LMFDB - Embedded newform 507.4.b.j.337.17

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} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + \cdots + 26460736 $ x^18 + 97x^16 + 3906x^14 + 84743x^12 + 1077128x^10 + 8187552x^8 + 36483705x^6 + 88861676x^4 + 98825392x^2 + 26460736
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.17
Root			$3.76649i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.j.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.76649i q^{2} -3.00000 q^{3} -14.7194 q^{4} +18.8390i q^{5} -14.2995i q^{6} +23.8593i q^{7} -32.0282i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.76649i q^{2} -3.00000 q^{3} -14.7194 q^{4} +18.8390i q^{5} -14.2995i q^{6} +23.8593i q^{7} -32.0282i q^{8} +9.00000 q^{9} -89.7961 q^{10} -60.2327i q^{11} +44.1583 q^{12} -113.725 q^{14} -56.5171i q^{15} +34.9065 q^{16} +1.17630 q^{17} +42.8984i q^{18} +29.9930i q^{19} -277.300i q^{20} -71.5780i q^{21} +287.099 q^{22} -159.182 q^{23} +96.0845i q^{24} -229.909 q^{25} -27.0000 q^{27} -351.196i q^{28} +20.8211 q^{29} +269.388 q^{30} -67.2170i q^{31} -89.8439i q^{32} +180.698i q^{33} +5.60681i q^{34} -449.487 q^{35} -132.475 q^{36} -138.799i q^{37} -142.961 q^{38} +603.380 q^{40} -113.297i q^{41} +341.176 q^{42} -32.9644 q^{43} +886.592i q^{44} +169.551i q^{45} -758.740i q^{46} +520.256i q^{47} -104.720 q^{48} -226.268 q^{49} -1095.86i q^{50} -3.52889 q^{51} +467.189 q^{53} -128.695i q^{54} +1134.73 q^{55} +764.171 q^{56} -89.9789i q^{57} +99.2437i q^{58} +409.028i q^{59} +831.900i q^{60} +74.9067 q^{61} +320.389 q^{62} +214.734i q^{63} +707.492 q^{64} -861.296 q^{66} +305.693i q^{67} -17.3144 q^{68} +477.546 q^{69} -2142.47i q^{70} -318.757i q^{71} -288.254i q^{72} -867.378i q^{73} +661.585 q^{74} +689.727 q^{75} -441.480i q^{76} +1437.11 q^{77} -626.973 q^{79} +657.605i q^{80} +81.0000 q^{81} +540.031 q^{82} +1212.06i q^{83} +1053.59i q^{84} +22.1603i q^{85} -157.125i q^{86} -62.4634 q^{87} -1929.14 q^{88} +679.991i q^{89} -808.165 q^{90} +2343.07 q^{92} +201.651i q^{93} -2479.80 q^{94} -565.038 q^{95} +269.532i q^{96} -491.109i q^{97} -1078.51i q^{98} -542.094i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * 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 $

$2$	4.76649i	1.68521i	0.538533	+	0.842605i	$0.318979\pi$
$2$	4.76649i	1.68521i	−0.538533	+	0.842605i	$0.681021\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−14.7194	−1.83993
$5$	18.8390i	1.68501i	0.538686	+	0.842507i	$0.318921\pi$
$5$	18.8390i	1.68501i	−0.538686	+	0.842507i	$0.681079\pi$
$6$	− 14.2995i	− 0.972956i
$7$	23.8593i	1.28828i	0.764906	+	0.644142i	$0.222785\pi$
$7$	23.8593i	1.28828i	−0.764906	+	0.644142i	$0.777215\pi$
$8$	− 32.0282i	− 1.41546i
$9$	9.00000	0.333333
$10$	−89.7961	−2.83960
$11$	− 60.2327i	− 1.65099i	−0.564412	−	0.825493i	$-0.690897\pi$
$11$	− 60.2327i	− 1.65099i	0.564412	−	0.825493i	$-0.309103\pi$
$12$	44.1583	1.06228
$13$	0	0
$13$	0	0
$14$	−113.725	−2.17103
$15$	− 56.5171i	− 0.972843i
$16$	34.9065	0.545414
$17$	1.17630	0.0167820	0.00839099	−	0.999965i	$-0.497329\pi$
$17$	1.17630	0.0167820	0.00839099	+	0.999965i	$0.497329\pi$
$18$	42.8984i	0.561736i
$19$	29.9930i	0.362150i	0.983469	+	0.181075i	$0.0579577\pi$
$19$	29.9930i	0.362150i	−0.983469	+	0.181075i	$0.942042\pi$
$20$	− 277.300i	− 3.10031i
$21$	− 71.5780i	− 0.743791i
$22$	287.099	2.78226
$23$	−159.182	−1.44312	−0.721560	−	0.692352i	$-0.756575\pi$
$23$	−159.182	−1.44312	−0.721560	+	0.692352i	$0.756575\pi$
$24$	96.0845i	0.817216i
$25$	−229.909	−1.83927
$26$	0	0
$27$	−27.0000	−0.192450
$28$	− 351.196i	− 2.37035i
$29$	20.8211	0.133324	0.0666618	−	0.997776i	$-0.478765\pi$
$29$	20.8211	0.133324	0.0666618	+	0.997776i	$0.478765\pi$
$30$	269.388	1.63944
$31$	− 67.2170i	− 0.389436i	−0.980859	−	0.194718i	$-0.937621\pi$
$31$	− 67.2170i	− 0.389436i	0.980859	−	0.194718i	$-0.0623792\pi$
$32$	− 89.8439i	− 0.496322i
$33$	180.698i	0.953197i
$34$	5.60681i	0.0282812i
$35$	−449.487	−2.17077
$36$	−132.475	−0.613310
$37$	− 138.799i	− 0.616714i	−0.951271	−	0.308357i	$-0.900221\pi$
$37$	− 138.799i	− 0.616714i	0.951271	−	0.308357i	$-0.0997792\pi$
$38$	−142.961	−0.610299
$39$	0	0
$40$	603.380	2.38507
$41$	− 113.297i	− 0.431563i	−0.976442	−	0.215781i	$-0.930770\pi$
$41$	− 113.297i	− 0.431563i	0.976442	−	0.215781i	$-0.0692299\pi$
$42$	341.176	1.25344
$43$	−32.9644	−0.116908	−0.0584538	−	0.998290i	$-0.518617\pi$
$43$	−32.9644	−0.116908	−0.0584538	+	0.998290i	$0.518617\pi$
$44$	886.592i	3.03770i
$45$	169.551i	0.561671i
$46$	− 758.740i	− 2.43196i
$47$	520.256i	1.61462i	0.590127	+	0.807310i	$0.299078\pi$
$47$	520.256i	1.61462i	−0.590127	+	0.807310i	$0.700922\pi$
$48$	−104.720	−0.314895
$49$	−226.268	−0.659674
$50$	− 1095.86i	− 3.09956i
$51$	−3.52889	−0.00968909
$52$	0	0
$53$	467.189	1.21082	0.605409	−	0.795915i	$-0.293010\pi$
$53$	467.189	1.21082	0.605409	+	0.795915i	$0.293010\pi$
$54$	− 128.695i	− 0.324319i
$55$	1134.73	2.78193
$56$	764.171	1.82351
$57$	− 89.9789i	− 0.209088i
$58$	99.2437i	0.224678i
$59$	409.028i	0.902558i	0.892383	+	0.451279i	$0.149032\pi$
$59$	409.028i	0.902558i	−0.892383	+	0.451279i	$0.850968\pi$
$60$	831.900i	1.78996i
$61$	74.9067	0.157226	0.0786132	−	0.996905i	$-0.474951\pi$
$61$	74.9067	0.157226	0.0786132	+	0.996905i	$0.474951\pi$
$62$	320.389	0.656282
$63$	214.734i	0.429428i
$64$	707.492	1.38182
$65$	0	0
$66$	−861.296	−1.60634
$67$	305.693i	0.557408i	0.960377	+	0.278704i	$0.0899049\pi$
$67$	305.693i	0.557408i	−0.960377	+	0.278704i	$0.910095\pi$
$68$	−17.3144	−0.0308777
$69$	477.546	0.833186
$70$	− 2142.47i	− 3.65821i
$71$	− 318.757i	− 0.532810i	−0.963861	−	0.266405i	$-0.914164\pi$
$71$	− 318.757i	− 0.532810i	0.963861	−	0.266405i	$-0.0858359\pi$
