LMFDB - Embedded newform 338.4.a.g.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{217}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 54 $ x^2 - x - 54
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.86546$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -8.86546 q^{3} +4.00000 q^{4} -3.86546 q^{5} +17.7309 q^{6} +15.1345 q^{7} -8.00000 q^{8} +51.5964 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -8.86546 q^{3} +4.00000 q^{4} -3.86546 q^{5} +17.7309 q^{6} +15.1345 q^{7} -8.00000 q^{8} +51.5964 q^{9} +7.73092 q^{10} -54.0582 q^{11} -35.4618 q^{12} -30.2691 q^{14} +34.2691 q^{15} +16.0000 q^{16} -123.924 q^{17} -103.193 q^{18} -8.86546 q^{19} -15.4618 q^{20} -134.175 q^{21} +108.116 q^{22} +38.5964 q^{23} +70.9237 q^{24} -110.058 q^{25} -218.058 q^{27} +60.5382 q^{28} -187.386 q^{29} -68.5382 q^{30} +36.7710 q^{31} -32.0000 q^{32} +479.251 q^{33} +247.847 q^{34} -58.5020 q^{35} +206.386 q^{36} +321.538 q^{37} +17.7309 q^{38} +30.9237 q^{40} +26.3092 q^{41} +268.349 q^{42} +137.636 q^{43} -216.233 q^{44} -199.444 q^{45} -77.1928 q^{46} +300.466 q^{47} -141.847 q^{48} -113.946 q^{49} +220.116 q^{50} +1098.64 q^{51} -260.135 q^{53} +436.116 q^{54} +208.960 q^{55} -121.076 q^{56} +78.5964 q^{57} +374.771 q^{58} +246.058 q^{59} +137.076 q^{60} +91.3053 q^{61} -73.5421 q^{62} +780.887 q^{63} +64.0000 q^{64} -958.502 q^{66} -410.524 q^{67} -495.695 q^{68} -342.175 q^{69} +117.004 q^{70} +424.444 q^{71} -412.771 q^{72} +421.982 q^{73} -643.076 q^{74} +975.717 q^{75} -35.4618 q^{76} -818.146 q^{77} +733.542 q^{79} -61.8474 q^{80} +540.084 q^{81} -52.6184 q^{82} +616.843 q^{83} -536.699 q^{84} +479.022 q^{85} -275.273 q^{86} +1661.26 q^{87} +432.466 q^{88} +207.171 q^{89} +398.887 q^{90} +154.386 q^{92} -325.992 q^{93} -600.932 q^{94} +34.2691 q^{95} +283.695 q^{96} +741.484 q^{97} +227.891 q^{98} -2789.21 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 7 q^{5} + 6 q^{6} + 45 q^{7} - 16 q^{8} + 59 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 - 3 * q^3 + 8 * q^4 + 7 * q^5 + 6 * q^6 + 45 * q^7 - 16 * q^8 + 59 * q^9	\( 2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 7 q^{5} + 6 q^{6} + 45 q^{7} - 16 q^{8} + 59 q^{9} - 14 q^{10} - 5 q^{11} - 12 q^{12} - 90 q^{14} + 98 q^{15} + 32 q^{16} - 130 q^{17} - 118 q^{18} - 3 q^{19} + 28 q^{20} + 41 q^{21} + 10 q^{22} + 33 q^{23} + 24 q^{24} - 117 q^{25} - 333 q^{27} + 180 q^{28} - 198 q^{29} - 196 q^{30} - 280 q^{31} - 64 q^{32} + 767 q^{33} + 260 q^{34} + 266 q^{35} + 236 q^{36} + 702 q^{37} + 6 q^{38} - 56 q^{40} - 242 q^{41} - 82 q^{42} - 93 q^{43} - 20 q^{44} - 119 q^{45} - 66 q^{46} - 224 q^{47} - 48 q^{48} + 435 q^{49} + 234 q^{50} + 1063 q^{51} - 535 q^{53} + 666 q^{54} + 742 q^{55} - 360 q^{56} + 113 q^{57} + 396 q^{58} + 389 q^{59} + 392 q^{60} + 654 q^{61} + 560 q^{62} + 1002 q^{63} + 128 q^{64} - 1534 q^{66} + 107 q^{67} - 520 q^{68} - 375 q^{69} - 532 q^{70} + 569 q^{71} - 472 q^{72} + 623 q^{73} - 1404 q^{74} + 935 q^{75} - 12 q^{76} + 647 q^{77} + 760 q^{79} + 112 q^{80} - 334 q^{81} + 484 q^{82} + 1764 q^{83} + 164 q^{84} + 413 q^{85} + 186 q^{86} + 1599 q^{87} + 40 q^{88} + 871 q^{89} + 238 q^{90} + 132 q^{92} - 2184 q^{93} + 448 q^{94} + 98 q^{95} + 96 q^{96} + 879 q^{97} - 870 q^{98} - 2426 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 3 * q^3 + 8 * q^4 + 7 * q^5 + 6 * q^6 + 45 * q^7 - 16 * q^8 + 59 * q^9 - 14 * q^10 - 5 * q^11 - 12 * q^12 - 90 * q^14 + 98 * q^15 + 32 * q^16 - 130 * q^17 - 118 * q^18 - 3 * q^19 + 28 * q^20 + 41 * q^21 + 10 * q^22 + 33 * q^23 + 24 * q^24 - 117 * q^25 - 333 * q^27 + 180 * q^28 - 198 * q^29 - 196 * q^30 - 280 * q^31 - 64 * q^32 + 767 * q^33 + 260 * q^34 + 266 * q^35 + 236 * q^36 + 702 * q^37 + 6 * q^38 - 56 * q^40 - 242 * q^41 - 82 * q^42 - 93 * q^43 - 20 * q^44 - 119 * q^45 - 66 * q^46 - 224 * q^47 - 48 * q^48 + 435 * q^49 + 234 * q^50 + 1063 * q^51 - 535 * q^53 + 666 * q^54 + 742 * q^55 - 360 * q^56 + 113 * q^57 + 396 * q^58 + 389 * q^59 + 392 * q^60 + 654 * q^61 + 560 * q^62 + 1002 * q^63 + 128 * q^64 - 1534 * q^66 + 107 * q^67 - 520 * q^68 - 375 * q^69 - 532 * q^70 + 569 * q^71 - 472 * q^72 + 623 * q^73 - 1404 * q^74 + 935 * q^75 - 12 * q^76 + 647 * q^77 + 760 * q^79 + 112 * q^80 - 334 * q^81 + 484 * q^82 + 1764 * q^83 + 164 * q^84 + 413 * q^85 + 186 * q^86 + 1599 * q^87 + 40 * q^88 + 871 * q^89 + 238 * q^90 + 132 * q^92 - 2184 * q^93 + 448 * q^94 + 98 * q^95 + 96 * q^96 + 879 * q^97 - 870 * q^98 - 2426 * 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
$3$	−8.86546	−1.70616	−0.853079	−	0.521781i	$-0.825268\pi$
$3$	−8.86546	−1.70616	−0.853079	+	0.521781i	$0.825268\pi$
$4$	4.00000	0.500000
$5$	−3.86546	−0.345737	−0.172869	−	0.984945i	$-0.555304\pi$
$5$	−3.86546	−0.345737	−0.172869	+	0.984945i	$0.555304\pi$
$6$	17.7309	1.20644
$7$	15.1345	0.817188	0.408594	−	0.912716i	$-0.366019\pi$
$7$	15.1345	0.817188	0.408594	+	0.912716i	$0.366019\pi$
$8$	−8.00000	−0.353553
$9$	51.5964	1.91098
$10$	7.73092	0.244473
$11$	−54.0582	−1.48174	−0.740871	−	0.671647i	$-0.765587\pi$
$11$	−54.0582	−1.48174	−0.740871	+	0.671647i	$0.765587\pi$
$12$	−35.4618	−0.853079
$13$	0	0
$13$	0	0
$14$	−30.2691	−0.577839
$15$	34.2691	0.589883
$16$	16.0000	0.250000
$17$	−123.924	−1.76799	−0.883997	−	0.467492i	$-0.845158\pi$
$17$	−123.924	−1.76799	−0.883997	+	0.467492i	$0.845158\pi$
$18$	−103.193	−1.35126
$19$	−8.86546	−0.107046	−0.0535231	−	0.998567i	$-0.517045\pi$
$19$	−8.86546	−0.107046	−0.0535231	+	0.998567i	$0.517045\pi$
$20$	−15.4618	−0.172869
$21$	−134.175	−1.39425
$22$	108.116	1.04775
$23$	38.5964	0.349909	0.174954	−	0.984577i	$-0.444022\pi$
$23$	38.5964	0.349909	0.174954	+	0.984577i	$0.444022\pi$
$24$	70.9237	0.603218
$25$	−110.058	−0.880466
$26$	0	0
$27$	−218.058	−1.55427
$28$	60.5382	0.408594
$29$	−187.386	−1.19988	−0.599942	−	0.800044i	$-0.704810\pi$
$29$	−187.386	−1.19988	−0.599942	+	0.800044i	$0.704810\pi$
$30$	−68.5382	−0.417110
$31$	36.7710	0.213041	0.106521	−	0.994311i	$-0.466029\pi$
$31$	36.7710	0.213041	0.106521	+	0.994311i	$0.466029\pi$
$32$	−32.0000	−0.176777
$33$	479.251	2.52809
$34$	247.847	1.25016
