LMFDB - Embedded newform 507.4.b.k.337.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + 6703200 x^{4} + 137781 x^{2} + 729 $ x^18 + 112x^16 + 5026x^14 + 114847x^12 + 1397921x^10 + 8545747x^8 + 21033277x^6 + 6703200x^4 + 137781x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.12
Root			$0.100291i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.k.337.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34727i q^{2} +3.00000 q^{3} +2.49032 q^{4} +15.3991i q^{5} +7.04181i q^{6} +10.1317i q^{7} +24.6236i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.34727i q^{2} +3.00000 q^{3} +2.49032 q^{4} +15.3991i q^{5} +7.04181i q^{6} +10.1317i q^{7} +24.6236i q^{8} +9.00000 q^{9} -36.1458 q^{10} -15.0669i q^{11} +7.47096 q^{12} -23.7819 q^{14} +46.1972i q^{15} -37.8757 q^{16} -90.8352 q^{17} +21.1254i q^{18} +114.640i q^{19} +38.3486i q^{20} +30.3952i q^{21} +35.3661 q^{22} -75.7635 q^{23} +73.8709i q^{24} -112.132 q^{25} +27.0000 q^{27} +25.2313i q^{28} +214.817 q^{29} -108.437 q^{30} -284.476i q^{31} +108.084i q^{32} -45.2007i q^{33} -213.215i q^{34} -156.019 q^{35} +22.4129 q^{36} -358.878i q^{37} -269.091 q^{38} -379.181 q^{40} +313.154i q^{41} -71.3458 q^{42} +296.702 q^{43} -37.5214i q^{44} +138.592i q^{45} -177.837i q^{46} +316.691i q^{47} -113.627 q^{48} +240.348 q^{49} -263.203i q^{50} -272.506 q^{51} +163.911 q^{53} +63.3763i q^{54} +232.016 q^{55} -249.480 q^{56} +343.920i q^{57} +504.233i q^{58} -254.149i q^{59} +115.046i q^{60} -935.247 q^{61} +667.742 q^{62} +91.1856i q^{63} -556.709 q^{64} +106.098 q^{66} -240.494i q^{67} -226.209 q^{68} -227.291 q^{69} -366.220i q^{70} +947.455i q^{71} +221.613i q^{72} +430.712i q^{73} +842.384 q^{74} -336.395 q^{75} +285.490i q^{76} +152.654 q^{77} -496.620 q^{79} -583.251i q^{80} +81.0000 q^{81} -735.058 q^{82} +392.527i q^{83} +75.6938i q^{84} -1398.78i q^{85} +696.439i q^{86} +644.451 q^{87} +371.002 q^{88} -979.895i q^{89} -325.312 q^{90} -188.675 q^{92} -853.428i q^{93} -743.360 q^{94} -1765.35 q^{95} +324.253i q^{96} -553.356i q^{97} +564.162i q^{98} -135.602i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.34727i	0.829886i	0.909848	+	0.414943i	$0.136198\pi$
$2$	2.34727i	0.829886i	−0.909848	+	0.414943i	$0.863802\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	2.49032	0.311290
$5$	15.3991i	1.37734i	0.725077	+	0.688668i	$0.241804\pi$
$5$	15.3991i	1.37734i	−0.725077	+	0.688668i	$0.758196\pi$
$6$	7.04181i	0.479135i
$7$	10.1317i	0.547062i	0.961863	+	0.273531i	$0.0881917\pi$
$7$	10.1317i	0.547062i	−0.961863	+	0.273531i	$0.911808\pi$
$8$	24.6236i	1.08822i
$9$	9.00000	0.333333
$10$	−36.1458	−1.14303
$11$	− 15.0669i	− 0.412985i	−0.978448	−	0.206493i	$-0.933795\pi$
$11$	− 15.0669i	− 0.412985i	0.978448	−	0.206493i	$-0.0662050\pi$
$12$	7.47096	0.179723
$13$	0	0
$13$	0	0
$14$	−23.7819	−0.453999
$15$	46.1972i	0.795205i
$16$	−37.8757	−0.591809
$17$	−90.8352	−1.29593	−0.647964	−	0.761671i	$-0.724379\pi$
$17$	−90.8352	−1.29593	−0.647964	+	0.761671i	$0.724379\pi$
$18$	21.1254i	0.276629i
$19$	114.640i	1.38422i	0.721792	+	0.692110i	$0.243319\pi$
$19$	114.640i	1.38422i	−0.721792	+	0.692110i	$0.756681\pi$
$20$	38.3486i	0.428751i
$21$	30.3952i	0.315847i
$22$	35.3661	0.342731
$23$	−75.7635	−0.686860	−0.343430	−	0.939178i	$-0.611589\pi$
$23$	−75.7635	−0.686860	−0.343430	+	0.939178i	$0.611589\pi$
$24$	73.8709i	0.628284i
$25$	−112.132	−0.897052
$26$	0	0
$27$	27.0000	0.192450
$28$	25.2313i	0.170295i
$29$	214.817	1.37553	0.687767	−	0.725931i	$-0.258591\pi$
$29$	214.817	1.37553	0.687767	+	0.725931i	$0.258591\pi$
$30$	−108.437	−0.659929
$31$	− 284.476i	− 1.64817i	−0.566463	−	0.824087i	$-0.691689\pi$
$31$	− 284.476i	− 1.64817i	0.566463	−	0.824087i	$-0.308311\pi$
$32$	108.084i	0.597087i
$33$	− 45.2007i	− 0.238437i
$34$	− 213.215i	− 1.07547i
$35$	−156.019	−0.753488
$36$	22.4129	0.103763
$37$	− 358.878i	− 1.59457i	−0.603602	−	0.797286i	$-0.706268\pi$
$37$	− 358.878i	− 1.59457i	0.603602	−	0.797286i	$-0.293732\pi$
$38$	−269.091	−1.14874
$39$	0	0
$40$	−379.181	−1.49884
$41$	313.154i	1.19284i	0.802672	+	0.596421i	$0.203411\pi$
$41$	313.154i	1.19284i	−0.802672	+	0.596421i	$0.796589\pi$
$42$	−71.3458	−0.262116
$43$	296.702	1.05225	0.526123	−	0.850409i	$-0.323645\pi$
$43$	296.702	1.05225	0.526123	+	0.850409i	$0.323645\pi$
$44$	− 37.5214i	− 0.128558i
$45$	138.592i	0.459112i
$46$	− 177.837i	− 0.570015i
$47$	316.691i	0.982854i	0.870919	+	0.491427i	$0.163525\pi$
$47$	316.691i	0.982854i	−0.870919	+	0.491427i	$0.836475\pi$
$48$	−113.627	−0.341681
$49$	240.348	0.700723
$50$	− 263.203i	− 0.744451i
$51$	−272.506	−0.748204
$52$	0	0
$53$	163.911	0.424810	0.212405	−	0.977182i	$-0.431870\pi$
$53$	163.911	0.424810	0.212405	+	0.977182i	$0.431870\pi$
$54$	63.3763i	0.159712i
$55$	232.016	0.568819
$56$	−249.480	−0.595324
$57$	343.920i	0.799180i
$58$	504.233i	1.14154i
$59$	− 254.149i	− 0.560803i	−0.959883	−	0.280401i	$-0.909532\pi$
$59$	− 254.149i	− 0.560803i	0.959883	−	0.280401i	$-0.0904675\pi$
$60$	115.046i	0.247539i
$61$	−935.247	−1.96305	−0.981526	−	0.191330i	$-0.938720\pi$
$61$	−935.247	−1.96305	−0.981526	+	0.191330i	$0.938720\pi$
$62$	667.742	1.36780
$63$	91.1856i	0.182354i
$64$	−556.709	−1.08732
$65$	0	0
$66$	106.098	0.197876
$67$	− 240.494i	− 0.438522i	−0.975666	−	0.219261i	$-0.929635\pi$
$67$	− 240.494i	− 0.438522i	0.975666	−	0.219261i	$-0.0703646\pi$
$68$	−226.209	−0.403409
$69$	−227.291	−0.396559
$70$	− 366.220i	− 0.625309i
$71$	947.455i	1.58369i	0.610720	+	0.791847i	$0.290880\pi$
$71$	947.455i	1.58369i	−0.610720	+	0.791847i	$0.709120\pi$
$72$	221.613i	0.362740i
$73$	430.712i	0.690562i	0.938499	+	0.345281i	$0.112216\pi$