$72$	− 288.254i	− 0.471820i
$73$	− 867.378i	− 1.39067i	−0.718686	−	0.695335i	$-0.755256\pi$
$73$	− 867.378i	− 1.39067i	0.718686	−	0.695335i	$-0.244744\pi$
$74$	661.585	1.03929
$75$	689.727	1.06190
$76$	− 441.480i	− 0.666332i
$77$	1437.11	2.12694
$78$	0	0
$79$	−626.973	−0.892911	−0.446456	−	0.894806i	$-0.647314\pi$
$79$	−626.973	−0.892911	−0.446456	+	0.894806i	$0.647314\pi$
$80$	657.605i	0.919030i
$81$	81.0000	0.111111
$82$	540.031	0.727274
$83$	1212.06i	1.60291i	0.598056	+	0.801454i	$0.295940\pi$
$83$	1212.06i	1.60291i	−0.598056	+	0.801454i	$0.704060\pi$
$84$	1053.59i	1.36852i
$85$	22.1603i	0.0282779i
$86$	− 157.125i	− 0.197014i
$87$	−62.4634	−0.0769744
$88$	−1929.14	−2.33690
$89$	679.991i	0.809875i	0.914344	+	0.404938i	$0.132707\pi$
$89$	679.991i	0.809875i	−0.914344	+	0.404938i	$0.867293\pi$
$90$	−808.165	−0.946534
$91$	0	0
$92$	2343.07	2.65524
$93$	201.651i	0.224841i
$94$	−2479.80	−2.72097
$95$	−565.038	−0.610228
$96$	269.532i	0.286552i
$97$	− 491.109i	− 0.514068i	−0.966402	−	0.257034i	$-0.917255\pi$
$97$	− 491.109i	− 0.514068i	0.966402	−	0.257034i	$-0.0827452\pi$
$98$	− 1078.51i	− 1.11169i
$99$	− 542.094i	− 0.550329i
$100$	3384.13	3.38413
$101$	−707.245	−0.696767	−0.348384	−	0.937352i	$-0.613269\pi$
$101$	−707.245	−0.696767	−0.348384	+	0.937352i	$0.613269\pi$
$102$	− 16.8204i	− 0.0163281i
$103$	1656.38	1.58454	0.792269	−	0.610171i	$-0.208899\pi$
$103$	1656.38	1.58454	0.792269	+	0.610171i	$0.208899\pi$
$104$	0	0
$105$	1348.46	1.25330
$106$	2226.85i	2.04048i
$107$	−1416.79	−1.28006	−0.640029	−	0.768351i	$-0.721078\pi$
$107$	−1416.79	−1.28006	−0.640029	+	0.768351i	$0.721078\pi$
$108$	397.425	0.354095
$109$	− 855.136i	− 0.751442i	−0.926733	−	0.375721i	$-0.877395\pi$
$109$	− 855.136i	− 0.751442i	0.926733	−	0.375721i	$-0.122605\pi$
$110$	5408.66i	4.68814i
$111$	416.397i	0.356060i
$112$	832.846i	0.702648i
$113$	−2258.04	−1.87981	−0.939905	−	0.341437i	$-0.889086\pi$
$113$	−2258.04	−1.87981	−0.939905	+	0.341437i	$0.889086\pi$
$114$	428.884	0.352356
$115$	− 2998.84i	− 2.43168i
$116$	−306.475	−0.245306
$117$	0	0
$118$	−1949.63	−1.52100
$119$	28.0657i	0.0216200i
$120$	−1810.14	−1.37702
$121$	−2296.98	−1.72576
$122$	357.042i	0.264960i
$123$	339.892i	0.249163i
$124$	989.397i	0.716536i
$125$	− 1976.38i	− 1.41418i
$126$	−1023.53	−0.723676
$127$	111.921	0.0781998	0.0390999	−	0.999235i	$-0.487551\pi$
$127$	111.921	0.0781998	0.0390999	+	0.999235i	$0.487551\pi$
$128$	2653.50i	1.83234i
$129$	98.8932	0.0674966
$130$	0	0
$131$	−2994.35	−1.99708	−0.998540	−	0.0540104i	$-0.982800\pi$
$131$	−2994.35	−1.99708	−0.998540	+	0.0540104i	$0.982800\pi$
$132$	− 2659.78i	− 1.75382i
$133$	−715.612	−0.466552
$134$	−1457.08	−0.939350
$135$	− 508.654i	− 0.324281i
$136$	− 37.6746i	− 0.0237542i
$137$	− 559.705i	− 0.349043i	−0.984653	−	0.174521i	$-0.944162\pi$
$137$	− 559.705i	− 0.349043i	0.984653	−	0.174521i	$-0.0558378\pi$
$138$	2276.22i	1.40409i
$139$	436.351	0.266265	0.133132	−	0.991098i	$-0.457496\pi$
$139$	436.351	0.266265	0.133132	+	0.991098i	$0.457496\pi$
$140$	6616.20	3.99408
$141$	− 1560.77i	− 0.932201i
$142$	1519.35	0.897897
$143$	0	0
$144$	314.159	0.181805
$145$	392.250i	0.224652i
$146$	4134.35	2.34357
$147$	678.804	0.380863
$148$	2043.05i	1.13471i
$149$	489.586i	0.269184i	0.990901	+	0.134592i	$0.0429724\pi$
$149$	489.586i	0.269184i	−0.990901	+	0.134592i	$0.957028\pi$
$150$	3287.58i	1.78953i
$151$	1817.82i	0.979682i	0.871812	+	0.489841i	$0.162945\pi$
$151$	1817.82i	0.979682i	−0.871812	+	0.489841i	$0.837055\pi$
$152$	960.620	0.512609
$153$	10.5867	0.00559400
$154$	6849.99i	3.58434i
$155$	1266.30	0.656206
$156$	0	0
$157$	1866.37	0.948740	0.474370	−	0.880325i	$-0.342676\pi$
$157$	1866.37	0.948740	0.474370	+	0.880325i	$0.342676\pi$
$158$	− 2988.46i	− 1.50474i
$159$	−1401.57	−0.699066
$160$	1692.57	0.836309
$161$	− 3797.98i	− 1.85915i
$162$	386.086i	0.187245i
$163$	1044.58i	0.501948i	0.967994	+	0.250974i	$0.0807509\pi$
$163$	1044.58i	0.501948i	−0.967994	+	0.250974i	$0.919249\pi$
$164$	1667.67i	0.794046i
$165$	−3404.18	−1.60615
$166$	−5777.30	−2.70124
$167$	− 936.877i	− 0.434118i	−0.976158	−	0.217059i	$-0.930354\pi$
$167$	− 936.877i	− 0.434118i	0.976158	−	0.217059i	$-0.0696464\pi$
$168$	−2292.51	−1.05281
$169$	0	0
$170$	−105.627	−0.0476542
$171$	269.937i	0.120717i
$172$	485.218	0.215102
$173$	2989.51	1.31380	0.656902	−	0.753976i	$-0.271866\pi$
$173$	2989.51	1.31380	0.656902	+	0.753976i	$0.271866\pi$
$174$	− 297.731i	− 0.129718i
$175$	− 5485.47i	− 2.36950i
$176$	− 2102.51i	− 0.900471i
$177$	− 1227.08i	− 0.521092i
$178$	−3241.17	−1.36481
$179$	438.872	0.183256	0.0916281	−	0.995793i	$-0.470793\pi$
$179$	438.872	0.183256	0.0916281	+	0.995793i	$0.470793\pi$
$180$	− 2495.70i	− 1.03344i
$181$	−991.289	−0.407082	−0.203541	−	0.979066i	$-0.565245\pi$
$181$	−991.289	−0.407082	−0.203541	+	0.979066i	$0.565245\pi$
$182$	0	0
$183$	−224.720	−0.0907747
$184$	5098.31i	2.04268i
$185$	2614.84	1.03917
$186$	−961.168	−0.378905
$187$	− 70.8515i	− 0.0277068i
$188$	− 7657.88i	− 2.97079i
$189$	− 644.202i	− 0.247930i
$190$	− 2693.25i	− 1.02836i
$191$	−1626.28	−0.616093	−0.308046	−	0.951371i	$-0.599675\pi$
$191$	−1626.28	−0.616093	−0.308046	+	0.951371i	$0.599675\pi$
$192$	−2122.48	−0.797794
$193$	− 5282.91i	− 1.97032i	−0.171638	−	0.985160i	$-0.554906\pi$
$193$	− 5282.91i	− 1.97032i	0.171638	−	0.985160i	$-0.445094\pi$
$194$	2340.87	0.866312
$195$	0	0
$196$	3330.54	1.21375
$197$	− 2285.76i	− 0.826670i	−0.910579	−	0.413335i	$-0.864364\pi$
$197$	− 2285.76i	− 0.826670i	0.910579	−	0.413335i	$-0.135636\pi$
$198$	2583.89	0.927419
$199$	−567.192	−0.202046	−0.101023	−	0.994884i	$-0.532212\pi$
$199$	−567.192	−0.202046	−0.101023	+	0.994884i	$0.532212\pi$