$35$	−58.5020	−0.282532
$36$	206.386	0.955489
$37$	321.538	1.42866	0.714332	−	0.699807i	$-0.246731\pi$
$37$	321.538	1.42866	0.714332	+	0.699807i	$0.246731\pi$
$38$	17.7309	0.0756930
$39$	0	0
$40$	30.9237	0.122237
$41$	26.3092	0.100215	0.0501074	−	0.998744i	$-0.484044\pi$
$41$	26.3092	0.100215	0.0501074	+	0.998744i	$0.484044\pi$
$42$	268.349	0.985886
$43$	137.636	0.488125	0.244062	−	0.969760i	$-0.421520\pi$
$43$	137.636	0.488125	0.244062	+	0.969760i	$0.421520\pi$
$44$	−216.233	−0.740871
$45$	−199.444	−0.660696
$46$	−77.1928	−0.247423
$47$	300.466	0.932499	0.466249	−	0.884653i	$-0.345605\pi$
$47$	300.466	0.932499	0.466249	+	0.884653i	$0.345605\pi$
$48$	−141.847	−0.426540
$49$	−113.946	−0.332203
$50$	220.116	0.622583
$51$	1098.64	3.01648
$52$	0	0
$53$	−260.135	−0.674193	−0.337096	−	0.941470i	$-0.609445\pi$
$53$	−260.135	−0.674193	−0.337096	+	0.941470i	$0.609445\pi$
$54$	436.116	1.09904
$55$	208.960	0.512294
$56$	−121.076	−0.288920
$57$	78.5964	0.182638
$58$	374.771	0.848446
$59$	246.058	0.542950	0.271475	−	0.962445i	$-0.412489\pi$
$59$	246.058	0.542950	0.271475	+	0.962445i	$0.412489\pi$
$60$	137.076	0.294941
$61$	91.3053	0.191647	0.0958233	−	0.995398i	$-0.469452\pi$
$61$	91.3053	0.191647	0.0958233	+	0.995398i	$0.469452\pi$
$62$	−73.5421	−0.150643
$63$	780.887	1.56163
$64$	64.0000	0.125000
$65$	0	0
$66$	−958.502	−1.78763
$67$	−410.524	−0.748559	−0.374280	−	0.927316i	$-0.622110\pi$
$67$	−410.524	−0.748559	−0.374280	+	0.927316i	$0.622110\pi$
$68$	−495.695	−0.883997
$69$	−342.175	−0.597000
$70$	117.004	0.199781
$71$	424.444	0.709468	0.354734	−	0.934967i	$-0.384571\pi$
$71$	424.444	0.709468	0.354734	+	0.934967i	$0.384571\pi$
$72$	−412.771	−0.675632
$73$	421.982	0.676565	0.338283	−	0.941045i	$-0.390154\pi$
$73$	421.982	0.676565	0.338283	+	0.941045i	$0.390154\pi$
$74$	−643.076	−1.01022
$75$	975.717	1.50221
$76$	−35.4618	−0.0535231
$77$	−818.146	−1.21086
$78$	0	0
$79$	733.542	1.04468	0.522341	−	0.852736i	$-0.325059\pi$
$79$	733.542	1.04468	0.522341	+	0.852736i	$0.325059\pi$
$80$	−61.8474	−0.0864343
$81$	540.084	0.740856
$82$	−52.6184	−0.0708626
$83$	616.843	0.815751	0.407876	−	0.913037i	$-0.366270\pi$
$83$	616.843	0.815751	0.407876	+	0.913037i	$0.366270\pi$
$84$	−536.699	−0.697126
$85$	479.022	0.611262
$86$	−275.273	−0.345156
$87$	1661.26	2.04719
$88$	432.466	0.523875
$89$	207.171	0.246742	0.123371	−	0.992361i	$-0.460629\pi$
$89$	207.171	0.246742	0.123371	+	0.992361i	$0.460629\pi$
$90$	398.887	0.467183
$91$	0	0
$92$	154.386	0.174954
$93$	−325.992	−0.363482
$94$	−600.932	−0.659376
$95$	34.2691	0.0370098
$96$	283.695	0.301609
$97$	741.484	0.776147	0.388074	−	0.921628i	$-0.373141\pi$
$97$	741.484	0.776147	0.388074	+	0.921628i	$0.373141\pi$
$98$	227.891	0.234903
$99$	−2789.21	−2.83158
$100$	−440.233	−0.440233
$101$	−413.305	−0.407182	−0.203591	−	0.979056i	$-0.565261\pi$
$101$	−413.305	−0.407182	−0.203591	+	0.979056i	$0.565261\pi$
$102$	−2197.28	−2.13297
$103$	−3.76712	−0.00360374	−0.00180187	−	0.999998i	$-0.500574\pi$
$103$	−3.76712	−0.00360374	−0.00180187	+	0.999998i	$0.500574\pi$
$104$	0	0
$105$	518.647	0.482045
$106$	520.269	0.476726
$107$	113.825	0.102840	0.0514201	−	0.998677i	$-0.483625\pi$
$107$	113.825	0.102840	0.0514201	+	0.998677i	$0.483625\pi$
$108$	−872.233	−0.777136
$109$	−1935.63	−1.70092	−0.850458	−	0.526044i	$-0.823675\pi$
$109$	−1935.63	−1.70092	−0.850458	+	0.526044i	$0.823675\pi$
$110$	−417.920	−0.362246
$111$	−2850.58	−2.43753
$112$	242.153	0.204297
$113$	1898.78	1.58073	0.790365	−	0.612636i	$-0.209891\pi$
$113$	1898.78	1.58073	0.790365	+	0.612636i	$0.209891\pi$
$114$	−157.193	−0.129144
$115$	−149.193	−0.120976
$116$	−749.542	−0.599942
$117$	0	0
$118$	−492.116	−0.383924
$119$	−1875.53	−1.44478
$120$	−274.153	−0.208555
$121$	1591.29	1.19556
$122$	−182.611	−0.135515
$123$	−233.243	−0.170982
$124$	147.084	0.106521
$125$	908.608	0.650147
$126$	−1561.77	−1.10424
$127$	244.604	0.170906	0.0854532	−	0.996342i	$-0.472766\pi$
$127$	244.604	0.170906	0.0854532	+	0.996342i	$0.472766\pi$
$128$	−128.000	−0.0883883
$129$	−1220.21	−0.832818
$130$	0	0
$131$	1615.62	1.07754	0.538769	−	0.842453i	$-0.318889\pi$
$131$	1615.62	1.07754	0.538769	+	0.842453i	$0.318889\pi$
$132$	1917.00	1.26404
$133$	−134.175	−0.0874768
$134$	821.048	0.529311
$135$	842.895	0.537369
$136$	991.389	0.625080
$137$	2301.95	1.43554	0.717769	−	0.696282i	$-0.245164\pi$
$137$	2301.95	1.43554	0.717769	+	0.696282i	$0.245164\pi$
$138$	684.349	0.422143
$139$	−2809.07	−1.71412	−0.857058	−	0.515219i	$-0.827710\pi$
$139$	−2809.07	−1.71412	−0.857058	+	0.515219i	$0.827710\pi$
$140$	−234.008	−0.141266
$141$	−2663.77	−1.59099
$142$	−848.887	−0.501669
$143$	0	0
$144$	825.542	0.477744
$145$	724.331	0.414844
$146$	−843.964	−0.478404
$147$	1010.18	0.566791
$148$	1286.15	0.714332
$149$	−2533.17	−1.39279	−0.696393	−	0.717660i	$-0.745213\pi$
$149$	−2533.17	−1.39279	−0.696393	+	0.717660i	$0.745213\pi$
$150$	−1951.43	−1.06223
$151$	2391.10	1.28864	0.644321	−	0.764755i	$-0.277140\pi$
$151$	2391.10	1.28864	0.644321	+	0.764755i	$0.277140\pi$
$152$	70.9237	0.0378465
$153$	−6394.01	−3.37860
$154$	1636.29	0.856209
$155$	−142.137	−0.0736562
$156$	0	0
$157$	1927.49	0.979811	0.489905	−	0.871776i	$-0.337031\pi$
$157$	1927.49	0.979811	0.489905	+	0.871776i	$0.337031\pi$
$158$	−1467.08	−0.738702
$159$	2306.21	1.15028
$160$	123.695	0.0611183
$161$	584.138	0.285941
$162$	−1080.17	−0.523864
$163$	2218.52	1.06606	0.533031	−	0.846096i	$-0.321053\pi$
$163$	2218.52	1.06606	0.533031	+	0.846096i	$0.321053\pi$
$164$	105.237	0.0501074
$165$	−1852.53	−0.874054
$166$	−1233.69	−0.576823
$167$	1508.27	0.698883	0.349442	−	0.936958i	$-0.386371\pi$
$167$	1508.27	0.698883	0.349442	+	0.936958i	$0.386371\pi$
$168$	1073.40	0.492943
$169$	0	0
$170$	−958.044	−0.432227
$171$	−457.426	−0.204563
$172$	550.546	0.244062
$173$	−1706.47	−0.749946	−0.374973	−	0.927036i	$-0.622348\pi$
$173$	−1706.47	−0.749946	−0.374973	+	0.927036i	$0.622348\pi$
$174$	−3322.52	−1.44758
$175$	−1665.68	−0.719506
$176$	−864.932	−0.370436
$177$	−2181.42	−0.926359
$178$	−414.341	−0.174473
$179$	595.223	0.248542	0.124271	−	0.992248i	$-0.460341\pi$
