LMFDB - Embedded newform 507.4.b.h.337.3

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 $ x^8 + 54x^6 + 889x^4 + 4584*x^2 + 5776
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 13^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-1.36176i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.h.337.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36176i q^{2} -3.00000 q^{3} +2.42208 q^{4} -6.42208i q^{5} +7.08529i q^{6} -29.4938i q^{7} -24.6145i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.36176i q^{2} -3.00000 q^{3} +2.42208 q^{4} -6.42208i q^{5} +7.08529i q^{6} -29.4938i q^{7} -24.6145i q^{8} +9.00000 q^{9} -15.1674 q^{10} -0.624715i q^{11} -7.26623 q^{12} -69.6575 q^{14} +19.2662i q^{15} -38.7569 q^{16} -87.7291 q^{17} -21.2559i q^{18} -82.8018i q^{19} -15.5548i q^{20} +88.4815i q^{21} -1.47543 q^{22} +74.7977 q^{23} +73.8434i q^{24} +83.7569 q^{25} -27.0000 q^{27} -71.4364i q^{28} +226.329 q^{29} +45.5023 q^{30} -173.660i q^{31} -105.381i q^{32} +1.87415i q^{33} +207.195i q^{34} -189.412 q^{35} +21.7987 q^{36} +112.020i q^{37} -195.558 q^{38} -158.076 q^{40} +267.011i q^{41} +208.972 q^{42} -383.450 q^{43} -1.51311i q^{44} -57.7987i q^{45} -176.654i q^{46} +337.380i q^{47} +116.271 q^{48} -526.887 q^{49} -197.814i q^{50} +263.187 q^{51} -146.354 q^{53} +63.7676i q^{54} -4.01197 q^{55} -725.975 q^{56} +248.406i q^{57} -534.536i q^{58} -529.173i q^{59} +46.6643i q^{60} +203.272 q^{61} -410.144 q^{62} -265.445i q^{63} -558.941 q^{64} +4.42629 q^{66} -121.497i q^{67} -212.487 q^{68} -224.393 q^{69} +447.346i q^{70} +661.314i q^{71} -221.530i q^{72} +167.341i q^{73} +264.565 q^{74} -251.271 q^{75} -200.552i q^{76} -18.4253 q^{77} -101.399 q^{79} +248.900i q^{80} +81.0000 q^{81} +630.617 q^{82} +506.985i q^{83} +214.309i q^{84} +563.403i q^{85} +905.617i q^{86} -678.988 q^{87} -15.3770 q^{88} +1402.33i q^{89} -136.507 q^{90} +181.166 q^{92} +520.981i q^{93} +796.810 q^{94} -531.760 q^{95} +316.143i q^{96} -1902.89i q^{97} +1244.38i q^{98} -5.62244i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10}) $ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9	$ 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100}) $ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 + 124 * q^10 + 132 * q^12 + 80 * q^14 + 244 * q^16 - 196 * q^17 + 440 * q^22 - 208 * q^23 + 116 * q^25 - 216 * q^27 + 388 * q^29 - 372 * q^30 + 176 * q^35 - 396 * q^36 - 664 * q^38 - 1996 * q^40 - 240 * q^42 - 900 * q^43 - 732 * q^48 - 2140 * q^49 + 588 * q^51 + 524 * q^53 + 408 * q^55 - 4328 * q^56 - 1856 * q^61 - 5560 * q^62 - 2052 * q^64 - 1320 * q^66 + 3572 * q^68 + 624 * q^69 + 2316 * q^74 - 348 * q^75 - 5016 * q^77 - 1492 * q^79 + 648 * q^81 - 3468 * q^82 - 1164 * q^87 - 6120 * q^88 + 1116 * q^90 + 664 * q^92 - 1544 * q^94 - 4408 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.36176i	− 0.835009i	−0.908675	−	0.417505i	$-0.862905\pi$
$2$	− 2.36176i	− 0.835009i	0.908675	−	0.417505i	$-0.137095\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	2.42208	0.302760
$5$	− 6.42208i	− 0.574408i	−0.957869	−	0.287204i	$-0.907274\pi$
$5$	− 6.42208i	− 0.574408i	0.957869	−	0.287204i	$-0.0927258\pi$
$6$	7.08529i	0.482093i
$7$	− 29.4938i	− 1.59252i	−0.604956	−	0.796259i	$-0.706809\pi$
$7$	− 29.4938i	− 1.59252i	0.604956	−	0.796259i	$-0.293191\pi$
$8$	− 24.6145i	− 1.08782i
$9$	9.00000	0.333333
$10$	−15.1674	−0.479636
$11$	− 0.624715i	− 0.0171235i	−0.999963	−	0.00856176i	$-0.997275\pi$
$11$	− 0.624715i	− 0.0171235i	0.999963	−	0.00856176i	$-0.00272533\pi$
$12$	−7.26623	−0.174798
$13$	0	0
$13$	0	0
$14$	−69.6575	−1.32977
$15$	19.2662i	0.331635i
$16$	−38.7569	−0.605577
$17$	−87.7291	−1.25161	−0.625807	−	0.779978i	$-0.715230\pi$
$17$	−87.7291	−1.25161	−0.625807	+	0.779978i	$0.715230\pi$
$18$	− 21.2559i	− 0.278336i
$19$	− 82.8018i	− 0.999792i	−0.866085	−	0.499896i	$-0.833372\pi$
$19$	− 82.8018i	− 0.999792i	0.866085	−	0.499896i	$-0.166628\pi$
$20$	− 15.5548i	− 0.173908i
$21$	88.4815i	0.919441i
$22$	−1.47543	−0.0142983
$23$	74.7977	0.678104	0.339052	−	0.940768i	$-0.389894\pi$
$23$	74.7977	0.678104	0.339052	+	0.940768i	$0.389894\pi$
$24$	73.8434i	0.628051i
$25$	83.7569	0.670055
$26$	0	0
$27$	−27.0000	−0.192450
$28$	− 71.4364i	− 0.482150i
$29$	226.329	1.44925	0.724625	−	0.689143i	$-0.242013\pi$
$29$	226.329	1.44925	0.724625	+	0.689143i	$0.242013\pi$
$30$	45.5023	0.276918
$31$	− 173.660i	− 1.00614i	−0.864246	−	0.503070i	$-0.832204\pi$
$31$	− 173.660i	− 1.00614i	0.864246	−	0.503070i	$-0.167796\pi$
$32$	− 105.381i	− 0.582154i
$33$	1.87415i	0.00988627i
$34$	207.195i	1.04511i
$35$	−189.412	−0.914755
$36$	21.7987	0.100920
$37$	112.020i	0.497730i	0.968538	+	0.248865i	$0.0800576\pi$
$37$	112.020i	0.497730i	−0.968538	+	0.248865i	$0.919942\pi$
$38$	−195.558	−0.834835
$39$	0	0
$40$	−158.076	−0.624850
$41$	267.011i	1.01708i	0.861040	+	0.508538i	$0.169814\pi$
$41$	267.011i	1.01708i	−0.861040	+	0.508538i	$0.830186\pi$
$42$	208.972	0.767741
$43$	−383.450	−1.35990	−0.679948	−	0.733260i	$-0.737998\pi$
$43$	−383.450	−1.35990	−0.679948	+	0.733260i	$0.737998\pi$
$44$	− 1.51311i	− 0.00518431i
$45$	− 57.7987i	− 0.191469i
$46$	− 176.654i	− 0.566223i
$47$	337.380i	1.04706i	0.852007	+	0.523530i	$0.175385\pi$
$47$	337.380i	1.04706i	−0.852007	+	0.523530i	$0.824615\pi$
$48$	116.271	0.349630
$49$	−526.887	−1.53611
$50$	− 197.814i	− 0.559502i
$51$	263.187	0.722619
$52$	0	0
$53$	−146.354	−0.379308	−0.189654	−	0.981851i	$-0.560737\pi$
$53$	−146.354	−0.379308	−0.189654	+	0.981851i	$0.560737\pi$
$54$	63.7676i	0.160698i
$55$	−4.01197	−0.00983589
$56$	−725.975	−1.73237
$57$	248.406i	0.577230i
$58$	− 534.536i	− 1.21014i
$59$	− 529.173i	− 1.16767i	−0.811873	−	0.583834i	$-0.801552\pi$
$59$	− 529.173i	− 1.16767i	0.811873	−	0.583834i	$-0.198448\pi$
$60$	46.6643i	0.100406i
$61$	203.272	0.426660	0.213330	−	0.976980i	$-0.431569\pi$
$61$	203.272	0.426660	0.213330	+	0.976980i	$0.431569\pi$
$62$	−410.144	−0.840135
