LMFDB - Embedded newform 507.4.b.i.337.8

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 $ x^10 + 70x^8 + 1645x^6 + 14700x^4 + 44100x^2 + 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.8
Root			$3.27897i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.i.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27897i q^{2} +3.00000 q^{3} -2.75167 q^{4} -17.5414i q^{5} +9.83692i q^{6} -26.6999i q^{7} +17.2091i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.27897i q^{2} +3.00000 q^{3} -2.75167 q^{4} -17.5414i q^{5} +9.83692i q^{6} -26.6999i q^{7} +17.2091i q^{8} +9.00000 q^{9} +57.5178 q^{10} -21.4026i q^{11} -8.25501 q^{12} +87.5483 q^{14} -52.6242i q^{15} -78.4417 q^{16} -83.9630 q^{17} +29.5108i q^{18} +77.1142i q^{19} +48.2682i q^{20} -80.0997i q^{21} +70.1785 q^{22} -142.119 q^{23} +51.6274i q^{24} -182.701 q^{25} +27.0000 q^{27} +73.4693i q^{28} +134.223 q^{29} +172.553 q^{30} -122.559i q^{31} -119.535i q^{32} -64.2077i q^{33} -275.312i q^{34} -468.354 q^{35} -24.7650 q^{36} -222.587i q^{37} -252.855 q^{38} +301.873 q^{40} -198.321i q^{41} +262.645 q^{42} -154.656 q^{43} +58.8928i q^{44} -157.873i q^{45} -466.006i q^{46} -78.7956i q^{47} -235.325 q^{48} -369.884 q^{49} -599.072i q^{50} -251.889 q^{51} -477.088 q^{53} +88.5323i q^{54} -375.431 q^{55} +459.482 q^{56} +231.342i q^{57} +440.114i q^{58} -42.9282i q^{59} +144.804i q^{60} +496.539 q^{61} +401.869 q^{62} -240.299i q^{63} -235.581 q^{64} +210.535 q^{66} +484.659i q^{67} +231.038 q^{68} -426.358 q^{69} -1535.72i q^{70} -382.432i q^{71} +154.882i q^{72} +193.622i q^{73} +729.858 q^{74} -548.103 q^{75} -212.193i q^{76} -571.447 q^{77} +1049.60 q^{79} +1375.98i q^{80} +81.0000 q^{81} +650.289 q^{82} -861.900i q^{83} +220.408i q^{84} +1472.83i q^{85} -507.112i q^{86} +402.669 q^{87} +368.320 q^{88} +967.645i q^{89} +517.660 q^{90} +391.065 q^{92} -367.678i q^{93} +258.369 q^{94} +1352.69 q^{95} -358.605i q^{96} -591.470i q^{97} -1212.84i q^{98} -192.623i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10}) $ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9	$ 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) $ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * 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 $

$2$	3.27897i	1.15929i	0.814868	+	0.579646i	$0.196809\pi$
$2$	3.27897i	1.15929i	−0.814868	+	0.579646i	$0.803191\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−2.75167	−0.343959
$5$	− 17.5414i	− 1.56895i	−0.620160	−	0.784476i	$-0.712932\pi$
$5$	− 17.5414i	− 1.56895i	0.620160	−	0.784476i	$-0.287068\pi$
$6$	9.83692i	0.669318i
$7$	− 26.6999i	− 1.44166i	−0.693112	−	0.720829i	$-0.743761\pi$
$7$	− 26.6999i	− 1.44166i	0.693112	−	0.720829i	$-0.256239\pi$
$8$	17.2091i	0.760544i
$9$	9.00000	0.333333
$10$	57.5178	1.81887
$11$	− 21.4026i	− 0.586647i	−0.956013	−	0.293324i	$-0.905239\pi$
$11$	− 21.4026i	− 0.586647i	0.956013	−	0.293324i	$-0.0947613\pi$
$12$	−8.25501	−0.198585
$13$	0	0
$13$	0	0
$14$	87.5483	1.67130
$15$	− 52.6242i	− 0.905834i
$16$	−78.4417	−1.22565
$17$	−83.9630	−1.19788	−0.598942	−	0.800793i	$-0.704412\pi$
$17$	−83.9630	−1.19788	−0.598942	+	0.800793i	$0.704412\pi$
$18$	29.5108i	0.386431i
$19$	77.1142i	0.931116i	0.885017	+	0.465558i	$0.154146\pi$
$19$	77.1142i	0.931116i	−0.885017	+	0.465558i	$0.845854\pi$
$20$	48.2682i	0.539654i
$21$	− 80.0997i	− 0.832342i
$22$	70.1785	0.680096
$23$	−142.119	−1.28843	−0.644216	−	0.764844i	$-0.722816\pi$
$23$	−142.119	−1.28843	−0.644216	+	0.764844i	$0.722816\pi$
$24$	51.6274i	0.439100i
$25$	−182.701	−1.46161
$26$	0	0
$27$	27.0000	0.192450
$28$	73.4693i	0.495871i
$29$	134.223	0.859469	0.429734	−	0.902955i	$-0.358607\pi$
$29$	134.223	0.859469	0.429734	+	0.902955i	$0.358607\pi$
$30$	172.553	1.05013
$31$	− 122.559i	− 0.710074i	−0.934852	−	0.355037i	$-0.884468\pi$
$31$	− 122.559i	− 0.710074i	0.934852	−	0.355037i	$-0.115532\pi$
$32$	− 119.535i	− 0.660344i
$33$	− 64.2077i	− 0.338701i
$34$	− 275.312i	− 1.38870i
$35$	−468.354	−2.26189
$36$	−24.7650	−0.114653
$37$	− 222.587i	− 0.989003i	−0.869177	−	0.494502i	$-0.835351\pi$
$37$	− 222.587i	− 0.989003i	0.869177	−	0.494502i	$-0.164649\pi$
$38$	−252.855	−1.07944
$39$	0	0
$40$	301.873	1.19326
$41$	− 198.321i	− 0.755427i	−0.925923	−	0.377713i	$-0.876710\pi$
$41$	− 198.321i	− 0.755427i	0.925923	−	0.377713i	$-0.123290\pi$
$42$	262.645	0.964928
$43$	−154.656	−0.548483	−0.274242	−	0.961661i	$-0.588427\pi$
$43$	−154.656	−0.548483	−0.274242	+	0.961661i	$0.588427\pi$
$44$	58.8928i	0.201782i
$45$	− 157.873i	− 0.522984i
$46$	− 466.006i	− 1.49367i
$47$	− 78.7956i	− 0.244543i	−0.992497	−	0.122271i	$-0.960982\pi$
$47$	− 78.7956i	− 0.244543i	0.992497	−	0.122271i	$-0.0390178\pi$
$48$	−235.325	−0.707630
$49$	−369.884	−1.07838
$50$	− 599.072i	− 1.69443i
$51$	−251.889	−0.691598
$52$	0	0
$53$	−477.088	−1.23647	−0.618237	−	0.785992i	$-0.712153\pi$
$53$	−477.088	−1.23647	−0.618237	+	0.785992i	$0.712153\pi$
$54$	88.5323i	0.223106i
$55$	−375.431	−0.920421
$56$	459.482	1.09644
$57$	231.342i	0.537580i
$58$	440.114i	0.996376i
$59$	− 42.9282i	− 0.0947249i	−0.998878	−	0.0473625i	$-0.984918\pi$
$59$	− 42.9282i	− 0.0947249i	0.998878	−	0.0473625i	$-0.0150816\pi$
$60$	144.804i	0.311570i
$61$	496.539	1.04222	0.521109	−	0.853490i	$-0.325519\pi$
$61$	496.539	1.04222	0.521109	+	0.853490i	$0.325519\pi$
$62$	401.869	0.823183
$63$	− 240.299i	− 0.480553i
$64$	−235.581	−0.460119
$65$	0	0
$66$	210.535	0.392653
$67$	484.659i	0.883739i	0.897079	+	0.441869i	$0.145685\pi$
$67$	484.659i	0.883739i	−0.897079	+	0.441869i	$0.854315\pi$
$68$	231.038	0.412022
$69$	−426.358	−0.743876
$70$	− 1535.72i	− 2.62219i
$71$	− 382.432i	− 0.639245i	−0.947545	−	0.319622i	$-0.896444\pi$
$71$	− 382.432i	− 0.639245i	0.947545	−	0.319622i	$-0.103556\pi$