$73$	430.712i	0.690562i	−0.938499	+	0.345281i	$0.887784\pi$
$74$	842.384	1.32331
$75$	−336.395	−0.517913
$76$	285.490i	0.430894i
$77$	152.654	0.225929
$78$	0	0
$79$	−496.620	−0.707268	−0.353634	−	0.935384i	$-0.615054\pi$
$79$	−496.620	−0.707268	−0.353634	+	0.935384i	$0.615054\pi$
$80$	− 583.251i	− 0.815119i
$81$	81.0000	0.111111
$82$	−735.058	−0.989922
$83$	392.527i	0.519102i	0.965729	+	0.259551i	$0.0835746\pi$
$83$	392.527i	0.519102i	−0.965729	+	0.259551i	$0.916425\pi$
$84$	75.6938i	0.0983199i
$85$	− 1398.78i	− 1.78493i
$86$	696.439i	0.873244i
$87$	644.451	0.794165
$88$	371.002	0.449419
$89$	− 979.895i	− 1.16706i	−0.812090	−	0.583532i	$-0.801670\pi$
$89$	− 979.895i	− 1.16706i	0.812090	−	0.583532i	$-0.198330\pi$
$90$	−325.312	−0.381010
$91$	0	0
$92$	−188.675	−0.213813
$93$	− 853.428i	− 0.951574i
$94$	−743.360	−0.815656
$95$	−1765.35	−1.90654
$96$	324.253i	0.344729i
$97$	− 553.356i	− 0.579225i	−0.957144	−	0.289613i	$-0.906474\pi$
$97$	− 553.356i	− 0.579225i	0.957144	−	0.289613i	$-0.0935265\pi$
$98$	564.162i	0.581520i
$99$	− 135.602i	− 0.137662i
$100$	−279.243	−0.279243
$101$	763.951	0.752633	0.376317	−	0.926491i	$-0.377190\pi$
$101$	763.951	0.752633	0.376317	+	0.926491i	$0.377190\pi$
$102$	− 639.645i	− 0.620924i
$103$	−182.518	−0.174602	−0.0873010	−	0.996182i	$-0.527824\pi$
$103$	−182.518	−0.174602	−0.0873010	+	0.996182i	$0.527824\pi$
$104$	0	0
$105$	−468.058	−0.435027
$106$	384.744i	0.352544i
$107$	183.699	0.165971	0.0829855	−	0.996551i	$-0.473554\pi$
$107$	183.699	0.165971	0.0829855	+	0.996551i	$0.473554\pi$
$108$	67.2386	0.0599078
$109$	1774.42i	1.55925i	0.626244	+	0.779627i	$0.284592\pi$
$109$	1774.42i	1.55925i	−0.626244	+	0.779627i	$0.715408\pi$
$110$	544.605i	0.472055i
$111$	− 1076.63i	− 0.920627i
$112$	− 383.747i	− 0.323756i
$113$	417.288	0.347391	0.173695	−	0.984799i	$-0.444429\pi$
$113$	417.288	0.347391	0.173695	+	0.984799i	$0.444429\pi$
$114$	−807.273	−0.663228
$115$	− 1166.69i	− 0.946037i
$116$	534.963	0.428190
$117$	0	0
$118$	596.556	0.465402
$119$	− 920.318i	− 0.708953i
$120$	−1137.54	−0.865358
$121$	1103.99	0.829443
$122$	− 2195.28i	− 1.62911i
$123$	939.463i	0.688687i
$124$	− 708.436i	− 0.513060i
$125$	198.162i	0.141793i
$126$	−214.037	−0.151333
$127$	1951.69	1.36366	0.681828	−	0.731513i	$-0.261185\pi$
$127$	1951.69	1.36366	0.681828	+	0.731513i	$0.261185\pi$
$128$	− 442.072i	− 0.305266i
$129$	890.105	0.607514
$130$	0	0
$131$	−1475.61	−0.984159	−0.492080	−	0.870550i	$-0.663763\pi$
$131$	−1475.61	−0.984159	−0.492080	+	0.870550i	$0.663763\pi$
$132$	− 112.564i	− 0.0742231i
$133$	−1161.50	−0.757255
$134$	564.503	0.363923
$135$	415.775i	0.265068i
$136$	− 2236.69i	− 1.41026i
$137$	1900.71i	1.18532i	0.805453	+	0.592660i	$0.201922\pi$
$137$	1900.71i	1.18532i	−0.805453	+	0.592660i	$0.798078\pi$
$138$	− 533.512i	− 0.329099i
$139$	−2326.76	−1.41981	−0.709903	−	0.704299i	$-0.751261\pi$
$139$	−2326.76	−1.41981	−0.709903	+	0.704299i	$0.751261\pi$
$140$	−388.538	−0.234553
$141$	950.073i	0.567451i
$142$	−2223.93	−1.31428
$143$	0	0
$144$	−340.882	−0.197270
$145$	3307.98i	1.89457i
$146$	−1011.00	−0.573088
$147$	721.044	0.404563
$148$	− 893.721i	− 0.496374i
$149$	− 1370.92i	− 0.753761i	−0.926262	−	0.376881i	$-0.876997\pi$
$149$	− 1370.92i	− 0.753761i	0.926262	−	0.376881i	$-0.123003\pi$
$150$	− 789.609i	− 0.429809i
$151$	− 1177.57i	− 0.634628i	−0.948320	−	0.317314i	$-0.897219\pi$
$151$	− 1177.57i	− 0.634628i	0.948320	−	0.317314i	$-0.102781\pi$
$152$	−2822.85	−1.50634
$153$	−817.517	−0.431976
$154$	358.320i	0.187495i
$155$	4380.67	2.27009
$156$	0	0
$157$	1621.57	0.824302	0.412151	−	0.911116i	$-0.364778\pi$
$157$	1621.57	0.824302	0.412151	+	0.911116i	$0.364778\pi$
$158$	− 1165.70i	− 0.586951i
$159$	491.734	0.245264
$160$	−1664.40	−0.822389
$161$	− 767.616i	− 0.375755i
$162$	190.129i	0.0922095i
$163$	133.130i	0.0639727i	0.999488	+	0.0319864i	$0.0101833\pi$
$163$	133.130i	0.0639727i	−0.999488	+	0.0319864i	$0.989817\pi$
$164$	779.854i	0.371320i
$165$	696.049	0.328408
$166$	−921.368	−0.430795
$167$	490.724i	0.227385i	0.993516	+	0.113693i	$0.0362679\pi$
$167$	490.724i	0.227385i	−0.993516	+	0.113693i	$0.963732\pi$
$168$	−748.440	−0.343711
$169$	0	0
$170$	3283.31	1.48129
$171$	1031.76i	0.461407i
$172$	738.882	0.327554
$173$	−2008.13	−0.882518	−0.441259	−	0.897380i	$-0.645468\pi$
$173$	−2008.13	−0.882518	−0.441259	+	0.897380i	$0.645468\pi$
$174$	1512.70i	0.659066i
$175$	− 1136.09i	− 0.490743i
$176$	570.670i	0.244408i
$177$	− 762.446i	− 0.323779i
$178$	2300.08	0.968529
$179$	2152.60	0.898843	0.449421	−	0.893320i	$-0.351630\pi$
$179$	2152.60	0.898843	0.449421	+	0.893320i	$0.351630\pi$
$180$	345.138i	0.142917i
$181$	−834.690	−0.342774	−0.171387	−	0.985204i	$-0.554825\pi$
$181$	−834.690	−0.342774	−0.171387	+	0.985204i	$0.554825\pi$
$182$	0	0
$183$	−2805.74	−1.13337
$184$	− 1865.57i	− 0.747455i
$185$	5526.39	2.19626
$186$	2003.23	0.789697
$187$	1368.60i	0.535200i
$188$	788.662i	0.305953i
$189$	273.557i	0.105282i
$190$	− 4143.75i	− 1.58221i
$191$	4464.71	1.69139	0.845695	−	0.533667i	$-0.179186\pi$
$191$	4464.71	1.69139	0.845695	+	0.533667i	$0.179186\pi$
$192$	−1670.13	−0.627766
$193$	3299.84i	1.23071i	0.788248	+	0.615357i	$0.210988\pi$
$193$	3299.84i	1.23071i	−0.788248	+	0.615357i	$0.789012\pi$
$194$	1298.88	0.480691
$195$	0	0
$196$	598.543	0.218128
$197$	2973.59i	1.07543i	0.843127	+	0.537714i	$0.180712\pi$
$197$	2973.59i	1.07543i	−0.843127	+	0.537714i	$0.819288\pi$
$198$	318.295	0.114244
$199$	5430.82	1.93457	0.967287	−	0.253684i	$-0.0816422\pi$
$199$	5430.82	1.93457	0.967287	+	0.253684i	$0.0816422\pi$
$200$	− 2761.08i	− 0.976191i
$201$	− 721.481i	− 0.253181i
$202$	1793.20i	0.624600i
$203$	2176.47i	0.752503i
