LMFDB - Embedded newform 675.4.a.s.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,5,0,17,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.654334$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.45876 q^{2} +21.7980 q^{4} +11.8065 q^{7} +75.3201 q^{8} +56.2376 q^{11} -34.5961 q^{13} +64.4489 q^{14} +236.770 q^{16} +39.2675 q^{17} -146.561 q^{19} +306.987 q^{22} -23.5777 q^{23} -188.851 q^{26} +257.359 q^{28} -161.003 q^{29} -29.5465 q^{31} +689.908 q^{32} +214.352 q^{34} +217.688 q^{37} -800.039 q^{38} -142.290 q^{41} +468.030 q^{43} +1225.87 q^{44} -128.705 q^{46} +394.318 q^{47} -203.606 q^{49} -754.126 q^{52} -134.780 q^{53} +889.268 q^{56} -878.875 q^{58} +131.195 q^{59} +259.801 q^{61} -161.287 q^{62} +1871.88 q^{64} -445.244 q^{67} +855.954 q^{68} -560.841 q^{71} +88.6681 q^{73} +1188.30 q^{74} -3194.74 q^{76} +663.970 q^{77} +450.342 q^{79} -776.726 q^{82} +284.295 q^{83} +2554.86 q^{86} +4235.82 q^{88} -625.305 q^{89} -408.459 q^{91} -513.948 q^{92} +2152.49 q^{94} +193.261 q^{97} -1111.44 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{2} + 17 q^{4} + 4 q^{7} + 75 q^{8} - 5 q^{11} - 7 q^{13} - 60 q^{14} + 161 q^{16} + 155 q^{17} - 50 q^{19} + 229 q^{22} + 285 q^{23} - 185 q^{26} + 334 q^{28} - 115 q^{29} - 115 q^{31} + 775 q^{32}+ \cdots - 305 q^{98}+O(q^{100}) $ 3 * q + 5 * q^2 + 17 * q^4 + 4 * q^7 + 75 * q^8 - 5 * q^11 - 7 * q^13 - 60 * q^14 + 161 * q^16 + 155 * q^17 - 50 * q^19 + 229 * q^22 + 285 * q^23 - 185 * q^26 + 334 * q^28 - 115 * q^29 - 115 * q^31 + 775 * q^32 + 413 * q^34 + 384 * q^37 - 1150 * q^38 - 580 * q^41 + 797 * q^43 + 1415 * q^44 - 285 * q^46 - 145 * q^47 + 577 * q^49 - 825 * q^52 - 400 * q^53 + 2190 * q^56 + 59 * q^58 - 380 * q^59 - 152 * q^61 - 1005 * q^62 + 2937 * q^64 - 2 * q^67 + 475 * q^68 - 40 * q^71 + 980 * q^73 + 2720 * q^74 - 3276 * q^76 + 1950 * q^77 + 1013 * q^79 - 4 * q^82 + 270 * q^83 + 1555 * q^86 + 5193 * q^88 - 1020 * q^89 - 632 * q^91 - 1215 * q^92 + 3833 * q^94 - 720 * q^97 - 305 * q^98

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 $

$2$	5.45876	1.92996	0.964981	−	0.262320i	$-0.0844875\pi$
$2$	5.45876	1.92996	0.964981	+	0.262320i	$0.0844875\pi$
$3$	0	0
$3$	0	0
$4$	21.7980	2.72475
$5$	0	0
$5$	0	0
$6$	0	0
$7$	11.8065	0.637492	0.318746	−	0.947840i	$-0.396738\pi$
$7$	11.8065	0.637492	0.318746	+	0.947840i	$0.396738\pi$
$8$	75.3201	3.32871
$9$	0	0
$10$	0	0
$11$	56.2376	1.54148	0.770740	−	0.637150i	$-0.219887\pi$
$11$	56.2376	1.54148	0.770740	+	0.637150i	$0.219887\pi$
$12$	0	0
$13$	−34.5961	−0.738094	−0.369047	−	0.929411i	$-0.620316\pi$
$13$	−34.5961	−0.738094	−0.369047	+	0.929411i	$0.620316\pi$
$14$	64.4489	1.23034
$15$	0	0
$16$	236.770	3.69953
$17$	39.2675	0.560221	0.280111	−	0.959968i	$-0.409629\pi$
$17$	39.2675	0.560221	0.280111	+	0.959968i	$0.409629\pi$
$18$	0	0
$19$	−146.561	−1.76965	−0.884825	−	0.465924i	$-0.845722\pi$
$19$	−146.561	−1.76965	−0.884825	+	0.465924i	$0.845722\pi$
$20$	0	0
$21$	0	0
$22$	306.987	2.97500
$23$	−23.5777	−0.213752	−0.106876	−	0.994272i	$-0.534085\pi$
$23$	−23.5777	−0.213752	−0.106876	+	0.994272i	$0.534085\pi$
$24$	0	0
$25$	0	0
$26$	−188.851	−1.42449
$27$	0	0
$28$	257.359	1.73701
$29$	−161.003	−1.03095	−0.515473	−	0.856906i	$-0.672384\pi$
$29$	−161.003	−1.03095	−0.515473	+	0.856906i	$0.672384\pi$
$30$	0	0
$31$	−29.5465	−0.171184	−0.0855921	−	0.996330i	$-0.527278\pi$
$31$	−29.5465	−0.171184	−0.0855921	+	0.996330i	$0.527278\pi$
$32$	689.908	3.81124
$33$	0	0
$34$	214.352	1.08121
$35$	0	0
$36$	0	0
$37$	217.688	0.967233	0.483617	−	0.875280i	$-0.339323\pi$
$37$	217.688	0.967233	0.483617	+	0.875280i	$0.339323\pi$
$38$	−800.039	−3.41536
$39$	0	0
$40$	0	0
$41$	−142.290	−0.541999	−0.270999	−	0.962580i	$-0.587354\pi$
$41$	−142.290	−0.541999	−0.270999	+	0.962580i	$0.587354\pi$
$42$	0	0
$43$	468.030	1.65986	0.829929	−	0.557869i	$-0.188381\pi$
$43$	468.030	1.65986	0.829929	+	0.557869i	$0.188381\pi$
$44$	1225.87	4.20015
$45$	0	0
$46$	−128.705	−0.412533
$47$	394.318	1.22377	0.611886	−	0.790946i	$-0.290411\pi$
$47$	394.318	1.22377	0.611886	+	0.790946i	$0.290411\pi$
$48$	0	0
$49$	−203.606	−0.593604
$50$	0	0
$51$	0	0
$52$	−754.126	−2.01112
$53$	−134.780	−0.349311	−0.174655	−	0.984630i	$-0.555881\pi$
$53$	−134.780	−0.349311	−0.174655	+	0.984630i	$0.555881\pi$
$54$	0	0
$55$	0	0
$56$	889.268	2.12203
$57$	0	0
$58$	−878.875	−1.98969
$59$	131.195	0.289495	0.144747	−	0.989469i	$-0.453763\pi$
$59$	131.195	0.289495	0.144747	+	0.989469i	$0.453763\pi$
$60$	0	0
$61$	259.801	0.545313	0.272657	−	0.962111i	$-0.412098\pi$
$61$	259.801	0.545313	0.272657	+	0.962111i	$0.412098\pi$
$62$	−161.287	−0.330379
$63$	0	0
$64$	1871.88	3.65602
$65$	0	0
$66$	0	0
$67$	−445.244	−0.811869	−0.405935	−	0.913902i	$-0.633054\pi$
$67$	−445.244	−0.811869	−0.405935	+	0.913902i	$0.633054\pi$
