LMFDB - Embedded newform 336.2.h.b.239.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,2,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, 2, names="a")

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

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
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:	$2.68297350792$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.2
Root			$0.500000 + 2.19293i$ of defining polynomial
Character	$\chi$	$=$	336.239
Dual form			336.2.h.b.239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.69293 + 0.366025i) q^{3} +3.38587i q^{5} -1.00000i q^{7} +(2.73205 - 1.23931i) q^{9} +O(q^{10})$	\(q+(-1.69293 + 0.366025i) q^{3} +3.38587i q^{5} -1.00000i q^{7} +(2.73205 - 1.23931i) q^{9} -2.47863 q^{11} -6.19615 q^{13} +(-1.23931 - 5.73205i) q^{15} +2.47863i q^{17} +0.732051i q^{19} +(0.366025 + 1.69293i) q^{21} -6.77174 q^{23} -6.46410 q^{25} +(-4.17156 + 3.09808i) q^{27} -2.47863i q^{29} +9.46410i q^{31} +(4.19615 - 0.907241i) q^{33} +3.38587 q^{35} +4.53590 q^{37} +(10.4897 - 2.26795i) q^{39} -9.25036i q^{41} +2.00000i q^{43} +(4.19615 + 9.25036i) q^{45} -1.00000 q^{49} +(-0.907241 - 4.19615i) q^{51} +9.25036i q^{53} -8.39230i q^{55} +(-0.267949 - 1.23931i) q^{57} +8.34312 q^{59} -4.73205 q^{61} +(-1.23931 - 2.73205i) q^{63} -20.9794i q^{65} -3.46410i q^{67} +(11.4641 - 2.47863i) q^{69} +9.25036 q^{71} -4.53590 q^{73} +(10.9433 - 2.36603i) q^{75} +2.47863i q^{77} +12.0000i q^{79} +(5.92820 - 6.77174i) q^{81} +8.34312 q^{83} -8.39230 q^{85} +(0.907241 + 4.19615i) q^{87} +14.2076i q^{89} +6.19615i q^{91} +(-3.46410 - 16.0221i) q^{93} -2.47863 q^{95} +8.92820 q^{97} +(-6.77174 + 3.07180i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} - 8 q^{13} - 4 q^{21} - 24 q^{25} - 8 q^{33} + 64 q^{37} - 8 q^{45} - 8 q^{49} - 16 q^{57} - 24 q^{61} + 64 q^{69} - 64 q^{73} - 8 q^{81} + 16 q^{85} + 16 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 - 8 * q^13 - 4 * q^21 - 24 * q^25 - 8 * q^33 + 64 * q^37 - 8 * q^45 - 8 * q^49 - 16 * q^57 - 24 * q^61 + 64 * q^69 - 64 * q^73 - 8 * q^81 + 16 * q^85 + 16 * 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
$3$	−1.69293	+	0.366025i	−0.977416	+	0.211325i
$3$	−1.69293	+	0.366025i	−0.977416	+	0.211325i
$4$		0			0
$5$			3.38587i			1.51421i	0.653295	+	0.757103i	$0.273386\pi$
$5$			3.38587i			1.51421i	−0.653295	+	0.757103i	$0.726614\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	2.73205	−	1.23931i	0.910684	−	0.413105i
$10$		0			0
$11$	−2.47863			−0.747334			−0.373667	−	0.927563i	$-0.621900\pi$
$11$	−2.47863			−0.747334			−0.373667	+	0.927563i	$0.621900\pi$
$12$		0			0
$13$	−6.19615			−1.71850			−0.859252	−	0.511553i	$-0.829070\pi$
$13$	−6.19615			−1.71850			−0.859252	+	0.511553i	$0.829070\pi$
$14$		0			0
$15$	−1.23931	−	5.73205i	−0.319989	−	1.48001i
$16$		0			0
$17$			2.47863i			0.601155i	0.953757	+	0.300578i	$0.0971795\pi$
$17$			2.47863i			0.601155i	−0.953757	+	0.300578i	$0.902821\pi$
$18$		0			0
$19$			0.732051i			0.167944i	0.996468	+	0.0839720i	$0.0267606\pi$
$19$			0.732051i			0.167944i	−0.996468	+	0.0839720i	$0.973239\pi$
$20$		0			0
$21$	0.366025	+	1.69293i	0.0798733	+	0.369428i
$22$		0			0
$23$	−6.77174			−1.41200			−0.706002	−	0.708210i	$-0.749503\pi$
$23$	−6.77174			−1.41200			−0.706002	+	0.708210i	$0.749503\pi$
$24$		0			0
$25$	−6.46410			−1.29282
$26$		0			0
$27$	−4.17156	+	3.09808i	−0.802817	+	0.596225i
$28$		0			0
$29$		−	2.47863i		−	0.460270i	−0.973159	−	0.230135i	$-0.926083\pi$
$29$		−	2.47863i		−	0.460270i	0.973159	−	0.230135i	$-0.0739167\pi$
$30$		0			0
$31$			9.46410i			1.69980i	0.526942	+	0.849901i	$0.323339\pi$
$31$			9.46410i			1.69980i	−0.526942	+	0.849901i	$0.676661\pi$
$32$		0			0
$33$	4.19615	−	0.907241i	0.730456	−	0.157930i
$34$		0			0
$35$	3.38587			0.572316
$36$		0			0
$37$	4.53590			0.745697			0.372849	−	0.927892i	$-0.378381\pi$
$37$	4.53590			0.745697			0.372849	+	0.927892i	$0.378381\pi$
$38$		0			0
$39$	10.4897	−	2.26795i	1.67969	−	0.363163i
$40$		0			0
$41$		−	9.25036i		−	1.44466i	−0.691546	−	0.722332i	$-0.743070\pi$
$41$		−	9.25036i		−	1.44466i	0.691546	−	0.722332i	$-0.256930\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$	4.19615	+	9.25036i	0.625525	+	1.37896i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−0.907241	−	4.19615i	−0.127039	−	0.587579i
$52$		0			0
$53$			9.25036i			1.27064i	0.772251	+	0.635318i	$0.219131\pi$
$53$			9.25036i			1.27064i	−0.772251	+	0.635318i	$0.780869\pi$
$54$		0			0
$55$		−	8.39230i		−	1.13162i
$56$		0			0
$57$	−0.267949	−	1.23931i	−0.0354907	−	0.164151i
$58$		0			0
$59$	8.34312			1.08618			0.543091	−	0.839674i	$-0.317254\pi$
$59$	8.34312			1.08618			0.543091	+	0.839674i	$0.317254\pi$
$60$		0			0
$61$	−4.73205			−0.605877			−0.302939	−	0.953010i	$-0.597968\pi$
$61$	−4.73205			−0.605877			−0.302939	+	0.953010i	$0.597968\pi$
$62$		0			0
$63$	−1.23931	−	2.73205i	−0.156139	−	0.344206i
$64$		0			0
$65$		−	20.9794i		−	2.60217i
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$	11.4641	−	2.47863i	1.38012	−	0.298392i
$70$		0			0
$71$	9.25036			1.09782			0.548908	−	0.835883i	$-0.315044\pi$
$71$	9.25036			1.09782			0.548908	+	0.835883i	$0.315044\pi$
$72$		0			0
$73$	−4.53590			−0.530887			−0.265443	−	0.964126i	$-0.585518\pi$
$73$	−4.53590			−0.530887			−0.265443	+	0.964126i	$0.585518\pi$
$74$		0			0
$75$	10.9433	−	2.36603i	1.26362	−	0.273205i
$76$		0			0
$77$			2.47863i			0.282466i
$78$		0			0
$79$			12.0000i			1.35011i	0.737769	+	0.675053i	$0.235879\pi$
$79$			12.0000i			1.35011i	−0.737769	+	0.675053i	$0.764121\pi$
$80$		0			0
$81$	5.92820	−	6.77174i	0.658689	−	0.752415i
