LMFDB - Embedded newform 338.4.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(1,338)]

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([0]))

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	338.a (trivial)

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:	yes
Analytic conductor:	$19.9426455819$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +0.405813 q^{3} +4.00000 q^{4} +6.36227 q^{5} -0.811626 q^{6} -2.55065 q^{7} -8.00000 q^{8} -26.8353 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +0.405813 q^{3} +4.00000 q^{4} +6.36227 q^{5} -0.811626 q^{6} -2.55065 q^{7} -8.00000 q^{8} -26.8353 q^{9} -12.7245 q^{10} +26.1209 q^{11} +1.62325 q^{12} +5.10129 q^{14} +2.58189 q^{15} +16.0000 q^{16} -93.7085 q^{17} +53.6706 q^{18} -37.2329 q^{19} +25.4491 q^{20} -1.03509 q^{21} -52.2419 q^{22} -104.926 q^{23} -3.24651 q^{24} -84.5215 q^{25} -21.8471 q^{27} -10.2026 q^{28} +249.544 q^{29} -5.16379 q^{30} +278.993 q^{31} -32.0000 q^{32} +10.6002 q^{33} +187.417 q^{34} -16.2279 q^{35} -107.341 q^{36} -10.9001 q^{37} +74.4658 q^{38} -50.8982 q^{40} -371.375 q^{41} +2.07017 q^{42} -413.718 q^{43} +104.484 q^{44} -170.734 q^{45} +209.852 q^{46} +238.631 q^{47} +6.49301 q^{48} -336.494 q^{49} +169.043 q^{50} -38.0282 q^{51} -424.907 q^{53} +43.6942 q^{54} +166.188 q^{55} +20.4052 q^{56} -15.1096 q^{57} -499.089 q^{58} -774.453 q^{59} +10.3276 q^{60} -123.423 q^{61} -557.985 q^{62} +68.4474 q^{63} +64.0000 q^{64} -21.2004 q^{66} -881.604 q^{67} -374.834 q^{68} -42.5804 q^{69} +32.4558 q^{70} +118.765 q^{71} +214.683 q^{72} +209.319 q^{73} +21.8002 q^{74} -34.2999 q^{75} -148.932 q^{76} -66.6253 q^{77} -532.358 q^{79} +101.796 q^{80} +715.688 q^{81} +742.751 q^{82} -376.511 q^{83} -4.14034 q^{84} -596.199 q^{85} +827.436 q^{86} +101.268 q^{87} -208.967 q^{88} +42.6336 q^{89} +341.467 q^{90} -419.704 q^{92} +113.219 q^{93} -477.263 q^{94} -236.886 q^{95} -12.9860 q^{96} +639.319 q^{97} +672.988 q^{98} -700.963 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9	\( 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9} - 24 q^{10} + 82 q^{11} - 48 q^{12} + 54 q^{14} - 90 q^{15} + 48 q^{16} - 90 q^{17} - 18 q^{18} + 130 q^{19} + 48 q^{20} + 234 q^{21} - 164 q^{22} + 19 q^{23} + 96 q^{24} - 61 q^{25} - 69 q^{27} - 108 q^{28} - 101 q^{29} + 180 q^{30} + 519 q^{31} - 96 q^{32} - 146 q^{33} + 180 q^{34} - 458 q^{35} + 36 q^{36} + 84 q^{37} - 260 q^{38} - 96 q^{40} + 187 q^{41} - 468 q^{42} - 1205 q^{43} + 328 q^{44} + 645 q^{45} - 38 q^{46} - 536 q^{47} - 192 q^{48} - 184 q^{49} + 122 q^{50} - 207 q^{51} - 1095 q^{53} + 138 q^{54} - 526 q^{55} + 216 q^{56} - 1409 q^{57} + 202 q^{58} - 1413 q^{59} - 360 q^{60} - 1108 q^{61} - 1038 q^{62} - 1404 q^{63} + 192 q^{64} + 292 q^{66} - 1605 q^{67} - 360 q^{68} - 314 q^{69} + 916 q^{70} - 909 q^{71} - 72 q^{72} + 287 q^{73} - 168 q^{74} - 505 q^{75} + 520 q^{76} + 480 q^{77} - 1961 q^{79} + 192 q^{80} + 915 q^{81} - 374 q^{82} + 191 q^{83} + 936 q^{84} + 67 q^{85} + 2410 q^{86} + 1636 q^{87} - 656 q^{88} + 1091 q^{89} - 1290 q^{90} + 76 q^{92} - 1614 q^{93} + 1072 q^{94} + 1829 q^{95} + 384 q^{96} - 947 q^{97} + 368 q^{98} - 2057 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9 - 24 * q^10 + 82 * q^11 - 48 * q^12 + 54 * q^14 - 90 * q^15 + 48 * q^16 - 90 * q^17 - 18 * q^18 + 130 * q^19 + 48 * q^20 + 234 * q^21 - 164 * q^22 + 19 * q^23 + 96 * q^24 - 61 * q^25 - 69 * q^27 - 108 * q^28 - 101 * q^29 + 180 * q^30 + 519 * q^31 - 96 * q^32 - 146 * q^33 + 180 * q^34 - 458 * q^35 + 36 * q^36 + 84 * q^37 - 260 * q^38 - 96 * q^40 + 187 * q^41 - 468 * q^42 - 1205 * q^43 + 328 * q^44 + 645 * q^45 - 38 * q^46 - 536 * q^47 - 192 * q^48 - 184 * q^49 + 122 * q^50 - 207 * q^51 - 1095 * q^53 + 138 * q^54 - 526 * q^55 + 216 * q^56 - 1409 * q^57 + 202 * q^58 - 1413 * q^59 - 360 * q^60 - 1108 * q^61 - 1038 * q^62 - 1404 * q^63 + 192 * q^64 + 292 * q^66 - 1605 * q^67 - 360 * q^68 - 314 * q^69 + 916 * q^70 - 909 * q^71 - 72 * q^72 + 287 * q^73 - 168 * q^74 - 505 * q^75 + 520 * q^76 + 480 * q^77 - 1961 * q^79 + 192 * q^80 + 915 * q^81 - 374 * q^82 + 191 * q^83 + 936 * q^84 + 67 * q^85 + 2410 * q^86 + 1636 * q^87 - 656 * q^88 + 1091 * q^89 - 1290 * q^90 + 76 * q^92 - 1614 * q^93 + 1072 * q^94 + 1829 * q^95 + 384 * q^96 - 947 * q^97 + 368 * q^98 - 2057 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0.405813	0.0780988	0.0390494	−	0.999237i	$-0.487567\pi$
$3$	0.405813	0.0780988	0.0390494	+	0.999237i	$0.487567\pi$
$4$	4.00000	0.500000
$5$	6.36227	0.569059	0.284529	−	0.958667i	$-0.408163\pi$
$5$	6.36227	0.569059	0.284529	+	0.958667i	$0.408163\pi$
$6$	−0.811626	−0.0552242
$7$	−2.55065	−0.137722	−0.0688610	−	0.997626i	$-0.521936\pi$
$7$	−2.55065	−0.137722	−0.0688610	+	0.997626i	$0.521936\pi$
$8$	−8.00000	−0.353553
$9$	−26.8353	−0.993901
$10$	−12.7245	−0.402385
$11$	26.1209	0.715978	0.357989	−	0.933726i	$-0.383463\pi$
$11$	26.1209	0.715978	0.357989	+	0.933726i	$0.383463\pi$
$12$	1.62325	0.0390494
$13$	0	0
$13$	0	0
$14$	5.10129	0.0973841
$15$	2.58189	0.0444428
$16$	16.0000	0.250000
$17$	−93.7085	−1.33692	−0.668461	−	0.743748i	$-0.733046\pi$
$17$	−93.7085	−1.33692	−0.668461	+	0.743748i	$0.733046\pi$
$18$	53.6706	0.702794
$19$	−37.2329	−0.449569	−0.224784	−	0.974409i	$-0.572168\pi$
$19$	−37.2329	−0.449569	−0.224784	+	0.974409i	$0.572168\pi$
$20$	25.4491	0.284529
$21$	−1.03509	−0.0107559
$22$	−52.2419	−0.506273
$23$	−104.926	−0.951243	−0.475622	−	0.879650i	$-0.657777\pi$
$23$	−104.926	−0.951243	−0.475622	+	0.879650i	$0.657777\pi$
$24$	−3.24651	−0.0276121
$25$	−84.5215	−0.676172
$26$	0	0
$27$	−21.8471	−0.155721
$28$	−10.2026	−0.0688610
$29$	249.544	1.59791	0.798953	−	0.601394i	$-0.205388\pi$
$29$	249.544	1.59791	0.798953	+	0.601394i	$0.205388\pi$
$30$	−5.16379	−0.0314258
$31$	278.993	1.61641	0.808203	−	0.588904i	$-0.200441\pi$
$31$	278.993	1.61641	0.808203	+	0.588904i	$0.200441\pi$
$32$	−32.0000	−0.176777
$33$	10.6002	0.0559170
$34$	187.417	0.945346
$35$	−16.2279	−0.0783719
$36$	−107.341	−0.496950
$37$	−10.9001	−0.0484315	−0.0242157	−	0.999707i	$-0.507709\pi$
