LMFDB - Embedded newform 338.4.b.f.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(337,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, 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("338.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	338.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:	$19.9426455819$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$0.445042i$ of defining polynomial
Character	$\chi$	$=$	338.337
Dual form			338.4.b.f.337.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -3.66487 q^{3} -4.00000 q^{4} -8.53079i q^{5} +7.32975i q^{6} -4.20105i q^{7} +8.00000i q^{8} -13.5687 q^{9} -17.0616 q^{10} -65.3726i q^{11} +14.6595 q^{12} -8.40209 q^{14} +31.2643i q^{15} +16.0000 q^{16} +26.9019 q^{17} +27.1374i q^{18} +13.3730i q^{19} +34.1232i q^{20} +15.3963i q^{21} -130.745 q^{22} -159.929 q^{23} -29.3190i q^{24} +52.2255 q^{25} +148.679 q^{27} +16.8042i q^{28} -301.288 q^{29} +62.5286 q^{30} +73.0232i q^{31} -32.0000i q^{32} +239.582i q^{33} -53.8038i q^{34} -35.8383 q^{35} +54.2748 q^{36} -118.781i q^{37} +26.7459 q^{38} +68.2464 q^{40} +432.901i q^{41} +30.7926 q^{42} +356.508 q^{43} +261.490i q^{44} +115.752i q^{45} +319.858i q^{46} +588.614i q^{47} -58.6380 q^{48} +325.351 q^{49} -104.451i q^{50} -98.5921 q^{51} -269.462 q^{53} -297.358i q^{54} -557.680 q^{55} +33.6084 q^{56} -49.0102i q^{57} +602.576i q^{58} +230.340i q^{59} -125.057i q^{60} -380.816 q^{61} +146.046 q^{62} +57.0027i q^{63} -64.0000 q^{64} +479.164 q^{66} -435.848i q^{67} -107.608 q^{68} +586.120 q^{69} +71.6765i q^{70} -65.9622i q^{71} -108.550i q^{72} +885.517i q^{73} -237.562 q^{74} -191.400 q^{75} -53.4918i q^{76} -274.633 q^{77} -385.463 q^{79} -136.493i q^{80} -178.536 q^{81} +865.803 q^{82} +254.207i q^{83} -61.5852i q^{84} -229.495i q^{85} -713.016i q^{86} +1104.18 q^{87} +522.980 q^{88} -372.612i q^{89} +231.504 q^{90} +639.716 q^{92} -267.621i q^{93} +1177.23 q^{94} +114.082 q^{95} +117.276i q^{96} -1313.88i q^{97} -650.702i q^{98} +887.020i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24 q^{3} - 24 q^{4} + 18 q^{9} + 48 q^{10} + 96 q^{12} + 108 q^{14} + 96 q^{16} + 180 q^{17} - 328 q^{22} - 38 q^{23} + 122 q^{25} - 138 q^{27} - 202 q^{29} - 360 q^{30} - 916 q^{35} - 72 q^{36} + 520 q^{38}+ \cdots - 3658 q^{95}+O(q^{100}) $ 6 * q - 24 * q^3 - 24 * q^4 + 18 * q^9 + 48 * q^10 + 96 * q^12 + 108 * q^14 + 96 * q^16 + 180 * q^17 - 328 * q^22 - 38 * q^23 + 122 * q^25 - 138 * q^27 - 202 * q^29 - 360 * q^30 - 916 * q^35 - 72 * q^36 + 520 * q^38 - 192 * q^40 - 936 * q^42 + 2410 * q^43 - 384 * q^48 + 368 * q^49 + 414 * q^51 - 2190 * q^53 - 1052 * q^55 - 432 * q^56 - 2216 * q^61 + 2076 * q^62 - 384 * q^64 + 584 * q^66 - 720 * q^68 + 628 * q^69 - 336 * q^74 + 1010 * q^75 - 960 * q^77 - 3922 * q^79 + 1830 * q^81 + 748 * q^82 + 3272 * q^87 + 1312 * q^88 + 2580 * q^90 + 152 * q^92 + 2144 * q^94 - 3658 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$.

$n$	$171$
$\chi(n)$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	−3.66487	−0.705305	−0.352653	−	0.935754i	$-0.614720\pi$
$3$	−3.66487	−0.705305	−0.352653	+	0.935754i	$0.614720\pi$
$4$	−4.00000	−0.500000
$5$	− 8.53079i	− 0.763017i	−0.924365	−	0.381509i	$-0.875405\pi$
$5$	− 8.53079i	− 0.763017i	0.924365	−	0.381509i	$-0.124595\pi$
$6$	7.32975i	0.498726i
$7$	− 4.20105i	− 0.226835i	−0.993547	−	0.113418i	$-0.963820\pi$
$7$	− 4.20105i	− 0.226835i	0.993547	−	0.113418i	$-0.0361798\pi$
$8$	8.00000i	0.353553i
$9$	−13.5687	−0.502544
$10$	−17.0616	−0.539535
$11$	− 65.3726i	− 1.79187i	−0.444186	−	0.895935i	$-0.646507\pi$
$11$	− 65.3726i	− 1.79187i	0.444186	−	0.895935i	$-0.353493\pi$
$12$	14.6595	0.352653
$13$	0	0
$13$	0	0
$14$	−8.40209	−0.160397
$15$	31.2643i	0.538160i
$16$	16.0000	0.250000
$17$	26.9019	0.383804	0.191902	−	0.981414i	$-0.438534\pi$
$17$	26.9019	0.383804	0.191902	+	0.981414i	$0.438534\pi$
$18$	27.1374i	0.355352i
$19$	13.3730i	0.161472i	0.996736	+	0.0807360i	$0.0257271\pi$
$19$	13.3730i	0.161472i	−0.996736	+	0.0807360i	$0.974273\pi$
$20$	34.1232i	0.381509i
$21$	15.3963i	0.159988i
$22$	−130.745	−1.26704
$23$	−159.929	−1.44989	−0.724946	−	0.688806i	$-0.758135\pi$
$23$	−159.929	−1.44989	−0.724946	+	0.688806i	$0.758135\pi$
$24$	− 29.3190i	− 0.249363i
$25$	52.2255	0.417804
$26$	0	0
$27$	148.679	1.05975
$28$	16.8042i	0.113418i
$29$	−301.288	−1.92923	−0.964616	−	0.263658i	$-0.915071\pi$
$29$	−301.288	−1.92923	−0.964616	+	0.263658i	$0.915071\pi$
$30$	62.5286	0.380537
$31$	73.0232i	0.423076i	0.977370	+	0.211538i	$0.0678472\pi$
$31$	73.0232i	0.423076i	−0.977370	+	0.211538i	$0.932153\pi$
$32$	− 32.0000i	− 0.176777i
$33$	239.582i	1.26382i
$34$	− 53.8038i	− 0.271390i
$35$	−35.8383	−0.173079
$36$	54.2748	0.251272
$37$	− 118.781i	− 0.527769i	−0.964554	−	0.263885i	$-0.914996\pi$
$37$	− 118.781i	− 0.527769i	0.964554	−	0.263885i	$-0.0850038\pi$
$38$	26.7459	0.114178
$39$	0	0
$40$	68.2464	0.269767
$41$	432.901i	1.64897i	0.565883	+	0.824486i	$0.308535\pi$
$41$	432.901i	1.64897i	−0.565883	+	0.824486i	$0.691465\pi$
$42$	30.7926	0.113129
$43$	356.508	1.26435	0.632174	−	0.774827i	$-0.282163\pi$
$43$	356.508	1.26435	0.632174	+	0.774827i	$0.282163\pi$
$44$	261.490i	0.895935i
$45$	115.752i	0.383450i
$46$	319.858i	1.02523i
$47$	588.614i	1.82677i	0.407096	+	0.913385i	$0.366541\pi$
$47$	588.614i	1.82677i	−0.407096	+	0.913385i	$0.633459\pi$
$48$	−58.6380	−0.176326
$49$	325.351	0.948546
$50$	− 104.451i	− 0.295432i
$51$	−98.5921	−0.270699
$52$	0	0
$53$	−269.462	−0.698366	−0.349183	−	0.937054i	$-0.613541\pi$
$53$	−269.462	−0.698366	−0.349183	+	0.937054i	$0.613541\pi$
$54$	− 297.358i	− 0.749358i
$55$	−557.680	−1.36723
$56$	33.6084	0.0801983
$57$	− 49.0102i	− 0.113887i
$58$	602.576i	1.36417i
$59$	230.340i	0.508265i	0.967169	+	0.254133i	$0.0817900\pi$
$59$	230.340i	0.508265i	−0.967169	+	0.254133i	$0.918210\pi$
$60$	− 125.057i	− 0.269080i
$61$	−380.816	−0.799319	−0.399659	−	0.916664i	$-0.630872\pi$