$179$	595.223	0.248542	0.124271	+	0.992248i	$0.460341\pi$
$180$	−797.775	−0.330348
$181$	403.006	0.165498	0.0827492	−	0.996570i	$-0.473630\pi$
$181$	403.006	0.165498	0.0827492	+	0.996570i	$0.473630\pi$
$182$	0	0
$183$	−809.463	−0.326980
$184$	−308.771	−0.123711
$185$	−1242.89	−0.493942
$186$	651.984	0.257020
$187$	6699.09	2.61971
$188$	1201.86	0.466249
$189$	−3300.21	−1.27013
$190$	−68.5382	−0.0261699
$191$	−613.696	−0.232490	−0.116245	−	0.993221i	$-0.537086\pi$
$191$	−613.696	−0.232490	−0.116245	+	0.993221i	$0.537086\pi$
$192$	−567.389	−0.213270
$193$	−2005.33	−0.747910	−0.373955	−	0.927447i	$-0.621999\pi$
$193$	−2005.33	−0.747910	−0.373955	+	0.927447i	$0.621999\pi$
$194$	−1482.97	−0.548819
$195$	0	0
$196$	−455.783	−0.166102
$197$	557.500	0.201625	0.100813	−	0.994905i	$-0.467856\pi$
$197$	557.500	0.201625	0.100813	+	0.994905i	$0.467856\pi$
$198$	5578.42	2.00223
$199$	5533.97	1.97132	0.985661	−	0.168739i	$-0.0539697\pi$
$199$	5533.97	1.97132	0.985661	+	0.168739i	$0.0539697\pi$
$200$	880.466	0.311292
$201$	3639.48	1.27716
$202$	826.611	0.287921
$203$	−2835.99	−0.980531
$204$	4394.56	1.50824
$205$	−101.697	−0.0346480
$206$	7.53424	0.00254823
$207$	1991.43	0.668668
$208$	0	0
$209$	479.251	0.158615
$210$	−1037.29	−0.340857
$211$	−4683.73	−1.52816	−0.764079	−	0.645122i	$-0.776806\pi$
$211$	−4683.73	−1.52816	−0.764079	+	0.645122i	$0.776806\pi$
$212$	−1040.54	−0.337096
$213$	−3762.89	−1.21046
$214$	−227.651	−0.0727191
$215$	−532.028	−0.168763
$216$	1744.47	0.549518
$217$	556.513	0.174095
$218$	3871.26	1.20273
$219$	−3741.06	−1.15433
$220$	835.840	0.256147
$221$	0	0
$222$	5701.17	1.72359
$223$	2383.86	0.715853	0.357926	−	0.933750i	$-0.383484\pi$
$223$	2383.86	0.715853	0.357926	+	0.933750i	$0.383484\pi$
$224$	−484.305	−0.144460
$225$	−5678.61	−1.68255
$226$	−3797.57	−1.11774
$227$	1554.07	0.454394	0.227197	−	0.973849i	$-0.427044\pi$
$227$	1554.07	0.454394	0.227197	+	0.973849i	$0.427044\pi$
$228$	314.386	0.0913188
$229$	2915.60	0.841346	0.420673	−	0.907212i	$-0.361794\pi$
$229$	2915.60	0.841346	0.420673	+	0.907212i	$0.361794\pi$
$230$	298.386	0.0855433
$231$	7253.24	2.06592
$232$	1499.08	0.424223
$233$	−4233.93	−1.19045	−0.595223	−	0.803561i	$-0.702936\pi$
$233$	−4233.93	−1.19045	−0.595223	+	0.803561i	$0.702936\pi$
$234$	0	0
$235$	−1161.44	−0.322400
$236$	984.233	0.271475
$237$	−6503.19	−1.78239
$238$	3751.06	1.02162
$239$	−2372.55	−0.642124	−0.321062	−	0.947058i	$-0.604040\pi$
$239$	−2372.55	−0.642124	−0.321062	+	0.947058i	$0.604040\pi$
$240$	548.305	0.147471
$241$	−325.763	−0.0870716	−0.0435358	−	0.999052i	$-0.513862\pi$
$241$	−325.763	−0.0870716	−0.0435358	+	0.999052i	$0.513862\pi$
$242$	−3182.58	−0.845389
$243$	1099.48	0.290253
$244$	365.221	0.0958233
$245$	440.453	0.114855
$246$	466.486	0.120903
$247$	0	0
$248$	−294.168	−0.0753214
$249$	−5468.60	−1.39180
$250$	−1817.22	−0.459723
$251$	−2601.77	−0.654272	−0.327136	−	0.944977i	$-0.606083\pi$
$251$	−2601.77	−0.654272	−0.327136	+	0.944977i	$0.606083\pi$
$252$	3123.55	0.780814
$253$	−2086.45	−0.518475
$254$	−489.208	−0.120849
$255$	−4246.75	−1.04291
$256$	256.000	0.0625000
$257$	4217.10	1.02356	0.511781	−	0.859116i	$-0.328986\pi$
$257$	4217.10	1.02356	0.511781	+	0.859116i	$0.328986\pi$
$258$	2440.42	0.588892
$259$	4866.33	1.16749
$260$	0	0
$261$	−9668.41	−2.29295
$262$	−3231.24	−0.761935
$263$	6474.32	1.51796	0.758981	−	0.651113i	$-0.225698\pi$
$263$	6474.32	1.51796	0.758981	+	0.651113i	$0.225698\pi$
$264$	−3834.01	−0.893814
$265$	1005.54	0.233094
$266$	268.349	0.0618555
$267$	−1836.66	−0.420981
$268$	−1642.10	−0.374280
$269$	−1301.03	−0.294890	−0.147445	−	0.989070i	$-0.547105\pi$
$269$	−1301.03	−0.294890	−0.147445	+	0.989070i	$0.547105\pi$
$270$	−1685.79	−0.379978
$271$	3079.64	0.690314	0.345157	−	0.938545i	$-0.387826\pi$
$271$	3079.64	0.690314	0.345157	+	0.938545i	$0.387826\pi$
$272$	−1982.78	−0.441999
$273$	0	0
$274$	−4603.89	−1.01508
$275$	5949.55	1.30462
$276$	−1368.70	−0.298500
$277$	−6287.90	−1.36391	−0.681955	−	0.731394i	$-0.738870\pi$
$277$	−6287.90	−1.36391	−0.681955	+	0.731394i	$0.738870\pi$
$278$	5618.14	1.21206
$279$	1897.25	0.407117
$280$	468.016	0.0998903
$281$	3226.00	0.684864	0.342432	−	0.939543i	$-0.388749\pi$
$281$	3226.00	0.684864	0.342432	+	0.939543i	$0.388749\pi$
$282$	5327.53	1.12500
$283$	−386.584	−0.0812015	−0.0406007	−	0.999175i	$-0.512927\pi$
$283$	−386.584	−0.0812015	−0.0406007	+	0.999175i	$0.512927\pi$
$284$	1697.77	0.354734
$285$	−303.811	−0.0631446
$286$	0	0
$287$	398.178	0.0818944
$288$	−1651.08	−0.337816
$289$	10444.1	2.12580
$290$	−1448.66	−0.293339
$291$	−6573.60	−1.32423
$292$	1687.93	0.338283
$293$	−2192.82	−0.437222	−0.218611	−	0.975812i	$-0.570153\pi$
$293$	−2192.82	−0.437222	−0.218611	+	0.975812i	$0.570153\pi$
$294$	−2020.36	−0.400782
$295$	−951.128	−0.187718
$296$	−2572.31	−0.505109
$297$	11787.8	2.30303
$298$	5066.34	0.984849
$299$	0	0
$300$	3902.87	0.751107
$301$	2083.07	0.398890
$302$	−4782.20	−0.911208
$303$	3664.14	0.694718
$304$	−141.847	−0.0267615
$305$	−352.937	−0.0662594
$306$	12788.0	2.38903
$307$	8083.96	1.50285	0.751427	−	0.659817i	$-0.229366\pi$
$307$	8083.96	1.50285	0.751427	+	0.659817i	$0.229366\pi$
$308$	−3272.59	−0.605431
$309$	33.3973	0.00614856
$310$	284.274	0.0520828
$311$	244.409	0.0445632	0.0222816	−	0.999752i	$-0.492907\pi$
$311$	244.409	0.0445632	0.0222816	+	0.999752i	$0.492907\pi$
$312$	0	0
$313$	4444.13	0.802546	0.401273	−	0.915958i	$-0.368568\pi$
$313$	4444.13	0.802546	0.401273	+	0.915958i	$0.368568\pi$
$314$	−3854.98	−0.692831
$315$	−3018.49	−0.539913
$316$	2934.17	0.522341
$317$	−930.597	−0.164882	−0.0824409	−	0.996596i	$-0.526272\pi$
$317$	−930.597	−0.164882	−0.0824409	+	0.996596i	$0.526272\pi$
$318$	−4612.42	−0.813371
$319$	10129.7	1.77792
$320$	−247.389	−0.0432172
$321$	−1009.11	−0.175462
$322$	−1168.28	−0.202191
$323$	1098.64	0.189257
$324$	2160.34	0.370428
$325$	0	0
$326$	−4437.05	−0.753820
$327$	17160.3	2.90203
$328$	−210.474	−0.0354313
$329$	4547.41	0.762027
$330$	3705.05	0.618050
$331$	−4827.35	−0.801618	−0.400809	−	0.916162i	$-0.631271\pi$
$331$	−4827.35	−0.801618	−0.400809	+	0.916162i	$0.631271\pi$
$332$	2467.37	0.407876