$82$		0			0
$83$	8.34312			0.915777			0.457888	−	0.889010i	$-0.348606\pi$
$83$	8.34312			0.915777			0.457888	+	0.889010i	$0.348606\pi$
$84$		0			0
$85$	−8.39230			−0.910273
$86$		0			0
$87$	0.907241	+	4.19615i	0.0972664	+	0.449875i
$88$		0			0
$89$			14.2076i			1.50600i	0.658018	+	0.753002i	$0.271395\pi$
$89$			14.2076i			1.50600i	−0.658018	+	0.753002i	$0.728605\pi$
$90$		0			0
$91$			6.19615i			0.649533i
$92$		0			0
$93$	−3.46410	−	16.0221i	−0.359211	−	1.66141i
$94$		0			0
$95$	−2.47863			−0.254302
$96$		0			0
$97$	8.92820			0.906522			0.453261	−	0.891378i	$-0.350261\pi$
$97$	8.92820			0.906522			0.453261	+	0.891378i	$0.350261\pi$
$98$		0			0
$99$	−6.77174	+	3.07180i	−0.680585	+	0.308727i
$100$		0			0
$101$			8.34312i			0.830172i	0.909782	+	0.415086i	$0.136248\pi$
$101$			8.34312i			0.830172i	−0.909782	+	0.415086i	$0.863752\pi$
$102$		0			0
$103$			14.9282i			1.47092i	0.677568	+	0.735460i	$0.263034\pi$
$103$			14.9282i			1.47092i	−0.677568	+	0.735460i	$0.736966\pi$
$104$		0			0
$105$	−5.73205	+	1.23931i	−0.559391	+	0.120945i
$106$		0			0
$107$	−9.25036			−0.894266			−0.447133	−	0.894467i	$-0.647555\pi$
$107$	−9.25036			−0.894266			−0.447133	+	0.894467i	$0.647555\pi$
$108$		0			0
$109$	6.39230			0.612272			0.306136	−	0.951988i	$-0.400964\pi$
$109$	6.39230			0.612272			0.306136	+	0.951988i	$0.400964\pi$
$110$		0			0
$111$	−7.67898	+	1.66025i	−0.728856	+	0.157584i
$112$		0			0
$113$		−	11.0648i		−	1.04089i	−0.853894	−	0.520446i	$-0.825766\pi$
$113$		−	11.0648i		−	1.04089i	0.853894	−	0.520446i	$-0.174234\pi$
$114$		0			0
$115$		−	22.9282i		−	2.13807i
$116$		0			0
$117$	−16.9282	+	7.67898i	−1.56501	+	0.709922i
$118$		0			0
$119$	2.47863			0.227215
$120$		0			0
$121$	−4.85641			−0.441491
$122$		0			0
$123$	3.38587	+	15.6603i	0.305293	+	1.41204i
$124$		0			0
$125$		−	4.95725i		−	0.443390i
$126$		0			0
$127$			6.39230i			0.567225i	0.958939	+	0.283613i	$0.0915330\pi$
$127$			6.39230i			0.567225i	−0.958939	+	0.283613i	$0.908467\pi$
$128$		0			0
$129$	−0.732051	−	3.38587i	−0.0644535	−	0.298109i
$130$		0			0
$131$	5.20035			0.454357			0.227178	−	0.973853i	$-0.427050\pi$
$131$	5.20035			0.454357			0.227178	+	0.973853i	$0.427050\pi$
$132$		0			0
$133$	0.732051			0.0634769
$134$		0			0
$135$	−10.4897	−	14.1244i	−0.902808	−	1.21563i
$136$		0			0
$137$			13.5435i			1.15710i	0.815648	+	0.578548i	$0.196381\pi$
$137$			13.5435i			1.15710i	−0.815648	+	0.578548i	$0.803619\pi$
$138$		0			0
$139$		−	13.1244i		−	1.11319i	−0.830783	−	0.556597i	$-0.812107\pi$
$139$		−	13.1244i		−	1.11319i	0.830783	−	0.556597i	$-0.187893\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	15.3580			1.28430
$144$		0			0
$145$	8.39230			0.696943
$146$		0			0
$147$	1.69293	−	0.366025i	0.139631	−	0.0301893i
$148$		0			0
$149$		−	0.664146i		−	0.0544090i	−0.999630	−	0.0272045i	$-0.991339\pi$
$149$		−	0.664146i		−	0.0544090i	0.999630	−	0.0272045i	$-0.00866053\pi$
$150$		0			0
$151$		−	7.46410i		−	0.607420i	−0.952765	−	0.303710i	$-0.901775\pi$
$151$		−	7.46410i		−	0.607420i	0.952765	−	0.303710i	$-0.0982254\pi$
$152$		0			0
$153$	3.07180	+	6.77174i	0.248340	+	0.547462i
$154$		0			0
$155$	−32.0442			−2.57385
$156$		0			0
$157$	−4.73205			−0.377659			−0.188829	−	0.982010i	$-0.560469\pi$
$157$	−4.73205			−0.377659			−0.188829	+	0.982010i	$0.560469\pi$
$158$		0			0
$159$	−3.38587	−	15.6603i	−0.268517	−	1.24194i
$160$		0			0
$161$			6.77174i			0.533688i
$162$		0			0
$163$		−	7.07180i		−	0.553906i	−0.960883	−	0.276953i	$-0.910675\pi$
$163$		−	7.07180i		−	0.553906i	0.960883	−	0.276953i	$-0.0893246\pi$
$164$		0			0
$165$	3.07180	+	14.2076i	0.239139	+	1.10606i
$166$		0			0
$167$	−25.2725			−1.95564			−0.977821	−	0.209443i	$-0.932835\pi$
$167$	−25.2725			−1.95564			−0.977821	+	0.209443i	$0.932835\pi$
$168$		0			0
$169$	25.3923			1.95325
$170$		0			0
$171$	0.907241	+	2.00000i	0.0693784	+	0.152944i
$172$		0			0
$173$		−	10.1576i		−	0.772268i	−0.922443	−	0.386134i	$-0.873810\pi$
$173$		−	10.1576i		−	0.772268i	0.922443	−	0.386134i	$-0.126190\pi$
$174$		0			0
$175$			6.46410i			0.488640i
$176$		0			0
$177$	−14.1244	+	3.05379i	−1.06165	+	0.229537i
$178$		0			0
$179$	0.664146			0.0496406			0.0248203	−	0.999692i	$-0.492099\pi$
$179$	0.664146			0.0496406			0.0248203	+	0.999692i	$0.492099\pi$
$180$		0			0
$181$	−7.26795			−0.540222			−0.270111	−	0.962829i	$-0.587060\pi$
$181$	−7.26795			−0.540222			−0.270111	+	0.962829i	$0.587060\pi$
$182$		0			0
$183$	8.01105	−	1.73205i	0.592194	−	0.128037i
$184$		0			0
$185$			15.3580i			1.12914i
$186$		0			0
$187$		−	6.14359i		−	0.449264i
$188$		0			0
$189$	3.09808	+	4.17156i	0.225352	+	0.303436i
$190$		0			0
$191$	−4.29311			−0.310638			−0.155319	−	0.987864i	$-0.549641\pi$
$191$	−4.29311			−0.310638			−0.155319	+	0.987864i	$0.549641\pi$
$192$		0			0
$193$	−9.46410			−0.681241			−0.340620	−	0.940201i	$-0.610637\pi$
$193$	−9.46410			−0.681241			−0.340620	+	0.940201i	$0.610637\pi$
$194$		0			0
$195$	7.67898	+	35.5167i	0.549903	+	2.54340i
$196$		0			0
$197$			4.29311i			0.305871i	0.988236	+	0.152936i	$0.0488727\pi$
$197$			4.29311i			0.305871i	−0.988236	+	0.152936i	$0.951127\pi$
$198$		0			0
$199$		−	8.00000i		−	0.567105i	−0.958957	−	0.283552i	$-0.908487\pi$
$199$		−	8.00000i		−	0.567105i	0.958957	−	0.283552i	$-0.0915130\pi$
$200$		0			0
$201$	1.26795	+	5.86450i	0.0894342	+	0.413650i
$202$		0			0
$203$	−2.47863			−0.173966
$204$		0			0
$205$	31.3205			2.18752
$206$		0			0
$207$	−18.5007	+	8.39230i	−1.28589	+	0.583306i
$208$		0			0
$209$		−	1.81448i		−	0.125510i
$210$		0			0
$211$			7.46410i			0.513850i	0.966431	+	0.256925i	$0.0827093\pi$
