LMFDB - Embedded newform 507.4.b.d.337.1

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.d.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +3.00000 q^{3} +7.00000 q^{4} +7.00000i q^{5} -3.00000i q^{6} +10.0000i q^{7} -15.0000i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +3.00000 q^{3} +7.00000 q^{4} +7.00000i q^{5} -3.00000i q^{6} +10.0000i q^{7} -15.0000i q^{8} +9.00000 q^{9} +7.00000 q^{10} +22.0000i q^{11} +21.0000 q^{12} +10.0000 q^{14} +21.0000i q^{15} +41.0000 q^{16} -37.0000 q^{17} -9.00000i q^{18} +30.0000i q^{19} +49.0000i q^{20} +30.0000i q^{21} +22.0000 q^{22} +162.000 q^{23} -45.0000i q^{24} +76.0000 q^{25} +27.0000 q^{27} +70.0000i q^{28} -113.000 q^{29} +21.0000 q^{30} +196.000i q^{31} -161.000i q^{32} +66.0000i q^{33} +37.0000i q^{34} -70.0000 q^{35} +63.0000 q^{36} -13.0000i q^{37} +30.0000 q^{38} +105.000 q^{40} +285.000i q^{41} +30.0000 q^{42} +246.000 q^{43} +154.000i q^{44} +63.0000i q^{45} -162.000i q^{46} +462.000i q^{47} +123.000 q^{48} +243.000 q^{49} -76.0000i q^{50} -111.000 q^{51} -537.000 q^{53} -27.0000i q^{54} -154.000 q^{55} +150.000 q^{56} +90.0000i q^{57} +113.000i q^{58} -576.000i q^{59} +147.000i q^{60} -635.000 q^{61} +196.000 q^{62} +90.0000i q^{63} +167.000 q^{64} +66.0000 q^{66} +202.000i q^{67} -259.000 q^{68} +486.000 q^{69} +70.0000i q^{70} -1086.00i q^{71} -135.000i q^{72} +805.000i q^{73} -13.0000 q^{74} +228.000 q^{75} +210.000i q^{76} -220.000 q^{77} +884.000 q^{79} +287.000i q^{80} +81.0000 q^{81} +285.000 q^{82} +518.000i q^{83} +210.000i q^{84} -259.000i q^{85} -246.000i q^{86} -339.000 q^{87} +330.000 q^{88} -194.000i q^{89} +63.0000 q^{90} +1134.00 q^{92} +588.000i q^{93} +462.000 q^{94} -210.000 q^{95} -483.000i q^{96} -1202.00i q^{97} -243.000i q^{98} +198.000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 14 q^{4} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 14 * q^4 + 18 * q^9	$ 2 q + 6 q^{3} + 14 q^{4} + 18 q^{9} + 14 q^{10} + 42 q^{12} + 20 q^{14} + 82 q^{16} - 74 q^{17} + 44 q^{22} + 324 q^{23} + 152 q^{25} + 54 q^{27} - 226 q^{29} + 42 q^{30} - 140 q^{35} + 126 q^{36} + 60 q^{38} + 210 q^{40} + 60 q^{42} + 492 q^{43} + 246 q^{48} + 486 q^{49} - 222 q^{51} - 1074 q^{53} - 308 q^{55} + 300 q^{56} - 1270 q^{61} + 392 q^{62} + 334 q^{64} + 132 q^{66} - 518 q^{68} + 972 q^{69} - 26 q^{74} + 456 q^{75} - 440 q^{77} + 1768 q^{79} + 162 q^{81} + 570 q^{82} - 678 q^{87} + 660 q^{88} + 126 q^{90} + 2268 q^{92} + 924 q^{94} - 420 q^{95}+O(q^{100}) $ 2 * q + 6 * q^3 + 14 * q^4 + 18 * q^9 + 14 * q^10 + 42 * q^12 + 20 * q^14 + 82 * q^16 - 74 * q^17 + 44 * q^22 + 324 * q^23 + 152 * q^25 + 54 * q^27 - 226 * q^29 + 42 * q^30 - 140 * q^35 + 126 * q^36 + 60 * q^38 + 210 * q^40 + 60 * q^42 + 492 * q^43 + 246 * q^48 + 486 * q^49 - 222 * q^51 - 1074 * q^53 - 308 * q^55 + 300 * q^56 - 1270 * q^61 + 392 * q^62 + 334 * q^64 + 132 * q^66 - 518 * q^68 + 972 * q^69 - 26 * q^74 + 456 * q^75 - 440 * q^77 + 1768 * q^79 + 162 * q^81 + 570 * q^82 - 678 * q^87 + 660 * q^88 + 126 * q^90 + 2268 * q^92 + 924 * q^94 - 420 * 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 $

$2$	− 1.00000i	− 0.353553i	−0.984251	−	0.176777i	$-0.943433\pi$
$2$	− 1.00000i	− 0.353553i	0.984251	−	0.176777i	$-0.0565670\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	7.00000	0.875000
$5$	7.00000i	0.626099i	0.949737	+	0.313050i	$0.101351\pi$
$5$	7.00000i	0.626099i	−0.949737	+	0.313050i	$0.898649\pi$
$6$	− 3.00000i	− 0.204124i
$7$	10.0000i	0.539949i	0.962867	+	0.269975i	$0.0870153\pi$
$7$	10.0000i	0.539949i	−0.962867	+	0.269975i	$0.912985\pi$
$8$	− 15.0000i	− 0.662913i
$9$	9.00000	0.333333
$10$	7.00000	0.221359
$11$	22.0000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	22.0000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	21.0000	0.505181
$13$	0	0
$13$	0	0
$14$	10.0000	0.190901
$15$	21.0000i	0.361478i
$16$	41.0000	0.640625
$17$	−37.0000	−0.527872	−0.263936	−	0.964540i	$-0.585021\pi$
$17$	−37.0000	−0.527872	−0.263936	+	0.964540i	$0.585021\pi$
$18$	− 9.00000i	− 0.117851i
$19$	30.0000i	0.362235i	0.983461	+	0.181118i	$0.0579715\pi$
$19$	30.0000i	0.362235i	−0.983461	+	0.181118i	$0.942029\pi$
$20$	49.0000i	0.547837i
$21$	30.0000i	0.311740i
$22$	22.0000	0.213201
$23$	162.000	1.46867	0.734333	−	0.678789i	$-0.237495\pi$
$23$	162.000	1.46867	0.734333	+	0.678789i	$0.237495\pi$
$24$	− 45.0000i	− 0.382733i
$25$	76.0000	0.608000
$26$	0	0
$27$	27.0000	0.192450
$28$	70.0000i	0.472456i
$29$	−113.000	−0.723571	−0.361786	−	0.932261i	$-0.617833\pi$
$29$	−113.000	−0.723571	−0.361786	+	0.932261i	$0.617833\pi$
$30$	21.0000	0.127802
$31$	196.000i	1.13557i	0.823177	+	0.567785i	$0.192199\pi$
$31$	196.000i	1.13557i	−0.823177	+	0.567785i	$0.807801\pi$
$32$	− 161.000i	− 0.889408i
$33$	66.0000i	0.348155i
$34$	37.0000i	0.186631i
$35$	−70.0000	−0.338062
$36$	63.0000	0.291667
$37$	− 13.0000i	− 0.0577618i	−0.999583	−	0.0288809i	$-0.990806\pi$
$37$	− 13.0000i	− 0.0577618i	0.999583	−	0.0288809i	$-0.00919436\pi$
$38$	30.0000	0.128070
$39$	0	0
$40$	105.000	0.415049
$41$	285.000i	1.08560i	0.839863	+	0.542799i	$0.182635\pi$
$41$	285.000i	1.08560i	−0.839863	+	0.542799i	$0.817365\pi$
$42$	30.0000	0.110217
$43$	246.000	0.872434	0.436217	−	0.899842i	$-0.356318\pi$
$43$	246.000	0.872434	0.436217	+	0.899842i	$0.356318\pi$
$44$	154.000i	0.527645i
$45$	63.0000i	0.208700i
$46$	− 162.000i	− 0.519252i
$47$	462.000i	1.43382i	0.697165	+	0.716911i	$0.254445\pi$
$47$	462.000i	1.43382i	−0.697165	+	0.716911i	$0.745555\pi$
$48$	123.000	0.369865
$49$	243.000	0.708455
$50$	− 76.0000i	− 0.214960i
$51$	−111.000	−0.304767
$52$	0	0
$53$	−537.000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$53$	−537.000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$54$	− 27.0000i	− 0.0680414i
$55$	−154.000	−0.377552
$56$	150.000	0.357939
$57$	90.0000i	0.209137i
$58$	113.000i	0.255821i
$59$	− 576.000i	− 1.27100i	−0.772102	−	0.635498i	$-0.780795\pi$
$59$	− 576.000i	− 1.27100i	0.772102	−	0.635498i	$-0.219205\pi$