$63$	− 265.445i	− 0.530839i
$64$	−558.941	−1.09168
$65$	0	0
$66$	4.42629	0.00825513
$67$	− 121.497i	− 0.221540i	−0.993846	−	0.110770i	$-0.964668\pi$
$67$	− 121.497i	− 0.221540i	0.993846	−	0.110770i	$-0.0353317\pi$
$68$	−212.487	−0.378938
$69$	−224.393	−0.391504
$70$	447.346i	0.763829i
$71$	661.314i	1.10540i	0.833380	+	0.552701i	$0.186403\pi$
$71$	661.314i	1.10540i	−0.833380	+	0.552701i	$0.813597\pi$
$72$	− 221.530i	− 0.362605i
$73$	167.341i	0.268299i	0.990961	+	0.134150i	$0.0428302\pi$
$73$	167.341i	0.268299i	−0.990961	+	0.134150i	$0.957170\pi$
$74$	264.565	0.415609
$75$	−251.271	−0.386857
$76$	− 200.552i	− 0.302697i
$77$	−18.4253	−0.0272695
$78$	0	0
$79$	−101.399	−0.144408	−0.0722042	−	0.997390i	$-0.523003\pi$
$79$	−101.399	−0.144408	−0.0722042	+	0.997390i	$0.523003\pi$
$80$	248.900i	0.347848i
$81$	81.0000	0.111111
$82$	630.617	0.849268
$83$	506.985i	0.670468i	0.942135	+	0.335234i	$0.108815\pi$
$83$	506.985i	0.670468i	−0.942135	+	0.335234i	$0.891185\pi$
$84$	214.309i	0.278370i
$85$	563.403i	0.718937i
$86$	905.617i	1.13553i
$87$	−678.988	−0.836725
$88$	−15.3770	−0.0186272
$89$	1402.33i	1.67019i	0.550106	+	0.835095i	$0.314587\pi$
$89$	1402.33i	1.67019i	−0.550106	+	0.835095i	$0.685413\pi$
$90$	−136.507	−0.159879
$91$	0	0
$92$	181.166	0.205303
$93$	520.981i	0.580895i
$94$	796.810	0.874306
$95$	−531.760	−0.574289
$96$	316.143i	0.336107i
$97$	− 1902.89i	− 1.99185i	−0.0901969	−	0.995924i	$-0.528750\pi$
$97$	− 1902.89i	− 1.99185i	0.0901969	−	0.995924i	$-0.471250\pi$
$98$	1244.38i	1.28267i
$99$	− 5.62244i	− 0.00570784i
$100$	202.866	0.202866
$101$	−1833.09	−1.80594	−0.902968	−	0.429708i	$-0.858617\pi$
$101$	−1833.09	−1.80594	−0.902968	+	0.429708i	$0.858617\pi$
$102$	− 621.586i	− 0.603394i
$103$	−1446.99	−1.38423	−0.692115	−	0.721787i	$-0.743321\pi$
$103$	−1446.99	−1.38423	−0.692115	+	0.721787i	$0.743321\pi$
$104$	0	0
$105$	568.235	0.528134
$106$	345.654i	0.316725i
$107$	−369.286	−0.333647	−0.166823	−	0.985987i	$-0.553351\pi$
$107$	−369.286	−0.333647	−0.166823	+	0.985987i	$0.553351\pi$
$108$	−65.3961	−0.0582661
$109$	815.694i	0.716782i	0.933572	+	0.358391i	$0.116675\pi$
$109$	815.694i	0.716782i	−0.933572	+	0.358391i	$0.883325\pi$
$110$	9.47532i	0.00821306i
$111$	− 336.061i	− 0.287365i
$112$	1143.09i	0.964392i
$113$	1790.56	1.49064	0.745319	−	0.666708i	$-0.232297\pi$
$113$	1790.56	1.49064	0.745319	+	0.666708i	$0.232297\pi$
$114$	586.675	0.481992
$115$	− 480.357i	− 0.389509i
$116$	548.187	0.438775
$117$	0	0
$118$	−1249.78	−0.975014
$119$	2587.47i	1.99322i
$120$	474.228	0.360758
$121$	1330.61	0.999707
$122$	− 480.079i	− 0.356265i
$123$	− 801.033i	− 0.587209i
$124$	− 420.619i	− 0.304618i
$125$	− 1340.65i	− 0.959293i
$126$	−626.917	−0.443256
$127$	45.2900	0.0316444	0.0158222	−	0.999875i	$-0.494963\pi$
$127$	45.2900	0.0316444	0.0158222	+	0.999875i	$0.494963\pi$
$128$	477.036i	0.329410i
$129$	1150.35	0.785137
$130$	0	0
$131$	1051.82	0.701511	0.350756	−	0.936467i	$-0.385925\pi$
$131$	1051.82	0.701511	0.350756	+	0.936467i	$0.385925\pi$
$132$	4.53933i	0.00299316i
$133$	−2442.14	−1.59219
$134$	−286.946	−0.184988
$135$	173.396i	0.110545i
$136$	2159.41i	1.36153i
$137$	− 1542.94i	− 0.962208i	−0.876664	−	0.481104i	$-0.840236\pi$
$137$	− 1542.94i	− 0.962208i	0.876664	−	0.481104i	$-0.159764\pi$
$138$	529.963i	0.326909i
$139$	37.8644	0.0231052	0.0115526	−	0.999933i	$-0.496323\pi$
$139$	37.8644	0.0231052	0.0115526	+	0.999933i	$0.496323\pi$
$140$	−458.770	−0.276951
$141$	− 1012.14i	− 0.604521i
$142$	1561.87	0.923020
$143$	0	0
$144$	−348.812	−0.201859
$145$	− 1453.50i	− 0.832461i
$146$	395.221	0.224032
$147$	1580.66	0.886875
$148$	271.322i	0.150693i
$149$	1822.40i	1.00199i	0.865450	+	0.500995i	$0.167033\pi$
$149$	1822.40i	1.00199i	−0.865450	+	0.500995i	$0.832967\pi$
$150$	593.442i	0.323029i
$151$	− 3239.36i	− 1.74580i	−0.487899	−	0.872900i	$-0.662237\pi$
$151$	− 3239.36i	− 1.74580i	0.487899	−	0.872900i	$-0.337763\pi$
$152$	−2038.12	−1.08759
$153$	−789.562	−0.417205
$154$	43.5161i	0.0227703i
$155$	−1115.26	−0.577934
$156$	0	0
$157$	830.565	0.422206	0.211103	−	0.977464i	$-0.432295\pi$
$157$	830.565	0.422206	0.211103	+	0.977464i	$0.432295\pi$
$158$	239.480i	0.120582i
$159$	439.063	0.218993
$160$	−676.766	−0.334394
$161$	− 2206.07i	− 1.07989i
$162$	− 191.303i	− 0.0927788i
$163$	− 2079.90i	− 0.999451i	−0.866184	−	0.499725i	$-0.833434\pi$
$163$	− 2079.90i	− 0.999451i	0.866184	−	0.499725i	$-0.166566\pi$
$164$	646.721i	0.307930i
$165$	12.0359	0.00567875
$166$	1197.38	0.559847
$167$	85.9790i	0.0398398i	0.999802	+	0.0199199i	$0.00634113\pi$
$167$	85.9790i	0.0398398i	−0.999802	+	0.0199199i	$0.993659\pi$
$168$	2177.93	1.00018
$169$	0	0
$170$	1330.62	0.600319
$171$	− 745.217i	− 0.333264i
$172$	−928.745	−0.411722
$173$	−2706.47	−1.18942	−0.594708	−	0.803942i	$-0.702732\pi$
$173$	−2706.47	−1.18942	−0.594708	+	0.803942i	$0.702732\pi$
$174$	1603.61i	0.698673i
$175$	− 2470.31i	− 1.06708i
$176$	24.2120i	0.0103696i
$177$	1587.52i	0.674154i
$178$	3311.97	1.39462
$179$	4402.10	1.83815	0.919074	−	0.394085i	$-0.128938\pi$
$179$	4402.10	1.83815	0.919074	+	0.394085i	$0.128938\pi$
$180$	− 139.993i	− 0.0579692i
$181$	1673.98	0.687435	0.343718	−	0.939073i	$-0.388314\pi$
$181$	1673.98	0.687435	0.343718	+	0.939073i	$0.388314\pi$
$182$	0	0
$183$	−609.815	−0.246332
$184$	− 1841.11i	− 0.737653i
$185$	719.403	0.285900
$186$	1230.43	0.485052
$187$	54.8057i	0.0214320i
$188$	817.159i	0.317008i
$189$	796.334i	0.306480i
$190$	1255.89i	0.479536i
$191$	−290.117	−0.109907	−0.0549533	−	0.998489i	$-0.517501\pi$
$191$	−290.117	−0.109907	−0.0549533	+	0.998489i	$0.517501\pi$
$192$	1676.82	0.630282
$193$	1039.50i	0.387693i	0.981032	+	0.193847i	$0.0620964\pi$
$193$	1039.50i	0.387693i	−0.981032	+	0.193847i	$0.937904\pi$
$194$	−4494.18	−1.66321
$195$	0	0
$196$	−1276.16	−0.465073
$197$	− 1418.70i	− 0.513089i	−0.966532	−	0.256544i	$-0.917416\pi$
$197$	− 1418.70i	− 0.513089i	0.966532	−	0.256544i	$-0.0825840\pi$