$68$	855.954	1.52647
$69$	0	0
$70$	0	0
$71$	−560.841	−0.937459	−0.468729	−	0.883342i	$-0.655288\pi$
$71$	−560.841	−0.937459	−0.468729	+	0.883342i	$0.655288\pi$
$72$	0	0
$73$	88.6681	0.142162	0.0710809	−	0.997471i	$-0.477355\pi$
$73$	88.6681	0.142162	0.0710809	+	0.997471i	$0.477355\pi$
$74$	1188.30	1.86672
$75$	0	0
$76$	−3194.74	−4.82186
$77$	663.970	0.982681
$78$	0	0
$79$	450.342	0.641360	0.320680	−	0.947188i	$-0.396089\pi$
$79$	450.342	0.641360	0.320680	+	0.947188i	$0.396089\pi$
$80$	0	0
$81$	0	0
$82$	−776.726	−1.04604
$83$	284.295	0.375969	0.187985	−	0.982172i	$-0.439804\pi$
$83$	284.295	0.375969	0.187985	+	0.982172i	$0.439804\pi$
$84$	0	0
$85$	0	0
$86$	2554.86	3.20346
$87$	0	0
$88$	4235.82	5.13114
$89$	−625.305	−0.744744	−0.372372	−	0.928083i	$-0.621455\pi$
$89$	−625.305	−0.744744	−0.372372	+	0.928083i	$0.621455\pi$
$90$	0	0
$91$	−408.459	−0.470529
$92$	−513.948	−0.582421
$93$	0	0
$94$	2152.49	2.36183
$95$	0	0
$96$	0	0
$97$	193.261	0.202296	0.101148	−	0.994871i	$-0.467748\pi$
$97$	193.261	0.202296	0.101148	+	0.994871i	$0.467748\pi$
$98$	−1111.44	−1.14563
$99$	0	0
$100$	0	0
$101$	−1374.86	−1.35449	−0.677245	−	0.735758i	$-0.736826\pi$
$101$	−1374.86	−1.35449	−0.677245	+	0.735758i	$0.736826\pi$
$102$	0	0
$103$	−2029.60	−1.94158	−0.970789	−	0.239935i	$-0.922874\pi$
$103$	−2029.60	−1.94158	−0.970789	+	0.239935i	$0.922874\pi$
$104$	−2605.78	−2.45690
$105$	0	0
$106$	−735.732	−0.674156
$107$	−823.062	−0.743630	−0.371815	−	0.928307i	$-0.621264\pi$
$107$	−823.062	−0.743630	−0.371815	+	0.928307i	$0.621264\pi$
$108$	0	0
$109$	−829.868	−0.729238	−0.364619	−	0.931157i	$-0.618801\pi$
$109$	−829.868	−0.729238	−0.364619	+	0.931157i	$0.618801\pi$
$110$	0	0
$111$	0	0
$112$	2795.43	2.35842
$113$	−1503.37	−1.25155	−0.625773	−	0.780005i	$-0.715216\pi$
$113$	−1503.37	−1.25155	−0.625773	+	0.780005i	$0.715216\pi$
$114$	0	0
$115$	0	0
$116$	−3509.54	−2.80908
$117$	0	0
$118$	716.164	0.558714
$119$	463.612	0.357137
$120$	0	0
$121$	1831.67	1.37616
$122$	1418.19	1.05243
$123$	0	0
$124$	−644.056	−0.466435
$125$	0	0
$126$	0	0
$127$	576.348	0.402698	0.201349	−	0.979520i	$-0.435468\pi$
$127$	576.348	0.402698	0.201349	+	0.979520i	$0.435468\pi$
$128$	4698.89	3.24474
$129$	0	0
$130$	0	0
$131$	−2390.04	−1.59403	−0.797017	−	0.603957i	$-0.793590\pi$
$131$	−2390.04	−1.59403	−0.797017	+	0.603957i	$0.793590\pi$
$132$	0	0
$133$	−1730.37	−1.12814
$134$	−2430.48	−1.56688
$135$	0	0
$136$	2957.63	1.86481
$137$	1002.46	0.625152	0.312576	−	0.949893i	$-0.398808\pi$
$137$	1002.46	0.625152	0.312576	+	0.949893i	$0.398808\pi$
$138$	0	0
$139$	131.817	0.0804356	0.0402178	−	0.999191i	$-0.487195\pi$
$139$	131.817	0.0804356	0.0402178	+	0.999191i	$0.487195\pi$
$140$	0	0
$141$	0	0
$142$	−3061.50	−1.80926
$143$	−1945.60	−1.13776
$144$	0	0
$145$	0	0
$146$	484.017	0.274367
$147$	0	0
$148$	4745.16	2.63547
$149$	−1019.49	−0.560533	−0.280267	−	0.959922i	$-0.590423\pi$
$149$	−1019.49	−0.560533	−0.280267	+	0.959922i	$0.590423\pi$
$150$	0	0
$151$	2822.38	1.52107	0.760537	−	0.649295i	$-0.224936\pi$
$151$	2822.38	1.52107	0.760537	+	0.649295i	$0.224936\pi$
$152$	−11039.0	−5.89065
$153$	0	0
$154$	3624.45	1.89654
$155$	0	0
$156$	0	0
$157$	−476.499	−0.242222	−0.121111	−	0.992639i	$-0.538646\pi$
$157$	−476.499	−0.242222	−0.121111	+	0.992639i	$0.538646\pi$
$158$	2458.31	1.23780
$159$	0	0
$160$	0	0
$161$	−278.371	−0.136265
$162$	0	0
$163$	2242.26	1.07747	0.538734	−	0.842476i	$-0.318903\pi$
$163$	2242.26	1.07747	0.538734	+	0.842476i	$0.318903\pi$
$164$	−3101.64	−1.47681
$165$	0	0
$166$	1551.90	0.725607
$167$	95.0390	0.0440380	0.0220190	−	0.999758i	$-0.492991\pi$
$167$	95.0390	0.0440380	0.0220190	+	0.999758i	$0.492991\pi$
$168$	0	0
$169$	−1000.11	−0.455217
$170$	0	0
$171$	0	0
$172$	10202.1	4.52270
$173$	−2133.76	−0.937727	−0.468864	−	0.883271i	$-0.655336\pi$
$173$	−2133.76	−0.937727	−0.468864	+	0.883271i	$0.655336\pi$
$174$	0	0
$175$	0	0
$176$	13315.4	5.70275
$177$	0	0
$178$	−3413.39	−1.43733
$179$	1704.68	0.711808	0.355904	−	0.934523i	$-0.384173\pi$
$179$	1704.68	0.711808	0.355904	+	0.934523i	$0.384173\pi$
$180$	0	0
$181$	−1360.98	−0.558902	−0.279451	−	0.960160i	$-0.590152\pi$
$181$	−1360.98	−0.558902	−0.279451	+	0.960160i	$0.590152\pi$
$182$	−2229.68	−0.908103
$183$	0	0
$184$	−1775.88	−0.711518
$185$	0	0
$186$	0	0
$187$	2208.31	0.863570
$188$	8595.36	3.33447
$189$	0	0
$190$	0	0
$191$	1096.84	0.415522	0.207761	−	0.978180i	$-0.433382\pi$
$191$	1096.84	0.415522	0.207761	+	0.978180i	$0.433382\pi$
$192$	0	0
$193$	2867.27	1.06938	0.534691	−	0.845048i	$-0.320428\pi$
$193$	2867.27	1.06938	0.534691	+	0.845048i	$0.320428\pi$
$194$	1054.97	0.390424
$195$	0	0
$196$	−4438.21	−1.61742
$197$	724.139	0.261892	0.130946	−	0.991389i	$-0.458199\pi$
$197$	724.139	0.261892	0.130946	+	0.991389i	$0.458199\pi$
$198$	0	0
