LMFDB - Embedded newform 1150.4.a.o.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(1,1150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1150.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-8,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 84x^{2} - 11x + 1242 $ x^4 - 84x^2 - 11x + 1242
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.92791$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -8.92791 q^{3} +4.00000 q^{4} +17.8558 q^{6} -23.5524 q^{7} -8.00000 q^{8} +52.7075 q^{9} +1.63537 q^{11} -35.7116 q^{12} +88.6599 q^{13} +47.1048 q^{14} +16.0000 q^{16} +7.31435 q^{17} -105.415 q^{18} -6.53738 q^{19} +210.274 q^{21} -3.27074 q^{22} -23.0000 q^{23} +71.4232 q^{24} -177.320 q^{26} -229.514 q^{27} -94.2096 q^{28} +119.240 q^{29} -156.456 q^{31} -32.0000 q^{32} -14.6004 q^{33} -14.6287 q^{34} +210.830 q^{36} +293.173 q^{37} +13.0748 q^{38} -791.547 q^{39} +74.3404 q^{41} -420.547 q^{42} -468.081 q^{43} +6.54148 q^{44} +46.0000 q^{46} +393.971 q^{47} -142.846 q^{48} +211.715 q^{49} -65.3018 q^{51} +354.639 q^{52} -233.171 q^{53} +459.028 q^{54} +188.419 q^{56} +58.3651 q^{57} -238.481 q^{58} -766.648 q^{59} +178.365 q^{61} +312.912 q^{62} -1241.39 q^{63} +64.0000 q^{64} +29.2009 q^{66} -246.904 q^{67} +29.2574 q^{68} +205.342 q^{69} -650.678 q^{71} -421.660 q^{72} -695.444 q^{73} -586.346 q^{74} -26.1495 q^{76} -38.5169 q^{77} +1583.09 q^{78} +660.717 q^{79} +625.978 q^{81} -148.681 q^{82} +1328.39 q^{83} +841.094 q^{84} +936.163 q^{86} -1064.57 q^{87} -13.0830 q^{88} -824.702 q^{89} -2088.15 q^{91} -92.0000 q^{92} +1396.82 q^{93} -787.942 q^{94} +285.693 q^{96} -383.833 q^{97} -423.430 q^{98} +86.1963 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} - 26 q^{7} - 32 q^{8} + 64 q^{9} + 93 q^{11} - 16 q^{12} - 32 q^{13} + 52 q^{14} + 64 q^{16} - 108 q^{17} - 128 q^{18} + 185 q^{19} + 302 q^{21} - 186 q^{22}+ \cdots + 2013 q^{99}+O(q^{100}) $ 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 - 26 * q^7 - 32 * q^8 + 64 * q^9 + 93 * q^11 - 16 * q^12 - 32 * q^13 + 52 * q^14 + 64 * q^16 - 108 * q^17 - 128 * q^18 + 185 * q^19 + 302 * q^21 - 186 * q^22 - 92 * q^23 + 32 * q^24 + 64 * q^26 - 259 * q^27 - 104 * q^28 + 294 * q^29 - 211 * q^31 - 128 * q^32 + 237 * q^33 + 216 * q^34 + 256 * q^36 - 5 * q^37 - 370 * q^38 - 421 * q^39 - 369 * q^41 - 604 * q^42 + 100 * q^43 + 372 * q^44 + 184 * q^46 + 363 * q^47 - 64 * q^48 + 600 * q^49 - 315 * q^51 - 128 * q^52 - 21 * q^53 + 518 * q^54 + 208 * q^56 + 160 * q^57 - 588 * q^58 - 33 * q^59 - 307 * q^61 + 422 * q^62 - 167 * q^63 + 256 * q^64 - 474 * q^66 - 725 * q^67 - 432 * q^68 + 92 * q^69 - 1257 * q^71 - 512 * q^72 - 509 * q^73 + 10 * q^74 + 740 * q^76 + 1962 * q^77 + 842 * q^78 + 1202 * q^79 - 992 * q^81 + 738 * q^82 + 1377 * q^83 + 1208 * q^84 - 200 * q^86 + 2829 * q^87 - 744 * q^88 - 984 * q^89 - 995 * q^91 - 368 * q^92 + 1843 * q^93 - 726 * q^94 + 128 * q^96 - 137 * q^97 - 1200 * q^98 + 2013 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−8.92791	−1.71818	−0.859088	−	0.511828i	$-0.828969\pi$
$3$	−8.92791	−1.71818	−0.859088	+	0.511828i	$0.828969\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	17.8558	1.21493
$7$	−23.5524	−1.27171	−0.635855	−	0.771809i	$-0.719352\pi$
$7$	−23.5524	−1.27171	−0.635855	+	0.771809i	$0.719352\pi$
$8$	−8.00000	−0.353553
$9$	52.7075	1.95213
$10$	0	0
$11$	1.63537	0.0448257	0.0224129	−	0.999749i	$-0.492865\pi$
$11$	1.63537	0.0448257	0.0224129	+	0.999749i	$0.492865\pi$
$12$	−35.7116	−0.859088
$13$	88.6599	1.89152	0.945762	−	0.324860i	$-0.105317\pi$
$13$	88.6599	1.89152	0.945762	+	0.324860i	$0.105317\pi$
$14$	47.1048	0.899234
$15$	0	0
$16$	16.0000	0.250000
$17$	7.31435	0.104352	0.0521762	−	0.998638i	$-0.483384\pi$
$17$	7.31435	0.104352	0.0521762	+	0.998638i	$0.483384\pi$
$18$	−105.415	−1.38036
$19$	−6.53738	−0.0789357	−0.0394678	−	0.999221i	$-0.512566\pi$
$19$	−6.53738	−0.0789357	−0.0394678	+	0.999221i	$0.512566\pi$
$20$	0	0
$21$	210.274	2.18502
$22$	−3.27074	−0.0316966
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	71.4232	0.607467
$25$	0	0
$26$	−177.320	−1.33751
$27$	−229.514	−1.63593
$28$	−94.2096	−0.635855
$29$	119.240	0.763530	0.381765	−	0.924259i	$-0.375316\pi$
$29$	119.240	0.763530	0.381765	+	0.924259i	$0.375316\pi$
$30$	0	0
$31$	−156.456	−0.906462	−0.453231	−	0.891393i	$-0.649729\pi$
$31$	−156.456	−0.906462	−0.453231	+	0.891393i	$0.649729\pi$
$32$	−32.0000	−0.176777
$33$	−14.6004	−0.0770185
$34$	−14.6287	−0.0737883
$35$	0	0
$36$	210.830	0.976065
$37$	293.173	1.30263	0.651315	−	0.758807i	$-0.274218\pi$
$37$	293.173	1.30263	0.651315	+	0.758807i	$0.274218\pi$
$38$	13.0748	0.0558159
$39$	−791.547	−3.24997
$40$	0	0
$41$	74.3404	0.283171	0.141586	−	0.989926i	$-0.454780\pi$
$41$	74.3404	0.283171	0.141586	+	0.989926i	$0.454780\pi$
$42$	−420.547	−1.54504
$43$	−468.081	−1.66004	−0.830020	−	0.557733i	$-0.811671\pi$
$43$	−468.081	−1.66004	−0.830020	+	0.557733i	$0.811671\pi$
$44$	6.54148	0.0224129
$45$	0	0
$46$	46.0000	0.147442
$47$	393.971	1.22269	0.611346	−	0.791363i	$-0.290628\pi$
$47$	393.971	1.22269	0.611346	+	0.791363i	$0.290628\pi$
$48$	−142.846	−0.429544
$49$	211.715	0.617245
$50$	0	0
$51$	−65.3018	−0.179296
$52$	354.639	0.945762
$53$	−233.171	−0.604311	−0.302155	−	0.953259i	$-0.597706\pi$
$53$	−233.171	−0.604311	−0.302155	+	0.953259i	$0.597706\pi$
$54$	459.028	1.15677
$55$	0	0
$56$	188.419	0.449617
$57$	58.3651	0.135625
$58$	−238.481	−0.539897