$37$	−10.9001	−0.0484315	−0.0242157	+	0.999707i	$0.507709\pi$
$38$	74.4658	0.317893
$39$	0	0
$40$	−50.8982	−0.201193
$41$	−371.375	−1.41461	−0.707306	−	0.706907i	$-0.750090\pi$
$41$	−371.375	−1.41461	−0.707306	+	0.706907i	$0.750090\pi$
$42$	2.07017	0.00760558
$43$	−413.718	−1.46724	−0.733621	−	0.679559i	$-0.762171\pi$
$43$	−413.718	−1.46724	−0.733621	+	0.679559i	$0.762171\pi$
$44$	104.484	0.357989
$45$	−170.734	−0.565588
$46$	209.852	0.672630
$47$	238.631	0.740595	0.370297	−	0.928913i	$-0.379256\pi$
$47$	238.631	0.740595	0.370297	+	0.928913i	$0.379256\pi$
$48$	6.49301	0.0195247
$49$	−336.494	−0.981033
$50$	169.043	0.478126
$51$	−38.0282	−0.104412
$52$	0	0
$53$	−424.907	−1.10124	−0.550618	−	0.834757i	$-0.685608\pi$
$53$	−424.907	−1.10124	−0.550618	+	0.834757i	$0.685608\pi$
$54$	43.6942	0.110112
$55$	166.188	0.407434
$56$	20.4052	0.0486921
$57$	−15.1096	−0.0351108
$58$	−499.089	−1.12989
$59$	−774.453	−1.70890	−0.854451	−	0.519533i	$-0.826106\pi$
$59$	−774.453	−1.70890	−0.854451	+	0.519533i	$0.826106\pi$
$60$	10.3276	0.0222214
$61$	−123.423	−0.259060	−0.129530	−	0.991576i	$-0.541347\pi$
$61$	−123.423	−0.259060	−0.129530	+	0.991576i	$0.541347\pi$
$62$	−557.985	−1.14297
$63$	68.4474	0.136882
$64$	64.0000	0.125000
$65$	0	0
$66$	−21.2004	−0.0395393
$67$	−881.604	−1.60754	−0.803769	−	0.594941i	$-0.797175\pi$
$67$	−881.604	−1.60754	−0.803769	+	0.594941i	$0.797175\pi$
$68$	−374.834	−0.668461
$69$	−42.5804	−0.0742909
$70$	32.4558	0.0554173
$71$	118.765	0.198519	0.0992593	−	0.995062i	$-0.468353\pi$
$71$	118.765	0.198519	0.0992593	+	0.995062i	$0.468353\pi$
$72$	214.683	0.351397
$73$	209.319	0.335602	0.167801	−	0.985821i	$-0.446333\pi$
$73$	209.319	0.335602	0.167801	+	0.985821i	$0.446333\pi$
$74$	21.8002	0.0342462
$75$	−34.2999	−0.0528082
$76$	−148.932	−0.224784
$77$	−66.6253	−0.0986059
$78$	0	0
$79$	−532.358	−0.758164	−0.379082	−	0.925363i	$-0.623760\pi$
$79$	−532.358	−0.758164	−0.379082	+	0.925363i	$0.623760\pi$
$80$	101.796	0.142265
$81$	715.688	0.981739
$82$	742.751	1.00028
$83$	−376.511	−0.497921	−0.248960	−	0.968514i	$-0.580089\pi$
$83$	−376.511	−0.497921	−0.248960	+	0.968514i	$0.580089\pi$
$84$	−4.14034	−0.00537796
$85$	−596.199	−0.760787
$86$	827.436	1.03750
$87$	101.268	0.124794
$88$	−208.967	−0.253136
$89$	42.6336	0.0507770	0.0253885	−	0.999678i	$-0.491918\pi$
$89$	42.6336	0.0507770	0.0253885	+	0.999678i	$0.491918\pi$
$90$	341.467	0.399931
$91$	0	0
$92$	−419.704	−0.475622
$93$	113.219	0.126239
$94$	−477.263	−0.523679
$95$	−236.886	−0.255831
$96$	−12.9860	−0.0138060
$97$	639.319	0.669206	0.334603	−	0.942359i	$-0.391398\pi$
$97$	639.319	0.669206	0.334603	+	0.942359i	$0.391398\pi$
$98$	672.988	0.693695
$99$	−700.963	−0.711611
$100$	−338.086	−0.338086
$101$	474.839	0.467804	0.233902	−	0.972260i	$-0.424851\pi$
$101$	474.839	0.467804	0.233902	+	0.972260i	$0.424851\pi$
$102$	76.0563	0.0738304
$103$	−1100.53	−1.05280	−0.526402	−	0.850236i	$-0.676459\pi$
$103$	−1100.53	−1.05280	−0.526402	+	0.850236i	$0.676459\pi$
$104$	0	0
$105$	−6.58550	−0.00612075
$106$	849.815	0.778692
$107$	−570.383	−0.515336	−0.257668	−	0.966233i	$-0.582954\pi$
$107$	−570.383	−0.515336	−0.257668	+	0.966233i	$0.582954\pi$
$108$	−87.3883	−0.0778606
$109$	−593.029	−0.521118	−0.260559	−	0.965458i	$-0.583907\pi$
$109$	−593.029	−0.521118	−0.260559	+	0.965458i	$0.583907\pi$
$110$	−332.377	−0.288099
$111$	−4.42340	−0.00378244
$112$	−40.8103	−0.0344305
$113$	1839.12	1.53106	0.765529	−	0.643401i	$-0.222477\pi$
$113$	1839.12	1.53106	0.765529	+	0.643401i	$0.222477\pi$
$114$	30.2192	0.0248271
$115$	−667.568	−0.541313
$116$	998.178	0.798953
$117$	0	0
$118$	1548.91	1.20838
$119$	239.017	0.184123
$120$	−20.6552	−0.0157129
$121$	−648.697	−0.487376
$122$	246.845	0.183183
$123$	−150.709	−0.110480
$124$	1115.97	0.808203
$125$	−1333.03	−0.953841
$126$	−136.895	−0.0967901
$127$	2468.51	1.72476	0.862382	−	0.506258i	$-0.168972\pi$
$127$	2468.51	1.72476	0.862382	+	0.506258i	$0.168972\pi$
$128$	−128.000	−0.0883883
$129$	−167.892	−0.114590
$130$	0	0
$131$	2279.82	1.52053	0.760264	−	0.649615i	$-0.225070\pi$
$131$	2279.82	1.52053	0.760264	+	0.649615i	$0.225070\pi$
$132$	42.4009	0.0279585
$133$	94.9679	0.0619155
$134$	1763.21	1.13670
$135$	−138.997	−0.0886146
$136$	749.668	0.472673
$137$	−439.309	−0.273961	−0.136981	−	0.990574i	$-0.543740\pi$
$137$	−439.309	−0.273961	−0.136981	+	0.990574i	$0.543740\pi$
$138$	85.1607	0.0525316
$139$	−2178.94	−1.32961	−0.664803	−	0.747019i	$-0.731484\pi$
$139$	−2178.94	−1.32961	−0.664803	+	0.747019i	$0.731484\pi$
$140$	−64.9116	−0.0391860
$141$	96.8397	0.0578395
$142$	−237.530	−0.140374
$143$	0	0
$144$	−429.365	−0.248475
$145$	1587.67	0.909302
$146$	−418.638	−0.237307
$147$	−136.554	−0.0766175
$148$	−43.6004	−0.0242157
$149$	2868.47	1.57714	0.788572	−	0.614942i	$-0.210821\pi$
$149$	2868.47	1.57714	0.788572	+	0.614942i	$0.210821\pi$
$150$	68.5999	0.0373410
$151$	−2202.00	−1.18673	−0.593366	−	0.804933i	$-0.702201\pi$
$151$	−2202.00	−1.18673	−0.593366	+	0.804933i	$0.702201\pi$
$152$	297.863	0.158947
$153$	2514.70	1.32877
$154$	133.251	0.0697249
$155$	1775.03	0.919830
$156$	0	0
$157$	865.139	0.439781	0.219890	−	0.975525i	$-0.429430\pi$
$157$	865.139	0.439781	0.219890	+	0.975525i	$0.429430\pi$
$158$	1064.72	0.536103
$159$	−172.433	−0.0860052
$160$	−203.593	−0.100596
$161$	267.629	0.131007
$162$	−1431.38	−0.694194
$163$	1060.84	0.509762	0.254881	−	0.966972i	$-0.417964\pi$
$163$	1060.84	0.509762	0.254881	+	0.966972i	$0.417964\pi$
$164$	−1485.50	−0.707306
$165$	67.4415	0.0318201
$166$	753.022	0.352083
$167$	1404.09	0.650608	0.325304	−	0.945610i	$-0.394533\pi$
$167$	1404.09	0.650608	0.325304	+	0.945610i	$0.394533\pi$
$168$	8.28069	0.00380279
$169$	0	0
$170$	1192.40	0.537958
$171$	999.156	0.446827
$172$	−1654.87	−0.733621
$173$	1022.22	0.449237	0.224618	−	0.974447i	$-0.427886\pi$
$173$	1022.22	0.449237	0.224618	+	0.974447i	$0.427886\pi$
$174$	−202.537	−0.0882430
$175$	215.584	0.0931237
$176$	417.935	0.178994
$177$	−314.283	−0.133463
$178$	−85.2673	−0.0359048
$179$	418.223	0.174634	0.0873169	−	0.996181i	$-0.472171\pi$
$179$	418.223	0.174634	0.0873169	+	0.996181i	$0.472171\pi$
