LMFDB - Embedded newform 289.4.b.d.288.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(288,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("289.288");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	289.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:	$17.0515519917$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 43x^{6} + 505x^{4} + 1528x^{2} + 1296 $ x^8 + 43x^6 + 505x^4 + 1528*x^2 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.1
Root			$-5.04171i$ of defining polynomial
Character	$\chi$	$=$	289.288
Dual form			289.4.b.d.288.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.04171 q^{2} -2.60975i q^{3} +17.4188 q^{4} -13.4166i q^{5} +13.1576i q^{6} +22.2015i q^{7} -47.4869 q^{8} +20.1892 q^{9} +O(q^{10})$	\(q-5.04171 q^{2} -2.60975i q^{3} +17.4188 q^{4} -13.4166i q^{5} +13.1576i q^{6} +22.2015i q^{7} -47.4869 q^{8} +20.1892 q^{9} +67.6428i q^{10} -33.6445i q^{11} -45.4588i q^{12} -73.4255 q^{13} -111.933i q^{14} -35.0141 q^{15} +100.065 q^{16} -101.788 q^{18} +42.5131 q^{19} -233.702i q^{20} +57.9404 q^{21} +169.626i q^{22} -59.2553i q^{23} +123.929i q^{24} -55.0063 q^{25} +370.190 q^{26} -123.152i q^{27} +386.724i q^{28} +21.3559i q^{29} +176.531 q^{30} -42.2667i q^{31} -124.601 q^{32} -87.8038 q^{33} +297.869 q^{35} +351.672 q^{36} -265.496i q^{37} -214.339 q^{38} +191.622i q^{39} +637.115i q^{40} +80.0523i q^{41} -292.118 q^{42} -353.946 q^{43} -586.048i q^{44} -270.871i q^{45} +298.748i q^{46} +52.4119 q^{47} -261.144i q^{48} -149.906 q^{49} +277.325 q^{50} -1278.98 q^{52} -551.066 q^{53} +620.897i q^{54} -451.396 q^{55} -1054.28i q^{56} -110.949i q^{57} -107.670i q^{58} -508.032 q^{59} -609.904 q^{60} +671.496i q^{61} +213.096i q^{62} +448.230i q^{63} -172.314 q^{64} +985.123i q^{65} +442.681 q^{66} -859.787 q^{67} -154.642 q^{69} -1501.77 q^{70} -147.724i q^{71} -958.723 q^{72} -522.518i q^{73} +1338.55i q^{74} +143.553i q^{75} +740.528 q^{76} +746.958 q^{77} -966.103i q^{78} -245.472i q^{79} -1342.53i q^{80} +223.712 q^{81} -403.600i q^{82} +293.554 q^{83} +1009.25 q^{84} +1784.49 q^{86} +55.7336 q^{87} +1597.67i q^{88} -72.0418 q^{89} +1365.65i q^{90} -1630.15i q^{91} -1032.16i q^{92} -110.305 q^{93} -264.245 q^{94} -570.383i q^{95} +325.178i q^{96} +1386.68i q^{97} +755.782 q^{98} -679.256i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9	$ 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9} - 44 q^{13} - 108 q^{15} + 126 q^{16} - 668 q^{18} - 44 q^{19} - 704 q^{21} - 756 q^{25} + 896 q^{26} + 626 q^{30} - 662 q^{32} + 188 q^{33} - 484 q^{35} - 282 q^{36} - 1048 q^{38} - 2910 q^{42} + 228 q^{43} + 20 q^{47} - 2012 q^{49} + 1610 q^{50} - 3074 q^{52} - 100 q^{53} - 2632 q^{55} - 1992 q^{59} + 434 q^{60} - 300 q^{64} + 2180 q^{66} - 1736 q^{67} - 2256 q^{69} - 2104 q^{70} - 78 q^{72} + 1746 q^{76} + 1788 q^{77} + 2160 q^{81} - 1700 q^{83} + 886 q^{84} + 4822 q^{86} + 768 q^{87} + 1568 q^{89} + 3100 q^{93} - 2238 q^{94} - 3754 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9 - 44 * q^13 - 108 * q^15 + 126 * q^16 - 668 * q^18 - 44 * q^19 - 704 * q^21 - 756 * q^25 + 896 * q^26 + 626 * q^30 - 662 * q^32 + 188 * q^33 - 484 * q^35 - 282 * q^36 - 1048 * q^38 - 2910 * q^42 + 228 * q^43 + 20 * q^47 - 2012 * q^49 + 1610 * q^50 - 3074 * q^52 - 100 * q^53 - 2632 * q^55 - 1992 * q^59 + 434 * q^60 - 300 * q^64 + 2180 * q^66 - 1736 * q^67 - 2256 * q^69 - 2104 * q^70 - 78 * q^72 + 1746 * q^76 + 1788 * q^77 + 2160 * q^81 - 1700 * q^83 + 886 * q^84 + 4822 * q^86 + 768 * q^87 + 1568 * q^89 + 3100 * q^93 - 2238 * q^94 - 3754 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−5.04171	−1.78251	−0.891256	−	0.453500i	$-0.850175\pi$
$2$	−5.04171	−1.78251	−0.891256	+	0.453500i	$0.850175\pi$
$3$	− 2.60975i	− 0.502247i	−0.967955	−	0.251123i	$-0.919200\pi$
$3$	− 2.60975i	− 0.502247i	0.967955	−	0.251123i	$-0.0808000\pi$
$4$	17.4188	2.17735
$5$	− 13.4166i	− 1.20002i	−0.799992	−	0.600010i	$-0.795163\pi$
$5$	− 13.4166i	− 1.20002i	0.799992	−	0.600010i	$-0.204837\pi$
$6$	13.1576i	0.895262i
$7$	22.2015i	1.19877i	0.800462	+	0.599384i	$0.204588\pi$
$7$	22.2015i	1.19877i	−0.800462	+	0.599384i	$0.795412\pi$
$8$	−47.4869	−2.09865
$9$	20.1892	0.747748
$10$	67.6428i	2.13905i
$11$	− 33.6445i	− 0.922200i	−0.887348	−	0.461100i	$-0.847455\pi$
$11$	− 33.6445i	− 0.922200i	0.887348	−	0.461100i	$-0.152545\pi$
$12$	− 45.4588i	− 1.09357i
$13$	−73.4255	−1.56650	−0.783252	−	0.621704i	$-0.786441\pi$
$13$	−73.4255	−1.56650	−0.783252	+	0.621704i	$0.786441\pi$
$14$	− 111.933i	− 2.13682i
$15$	−35.0141	−0.602707
$16$	100.065	1.56351
$17$	0	0
$17$	0	0
$18$	−101.788	−1.33287
$19$	42.5131	0.513325	0.256663	−	0.966501i	$-0.417377\pi$
$19$	42.5131	0.513325	0.256663	+	0.966501i	$0.417377\pi$
$20$	− 233.702i	− 2.61287i
$21$	57.9404	0.602077
$22$	169.626i	1.64383i
$23$	− 59.2553i	− 0.537199i	−0.963252	−	0.268599i	$-0.913439\pi$
$23$	− 59.2553i	− 0.537199i	0.963252	−	0.268599i	$-0.0865608\pi$
$24$	123.929i	1.05404i
$25$	−55.0063	−0.440050
$26$	370.190	2.79231
$27$	− 123.152i	− 0.877801i
$28$	386.724i	2.61014i
$29$	21.3559i	0.136748i	0.997660	+	0.0683740i	$0.0217811\pi$
$29$	21.3559i	0.136748i	−0.997660	+	0.0683740i	$0.978219\pi$
$30$	176.531	1.07433
$31$	− 42.2667i	− 0.244881i	−0.992476	−	0.122441i	$-0.960928\pi$
$31$	− 42.2667i	− 0.244881i	0.992476	−	0.122441i	$-0.0390721\pi$
$32$	−124.601	−0.688331
$33$	−87.8038	−0.463172
$34$	0	0
$35$	297.869	1.43855
$36$	351.672	1.62811
$37$	− 265.496i	− 1.17965i	−0.807530	−	0.589827i	$-0.799196\pi$
$37$	− 265.496i	− 1.17965i	0.807530	−	0.589827i	$-0.200804\pi$
$38$	−214.339	−0.915008
$39$	191.622i	0.786772i
$40$	637.115i	2.51842i
$41$	80.0523i	0.304929i	0.988309	+	0.152464i	$0.0487209\pi$
$41$	80.0523i	0.304929i	−0.988309	+	0.152464i	$0.951279\pi$
$42$	−292.118	−1.07321
$43$	−353.946	−1.25526	−0.627632	−	0.778510i	$-0.715976\pi$
$43$	−353.946	−1.25526	−0.627632	+	0.778510i	$0.715976\pi$
$44$	− 586.048i	− 2.00795i
$45$	− 270.871i	− 0.897313i
$46$	298.748i	0.957564i