$211$			7.46410i			0.513850i	−0.966431	+	0.256925i	$0.917291\pi$
$212$		0			0
$213$	−15.6603	+	3.38587i	−1.07302	+	0.231996i
$214$		0			0
$215$	−6.77174			−0.461829
$216$		0			0
$217$	9.46410			0.642465
$218$		0			0
$219$	7.67898	−	1.66025i	0.518897	−	0.112190i
$220$		0			0
$221$		−	15.3580i		−	1.03309i
$222$		0			0
$223$		−	7.32051i		−	0.490217i	−0.969496	−	0.245109i	$-0.921176\pi$
$223$		−	7.32051i		−	0.490217i	0.969496	−	0.245109i	$-0.0788237\pi$
$224$		0			0
$225$	−17.6603	+	8.01105i	−1.17735	+	0.534070i
$226$		0			0
$227$	−20.0721			−1.33223			−0.666116	−	0.745848i	$-0.732045\pi$
$227$	−20.0721			−1.33223			−0.666116	+	0.745848i	$0.732045\pi$
$228$		0			0
$229$	3.26795			0.215952			0.107976	−	0.994153i	$-0.465563\pi$
$229$	3.26795			0.215952			0.107976	+	0.994153i	$0.465563\pi$
$230$		0			0
$231$	−0.907241	−	4.19615i	−0.0596920	−	0.276087i
$232$		0			0
$233$		−	13.5435i		−	0.887262i	−0.896209	−	0.443631i	$-0.853690\pi$
$233$		−	13.5435i		−	0.887262i	0.896209	−	0.443631i	$-0.146310\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−4.39230	−	20.3152i	−0.285311	−	1.31961i
$238$		0			0
$239$	25.2725			1.63474			0.817370	−	0.576113i	$-0.195431\pi$
$239$	25.2725			1.63474			0.817370	+	0.576113i	$0.195431\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$	−7.55743	+	13.6340i	−0.484809	+	0.874620i
$244$		0			0
$245$		−	3.38587i		−	0.216315i
$246$		0			0
$247$		−	4.53590i		−	0.288612i
$248$		0			0
$249$	−14.1244	+	3.05379i	−0.895095	+	0.193526i
$250$		0			0
$251$	−1.57139			−0.0991851			−0.0495925	−	0.998770i	$-0.515792\pi$
$251$	−1.57139			−0.0991851			−0.0495925	+	0.998770i	$0.515792\pi$
$252$		0			0
$253$	16.7846			1.05524
$254$		0			0
$255$	14.2076	−	3.07180i	0.889716	−	0.192363i
$256$		0			0
$257$			11.0648i			0.690206i	0.938565	+	0.345103i	$0.112156\pi$
$257$			11.0648i			0.690206i	−0.938565	+	0.345103i	$0.887844\pi$
$258$		0			0
$259$		−	4.53590i		−	0.281847i
$260$		0			0
$261$	−3.07180	−	6.77174i	−0.190139	−	0.419160i
$262$		0			0
$263$	22.7938			1.40553			0.702764	−	0.711423i	$-0.251949\pi$
$263$	22.7938			1.40553			0.702764	+	0.711423i	$0.251949\pi$
$264$		0			0
$265$	−31.3205			−1.92400
$266$		0			0
$267$	−5.20035	−	24.0526i	−0.318256	−	1.47199i
$268$		0			0
$269$		−	23.7011i		−	1.44508i	−0.691329	−	0.722540i	$-0.742974\pi$
$269$		−	23.7011i		−	1.44508i	0.691329	−	0.722540i	$-0.257026\pi$
$270$		0			0
$271$		−	19.3205i		−	1.17364i	−0.809718	−	0.586819i	$-0.800380\pi$
$271$		−	19.3205i		−	1.17364i	0.809718	−	0.586819i	$-0.199620\pi$
$272$		0			0
$273$	−2.26795	−	10.4897i	−0.137263	−	0.634864i
$274$		0			0
$275$	16.0221			0.966169
$276$		0			0
$277$	−3.07180			−0.184566			−0.0922832	−	0.995733i	$-0.529417\pi$
$277$	−3.07180			−0.184566			−0.0922832	+	0.995733i	$0.529417\pi$
$278$		0			0
$279$	11.7290	+	25.8564i	0.702196	+	1.54798i
$280$		0			0
$281$			27.0869i			1.61587i	0.589271	+	0.807936i	$0.299415\pi$
$281$			27.0869i			1.61587i	−0.589271	+	0.807936i	$0.700585\pi$
$282$		0			0
$283$			12.3397i			0.733522i	0.930315	+	0.366761i	$0.119533\pi$
$283$			12.3397i			0.733522i	−0.930315	+	0.366761i	$0.880467\pi$
$284$		0			0
$285$	4.19615	−	0.907241i	0.248559	−	0.0537403i
$286$		0			0
$287$	−9.25036			−0.546032
$288$		0			0
$289$	10.8564			0.638612
$290$		0			0
$291$	−15.1149	+	3.26795i	−0.886049	+	0.191571i
$292$		0			0
$293$			15.1149i			0.883019i	0.897256	+	0.441510i	$0.145557\pi$
$293$			15.1149i			0.883019i	−0.897256	+	0.441510i	$0.854443\pi$
$294$		0			0
$295$			28.2487i			1.64470i
$296$		0			0
$297$	10.3397	−	7.67898i	0.599973	−	0.445579i
$298$		0			0
$299$	41.9587			2.42653
$300$		0			0
$301$	2.00000			0.115278
$302$		0			0
$303$	−3.05379	−	14.1244i	−0.175436	−	0.811423i
$304$		0			0
$305$		−	16.0221i		−	0.917423i
$306$		0			0
$307$		−	33.1244i		−	1.89051i	−0.326337	−	0.945253i	$-0.605814\pi$
$307$		−	33.1244i		−	1.89051i	0.326337	−	0.945253i	$-0.394186\pi$
$308$		0			0
$309$	−5.46410	−	25.2725i	−0.310842	−	1.43770i
$310$		0			0
$311$	3.14277			0.178210			0.0891052	−	0.996022i	$-0.471599\pi$
$311$	3.14277			0.178210			0.0891052	+	0.996022i	$0.471599\pi$
$312$		0			0
$313$	15.4641			0.874083			0.437041	−	0.899441i	$-0.356026\pi$
$313$	15.4641			0.874083			0.437041	+	0.899441i	$0.356026\pi$
$314$		0			0
$315$	9.25036	−	4.19615i	0.521199	−	0.236426i
$316$		0			0
$317$			19.1649i			1.07641i	0.842815	+	0.538203i	$0.180897\pi$
$317$			19.1649i			1.07641i	−0.842815	+	0.538203i	$0.819103\pi$
$318$		0			0
$319$			6.14359i			0.343975i
$320$		0			0
$321$	15.6603	−	3.38587i	0.874070	−	0.188981i
$322$		0			0
$323$	−1.81448			−0.100960
$324$		0			0
$325$	40.0526			2.22172
$326$		0			0
$327$	−10.8217	+	2.33975i	−0.598444	+	0.129388i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			30.7846i			1.69208i	0.533123	+	0.846038i	$0.321018\pi$
$331$			30.7846i			1.69208i	−0.533123	+	0.846038i	$0.678982\pi$
$332$		0			0
$333$	12.3923	−	5.62140i	0.679094	−	0.308051i
$334$		0			0
$335$	11.7290			0.640823
$336$		0			0
$337$	−0.392305			−0.0213702			−0.0106851	−	0.999943i	$-0.503401\pi$
$337$	−0.392305			−0.0213702			−0.0106851	+	0.999943i	$0.503401\pi$
$338$		0			0
$339$	4.05001	+	18.7321i	0.219967	+	1.01739i
$340$		0			0
$341$		−	23.4580i		−	1.27032i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	8.39230	+	38.8159i	0.451827	+	2.08978i
$346$		0			0
$347$	16.0221			0.860111			0.430056	−	0.902802i	$-0.358494\pi$
$347$	16.0221			0.860111			0.430056	+	0.902802i	$0.358494\pi$
$348$		0			0
$349$	−10.8756			−0.582160			−0.291080	−	0.956699i	$-0.594015\pi$
$349$	−10.8756			−0.582160			−0.291080	+	0.956699i	$0.594015\pi$
$350$		0			0