$72$	154.882i	0.253515i
$73$	193.622i	0.310435i	0.987880	+	0.155217i	$0.0496078\pi$
$73$	193.622i	0.310435i	−0.987880	+	0.155217i	$0.950392\pi$
$74$	729.858	1.14654
$75$	−548.103	−0.843860
$76$	− 212.193i	− 0.320265i
$77$	−571.447	−0.845745
$78$	0	0
$79$	1049.60	1.49480	0.747399	−	0.664375i	$-0.231302\pi$
$79$	1049.60	1.49480	0.747399	+	0.664375i	$0.231302\pi$
$80$	1375.98i	1.92299i
$81$	81.0000	0.111111
$82$	650.289	0.875761
$83$	− 861.900i	− 1.13983i	−0.821704	−	0.569914i	$-0.806976\pi$
$83$	− 861.900i	− 1.13983i	0.821704	−	0.569914i	$-0.193024\pi$
$84$	220.408i	0.286291i
$85$	1472.83i	1.87942i
$86$	− 507.112i	− 0.635853i
$87$	402.669	0.496215
$88$	368.320	0.446171
$89$	967.645i	1.15247i	0.817283	+	0.576237i	$0.195479\pi$
$89$	967.645i	1.15247i	−0.817283	+	0.576237i	$0.804521\pi$
$90$	517.660	0.606291
$91$	0	0
$92$	391.065	0.443167
$93$	− 367.678i	− 0.409961i
$94$	258.369	0.283497
$95$	1352.69	1.46088
$96$	− 358.605i	− 0.381250i
$97$	− 591.470i	− 0.619120i	−0.950880	−	0.309560i	$-0.899818\pi$
$97$	− 591.470i	− 0.619120i	0.950880	−	0.309560i	$-0.100182\pi$
$98$	− 1212.84i	− 1.25016i
$99$	− 192.623i	− 0.195549i
$100$	502.733	0.502733
$101$	−255.110	−0.251331	−0.125665	−	0.992073i	$-0.540107\pi$
$101$	−255.110	−0.251331	−0.125665	+	0.992073i	$0.540107\pi$
$102$	− 825.937i	− 0.801764i
$103$	247.355	0.236627	0.118313	−	0.992976i	$-0.462251\pi$
$103$	247.355	0.236627	0.118313	+	0.992976i	$0.462251\pi$
$104$	0	0
$105$	−1405.06	−1.30590
$106$	− 1564.36i	− 1.43343i
$107$	683.484	0.617522	0.308761	−	0.951140i	$-0.400086\pi$
$107$	683.484	0.617522	0.308761	+	0.951140i	$0.400086\pi$
$108$	−74.2951	−0.0661949
$109$	− 1697.76i	− 1.49189i	−0.666006	−	0.745946i	$-0.731998\pi$
$109$	− 1697.76i	− 1.49189i	0.666006	−	0.745946i	$-0.268002\pi$
$110$	− 1231.03i	− 1.06704i
$111$	− 667.762i	− 0.571001i
$112$	2094.38i	1.76697i
$113$	−380.709	−0.316939	−0.158469	−	0.987364i	$-0.550656\pi$
$113$	−380.709	−0.316939	−0.158469	+	0.987364i	$0.550656\pi$
$114$	−758.566	−0.623212
$115$	2492.97i	2.02149i
$116$	−369.337	−0.295622
$117$	0	0
$118$	140.760	0.109814
$119$	2241.80i	1.72694i
$120$	905.618	0.688927
$121$	872.930	0.655845
$122$	1628.14i	1.20824i
$123$	− 594.962i	− 0.436146i
$124$	337.243i	0.244236i
$125$	1012.16i	0.724241i
$126$	787.934	0.557101
$127$	−123.231	−0.0861025	−0.0430513	−	0.999073i	$-0.513708\pi$
$127$	−123.231	−0.0861025	−0.0430513	+	0.999073i	$0.513708\pi$
$128$	− 1728.74i	− 1.19376i
$129$	−463.967	−0.316667
$130$	0	0
$131$	−1218.41	−0.812616	−0.406308	−	0.913736i	$-0.633184\pi$
$131$	−1218.41	−0.812616	−0.406308	+	0.913736i	$0.633184\pi$
$132$	176.678i	0.116499i
$133$	2058.94	1.34235
$134$	−1589.18	−1.02451
$135$	− 473.618i	− 0.301945i
$136$	− 1444.93i	− 0.911042i
$137$	2728.83i	1.70175i	0.525369	+	0.850875i	$0.323927\pi$
$137$	2728.83i	1.70175i	−0.525369	+	0.850875i	$0.676073\pi$
$138$	− 1398.02i	− 0.862370i
$139$	3112.78	1.89944	0.949722	−	0.313094i	$-0.101366\pi$
$139$	3112.78	1.89944	0.949722	+	0.313094i	$0.101366\pi$
$140$	1288.75	0.777998
$141$	− 236.387i	− 0.141187i
$142$	1253.99	0.741071
$143$	0	0
$144$	−705.975	−0.408550
$145$	− 2354.46i	− 1.34846i
$146$	−634.881	−0.359885
$147$	−1109.65	−0.622603
$148$	612.487i	0.340176i
$149$	− 1370.57i	− 0.753567i	−0.926301	−	0.376784i	$-0.877030\pi$
$149$	− 1370.57i	− 0.753567i	0.926301	−	0.376784i	$-0.122970\pi$
$150$	− 1797.22i	− 0.978280i
$151$	− 2847.56i	− 1.53464i	−0.641263	−	0.767321i	$-0.721589\pi$
$151$	− 2847.56i	− 1.53464i	0.641263	−	0.767321i	$-0.278411\pi$
$152$	−1327.07	−0.708154
$153$	−755.667	−0.399294
$154$	− 1873.76i	− 0.980466i
$155$	−2149.86	−1.11407
$156$	0	0
$157$	3354.00	1.70496	0.852479	−	0.522761i	$-0.175098\pi$
$157$	3354.00	1.70496	0.852479	+	0.522761i	$0.175098\pi$
$158$	3441.61i	1.73291i
$159$	−1431.26	−0.713878
$160$	−2096.81	−1.03605
$161$	3794.57i	1.85748i
$162$	265.597i	0.128810i
$163$	− 2196.18i	− 1.05533i	−0.849453	−	0.527664i	$-0.823068\pi$
$163$	− 2196.18i	− 1.05533i	0.849453	−	0.527664i	$-0.176932\pi$
$164$	545.713i	0.259836i
$165$	−1126.29	−0.531405
$166$	2826.15	1.32139
$167$	912.535i	0.422839i	0.977395	+	0.211419i	$0.0678086\pi$
$167$	912.535i	0.422839i	−0.977395	+	0.211419i	$0.932191\pi$
$168$	1378.45	0.633033
$169$	0	0
$170$	−4829.37	−2.17880
$171$	694.027i	0.310372i
$172$	425.562	0.188656
$173$	−899.636	−0.395364	−0.197682	−	0.980266i	$-0.563341\pi$
$173$	−899.636	−0.395364	−0.197682	+	0.980266i	$0.563341\pi$
$174$	1320.34i	0.575258i
$175$	4878.10i	2.10714i
$176$	1678.85i	0.719025i
$177$	− 128.784i	− 0.0546895i
$178$	−3172.88	−1.33605
$179$	−313.278	−0.130813	−0.0654064	−	0.997859i	$-0.520834\pi$
$179$	−313.278	−0.130813	−0.0654064	+	0.997859i	$0.520834\pi$
$180$	434.413i	0.179885i
$181$	2745.06	1.12728	0.563642	−	0.826019i	$-0.309400\pi$
$181$	2745.06	1.12728	0.563642	+	0.826019i	$0.309400\pi$
$182$	0	0
$183$	1489.62	0.601725
$184$	− 2445.75i	− 0.979909i
$185$	−3904.50	−1.55170
$186$	1205.61	0.475265
$187$	1797.02i	0.702735i
$188$	216.819i	0.0841126i
$189$	− 720.897i	− 0.277447i
$190$	4435.44i	1.69358i
$191$	89.8679	0.0340451	0.0170226	−	0.999855i	$-0.494581\pi$
$191$	89.8679	0.0340451	0.0170226	+	0.999855i	$0.494581\pi$
$192$	−706.743	−0.265650
$193$	848.954i	0.316627i	0.987389	+	0.158314i	$0.0506057\pi$
$193$	848.954i	0.316627i	−0.987389	+	0.158314i	$0.949394\pi$
$194$	1939.41	0.717741
$195$	0	0
$196$	1017.80	0.370918
$197$	− 4343.86i	− 1.57100i	−0.618860	−	0.785501i	$-0.712405\pi$
$197$	− 4343.86i	− 1.57100i	0.618860	−	0.785501i	$-0.287595\pi$
$198$	631.606	0.226699
$199$	3328.41	1.18565	0.592825	−	0.805331i	$-0.298012\pi$
$199$	3328.41	1.18565	0.592825	+	0.805331i	$0.298012\pi$
$200$	− 3144.13i	− 1.11162i
$201$	1453.98i	0.510227i
$202$	− 836.499i	− 0.291366i
