LMFDB - Embedded newform 117.4.b.e.64.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, 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("117.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.b (of order $2$, degree $1$, 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:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.1362828.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 23x^{2} + 48 $ x^4 + 23*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.3
Root			$1.52356i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.4.b.e.64.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.52356i q^{2} +5.67878 q^{4} -9.65841i q^{5} -22.3639i q^{7} +20.8404i q^{8} +14.7151 q^{10} -50.3050i q^{11} +(-39.7151 - 24.8940i) q^{13} +34.0727 q^{14} +13.6788 q^{16} +86.1454 q^{17} +116.880i q^{19} -54.8480i q^{20} +76.6424 q^{22} +72.0000 q^{23} +31.7151 q^{25} +(37.9273 - 60.5082i) q^{26} -127.000i q^{28} -14.1454 q^{29} -196.215i q^{31} +187.563i q^{32} +131.247i q^{34} -216.000 q^{35} +154.424i q^{37} -178.073 q^{38} +201.285 q^{40} +265.726i q^{41} -211.855 q^{43} -285.671i q^{44} +109.696i q^{46} +67.5535i q^{47} -157.145 q^{49} +48.3197i q^{50} +(-225.533 - 141.367i) q^{52} -686.581 q^{53} -485.866 q^{55} +466.073 q^{56} -21.5512i q^{58} -91.9304i q^{59} +329.006 q^{61} +298.945 q^{62} -176.333 q^{64} +(-240.436 + 383.585i) q^{65} +768.370i q^{67} +489.201 q^{68} -329.088i q^{70} -264.969i q^{71} -771.306i q^{73} -235.273 q^{74} +663.734i q^{76} -1125.02 q^{77} +1226.86 q^{79} -132.115i q^{80} -404.849 q^{82} +514.019i q^{83} -832.027i q^{85} -322.772i q^{86} +1048.38 q^{88} +527.889i q^{89} +(-556.727 + 888.186i) q^{91} +408.872 q^{92} -102.921 q^{94} +1128.87 q^{95} -74.2755i q^{97} -239.420i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{4} - 88 q^{10} - 12 q^{13} - 84 q^{14} + 18 q^{16} - 96 q^{17} + 380 q^{22} + 288 q^{23} - 20 q^{25} + 372 q^{26} + 384 q^{29} - 864 q^{35} - 492 q^{38} + 952 q^{40} - 1288 q^{43} - 188 q^{49}+ \cdots + 1872 q^{95}+O(q^{100}) $ 4 * q - 14 * q^4 - 88 * q^10 - 12 * q^13 - 84 * q^14 + 18 * q^16 - 96 * q^17 + 380 * q^22 + 288 * q^23 - 20 * q^25 + 372 * q^26 + 384 * q^29 - 864 * q^35 - 492 * q^38 + 952 * q^40 - 1288 * q^43 - 188 * q^49 - 1306 * q^52 - 984 * q^53 - 328 * q^55 + 1644 * q^56 + 288 * q^61 - 1668 * q^62 + 1314 * q^64 + 360 * q^65 + 4380 * q^68 - 3144 * q^74 - 1416 * q^77 + 4320 * q^79 - 3088 * q^82 + 1036 * q^88 - 24 * q^91 - 1008 * q^92 - 1660 * q^94 + 1872 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

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 $

$2$			1.52356i			0.538658i	0.963048	+	0.269329i	$0.0868019\pi$
$2$			1.52356i			0.538658i	−0.963048	+	0.269329i	$0.913198\pi$
$3$		0			0
$3$		0			0
$4$	5.67878			0.709847
$5$		−	9.65841i		−	0.863874i	−0.901904	−	0.431937i	$-0.857830\pi$
$5$		−	9.65841i		−	0.863874i	0.901904	−	0.431937i	$-0.142170\pi$
$6$		0			0
$7$		−	22.3639i		−	1.20754i	−0.797159	−	0.603769i	$-0.793665\pi$
$7$		−	22.3639i		−	1.20754i	0.797159	−	0.603769i	$-0.206335\pi$
$8$			20.8404i			0.921023i
$9$		0			0
$10$	14.7151			0.465333
$11$		−	50.3050i		−	1.37887i	−0.724349	−	0.689433i	$-0.757860\pi$
$11$		−	50.3050i		−	1.37887i	0.724349	−	0.689433i	$-0.242140\pi$
$12$		0			0
$13$	−39.7151	−	24.8940i	−0.847307	−	0.531103i
$13$	−39.7151	−	24.8940i	−0.847307	−	0.531103i
$14$	34.0727			0.650450
$15$		0			0
$16$	13.6788			0.213731
$17$	86.1454			1.22902			0.614509	−	0.788910i	$-0.289354\pi$
$17$	86.1454			1.22902			0.614509	+	0.788910i	$0.289354\pi$
$18$		0			0
$19$			116.880i			1.41127i	0.708578	+	0.705633i	$0.249337\pi$
$19$			116.880i			1.41127i	−0.708578	+	0.705633i	$0.750663\pi$
$20$		−	54.8480i		−	0.613219i
$21$		0			0
$22$	76.6424			0.742737
$23$	72.0000			0.652741			0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000			0.652741			0.326370	+	0.945242i	$0.394174\pi$
$24$		0			0
$25$	31.7151			0.253721
$26$	37.9273	−	60.5082i	0.286083	−	0.456409i
$27$		0			0
$28$		−	127.000i		−	0.857168i
$29$	−14.1454			−0.0905768			−0.0452884	−	0.998974i	$-0.514421\pi$
$29$	−14.1454			−0.0905768			−0.0452884	+	0.998974i	$0.514421\pi$
$30$		0			0
$31$		−	196.215i		−	1.13682i	−0.822747	−	0.568408i	$-0.807559\pi$
$31$		−	196.215i		−	1.13682i	0.822747	−	0.568408i	$-0.192441\pi$
$32$			187.563i			1.03615i
$33$		0			0
$34$			131.247i			0.662021i
$35$	−216.000			−1.04316
$36$		0			0
$37$			154.424i			0.686138i	0.939310	+	0.343069i	$0.111467\pi$
$37$			154.424i			0.686138i	−0.939310	+	0.343069i	$0.888533\pi$
$38$	−178.073			−0.760190
$39$		0			0
$40$	201.285			0.795648
$41$			265.726i			1.01218i	0.862480	+	0.506091i	$0.168910\pi$
$41$			265.726i			1.01218i	−0.862480	+	0.506091i	$0.831090\pi$
$42$		0			0
$43$	−211.855			−0.751338			−0.375669	−	0.926754i	$-0.622587\pi$
$43$	−211.855			−0.751338			−0.375669	+	0.926754i	$0.622587\pi$
$44$		−	285.671i		−	0.978785i
$45$		0			0
$46$			109.696i			0.351604i
$47$			67.5535i			0.209653i	0.994491	+	0.104827i	$0.0334287\pi$
$47$			67.5535i			0.209653i	−0.994491	+	0.104827i	$0.966571\pi$
$48$		0			0
$49$	−157.145			−0.458150
$50$			48.3197i			0.136669i
$51$		0			0
$52$	−225.533	−	141.367i	−0.601459	−	0.377002i
$53$	−686.581			−1.77942			−0.889710	−	0.456527i	$-0.849093\pi$
$53$	−686.581			−1.77942			−0.889710	+	0.456527i	$0.849093\pi$
$54$		0			0
$55$	−485.866			−1.19117
$56$	466.073			1.11217
$57$		0			0
$58$		−	21.5512i		−	0.0487899i
$59$		−	91.9304i		−	0.202853i	−0.994843	−	0.101426i	$-0.967659\pi$
$59$		−	91.9304i		−	0.202853i	0.994843	−	0.101426i	$-0.0323406\pi$
$60$		0			0
$61$	329.006			0.690572			0.345286	−	0.938498i	$-0.387782\pi$
$61$	329.006			0.690572			0.345286	+	0.938498i	$0.387782\pi$
$62$	298.945			0.612355
$63$		0			0
$64$	−176.333			−0.344400
$65$	−240.436	+	383.585i	−0.458807	+	0.731967i
$66$		0			0
