LMFDB - Embedded newform 768.2.f.c.383.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,2,Mod(383,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.383"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$768 = 2^{8} \cdot 3$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	768.f (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$6.13251087523$
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:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.1
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.2.f.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)

$f(q)$	$=$	$q+(-1.61803 - 0.618034i) q^{3} +3.23607 q^{5} -1.23607i q^{7} +(2.23607 + 2.00000i) q^{9} +5.23607i q^{11} +4.47214i q^{13} +(-5.23607 - 2.00000i) q^{15} -2.47214i q^{17} +0.763932 q^{19} +(-0.763932 + 2.00000i) q^{21} +2.47214 q^{23} +5.47214 q^{25} +(-2.38197 - 4.61803i) q^{27} -4.76393 q^{29} +5.23607i q^{31} +(3.23607 - 8.47214i) q^{33} -4.00000i q^{35} +8.47214i q^{37} +(2.76393 - 7.23607i) q^{39} -6.47214i q^{41} +7.23607 q^{43} +(7.23607 + 6.47214i) q^{45} +8.00000 q^{47} +5.47214 q^{49} +(-1.52786 + 4.00000i) q^{51} +3.23607 q^{53} +16.9443i q^{55} +(-1.23607 - 0.472136i) q^{57} -1.23607i q^{59} -0.472136i q^{61} +(2.47214 - 2.76393i) q^{63} +14.4721i q^{65} +9.70820 q^{67} +(-4.00000 - 1.52786i) q^{69} -15.4164 q^{71} +2.00000 q^{73} +(-8.85410 - 3.38197i) q^{75} +6.47214 q^{77} +0.291796i q^{79} +(1.00000 + 8.94427i) q^{81} -2.76393i q^{83} -8.00000i q^{85} +(7.70820 + 2.94427i) q^{87} -4.00000i q^{89} +5.52786 q^{91} +(3.23607 - 8.47214i) q^{93} +2.47214 q^{95} +0.472136 q^{97} +(-10.4721 + 11.7082i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{3} + 4 q^{5} - 12 q^{15} + 12 q^{19} - 12 q^{21} - 8 q^{23} + 4 q^{25} - 14 q^{27} - 28 q^{29} + 4 q^{33} + 20 q^{39} + 20 q^{43} + 20 q^{45} + 32 q^{47} + 4 q^{49} - 24 q^{51} + 4 q^{53} + 4 q^{57}+ \cdots - 24 q^{99}+O(q^{100})$ 4 * q - 2 * q^3 + 4 * q^5 - 12 * q^15 + 12 * q^19 - 12 * q^21 - 8 * q^23 + 4 * q^25 - 14 * q^27 - 28 * q^29 + 4 * q^33 + 20 * q^39 + 20 * q^43 + 20 * q^45 + 32 * q^47 + 4 * q^49 - 24 * q^51 + 4 * q^53 + 4 * q^57 - 8 * q^63 + 12 * q^67 - 16 * q^69 - 8 * q^71 + 8 * q^73 - 22 * q^75 + 8 * q^77 + 4 * q^81 + 4 * q^87 + 40 * q^91 + 4 * q^93 - 8 * q^95 - 16 * q^97 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$ .

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−1.61803	−	0.618034i	−0.934172	−	0.356822i
$3$	−1.61803	−	0.618034i	−0.934172	−	0.356822i
$4$		0			0
$5$	3.23607			1.44721			0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607			1.44721			0.723607	+	0.690212i	$0.242483\pi$
$6$		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
$11$			5.23607i			1.57873i	0.613922	+	0.789367i	$0.289591\pi$
$11$			5.23607i			1.57873i	−0.613922	+	0.789367i	$0.710409\pi$
$12$		0			0
$13$			4.47214i			1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$			4.47214i			1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$		0			0
$15$	−5.23607	−	2.00000i	−1.35195	−	0.516398i
$16$		0			0
$17$		−	2.47214i		−	0.599581i	−0.954005	−	0.299791i	$-0.903083\pi$
$17$		−	2.47214i		−	0.599581i	0.954005	−	0.299791i	$-0.0969168\pi$
$18$		0			0
$19$	0.763932			0.175258			0.0876290	−	0.996153i	$-0.472071\pi$
$19$	0.763932			0.175258			0.0876290	+	0.996153i	$0.472071\pi$
$20$		0			0
$21$	−0.763932	+	2.00000i	−0.166704	+	0.436436i
$22$		0			0
$23$	2.47214			0.515476			0.257738	−	0.966215i	$-0.417023\pi$
$23$	2.47214			0.515476			0.257738	+	0.966215i	$0.417023\pi$
$24$		0			0
$25$	5.47214			1.09443
$26$		0			0
$27$	−2.38197	−	4.61803i	−0.458410	−	0.888741i
$28$		0			0
$29$	−4.76393			−0.884640			−0.442320	−	0.896857i	$-0.645844\pi$
$29$	−4.76393			−0.884640			−0.442320	+	0.896857i	$0.645844\pi$
$30$		0			0
$31$			5.23607i			0.940426i	0.882553	+	0.470213i	$0.155823\pi$
$31$			5.23607i			0.940426i	−0.882553	+	0.470213i	$0.844177\pi$
$32$		0			0
$33$	3.23607	−	8.47214i	0.563327	−	1.47481i
$34$		0			0
$35$		−	4.00000i		−	0.676123i
$36$		0			0
$37$			8.47214i			1.39281i	0.717649	+	0.696405i	$0.245218\pi$
$37$			8.47214i			1.39281i	−0.717649	+	0.696405i	$0.754782\pi$
$38$		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
$43$	7.23607			1.10349			0.551745	−	0.834013i	$-0.313962\pi$
$43$	7.23607			1.10349			0.551745	+	0.834013i	$0.313962\pi$
$44$		0			0
$45$	7.23607	+	6.47214i	1.07869	+	0.964809i
$46$		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
$51$	−1.52786	+	4.00000i	−0.213944	+	0.560112i
$52$		0			0
$53$	3.23607			0.444508			0.222254	−	0.974989i	$-0.428659\pi$
$53$	3.23607			0.444508			0.222254	+	0.974989i	$0.428659\pi$
$54$		0			0
$55$			16.9443i			2.28477i
$56$		0			0
$57$	−1.23607	−	0.472136i	−0.163721	−	0.0625359i
$58$		0			0