$333$	16590.2	2.73014
$334$	−3016.54	−0.494185
$335$	1586.86	0.258805
$336$	−2146.79	−0.348563
$337$	10709.7	1.73115	0.865573	−	0.500782i	$-0.166954\pi$
$337$	10709.7	1.73115	0.865573	+	0.500782i	$0.166954\pi$
$338$	0	0
$339$	−16833.6	−2.69698
$340$	1916.09	0.305631
$341$	−1987.78	−0.315672
$342$	914.851	0.144648
$343$	−6915.66	−1.08866
$344$	−1101.09	−0.172578
$345$	1322.66	0.206405
$346$	3412.94	0.530292
$347$	6401.09	0.990284	0.495142	−	0.868812i	$-0.335116\pi$
$347$	6401.09	0.990284	0.495142	+	0.868812i	$0.335116\pi$
$348$	6645.04	1.02360
$349$	2430.02	0.372711	0.186355	−	0.982482i	$-0.440332\pi$
$349$	2430.02	0.372711	0.186355	+	0.982482i	$0.440332\pi$
$350$	3331.36	0.508768
$351$	0	0
$352$	1729.86	0.261938
$353$	−8080.03	−1.21829	−0.609145	−	0.793059i	$-0.708487\pi$
$353$	−8080.03	−1.21829	−0.609145	+	0.793059i	$0.708487\pi$
$354$	4362.84	0.655035
$355$	−1640.67	−0.245289
$356$	828.683	0.123371
$357$	16627.4	2.46503
$358$	−1190.45	−0.175746
$359$	−8715.23	−1.28126	−0.640630	−	0.767850i	$-0.721327\pi$
$359$	−8715.23	−1.28126	−0.640630	+	0.767850i	$0.721327\pi$
$360$	1595.55	0.233591
$361$	−6780.40	−0.988541
$362$	−806.013	−0.117025
$363$	−14107.5	−2.03982
$364$	0	0
$365$	−1631.15	−0.233914
$366$	1618.93	0.231209
$367$	2820.88	0.401222	0.200611	−	0.979671i	$-0.435707\pi$
$367$	2820.88	0.401222	0.200611	+	0.979671i	$0.435707\pi$
$368$	617.542	0.0874772
$369$	1357.46	0.191508
$370$	2485.79	0.349270
$371$	−3937.02	−0.550943
$372$	−1303.97	−0.181741
$373$	−12929.7	−1.79484	−0.897418	−	0.441181i	$-0.854560\pi$
$373$	−12929.7	−1.79484	−0.897418	+	0.441181i	$0.854560\pi$
$374$	−13398.2	−1.85242
$375$	−8055.23	−1.10925
$376$	−2403.73	−0.329688
$377$	0	0
$378$	6600.42	0.898119
$379$	2288.05	0.310103	0.155052	−	0.987906i	$-0.450446\pi$
$379$	2288.05	0.310103	0.155052	+	0.987906i	$0.450446\pi$
$380$	137.076	0.0185049
$381$	−2168.53	−0.291593
$382$	1227.39	0.164395
$383$	−12156.2	−1.62181	−0.810904	−	0.585179i	$-0.801024\pi$
$383$	−12156.2	−1.62181	−0.810904	+	0.585179i	$0.801024\pi$
$384$	1134.78	0.150805
$385$	3162.51	0.418640
$386$	4010.66	0.528853
$387$	7101.55	0.932795
$388$	2965.94	0.388074
$389$	−1479.71	−0.192864	−0.0964322	−	0.995340i	$-0.530743\pi$
$389$	−1479.71	−0.192864	−0.0964322	+	0.995340i	$0.530743\pi$
$390$	0	0
$391$	−4783.01	−0.618637
$392$	911.566	0.117452
$393$	−14323.2	−1.83845
$394$	−1115.00	−0.142571
$395$	−2835.48	−0.361186
$396$	−11156.8	−1.41579
$397$	−15271.3	−1.93058	−0.965292	−	0.261172i	$-0.915891\pi$
$397$	−15271.3	−1.93058	−0.965292	+	0.261172i	$0.915891\pi$
$398$	−11067.9	−1.39393
$399$	1189.52	0.149249
$400$	−1760.93	−0.220116
$401$	−2564.54	−0.319369	−0.159685	−	0.987168i	$-0.551048\pi$
$401$	−2564.54	−0.319369	−0.159685	+	0.987168i	$0.551048\pi$
$402$	−7278.97	−0.903089
$403$	0	0
$404$	−1653.22	−0.203591
$405$	−2087.67	−0.256142
$406$	5671.99	0.693340
$407$	−17381.8	−2.11691
$408$	−8789.12	−1.06649
$409$	−1052.47	−0.127240	−0.0636201	−	0.997974i	$-0.520265\pi$
$409$	−1052.47	−0.127240	−0.0636201	+	0.997974i	$0.520265\pi$
$410$	203.394	0.0244998
$411$	−20407.8	−2.44925
$412$	−15.0685	−0.00180187
$413$	3723.98	0.443692
$414$	−3982.87	−0.472819
$415$	−2384.38	−0.282036
$416$	0	0
$417$	24903.7	2.92455
$418$	−958.502	−0.112158
$419$	8711.62	1.01573	0.507864	−	0.861437i	$-0.330435\pi$
$419$	8711.62	1.01573	0.507864	+	0.861437i	$0.330435\pi$
$420$	2074.59	0.241023
$421$	−213.335	−0.0246967	−0.0123484	−	0.999924i	$-0.503931\pi$
$421$	−213.335	−0.0246967	−0.0123484	+	0.999924i	$0.503931\pi$
$422$	9367.46	1.08057
$423$	15502.9	1.78198
$424$	2081.08	0.238363
$425$	13638.8	1.55666
$426$	7525.78	0.855928
$427$	1381.86	0.156611
$428$	455.301	0.0514201
$429$	0	0
$430$	1064.06	0.119333
$431$	−8953.45	−1.00063	−0.500316	−	0.865843i	$-0.666783\pi$
$431$	−8953.45	−1.00063	−0.500316	+	0.865843i	$0.666783\pi$
$432$	−3488.93	−0.388568
$433$	5861.44	0.650538	0.325269	−	0.945622i	$-0.394545\pi$
$433$	5861.44	0.650538	0.325269	+	0.945622i	$0.394545\pi$
$434$	−1113.03	−0.123104
$435$	−6421.53	−0.707790
$436$	−7742.52	−0.850458
$437$	−342.175	−0.0374564
$438$	7482.13	0.816233
$439$	−10443.5	−1.13540	−0.567702	−	0.823234i	$-0.692167\pi$
$439$	−10443.5	−1.13540	−0.567702	+	0.823234i	$0.692167\pi$
$440$	−1671.68	−0.181123
$441$	−5879.19	−0.634833
$442$	0	0
$443$	8789.73	0.942692	0.471346	−	0.881948i	$-0.343768\pi$
$443$	8789.73	0.942692	0.471346	+	0.881948i	$0.343768\pi$
$444$	−11402.3	−1.21876
$445$	−800.810	−0.0853080
$446$	−4767.72	−0.506184
$447$	22457.7	2.37632
$448$	968.611	0.102149
$449$	−8636.11	−0.907714	−0.453857	−	0.891075i	$-0.649952\pi$
$449$	−8636.11	−0.907714	−0.453857	+	0.891075i	$0.649952\pi$
$450$	11357.2	1.18974
$451$	−1422.23	−0.148493
$452$	7595.13	0.790365
$453$	−21198.2	−2.19863
$454$	−3108.15	−0.321305
$455$	0	0
$456$	−628.771	−0.0645722
$457$	−4876.96	−0.499200	−0.249600	−	0.968349i	$-0.580299\pi$
$457$	−4876.96	−0.499200	−0.249600	+	0.968349i	$0.580299\pi$
$458$	−5831.20	−0.594921
$459$	27022.6	2.74794
$460$	−596.771	−0.0604882
$461$	11803.9	1.19254	0.596272	−	0.802782i	$-0.296648\pi$
$461$	11803.9	1.19254	0.596272	+	0.802782i	$0.296648\pi$
$462$	−14506.5	−1.46083
$463$	4797.53	0.481555	0.240777	−	0.970580i	$-0.422598\pi$
$463$	4797.53	0.481555	0.240777	+	0.970580i	$0.422598\pi$
$464$	−2998.17	−0.299971
$465$	1260.11	0.125669
$466$	8467.86	0.841772
$467$	−13188.7	−1.30686	−0.653428	−	0.756989i	$-0.726670\pi$
$467$	−13188.7	−1.30686	−0.653428	+	0.756989i	$0.726670\pi$
$468$	0	0
$469$	−6213.09	−0.611714
$470$	2322.88	0.227971
$471$	−17088.1	−1.67171
$472$	−1968.47	−0.191962
$473$	−7440.38	−0.723275
$474$	13006.4	1.26034
$475$	975.717	0.0942504
$476$	−7502.11	−0.722392
$477$	−13422.0	−1.28837
$478$	4745.11	0.454051
$479$	−18048.8	−1.72165	−0.860827	−	0.508898i	$-0.830053\pi$
$479$	−18048.8	−1.72165	−0.860827	+	0.508898i	$0.830053\pi$
$480$	−1096.61	−0.104277
$481$	0	0
$482$	651.526	0.0615689
$483$	−5178.66	−0.487861
$484$	6365.16	0.597780
$485$	−2866.18	−0.268343
$486$	−2198.96	−0.205240
$487$	−11501.2	−1.07017	−0.535084	−	0.844799i	$-0.679720\pi$
$487$	−11501.2	−1.07017	−0.535084	+	0.844799i	$0.679720\pi$