$60$	147.000i	0.316294i
$61$	−635.000	−1.33284	−0.666421	−	0.745575i	$-0.732175\pi$
$61$	−635.000	−1.33284	−0.666421	+	0.745575i	$0.732175\pi$
$62$	196.000	0.401484
$63$	90.0000i	0.179983i
$64$	167.000	0.326172
$65$	0	0
$66$	66.0000	0.123091
$67$	202.000i	0.368332i	0.982895	+	0.184166i	$0.0589584\pi$
$67$	202.000i	0.368332i	−0.982895	+	0.184166i	$0.941042\pi$
$68$	−259.000	−0.461888
$69$	486.000	0.847935
$70$	70.0000i	0.119523i
$71$	− 1086.00i	− 1.81527i	−0.419755	−	0.907637i	$-0.637884\pi$
$71$	− 1086.00i	− 1.81527i	0.419755	−	0.907637i	$-0.362116\pi$
$72$	− 135.000i	− 0.220971i
$73$	805.000i	1.29066i	0.763904	+	0.645330i	$0.223280\pi$
$73$	805.000i	1.29066i	−0.763904	+	0.645330i	$0.776720\pi$
$74$	−13.0000	−0.0204219
$75$	228.000	0.351029
$76$	210.000i	0.316956i
$77$	−220.000	−0.325602
$78$	0	0
$79$	884.000	1.25896	0.629480	−	0.777017i	$-0.283268\pi$
$79$	884.000	1.25896	0.629480	+	0.777017i	$0.283268\pi$
$80$	287.000i	0.401095i
$81$	81.0000	0.111111
$82$	285.000	0.383817
$83$	518.000i	0.685035i	0.939511	+	0.342517i	$0.111280\pi$
$83$	518.000i	0.685035i	−0.939511	+	0.342517i	$0.888720\pi$
$84$	210.000i	0.272772i
$85$	− 259.000i	− 0.330500i
$86$	− 246.000i	− 0.308452i
$87$	−339.000	−0.417754
$88$	330.000	0.399751
$89$	− 194.000i	− 0.231056i	−0.993304	−	0.115528i	$-0.963144\pi$
$89$	− 194.000i	− 0.231056i	0.993304	−	0.115528i	$-0.0368560\pi$
$90$	63.0000	0.0737865
$91$	0	0
$92$	1134.00	1.28508
$93$	588.000i	0.655621i
$94$	462.000	0.506933
$95$	−210.000	−0.226795
$96$	− 483.000i	− 0.513500i
$97$	− 1202.00i	− 1.25819i	−0.777328	−	0.629096i	$-0.783425\pi$
$97$	− 1202.00i	− 1.25819i	0.777328	−	0.629096i	$-0.216575\pi$
$98$	− 243.000i	− 0.250477i
$99$	198.000i	0.201008i
$100$	532.000	0.532000
$101$	429.000	0.422645	0.211322	−	0.977416i	$-0.432223\pi$
$101$	429.000	0.422645	0.211322	+	0.977416i	$0.432223\pi$
$102$	111.000i	0.107751i
$103$	1302.00	1.24553	0.622766	−	0.782408i	$-0.286009\pi$
$103$	1302.00	1.24553	0.622766	+	0.782408i	$0.286009\pi$
$104$	0	0
$105$	−210.000	−0.195180
$106$	537.000i	0.492057i
$107$	−1338.00	−1.20887	−0.604436	−	0.796654i	$-0.706602\pi$
$107$	−1338.00	−1.20887	−0.604436	+	0.796654i	$0.706602\pi$
$108$	189.000	0.168394
$109$	− 1034.00i	− 0.908617i	−0.890844	−	0.454308i	$-0.849886\pi$
$109$	− 1034.00i	− 0.908617i	0.890844	−	0.454308i	$-0.150114\pi$
$110$	154.000i	0.133485i
$111$	− 39.0000i	− 0.0333488i
$112$	410.000i	0.345905i
$113$	1077.00	0.896599	0.448299	−	0.893884i	$-0.352030\pi$
$113$	1077.00	0.896599	0.448299	+	0.893884i	$0.352030\pi$
$114$	90.0000	0.0739410
$115$	1134.00i	0.919531i
$116$	−791.000	−0.633125
$117$	0	0
$118$	−576.000	−0.449365
$119$	− 370.000i	− 0.285024i
$120$	315.000	0.239629
$121$	847.000	0.636364
$122$	635.000i	0.471231i
$123$	855.000i	0.626770i
$124$	1372.00i	0.993623i
$125$	1407.00i	1.00677i
$126$	90.0000	0.0636336
$127$	988.000	0.690321	0.345161	−	0.938544i	$-0.387824\pi$
$127$	988.000	0.690321	0.345161	+	0.938544i	$0.387824\pi$
$128$	− 1455.00i	− 1.00473i
$129$	738.000	0.503700
$130$	0	0
$131$	560.000	0.373492	0.186746	−	0.982408i	$-0.440206\pi$
$131$	560.000	0.373492	0.186746	+	0.982408i	$0.440206\pi$
$132$	462.000i	0.304636i
$133$	−300.000	−0.195589
$134$	202.000	0.130225
$135$	189.000i	0.120493i
$136$	555.000i	0.349933i
$137$	519.000i	0.323658i	0.986819	+	0.161829i	$0.0517393\pi$
$137$	519.000i	0.323658i	−0.986819	+	0.161829i	$0.948261\pi$
$138$	− 486.000i	− 0.299790i
$139$	−348.000	−0.212352	−0.106176	−	0.994347i	$-0.533861\pi$
$139$	−348.000	−0.212352	−0.106176	+	0.994347i	$0.533861\pi$
$140$	−490.000	−0.295804
$141$	1386.00i	0.827817i
$142$	−1086.00	−0.641796
$143$	0	0
$144$	369.000	0.213542
$145$	− 791.000i	− 0.453027i
$146$	805.000	0.456317
$147$	729.000	0.409027
$148$	− 91.0000i	− 0.0505416i
$149$	− 645.000i	− 0.354634i	−0.984154	−	0.177317i	$-0.943258\pi$
$149$	− 645.000i	− 0.354634i	0.984154	−	0.177317i	$-0.0567418\pi$
$150$	− 228.000i	− 0.124107i
$151$	− 2914.00i	− 1.57045i	−0.619211	−	0.785225i	$-0.712547\pi$
$151$	− 2914.00i	− 1.57045i	0.619211	−	0.785225i	$-0.287453\pi$
$152$	450.000	0.240130
$153$	−333.000	−0.175957
$154$	220.000i	0.115118i
$155$	−1372.00	−0.710979
$156$	0	0
$157$	−2079.00	−1.05683	−0.528415	−	0.848986i	$-0.677213\pi$
$157$	−2079.00	−1.05683	−0.528415	+	0.848986i	$0.677213\pi$
$158$	− 884.000i	− 0.445109i
$159$	−1611.00	−0.803526
$160$	1127.00	0.556857
$161$	1620.00i	0.793006i
$162$	− 81.0000i	− 0.0392837i
$163$	− 1700.00i	− 0.816897i	−0.912781	−	0.408449i	$-0.866070\pi$
$163$	− 1700.00i	− 0.816897i	0.912781	−	0.408449i	$-0.133930\pi$
$164$	1995.00i	0.949898i
$165$	−462.000	−0.217980
$166$	518.000	0.242196
$167$	− 3680.00i	− 1.70519i	−0.522571	−	0.852596i	$-0.675027\pi$
$167$	− 3680.00i	− 1.70519i	0.522571	−	0.852596i	$-0.324973\pi$
$168$	450.000	0.206656
$169$	0	0
$170$	−259.000	−0.116849
$171$	270.000i	0.120745i
$172$	1722.00	0.763379
$173$	−4146.00	−1.82205	−0.911025	−	0.412352i	$-0.864707\pi$
$173$	−4146.00	−1.82205	−0.911025	+	0.412352i	$0.864707\pi$
$174$	339.000i	0.147698i
$175$	760.000i	0.328289i
$176$	902.000i	0.386311i
$177$	− 1728.00i	− 0.733810i
$178$	−194.000	−0.0816905
$179$	−3674.00	−1.53412	−0.767060	−	0.641575i	$-0.778281\pi$
$179$	−3674.00	−1.53412	−0.767060	+	0.641575i	$0.778281\pi$
$180$	441.000i	0.182612i
$181$	3283.00	1.34820	0.674098	−	0.738642i	$-0.264533\pi$
$181$	3283.00	1.34820	0.674098	+	0.738642i	$0.264533\pi$
$182$	0	0
$183$	−1905.00	−0.769517
$184$	− 2430.00i	− 0.973598i
$185$	91.0000	0.0361646
$186$	588.000	0.231797
$187$	− 814.000i	− 0.318319i
$188$	3234.00i	1.25459i
$189$	270.000i	0.103913i
$190$	210.000i	0.0801842i
$191$	−596.000	−0.225786	−0.112893	−	0.993607i	$-0.536012\pi$
$191$	−596.000	−0.225786	−0.112893	+	0.993607i	$0.536012\pi$
$192$	501.000	0.188315
$193$	393.000i	0.146574i	0.997311	+	0.0732869i	$0.0233489\pi$
$193$	393.000i	0.146574i	−0.997311	+	0.0732869i	$0.976651\pi$
$194$	−1202.00	−0.444838
$195$	0	0
$196$	1701.00	0.619898
$197$	− 3522.00i	− 1.27377i	−0.770960	−	0.636884i	$-0.780223\pi$
