LMFDB - Embedded newform 507.4.b.k.337.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			337.6
Root			$-5.06791i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.k.337.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82093i q^{2} +3.00000 q^{3} +0.0423641 q^{4} -3.41089i q^{5} -8.46278i q^{6} -13.3442i q^{7} -22.6869i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.82093i q^{2} +3.00000 q^{3} +0.0423641 q^{4} -3.41089i q^{5} -8.46278i q^{6} -13.3442i q^{7} -22.6869i q^{8} +9.00000 q^{9} -9.62187 q^{10} -35.4529i q^{11} +0.127092 q^{12} -37.6430 q^{14} -10.2327i q^{15} -63.6593 q^{16} -69.6526 q^{17} -25.3884i q^{18} +12.4014i q^{19} -0.144499i q^{20} -40.0325i q^{21} -100.010 q^{22} +126.251 q^{23} -68.0608i q^{24} +113.366 q^{25} +27.0000 q^{27} -0.565314i q^{28} -179.060 q^{29} -28.8656 q^{30} +255.935i q^{31} -1.91716i q^{32} -106.359i q^{33} +196.485i q^{34} -45.5155 q^{35} +0.381277 q^{36} -207.235i q^{37} +34.9833 q^{38} -77.3825 q^{40} -117.701i q^{41} -112.929 q^{42} -553.224 q^{43} -1.50193i q^{44} -30.6980i q^{45} -356.146i q^{46} -62.9185i q^{47} -190.978 q^{48} +164.933 q^{49} -319.797i q^{50} -208.958 q^{51} -147.031 q^{53} -76.1651i q^{54} -120.926 q^{55} -302.738 q^{56} +37.2041i q^{57} +505.115i q^{58} -274.087i q^{59} -0.433498i q^{60} +603.039 q^{61} +721.974 q^{62} -120.098i q^{63} -514.683 q^{64} -300.031 q^{66} -741.019i q^{67} -2.95077 q^{68} +378.754 q^{69} +128.396i q^{70} +572.574i q^{71} -204.182i q^{72} -26.7155i q^{73} -584.595 q^{74} +340.098 q^{75} +0.525372i q^{76} -473.090 q^{77} -207.798 q^{79} +217.135i q^{80} +81.0000 q^{81} -332.026 q^{82} -1031.37i q^{83} -1.69594i q^{84} +237.577i q^{85} +1560.60i q^{86} -537.179 q^{87} -804.318 q^{88} -1229.66i q^{89} -86.5968 q^{90} +5.34852 q^{92} +767.805i q^{93} -177.488 q^{94} +42.2996 q^{95} -5.75148i q^{96} +1795.12i q^{97} -465.264i q^{98} -319.076i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	− 2.82093i	− 0.997349i	−0.866789	−	0.498674i	$-0.833820\pi$
$2$	− 2.82093i	− 0.997349i	0.866789	−	0.498674i	$-0.166180\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0.0423641	0.00529552
$5$	− 3.41089i	− 0.305079i	−0.988297	−	0.152539i	$-0.951255\pi$
$5$	− 3.41089i	− 0.305079i	0.988297	−	0.152539i	$-0.0487451\pi$
$6$	− 8.46278i	− 0.575820i
$7$	− 13.3442i	− 0.720518i	−0.932852	−	0.360259i	$-0.882688\pi$
$7$	− 13.3442i	− 0.720518i	0.932852	−	0.360259i	$-0.117312\pi$
$8$	− 22.6869i	− 1.00263i
$9$	9.00000	0.333333
$10$	−9.62187	−0.304270
$11$	− 35.4529i	− 0.971769i	−0.874023	−	0.485885i	$-0.838498\pi$
$11$	− 35.4529i	− 0.971769i	0.874023	−	0.485885i	$-0.161502\pi$
$12$	0.127092	0.00305737
$13$	0	0
$13$	0	0
$14$	−37.6430	−0.718607
$15$	− 10.2327i	− 0.176137i
$16$	−63.6593	−0.994676
$17$	−69.6526	−0.993720	−0.496860	−	0.867831i	$-0.665514\pi$
$17$	−69.6526	−0.993720	−0.496860	+	0.867831i	$0.665514\pi$
$18$	− 25.3884i	− 0.332450i
$19$	12.4014i	0.149740i	0.997193	+	0.0748701i	$0.0238542\pi$
$19$	12.4014i	0.149740i	−0.997193	+	0.0748701i	$0.976146\pi$
$20$	− 0.144499i	− 0.00161555i
$21$	− 40.0325i	− 0.415991i
$22$	−100.010	−0.969193
$23$	126.251	1.14457	0.572287	−	0.820053i	$-0.306056\pi$
$23$	126.251	1.14457	0.572287	+	0.820053i	$0.306056\pi$
$24$	− 68.0608i	− 0.578869i
$25$	113.366	0.906927
$26$	0	0
$27$	27.0000	0.192450
$28$	− 0.565314i	− 0.00381551i
$29$	−179.060	−1.14657	−0.573286	−	0.819356i	$-0.694331\pi$
$29$	−179.060	−1.14657	−0.573286	+	0.819356i	$0.694331\pi$
$30$	−28.8656	−0.175670
$31$	255.935i	1.48281i	0.671055	+	0.741407i	$0.265841\pi$
$31$	255.935i	1.48281i	−0.671055	+	0.741407i	$0.734159\pi$
$32$	− 1.91716i	− 0.0105909i
$33$	− 106.359i	− 0.561051i
$34$	196.485i	0.991085i
$35$	−45.5155	−0.219815
$36$	0.381277	0.00176517
$37$	− 207.235i	− 0.920790i	−0.887714	−	0.460395i	$-0.847708\pi$
$37$	− 207.235i	− 0.920790i	0.887714	−	0.460395i	$-0.152292\pi$
$38$	34.9833	0.149343
$39$	0	0
$40$	−77.3825	−0.305881
$41$	− 117.701i	− 0.448337i	−0.974550	−	0.224169i	$-0.928033\pi$
$41$	− 117.701i	− 0.448337i	0.974550	−	0.224169i	$-0.0719666\pi$
$42$	−112.929	−0.414888
$43$	−553.224	−1.96200	−0.980998	−	0.194016i	$-0.937849\pi$
$43$	−553.224	−1.96200	−0.980998	+	0.194016i	$0.937849\pi$
$44$	− 1.50193i	− 0.00514602i
$45$	− 30.6980i	− 0.101693i
$46$	− 356.146i	− 1.14154i
$47$	− 62.9185i	− 0.195268i	−0.995222	−	0.0976341i	$-0.968873\pi$
$47$	− 62.9185i	− 0.195268i	0.995222	−	0.0976341i	$-0.0311275\pi$
$48$	−190.978	−0.574277
$49$	164.933	0.480854
$50$	− 319.797i	− 0.904522i
$51$	−208.958	−0.573724
$52$	0	0
$53$	−147.031	−0.381061	−0.190531	−	0.981681i	$-0.561021\pi$
$53$	−147.031	−0.381061	−0.190531	+	0.981681i	$0.561021\pi$
$54$	− 76.1651i	− 0.191940i
$55$	−120.926	−0.296466
$56$	−302.738	−0.722413
$57$	37.2041i	0.0864526i
$58$	505.115i	1.14353i
$59$	− 274.087i	− 0.604798i	−0.953181	−	0.302399i	$-0.902213\pi$
$59$	− 274.087i	− 0.604798i	0.953181	−	0.302399i	$-0.0977875\pi$
$60$	− 0.433498i	0 0.000932738i
$61$	603.039	1.26576	0.632879	−	0.774251i	$-0.281873\pi$
$61$	603.039	1.26576	0.632879	+	0.774251i	$0.281873\pi$
$62$	721.974	1.47888
$63$	− 120.098i	− 0.240173i
$64$	−514.683	−1.00524
$65$	0	0
$66$	−300.031	−0.559564
$67$	− 741.019i	− 1.35119i	−0.737272	−	0.675596i	$-0.763887\pi$
$67$	− 741.019i	− 1.35119i	0.737272	−	0.675596i	$-0.236113\pi$
$68$	−2.95077	−0.00526226
$69$	378.754	0.660820
$70$	128.396i	0.219232i
$71$	572.574i	0.957071i	0.878068	+	0.478536i	$0.158832\pi$
$71$	572.574i	0.957071i	−0.878068	+	0.478536i	$0.841168\pi$
$72$	− 204.182i	− 0.334210i
$73$	− 26.7155i	− 0.0428330i	−0.999771	−	0.0214165i	$-0.993182\pi$
$73$	− 26.7155i	− 0.0428330i	0.999771	−	0.0214165i	$-0.00681761\pi$
$74$	−584.595	−0.918349
$75$	340.098	0.523614
