LMFDB - Embedded newform 3675.2.a.bv.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3675,2,Mod(1,3675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3675.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4,8,0,0,0,0,4,0,0,8,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.3450227428$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.54336$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.49721 q^{2} +1.00000 q^{3} +4.23607 q^{4} +2.49721 q^{6} +5.58394 q^{8} +1.00000 q^{9} -4.47214 q^{11} +4.23607 q^{12} +5.23607 q^{13} +5.47214 q^{16} +0.763932 q^{17} +2.49721 q^{18} +8.08115 q^{19} -11.1679 q^{22} +3.08672 q^{23} +5.58394 q^{24} +13.0756 q^{26} +1.00000 q^{27} +2.00000 q^{29} -1.90770 q^{31} +2.49721 q^{32} -4.47214 q^{33} +1.90770 q^{34} +4.23607 q^{36} -6.17345 q^{37} +20.1803 q^{38} +5.23607 q^{39} +6.90212 q^{41} -9.98885 q^{43} -18.9443 q^{44} +7.70820 q^{46} -4.94427 q^{47} +5.47214 q^{48} +0.763932 q^{51} +22.1803 q^{52} +1.90770 q^{53} +2.49721 q^{54} +8.08115 q^{57} +4.99442 q^{58} -9.98885 q^{59} +3.81540 q^{61} -4.76393 q^{62} -4.70820 q^{64} -11.1679 q^{66} +6.17345 q^{67} +3.23607 q^{68} +3.08672 q^{69} +0.472136 q^{71} +5.58394 q^{72} +2.76393 q^{73} -15.4164 q^{74} +34.2323 q^{76} +13.0756 q^{78} +12.9443 q^{79} +1.00000 q^{81} +17.2361 q^{82} +6.47214 q^{83} -24.9443 q^{86} +2.00000 q^{87} -24.9721 q^{88} -9.26017 q^{89} +13.0756 q^{92} -1.90770 q^{93} -12.3469 q^{94} +2.49721 q^{96} +10.1803 q^{97} -4.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 8 q^{4} + 4 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{16} + 12 q^{17} + 4 q^{27} + 8 q^{29} + 8 q^{36} + 36 q^{38} + 12 q^{39} - 40 q^{44} + 4 q^{46} + 16 q^{47} + 4 q^{48} + 12 q^{51} + 44 q^{52}+ \cdots - 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 8 * q^4 + 4 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^16 + 12 * q^17 + 4 * q^27 + 8 * q^29 + 8 * q^36 + 36 * q^38 + 12 * q^39 - 40 * q^44 + 4 * q^46 + 16 * q^47 + 4 * q^48 + 12 * q^51 + 44 * q^52 - 28 * q^62 + 8 * q^64 + 4 * q^68 - 16 * q^71 + 20 * q^73 - 8 * q^74 + 16 * q^79 + 4 * q^81 + 60 * q^82 + 8 * q^83 - 64 * q^86 + 8 * q^87 - 4 * q^97

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 $

$2$	2.49721	1.76580	0.882898	−	0.469565i	$-0.155589\pi$
$2$	2.49721	1.76580	0.882898	+	0.469565i	$0.155589\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.23607	2.11803
$5$	0	0
$5$	0	0
$6$	2.49721	1.01948
$7$	0	0
$7$	0	0
$8$	5.58394	1.97422
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	4.23607	1.22285
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	0	0
$15$	0	0
$16$	5.47214	1.36803
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	2.49721	0.588599
$19$	8.08115	1.85394	0.926971	−	0.375132i	$-0.122403\pi$
$19$	8.08115	1.85394	0.926971	+	0.375132i	$0.122403\pi$
$20$	0	0
$21$	0	0
$22$	−11.1679	−2.38100
$23$	3.08672	0.643626	0.321813	−	0.946803i	$-0.395708\pi$
$23$	3.08672	0.643626	0.321813	+	0.946803i	$0.395708\pi$
$24$	5.58394	1.13982
$25$	0	0
$26$	13.0756	2.56433
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−1.90770	−0.342633	−0.171317	−	0.985216i	$-0.554802\pi$
$31$	−1.90770	−0.342633	−0.171317	+	0.985216i	$0.554802\pi$
$32$	2.49721	0.441449
$33$	−4.47214	−0.778499
$34$	1.90770	0.327168
$35$	0	0
$36$	4.23607	0.706011
$37$	−6.17345	−1.01491	−0.507454	−	0.861679i	$-0.669413\pi$
$37$	−6.17345	−1.01491	−0.507454	+	0.861679i	$0.669413\pi$
$38$	20.1803	3.27368
$39$	5.23607	0.838442
$40$	0	0
$41$	6.90212	1.07793	0.538965	−	0.842328i	$-0.318815\pi$
$41$	6.90212	1.07793	0.538965	+	0.842328i	$0.318815\pi$
$42$	0	0
$43$	−9.98885	−1.52329	−0.761643	−	0.647997i	$-0.775607\pi$
$43$	−9.98885	−1.52329	−0.761643	+	0.647997i	$0.775607\pi$
$44$	−18.9443	−2.85596
$45$	0	0
$46$	7.70820	1.13651
$47$	−4.94427	−0.721196	−0.360598	−	0.932721i	$-0.617427\pi$
$47$	−4.94427	−0.721196	−0.360598	+	0.932721i	$0.617427\pi$
$48$	5.47214	0.789835
$49$	0	0
$50$	0	0
$51$	0.763932	0.106972
$52$	22.1803	3.07586
$53$	1.90770	0.262043	0.131021	−	0.991380i	$-0.458174\pi$
$53$	1.90770	0.262043	0.131021	+	0.991380i	$0.458174\pi$
$54$	2.49721	0.339828
$55$	0	0
$56$	0	0
$57$	8.08115	1.07037
$58$	4.99442	0.655800
$59$	−9.98885	−1.30044	−0.650219	−	0.759747i	$-0.725323\pi$
$59$	−9.98885	−1.30044	−0.650219	+	0.759747i	$0.725323\pi$
$60$	0	0
$61$	3.81540	0.488512	0.244256	−	0.969711i	$-0.421456\pi$
$61$	3.81540	0.488512	0.244256	+	0.969711i	$0.421456\pi$
$62$	−4.76393	−0.605020
$63$	0	0
$64$	−4.70820	−0.588525
$65$	0	0
$66$	−11.1679	−1.37467
$67$	6.17345	0.754207	0.377103	−	0.926171i	$-0.376920\pi$
$67$	6.17345	0.754207	0.377103	+	0.926171i	$0.376920\pi$
$68$	3.23607	0.392431
$69$	3.08672	0.371598
$70$	0	0
$71$	0.472136	0.0560322	0.0280161	−	0.999607i	$-0.491081\pi$
$71$	0.472136	0.0560322	0.0280161	+	0.999607i	$0.491081\pi$
$72$	5.58394	0.658073
$73$	2.76393	0.323494	0.161747	−	0.986832i	$-0.448287\pi$
$73$	2.76393	0.323494	0.161747	+	0.986832i	$0.448287\pi$