$488$	−730.442	−0.0677573
$489$	−19668.2	−1.81887
$490$	−880.905	−0.0812148
$491$	−4000.65	−0.367713	−0.183856	−	0.982953i	$-0.558858\pi$
$491$	−4000.65	−0.367713	−0.183856	+	0.982953i	$0.558858\pi$
$492$	−932.973	−0.0854912
$493$	23221.5	2.12139
$494$	0	0
$495$	10781.6	0.978981
$496$	588.337	0.0532603
$497$	6423.76	0.579769
$498$	10937.2	0.984152
$499$	8322.72	0.746645	0.373323	−	0.927702i	$-0.378218\pi$
$499$	8322.72	0.746645	0.373323	+	0.927702i	$0.378218\pi$
$500$	3634.43	0.325074
$501$	−13371.5	−1.19241
$502$	5203.54	0.462640
$503$	−7127.48	−0.631806	−0.315903	−	0.948791i	$-0.602307\pi$
$503$	−7127.48	−0.631806	−0.315903	+	0.948791i	$0.602307\pi$
$504$	−6247.10	−0.552119
$505$	1597.62	0.140778
$506$	4172.90	0.366617
$507$	0	0
$508$	978.417	0.0854532
$509$	14348.9	1.24952	0.624759	−	0.780817i	$-0.285197\pi$
$509$	14348.9	1.24952	0.624759	+	0.780817i	$0.285197\pi$
$510$	8493.50	0.737448
$511$	6386.50	0.552881
$512$	−512.000	−0.0441942
$513$	1933.19	0.166379
$514$	−8434.19	−0.723767
$515$	14.5617	0.00124595
$516$	−4880.84	−0.416409
$517$	−16242.6	−1.38172
$518$	−9732.66	−0.825538
$519$	15128.7	1.27953
$520$	0	0
$521$	3535.86	0.297329	0.148665	−	0.988888i	$-0.452502\pi$
$521$	3535.86	0.297329	0.148665	+	0.988888i	$0.452502\pi$
$522$	19336.8	1.62136
$523$	13964.9	1.16757	0.583787	−	0.811907i	$-0.301570\pi$
$523$	13964.9	1.16757	0.583787	+	0.811907i	$0.301570\pi$
$524$	6462.49	0.538769
$525$	14767.0	1.22759
$526$	−12948.6	−1.07336
$527$	−4556.80	−0.376655
$528$	7668.02	0.632022
$529$	−10677.3	−0.877564
$530$	−2011.08	−0.164822
$531$	12695.7	1.03757
$532$	−536.699	−0.0437384
$533$	0	0
$534$	3673.33	0.297679
$535$	−439.987	−0.0355557
$536$	3284.19	0.264656
$537$	−5276.92	−0.424052
$538$	2602.07	0.208519
$539$	6159.70	0.492240
$540$	3371.58	0.268685
$541$	10661.6	0.847277	0.423638	−	0.905831i	$-0.360753\pi$
$541$	10661.6	0.847277	0.423638	+	0.905831i	$0.360753\pi$
$542$	−6159.29	−0.488126
$543$	−3572.84	−0.282367
$544$	3965.56	0.312540
$545$	7482.10	0.588070
$546$	0	0
$547$	−3393.59	−0.265264	−0.132632	−	0.991165i	$-0.542343\pi$
$547$	−3393.59	−0.265264	−0.132632	+	0.991165i	$0.542343\pi$
$548$	9207.79	0.717769
$549$	4711.02	0.366232
$550$	−11899.1	−0.922508
$551$	1661.26	0.128443
$552$	2737.40	0.211071
$553$	11101.8	0.853703
$554$	12575.8	0.964430
$555$	11018.8	0.842744
$556$	−11236.3	−0.857058
$557$	8249.03	0.627509	0.313754	−	0.949504i	$-0.398413\pi$
$557$	8249.03	0.627509	0.313754	+	0.949504i	$0.398413\pi$
$558$	−3794.50	−0.287875
$559$	0	0
$560$	−936.031	−0.0706331
$561$	−59390.5	−4.46964
$562$	−6452.00	−0.484272
$563$	8868.79	0.663898	0.331949	−	0.943297i	$-0.392294\pi$
$563$	8868.79	0.663898	0.331949	+	0.943297i	$0.392294\pi$
$564$	−10655.1	−0.795495
$565$	−7339.67	−0.546517
$566$	773.167	0.0574181
$567$	8173.93	0.605419
$568$	−3395.55	−0.250835
$569$	−6408.49	−0.472158	−0.236079	−	0.971734i	$-0.575862\pi$
$569$	−6408.49	−0.472158	−0.236079	+	0.971734i	$0.575862\pi$
$570$	607.622	0.0446500
$571$	10045.2	0.736212	0.368106	−	0.929784i	$-0.380006\pi$
$571$	10045.2	0.736212	0.368106	+	0.929784i	$0.380006\pi$
$572$	0	0
$573$	5440.70	0.396664
$574$	−796.355	−0.0579081
$575$	−4247.85	−0.308083
$576$	3302.17	0.238872
$577$	24528.9	1.76976	0.884880	−	0.465818i	$-0.154240\pi$
$577$	24528.9	1.76976	0.884880	+	0.465818i	$0.154240\pi$
$578$	−20888.2	−1.50317
$579$	17778.2	1.27605
$580$	2897.32	0.207422
$581$	9335.64	0.666623
$582$	13147.2	0.936372
$583$	14062.4	0.998980
$584$	−3375.86	−0.239202
$585$	0	0
$586$	4385.64	0.309163
$587$	15903.8	1.11826	0.559130	−	0.829080i	$-0.311135\pi$
$587$	15903.8	1.11826	0.559130	+	0.829080i	$0.311135\pi$
$588$	4040.72	0.283396
$589$	−325.992	−0.0228052
$590$	1902.26	0.132737
$591$	−4942.49	−0.344005
$592$	5144.61	0.357166
$593$	5436.51	0.376477	0.188238	−	0.982123i	$-0.439722\pi$
$593$	5436.51	0.376477	0.188238	+	0.982123i	$0.439722\pi$
$594$	−23575.7	−1.62849
$595$	7249.78	0.499516
$596$	−10132.7	−0.696393
$597$	−49061.2	−3.36339
$598$	0	0
$599$	6872.46	0.468783	0.234392	−	0.972142i	$-0.424690\pi$
$599$	6872.46	0.468783	0.234392	+	0.972142i	$0.424690\pi$
$600$	−7805.73	−0.531113
$601$	−1417.82	−0.0962299	−0.0481149	−	0.998842i	$-0.515321\pi$
$601$	−1417.82	−0.0962299	−0.0481149	+	0.998842i	$0.515321\pi$
$602$	−4166.13	−0.282058
$603$	−21181.6	−1.43048
$604$	9564.40	0.644321
$605$	−6151.07	−0.413350
$606$	−7328.28	−0.491240
$607$	−13593.3	−0.908957	−0.454478	−	0.890758i	$-0.650174\pi$
$607$	−13593.3	−0.908957	−0.454478	+	0.890758i	$0.650174\pi$
$608$	283.695	0.0189233
$609$	25142.4	1.67294
$610$	705.874	0.0468525
$611$	0	0
$612$	−25576.1	−1.68930
$613$	17279.4	1.13851	0.569257	−	0.822160i	$-0.307231\pi$
$613$	17279.4	1.13851	0.569257	+	0.822160i	$0.307231\pi$
$614$	−16167.9	−1.06268
$615$	901.592	0.0591150
$616$	6545.17	0.428105
$617$	2177.07	0.142051	0.0710256	−	0.997474i	$-0.477373\pi$
$617$	2177.07	0.142051	0.0710256	+	0.997474i	$0.477373\pi$
$618$	−66.7945	−0.00434769
$619$	−17067.5	−1.10824	−0.554121	−	0.832436i	$-0.686945\pi$
$619$	−17067.5	−1.10824	−0.554121	+	0.832436i	$0.686945\pi$
$620$	−568.548	−0.0368281
$621$	−8416.26	−0.543853
$622$	−488.818	−0.0315110
$623$	3135.43	0.201635
$624$	0	0
$625$	10245.1	0.655686
$626$	−8888.25	−0.567486
$627$	−4248.78	−0.270622
$628$	7709.95	0.489905
$629$	−39846.2	−2.52587
$630$	6036.98	0.381776
$631$	7929.96	0.500296	0.250148	−	0.968208i	$-0.419521\pi$
$631$	7929.96	0.500296	0.250148	+	0.968208i	$0.419521\pi$
$632$	−5868.34	−0.369351
$633$	41523.4	2.60728
$634$	1861.19	0.116589
$635$	−945.508	−0.0590887
$636$	9224.85	0.575140
$637$	0	0
$638$	−20259.5	−1.25718
$639$	21899.8	1.35578
$640$	494.779	0.0305591
$641$	−5839.06	−0.359796	−0.179898	−	0.983685i	$-0.557577\pi$
$641$	−5839.06	−0.359796	−0.179898	+	0.983685i	$0.557577\pi$
$642$	2018.23	0.124070
$643$	17022.8	1.04404	0.522018	−	0.852935i	$-0.325179\pi$
$643$	17022.8	1.04404	0.522018	+	0.852935i	$0.325179\pi$
$644$	2336.55	0.142971
$645$	4716.68	0.287936
$646$	−2197.28	−0.133825
$647$	21779.3	1.32339	0.661694	−	0.749774i	$-0.269838\pi$
$647$	21779.3	1.32339	0.661694	+	0.749774i	$0.269838\pi$