$180$	−682.934	−0.282794
$181$	−2816.05	−1.15644	−0.578219	−	0.815882i	$-0.696252\pi$
$181$	−2816.05	−1.15644	−0.578219	+	0.815882i	$0.696252\pi$
$182$	0	0
$183$	−50.0865	−0.0202322
$184$	839.408	0.336315
$185$	−69.3494	−0.0275604
$186$	−226.438	−0.0892647
$187$	−2447.75	−0.957206
$188$	954.525	0.370297
$189$	55.7242	0.0214462
$190$	473.772	0.180900
$191$	1614.11	0.611480	0.305740	−	0.952115i	$-0.401096\pi$
$191$	1614.11	0.611480	0.305740	+	0.952115i	$0.401096\pi$
$192$	25.9720	0.00976235
$193$	−2461.12	−0.917902	−0.458951	−	0.888462i	$-0.651775\pi$
$193$	−2461.12	−0.917902	−0.458951	+	0.888462i	$0.651775\pi$
$194$	−1278.64	−0.473200
$195$	0	0
$196$	−1345.98	−0.490516
$197$	3445.91	1.24625	0.623123	−	0.782123i	$-0.285863\pi$
$197$	3445.91	1.24625	0.623123	+	0.782123i	$0.285863\pi$
$198$	1401.93	0.503185
$199$	2205.02	0.785477	0.392739	−	0.919650i	$-0.371528\pi$
$199$	2205.02	0.785477	0.392739	+	0.919650i	$0.371528\pi$
$200$	676.172	0.239063
$201$	−357.767	−0.125547
$202$	−949.677	−0.330787
$203$	−636.500	−0.220067
$204$	−152.113	−0.0522060
$205$	−2362.79	−0.804998
$206$	2201.07	0.744444
$207$	2815.72	0.945441
$208$	0	0
$209$	−972.558	−0.321881
$210$	13.1710	0.00432802
$211$	−933.193	−0.304472	−0.152236	−	0.988344i	$-0.548647\pi$
$211$	−933.193	−0.304472	−0.152236	+	0.988344i	$0.548647\pi$
$212$	−1699.63	−0.550618
$213$	48.1964	0.0155041
$214$	1140.77	0.364398
$215$	−2632.19	−0.834947
$216$	174.777	0.0550558
$217$	−711.612	−0.222614
$218$	1186.06	0.368486
$219$	84.9445	0.0262101
$220$	664.754	0.203717
$221$	0	0
$222$	8.84681	0.00267459
$223$	−6363.62	−1.91094	−0.955469	−	0.295090i	$-0.904650\pi$
$223$	−6363.62	−1.91094	−0.955469	+	0.295090i	$0.904650\pi$
$224$	81.6207	0.0243460
$225$	2268.16	0.672048
$226$	−3678.23	−1.08262
$227$	−829.510	−0.242540	−0.121270	−	0.992620i	$-0.538697\pi$
$227$	−829.510	−0.242540	−0.121270	+	0.992620i	$0.538697\pi$
$228$	−60.4384	−0.0175554
$229$	5789.35	1.67062	0.835308	−	0.549782i	$-0.185289\pi$
$229$	5789.35	1.67062	0.835308	+	0.549782i	$0.185289\pi$
$230$	1335.14	0.382766
$231$	−27.0374	−0.00770100
$232$	−1996.36	−0.564945
$233$	320.751	0.0901849	0.0450924	−	0.998983i	$-0.485642\pi$
$233$	320.751	0.0901849	0.0450924	+	0.998983i	$0.485642\pi$
$234$	0	0
$235$	1518.24	0.421442
$236$	−3097.81	−0.854451
$237$	−216.038	−0.0592117
$238$	−478.035	−0.130195
$239$	3582.11	0.969487	0.484744	−	0.874656i	$-0.338913\pi$
$239$	3582.11	0.969487	0.484744	+	0.874656i	$0.338913\pi$
$240$	41.3103	0.0111107
$241$	4880.24	1.30441	0.652207	−	0.758041i	$-0.273843\pi$
$241$	4880.24	1.30441	0.652207	+	0.758041i	$0.273843\pi$
$242$	1297.39	0.344627
$243$	880.307	0.232394
$244$	−493.690	−0.129530
$245$	−2140.87	−0.558265
$246$	301.418	0.0781208
$247$	0	0
$248$	−2231.94	−0.571486
$249$	−152.793	−0.0388870
$250$	2666.07	0.674467
$251$	−2980.22	−0.749442	−0.374721	−	0.927138i	$-0.622261\pi$
$251$	−2980.22	−0.749442	−0.374721	+	0.927138i	$0.622261\pi$
$252$	273.790	0.0684410
$253$	−2740.77	−0.681069
$254$	−4937.02	−1.21959
$255$	−241.946	−0.0594165
$256$	256.000	0.0625000
$257$	−3855.93	−0.935900	−0.467950	−	0.883755i	$-0.655007\pi$
$257$	−3855.93	−0.935900	−0.467950	+	0.883755i	$0.655007\pi$
$258$	335.784	0.0810272
$259$	27.8023	0.00667008
$260$	0	0
$261$	−6696.60	−1.58816
$262$	−4559.65	−1.07518
$263$	−4431.18	−1.03893	−0.519464	−	0.854492i	$-0.673868\pi$
$263$	−4431.18	−1.03893	−0.519464	+	0.854492i	$0.673868\pi$
$264$	−84.8018	−0.0197696
$265$	−2703.38	−0.626668
$266$	−189.936	−0.0437809
$267$	17.3013	0.00396562
$268$	−3526.42	−0.803769
$269$	1261.33	0.285892	0.142946	−	0.989731i	$-0.454343\pi$
$269$	1261.33	0.285892	0.142946	+	0.989731i	$0.454343\pi$
$270$	277.994	0.0626600
$271$	6200.75	1.38992	0.694960	−	0.719048i	$-0.255422\pi$
$271$	6200.75	1.38992	0.694960	+	0.719048i	$0.255422\pi$
$272$	−1499.34	−0.334230
$273$	0	0
$274$	878.618	0.193720
$275$	−2207.78	−0.484124
$276$	−170.321	−0.0371455
$277$	−1157.57	−0.251089	−0.125544	−	0.992088i	$-0.540068\pi$
$277$	−1157.57	−0.251089	−0.125544	+	0.992088i	$0.540068\pi$
$278$	4357.88	0.940173
$279$	−7486.86	−1.60655
$280$	129.823	0.0277087
$281$	6322.71	1.34228	0.671141	−	0.741330i	$-0.265805\pi$
$281$	6322.71	1.34228	0.671141	+	0.741330i	$0.265805\pi$
$282$	−193.679	−0.0408987
$283$	−9308.35	−1.95521	−0.977604	−	0.210454i	$-0.932506\pi$
$283$	−9308.35	−1.95521	−0.977604	+	0.210454i	$0.932506\pi$
$284$	475.060	0.0992593
$285$	−96.1314	−0.0199801
$286$	0	0
$287$	947.247	0.194823
$288$	858.730	0.175698
$289$	3868.29	0.787358
$290$	−3175.34	−0.642974
$291$	259.444	0.0522642
$292$	837.277	0.167801
$293$	−506.196	−0.100929	−0.0504646	−	0.998726i	$-0.516070\pi$
$293$	−506.196	−0.100929	−0.0504646	+	0.998726i	$0.516070\pi$
$294$	273.108	0.0541767
$295$	−4927.28	−0.972466
$296$	87.2008	0.0171231
$297$	−570.666	−0.111493
$298$	−5736.95	−1.11521
$299$	0	0
$300$	−137.200	−0.0264041
$301$	1055.25	0.202071
$302$	4404.01	0.839146
$303$	192.696	0.0365349
$304$	−595.726	−0.112392
$305$	−785.248	−0.147420
$306$	−5029.40	−0.939580
$307$	4094.64	0.761217	0.380608	−	0.924736i	$-0.375715\pi$
$307$	4094.64	0.761217	0.380608	+	0.924736i	$0.375715\pi$
$308$	−266.501	−0.0493029
$309$	−446.611	−0.0822227
$310$	−3550.05	−0.650418
$311$	8705.00	1.58719	0.793594	−	0.608448i	$-0.208208\pi$
$311$	8705.00	1.58719	0.793594	+	0.608448i	$0.208208\pi$
$312$	0	0
$313$	−3144.78	−0.567903	−0.283951	−	0.958839i	$-0.591645\pi$
$313$	−3144.78	−0.567903	−0.283951	+	0.958839i	$0.591645\pi$
$314$	−1730.28	−0.310972
$315$	435.481	0.0778939
$316$	−2129.43	−0.379082
$317$	9902.03	1.75443	0.877214	−	0.480100i	$-0.159400\pi$
$317$	9902.03	1.75443	0.877214	+	0.480100i	$0.159400\pi$
$318$	344.866	0.0608149
$319$	6518.33	1.14406
$320$	407.185	0.0711324
$321$	−231.469	−0.0402471
$322$	−535.258	−0.0926360
$323$	3489.04	0.601038
$324$	2862.75	0.490869
$325$	0	0
$326$	−2121.67	−0.360456
$327$	−240.659	−0.0406987
$328$	2971.00	0.500141
$329$	−608.664	−0.101996
$330$	−134.883	−0.0225002
$331$	1966.37	0.326530	0.163265	−	0.986582i	$-0.447797\pi$
$331$	1966.37	0.326530	0.163265	+	0.986582i	$0.447797\pi$