$199$	−1693.65	−0.603315	−0.301658	−	0.953416i	$-0.597540\pi$
$199$	−1693.65	−0.603315	−0.301658	+	0.953416i	$0.597540\pi$
$200$	0	0
$201$	0	0
$202$	−7505.02	−2.61411
$203$	−1900.88	−0.657220
$204$	0	0
$205$	0	0
$206$	−11079.1	−3.74717
$207$	0	0
$208$	−8191.30	−2.73060
$209$	−8242.22	−2.72788
$210$	0	0
$211$	947.452	0.309124	0.154562	−	0.987983i	$-0.450603\pi$
$211$	947.452	0.309124	0.154562	+	0.987983i	$0.450603\pi$
$212$	−2937.94	−0.951785
$213$	0	0
$214$	−4492.89	−1.43518
$215$	0	0
$216$	0	0
$217$	−348.841	−0.109129
$218$	−4530.05	−1.40740
$219$	0	0
$220$	0	0
$221$	−1358.50	−0.413496
$222$	0	0
$223$	−111.866	−0.0335923	−0.0167961	−	0.999859i	$-0.505347\pi$
$223$	−111.866	−0.0335923	−0.0167961	+	0.999859i	$0.505347\pi$
$224$	8145.41	2.42964
$225$	0	0
$226$	−8206.51	−2.41544
$227$	1200.70	0.351072	0.175536	−	0.984473i	$-0.443834\pi$
$227$	1200.70	0.351072	0.175536	+	0.984473i	$0.443834\pi$
$228$	0	0
$229$	822.380	0.237312	0.118656	−	0.992935i	$-0.462141\pi$
$229$	822.380	0.237312	0.118656	+	0.992935i	$0.462141\pi$
$230$	0	0
$231$	0	0
$232$	−12126.7	−3.43172
$233$	−5329.21	−1.49840	−0.749202	−	0.662341i	$-0.769563\pi$
$233$	−5329.21	−1.49840	−0.749202	+	0.662341i	$0.769563\pi$
$234$	0	0
$235$	0	0
$236$	2859.80	0.788802
$237$	0	0
$238$	2530.75	0.689260
$239$	7085.61	1.91770	0.958850	−	0.283914i	$-0.0916331\pi$
$239$	7085.61	1.91770	0.958850	+	0.283914i	$0.0916331\pi$
$240$	0	0
$241$	−6560.09	−1.75341	−0.876707	−	0.481025i	$-0.840265\pi$
$241$	−6560.09	−1.75341	−0.876707	+	0.481025i	$0.840265\pi$
$242$	9998.63	2.65593
$243$	0	0
$244$	5663.15	1.48584
$245$	0	0
$246$	0	0
$247$	5070.42	1.30617
$248$	−2225.45	−0.569822
$249$	0	0
$250$	0	0
$251$	−714.222	−0.179607	−0.0898033	−	0.995960i	$-0.528624\pi$
$251$	−714.222	−0.179607	−0.0898033	+	0.995960i	$0.528624\pi$
$252$	0	0
$253$	−1325.95	−0.329494
$254$	3146.14	0.777191
$255$	0	0
$256$	10675.0	2.60621
$257$	4396.59	1.06713	0.533563	−	0.845760i	$-0.320853\pi$
$257$	4396.59	1.06713	0.533563	+	0.845760i	$0.320853\pi$
$258$	0	0
$259$	2570.13	0.616604
$260$	0	0
$261$	0	0
$262$	−13046.6	−3.07643
$263$	−7550.31	−1.77024	−0.885118	−	0.465367i	$-0.845922\pi$
$263$	−7550.31	−1.77024	−0.885118	+	0.465367i	$0.845922\pi$
$264$	0	0
$265$	0	0
$266$	−9445.68	−2.17726
$267$	0	0
$268$	−9705.45	−2.21214
$269$	5536.86	1.25497	0.627487	−	0.778627i	$-0.284083\pi$
$269$	5536.86	1.25497	0.627487	+	0.778627i	$0.284083\pi$
$270$	0	0
$271$	3058.25	0.685518	0.342759	−	0.939423i	$-0.388639\pi$
$271$	3058.25	0.685518	0.342759	+	0.939423i	$0.388639\pi$
$272$	9297.36	2.07256
$273$	0	0
$274$	5472.18	1.20652
$275$	0	0
$276$	0	0
$277$	4070.19	0.882865	0.441433	−	0.897294i	$-0.354470\pi$
$277$	4070.19	0.882865	0.441433	+	0.897294i	$0.354470\pi$
$278$	719.556	0.155238
$279$	0	0
$280$	0	0
$281$	−7446.19	−1.58079	−0.790396	−	0.612597i	$-0.790125\pi$
$281$	−7446.19	−1.58079	−0.790396	+	0.612597i	$0.790125\pi$
$282$	0	0
$283$	774.651	0.162715	0.0813573	−	0.996685i	$-0.474075\pi$
$283$	774.651	0.162715	0.0813573	+	0.996685i	$0.474075\pi$
$284$	−12225.2	−2.55434
$285$	0	0
$286$	−10620.6	−2.19583
$287$	−1679.95	−0.345520
$288$	0	0
$289$	−3371.06	−0.686152
$290$	0	0
$291$	0	0
$292$	1932.79	0.387356
$293$	6749.23	1.34571	0.672857	−	0.739772i	$-0.265067\pi$
$293$	6749.23	1.34571	0.672857	+	0.739772i	$0.265067\pi$
$294$	0	0
$295$	0	0
$296$	16396.3	3.21964
$297$	0	0
$298$	−5565.12	−1.08181
$299$	815.696	0.157769
$300$	0	0
$301$	5525.80	1.05815
$302$	15406.7	2.93561
$303$	0	0
$304$	−34701.2	−6.54687
$305$	0	0
$306$	0	0
$307$	2204.39	0.409808	0.204904	−	0.978782i	$-0.434312\pi$
$307$	2204.39	0.409808	0.204904	+	0.978782i	$0.434312\pi$
$308$	14473.2	2.67756
$309$	0	0
$310$	0	0
$311$	6032.98	1.10000	0.549998	−	0.835166i	$-0.314628\pi$
$311$	6032.98	1.10000	0.549998	+	0.835166i	$0.314628\pi$
$312$	0	0
$313$	5772.96	1.04251	0.521257	−	0.853400i	$-0.325463\pi$
$313$	5772.96	1.04251	0.521257	+	0.853400i	$0.325463\pi$
$314$	−2601.09	−0.467479
$315$	0	0
$316$	9816.57	1.74755
$317$	−3302.07	−0.585057	−0.292528	−	0.956257i	$-0.594497\pi$
$317$	−3302.07	−0.585057	−0.292528	+	0.956257i	$0.594497\pi$
$318$	0	0
$319$	−9054.41	−1.58918
$320$	0	0
$321$	0	0
$322$	−1519.56	−0.262986
$323$	−5755.07	−0.991395
$324$	0	0
$325$	0	0
$326$	12239.9	2.07947
$327$	0	0
$328$	−10717.3	−1.80416
$329$	4655.53	0.780144
$330$	0	0
$331$	8053.15	1.33728	0.668642	−	0.743584i	$-0.266876\pi$
$331$	8053.15	1.33728	0.668642	+	0.743584i	$0.266876\pi$
$332$	6197.08	1.02442
$333$	0	0
$334$	518.795	0.0849916
$335$	0	0
$336$	0	0
$337$	−3460.33	−0.559335	−0.279668	−	0.960097i	$-0.590224\pi$
$337$	−3460.33	−0.559335	−0.279668	+	0.960097i	$0.590224\pi$
$338$	−5459.37	−0.878552
$339$	0	0
$340$	0	0
$341$	−1661.62	−0.263877
$342$	0	0
$343$	−6453.51	−1.01591
$344$	35252.0	5.52518
$345$	0	0