$74$	−15.4164	−1.79212
$75$	0	0
$76$	34.2323	3.92671
$77$	0	0
$78$	13.0756	1.48052
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	17.2361	1.90341
$83$	6.47214	0.710409	0.355205	−	0.934789i	$-0.384411\pi$
$83$	6.47214	0.710409	0.355205	+	0.934789i	$0.384411\pi$
$84$	0	0
$85$	0	0
$86$	−24.9443	−2.68981
$87$	2.00000	0.214423
$88$	−24.9721	−2.66204
$89$	−9.26017	−0.981576	−0.490788	−	0.871279i	$-0.663291\pi$
$89$	−9.26017	−0.981576	−0.490788	+	0.871279i	$0.663291\pi$
$90$	0	0
$91$	0	0
$92$	13.0756	1.36322
$93$	−1.90770	−0.197819
$94$	−12.3469	−1.27349
$95$	0	0
$96$	2.49721	0.254871
$97$	10.1803	1.03366	0.516828	−	0.856089i	$-0.327113\pi$
$97$	10.1803	1.03366	0.516828	+	0.856089i	$0.327113\pi$
$98$	0	0
$99$	−4.47214	−0.449467
$100$	0	0
$101$	−13.0756	−1.30107	−0.650534	−	0.759477i	$-0.725455\pi$
$101$	−13.0756	−1.30107	−0.650534	+	0.759477i	$0.725455\pi$
$102$	1.90770	0.188890
$103$	1.52786	0.150545	0.0752725	−	0.997163i	$-0.476017\pi$
$103$	1.52786	0.150545	0.0752725	+	0.997163i	$0.476017\pi$
$104$	29.2379	2.86701
$105$	0	0
$106$	4.76393	0.462714
$107$	−6.90212	−0.667254	−0.333627	−	0.942705i	$-0.608273\pi$
$107$	−6.90212	−0.667254	−0.333627	+	0.942705i	$0.608273\pi$
$108$	4.23607	0.407616
$109$	12.4721	1.19461	0.597307	−	0.802013i	$-0.296237\pi$
$109$	12.4721	1.19461	0.597307	+	0.802013i	$0.296237\pi$
$110$	0	0
$111$	−6.17345	−0.585958
$112$	0	0
$113$	−4.26575	−0.401288	−0.200644	−	0.979664i	$-0.564303\pi$
$113$	−4.26575	−0.401288	−0.200644	+	0.979664i	$0.564303\pi$
$114$	20.1803	1.89006
$115$	0	0
$116$	8.47214	0.786618
$117$	5.23607	0.484075
$118$	−24.9443	−2.29631
$119$	0	0
$120$	0	0
$121$	9.00000	0.818182
$122$	9.52786	0.862612
$123$	6.90212	0.622344
$124$	−8.08115	−0.725709
$125$	0	0
$126$	0	0
$127$	−3.81540	−0.338562	−0.169281	−	0.985568i	$-0.554145\pi$
$127$	−3.81540	−0.338562	−0.169281	+	0.985568i	$0.554145\pi$
$128$	−16.7518	−1.48066
$129$	−9.98885	−0.879469
$130$	0	0
$131$	9.98885	0.872730	0.436365	−	0.899770i	$-0.356266\pi$
$131$	9.98885	0.872730	0.436365	+	0.899770i	$0.356266\pi$
$132$	−18.9443	−1.64889
$133$	0	0
$134$	15.4164	1.33177
$135$	0	0
$136$	4.26575	0.365785
$137$	−8.08115	−0.690419	−0.345210	−	0.938526i	$-0.612192\pi$
$137$	−8.08115	−0.690419	−0.345210	+	0.938526i	$0.612192\pi$
$138$	7.70820	0.656166
$139$	−14.2546	−1.20906	−0.604530	−	0.796583i	$-0.706639\pi$
$139$	−14.2546	−1.20906	−0.604530	+	0.796583i	$0.706639\pi$
$140$	0	0
$141$	−4.94427	−0.416383
$142$	1.17902	0.0989415
$143$	−23.4164	−1.95818
$144$	5.47214	0.456011
$145$	0	0
$146$	6.90212	0.571224
$147$	0	0
$148$	−26.1511	−2.14961
$149$	−18.9443	−1.55198	−0.775988	−	0.630748i	$-0.782748\pi$
$149$	−18.9443	−1.55198	−0.775988	+	0.630748i	$0.782748\pi$
$150$	0	0
$151$	−8.94427	−0.727875	−0.363937	−	0.931423i	$-0.618568\pi$
$151$	−8.94427	−0.727875	−0.363937	+	0.931423i	$0.618568\pi$
$152$	45.1246	3.66009
$153$	0.763932	0.0617602
$154$	0	0
$155$	0	0
$156$	22.1803	1.77585
$157$	−8.65248	−0.690543	−0.345271	−	0.938503i	$-0.612213\pi$
$157$	−8.65248	−0.690543	−0.345271	+	0.938503i	$0.612213\pi$
$158$	32.3246	2.57161
$159$	1.90770	0.151290
$160$	0	0
$161$	0	0
$162$	2.49721	0.196200
$163$	3.81540	0.298845	0.149423	−	0.988773i	$-0.452259\pi$
$163$	3.81540	0.298845	0.149423	+	0.988773i	$0.452259\pi$
$164$	29.2379	2.28309
$165$	0	0
$166$	16.1623	1.25444
$167$	−15.4164	−1.19296	−0.596479	−	0.802629i	$-0.703434\pi$
$167$	−15.4164	−1.19296	−0.596479	+	0.802629i	$0.703434\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	8.08115	0.617981
$172$	−42.3134	−3.22637
$173$	19.2361	1.46249	0.731246	−	0.682114i	$-0.238939\pi$
$173$	19.2361	1.46249	0.731246	+	0.682114i	$0.238939\pi$
$174$	4.99442	0.378626
$175$	0	0
$176$	−24.4721	−1.84466
$177$	−9.98885	−0.750808
$178$	−23.1246	−1.73326
$179$	−7.52786	−0.562659	−0.281329	−	0.959611i	$-0.590775\pi$
$179$	−7.52786	−0.562659	−0.281329	+	0.959611i	$0.590775\pi$
$180$	0	0
$181$	−9.98885	−0.742465	−0.371233	−	0.928540i	$-0.621065\pi$
$181$	−9.98885	−0.742465	−0.371233	+	0.928540i	$0.621065\pi$
$182$	0	0
$183$	3.81540	0.282043
$184$	17.2361	1.27066
$185$	0	0
$186$	−4.76393	−0.349308
$187$	−3.41641	−0.249832
$188$	−20.9443	−1.52752
$189$	0	0
$190$	0	0
$191$	−3.52786	−0.255267	−0.127634	−	0.991821i	$-0.540738\pi$
$191$	−3.52786	−0.255267	−0.127634	+	0.991821i	$0.540738\pi$
$192$	−4.70820	−0.339785
$193$	−13.8042	−0.993652	−0.496826	−	0.867850i	$-0.665501\pi$
$193$	−13.8042	−0.993652	−0.496826	+	0.867850i	$0.665501\pi$
$194$	25.4225	1.82523
$195$	0	0
$196$	0	0
$197$	−1.90770	−0.135918	−0.0679590	−	0.997688i	$-0.521649\pi$
$197$	−1.90770	−0.135918	−0.0679590	+	0.997688i	$0.521649\pi$
$198$	−11.1679	−0.793666
$199$	−11.8965	−0.843324	−0.421662	−	0.906753i	$-0.638553\pi$
$199$	−11.8965	−0.843324	−0.421662	+	0.906753i	$0.638553\pi$
$200$	0	0
$201$	6.17345	0.435441
$202$	−32.6525	−2.29742
$203$	0	0
$204$	3.23607	0.226570
$205$	0	0
$206$	3.81540	0.265832
$207$	3.08672	0.214542
$208$	28.6525	1.98669