$198$	−13.2789	−0.00476610
$199$	2388.39	0.850795	0.425398	−	0.905007i	$-0.360134\pi$
$199$	2388.39	0.850795	0.425398	+	0.905007i	$0.360134\pi$
$200$	− 2061.63i	− 0.728897i
$201$	364.490i	0.127906i
$202$	4329.33i	1.50797i
$203$	− 6675.32i	− 2.30796i
$204$	637.460	0.218780
$205$	1714.77	0.584217
$206$	3417.44i	1.15584i
$207$	673.179	0.226035
$208$	0	0
$209$	−51.7276	−0.0171200
$210$	− 1342.04i	− 0.440997i
$211$	4341.45	1.41648	0.708241	−	0.705971i	$-0.249489\pi$
$211$	4341.45	1.41648	0.708241	+	0.705971i	$0.249489\pi$
$212$	−354.481	−0.114839
$213$	− 1983.94i	− 0.638204i
$214$	872.165i	0.278598i
$215$	2462.54i	0.781136i
$216$	664.591i	0.209350i
$217$	−5121.91	−1.60229
$218$	1926.47	0.598520
$219$	− 502.024i	− 0.154903i
$220$	−9.71730	−0.00297791
$221$	0	0
$222$	−793.696	−0.239952
$223$	− 4615.37i	− 1.38596i	−0.720959	−	0.692978i	$-0.756298\pi$
$223$	− 4615.37i	− 1.38596i	0.720959	−	0.692978i	$-0.243702\pi$
$224$	−3108.09	−0.927091
$225$	753.812	0.223352
$226$	− 4228.89i	− 1.24470i
$227$	2163.44i	0.632565i	0.948665	+	0.316283i	$0.102435\pi$
$227$	2163.44i	0.632565i	−0.948665	+	0.316283i	$0.897565\pi$
$228$	601.657i	0.174762i
$229$	1859.48i	0.536584i	0.963338	+	0.268292i	$0.0864592\pi$
$229$	1859.48i	0.536584i	−0.963338	+	0.268292i	$0.913541\pi$
$230$	−1134.49	−0.325243
$231$	55.2758	0.0157441
$232$	− 5570.97i	− 1.57652i
$233$	−2866.87	−0.806073	−0.403037	−	0.915184i	$-0.632045\pi$
$233$	−2866.87	−0.806073	−0.403037	+	0.915184i	$0.632045\pi$
$234$	0	0
$235$	2166.68	0.601440
$236$	− 1281.70i	− 0.353523i
$237$	304.197	0.0833743
$238$	6110.99	1.66435
$239$	− 1893.55i	− 0.512485i	−0.966613	−	0.256242i	$-0.917516\pi$
$239$	− 1893.55i	− 0.512485i	0.966613	−	0.256242i	$-0.0824845\pi$
$240$	− 746.700i	− 0.200830i
$241$	− 1813.58i	− 0.484741i	−0.970184	−	0.242371i	$-0.922075\pi$
$241$	− 1813.58i	− 0.484741i	0.970184	−	0.242371i	$-0.0779250\pi$
$242$	− 3142.58i	− 0.834764i
$243$	−243.000	−0.0641500
$244$	492.340	0.129176
$245$	3383.71i	0.882356i
$246$	−1891.85	−0.490325
$247$	0	0
$248$	−4274.56	−1.09449
$249$	− 1520.96i	− 0.387095i
$250$	−3166.30	−0.801019
$251$	−4162.32	−1.04671	−0.523353	−	0.852116i	$-0.675319\pi$
$251$	−4162.32	−1.04671	−0.523353	+	0.852116i	$0.675319\pi$
$252$	− 642.927i	− 0.160717i
$253$	− 46.7273i	− 0.0116115i
$254$	− 106.964i	− 0.0264234i
$255$	− 1690.21i	− 0.415078i
$256$	−3344.88	−0.816621
$257$	−5985.30	−1.45274	−0.726368	−	0.687306i	$-0.758793\pi$
$257$	−5985.30	−1.45274	−0.726368	+	0.687306i	$0.758793\pi$
$258$	− 2716.85i	− 0.655596i
$259$	3303.91	0.792645
$260$	0	0
$261$	2036.96	0.483084
$262$	− 2484.15i	− 0.585768i
$263$	574.618	0.134724	0.0673621	−	0.997729i	$-0.478542\pi$
$263$	574.618	0.134724	0.0673621	+	0.997729i	$0.478542\pi$
$264$	46.1311	0.0107544
$265$	939.899i	0.217877i
$266$	5767.77i	1.32949i
$267$	− 4206.99i	− 0.964284i
$268$	− 294.275i	− 0.0670734i
$269$	6348.61	1.43896	0.719482	−	0.694511i	$-0.244379\pi$
$269$	6348.61	1.43896	0.719482	+	0.694511i	$0.244379\pi$
$270$	409.520	0.0923060
$271$	− 3278.38i	− 0.734861i	−0.930051	−	0.367431i	$-0.880238\pi$
$271$	− 3278.38i	− 0.734861i	0.930051	−	0.367431i	$-0.119762\pi$
$272$	3400.11	0.757948
$273$	0	0
$274$	−3644.06	−0.803452
$275$	− 52.3242i	− 0.0114737i
$276$	−543.498	−0.118532
$277$	−3952.17	−0.857267	−0.428633	−	0.903478i	$-0.641005\pi$
$277$	−3952.17	−0.857267	−0.428633	+	0.903478i	$0.641005\pi$
$278$	− 89.4267i	− 0.0192930i
$279$	− 1562.94i	− 0.335380i
$280$	4662.27i	0.995085i
$281$	− 411.389i	− 0.0873360i	−0.999046	−	0.0436680i	$-0.986096\pi$
$281$	− 411.389i	− 0.0873360i	0.999046	−	0.0436680i	$-0.0139044\pi$
$282$	−2390.43	−0.504781
$283$	5872.78	1.23357	0.616785	−	0.787131i	$-0.288435\pi$
$283$	5872.78	1.23357	0.616785	+	0.787131i	$0.288435\pi$
$284$	1601.75i	0.334671i
$285$	1595.28	0.331566
$286$	0	0
$287$	7875.18	1.61971
$288$	− 948.430i	− 0.194051i
$289$	2783.39	0.566537
$290$	−3432.83	−0.695113
$291$	5708.67i	1.14999i
$292$	405.314i	0.0812301i
$293$	− 500.957i	− 0.0998847i	−0.998752	−	0.0499423i	$-0.984096\pi$
$293$	− 500.957i	− 0.0998847i	0.998752	−	0.0499423i	$-0.0159038\pi$
$294$	− 3733.14i	− 0.740549i
$295$	−3398.39	−0.670718
$296$	2757.32	0.541439
$297$	16.8673i	0.00329542i
$298$	4304.07	0.836671
$299$	0	0
$300$	−608.597	−0.117125
$301$	11309.4i	2.16566i
$302$	−7650.61	−1.45776
$303$	5499.28	1.04266
$304$	3209.14i	0.605451i
$305$	− 1305.43i	− 0.245077i
$306$	1864.76i	0.348370i
$307$	− 5975.57i	− 1.11089i	−0.831553	−	0.555446i	$-0.812548\pi$
$307$	− 5975.57i	− 1.11089i	0.831553	−	0.555446i	$-0.187452\pi$
$308$	−44.6274	−0.00825611
$309$	4340.96	0.799185
$310$	2633.98i	0.482581i
$311$	−44.4925	−0.00811234	−0.00405617	−	0.999992i	$-0.501291\pi$
$311$	−44.4925	−0.00811234	−0.00405617	+	0.999992i	$0.501291\pi$
$312$	0	0
$313$	9957.78	1.79823	0.899117	−	0.437709i	$-0.144210\pi$
$313$	9957.78	1.79823	0.899117	+	0.437709i	$0.144210\pi$
$314$	− 1961.60i	− 0.352546i
$315$	−1704.71	−0.304918
$316$	−245.596	−0.0437211
$317$	− 7752.29i	− 1.37354i	−0.726875	−	0.686770i	$-0.759028\pi$
$317$	− 7752.29i	− 1.37354i	0.726875	−	0.686770i	$-0.240972\pi$
$318$	− 1036.96i	− 0.182862i
$319$	− 141.391i	− 0.0248163i
$320$	3589.56i	0.627070i
$321$	1107.86	0.192631
$322$	−5210.22	−0.901721
$323$	7264.13i	1.25135i
$324$	196.188	0.0336400
$325$	0	0
$326$	−4912.23	−0.834550
$327$	− 2447.08i	− 0.413834i
$328$	6572.34	1.10639
$329$	9950.62	1.66746
$330$	− 28.4260i	− 0.00474181i
$331$	1338.85i	0.222326i	0.993802	+	0.111163i	$0.0354576\pi$
$331$	1338.85i	0.222326i	−0.993802	+	0.111163i	$0.964542\pi$
$332$	1227.96i	0.202991i
$333$	1008.18i	0.165910i
$334$	203.062	0.0332666
$335$	−780.261	−0.127254
$336$	− 3429.27i	− 0.556792i
$337$	−3788.95	−0.612454	−0.306227	−	0.951958i	$-0.599067\pi$
$337$	−3788.95	−0.612454	−0.306227	+	0.951958i	$0.599067\pi$
$338$	0	0
$339$	−5371.69	−0.860621
$340$	1364.61i	0.217665i
$341$	−108.488	−0.0172286
$342$	−1760.02	−0.278278
$343$	5423.53i	0.853770i