$47$	52.4119	0.162661	0.0813304	−	0.996687i	$-0.474083\pi$
$47$	52.4119	0.162661	0.0813304	+	0.996687i	$0.474083\pi$
$48$	− 261.144i	− 0.785268i
$49$	−149.906	−0.437044
$50$	277.325	0.784395
$51$	0	0
$52$	−1278.98	−3.41083
$53$	−551.066	−1.42820	−0.714101	−	0.700043i	$-0.753164\pi$
$53$	−551.066	−1.42820	−0.714101	+	0.700043i	$0.753164\pi$
$54$	620.897i	1.56469i
$55$	−451.396	−1.10666
$56$	− 1054.28i	− 2.51579i
$57$	− 110.949i	− 0.257816i
$58$	− 107.670i	− 0.243755i
$59$	−508.032	−1.12102	−0.560510	−	0.828148i	$-0.689395\pi$
$59$	−508.032	−1.12102	−0.560510	+	0.828148i	$0.689395\pi$
$60$	−609.904	−1.31230
$61$	671.496i	1.40945i	0.709482	+	0.704723i	$0.248929\pi$
$61$	671.496i	1.40945i	−0.709482	+	0.704723i	$0.751071\pi$
$62$	213.096i	0.436504i
$63$	448.230i	0.896376i
$64$	−172.314	−0.336551
$65$	985.123i	1.87984i
$66$	442.681	0.825611
$67$	−859.787	−1.56776	−0.783878	−	0.620915i	$-0.786761\pi$
$67$	−859.787	−1.56776	−0.783878	+	0.620915i	$0.786761\pi$
$68$	0	0
$69$	−154.642	−0.269807
$70$	−1501.77	−2.56423
$71$	− 147.724i	− 0.246924i	−0.992349	−	0.123462i	$-0.960600\pi$
$71$	− 147.724i	− 0.246924i	0.992349	−	0.123462i	$-0.0393997\pi$
$72$	−958.723	−1.56926
$73$	− 522.518i	− 0.837755i	−0.908043	−	0.418878i	$-0.862424\pi$
$73$	− 522.518i	− 0.837755i	0.908043	−	0.418878i	$-0.137576\pi$
$74$	1338.55i	2.10275i
$75$	143.553i	0.221014i
$76$	740.528	1.11769
$77$	746.958	1.10550
$78$	− 966.103i	− 1.40243i
$79$	− 245.472i	− 0.349592i	−0.984605	−	0.174796i	$-0.944073\pi$
$79$	− 245.472i	− 0.349592i	0.984605	−	0.174796i	$-0.0559266\pi$
$80$	− 1342.53i	− 1.87624i
$81$	223.712	0.306875
$82$	− 403.600i	− 0.543539i
$83$	293.554	0.388213	0.194107	−	0.980980i	$-0.437819\pi$
$83$	293.554	0.388213	0.194107	+	0.980980i	$0.437819\pi$
$84$	1009.25	1.31093
$85$	0	0
$86$	1784.49	2.23752
$87$	55.7336	0.0686813
$88$	1597.67i	1.93537i
$89$	−72.0418	−0.0858024	−0.0429012	−	0.999079i	$-0.513660\pi$
$89$	−72.0418	−0.0858024	−0.0429012	+	0.999079i	$0.513660\pi$
$90$	1365.65i	1.59947i
$91$	− 1630.15i	− 1.87787i
$92$	− 1032.16i	− 1.16967i
$93$	−110.305	−0.122991
$94$	−264.245	−0.289945
$95$	− 570.383i	− 0.616001i
$96$	325.178i	0.345712i
$97$	1386.68i	1.45151i	0.687955	+	0.725754i	$0.258509\pi$
$97$	1386.68i	1.45151i	−0.687955	+	0.725754i	$0.741491\pi$
$98$	755.782	0.779036
$99$	− 679.256i	− 0.689574i
$100$	−958.144	−0.958144
$101$	−1455.68	−1.43412	−0.717059	−	0.697012i	$-0.754512\pi$
$101$	−1455.68	−1.43412	−0.717059	+	0.697012i	$0.754512\pi$
$102$	0	0
$103$	−349.197	−0.334052	−0.167026	−	0.985952i	$-0.553416\pi$
$103$	−349.197	−0.334052	−0.167026	+	0.985952i	$0.553416\pi$
$104$	3486.75	3.28754
$105$	− 777.365i	− 0.722505i
$106$	2778.31	2.54579
$107$	1213.83i	1.09669i	0.836253	+	0.548344i	$0.184742\pi$
$107$	1213.83i	1.09669i	−0.836253	+	0.548344i	$0.815258\pi$
$108$	− 2145.16i	− 1.91128i
$109$	981.173i	0.862196i	0.902305	+	0.431098i	$0.141874\pi$
$109$	981.173i	0.862196i	−0.902305	+	0.431098i	$0.858126\pi$
$110$	2275.81	1.97264
$111$	−692.877	−0.592478
$112$	2221.58i	1.87428i
$113$	1354.85i	1.12791i	0.825807	+	0.563953i	$0.190720\pi$
$113$	1354.85i	1.12791i	−0.825807	+	0.563953i	$0.809280\pi$
$114$	559.371i	0.459560i
$115$	−795.007	−0.644650
$116$	371.995i	0.297749i
$117$	−1482.40	−1.17135
$118$	2561.35	1.99823
$119$	0	0
$120$	1662.71	1.26487
$121$	199.046	0.149546
$122$	− 3385.48i	− 2.51236i
$123$	208.917	0.153149
$124$	− 736.235i	− 0.533193i
$125$	− 939.081i	− 0.671952i
$126$	− 2259.85i	− 1.59780i
$127$	−2377.49	−1.66116	−0.830582	−	0.556897i	$-0.811992\pi$
$127$	−2377.49	−1.66116	−0.830582	+	0.556897i	$0.811992\pi$
$128$	1865.57	1.28824
$129$	923.712i	0.630452i
$130$	− 4966.70i	− 3.35084i
$131$	− 2397.58i	− 1.59907i	−0.600622	−	0.799533i	$-0.705080\pi$
$131$	− 2397.58i	− 1.59907i	0.600622	−	0.799533i	$-0.294920\pi$
$132$	−1529.44	−1.00849
$133$	943.854i	0.615357i
$134$	4334.79	2.79455
$135$	−1652.29	−1.05338
$136$	0	0
$137$	1595.27	0.994838	0.497419	−	0.867510i	$-0.334281\pi$
$137$	1595.27	0.994838	0.497419	+	0.867510i	$0.334281\pi$
$138$	779.657	0.480934
$139$	362.559i	0.221236i	0.993863	+	0.110618i	$0.0352831\pi$
$139$	362.559i	0.221236i	−0.993863	+	0.110618i	$0.964717\pi$
$140$	5188.53	3.13222
$141$	− 136.782i	− 0.0816959i
$142$	744.781i	0.440145i
$143$	2470.36i	1.44463i
$144$	2020.22	1.16911
$145$	286.525	0.164101
$146$	2634.38i	1.49331i
$147$	391.218i	0.219504i
$148$	− 4624.62i	− 2.56852i
$149$	−520.649	−0.286263	−0.143132	−	0.989704i	$-0.545717\pi$
$149$	−520.649	−0.286263	−0.143132	+	0.989704i	$0.545717\pi$
$150$	− 723.751i	− 0.393960i
$151$	1157.02	0.623555	0.311778	−	0.950155i	$-0.399076\pi$
$151$	1157.02	0.623555	0.311778	+	0.950155i	$0.399076\pi$
$152$	−2018.82	−1.07729
$153$	0	0
$154$	−3765.95	−1.97058
$155$	−567.077	−0.293863
$156$	3337.83i	1.71308i
$157$	−2287.62	−1.16288	−0.581440	−	0.813590i	$-0.697510\pi$
$157$	−2287.62	−1.16288	−0.581440	+	0.813590i	$0.697510\pi$
$158$	1237.60i	0.623152i
$159$	1438.14i	0.717310i
$160$	1671.73i	0.826012i
$161$	1315.55	0.643977
$162$	−1127.89	−0.547009
$163$	3471.88i	1.66834i	0.551510	+	0.834168i	$0.314052\pi$
$163$	3471.88i	1.66834i	−0.551510	+	0.834168i	$0.685948\pi$
$164$	1394.42i	0.663937i
$165$	1178.03i	0.555816i
$166$	−1480.01	−0.691995
$167$	− 611.808i	− 0.283492i	−0.989903	−	0.141746i	$-0.954728\pi$
$167$	− 611.808i	− 0.283492i	0.989903	−	0.141746i	$-0.0452716\pi$
$168$	−2751.41	−1.26355
$169$	3194.30	1.45394
$170$	0	0
$171$	858.305	0.383838
$172$	−6165.33	−2.73315
$173$	1074.13i	0.472049i	0.971747	+	0.236024i	$0.0758446\pi$
$173$	1074.13i	0.472049i	−0.971747	+	0.236024i	$0.924155\pi$
$174$	−280.993	−0.122425
$175$	− 1221.22i	− 0.527518i
$176$	− 3366.63i	− 1.44187i
$177$	1325.84i	0.563029i
$178$	363.214	0.152944
$179$	−1326.09	−0.553724	−0.276862	−	0.960910i	$-0.589294\pi$
$179$	−1326.09	−0.553724	−0.276862	+	0.960910i	$0.589294\pi$
$180$	− 4718.26i	− 1.95377i
$181$	− 2450.13i	− 1.00617i	−0.864237	−	0.503085i	$-0.832198\pi$
$181$	− 2450.13i	− 1.00617i	0.864237	−	0.503085i	$-0.167802\pi$
$182$	8218.76i	3.34734i