$332$	−1506.04	−0.248960
$333$	292.508	0.0481361
$334$	−2808.17	−0.460049
$335$	−5609.00	−0.914784
$336$	−16.5614	−0.00268898
$337$	−9570.83	−1.54705	−0.773526	−	0.633765i	$-0.781509\pi$
$337$	−9570.83	−1.54705	−0.773526	+	0.633765i	$0.781509\pi$
$338$	0	0
$339$	746.338	0.119574
$340$	−2384.80	−0.380393
$341$	7287.55	1.15731
$342$	−1998.31	−0.315954
$343$	1733.15	0.272832
$344$	3309.74	0.518748
$345$	−270.908	−0.0422759
$346$	−2044.44	−0.317658
$347$	−6454.21	−0.998502	−0.499251	−	0.866457i	$-0.666392\pi$
$347$	−6454.21	−0.998502	−0.499251	+	0.866457i	$0.666392\pi$
$348$	405.074	0.0623972
$349$	−2779.52	−0.426316	−0.213158	−	0.977018i	$-0.568375\pi$
$349$	−2779.52	−0.426316	−0.213158	+	0.977018i	$0.568375\pi$
$350$	−431.169	−0.0658484
$351$	0	0
$352$	−835.870	−0.126568
$353$	11147.3	1.68077	0.840387	−	0.541987i	$-0.182328\pi$
$353$	11147.3	1.68077	0.840387	+	0.541987i	$0.182328\pi$
$354$	628.566	0.0943727
$355$	755.616	0.112969
$356$	170.535	0.0253885
$357$	96.9964	0.0143798
$358$	−836.446	−0.123485
$359$	−4313.27	−0.634111	−0.317056	−	0.948407i	$-0.602694\pi$
$359$	−4313.27	−0.634111	−0.317056	+	0.948407i	$0.602694\pi$
$360$	1365.87	0.199966
$361$	−5472.71	−0.797888
$362$	5632.10	0.817725
$363$	−263.250	−0.0380634
$364$	0	0
$365$	1331.75	0.190977
$366$	100.173	0.0143064
$367$	−6909.51	−0.982762	−0.491381	−	0.870945i	$-0.663508\pi$
$367$	−6909.51	−0.982762	−0.491381	+	0.870945i	$0.663508\pi$
$368$	−1678.82	−0.237811
$369$	9965.98	1.40598
$370$	138.699	0.0194881
$371$	1083.79	0.151664
$372$	452.876	0.0631196
$373$	2316.19	0.321523	0.160761	−	0.986993i	$-0.448605\pi$
$373$	2316.19	0.321523	0.160761	+	0.986993i	$0.448605\pi$
$374$	4895.51	0.676847
$375$	−540.962	−0.0744938
$376$	−1909.05	−0.261840
$377$	0	0
$378$	−111.448	−0.0151648
$379$	−11393.1	−1.54412	−0.772062	−	0.635547i	$-0.780775\pi$
$379$	−11393.1	−1.54412	−0.772062	+	0.635547i	$0.780775\pi$
$380$	−947.543	−0.127916
$381$	1001.75	0.134702
$382$	−3228.21	−0.432381
$383$	−11952.8	−1.59467	−0.797335	−	0.603537i	$-0.793757\pi$
$383$	−11952.8	−1.59467	−0.797335	+	0.603537i	$0.793757\pi$
$384$	−51.9441	−0.00690302
$385$	−423.888	−0.0561126
$386$	4922.23	0.649055
$387$	11102.2	1.45829
$388$	2557.28	0.334603
$389$	5282.24	0.688484	0.344242	−	0.938881i	$-0.388136\pi$
$389$	5282.24	0.688484	0.344242	+	0.938881i	$0.388136\pi$
$390$	0	0
$391$	9832.46	1.27174
$392$	2691.95	0.346847
$393$	925.182	0.118751
$394$	−6891.81	−0.881230
$395$	−3387.01	−0.431440
$396$	−2803.85	−0.355805
$397$	9111.58	1.15188	0.575941	−	0.817491i	$-0.304636\pi$
$397$	9111.58	1.15188	0.575941	+	0.817491i	$0.304636\pi$
$398$	−4410.05	−0.555416
$399$	38.5392	0.00483553
$400$	−1352.34	−0.169043
$401$	−499.562	−0.0622118	−0.0311059	−	0.999516i	$-0.509903\pi$
$401$	−499.562	−0.0622118	−0.0311059	+	0.999516i	$0.509903\pi$
$402$	715.533	0.0887750
$403$	0	0
$404$	1899.35	0.233902
$405$	4553.40	0.558667
$406$	1273.00	0.155611
$407$	−284.721	−0.0346759
$408$	304.225	0.0369152
$409$	12803.2	1.54787	0.773934	−	0.633266i	$-0.218286\pi$
$409$	12803.2	1.54787	0.773934	+	0.633266i	$0.218286\pi$
$410$	4725.58	0.569219
$411$	−178.277	−0.0213960
$412$	−4402.13	−0.526402
$413$	1975.36	0.235353
$414$	−5631.45	−0.668528
$415$	−2395.47	−0.283346
$416$	0	0
$417$	−884.242	−0.103841
$418$	1945.12	0.227605
$419$	−7368.27	−0.859102	−0.429551	−	0.903043i	$-0.641328\pi$
$419$	−7368.27	−0.859102	−0.429551	+	0.903043i	$0.641328\pi$
$420$	−26.3420	−0.00306038
$421$	−15266.9	−1.76737	−0.883686	−	0.468080i	$-0.844946\pi$
$421$	−15266.9	−1.76737	−0.883686	+	0.468080i	$0.844946\pi$
$422$	1866.39	0.215294
$423$	−6403.75	−0.736077
$424$	3399.26	0.389346
$425$	7920.38	0.903988
$426$	−96.3929	−0.0109630
$427$	314.807	0.0356782
$428$	−2281.53	−0.257668
$429$	0	0
$430$	5264.37	0.590397
$431$	−20.9424	−0.00234051	−0.00117026	−	0.999999i	$-0.500373\pi$
$431$	−20.9424	−0.00234051	−0.00117026	+	0.999999i	$0.500373\pi$
$432$	−349.553	−0.0389303
$433$	7458.11	0.827746	0.413873	−	0.910335i	$-0.364176\pi$
$433$	7458.11	0.827746	0.413873	+	0.910335i	$0.364176\pi$
$434$	1423.22	0.157412
$435$	644.297	0.0710154
$436$	−2372.12	−0.260559
$437$	3906.70	0.427649
$438$	−169.889	−0.0185334
$439$	9014.34	0.980025	0.490012	−	0.871715i	$-0.336992\pi$
$439$	9014.34	0.980025	0.490012	+	0.871715i	$0.336992\pi$
$440$	−1329.51	−0.144050
$441$	9029.93	0.975049
$442$	0	0
$443$	1787.12	0.191668	0.0958339	−	0.995397i	$-0.469448\pi$
$443$	1787.12	0.191668	0.0958339	+	0.995397i	$0.469448\pi$
$444$	−17.6936	−0.00189122
$445$	271.247	0.0288951
$446$	12727.2	1.35124
$447$	1164.06	0.123173
$448$	−163.241	−0.0172152
$449$	−4540.95	−0.477284	−0.238642	−	0.971108i	$-0.576702\pi$
$449$	−4540.95	−0.477284	−0.238642	+	0.971108i	$0.576702\pi$
$450$	−4536.32	−0.475209
$451$	−9700.67	−1.01283
$452$	7356.47	0.765529
$453$	−893.602	−0.0926823
$454$	1659.02	0.171501
$455$	0	0
$456$	120.877	0.0124135
$457$	6873.87	0.703602	0.351801	−	0.936075i	$-0.385569\pi$
$457$	6873.87	0.703602	0.351801	+	0.936075i	$0.385569\pi$
$458$	−11578.7	−1.18130
$459$	2047.26	0.208187
$460$	−2670.27	−0.270657
$461$	9308.59	0.940443	0.470221	−	0.882549i	$-0.344174\pi$
$461$	9308.59	0.940443	0.470221	+	0.882549i	$0.344174\pi$
$462$	54.0748	0.00544543
$463$	−14763.5	−1.48190	−0.740950	−	0.671560i	$-0.765625\pi$
$463$	−14763.5	−1.48190	−0.740950	+	0.671560i	$0.765625\pi$
$464$	3992.71	0.399476
$465$	720.330	0.0718376
$466$	−641.501	−0.0637703
$467$	7397.81	0.733040	0.366520	−	0.930410i	$-0.380549\pi$
$467$	7397.81	0.733040	0.366520	+	0.930410i	$0.380549\pi$
$468$	0	0
$469$	2248.66	0.221393
$470$	−3036.47	−0.298005
$471$	351.085	0.0343463
$472$	6195.62	0.604188
$473$	−10806.7	−1.05051
$474$	432.076	0.0418690
$475$	3146.98	0.303986
$476$	956.069	0.0920617
$477$	11402.5	1.09452
$478$	−7164.22	−0.685531
$479$	−12365.0	−1.17948	−0.589740	−	0.807593i	$-0.700770\pi$
$479$	−12365.0	−1.17948	−0.589740	+	0.807593i	$0.700770\pi$
$480$	−82.6206	−0.00785645
$481$	0	0
$482$	−9760.48	−0.922360
$483$	108.607	0.0102315
$484$	−2594.79	−0.243688
$485$	4067.52	0.380818
$486$	−1760.61	−0.164327
$487$	8447.36	0.786010	0.393005	−	0.919536i	$-0.371436\pi$