$648$	−4320.67	−0.261932
$649$	−13301.5	−0.804512
$650$	0	0
$651$	−4933.74	−0.297033
$652$	8874.10	0.533031
$653$	17041.6	1.02127	0.510636	−	0.859797i	$-0.329410\pi$
$653$	17041.6	1.02127	0.510636	+	0.859797i	$0.329410\pi$
$654$	−34320.5	−2.05205
$655$	−6245.12	−0.372545
$656$	420.947	0.0250537
$657$	21772.7	1.29290
$658$	−9094.82	−0.538834
$659$	30765.8	1.81861	0.909306	−	0.416127i	$-0.136613\pi$
$659$	30765.8	1.81861	0.909306	+	0.416127i	$0.136613\pi$
$660$	−7410.10	−0.437027
$661$	−7744.70	−0.455724	−0.227862	−	0.973693i	$-0.573174\pi$
$661$	−7744.70	−0.455724	−0.227862	+	0.973693i	$0.573174\pi$
$662$	9654.71	0.566829
$663$	0	0
$664$	−4934.75	−0.288412
$665$	518.647	0.0302440
$666$	−33180.4	−1.93050
$667$	−7232.40	−0.419850
$668$	6033.08	0.349442
$669$	−21134.0	−1.22136
$670$	−3173.73	−0.183003
$671$	−4935.80	−0.283971
$672$	4293.59	0.246471
$673$	−2302.45	−0.131877	−0.0659384	−	0.997824i	$-0.521004\pi$
$673$	−2302.45	−0.131877	−0.0659384	+	0.997824i	$0.521004\pi$
$674$	−21419.5	−1.22411
$675$	23999.1	1.36848
$676$	0	0
$677$	−15932.6	−0.904488	−0.452244	−	0.891894i	$-0.649376\pi$
$677$	−15932.6	−0.904488	−0.452244	+	0.891894i	$0.649376\pi$
$678$	33667.2	1.90705
$679$	11222.0	0.634258
$680$	−3832.18	−0.216114
$681$	−13777.6	−0.775269
$682$	3975.55	0.223214
$683$	−18458.6	−1.03411	−0.517055	−	0.855952i	$-0.672972\pi$
$683$	−18458.6	−1.03411	−0.517055	+	0.855952i	$0.672972\pi$
$684$	−1829.70	−0.102281
$685$	−8898.08	−0.496319
$686$	13831.3	0.769800
$687$	−25848.1	−1.43547
$688$	2202.18	0.122031
$689$	0	0
$690$	−2645.32	−0.145950
$691$	−22979.4	−1.26509	−0.632544	−	0.774524i	$-0.717989\pi$
$691$	−22979.4	−1.26509	−0.632544	+	0.774524i	$0.717989\pi$
$692$	−6825.89	−0.374973
$693$	−42213.4	−2.31393
$694$	−12802.2	−0.700236
$695$	10858.3	0.592634
$696$	−13290.1	−0.723791
$697$	−3260.33	−0.177179
$698$	−4860.04	−0.263546
$699$	37535.7	2.03109
$700$	−6662.72	−0.359753
$701$	8633.81	0.465185	0.232592	−	0.972574i	$-0.425279\pi$
$701$	8633.81	0.465185	0.232592	+	0.972574i	$0.425279\pi$
$702$	0	0
$703$	−2850.58	−0.152933
$704$	−3459.73	−0.185218
$705$	10296.7	0.550065
$706$	16160.1	0.861461
$707$	−6255.19	−0.332745
$708$	−8725.68	−0.463179
$709$	25622.5	1.35722	0.678612	−	0.734497i	$-0.262582\pi$
$709$	25622.5	1.35722	0.678612	+	0.734497i	$0.262582\pi$
$710$	3281.34	0.173446
$711$	37848.1	1.99636
$712$	−1657.37	−0.0872365
$713$	1419.23	0.0745449
$714$	−33254.8	−1.74304
$715$	0	0
$716$	2380.89	0.124271
$717$	21033.8	1.09557
$718$	17430.5	0.905987
$719$	34954.8	1.81306	0.906532	−	0.422137i	$-0.138720\pi$
$719$	34954.8	1.81306	0.906532	+	0.422137i	$0.138720\pi$
$720$	−3191.10	−0.165174
$721$	−57.0137	−0.00294494
$722$	13560.8	0.699004
$723$	2888.04	0.148558
$724$	1612.03	0.0827492
$725$	20623.3	1.05646
$726$	28215.1	1.44237
$727$	−23397.0	−1.19360	−0.596800	−	0.802390i	$-0.703562\pi$
$727$	−23397.0	−1.19360	−0.596800	+	0.802390i	$0.703562\pi$
$728$	0	0
$729$	−24329.6	−1.23607
$730$	3262.31	0.165402
$731$	−17056.4	−0.863002
$732$	−3237.85	−0.163490
$733$	3541.17	0.178440	0.0892198	−	0.996012i	$-0.471563\pi$
$733$	3541.17	0.178440	0.0892198	+	0.996012i	$0.471563\pi$
$734$	−5641.76	−0.283707
$735$	−3904.81	−0.195961
$736$	−1235.08	−0.0618557
$737$	22192.2	1.10917
$738$	−2714.92	−0.135417
$739$	39233.0	1.95292	0.976460	−	0.215697i	$-0.0692024\pi$
$739$	39233.0	1.95292	0.976460	+	0.215697i	$0.0692024\pi$
$740$	−4971.57	−0.246971
$741$	0	0
$742$	7874.03	0.389575
$743$	−38198.7	−1.88610	−0.943052	−	0.332646i	$-0.892058\pi$
$743$	−38198.7	−1.88610	−0.943052	+	0.332646i	$0.892058\pi$
$744$	2607.94	0.128510
$745$	9791.86	0.481538
$746$	25859.4	1.26914
$747$	31826.9	1.55888
$748$	26796.4	1.30986
$749$	1722.69	0.0840399
$750$	16110.5	0.784361
$751$	−1670.75	−0.0811807	−0.0405903	−	0.999176i	$-0.512924\pi$
$751$	−1670.75	−0.0811807	−0.0405903	+	0.999176i	$0.512924\pi$
$752$	4807.45	0.233125
$753$	23065.9	1.11629
$754$	0	0
$755$	−9242.70	−0.445532
$756$	−13200.8	−0.635066
$757$	−37948.6	−1.82201	−0.911007	−	0.412392i	$-0.864694\pi$
$757$	−37948.6	−1.82201	−0.911007	+	0.412392i	$0.864694\pi$
$758$	−4576.09	−0.219276
$759$	18497.4	0.884600
$760$	−274.153	−0.0130849
$761$	37772.7	1.79929	0.899645	−	0.436622i	$-0.143825\pi$
$761$	37772.7	1.79929	0.899645	+	0.436622i	$0.143825\pi$
$762$	4337.06	0.206188
$763$	−29294.9	−1.38997
$764$	−2454.78	−0.116245
$765$	24715.8	1.16811
$766$	24312.4	1.14679
$767$	0	0
$768$	−2269.56	−0.106635
$769$	13073.3	0.613049	0.306524	−	0.951863i	$-0.400834\pi$
$769$	13073.3	0.613049	0.306524	+	0.951863i	$0.400834\pi$
$770$	−6325.02	−0.296023
$771$	−37386.5	−1.74636
$772$	−8021.32	−0.373955
$773$	25143.1	1.16990	0.584951	−	0.811069i	$-0.301114\pi$
$773$	25143.1	1.16990	0.584951	+	0.811069i	$0.301114\pi$
$774$	−14203.1	−0.659586
$775$	−4046.96	−0.187575
$776$	−5931.87	−0.274409
$777$	−43142.3	−1.99192
$778$	2959.42	0.136376
$779$	−233.243	−0.0107276
$780$	0	0
$781$	−22944.7	−1.05125
$782$	9566.01	0.437442
$783$	40861.0	1.86494
$784$	−1823.13	−0.0830508
$785$	−7450.63	−0.338757
$786$	28646.5	1.29998
$787$	−19164.6	−0.868035	−0.434017	−	0.900904i	$-0.642904\pi$
$787$	−19164.6	−0.868035	−0.434017	+	0.900904i	$0.642904\pi$
$788$	2230.00	0.100813
$789$	−57397.8	−2.58988
$790$	5670.96	0.255397
$791$	28737.2	1.29175
$792$	22313.7	1.00111
$793$	0	0
$794$	30542.5	1.36513
$795$	−8914.57	−0.397695
$796$	22135.9	0.985661
$797$	28991.4	1.28849	0.644246	−	0.764818i	$-0.277171\pi$
$797$	28991.4	1.28849	0.644246	+	0.764818i	$0.277171\pi$
$798$	−2379.04	−0.105535
$799$	−37234.8	−1.64865
$800$	3521.86	0.155646
$801$	10689.3	0.471519
$802$	5129.08	0.225828
$803$	−22811.6	−1.00250
$804$	14557.9	0.638581
$805$	−2257.96	−0.0988606
$806$	0	0
$807$	11534.3	0.503129
$808$	3306.44	0.143961
$809$	−16892.8	−0.734141	−0.367070	−	0.930193i	$-0.619639\pi$
$809$	−16892.8	−0.734141	−0.367070	+	0.930193i	$0.619639\pi$
$810$	4175.35	0.181119
$811$	42182.9	1.82644	0.913220	−	0.407466i	$-0.133587\pi$
$811$	42182.9	1.82644	0.913220	+	0.407466i	$0.133587\pi$
$812$	−11344.0	−0.490265
$813$	−27302.5	−1.17778
$814$	34763.6	1.49688
$815$	−8575.62	−0.368578
$816$	17578.2	0.754120
