LMFDB - Embedded newform 384.2.c.c.383.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,2,Mod(383,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("384.383");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	384.c (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:	$3.06625543762$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.2.c.c.383.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.618034 - 1.61803i) q^{3} -3.23607i q^{5} -1.23607i q^{7} +(-2.23607 + 2.00000i) q^{9} -5.23607 q^{11} +4.47214 q^{13} +(-5.23607 + 2.00000i) q^{15} +2.47214i q^{17} +0.763932i q^{19} +(-2.00000 + 0.763932i) q^{21} -2.47214 q^{23} -5.47214 q^{25} +(4.61803 + 2.38197i) q^{27} -4.76393i q^{29} -5.23607i q^{31} +(3.23607 + 8.47214i) q^{33} -4.00000 q^{35} -8.47214 q^{37} +(-2.76393 - 7.23607i) q^{39} -6.47214i q^{41} -7.23607i q^{43} +(6.47214 + 7.23607i) q^{45} +8.00000 q^{47} +5.47214 q^{49} +(4.00000 - 1.52786i) q^{51} -3.23607i q^{53} +16.9443i q^{55} +(1.23607 - 0.472136i) q^{57} +1.23607 q^{59} -0.472136 q^{61} +(2.47214 + 2.76393i) q^{63} -14.4721i q^{65} +9.70820i q^{67} +(1.52786 + 4.00000i) q^{69} +15.4164 q^{71} -2.00000 q^{73} +(3.38197 + 8.85410i) q^{75} +6.47214i q^{77} -0.291796i q^{79} +(1.00000 - 8.94427i) q^{81} -2.76393 q^{83} +8.00000 q^{85} +(-7.70820 + 2.94427i) q^{87} -4.00000i q^{89} -5.52786i q^{91} +(-8.47214 + 3.23607i) q^{93} +2.47214 q^{95} +0.472136 q^{97} +(11.7082 - 10.4721i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 12 q^{11} - 12 q^{15} - 8 q^{21} + 8 q^{23} - 4 q^{25} + 14 q^{27} + 4 q^{33} - 16 q^{35} - 16 q^{37} - 20 q^{39} + 8 q^{45} + 32 q^{47} + 4 q^{49} + 16 q^{51} - 4 q^{57} - 4 q^{59} + 16 q^{61}+ \cdots + 20 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 12 * q^11 - 12 * q^15 - 8 * q^21 + 8 * q^23 - 4 * q^25 + 14 * q^27 + 4 * q^33 - 16 * q^35 - 16 * q^37 - 20 * q^39 + 8 * q^45 + 32 * q^47 + 4 * q^49 + 16 * q^51 - 4 * q^57 - 4 * q^59 + 16 * q^61 - 8 * q^63 + 24 * q^69 + 8 * q^71 - 8 * q^73 + 18 * q^75 + 4 * q^81 - 20 * q^83 + 32 * q^85 - 4 * q^87 - 16 * q^93 - 8 * q^95 - 16 * q^97 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−0.618034	−	1.61803i	−0.356822	−	0.934172i
$3$	−0.618034	−	1.61803i	−0.356822	−	0.934172i
$4$		0			0
$5$		−	3.23607i		−	1.44721i	−0.690212	−	0.723607i	$-0.742483\pi$
$5$		−	3.23607i		−	1.44721i	0.690212	−	0.723607i	$-0.257517\pi$
$6$		0			0
$7$		−	1.23607i		−	0.467190i	−0.972334	−	0.233595i	$-0.924951\pi$
$7$		−	1.23607i		−	0.467190i	0.972334	−	0.233595i	$-0.0750489\pi$
$8$		0			0
$9$	−2.23607	+	2.00000i	−0.745356	+	0.666667i
$10$		0			0
$11$	−5.23607			−1.57873			−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607			−1.57873			−0.789367	+	0.613922i	$0.789591\pi$
$12$		0			0
$13$	4.47214			1.24035			0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214			1.24035			0.620174	+	0.784465i	$0.287062\pi$
$14$		0			0
$15$	−5.23607	+	2.00000i	−1.35195	+	0.516398i
$16$		0			0
$17$			2.47214i			0.599581i	0.954005	+	0.299791i	$0.0969168\pi$
$17$			2.47214i			0.599581i	−0.954005	+	0.299791i	$0.903083\pi$
$18$		0			0
$19$			0.763932i			0.175258i	0.996153	+	0.0876290i	$0.0279290\pi$
$19$			0.763932i			0.175258i	−0.996153	+	0.0876290i	$0.972071\pi$
$20$		0			0
$21$	−2.00000	+	0.763932i	−0.436436	+	0.166704i
$22$		0			0
$23$	−2.47214			−0.515476			−0.257738	−	0.966215i	$-0.582977\pi$
$23$	−2.47214			−0.515476			−0.257738	+	0.966215i	$0.582977\pi$
$24$		0			0
$25$	−5.47214			−1.09443
$26$		0			0
$27$	4.61803	+	2.38197i	0.888741	+	0.458410i
$28$		0			0
$29$		−	4.76393i		−	0.884640i	−0.896857	−	0.442320i	$-0.854156\pi$
$29$		−	4.76393i		−	0.884640i	0.896857	−	0.442320i	$-0.145844\pi$
$30$		0			0
$31$		−	5.23607i		−	0.940426i	−0.882553	−	0.470213i	$-0.844177\pi$
$31$		−	5.23607i		−	0.940426i	0.882553	−	0.470213i	$-0.155823\pi$
$32$		0			0
$33$	3.23607	+	8.47214i	0.563327	+	1.47481i
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$	−8.47214			−1.39281			−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214			−1.39281			−0.696405	+	0.717649i	$0.745218\pi$
$38$		0			0
$39$	−2.76393	−	7.23607i	−0.442583	−	1.15870i
$40$		0			0
$41$		−	6.47214i		−	1.01078i	−0.862892	−	0.505389i	$-0.831349\pi$
$41$		−	6.47214i		−	1.01078i	0.862892	−	0.505389i	$-0.168651\pi$
$42$		0			0
$43$		−	7.23607i		−	1.10349i	−0.834013	−	0.551745i	$-0.813962\pi$
$43$		−	7.23607i		−	1.10349i	0.834013	−	0.551745i	$-0.186038\pi$
$44$		0			0
$45$	6.47214	+	7.23607i	0.964809	+	1.07869i
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	5.47214			0.781734
$50$		0			0
$51$	4.00000	−	1.52786i	0.560112	−	0.213944i
$52$		0			0
$53$		−	3.23607i		−	0.444508i	−0.974989	−	0.222254i	$-0.928659\pi$
$53$		−	3.23607i		−	0.444508i	0.974989	−	0.222254i	$-0.0713414\pi$
$54$		0			0
$55$			16.9443i			2.28477i
$56$		0			0
$57$	1.23607	−	0.472136i	0.163721	−	0.0625359i
$58$		0			0
$59$	1.23607			0.160922			0.0804612	−	0.996758i	$-0.474361\pi$
$59$	1.23607			0.160922			0.0804612	+	0.996758i	$0.474361\pi$
$60$		0			0
$61$	−0.472136			−0.0604508			−0.0302254	−	0.999543i	$-0.509623\pi$
$61$	−0.472136			−0.0604508			−0.0302254	+	0.999543i	$0.509623\pi$
$62$		0			0
$63$	2.47214	+	2.76393i	0.311460	+	0.348223i
$64$		0			0
$65$		−	14.4721i		−	1.79505i
$66$		0			0
$67$			9.70820i			1.18605i	0.805186	+	0.593023i	$0.202066\pi$
$67$			9.70820i			1.18605i	−0.805186	+	0.593023i	$0.797934\pi$
$68$		0			0
$69$	1.52786	+	4.00000i	0.183933	+	0.481543i
$70$		0			0
$71$	15.4164			1.82959			0.914796	−	0.403917i	$-0.132352\pi$
$71$	15.4164			1.82959			0.914796	+	0.403917i	$0.132352\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$	3.38197	+	8.85410i	0.390516	+	1.02238i
$76$		0			0
$77$			6.47214i			0.737568i
