LMFDB - Embedded newform 384.2.c.a.383.4

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			383.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.2.c.a.383.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.618034 + 1.61803i) q^{3} +3.23607i q^{5} -1.23607i q^{7} +(-2.23607 + 2.00000i) q^{9} +5.23607 q^{11} -4.47214 q^{13} +(-5.23607 + 2.00000i) q^{15} +2.47214i q^{17} -0.763932i q^{19} +(2.00000 - 0.763932i) q^{21} -2.47214 q^{23} -5.47214 q^{25} +(-4.61803 - 2.38197i) q^{27} +4.76393i q^{29} -5.23607i q^{31} +(3.23607 + 8.47214i) q^{33} +4.00000 q^{35} +8.47214 q^{37} +(-2.76393 - 7.23607i) q^{39} -6.47214i q^{41} +7.23607i q^{43} +(-6.47214 - 7.23607i) q^{45} +8.00000 q^{47} +5.47214 q^{49} +(-4.00000 + 1.52786i) q^{51} +3.23607i q^{53} +16.9443i q^{55} +(1.23607 - 0.472136i) q^{57} -1.23607 q^{59} +0.472136 q^{61} +(2.47214 + 2.76393i) q^{63} -14.4721i q^{65} -9.70820i q^{67} +(-1.52786 - 4.00000i) q^{69} +15.4164 q^{71} -2.00000 q^{73} +(-3.38197 - 8.85410i) q^{75} -6.47214i q^{77} -0.291796i q^{79} +(1.00000 - 8.94427i) q^{81} +2.76393 q^{83} -8.00000 q^{85} +(-7.70820 + 2.94427i) q^{87} -4.00000i q^{89} +5.52786i q^{91} +(8.47214 - 3.23607i) q^{93} +2.47214 q^{95} +0.472136 q^{97} +(-11.7082 + 10.4721i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.257517\pi$
$5$			3.23607i			1.44721i	−0.690212	+	0.723607i	$0.742483\pi$
$6$		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	$-0.210409\pi$
$11$	5.23607			1.57873			0.789367	+	0.613922i	$0.210409\pi$
$12$		0			0
$13$	−4.47214			−1.24035			−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214			−1.24035			−0.620174	+	0.784465i	$0.712938\pi$
$14$		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	$-0.972071\pi$
$19$		−	0.763932i		−	0.175258i	0.996153	−	0.0876290i	$-0.0279290\pi$
$20$		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	$0.145844\pi$
$29$			4.76393i			0.884640i	−0.896857	+	0.442320i	$0.854156\pi$
$30$		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	$-0.254782\pi$
$37$	8.47214			1.39281			0.696405	+	0.717649i	$0.254782\pi$
$38$		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	$0.186038\pi$
$43$			7.23607i			1.10349i	−0.834013	+	0.551745i	$0.813962\pi$
$44$		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	$0.0713414\pi$
$53$			3.23607i			0.444508i	−0.974989	+	0.222254i	$0.928659\pi$
$54$		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	$-0.525639\pi$
$59$	−1.23607			−0.160922			−0.0804612	+	0.996758i	$0.525639\pi$
$60$		0			0
$61$	0.472136			0.0604508			0.0302254	−	0.999543i	$-0.490377\pi$
$61$	0.472136			0.0604508			0.0302254	+	0.999543i	$0.490377\pi$
$62$		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	$-0.797934\pi$
$67$		−	9.70820i		−	1.18605i	0.805186	−	0.593023i	$-0.202066\pi$
$68$		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	$-0.451528\pi$
$83$	2.76393			0.303381			0.151690	+	0.988428i	$0.451528\pi$
$84$		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	$-0.972915\pi$
$101$		−	1.70820i		−	0.169973i	0.996382	−	0.0849863i	$-0.0270847\pi$
$102$		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	$-0.740387\pi$
$107$	−14.1803			−1.37087			−0.685433	+	0.728136i	$0.740387\pi$
$108$		0			0
$109$	8.47214			0.811483			0.405742	−	0.913988i	$-0.367013\pi$
$109$	8.47214			0.811483			0.405742	+	0.913988i	$0.367013\pi$
$110$		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	$-0.595481\pi$
$131$	−6.76393			−0.590967			−0.295484	+	0.955348i	$0.595481\pi$
$132$		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	$-0.864928\pi$
$139$		−	9.70820i		−	0.823439i	0.911311	−	0.411720i	$-0.135072\pi$
$140$		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	$-0.869821\pi$
$149$		−	9.70820i		−	0.795327i	0.917531	−	0.397664i	$-0.130179\pi$
$150$		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	$-0.557111\pi$