$61$	−380.816	−0.799319	−0.399659	+	0.916664i	$0.630872\pi$
$62$	146.046	0.299160
$63$	57.0027i	0.113995i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	479.164	0.893652
$67$	− 435.848i	− 0.794736i	−0.917659	−	0.397368i	$-0.869924\pi$
$67$	− 435.848i	− 0.794736i	0.917659	−	0.397368i	$-0.130076\pi$
$68$	−107.608	−0.191902
$69$	586.120	1.02262
$70$	71.6765i	0.122385i
$71$	− 65.9622i	− 0.110257i	−0.998479	−	0.0551287i	$-0.982443\pi$
$71$	− 65.9622i	− 0.110257i	0.998479	−	0.0551287i	$-0.0175569\pi$
$72$	− 108.550i	− 0.177676i
$73$	885.517i	1.41975i	0.704326	+	0.709876i	$0.251249\pi$
$73$	885.517i	1.41975i	−0.704326	+	0.709876i	$0.748751\pi$
$74$	−237.562	−0.373189
$75$	−191.400	−0.294680
$76$	− 53.4918i	− 0.0807360i
$77$	−274.633	−0.406459
$78$	0	0
$79$	−385.463	−0.548962	−0.274481	−	0.961592i	$-0.588506\pi$
$79$	−385.463	−0.548962	−0.274481	+	0.961592i	$0.588506\pi$
$80$	− 136.493i	− 0.190754i
$81$	−178.536	−0.244905
$82$	865.803	1.16600
$83$	254.207i	0.336179i	0.985772	+	0.168089i	$0.0537597\pi$
$83$	254.207i	0.336179i	−0.985772	+	0.168089i	$0.946240\pi$
$84$	− 61.5852i	− 0.0799940i
$85$	− 229.495i	− 0.292849i
$86$	− 713.016i	− 0.894029i
$87$	1104.18	1.36070
$88$	522.980	0.633522
$89$	− 372.612i	− 0.443785i	−0.975071	−	0.221892i	$-0.928777\pi$
$89$	− 372.612i	− 0.443785i	0.975071	−	0.221892i	$-0.0712234\pi$
$90$	231.504	0.271140
$91$	0	0
$92$	639.716	0.724946
$93$	− 267.621i	− 0.298398i
$94$	1177.23	1.29172
$95$	114.082	0.123206
$96$	117.276i	0.124682i
$97$	− 1313.88i	− 1.37531i	−0.726039	−	0.687653i	$-0.758641\pi$
$97$	− 1313.88i	− 1.37531i	0.726039	−	0.687653i	$-0.241359\pi$
$98$	− 650.702i	− 0.670723i
$99$	887.020i	0.900494i
$100$	−208.902	−0.208902
$101$	−1463.72	−1.44204	−0.721019	−	0.692915i	$-0.756326\pi$
$101$	−1463.72	−1.44204	−0.721019	+	0.692915i	$0.756326\pi$
$102$	197.184i	0.191413i
$103$	−210.886	−0.201740	−0.100870	−	0.994900i	$-0.532163\pi$
$103$	−210.886	−0.201740	−0.100870	+	0.994900i	$0.532163\pi$
$104$	0	0
$105$	131.343	0.122074
$106$	538.924i	0.493820i
$107$	391.061	0.353321	0.176660	−	0.984272i	$-0.443471\pi$
$107$	391.061	0.353321	0.176660	+	0.984272i	$0.443471\pi$
$108$	−594.717	−0.529876
$109$	− 1331.40i	− 1.16996i	−0.811049	−	0.584978i	$-0.801103\pi$
$109$	− 1331.40i	− 1.16996i	0.811049	−	0.584978i	$-0.198897\pi$
$110$	1115.36i	0.966776i
$111$	435.317i	0.372238i
$112$	− 67.2167i	− 0.0567088i
$113$	−711.912	−0.592665	−0.296332	−	0.955085i	$-0.595764\pi$
$113$	−711.912	−0.592665	−0.296332	+	0.955085i	$0.595764\pi$
$114$	−98.0204	−0.0805303
$115$	1364.32i	1.10629i
$116$	1205.15	0.964616
$117$	0	0
$118$	460.679	0.359398
$119$	− 113.016i	− 0.0870603i
$120$	−250.114	−0.190268
$121$	−2942.57	−2.21080
$122$	761.631i	0.565204i
$123$	− 1586.53i	− 1.16303i
$124$	− 292.093i	− 0.211538i
$125$	− 1511.87i	− 1.08181i
$126$	114.005	0.0806064
$127$	1171.70	0.818677	0.409338	−	0.912383i	$-0.365760\pi$
$127$	1171.70	0.818677	0.409338	+	0.912383i	$0.365760\pi$
$128$	128.000i	0.0883883i
$129$	−1306.56	−0.891751
$130$	0	0
$131$	−1128.61	−0.752726	−0.376363	−	0.926472i	$-0.622825\pi$
$131$	−1128.61	−0.752726	−0.376363	+	0.926472i	$0.622825\pi$
$132$	− 958.329i	− 0.631908i
$133$	56.1804	0.0366275
$134$	−871.696	−0.561963
$135$	− 1268.35i	− 0.808610i
$136$	215.215i	0.135695i
$137$	− 1747.93i	− 1.09004i	−0.838422	−	0.545022i	$-0.816521\pi$
$137$	− 1747.93i	− 1.09004i	0.838422	−	0.545022i	$-0.183479\pi$
$138$	− 1172.24i	− 0.723099i
$139$	1676.54	1.02304	0.511518	−	0.859273i	$-0.329083\pi$
$139$	1676.54	1.02304	0.511518	+	0.859273i	$0.329083\pi$
$140$	143.353	0.0865396
$141$	− 2157.20i	− 1.28843i
$142$	−131.924	−0.0779637
$143$	0	0
$144$	−217.099	−0.125636
$145$	2570.22i	1.47204i
$146$	1771.03	1.00392
$147$	−1192.37	−0.669014
$148$	475.124i	0.263885i
$149$	896.961i	0.493167i	0.969122	+	0.246584i	$0.0793080\pi$
$149$	896.961i	0.493167i	−0.969122	+	0.246584i	$0.920692\pi$
$150$	382.800i	0.208370i
$151$	− 2078.28i	− 1.12005i	−0.828475	−	0.560027i	$-0.810791\pi$
$151$	− 2078.28i	− 1.12005i	0.828475	−	0.560027i	$-0.189209\pi$
$152$	−106.984	−0.0570890
$153$	−365.024	−0.192879
$154$	549.266i	0.287410i
$155$	622.946	0.322814
$156$	0	0
$157$	−3494.75	−1.77650	−0.888252	−	0.459357i	$-0.848080\pi$
$157$	−3494.75	−1.77650	−0.888252	+	0.459357i	$0.848080\pi$
$158$	770.927i	0.388175i
$159$	987.544	0.492562
$160$	−272.985	−0.134884
$161$	671.869i	0.328887i
$162$	357.071i	0.173174i
$163$	− 451.254i	− 0.216840i	−0.994105	−	0.108420i	$-0.965421\pi$
$163$	− 451.254i	− 0.216840i	0.994105	−	0.108420i	$-0.0345792\pi$
$164$	− 1731.61i	− 0.824486i
$165$	2043.83	0.964313
$166$	508.414	0.237714
$167$	3203.35i	1.48433i	0.670218	+	0.742164i	$0.266200\pi$
$167$	3203.35i	1.48433i	−0.670218	+	0.742164i	$0.733800\pi$
$168$	−123.170	−0.0565643
$169$	0	0
$170$	−458.989	−0.207076
$171$	− 181.454i	− 0.0811468i
$172$	−1426.03	−0.632174
$173$	−1899.38	−0.834725	−0.417362	−	0.908740i	$-0.637045\pi$
$173$	−1899.38	−0.834725	−0.417362	+	0.908740i	$0.637045\pi$
$174$	− 2208.36i	− 0.962159i
$175$	− 219.402i	− 0.0947727i
$176$	− 1045.96i	− 0.447967i
$177$	− 844.165i	− 0.358482i
$178$	−745.225	−0.313803
$179$	−2650.89	−1.10691	−0.553455	−	0.832879i	$-0.686691\pi$
$179$	−2650.89	−1.10691	−0.553455	+	0.832879i	$0.686691\pi$
$180$	− 463.007i	− 0.191725i
$181$	2289.94	0.940387	0.470194	−	0.882563i	$-0.344184\pi$
$181$	2289.94	0.940387	0.470194	+	0.882563i	$0.344184\pi$
$182$	0	0
$183$	1395.64	0.563764
$184$	− 1279.43i	− 0.512614i
$185$	−1013.30	−0.402697
$186$	−535.242	−0.210999
$187$	− 1758.65i	− 0.687727i
$188$	− 2354.46i	− 0.913385i
$189$	− 624.608i	− 0.240389i
$190$	− 228.164i	− 0.0871198i
$191$	−338.313	−0.128165	−0.0640825	−	0.997945i	$-0.520412\pi$
$191$	−338.313	−0.128165	−0.0640825	+	0.997945i	$0.520412\pi$
$192$	234.552	0.0881632
$193$	2684.05i	1.00105i	0.865723	+	0.500524i	$0.166859\pi$
$193$	2684.05i	1.00105i	−0.865723	+	0.500524i	$0.833141\pi$
$194$	−2627.77	−0.972489