$203$	− 3583.74i	− 1.23906i
$204$	693.115	0.237881
$205$	−3478.83	−1.18523
$206$	811.069i	0.274320i
$207$	−1279.07	−0.429477
$208$	0	0
$209$	1650.44	0.546236
$210$	− 4607.16i	− 1.51392i
$211$	−4599.88	−1.50080	−0.750401	−	0.660983i	$-0.770139\pi$
$211$	−4599.88	−1.50080	−0.750401	+	0.660983i	$0.770139\pi$
$212$	1312.79	0.425296
$213$	− 1147.30i	− 0.369068i
$214$	2241.13i	0.715889i
$215$	2712.88i	0.860544i
$216$	464.647i	0.146367i
$217$	−3272.32	−1.02368
$218$	5566.92	1.72954
$219$	580.866i	0.179230i
$220$	1033.06	0.316587
$221$	0	0
$222$	2189.57	0.661958
$223$	2529.58i	0.759611i	0.925066	+	0.379806i	$0.124009\pi$
$223$	2529.58i	0.759611i	−0.925066	+	0.379806i	$0.875991\pi$
$224$	−3191.57	−0.951991
$225$	−1644.31	−0.487203
$226$	− 1248.33i	− 0.367425i
$227$	− 37.2670i	− 0.0108965i	−0.999985	−	0.00544823i	$-0.998266\pi$
$227$	− 37.2670i	− 0.0108965i	0.999985	−	0.00544823i	$-0.00173423\pi$
$228$	− 636.578i	− 0.184905i
$229$	− 4094.45i	− 1.18152i	−0.806846	−	0.590762i	$-0.798827\pi$
$229$	− 4094.45i	− 1.18152i	0.806846	−	0.590762i	$-0.201173\pi$
$230$	−8174.40	−2.34349
$231$	−1714.34	−0.488291
$232$	2309.86i	0.653664i
$233$	−1466.04	−0.412205	−0.206103	−	0.978530i	$-0.566078\pi$
$233$	−1466.04	−0.412205	−0.206103	+	0.978530i	$0.566078\pi$
$234$	0	0
$235$	−1382.19	−0.383676
$236$	118.124i	0.0325815i
$237$	3148.80	0.863023
$238$	−7350.81	−2.00203
$239$	5520.53i	1.49412i	0.664759	+	0.747058i	$0.268534\pi$
$239$	5520.53i	1.49412i	−0.664759	+	0.747058i	$0.731466\pi$
$240$	4127.93i	1.11024i
$241$	2665.63i	0.712483i	0.934394	+	0.356241i	$0.115942\pi$
$241$	2665.63i	0.712483i	−0.934394	+	0.356241i	$0.884058\pi$
$242$	2862.31i	0.760316i
$243$	243.000	0.0641500
$244$	−1366.31	−0.358480
$245$	6488.30i	1.69193i
$246$	1950.87	0.505621
$247$	0	0
$248$	2109.14	0.540042
$249$	− 2585.70i	− 0.658080i
$250$	−3318.84	−0.839607
$251$	−1579.21	−0.397127	−0.198564	−	0.980088i	$-0.563628\pi$
$251$	−1579.21	−0.397127	−0.198564	+	0.980088i	$0.563628\pi$
$252$	661.224i	0.165290i
$253$	3041.72i	0.755855i
$254$	− 404.073i	− 0.0998180i
$255$	4418.49i	1.08508i
$256$	3783.86	0.923794
$257$	2663.91	0.646577	0.323288	−	0.946300i	$-0.395212\pi$
$257$	2663.91	0.646577	0.323288	+	0.946300i	$0.395212\pi$
$258$	− 1521.34i	− 0.367110i
$259$	−5943.06	−1.42581
$260$	0	0
$261$	1208.01	0.286490
$262$	− 3995.12i	− 0.942059i
$263$	−2436.93	−0.571360	−0.285680	−	0.958325i	$-0.592219\pi$
$263$	−2436.93	−0.571360	−0.285680	+	0.958325i	$0.592219\pi$
$264$	1104.96	0.257597
$265$	8368.80i	1.93997i
$266$	6751.21i	1.55618i
$267$	2902.94i	0.665381i
$268$	− 1333.62i	− 0.303970i
$269$	−2683.18	−0.608165	−0.304083	−	0.952646i	$-0.598350\pi$
$269$	−2683.18	−0.608165	−0.304083	+	0.952646i	$0.598350\pi$
$270$	1552.98	0.350042
$271$	− 3857.14i	− 0.864592i	−0.901732	−	0.432296i	$-0.857704\pi$
$271$	− 3857.14i	− 0.864592i	0.901732	−	0.432296i	$-0.142296\pi$
$272$	6586.20	1.46819
$273$	0	0
$274$	−8947.76	−1.97282
$275$	3910.27i	0.857448i
$276$	1173.20	0.255863
$277$	−1104.91	−0.239666	−0.119833	−	0.992794i	$-0.538236\pi$
$277$	−1104.91	−0.239666	−0.119833	+	0.992794i	$0.538236\pi$
$278$	10206.7i	2.20201i
$279$	− 1103.03i	− 0.236691i
$280$	− 8059.97i	− 1.72027i
$281$	4982.58i	1.05778i	0.848691	+	0.528890i	$0.177391\pi$
$281$	4982.58i	1.05778i	−0.848691	+	0.528890i	$0.822609\pi$
$282$	775.106	0.163677
$283$	−2584.86	−0.542947	−0.271473	−	0.962446i	$-0.587511\pi$
$283$	−2584.86	−0.542947	−0.271473	+	0.962446i	$0.587511\pi$
$284$	1052.33i	0.219874i
$285$	4058.07	0.843437
$286$	0	0
$287$	−5295.14	−1.08907
$288$	− 1075.82i	− 0.220115i
$289$	2136.78	0.434924
$290$	7720.22	1.56326
$291$	− 1774.41i	− 0.357449i
$292$	− 532.784i	− 0.106777i
$293$	− 92.5482i	− 0.0184530i	−0.999957	−	0.00922649i	$-0.997063\pi$
$293$	− 92.5482i	− 0.0184530i	0.999957	−	0.00922649i	$-0.00293692\pi$
$294$	− 3638.52i	− 0.721779i
$295$	−753.020	−0.148619
$296$	3830.54	0.752180
$297$	− 577.870i	− 0.112900i
$298$	4494.07	0.873605
$299$	0	0
$300$	1508.20	0.290253
$301$	4129.29i	0.790726i
$302$	9337.07	1.77910
$303$	−765.330	−0.145106
$304$	− 6048.96i	− 1.14122i
$305$	− 8709.99i	− 1.63519i
$306$	− 2477.81i	− 0.462899i
$307$	− 3979.46i	− 0.739803i	−0.929071	−	0.369901i	$-0.879391\pi$
$307$	− 3979.46i	− 0.739803i	0.929071	−	0.369901i	$-0.120609\pi$
$308$	1572.43	0.290901
$309$	742.064	0.136617
$310$	− 7049.34i	− 1.29153i
$311$	−3450.91	−0.629207	−0.314604	−	0.949223i	$-0.601872\pi$
$311$	−3450.91	−0.629207	−0.314604	+	0.949223i	$0.601872\pi$
$312$	0	0
$313$	−6189.03	−1.11765	−0.558825	−	0.829285i	$-0.688748\pi$
$313$	−6189.03	−1.11765	−0.558825	+	0.829285i	$0.688748\pi$
$314$	10997.7i	1.97654i
$315$	−4215.18	−0.753964
$316$	−2888.15	−0.514149
$317$	− 5437.78i	− 0.963459i	−0.876320	−	0.481729i	$-0.840009\pi$
$317$	− 5437.78i	− 0.963459i	0.876320	−	0.481729i	$-0.159991\pi$
$318$	− 4693.08i	− 0.827593i
$319$	− 2872.72i	− 0.504205i
$320$	4132.42i	0.721904i
$321$	2050.45	0.356527
$322$	−12442.3	−2.15336
$323$	− 6474.73i	− 1.11537i
$324$	−222.885	−0.0382176
$325$	0	0
$326$	7201.23	1.22343
$327$	− 5093.29i	− 0.861344i
$328$	3412.93	0.574535
$329$	−2103.83	−0.352547
$330$	− 3693.09i	− 0.616054i
$331$	− 6626.64i	− 1.10040i	−0.835032	−	0.550201i	$-0.814551\pi$
$331$	− 6626.64i	− 1.10040i	0.835032	−	0.550201i	$-0.185449\pi$
$332$	2371.66i	0.392054i
$333$	− 2003.29i	− 0.329668i
$334$	−2992.18	−0.490194
$335$	8501.60	1.38654
$336$	6283.15i	1.02016i
$337$	5538.63	0.895277	0.447638	−	0.894215i	$-0.352265\pi$
$337$	5538.63	0.895277	0.447638	+	0.894215i	$0.352265\pi$
$338$	0	0
$339$	−1142.13	−0.182985
$340$	− 4052.74i	− 0.646443i
$341$	−2623.08	−0.416563
$342$	−2275.70	−0.359812
$343$	717.813i	0.112998i
$344$	− 2661.49i	− 0.417146i
$345$	7478.92i	1.16711i
$346$	− 2949.88i	− 0.458343i