$344$	9438.42i	1.47932i
$345$	1441.07i	0.224883i
$346$	6392.03i	0.993173i
$347$	7795.15	1.20595	0.602977	−	0.797759i	$-0.293981\pi$
$347$	7795.15	1.20595	0.602977	+	0.797759i	$0.293981\pi$
$348$	−1644.56	−0.253327
$349$	134.533i	0.0206344i	0.999947	+	0.0103172i	$0.00328412\pi$
$349$	134.533i	0.0206344i	−0.999947	+	0.0103172i	$0.996716\pi$
$350$	−5834.29	−0.891018
$351$	0	0
$352$	−65.8332	−0.00996853
$353$	2973.71i	0.448370i	0.974547	+	0.224185i	$0.0719719\pi$
$353$	2973.71i	0.448370i	−0.974547	+	0.224185i	$0.928028\pi$
$354$	3749.34	0.562925
$355$	4247.01	0.634952
$356$	3396.56i	0.505666i
$357$	− 7762.40i	− 1.15078i
$358$	− 10396.7i	− 1.53487i
$359$	8671.60i	1.27485i	0.770514	+	0.637423i	$0.219999\pi$
$359$	8671.60i	1.27485i	−0.770514	+	0.637423i	$0.780001\pi$
$360$	−1422.68	−0.208283
$361$	2.85364	0.000416043 0
$362$	− 3953.54i	− 0.574015i
$363$	−3991.83	−0.577181
$364$	0	0
$365$	1074.68	0.154113
$366$	1440.24i	0.205690i
$367$	4514.32	0.642086	0.321043	−	0.947065i	$-0.395967\pi$
$367$	4514.32	0.642086	0.321043	+	0.947065i	$0.395967\pi$
$368$	−2898.93	−0.410644
$369$	2403.10i	0.339025i
$370$	− 1699.06i	− 0.238729i
$371$	4316.55i	0.604054i
$372$	1261.86i	0.175871i
$373$	6570.60	0.912099	0.456049	−	0.889955i	$-0.349264\pi$
$373$	6570.60	0.912099	0.456049	+	0.889955i	$0.349264\pi$
$374$	129.438	0.0178959
$375$	4021.96i	0.553848i
$376$	8304.42	1.13901
$377$	0	0
$378$	1880.75	0.255914
$379$	5490.38i	0.744121i	0.928209	+	0.372060i	$0.121349\pi$
$379$	5490.38i	0.744121i	−0.928209	+	0.372060i	$0.878651\pi$
$380$	−1287.96	−0.173871
$381$	−135.870	−0.0182699
$382$	685.188i	0.0917730i
$383$	− 10187.3i	− 1.35912i	−0.733618	−	0.679562i	$-0.762170\pi$
$383$	− 10187.3i	− 1.35912i	0.733618	−	0.679562i	$-0.237830\pi$
$384$	− 1431.11i	− 0.190185i
$385$	118.328i	0.0156638i
$386$	2455.05	0.323728
$387$	−3451.05	−0.453299
$388$	− 4608.95i	− 0.603051i
$389$	−4883.97	−0.636573	−0.318287	−	0.947995i	$-0.603107\pi$
$389$	−4883.97	−0.636573	−0.318287	+	0.947995i	$0.603107\pi$
$390$	0	0
$391$	−6561.94	−0.848725
$392$	12969.0i	1.67101i
$393$	−3155.46	−0.405018
$394$	−3350.64	−0.428434
$395$	651.192i	0.0829494i
$396$	− 13.6180i	− 0.00172810i
$397$	− 2115.04i	− 0.267382i	−0.991023	−	0.133691i	$-0.957317\pi$
$397$	− 2115.04i	− 0.267382i	0.991023	−	0.133691i	$-0.0426830\pi$
$398$	− 5640.80i	− 0.710422i
$399$	7326.43	0.919249
$400$	−3246.16	−0.405770
$401$	− 674.254i	− 0.0839667i	−0.999118	−	0.0419834i	$-0.986632\pi$
$401$	− 674.254i	− 0.0839667i	0.999118	−	0.0419834i	$-0.0133676\pi$
$402$	860.839	0.106803
$403$	0	0
$404$	−4439.89	−0.546765
$405$	− 520.188i	− 0.0638231i
$406$	−15765.5	−1.92717
$407$	69.9808	0.00852290
$408$	− 6478.22i	− 0.786077i
$409$	− 4673.33i	− 0.564991i	−0.959269	−	0.282496i	$-0.908838\pi$
$409$	− 4673.33i	− 0.564991i	0.959269	−	0.282496i	$-0.0911623\pi$
$410$	− 4049.87i	− 0.487826i
$411$	4628.83i	0.555531i
$412$	−3504.71	−0.419089
$413$	−15607.3	−1.85953
$414$	− 1589.89i	− 0.188741i
$415$	3255.90	0.385122
$416$	0	0
$417$	−113.593	−0.0133398
$418$	122.168i	0.0142953i
$419$	−256.853	−0.0299477	−0.0149739	−	0.999888i	$-0.504767\pi$
$419$	−256.853	−0.0299477	−0.0149739	+	0.999888i	$0.504767\pi$
$420$	1376.31	0.159898
$421$	8746.82i	1.01257i	0.862365	+	0.506287i	$0.168982\pi$
$421$	8746.82i	1.01257i	−0.862365	+	0.506287i	$0.831018\pi$
$422$	− 10253.5i	− 1.18278i
$423$	3036.42i	0.349020i
$424$	3602.43i	0.412617i
$425$	−7347.92	−0.838650
$426$	−4685.60	−0.532906
$427$	− 5995.26i	− 0.679464i
$428$	−894.439	−0.101015
$429$	0	0
$430$	5815.95	0.652255
$431$	− 8762.72i	− 0.979316i	−0.871914	−	0.489658i	$-0.837122\pi$
$431$	− 8762.72i	− 0.979316i	0.871914	−	0.489658i	$-0.162878\pi$
$432$	1046.44	0.116543
$433$	−6425.50	−0.713140	−0.356570	−	0.934269i	$-0.616054\pi$
$433$	−6425.50	−0.713140	−0.356570	+	0.934269i	$0.616054\pi$
$434$	12096.7i	1.33793i
$435$	4360.51i	0.480622i
$436$	1975.67i	0.217013i
$437$	− 6193.39i	− 0.677963i
$438$	−1185.66	−0.129345
$439$	−6820.28	−0.741490	−0.370745	−	0.928735i	$-0.620898\pi$
$439$	−6820.28	−0.741490	−0.370745	+	0.928735i	$0.620898\pi$
$440$	98.7525i	0.0106996i
$441$	−4741.98	−0.512038
$442$	0	0
$443$	5062.48	0.542948	0.271474	−	0.962446i	$-0.412489\pi$
$443$	5062.48	0.542948	0.271474	+	0.962446i	$0.412489\pi$
$444$	− 813.966i	− 0.0870025i
$445$	9005.88	0.959370
$446$	−10900.4	−1.15729
$447$	− 5467.19i	− 0.578500i
$448$	16485.3i	1.73852i
$449$	− 6590.82i	− 0.692740i	−0.938098	−	0.346370i	$-0.887414\pi$
$449$	− 6590.82i	− 0.692740i	0.938098	−	0.346370i	$-0.112586\pi$
$450$	− 1780.33i	− 0.186501i
$451$	166.806	0.0174159
$452$	4336.89	0.451305
$453$	9718.09i	1.00794i
$454$	5109.52	0.528198
$455$	0	0
$456$	6114.37	0.627920
$457$	2254.75i	0.230793i	0.993319	+	0.115397i	$0.0368139\pi$
$457$	2254.75i	0.230793i	−0.993319	+	0.115397i	$0.963186\pi$
$458$	4391.65	0.448053
$459$	2368.69	0.240873
$460$	− 1163.46i	− 0.117928i
$461$	− 7358.79i	− 0.743456i	−0.928342	−	0.371728i	$-0.878765\pi$
$461$	− 7358.79i	− 0.743456i	0.928342	−	0.371728i	$-0.121235\pi$
$462$	− 130.548i	− 0.0131464i
$463$	11598.3i	1.16419i	0.813120	+	0.582096i	$0.197767\pi$
$463$	11598.3i	1.16419i	−0.813120	+	0.582096i	$0.802233\pi$
$464$	−8771.82	−0.877633
$465$	3345.78	0.333671
$466$	6770.87i	0.673079i
$467$	302.150	0.0299397	0.0149698	−	0.999888i	$-0.495235\pi$
$467$	302.150	0.0299397	0.0149698	+	0.999888i	$0.495235\pi$
$468$	0	0
$469$	−3583.41	−0.352807
$470$	− 5117.18i	− 0.502208i
$471$	−2491.69	−0.243760
$472$	−13025.3	−1.27021
$473$	239.547i	0.0232862i
$474$	− 718.441i	− 0.0696183i
$475$	− 6935.23i	− 0.669916i
$476$	6267.05i	0.603466i
$477$	−1317.19	−0.126436
$478$	−4472.12	−0.427929
$479$	− 3146.15i	− 0.300107i	−0.988678	−	0.150053i	$-0.952055\pi$
$479$	− 3146.15i	− 0.300107i	0.988678	−	0.150053i	$-0.0479446\pi$
$480$	2030.30	0.193062
$481$	0	0
$482$	−4283.24	−0.404764
$483$	6618.22i	0.623477i
$484$	3222.84	0.302671
$485$	−12220.5	−1.14413
$486$	573.908i	0.0535659i
