LMFDB - Embedded newform 1150.4.b.m.599.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(599,1150)] 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("1150.599"); 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([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-32,0,16,0,0,-128,0,186] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 168x^{6} + 9540x^{4} + 208777x^{2} + 1542564 $ x^8 + 168x^6 + 9540x^4 + 208777*x^2 + 1542564
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.4
Root			$7.92791i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.m.599.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} +8.92791i q^{3} -4.00000 q^{4} +17.8558 q^{6} -23.5524i q^{7} +8.00000i q^{8} -52.7075 q^{9} +1.63537 q^{11} -35.7116i q^{12} -88.6599i q^{13} -47.1048 q^{14} +16.0000 q^{16} +7.31435i q^{17} +105.415i q^{18} +6.53738 q^{19} +210.274 q^{21} -3.27074i q^{22} +23.0000i q^{23} -71.4232 q^{24} -177.320 q^{26} -229.514i q^{27} +94.2096i q^{28} -119.240 q^{29} -156.456 q^{31} -32.0000i q^{32} +14.6004i q^{33} +14.6287 q^{34} +210.830 q^{36} +293.173i q^{37} -13.0748i q^{38} +791.547 q^{39} +74.3404 q^{41} -420.547i q^{42} +468.081i q^{43} -6.54148 q^{44} +46.0000 q^{46} +393.971i q^{47} +142.846i q^{48} -211.715 q^{49} -65.3018 q^{51} +354.639i q^{52} +233.171i q^{53} -459.028 q^{54} +188.419 q^{56} +58.3651i q^{57} +238.481i q^{58} +766.648 q^{59} +178.365 q^{61} +312.912i q^{62} +1241.39i q^{63} -64.0000 q^{64} +29.2009 q^{66} -246.904i q^{67} -29.2574i q^{68} -205.342 q^{69} -650.678 q^{71} -421.660i q^{72} +695.444i q^{73} +586.346 q^{74} -26.1495 q^{76} -38.5169i q^{77} -1583.09i q^{78} -660.717 q^{79} +625.978 q^{81} -148.681i q^{82} -1328.39i q^{83} -841.094 q^{84} +936.163 q^{86} -1064.57i q^{87} +13.0830i q^{88} +824.702 q^{89} -2088.15 q^{91} -92.0000i q^{92} -1396.82i q^{93} +787.942 q^{94} +285.693 q^{96} -383.833i q^{97} +423.430i q^{98} -86.1963 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4} + 16 q^{6} - 128 q^{9} + 186 q^{11} - 104 q^{14} + 128 q^{16} - 370 q^{19} + 604 q^{21} - 64 q^{24} + 128 q^{26} - 588 q^{29} - 422 q^{31} - 432 q^{34} + 512 q^{36} + 842 q^{39} - 738 q^{41}+ \cdots - 4026 q^{99}+O(q^{100}) $ 8 * q - 32 * q^4 + 16 * q^6 - 128 * q^9 + 186 * q^11 - 104 * q^14 + 128 * q^16 - 370 * q^19 + 604 * q^21 - 64 * q^24 + 128 * q^26 - 588 * q^29 - 422 * q^31 - 432 * q^34 + 512 * q^36 + 842 * q^39 - 738 * q^41 - 744 * q^44 + 368 * q^46 - 1200 * q^49 - 630 * q^51 - 1036 * q^54 + 416 * q^56 + 66 * q^59 - 614 * q^61 - 512 * q^64 - 948 * q^66 - 184 * q^69 - 2514 * q^71 - 20 * q^74 + 1480 * q^76 - 2404 * q^79 - 1984 * q^81 - 2416 * q^84 - 400 * q^86 + 1968 * q^89 - 1990 * q^91 + 1452 * q^94 + 256 * q^96 - 4026 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	8.92791i	1.71818i	0.511828	+	0.859088i	$0.328969\pi$
$3$	8.92791i	1.71818i	−0.511828	+	0.859088i	$0.671031\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	17.8558	1.21493
$7$	− 23.5524i	− 1.27171i	−0.771809	−	0.635855i	$-0.780648\pi$
$7$	− 23.5524i	− 1.27171i	0.771809	−	0.635855i	$-0.219352\pi$
$8$	8.00000i	0.353553i
$9$	−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	$0.492865\pi$
$12$	− 35.7116i	− 0.859088i
$13$	− 88.6599i	− 1.89152i	−0.324860	−	0.945762i	$-0.605317\pi$
$13$	− 88.6599i	− 1.89152i	0.324860	−	0.945762i	$-0.394683\pi$
$14$	−47.1048	−0.899234
$15$	0	0
$16$	16.0000	0.250000
$17$	7.31435i	0.104352i	0.998638	+	0.0521762i	$0.0166157\pi$
$17$	7.31435i	0.104352i	−0.998638	+	0.0521762i	$0.983384\pi$
$18$	105.415i	1.38036i
$19$	6.53738	0.0789357	0.0394678	−	0.999221i	$-0.487434\pi$
$19$	6.53738	0.0789357	0.0394678	+	0.999221i	$0.487434\pi$
$20$	0	0
$21$	210.274	2.18502
$22$	− 3.27074i	− 0.0316966i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	−71.4232	−0.607467
$25$	0	0
$26$	−177.320	−1.33751
$27$	− 229.514i	− 1.63593i
$28$	94.2096i	0.635855i
$29$	−119.240	−0.763530	−0.381765	−	0.924259i	$-0.624684\pi$
$29$	−119.240	−0.763530	−0.381765	+	0.924259i	$0.624684\pi$
$30$	0	0
$31$	−156.456	−0.906462	−0.453231	−	0.891393i	$-0.649729\pi$
$31$	−156.456	−0.906462	−0.453231	+	0.891393i	$0.649729\pi$
$32$	− 32.0000i	− 0.176777i
$33$	14.6004i	0.0770185i
$34$	14.6287	0.0737883
$35$	0	0
$36$	210.830	0.976065
$37$	293.173i	1.30263i	0.758807	+	0.651315i	$0.225782\pi$
$37$	293.173i	1.30263i	−0.758807	+	0.651315i	$0.774218\pi$
$38$	− 13.0748i	− 0.0558159i
$39$	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	$0.454780\pi$
$42$	− 420.547i	− 1.54504i
$43$	468.081i	1.66004i	0.557733	+	0.830020i	$0.311671\pi$
$43$	468.081i	1.66004i	−0.557733	+	0.830020i	$0.688329\pi$
$44$	−6.54148	−0.0224129
$45$	0	0
$46$	46.0000	0.147442
$47$	393.971i	1.22269i	0.791363	+	0.611346i	$0.209372\pi$
$47$	393.971i	1.22269i	−0.791363	+	0.611346i	$0.790628\pi$
$48$	142.846i	0.429544i
$49$	−211.715	−0.617245
$50$	0	0
$51$	−65.3018	−0.179296
$52$	354.639i	0.945762i
$53$	233.171i	0.604311i	0.953259	+	0.302155i	$0.0977061\pi$
$53$	233.171i	0.604311i	−0.953259	+	0.302155i	$0.902294\pi$
$54$	−459.028	−1.15677
$55$	0	0
$56$	188.419	0.449617
$57$	58.3651i	0.135625i
$58$	238.481i	0.539897i
$59$	766.648	1.69168	0.845840	−	0.533437i	$-0.179100\pi$