$347$	9398.94	1.45407	0.727034	−	0.686602i	$-0.240898\pi$
$347$	9398.94	1.45407	0.727034	+	0.686602i	$0.240898\pi$
$348$	−1108.01	−0.170677
$349$	5757.33i	0.883045i	0.897250	+	0.441522i	$0.145561\pi$
$349$	5757.33i	0.883045i	−0.897250	+	0.441522i	$0.854439\pi$
$350$	−15995.2	−2.44279
$351$	0	0
$352$	−2558.36	−0.387389
$353$	− 3457.82i	− 0.521364i	−0.965425	−	0.260682i	$-0.916053\pi$
$353$	− 3457.82i	− 0.521364i	0.965425	−	0.260682i	$-0.0839474\pi$
$354$	422.281	0.0634011
$355$	−6708.40	−1.00294
$356$	− 2662.64i	− 0.396403i
$357$	6725.41i	0.997049i
$358$	− 1027.23i	− 0.151650i
$359$	− 7168.96i	− 1.05394i	−0.849885	−	0.526968i	$-0.823329\pi$
$359$	− 7168.96i	− 1.05394i	0.849885	−	0.526968i	$-0.176671\pi$
$360$	2716.85	0.397752
$361$	912.407	0.133023
$362$	9000.97i	1.30685i
$363$	2618.79	0.378652
$364$	0	0
$365$	3396.40	0.487057
$366$	4884.41i	0.697575i
$367$	3910.11	0.556147	0.278073	−	0.960560i	$-0.410304\pi$
$367$	3910.11	0.556147	0.278073	+	0.960560i	$0.410304\pi$
$368$	11148.1	1.57917
$369$	− 1784.89i	− 0.251809i
$370$	− 12802.7i	− 1.79887i
$371$	12738.2i	1.78257i
$372$	1011.73i	0.141010i
$373$	11377.6	1.57938	0.789691	−	0.613505i	$-0.210241\pi$
$373$	11377.6	1.57938	0.789691	+	0.613505i	$0.210241\pi$
$374$	−5892.39	−0.814675
$375$	3036.47i	0.418141i
$376$	1356.00	0.185986
$377$	0	0
$378$	2363.80	0.321643
$379$	4032.22i	0.546494i	0.961944	+	0.273247i	$0.0880976\pi$
$379$	4032.22i	0.546494i	−0.961944	+	0.273247i	$0.911902\pi$
$380$	−3722.16	−0.502481
$381$	−369.694	−0.0497113
$382$	294.675i	0.0394682i
$383$	1990.96i	0.265622i	0.991141	+	0.132811i	$0.0424003\pi$
$383$	1990.96i	0.265622i	−0.991141	+	0.132811i	$0.957600\pi$
$384$	− 5186.23i	− 0.689216i
$385$	10024.0i	1.32693i
$386$	−2783.70	−0.367063
$387$	−1391.90	−0.182828
$388$	1627.53i	0.212952i
$389$	11122.4	1.44969	0.724846	−	0.688911i	$-0.241911\pi$
$389$	11122.4	1.44969	0.724846	+	0.688911i	$0.241911\pi$
$390$	0	0
$391$	11932.8	1.54339
$392$	− 6365.39i	− 0.820155i
$393$	−3655.22	−0.469164
$394$	14243.4	1.82125
$395$	− 18411.4i	− 2.34527i
$396$	530.035i	0.0672608i
$397$	10778.1i	1.36256i	0.732021	+	0.681282i	$0.238577\pi$
$397$	10778.1i	1.36256i	−0.732021	+	0.681282i	$0.761423\pi$
$398$	10913.8i	1.37452i
$399$	6176.82	0.775007
$400$	14331.4	1.79142
$401$	− 4711.58i	− 0.586746i	−0.955998	−	0.293373i	$-0.905222\pi$
$401$	− 4711.58i	− 0.586746i	0.955998	−	0.293373i	$-0.0947778\pi$
$402$	−4767.55	−0.591502
$403$	0	0
$404$	701.978	0.0864473
$405$	− 1420.85i	− 0.174328i
$406$	11751.0	1.43643
$407$	−4763.94	−0.580196
$408$	− 4334.79i	− 0.525991i
$409$	− 1184.78i	− 0.143236i	−0.997432	−	0.0716179i	$-0.977184\pi$
$409$	− 1184.78i	− 0.143236i	0.997432	−	0.0716179i	$-0.0228162\pi$
$410$	− 11407.0i	− 1.37403i
$411$	8186.49i	0.982505i
$412$	−680.638	−0.0813899
$413$	−1146.18	−0.136561
$414$	− 4194.05i	− 0.497890i
$415$	−15118.9	−1.78834
$416$	0	0
$417$	9338.35	1.09664
$418$	5411.75i	0.633248i
$419$	−6168.30	−0.719191	−0.359596	−	0.933108i	$-0.617085\pi$
$419$	−6168.30	−0.719191	−0.359596	+	0.933108i	$0.617085\pi$
$420$	3866.26	0.449177
$421$	10328.8i	1.19571i	0.801603	+	0.597857i	$0.203981\pi$
$421$	10328.8i	1.19571i	−0.801603	+	0.597857i	$0.796019\pi$
$422$	− 15082.9i	− 1.73987i
$423$	− 709.160i	− 0.0815143i
$424$	− 8210.27i	− 0.940392i
$425$	15340.1	1.75084
$426$	3761.96	0.427858
$427$	− 13257.5i	− 1.50252i
$428$	−1880.72	−0.212402
$429$	0	0
$430$	−8895.46	−0.997622
$431$	11312.5i	1.26427i	0.774857	+	0.632137i	$0.217822\pi$
$431$	11312.5i	1.26427i	−0.774857	+	0.632137i	$0.782178\pi$
$432$	−2117.93	−0.235877
$433$	10475.7	1.16266	0.581331	−	0.813667i	$-0.302532\pi$
$433$	10475.7	1.16266	0.581331	+	0.813667i	$0.302532\pi$
$434$	− 10729.8i	− 1.18675i
$435$	− 7063.38i	− 0.778536i
$436$	4671.68i	0.513149i
$437$	− 10959.4i	− 1.19968i
$438$	−1904.64	−0.207779
$439$	2040.62	0.221853	0.110926	−	0.993829i	$-0.464618\pi$
$439$	2040.62	0.221853	0.110926	+	0.993829i	$0.464618\pi$
$440$	− 6460.85i	− 0.700020i
$441$	−3328.96	−0.359460
$442$	0	0
$443$	−4089.28	−0.438572	−0.219286	−	0.975661i	$-0.570373\pi$
$443$	−4089.28	−0.438572	−0.219286	+	0.975661i	$0.570373\pi$
$444$	1837.46i	0.196401i
$445$	16973.9	1.80818
$446$	−8294.43	−0.880611
$447$	− 4111.71i	− 0.435072i
$448$	6289.99i	0.663335i
$449$	− 15217.9i	− 1.59951i	−0.600330	−	0.799753i	$-0.704964\pi$
$449$	− 15217.9i	− 1.59951i	0.600330	−	0.799753i	$-0.295036\pi$
$450$	− 5391.65i	− 0.564810i
$451$	−4244.58	−0.443169
$452$	1047.58	0.109014
$453$	− 8542.67i	− 0.886026i
$454$	122.197	0.0126322
$455$	0	0
$456$	−3981.20	−0.408853
$457$	− 876.316i	− 0.0896988i	−0.998994	−	0.0448494i	$-0.985719\pi$
$457$	− 876.316i	− 0.0896988i	0.998994	−	0.0448494i	$-0.0142808\pi$
$458$	13425.6	1.36973
$459$	−2267.00	−0.230533
$460$	− 6859.84i	− 0.695308i
$461$	16293.4i	1.64611i	0.567959	+	0.823057i	$0.307733\pi$
$461$	16293.4i	1.64611i	−0.567959	+	0.823057i	$0.692267\pi$
$462$	− 5621.27i	− 0.566072i
$463$	11704.8i	1.17488i	0.809269	+	0.587438i	$0.199863\pi$
$463$	11704.8i	1.17488i	−0.809269	+	0.587438i	$0.800137\pi$
$464$	−10528.7	−1.05341
$465$	−6449.59	−0.643209
$466$	− 4807.12i	− 0.477866i
$467$	15616.1	1.54738	0.773688	−	0.633567i	$-0.218410\pi$
$467$	15616.1	1.54738	0.773688	+	0.633567i	$0.218410\pi$
$468$	0	0
$469$	12940.3	1.27405
$470$	− 4532.15i	− 0.444792i
$471$	10062.0	0.984358
$472$	738.757	0.0720424
$473$	3310.03i	0.321766i
$474$	10324.8i	1.00050i
$475$	− 14088.8i	− 1.36093i
$476$	− 6168.70i	− 0.593996i
$477$	−4293.79	−0.412158
$478$	−18101.7	−1.73212
$479$	10376.0i	0.989755i	0.868963	+	0.494877i	$0.164787\pi$
$479$	10376.0i	0.989755i	−0.868963	+	0.494877i	$0.835213\pi$
$480$	−6290.44	−0.598163
$481$	0	0
$482$	−8740.53	−0.825976
$483$	11383.7i	1.07242i
$484$	−2402.01	−0.225584