$487$	− 3068.16i	− 0.285486i	−0.989760	−	0.142743i	$-0.954408\pi$
$487$	− 3068.16i	− 0.285486i	0.989760	−	0.142743i	$-0.0455922\pi$
$488$	− 5003.43i	− 0.464128i
$489$	6239.70i	0.577033i
$490$	7991.51	0.736775
$491$	100.558	0.00924262	0.00462131	−	0.999989i	$-0.498529\pi$
$491$	100.558	0.00924262	0.00462131	+	0.999989i	$0.498529\pi$
$492$	− 1940.16i	− 0.177783i
$493$	−19855.7	−1.81390
$494$	0	0
$495$	−36.1077	−0.00327863
$496$	6730.54i	0.609295i
$497$	19504.7	1.76037
$498$	−3592.14	−0.323228
$499$	− 3616.55i	− 0.324447i	−0.986754	−	0.162223i	$-0.948133\pi$
$499$	− 3616.55i	− 0.324447i	0.986754	−	0.162223i	$-0.0518665\pi$
$500$	− 3247.17i	− 0.290435i
$501$	− 257.937i	− 0.0230015i
$502$	9830.40i	0.874009i
$503$	−5372.64	−0.476251	−0.238125	−	0.971234i	$-0.576533\pi$
$503$	−5372.64	−0.476251	−0.238125	+	0.971234i	$0.576533\pi$
$504$	−6533.78	−0.577456
$505$	11772.3i	1.03734i
$506$	−110.359	−0.00969574
$507$	0	0
$508$	109.696	0.00958065
$509$	11314.7i	0.985298i	0.870228	+	0.492649i	$0.163971\pi$
$509$	11314.7i	0.985298i	−0.870228	+	0.492649i	$0.836029\pi$
$510$	−3991.87	−0.346594
$511$	4935.54	0.427271
$512$	11716.1i	1.01130i
$513$	2235.65i	0.192410i
$514$	14135.9i	1.21305i
$515$	9292.65i	0.795113i
$516$	2786.24	0.237708
$517$	210.766	0.0179294
$518$	− 7803.05i	− 0.661865i
$519$	8119.40	0.686709
$520$	0	0
$521$	−18470.9	−1.55321	−0.776606	−	0.629986i	$-0.783061\pi$
$521$	−18470.9	−1.55321	−0.776606	+	0.629986i	$0.783061\pi$
$522$	− 4810.82i	− 0.403379i
$523$	10891.0	0.910576	0.455288	−	0.890344i	$-0.349536\pi$
$523$	10891.0	0.910576	0.455288	+	0.890344i	$0.349536\pi$
$524$	2547.59	0.212389
$525$	7410.94i	0.616076i
$526$	− 1357.11i	− 0.112496i
$527$	15235.1i	1.25930i
$528$	− 72.6361i	− 0.00598690i
$529$	−6572.30	−0.540174
$530$	2219.82	0.181930
$531$	− 4762.56i	− 0.389223i
$532$	−5915.06	−0.482050
$533$	0	0
$534$	−9935.92	−0.805186
$535$	2371.58i	0.191649i
$536$	−2990.58	−0.240995
$537$	−13206.3	−1.06126
$538$	− 14993.9i	− 1.20155i
$539$	329.154i	0.0263037i
$540$	419.979i	0.0334685i
$541$	− 13416.3i	− 1.06620i	−0.846053	−	0.533099i	$-0.821027\pi$
$541$	− 13416.3i	− 1.06620i	0.846053	−	0.533099i	$-0.178973\pi$
$542$	−7742.75	−0.613616
$543$	−5021.93	−0.396891
$544$	9244.99i	0.728632i
$545$	5238.45	0.411726
$546$	0	0
$547$	−17849.0	−1.39519	−0.697593	−	0.716495i	$-0.745745\pi$
$547$	−17849.0	−1.39519	−0.697593	+	0.716495i	$0.745745\pi$
$548$	− 3737.13i	− 0.291318i
$549$	1829.45	0.142220
$550$	−123.577	−0.00958065
$551$	− 18740.5i	− 1.44895i
$552$	5523.32i	0.425884i
$553$	2990.64i	0.229973i
$554$	9334.09i	0.715826i
$555$	−2158.21	−0.165065
$556$	91.7105	0.00699531
$557$	− 20131.4i	− 1.53141i	−0.643195	−	0.765703i	$-0.722391\pi$
$557$	− 20131.4i	− 1.53141i	0.643195	−	0.765703i	$-0.277609\pi$
$558$	−3691.30	−0.280045
$559$	0	0
$560$	7341.02	0.553955
$561$	− 164.417i	− 0.0123738i
$562$	−971.603	−0.0729263
$563$	22345.8	1.67276	0.836380	−	0.548150i	$-0.184668\pi$
$563$	22345.8	1.67276	0.836380	+	0.548150i	$0.184668\pi$
$564$	− 2451.48i	− 0.183025i
$565$	− 11499.1i	− 0.856235i
$566$	− 13870.1i	− 1.03004i
$567$	− 2389.00i	− 0.176946i
$568$	16277.9	1.20247
$569$	8455.72	0.622992	0.311496	−	0.950248i	$-0.399170\pi$
$569$	8455.72	0.622992	0.311496	+	0.950248i	$0.399170\pi$
$570$	− 3767.67i	− 0.276860i
$571$	−12813.0	−0.939069	−0.469534	−	0.882914i	$-0.655578\pi$
$571$	−12813.0	−0.939069	−0.469534	+	0.882914i	$0.655578\pi$
$572$	0	0
$573$	870.352	0.0634546
$574$	− 18599.3i	− 1.35247i
$575$	6264.83	0.454367
$576$	−5030.47	−0.363894
$577$	− 1971.59i	− 0.142251i	−0.997467	−	0.0711253i	$-0.977341\pi$
$577$	− 1971.59i	− 0.142251i	0.997467	−	0.0711253i	$-0.0226590\pi$
$578$	− 6573.72i	− 0.473063i
$579$	− 3118.50i	− 0.223835i
$580$	− 3520.50i	− 0.252036i
$581$	14952.9	1.06773
$582$	13482.5	0.960255
$583$	91.4298i	0.00649508i
$584$	4119.02	0.291860
$585$	0	0
$586$	−1183.14	−0.0834046
$587$	− 8585.69i	− 0.603696i	−0.953356	−	0.301848i	$-0.902397\pi$
$587$	− 8585.69i	− 0.603696i	0.953356	−	0.301848i	$-0.0976035\pi$
$588$	3828.48	0.268510
$589$	−14379.4	−1.00593
$590$	8026.19i	0.560056i
$591$	4256.11i	0.296232i
$592$	− 4341.56i	− 0.301414i
$593$	− 1746.73i	− 0.120961i	−0.998169	−	0.0604803i	$-0.980737\pi$
$593$	− 1746.73i	− 0.120961i	0.998169	−	0.0604803i	$-0.0192632\pi$
$594$	39.8366	0.00275171
$595$	16616.9	1.14492
$596$	4413.99i	0.303362i
$597$	−7165.16	−0.491207
$598$	0	0
$599$	27531.1	1.87794	0.938972	−	0.343994i	$-0.111780\pi$
$599$	27531.1	1.87794	0.938972	+	0.343994i	$0.111780\pi$
$600$	6184.90i	0.420829i
$601$	−17539.1	−1.19041	−0.595203	−	0.803575i	$-0.702928\pi$
$601$	−17539.1	−1.19041	−0.595203	+	0.803575i	$0.702928\pi$
$602$	26710.1	1.80835
$603$	− 1093.47i	− 0.0738467i
$604$	− 7845.99i	− 0.528558i
$605$	− 8545.28i	− 0.574240i
$606$	− 12988.0i	− 0.870629i
$607$	−22691.1	−1.51730	−0.758651	−	0.651497i	$-0.774141\pi$
$607$	−22691.1	−1.51730	−0.758651	+	0.651497i	$0.774141\pi$
$608$	−8725.75	−0.582033
$609$	20026.0i	1.33250i
$610$	−3083.11	−0.204642
$611$	0	0
$612$	−1912.38	−0.126313
$613$	− 7215.30i	− 0.475405i	−0.971338	−	0.237702i	$-0.923606\pi$
$613$	− 7215.30i	− 0.475405i	0.971338	−	0.237702i	$-0.0763943\pi$
$614$	−14112.9	−0.927605
$615$	−5144.30	−0.337298
$616$	453.528i	0.0296642i
$617$	16870.4i	1.10077i	0.834909	+	0.550387i	$0.185520\pi$
$617$	16870.4i	1.10077i	−0.834909	+	0.550387i	$0.814480\pi$
$618$	− 10252.3i	− 0.667327i
$619$	2244.53i	0.145743i	0.997341	+	0.0728717i	$0.0232164\pi$
$619$	2244.53i	0.145743i	−0.997341	+	0.0728717i	$0.976784\pi$
$620$	−2701.25	−0.174975
$621$	−2019.54	−0.130501
$622$	105.081i	0.00677388i
$623$	41360.1	2.65981
$624$	0	0
$625$	1859.84	0.119030
$626$	− 23517.9i	− 1.50154i
$627$	155.183	0.00988421
$628$	2011.69	0.127827
$629$	− 9827.44i	− 0.622966i
$630$	4026.11i	0.254610i
$631$	− 3669.22i	− 0.231488i	−0.993279	−	0.115744i	$-0.963075\pi$
$631$	− 3669.22i	− 0.231488i	0.993279	−	0.115744i	$-0.0369253\pi$