$59$	766.648	1.69168	0.845840	+	0.533437i	$0.179100\pi$
$60$	0	0
$61$	178.365	0.374383	0.187191	−	0.982323i	$-0.440062\pi$
$61$	178.365	0.374383	0.187191	+	0.982323i	$0.440062\pi$
$62$	312.912i	0.640965i
$63$	1241.39i	2.48254i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	29.2009	0.0544603
$67$	− 246.904i	− 0.450211i	−0.974334	−	0.225105i	$-0.927727\pi$
$67$	− 246.904i	− 0.450211i	0.974334	−	0.225105i	$-0.0722727\pi$
$68$	− 29.2574i	− 0.0521762i
$69$	−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	$0.683020\pi$
$72$	− 421.660i	− 0.690182i
$73$	695.444i	1.11501i	0.830174	+	0.557504i	$0.188241\pi$
$73$	695.444i	1.11501i	−0.830174	+	0.557504i	$0.811759\pi$
$74$	586.346	0.921099
$75$	0	0
$76$	−26.1495	−0.0394678
$77$	− 38.5169i	− 0.0570053i
$78$	− 1583.09i	− 2.29808i
$79$	−660.717	−0.940967	−0.470484	−	0.882409i	$-0.655921\pi$
$79$	−660.717	−0.940967	−0.470484	+	0.882409i	$0.655921\pi$
$80$	0	0
$81$	625.978	0.858681
$82$	− 148.681i	− 0.200232i
$83$	− 1328.39i	− 1.75674i	−0.477977	−	0.878372i	$-0.658630\pi$
$83$	− 1328.39i	− 1.75674i	0.477977	−	0.878372i	$-0.341370\pi$
$84$	−841.094	−1.09251
$85$	0	0
$86$	936.163	1.17383
$87$	− 1064.57i	− 1.31188i
$88$	13.0830i	0.0158483i
$89$	824.702	0.982227	0.491113	−	0.871096i	$-0.336590\pi$
$89$	824.702	0.982227	0.491113	+	0.871096i	$0.336590\pi$
$90$	0	0
$91$	−2088.15	−2.40547
$92$	− 92.0000i	− 0.104257i
$93$	− 1396.82i	− 1.55746i
$94$	787.942	0.864574
$95$	0	0
$96$	285.693	0.303734
$97$	− 383.833i	− 0.401776i	−0.979614	−	0.200888i	$-0.935617\pi$
$97$	− 383.833i	− 0.401776i	0.979614	−	0.200888i	$-0.0643828\pi$
$98$	423.430i	0.436458i
$99$	−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	$0.607822\pi$
$102$	130.604i	0.126781i
$103$	169.297i	0.161955i	0.996716	+	0.0809773i	$0.0258041\pi$
$103$	169.297i	0.161955i	−0.996716	+	0.0809773i	$0.974196\pi$
$104$	709.279	0.668755
$105$	0	0
$106$	466.342	0.427312
$107$	1533.31i	1.38533i	0.721259	+	0.692665i	$0.243564\pi$
$107$	1533.31i	1.38533i	−0.721259	+	0.692665i	$0.756436\pi$
$108$	918.057i	0.817963i
$109$	−1239.75	−1.08941	−0.544707	−	0.838626i	$-0.683359\pi$
$109$	−1239.75	−1.08941	−0.544707	+	0.838626i	$0.683359\pi$
$110$	0	0
$111$	−2617.42	−2.23815
$112$	− 376.838i	− 0.317927i
$113$	2299.49i	1.91432i	0.289561	+	0.957159i	$0.406491\pi$
$113$	2299.49i	1.91432i	−0.289561	+	0.957159i	$0.593509\pi$
$114$	116.730	0.0959016
$115$	0	0
$116$	476.961	0.381765
$117$	4673.04i	3.69250i
$118$	− 1533.30i	− 1.19620i
$119$	172.270	0.132706
$120$	0	0
$121$	−1328.33	−0.997991
$122$	− 356.731i	− 0.264729i
$123$	663.704i	0.486538i
$124$	625.824	0.453231
$125$	0	0
$126$	2482.78	1.75542
$127$	1177.82i	0.822948i	0.911422	+	0.411474i	$0.134986\pi$
$127$	1177.82i	0.822948i	−0.911422	+	0.411474i	$0.865014\pi$
$128$	128.000i	0.0883883i
$129$	−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	$0.207195\pi$
$132$	− 58.4017i	− 0.0385092i
$133$	− 153.971i	− 0.100383i
$134$	−493.808	−0.318347
$135$	0	0
$136$	−58.5148	−0.0368941
$137$	− 1123.22i	− 0.700460i	−0.936664	−	0.350230i	$-0.886103\pi$
$137$	− 1123.22i	− 0.700460i	0.936664	−	0.350230i	$-0.113897\pi$
$138$	410.684i	0.253331i
$139$	1801.75	1.09944	0.549721	−	0.835348i	$-0.314734\pi$
$139$	1801.75	1.09944	0.549721	+	0.835348i	$0.314734\pi$
$140$	0	0
$141$	−3517.33	−2.10080
$142$	1301.36i	0.769067i
$143$	− 144.992i	− 0.0847889i
$144$	−843.320	−0.488032
$145$	0	0
$146$	1390.89	0.788429
$147$	− 1890.17i	− 1.06054i
$148$	− 1172.69i	− 0.651315i
$149$	−663.138	−0.364607	−0.182303	−	0.983242i	$-0.558355\pi$
$149$	−663.138	−0.364607	−0.182303	+	0.983242i	$0.558355\pi$
$150$	0	0
$151$	1617.66	0.871808	0.435904	−	0.899993i	$-0.356429\pi$
$151$	1617.66	0.871808	0.435904	+	0.899993i	$0.356429\pi$
$152$	52.2990i	0.0279080i
$153$	− 385.521i	− 0.203709i
$154$	−77.0338	−0.0403088
$155$	0	0
$156$	−3166.19	−1.62499
$157$	− 875.080i	− 0.444834i	−0.974952	−	0.222417i	$-0.928605\pi$
$157$	− 875.080i	− 0.444834i	0.974952	−	0.222417i	$-0.0713946\pi$
$158$	1321.43i	0.665364i
$159$	−2081.73	−1.03831
$160$	0	0
$161$	541.705	0.265170
$162$	− 1251.96i	− 0.607179i
$163$	2532.39i	1.21689i	0.793597	+	0.608443i	$0.208206\pi$
$163$	2532.39i	1.21689i	−0.793597	+	0.608443i	$0.791794\pi$
$164$	−297.361	−0.141586
$165$	0	0
$166$	−2656.78	−1.24221
$167$	1131.65i	0.524371i	0.965017	+	0.262186i	$0.0844432\pi$
$167$	1131.65i	0.524371i	−0.965017	+	0.262186i	$0.915557\pi$
$168$	1682.19i	0.772522i
$169$	−5663.57	−2.57786
$170$	0	0
$171$	−344.569	−0.154093
$172$	− 1872.33i	− 0.830020i
$173$	− 3186.23i	− 1.40026i	−0.714017	−	0.700128i	$-0.753126\pi$
$173$	− 3186.23i	− 1.40026i	0.714017	−	0.700128i	$-0.246874\pi$
$174$	−2129.13	−0.927639
$175$	0	0
$176$	26.1659	0.0112064
$177$	6844.56i	2.90660i
$178$	− 1649.40i	− 0.694539i
$179$	−1663.48	−0.694606	−0.347303	−	0.937753i	$-0.612902\pi$
$179$	−1663.48	−0.694606	−0.347303	+	0.937753i	$0.612902\pi$
$180$	0	0
$181$	−3747.35	−1.53889	−0.769443	−	0.638716i	$-0.779466\pi$
$181$	−3747.35	−1.53889	−0.769443	+	0.638716i	$0.779466\pi$
$182$	4176.30i	1.70092i
$183$	1592.43i	0.643256i
$184$	−184.000	−0.0737210
$185$	0	0
$186$	−2793.65	−1.10129
$187$	11.9617i	0.00467767i
$188$	− 1575.88i	− 0.611346i
$189$	−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	$0.358991\pi$