$200$	7363.56i	2.60341i
$201$	− 917.079i	− 0.321820i
$202$	− 3371.08i	− 1.17420i
$203$	496.778i	0.171759i
$204$	51.9433	0.0178272
$205$	2134.41	0.727189
$206$	7895.10i	2.67028i
$207$	−1432.64	−0.481040
$208$	0	0
$209$	1806.56	0.597905
$210$	6427.42i	2.11207i
$211$	−3285.95	−1.07210	−0.536052	−	0.844185i	$-0.680085\pi$
$211$	−3285.95	−1.07210	−0.536052	+	0.844185i	$0.680085\pi$
$212$	−6876.76	−2.22782
$213$	956.272i	0.307618i
$214$	− 6753.11i	− 2.15716i
$215$	− 621.017i	− 0.196991i
$216$	864.761i	0.272405i
$217$	1603.75	0.501704
$218$	4076.00	1.26634
$219$	2602.13i	0.802903i
$220$	−16702.5	−5.11857
$221$	0	0
$222$	−1984.75	−0.600036
$223$	− 3801.11i	− 1.14144i	−0.821145	−	0.570720i	$-0.806664\pi$
$223$	− 3801.11i	− 1.14144i	0.821145	−	0.570720i	$-0.193336\pi$
$224$	2143.62	0.639403
$225$	−2069.18	−0.613090
$226$	− 10762.9i	− 3.16787i
$227$	− 3053.98i	− 0.892951i	−0.894796	−	0.446476i	$-0.852679\pi$
$227$	− 3053.98i	− 0.892951i	0.894796	−	0.446476i	$-0.147321\pi$
$228$	1324.44i	0.384707i
$229$	1338.32i	0.386195i	0.981180	+	0.193098i	$0.0618534\pi$
$229$	1338.32i	0.386195i	−0.981180	+	0.193098i	$0.938147\pi$
$230$	14293.9	4.09789
$231$	−4311.34	−1.22799
$232$	− 666.863i	− 0.188714i
$233$	−1979.59	−0.556599	−0.278299	−	0.960494i	$-0.589771\pi$
$233$	−1979.59	−0.556599	−0.278299	+	0.960494i	$0.589771\pi$
$234$	0	0
$235$	−9801.12	−2.72066
$236$	− 6020.67i	− 1.66064i
$237$	1880.92	0.515523
$238$	−133.775	−0.0364342
$239$	5326.50i	1.44160i	0.693143	+	0.720800i	$0.256226\pi$
$239$	5326.50i	1.44160i	−0.693143	+	0.720800i	$0.743774\pi$
$240$	− 1972.81i	− 0.530602i
$241$	2663.04i	0.711791i	0.934526	+	0.355895i	$0.115824\pi$
$241$	2663.04i	0.711791i	−0.934526	+	0.355895i	$0.884176\pi$
$242$	− 10948.5i	− 2.90826i
$243$	−243.000	−0.0641500
$244$	−1102.58	−0.289286
$245$	− 4262.67i	− 1.11156i
$246$	−1620.09	−0.419892
$247$	0	0
$248$	−2152.84	−0.551231
$249$	− 3636.19i	− 0.925440i
$250$	9420.40	2.38319
$251$	2127.31	0.534959	0.267479	−	0.963564i	$-0.413809\pi$
$251$	2127.31	0.534959	0.267479	+	0.963564i	$0.413809\pi$
$252$	− 3160.77i	− 0.790117i
$253$	9587.97i	2.38257i
$254$	533.470i	0.131783i
$255$	− 66.4808i	− 0.0163262i
$256$	−6987.97	−1.70605
$257$	−399.185	−0.0968891	−0.0484446	−	0.998826i	$-0.515426\pi$
$257$	−399.185	−0.0968891	−0.0484446	+	0.998826i	$0.515426\pi$
$258$	471.374i	0.113746i
$259$	3311.66	0.794503
$260$	0	0
$261$	187.390	0.0444412
$262$	− 14272.5i	− 3.36550i
$263$	1901.20	0.445752	0.222876	−	0.974847i	$-0.428455\pi$
$263$	1901.20	0.445752	0.222876	+	0.974847i	$0.428455\pi$
$264$	5787.43	1.34921
$265$	8801.38i	2.04024i
$266$	− 3410.96i	− 0.786238i
$267$	− 2039.97i	− 0.467582i
$268$	− 4499.63i	− 1.02559i
$269$	−3541.04	−0.802606	−0.401303	−	0.915945i	$-0.631443\pi$
$269$	−3541.04	−0.802606	−0.401303	+	0.915945i	$0.631443\pi$
$270$	2424.49	0.546481
$271$	− 3421.98i	− 0.767051i	−0.923530	−	0.383525i	$-0.874710\pi$
$271$	− 3421.98i	− 0.767051i	0.923530	−	0.383525i	$-0.125290\pi$
$272$	41.0604	0.00915314
$273$	0	0
$274$	2667.83	0.588210
$275$	13848.0i	3.03661i
$276$	−7029.22	−1.53300
$277$	−5899.10	−1.27958	−0.639788	−	0.768552i	$-0.720978\pi$
$277$	−5899.10	−1.27958	−0.639788	+	0.768552i	$0.720978\pi$
$278$	2079.86i	0.448712i
$279$	− 604.953i	− 0.129812i
$280$	14396.2i	3.07264i
$281$	− 7550.89i	− 1.60302i	−0.597982	−	0.801509i	$-0.704031\pi$
$281$	− 7550.89i	− 1.60302i	0.597982	−	0.801509i	$-0.295969\pi$
$282$	7439.39	1.57095
$283$	4533.54	0.952266	0.476133	−	0.879373i	$-0.342038\pi$
$283$	4533.54	0.952266	0.476133	+	0.879373i	$0.342038\pi$
$284$	4691.93i	0.980334i
$285$	1695.11	0.352315
$286$	0	0
$287$	2703.20	0.555975
$288$	− 808.595i	− 0.165441i
$289$	−4911.62	−0.999718
$290$	−1869.65	−0.378586
$291$	1473.33i	0.296797i
$292$	12767.3i	2.55874i
$293$	− 1847.62i	− 0.368393i	−0.982889	−	0.184197i	$-0.941032\pi$
$293$	− 1847.62i	− 0.368393i	0.982889	−	0.184197i	$-0.0589683\pi$
$294$	3235.52i	0.641834i
$295$	−7705.69	−1.52082
$296$	−4445.48	−0.872934
$297$	1626.28i	0.317732i
$298$	−2333.61	−0.453631
$299$	0	0
$300$	−10152.4	−1.95383
$301$	− 786.509i	− 0.150610i
$302$	−8664.62	−1.65097
$303$	2121.74	0.402279
$304$	1046.95i	0.197522i
$305$	1411.17i	0.264929i
$306$	50.4613i	0.00942706i
$307$	− 370.739i	− 0.0689225i	−0.999406	−	0.0344612i	$-0.989028\pi$
$307$	− 370.739i	− 0.0689225i	0.999406	−	0.0344612i	$-0.0109715\pi$
$308$	−21153.5	−3.91342
$309$	−4969.13	−0.914834
$310$	6035.82i	1.10584i
$311$	−5288.17	−0.964195	−0.482097	−	0.876118i	$-0.660125\pi$
$311$	−5288.17	−0.964195	−0.482097	+	0.876118i	$0.660125\pi$
$312$	0	0
$313$	5220.25	0.942703	0.471351	−	0.881945i	$-0.343766\pi$
$313$	5220.25	0.942703	0.471351	+	0.881945i	$0.343766\pi$
$314$	8896.02i	1.59883i
$315$	−4045.38	−0.723592
$316$	9228.70	1.64290
$317$	8106.97i	1.43638i	0.695846	+	0.718191i	$0.255029\pi$
$317$	8106.97i	1.43638i	−0.695846	+	0.718191i	$0.744971\pi$
$318$	− 6680.55i	− 1.17807i
$319$	− 1254.11i	− 0.220115i
$320$	13328.5i	2.32839i
$321$	4250.37	0.739041
$322$	18103.0	3.13305
$323$	35.2806i	0.00607760i
$324$	−1192.28	−0.204437
$325$	0	0
$326$	−4978.97	−0.845888
$327$	2565.41i	0.433845i
$328$	−3628.71	−0.610859
$329$	−12413.0	−2.08009
$330$	− 16226.0i	− 2.70670i
$331$	− 8131.30i	− 1.35026i	−0.737698	−	0.675131i	$-0.764087\pi$
$331$	− 8131.30i	− 1.35026i	0.737698	−	0.675131i	$-0.235913\pi$
$332$	− 17840.9i	− 2.94924i
$333$	− 1249.19i	− 0.205571i
$334$	4465.62	0.731580
$335$	−5758.96	−0.939240
$336$	− 2498.54i	− 0.405674i
$337$	3109.19	0.502577	0.251288	−	0.967912i	$-0.419146\pi$
$337$	3109.19	0.502577	0.251288	+	0.967912i	$0.419146\pi$
$338$	0	0
$339$	6774.12	1.08531
$340$	− 326.187i	− 0.0520293i