$76$	0.525372i	0 0.000792952i
$77$	−473.090	−0.700177
$78$	0	0
$79$	−207.798	−0.295938	−0.147969	−	0.988992i	$-0.547274\pi$
$79$	−207.798	−0.295938	−0.147969	+	0.988992i	$0.547274\pi$
$80$	217.135i	0.303455i
$81$	81.0000	0.111111
$82$	−332.026	−0.447149
$83$	− 1031.37i	− 1.36395i	−0.731374	−	0.681976i	$-0.761121\pi$
$83$	− 1031.37i	− 1.36395i	0.731374	−	0.681976i	$-0.238879\pi$
$84$	− 1.69594i	− 0.00220289i
$85$	237.577i	0.303163i
$86$	1560.60i	1.95679i
$87$	−537.179	−0.661973
$88$	−804.318	−0.974325
$89$	− 1229.66i	− 1.46453i	−0.681019	−	0.732266i	$-0.738463\pi$
$89$	− 1229.66i	− 1.46453i	0.681019	−	0.732266i	$-0.261537\pi$
$90$	−86.5968	−0.101423
$91$	0	0
$92$	5.34852	0.00606111
$93$	767.805i	0.856103i
$94$	−177.488	−0.194750
$95$	42.2996	0.0456826
$96$	− 5.75148i	− 0.00611467i
$97$	1795.12i	1.87903i	0.342502	+	0.939517i	$0.388726\pi$
$97$	1795.12i	1.87903i	−0.342502	+	0.939517i	$0.611274\pi$
$98$	− 465.264i	− 0.479579i
$99$	− 319.076i	− 0.323923i
$100$	4.80265	0.00480265
$101$	769.697	0.758295	0.379147	−	0.925336i	$-0.376217\pi$
$101$	769.697	0.758295	0.379147	+	0.925336i	$0.376217\pi$
$102$	589.455i	0.572203i
$103$	−1543.66	−1.47671	−0.738357	−	0.674410i	$-0.764398\pi$
$103$	−1543.66	−1.47671	−0.738357	+	0.674410i	$0.764398\pi$
$104$	0	0
$105$	−136.546	−0.126910
$106$	414.764i	0.380051i
$107$	2027.64	1.83196	0.915978	−	0.401228i	$-0.131417\pi$
$107$	2027.64	1.83196	0.915978	+	0.401228i	$0.131417\pi$
$108$	1.14383	0.00101912
$109$	1644.97i	1.44550i	0.691108	+	0.722751i	$0.257123\pi$
$109$	1644.97i	1.44550i	−0.691108	+	0.722751i	$0.742877\pi$
$110$	341.123i	0.295680i
$111$	− 621.705i	− 0.531618i
$112$	849.481i	0.716682i
$113$	−654.514	−0.544880	−0.272440	−	0.962173i	$-0.587831\pi$
$113$	−654.514	−0.544880	−0.272440	+	0.962173i	$0.587831\pi$
$114$	104.950	0.0862234
$115$	− 430.629i	− 0.349186i
$116$	−7.58571	−0.00607169
$117$	0	0
$118$	−773.179	−0.603194
$119$	929.456i	0.715993i
$120$	−232.148	−0.176601
$121$	74.0896	0.0556646
$122$	− 1701.13i	− 1.26240i
$123$	− 353.103i	− 0.258848i
$124$	10.8425i	0.00785227i
$125$	− 813.039i	− 0.581763i
$126$	−338.787	−0.239536
$127$	−1325.54	−0.926161	−0.463080	−	0.886316i	$-0.653256\pi$
$127$	−1325.54	−0.926161	−0.463080	+	0.886316i	$0.653256\pi$
$128$	1436.55i	0.991983i
$129$	−1659.67	−1.13276
$130$	0	0
$131$	930.114	0.620339	0.310170	−	0.950681i	$-0.399614\pi$
$131$	930.114	0.620339	0.310170	+	0.950681i	$0.399614\pi$
$132$	− 4.50580i	− 0.00297106i
$133$	165.486	0.107891
$134$	−2090.36	−1.34761
$135$	− 92.0939i	− 0.0587125i
$136$	1580.20i	0.996333i
$137$	2751.03i	1.71559i	0.513989	+	0.857797i	$0.328167\pi$
$137$	2751.03i	1.71559i	−0.513989	+	0.857797i	$0.671833\pi$
$138$	− 1068.44i	− 0.659068i
$139$	2436.65	1.48686	0.743432	−	0.668812i	$-0.233197\pi$
$139$	2436.65	1.48686	0.743432	+	0.668812i	$0.233197\pi$
$140$	−1.92822	−0.00116403
$141$	− 188.755i	− 0.112738i
$142$	1615.19	0.954534
$143$	0	0
$144$	−572.934	−0.331559
$145$	610.753i	0.349795i
$146$	−75.3624	−0.0427194
$147$	494.799	0.277621
$148$	− 8.77933i	− 0.00487606i
$149$	− 1517.58i	− 0.834397i	−0.908815	−	0.417198i	$-0.863012\pi$
$149$	− 1517.58i	− 0.834397i	0.908815	−	0.417198i	$-0.136988\pi$
$150$	− 959.391i	− 0.522226i
$151$	− 583.642i	− 0.314544i	−0.987555	−	0.157272i	$-0.949730\pi$
$151$	− 583.642i	− 0.314544i	0.987555	−	0.157272i	$-0.0502699\pi$
$152$	281.349	0.150134
$153$	−626.873	−0.331240
$154$	1334.55i	0.698321i
$155$	872.965	0.452376
$156$	0	0
$157$	26.0932	0.0132641	0.00663206	−	0.999978i	$-0.497889\pi$
$157$	26.0932	0.0132641	0.00663206	+	0.999978i	$0.497889\pi$
$158$	586.184i	0.295154i
$159$	−441.093	−0.220006
$160$	−6.53922	−0.00323107
$161$	− 1684.72i	− 0.824686i
$162$	− 228.495i	− 0.110817i
$163$	− 17.0905i	− 0.00821246i	−0.999992	−	0.00410623i	$-0.998693\pi$
$163$	− 17.0905i	− 0.00821246i	0.999992	−	0.00410623i	$-0.00130706\pi$
$164$	− 4.98631i	− 0.00237418i
$165$	−362.778	−0.171165
$166$	−2909.43	−1.36034
$167$	− 3919.48i	− 1.81616i	−0.418799	−	0.908079i	$-0.637549\pi$
$167$	− 3919.48i	− 1.81616i	0.418799	−	0.908079i	$-0.362451\pi$
$168$	−908.215	−0.417085
$169$	0	0
$170$	670.188	0.302359
$171$	111.612i	0.0499134i
$172$	−23.4368	−0.0103898
$173$	1976.35	0.868551	0.434276	−	0.900780i	$-0.357004\pi$
$173$	1976.35	0.868551	0.434276	+	0.900780i	$0.357004\pi$
$174$	1515.34i	0.660218i
$175$	− 1512.77i	− 0.653457i
$176$	2256.91i	0.966596i
$177$	− 822.260i	− 0.349180i
$178$	−3468.77	−1.46065
$179$	2784.34	1.16263	0.581316	−	0.813678i	$-0.302538\pi$
$179$	2784.34	1.16263	0.581316	+	0.813678i	$0.302538\pi$
$180$	− 1.30049i	0 0.000538517i
$181$	1886.53	0.774723	0.387361	−	0.921928i	$-0.373387\pi$
$181$	1886.53	0.774723	0.387361	+	0.921928i	$0.373387\pi$
$182$	0	0
$183$	1809.12	0.730786
$184$	− 2864.25i	− 1.14758i
$185$	−706.855	−0.280914
$186$	2165.92	0.853834
$187$	2469.39i	0.965666i
$188$	− 2.66549i	− 0.00103405i
$189$	− 360.293i	− 0.138664i
$190$	− 119.324i	− 0.0455615i
$191$	2447.94	0.927364	0.463682	−	0.886002i	$-0.346528\pi$
$191$	2447.94	0.927364	0.463682	+	0.886002i	$0.346528\pi$
$192$	−1544.05	−0.580375
$193$	1925.66i	0.718196i	0.933300	+	0.359098i	$0.116916\pi$
$193$	1925.66i	0.718196i	−0.933300	+	0.359098i	$0.883084\pi$
$194$	5063.89	1.87405
$195$	0	0
$196$	6.98724	0.00254637
$197$	− 1819.95i	− 0.658203i	−0.944294	−	0.329102i	$-0.893254\pi$
$197$	− 1819.95i	− 0.658203i	0.944294	−	0.329102i	$-0.106746\pi$
$198$	−900.092	−0.323064
$199$	1273.97	0.453816	0.226908	−	0.973916i	$-0.427138\pi$
$199$	1273.97	0.453816	0.226908	+	0.973916i	$0.427138\pi$
$200$	− 2571.92i	− 0.909312i
$201$	− 2223.06i	− 0.780111i
$202$	− 2171.26i	− 0.756284i
$203$	2389.40i	0.826125i
$204$	−8.85231	−0.00303817
