LMFDB - Embedded newform 336.4.h.b.239.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,Mod(239,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.h (of order $2$, degree $1$, 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:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.16
Character	$\chi$	$=$	336.239
Dual form			336.4.h.b.239.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.99829 + 4.79655i) q^{3} +16.7287i q^{5} +7.00000i q^{7} +(-19.0137 + 19.1697i) q^{9} +O(q^{10})$	\(q+(1.99829 + 4.79655i) q^{3} +16.7287i q^{5} +7.00000i q^{7} +(-19.0137 + 19.1697i) q^{9} +60.9218 q^{11} +73.9074 q^{13} +(-80.2398 + 33.4286i) q^{15} -16.3004i q^{17} +153.871i q^{19} +(-33.5758 + 13.9880i) q^{21} -111.882 q^{23} -154.848 q^{25} +(-129.943 - 52.8935i) q^{27} -155.121i q^{29} -53.7611i q^{31} +(121.739 + 292.214i) q^{33} -117.101 q^{35} -63.0884 q^{37} +(147.688 + 354.500i) q^{39} -458.858i q^{41} +95.5852i q^{43} +(-320.684 - 318.074i) q^{45} -70.5132 q^{47} -49.0000 q^{49} +(78.1857 - 32.5729i) q^{51} -417.021i q^{53} +1019.14i q^{55} +(-738.050 + 307.478i) q^{57} +328.024 q^{59} -637.654 q^{61} +(-134.188 - 133.096i) q^{63} +1236.37i q^{65} +201.474i q^{67} +(-223.572 - 536.647i) q^{69} -394.426 q^{71} +977.889 q^{73} +(-309.430 - 742.735i) q^{75} +426.453i q^{77} +514.984i q^{79} +(-5.95806 - 728.976i) q^{81} +1133.84 q^{83} +272.684 q^{85} +(744.045 - 309.976i) q^{87} -1035.66i q^{89} +517.352i q^{91} +(257.867 - 107.430i) q^{93} -2574.06 q^{95} +995.638 q^{97} +(-1158.35 + 1167.86i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 136 q^{9}+O(q^{10}) $ 24 * q - 136 * q^9	$ 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) $ 24 * q - 136 * q^9 + 96 * q^13 - 112 * q^21 + 168 * q^25 - 320 * q^33 - 864 * q^37 + 592 * q^45 - 1176 * q^49 - 1120 * q^57 - 2304 * q^61 + 1168 * q^69 + 3312 * q^73 - 968 * q^81 - 5808 * q^85 + 1776 * q^93 + 6480 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.99829	+	4.79655i	0.384570	+	0.923096i
$3$	1.99829	+	4.79655i	0.384570	+	0.923096i
$4$		0			0
$5$			16.7287i			1.49626i	0.663554	+	0.748128i	$0.269047\pi$
$5$			16.7287i			1.49626i	−0.663554	+	0.748128i	$0.730953\pi$
$6$		0			0
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$		0			0
$9$	−19.0137	+	19.1697i	−0.704211	+	0.709990i
$10$		0			0
$11$	60.9218			1.66987			0.834937	−	0.550345i	$-0.185504\pi$
$11$	60.9218			1.66987			0.834937	+	0.550345i	$0.185504\pi$
$12$		0			0
$13$	73.9074			1.57679			0.788393	−	0.615171i	$-0.210913\pi$
$13$	73.9074			1.57679			0.788393	+	0.615171i	$0.210913\pi$
$14$		0			0
$15$	−80.2398	+	33.4286i	−1.38119	+	0.575416i
$16$		0			0
$17$		−	16.3004i		−	0.232555i	−0.993217	−	0.116277i	$-0.962904\pi$
$17$		−	16.3004i		−	0.232555i	0.993217	−	0.116277i	$-0.0370961\pi$
$18$		0			0
$19$			153.871i			1.85792i	0.370182	+	0.928959i	$0.379295\pi$
$19$			153.871i			1.85792i	−0.370182	+	0.928959i	$0.620705\pi$
$20$		0			0
$21$	−33.5758	+	13.9880i	−0.348897	+	0.145354i
$22$		0			0
$23$	−111.882			−1.01430			−0.507152	−	0.861856i	$-0.669302\pi$
$23$	−111.882			−1.01430			−0.507152	+	0.861856i	$0.669302\pi$
$24$		0			0
$25$	−154.848			−1.23878
$26$		0			0
$27$	−129.943	−	52.8935i	−0.926208	−	0.377013i
$28$		0			0
$29$		−	155.121i		−	0.993284i	−0.867955	−	0.496642i	$-0.834566\pi$
$29$		−	155.121i		−	0.993284i	0.867955	−	0.496642i	$-0.165434\pi$
$30$		0			0
$31$		−	53.7611i		−	0.311477i	−0.987798	−	0.155738i	$-0.950224\pi$
$31$		−	53.7611i		−	0.311477i	0.987798	−	0.155738i	$-0.0497756\pi$
$32$		0			0
$33$	121.739	+	292.214i	0.642184	+	1.54145i
$34$		0			0
$35$	−117.101			−0.565532
$36$		0			0
$37$	−63.0884			−0.280315			−0.140158	−	0.990129i	$-0.544761\pi$
$37$	−63.0884			−0.280315			−0.140158	+	0.990129i	$0.544761\pi$
$38$		0			0
$39$	147.688	+	354.500i	0.606385	+	1.45553i
$40$		0			0
$41$		−	458.858i		−	1.74784i	−0.486067	−	0.873921i	$-0.661569\pi$
$41$		−	458.858i		−	1.74784i	0.486067	−	0.873921i	$-0.338431\pi$
$42$		0			0
$43$			95.5852i			0.338991i	0.985531	+	0.169495i	$0.0542138\pi$
$43$			95.5852i			0.338991i	−0.985531	+	0.169495i	$0.945786\pi$
$44$		0			0
$45$	−320.684	−	318.074i	−1.06233	−	1.05368i
$46$		0			0
$47$	−70.5132			−0.218839			−0.109419	−	0.993996i	$-0.534899\pi$
$47$	−70.5132			−0.218839			−0.109419	+	0.993996i	$0.534899\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	78.1857	−	32.5729i	0.214670	−	0.0894336i
$52$		0			0
$53$		−	417.021i		−	1.08080i	−0.841410	−	0.540398i	$-0.818274\pi$
$53$		−	417.021i		−	1.08080i	0.841410	−	0.540398i	$-0.181726\pi$
$54$		0			0
$55$			1019.14i			2.49856i
$56$		0			0
$57$	−738.050	+	307.478i	−1.71504	+	0.714500i
$58$		0			0
$59$	328.024			0.723815			0.361907	−	0.932214i	$-0.382126\pi$
$59$	328.024			0.723815			0.361907	+	0.932214i	$0.382126\pi$
$60$		0			0
$61$	−637.654			−1.33841			−0.669207	−	0.743076i	$-0.733366\pi$
$61$	−637.654			−1.33841			−0.669207	+	0.743076i	$0.733366\pi$
$62$		0			0
$63$	−134.188	−	133.096i	−0.268351	−	0.266167i
$64$		0			0
$65$			1236.37i			2.35928i
$66$		0			0
$67$			201.474i			0.367372i	0.982985	+	0.183686i	$0.0588030\pi$
$67$			201.474i			0.367372i	−0.982985	+	0.183686i	$0.941197\pi$
$68$		0			0
$69$	−223.572	−	536.647i	−0.390071	−	0.936300i
$70$		0			0
$71$	−394.426			−0.659293			−0.329646	−	0.944105i	$-0.606929\pi$
$71$	−394.426			−0.659293			−0.329646	+	0.944105i	$0.606929\pi$
$72$		0			0
$73$	977.889			1.56785			0.783926	−	0.620854i	$-0.213214\pi$
$73$	977.889			1.56785			0.783926	+	0.620854i	$0.213214\pi$
$74$		0			0
$75$	−309.430	−	742.735i	−0.476399	−	1.14352i
$76$		0			0
$77$			426.453i			0.631153i
$78$		0			0