$209$	−36.1400	−2.49986
$210$	0	0
$211$	−12.9443	−0.891120	−0.445560	−	0.895252i	$-0.646995\pi$
$211$	−12.9443	−0.891120	−0.445560	+	0.895252i	$0.646995\pi$
$212$	8.08115	0.555016
$213$	0.472136	0.0323502
$214$	−17.2361	−1.17823
$215$	0	0
$216$	5.58394	0.379939
$217$	0	0
$218$	31.1456	2.10944
$219$	2.76393	0.186769
$220$	0	0
$221$	4.00000	0.269069
$222$	−15.4164	−1.03468
$223$	−17.8885	−1.19791	−0.598953	−	0.800784i	$-0.704416\pi$
$223$	−17.8885	−1.19791	−0.598953	+	0.800784i	$0.704416\pi$
$224$	0	0
$225$	0	0
$226$	−10.6525	−0.708592
$227$	−1.52786	−0.101408	−0.0507039	−	0.998714i	$-0.516146\pi$
$227$	−1.52786	−0.101408	−0.0507039	+	0.998714i	$0.516146\pi$
$228$	34.2323	2.26709
$229$	22.3357	1.47599	0.737994	−	0.674808i	$-0.235773\pi$
$229$	22.3357	1.47599	0.737994	+	0.674808i	$0.235773\pi$
$230$	0	0
$231$	0	0
$232$	11.1679	0.733207
$233$	−11.8965	−0.779369	−0.389684	−	0.920948i	$-0.627416\pi$
$233$	−11.8965	−0.779369	−0.389684	+	0.920948i	$0.627416\pi$
$234$	13.0756	0.854777
$235$	0	0
$236$	−42.3134	−2.75437
$237$	12.9443	0.840821
$238$	0	0
$239$	−7.52786	−0.486937	−0.243469	−	0.969909i	$-0.578285\pi$
$239$	−7.52786	−0.486937	−0.243469	+	0.969909i	$0.578285\pi$
$240$	0	0
$241$	−6.17345	−0.397667	−0.198833	−	0.980033i	$-0.563715\pi$
$241$	−6.17345	−0.397667	−0.198833	+	0.980033i	$0.563715\pi$
$242$	22.4749	1.44474
$243$	1.00000	0.0641500
$244$	16.1623	1.03468
$245$	0	0
$246$	17.2361	1.09893
$247$	42.3134	2.69234
$248$	−10.6525	−0.676433
$249$	6.47214	0.410155
$250$	0	0
$251$	2.35805	0.148839	0.0744193	−	0.997227i	$-0.476290\pi$
$251$	2.35805	0.148839	0.0744193	+	0.997227i	$0.476290\pi$
$252$	0	0
$253$	−13.8042	−0.867866
$254$	−9.52786	−0.597831
$255$	0	0
$256$	−32.4164	−2.02603
$257$	21.7082	1.35412	0.677060	−	0.735928i	$-0.263254\pi$
$257$	21.7082	1.35412	0.677060	+	0.735928i	$0.263254\pi$
$258$	−24.9443	−1.55296
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	24.9443	1.54106
$263$	−15.4336	−0.951678	−0.475839	−	0.879532i	$-0.657855\pi$
$263$	−15.4336	−0.951678	−0.475839	+	0.879532i	$0.657855\pi$
$264$	−24.9721	−1.53693
$265$	0	0
$266$	0	0
$267$	−9.26017	−0.566713
$268$	26.1511	1.59744
$269$	−6.90212	−0.420830	−0.210415	−	0.977612i	$-0.567482\pi$
$269$	−6.90212	−0.420830	−0.210415	+	0.977612i	$0.567482\pi$
$270$	0	0
$271$	−18.0700	−1.09767	−0.548837	−	0.835929i	$-0.684929\pi$
$271$	−18.0700	−1.09767	−0.548837	+	0.835929i	$0.684929\pi$
$272$	4.18034	0.253470
$273$	0	0
$274$	−20.1803	−1.21914
$275$	0	0
$276$	13.0756	0.787057
$277$	9.98885	0.600172	0.300086	−	0.953912i	$-0.402985\pi$
$277$	9.98885	0.600172	0.300086	+	0.953912i	$0.402985\pi$
$278$	−35.5967	−2.13495
$279$	−1.90770	−0.114211
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	−12.3469	−0.735247
$283$	12.3607	0.734766	0.367383	−	0.930070i	$-0.380254\pi$
$283$	12.3607	0.734766	0.367383	+	0.930070i	$0.380254\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	−58.4757	−3.45774
$287$	0	0
$288$	2.49721	0.147150
$289$	−16.4164	−0.965671
$290$	0	0
$291$	10.1803	0.596782
$292$	11.7082	0.685171
$293$	20.1803	1.17895	0.589474	−	0.807787i	$-0.299335\pi$
$293$	20.1803	1.17895	0.589474	+	0.807787i	$0.299335\pi$
$294$	0	0
$295$	0	0
$296$	−34.4721	−2.00365
$297$	−4.47214	−0.259500
$298$	−47.3079	−2.74047
$299$	16.1623	0.934690
$300$	0	0
$301$	0	0
$302$	−22.3357	−1.28528
$303$	−13.0756	−0.751172
$304$	44.2211	2.53626
$305$	0	0
$306$	1.90770	0.109056
$307$	2.47214	0.141092	0.0705461	−	0.997509i	$-0.477526\pi$
$307$	2.47214	0.141092	0.0705461	+	0.997509i	$0.477526\pi$
$308$	0	0
$309$	1.52786	0.0869171
$310$	0	0
$311$	−3.81540	−0.216352	−0.108176	−	0.994132i	$-0.534501\pi$
$311$	−3.81540	−0.216352	−0.108176	+	0.994132i	$0.534501\pi$
$312$	29.2379	1.65527
$313$	24.6525	1.39344	0.696720	−	0.717343i	$-0.254642\pi$
$313$	24.6525	1.39344	0.696720	+	0.717343i	$0.254642\pi$
$314$	−21.6071	−1.21936
$315$	0	0
$316$	54.8328	3.08459
$317$	−10.4392	−0.586324	−0.293162	−	0.956063i	$-0.594707\pi$
$317$	−10.4392	−0.586324	−0.293162	+	0.956063i	$0.594707\pi$
$318$	4.76393	0.267148
$319$	−8.94427	−0.500783
$320$	0	0
$321$	−6.90212	−0.385239
$322$	0	0
$323$	6.17345	0.343500
$324$	4.23607	0.235337
$325$	0	0
$326$	9.52786	0.527700
$327$	12.4721	0.689711
$328$	38.5410	2.12807
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	27.4164	1.50467
$333$	−6.17345	−0.338303
$334$	−38.4980	−2.10652
$335$	0	0
$336$	0	0
$337$	−3.81540	−0.207838	−0.103919	−	0.994586i	$-0.533138\pi$
$337$	−3.81540	−0.207838	−0.103919	+	0.994586i	$0.533138\pi$
$338$	36.0008	1.95819
$339$	−4.26575	−0.231684
$340$	0	0
$341$	8.53149	0.462006
$342$	20.1803	1.09123
$343$	0	0
$344$	−55.7771	−3.00730
$345$	0	0
$346$	48.0365	2.58246
$347$	25.4225	1.36475	0.682375	−	0.731003i	$-0.260947\pi$
$347$	25.4225	1.36475	0.682375	+	0.731003i	$0.260947\pi$
$348$	8.47214	0.454154
$349$	8.53149	0.456680	0.228340	−	0.973581i	$-0.426670\pi$
$349$	8.53149	0.456680	0.228340	+	0.973581i	$0.426670\pi$
$350$	0	0
$351$	5.23607	0.279481