$197$	− 3522.00i	− 1.27377i	0.770960	−	0.636884i	$-0.219777\pi$
$198$	198.000	0.0710669
$199$	−2018.00	−0.718855	−0.359428	−	0.933173i	$-0.617028\pi$
$199$	−2018.00	−0.718855	−0.359428	+	0.933173i	$0.617028\pi$
$200$	− 1140.00i	− 0.403051i
$201$	606.000i	0.212656i
$202$	− 429.000i	− 0.149427i
$203$	− 1130.00i	− 0.390692i
$204$	−777.000	−0.266671
$205$	−1995.00	−0.679692
$206$	− 1302.00i	− 0.440362i
$207$	1458.00	0.489556
$208$	0	0
$209$	−660.000	−0.218436
$210$	210.000i	0.0690066i
$211$	160.000	0.0522031	0.0261016	−	0.999659i	$-0.491691\pi$
$211$	160.000	0.0522031	0.0261016	+	0.999659i	$0.491691\pi$
$212$	−3759.00	−1.21778
$213$	− 3258.00i	− 1.04805i
$214$	1338.00i	0.427401i
$215$	1722.00i	0.546230i
$216$	− 405.000i	− 0.127578i
$217$	−1960.00	−0.613150
$218$	−1034.00	−0.321245
$219$	2415.00i	0.745162i
$220$	−1078.00	−0.330358
$221$	0	0
$222$	−39.0000	−0.0117906
$223$	4072.00i	1.22279i	0.791327	+	0.611393i	$0.209391\pi$
$223$	4072.00i	1.22279i	−0.791327	+	0.611393i	$0.790609\pi$
$224$	1610.00	0.480235
$225$	684.000	0.202667
$226$	− 1077.00i	− 0.316995i
$227$	− 5794.00i	− 1.69410i	−0.531511	−	0.847051i	$-0.678376\pi$
$227$	− 5794.00i	− 1.69410i	0.531511	−	0.847051i	$-0.321624\pi$
$228$	630.000i	0.182995i
$229$	− 6482.00i	− 1.87049i	−0.353999	−	0.935246i	$-0.615178\pi$
$229$	− 6482.00i	− 1.87049i	0.353999	−	0.935246i	$-0.384822\pi$
$230$	1134.00	0.325103
$231$	−660.000	−0.187986
$232$	1695.00i	0.479665i
$233$	−6890.00	−1.93725	−0.968624	−	0.248530i	$-0.920053\pi$
$233$	−6890.00	−1.93725	−0.968624	+	0.248530i	$0.920053\pi$
$234$	0	0
$235$	−3234.00	−0.897714
$236$	− 4032.00i	− 1.11212i
$237$	2652.00	0.726860
$238$	−370.000	−0.100771
$239$	2466.00i	0.667415i	0.942677	+	0.333708i	$0.108300\pi$
$239$	2466.00i	0.667415i	−0.942677	+	0.333708i	$0.891700\pi$
$240$	861.000i	0.231572i
$241$	3617.00i	0.966770i	0.875408	+	0.483385i	$0.160593\pi$
$241$	3617.00i	0.966770i	−0.875408	+	0.483385i	$0.839407\pi$
$242$	− 847.000i	− 0.224989i
$243$	243.000	0.0641500
$244$	−4445.00	−1.16624
$245$	1701.00i	0.443563i
$246$	855.000	0.221597
$247$	0	0
$248$	2940.00	0.752783
$249$	1554.00i	0.395505i
$250$	1407.00	0.355946
$251$	−4860.00	−1.22215	−0.611077	−	0.791571i	$-0.709263\pi$
$251$	−4860.00	−1.22215	−0.611077	+	0.791571i	$0.709263\pi$
$252$	630.000i	0.157485i
$253$	3564.00i	0.885639i
$254$	− 988.000i	− 0.244065i
$255$	− 777.000i	− 0.190814i
$256$	−119.000	−0.0290527
$257$	−565.000	−0.137135	−0.0685676	−	0.997646i	$-0.521843\pi$
$257$	−565.000	−0.137135	−0.0685676	+	0.997646i	$0.521843\pi$
$258$	− 738.000i	− 0.178085i
$259$	130.000	0.0311884
$260$	0	0
$261$	−1017.00	−0.241190
$262$	− 560.000i	− 0.132049i
$263$	−498.000	−0.116760	−0.0583802	−	0.998294i	$-0.518594\pi$
$263$	−498.000	−0.116760	−0.0583802	+	0.998294i	$0.518594\pi$
$264$	990.000	0.230797
$265$	− 3759.00i	− 0.871372i
$266$	300.000i	0.0691511i
$267$	− 582.000i	− 0.133400i
$268$	1414.00i	0.322290i
$269$	5546.00	1.25705	0.628523	−	0.777791i	$-0.283660\pi$
$269$	5546.00	1.25705	0.628523	+	0.777791i	$0.283660\pi$
$270$	189.000	0.0426006
$271$	2256.00i	0.505691i	0.967507	+	0.252845i	$0.0813664\pi$
$271$	2256.00i	0.505691i	−0.967507	+	0.252845i	$0.918634\pi$
$272$	−1517.00	−0.338168
$273$	0	0
$274$	519.000	0.114430
$275$	1672.00i	0.366638i
$276$	3402.00	0.741943
$277$	−2309.00	−0.500846	−0.250423	−	0.968137i	$-0.580570\pi$
$277$	−2309.00	−0.500846	−0.250423	+	0.968137i	$0.580570\pi$
$278$	348.000i	0.0750779i
$279$	1764.00i	0.378523i
$280$	1050.00i	0.224105i
$281$	− 5833.00i	− 1.23832i	−0.785265	−	0.619159i	$-0.787473\pi$
$281$	− 5833.00i	− 1.23832i	0.785265	−	0.619159i	$-0.212527\pi$
$282$	1386.00	0.292678
$283$	−1650.00	−0.346581	−0.173290	−	0.984871i	$-0.555440\pi$
$283$	−1650.00	−0.346581	−0.173290	+	0.984871i	$0.555440\pi$
$284$	− 7602.00i	− 1.58837i
$285$	−630.000	−0.130940
$286$	0	0
$287$	−2850.00	−0.586168
$288$	− 1449.00i	− 0.296469i
$289$	−3544.00	−0.721352
$290$	−791.000	−0.160169
$291$	− 3606.00i	− 0.726417i
$292$	5635.00i	1.12933i
$293$	− 2991.00i	− 0.596369i	−0.954508	−	0.298184i	$-0.903619\pi$
$293$	− 2991.00i	− 0.596369i	0.954508	−	0.298184i	$-0.0963811\pi$
$294$	− 729.000i	− 0.144613i
$295$	4032.00	0.795770
$296$	−195.000	−0.0382910
$297$	594.000i	0.116052i
$298$	−645.000	−0.125382
$299$	0	0
$300$	1596.00	0.307150
$301$	2460.00i	0.471070i
$302$	−2914.00	−0.555238
$303$	1287.00	0.244014
$304$	1230.00i	0.232057i
$305$	− 4445.00i	− 0.834492i
$306$	333.000i	0.0622103i
$307$	2422.00i	0.450263i	0.974328	+	0.225132i	$0.0722812\pi$
$307$	2422.00i	0.450263i	−0.974328	+	0.225132i	$0.927719\pi$
$308$	−1540.00	−0.284901
$309$	3906.00	0.719109
$310$	1372.00i	0.251369i
$311$	3402.00	0.620288	0.310144	−	0.950690i	$-0.399623\pi$
$311$	3402.00	0.620288	0.310144	+	0.950690i	$0.399623\pi$
$312$	0	0
$313$	2310.00	0.417153	0.208577	−	0.978006i	$-0.433117\pi$
$313$	2310.00	0.417153	0.208577	+	0.978006i	$0.433117\pi$
$314$	2079.00i	0.373646i
$315$	−630.000	−0.112687
$316$	6188.00	1.10159
$317$	− 257.000i	− 0.0455349i	−0.999741	−	0.0227674i	$-0.992752\pi$
$317$	− 257.000i	− 0.0455349i	0.999741	−	0.0227674i	$-0.00724773\pi$
$318$	1611.00i	0.284089i
$319$	− 2486.00i	− 0.436330i
$320$	1169.00i	0.204216i
$321$	−4014.00	−0.697943
$322$	1620.00	0.280370
$323$	− 1110.00i	− 0.191214i
$324$	567.000	0.0972222
$325$	0	0
$326$	−1700.00	−0.288817
$327$	− 3102.00i	− 0.524590i
$328$	4275.00	0.719657
$329$	−4620.00	−0.774191
$330$	462.000i	0.0770675i
$331$	1028.00i	0.170707i	0.996351	+	0.0853535i	$0.0272019\pi$
$331$	1028.00i	0.170707i	−0.996351	+	0.0853535i	$0.972798\pi$
$332$	3626.00i	0.599405i
$333$	− 117.000i	− 0.0192539i
$334$	−3680.00	−0.602876
$335$	−1414.00	−0.230612
$336$	1230.00i	0.199708i
$337$	−2487.00	−0.402005	−0.201002	−	0.979591i	$-0.564420\pi$
$337$	−2487.00	−0.402005	−0.201002	+	0.979591i	$0.564420\pi$
$338$	0	0
$339$	3231.00	0.517651
$340$	− 1813.00i	− 0.289187i
$341$	−4312.00	−0.684774
$342$	270.000	0.0426898
$343$	5860.00i	0.922479i
$344$	− 3690.00i	− 0.578347i