$632$	2495.88i	0.157090i
$633$	−13024.3	−0.817806
$634$	−18309.1	−1.14692
$635$	− 290.856i	− 0.0181768i
$636$	1063.44	0.0663024
$637$	0	0
$638$	−333.933	−0.0207218
$639$	5951.82i	0.368467i
$640$	3063.56	0.189215
$641$	−14678.6	−0.904477	−0.452239	−	0.891897i	$-0.649374\pi$
$641$	−14678.6	−0.904477	−0.452239	+	0.891897i	$0.649374\pi$
$642$	− 2616.50i	− 0.160849i
$643$	− 5519.72i	− 0.338533i	−0.985570	−	0.169266i	$-0.945860\pi$
$643$	− 5519.72i	− 0.338533i	0.985570	−	0.169266i	$-0.0541398\pi$
$644$	− 5343.28i	− 0.326948i
$645$	− 7387.63i	− 0.450989i
$646$	17156.2	1.04489
$647$	11326.9	0.688261	0.344131	−	0.938922i	$-0.388174\pi$
$647$	11326.9	0.688261	0.344131	+	0.938922i	$0.388174\pi$
$648$	− 1993.77i	− 0.120868i
$649$	−330.582	−0.0199946
$650$	0	0
$651$	15365.7	0.925085
$652$	− 5037.68i	− 0.302593i
$653$	3902.89	0.233893	0.116946	−	0.993138i	$-0.462689\pi$
$653$	3902.89	0.233893	0.116946	+	0.993138i	$0.462689\pi$
$654$	−5779.42	−0.345556
$655$	− 6754.87i	− 0.402954i
$656$	− 10348.5i	− 0.615918i
$657$	1506.07i	0.0894330i
$658$	− 23501.0i	− 1.39235i
$659$	8022.47	0.474220	0.237110	−	0.971483i	$-0.423800\pi$
$659$	8022.47	0.474220	0.237110	+	0.971483i	$0.423800\pi$
$660$	29.1519	0.00171930
$661$	− 5168.76i	− 0.304147i	−0.988369	−	0.152074i	$-0.951405\pi$
$661$	− 5168.76i	− 0.304147i	0.988369	−	0.152074i	$-0.0485951\pi$
$662$	3162.05	0.185645
$663$	0	0
$664$	12479.2	0.729346
$665$	15683.6i	0.914565i
$666$	2381.09	0.138536
$667$	16928.9	0.982743
$668$	208.248i	0.0120619i
$669$	13846.1i	0.800182i
$670$	1842.79i	0.106259i
$671$	− 126.987i	− 0.00730593i
$672$	9324.28	0.535256
$673$	6654.10	0.381124	0.190562	−	0.981675i	$-0.438969\pi$
$673$	6654.10	0.381124	0.190562	+	0.981675i	$0.438969\pi$
$674$	8948.59i	0.511405i
$675$	−2261.44	−0.128952
$676$	0	0
$677$	−20649.4	−1.17226	−0.586130	−	0.810217i	$-0.699349\pi$
$677$	−20649.4	−1.17226	−0.586130	+	0.810217i	$0.699349\pi$
$678$	12686.7i	0.718626i
$679$	−56123.6	−3.17205
$680$	13867.9	0.782071
$681$	− 6490.31i	− 0.365212i
$682$	256.223i	0.0143861i
$683$	28475.6i	1.59530i	0.603122	+	0.797649i	$0.293923\pi$
$683$	28475.6i	1.59530i	−0.603122	+	0.797649i	$0.706077\pi$
$684$	− 1804.97i	− 0.100899i
$685$	−9908.90	−0.552700
$686$	12809.1	0.712905
$687$	− 5578.43i	− 0.309797i
$688$	14861.3	0.823522
$689$	0	0
$690$	3403.47	0.187779
$691$	− 10610.8i	− 0.584160i	−0.956394	−	0.292080i	$-0.905653\pi$
$691$	− 10610.8i	− 0.584160i	0.956394	−	0.292080i	$-0.0943473\pi$
$692$	−6555.27	−0.360107
$693$	−165.827	−0.00908984
$694$	− 18410.3i	− 1.00698i
$695$	− 243.168i	− 0.0132718i
$696$	16712.9i	0.910203i
$697$	− 23424.6i	− 1.27299i
$698$	317.736	0.0172299
$699$	8600.62	0.465387
$700$	− 5983.29i	− 0.323067i
$701$	13518.9	0.728390	0.364195	−	0.931323i	$-0.381344\pi$
$701$	13518.9	0.728390	0.364195	+	0.931323i	$0.381344\pi$
$702$	0	0
$703$	9275.49	0.497627
$704$	349.179i	0.0186934i
$705$	−6500.03	−0.347242
$706$	7023.19	0.374393
$707$	54064.9i	2.87599i
$708$	3845.09i	0.204107i
$709$	14845.4i	0.786361i	0.919461	+	0.393180i	$0.128625\pi$
$709$	14845.4i	0.786361i	−0.919461	+	0.393180i	$0.871375\pi$
$710$	− 10030.4i	− 0.530190i
$711$	−912.590	−0.0481362
$712$	34517.7	1.81686
$713$	− 12989.4i	− 0.682267i
$714$	−18333.0	−0.960915
$715$	0	0
$716$	10662.2	0.556517
$717$	5680.66i	0.295883i
$718$	20480.3	1.06451
$719$	35900.1	1.86210	0.931049	−	0.364893i	$-0.118894\pi$
$719$	35900.1	1.86210	0.931049	+	0.364893i	$0.118894\pi$
$720$	2240.10i	0.115949i
$721$	42677.2i	2.20441i
$722$	− 6.73962i	0 0.000347400i
$723$	5440.73i	0.279866i
$724$	4054.50	0.208128
$725$	18956.6	0.971078
$726$	9427.75i	0.481951i
$727$	−12951.4	−0.660715	−0.330357	−	0.943856i	$-0.607169\pi$
$727$	−12951.4	−0.660715	−0.330357	+	0.943856i	$0.607169\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 2538.14i	− 0.128686i
$731$	33639.7	1.70206
$732$	−1477.02	−0.0745795
$733$	1105.36i	0.0556989i	0.999612	+	0.0278494i	$0.00886590\pi$
$733$	1105.36i	0.0556989i	−0.999612	+	0.0278494i	$0.991134\pi$
$734$	− 10661.8i	− 0.536148i
$735$	− 10151.1i	− 0.509428i
$736$	− 7882.27i	− 0.394761i
$737$	−75.9009	−0.00379355
$738$	5675.55	0.283089
$739$	13638.1i	0.678869i	0.940630	+	0.339434i	$0.110236\pi$
$739$	13638.1i	0.678869i	−0.940630	+	0.339434i	$0.889764\pi$
$740$	1742.45	0.0865591
$741$	0	0
$742$	10194.7	0.504391
$743$	− 10154.9i	− 0.501408i	−0.968064	−	0.250704i	$-0.919338\pi$
$743$	− 10154.9i	− 0.501408i	0.968064	−	0.250704i	$-0.0806621\pi$
$744$	12823.7	0.631907
$745$	11703.6	0.575552
$746$	− 15518.2i	− 0.761611i
$747$	4562.87i	0.223489i
$748$	132.744i	0.00648876i
$749$	10891.7i	0.531338i
$750$	9498.91	0.462468
$751$	−26579.5	−1.29148	−0.645738	−	0.763559i	$-0.723450\pi$
$751$	−26579.5	−1.29148	−0.645738	+	0.763559i	$0.723450\pi$
$752$	− 13075.8i	− 0.634076i
$753$	12487.0	0.604316
$754$	0	0
$755$	−20803.5	−1.00280
$756$	1928.78i	0.0927898i
$757$	−13677.7	−0.656705	−0.328352	−	0.944555i	$-0.606493\pi$
$757$	−13677.7	−0.656705	−0.328352	+	0.944555i	$0.606493\pi$
$758$	12967.0	0.621348
$759$	140.182i	0.00670392i
$760$	13089.0i	0.624720i
$761$	17996.7i	0.857268i	0.903478	+	0.428634i	$0.141005\pi$
$761$	17996.7i	0.857268i	−0.903478	+	0.428634i	$0.858995\pi$
$762$	320.893i	0.0152555i
$763$	24057.9	1.14149
$764$	−702.687	−0.0332753
$765$	5070.63i	0.239646i
$766$	−24059.9	−1.13488
$767$	0	0
$768$	10034.6	0.471476
$769$	− 2995.04i	− 0.140447i	−0.997531	−	0.0702236i	$-0.977629\pi$
$769$	− 2995.04i	− 0.140447i	0.997531	−	0.0702236i	$-0.0223713\pi$
$770$	279.464	0.0130794
$771$	17955.9	0.838737
$772$	2517.75i	0.117378i
$773$	26055.4i	1.21235i	0.795330	+	0.606177i	$0.207298\pi$
$773$	26055.4i	1.21235i	−0.795330	+	0.606177i	$0.792702\pi$
$774$	8150.56i	0.378509i
$775$	− 14545.3i	− 0.674169i
$776$	−46838.6	−2.16676
$777$	−9911.73	−0.457634
$778$	11534.8i	0.531545i
$779$	22109.0	1.01686
$780$	0	0
$781$	413.133	0.0189284
$782$	15497.7i	0.708693i
$783$	−6110.89	−0.278908
$784$	20420.5	0.930235