$59$	−766.648	−1.69168	−0.845840	−	0.533437i	$-0.820900\pi$
$59$	−766.648	−1.69168	−0.845840	+	0.533437i	$0.820900\pi$
$60$	0	0
$61$	178.365	0.374383	0.187191	−	0.982323i	$-0.440062\pi$
$61$	178.365	0.374383	0.187191	+	0.982323i	$0.440062\pi$
$62$	312.912	0.640965
$63$	−1241.39	−2.48254
$64$	64.0000	0.125000
$65$	0	0
$66$	29.2009	0.0544603
$67$	−246.904	−0.450211	−0.225105	−	0.974334i	$-0.572273\pi$
$67$	−246.904	−0.450211	−0.225105	+	0.974334i	$0.572273\pi$
$68$	29.2574	0.0521762
$69$	205.342	0.358265
$70$	0	0
$71$	−650.678	−1.08762	−0.543812	−	0.839207i	$-0.683020\pi$
$71$	−650.678	−1.08762	−0.543812	+	0.839207i	$0.683020\pi$
$72$	−421.660	−0.690182
$73$	−695.444	−1.11501	−0.557504	−	0.830174i	$-0.688241\pi$
$73$	−695.444	−1.11501	−0.557504	+	0.830174i	$0.688241\pi$
$74$	−586.346	−0.921099
$75$	0	0
$76$	−26.1495	−0.0394678
$77$	−38.5169	−0.0570053
$78$	1583.09	2.29808
$79$	660.717	0.940967	0.470484	−	0.882409i	$-0.344079\pi$
$79$	660.717	0.940967	0.470484	+	0.882409i	$0.344079\pi$
$80$	0	0
$81$	625.978	0.858681
$82$	−148.681	−0.200232
$83$	1328.39	1.75674	0.878372	−	0.477977i	$-0.158630\pi$
$83$	1328.39	1.75674	0.878372	+	0.477977i	$0.158630\pi$
$84$	841.094	1.09251
$85$	0	0
$86$	936.163	1.17383
$87$	−1064.57	−1.31188
$88$	−13.0830	−0.0158483
$89$	−824.702	−0.982227	−0.491113	−	0.871096i	$-0.663410\pi$
$89$	−824.702	−0.982227	−0.491113	+	0.871096i	$0.663410\pi$
$90$	0	0
$91$	−2088.15	−2.40547
$92$	−92.0000	−0.104257
$93$	1396.82	1.55746
$94$	−787.942	−0.864574
$95$	0	0
$96$	285.693	0.303734
$97$	−383.833	−0.401776	−0.200888	−	0.979614i	$-0.564383\pi$
$97$	−383.833	−0.401776	−0.200888	+	0.979614i	$0.564383\pi$
$98$	−423.430	−0.436458
$99$	86.1963	0.0875056
$100$	0	0
$101$	−674.579	−0.664586	−0.332293	−	0.943176i	$-0.607822\pi$
$101$	−674.579	−0.664586	−0.332293	+	0.943176i	$0.607822\pi$
$102$	130.604	0.126781
$103$	−169.297	−0.161955	−0.0809773	−	0.996716i	$-0.525804\pi$
$103$	−169.297	−0.161955	−0.0809773	+	0.996716i	$0.525804\pi$
$104$	−709.279	−0.668755
$105$	0	0
$106$	466.342	0.427312
$107$	1533.31	1.38533	0.692665	−	0.721259i	$-0.256436\pi$
$107$	1533.31	1.38533	0.692665	+	0.721259i	$0.256436\pi$
$108$	−918.057	−0.817963
$109$	1239.75	1.08941	0.544707	−	0.838626i	$-0.316641\pi$
$109$	1239.75	1.08941	0.544707	+	0.838626i	$0.316641\pi$
$110$	0	0
$111$	−2617.42	−2.23815
$112$	−376.838	−0.317927
$113$	−2299.49	−1.91432	−0.957159	−	0.289561i	$-0.906491\pi$
$113$	−2299.49	−1.91432	−0.957159	+	0.289561i	$0.906491\pi$
$114$	−116.730	−0.0959016
$115$	0	0
$116$	476.961	0.381765
$117$	4673.04	3.69250
$118$	1533.30	1.19620
$119$	−172.270	−0.132706
$120$	0	0
$121$	−1328.33	−0.997991
$122$	−356.731	−0.264729
$123$	−663.704	−0.486538
$124$	−625.824	−0.453231
$125$	0	0
$126$	2482.78	1.75542
$127$	1177.82	0.822948	0.411474	−	0.911422i	$-0.365014\pi$
$127$	1177.82	0.822948	0.411474	+	0.911422i	$0.365014\pi$
$128$	−128.000	−0.0883883
$129$	4178.99	2.85224
$130$	0	0
$131$	2385.56	1.59105	0.795525	−	0.605921i	$-0.207195\pi$
$131$	2385.56	1.59105	0.795525	+	0.605921i	$0.207195\pi$
$132$	−58.4017	−0.0385092
$133$	153.971	0.100383
$134$	493.808	0.318347
$135$	0	0
$136$	−58.5148	−0.0368941
$137$	−1123.22	−0.700460	−0.350230	−	0.936664i	$-0.613897\pi$
$137$	−1123.22	−0.700460	−0.350230	+	0.936664i	$0.613897\pi$
$138$	−410.684	−0.253331
$139$	−1801.75	−1.09944	−0.549721	−	0.835348i	$-0.685266\pi$
$139$	−1801.75	−1.09944	−0.549721	+	0.835348i	$0.685266\pi$
$140$	0	0
$141$	−3517.33	−2.10080
$142$	1301.36	0.769067
$143$	144.992	0.0847889
$144$	843.320	0.488032
$145$	0	0
$146$	1390.89	0.788429
$147$	−1890.17	−1.06054
$148$	1172.69	0.651315
$149$	663.138	0.364607	0.182303	−	0.983242i	$-0.441645\pi$
$149$	663.138	0.364607	0.182303	+	0.983242i	$0.441645\pi$
$150$	0	0
$151$	1617.66	0.871808	0.435904	−	0.899993i	$-0.356429\pi$
$151$	1617.66	0.871808	0.435904	+	0.899993i	$0.356429\pi$
$152$	52.2990	0.0279080
$153$	385.521	0.203709
$154$	77.0338	0.0403088
$155$	0	0
$156$	−3166.19	−1.62499
$157$	−875.080	−0.444834	−0.222417	−	0.974952i	$-0.571395\pi$
$157$	−875.080	−0.444834	−0.222417	+	0.974952i	$0.571395\pi$
$158$	−1321.43	−0.665364
$159$	2081.73	1.03831
$160$	0	0
$161$	541.705	0.265170
$162$	−1251.96	−0.607179
$163$	−2532.39	−1.21689	−0.608443	−	0.793597i	$-0.708206\pi$
$163$	−2532.39	−1.21689	−0.608443	+	0.793597i	$0.708206\pi$
$164$	297.361	0.141586
$165$	0	0
$166$	−2656.78	−1.24221
$167$	1131.65	0.524371	0.262186	−	0.965017i	$-0.415557\pi$
$167$	1131.65	0.524371	0.262186	+	0.965017i	$0.415557\pi$
$168$	−1682.19	−0.772522
$169$	5663.57	2.57786
$170$	0	0
$171$	−344.569	−0.154093
$172$	−1872.33	−0.830020
$173$	3186.23	1.40026	0.700128	−	0.714017i	$-0.253126\pi$
$173$	3186.23	1.40026	0.700128	+	0.714017i	$0.253126\pi$
$174$	2129.13	0.927639
$175$	0	0
$176$	26.1659	0.0112064
$177$	6844.56	2.90660
$178$	1649.40	0.694539
$179$	1663.48	0.694606	0.347303	−	0.937753i	$-0.387098\pi$
$179$	1663.48	0.694606	0.347303	+	0.937753i	$0.387098\pi$
$180$	0	0
$181$	−3747.35	−1.53889	−0.769443	−	0.638716i	$-0.779466\pi$
$181$	−3747.35	−1.53889	−0.769443	+	0.638716i	$0.779466\pi$
$182$	4176.30	1.70092
$183$	−1592.43	−0.643256
$184$	184.000	0.0737210
$185$	0	0
$186$	−2793.65	−1.10129
$187$	11.9617	0.00467767
$188$	1575.88	0.611346
$189$	5405.61	2.08042
$190$	0	0
$191$	2262.97	0.857291	0.428645	−	0.903473i	$-0.358991\pi$
$191$	2262.97	0.857291	0.428645	+	0.903473i	$0.358991\pi$
$192$	−571.386	−0.214772