$487$	8447.36	0.786010	0.393005	+	0.919536i	$0.371436\pi$
$488$	987.380	0.0915914
$489$	430.502	0.0398118
$490$	4281.74	0.394753
$491$	14183.0	1.30360	0.651801	−	0.758390i	$-0.274014\pi$
$491$	14183.0	1.30360	0.651801	+	0.758390i	$0.274014\pi$
$492$	−602.836	−0.0552398
$493$	−23384.4	−2.13627
$494$	0	0
$495$	−4459.72	−0.404949
$496$	4463.88	0.404101
$497$	−302.928	−0.0273404
$498$	305.586	0.0274973
$499$	−14029.9	−1.25865	−0.629324	−	0.777143i	$-0.716668\pi$
$499$	−14029.9	−1.25865	−0.629324	+	0.777143i	$0.716668\pi$
$500$	−5332.13	−0.476920
$501$	569.797	0.0508117
$502$	5960.44	0.529935
$503$	9748.44	0.864139	0.432069	−	0.901840i	$-0.357784\pi$
$503$	9748.44	0.864139	0.432069	+	0.901840i	$0.357784\pi$
$504$	−547.579	−0.0483951
$505$	3021.05	0.266208
$506$	5481.53	0.481589
$507$	0	0
$508$	9874.05	0.862382
$509$	11469.7	0.998793	0.499397	−	0.866373i	$-0.333555\pi$
$509$	11469.7	0.998793	0.499397	+	0.866373i	$0.333555\pi$
$510$	483.891	0.0420138
$511$	−533.899	−0.0462198
$512$	−512.000	−0.0441942
$513$	813.430	0.0700074
$514$	7711.86	0.661781
$515$	−7001.89	−0.599107
$516$	−671.569	−0.0572949
$517$	6233.27	0.530249
$518$	−55.6046	−0.00471646
$519$	414.830	0.0350848
$520$	0	0
$521$	−8428.10	−0.708717	−0.354359	−	0.935110i	$-0.615301\pi$
$521$	−8428.10	−0.708717	−0.354359	+	0.935110i	$0.615301\pi$
$522$	13393.2	1.12300
$523$	−17172.7	−1.43577	−0.717885	−	0.696162i	$-0.754890\pi$
$523$	−17172.7	−1.43577	−0.717885	+	0.696162i	$0.754890\pi$
$524$	9119.29	0.760264
$525$	87.4870	0.00727285
$526$	8862.35	0.734633
$527$	−26144.0	−2.16101
$528$	169.604	0.0139793
$529$	−1157.53	−0.0951366
$530$	5406.75	0.443121
$531$	20782.7	1.69848
$532$	379.872	0.0309578
$533$	0	0
$534$	−34.6026	−0.00280412
$535$	−3628.93	−0.293257
$536$	7052.83	0.568351
$537$	169.720	0.0136387
$538$	−2522.67	−0.202156
$539$	−8789.54	−0.702398
$540$	−555.988	−0.0443073
$541$	−3228.20	−0.256546	−0.128273	−	0.991739i	$-0.540943\pi$
$541$	−3228.20	−0.256546	−0.128273	+	0.991739i	$0.540943\pi$
$542$	−12401.5	−0.982823
$543$	−1142.79	−0.0903164
$544$	2998.67	0.236336
$545$	−3773.01	−0.296547
$546$	0	0
$547$	6923.31	0.541169	0.270584	−	0.962696i	$-0.412783\pi$
$547$	6923.31	0.541169	0.270584	+	0.962696i	$0.412783\pi$
$548$	−1757.24	−0.136981
$549$	3312.08	0.257480
$550$	4415.56	0.342327
$551$	−9291.26	−0.718368
$552$	340.643	0.0262658
$553$	1357.86	0.104416
$554$	2315.14	0.177547
$555$	−28.1429	−0.00215243
$556$	−8715.75	−0.664803
$557$	−17673.8	−1.34446	−0.672229	−	0.740343i	$-0.734663\pi$
$557$	−17673.8	−1.34446	−0.672229	+	0.740343i	$0.734663\pi$
$558$	14973.7	1.13600
$559$	0	0
$560$	−259.647	−0.0195930
$561$	−993.331	−0.0747566
$562$	−12645.4	−0.949136
$563$	−2442.19	−0.182817	−0.0914085	−	0.995813i	$-0.529137\pi$
$563$	−2442.19	−0.182817	−0.0914085	+	0.995813i	$0.529137\pi$
$564$	387.359	0.0289198
$565$	11701.0	0.871262
$566$	18616.7	1.38254
$567$	−1825.47	−0.135207
$568$	−950.121	−0.0701869
$569$	6974.12	0.513832	0.256916	−	0.966434i	$-0.417294\pi$
$569$	6974.12	0.513832	0.256916	+	0.966434i	$0.417294\pi$
$570$	192.263	0.0141281
$571$	5061.39	0.370950	0.185475	−	0.982649i	$-0.440618\pi$
$571$	5061.39	0.370950	0.185475	+	0.982649i	$0.440618\pi$
$572$	0	0
$573$	655.026	0.0477558
$574$	−1894.49	−0.137761
$575$	8868.50	0.643204
$576$	−1717.46	−0.124238
$577$	3849.90	0.277771	0.138885	−	0.990308i	$-0.455648\pi$
$577$	3849.90	0.277771	0.138885	+	0.990308i	$0.455648\pi$
$578$	−7736.58	−0.556746
$579$	−998.754	−0.0716870
$580$	6350.68	0.454651
$581$	960.346	0.0685746
$582$	−518.888	−0.0369564
$583$	−11099.0	−0.788461
$584$	−1674.55	−0.118653
$585$	0	0
$586$	1012.39	0.0713677
$587$	−11682.1	−0.821418	−0.410709	−	0.911766i	$-0.634719\pi$
$587$	−11682.1	−0.821418	−0.410709	+	0.911766i	$0.634719\pi$
$588$	−546.215	−0.0383087
$589$	−10387.7	−0.726686
$590$	9854.56	0.687637
$591$	1398.39	0.0973304
$592$	−174.402	−0.0121079
$593$	−14539.7	−1.00687	−0.503435	−	0.864033i	$-0.667931\pi$
$593$	−14539.7	−1.00687	−0.503435	+	0.864033i	$0.667931\pi$
$594$	1141.33	0.0788374
$595$	1520.69	0.104777
$596$	11473.9	0.788572
$597$	894.827	0.0613448
$598$	0	0
$599$	8414.92	0.573997	0.286998	−	0.957931i	$-0.407343\pi$
$599$	8414.92	0.573997	0.286998	+	0.957931i	$0.407343\pi$
$600$	274.399	0.0186705
$601$	18270.7	1.24006	0.620032	−	0.784576i	$-0.287120\pi$
$601$	18270.7	1.24006	0.620032	+	0.784576i	$0.287120\pi$
$602$	−2110.50	−0.142886
$603$	23658.1	1.59773
$604$	−8808.01	−0.593366
$605$	−4127.19	−0.277345
$606$	−385.392	−0.0258341
$607$	−5667.19	−0.378952	−0.189476	−	0.981885i	$-0.560679\pi$
$607$	−5667.19	−0.378952	−0.189476	+	0.981885i	$0.560679\pi$
$608$	1191.45	0.0794733
$609$	−258.300	−0.0171869
$610$	1570.50	0.104242
$611$	0	0
$612$	10058.8	0.664383
$613$	−14532.6	−0.957533	−0.478767	−	0.877942i	$-0.658916\pi$
$613$	−14532.6	−0.957533	−0.478767	+	0.877942i	$0.658916\pi$
$614$	−8189.28	−0.538261
$615$	−958.852	−0.0628694
$616$	533.002	0.0348624
$617$	−11454.1	−0.747367	−0.373684	−	0.927556i	$-0.621905\pi$
$617$	−11454.1	−0.747367	−0.373684	+	0.927556i	$0.621905\pi$
$618$	893.222	0.0581402
$619$	10052.7	0.652751	0.326376	−	0.945240i	$-0.394173\pi$
$619$	10052.7	0.652751	0.326376	+	0.945240i	$0.394173\pi$
$620$	7100.11	0.459915
$621$	2292.33	0.148129
$622$	−17410.0	−1.12231
$623$	−108.743	−0.00699311
$624$	0	0
$625$	2084.07	0.133380
$626$	6289.56	0.401568
$627$	−394.677	−0.0251386
$628$	3460.55	0.219890
$629$	1021.43	0.0647491
$630$	−870.962	−0.0550793
$631$	12047.6	0.760078	0.380039	−	0.924970i	$-0.375911\pi$
$631$	12047.6	0.760078	0.380039	+	0.924970i	$0.375911\pi$
$632$	4258.86	0.268051
$633$	−378.702	−0.0237789
$634$	−19804.1	−1.24057
$635$	15705.3	0.981492
$636$	−689.732	−0.0430026
$637$	0	0
$638$	−13036.7	−0.808976
$639$	−3187.10	−0.197308
$640$	−814.371	−0.0502982
$641$	−21150.8	−1.30329	−0.651643	−	0.758526i	$-0.725920\pi$
$641$	−21150.8	−1.30329	−0.651643	+	0.758526i	$0.725920\pi$
$642$	462.938	0.0284590
$643$	15329.4	0.940177	0.470089	−	0.882619i	$-0.344222\pi$
$643$	15329.4	0.940177	0.470089	+	0.882619i	$0.344222\pi$
$644$	1070.52	0.0655035