$351$	25.8476	−	19.1962i	1.37964	−	1.02461i
$352$		0			0
$353$			2.47863i			0.131924i	0.997822	+	0.0659620i	$0.0210116\pi$
$353$			2.47863i			0.131924i	−0.997822	+	0.0659620i	$0.978988\pi$
$354$		0			0
$355$			31.3205i			1.66232i
$356$		0			0
$357$	−4.19615	+	0.907241i	−0.222084	+	0.0480163i
$358$		0			0
$359$	6.77174			0.357399			0.178699	−	0.983904i	$-0.442811\pi$
$359$	6.77174			0.357399			0.178699	+	0.983904i	$0.442811\pi$
$360$		0			0
$361$	18.4641			0.971795
$362$		0			0
$363$	8.22158	−	1.77757i	0.431521	−	0.0932981i
$364$		0			0
$365$		−	15.3580i		−	0.803872i
$366$		0			0
$367$		−	1.46410i		−	0.0764255i	−0.999270	−	0.0382127i	$-0.987834\pi$
$367$		−	1.46410i		−	0.0764255i	0.999270	−	0.0382127i	$-0.0121665\pi$
$368$		0			0
$369$	−11.4641	−	25.2725i	−0.596797	−	1.31563i
$370$		0			0
$371$	9.25036			0.480255
$372$		0			0
$373$	−30.7846			−1.59397			−0.796983	−	0.604001i	$-0.793572\pi$
$373$	−30.7846			−1.59397			−0.796983	+	0.604001i	$0.793572\pi$
$374$		0			0
$375$	1.81448	+	8.39230i	0.0936994	+	0.433377i
$376$		0			0
$377$			15.3580i			0.790975i
$378$		0			0
$379$		−	26.7846i		−	1.37583i	−0.725790	−	0.687916i	$-0.758526\pi$
$379$		−	26.7846i		−	1.37583i	0.725790	−	0.687916i	$-0.241474\pi$
$380$		0			0
$381$	−2.33975	−	10.8217i	−0.119869	−	0.554415i
$382$		0			0
$383$	−35.6732			−1.82281			−0.911407	−	0.411507i	$-0.865003\pi$
$383$	−35.6732			−1.82281			−0.911407	+	0.411507i	$0.865003\pi$
$384$		0			0
$385$	−8.39230			−0.427711
$386$		0			0
$387$	2.47863	+	5.46410i	0.125996	+	0.277756i
$388$		0			0
$389$		−	20.9794i		−	1.06370i	−0.846840	−	0.531848i	$-0.821498\pi$
$389$		−	20.9794i		−	1.06370i	0.846840	−	0.531848i	$-0.178502\pi$
$390$		0			0
$391$		−	16.7846i		−	0.848834i
$392$		0			0
$393$	−8.80385	+	1.90346i	−0.444095	+	0.0960168i
$394$		0			0
$395$	−40.6304			−2.04434
$396$		0			0
$397$	−0.732051			−0.0367406			−0.0183703	−	0.999831i	$-0.505848\pi$
$397$	−0.732051			−0.0367406			−0.0183703	+	0.999831i	$0.505848\pi$
$398$		0			0
$399$	−1.23931	+	0.267949i	−0.0620433	+	0.0134142i
$400$		0			0
$401$			25.9366i			1.29521i	0.761975	+	0.647606i	$0.224230\pi$
$401$			25.9366i			1.29521i	−0.761975	+	0.647606i	$0.775770\pi$
$402$		0			0
$403$		−	58.6410i		−	2.92112i
$404$		0			0
$405$	22.9282	+	20.0721i	1.13931	+	0.997391i
$406$		0			0
$407$	−11.2428			−0.557285
$408$		0			0
$409$	−17.3205			−0.856444			−0.428222	−	0.903674i	$-0.640860\pi$
$409$	−17.3205			−0.856444			−0.428222	+	0.903674i	$0.640860\pi$
$410$		0			0
$411$	−4.95725	−	22.9282i	−0.244523	−	1.13096i
$412$		0			0
$413$		−	8.34312i		−	0.410538i
$414$		0			0
$415$			28.2487i			1.38667i
$416$		0			0
$417$	4.80385	+	22.2187i	0.235245	+	1.08805i
$418$		0			0
$419$	13.7866			0.673518			0.336759	−	0.941591i	$-0.390669\pi$
$419$	13.7866			0.673518			0.336759	+	0.941591i	$0.390669\pi$
$420$		0			0
$421$	−23.8564			−1.16269			−0.581345	−	0.813657i	$-0.697473\pi$
$421$	−23.8564			−1.16269			−0.581345	+	0.813657i	$0.697473\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	16.0221i		−	0.777186i
$426$		0			0
$427$			4.73205i			0.229000i
$428$		0			0
$429$	−26.0000	+	5.62140i	−1.25529	+	0.271404i
$430$		0			0
$431$	−1.81448			−0.0874005			−0.0437002	−	0.999045i	$-0.513915\pi$
$431$	−1.81448			−0.0874005			−0.0437002	+	0.999045i	$0.513915\pi$
$432$		0			0
$433$	18.7846			0.902731			0.451365	−	0.892339i	$-0.350937\pi$
$433$	18.7846			0.902731			0.451365	+	0.892339i	$0.350937\pi$
$434$		0			0
$435$	−14.2076	+	3.07180i	−0.681203	+	0.147281i
$436$		0			0
$437$		−	4.95725i		−	0.237138i
$438$		0			0
$439$			25.8564i			1.23406i	0.786940	+	0.617029i	$0.211664\pi$
$439$			25.8564i			1.23406i	−0.786940	+	0.617029i	$0.788336\pi$
$440$		0			0
$441$	−2.73205	+	1.23931i	−0.130098	+	0.0590149i
$442$		0			0
$443$	36.3373			1.72644			0.863219	−	0.504830i	$-0.168445\pi$
$443$	36.3373			1.72644			0.863219	+	0.504830i	$0.168445\pi$
$444$		0			0
$445$	−48.1051			−2.28040
$446$		0			0
$447$	0.243094	+	1.12436i	0.0114980	+	0.0531802i
$448$		0			0
$449$		−	23.4580i		−	1.10705i	−0.832832	−	0.553525i	$-0.813282\pi$
$449$		−	23.4580i		−	1.10705i	0.832832	−	0.553525i	$-0.186718\pi$
$450$		0			0
$451$			22.9282i			1.07965i
$452$		0			0
$453$	2.73205	+	12.6362i	0.128363	+	0.593702i
$454$		0			0
$455$	−20.9794			−0.983527
$456$		0			0
$457$	−7.60770			−0.355873			−0.177936	−	0.984042i	$-0.556942\pi$
$457$	−7.60770			−0.355873			−0.177936	+	0.984042i	$0.556942\pi$
$458$		0			0
$459$	−7.67898	−	10.3397i	−0.358424	−	0.482618i
$460$		0			0
$461$			5.20035i			0.242204i	0.992640	+	0.121102i	$0.0386429\pi$
$461$			5.20035i			0.242204i	−0.992640	+	0.121102i	$0.961357\pi$
$462$		0			0
$463$			23.7128i			1.10203i	0.834496	+	0.551014i	$0.185759\pi$
$463$			23.7128i			1.10203i	−0.834496	+	0.551014i	$0.814241\pi$
$464$		0			0
$465$	54.2487	−	11.7290i	2.51572	−	0.543919i
$466$		0			0
$467$	−20.0721			−0.928827			−0.464413	−	0.885619i	$-0.653735\pi$
$467$	−20.0721			−0.928827			−0.464413	+	0.885619i	$0.653735\pi$
$468$		0			0
$469$	−3.46410			−0.159957
$470$		0			0
$471$	8.01105	−	1.73205i	0.369130	−	0.0798087i
$472$		0			0
$473$		−	4.95725i		−	0.227935i
$474$		0			0
$475$		−	4.73205i		−	0.217121i
$476$		0			0
$477$	11.4641	+	25.2725i	0.524905	+	1.15715i
$478$		0			0
$479$	−18.5007			−0.845320			−0.422660	−	0.906288i	$-0.638904\pi$
$479$	−18.5007			−0.845320			−0.422660	+	0.906288i	$0.638904\pi$
$480$		0			0
$481$	−28.1051			−1.28148
$482$		0			0
$483$	−2.47863	−	11.4641i	−0.112781	−	0.521635i
$484$		0			0
$485$			30.2297i			1.37266i
$486$		0			0
$487$		−	2.67949i		−	0.121419i	−0.998155	−	0.0607097i	$-0.980664\pi$