$817$	−1220.21	−0.0522519
$818$	2104.94	0.0899725
$819$	0	0
$820$	−406.789	−0.0173240
$821$	18500.8	0.786457	0.393229	−	0.919441i	$-0.371358\pi$
$821$	18500.8	0.786457	0.393229	+	0.919441i	$0.371358\pi$
$822$	40815.6	1.73188
$823$	−27868.7	−1.18037	−0.590184	−	0.807269i	$-0.700945\pi$
$823$	−27868.7	−1.18037	−0.590184	+	0.807269i	$0.700945\pi$
$824$	30.1370	0.00127412
$825$	−52745.5	−2.22589
$826$	−7447.96	−0.313738
$827$	−2541.96	−0.106884	−0.0534418	−	0.998571i	$-0.517019\pi$
$827$	−2541.96	−0.106884	−0.0534418	+	0.998571i	$0.517019\pi$
$828$	7965.73	0.334334
$829$	29570.0	1.23885	0.619427	−	0.785054i	$-0.287365\pi$
$829$	29570.0	1.23885	0.619427	+	0.785054i	$0.287365\pi$
$830$	4768.77	0.199429
$831$	55745.1	2.32705
$832$	0	0
$833$	14120.6	0.587333
$834$	−49807.4	−2.06797
$835$	−5830.16	−0.241630
$836$	1917.00	0.0793074
$837$	−8018.23	−0.331124
$838$	−17423.2	−0.718229
$839$	−21607.5	−0.889122	−0.444561	−	0.895749i	$-0.646640\pi$
$839$	−21607.5	−0.889122	−0.444561	+	0.895749i	$0.646640\pi$
$840$	−4149.17	−0.170429
$841$	10724.3	0.439720
$842$	426.670	0.0174632
$843$	−28600.0	−1.16849
$844$	−18734.9	−0.764079
$845$	0	0
$846$	−31005.9	−1.26005
$847$	24083.5	0.976998
$848$	−4162.15	−0.168548
$849$	3427.24	0.138543
$850$	−27277.6	−1.10072
$851$	12410.2	0.499902
$852$	−15051.6	−0.605232
$853$	28695.1	1.15182	0.575910	−	0.817513i	$-0.304648\pi$
$853$	28695.1	1.15182	0.575910	+	0.817513i	$0.304648\pi$
$854$	−2763.73	−0.110741
$855$	1768.16	0.0707249
$856$	−910.603	−0.0363595
$857$	8720.01	0.347573	0.173786	−	0.984783i	$-0.444400\pi$
$857$	8720.01	0.347573	0.173786	+	0.984783i	$0.444400\pi$
$858$	0	0
$859$	−1750.23	−0.0695191	−0.0347596	−	0.999396i	$-0.511067\pi$
$859$	−1750.23	−0.0695191	−0.0347596	+	0.999396i	$0.511067\pi$
$860$	−2128.11	−0.0843815
$861$	−3530.03	−0.139725
$862$	17906.9	0.707554
$863$	35493.3	1.40001	0.700004	−	0.714139i	$-0.253181\pi$
$863$	35493.3	1.40001	0.700004	+	0.714139i	$0.253181\pi$
$864$	6977.86	0.274759
$865$	6596.30	0.259284
$866$	−11722.9	−0.460000
$867$	−92591.6	−3.62696
$868$	2226.05	0.0870473
$869$	−39654.0	−1.54795
$870$	12843.1	0.500483
$871$	0	0
$872$	15485.0	0.601364
$873$	38257.9	1.48320
$874$	684.349	0.0264857
$875$	13751.4	0.531293
$876$	−14964.3	−0.577164
$877$	26427.8	1.01756	0.508782	−	0.860895i	$-0.330096\pi$
$877$	26427.8	1.01756	0.508782	+	0.860895i	$0.330096\pi$
$878$	20887.1	0.802852
$879$	19440.4	0.745970
$880$	3343.36	0.128073
$881$	766.125	0.0292979	0.0146489	−	0.999893i	$-0.495337\pi$
$881$	766.125	0.0292979	0.0146489	+	0.999893i	$0.495337\pi$
$882$	11758.4	0.448894
$883$	6521.89	0.248561	0.124280	−	0.992247i	$-0.460338\pi$
$883$	6521.89	0.248561	0.124280	+	0.992247i	$0.460338\pi$
$884$	0	0
$885$	8432.19	0.320277
$886$	−17579.5	−0.666584
$887$	25351.7	0.959668	0.479834	−	0.877359i	$-0.340697\pi$
$887$	25351.7	0.959668	0.479834	+	0.877359i	$0.340697\pi$
$888$	22804.7	0.861796
$889$	3701.97	0.139663
$890$	1601.62	0.0603218
$891$	−29196.0	−1.09776
$892$	9535.45	0.357926
$893$	−2663.77	−0.0998203
$894$	−44915.4	−1.68031
$895$	−2300.81	−0.0859302
$896$	−1937.22	−0.0722299
$897$	0	0
$898$	17272.2	0.641851
$899$	−6890.36	−0.255624
$900$	−22714.4	−0.841275
$901$	32236.8	1.19197
$902$	2844.46	0.105000
$903$	−18467.3	−0.680570
$904$	−15190.3	−0.558872
$905$	−1557.80	−0.0572190
$906$	42396.4	1.55466
$907$	−9504.57	−0.347954	−0.173977	−	0.984750i	$-0.555662\pi$
$907$	−9504.57	−0.347954	−0.173977	+	0.984750i	$0.555662\pi$
$908$	6216.30	0.227197
$909$	−21325.1	−0.778116
$910$	0	0
$911$	6435.82	0.234059	0.117030	−	0.993128i	$-0.462663\pi$
$911$	6435.82	0.234059	0.117030	+	0.993128i	$0.462663\pi$
$912$	1257.54	0.0456594
$913$	−33345.5	−1.20873
$914$	9753.92	0.352988
$915$	3128.95	0.113049
$916$	11662.4	0.420673
$917$	24451.7	0.880552
$918$	−54045.2	−1.94309
$919$	14953.9	0.536762	0.268381	−	0.963313i	$-0.413511\pi$
$919$	14953.9	0.536762	0.268381	+	0.963313i	$0.413511\pi$
$920$	1193.54	0.0427716
$921$	−71668.0	−2.56411
$922$	−23607.8	−0.843256
$923$	0	0
$924$	29013.0	1.03296
$925$	−35387.9	−1.25789
$926$	−9595.05	−0.340511
$927$	−194.370	−0.00688667
$928$	5996.34	0.212111
$929$	38061.8	1.34421	0.672103	−	0.740458i	$-0.265391\pi$
$929$	38061.8	1.34421	0.672103	+	0.740458i	$0.265391\pi$
$930$	−2520.22	−0.0888616
$931$	1010.18	0.0355611
$932$	−16935.7	−0.595223
$933$	−2166.80	−0.0760319
$934$	26377.5	0.924087
$935$	−25895.1	−0.905732
$936$	0	0
$937$	14572.5	0.508072	0.254036	−	0.967195i	$-0.418242\pi$
$937$	14572.5	0.508072	0.254036	+	0.967195i	$0.418242\pi$
$938$	12426.2	0.432547
$939$	−39399.2	−1.36927
$940$	−4645.75	−0.161200
$941$	−32190.2	−1.11516	−0.557582	−	0.830122i	$-0.688271\pi$
$941$	−32190.2	−1.11516	−0.557582	+	0.830122i	$0.688271\pi$
$942$	34176.1	1.18208
$943$	1015.44	0.0350660
$944$	3936.93	0.135738
$945$	12756.8	0.439132
$946$	14880.8	0.511433
$947$	42231.0	1.44913	0.724563	−	0.689208i	$-0.242041\pi$
$947$	42231.0	1.44913	0.724563	+	0.689208i	$0.242041\pi$
$948$	−26012.8	−0.891197
$949$	0	0
$950$	−1951.43	−0.0666451
$951$	8250.17	0.281315
$952$	15004.2	0.510808
$953$	−8168.44	−0.277651	−0.138826	−	0.990317i	$-0.544333\pi$
$953$	−8168.44	−0.277651	−0.138826	+	0.990317i	$0.544333\pi$
$954$	26844.0	0.911013
$955$	2372.22	0.0803803
$956$	−9490.22	−0.321062
$957$	−89804.7	−3.03341
$958$	36097.7	1.21739
$959$	34838.9	1.17310
$960$	2193.22	0.0737353
$961$	−28438.9	−0.954613
$962$	0	0
$963$	5872.98	0.196525
$964$	−1303.05	−0.0435358
$965$	7751.52	0.258581
$966$	10357.3	0.344970
$967$	−52022.1	−1.73001	−0.865003	−	0.501766i	$-0.832684\pi$
$967$	−52022.1	−1.73001	−0.865003	+	0.501766i	$0.832684\pi$
$968$	−12730.3	−0.422694
$969$	−9739.95	−0.322902
$970$	5732.35	0.189747
$971$	−11944.9	−0.394780	−0.197390	−	0.980325i	$-0.563247\pi$
$971$	−11944.9	−0.394780	−0.197390	+	0.980325i	$0.563247\pi$
$972$	4397.91	0.145127
$973$	−42514.0	−1.40076
$974$	23002.5	0.756722
$975$	0	0
$976$	1460.88	0.0479117
$977$	−18989.9	−0.621843	−0.310922	−	0.950436i	$-0.600638\pi$
$977$	−18989.9	−0.621843	−0.310922	+	0.950436i	$0.600638\pi$
$978$	39336.5	1.28614
$979$	−11199.3	−0.365608
$980$	1761.81	0.0574275