$205$	−401.465	−0.136778
$206$	4354.56i	1.47280i
$207$	1136.26	0.381525
$208$	0	0
$209$	439.664	0.145513
$210$	385.188i	0.126574i
$211$	−1319.44	−0.430492	−0.215246	−	0.976560i	$-0.569055\pi$
$211$	−1319.44	−0.430492	−0.215246	+	0.976560i	$0.569055\pi$
$212$	−6.22884	−0.00201792
$213$	1717.72i	0.552565i
$214$	− 5719.83i	− 1.82710i
$215$	1886.98i	0.598564i
$216$	− 612.547i	− 0.192956i
$217$	3415.24	1.06839
$218$	4640.35	1.44167
$219$	− 80.1464i	− 0.0247296i
$220$	−5.12292	−0.00156994
$221$	0	0
$222$	−1753.79	−0.530209
$223$	1203.45i	0.361384i	0.983540	+	0.180692i	$0.0578338\pi$
$223$	1203.45i	0.361384i	−0.983540	+	0.180692i	$0.942166\pi$
$224$	−25.5829	−0.00763095
$225$	1020.29	0.302309
$226$	1846.34i	0.543436i
$227$	− 4361.23i	− 1.27518i	−0.770377	−	0.637589i	$-0.779932\pi$
$227$	− 4361.23i	− 1.27518i	0.770377	−	0.637589i	$-0.220068\pi$
$228$	1.57612i	0 0.000457811i
$229$	− 5384.29i	− 1.55373i	−0.629669	−	0.776864i	$-0.716809\pi$
$229$	− 5384.29i	− 1.55373i	0.629669	−	0.776864i	$-0.283191\pi$
$230$	−1214.77	−0.348260
$231$	−1419.27	−0.404247
$232$	4062.32i	1.14959i
$233$	4913.60	1.38155	0.690774	−	0.723070i	$-0.257270\pi$
$233$	4913.60	1.38155	0.690774	+	0.723070i	$0.257270\pi$
$234$	0	0
$235$	−214.608	−0.0595722
$236$	− 11.6114i	− 0.00320272i
$237$	−623.394	−0.170860
$238$	2621.93	0.714094
$239$	963.718i	0.260827i	0.991460	+	0.130414i	$0.0416306\pi$
$239$	963.718i	0.260827i	−0.991460	+	0.130414i	$0.958369\pi$
$240$	651.404i	0.175200i
$241$	− 1544.48i	− 0.412817i	−0.978466	−	0.206409i	$-0.933822\pi$
$241$	− 1544.48i	− 0.412817i	0.978466	−	0.206409i	$-0.0661776\pi$
$242$	− 209.001i	− 0.0555170i
$243$	243.000	0.0641500
$244$	25.5472	0.00670284
$245$	− 562.568i	− 0.146698i
$246$	−996.079	−0.258161
$247$	0	0
$248$	5806.38	1.48671
$249$	− 3094.12i	− 0.787478i
$250$	−2293.52	−0.580221
$251$	−3768.17	−0.947588	−0.473794	−	0.880636i	$-0.657116\pi$
$251$	−3768.17	−0.947588	−0.473794	+	0.880636i	$0.657116\pi$
$252$	− 5.08783i	− 0.00127184i
$253$	− 4475.98i	− 1.11226i
$254$	3739.25i	0.923705i
$255$	712.731i	0.175031i
$256$	−65.0695	−0.0158861
$257$	−1282.68	−0.311327	−0.155664	−	0.987810i	$-0.549752\pi$
$257$	−1282.68	−0.311327	−0.155664	+	0.987810i	$0.549752\pi$
$258$	4681.81i	1.12976i
$259$	−2765.38	−0.663445
$260$	0	0
$261$	−1611.54	−0.382190
$262$	− 2623.78i	− 0.618694i
$263$	5029.74	1.17927	0.589633	−	0.807671i	$-0.299272\pi$
$263$	5029.74	1.17927	0.589633	+	0.807671i	$0.299272\pi$
$264$	−2412.95	−0.562527
$265$	501.506i	0.116254i
$266$	− 466.824i	− 0.107604i
$267$	− 3688.97i	− 0.845548i
$268$	− 31.3926i	− 0.00715526i
$269$	5628.46	1.27574	0.637868	−	0.770146i	$-0.279816\pi$
$269$	5628.46	1.27574	0.637868	+	0.770146i	$0.279816\pi$
$270$	−259.790	−0.0585568
$271$	− 3368.39i	− 0.755037i	−0.926002	−	0.377518i	$-0.876778\pi$
$271$	− 3368.39i	− 0.755037i	0.926002	−	0.377518i	$-0.123222\pi$
$272$	4434.03	0.988429
$273$	0	0
$274$	7760.45	1.71104
$275$	− 4019.15i	− 0.881324i
$276$	16.0456	0.00349938
$277$	6507.75	1.41160	0.705799	−	0.708412i	$-0.250588\pi$
$277$	6507.75	1.41160	0.705799	+	0.708412i	$0.250588\pi$
$278$	− 6873.62i	− 1.48292i
$279$	2303.41i	0.494272i
$280$	1032.61i	0.220393i
$281$	3625.51i	0.769680i	0.922983	+	0.384840i	$0.125743\pi$
$281$	3625.51i	0.769680i	−0.922983	+	0.384840i	$0.874257\pi$
$282$	−532.465	−0.112439
$283$	4635.41	0.973662	0.486831	−	0.873496i	$-0.338153\pi$
$283$	4635.41	0.973662	0.486831	+	0.873496i	$0.338153\pi$
$284$	24.2566i	0.00506819i
$285$	126.899	0.0263749
$286$	0	0
$287$	−1570.62	−0.323035
$288$	− 17.2545i	− 0.00353031i
$289$	−61.5178	−0.0125214
$290$	1722.89	0.348867
$291$	5385.35i	1.08486i
$292$	− 1.13178i	0 0.000226823i
$293$	2907.07i	0.579634i	0.957082	+	0.289817i	$0.0935945\pi$
$293$	2907.07i	0.579634i	−0.957082	+	0.289817i	$0.906406\pi$
$294$	− 1395.79i	− 0.276885i
$295$	−934.879	−0.184511
$296$	−4701.53	−0.923212
$297$	− 957.229i	− 0.187017i
$298$	−4280.99	−0.832185
$299$	0	0
$300$	14.4079	0.00277281
$301$	7382.32i	1.41365i
$302$	−1646.41	−0.313710
$303$	2309.09	0.437802
$304$	− 789.461i	− 0.148943i
$305$	− 2056.90i	− 0.386156i
$306$	1768.36i	0.330362i
$307$	− 933.950i	− 0.173627i	−0.996225	−	0.0868133i	$-0.972332\pi$
$307$	− 933.950i	− 0.173627i	0.996225	−	0.0868133i	$-0.0276683\pi$
$308$	−20.0421	−0.00370780
$309$	−4630.98	−0.852581
$310$	− 2462.57i	− 0.451176i
$311$	3633.55	0.662508	0.331254	−	0.943542i	$-0.392528\pi$
$311$	3633.55	0.662508	0.331254	+	0.943542i	$0.392528\pi$
$312$	0	0
$313$	−7507.26	−1.35570	−0.677852	−	0.735198i	$-0.737089\pi$
$313$	−7507.26	−1.35570	−0.677852	+	0.735198i	$0.737089\pi$
$314$	− 73.6071i	− 0.0132289i
$315$	−409.639	−0.0732716
$316$	−8.80319	−0.00156715
$317$	1214.60i	0.215202i	0.994194	+	0.107601i	$0.0343168\pi$
$317$	1214.60i	0.215202i	−0.994194	+	0.107601i	$0.965683\pi$
$318$	1244.29i	0.219423i
$319$	6348.19i	1.11420i
$320$	1755.52i	0.306677i
$321$	6082.92	1.05768
$322$	−4752.47	−0.822500
$323$	− 863.786i	− 0.148800i
$324$	3.43149	0.000588391 0
$325$	0	0
$326$	−48.2111	−0.00819069
$327$	4934.92i	0.834561i
$328$	−2670.28	−0.449517
$329$	−839.595	−0.140694
$330$	1023.37i	0.170711i
$331$	2443.20i	0.405712i	0.979209	+	0.202856i	$0.0650223\pi$
$331$	2443.20i	0.405712i	−0.979209	+	0.202856i	$0.934978\pi$
$332$	− 43.6933i	− 0.00722283i
$333$	− 1865.12i	− 0.306930i
$334$	−11056.6	−1.81134
$335$	−2527.53	−0.412220
$336$	2548.44i	0.413777i
$337$	−1890.96	−0.305660	−0.152830	−	0.988253i	$-0.548839\pi$
$337$	−1890.96	−0.305660	−0.152830	+	0.988253i	$0.548839\pi$
$338$	0	0
$339$	−1963.54	−0.314587
$340$	10.0647i	0.00160540i
$341$	9073.64	1.44095
$342$	314.850	0.0497811
$343$	− 6777.95i	− 1.06698i
$344$	12551.0i	1.96716i
$345$	− 1291.89i	− 0.201602i
$346$	− 5575.15i	− 0.866248i