$192$	− 571.386i	− 0.214772i
$193$	− 2001.25i	− 0.746390i	−0.927753	−	0.373195i	$-0.878262\pi$
$193$	− 2001.25i	− 0.746390i	0.927753	−	0.373195i	$-0.121738\pi$
$194$	−767.665	−0.284099
$195$	0	0
$196$	846.860	0.308622
$197$	4804.08i	1.73744i	0.495300	+	0.868722i	$0.335058\pi$
$197$	4804.08i	1.73744i	−0.495300	+	0.868722i	$0.664942\pi$
$198$	172.393i	0.0618758i
$199$	−3885.03	−1.38393	−0.691967	−	0.721929i	$-0.743256\pi$
$199$	−3885.03	−1.38393	−0.691967	+	0.721929i	$0.743256\pi$
$200$	0	0
$201$	2204.34	0.773542
$202$	1349.16i	0.469933i
$203$	2808.39i	0.970989i
$204$	261.207	0.0896479
$205$	0	0
$206$	338.594	0.114519
$207$	− 1212.27i	− 0.407047i
$208$	− 1418.56i	− 0.472881i
$209$	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	$0.232324\pi$
$212$	− 932.683i	− 0.302155i
$213$	− 5809.20i	− 1.86873i
$214$	3066.61	0.979576
$215$	0	0
$216$	1836.11	0.578387
$217$	3684.91i	1.15276i
$218$	2479.49i	0.770332i
$219$	−6208.86	−1.91578
$220$	0	0
$221$	648.489	0.197385
$222$	5234.84i	1.58261i
$223$	3081.07i	0.925218i	0.886563	+	0.462609i	$0.153087\pi$
$223$	3081.07i	0.925218i	−0.886563	+	0.462609i	$0.846913\pi$
$224$	−753.676	−0.224809
$225$	0	0
$226$	4598.98	1.35363
$227$	− 678.971i	− 0.198524i	−0.995061	−	0.0992618i	$-0.968352\pi$
$227$	− 678.971i	− 0.198524i	0.995061	−	0.0992618i	$-0.0316481\pi$
$228$	− 233.460i	− 0.0678127i
$229$	−6399.27	−1.84662	−0.923310	−	0.384056i	$-0.874527\pi$
$229$	−6399.27	−1.84662	−0.923310	+	0.384056i	$0.874527\pi$
$230$	0	0
$231$	343.875	0.0979451
$232$	− 953.923i	− 0.269949i
$233$	6083.30i	1.71043i	0.518274	+	0.855215i	$0.326575\pi$
$233$	6083.30i	1.71043i	−0.518274	+	0.855215i	$0.673425\pi$
$234$	9346.08	2.61099
$235$	0	0
$236$	−3066.59	−0.845840
$237$	− 5898.82i	− 1.61675i
$238$	− 344.541i	− 0.0938372i
$239$	−438.313	−0.118628	−0.0593140	−	0.998239i	$-0.518891\pi$
$239$	−438.313	−0.118628	−0.0593140	+	0.998239i	$0.518891\pi$
$240$	0	0
$241$	2854.41	0.762941	0.381471	−	0.924381i	$-0.375418\pi$
$241$	2854.41	0.762941	0.381471	+	0.924381i	$0.375418\pi$
$242$	2656.65i	0.705686i
$243$	− 608.207i	− 0.160562i
$244$	−713.462	−0.187191
$245$	0	0
$246$	1327.41	0.344034
$247$	− 579.603i	− 0.149309i
$248$	− 1251.65i	− 0.320483i
$249$	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	$0.593091\pi$
$252$	− 4965.55i	− 1.24127i
$253$	37.6135i	0.00934681i
$254$	2355.63	0.581912
$255$	0	0
$256$	256.000	0.0625000
$257$	− 3547.59i	− 0.861060i	−0.902576	−	0.430530i	$-0.858327\pi$
$257$	− 3547.59i	− 0.861060i	0.902576	−	0.430530i	$-0.141673\pi$
$258$	8357.97i	2.01684i
$259$	6904.92	1.65657
$260$	0	0
$261$	6284.86	1.49051
$262$	− 4771.12i	− 1.12504i
$263$	− 2278.35i	− 0.534178i	−0.963672	−	0.267089i	$-0.913938\pi$
$263$	− 2278.35i	− 0.534178i	0.963672	−	0.267089i	$-0.0860618\pi$
$264$	−116.803	−0.0272301
$265$	0	0
$266$	−307.942	−0.0709817
$267$	7362.86i	1.68764i
$268$	987.616i	0.225105i
$269$	−3915.36	−0.887450	−0.443725	−	0.896163i	$-0.646343\pi$
$269$	−3915.36	−0.887450	−0.443725	+	0.896163i	$0.646343\pi$
$270$	0	0
$271$	6252.71	1.40157	0.700784	−	0.713374i	$-0.252834\pi$
$271$	6252.71	1.40157	0.700784	+	0.713374i	$0.252834\pi$
$272$	117.030i	0.0260881i
$273$	− 18642.8i	− 4.13302i
$274$	−2246.44	−0.495300
$275$	0	0
$276$	821.367	0.179132
$277$	4700.53i	1.01959i	0.860295	+	0.509797i	$0.170279\pi$
$277$	4700.53i	1.01959i	−0.860295	+	0.509797i	$0.829721\pi$
$278$	− 3603.50i	− 0.777423i
$279$	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	$0.301224\pi$
$282$	7034.67i	1.48549i
$283$	6639.20i	1.39456i	0.716801	+	0.697278i	$0.245606\pi$
$283$	6639.20i	1.39456i	−0.716801	+	0.697278i	$0.754394\pi$
$284$	2602.71	0.543812
$285$	0	0
$286$	−289.983	−0.0599548
$287$	− 1750.89i	− 0.360111i
$288$	1686.64i	0.345091i
$289$	4859.50	0.989111
$290$	0	0
$291$	3426.82	0.690323
$292$	− 2781.78i	− 0.557504i
$293$	− 2120.85i	− 0.422872i	−0.977392	−	0.211436i	$-0.932186\pi$
$293$	− 2120.85i	− 0.422872i	0.977392	−	0.211436i	$-0.0678140\pi$
$294$	−3780.34	−0.749912
$295$	0	0
$296$	−2345.38	−0.460550
$297$	− 375.341i	− 0.0733316i
$298$	1326.28i	0.257816i
$299$	2039.18	0.394410
$300$	0	0
$301$	11024.4	2.11109
$302$	− 3235.31i	− 0.616461i
$303$	− 6022.58i	− 1.14188i
$304$	104.598	0.0197339
$305$	0	0
$306$	−771.042	−0.144044
$307$	− 7422.87i	− 1.37995i	−0.723832	−	0.689977i	$-0.757621\pi$
$307$	− 7422.87i	− 1.37995i	0.723832	−	0.689977i	$-0.242379\pi$
$308$	154.068i	0.0285026i
$309$	−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	$0.575098\pi$
$312$	6332.37i	1.14904i
$313$	9403.53i	1.69814i	0.528277	+	0.849072i	$0.322838\pi$
$313$	9403.53i	1.69814i	−0.528277	+	0.849072i	$0.677162\pi$
$314$	−1750.16	−0.314545
$315$	0	0
$316$	2642.87	0.470484
$317$	− 407.110i	− 0.0721312i	−0.999349	−	0.0360656i	$-0.988517\pi$
$317$	− 407.110i	− 0.0721312i	0.999349	−	0.0360656i	$-0.0114825\pi$
$318$	4163.45i	0.734198i
$319$	−195.002	−0.0342258
$320$	0	0
$321$	−13689.2	−2.38024
$322$	− 1083.41i	− 0.187503i
$323$	47.8167i	0.00823713i
$324$	−2503.91	−0.429340
$325$	0	0
$326$	5064.79	0.860469
$327$	− 11068.3i	− 1.87180i
$328$	594.723i	0.100116i
$329$	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	$0.701408\pi$
$332$	5313.56i	0.878372i
$333$	− 15452.4i	− 2.54290i