$487$		−	2.67949i		−	0.121419i	0.998155	−	0.0607097i	$-0.0193364\pi$
$488$		0			0
$489$	2.58846	+	11.9721i	0.117054	+	0.541396i
$490$		0			0
$491$	0.664146			0.0299725			0.0149862	−	0.999888i	$-0.495230\pi$
$491$	0.664146			0.0299725			0.0149862	+	0.999888i	$0.495230\pi$
$492$		0			0
$493$	6.14359			0.276694
$494$		0			0
$495$	−10.4007	−	22.9282i	−0.467477	−	1.03055i
$496$		0			0
$497$		−	9.25036i		−	0.414935i
$498$		0			0
$499$			25.3205i			1.13350i	0.823889	+	0.566751i	$0.191800\pi$
$499$			25.3205i			1.13350i	−0.823889	+	0.566751i	$0.808200\pi$
$500$		0			0
$501$	42.7846	−	9.25036i	1.91148	−	0.413276i
$502$		0			0
$503$	−20.3152			−0.905810			−0.452905	−	0.891559i	$-0.649612\pi$
$503$	−20.3152			−0.905810			−0.452905	+	0.891559i	$0.649612\pi$
$504$		0			0
$505$	−28.2487			−1.25705
$506$		0			0
$507$	−42.9875	+	9.29423i	−1.90914	+	0.412771i
$508$		0			0
$509$			8.34312i			0.369802i	0.982757	+	0.184901i	$0.0591965\pi$
$509$			8.34312i			0.369802i	−0.982757	+	0.184901i	$0.940803\pi$
$510$		0			0
$511$			4.53590i			0.200656i
$512$		0			0
$513$	−2.26795	−	3.05379i	−0.100132	−	0.134828i
$514$		0			0
$515$	−50.5449			−2.22728
$516$		0			0
$517$		0			0
$518$		0			0
$519$	3.71794	+	17.1962i	0.163199	+	0.754827i
$520$		0			0
$521$		−	9.25036i		−	0.405266i	−0.979255	−	0.202633i	$-0.935050\pi$
$521$		−	9.25036i		−	0.405266i	0.979255	−	0.202633i	$-0.0649498\pi$
$522$		0			0
$523$		−	1.12436i		−	0.0491646i	−0.999698	−	0.0245823i	$-0.992174\pi$
$523$		−	1.12436i		−	0.0491646i	0.999698	−	0.0245823i	$-0.00782558\pi$
$524$		0			0
$525$	−2.36603	−	10.9433i	−0.103262	−	0.477605i
$526$		0			0
$527$	−23.4580			−1.02185
$528$		0			0
$529$	22.8564			0.993757
$530$		0			0
$531$	22.7938	−	10.3397i	0.989168	−	0.448707i
$532$		0			0
$533$			57.3167i			2.48266i
$534$		0			0
$535$		−	31.3205i		−	1.35410i
$536$		0			0
$537$	−1.12436	+	0.243094i	−0.0485195	+	0.0104903i
$538$		0			0
$539$	2.47863			0.106762
$540$		0			0
$541$	16.9282			0.727800			0.363900	−	0.931438i	$-0.381445\pi$
$541$	16.9282			0.727800			0.363900	+	0.931438i	$0.381445\pi$
$542$		0			0
$543$	12.3042	−	2.66025i	0.528022	−	0.114162i
$544$		0			0
$545$			21.6435i			0.927106i
$546$		0			0
$547$		−	7.07180i		−	0.302368i	−0.988506	−	0.151184i	$-0.951691\pi$
$547$		−	7.07180i		−	0.302368i	0.988506	−	0.151184i	$-0.0483086\pi$
$548$		0			0
$549$	−12.9282	+	5.86450i	−0.551762	+	0.250291i
$550$		0			0
$551$	1.81448			0.0772995
$552$		0			0
$553$	12.0000			0.510292
$554$		0			0
$555$	−5.62140	−	26.0000i	−0.238615	−	1.10364i
$556$		0			0
$557$			7.92207i			0.335669i	0.985815	+	0.167834i	$0.0536774\pi$
$557$			7.92207i			0.335669i	−0.985815	+	0.167834i	$0.946323\pi$
$558$		0			0
$559$		−	12.3923i		−	0.524139i
$560$		0			0
$561$	2.24871	+	10.4007i	0.0949407	+	0.439118i
$562$		0			0
$563$	21.8866			0.922410			0.461205	−	0.887294i	$-0.347417\pi$
$563$	21.8866			0.922410			0.461205	+	0.887294i	$0.347417\pi$
$564$		0			0
$565$	37.4641			1.57613
$566$		0			0
$567$	−6.77174	−	5.92820i	−0.284386	−	0.248961i
$568$		0			0
$569$			30.8939i			1.29514i	0.762007	+	0.647569i	$0.224214\pi$
$569$			30.8939i			1.29514i	−0.762007	+	0.647569i	$0.775786\pi$
$570$		0			0
$571$			26.0000i			1.08807i	0.839064	+	0.544033i	$0.183103\pi$
$571$			26.0000i			1.08807i	−0.839064	+	0.544033i	$0.816897\pi$
$572$		0			0
$573$	7.26795	−	1.57139i	0.303623	−	0.0656456i
$574$		0			0
$575$	43.7732			1.82547
$576$		0			0
$577$	−16.9282			−0.704730			−0.352365	−	0.935863i	$-0.614622\pi$
$577$	−16.9282			−0.704730			−0.352365	+	0.935863i	$0.614622\pi$
$578$		0			0
$579$	16.0221	−	3.46410i	0.665856	−	0.143963i
$580$		0			0
$581$		−	8.34312i		−	0.346131i
$582$		0			0
$583$		−	22.9282i		−	0.949589i
$584$		0			0
$585$	−26.0000	−	57.3167i	−1.07497	−	2.36975i
$586$		0			0
$587$	26.8438			1.10796			0.553982	−	0.832529i	$-0.313108\pi$
$587$	26.8438			1.10796			0.553982	+	0.832529i	$0.313108\pi$
$588$		0			0
$589$	−6.92820			−0.285472
$590$		0			0
$591$	−1.57139	−	7.26795i	−0.0646382	−	0.298963i
$592$		0			0
$593$		−	25.9366i		−	1.06509i	−0.846402	−	0.532544i	$-0.821236\pi$
$593$		−	25.9366i		−	1.06509i	0.846402	−	0.532544i	$-0.178764\pi$
$594$		0			0
$595$			8.39230i			0.344051i
$596$		0			0
$597$	2.92820	+	13.5435i	0.119843	+	0.554297i
$598$		0			0
$599$	4.29311			0.175412			0.0877058	−	0.996146i	$-0.472046\pi$
$599$	4.29311			0.175412			0.0877058	+	0.996146i	$0.472046\pi$
$600$		0			0
$601$	−28.2487			−1.15229			−0.576144	−	0.817348i	$-0.695443\pi$
$601$	−28.2487			−1.15229			−0.576144	+	0.817348i	$0.695443\pi$
$602$		0			0
$603$	−4.29311	−	9.46410i	−0.174829	−	0.385408i
$604$		0			0
$605$		−	16.4432i		−	0.668509i
$606$		0			0
$607$		−	8.39230i		−	0.340633i	−0.985389	−	0.170317i	$-0.945521\pi$
$607$		−	8.39230i		−	0.340633i	0.985389	−	0.170317i	$-0.0544790\pi$
$608$		0			0
$609$	4.19615	−	0.907241i	0.170037	−	0.0367632i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−18.3923			−0.742858			−0.371429	−	0.928461i	$-0.621132\pi$
$613$	−18.3923			−0.742858			−0.371429	+	0.928461i	$0.621132\pi$
$614$		0			0
$615$	−53.0236	+	11.4641i	−2.13812	+	0.462277i
$616$		0			0
$617$		−	6.10759i		−	0.245882i	−0.992414	−	0.122941i	$-0.960767\pi$
$617$		−	6.10759i		−	0.245882i	0.992414	−	0.122941i	$-0.0392326\pi$
$618$		0			0
$619$		−	8.05256i		−	0.323660i	−0.986819	−	0.161830i	$-0.948260\pi$
$619$		−	8.05256i		−	0.323660i	0.986819	−	0.161830i	$-0.0517396\pi$
$620$		0			0
$621$	28.2487	−	20.9794i	1.13358	−	0.841872i
$622$		0			0
$623$	14.2076			0.569216
$624$		0			0
$625$	−15.5359			−0.621436
$626$		0			0
$627$	0.664146	+	3.07180i	0.0265234	+	0.122676i
$628$		0			0