$352$	−11.1679	−0.595250
$353$	−3.23607	−0.172239	−0.0861193	−	0.996285i	$-0.527447\pi$
$353$	−3.23607	−0.172239	−0.0861193	+	0.996285i	$0.527447\pi$
$354$	−24.9443	−1.32577
$355$	0	0
$356$	−39.2267	−2.07901
$357$	0	0
$358$	−18.7987	−0.993541
$359$	20.4721	1.08048	0.540239	−	0.841512i	$-0.318334\pi$
$359$	20.4721	1.08048	0.540239	+	0.841512i	$0.318334\pi$
$360$	0	0
$361$	46.3050	2.43710
$362$	−24.9443	−1.31104
$363$	9.00000	0.472377
$364$	0	0
$365$	0	0
$366$	9.52786	0.498029
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	16.8910	0.880503
$369$	6.90212	0.359310
$370$	0	0
$371$	0	0
$372$	−8.08115	−0.418988
$373$	32.3246	1.67370	0.836852	−	0.547429i	$-0.184393\pi$
$373$	32.3246	1.67370	0.836852	+	0.547429i	$0.184393\pi$
$374$	−8.53149	−0.441153
$375$	0	0
$376$	−27.6085	−1.42380
$377$	10.4721	0.539342
$378$	0	0
$379$	3.05573	0.156962	0.0784811	−	0.996916i	$-0.474993\pi$
$379$	3.05573	0.156962	0.0784811	+	0.996916i	$0.474993\pi$
$380$	0	0
$381$	−3.81540	−0.195469
$382$	−8.80982	−0.450750
$383$	10.4721	0.535101	0.267551	−	0.963544i	$-0.413786\pi$
$383$	10.4721	0.535101	0.267551	+	0.963544i	$0.413786\pi$
$384$	−16.7518	−0.854862
$385$	0	0
$386$	−34.4721	−1.75459
$387$	−9.98885	−0.507762
$388$	43.1246	2.18932
$389$	−11.8885	−0.602773	−0.301387	−	0.953502i	$-0.597449\pi$
$389$	−11.8885	−0.602773	−0.301387	+	0.953502i	$0.597449\pi$
$390$	0	0
$391$	2.35805	0.119252
$392$	0	0
$393$	9.98885	0.503871
$394$	−4.76393	−0.240003
$395$	0	0
$396$	−18.9443	−0.951985
$397$	−9.23607	−0.463545	−0.231772	−	0.972770i	$-0.574452\pi$
$397$	−9.23607	−0.463545	−0.231772	+	0.972770i	$0.574452\pi$
$398$	−29.7082	−1.48914
$399$	0	0
$400$	0	0
$401$	36.8328	1.83934	0.919672	−	0.392689i	$-0.128455\pi$
$401$	36.8328	1.83934	0.919672	+	0.392689i	$0.128455\pi$
$402$	15.4164	0.768901
$403$	−9.98885	−0.497580
$404$	−55.3890	−2.75571
$405$	0	0
$406$	0	0
$407$	27.6085	1.36850
$408$	4.26575	0.211186
$409$	−33.7819	−1.67041	−0.835205	−	0.549939i	$-0.814651\pi$
$409$	−33.7819	−1.67041	−0.835205	+	0.549939i	$0.814651\pi$
$410$	0	0
$411$	−8.08115	−0.398614
$412$	6.47214	0.318859
$413$	0	0
$414$	7.70820	0.378838
$415$	0	0
$416$	13.0756	0.641083
$417$	−14.2546	−0.698051
$418$	−90.2492	−4.41423
$419$	−1.45735	−0.0711964	−0.0355982	−	0.999366i	$-0.511334\pi$
$419$	−1.45735	−0.0711964	−0.0355982	+	0.999366i	$0.511334\pi$
$420$	0	0
$421$	16.4721	0.802803	0.401401	−	0.915902i	$-0.368523\pi$
$421$	16.4721	0.802803	0.401401	+	0.915902i	$0.368523\pi$
$422$	−32.3246	−1.57354
$423$	−4.94427	−0.240399
$424$	10.6525	0.517330
$425$	0	0
$426$	1.17902	0.0571239
$427$	0	0
$428$	−29.2379	−1.41327
$429$	−23.4164	−1.13055
$430$	0	0
$431$	−4.47214	−0.215415	−0.107708	−	0.994183i	$-0.534351\pi$
$431$	−4.47214	−0.215415	−0.107708	+	0.994183i	$0.534351\pi$
$432$	5.47214	0.263278
$433$	35.7082	1.71603	0.858013	−	0.513627i	$-0.171699\pi$
$433$	35.7082	1.71603	0.858013	+	0.513627i	$0.171699\pi$
$434$	0	0
$435$	0	0
$436$	52.8328	2.53023
$437$	24.9443	1.19325
$438$	6.90212	0.329796
$439$	−24.2434	−1.15708	−0.578538	−	0.815655i	$-0.696377\pi$
$439$	−24.2434	−1.15708	−0.578538	+	0.815655i	$0.696377\pi$
$440$	0	0
$441$	0	0
$442$	9.98885	0.475121
$443$	−31.5959	−1.50117	−0.750584	−	0.660775i	$-0.770228\pi$
$443$	−31.5959	−1.50117	−0.750584	+	0.660775i	$0.770228\pi$
$444$	−26.1511	−1.24108
$445$	0	0
$446$	−44.6715	−2.11526
$447$	−18.9443	−0.896033
$448$	0	0
$449$	−17.0557	−0.804910	−0.402455	−	0.915440i	$-0.631843\pi$
$449$	−17.0557	−0.804910	−0.402455	+	0.915440i	$0.631843\pi$
$450$	0	0
$451$	−30.8672	−1.45348
$452$	−18.0700	−0.849941
$453$	−8.94427	−0.420239
$454$	−3.81540	−0.179066
$455$	0	0
$456$	45.1246	2.11315
$457$	23.7931	1.11299	0.556497	−	0.830850i	$-0.312145\pi$
$457$	23.7931	1.11299	0.556497	+	0.830850i	$0.312145\pi$
$458$	55.7771	2.60629
$459$	0.763932	0.0356573
$460$	0	0
$461$	−36.8687	−1.71715	−0.858573	−	0.512692i	$-0.828648\pi$
$461$	−36.8687	−1.71715	−0.858573	+	0.512692i	$0.828648\pi$
$462$	0	0
$463$	28.5092	1.32493	0.662467	−	0.749091i	$-0.269509\pi$
$463$	28.5092	1.32493	0.662467	+	0.749091i	$0.269509\pi$
$464$	10.9443	0.508075
$465$	0	0
$466$	−29.7082	−1.37621
$467$	−27.4164	−1.26868	−0.634340	−	0.773054i	$-0.718728\pi$
$467$	−27.4164	−1.26868	−0.634340	+	0.773054i	$0.718728\pi$
$468$	22.1803	1.02529
$469$	0	0
$470$	0	0
$471$	−8.65248	−0.398685
$472$	−55.7771	−2.56735
$473$	44.6715	2.05400
$474$	32.3246	1.48472
$475$	0	0
$476$	0	0
$477$	1.90770	0.0873476
$478$	−18.7987	−0.859831
$479$	−28.5092	−1.30262	−0.651309	−	0.758813i	$-0.725780\pi$
$479$	−28.5092	−1.30262	−0.651309	+	0.758813i	$0.725780\pi$
$480$	0	0
$481$	−32.3246	−1.47387
$482$	−15.4164	−0.702198
$483$	0	0
$484$	38.1246	1.73294
$485$	0	0
$486$	2.49721	0.113276
$487$	9.98885	0.452638	0.226319	−	0.974053i	$-0.427331\pi$
$487$	9.98885	0.452638	0.226319	+	0.974053i	$0.427331\pi$
$488$	21.3050	0.964430
$489$	3.81540	0.172538
$490$	0	0
$491$	18.3607	0.828606	0.414303	−	0.910139i	$-0.364025\pi$