$341$	−4048.66	−0.642954
$342$	−1286.65	−0.203433
$343$	2785.15i	0.438436i
$344$	1055.79i	0.165478i
$345$	8996.51i	1.40393i
$346$	14249.5i	2.21404i
$347$	851.124	0.131674	0.0658368	−	0.997830i	$-0.479028\pi$
$347$	851.124	0.131674	0.0658368	+	0.997830i	$0.479028\pi$
$348$	919.426	0.141628
$349$	4114.16i	0.631019i	0.948922	+	0.315510i	$0.102175\pi$
$349$	4114.16i	0.631019i	−0.948922	+	0.315510i	$0.897825\pi$
$350$	26146.5	3.99311
$351$	0	0
$352$	−5411.54	−0.819421
$353$	4659.80i	0.702595i	0.936264	+	0.351297i	$0.114259\pi$
$353$	4659.80i	0.702595i	−0.936264	+	0.351297i	$0.885741\pi$
$354$	5848.89	0.878149
$355$	6005.08	0.897793
$356$	− 10009.1i	− 1.49011i
$357$	− 84.1970i	− 0.0124823i
$358$	2091.88i	0.308825i
$359$	− 2811.33i	− 0.413305i	−0.978414	−	0.206653i	$-0.933743\pi$
$359$	− 2811.33i	− 0.413305i	0.978414	−	0.206653i	$-0.0662570\pi$
$360$	5430.42	0.795023
$361$	5959.42	0.868847
$362$	− 4724.97i	− 0.686019i
$363$	6890.94	0.996365
$364$	0	0
$365$	16340.5	2.34330
$366$	− 1071.13i	− 0.152974i
$367$	−1159.77	−0.164958	−0.0824791	−	0.996593i	$-0.526284\pi$
$367$	−1159.77	−0.164958	−0.0824791	+	0.996593i	$0.526284\pi$
$368$	−5556.49	−0.787098
$369$	− 1019.68i	− 0.143854i
$370$	12463.6i	1.75122i
$371$	11146.8i	1.55988i
$372$	− 2968.19i	− 0.413692i
$373$	8876.80	1.23223	0.616117	−	0.787655i	$-0.288705\pi$
$373$	8876.80	1.23223	0.616117	+	0.787655i	$0.288705\pi$
$374$	337.713	0.0466918
$375$	5929.14i	0.816479i
$376$	16662.9	2.28543
$377$	0	0
$378$	3070.58	0.417814
$379$	2256.21i	0.305788i	0.988243	+	0.152894i	$0.0488593\pi$
$379$	2256.21i	0.305788i	−0.988243	+	0.152894i	$0.951141\pi$
$380$	8317.05	1.12278
$381$	−335.763	−0.0451487
$382$	− 7751.67i	− 1.03825i
$383$	10331.9i	1.37843i	0.724559	+	0.689213i	$0.242044\pi$
$383$	10331.9i	1.37843i	−0.724559	+	0.689213i	$0.757956\pi$
$384$	− 7960.51i	− 1.05790i
$385$	27073.8i	3.58392i
$386$	25180.9	3.32040
$387$	−296.680	−0.0389692
$388$	7228.85i	0.945849i
$389$	−5548.06	−0.723131	−0.361565	−	0.932347i	$-0.617758\pi$
$389$	−5548.06	−0.723131	−0.361565	+	0.932347i	$0.617758\pi$
$390$	0	0
$391$	−187.245	−0.0242184
$392$	7246.96i	0.933741i
$393$	8983.05	1.15302
$394$	10895.1	1.39311
$395$	− 11811.6i	− 1.50457i
$396$	7979.33i	1.01257i
$397$	− 15168.9i	− 1.91764i	−0.284010	−	0.958821i	$-0.591665\pi$
$397$	− 15168.9i	− 1.91764i	0.284010	−	0.958821i	$-0.408335\pi$
$398$	− 2703.52i	− 0.340490i
$399$	2146.84	0.269364
$400$	−8025.32	−1.00316
$401$	− 4305.72i	− 0.536204i	−0.963391	−	0.268102i	$-0.913604\pi$
$401$	− 4305.72i	− 0.536204i	0.963391	−	0.268102i	$-0.0863963\pi$
$402$	4371.25	0.542334
$403$	0	0
$404$	10410.3	1.28200
$405$	1525.96i	0.187224i
$406$	−2367.89	−0.289449
$407$	−8360.25	−1.01819
$408$	113.024i	0.0137145i
$409$	5100.24i	0.616603i	0.951289	+	0.308301i	$0.0997606\pi$
$409$	5100.24i	0.616603i	−0.951289	+	0.308301i	$0.900239\pi$
$410$	10173.7i	1.22547i
$411$	1679.12i	0.201520i
$412$	−24380.9	−2.91544
$413$	−9759.14	−1.16275
$414$	− 6828.66i	− 0.810653i
$415$	−22834.1	−2.70092
$416$	0	0
$417$	−1309.05	−0.153728
$418$	8610.94i	1.00760i
$419$	−10210.2	−1.19045	−0.595226	−	0.803558i	$-0.702938\pi$
$419$	−10210.2	−1.19045	−0.595226	+	0.803558i	$0.702938\pi$
$420$	−19848.6	−2.30598
$421$	− 14590.5i	− 1.68906i	−0.535506	−	0.844531i	$-0.679879\pi$
$421$	− 14590.5i	− 1.68906i	0.535506	−	0.844531i	$-0.320121\pi$
$422$	− 15662.4i	− 1.80672i
$423$	4682.31i	0.538207i
$424$	− 14963.2i	− 1.71386i
$425$	−270.441	−0.0308666
$426$	−4558.06	−0.518401
$427$	1787.22i	0.202552i
$428$	20854.3	2.35522
$429$	0	0
$430$	2960.07	0.331971
$431$	− 2653.77i	− 0.296584i	−0.988944	−	0.148292i	$-0.952622\pi$
$431$	− 2653.77i	− 0.296584i	0.988944	−	0.148292i	$-0.0473776\pi$
$432$	−942.476	−0.104965
$433$	4081.18	0.452953	0.226477	−	0.974017i	$-0.427279\pi$
$433$	4081.18	0.452953	0.226477	+	0.974017i	$0.427279\pi$
$434$	7644.28i	0.845477i
$435$	− 1176.75i	− 0.129703i
$436$	12587.1i	1.38260i
$437$	− 4774.34i	− 0.522627i
$438$	−12403.0	−1.35306
$439$	−13002.4	−1.41360	−0.706802	−	0.707411i	$-0.749863\pi$
$439$	−13002.4	−1.41360	−0.706802	+	0.707411i	$0.749863\pi$
$440$	− 36343.2i	− 3.93771i
$441$	−2036.41	−0.219891
$442$	0	0
$443$	−2681.37	−0.287575	−0.143787	−	0.989609i	$-0.545928\pi$
$443$	−2681.37	−0.287575	−0.143787	+	0.989609i	$0.545928\pi$
$444$	− 6129.14i	− 0.655126i
$445$	−12810.4	−1.36465
$446$	18118.0	1.92357
$447$	− 1468.76i	− 0.155413i
$448$	16880.3i	1.78018i
$449$	− 14525.2i	− 1.52669i	−0.645990	−	0.763346i	$-0.723555\pi$
$449$	− 14525.2i	− 1.52669i	0.645990	−	0.763346i	$-0.276445\pi$
$450$	− 9862.73i	− 1.03319i
$451$	−6824.21	−0.712504
$452$	33237.1	3.45872
$453$	− 5453.46i	− 0.565620i
$454$	14556.8	1.50481
$455$	0	0
$456$	−2881.86	−0.295955
$457$	11054.8i	1.13156i	0.824557	+	0.565779i	$0.191424\pi$
$457$	11054.8i	1.13156i	−0.824557	+	0.565779i	$0.808576\pi$
$458$	−6379.09	−0.650819
$459$	−31.7600	−0.00322970
$460$	44141.2i	4.47412i
$461$	− 2168.26i	− 0.219059i	−0.993984	−	0.109529i	$-0.965066\pi$
$461$	− 2168.26i	− 0.219059i	0.993984	−	0.109529i	$-0.0349344\pi$
$462$	− 20550.0i	− 2.06942i
$463$	− 16618.0i	− 1.66804i	−0.551733	−	0.834021i	$-0.686033\pi$
$463$	− 16618.0i	− 1.66804i	0.551733	−	0.834021i	$-0.313967\pi$
$464$	726.793	0.0727166
$465$	−3798.91	−0.378861
$466$	− 9435.72i	− 0.937985i
$467$	−6857.86	−0.679537	−0.339769	−	0.940509i	$-0.610349\pi$
$467$	−6857.86	−0.679537	−0.339769	+	0.940509i	$0.610349\pi$
$468$	0	0
$469$	−7293.63	−0.718100
$470$	− 46717.0i	− 4.58488i
$471$	−5599.10	−0.547755
$472$	13100.4	1.27753
$473$	1985.53i	0.193013i
$474$	8965.39i	0.868764i
$475$	− 6895.65i	− 0.666093i
$476$	− 413.111i	− 0.0397792i
$477$	4204.70	0.403606
$478$	−25388.7	−2.42940