$345$	3402.00i	0.530891i
$346$	4146.00i	0.644192i
$347$	−2850.00	−0.440911	−0.220455	−	0.975397i	$-0.570754\pi$
$347$	−2850.00	−0.440911	−0.220455	+	0.975397i	$0.570754\pi$
$348$	−2373.00	−0.365535
$349$	2018.00i	0.309516i	0.987952	+	0.154758i	$0.0494598\pi$
$349$	2018.00i	0.309516i	−0.987952	+	0.154758i	$0.950540\pi$
$350$	760.000	0.116068
$351$	0	0
$352$	3542.00	0.536333
$353$	− 5287.00i	− 0.797163i	−0.917133	−	0.398582i	$-0.869503\pi$
$353$	− 5287.00i	− 0.797163i	0.917133	−	0.398582i	$-0.130497\pi$
$354$	−1728.00	−0.259441
$355$	7602.00	1.13654
$356$	− 1358.00i	− 0.202174i
$357$	− 1110.00i	− 0.164559i
$358$	3674.00i	0.542394i
$359$	7278.00i	1.06997i	0.844863	+	0.534983i	$0.179682\pi$
$359$	7278.00i	1.06997i	−0.844863	+	0.534983i	$0.820318\pi$
$360$	945.000	0.138350
$361$	5959.00	0.868786
$362$	− 3283.00i	− 0.476659i
$363$	2541.00	0.367405
$364$	0	0
$365$	−5635.00	−0.808080
$366$	1905.00i	0.272065i
$367$	−4202.00	−0.597664	−0.298832	−	0.954306i	$-0.596597\pi$
$367$	−4202.00	−0.597664	−0.298832	+	0.954306i	$0.596597\pi$
$368$	6642.00	0.940865
$369$	2565.00i	0.361866i
$370$	− 91.0000i	− 0.0127861i
$371$	− 5370.00i	− 0.751473i
$372$	4116.00i	0.573668i
$373$	−1583.00	−0.219744	−0.109872	−	0.993946i	$-0.535044\pi$
$373$	−1583.00	−0.219744	−0.109872	+	0.993946i	$0.535044\pi$
$374$	−814.000	−0.112543
$375$	4221.00i	0.581257i
$376$	6930.00	0.950499
$377$	0	0
$378$	270.000	0.0367389
$379$	− 2052.00i	− 0.278111i	−0.990285	−	0.139056i	$-0.955593\pi$
$379$	− 2052.00i	− 0.278111i	0.990285	−	0.139056i	$-0.0444067\pi$
$380$	−1470.00	−0.198446
$381$	2964.00	0.398557
$382$	596.000i	0.0798273i
$383$	− 6872.00i	− 0.916822i	−0.888740	−	0.458411i	$-0.848419\pi$
$383$	− 6872.00i	− 0.916822i	0.888740	−	0.458411i	$-0.151581\pi$
$384$	− 4365.00i	− 0.580079i
$385$	− 1540.00i	− 0.203859i
$386$	393.000	0.0518217
$387$	2214.00	0.290811
$388$	− 8414.00i	− 1.10092i
$389$	11653.0	1.51884	0.759422	−	0.650598i	$-0.225482\pi$
$389$	11653.0	1.51884	0.759422	+	0.650598i	$0.225482\pi$
$390$	0	0
$391$	−5994.00	−0.775268
$392$	− 3645.00i	− 0.469644i
$393$	1680.00	0.215636
$394$	−3522.00	−0.450345
$395$	6188.00i	0.788233i
$396$	1386.00i	0.175882i
$397$	− 6134.00i	− 0.775458i	−0.921774	−	0.387729i	$-0.873260\pi$
$397$	− 6134.00i	− 0.775458i	0.921774	−	0.387729i	$-0.126740\pi$
$398$	2018.00i	0.254154i
$399$	−900.000	−0.112923
$400$	3116.00	0.389500
$401$	10795.0i	1.34433i	0.740401	+	0.672165i	$0.234636\pi$
$401$	10795.0i	1.34433i	−0.740401	+	0.672165i	$0.765364\pi$
$402$	606.000	0.0751854
$403$	0	0
$404$	3003.00	0.369814
$405$	567.000i	0.0695666i
$406$	−1130.00	−0.138130
$407$	286.000	0.0348317
$408$	1665.00i	0.202034i
$409$	− 8489.00i	− 1.02629i	−0.858301	−	0.513147i	$-0.828480\pi$
$409$	− 8489.00i	− 1.02629i	0.858301	−	0.513147i	$-0.171520\pi$
$410$	1995.00i	0.240307i
$411$	1557.00i	0.186864i
$412$	9114.00	1.08984
$413$	5760.00	0.686274
$414$	− 1458.00i	− 0.173084i
$415$	−3626.00	−0.428900
$416$	0	0
$417$	−1044.00	−0.122602
$418$	660.000i	0.0772288i
$419$	1496.00	0.174426	0.0872129	−	0.996190i	$-0.472204\pi$
$419$	1496.00	0.174426	0.0872129	+	0.996190i	$0.472204\pi$
$420$	−1470.00	−0.170783
$421$	− 11695.0i	− 1.35387i	−0.736043	−	0.676935i	$-0.763308\pi$
$421$	− 11695.0i	− 1.35387i	0.736043	−	0.676935i	$-0.236692\pi$
$422$	− 160.000i	− 0.0184566i
$423$	4158.00i	0.477941i
$424$	8055.00i	0.922607i
$425$	−2812.00	−0.320946
$426$	−3258.00	−0.370541
$427$	− 6350.00i	− 0.719668i
$428$	−9366.00	−1.05776
$429$	0	0
$430$	1722.00	0.193121
$431$	10590.0i	1.18353i	0.806110	+	0.591766i	$0.201569\pi$
$431$	10590.0i	1.18353i	−0.806110	+	0.591766i	$0.798431\pi$
$432$	1107.00	0.123288
$433$	13949.0	1.54814	0.774072	−	0.633098i	$-0.218217\pi$
$433$	13949.0	1.54814	0.774072	+	0.633098i	$0.218217\pi$
$434$	1960.00i	0.216781i
$435$	− 2373.00i	− 0.261555i
$436$	− 7238.00i	− 0.795040i
$437$	4860.00i	0.532003i
$438$	2415.00	0.263455
$439$	10726.0	1.16611	0.583057	−	0.812431i	$-0.301856\pi$
$439$	10726.0	1.16611	0.583057	+	0.812431i	$0.301856\pi$
$440$	2310.00i	0.250284i
$441$	2187.00	0.236152
$442$	0	0
$443$	16228.0	1.74044	0.870221	−	0.492662i	$-0.163976\pi$
$443$	16228.0	1.74044	0.870221	+	0.492662i	$0.163976\pi$
$444$	− 273.000i	− 0.0291802i
$445$	1358.00	0.144664
$446$	4072.00	0.432320
$447$	− 1935.00i	− 0.204748i
$448$	1670.00i	0.176116i
$449$	− 7538.00i	− 0.792294i	−0.918187	−	0.396147i	$-0.870347\pi$
$449$	− 7538.00i	− 0.792294i	0.918187	−	0.396147i	$-0.129653\pi$
$450$	− 684.000i	− 0.0716535i
$451$	−6270.00	−0.654640
$452$	7539.00	0.784524
$453$	− 8742.00i	− 0.906700i
$454$	−5794.00	−0.598956
$455$	0	0
$456$	1350.00	0.138639
$457$	15539.0i	1.59056i	0.606245	+	0.795278i	$0.292675\pi$
$457$	15539.0i	1.59056i	−0.606245	+	0.795278i	$0.707325\pi$
$458$	−6482.00	−0.661319
$459$	−999.000	−0.101589
$460$	7938.00i	0.804589i
$461$	4811.00i	0.486053i	0.970020	+	0.243027i	$0.0781403\pi$
$461$	4811.00i	0.486053i	−0.970020	+	0.243027i	$0.921860\pi$
$462$	660.000i	0.0664632i
$463$	− 562.000i	− 0.0564111i	−0.999602	−	0.0282056i	$-0.991021\pi$
$463$	− 562.000i	− 0.0564111i	0.999602	−	0.0282056i	$-0.00897930\pi$
$464$	−4633.00	−0.463538
$465$	−4116.00	−0.410484
$466$	6890.00i	0.684921i
$467$	−4914.00	−0.486922	−0.243461	−	0.969911i	$-0.578283\pi$
$467$	−4914.00	−0.486922	−0.243461	+	0.969911i	$0.578283\pi$
$468$	0	0
$469$	−2020.00	−0.198880
$470$	3234.00i	0.317390i
$471$	−6237.00	−0.610161
$472$	−8640.00	−0.842560
$473$	5412.00i	0.526097i
$474$	− 2652.00i	− 0.256984i
$475$	2280.00i	0.220239i
$476$	− 2590.00i	− 0.249396i
$477$	−4833.00	−0.463916
$478$	2466.00	0.235967
$479$	3600.00i	0.343399i	0.985149	+	0.171700i	$0.0549258\pi$
$479$	3600.00i	0.343399i	−0.985149	+	0.171700i	$0.945074\pi$
$480$	3381.00	0.321502
$481$	0	0
$482$	3617.00	0.341805
$483$	4860.00i	0.457842i
$484$	5929.00	0.556818
$485$	8414.00	0.787753
$486$	− 243.000i	− 0.0226805i
$487$	− 17130.0i	− 1.59391i	−0.604038	−	0.796955i	$-0.706443\pi$
$487$	− 17130.0i	− 1.59391i	0.604038	−	0.796955i	$-0.293557\pi$