$491$	18.3607	0.828606	0.414303	+	0.910139i	$0.364025\pi$
$492$	29.2379	1.31814
$493$	1.52786	0.0688115
$494$	105.666	4.75412
$495$	0	0
$496$	−10.4392	−0.468734
$497$	0	0
$498$	16.1623	0.724250
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	−15.4164	−0.688754
$502$	5.88854	0.262819
$503$	29.5279	1.31658	0.658291	−	0.752763i	$-0.271280\pi$
$503$	29.5279	1.31658	0.658291	+	0.752763i	$0.271280\pi$
$504$	0	0
$505$	0	0
$506$	−34.4721	−1.53247
$507$	14.4164	0.640255
$508$	−16.1623	−0.717086
$509$	−13.0756	−0.579565	−0.289782	−	0.957093i	$-0.593583\pi$
$509$	−13.0756	−0.579565	−0.289782	+	0.957093i	$0.593583\pi$
$510$	0	0
$511$	0	0
$512$	−47.4470	−2.09688
$513$	8.08115	0.356791
$514$	54.2100	2.39110
$515$	0	0
$516$	−42.3134	−1.86275
$517$	22.1115	0.972461
$518$	0	0
$519$	19.2361	0.844370
$520$	0	0
$521$	39.2267	1.71855	0.859277	−	0.511511i	$-0.170914\pi$
$521$	39.2267	1.71855	0.859277	+	0.511511i	$0.170914\pi$
$522$	4.99442	0.218600
$523$	15.0557	0.658341	0.329171	−	0.944270i	$-0.393231\pi$
$523$	15.0557	0.658341	0.329171	+	0.944270i	$0.393231\pi$
$524$	42.3134	1.84847
$525$	0	0
$526$	−38.5410	−1.68047
$527$	−1.45735	−0.0634833
$528$	−24.4721	−1.06501
$529$	−13.4721	−0.585745
$530$	0	0
$531$	−9.98885	−0.433479
$532$	0	0
$533$	36.1400	1.56540
$534$	−23.1246	−1.00070
$535$	0	0
$536$	34.4721	1.48897
$537$	−7.52786	−0.324851
$538$	−17.2361	−0.743100
$539$	0	0
$540$	0	0
$541$	−24.4721	−1.05214	−0.526070	−	0.850441i	$-0.676335\pi$
$541$	−24.4721	−1.05214	−0.526070	+	0.850441i	$0.676335\pi$
$542$	−45.1246	−1.93827
$543$	−9.98885	−0.428663
$544$	1.90770	0.0817920
$545$	0	0
$546$	0	0
$547$	−38.4980	−1.64606	−0.823029	−	0.568000i	$-0.807717\pi$
$547$	−38.4980	−1.64606	−0.823029	+	0.568000i	$0.807717\pi$
$548$	−34.2323	−1.46233
$549$	3.81540	0.162837
$550$	0	0
$551$	16.1623	0.688537
$552$	17.2361	0.733616
$553$	0	0
$554$	24.9443	1.05978
$555$	0	0
$556$	−60.3834	−2.56083
$557$	−30.4169	−1.28881	−0.644403	−	0.764686i	$-0.722894\pi$
$557$	−30.4169	−1.28881	−0.644403	+	0.764686i	$0.722894\pi$
$558$	−4.76393	−0.201673
$559$	−52.3023	−2.21215
$560$	0	0
$561$	−3.41641	−0.144241
$562$	−54.9387	−2.31745
$563$	13.8885	0.585332	0.292666	−	0.956215i	$-0.405458\pi$
$563$	13.8885	0.585332	0.292666	+	0.956215i	$0.405458\pi$
$564$	−20.9443	−0.881913
$565$	0	0
$566$	30.8672	1.29745
$567$	0	0
$568$	2.63638	0.110620
$569$	36.8328	1.54411	0.772056	−	0.635555i	$-0.219228\pi$
$569$	36.8328	1.54411	0.772056	+	0.635555i	$0.219228\pi$
$570$	0	0
$571$	−42.8328	−1.79250	−0.896249	−	0.443552i	$-0.853718\pi$
$571$	−42.8328	−1.79250	−0.896249	+	0.443552i	$0.853718\pi$
$572$	−99.1935	−4.14749
$573$	−3.52786	−0.147379
$574$	0	0
$575$	0	0
$576$	−4.70820	−0.196175
$577$	16.6525	0.693252	0.346626	−	0.938003i	$-0.387327\pi$
$577$	16.6525	0.693252	0.346626	+	0.938003i	$0.387327\pi$
$578$	−40.9953	−1.70518
$579$	−13.8042	−0.573685
$580$	0	0
$581$	0	0
$582$	25.4225	1.05380
$583$	−8.53149	−0.353338
$584$	15.4336	0.638648
$585$	0	0
$586$	50.3946	2.08178
$587$	8.94427	0.369170	0.184585	−	0.982817i	$-0.440906\pi$
$587$	8.94427	0.369170	0.184585	+	0.982817i	$0.440906\pi$
$588$	0	0
$589$	−15.4164	−0.635222
$590$	0	0
$591$	−1.90770	−0.0784723
$592$	−33.7819	−1.38843
$593$	−17.7082	−0.727189	−0.363594	−	0.931557i	$-0.618451\pi$
$593$	−17.7082	−0.727189	−0.363594	+	0.931557i	$0.618451\pi$
$594$	−11.1679	−0.458223
$595$	0	0
$596$	−80.2492	−3.28714
$597$	−11.8965	−0.486893
$598$	40.3607	1.65047
$599$	9.41641	0.384744	0.192372	−	0.981322i	$-0.438382\pi$
$599$	9.41641	0.384744	0.192372	+	0.981322i	$0.438382\pi$
$600$	0	0
$601$	−12.3469	−0.503640	−0.251820	−	0.967774i	$-0.581029\pi$
$601$	−12.3469	−0.503640	−0.251820	+	0.967774i	$0.581029\pi$
$602$	0	0
$603$	6.17345	0.251402
$604$	−37.8885	−1.54166
$605$	0	0
$606$	−32.6525	−1.32642
$607$	38.8328	1.57618	0.788088	−	0.615563i	$-0.211071\pi$
$607$	38.8328	1.57618	0.788088	+	0.615563i	$0.211071\pi$
$608$	20.1803	0.818421
$609$	0	0
$610$	0	0
$611$	−25.8885	−1.04734
$612$	3.23607	0.130810
$613$	44.6715	1.80426	0.902132	−	0.431460i	$-0.142001\pi$
$613$	44.6715	1.80426	0.902132	+	0.431460i	$0.142001\pi$
$614$	6.17345	0.249140
$615$	0	0
$616$	0	0
$617$	20.4280	0.822402	0.411201	−	0.911545i	$-0.365109\pi$
$617$	20.4280	0.822402	0.411201	+	0.911545i	$0.365109\pi$
$618$	3.81540	0.153478
$619$	22.7861	0.915850	0.457925	−	0.888991i	$-0.348593\pi$
$619$	22.7861	0.915850	0.457925	+	0.888991i	$0.348593\pi$
$620$	0	0
$621$	3.08672	0.123866
$622$	−9.52786	−0.382033
$623$	0	0
$624$	28.6525	1.14702
$625$	0	0
$626$	61.5625	2.46053
$627$	−36.1400	−1.44329
$628$	−36.6525	−1.46259
$629$	−4.71609	−0.188043
$630$	0	0
$631$	27.0557	1.07707	0.538536	−	0.842603i	$-0.318978\pi$
$631$	27.0557	1.07707	0.538536	+	0.842603i	$0.318978\pi$
$632$	72.2800	2.87514
$633$	−12.9443	−0.514489
$634$	−26.0689	−1.03533
$635$	0	0
$636$	8.08115	0.320438
$637$	0	0
$638$	−22.3357	−0.884281
$639$	0.472136	0.0186774
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−17.2361	−0.680253