$78$		0			0
$79$		−	0.291796i		−	0.0328296i	−0.999865	−	0.0164148i	$-0.994775\pi$
$79$		−	0.291796i		−	0.0328296i	0.999865	−	0.0164148i	$-0.00522523\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	−2.76393			−0.303381			−0.151690	−	0.988428i	$-0.548472\pi$
$83$	−2.76393			−0.303381			−0.151690	+	0.988428i	$0.548472\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$	−7.70820	+	2.94427i	−0.826406	+	0.315659i
$88$		0			0
$89$		−	4.00000i		−	0.423999i	−0.977270	−	0.212000i	$-0.932002\pi$
$89$		−	4.00000i		−	0.423999i	0.977270	−	0.212000i	$-0.0679975\pi$
$90$		0			0
$91$		−	5.52786i		−	0.579478i
$92$		0			0
$93$	−8.47214	+	3.23607i	−0.878520	+	0.335565i
$94$		0			0
$95$	2.47214			0.253636
$96$		0			0
$97$	0.472136			0.0479381			0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136			0.0479381			0.0239691	+	0.999713i	$0.492370\pi$
$98$		0			0
$99$	11.7082	−	10.4721i	1.17672	−	1.05249i
$100$		0			0
$101$			1.70820i			0.169973i	0.996382	+	0.0849863i	$0.0270847\pi$
$101$			1.70820i			0.169973i	−0.996382	+	0.0849863i	$0.972915\pi$
$102$		0			0
$103$		−	14.1803i		−	1.39723i	−0.715498	−	0.698615i	$-0.753800\pi$
$103$		−	14.1803i		−	1.39723i	0.715498	−	0.698615i	$-0.246200\pi$
$104$		0			0
$105$	2.47214	+	6.47214i	0.241256	+	0.631616i
$106$		0			0
$107$	14.1803			1.37087			0.685433	−	0.728136i	$-0.259613\pi$
$107$	14.1803			1.37087			0.685433	+	0.728136i	$0.259613\pi$
$108$		0			0
$109$	−8.47214			−0.811483			−0.405742	−	0.913988i	$-0.632987\pi$
$109$	−8.47214			−0.811483			−0.405742	+	0.913988i	$0.632987\pi$
$110$		0			0
$111$	5.23607	+	13.7082i	0.496986	+	1.30113i
$112$		0			0
$113$			8.00000i			0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$			8.00000i			0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$		0			0
$115$			8.00000i			0.746004i
$116$		0			0
$117$	−10.0000	+	8.94427i	−0.924500	+	0.826898i
$118$		0			0
$119$	3.05573			0.280118
$120$		0			0
$121$	16.4164			1.49240
$122$		0			0
$123$	−10.4721	+	4.00000i	−0.944241	+	0.360668i
$124$		0			0
$125$			1.52786i			0.136656i
$126$		0			0
$127$		−	13.2361i		−	1.17451i	−0.809402	−	0.587256i	$-0.800208\pi$
$127$		−	13.2361i		−	1.17451i	0.809402	−	0.587256i	$-0.199792\pi$
$128$		0			0
$129$	−11.7082	+	4.47214i	−1.03085	+	0.393750i
$130$		0			0
$131$	6.76393			0.590967			0.295484	−	0.955348i	$-0.404519\pi$
$131$	6.76393			0.590967			0.295484	+	0.955348i	$0.404519\pi$
$132$		0			0
$133$	0.944272			0.0818788
$134$		0			0
$135$	7.70820	−	14.9443i	0.663417	−	1.28620i
$136$		0			0
$137$		−	1.52786i		−	0.130534i	−0.997868	−	0.0652671i	$-0.979210\pi$
$137$		−	1.52786i		−	0.130534i	0.997868	−	0.0652671i	$-0.0207899\pi$
$138$		0			0
$139$			9.70820i			0.823439i	0.911311	+	0.411720i	$0.135072\pi$
$139$			9.70820i			0.823439i	−0.911311	+	0.411720i	$0.864928\pi$
$140$		0			0
$141$	−4.94427	−	12.9443i	−0.416383	−	1.09010i
$142$		0			0
$143$	−23.4164			−1.95818
$144$		0			0
$145$	−15.4164			−1.28026
$146$		0			0
$147$	−3.38197	−	8.85410i	−0.278940	−	0.730274i
$148$		0			0
$149$			9.70820i			0.795327i	0.917531	+	0.397664i	$0.130179\pi$
$149$			9.70820i			0.795327i	−0.917531	+	0.397664i	$0.869821\pi$
$150$		0			0
$151$			19.7082i			1.60383i	0.597438	+	0.801915i	$0.296186\pi$
$151$			19.7082i			1.60383i	−0.597438	+	0.801915i	$0.703814\pi$
$152$		0			0
$153$	−4.94427	−	5.52786i	−0.399721	−	0.446901i
$154$		0			0
$155$	−16.9443			−1.36100
$156$		0			0
$157$	4.47214			0.356915			0.178458	−	0.983948i	$-0.442889\pi$
$157$	4.47214			0.356915			0.178458	+	0.983948i	$0.442889\pi$
$158$		0			0
$159$	−5.23607	+	2.00000i	−0.415247	+	0.158610i
$160$		0			0
$161$			3.05573i			0.240825i
$162$		0			0
$163$		−	4.18034i		−	0.327429i	−0.986508	−	0.163715i	$-0.947652\pi$
$163$		−	4.18034i		−	0.327429i	0.986508	−	0.163715i	$-0.0523477\pi$
$164$		0			0
$165$	27.4164	−	10.4721i	2.13436	−	0.815255i
$166$		0			0
$167$	−13.5279			−1.04682			−0.523409	−	0.852082i	$-0.675340\pi$
$167$	−13.5279			−1.04682			−0.523409	+	0.852082i	$0.675340\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$	−1.52786	−	1.70820i	−0.116839	−	0.130630i
$172$		0			0
$173$		−	22.6525i		−	1.72224i	−0.508405	−	0.861118i	$-0.669765\pi$
$173$		−	22.6525i		−	1.72224i	0.508405	−	0.861118i	$-0.330235\pi$
$174$		0			0
$175$			6.76393i			0.511305i
$176$		0			0
$177$	−0.763932	−	2.00000i	−0.0574206	−	0.150329i
$178$		0			0
$179$	−6.18034			−0.461940			−0.230970	−	0.972961i	$-0.574190\pi$
$179$	−6.18034			−0.461940			−0.230970	+	0.972961i	$0.574190\pi$
$180$		0			0
$181$	20.4721			1.52168			0.760841	−	0.648938i	$-0.224787\pi$
$181$	20.4721			1.52168			0.760841	+	0.648938i	$0.224787\pi$
$182$		0			0
$183$	0.291796	+	0.763932i	0.0215702	+	0.0564715i
$184$		0			0
$185$			27.4164i			2.01569i
$186$		0			0
$187$		−	12.9443i		−	0.946579i
$188$		0			0
$189$	2.94427	−	5.70820i	0.214164	−	0.415211i
$190$		0			0
$191$	12.9443			0.936615			0.468307	−	0.883566i	$-0.344864\pi$
$191$	12.9443			0.936615			0.468307	+	0.883566i	$0.344864\pi$
$192$		0			0
$193$	−11.8885			−0.855756			−0.427878	−	0.903836i	$-0.640739\pi$
$193$	−11.8885			−0.855756			−0.427878	+	0.903836i	$0.640739\pi$
$194$		0			0
$195$	−23.4164	+	8.94427i	−1.67688	+	0.640513i
$196$		0			0
$197$			4.76393i			0.339416i	0.985494	+	0.169708i	$0.0542825\pi$
$197$			4.76393i			0.339416i	−0.985494	+	0.169708i	$0.945718\pi$
$198$		0			0
$199$		−	6.18034i		−	0.438113i	−0.975712	−	0.219056i	$-0.929702\pi$
$199$		−	6.18034i		−	0.438113i	0.975712	−	0.219056i	$-0.0702979\pi$
$200$		0			0
$201$	15.7082	−	6.00000i	1.10797	−	0.423207i
$202$		0			0
$203$	−5.88854			−0.413295
$204$		0			0
$205$	−20.9443			−1.46281
$206$		0			0
$207$	5.52786	−	4.94427i	0.384213	−	0.343651i
$208$		0			0