$981$	−99871.5	−3.25041
$982$	8001.31	0.260012
$983$	47187.4	1.53107	0.765537	−	0.643392i	$-0.222474\pi$
$983$	47187.4	1.53107	0.765537	+	0.643392i	$0.222474\pi$
$984$	1865.95	0.0604514
$985$	−2154.99	−0.0697094
$986$	−46443.0	−1.50005
$987$	−40314.9	−1.30014
$988$	0	0
$989$	5312.27	0.170799
$990$	−21563.1	−0.692244
$991$	−30759.5	−0.985981	−0.492991	−	0.870035i	$-0.664096\pi$
$991$	−30759.5	−0.985981	−0.492991	+	0.870035i	$0.664096\pi$
$992$	−1176.67	−0.0376607
$993$	42796.7	1.36769
$994$	−12847.5	−0.409958
$995$	−21391.4	−0.681559
$996$	−21874.4	−0.695901
$997$	49332.2	1.56707	0.783533	−	0.621350i	$-0.213415\pi$
$997$	49332.2	1.56707	0.783533	+	0.621350i	$0.213415\pi$
$998$	−16645.4	−0.527958
$999$	−70114.0	−2.22053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.a.g.1.1		2
13.2	odd	12		338.4.e.f.147.2		8
13.3	even	3		338.4.c.j.191.2		4
13.4	even	6		26.4.c.b.3.2	✓	4
13.5	odd	4		338.4.b.e.337.3		4
13.6	odd	12		338.4.e.f.23.4		8
13.7	odd	12		338.4.e.f.23.2		8
13.8	odd	4		338.4.b.e.337.1		4
13.9	even	3		338.4.c.j.315.2		4
13.10	even	6		26.4.c.b.9.2	yes	4
13.11	odd	12		338.4.e.f.147.4		8
13.12	even	2		338.4.a.h.1.1		2
39.17	odd	6		234.4.h.h.55.1		4
39.23	odd	6		234.4.h.h.217.1		4
52.23	odd	6		208.4.i.d.113.1		4
52.43	odd	6		208.4.i.d.81.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.4.c.b.3.2	✓	4	13.4	even	6
26.4.c.b.9.2	yes	4	13.10	even	6
208.4.i.d.81.1		4	52.43	odd	6
208.4.i.d.113.1		4	52.23	odd	6
234.4.h.h.55.1		4	39.17	odd	6
234.4.h.h.217.1		4	39.23	odd	6
338.4.a.g.1.1		2	1.1	even	1	trivial
338.4.a.h.1.1		2	13.12	even	2
338.4.b.e.337.1		4	13.8	odd	4
338.4.b.e.337.3		4	13.5	odd	4
338.4.c.j.191.2		4	13.3	even	3
338.4.c.j.315.2		4	13.9	even	3
338.4.e.f.23.2		8	13.7	odd	12
338.4.e.f.23.4		8	13.6	odd	12
338.4.e.f.147.2		8	13.2	odd	12
338.4.e.f.147.4		8	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -8.86546 q^{3} +4.00000 q^{4} -3.86546 q^{5} +17.7309 q^{6} +15.1345 q^{7} -8.00000 q^{8} +51.5964 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -8.86546 q^{3} +4.00000 q^{4} -3.86546 q^{5} +17.7309 q^{6} +15.1345 q^{7} -8.00000 q^{8} +51.5964 q^{9} +7.73092 q^{10} -54.0582 q^{11} -35.4618 q^{12} -30.2691 q^{14} +34.2691 q^{15} +16.0000 q^{16} -123.924 q^{17} -103.193 q^{18} -8.86546 q^{19} -15.4618 q^{20} -134.175 q^{21} +108.116 q^{22} +38.5964 q^{23} +70.9237 q^{24} -110.058 q^{25} -218.058 q^{27} +60.5382 q^{28} -187.386 q^{29} -68.5382 q^{30} +36.7710 q^{31} -32.0000 q^{32} +479.251 q^{33} +247.847 q^{34} -58.5020 q^{35} +206.386 q^{36} +321.538 q^{37} +17.7309 q^{38} +30.9237 q^{40} +26.3092 q^{41} +268.349 q^{42} +137.636 q^{43} -216.233 q^{44} -199.444 q^{45} -77.1928 q^{46} +300.466 q^{47} -141.847 q^{48} -113.946 q^{49} +220.116 q^{50} +1098.64 q^{51} -260.135 q^{53} +436.116 q^{54} +208.960 q^{55} -121.076 q^{56} +78.5964 q^{57} +374.771 q^{58} +246.058 q^{59} +137.076 q^{60} +91.3053 q^{61} -73.5421 q^{62} +780.887 q^{63} +64.0000 q^{64} -958.502 q^{66} -410.524 q^{67} -495.695 q^{68} -342.175 q^{69} +117.004 q^{70} +424.444 q^{71} -412.771 q^{72} +421.982 q^{73} -643.076 q^{74} +975.717 q^{75} -35.4618 q^{76} -818.146 q^{77} +733.542 q^{79} -61.8474 q^{80} +540.084 q^{81} -52.6184 q^{82} +616.843 q^{83} -536.699 q^{84} +479.022 q^{85} -275.273 q^{86} +1661.26 q^{87} +432.466 q^{88} +207.171 q^{89} +398.887 q^{90} +154.386 q^{92} -325.992 q^{93} -600.932 q^{94} +34.2691 q^{95} +283.695 q^{96} +741.484 q^{97} +227.891 q^{98} -2789.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 7 q^{5} + 6 q^{6} + 45 q^{7} - 16 q^{8} + 59 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 - 3 * q^3 + 8 * q^4 + 7 * q^5 + 6 * q^6 + 45 * q^7 - 16 * q^8 + 59 * q^9	\( 2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 7 q^{5} + 6 q^{6} + 45 q^{7} - 16 q^{8} + 59 q^{9} - 14 q^{10} - 5 q^{11} - 12 q^{12} - 90 q^{14} + 98 q^{15} + 32 q^{16} - 130 q^{17} - 118 q^{18} - 3 q^{19} + 28 q^{20} + 41 q^{21} + 10 q^{22} + 33 q^{23} + 24 q^{24} - 117 q^{25} - 333 q^{27} + 180 q^{28} - 198 q^{29} - 196 q^{30} - 280 q^{31} - 64 q^{32} + 767 q^{33} + 260 q^{34} + 266 q^{35} + 236 q^{36} + 702 q^{37} + 6 q^{38} - 56 q^{40} - 242 q^{41} - 82 q^{42} - 93 q^{43} - 20 q^{44} - 119 q^{45} - 66 q^{46} - 224 q^{47} - 48 q^{48} + 435 q^{49} + 234 q^{50} + 1063 q^{51} - 535 q^{53} + 666 q^{54} + 742 q^{55} - 360 q^{56} + 113 q^{57} + 396 q^{58} + 389 q^{59} + 392 q^{60} + 654 q^{61} + 560 q^{62} + 1002 q^{63} + 128 q^{64} - 1534 q^{66} + 107 q^{67} - 520 q^{68} - 375 q^{69} - 532 q^{70} + 569 q^{71} - 472 q^{72} + 623 q^{73} - 1404 q^{74} + 935 q^{75} - 12 q^{76} + 647 q^{77} + 760 q^{79} + 112 q^{80} - 334 q^{81} + 484 q^{82} + 1764 q^{83} + 164 q^{84} + 413 q^{85} + 186 q^{86} + 1599 q^{87} + 40 q^{88} + 871 q^{89} + 238 q^{90} + 132 q^{92} - 2184 q^{93} + 448 q^{94} + 98 q^{95} + 96 q^{96} + 879 q^{97} - 870 q^{98} - 2426 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 3 * q^3 + 8 * q^4 + 7 * q^5 + 6 * q^6 + 45 * q^7 - 16 * q^8 + 59 * q^9 - 14 * q^10 - 5 * q^11 - 12 * q^12 - 90 * q^14 + 98 * q^15 + 32 * q^16 - 130 * q^17 - 118 * q^18 - 3 * q^19 + 28 * q^20 + 41 * q^21 + 10 * q^22 + 33 * q^23 + 24 * q^24 - 117 * q^25 - 333 * q^27 + 180 * q^28 - 198 * q^29 - 196 * q^30 - 280 * q^31 - 64 * q^32 + 767 * q^33 + 260 * q^34 + 266 * q^35 + 236 * q^36 + 702 * q^37 + 6 * q^38 - 56 * q^40 - 242 * q^41 - 82 * q^42 - 93 * q^43 - 20 * q^44 - 119 * q^45 - 66 * q^46 - 224 * q^47 - 48 * q^48 + 435 * q^49 + 234 * q^50 + 1063 * q^51 - 535 * q^53 + 666 * q^54 + 742 * q^55 - 360 * q^56 + 113 * q^57 + 396 * q^58 + 389 * q^59 + 392 * q^60 + 654 * q^61 + 560 * q^62 + 1002 * q^63 + 128 * q^64 - 1534 * q^66 + 107 * q^67 - 520 * q^68 - 375 * q^69 - 532 * q^70 + 569 * q^71 - 472 * q^72 + 623 * q^73 - 1404 * q^74 + 935 * q^75 - 12 * q^76 + 647 * q^77 + 760 * q^79 + 112 * q^80 - 334 * q^81 + 484 * q^82 + 1764 * q^83 + 164 * q^84 + 413 * q^85 + 186 * q^86 + 1599 * q^87 + 40 * q^88 + 871 * q^89 + 238 * q^90 + 132 * q^92 - 2184 * q^93 + 448 * q^94 + 98 * q^95 + 96 * q^96 + 879 * q^97 - 870 * q^98 - 2426 * q^99

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