LMFDB - Embedded newform 338.4.b.e.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.9426455819$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{217})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 109x^{2} + 2916 $ x^4 + 109*x^2 + 2916
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$6.86546i$ of defining polynomial
Character	$\chi$	$=$	338.337
Dual form			338.4.b.e.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -8.86546 q^{3} -4.00000 q^{4} -3.86546i q^{5} +17.7309i q^{6} -15.1345i q^{7} +8.00000i q^{8} +51.5964 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} -8.86546 q^{3} -4.00000 q^{4} -3.86546i q^{5} +17.7309i q^{6} -15.1345i q^{7} +8.00000i q^{8} +51.5964 q^{9} -7.73092 q^{10} +54.0582i q^{11} +35.4618 q^{12} -30.2691 q^{14} +34.2691i q^{15} +16.0000 q^{16} +123.924 q^{17} -103.193i q^{18} -8.86546i q^{19} +15.4618i q^{20} +134.175i q^{21} +108.116 q^{22} -38.5964 q^{23} -70.9237i q^{24} +110.058 q^{25} -218.058 q^{27} +60.5382i q^{28} -187.386 q^{29} +68.5382 q^{30} +36.7710i q^{31} -32.0000i q^{32} -479.251i q^{33} -247.847i q^{34} -58.5020 q^{35} -206.386 q^{36} -321.538i q^{37} -17.7309 q^{38} +30.9237 q^{40} +26.3092i q^{41} +268.349 q^{42} -137.636 q^{43} -216.233i q^{44} -199.444i q^{45} +77.1928i q^{46} -300.466i q^{47} -141.847 q^{48} +113.946 q^{49} -220.116i q^{50} -1098.64 q^{51} -260.135 q^{53} +436.116i q^{54} +208.960 q^{55} +121.076 q^{56} +78.5964i q^{57} +374.771i q^{58} -246.058i q^{59} -137.076i q^{60} +91.3053 q^{61} +73.5421 q^{62} -780.887i q^{63} -64.0000 q^{64} -958.502 q^{66} -410.524i q^{67} -495.695 q^{68} +342.175 q^{69} +117.004i q^{70} +424.444i q^{71} +412.771i q^{72} -421.982i q^{73} -643.076 q^{74} -975.717 q^{75} +35.4618i q^{76} +818.146 q^{77} +733.542 q^{79} -61.8474i q^{80} +540.084 q^{81} +52.6184 q^{82} +616.843i q^{83} -536.699i q^{84} -479.022i q^{85} +275.273i q^{86} +1661.26 q^{87} -432.466 q^{88} -207.171i q^{89} -398.887 q^{90} +154.386 q^{92} -325.992i q^{93} -600.932 q^{94} -34.2691 q^{95} +283.695i q^{96} +741.484i q^{97} -227.891i q^{98} +2789.21i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{3} - 16 q^{4} + 118 q^{9}+O(q^{10}) $ 4 * q - 6 * q^3 - 16 * q^4 + 118 * q^9	$ 4 q - 6 q^{3} - 16 q^{4} + 118 q^{9} + 28 q^{10} + 24 q^{12} - 180 q^{14} + 64 q^{16} + 260 q^{17} + 20 q^{22} - 66 q^{23} + 234 q^{25} - 666 q^{27} - 396 q^{29} + 392 q^{30} + 532 q^{35} - 472 q^{36} - 12 q^{38} - 112 q^{40} - 164 q^{42} + 186 q^{43} - 96 q^{48} - 870 q^{49} - 2126 q^{51} - 1070 q^{53} + 1484 q^{55} + 720 q^{56} + 1308 q^{61} - 1120 q^{62} - 256 q^{64} - 3068 q^{66} - 1040 q^{68} + 750 q^{69} - 2808 q^{74} - 1870 q^{75} - 1294 q^{77} + 1520 q^{79} - 668 q^{81} - 968 q^{82} + 3198 q^{87} - 80 q^{88} - 476 q^{90} + 264 q^{92} + 896 q^{94} - 196 q^{95}+O(q^{100}) $ 4 * q - 6 * q^3 - 16 * q^4 + 118 * q^9 + 28 * q^10 + 24 * q^12 - 180 * q^14 + 64 * q^16 + 260 * q^17 + 20 * q^22 - 66 * q^23 + 234 * q^25 - 666 * q^27 - 396 * q^29 + 392 * q^30 + 532 * q^35 - 472 * q^36 - 12 * q^38 - 112 * q^40 - 164 * q^42 + 186 * q^43 - 96 * q^48 - 870 * q^49 - 2126 * q^51 - 1070 * q^53 + 1484 * q^55 + 720 * q^56 + 1308 * q^61 - 1120 * q^62 - 256 * q^64 - 3068 * q^66 - 1040 * q^68 + 750 * q^69 - 2808 * q^74 - 1870 * q^75 - 1294 * q^77 + 1520 * q^79 - 668 * q^81 - 968 * q^82 + 3198 * q^87 - 80 * q^88 - 476 * q^90 + 264 * q^92 + 896 * q^94 - 196 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	−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
$5$	− 3.86546i	− 0.345737i	−0.984945	−	0.172869i	$-0.944696\pi$
$5$	− 3.86546i	− 0.345737i	0.984945	−	0.172869i	$-0.0553036\pi$
$6$	17.7309i	1.20644i
$7$	− 15.1345i	− 0.817188i	−0.912716	−	0.408594i	$-0.866019\pi$
$7$	− 15.1345i	− 0.817188i	0.912716	−	0.408594i	$-0.133981\pi$
$8$	8.00000i	0.353553i
$9$	51.5964	1.91098
$10$	−7.73092	−0.244473
$11$	54.0582i	1.48174i	0.671647	+	0.740871i	$0.265587\pi$
$11$	54.0582i	1.48174i	−0.671647	+	0.740871i	$0.734413\pi$
$12$	35.4618	0.853079
$13$	0	0
$13$	0	0
$14$	−30.2691	−0.577839
$15$	34.2691i	0.589883i
$16$	16.0000	0.250000
$17$	123.924	1.76799	0.883997	−	0.467492i	$-0.154842\pi$
$17$	123.924	1.76799	0.883997	+	0.467492i	$0.154842\pi$
$18$	− 103.193i	− 1.35126i
$19$	− 8.86546i	− 0.107046i	−0.998567	−	0.0535231i	$-0.982955\pi$
$19$	− 8.86546i	− 0.107046i	0.998567	−	0.0535231i	$-0.0170451\pi$
$20$	15.4618i	0.172869i
$21$	134.175i	1.39425i
$22$	108.116	1.04775
$23$	−38.5964	−0.349909	−0.174954	−	0.984577i	$-0.555978\pi$
$23$	−38.5964	−0.349909	−0.174954	+	0.984577i	$0.555978\pi$
$24$	− 70.9237i	− 0.603218i
$25$	110.058	0.880466
$26$	0	0
$27$	−218.058	−1.55427
$28$	60.5382i	0.408594i
$29$	−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
$31$	36.7710i	0.213041i	0.994311	+	0.106521i	$0.0339710\pi$
$31$	36.7710i	0.213041i	−0.994311	+	0.106521i	$0.966029\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 479.251i	− 2.52809i
$34$	− 247.847i	− 1.25016i
$35$	−58.5020	−0.282532
$36$	−206.386	−0.955489
$37$	− 321.538i	− 1.42866i	−0.699807	−	0.714332i	$-0.746731\pi$
$37$	− 321.538i	− 1.42866i	0.699807	−	0.714332i	$-0.253269\pi$
$38$	−17.7309	−0.0756930
$39$	0	0
$40$	30.9237	0.122237
$41$	26.3092i	0.100215i	0.998744	+	0.0501074i	$0.0159564\pi$
$41$	26.3092i	0.100215i	−0.998744	+	0.0501074i	$0.984044\pi$
$42$	268.349	0.985886
$43$	−137.636	−0.488125	−0.244062	−	0.969760i	$-0.578480\pi$
$43$	−137.636	−0.488125	−0.244062	+	0.969760i	$0.578480\pi$
$44$	− 216.233i	− 0.740871i
$45$	− 199.444i	− 0.660696i
$46$	77.1928i	0.247423i
$47$	− 300.466i	− 0.932499i	−0.884653	−	0.466249i	$-0.845605\pi$