$193$	2001.25	0.746390	0.373195	−	0.927753i	$-0.378262\pi$
$193$	2001.25	0.746390	0.373195	+	0.927753i	$0.378262\pi$
$194$	767.665	0.284099
$195$	0	0
$196$	846.860	0.308622
$197$	4804.08	1.73744	0.868722	−	0.495300i	$-0.164942\pi$
$197$	4804.08	1.73744	0.868722	+	0.495300i	$0.164942\pi$
$198$	−172.393	−0.0618758
$199$	3885.03	1.38393	0.691967	−	0.721929i	$-0.256744\pi$
$199$	3885.03	1.38393	0.691967	+	0.721929i	$0.256744\pi$
$200$	0	0
$201$	2204.34	0.773542
$202$	1349.16	0.469933
$203$	−2808.39	−0.970989
$204$	−261.207	−0.0896479
$205$	0	0
$206$	338.594	0.114519
$207$	−1212.27	−0.407047
$208$	1418.56	0.472881
$209$	−10.6910	−0.00353835
$210$	0	0
$211$	4568.39	1.49052	0.745262	−	0.666772i	$-0.232324\pi$
$211$	4568.39	1.49052	0.745262	+	0.666772i	$0.232324\pi$
$212$	−932.683	−0.302155
$213$	5809.20	1.86873
$214$	−3066.61	−0.979576
$215$	0	0
$216$	1836.11	0.578387
$217$	3684.91	1.15276
$218$	−2479.49	−0.770332
$219$	6208.86	1.91578
$220$	0	0
$221$	648.489	0.197385
$222$	5234.84	1.58261
$223$	−3081.07	−0.925218	−0.462609	−	0.886563i	$-0.653087\pi$
$223$	−3081.07	−0.925218	−0.462609	+	0.886563i	$0.653087\pi$
$224$	753.676	0.224809
$225$	0	0
$226$	4598.98	1.35363
$227$	−678.971	−0.198524	−0.0992618	−	0.995061i	$-0.531648\pi$
$227$	−678.971	−0.198524	−0.0992618	+	0.995061i	$0.531648\pi$
$228$	233.460	0.0678127
$229$	6399.27	1.84662	0.923310	−	0.384056i	$-0.125473\pi$
$229$	6399.27	1.84662	0.923310	+	0.384056i	$0.125473\pi$
$230$	0	0
$231$	343.875	0.0979451
$232$	−953.923	−0.269949
$233$	−6083.30	−1.71043	−0.855215	−	0.518274i	$-0.826575\pi$
$233$	−6083.30	−1.71043	−0.855215	+	0.518274i	$0.826575\pi$
$234$	−9346.08	−2.61099
$235$	0	0
$236$	−3066.59	−0.845840
$237$	−5898.82	−1.61675
$238$	344.541	0.0938372
$239$	438.313	0.118628	0.0593140	−	0.998239i	$-0.481109\pi$
$239$	438.313	0.118628	0.0593140	+	0.998239i	$0.481109\pi$
$240$	0	0
$241$	2854.41	0.762941	0.381471	−	0.924381i	$-0.375418\pi$
$241$	2854.41	0.762941	0.381471	+	0.924381i	$0.375418\pi$
$242$	2656.65	0.705686
$243$	608.207	0.160562
$244$	713.462	0.187191
$245$	0	0
$246$	1327.41	0.344034
$247$	−579.603	−0.149309
$248$	1251.65	0.320483
$249$	−11859.7	−3.01840
$250$	0	0
$251$	−2292.93	−0.576607	−0.288303	−	0.957539i	$-0.593091\pi$
$251$	−2292.93	−0.576607	−0.288303	+	0.957539i	$0.593091\pi$
$252$	−4965.55	−1.24127
$253$	−37.6135	−0.00934681
$254$	−2355.63	−0.581912
$255$	0	0
$256$	256.000	0.0625000
$257$	−3547.59	−0.861060	−0.430530	−	0.902576i	$-0.641673\pi$
$257$	−3547.59	−0.861060	−0.430530	+	0.902576i	$0.641673\pi$
$258$	−8357.97	−2.01684
$259$	−6904.92	−1.65657
$260$	0	0
$261$	6284.86	1.49051
$262$	−4771.12	−1.12504
$263$	2278.35	0.534178	0.267089	−	0.963672i	$-0.413938\pi$
$263$	2278.35	0.534178	0.267089	+	0.963672i	$0.413938\pi$
$264$	116.803	0.0272301
$265$	0	0
$266$	−307.942	−0.0709817
$267$	7362.86	1.68764
$268$	−987.616	−0.225105
$269$	3915.36	0.887450	0.443725	−	0.896163i	$-0.353657\pi$
$269$	3915.36	0.887450	0.443725	+	0.896163i	$0.353657\pi$
$270$	0	0
$271$	6252.71	1.40157	0.700784	−	0.713374i	$-0.252834\pi$
$271$	6252.71	1.40157	0.700784	+	0.713374i	$0.252834\pi$
$272$	117.030	0.0260881
$273$	18642.8	4.13302
$274$	2246.44	0.495300
$275$	0	0
$276$	821.367	0.179132
$277$	4700.53	1.01959	0.509797	−	0.860295i	$-0.329721\pi$
$277$	4700.53	1.01959	0.509797	+	0.860295i	$0.329721\pi$
$278$	3603.50	0.777423
$279$	−8246.40	−1.76953
$280$	0	0
$281$	5508.07	1.16934	0.584669	−	0.811272i	$-0.301224\pi$
$281$	5508.07	1.16934	0.584669	+	0.811272i	$0.301224\pi$
$282$	7034.67	1.48549
$283$	−6639.20	−1.39456	−0.697278	−	0.716801i	$-0.745606\pi$
$283$	−6639.20	−1.39456	−0.697278	+	0.716801i	$0.745606\pi$
$284$	−2602.71	−0.543812
$285$	0	0
$286$	−289.983	−0.0599548
$287$	−1750.89	−0.360111
$288$	−1686.64	−0.345091
$289$	−4859.50	−0.989111
$290$	0	0
$291$	3426.82	0.690323
$292$	−2781.78	−0.557504
$293$	2120.85	0.422872	0.211436	−	0.977392i	$-0.432186\pi$
$293$	2120.85	0.422872	0.211436	+	0.977392i	$0.432186\pi$
$294$	3780.34	0.749912
$295$	0	0
$296$	−2345.38	−0.460550
$297$	−375.341	−0.0733316
$298$	−1326.28	−0.257816
$299$	−2039.18	−0.394410
$300$	0	0
$301$	11024.4	2.11109
$302$	−3235.31	−0.616461
$303$	6022.58	1.14188
$304$	−104.598	−0.0197339
$305$	0	0
$306$	−771.042	−0.144044
$307$	−7422.87	−1.37995	−0.689977	−	0.723832i	$-0.742379\pi$
$307$	−7422.87	−1.37995	−0.689977	+	0.723832i	$0.742379\pi$
$308$	−154.068	−0.0285026
$309$	1511.47	0.278266
$310$	0	0
$311$	−2563.97	−0.467490	−0.233745	−	0.972298i	$-0.575098\pi$
$311$	−2563.97	−0.467490	−0.233745	+	0.972298i	$0.575098\pi$
$312$	6332.37	1.14904
$313$	−9403.53	−1.69814	−0.849072	−	0.528277i	$-0.822838\pi$
$313$	−9403.53	−1.69814	−0.849072	+	0.528277i	$0.822838\pi$
$314$	1750.16	0.314545
$315$	0	0
$316$	2642.87	0.470484
$317$	−407.110	−0.0721312	−0.0360656	−	0.999349i	$-0.511483\pi$
$317$	−407.110	−0.0721312	−0.0360656	+	0.999349i	$0.511483\pi$
$318$	−4163.45	−0.734198
$319$	195.002	0.0342258
$320$	0	0
$321$	−13689.2	−2.38024
$322$	−1083.41	−0.187503
$323$	−47.8167	−0.00823713
$324$	2503.91	0.429340
$325$	0	0
$326$	5064.79	0.860469
$327$	−11068.3	−1.87180
$328$	−594.723	−0.100116
$329$	−9278.95	−1.55491
$330$	0	0
$331$	−7122.33	−1.18272	−0.591358	−	0.806409i	$-0.701408\pi$
$331$	−7122.33	−1.18272	−0.591358	+	0.806409i	$0.701408\pi$
$332$	5313.56	0.878372
$333$	15452.4	2.54290
$334$	−2263.31	−0.370786
$335$	0	0
$336$	3364.38	0.546255
$337$	8267.78	1.33642	0.668212	−	0.743971i	$-0.267060\pi$