$643$	28.3607	1.11844	0.559218	−	0.829021i	$-0.311101\pi$
$643$	28.3607	1.11844	0.559218	+	0.829021i	$0.311101\pi$
$644$	0	0
$645$	0	0
$646$	15.4164	0.606550
$647$	−47.4164	−1.86413	−0.932066	−	0.362289i	$-0.881995\pi$
$647$	−47.4164	−1.86413	−0.932066	+	0.362289i	$0.881995\pi$
$648$	5.58394	0.219358
$649$	44.6715	1.75351
$650$	0	0
$651$	0	0
$652$	16.1623	0.632964
$653$	42.7638	1.67348	0.836738	−	0.547603i	$-0.184460\pi$
$653$	42.7638	1.67348	0.836738	+	0.547603i	$0.184460\pi$
$654$	31.1456	1.21789
$655$	0	0
$656$	37.7694	1.47465
$657$	2.76393	0.107831
$658$	0	0
$659$	−8.47214	−0.330028	−0.165014	−	0.986291i	$-0.552767\pi$
$659$	−8.47214	−0.330028	−0.165014	+	0.986291i	$0.552767\pi$
$660$	0	0
$661$	−3.81540	−0.148402	−0.0742009	−	0.997243i	$-0.523641\pi$
$661$	−3.81540	−0.148402	−0.0742009	+	0.997243i	$0.523641\pi$
$662$	19.9777	0.776455
$663$	4.00000	0.155347
$664$	36.1400	1.40250
$665$	0	0
$666$	−15.4164	−0.597374
$667$	6.17345	0.239037
$668$	−65.3050	−2.52672
$669$	−17.8885	−0.691611
$670$	0	0
$671$	−17.0630	−0.658709
$672$	0	0
$673$	−30.8672	−1.18984	−0.594922	−	0.803783i	$-0.702817\pi$
$673$	−30.8672	−1.18984	−0.594922	+	0.803783i	$0.702817\pi$
$674$	−9.52786	−0.367000
$675$	0	0
$676$	61.0689	2.34880
$677$	31.5967	1.21436	0.607181	−	0.794564i	$-0.292300\pi$
$677$	31.5967	1.21436	0.607181	+	0.794564i	$0.292300\pi$
$678$	−10.6525	−0.409106
$679$	0	0
$680$	0	0
$681$	−1.52786	−0.0585479
$682$	21.3050	0.815809
$683$	−26.8798	−1.02853	−0.514264	−	0.857632i	$-0.671935\pi$
$683$	−26.8798	−1.02853	−0.514264	+	0.857632i	$0.671935\pi$
$684$	34.2323	1.30890
$685$	0	0
$686$	0	0
$687$	22.3357	0.852162
$688$	−54.6603	−2.08391
$689$	9.98885	0.380545
$690$	0	0
$691$	34.2323	1.30226	0.651129	−	0.758967i	$-0.274296\pi$
$691$	34.2323	1.30226	0.651129	+	0.758967i	$0.274296\pi$
$692$	81.4853	3.09761
$693$	0	0
$694$	63.4853	2.40987
$695$	0	0
$696$	11.1679	0.423317
$697$	5.27275	0.199720
$698$	21.3050	0.806404
$699$	−11.8965	−0.449969
$700$	0	0
$701$	8.83282	0.333611	0.166805	−	0.985990i	$-0.446655\pi$
$701$	8.83282	0.333611	0.166805	+	0.985990i	$0.446655\pi$
$702$	13.0756	0.493506
$703$	−49.8885	−1.88158
$704$	21.0557	0.793568
$705$	0	0
$706$	−8.08115	−0.304138
$707$	0	0
$708$	−42.3134	−1.59024
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	12.9443	0.485448
$712$	−51.7082	−1.93785
$713$	−5.88854	−0.220528
$714$	0	0
$715$	0	0
$716$	−31.8885	−1.19173
$717$	−7.52786	−0.281133
$718$	51.1233	1.90790
$719$	23.7931	0.887333	0.443666	−	0.896192i	$-0.353678\pi$
$719$	23.7931	0.887333	0.443666	+	0.896192i	$0.353678\pi$
$720$	0	0
$721$	0	0
$722$	115.633	4.30343
$723$	−6.17345	−0.229593
$724$	−42.3134	−1.57257
$725$	0	0
$726$	22.4749	0.834122
$727$	−1.52786	−0.0566653	−0.0283327	−	0.999599i	$-0.509020\pi$
$727$	−1.52786	−0.0566653	−0.0283327	+	0.999599i	$0.509020\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.63080	−0.282235
$732$	16.1623	0.597376
$733$	34.1803	1.26248	0.631240	−	0.775588i	$-0.282546\pi$
$733$	34.1803	1.26248	0.631240	+	0.775588i	$0.282546\pi$
$734$	−39.9554	−1.47478
$735$	0	0
$736$	7.70820	0.284128
$737$	−27.6085	−1.01697
$738$	17.2361	0.634468
$739$	24.9443	0.917590	0.458795	−	0.888542i	$-0.348281\pi$
$739$	24.9443	0.917590	0.458795	+	0.888542i	$0.348281\pi$
$740$	0	0
$741$	42.3134	1.55442
$742$	0	0
$743$	3.08672	0.113241	0.0566205	−	0.998396i	$-0.481968\pi$
$743$	3.08672	0.113241	0.0566205	+	0.998396i	$0.481968\pi$
$744$	−10.6525	−0.390539
$745$	0	0
$746$	80.7214	2.95542
$747$	6.47214	0.236803
$748$	−14.4721	−0.529154
$749$	0	0
$750$	0	0
$751$	−10.8328	−0.395295	−0.197648	−	0.980273i	$-0.563330\pi$
$751$	−10.8328	−0.395295	−0.197648	+	0.980273i	$0.563330\pi$
$752$	−27.0557	−0.986621
$753$	2.35805	0.0859320
$754$	26.1511	0.952368
$755$	0	0
$756$	0	0
$757$	−28.5092	−1.03618	−0.518092	−	0.855325i	$-0.673358\pi$
$757$	−28.5092	−1.03618	−0.518092	+	0.855325i	$0.673358\pi$
$758$	7.63080	0.277163
$759$	−13.8042	−0.501062
$760$	0	0
$761$	4.54408	0.164723	0.0823613	−	0.996603i	$-0.473754\pi$
$761$	4.54408	0.164723	0.0823613	+	0.996603i	$0.473754\pi$
$762$	−9.52786	−0.345158
$763$	0	0
$764$	−14.9443	−0.540665
$765$	0	0
$766$	26.1511	0.944879
$767$	−52.3023	−1.88853
$768$	−32.4164	−1.16973
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	21.7082	0.781802
$772$	−58.4757	−2.10459
$773$	14.6525	0.527013	0.263506	−	0.964658i	$-0.415121\pi$
$773$	14.6525	0.527013	0.263506	+	0.964658i	$0.415121\pi$
$774$	−24.9443	−0.896603
$775$	0	0
$776$	56.8464	2.04067
$777$	0	0
$778$	−29.6882	−1.06437
$779$	55.7771	1.99842
$780$	0	0
$781$	−2.11146	−0.0755538
$782$	5.88854	0.210574
$783$	2.00000	0.0714742
$784$	0	0
$785$	0	0
$786$	24.9443	0.889733
$787$	−40.9443	−1.45951	−0.729753	−	0.683711i	$-0.760365\pi$
$787$	−40.9443	−1.45951	−0.729753	+	0.683711i	$0.760365\pi$
$788$	−8.08115	−0.287879
$789$	−15.4336	−0.549451
$790$	0	0
$791$	0	0
$792$	−24.9721	−0.887346
$793$	19.9777	0.709429
$794$	−23.0644	−0.818526
$795$	0	0
$796$	−50.3946	−1.78619