$347$	−2791.57	−0.431872	−0.215936	−	0.976408i	$-0.569280\pi$
$347$	−2791.57	−0.431872	−0.215936	+	0.976408i	$0.569280\pi$
$348$	−22.7571	−0.00350549
$349$	6917.33i	1.06096i	0.847696	+	0.530482i	$0.177989\pi$
$349$	6917.33i	1.06096i	−0.847696	+	0.530482i	$0.822011\pi$
$350$	−4267.43	−0.651724
$351$	0	0
$352$	−67.9690	−0.0102919
$353$	7638.37i	1.15170i	0.817556	+	0.575849i	$0.195328\pi$
$353$	7638.37i	1.15170i	−0.817556	+	0.575849i	$0.804672\pi$
$354$	−2319.54	−0.348254
$355$	1952.99	0.291982
$356$	− 52.0933i	− 0.00775545i
$357$	2788.37i	0.413379i
$358$	− 7854.42i	− 1.15955i
$359$	5490.97i	0.807249i	0.914925	+	0.403625i	$0.132250\pi$
$359$	5490.97i	0.807249i	−0.914925	+	0.403625i	$0.867750\pi$
$360$	−696.443	−0.101960
$361$	6705.21	0.977578
$362$	− 5321.77i	− 0.772669i
$363$	222.269	0.0321380
$364$	0	0
$365$	−91.1234	−0.0130674
$366$	− 5103.39i	− 0.728848i
$367$	6737.43	0.958287	0.479143	−	0.877737i	$-0.340947\pi$
$367$	6737.43	0.958287	0.479143	+	0.877737i	$0.340947\pi$
$368$	−8037.07	−1.13848
$369$	− 1059.31i	− 0.149446i
$370$	1993.99i	0.280169i
$371$	1962.01i	0.274561i
$372$	32.5274i	0.00453351i
$373$	−3930.42	−0.545602	−0.272801	−	0.962070i	$-0.587950\pi$
$373$	−3930.42	−0.545602	−0.272801	+	0.962070i	$0.587950\pi$
$374$	6965.97	0.963106
$375$	− 2439.12i	− 0.335881i
$376$	−1427.43	−0.195782
$377$	0	0
$378$	−1016.36	−0.138296
$379$	11897.7i	1.61252i	0.591563	+	0.806258i	$0.298511\pi$
$379$	11897.7i	1.61252i	−0.591563	+	0.806258i	$0.701489\pi$
$380$	1.79199	0.000241913 0
$381$	−3976.61	−0.534719
$382$	− 6905.46i	− 0.924906i
$383$	4748.31i	0.633492i	0.948510	+	0.316746i	$0.102590\pi$
$383$	4748.31i	0.633492i	−0.948510	+	0.316746i	$0.897410\pi$
$384$	4309.64i	0.572722i
$385$	1613.66i	0.213609i
$386$	5432.14	0.716292
$387$	−4979.02	−0.653999
$388$	76.0485i	0.00995046i
$389$	−11299.5	−1.47277	−0.736386	−	0.676561i	$-0.763469\pi$
$389$	−11299.5	−1.47277	−0.736386	+	0.676561i	$0.763469\pi$
$390$	0	0
$391$	−8793.73	−1.13739
$392$	− 3741.82i	− 0.482119i
$393$	2790.34	0.358153
$394$	−5133.95	−0.656458
$395$	708.776i	0.0902845i
$396$	− 13.5174i	− 0.00171534i
$397$	− 13853.0i	− 1.75129i	−0.482953	−	0.875646i	$-0.660436\pi$
$397$	− 13853.0i	− 1.75129i	0.482953	−	0.875646i	$-0.339564\pi$
$398$	− 3593.78i	− 0.452613i
$399$	496.457	0.0622906
$400$	−7216.79	−0.902099
$401$	− 5434.64i	− 0.676790i	−0.941004	−	0.338395i	$-0.890116\pi$
$401$	− 5434.64i	− 0.676790i	0.941004	−	0.338395i	$-0.109884\pi$
$402$	−6271.08	−0.778042
$403$	0	0
$404$	32.6076	0.00401556
$405$	− 276.282i	− 0.0338977i
$406$	6740.34	0.823935
$407$	−7347.09	−0.894795
$408$	4740.61i	0.575233i
$409$	− 3472.43i	− 0.419806i	−0.977722	−	0.209903i	$-0.932685\pi$
$409$	− 3472.43i	− 0.419806i	0.977722	−	0.209903i	$-0.0673148\pi$
$410$	1132.50i	0.136416i
$411$	8253.09i	0.990498i
$412$	−65.3959	−0.00781996
$413$	−3657.46	−0.435767
$414$	− 3205.31i	− 0.380513i
$415$	−3517.90	−0.416113
$416$	0	0
$417$	7309.95	0.858441
$418$	− 1240.26i	− 0.145127i
$419$	−4261.65	−0.496886	−0.248443	−	0.968647i	$-0.579919\pi$
$419$	−4261.65	−0.496886	−0.248443	+	0.968647i	$0.579919\pi$
$420$	−5.78467	−0.000672055 0
$421$	6235.06i	0.721801i	0.932604	+	0.360900i	$0.117531\pi$
$421$	6235.06i	0.721801i	−0.932604	+	0.360900i	$0.882469\pi$
$422$	3722.04i	0.429351i
$423$	− 566.266i	− 0.0650894i
$424$	3335.68i	0.382064i
$425$	−7896.22	−0.901231
$426$	4845.57	0.551100
$427$	− 8047.06i	− 0.912001i
$428$	85.8992	0.00970115
$429$	0	0
$430$	5323.05	0.596977
$431$	2520.58i	0.281698i	0.990031	+	0.140849i	$0.0449833\pi$
$431$	2520.58i	0.281698i	−0.990031	+	0.140849i	$0.955017\pi$
$432$	−1718.80	−0.191426
$433$	1283.89	0.142494	0.0712468	−	0.997459i	$-0.477302\pi$
$433$	1283.89	0.142494	0.0712468	+	0.997459i	$0.477302\pi$
$434$	− 9634.15i	− 1.06556i
$435$	1832.26i	0.201954i
$436$	69.6878i	0.00765468i
$437$	1565.69i	0.171389i
$438$	−226.087	−0.0246641
$439$	16942.0	1.84190	0.920951	−	0.389677i	$-0.127414\pi$
$439$	16942.0	1.84190	0.920951	+	0.389677i	$0.127414\pi$
$440$	2743.44i	0.297246i
$441$	1484.40	0.160285
$442$	0	0
$443$	−2163.00	−0.231980	−0.115990	−	0.993250i	$-0.537004\pi$
$443$	−2163.00	−0.231980	−0.115990	+	0.993250i	$0.537004\pi$
$444$	− 26.3380i	− 0.00281519i
$445$	−4194.22	−0.446798
$446$	3394.83	0.360426
$447$	− 4552.75i	− 0.481739i
$448$	6868.01i	0.724293i
$449$	1673.35i	0.175880i	0.996126	+	0.0879400i	$0.0280284\pi$
$449$	1673.35i	0.175880i	−0.996126	+	0.0879400i	$0.971972\pi$
$450$	− 2878.17i	− 0.301507i
$451$	−4172.85	−0.435680
$452$	−27.7279	−0.00288542
$453$	− 1750.92i	− 0.181602i
$454$	−12302.7	−1.27180
$455$	0	0
$456$	844.046	0.0866800
$457$	− 4652.38i	− 0.476212i	−0.971239	−	0.238106i	$-0.923473\pi$
$457$	− 4652.38i	− 0.476212i	0.971239	−	0.238106i	$-0.0765266\pi$
$458$	−15188.7	−1.54961
$459$	−1880.62	−0.191241
$460$	− 18.2432i	− 0.00184912i
$461$	3460.07i	0.349570i	0.984607	+	0.174785i	$0.0559230\pi$
$461$	3460.07i	0.349570i	−0.984607	+	0.174785i	$0.944077\pi$
$462$	4003.66i	0.403176i
$463$	2105.64i	0.211355i	0.994400	+	0.105677i	$0.0337011\pi$
$463$	2105.64i	0.211355i	−0.994400	+	0.105677i	$0.966299\pi$
$464$	11398.8	1.14047
$465$	2618.89	0.261179
$466$	− 13860.9i	− 1.37789i
$467$	1415.88	0.140298	0.0701489	−	0.997537i	$-0.477653\pi$
$467$	1415.88	0.140298	0.0701489	+	0.997537i	$0.477653\pi$
$468$	0	0
$469$	−9888.28	−0.973557
$470$	605.393i	0.0594143i
$471$	78.2797	0.00765804
$472$	−6218.19	−0.606388
$473$	19613.4i	1.90661i
$474$	1758.55i	0.170407i
$475$	1405.89i	0.135803i
$476$	39.3756i	0.00379155i
$477$	−1323.28	−0.127020
$478$	2718.58	0.260136
$479$	985.594i	0.0940145i	0.998895	+	0.0470073i	$0.0149684\pi$
$479$	985.594i	0.0940145i	−0.998895	+	0.0470073i	$0.985032\pi$