$79$			514.984i			0.733420i	0.930335	+	0.366710i	$0.119516\pi$
$79$			514.984i			0.733420i	−0.930335	+	0.366710i	$0.880484\pi$
$80$		0			0
$81$	−5.95806	−	728.976i	−0.00817293	−	0.999967i
$82$		0			0
$83$	1133.84			1.49946			0.749728	−	0.661746i	$-0.230184\pi$
$83$	1133.84			1.49946			0.749728	+	0.661746i	$0.230184\pi$
$84$		0			0
$85$	272.684			0.347961
$86$		0			0
$87$	744.045	−	309.976i	0.916897	−	0.381988i
$88$		0			0
$89$		−	1035.66i		−	1.23348i	−0.787167	−	0.616740i	$-0.788453\pi$
$89$		−	1035.66i		−	1.23348i	0.787167	−	0.616740i	$-0.211547\pi$
$90$		0			0
$91$			517.352i			0.595969i
$92$		0			0
$93$	257.867	−	107.430i	0.287523	−	0.119785i
$94$		0			0
$95$	−2574.06			−2.77992
$96$		0			0
$97$	995.638			1.04218			0.521092	−	0.853501i	$-0.325525\pi$
$97$	995.638			1.04218			0.521092	+	0.853501i	$0.325525\pi$
$98$		0			0
$99$	−1158.35	+	1167.86i	−1.17594	+	1.18560i
$100$		0			0
$101$			1457.31i			1.43572i	0.696190	+	0.717858i	$0.254877\pi$
$101$			1457.31i			1.43572i	−0.696190	+	0.717858i	$0.745123\pi$
$102$		0			0
$103$		−	359.018i		−	0.343448i	−0.985145	−	0.171724i	$-0.945066\pi$
$103$		−	359.018i		−	0.343448i	0.985145	−	0.171724i	$-0.0549337\pi$
$104$		0			0
$105$	−234.000	−	561.678i	−0.217487	−	0.522040i
$106$		0			0
$107$	−765.314			−0.691455			−0.345728	−	0.938335i	$-0.612368\pi$
$107$	−765.314			−0.691455			−0.345728	+	0.938335i	$0.612368\pi$
$108$		0			0
$109$	736.101			0.646842			0.323421	−	0.946255i	$-0.395167\pi$
$109$	736.101			0.646842			0.323421	+	0.946255i	$0.395167\pi$
$110$		0			0
$111$	−126.069	−	302.606i	−0.107801	−	0.258758i
$112$		0			0
$113$			1228.45i			1.02268i	0.859380	+	0.511338i	$0.170850\pi$
$113$			1228.45i			1.02268i	−0.859380	+	0.511338i	$0.829150\pi$
$114$		0			0
$115$		−	1871.63i		−	1.51766i
$116$		0			0
$117$	−1405.25	+	1416.79i	−1.11039	+	1.11950i
$118$		0			0
$119$	114.103			0.0878974
$120$		0			0
$121$	2380.47			1.78848
$122$		0			0
$123$	2200.93	−	916.929i	1.61343	−	0.672168i
$124$		0			0
$125$		−	499.315i		−	0.357281i
$126$		0			0
$127$			127.914i			0.0893745i	0.999001	+	0.0446872i	$0.0142291\pi$
$127$			127.914i			0.0893745i	−0.999001	+	0.0446872i	$0.985771\pi$
$128$		0			0
$129$	−458.479	+	191.007i	−0.312921	+	0.130366i
$130$		0			0
$131$	1142.47			0.761968			0.380984	−	0.924582i	$-0.375585\pi$
$131$	1142.47			0.761968			0.380984	+	0.924582i	$0.375585\pi$
$132$		0			0
$133$	−1077.10			−0.702227
$134$		0			0
$135$	884.837	−	2173.78i	0.564108	−	1.38584i
$136$		0			0
$137$		−	660.795i		−	0.412084i	−0.978543	−	0.206042i	$-0.933942\pi$
$137$		−	660.795i		−	0.412084i	0.978543	−	0.206042i	$-0.0660583\pi$
$138$		0			0
$139$		−	502.841i		−	0.306837i	−0.988161	−	0.153419i	$-0.950972\pi$
$139$		−	502.841i		−	0.306837i	0.988161	−	0.153419i	$-0.0490283\pi$
$140$		0			0
$141$	−140.906	−	338.220i	−0.0841588	−	0.202009i
$142$		0			0
$143$	4502.57			2.63304
$144$		0			0
$145$	2594.97			1.48621
$146$		0			0
$147$	−97.9160	−	235.031i	−0.0549386	−	0.131871i
$148$		0			0
$149$		−	470.137i		−	0.258491i	−0.991613	−	0.129245i	$-0.958745\pi$
$149$		−	470.137i		−	0.258491i	0.991613	−	0.129245i	$-0.0412555\pi$
$150$		0			0
$151$			833.149i			0.449011i	0.974473	+	0.224506i	$0.0720767\pi$
$151$			833.149i			0.449011i	−0.974473	+	0.224506i	$0.927923\pi$
$152$		0			0
$153$	312.475	+	309.931i	0.165112	+	0.163768i
$154$		0			0
$155$	899.350			0.466049
$156$		0			0
$157$	−1091.24			−0.554715			−0.277358	−	0.960767i	$-0.589459\pi$
$157$	−1091.24			−0.554715			−0.277358	+	0.960767i	$0.589459\pi$
$158$		0			0
$159$	2000.26	−	833.326i	0.997678	−	0.415642i
$160$		0			0
$161$		−	783.174i		−	0.383371i
$162$		0			0
$163$		−	26.3933i		−	0.0126827i	−0.999980	−	0.00634135i	$-0.997981\pi$
$163$		−	26.3933i		−	0.0126827i	0.999980	−	0.00634135i	$-0.00201853\pi$
$164$		0			0
$165$	−4888.35	+	2036.53i	−2.30641	+	0.960872i
$166$		0			0
$167$	−2067.92			−0.958206			−0.479103	−	0.877759i	$-0.659038\pi$
$167$	−2067.92			−0.958206			−0.479103	+	0.877759i	$0.659038\pi$
$168$		0			0
$169$	3265.31			1.48626
$170$		0			0
$171$	−2949.67	−	2925.66i	−1.31910	−	1.30837i
$172$		0			0
$173$		−	2289.25i		−	1.00606i	−0.864269	−	0.503030i	$-0.832219\pi$
$173$		−	2289.25i		−	1.00606i	0.864269	−	0.503030i	$-0.167781\pi$
$174$		0			0
$175$		−	1083.94i		−	0.468216i
$176$		0			0
$177$	655.485	+	1573.38i	0.278358	+	0.668150i
$178$		0			0
$179$	108.183			0.0451731			0.0225865	−	0.999745i	$-0.492810\pi$
$179$	108.183			0.0451731			0.0225865	+	0.999745i	$0.492810\pi$
$180$		0			0
$181$	752.631			0.309075			0.154538	−	0.987987i	$-0.450611\pi$
$181$	752.631			0.309075			0.154538	+	0.987987i	$0.450611\pi$
$182$		0			0
$183$	−1274.22	−	3058.54i	−0.514714	−	1.23548i
$184$		0			0
$185$		−	1055.38i		−	0.419423i
$186$		0			0
$187$		−	993.051i		−	0.388337i
$188$		0			0
$189$	370.254	−	909.604i	0.142498	−	0.350074i
$190$		0			0
$191$	2279.55			0.863572			0.431786	−	0.901976i	$-0.357884\pi$
$191$	2279.55			0.863572			0.431786	+	0.901976i	$0.357884\pi$
$192$		0			0
$193$	−766.149			−0.285744			−0.142872	−	0.989741i	$-0.545634\pi$
$193$	−766.149			−0.285744			−0.142872	+	0.989741i	$0.545634\pi$
$194$		0			0
$195$	−5930.31	+	2470.62i	−2.17784	+	0.907308i
$196$		0			0
$197$		−	3890.03i		−	1.40687i	−0.710760	−	0.703435i	$-0.751649\pi$
$197$		−	3890.03i		−	1.40687i	0.710760	−	0.703435i	$-0.248351\pi$
$198$		0			0
$199$		−	2577.77i		−	0.918257i	−0.888370	−	0.459129i	$-0.848162\pi$
$199$		−	2577.77i		−	0.918257i	0.888370	−	0.459129i	$-0.151838\pi$
$200$		0			0
$201$	−966.378	+	402.602i	−0.339119	+	0.141280i
$202$		0			0
$203$	1085.85			0.375426
$204$		0			0
$205$	7676.07			2.61522