$47$	− 300.466i	− 0.932499i	0.884653	−	0.466249i	$-0.154395\pi$
$48$	−141.847	−0.426540
$49$	113.946	0.332203
$50$	− 220.116i	− 0.622583i
$51$	−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	$0.609445\pi$
$54$	436.116i	1.09904i
$55$	208.960	0.512294
$56$	121.076	0.288920
$57$	78.5964i	0.182638i
$58$	374.771i	0.848446i
$59$	− 246.058i	− 0.542950i	−0.962445	−	0.271475i	$-0.912489\pi$
$59$	− 246.058i	− 0.542950i	0.962445	−	0.271475i	$-0.0875114\pi$
$60$	− 137.076i	− 0.294941i
$61$	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
$63$	− 780.887i	− 1.56163i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−958.502	−1.78763
$67$	− 410.524i	− 0.748559i	−0.927316	−	0.374280i	$-0.877890\pi$
$67$	− 410.524i	− 0.748559i	0.927316	−	0.374280i	$-0.122110\pi$
$68$	−495.695	−0.883997
$69$	342.175	0.597000
$70$	117.004i	0.199781i
$71$	424.444i	0.709468i	0.934967	+	0.354734i	$0.115429\pi$
$71$	424.444i	0.709468i	−0.934967	+	0.354734i	$0.884571\pi$
$72$	412.771i	0.675632i
$73$	− 421.982i	− 0.676565i	−0.941045	−	0.338283i	$-0.890154\pi$
$73$	− 421.982i	− 0.676565i	0.941045	−	0.338283i	$-0.109846\pi$
$74$	−643.076	−1.01022
$75$	−975.717	−1.50221
$76$	35.4618i	0.0535231i
$77$	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	$0.325059\pi$
$80$	− 61.8474i	− 0.0864343i
$81$	540.084	0.740856
$82$	52.6184	0.0708626
$83$	616.843i	0.815751i	0.913037	+	0.407876i	$0.133730\pi$
$83$	616.843i	0.815751i	−0.913037	+	0.407876i	$0.866270\pi$
$84$	− 536.699i	− 0.697126i
$85$	− 479.022i	− 0.611262i
$86$	275.273i	0.345156i
$87$	1661.26	2.04719
$88$	−432.466	−0.523875
$89$	− 207.171i	− 0.246742i	−0.992361	−	0.123371i	$-0.960629\pi$
$89$	− 207.171i	− 0.246742i	0.992361	−	0.123371i	$-0.0393705\pi$
$90$	−398.887	−0.467183
$91$	0	0
$92$	154.386	0.174954
$93$	− 325.992i	− 0.363482i
$94$	−600.932	−0.659376
$95$	−34.2691	−0.0370098
$96$	283.695i	0.301609i
$97$	741.484i	0.776147i	0.921628	+	0.388074i	$0.126859\pi$
$97$	741.484i	0.776147i	−0.921628	+	0.388074i	$0.873141\pi$
$98$	− 227.891i	− 0.234903i
$99$	2789.21i	2.83158i
$100$	−440.233	−0.440233
$101$	413.305	0.407182	0.203591	−	0.979056i	$-0.434739\pi$
$101$	413.305	0.407182	0.203591	+	0.979056i	$0.434739\pi$
$102$	2197.28i	2.13297i
$103$	3.76712	0.00360374	0.00180187	−	0.999998i	$-0.499426\pi$
$103$	3.76712	0.00360374	0.00180187	+	0.999998i	$0.499426\pi$
$104$	0	0
$105$	518.647	0.482045
$106$	520.269i	0.476726i
$107$	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
$109$	− 1935.63i	− 1.70092i	−0.526044	−	0.850458i	$-0.676325\pi$
$109$	− 1935.63i	− 1.70092i	0.526044	−	0.850458i	$-0.323675\pi$
$110$	− 417.920i	− 0.362246i
$111$	2850.58i	2.43753i
$112$	− 242.153i	− 0.204297i
$113$	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
$115$	149.193i	0.120976i
$116$	749.542	0.599942
$117$	0	0
$118$	−492.116	−0.383924
$119$	− 1875.53i	− 1.44478i
$120$	−274.153	−0.208555
$121$	−1591.29	−1.19556
$122$	− 182.611i	− 0.135515i
$123$	− 233.243i	− 0.170982i
$124$	− 147.084i	− 0.106521i
$125$	− 908.608i	− 0.650147i
$126$	−1561.77	−1.10424
$127$	−244.604	−0.170906	−0.0854532	−	0.996342i	$-0.527234\pi$
$127$	−244.604	−0.170906	−0.0854532	+	0.996342i	$0.527234\pi$
$128$	128.000i	0.0883883i
$129$	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	$0.318889\pi$
$132$	1917.00i	1.26404i
$133$	−134.175	−0.0874768
$134$	−821.048	−0.529311
$135$	842.895i	0.537369i
$136$	991.389i	0.625080i
$137$	− 2301.95i	− 1.43554i	−0.696282	−	0.717769i	$-0.745164\pi$
$137$	− 2301.95i	− 1.43554i	0.696282	−	0.717769i	$-0.254836\pi$
$138$	− 684.349i	− 0.422143i
$139$	−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
$141$	2663.77i	1.59099i
$142$	848.887	0.501669
$143$	0	0
$144$	825.542	0.477744
$145$	724.331i	0.414844i
$146$	−843.964	−0.478404
$147$	−1010.18	−0.566791
$148$	1286.15i	0.714332i
$149$	− 2533.17i	− 1.39279i	−0.717660	−	0.696393i	$-0.754787\pi$
$149$	− 2533.17i	− 1.39279i	0.717660	−	0.696393i	$-0.245213\pi$
$150$	1951.43i	1.06223i
$151$	− 2391.10i	− 1.28864i	−0.764755	−	0.644321i	$-0.777140\pi$
$151$	− 2391.10i	− 1.28864i	0.764755	−	0.644321i	$-0.222860\pi$
$152$	70.9237	0.0378465
$153$	6394.01	3.37860
$154$	− 1636.29i	− 0.856209i
$155$	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	$0.337031\pi$
$158$	− 1467.08i	− 0.738702i
$159$	2306.21	1.15028
$160$	−123.695	−0.0611183
$161$	584.138i	0.285941i
$162$	− 1080.17i	− 0.523864i
$163$	− 2218.52i	− 1.06606i	−0.846096	−	0.533031i	$-0.821053\pi$
$163$	− 2218.52i	− 1.06606i	0.846096	−	0.533031i	$-0.178947\pi$
$164$	− 105.237i	− 0.0501074i
$165$	−1852.53	−0.874054
$166$	1233.69	0.576823
$167$	− 1508.27i	− 0.698883i	−0.936958	−	0.349442i	$-0.886371\pi$
$167$	− 1508.27i	− 0.698883i	0.936958	−	0.349442i	$-0.113629\pi$
$168$	−1073.40	−0.492943
$169$	0	0
$170$	−958.044	−0.432227
$171$	− 457.426i	− 0.204563i
$172$	550.546	0.244062
$173$	1706.47	0.749946	0.374973	−	0.927036i	$-0.377652\pi$
$173$	1706.47	0.749946	0.374973	+	0.927036i	$0.377652\pi$
$174$	− 3322.52i	− 1.44758i
$175$	− 1665.68i	− 0.719506i
$176$	864.932i	0.370436i
$177$	2181.42i	0.926359i
$178$	−414.341	−0.174473
$179$	−595.223	−0.248542	−0.124271	−	0.992248i	$-0.539659\pi$
$179$	−595.223	−0.248542	−0.124271	+	0.992248i	$0.539659\pi$
$180$	797.775i	0.330348i
$181$	−403.006	−0.165498	−0.0827492	−	0.996570i	$-0.526370\pi$
$181$	−403.006	−0.165498	−0.0827492	+	0.996570i	$0.526370\pi$
$182$	0	0
$183$	−809.463	−0.326980
$184$	− 308.771i	− 0.123711i
$185$	−1242.89	−0.493942
$186$	−651.984	−0.257020
$187$	6699.09i	2.61971i
$188$	1201.86i	0.466249i
$189$	3300.21i	1.27013i
$190$	68.5382i	0.0261699i