$204$	−678.626	−0.232909
$205$	−4822.29	−1.64294
$206$	− 428.419i	− 0.144900i
$207$	−681.872	−0.228953
$208$	0	0
$209$	1727.27	0.571663
$210$	− 1098.66i	− 0.361022i
$211$	1228.25	0.400742	0.200371	−	0.979720i	$-0.435785\pi$
$211$	1228.25	0.400742	0.200371	+	0.979720i	$0.435785\pi$
$212$	408.192	0.132239
$213$	2842.37i	0.914346i
$214$	431.192i	0.137737i
$215$	4568.93i	1.44930i
$216$	664.838i	0.209428i
$217$	2882.24	0.901654
$218$	−4165.05	−1.29400
$219$	1292.14i	0.398696i
$220$	577.795	0.177068
$221$	0	0
$222$	2527.15	0.764015
$223$	− 685.256i	− 0.205776i	−0.994693	−	0.102888i	$-0.967192\pi$
$223$	− 685.256i	− 0.205776i	0.994693	−	0.102888i	$-0.0328084\pi$
$224$	−1095.08	−0.326644
$225$	−1009.18	−0.299017
$226$	979.487i	0.288294i
$227$	287.067i	0.0839354i	0.999119	+	0.0419677i	$0.0133627\pi$
$227$	287.067i	0.0839354i	−0.999119	+	0.0419677i	$0.986637\pi$
$228$	856.470i	0.248777i
$229$	2302.23i	0.664347i	0.943218	+	0.332174i	$0.107782\pi$
$229$	2302.23i	0.664347i	−0.943218	+	0.332174i	$0.892218\pi$
$230$	2738.53	0.785102
$231$	457.961	0.130440
$232$	5289.57i	1.49688i
$233$	970.620	0.272908	0.136454	−	0.990646i	$-0.456429\pi$
$233$	970.620	0.272908	0.136454	+	0.990646i	$0.456429\pi$
$234$	0	0
$235$	−4876.75	−1.35372
$236$	− 632.912i	− 0.174572i
$237$	−1489.86	−0.408341
$238$	2160.24	0.588350
$239$	− 5007.40i	− 1.35524i	−0.735413	−	0.677619i	$-0.763012\pi$
$239$	− 5007.40i	− 1.35524i	0.735413	−	0.677619i	$-0.236988\pi$
$240$	− 1749.75i	− 0.470609i
$241$	540.092i	0.144358i	0.997392	+	0.0721792i	$0.0229953\pi$
$241$	540.092i	0.144358i	−0.997392	+	0.0721792i	$0.977005\pi$
$242$	2591.36i	0.688343i
$243$	243.000	0.0641500
$244$	−2329.07	−0.611078
$245$	3701.14i	0.965130i
$246$	−2205.17	−0.571531
$247$	0	0
$248$	7004.83	1.79358
$249$	1177.58i	0.299704i
$250$	−465.141	−0.117672
$251$	6087.80	1.53091	0.765455	−	0.643489i	$-0.222514\pi$
$251$	6087.80	1.53091	0.765455	+	0.643489i	$0.222514\pi$
$252$	227.081i	0.0567650i
$253$	1141.52i	0.283663i
$254$	4581.14i	1.13168i
$255$	− 4196.34i	− 1.03053i
$256$	−3416.01	−0.833987
$257$	5096.34	1.23697	0.618484	−	0.785797i	$-0.287747\pi$
$257$	5096.34	1.23697	0.618484	+	0.785797i	$0.287747\pi$
$258$	2089.32i	0.504167i
$259$	3636.06	0.872330
$260$	0	0
$261$	1933.35	0.458511
$262$	− 3463.66i	− 0.816739i
$263$	3405.50	0.798450	0.399225	−	0.916853i	$-0.369279\pi$
$263$	3405.50	0.798450	0.399225	+	0.916853i	$0.369279\pi$
$264$	1113.00	0.259472
$265$	2524.08i	0.585106i
$266$	− 2726.36i	− 0.628435i
$267$	− 2939.68i	− 0.673804i
$268$	− 598.906i	− 0.136507i
$269$	−2720.44	−0.616611	−0.308306	−	0.951287i	$-0.599762\pi$
$269$	−2720.44	−0.616611	−0.308306	+	0.951287i	$0.599762\pi$
$270$	−975.937	−0.219976
$271$	− 6954.27i	− 1.55883i	−0.626511	−	0.779413i	$-0.715518\pi$
$271$	− 6954.27i	− 1.55883i	0.626511	−	0.779413i	$-0.284482\pi$
$272$	3440.45	0.766941
$273$	0	0
$274$	−4461.49	−0.983680
$275$	1689.47i	0.370470i
$276$	−566.026	−0.123445
$277$	6563.96	1.42379	0.711895	−	0.702286i	$-0.247837\pi$
$277$	6563.96	1.42379	0.711895	+	0.702286i	$0.247837\pi$
$278$	− 5461.53i	− 1.17828i
$279$	− 2560.28i	− 0.549392i
$280$	− 3841.76i	− 0.819961i
$281$	652.800i	0.138586i	0.997596	+	0.0692932i	$0.0220744\pi$
$281$	652.800i	0.138586i	−0.997596	+	0.0692932i	$0.977926\pi$
$282$	−2230.08	−0.470919
$283$	−6010.62	−1.26252	−0.631262	−	0.775570i	$-0.717463\pi$
$283$	−6010.62	−1.26252	−0.631262	+	0.775570i	$0.717463\pi$
$284$	2359.47i	0.492988i
$285$	−5296.05	−1.10074
$286$	0	0
$287$	−3172.80	−0.652558
$288$	972.759i	0.199029i
$289$	3338.04	0.679430
$290$	−7764.73	−1.57228
$291$	− 1660.07i	− 0.334416i
$292$	1072.61i	0.214965i
$293$	1912.72i	0.381372i	0.981651	+	0.190686i	$0.0610713\pi$
$293$	1912.72i	0.381372i	−0.981651	+	0.190686i	$0.938929\pi$
$294$	1692.49i	0.335741i
$295$	3913.66	0.772413
$296$	8836.87	1.73525
$297$	− 406.806i	− 0.0794791i
$298$	3217.93	0.625535
$299$	0	0
$300$	−837.730	−0.161221
$301$	3006.10i	0.575644i
$302$	2764.06	0.526669
$303$	2291.85	0.434533
$304$	− 4342.07i	− 0.819194i
$305$	− 14401.9i	− 2.70378i
$306$	− 1918.93i	− 0.358491i
$307$	1983.07i	0.368665i	0.982864	+	0.184332i	$0.0590123\pi$
$307$	1983.07i	0.368665i	−0.982864	+	0.184332i	$0.940988\pi$
$308$	380.157	0.0703294
$309$	−547.553	−0.100807
$310$	10282.6i	1.88391i
$311$	−1893.76	−0.345291	−0.172645	−	0.984984i	$-0.555231\pi$
$311$	−1893.76	−0.345291	−0.172645	+	0.984984i	$0.555231\pi$
$312$	0	0
$313$	4574.19	0.826034	0.413017	−	0.910723i	$-0.364475\pi$
$313$	4574.19	0.826034	0.413017	+	0.910723i	$0.364475\pi$
$314$	3806.26i	0.684076i
$315$	−1404.17	−0.251163
$316$	−1236.74	−0.220165
$317$	7594.53i	1.34559i	0.739830	+	0.672794i	$0.234906\pi$
$317$	7594.53i	1.34559i	−0.739830	+	0.672794i	$0.765094\pi$
$318$	1154.23i	0.203541i
$319$	− 3236.62i	− 0.568076i
$320$	− 8572.81i	− 1.49761i
$321$	551.098	0.0958234
$322$	1801.80	0.311834
$323$	− 10413.3i	− 1.79385i
$324$	201.716	0.0345878
$325$	0	0
$326$	−312.492	−0.0530901
$327$	5323.27i	0.900236i
$328$	−7710.99	−1.29807
$329$	−3208.63	−0.537682
$330$	1633.82i	0.272541i
$331$	− 1738.58i	− 0.288705i	−0.989526	−	0.144352i	$-0.953890\pi$
$331$	− 1738.58i	− 0.288705i	0.989526	−	0.144352i	$-0.0461099\pi$
$332$	977.519i	0.161591i
$333$	− 3229.90i	− 0.531524i
$334$	−1151.86	−0.188704
$335$	3703.38	0.603992
$336$	− 1151.24i	− 0.186921i
$337$	2710.61	0.438149	0.219074	−	0.975708i	$-0.429696\pi$
$337$	2710.61	0.438149	0.219074	+	0.975708i	$0.429696\pi$
$338$	0	0
$339$	1251.86	0.200566
$340$	− 3483.41i	− 0.555630i
$341$	−4286.17	−0.680672
$342$	−2421.82	−0.382915
$343$	5910.33i	0.930401i
$344$	7305.87i	1.14508i
$345$	− 3500.06i	− 0.546195i