$206$		0			0
$207$	2127.29	−	2144.75i	0.714285	−	0.720147i
$208$		0			0
$209$			9374.11i			3.10249i
$210$		0			0
$211$			4839.87i			1.57910i	0.613685	+	0.789551i	$0.289687\pi$
$211$			4839.87i			1.57910i	−0.613685	+	0.789551i	$0.710313\pi$
$212$		0			0
$213$	−788.176	−	1891.88i	−0.253544	−	0.608590i
$214$		0			0
$215$	−1599.01			−0.507217
$216$		0			0
$217$	376.327			0.117727
$218$		0			0
$219$	1954.10	+	4690.49i	0.602950	+	1.44728i
$220$		0			0
$221$		−	1204.72i		−	0.366689i
$222$		0			0
$223$		−	1258.94i		−	0.378049i	−0.981972	−	0.189024i	$-0.939467\pi$
$223$		−	1258.94i		−	0.378049i	0.981972	−	0.189024i	$-0.0605325\pi$
$224$		0			0
$225$	2944.23	−	2968.39i	0.872365	−	0.879524i
$226$		0			0
$227$	−3028.13			−0.885393			−0.442696	−	0.896672i	$-0.645978\pi$
$227$	−3028.13			−0.885393			−0.442696	+	0.896672i	$0.645978\pi$
$228$		0			0
$229$	−4162.54			−1.20117			−0.600585	−	0.799561i	$-0.705066\pi$
$229$	−4162.54			−1.20117			−0.600585	+	0.799561i	$0.705066\pi$
$230$		0			0
$231$	−2045.50	+	852.175i	−0.582615	+	0.242723i
$232$		0			0
$233$		−	631.397i		−	0.177529i	−0.996053	−	0.0887644i	$-0.971708\pi$
$233$		−	631.397i		−	0.177529i	0.996053	−	0.0887644i	$-0.0282918\pi$
$234$		0			0
$235$		−	1179.59i		−	0.327439i
$236$		0			0
$237$	−2470.14	+	1029.08i	−0.677017	+	0.282052i
$238$		0			0
$239$	1634.45			0.442358			0.221179	−	0.975233i	$-0.429010\pi$
$239$	1634.45			0.442358			0.221179	+	0.975233i	$0.429010\pi$
$240$		0			0
$241$	−3537.63			−0.945555			−0.472777	−	0.881182i	$-0.656748\pi$
$241$	−3537.63			−0.945555			−0.472777	+	0.881182i	$0.656748\pi$
$242$		0			0
$243$	3484.66	−	1485.28i	0.919922	−	0.392102i
$244$		0			0
$245$		−	819.704i		−	0.213751i
$246$		0			0
$247$			11372.2i			2.92954i
$248$		0			0
$249$	2265.73	+	5438.50i	0.576646	+	1.38414i
$250$		0			0
$251$	3866.21			0.972244			0.486122	−	0.873891i	$-0.338411\pi$
$251$	3866.21			0.972244			0.486122	+	0.873891i	$0.338411\pi$
$252$		0			0
$253$	−6816.05			−1.69376
$254$		0			0
$255$	544.901	+	1307.94i	0.133816	+	0.321202i
$256$		0			0
$257$			2615.37i			0.634794i	0.948293	+	0.317397i	$0.102809\pi$
$257$			2615.37i			0.634794i	−0.948293	+	0.317397i	$0.897191\pi$
$258$		0			0
$259$		−	441.619i		−	0.105949i
$260$		0			0
$261$	2973.63	+	2949.43i	0.705223	+	0.699482i
$262$		0			0
$263$	−7004.30			−1.64222			−0.821110	−	0.570770i	$-0.806645\pi$
$263$	−7004.30			−1.64222			−0.821110	+	0.570770i	$0.806645\pi$
$264$		0			0
$265$	6976.19			1.61715
$266$		0			0
$267$	4967.59	−	2069.54i	1.13862	−	0.474360i
$268$		0			0
$269$			404.298i			0.0916374i	0.998950	+	0.0458187i	$0.0145897\pi$
$269$			404.298i			0.0916374i	−0.998950	+	0.0458187i	$0.985410\pi$
$270$		0			0
$271$			758.555i			0.170033i	0.996380	+	0.0850165i	$0.0270943\pi$
$271$			758.555i			0.170033i	−0.996380	+	0.0850165i	$0.972906\pi$
$272$		0			0
$273$	−2481.50	+	1033.82i	−0.550137	+	0.229192i
$274$		0			0
$275$	−9433.62			−2.06861
$276$		0			0
$277$	−1084.77			−0.235298			−0.117649	−	0.993055i	$-0.537536\pi$
$277$	−1084.77			−0.235298			−0.117649	+	0.993055i	$0.537536\pi$
$278$		0			0
$279$	1030.59	+	1022.20i	0.221145	+	0.219345i
$280$		0			0
$281$			4094.10i			0.869159i	0.900633	+	0.434580i	$0.143103\pi$
$281$			4094.10i			0.869159i	−0.900633	+	0.434580i	$0.856897\pi$
$282$		0			0
$283$			5030.95i			1.05675i	0.849013	+	0.528373i	$0.177198\pi$
$283$			5030.95i			1.05675i	−0.849013	+	0.528373i	$0.822802\pi$
$284$		0			0
$285$	−5143.70	−	12346.6i	−1.06908	−	2.56613i
$286$		0			0
$287$	3212.00			0.660622
$288$		0			0
$289$	4647.30			0.945918
$290$		0			0
$291$	1989.57	+	4775.62i	0.400793	+	0.962035i
$292$		0			0
$293$			2585.51i			0.515519i	0.966209	+	0.257759i	$0.0829842\pi$
$293$			2585.51i			0.515519i	−0.966209	+	0.257759i	$0.917016\pi$
$294$		0			0
$295$			5487.40i			1.08301i
$296$		0			0
$297$	−7916.39	−	3222.37i	−1.54665	−	0.629565i
$298$		0			0
$299$	−8268.91			−1.59934
$300$		0			0
$301$	−669.097			−0.128127
$302$		0			0
$303$	−6990.03	+	2912.11i	−1.32530	+	0.552134i
$304$		0			0
$305$		−	10667.1i		−	2.00261i
$306$		0			0
$307$			766.680i			0.142530i	0.997457	+	0.0712651i	$0.0227036\pi$
$307$			766.680i			0.142530i	−0.997457	+	0.0712651i	$0.977296\pi$
$308$		0			0
$309$	1722.05	−	717.421i	0.317035	−	0.132080i
$310$		0			0
$311$	5681.41			1.03589			0.517947	−	0.855412i	$-0.326696\pi$
$311$	5681.41			1.03589			0.517947	+	0.855412i	$0.326696\pi$
$312$		0			0
$313$	−9636.05			−1.74013			−0.870067	−	0.492934i	$-0.835924\pi$
$313$	−9636.05			−1.74013			−0.870067	+	0.492934i	$0.835924\pi$
$314$		0			0
$315$	2226.52	−	2244.79i	0.398254	−	0.401522i
$316$		0			0
$317$			1585.82i			0.280972i	0.990083	+	0.140486i	$0.0448666\pi$
$317$			1585.82i			0.280972i	−0.990083	+	0.140486i	$0.955133\pi$
$318$		0			0
$319$		−	9450.26i		−	1.65866i
$320$		0			0
$321$	−1529.32	−	3670.86i	−0.265913	−	0.638279i
$322$		0			0
$323$	2508.16			0.432068
$324$		0			0
$325$	−11444.4			−1.95330
$326$		0			0
$327$	1470.94	+	3530.74i	0.248756	+	0.597097i
$328$		0			0
$329$		−	493.593i		−	0.0827132i
$330$		0			0
$331$		−	3330.99i		−	0.553135i	−0.960995	−	0.276567i	$-0.910803\pi$
$331$		−	3330.99i		−	0.553135i	0.960995	−	0.276567i	$-0.0891969\pi$
$332$		0			0
$333$	1199.54	−	1209.39i	0.197401	−	0.199021i
$334$		0			0
$335$	−3370.38			−0.549683
$336$		0			0
$337$	10234.2			1.65428			0.827140	−	0.561996i	$-0.189967\pi$
$337$	10234.2			1.65428			0.827140	+	0.561996i	$0.189967\pi$
$338$		0			0
$339$	−5892.30	+	2454.79i	−0.944028	+	0.393291i
$340$		0			0
$341$		−	3275.22i		−	0.520127i
$342$		0			0
$343$		−	343.000i		−	0.0539949i
$344$		0			0