$629$			11.2428i			0.448280i
$630$		0			0
$631$		−	18.9282i		−	0.753520i	−0.926311	−	0.376760i	$-0.877038\pi$
$631$		−	18.9282i		−	0.753520i	0.926311	−	0.376760i	$-0.122962\pi$
$632$		0			0
$633$	−2.73205	−	12.6362i	−0.108589	−	0.502245i
$634$		0			0
$635$	−21.6435			−0.858896
$636$		0			0
$637$	6.19615			0.245500
$638$		0			0
$639$	25.2725	−	11.4641i	0.999763	−	0.453513i
$640$		0			0
$641$			20.9794i			0.828635i	0.910132	+	0.414317i	$0.135980\pi$
$641$			20.9794i			0.828635i	−0.910132	+	0.414317i	$0.864020\pi$
$642$		0			0
$643$			44.0526i			1.73726i	0.495458	+	0.868632i	$0.335000\pi$
$643$			44.0526i			1.73726i	−0.495458	+	0.868632i	$0.665000\pi$
$644$		0			0
$645$	11.4641	−	2.47863i	0.451399	−	0.0975959i
$646$		0			0
$647$	30.2297			1.18845			0.594226	−	0.804298i	$-0.297458\pi$
$647$	30.2297			1.18845			0.594226	+	0.804298i	$0.297458\pi$
$648$		0			0
$649$	−20.6795			−0.811741
$650$		0			0
$651$	−16.0221	+	3.46410i	−0.627956	+	0.135769i
$652$		0			0
$653$			19.6511i			0.769005i	0.923124	+	0.384503i	$0.125627\pi$
$653$			19.6511i			0.769005i	−0.923124	+	0.384503i	$0.874373\pi$
$654$		0			0
$655$			17.6077i			0.687990i
$656$		0			0
$657$	−12.3923	+	5.62140i	−0.483470	+	0.219312i
$658$		0			0
$659$	−43.1090			−1.67929			−0.839645	−	0.543136i	$-0.817237\pi$
$659$	−43.1090			−1.67929			−0.839645	+	0.543136i	$0.817237\pi$
$660$		0			0
$661$	−20.4449			−0.795213			−0.397607	−	0.917556i	$-0.630159\pi$
$661$	−20.4449			−0.795213			−0.397607	+	0.917556i	$0.630159\pi$
$662$		0			0
$663$	5.62140	+	26.0000i	0.218317	+	1.00976i
$664$		0			0
$665$			2.47863i			0.0961170i
$666$		0			0
$667$			16.7846i			0.649903i
$668$		0			0
$669$	2.67949	+	12.3931i	0.103595	+	0.479146i
$670$		0			0
$671$	11.7290			0.452793
$672$		0			0
$673$	37.7128			1.45372			0.726861	−	0.686785i	$-0.240978\pi$
$673$	37.7128			1.45372			0.726861	+	0.686785i	$0.240978\pi$
$674$		0			0
$675$	26.9654	−	20.0263i	1.03790	−	0.770812i
$676$		0			0
$677$		−	18.7438i		−	0.720384i	−0.932878	−	0.360192i	$-0.882711\pi$
$677$		−	18.7438i		−	0.720384i	0.932878	−	0.360192i	$-0.117289\pi$
$678$		0			0
$679$		−	8.92820i		−	0.342633i
$680$		0			0
$681$	33.9808	−	7.34690i	1.30215	−	0.281534i
$682$		0			0
$683$	7.92207			0.303130			0.151565	−	0.988447i	$-0.451569\pi$
$683$	7.92207			0.303130			0.151565	+	0.988447i	$0.451569\pi$
$684$		0			0
$685$	−45.8564			−1.75208
$686$		0			0
$687$	−5.53242	+	1.19615i	−0.211075	+	0.0456361i
$688$		0			0
$689$		−	57.3167i		−	2.18359i
$690$		0			0
$691$		−	0.339746i		−	0.0129245i	−0.999979	−	0.00646227i	$-0.997943\pi$
$691$		−	0.339746i		−	0.0129245i	0.999979	−	0.00646227i	$-0.00205702\pi$
$692$		0			0
$693$	3.07180	+	6.77174i	0.116688	+	0.257237i
$694$		0			0
$695$	44.4373			1.68560
$696$		0			0
$697$	22.9282			0.868468
$698$		0			0
$699$	4.95725	+	22.9282i	0.187501	+	0.867224i
$700$		0			0
$701$		−	14.6938i		−	0.554977i	−0.960729	−	0.277489i	$-0.910498\pi$
$701$		−	14.6938i		−	0.554977i	0.960729	−	0.277489i	$-0.0895021\pi$
$702$		0			0
$703$			3.32051i			0.125235i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.34312			0.313775
$708$		0			0
$709$	−15.4641			−0.580767			−0.290383	−	0.956910i	$-0.593783\pi$
$709$	−15.4641			−0.580767			−0.290383	+	0.956910i	$0.593783\pi$
$710$		0			0
$711$	14.8718	+	32.7846i	0.557735	+	1.22952i
$712$		0			0
$713$		−	64.0884i		−	2.40013i
$714$		0			0
$715$			52.0000i			1.94469i
$716$		0			0
$717$	−42.7846	+	9.25036i	−1.59782	+	0.345461i
$718$		0			0
$719$	−4.95725			−0.184874			−0.0924372	−	0.995719i	$-0.529466\pi$
$719$	−4.95725			−0.184874			−0.0924372	+	0.995719i	$0.529466\pi$
$720$		0			0
$721$	14.9282			0.555955
$722$		0			0
$723$	23.7011	−	5.12436i	0.881452	−	0.190577i
$724$		0			0
$725$			16.0221i			0.595046i
$726$		0			0
$727$			2.92820i			0.108601i	0.998525	+	0.0543005i	$0.0172929\pi$
$727$			2.92820i			0.108601i	−0.998525	+	0.0543005i	$0.982707\pi$
$728$		0			0
$729$	7.80385	−	25.8476i	0.289031	−	0.957320i
$730$		0			0
$731$	−4.95725			−0.183351
$732$		0			0
$733$	20.7321			0.765756			0.382878	−	0.923799i	$-0.374933\pi$
$733$	20.7321			0.765756			0.382878	+	0.923799i	$0.374933\pi$
$734$		0			0
$735$	1.23931	+	5.73205i	0.0457128	+	0.211430i
$736$		0			0
$737$			8.58622i			0.316277i
$738$		0			0
$739$			32.9282i			1.21128i	0.795737	+	0.605642i	$0.207084\pi$
$739$			32.9282i			1.21128i	−0.795737	+	0.605642i	$0.792916\pi$
$740$		0			0
$741$	1.66025	+	7.67898i	0.0609910	+	0.282094i
$742$		0			0
$743$	40.1442			1.47275			0.736374	−	0.676574i	$-0.236536\pi$
$743$	40.1442			1.47275			0.736374	+	0.676574i	$0.236536\pi$
$744$		0			0
$745$	2.24871			0.0823864
$746$		0			0
$747$	22.7938	−	10.3397i	0.833983	−	0.378312i
$748$		0			0
$749$			9.25036i			0.338001i
$750$		0			0
$751$		−	16.2487i		−	0.592924i	−0.955045	−	0.296462i	$-0.904193\pi$
$751$		−	16.2487i		−	0.592924i	0.955045	−	0.296462i	$-0.0958068\pi$
$752$		0			0
$753$	2.66025	−	0.575167i	0.0969450	−	0.0209603i
$754$		0			0
$755$	25.2725			0.919759
$756$		0			0
$757$	45.0333			1.63676			0.818382	−	0.574675i	$-0.194871\pi$
$757$	45.0333			1.63676			0.818382	+	0.574675i	$0.194871\pi$
$758$		0			0
$759$	−28.4152	+	6.14359i	−1.03141	+	0.222998i
$760$		0			0
$761$		−	24.1221i		−	0.874426i	−0.899358	−	0.437213i	$-0.855966\pi$
$761$		−	24.1221i		−	0.874426i	0.899358	−	0.437213i	$-0.144034\pi$
$762$		0			0
$763$		−	6.39230i		−	0.231417i
$764$		0			0
$765$	−22.9282	+	10.4007i	−0.828971	+	0.376038i
$766$		0			0
$767$	−51.6953			−1.86661
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$	−4.05001	−	18.7321i	−0.145858	−	0.674618i
$772$		0			0
$773$			3.38587i			0.121781i	0.998144	+	0.0608906i	$0.0193941\pi$