$183$	1752.44	0.707890
$184$	2813.85i	1.12739i
$185$	−3562.06	−1.41561
$186$	556.128	0.219233
$187$	0	0
$188$	912.953	0.354170
$189$	2734.16	1.05228
$190$	2875.70i	1.09803i
$191$	−3177.92	−1.20391	−0.601953	−	0.798532i	$-0.705610\pi$
$191$	−3177.92	−1.20391	−0.601953	+	0.798532i	$0.705610\pi$
$192$	449.697i	0.169032i
$193$	504.377i	0.188113i	0.995567	+	0.0940566i	$0.0299835\pi$
$193$	504.377i	0.188113i	−0.995567	+	0.0940566i	$0.970017\pi$
$194$	− 6991.24i	− 2.58733i
$195$	2570.93	0.944143
$196$	−2611.19	−0.951598
$197$	− 4707.42i	− 1.70249i	−0.524770	−	0.851244i	$-0.675849\pi$
$197$	− 4707.42i	− 1.70249i	0.524770	−	0.851244i	$-0.324151\pi$
$198$	3424.61i	1.22917i
$199$	− 3864.28i	− 1.37654i	−0.725455	−	0.688269i	$-0.758371\pi$
$199$	− 3864.28i	− 1.37654i	0.725455	−	0.688269i	$-0.241629\pi$
$200$	2612.08	0.923509
$201$	2243.83i	0.787401i
$202$	7339.13	2.55634
$203$	−474.133	−0.163929
$204$	0	0
$205$	1074.03	0.365921
$206$	1760.55	0.595452
$207$	− 1196.32i	− 0.401689i
$208$	−7347.29	−2.44924
$209$	− 1430.33i	− 0.473389i
$210$	3919.25i	1.28788i
$211$	− 2431.92i	− 0.793462i	−0.917935	−	0.396731i	$-0.870145\pi$
$211$	− 2431.92i	− 0.793462i	0.917935	−	0.396731i	$-0.129855\pi$
$212$	−9598.91	−3.10970
$213$	−385.523	−0.124017
$214$	− 6119.79i	− 1.95486i
$215$	4748.77i	1.50634i
$216$	5848.11i	1.84219i
$217$	938.383	0.293556
$218$	− 4946.79i	− 1.53688i
$219$	−1363.64	−0.420760
$220$	−7862.79	−2.40959
$221$	0	0
$222$	3493.29	1.05610
$223$	−584.515	−0.175525	−0.0877624	−	0.996141i	$-0.527972\pi$
$223$	−584.515	−0.175525	−0.0877624	+	0.996141i	$0.527972\pi$
$224$	− 2766.33i	− 0.825149i
$225$	−1110.53	−0.329047
$226$	− 6830.75i	− 2.01051i
$227$	− 3235.58i	− 0.946048i	−0.881049	−	0.473024i	$-0.843162\pi$
$227$	− 3235.58i	− 0.946048i	0.881049	−	0.473024i	$-0.156838\pi$
$228$	− 1932.59i	− 0.561356i
$229$	3039.00	0.876955	0.438477	−	0.898742i	$-0.355518\pi$
$229$	3039.00	0.876955	0.438477	+	0.898742i	$0.355518\pi$
$230$	4008.19	1.14910
$231$	− 1949.38i	− 0.555236i
$232$	− 1014.13i	− 0.286986i
$233$	− 5387.38i	− 1.51476i	−0.652974	−	0.757380i	$-0.726479\pi$
$233$	− 5387.38i	− 1.51476i	0.652974	−	0.757380i	$-0.273521\pi$
$234$	7473.83	2.08795
$235$	− 703.191i	− 0.195196i
$236$	−8849.32	−2.44085
$237$	−640.621	−0.175582
$238$	0	0
$239$	−6534.43	−1.76852	−0.884262	−	0.466991i	$-0.845338\pi$
$239$	−6534.43	−1.76852	−0.884262	+	0.466991i	$0.845338\pi$
$240$	−3503.67	−0.942338
$241$	2242.97i	0.599511i	0.954016	+	0.299756i	$0.0969052\pi$
$241$	2242.97i	0.599511i	−0.954016	+	0.299756i	$0.903095\pi$
$242$	−1003.53	−0.266568
$243$	− 3908.94i	− 1.03193i
$244$	11696.7i	3.06886i
$245$	2011.24i	0.524462i
$246$	−1053.30	−0.272991
$247$	−3121.54	−0.804126
$248$	2007.11i	0.513919i
$249$	− 766.102i	− 0.194979i
$250$	4734.57i	1.19776i
$251$	1846.41	0.464319	0.232160	−	0.972678i	$-0.425421\pi$
$251$	1846.41	0.464319	0.232160	+	0.972678i	$0.425421\pi$
$252$	7807.64i	1.95173i
$253$	−1993.61	−0.495405
$254$	11986.6	2.96105
$255$	0	0
$256$	−8027.13	−1.95975
$257$	1556.43	0.377772	0.188886	−	0.981999i	$-0.439512\pi$
$257$	1556.43	0.377772	0.188886	+	0.981999i	$0.439512\pi$
$258$	− 4657.09i	− 1.12379i
$259$	5894.40	1.41413
$260$	17159.7i	4.09307i
$261$	431.159i	0.102253i
$262$	12087.9i	2.85036i
$263$	2926.03	0.686033	0.343017	−	0.939329i	$-0.388551\pi$
$263$	2926.03	0.686033	0.343017	+	0.939329i	$0.388551\pi$
$264$	4169.53	0.972034
$265$	7393.45i	1.71387i
$266$	− 4758.64i	− 1.09688i
$267$	188.011i	0.0430940i
$268$	−14976.5	−3.41356
$269$	− 6835.44i	− 1.54931i	−0.632384	−	0.774655i	$-0.717923\pi$
$269$	− 6835.44i	− 1.54931i	0.632384	−	0.774655i	$-0.282077\pi$
$270$	8330.35	1.87766
$271$	−1404.62	−0.314850	−0.157425	−	0.987531i	$-0.550319\pi$
$271$	−1404.62	−0.314850	−0.157425	+	0.987531i	$0.550319\pi$
$272$	0	0
$273$	−4254.30	−0.943157
$274$	−8042.87	−1.77331
$275$	1850.66i	0.405814i
$276$	−2693.67	−0.587464
$277$	− 2147.18i	− 0.465745i	−0.972507	−	0.232872i	$-0.925188\pi$
$277$	− 2147.18i	− 0.465745i	0.972507	−	0.232872i	$-0.0748124\pi$
$278$	− 1827.92i	− 0.394357i
$279$	− 853.330i	− 0.183109i
$280$	−14144.9	−3.01900
$281$	437.112	0.0927968	0.0463984	−	0.998923i	$-0.485226\pi$
$281$	437.112	0.0927968	0.0463984	+	0.998923i	$0.485226\pi$
$282$	689.615i	0.145624i
$283$	6747.75i	1.41736i	0.705532	+	0.708678i	$0.250708\pi$
$283$	6747.75i	1.41736i	−0.705532	+	0.708678i	$0.749292\pi$
$284$	− 2573.18i	− 0.537641i
$285$	−1488.56	−0.309384
$286$	− 12454.9i	− 2.57507i
$287$	−1777.28	−0.365539
$288$	−2515.60	−0.514698
$289$	0	0
$290$	−1444.57	−0.292511
$291$	3618.89	0.729015
$292$	− 9101.65i	− 1.82409i
$293$	1219.13	0.243081	0.121540	−	0.992586i	$-0.461217\pi$
$293$	1219.13	0.243081	0.121540	+	0.992586i	$0.461217\pi$
$294$	− 1972.40i	− 0.391269i
$295$	6816.09i	1.34525i
$296$	12607.6i	2.47567i
$297$	−4143.39	−0.809509
$298$	2624.96	0.510268
$299$	4350.84i	0.841524i
$300$	2500.52i	0.481225i
$301$	− 7858.14i	− 1.50477i
$302$	−5833.35	−1.11150
$303$	3798.97i	0.720282i
$304$	4254.06	0.802589
$305$	9009.22	1.69136
$306$	0	0
$307$	9237.79	1.71736	0.858679	−	0.512514i	$-0.171286\pi$
$307$	9237.79	1.71736	0.858679	+	0.512514i	$0.171286\pi$
$308$	13011.1	2.40707
$309$	911.316i	0.167777i
$310$	2859.03	0.523814
$311$	741.272i	0.135157i	0.997714	+	0.0675783i	$0.0215272\pi$
$311$	741.272i	0.135157i	−0.997714	+	0.0675783i	$0.978473\pi$
$312$	− 9099.55i	− 1.65116i
$313$	− 188.399i	− 0.0340221i	−0.999855	−	0.0170110i	$-0.994585\pi$
$313$	− 188.399i	− 0.0340221i	0.999855	−	0.0170110i	$-0.00541505\pi$
$314$	11533.5	2.07285
$315$	6013.74	1.07567
$316$	− 4275.83i	− 0.761185i
$317$	− 6282.92i	− 1.11320i	−0.830781	−	0.556599i	$-0.812106\pi$
$317$	− 6282.92i	− 1.11320i	0.830781	−	0.556599i	$-0.187894\pi$
$318$	− 7250.70i	− 1.27861i
$319$	718.510	0.126109
$320$	2311.87i	0.403868i
$321$	3167.80	0.550808
$322$	−6632.64	−1.14790
$323$	0	0
$324$	3896.80	0.668175
$325$	4038.86	0.689340
$326$	− 17504.2i	− 2.97383i
$327$	2560.62	0.433035
$328$	− 3801.44i	− 0.639937i
$329$	1163.62i	0.194993i