$645$	−1068.18	−0.0652083
$646$	−6978.08	−0.424998
$647$	−14519.9	−0.882284	−0.441142	−	0.897437i	$-0.645426\pi$
$647$	−14519.9	−0.882284	−0.441142	+	0.897437i	$0.645426\pi$
$648$	−5725.50	−0.347097
$649$	−20229.4	−1.22354
$650$	0	0
$651$	−288.781	−0.0173859
$652$	4243.35	0.254881
$653$	27615.6	1.65495	0.827475	−	0.561502i	$-0.189776\pi$
$653$	27615.6	1.65495	0.827475	+	0.561502i	$0.189776\pi$
$654$	481.318	0.0287783
$655$	14504.9	0.865270
$656$	−5942.01	−0.353653
$657$	−5617.15	−0.333555
$658$	1217.33	0.0721222
$659$	−13898.5	−0.821560	−0.410780	−	0.911735i	$-0.634744\pi$
$659$	−13898.5	−0.821560	−0.410780	+	0.911735i	$0.634744\pi$
$660$	269.766	0.0159100
$661$	17344.3	1.02060	0.510298	−	0.859998i	$-0.329535\pi$
$661$	17344.3	1.02060	0.510298	+	0.859998i	$0.329535\pi$
$662$	−3932.74	−0.230892
$663$	0	0
$664$	3012.09	0.176042
$665$	604.212	0.0352336
$666$	−585.015	−0.0340374
$667$	−26183.7	−1.52000
$668$	5616.35	0.325304
$669$	−2582.44	−0.149242
$670$	11218.0	0.646850
$671$	−3223.91	−0.185481
$672$	33.1228	0.00190140
$673$	−2655.99	−0.152126	−0.0760631	−	0.997103i	$-0.524235\pi$
$673$	−2655.99	−0.152126	−0.0760631	+	0.997103i	$0.524235\pi$
$674$	19141.7	1.09393
$675$	1846.55	0.105294
$676$	0	0
$677$	−18853.1	−1.07029	−0.535143	−	0.844761i	$-0.679742\pi$
$677$	−18853.1	−1.07029	−0.535143	+	0.844761i	$0.679742\pi$
$678$	−1492.68	−0.0845514
$679$	−1630.68	−0.0921644
$680$	4769.59	0.268979
$681$	−336.626	−0.0189421
$682$	−14575.1	−0.818342
$683$	−1680.25	−0.0941333	−0.0470666	−	0.998892i	$-0.514987\pi$
$683$	−1680.25	−0.0941333	−0.0470666	+	0.998892i	$0.514987\pi$
$684$	3996.62	0.223413
$685$	−2795.00	−0.155900
$686$	−3466.30	−0.192921
$687$	2349.39	0.130473
$688$	−6619.49	−0.366810
$689$	0	0
$690$	541.816	0.0298936
$691$	−10938.4	−0.602195	−0.301098	−	0.953593i	$-0.597353\pi$
$691$	−10938.4	−0.602195	−0.301098	+	0.953593i	$0.597353\pi$
$692$	4088.88	0.224618
$693$	1787.91	0.0980044
$694$	12908.4	0.706048
$695$	−13863.0	−0.756624
$696$	−810.148	−0.0441215
$697$	34801.0	1.89122
$698$	5559.05	0.301451
$699$	130.165	0.00704333
$700$	862.338	0.0465619
$701$	19632.4	1.05778	0.528892	−	0.848689i	$-0.322608\pi$
$701$	19632.4	1.05778	0.528892	+	0.848689i	$0.322608\pi$
$702$	0	0
$703$	405.842	0.0217733
$704$	1671.74	0.0894972
$705$	616.121	0.0329141
$706$	−22294.7	−1.18849
$707$	−1211.15	−0.0644269
$708$	−1257.13	−0.0667316
$709$	−19477.8	−1.03174	−0.515870	−	0.856667i	$-0.672531\pi$
$709$	−19477.8	−1.03174	−0.515870	+	0.856667i	$0.672531\pi$
$710$	−1511.23	−0.0798810
$711$	14286.0	0.753539
$712$	−341.069	−0.0179524
$713$	−29273.6	−1.53759
$714$	−193.993	−0.0101681
$715$	0	0
$716$	1672.89	0.0873169
$717$	1453.67	0.0757158
$718$	8626.55	0.448384
$719$	−403.222	−0.0209147	−0.0104573	−	0.999945i	$-0.503329\pi$
$719$	−403.222	−0.0209147	−0.0104573	+	0.999945i	$0.503329\pi$
$720$	−2731.74	−0.141397
$721$	2807.07	0.144994
$722$	10945.4	0.564192
$723$	1980.47	0.101873
$724$	−11264.2	−0.578219
$725$	−21091.9	−1.08046
$726$	526.500	0.0269149
$727$	−29051.3	−1.48206	−0.741028	−	0.671474i	$-0.765661\pi$
$727$	−29051.3	−1.48206	−0.741028	+	0.671474i	$0.765661\pi$
$728$	0	0
$729$	−18966.3	−0.963589
$730$	−2663.49	−0.135041
$731$	38768.9	1.96159
$732$	−200.346	−0.0101161
$733$	25289.5	1.27434	0.637169	−	0.770724i	$-0.280105\pi$
$733$	25289.5	1.27434	0.637169	+	0.770724i	$0.280105\pi$
$734$	13819.0	0.694918
$735$	−868.792	−0.0435999
$736$	3357.63	0.168158
$737$	−23028.3	−1.15096
$738$	−19932.0	−0.994181
$739$	−253.911	−0.0126391	−0.00631953	−	0.999980i	$-0.502012\pi$
$739$	−253.911	−0.0126391	−0.00631953	+	0.999980i	$0.502012\pi$
$740$	−277.398	−0.0137802
$741$	0	0
$742$	−2167.58	−0.107243
$743$	−12903.6	−0.637127	−0.318564	−	0.947901i	$-0.603200\pi$
$743$	−12903.6	−0.637127	−0.318564	+	0.947901i	$0.603200\pi$
$744$	−905.751	−0.0446323
$745$	18250.0	0.897488
$746$	−4632.39	−0.227351
$747$	10103.8	0.494884
$748$	−9791.02	−0.478603
$749$	1454.84	0.0709731
$750$	1081.92	0.0526751
$751$	−33825.2	−1.64354	−0.821771	−	0.569818i	$-0.807014\pi$
$751$	−33825.2	−1.64354	−0.821771	+	0.569818i	$0.807014\pi$
$752$	3818.10	0.185149
$753$	−1209.41	−0.0585305
$754$	0	0
$755$	−14009.7	−0.675320
$756$	222.897	0.0107231
$757$	−33857.6	−1.62559	−0.812796	−	0.582548i	$-0.802056\pi$
$757$	−33857.6	−1.62559	−0.812796	+	0.582548i	$0.802056\pi$
$758$	22786.2	1.09186
$759$	−1112.24	−0.0531907
$760$	1895.09	0.0904500
$761$	11841.6	0.564072	0.282036	−	0.959404i	$-0.408990\pi$
$761$	11841.6	0.564072	0.282036	+	0.959404i	$0.408990\pi$
$762$	−2003.51	−0.0952487
$763$	1512.61	0.0717694
$764$	6456.43	0.305740
$765$	15999.2	0.756147
$766$	23905.6	1.12760
$767$	0	0
$768$	103.888	0.00488117
$769$	−9809.13	−0.459982	−0.229991	−	0.973193i	$-0.573870\pi$
$769$	−9809.13	−0.459982	−0.229991	+	0.973193i	$0.573870\pi$
$770$	847.776	0.0396776
$771$	−1564.79	−0.0730926
$772$	−9844.47	−0.458951
$773$	−27121.7	−1.26196	−0.630982	−	0.775797i	$-0.717348\pi$
$773$	−27121.7	−1.26196	−0.630982	+	0.775797i	$0.717348\pi$
$774$	−22204.5	−1.03117
$775$	−23580.9	−1.09297
$776$	−5114.55	−0.236600
$777$	11.2825	0.000520925 0
$778$	−10564.5	−0.486832
$779$	13827.4	0.635966
$780$	0	0
$781$	3102.26	0.142135
$782$	−19664.9	−0.899254
$783$	−5451.82	−0.248828
$784$	−5383.91	−0.245258
$785$	5504.25	0.250261
$786$	−1850.36	−0.0839699
$787$	−18455.4	−0.835914	−0.417957	−	0.908467i	$-0.637254\pi$
$787$	−18455.4	−0.835914	−0.417957	+	0.908467i	$0.637254\pi$
$788$	13783.6	0.623123
$789$	−1798.23	−0.0811390
$790$	6774.01	0.305074
$791$	−4690.94	−0.210860
$792$	5607.71	0.251592
$793$	0	0
$794$	−18223.2	−0.814504
$795$	−1097.07	−0.0489420
$796$	8820.09	0.392739
$797$	−14051.4	−0.624498	−0.312249	−	0.950000i	$-0.601082\pi$
$797$	−14051.4	−0.624498	−0.312249	+	0.950000i	$0.601082\pi$
$798$	−77.0785	−0.00341923
$799$	−22361.8	−0.990117
$800$	2704.69	0.119531
$801$	−1144.09	−0.0504673
$802$	999.123	0.0439904
$803$	5467.61	0.240284
$804$	−1431.07	−0.0627734
$805$	1702.73	0.0745507
$806$	0	0
$807$	511.866	0.0223278
$808$	−3798.71	−0.165394
$809$	−24386.4	−1.05980	−0.529902	−	0.848059i	$-0.677771\pi$
