LMFDB - Embedded newform 387.5.b.c.343.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,5,Mod(343,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("387.343");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	387.b (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:	$40.0041757134$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 142x^{10} + 7173x^{8} + 157368x^{6} + 1510016x^{4} + 5098688x^{2} + 90352 $ x^12 + 142x^10 + 7173x^8 + 157368x^6 + 1510016x^4 + 5098688*x^2 + 90352
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			343.1
Root			$-7.49282i$ of defining polynomial
Character	$\chi$	$=$	387.343
Dual form			387.5.b.c.343.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.49282i q^{2} -40.1424 q^{4} +9.40180i q^{5} -53.5326i q^{7} +180.895i q^{8} +O(q^{10})$	\(q-7.49282i q^{2} -40.1424 q^{4} +9.40180i q^{5} -53.5326i q^{7} +180.895i q^{8} +70.4460 q^{10} -151.146 q^{11} -319.508 q^{13} -401.110 q^{14} +713.133 q^{16} -32.2585 q^{17} -304.131i q^{19} -377.411i q^{20} +1132.51i q^{22} +199.312 q^{23} +536.606 q^{25} +2394.02i q^{26} +2148.92i q^{28} +268.672i q^{29} +665.439 q^{31} -2449.06i q^{32} +241.707i q^{34} +503.303 q^{35} +1176.50i q^{37} -2278.80 q^{38} -1700.74 q^{40} -212.488 q^{41} +(1352.49 + 1260.78i) q^{43} +6067.35 q^{44} -1493.41i q^{46} +3100.78 q^{47} -464.736 q^{49} -4020.69i q^{50} +12825.8 q^{52} -2662.72 q^{53} -1421.04i q^{55} +9683.75 q^{56} +2013.11 q^{58} -2748.48 q^{59} -5880.73i q^{61} -4986.02i q^{62} -6940.28 q^{64} -3003.95i q^{65} -1093.36 q^{67} +1294.93 q^{68} -3771.16i q^{70} +5841.22i q^{71} +663.511i q^{73} +8815.31 q^{74} +12208.6i q^{76} +8091.21i q^{77} +6328.79 q^{79} +6704.74i q^{80} +1592.13i q^{82} -8356.34 q^{83} -303.288i q^{85} +(9446.83 - 10134.0i) q^{86} -27341.4i q^{88} +4257.21i q^{89} +17104.1i q^{91} -8000.85 q^{92} -23233.6i q^{94} +2859.38 q^{95} -3585.40 q^{97} +3482.18i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 92 q^{4}+O(q^{10}) $ 12 * q - 92 * q^4	$ 12 q - 92 q^{4} + 182 q^{10} + 180 q^{11} - 216 q^{13} - 732 q^{14} + 1076 q^{16} - 678 q^{17} - 1566 q^{23} - 174 q^{25} + 5710 q^{31} - 936 q^{35} - 1242 q^{38} - 2618 q^{40} - 4878 q^{41} - 1108 q^{43} + 15168 q^{44} + 5526 q^{47} - 8544 q^{49} + 24084 q^{52} - 1212 q^{53} + 10152 q^{56} - 4666 q^{58} - 14016 q^{59} - 15580 q^{64} - 1088 q^{67} - 15186 q^{68} + 7674 q^{74} + 24302 q^{79} + 7032 q^{83} + 14412 q^{86} - 48354 q^{92} - 606 q^{95} - 5842 q^{97}+O(q^{100}) $ 12 * q - 92 * q^4 + 182 * q^10 + 180 * q^11 - 216 * q^13 - 732 * q^14 + 1076 * q^16 - 678 * q^17 - 1566 * q^23 - 174 * q^25 + 5710 * q^31 - 936 * q^35 - 1242 * q^38 - 2618 * q^40 - 4878 * q^41 - 1108 * q^43 + 15168 * q^44 + 5526 * q^47 - 8544 * q^49 + 24084 * q^52 - 1212 * q^53 + 10152 * q^56 - 4666 * q^58 - 14016 * q^59 - 15580 * q^64 - 1088 * q^67 - 15186 * q^68 + 7674 * q^74 + 24302 * q^79 + 7032 * q^83 + 14412 * q^86 - 48354 * q^92 - 606 * q^95 - 5842 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/387\mathbb{Z}\right)^\times$.