$209$		−	4.00000i		−	0.276686i
$210$		0			0
$211$		−	11.2361i		−	0.773523i	−0.922180	−	0.386761i	$-0.873594\pi$
$211$		−	11.2361i		−	0.773523i	0.922180	−	0.386761i	$-0.126406\pi$
$212$		0			0
$213$	−9.52786	−	24.9443i	−0.652838	−	1.70915i
$214$		0			0
$215$	−23.4164			−1.59699
$216$		0			0
$217$	−6.47214			−0.439357
$218$		0			0
$219$	1.23607	+	3.23607i	0.0835257	+	0.218673i
$220$		0			0
$221$			11.0557i			0.743689i
$222$		0			0
$223$		−	10.1803i		−	0.681726i	−0.940113	−	0.340863i	$-0.889281\pi$
$223$		−	10.1803i		−	0.681726i	0.940113	−	0.340863i	$-0.110719\pi$
$224$		0			0
$225$	12.2361	−	10.9443i	0.815738	−	0.729618i
$226$		0			0
$227$	2.18034			0.144714			0.0723571	−	0.997379i	$-0.476948\pi$
$227$	2.18034			0.144714			0.0723571	+	0.997379i	$0.476948\pi$
$228$		0			0
$229$	−11.5279			−0.761783			−0.380891	−	0.924620i	$-0.624383\pi$
$229$	−11.5279			−0.761783			−0.380891	+	0.924620i	$0.624383\pi$
$230$		0			0
$231$	10.4721	−	4.00000i	0.689016	−	0.263181i
$232$		0			0
$233$			24.9443i			1.63415i	0.576529	+	0.817077i	$0.304407\pi$
$233$			24.9443i			1.63415i	−0.576529	+	0.817077i	$0.695593\pi$
$234$		0			0
$235$		−	25.8885i		−	1.68878i
$236$		0			0
$237$	−0.472136	+	0.180340i	−0.0306685	+	0.0117143i
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		0			0
$243$	−15.0902	+	3.90983i	−0.968035	+	0.250816i
$244$		0			0
$245$		−	17.7082i		−	1.13134i
$246$		0			0
$247$			3.41641i			0.217381i
$248$		0			0
$249$	1.70820	+	4.47214i	0.108253	+	0.283410i
$250$		0			0
$251$	23.7082			1.49645			0.748224	−	0.663446i	$-0.230907\pi$
$251$	23.7082			1.49645			0.748224	+	0.663446i	$0.230907\pi$
$252$		0			0
$253$	12.9443			0.813799
$254$		0			0
$255$	−4.94427	−	12.9443i	−0.309622	−	0.810602i
$256$		0			0
$257$			20.9443i			1.30647i	0.757156	+	0.653234i	$0.226588\pi$
$257$			20.9443i			1.30647i	−0.757156	+	0.653234i	$0.773412\pi$
$258$		0			0
$259$			10.4721i			0.650707i
$260$		0			0
$261$	9.52786	+	10.6525i	0.589760	+	0.659372i
$262$		0			0
$263$	−10.4721			−0.645740			−0.322870	−	0.946443i	$-0.604648\pi$
$263$	−10.4721			−0.645740			−0.322870	+	0.946443i	$0.604648\pi$
$264$		0			0
$265$	−10.4721			−0.643298
$266$		0			0
$267$	−6.47214	+	2.47214i	−0.396088	+	0.151292i
$268$		0			0
$269$			16.1803i			0.986533i	0.869878	+	0.493266i	$0.164197\pi$
$269$			16.1803i			0.986533i	−0.869878	+	0.493266i	$0.835803\pi$
$270$		0			0
$271$		−	0.291796i		−	0.0177253i	−0.999961	−	0.00886267i	$-0.997179\pi$
$271$		−	0.291796i		−	0.0177253i	0.999961	−	0.00886267i	$-0.00282111\pi$
$272$		0			0
$273$	−8.94427	+	3.41641i	−0.541332	+	0.206770i
$274$		0			0
$275$	28.6525			1.72781
$276$		0			0
$277$	7.52786			0.452306			0.226153	−	0.974092i	$-0.427385\pi$
$277$	7.52786			0.452306			0.226153	+	0.974092i	$0.427385\pi$
$278$		0			0
$279$	10.4721	+	11.7082i	0.626950	+	0.700952i
$280$		0			0
$281$		−	24.9443i		−	1.48805i	−0.668151	−	0.744025i	$-0.732914\pi$
$281$		−	24.9443i		−	1.48805i	0.668151	−	0.744025i	$-0.267086\pi$
$282$		0			0
$283$			20.7639i			1.23429i	0.786850	+	0.617144i	$0.211710\pi$
$283$			20.7639i			1.23429i	−0.786850	+	0.617144i	$0.788290\pi$
$284$		0			0
$285$	−1.52786	−	4.00000i	−0.0905029	−	0.236940i
$286$		0			0
$287$	−8.00000			−0.472225
$288$		0			0
$289$	10.8885			0.640503
$290$		0			0
$291$	−0.291796	−	0.763932i	−0.0171054	−	0.0447825i
$292$		0			0
$293$			20.7639i			1.21304i	0.795068	+	0.606521i	$0.207435\pi$
$293$			20.7639i			1.21304i	−0.795068	+	0.606521i	$0.792565\pi$
$294$		0			0
$295$		−	4.00000i		−	0.232889i
$296$		0			0
$297$	−24.1803	−	12.4721i	−1.40309	−	0.723707i
$298$		0			0
$299$	−11.0557			−0.639369
$300$		0			0
$301$	−8.94427			−0.515539
$302$		0			0
$303$	2.76393	−	1.05573i	0.158784	−	0.0606500i
$304$		0			0
$305$			1.52786i			0.0874852i
$306$		0			0
$307$			27.5967i			1.57503i	0.616296	+	0.787515i	$0.288633\pi$
$307$			27.5967i			1.57503i	−0.616296	+	0.787515i	$0.711367\pi$
$308$		0			0
$309$	−22.9443	+	8.76393i	−1.30525	+	0.498563i
$310$		0			0
$311$	−5.52786			−0.313456			−0.156728	−	0.987642i	$-0.550095\pi$
$311$	−5.52786			−0.313456			−0.156728	+	0.987642i	$0.550095\pi$
$312$		0			0
$313$	−0.472136			−0.0266867			−0.0133434	−	0.999911i	$-0.504247\pi$
$313$	−0.472136			−0.0266867			−0.0133434	+	0.999911i	$0.504247\pi$
$314$		0			0
$315$	8.94427	−	8.00000i	0.503953	−	0.450749i
$316$		0			0
$317$			13.1246i			0.737152i	0.929598	+	0.368576i	$0.120155\pi$
$317$			13.1246i			0.737152i	−0.929598	+	0.368576i	$0.879845\pi$
$318$		0			0
$319$			24.9443i			1.39661i
$320$		0			0
$321$	−8.76393	−	22.9443i	−0.489155	−	1.28062i
$322$		0			0
$323$	−1.88854			−0.105081
$324$		0			0
$325$	−24.4721			−1.35747
$326$		0			0
$327$	5.23607	+	13.7082i	0.289555	+	0.758065i
$328$		0			0
$329$		−	9.88854i		−	0.545173i
$330$		0			0
$331$			32.5410i			1.78862i	0.447452	+	0.894308i	$0.352332\pi$
$331$			32.5410i			1.78862i	−0.447452	+	0.894308i	$0.647668\pi$
$332$		0			0
$333$	18.9443	−	16.9443i	1.03814	−	0.928540i
$334$		0			0
$335$	31.4164			1.71646
$336$		0			0
$337$	22.3607			1.21806			0.609032	−	0.793146i	$-0.291558\pi$
$337$	22.3607			1.21806			0.609032	+	0.793146i	$0.291558\pi$
$338$		0			0
$339$	12.9443	−	4.94427i	0.703036	−	0.268536i
$340$		0			0
$341$			27.4164i			1.48468i
$342$		0			0
$343$		−	15.4164i		−	0.832408i
$344$		0			0
$345$	12.9443	−	4.94427i	0.696896	−	0.266191i
$346$		0			0
$347$	23.7082			1.27272			0.636362	−	0.771391i	$-0.280439\pi$
$347$	23.7082			1.27272			0.636362	+	0.771391i	$0.280439\pi$
$348$		0			0
$349$	28.4721			1.52408			0.762039	−	0.647531i	$-0.224198\pi$
$349$	28.4721			1.52408			0.762039	+	0.647531i	$0.224198\pi$
$350$		0			0
$351$	20.6525	+	10.6525i	1.10235	+	0.568587i
$352$		0			0