$67$			768.370i			1.40106i	0.713621	+	0.700532i	$0.247054\pi$
$67$			768.370i			1.40106i	−0.713621	+	0.700532i	$0.752946\pi$
$68$	489.201			0.872416
$69$		0			0
$70$		−	329.088i		−	0.561908i
$71$		−	264.969i		−	0.442902i	−0.975171	−	0.221451i	$-0.928921\pi$
$71$		−	264.969i		−	0.442902i	0.975171	−	0.221451i	$-0.0710793\pi$
$72$		0			0
$73$		−	771.306i		−	1.23664i	−0.785927	−	0.618319i	$-0.787814\pi$
$73$		−	771.306i		−	1.23664i	0.785927	−	0.618319i	$-0.212186\pi$
$74$	−235.273			−0.369594
$75$		0			0
$76$			663.734i			1.00178i
$77$	−1125.02			−1.66503
$78$		0			0
$79$	1226.86			1.74725			0.873624	−	0.486602i	$-0.161764\pi$
$79$	1226.86			1.74725			0.873624	+	0.486602i	$0.161764\pi$
$80$		−	132.115i		−	0.184637i
$81$		0			0
$82$	−404.849			−0.545220
$83$			514.019i			0.679771i	0.940467	+	0.339885i	$0.110388\pi$
$83$			514.019i			0.679771i	−0.940467	+	0.339885i	$0.889612\pi$
$84$		0			0
$85$		−	832.027i		−	1.06172i
$86$		−	322.772i		−	0.404714i
$87$		0			0
$88$	1048.38			1.26997
$89$			527.889i			0.628720i	0.949304	+	0.314360i	$0.101790\pi$
$89$			527.889i			0.628720i	−0.949304	+	0.314360i	$0.898210\pi$
$90$		0			0
$91$	−556.727	+	888.186i	−0.641328	+	1.02316i
$92$	408.872			0.463346
$93$		0			0
$94$	−102.921			−0.112931
$95$	1128.87			1.21916
$96$		0			0
$97$		−	74.2755i		−	0.0777478i	−0.999244	−	0.0388739i	$-0.987623\pi$
$97$		−	74.2755i		−	0.0777478i	0.999244	−	0.0388739i	$-0.0123771\pi$
$98$		−	239.420i		−	0.246786i
$99$		0			0
$100$	180.103			0.180103
$101$	609.419			0.600390			0.300195	−	0.953878i	$-0.402948\pi$
$101$	609.419			0.600390			0.300195	+	0.953878i	$0.402948\pi$
$102$		0			0
$103$	32.4361			0.0310293			0.0155147	−	0.999880i	$-0.495061\pi$
$103$	32.4361			0.0310293			0.0155147	+	0.999880i	$0.495061\pi$
$104$	518.799	−	827.678i	0.489158	−	0.780389i
$105$		0			0
$106$		−	1046.04i		−	0.958498i
$107$	1725.45			1.55893			0.779467	−	0.626444i	$-0.215490\pi$
$107$	1725.45			1.55893			0.779467	+	0.626444i	$0.215490\pi$
$108$		0			0
$109$			1273.43i			1.11902i	0.828825	+	0.559508i	$0.189010\pi$
$109$			1273.43i			1.11902i	−0.828825	+	0.559508i	$0.810990\pi$
$110$		−	740.244i		−	0.641632i
$111$		0			0
$112$		−	305.911i		−	0.258088i
$113$	−1114.73			−0.928006			−0.464003	−	0.885834i	$-0.653587\pi$
$113$	−1114.73			−0.928006			−0.464003	+	0.885834i	$0.653587\pi$
$114$		0			0
$115$		−	695.406i		−	0.563886i
$116$	−80.3284			−0.0642957
$117$		0			0
$118$	140.061			0.109268
$119$		−	1926.55i		−	1.48409i
$120$		0			0
$121$	−1199.59			−0.901272
$122$			501.259i			0.371982i
$123$		0			0
$124$		−	1114.26i		−	0.806966i
$125$		−	1513.62i		−	1.08306i
$126$		0			0
$127$	−1174.01			−0.820289			−0.410144	−	0.912021i	$-0.634522\pi$
$127$	−1174.01			−0.820289			−0.410144	+	0.912021i	$0.634522\pi$
$128$			1231.85i			0.850637i
$129$		0			0
$130$	−584.413	−	366.318i	−0.394280	−	0.247140i
$131$	1445.16			0.963851			0.481925	−	0.876212i	$-0.339938\pi$
$131$	1445.16			0.963851			0.481925	+	0.876212i	$0.339938\pi$
$132$		0			0
$133$	2613.89			1.70416
$134$	−1170.65			−0.754695
$135$		0			0
$136$			1795.30i			1.13195i
$137$			508.793i			0.317293i	0.987335	+	0.158646i	$0.0507130\pi$
$137$			508.793i			0.317293i	−0.987335	+	0.158646i	$0.949287\pi$
$138$		0			0
$139$	−757.018			−0.461938			−0.230969	−	0.972961i	$-0.574190\pi$
$139$	−757.018			−0.461938			−0.230969	+	0.972961i	$0.574190\pi$
$140$	−1226.62			−0.740486
$141$		0			0
$142$	403.695			0.238573
$143$	−1252.29	+	1997.87i	−0.732320	+	1.16832i
$144$		0			0
$145$			136.622i			0.0782470i
$146$	1175.13			0.666125
$147$		0			0
$148$			876.939i			0.487054i
$149$			3247.79i			1.78570i	0.450352	+	0.892851i	$0.351298\pi$
$149$			3247.79i			1.78570i	−0.450352	+	0.892851i	$0.648702\pi$
$150$		0			0
$151$		−	795.296i		−	0.428611i	−0.976767	−	0.214305i	$-0.931251\pi$
$151$		−	795.296i		−	0.428611i	0.976767	−	0.214305i	$-0.0687488\pi$
$152$	−2435.82			−1.29981
$153$		0			0
$154$		−	1714.03i		−	0.896884i
$155$	−1895.13			−0.982067
$156$		0			0
$157$	−65.2732			−0.0331807			−0.0165903	−	0.999862i	$-0.505281\pi$
$157$	−65.2732			−0.0331807			−0.0165903	+	0.999862i	$0.505281\pi$
$158$			1869.19i			0.941169i
$159$		0			0
$160$	1811.56			0.895104
$161$		−	1610.20i		−	0.788210i
$162$		0			0
$163$			1855.39i			0.891568i	0.895140	+	0.445784i	$0.147075\pi$
$163$			1855.39i			0.891568i	−0.895140	+	0.445784i	$0.852925\pi$
$164$			1509.00i			0.718495i
$165$		0			0
$166$	−783.137			−0.366164
$167$		−	3532.54i		−	1.63686i	−0.574604	−	0.818432i	$-0.694844\pi$
$167$		−	3532.54i		−	1.63686i	0.574604	−	0.818432i	$-0.305156\pi$
$168$		0			0
$169$	957.581	+	1977.33i	0.435859	+	0.900015i
$170$	1267.64			0.571903
$171$		0			0
$172$	−1203.08			−0.533335
$173$	−3178.36			−1.39680			−0.698400	−	0.715708i	$-0.746104\pi$
$173$	−3178.36			−1.39680			−0.698400	+	0.715708i	$0.746104\pi$
$174$		0			0
$175$		−	709.275i		−	0.306378i
$176$		−	688.111i		−	0.294706i
$177$		0			0
$178$	−804.267			−0.338665
$179$	2741.16			1.14460			0.572302	−	0.820043i	$-0.306050\pi$
$179$	2741.16			1.14460			0.572302	+	0.820043i	$0.306050\pi$
$180$		0			0
$181$	−3871.09			−1.58970			−0.794850	−	0.606806i	$-0.792450\pi$
$181$	−3871.09			−1.58970			−0.794850	+	0.606806i	$0.792450\pi$
$182$	−1353.20	−	848.204i	−0.551131	−	0.345456i
$183$		0			0
$184$			1500.51i			0.601189i
$185$	1491.49			0.592737
$186$		0			0
$187$		−	4333.54i		−	1.69465i
$188$			383.621i			0.148822i
$189$		0			0
$190$			1719.90i			0.656708i
$191$	928.291			0.351669			0.175834	−	0.984420i	$-0.443738\pi$
$191$	928.291			0.351669			0.175834	+	0.984420i	$0.443738\pi$
$192$		0			0
$193$			2261.51i			0.843456i	0.906722	+	0.421728i	$0.138576\pi$
$193$			2261.51i			0.843456i	−0.906722	+	0.421728i	$0.861424\pi$
$194$	113.163			0.0418795
$195$		0			0
$196$	−892.394			−0.325216
$197$			2265.59i			0.819374i	0.912226	+	0.409687i	$0.134362\pi$