$59$		−	1.23607i		−	0.160922i	−0.996758	−	0.0804612i	$-0.974361\pi$
$59$		−	1.23607i		−	0.160922i	0.996758	−	0.0804612i	$-0.0256393\pi$
$60$		0			0
$61$		−	0.472136i		−	0.0604508i	−0.999543	−	0.0302254i	$-0.990377\pi$
$61$		−	0.472136i		−	0.0604508i	0.999543	−	0.0302254i	$-0.00962251\pi$
$62$		0			0
$63$	2.47214	−	2.76393i	0.311460	−	0.348223i
$64$		0			0
$65$			14.4721i			1.79505i
$66$		0			0
$67$	9.70820			1.18605			0.593023	−	0.805186i	$-0.297934\pi$
$67$	9.70820			1.18605			0.593023	+	0.805186i	$0.297934\pi$
$68$		0			0
$69$	−4.00000	−	1.52786i	−0.481543	−	0.183933i
$70$		0			0
$71$	−15.4164			−1.82959			−0.914796	−	0.403917i	$-0.867648\pi$
$71$	−15.4164			−1.82959			−0.914796	+	0.403917i	$0.867648\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$	−8.85410	−	3.38197i	−1.02238	−	0.390516i
$76$		0			0
$77$	6.47214			0.737568
$78$		0			0
$79$			0.291796i			0.0328296i	0.999865	+	0.0164148i	$0.00522523\pi$
$79$			0.291796i			0.0328296i	−0.999865	+	0.0164148i	$0.994775\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$		−	2.76393i		−	0.303381i	−0.988428	−	0.151690i	$-0.951528\pi$
$83$		−	2.76393i		−	0.303381i	0.988428	−	0.151690i	$-0.0484717\pi$
$84$		0			0
$85$		−	8.00000i		−	0.867722i
$86$		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
$91$	5.52786			0.579478
$92$		0			0
$93$	3.23607	−	8.47214i	0.335565	−	0.878520i
$94$		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
$99$	−10.4721	+	11.7082i	−1.05249	+	1.17672i
$100$		0			0
$101$	−1.70820			−0.169973			−0.0849863	−	0.996382i	$-0.527085\pi$
$101$	−1.70820			−0.169973			−0.0849863	+	0.996382i	$0.527085\pi$
$102$		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
$107$		−	14.1803i		−	1.37087i	−0.728136	−	0.685433i	$-0.759613\pi$
$107$		−	14.1803i		−	1.37087i	0.728136	−	0.685433i	$-0.240387\pi$
$108$		0			0
$109$		−	8.47214i		−	0.811483i	−0.913988	−	0.405742i	$-0.867013\pi$
$109$		−	8.47214i		−	0.811483i	0.913988	−	0.405742i	$-0.132987\pi$
$110$		0			0
$111$	5.23607	−	13.7082i	0.496986	−	1.30113i
$112$		0			0
$113$		−	8.00000i		−	0.752577i	−0.926503	−	0.376288i	$-0.877200\pi$
$113$		−	8.00000i		−	0.752577i	0.926503	−	0.376288i	$-0.122800\pi$
$114$		0			0
$115$	8.00000			0.746004
$116$		0			0
$117$	−8.94427	+	10.0000i	−0.826898	+	0.924500i
$118$		0			0
$119$	−3.05573			−0.280118
$120$		0			0
$121$	−16.4164			−1.49240
$122$		0			0
$123$	−4.00000	+	10.4721i	−0.360668	+	0.944241i
$124$		0			0
$125$	1.52786			0.136656
$126$		0			0
$127$			13.2361i			1.17451i	0.809402	+	0.587256i	$0.199792\pi$
$127$			13.2361i			1.17451i	−0.809402	+	0.587256i	$0.800208\pi$
$128$		0			0
$129$	−11.7082	−	4.47214i	−1.03085	−	0.393750i
$130$		0			0
$131$			6.76393i			0.590967i	0.955348	+	0.295484i	$0.0954808\pi$
$131$			6.76393i			0.590967i	−0.955348	+	0.295484i	$0.904519\pi$
$132$		0			0
$133$		−	0.944272i		−	0.0818788i
$134$		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
$139$	−9.70820			−0.823439			−0.411720	−	0.911311i	$-0.635072\pi$
$139$	−9.70820			−0.823439			−0.411720	+	0.911311i	$0.635072\pi$
$140$		0			0
$141$	−12.9443	−	4.94427i	−1.09010	−	0.416383i
$142$		0			0
$143$	−23.4164			−1.95818
$144$		0			0
$145$	−15.4164			−1.28026
$146$		0			0
$147$	−8.85410	−	3.38197i	−0.730274	−	0.278940i
$148$		0			0
$149$	−9.70820			−0.795327			−0.397664	−	0.917531i	$-0.630179\pi$
$149$	−9.70820			−0.795327			−0.397664	+	0.917531i	$0.630179\pi$
$150$		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
$155$			16.9443i			1.36100i
$156$		0			0
$157$			4.47214i			0.356915i	0.983948	+	0.178458i	$0.0571108\pi$
$157$			4.47214i			0.356915i	−0.983948	+	0.178458i	$0.942889\pi$
$158$		0			0
$159$	−5.23607	−	2.00000i	−0.415247	−	0.158610i
$160$		0			0
$161$		−	3.05573i		−	0.240825i
$162$		0			0
$163$	−4.18034			−0.327429			−0.163715	−	0.986508i	$-0.552348\pi$
$163$	−4.18034			−0.327429			−0.163715	+	0.986508i	$0.552348\pi$
$164$		0			0
$165$	10.4721	−	27.4164i	0.815255	−	2.13436i
$166$		0			0
$167$	13.5279			1.04682			0.523409	−	0.852082i	$-0.324660\pi$
$167$	13.5279			1.04682			0.523409	+	0.852082i	$0.324660\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$	1.70820	+	1.52786i	0.130630	+	0.116839i
$172$		0			0
$173$	−22.6525			−1.72224			−0.861118	−	0.508405i	$-0.830235\pi$
$173$	−22.6525			−1.72224			−0.861118	+	0.508405i	$0.830235\pi$
$174$		0			0
$175$		−	6.76393i		−	0.511305i
$176$		0			0
$177$	−0.763932	+	2.00000i	−0.0574206	+	0.150329i
$178$		0			0
$179$		−	6.18034i		−	0.461940i	−0.972961	−	0.230970i	$-0.925810\pi$
$179$		−	6.18034i		−	0.461940i	0.972961	−	0.230970i	$-0.0741900\pi$
$180$		0			0
$181$		−	20.4721i		−	1.52168i	−0.648938	−	0.760841i	$-0.724787\pi$
$181$		−	20.4721i		−	1.52168i	0.648938	−	0.760841i	$-0.275213\pi$
$182$		0			0