$785$	− 5333.95i	− 0.242518i
$786$	7452.45i	0.338193i
$787$	− 22992.0i	− 1.04139i	−0.853741	−	0.520697i	$-0.825672\pi$
$787$	− 22992.0i	− 1.04139i	0.853741	−	0.520697i	$-0.174328\pi$
$788$	− 3436.21i	− 0.155343i
$789$	−1723.85	−0.0777830
$790$	1537.96	0.0692635
$791$	− 52810.6i	− 2.37387i
$792$	−138.393	−0.00620908
$793$	0	0
$794$	−4995.22	−0.223267
$795$	− 2819.70i	− 0.125792i
$796$	5784.86	0.257587
$797$	25826.3	1.14782	0.573910	−	0.818918i	$-0.305426\pi$
$797$	25826.3	1.14782	0.573910	+	0.818918i	$0.305426\pi$
$798$	− 17303.3i	− 0.767582i
$799$	− 29598.0i	− 1.31052i
$800$	− 8826.40i	− 0.390075i
$801$	12621.0i	0.556730i
$802$	−1592.43	−0.0701130
$803$	104.541	0.00459423
$804$	882.824i	0.0387249i
$805$	−14167.6	−0.620299
$806$	0	0
$807$	−19045.8	−0.830787
$808$	45120.6i	1.96453i
$809$	−28495.8	−1.23839	−0.619195	−	0.785237i	$-0.712541\pi$
$809$	−28495.8	−1.23839	−0.619195	+	0.785237i	$0.712541\pi$
$810$	−1228.56	−0.0532929
$811$	− 6992.41i	− 0.302758i	−0.988476	−	0.151379i	$-0.951629\pi$
$811$	− 6992.41i	− 0.302758i	0.988476	−	0.151379i	$-0.0483714\pi$
$812$	− 16168.1i	− 0.698757i
$813$	9835.14i	0.424272i
$814$	− 165.278i	− 0.00711670i
$815$	−13357.3	−0.574093
$816$	−10200.3	−0.437602
$817$	31750.4i	1.35961i
$818$	−11037.3	−0.471773
$819$	0	0
$820$	4153.29	0.176877
$821$	− 31512.7i	− 1.33959i	−0.742547	−	0.669793i	$-0.766383\pi$
$821$	− 31512.7i	− 1.33959i	0.742547	−	0.669793i	$-0.233617\pi$
$822$	10932.2	0.463873
$823$	−39159.6	−1.65859	−0.829293	−	0.558814i	$-0.811257\pi$
$823$	−39159.6	−1.65859	−0.829293	+	0.558814i	$0.811257\pi$
$824$	35616.8i	1.50579i
$825$	156.973i	0.00662435i
$826$	36860.8i	1.55273i
$827$	− 36557.6i	− 1.53716i	−0.639752	−	0.768581i	$-0.720963\pi$
$827$	− 36557.6i	− 1.53716i	0.639752	−	0.768581i	$-0.279037\pi$
$828$	1630.49	0.0684342
$829$	−14675.2	−0.614825	−0.307413	−	0.951576i	$-0.599463\pi$
$829$	−14675.2	−0.614825	−0.307413	+	0.951576i	$0.599463\pi$
$830$	− 7689.66i	− 0.321581i
$831$	11856.5	0.494943
$832$	0	0
$833$	46223.3	1.92262
$834$	268.280i	0.0111388i
$835$	552.164	0.0228843
$836$	−125.288	−0.00518323
$837$	4688.83i	0.193632i
$838$	606.626i	0.0250066i
$839$	− 8515.56i	− 0.350405i	−0.984532	−	0.175202i	$-0.943942\pi$
$839$	− 8515.56i	− 0.350405i	0.984532	−	0.175202i	$-0.0560580\pi$
$840$	− 13986.8i	− 0.574513i
$841$	26835.9	1.10033
$842$	20657.9	0.845509
$843$	1234.17i	0.0504234i
$844$	10515.3	0.428854
$845$	0	0
$846$	7171.29	0.291435
$847$	− 39244.8i	− 1.59205i
$848$	5672.24	0.229700
$849$	−17618.3	−0.712202
$850$	17354.0i	0.700281i
$851$	8378.86i	0.337513i
$852$	− 4805.26i	− 0.193222i
$853$	9645.93i	0.387187i	0.981082	+	0.193593i	$0.0620142\pi$
$853$	9645.93i	0.387187i	−0.981082	+	0.193593i	$0.937986\pi$
$854$	−14159.4	−0.567359
$855$	−4785.84	−0.191430
$856$	9089.77i	0.362946i
$857$	−36139.6	−1.44050	−0.720248	−	0.693717i	$-0.755972\pi$
$857$	−36139.6	−1.44050	−0.720248	+	0.693717i	$0.755972\pi$
$858$	0	0
$859$	−7108.04	−0.282332	−0.141166	−	0.989986i	$-0.545085\pi$
$859$	−7108.04	−0.282332	−0.141166	+	0.989986i	$0.545085\pi$
$860$	5964.47i	0.236496i
$861$	−23625.5	−0.935141
$862$	−20695.5	−0.817738
$863$	− 16225.8i	− 0.640016i	−0.947415	−	0.320008i	$-0.896314\pi$
$863$	− 16225.8i	− 0.640016i	0.947415	−	0.320008i	$-0.103686\pi$
$864$	2845.29i	0.112036i
$865$	17381.1i	0.683210i
$866$	15175.5i	0.595478i
$867$	−8350.18	−0.327090
$868$	−12405.7	−0.485110
$869$	63.3455i	0.00247278i
$870$	10298.5	0.401324
$871$	0	0
$872$	20077.9	0.779727
$873$	− 17126.0i	− 0.663949i
$874$	−14627.3	−0.566106
$875$	−39541.0	−1.52769
$876$	− 1215.94i	− 0.0468982i
$877$	30983.0i	1.19295i	0.802630	+	0.596477i	$0.203433\pi$
$877$	30983.0i	1.19295i	−0.802630	+	0.596477i	$0.796567\pi$
$878$	16107.9i	0.619151i
$879$	1502.87i	0.0576684i
$880$	155.492	0.00595639
$881$	7670.76	0.293342	0.146671	−	0.989185i	$-0.453144\pi$
$881$	7670.76	0.293342	0.146671	+	0.989185i	$0.453144\pi$
$882$	11199.4i	0.427556i
$883$	34340.6	1.30878	0.654390	−	0.756157i	$-0.272925\pi$
$883$	34340.6	1.30878	0.654390	+	0.756157i	$0.272925\pi$
$884$	0	0
$885$	10195.2	0.387239
$886$	− 11956.4i	− 0.453366i
$887$	−19208.3	−0.727118	−0.363559	−	0.931571i	$-0.618438\pi$
$887$	−19208.3	−0.727118	−0.363559	+	0.931571i	$0.618438\pi$
$888$	−8271.96	−0.312600
$889$	− 1335.78i	− 0.0503943i
$890$	− 21269.8i	− 0.801083i
$891$	− 50.6019i	− 0.00190261i
$892$	− 11178.8i	− 0.419611i
$893$	27935.7	1.04684
$894$	−12912.2	−0.483052
$895$	− 28270.6i	− 1.05585i
$896$	14069.6	0.524591
$897$	0	0
$898$	−15566.0	−0.578444
$899$	− 39304.4i	− 1.45815i
$900$	1825.79	0.0676219
$901$	12839.5	0.474747
$902$	− 393.956i	− 0.0145425i
$903$	− 33928.2i	− 1.25034i
$904$	− 44073.8i	− 1.62154i
$905$	− 10750.4i	− 0.394868i
$906$	22951.8	0.841637
$907$	46481.0	1.70163	0.850813	−	0.525468i	$-0.176110\pi$
$907$	46481.0	1.70163	0.850813	+	0.525468i	$0.176110\pi$
$908$	5240.01i	0.191515i
$909$	−16497.8	−0.601979
$910$	0	0
$911$	−34109.3	−1.24050	−0.620248	−	0.784406i	$-0.712968\pi$
$911$	−34109.3	−1.24050	−0.620248	+	0.784406i	$0.712968\pi$
$912$	− 9627.43i	− 0.349557i
$913$	316.721	0.0114808
$914$	5325.18	0.192715
$915$	3916.28i	0.141495i
$916$	4503.80i	0.162456i
$917$	− 31022.2i	− 1.11717i
$918$	− 5594.27i	− 0.201131i
$919$	−37533.6	−1.34725	−0.673623	−	0.739075i	$-0.735263\pi$
$919$	−37533.6	−1.34725	−0.673623	+	0.739075i	$0.735263\pi$
$920$	−11823.7	−0.423714
$921$	17926.7i	0.641373i
$922$	−17379.7	−0.620793
$923$	0	0
$924$	133.882	0.00476667
$925$	9382.48i	0.333507i
$926$	27392.5	0.972111
$927$	−13022.9	−0.461410
$928$	− 23850.8i	− 0.843687i
$929$	− 4966.75i	− 0.175408i	−0.996147	−	0.0877038i	$-0.972047\pi$
$929$	− 4966.75i	− 0.175408i	0.996147	−	0.0877038i	$-0.0279529\pi$
$930$	− 7901.94i	− 0.278618i
$931$	43627.2i	1.53579i
$932$	−6943.79	−0.244047
$933$	133.477	0.00468366
$934$	− 713.606i	− 0.0249999i
$935$	351.966	0.0123107
$936$	0	0
$937$	−5096.90	−0.177704	−0.0888519	−	0.996045i	$-0.528320\pi$