$197$			2265.59i			0.819374i	−0.912226	+	0.409687i	$0.865638\pi$
$198$		0			0
$199$	260.895			0.0929366			0.0464683	−	0.998920i	$-0.485203\pi$
$199$	260.895			0.0929366			0.0464683	+	0.998920i	$0.485203\pi$
$200$			660.955i			0.233683i
$201$		0			0
$202$			928.483i			0.323405i
$203$			316.346i			0.109375i
$204$		0			0
$205$	2566.49			0.874399
$206$			49.4181i			0.0167142i
$207$		0			0
$208$	−543.254	−	340.519i	−0.181096	−	0.113513i
$209$	5879.63			1.94595
$210$		0			0
$211$	5851.22			1.90907			0.954537	−	0.298092i	$-0.0963502\pi$
$211$	5851.22			1.90907			0.954537	+	0.298092i	$0.0963502\pi$
$212$	−3898.94			−1.26312
$213$		0			0
$214$			2628.82i			0.839732i
$215$			2046.18i			0.649062i
$216$		0			0
$217$	−4388.15			−1.37275
$218$	−1940.15			−0.602767
$219$		0			0
$220$	−2759.13			−0.845547
$221$	−3421.27	−	2144.50i	−1.04136	−	0.652736i
$222$		0			0
$223$		−	3463.60i		−	1.04009i	−0.854139	−	0.520045i	$-0.825915\pi$
$223$		−	3463.60i		−	1.04009i	0.854139	−	0.520045i	$-0.174085\pi$
$224$	4194.65			1.25119
$225$		0			0
$226$		−	1698.35i		−	0.499878i
$227$		−	5329.15i		−	1.55819i	−0.626908	−	0.779093i	$-0.715680\pi$
$227$		−	5329.15i		−	1.55819i	0.626908	−	0.779093i	$-0.284320\pi$
$228$		0			0
$229$		−	4773.95i		−	1.37761i	−0.724949	−	0.688803i	$-0.758137\pi$
$229$		−	4773.95i		−	1.37761i	0.724949	−	0.688803i	$-0.241863\pi$
$230$	1059.49			0.303742
$231$		0			0
$232$		−	294.795i		−	0.0834233i
$233$	−4813.78			−1.35348			−0.676741	−	0.736221i	$-0.736608\pi$
$233$	−4813.78			−1.35348			−0.676741	+	0.736221i	$0.736608\pi$
$234$		0			0
$235$	652.459			0.181114
$236$		−	522.052i		−	0.143995i
$237$		0			0
$238$	2935.20			0.799416
$239$		−	1683.19i		−	0.455549i	−0.973714	−	0.227775i	$-0.926855\pi$
$239$		−	1683.19i		−	0.455549i	0.973714	−	0.227775i	$-0.0731449\pi$
$240$		0			0
$241$			664.861i			0.177707i	0.996045	+	0.0888537i	$0.0283204\pi$
$241$			664.861i			0.177707i	−0.996045	+	0.0888537i	$0.971680\pi$
$242$		−	1827.65i		−	0.485477i
$243$		0			0
$244$	1868.35			0.490201
$245$			1517.77i			0.395784i
$246$		0			0
$247$	2909.60	−	4641.89i	0.749528	−	1.19578i
$248$	4089.20			1.04703
$249$		0			0
$250$	2306.08			0.583398
$251$	2142.04			0.538662			0.269331	−	0.963048i	$-0.413198\pi$
$251$	2142.04			0.538662			0.269331	+	0.963048i	$0.413198\pi$
$252$		0			0
$253$		−	3621.96i		−	0.900042i
$254$		−	1788.67i		−	0.441855i
$255$		0			0
$256$	−3287.46			−0.802603
$257$	3152.62			0.765194			0.382597	−	0.923915i	$-0.375030\pi$
$257$	3152.62			0.765194			0.382597	+	0.923915i	$0.375030\pi$
$258$		0			0
$259$	3453.52			0.828539
$260$	−1365.38	+	2178.29i	−0.325683	+	0.519585i
$261$		0			0
$262$			2201.79i			0.519186i
$263$	2167.71			0.508238			0.254119	−	0.967173i	$-0.418214\pi$
$263$	2167.71			0.508238			0.254119	+	0.967173i	$0.418214\pi$
$264$		0			0
$265$			6631.28i			1.53719i
$266$			3982.40i			0.917958i
$267$		0			0
$268$			4363.40i			0.994542i
$269$	3248.98			0.736409			0.368204	−	0.929745i	$-0.379973\pi$
$269$	3248.98			0.736409			0.368204	+	0.929745i	$0.379973\pi$
$270$		0			0
$271$			4897.70i			1.09784i	0.835876	+	0.548919i	$0.184960\pi$
$271$			4897.70i			1.09784i	−0.835876	+	0.548919i	$0.815040\pi$
$272$	1178.36			0.262679
$273$		0			0
$274$	−775.175			−0.170912
$275$		−	1595.43i		−	0.349847i
$276$		0			0
$277$	−2900.33			−0.629111			−0.314555	−	0.949239i	$-0.601855\pi$
$277$	−2900.33			−0.629111			−0.314555	+	0.949239i	$0.601855\pi$
$278$		−	1153.36i		−	0.248827i
$279$		0			0
$280$		−	4501.52i		−	0.960776i
$281$			4396.53i			0.933364i	0.884425	+	0.466682i	$0.154551\pi$
$281$			4396.53i			0.933364i	−0.884425	+	0.466682i	$0.845449\pi$
$282$		0			0
$283$	559.151			0.117449			0.0587245	−	0.998274i	$-0.481297\pi$
$283$	559.151			0.117449			0.0587245	+	0.998274i	$0.481297\pi$
$284$		−	1504.70i		−	0.314393i
$285$		0			0
$286$	−3043.86	−	1907.93i	−0.629327	−	0.394470i
$287$	5942.69			1.22225
$288$		0			0
$289$	2508.02			0.510487
$290$	−208.151			−0.0421484
$291$		0			0
$292$		−	4380.08i		−	0.877825i
$293$		−	6918.65i		−	1.37950i	−0.724050	−	0.689748i	$-0.757722\pi$
$293$		−	6918.65i		−	1.37950i	0.724050	−	0.689748i	$-0.242278\pi$
$294$		0			0
$295$	−887.901			−0.175239
$296$	−3218.25			−0.631949
$297$		0			0
$298$	−4948.19			−0.961883
$299$	−2859.49	−	1792.37i	−0.553072	−	0.346673i
$300$		0			0
$301$			4737.90i			0.907270i
$302$	1211.68			0.230875
$303$		0			0
$304$			1598.77i			0.301631i
$305$		−	3177.67i		−	0.596567i
$306$		0			0
$307$			8980.94i			1.66961i	0.550548	+	0.834803i	$0.314419\pi$
$307$			8980.94i			1.66961i	−0.550548	+	0.834803i	$0.685581\pi$
$308$	−6388.73			−1.18192
$309$		0			0
$310$		−	2887.33i		−	0.528998i
$311$	−7943.13			−1.44827			−0.724137	−	0.689656i	$-0.757762\pi$
$311$	−7943.13			−1.44827			−0.724137	+	0.689656i	$0.757762\pi$
$312$		0			0
$313$	−5059.57			−0.913686			−0.456843	−	0.889547i	$-0.651020\pi$
$313$	−5059.57			−0.913686			−0.456843	+	0.889547i	$0.651020\pi$
$314$		−	99.4473i		−	0.0178730i
$315$		0			0
$316$	6967.07			1.24028
$317$			8702.12i			1.54183i	0.636939	+	0.770914i	$0.280200\pi$
$317$			8702.12i			1.54183i	−0.636939	+	0.770914i	$0.719800\pi$
$318$		0			0
$319$			711.582i			0.124893i
$320$			1703.10i			0.297519i
$321$		0			0
$322$	2453.23			0.424576
$323$			10068.6i			1.73447i
$324$		0			0
$325$	−1259.57	−	789.515i	−0.214980	−	0.134752i
$326$	−2826.79			−0.480250
$327$		0			0
$328$	−5537.84			−0.932244
$329$	1510.76			0.253164
$330$		0			0
$331$		−	1737.65i		−	0.288549i	−0.989538	−	0.144275i	$-0.953915\pi$
$331$		−	1737.65i		−	0.288549i	0.989538	−	0.144275i	$-0.0460848\pi$
$332$			2919.00i			0.482533i
$333$		0			0
$334$	5382.02			0.881710
$335$	7421.23			1.21034
$336$		0			0
$337$	−2917.47			−0.471586			−0.235793	−	0.971803i	$-0.575769\pi$
$337$	−2917.47			−0.471586			−0.235793	+	0.971803i	$0.575769\pi$
$338$	−3012.58	+	1458.93i	−0.484800	+	0.234779i