$183$	−0.291796	+	0.763932i	−0.0215702	+	0.0564715i
$184$		0			0
$185$			27.4164i			2.01569i
$186$		0			0
$187$	12.9443			0.946579
$188$		0			0
$189$	−5.70820	+	2.94427i	−0.415211	+	0.214164i
$190$		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
$195$	8.94427	−	23.4164i	0.640513	−	1.67688i
$196$		0			0
$197$	−4.76393			−0.339416			−0.169708	−	0.985494i	$-0.554282\pi$
$197$	−4.76393			−0.339416			−0.169708	+	0.985494i	$0.554282\pi$
$198$		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
$203$			5.88854i			0.413295i
$204$		0			0
$205$		−	20.9443i		−	1.46281i
$206$		0			0
$207$	5.52786	+	4.94427i	0.384213	+	0.343651i
$208$		0			0
$209$			4.00000i			0.276686i
$210$		0			0
$211$	−11.2361			−0.773523			−0.386761	−	0.922180i	$-0.626406\pi$
$211$	−11.2361			−0.773523			−0.386761	+	0.922180i	$0.626406\pi$
$212$		0			0
$213$	24.9443	+	9.52786i	1.70915	+	0.652838i
$214$		0			0
$215$	23.4164			1.59699
$216$		0			0
$217$	6.47214			0.439357
$218$		0			0
$219$	−3.23607	−	1.23607i	−0.218673	−	0.0835257i
$220$		0			0
$221$	11.0557			0.743689
$222$		0			0
$223$			10.1803i			0.681726i	0.940113	+	0.340863i	$0.110719\pi$
$223$			10.1803i			0.681726i	−0.940113	+	0.340863i	$0.889281\pi$
$224$		0			0
$225$	12.2361	+	10.9443i	0.815738	+	0.729618i
$226$		0			0
$227$			2.18034i			0.144714i	0.997379	+	0.0723571i	$0.0230521\pi$
$227$			2.18034i			0.144714i	−0.997379	+	0.0723571i	$0.976948\pi$
$228$		0			0
$229$			11.5279i			0.761783i	0.924620	+	0.380891i	$0.124383\pi$
$229$			11.5279i			0.761783i	−0.924620	+	0.380891i	$0.875617\pi$
$230$		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
$235$	25.8885			1.68878
$236$		0			0
$237$	0.180340	−	0.472136i	0.0117143	−	0.0306685i
$238$		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
$243$	3.90983	−	15.0902i	0.250816	−	0.968035i
$244$		0			0
$245$	17.7082			1.13134
$246$		0			0
$247$			3.41641i			0.217381i
$248$		0			0
$249$	−1.70820	+	4.47214i	−0.108253	+	0.283410i
$250$		0			0
$251$		−	23.7082i		−	1.49645i	−0.663446	−	0.748224i	$-0.730907\pi$
$251$		−	23.7082i		−	1.49645i	0.663446	−	0.748224i	$-0.269093\pi$
$252$		0			0
$253$			12.9443i			0.813799i
$254$		0			0
$255$	−4.94427	+	12.9443i	−0.309622	+	0.810602i
$256$		0			0
$257$		−	20.9443i		−	1.30647i	−0.757156	−	0.653234i	$-0.773412\pi$
$257$		−	20.9443i		−	1.30647i	0.757156	−	0.653234i	$-0.226588\pi$
$258$		0			0
$259$	10.4721			0.650707
$260$		0			0
$261$	−10.6525	−	9.52786i	−0.659372	−	0.589760i
$262$		0			0
$263$	10.4721			0.645740			0.322870	−	0.946443i	$-0.395352\pi$
$263$	10.4721			0.645740			0.322870	+	0.946443i	$0.395352\pi$
$264$		0			0
$265$	10.4721			0.643298
$266$		0			0
$267$	−2.47214	+	6.47214i	−0.151292	+	0.396088i
$268$		0			0
$269$	16.1803			0.986533			0.493266	−	0.869878i	$-0.335803\pi$
$269$	16.1803			0.986533			0.493266	+	0.869878i	$0.335803\pi$
$270$		0			0
$271$			0.291796i			0.0177253i	0.999961	+	0.00886267i	$0.00282111\pi$
$271$			0.291796i			0.0177253i	−0.999961	+	0.00886267i	$0.997179\pi$
$272$		0			0
$273$	−8.94427	−	3.41641i	−0.541332	−	0.206770i
$274$		0			0
$275$			28.6525i			1.72781i
$276$		0			0
$277$		−	7.52786i		−	0.452306i	−0.974092	−	0.226153i	$-0.927385\pi$
$277$		−	7.52786i		−	0.452306i	0.974092	−	0.226153i	$-0.0726149\pi$
$278$		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
$283$	−20.7639			−1.23429			−0.617144	−	0.786850i	$-0.711710\pi$
$283$	−20.7639			−1.23429			−0.617144	+	0.786850i	$0.711710\pi$
$284$		0			0
$285$	−4.00000	−	1.52786i	−0.236940	−	0.0905029i
$286$		0			0
$287$	−8.00000			−0.472225
$288$		0			0
$289$	10.8885			0.640503
$290$		0			0
$291$	−0.763932	−	0.291796i	−0.0447825	−	0.0171054i
$292$		0			0
$293$	−20.7639			−1.21304			−0.606521	−	0.795068i	$-0.707435\pi$
$293$	−20.7639			−1.21304			−0.606521	+	0.795068i	$0.707435\pi$
$294$		0			0
$295$		−	4.00000i		−	0.232889i
$296$		0			0
$297$	24.1803	−	12.4721i	1.40309	−	0.723707i
$298$		0			0
$299$			11.0557i			0.639369i
$300$		0			0
$301$		−	8.94427i		−	0.515539i
$302$		0			0
$303$	2.76393	+	1.05573i	0.158784	+	0.0606500i
$304$		0			0
$305$		−	1.52786i		−	0.0874852i
$306$		0			0
$307$	27.5967			1.57503			0.787515	−	0.616296i	$-0.211367\pi$
$307$	27.5967			1.57503			0.787515	+	0.616296i	$0.211367\pi$
$308$		0			0
$309$	−8.76393	+	22.9443i	−0.498563	+	1.30525i
$310$		0			0
$311$	5.52786			0.313456			0.156728	−	0.987642i	$-0.449905\pi$
$311$	5.52786			0.313456			0.156728	+	0.987642i	$0.449905\pi$
$312$		0			0
$313$	0.472136			0.0266867			0.0133434	−	0.999911i	$-0.495753\pi$
$313$	0.472136			0.0266867			0.0133434	+	0.999911i	$0.495753\pi$
$314$		0			0
$315$	8.00000	−	8.94427i	0.450749	−	0.503953i
$316$		0			0
$317$	13.1246			0.737152			0.368576	−	0.929598i	$-0.379845\pi$
$317$	13.1246			0.737152			0.368576	+	0.929598i	$0.379845\pi$