$337$	8267.78	1.33642	0.668212	+	0.743971i	$0.267060\pi$
$338$	−11327.1	−1.82283
$339$	20529.7	3.28914
$340$	0	0
$341$	−255.864	−0.0406328
$342$	689.138	0.108960
$343$	3092.08	0.486753
$344$	3744.65	0.586913
$345$	0	0
$346$	−6372.45	−0.990130
$347$	−1048.83	−0.162260	−0.0811302	−	0.996704i	$-0.525853\pi$
$347$	−1048.83	−0.162260	−0.0811302	+	0.996704i	$0.525853\pi$
$348$	−4258.27	−0.655940
$349$	−9135.87	−1.40124	−0.700619	−	0.713535i	$-0.747093\pi$
$349$	−9135.87	−1.40124	−0.700619	+	0.713535i	$0.747093\pi$
$350$	0	0
$351$	−20348.7	−3.09440
$352$	−52.3319	−0.00792414
$353$	−5253.06	−0.792046	−0.396023	−	0.918240i	$-0.629610\pi$
$353$	−5253.06	−0.792046	−0.396023	+	0.918240i	$0.629610\pi$
$354$	−13689.1	−2.05528
$355$	0	0
$356$	−3298.81	−0.491113
$357$	1538.01	0.228012
$358$	−3326.96	−0.491160
$359$	−11269.7	−1.65680	−0.828400	−	0.560137i	$-0.810748\pi$
$359$	−11269.7	−1.65680	−0.828400	+	0.560137i	$0.810748\pi$
$360$	0	0
$361$	−6816.26	−0.993769
$362$	7494.70	1.08816
$363$	11859.2	1.71472
$364$	−8352.60	−1.20273
$365$	0	0
$366$	3184.86	0.454850
$367$	5675.46	0.807238	0.403619	−	0.914927i	$-0.367752\pi$
$367$	5675.46	0.807238	0.403619	+	0.914927i	$0.367752\pi$
$368$	−368.000	−0.0521286
$369$	3918.29	0.552787
$370$	0	0
$371$	5491.73	0.768508
$372$	5587.30	0.778731
$373$	−3662.20	−0.508369	−0.254185	−	0.967156i	$-0.581807\pi$
$373$	−3662.20	−0.508369	−0.254185	+	0.967156i	$0.581807\pi$
$374$	−23.9233	−0.00330761
$375$	0	0
$376$	−3151.77	−0.432287
$377$	10571.8	1.44424
$378$	−10811.2	−1.47108
$379$	5520.89	0.748256	0.374128	−	0.927377i	$-0.377942\pi$
$379$	5520.89	0.748256	0.374128	+	0.927377i	$0.377942\pi$
$380$	0	0
$381$	−10515.4	−1.41397
$382$	−4525.93	−0.606196
$383$	2309.13	0.308071	0.154036	−	0.988065i	$-0.450773\pi$
$383$	2309.13	0.308071	0.154036	+	0.988065i	$0.450773\pi$
$384$	1142.77	0.151867
$385$	0	0
$386$	−4002.50	−0.527777
$387$	−24671.4	−3.24061
$388$	−1535.33	−0.200888
$389$	−8194.51	−1.06807	−0.534034	−	0.845463i	$-0.679324\pi$
$389$	−8194.51	−1.06807	−0.534034	+	0.845463i	$0.679324\pi$
$390$	0	0
$391$	−168.230	−0.0217590
$392$	−1693.72	−0.218229
$393$	−21298.1	−2.73370
$394$	−9608.16	−1.22856
$395$	0	0
$396$	344.785	0.0437528
$397$	−7273.90	−0.919564	−0.459782	−	0.888032i	$-0.652072\pi$
$397$	−7273.90	−0.919564	−0.459782	+	0.888032i	$0.652072\pi$
$398$	−7770.07	−0.978589
$399$	−1374.64	−0.172476
$400$	0	0
$401$	−1835.81	−0.228618	−0.114309	−	0.993445i	$-0.536465\pi$
$401$	−1835.81	−0.228618	−0.114309	+	0.993445i	$0.536465\pi$
$402$	−4408.67	−0.546977
$403$	−13871.4	−1.71460
$404$	−2698.32	−0.332293
$405$	0	0
$406$	5616.79	0.686593
$407$	479.446	0.0583914
$408$	522.415	0.0633906
$409$	−2865.80	−0.346466	−0.173233	−	0.984881i	$-0.555421\pi$
$409$	−2865.80	−0.346466	−0.173233	+	0.984881i	$0.555421\pi$
$410$	0	0
$411$	10028.0	1.20351
$412$	−677.187	−0.0809773
$413$	18056.4	2.15132
$414$	2424.55	0.287826
$415$	0	0
$416$	−2837.12	−0.334377
$417$	16085.9	1.88904
$418$	21.3821	0.00250199
$419$	11044.0	1.28767	0.643836	−	0.765164i	$-0.277342\pi$
$419$	11044.0	1.28767	0.643836	+	0.765164i	$0.277342\pi$
$420$	0	0
$421$	6343.24	0.734324	0.367162	−	0.930157i	$-0.380329\pi$
$421$	6343.24	0.734324	0.367162	+	0.930157i	$0.380329\pi$
$422$	−9136.77	−1.05396
$423$	20765.2	2.38685
$424$	1865.37	0.213656
$425$	0	0
$426$	−11618.4	−1.32139
$427$	−4200.93	−0.476106
$428$	6133.23	0.692665
$429$	−1294.47	−0.145682
$430$	0	0
$431$	−4117.01	−0.460115	−0.230058	−	0.973177i	$-0.573891\pi$
$431$	−4117.01	−0.460115	−0.230058	+	0.973177i	$0.573891\pi$
$432$	−3672.23	−0.408982
$433$	758.791	0.0842151	0.0421076	−	0.999113i	$-0.486593\pi$
$433$	758.791	0.0842151	0.0421076	+	0.999113i	$0.486593\pi$
$434$	−7369.82	−0.815122
$435$	0	0
$436$	4958.98	0.544707
$437$	150.360	0.0164592
$438$	−12417.7	−1.35466
$439$	4150.10	0.451193	0.225596	−	0.974221i	$-0.427567\pi$
$439$	4150.10	0.451193	0.225596	+	0.974221i	$0.427567\pi$
$440$	0	0
$441$	11159.0	1.20494
$442$	−1296.98	−0.139572
$443$	4597.24	0.493051	0.246525	−	0.969136i	$-0.420711\pi$
$443$	4597.24	0.493051	0.246525	+	0.969136i	$0.420711\pi$
$444$	−10469.7	−1.11907
$445$	0	0
$446$	6162.13	0.654228
$447$	−5920.43	−0.626458
$448$	−1507.35	−0.158964
$449$	6402.58	0.672954	0.336477	−	0.941692i	$-0.390765\pi$
$449$	6402.58	0.672954	0.336477	+	0.941692i	$0.390765\pi$
$450$	0	0
$451$	121.574	0.0126933
$452$	−9197.97	−0.957159
$453$	−14442.3	−1.49792
$454$	1357.94	0.140377
$455$	0	0
$456$	−466.921	−0.0479508
$457$	−551.499	−0.0564508	−0.0282254	−	0.999602i	$-0.508986\pi$
$457$	−551.499	−0.0564508	−0.0282254	+	0.999602i	$0.508986\pi$
$458$	−12798.5	−1.30576
$459$	−1678.75	−0.170713
$460$	0	0
$461$	−3901.96	−0.394213	−0.197107	−	0.980382i	$-0.563154\pi$
$461$	−3901.96	−0.394213	−0.197107	+	0.980382i	$0.563154\pi$
$462$	−687.750	−0.0692577
$463$	−4828.15	−0.484628	−0.242314	−	0.970198i	$-0.577906\pi$
$463$	−4828.15	−0.484628	−0.242314	+	0.970198i	$0.577906\pi$
$464$	1907.85	0.190883
$465$	0	0
$466$	12166.6	1.20946
$467$	14136.3	1.40075	0.700375	−	0.713775i	$-0.253016\pi$
$467$	14136.3	1.40075	0.700375	+	0.713775i	$0.253016\pi$
$468$	18692.2	1.84625
$469$	5815.18	0.572538
$470$	0	0
$471$	7812.63	0.764303
$472$	6133.19	0.598099
$473$	−765.487	−0.0744125
$474$	11797.6	1.14321
$475$	0	0
$476$	−689.082	−0.0663530
$477$	−12289.8	−1.17969
$478$	−876.626	−0.0838827
$479$	15246.1	1.45430	0.727152	−	0.686476i	$-0.240843\pi$