$334$	2263.31	0.370786
$335$	0	0
$336$	3364.38	0.546255
$337$	8267.78i	1.33642i	0.743971	+	0.668212i	$0.232940\pi$
$337$	8267.78i	1.33642i	−0.743971	+	0.668212i	$0.767060\pi$
$338$	11327.1i	1.82283i
$339$	−20529.7	−3.28914
$340$	0	0
$341$	−255.864	−0.0406328
$342$	689.138i	0.108960i
$343$	− 3092.08i	− 0.486753i
$344$	−3744.65	−0.586913
$345$	0	0
$346$	−6372.45	−0.990130
$347$	− 1048.83i	− 0.162260i	−0.996704	−	0.0811302i	$-0.974147\pi$
$347$	− 1048.83i	− 0.162260i	0.996704	−	0.0811302i	$-0.0258530\pi$
$348$	4258.27i	0.655940i
$349$	9135.87	1.40124	0.700619	−	0.713535i	$-0.252907\pi$
$349$	9135.87	1.40124	0.700619	+	0.713535i	$0.252907\pi$
$350$	0	0
$351$	−20348.7	−3.09440
$352$	− 52.3319i	− 0.00792414i
$353$	5253.06i	0.792046i	0.918240	+	0.396023i	$0.129610\pi$
$353$	5253.06i	0.792046i	−0.918240	+	0.396023i	$0.870390\pi$
$354$	13689.1	2.05528
$355$	0	0
$356$	−3298.81	−0.491113
$357$	1538.01i	0.228012i
$358$	3326.96i	0.491160i
$359$	11269.7	1.65680	0.828400	−	0.560137i	$-0.189252\pi$
$359$	11269.7	1.65680	0.828400	+	0.560137i	$0.189252\pi$
$360$	0	0
$361$	−6816.26	−0.993769
$362$	7494.70i	1.08816i
$363$	− 11859.2i	− 1.71472i
$364$	8352.60	1.20273
$365$	0	0
$366$	3184.86	0.454850
$367$	5675.46i	0.807238i	0.914927	+	0.403619i	$0.132248\pi$
$367$	5675.46i	0.807238i	−0.914927	+	0.403619i	$0.867752\pi$
$368$	368.000i	0.0521286i
$369$	−3918.29	−0.552787
$370$	0	0
$371$	5491.73	0.768508
$372$	5587.30i	0.778731i
$373$	3662.20i	0.508369i	0.967156	+	0.254185i	$0.0818071\pi$
$373$	3662.20i	0.508369i	−0.967156	+	0.254185i	$0.918193\pi$
$374$	23.9233	0.00330761
$375$	0	0
$376$	−3151.77	−0.432287
$377$	10571.8i	1.44424i
$378$	10811.2i	1.47108i
$379$	−5520.89	−0.748256	−0.374128	−	0.927377i	$-0.622058\pi$
$379$	−5520.89	−0.748256	−0.374128	+	0.927377i	$0.622058\pi$
$380$	0	0
$381$	−10515.4	−1.41397
$382$	− 4525.93i	− 0.606196i
$383$	− 2309.13i	− 0.308071i	−0.988065	−	0.154036i	$-0.950773\pi$
$383$	− 2309.13i	− 0.308071i	0.988065	−	0.154036i	$-0.0492271\pi$
$384$	−1142.77	−0.151867
$385$	0	0
$386$	−4002.50	−0.527777
$387$	− 24671.4i	− 3.24061i
$388$	1535.33i	0.200888i
$389$	8194.51	1.06807	0.534034	−	0.845463i	$-0.320676\pi$
$389$	8194.51	1.06807	0.534034	+	0.845463i	$0.320676\pi$
$390$	0	0
$391$	−168.230	−0.0217590
$392$	− 1693.72i	− 0.218229i
$393$	21298.1i	2.73370i
$394$	9608.16	1.22856
$395$	0	0
$396$	344.785	0.0437528
$397$	− 7273.90i	− 0.919564i	−0.888032	−	0.459782i	$-0.847928\pi$
$397$	− 7273.90i	− 0.919564i	0.888032	−	0.459782i	$-0.152072\pi$
$398$	7770.07i	0.978589i
$399$	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	$0.536465\pi$
$402$	− 4408.67i	− 0.546977i
$403$	13871.4i	1.71460i
$404$	2698.32	0.332293
$405$	0	0
$406$	5616.79	0.686593
$407$	479.446i	0.0583914i
$408$	− 522.415i	− 0.0633906i
$409$	2865.80	0.346466	0.173233	−	0.984881i	$-0.444579\pi$
$409$	2865.80	0.346466	0.173233	+	0.984881i	$0.444579\pi$
$410$	0	0
$411$	10028.0	1.20351
$412$	− 677.187i	− 0.0809773i
$413$	− 18056.4i	− 2.15132i
$414$	−2424.55	−0.287826
$415$	0	0
$416$	−2837.12	−0.334377
$417$	16085.9i	1.88904i
$418$	− 21.3821i	− 0.00250199i
$419$	−11044.0	−1.28767	−0.643836	−	0.765164i	$-0.722658\pi$
$419$	−11044.0	−1.28767	−0.643836	+	0.765164i	$0.722658\pi$
$420$	0	0
$421$	6343.24	0.734324	0.367162	−	0.930157i	$-0.380329\pi$
$421$	6343.24	0.734324	0.367162	+	0.930157i	$0.380329\pi$
$422$	− 9136.77i	− 1.05396i
$423$	− 20765.2i	− 2.38685i
$424$	−1865.37	−0.213656
$425$	0	0
$426$	−11618.4	−1.32139
$427$	− 4200.93i	− 0.476106i
$428$	− 6133.23i	− 0.692665i
$429$	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	$0.573891\pi$
$432$	− 3672.23i	− 0.408982i
$433$	− 758.791i	− 0.0842151i	−0.999113	−	0.0421076i	$-0.986593\pi$
$433$	− 758.791i	− 0.0842151i	0.999113	−	0.0421076i	$-0.0134072\pi$
$434$	7369.82	0.815122
$435$	0	0
$436$	4958.98	0.544707
$437$	150.360i	0.0164592i
$438$	12417.7i	1.35466i
$439$	−4150.10	−0.451193	−0.225596	−	0.974221i	$-0.572433\pi$
$439$	−4150.10	−0.451193	−0.225596	+	0.974221i	$0.572433\pi$
$440$	0	0
$441$	11159.0	1.20494
$442$	− 1296.98i	− 0.139572i
$443$	− 4597.24i	− 0.493051i	−0.969136	−	0.246525i	$-0.920711\pi$
$443$	− 4597.24i	− 0.493051i	0.969136	−	0.246525i	$-0.0792889\pi$
$444$	10469.7	1.11907
$445$	0	0
$446$	6162.13	0.654228
$447$	− 5920.43i	− 0.626458i
$448$	1507.35i	0.158964i
$449$	−6402.58	−0.672954	−0.336477	−	0.941692i	$-0.609235\pi$
$449$	−6402.58	−0.672954	−0.336477	+	0.941692i	$0.609235\pi$
$450$	0	0
$451$	121.574	0.0126933
$452$	− 9197.97i	− 0.957159i
$453$	14442.3i	1.49792i
$454$	−1357.94	−0.140377
$455$	0	0
$456$	−466.921	−0.0479508
$457$	− 551.499i	− 0.0564508i	−0.999602	−	0.0282254i	$-0.991014\pi$
$457$	− 551.499i	− 0.0564508i	0.999602	−	0.0282254i	$-0.00898562\pi$
$458$	12798.5i	1.30576i
$459$	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	$0.563154\pi$
$462$	− 687.750i	− 0.0692577i
$463$	4828.15i	0.484628i	0.970198	+	0.242314i	$0.0779065\pi$
$463$	4828.15i	0.484628i	−0.970198	+	0.242314i	$0.922094\pi$
$464$	−1907.85	−0.190883
$465$	0	0
$466$	12166.6	1.20946
$467$	14136.3i	1.40075i	0.713775	+	0.700375i	$0.246984\pi$
$467$	14136.3i	1.40075i	−0.713775	+	0.700375i	$0.753016\pi$
$468$	− 18692.2i	− 1.84625i
$469$	−5815.18	−0.572538
$470$	0	0
