255	0.195678	0.0978388	−	0.995202i	$-0.468807\pi$
$941$	6.00255	0.195678	0.0978388	+	0.995202i	$0.468807\pi$
$942$	0	0
$943$	4.96225	0.161593
$944$	3.21096	0.104508
$945$	0	0
$946$	0	0
$947$	20.2935	0.659450	0.329725	−	0.944077i	$-0.393044\pi$
$947$	20.2935	0.659450	0.329725	+	0.944077i	$0.393044\pi$
$948$	0	0
$949$	−48.4052	−1.57130
$950$	−0.170647	−0.00553653
$951$	0	0
$952$	11.4959	0.372584
$953$	25.6882	0.832122	0.416061	−	0.909337i	$-0.363410\pi$
$953$	25.6882	0.832122	0.416061	+	0.909337i	$0.363410\pi$
$954$	0	0
$955$	24.2093	0.783394
$956$	41.8741	1.35431
$957$	0	0
$958$	3.81820	0.123360
$959$	19.0993	0.616747
$960$	0	0
$961$	34.8111	1.12294
$962$	19.9291	0.642541
$963$	0	0
$964$	8.89657	0.286539
$965$	29.8579	0.961158
$966$	0	0
$967$	−3.86788	−0.124383	−0.0621914	−	0.998064i	$-0.519809\pi$
$967$	−3.86788	−0.124383	−0.0621914	+	0.998064i	$0.519809\pi$
$968$	0	0
$969$	0	0
$970$	−5.31210	−0.170561
$971$	25.8216	0.828656	0.414328	−	0.910128i	$-0.364017\pi$
$971$	25.8216	0.828656	0.414328	+	0.910128i	$0.364017\pi$
$972$	0	0
$973$	−3.93905	−0.126280
$974$	−0.438813	−0.0140605
$975$	0	0
$976$	−20.4988	−0.656152
$977$	52.2227	1.67075	0.835376	−	0.549679i	$-0.185250\pi$
$977$	52.2227	1.67075	0.835376	+	0.549679i	$0.185250\pi$
$978$	0	0
$979$	0	0
$980$	4.24055	0.135459
$981$	0	0
$982$	−8.28740	−0.264462
$983$	32.4955	1.03645	0.518223	−	0.855246i	$-0.326594\pi$
$983$	32.4955	1.03645	0.518223	+	0.855246i	$0.326594\pi$
$984$	0	0
$985$	1.53143	0.0487953
$986$	3.37128	0.107363
$987$	0	0
$988$	11.1299	0.354090
$989$	42.5203	1.35207
$990$	0	0
$991$	1.37062	0.0435391	0.0217695	−	0.999763i	$-0.493070\pi$
$991$	1.37062	0.0435391	0.0217695	+	0.999763i	$0.493070\pi$
$992$	−34.9342	−1.10916
$993$	0	0
$994$	−3.37730	−0.107122
$995$	−30.7915	−0.976157
$996$	0	0
$997$	8.64360	0.273746	0.136873	−	0.990589i	$-0.456295\pi$
$997$	8.64360	0.273746	0.136873	+	0.990589i	$0.456295\pi$
$998$	9.55712	0.302525
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.dc.1.9		16
3.2	odd	2	inner	7623.2.a.dc.1.8		16
11.3	even	5		693.2.m.k.64.5	yes	32
11.4	even	5		693.2.m.k.379.5	yes	32
11.10	odd	2		7623.2.a.db.1.8		16
33.14	odd	10		693.2.m.k.64.4	✓	32
33.26	odd	10		693.2.m.k.379.4	yes	32
33.32	even	2		7623.2.a.db.1.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.k.64.4	✓	32	33.14	odd	10
693.2.m.k.64.5	yes	32	11.3	even	5
693.2.m.k.379.4	yes	32	33.26	odd	10
693.2.m.k.379.5	yes	32	11.4	even	5
7623.2.a.db.1.8		16	11.10	odd	2
7623.2.a.db.1.9		16	33.32	even	2
7623.2.a.dc.1.8		16	3.2	odd	2	inner
7623.2.a.dc.1.9		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.400970 q^{2} -1.83922 q^{4} -2.30562 q^{5} +1.00000 q^{7} -1.53941 q^{8} +O(q^{10})\)	\(q+0.400970 q^{2} -1.83922 q^{4} -2.30562 q^{5} +1.00000 q^{7} -1.53941 q^{8} -0.924485 q^{10} +4.49155 q^{13} +0.400970 q^{14} +3.06119 q^{16} -7.46771 q^{17} -1.34729 q^{19} +4.24055 q^{20} +4.27146 q^{23} +0.315883 q^{25} +1.80098 q^{26} -1.83922 q^{28} -1.12589 q^{29} -8.11240 q^{31} +4.30627 q^{32} -2.99433 q^{34} -2.30562 q^{35} +11.0657 q^{37} -0.540224 q^{38} +3.54930 q^{40} +1.16172 q^{41} +9.95451 q^{43} +1.71273 q^{46} -12.8441 q^{47} +1.00000 q^{49} +0.126660 q^{50} -8.26096 q^{52} -11.7297 q^{53} -1.53941 q^{56} -0.451447 q^{58} +1.04893 q^{59} -6.69637 q^{61} -3.25283 q^{62} -4.39569 q^{64} -10.3558 q^{65} -2.46445 q^{67} +13.7348 q^{68} -0.924485 q^{70} -8.42283 q^{71} -10.7770 q^{73} +4.43703 q^{74} +2.47797 q^{76} +2.71147 q^{79} -7.05793 q^{80} +0.465816 q^{82} +9.12912 q^{83} +17.2177 q^{85} +3.99146 q^{86} -5.28542 q^{89} +4.49155 q^{91} -7.85617 q^{92} -5.15009 q^{94} +3.10634 q^{95} +5.74601 q^{97} +0.400970 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 16 * q^7	\( 16 q + 20 q^{4} + 16 q^{7} - 6 q^{10} + 32 q^{16} + 10 q^{19} + 44 q^{25} + 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} - 68 q^{40} + 16 q^{43} + 80 q^{46} + 16 q^{49} + 2 q^{52} + 18 q^{58} - 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} - 14 q^{73} - 14 q^{76} + 54 q^{79} + 64 q^{82} + 30 q^{85} - 60 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 16 * q^7 - 6 * q^10 + 32 * q^16 + 10 * q^19 + 44 * q^25 + 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 - 68 * q^40 + 16 * q^43 + 80 * q^46 + 16 * q^49 + 2 * q^52 + 18 * q^58 - 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 - 14 * q^73 - 14 * q^76 + 54 * q^79 + 64 * q^82 + 30 * q^85 - 60 * q^94 + 108 * q^97

Introduction

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