$479$	15246.1	1.45430	0.727152	+	0.686476i	$0.240843\pi$
$480$	0	0
$481$	25992.7	2.46396
$482$	−5708.83	−0.539481
$483$	−4836.29	−0.455608
$484$	−5313.30	−0.498995
$485$	0	0
$486$	−1216.41	−0.113534
$487$	−3876.39	−0.360690	−0.180345	−	0.983603i	$-0.557721\pi$
$487$	−3876.39	−0.360690	−0.180345	+	0.983603i	$0.557721\pi$
$488$	−1426.92	−0.132364
$489$	22609.0	2.09083
$490$	0	0
$491$	−5027.14	−0.462061	−0.231030	−	0.972947i	$-0.574210\pi$
$491$	−5027.14	−0.462061	−0.231030	+	0.972947i	$0.574210\pi$
$492$	−2654.82	−0.243269
$493$	872.166	0.0796762
$494$	1159.21	0.105577
$495$	0	0
$496$	−2503.30	−0.226616
$497$	15325.0	1.38314
$498$	23719.5	2.13433
$499$	7178.10	0.643960	0.321980	−	0.946747i	$-0.395652\pi$
$499$	7178.10	0.643960	0.321980	+	0.946747i	$0.395652\pi$
$500$	0	0
$501$	−10103.3	−0.900962
$502$	4585.86	0.407723
$503$	18169.6	1.61063	0.805313	−	0.592850i	$-0.201997\pi$
$503$	18169.6	1.61063	0.805313	+	0.592850i	$0.201997\pi$
$504$	9931.10	0.877711
$505$	0	0
$506$	75.2271	0.00660919
$507$	−50563.8	−4.42923
$508$	4711.27	0.411474
$509$	2296.14	0.199950	0.0999750	−	0.994990i	$-0.468124\pi$
$509$	2296.14	0.199950	0.0999750	+	0.994990i	$0.468124\pi$
$510$	0	0
$511$	16379.4	1.41797
$512$	−512.000	−0.0441942
$513$	1500.42	0.129133
$514$	7095.18	0.608862
$515$	0	0
$516$	16715.9	1.42612
$517$	644.288	0.0548081
$518$	13809.8	1.17137
$519$	−28446.3	−2.40589
$520$	0	0
$521$	12945.5	1.08859	0.544293	−	0.838895i	$-0.316798\pi$
$521$	12945.5	1.08859	0.544293	+	0.838895i	$0.316798\pi$
$522$	−12569.7	−1.05395
$523$	16093.6	1.34555	0.672775	−	0.739847i	$-0.265102\pi$
$523$	16093.6	1.34555	0.672775	+	0.739847i	$0.265102\pi$
$524$	9542.25	0.795525
$525$	0	0
$526$	−4556.69	−0.377721
$527$	−1144.37	−0.0945915
$528$	−233.607	−0.0192546
$529$	529.000	0.0434783
$530$	0	0
$531$	−40408.1	−3.30238
$532$	615.884	0.0501916
$533$	6591.01	0.535625
$534$	−14725.7	−1.19334
$535$	0	0
$536$	1975.23	0.159174
$537$	−14851.4	−1.19346
$538$	−7830.73	−0.627522
$539$	346.232	0.0276684
$540$	0	0
$541$	−15424.7	−1.22580	−0.612900	−	0.790161i	$-0.709997\pi$
$541$	−15424.7	−1.22580	−0.612900	+	0.790161i	$0.709997\pi$
$542$	−12505.4	−0.991058
$543$	33456.0	2.64408
$544$	−234.059	−0.0184471
$545$	0	0
$546$	−37285.6	−2.92249
$547$	−5307.37	−0.414857	−0.207428	−	0.978250i	$-0.566509\pi$
$547$	−5307.37	−0.414857	−0.207428	+	0.978250i	$0.566509\pi$
$548$	−4492.87	−0.350230
$549$	9401.20	0.730844
$550$	0	0
$551$	−779.519	−0.0602698
$552$	−1642.73	−0.126666
$553$	−15561.5	−1.19664
$554$	−9401.06	−0.720961
$555$	0	0
$556$	−7207.00	−0.549721
$557$	13530.9	1.02930	0.514651	−	0.857400i	$-0.327921\pi$
$557$	13530.9	1.02930	0.514651	+	0.857400i	$0.327921\pi$
$558$	16492.8	1.25125
$559$	−41500.0	−3.14001
$560$	0	0
$561$	−106.793	−0.00803706
$562$	−11016.1	−0.826847
$563$	18109.0	1.35560	0.677799	−	0.735248i	$-0.262934\pi$
$563$	18109.0	1.35560	0.677799	+	0.735248i	$0.262934\pi$
$564$	−14069.3	−1.05040
$565$	0	0
$566$	13278.4	0.986100
$567$	−14743.3	−1.09199
$568$	5205.43	0.384533
$569$	−908.991	−0.0669717	−0.0334858	−	0.999439i	$-0.510661\pi$
$569$	−908.991	−0.0669717	−0.0334858	+	0.999439i	$0.510661\pi$
$570$	0	0
$571$	−8894.71	−0.651895	−0.325948	−	0.945388i	$-0.605683\pi$
$571$	−8894.71	−0.651895	−0.325948	+	0.945388i	$0.605683\pi$
$572$	579.967	0.0423945
$573$	−20203.6	−1.47298
$574$	3501.79	0.254637
$575$	0	0
$576$	3373.28	0.244016
$577$	12947.3	0.934148	0.467074	−	0.884218i	$-0.345308\pi$
$577$	12947.3	0.934148	0.467074	+	0.884218i	$0.345308\pi$
$578$	9719.00	0.699407
$579$	−17867.0	−1.28243
$580$	0	0
$581$	−31286.8	−2.23407
$582$	−6853.65	−0.488132
$583$	−381.321	−0.0270887
$584$	5563.55	0.394215
$585$	0	0
$586$	−4241.71	−0.299016
$587$	19404.1	1.36438	0.682190	−	0.731175i	$-0.261028\pi$
$587$	19404.1	1.36438	0.682190	+	0.731175i	$0.261028\pi$
$588$	−7560.69	−0.530268
$589$	1022.81	0.0715522
$590$	0	0
$591$	−42890.4	−2.98524
$592$	4690.77	0.325658
$593$	13633.9	0.944141	0.472070	−	0.881561i	$-0.343507\pi$
$593$	13633.9	0.944141	0.472070	+	0.881561i	$0.343507\pi$
$594$	750.682	0.0518533
$595$	0	0
$596$	2652.55	0.182303
$597$	−34685.2	−2.37784
$598$	4078.35	0.278890
$599$	11276.5	0.769191	0.384595	−	0.923085i	$-0.374341\pi$
$599$	11276.5	0.769191	0.384595	+	0.923085i	$0.374341\pi$
$600$	0	0
$601$	−380.766	−0.0258432	−0.0129216	−	0.999917i	$-0.504113\pi$
$601$	−380.766	−0.0258432	−0.0129216	+	0.999917i	$0.504113\pi$
$602$	−22048.9	−1.49277
$603$	−13013.7	−0.878870
$604$	6470.63	0.435904
$605$	0	0
$606$	−12045.2	−0.807428
$607$	21669.2	1.44897	0.724485	−	0.689290i	$-0.242077\pi$
$607$	21669.2	1.44897	0.724485	+	0.689290i	$0.242077\pi$
$608$	209.196	0.0139540
$609$	25073.1	1.66833
$610$	0	0
$611$	34929.4	2.31275
$612$	1542.08	0.101855
$613$	936.980	0.0617362	0.0308681	−	0.999523i	$-0.490173\pi$
$613$	936.980	0.0617362	0.0308681	+	0.999523i	$0.490173\pi$
$614$	14845.7	0.975774
$615$	0	0
$616$	308.135	0.0201544
$617$	17539.2	1.14441	0.572205	−	0.820111i	$-0.306088\pi$
$617$	17539.2	1.14441	0.572205	+	0.820111i	$0.306088\pi$
$618$	−3022.93	−0.196764
$619$	−4740.20	−0.307794	−0.153897	−	0.988087i	$-0.549182\pi$
$619$	−4740.20	−0.307794	−0.153897	+	0.988087i	$0.549182\pi$
$620$	0	0
$621$	5278.83	0.341114
$622$	5127.94	0.330565
$623$	19423.7	1.24911
$624$	−12664.7	−0.812493
$625$	0	0
$626$	18807.1	1.20077
$627$	95.4486	0.00607950
$628$	−3500.32	−0.222417
$629$	2144.37	0.135933
$630$	0	0
$631$	−18505.6	−1.16751	−0.583754	−	0.811930i	$-0.698417\pi$