$345$	8977.38	−	3740.06i	1.40095	−	0.583647i
$346$		0			0
$347$	4657.40			0.720525			0.360263	−	0.932851i	$-0.382687\pi$
$347$	4657.40			0.720525			0.360263	+	0.932851i	$0.382687\pi$
$348$		0			0
$349$	3280.13			0.503098			0.251549	−	0.967845i	$-0.419060\pi$
$349$	3280.13			0.503098			0.251549	+	0.967845i	$0.419060\pi$
$350$		0			0
$351$	−9603.78	−	3909.22i	−1.46043	−	0.594469i
$352$		0			0
$353$		−	6977.11i		−	1.05200i	−0.850486	−	0.525998i	$-0.823692\pi$
$353$		−	6977.11i		−	1.05200i	0.850486	−	0.525998i	$-0.176308\pi$
$354$		0			0
$355$		−	6598.22i		−	0.986471i
$356$		0			0
$357$	228.010	+	547.300i	0.0338027	+	0.0811377i
$358$		0			0
$359$	457.102			0.0672004			0.0336002	−	0.999435i	$-0.489303\pi$
$359$	457.102			0.0672004			0.0336002	+	0.999435i	$0.489303\pi$
$360$		0			0
$361$	−16817.3			−2.45186
$362$		0			0
$363$	4756.86	+	11418.0i	0.687797	+	1.65094i
$364$		0			0
$365$			16358.8i			2.34591i
$366$		0			0
$367$			6117.95i			0.870175i	0.900388	+	0.435088i	$0.143283\pi$
$367$			6117.95i			0.870175i	−0.900388	+	0.435088i	$0.856717\pi$
$368$		0			0
$369$	8796.19	+	8724.59i	1.24095	+	1.23085i
$370$		0			0
$371$	2919.14			0.408502
$372$		0			0
$373$	−2466.70			−0.342415			−0.171208	−	0.985235i	$-0.554767\pi$
$373$	−2466.70			−0.342415			−0.171208	+	0.985235i	$0.554767\pi$
$374$		0			0
$375$	2394.99	−	997.775i	0.329804	−	0.137400i
$376$		0			0
$377$		−	11464.6i		−	1.56620i
$378$		0			0
$379$		−	6021.87i		−	0.816155i	−0.912947	−	0.408078i	$-0.866199\pi$
$379$		−	6021.87i		−	0.816155i	0.912947	−	0.408078i	$-0.133801\pi$
$380$		0			0
$381$	−613.547	+	255.609i	−0.0825012	+	0.0343708i
$382$		0			0
$383$	4212.17			0.561963			0.280982	−	0.959713i	$-0.409340\pi$
$383$	4212.17			0.561963			0.280982	+	0.959713i	$0.409340\pi$
$384$		0			0
$385$	−7133.98			−0.944367
$386$		0			0
$387$	−1832.34	−	1817.43i	−0.240680	−	0.238721i
$388$		0			0
$389$			2689.11i			0.350497i	0.984524	+	0.175248i	$0.0560729\pi$
$389$			2689.11i			0.350497i	−0.984524	+	0.175248i	$0.943927\pi$
$390$		0			0
$391$			1823.72i			0.235881i
$392$		0			0
$393$	2282.98	+	5479.90i	0.293030	+	0.703369i
$394$		0			0
$395$	−8614.98			−1.09738
$396$		0			0
$397$	−8370.78			−1.05823			−0.529115	−	0.848550i	$-0.677476\pi$
$397$	−8370.78			−1.05823			−0.529115	+	0.848550i	$0.677476\pi$
$398$		0			0
$399$	−2152.35	−	5166.35i	−0.270056	−	0.648223i
$400$		0			0
$401$			4771.81i			0.594247i	0.954839	+	0.297123i	$0.0960272\pi$
$401$			4771.81i			0.594247i	−0.954839	+	0.297123i	$0.903973\pi$
$402$		0			0
$403$		−	3973.34i		−	0.491132i
$404$		0			0
$405$	12194.8	−	99.6704i	1.49621	−	0.0122288i
$406$		0			0
$407$	−3843.46			−0.468091
$408$		0			0
$409$	−3505.84			−0.423846			−0.211923	−	0.977286i	$-0.567973\pi$
$409$	−3505.84			−0.423846			−0.211923	+	0.977286i	$0.567973\pi$
$410$		0			0
$411$	3169.53	−	1320.46i	0.380393	−	0.158475i
$412$		0			0
$413$			2296.17i			0.273576i
$414$		0			0
$415$			18967.6i			2.24357i
$416$		0			0
$417$	2411.90	−	1004.82i	0.283240	−	0.118001i
$418$		0			0
$419$	13302.4			1.55099			0.775496	−	0.631352i	$-0.217500\pi$
$419$	13302.4			1.55099			0.775496	+	0.631352i	$0.217500\pi$
$420$		0			0
$421$	5120.87			0.592816			0.296408	−	0.955061i	$-0.404211\pi$
$421$	5120.87			0.592816			0.296408	+	0.955061i	$0.404211\pi$
$422$		0			0
$423$	1340.72	−	1351.72i	0.154109	−	0.155373i
$424$		0			0
$425$			2524.08i			0.288085i
$426$		0			0
$427$		−	4463.58i		−	0.505873i
$428$		0			0
$429$	8997.43	+	21596.8i	1.01259	+	2.43054i
$430$		0			0
$431$	17758.5			1.98468			0.992339	−	0.123544i	$-0.0394259\pi$
$431$	17758.5			1.98468			0.992339	+	0.123544i	$0.0394259\pi$
$432$		0			0
$433$	17278.3			1.91765			0.958826	−	0.283994i	$-0.0916595\pi$
$433$	17278.3			1.91765			0.958826	+	0.283994i	$0.0916595\pi$
$434$		0			0
$435$	5185.48	+	12446.9i	0.571552	+	1.37191i
$436$		0			0
$437$		−	17215.4i		−	1.88449i
$438$		0			0
$439$			4721.22i			0.513284i	0.966507	+	0.256642i	$0.0826162\pi$
$439$			4721.22i			0.513284i	−0.966507	+	0.256642i	$0.917384\pi$
$440$		0			0
$441$	931.672	−	939.317i	0.100602	−	0.101427i
$442$		0			0
$443$	−2733.59			−0.293176			−0.146588	−	0.989198i	$-0.546829\pi$
$443$	−2733.59			−0.293176			−0.146588	+	0.989198i	$0.546829\pi$
$444$		0			0
$445$	17325.2			1.84560
$446$		0			0
$447$	2255.03	−	939.467i	0.238611	−	0.0994078i
$448$		0			0
$449$			2204.08i			0.231663i	0.993269	+	0.115832i	$0.0369533\pi$
$449$			2204.08i			0.231663i	−0.993269	+	0.115832i	$0.963047\pi$
$450$		0			0
$451$		−	27954.5i		−	2.91868i
$452$		0			0
$453$	−3996.24	+	1664.87i	−0.414480	+	0.172676i
$454$		0			0
$455$	−8654.60			−0.891723
$456$		0			0
$457$	−1401.18			−0.143423			−0.0717116	−	0.997425i	$-0.522846\pi$
$457$	−1401.18			−0.143423			−0.0717116	+	0.997425i	$0.522846\pi$
$458$		0			0
$459$	−862.185	+	2118.13i	−0.0876762	+	0.215394i
$460$		0			0
$461$		−	5775.00i		−	0.583446i	−0.956503	−	0.291723i	$-0.905772\pi$
$461$		−	5775.00i		−	0.583446i	0.956503	−	0.291723i	$-0.0942285\pi$
$462$		0			0
$463$		−	15866.2i		−	1.59258i	−0.604916	−	0.796289i	$-0.706793\pi$
$463$		−	15866.2i		−	1.59258i	0.604916	−	0.796289i	$-0.293207\pi$
$464$		0			0
$465$	1797.16	+	4313.77i	0.179229	+	0.430208i
$466$		0			0
$467$	4802.77			0.475900			0.237950	−	0.971277i	$-0.423524\pi$
$467$	4802.77			0.475900			0.237950	+	0.971277i	$0.423524\pi$
$468$		0			0
$469$	−1410.32			−0.138854
$470$		0			0
$471$	−2180.61	−	5234.17i	−0.213327	−	0.512055i
$472$		0			0
$473$			5823.23i			0.566072i
$474$		0			0
$475$		−	23826.6i		−	2.30156i
$476$		0			0
$477$	7994.18	+	7929.11i	0.767355	+	0.761109i
$478$		0			0
$479$	−5778.37			−0.551191			−0.275595	−	0.961274i	$-0.588875\pi$