$n$	$46$	$173$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	7.49282i		−	1.87321i	−0.350393	−	0.936603i	$-0.613952\pi$
$2$		−	7.49282i		−	1.87321i	0.350393	−	0.936603i	$-0.386048\pi$
$3$		0			0
$3$		0			0
$4$	−40.1424			−2.50890
$5$			9.40180i			0.376072i	0.982162	+	0.188036i	$0.0602122\pi$
$5$			9.40180i			0.376072i	−0.982162	+	0.188036i	$0.939788\pi$
$6$		0			0
$7$		−	53.5326i		−	1.09250i	−0.837622	−	0.546251i	$-0.816055\pi$
$7$		−	53.5326i		−	1.09250i	0.837622	−	0.546251i	$-0.183945\pi$
$8$			180.895i			2.82648i
$9$		0			0
$10$	70.4460			0.704460
$11$	−151.146			−1.24914			−0.624569	−	0.780970i	$-0.714725\pi$
$11$	−151.146			−1.24914			−0.624569	+	0.780970i	$0.714725\pi$
$12$		0			0
$13$	−319.508			−1.89058			−0.945289	−	0.326233i	$-0.894221\pi$
$13$	−319.508			−1.89058			−0.945289	+	0.326233i	$0.894221\pi$
$14$	−401.110			−2.04648
$15$		0			0
$16$	713.133			2.78568
$17$	−32.2585			−0.111621			−0.0558106	−	0.998441i	$-0.517774\pi$
$17$	−32.2585			−0.111621			−0.0558106	+	0.998441i	$0.517774\pi$
$18$		0			0
$19$		−	304.131i		−	0.842469i	−0.906952	−	0.421235i	$-0.861597\pi$
$19$		−	304.131i		−	0.842469i	0.906952	−	0.421235i	$-0.138403\pi$
$20$		−	377.411i		−	0.943527i
$21$		0			0
$22$			1132.51i			2.33989i
$23$	199.312			0.376771			0.188386	−	0.982095i	$-0.439675\pi$
$23$	199.312			0.376771			0.188386	+	0.982095i	$0.439675\pi$
$24$		0			0
$25$	536.606			0.858570
$26$			2394.02i			3.54144i
$27$		0			0
$28$			2148.92i			2.74098i
$29$			268.672i			0.319467i	0.987160	+	0.159734i	$0.0510635\pi$
$29$			268.672i			0.319467i	−0.987160	+	0.159734i	$0.948937\pi$
$30$		0			0
$31$	665.439			0.692444			0.346222	−	0.938153i	$-0.387464\pi$
$31$	665.439			0.692444			0.346222	+	0.938153i	$0.387464\pi$
$32$		−	2449.06i		−	2.39166i
$33$		0			0
$34$			241.707i			0.209089i
$35$	503.303			0.410859
$36$		0			0
$37$			1176.50i			0.859387i	0.902975	+	0.429693i	$0.141378\pi$
$37$			1176.50i			0.859387i	−0.902975	+	0.429693i	$0.858622\pi$
$38$	−2278.80			−1.57812
$39$		0			0
$40$	−1700.74			−1.06296
$41$	−212.488			−0.126406			−0.0632028	−	0.998001i	$-0.520132\pi$
$41$	−212.488			−0.126406			−0.0632028	+	0.998001i	$0.520132\pi$
$42$		0			0
$43$	1352.49	+	1260.78i	0.731470	+	0.681873i
$43$	1352.49	+	1260.78i	0.731470	+	0.681873i
$44$	6067.35			3.13396
$45$		0			0
$46$		−	1493.41i		−	0.705770i
$47$	3100.78			1.40371			0.701853	−	0.712322i	$-0.252357\pi$
$47$	3100.78			1.40371			0.701853	+	0.712322i	$0.252357\pi$
$48$		0			0
$49$	−464.736			−0.193559
$50$		−	4020.69i		−	1.60828i
$51$		0			0
$52$	12825.8			4.74327
$53$	−2662.72			−0.947924			−0.473962	−	0.880545i	$-0.657177\pi$