$631$	−18505.6	−1.16751	−0.583754	+	0.811930i	$0.698417\pi$
$632$	−5285.73	−0.332682
$633$	−40786.1	−2.56098
$634$	814.221	0.0510045
$635$	0	0
$636$	8326.91	0.519156
$637$	18770.6	1.16753
$638$	−390.004	−0.0242013
$639$	−34295.6	−2.12318
$640$	0	0
$641$	10163.6	0.626266	0.313133	−	0.949709i	$-0.398621\pi$
$641$	10163.6	0.626266	0.313133	+	0.949709i	$0.398621\pi$
$642$	27378.4	1.68308
$643$	−7132.33	−0.437436	−0.218718	−	0.975788i	$-0.570188\pi$
$643$	−7132.33	−0.437436	−0.218718	+	0.975788i	$0.570188\pi$
$644$	2166.82	0.132585
$645$	0	0
$646$	95.6334	0.00582453
$647$	−14536.9	−0.883315	−0.441657	−	0.897184i	$-0.645609\pi$
$647$	−14536.9	−0.883315	−0.441657	+	0.897184i	$0.645609\pi$
$648$	−5007.83	−0.303590
$649$	−1253.75	−0.0758307
$650$	0	0
$651$	−32898.5	−1.98064
$652$	−10129.6	−0.608443
$653$	−3318.33	−0.198861	−0.0994305	−	0.995045i	$-0.531702\pi$
$653$	−3318.33	−0.198861	−0.0994305	+	0.995045i	$0.531702\pi$
$654$	22136.7	1.32357
$655$	0	0
$656$	1189.45	0.0707928
$657$	−36655.1	−2.17664
$658$	18557.9	1.09949
$659$	3088.17	0.182546	0.0912730	−	0.995826i	$-0.470906\pi$
$659$	3088.17	0.182546	0.0912730	+	0.995826i	$0.470906\pi$
$660$	0	0
$661$	27602.0	1.62420	0.812098	−	0.583521i	$-0.198325\pi$
$661$	27602.0	1.62420	0.812098	+	0.583521i	$0.198325\pi$
$662$	14244.7	0.836306
$663$	−5789.65	−0.339142
$664$	−10627.1	−0.621103
$665$	0	0
$666$	−30904.8	−1.79810
$667$	−2742.53	−0.159207
$668$	4526.61	0.262186
$669$	27507.5	1.58969
$670$	0	0
$671$	291.694	0.0167820
$672$	−6728.75	−0.386261
$673$	−8091.33	−0.463444	−0.231722	−	0.972782i	$-0.574436\pi$
$673$	−8091.33	−0.463444	−0.231722	+	0.972782i	$0.574436\pi$
$674$	−16535.6	−0.944994
$675$	0	0
$676$	22654.3	1.28893
$677$	9515.37	0.540185	0.270093	−	0.962834i	$-0.412946\pi$
$677$	9515.37	0.540185	0.270093	+	0.962834i	$0.412946\pi$
$678$	−41059.3	−2.32577
$679$	9040.18	0.510943
$680$	0	0
$681$	6061.79	0.341099
$682$	511.727	0.0287317
$683$	630.346	0.0353141	0.0176570	−	0.999844i	$-0.494379\pi$
$683$	630.346	0.0353141	0.0176570	+	0.999844i	$0.494379\pi$
$684$	−1378.28	−0.0770463
$685$	0	0
$686$	−6184.15	−0.344187
$687$	−57132.1	−3.17282
$688$	−7489.30	−0.415010
$689$	−20672.9	−1.14307
$690$	0	0
$691$	−6515.92	−0.358723	−0.179361	−	0.983783i	$-0.557403\pi$
$691$	−6515.92	−0.358723	−0.179361	+	0.983783i	$0.557403\pi$
$692$	12744.9	0.700128
$693$	−2030.13	−0.111282
$694$	2097.67	0.114735
$695$	0	0
$696$	8516.53	0.463819
$697$	543.751	0.0295496
$698$	18271.7	0.990825
$699$	54311.1	2.93882
$700$	0	0
$701$	−32850.2	−1.76995	−0.884975	−	0.465638i	$-0.845825\pi$
$701$	−32850.2	−1.76995	−0.884975	+	0.465638i	$0.845825\pi$
$702$	40697.4	2.18807
$703$	−1916.58	−0.102824
$704$	104.664	0.00560321
$705$	0	0
$706$	10506.1	0.560061
$707$	15888.0	0.845160
$708$	27378.3	1.45330
$709$	−21841.2	−1.15693	−0.578466	−	0.815707i	$-0.696348\pi$
$709$	−21841.2	−1.15693	−0.578466	+	0.815707i	$0.696348\pi$
$710$	0	0
$711$	34824.7	1.83689
$712$	6597.61	0.347270
$713$	3598.49	0.189010
$714$	−3076.03	−0.161229
$715$	0	0
$716$	6653.93	0.347303
$717$	−3913.22	−0.203824
$718$	22539.4	1.17153
$719$	25807.3	1.33859	0.669296	−	0.742996i	$-0.266596\pi$
$719$	25807.3	1.33859	0.669296	+	0.742996i	$0.266596\pi$
$720$	0	0
$721$	3987.34	0.205959
$722$	13632.5	0.702701
$723$	−25483.9	−1.31087
$724$	−14989.4	−0.769443
$725$	0	0
$726$	−23718.3	−1.21249
$727$	−14264.0	−0.727679	−0.363839	−	0.931462i	$-0.618534\pi$
$727$	−14264.0	−0.727679	−0.363839	+	0.931462i	$0.618534\pi$
$728$	16705.2	0.850462
$729$	−22331.4	−1.13455
$730$	0	0
$731$	−3423.71	−0.173229
$732$	−6369.72	−0.321628
$733$	−811.706	−0.0409018	−0.0204509	−	0.999791i	$-0.506510\pi$
$733$	−811.706	−0.0409018	−0.0204509	+	0.999791i	$0.506510\pi$
$734$	−11350.9	−0.570804
$735$	0	0
$736$	736.000	0.0368605
$737$	−403.780	−0.0201810
$738$	−7836.59	−0.390879
$739$	−30696.9	−1.52802	−0.764008	−	0.645207i	$-0.776771\pi$
$739$	−30696.9	−1.52802	−0.764008	+	0.645207i	$0.776771\pi$
$740$	0	0
$741$	5174.64	0.256539
$742$	−10983.5	−0.543417
$743$	26935.6	1.32998	0.664988	−	0.746854i	$-0.268437\pi$
$743$	26935.6	1.32998	0.664988	+	0.746854i	$0.268437\pi$
$744$	−11174.6	−0.550646
$745$	0	0
$746$	7324.40	0.359471
$747$	70016.2	3.42939
$748$	47.8467	0.00233883
$749$	−36113.0	−1.76174
$750$	0	0
$751$	22371.2	1.08700	0.543501	−	0.839409i	$-0.317098\pi$
$751$	22371.2	1.08700	0.543501	+	0.839409i	$0.317098\pi$
$752$	6303.53	0.305673
$753$	20471.0	0.990712
$754$	−21143.7	−1.02123
$755$	0	0
$756$	21622.4	1.04021
$757$	−697.182	−0.0334736	−0.0167368	−	0.999860i	$-0.505328\pi$
$757$	−697.182	−0.0334736	−0.0167368	+	0.999860i	$0.505328\pi$
$758$	−11041.8	−0.529097
$759$	335.810	0.0160595
$760$	0	0
$761$	39756.9	1.89380	0.946902	−	0.321522i	$-0.104194\pi$
$761$	39756.9	1.89380	0.946902	+	0.321522i	$0.104194\pi$
$762$	21030.9	0.999827
$763$	−29199.0	−1.38542
$764$	9051.87	0.428645
$765$	0	0
$766$	−4618.27	−0.217839
$767$	−67970.9	−3.19985
$768$	−2285.54	−0.107386
$769$	−12574.8	−0.589673	−0.294836	−	0.955548i	$-0.595265\pi$
$769$	−12574.8	−0.589673	−0.294836	+	0.955548i	$0.595265\pi$
$770$	0	0
$771$	31672.5	1.47945
$772$	8005.00	0.373195
$773$	−2478.65	−0.115331	−0.0576654	−	0.998336i	$-0.518366\pi$
$773$	−2478.65	−0.115331	−0.0576654	+	0.998336i	$0.518366\pi$
$774$	49342.8	2.29146
$775$	0	0
$776$	3070.66	0.142049
$777$	61646.5	2.84628
$778$	16389.0	0.755238
$779$	−485.991	−0.0223523
$780$	0	0
$781$	−1064.10	−0.0487535