$479$	17395.1i	1.65930i	0.558287	+	0.829648i	$0.311459\pi$
$479$	17395.1i	1.65930i	−0.558287	+	0.829648i	$0.688541\pi$
$480$	−5077.71	−0.482843
$481$	0	0
$482$	−12693.4	−1.19952
$483$	11393.9i	1.07338i
$484$	33810.3	3.17527
$485$	9252.01	0.866211
$486$	− 1158.26i	− 0.108106i
$487$	− 11698.7i	− 1.08854i	−0.838910	−	0.544270i	$-0.816807\pi$
$487$	− 11698.7i	− 1.08854i	0.838910	−	0.544270i	$-0.183193\pi$
$488$	− 2399.12i	− 0.222548i
$489$	− 3133.73i	− 0.289800i
$490$	20318.0	1.87321
$491$	−6600.47	−0.606670	−0.303335	−	0.952884i	$-0.598100\pi$
$491$	−6600.47	−0.606670	−0.303335	+	0.952884i	$0.598100\pi$
$492$	− 5003.02i	− 0.458442i
$493$	24.4918	0.00223744
$494$	0	0
$495$	10212.5	0.927311
$496$	− 2346.31i	− 0.212404i
$497$	7605.34	0.686411
$498$	17331.9	1.55956
$499$	21149.0i	1.89731i	0.316308	+	0.948657i	$0.397557\pi$
$499$	21149.0i	1.89731i	−0.316308	+	0.948657i	$0.602443\pi$
$500$	29091.2i	2.60200i
$501$	2810.63i	0.250638i
$502$	10139.8i	0.901517i
$503$	9591.55	0.850231	0.425116	−	0.905139i	$-0.360233\pi$
$503$	9591.55	0.850231	0.425116	+	0.905139i	$0.360233\pi$
$504$	6877.54	0.607837
$505$	− 13323.8i	− 1.17406i
$506$	−45701.0	−4.01513
$507$	0	0
$508$	−1647.41	−0.143882
$509$	− 2306.28i	− 0.200833i	−0.994946	−	0.100416i	$-0.967983\pi$
$509$	− 2306.28i	− 0.200833i	0.994946	−	0.100416i	$-0.0320175\pi$
$510$	316.880	0.0275131
$511$	20695.1	1.79158
$512$	− 12080.1i	− 1.04271i
$513$	− 809.810i	− 0.0696959i
$514$	− 1902.71i	− 0.163278i
$515$	31204.5i	2.66997i
$516$	−1455.65	−0.124189
$517$	31336.4	2.66572
$518$	15785.0i	1.33890i
$519$	−8968.53	−0.758525
$520$	0	0
$521$	−12511.5	−1.05209	−0.526045	−	0.850457i	$-0.676326\pi$
$521$	−12511.5	−1.05209	−0.526045	+	0.850457i	$0.676326\pi$
$522$	893.193i	0.0748927i
$523$	−12924.3	−1.08058	−0.540289	−	0.841480i	$-0.681685\pi$
$523$	−12924.3	−1.08058	−0.540289	+	0.841480i	$0.681685\pi$
$524$	44075.2	3.67449
$525$	16456.4i	1.36803i
$526$	9062.04i	0.751186i
$527$	− 79.0671i	− 0.00653552i
$528$	6307.54i	0.519887i
$529$	13172.0	1.08260
$530$	−41951.7	−3.43824
$531$	3681.25i	0.300853i
$532$	10533.4	0.858424
$533$	0	0
$534$	9723.51	0.787973
$535$	− 26690.9i	− 2.15691i
$536$	9790.79	0.788988
$537$	−1316.62	−0.105803
$538$	− 16878.3i	− 1.35256i
$539$	13628.7i	1.08911i
$540$	7487.10i	0.596655i
$541$	− 15384.3i	− 1.22260i	−0.791400	−	0.611298i	$-0.790648\pi$
$541$	− 15384.3i	− 1.22260i	0.791400	−	0.611298i	$-0.209352\pi$
$542$	16310.9	1.29264
$543$	2973.87	0.235029
$544$	− 105.683i	− 0.00832927i
$545$	16109.9	1.26619
$546$	0	0
$547$	7658.37	0.598626	0.299313	−	0.954155i	$-0.403243\pi$
$547$	7658.37	0.598626	0.299313	+	0.954155i	$0.403243\pi$
$548$	8238.55i	0.642215i
$549$	674.160	0.0524088
$550$	−66006.5	−5.11732
$551$	624.487i	0.0482832i
$552$	− 15294.9i	− 1.17934i
$553$	− 14959.2i	− 1.15032i
$554$	− 28118.0i	− 2.15635i
$555$	−7844.52	−0.599966
$556$	−6422.84	−0.489909
$557$	13922.9i	1.05913i	0.848271	+	0.529563i	$0.177644\pi$
$557$	13922.9i	1.05913i	−0.848271	+	0.529563i	$0.822356\pi$
$558$	2883.50	0.218761
$559$	0	0
$560$	−15690.0	−1.18397
$561$	212.555i	0.0159965i
$562$	35991.2	2.70142
$563$	−20611.9	−1.54296	−0.771481	−	0.636252i	$-0.780484\pi$
$563$	−20611.9	−1.54296	−0.771481	+	0.636252i	$0.780484\pi$
$564$	22973.6i	1.71519i
$565$	− 42539.3i	− 3.16750i
$566$	21609.1i	1.60477i
$567$	1932.61i	0.143143i
$568$	−10209.2	−0.754171
$569$	−2204.50	−0.162421	−0.0812105	−	0.996697i	$-0.525879\pi$
$569$	−2204.50	−0.162421	−0.0812105	+	0.996697i	$0.525879\pi$
$570$	8079.75i	0.593725i
$571$	−1018.86	−0.0746723	−0.0373362	−	0.999303i	$-0.511887\pi$
$571$	−1018.86	−0.0746723	−0.0373362	+	0.999303i	$0.511887\pi$
$572$	0	0
$573$	4878.85	0.355701
$574$	12884.8i	0.936935i
$575$	36597.4	2.65429
$576$	6367.43	0.460607
$577$	20426.8i	1.47379i	0.676006	+	0.736896i	$0.263709\pi$
$577$	20426.8i	1.47379i	−0.676006	+	0.736896i	$0.736291\pi$
$578$	− 23411.2i	− 1.68473i
$579$	15848.7i	1.13756i
$580$	− 5773.70i	− 0.413344i
$581$	−28919.1	−2.06500
$582$	−7022.60	−0.500165
$583$	− 28140.0i	− 1.99904i
$584$	−27780.5	−1.96844
$585$	0	0
$586$	8806.68	0.620820
$587$	11674.0i	0.820846i	0.911895	+	0.410423i	$0.134619\pi$
$587$	11674.0i	0.820846i	−0.911895	+	0.410423i	$0.865381\pi$
$588$	−9991.62	−0.700761
$589$	2016.04	0.141035
$590$	− 36729.1i	− 2.56290i
$591$	6857.29i	0.477278i
$592$	− 4844.99i	− 0.336365i
$593$	21447.4i	1.48523i	0.669721	+	0.742613i	$0.266414\pi$
$593$	21447.4i	1.48523i	−0.669721	+	0.742613i	$0.733586\pi$
$594$	−7751.67	−0.535446
$595$	−528.730	−0.0364299
$596$	− 7206.43i	− 0.495280i
$597$	1701.58	0.116651
$598$	0	0
$599$	11316.8	0.771937	0.385968	−	0.922512i	$-0.373867\pi$
$599$	11316.8	0.771937	0.385968	+	0.922512i	$0.373867\pi$
$600$	− 22090.7i	− 1.50308i
$601$	−9753.39	−0.661979	−0.330989	−	0.943635i	$-0.607382\pi$
$601$	−9753.39	−0.661979	−0.330989	+	0.943635i	$0.607382\pi$
$602$	3748.89	0.253809
$603$	2751.24i	0.185803i
$604$	− 26757.3i	− 1.80255i
$605$	− 43272.9i	− 2.90792i
$606$	10113.2i	0.677924i
$607$	17947.7	1.20012	0.600061	−	0.799954i	$-0.295143\pi$
$607$	17947.7	1.20012	0.600061	+	0.799954i	$0.295143\pi$
$608$	2694.68	0.179743
$609$	− 1490.33i	− 0.0991649i
$610$	−6726.32	−0.446460
$611$	0	0
$612$	−155.830	−0.0102926
$613$	6494.46i	0.427910i	0.976844	+	0.213955i	$0.0686345\pi$
$613$	6494.46i	0.427910i	−0.976844	+	0.213955i	$0.931365\pi$
$614$	1767.13	0.116149
$615$	−6403.24	−0.419843
$616$	− 46028.1i	− 3.01059i
$617$	2746.15i	0.179183i	0.995979	+	0.0895914i	$0.0285561\pi$
$617$	2746.15i	0.179183i	−0.995979	+	0.0895914i	$0.971444\pi$
$618$	− 23685.3i	− 1.54169i
$619$	16319.9i	1.05969i	0.848093	+	0.529847i	$0.177751\pi$
$619$	16319.9i	1.05969i	−0.848093	+	0.529847i	$0.822249\pi$