$353$			9.88854i			0.526314i	0.964753	+	0.263157i	$0.0847637\pi$
$353$			9.88854i			0.526314i	−0.964753	+	0.263157i	$0.915236\pi$
$354$		0			0
$355$		−	49.8885i		−	2.64781i
$356$		0			0
$357$	−1.88854	−	4.94427i	−0.0999523	−	0.261679i
$358$		0			0
$359$	31.4164			1.65809			0.829047	−	0.559178i	$-0.188883\pi$
$359$	31.4164			1.65809			0.829047	+	0.559178i	$0.188883\pi$
$360$		0			0
$361$	18.4164			0.969285
$362$		0			0
$363$	−10.1459	−	26.5623i	−0.532522	−	1.39416i
$364$		0			0
$365$			6.47214i			0.338767i
$366$		0			0
$367$		−	5.23607i		−	0.273321i	−0.990618	−	0.136660i	$-0.956363\pi$
$367$		−	5.23607i		−	0.273321i	0.990618	−	0.136660i	$-0.0436369\pi$
$368$		0			0
$369$	12.9443	+	14.4721i	0.673852	+	0.753389i
$370$		0			0
$371$	−4.00000			−0.207670
$372$		0			0
$373$	2.58359			0.133773			0.0668867	−	0.997761i	$-0.478693\pi$
$373$	2.58359			0.133773			0.0668867	+	0.997761i	$0.478693\pi$
$374$		0			0
$375$	2.47214	−	0.944272i	0.127661	−	0.0487620i
$376$		0			0
$377$		−	21.3050i		−	1.09726i
$378$		0			0
$379$			10.6525i			0.547181i	0.961846	+	0.273590i	$0.0882113\pi$
$379$			10.6525i			0.547181i	−0.961846	+	0.273590i	$0.911789\pi$
$380$		0			0
$381$	−21.4164	+	8.18034i	−1.09720	+	0.419092i
$382$		0			0
$383$	−28.9443			−1.47898			−0.739492	−	0.673166i	$-0.764934\pi$
$383$	−28.9443			−1.47898			−0.739492	+	0.673166i	$0.764934\pi$
$384$		0			0
$385$	20.9443			1.06742
$386$		0			0
$387$	14.4721	+	16.1803i	0.735660	+	0.822493i
$388$		0			0
$389$		−	32.1803i		−	1.63161i	−0.578328	−	0.815804i	$-0.696295\pi$
$389$		−	32.1803i		−	1.63161i	0.578328	−	0.815804i	$-0.303705\pi$
$390$		0			0
$391$		−	6.11146i		−	0.309070i
$392$		0			0
$393$	−4.18034	−	10.9443i	−0.210870	−	0.552065i
$394$		0			0
$395$	−0.944272			−0.0475115
$396$		0			0
$397$	17.4164			0.874104			0.437052	−	0.899436i	$-0.356022\pi$
$397$	17.4164			0.874104			0.437052	+	0.899436i	$0.356022\pi$
$398$		0			0
$399$	−0.583592	−	1.52786i	−0.0292161	−	0.0764889i
$400$		0			0
$401$		−	28.3607i		−	1.41626i	−0.706080	−	0.708132i	$-0.749538\pi$
$401$		−	28.3607i		−	1.41626i	0.706080	−	0.708132i	$-0.250462\pi$
$402$		0			0
$403$		−	23.4164i		−	1.16645i
$404$		0			0
$405$	−28.9443	−	3.23607i	−1.43825	−	0.160802i
$406$		0			0
$407$	44.3607			2.19888
$408$		0			0
$409$	−31.8885			−1.57679			−0.788394	−	0.615171i	$-0.789087\pi$
$409$	−31.8885			−1.57679			−0.788394	+	0.615171i	$0.789087\pi$
$410$		0			0
$411$	−2.47214	+	0.944272i	−0.121941	+	0.0465775i
$412$		0			0
$413$		−	1.52786i		−	0.0751813i
$414$		0			0
$415$			8.94427i			0.439057i
$416$		0			0
$417$	15.7082	−	6.00000i	0.769234	−	0.293821i
$418$		0			0
$419$	10.1803			0.497342			0.248671	−	0.968588i	$-0.420006\pi$
$419$	10.1803			0.497342			0.248671	+	0.968588i	$0.420006\pi$
$420$		0			0
$421$	−21.4164			−1.04377			−0.521886	−	0.853015i	$-0.674771\pi$
$421$	−21.4164			−1.04377			−0.521886	+	0.853015i	$0.674771\pi$
$422$		0			0
$423$	−17.8885	+	16.0000i	−0.869771	+	0.777947i
$424$		0			0
$425$		−	13.5279i		−	0.656198i
$426$		0			0
$427$			0.583592i			0.0282420i
$428$		0			0
$429$	14.4721	+	37.8885i	0.698721	+	1.82928i
$430$		0			0
$431$	−8.00000			−0.385346			−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000			−0.385346			−0.192673	+	0.981263i	$0.561716\pi$
$432$		0			0
$433$	16.4721			0.791600			0.395800	−	0.918337i	$-0.370467\pi$
$433$	16.4721			0.791600			0.395800	+	0.918337i	$0.370467\pi$
$434$		0			0
$435$	9.52786	+	24.9443i	0.456826	+	1.19599i
$436$		0			0
$437$		−	1.88854i		−	0.0903413i
$438$		0			0
$439$		−	22.1803i		−	1.05861i	−0.848432	−	0.529305i	$-0.822453\pi$
$439$		−	22.1803i		−	1.05861i	0.848432	−	0.529305i	$-0.177547\pi$
$440$		0			0
$441$	−12.2361	+	10.9443i	−0.582670	+	0.521156i
$442$		0			0
$443$	−16.2918			−0.774047			−0.387023	−	0.922070i	$-0.626497\pi$
$443$	−16.2918			−0.774047			−0.387023	+	0.922070i	$0.626497\pi$
$444$		0			0
$445$	−12.9443			−0.613617
$446$		0			0
$447$	15.7082	−	6.00000i	0.742973	−	0.283790i
$448$		0			0
$449$		−	0.583592i		−	0.0275414i	−0.999905	−	0.0137707i	$-0.995617\pi$
$449$		−	0.583592i		−	0.0275414i	0.999905	−	0.0137707i	$-0.00438349\pi$
$450$		0			0
$451$			33.8885i			1.59575i
$452$		0			0
$453$	31.8885	−	12.1803i	1.49825	−	0.572282i
$454$		0			0
$455$	−17.8885			−0.838628
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$		0			0
$459$	−5.88854	+	11.4164i	−0.274854	+	0.532872i
$460$		0			0
$461$			8.18034i			0.380996i	0.981688	+	0.190498i	$0.0610103\pi$
$461$			8.18034i			0.380996i	−0.981688	+	0.190498i	$0.938990\pi$
$462$		0			0
$463$			34.7639i			1.61562i	0.589445	+	0.807808i	$0.299346\pi$
$463$			34.7639i			1.61562i	−0.589445	+	0.807808i	$0.700654\pi$
$464$		0			0
$465$	10.4721	+	27.4164i	0.485634	+	1.27141i
$466$		0			0
$467$	−10.7639			−0.498095			−0.249048	−	0.968491i	$-0.580118\pi$
$467$	−10.7639			−0.498095			−0.249048	+	0.968491i	$0.580118\pi$
$468$		0			0
$469$	12.0000			0.554109
$470$		0			0
$471$	−2.76393	−	7.23607i	−0.127355	−	0.333420i
$472$		0			0
$473$			37.8885i			1.74212i
$474$		0			0
$475$		−	4.18034i		−	0.191807i
$476$		0			0
$477$	6.47214	+	7.23607i	0.296339	+	0.331317i
$478$		0			0
$479$	6.11146			0.279240			0.139620	−	0.990205i	$-0.455412\pi$
$479$	6.11146			0.279240			0.139620	+	0.990205i	$0.455412\pi$
$480$		0			0
$481$	−37.8885			−1.72757
$482$		0			0
$483$	4.94427	−	1.88854i	0.224972	−	0.0859317i
$484$		0			0
$485$		−	1.52786i		−	0.0693767i
$486$		0			0
$487$			32.6525i			1.47962i	0.672813	+	0.739812i	$0.265086\pi$
$487$			32.6525i			1.47962i	−0.672813	+	0.739812i	$0.734914\pi$
$488$		0			0
$489$	−6.76393	+	2.58359i	−0.305876	+	0.116834i
$490$		0			0
$491$	32.0689			1.44725			0.723624	−	0.690194i	$-0.242475\pi$
$491$	32.0689			1.44725			0.723624	+	0.690194i	$0.242475\pi$