$346$	−11647.7	−1.80978
$347$	−9328.27	−1.44314	−0.721568	−	0.692344i	$-0.756578\pi$
$347$	−9328.27	−1.44314	−0.721568	+	0.692344i	$0.756578\pi$
$348$	0	0
$349$	8899.42	1.36497	0.682486	−	0.730899i	$-0.260899\pi$
$349$	8899.42	1.36497	0.682486	+	0.730899i	$0.260899\pi$
$350$	0	0
$351$	0	0
$352$	38798.8	5.87495
$353$	3722.45	0.561264	0.280632	−	0.959816i	$-0.409456\pi$
$353$	3722.45	0.561264	0.280632	+	0.959816i	$0.409456\pi$
$354$	0	0
$355$	0	0
$356$	−13630.4	−2.02924
$357$	0	0
$358$	9305.42	1.37376
$359$	−11029.3	−1.62145	−0.810727	−	0.585425i	$-0.800928\pi$
$359$	−11029.3	−1.62145	−0.810727	+	0.585425i	$0.800928\pi$
$360$	0	0
$361$	14621.1	2.13166
$362$	−7429.29	−1.07866
$363$	0	0
$364$	−8903.60	−1.28208
$365$	0	0
$366$	0	0
$367$	4853.11	0.690274	0.345137	−	0.938552i	$-0.387833\pi$
$367$	4853.11	0.690274	0.345137	+	0.938552i	$0.387833\pi$
$368$	−5582.49	−0.790781
$369$	0	0
$370$	0	0
$371$	−1591.28	−0.222683
$372$	0	0
$373$	12373.8	1.71767	0.858834	−	0.512254i	$-0.171189\pi$
$373$	12373.8	1.71767	0.858834	+	0.512254i	$0.171189\pi$
$374$	12054.6	1.66666
$375$	0	0
$376$	29700.1	4.07358
$377$	5570.06	0.760935
$378$	0	0
$379$	11150.6	1.51127	0.755634	−	0.654994i	$-0.227329\pi$
$379$	11150.6	1.51127	0.755634	+	0.654994i	$0.227329\pi$
$380$	0	0
$381$	0	0
$382$	5987.39	0.801941
$383$	2199.59	0.293457	0.146728	−	0.989177i	$-0.453126\pi$
$383$	2199.59	0.293457	0.146728	+	0.989177i	$0.453126\pi$
$384$	0	0
$385$	0	0
$386$	15651.7	2.06386
$387$	0	0
$388$	4212.72	0.551207
$389$	−9218.92	−1.20159	−0.600794	−	0.799404i	$-0.705149\pi$
$389$	−9218.92	−1.20159	−0.600794	+	0.799404i	$0.705149\pi$
$390$	0	0
$391$	−925.838	−0.119748
$392$	−15335.6	−1.97594
$393$	0	0
$394$	3952.90	0.505442
$395$	0	0
$396$	0	0
$397$	−1119.36	−0.141509	−0.0707544	−	0.997494i	$-0.522541\pi$
$397$	−1119.36	−0.141509	−0.0707544	+	0.997494i	$0.522541\pi$
$398$	−9245.23	−1.16438
$399$	0	0
$400$	0	0
$401$	−12296.9	−1.53137	−0.765683	−	0.643218i	$-0.777599\pi$
$401$	−12296.9	−1.53137	−0.765683	+	0.643218i	$0.777599\pi$
$402$	0	0
$403$	1022.19	0.126350
$404$	−29969.2	−3.69065
$405$	0	0
$406$	−10376.4	−1.26841
$407$	12242.2	1.49097
$408$	0	0
$409$	2500.22	0.302269	0.151134	−	0.988513i	$-0.451707\pi$
$409$	2500.22	0.302269	0.151134	+	0.988513i	$0.451707\pi$
$410$	0	0
$411$	0	0
$412$	−44241.3	−5.29032
$413$	1548.96	0.184551
$414$	0	0
$415$	0	0
$416$	−23868.1	−2.81305
$417$	0	0
$418$	−44992.3	−5.26470
$419$	8332.97	0.971581	0.485790	−	0.874075i	$-0.338532\pi$
$419$	8332.97	0.971581	0.485790	+	0.874075i	$0.338532\pi$
$420$	0	0
$421$	11374.2	1.31673	0.658367	−	0.752697i	$-0.271248\pi$
$421$	11374.2	1.31673	0.658367	+	0.752697i	$0.271248\pi$
$422$	5171.91	0.596598
$423$	0	0
$424$	−10151.6	−1.16275
$425$	0	0
$426$	0	0
$427$	3067.35	0.347633
$428$	−17941.1	−2.02621
$429$	0	0
$430$	0	0
$431$	10030.2	1.12097	0.560484	−	0.828165i	$-0.310615\pi$
$431$	10030.2	1.12097	0.560484	+	0.828165i	$0.310615\pi$
$432$	0	0
$433$	−7609.38	−0.844535	−0.422267	−	0.906471i	$-0.638766\pi$
$433$	−7609.38	−0.844535	−0.422267	+	0.906471i	$0.638766\pi$
$434$	−1904.24	−0.210614
$435$	0	0
$436$	−18089.5	−1.98699
$437$	3455.57	0.378266
$438$	0	0
$439$	−12370.5	−1.34490	−0.672450	−	0.740143i	$-0.734758\pi$
$439$	−12370.5	−1.34490	−0.672450	+	0.740143i	$0.734758\pi$
$440$	0	0
$441$	0	0
$442$	−7415.72	−0.798032
$443$	−12084.4	−1.29605	−0.648023	−	0.761621i	$-0.724404\pi$
$443$	−12084.4	−1.29605	−0.648023	+	0.761621i	$0.724404\pi$
$444$	0	0
$445$	0	0
$446$	−610.647	−0.0648318
$447$	0	0
$448$	22100.4	2.33068
$449$	625.550	0.0657495	0.0328747	−	0.999459i	$-0.489534\pi$
$449$	625.550	0.0657495	0.0328747	+	0.999459i	$0.489534\pi$
$450$	0	0
$451$	−8002.04	−0.835480
$452$	−32770.4	−3.41016
$453$	0	0
$454$	6554.33	0.677555
$455$	0	0
$456$	0	0
$457$	−1811.22	−0.185395	−0.0926974	−	0.995694i	$-0.529549\pi$
$457$	−1811.22	−0.185395	−0.0926974	+	0.995694i	$0.529549\pi$
$458$	4489.17	0.458003
$459$	0	0
$460$	0	0
$461$	11625.0	1.17447	0.587233	−	0.809418i	$-0.300217\pi$
$461$	11625.0	1.17447	0.587233	+	0.809418i	$0.300217\pi$
$462$	0	0
$463$	−7291.88	−0.731928	−0.365964	−	0.930629i	$-0.619261\pi$
$463$	−7291.88	−0.731928	−0.365964	+	0.930629i	$0.619261\pi$
$464$	−38120.6	−3.81402
$465$	0	0
$466$	−29090.9	−2.89186
$467$	−11637.6	−1.15316	−0.576579	−	0.817042i	$-0.695613\pi$
$467$	−11637.6	−1.15316	−0.576579	+	0.817042i	$0.695613\pi$
$468$	0	0
$469$	−5256.78	−0.517560
$470$	0	0
$471$	0	0
$472$	9881.66	0.963644
$473$	26320.9	2.55864
$474$	0	0
$475$	0	0
$476$	10105.8	0.973109
$477$	0	0
$478$	38678.6	3.70109
$479$	−12041.4	−1.14862	−0.574309	−	0.818639i	$-0.694729\pi$
$479$	−12041.4	−1.14862	−0.574309	+	0.818639i	$0.694729\pi$
$480$	0	0
$481$	−7531.14	−0.713909
$482$	−35810.0	−3.38402
$483$	0	0
$484$	39926.7	3.74969
$485$	0	0
$486$	0	0