$195$	0	0
$196$	−1301.40	−0.474273
$197$	− 1897.24i	− 0.686156i	−0.939307	−	0.343078i	$-0.888530\pi$
$197$	− 1897.24i	− 0.686156i	0.939307	−	0.343078i	$-0.111470\pi$
$198$	1774.04	0.636745
$199$	2415.60	0.860491	0.430245	−	0.902712i	$-0.358427\pi$
$199$	2415.60	0.860491	0.430245	+	0.902712i	$0.358427\pi$
$200$	417.804i	0.147716i
$201$	1597.33i	0.560532i
$202$	2927.44i	1.01967i
$203$	1265.72i	0.437618i
$204$	394.368	0.135350
$205$	3692.99	1.25819
$206$	421.772i	0.142652i
$207$	2170.03	0.728635
$208$	0	0
$209$	874.225	0.289337
$210$	− 262.685i	− 0.0863191i
$211$	2668.42	0.870623	0.435312	−	0.900280i	$-0.356638\pi$
$211$	2668.42	0.870623	0.435312	+	0.900280i	$0.356638\pi$
$212$	1077.85	0.349183
$213$	241.743i	0.0777651i
$214$	− 782.122i	− 0.249835i
$215$	− 3041.30i	− 0.964719i
$216$	1189.43i	0.374679i
$217$	306.774	0.0959685
$218$	−2662.81	−0.827284
$219$	− 3245.31i	− 1.00136i
$220$	2230.72	0.683614
$221$	0	0
$222$	870.634	0.263212
$223$	286.358i	0.0859907i	0.999075	+	0.0429953i	$0.0136901\pi$
$223$	286.358i	0.0859907i	−0.999075	+	0.0429953i	$0.986310\pi$
$224$	−134.433	−0.0400992
$225$	−708.632	−0.209965
$226$	1423.82i	0.419077i
$227$	− 5201.24i	− 1.52079i	−0.649463	−	0.760393i	$-0.725006\pi$
$227$	− 5201.24i	− 1.52079i	0.649463	−	0.760393i	$-0.274994\pi$
$228$	196.041i	0.0569435i
$229$	890.458i	0.256957i	0.991712	+	0.128478i	$0.0410093\pi$
$229$	890.458i	0.256957i	−0.991712	+	0.128478i	$0.958991\pi$
$230$	2728.64	0.782267
$231$	1006.50	0.286678
$232$	− 2410.30i	− 0.682087i
$233$	−4753.11	−1.33642	−0.668212	−	0.743971i	$-0.732940\pi$
$233$	−4753.11	−1.33642	−0.668212	+	0.743971i	$0.732940\pi$
$234$	0	0
$235$	5021.35	1.39386
$236$	− 921.358i	− 0.254133i
$237$	1412.67	0.387186
$238$	−226.032	−0.0615609
$239$	2292.62i	0.620491i	0.950656	+	0.310245i	$0.100411\pi$
$239$	2292.62i	0.620491i	−0.950656	+	0.310245i	$0.899589\pi$
$240$	500.229i	0.134540i
$241$	1975.21i	0.527943i	0.964530	+	0.263972i	$0.0850325\pi$
$241$	1975.21i	0.527943i	−0.964530	+	0.263972i	$0.914967\pi$
$242$	5885.14i	1.56327i
$243$	−3360.03	−0.887020
$244$	1523.26	0.399659
$245$	− 2775.50i	− 0.723757i
$246$	−3173.06	−0.822385
$247$	0	0
$248$	−584.186	−0.149580
$249$	− 931.637i	− 0.237109i
$250$	−3023.75	−0.764955
$251$	7465.74	1.87742	0.938711	−	0.344705i	$-0.112021\pi$
$251$	7465.74	1.87742	0.938711	+	0.344705i	$0.112021\pi$
$252$	− 228.011i	− 0.0569974i
$253$	10455.0i	2.59802i
$254$	− 2343.41i	− 0.578892i
$255$	841.069i	0.206548i
$256$	256.000	0.0625000
$257$	554.966	0.134700	0.0673499	−	0.997729i	$-0.478546\pi$
$257$	554.966	0.134700	0.0673499	+	0.997729i	$0.478546\pi$
$258$	2613.11i	0.630563i
$259$	−499.004	−0.119717
$260$	0	0
$261$	4088.08	0.969525
$262$	2257.22i	0.532258i
$263$	−1993.07	−0.467292	−0.233646	−	0.972322i	$-0.575066\pi$
$263$	−1993.07	−0.467292	−0.233646	+	0.972322i	$0.575066\pi$
$264$	−1916.66	−0.446826
$265$	2298.72i	0.532866i
$266$	− 112.361i	− 0.0258996i
$267$	1365.58i	0.313004i
$268$	1743.39i	0.397368i
$269$	−5207.76	−1.18038	−0.590191	−	0.807263i	$-0.700948\pi$
$269$	−5207.76	−1.18038	−0.590191	+	0.807263i	$0.700948\pi$
$270$	−2536.70	−0.571773
$271$	4084.90i	0.915646i	0.889043	+	0.457823i	$0.151371\pi$
$271$	4084.90i	0.915646i	−0.889043	+	0.457823i	$0.848629\pi$
$272$	430.430	0.0959510
$273$	0	0
$274$	−3495.87	−0.770778
$275$	− 3414.12i	− 0.748651i
$276$	−2344.48	−0.511308
$277$	−437.993	−0.0950051	−0.0475026	−	0.998871i	$-0.515126\pi$
$277$	−437.993	−0.0950051	−0.0475026	+	0.998871i	$0.515126\pi$
$278$	− 3353.07i	− 0.723395i
$279$	− 990.829i	− 0.212614i
$280$	− 286.706i	− 0.0611927i
$281$	5462.34i	1.15963i	0.814748	+	0.579815i	$0.196875\pi$
$281$	5462.34i	1.15963i	−0.814748	+	0.579815i	$0.803125\pi$
$282$	−4314.40	−0.911058
$283$	796.676	0.167341	0.0836704	−	0.996493i	$-0.473336\pi$
$283$	796.676	0.167341	0.0836704	+	0.996493i	$0.473336\pi$
$284$	263.849i	0.0551287i
$285$	−418.096	−0.0868978
$286$	0	0
$287$	1818.64	0.374045
$288$	434.198i	0.0888381i
$289$	−4189.29	−0.852694
$290$	5140.45	1.04089
$291$	4815.22i	0.970011i
$292$	− 3542.07i	− 0.709876i
$293$	1697.32i	0.338426i	0.985580	+	0.169213i	$0.0541225\pi$
$293$	1697.32i	0.338426i	−0.985580	+	0.169213i	$0.945877\pi$
$294$	2384.74i	0.473065i
$295$	1964.98	0.387815
$296$	950.247	0.186595
$297$	− 9719.54i	− 1.89894i
$298$	1793.92	0.348722
$299$	0	0
$300$	765.600	0.147340
$301$	− 1497.71i	− 0.286798i
$302$	−4156.56	−0.791997
$303$	5364.36	1.01708
$304$	213.967i	0.0403680i
$305$	3248.66i	0.609894i
$306$	730.047i	0.136386i
$307$	− 9385.86i	− 1.74488i	−0.488718	−	0.872442i	$-0.662535\pi$
$307$	− 9385.86i	− 1.74488i	0.488718	−	0.872442i	$-0.337465\pi$
$308$	1098.53	0.203230
$309$	772.870	0.142288
$310$	− 1245.89i	− 0.228264i
$311$	3282.65	0.598527	0.299264	−	0.954170i	$-0.403259\pi$
$311$	3282.65	0.598527	0.299264	+	0.954170i	$0.403259\pi$
$312$	0	0
$313$	−5924.67	−1.06991	−0.534955	−	0.844880i	$-0.679672\pi$
$313$	−5924.67	−1.06991	−0.534955	+	0.844880i	$0.679672\pi$
$314$	6989.49i	1.25618i
$315$	486.278	0.0869800
$316$	1541.85	0.274481
$317$	6467.88i	1.14597i	0.819566	+	0.572985i	$0.194215\pi$
$317$	6467.88i	1.14597i	−0.819566	+	0.572985i	$0.805785\pi$
$318$	− 1975.09i	− 0.348294i
$319$	19696.0i	3.45693i
$320$	545.971i	0.0953772i
$321$	−1433.19	−0.249199
$322$	1343.74	0.232558
$323$	359.758i	0.0619736i
$324$	714.143	0.122452
$325$	0	0
$326$	−902.508	−0.153329
$327$	4879.42i	0.825177i
$328$	−3463.21	−0.582999
$329$	2472.80	0.414376
$330$	− 4087.65i	− 0.681872i
$331$	3509.24i	0.582735i	0.956611	+	0.291368i	$0.0941103\pi$
$331$	3509.24i	0.582735i	−0.956611	+	0.291368i	$0.905890\pi$
$332$	− 1016.83i	− 0.168089i
$333$	1611.70i	0.265227i
$334$	6406.70	1.04958
$335$	−3718.13	−0.606398
$336$	246.341i	0.0399970i
$337$	1834.82	0.296584	0.148292	−	0.988944i	$-0.452622\pi$
$337$	1834.82	0.296584	0.148292	+	0.988944i	$0.452622\pi$
$338$	0	0
$339$	2609.07	0.418010
$340$	917.978i	0.146425i
$341$	4773.71	0.758097