$797$	27.2361	0.964751	0.482376	−	0.875965i	$-0.339774\pi$
$797$	27.2361	0.964751	0.482376	+	0.875965i	$0.339774\pi$
$798$	0	0
$799$	−3.77709	−0.133624
$800$	0	0
$801$	−9.26017	−0.327192
$802$	91.9794	3.24790
$803$	−12.3607	−0.436199
$804$	26.1511	0.922280
$805$	0	0
$806$	−24.9443	−0.878625
$807$	−6.90212	−0.242966
$808$	−73.0132	−2.56859
$809$	−30.9443	−1.08794	−0.543971	−	0.839104i	$-0.683080\pi$
$809$	−30.9443	−1.08794	−0.543971	+	0.839104i	$0.683080\pi$
$810$	0	0
$811$	30.4169	1.06808	0.534041	−	0.845459i	$-0.320673\pi$
$811$	30.4169	1.06808	0.534041	+	0.845459i	$0.320673\pi$
$812$	0	0
$813$	−18.0700	−0.633742
$814$	68.9443	2.41650
$815$	0	0
$816$	4.18034	0.146341
$817$	−80.7214	−2.82408
$818$	−84.3607	−2.94960
$819$	0	0
$820$	0	0
$821$	7.88854	0.275312	0.137656	−	0.990480i	$-0.456043\pi$
$821$	7.88854	0.275312	0.137656	+	0.990480i	$0.456043\pi$
$822$	−20.1803	−0.703870
$823$	11.4462	0.398990	0.199495	−	0.979899i	$-0.436070\pi$
$823$	11.4462	0.398990	0.199495	+	0.979899i	$0.436070\pi$
$824$	8.53149	0.297209
$825$	0	0
$826$	0	0
$827$	−46.8575	−1.62940	−0.814698	−	0.579886i	$-0.803097\pi$
$827$	−46.8575	−1.62940	−0.814698	+	0.579886i	$0.803097\pi$
$828$	13.0756	0.454408
$829$	17.6196	0.611956	0.305978	−	0.952039i	$-0.401017\pi$
$829$	17.6196	0.611956	0.305978	+	0.952039i	$0.401017\pi$
$830$	0	0
$831$	9.98885	0.346509
$832$	−24.6525	−0.854671
$833$	0	0
$834$	−35.5967	−1.23261
$835$	0	0
$836$	−153.091	−5.29478
$837$	−1.90770	−0.0659398
$838$	−3.63932	−0.125718
$839$	52.3023	1.80568	0.902838	−	0.429981i	$-0.141480\pi$
$839$	52.3023	1.80568	0.902838	+	0.429981i	$0.141480\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	41.1344	1.41759
$843$	−22.0000	−0.757720
$844$	−54.8328	−1.88742
$845$	0	0
$846$	−12.3469	−0.424495
$847$	0	0
$848$	10.4392	0.358483
$849$	12.3607	0.424217
$850$	0	0
$851$	−19.0557	−0.653222
$852$	2.00000	0.0685189
$853$	4.87539	0.166930	0.0834651	−	0.996511i	$-0.473401\pi$
$853$	4.87539	0.166930	0.0834651	+	0.996511i	$0.473401\pi$
$854$	0	0
$855$	0	0
$856$	−38.5410	−1.31730
$857$	45.1246	1.54143	0.770714	−	0.637182i	$-0.219900\pi$
$857$	45.1246	1.54143	0.770714	+	0.637182i	$0.219900\pi$
$858$	−58.4757	−1.99633
$859$	−0.450347	−0.0153656	−0.00768282	−	0.999970i	$-0.502446\pi$
$859$	−0.450347	−0.0153656	−0.00768282	+	0.999970i	$0.502446\pi$
$860$	0	0
$861$	0	0
$862$	−11.1679	−0.380379
$863$	10.7175	0.364829	0.182414	−	0.983222i	$-0.441609\pi$
$863$	10.7175	0.364829	0.182414	+	0.983222i	$0.441609\pi$
$864$	2.49721	0.0849569
$865$	0	0
$866$	89.1710	3.03015
$867$	−16.4164	−0.557530
$868$	0	0
$869$	−57.8885	−1.96373
$870$	0	0
$871$	32.3246	1.09528
$872$	69.6436	2.35843
$873$	10.1803	0.344552
$874$	62.2911	2.10703
$875$	0	0
$876$	11.7082	0.395584
$877$	15.2616	0.515348	0.257674	−	0.966232i	$-0.417044\pi$
$877$	15.2616	0.515348	0.257674	+	0.966232i	$0.417044\pi$
$878$	−60.5410	−2.04316
$879$	20.1803	0.680666
$880$	0	0
$881$	41.5848	1.40103	0.700513	−	0.713640i	$-0.252955\pi$
$881$	41.5848	1.40103	0.700513	+	0.713640i	$0.252955\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	16.9443	0.569898
$885$	0	0
$886$	−78.9017	−2.65075
$887$	35.7771	1.20128	0.600639	−	0.799521i	$-0.294913\pi$
$887$	35.7771	1.20128	0.600639	+	0.799521i	$0.294913\pi$
$888$	−34.4721	−1.15681
$889$	0	0
$890$	0	0
$891$	−4.47214	−0.149822
$892$	−75.7771	−2.53720
$893$	−39.9554	−1.33706
$894$	−47.3079	−1.58221
$895$	0	0
$896$	0	0
$897$	16.1623	0.539643
$898$	−42.5918	−1.42131
$899$	−3.81540	−0.127251
$900$	0	0
$901$	1.45735	0.0485515
$902$	−77.0820	−2.56655
$903$	0	0
$904$	−23.8197	−0.792230
$905$	0	0
$906$	−22.3357	−0.742055
$907$	39.9554	1.32670	0.663349	−	0.748311i	$-0.269135\pi$
$907$	39.9554	1.32670	0.663349	+	0.748311i	$0.269135\pi$
$908$	−6.47214	−0.214785
$909$	−13.0756	−0.433689
$910$	0	0
$911$	25.4164	0.842083	0.421042	−	0.907041i	$-0.361665\pi$
$911$	25.4164	0.842083	0.421042	+	0.907041i	$0.361665\pi$
$912$	44.2211	1.46431
$913$	−28.9443	−0.957916
$914$	59.4164	1.96532
$915$	0	0
$916$	94.6157	3.12619
$917$	0	0
$918$	1.90770	0.0629635
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	2.47214	0.0814596
$922$	−92.0689	−3.03213
$923$	2.47214	0.0813713
$924$	0	0
$925$	0	0
$926$	71.1935	2.33956
$927$	1.52786	0.0501816
$928$	4.99442	0.163950
$929$	−3.08672	−0.101272	−0.0506361	−	0.998717i	$-0.516125\pi$
$929$	−3.08672	−0.101272	−0.0506361	+	0.998717i	$0.516125\pi$
$930$	0	0
$931$	0	0
$932$	−50.3946	−1.65073
$933$	−3.81540	−0.124911
$934$	−68.4646	−2.24023
$935$	0	0
$936$	29.2379	0.955670
$937$	−26.5410	−0.867057	−0.433529	−	0.901140i	$-0.642732\pi$
$937$	−26.5410	−0.867057	−0.433529	+	0.901140i	$0.642732\pi$
$938$	0	0
$939$	24.6525	0.804503
$940$	0	0
$941$	−35.4113	−1.15438	−0.577188	−	0.816611i	$-0.695850\pi$
$941$	−35.4113	−1.15438	−0.577188	+	0.816611i	$0.695850\pi$
$942$	−21.6071	−0.703996
$943$	21.3050	0.693785
$944$	−54.6603	−1.77904
$945$	0	0
$946$	111.554	3.62694
$947$	14.5329	0.472257	0.236128	−	0.971722i	$-0.424121\pi$