$620$	−18639.3	−1.20737
$621$	4297.92	0.277729
$622$	− 25206.0i	− 1.62487i
$623$	−16224.1	−1.04335
$624$	0	0
$625$	8494.48	0.543647
$626$	24882.3i	1.58865i
$627$	−5419.67	−0.345201
$628$	−27471.9	−1.74562
$629$	− 163.269i	− 0.0103497i
$630$	− 19282.3i	− 1.21940i
$631$	1381.57i	0.0871621i	0.999050	+	0.0435810i	$0.0138767\pi$
$631$	1381.57i	0.0871621i	−0.999050	+	0.0435810i	$0.986123\pi$
$632$	20080.8i	1.26388i
$633$	9857.84	0.618979
$634$	−38641.8	−2.42060
$635$	2108.48i	0.131768i
$636$	20630.3	1.28623
$637$	0	0
$638$	5977.72	0.370941
$639$	− 2868.82i	− 0.177603i
$640$	−49989.4	−3.08751
$641$	15055.0	0.927672	0.463836	−	0.885921i	$-0.346473\pi$
$641$	15055.0	0.927672	0.463836	+	0.885921i	$0.346473\pi$
$642$	20259.3i	1.24544i
$643$	− 5942.18i	− 0.364443i	−0.983258	−	0.182221i	$-0.941671\pi$
$643$	− 5942.18i	− 0.364443i	0.983258	−	0.182221i	$-0.0583287\pi$
$644$	55904.2i	3.42070i
$645$	1863.05i	0.113733i
$646$	−168.165	−0.0102420
$647$	26854.0	1.63175	0.815874	−	0.578229i	$-0.196256\pi$
$647$	26854.0	1.63175	0.815874	+	0.578229i	$0.196256\pi$
$648$	− 2594.28i	− 0.157273i
$649$	24636.9	1.49011
$650$	0	0
$651$	−4811.26	−0.289659
$652$	− 15375.6i	− 0.923550i
$653$	−6885.76	−0.412650	−0.206325	−	0.978483i	$-0.566150\pi$
$653$	−6885.76	−0.412650	−0.206325	+	0.978483i	$0.566150\pi$
$654$	−12228.0	−0.731120
$655$	− 56410.6i	− 3.36511i
$656$	− 3954.82i	− 0.235380i
$657$	− 7806.40i	− 0.463556i
$658$	− 59166.3i	− 3.50538i
$659$	−14745.5	−0.871627	−0.435813	−	0.900037i	$-0.643539\pi$
$659$	−14745.5	−0.871627	−0.435813	+	0.900037i	$0.643539\pi$
$660$	50107.6	2.95521
$661$	28891.3i	1.70006i	0.526733	+	0.850031i	$0.323417\pi$
$661$	28891.3i	1.70006i	−0.526733	+	0.850031i	$0.676583\pi$
$662$	38757.8	2.27547
$663$	0	0
$664$	38820.2	2.26885
$665$	− 13481.4i	− 0.786147i
$666$	5954.26	0.346431
$667$	−3314.35	−0.192402
$668$	13790.3i	0.798747i
$669$	11403.3i	0.659011i
$670$	− 27450.0i	− 1.58282i
$671$	− 4511.83i	− 0.259579i
$672$	−6430.85	−0.369160
$673$	18362.8	1.05176	0.525880	−	0.850559i	$-0.323736\pi$
$673$	18362.8	1.05176	0.525880	+	0.850559i	$0.323736\pi$
$674$	14819.9i	0.846947i
$675$	6207.54	0.353968
$676$	0	0
$677$	−25937.6	−1.47247	−0.736237	−	0.676724i	$-0.763399\pi$
$677$	−25937.6	−1.47247	−0.736237	+	0.676724i	$0.763399\pi$
$678$	32288.8i	1.82897i
$679$	11717.5	0.662265
$680$	709.754	0.0400262
$681$	9161.95i	0.515546i
$682$	− 19297.9i	− 1.08351i
$683$	34538.1i	1.93494i	0.252989	+	0.967469i	$0.418586\pi$
$683$	34538.1i	1.93494i	−0.252989	+	0.967469i	$0.581414\pi$
$684$	− 3973.32i	− 0.222111i
$685$	10544.3	0.588142
$686$	−13275.4	−0.738857
$687$	− 4014.96i	− 0.222970i
$688$	−1150.67	−0.0637630
$689$	0	0
$690$	−42881.8	−2.36592
$691$	− 10009.5i	− 0.551056i	−0.961293	−	0.275528i	$-0.911147\pi$
$691$	− 10009.5i	− 0.551056i	0.961293	−	0.275528i	$-0.0888527\pi$
$692$	−44003.9	−2.41731
$693$	12934.0	0.708979
$694$	4056.87i	0.221897i
$695$	8220.42i	0.448660i
$696$	2000.59i	0.108954i
$697$	− 133.271i	− 0.00724248i
$698$	−19610.1	−1.06340
$699$	5938.78	0.321352
$700$	80743.1i	4.35972i
$701$	−7115.37	−0.383372	−0.191686	−	0.981456i	$-0.561395\pi$
$701$	−7115.37	−0.383372	−0.191686	+	0.981456i	$0.561395\pi$
$702$	0	0
$703$	4163.00	0.223343
$704$	− 42614.2i	− 2.28137i
$705$	29403.4	1.57077
$706$	−22210.9	−1.18402
$707$	− 16874.4i	− 0.897634i
$708$	18062.0i	0.958773i
$709$	19728.3i	1.04501i	0.852636	+	0.522505i	$0.175002\pi$
$709$	19728.3i	1.04501i	−0.852636	+	0.522505i	$0.824998\pi$
$710$	28623.2i	1.51297i
$711$	−5642.76	−0.297637
$712$	21778.9	1.14635
$713$	10699.7i	0.562004i
$714$	401.324	0.0210353
$715$	0	0
$716$	−6459.96	−0.337179
$717$	− 15979.5i	− 0.832309i
$718$	13400.2	0.696506
$719$	−28248.7	−1.46523	−0.732614	−	0.680644i	$-0.761700\pi$
$719$	−28248.7	−1.46523	−0.732614	+	0.680644i	$0.761700\pi$
$720$	5918.44i	0.306343i
$721$	39520.0i	2.04134i
$722$	28405.5i	1.46419i
$723$	− 7989.12i	− 0.410952i
$724$	14591.2	0.749003
$725$	−4786.96	−0.245218
$726$	32845.6i	1.67908i
$727$	−15082.9	−0.769457	−0.384728	−	0.923030i	$-0.625705\pi$
$727$	−15082.9	−0.769457	−0.384728	+	0.923030i	$0.625705\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	77887.1i	3.94895i
$731$	−38.7759	−0.00196194
$732$	3307.75	0.167019
$733$	− 13245.3i	− 0.667432i	−0.942674	−	0.333716i	$-0.891697\pi$
$733$	− 13245.3i	− 0.667432i	0.942674	−	0.333716i	$-0.108303\pi$
$734$	− 5528.05i	− 0.277989i
$735$	12788.0i	0.641759i
$736$	14301.5i	0.716252i
$737$	18412.7	0.920273
$738$	4860.28	0.242425
$739$	9334.12i	0.464629i	0.972641	+	0.232315i	$0.0746299\pi$
$739$	9334.12i	0.464629i	−0.972641	+	0.232315i	$0.925370\pi$
$740$	−38489.0	−1.91200
$741$	0	0
$742$	−53131.2	−2.62872
$743$	29628.9i	1.46296i	0.681863	+	0.731480i	$0.261170\pi$
$743$	29628.9i	1.46296i	−0.681863	+	0.731480i	$0.738830\pi$
$744$	6458.51	0.318254
$745$	−9223.32	−0.453579
$746$	42311.2i	2.07657i
$747$	10908.6i	0.534303i
$748$	1042.90i	0.0509786i
$749$	− 33803.6i	− 1.64908i
$750$	−28261.2	−1.37594
$751$	−1682.85	−0.0817682	−0.0408841	−	0.999164i	$-0.513017\pi$
$751$	−1682.85	−0.0817682	−0.0408841	+	0.999164i	$0.513017\pi$
$752$	18160.3i	0.880637i
$753$	−6381.93	−0.308859
$754$	0	0
$755$	−34245.9	−1.65078
$756$	9482.30i	0.456174i
$757$	−34195.3	−1.64181	−0.820905	−	0.571064i	$-0.806531\pi$
$757$	−34195.3	−1.64181	−0.820905	+	0.571064i	$0.806531\pi$
$758$	−10754.2	−0.515316
$759$	− 28763.9i	− 1.37558i
$760$	18097.1i	0.863753i
$761$	36265.5i	1.72750i	0.503924	+	0.863748i	$0.331889\pi$
$761$	36265.5i	1.72750i	−0.503924	+	0.863748i	$0.668111\pi$
$762$	− 1600.41i	− 0.0760850i
$763$	20403.0	0.968070
$764$	23938.0	1.13357
$765$	199.443i	0.00942596i