$342$	−362.907	−0.0573795
$343$	− 2807.77i	− 0.441999i
$344$	2852.06i	0.447014i
$345$	− 5000.07i	− 0.780274i
$346$	3798.76i	0.590239i
$347$	−7259.30	−1.12305	−0.561527	−	0.827459i	$-0.689786\pi$
$347$	−7259.30	−1.12305	−0.561527	+	0.827459i	$0.689786\pi$
$348$	−4416.73	−0.680349
$349$	1795.24i	0.275349i	0.990478	+	0.137674i	$0.0439628\pi$
$349$	1795.24i	0.275349i	−0.990478	+	0.137674i	$0.956037\pi$
$350$	−438.804	−0.0670144
$351$	0	0
$352$	−2091.92	−0.316761
$353$	− 4552.38i	− 0.686398i	−0.939263	−	0.343199i	$-0.888489\pi$
$353$	− 4552.38i	− 0.686398i	0.939263	−	0.343199i	$-0.111511\pi$
$354$	−1688.33	−0.253485
$355$	−562.710	−0.0841283
$356$	1490.45i	0.221892i
$357$	414.190i	0.0614041i
$358$	5301.78i	0.782703i
$359$	− 10165.4i	− 1.49445i	−0.664569	−	0.747227i	$-0.731385\pi$
$359$	− 10165.4i	− 1.49445i	0.664569	−	0.747227i	$-0.268615\pi$
$360$	−926.014	−0.135570
$361$	6680.16	0.973927
$362$	− 4579.88i	− 0.664954i
$363$	10784.2	1.55929
$364$	0	0
$365$	7554.16	1.08330
$366$	− 2791.28i	− 0.398641i
$367$	−10401.6	−1.47945	−0.739723	−	0.672911i	$-0.765044\pi$
$367$	−10401.6	−1.47945	−0.739723	+	0.672911i	$0.765044\pi$
$368$	−2558.87	−0.362473
$369$	− 5873.91i	− 0.828681i
$370$	2026.59i	0.284750i
$371$	1132.02i	0.158414i
$372$	1070.48i	0.149199i
$373$	−7953.41	−1.10405	−0.552027	−	0.833826i	$-0.686145\pi$
$373$	−7953.41	−1.10405	−0.552027	+	0.833826i	$0.686145\pi$
$374$	−3517.29	−0.486296
$375$	5540.83i	0.763006i
$376$	−4708.91	−0.645861
$377$	0	0
$378$	−1249.22	−0.169981
$379$	9014.81i	1.22179i	0.791710	+	0.610897i	$0.209191\pi$
$379$	9014.81i	1.22179i	−0.791710	+	0.610897i	$0.790809\pi$
$380$	−456.328	−0.0616030
$381$	−4294.15	−0.577417
$382$	676.627i	0.0906263i
$383$	− 2496.24i	− 0.333034i	−0.986039	−	0.166517i	$-0.946748\pi$
$383$	− 2496.24i	− 0.333034i	0.986039	−	0.166517i	$-0.0532520\pi$
$384$	− 469.104i	− 0.0623408i
$385$	2342.84i	0.310135i
$386$	5368.10	0.707848
$387$	−4837.35	−0.635391
$388$	5255.54i	0.687653i
$389$	−10896.9	−1.42029	−0.710145	−	0.704056i	$-0.751371\pi$
$389$	−10896.9	−1.42029	−0.710145	+	0.704056i	$0.751371\pi$
$390$	0	0
$391$	−4302.40	−0.556475
$392$	2602.81i	0.335362i
$393$	4136.21	0.530902
$394$	−3794.48	−0.485186
$395$	3288.31i	0.418868i
$396$	− 3548.08i	− 0.450247i
$397$	− 10887.1i	− 1.37635i	−0.725546	−	0.688174i	$-0.758413\pi$
$397$	− 10887.1i	− 1.37635i	0.725546	−	0.688174i	$-0.241587\pi$
$398$	− 4831.21i	− 0.608459i
$399$	−205.894	−0.0258336
$400$	835.609	0.104451
$401$	− 9968.76i	− 1.24144i	−0.784034	−	0.620718i	$-0.786841\pi$
$401$	− 9968.76i	− 1.24144i	0.784034	−	0.620718i	$-0.213159\pi$
$402$	3194.66	0.396356
$403$	0	0
$404$	5854.89	0.721019
$405$	1523.05i	0.186867i
$406$	2531.45	0.309442
$407$	−7765.01	−0.945693
$408$	− 788.737i	− 0.0957066i
$409$	4036.32i	0.487979i	0.969778	+	0.243989i	$0.0784562\pi$
$409$	4036.32i	0.487979i	−0.969778	+	0.243989i	$0.921544\pi$
$410$	− 7385.98i	− 0.889677i
$411$	6405.96i	0.768814i
$412$	843.543	0.100870
$413$	967.667	0.115292
$414$	− 4340.06i	− 0.515223i
$415$	2168.59	0.256510
$416$	0	0
$417$	−6144.29	−0.721552
$418$	− 1748.45i	− 0.204592i
$419$	15194.9	1.77164	0.885821	−	0.464028i	$-0.153596\pi$
$419$	15194.9	1.77164	0.885821	+	0.464028i	$0.153596\pi$
$420$	−525.371	−0.0610368
$421$	10154.8i	1.17556i	0.809019	+	0.587782i	$0.199999\pi$
$421$	10154.8i	1.17556i	−0.809019	+	0.587782i	$0.800001\pi$
$422$	− 5336.84i	− 0.615624i
$423$	− 7986.73i	− 0.918033i
$424$	− 2155.69i	− 0.246910i
$425$	1404.97	0.160355
$426$	483.486	0.0549882
$427$	1599.82i	0.181314i
$428$	−1564.24	−0.176660
$429$	0	0
$430$	−6082.59	−0.682159
$431$	− 6616.84i	− 0.739494i	−0.929133	−	0.369747i	$-0.879444\pi$
$431$	− 6616.84i	− 0.739494i	0.929133	−	0.369747i	$-0.120556\pi$
$432$	2378.87	0.264938
$433$	−5697.60	−0.632354	−0.316177	−	0.948700i	$-0.602399\pi$
$433$	−5697.60	−0.632354	−0.316177	+	0.948700i	$0.602399\pi$
$434$	− 613.548i	− 0.0678600i
$435$	− 9419.55i	− 1.03824i
$436$	5325.61i	0.584978i
$437$	− 2138.73i	− 0.234117i
$438$	−6490.62	−0.708068
$439$	−436.007	−0.0474020	−0.0237010	−	0.999719i	$-0.507545\pi$
$439$	−436.007	−0.0474020	−0.0237010	+	0.999719i	$0.507545\pi$
$440$	− 4461.44i	− 0.483388i
$441$	−4414.59	−0.476686
$442$	0	0
$443$	6609.58	0.708873	0.354436	−	0.935080i	$-0.384673\pi$
$443$	6609.58	0.708873	0.354436	+	0.935080i	$0.384673\pi$
$444$	− 1741.27i	− 0.186119i
$445$	−3178.68	−0.338616
$446$	572.715	0.0608046
$447$	− 3287.25i	− 0.347833i
$448$	268.867i	0.0283544i
$449$	− 8015.31i	− 0.842463i	−0.906953	−	0.421231i	$-0.861598\pi$
$449$	− 8015.31i	− 0.842463i	0.906953	−	0.421231i	$-0.138402\pi$
$450$	1417.26i	0.148468i
$451$	28299.9	2.95474
$452$	2847.65	0.296332
$453$	7616.64i	0.789980i
$454$	−10402.5	−1.07536
$455$	0	0
$456$	392.082	0.0402652
$457$	− 4507.03i	− 0.461335i	−0.973033	−	0.230668i	$-0.925909\pi$
$457$	− 4507.03i	− 0.461335i	0.973033	−	0.230668i	$-0.0740910\pi$
$458$	1780.92	0.181696
$459$	3999.75	0.406737
$460$	− 5457.29i	− 0.553146i
$461$	2103.55i	0.212521i	0.994338	+	0.106260i	$0.0338877\pi$
$461$	2103.55i	0.212521i	−0.994338	+	0.106260i	$0.966112\pi$
$462$	− 2012.99i	− 0.202712i
$463$	− 5468.28i	− 0.548883i	−0.961604	−	0.274441i	$-0.911507\pi$
$463$	− 5468.28i	− 0.548883i	0.961604	−	0.274441i	$-0.0884929\pi$
$464$	−4820.60	−0.482308
$465$	−2283.02	−0.227683
$466$	9506.22i	0.944994i
$467$	6043.79	0.598872	0.299436	−	0.954116i	$-0.403201\pi$
$467$	6043.79	0.598872	0.299436	+	0.954116i	$0.403201\pi$
$468$	0	0
$469$	−1831.02	−0.180274
$470$	− 10042.7i	− 0.985606i
$471$	12807.8	1.25298
$472$	−1842.72	−0.179699
$473$	− 23305.8i	− 2.26555i
$474$	− 2825.35i	− 0.273782i
$475$	698.410i	0.0674637i
$476$	452.065i	0.0435301i
$477$	3656.25	0.350960
$478$	4585.24	0.438753
$479$	8481.30i	0.809020i	0.914534	+	0.404510i	$0.132558\pi$
$479$	8481.30i	0.809020i	−0.914534	+	0.404510i	$0.867442\pi$
$480$	1000.46	0.0951342
$481$	0	0
$482$	3950.42	0.373312
$483$	− 2462.32i	− 0.231965i