$487$	−7037.81	−0.654853	−0.327427	−	0.944877i	$-0.606181\pi$
$487$	−7037.81	−0.654853	−0.327427	+	0.944877i	$0.606181\pi$
$488$	19568.2	1.81519
$489$	0	0
$490$	0	0
$491$	−6603.07	−0.606909	−0.303454	−	0.952846i	$-0.598140\pi$
$491$	−6603.07	−0.606909	−0.303454	+	0.952846i	$0.598140\pi$
$492$	0	0
$493$	−6322.17	−0.577558
$494$	27678.2	2.52085
$495$	0	0
$496$	−6995.72	−0.633301
$497$	−6621.58	−0.597623
$498$	0	0
$499$	−36.7047	−0.00329284	−0.00164642	−	0.999999i	$-0.500524\pi$
$499$	−36.7047	−0.00329284	−0.00164642	+	0.999999i	$0.500524\pi$
$500$	0	0
$501$	0	0
$502$	−3898.76	−0.346634
$503$	21242.5	1.88302	0.941508	−	0.336990i	$-0.109409\pi$
$503$	21242.5	1.88302	0.941508	+	0.336990i	$0.109409\pi$
$504$	0	0
$505$	0	0
$506$	−7238.06	−0.635911
$507$	0	0
$508$	12563.2	1.09725
$509$	11280.4	0.982309	0.491155	−	0.871072i	$-0.336575\pi$
$509$	11280.4	0.982309	0.491155	+	0.871072i	$0.336575\pi$
$510$	0	0
$511$	1046.86	0.0906270
$512$	20681.3	1.78514
$513$	0	0
$514$	23999.9	2.05951
$515$	0	0
$516$	0	0
$517$	22175.5	1.88642
$518$	14029.7	1.19002
$519$	0	0
$520$	0	0
$521$	10239.3	0.861023	0.430511	−	0.902585i	$-0.358333\pi$
$521$	10239.3	0.861023	0.430511	+	0.902585i	$0.358333\pi$
$522$	0	0
$523$	2822.00	0.235942	0.117971	−	0.993017i	$-0.462361\pi$
$523$	2822.00	0.235942	0.117971	+	0.993017i	$0.462361\pi$
$524$	−52098.1	−4.34335
$525$	0	0
$526$	−41215.3	−3.41649
$527$	−1160.22	−0.0959010
$528$	0	0
$529$	−11611.1	−0.954310
$530$	0	0
$531$	0	0
$532$	−37718.7	−3.07390
$533$	4922.67	0.400046
$534$	0	0
$535$	0	0
$536$	−33535.8	−2.70248
$537$	0	0
$538$	30224.4	2.42205
$539$	−11450.3	−0.915028
$540$	0	0
$541$	−9409.63	−0.747785	−0.373892	−	0.927472i	$-0.621977\pi$
$541$	−9409.63	−0.747785	−0.373892	+	0.927472i	$0.621977\pi$
$542$	16694.2	1.32302
$543$	0	0
$544$	27091.0	2.13514
$545$	0	0
$546$	0	0
$547$	3836.43	0.299879	0.149940	−	0.988695i	$-0.452092\pi$
$547$	3836.43	0.299879	0.149940	+	0.988695i	$0.452092\pi$
$548$	21851.6	1.70339
$549$	0	0
$550$	0	0
$551$	23596.7	1.82441
$552$	0	0
$553$	5316.97	0.408862
$554$	22218.2	1.70390
$555$	0	0
$556$	2873.35	0.219167
$557$	6145.92	0.467524	0.233762	−	0.972294i	$-0.424896\pi$
$557$	6145.92	0.467524	0.233762	+	0.972294i	$0.424896\pi$
$558$	0	0
$559$	−16192.0	−1.22513
$560$	0	0
$561$	0	0
$562$	−40646.9	−3.05087
$563$	13247.9	0.991707	0.495854	−	0.868406i	$-0.334855\pi$
$563$	13247.9	0.991707	0.495854	+	0.868406i	$0.334855\pi$
$564$	0	0
$565$	0	0
$566$	4228.63	0.314033
$567$	0	0
$568$	−42242.6	−3.12053
$569$	−6544.89	−0.482208	−0.241104	−	0.970499i	$-0.577509\pi$
$569$	−6544.89	−0.482208	−0.241104	+	0.970499i	$0.577509\pi$
$570$	0	0
$571$	20362.1	1.49234	0.746170	−	0.665755i	$-0.231890\pi$
$571$	20362.1	1.49234	0.746170	+	0.665755i	$0.231890\pi$
$572$	−42410.2	−3.10011
$573$	0	0
$574$	−9170.43	−0.666840
$575$	0	0
$576$	0	0
$577$	26247.4	1.89375	0.946876	−	0.321600i	$-0.104221\pi$
$577$	26247.4	1.89375	0.946876	+	0.321600i	$0.104221\pi$
$578$	−18401.8	−1.32425
$579$	0	0
$580$	0	0
$581$	3356.54	0.239677
$582$	0	0
$583$	−7579.71	−0.538455
$584$	6678.49	0.473215
$585$	0	0
$586$	36842.4	2.59718
$587$	−14098.2	−0.991301	−0.495650	−	0.868522i	$-0.665070\pi$
$587$	−14098.2	−0.991301	−0.495650	+	0.868522i	$0.665070\pi$
$588$	0	0
$589$	4330.36	0.302936
$590$	0	0
$591$	0	0
$592$	51541.9	3.57831
$593$	3476.71	0.240761	0.120380	−	0.992728i	$-0.461589\pi$
$593$	3476.71	0.240761	0.120380	+	0.992728i	$0.461589\pi$
$594$	0	0
$595$	0	0
$596$	−22222.8	−1.52732
$597$	0	0
$598$	4452.69	0.304488
$599$	16179.0	1.10360	0.551798	−	0.833978i	$-0.313942\pi$
$599$	16179.0	1.10360	0.551798	+	0.833978i	$0.313942\pi$
$600$	0	0
$601$	8112.16	0.550586	0.275293	−	0.961360i	$-0.411225\pi$
$601$	8112.16	0.550586	0.275293	+	0.961360i	$0.411225\pi$
$602$	30164.0	2.04218
$603$	0	0
$604$	61522.3	4.14455
$605$	0	0
$606$	0	0
$607$	−21139.2	−1.41353	−0.706765	−	0.707449i	$-0.749846\pi$
$607$	−21139.2	−1.41353	−0.706765	+	0.707449i	$0.749846\pi$
$608$	−101113.	−6.74456
$609$	0	0
$610$	0	0
$611$	−13641.9	−0.903258
$612$	0	0
$613$	11440.3	0.753785	0.376892	−	0.926257i	$-0.376993\pi$
$613$	11440.3	0.753785	0.376892	+	0.926257i	$0.376993\pi$
$614$	12033.2	0.790915
$615$	0	0
$616$	50010.3	3.27106
$617$	21566.2	1.40717	0.703584	−	0.710612i	$-0.251582\pi$
$617$	21566.2	1.40717	0.703584	+	0.710612i	$0.251582\pi$
$618$	0	0
$619$	−15198.3	−0.986866	−0.493433	−	0.869784i	$-0.664258\pi$
$619$	−15198.3	−0.986866	−0.493433	+	0.869784i	$0.664258\pi$
$620$	0	0
$621$	0	0
$622$	32932.6	2.12295
$623$	−7382.68	−0.474769
$624$	0	0
$625$	0	0
$626$	31513.2	2.01201
$627$	0	0
$628$	−10386.7	−0.659994
$629$	8548.05	0.541865
$630$	0	0
$631$	−4929.66	−0.311009	−0.155504	−	0.987835i	$-0.549700\pi$
$631$	−4929.66	−0.311009	−0.155504	+	0.987835i	$0.549700\pi$
$632$	33919.8	2.13490
$633$	0	0