$318$		0			0
$319$		−	24.9443i		−	1.39661i
$320$		0			0
$321$	−8.76393	+	22.9443i	−0.489155	+	1.28062i
$322$		0			0
$323$		−	1.88854i		−	0.105081i
$324$		0			0
$325$			24.4721i			1.35747i
$326$		0			0
$327$	−5.23607	+	13.7082i	−0.289555	+	0.758065i
$328$		0			0
$329$		−	9.88854i		−	0.545173i
$330$		0			0
$331$	−32.5410			−1.78862			−0.894308	−	0.447452i	$-0.852332\pi$
$331$	−32.5410			−1.78862			−0.894308	+	0.447452i	$0.852332\pi$
$332$		0			0
$333$	−16.9443	+	18.9443i	−0.928540	+	1.03814i
$334$		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
$339$	−4.94427	+	12.9443i	−0.268536	+	0.703036i
$340$		0			0
$341$	−27.4164			−1.48468
$342$		0			0
$343$		−	15.4164i		−	0.832408i
$344$		0			0
$345$	−12.9443	−	4.94427i	−0.696896	−	0.266191i
$346$		0			0
$347$		−	23.7082i		−	1.27272i	−0.771391	−	0.636362i	$-0.780439\pi$
$347$		−	23.7082i		−	1.27272i	0.771391	−	0.636362i	$-0.219561\pi$
$348$		0			0
$349$			28.4721i			1.52408i	0.647531	+	0.762039i	$0.275802\pi$
$349$			28.4721i			1.52408i	−0.647531	+	0.762039i	$0.724198\pi$
$350$		0			0
$351$	20.6525	−	10.6525i	1.10235	−	0.568587i
$352$		0			0
$353$		−	9.88854i		−	0.526314i	−0.964753	−	0.263157i	$-0.915236\pi$
$353$		−	9.88854i		−	0.526314i	0.964753	−	0.263157i	$-0.0847637\pi$
$354$		0			0
$355$	−49.8885			−2.64781
$356$		0			0
$357$	4.94427	+	1.88854i	0.261679	+	0.0999523i
$358$		0			0
$359$	−31.4164			−1.65809			−0.829047	−	0.559178i	$-0.811117\pi$
$359$	−31.4164			−1.65809			−0.829047	+	0.559178i	$0.811117\pi$
$360$		0			0
$361$	−18.4164			−0.969285
$362$		0			0
$363$	26.5623	+	10.1459i	1.39416	+	0.532522i
$364$		0			0
$365$	6.47214			0.338767
$366$		0			0
$367$			5.23607i			0.273321i	0.990618	+	0.136660i	$0.0436369\pi$
$367$			5.23607i			0.273321i	−0.990618	+	0.136660i	$0.956363\pi$
$368$		0			0
$369$	12.9443	−	14.4721i	0.673852	−	0.753389i
$370$		0			0
$371$		−	4.00000i		−	0.207670i
$372$		0			0
$373$		−	2.58359i		−	0.133773i	−0.997761	−	0.0668867i	$-0.978693\pi$
$373$		−	2.58359i		−	0.133773i	0.997761	−	0.0668867i	$-0.0213066\pi$
$374$		0			0
$375$	−2.47214	−	0.944272i	−0.127661	−	0.0487620i
$376$		0			0
$377$		−	21.3050i		−	1.09726i
$378$		0			0
$379$	−10.6525			−0.547181			−0.273590	−	0.961846i	$-0.588211\pi$
$379$	−10.6525			−0.547181			−0.273590	+	0.961846i	$0.588211\pi$
$380$		0			0
$381$	8.18034	−	21.4164i	0.419092	−	1.09720i
$382$		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
$387$	16.1803	+	14.4721i	0.822493	+	0.735660i
$388$		0			0
$389$	32.1803			1.63161			0.815804	−	0.578328i	$-0.196295\pi$
$389$	32.1803			1.63161			0.815804	+	0.578328i	$0.196295\pi$
$390$		0			0
$391$		−	6.11146i		−	0.309070i
$392$		0			0
$393$	4.18034	−	10.9443i	0.210870	−	0.552065i
$394$		0			0
$395$			0.944272i			0.0475115i
$396$		0			0
$397$			17.4164i			0.874104i	0.899436	+	0.437052i	$0.143978\pi$
$397$			17.4164i			0.874104i	−0.899436	+	0.437052i	$0.856022\pi$
$398$		0			0
$399$	−0.583592	+	1.52786i	−0.0292161	+	0.0764889i
$400$		0			0
$401$			28.3607i			1.41626i	0.706080	+	0.708132i	$0.250462\pi$
$401$			28.3607i			1.41626i	−0.706080	+	0.708132i	$0.749538\pi$
$402$		0			0
$403$	−23.4164			−1.16645
$404$		0			0
$405$	3.23607	+	28.9443i	0.160802	+	1.43825i
$406$		0			0
$407$	−44.3607			−2.19888
$408$		0			0
$409$	31.8885			1.57679			0.788394	−	0.615171i	$-0.210913\pi$
$409$	31.8885			1.57679			0.788394	+	0.615171i	$0.210913\pi$
$410$		0			0
$411$	−0.944272	+	2.47214i	−0.0465775	+	0.121941i
$412$		0			0
$413$	−1.52786			−0.0751813
$414$		0			0
$415$		−	8.94427i		−	0.439057i
$416$		0			0
$417$	15.7082	+	6.00000i	0.769234	+	0.293821i
$418$		0			0
$419$			10.1803i			0.497342i	0.968588	+	0.248671i	$0.0799938\pi$
$419$			10.1803i			0.497342i	−0.968588	+	0.248671i	$0.920006\pi$
$420$		0			0
$421$			21.4164i			1.04377i	0.853015	+	0.521886i	$0.174771\pi$
$421$			21.4164i			1.04377i	−0.853015	+	0.521886i	$0.825229\pi$
$422$		0			0
$423$	17.8885	+	16.0000i	0.869771	+	0.777947i
$424$		0			0
$425$		−	13.5279i		−	0.656198i
$426$		0			0
$427$	−0.583592			−0.0282420
$428$		0			0
$429$	37.8885	+	14.4721i	1.82928	+	0.698721i
$430$		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
$435$	24.9443	+	9.52786i	1.19599	+	0.456826i
$436$		0			0
$437$	1.88854			0.0903413
$438$		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
$443$			16.2918i			0.774047i	0.922070	+	0.387023i	$0.126497\pi$
$443$			16.2918i			0.774047i	−0.922070	+	0.387023i	$0.873503\pi$
$444$		0			0
$445$		−	12.9443i		−	0.613617i
$446$		0			0
$447$	15.7082	+	6.00000i	0.742973	+	0.283790i
$448$		0			0
$449$			0.583592i			0.0275414i	0.999905	+	0.0137707i	$0.00438349\pi$
$449$			0.583592i			0.0275414i	−0.999905	+	0.0137707i	$0.995617\pi$
$450$		0			0
$451$	33.8885			1.59575
$452$		0			0
$453$	12.1803	−	31.8885i	0.572282	−	1.49825i