$346$	− 4713.63i	− 0.732389i
$347$	−2506.81	−0.387817	−0.193908	−	0.981020i	$-0.562116\pi$
$347$	−2506.81	−0.387817	−0.193908	+	0.981020i	$0.562116\pi$
$348$	1604.89	0.247216
$349$	− 7536.48i	− 1.15593i	−0.816063	−	0.577963i	$-0.803848\pi$
$349$	− 7536.48i	− 1.15593i	0.816063	−	0.577963i	$-0.196152\pi$
$350$	2666.70	0.407261
$351$	0	0
$352$	1628.50	0.246588
$353$	− 8992.88i	− 1.35593i	−0.735095	−	0.677964i	$-0.762862\pi$
$353$	− 8992.88i	− 1.35593i	0.735095	−	0.677964i	$-0.237138\pi$
$354$	1789.67	0.268700
$355$	−14589.9	−2.18128
$356$	− 2440.25i	− 0.363295i
$357$	− 2760.96i	− 0.409314i
$358$	5052.74i	0.745937i
$359$	4566.64i	0.671359i	0.941976	+	0.335679i	$0.108966\pi$
$359$	4566.64i	0.671359i	−0.941976	+	0.335679i	$0.891034\pi$
$360$	−3412.63	−0.499615
$361$	−6283.31	−0.916068
$362$	− 1959.24i	− 0.284463i
$363$	3311.97	0.478879
$364$	0	0
$365$	−6632.57	−0.951136
$366$	− 6585.84i	− 0.940566i
$367$	−6449.50	−0.917332	−0.458666	−	0.888609i	$-0.651673\pi$
$367$	−6449.50	−0.917332	−0.458666	+	0.888609i	$0.651673\pi$
$368$	2869.60	0.406490
$369$	2818.39i	0.397614i
$370$	12971.9i	1.82264i
$371$	1660.71i	0.232398i
$372$	− 2125.31i	− 0.296215i
$373$	7648.89	1.06178	0.530891	−	0.847440i	$-0.321857\pi$
$373$	7648.89	1.06178	0.530891	+	0.847440i	$0.321857\pi$
$374$	−3212.49	−0.444154
$375$	594.487i	0.0818645i
$376$	−7798.08	−1.06956
$377$	0	0
$378$	−642.112	−0.0873722
$379$	− 10297.8i	− 1.39567i	−0.716256	−	0.697837i	$-0.754146\pi$
$379$	− 10297.8i	− 1.39567i	0.716256	−	0.697837i	$-0.245854\pi$
$380$	−4396.28	−0.593486
$381$	5855.06	0.787307
$382$	10479.9i	1.40366i
$383$	10258.0i	1.36856i	0.729217	+	0.684282i	$0.239884\pi$
$383$	10258.0i	1.36856i	−0.729217	+	0.684282i	$0.760116\pi$
$384$	− 1326.22i	− 0.176245i
$385$	2350.73i	0.311180i
$386$	−7745.63	−1.02135
$387$	2670.31	0.350749
$388$	− 1378.03i	− 0.180307i
$389$	4771.57	0.621924	0.310962	−	0.950422i	$-0.399349\pi$
$389$	4771.57	0.621924	0.310962	+	0.950422i	$0.399349\pi$
$390$	0	0
$391$	6882.00	0.890122
$392$	5918.24i	0.762541i
$393$	−4426.84	−0.568205
$394$	−6979.82	−0.892483
$395$	− 7647.50i	− 0.974145i
$396$	− 337.693i	− 0.0428528i
$397$	− 2291.22i	− 0.289655i	−0.989457	−	0.144827i	$-0.953737\pi$
$397$	− 2291.22i	− 0.289655i	0.989457	−	0.144827i	$-0.0462627\pi$
$398$	12747.6i	1.60548i
$399$	−3484.50	−0.437201
$400$	4247.07	0.530883
$401$	− 7534.63i	− 0.938308i	−0.883116	−	0.469154i	$-0.844559\pi$
$401$	− 7534.63i	− 0.938308i	0.883116	−	0.469154i	$-0.155441\pi$
$402$	1693.51	0.210111
$403$	0	0
$404$	1902.48	0.234287
$405$	1247.33i	0.153037i
$406$	−5108.76	−0.624491
$407$	−5407.18	−0.658535
$408$	− 6710.08i	− 0.814211i
$409$	− 1517.97i	− 0.183517i	−0.995781	−	0.0917587i	$-0.970751\pi$
$409$	− 1517.97i	− 0.183517i	0.995781	−	0.0917587i	$-0.0292488\pi$
$410$	− 11319.2i	− 1.36345i
$411$	5702.14i	0.684345i
$412$	−454.528	−0.0543519
$413$	2574.97	0.306794
$414$	− 1600.54i	− 0.190005i
$415$	−6044.56	−0.714978
$416$	0	0
$417$	−6980.28	−0.819726
$418$	4054.36i	0.474415i
$419$	8080.97	0.942199	0.471100	−	0.882080i	$-0.343857\pi$
$419$	8080.97	0.942199	0.471100	+	0.882080i	$0.343857\pi$
$420$	−1165.61	−0.135419
$421$	8073.13i	0.934585i	0.884103	+	0.467293i	$0.154771\pi$
$421$	8073.13i	0.934585i	−0.884103	+	0.467293i	$0.845229\pi$
$422$	2883.05i	0.332570i
$423$	2850.22i	0.327618i
$424$	4036.09i	0.462287i
$425$	10185.5	1.16252
$426$	−6671.80	−0.758803
$427$	− 9475.68i	− 1.07391i
$428$	457.470	0.0516651
$429$	0	0
$430$	−10724.5	−1.20275
$431$	2241.39i	0.250496i	0.992125	+	0.125248i	$0.0399726\pi$
$431$	2241.39i	0.250496i	−0.992125	+	0.125248i	$0.960027\pi$
$432$	−1022.65	−0.113894
$433$	−10237.9	−1.13626	−0.568130	−	0.822939i	$-0.692333\pi$
$433$	−10237.9	−1.13626	−0.568130	+	0.822939i	$0.692333\pi$
$434$	6765.39i	0.748270i
$435$	9923.94i	1.09383i
$436$	4418.88i	0.485380i
$437$	− 8685.52i	− 0.950766i
$438$	−3033.00	−0.330872
$439$	5416.69	0.588894	0.294447	−	0.955668i	$-0.404865\pi$
$439$	5416.69	0.588894	0.294447	+	0.955668i	$0.404865\pi$
$440$	5713.08i	0.619001i
$441$	2163.13	0.233574
$442$	0	0
$443$	−2537.44	−0.272138	−0.136069	−	0.990699i	$-0.543447\pi$
$443$	−2537.44	−0.272138	−0.136069	+	0.990699i	$0.543447\pi$
$444$	− 2681.16i	− 0.286582i
$445$	15089.5	1.60744
$446$	1608.48	0.170771
$447$	− 4112.77i	− 0.435184i
$448$	− 5640.43i	− 0.594833i
$449$	− 7790.61i	− 0.818846i	−0.912345	−	0.409423i	$-0.865730\pi$
$449$	− 7790.61i	− 0.818846i	0.912345	−	0.409423i	$-0.134270\pi$
$450$	− 2368.83i	− 0.248150i
$451$	4718.26	0.492626
$452$	1039.18	0.108139
$453$	− 3532.70i	− 0.366403i
$454$	−673.825	−0.0696568
$455$	0	0
$456$	−8468.55	−0.869685
$457$	6599.71i	0.675539i	0.941229	+	0.337770i	$0.109672\pi$
$457$	6599.71i	0.675539i	−0.941229	+	0.337770i	$0.890328\pi$
$458$	−5403.95	−0.551332
$459$	−2452.55	−0.249401
$460$	− 2905.43i	− 0.294492i
$461$	4482.57i	0.452872i	0.974026	+	0.226436i	$0.0727074\pi$
$461$	4482.57i	0.452872i	−0.974026	+	0.226436i	$0.927293\pi$
$462$	1074.96i	0.108250i
$463$	3805.86i	0.382016i	0.981589	+	0.191008i	$0.0611756\pi$
$463$	3805.86i	0.382016i	−0.981589	+	0.191008i	$0.938824\pi$
$464$	−8136.35	−0.814053
$465$	13142.0	1.31064
$466$	2278.31i	0.226482i
$467$	−14778.8	−1.46441	−0.732207	−	0.681082i	$-0.761510\pi$
$467$	−14778.8	−1.46441	−0.732207	+	0.681082i	$0.761510\pi$
$468$	0	0
$469$	2436.62	0.239899
$470$	− 11447.0i	− 1.12343i
$471$	4864.71	0.475911
$472$	6258.06	0.610277
$473$	− 4470.37i	− 0.434562i
$474$	− 3497.11i	− 0.338877i
$475$	− 12854.8i	− 1.24172i
$476$	− 2291.89i	− 0.220690i
$477$	1475.20	0.141603
$478$	11753.7	1.12469
$479$	− 11171.2i	− 1.06560i	−0.846240	−	0.532801i	$-0.821139\pi$