$634$	−18025.2	−1.12914
$635$	0	0
$636$	0	0
$637$	7043.97	0.438135
$638$	−49425.8	−3.06706
$639$	0	0
$640$	0	0
$641$	7535.73	0.464342	0.232171	−	0.972675i	$-0.425417\pi$
$641$	7535.73	0.464342	0.232171	+	0.972675i	$0.425417\pi$
$642$	0	0
$643$	15997.2	0.981135	0.490568	−	0.871403i	$-0.336789\pi$
$643$	15997.2	0.981135	0.490568	+	0.871403i	$0.336789\pi$
$644$	−6067.93	−0.371289
$645$	0	0
$646$	−31415.5	−1.91336
$647$	6020.46	0.365825	0.182913	−	0.983129i	$-0.441447\pi$
$647$	6020.46	0.365825	0.182913	+	0.983129i	$0.441447\pi$
$648$	0	0
$649$	7378.12	0.446250
$650$	0	0
$651$	0	0
$652$	48876.8	2.93583
$653$	10948.7	0.656133	0.328067	−	0.944655i	$-0.393603\pi$
$653$	10948.7	0.656133	0.328067	+	0.944655i	$0.393603\pi$
$654$	0	0
$655$	0	0
$656$	−33690.0	−2.00514
$657$	0	0
$658$	25413.4	1.50565
$659$	12338.3	0.729334	0.364667	−	0.931138i	$-0.381183\pi$
$659$	12338.3	0.729334	0.364667	+	0.931138i	$0.381183\pi$
$660$	0	0
$661$	20016.8	1.17786	0.588928	−	0.808186i	$-0.299550\pi$
$661$	20016.8	1.17786	0.588928	+	0.808186i	$0.299550\pi$
$662$	43960.2	2.58091
$663$	0	0
$664$	21413.1	1.25149
$665$	0	0
$666$	0	0
$667$	3796.08	0.220367
$668$	2071.66	0.119993
$669$	0	0
$670$	0	0
$671$	14610.6	0.840589
$672$	0	0
$673$	8419.22	0.482225	0.241112	−	0.970497i	$-0.422488\pi$
$673$	8419.22	0.482225	0.241112	+	0.970497i	$0.422488\pi$
$674$	−18889.1	−1.07950
$675$	0	0
$676$	−21800.5	−1.24036
$677$	−25707.1	−1.45938	−0.729692	−	0.683776i	$-0.760337\pi$
$677$	−25707.1	−1.45938	−0.729692	+	0.683776i	$0.760337\pi$
$678$	0	0
$679$	2281.74	0.128962
$680$	0	0
$681$	0	0
$682$	−9070.41	−0.509272
$683$	19624.3	1.09942	0.549708	−	0.835357i	$-0.314739\pi$
$683$	19624.3	1.09942	0.549708	+	0.835357i	$0.314739\pi$
$684$	0	0
$685$	0	0
$686$	−35228.2	−1.96067
$687$	0	0
$688$	110815.	6.14069
$689$	4662.86	0.257824
$690$	0	0
$691$	7273.23	0.400415	0.200207	−	0.979754i	$-0.435838\pi$
$691$	7273.23	0.400415	0.200207	+	0.979754i	$0.435838\pi$
$692$	−46511.8	−2.55508
$693$	0	0
$694$	−50920.8	−2.78520
$695$	0	0
$696$	0	0
$697$	−5587.37	−0.303639
$698$	48579.8	2.63434
$699$	0	0
$700$	0	0
$701$	17644.3	0.950664	0.475332	−	0.879807i	$-0.342328\pi$
$701$	17644.3	0.950664	0.475332	+	0.879807i	$0.342328\pi$
$702$	0	0
$703$	−31904.5	−1.71166
$704$	105270.	5.63568
$705$	0	0
$706$	20320.0	1.08322
$707$	−16232.3	−0.863476
$708$	0	0
$709$	24304.4	1.28741	0.643703	−	0.765276i	$-0.277397\pi$
$709$	24304.4	1.28741	0.643703	+	0.765276i	$0.277397\pi$
$710$	0	0
$711$	0	0
$712$	−47098.1	−2.47904
$713$	696.639	0.0365909
$714$	0	0
$715$	0	0
$716$	37158.6	1.93950
$717$	0	0
$718$	−60206.0	−3.12934
$719$	−15170.2	−0.786863	−0.393431	−	0.919354i	$-0.628712\pi$
$719$	−15170.2	−0.786863	−0.393431	+	0.919354i	$0.628712\pi$
$720$	0	0
$721$	−23962.5	−1.23774
$722$	79812.8	4.11402
$723$	0	0
$724$	−29666.8	−1.52287
$725$	0	0
$726$	0	0
$727$	17487.0	0.892102	0.446051	−	0.895008i	$-0.352830\pi$
$727$	17487.0	0.892102	0.446051	+	0.895008i	$0.352830\pi$
$728$	−30765.2	−1.56625
$729$	0	0
$730$	0	0
$731$	18378.4	0.929888
$732$	0	0
$733$	−18698.0	−0.942194	−0.471097	−	0.882082i	$-0.656142\pi$
$733$	−18698.0	−0.942194	−0.471097	+	0.882082i	$0.656142\pi$
$734$	26492.0	1.33220
$735$	0	0
$736$	−16266.5	−0.814660
$737$	−25039.5	−1.25148
$738$	0	0
$739$	−35250.3	−1.75467	−0.877336	−	0.479876i	$-0.840682\pi$
$739$	−35250.3	−1.75467	−0.877336	+	0.479876i	$0.840682\pi$
$740$	0	0
$741$	0	0
$742$	−8686.43	−0.429769
$743$	11133.6	0.549731	0.274866	−	0.961483i	$-0.411367\pi$
$743$	11133.6	0.549731	0.274866	+	0.961483i	$0.411367\pi$
$744$	0	0
$745$	0	0
$746$	67545.5	3.31503
$747$	0	0
$748$	48136.8	2.35301
$749$	−9717.49	−0.474058
$750$	0	0
$751$	17197.6	0.835619	0.417809	−	0.908535i	$-0.362798\pi$
$751$	17197.6	0.835619	0.417809	+	0.908535i	$0.362798\pi$
$752$	93362.7	4.52738
$753$	0	0
$754$	30405.6	1.46858
$755$	0	0
$756$	0	0
$757$	−804.647	−0.0386333	−0.0193166	−	0.999813i	$-0.506149\pi$
$757$	−804.647	−0.0386333	−0.0193166	+	0.999813i	$0.506149\pi$
$758$	60868.7	2.91669
$759$	0	0
$760$	0	0
$761$	−26208.9	−1.24845	−0.624225	−	0.781245i	$-0.714585\pi$
$761$	−26208.9	−1.24845	−0.624225	+	0.781245i	$0.714585\pi$
$762$	0	0
$763$	−9797.85	−0.464883
$764$	23909.0	1.13219
$765$	0	0
$766$	12007.0	0.566360
$767$	−4538.85	−0.213674
$768$	0	0
$769$	36544.2	1.71367	0.856837	−	0.515587i	$-0.172426\pi$
$769$	36544.2	1.71367	0.856837	+	0.515587i	$0.172426\pi$
$770$	0	0
$771$	0	0
$772$	62500.8	2.91380
$773$	−42387.4	−1.97228	−0.986139	−	0.165923i	$-0.946940\pi$
$773$	−42387.4	−1.97228	−0.986139	+	0.165923i	$0.946940\pi$
$774$	0	0
$775$	0	0
$776$	14556.5	0.673385
$777$	0	0
$778$	−50323.8	−2.31902
$779$	20854.1	0.959148
$780$	0	0
$781$	−31540.4	−1.44507
$782$	−5053.92	−0.231110
$783$	0	0
$784$	−48207.8	−2.19606
$785$	0	0
$786$	0	0