$339$		0			0
$340$		−	4724.90i		−	0.753658i
$341$	−9870.61			−1.56752
$342$		0			0
$343$		−	4156.44i		−	0.654305i
$344$		−	4415.13i		−	0.692000i
$345$		0			0
$346$		−	4842.41i		−	0.752397i
$347$	−5081.23			−0.786095			−0.393047	−	0.919518i	$-0.628579\pi$
$347$	−5081.23			−0.786095			−0.393047	+	0.919518i	$0.628579\pi$
$348$		0			0
$349$		−	4266.14i		−	0.654330i	−0.944967	−	0.327165i	$-0.893907\pi$
$349$		−	4266.14i		−	0.654330i	0.944967	−	0.327165i	$-0.106093\pi$
$350$	1080.62			0.165033
$351$		0			0
$352$	9435.38			1.42871
$353$		−	3264.49i		−	0.492213i	−0.969243	−	0.246106i	$-0.920849\pi$
$353$		−	3264.49i		−	0.492213i	0.969243	−	0.246106i	$-0.0791513\pi$
$354$		0			0
$355$	−2559.18			−0.382612
$356$			2997.76i			0.446295i
$357$		0			0
$358$			4176.31i			0.616550i
$359$		−	4416.98i		−	0.649357i	−0.945824	−	0.324679i	$-0.894744\pi$
$359$		−	4416.98i		−	0.649357i	0.945824	−	0.324679i	$-0.105256\pi$
$360$		0			0
$361$	−6801.87			−0.991670
$362$		−	5897.82i		−	0.856305i
$363$		0			0
$364$	−3161.53	+	5043.81i	−0.455245	+	0.726285i
$365$	−7449.59			−1.06830
$366$		0			0
$367$	−1740.22			−0.247516			−0.123758	−	0.992312i	$-0.539495\pi$
$367$	−1740.22			−0.247516			−0.123758	+	0.992312i	$0.539495\pi$
$368$	984.872			0.139511
$369$		0			0
$370$			2272.37i			0.319283i
$371$			15354.7i			2.14872i
$372$		0			0
$373$	1176.28			0.163285			0.0816427	−	0.996662i	$-0.473983\pi$
$373$	1176.28			0.163285			0.0816427	+	0.996662i	$0.473983\pi$
$374$	6602.39			0.912838
$375$		0			0
$376$	−1407.84			−0.193095
$377$	561.785	+	352.134i	0.0767464	+	0.0481056i
$378$		0			0
$379$		−	7135.54i		−	0.967092i	−0.875319	−	0.483546i	$-0.839349\pi$
$379$		−	7135.54i		−	0.967092i	0.875319	−	0.483546i	$-0.160651\pi$
$380$	6410.62			0.865415
$381$		0			0
$382$			1414.30i			0.189429i
$383$		−	1942.87i		−	0.259207i	−0.991566	−	0.129603i	$-0.958630\pi$
$383$		−	1942.87i		−	0.259207i	0.991566	−	0.129603i	$-0.0413704\pi$
$384$		0			0
$385$			10865.9i			1.43838i
$386$	−3445.53			−0.454334
$387$		0			0
$388$		−	421.794i		−	0.0551891i
$389$	−7545.92			−0.983531			−0.491766	−	0.870728i	$-0.663648\pi$
$389$	−7545.92			−0.983531			−0.491766	+	0.870728i	$0.663648\pi$
$390$		0			0
$391$	6202.47			0.802231
$392$		−	3274.97i		−	0.421967i
$393$		0			0
$394$	−3451.75			−0.441362
$395$		−	11849.5i		−	1.50940i
$396$		0			0
$397$			415.922i			0.0525806i	0.999654	+	0.0262903i	$0.00836943\pi$
$397$			415.922i			0.0525806i	−0.999654	+	0.0262903i	$0.991631\pi$
$398$			397.489i			0.0500611i
$399$		0			0
$400$	433.824			0.0542280
$401$			958.178i			0.119324i	0.998219	+	0.0596622i	$0.0190024\pi$
$401$			958.178i			0.119324i	−0.998219	+	0.0596622i	$0.980998\pi$
$402$		0			0
$403$	−4884.58	+	7792.71i	−0.603767	+	0.963233i
$404$	3460.75			0.426185
$405$		0			0
$406$	−481.970			−0.0589157
$407$	7768.29			0.946093
$408$		0			0
$409$		−	4284.83i		−	0.518022i	−0.965874	−	0.259011i	$-0.916603\pi$
$409$		−	4284.83i		−	0.518022i	0.965874	−	0.259011i	$-0.0833966\pi$
$410$			3910.20i			0.471002i
$411$		0			0
$412$	184.197			0.0220261
$413$	−2055.92			−0.244953
$414$		0			0
$415$	4964.61			0.587236
$416$	4669.20	−	7449.10i	0.550303	−	0.877938i
$417$		0			0
$418$			8957.95i			1.04820i
$419$	9949.01			1.16000			0.580001	−	0.814616i	$-0.303052\pi$
$419$	9949.01			1.16000			0.580001	+	0.814616i	$0.303052\pi$
$420$		0			0
$421$			377.250i			0.0436724i	0.999762	+	0.0218362i	$0.00695122\pi$
$421$			377.250i			0.0436724i	−0.999762	+	0.0218362i	$0.993049\pi$
$422$			8914.66i			1.02834i
$423$		0			0
$424$		−	14308.6i		−	1.63889i
$425$	2732.11			0.311828
$426$		0			0
$427$		−	7357.86i		−	0.833892i
$428$	9798.47			1.10661
$429$		0			0
$430$	−3117.47			−0.349622
$431$			2437.13i			0.272373i	0.990683	+	0.136186i	$0.0434846\pi$
$431$			2437.13i			0.272373i	−0.990683	+	0.136186i	$0.956515\pi$
$432$		0			0
$433$	−11215.1			−1.24471			−0.622357	−	0.782733i	$-0.713825\pi$
$433$	−11215.1			−1.24471			−0.622357	+	0.782733i	$0.713825\pi$
$434$		−	6685.58i		−	0.739443i
$435$		0			0
$436$			7231.55i			0.794331i
$437$			8415.34i			0.921191i
$438$		0			0
$439$	1835.19			0.199519			0.0997596	−	0.995012i	$-0.468193\pi$
$439$	1835.19			0.199519			0.0997596	+	0.995012i	$0.468193\pi$
$440$		−	10125.6i		−	1.09709i
$441$		0			0
$442$	3267.26	−	5212.50i	0.351601	−	0.560935i
$443$	−11610.1			−1.24518			−0.622588	−	0.782550i	$-0.713919\pi$
$443$	−11610.1			−1.24518			−0.622588	+	0.782550i	$0.713919\pi$
$444$		0			0
$445$	5098.56			0.543135
$446$	5276.99			0.560253
$447$		0			0
$448$			3943.50i			0.415877i
$449$		−	14087.0i		−	1.48064i	−0.672255	−	0.740319i	$-0.734674\pi$
$449$		−	14087.0i		−	1.48064i	0.672255	−	0.740319i	$-0.265326\pi$
$450$		0			0
$451$	13367.4			1.39566
$452$	−6330.29			−0.658743
$453$		0			0
$454$	8119.26			0.839330
$455$	8578.47	+	5377.10i	0.883878	+	0.554027i
$456$		0			0
$457$			2375.01i			0.243103i	0.992585	+	0.121552i	$0.0387870\pi$
$457$			2375.01i			0.243103i	−0.992585	+	0.121552i	$0.961213\pi$
$458$	7273.38			0.742058
$459$		0			0
$460$		−	3949.05i		−	0.400273i
$461$			6372.06i			0.643766i	0.946779	+	0.321883i	$0.104316\pi$
$461$			6372.06i			0.643766i	−0.946779	+	0.321883i	$0.895684\pi$
$462$		0			0
$463$			63.4732i			0.00637117i	0.999995	+	0.00318558i	$0.00101400\pi$
$463$			63.4732i			0.00637117i	−0.999995	+	0.00318558i	$0.998986\pi$
$464$	−193.491			−0.0193591
$465$		0			0
$466$		−	7334.06i		−	0.729064i
$467$	−7855.78			−0.778420			−0.389210	−	0.921149i	$-0.627252\pi$
$467$	−7855.78			−0.778420			−0.389210	+	0.921149i	$0.627252\pi$
$468$		0			0
$469$	17183.8			1.69184
$470$			994.058i			0.0975584i
$471$		0			0
$472$	1915.86			0.186832
$473$			10657.3i			1.03599i
$474$		0			0
$475$			3706.85i			0.358068i
$476$		−	10940.4i		−	1.05348i
$477$		0			0
$478$	2564.43			0.245385
$479$		−	13033.3i		−	1.24323i	−0.783324	−	0.621613i	$-0.786477\pi$