$766$	−49247.1	−2.32294
$767$	0	0
$768$	20963.9	0.984987
$769$	4632.34i	0.217226i	0.994084	+	0.108613i	$0.0346409\pi$
$769$	4632.34i	0.217226i	−0.994084	+	0.108613i	$0.965359\pi$
$770$	−129047.	−6.03965
$771$	1197.56	0.0559390
$772$	77761.4i	3.62525i
$773$	− 243.045i	− 0.0113088i	−0.999984	−	0.00565442i	$-0.998200\pi$
$773$	− 243.045i	− 0.0113088i	0.999984	−	0.00565442i	$-0.00179987\pi$
$774$	− 1414.12i	− 0.0656712i
$775$	15453.8i	0.716279i
$776$	−15729.3	−0.727642
$777$	−9934.97	−0.458706
$778$	− 26444.8i	− 1.21863i
$779$	3398.12	0.156291
$780$	0	0
$781$	−19199.6	−0.879662
$782$	− 892.504i	− 0.0408131i
$783$	−562.170	−0.0256581
$784$	−7898.23	−0.359796
$785$	35160.5i	1.59864i
$786$	42817.6i	1.94307i
$787$	− 17245.4i	− 0.781108i	−0.920580	−	0.390554i	$-0.872284\pi$
$787$	− 17245.4i	− 0.781108i	0.920580	−	0.390554i	$-0.127716\pi$
$788$	33645.2i	1.52102i
$789$	−5703.59	−0.257355
$790$	56299.7	2.53551
$791$	− 53875.3i	− 2.42173i
$792$	−17362.3	−0.778968
$793$	0	0
$794$	72302.3	3.23163
$795$	− 26404.1i	− 1.17794i
$796$	8348.75	0.371751
$797$	36984.6	1.64374	0.821871	−	0.569673i	$-0.192930\pi$
$797$	36984.6	1.64374	0.821871	+	0.569673i	$0.192930\pi$
$798$	10232.9i	0.453935i
$799$	611.975i	0.0270965i
$800$	20655.9i	0.912871i
$801$	6119.92i	0.269958i
$802$	20523.2	0.903615
$803$	−52244.5	−2.29598
$804$	13498.9i	0.592126i
$805$	71550.3	3.13269
$806$	0	0
$807$	10623.1	0.463385
$808$	22651.8i	0.986246i
$809$	26646.8	1.15804	0.579019	−	0.815314i	$-0.303436\pi$
$809$	26646.8	1.15804	0.579019	+	0.815314i	$0.303436\pi$
$810$	−7273.48	−0.315511
$811$	− 18331.3i	− 0.793709i	−0.917881	−	0.396855i	$-0.870102\pi$
$811$	− 18331.3i	− 0.793709i	0.917881	−	0.396855i	$-0.129898\pi$
$812$	− 7312.30i	− 0.316024i
$813$	10266.0i	0.442857i
$814$	− 39849.0i	− 1.71586i
$815$	−19678.8	−0.845790
$816$	−123.181	−0.00528457
$817$	− 988.700i	− 0.0423381i
$818$	−24310.3	−1.03911
$819$	0	0
$820$	−31417.4	−1.33798
$821$	28227.4i	1.19993i	0.800026	+	0.599966i	$0.204819\pi$
$821$	28227.4i	1.19993i	−0.800026	+	0.599966i	$0.795181\pi$
$822$	−8003.49	−0.339603
$823$	−27183.2	−1.15133	−0.575666	−	0.817685i	$-0.695257\pi$
$823$	−27183.2	−1.15133	−0.575666	+	0.817685i	$0.695257\pi$
$824$	− 53050.7i	− 2.24285i
$825$	− 41544.1i	− 1.75319i
$826$	− 46516.9i	− 1.95948i
$827$	− 32419.4i	− 1.36316i	−0.731744	−	0.681580i	$-0.761293\pi$
$827$	− 32419.4i	− 1.36316i	0.731744	−	0.681580i	$-0.238707\pi$
$828$	21087.7	0.885081
$829$	−22920.2	−0.960255	−0.480128	−	0.877199i	$-0.659410\pi$
$829$	−22920.2	−0.960255	−0.480128	+	0.877199i	$0.659410\pi$
$830$	− 108839.i	− 4.55162i
$831$	17697.3	0.738763
$832$	0	0
$833$	−266.158	−0.0110706
$834$	− 6239.59i	− 0.259064i
$835$	17649.9	0.731495
$836$	−26591.5	−1.10010
$837$	1814.86i	0.0749471i
$838$	− 48666.7i	− 2.00616i
$839$	958.186i	0.0394282i	0.999806	+	0.0197141i	$0.00627560\pi$
$839$	958.186i	0.0394282i	−0.999806	+	0.0197141i	$0.993724\pi$
$840$	− 43188.7i	− 1.77399i
$841$	−23955.5	−0.982225
$842$	69545.3	2.84642
$843$	22652.7i	0.925503i
$844$	48367.3	1.97260
$845$	0	0
$846$	−22318.2	−0.906991
$847$	− 54804.4i	− 2.22326i
$848$	16307.9	0.660397
$849$	−13600.6	−0.549791
$850$	− 1289.05i	− 0.0520167i
$851$	22094.3i	0.889993i
$852$	− 14075.8i	− 0.565996i
$853$	− 40410.6i	− 1.62208i	−0.584991	−	0.811040i	$-0.698902\pi$
$853$	− 40410.6i	− 1.62208i	0.584991	−	0.811040i	$-0.301098\pi$
$854$	−8518.79	−0.341343
$855$	−5085.34	−0.203409
$856$	45377.2i	1.81187i
$857$	5170.15	0.206078	0.103039	−	0.994677i	$-0.467143\pi$
$857$	5170.15	0.206078	0.103039	+	0.994677i	$0.467143\pi$
$858$	0	0
$859$	−1674.04	−0.0664931	−0.0332465	−	0.999447i	$-0.510585\pi$
$859$	−1674.04	−0.0664931	−0.0332465	+	0.999447i	$0.510585\pi$
$860$	9141.03i	0.362449i
$861$	−8109.60	−0.320992
$862$	12649.2	0.499806
$863$	− 43213.2i	− 1.70451i	−0.523124	−	0.852257i	$-0.675234\pi$
$863$	− 43213.2i	− 1.70451i	0.523124	−	0.852257i	$-0.324766\pi$
$864$	2425.78i	0.0955172i
$865$	56319.4i	2.21378i
$866$	19452.9i	0.763321i
$867$	14734.8	0.577188
$868$	−23606.4	−0.923101
$869$	37764.3i	1.47418i
$870$	5608.96	0.218577
$871$	0	0
$872$	−27388.5	−1.06364
$873$	− 4419.98i	− 0.171356i
$874$	22756.9	0.880735
$875$	47155.1	1.82187
$876$	− 38301.9i	− 1.47729i
$877$	3665.73i	0.141144i	0.997507	+	0.0705718i	$0.0224824\pi$
$877$	3665.73i	0.141144i	−0.997507	+	0.0705718i	$0.977518\pi$
$878$	− 61976.0i	− 2.38222i
$879$	5542.87i	0.212692i
$880$	39609.3	1.51731
$881$	24997.5	0.955944	0.477972	−	0.878375i	$-0.341372\pi$
$881$	24997.5	0.955944	0.477972	+	0.878375i	$0.341372\pi$
$882$	− 9706.55i	− 0.370563i
$883$	−28259.1	−1.07700	−0.538501	−	0.842625i	$-0.681009\pi$
$883$	−28259.1	−1.07700	−0.538501	+	0.842625i	$0.681009\pi$
$884$	0	0
$885$	23117.1	0.878047
$886$	− 12780.7i	− 0.484623i
$887$	−32725.2	−1.23879	−0.619394	−	0.785081i	$-0.712622\pi$
$887$	−32725.2	−1.23879	−0.619394	+	0.785081i	$0.712622\pi$
$888$	13336.4	0.503989
$889$	2670.36i	0.100744i
$890$	− 61060.5i	− 2.29972i
$891$	− 4878.85i	− 0.183443i
$892$	55950.2i	2.10017i
$893$	−15604.0	−0.584735
$894$	7000.82	0.261904
$895$	8267.92i	0.308789i
$896$	−63310.9	−2.36057
$897$	0	0
$898$	69234.1	2.57280
$899$	− 1399.53i	− 0.0519211i
$900$	30457.2	1.12804
$901$	549.553	0.0203199
$902$	− 32527.5i	− 1.20072i
$903$	2359.53i	0.0869547i
$904$	72320.9i	2.66079i
$905$	− 18674.9i	− 0.685939i
$906$	25993.9	0.953188
$907$	−41341.4	−1.51347	−0.756736	−	0.653721i	$-0.773207\pi$
$907$	−41341.4	−1.51347	−0.756736	+	0.653721i	$0.773207\pi$
$908$	44952.9i	1.64297i
$909$	−6365.21	−0.232256
$910$	0	0
$911$	−12496.5	−0.454475	−0.227237	−	0.973839i	$-0.572969\pi$
$911$	−12496.5	−0.454475	−0.227237	+	0.973839i	$0.572969\pi$
$912$	− 3140.85i	− 0.114039i