$330$	− 5939.30i	− 0.990750i
$331$	252.673	0.0419582	0.0209791	−	0.999780i	$-0.493322\pi$
$331$	252.673	0.0419582	0.0209791	+	0.999780i	$0.493322\pi$
$332$	5113.36	0.845277
$333$	− 5360.14i	− 0.882084i
$334$	3084.56i	0.505328i
$335$	11535.5i	1.88134i
$336$	5797.78	0.941354
$337$	5809.35i	0.939038i	0.882922	+	0.469519i	$0.155573\pi$
$337$	5809.35i	0.939038i	−0.882922	+	0.469519i	$0.844427\pi$
$338$	−16104.7	−2.59166
$339$	3535.82	0.566487
$340$	0	0
$341$	−1422.04	−0.225830
$342$	−4327.33	−0.684196
$343$	4286.97i	0.674854i
$344$	16807.8	2.63435
$345$	2074.77i	0.323773i
$346$	− 5415.44i	− 0.841433i
$347$	− 5961.59i	− 0.922291i	−0.887324	−	0.461146i	$-0.847439\pi$
$347$	− 5961.59i	− 0.922291i	0.887324	−	0.461146i	$-0.152561\pi$
$348$	970.814	0.149543
$349$	600.801	0.0921494	0.0460747	−	0.998938i	$-0.485329\pi$
$349$	600.801	0.0921494	0.0460747	+	0.998938i	$0.485329\pi$
$350$	6157.04i	0.940307i
$351$	9042.50i	1.37508i
$352$	4192.15i	0.634779i
$353$	9716.67	1.46506	0.732530	−	0.680735i	$-0.238339\pi$
$353$	9716.67	1.46506	0.732530	+	0.680735i	$0.238339\pi$
$354$	− 6684.49i	− 1.00361i
$355$	−1981.96	−0.296314
$356$	−1254.88	−0.186822
$357$	0	0
$358$	6685.76	0.987020
$359$	7176.43	1.05504	0.527518	−	0.849544i	$-0.323123\pi$
$359$	7176.43	1.05504	0.527518	+	0.849544i	$0.323123\pi$
$360$	12862.8i	1.88314i
$361$	−5051.64	−0.736497
$362$	12352.9i	1.79351i
$363$	− 519.461i	− 0.0751092i
$364$	− 28395.4i	− 4.08879i
$365$	−7010.44	−1.00532
$366$	−8835.28	−1.26182
$367$	4524.36i	0.643513i	0.946822	+	0.321757i	$0.104273\pi$
$367$	4524.36i	0.643513i	−0.946822	+	0.321757i	$0.895727\pi$
$368$	− 5929.35i	− 0.839916i
$369$	1616.19i	0.228010i
$370$	17958.9	2.52334
$371$	− 12234.5i	− 1.71208i
$372$	−1921.39	−0.267794
$373$	−2266.68	−0.314650	−0.157325	−	0.987547i	$-0.550287\pi$
$373$	−2266.68	−0.314650	−0.157325	+	0.987547i	$0.550287\pi$
$374$	0	0
$375$	−2450.77	−0.337486
$376$	−2488.88	−0.341367
$377$	− 1568.07i	− 0.214216i
$378$	−13784.8	−1.87570
$379$	5930.46i	0.803765i	0.915691	+	0.401883i	$0.131644\pi$
$379$	5930.46i	0.803765i	−0.915691	+	0.401883i	$0.868356\pi$
$380$	− 9935.40i	− 1.34125i
$381$	6204.65i	0.834314i
$382$	16022.1	2.14598
$383$	−11373.3	−1.51736	−0.758678	−	0.651466i	$-0.774154\pi$
$383$	−11373.3	−1.51736	−0.758678	+	0.651466i	$0.774154\pi$
$384$	− 4868.67i	− 0.647013i
$385$	− 10021.7i	− 1.32663i
$386$	− 2542.92i	− 0.335314i
$387$	−7145.89	−0.938621
$388$	24154.4i	3.16044i
$389$	12707.2	1.65625	0.828123	−	0.560546i	$-0.189409\pi$
$389$	12707.2	1.65625	0.828123	+	0.560546i	$0.189409\pi$
$390$	−12961.9	−1.68295
$391$	0	0
$392$	7118.58	0.917200
$393$	−6257.09	−0.803126
$394$	23733.5i	3.03471i
$395$	−3293.41	−0.419518
$396$	− 11831.8i	− 1.50144i
$397$	− 12102.6i	− 1.53000i	−0.644028	−	0.765002i	$-0.722738\pi$
$397$	− 12102.6i	− 1.53000i	0.644028	−	0.765002i	$-0.277262\pi$
$398$	19482.5i	2.45370i
$399$	2463.23	0.309061
$400$	−5504.18	−0.688023
$401$	− 9678.17i	− 1.20525i	−0.798025	−	0.602624i	$-0.794122\pi$
$401$	− 9678.17i	− 1.20525i	0.798025	−	0.602624i	$-0.205878\pi$
$402$	− 11312.7i	− 1.40355i
$403$	3103.45i	0.383607i
$404$	−25356.3	−3.12258
$405$	− 3001.46i	− 0.368257i
$406$	2390.44	0.292206
$407$	−8932.47	−1.08788
$408$	0	0
$409$	4382.11	0.529784	0.264892	−	0.964278i	$-0.414664\pi$
$409$	4382.11	0.529784	0.264892	+	0.964278i	$0.414664\pi$
$410$	−5414.96	−0.652258
$411$	− 4163.25i	− 0.499655i
$412$	−6082.59	−0.727349
$413$	− 11279.1i	− 1.34384i
$414$	6031.48i	0.716017i
$415$	− 3938.51i	− 0.465864i
$416$	9148.90	1.07827
$417$	946.189	0.111115
$418$	7211.32i	0.843821i
$419$	− 2651.71i	− 0.309176i	−0.987979	−	0.154588i	$-0.950595\pi$
$419$	− 2651.71i	− 0.309176i	0.987979	−	0.154588i	$-0.0494050\pi$
$420$	− 13540.8i	− 1.57315i
$421$	−1133.25	−0.131191	−0.0655954	−	0.997846i	$-0.520895\pi$
$421$	−1133.25	−0.131191	−0.0655954	+	0.997846i	$0.520895\pi$
$422$	12261.0i	1.41436i
$423$	1058.15	0.121629
$424$	26168.4	2.99729
$425$	0	0
$426$	1943.69	0.221062
$427$	−14908.2	−1.68960
$428$	21143.5i	2.38788i
$429$	6447.04	0.725562
$430$	− 23941.9i	− 2.68507i
$431$	16138.0i	1.80357i	0.432180	+	0.901787i	$0.357744\pi$
$431$	16138.0i	1.80357i	−0.432180	+	0.901787i	$0.642256\pi$
$432$	− 12323.2i	− 1.37245i
$433$	1273.94	0.141390	0.0706950	−	0.997498i	$-0.477478\pi$
$433$	1273.94	0.141390	0.0706950	+	0.997498i	$0.477478\pi$
$434$	−4731.05	−0.523267
$435$	− 747.758i	− 0.0824190i
$436$	17090.9i	1.87730i
$437$	− 2519.12i	− 0.275758i
$438$	6875.09	0.750010
$439$	5326.71i	0.579112i	0.957161	+	0.289556i	$0.0935076\pi$
$439$	5326.71i	0.579112i	−0.957161	+	0.289556i	$0.906492\pi$
$440$	21435.4	2.32249
$441$	−3026.48	−0.326799
$442$	0	0
$443$	−2382.90	−0.255564	−0.127782	−	0.991802i	$-0.540786\pi$
$443$	−2382.90	−0.255564	−0.127782	+	0.991802i	$0.540786\pi$
$444$	−12069.1	−1.29003
$445$	966.559i	0.102965i
$446$	2946.96	0.312875
$447$	1358.76i	0.143775i
$448$	− 3825.63i	− 0.403446i
$449$	− 8253.53i	− 0.867501i	−0.901033	−	0.433750i	$-0.857190\pi$
$449$	− 8253.53i	− 0.867501i	0.901033	−	0.433750i	$-0.142810\pi$
$450$	5598.98	0.586530
$451$	2693.32	0.281205
$452$	23599.9i	2.45585i
$453$	− 3019.53i	− 0.313179i
$454$	16312.8i	1.68634i
$455$	−21871.2	−2.25349
$456$	5268.61i	0.541064i
$457$	4710.14	0.482125	0.241063	−	0.970510i	$-0.422504\pi$
$457$	4710.14	0.482125	0.241063	+	0.970510i	$0.422504\pi$
$458$	−15321.7	−1.56318
$459$	0	0
$460$	−13848.1	−1.40363
$461$	−10294.3	−1.04003	−0.520016	−	0.854157i	$-0.674074\pi$
$461$	−10294.3	−1.04003	−0.520016	+	0.854157i	$0.674074\pi$
$462$	9828.18i	0.989715i
$463$	3454.44	0.346742	0.173371	−	0.984857i	$-0.444534\pi$
$463$	3454.44	0.346742	0.173371	+	0.984857i	$0.444534\pi$
$464$	2136.97i	0.213807i
$465$	1479.93i	0.147592i
$466$	27161.6i	2.70008i
$467$	−2128.17	−0.210878	−0.105439	−	0.994426i	$-0.533625\pi$
$467$	−2128.17	−0.210878	−0.105439	+	0.994426i	$0.533625\pi$
$468$	−25821.7	−2.55044
$469$	− 19088.5i	− 1.87938i
$470$	3545.29i	0.347940i
$471$	5970.12i	0.584053i
$472$	24124.9	2.35262
$473$	11908.4i	1.15760i