$492$		0			0
$493$	11.7771			0.530413
$494$		0			0
$495$	−33.8885	−	37.8885i	−1.52318	−	1.70296i
$496$		0			0
$497$		−	19.0557i		−	0.854766i
$498$		0			0
$499$		−	27.2361i		−	1.21925i	−0.792688	−	0.609627i	$-0.791319\pi$
$499$		−	27.2361i		−	1.21925i	0.792688	−	0.609627i	$-0.208681\pi$
$500$		0			0
$501$	8.36068	+	21.8885i	0.373528	+	0.977908i
$502$		0			0
$503$	−21.5279			−0.959880			−0.479940	−	0.877301i	$-0.659342\pi$
$503$	−21.5279			−0.959880			−0.479940	+	0.877301i	$0.659342\pi$
$504$		0			0
$505$	5.52786			0.245987
$506$		0			0
$507$	−4.32624	−	11.3262i	−0.192135	−	0.503016i
$508$		0			0
$509$		−	25.7082i		−	1.13950i	−0.821819	−	0.569748i	$-0.807041\pi$
$509$		−	25.7082i		−	1.13950i	0.821819	−	0.569748i	$-0.192959\pi$
$510$		0			0
$511$			2.47214i			0.109361i
$512$		0			0
$513$	−1.81966	+	3.52786i	−0.0803400	+	0.155759i
$514$		0			0
$515$	−45.8885			−2.02209
$516$		0			0
$517$	−41.8885			−1.84226
$518$		0			0
$519$	−36.6525	+	14.0000i	−1.60887	+	0.614532i
$520$		0			0
$521$			24.3607i			1.06726i	0.845718	+	0.533630i	$0.179173\pi$
$521$			24.3607i			1.06726i	−0.845718	+	0.533630i	$0.820827\pi$
$522$		0			0
$523$			11.8197i			0.516838i	0.966033	+	0.258419i	$0.0832015\pi$
$523$			11.8197i			0.516838i	−0.966033	+	0.258419i	$0.916799\pi$
$524$		0			0
$525$	10.9443	−	4.18034i	0.477647	−	0.182445i
$526$		0			0
$527$	12.9443			0.563861
$528$		0			0
$529$	−16.8885			−0.734285
$530$		0			0
$531$	−2.76393	+	2.47214i	−0.119944	+	0.107282i
$532$		0			0
$533$		−	28.9443i		−	1.25372i
$534$		0			0
$535$		−	45.8885i		−	1.98393i
$536$		0			0
$537$	3.81966	+	10.0000i	0.164831	+	0.431532i
$538$		0			0
$539$	−28.6525			−1.23415
$540$		0			0
$541$	−26.3607			−1.13333			−0.566667	−	0.823947i	$-0.691767\pi$
$541$	−26.3607			−1.13333			−0.566667	+	0.823947i	$0.691767\pi$
$542$		0			0
$543$	−12.6525	−	33.1246i	−0.542970	−	1.42151i
$544$		0			0
$545$			27.4164i			1.17439i
$546$		0			0
$547$		−	7.23607i		−	0.309392i	−0.987962	−	0.154696i	$-0.950560\pi$
$547$		−	7.23607i		−	0.309392i	0.987962	−	0.154696i	$-0.0494398\pi$
$548$		0			0
$549$	1.05573	−	0.944272i	0.0450574	−	0.0403005i
$550$		0			0
$551$	3.63932			0.155040
$552$		0			0
$553$	−0.360680			−0.0153377
$554$		0			0
$555$	44.3607	−	16.9443i	1.88301	−	0.719244i
$556$		0			0
$557$		−	14.6525i		−	0.620845i	−0.950599	−	0.310423i	$-0.899529\pi$
$557$		−	14.6525i		−	0.620845i	0.950599	−	0.310423i	$-0.100471\pi$
$558$		0			0
$559$		−	32.3607i		−	1.36871i
$560$		0			0
$561$	−20.9443	+	8.00000i	−0.884268	+	0.337760i
$562$		0			0
$563$	−30.5410			−1.28715			−0.643575	−	0.765383i	$-0.722550\pi$
$563$	−30.5410			−1.28715			−0.643575	+	0.765383i	$0.722550\pi$
$564$		0			0
$565$	25.8885			1.08914
$566$		0			0
$567$	−11.0557	−	1.23607i	−0.464297	−	0.0519100i
$568$		0			0
$569$		−	46.4721i		−	1.94821i	−0.226089	−	0.974107i	$-0.572594\pi$
$569$		−	46.4721i		−	1.94821i	0.226089	−	0.974107i	$-0.427406\pi$
$570$		0			0
$571$		−	5.12461i		−	0.214458i	−0.994234	−	0.107229i	$-0.965802\pi$
$571$		−	5.12461i		−	0.214458i	0.994234	−	0.107229i	$-0.0341979\pi$
$572$		0			0
$573$	−8.00000	−	20.9443i	−0.334205	−	0.874960i
$574$		0			0
$575$	13.5279			0.564151
$576$		0			0
$577$	28.4721			1.18531			0.592655	−	0.805456i	$-0.298080\pi$
$577$	28.4721			1.18531			0.592655	+	0.805456i	$0.298080\pi$
$578$		0			0
$579$	7.34752	+	19.2361i	0.305353	+	0.799424i
$580$		0			0
$581$			3.41641i			0.141736i
$582$		0			0
$583$			16.9443i			0.701760i
$584$		0			0
$585$	28.9443	+	32.3607i	1.19670	+	1.33795i
$586$		0			0
$587$	−21.5967			−0.891393			−0.445697	−	0.895184i	$-0.647044\pi$
$587$	−21.5967			−0.891393			−0.445697	+	0.895184i	$0.647044\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$		0			0
$591$	7.70820	−	2.94427i	0.317073	−	0.121111i
$592$		0			0
$593$			28.9443i			1.18860i	0.804244	+	0.594299i	$0.202571\pi$
$593$			28.9443i			1.18860i	−0.804244	+	0.594299i	$0.797429\pi$
$594$		0			0
$595$		−	9.88854i		−	0.405391i
$596$		0			0
$597$	−10.0000	+	3.81966i	−0.409273	+	0.156328i
$598$		0			0
$599$	7.41641			0.303026			0.151513	−	0.988455i	$-0.451585\pi$
$599$	7.41641			0.303026			0.151513	+	0.988455i	$0.451585\pi$
$600$		0			0
$601$	33.7771			1.37780			0.688898	−	0.724858i	$-0.258095\pi$
$601$	33.7771			1.37780			0.688898	+	0.724858i	$0.258095\pi$
$602$		0			0
$603$	−19.4164	−	21.7082i	−0.790697	−	0.884026i
$604$		0			0
$605$		−	53.1246i		−	2.15982i
$606$		0			0
$607$			34.7639i			1.41102i	0.708698	+	0.705512i	$0.249283\pi$
$607$			34.7639i			1.41102i	−0.708698	+	0.705512i	$0.750717\pi$
$608$		0			0
$609$	3.63932	+	9.52786i	0.147473	+	0.386089i
$610$		0			0
$611$	35.7771			1.44739
$612$		0			0
$613$	19.3050			0.779720			0.389860	−	0.920874i	$-0.372523\pi$
$613$	19.3050			0.779720			0.389860	+	0.920874i	$0.372523\pi$
$614$		0			0
$615$	12.9443	+	33.8885i	0.521963	+	1.36652i
$616$		0			0
$617$		−	21.8885i		−	0.881200i	−0.897704	−	0.440600i	$-0.854766\pi$
$617$		−	21.8885i		−	0.881200i	0.897704	−	0.440600i	$-0.145234\pi$
$618$		0			0
$619$		−	29.1246i		−	1.17062i	−0.810811	−	0.585308i	$-0.800973\pi$
$619$		−	29.1246i		−	1.17062i	0.810811	−	0.585308i	$-0.199027\pi$
$620$		0			0
$621$	−11.4164	−	5.88854i	−0.458125	−	0.236299i
$622$		0			0
$623$	−4.94427			−0.198088
$624$		0			0
$625$	−22.4164			−0.896656
$626$		0			0
$627$	−6.47214	+	2.47214i	−0.258472	+	0.0987276i
$628$		0			0
$629$		−	20.9443i		−	0.835103i
$630$		0			0
$631$		−	19.1246i		−	0.761339i	−0.924711	−	0.380669i	$-0.875694\pi$
$631$		−	19.1246i		−	0.761339i	0.924711	−	0.380669i	$-0.124306\pi$
$632$		0			0
$633$	−18.1803	+	6.94427i	−0.722604	+	0.276010i
$634$		0			0
$635$	−42.8328			−1.69977
$636$		0			0
$637$	24.4721			0.969621
$638$		0			0