$479$		−	13033.3i		−	1.24323i	0.783324	−	0.621613i	$-0.213523\pi$
$480$		0			0
$481$	3844.22	−	6132.96i	0.364410	−	0.581370i
$482$	−1012.95			−0.0957235
$483$		0			0
$484$	−6812.23			−0.639766
$485$	−717.384			−0.0671643
$486$		0			0
$487$		−	69.8976i		−	0.00650383i	−0.999995	−	0.00325191i	$-0.998965\pi$
$487$		−	69.8976i		−	0.00650383i	0.999995	−	0.00325191i	$-0.00103512\pi$
$488$			6856.60i			0.636033i
$489$		0			0
$490$	−2312.41			−0.213192
$491$	−2625.66			−0.241333			−0.120667	−	0.992693i	$-0.538503\pi$
$491$	−2625.66			−0.241333			−0.120667	+	0.992693i	$0.538503\pi$
$492$		0			0
$493$	−1218.56			−0.111321
$494$	7072.18	+	4432.93i	0.644114	+	0.403739i
$495$		0			0
$496$		−	2683.99i		−	0.242973i
$497$	−5925.75			−0.534821
$498$		0			0
$499$			7631.34i			0.684621i	0.939587	+	0.342310i	$0.111209\pi$
$499$			7631.34i			0.684621i	−0.939587	+	0.342310i	$0.888791\pi$
$500$		−	8595.51i		−	0.768806i
$501$		0			0
$502$			3263.51i			0.290154i
$503$	4320.14			0.382953			0.191477	−	0.981497i	$-0.438672\pi$
$503$	4320.14			0.382953			0.191477	+	0.981497i	$0.438672\pi$
$504$		0			0
$505$		−	5886.01i		−	0.518662i
$506$	5518.26			0.484815
$507$		0			0
$508$	−6666.95			−0.582280
$509$			12450.7i			1.08422i	0.840308	+	0.542109i	$0.182374\pi$
$509$			12450.7i			1.08422i	−0.840308	+	0.542109i	$0.817626\pi$
$510$		0			0
$511$	−17249.4			−1.49329
$512$			4846.21i			0.418309i
$513$		0			0
$514$			4803.18i			0.412178i
$515$		−	313.281i		−	0.0268054i
$516$		0			0
$517$	3398.28			0.289083
$518$			5261.63i			0.446299i
$519$		0			0
$520$	−7994.05	−	5010.78i	−0.674158	−	0.422571i
$521$	−14373.1			−1.20863			−0.604314	−	0.796746i	$-0.706553\pi$
$521$	−14373.1			−1.20863			−0.604314	+	0.796746i	$0.706553\pi$
$522$		0			0
$523$	−16946.7			−1.41688			−0.708439	−	0.705772i	$-0.750600\pi$
$523$	−16946.7			−1.41688			−0.708439	+	0.705772i	$0.750600\pi$
$524$	8206.76			0.684187
$525$		0			0
$526$			3302.62i			0.273767i
$527$		−	16903.0i		−	1.39717i
$528$		0			0
$529$	−6983.00			−0.573929
$530$	−10103.1			−0.828022
$531$		0			0
$532$	14843.7			1.20969
$533$	6614.98	−	10553.4i	0.537574	−	0.857630i
$534$		0			0
$535$		−	16665.1i		−	1.34672i
$536$	−16013.1			−1.29041
$537$		0			0
$538$			4950.00i			0.396673i
$539$			7905.20i			0.631727i
$540$		0			0
$541$		−	815.667i		−	0.0648212i	−0.999475	−	0.0324106i	$-0.989682\pi$
$541$		−	815.667i		−	0.0648212i	0.999475	−	0.0324106i	$-0.0103184\pi$
$542$	−7461.92			−0.591359
$543$		0			0
$544$			16157.7i			1.27345i
$545$	12299.3			0.966689
$546$		0			0
$547$	−17971.4			−1.40476			−0.702378	−	0.711804i	$-0.747878\pi$
$547$	−17971.4			−1.40476			−0.702378	+	0.711804i	$0.747878\pi$
$548$			2889.32i			0.225230i
$549$		0			0
$550$	2430.72			0.188448
$551$		−	1653.31i		−	0.127828i
$552$		0			0
$553$		−	27437.4i		−	2.10987i
$554$		−	4418.81i		−	0.338876i
$555$		0			0
$556$	−4298.94			−0.327906
$557$			6760.83i			0.514301i	0.966371	+	0.257151i	$0.0827836\pi$
$557$			6760.83i			0.514301i	−0.966371	+	0.257151i	$0.917216\pi$
$558$		0			0
$559$	8413.83	+	5273.90i	0.636614	+	0.399038i
$560$	−2954.62			−0.222956
$561$		0			0
$562$	−6698.36			−0.502764
$563$	−12962.7			−0.970359			−0.485179	−	0.874415i	$-0.661246\pi$
$563$	−12962.7			−0.970359			−0.485179	+	0.874415i	$0.661246\pi$
$564$		0			0
$565$			10766.5i			0.801681i
$566$			851.898i			0.0632649i
$567$		0			0
$568$	5522.05			0.407923
$569$	−164.757			−0.0121388			−0.00606938	−	0.999982i	$-0.501932\pi$
$569$	−164.757			−0.0121388			−0.00606938	+	0.999982i	$0.501932\pi$
$570$		0			0
$571$	−5216.55			−0.382322			−0.191161	−	0.981559i	$-0.561225\pi$
$571$	−5216.55			−0.382322			−0.191161	+	0.981559i	$0.561225\pi$
$572$	−7111.48	+	11345.5i	−0.519836	+	0.829331i
$573$		0			0
$574$			9054.01i			0.658375i
$575$	2283.49			0.165614
$576$		0			0
$577$			12753.3i			0.920151i	0.887880	+	0.460076i	$0.152178\pi$
$577$			12753.3i			0.920151i	−0.887880	+	0.460076i	$0.847822\pi$
$578$			3821.11i			0.274978i
$579$		0			0
$580$			775.844i			0.0555434i
$581$	11495.5			0.820849
$582$		0			0
$583$			34538.5i			2.45358i
$584$	16074.3			1.13897
$585$		0			0
$586$	10540.9			0.743076
$587$		−	1575.38i		−	0.110771i	−0.998465	−	0.0553857i	$-0.982361\pi$
$587$		−	1575.38i		−	0.110771i	0.998465	−	0.0553857i	$-0.0176388\pi$
$588$		0			0
$589$	22933.6			1.60435
$590$		−	1352.77i		−	0.0943941i
$591$		0			0
$592$			2112.33i			0.146649i
$593$			3845.95i			0.266331i	0.991094	+	0.133165i	$0.0425142\pi$
$593$			3845.95i			0.266331i	−0.991094	+	0.133165i	$0.957486\pi$
$594$		0			0
$595$	−18607.4			−1.28207
$596$			18443.5i			1.26758i
$597$		0			0
$598$	2730.77	−	4356.59i	0.186738	−	0.297917i
$599$	−6107.20			−0.416583			−0.208292	−	0.978067i	$-0.566790\pi$
$599$	−6107.20			−0.416583			−0.208292	+	0.978067i	$0.566790\pi$
$600$		0			0
$601$	9638.90			0.654208			0.327104	−	0.944988i	$-0.393927\pi$
$601$	9638.90			0.654208			0.327104	+	0.944988i	$0.393927\pi$
$602$	−7218.46			−0.488708
$603$		0			0
$604$		−	4516.31i		−	0.304248i
$605$			11586.2i			0.778586i
$606$		0			0
$607$	11821.7			0.790489			0.395244	−	0.918576i	$-0.370660\pi$
$607$	11821.7			0.790489			0.395244	+	0.918576i	$0.370660\pi$
$608$	−21922.4			−1.46228
$609$		0			0
$610$	4841.36			0.321346
$611$	1681.67	−	2682.90i	0.111347	−	0.177640i
$612$		0			0
$613$			8107.86i			0.534214i	0.963667	+	0.267107i	$0.0860677\pi$
$613$			8107.86i			0.534214i	−0.963667	+	0.267107i	$0.913932\pi$
$614$	−13683.0			−0.899347
$615$		0			0
$616$		−	23445.8i		−	1.53354i
$617$		−	27647.6i		−	1.80397i	−0.431768	−	0.901985i	$-0.642110\pi$
$617$		−	27647.6i		−	1.80397i	0.431768	−	0.901985i	$-0.357890\pi$
$618$		0			0
$619$		−	29181.9i		−	1.89486i	−0.319956	−	0.947432i	$-0.603668\pi$
$619$		−	29181.9i		−	1.89486i	0.319956	−	0.947432i	$-0.396332\pi$
$620$	−10762.0			−0.697118
$621$		0			0