$157$	−4.47214			−0.356915			−0.178458	+	0.983948i	$0.557111\pi$
$158$		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	$0.0523477\pi$
$163$			4.18034i			0.327429i	−0.986508	+	0.163715i	$0.947652\pi$
$164$		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	$0.330235\pi$
$173$			22.6525i			1.72224i	−0.508405	+	0.861118i	$0.669765\pi$
$174$		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	$-0.425810\pi$
$179$	6.18034			0.461940			0.230970	+	0.972961i	$0.425810\pi$
$180$		0			0
$181$	−20.4721			−1.52168			−0.760841	−	0.648938i	$-0.775213\pi$
$181$	−20.4721			−1.52168			−0.760841	+	0.648938i	$0.775213\pi$
$182$		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	$-0.945718\pi$
$197$		−	4.76393i		−	0.339416i	0.985494	−	0.169708i	$-0.0542825\pi$
$198$		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	$0.126406\pi$
$211$			11.2361i			0.773523i	−0.922180	+	0.386761i	$0.873594\pi$
$212$		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	$-0.523052\pi$
$227$	−2.18034			−0.144714			−0.0723571	+	0.997379i	$0.523052\pi$
$228$		0			0
$229$	11.5279			0.761783			0.380891	−	0.924620i	$-0.375617\pi$
$229$	11.5279			0.761783			0.380891	+	0.924620i	$0.375617\pi$
$230$		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	$-0.769093\pi$
$251$	−23.7082			−1.49645			−0.748224	+	0.663446i	$0.769093\pi$
$252$		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	$-0.835803\pi$
$269$		−	16.1803i		−	0.986533i	0.869878	−	0.493266i	$-0.164197\pi$
$270$		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	$-0.572615\pi$
$277$	−7.52786			−0.452306			−0.226153	+	0.974092i	$0.572615\pi$
$278$		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	$-0.788290\pi$
$283$		−	20.7639i		−	1.23429i	0.786850	−	0.617144i	$-0.211710\pi$
$284$		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	$-0.792565\pi$
$293$		−	20.7639i		−	1.21304i	0.795068	−	0.606521i	$-0.207435\pi$
$294$		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	$-0.711367\pi$
$307$		−	27.5967i		−	1.57503i	0.616296	−	0.787515i	$-0.288633\pi$
$308$		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	$-0.879845\pi$
$317$		−	13.1246i		−	0.737152i	0.929598	−	0.368576i	$-0.120155\pi$
$318$		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	$-0.647668\pi$
$331$		−	32.5410i		−	1.78862i	0.447452	−	0.894308i	$-0.352332\pi$
$332$		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	$-0.719561\pi$
$347$	−23.7082			−1.27272			−0.636362	+	0.771391i	$0.719561\pi$
$348$		0			0
$349$	−28.4721			−1.52408			−0.762039	−	0.647531i	$-0.775802\pi$
$349$	−28.4721			−1.52408			−0.762039	+	0.647531i	$0.775802\pi$
$350$		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	$-0.521307\pi$
$373$	−2.58359			−0.133773			−0.0668867	+	0.997761i	$0.521307\pi$
$374$		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	$-0.911789\pi$
$379$		−	10.6525i		−	0.547181i	0.961846	−	0.273590i	$-0.0882113\pi$
$380$		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	$0.303705\pi$
$389$			32.1803i			1.63161i	−0.578328	+	0.815804i	$0.696295\pi$
$390$		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	$-0.643978\pi$
$397$	−17.4164			−0.874104			−0.437052	+	0.899436i	$0.643978\pi$
$398$		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	$-0.579994\pi$
$419$	−10.1803			−0.497342			−0.248671	+	0.968588i	$0.579994\pi$
$420$		0			0
$421$	21.4164			1.04377			0.521886	−	0.853015i	$-0.325229\pi$
$421$	21.4164			1.04377			0.521886	+	0.853015i	$0.325229\pi$
$422$		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	$-0.373503\pi$
$443$	16.2918			0.774047			0.387023	+	0.922070i	$0.373503\pi$
$444$		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	$-0.938990\pi$
$461$		−	8.18034i		−	0.380996i	0.981688	−	0.190498i	$-0.0610103\pi$
$462$		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	$-0.419882\pi$
$467$	10.7639			0.498095			0.249048	+	0.968491i	$0.419882\pi$