$488$	9525.00i	0.883558i
$489$	− 5100.00i	− 0.471636i
$490$	1701.00	0.156823
$491$	11838.0	1.08807	0.544034	−	0.839063i	$-0.316896\pi$
$491$	11838.0	1.08807	0.544034	+	0.839063i	$0.316896\pi$
$492$	5985.00i	0.548424i
$493$	4181.00	0.381953
$494$	0	0
$495$	−1386.00	−0.125851
$496$	8036.00i	0.727474i
$497$	10860.0	0.980156
$498$	1554.00	0.139832
$499$	8976.00i	0.805252i	0.915364	+	0.402626i	$0.131903\pi$
$499$	8976.00i	0.805252i	−0.915364	+	0.402626i	$0.868097\pi$
$500$	9849.00i	0.880921i
$501$	− 11040.0i	− 0.984493i
$502$	4860.00i	0.432096i
$503$	1682.00	0.149099	0.0745494	−	0.997217i	$-0.476248\pi$
$503$	1682.00	0.149099	0.0745494	+	0.997217i	$0.476248\pi$
$504$	1350.00	0.119313
$505$	3003.00i	0.264617i
$506$	3564.00	0.313121
$507$	0	0
$508$	6916.00	0.604031
$509$	15167.0i	1.32076i	0.750933	+	0.660379i	$0.229604\pi$
$509$	15167.0i	1.32076i	−0.750933	+	0.660379i	$0.770396\pi$
$510$	−777.000	−0.0674630
$511$	−8050.00	−0.696890
$512$	− 11521.0i	− 0.994455i
$513$	810.000i	0.0697122i
$514$	565.000i	0.0484846i
$515$	9114.00i	0.779827i
$516$	5166.00	0.440737
$517$	−10164.0	−0.864627
$518$	− 130.000i	− 0.0110268i
$519$	−12438.0	−1.05196
$520$	0	0
$521$	−6783.00	−0.570381	−0.285191	−	0.958471i	$-0.592057\pi$
$521$	−6783.00	−0.570381	−0.285191	+	0.958471i	$0.592057\pi$
$522$	1017.00i	0.0852737i
$523$	−13918.0	−1.16366	−0.581828	−	0.813312i	$-0.697662\pi$
$523$	−13918.0	−1.16366	−0.581828	+	0.813312i	$0.697662\pi$
$524$	3920.00	0.326805
$525$	2280.00i	0.189538i
$526$	498.000i	0.0412810i
$527$	− 7252.00i	− 0.599435i
$528$	2706.00i	0.223037i
$529$	14077.0	1.15698
$530$	−3759.00	−0.308076
$531$	− 5184.00i	− 0.423666i
$532$	−2100.00	−0.171140
$533$	0	0
$534$	−582.000	−0.0471641
$535$	− 9366.00i	− 0.756874i
$536$	3030.00	0.244172
$537$	−11022.0	−0.885725
$538$	− 5546.00i	− 0.444433i
$539$	5346.00i	0.427214i
$540$	1323.00i	0.105431i
$541$	1335.00i	0.106093i	0.998592	+	0.0530463i	$0.0168931\pi$
$541$	1335.00i	0.106093i	−0.998592	+	0.0530463i	$0.983107\pi$
$542$	2256.00	0.178789
$543$	9849.00	0.778381
$544$	5957.00i	0.469493i
$545$	7238.00	0.568884
$546$	0	0
$547$	−3806.00	−0.297501	−0.148750	−	0.988875i	$-0.547525\pi$
$547$	−3806.00	−0.297501	−0.148750	+	0.988875i	$0.547525\pi$
$548$	3633.00i	0.283201i
$549$	−5715.00	−0.444281
$550$	1672.00	0.129626
$551$	− 3390.00i	− 0.262103i
$552$	− 7290.00i	− 0.562107i
$553$	8840.00i	0.679774i
$554$	2309.00i	0.177076i
$555$	273.000	0.0208796
$556$	−2436.00	−0.185808
$557$	1905.00i	0.144915i	0.997372	+	0.0724573i	$0.0230841\pi$
$557$	1905.00i	0.144915i	−0.997372	+	0.0724573i	$0.976916\pi$
$558$	1764.00	0.133828
$559$	0	0
$560$	−2870.00	−0.216571
$561$	− 2442.00i	− 0.183781i
$562$	−5833.00	−0.437812
$563$	4800.00	0.359318	0.179659	−	0.983729i	$-0.442501\pi$
$563$	4800.00	0.359318	0.179659	+	0.983729i	$0.442501\pi$
$564$	9702.00i	0.724340i
$565$	7539.00i	0.561359i
$566$	1650.00i	0.122535i
$567$	810.000i	0.0599944i
$568$	−16290.0	−1.20337
$569$	−14678.0	−1.08143	−0.540715	−	0.841206i	$-0.681846\pi$
$569$	−14678.0	−1.08143	−0.540715	+	0.841206i	$0.681846\pi$
$570$	630.000i	0.0462944i
$571$	586.000	0.0429481	0.0214740	−	0.999769i	$-0.493164\pi$
$571$	586.000	0.0429481	0.0214740	+	0.999769i	$0.493164\pi$
$572$	0	0
$573$	−1788.00	−0.130357
$574$	2850.00i	0.207242i
$575$	12312.0	0.892949
$576$	1503.00	0.108724
$577$	8939.00i	0.644949i	0.946578	+	0.322474i	$0.104515\pi$
$577$	8939.00i	0.644949i	−0.946578	+	0.322474i	$0.895485\pi$
$578$	3544.00i	0.255036i
$579$	1179.00i	0.0846245i
$580$	− 5537.00i	− 0.396399i
$581$	−5180.00	−0.369884
$582$	−3606.00	−0.256827
$583$	− 11814.0i	− 0.839255i
$584$	12075.0	0.855594
$585$	0	0
$586$	−2991.00	−0.210848
$587$	− 13792.0i	− 0.969773i	−0.874577	−	0.484887i	$-0.838861\pi$
$587$	− 13792.0i	− 0.969773i	0.874577	−	0.484887i	$-0.161139\pi$
$588$	5103.00	0.357898
$589$	−5880.00	−0.411343
$590$	− 4032.00i	− 0.281347i
$591$	− 10566.0i	− 0.735410i
$592$	− 533.000i	− 0.0370037i
$593$	− 9569.00i	− 0.662650i	−0.943517	−	0.331325i	$-0.892504\pi$
$593$	− 9569.00i	− 0.662650i	0.943517	−	0.331325i	$-0.107496\pi$
$594$	594.000	0.0410305
$595$	2590.00	0.178453
$596$	− 4515.00i	− 0.310305i
$597$	−6054.00	−0.415031
$598$	0	0
$599$	−5192.00	−0.354156	−0.177078	−	0.984197i	$-0.556664\pi$
$599$	−5192.00	−0.354156	−0.177078	+	0.984197i	$0.556664\pi$
$600$	− 3420.00i	− 0.232702i
$601$	−3677.00	−0.249564	−0.124782	−	0.992184i	$-0.539823\pi$
$601$	−3677.00	−0.249564	−0.124782	+	0.992184i	$0.539823\pi$
$602$	2460.00	0.166548
$603$	1818.00i	0.122777i
$604$	− 20398.0i	− 1.37414i
$605$	5929.00i	0.398427i
$606$	− 1287.00i	− 0.0862719i
$607$	−10960.0	−0.732871	−0.366435	−	0.930443i	$-0.619422\pi$
$607$	−10960.0	−0.732871	−0.366435	+	0.930443i	$0.619422\pi$
$608$	4830.00	0.322175
$609$	− 3390.00i	− 0.225566i
$610$	−4445.00	−0.295037
$611$	0	0
$612$	−2331.00	−0.153963
$613$	− 26027.0i	− 1.71488i	−0.514585	−	0.857439i	$-0.672054\pi$
$613$	− 26027.0i	− 1.71488i	0.514585	−	0.857439i	$-0.327946\pi$
$614$	2422.00	0.159192
$615$	−5985.00	−0.392420
$616$	3300.00i	0.215845i
$617$	17681.0i	1.15366i	0.816863	+	0.576832i	$0.195711\pi$
$617$	17681.0i	1.15366i	−0.816863	+	0.576832i	$0.804289\pi$
$618$	− 3906.00i	− 0.254243i
$619$	− 3192.00i	− 0.207265i	−0.994616	−	0.103633i	$-0.966953\pi$
$619$	− 3192.00i	− 0.207265i	0.994616	−	0.103633i	$-0.0330467\pi$
$620$	−9604.00	−0.622106
$621$	4374.00	0.282645
$622$	− 3402.00i	− 0.219305i
$623$	1940.00	0.124758
$624$	0	0
$625$	−349.000	−0.0223360
$626$	− 2310.00i	− 0.147486i
$627$	−1980.00	−0.126114
$628$	−14553.0	−0.924726
$629$	481.000i	0.0304908i
$630$	630.000i	0.0398410i
$631$	− 7580.00i	− 0.478217i	−0.970993	−	0.239109i	$-0.923145\pi$
$631$	− 7580.00i	− 0.478217i	0.970993	−	0.239109i	$-0.0768552\pi$
$632$	− 13260.0i	− 0.834580i
$633$	480.000	0.0301395
$634$	−257.000	−0.0160990
$635$	6916.00i	0.432210i
$636$	−11277.0	−0.703085
$637$	0	0
$638$	−2486.00	−0.154266
$639$	− 9774.00i	− 0.605091i
$640$	10185.0	0.629059
$641$	27707.0	1.70727	0.853635	−	0.520871i	$-0.174393\pi$