$484$	11770.3	1.10540
$485$	−11208.5	−1.04938
$486$	6720.05i	0.627218i
$487$	− 6623.36i	− 0.616289i	−0.951340	−	0.308145i	$-0.900292\pi$
$487$	− 6623.36i	− 0.616289i	0.951340	−	0.308145i	$-0.0997081\pi$
$488$	− 3046.53i	− 0.282602i
$489$	1653.79i	0.152939i
$490$	−5551.01	−0.511774
$491$	−16379.4	−1.50548	−0.752739	−	0.658319i	$-0.771268\pi$
$491$	−16379.4	−1.50548	−0.752739	+	0.658319i	$0.771268\pi$
$492$	6346.12i	0.581514i
$493$	−8105.21	−0.740447
$494$	0	0
$495$	7566.99	0.687093
$496$	1168.37i	0.105769i
$497$	−277.110	−0.0250102
$498$	−1863.27	−0.167661
$499$	− 7915.70i	− 0.710131i	−0.934841	−	0.355065i	$-0.884459\pi$
$499$	− 7915.70i	− 0.710131i	0.934841	−	0.355065i	$-0.115541\pi$
$500$	6047.50i	0.540905i
$501$	− 11739.9i	− 1.04690i
$502$	− 14931.5i	− 1.32754i
$503$	−479.511	−0.0425056	−0.0212528	−	0.999774i	$-0.506765\pi$
$503$	−479.511	−0.0425056	−0.0212528	+	0.999774i	$0.506765\pi$
$504$	−456.022	−0.0403032
$505$	12486.7i	1.10030i
$506$	20909.9	1.83708
$507$	0	0
$508$	−4686.82	−0.409338
$509$	1192.33i	0.103829i	0.998652	+	0.0519145i	$0.0165323\pi$
$509$	1192.33i	0.103829i	−0.998652	+	0.0519145i	$0.983468\pi$
$510$	1682.14	0.146052
$511$	3720.10	0.322050
$512$	− 512.000i	− 0.0441942i
$513$	1988.28i	0.171120i
$514$	− 1109.93i	− 0.0952471i
$515$	1799.02i	0.153931i
$516$	5226.23	0.445876
$517$	38479.2	3.27333
$518$	998.008i	0.0846524i
$519$	6961.00	0.588736
$520$	0	0
$521$	−23238.5	−1.95412	−0.977062	−	0.212955i	$-0.931691\pi$
$521$	−23238.5	−1.95412	−0.977062	+	0.212955i	$0.931691\pi$
$522$	− 8176.16i	− 0.685557i
$523$	−11026.3	−0.921884	−0.460942	−	0.887430i	$-0.652488\pi$
$523$	−11026.3	−0.921884	−0.460942	+	0.887430i	$0.652488\pi$
$524$	4514.44	0.376363
$525$	804.080i	0.0668437i
$526$	3986.14i	0.330425i
$527$	1964.46i	0.162378i
$528$	3833.32i	0.315954i
$529$	13410.3	1.10219
$530$	4597.45	0.376793
$531$	− 3125.41i	− 0.255426i
$532$	−224.722	−0.0183138
$533$	0	0
$534$	2731.16	0.221327
$535$	− 3336.06i	− 0.269590i
$536$	3486.79	0.280982
$537$	9715.18	0.780709
$538$	10415.5i	0.834657i
$539$	− 21269.0i	− 1.69967i
$540$	5073.41i	0.404305i
$541$	− 8987.70i	− 0.714254i	−0.934056	−	0.357127i	$-0.883756\pi$
$541$	− 8987.70i	− 0.714254i	0.934056	−	0.357127i	$-0.116244\pi$
$542$	8169.80	0.647459
$543$	−8392.35	−0.663260
$544$	− 860.861i	− 0.0678476i
$545$	−11357.9	−0.892697
$546$	0	0
$547$	−10734.8	−0.839101	−0.419550	−	0.907732i	$-0.637812\pi$
$547$	−10734.8	−0.839101	−0.419550	+	0.907732i	$0.637812\pi$
$548$	6991.74i	0.545022i
$549$	5167.17	0.401693
$550$	−6828.23	−0.529376
$551$	− 4029.11i	− 0.311517i
$552$	4688.96i	0.361550i
$553$	1619.35i	0.124524i
$554$	875.985i	0.0671788i
$555$	3713.60	0.284024
$556$	−6706.14	−0.511518
$557$	− 13228.3i	− 1.00628i	−0.864204	−	0.503141i	$-0.832178\pi$
$557$	− 13228.3i	− 1.00628i	0.864204	−	0.503141i	$-0.167822\pi$
$558$	−1981.66	−0.150341
$559$	0	0
$560$	−573.412	−0.0432698
$561$	6445.22i	0.485058i
$562$	10924.7	0.819983
$563$	−10734.0	−0.803522	−0.401761	−	0.915745i	$-0.631602\pi$
$563$	−10734.0	−0.803522	−0.401761	+	0.915745i	$0.631602\pi$
$564$	8628.79i	0.644216i
$565$	6073.18i	0.452213i
$566$	− 1593.35i	− 0.118328i
$567$	750.037i	0.0555531i
$568$	527.698	0.0389819
$569$	−5313.45	−0.391479	−0.195739	−	0.980656i	$-0.562711\pi$
$569$	−5313.45	−0.391479	−0.195739	+	0.980656i	$0.562711\pi$
$570$	836.192i	0.0614460i
$571$	−4629.37	−0.339287	−0.169644	−	0.985505i	$-0.554262\pi$
$571$	−4629.37	−0.339287	−0.169644	+	0.985505i	$0.554262\pi$
$572$	0	0
$573$	1239.88	0.0903954
$574$	− 3637.28i	− 0.264490i
$575$	−8352.38	−0.605771
$576$	868.397	0.0628180
$577$	− 504.750i	− 0.0364177i	−0.999834	−	0.0182088i	$-0.994204\pi$
$577$	− 504.750i	− 0.0364177i	0.999834	−	0.0182088i	$-0.00579638\pi$
$578$	8378.58i	0.602946i
$579$	− 9836.71i	− 0.706045i
$580$	− 10280.9i	− 0.736019i
$581$	1067.94	0.0762572
$582$	9630.44	0.685901
$583$	17615.4i	1.25138i
$584$	−7084.14	−0.501958
$585$	0	0
$586$	3394.65	0.239303
$587$	5532.93i	0.389044i	0.980898	+	0.194522i	$0.0623155\pi$
$587$	5532.93i	0.389044i	−0.980898	+	0.194522i	$0.937684\pi$
$588$	4769.49	0.334507
$589$	−976.536	−0.0683149
$590$	− 3929.96i	− 0.274227i
$591$	6953.15i	0.483950i
$592$	− 1900.49i	− 0.131942i
$593$	18079.8i	1.25202i	0.779816	+	0.626009i	$0.215313\pi$
$593$	18079.8i	1.25202i	−0.779816	+	0.626009i	$0.784687\pi$
$594$	−19439.1	−1.34275
$595$	−964.117	−0.0664285
$596$	− 3587.84i	− 0.246584i
$597$	−8852.88	−0.606909
$598$	0	0
$599$	−1837.55	−0.125342	−0.0626712	−	0.998034i	$-0.519962\pi$
$599$	−1837.55	−0.125342	−0.0626712	+	0.998034i	$0.519962\pi$
$600$	− 1531.20i	− 0.104185i
$601$	−23487.3	−1.59412	−0.797060	−	0.603901i	$-0.793612\pi$
$601$	−23487.3	−1.59412	−0.797060	+	0.603901i	$0.793612\pi$
$602$	−2995.41	−0.202797
$603$	5913.89i	0.399390i
$604$	8313.12i	0.560027i
$605$	25102.5i	1.68688i
$606$	− 10728.7i	− 0.719182i
$607$	−16310.8	−1.09067	−0.545335	−	0.838218i	$-0.683598\pi$
$607$	−16310.8	−1.09067	−0.545335	+	0.838218i	$0.683598\pi$
$608$	427.935	0.0285445
$609$	− 4638.72i	− 0.308654i
$610$	6497.32	0.431260
$611$	0	0
$612$	1460.09	0.0964393
$613$	8453.11i	0.556962i	0.960442	+	0.278481i	$0.0898310\pi$
$613$	8453.11i	0.556962i	−0.960442	+	0.278481i	$0.910169\pi$
$614$	−18771.7	−1.23382
$615$	−13534.4	−0.887411
$616$	− 2197.06i	− 0.143705i
$617$	− 13410.3i	− 0.875008i	−0.899216	−	0.437504i	$-0.855863\pi$
$617$	− 13410.3i	− 0.875008i	0.899216	−	0.437504i	$-0.144137\pi$
$618$	− 1545.74i	− 0.100613i
$619$	890.135i	0.0577990i	0.999582	+	0.0288995i	$0.00920027\pi$
$619$	890.135i	0.0577990i	−0.999582	+	0.0288995i	$0.990800\pi$
$620$	−2491.78	−0.161407
$621$	−23778.1	−1.53653
$622$	− 6565.30i	− 0.423223i
$623$	−1565.36	−0.100666
$624$	0	0
$625$	−6369.30	−0.407635
$626$	11849.3i	0.756541i
$627$	−3203.92	−0.204071
$628$	13979.0	0.888252
$629$	− 3195.43i	− 0.202560i
$630$	− 972.557i	− 0.0615041i
$631$	10582.4i	0.667639i	0.942637	+	0.333819i	$0.108338\pi$