$479$	−5778.37			−0.551191			−0.275595	+	0.961274i	$0.588875\pi$
$480$		0			0
$481$	−4662.70			−0.441997
$482$		0			0
$483$	3756.53	−	1565.01i	0.353888	−	0.147433i
$484$		0			0
$485$			16655.7i			1.55937i
$486$		0			0
$487$		−	8916.11i		−	0.829625i	−0.909907	−	0.414813i	$-0.863847\pi$
$487$		−	8916.11i		−	0.829625i	0.909907	−	0.414813i	$-0.136153\pi$
$488$		0			0
$489$	126.597	−	52.7413i	0.0117074	−	0.00487739i
$490$		0			0
$491$	−9826.22			−0.903159			−0.451579	−	0.892231i	$-0.649139\pi$
$491$	−9826.22			−0.903159			−0.451579	+	0.892231i	$0.649139\pi$
$492$		0			0
$493$	−2528.54			−0.230993
$494$		0			0
$495$	−19536.7	−	19377.6i	−1.77395	−	1.75951i
$496$		0			0
$497$		−	2760.98i		−	0.249189i
$498$		0			0
$499$			14608.3i			1.31053i	0.755398	+	0.655266i	$0.227443\pi$
$499$			14608.3i			1.31053i	−0.755398	+	0.655266i	$0.772557\pi$
$500$		0			0
$501$	−4132.29	−	9918.87i	−0.368497	−	0.884515i
$502$		0			0
$503$	−10453.8			−0.926662			−0.463331	−	0.886185i	$-0.653346\pi$
$503$	−10453.8			−0.926662			−0.463331	+	0.886185i	$0.653346\pi$
$504$		0			0
$505$	−24378.8			−2.14820
$506$		0			0
$507$	6525.02	+	15662.2i	0.571570	+	1.37196i
$508$		0			0
$509$		−	9286.13i		−	0.808645i	−0.914616	−	0.404323i	$-0.867507\pi$
$509$		−	9286.13i		−	0.808645i	0.914616	−	0.404323i	$-0.132493\pi$
$510$		0			0
$511$			6845.22i			0.592593i
$512$		0			0
$513$	8138.78	−	19994.5i	0.700460	−	1.72082i
$514$		0			0
$515$	6005.89			0.513886
$516$		0			0
$517$	−4295.80			−0.365433
$518$		0			0
$519$	10980.5	−	4574.57i	0.928689	−	0.386901i
$520$		0			0
$521$			2764.21i			0.232442i	0.993223	+	0.116221i	$0.0370781\pi$
$521$			2764.21i			0.232442i	−0.993223	+	0.116221i	$0.962922\pi$
$522$		0			0
$523$		−	11042.1i		−	0.923209i	−0.887086	−	0.461605i	$-0.847274\pi$
$523$		−	11042.1i		−	0.923209i	0.887086	−	0.461605i	$-0.152726\pi$
$524$		0			0
$525$	5199.15	−	2166.01i	0.432208	−	0.180062i
$526$		0			0
$527$	−876.327			−0.0724353
$528$		0			0
$529$	350.576			0.0288137
$530$		0			0
$531$	−6236.95	+	6288.13i	−0.509718	+	0.513902i
$532$		0			0
$533$		−	33913.0i		−	2.75598i
$534$		0			0
$535$		−	12802.7i		−	1.03459i
$536$		0			0
$537$	216.181	+	518.905i	0.0173722	+	0.0416991i
$538$		0			0
$539$	−2985.17			−0.238554
$540$		0			0
$541$	23666.7			1.88080			0.940398	−	0.340075i	$-0.110452\pi$
$541$	23666.7			1.88080			0.940398	+	0.340075i	$0.110452\pi$
$542$		0			0
$543$	1503.97	+	3610.03i	0.118861	+	0.285306i
$544$		0			0
$545$			12314.0i			0.967841i
$546$		0			0
$547$			6867.33i			0.536793i	0.963309	+	0.268397i	$0.0864938\pi$
$547$			6867.33i			0.536793i	−0.963309	+	0.268397i	$0.913506\pi$
$548$		0			0
$549$	12124.2	−	12223.7i	0.942526	−	0.950261i
$550$		0			0
$551$	23868.6			1.84544
$552$		0			0
$553$	−3604.88			−0.277207
$554$		0			0
$555$	5062.19	−	2108.96i	0.387168	−	0.161298i
$556$		0			0
$557$			17973.1i			1.36722i	0.729845	+	0.683612i	$0.239592\pi$
$557$			17973.1i			1.36722i	−0.729845	+	0.683612i	$0.760408\pi$
$558$		0			0
$559$			7064.46i			0.534516i
$560$		0			0
$561$	4763.21	−	1984.40i	0.358472	−	0.149343i
$562$		0			0
$563$	19148.6			1.43342			0.716711	−	0.697370i	$-0.245647\pi$
$563$	19148.6			1.43342			0.716711	+	0.697370i	$0.245647\pi$
$564$		0			0
$565$	−20550.2			−1.53019
$566$		0			0
$567$	5102.83	−	41.7064i	0.377952	−	0.00308908i
$568$		0			0
$569$			13679.2i			1.00784i	0.863750	+	0.503921i	$0.168110\pi$
$569$			13679.2i			1.00784i	−0.863750	+	0.503921i	$0.831890\pi$
$570$		0			0
$571$		−	10288.3i		−	0.754033i	−0.926207	−	0.377017i	$-0.876950\pi$
$571$		−	10288.3i		−	0.754033i	0.926207	−	0.377017i	$-0.123050\pi$
$572$		0			0
$573$	4555.19	+	10934.0i	0.332104	+	0.797160i
$574$		0			0
$575$	17324.7			1.25650
$576$		0			0
$577$	−13939.4			−1.00573			−0.502864	−	0.864365i	$-0.667720\pi$
$577$	−13939.4			−1.00573			−0.502864	+	0.864365i	$0.667720\pi$
$578$		0			0
$579$	−1530.99	−	3674.87i	−0.109889	−	0.263769i
$580$		0			0
$581$			7936.86i			0.566741i
$582$		0			0
$583$		−	25405.7i		−	1.80479i
$584$		0			0
$585$	−23700.9	−	23508.0i	−1.67506	−	1.66143i
$586$		0			0
$587$	−8490.79			−0.597023			−0.298511	−	0.954406i	$-0.596490\pi$
$587$	−8490.79			−0.597023			−0.298511	+	0.954406i	$0.596490\pi$
$588$		0			0
$589$	8272.27			0.578698
$590$		0			0
$591$	18658.7	−	7773.40i	1.29868	−	0.541040i
$592$		0			0
$593$		−	3335.30i		−	0.230968i	−0.993309	−	0.115484i	$-0.963158\pi$
$593$		−	3335.30i		−	0.230968i	0.993309	−	0.115484i	$-0.0368420\pi$
$594$		0			0
$595$			1908.79i			0.131517i
$596$		0			0
$597$	12364.4	−	5151.12i	0.847639	−	0.353134i
$598$		0			0
$599$	−14950.3			−1.01978			−0.509892	−	0.860238i	$-0.670315\pi$
$599$	−14950.3			−1.01978			−0.509892	+	0.860238i	$0.670315\pi$
$600$		0			0
$601$	−15554.3			−1.05570			−0.527849	−	0.849338i	$-0.677001\pi$
$601$	−15554.3			−1.05570			−0.527849	+	0.849338i	$0.677001\pi$
$602$		0			0
$603$	−3862.20	−	3830.76i	−0.260831	−	0.258707i
$604$		0			0
$605$			39822.0i			2.67603i
$606$		0			0
$607$			23162.9i			1.54885i	0.632663	+	0.774427i	$0.281962\pi$
$607$			23162.9i			1.54885i	−0.632663	+	0.774427i	$0.718038\pi$
$608$		0			0
$609$	2169.83	+	5208.32i	0.144378	+	0.346554i
$610$		0			0
$611$	−5211.45			−0.345062
$612$		0			0
$613$	−517.112			−0.0340717			−0.0170359	−	0.999855i	$-0.505423\pi$
$613$	−517.112			−0.0340717			−0.0170359	+	0.999855i	$0.505423\pi$
$614$		0			0
$615$	15339.0	+	36818.6i	1.00574	+	2.41410i
$616$		0			0
$617$			3318.03i			0.216497i	0.994124	+	0.108249i	$0.0345242\pi$
$617$			3318.03i			0.216497i	−0.994124	+	0.108249i	$0.965476\pi$
$618$		0			0
$619$			6295.88i			0.408809i	0.978886	+	0.204405i	$0.0655258\pi$