$454$		0			0
$455$	17.8885			0.838628
$456$		0			0
$457$	22.0000			1.02912			0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000			1.02912			0.514558	+	0.857455i	$0.327956\pi$
$458$		0			0
$459$	−11.4164	+	5.88854i	−0.532872	+	0.274854i
$460$		0			0
$461$	8.18034			0.380996			0.190498	−	0.981688i	$-0.438990\pi$
$461$	8.18034			0.380996			0.190498	+	0.981688i	$0.438990\pi$
$462$		0			0
$463$		−	34.7639i		−	1.61562i	−0.589445	−	0.807808i	$-0.700654\pi$
$463$		−	34.7639i		−	1.61562i	0.589445	−	0.807808i	$-0.299346\pi$
$464$		0			0
$465$	10.4721	−	27.4164i	0.485634	−	1.27141i
$466$		0			0
$467$		−	10.7639i		−	0.498095i	−0.968491	−	0.249048i	$-0.919882\pi$
$467$		−	10.7639i		−	0.498095i	0.968491	−	0.249048i	$-0.0801176\pi$
$468$		0			0
$469$		−	12.0000i		−	0.554109i
$470$		0			0
$471$	2.76393	−	7.23607i	0.127355	−	0.333420i
$472$		0			0
$473$			37.8885i			1.74212i
$474$		0			0
$475$	4.18034			0.191807
$476$		0			0
$477$	7.23607	+	6.47214i	0.331317	+	0.296339i
$478$		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
$483$	−1.88854	+	4.94427i	−0.0859317	+	0.224972i
$484$		0			0
$485$	1.52786			0.0693767
$486$		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
$491$		−	32.0689i		−	1.44725i	−0.690194	−	0.723624i	$-0.742475\pi$
$491$		−	32.0689i		−	1.44725i	0.690194	−	0.723624i	$-0.257525\pi$
$492$		0			0
$493$			11.7771i			0.530413i
$494$		0			0
$495$	−33.8885	+	37.8885i	−1.52318	+	1.70296i
$496$		0			0
$497$			19.0557i			0.854766i
$498$		0			0
$499$	−27.2361			−1.21925			−0.609627	−	0.792688i	$-0.708681\pi$
$499$	−27.2361			−1.21925			−0.609627	+	0.792688i	$0.708681\pi$
$500$		0			0
$501$	−21.8885	−	8.36068i	−0.977908	−	0.373528i
$502$		0			0
$503$	21.5279			0.959880			0.479940	−	0.877301i	$-0.340658\pi$
$503$	21.5279			0.959880			0.479940	+	0.877301i	$0.340658\pi$
$504$		0			0
$505$	−5.52786			−0.245987
$506$		0			0
$507$	11.3262	+	4.32624i	0.503016	+	0.192135i
$508$		0			0
$509$	−25.7082			−1.13950			−0.569748	−	0.821819i	$-0.692959\pi$
$509$	−25.7082			−1.13950			−0.569748	+	0.821819i	$0.692959\pi$
$510$		0			0
$511$		−	2.47214i		−	0.109361i
$512$		0			0
$513$	−1.81966	−	3.52786i	−0.0803400	−	0.155759i
$514$		0			0
$515$		−	45.8885i		−	2.02209i
$516$		0			0
$517$			41.8885i			1.84226i
$518$		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
$523$	−11.8197			−0.516838			−0.258419	−	0.966033i	$-0.583201\pi$
$523$	−11.8197			−0.516838			−0.258419	+	0.966033i	$0.583201\pi$
$524$		0			0
$525$	−4.18034	+	10.9443i	−0.182445	+	0.477647i
$526$		0			0
$527$	12.9443			0.563861
$528$		0			0
$529$	−16.8885			−0.734285
$530$		0			0
$531$	2.47214	−	2.76393i	0.107282	−	0.119944i
$532$		0			0
$533$	28.9443			1.25372
$534$		0			0
$535$		−	45.8885i		−	1.98393i
$536$		0			0
$537$	−3.81966	+	10.0000i	−0.164831	+	0.431532i
$538$		0			0
$539$			28.6525i			1.23415i
$540$		0			0
$541$		−	26.3607i		−	1.13333i	−0.823947	−	0.566667i	$-0.808233\pi$
$541$		−	26.3607i		−	1.13333i	0.823947	−	0.566667i	$-0.191767\pi$
$542$		0			0
$543$	−12.6525	+	33.1246i	−0.542970	+	1.42151i
$544$		0			0
$545$		−	27.4164i		−	1.17439i
$546$		0			0
$547$	−7.23607			−0.309392			−0.154696	−	0.987962i	$-0.549440\pi$
$547$	−7.23607			−0.309392			−0.154696	+	0.987962i	$0.549440\pi$
$548$		0			0
$549$	0.944272	−	1.05573i	0.0403005	−	0.0450574i
$550$		0			0
$551$	−3.63932			−0.155040
$552$		0			0
$553$	0.360680			0.0153377
$554$		0			0
$555$	16.9443	−	44.3607i	0.719244	−	1.88301i
$556$		0			0
$557$	−14.6525			−0.620845			−0.310423	−	0.950599i	$-0.600471\pi$
$557$	−14.6525			−0.620845			−0.310423	+	0.950599i	$0.600471\pi$
$558$		0			0
$559$			32.3607i			1.36871i
$560$		0			0
$561$	−20.9443	−	8.00000i	−0.884268	−	0.337760i
$562$		0			0
$563$		−	30.5410i		−	1.28715i	−0.765383	−	0.643575i	$-0.777450\pi$
$563$		−	30.5410i		−	1.28715i	0.765383	−	0.643575i	$-0.222550\pi$
$564$		0			0
$565$		−	25.8885i		−	1.08914i
$566$		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
$571$	5.12461			0.214458			0.107229	−	0.994234i	$-0.465802\pi$
$571$	5.12461			0.214458			0.107229	+	0.994234i	$0.465802\pi$
$572$		0			0
$573$	−20.9443	−	8.00000i	−0.874960	−	0.334205i
$574$		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
$579$	19.2361	+	7.34752i	0.799424	+	0.305353i
$580$		0			0
$581$	−3.41641			−0.141736
$582$		0			0
$583$			16.9443i			0.701760i
$584$		0			0
$585$	−28.9443	+	32.3607i	−1.19670	+	1.33795i
$586$		0			0
$587$			21.5967i			0.891393i	0.895184	+	0.445697i	$0.147044\pi$
$587$			21.5967i			0.891393i	−0.895184	+	0.445697i	$0.852956\pi$
$588$		0			0
$589$			4.00000i			0.164817i
$590$		0			0
$591$	7.70820	+	2.94427i	0.317073	+	0.121111i
$592$		0			0
$593$		−	28.9443i		−	1.18860i	−0.804244	−	0.594299i	$-0.797429\pi$