$471$	7812.63	0.764303
$472$	6133.19i	0.598099i
$473$	765.487i	0.0744125i
$474$	−11797.6	−1.14321
$475$	0	0
$476$	−689.082	−0.0663530
$477$	− 12289.8i	− 1.17969i
$478$	876.626i	0.0838827i
$479$	−15246.1	−1.45430	−0.727152	−	0.686476i	$-0.759157\pi$
$479$	−15246.1	−1.45430	−0.727152	+	0.686476i	$0.759157\pi$
$480$	0	0
$481$	25992.7	2.46396
$482$	− 5708.83i	− 0.539481i
$483$	4836.29i	0.455608i
$484$	5313.30	0.498995
$485$	0	0
$486$	−1216.41	−0.113534
$487$	− 3876.39i	− 0.360690i	−0.983603	−	0.180345i	$-0.942279\pi$
$487$	− 3876.39i	− 0.360690i	0.983603	−	0.180345i	$-0.0577214\pi$
$488$	1426.92i	0.132364i
$489$	−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	$0.574210\pi$
$492$	− 2654.82i	− 0.243269i
$493$	− 872.166i	− 0.0796762i
$494$	−1159.21	−0.105577
$495$	0	0
$496$	−2503.30	−0.226616
$497$	15325.0i	1.38314i
$498$	− 23719.5i	− 2.13433i
$499$	−7178.10	−0.643960	−0.321980	−	0.946747i	$-0.604348\pi$
$499$	−7178.10	−0.643960	−0.321980	+	0.946747i	$0.604348\pi$
$500$	0	0
$501$	−10103.3	−0.900962
$502$	4585.86i	0.407723i
$503$	− 18169.6i	− 1.61063i	−0.592850	−	0.805313i	$-0.701997\pi$
$503$	− 18169.6i	− 1.61063i	0.592850	−	0.805313i	$-0.298003\pi$
$504$	−9931.10	−0.877711
$505$	0	0
$506$	75.2271	0.00660919
$507$	− 50563.8i	− 4.42923i
$508$	− 4711.27i	− 0.411474i
$509$	−2296.14	−0.199950	−0.0999750	−	0.994990i	$-0.531876\pi$
$509$	−2296.14	−0.199950	−0.0999750	+	0.994990i	$0.531876\pi$
$510$	0	0
$511$	16379.4	1.41797
$512$	− 512.000i	− 0.0441942i
$513$	− 1500.42i	− 0.129133i
$514$	−7095.18	−0.608862
$515$	0	0
$516$	16715.9	1.42612
$517$	644.288i	0.0548081i
$518$	− 13809.8i	− 1.17137i
$519$	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	$0.316798\pi$
$522$	− 12569.7i	− 1.05395i
$523$	− 16093.6i	− 1.34555i	−0.739847	−	0.672775i	$-0.765102\pi$
$523$	− 16093.6i	− 1.34555i	0.739847	−	0.672775i	$-0.234898\pi$
$524$	−9542.25	−0.795525
$525$	0	0
$526$	−4556.69	−0.377721
$527$	− 1144.37i	− 0.0945915i
$528$	233.607i	0.0192546i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	−40408.1	−3.30238
$532$	615.884i	0.0501916i
$533$	− 6591.01i	− 0.535625i
$534$	14725.7	1.19334
$535$	0	0
$536$	1975.23	0.159174
$537$	− 14851.4i	− 1.19346i
$538$	7830.73i	0.627522i
$539$	−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	$0.709997\pi$
$542$	− 12505.4i	− 0.991058i
$543$	− 33456.0i	− 2.64408i
$544$	234.059	0.0184471
$545$	0	0
$546$	−37285.6	−2.92249
$547$	− 5307.37i	− 0.414857i	−0.978250	−	0.207428i	$-0.933491\pi$
$547$	− 5307.37i	− 0.414857i	0.978250	−	0.207428i	$-0.0665094\pi$
$548$	4492.87i	0.350230i
$549$	−9401.20	−0.730844
$550$	0	0
$551$	−779.519	−0.0602698
$552$	− 1642.73i	− 0.126666i
$553$	15561.5i	1.19664i
$554$	9401.06	0.720961
$555$	0	0
$556$	−7207.00	−0.549721
$557$	13530.9i	1.02930i	0.857400	+	0.514651i	$0.172079\pi$
$557$	13530.9i	1.02930i	−0.857400	+	0.514651i	$0.827921\pi$
$558$	− 16492.8i	− 1.25125i
$559$	41500.0	3.14001
$560$	0	0
$561$	−106.793	−0.00803706
$562$	− 11016.1i	− 0.826847i
$563$	− 18109.0i	− 1.35560i	−0.735248	−	0.677799i	$-0.762934\pi$
$563$	− 18109.0i	− 1.35560i	0.735248	−	0.677799i	$-0.237066\pi$
$564$	14069.3	1.05040
$565$	0	0
$566$	13278.4	0.986100
$567$	− 14743.3i	− 1.09199i
$568$	− 5205.43i	− 0.384533i
$569$	908.991	0.0669717	0.0334858	−	0.999439i	$-0.489339\pi$
$569$	908.991	0.0669717	0.0334858	+	0.999439i	$0.489339\pi$
$570$	0	0
$571$	−8894.71	−0.651895	−0.325948	−	0.945388i	$-0.605683\pi$
$571$	−8894.71	−0.651895	−0.325948	+	0.945388i	$0.605683\pi$
$572$	579.967i	0.0423945i
$573$	20203.6i	1.47298i
$574$	−3501.79	−0.254637
$575$	0	0
$576$	3373.28	0.244016
$577$	12947.3i	0.934148i	0.884218	+	0.467074i	$0.154692\pi$
$577$	12947.3i	0.934148i	−0.884218	+	0.467074i	$0.845308\pi$
$578$	− 9719.00i	− 0.699407i
$579$	17867.0	1.28243
$580$	0	0
$581$	−31286.8	−2.23407
$582$	− 6853.65i	− 0.488132i
$583$	381.321i	0.0270887i
$584$	−5563.55	−0.394215
$585$	0	0
$586$	−4241.71	−0.299016
$587$	19404.1i	1.36438i	0.731175	+	0.682190i	$0.238972\pi$
$587$	19404.1i	1.36438i	−0.731175	+	0.682190i	$0.761028\pi$
$588$	7560.69i	0.530268i
$589$	−1022.81	−0.0715522
$590$	0	0
$591$	−42890.4	−2.98524
$592$	4690.77i	0.325658i
$593$	− 13633.9i	− 0.944141i	−0.881561	−	0.472070i	$-0.843507\pi$
$593$	− 13633.9i	− 0.944141i	0.881561	−	0.472070i	$-0.156493\pi$
$594$	−750.682	−0.0518533
$595$	0	0
$596$	2652.55	0.182303
$597$	− 34685.2i	− 2.37784i
$598$	− 4078.35i	− 0.278890i
$599$	−11276.5	−0.769191	−0.384595	−	0.923085i	$-0.625659\pi$
$599$	−11276.5	−0.769191	−0.384595	+	0.923085i	$0.625659\pi$
$600$	0	0
$601$	−380.766	−0.0258432	−0.0129216	−	0.999917i	$-0.504113\pi$
$601$	−380.766	−0.0258432	−0.0129216	+	0.999917i	$0.504113\pi$
$602$	− 22048.9i	− 1.49277i
$603$	13013.7i	0.878870i
$604$	−6470.63	−0.435904
$605$	0	0
$606$	−12045.2	−0.807428
$607$	21669.2i	1.44897i	0.689290	+	0.724485i	$0.257923\pi$
$607$	21669.2i	1.44897i	−0.689290	+	0.724485i	$0.742077\pi$
$608$	− 209.196i	− 0.0139540i
$609$	−25073.1	−1.66833
$610$	0	0
$611$	34929.4	2.31275
$612$	1542.08i	0.101855i
$613$	− 936.980i	− 0.0617362i	−0.999523	−	0.0308681i	$-0.990173\pi$
$613$	− 936.980i	− 0.0617362i	0.999523	−	0.0308681i	$-0.00982718\pi$
$614$	−14845.7	−0.975774
$615$	0	0
$616$	308.135	0.0201544