$641$	27707.0	1.70727	0.853635	+	0.520871i	$0.174393\pi$
$642$	4014.00i	0.246760i
$643$	11216.0i	0.687894i	0.938989	+	0.343947i	$0.111764\pi$
$643$	11216.0i	0.687894i	−0.938989	+	0.343947i	$0.888236\pi$
$644$	11340.0i	0.693880i
$645$	5166.00i	0.315366i
$646$	−1110.00	−0.0676043
$647$	2536.00	0.154097	0.0770483	−	0.997027i	$-0.475450\pi$
$647$	2536.00	0.154097	0.0770483	+	0.997027i	$0.475450\pi$
$648$	− 1215.00i	− 0.0736570i
$649$	12672.0	0.766440
$650$	0	0
$651$	−5880.00	−0.354002
$652$	− 11900.0i	− 0.714785i
$653$	17730.0	1.06252	0.531262	−	0.847207i	$-0.321718\pi$
$653$	17730.0	1.06252	0.531262	+	0.847207i	$0.321718\pi$
$654$	−3102.00	−0.185471
$655$	3920.00i	0.233843i
$656$	11685.0i	0.695461i
$657$	7245.00i	0.430220i
$658$	4620.00i	0.273718i
$659$	18920.0	1.11839	0.559195	−	0.829036i	$-0.311110\pi$
$659$	18920.0	1.11839	0.559195	+	0.829036i	$0.311110\pi$
$660$	−3234.00	−0.190732
$661$	− 5241.00i	− 0.308398i	−0.988040	−	0.154199i	$-0.950720\pi$
$661$	− 5241.00i	− 0.308398i	0.988040	−	0.154199i	$-0.0492797\pi$
$662$	1028.00	0.0603540
$663$	0	0
$664$	7770.00	0.454118
$665$	− 2100.00i	− 0.122458i
$666$	−117.000	−0.00680729
$667$	−18306.0	−1.06269
$668$	− 25760.0i	− 1.49204i
$669$	12216.0i	0.705976i
$670$	1414.00i	0.0815337i
$671$	− 13970.0i	− 0.803735i
$672$	4830.00	0.277264
$673$	−20467.0	−1.17228	−0.586140	−	0.810210i	$-0.699353\pi$
$673$	−20467.0	−1.17228	−0.586140	+	0.810210i	$0.699353\pi$
$674$	2487.00i	0.142130i
$675$	2052.00	0.117010
$676$	0	0
$677$	−70.0000	−0.00397388	−0.00198694	−	0.999998i	$-0.500632\pi$
$677$	−70.0000	−0.00397388	−0.00198694	+	0.999998i	$0.500632\pi$
$678$	− 3231.00i	− 0.183017i
$679$	12020.0	0.679360
$680$	−3885.00	−0.219093
$681$	− 17382.0i	− 0.978091i
$682$	4312.00i	0.242104i
$683$	− 6432.00i	− 0.360342i	−0.983635	−	0.180171i	$-0.942335\pi$
$683$	− 6432.00i	− 0.360342i	0.983635	−	0.180171i	$-0.0576651\pi$
$684$	1890.00i	0.105652i
$685$	−3633.00	−0.202642
$686$	5860.00	0.326146
$687$	− 19446.0i	− 1.07993i
$688$	10086.0	0.558903
$689$	0	0
$690$	3402.00	0.187698
$691$	− 6666.00i	− 0.366985i	−0.983021	−	0.183492i	$-0.941260\pi$
$691$	− 6666.00i	− 0.366985i	0.983021	−	0.183492i	$-0.0587403\pi$
$692$	−29022.0	−1.59429
$693$	−1980.00	−0.108534
$694$	2850.00i	0.155885i
$695$	− 2436.00i	− 0.132954i
$696$	5085.00i	0.276935i
$697$	− 10545.0i	− 0.573056i
$698$	2018.00	0.109430
$699$	−20670.0	−1.11847
$700$	5320.00i	0.287253i
$701$	14054.0	0.757221	0.378611	−	0.925556i	$-0.376402\pi$
$701$	14054.0	0.757221	0.378611	+	0.925556i	$0.376402\pi$
$702$	0	0
$703$	390.000	0.0209234
$704$	3674.00i	0.196689i
$705$	−9702.00	−0.518296
$706$	−5287.00	−0.281840
$707$	4290.00i	0.228207i
$708$	− 12096.0i	− 0.642084i
$709$	71.0000i	0.00376088i	0.999998	+	0.00188044i	$0.000598562\pi$
$709$	71.0000i	0.00376088i	−0.999998	+	0.00188044i	$0.999401\pi$
$710$	− 7602.00i	− 0.401828i
$711$	7956.00	0.419653
$712$	−2910.00	−0.153170
$713$	31752.0i	1.66777i
$714$	−1110.00	−0.0581803
$715$	0	0
$716$	−25718.0	−1.34236
$717$	7398.00i	0.385332i
$718$	7278.00	0.378290
$719$	−3936.00	−0.204156	−0.102078	−	0.994776i	$-0.532549\pi$
$719$	−3936.00	−0.204156	−0.102078	+	0.994776i	$0.532549\pi$
$720$	2583.00i	0.133698i
$721$	13020.0i	0.672524i
$722$	− 5959.00i	− 0.307162i
$723$	10851.0i	0.558165i
$724$	22981.0	1.17967
$725$	−8588.00	−0.439931
$726$	− 2541.00i	− 0.129897i
$727$	−34202.0	−1.74482	−0.872409	−	0.488777i	$-0.837443\pi$
$727$	−34202.0	−1.74482	−0.872409	+	0.488777i	$0.837443\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	5635.00i	0.285700i
$731$	−9102.00	−0.460533
$732$	−13335.0	−0.673328
$733$	− 27363.0i	− 1.37882i	−0.724371	−	0.689410i	$-0.757870\pi$
$733$	− 27363.0i	− 1.37882i	0.724371	−	0.689410i	$-0.242130\pi$
$734$	4202.00i	0.211306i
$735$	5103.00i	0.256091i
$736$	− 26082.0i	− 1.30624i
$737$	−4444.00	−0.222112
$738$	2565.00	0.127939
$739$	− 21776.0i	− 1.08396i	−0.840393	−	0.541978i	$-0.817676\pi$
$739$	− 21776.0i	− 1.08396i	0.840393	−	0.541978i	$-0.182324\pi$
$740$	637.000	0.0316440
$741$	0	0
$742$	−5370.00	−0.265686
$743$	2484.00i	0.122650i	0.998118	+	0.0613251i	$0.0195326\pi$
$743$	2484.00i	0.122650i	−0.998118	+	0.0613251i	$0.980467\pi$
$744$	8820.00	0.434619
$745$	4515.00	0.222036
$746$	1583.00i	0.0776914i
$747$	4662.00i	0.228345i
$748$	− 5698.00i	− 0.278529i
$749$	− 13380.0i	− 0.652730i
$750$	4221.00	0.205506
$751$	−32906.0	−1.59888	−0.799439	−	0.600748i	$-0.794870\pi$
$751$	−32906.0	−1.59888	−0.799439	+	0.600748i	$0.794870\pi$
$752$	18942.0i	0.918542i
$753$	−14580.0	−0.705611
$754$	0	0
$755$	20398.0	0.983257
$756$	1890.00i	0.0909241i
$757$	−3914.00	−0.187922	−0.0939609	−	0.995576i	$-0.529953\pi$
$757$	−3914.00	−0.187922	−0.0939609	+	0.995576i	$0.529953\pi$
$758$	−2052.00	−0.0983272
$759$	10692.0i	0.511324i
$760$	3150.00i	0.150345i
$761$	33038.0i	1.57375i	0.617110	+	0.786877i	$0.288303\pi$
$761$	33038.0i	1.57375i	−0.617110	+	0.786877i	$0.711697\pi$
$762$	− 2964.00i	− 0.140911i
$763$	10340.0	0.490607
$764$	−4172.00	−0.197562
$765$	− 2331.00i	− 0.110167i
$766$	−6872.00	−0.324145
$767$	0	0
$768$	−357.000	−0.0167736
$769$	− 17586.0i	− 0.824665i	−0.911033	−	0.412332i	$-0.864714\pi$
$769$	− 17586.0i	− 0.824665i	0.911033	−	0.412332i	$-0.135286\pi$
$770$	−1540.00	−0.0720750
$771$	−1695.00	−0.0791750
$772$	2751.00i	0.128252i
$773$	18314.0i	0.852146i	0.904689	+	0.426073i	$0.140103\pi$
$773$	18314.0i	0.852146i	−0.904689	+	0.426073i	$0.859897\pi$
$774$	− 2214.00i	− 0.102817i
$775$	14896.0i	0.690426i
$776$	−18030.0	−0.834071
$777$	390.000	0.0180067
$778$	− 11653.0i	− 0.536993i
$779$	−8550.00	−0.393242
$780$	0	0
$781$	23892.0	1.09465
$782$	5994.00i	0.274098i
$783$	−3051.00	−0.139251
$784$	9963.00	0.453854
$785$	− 14553.0i	− 0.661680i
$786$	− 1680.00i	− 0.0762387i
$787$	42068.0i	1.90542i	0.303888	+	0.952708i	$0.401715\pi$
$787$	42068.0i	1.90542i	−0.303888	+	0.952708i	$0.598285\pi$
$788$	− 24654.0i	− 1.11455i
$789$	−1494.00	−0.0674117
$790$	6188.00	0.278682
$791$	10770.0i	0.484118i
$792$	2970.00	0.133250
$793$	0	0