$485$	−10375.2	−0.971369
$486$	796.791i	0.0743686i
$487$	3994.27i	0.371658i	0.982582	+	0.185829i	$0.0594970\pi$
$487$	3994.27i	0.371658i	−0.982582	+	0.185829i	$0.940503\pi$
$488$	8545.01i	0.792652i
$489$	− 6588.55i	− 0.609294i
$490$	−21275.0	−1.96144
$491$	−12267.3	−1.12753	−0.563763	−	0.825936i	$-0.690647\pi$
$491$	−12267.3	−1.12753	−0.563763	+	0.825936i	$0.690647\pi$
$492$	1637.14i	0.150016i
$493$	−11269.8	−1.02954
$494$	0	0
$495$	−3378.88	−0.306807
$496$	9613.75i	0.870303i
$497$	−10210.9	−0.921573
$498$	8478.44	0.762907
$499$	3578.76i	0.321056i	0.987031	+	0.160528i	$0.0513198\pi$
$499$	3578.76i	0.321056i	−0.987031	+	0.160528i	$0.948680\pi$
$500$	− 2785.12i	− 0.249109i
$501$	2737.61i	0.244126i
$502$	− 5178.19i	− 0.460386i
$503$	−1296.82	−0.114955	−0.0574774	−	0.998347i	$-0.518306\pi$
$503$	−1296.82	−0.114955	−0.0574774	+	0.998347i	$0.518306\pi$
$504$	4135.34	0.365482
$505$	4474.99i	0.394325i
$506$	−9973.72	−0.876257
$507$	0	0
$508$	339.092	0.0296157
$509$	− 4728.60i	− 0.411771i	−0.978576	−	0.205886i	$-0.933993\pi$
$509$	− 4728.60i	− 0.411771i	0.978576	−	0.205886i	$-0.0660075\pi$
$510$	−14488.1	−1.25793
$511$	5169.69	0.447541
$512$	− 1422.78i	− 0.122810i
$513$	2082.08i	0.179193i
$514$	8734.90i	0.749571i
$515$	− 4338.95i	− 0.371256i
$516$	1276.68	0.108920
$517$	−1686.43	−0.143460
$518$	− 19487.1i	− 1.65293i
$519$	−2698.91	−0.228264
$520$	0	0
$521$	−9220.74	−0.775370	−0.387685	−	0.921792i	$-0.626725\pi$
$521$	−9220.74	−0.775370	−0.387685	+	0.921792i	$0.626725\pi$
$522$	3961.03i	0.332125i
$523$	−12102.7	−1.01188	−0.505939	−	0.862569i	$-0.668854\pi$
$523$	−12102.7	−1.01188	−0.505939	+	0.862569i	$0.668854\pi$
$524$	3352.65	0.279506
$525$	14634.3i	1.21656i
$526$	− 7990.64i	− 0.662373i
$527$	10290.4i	0.850585i
$528$	5036.56i	0.415129i
$529$	8030.91	0.660057
$530$	−27441.1	−2.24899
$531$	− 386.353i	− 0.0315750i
$532$	−5665.52	−0.461713
$533$	0	0
$534$	−9518.65	−0.771371
$535$	− 11989.3i	− 0.968862i
$536$	−8340.56	−0.672122
$537$	−939.833	−0.0755248
$538$	− 8798.08i	− 0.705041i
$539$	7916.48i	0.632629i
$540$	1303.24i	0.103857i
$541$	− 12801.3i	− 1.01732i	−0.860968	−	0.508659i	$-0.830141\pi$
$541$	− 12801.3i	− 1.01732i	0.860968	−	0.508659i	$-0.169859\pi$
$542$	12647.5	1.00231
$543$	8235.17	0.650838
$544$	10036.5i	0.791015i
$545$	−29781.2	−2.34071
$546$	0	0
$547$	400.693	0.0313207	0.0156603	−	0.999877i	$-0.495015\pi$
$547$	400.693	0.0313207	0.0156603	+	0.999877i	$0.495015\pi$
$548$	− 7508.84i	− 0.585331i
$549$	4468.85	0.347406
$550$	−12821.7	−0.994033
$551$	10350.5i	0.800265i
$552$	− 7337.25i	− 0.565751i
$553$	− 28024.2i	− 2.15499i
$554$	− 3622.96i	− 0.277843i
$555$	−11713.5	−0.895873
$556$	−8565.35	−0.653330
$557$	− 14475.5i	− 1.10116i	−0.834781	−	0.550582i	$-0.814406\pi$
$557$	− 14475.5i	− 1.10116i	0.834781	−	0.550582i	$-0.185594\pi$
$558$	3616.82	0.274394
$559$	0	0
$560$	36738.5	2.77229
$561$	5391.07i	0.405724i
$562$	−16337.8	−1.22628
$563$	14776.4	1.10613	0.553063	−	0.833139i	$-0.313459\pi$
$563$	14776.4	1.10613	0.553063	+	0.833139i	$0.313459\pi$
$564$	650.458i	0.0485625i
$565$	6678.17i	0.497261i
$566$	− 8475.68i	− 0.629434i
$567$	− 2162.69i	− 0.160184i
$568$	6581.33	0.486173
$569$	6868.88	0.506078	0.253039	−	0.967456i	$-0.418570\pi$
$569$	6868.88	0.506078	0.253039	+	0.967456i	$0.418570\pi$
$570$	13306.3i	0.977790i
$571$	−3011.00	−0.220677	−0.110338	−	0.993894i	$-0.535193\pi$
$571$	−3011.00	−0.220677	−0.110338	+	0.993894i	$0.535193\pi$
$572$	0	0
$573$	269.604	0.0196559
$574$	− 17362.6i	− 1.26255i
$575$	25965.4	1.88318
$576$	−2120.23	−0.153373
$577$	− 23106.6i	− 1.66714i	−0.552411	−	0.833572i	$-0.686292\pi$
$577$	− 23106.6i	− 1.66714i	0.552411	−	0.833572i	$-0.313708\pi$
$578$	7006.45i	0.504204i
$579$	2546.86i	0.182805i
$580$	6478.70i	0.463816i
$581$	−23012.6	−1.64324
$582$	5818.24	0.414388
$583$	10210.9i	0.725374i
$584$	−3332.07	−0.236099
$585$	0	0
$586$	303.463	0.0213924
$587$	3024.81i	0.212687i	0.994329	+	0.106343i	$0.0339143\pi$
$587$	3024.81i	0.212687i	−0.994329	+	0.106343i	$0.966086\pi$
$588$	3053.40	0.214150
$589$	9451.05	0.661161
$590$	− 2469.13i	− 0.172293i
$591$	− 13031.6i	− 0.907019i
$592$	17460.1i	1.21217i
$593$	6396.07i	0.442926i	0.975169	+	0.221463i	$0.0710832\pi$
$593$	6396.07i	0.442926i	−0.975169	+	0.221463i	$0.928917\pi$
$594$	1894.82	0.130884
$595$	39324.4	2.70948
$596$	3771.36i	0.259196i
$597$	9985.22	0.684536
$598$	0	0
$599$	12095.9	0.825084	0.412542	−	0.910939i	$-0.364641\pi$
$599$	12095.9	0.825084	0.412542	+	0.910939i	$0.364641\pi$
$600$	− 9432.38i	− 0.641792i
$601$	−11816.5	−0.802005	−0.401003	−	0.916077i	$-0.631338\pi$
$601$	−11816.5	−0.802005	−0.401003	+	0.916077i	$0.631338\pi$
$602$	−13539.8	−0.916682
$603$	4361.93i	0.294580i
$604$	7835.54i	0.527853i
$605$	− 15312.4i	− 1.02899i
$606$	− 2509.50i	− 0.168220i
$607$	−25164.0	−1.68266	−0.841329	−	0.540523i	$-0.818226\pi$
$607$	−25164.0	−1.68266	−0.841329	+	0.540523i	$0.818226\pi$
$608$	9217.85	0.614857
$609$	− 10751.2i	− 0.715372i
$610$	28559.8	1.89566
$611$	0	0
$612$	2079.35	0.137341
$613$	19583.0i	1.29030i	0.764058	+	0.645148i	$0.223204\pi$
$613$	19583.0i	1.29030i	−0.764058	+	0.645148i	$0.776796\pi$
$614$	13048.5	0.857648
$615$	−10436.5	−0.684292
$616$	− 9834.10i	− 0.643226i
$617$	19677.1i	1.28390i	0.766745	+	0.641952i	$0.221875\pi$
$617$	19677.1i	1.28390i	−0.766745	+	0.641952i	$0.778125\pi$
$618$	2433.21i	0.158379i
$619$	4394.05i	0.285318i	0.989772	+	0.142659i	$0.0455652\pi$
$619$	4394.05i	0.285318i	−0.989772	+	0.142659i	$0.954435\pi$
$620$	5915.71	0.383194
$621$	−3837.22	−0.247959
$622$	− 11315.5i	− 0.729435i
$623$	25836.0	1.66147
$624$	0	0
$625$	−5082.96	−0.325310
$626$	− 20293.7i	− 1.29568i
$627$	4951.32	0.315370
$628$	−9229.10	−0.586435
$629$	18689.1i	1.18471i
$630$	− 13821.5i	− 0.874065i