$773$			3.38587i			0.121781i	−0.998144	+	0.0608906i	$0.980606\pi$
$774$		0			0
$775$		−	61.1769i		−	2.19754i
$776$		0			0
$777$	1.66025	+	7.67898i	0.0595613	+	0.275482i
$778$		0			0
$779$	6.77174			0.242623
$780$		0			0
$781$	−22.9282			−0.820436
$782$		0			0
$783$	7.67898	+	10.3397i	0.274424	+	0.369512i
$784$		0			0
$785$		−	16.0221i		−	0.571853i
$786$		0			0
$787$		−	30.5885i		−	1.09036i	−0.838319	−	0.545180i	$-0.816461\pi$
$787$		−	30.5885i		−	1.09036i	0.838319	−	0.545180i	$-0.183539\pi$
$788$		0			0
$789$	−38.5885	+	8.34312i	−1.37379	+	0.297023i
$790$		0			0
$791$	−11.0648			−0.393421
$792$		0			0
$793$	29.3205			1.04120
$794$		0			0
$795$	53.0236	−	11.4641i	1.88055	−	0.406590i
$796$		0			0
$797$			13.7866i			0.488345i	0.969732	+	0.244173i	$0.0785164\pi$
$797$			13.7866i			0.488345i	−0.969732	+	0.244173i	$0.921484\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	17.6077	+	38.8159i	0.622137	+	1.37149i
$802$		0			0
$803$	11.2428			0.396750
$804$		0			0
$805$	−22.9282			−0.808113
$806$		0			0
$807$	8.67520	+	40.1244i	0.305381	+	1.41244i
$808$		0			0
$809$		−	19.6511i		−	0.690895i	−0.938438	−	0.345447i	$-0.887727\pi$
$809$		−	19.6511i		−	0.690895i	0.938438	−	0.345447i	$-0.112273\pi$
$810$		0			0
$811$			12.3397i			0.433307i	0.976249	+	0.216654i	$0.0695142\pi$
$811$			12.3397i			0.433307i	−0.976249	+	0.216654i	$0.930486\pi$
$812$		0			0
$813$	7.07180	+	32.7083i	0.248019	+	1.14713i
$814$		0			0
$815$	23.9442			0.838728
$816$		0			0
$817$	−1.46410			−0.0512224
$818$		0			0
$819$	7.67898	+	16.9282i	0.268325	+	0.591519i
$820$		0			0
$821$		−	46.2518i		−	1.61420i	−0.590415	−	0.807100i	$-0.701036\pi$
$821$		−	46.2518i		−	1.61420i	0.590415	−	0.807100i	$-0.298964\pi$
$822$		0			0
$823$		−	48.0000i		−	1.67317i	−0.547833	−	0.836587i	$-0.684547\pi$
$823$		−	48.0000i		−	1.67317i	0.547833	−	0.836587i	$-0.315453\pi$
$824$		0			0
$825$	−27.1244	+	5.86450i	−0.944349	+	0.204176i
$826$		0			0
$827$	−4.29311			−0.149286			−0.0746430	−	0.997210i	$-0.523782\pi$
$827$	−4.29311			−0.149286			−0.0746430	+	0.997210i	$0.523782\pi$
$828$		0			0
$829$	27.2679			0.947055			0.473528	−	0.880779i	$-0.342980\pi$
$829$	27.2679			0.947055			0.473528	+	0.880779i	$0.342980\pi$
$830$		0			0
$831$	5.20035	−	1.12436i	0.180398	−	0.0390035i
$832$		0			0
$833$		−	2.47863i		−	0.0858793i
$834$		0			0
$835$		−	85.5692i		−	2.96124i
$836$		0			0
$837$	−29.3205	−	39.4801i	−1.01347	−	1.36463i
$838$		0			0
$839$	16.6862			0.576073			0.288037	−	0.957619i	$-0.406998\pi$
$839$	16.6862			0.576073			0.288037	+	0.957619i	$0.406998\pi$
$840$		0			0
$841$	22.8564			0.788152
$842$		0			0
$843$	−9.91451	−	45.8564i	−0.341474	−	1.57938i
$844$		0			0
$845$			85.9750i			2.95763i
$846$		0			0
$847$			4.85641i			0.166868i
$848$		0			0
$849$	−4.51666	−	20.8904i	−0.155011	−	0.716956i
$850$		0			0
$851$	−30.7159			−1.05293
$852$		0			0
$853$	44.0526			1.50833			0.754165	−	0.656684i	$-0.228042\pi$
$853$	44.0526			1.50833			0.754165	+	0.656684i	$0.228042\pi$
$854$		0			0
$855$	−6.77174	+	3.07180i	−0.231588	+	0.105053i
$856$		0			0
$857$			14.2076i			0.485323i	0.970111	+	0.242661i	$0.0780204\pi$
$857$			14.2076i			0.485323i	−0.970111	+	0.242661i	$0.921980\pi$
$858$		0			0
$859$			12.0526i			0.411228i	0.978633	+	0.205614i	$0.0659191\pi$
$859$			12.0526i			0.411228i	−0.978633	+	0.205614i	$0.934081\pi$
$860$		0			0
$861$	15.6603	−	3.38587i	0.533700	−	0.115390i
$862$		0			0
$863$	−29.0794			−0.989874			−0.494937	−	0.868929i	$-0.664809\pi$
$863$	−29.0794			−0.989874			−0.494937	+	0.868929i	$0.664809\pi$
$864$		0			0
$865$	34.3923			1.16937
$866$		0			0
$867$	−18.3792	+	3.97372i	−0.624190	+	0.134955i
$868$		0			0
$869$		−	29.7435i		−	1.00898i
$870$		0			0
$871$			21.4641i			0.727283i
$872$		0			0
$873$	24.3923	−	11.0648i	0.825554	−	0.374488i
$874$		0			0
$875$	−4.95725			−0.167586
$876$		0			0
$877$	26.3923			0.891205			0.445602	−	0.895231i	$-0.352990\pi$
$877$	26.3923			0.891205			0.445602	+	0.895231i	$0.352990\pi$
$878$		0			0
$879$	−5.53242	−	25.5885i	−0.186604	−	0.863077i
$880$		0			0
$881$			24.6083i			0.829075i	0.910032	+	0.414538i	$0.136057\pi$
$881$			24.6083i			0.829075i	−0.910032	+	0.414538i	$0.863943\pi$
$882$		0			0
$883$			20.5359i			0.691088i	0.938403	+	0.345544i	$0.112306\pi$
$883$			20.5359i			0.691088i	−0.938403	+	0.345544i	$0.887694\pi$
$884$		0			0
$885$	−10.3397	−	47.8232i	−0.347567	−	1.60756i
$886$		0			0
$887$	1.81448			0.0609243			0.0304622	−	0.999536i	$-0.490302\pi$
$887$	1.81448			0.0609243			0.0304622	+	0.999536i	$0.490302\pi$
$888$		0			0
$889$	6.39230			0.214391
$890$		0			0
$891$	−14.6938	+	16.7846i	−0.492261	+	0.562306i
$892$		0			0
$893$		0			0
$894$		0			0
$895$			2.24871i			0.0751661i
$896$		0			0
$897$	−71.0333	+	15.3580i	−2.37173	+	0.512787i
$898$		0			0
$899$	23.4580			0.782367
$900$		0			0
$901$	−22.9282			−0.763849
$902$		0			0
$903$	−3.38587	+	0.732051i	−0.112675	+	0.0243611i
$904$		0			0
$905$		−	24.6083i		−	0.818008i
$906$		0			0
$907$			28.2487i			0.937983i	0.883203	+	0.468992i	$0.155383\pi$
$907$			28.2487i			0.937983i	−0.883203	+	0.468992i	$0.844617\pi$
$908$		0			0
$909$	10.3397	+	22.7938i	0.342948	+	0.756024i
$910$		0			0
$911$	−21.6435			−0.717081			−0.358541	−	0.933514i	$-0.616726\pi$
$911$	−21.6435			−0.717081			−0.358541	+	0.933514i	$0.616726\pi$
$912$		0			0
$913$	−20.6795			−0.684391
$914$		0			0
$915$	5.86450	+	27.1244i	0.193874	+	0.896704i
$916$		0			0
$917$		−	5.20035i		−	0.171731i
$918$		0			0
$919$			19.7128i			0.650266i	0.945668	+	0.325133i	$0.105409\pi$
$919$			19.7128i			0.650266i	−0.945668	+	0.325133i	$0.894591\pi$