$794$	−6134.00	−0.274166
$795$	− 11277.0i	− 0.503087i
$796$	−14126.0	−0.628998
$797$	4282.00	0.190309	0.0951545	−	0.995463i	$-0.469665\pi$
$797$	4282.00	0.190309	0.0951545	+	0.995463i	$0.469665\pi$
$798$	900.000i	0.0399244i
$799$	− 17094.0i	− 0.756874i
$800$	− 12236.0i	− 0.540760i
$801$	− 1746.00i	− 0.0770186i
$802$	10795.0	0.475293
$803$	−17710.0	−0.778297
$804$	4242.00i	0.186074i
$805$	−11340.0	−0.496500
$806$	0	0
$807$	16638.0	0.725756
$808$	− 6435.00i	− 0.280176i
$809$	40221.0	1.74795	0.873977	−	0.485967i	$-0.161532\pi$
$809$	40221.0	1.74795	0.873977	+	0.485967i	$0.161532\pi$
$810$	567.000	0.0245955
$811$	− 7084.00i	− 0.306724i	−0.988170	−	0.153362i	$-0.950990\pi$
$811$	− 7084.00i	− 0.306724i	0.988170	−	0.153362i	$-0.0490100\pi$
$812$	− 7910.00i	− 0.341855i
$813$	6768.00i	0.291961i
$814$	− 286.000i	− 0.0123149i
$815$	11900.0	0.511459
$816$	−4551.00	−0.195241
$817$	7380.00i	0.316026i
$818$	−8489.00	−0.362850
$819$	0	0
$820$	−13965.0	−0.594730
$821$	17338.0i	0.737028i	0.929622	+	0.368514i	$0.120133\pi$
$821$	17338.0i	0.737028i	−0.929622	+	0.368514i	$0.879867\pi$
$822$	1557.00	0.0660664
$823$	−35496.0	−1.50342	−0.751709	−	0.659495i	$-0.770770\pi$
$823$	−35496.0	−1.50342	−0.751709	+	0.659495i	$0.770770\pi$
$824$	− 19530.0i	− 0.825679i
$825$	5016.00i	0.211678i
$826$	− 5760.00i	− 0.242634i
$827$	14992.0i	0.630378i	0.949029	+	0.315189i	$0.102068\pi$
$827$	14992.0i	0.630378i	−0.949029	+	0.315189i	$0.897932\pi$
$828$	10206.0	0.428361
$829$	20659.0	0.865521	0.432760	−	0.901509i	$-0.357540\pi$
$829$	20659.0	0.865521	0.432760	+	0.901509i	$0.357540\pi$
$830$	3626.00i	0.151639i
$831$	−6927.00	−0.289164
$832$	0	0
$833$	−8991.00	−0.373973
$834$	1044.00i	0.0433462i
$835$	25760.0	1.06762
$836$	−4620.00	−0.191132
$837$	5292.00i	0.218540i
$838$	− 1496.00i	− 0.0616688i
$839$	− 28716.0i	− 1.18163i	−0.806808	−	0.590814i	$-0.798807\pi$
$839$	− 28716.0i	− 1.18163i	0.806808	−	0.590814i	$-0.201193\pi$
$840$	3150.00i	0.129387i
$841$	−11620.0	−0.476444
$842$	−11695.0	−0.478665
$843$	− 17499.0i	− 0.714944i
$844$	1120.00	0.0456777
$845$	0	0
$846$	4158.00	0.168978
$847$	8470.00i	0.343604i
$848$	−22017.0	−0.891588
$849$	−4950.00	−0.200098
$850$	2812.00i	0.113472i
$851$	− 2106.00i	− 0.0848328i
$852$	− 22806.0i	− 0.917043i
$853$	− 13377.0i	− 0.536952i	−0.963286	−	0.268476i	$-0.913480\pi$
$853$	− 13377.0i	− 0.536952i	0.963286	−	0.268476i	$-0.0865199\pi$
$854$	−6350.00	−0.254441
$855$	−1890.00	−0.0755984
$856$	20070.0i	0.801377i
$857$	27419.0	1.09290	0.546450	−	0.837492i	$-0.315979\pi$
$857$	27419.0	1.09290	0.546450	+	0.837492i	$0.315979\pi$
$858$	0	0
$859$	2422.00	0.0962021	0.0481010	−	0.998842i	$-0.484683\pi$
$859$	2422.00	0.0962021	0.0481010	+	0.998842i	$0.484683\pi$
$860$	12054.0i	0.477951i
$861$	−8550.00	−0.338424
$862$	10590.0	0.418442
$863$	− 34522.0i	− 1.36169i	−0.732425	−	0.680847i	$-0.761612\pi$
$863$	− 34522.0i	− 1.36169i	0.732425	−	0.680847i	$-0.238388\pi$
$864$	− 4347.00i	− 0.171167i
$865$	− 29022.0i	− 1.14078i
$866$	− 13949.0i	− 0.547351i
$867$	−10632.0	−0.416472
$868$	−13720.0	−0.536506
$869$	19448.0i	0.759181i
$870$	−2373.00	−0.0924738
$871$	0	0
$872$	−15510.0	−0.602334
$873$	− 10818.0i	− 0.419397i
$874$	4860.00	0.188091
$875$	−14070.0	−0.543603
$876$	16905.0i	0.652017i
$877$	13733.0i	0.528769i	0.964417	+	0.264385i	$0.0851688\pi$
$877$	13733.0i	0.528769i	−0.964417	+	0.264385i	$0.914831\pi$
$878$	− 10726.0i	− 0.412284i
$879$	− 8973.00i	− 0.344314i
$880$	−6314.00	−0.241869
$881$	22759.0	0.870341	0.435170	−	0.900348i	$-0.356688\pi$
$881$	22759.0	0.870341	0.435170	+	0.900348i	$0.356688\pi$
$882$	− 2187.00i	− 0.0834922i
$883$	2168.00	0.0826263	0.0413131	−	0.999146i	$-0.486846\pi$
$883$	2168.00	0.0826263	0.0413131	+	0.999146i	$0.486846\pi$
$884$	0	0
$885$	12096.0	0.459438
$886$	− 16228.0i	− 0.615339i
$887$	15888.0	0.601428	0.300714	−	0.953714i	$-0.402775\pi$
$887$	15888.0	0.601428	0.300714	+	0.953714i	$0.402775\pi$
$888$	−585.000	−0.0221073
$889$	9880.00i	0.372739i
$890$	− 1358.00i	− 0.0511464i
$891$	1782.00i	0.0670025i
$892$	28504.0i	1.06994i
$893$	−13860.0	−0.519381
$894$	−1935.00	−0.0723894
$895$	− 25718.0i	− 0.960512i
$896$	14550.0	0.542502
$897$	0	0
$898$	−7538.00	−0.280118
$899$	− 22148.0i	− 0.821665i
$900$	4788.00	0.177333
$901$	19869.0	0.734664
$902$	6270.00i	0.231450i
$903$	7380.00i	0.271972i
$904$	− 16155.0i	− 0.594366i
$905$	22981.0i	0.844104i
$906$	−8742.00	−0.320567
$907$	11628.0	0.425691	0.212845	−	0.977086i	$-0.431727\pi$
$907$	11628.0	0.425691	0.212845	+	0.977086i	$0.431727\pi$
$908$	− 40558.0i	− 1.48234i
$909$	3861.00	0.140882
$910$	0	0
$911$	−12584.0	−0.457658	−0.228829	−	0.973467i	$-0.573490\pi$
$911$	−12584.0	−0.457658	−0.228829	+	0.973467i	$0.573490\pi$
$912$	3690.00i	0.133978i
$913$	−11396.0	−0.413092
$914$	15539.0	0.562346
$915$	− 13335.0i	− 0.481794i
$916$	− 45374.0i	− 1.63668i
$917$	5600.00i	0.201667i
$918$	999.000i	0.0359171i
$919$	17184.0	0.616809	0.308405	−	0.951255i	$-0.400205\pi$
$919$	17184.0	0.616809	0.308405	+	0.951255i	$0.400205\pi$
$920$	17010.0	0.609569
$921$	7266.00i	0.259960i
$922$	4811.00	0.171846
$923$	0	0
$924$	−4620.00	−0.164488
$925$	− 988.000i	− 0.0351192i
$926$	−562.000	−0.0199443
$927$	11718.0	0.415178
$928$	18193.0i	0.643550i
$929$	12777.0i	0.451238i	0.974216	+	0.225619i	$0.0724404\pi$
$929$	12777.0i	0.451238i	−0.974216	+	0.225619i	$0.927560\pi$
$930$	4116.00i	0.145128i
$931$	7290.00i	0.256627i
$932$	−48230.0	−1.69509
$933$	10206.0	0.358124
$934$	4914.00i	0.172153i
$935$	5698.00	0.199299
$936$	0	0
$937$	9191.00	0.320445	0.160222	−	0.987081i	$-0.448779\pi$
$937$	9191.00	0.320445	0.160222	+	0.987081i	$0.448779\pi$
$938$	2020.00i	0.0703149i
$939$	6930.00	0.240843
$940$	−22638.0	−0.785500
$941$	50498.0i	1.74940i	0.484662	+	0.874701i	$0.338942\pi$
$941$	50498.0i	1.74940i	−0.484662	+	0.874701i	$0.661058\pi$
$942$	6237.00i	0.215724i
$943$	46170.0i	1.59438i
$944$	− 23616.0i	− 0.814232i
$945$	−1890.00	−0.0650600
$946$	5412.00	0.186003
$947$	− 1560.00i	− 0.0535303i	−0.999642	−	0.0267651i	$-0.991479\pi$