$631$	23887.7i	1.50706i	0.657415	+	0.753529i	$0.271650\pi$
$631$	23887.7i	1.50706i	−0.657415	+	0.753529i	$0.728350\pi$
$632$	18062.7i	1.13686i
$633$	−13799.7	−0.866488
$634$	17830.4	1.11693
$635$	2161.65i	0.135091i
$636$	3938.37	0.245545
$637$	0	0
$638$	9419.57	0.584521
$639$	− 3441.89i	− 0.213082i
$640$	−30324.6	−1.87295
$641$	5443.62	0.335429	0.167714	−	0.985836i	$-0.446361\pi$
$641$	5443.62	0.335429	0.167714	+	0.985836i	$0.446361\pi$
$642$	6723.38i	0.413318i
$643$	5839.00i	0.358115i	0.983839	+	0.179057i	$0.0573048\pi$
$643$	5839.00i	0.358115i	−0.983839	+	0.179057i	$0.942695\pi$
$644$	− 10441.4i	− 0.638896i
$645$	8138.64i	0.496835i
$646$	21230.5	1.29304
$647$	−8708.86	−0.529182	−0.264591	−	0.964361i	$-0.585237\pi$
$647$	−8708.86	−0.529182	−0.264591	+	0.964361i	$0.585237\pi$
$648$	1393.94i	0.0845049i
$649$	−918.773	−0.0555701
$650$	0	0
$651$	−9816.96	−0.591024
$652$	6043.17i	0.362989i
$653$	2794.93	0.167495	0.0837475	−	0.996487i	$-0.473311\pi$
$653$	2794.93	0.167495	0.0837475	+	0.996487i	$0.473311\pi$
$654$	16700.8	0.998550
$655$	21372.6i	1.27495i
$656$	15556.6i	0.925890i
$657$	1742.60i	0.103478i
$658$	− 6898.41i	− 0.408705i
$659$	31389.9	1.85550	0.927752	−	0.373197i	$-0.121738\pi$
$659$	31389.9	1.85550	0.927752	+	0.373197i	$0.121738\pi$
$660$	3099.19	0.182781
$661$	20597.2i	1.21201i	0.795461	+	0.606005i	$0.207229\pi$
$661$	20597.2i	1.21201i	−0.795461	+	0.606005i	$0.792771\pi$
$662$	21728.6	1.27569
$663$	0	0
$664$	14832.5	0.866889
$665$	− 36116.7i	− 2.10608i
$666$	6568.72	0.382181
$667$	−19075.7	−1.10737
$668$	− 2511.00i	− 0.145439i
$669$	7588.74i	0.438562i
$670$	27876.5i	1.60741i
$671$	− 10627.2i	− 0.611414i
$672$	−9574.72	−0.549632
$673$	17935.8	1.02730	0.513651	−	0.857999i	$-0.328293\pi$
$673$	17935.8	1.02730	0.513651	+	0.857999i	$0.328293\pi$
$674$	18161.0i	1.03789i
$675$	−4932.93	−0.281287
$676$	0	0
$677$	−24104.0	−1.36838	−0.684188	−	0.729305i	$-0.739843\pi$
$677$	−24104.0	−1.36838	−0.684188	+	0.729305i	$0.739843\pi$
$678$	− 3745.00i	− 0.212133i
$679$	−15792.2	−0.892560
$680$	−25346.1	−1.42938
$681$	− 111.801i	− 0.00629107i
$682$	− 8601.02i	− 0.482918i
$683$	− 6744.25i	− 0.377835i	−0.981993	−	0.188918i	$-0.939502\pi$
$683$	− 6744.25i	− 0.377835i	0.981993	−	0.188918i	$-0.0604979\pi$
$684$	− 1909.73i	− 0.106755i
$685$	47867.5	2.66996
$686$	−2353.69	−0.130998
$687$	− 12283.3i	− 0.682153i
$688$	12131.5	0.672249
$689$	0	0
$690$	−24523.2	−1.35302
$691$	30844.4i	1.69809i	0.528323	+	0.849043i	$0.322821\pi$
$691$	30844.4i	1.69809i	−0.528323	+	0.849043i	$0.677179\pi$
$692$	2475.50	0.135989
$693$	−5143.02	−0.281915
$694$	30818.9i	1.68569i
$695$	− 54602.6i	− 2.98014i
$696$	6929.59i	0.377393i
$697$	16651.6i	0.904913i
$698$	−18878.1	−1.02371
$699$	−4398.13	−0.237987
$700$	− 13422.9i	− 0.724769i
$701$	21007.6	1.13188	0.565940	−	0.824447i	$-0.308514\pi$
$701$	21007.6	1.13188	0.565940	+	0.824447i	$0.308514\pi$
$702$	0	0
$703$	17164.6	0.920877
$704$	5042.04i	0.269928i
$705$	−4146.56	−0.221515
$706$	11338.1	0.604413
$707$	6811.41i	0.362333i
$708$	354.372i	0.0188109i
$709$	− 14763.4i	− 0.782017i	−0.920387	−	0.391008i	$-0.872126\pi$
$709$	− 14763.4i	− 0.782017i	0.920387	−	0.391008i	$-0.127874\pi$
$710$	− 21996.7i	− 1.16271i
$711$	9446.39	0.498266
$712$	−16652.3	−0.876507
$713$	17418.0i	0.914882i
$714$	−22052.4	−1.15587
$715$	0	0
$716$	862.037	0.0449942
$717$	16561.6i	0.862628i
$718$	23506.8	1.22182
$719$	26186.3	1.35825	0.679126	−	0.734022i	$-0.262359\pi$
$719$	26186.3	1.35825	0.679126	+	0.734022i	$0.262359\pi$
$720$	12383.8i	0.640996i
$721$	− 6604.34i	− 0.341135i
$722$	2991.76i	0.154213i
$723$	7996.89i	0.411352i
$724$	−7553.49	−0.387739
$725$	−24522.7	−1.25621
$726$	8586.94i	0.438969i
$727$	−20044.0	−1.02254	−0.511272	−	0.859419i	$-0.670825\pi$
$727$	−20044.0	−1.02254	−0.511272	+	0.859419i	$0.670825\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	11136.7i	0.564641i
$731$	12985.4	0.657019
$732$	−4098.93	−0.206968
$733$	− 25555.5i	− 1.28774i	−0.765134	−	0.643871i	$-0.777327\pi$
$733$	− 25555.5i	− 1.28774i	0.765134	−	0.643871i	$-0.222673\pi$
$734$	12821.1i	0.644737i
$735$	19464.9i	0.976834i
$736$	16988.2i	0.850809i
$737$	10372.9	0.518443
$738$	5852.60	0.291920
$739$	11108.5i	0.552953i	0.961021	+	0.276477i	$0.0891669\pi$
$739$	11108.5i	0.552953i	−0.961021	+	0.276477i	$0.910833\pi$
$740$	10743.9	0.533720
$741$	0	0
$742$	−41768.2	−2.06652
$743$	− 28188.2i	− 1.39182i	−0.718128	−	0.695911i	$-0.755001\pi$
$743$	− 28188.2i	− 1.39182i	0.718128	−	0.695911i	$-0.244999\pi$
$744$	6327.42	0.311793
$745$	−24041.7	−1.18231
$746$	37306.8i	1.83096i
$747$	− 7757.10i	− 0.379943i
$748$	− 4944.82i	− 0.241712i
$749$	− 18248.9i	− 0.890256i
$750$	−9956.51	−0.484747
$751$	−17176.6	−0.834596	−0.417298	−	0.908770i	$-0.637023\pi$
$751$	−17176.6	−0.834596	−0.417298	+	0.908770i	$0.637023\pi$
$752$	6180.86i	0.299724i
$753$	−4737.63	−0.229281
$754$	0	0
$755$	−49950.2	−2.40778
$756$	1983.67i	0.0954304i
$757$	−4409.96	−0.211734	−0.105867	−	0.994380i	$-0.533762\pi$
$757$	−4409.96	−0.211734	−0.105867	+	0.994380i	$0.533762\pi$
$758$	−13221.5	−0.633546
$759$	9125.16i	0.436393i
$760$	23278.6i	1.11106i
$761$	− 32443.6i	− 1.54544i	−0.634746	−	0.772721i	$-0.718895\pi$
$761$	− 32443.6i	− 1.54544i	0.634746	−	0.772721i	$-0.281105\pi$
$762$	− 1212.22i	− 0.0576300i
$763$	−45330.1	−2.15080
$764$	−247.287	−0.0117101
$765$	13255.5i	0.626473i
$766$	−6528.30	−0.307934
$767$	0	0
$768$	11351.6	0.533353
$769$	− 31994.1i	− 1.50031i	−0.661263	−	0.750155i	$-0.729979\pi$
$769$	− 31994.1i	− 1.50031i	0.661263	−	0.750155i	$-0.270021\pi$
$770$	−32868.4	−1.53830
$771$	7991.73	0.373301
$772$	− 2336.04i	− 0.108907i
$773$	− 8979.39i	− 0.417809i	−0.977936	−	0.208904i	$-0.933010\pi$
$773$	− 8979.39i	− 0.417809i	0.977936	−	0.208904i	$-0.0669898\pi$