$191$	−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
$193$	2005.33i	0.747910i	0.927447	+	0.373955i	$0.121999\pi$
$193$	2005.33i	0.747910i	−0.927447	+	0.373955i	$0.878001\pi$
$194$	1482.97	0.548819
$195$	0	0
$196$	−455.783	−0.166102
$197$	557.500i	0.201625i	0.994905	+	0.100813i	$0.0321443\pi$
$197$	557.500i	0.201625i	−0.994905	+	0.100813i	$0.967856\pi$
$198$	5578.42	2.00223
$199$	−5533.97	−1.97132	−0.985661	−	0.168739i	$-0.946030\pi$
$199$	−5533.97	−1.97132	−0.985661	+	0.168739i	$0.946030\pi$
$200$	880.466i	0.311292i
$201$	3639.48i	1.27716i
$202$	− 826.611i	− 0.287921i
$203$	2835.99i	0.980531i
$204$	4394.56	1.50824
$205$	101.697	0.0346480
$206$	− 7.53424i	− 0.00254823i
$207$	−1991.43	−0.668668
$208$	0	0
$209$	479.251	0.158615
$210$	− 1037.29i	− 0.340857i
$211$	−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
$213$	− 3762.89i	− 1.21046i
$214$	− 227.651i	− 0.0727191i
$215$	532.028i	0.168763i
$216$	− 1744.47i	− 0.549518i
$217$	556.513	0.174095
$218$	−3871.26	−1.20273
$219$	3741.06i	1.15433i
$220$	−835.840	−0.256147
$221$	0	0
$222$	5701.17	1.72359
$223$	2383.86i	0.715853i	0.933750	+	0.357926i	$0.116516\pi$
$223$	2383.86i	0.715853i	−0.933750	+	0.357926i	$0.883484\pi$
$224$	−484.305	−0.144460
$225$	5678.61	1.68255
$226$	− 3797.57i	− 1.11774i
$227$	1554.07i	0.454394i	0.973849	+	0.227197i	$0.0729562\pi$
$227$	1554.07i	0.454394i	−0.973849	+	0.227197i	$0.927044\pi$
$228$	− 314.386i	− 0.0913188i
$229$	− 2915.60i	− 0.841346i	−0.907212	−	0.420673i	$-0.861794\pi$
$229$	− 2915.60i	− 0.841346i	0.907212	−	0.420673i	$-0.138206\pi$
$230$	298.386	0.0855433
$231$	−7253.24	−2.06592
$232$	− 1499.08i	− 0.424223i
$233$	4233.93	1.19045	0.595223	−	0.803561i	$-0.297064\pi$
$233$	4233.93	1.19045	0.595223	+	0.803561i	$0.297064\pi$
$234$	0	0
$235$	−1161.44	−0.322400
$236$	984.233i	0.271475i
$237$	−6503.19	−1.78239
$238$	−3751.06	−1.02162
$239$	− 2372.55i	− 0.642124i	−0.947058	−	0.321062i	$-0.895960\pi$
$239$	− 2372.55i	− 0.642124i	0.947058	−	0.321062i	$-0.104040\pi$
$240$	548.305i	0.147471i
$241$	325.763i	0.0870716i	0.999052	+	0.0435358i	$0.0138623\pi$
$241$	325.763i	0.0870716i	−0.999052	+	0.0435358i	$0.986138\pi$
$242$	3182.58i	0.845389i
$243$	1099.48	0.290253
$244$	−365.221	−0.0958233
$245$	− 440.453i	− 0.114855i
$246$	−466.486	−0.120903
$247$	0	0
$248$	−294.168	−0.0753214
$249$	− 5468.60i	− 1.39180i
$250$	−1817.22	−0.459723
$251$	2601.77	0.654272	0.327136	−	0.944977i	$-0.393917\pi$
$251$	2601.77	0.654272	0.327136	+	0.944977i	$0.393917\pi$
$252$	3123.55i	0.780814i
$253$	− 2086.45i	− 0.518475i
$254$	489.208i	0.120849i
$255$	4246.75i	1.04291i
$256$	256.000	0.0625000
$257$	−4217.10	−1.02356	−0.511781	−	0.859116i	$-0.671014\pi$
$257$	−4217.10	−1.02356	−0.511781	+	0.859116i	$0.671014\pi$
$258$	− 2440.42i	− 0.588892i
$259$	−4866.33	−1.16749
$260$	0	0
$261$	−9668.41	−2.29295
$262$	− 3231.24i	− 0.761935i
$263$	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
$265$	1005.54i	0.233094i
$266$	268.349i	0.0618555i
$267$	1836.66i	0.420981i
$268$	1642.10i	0.374280i
$269$	−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
$271$	− 3079.64i	− 0.690314i	−0.938545	−	0.345157i	$-0.887826\pi$
$271$	− 3079.64i	− 0.690314i	0.938545	−	0.345157i	$-0.112174\pi$
$272$	1982.78	0.441999
$273$	0	0
$274$	−4603.89	−1.01508
$275$	5949.55i	1.30462i
$276$	−1368.70	−0.298500
$277$	6287.90	1.36391	0.681955	−	0.731394i	$-0.261130\pi$
$277$	6287.90	1.36391	0.681955	+	0.731394i	$0.261130\pi$
$278$	5618.14i	1.21206i
$279$	1897.25i	0.407117i
$280$	− 468.016i	− 0.0998903i
$281$	− 3226.00i	− 0.684864i	−0.939543	−	0.342432i	$-0.888749\pi$
$281$	− 3226.00i	− 0.684864i	0.939543	−	0.342432i	$-0.111251\pi$
$282$	5327.53	1.12500
$283$	386.584	0.0812015	0.0406007	−	0.999175i	$-0.487073\pi$
$283$	386.584	0.0812015	0.0406007	+	0.999175i	$0.487073\pi$
$284$	− 1697.77i	− 0.354734i
$285$	303.811	0.0631446
$286$	0	0
$287$	398.178	0.0818944
$288$	− 1651.08i	− 0.337816i
$289$	10444.1	2.12580
$290$	1448.66	0.293339
$291$	− 6573.60i	− 1.32423i
$292$	1687.93i	0.338283i
$293$	2192.82i	0.437222i	0.975812	+	0.218611i	$0.0701526\pi$
$293$	2192.82i	0.437222i	−0.975812	+	0.218611i	$0.929847\pi$
$294$	2020.36i	0.400782i
$295$	−951.128	−0.187718
$296$	2572.31	0.505109
$297$	− 11787.8i	− 2.30303i
$298$	−5066.34	−0.984849
$299$	0	0
$300$	3902.87	0.751107
$301$	2083.07i	0.398890i
$302$	−4782.20	−0.911208
$303$	−3664.14	−0.694718
$304$	− 141.847i	− 0.0267615i
$305$	− 352.937i	− 0.0662594i
$306$	− 12788.0i	− 2.38903i
$307$	− 8083.96i	− 1.50285i	−0.659817	−	0.751427i	$-0.729366\pi$
$307$	− 8083.96i	− 1.50285i	0.659817	−	0.751427i	$-0.270634\pi$
$308$	−3272.59	−0.605431
$309$	−33.3973	−0.00614856
$310$	− 284.274i	− 0.0520828i
$311$	−244.409	−0.0445632	−0.0222816	−	0.999752i	$-0.507093\pi$
$311$	−244.409	−0.0445632	−0.0222816	+	0.999752i	$0.507093\pi$
$312$	0	0
$313$	4444.13	0.802546	0.401273	−	0.915958i	$-0.368568\pi$
$313$	4444.13	0.802546	0.401273	+	0.915958i	$0.368568\pi$
$314$	− 3854.98i	− 0.692831i
$315$	−3018.49	−0.539913
$316$	−2934.17	−0.522341
$317$	− 930.597i	− 0.164882i	−0.996596	−	0.0824409i	$-0.973728\pi$
$317$	− 930.597i	− 0.164882i	0.996596	−	0.0824409i	$-0.0262716\pi$
$318$	− 4612.42i	− 0.813371i
$319$	− 10129.7i	− 1.77792i
$320$	247.389i	0.0432172i
$321$	−1009.11	−0.175462
$322$	1168.28	0.202191
$323$	− 1098.64i	− 0.189257i
$324$	−2160.34	−0.370428
$325$	0	0
$326$	−4437.05	−0.753820
$327$	17160.3i	2.90203i
$328$	−210.474	−0.0354313
$329$	−4547.41	−0.762027
$330$	3705.05i	0.618050i
$331$	− 4827.35i	− 0.801618i	−0.916162	−	0.400809i	$-0.868729\pi$
$331$	− 4827.35i	− 0.801618i	0.916162	−	0.400809i	$-0.131271\pi$