$947$	− 1560.00i	− 0.0535303i	0.999642	−	0.0267651i	$-0.00852063\pi$
$948$	18564.0	0.636003
$949$	0	0
$950$	2280.00	0.0778663
$951$	− 771.000i	− 0.0262896i
$952$	−5550.00	−0.188946
$953$	21498.0	0.730733	0.365366	−	0.930864i	$-0.380944\pi$
$953$	21498.0	0.730733	0.365366	+	0.930864i	$0.380944\pi$
$954$	4833.00i	0.164019i
$955$	− 4172.00i	− 0.141364i
$956$	17262.0i	0.583988i
$957$	− 7458.00i	− 0.251915i
$958$	3600.00	0.121410
$959$	−5190.00	−0.174759
$960$	3507.00i	0.117904i
$961$	−8625.00	−0.289517
$962$	0	0
$963$	−12042.0	−0.402957
$964$	25319.0i	0.845923i
$965$	−2751.00	−0.0917698
$966$	4860.00	0.161872
$967$	− 418.000i	− 0.0139007i	−0.999976	−	0.00695035i	$-0.997788\pi$
$967$	− 418.000i	− 0.0139007i	0.999976	−	0.00695035i	$-0.00221238\pi$
$968$	− 12705.0i	− 0.421853i
$969$	− 3330.00i	− 0.110397i
$970$	− 8414.00i	− 0.278513i
$971$	18132.0	0.599262	0.299631	−	0.954055i	$-0.403136\pi$
$971$	18132.0	0.599262	0.299631	+	0.954055i	$0.403136\pi$
$972$	1701.00	0.0561313
$973$	− 3480.00i	− 0.114659i
$974$	−17130.0	−0.563532
$975$	0	0
$976$	−26035.0	−0.853853
$977$	12501.0i	0.409358i	0.978829	+	0.204679i	$0.0656150\pi$
$977$	12501.0i	0.409358i	−0.978829	+	0.204679i	$0.934385\pi$
$978$	−5100.00	−0.166748
$979$	4268.00	0.139332
$980$	11907.0i	0.388118i
$981$	− 9306.00i	− 0.302872i
$982$	− 11838.0i	− 0.384690i
$983$	43708.0i	1.41818i	0.705119	+	0.709089i	$0.250894\pi$
$983$	43708.0i	1.41818i	−0.705119	+	0.709089i	$0.749106\pi$
$984$	12825.0	0.415494
$985$	24654.0	0.797504
$986$	− 4181.00i	− 0.135041i
$987$	−13860.0	−0.446979
$988$	0	0
$989$	39852.0	1.28131
$990$	1386.00i	0.0444949i
$991$	−39614.0	−1.26981	−0.634904	−	0.772591i	$-0.718960\pi$
$991$	−39614.0	−1.26981	−0.634904	+	0.772591i	$0.718960\pi$
$992$	31556.0	1.00998
$993$	3084.00i	0.0985577i
$994$	− 10860.0i	− 0.346538i
$995$	− 14126.0i	− 0.450075i
$996$	10878.0i	0.346067i
$997$	−36503.0	−1.15954	−0.579770	−	0.814780i	$-0.696858\pi$
$997$	−36503.0	−1.15954	−0.579770	+	0.814780i	$0.696858\pi$
$998$	8976.00	0.284700
$999$	− 351.000i	− 0.0111163i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.d.337.1		2
13.2	odd	12		39.4.e.b.22.1	yes	2
13.5	odd	4		507.4.a.b.1.1		1
13.6	odd	12		39.4.e.b.16.1	✓	2
13.8	odd	4		507.4.a.d.1.1		1
13.12	even	2	inner	507.4.b.d.337.2		2
39.2	even	12		117.4.g.a.100.1		2
39.5	even	4		1521.4.a.h.1.1		1
39.8	even	4		1521.4.a.e.1.1		1
39.32	even	12		117.4.g.a.55.1		2
52.15	even	12		624.4.q.c.529.1		2
52.19	even	12		624.4.q.c.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.b.16.1	✓	2	13.6	odd	12
39.4.e.b.22.1	yes	2	13.2	odd	12
117.4.g.a.55.1		2	39.32	even	12
117.4.g.a.100.1		2	39.2	even	12
507.4.a.b.1.1		1	13.5	odd	4
507.4.a.d.1.1		1	13.8	odd	4
507.4.b.d.337.1		2	1.1	even	1	trivial
507.4.b.d.337.2		2	13.12	even	2	inner
624.4.q.c.289.1		2	52.19	even	12
624.4.q.c.529.1		2	52.15	even	12
1521.4.a.e.1.1		1	39.8	even	4
1521.4.a.h.1.1		1	39.5	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +3.00000 q^{3} +7.00000 q^{4} +7.00000i q^{5} -3.00000i q^{6} +10.0000i q^{7} -15.0000i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +3.00000 q^{3} +7.00000 q^{4} +7.00000i q^{5} -3.00000i q^{6} +10.0000i q^{7} -15.0000i q^{8} +9.00000 q^{9} +7.00000 q^{10} +22.0000i q^{11} +21.0000 q^{12} +10.0000 q^{14} +21.0000i q^{15} +41.0000 q^{16} -37.0000 q^{17} -9.00000i q^{18} +30.0000i q^{19} +49.0000i q^{20} +30.0000i q^{21} +22.0000 q^{22} +162.000 q^{23} -45.0000i q^{24} +76.0000 q^{25} +27.0000 q^{27} +70.0000i q^{28} -113.000 q^{29} +21.0000 q^{30} +196.000i q^{31} -161.000i q^{32} +66.0000i q^{33} +37.0000i q^{34} -70.0000 q^{35} +63.0000 q^{36} -13.0000i q^{37} +30.0000 q^{38} +105.000 q^{40} +285.000i q^{41} +30.0000 q^{42} +246.000 q^{43} +154.000i q^{44} +63.0000i q^{45} -162.000i q^{46} +462.000i q^{47} +123.000 q^{48} +243.000 q^{49} -76.0000i q^{50} -111.000 q^{51} -537.000 q^{53} -27.0000i q^{54} -154.000 q^{55} +150.000 q^{56} +90.0000i q^{57} +113.000i q^{58} -576.000i q^{59} +147.000i q^{60} -635.000 q^{61} +196.000 q^{62} +90.0000i q^{63} +167.000 q^{64} +66.0000 q^{66} +202.000i q^{67} -259.000 q^{68} +486.000 q^{69} +70.0000i q^{70} -1086.00i q^{71} -135.000i q^{72} +805.000i q^{73} -13.0000 q^{74} +228.000 q^{75} +210.000i q^{76} -220.000 q^{77} +884.000 q^{79} +287.000i q^{80} +81.0000 q^{81} +285.000 q^{82} +518.000i q^{83} +210.000i q^{84} -259.000i q^{85} -246.000i q^{86} -339.000 q^{87} +330.000 q^{88} -194.000i q^{89} +63.0000 q^{90} +1134.00 q^{92} +588.000i q^{93} +462.000 q^{94} -210.000 q^{95} -483.000i q^{96} -1202.00i q^{97} -243.000i q^{98} +198.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 14 q^{4} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 14 * q^4 + 18 * q^9	\( 2 q + 6 q^{3} + 14 q^{4} + 18 q^{9} + 14 q^{10} + 42 q^{12} + 20 q^{14} + 82 q^{16} - 74 q^{17} + 44 q^{22} + 324 q^{23} + 152 q^{25} + 54 q^{27} - 226 q^{29} + 42 q^{30} - 140 q^{35} + 126 q^{36} + 60 q^{38} + 210 q^{40} + 60 q^{42} + 492 q^{43} + 246 q^{48} + 486 q^{49} - 222 q^{51} - 1074 q^{53} - 308 q^{55} + 300 q^{56} - 1270 q^{61} + 392 q^{62} + 334 q^{64} + 132 q^{66} - 518 q^{68} + 972 q^{69} - 26 q^{74} + 456 q^{75} - 440 q^{77} + 1768 q^{79} + 162 q^{81} + 570 q^{82} - 678 q^{87} + 660 q^{88} + 126 q^{90} + 2268 q^{92} + 924 q^{94} - 420 q^{95}+O(q^{100}) \) 2 * q + 6 * q^3 + 14 * q^4 + 18 * q^9 + 14 * q^10 + 42 * q^12 + 20 * q^14 + 82 * q^16 - 74 * q^17 + 44 * q^22 + 324 * q^23 + 152 * q^25 + 54 * q^27 - 226 * q^29 + 42 * q^30 - 140 * q^35 + 126 * q^36 + 60 * q^38 + 210 * q^40 + 60 * q^42 + 492 * q^43 + 246 * q^48 + 486 * q^49 - 222 * q^51 - 1074 * q^53 - 308 * q^55 + 300 * q^56 - 1270 * q^61 + 392 * q^62 + 334 * q^64 + 132 * q^66 - 518 * q^68 + 972 * q^69 - 26 * q^74 + 456 * q^75 - 440 * q^77 + 1768 * q^79 + 162 * q^81 + 570 * q^82 - 678 * q^87 + 660 * q^88 + 126 * q^90 + 2268 * q^92 + 924 * q^94 - 420 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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