$617$	17539.2i	1.14441i	0.820111	+	0.572205i	$0.193912\pi$
$617$	17539.2i	1.14441i	−0.820111	+	0.572205i	$0.806088\pi$
$618$	3022.93i	0.196764i
$619$	4740.20	0.307794	0.153897	−	0.988087i	$-0.450818\pi$
$619$	4740.20	0.307794	0.153897	+	0.988087i	$0.450818\pi$
$620$	0	0
$621$	5278.83	0.341114
$622$	5127.94i	0.330565i
$623$	− 19423.7i	− 1.24911i
$624$	12664.7	0.812493
$625$	0	0
$626$	18807.1	1.20077
$627$	95.4486i	0.00607950i
$628$	3500.32i	0.222417i
$629$	−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	$0.698417\pi$
$632$	− 5285.73i	− 0.332682i
$633$	40786.1i	2.56098i
$634$	−814.221	−0.0510045
$635$	0	0
$636$	8326.91	0.519156
$637$	18770.6i	1.16753i
$638$	390.004i	0.0242013i
$639$	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	$0.398621\pi$
$642$	27378.4i	1.68308i
$643$	7132.33i	0.437436i	0.975788	+	0.218718i	$0.0701875\pi$
$643$	7132.33i	0.437436i	−0.975788	+	0.218718i	$0.929812\pi$
$644$	−2166.82	−0.132585
$645$	0	0
$646$	95.6334	0.00582453
$647$	− 14536.9i	− 0.883315i	−0.897184	−	0.441657i	$-0.854391\pi$
$647$	− 14536.9i	− 0.883315i	0.897184	−	0.441657i	$-0.145609\pi$
$648$	5007.83i	0.303590i
$649$	1253.75	0.0758307
$650$	0	0
$651$	−32898.5	−1.98064
$652$	− 10129.6i	− 0.608443i
$653$	3318.33i	0.198861i	0.995045	+	0.0994305i	$0.0317021\pi$
$653$	3318.33i	0.198861i	−0.995045	+	0.0994305i	$0.968298\pi$
$654$	−22136.7	−1.32357
$655$	0	0
$656$	1189.45	0.0707928
$657$	− 36655.1i	− 2.17664i
$658$	− 18557.9i	− 1.09949i
$659$	−3088.17	−0.182546	−0.0912730	−	0.995826i	$-0.529094\pi$
$659$	−3088.17	−0.182546	−0.0912730	+	0.995826i	$0.529094\pi$
$660$	0	0
$661$	27602.0	1.62420	0.812098	−	0.583521i	$-0.198325\pi$
$661$	27602.0	1.62420	0.812098	+	0.583521i	$0.198325\pi$
$662$	14244.7i	0.836306i
$663$	5789.65i	0.339142i
$664$	10627.1	0.621103
$665$	0	0
$666$	−30904.8	−1.79810
$667$	− 2742.53i	− 0.159207i
$668$	− 4526.61i	− 0.262186i
$669$	−27507.5	−1.58969
$670$	0	0
$671$	291.694	0.0167820
$672$	− 6728.75i	− 0.386261i
$673$	8091.33i	0.463444i	0.972782	+	0.231722i	$0.0744360\pi$
$673$	8091.33i	0.463444i	−0.972782	+	0.231722i	$0.925564\pi$
$674$	16535.6	0.944994
$675$	0	0
$676$	22654.3	1.28893
$677$	9515.37i	0.540185i	0.962834	+	0.270093i	$0.0870543\pi$
$677$	9515.37i	0.540185i	−0.962834	+	0.270093i	$0.912946\pi$
$678$	41059.3i	2.32577i
$679$	−9040.18	−0.510943
$680$	0	0
$681$	6061.79	0.341099
$682$	511.727i	0.0287317i
$683$	− 630.346i	− 0.0353141i	−0.999844	−	0.0176570i	$-0.994379\pi$
$683$	− 630.346i	− 0.0353141i	0.999844	−	0.0176570i	$-0.00562070\pi$
$684$	1378.28	0.0770463
$685$	0	0
$686$	−6184.15	−0.344187
$687$	− 57132.1i	− 3.17282i
$688$	7489.30i	0.415010i
$689$	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	$0.557403\pi$
$692$	12744.9i	0.700128i
$693$	2030.13i	0.111282i
$694$	−2097.67	−0.114735
$695$	0	0
$696$	8516.53	0.463819
$697$	543.751i	0.0295496i
$698$	− 18271.7i	− 0.990825i
$699$	−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	$0.845825\pi$
$702$	40697.4i	2.18807i
$703$	1916.58i	0.102824i
$704$	−104.664	−0.00560321
$705$	0	0
$706$	10506.1	0.560061
$707$	15888.0i	0.845160i
$708$	− 27378.3i	− 1.45330i
$709$	21841.2	1.15693	0.578466	−	0.815707i	$-0.303652\pi$
$709$	21841.2	1.15693	0.578466	+	0.815707i	$0.303652\pi$
$710$	0	0
$711$	34824.7	1.83689
$712$	6597.61i	0.347270i
$713$	− 3598.49i	− 0.189010i
$714$	3076.03	0.161229
$715$	0	0
$716$	6653.93	0.347303
$717$	− 3913.22i	− 0.203824i
$718$	− 22539.4i	− 1.17153i
$719$	−25807.3	−1.33859	−0.669296	−	0.742996i	$-0.733404\pi$
$719$	−25807.3	−1.33859	−0.669296	+	0.742996i	$0.733404\pi$
$720$	0	0
$721$	3987.34	0.205959
$722$	13632.5i	0.702701i
$723$	25483.9i	1.31087i
$724$	14989.4	0.769443
$725$	0	0
$726$	−23718.3	−1.21249
$727$	− 14264.0i	− 0.727679i	−0.931462	−	0.363839i	$-0.881466\pi$
$727$	− 14264.0i	− 0.727679i	0.931462	−	0.363839i	$-0.118534\pi$
$728$	− 16705.2i	− 0.850462i
$729$	22331.4	1.13455
$730$	0	0
$731$	−3423.71	−0.173229
$732$	− 6369.72i	− 0.321628i
$733$	811.706i	0.0409018i	0.999791	+	0.0204509i	$0.00651018\pi$
$733$	811.706i	0.0409018i	−0.999791	+	0.0204509i	$0.993490\pi$
$734$	11350.9	0.570804
$735$	0	0
$736$	736.000	0.0368605
$737$	− 403.780i	− 0.0201810i
$738$	7836.59i	0.390879i
$739$	30696.9	1.52802	0.764008	−	0.645207i	$-0.223229\pi$
$739$	30696.9	1.52802	0.764008	+	0.645207i	$0.223229\pi$
$740$	0	0
$741$	5174.64	0.256539
$742$	− 10983.5i	− 0.543417i
$743$	− 26935.6i	− 1.32998i	−0.746854	−	0.664988i	$-0.768437\pi$
$743$	− 26935.6i	− 1.32998i	0.746854	−	0.664988i	$-0.231563\pi$
$744$	11174.6	0.550646
$745$	0	0
$746$	7324.40	0.359471
$747$	70016.2i	3.42939i
$748$	− 47.8467i	− 0.00233883i
$749$	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	$0.317098\pi$
$752$	6303.53i	0.305673i
$753$	− 20471.0i	− 0.990712i
$754$	21143.7	1.02123
$755$	0	0
$756$	21622.4	1.04021
$757$	− 697.182i	− 0.0334736i	−0.999860	−	0.0167368i	$-0.994672\pi$
$757$	− 697.182i	− 0.0334736i	0.999860	−	0.0167368i	$-0.00532774\pi$
$758$	11041.8i	0.529097i
$759$	−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	$0.104194\pi$
$762$	21030.9i	0.999827i
$763$	29199.0i	1.38542i
$764$	−9051.87	−0.428645
$765$	0	0
$766$	−4618.27	−0.217839
$767$	− 67970.9i	− 3.19985i
$768$	2285.54i	0.107386i