$332$	− 2467.37i	− 0.407876i
$333$	− 16590.2i	− 2.73014i
$334$	−3016.54	−0.494185
$335$	−1586.86	−0.258805
$336$	2146.79i	0.348563i
$337$	−10709.7	−1.73115	−0.865573	−	0.500782i	$-0.833046\pi$
$337$	−10709.7	−1.73115	−0.865573	+	0.500782i	$0.833046\pi$
$338$	0	0
$339$	−16833.6	−2.69698
$340$	1916.09i	0.305631i
$341$	−1987.78	−0.315672
$342$	−914.851	−0.144648
$343$	− 6915.66i	− 1.08866i
$344$	− 1101.09i	− 0.172578i
$345$	− 1322.66i	− 0.206405i
$346$	− 3412.94i	− 0.530292i
$347$	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
$349$	− 2430.02i	− 0.372711i	−0.982482	−	0.186355i	$-0.940332\pi$
$349$	− 2430.02i	− 0.372711i	0.982482	−	0.186355i	$-0.0596676\pi$
$350$	−3331.36	−0.508768
$351$	0	0
$352$	1729.86	0.261938
$353$	− 8080.03i	− 1.21829i	−0.793059	−	0.609145i	$-0.791513\pi$
$353$	− 8080.03i	− 1.21829i	0.793059	−	0.609145i	$-0.208487\pi$
$354$	4362.84	0.655035
$355$	1640.67	0.245289
$356$	828.683i	0.123371i
$357$	16627.4i	2.46503i
$358$	1190.45i	0.175746i
$359$	8715.23i	1.28126i	0.767850	+	0.640630i	$0.221327\pi$
$359$	8715.23i	1.28126i	−0.767850	+	0.640630i	$0.778673\pi$
$360$	1595.55	0.233591
$361$	6780.40	0.988541
$362$	806.013i	0.117025i
$363$	14107.5	2.03982
$364$	0	0
$365$	−1631.15	−0.233914
$366$	1618.93i	0.231209i
$367$	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
$369$	1357.46i	0.191508i
$370$	2485.79i	0.349270i
$371$	3937.02i	0.550943i
$372$	1303.97i	0.181741i
$373$	−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
$375$	8055.23i	1.10925i
$376$	2403.73	0.329688
$377$	0	0
$378$	6600.42	0.898119
$379$	2288.05i	0.310103i	0.987906	+	0.155052i	$0.0495544\pi$
$379$	2288.05i	0.310103i	−0.987906	+	0.155052i	$0.950446\pi$
$380$	137.076	0.0185049
$381$	2168.53	0.291593
$382$	1227.39i	0.164395i
$383$	− 12156.2i	− 1.62181i	−0.585179	−	0.810904i	$-0.698976\pi$
$383$	− 12156.2i	− 1.62181i	0.585179	−	0.810904i	$-0.301024\pi$
$384$	− 1134.78i	− 0.150805i
$385$	− 3162.51i	− 0.418640i
$386$	4010.66	0.528853
$387$	−7101.55	−0.932795
$388$	− 2965.94i	− 0.388074i
$389$	1479.71	0.192864	0.0964322	−	0.995340i	$-0.469257\pi$
$389$	1479.71	0.192864	0.0964322	+	0.995340i	$0.469257\pi$
$390$	0	0
$391$	−4783.01	−0.618637
$392$	911.566i	0.117452i
$393$	−14323.2	−1.83845
$394$	1115.00	0.142571
$395$	− 2835.48i	− 0.361186i
$396$	− 11156.8i	− 1.41579i
$397$	15271.3i	1.93058i	0.261172	+	0.965292i	$0.415891\pi$
$397$	15271.3i	1.93058i	−0.261172	+	0.965292i	$0.584109\pi$
$398$	11067.9i	1.39393i
$399$	1189.52	0.149249
$400$	1760.93	0.220116
$401$	2564.54i	0.319369i	0.987168	+	0.159685i	$0.0510477\pi$
$401$	2564.54i	0.319369i	−0.987168	+	0.159685i	$0.948952\pi$
$402$	7278.97	0.903089
$403$	0	0
$404$	−1653.22	−0.203591
$405$	− 2087.67i	− 0.256142i
$406$	5671.99	0.693340
$407$	17381.8	2.11691
$408$	− 8789.12i	− 1.06649i
$409$	− 1052.47i	− 0.127240i	−0.997974	−	0.0636201i	$-0.979735\pi$
$409$	− 1052.47i	− 0.127240i	0.997974	−	0.0636201i	$-0.0202646\pi$
$410$	− 203.394i	− 0.0244998i
$411$	20407.8i	2.44925i
$412$	−15.0685	−0.00180187
$413$	−3723.98	−0.443692
$414$	3982.87i	0.472819i
$415$	2384.38	0.282036
$416$	0	0
$417$	24903.7	2.92455
$418$	− 958.502i	− 0.112158i
$419$	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
$421$	− 213.335i	− 0.0246967i	−0.999924	−	0.0123484i	$-0.996069\pi$
$421$	− 213.335i	− 0.0246967i	0.999924	−	0.0123484i	$-0.00393070\pi$
$422$	9367.46i	1.08057i
$423$	− 15502.9i	− 1.78198i
$424$	− 2081.08i	− 0.238363i
$425$	13638.8	1.55666
$426$	−7525.78	−0.855928
$427$	− 1381.86i	− 0.156611i
$428$	−455.301	−0.0514201
$429$	0	0
$430$	1064.06	0.119333
$431$	− 8953.45i	− 1.00063i	−0.865843	−	0.500316i	$-0.833217\pi$
$431$	− 8953.45i	− 1.00063i	0.865843	−	0.500316i	$-0.166783\pi$
$432$	−3488.93	−0.388568
$433$	−5861.44	−0.650538	−0.325269	−	0.945622i	$-0.605455\pi$
$433$	−5861.44	−0.650538	−0.325269	+	0.945622i	$0.605455\pi$
$434$	− 1113.03i	− 0.123104i
$435$	− 6421.53i	− 0.707790i
$436$	7742.52i	0.850458i
$437$	342.175i	0.0374564i
$438$	7482.13	0.816233
$439$	10443.5	1.13540	0.567702	−	0.823234i	$-0.307833\pi$
$439$	10443.5	1.13540	0.567702	+	0.823234i	$0.307833\pi$
$440$	1671.68i	0.181123i
$441$	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	$0.343768\pi$
$444$	− 11402.3i	− 1.21876i
$445$	−800.810	−0.0853080
$446$	4767.72	0.506184
$447$	22457.7i	2.37632i
$448$	968.611i	0.102149i
$449$	8636.11i	0.907714i	0.891075	+	0.453857i	$0.149952\pi$
$449$	8636.11i	0.907714i	−0.891075	+	0.453857i	$0.850048\pi$
$450$	− 11357.2i	− 1.18974i
$451$	−1422.23	−0.148493
$452$	−7595.13	−0.790365
$453$	21198.2i	2.19863i
$454$	3108.15	0.321305
$455$	0	0
$456$	−628.771	−0.0645722
$457$	− 4876.96i	− 0.499200i	−0.968349	−	0.249600i	$-0.919701\pi$
$457$	− 4876.96i	− 0.499200i	0.968349	−	0.249600i	$-0.0802992\pi$
$458$	−5831.20	−0.594921
$459$	−27022.6	−2.74794
$460$	− 596.771i	− 0.0604882i
$461$	11803.9i	1.19254i	0.802782	+	0.596272i	$0.203352\pi$
$461$	11803.9i	1.19254i	−0.802782	+	0.596272i	$0.796648\pi$
$462$	14506.5i	1.46083i
$463$	− 4797.53i	− 0.481555i	−0.970580	−	0.240777i	$-0.922598\pi$
$463$	− 4797.53i	− 0.481555i	0.970580	−	0.240777i	$-0.0774024\pi$
$464$	−2998.17	−0.299971
$465$	−1260.11	−0.125669
$466$	− 8467.86i	− 0.841772i
$467$	13188.7	1.30686	0.653428	−	0.756989i	$-0.273330\pi$
$467$	13188.7	1.30686	0.653428	+	0.756989i	$0.273330\pi$
$468$	0	0
$469$	−6213.09	−0.611714
$470$	2322.88i	0.227971i
$471$	−17088.1	−1.67171
$472$	1968.47	0.191962
$473$	− 7440.38i	− 0.723275i
$474$	13006.4i	1.26034i
$475$	− 975.717i	− 0.0942504i
$476$	7502.11i	0.722392i
$477$	−13422.0	−1.28837
$478$	−4745.11	−0.454051