$809$	−24386.4	−1.05980	−0.529902	+	0.848059i	$0.677771\pi$
$810$	−9106.80	−0.395037
$811$	−34306.9	−1.48542	−0.742711	−	0.669612i	$-0.766460\pi$
$811$	−34306.9	−1.48542	−0.742711	+	0.669612i	$0.766460\pi$
$812$	−2546.00	−0.110033
$813$	2516.35	0.108551
$814$	569.442	0.0245196
$815$	6749.34	0.290085
$816$	−608.451	−0.0261030
$817$	15403.9	0.659626
$818$	−25606.4	−1.09451
$819$	0	0
$820$	−9451.17	−0.402499
$821$	3274.55	0.139199	0.0695995	−	0.997575i	$-0.477828\pi$
$821$	3274.55	0.139199	0.0695995	+	0.997575i	$0.477828\pi$
$822$	356.555	0.0151293
$823$	17214.8	0.729125	0.364562	−	0.931179i	$-0.381219\pi$
$823$	17214.8	0.729125	0.364562	+	0.931179i	$0.381219\pi$
$824$	8804.26	0.372222
$825$	−895.946	−0.0378095
$826$	−3950.71	−0.166420
$827$	40718.9	1.71213	0.856067	−	0.516864i	$-0.172901\pi$
$827$	40718.9	1.71213	0.856067	+	0.516864i	$0.172901\pi$
$828$	11262.9	0.472721
$829$	1627.26	0.0681751	0.0340876	−	0.999419i	$-0.489147\pi$
$829$	1627.26	0.0681751	0.0340876	+	0.999419i	$0.489147\pi$
$830$	4790.93	0.200356
$831$	−469.757	−0.0196097
$832$	0	0
$833$	31532.4	1.31156
$834$	1768.48	0.0734264
$835$	8933.18	0.370234
$836$	−3890.23	−0.160941
$837$	−6095.18	−0.251709
$838$	14736.5	0.607477
$839$	17121.2	0.704516	0.352258	−	0.935903i	$-0.385414\pi$
$839$	17121.2	0.704516	0.352258	+	0.935903i	$0.385414\pi$
$840$	52.6840	0.00216401
$841$	37883.4	1.55330
$842$	30533.8	1.24972
$843$	2565.84	0.104831
$844$	−3732.77	−0.152236
$845$	0	0
$846$	12807.5	0.520485
$847$	1654.60	0.0671223
$848$	−6798.52	−0.275309
$849$	−3777.45	−0.152699
$850$	−15840.8	−0.639216
$851$	1143.70	0.0460701
$852$	192.786	0.00775203
$853$	4791.65	0.192336	0.0961682	−	0.995365i	$-0.469341\pi$
$853$	4791.65	0.192336	0.0961682	+	0.995365i	$0.469341\pi$
$854$	−629.614	−0.0252283
$855$	6356.90	0.254271
$856$	4563.06	0.182199
$857$	12689.7	0.505802	0.252901	−	0.967492i	$-0.418615\pi$
$857$	12689.7	0.505802	0.252901	+	0.967492i	$0.418615\pi$
$858$	0	0
$859$	−35646.1	−1.41586	−0.707932	−	0.706280i	$-0.750372\pi$
$859$	−35646.1	−1.41586	−0.707932	+	0.706280i	$0.750372\pi$
$860$	−10528.7	−0.417473
$861$	384.406	0.0152155
$862$	41.8849	0.00165499
$863$	−10664.4	−0.420650	−0.210325	−	0.977631i	$-0.567452\pi$
$863$	−10664.4	−0.420650	−0.210325	+	0.977631i	$0.567452\pi$
$864$	699.107	0.0275279
$865$	6503.64	0.255642
$866$	−14916.2	−0.585305
$867$	1569.80	0.0614917
$868$	−2846.45	−0.111307
$869$	−13905.7	−0.542828
$870$	−1288.59	−0.0502155
$871$	0	0
$872$	4744.23	0.184243
$873$	−17156.3	−0.665124
$874$	−7813.40	−0.302394
$875$	3400.10	0.131365
$876$	339.778	0.0131051
$877$	11014.5	0.424097	0.212048	−	0.977259i	$-0.431987\pi$
$877$	11014.5	0.424097	0.212048	+	0.977259i	$0.431987\pi$
$878$	−18028.7	−0.692982
$879$	−205.421	−0.00788245
$880$	2659.02	0.101858
$881$	−4543.01	−0.173732	−0.0868660	−	0.996220i	$-0.527685\pi$
$881$	−4543.01	−0.173732	−0.0868660	+	0.996220i	$0.527685\pi$
$882$	−18059.9	−0.689464
$883$	28212.2	1.07522	0.537608	−	0.843195i	$-0.319328\pi$
$883$	28212.2	1.07522	0.537608	+	0.843195i	$0.319328\pi$
$884$	0	0
$885$	−1999.56	−0.0759484
$886$	−3574.25	−0.135530
$887$	−30696.0	−1.16197	−0.580987	−	0.813913i	$-0.697333\pi$
$887$	−30696.0	−1.16197	−0.580987	+	0.813913i	$0.697333\pi$
$888$	35.3872	0.00133729
$889$	−6296.30	−0.237538
$890$	−542.494	−0.0204319
$891$	18694.4	0.702903
$892$	−25454.5	−0.955469
$893$	−8884.93	−0.332948
$894$	−2328.13	−0.0870965
$895$	2660.85	0.0993770
$896$	326.483	0.0121730
$897$	0	0
$898$	9081.89	0.337491
$899$	69621.1	2.58286
$900$	9072.64	0.336024
$901$	39817.4	1.47227
$902$	19401.3	0.716180
$903$	428.234	0.0157815
$904$	−14712.9	−0.541311
$905$	−17916.5	−0.658082
$906$	1787.20	0.0655363
$907$	30277.2	1.10842	0.554211	−	0.832376i	$-0.313020\pi$
$907$	30277.2	1.10842	0.554211	+	0.832376i	$0.313020\pi$
$908$	−3318.04	−0.121270
$909$	−12742.4	−0.464951
$910$	0	0
$911$	18833.6	0.684946	0.342473	−	0.939528i	$-0.388736\pi$
$911$	18833.6	0.684946	0.342473	+	0.939528i	$0.388736\pi$
$912$	−241.754	−0.00877770
$913$	−9834.82	−0.356500
$914$	−13747.7	−0.497522
$915$	−318.664	−0.0115133
$916$	23157.4	0.835308
$917$	−5815.02	−0.209410
$918$	−4094.52	−0.147210
$919$	−2558.65	−0.0918411	−0.0459206	−	0.998945i	$-0.514622\pi$
$919$	−2558.65	−0.0918411	−0.0459206	+	0.998945i	$0.514622\pi$
$920$	5340.54	0.191383
$921$	1661.66	0.0594501
$922$	−18617.2	−0.664993
$923$	0	0
$924$	−108.150	−0.00385050
$925$	921.293	0.0327480
$926$	29527.1	1.04786
$927$	29533.1	1.04638
$928$	−7985.42	−0.282472
$929$	16392.9	0.578938	0.289469	−	0.957187i	$-0.406521\pi$
$929$	16392.9	0.578938	0.289469	+	0.957187i	$0.406521\pi$
$930$	−1440.66	−0.0507969
$931$	12528.7	0.441042
$932$	1283.00	0.0450924
$933$	3532.60	0.123957
$934$	−14795.6	−0.518338
$935$	−15573.3	−0.544707
$936$	0	0
$937$	24289.2	0.846845	0.423422	−	0.905932i	$-0.360829\pi$
$937$	24289.2	0.846845	0.423422	+	0.905932i	$0.360829\pi$
$938$	−4497.32	−0.156549
$939$	−1276.19	−0.0443525
$940$	6072.95	0.210721
$941$	−31263.6	−1.08306	−0.541532	−	0.840680i	$-0.682156\pi$
$941$	−31263.6	−1.08306	−0.541532	+	0.840680i	$0.682156\pi$
$942$	−702.169	−0.0242865
$943$	38967.0	1.34564
$944$	−12391.2	−0.427225
$945$	354.532	0.0122042
$946$	21613.4	0.742825
$947$	−44085.9	−1.51278	−0.756389	−	0.654122i	$-0.773038\pi$
$947$	−44085.9	−1.51278	−0.756389	+	0.654122i	$0.773038\pi$
$948$	−864.151	−0.0296058
$949$	0	0
$950$	−6293.96	−0.214950
$951$	4018.37	0.137019
$952$	−1912.14	−0.0650974
$953$	−634.082	−0.0215529	−0.0107765	−	0.999942i	$-0.503430\pi$
$953$	−634.082	−0.0215529	−0.0107765	+	0.999942i	$0.503430\pi$
$954$	−22805.0	−0.773942
$955$	10269.4	0.347968
$956$	14328.4	0.484744
$957$	2645.23	0.0893501
$958$	24730.0	0.834018
$959$	1120.52	0.0377305
$960$	165.241	0.00555535
$961$	48045.9	1.61277
$962$	0	0
$963$	15306.4	0.512193
$964$	19521.0	0.652207
$965$	−15658.3	−0.522340
$966$	−217.215	−0.00723476
$967$	58557.1	1.94733	0.973666	−	0.227981i	$-0.0732125\pi$
$967$	58557.1	1.94733	0.973666	+	0.227981i	$0.0732125\pi$
$968$	5189.57	0.172313
$969$	1415.90	0.0469404
$970$	−8135.04	−0.269279
$971$	−3335.69	−0.110244	−0.0551222	−	0.998480i	$-0.517555\pi$