$937$	−5096.90	−0.177704	−0.0888519	+	0.996045i	$0.528320\pi$
$938$	8463.15i	0.294597i
$939$	−29873.3	−1.03821
$940$	5247.86	0.182092
$941$	− 54774.8i	− 1.89756i	−0.315933	−	0.948781i	$-0.602318\pi$
$941$	− 54774.8i	− 1.89756i	0.315933	−	0.948781i	$-0.397682\pi$
$942$	5884.79i	0.203542i
$943$	19971.8i	0.689684i
$944$	20509.1i	0.707113i
$945$	5114.12	0.176045
$946$	565.753	0.0194442
$947$	− 13768.5i	− 0.472456i	−0.971698	−	0.236228i	$-0.924089\pi$
$947$	− 13768.5i	− 0.472456i	0.971698	−	0.236228i	$-0.0759112\pi$
$948$	736.788	0.0252424
$949$	0	0
$950$	−16379.4	−0.559386
$951$	23256.9i	0.793013i
$952$	63689.2	2.16825
$953$	−35866.6	−1.21913	−0.609566	−	0.792735i	$-0.708656\pi$
$953$	−35866.6	−1.21913	−0.609566	+	0.792735i	$0.708656\pi$
$954$	3110.89i	0.105575i
$955$	1863.16i	0.0631312i
$956$	− 4586.33i	− 0.155160i
$957$	424.174i	0.0143277i
$958$	−7430.46	−0.250592
$959$	−45507.3	−1.53233
$960$	− 10768.7i	− 0.362039i
$961$	−366.899	−0.0123158
$962$	0	0
$963$	−3323.57	−0.111216
$964$	− 4392.62i	− 0.146760i
$965$	6675.75	0.222694
$966$	15630.7	0.520609
$967$	24476.5i	0.813972i	0.913434	+	0.406986i	$0.133420\pi$
$967$	24476.5i	0.813972i	−0.913434	+	0.406986i	$0.866580\pi$
$968$	− 32752.3i	− 1.08750i
$969$	− 21792.4i	− 0.722469i
$970$	28861.9i	0.955362i
$971$	−8976.07	−0.296659	−0.148329	−	0.988938i	$-0.547390\pi$
$971$	−8976.07	−0.296659	−0.148329	+	0.988938i	$0.547390\pi$
$972$	−588.565	−0.0194220
$973$	− 1116.77i	− 0.0367954i
$974$	−7246.26	−0.238383
$975$	0	0
$976$	−7878.18	−0.258376
$977$	42002.8i	1.37542i	0.725984	+	0.687711i	$0.241385\pi$
$977$	42002.8i	1.37542i	−0.725984	+	0.687711i	$0.758615\pi$
$978$	14736.7	0.481828
$979$	876.058	0.0285995
$980$	8195.60i	0.267142i
$981$	7341.24i	0.238927i
$982$	− 237.495i	− 0.00771767i
$983$	43240.7i	1.40301i	0.712662	+	0.701507i	$0.247489\pi$
$983$	43240.7i	1.40301i	−0.712662	+	0.701507i	$0.752511\pi$
$984$	−19717.0	−0.638776
$985$	−9111.03	−0.294722
$986$	46894.3i	1.51462i
$987$	−29851.9	−0.962710
$988$	0	0
$989$	−28681.2	−0.922152
$990$	85.2779i	0.00273769i
$991$	37916.5	1.21540	0.607698	−	0.794168i	$-0.292093\pi$
$991$	37916.5	1.21540	0.607698	+	0.794168i	$0.292093\pi$
$992$	−18300.5	−0.585728
$993$	− 4016.56i	− 0.128360i
$994$	− 46065.4i	− 1.46993i
$995$	− 15338.4i	− 0.488704i
$996$	− 3683.87i	− 0.117197i
$997$	6634.86	0.210760	0.105380	−	0.994432i	$-0.466394\pi$
$997$	6634.86	0.210760	0.105380	+	0.994432i	$0.466394\pi$
$998$	−8541.43	−0.270916
$999$	− 3024.55i	− 0.0957883i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.h.337.3		8
13.5	odd	4		507.4.a.i.1.2		4
13.7	odd	12		39.4.e.c.16.2	✓	8
13.8	odd	4		507.4.a.m.1.3		4
13.11	odd	12		39.4.e.c.22.2	yes	8
13.12	even	2	inner	507.4.b.h.337.6		8
39.5	even	4		1521.4.a.bb.1.3		4
39.8	even	4		1521.4.a.v.1.2		4
39.11	even	12		117.4.g.e.100.3		8
39.20	even	12		117.4.g.e.55.3		8
52.7	even	12		624.4.q.i.289.3		8
52.11	even	12		624.4.q.i.529.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.2	✓	8	13.7	odd	12
39.4.e.c.22.2	yes	8	13.11	odd	12
117.4.g.e.55.3		8	39.20	even	12
117.4.g.e.100.3		8	39.11	even	12
507.4.a.i.1.2		4	13.5	odd	4
507.4.a.m.1.3		4	13.8	odd	4
507.4.b.h.337.3		8	1.1	even	1	trivial
507.4.b.h.337.6		8	13.12	even	2	inner
624.4.q.i.289.3		8	52.7	even	12
624.4.q.i.529.3		8	52.11	even	12
1521.4.a.v.1.2		4	39.8	even	4
1521.4.a.bb.1.3		4	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)

\(f(q)\)	\(=\)	\(q-2.36176i q^{2} -3.00000 q^{3} +2.42208 q^{4} -6.42208i q^{5} +7.08529i q^{6} -29.4938i q^{7} -24.6145i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.36176i q^{2} -3.00000 q^{3} +2.42208 q^{4} -6.42208i q^{5} +7.08529i q^{6} -29.4938i q^{7} -24.6145i q^{8} +9.00000 q^{9} -15.1674 q^{10} -0.624715i q^{11} -7.26623 q^{12} -69.6575 q^{14} +19.2662i q^{15} -38.7569 q^{16} -87.7291 q^{17} -21.2559i q^{18} -82.8018i q^{19} -15.5548i q^{20} +88.4815i q^{21} -1.47543 q^{22} +74.7977 q^{23} +73.8434i q^{24} +83.7569 q^{25} -27.0000 q^{27} -71.4364i q^{28} +226.329 q^{29} +45.5023 q^{30} -173.660i q^{31} -105.381i q^{32} +1.87415i q^{33} +207.195i q^{34} -189.412 q^{35} +21.7987 q^{36} +112.020i q^{37} -195.558 q^{38} -158.076 q^{40} +267.011i q^{41} +208.972 q^{42} -383.450 q^{43} -1.51311i q^{44} -57.7987i q^{45} -176.654i q^{46} +337.380i q^{47} +116.271 q^{48} -526.887 q^{49} -197.814i q^{50} +263.187 q^{51} -146.354 q^{53} +63.7676i q^{54} -4.01197 q^{55} -725.975 q^{56} +248.406i q^{57} -534.536i q^{58} -529.173i q^{59} +46.6643i q^{60} +203.272 q^{61} -410.144 q^{62} -265.445i q^{63} -558.941 q^{64} +4.42629 q^{66} -121.497i q^{67} -212.487 q^{68} -224.393 q^{69} +447.346i q^{70} +661.314i q^{71} -221.530i q^{72} +167.341i q^{73} +264.565 q^{74} -251.271 q^{75} -200.552i q^{76} -18.4253 q^{77} -101.399 q^{79} +248.900i q^{80} +81.0000 q^{81} +630.617 q^{82} +506.985i q^{83} +214.309i q^{84} +563.403i q^{85} +905.617i q^{86} -678.988 q^{87} -15.3770 q^{88} +1402.33i q^{89} -136.507 q^{90} +181.166 q^{92} +520.981i q^{93} +796.810 q^{94} -531.760 q^{95} +316.143i q^{96} -1902.89i q^{97} +1244.38i q^{98} -5.62244i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10}) \) 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9	\( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100}) \) 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 + 124 * q^10 + 132 * q^12 + 80 * q^14 + 244 * q^16 - 196 * q^17 + 440 * q^22 - 208 * q^23 + 116 * q^25 - 216 * q^27 + 388 * q^29 - 372 * q^30 + 176 * q^35 - 396 * q^36 - 664 * q^38 - 1996 * q^40 - 240 * q^42 - 900 * q^43 - 732 * q^48 - 2140 * q^49 + 588 * q^51 + 524 * q^53 + 408 * q^55 - 4328 * q^56 - 1856 * q^61 - 5560 * q^62 - 2052 * q^64 - 1320 * q^66 + 3572 * q^68 + 624 * q^69 + 2316 * q^74 - 348 * q^75 - 5016 * q^77 - 1492 * q^79 + 648 * q^81 - 3468 * q^82 - 1164 * q^87 - 6120 * q^88 + 1116 * q^90 + 664 * q^92 - 1544 * q^94 - 4408 * q^95

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