$479$	18048.8i	1.72165i	0.508898	+	0.860827i	$0.330053\pi$
$479$	18048.8i	1.72165i	−0.508898	+	0.860827i	$0.669947\pi$
$480$	1096.61	0.104277
$481$	0	0
$482$	651.526	0.0615689
$483$	− 5178.66i	− 0.487861i
$484$	6365.16	0.597780
$485$	2866.18	0.268343
$486$	− 2198.96i	− 0.205240i
$487$	− 11501.2i	− 1.07017i	−0.844799	−	0.535084i	$-0.820280\pi$
$487$	− 11501.2i	− 1.07017i	0.844799	−	0.535084i	$-0.179720\pi$
$488$	730.442i	0.0677573i
$489$	19668.2i	1.81887i
$490$	−880.905	−0.0812148
$491$	4000.65	0.367713	0.183856	−	0.982953i	$-0.441142\pi$
$491$	4000.65	0.367713	0.183856	+	0.982953i	$0.441142\pi$
$492$	932.973i	0.0854912i
$493$	−23221.5	−2.12139
$494$	0	0
$495$	10781.6	0.978981
$496$	588.337i	0.0532603i
$497$	6423.76	0.579769
$498$	−10937.2	−0.984152
$499$	8322.72i	0.746645i	0.927702	+	0.373323i	$0.121782\pi$
$499$	8322.72i	0.746645i	−0.927702	+	0.373323i	$0.878218\pi$
$500$	3634.43i	0.325074i
$501$	13371.5i	1.19241i
$502$	− 5203.54i	− 0.462640i
$503$	−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
$505$	− 1597.62i	− 0.140778i
$506$	−4172.90	−0.366617
$507$	0	0
$508$	978.417	0.0854532
$509$	14348.9i	1.24952i	0.780817	+	0.624759i	$0.214803\pi$
$509$	14348.9i	1.24952i	−0.780817	+	0.624759i	$0.785197\pi$
$510$	8493.50	0.737448
$511$	−6386.50	−0.552881
$512$	− 512.000i	− 0.0441942i
$513$	1933.19i	0.166379i
$514$	8434.19i	0.723767i
$515$	− 14.5617i	− 0.00124595i
$516$	−4880.84	−0.416409
$517$	16242.6	1.38172
$518$	9732.66i	0.825538i
$519$	−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	$0.452502\pi$
$522$	19336.8i	1.62136i
$523$	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
$525$	14767.0i	1.22759i
$526$	− 12948.6i	− 1.07336i
$527$	4556.80i	0.376655i
$528$	− 7668.02i	− 0.632022i
$529$	−10677.3	−0.877564
$530$	2011.08	0.164822
$531$	− 12695.7i	− 1.03757i
$532$	536.699	0.0437384
$533$	0	0
$534$	3673.33	0.297679
$535$	− 439.987i	− 0.0355557i
$536$	3284.19	0.264656
$537$	5276.92	0.424052
$538$	2602.07i	0.208519i
$539$	6159.70i	0.492240i
$540$	− 3371.58i	− 0.268685i
$541$	− 10661.6i	− 0.847277i	−0.905831	−	0.423638i	$-0.860753\pi$
$541$	− 10661.6i	− 0.847277i	0.905831	−	0.423638i	$-0.139247\pi$
$542$	−6159.29	−0.488126
$543$	3572.84	0.282367
$544$	− 3965.56i	− 0.312540i
$545$	−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	$0.542343\pi$
$548$	9207.79i	0.717769i
$549$	4711.02	0.366232
$550$	11899.1	0.922508
$551$	1661.26i	0.128443i
$552$	2737.40i	0.211071i
$553$	− 11101.8i	− 0.853703i
$554$	− 12575.8i	− 0.964430i
$555$	11018.8	0.842744
$556$	11236.3	0.857058
$557$	− 8249.03i	− 0.627509i	−0.949504	−	0.313754i	$-0.898413\pi$
$557$	− 8249.03i	− 0.627509i	0.949504	−	0.313754i	$-0.101587\pi$
$558$	3794.50	0.287875
$559$	0	0
$560$	−936.031	−0.0706331
$561$	− 59390.5i	− 4.46964i
$562$	−6452.00	−0.484272
$563$	−8868.79	−0.663898	−0.331949	−	0.943297i	$-0.607706\pi$
$563$	−8868.79	−0.663898	−0.331949	+	0.943297i	$0.607706\pi$
$564$	− 10655.1i	− 0.795495i
$565$	− 7339.67i	− 0.546517i
$566$	− 773.167i	− 0.0574181i
$567$	− 8173.93i	− 0.605419i
$568$	−3395.55	−0.250835
$569$	6408.49	0.472158	0.236079	−	0.971734i	$-0.424138\pi$
$569$	6408.49	0.472158	0.236079	+	0.971734i	$0.424138\pi$
$570$	− 607.622i	− 0.0446500i
$571$	−10045.2	−0.736212	−0.368106	−	0.929784i	$-0.619994\pi$
$571$	−10045.2	−0.736212	−0.368106	+	0.929784i	$0.619994\pi$
$572$	0	0
$573$	5440.70	0.396664
$574$	− 796.355i	− 0.0579081i
$575$	−4247.85	−0.308083
$576$	−3302.17	−0.238872
$577$	24528.9i	1.76976i	0.465818	+	0.884880i	$0.345760\pi$
$577$	24528.9i	1.76976i	−0.465818	+	0.884880i	$0.654240\pi$
$578$	− 20888.2i	− 1.50317i
$579$	− 17778.2i	− 1.27605i
$580$	− 2897.32i	− 0.207422i
$581$	9335.64	0.666623
$582$	−13147.2	−0.936372
$583$	− 14062.4i	− 0.998980i
$584$	3375.86	0.239202
$585$	0	0
$586$	4385.64	0.309163
$587$	15903.8i	1.11826i	0.829080	+	0.559130i	$0.188865\pi$
$587$	15903.8i	1.11826i	−0.829080	+	0.559130i	$0.811135\pi$
$588$	4040.72	0.283396
$589$	325.992	0.0228052
$590$	1902.26i	0.132737i
$591$	− 4942.49i	− 0.344005i
$592$	− 5144.61i	− 0.357166i
$593$	− 5436.51i	− 0.376477i	−0.982123	−	0.188238i	$-0.939722\pi$
$593$	− 5436.51i	− 0.376477i	0.982123	−	0.188238i	$-0.0602778\pi$
$594$	−23575.7	−1.62849
$595$	−7249.78	−0.499516
$596$	10132.7i	0.696393i
$597$	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	$0.424690\pi$
$600$	− 7805.73i	− 0.531113i
$601$	−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
$603$	− 21181.6i	− 1.43048i
$604$	9564.40i	0.644321i
$605$	6151.07i	0.413350i
$606$	7328.28i	0.491240i
$607$	−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
$609$	− 25142.4i	− 1.67294i
$610$	−705.874	−0.0468525
$611$	0	0
$612$	−25576.1	−1.68930
$613$	17279.4i	1.13851i	0.822160	+	0.569257i	$0.192769\pi$
$613$	17279.4i	1.13851i	−0.822160	+	0.569257i	$0.807231\pi$
$614$	−16167.9	−1.06268
$615$	−901.592	−0.0591150
$616$	6545.17i	0.428105i
$617$	2177.07i	0.142051i	0.997474	+	0.0710256i	$0.0226272\pi$
$617$	2177.07i	0.142051i	−0.997474	+	0.0710256i	$0.977373\pi$
$618$	66.7945i	0.00434769i
$619$	17067.5i	1.10824i	0.832436	+	0.554121i	$0.186945\pi$
$619$	17067.5i	1.10824i	−0.832436	+	0.554121i	$0.813055\pi$
$620$	−568.548	−0.0368281
$621$	8416.26	0.543853
$622$	488.818i	0.0315110i
$623$	−3135.43	−0.201635
$624$	0	0
$625$	10245.1	0.655686
$626$	− 8888.25i	− 0.567486i
$627$	−4248.78	−0.270622
$628$	−7709.95	−0.489905
$629$	− 39846.2i	− 2.52587i
$630$	6036.98i	0.381776i
$631$	− 7929.96i	− 0.500296i	−0.968208	−	0.250148i	$-0.919521\pi$
$631$	− 7929.96i	− 0.500296i	0.968208	−	0.250148i	$-0.0804793\pi$
$632$	5868.34i	0.369351i
$633$	41523.4	2.60728
$634$	−1861.19	−0.116589