$468$		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	$-0.757525\pi$
$491$	−32.0689			−1.44725			−0.723624	+	0.690194i	$0.757525\pi$
$492$		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	$0.208681\pi$
$499$			27.2361i			1.21925i	−0.792688	+	0.609627i	$0.791319\pi$
$500$		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	$0.192959\pi$
$509$			25.7082i			1.13950i	−0.821819	+	0.569748i	$0.807041\pi$
$510$		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	$-0.916799\pi$
$523$		−	11.8197i		−	0.516838i	0.966033	−	0.258419i	$-0.0832015\pi$
$524$		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	$-0.308233\pi$
$541$	26.3607			1.13333			0.566667	+	0.823947i	$0.308233\pi$
$542$		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	$0.0494398\pi$
$547$			7.23607i			0.309392i	−0.987962	+	0.154696i	$0.950560\pi$
$548$		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	$0.100471\pi$
$557$			14.6525i			0.620845i	−0.950599	+	0.310423i	$0.899529\pi$
$558$		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	$-0.277450\pi$
$563$	30.5410			1.28715			0.643575	+	0.765383i	$0.277450\pi$
$564$		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	$0.0341979\pi$
$571$			5.12461i			0.214458i	−0.994234	+	0.107229i	$0.965802\pi$
$572$		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	$-0.352956\pi$
$587$	21.5967			0.891393			0.445697	+	0.895184i	$0.352956\pi$
$588$		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	$-0.627477\pi$
$613$	−19.3050			−0.779720			−0.389860	+	0.920874i	$0.627477\pi$
$614$		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	$0.199027\pi$
$619$			29.1246i			1.17062i	−0.810811	+	0.585308i	$0.800973\pi$
$620$		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	$0.0847738\pi$
$643$			13.3475i			0.526375i	−0.964745	+	0.263187i	$0.915226\pi$
$644$		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	$-0.909785\pi$
$653$		−	14.2918i		−	0.559281i	0.960105	−	0.279641i	$-0.0902153\pi$
$654$		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	$-0.314229\pi$
$659$	28.2918			1.10209			0.551046	+	0.834475i	$0.314229\pi$
$660$		0			0
$661$	−51.3050			−1.99553			−0.997766	−	0.0668107i	$-0.978718\pi$
$661$	−51.3050			−1.99553			−0.997766	+	0.0668107i	$0.978718\pi$
$662$		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	$0.0885584\pi$
$677$			14.2918i			0.549278i	−0.961547	+	0.274639i	$0.911442\pi$
$678$		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	$-0.418506\pi$
$683$	13.2361			0.506464			0.253232	+	0.967406i	$0.418506\pi$
$684$		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	$-0.808847\pi$
$691$		−	29.7082i		−	1.13015i	0.825038	−	0.565077i	$-0.191153\pi$
$692$		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	$-0.913010\pi$
$701$		−	14.2918i		−	0.539794i	0.962889	−	0.269897i	$-0.0869896\pi$
$702$		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	$-0.545147\pi$
$709$	−7.52786			−0.282715			−0.141357	+	0.989959i	$0.545147\pi$
$710$		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	$-0.317190\pi$
$733$	29.4164			1.08652			0.543260	+	0.839565i	$0.317190\pi$
$734$		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	$0.0776060\pi$
$739$			13.1246i			0.482797i	−0.970426	+	0.241398i	$0.922394\pi$
$740$		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	$-0.468618\pi$
$757$	5.41641			0.196863			0.0984313	+	0.995144i	$0.468618\pi$
$758$		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	$-0.955088\pi$
$773$		−	7.81966i		−	0.281254i	0.990063	−	0.140627i	$-0.0449118\pi$
$774$		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	$0.167498\pi$
$787$			28.1803i			1.00452i	−0.864716	+	0.502260i	$0.832502\pi$
$788$		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	−

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

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