$480$	−19.6177	−0.00186546
$481$	0	0
$482$	−4356.88	−0.411723
$483$	− 5054.16i	− 0.476133i
$484$	3.13874	0.000294773 0
$485$	6122.93	0.573254
$486$	− 685.486i	− 0.0639800i
$487$	14724.7i	1.37011i	0.728494	+	0.685053i	$0.240221\pi$
$487$	14724.7i	1.37011i	−0.728494	+	0.685053i	$0.759779\pi$
$488$	− 13681.1i	− 1.26909i
$489$	− 51.2715i	− 0.00474147i
$490$	−1586.96	−0.146310
$491$	16301.2	1.49830	0.749149	−	0.662401i	$-0.230463\pi$
$491$	16301.2	1.49830	0.749149	+	0.662401i	$0.230463\pi$
$492$	− 14.9589i	− 0.00137073i
$493$	12472.0	1.13937
$494$	0	0
$495$	−1088.33	−0.0988221
$496$	− 16292.6i	− 1.47492i
$497$	7640.53	0.689587
$498$	−8728.30	−0.785391
$499$	3230.42i	0.289806i	0.989446	+	0.144903i	$0.0462870\pi$
$499$	3230.42i	0.289806i	−0.989446	+	0.144903i	$0.953713\pi$
$500$	− 34.4437i	− 0.00308074i
$501$	− 11758.4i	− 1.04856i
$502$	10629.7i	0.945076i
$503$	−3577.57	−0.317129	−0.158565	−	0.987349i	$-0.550687\pi$
$503$	−3577.57	−0.317129	−0.158565	+	0.987349i	$0.550687\pi$
$504$	−2724.65	−0.240804
$505$	− 2625.35i	− 0.231340i
$506$	−12626.4	−1.10931
$507$	0	0
$508$	−56.1552	−0.00490450
$509$	− 13864.7i	− 1.20735i	−0.797231	−	0.603674i	$-0.793703\pi$
$509$	− 13864.7i	− 1.20735i	0.797231	−	0.603674i	$-0.206297\pi$
$510$	2010.56	0.174567
$511$	−356.496	−0.0308619
$512$	11675.9i	1.00783i
$513$	334.836i	0.0288175i
$514$	3618.34i	0.310502i
$515$	5265.25i	0.450514i
$516$	−70.3105	−0.00599854
$517$	−2230.64	−0.189756
$518$	7800.94i	0.661686i
$519$	5929.06	0.501458
$520$	0	0
$521$	−9554.66	−0.803449	−0.401725	−	0.915760i	$-0.631589\pi$
$521$	−9554.66	−0.803449	−0.401725	+	0.915760i	$0.631589\pi$
$522$	4546.03i	0.381177i
$523$	−19809.1	−1.65620	−0.828098	−	0.560583i	$-0.810577\pi$
$523$	−19809.1	−1.65620	−0.828098	+	0.560583i	$0.810577\pi$
$524$	39.4035	0.00328502
$525$	− 4538.32i	− 0.377274i
$526$	− 14188.5i	− 1.17614i
$527$	− 17826.5i	− 1.47350i
$528$	6770.73i	0.558064i
$529$	3772.38	0.310050
$530$	1414.71	0.115946
$531$	− 2466.78i	− 0.201599i
$532$	7.01066	0.000571336 0
$533$	0	0
$534$	−10406.3	−0.843306
$535$	− 6916.05i	− 0.558891i
$536$	−16811.4	−1.35475
$537$	8353.02	0.671246
$538$	− 15877.5i	− 1.27235i
$539$	− 5847.36i	− 0.467279i
$540$	− 3.90148i	0 0.000310913i
$541$	1011.14i	0.0803551i	0.999193	+	0.0401775i	$0.0127923\pi$
$541$	1011.14i	0.0803551i	−0.999193	+	0.0401775i	$0.987208\pi$
$542$	−9501.98	−0.753035
$543$	5659.59	0.447286
$544$	133.535i	0.0105244i
$545$	5610.81	0.440992
$546$	0	0
$547$	15745.2	1.23074	0.615372	−	0.788237i	$-0.289006\pi$
$547$	15745.2	1.23074	0.615372	+	0.788237i	$0.289006\pi$
$548$	116.545i	0.00908495i
$549$	5427.35	0.421919
$550$	−11337.7	−0.878987
$551$	− 2220.58i	− 0.171688i
$552$	− 8592.76i	− 0.662558i
$553$	2772.89i	0.213229i
$554$	− 18357.9i	− 1.40785i
$555$	−2120.57	−0.162186
$556$	103.227	0.00787371
$557$	− 8510.94i	− 0.647433i	−0.946154	−	0.323716i	$-0.895068\pi$
$557$	− 8510.94i	− 0.647433i	0.946154	−	0.323716i	$-0.104932\pi$
$558$	6497.76	0.492961
$559$	0	0
$560$	2897.48	0.218645
$561$	7408.16i	0.557528i
$562$	10227.3	0.767639
$563$	−16764.1	−1.25493	−0.627463	−	0.778646i	$-0.715907\pi$
$563$	−16764.1	−1.25493	−0.627463	+	0.778646i	$0.715907\pi$
$564$	− 7.99646i	0 0.000597006i
$565$	2232.47i	0.166232i
$566$	− 13076.1i	− 0.971080i
$567$	− 1080.88i	− 0.0800575i
$568$	12990.0	0.959589
$569$	−20755.4	−1.52919	−0.764597	−	0.644508i	$-0.777062\pi$
$569$	−20755.4	−1.52919	−0.764597	+	0.644508i	$0.777062\pi$
$570$	− 357.972i	− 0.0263049i
$571$	23408.6	1.71562	0.857810	−	0.513968i	$-0.171825\pi$
$571$	23408.6	1.71562	0.857810	+	0.513968i	$0.171825\pi$
$572$	0	0
$573$	7343.81	0.535414
$574$	4430.62i	0.322179i
$575$	14312.6	1.03805
$576$	−4632.14	−0.335080
$577$	10387.2i	0.749436i	0.927139	+	0.374718i	$0.122261\pi$
$577$	10387.2i	0.749436i	−0.927139	+	0.374718i	$0.877739\pi$
$578$	173.537i	0.0124882i
$579$	5776.97i	0.414650i
$580$	25.8740i	0.00185234i
$581$	−13762.8	−0.982752
$582$	15191.7	1.08198
$583$	5212.68i	0.370304i
$584$	−606.092	−0.0429457
$585$	0	0
$586$	8200.63	0.578097
$587$	− 2898.42i	− 0.203800i	−0.994795	−	0.101900i	$-0.967508\pi$
$587$	− 2898.42i	− 0.203800i	0.994795	−	0.101900i	$-0.0324921\pi$
$588$	20.9617	0.00147015
$589$	−3173.94	−0.222037
$590$	2637.23i	0.184022i
$591$	− 5459.85i	− 0.380014i
$592$	13192.4i	0.915888i
$593$	− 8805.33i	− 0.609766i	−0.952390	−	0.304883i	$-0.901383\pi$
$593$	− 8805.33i	− 0.609766i	0.952390	−	0.304883i	$-0.0986174\pi$
$594$	−2700.27	−0.186521
$595$	3170.27	0.218434
$596$	− 64.2910i	− 0.00441856i
$597$	3821.91	0.262011
$598$	0	0
$599$	−20236.8	−1.38039	−0.690194	−	0.723624i	$-0.742475\pi$
$599$	−20236.8	−1.38039	−0.690194	+	0.723624i	$0.742475\pi$
$600$	− 7715.77i	− 0.524992i
$601$	−9884.92	−0.670906	−0.335453	−	0.942057i	$-0.608889\pi$
$601$	−9884.92	−0.670906	−0.335453	+	0.942057i	$0.608889\pi$
$602$	20825.0	1.40991
$603$	− 6669.17i	− 0.450397i
$604$	− 24.7255i	− 0.00166567i
$605$	− 252.711i	− 0.0169821i
$606$	− 6513.78i	− 0.436641i
$607$	22919.3	1.53256	0.766281	−	0.642505i	$-0.222105\pi$
$607$	22919.3	1.53256	0.766281	+	0.642505i	$0.222105\pi$
$608$	23.7754	0.00158589
$609$	7168.21i	0.476963i
$610$	−5802.36	−0.385132
$611$	0	0
$612$	−26.5569	−0.00175409
$613$	916.052i	0.0603572i	0.999545	+	0.0301786i	$0.00960761\pi$
$613$	916.052i	0.0603572i	−0.999545	+	0.0301786i	$0.990392\pi$
$614$	−2634.61	−0.173166
$615$	−1204.40	−0.0789690
$616$	10733.0i	0.702019i
$617$	− 30179.4i	− 1.96917i	−0.174915	−	0.984584i	$-0.555965\pi$
$617$	− 30179.4i	− 1.96917i	0.174915	−	0.984584i	$-0.444035\pi$
$618$	13063.7i	0.850320i
$619$	− 5571.12i	− 0.361748i	−0.983506	−	0.180874i	$-0.942107\pi$
$619$	− 5571.12i	− 0.361748i	0.983506	−	0.180874i	$-0.0578927\pi$
$620$	36.9824	0.00239556