$913$	73006.0	2.64638
$914$	−52692.6	−1.90691
$915$	− 4233.51i	− 0.152957i
$916$	− 19699.3i	− 0.710572i
$917$	− 71443.2i	− 2.57281i
$918$	− 151.384i	− 0.00544271i
$919$	−19871.5	−0.713274	−0.356637	−	0.934243i	$-0.616077\pi$
$919$	−19871.5	−0.713274	−0.356637	+	0.934243i	$0.616077\pi$
$920$	−96047.3	−3.44194
$921$	1112.22i	0.0397924i
$922$	10335.0	0.369160
$923$	0	0
$924$	63460.5	2.25941
$925$	31911.1i	1.13430i
$926$	79209.5	2.81100
$927$	14907.4	0.528180
$928$	− 1870.65i	− 0.0661714i
$929$	− 30850.8i	− 1.08954i	−0.838586	−	0.544769i	$-0.816617\pi$
$929$	− 30850.8i	− 1.08954i	0.838586	−	0.544769i	$-0.183383\pi$
$930$	− 18107.5i	− 0.638459i
$931$	− 6786.45i	− 0.238901i
$932$	29138.5	1.02410
$933$	15864.5	0.556678
$934$	− 32687.9i	− 1.14516i
$935$	1334.77	0.0466864
$936$	0	0
$937$	−31212.1	−1.08821	−0.544106	−	0.839017i	$-0.683131\pi$
$937$	−31212.1	−1.08821	−0.544106	+	0.839017i	$0.683131\pi$
$938$	− 34765.1i	− 1.21015i
$939$	−15660.8	−0.544270
$940$	144267.	5.00582
$941$	1150.30i	0.0398500i	0.999801	+	0.0199250i	$0.00634275\pi$
$941$	1150.30i	0.0398500i	−0.999801	+	0.0199250i	$0.993657\pi$
$942$	− 26688.1i	− 0.923083i
$943$	18034.9i	0.622797i
$944$	14277.7i	0.492268i
$945$	12136.1	0.417766
$946$	−9464.04	−0.325267
$947$	− 39077.9i	− 1.34093i	−0.741941	−	0.670465i	$-0.766095\pi$
$947$	− 39077.9i	− 1.34093i	0.741941	−	0.670465i	$-0.233905\pi$
$948$	−27686.1	−0.948526
$949$	0	0
$950$	32868.1	1.12251
$951$	− 24320.9i	− 0.829295i
$952$	898.892	0.0306022
$953$	−51475.3	−1.74968	−0.874841	−	0.484409i	$-0.839035\pi$
$953$	−51475.3	−1.74968	−0.874841	+	0.484409i	$0.839035\pi$
$954$	20041.7i	0.680160i
$955$	− 30637.6i	− 1.03812i
$956$	− 78403.1i	− 2.65245i
$957$	3762.34i	0.127084i
$958$	−82913.6	−2.79626
$959$	13354.2	0.449666
$960$	− 39985.4i	− 1.34429i
$961$	25272.9	0.848339
$962$	0	0
$963$	−12751.1	−0.426686
$964$	− 39198.5i	− 1.30965i
$965$	99524.8	3.32002
$966$	−54309.1	−1.80887
$967$	19803.2i	0.658562i	0.944232	+	0.329281i	$0.106806\pi$
$967$	19803.2i	0.658562i	−0.944232	+	0.329281i	$0.893194\pi$
$968$	73568.1i	2.44274i
$969$	− 105.842i	− 0.00350891i
$970$	44099.6i	1.45975i
$971$	−9889.70	−0.326854	−0.163427	−	0.986555i	$-0.552255\pi$
$971$	−9889.70	−0.326854	−0.163427	+	0.986555i	$0.552255\pi$
$972$	3576.83	0.118032
$973$	10411.0i	0.343024i
$974$	55761.7	1.83442
$975$	0	0
$976$	2614.73	0.0857535
$977$	28202.9i	0.923532i	0.887002	+	0.461766i	$0.152784\pi$
$977$	28202.9i	0.923532i	−0.887002	+	0.461766i	$0.847216\pi$
$978$	14936.9	0.488374
$979$	40957.7	1.33709
$980$	62744.2i	2.04519i
$981$	− 7696.22i	− 0.250481i
$982$	− 31461.1i	− 1.02237i
$983$	− 5757.36i	− 0.186807i	−0.995628	−	0.0934035i	$-0.970225\pi$
$983$	− 5757.36i	− 0.186807i	0.995628	−	0.0934035i	$-0.0297746\pi$
$984$	10886.1	0.352680
$985$	43061.6	1.39295
$986$	116.740i	0.00377055i
$987$	37238.9	1.20094
$988$	0	0
$989$	5247.34	0.168712
$990$	48677.9i	1.56271i
$991$	24311.1	0.779281	0.389641	−	0.920967i	$-0.372599\pi$
$991$	24311.1	0.779281	0.389641	+	0.920967i	$0.372599\pi$
$992$	−6039.04	−0.193286
$993$	24393.9i	0.779574i
$994$	36250.8i	1.15675i
$995$	− 10685.3i	− 0.340450i
$996$	53522.8i	1.70274i
$997$	−7390.90	−0.234776	−0.117388	−	0.993086i	$-0.537452\pi$
$997$	−7390.90	−0.234776	−0.117388	+	0.993086i	$0.537452\pi$
$998$	−100807.	−3.19737
$999$	3747.58i	0.118687i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.j.337.17		18
13.5	odd	4		507.4.a.q.1.8	yes	9
13.8	odd	4		507.4.a.n.1.2	✓	9
13.12	even	2	inner	507.4.b.j.337.2		18
39.5	even	4		1521.4.a.be.1.2		9
39.8	even	4		1521.4.a.bj.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.2	✓	9	13.8	odd	4
507.4.a.q.1.8	yes	9	13.5	odd	4
507.4.b.j.337.2		18	13.12	even	2	inner
507.4.b.j.337.17		18	1.1	even	1	trivial
1521.4.a.be.1.2		9	39.5	even	4
1521.4.a.bj.1.8		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)

\(f(q)\)	\(=\)	\(q+4.76649i q^{2} -3.00000 q^{3} -14.7194 q^{4} +18.8390i q^{5} -14.2995i q^{6} +23.8593i q^{7} -32.0282i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.76649i q^{2} -3.00000 q^{3} -14.7194 q^{4} +18.8390i q^{5} -14.2995i q^{6} +23.8593i q^{7} -32.0282i q^{8} +9.00000 q^{9} -89.7961 q^{10} -60.2327i q^{11} +44.1583 q^{12} -113.725 q^{14} -56.5171i q^{15} +34.9065 q^{16} +1.17630 q^{17} +42.8984i q^{18} +29.9930i q^{19} -277.300i q^{20} -71.5780i q^{21} +287.099 q^{22} -159.182 q^{23} +96.0845i q^{24} -229.909 q^{25} -27.0000 q^{27} -351.196i q^{28} +20.8211 q^{29} +269.388 q^{30} -67.2170i q^{31} -89.8439i q^{32} +180.698i q^{33} +5.60681i q^{34} -449.487 q^{35} -132.475 q^{36} -138.799i q^{37} -142.961 q^{38} +603.380 q^{40} -113.297i q^{41} +341.176 q^{42} -32.9644 q^{43} +886.592i q^{44} +169.551i q^{45} -758.740i q^{46} +520.256i q^{47} -104.720 q^{48} -226.268 q^{49} -1095.86i q^{50} -3.52889 q^{51} +467.189 q^{53} -128.695i q^{54} +1134.73 q^{55} +764.171 q^{56} -89.9789i q^{57} +99.2437i q^{58} +409.028i q^{59} +831.900i q^{60} +74.9067 q^{61} +320.389 q^{62} +214.734i q^{63} +707.492 q^{64} -861.296 q^{66} +305.693i q^{67} -17.3144 q^{68} +477.546 q^{69} -2142.47i q^{70} -318.757i q^{71} -288.254i q^{72} -867.378i q^{73} +661.585 q^{74} +689.727 q^{75} -441.480i q^{76} +1437.11 q^{77} -626.973 q^{79} +657.605i q^{80} +81.0000 q^{81} +540.031 q^{82} +1212.06i q^{83} +1053.59i q^{84} +22.1603i q^{85} -157.125i q^{86} -62.4634 q^{87} -1929.14 q^{88} +679.991i q^{89} -808.165 q^{90} +2343.07 q^{92} +201.651i q^{93} -2479.80 q^{94} -565.038 q^{95} +269.532i q^{96} -491.109i q^{97} -1078.51i q^{98} -542.094i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * q^95

Introduction

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