$474$	3229.83	0.312976
$475$	−2338.49	−0.225889
$476$	0	0
$477$	−11125.6	−1.06793
$478$	32944.7	3.15242
$479$	− 12574.1i	− 1.19942i	−0.800216	−	0.599712i	$-0.795282\pi$
$479$	− 12574.1i	− 1.19942i	0.800216	−	0.599712i	$-0.204718\pi$
$480$	4362.80	0.414862
$481$	19494.1i	1.84793i
$482$	− 11308.4i	− 1.06864i
$483$	− 3433.27i	− 0.323435i
$484$	3467.15	0.325615
$485$	18604.6	1.74184
$486$	19707.7i	1.83943i
$487$	− 20485.6i	− 1.90615i	−0.302740	−	0.953073i	$-0.597901\pi$
$487$	− 20485.6i	− 1.90615i	0.302740	−	0.953073i	$-0.402099\pi$
$488$	− 31887.3i	− 2.95793i
$489$	9060.75	0.837917
$490$	− 10140.1i	− 0.934860i
$491$	−10450.5	−0.960539	−0.480270	−	0.877121i	$-0.659461\pi$
$491$	−10450.5	−0.960539	−0.480270	+	0.877121i	$0.659461\pi$
$492$	3639.08	0.333460
$493$	0	0
$494$	15737.9	1.43336
$495$	−9113.33	−0.827503
$496$	− 4229.40i	− 0.382874i
$497$	3279.69	0.296005
$498$	3862.46i	0.347553i
$499$	− 6819.20i	− 0.611762i	−0.952070	−	0.305881i	$-0.901049\pi$
$499$	− 6819.20i	− 0.611762i	0.952070	−	0.305881i	$-0.0989510\pi$
$500$	− 16357.7i	− 1.46308i
$501$	−1596.67	−0.142383
$502$	−9309.04	−0.827655
$503$	8988.46i	0.796771i	0.917218	+	0.398386i	$0.130429\pi$
$503$	8988.46i	0.796771i	−0.917218	+	0.398386i	$0.869571\pi$
$504$	− 21285.1i	− 1.88118i
$505$	19530.4i	1.72097i
$506$	10051.2	0.883066
$507$	− 8336.32i	− 0.730235i
$508$	−41413.0	−3.61694
$509$	4660.74	0.405862	0.202931	−	0.979193i	$-0.434953\pi$
$509$	4660.74	0.405862	0.202931	+	0.979193i	$0.434953\pi$
$510$	0	0
$511$	11600.7	1.00427
$512$	25545.9	2.20504
$513$	− 5235.58i	− 0.450597i
$514$	−7847.06	−0.673383
$515$	4685.05i	0.400869i
$516$	16090.0i	1.37272i
$517$	− 1763.37i	− 0.150006i
$518$	−29717.8	−2.52071
$519$	2803.21	0.237085
$520$	− 46780.5i	− 3.94511i
$521$	2931.59i	0.246517i	0.992375	+	0.123258i	$0.0393344\pi$
$521$	2931.59i	0.246517i	−0.992375	+	0.123258i	$0.960666\pi$
$522$	− 2173.78i	− 0.182267i
$523$	−4855.55	−0.405963	−0.202981	−	0.979183i	$-0.565063\pi$
$523$	−4855.55	−0.405963	−0.202981	+	0.979183i	$0.565063\pi$
$524$	− 41763.0i	− 3.48173i
$525$	−3187.08	−0.264944
$526$	−14752.2	−1.22286
$527$	0	0
$528$	−8786.06	−0.724174
$529$	8655.81	0.711417
$530$	− 37275.6i	− 3.05500i
$531$	−10256.8	−0.838240
$532$	16440.8i	1.33985i
$533$	− 5877.88i	− 0.477672i
$534$	− 947.897i	− 0.0768156i
$535$	16285.6	1.31605
$536$	40828.6	3.29016
$537$	3460.76i	0.278106i
$538$	34462.3i	2.76167i
$539$	5043.52i	0.403042i
$540$	−28780.9	−2.29358
$541$	14604.2i	1.16060i	0.814403	+	0.580300i	$0.197065\pi$
$541$	14604.2i	1.16060i	−0.814403	+	0.580300i	$0.802935\pi$
$542$	7081.67	0.561225
$543$	−6394.24	−0.505346
$544$	0	0
$545$	13164.0	1.03465
$546$	21448.9	1.68119
$547$	− 10928.1i	− 0.854209i	−0.904202	−	0.427105i	$-0.859534\pi$
$547$	− 10928.1i	− 0.854209i	0.904202	−	0.427105i	$-0.140466\pi$
$548$	27787.7	2.16611
$549$	13557.0i	1.05391i
$550$	− 9330.48i	− 0.723369i
$551$	907.906i	0.0701962i
$552$	7343.45	0.566228
$553$	5449.85	0.419080
$554$	10825.4i	0.830196i
$555$	9296.09i	0.710985i
$556$	6315.35i	0.481710i
$557$	20141.0	1.53214	0.766071	−	0.642756i	$-0.222209\pi$
$557$	20141.0	1.53214	0.766071	+	0.642756i	$0.222209\pi$
$558$	4302.24i	0.326395i
$559$	25988.7	1.96638
$560$	29806.2	2.24918
$561$	0	0
$562$	−2203.79	−0.165411
$563$	−7089.55	−0.530708	−0.265354	−	0.964151i	$-0.585489\pi$
$563$	−7089.55	−0.530708	−0.265354	+	0.964151i	$0.585489\pi$
$564$	− 2382.58i	− 0.177881i
$565$	18177.5	1.35351
$566$	− 34020.2i	− 2.52646i
$567$	4966.74i	0.367872i
$568$	7014.96i	0.518206i
$569$	−22613.5	−1.66609	−0.833046	−	0.553204i	$-0.813405\pi$
$569$	−22613.5	−1.66609	−0.833046	+	0.553204i	$0.813405\pi$
$570$	7504.88	0.551482
$571$	1483.70i	0.108741i	0.998521	+	0.0543705i	$0.0173152\pi$
$571$	1483.70i	0.108741i	−0.998521	+	0.0543705i	$0.982685\pi$
$572$	43030.8i	3.14547i
$573$	8293.57i	0.604658i
$574$	8960.53	0.651577
$575$	3259.41i	0.236394i
$576$	−3478.88	−0.251655
$577$	23288.0	1.68023	0.840115	−	0.542408i	$-0.182487\pi$
$577$	23288.0	1.68023	0.840115	+	0.542408i	$0.182487\pi$
$578$	0	0
$579$	1316.30	0.0944792
$580$	4990.92	0.357305
$581$	6517.33i	0.465378i
$582$	−18245.4	−1.29948
$583$	18540.3i	1.31709i
$584$	24812.8i	1.75815i
$585$	19888.8i	1.40565i
$586$	−6146.52	−0.433294
$587$	−3330.73	−0.234197	−0.117099	−	0.993120i	$-0.537359\pi$
$587$	−3330.73	−0.234197	−0.117099	+	0.993120i	$0.537359\pi$
$588$	6814.55i	0.477937i
$589$	− 1796.89i	− 0.125704i
$590$	− 34364.7i	− 2.39792i
$591$	−12285.2	−0.855069
$592$	− 26566.7i	− 1.84440i
$593$	5795.34	0.401326	0.200663	−	0.979660i	$-0.435690\pi$
$593$	5795.34	0.401326	0.200663	+	0.979660i	$0.435690\pi$
$594$	20889.8	1.44296
$595$	0	0
$596$	−9069.08	−0.623295
$597$	−10084.8	−0.691362
$598$	− 21935.7i	− 1.50003i
$599$	9822.45	0.670007	0.335004	−	0.942217i	$-0.391262\pi$
$599$	9822.45	0.670007	0.335004	+	0.942217i	$0.391262\pi$
$600$	− 6816.87i	− 0.463830i
$601$	− 24976.8i	− 1.69522i	−0.530622	−	0.847609i	$-0.678042\pi$
$601$	− 24976.8i	− 1.69522i	0.530622	−	0.847609i	$-0.321958\pi$
$602$	39618.4i	2.68227i
$603$	−17358.4	−1.17229
$604$	20153.9	1.35770
$605$	− 2670.53i	− 0.179459i
$606$	− 19153.3i	− 1.28391i
$607$	− 12602.3i	− 0.842689i	−0.906901	−	0.421344i	$-0.861558\pi$
$607$	− 12602.3i	− 0.842689i	0.906901	−	0.421344i	$-0.138442\pi$
$608$	−5297.19	−0.353338
$609$	1237.37i	0.0823329i
$610$	−45421.8	−3.01488
$611$	−3848.37	−0.254809
$612$	0	0
$613$	−6503.06	−0.428476	−0.214238	−	0.976781i	$-0.568727\pi$
$613$	−6503.06	−0.428476	−0.214238	+	0.976781i	$0.568727\pi$
$614$	−46574.3	−3.06121
$615$	− 2802.96i	− 0.183783i
$616$	−35470.8	−2.32006
$617$	− 15511.7i	− 1.01212i	−0.862498	−	0.506061i	$-0.831101\pi$
$617$	− 15511.7i	− 1.01212i	0.862498	−	0.506061i	$-0.168899\pi$
$618$	− 4594.59i	− 0.299064i
$619$	− 27666.3i	− 1.79645i	−0.439534	−	0.898226i	$-0.644856\pi$
$619$	− 27666.3i	− 1.79645i	0.439534	−	0.898226i	$-0.355144\pi$
$620$	−9877.80	−0.639842
$621$	−7297.41	−0.471554
$622$	− 3737.28i	− 0.240918i
$623$	− 1599.43i	− 0.102857i
$624$	19174.6i	1.23013i