$635$	945.508i	0.0590887i
$636$	−9224.85	−0.575140
$637$	0	0
$638$	−20259.5	−1.25718
$639$	21899.8i	1.35578i
$640$	494.779	0.0305591
$641$	5839.06	0.359796	0.179898	−	0.983685i	$-0.442423\pi$
$641$	5839.06	0.359796	0.179898	+	0.983685i	$0.442423\pi$
$642$	2018.23i	0.124070i
$643$	17022.8i	1.04404i	0.852935	+	0.522018i	$0.174821\pi$
$643$	17022.8i	1.04404i	−0.852935	+	0.522018i	$0.825179\pi$
$644$	− 2336.55i	− 0.142971i
$645$	− 4716.68i	− 0.287936i
$646$	−2197.28	−0.133825
$647$	−21779.3	−1.32339	−0.661694	−	0.749774i	$-0.730162\pi$
$647$	−21779.3	−1.32339	−0.661694	+	0.749774i	$0.730162\pi$
$648$	4320.67i	0.261932i
$649$	13301.5	0.804512
$650$	0	0
$651$	−4933.74	−0.297033
$652$	8874.10i	0.533031i
$653$	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
$655$	− 6245.12i	− 0.372545i
$656$	420.947i	0.0250537i
$657$	− 21772.7i	− 1.29290i
$658$	9094.82i	0.538834i
$659$	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
$661$	7744.70i	0.455724i	0.973693	+	0.227862i	$0.0731736\pi$
$661$	7744.70i	0.455724i	−0.973693	+	0.227862i	$0.926826\pi$
$662$	−9654.71	−0.566829
$663$	0	0
$664$	−4934.75	−0.288412
$665$	518.647i	0.0302440i
$666$	−33180.4	−1.93050
$667$	7232.40	0.419850
$668$	6033.08i	0.349442i
$669$	− 21134.0i	− 1.22136i
$670$	3173.73i	0.183003i
$671$	4935.80i	0.283971i
$672$	4293.59	0.246471
$673$	2302.45	0.131877	0.0659384	−	0.997824i	$-0.478996\pi$
$673$	2302.45	0.131877	0.0659384	+	0.997824i	$0.478996\pi$
$674$	21419.5i	1.22411i
$675$	−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	$0.649376\pi$
$678$	33667.2i	1.90705i
$679$	11222.0	0.634258
$680$	3832.18	0.216114
$681$	− 13777.6i	− 0.775269i
$682$	3975.55i	0.223214i
$683$	18458.6i	1.03411i	0.855952	+	0.517055i	$0.172972\pi$
$683$	18458.6i	1.03411i	−0.855952	+	0.517055i	$0.827028\pi$
$684$	1829.70i	0.102281i
$685$	−8898.08	−0.496319
$686$	−13831.3	−0.769800
$687$	25848.1i	1.43547i
$688$	−2202.18	−0.122031
$689$	0	0
$690$	−2645.32	−0.145950
$691$	− 22979.4i	− 1.26509i	−0.774524	−	0.632544i	$-0.782011\pi$
$691$	− 22979.4i	− 1.26509i	0.774524	−	0.632544i	$-0.217989\pi$
$692$	−6825.89	−0.374973
$693$	42213.4	2.31393
$694$	− 12802.2i	− 0.700236i
$695$	10858.3i	0.592634i
$696$	13290.1i	0.723791i
$697$	3260.33i	0.177179i
$698$	−4860.04	−0.263546
$699$	−37535.7	−2.03109
$700$	6662.72i	0.359753i
$701$	−8633.81	−0.465185	−0.232592	−	0.972574i	$-0.574721\pi$
$701$	−8633.81	−0.465185	−0.232592	+	0.972574i	$0.574721\pi$
$702$	0	0
$703$	−2850.58	−0.152933
$704$	− 3459.73i	− 0.185218i
$705$	10296.7	0.550065
$706$	−16160.1	−0.861461
$707$	− 6255.19i	− 0.332745i
$708$	− 8725.68i	− 0.463179i
$709$	− 25622.5i	− 1.35722i	−0.734497	−	0.678612i	$-0.762582\pi$
$709$	− 25622.5i	− 1.35722i	0.734497	−	0.678612i	$-0.237418\pi$
$710$	− 3281.34i	− 0.173446i
$711$	37848.1	1.99636
$712$	1657.37	0.0872365
$713$	− 1419.23i	− 0.0745449i
$714$	33254.8	1.74304
$715$	0	0
$716$	2380.89	0.124271
$717$	21033.8i	1.09557i
$718$	17430.5	0.905987
$719$	−34954.8	−1.81306	−0.906532	−	0.422137i	$-0.861280\pi$
$719$	−34954.8	−1.81306	−0.906532	+	0.422137i	$0.861280\pi$
$720$	− 3191.10i	− 0.165174i
$721$	− 57.0137i	− 0.00294494i
$722$	− 13560.8i	− 0.699004i
$723$	− 2888.04i	− 0.148558i
$724$	1612.03	0.0827492
$725$	−20623.3	−1.05646
$726$	− 28215.1i	− 1.44237i
$727$	23397.0	1.19360	0.596800	−	0.802390i	$-0.296438\pi$
$727$	23397.0	1.19360	0.596800	+	0.802390i	$0.296438\pi$
$728$	0	0
$729$	−24329.6	−1.23607
$730$	3262.31i	0.165402i
$731$	−17056.4	−0.863002
$732$	3237.85	0.163490
$733$	3541.17i	0.178440i	0.996012	+	0.0892198i	$0.0284374\pi$
$733$	3541.17i	0.178440i	−0.996012	+	0.0892198i	$0.971563\pi$
$734$	− 5641.76i	− 0.283707i
$735$	3904.81i	0.195961i
$736$	1235.08i	0.0618557i
$737$	22192.2	1.10917
$738$	2714.92	0.135417
$739$	− 39233.0i	− 1.95292i	−0.215697	−	0.976460i	$-0.569202\pi$
$739$	− 39233.0i	− 1.95292i	0.215697	−	0.976460i	$-0.430798\pi$
$740$	4971.57	0.246971
$741$	0	0
$742$	7874.03	0.389575
$743$	− 38198.7i	− 1.88610i	−0.332646	−	0.943052i	$-0.607942\pi$
$743$	− 38198.7i	− 1.88610i	0.332646	−	0.943052i	$-0.392058\pi$
$744$	2607.94	0.128510
$745$	−9791.86	−0.481538
$746$	25859.4i	1.26914i
$747$	31826.9i	1.55888i
$748$	− 26796.4i	− 1.30986i
$749$	− 1722.69i	− 0.0840399i
$750$	16110.5	0.784361
$751$	1670.75	0.0811807	0.0405903	−	0.999176i	$-0.487076\pi$
$751$	1670.75	0.0811807	0.0405903	+	0.999176i	$0.487076\pi$
$752$	− 4807.45i	− 0.233125i
$753$	−23065.9	−1.11629
$754$	0	0
$755$	−9242.70	−0.445532
$756$	− 13200.8i	− 0.635066i
$757$	−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
$759$	18497.4i	0.884600i
$760$	− 274.153i	− 0.0130849i
$761$	− 37772.7i	− 1.79929i	−0.436622	−	0.899645i	$-0.643825\pi$
$761$	− 37772.7i	− 1.79929i	0.436622	−	0.899645i	$-0.356175\pi$
$762$	− 4337.06i	− 0.206188i
$763$	−29294.9	−1.38997
$764$	2454.78	0.116245
$765$	− 24715.8i	− 1.16811i
$766$	−24312.4	−1.14679
$767$	0	0
$768$	−2269.56	−0.106635
$769$	13073.3i	0.613049i	0.951863	+	0.306524i	$0.0991661\pi$
$769$	13073.3i	0.613049i	−0.951863	+	0.306524i	$0.900834\pi$
$770$	−6325.02	−0.296023
$771$	37386.5	1.74636
$772$	− 8021.32i	− 0.373955i
$773$	25143.1i	1.16990i	0.811069	+	0.584951i	$0.198886\pi$
$773$	25143.1i	1.16990i	−0.811069	+	0.584951i	$0.801114\pi$
$774$	14203.1i	0.659586i
$775$	4046.96i	0.187575i
$776$	−5931.87	−0.274409
$777$	43142.3	1.99192
$778$	− 2959.42i	− 0.136376i
$779$	233.243	0.0107276
$780$	0	0
$781$	−22944.7	−1.05125
$782$	9566.01i	0.437442i
$783$	40861.0	1.86494
$784$	1823.13	0.0830508
$785$	− 7450.63i	− 0.338757i
$786$	28646.5i	1.29998i
$787$	19164.6i	0.868035i	0.900904	+	0.434017i	$0.142904\pi$