$947$	14.5329	0.472257	0.236128	+	0.971722i	$0.424121\pi$
$948$	54.8328	1.78089
$949$	14.4721	0.469785
$950$	0	0
$951$	−10.4392	−0.338514
$952$	0	0
$953$	−51.8519	−1.67965	−0.839825	−	0.542858i	$-0.817342\pi$
$953$	−51.8519	−1.67965	−0.839825	+	0.542858i	$0.817342\pi$
$954$	4.76393	0.154238
$955$	0	0
$956$	−31.8885	−1.03135
$957$	−8.94427	−0.289127
$958$	−71.1935	−2.30016
$959$	0	0
$960$	0	0
$961$	−27.3607	−0.882603
$962$	−80.7214	−2.60256
$963$	−6.90212	−0.222418
$964$	−26.1511	−0.842272
$965$	0	0
$966$	0	0
$967$	26.1511	0.840964	0.420482	−	0.907301i	$-0.361861\pi$
$967$	26.1511	0.840964	0.420482	+	0.907301i	$0.361861\pi$
$968$	50.2554	1.61527
$969$	6.17345	0.198320
$970$	0	0
$971$	−58.4757	−1.87658	−0.938288	−	0.345855i	$-0.887589\pi$
$971$	−58.4757	−1.87658	−0.938288	+	0.345855i	$0.887589\pi$
$972$	4.23607	0.135872
$973$	0	0
$974$	24.9443	0.799266
$975$	0	0
$976$	20.8784	0.668301
$977$	−24.2434	−0.775616	−0.387808	−	0.921740i	$-0.626768\pi$
$977$	−24.2434	−0.775616	−0.387808	+	0.921740i	$0.626768\pi$
$978$	9.52786	0.304667
$979$	41.4127	1.32356
$980$	0	0
$981$	12.4721	0.398205
$982$	45.8505	1.46315
$983$	−38.8328	−1.23857	−0.619287	−	0.785165i	$-0.712578\pi$
$983$	−38.8328	−1.23857	−0.619287	+	0.785165i	$0.712578\pi$
$984$	38.5410	1.22864
$985$	0	0
$986$	3.81540	0.121507
$987$	0	0
$988$	179.243	5.70247
$989$	−30.8328	−0.980427
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	−4.76393	−0.151255
$993$	8.00000	0.253872
$994$	0	0
$995$	0	0
$996$	27.4164	0.868722
$997$	27.7082	0.877528	0.438764	−	0.898602i	$-0.355416\pi$
$997$	27.7082	0.877528	0.438764	+	0.898602i	$0.355416\pi$
$998$	19.9777	0.632383
$999$	−6.17345	−0.195319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bv.1.4		4
5.2	odd	4		735.2.d.c.589.7	yes	8
5.3	odd	4		735.2.d.c.589.2	yes	8
5.4	even	2		3675.2.a.bt.1.1		4
7.6	odd	2		3675.2.a.bt.1.4		4
15.2	even	4		2205.2.d.m.1324.1		8
15.8	even	4		2205.2.d.m.1324.7		8
35.2	odd	12		735.2.q.h.214.8		16
35.3	even	12		735.2.q.h.79.7		16
35.12	even	12		735.2.q.h.214.7		16
35.13	even	4		735.2.d.c.589.1	✓	8
35.17	even	12		735.2.q.h.79.2		16
35.18	odd	12		735.2.q.h.79.8		16
35.23	odd	12		735.2.q.h.214.1		16
35.27	even	4		735.2.d.c.589.8	yes	8
35.32	odd	12		735.2.q.h.79.1		16
35.33	even	12		735.2.q.h.214.2		16
35.34	odd	2	inner	3675.2.a.bv.1.1		4
105.62	odd	4		2205.2.d.m.1324.2		8
105.83	odd	4		2205.2.d.m.1324.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.d.c.589.1	✓	8	35.13	even	4
735.2.d.c.589.2	yes	8	5.3	odd	4
735.2.d.c.589.7	yes	8	5.2	odd	4
735.2.d.c.589.8	yes	8	35.27	even	4
735.2.q.h.79.1		16	35.32	odd	12
735.2.q.h.79.2		16	35.17	even	12
735.2.q.h.79.7		16	35.3	even	12
735.2.q.h.79.8		16	35.18	odd	12
735.2.q.h.214.1		16	35.23	odd	12
735.2.q.h.214.2		16	35.33	even	12
735.2.q.h.214.7		16	35.12	even	12
735.2.q.h.214.8		16	35.2	odd	12
2205.2.d.m.1324.1		8	15.2	even	4
2205.2.d.m.1324.2		8	105.62	odd	4
2205.2.d.m.1324.7		8	15.8	even	4
2205.2.d.m.1324.8		8	105.83	odd	4
3675.2.a.bt.1.1		4	5.4	even	2
3675.2.a.bt.1.4		4	7.6	odd	2
3675.2.a.bv.1.1		4	35.34	odd	2	inner
3675.2.a.bv.1.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49721 q^{2} +1.00000 q^{3} +4.23607 q^{4} +2.49721 q^{6} +5.58394 q^{8} +1.00000 q^{9} -4.47214 q^{11} +4.23607 q^{12} +5.23607 q^{13} +5.47214 q^{16} +0.763932 q^{17} +2.49721 q^{18} +8.08115 q^{19} -11.1679 q^{22} +3.08672 q^{23} +5.58394 q^{24} +13.0756 q^{26} +1.00000 q^{27} +2.00000 q^{29} -1.90770 q^{31} +2.49721 q^{32} -4.47214 q^{33} +1.90770 q^{34} +4.23607 q^{36} -6.17345 q^{37} +20.1803 q^{38} +5.23607 q^{39} +6.90212 q^{41} -9.98885 q^{43} -18.9443 q^{44} +7.70820 q^{46} -4.94427 q^{47} +5.47214 q^{48} +0.763932 q^{51} +22.1803 q^{52} +1.90770 q^{53} +2.49721 q^{54} +8.08115 q^{57} +4.99442 q^{58} -9.98885 q^{59} +3.81540 q^{61} -4.76393 q^{62} -4.70820 q^{64} -11.1679 q^{66} +6.17345 q^{67} +3.23607 q^{68} +3.08672 q^{69} +0.472136 q^{71} +5.58394 q^{72} +2.76393 q^{73} -15.4164 q^{74} +34.2323 q^{76} +13.0756 q^{78} +12.9443 q^{79} +1.00000 q^{81} +17.2361 q^{82} +6.47214 q^{83} -24.9443 q^{86} +2.00000 q^{87} -24.9721 q^{88} -9.26017 q^{89} +13.0756 q^{92} -1.90770 q^{93} -12.3469 q^{94} +2.49721 q^{96} +10.1803 q^{97} -4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 8 q^{4} + 4 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{16} + 12 q^{17} + 4 q^{27} + 8 q^{29} + 8 q^{36} + 36 q^{38} + 12 q^{39} - 40 q^{44} + 4 q^{46} + 16 q^{47} + 4 q^{48} + 12 q^{51} + 44 q^{52}+ \cdots - 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 8 * q^4 + 4 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^16 + 12 * q^17 + 4 * q^27 + 8 * q^29 + 8 * q^36 + 36 * q^38 + 12 * q^39 - 40 * q^44 + 4 * q^46 + 16 * q^47 + 4 * q^48 + 12 * q^51 + 44 * q^52 - 28 * q^62 + 8 * q^64 + 4 * q^68 - 16 * q^71 + 20 * q^73 - 8 * q^74 + 16 * q^79 + 4 * q^81 + 60 * q^82 + 8 * q^83 - 64 * q^86 + 8 * q^87 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more