$782$	336.460	0.0153859
$783$	−27367.3	−1.24908
$784$	3387.44	0.154311
$785$	0	0
$786$	42596.1	1.93302
$787$	7716.40	0.349504	0.174752	−	0.984612i	$-0.444088\pi$
$787$	7716.40	0.349504	0.174752	+	0.984612i	$0.444088\pi$
$788$	19216.3	0.868722
$789$	−20340.9	−0.917812
$790$	0	0
$791$	54158.5	2.43446
$792$	−689.570	−0.0309379
$793$	15813.9	0.708154
$794$	14547.8	0.650230
$795$	0	0
$796$	15540.1	0.691967
$797$	26798.1	1.19101	0.595507	−	0.803350i	$-0.296951\pi$
$797$	26798.1	1.19101	0.595507	+	0.803350i	$0.296951\pi$
$798$	2749.28	0.121959
$799$	2881.64	0.127591
$800$	0	0
$801$	−43468.0	−1.91743
$802$	3671.61	0.161657
$803$	−1137.31	−0.0499810
$804$	8817.35	0.386771
$805$	0	0
$806$	27742.7	1.21240
$807$	−34956.0	−1.52479
$808$	5396.63	0.234967
$809$	45278.0	1.96772	0.983862	−	0.178927i	$-0.0572625\pi$
$809$	45278.0	1.96772	0.983862	+	0.178927i	$0.0572625\pi$
$810$	0	0
$811$	−24889.6	−1.07767	−0.538836	−	0.842411i	$-0.681136\pi$
$811$	−24889.6	−1.07767	−0.538836	+	0.842411i	$0.681136\pi$
$812$	−11233.6	−0.485494
$813$	−55823.6	−2.40814
$814$	−958.893	−0.0412889
$815$	0	0
$816$	−1044.83	−0.0448239
$817$	3060.03	0.131036
$818$	5731.60	0.244989
$819$	−110061.	−4.69579
$820$	0	0
$821$	19274.0	0.819326	0.409663	−	0.912237i	$-0.365646\pi$
$821$	19274.0	0.819326	0.409663	+	0.912237i	$0.365646\pi$
$822$	−20056.0	−0.851013
$823$	21620.8	0.915740	0.457870	−	0.889019i	$-0.348613\pi$
$823$	21620.8	0.915740	0.457870	+	0.889019i	$0.348613\pi$
$824$	1354.37	0.0572596
$825$	0	0
$826$	−36112.8	−1.52122
$827$	9012.81	0.378967	0.189484	−	0.981884i	$-0.439319\pi$
$827$	9012.81	0.378967	0.189484	+	0.981884i	$0.439319\pi$
$828$	−4849.09	−0.203524
$829$	26574.7	1.11336	0.556682	−	0.830726i	$-0.312074\pi$
$829$	26574.7	1.11336	0.556682	+	0.830726i	$0.312074\pi$
$830$	0	0
$831$	−41965.9	−1.75184
$832$	5674.23	0.236441
$833$	1548.56	0.0644110
$834$	−32171.7	−1.33575
$835$	0	0
$836$	−42.7642	−0.00176917
$837$	35908.9	1.48291
$838$	−22088.0	−0.910521
$839$	−1261.30	−0.0519009	−0.0259504	−	0.999663i	$-0.508261\pi$
$839$	−1261.30	−0.0519009	−0.0259504	+	0.999663i	$0.508261\pi$
$840$	0	0
$841$	−10170.7	−0.417022
$842$	−12686.5	−0.519245
$843$	−49175.6	−2.00913
$844$	18273.5	0.745262
$845$	0	0
$846$	−41530.4	−1.68776
$847$	31285.2	1.26915
$848$	−3730.73	−0.151078
$849$	59274.1	2.39609
$850$	0	0
$851$	−6742.98	−0.271617
$852$	23236.8	0.934365
$853$	−32765.1	−1.31519	−0.657594	−	0.753373i	$-0.728426\pi$
$853$	−32765.1	−1.31519	−0.657594	+	0.753373i	$0.728426\pi$
$854$	8401.86	0.336658
$855$	0	0
$856$	−12266.5	−0.489788
$857$	−722.029	−0.0287795	−0.0143898	−	0.999896i	$-0.504581\pi$
$857$	−722.029	−0.0287795	−0.0143898	+	0.999896i	$0.504581\pi$
$858$	2588.94	0.103013
$859$	−36626.9	−1.45482	−0.727411	−	0.686202i	$-0.759277\pi$
$859$	−36626.9	−1.45482	−0.727411	+	0.686202i	$0.759277\pi$
$860$	0	0
$861$	15631.8	0.618735
$862$	8234.03	0.325351
$863$	33207.9	1.30986	0.654931	−	0.755688i	$-0.272698\pi$
$863$	33207.9	1.30986	0.654931	+	0.755688i	$0.272698\pi$
$864$	7344.45	0.289194
$865$	0	0
$866$	−1517.58	−0.0595491
$867$	43385.2	1.69947
$868$	14739.6	0.576378
$869$	1080.52	0.0421795
$870$	0	0
$871$	−21890.5	−0.851585
$872$	−9917.97	−0.385166
$873$	−20230.9	−0.784320
$874$	−300.719	−0.0116384
$875$	0	0
$876$	24835.4	0.957890
$877$	39507.6	1.52118	0.760592	−	0.649230i	$-0.224909\pi$
$877$	39507.6	1.52118	0.760592	+	0.649230i	$0.224909\pi$
$878$	−8300.21	−0.319041
$879$	−18934.8	−0.726569
$880$	0	0
$881$	47928.6	1.83287	0.916434	−	0.400187i	$-0.131055\pi$
$881$	47928.6	1.83287	0.916434	+	0.400187i	$0.131055\pi$
$882$	−22317.9	−0.852023
$883$	−48176.2	−1.83608	−0.918040	−	0.396488i	$-0.870229\pi$
$883$	−48176.2	−1.83608	−0.918040	+	0.396488i	$0.870229\pi$
$884$	2593.96	0.0986925
$885$	0	0
$886$	−9194.48	−0.348640
$887$	−31421.7	−1.18944	−0.594722	−	0.803932i	$-0.702738\pi$
$887$	−31421.7	−1.18944	−0.594722	+	0.803932i	$0.702738\pi$
$888$	20939.4	0.791305
$889$	−27740.4	−1.04655
$890$	0	0
$891$	1023.71	0.0384910
$892$	−12324.3	−0.462609
$893$	−2575.54	−0.0965141
$894$	11840.9	0.442973
$895$	0	0
$896$	3014.71	0.112404
$897$	18205.6	0.677666
$898$	−12805.2	−0.475851
$899$	−18655.9	−0.692111
$900$	0	0
$901$	−1705.49	−0.0630613
$902$	−243.148	−0.00897555
$903$	−98425.1	−3.62722
$904$	18395.9	0.676814
$905$	0	0
$906$	28884.6	1.05919
$907$	−27694.1	−1.01385	−0.506927	−	0.861989i	$-0.669219\pi$
$907$	−27694.1	−1.01385	−0.506927	+	0.861989i	$0.669219\pi$
$908$	−2715.88	−0.0992618
$909$	−35555.4	−1.29736
$910$	0	0
$911$	−27325.1	−0.993764	−0.496882	−	0.867818i	$-0.665522\pi$
$911$	−27325.1	−0.993764	−0.496882	+	0.867818i	$0.665522\pi$
$912$	933.842	0.0339063
$913$	2172.41	0.0787473
$914$	1103.00	0.0399168
$915$	0	0
$916$	25597.1	0.923310
$917$	−56185.7	−2.02335
$918$	3357.49	0.120712
$919$	28135.6	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$919$	28135.6	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$920$	0	0
$921$	66270.7	2.37100
$922$	7803.91	0.278751
$923$	−57689.1	−2.05727
$924$	1375.50	0.0489726
$925$	0	0
$926$	9656.29	0.342684
$927$	−8923.21	−0.316156
$928$	−3815.69	−0.134974
$929$	32497.6	1.14770	0.573850	−	0.818961i	$-0.305449\pi$
$929$	32497.6	1.14770	0.573850	+	0.818961i	$0.305449\pi$
$930$	0	0
$931$	−1384.06	−0.0487226
$932$	−24333.2	−0.855215
$933$	22890.9	0.803230
$934$	−28272.6	−0.990479
$935$	0	0
$936$	−37384.3	−1.30550
$937$	12209.3	0.425678	0.212839	−	0.977087i	$-0.431729\pi$