$787$	19164.6i	0.868035i	−0.900904	+	0.434017i	$0.857096\pi$
$788$	− 2230.00i	− 0.100813i
$789$	−57397.8	−2.58988
$790$	−5670.96	−0.255397
$791$	− 28737.2i	− 1.29175i
$792$	−22313.7	−1.00111
$793$	0	0
$794$	30542.5	1.36513
$795$	− 8914.57i	− 0.397695i
$796$	22135.9	0.985661
$797$	−28991.4	−1.28849	−0.644246	−	0.764818i	$-0.722829\pi$
$797$	−28991.4	−1.28849	−0.644246	+	0.764818i	$0.722829\pi$
$798$	− 2379.04i	− 0.105535i
$799$	− 37234.8i	− 1.64865i
$800$	− 3521.86i	− 0.155646i
$801$	− 10689.3i	− 0.471519i
$802$	5129.08	0.225828
$803$	22811.6	1.00250
$804$	− 14557.9i	− 0.638581i
$805$	2257.96	0.0988606
$806$	0	0
$807$	11534.3	0.503129
$808$	3306.44i	0.143961i
$809$	−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
$811$	42182.9i	1.82644i	0.407466	+	0.913220i	$0.366413\pi$
$811$	42182.9i	1.82644i	−0.407466	+	0.913220i	$0.633587\pi$
$812$	− 11344.0i	− 0.490265i
$813$	27302.5i	1.17778i
$814$	− 34763.6i	− 1.49688i
$815$	−8575.62	−0.368578
$816$	−17578.2	−0.754120
$817$	1220.21i	0.0522519i
$818$	−2104.94	−0.0899725
$819$	0	0
$820$	−406.789	−0.0173240
$821$	18500.8i	0.786457i	0.919441	+	0.393229i	$0.128642\pi$
$821$	18500.8i	0.786457i	−0.919441	+	0.393229i	$0.871358\pi$
$822$	40815.6	1.73188
$823$	27868.7	1.18037	0.590184	−	0.807269i	$-0.299055\pi$
$823$	27868.7	1.18037	0.590184	+	0.807269i	$0.299055\pi$
$824$	30.1370i	0.00127412i
$825$	− 52745.5i	− 2.22589i
$826$	7447.96i	0.313738i
$827$	2541.96i	0.106884i	0.998571	+	0.0534418i	$0.0170192\pi$
$827$	2541.96i	0.106884i	−0.998571	+	0.0534418i	$0.982981\pi$
$828$	7965.73	0.334334
$829$	−29570.0	−1.23885	−0.619427	−	0.785054i	$-0.712635\pi$
$829$	−29570.0	−1.23885	−0.619427	+	0.785054i	$0.712635\pi$
$830$	− 4768.77i	− 0.199429i
$831$	−55745.1	−2.32705
$832$	0	0
$833$	14120.6	0.587333
$834$	− 49807.4i	− 2.06797i
$835$	−5830.16	−0.241630
$836$	−1917.00	−0.0793074
$837$	− 8018.23i	− 0.331124i
$838$	− 17423.2i	− 0.718229i
$839$	21607.5i	0.889122i	0.895749	+	0.444561i	$0.146640\pi$
$839$	21607.5i	0.889122i	−0.895749	+	0.444561i	$0.853360\pi$
$840$	4149.17i	0.170429i
$841$	10724.3	0.439720
$842$	−426.670	−0.0174632
$843$	28600.0i	1.16849i
$844$	18734.9	0.764079
$845$	0	0
$846$	−31005.9	−1.26005
$847$	24083.5i	0.976998i
$848$	−4162.15	−0.168548
$849$	−3427.24	−0.138543
$850$	− 27277.6i	− 1.10072i
$851$	12410.2i	0.499902i
$852$	15051.6i	0.605232i
$853$	− 28695.1i	− 1.15182i	−0.817513	−	0.575910i	$-0.804648\pi$
$853$	− 28695.1i	− 1.15182i	0.817513	−	0.575910i	$-0.195352\pi$
$854$	−2763.73	−0.110741
$855$	−1768.16	−0.0707249
$856$	910.603i	0.0363595i
$857$	−8720.01	−0.347573	−0.173786	−	0.984783i	$-0.555600\pi$
$857$	−8720.01	−0.347573	−0.173786	+	0.984783i	$0.555600\pi$
$858$	0	0
$859$	−1750.23	−0.0695191	−0.0347596	−	0.999396i	$-0.511067\pi$
$859$	−1750.23	−0.0695191	−0.0347596	+	0.999396i	$0.511067\pi$
$860$	− 2128.11i	− 0.0843815i
$861$	−3530.03	−0.139725
$862$	−17906.9	−0.707554
$863$	35493.3i	1.40001i	0.714139	+	0.700004i	$0.246819\pi$
$863$	35493.3i	1.40001i	−0.714139	+	0.700004i	$0.753181\pi$
$864$	6977.86i	0.274759i
$865$	− 6596.30i	− 0.259284i
$866$	11722.9i	0.460000i
$867$	−92591.6	−3.62696
$868$	−2226.05	−0.0870473
$869$	39654.0i	1.54795i
$870$	−12843.1	−0.500483
$871$	0	0
$872$	15485.0	0.601364
$873$	38257.9i	1.48320i
$874$	684.349	0.0264857
$875$	−13751.4	−0.531293
$876$	− 14964.3i	− 0.577164i
$877$	26427.8i	1.01756i	0.860895	+	0.508782i	$0.169904\pi$
$877$	26427.8i	1.01756i	−0.860895	+	0.508782i	$0.830096\pi$
$878$	− 20887.1i	− 0.802852i
$879$	− 19440.4i	− 0.745970i
$880$	3343.36	0.128073
$881$	−766.125	−0.0292979	−0.0146489	−	0.999893i	$-0.504663\pi$
$881$	−766.125	−0.0292979	−0.0146489	+	0.999893i	$0.504663\pi$
$882$	− 11758.4i	− 0.448894i
$883$	−6521.89	−0.248561	−0.124280	−	0.992247i	$-0.539662\pi$
$883$	−6521.89	−0.248561	−0.124280	+	0.992247i	$0.539662\pi$
$884$	0	0
$885$	8432.19	0.320277
$886$	− 17579.5i	− 0.666584i
$887$	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
$889$	3701.97i	0.139663i
$890$	1601.62i	0.0603218i
$891$	29196.0i	1.09776i
$892$	− 9535.45i	− 0.357926i
$893$	−2663.77	−0.0998203
$894$	44915.4	1.68031
$895$	2300.81i	0.0859302i
$896$	1937.22	0.0722299
$897$	0	0
$898$	17272.2	0.641851
$899$	− 6890.36i	− 0.255624i
$900$	−22714.4	−0.841275
$901$	−32236.8	−1.19197
$902$	2844.46i	0.105000i
$903$	− 18467.3i	− 0.680570i
$904$	15190.3i	0.558872i
$905$	1557.80i	0.0572190i
$906$	42396.4	1.55466
$907$	9504.57	0.347954	0.173977	−	0.984750i	$-0.444338\pi$
$907$	9504.57	0.347954	0.173977	+	0.984750i	$0.444338\pi$
$908$	− 6216.30i	− 0.227197i
$909$	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	$0.462663\pi$
$912$	1257.54i	0.0456594i
$913$	−33345.5	−1.20873
$914$	−9753.92	−0.352988
$915$	3128.95i	0.113049i
$916$	11662.4i	0.420673i
$917$	− 24451.7i	− 0.880552i
$918$	54045.2i	1.94309i
$919$	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
$921$	71668.0i	2.56411i
$922$	23607.8	0.843256
$923$	0	0
$924$	29013.0	1.03296
$925$	− 35387.9i	− 1.25789i
$926$	−9595.05	−0.340511
$927$	194.370	0.00688667
$928$	5996.34i	0.212111i
$929$	38061.8i	1.34421i	0.740458	+	0.672103i	$0.234609\pi$
$929$	38061.8i	1.34421i	−0.740458	+	0.672103i	$0.765391\pi$
$930$	2520.22i	0.0888616i
$931$	− 1010.18i	− 0.0355611i
$932$	−16935.7	−0.595223
$933$	2166.80	0.0760319
$934$	− 26377.5i	− 0.924087i
$935$	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	$0.418242\pi$
$938$	12426.2i	0.432547i
$939$	−39399.2	−1.36927
$940$	4645.75	0.161200
$941$	− 32190.2i	− 1.11516i	−0.830122	−	0.557582i	$-0.811729\pi$
$941$	− 32190.2i	− 1.11516i	0.830122	−	0.557582i	$-0.188271\pi$