$619$			6295.88i			0.408809i	−0.978886	+	0.204405i	$0.934474\pi$
$620$		0			0
$621$	14538.3	+	5917.83i	0.939457	+	0.382406i
$622$		0			0
$623$	7249.61			0.466211
$624$		0			0
$625$	−11003.1			−0.704199
$626$		0			0
$627$	−44963.3	+	18732.1i	−2.86390	+	1.19313i
$628$		0			0
$629$			1028.37i			0.0651886i
$630$		0			0
$631$			16754.4i			1.05702i	0.848926	+	0.528511i	$0.177250\pi$
$631$			16754.4i			1.05702i	−0.848926	+	0.528511i	$0.822750\pi$
$632$		0			0
$633$	−23214.7	+	9671.45i	−1.45766	+	0.607276i
$634$		0			0
$635$	−2139.83			−0.133727
$636$		0			0
$637$	−3621.46			−0.225255
$638$		0			0
$639$	7499.50	−	7561.05i	0.464281	−	0.468091i
$640$		0			0
$641$		−	5871.80i		−	0.361813i	−0.983500	−	0.180907i	$-0.942097\pi$
$641$		−	5871.80i		−	0.361813i	0.983500	−	0.180907i	$-0.0579032\pi$
$642$		0			0
$643$		−	12954.8i		−	0.794539i	−0.917702	−	0.397269i	$-0.869958\pi$
$643$		−	12954.8i		−	0.794539i	0.917702	−	0.397269i	$-0.130042\pi$
$644$		0			0
$645$	−3195.28	−	7669.74i	−0.195061	−	0.468210i
$646$		0			0
$647$	−20113.4			−1.22216			−0.611080	−	0.791569i	$-0.709265\pi$
$647$	−20113.4			−1.22216			−0.611080	+	0.791569i	$0.709265\pi$
$648$		0			0
$649$	19983.8			1.20868
$650$		0			0
$651$	752.010	+	1805.07i	0.0452743	+	0.108673i
$652$		0			0
$653$		−	31517.3i		−	1.88877i	−0.328844	−	0.944384i	$-0.606659\pi$
$653$		−	31517.3i		−	1.88877i	0.328844	−	0.944384i	$-0.393341\pi$
$654$		0			0
$655$			19111.9i			1.14010i
$656$		0			0
$657$	−18593.3	+	18745.9i	−1.10410	+	1.11316i
$658$		0			0
$659$	−19556.1			−1.15599			−0.577994	−	0.816041i	$-0.696164\pi$
$659$	−19556.1			−1.15599			−0.577994	+	0.816041i	$0.696164\pi$
$660$		0			0
$661$	26582.3			1.56419			0.782097	−	0.623157i	$-0.214150\pi$
$661$	26582.3			1.56419			0.782097	+	0.623157i	$0.214150\pi$
$662$		0			0
$663$	5778.50	−	2407.38i	0.338489	−	0.141018i
$664$		0			0
$665$		−	18018.4i		−	1.05071i
$666$		0			0
$667$			17355.2i			1.00749i
$668$		0			0
$669$	6038.57	−	2515.72i	0.348975	−	0.145386i
$670$		0			0
$671$	−38847.1			−2.23498
$672$		0			0
$673$	201.721			0.0115539			0.00577695	−	0.999983i	$-0.498161\pi$
$673$	201.721			0.0115539			0.00577695	+	0.999983i	$0.498161\pi$
$674$		0			0
$675$	20121.5	+	8190.44i	1.14737	+	0.467038i
$676$		0			0
$677$			4804.24i			0.272736i	0.990658	+	0.136368i	$0.0435429\pi$
$677$			4804.24i			0.272736i	−0.990658	+	0.136368i	$0.956457\pi$
$678$		0			0
$679$			6969.47i			0.393908i
$680$		0			0
$681$	−6051.07	−	14524.6i	−0.340496	−	0.817302i
$682$		0			0
$683$	−3043.76			−0.170521			−0.0852607	−	0.996359i	$-0.527172\pi$
$683$	−3043.76			−0.170521			−0.0852607	+	0.996359i	$0.527172\pi$
$684$		0			0
$685$	11054.2			0.616583
$686$		0			0
$687$	−8317.94	−	19965.8i	−0.461935	−	1.10880i
$688$		0			0
$689$		−	30820.9i		−	1.70418i
$690$		0			0
$691$			8025.16i			0.441811i	0.975295	+	0.220906i	$0.0709013\pi$
$691$			8025.16i			0.441811i	−0.975295	+	0.220906i	$0.929099\pi$
$692$		0			0
$693$	−8174.99	−	8108.45i	−0.448113	−	0.444465i
$694$		0			0
$695$	8411.85			0.459108
$696$		0			0
$697$	−7479.57			−0.406469
$698$		0			0
$699$	3028.52	−	1261.71i	0.163876	−	0.0682723i
$700$		0			0
$701$		−	9256.11i		−	0.498714i	−0.968412	−	0.249357i	$-0.919781\pi$
$701$		−	9256.11i		−	0.498714i	0.968412	−	0.249357i	$-0.0802192\pi$
$702$		0			0
$703$		−	9707.47i		−	0.520803i
$704$		0			0
$705$	5657.97	−	2357.16i	0.302257	−	0.125923i
$706$		0			0
$707$	−10201.1			−0.542650
$708$		0			0
$709$	−10786.1			−0.571342			−0.285671	−	0.958328i	$-0.592217\pi$
$709$	−10786.1			−0.571342			−0.285671	+	0.958328i	$0.592217\pi$
$710$		0			0
$711$	−9872.10	−	9791.74i	−0.520721	−	0.516483i
$712$		0			0
$713$			6014.89i			0.315932i
$714$		0			0
$715$			75322.0i			3.93970i
$716$		0			0
$717$	3266.09	+	7839.69i	0.170118	+	0.408338i
$718$		0			0
$719$	−16903.4			−0.876759			−0.438379	−	0.898790i	$-0.644447\pi$
$719$	−16903.4			−0.876759			−0.438379	+	0.898790i	$0.644447\pi$
$720$		0			0
$721$	2513.13			0.129811
$722$		0			0
$723$	−7069.19	−	16968.4i	−0.363632	−	0.872837i
$724$		0			0
$725$			24020.2i			1.23046i
$726$		0			0
$727$		−	17968.4i		−	0.916661i	−0.888782	−	0.458330i	$-0.848448\pi$
$727$		−	17968.4i		−	0.916661i	0.888782	−	0.458330i	$-0.151552\pi$
$728$		0			0
$729$	14087.6	+	13746.3i	0.715722	+	0.698385i
$730$		0			0
$731$	1558.08			0.0788339
$732$		0			0
$733$	26141.6			1.31728			0.658638	−	0.752460i	$-0.271133\pi$
$733$	26141.6			1.31728			0.658638	+	0.752460i	$0.271133\pi$
$734$		0			0
$735$	3931.75	−	1638.00i	0.197313	−	0.0822023i
$736$		0			0
$737$			12274.1i			0.613465i
$738$		0			0
$739$		−	20364.7i		−	1.01370i	−0.862033	−	0.506851i	$-0.830809\pi$
$739$		−	20364.7i		−	1.01370i	0.862033	−	0.506851i	$-0.169191\pi$
$740$		0			0
$741$	−54547.4	+	22724.9i	−2.70425	+	1.12661i
$742$		0			0
$743$	5300.06			0.261696			0.130848	−	0.991402i	$-0.458230\pi$
$743$	5300.06			0.261696			0.130848	+	0.991402i	$0.458230\pi$
$744$		0			0
$745$	7864.75			0.386768
$746$		0			0
$747$	−21558.5	+	21735.4i	−1.05593	+	1.06460i
$748$		0			0
$749$		−	5357.20i		−	0.261346i
$750$		0			0
$751$			38359.6i			1.86386i	0.362634	+	0.931932i	$0.381877\pi$
$751$			38359.6i			1.86386i	−0.362634	+	0.931932i	$0.618123\pi$
$752$		0			0
$753$	7725.80	+	18544.5i	0.373896	+	0.897474i
$754$		0			0
$755$	−13937.5			−0.671836
$756$		0			0
$757$	−29775.4			−1.42960			−0.714799	−	0.699330i	$-0.753482\pi$
$757$	−29775.4			−1.42960			−0.714799	+	0.699330i	$0.753482\pi$
$758$		0			0
$759$	−13620.4	−	32693.5i	−0.651370	−	1.56350i
$760$		0			0
$761$		−	24451.7i		−	1.16475i	−0.812921	−	0.582373i	$-0.802124\pi$