$787$	−26849.4	−1.21611	−0.608054	−	0.793896i	$-0.708050\pi$
$787$	−26849.4	−1.21611	−0.608054	+	0.793896i	$0.708050\pi$
$788$	15784.8	0.713592
$789$	0	0
$790$	0	0
$791$	−17749.5	−0.797851
$792$	0	0
$793$	−8988.09	−0.402492
$794$	−6110.31	−0.273107
$795$	0	0
$796$	−36918.3	−1.64389
$797$	−31307.0	−1.39141	−0.695704	−	0.718328i	$-0.744908\pi$
$797$	−31307.0	−1.39141	−0.695704	+	0.718328i	$0.744908\pi$
$798$	0	0
$799$	15483.9	0.685583
$800$	0	0
$801$	0	0
$802$	−67125.7	−2.95548
$803$	4986.48	0.219140
$804$	0	0
$805$	0	0
$806$	5579.90	0.243851
$807$	0	0
$808$	−103554.	−4.50870
$809$	10011.9	0.435106	0.217553	−	0.976049i	$-0.430193\pi$
$809$	10011.9	0.435106	0.217553	+	0.976049i	$0.430193\pi$
$810$	0	0
$811$	4603.68	0.199331	0.0996653	−	0.995021i	$-0.468223\pi$
$811$	4603.68	0.199331	0.0996653	+	0.995021i	$0.468223\pi$
$812$	−41435.5	−1.79076
$813$	0	0
$814$	66827.4	2.87752
$815$	0	0
$816$	0	0
$817$	−68594.8	−2.93737
$818$	13648.1	0.583367
$819$	0	0
$820$	0	0
$821$	35429.0	1.50607	0.753033	−	0.657983i	$-0.228590\pi$
$821$	35429.0	1.50607	0.753033	+	0.657983i	$0.228590\pi$
$822$	0	0
$823$	28297.6	1.19853	0.599265	−	0.800550i	$-0.295459\pi$
$823$	28297.6	1.19853	0.599265	+	0.800550i	$0.295459\pi$
$824$	−152870.	−6.46295
$825$	0	0
$826$	8455.40	0.356176
$827$	41059.9	1.72647	0.863236	−	0.504800i	$-0.168434\pi$
$827$	41059.9	1.72647	0.863236	+	0.504800i	$0.168434\pi$
$828$	0	0
$829$	9030.68	0.378345	0.189173	−	0.981944i	$-0.439419\pi$
$829$	9030.68	0.378345	0.189173	+	0.981944i	$0.439419\pi$
$830$	0	0
$831$	0	0
$832$	−64759.8	−2.69849
$833$	−7995.10	−0.332550
$834$	0	0
$835$	0	0
$836$	−179664.	−7.43280
$837$	0	0
$838$	45487.7	1.87511
$839$	−5227.81	−0.215118	−0.107559	−	0.994199i	$-0.534303\pi$
$839$	−5227.81	−0.215118	−0.107559	+	0.994199i	$0.534303\pi$
$840$	0	0
$841$	1532.87	0.0628508
$842$	62089.0	2.54125
$843$	0	0
$844$	20652.6	0.842288
$845$	0	0
$846$	0	0
$847$	21625.6	0.877290
$848$	−31911.8	−1.29228
$849$	0	0
$850$	0	0
$851$	−5132.58	−0.206748
$852$	0	0
$853$	30002.9	1.20432	0.602158	−	0.798377i	$-0.294308\pi$
$853$	30002.9	1.20432	0.602158	+	0.798377i	$0.294308\pi$
$854$	16743.9	0.670918
$855$	0	0
$856$	−61993.1	−2.47533
$857$	15671.7	0.624662	0.312331	−	0.949973i	$-0.398890\pi$
$857$	15671.7	0.624662	0.312331	+	0.949973i	$0.398890\pi$
$858$	0	0
$859$	−31306.8	−1.24351	−0.621755	−	0.783212i	$-0.713580\pi$
$859$	−31306.8	−1.24351	−0.621755	+	0.783212i	$0.713580\pi$
$860$	0	0
$861$	0	0
$862$	54752.3	2.16342
$863$	13212.3	0.521150	0.260575	−	0.965454i	$-0.416088\pi$
$863$	13212.3	0.521150	0.260575	+	0.965454i	$0.416088\pi$
$864$	0	0
$865$	0	0
$866$	−41537.8	−1.62992
$867$	0	0
$868$	−7604.05	−0.297348
$869$	25326.2	0.988643
$870$	0	0
$871$	15403.7	0.599236
$872$	−62505.7	−2.42742
$873$	0	0
$874$	18863.1	0.730039
$875$	0	0
$876$	0	0
$877$	−15440.6	−0.594519	−0.297260	−	0.954797i	$-0.596073\pi$
$877$	−15440.6	−0.594519	−0.297260	+	0.954797i	$0.596073\pi$
$878$	−67527.5	−2.59561
$879$	0	0
$880$	0	0
$881$	27563.1	1.05406	0.527029	−	0.849847i	$-0.323306\pi$
$881$	27563.1	1.05406	0.527029	+	0.849847i	$0.323306\pi$
$882$	0	0
$883$	−19897.7	−0.758337	−0.379169	−	0.925328i	$-0.623790\pi$
$883$	−19897.7	−0.758337	−0.379169	+	0.925328i	$0.623790\pi$
$884$	−29612.6	−1.12667
$885$	0	0
$886$	−65965.9	−2.50132
$887$	40061.6	1.51650	0.758251	−	0.651963i	$-0.226054\pi$
$887$	40061.6	1.51650	0.758251	+	0.651963i	$0.226054\pi$
$888$	0	0
$889$	6804.66	0.256716
$890$	0	0
$891$	0	0
$892$	−2438.45	−0.0915306
$893$	−57791.6	−2.16565
$894$	0	0
$895$	0	0
$896$	55477.5	2.06850
$897$	0	0
$898$	3414.72	0.126894
$899$	4757.07	0.176482
$900$	0	0
$901$	−5292.47	−0.195691
$902$	−43681.2	−1.61244
$903$	0	0
$904$	−113234.	−4.16603
$905$	0	0
$906$	0	0
$907$	−27839.6	−1.01918	−0.509591	−	0.860417i	$-0.670203\pi$
$907$	−27839.6	−1.01918	−0.509591	+	0.860417i	$0.670203\pi$
$908$	26172.9	0.956584
$909$	0	0
$910$	0	0
$911$	−22251.0	−0.809232	−0.404616	−	0.914487i	$-0.632595\pi$
$911$	−22251.0	−0.809232	−0.404616	+	0.914487i	$0.632595\pi$
$912$	0	0
$913$	15988.1	0.579549
$914$	−9887.02	−0.357805
$915$	0	0
$916$	17926.3	0.646616
$917$	−28218.0	−1.01618
$918$	0	0
$919$	29480.5	1.05819	0.529093	−	0.848564i	$-0.322532\pi$
$919$	29480.5	1.05819	0.529093	+	0.848564i	$0.322532\pi$
$920$	0	0
$921$	0	0
$922$	63457.9	2.26668
$923$	19402.9	0.691933
$924$	0	0
$925$	0	0
$926$	−39804.6	−1.41259
$927$	0	0
$928$	−111077.	−3.92919
$929$	−42351.8	−1.49571	−0.747857	−	0.663860i	$-0.768917\pi$
$929$	−42351.8	−1.49571	−0.747857	+	0.663860i	$0.768917\pi$
$930$	0	0
$931$	29840.7	1.05047
$932$	−116166.	−4.08278
$933$	0	0
$934$	−63526.9	−2.22555
$935$	0	0
$936$	0	0
$937$	35930.6	1.25272	0.626362	−	0.779532i	$-0.284543\pi$
$937$	35930.6	1.25272	0.626362	+	0.779532i	$0.284543\pi$
$938$	−28695.5	−0.998872
$939$	0	0