$631$	10582.4i	0.667639i	−0.942637	+	0.333819i	$0.891662\pi$
$632$	− 3083.71i	− 0.194087i
$633$	−9779.42	−0.614055
$634$	12935.8	0.810323
$635$	− 9995.57i	− 0.624665i
$636$	−3950.17	−0.246281
$637$	0	0
$638$	39391.9	2.44442
$639$	895.021i	0.0554092i
$640$	1091.94	0.0674419
$641$	−26884.2	−1.65657	−0.828286	−	0.560305i	$-0.810684\pi$
$641$	−26884.2	−1.65657	−0.828286	+	0.560305i	$0.810684\pi$
$642$	2866.38i	0.176210i
$643$	5691.12i	0.349045i	0.984653	+	0.174523i	$0.0558382\pi$
$643$	5691.12i	0.349045i	−0.984653	+	0.174523i	$0.944162\pi$
$644$	− 2687.48i	− 0.164443i
$645$	11146.0i	0.680422i
$646$	719.516	0.0438220
$647$	−1809.79	−0.109969	−0.0549847	−	0.998487i	$-0.517511\pi$
$647$	−1809.79	−0.109969	−0.0549847	+	0.998487i	$0.517511\pi$
$648$	− 1428.29i	− 0.0865870i
$649$	15057.9	0.910745
$650$	0	0
$651$	−1124.29	−0.0676871
$652$	1805.02i	0.108420i
$653$	9459.16	0.566869	0.283435	−	0.958992i	$-0.408526\pi$
$653$	9459.16	0.566869	0.283435	+	0.958992i	$0.408526\pi$
$654$	9758.85	0.583488
$655$	9627.94i	0.574343i
$656$	6926.42i	0.412243i
$657$	− 12015.3i	− 0.713488i
$658$	− 4945.59i	− 0.293008i
$659$	1015.32	0.0600171	0.0300085	−	0.999550i	$-0.490447\pi$
$659$	1015.32	0.0600171	0.0300085	+	0.999550i	$0.490447\pi$
$660$	−8175.31	−0.482157
$661$	− 23648.4i	− 1.39156i	−0.718257	−	0.695778i	$-0.755060\pi$
$661$	− 23648.4i	− 1.39156i	0.718257	−	0.695778i	$-0.244940\pi$
$662$	7018.49	0.412056
$663$	0	0
$664$	−2033.66	−0.118857
$665$	− 479.264i	− 0.0279474i
$666$	3223.40	0.187544
$667$	48184.7	2.79718
$668$	− 12813.4i	− 0.742164i
$669$	− 1049.46i	− 0.0606497i
$670$	7436.26i	0.428788i
$671$	24894.9i	1.43228i
$672$	492.682	0.0282822
$673$	−17747.0	−1.01649	−0.508243	−	0.861214i	$-0.669705\pi$
$673$	−17747.0	−1.01649	−0.508243	+	0.861214i	$0.669705\pi$
$674$	− 3669.63i	− 0.209717i
$675$	7764.85	0.442769
$676$	0	0
$677$	10754.2	0.610511	0.305256	−	0.952270i	$-0.401258\pi$
$677$	10754.2	0.610511	0.305256	+	0.952270i	$0.401258\pi$
$678$	− 5218.14i	− 0.295577i
$679$	−5519.69	−0.311968
$680$	1835.96	0.103538
$681$	19061.9i	1.07262i
$682$	− 9547.42i	− 0.536055i
$683$	23981.7i	1.34354i	0.740762	+	0.671768i	$0.234465\pi$
$683$	23981.7i	1.34354i	−0.740762	+	0.671768i	$0.765535\pi$
$684$	725.815i	0.0405734i
$685$	−14911.3	−0.831723
$686$	−5615.55	−0.312540
$687$	− 3263.42i	− 0.181233i
$688$	5704.13	0.316087
$689$	0	0
$690$	−10000.1	−0.551737
$691$	29001.4i	1.59662i	0.602244	+	0.798312i	$0.294273\pi$
$691$	29001.4i	1.59662i	−0.602244	+	0.798312i	$0.705727\pi$
$692$	7597.53	0.417362
$693$	3726.41	0.204264
$694$	14518.6i	0.794119i
$695$	− 14302.2i	− 0.780594i
$696$	8833.46i	0.481079i
$697$	11645.9i	0.632882i
$698$	3590.47	0.194701
$699$	17419.6	0.942587
$700$	877.608i	0.0473864i
$701$	31031.5	1.67196	0.835979	−	0.548761i	$-0.184900\pi$
$701$	31031.5	1.67196	0.835979	+	0.548761i	$0.184900\pi$
$702$	0	0
$703$	1588.45	0.0852199
$704$	4183.84i	0.223984i
$705$	−18402.6	−0.983096
$706$	−9104.75	−0.485357
$707$	6149.16i	0.327105i
$708$	3376.66i	0.179241i
$709$	4382.59i	0.232146i	0.993241	+	0.116073i	$0.0370307\pi$
$709$	4382.59i	0.232146i	−0.993241	+	0.116073i	$0.962969\pi$
$710$	1125.42i	0.0594877i
$711$	5230.23	0.275878
$712$	2980.90	0.156902
$713$	− 11678.5i	− 0.613414i
$714$	828.380	0.0434192
$715$	0	0
$716$	10603.6	0.553455
$717$	− 8402.17i	− 0.437636i
$718$	−20330.8	−1.05674
$719$	28764.6	1.49199	0.745994	−	0.665953i	$-0.231975\pi$
$719$	28764.6	1.49199	0.745994	+	0.665953i	$0.231975\pi$
$720$	1852.03i	0.0958625i
$721$	885.941i	0.0457617i
$722$	− 13360.3i	− 0.688670i
$723$	− 7238.89i	− 0.372361i
$724$	−9159.77	−0.470194
$725$	−15734.9	−0.806042
$726$	− 21568.3i	− 1.10258i
$727$	−25408.5	−1.29621	−0.648107	−	0.761549i	$-0.724439\pi$
$727$	−25408.5	−1.29621	−0.648107	+	0.761549i	$0.724439\pi$
$728$	0	0
$729$	17134.5	0.870525
$730$	− 15108.3i	− 0.766006i
$731$	9590.74	0.485262
$732$	−5582.57	−0.281882
$733$	− 22341.8i	− 1.12580i	−0.826524	−	0.562902i	$-0.809685\pi$
$733$	− 22341.8i	− 1.12580i	0.826524	−	0.562902i	$-0.190315\pi$
$734$	20803.1i	1.04613i
$735$	10171.9i	0.510470i
$736$	5117.73i	0.256307i
$737$	−28492.5	−1.42406
$738$	−11747.8	−0.585966
$739$	8210.90i	0.408718i	0.978896	+	0.204359i	$0.0655110\pi$
$739$	8210.90i	0.408718i	−0.978896	+	0.204359i	$0.934489\pi$
$740$	4053.18	0.201349
$741$	0	0
$742$	2264.04	0.112016
$743$	− 32752.3i	− 1.61718i	−0.588373	−	0.808590i	$-0.700231\pi$
$743$	− 32752.3i	− 1.61718i	0.588373	−	0.808590i	$-0.299769\pi$
$744$	2140.97	0.105500
$745$	7651.79	0.376295
$746$	15906.8i	0.780684i
$747$	− 3449.26i	− 0.168945i
$748$	7034.58i	0.343863i
$749$	− 1642.87i	− 0.0801455i
$750$	11081.7	0.539527
$751$	−8907.41	−0.432804	−0.216402	−	0.976304i	$-0.569432\pi$
$751$	−8907.41	−0.432804	−0.216402	+	0.976304i	$0.569432\pi$
$752$	9417.83i	0.456693i
$753$	−27361.0	−1.32416
$754$	0	0
$755$	−17729.4	−0.854620
$756$	2498.43i	0.120195i
$757$	−10744.6	−0.515879	−0.257940	−	0.966161i	$-0.583044\pi$
$757$	−10744.6	−0.515879	−0.257940	+	0.966161i	$0.583044\pi$
$758$	18029.6	0.863938
$759$	− 38316.2i	− 1.83240i
$760$	912.656i	0.0435599i
$761$	16997.8i	0.809683i	0.914387	+	0.404841i	$0.132673\pi$
$761$	16997.8i	0.809683i	−0.914387	+	0.404841i	$0.867327\pi$
$762$	8588.30i	0.408296i
$763$	−5593.28	−0.265387
$764$	1353.25	0.0640825
$765$	3113.94i	0.147170i
$766$	−4992.48	−0.235490
$767$	0	0
$768$	−938.208	−0.0440816
$769$	19032.4i	0.892492i	0.894910	+	0.446246i	$0.147239\pi$
$769$	19032.4i	0.892492i	−0.894910	+	0.446246i	$0.852761\pi$
$770$	4685.68	0.219299
$771$	−2033.88	−0.0950045
$772$	− 10736.2i	− 0.500524i
$773$	23750.6i	1.10511i	0.833477	+	0.552555i	$0.186347\pi$
$773$	23750.6i	1.10511i	−0.833477	+	0.552555i	$0.813653\pi$
$774$	9674.69i	0.449289i
$775$	3813.68i	0.176763i
$776$	10511.1	0.486244
$777$	1828.79	0.0844368
$778$	21793.7i	1.00430i
$779$	−5789.17	−0.266263
$780$	0	0
$781$	−4312.12	−0.197567
$782$	8604.79i	0.393487i
$783$	−44795.2	−2.04451
$784$	5205.62	0.237136