$625$	−19475.1	−1.24641
$626$	949.850i	0.0606448i
$627$	−3732.81	−0.237758
$628$	−39847.7	−2.53200
$629$	0	0
$630$	−30319.5	−1.91740
$631$	−5269.63	−0.332458	−0.166229	−	0.986087i	$-0.553159\pi$
$631$	−5269.63	−0.332458	−0.166229	+	0.986087i	$0.553159\pi$
$632$	11656.7i	0.733670i
$633$	−6346.72	−0.398514
$634$	31676.6i	1.98429i
$635$	31897.9i	1.99343i
$636$	25050.8i	1.56184i
$637$	11006.9	0.684631
$638$	−3622.51	−0.224791
$639$	− 2982.43i	− 0.184637i
$640$	− 25029.6i	− 1.54591i
$641$	8458.94i	0.521230i	0.965443	+	0.260615i	$0.0839252\pi$
$641$	8458.94i	0.521230i	−0.965443	+	0.260615i	$0.916075\pi$
$642$	−15971.1	−0.981823
$643$	− 25746.8i	− 1.57909i	−0.613693	−	0.789545i	$-0.710317\pi$
$643$	− 25746.8i	− 1.57909i	0.613693	−	0.789545i	$-0.289683\pi$
$644$	22915.4	1.40216
$645$	12393.1	0.756556
$646$	0	0
$647$	−13724.8	−0.833968	−0.416984	−	0.908914i	$-0.636913\pi$
$647$	−13724.8	−0.833968	−0.416984	+	0.908914i	$0.636913\pi$
$648$	−10623.4	−0.644022
$649$	17092.5i	1.03380i
$650$	−20362.7	−1.22876
$651$	− 2448.95i	− 0.147437i
$652$	60476.1i	3.63256i
$653$	14415.2i	0.863877i	0.901903	+	0.431939i	$0.142170\pi$
$653$	14415.2i	0.863877i	−0.901903	+	0.431939i	$0.857830\pi$
$654$	−12909.9	−0.771891
$655$	−32167.5	−1.91891
$656$	8010.41i	0.476759i
$657$	− 10549.2i	− 0.626430i
$658$	− 5866.64i	− 0.347577i
$659$	−7405.85	−0.437771	−0.218885	−	0.975751i	$-0.570242\pi$
$659$	−7405.85	−0.437771	−0.218885	+	0.975751i	$0.570242\pi$
$660$	20519.9i	1.21021i
$661$	19937.3	1.17318	0.586589	−	0.809885i	$-0.300470\pi$
$661$	19937.3	1.17318	0.586589	+	0.809885i	$0.300470\pi$
$662$	−1273.90	−0.0747910
$663$	0	0
$664$	−13940.0	−0.814722
$665$	12663.4	0.738442
$666$	27024.3i	1.57233i
$667$	1265.45	0.0734609
$668$	− 10657.0i	− 0.617262i
$669$	1525.44i	0.0881568i
$670$	− 58158.4i	− 3.35351i
$671$	22592.2	1.29979
$672$	−7219.44	−0.414429
$673$	13809.6i	0.790965i	0.918473	+	0.395483i	$0.129423\pi$
$673$	13809.6i	0.790965i	−0.918473	+	0.395483i	$0.870577\pi$
$674$	− 29289.1i	− 1.67385i
$675$	6774.14i	0.386276i
$676$	55640.9	3.16573
$677$	− 6869.29i	− 0.389968i	−0.980806	−	0.194984i	$-0.937535\pi$
$677$	− 6869.29i	− 0.389968i	0.980806	−	0.194984i	$-0.0624655\pi$
$678$	−17826.6	−1.00977
$679$	−30786.4	−1.74002
$680$	0	0
$681$	−8444.06	−0.475150
$682$	7169.52	0.402544
$683$	4911.19i	0.275141i	0.990492	+	0.137571i	$0.0439294\pi$
$683$	4911.19i	0.275141i	−0.990492	+	0.137571i	$0.956071\pi$
$684$	14950.7	0.835750
$685$	− 21403.1i	− 1.19383i
$686$	− 21613.7i	− 1.20294i
$687$	− 7931.03i	− 0.440448i
$688$	−35417.5	−1.96262
$689$	40462.2	2.23728
$690$	− 10460.4i	− 0.577130i
$691$	30039.7i	1.65378i	0.562363	+	0.826891i	$0.309892\pi$
$691$	30039.7i	1.65378i	−0.562363	+	0.826891i	$0.690108\pi$
$692$	18710.0i	1.02782i
$693$	15080.5	0.826638
$694$	30056.6i	1.64400i
$695$	4864.33	0.265488
$696$	−2646.62	−0.144138
$697$	0	0
$698$	−3029.06	−0.164257
$699$	−14059.7	−0.760784
$700$	− 21272.2i	− 1.14859i
$701$	−4144.68	−0.223313	−0.111657	−	0.993747i	$-0.535616\pi$
$701$	−4144.68	−0.223313	−0.111657	+	0.993747i	$0.535616\pi$
$702$	− 45589.6i	− 2.45110i
$703$	− 11287.0i	− 0.605546i
$704$	5797.42i	0.310367i
$705$	−1835.16	−0.0980368
$706$	−48988.6	−2.61149
$707$	− 32318.4i	− 1.71918i
$708$	23094.5i	1.22591i
$709$	18027.8i	0.954935i	0.878649	+	0.477468i	$0.158445\pi$
$709$	18027.8i	0.954935i	−0.878649	+	0.477468i	$0.841555\pi$
$710$	9992.46	0.528184
$711$	− 4955.89i	− 0.261407i
$712$	3421.04	0.180069
$713$	−2504.52	−0.131550
$714$	0	0
$715$	33144.0	1.73359
$716$	−23098.9	−1.20565
$717$	17053.2i	0.888236i
$718$	−36181.5	−1.88061
$719$	− 4482.73i	− 0.232514i	−0.993219	−	0.116257i	$-0.962910\pi$
$719$	− 4482.73i	− 0.232514i	0.993219	−	0.116257i	$-0.0370896\pi$
$720$	− 27104.6i	− 1.40296i
$721$	− 7752.68i	− 0.400451i
$722$	25468.9	1.31282
$723$	5853.59	0.301103
$724$	− 42678.4i	− 2.19079i
$725$	− 1174.71i	− 0.0601760i
$726$	2618.97i	0.133883i
$727$	15342.7	0.782709	0.391355	−	0.920240i	$-0.372007\pi$
$727$	15342.7	0.782709	0.391355	+	0.920240i	$0.372007\pi$
$728$	77411.0i	3.94099i
$729$	−4161.14	−0.211408
$730$	35344.6	1.79200
$731$	0	0
$732$	30525.4	1.54133
$733$	35022.1	1.76476	0.882381	−	0.470536i	$-0.155939\pi$
$733$	35022.1	1.76476	0.882381	+	0.470536i	$0.155939\pi$
$734$	− 22810.5i	− 1.14707i
$735$	5248.83	0.263409
$736$	7383.28i	0.369771i
$737$	28927.1i	1.44579i
$738$	− 8148.37i	− 0.406430i
$739$	8292.46	0.412778	0.206389	−	0.978470i	$-0.433829\pi$
$739$	8292.46	0.412778	0.206389	+	0.978470i	$0.433829\pi$
$740$	−62046.8	−3.08228
$741$	8146.45i	0.403870i
$742$	61682.7i	3.05181i
$743$	− 29918.5i	− 1.47726i	−0.674111	−	0.738630i	$-0.735473\pi$
$743$	− 29918.5i	− 1.47726i	0.674111	−	0.738630i	$-0.264527\pi$
$744$	5238.07	0.258114
$745$	6985.36i	0.343522i
$746$	11428.0	0.560867
$747$	5926.61	0.290286
$748$	0	0
$749$	−26948.9	−1.31467
$750$	12356.1	0.601572
$751$	3090.08i	0.150145i	0.997178	+	0.0750723i	$0.0239188\pi$
$751$	3090.08i	0.150145i	−0.997178	+	0.0750723i	$0.976081\pi$
$752$	5244.58	0.254322
$753$	− 4818.66i	− 0.233203i
$754$	7905.74i	0.381844i
$755$	− 15523.3i	− 0.748279i
$756$	47625.8	2.29118
$757$	8344.86	0.400659	0.200330	−	0.979729i	$-0.435799\pi$
$757$	8344.86	0.400659	0.200330	+	0.979729i	$0.435799\pi$
$758$	− 29899.6i	− 1.43272i
$759$	5202.84i	0.248816i
$760$	27085.7i	1.29277i
$761$	−4825.17	−0.229845	−0.114923	−	0.993374i	$-0.536662\pi$
$761$	−4825.17	−0.229845	−0.114923	+	0.993374i	$0.536662\pi$
$762$	− 31282.0i	− 1.48718i
$763$	−21783.5	−1.03357
$764$	−55355.5	−2.62133
$765$	0	0
$766$	57340.7	2.70471
$767$	37302.5	1.75608
$768$	20948.8i	0.984278i
$769$	−14268.8	−0.669110	−0.334555	−	0.942376i	$-0.608586\pi$
$769$	−14268.8	−0.669110	−0.334555	+	0.942376i	$0.608586\pi$
$770$	50526.3i	2.36473i
$771$	− 4061.89i	− 0.189735i
$772$	8785.65i	0.409588i
$773$	−4746.48	−0.220853	−0.110426	−	0.993884i	$-0.535222\pi$
$773$	−4746.48	−0.220853	−0.110426	+	0.993884i	$0.535222\pi$
$774$	36027.5	1.67310
$775$	2324.93i	0.107760i
$776$	− 65849.2i	− 3.04620i
$777$	− 15382.9i	− 0.710243i