$940$	0	0
$941$	21564.0	0.747041	0.373521	−	0.927622i	$-0.378151\pi$
$941$	21564.0	0.747041	0.373521	+	0.927622i	$0.378151\pi$
$942$	0	0
$943$	3354.87	0.115853
$944$	31063.1	1.07099
$945$	0	0
$946$	143679.	4.93807
$947$	−16632.4	−0.570729	−0.285364	−	0.958419i	$-0.592115\pi$
$947$	−16632.4	−0.570729	−0.285364	+	0.958419i	$0.592115\pi$
$948$	0	0
$949$	−3067.57	−0.104929
$950$	0	0
$951$	0	0
$952$	34919.3	1.18880
$953$	39557.8	1.34460	0.672299	−	0.740279i	$-0.265307\pi$
$953$	39557.8	1.34460	0.672299	+	0.740279i	$0.265307\pi$
$954$	0	0
$955$	0	0
$956$	154452.	5.22526
$957$	0	0
$958$	−65731.4	−2.21679
$959$	11835.5	0.398529
$960$	0	0
$961$	−28918.0	−0.970696
$962$	−41110.6	−1.37782
$963$	0	0
$964$	−142997.	−4.77762
$965$	0	0
$966$	0	0
$967$	10666.2	0.354707	0.177354	−	0.984147i	$-0.443246\pi$
$967$	10666.2	0.354707	0.177354	+	0.984147i	$0.443246\pi$
$968$	137961.	4.58083
$969$	0	0
$970$	0	0
$971$	−31821.8	−1.05171	−0.525855	−	0.850574i	$-0.676254\pi$
$971$	−31821.8	−1.05171	−0.525855	+	0.850574i	$0.676254\pi$
$972$	0	0
$973$	1556.30	0.0512771
$974$	−38417.7	−1.26384
$975$	0	0
$976$	61513.0	2.01740
$977$	11126.5	0.364347	0.182173	−	0.983266i	$-0.441687\pi$
$977$	11126.5	0.364347	0.182173	+	0.983266i	$0.441687\pi$
$978$	0	0
$979$	−35165.7	−1.14801
$980$	0	0
$981$	0	0
$982$	−36044.5	−1.17131
$983$	−991.225	−0.0321619	−0.0160810	−	0.999871i	$-0.505119\pi$
$983$	−991.225	−0.0321619	−0.0160810	+	0.999871i	$0.505119\pi$
$984$	0	0
$985$	0	0
$986$	−34511.2	−1.11467
$987$	0	0
$988$	110525.	3.55898
$989$	−11035.1	−0.354798
$990$	0	0
$991$	−48714.9	−1.56153	−0.780767	−	0.624822i	$-0.785171\pi$
$991$	−48714.9	−1.56153	−0.780767	+	0.624822i	$0.785171\pi$
$992$	−20384.4	−0.652424
$993$	0	0
$994$	−36145.6	−1.15339
$995$	0	0
$996$	0	0
$997$	42207.2	1.34074	0.670369	−	0.742028i	$-0.266136\pi$
$997$	42207.2	1.34074	0.670369	+	0.742028i	$0.266136\pi$
$998$	−200.362	−0.00635505
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.s.1.3		3
3.2	odd	2		675.4.a.p.1.1		3
5.2	odd	4		675.4.b.m.649.6		6
5.3	odd	4		675.4.b.m.649.1		6
5.4	even	2		135.4.a.e.1.1	✓	3
15.2	even	4		675.4.b.n.649.1		6
15.8	even	4		675.4.b.n.649.6		6
15.14	odd	2		135.4.a.h.1.3	yes	3
20.19	odd	2		2160.4.a.bi.1.2		3
45.4	even	6		405.4.e.v.136.3		6
45.14	odd	6		405.4.e.q.136.1		6
45.29	odd	6		405.4.e.q.271.1		6
45.34	even	6		405.4.e.v.271.3		6
60.59	even	2		2160.4.a.bq.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.1	✓	3	5.4	even	2
135.4.a.h.1.3	yes	3	15.14	odd	2
405.4.e.q.136.1		6	45.14	odd	6
405.4.e.q.271.1		6	45.29	odd	6
405.4.e.v.136.3		6	45.4	even	6
405.4.e.v.271.3		6	45.34	even	6
675.4.a.p.1.1		3	3.2	odd	2
675.4.a.s.1.3		3	1.1	even	1	trivial
675.4.b.m.649.1		6	5.3	odd	4
675.4.b.m.649.6		6	5.2	odd	4
675.4.b.n.649.1		6	15.2	even	4
675.4.b.n.649.6		6	15.8	even	4
2160.4.a.bi.1.2		3	20.19	odd	2
2160.4.a.bq.1.2		3	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q+5.45876 q^{2} +21.7980 q^{4} +11.8065 q^{7} +75.3201 q^{8} +56.2376 q^{11} -34.5961 q^{13} +64.4489 q^{14} +236.770 q^{16} +39.2675 q^{17} -146.561 q^{19} +306.987 q^{22} -23.5777 q^{23} -188.851 q^{26} +257.359 q^{28} -161.003 q^{29} -29.5465 q^{31} +689.908 q^{32} +214.352 q^{34} +217.688 q^{37} -800.039 q^{38} -142.290 q^{41} +468.030 q^{43} +1225.87 q^{44} -128.705 q^{46} +394.318 q^{47} -203.606 q^{49} -754.126 q^{52} -134.780 q^{53} +889.268 q^{56} -878.875 q^{58} +131.195 q^{59} +259.801 q^{61} -161.287 q^{62} +1871.88 q^{64} -445.244 q^{67} +855.954 q^{68} -560.841 q^{71} +88.6681 q^{73} +1188.30 q^{74} -3194.74 q^{76} +663.970 q^{77} +450.342 q^{79} -776.726 q^{82} +284.295 q^{83} +2554.86 q^{86} +4235.82 q^{88} -625.305 q^{89} -408.459 q^{91} -513.948 q^{92} +2152.49 q^{94} +193.261 q^{97} -1111.44 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{2} + 17 q^{4} + 4 q^{7} + 75 q^{8} - 5 q^{11} - 7 q^{13} - 60 q^{14} + 161 q^{16} + 155 q^{17} - 50 q^{19} + 229 q^{22} + 285 q^{23} - 185 q^{26} + 334 q^{28} - 115 q^{29} - 115 q^{31} + 775 q^{32}+ \cdots - 305 q^{98}+O(q^{100}) \) 3 * q + 5 * q^2 + 17 * q^4 + 4 * q^7 + 75 * q^8 - 5 * q^11 - 7 * q^13 - 60 * q^14 + 161 * q^16 + 155 * q^17 - 50 * q^19 + 229 * q^22 + 285 * q^23 - 185 * q^26 + 334 * q^28 - 115 * q^29 - 115 * q^31 + 775 * q^32 + 413 * q^34 + 384 * q^37 - 1150 * q^38 - 580 * q^41 + 797 * q^43 + 1415 * q^44 - 285 * q^46 - 145 * q^47 + 577 * q^49 - 825 * q^52 - 400 * q^53 + 2190 * q^56 + 59 * q^58 - 380 * q^59 - 152 * q^61 - 1005 * q^62 + 2937 * q^64 - 2 * q^67 + 475 * q^68 - 40 * q^71 + 980 * q^73 + 2720 * q^74 - 3276 * q^76 + 1950 * q^77 + 1013 * q^79 - 4 * q^82 + 270 * q^83 + 1555 * q^86 + 5193 * q^88 - 1020 * q^89 - 632 * q^91 - 1215 * q^92 + 3833 * q^94 - 720 * q^97 - 305 * q^98

Introduction

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