$479$	− 11171.2i	− 1.06560i	0.846240	−	0.532801i	$-0.178861\pi$
$480$	−4993.20	−0.474807
$481$	0	0
$482$	−1267.74	−0.119801
$483$	− 2302.85i	− 0.216942i
$484$	2749.29	0.258197
$485$	8521.18	0.797787
$486$	570.387i	0.0532372i
$487$	6046.57i	0.562621i	0.959617	+	0.281310i	$0.0907690\pi$
$487$	6046.57i	0.562621i	−0.959617	+	0.281310i	$0.909231\pi$
$488$	− 23029.2i	− 2.13623i
$489$	399.390i	0.0369347i
$490$	−8687.57	−0.800948
$491$	1035.04	0.0951338	0.0475669	−	0.998868i	$-0.484853\pi$
$491$	1035.04	0.0951338	0.0475669	+	0.998868i	$0.484853\pi$
$492$	2339.56i	0.214381i
$493$	−19512.9	−1.78259
$494$	0	0
$495$	2088.15	0.189606
$496$	10774.7i	0.975404i
$497$	−9599.37	−0.866379
$498$	−2764.10	−0.248720
$499$	− 11698.3i	− 1.04947i	−0.851265	−	0.524736i	$-0.824164\pi$
$499$	− 11698.3i	− 1.04947i	0.851265	−	0.524736i	$-0.175836\pi$
$500$	493.488i	0.0441389i
$501$	1472.17i	0.131281i
$502$	14289.7i	1.27048i
$503$	−13552.0	−1.20130	−0.600651	−	0.799511i	$-0.705092\pi$
$503$	−13552.0	−1.20130	−0.600651	+	0.799511i	$0.705092\pi$
$504$	−2245.32	−0.198441
$505$	11764.1i	1.03663i
$506$	−2679.46	−0.235408
$507$	0	0
$508$	4860.33	0.424492
$509$	5076.46i	0.442064i	0.975267	+	0.221032i	$0.0709425\pi$
$509$	5076.46i	0.442064i	−0.975267	+	0.221032i	$0.929058\pi$
$510$	9849.94	0.855221
$511$	−4363.86	−0.377781
$512$	− 11554.9i	− 0.997380i
$513$	3095.28i	0.266393i
$514$	11962.5i	1.02654i
$515$	− 2810.60i	− 0.240486i
$516$	2216.65	0.189113
$517$	4771.55	0.405904
$518$	8534.81i	0.723934i
$519$	−6024.40	−0.509522
$520$	0	0
$521$	−8493.73	−0.714236	−0.357118	−	0.934059i	$-0.616241\pi$
$521$	−8493.73	−0.714236	−0.357118	+	0.934059i	$0.616241\pi$
$522$	4538.10i	0.380512i
$523$	−1384.53	−0.115758	−0.0578789	−	0.998324i	$-0.518434\pi$
$523$	−1384.53	−0.115758	−0.0578789	+	0.998324i	$0.518434\pi$
$524$	−3674.75	−0.306359
$525$	− 3408.26i	− 0.283331i
$526$	7993.64i	0.662622i
$527$	25840.4i	2.13592i
$528$	1712.01i	0.141109i
$529$	−6426.89	−0.528223
$530$	−5924.71	−0.485571
$531$	− 2287.34i	− 0.186934i
$532$	−2892.51	−0.235726
$533$	0	0
$534$	6900.24	0.559181
$535$	2828.80i	0.228598i
$536$	5921.82	0.477208
$537$	6457.80	0.518947
$538$	− 6385.62i	− 0.511717i
$539$	− 3621.30i	− 0.289388i
$540$	1035.41i	0.0825131i
$541$	− 5026.83i	− 0.399483i	−0.979849	−	0.199742i	$-0.935990\pi$
$541$	− 5026.83i	− 0.399483i	0.979849	−	0.199742i	$-0.0640103\pi$
$542$	16323.6	1.29365
$543$	−2504.07	−0.197900
$544$	− 9817.87i	− 0.773782i
$545$	−27324.5	−2.14762
$546$	0	0
$547$	1540.36	0.120404	0.0602021	−	0.998186i	$-0.480825\pi$
$547$	1540.36	0.120404	0.0602021	+	0.998186i	$0.480825\pi$
$548$	4733.38i	0.368978i
$549$	−8417.23	−0.654351
$550$	−3965.65	−0.307447
$551$	24626.6i	1.90404i
$552$	− 5596.72i	− 0.431544i
$553$	− 5031.63i	− 0.386920i
$554$	15407.4i	1.18158i
$555$	16579.2	1.26801
$556$	−5794.38	−0.441972
$557$	8550.51i	0.650443i	0.945638	+	0.325221i	$0.105439\pi$
$557$	8550.51i	0.650443i	−0.945638	+	0.325221i	$0.894561\pi$
$558$	6009.68	0.455932
$559$	0	0
$560$	5909.35	0.445921
$561$	4105.81i	0.308998i
$562$	−1532.30	−0.115011
$563$	6569.35	0.491768	0.245884	−	0.969299i	$-0.420922\pi$
$563$	6569.35	0.491768	0.245884	+	0.969299i	$0.420922\pi$
$564$	2365.99i	0.176642i
$565$	6425.85i	0.478473i
$566$	− 14108.5i	− 1.04775i
$567$	820.671i	0.0607847i
$568$	−23329.8	−1.72341
$569$	−23766.1	−1.75102	−0.875508	−	0.483204i	$-0.839473\pi$
$569$	−23766.1	−1.75102	−0.875508	+	0.483204i	$0.839473\pi$
$570$	− 12431.3i	− 0.913488i
$571$	24971.7	1.83018	0.915091	−	0.403248i	$-0.132119\pi$
$571$	24971.7	1.83018	0.915091	+	0.403248i	$0.132119\pi$
$572$	0	0
$573$	13394.1	0.976524
$574$	− 7447.41i	− 0.541549i
$575$	8495.48	0.616150
$576$	−5010.38	−0.362441
$577$	− 16643.4i	− 1.20082i	−0.799693	−	0.600409i	$-0.795004\pi$
$577$	− 16643.4i	− 1.20082i	0.799693	−	0.600409i	$-0.204996\pi$
$578$	7835.28i	0.563849i
$579$	9899.53i	0.710553i
$580$	8237.93i	0.589761i
$581$	−3976.98	−0.283981
$582$	3896.63	0.277527
$583$	− 2469.64i	− 0.175441i
$584$	−10605.7	−0.751484
$585$	0	0
$586$	−4489.66	−0.316495
$587$	− 6720.67i	− 0.472558i	−0.971685	−	0.236279i	$-0.924072\pi$
$587$	− 6720.67i	− 0.472558i	0.971685	−	0.236279i	$-0.0759280\pi$
$588$	1795.63	0.125936
$589$	32612.3	2.28144
$590$	9186.41i	0.641014i
$591$	8920.77i	0.620899i
$592$	13592.8i	0.943681i
$593$	− 26349.8i	− 1.82471i	−0.409395	−	0.912357i	$-0.634260\pi$
$593$	− 26349.8i	− 1.82471i	0.409395	−	0.912357i	$-0.365740\pi$
$594$	954.884	0.0659586
$595$	14172.1	0.976466
$596$	− 3414.04i	− 0.234638i
$597$	16292.4	1.11693
$598$	0	0
$599$	−6136.81	−0.418603	−0.209302	−	0.977851i	$-0.567119\pi$
$599$	−6136.81	−0.418603	−0.209302	+	0.977851i	$0.567119\pi$
$600$	− 8283.25i	− 0.563604i
$601$	12493.7	0.847966	0.423983	−	0.905670i	$-0.360632\pi$
$601$	12493.7	0.847966	0.423983	+	0.905670i	$0.360632\pi$
$602$	−7056.14	−0.477719
$603$	− 2164.44i	− 0.146174i
$604$	− 2932.51i	− 0.197553i
$605$	17000.4i	1.14242i
$606$	5379.60i	0.360613i
$607$	18696.6	1.25020	0.625099	−	0.780545i	$-0.285059\pi$
$607$	18696.6	1.25020	0.625099	+	0.780545i	$0.285059\pi$
$608$	−12390.8	−0.826501
$609$	6529.40i	0.434458i
$610$	33805.3	2.24383
$611$	0	0
$612$	−2035.88	−0.134470
$613$	1238.64i	0.0816123i	0.999167	+	0.0408062i	$0.0129926\pi$
$613$	1238.64i	0.0816123i	−0.999167	+	0.0408062i	$0.987007\pi$
$614$	−4654.81	−0.305949
$615$	−14466.9	−0.948553
$616$	3758.89i	0.245860i
$617$	16549.5i	1.07983i	0.841718	+	0.539917i	$0.181544\pi$
$617$	16549.5i	1.07983i	−0.841718	+	0.539917i	$0.818456\pi$
$618$	− 1285.26i	− 0.0836579i
$619$	− 13945.4i	− 0.905513i	−0.891634	−	0.452756i	$-0.850441\pi$
$619$	− 13945.4i	− 0.905513i	0.891634	−	0.452756i	$-0.149559\pi$