$53$	−2662.72			−0.947924			−0.473962	+	0.880545i	$0.657177\pi$
$54$		0			0
$55$		−	1421.04i		−	0.469766i
$56$	9683.75			3.08793
$57$		0			0
$58$	2013.11			0.598427
$59$	−2748.48			−0.789565			−0.394783	−	0.918775i	$-0.629180\pi$
$59$	−2748.48			−0.789565			−0.394783	+	0.918775i	$0.629180\pi$
$60$		0			0
$61$		−	5880.73i		−	1.58042i	−0.612839	−	0.790208i	$-0.709972\pi$
$61$		−	5880.73i		−	1.58042i	0.612839	−	0.790208i	$-0.290028\pi$
$62$		−	4986.02i		−	1.29709i
$63$		0			0
$64$	−6940.28			−1.69440
$65$		−	3003.95i		−	0.710994i
$66$		0			0
$67$	−1093.36			−0.243565			−0.121782	−	0.992557i	$-0.538861\pi$
$67$	−1093.36			−0.243565			−0.121782	+	0.992557i	$0.538861\pi$
$68$	1294.93			0.280046
$69$		0			0
$70$		−	3771.16i		−	0.769624i
$71$			5841.22i			1.15874i	0.815064	+	0.579371i	$0.196702\pi$
$71$			5841.22i			1.15874i	−0.815064	+	0.579371i	$0.803298\pi$
$72$		0			0
$73$			663.511i			0.124509i	0.998060	+	0.0622547i	$0.0198291\pi$
$73$			663.511i			0.124509i	−0.998060	+	0.0622547i	$0.980171\pi$
$74$	8815.31			1.60981
$75$		0			0
$76$			12208.6i			2.11367i
$77$			8091.21i			1.36468i
$78$		0			0
$79$	6328.79			1.01407			0.507034	−	0.861926i	$-0.330742\pi$
$79$	6328.79			1.01407			0.507034	+	0.861926i	$0.330742\pi$
$80$			6704.74i			1.04762i
$81$		0			0
$82$			1592.13i			0.236784i
$83$	−8356.34			−1.21300			−0.606499	−	0.795084i	$-0.707427\pi$
$83$	−8356.34			−1.21300			−0.606499	+	0.795084i	$0.707427\pi$
$84$		0			0
$85$		−	303.288i		−	0.0419776i
$86$	9446.83	−	10134.0i	1.27729	−	1.37019i
$87$		0			0
$88$		−	27341.4i		−	3.53066i
$89$			4257.21i			0.537459i	0.963216	+	0.268729i	$0.0866037\pi$
$89$			4257.21i			0.537459i	−0.963216	+	0.268729i	$0.913396\pi$
$90$		0			0
$91$			17104.1i			2.06546i
$92$	−8000.85			−0.945281
$93$		0			0
$94$		−	23233.6i		−	2.62943i
$95$	2859.38			0.316829
$96$		0			0
$97$	−3585.40			−0.381061			−0.190530	−	0.981681i	$-0.561021\pi$
$97$	−3585.40			−0.381061			−0.190530	+	0.981681i	$0.561021\pi$
$98$			3482.18i			0.362576i
$99$		0			0
$100$	−21540.6			−2.15406
$101$	−3138.84			−0.307699			−0.153850	−	0.988094i	$-0.549167\pi$
$101$	−3138.84			−0.307699			−0.153850	+	0.988094i	$0.549167\pi$
$102$		0			0
$103$	−7225.57			−0.681079			−0.340539	−	0.940230i	$-0.610610\pi$
$103$	−7225.57			−0.681079			−0.340539	+	0.940230i	$0.610610\pi$
$104$		−	57797.2i		−	5.34368i
$105$		0			0
$106$			19951.3i			1.77566i
$107$	11372.2			0.993291			0.496645	−	0.867954i	$-0.334565\pi$
$107$	11372.2			0.993291			0.496645	+	0.867954i	$0.334565\pi$
$108$		0			0
$109$	6830.22			0.574886			0.287443	−	0.957798i	$-0.407195\pi$