$778$	−64065.9	−2.95228
$779$	3403.27i	0.156527i
$780$	44782.5	2.05573
$781$	−4970.10	−0.227714
$782$	0	0
$783$	2630.03	0.120038
$784$	−15000.3	−0.683322
$785$	30692.2i	1.39548i
$786$	31546.4	1.43158
$787$	− 4925.33i	− 0.223086i	−0.993760	−	0.111543i	$-0.964421\pi$
$787$	− 4925.33i	− 0.223086i	0.993760	−	0.111543i	$-0.0355793\pi$
$788$	− 81997.8i	− 3.70691i
$789$	− 7636.21i	− 0.344558i
$790$	16604.4	0.747796
$791$	−30079.6	−1.35210
$792$	32255.8i	1.44717i
$793$	− 49304.9i	− 2.20790i
$794$	61017.7i	2.72725i
$795$	19295.1	0.860787
$796$	− 67311.1i	− 2.99721i
$797$	−38377.6	−1.70565	−0.852826	−	0.522195i	$-0.825113\pi$
$797$	−38377.6	−1.70565	−0.852826	+	0.522195i	$0.825113\pi$
$798$	−12418.9	−0.550906
$799$	0	0
$800$	6853.85	0.302900
$801$	−1454.47	−0.0641586
$802$	48794.5i	2.14837i
$803$	−17579.9	−0.772578
$804$	39084.9i	1.71445i
$805$	− 17650.3i	− 0.772785i
$806$	− 15646.7i	− 0.683785i
$807$	−17838.8	−0.778136
$808$	69126.0	3.00971
$809$	9966.44i	0.433129i	0.976268	+	0.216565i	$0.0694852\pi$
$809$	9966.44i	0.433129i	−0.976268	+	0.216565i	$0.930515\pi$
$810$	15132.5i	0.656422i
$811$	− 35469.1i	− 1.53574i	−0.640604	−	0.767871i	$-0.721316\pi$
$811$	− 35469.1i	− 1.53574i	0.640604	−	0.767871i	$-0.278684\pi$
$812$	−8258.84	−0.356931
$813$	3665.70i	0.158133i
$814$	45034.9	1.93916
$815$	46581.0	2.00204
$816$	0	0
$817$	−15047.4	−0.644358
$818$	−22093.3	−0.944346
$819$	− 32911.5i	− 1.40418i
$820$	18708.4	0.796738
$821$	19408.8i	0.825058i	0.910944	+	0.412529i	$0.135354\pi$
$821$	19408.8i	0.825058i	−0.910944	+	0.412529i	$0.864646\pi$
$822$	20989.9i	0.890641i
$823$	19804.6i	0.838815i	0.907798	+	0.419407i	$0.137762\pi$
$823$	19804.6i	0.838815i	−0.907798	+	0.419407i	$0.862238\pi$
$824$	16582.3	0.701057
$825$	4829.76	0.203819
$826$	56865.8i	2.39542i
$827$	30581.1i	1.28586i	0.765923	+	0.642932i	$0.222282\pi$
$827$	30581.1i	1.28586i	−0.765923	+	0.642932i	$0.777718\pi$
$828$	− 20838.4i	− 0.874619i
$829$	15945.5	0.668046	0.334023	−	0.942565i	$-0.391594\pi$
$829$	15945.5	0.668046	0.334023	+	0.942565i	$0.391594\pi$
$830$	19856.8i	0.830409i
$831$	−5603.60	−0.233919
$832$	12652.2	0.527208
$833$	0	0
$834$	−4770.41	−0.198065
$835$	−8208.41	−0.340196
$836$	− 24914.7i	− 1.03073i
$837$	−5205.23	−0.214957
$838$	13369.2i	0.551110i
$839$	20434.3i	0.840848i	0.907328	+	0.420424i	$0.138119\pi$
$839$	20434.3i	0.840848i	−0.907328	+	0.420424i	$0.861881\pi$
$840$	36914.7i	1.51628i
$841$	23932.9	0.981300
$842$	5713.53	0.233849
$843$	− 1140.75i	− 0.0466069i
$844$	− 42361.2i	− 1.72765i
$845$	− 42856.7i	− 1.74475i
$846$	−5334.90	−0.216806
$847$	4419.12i	0.179271i
$848$	−55142.2	−2.23301
$849$	17609.9	0.711863
$850$	0	0
$851$	−15732.0	−0.633709
$852$	−6715.35	−0.270028
$853$	− 7508.30i	− 0.301382i	−0.988581	−	0.150691i	$-0.951850\pi$
$853$	− 7508.30i	− 0.301382i	0.988581	−	0.150691i	$-0.0481499\pi$
$854$	75162.8	3.01173
$855$	− 11515.6i	− 0.460613i
$856$	− 57641.2i	− 2.30156i
$857$	4013.00i	0.159955i	0.996797	+	0.0799776i	$0.0254849\pi$
$857$	4013.00i	0.159955i	−0.996797	+	0.0799776i	$0.974515\pi$
$858$	−32504.1	−1.29332
$859$	−32674.1	−1.29782	−0.648910	−	0.760865i	$-0.724775\pi$
$859$	−32674.1	−1.29782	−0.648910	+	0.760865i	$0.724775\pi$
$860$	82718.0i	3.27984i
$861$	4638.26i	0.183591i
$862$	− 81363.1i	− 3.21489i
$863$	−9829.13	−0.387703	−0.193851	−	0.981031i	$-0.562098\pi$
$863$	−9829.13	−0.387703	−0.193851	+	0.981031i	$0.562098\pi$
$864$	15344.9i	0.604218i
$865$	14411.2	0.566469
$866$	−6422.85	−0.252029
$867$	0	0
$868$	16345.5	0.639174
$869$	−8258.79	−0.322394
$870$	3769.98i	0.146913i
$871$	63130.2	2.45590
$872$	− 46592.9i	− 1.80944i
$873$	27996.0i	1.08536i
$874$	12700.7i	0.491542i
$875$	20849.0	0.805514
$876$	−23753.0	−0.916143
$877$	17731.1i	0.682711i	0.939934	+	0.341356i	$0.110886\pi$
$877$	17731.1i	0.682711i	−0.939934	+	0.341356i	$0.889114\pi$
$878$	− 26855.7i	− 1.03227i
$879$	− 3181.64i	− 0.122086i
$880$	−45168.8	−1.73027
$881$	− 18721.4i	− 0.715937i	−0.933734	−	0.357969i	$-0.883470\pi$
$881$	− 18721.4i	− 0.715937i	0.933734	−	0.357969i	$-0.116530\pi$
$882$	15258.6	0.582523
$883$	−19025.1	−0.725079	−0.362539	−	0.931968i	$-0.618090\pi$
$883$	−19025.1	−0.725079	−0.362539	+	0.931968i	$0.618090\pi$
$884$	0	0
$885$	17788.3	0.675646
$886$	12013.9	0.455546
$887$	− 30933.2i	− 1.17095i	−0.810689	−	0.585476i	$-0.800907\pi$
$887$	− 30933.2i	− 1.17095i	0.810689	−	0.585476i	$-0.199093\pi$
$888$	32902.6	1.24340
$889$	− 52783.7i	− 1.99135i
$890$	− 4873.11i	− 0.183536i
$891$	− 7526.68i	− 0.283000i
$892$	−10181.6	−0.382179
$893$	2228.19	0.0834979
$894$	− 6850.49i	− 0.256280i
$895$	17791.7i	0.664480i
$896$	41418.4i	1.54430i
$897$	11354.6	0.422653
$898$	41611.9i	1.54633i
$899$	902.643	0.0334870
$900$	−19344.2	−0.716450
$901$	0	0
$902$	−13578.9	−0.501252
$903$	−20507.8	−0.755766
$904$	− 64337.6i	− 2.36707i
$905$	−32872.6	−1.20743
$906$	15223.6i	0.558245i
$907$	− 17462.3i	− 0.639280i	−0.947539	−	0.319640i	$-0.896438\pi$
$907$	− 17462.3i	− 0.639280i	0.947539	−	0.319640i	$-0.103562\pi$
$908$	− 56360.0i	− 2.05988i
$909$	−29389.1	−1.07236
$910$	110268.	4.01687
$911$	50841.6i	1.84902i	0.381158	+	0.924510i	$0.375525\pi$
$911$	50841.6i	1.84902i	−0.381158	+	0.924510i	$0.624475\pi$
$912$	− 11102.0i	− 0.403098i
$913$	− 9876.48i	− 0.358011i
$914$	−23747.2	−0.859394
$915$	− 23511.8i	− 0.849483i
$916$	52935.7	1.90944
$917$	53229.9	1.91691
$918$	0	0
$919$	−2252.46	−0.0808508	−0.0404254	−	0.999183i	$-0.512871\pi$
$919$	−2252.46	−0.0808508	−0.0404254	+	0.999183i	$0.512871\pi$
$920$	37752.4	1.35289
$921$	− 24108.3i	− 0.862538i
$922$	51901.0	1.85387
$923$	10846.7i	0.386808i
$924$	− 33955.8i	− 1.20894i
$925$	14603.9i	0.519107i
$926$	−17416.3	−0.618073
$927$	−7050.00	−0.249787
$928$	− 2660.97i	− 0.0941279i
$929$	15783.0i	0.557398i	0.960379	+	0.278699i	$0.0899031\pi$
$929$	15783.0i	0.557398i	−0.960379	+	0.278699i	$0.910097\pi$
$930$	− 7461.37i	− 0.263084i
$931$	−6372.97	−0.224346
$932$	− 93841.8i	− 3.29817i
$933$	1934.54	0.0678820
$934$	10729.6	0.375892
$935$	0	0
$936$	70394.6	2.45825