$593$		−	28.9443i		−	1.18860i	0.804244	−	0.594299i	$-0.202571\pi$
$594$		0			0
$595$	−9.88854			−0.405391
$596$		0			0
$597$	−3.81966	+	10.0000i	−0.156328	+	0.409273i
$598$		0			0
$599$	−7.41641			−0.303026			−0.151513	−	0.988455i	$-0.548415\pi$
$599$	−7.41641			−0.303026			−0.151513	+	0.988455i	$0.548415\pi$
$600$		0			0
$601$	−33.7771			−1.37780			−0.688898	−	0.724858i	$-0.741905\pi$
$601$	−33.7771			−1.37780			−0.688898	+	0.724858i	$0.741905\pi$
$602$		0			0
$603$	21.7082	+	19.4164i	0.884026	+	0.790697i
$604$		0			0
$605$	−53.1246			−2.15982
$606$		0			0
$607$		−	34.7639i		−	1.41102i	−0.708698	−	0.705512i	$-0.750717\pi$
$607$		−	34.7639i		−	1.41102i	0.708698	−	0.705512i	$-0.249283\pi$
$608$		0			0
$609$	3.63932	−	9.52786i	0.147473	−	0.386089i
$610$		0			0
$611$			35.7771i			1.44739i
$612$		0			0
$613$		−	19.3050i		−	0.779720i	−0.920874	−	0.389860i	$-0.872523\pi$
$613$		−	19.3050i		−	0.779720i	0.920874	−	0.389860i	$-0.127477\pi$
$614$		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
$619$	29.1246			1.17062			0.585308	−	0.810811i	$-0.300973\pi$
$619$	29.1246			1.17062			0.585308	+	0.810811i	$0.300973\pi$
$620$		0			0
$621$	−5.88854	−	11.4164i	−0.236299	−	0.458125i
$622$		0			0
$623$	−4.94427			−0.198088
$624$		0			0
$625$	−22.4164			−0.896656
$626$		0			0
$627$	2.47214	−	6.47214i	0.0987276	−	0.258472i
$628$		0			0
$629$	20.9443			0.835103
$630$		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
$635$			42.8328i			1.69977i
$636$		0			0
$637$			24.4721i			0.969621i
$638$		0			0
$639$	−34.4721	−	30.8328i	−1.36370	−	1.21973i
$640$		0			0
$641$			25.3050i			0.999485i	0.866174	+	0.499743i	$0.166572\pi$
$641$			25.3050i			0.999485i	−0.866174	+	0.499743i	$0.833428\pi$
$642$		0			0
$643$	−13.3475			−0.526375			−0.263187	−	0.964745i	$-0.584774\pi$
$643$	−13.3475			−0.526375			−0.263187	+	0.964745i	$0.584774\pi$
$644$		0			0
$645$	−37.8885	−	14.4721i	−1.49186	−	0.569840i
$646$		0			0
$647$	33.3050			1.30935			0.654676	−	0.755909i	$-0.272805\pi$
$647$	33.3050			1.30935			0.654676	+	0.755909i	$0.272805\pi$
$648$		0			0
$649$	6.47214			0.254054
$650$		0			0
$651$	−10.4721	−	4.00000i	−0.410435	−	0.156772i
$652$		0			0
$653$	14.2918			0.559281			0.279641	−	0.960105i	$-0.409785\pi$
$653$	14.2918			0.559281			0.279641	+	0.960105i	$0.409785\pi$
$654$		0			0
$655$			21.8885i			0.855256i
$656$		0			0
$657$	4.47214	+	4.00000i	0.174475	+	0.156055i
$658$		0			0
$659$		−	28.2918i		−	1.10209i	−0.834475	−	0.551046i	$-0.814229\pi$
$659$		−	28.2918i		−	1.10209i	0.834475	−	0.551046i	$-0.185771\pi$
$660$		0			0
$661$		−	51.3050i		−	1.99553i	−0.0668107	−	0.997766i	$-0.521282\pi$
$661$		−	51.3050i		−	1.99553i	0.0668107	−	0.997766i	$-0.478718\pi$
$662$		0			0
$663$	−17.8885	−	6.83282i	−0.694733	−	0.265365i
$664$		0			0
$665$		−	3.05573i		−	0.118496i
$666$		0			0
$667$	−11.7771			−0.456011
$668$		0			0
$669$	6.29180	−	16.4721i	0.243255	−	0.636850i
$670$		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
$675$	−13.0344	−	25.2705i	−0.501696	−	0.972662i
$676$		0			0
$677$	14.2918			0.549278			0.274639	−	0.961547i	$-0.411442\pi$
$677$	14.2918			0.549278			0.274639	+	0.961547i	$0.411442\pi$
$678$		0			0
$679$		−	0.583592i		−	0.0223962i
$680$		0			0
$681$	1.34752	−	3.52786i	0.0516372	−	0.135188i
$682$		0			0
$683$			13.2361i			0.506464i	0.967406	+	0.253232i	$0.0814936\pi$
$683$			13.2361i			0.506464i	−0.967406	+	0.253232i	$0.918506\pi$
$684$		0			0
$685$		−	4.94427i		−	0.188911i
$686$		0			0
$687$	7.12461	−	18.6525i	0.271821	−	0.711636i
$688$		0			0
$689$			14.4721i			0.551344i
$690$		0			0
$691$	29.7082			1.13015			0.565077	−	0.825038i	$-0.308847\pi$
$691$	29.7082			1.13015			0.565077	+	0.825038i	$0.308847\pi$
$692$		0			0
$693$	14.4721	+	12.9443i	0.549751	+	0.491712i
$694$		0			0
$695$	−31.4164			−1.19169
$696$		0			0
$697$	−16.0000			−0.606043
$698$		0			0
$699$	15.4164	−	40.3607i	0.583102	−	1.52658i
$700$		0			0
$701$	14.2918			0.539794			0.269897	−	0.962889i	$-0.413010\pi$
$701$	14.2918			0.539794			0.269897	+	0.962889i	$0.413010\pi$
$702$		0			0
$703$			6.47214i			0.244101i
$704$		0			0
$705$	−41.8885	−	16.0000i	−1.57761	−	0.602595i
$706$		0			0
$707$			2.11146i			0.0794095i
$708$		0			0
$709$		−	7.52786i		−	0.282715i	−0.989959	−	0.141357i	$-0.954853\pi$
$709$		−	7.52786i		−	0.282715i	0.989959	−	0.141357i	$-0.0451467\pi$
$710$		0			0
$711$	−0.583592	+	0.652476i	−0.0218864	+	0.0244698i
$712$		0			0
$713$			12.9443i			0.484767i
$714$		0			0
$715$	−75.7771			−2.83390
$716$		0			0
$717$	12.9443	+	4.94427i	0.483413	+	0.184647i
$718$		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
$723$	3.23607	+	1.23607i	0.120351	+	0.0459699i
$724$		0			0
$725$	−26.0689			−0.968174
$726$		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
$731$		−	17.8885i		−	0.661632i