$639$	−34.4721	+	30.8328i	−1.36370	+	1.21973i
$640$		0			0
$641$		−	25.3050i		−	0.999485i	−0.866174	−	0.499743i	$-0.833428\pi$
$641$		−	25.3050i		−	0.999485i	0.866174	−	0.499743i	$-0.166572\pi$
$642$		0			0
$643$		−	13.3475i		−	0.526375i	−0.964745	−	0.263187i	$-0.915226\pi$
$643$		−	13.3475i		−	0.526375i	0.964745	−	0.263187i	$-0.0847738\pi$
$644$		0			0
$645$	14.4721	+	37.8885i	0.569840	+	1.49186i
$646$		0			0
$647$	−33.3050			−1.30935			−0.654676	−	0.755909i	$-0.727195\pi$
$647$	−33.3050			−1.30935			−0.654676	+	0.755909i	$0.727195\pi$
$648$		0			0
$649$	−6.47214			−0.254054
$650$		0			0
$651$	4.00000	+	10.4721i	0.156772	+	0.410435i
$652$		0			0
$653$			14.2918i			0.559281i	0.960105	+	0.279641i	$0.0902153\pi$
$653$			14.2918i			0.559281i	−0.960105	+	0.279641i	$0.909785\pi$
$654$		0			0
$655$		−	21.8885i		−	0.855256i
$656$		0			0
$657$	4.47214	−	4.00000i	0.174475	−	0.156055i
$658$		0			0
$659$	−28.2918			−1.10209			−0.551046	−	0.834475i	$-0.685771\pi$
$659$	−28.2918			−1.10209			−0.551046	+	0.834475i	$0.685771\pi$
$660$		0			0
$661$	51.3050			1.99553			0.997766	−	0.0668107i	$-0.0212824\pi$
$661$	51.3050			1.99553			0.997766	+	0.0668107i	$0.0212824\pi$
$662$		0			0
$663$	17.8885	−	6.83282i	0.694733	−	0.265365i
$664$		0			0
$665$		−	3.05573i		−	0.118496i
$666$		0			0
$667$			11.7771i			0.456011i
$668$		0			0
$669$	−16.4721	+	6.29180i	−0.636850	+	0.243255i
$670$		0			0
$671$	2.47214			0.0954358
$672$		0			0
$673$	−25.4164			−0.979731			−0.489865	−	0.871798i	$-0.662954\pi$
$673$	−25.4164			−0.979731			−0.489865	+	0.871798i	$0.662954\pi$
$674$		0			0
$675$	−25.2705	−	13.0344i	−0.972662	−	0.501696i
$676$		0			0
$677$		−	14.2918i		−	0.549278i	−0.961547	−	0.274639i	$-0.911442\pi$
$677$		−	14.2918i		−	0.549278i	0.961547	−	0.274639i	$-0.0885584\pi$
$678$		0			0
$679$		−	0.583592i		−	0.0223962i
$680$		0			0
$681$	−1.34752	−	3.52786i	−0.0516372	−	0.135188i
$682$		0			0
$683$	−13.2361			−0.506464			−0.253232	−	0.967406i	$-0.581494\pi$
$683$	−13.2361			−0.506464			−0.253232	+	0.967406i	$0.581494\pi$
$684$		0			0
$685$	−4.94427			−0.188911
$686$		0			0
$687$	7.12461	+	18.6525i	0.271821	+	0.711636i
$688$		0			0
$689$		−	14.4721i		−	0.551344i
$690$		0			0
$691$			29.7082i			1.13015i	0.825038	+	0.565077i	$0.191153\pi$
$691$			29.7082i			1.13015i	−0.825038	+	0.565077i	$0.808847\pi$
$692$		0			0
$693$	−12.9443	−	14.4721i	−0.491712	−	0.549751i
$694$		0			0
$695$	31.4164			1.19169
$696$		0			0
$697$	16.0000			0.606043
$698$		0			0
$699$	40.3607	−	15.4164i	1.52658	−	0.583102i
$700$		0			0
$701$			14.2918i			0.539794i	0.962889	+	0.269897i	$0.0869896\pi$
$701$			14.2918i			0.539794i	−0.962889	+	0.269897i	$0.913010\pi$
$702$		0			0
$703$		−	6.47214i		−	0.244101i
$704$		0			0
$705$	−41.8885	+	16.0000i	−1.57761	+	0.602595i
$706$		0			0
$707$	2.11146			0.0794095
$708$		0			0
$709$	7.52786			0.282715			0.141357	−	0.989959i	$-0.454853\pi$
$709$	7.52786			0.282715			0.141357	+	0.989959i	$0.454853\pi$
$710$		0			0
$711$	0.583592	+	0.652476i	0.0218864	+	0.0244698i
$712$		0			0
$713$			12.9443i			0.484767i
$714$		0			0
$715$			75.7771i			2.83390i
$716$		0			0
$717$	4.94427	+	12.9443i	0.184647	+	0.483413i
$718$		0			0
$719$	30.8328			1.14987			0.574935	−	0.818199i	$-0.305027\pi$
$719$	30.8328			1.14987			0.574935	+	0.818199i	$0.305027\pi$
$720$		0			0
$721$	−17.5279			−0.652772
$722$		0			0
$723$	1.23607	+	3.23607i	0.0459699	+	0.120351i
$724$		0			0
$725$			26.0689i			0.968174i
$726$		0			0
$727$		−	14.1803i		−	0.525920i	−0.964807	−	0.262960i	$-0.915301\pi$
$727$		−	14.1803i		−	0.525920i	0.964807	−	0.262960i	$-0.0846987\pi$
$728$		0			0
$729$	15.6525	+	22.0000i	0.579721	+	0.814815i
$730$		0			0
$731$	17.8885			0.661632
$732$		0			0
$733$	−29.4164			−1.08652			−0.543260	−	0.839565i	$-0.682810\pi$
$733$	−29.4164			−1.08652			−0.543260	+	0.839565i	$0.682810\pi$
$734$		0			0
$735$	−28.6525	+	10.9443i	−1.05686	+	0.403686i
$736$		0			0
$737$		−	50.8328i		−	1.87245i
$738$		0			0
$739$		−	13.1246i		−	0.482797i	−0.970426	−	0.241398i	$-0.922394\pi$
$739$		−	13.1246i		−	0.482797i	0.970426	−	0.241398i	$-0.0776060\pi$
$740$		0			0
$741$	5.52786	−	2.11146i	0.203071	−	0.0775663i
$742$		0			0
$743$	−7.41641			−0.272082			−0.136041	−	0.990703i	$-0.543438\pi$
$743$	−7.41641			−0.272082			−0.136041	+	0.990703i	$0.543438\pi$
$744$		0			0
$745$	31.4164			1.15101
$746$		0			0
$747$	6.18034	−	5.52786i	0.226127	−	0.202254i
$748$		0			0
$749$		−	17.5279i		−	0.640454i
$750$		0			0
$751$			17.5967i			0.642114i	0.947060	+	0.321057i	$0.104038\pi$
$751$			17.5967i			0.642114i	−0.947060	+	0.321057i	$0.895962\pi$
$752$		0			0
$753$	−14.6525	−	38.3607i	−0.533966	−	1.39794i
$754$		0			0
$755$	63.7771			2.32109
$756$		0			0
$757$	−5.41641			−0.196863			−0.0984313	−	0.995144i	$-0.531382\pi$
$757$	−5.41641			−0.196863			−0.0984313	+	0.995144i	$0.531382\pi$
$758$		0			0
$759$	−8.00000	−	20.9443i	−0.290382	−	0.760229i
$760$		0			0
$761$			42.2492i			1.53153i	0.643119	+	0.765767i	$0.277640\pi$
$761$			42.2492i			1.53153i	−0.643119	+	0.765767i	$0.722360\pi$
$762$		0			0
$763$			10.4721i			0.379117i
$764$		0			0
$765$	−17.8885	+	16.0000i	−0.646762	+	0.578481i
$766$		0			0
$767$	5.52786			0.199600
$768$		0			0
$769$	49.7771			1.79501			0.897504	−	0.441007i	$-0.145378\pi$
$769$	49.7771			1.79501			0.897504	+	0.441007i	$0.145378\pi$
$770$		0			0
$771$	33.8885	−	12.9443i	1.22047	−	0.466177i
$772$		0			0
$773$			7.81966i			0.281254i	0.990063	+	0.140627i	$0.0449118\pi$
$773$			7.81966i			0.281254i	−0.990063	+	0.140627i	$0.955088\pi$
$774$		0			0
$775$			28.6525i			1.02923i
$776$		0			0
$777$	16.9443	−	6.47214i	0.607872	−	0.232187i
$778$		0			0
$779$	4.94427			0.177147
$780$		0			0