$937$	51506.8	1.79579	0.897894	−	0.440211i	$-0.145096\pi$
$937$	51506.8	1.79579	0.897894	+	0.440211i	$0.145096\pi$
$938$	96238.9i	3.35001i
$939$	−491.673	−0.0170875
$940$	− 12248.8i	− 0.425011i
$941$	− 7103.99i	− 0.246104i	−0.992400	−	0.123052i	$-0.960732\pi$
$941$	− 7103.99i	− 0.246104i	0.992400	−	0.123052i	$-0.0392681\pi$
$942$	− 30099.6i	− 1.04108i
$943$	4743.52	0.163807
$944$	−50836.1	−1.75273
$945$	− 36683.2i	− 1.26276i
$946$	− 60038.5i	− 2.06344i
$947$	24874.2i	0.853542i	0.904360	+	0.426771i	$0.140349\pi$
$947$	24874.2i	0.853542i	−0.904360	+	0.426771i	$0.859651\pi$
$948$	−11158.9	−0.382303
$949$	38366.1i	1.31235i
$950$	11790.0	0.402650
$951$	−16396.9	−0.559101
$952$	0	0
$953$	15169.7	0.515630	0.257815	−	0.966194i	$-0.416997\pi$
$953$	15169.7	0.515630	0.257815	+	0.966194i	$0.416997\pi$
$954$	56091.9	1.90361
$955$	42637.0i	1.44471i
$956$	−113822.	−3.85070
$957$	− 1875.13i	− 0.0633379i
$958$	63394.8i	2.13799i
$959$	35417.3i	1.19258i
$960$	6033.42	0.202841
$961$	28004.5	0.940033
$962$	− 98283.7i	− 3.29396i
$963$	24506.3i	0.820047i
$964$	39069.8i	1.30535i
$965$	6767.04	0.225740
$966$	17309.6i	0.576528i
$967$	−44604.0	−1.48332	−0.741659	−	0.670777i	$-0.765961\pi$
$967$	−44604.0	−1.48332	−0.741659	+	0.670777i	$0.765961\pi$
$968$	−9452.09	−0.313845
$969$	0	0
$970$	−93799.0	−3.10485
$971$	7770.42	0.256812	0.128406	−	0.991722i	$-0.459014\pi$
$971$	7770.42	0.256812	0.128406	+	0.991722i	$0.459014\pi$
$972$	− 68089.1i	− 2.24687i
$973$	−8049.35	−0.265211
$974$	103283.i	3.39773i
$975$	− 10540.4i	− 0.346219i
$976$	67193.0i	2.20368i
$977$	40455.1	1.32474	0.662371	−	0.749176i	$-0.269550\pi$
$977$	40455.1	1.32474	0.662371	+	0.749176i	$0.269550\pi$
$978$	−45681.7	−1.49360
$979$	2423.81i	0.0791270i
$980$	35033.3i	1.14194i
$981$	19809.1i	0.644705i
$982$	52688.4	1.71217
$983$	− 16755.2i	− 0.543649i	−0.962347	−	0.271825i	$-0.912373\pi$
$983$	− 16755.2i	− 0.543649i	0.962347	−	0.271825i	$-0.0876271\pi$
$984$	−9920.81	−0.321406
$985$	−63157.8	−2.04302
$986$	0	0
$987$	3036.76	0.0979344
$988$	−54373.6	−1.75087
$989$	20973.2i	0.674326i
$990$	45946.8	1.47503
$991$	− 7953.79i	− 0.254955i	−0.991841	−	0.127478i	$-0.959312\pi$
$991$	− 7953.79i	− 0.254955i	0.991841	−	0.127478i	$-0.0406881\pi$
$992$	5266.48i	0.168559i
$993$	− 659.414i	− 0.0210734i
$994$	−16535.2	−0.527632
$995$	−51845.6	−1.65188
$996$	− 13344.6i	− 0.424538i
$997$	− 27528.7i	− 0.874466i	−0.899348	−	0.437233i	$-0.855959\pi$
$997$	− 27528.7i	− 0.874466i	0.899348	−	0.437233i	$-0.144041\pi$
$998$	34380.4i	1.09047i
$999$	−32696.3	−1.03550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.d.288.1		8
17.4	even	4		289.4.a.d.1.4	yes	4
17.13	even	4		289.4.a.c.1.4	✓	4
17.16	even	2	inner	289.4.b.d.288.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.c.1.4	✓	4	17.13	even	4
289.4.a.d.1.4	yes	4	17.4	even	4
289.4.b.d.288.1		8	1.1	even	1	trivial
289.4.b.d.288.2		8	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	289.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.04171 q^{2} -2.60975i q^{3} +17.4188 q^{4} -13.4166i q^{5} +13.1576i q^{6} +22.2015i q^{7} -47.4869 q^{8} +20.1892 q^{9} +O(q^{10})\)	\(q-5.04171 q^{2} -2.60975i q^{3} +17.4188 q^{4} -13.4166i q^{5} +13.1576i q^{6} +22.2015i q^{7} -47.4869 q^{8} +20.1892 q^{9} +67.6428i q^{10} -33.6445i q^{11} -45.4588i q^{12} -73.4255 q^{13} -111.933i q^{14} -35.0141 q^{15} +100.065 q^{16} -101.788 q^{18} +42.5131 q^{19} -233.702i q^{20} +57.9404 q^{21} +169.626i q^{22} -59.2553i q^{23} +123.929i q^{24} -55.0063 q^{25} +370.190 q^{26} -123.152i q^{27} +386.724i q^{28} +21.3559i q^{29} +176.531 q^{30} -42.2667i q^{31} -124.601 q^{32} -87.8038 q^{33} +297.869 q^{35} +351.672 q^{36} -265.496i q^{37} -214.339 q^{38} +191.622i q^{39} +637.115i q^{40} +80.0523i q^{41} -292.118 q^{42} -353.946 q^{43} -586.048i q^{44} -270.871i q^{45} +298.748i q^{46} +52.4119 q^{47} -261.144i q^{48} -149.906 q^{49} +277.325 q^{50} -1278.98 q^{52} -551.066 q^{53} +620.897i q^{54} -451.396 q^{55} -1054.28i q^{56} -110.949i q^{57} -107.670i q^{58} -508.032 q^{59} -609.904 q^{60} +671.496i q^{61} +213.096i q^{62} +448.230i q^{63} -172.314 q^{64} +985.123i q^{65} +442.681 q^{66} -859.787 q^{67} -154.642 q^{69} -1501.77 q^{70} -147.724i q^{71} -958.723 q^{72} -522.518i q^{73} +1338.55i q^{74} +143.553i q^{75} +740.528 q^{76} +746.958 q^{77} -966.103i q^{78} -245.472i q^{79} -1342.53i q^{80} +223.712 q^{81} -403.600i q^{82} +293.554 q^{83} +1009.25 q^{84} +1784.49 q^{86} +55.7336 q^{87} +1597.67i q^{88} -72.0418 q^{89} +1365.65i q^{90} -1630.15i q^{91} -1032.16i q^{92} -110.305 q^{93} -264.245 q^{94} -570.383i q^{95} +325.178i q^{96} +1386.68i q^{97} +755.782 q^{98} -679.256i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9	\( 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9} - 44 q^{13} - 108 q^{15} + 126 q^{16} - 668 q^{18} - 44 q^{19} - 704 q^{21} - 756 q^{25} + 896 q^{26} + 626 q^{30} - 662 q^{32} + 188 q^{33} - 484 q^{35} - 282 q^{36} - 1048 q^{38} - 2910 q^{42} + 228 q^{43} + 20 q^{47} - 2012 q^{49} + 1610 q^{50} - 3074 q^{52} - 100 q^{53} - 2632 q^{55} - 1992 q^{59} + 434 q^{60} - 300 q^{64} + 2180 q^{66} - 1736 q^{67} - 2256 q^{69} - 2104 q^{70} - 78 q^{72} + 1746 q^{76} + 1788 q^{77} + 2160 q^{81} - 1700 q^{83} + 886 q^{84} + 4822 q^{86} + 768 q^{87} + 1568 q^{89} + 3100 q^{93} - 2238 q^{94} - 3754 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9 - 44 * q^13 - 108 * q^15 + 126 * q^16 - 668 * q^18 - 44 * q^19 - 704 * q^21 - 756 * q^25 + 896 * q^26 + 626 * q^30 - 662 * q^32 + 188 * q^33 - 484 * q^35 - 282 * q^36 - 1048 * q^38 - 2910 * q^42 + 228 * q^43 + 20 * q^47 - 2012 * q^49 + 1610 * q^50 - 3074 * q^52 - 100 * q^53 - 2632 * q^55 - 1992 * q^59 + 434 * q^60 - 300 * q^64 + 2180 * q^66 - 1736 * q^67 - 2256 * q^69 - 2104 * q^70 - 78 * q^72 + 1746 * q^76 + 1788 * q^77 + 2160 * q^81 - 1700 * q^83 + 886 * q^84 + 4822 * q^86 + 768 * q^87 + 1568 * q^89 + 3100 * q^93 - 2238 * q^94 - 3754 * q^98

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