$732$		0			0
$733$		−	29.4164i		−	1.08652i	−0.839565	−	0.543260i	$-0.817190\pi$
$733$		−	29.4164i		−	1.08652i	0.839565	−	0.543260i	$-0.182810\pi$
$734$		0			0
$735$	−28.6525	−	10.9443i	−1.05686	−	0.403686i
$736$		0			0
$737$			50.8328i			1.87245i
$738$		0			0
$739$	−13.1246			−0.482797			−0.241398	−	0.970426i	$-0.577606\pi$
$739$	−13.1246			−0.482797			−0.241398	+	0.970426i	$0.577606\pi$
$740$		0			0
$741$	2.11146	−	5.52786i	0.0775663	−	0.203071i
$742$		0			0
$743$	7.41641			0.272082			0.136041	−	0.990703i	$-0.456562\pi$
$743$	7.41641			0.272082			0.136041	+	0.990703i	$0.456562\pi$
$744$		0			0
$745$	−31.4164			−1.15101
$746$		0			0
$747$	5.52786	−	6.18034i	0.202254	−	0.226127i
$748$		0			0
$749$	−17.5279			−0.640454
$750$		0			0
$751$		−	17.5967i		−	0.642114i	−0.947060	−	0.321057i	$-0.895962\pi$
$751$		−	17.5967i		−	0.642114i	0.947060	−	0.321057i	$-0.104038\pi$
$752$		0			0
$753$	−14.6525	+	38.3607i	−0.533966	+	1.39794i
$754$		0			0
$755$			63.7771i			2.32109i
$756$		0			0
$757$			5.41641i			0.196863i	0.995144	+	0.0984313i	$0.0313825\pi$
$757$			5.41641i			0.196863i	−0.995144	+	0.0984313i	$0.968618\pi$
$758$		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
$763$	−10.4721			−0.379117
$764$		0			0
$765$	16.0000	−	17.8885i	0.578481	−	0.646762i
$766$		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
$771$	−12.9443	+	33.8885i	−0.466177	+	1.22047i
$772$		0			0
$773$	−7.81966			−0.281254			−0.140627	−	0.990063i	$-0.544912\pi$
$773$	−7.81966			−0.281254			−0.140627	+	0.990063i	$0.544912\pi$
$774$		0			0
$775$			28.6525i			1.02923i
$776$		0			0
$777$	−16.9443	−	6.47214i	−0.607872	−	0.232187i
$778$		0			0
$779$		−	4.94427i		−	0.177147i
$780$		0			0
$781$		−	80.7214i		−	2.88844i
$782$		0			0
$783$	11.3475	+	22.0000i	0.405527	+	0.786216i
$784$		0			0
$785$			14.4721i			0.516533i
$786$		0			0
$787$	−28.1803			−1.00452			−0.502260	−	0.864716i	$-0.667498\pi$
$787$	−28.1803			−1.00452			−0.502260	+	0.864716i	$0.667498\pi$
$788$		0			0
$789$	−16.9443	−	6.47214i	−0.603232	−	0.230414i
$790$		0			0
$791$	−9.88854			−0.351596
$792$		0			0
$793$	2.11146			0.0749800
$794$		0			0
$795$	−16.9443	−	6.47214i	−0.600951	−	0.229543i
$796$		0			0
$797$	50.0689			1.77353			0.886765	−	0.462220i	$-0.152947\pi$
$797$	50.0689			1.77353			0.886765	+	0.462220i	$0.152947\pi$
$798$		0			0
$799$		−	19.7771i		−	0.699663i
$800$		0			0
$801$	8.00000	−	8.94427i	0.282666	−	0.316030i
$802$		0			0
$803$			10.4721i			0.369554i
$804$		0			0
$805$		−	9.88854i		−	0.348525i
$806$		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
$811$	52.1803			1.83230			0.916150	−	0.400836i	$-0.131280\pi$
$811$	52.1803			1.83230			0.916150	+	0.400836i	$0.131280\pi$
$812$		0			0
$813$	0.180340	−	0.472136i	0.00632480	−	0.0165585i
$814$		0			0
$815$	−13.5279			−0.473860
$816$		0			0
$817$	5.52786			0.193395
$818$		0			0
$819$	12.3607	+	11.0557i	0.431917	+	0.386318i
$820$		0			0
$821$	−50.4296			−1.76000			−0.880002	−	0.474970i	$-0.842459\pi$
$821$	−50.4296			−1.76000			−0.880002	+	0.474970i	$0.842459\pi$
$822$		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
$827$		−	19.1246i		−	0.665028i	−0.943098	−	0.332514i	$-0.892103\pi$
$827$		−	19.1246i		−	0.665028i	0.943098	−	0.332514i	$-0.107897\pi$
$828$		0			0
$829$			4.47214i			0.155324i	0.996980	+	0.0776619i	$0.0247455\pi$
$829$			4.47214i			0.155324i	−0.996980	+	0.0776619i	$0.975255\pi$
$830$		0			0
$831$	−4.65248	+	12.1803i	−0.161393	+	0.422531i
$832$		0			0
$833$		−	13.5279i		−	0.468713i
$834$		0			0
$835$	43.7771			1.51497
$836$		0			0
$837$	24.1803	−	12.4721i	0.835795	−	0.431100i
$838$		0			0
$839$	−31.4164			−1.08461			−0.542307	−	0.840180i	$-0.682449\pi$
$839$	−31.4164			−1.08461			−0.542307	+	0.840180i	$0.682449\pi$
$840$		0			0
$841$	−6.30495			−0.217412
$842$		0			0
$843$	−15.4164	+	40.3607i	−0.530969	+	1.39010i
$844$		0			0
$845$	−22.6525			−0.779269
$846$		0			0
$847$			20.2918i			0.697234i
$848$		0			0
$849$	33.5967	+	12.8328i	1.15304	+	0.440421i
$850$		0			0
$851$			20.9443i			0.717960i
$852$		0			0
$853$			39.3050i			1.34578i	0.739744	+	0.672888i	$0.234946\pi$
$853$			39.3050i			1.34578i	−0.739744	+	0.672888i	$0.765054\pi$
$854$		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
$859$	−16.7639			−0.571978			−0.285989	−	0.958233i	$-0.592322\pi$
$859$	−16.7639			−0.571978			−0.285989	+	0.958233i	$0.592322\pi$
$860$		0			0
$861$	12.9443	+	4.94427i	0.441140	+	0.168500i
$862$		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
$867$	−17.6180	−	6.72949i	−0.598340	−	0.228545i
$868$		0			0
$869$	−1.52786			−0.0518292
$870$		0			0
$871$			43.4164i			1.47111i
$872$		0			0
$873$	1.05573	+	0.944272i	0.0357310	+	0.0319588i
$874$		0			0
$875$		−	1.88854i		−	0.0638444i
$876$		0			0