$781$	−80.7214			−2.88844
$782$		0			0
$783$	11.3475	−	22.0000i	0.405527	−	0.786216i
$784$		0			0
$785$		−	14.4721i		−	0.516533i
$786$		0			0
$787$		−	28.1803i		−	1.00452i	−0.864716	−	0.502260i	$-0.832502\pi$
$787$		−	28.1803i		−	1.00452i	0.864716	−	0.502260i	$-0.167498\pi$
$788$		0			0
$789$	6.47214	+	16.9443i	0.230414	+	0.603232i
$790$		0			0
$791$	9.88854			0.351596
$792$		0			0
$793$	−2.11146			−0.0749800
$794$		0			0
$795$	6.47214	+	16.9443i	0.229543	+	0.600951i
$796$		0			0
$797$			50.0689i			1.77353i	0.462220	+	0.886765i	$0.347053\pi$
$797$			50.0689i			1.77353i	−0.462220	+	0.886765i	$0.652947\pi$
$798$		0			0
$799$			19.7771i			0.699663i
$800$		0			0
$801$	8.00000	+	8.94427i	0.282666	+	0.316030i
$802$		0			0
$803$	10.4721			0.369554
$804$		0			0
$805$	9.88854			0.348525
$806$		0			0
$807$	26.1803	−	10.0000i	0.921592	−	0.352017i
$808$		0			0
$809$			3.41641i			0.120115i	0.998195	+	0.0600573i	$0.0191283\pi$
$809$			3.41641i			0.120115i	−0.998195	+	0.0600573i	$0.980872\pi$
$810$		0			0
$811$		−	52.1803i		−	1.83230i	−0.400836	−	0.916150i	$-0.631280\pi$
$811$		−	52.1803i		−	1.83230i	0.400836	−	0.916150i	$-0.368720\pi$
$812$		0			0
$813$	−0.472136	+	0.180340i	−0.0165585	+	0.00632480i
$814$		0			0
$815$	−13.5279			−0.473860
$816$		0			0
$817$	5.52786			0.193395
$818$		0			0
$819$	11.0557	+	12.3607i	0.386318	+	0.431917i
$820$		0			0
$821$			50.4296i			1.76000i	0.474970	+	0.880002i	$0.342459\pi$
$821$			50.4296i			1.76000i	−0.474970	+	0.880002i	$0.657541\pi$
$822$		0			0
$823$		−	51.1246i		−	1.78209i	−0.453913	−	0.891046i	$-0.649972\pi$
$823$		−	51.1246i		−	1.78209i	0.453913	−	0.891046i	$-0.350028\pi$
$824$		0			0
$825$	−17.7082	−	46.3607i	−0.616521	−	1.61407i
$826$		0			0
$827$	19.1246			0.665028			0.332514	−	0.943098i	$-0.392103\pi$
$827$	19.1246			0.665028			0.332514	+	0.943098i	$0.392103\pi$
$828$		0			0
$829$	4.47214			0.155324			0.0776619	−	0.996980i	$-0.475255\pi$
$829$	4.47214			0.155324			0.0776619	+	0.996980i	$0.475255\pi$
$830$		0			0
$831$	−4.65248	−	12.1803i	−0.161393	−	0.422531i
$832$		0			0
$833$			13.5279i			0.468713i
$834$		0			0
$835$			43.7771i			1.51497i
$836$		0			0
$837$	12.4721	−	24.1803i	0.431100	−	0.835795i
$838$		0			0
$839$	31.4164			1.08461			0.542307	−	0.840180i	$-0.317551\pi$
$839$	31.4164			1.08461			0.542307	+	0.840180i	$0.317551\pi$
$840$		0			0
$841$	6.30495			0.217412
$842$		0			0
$843$	−40.3607	+	15.4164i	−1.39010	+	0.530969i
$844$		0			0
$845$		−	22.6525i		−	0.779269i
$846$		0			0
$847$		−	20.2918i		−	0.697234i
$848$		0			0
$849$	33.5967	−	12.8328i	1.15304	−	0.440421i
$850$		0			0
$851$	20.9443			0.717960
$852$		0			0
$853$	−39.3050			−1.34578			−0.672888	−	0.739744i	$-0.734946\pi$
$853$	−39.3050			−1.34578			−0.672888	+	0.739744i	$0.734946\pi$
$854$		0			0
$855$	−5.52786	+	4.94427i	−0.189049	+	0.169091i
$856$		0			0
$857$			7.63932i			0.260954i	0.991451	+	0.130477i	$0.0416509\pi$
$857$			7.63932i			0.260954i	−0.991451	+	0.130477i	$0.958349\pi$
$858$		0			0
$859$			16.7639i			0.571978i	0.958233	+	0.285989i	$0.0923221\pi$
$859$			16.7639i			0.571978i	−0.958233	+	0.285989i	$0.907678\pi$
$860$		0			0
$861$	4.94427	+	12.9443i	0.168500	+	0.441140i
$862$		0			0
$863$	22.8328			0.777238			0.388619	−	0.921399i	$-0.372952\pi$
$863$	22.8328			0.777238			0.388619	+	0.921399i	$0.372952\pi$
$864$		0			0
$865$	−73.3050			−2.49244
$866$		0			0
$867$	−6.72949	−	17.6180i	−0.228545	−	0.598340i
$868$		0			0
$869$			1.52786i			0.0518292i
$870$		0			0
$871$			43.4164i			1.47111i
$872$		0			0
$873$	−1.05573	+	0.944272i	−0.0357310	+	0.0319588i
$874$		0			0
$875$	1.88854			0.0638444
$876$		0			0
$877$	18.5836			0.627523			0.313762	−	0.949502i	$-0.398411\pi$
$877$	18.5836			0.627523			0.313762	+	0.949502i	$0.398411\pi$
$878$		0			0
$879$	33.5967	−	12.8328i	1.13319	−	0.432840i
$880$		0			0
$881$			24.0000i			0.808581i	0.914631	+	0.404290i	$0.132481\pi$
$881$			24.0000i			0.808581i	−0.914631	+	0.404290i	$0.867519\pi$
$882$		0			0
$883$			0.763932i			0.0257084i	0.999917	+	0.0128542i	$0.00409173\pi$
$883$			0.763932i			0.0257084i	−0.999917	+	0.0128542i	$0.995908\pi$
$884$		0			0
$885$	−6.47214	+	2.47214i	−0.217558	+	0.0830999i
$886$		0			0
$887$	20.3607			0.683645			0.341822	−	0.939765i	$-0.388956\pi$
$887$	20.3607			0.683645			0.341822	+	0.939765i	$0.388956\pi$
$888$		0			0
$889$	−16.3607			−0.548720
$890$		0			0
$891$	−5.23607	+	46.8328i	−0.175415	+	1.56896i
$892$		0			0
$893$			6.11146i			0.204512i
$894$		0			0
$895$			20.0000i			0.668526i
$896$		0			0
$897$	6.83282	+	17.8885i	0.228141	+	0.597281i
$898$		0			0
$899$	−24.9443			−0.831938
$900$		0			0
$901$	8.00000			0.266519
$902$		0			0
$903$	5.52786	+	14.4721i	0.183956	+	0.481603i
$904$		0			0
$905$		−	66.2492i		−	2.20220i
$906$		0			0
$907$		−	39.2361i		−	1.30281i	−0.758729	−	0.651406i	$-0.774179\pi$
$907$		−	39.2361i		−	1.30281i	0.758729	−	0.651406i	$-0.225821\pi$
$908$		0			0
$909$	−3.41641	−	3.81966i	−0.113315	−	0.126690i
$910$		0			0
$911$	−33.8885			−1.12278			−0.561389	−	0.827552i	$-0.689733\pi$
$911$	−33.8885			−1.12278			−0.561389	+	0.827552i	$0.689733\pi$
$912$		0			0
$913$	14.4721			0.478958
$914$		0			0
$915$	2.47214	−	0.944272i	0.0817263	−	0.0312167i
$916$		0			0
$917$		−	8.36068i		−	0.276094i
$918$		0			0
$919$			58.5410i			1.93109i	0.260236	+	0.965545i	$0.416200\pi$
$919$			58.5410i			1.93109i	−0.260236	+	0.965545i	$0.583800\pi$
$920$		0			0
$921$	44.6525	−	17.0557i	1.47135	−	0.562005i
$922$		0			0
$923$	68.9443			2.26933
$924$		0			0
$925$	46.3607			1.52433
$926$		0			0
$927$	28.3607	+	31.7082i	0.931487	+	1.04143i
$928$		0			0
$929$		−	47.4164i		−	1.55568i	−0.628461	−	0.777841i	$-0.716315\pi$
$929$		−	47.4164i		−	1.55568i	0.628461	−	0.777841i	$-0.283685\pi$