$620$	10909.3	0.706656
$621$	−2045.62	−0.132186
$622$	− 4445.17i	− 0.286552i
$623$	9928.03	0.638456
$624$	0	0
$625$	−17068.0	−1.09235
$626$	10736.9i	0.685513i
$627$	5181.80	0.330050
$628$	4038.23	0.256597
$629$	32598.8i	2.06645i
$630$	− 3295.98i	− 0.208436i
$631$	− 15343.1i	− 0.967987i	−0.875072	−	0.483994i	$-0.839186\pi$
$631$	− 15343.1i	− 0.967987i	0.875072	−	0.483994i	$-0.160814\pi$
$632$	− 12228.6i	− 0.769663i
$633$	3684.76	0.231368
$634$	−17826.4	−1.11668
$635$	30054.2i	1.87821i
$636$	1224.58	0.0763484
$637$	0	0
$638$	7597.23	0.471438
$639$	8527.10i	0.527898i
$640$	6807.51	0.420454
$641$	−11067.7	−0.681979	−0.340989	−	0.940067i	$-0.610762\pi$
$641$	−11067.7	−0.681979	−0.340989	+	0.940067i	$0.610762\pi$
$642$	1293.58i	0.0795224i
$643$	− 25118.7i	− 1.54057i	−0.637701	−	0.770284i	$-0.720115\pi$
$643$	− 25118.7i	− 1.54057i	0.637701	−	0.770284i	$-0.279885\pi$
$644$	− 1911.61i	− 0.116969i
$645$	13706.8i	0.836751i
$646$	24442.9	1.48869
$647$	4447.03	0.270217	0.135109	−	0.990831i	$-0.456862\pi$
$647$	4447.03	0.270217	0.135109	+	0.990831i	$0.456862\pi$
$648$	1994.51i	0.120913i
$649$	−3829.23	−0.231603
$650$	0	0
$651$	8646.71	0.520570
$652$	331.537i	0.0199141i
$653$	−19426.2	−1.16418	−0.582088	−	0.813126i	$-0.697764\pi$
$653$	−19426.2	−1.16418	−0.582088	+	0.813126i	$0.697764\pi$
$654$	−12495.1	−0.747093
$655$	− 22723.1i	− 1.35552i
$656$	− 11861.0i	− 0.705934i
$657$	3876.41i	0.230187i
$658$	− 7531.52i	− 0.446215i
$659$	−14099.4	−0.833438	−0.416719	−	0.909035i	$-0.636820\pi$
$659$	−14099.4	−0.833438	−0.416719	+	0.909035i	$0.636820\pi$
$660$	1733.38	0.102230
$661$	− 2754.25i	− 0.162070i	−0.996711	−	0.0810348i	$-0.974177\pi$
$661$	− 2754.25i	− 0.162070i	0.996711	−	0.0810348i	$-0.0258225\pi$
$662$	4080.93	0.239592
$663$	0	0
$664$	−9665.44	−0.564898
$665$	− 17886.0i	− 1.04299i
$666$	7581.45	0.441104
$667$	−16275.3	−0.944800
$668$	1222.06i	0.0707828i
$669$	− 2055.77i	− 0.118805i
$670$	8692.83i	0.501244i
$671$	14091.3i	0.810712i
$672$	−3285.25	−0.188588
$673$	−11936.8	−0.683700	−0.341850	−	0.939754i	$-0.611054\pi$
$673$	−11936.8	−0.683700	−0.341850	+	0.939754i	$0.611054\pi$
$674$	6362.53i	0.363613i
$675$	−3027.55	−0.172638
$676$	0	0
$677$	−10255.2	−0.582186	−0.291093	−	0.956695i	$-0.594019\pi$
$677$	−10255.2	−0.582186	−0.291093	+	0.956695i	$0.594019\pi$
$678$	2938.46i	0.166447i
$679$	5606.46	0.316872
$680$	34443.0	1.94239
$681$	861.202i	0.0484601i
$682$	− 10060.8i	− 0.564880i
$683$	− 10605.7i	− 0.594169i	−0.954851	−	0.297085i	$-0.903986\pi$
$683$	− 10605.7i	− 0.594169i	0.954851	−	0.297085i	$-0.0960143\pi$
$684$	2569.41i	0.143631i
$685$	−29269.2	−1.63258
$686$	−13873.1	−0.772127
$687$	6906.69i	0.383561i
$688$	−11237.8	−0.622728
$689$	0	0
$690$	8215.60	0.453279
$691$	4660.14i	0.256556i	0.991738	+	0.128278i	$0.0409449\pi$
$691$	4660.14i	0.256556i	−0.991738	+	0.128278i	$0.959055\pi$
$692$	−5000.90	−0.274719
$693$	1373.88	0.0753096
$694$	− 5884.15i	− 0.321844i
$695$	− 35829.9i	− 1.95555i
$696$	15868.7i	0.864227i
$697$	− 28445.4i	− 1.54584i
$698$	17690.2	0.959287
$699$	2911.86	0.157563
$700$	− 2829.22i	− 0.152764i
$701$	24016.9	1.29402	0.647008	−	0.762483i	$-0.276020\pi$
$701$	24016.9	1.29402	0.647008	+	0.762483i	$0.276020\pi$
$702$	0	0
$703$	41141.7	2.20724
$704$	8387.88i	0.449048i
$705$	−14630.2	−0.781570
$706$	21108.7	1.12527
$707$	7740.15i	0.411737i
$708$	− 1898.74i	− 0.100789i
$709$	7252.56i	0.384169i	0.981378	+	0.192084i	$0.0615247\pi$
$709$	7252.56i	0.384169i	−0.981378	+	0.192084i	$0.938475\pi$
$710$	− 34246.5i	− 1.81021i
$711$	−4469.58	−0.235756
$712$	24128.6	1.27002
$713$	21552.9i	1.13207i
$714$	6480.71	0.339684
$715$	0	0
$716$	5360.66	0.279801
$717$	− 15022.2i	− 0.782447i
$718$	−10719.1	−0.557151
$719$	−10951.7	−0.568050	−0.284025	−	0.958817i	$-0.591670\pi$
$719$	−10951.7	−0.568050	−0.284025	+	0.958817i	$0.591670\pi$
$720$	− 5249.26i	− 0.271706i
$721$	− 1849.22i	− 0.0955182i
$722$	− 14748.6i	− 0.760231i
$723$	1620.28i	0.0833454i
$724$	−2078.65	−0.106702
$725$	−24087.7	−1.23393
$726$	7774.08i	0.397415i
$727$	27856.0	1.42108	0.710538	−	0.703658i	$-0.248451\pi$
$727$	27856.0	1.42108	0.710538	+	0.703658i	$0.248451\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 15568.4i	− 0.789334i
$731$	−26951.0	−1.36364
$732$	−6987.20	−0.352806
$733$	− 31213.8i	− 1.57286i	−0.617678	−	0.786431i	$-0.711927\pi$
$733$	− 31213.8i	− 1.57286i	0.617678	−	0.786431i	$-0.288073\pi$
$734$	− 15138.7i	− 0.761281i
$735$	11103.4i	0.557218i
$736$	− 8188.85i	− 0.410116i
$737$	−3623.49	−0.181103
$738$	−6615.52	−0.329974
$739$	− 14423.1i	− 0.717946i	−0.933348	−	0.358973i	$-0.883127\pi$
$739$	− 14423.1i	− 0.717946i	0.933348	−	0.358973i	$-0.116873\pi$
$740$	13762.5	0.683674
$741$	0	0
$742$	−3898.13	−0.192864
$743$	13469.7i	0.665079i	0.943089	+	0.332539i	$0.107905\pi$
$743$	13469.7i	0.665079i	−0.943089	+	0.332539i	$0.892095\pi$
$744$	21014.5	1.03552
$745$	21111.0	1.03818
$746$	17954.0i	0.881158i
$747$	3532.75i	0.173034i
$748$	3408.26i	0.166602i
$749$	1861.19i	0.0907965i
$750$	−1395.42	−0.0679381
$751$	32033.6	1.55649	0.778245	−	0.627961i	$-0.216110\pi$
$751$	32033.6	1.55649	0.778245	+	0.627961i	$0.216110\pi$
$752$	− 11994.9i	− 0.581661i
$753$	18263.4	0.883871
$754$	0	0
$755$	18133.4	0.874096
$756$	681.244i	0.0327733i
$757$	−26097.2	−1.25300	−0.626498	−	0.779423i	$-0.715512\pi$
$757$	−26097.2	−1.25300	−0.626498	+	0.779423i	$0.715512\pi$
$758$	24171.6	1.15825
$759$	3424.56i	0.163773i
$760$	− 43469.3i	− 2.07473i
$761$	− 18238.2i	− 0.868772i	−0.900727	−	0.434386i	$-0.856965\pi$
$761$	− 18238.2i	− 0.868772i	0.900727	−	0.434386i	$-0.143035\pi$
$762$	13743.4i	0.653375i
$763$	−17978.0	−0.853009
$764$	11118.6	0.526513