$937$	12209.3	0.425678	0.212839	+	0.977087i	$0.431729\pi$
$938$	−11630.4	−0.404845
$939$	83953.8	2.91771
$940$	0	0
$941$	−1317.78	−0.0456518	−0.0228259	−	0.999739i	$-0.507266\pi$
$941$	−1317.78	−0.0456518	−0.0228259	+	0.999739i	$0.507266\pi$
$942$	−15625.3	−0.540444
$943$	−1709.83	−0.0590453
$944$	−12266.4	−0.422920
$945$	0	0
$946$	1530.97	0.0526176
$947$	11954.8	0.410219	0.205110	−	0.978739i	$-0.434245\pi$
$947$	11954.8	0.410219	0.205110	+	0.978739i	$0.434245\pi$
$948$	−23595.3	−0.808374
$949$	−61657.9	−2.10906
$950$	0	0
$951$	3634.64	0.123934
$952$	1378.16	0.0469186
$953$	19658.8	0.668216	0.334108	−	0.942535i	$-0.391565\pi$
$953$	19658.8	0.668216	0.334108	+	0.942535i	$0.391565\pi$
$954$	24579.7	0.834169
$955$	0	0
$956$	1753.25	0.0593140
$957$	−1740.96	−0.0588059
$958$	−30492.2	−1.02835
$959$	26454.5	0.890782
$960$	0	0
$961$	−5312.52	−0.178326
$962$	−51985.3	−1.74228
$963$	80816.8	2.70434
$964$	11417.7	0.381471
$965$	0	0
$966$	9672.58	0.322164
$967$	7604.58	0.252892	0.126446	−	0.991973i	$-0.459643\pi$
$967$	7604.58	0.252892	0.126446	+	0.991973i	$0.459643\pi$
$968$	10626.6	0.352843
$969$	426.903	0.0141528
$970$	0	0
$971$	−14731.2	−0.486866	−0.243433	−	0.969918i	$-0.578274\pi$
$971$	−14731.2	−0.486866	−0.243433	+	0.969918i	$0.578274\pi$
$972$	2432.83	0.0802809
$973$	42435.5	1.39817
$974$	7752.77	0.255046
$975$	0	0
$976$	2853.85	0.0935957
$977$	50584.6	1.65644	0.828222	−	0.560400i	$-0.189353\pi$
$977$	50584.6	1.65644	0.828222	+	0.560400i	$0.189353\pi$
$978$	−45218.0	−1.47844
$979$	−1348.69	−0.0440290
$980$	0	0
$981$	65343.9	2.12668
$982$	10054.3	0.326726
$983$	−15625.9	−0.507007	−0.253503	−	0.967334i	$-0.581583\pi$
$983$	−15625.9	−0.507007	−0.253503	+	0.967334i	$0.581583\pi$
$984$	5309.63	0.172017
$985$	0	0
$986$	−1744.33	−0.0563396
$987$	82841.6	2.67161
$988$	−2318.41	−0.0746544
$989$	10765.9	0.346142
$990$	0	0
$991$	18267.8	0.585567	0.292783	−	0.956179i	$-0.405419\pi$
$991$	18267.8	0.585567	0.292783	+	0.956179i	$0.405419\pi$
$992$	5006.59	0.160241
$993$	63587.5	2.03211
$994$	−30650.1	−0.978029
$995$	0	0
$996$	−47439.0	−1.50920
$997$	−21580.5	−0.685519	−0.342760	−	0.939423i	$-0.611362\pi$
$997$	−21580.5	−0.685519	−0.342760	+	0.939423i	$0.611362\pi$
$998$	−14356.2	−0.455348
$999$	−67287.3	−2.13101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.o.1.1		4
5.2	odd	4		1150.4.b.m.599.4		8
5.3	odd	4		1150.4.b.m.599.5		8
5.4	even	2		230.4.a.i.1.4	✓	4
15.14	odd	2		2070.4.a.bi.1.3		4
20.19	odd	2		1840.4.a.l.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.4	✓	4	5.4	even	2
1150.4.a.o.1.1		4	1.1	even	1	trivial
1150.4.b.m.599.4		8	5.2	odd	4
1150.4.b.m.599.5		8	5.3	odd	4
1840.4.a.l.1.1		4	20.19	odd	2
2070.4.a.bi.1.3		4	15.14	odd	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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -8.92791 q^{3} +4.00000 q^{4} +17.8558 q^{6} -23.5524 q^{7} -8.00000 q^{8} +52.7075 q^{9} +1.63537 q^{11} -35.7116 q^{12} +88.6599 q^{13} +47.1048 q^{14} +16.0000 q^{16} +7.31435 q^{17} -105.415 q^{18} -6.53738 q^{19} +210.274 q^{21} -3.27074 q^{22} -23.0000 q^{23} +71.4232 q^{24} -177.320 q^{26} -229.514 q^{27} -94.2096 q^{28} +119.240 q^{29} -156.456 q^{31} -32.0000 q^{32} -14.6004 q^{33} -14.6287 q^{34} +210.830 q^{36} +293.173 q^{37} +13.0748 q^{38} -791.547 q^{39} +74.3404 q^{41} -420.547 q^{42} -468.081 q^{43} +6.54148 q^{44} +46.0000 q^{46} +393.971 q^{47} -142.846 q^{48} +211.715 q^{49} -65.3018 q^{51} +354.639 q^{52} -233.171 q^{53} +459.028 q^{54} +188.419 q^{56} +58.3651 q^{57} -238.481 q^{58} -766.648 q^{59} +178.365 q^{61} +312.912 q^{62} -1241.39 q^{63} +64.0000 q^{64} +29.2009 q^{66} -246.904 q^{67} +29.2574 q^{68} +205.342 q^{69} -650.678 q^{71} -421.660 q^{72} -695.444 q^{73} -586.346 q^{74} -26.1495 q^{76} -38.5169 q^{77} +1583.09 q^{78} +660.717 q^{79} +625.978 q^{81} -148.681 q^{82} +1328.39 q^{83} +841.094 q^{84} +936.163 q^{86} -1064.57 q^{87} -13.0830 q^{88} -824.702 q^{89} -2088.15 q^{91} -92.0000 q^{92} +1396.82 q^{93} -787.942 q^{94} +285.693 q^{96} -383.833 q^{97} -423.430 q^{98} +86.1963 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} - 26 q^{7} - 32 q^{8} + 64 q^{9} + 93 q^{11} - 16 q^{12} - 32 q^{13} + 52 q^{14} + 64 q^{16} - 108 q^{17} - 128 q^{18} + 185 q^{19} + 302 q^{21} - 186 q^{22}+ \cdots + 2013 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 - 26 * q^7 - 32 * q^8 + 64 * q^9 + 93 * q^11 - 16 * q^12 - 32 * q^13 + 52 * q^14 + 64 * q^16 - 108 * q^17 - 128 * q^18 + 185 * q^19 + 302 * q^21 - 186 * q^22 - 92 * q^23 + 32 * q^24 + 64 * q^26 - 259 * q^27 - 104 * q^28 + 294 * q^29 - 211 * q^31 - 128 * q^32 + 237 * q^33 + 216 * q^34 + 256 * q^36 - 5 * q^37 - 370 * q^38 - 421 * q^39 - 369 * q^41 - 604 * q^42 + 100 * q^43 + 372 * q^44 + 184 * q^46 + 363 * q^47 - 64 * q^48 + 600 * q^49 - 315 * q^51 - 128 * q^52 - 21 * q^53 + 518 * q^54 + 208 * q^56 + 160 * q^57 - 588 * q^58 - 33 * q^59 - 307 * q^61 + 422 * q^62 - 167 * q^63 + 256 * q^64 - 474 * q^66 - 725 * q^67 - 432 * q^68 + 92 * q^69 - 1257 * q^71 - 512 * q^72 - 509 * q^73 + 10 * q^74 + 740 * q^76 + 1962 * q^77 + 842 * q^78 + 1202 * q^79 - 992 * q^81 + 738 * q^82 + 1377 * q^83 + 1208 * q^84 - 200 * q^86 + 2829 * q^87 - 744 * q^88 - 984 * q^89 - 995 * q^91 - 368 * q^92 + 1843 * q^93 - 726 * q^94 + 128 * q^96 - 137 * q^97 - 1200 * q^98 + 2013 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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