$785$	29813.0i	1.35550i
$786$	− 8272.42i	− 0.375404i
$787$	− 17825.9i	− 0.807403i	−0.914891	−	0.403702i	$-0.867723\pi$
$787$	− 17825.9i	− 0.807403i	0.914891	−	0.403702i	$-0.132277\pi$
$788$	7588.96i	0.343078i
$789$	7304.34	0.329584
$790$	6576.62	0.296184
$791$	2990.78i	0.134437i
$792$	−7096.16	−0.318373
$793$	0	0
$794$	−21774.3	−0.973225
$795$	− 8424.53i	− 0.375833i
$796$	−9662.42	−0.430245
$797$	26461.2	1.17604	0.588020	−	0.808846i	$-0.299908\pi$
$797$	26461.2	1.17604	0.588020	+	0.808846i	$0.299908\pi$
$798$	411.788i	0.0182671i
$799$	15834.8i	0.701122i
$800$	− 1671.22i	− 0.0738581i
$801$	5055.87i	0.223021i
$802$	−19937.5	−0.877828
$803$	57888.5	2.54401
$804$	− 6389.32i	− 0.280266i
$805$	5731.58	0.250946
$806$	0	0
$807$	19085.8	0.832530
$808$	− 11709.8i	− 0.509837i
$809$	21873.3	0.950588	0.475294	−	0.879827i	$-0.342342\pi$
$809$	21873.3	0.950588	0.475294	+	0.879827i	$0.342342\pi$
$810$	3046.10	0.132135
$811$	39113.8i	1.69355i	0.531949	+	0.846776i	$0.321460\pi$
$811$	39113.8i	1.69355i	−0.531949	+	0.846776i	$0.678540\pi$
$812$	− 5062.90i	− 0.218809i
$813$	− 14970.6i	− 0.645810i
$814$	15530.0i	0.668706i
$815$	−3849.56	−0.165453
$816$	−1577.47	−0.0676748
$817$	4767.57i	0.204157i
$818$	8072.65	0.345053
$819$	0	0
$820$	−14772.0	−0.629097
$821$	− 26296.8i	− 1.11786i	−0.829214	−	0.558931i	$-0.811212\pi$
$821$	− 26296.8i	− 1.11786i	0.829214	−	0.558931i	$-0.188788\pi$
$822$	12811.9	0.543634
$823$	−24681.1	−1.04536	−0.522678	−	0.852530i	$-0.675067\pi$
$823$	−24681.1	−1.04536	−0.522678	+	0.852530i	$0.675067\pi$
$824$	− 1687.09i	− 0.0713258i
$825$	12512.3i	0.528027i
$826$	− 1935.33i	− 0.0815241i
$827$	3081.68i	0.129578i	0.997899	+	0.0647888i	$0.0206374\pi$
$827$	3081.68i	0.129578i	−0.997899	+	0.0647888i	$0.979363\pi$
$828$	−8680.12	−0.364317
$829$	8498.53	0.356051	0.178026	−	0.984026i	$-0.443029\pi$
$829$	8498.53	0.356051	0.178026	+	0.984026i	$0.443029\pi$
$830$	− 4337.18i	− 0.181380i
$831$	1605.19	0.0670076
$832$	0	0
$833$	8752.57	0.364056
$834$	12288.6i	0.510215i
$835$	27327.1	1.13257
$836$	−3496.90	−0.144668
$837$	10857.0i	0.448356i
$838$	− 30389.7i	− 1.25274i
$839$	− 18680.1i	− 0.768664i	−0.923195	−	0.384332i	$-0.874432\pi$
$839$	− 18680.1i	− 0.768664i	0.923195	−	0.384332i	$-0.125568\pi$
$840$	1050.74i	0.0431596i
$841$	66385.3	2.72194
$842$	20309.5	0.831249
$843$	− 20018.8i	− 0.817894i
$844$	−10673.7	−0.435312
$845$	0	0
$846$	−15973.5	−0.649147
$847$	12361.9i	0.501486i
$848$	−4311.39	−0.174592
$849$	−2919.72	−0.118026
$850$	− 2809.93i	− 0.113388i
$851$	18996.5i	0.765208i
$852$	− 966.973i	− 0.0388826i
$853$	− 24129.9i	− 0.968574i	−0.874909	−	0.484287i	$-0.839079\pi$
$853$	− 24129.9i	− 0.968574i	0.874909	−	0.484287i	$-0.160921\pi$
$854$	3199.65	0.128208
$855$	−1547.94	−0.0619164
$856$	3128.49i	0.124918i
$857$	7739.30	0.308482	0.154241	−	0.988033i	$-0.450707\pi$
$857$	7739.30	0.308482	0.154241	+	0.988033i	$0.450707\pi$
$858$	0	0
$859$	4302.59	0.170899	0.0854496	−	0.996342i	$-0.472767\pi$
$859$	4302.59	0.170899	0.0854496	+	0.996342i	$0.472767\pi$
$860$	12165.2i	0.482360i
$861$	−6665.08	−0.263816
$862$	−13233.7	−0.522901
$863$	− 16253.2i	− 0.641096i	−0.947232	−	0.320548i	$-0.896133\pi$
$863$	− 16253.2i	− 0.641096i	0.947232	−	0.320548i	$-0.103867\pi$
$864$	− 4757.73i	− 0.187340i
$865$	16203.2i	0.636910i
$866$	11395.2i	0.447142i
$867$	15353.2	0.601410
$868$	−1227.10	−0.0479842
$869$	25198.7i	0.983669i
$870$	−18839.1	−0.734144
$871$	0	0
$872$	10651.2	0.413642
$873$	17827.7i	0.691152i
$874$	−4277.45	−0.165546
$875$	−6351.46	−0.245392
$876$	12981.2i	0.500680i
$877$	− 9441.33i	− 0.363525i	−0.983342	−	0.181762i	$-0.941820\pi$
$877$	− 9441.33i	− 0.363525i	0.983342	−	0.181762i	$-0.0581802\pi$
$878$	872.014i	0.0335183i
$879$	− 6220.48i	− 0.238693i
$880$	−8922.88	−0.341807
$881$	11872.4	0.454021	0.227011	−	0.973892i	$-0.427105\pi$
$881$	11872.4	0.454021	0.227011	+	0.973892i	$0.427105\pi$
$882$	8829.18i	0.337068i
$883$	493.891	0.0188231	0.00941153	−	0.999956i	$-0.497004\pi$
$883$	493.891	0.0188231	0.00941153	+	0.999956i	$0.497004\pi$
$884$	0	0
$885$	−7201.40	−0.273528
$886$	− 13219.2i	− 0.501249i
$887$	20543.6	0.777664	0.388832	−	0.921309i	$-0.372879\pi$
$887$	20543.6	0.777664	0.388832	+	0.921309i	$0.372879\pi$
$888$	−3482.54	−0.131606
$889$	− 4922.38i	− 0.185705i
$890$	6357.36i	0.239437i
$891$	11671.3i	0.438838i
$892$	− 1145.43i	− 0.0429953i
$893$	−7871.52	−0.294972
$894$	−6574.50	−0.245955
$895$	22614.2i	0.844592i
$896$	537.734	0.0200496
$897$	0	0
$898$	−16030.6	−0.595711
$899$	− 22001.0i	− 0.816212i
$900$	2834.53	0.104983
$901$	−7249.03	−0.268036
$902$	− 56599.7i	− 2.08932i
$903$	5488.90i	0.202280i
$904$	− 5695.30i	− 0.209539i
$905$	− 19535.0i	− 0.717532i
$906$	15233.3	0.558600
$907$	5386.61	0.197199	0.0985994	−	0.995127i	$-0.468564\pi$
$907$	5386.61	0.197199	0.0985994	+	0.995127i	$0.468564\pi$
$908$	20805.0i	0.760393i
$909$	19860.8	0.724688
$910$	0	0
$911$	31793.5	1.15627	0.578137	−	0.815940i	$-0.303780\pi$
$911$	31793.5	1.15627	0.578137	+	0.815940i	$0.303780\pi$
$912$	− 784.164i	− 0.0284718i
$913$	16618.2	0.602389
$914$	−9014.07	−0.326213
$915$	− 11905.9i	− 0.430162i
$916$	− 3561.83i	− 0.128478i
$917$	4741.34i	0.170745i
$918$	− 7999.51i	− 0.287607i
$919$	−42991.4	−1.54315	−0.771575	−	0.636139i	$-0.780531\pi$
$919$	−42991.4	−1.54315	−0.771575	+	0.636139i	$0.780531\pi$
$920$	−10914.6	−0.391134
$921$	34398.0i	1.23068i
$922$	4207.10	0.150275
$923$	0	0
$924$	−4025.98	−0.143339
$925$	− 6203.40i	− 0.220504i
$926$	−10936.6	−0.388119
$927$	2861.45	0.101383
$928$	9641.21i	0.341043i
$929$	− 40676.7i	− 1.43655i	−0.695757	−	0.718277i	$-0.744931\pi$
$929$	− 40676.7i	− 1.43655i	0.695757	−	0.718277i	$-0.255069\pi$
$930$	4566.04i	0.160996i
$931$	4350.91i	0.153164i
$932$	19012.4	0.668212
$933$	−12030.5	−0.422145
$934$	− 12087.6i	− 0.423467i
$935$	−15002.6	−0.524748
$936$	0	0
$937$	21008.3	0.732456	0.366228	−	0.930525i	$-0.380649\pi$
$937$	21008.3	0.732456	0.366228	+	0.930525i	$0.380649\pi$