$920$		0			0
$921$	12.1244	+	56.0773i	0.399511	+	1.84781i
$922$		0			0
$923$	−57.3167			−1.88660
$924$		0			0
$925$	−29.3205			−0.964052
$926$		0			0
$927$	18.5007	+	40.7846i	0.607644	+	1.33954i
$928$		0			0
$929$			29.5656i			0.970015i	0.874510	+	0.485007i	$0.161183\pi$
$929$			29.5656i			0.970015i	−0.874510	+	0.485007i	$0.838817\pi$
$930$		0			0
$931$		−	0.732051i		−	0.0239920i
$932$		0			0
$933$	−5.32051	+	1.15033i	−0.174186	+	0.0376603i
$934$		0			0
$935$	20.8014			0.680278
$936$		0			0
$937$	51.1769			1.67188			0.835938	−	0.548823i	$-0.184924\pi$
$937$	51.1769			1.67188			0.835938	+	0.548823i	$0.184924\pi$
$938$		0			0
$939$	−26.1797	+	5.66025i	−0.854342	+	0.184715i
$940$		0			0
$941$			28.1721i			0.918386i	0.888337	+	0.459193i	$0.151861\pi$
$941$			28.1721i			0.918386i	−0.888337	+	0.459193i	$0.848139\pi$
$942$		0			0
$943$			62.6410i			2.03987i
$944$		0			0
$945$	−14.1244	+	10.4897i	−0.459465	+	0.341229i
$946$		0			0
$947$	−39.4801			−1.28293			−0.641465	−	0.767152i	$-0.721673\pi$
$947$	−39.4801			−1.28293			−0.641465	+	0.767152i	$0.721673\pi$
$948$		0			0
$949$	28.1051			0.912331
$950$		0			0
$951$	−7.01483	−	32.4449i	−0.227471	−	1.05210i
$952$		0			0
$953$			55.5022i			1.79789i	0.438060	+	0.898946i	$0.355666\pi$
$953$			55.5022i			1.79789i	−0.438060	+	0.898946i	$0.644334\pi$
$954$		0			0
$955$		−	14.5359i		−	0.470371i
$956$		0			0
$957$	−2.24871	−	10.4007i	−0.0726905	−	0.336207i
$958$		0			0
$959$	13.5435			0.437342
$960$		0			0
$961$	−58.5692			−1.88933
$962$		0			0
$963$	−25.2725	+	11.4641i	−0.814394	+	0.369426i
$964$		0			0
$965$		−	32.0442i		−	1.03154i
$966$		0			0
$967$		−	43.4641i		−	1.39771i	−0.715263	−	0.698856i	$-0.753693\pi$
$967$		−	43.4641i		−	1.39771i	0.715263	−	0.698856i	$-0.246307\pi$
$968$		0			0
$969$	3.07180	−	0.664146i	0.0986803	−	0.0213354i
$970$		0			0
$971$	25.5156			0.818833			0.409417	−	0.912348i	$-0.365732\pi$
$971$	25.5156			0.818833			0.409417	+	0.912348i	$0.365732\pi$
$972$		0			0
$973$	−13.1244			−0.420748
$974$		0			0
$975$	−67.8063	+	14.6603i	−2.17154	+	0.469504i
$976$		0			0
$977$			18.5007i			0.591891i	0.955205	+	0.295945i	$0.0956346\pi$
$977$			18.5007i			0.591891i	−0.955205	+	0.295945i	$0.904365\pi$
$978$		0			0
$979$		−	35.2154i		−	1.12549i
$980$		0			0
$981$	17.4641	−	7.92207i	0.557586	−	0.252932i
$982$		0			0
$983$	−25.2725			−0.806066			−0.403033	−	0.915185i	$-0.632044\pi$
$983$	−25.2725			−0.806066			−0.403033	+	0.915185i	$0.632044\pi$
$984$		0			0
$985$	−14.5359			−0.463152
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	13.5435i		−	0.430657i
$990$		0			0
$991$		−	6.14359i		−	0.195158i	−0.995228	−	0.0975788i	$-0.968890\pi$
$991$		−	6.14359i		−	0.195158i	0.995228	−	0.0975788i	$-0.0311098\pi$
$992$		0			0
$993$	−11.2679	−	52.1163i	−0.357578	−	1.65386i
$994$		0			0
$995$	27.0869			0.858714
$996$		0			0
$997$	33.9090			1.07391			0.536954	−	0.843612i	$-0.319575\pi$
$997$	33.9090			1.07391			0.536954	+	0.843612i	$0.319575\pi$
$998$		0			0
$999$	−18.9218	+	14.0526i	−0.598659	+	0.444603i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.h.b.239.2	yes	8
3.2	odd	2	inner	336.2.h.b.239.8	yes	8
4.3	odd	2	inner	336.2.h.b.239.7	yes	8
7.6	odd	2		2352.2.h.o.2255.7		8
8.3	odd	2		1344.2.h.g.575.2		8
8.5	even	2		1344.2.h.g.575.7		8
12.11	even	2	inner	336.2.h.b.239.1	✓	8
21.20	even	2		2352.2.h.o.2255.1		8
24.5	odd	2		1344.2.h.g.575.1		8
24.11	even	2		1344.2.h.g.575.8		8
28.27	even	2		2352.2.h.o.2255.2		8
84.83	odd	2		2352.2.h.o.2255.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.h.b.239.1	✓	8	12.11	even	2	inner
336.2.h.b.239.2	yes	8	1.1	even	1	trivial
336.2.h.b.239.7	yes	8	4.3	odd	2	inner
336.2.h.b.239.8	yes	8	3.2	odd	2	inner
1344.2.h.g.575.1		8	24.5	odd	2
1344.2.h.g.575.2		8	8.3	odd	2
1344.2.h.g.575.7		8	8.5	even	2
1344.2.h.g.575.8		8	24.11	even	2
2352.2.h.o.2255.1		8	21.20	even	2
2352.2.h.o.2255.2		8	28.27	even	2
2352.2.h.o.2255.7		8	7.6	odd	2
2352.2.h.o.2255.8		8	84.83	odd	2

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.69293 + 0.366025i) q^{3} +3.38587i q^{5} -1.00000i q^{7} +(2.73205 - 1.23931i) q^{9} +O(q^{10})\)	\(q+(-1.69293 + 0.366025i) q^{3} +3.38587i q^{5} -1.00000i q^{7} +(2.73205 - 1.23931i) q^{9} -2.47863 q^{11} -6.19615 q^{13} +(-1.23931 - 5.73205i) q^{15} +2.47863i q^{17} +0.732051i q^{19} +(0.366025 + 1.69293i) q^{21} -6.77174 q^{23} -6.46410 q^{25} +(-4.17156 + 3.09808i) q^{27} -2.47863i q^{29} +9.46410i q^{31} +(4.19615 - 0.907241i) q^{33} +3.38587 q^{35} +4.53590 q^{37} +(10.4897 - 2.26795i) q^{39} -9.25036i q^{41} +2.00000i q^{43} +(4.19615 + 9.25036i) q^{45} -1.00000 q^{49} +(-0.907241 - 4.19615i) q^{51} +9.25036i q^{53} -8.39230i q^{55} +(-0.267949 - 1.23931i) q^{57} +8.34312 q^{59} -4.73205 q^{61} +(-1.23931 - 2.73205i) q^{63} -20.9794i q^{65} -3.46410i q^{67} +(11.4641 - 2.47863i) q^{69} +9.25036 q^{71} -4.53590 q^{73} +(10.9433 - 2.36603i) q^{75} +2.47863i q^{77} +12.0000i q^{79} +(5.92820 - 6.77174i) q^{81} +8.34312 q^{83} -8.39230 q^{85} +(0.907241 + 4.19615i) q^{87} +14.2076i q^{89} +6.19615i q^{91} +(-3.46410 - 16.0221i) q^{93} -2.47863 q^{95} +8.92820 q^{97} +(-6.77174 + 3.07180i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} - 8 q^{13} - 4 q^{21} - 24 q^{25} - 8 q^{33} + 64 q^{37} - 8 q^{45} - 8 q^{49} - 16 q^{57} - 24 q^{61} + 64 q^{69} - 64 q^{73} - 8 q^{81} + 16 q^{85} + 16 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 - 8 * q^13 - 4 * q^21 - 24 * q^25 - 8 * q^33 + 64 * q^37 - 8 * q^45 - 8 * q^49 - 16 * q^57 - 24 * q^61 + 64 * q^69 - 64 * q^73 - 8 * q^81 + 16 * q^85 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more