$942$	34176.1i	1.18208i
$943$	− 1015.44i	− 0.0350660i
$944$	− 3936.93i	− 0.135738i
$945$	12756.8	0.439132
$946$	−14880.8	−0.511433
$947$	− 42231.0i	− 1.44913i	−0.689208	−	0.724563i	$-0.742041\pi$
$947$	− 42231.0i	− 1.44913i	0.689208	−	0.724563i	$-0.257959\pi$
$948$	26012.8	0.891197
$949$	0	0
$950$	−1951.43	−0.0666451
$951$	8250.17i	0.281315i
$952$	15004.2	0.510808
$953$	8168.44	0.277651	0.138826	−	0.990317i	$-0.455667\pi$
$953$	8168.44	0.277651	0.138826	+	0.990317i	$0.455667\pi$
$954$	26844.0i	0.911013i
$955$	2372.22i	0.0803803i
$956$	9490.22i	0.321062i
$957$	89804.7i	3.03341i
$958$	36097.7	1.21739
$959$	−34838.9	−1.17310
$960$	− 2193.22i	− 0.0737353i
$961$	28438.9	0.954613
$962$	0	0
$963$	5872.98	0.196525
$964$	− 1303.05i	− 0.0435358i
$965$	7751.52	0.258581
$966$	−10357.3	−0.344970
$967$	− 52022.1i	− 1.73001i	−0.501766	−	0.865003i	$-0.667316\pi$
$967$	− 52022.1i	− 1.73001i	0.501766	−	0.865003i	$-0.332684\pi$
$968$	− 12730.3i	− 0.422694i
$969$	9739.95i	0.322902i
$970$	− 5732.35i	− 0.189747i
$971$	−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
$973$	42514.0i	1.40076i
$974$	−23002.5	−0.756722
$975$	0	0
$976$	1460.88	0.0479117
$977$	− 18989.9i	− 0.621843i	−0.950436	−	0.310922i	$-0.899362\pi$
$977$	− 18989.9i	− 0.621843i	0.950436	−	0.310922i	$-0.100638\pi$
$978$	39336.5	1.28614
$979$	11199.3	0.365608
$980$	1761.81i	0.0574275i
$981$	− 99871.5i	− 3.25041i
$982$	− 8001.31i	− 0.260012i
$983$	− 47187.4i	− 1.53107i	−0.643392	−	0.765537i	$-0.722474\pi$
$983$	− 47187.4i	− 1.53107i	0.643392	−	0.765537i	$-0.277526\pi$
$984$	1865.95	0.0604514
$985$	2154.99	0.0697094
$986$	46443.0i	1.50005i
$987$	40314.9	1.30014
$988$	0	0
$989$	5312.27	0.170799
$990$	− 21563.1i	− 0.692244i
$991$	−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
$993$	42796.7i	1.36769i
$994$	− 12847.5i	− 0.409958i
$995$	21391.4i	0.681559i
$996$	21874.4i	0.695901i
$997$	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
$999$	70114.0i	2.22053i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.4.c.b.3.2	✓	4	13.7	odd	12
26.4.c.b.9.2	yes	4	13.11	odd	12
208.4.i.d.81.1		4	52.7	even	12
208.4.i.d.113.1		4	52.11	even	12
234.4.h.h.55.1		4	39.20	even	12
234.4.h.h.217.1		4	39.11	even	12
338.4.a.g.1.1		2	13.5	odd	4
338.4.a.h.1.1		2	13.8	odd	4
338.4.b.e.337.1		4	1.1	even	1	trivial
338.4.b.e.337.3		4	13.12	even	2	inner
338.4.c.j.191.2		4	13.2	odd	12
338.4.c.j.315.2		4	13.6	odd	12
338.4.e.f.23.2		8	13.9	even	3
338.4.e.f.23.4		8	13.4	even	6
338.4.e.f.147.2		8	13.10	even	6
338.4.e.f.147.4		8	13.3	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -8.86546 q^{3} -4.00000 q^{4} -3.86546i q^{5} +17.7309i q^{6} -15.1345i q^{7} +8.00000i q^{8} +51.5964 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} -8.86546 q^{3} -4.00000 q^{4} -3.86546i q^{5} +17.7309i q^{6} -15.1345i q^{7} +8.00000i q^{8} +51.5964 q^{9} -7.73092 q^{10} +54.0582i q^{11} +35.4618 q^{12} -30.2691 q^{14} +34.2691i q^{15} +16.0000 q^{16} +123.924 q^{17} -103.193i q^{18} -8.86546i q^{19} +15.4618i q^{20} +134.175i q^{21} +108.116 q^{22} -38.5964 q^{23} -70.9237i q^{24} +110.058 q^{25} -218.058 q^{27} +60.5382i q^{28} -187.386 q^{29} +68.5382 q^{30} +36.7710i q^{31} -32.0000i q^{32} -479.251i q^{33} -247.847i q^{34} -58.5020 q^{35} -206.386 q^{36} -321.538i q^{37} -17.7309 q^{38} +30.9237 q^{40} +26.3092i q^{41} +268.349 q^{42} -137.636 q^{43} -216.233i q^{44} -199.444i q^{45} +77.1928i q^{46} -300.466i q^{47} -141.847 q^{48} +113.946 q^{49} -220.116i q^{50} -1098.64 q^{51} -260.135 q^{53} +436.116i q^{54} +208.960 q^{55} +121.076 q^{56} +78.5964i q^{57} +374.771i q^{58} -246.058i q^{59} -137.076i q^{60} +91.3053 q^{61} +73.5421 q^{62} -780.887i q^{63} -64.0000 q^{64} -958.502 q^{66} -410.524i q^{67} -495.695 q^{68} +342.175 q^{69} +117.004i q^{70} +424.444i q^{71} +412.771i q^{72} -421.982i q^{73} -643.076 q^{74} -975.717 q^{75} +35.4618i q^{76} +818.146 q^{77} +733.542 q^{79} -61.8474i q^{80} +540.084 q^{81} +52.6184 q^{82} +616.843i q^{83} -536.699i q^{84} -479.022i q^{85} +275.273i q^{86} +1661.26 q^{87} -432.466 q^{88} -207.171i q^{89} -398.887 q^{90} +154.386 q^{92} -325.992i q^{93} -600.932 q^{94} -34.2691 q^{95} +283.695i q^{96} +741.484i q^{97} -227.891i q^{98} +2789.21i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} - 16 q^{4} + 118 q^{9}+O(q^{10}) \) 4 * q - 6 * q^3 - 16 * q^4 + 118 * q^9	\( 4 q - 6 q^{3} - 16 q^{4} + 118 q^{9} + 28 q^{10} + 24 q^{12} - 180 q^{14} + 64 q^{16} + 260 q^{17} + 20 q^{22} - 66 q^{23} + 234 q^{25} - 666 q^{27} - 396 q^{29} + 392 q^{30} + 532 q^{35} - 472 q^{36} - 12 q^{38} - 112 q^{40} - 164 q^{42} + 186 q^{43} - 96 q^{48} - 870 q^{49} - 2126 q^{51} - 1070 q^{53} + 1484 q^{55} + 720 q^{56} + 1308 q^{61} - 1120 q^{62} - 256 q^{64} - 3068 q^{66} - 1040 q^{68} + 750 q^{69} - 2808 q^{74} - 1870 q^{75} - 1294 q^{77} + 1520 q^{79} - 668 q^{81} - 968 q^{82} + 3198 q^{87} - 80 q^{88} - 476 q^{90} + 264 q^{92} + 896 q^{94} - 196 q^{95}+O(q^{100}) \) 4 * q - 6 * q^3 - 16 * q^4 + 118 * q^9 + 28 * q^10 + 24 * q^12 - 180 * q^14 + 64 * q^16 + 260 * q^17 + 20 * q^22 - 66 * q^23 + 234 * q^25 - 666 * q^27 - 396 * q^29 + 392 * q^30 + 532 * q^35 - 472 * q^36 - 12 * q^38 - 112 * q^40 - 164 * q^42 + 186 * q^43 - 96 * q^48 - 870 * q^49 - 2126 * q^51 - 1070 * q^53 + 1484 * q^55 + 720 * q^56 + 1308 * q^61 - 1120 * q^62 - 256 * q^64 - 3068 * q^66 - 1040 * q^68 + 750 * q^69 - 2808 * q^74 - 1870 * q^75 - 1294 * q^77 + 1520 * q^79 - 668 * q^81 - 968 * q^82 + 3198 * q^87 - 80 * q^88 - 476 * q^90 + 264 * q^92 + 896 * q^94 - 196 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more