$622$		−	12101.8i		−	0.780125i
$623$	11805.7			0.759204
$624$		0			0
$625$	−10654.8			−0.681905
$626$		−	7708.53i		−	0.492165i
$627$		0			0
$628$	−370.672			−0.0235532
$629$			13302.9i			0.843277i
$630$		0			0
$631$			2209.34i			0.139386i	0.997569	+	0.0696928i	$0.0222019\pi$
$631$			2209.34i			0.139386i	−0.997569	+	0.0696928i	$0.977798\pi$
$632$			25568.2i			1.60926i
$633$		0			0
$634$	−13258.2			−0.830518
$635$			11339.1i			0.708627i
$636$		0			0
$637$	6241.05	+	3911.97i	0.388194	+	0.243325i
$638$	−1084.13			−0.0672748
$639$		0			0
$640$	11897.8			0.734844
$641$	−18256.0			−1.12491			−0.562455	−	0.826828i	$-0.690143\pi$
$641$	−18256.0			−1.12491			−0.562455	+	0.826828i	$0.690143\pi$
$642$		0			0
$643$			1281.61i			0.0786033i	0.999227	+	0.0393016i	$0.0125133\pi$
$643$			1281.61i			0.0786033i	−0.999227	+	0.0393016i	$0.987487\pi$
$644$		−	9143.99i		−	0.559509i
$645$		0			0
$646$	−15340.1			−0.934287
$647$	16393.2			0.996107			0.498054	−	0.867146i	$-0.334048\pi$
$647$	16393.2			0.996107			0.498054	+	0.867146i	$0.334048\pi$
$648$		0			0
$649$	−4624.56			−0.279707
$650$	1202.87	−	1919.02i	0.0725853	−	0.115800i
$651$		0			0
$652$			10536.4i			0.632878i
$653$	−16759.2			−1.00434			−0.502172	−	0.864768i	$-0.667466\pi$
$653$	−16759.2			−1.00434			−0.502172	+	0.864768i	$0.667466\pi$
$654$		0			0
$655$		−	13958.0i		−	0.832646i
$656$			3634.81i			0.216335i
$657$		0			0
$658$			2301.73i			0.136369i
$659$	29659.3			1.75320			0.876601	−	0.481217i	$-0.159805\pi$
$659$	29659.3			1.75320			0.876601	+	0.481217i	$0.159805\pi$
$660$		0			0
$661$			10386.0i			0.611147i	0.952169	+	0.305573i	$0.0988481\pi$
$661$			10386.0i			0.611147i	−0.952169	+	0.305573i	$0.901152\pi$
$662$	2647.40			0.155429
$663$		0			0
$664$	−10712.4			−0.626084
$665$		−	25246.0i		−	1.47218i
$666$		0			0
$667$	−1018.47			−0.0591232
$668$		−	20060.5i		−	1.16192i
$669$		0			0
$670$			11306.7i			0.651962i
$671$		−	16550.6i		−	0.952206i
$672$		0			0
$673$	19449.6			1.11400			0.557002	−	0.830511i	$-0.311951\pi$
$673$	19449.6			1.11400			0.557002	+	0.830511i	$0.311951\pi$
$674$		−	4444.92i		−	0.254024i
$675$		0			0
$676$	5437.89	+	11228.8i	0.309393	+	0.638873i
$677$	6629.48			0.376354			0.188177	−	0.982135i	$-0.439742\pi$
$677$	6629.48			0.376354			0.188177	+	0.982135i	$0.439742\pi$
$678$		0			0
$679$	−1661.09			−0.0938835
$680$	17339.8			0.977867
$681$		0			0
$682$		−	15038.4i		−	0.844356i
$683$		−	2526.40i		−	0.141537i	−0.997493	−	0.0707687i	$-0.977455\pi$
$683$		−	2526.40i		−	0.141537i	0.997493	−	0.0707687i	$-0.0225452\pi$
$684$		0			0
$685$	4914.13			0.274101
$686$	6332.57			0.352447
$687$		0			0
$688$	−2897.91			−0.160584
$689$	27267.7	+	17091.7i	1.50771	+	0.945055i
$690$		0			0
$691$			3808.76i			0.209685i	0.994489	+	0.104842i	$0.0334338\pi$
$691$			3808.76i			0.209685i	−0.994489	+	0.104842i	$0.966566\pi$
$692$	−18049.2			−0.991515
$693$		0			0
$694$		−	7741.54i		−	0.423436i
$695$			7311.59i			0.399056i
$696$		0			0
$697$			22891.1i			1.24399i
$698$	6499.69			0.352460
$699$		0			0
$700$		−	4027.81i		−	0.217482i
$701$	33617.8			1.81131			0.905655	−	0.424015i	$-0.139380\pi$
$701$	33617.8			1.81131			0.905655	+	0.424015i	$0.139380\pi$
$702$		0			0
$703$	−18049.0			−0.968323
$704$			8870.43i			0.474882i
$705$		0			0
$706$	4973.63			0.265134
$707$		−	13629.0i		−	0.724994i
$708$		0			0
$709$		−	26606.5i		−	1.40935i	−0.709530	−	0.704675i	$-0.751093\pi$
$709$		−	26606.5i		−	1.40935i	0.709530	−	0.704675i	$-0.248907\pi$
$710$		−	3899.05i		−	0.206097i
$711$		0			0
$712$	−11001.4			−0.579066
$713$		−	14127.5i		−	0.742046i
$714$		0			0
$715$	19296.2	+	12095.1i	1.00928	+	0.632633i
$716$	15566.5			0.812494
$717$		0			0
$718$	6729.51			0.349781
$719$	−16539.3			−0.857877			−0.428939	−	0.903334i	$-0.641112\pi$
$719$	−16539.3			−0.857877			−0.428939	+	0.903334i	$0.641112\pi$
$720$		0			0
$721$		−	725.398i		−	0.0374691i
$722$		−	10363.0i		−	0.534171i
$723$		0			0
$724$	−21983.1			−1.12844
$725$	−448.622			−0.0229812
$726$		0			0
$727$	12757.5			0.650823			0.325411	−	0.945573i	$-0.394497\pi$
$727$	12757.5			0.650823			0.325411	+	0.945573i	$0.394497\pi$
$728$	−18510.1	−	11602.4i	−0.942350	−	0.590678i
$729$		0			0
$730$		−	11349.9i		−	0.575448i
$731$	−18250.3			−0.923408
$732$		0			0
$733$		−	20523.4i		−	1.03417i	−0.855933	−	0.517087i	$-0.827016\pi$
$733$		−	20523.4i		−	1.03417i	0.855933	−	0.517087i	$-0.172984\pi$
$734$		−	2651.31i		−	0.133327i
$735$		0			0
$736$			13504.6i			0.676338i
$737$	38652.9			1.93188
$738$		0			0
$739$		−	7462.66i		−	0.371473i	−0.982600	−	0.185736i	$-0.940533\pi$
$739$		−	7462.66i		−	0.371473i	0.982600	−	0.185736i	$-0.0594670\pi$
$740$	8469.84			0.420753
$741$		0			0
$742$	−23393.7			−1.15742
$743$			18847.8i			0.930632i	0.885145	+	0.465316i	$0.154059\pi$
$743$			18847.8i			0.930632i	−0.885145	+	0.465316i	$0.845941\pi$
$744$		0			0
$745$	31368.5			1.54262
$746$			1792.13i			0.0879549i
$747$		0			0
$748$		−	24609.2i		−	1.20294i
$749$		−	38587.9i		−	1.88247i
$750$		0			0
$751$	20832.6			1.01224			0.506119	−	0.862464i	$-0.331080\pi$
$751$	20832.6			1.01224			0.506119	+	0.862464i	$0.331080\pi$
$752$			924.050i			0.0448093i
$753$		0			0
$754$	−536.496	+	855.910i	−0.0259125	+	0.0413400i
$755$	−7681.29			−0.370266
$756$		0			0
$757$	−20860.9			−1.00159			−0.500795	−	0.865566i	$-0.666959\pi$
$757$	−20860.9			−1.00159			−0.500795	+	0.865566i	$0.666959\pi$
$758$	10871.4			0.520932
$759$		0			0
$760$			23526.1i			1.12287i
$761$			4464.86i			0.212682i	0.994330	+	0.106341i	$0.0339135\pi$
$761$			4464.86i			0.212682i	−0.994330	+	0.106341i	$0.966086\pi$
$762$		0			0
$763$	28479.0			1.35126
$764$	5271.56			0.249631
$765$		0			0
$766$	2960.08			0.139624
$767$	−2288.51	+	3651.03i	−0.107736	+	0.171879i
$768$		0			0
$769$		−	23797.7i		−	1.11595i	−0.829857	−	0.557977i	$-0.811578\pi$
$769$		−	23797.7i		−	1.11595i	0.829857	−	0.557977i	$-0.188422\pi$