$774$	− 4564.01i	− 0.211951i
$775$	22391.7i	1.03785i
$776$	10178.7	0.470868
$777$	−17829.2	−0.823189
$778$	36470.2i	1.68062i
$779$	15293.3	0.703390
$780$	0	0
$781$	−8185.04	−0.375011
$782$	39127.2i	1.78924i
$783$	3624.02	0.165405
$784$	29014.4	1.32172
$785$	− 58833.9i	− 2.67500i
$786$	− 11985.4i	− 0.543898i
$787$	12570.5i	0.569366i	0.958622	+	0.284683i	$0.0918884\pi$
$787$	12570.5i	0.569366i	−0.958622	+	0.284683i	$0.908112\pi$
$788$	11952.9i	0.540360i
$789$	−7310.80	−0.329875
$790$	60370.6	2.71885
$791$	10164.9i	0.456917i
$792$	3314.88	0.148724
$793$	0	0
$794$	−35341.2	−1.57961
$795$	25106.4i	1.12004i
$796$	−9158.67	−0.407815
$797$	−37863.3	−1.68280	−0.841398	−	0.540416i	$-0.818267\pi$
$797$	−37863.3	−1.68280	−0.841398	+	0.540416i	$0.818267\pi$
$798$	20253.6i	0.898460i
$799$	6615.91i	0.292934i
$800$	21839.2i	0.965165i
$801$	8708.81i	0.384158i
$802$	15449.2	0.680210
$803$	4144.01	0.182116
$804$	− 4000.86i	− 0.175497i
$805$	66562.1	2.91429
$806$	0	0
$807$	−8049.54	−0.351124
$808$	− 4390.22i	− 0.191148i
$809$	2503.79	0.108812	0.0544058	−	0.998519i	$-0.482674\pi$
$809$	2503.79	0.108812	0.0544058	+	0.998519i	$0.482674\pi$
$810$	4658.94	0.202097
$811$	− 5409.55i	− 0.234223i	−0.993119	−	0.117112i	$-0.962636\pi$
$811$	− 5409.55i	− 0.234223i	0.993119	−	0.117112i	$-0.0373635\pi$
$812$	9861.27i	0.426186i
$813$	− 11571.4i	− 0.499172i
$814$	− 15620.8i	− 0.672617i
$815$	−38524.2	−1.65576
$816$	19758.6	0.847658
$817$	− 11926.1i	− 0.510702i
$818$	3884.85	0.166052
$819$	0	0
$820$	9572.58	0.407669
$821$	− 31381.6i	− 1.33401i	−0.745051	−	0.667007i	$-0.767575\pi$
$821$	− 31381.6i	− 1.33401i	0.745051	−	0.667007i	$-0.232425\pi$
$822$	−26843.3	−1.13901
$823$	33046.1	1.39965	0.699827	−	0.714312i	$-0.253260\pi$
$823$	33046.1	1.39965	0.699827	+	0.714312i	$0.253260\pi$
$824$	4256.76i	0.179965i
$825$	11730.8i	0.495048i
$826$	− 3758.29i	− 0.158314i
$827$	33653.7i	1.41506i	0.706684	+	0.707529i	$0.250190\pi$
$827$	33653.7i	1.41506i	−0.706684	+	0.707529i	$0.749810\pi$
$828$	3519.59	0.147722
$829$	−12898.5	−0.540390	−0.270195	−	0.962806i	$-0.587088\pi$
$829$	−12898.5	−0.540390	−0.270195	+	0.962806i	$0.587088\pi$
$830$	− 49574.6i	− 2.07320i
$831$	−3314.72	−0.138371
$832$	0	0
$833$	31056.6	1.29177
$834$	30620.2i	1.27133i
$835$	16007.2	0.663414
$836$	−4541.47	−0.187883
$837$	− 3309.10i	− 0.136654i
$838$	− 20225.7i	− 0.833753i
$839$	395.829i	0.0162879i	0.999967	+	0.00814395i	$0.00259233\pi$
$839$	395.829i	0.0162879i	−0.999967	+	0.00814395i	$0.997408\pi$
$840$	− 24179.9i	− 0.993197i
$841$	−6373.17	−0.261313
$842$	−33867.9	−1.38618
$843$	14947.7i	0.610709i
$844$	12657.4	0.516214
$845$	0	0
$846$	2325.32	0.0944989
$847$	− 23307.1i	− 0.945505i
$848$	37423.6	1.51548
$849$	−7754.57	−0.313470
$850$	50299.9i	2.02973i
$851$	31634.0i	1.27426i
$852$	3156.98i	0.126944i
$853$	21248.9i	0.852930i	0.904504	+	0.426465i	$0.140241\pi$
$853$	21248.9i	0.852930i	−0.904504	+	0.426465i	$0.859759\pi$
$854$	43471.1	1.74186
$855$	12174.2	0.486958
$856$	11762.2i	0.469653i
$857$	9920.37	0.395418	0.197709	−	0.980261i	$-0.436650\pi$
$857$	9920.37	0.395418	0.197709	+	0.980261i	$0.436650\pi$
$858$	0	0
$859$	20946.3	0.831990	0.415995	−	0.909367i	$-0.363433\pi$
$859$	20946.3	0.831990	0.415995	+	0.909367i	$0.363433\pi$
$860$	− 7464.95i	− 0.295991i
$861$	−15885.4	−0.628774
$862$	−37093.2	−1.46566
$863$	− 11271.4i	− 0.444594i	−0.974979	−	0.222297i	$-0.928645\pi$
$863$	− 11271.4i	− 0.444594i	0.974979	−	0.222297i	$-0.0713554\pi$
$864$	− 3227.45i	− 0.127083i
$865$	15780.9i	0.620308i
$866$	34349.7i	1.34786i
$867$	6410.34	0.251103
$868$	9004.34	0.352105
$869$	− 22464.1i	− 0.876920i
$870$	23160.7	0.902551
$871$	0	0
$872$	29217.0	1.13465
$873$	− 5323.23i	− 0.206373i
$874$	35935.6	1.39078
$875$	27024.5	1.04411
$876$	− 1598.35i	− 0.0616476i
$877$	− 9436.36i	− 0.363333i	−0.983360	−	0.181667i	$-0.941851\pi$
$877$	− 9436.36i	− 0.363333i	0.983360	−	0.181667i	$-0.0581491\pi$
$878$	6691.13i	0.257192i
$879$	− 277.645i	− 0.0106538i
$880$	29449.5	1.12811
$881$	20343.0	0.777949	0.388974	−	0.921249i	$-0.372829\pi$
$881$	20343.0	0.777949	0.388974	+	0.921249i	$0.372829\pi$
$882$	− 10915.6i	− 0.416719i
$883$	−46521.9	−1.77303	−0.886515	−	0.462699i	$-0.846881\pi$
$883$	−46521.9	−1.77303	−0.886515	+	0.462699i	$0.846881\pi$
$884$	0	0
$885$	−2259.06	−0.0858051
$886$	− 13408.6i	− 0.508433i
$887$	−19955.1	−0.755384	−0.377692	−	0.925931i	$-0.623282\pi$
$887$	−19955.1	−0.755384	−0.377692	+	0.925931i	$0.623282\pi$
$888$	11491.6	0.434272
$889$	3290.27i	0.124130i
$890$	55656.8i	2.09620i
$891$	− 1733.61i	− 0.0651830i
$892$	− 6960.57i	− 0.261275i
$893$	6076.25	0.227698
$894$	13482.2	0.504376
$895$	5495.33i	0.205239i
$896$	−46157.3	−1.72099
$897$	0	0
$898$	49899.1	1.85429
$899$	− 16450.3i	− 0.610286i
$900$	4524.60	0.167578
$901$	40057.7	1.48115
$902$	− 13917.9i	− 0.513762i
$903$	12387.9i	0.456526i
$904$	− 6551.67i	− 0.241046i
$905$	− 48152.2i	− 1.76866i
$906$	28011.2	1.02716
$907$	2653.95	0.0971587	0.0485793	−	0.998819i	$-0.484531\pi$
$907$	2653.95	0.0971587	0.0485793	+	0.998819i	$0.484531\pi$
$908$	102.546i	0.00374793i
$909$	−2295.99	−0.0837769
$910$	0	0
$911$	1797.50	0.0653720	0.0326860	−	0.999466i	$-0.489594\pi$
$911$	1797.50	0.0653720	0.0326860	+	0.999466i	$0.489594\pi$
$912$	− 18146.9i	− 0.658885i
$913$	−18446.9	−0.668677
$914$	2873.42	0.103987
$915$	− 26130.0i	− 0.944077i
$916$	11266.6i	0.406395i
$917$	32531.3i	1.17151i
$918$	− 7433.43i	− 0.267255i
$919$	−48642.0	−1.74597	−0.872987	−	0.487743i	$-0.837820\pi$
$919$	−48642.0	−1.74597	−0.872987	+	0.487743i	$0.837820\pi$
$920$	−42901.9	−1.53743
$921$	− 11938.4i	− 0.427125i
$922$	−53425.6	−1.90833
$923$	0	0
$924$	4717.30	0.167952
$925$	40666.9i	1.44554i
$926$	−38379.7	−1.36202
$927$	2226.19	0.0788756