$761$		−	24451.7i		−	1.16475i	0.812921	−	0.582373i	$-0.197876\pi$
$762$		0			0
$763$			5152.71i			0.244483i
$764$		0			0
$765$	−5184.73	+	5227.28i	−0.245038	+	0.247049i
$766$		0			0
$767$	24243.4			1.14130
$768$		0			0
$769$	20070.8			0.941187			0.470593	−	0.882350i	$-0.344040\pi$
$769$	20070.8			0.941187			0.470593	+	0.882350i	$0.344040\pi$
$770$		0			0
$771$	−12544.7	+	5226.25i	−0.585976	+	0.244123i
$772$		0			0
$773$			39134.5i			1.82092i	0.413597	+	0.910460i	$0.364272\pi$
$773$			39134.5i			1.82092i	−0.413597	+	0.910460i	$0.635728\pi$
$774$		0			0
$775$			8324.79i			0.385852i
$776$		0			0
$777$	2118.24	−	882.480i	0.0978012	−	0.0407449i
$778$		0			0
$779$	70605.0			3.24735
$780$		0			0
$781$	−24029.2			−1.10094
$782$		0			0
$783$	−8204.89	+	20156.9i	−0.374481	+	0.919988i
$784$		0			0
$785$		−	18254.9i		−	0.829996i
$786$		0			0
$787$			9041.56i			0.409526i	0.978812	+	0.204763i	$0.0656423\pi$
$787$			9041.56i			0.409526i	−0.978812	+	0.204763i	$0.934358\pi$
$788$		0			0
$789$	−13996.6	−	33596.5i	−0.631549	−	1.51593i
$790$		0			0
$791$	−8599.12			−0.386535
$792$		0			0
$793$	−47127.4			−2.11039
$794$		0			0
$795$	13940.4	+	33461.6i	0.621907	+	1.49278i
$796$		0			0
$797$		−	36193.9i		−	1.60860i	−0.594222	−	0.804301i	$-0.702540\pi$
$797$		−	36193.9i		−	1.60860i	0.594222	−	0.804301i	$-0.297460\pi$
$798$		0			0
$799$			1149.39i			0.0508919i
$800$		0			0
$801$	19853.3	+	19691.7i	0.875758	+	0.868630i
$802$		0			0
$803$	59574.8			2.61812
$804$		0			0
$805$	13101.4			0.573621
$806$		0			0
$807$	−1939.23	+	807.902i	−0.0845901	+	0.0352410i
$808$		0			0
$809$		−	13745.8i		−	0.597377i	−0.954351	−	0.298689i	$-0.903451\pi$
$809$		−	13745.8i		−	0.597377i	0.954351	−	0.298689i	$-0.0965492\pi$
$810$		0			0
$811$		−	38555.3i		−	1.66937i	−0.550728	−	0.834685i	$-0.685650\pi$
$811$		−	38555.3i		−	1.66937i	0.550728	−	0.834685i	$-0.314350\pi$
$812$		0			0
$813$	−3638.44	+	1515.81i	−0.156957	+	0.0653896i
$814$		0			0
$815$	441.524			0.0189766
$816$		0			0
$817$	−14707.8			−0.629817
$818$		0			0
$819$	−9917.50	−	9836.78i	−0.423133	−	0.419688i
$820$		0			0
$821$		−	36235.3i		−	1.54034i	−0.637838	−	0.770171i	$-0.720171\pi$
$821$		−	36235.3i		−	1.54034i	0.637838	−	0.770171i	$-0.279829\pi$
$822$		0			0
$823$			23707.3i			1.00411i	0.864835	+	0.502056i	$0.167423\pi$
$823$			23707.3i			1.00411i	−0.864835	+	0.502056i	$0.832577\pi$
$824$		0			0
$825$	−18851.1	−	45248.8i	−0.795527	−	1.90953i
$826$		0			0
$827$	−35174.4			−1.47900			−0.739500	−	0.673156i	$-0.764938\pi$
$827$	−35174.4			−1.47900			−0.739500	+	0.673156i	$0.764938\pi$
$828$		0			0
$829$	24029.0			1.00671			0.503355	−	0.864080i	$-0.332099\pi$
$829$	24029.0			1.00671			0.503355	+	0.864080i	$0.332099\pi$
$830$		0			0
$831$	−2167.68	−	5203.15i	−0.0904887	−	0.217203i
$832$		0			0
$833$			798.720i			0.0332221i
$834$		0			0
$835$		−	34593.5i		−	1.43372i
$836$		0			0
$837$	−2843.61	+	6985.89i	−0.117431	+	0.288492i
$838$		0			0
$839$	46306.1			1.90544			0.952720	−	0.303850i	$-0.0982722\pi$
$839$	46306.1			1.90544			0.952720	+	0.303850i	$0.0982722\pi$
$840$		0			0
$841$	326.470			0.0133860
$842$		0			0
$843$	−19637.6	+	8181.19i	−0.802317	+	0.334253i
$844$		0			0
$845$			54624.2i			2.22382i
$846$		0			0
$847$			16663.3i			0.675982i
$848$		0			0
$849$	−24131.2	+	10053.3i	−0.975477	+	0.406393i
$850$		0			0
$851$	7058.45			0.284325
$852$		0			0
$853$	44352.0			1.78028			0.890142	−	0.455684i	$-0.150605\pi$
$853$	44352.0			1.78028			0.890142	+	0.455684i	$0.150605\pi$
$854$		0			0
$855$	48942.3	−	49344.0i	1.95765	−	1.97372i
$856$		0			0
$857$		−	42349.5i		−	1.68802i	−0.536328	−	0.844009i	$-0.680189\pi$
$857$		−	42349.5i		−	1.68802i	0.536328	−	0.844009i	$-0.319811\pi$
$858$		0			0
$859$		−	41927.4i		−	1.66536i	−0.553755	−	0.832680i	$-0.686806\pi$
$859$		−	41927.4i		−	1.66536i	0.553755	−	0.832680i	$-0.313194\pi$
$860$		0			0
$861$	6418.50	+	15406.5i	0.254056	+	0.609818i
$862$		0			0
$863$	−28670.6			−1.13089			−0.565445	−	0.824786i	$-0.691296\pi$
$863$	−28670.6			−1.13089			−0.565445	+	0.824786i	$0.691296\pi$
$864$		0			0
$865$	38296.0			1.50532
$866$		0			0
$867$	9286.63	+	22291.0i	0.363772	+	0.873173i
$868$		0			0
$869$			31373.7i			1.22472i
$870$		0			0
$871$			14890.4i			0.579267i
$872$		0			0
$873$	−18930.8	+	19086.1i	−0.733917	+	0.739940i
$874$		0			0
$875$	3495.21			0.135039
$876$		0			0
$877$	−1035.04			−0.0398529			−0.0199264	−	0.999801i	$-0.506343\pi$
$877$	−1035.04			−0.0398529			−0.0199264	+	0.999801i	$0.506343\pi$
$878$		0			0
$879$	−12401.5	+	5166.58i	−0.475873	+	0.198253i
$880$		0			0
$881$			7978.04i			0.305093i	0.988296	+	0.152547i	$0.0487474\pi$
$881$			7978.04i			0.305093i	−0.988296	+	0.152547i	$0.951253\pi$
$882$		0			0
$883$		−	22682.9i		−	0.864484i	−0.901758	−	0.432242i	$-0.857723\pi$
$883$		−	22682.9i		−	0.864484i	0.901758	−	0.432242i	$-0.142277\pi$
$884$		0			0
$885$	−26320.6	+	10965.4i	−0.999724	+	0.416494i
$886$		0			0
$887$	−41695.5			−1.57835			−0.789175	−	0.614168i	$-0.789492\pi$
$887$	−41695.5			−1.57835			−0.789175	+	0.614168i	$0.789492\pi$
$888$		0			0
$889$	−895.400			−0.0337804
$890$		0			0
$891$	−362.976	−	44410.5i	−0.0136478	−	1.66982i
$892$		0			0
$893$		−	10849.9i		−	0.406584i
$894$		0			0
$895$			1809.76i			0.0675905i
$896$		0			0
$897$	−16523.6	−	39662.2i	−0.615060	−	1.47635i
$898$		0			0
$899$	−8339.47			−0.309385
$900$		0			0
$901$	−6797.61			−0.251344
$902$		0			0
$903$	−1337.05	−	3209.35i	−0.0492737	−	0.118273i
$904$		0			0
$905$			12590.5i			0.462456i
$906$		0			0
$907$			5918.76i			0.216680i	0.994114	+	0.108340i	$0.0345536\pi$
$907$			5918.76i			0.216680i	−0.994114	+	0.108340i	$0.965446\pi$