$938$	3662.04i	0.127473i
$939$	21713.2	0.754614
$940$	−20085.4	−0.696929
$941$	4525.50i	0.156777i	0.996923	+	0.0783885i	$0.0249775\pi$
$941$	4525.50i	0.156777i	−0.996923	+	0.0783885i	$0.975023\pi$
$942$	− 25615.6i	− 0.885989i
$943$	− 69233.5i	− 2.39083i
$944$	3685.43i	0.127066i
$945$	−5328.40	−0.183421
$946$	−46611.7	−1.60198
$947$	− 27401.6i	− 0.940267i	−0.882595	−	0.470134i	$-0.844206\pi$
$947$	− 27401.6i	− 0.940267i	0.882595	−	0.470134i	$-0.155794\pi$
$948$	−5650.70	−0.193593
$949$	0	0
$950$	1396.82	0.0477040
$951$	− 23704.0i	− 0.808259i
$952$	904.129	0.0307805
$953$	−26226.2	−0.891449	−0.445725	−	0.895170i	$-0.647054\pi$
$953$	−26226.2	−0.891449	−0.445725	+	0.895170i	$0.647054\pi$
$954$	− 7312.49i	− 0.248166i
$955$	2886.08i	0.0977921i
$956$	− 9170.48i	− 0.310245i
$957$	− 72183.2i	− 2.43819i
$958$	16962.6	0.572063
$959$	−7343.15	−0.247260
$960$	− 2000.91i	− 0.0672700i
$961$	24458.6	0.821007
$962$	0	0
$963$	−5306.19	−0.177559
$964$	− 7900.83i	− 0.263972i
$965$	22897.1	0.763817
$966$	−4924.63	−0.164024
$967$	− 20843.6i	− 0.693158i	−0.938021	−	0.346579i	$-0.887343\pi$
$967$	− 20843.6i	− 0.693158i	0.938021	−	0.346579i	$-0.112657\pi$
$968$	− 23540.6i	− 0.781635i
$969$	− 1318.47i	− 0.0437103i
$970$	22417.0i	0.742026i
$971$	32577.3	1.07668	0.538339	−	0.842729i	$-0.319052\pi$
$971$	32577.3	1.07668	0.538339	+	0.842729i	$0.319052\pi$
$972$	13440.1	0.443510
$973$	− 7043.20i	− 0.232060i
$974$	−13246.7	−0.435782
$975$	0	0
$976$	−6093.05	−0.199830
$977$	43864.1i	1.43637i	0.695851	+	0.718186i	$0.255027\pi$
$977$	43864.1i	1.43637i	−0.695851	+	0.718186i	$0.744973\pi$
$978$	3307.58	0.108144
$979$	−24358.6	−0.795204
$980$	11102.0i	0.361879i
$981$	18065.4i	0.587955i
$982$	32758.7i	1.06453i
$983$	− 14758.8i	− 0.478875i	−0.970912	−	0.239437i	$-0.923037\pi$
$983$	− 14758.8i	− 0.478875i	0.970912	−	0.239437i	$-0.0769630\pi$
$984$	12692.2	0.411193
$985$	−16185.0	−0.523549
$986$	16210.4i	0.523575i
$987$	−9062.49	−0.292262
$988$	0	0
$989$	−57016.0	−1.83317
$990$	− 15134.0i	− 0.485848i
$991$	48622.4	1.55857	0.779284	−	0.626671i	$-0.215583\pi$
$991$	48622.4	1.55857	0.779284	+	0.626671i	$0.215583\pi$
$992$	2336.74	0.0747900
$993$	− 12860.9i	− 0.411006i
$994$	554.220i	0.0176849i
$995$	− 20607.0i	− 0.656569i
$996$	3726.55i	0.118554i
$997$	−18812.3	−0.597586	−0.298793	−	0.954318i	$-0.596584\pi$
$997$	−18812.3	−0.597586	−0.298793	+	0.954318i	$0.596584\pi$
$998$	−15831.4	−0.502138
$999$	− 17660.2i	− 0.559305i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.b.f.337.2		6
13.2	odd	12		338.4.c.l.191.2		6
13.3	even	3		338.4.e.h.147.5		12
13.4	even	6		338.4.e.h.23.5		12
13.5	odd	4		338.4.a.j.1.2	✓	3
13.6	odd	12		338.4.c.l.315.2		6
13.7	odd	12		338.4.c.k.315.2		6
13.8	odd	4		338.4.a.k.1.2	yes	3
13.9	even	3		338.4.e.h.23.2		12
13.10	even	6		338.4.e.h.147.2		12
13.11	odd	12		338.4.c.k.191.2		6
13.12	even	2	inner	338.4.b.f.337.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.4.a.j.1.2	✓	3	13.5	odd	4
338.4.a.k.1.2	yes	3	13.8	odd	4
338.4.b.f.337.2		6	1.1	even	1	trivial
338.4.b.f.337.5		6	13.12	even	2	inner
338.4.c.k.191.2		6	13.11	odd	12
338.4.c.k.315.2		6	13.7	odd	12
338.4.c.l.191.2		6	13.2	odd	12
338.4.c.l.315.2		6	13.6	odd	12
338.4.e.h.23.2		12	13.9	even	3
338.4.e.h.23.5		12	13.4	even	6
338.4.e.h.147.2		12	13.10	even	6
338.4.e.h.147.5		12	13.3	even	3

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	338.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -3.66487 q^{3} -4.00000 q^{4} -8.53079i q^{5} +7.32975i q^{6} -4.20105i q^{7} +8.00000i q^{8} -13.5687 q^{9} -17.0616 q^{10} -65.3726i q^{11} +14.6595 q^{12} -8.40209 q^{14} +31.2643i q^{15} +16.0000 q^{16} +26.9019 q^{17} +27.1374i q^{18} +13.3730i q^{19} +34.1232i q^{20} +15.3963i q^{21} -130.745 q^{22} -159.929 q^{23} -29.3190i q^{24} +52.2255 q^{25} +148.679 q^{27} +16.8042i q^{28} -301.288 q^{29} +62.5286 q^{30} +73.0232i q^{31} -32.0000i q^{32} +239.582i q^{33} -53.8038i q^{34} -35.8383 q^{35} +54.2748 q^{36} -118.781i q^{37} +26.7459 q^{38} +68.2464 q^{40} +432.901i q^{41} +30.7926 q^{42} +356.508 q^{43} +261.490i q^{44} +115.752i q^{45} +319.858i q^{46} +588.614i q^{47} -58.6380 q^{48} +325.351 q^{49} -104.451i q^{50} -98.5921 q^{51} -269.462 q^{53} -297.358i q^{54} -557.680 q^{55} +33.6084 q^{56} -49.0102i q^{57} +602.576i q^{58} +230.340i q^{59} -125.057i q^{60} -380.816 q^{61} +146.046 q^{62} +57.0027i q^{63} -64.0000 q^{64} +479.164 q^{66} -435.848i q^{67} -107.608 q^{68} +586.120 q^{69} +71.6765i q^{70} -65.9622i q^{71} -108.550i q^{72} +885.517i q^{73} -237.562 q^{74} -191.400 q^{75} -53.4918i q^{76} -274.633 q^{77} -385.463 q^{79} -136.493i q^{80} -178.536 q^{81} +865.803 q^{82} +254.207i q^{83} -61.5852i q^{84} -229.495i q^{85} -713.016i q^{86} +1104.18 q^{87} +522.980 q^{88} -372.612i q^{89} +231.504 q^{90} +639.716 q^{92} -267.621i q^{93} +1177.23 q^{94} +114.082 q^{95} +117.276i q^{96} -1313.88i q^{97} -650.702i q^{98} +887.020i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24 q^{3} - 24 q^{4} + 18 q^{9} + 48 q^{10} + 96 q^{12} + 108 q^{14} + 96 q^{16} + 180 q^{17} - 328 q^{22} - 38 q^{23} + 122 q^{25} - 138 q^{27} - 202 q^{29} - 360 q^{30} - 916 q^{35} - 72 q^{36} + 520 q^{38}+ \cdots - 3658 q^{95}+O(q^{100}) \) 6 * q - 24 * q^3 - 24 * q^4 + 18 * q^9 + 48 * q^10 + 96 * q^12 + 108 * q^14 + 96 * q^16 + 180 * q^17 - 328 * q^22 - 38 * q^23 + 122 * q^25 - 138 * q^27 - 202 * q^29 - 360 * q^30 - 916 * q^35 - 72 * q^36 + 520 * q^38 - 192 * q^40 - 936 * q^42 + 2410 * q^43 - 384 * q^48 + 368 * q^49 + 414 * q^51 - 2190 * q^53 - 1052 * q^55 - 432 * q^56 - 2216 * q^61 + 2076 * q^62 - 384 * q^64 + 584 * q^66 - 720 * q^68 + 628 * q^69 - 336 * q^74 + 1010 * q^75 - 960 * q^77 - 3922 * q^79 + 1830 * q^81 + 748 * q^82 + 3272 * q^87 + 1312 * q^88 + 2580 * q^90 + 152 * q^92 + 2144 * q^94 - 3658 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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