$621$	3408.78	0.220273
$622$	− 10250.0i	− 0.660751i
$623$	−16408.8	−1.05522
$624$	0	0
$625$	11397.5	0.729443
$626$	21177.5i	1.35211i
$627$	1318.99	0.0840120
$628$	1.10542	7.02403e−5 0
$629$	14434.5i	0.915007i
$630$	1155.56i	0.0730773i
$631$	11360.1i	0.716703i	0.933587	+	0.358352i	$0.116661\pi$
$631$	11360.1i	0.716703i	−0.933587	+	0.358352i	$0.883339\pi$
$632$	4714.30i	0.296717i
$633$	−3958.32	−0.248545
$634$	3426.31	0.214631
$635$	4521.26i	0.282552i
$636$	−18.6865	−0.00116504
$637$	0	0
$638$	17907.8	1.11125
$639$	5153.17i	0.319024i
$640$	4899.89	0.302633
$641$	24709.8	1.52259	0.761293	−	0.648408i	$-0.224565\pi$
$641$	24709.8	1.52259	0.761293	+	0.648408i	$0.224565\pi$
$642$	− 17159.5i	− 1.05488i
$643$	15226.6i	0.933869i	0.884292	+	0.466934i	$0.154642\pi$
$643$	15226.6i	0.933869i	−0.884292	+	0.466934i	$0.845358\pi$
$644$	− 71.3717i	− 0.00436714i
$645$	5660.95i	0.345581i
$646$	−2436.68	−0.148405
$647$	−16706.6	−1.01515	−0.507576	−	0.861607i	$-0.669458\pi$
$647$	−16706.6	−1.01515	−0.507576	+	0.861607i	$0.669458\pi$
$648$	− 1837.64i	− 0.111403i
$649$	−9717.18	−0.587724
$650$	0	0
$651$	10245.7	0.616838
$652$	− 0.724024i	0 4.34892e-5i
$653$	−28205.8	−1.69032	−0.845158	−	0.534516i	$-0.820494\pi$
$653$	−28205.8	−1.69032	−0.845158	+	0.534516i	$0.820494\pi$
$654$	13921.0	0.832349
$655$	− 3172.51i	− 0.189252i
$656$	7492.77i	0.445951i
$657$	− 240.439i	− 0.0142777i
$658$	2368.44i	0.140321i
$659$	−11424.2	−0.675299	−0.337649	−	0.941272i	$-0.609632\pi$
$659$	−11424.2	−0.675299	−0.337649	+	0.941272i	$0.609632\pi$
$660$	−15.3688	−0.000906407 0
$661$	26518.2i	1.56042i	0.625516	+	0.780211i	$0.284888\pi$
$661$	26518.2i	1.56042i	−0.625516	+	0.780211i	$0.715112\pi$
$662$	6892.10	0.404636
$663$	0	0
$664$	−23398.7	−1.36754
$665$	− 564.453i	− 0.0329151i
$666$	−5261.36	−0.306116
$667$	−22606.5	−1.31234
$668$	− 166.045i	− 0.00961749i
$669$	3610.34i	0.208645i
$670$	7129.98i	0.411127i
$671$	− 21379.5i	− 1.23002i
$672$	−76.7488	−0.00440573
$673$	−22816.8	−1.30687	−0.653434	−	0.756984i	$-0.726672\pi$
$673$	−22816.8	−1.30687	−0.653434	+	0.756984i	$0.726672\pi$
$674$	5334.27i	0.304849i
$675$	3060.88	0.174538
$676$	0	0
$677$	17737.6	1.00696	0.503478	−	0.864008i	$-0.332053\pi$
$677$	17737.6	1.00696	0.503478	+	0.864008i	$0.332053\pi$
$678$	5539.01i	0.313753i
$679$	23954.3	1.35388
$680$	5389.89	0.303960
$681$	− 13083.7i	− 0.736224i
$682$	− 25596.1i	− 1.43713i
$683$	18878.4i	1.05763i	0.848737	+	0.528816i	$0.177364\pi$
$683$	18878.4i	1.05763i	−0.848737	+	0.528816i	$0.822636\pi$
$684$	4.72835i	0 0.000264317i
$685$	9383.45	0.523391
$686$	−19120.1	−1.06415
$687$	− 16152.9i	− 0.897045i
$688$	35217.8	1.95155
$689$	0	0
$690$	−3644.32	−0.201068
$691$	11692.8i	0.643726i	0.946786	+	0.321863i	$0.104309\pi$
$691$	11692.8i	0.643726i	−0.946786	+	0.321863i	$0.895691\pi$
$692$	83.7265	0.00459943
$693$	−4257.81	−0.233392
$694$	7874.82i	0.430726i
$695$	− 8311.14i	− 0.453611i
$696$	12186.9i	0.663714i
$697$	8198.19i	0.445522i
$698$	19513.3	1.05815
$699$	14740.8	0.797638
$700$	− 64.0873i	− 0.00346039i
$701$	−20658.1	−1.11304	−0.556522	−	0.830833i	$-0.687865\pi$
$701$	−20658.1	−1.11304	−0.556522	+	0.830833i	$0.687865\pi$
$702$	0	0
$703$	2569.99	0.137879
$704$	18247.0i	0.976861i
$705$	−643.823	−0.0343940
$706$	21547.3	1.14864
$707$	− 10271.0i	− 0.546365i
$708$	− 34.8343i	− 0.00184909i
$709$	20695.8i	1.09626i	0.836394	+	0.548128i	$0.184659\pi$
$709$	20695.8i	1.09626i	−0.836394	+	0.548128i	$0.815341\pi$
$710$	− 5509.23i	− 0.291208i
$711$	−1870.18	−0.0986461
$712$	−27897.1	−1.46838
$713$	32312.1i	1.69719i
$714$	7865.79	0.412283
$715$	0	0
$716$	117.956	0.00615674
$717$	2891.16i	0.150589i
$718$	15489.6	0.805109
$719$	26299.2	1.36411	0.682053	−	0.731302i	$-0.261087\pi$
$719$	26299.2	1.36411	0.682053	+	0.731302i	$0.261087\pi$
$720$	1954.21i	0.101152i
$721$	20598.9i	1.06400i
$722$	− 18914.9i	− 0.974986i
$723$	− 4633.45i	− 0.238340i
$724$	79.9213	0.00410256
$725$	−20299.3	−1.03986
$726$	− 627.004i	− 0.0320528i
$727$	16915.6	0.862952	0.431476	−	0.902125i	$-0.357993\pi$
$727$	16915.6	0.862952	0.431476	+	0.902125i	$0.357993\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	257.053i	0.0130328i
$731$	38533.5	1.94967
$732$	76.6417	0.00386989
$733$	− 13203.1i	− 0.665302i	−0.943050	−	0.332651i	$-0.892057\pi$
$733$	− 13203.1i	− 0.665302i	0.943050	−	0.332651i	$-0.107943\pi$
$734$	− 19005.8i	− 0.955746i
$735$	− 1687.70i	− 0.0846964i
$736$	− 242.044i	− 0.0121221i
$737$	−26271.3	−1.31305
$738$	−2988.24	−0.149050
$739$	16813.9i	0.836954i	0.908227	+	0.418477i	$0.137436\pi$
$739$	16813.9i	0.836954i	−0.908227	+	0.418477i	$0.862564\pi$
$740$	−29.9453	−0.00148758
$741$	0	0
$742$	5534.68	0.273834
$743$	− 32682.1i	− 1.61371i	−0.590747	−	0.806857i	$-0.701167\pi$
$743$	− 32682.1i	− 1.61371i	0.590747	−	0.806857i	$-0.298833\pi$
$744$	17419.1	0.858355
$745$	−5176.30	−0.254557
$746$	11087.4i	0.544156i
$747$	− 9282.37i	− 0.454651i
$748$	104.613i	0.00511370i
$749$	− 27057.2i	− 1.31996i
$750$	−6880.57	−0.334991
$751$	−2157.76	−0.104844	−0.0524219	−	0.998625i	$-0.516694\pi$
$751$	−2157.76	−0.104844	−0.0524219	+	0.998625i	$0.516694\pi$
$752$	4005.35i	0.194229i
$753$	−11304.5	−0.547090
$754$	0	0
$755$	−1990.74	−0.0959606
$756$	− 15.2635i	0 0.000734296i
$757$	13769.6	0.661117	0.330558	−	0.943786i	$-0.392763\pi$
$757$	13769.6	0.661117	0.330558	+	0.943786i	$0.392763\pi$
$758$	33562.6	1.60824
$759$	− 13427.9i	− 0.642165i
$760$	− 959.648i	− 0.0458028i
$761$	− 16905.2i	− 0.805272i	−0.915360	−	0.402636i	$-0.868094\pi$
$761$	− 16905.2i	− 0.805272i	0.915360	−	0.402636i	$-0.131906\pi$
$762$	11217.7i	0.533302i
$763$	21950.8	1.04151
$764$	103.705	0.00491087
$765$	2138.19i	0.101054i
$766$	13394.7	0.631813
$767$	0	0