$908$		0			0
$909$	−27936.2	−	27708.8i	−1.01934	−	1.01105i
$910$		0			0
$911$	−16275.8			−0.591921			−0.295961	−	0.955200i	$-0.595640\pi$
$911$	−16275.8			−0.591921			−0.295961	+	0.955200i	$0.595640\pi$
$912$		0			0
$913$	69075.5			2.50390
$914$		0			0
$915$	51165.2	−	21315.9i	1.84860	−	0.770145i
$916$		0			0
$917$			7997.27i			0.287997i
$918$		0			0
$919$		−	10367.7i		−	0.372141i	−0.982536	−	0.186070i	$-0.940425\pi$
$919$		−	10367.7i		−	0.372141i	0.982536	−	0.186070i	$-0.0595752\pi$
$920$		0			0
$921$	−3677.42	+	1532.05i	−0.131569	+	0.0548129i
$922$		0			0
$923$	−29151.0			−1.03956
$924$		0			0
$925$	9769.10			0.347250
$926$		0			0
$927$	6882.29	+	6826.27i	0.243845	+	0.241860i
$928$		0			0
$929$			38112.2i			1.34598i	0.739649	+	0.672992i	$0.234991\pi$
$929$			38112.2i			1.34598i	−0.739649	+	0.672992i	$0.765009\pi$
$930$		0			0
$931$		−	7539.68i		−	0.265417i
$932$		0			0
$933$	11353.1	+	27251.2i	0.398375	+	0.956230i
$934$		0			0
$935$	16612.4			0.581052
$936$		0			0
$937$	−36739.1			−1.28091			−0.640456	−	0.767995i	$-0.721255\pi$
$937$	−36739.1			−1.28091			−0.640456	+	0.767995i	$0.721255\pi$
$938$		0			0
$939$	−19255.6	−	46219.8i	−0.669204	−	1.60631i
$940$		0			0
$941$		−	31135.3i		−	1.07862i	−0.842107	−	0.539310i	$-0.818685\pi$
$941$		−	31135.3i		−	1.07862i	0.842107	−	0.539310i	$-0.181315\pi$
$942$		0			0
$943$			51337.9i			1.77284i
$944$		0			0
$945$	15216.4	+	6193.86i	0.523800	+	0.213213i
$946$		0			0
$947$	−16651.0			−0.571368			−0.285684	−	0.958324i	$-0.592221\pi$
$947$	−16651.0			−0.571368			−0.285684	+	0.958324i	$0.592221\pi$
$948$		0			0
$949$	72273.2			2.47217
$950$		0			0
$951$	−7606.44	+	3168.91i	−0.259364	+	0.108054i
$952$		0			0
$953$			21802.6i			0.741086i	0.928815	+	0.370543i	$0.120828\pi$
$953$			21802.6i			0.741086i	−0.928815	+	0.370543i	$0.879172\pi$
$954$		0			0
$955$			38133.8i			1.29213i
$956$		0			0
$957$	45328.6	−	18884.3i	1.53110	−	0.637872i
$958$		0			0
$959$	4625.56			0.155753
$960$		0			0
$961$	26900.7			0.902982
$962$		0			0
$963$	14551.5	−	14670.9i	0.486931	−	0.490927i
$964$		0			0
$965$		−	12816.6i		−	0.427546i
$966$		0			0
$967$		−	2820.93i		−	0.0938106i	−0.998899	−	0.0469053i	$-0.985064\pi$
$967$		−	2820.93i		−	0.0938106i	0.998899	−	0.0469053i	$-0.0149359\pi$
$968$		0			0
$969$	5012.02	+	12030.5i	0.166160	+	0.398840i
$970$		0			0
$971$	−40165.3			−1.32746			−0.663730	−	0.747972i	$-0.731028\pi$
$971$	−40165.3			−1.32746			−0.663730	+	0.747972i	$0.731028\pi$
$972$		0			0
$973$	3519.89			0.115974
$974$		0			0
$975$	−22869.2	−	54893.6i	−0.751180	−	1.80308i
$976$		0			0
$977$			18264.9i			0.598102i	0.954237	+	0.299051i	$0.0966701\pi$
$977$			18264.9i			0.598102i	−0.954237	+	0.299051i	$0.903330\pi$
$978$		0			0
$979$		−	63094.2i		−	2.05976i
$980$		0			0
$981$	−13996.0	+	14110.9i	−0.455513	+	0.459251i
$982$		0			0
$983$	1405.50			0.0456038			0.0228019	−	0.999740i	$-0.492741\pi$
$983$	1405.50			0.0456038			0.0228019	+	0.999740i	$0.492741\pi$
$984$		0			0
$985$	65075.0			2.10504
$986$		0			0
$987$	2367.54	−	986.339i	0.0763522	−	0.0318090i
$988$		0			0
$989$		−	10694.3i		−	0.343840i
$990$		0			0
$991$		−	32951.8i		−	1.05625i	−0.849165	−	0.528127i	$-0.822895\pi$
$991$		−	32951.8i		−	1.05625i	0.849165	−	0.528127i	$-0.177105\pi$
$992$		0			0
$993$	15977.2	−	6656.26i	0.510596	−	0.212719i
$994$		0			0
$995$	43122.6			1.37395
$996$		0			0
$997$	−19008.7			−0.603822			−0.301911	−	0.953336i	$-0.597625\pi$
$997$	−19008.7			−0.603822			−0.301911	+	0.953336i	$0.597625\pi$
$998$		0			0
$999$	8197.91	+	3336.96i	0.259630	+	0.105683i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.h.b.239.16	yes	24
3.2	odd	2	inner	336.4.h.b.239.10	yes	24
4.3	odd	2	inner	336.4.h.b.239.9	✓	24
12.11	even	2	inner	336.4.h.b.239.15	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.h.b.239.9	✓	24	4.3	odd	2	inner
336.4.h.b.239.10	yes	24	3.2	odd	2	inner
336.4.h.b.239.15	yes	24	12.11	even	2	inner
336.4.h.b.239.16	yes	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.99829 + 4.79655i) q^{3} +16.7287i q^{5} +7.00000i q^{7} +(-19.0137 + 19.1697i) q^{9} +O(q^{10})\)	\(q+(1.99829 + 4.79655i) q^{3} +16.7287i q^{5} +7.00000i q^{7} +(-19.0137 + 19.1697i) q^{9} +60.9218 q^{11} +73.9074 q^{13} +(-80.2398 + 33.4286i) q^{15} -16.3004i q^{17} +153.871i q^{19} +(-33.5758 + 13.9880i) q^{21} -111.882 q^{23} -154.848 q^{25} +(-129.943 - 52.8935i) q^{27} -155.121i q^{29} -53.7611i q^{31} +(121.739 + 292.214i) q^{33} -117.101 q^{35} -63.0884 q^{37} +(147.688 + 354.500i) q^{39} -458.858i q^{41} +95.5852i q^{43} +(-320.684 - 318.074i) q^{45} -70.5132 q^{47} -49.0000 q^{49} +(78.1857 - 32.5729i) q^{51} -417.021i q^{53} +1019.14i q^{55} +(-738.050 + 307.478i) q^{57} +328.024 q^{59} -637.654 q^{61} +(-134.188 - 133.096i) q^{63} +1236.37i q^{65} +201.474i q^{67} +(-223.572 - 536.647i) q^{69} -394.426 q^{71} +977.889 q^{73} +(-309.430 - 742.735i) q^{75} +426.453i q^{77} +514.984i q^{79} +(-5.95806 - 728.976i) q^{81} +1133.84 q^{83} +272.684 q^{85} +(744.045 - 309.976i) q^{87} -1035.66i q^{89} +517.352i q^{91} +(257.867 - 107.430i) q^{93} -2574.06 q^{95} +995.638 q^{97} +(-1158.35 + 1167.86i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 136 q^{9}+O(q^{10}) \) 24 * q - 136 * q^9	\( 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) \) 24 * q - 136 * q^9 + 96 * q^13 - 112 * q^21 + 168 * q^25 - 320 * q^33 - 864 * q^37 + 592 * q^45 - 1176 * q^49 - 1120 * q^57 - 2304 * q^61 + 1168 * q^69 + 3312 * q^73 - 968 * q^81 - 5808 * q^85 + 1776 * q^93 + 6480 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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