$765$	− 12589.0i	− 0.594976i
$766$	−24078.4	−1.13575
$767$	0	0
$768$	−10248.0	−0.481502
$769$	− 18817.0i	− 0.882390i	−0.897411	−	0.441195i	$-0.854555\pi$
$769$	− 18817.0i	− 0.882390i	0.897411	−	0.441195i	$-0.145445\pi$
$770$	−5517.79	−0.258243
$771$	15289.0	0.714164
$772$	8217.67i	0.383109i
$773$	− 13795.6i	− 0.641904i	−0.947095	−	0.320952i	$-0.895997\pi$
$773$	− 13795.6i	− 0.641904i	0.947095	−	0.320952i	$-0.104003\pi$
$774$	6267.95i	0.291081i
$775$	31898.7i	1.47850i
$776$	13625.6	0.630325
$777$	10908.2	0.503640
$778$	11200.2i	0.516126i
$779$	−35900.0	−1.65116
$780$	0	0
$781$	14275.2	0.654043
$782$	16153.9i	0.738699i
$783$	5800.05	0.264722
$784$	−9103.36	−0.414694
$785$	24970.7i	1.13534i
$786$	− 10391.0i	− 0.471545i
$787$	40545.4i	1.83645i	0.396055	+	0.918227i	$0.370379\pi$
$787$	40545.4i	1.83645i	−0.396055	+	0.918227i	$0.629621\pi$
$788$	7405.19i	0.334770i
$789$	10216.5	0.460985
$790$	17950.7	0.808429
$791$	4227.85i	0.190044i
$792$	3339.01	0.149806
$793$	0	0
$794$	5378.11	0.240380
$795$	7572.25i	0.337811i
$796$	13524.5	0.602214
$797$	31576.4	1.40338	0.701690	−	0.712482i	$-0.252429\pi$
$797$	31576.4	1.40338	0.701690	+	0.712482i	$0.252429\pi$
$798$	− 8179.07i	− 0.362827i
$799$	− 28766.7i	− 1.27371i
$800$	− 12119.7i	− 0.535619i
$801$	− 8819.05i	− 0.389021i
$802$	17685.8	0.778688
$803$	6489.50	0.285192
$804$	− 1796.72i	− 0.0788126i
$805$	11820.6	0.517541
$806$	0	0
$807$	−8161.33	−0.356001
$808$	18811.2i	0.819031i
$809$	31130.1	1.35287	0.676437	−	0.736501i	$-0.263523\pi$
$809$	31130.1	1.35287	0.676437	+	0.736501i	$0.263523\pi$
$810$	−2927.81	−0.127003
$811$	− 8733.71i	− 0.378153i	−0.981962	−	0.189076i	$-0.939451\pi$
$811$	− 8733.71i	− 0.378153i	0.981962	−	0.189076i	$-0.0605494\pi$
$812$	5420.10i	0.234247i
$813$	− 20862.8i	− 0.899988i
$814$	− 12692.1i	− 0.546509i
$815$	−2050.08	−0.0881119
$816$	10321.4	0.442794
$817$	34013.8i	1.45654i
$818$	3563.08	0.152298
$819$	0	0
$820$	−12009.0	−0.511431
$821$	− 36960.1i	− 1.57115i	−0.618765	−	0.785576i	$-0.712367\pi$
$821$	− 36960.1i	− 1.57115i	0.618765	−	0.785576i	$-0.287633\pi$
$822$	−13384.5	−0.567928
$823$	−20509.7	−0.868679	−0.434340	−	0.900749i	$-0.643018\pi$
$823$	−20509.7	−0.868679	−0.434340	+	0.900749i	$0.643018\pi$
$824$	− 4494.25i	− 0.190006i
$825$	5068.42i	0.213891i
$826$	6044.15i	0.254604i
$827$	11533.4i	0.484952i	0.970157	+	0.242476i	$0.0779596\pi$
$827$	11533.4i	0.484952i	−0.970157	+	0.242476i	$0.922040\pi$
$828$	−1698.08	−0.0712709
$829$	−34096.4	−1.42849	−0.714244	−	0.699897i	$-0.753229\pi$
$829$	−34096.4	−1.42849	−0.714244	+	0.699897i	$0.753229\pi$
$830$	− 14188.2i	− 0.593350i
$831$	19691.9	0.822026
$832$	0	0
$833$	−21832.1	−0.908087
$834$	− 16384.6i	− 0.680279i
$835$	−7556.69	−0.313186
$836$	4301.45	0.177953
$837$	− 7680.85i	− 0.317191i
$838$	18968.2i	0.781917i
$839$	1855.42i	0.0763484i	0.999271	+	0.0381742i	$0.0121542\pi$
$839$	1855.42i	0.0763484i	−0.999271	+	0.0381742i	$0.987846\pi$
$840$	− 11525.3i	− 0.473405i
$841$	21757.3	0.892094
$842$	−18949.8	−0.775599
$843$	1958.40i	0.0800129i
$844$	3058.75	0.124747
$845$	0	0
$846$	−6690.24	−0.271885
$847$	11185.3i	0.453757i
$848$	−6208.26	−0.251406
$849$	−18031.9	−0.728918
$850$	23908.1i	0.964755i
$851$	27189.9i	1.09525i
$852$	7078.40i	0.284627i
$853$	28668.7i	1.15076i	0.817886	+	0.575380i	$0.195146\pi$
$853$	28668.7i	1.15076i	−0.817886	+	0.575380i	$0.804854\pi$
$854$	22242.0	0.891224
$855$	−15888.1	−0.635512
$856$	4523.35i	0.180613i
$857$	−449.310	−0.0179091	−0.00895457	−	0.999960i	$-0.502850\pi$
$857$	−449.310	−0.0179091	−0.00895457	+	0.999960i	$0.502850\pi$
$858$	0	0
$859$	33466.9	1.32931	0.664654	−	0.747151i	$-0.268579\pi$
$859$	33466.9	1.32931	0.664654	+	0.747151i	$0.268579\pi$
$860$	11378.1i	0.451151i
$861$	−9518.39	−0.376755
$862$	−5261.14	−0.207883
$863$	− 25097.5i	− 0.989953i	−0.868906	−	0.494976i	$-0.835177\pi$
$863$	− 25097.5i	− 0.989953i	0.868906	−	0.494976i	$-0.164823\pi$
$864$	2918.28i	0.114910i
$865$	− 30923.4i	− 1.21552i
$866$	− 24031.0i	− 0.942965i
$867$	10014.1	0.392269
$868$	7177.69	0.280676
$869$	7482.53i	0.292091i
$870$	−23294.2	−0.907755
$871$	0	0
$872$	−43692.7	−1.69681
$873$	− 4980.21i	− 0.193075i
$874$	20387.3	0.789027
$875$	−2007.73	−0.0775698
$876$	3217.83i	0.124110i
$877$	27015.2i	1.04018i	0.854112	+	0.520090i	$0.174101\pi$
$877$	27015.2i	1.04018i	−0.854112	+	0.520090i	$0.825899\pi$
$878$	12714.4i	0.488715i
$879$	5738.15i	0.220185i
$880$	−8787.79	−0.336632
$881$	−48638.7	−1.86002	−0.930010	−	0.367534i	$-0.880202\pi$
$881$	−48638.7	−1.86002	−0.930010	+	0.367534i	$0.880202\pi$
$882$	5077.46i	0.193840i
$883$	25479.0	0.971048	0.485524	−	0.874223i	$-0.338629\pi$
$883$	25479.0	0.971048	0.485524	+	0.874223i	$0.338629\pi$
$884$	0	0
$885$	11741.0	0.445953
$886$	− 5956.05i	− 0.225843i
$887$	42359.7	1.60350	0.801748	−	0.597663i	$-0.203904\pi$
$887$	42359.7	1.60350	0.801748	+	0.597663i	$0.203904\pi$
$888$	26510.6	1.00184
$889$	19774.0i	0.746005i
$890$	35419.1i	1.33399i
$891$	− 1220.42i	− 0.0458873i
$892$	− 1706.51i	− 0.0640561i
$893$	−36305.4	−1.36049
$894$	9653.78	0.361153
$895$	33148.1i	1.23801i
$896$	4478.96	0.167000
$897$	0	0
$898$	18286.7	0.679548
$899$	− 61110.3i	− 2.26712i
$900$	−2513.19	−0.0930811
$901$	−14888.9	−0.550524
$902$	11075.0i	0.408823i
$903$	9018.31i	0.332348i
$904$	10275.1i	0.378038i
$905$	− 12853.5i	− 0.472114i
$906$	8292.19	0.304072
$907$	5314.91	0.194574	0.0972871	−	0.995256i	$-0.468984\pi$
$907$	5314.91	0.194574	0.0972871	+	0.995256i	$0.468984\pi$
$908$	714.890i	0.0261283i
$909$	6875.56	0.250878
$910$	0	0
$911$	2471.71	0.0898919	0.0449459	−	0.998989i	$-0.485688\pi$
$911$	2471.71	0.0898919	0.0449459	+	0.998989i	$0.485688\pi$