$768$	−195.208	−0.00917185
$769$	− 1153.35i	− 0.0540845i	−0.999634	−	0.0270422i	$-0.991391\pi$
$769$	− 1153.35i	− 0.0540845i	0.999634	−	0.0270422i	$-0.00860886\pi$
$770$	4552.01	0.213043
$771$	−3848.03	−0.179745
$772$	81.5787i	0.00380322i
$773$	20713.9i	0.963811i	0.876223	+	0.481905i	$0.160055\pi$
$773$	20713.9i	0.963811i	−0.876223	+	0.481905i	$0.839945\pi$
$774$	14045.4i	0.652265i
$775$	29014.3i	1.34480i
$776$	40725.7	1.88398
$777$	−8296.14	−0.383040
$778$	31875.1i	1.46887i
$779$	1459.65	0.0671341
$780$	0	0
$781$	20299.4	0.930053
$782$	24806.5i	1.13437i
$783$	−4834.61	−0.220658
$784$	−10499.5	−0.478294
$785$	− 89.0010i	− 0.00404660i
$786$	− 7871.35i	− 0.357203i
$787$	10870.2i	0.492353i	0.969225	+	0.246177i	$0.0791743\pi$
$787$	10870.2i	0.492353i	−0.969225	+	0.246177i	$0.920826\pi$
$788$	− 77.1006i	− 0.00348553i
$789$	15089.2	0.680850
$790$	1999.41	0.0900451
$791$	8733.95i	0.392596i
$792$	−7238.86	−0.324775
$793$	0	0
$794$	−39078.4	−1.74665
$795$	1504.52i	0.0671192i
$796$	53.9706	0.00240319
$797$	−12026.6	−0.534511	−0.267255	−	0.963626i	$-0.586117\pi$
$797$	−12026.6	−0.534511	−0.267255	+	0.963626i	$0.586117\pi$
$798$	− 1400.47i	− 0.0621255i
$799$	4382.43i	0.194042i
$800$	− 217.341i	− 0.00960519i
$801$	− 11066.9i	− 0.488177i
$802$	−15330.7	−0.674996
$803$	−947.142	−0.0416238
$804$	− 94.1778i	− 0.00413109i
$805$	−5746.39	−0.251594
$806$	0	0
$807$	16885.4	0.736547
$808$	− 17462.1i	− 0.760289i
$809$	−32384.0	−1.40737	−0.703685	−	0.710512i	$-0.748463\pi$
$809$	−32384.0	−1.40737	−0.703685	+	0.710512i	$0.748463\pi$
$810$	−779.371	−0.0338078
$811$	− 42506.8i	− 1.84046i	−0.391375	−	0.920231i	$-0.628000\pi$
$811$	− 42506.8i	− 1.84046i	0.391375	−	0.920231i	$-0.372000\pi$
$812$	101.225i	0.00437476i
$813$	− 10105.2i	− 0.435921i
$814$	20725.6i	0.892423i
$815$	−58.2937	−0.00250545
$816$	13302.1	0.570670
$817$	− 6860.72i	− 0.293790i
$818$	−9795.47	−0.418693
$819$	0	0
$820$	−17.0077	−0.000724312 0
$821$	20052.6i	0.852427i	0.904623	+	0.426213i	$0.140153\pi$
$821$	20052.6i	0.852427i	−0.904623	+	0.426213i	$0.859847\pi$
$822$	23281.4	0.987872
$823$	31625.5	1.33948	0.669742	−	0.742594i	$-0.266405\pi$
$823$	31625.5	1.33948	0.669742	+	0.742594i	$0.266405\pi$
$824$	35020.9i	1.48060i
$825$	− 12057.5i	− 0.508832i
$826$	10317.4i	0.434612i
$827$	5440.97i	0.228780i	0.993436	+	0.114390i	$0.0364913\pi$
$827$	5440.97i	0.228780i	−0.993436	+	0.114390i	$0.963509\pi$
$828$	48.1367	0.00202037
$829$	24845.8	1.04093	0.520465	−	0.853883i	$-0.325759\pi$
$829$	24845.8	1.04093	0.520465	+	0.853883i	$0.325759\pi$
$830$	9923.75i	0.415010i
$831$	19523.2	0.814986
$832$	0	0
$833$	−11488.0	−0.477834
$834$	− 20620.9i	− 0.856165i
$835$	−13368.9	−0.554071
$836$	18.6260	0.000770566 0
$837$	6910.24i	0.285368i
$838$	12021.8i	0.495568i
$839$	45887.8i	1.88823i	0.329621	+	0.944113i	$0.393079\pi$
$839$	45887.8i	1.88823i	−0.329621	+	0.944113i	$0.606921\pi$
$840$	3097.82i	0.127244i
$841$	7673.40	0.314625
$842$	17588.7	0.719887
$843$	10876.5i	0.444375i
$844$	−55.8969	−0.00227968
$845$	0	0
$846$	−1597.40	−0.0649168
$847$	− 988.665i	− 0.0401073i
$848$	9359.88	0.379033
$849$	13906.2	0.562144
$850$	22274.7i	0.898842i
$851$	− 26163.7i	− 1.05391i
$852$	72.7698i	0.00292612i
$853$	− 33655.3i	− 1.35092i	−0.737397	−	0.675460i	$-0.763945\pi$
$853$	− 33655.3i	− 1.35092i	0.737397	−	0.675460i	$-0.236055\pi$
$854$	−22700.2	−0.909583
$855$	380.696	0.0152275
$856$	− 46000.9i	− 1.83677i
$857$	−41365.6	−1.64880	−0.824399	−	0.566009i	$-0.808487\pi$
$857$	−41365.6	−1.64880	−0.824399	+	0.566009i	$0.808487\pi$
$858$	0	0
$859$	14866.0	0.590480	0.295240	−	0.955423i	$-0.404600\pi$
$859$	14866.0	0.590480	0.295240	+	0.955423i	$0.404600\pi$
$860$	79.9404i	0.00316970i
$861$	−4711.87	−0.186504
$862$	7110.37	0.280952
$863$	− 15469.1i	− 0.610167i	−0.952326	−	0.305083i	$-0.901316\pi$
$863$	− 15469.1i	− 0.610167i	0.952326	−	0.305083i	$-0.0986843\pi$
$864$	− 51.7634i	− 0.00203822i
$865$	− 6741.12i	− 0.264977i
$866$	− 3621.76i	− 0.142116i
$867$	−184.553	−0.00722925
$868$	144.684	0.00565770
$869$	7367.05i	0.287584i
$870$	5168.67	0.201419
$871$	0	0
$872$	37319.4	1.44930
$873$	16156.0i	0.626345i
$874$	4416.69	0.170934
$875$	−10849.3	−0.419171
$876$	− 3.39533i	0 0.000130956i
$877$	− 38100.7i	− 1.46701i	−0.679684	−	0.733505i	$-0.737883\pi$
$877$	− 38100.7i	− 1.46701i	0.679684	−	0.733505i	$-0.262117\pi$
$878$	− 47792.0i	− 1.83702i
$879$	8721.21i	0.334652i
$880$	7698.06	0.294888
$881$	−3879.84	−0.148371	−0.0741857	−	0.997244i	$-0.523636\pi$
$881$	−3879.84	−0.148371	−0.0741857	+	0.997244i	$0.523636\pi$
$882$	− 4187.38i	− 0.159860i
$883$	−13046.5	−0.497225	−0.248613	−	0.968603i	$-0.579975\pi$
$883$	−13046.5	−0.497225	−0.248613	+	0.968603i	$0.579975\pi$
$884$	0	0
$885$	−2804.64	−0.106528
$886$	6101.66i	0.231365i
$887$	−32833.2	−1.24288	−0.621438	−	0.783463i	$-0.713451\pi$
$887$	−32833.2	−1.24288	−0.621438	+	0.783463i	$0.713451\pi$
$888$	−14104.6	−0.533017
$889$	17688.2i	0.667315i
$890$	11831.6i	0.445613i
$891$	− 2871.69i	− 0.107974i
$892$	50.9829i	0.00191372i
$893$	780.274	0.0292395
$894$	−12843.0	−0.480462
$895$	− 9497.06i	− 0.354695i
$896$	19169.5	0.714741
$897$	0	0
$898$	4720.39	0.175414
$899$	− 45827.6i	− 1.70015i
$900$	43.2238	0.00160088
$901$	10241.1	0.378668
$902$	11771.3i	0.434525i
$903$	22147.0i	0.816173i
$904$	14848.9i	0.546314i
$905$	− 6434.74i	− 0.236352i
$906$	−4939.23	−0.181120
$907$	−21346.3	−0.781468	−0.390734	−	0.920504i	$-0.627779\pi$
$907$	−21346.3	−0.781468	−0.390734	+	0.920504i	$0.627779\pi$
$908$	− 184.760i	− 0.00675272i
$909$	6927.28	0.252765
$910$	0	0
$911$	7792.71	0.283407	0.141704	−	0.989909i	$-0.454742\pi$
$911$	7792.71	0.283407	0.141704	+	0.989909i	$0.454742\pi$
$912$	− 2368.38i	− 0.0859923i