$770$	−16554.8			−0.774795
$771$		0			0
$772$			12842.6i			0.598725i
$773$			29418.7i			1.36885i	0.729085	+	0.684423i	$0.239946\pi$
$773$			29418.7i			1.36885i	−0.729085	+	0.684423i	$0.760054\pi$
$774$		0			0
$775$		−	6222.99i		−	0.288434i
$776$	1547.93			0.0716075
$777$		0			0
$778$		−	11496.6i		−	0.529787i
$779$	−31058.0			−1.42846
$780$		0			0
$781$	−13329.3			−0.610703
$782$			9449.80i			0.432128i
$783$		0			0
$784$	−2149.56			−0.0979208
$785$			630.435i			0.0286640i
$786$		0			0
$787$			23896.0i			1.08234i	0.840913	+	0.541170i	$0.182018\pi$
$787$			23896.0i			1.08234i	−0.840913	+	0.541170i	$0.817982\pi$
$788$			12865.8i			0.581630i
$789$		0			0
$790$	18053.4			0.813052
$791$			24929.7i			1.12060i
$792$		0			0
$793$	−13066.5	−	8190.26i	−0.585126	−	0.366765i
$794$	−633.680			−0.0283230
$795$		0			0
$796$	1481.57			0.0659708
$797$	−1034.67			−0.0459847			−0.0229923	−	0.999736i	$-0.507319\pi$
$797$	−1034.67			−0.0459847			−0.0229923	+	0.999736i	$0.507319\pi$
$798$		0			0
$799$			5819.42i			0.257667i
$800$			5948.59i			0.262893i
$801$		0			0
$802$	−1459.84			−0.0642751
$803$	−38800.6			−1.70516
$804$		0			0
$805$	−15552.0			−0.680914
$806$	−11872.6	−	7441.92i	−0.518853	−	0.325224i
$807$		0			0
$808$			12700.5i			0.552973i
$809$	−29557.9			−1.28455			−0.642275	−	0.766474i	$-0.722009\pi$
$809$	−29557.9			−1.28455			−0.642275	+	0.766474i	$0.722009\pi$
$810$		0			0
$811$			30268.9i			1.31058i	0.755376	+	0.655292i	$0.227454\pi$
$811$			30268.9i			1.31058i	−0.755376	+	0.655292i	$0.772546\pi$
$812$			1796.46i			0.0776396i
$813$		0			0
$814$			11835.4i			0.509621i
$815$	17920.2			0.770203
$816$		0			0
$817$		−	24761.5i		−	1.06034i
$818$	6528.17			0.279037
$819$		0			0
$820$	14574.6			0.620690
$821$		−	39292.7i		−	1.67031i	−0.550013	−	0.835156i	$-0.685377\pi$
$821$		−	39292.7i		−	1.67031i	0.550013	−	0.835156i	$-0.314623\pi$
$822$		0			0
$823$	25988.8			1.10074			0.550372	−	0.834920i	$-0.314486\pi$
$823$	25988.8			1.10074			0.550372	+	0.834920i	$0.314486\pi$
$824$			675.980i			0.0285787i
$825$		0			0
$826$		−	3132.31i		−	0.131946i
$827$			19585.1i			0.823508i	0.911295	+	0.411754i	$0.135084\pi$
$827$			19585.1i			0.823508i	−0.911295	+	0.411754i	$0.864916\pi$
$828$		0			0
$829$	−666.472			−0.0279222			−0.0139611	−	0.999903i	$-0.504444\pi$
$829$	−666.472			−0.0279222			−0.0139611	+	0.999903i	$0.504444\pi$
$830$			7563.86i			0.316320i
$831$		0			0
$832$	7003.08	+	4389.62i	0.291813	+	0.182912i
$833$	−13537.3			−0.563075
$834$		0			0
$835$	−34118.7			−1.41404
$836$	33389.1			1.38133
$837$		0			0
$838$			15157.9i			0.624845i
$839$			36379.7i			1.49698i	0.663146	+	0.748490i	$0.269221\pi$
$839$			36379.7i			1.49698i	−0.663146	+	0.748490i	$0.730779\pi$
$840$		0			0
$841$	−24188.9			−0.991796
$842$	−574.762			−0.0235245
$843$		0			0
$844$	33227.8			1.35515
$845$	19097.9	−	9248.71i	0.777500	−	0.376527i
$846$		0			0
$847$			26827.6i			1.08832i
$848$	−9391.60			−0.380317
$849$		0			0
$850$			4162.52i			0.167969i
$851$			11118.5i			0.447871i
$852$		0			0
$853$		−	39951.4i		−	1.60364i	−0.597563	−	0.801822i	$-0.703864\pi$
$853$		−	39951.4i		−	1.60364i	0.597563	−	0.801822i	$-0.296136\pi$
$854$	11210.1			0.449183
$855$		0			0
$856$			35959.1i			1.43581i
$857$	17226.4			0.686629			0.343314	−	0.939221i	$-0.388450\pi$
$857$	17226.4			0.686629			0.343314	+	0.939221i	$0.388450\pi$
$858$		0			0
$859$	−33392.3			−1.32634			−0.663172	−	0.748467i	$-0.730790\pi$
$859$	−33392.3			−1.32634			−0.663172	+	0.748467i	$0.730790\pi$
$860$			11619.8i			0.460735i
$861$		0			0
$862$	−3713.11			−0.146716
$863$			17381.9i			0.685614i	0.939406	+	0.342807i	$0.111378\pi$
$863$			17381.9i			0.685614i	−0.939406	+	0.342807i	$0.888622\pi$
$864$		0			0
$865$			30697.9i			1.20666i
$866$		−	17086.8i		−	0.670476i
$867$		0			0
$868$	−24919.3			−0.974443
$869$		−	61717.2i		−	2.40922i
$870$		0			0
$871$	19127.8	−	30515.9i	0.744110	−	1.18713i
$872$	−26538.8			−1.03064
$873$		0			0
$874$	−12821.2			−0.496207
$875$	−33850.5			−1.30783
$876$		0			0
$877$			14335.6i			0.551970i	0.961162	+	0.275985i	$0.0890040\pi$
$877$			14335.6i			0.551970i	−0.961162	+	0.275985i	$0.910996\pi$
$878$			2796.02i			0.107473i
$879$		0			0
$880$	−6646.06			−0.254589
$881$	5436.53			0.207901			0.103951	−	0.994582i	$-0.466852\pi$
$881$	5436.53			0.207901			0.103951	+	0.994582i	$0.466852\pi$
$882$		0			0
$883$	−21185.9			−0.807430			−0.403715	−	0.914885i	$-0.632281\pi$
$883$	−21185.9			−0.807430			−0.403715	+	0.914885i	$0.632281\pi$
$884$	−19428.7	−	12178.1i	−0.739204	−	0.463343i
$885$		0			0
$886$		−	17688.6i		−	0.670724i
$887$	12661.4			0.479287			0.239644	−	0.970861i	$-0.422969\pi$
$887$	12661.4			0.479287			0.239644	+	0.970861i	$0.422969\pi$
$888$		0			0
$889$			26255.5i			0.990531i
$890$			7767.94i			0.292564i
$891$		0			0
$892$		−	19669.0i		−	0.738305i
$893$	−7895.63			−0.295876
$894$		0			0
$895$		−	26475.3i		−	0.988794i
$896$	27549.1			1.02718
$897$		0			0
$898$	21462.3			0.797558
$899$			2775.54i			0.102969i
$900$		0			0
$901$	−59145.8			−2.18694
$902$			20365.9i			0.751786i
$903$		0			0
$904$		−	23231.3i		−	0.854715i
$905$			37388.6i			1.37330i
$906$		0			0
$907$	−3545.27			−0.129789			−0.0648946	−	0.997892i	$-0.520671\pi$
$907$	−3545.27			−0.129789			−0.0648946	+	0.997892i	$0.520671\pi$
$908$		−	30263.1i		−	1.10607i
$909$		0			0
$910$	−8192.30	+	13069.8i	−0.298431	+	0.476108i
$911$	3913.88			0.142341			0.0711706	−	0.997464i	$-0.477327\pi$
$911$	3913.88			0.142341			0.0711706	+	0.997464i	$0.477327\pi$
$912$		0			0
$913$	25857.7			0.937313
$914$	−3618.46			−0.130950
$915$		0			0
$916$		−	27110.2i		−	0.977890i
$917$		−	32319.5i		−	1.16389i
$918$		0			0
$919$	6917.79			0.248310			0.124155	−	0.992263i	$-0.460378\pi$
$919$	6917.79			0.248310			0.124155	+	0.992263i	$0.460378\pi$
$920$	14492.5			0.519352
$921$		0			0