$769$	12574.8	0.589673	0.294836	−	0.955548i	$-0.404735\pi$
$769$	12574.8	0.589673	0.294836	+	0.955548i	$0.404735\pi$
$770$	0	0
$771$	31672.5	1.47945
$772$	8005.00i	0.373195i
$773$	2478.65i	0.115331i	0.998336	+	0.0576654i	$0.0183656\pi$
$773$	2478.65i	0.115331i	−0.998336	+	0.0576654i	$0.981634\pi$
$774$	−49342.8	−2.29146
$775$	0	0
$776$	3070.66	0.142049
$777$	61646.5i	2.84628i
$778$	− 16389.0i	− 0.755238i
$779$	485.991	0.0223523
$780$	0	0
$781$	−1064.10	−0.0487535
$782$	336.460i	0.0153859i
$783$	27367.3i	1.24908i
$784$	−3387.44	−0.154311
$785$	0	0
$786$	42596.1	1.93302
$787$	7716.40i	0.349504i	0.984612	+	0.174752i	$0.0559124\pi$
$787$	7716.40i	0.349504i	−0.984612	+	0.174752i	$0.944088\pi$
$788$	− 19216.3i	− 0.868722i
$789$	20340.9	0.917812
$790$	0	0
$791$	54158.5	2.43446
$792$	− 689.570i	− 0.0309379i
$793$	− 15813.9i	− 0.708154i
$794$	−14547.8	−0.650230
$795$	0	0
$796$	15540.1	0.691967
$797$	26798.1i	1.19101i	0.803350	+	0.595507i	$0.203049\pi$
$797$	26798.1i	1.19101i	−0.803350	+	0.595507i	$0.796951\pi$
$798$	− 2749.28i	− 0.121959i
$799$	−2881.64	−0.127591
$800$	0	0
$801$	−43468.0	−1.91743
$802$	3671.61i	0.161657i
$803$	1137.31i	0.0499810i
$804$	−8817.35	−0.386771
$805$	0	0
$806$	27742.7	1.21240
$807$	− 34956.0i	− 1.52479i
$808$	− 5396.63i	− 0.234967i
$809$	−45278.0	−1.96772	−0.983862	−	0.178927i	$-0.942738\pi$
$809$	−45278.0	−1.96772	−0.983862	+	0.178927i	$0.942738\pi$
$810$	0	0
$811$	−24889.6	−1.07767	−0.538836	−	0.842411i	$-0.681136\pi$
$811$	−24889.6	−1.07767	−0.538836	+	0.842411i	$0.681136\pi$
$812$	− 11233.6i	− 0.485494i
$813$	55823.6i	2.40814i
$814$	958.893	0.0412889
$815$	0	0
$816$	−1044.83	−0.0448239
$817$	3060.03i	0.131036i
$818$	− 5731.60i	− 0.244989i
$819$	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	$0.365646\pi$
$822$	− 20056.0i	− 0.851013i
$823$	− 21620.8i	− 0.915740i	−0.889019	−	0.457870i	$-0.848613\pi$
$823$	− 21620.8i	− 0.915740i	0.889019	−	0.457870i	$-0.151387\pi$
$824$	−1354.37	−0.0572596
$825$	0	0
$826$	−36112.8	−1.52122
$827$	9012.81i	0.378967i	0.981884	+	0.189484i	$0.0606814\pi$
$827$	9012.81i	0.378967i	−0.981884	+	0.189484i	$0.939319\pi$
$828$	4849.09i	0.203524i
$829$	−26574.7	−1.11336	−0.556682	−	0.830726i	$-0.687926\pi$
$829$	−26574.7	−1.11336	−0.556682	+	0.830726i	$0.687926\pi$
$830$	0	0
$831$	−41965.9	−1.75184
$832$	5674.23i	0.236441i
$833$	− 1548.56i	− 0.0644110i
$834$	32171.7	1.33575
$835$	0	0
$836$	−42.7642	−0.00176917
$837$	35908.9i	1.48291i
$838$	22088.0i	0.910521i
$839$	1261.30	0.0519009	0.0259504	−	0.999663i	$-0.491739\pi$
$839$	1261.30	0.0519009	0.0259504	+	0.999663i	$0.491739\pi$
$840$	0	0
$841$	−10170.7	−0.417022
$842$	− 12686.5i	− 0.519245i
$843$	49175.6i	2.00913i
$844$	−18273.5	−0.745262
$845$	0	0
$846$	−41530.4	−1.68776
$847$	31285.2i	1.26915i
$848$	3730.73i	0.151078i
$849$	−59274.1	−2.39609
$850$	0	0
$851$	−6742.98	−0.271617
$852$	23236.8i	0.934365i
$853$	32765.1i	1.31519i	0.753373	+	0.657594i	$0.228426\pi$
$853$	32765.1i	1.31519i	−0.753373	+	0.657594i	$0.771574\pi$
$854$	−8401.86	−0.336658
$855$	0	0
$856$	−12266.5	−0.489788
$857$	− 722.029i	− 0.0287795i	−0.999896	−	0.0143898i	$-0.995419\pi$
$857$	− 722.029i	− 0.0287795i	0.999896	−	0.0143898i	$-0.00458056\pi$
$858$	− 2588.94i	− 0.103013i
$859$	36626.9	1.45482	0.727411	−	0.686202i	$-0.240723\pi$
$859$	36626.9	1.45482	0.727411	+	0.686202i	$0.240723\pi$
$860$	0	0
$861$	15631.8	0.618735
$862$	8234.03i	0.325351i
$863$	− 33207.9i	− 1.30986i	−0.755688	−	0.654931i	$-0.772698\pi$
$863$	− 33207.9i	− 1.30986i	0.755688	−	0.654931i	$-0.227302\pi$
$864$	−7344.45	−0.289194
$865$	0	0
$866$	−1517.58	−0.0595491
$867$	43385.2i	1.69947i
$868$	− 14739.6i	− 0.576378i
$869$	−1080.52	−0.0421795
$870$	0	0
$871$	−21890.5	−0.851585
$872$	− 9917.97i	− 0.385166i
$873$	20230.9i	0.784320i
$874$	300.719	0.0116384
$875$	0	0
$876$	24835.4	0.957890
$877$	39507.6i	1.52118i	0.649230	+	0.760592i	$0.275091\pi$
$877$	39507.6i	1.52118i	−0.649230	+	0.760592i	$0.724909\pi$
$878$	8300.21i	0.319041i
$879$	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	$0.131055\pi$
$882$	− 22317.9i	− 0.852023i
$883$	48176.2i	1.83608i	0.396488	+	0.918040i	$0.370229\pi$
$883$	48176.2i	1.83608i	−0.396488	+	0.918040i	$0.629771\pi$
$884$	−2593.96	−0.0986925
$885$	0	0
$886$	−9194.48	−0.348640
$887$	− 31421.7i	− 1.18944i	−0.803932	−	0.594722i	$-0.797262\pi$
$887$	− 31421.7i	− 1.18944i	0.803932	−	0.594722i	$-0.202738\pi$
$888$	− 20939.4i	− 0.791305i
$889$	27740.4	1.04655
$890$	0	0
$891$	1023.71	0.0384910
$892$	− 12324.3i	− 0.462609i
$893$	2575.54i	0.0965141i
$894$	−11840.9	−0.442973
$895$	0	0
$896$	3014.71	0.112404
$897$	18205.6i	0.677666i
$898$	12805.2i	0.475851i
$899$	18655.9	0.692111
$900$	0	0
$901$	−1705.49	−0.0630613
$902$	− 243.148i	− 0.00897555i
$903$	98425.1i	3.62722i
$904$	−18395.9	−0.676814
$905$	0	0
$906$	28884.6	1.05919
$907$	− 27694.1i	− 1.01385i	−0.861989	−	0.506927i	$-0.830781\pi$
$907$	− 27694.1i	− 1.01385i	0.861989	−	0.506927i	$-0.169219\pi$
$908$	2715.88i	0.0992618i
$909$	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	$0.665522\pi$
$912$	933.842i	0.0339063i
$913$	− 2172.41i	− 0.0787473i
$914$	−1103.00	−0.0399168
$915$	0	0
$916$	25597.1	0.923310
$917$	− 56185.7i	− 2.02335i
$918$	− 3357.49i	− 0.120712i