$877$			18.5836i			0.627523i	0.949502	+	0.313762i	$0.101589\pi$
$877$			18.5836i			0.627523i	−0.949502	+	0.313762i	$0.898411\pi$
$878$		0			0
$879$	33.5967	+	12.8328i	1.13319	+	0.432840i
$880$		0			0
$881$		−	24.0000i		−	0.808581i	−0.914631	−	0.404290i	$-0.867519\pi$
$881$		−	24.0000i		−	0.808581i	0.914631	−	0.404290i	$-0.132481\pi$
$882$		0			0
$883$	0.763932			0.0257084			0.0128542	−	0.999917i	$-0.495908\pi$
$883$	0.763932			0.0257084			0.0128542	+	0.999917i	$0.495908\pi$
$884$		0			0
$885$	−2.47214	+	6.47214i	−0.0830999	+	0.217558i
$886$		0			0
$887$	−20.3607			−0.683645			−0.341822	−	0.939765i	$-0.611044\pi$
$887$	−20.3607			−0.683645			−0.341822	+	0.939765i	$0.611044\pi$
$888$		0			0
$889$	16.3607			0.548720
$890$		0			0
$891$	−46.8328	+	5.23607i	−1.56896	+	0.175415i
$892$		0			0
$893$	6.11146			0.204512
$894$		0			0
$895$		−	20.0000i		−	0.668526i
$896$		0			0
$897$	6.83282	−	17.8885i	0.228141	−	0.597281i
$898$		0			0
$899$		−	24.9443i		−	0.831938i
$900$		0			0
$901$		−	8.00000i		−	0.266519i
$902$		0			0
$903$	−5.52786	+	14.4721i	−0.183956	+	0.481603i
$904$		0			0
$905$		−	66.2492i		−	2.20220i
$906$		0			0
$907$	39.2361			1.30281			0.651406	−	0.758729i	$-0.274179\pi$
$907$	39.2361			1.30281			0.651406	+	0.758729i	$0.274179\pi$
$908$		0			0
$909$	−3.81966	−	3.41641i	−0.126690	−	0.113315i
$910$		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
$915$	−0.944272	+	2.47214i	−0.0312167	+	0.0817263i
$916$		0			0
$917$	8.36068			0.276094
$918$		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
$923$		−	68.9443i		−	2.26933i
$924$		0			0
$925$			46.3607i			1.52433i
$926$		0			0
$927$	28.3607	−	31.7082i	0.931487	−	1.04143i
$928$		0			0
$929$			47.4164i			1.55568i	0.628461	+	0.777841i	$0.283685\pi$
$929$			47.4164i			1.55568i	−0.628461	+	0.777841i	$0.716315\pi$
$930$		0			0
$931$	4.18034			0.137005
$932$		0			0
$933$	−8.94427	−	3.41641i	−0.292822	−	0.111848i
$934$		0			0
$935$	41.8885			1.36990
$936$		0			0
$937$	14.3607			0.469143			0.234571	−	0.972099i	$-0.424631\pi$
$937$	14.3607			0.469143			0.234571	+	0.972099i	$0.424631\pi$
$938$		0			0
$939$	−0.763932	−	0.291796i	−0.0249300	−	0.00952240i
$940$		0			0
$941$	−1.70820			−0.0556859			−0.0278429	−	0.999612i	$-0.508864\pi$
$941$	−1.70820			−0.0556859			−0.0278429	+	0.999612i	$0.508864\pi$
$942$		0			0
$943$		−	16.0000i		−	0.521032i
$944$		0			0
$945$	−18.4721	+	9.52786i	−0.600899	+	0.309941i
$946$		0			0
$947$		−	24.0689i		−	0.782134i	−0.920362	−	0.391067i	$-0.872106\pi$
$947$		−	24.0689i		−	0.782134i	0.920362	−	0.391067i	$-0.127894\pi$
$948$		0			0
$949$			8.94427i			0.290343i
$950$		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
$955$	41.8885			1.35548
$956$		0			0
$957$	−15.4164	+	40.3607i	−0.498342	+	1.30468i
$958$		0			0
$959$	−1.88854			−0.0609843
$960$		0			0
$961$	3.58359			0.115600
$962$		0			0
$963$	28.3607	−	31.7082i	0.913910	−	1.02178i
$964$		0			0
$965$	−38.4721			−1.23846
$966$		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
$971$			17.0132i			0.545978i	0.962017	+	0.272989i	$0.0880123\pi$
$971$			17.0132i			0.545978i	−0.962017	+	0.272989i	$0.911988\pi$
$972$		0			0
$973$			12.0000i			0.384702i
$974$		0			0
$975$	15.1246	−	39.5967i	0.484375	−	1.26811i
$976$		0			0
$977$		−	60.3607i		−	1.93111i	−0.260200	−	0.965555i	$-0.583789\pi$
$977$		−	60.3607i		−	1.93111i	0.260200	−	0.965555i	$-0.416211\pi$
$978$		0			0
$979$	20.9443			0.669382
$980$		0			0
$981$	16.9443	−	18.9443i	0.540989	−	0.604844i
$982$		0			0
$983$	21.5279			0.686632			0.343316	−	0.939220i	$-0.388450\pi$
$983$	21.5279			0.686632			0.343316	+	0.939220i	$0.388450\pi$
$984$		0			0
$985$	−15.4164			−0.491208
$986$		0			0
$987$	−6.11146	+	16.0000i	−0.194530	+	0.509286i
$988$		0			0
$989$	17.8885			0.568823
$990$		0			0
$991$			50.1803i			1.59403i	0.603959	+	0.797016i	$0.293589\pi$
$991$			50.1803i			1.59403i	−0.603959	+	0.797016i	$0.706411\pi$
$992$		0			0
$993$	52.6525	+	20.1115i	1.67088	+	0.638218i
$994$		0			0
$995$		−	20.0000i		−	0.634043i
$996$		0			0
$997$			18.3607i			0.581489i	0.956801	+	0.290744i	$0.0939029\pi$
$997$			18.3607i			0.581489i	−0.956801	+	0.290744i	$0.906097\pi$
$998$		0			0
$999$	39.1246	−	20.1803i	1.23785	−	0.638478i

Display

a_p

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p

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a_n

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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

)

Twists

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

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

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	768.2.f.c.383.1

Level	$768$
Weight	$2$
Character	768.383
Analytic conductor	$6.133$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$