$912$	− 13026.2i	− 0.472962i
$913$	5914.17	0.214382
$914$	−15491.3	−0.560620
$915$	− 43205.8i	− 1.56103i
$916$	5733.29i	0.206805i
$917$	− 14950.5i	− 0.538396i
$918$	− 5756.80i	− 0.206975i
$919$	−9636.13	−0.345883	−0.172942	−	0.984932i	$-0.555327\pi$
$919$	−9636.13	−0.345883	−0.172942	+	0.984932i	$0.555327\pi$
$920$	28728.1	1.02950
$921$	5949.22i	0.212849i
$922$	−10521.8	−0.375832
$923$	0	0
$924$	1140.47	0.0406047
$925$	40241.5i	1.43041i
$926$	−8933.38	−0.317029
$927$	−1642.66	−0.0582007
$928$	23218.3i	0.821314i
$929$	− 53249.1i	− 1.88057i	−0.340394	−	0.940283i	$-0.610561\pi$
$929$	− 53249.1i	− 1.88057i	0.340394	−	0.940283i	$-0.389439\pi$
$930$	30847.8i	1.08768i
$931$	27553.5i	0.969955i
$932$	2417.16	0.0849534
$933$	−5681.28	−0.199354
$934$	− 34689.8i	− 1.21530i
$935$	−21075.3	−0.737149
$936$	0	0
$937$	−41122.6	−1.43374	−0.716871	−	0.697205i	$-0.754427\pi$
$937$	−41122.6	−1.43374	−0.716871	+	0.697205i	$0.754427\pi$
$938$	5719.40i	0.199088i
$939$	13722.6	0.476911
$940$	−12144.7	−0.421399
$941$	− 8005.71i	− 0.277342i	−0.990339	−	0.138671i	$-0.955717\pi$
$941$	− 8005.71i	− 0.277342i	0.990339	−	0.138671i	$-0.0442831\pi$
$942$	11418.8i	0.394952i
$943$	− 23725.7i	− 0.819315i
$944$	9626.07i	0.331888i
$945$	−4212.52	−0.145009
$946$	10493.2	0.360637
$947$	17468.6i	0.599424i	0.954030	+	0.299712i	$0.0968905\pi$
$947$	17468.6i	0.599424i	−0.954030	+	0.299712i	$0.903109\pi$
$948$	−3710.23	−0.127113
$949$	0	0
$950$	30173.6	1.03048
$951$	22783.6i	0.776875i
$952$	22661.6	0.771498
$953$	−30681.4	−1.04288	−0.521442	−	0.853287i	$-0.674606\pi$
$953$	−30681.4	−1.04288	−0.521442	+	0.853287i	$0.674606\pi$
$954$	3462.70i	0.117515i
$955$	68752.5i	2.32961i
$956$	− 12470.0i	− 0.421872i
$957$	− 9709.87i	− 0.327979i
$958$	26221.8	0.884328
$959$	−19257.5	−0.648444
$960$	− 25718.4i	− 0.864644i
$961$	−51135.6	−1.71648
$962$	0	0
$963$	1653.29	0.0553237
$964$	1345.00i	0.0449373i
$965$	−50814.5	−1.69511
$966$	5405.41	0.180037
$967$	15616.0i	0.519313i	0.965701	+	0.259656i	$0.0836093\pi$
$967$	15616.0i	0.519313i	−0.965701	+	0.259656i	$0.916391\pi$
$968$	27184.2i	0.902617i
$969$	− 31240.0i	− 1.03568i
$970$	20001.5i	0.662072i
$971$	7185.98	0.237496	0.118748	−	0.992924i	$-0.462112\pi$
$971$	7185.98	0.237496	0.118748	+	0.992924i	$0.462112\pi$
$972$	605.148	0.0199693
$973$	− 23574.1i	− 0.776723i
$974$	−14192.9	−0.466911
$975$	0	0
$976$	35423.2	1.16175
$977$	33165.3i	1.08603i	0.839723	+	0.543015i	$0.182717\pi$
$977$	33165.3i	1.08603i	−0.839723	+	0.543015i	$0.817283\pi$
$978$	−937.477	−0.0306516
$979$	−14764.0	−0.481980
$980$	9217.01i	0.300435i
$981$	15969.8i	0.519752i
$982$	2429.52i	0.0789502i
$983$	11658.7i	0.378287i	0.981949	+	0.189143i	$0.0605711\pi$
$983$	11658.7i	0.378287i	−0.981949	+	0.189143i	$0.939429\pi$
$984$	−23133.0	−0.749444
$985$	−45790.5	−1.48123
$986$	− 45802.1i	− 1.47935i
$987$	−9625.89	−0.310431
$988$	0	0
$989$	−22479.2	−0.722746
$990$	4901.45i	0.157352i
$991$	−1957.32	−0.0627409	−0.0313704	−	0.999508i	$-0.509987\pi$
$991$	−1957.32	−0.0627409	−0.0313704	+	0.999508i	$0.509987\pi$
$992$	30747.4	0.984104
$993$	− 5215.75i	− 0.166684i
$994$	− 22532.3i	− 0.718996i
$995$	83629.5i	2.66456i
$996$	2932.56i	0.0932948i
$997$	−11434.8	−0.363233	−0.181617	−	0.983369i	$-0.558133\pi$
$997$	−11434.8	−0.363233	−0.181617	+	0.983369i	$0.558133\pi$
$998$	27459.0	0.870942
$999$	− 9689.70i	− 0.306876i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.k.337.12		18
13.5	odd	4		507.4.a.p.1.6	yes	9
13.8	odd	4		507.4.a.o.1.4	✓	9
13.12	even	2	inner	507.4.b.k.337.7		18
39.5	even	4		1521.4.a.bf.1.4		9
39.8	even	4		1521.4.a.bi.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.4	✓	9	13.8	odd	4
507.4.a.p.1.6	yes	9	13.5	odd	4
507.4.b.k.337.7		18	13.12	even	2	inner
507.4.b.k.337.12		18	1.1	even	1	trivial
1521.4.a.bf.1.4		9	39.5	even	4
1521.4.a.bi.1.6		9	39.8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.34727i q^{2} +3.00000 q^{3} +2.49032 q^{4} +15.3991i q^{5} +7.04181i q^{6} +10.1317i q^{7} +24.6236i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.34727i q^{2} +3.00000 q^{3} +2.49032 q^{4} +15.3991i q^{5} +7.04181i q^{6} +10.1317i q^{7} +24.6236i q^{8} +9.00000 q^{9} -36.1458 q^{10} -15.0669i q^{11} +7.47096 q^{12} -23.7819 q^{14} +46.1972i q^{15} -37.8757 q^{16} -90.8352 q^{17} +21.1254i q^{18} +114.640i q^{19} +38.3486i q^{20} +30.3952i q^{21} +35.3661 q^{22} -75.7635 q^{23} +73.8709i q^{24} -112.132 q^{25} +27.0000 q^{27} +25.2313i q^{28} +214.817 q^{29} -108.437 q^{30} -284.476i q^{31} +108.084i q^{32} -45.2007i q^{33} -213.215i q^{34} -156.019 q^{35} +22.4129 q^{36} -358.878i q^{37} -269.091 q^{38} -379.181 q^{40} +313.154i q^{41} -71.3458 q^{42} +296.702 q^{43} -37.5214i q^{44} +138.592i q^{45} -177.837i q^{46} +316.691i q^{47} -113.627 q^{48} +240.348 q^{49} -263.203i q^{50} -272.506 q^{51} +163.911 q^{53} +63.3763i q^{54} +232.016 q^{55} -249.480 q^{56} +343.920i q^{57} +504.233i q^{58} -254.149i q^{59} +115.046i q^{60} -935.247 q^{61} +667.742 q^{62} +91.1856i q^{63} -556.709 q^{64} +106.098 q^{66} -240.494i q^{67} -226.209 q^{68} -227.291 q^{69} -366.220i q^{70} +947.455i q^{71} +221.613i q^{72} +430.712i q^{73} +842.384 q^{74} -336.395 q^{75} +285.490i q^{76} +152.654 q^{77} -496.620 q^{79} -583.251i q^{80} +81.0000 q^{81} -735.058 q^{82} +392.527i q^{83} +75.6938i q^{84} -1398.78i q^{85} +696.439i q^{86} +644.451 q^{87} +371.002 q^{88} -979.895i q^{89} -325.312 q^{90} -188.675 q^{92} -853.428i q^{93} -743.360 q^{94} -1765.35 q^{95} +324.253i q^{96} -553.356i q^{97} +564.162i q^{98} -135.602i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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