$922$	−9708.18			−0.346770
$923$	−6596.13	+	10523.3i	−0.235227	+	0.375274i
$924$		0			0
$925$			4897.57i			0.174088i
$926$	−96.7050			−0.00343188
$927$		0			0
$928$		−	2653.15i		−	0.0938512i
$929$			3753.03i			0.132543i	0.997802	+	0.0662717i	$0.0211104\pi$
$929$			3753.03i			0.132543i	−0.997802	+	0.0662717i	$0.978890\pi$
$930$		0			0
$931$		−	18367.1i		−	0.646571i
$932$	−27336.4			−0.960765
$933$		0			0
$934$		−	11968.7i		−	0.419302i
$935$	−41855.1			−1.46397
$936$		0			0
$937$	48189.0			1.68011			0.840056	−	0.542499i	$-0.182522\pi$
$937$	48189.0			1.68011			0.840056	+	0.542499i	$0.182522\pi$
$938$			26180.4i			0.911323i
$939$		0			0
$940$	3705.17			0.128563
$941$		−	26656.4i		−	0.923456i	−0.887022	−	0.461728i	$-0.847230\pi$
$941$		−	26656.4i		−	0.923456i	0.887022	−	0.461728i	$-0.152770\pi$
$942$		0			0
$943$			19132.3i			0.660693i
$944$		−	1257.50i		−	0.0433559i
$945$		0			0
$946$	−16237.1			−0.558047
$947$			31258.9i			1.07263i	0.844018	+	0.536314i	$0.180184\pi$
$947$			31258.9i			1.07263i	−0.844018	+	0.536314i	$0.819816\pi$
$948$		0			0
$949$	−19200.9	+	30632.5i	−0.656783	+	1.04781i
$950$	−5647.60			−0.192876
$951$		0			0
$952$	40150.0			1.36688
$953$	10602.4			0.360382			0.180191	−	0.983632i	$-0.442328\pi$
$953$	10602.4			0.360382			0.180191	+	0.983632i	$0.442328\pi$
$954$		0			0
$955$		−	8965.81i		−	0.303798i
$956$		−	9558.44i		−	0.323370i
$957$		0			0
$958$	19856.9			0.669674
$959$	11378.6			0.383144
$960$		0			0
$961$	−8709.45			−0.292352
$962$	9343.90	+	5856.88i	0.313160	+	0.196293i
$963$		0			0
$964$			3775.60i			0.126145i
$965$	21842.6			0.728640
$966$		0			0
$967$		−	4815.93i		−	0.160155i	−0.996789	−	0.0800774i	$-0.974483\pi$
$967$		−	4815.93i		−	0.160155i	0.996789	−	0.0800774i	$-0.0255168\pi$
$968$		−	25000.0i		−	0.830092i
$969$		0			0
$970$		−	1092.97i		−	0.0361786i
$971$	−56729.5			−1.87491			−0.937454	−	0.348109i	$-0.886824\pi$
$971$	−56729.5			−1.87491			−0.937454	+	0.348109i	$0.886824\pi$
$972$		0			0
$973$			16929.9i			0.557808i
$974$	106.493			0.00350334
$975$		0			0
$976$	4500.40			0.147597
$977$			55246.2i			1.80909i	0.426375	+	0.904546i	$0.359790\pi$
$977$			55246.2i			1.80909i	−0.426375	+	0.904546i	$0.640210\pi$
$978$		0			0
$979$	26555.4			0.866921
$980$			8619.11i			0.280946i
$981$		0			0
$982$		−	4000.34i		−	0.129996i
$983$		−	22820.6i		−	0.740451i	−0.928942	−	0.370226i	$-0.879280\pi$
$983$		−	22820.6i		−	0.740451i	0.928942	−	0.370226i	$-0.120720\pi$
$984$		0			0
$985$	21882.0			0.707836
$986$		−	1856.54i		−	0.0599637i
$987$		0			0
$988$	16523.0	−	26360.3i	0.532050	−	0.848818i
$989$	−15253.5			−0.490429
$990$		0			0
$991$	−46490.4			−1.49023			−0.745115	−	0.666936i	$-0.767605\pi$
$991$	−46490.4			−1.49023			−0.745115	+	0.666936i	$0.767605\pi$
$992$	36802.8			1.17791
$993$		0			0
$994$		−	9028.21i		−	0.288086i
$995$		−	2519.84i		−	0.0802856i
$996$		0			0
$997$	24943.4			0.792344			0.396172	−	0.918176i	$-0.370338\pi$
$997$	24943.4			0.792344			0.396172	+	0.918176i	$0.370338\pi$
$998$	−11626.8			−0.368777
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.b.e.64.3		4
3.2	odd	2		39.4.b.b.25.2	✓	4
12.11	even	2		624.4.c.c.337.3		4
13.5	odd	4		1521.4.a.w.1.3		4
13.8	odd	4		1521.4.a.w.1.2		4
13.12	even	2	inner	117.4.b.e.64.2		4
39.5	even	4		507.4.a.l.1.2		4
39.8	even	4		507.4.a.l.1.3		4
39.38	odd	2		39.4.b.b.25.3	yes	4
156.155	even	2		624.4.c.c.337.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.b.25.2	✓	4	3.2	odd	2
39.4.b.b.25.3	yes	4	39.38	odd	2
117.4.b.e.64.2		4	13.12	even	2	inner
117.4.b.e.64.3		4	1.1	even	1	trivial
507.4.a.l.1.2		4	39.5	even	4
507.4.a.l.1.3		4	39.8	even	4
624.4.c.c.337.2		4	156.155	even	2
624.4.c.c.337.3		4	12.11	even	2
1521.4.a.w.1.2		4	13.8	odd	4
1521.4.a.w.1.3		4	13.5	odd	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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.52356i q^{2} +5.67878 q^{4} -9.65841i q^{5} -22.3639i q^{7} +20.8404i q^{8} +14.7151 q^{10} -50.3050i q^{11} +(-39.7151 - 24.8940i) q^{13} +34.0727 q^{14} +13.6788 q^{16} +86.1454 q^{17} +116.880i q^{19} -54.8480i q^{20} +76.6424 q^{22} +72.0000 q^{23} +31.7151 q^{25} +(37.9273 - 60.5082i) q^{26} -127.000i q^{28} -14.1454 q^{29} -196.215i q^{31} +187.563i q^{32} +131.247i q^{34} -216.000 q^{35} +154.424i q^{37} -178.073 q^{38} +201.285 q^{40} +265.726i q^{41} -211.855 q^{43} -285.671i q^{44} +109.696i q^{46} +67.5535i q^{47} -157.145 q^{49} +48.3197i q^{50} +(-225.533 - 141.367i) q^{52} -686.581 q^{53} -485.866 q^{55} +466.073 q^{56} -21.5512i q^{58} -91.9304i q^{59} +329.006 q^{61} +298.945 q^{62} -176.333 q^{64} +(-240.436 + 383.585i) q^{65} +768.370i q^{67} +489.201 q^{68} -329.088i q^{70} -264.969i q^{71} -771.306i q^{73} -235.273 q^{74} +663.734i q^{76} -1125.02 q^{77} +1226.86 q^{79} -132.115i q^{80} -404.849 q^{82} +514.019i q^{83} -832.027i q^{85} -322.772i q^{86} +1048.38 q^{88} +527.889i q^{89} +(-556.727 + 888.186i) q^{91} +408.872 q^{92} -102.921 q^{94} +1128.87 q^{95} -74.2755i q^{97} -239.420i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{4} - 88 q^{10} - 12 q^{13} - 84 q^{14} + 18 q^{16} - 96 q^{17} + 380 q^{22} + 288 q^{23} - 20 q^{25} + 372 q^{26} + 384 q^{29} - 864 q^{35} - 492 q^{38} + 952 q^{40} - 1288 q^{43} - 188 q^{49}+ \cdots + 1872 q^{95}+O(q^{100}) \) 4 * q - 14 * q^4 - 88 * q^10 - 12 * q^13 - 84 * q^14 + 18 * q^16 - 96 * q^17 + 380 * q^22 + 288 * q^23 - 20 * q^25 + 372 * q^26 + 384 * q^29 - 864 * q^35 - 492 * q^38 + 952 * q^40 - 1288 * q^43 - 188 * q^49 - 1306 * q^52 - 984 * q^53 - 328 * q^55 + 1644 * q^56 + 288 * q^61 - 1668 * q^62 + 1314 * q^64 + 360 * q^65 + 4380 * q^68 - 3144 * q^74 - 1416 * q^77 + 4320 * q^79 - 3088 * q^82 + 1036 * q^88 - 24 * q^91 - 1008 * q^92 - 1660 * q^94 + 1872 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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