$913$	−36565.2	−1.32545
$914$	−13124.0	−0.474950
$915$	− 6170.69i	− 0.222947i
$916$	− 228.101i	− 0.00822779i
$917$	− 12411.6i	− 0.446965i
$918$	5305.09i	0.190734i
$919$	−31352.0	−1.12536	−0.562681	−	0.826674i	$-0.690230\pi$
$919$	−31352.0	−1.12536	−0.562681	+	0.826674i	$0.690230\pi$
$920$	−9769.64	−0.350104
$921$	− 2801.85i	− 0.100243i
$922$	9760.61	0.348643
$923$	0	0
$924$	−60.1262	−0.00214070
$925$	− 23493.4i	− 0.835089i
$926$	5939.85	0.210794
$927$	−13893.0	−0.492238
$928$	343.286i	0.0121432i
$929$	12675.3i	0.447646i	0.974630	+	0.223823i	$0.0718537\pi$
$929$	12675.3i	0.447646i	−0.974630	+	0.223823i	$0.928146\pi$
$930$	− 7387.71i	− 0.260487i
$931$	2045.39i	0.0720032i
$932$	208.161	0.00731601
$933$	10900.7	0.382499
$934$	− 3994.09i	− 0.139926i
$935$	8422.80	0.294604
$936$	0	0
$937$	−12308.5	−0.429138	−0.214569	−	0.976709i	$-0.568835\pi$
$937$	−12308.5	−0.429138	−0.214569	+	0.976709i	$0.568835\pi$
$938$	27894.1i	0.970976i
$939$	−22521.8	−0.782717
$940$	−9.09167	−0.000315466 0
$941$	20465.2i	0.708978i	0.935060	+	0.354489i	$0.115345\pi$
$941$	20465.2i	0.708978i	−0.935060	+	0.354489i	$0.884655\pi$
$942$	− 220.821i	− 0.00763774i
$943$	− 14859.9i	− 0.513155i
$944$	17448.2i	0.601578i
$945$	−1228.92	−0.0423034
$946$	55328.0	1.90155
$947$	− 11170.8i	− 0.383318i	−0.981462	−	0.191659i	$-0.938613\pi$
$947$	− 11170.8i	− 0.383318i	0.981462	−	0.191659i	$-0.0613867\pi$
$948$	−26.4096	−0.000904792 0
$949$	0	0
$950$	3965.91	0.135443
$951$	3643.81i	0.124247i
$952$	21086.5	0.717876
$953$	−14109.4	−0.479587	−0.239794	−	0.970824i	$-0.577080\pi$
$953$	−14109.4	−0.479587	−0.239794	+	0.970824i	$0.577080\pi$
$954$	3732.87i	0.126684i
$955$	− 8349.64i	− 0.282919i
$956$	40.8271i	0.00138122i
$957$	19044.6i	0.643285i
$958$	2780.29	0.0937653
$959$	36710.2	1.23612
$960$	5266.57i	0.177060i
$961$	−35711.6	−1.19874
$962$	0	0
$963$	18248.8	0.610652
$964$	− 65.4307i	− 0.00218608i
$965$	6568.19	0.219106
$966$	−14257.4	−0.474870
$967$	− 40785.8i	− 1.35634i	−0.734904	−	0.678171i	$-0.762773\pi$
$967$	− 40785.8i	− 1.35634i	0.734904	−	0.678171i	$-0.237227\pi$
$968$	− 1680.87i	− 0.0558110i
$969$	− 2591.36i	− 0.0859096i
$970$	− 17272.4i	− 0.571734i
$971$	4230.29	0.139811	0.0699055	−	0.997554i	$-0.477730\pi$
$971$	4230.29	0.139811	0.0699055	+	0.997554i	$0.477730\pi$
$972$	10.2945	0.000339708 0
$973$	− 32515.1i	− 1.07131i
$974$	41537.4	1.36647
$975$	0	0
$976$	−38389.0	−1.25902
$977$	− 59925.0i	− 1.96230i	−0.193242	−	0.981151i	$-0.561900\pi$
$977$	− 59925.0i	− 1.96230i	0.193242	−	0.981151i	$-0.438100\pi$
$978$	−144.633	−0.00472890
$979$	−43594.9	−1.42319
$980$	− 23.8327i	0 0.000776844i
$981$	14804.8i	0.481834i
$982$	− 45984.6i	− 1.49433i
$983$	23333.3i	0.757087i	0.925584	+	0.378543i	$0.123575\pi$
$983$	23333.3i	0.757087i	−0.925584	+	0.378543i	$0.876425\pi$
$984$	−8010.83	−0.259528
$985$	−6207.64	−0.200804
$986$	− 35182.5i	− 1.13635i
$987$	−2518.79	−0.0812298
$988$	0	0
$989$	−69845.2	−2.24565
$990$	3070.11i	0.0985601i
$991$	−32523.5	−1.04253	−0.521263	−	0.853396i	$-0.674539\pi$
$991$	−32523.5	−1.04253	−0.521263	+	0.853396i	$0.674539\pi$
$992$	490.668	0.0157044
$993$	7329.61i	0.234238i
$994$	− 21553.4i	− 0.687759i
$995$	− 4345.37i	− 0.138450i
$996$	− 131.080i	− 0.00417010i
$997$	−7114.35	−0.225992	−0.112996	−	0.993595i	$-0.536045\pi$
$997$	−7114.35	−0.225992	−0.112996	+	0.993595i	$0.536045\pi$
$998$	9112.77	0.289038
$999$	− 5595.35i	− 0.177206i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.k.337.6		18
13.5	odd	4		507.4.a.p.1.3	yes	9
13.8	odd	4		507.4.a.o.1.7	✓	9
13.12	even	2	inner	507.4.b.k.337.13		18
39.5	even	4		1521.4.a.bf.1.7		9
39.8	even	4		1521.4.a.bi.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.7	✓	9	13.8	odd	4
507.4.a.p.1.3	yes	9	13.5	odd	4
507.4.b.k.337.6		18	1.1	even	1	trivial
507.4.b.k.337.13		18	13.12	even	2	inner
1521.4.a.bf.1.7		9	39.5	even	4
1521.4.a.bi.1.3		9	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.82093i q^{2} +3.00000 q^{3} +0.0423641 q^{4} -3.41089i q^{5} -8.46278i q^{6} -13.3442i q^{7} -22.6869i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.82093i q^{2} +3.00000 q^{3} +0.0423641 q^{4} -3.41089i q^{5} -8.46278i q^{6} -13.3442i q^{7} -22.6869i q^{8} +9.00000 q^{9} -9.62187 q^{10} -35.4529i q^{11} +0.127092 q^{12} -37.6430 q^{14} -10.2327i q^{15} -63.6593 q^{16} -69.6526 q^{17} -25.3884i q^{18} +12.4014i q^{19} -0.144499i q^{20} -40.0325i q^{21} -100.010 q^{22} +126.251 q^{23} -68.0608i q^{24} +113.366 q^{25} +27.0000 q^{27} -0.565314i q^{28} -179.060 q^{29} -28.8656 q^{30} +255.935i q^{31} -1.91716i q^{32} -106.359i q^{33} +196.485i q^{34} -45.5155 q^{35} +0.381277 q^{36} -207.235i q^{37} +34.9833 q^{38} -77.3825 q^{40} -117.701i q^{41} -112.929 q^{42} -553.224 q^{43} -1.50193i q^{44} -30.6980i q^{45} -356.146i q^{46} -62.9185i q^{47} -190.978 q^{48} +164.933 q^{49} -319.797i q^{50} -208.958 q^{51} -147.031 q^{53} -76.1651i q^{54} -120.926 q^{55} -302.738 q^{56} +37.2041i q^{57} +505.115i q^{58} -274.087i q^{59} -0.433498i q^{60} +603.039 q^{61} +721.974 q^{62} -120.098i q^{63} -514.683 q^{64} -300.031 q^{66} -741.019i q^{67} -2.95077 q^{68} +378.754 q^{69} +128.396i q^{70} +572.574i q^{71} -204.182i q^{72} -26.7155i q^{73} -584.595 q^{74} +340.098 q^{75} +0.525372i q^{76} -473.090 q^{77} -207.798 q^{79} +217.135i q^{80} +81.0000 q^{81} -332.026 q^{82} -1031.37i q^{83} -1.69594i q^{84} +237.577i q^{85} +1560.60i q^{86} -537.179 q^{87} -804.318 q^{88} -1229.66i q^{89} -86.5968 q^{90} +5.34852 q^{92} +767.805i q^{93} -177.488 q^{94} +42.2996 q^{95} -5.75148i q^{96} +1795.12i q^{97} -465.264i q^{98} -319.076i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

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