$928$	− 16044.4i	− 0.567545i
$929$	− 37745.3i	− 1.33303i	−0.745493	−	0.666513i	$-0.767786\pi$
$929$	− 37745.3i	− 1.33303i	0.745493	−	0.666513i	$-0.232214\pi$
$930$	− 21148.0i	− 0.745668i
$931$	− 28523.3i	− 1.00410i
$932$	4034.07	0.141782
$933$	−10352.7	−0.363273
$934$	51204.6i	1.79386i
$935$	31522.3	1.10256
$936$	0	0
$937$	2705.50	0.0943273	0.0471637	−	0.998887i	$-0.484982\pi$
$937$	2705.50	0.0943273	0.0471637	+	0.998887i	$0.484982\pi$
$938$	42431.0i	1.47700i
$939$	−18567.1	−0.645276
$940$	3803.32	0.131969
$941$	− 5189.27i	− 0.179772i	−0.995952	−	0.0898860i	$-0.971350\pi$
$941$	− 5189.27i	− 0.179772i	0.995952	−	0.0898860i	$-0.0286503\pi$
$942$	32993.0i	1.14116i
$943$	28185.2i	0.973316i
$944$	3367.36i	0.116100i
$945$	−12645.6	−0.435301
$946$	−10853.5	−0.373021
$947$	− 72.2711i	− 0.00247993i	−0.999999	−	0.00123997i	$-0.999605\pi$
$947$	− 72.2711i	− 0.00247993i	0.999999	−	0.00123997i	$-0.000394693\pi$
$948$	−8664.45	−0.296844
$949$	0	0
$950$	46196.9	1.57771
$951$	− 16313.4i	− 0.556253i
$952$	−38579.5	−1.31341
$953$	−44695.1	−1.51922	−0.759609	−	0.650380i	$-0.774610\pi$
$953$	−44695.1	−1.51922	−0.759609	+	0.650380i	$0.774610\pi$
$954$	− 14079.2i	− 0.477811i
$955$	− 1576.41i	− 0.0534151i
$956$	− 15190.7i	− 0.513914i
$957$	− 8618.16i	− 0.291103i
$958$	−34022.7	−1.14742
$959$	72859.5	2.45334
$960$	12397.3i	0.416792i
$961$	14770.2	0.495795
$962$	0	0
$963$	6151.35	0.205841
$964$	− 7334.93i	− 0.245065i
$965$	14891.8	0.496772
$966$	−37326.9	−1.24324
$967$	17936.9i	0.596496i	0.954488	+	0.298248i	$0.0964022\pi$
$967$	17936.9i	0.596496i	−0.954488	+	0.298248i	$0.903598\pi$
$968$	15022.4i	0.498799i
$969$	− 19424.2i	− 0.643958i
$970$	− 34020.1i	− 1.12610i
$971$	−40914.6	−1.35223	−0.676113	−	0.736798i	$-0.736337\pi$
$971$	−40914.6	−1.35223	−0.676113	+	0.736798i	$0.736337\pi$
$972$	−668.656	−0.0220650
$973$	− 83111.0i	− 2.73835i
$974$	−13097.1	−0.430860
$975$	0	0
$976$	−38949.3	−1.27740
$977$	− 24118.1i	− 0.789770i	−0.918731	−	0.394885i	$-0.870784\pi$
$977$	− 24118.1i	− 0.789770i	0.918731	−	0.394885i	$-0.129216\pi$
$978$	21603.7	0.706350
$979$	20710.1	0.676096
$980$	− 17853.6i	− 0.581953i
$981$	− 15279.9i	− 0.497297i
$982$	− 40224.2i	− 1.30713i
$983$	− 2928.61i	− 0.0950235i	−0.998871	−	0.0475118i	$-0.984871\pi$
$983$	− 2928.61i	− 0.0950235i	0.998871	−	0.0475118i	$-0.0151292\pi$
$984$	10238.8	0.331708
$985$	−76197.5	−2.46483
$986$	− 36953.3i	− 1.19354i
$987$	−6311.50	−0.203543
$988$	0	0
$989$	21979.6	0.706683
$990$	− 11079.3i	− 0.355679i
$991$	49809.9	1.59663	0.798317	−	0.602238i	$-0.205724\pi$
$991$	49809.9	1.59663	0.798317	+	0.602238i	$0.205724\pi$
$992$	−14650.1	−0.468893
$993$	− 19879.9i	− 0.635318i
$994$	− 33481.3i	− 1.06837i
$995$	− 58384.9i	− 1.86023i
$996$	7114.99i	0.226352i
$997$	−41151.7	−1.30721	−0.653604	−	0.756837i	$-0.726744\pi$
$997$	−41151.7	−1.30721	−0.653604	+	0.756837i	$0.726744\pi$
$998$	−11734.7	−0.372198
$999$	− 6009.86i	− 0.190334i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.i.337.8		10
13.5	odd	4		507.4.a.r.1.8		10
13.8	odd	4		507.4.a.r.1.3		10
13.9	even	3		39.4.j.c.10.4	yes	10
13.10	even	6		39.4.j.c.4.4	✓	10
13.12	even	2	inner	507.4.b.i.337.3		10
39.5	even	4		1521.4.a.bk.1.3		10
39.8	even	4		1521.4.a.bk.1.8		10
39.23	odd	6		117.4.q.e.82.2		10
39.35	odd	6		117.4.q.e.10.2		10
52.23	odd	6		624.4.bv.h.433.5		10
52.35	odd	6		624.4.bv.h.49.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.4	✓	10	13.10	even	6
39.4.j.c.10.4	yes	10	13.9	even	3
117.4.q.e.10.2		10	39.35	odd	6
117.4.q.e.82.2		10	39.23	odd	6
507.4.a.r.1.3		10	13.8	odd	4
507.4.a.r.1.8		10	13.5	odd	4
507.4.b.i.337.3		10	13.12	even	2	inner
507.4.b.i.337.8		10	1.1	even	1	trivial
624.4.bv.h.49.1		10	52.35	odd	6
624.4.bv.h.433.5		10	52.23	odd	6
1521.4.a.bk.1.3		10	39.5	even	4
1521.4.a.bk.1.8		10	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.27897i q^{2} +3.00000 q^{3} -2.75167 q^{4} -17.5414i q^{5} +9.83692i q^{6} -26.6999i q^{7} +17.2091i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.27897i q^{2} +3.00000 q^{3} -2.75167 q^{4} -17.5414i q^{5} +9.83692i q^{6} -26.6999i q^{7} +17.2091i q^{8} +9.00000 q^{9} +57.5178 q^{10} -21.4026i q^{11} -8.25501 q^{12} +87.5483 q^{14} -52.6242i q^{15} -78.4417 q^{16} -83.9630 q^{17} +29.5108i q^{18} +77.1142i q^{19} +48.2682i q^{20} -80.0997i q^{21} +70.1785 q^{22} -142.119 q^{23} +51.6274i q^{24} -182.701 q^{25} +27.0000 q^{27} +73.4693i q^{28} +134.223 q^{29} +172.553 q^{30} -122.559i q^{31} -119.535i q^{32} -64.2077i q^{33} -275.312i q^{34} -468.354 q^{35} -24.7650 q^{36} -222.587i q^{37} -252.855 q^{38} +301.873 q^{40} -198.321i q^{41} +262.645 q^{42} -154.656 q^{43} +58.8928i q^{44} -157.873i q^{45} -466.006i q^{46} -78.7956i q^{47} -235.325 q^{48} -369.884 q^{49} -599.072i q^{50} -251.889 q^{51} -477.088 q^{53} +88.5323i q^{54} -375.431 q^{55} +459.482 q^{56} +231.342i q^{57} +440.114i q^{58} -42.9282i q^{59} +144.804i q^{60} +496.539 q^{61} +401.869 q^{62} -240.299i q^{63} -235.581 q^{64} +210.535 q^{66} +484.659i q^{67} +231.038 q^{68} -426.358 q^{69} -1535.72i q^{70} -382.432i q^{71} +154.882i q^{72} +193.622i q^{73} +729.858 q^{74} -548.103 q^{75} -212.193i q^{76} -571.447 q^{77} +1049.60 q^{79} +1375.98i q^{80} +81.0000 q^{81} +650.289 q^{82} -861.900i q^{83} +220.408i q^{84} +1472.83i q^{85} -507.112i q^{86} +402.669 q^{87} +368.320 q^{88} +967.645i q^{89} +517.660 q^{90} +391.065 q^{92} -367.678i q^{93} +258.369 q^{94} +1352.69 q^{95} -358.605i q^{96} -591.470i q^{97} -1212.84i q^{98} -192.623i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10}) \) 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9	\( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * q^95

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