$919$	−28135.6	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$919$	−28135.6	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$920$	0	0
$921$	66270.7	2.37100
$922$	7803.91i	0.278751i
$923$	57689.1i	2.05727i
$924$	−1375.50	−0.0489726
$925$	0	0
$926$	9656.29	0.342684
$927$	− 8923.21i	− 0.316156i
$928$	3815.69i	0.134974i
$929$	−32497.6	−1.14770	−0.573850	−	0.818961i	$-0.694551\pi$
$929$	−32497.6	−1.14770	−0.573850	+	0.818961i	$0.694551\pi$
$930$	0	0
$931$	−1384.06	−0.0487226
$932$	− 24333.2i	− 0.855215i
$933$	− 22890.9i	− 0.803230i
$934$	28272.6	0.990479
$935$	0	0
$936$	−37384.3	−1.30550
$937$	12209.3i	0.425678i	0.977087	+	0.212839i	$0.0682710\pi$
$937$	12209.3i	0.425678i	−0.977087	+	0.212839i	$0.931729\pi$
$938$	11630.4i	0.404845i
$939$	−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	$0.507266\pi$
$942$	− 15625.3i	− 0.540444i
$943$	1709.83i	0.0590453i
$944$	12266.4	0.422920
$945$	0	0
$946$	1530.97	0.0526176
$947$	11954.8i	0.410219i	0.978739	+	0.205110i	$0.0657551\pi$
$947$	11954.8i	0.410219i	−0.978739	+	0.205110i	$0.934245\pi$
$948$	23595.3i	0.808374i
$949$	61657.9	2.10906
$950$	0	0
$951$	3634.64	0.123934
$952$	1378.16i	0.0469186i
$953$	− 19658.8i	− 0.668216i	−0.942535	−	0.334108i	$-0.891565\pi$
$953$	− 19658.8i	− 0.668216i	0.942535	−	0.334108i	$-0.108435\pi$
$954$	−24579.7	−0.834169
$955$	0	0
$956$	1753.25	0.0593140
$957$	− 1740.96i	− 0.0588059i
$958$	30492.2i	1.02835i
$959$	−26454.5	−0.890782
$960$	0	0
$961$	−5312.52	−0.178326
$962$	− 51985.3i	− 1.74228i
$963$	− 80816.8i	− 2.70434i
$964$	−11417.7	−0.381471
$965$	0	0
$966$	9672.58	0.322164
$967$	7604.58i	0.252892i	0.991973	+	0.126446i	$0.0403571\pi$
$967$	7604.58i	0.252892i	−0.991973	+	0.126446i	$0.959643\pi$
$968$	− 10626.6i	− 0.352843i
$969$	−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	$0.578274\pi$
$972$	2432.83i	0.0802809i
$973$	− 42435.5i	− 1.39817i
$974$	−7752.77	−0.255046
$975$	0	0
$976$	2853.85	0.0935957
$977$	50584.6i	1.65644i	0.560400	+	0.828222i	$0.310647\pi$
$977$	50584.6i	1.65644i	−0.560400	+	0.828222i	$0.689353\pi$
$978$	45218.0i	1.47844i
$979$	1348.69	0.0440290
$980$	0	0
$981$	65343.9	2.12668
$982$	10054.3i	0.326726i
$983$	15625.9i	0.507007i	0.967334	+	0.253503i	$0.0815829\pi$
$983$	15625.9i	0.507007i	−0.967334	+	0.253503i	$0.918417\pi$
$984$	−5309.63	−0.172017
$985$	0	0
$986$	−1744.33	−0.0563396
$987$	82841.6i	2.67161i
$988$	2318.41i	0.0746544i
$989$	−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	$0.405419\pi$
$992$	5006.59i	0.160241i
$993$	− 63587.5i	− 2.03211i
$994$	30650.1	0.978029
$995$	0	0
$996$	−47439.0	−1.50920
$997$	− 21580.5i	− 0.685519i	−0.939423	−	0.342760i	$-0.888638\pi$
$997$	− 21580.5i	− 0.685519i	0.939423	−	0.342760i	$-0.111362\pi$
$998$	14356.2i	0.455348i
$999$	67287.3	2.13101

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.4	✓	4	5.2	odd	4
1150.4.a.o.1.1		4	5.3	odd	4
1150.4.b.m.599.4		8	1.1	even	1	trivial
1150.4.b.m.599.5		8	5.4	even	2	inner
1840.4.a.l.1.1		4	20.7	even	4
2070.4.a.bi.1.3		4	15.2	even	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +8.92791i q^{3} -4.00000 q^{4} +17.8558 q^{6} -23.5524i q^{7} +8.00000i q^{8} -52.7075 q^{9} +1.63537 q^{11} -35.7116i q^{12} -88.6599i q^{13} -47.1048 q^{14} +16.0000 q^{16} +7.31435i q^{17} +105.415i q^{18} +6.53738 q^{19} +210.274 q^{21} -3.27074i q^{22} +23.0000i q^{23} -71.4232 q^{24} -177.320 q^{26} -229.514i q^{27} +94.2096i q^{28} -119.240 q^{29} -156.456 q^{31} -32.0000i q^{32} +14.6004i q^{33} +14.6287 q^{34} +210.830 q^{36} +293.173i q^{37} -13.0748i q^{38} +791.547 q^{39} +74.3404 q^{41} -420.547i q^{42} +468.081i q^{43} -6.54148 q^{44} +46.0000 q^{46} +393.971i q^{47} +142.846i q^{48} -211.715 q^{49} -65.3018 q^{51} +354.639i q^{52} +233.171i q^{53} -459.028 q^{54} +188.419 q^{56} +58.3651i q^{57} +238.481i q^{58} +766.648 q^{59} +178.365 q^{61} +312.912i q^{62} +1241.39i q^{63} -64.0000 q^{64} +29.2009 q^{66} -246.904i q^{67} -29.2574i q^{68} -205.342 q^{69} -650.678 q^{71} -421.660i q^{72} +695.444i q^{73} +586.346 q^{74} -26.1495 q^{76} -38.5169i q^{77} -1583.09i q^{78} -660.717 q^{79} +625.978 q^{81} -148.681i q^{82} -1328.39i q^{83} -841.094 q^{84} +936.163 q^{86} -1064.57i q^{87} +13.0830i q^{88} +824.702 q^{89} -2088.15 q^{91} -92.0000i q^{92} -1396.82i q^{93} +787.942 q^{94} +285.693 q^{96} -383.833i q^{97} +423.430i q^{98} -86.1963 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 16 q^{6} - 128 q^{9} + 186 q^{11} - 104 q^{14} + 128 q^{16} - 370 q^{19} + 604 q^{21} - 64 q^{24} + 128 q^{26} - 588 q^{29} - 422 q^{31} - 432 q^{34} + 512 q^{36} + 842 q^{39} - 738 q^{41}+ \cdots - 4026 q^{99}+O(q^{100}) \) 8 * q - 32 * q^4 + 16 * q^6 - 128 * q^9 + 186 * q^11 - 104 * q^14 + 128 * q^16 - 370 * q^19 + 604 * q^21 - 64 * q^24 + 128 * q^26 - 588 * q^29 - 422 * q^31 - 432 * q^34 + 512 * q^36 + 842 * q^39 - 738 * q^41 - 744 * q^44 + 368 * q^46 - 1200 * q^49 - 630 * q^51 - 1036 * q^54 + 416 * q^56 + 66 * q^59 - 614 * q^61 - 512 * q^64 - 948 * q^66 - 184 * q^69 - 2514 * q^71 - 20 * q^74 + 1480 * q^76 - 2404 * q^79 - 1984 * q^81 - 2416 * q^84 - 400 * q^86 + 1968 * q^89 - 1990 * q^91 + 1452 * q^94 + 256 * q^96 - 4026 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more