$971$	−3335.69	−0.110244	−0.0551222	+	0.998480i	$0.517555\pi$
$972$	3521.23	0.116197
$973$	5557.70	0.183116
$974$	−16894.7	−0.555793
$975$	0	0
$976$	−1974.76	−0.0647649
$977$	23707.7	0.776331	0.388166	−	0.921590i	$-0.373109\pi$
$977$	23707.7	0.776331	0.388166	+	0.921590i	$0.373109\pi$
$978$	−861.004	−0.0281512
$979$	1113.63	0.0363552
$980$	−8563.47	−0.279133
$981$	15914.1	0.517940
$982$	−28365.9	−0.921786
$983$	−34833.6	−1.13023	−0.565117	−	0.825011i	$-0.691169\pi$
$983$	−34833.6	−1.13023	−0.565117	+	0.825011i	$0.691169\pi$
$984$	1205.67	0.0390604
$985$	21923.8	0.709188
$986$	46768.9	1.51057
$987$	−247.004	−0.00796577
$988$	0	0
$989$	43409.8	1.39570
$990$	8919.44	0.286342
$991$	23957.5	0.767948	0.383974	−	0.923344i	$-0.374555\pi$
$991$	23957.5	0.767948	0.383974	+	0.923344i	$0.374555\pi$
$992$	−8927.77	−0.285743
$993$	797.979	0.0255016
$994$	605.856	0.0193326
$995$	14029.0	0.446983
$996$	−611.172	−0.0194435
$997$	−43361.9	−1.37742	−0.688708	−	0.725038i	$-0.741822\pi$
$997$	−43361.9	−1.37742	−0.688708	+	0.725038i	$0.741822\pi$
$998$	28059.8	0.889998
$999$	238.135	0.00754181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.a.j.1.3	✓	3
13.2	odd	12		338.4.e.h.147.1		12
13.3	even	3		338.4.c.l.191.1		6
13.4	even	6		338.4.c.k.315.1		6
13.5	odd	4		338.4.b.f.337.6		6
13.6	odd	12		338.4.e.h.23.4		12
13.7	odd	12		338.4.e.h.23.1		12
13.8	odd	4		338.4.b.f.337.3		6
13.9	even	3		338.4.c.l.315.1		6
13.10	even	6		338.4.c.k.191.1		6
13.11	odd	12		338.4.e.h.147.4		12
13.12	even	2		338.4.a.k.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.4.a.j.1.3	✓	3	1.1	even	1	trivial
338.4.a.k.1.3	yes	3	13.12	even	2
338.4.b.f.337.3		6	13.8	odd	4
338.4.b.f.337.6		6	13.5	odd	4
338.4.c.k.191.1		6	13.10	even	6
338.4.c.k.315.1		6	13.4	even	6
338.4.c.l.191.1		6	13.3	even	3
338.4.c.l.315.1		6	13.9	even	3
338.4.e.h.23.1		12	13.7	odd	12
338.4.e.h.23.4		12	13.6	odd	12
338.4.e.h.147.1		12	13.2	odd	12
338.4.e.h.147.4		12	13.11	odd	12

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.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +0.405813 q^{3} +4.00000 q^{4} +6.36227 q^{5} -0.811626 q^{6} -2.55065 q^{7} -8.00000 q^{8} -26.8353 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +0.405813 q^{3} +4.00000 q^{4} +6.36227 q^{5} -0.811626 q^{6} -2.55065 q^{7} -8.00000 q^{8} -26.8353 q^{9} -12.7245 q^{10} +26.1209 q^{11} +1.62325 q^{12} +5.10129 q^{14} +2.58189 q^{15} +16.0000 q^{16} -93.7085 q^{17} +53.6706 q^{18} -37.2329 q^{19} +25.4491 q^{20} -1.03509 q^{21} -52.2419 q^{22} -104.926 q^{23} -3.24651 q^{24} -84.5215 q^{25} -21.8471 q^{27} -10.2026 q^{28} +249.544 q^{29} -5.16379 q^{30} +278.993 q^{31} -32.0000 q^{32} +10.6002 q^{33} +187.417 q^{34} -16.2279 q^{35} -107.341 q^{36} -10.9001 q^{37} +74.4658 q^{38} -50.8982 q^{40} -371.375 q^{41} +2.07017 q^{42} -413.718 q^{43} +104.484 q^{44} -170.734 q^{45} +209.852 q^{46} +238.631 q^{47} +6.49301 q^{48} -336.494 q^{49} +169.043 q^{50} -38.0282 q^{51} -424.907 q^{53} +43.6942 q^{54} +166.188 q^{55} +20.4052 q^{56} -15.1096 q^{57} -499.089 q^{58} -774.453 q^{59} +10.3276 q^{60} -123.423 q^{61} -557.985 q^{62} +68.4474 q^{63} +64.0000 q^{64} -21.2004 q^{66} -881.604 q^{67} -374.834 q^{68} -42.5804 q^{69} +32.4558 q^{70} +118.765 q^{71} +214.683 q^{72} +209.319 q^{73} +21.8002 q^{74} -34.2999 q^{75} -148.932 q^{76} -66.6253 q^{77} -532.358 q^{79} +101.796 q^{80} +715.688 q^{81} +742.751 q^{82} -376.511 q^{83} -4.14034 q^{84} -596.199 q^{85} +827.436 q^{86} +101.268 q^{87} -208.967 q^{88} +42.6336 q^{89} +341.467 q^{90} -419.704 q^{92} +113.219 q^{93} -477.263 q^{94} -236.886 q^{95} -12.9860 q^{96} +639.319 q^{97} +672.988 q^{98} -700.963 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9	\( 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9} - 24 q^{10} + 82 q^{11} - 48 q^{12} + 54 q^{14} - 90 q^{15} + 48 q^{16} - 90 q^{17} - 18 q^{18} + 130 q^{19} + 48 q^{20} + 234 q^{21} - 164 q^{22} + 19 q^{23} + 96 q^{24} - 61 q^{25} - 69 q^{27} - 108 q^{28} - 101 q^{29} + 180 q^{30} + 519 q^{31} - 96 q^{32} - 146 q^{33} + 180 q^{34} - 458 q^{35} + 36 q^{36} + 84 q^{37} - 260 q^{38} - 96 q^{40} + 187 q^{41} - 468 q^{42} - 1205 q^{43} + 328 q^{44} + 645 q^{45} - 38 q^{46} - 536 q^{47} - 192 q^{48} - 184 q^{49} + 122 q^{50} - 207 q^{51} - 1095 q^{53} + 138 q^{54} - 526 q^{55} + 216 q^{56} - 1409 q^{57} + 202 q^{58} - 1413 q^{59} - 360 q^{60} - 1108 q^{61} - 1038 q^{62} - 1404 q^{63} + 192 q^{64} + 292 q^{66} - 1605 q^{67} - 360 q^{68} - 314 q^{69} + 916 q^{70} - 909 q^{71} - 72 q^{72} + 287 q^{73} - 168 q^{74} - 505 q^{75} + 520 q^{76} + 480 q^{77} - 1961 q^{79} + 192 q^{80} + 915 q^{81} - 374 q^{82} + 191 q^{83} + 936 q^{84} + 67 q^{85} + 2410 q^{86} + 1636 q^{87} - 656 q^{88} + 1091 q^{89} - 1290 q^{90} + 76 q^{92} - 1614 q^{93} + 1072 q^{94} + 1829 q^{95} + 384 q^{96} - 947 q^{97} + 368 q^{98} - 2057 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9 - 24 * q^10 + 82 * q^11 - 48 * q^12 + 54 * q^14 - 90 * q^15 + 48 * q^16 - 90 * q^17 - 18 * q^18 + 130 * q^19 + 48 * q^20 + 234 * q^21 - 164 * q^22 + 19 * q^23 + 96 * q^24 - 61 * q^25 - 69 * q^27 - 108 * q^28 - 101 * q^29 + 180 * q^30 + 519 * q^31 - 96 * q^32 - 146 * q^33 + 180 * q^34 - 458 * q^35 + 36 * q^36 + 84 * q^37 - 260 * q^38 - 96 * q^40 + 187 * q^41 - 468 * q^42 - 1205 * q^43 + 328 * q^44 + 645 * q^45 - 38 * q^46 - 536 * q^47 - 192 * q^48 - 184 * q^49 + 122 * q^50 - 207 * q^51 - 1095 * q^53 + 138 * q^54 - 526 * q^55 + 216 * q^56 - 1409 * q^57 + 202 * q^58 - 1413 * q^59 - 360 * q^60 - 1108 * q^61 - 1038 * q^62 - 1404 * q^63 + 192 * q^64 + 292 * q^66 - 1605 * q^67 - 360 * q^68 - 314 * q^69 + 916 * q^70 - 909 * q^71 - 72 * q^72 + 287 * q^73 - 168 * q^74 - 505 * q^75 + 520 * q^76 + 480 * q^77 - 1961 * q^79 + 192 * q^80 + 915 * q^81 - 374 * q^82 + 191 * q^83 + 936 * q^84 + 67 * q^85 + 2410 * q^86 + 1636 * q^87 - 656 * q^88 + 1091 * q^89 - 1290 * q^90 + 76 * q^92 - 1614 * q^93 + 1072 * q^94 + 1829 * q^95 + 384 * q^96 - 947 * q^97 + 368 * q^98 - 2057 * q^99

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