$930$		0			0
$931$			4.18034i			0.137005i
$932$		0			0
$933$	3.41641	+	8.94427i	0.111848	+	0.292822i
$934$		0			0
$935$	−41.8885			−1.36990
$936$		0			0
$937$	−14.3607			−0.469143			−0.234571	−	0.972099i	$-0.575369\pi$
$937$	−14.3607			−0.469143			−0.234571	+	0.972099i	$0.575369\pi$
$938$		0			0
$939$	0.291796	+	0.763932i	0.00952240	+	0.0249300i
$940$		0			0
$941$		−	1.70820i		−	0.0556859i	−0.999612	−	0.0278429i	$-0.991136\pi$
$941$		−	1.70820i		−	0.0556859i	0.999612	−	0.0278429i	$-0.00886382\pi$
$942$		0			0
$943$			16.0000i			0.521032i
$944$		0			0
$945$	−18.4721	−	9.52786i	−0.600899	−	0.309941i
$946$		0			0
$947$	−24.0689			−0.782134			−0.391067	−	0.920362i	$-0.627894\pi$
$947$	−24.0689			−0.782134			−0.391067	+	0.920362i	$0.627894\pi$
$948$		0			0
$949$	−8.94427			−0.290343
$950$		0			0
$951$	21.2361	−	8.11146i	0.688627	−	0.263032i
$952$		0			0
$953$		−	24.3607i		−	0.789120i	−0.918870	−	0.394560i	$-0.870897\pi$
$953$		−	24.3607i		−	0.789120i	0.918870	−	0.394560i	$-0.129103\pi$
$954$		0			0
$955$		−	41.8885i		−	1.35548i
$956$		0			0
$957$	40.3607	−	15.4164i	1.30468	−	0.498342i
$958$		0			0
$959$	−1.88854			−0.0609843
$960$		0			0
$961$	3.58359			0.115600
$962$		0			0
$963$	−31.7082	+	28.3607i	−1.02178	+	0.913910i
$964$		0			0
$965$			38.4721i			1.23846i
$966$		0			0
$967$		−	9.23607i		−	0.297012i	−0.988912	−	0.148506i	$-0.952554\pi$
$967$		−	9.23607i		−	0.297012i	0.988912	−	0.148506i	$-0.0474464\pi$
$968$		0			0
$969$	1.16718	+	3.05573i	0.0374954	+	0.0981641i
$970$		0			0
$971$	−17.0132			−0.545978			−0.272989	−	0.962017i	$-0.588012\pi$
$971$	−17.0132			−0.545978			−0.272989	+	0.962017i	$0.588012\pi$
$972$		0			0
$973$	12.0000			0.384702
$974$		0			0
$975$	15.1246	+	39.5967i	0.484375	+	1.26811i
$976$		0			0
$977$			60.3607i			1.93111i	0.260200	+	0.965555i	$0.416211\pi$
$977$			60.3607i			1.93111i	−0.260200	+	0.965555i	$0.583789\pi$
$978$		0			0
$979$			20.9443i			0.669382i
$980$		0			0
$981$	18.9443	−	16.9443i	0.604844	−	0.540989i
$982$		0			0
$983$	−21.5279			−0.686632			−0.343316	−	0.939220i	$-0.611550\pi$
$983$	−21.5279			−0.686632			−0.343316	+	0.939220i	$0.611550\pi$
$984$		0			0
$985$	15.4164			0.491208
$986$		0			0
$987$	−16.0000	+	6.11146i	−0.509286	+	0.194530i
$988$		0			0
$989$			17.8885i			0.568823i
$990$		0			0
$991$		−	50.1803i		−	1.59403i	−0.603959	−	0.797016i	$-0.706411\pi$
$991$		−	50.1803i		−	1.59403i	0.603959	−	0.797016i	$-0.293589\pi$
$992$		0			0
$993$	52.6525	−	20.1115i	1.67088	−	0.638218i
$994$		0			0
$995$	−20.0000			−0.634043
$996$		0			0
$997$	−18.3607			−0.581489			−0.290744	−	0.956801i	$-0.593903\pi$
$997$	−18.3607			−0.581489			−0.290744	+	0.956801i	$0.593903\pi$
$998$		0			0
$999$	−39.1246	−	20.1803i	−1.23785	−	0.638478i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.c.c.383.1	yes	4
3.2	odd	2		384.2.c.b.383.3	yes	4
4.3	odd	2		384.2.c.b.383.4	yes	4
8.3	odd	2		384.2.c.d.383.1	yes	4
8.5	even	2		384.2.c.a.383.4	yes	4
12.11	even	2	inner	384.2.c.c.383.2	yes	4
16.3	odd	4		768.2.f.b.383.2		4
16.5	even	4		768.2.f.c.383.2		4
16.11	odd	4		768.2.f.f.383.3		4
16.13	even	4		768.2.f.e.383.3		4
24.5	odd	2		384.2.c.d.383.2	yes	4
24.11	even	2		384.2.c.a.383.3	✓	4
48.5	odd	4		768.2.f.b.383.1		4
48.11	even	4		768.2.f.e.383.4		4
48.29	odd	4		768.2.f.f.383.4		4
48.35	even	4		768.2.f.c.383.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.c.a.383.3	✓	4	24.11	even	2
384.2.c.a.383.4	yes	4	8.5	even	2
384.2.c.b.383.3	yes	4	3.2	odd	2
384.2.c.b.383.4	yes	4	4.3	odd	2
384.2.c.c.383.1	yes	4	1.1	even	1	trivial
384.2.c.c.383.2	yes	4	12.11	even	2	inner
384.2.c.d.383.1	yes	4	8.3	odd	2
384.2.c.d.383.2	yes	4	24.5	odd	2
768.2.f.b.383.1		4	48.5	odd	4
768.2.f.b.383.2		4	16.3	odd	4
768.2.f.c.383.1		4	48.35	even	4
768.2.f.c.383.2		4	16.5	even	4
768.2.f.e.383.3		4	16.13	even	4
768.2.f.e.383.4		4	48.11	even	4
768.2.f.f.383.3		4	16.11	odd	4
768.2.f.f.383.4		4	48.29	odd	4

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.618034 - 1.61803i) q^{3} -3.23607i q^{5} -1.23607i q^{7} +(-2.23607 + 2.00000i) q^{9} -5.23607 q^{11} +4.47214 q^{13} +(-5.23607 + 2.00000i) q^{15} +2.47214i q^{17} +0.763932i q^{19} +(-2.00000 + 0.763932i) q^{21} -2.47214 q^{23} -5.47214 q^{25} +(4.61803 + 2.38197i) q^{27} -4.76393i q^{29} -5.23607i q^{31} +(3.23607 + 8.47214i) q^{33} -4.00000 q^{35} -8.47214 q^{37} +(-2.76393 - 7.23607i) q^{39} -6.47214i q^{41} -7.23607i q^{43} +(6.47214 + 7.23607i) q^{45} +8.00000 q^{47} +5.47214 q^{49} +(4.00000 - 1.52786i) q^{51} -3.23607i q^{53} +16.9443i q^{55} +(1.23607 - 0.472136i) q^{57} +1.23607 q^{59} -0.472136 q^{61} +(2.47214 + 2.76393i) q^{63} -14.4721i q^{65} +9.70820i q^{67} +(1.52786 + 4.00000i) q^{69} +15.4164 q^{71} -2.00000 q^{73} +(3.38197 + 8.85410i) q^{75} +6.47214i q^{77} -0.291796i q^{79} +(1.00000 - 8.94427i) q^{81} -2.76393 q^{83} +8.00000 q^{85} +(-7.70820 + 2.94427i) q^{87} -4.00000i q^{89} -5.52786i q^{91} +(-8.47214 + 3.23607i) q^{93} +2.47214 q^{95} +0.472136 q^{97} +(11.7082 - 10.4721i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 12 q^{11} - 12 q^{15} - 8 q^{21} + 8 q^{23} - 4 q^{25} + 14 q^{27} + 4 q^{33} - 16 q^{35} - 16 q^{37} - 20 q^{39} + 8 q^{45} + 32 q^{47} + 4 q^{49} + 16 q^{51} - 4 q^{57} - 4 q^{59} + 16 q^{61}+ \cdots + 20 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 12 * q^11 - 12 * q^15 - 8 * q^21 + 8 * q^23 - 4 * q^25 + 14 * q^27 + 4 * q^33 - 16 * q^35 - 16 * q^37 - 20 * q^39 + 8 * q^45 + 32 * q^47 + 4 * q^49 + 16 * q^51 - 4 * q^57 - 4 * q^59 + 16 * q^61 - 8 * q^63 + 24 * q^69 + 8 * q^71 - 8 * q^73 + 18 * q^75 + 4 * q^81 - 20 * q^83 + 32 * q^85 - 4 * q^87 - 16 * q^93 - 8 * q^95 - 16 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more