$109$	6830.22			0.574886			0.287443	+	0.957798i	$0.407195\pi$
$110$	−10647.6			−0.879968
$111$		0			0
$112$		−	38175.8i		−	3.04335i
$113$			14950.0i			1.17081i	0.810742	+	0.585403i	$0.199064\pi$
$113$			14950.0i			1.17081i	−0.810742	+	0.585403i	$0.800936\pi$
$114$		0			0
$115$			1873.89i			0.141693i
$116$		−	10785.1i		−	0.801511i
$117$		0			0
$118$			20593.8i			1.47902i
$119$			1726.88i			0.121946i
$120$		0			0
$121$	8204.00			0.560344
$122$	−44063.3			−2.96044
$123$		0			0
$124$	−26712.3			−1.73727
$125$			10921.2i			0.698956i
$126$		0			0
$127$	−11093.6			−0.687806			−0.343903	−	0.939005i	$-0.611749\pi$
$127$	−11093.6			−0.687806			−0.343903	+	0.939005i	$0.611749\pi$
$128$			12817.2i			0.782303i
$129$		0			0
$130$	−22508.1			−1.33184
$131$			7521.37i			0.438283i	0.975693	+	0.219141i	$0.0703256\pi$
$131$			7521.37i			0.438283i	−0.975693	+	0.219141i	$0.929674\pi$
$132$		0			0
$133$	−16280.9			−0.920399
$134$			8192.38i			0.456247i
$135$		0			0
$136$		−	5835.39i		−	0.315495i
$137$		−	15580.8i		−	0.830137i	−0.909790	−	0.415069i	$-0.863758\pi$
$137$		−	15580.8i		−	0.830137i	0.909790	−	0.415069i	$-0.136242\pi$
$138$		0			0
$139$	9394.31			0.486223			0.243111	−	0.969998i	$-0.421832\pi$
$139$	9394.31			0.486223			0.243111	+	0.969998i	$0.421832\pi$
$140$	−20203.8			−1.03080
$141$		0			0
$142$	43767.2			2.17056
$143$	48292.2			2.36159
$144$		0			0
$145$	−2526.00			−0.120143
$146$	4971.57			0.233232
$147$		0			0
$148$		−	47227.5i		−	2.15611i
$149$		−	2012.81i		−	0.0906630i	−0.998972	−	0.0453315i	$-0.985566\pi$
$149$		−	2012.81i		−	0.0906630i	0.998972	−	0.0453315i	$-0.0144344\pi$
$150$		0			0
$151$		−	7447.35i		−	0.326624i	−0.986574	−	0.163312i	$-0.947782\pi$
$151$		−	7447.35i		−	0.326624i	0.986574	−	0.163312i	$-0.0522177\pi$
$152$	55015.7			2.38122
$153$		0			0
$154$	60626.0			2.55633
$155$			6256.33i			0.260409i
$156$		0			0
$157$			16371.4i			0.664179i	0.943248	+	0.332090i	$0.107754\pi$
$157$			16371.4i			0.664179i	−0.943248	+	0.332090i	$0.892246\pi$
$158$		−	47420.5i		−	1.89956i
$159$		0			0
$160$	23025.6			0.899439
$161$		−	10669.7i		−	0.411623i
$162$		0			0
$163$			45112.4i			1.69793i	0.528445	+	0.848967i	$0.322775\pi$
$163$			45112.4i			1.69793i	−0.528445	+	0.848967i	$0.677225\pi$
$164$	8529.77			0.317139
$165$		0			0
$166$			62612.6i			2.27219i
$167$	24211.5			0.868139			0.434069	−	0.900879i	$-0.357077\pi$
$167$	24211.5			0.868139			0.434069	+	0.900879i	$0.357077\pi$
$168$		0			0
$169$	73524.2			2.57429
$170$	−2272.48			−0.0786327
$171$		0			0
$172$	−54292.1	−	50610.9i	−1.83518	−	1.71075i
$173$	−36697.5			−1.22615			−0.613076	−	0.790024i	$-0.710068\pi$
$173$	−36697.5			−1.22615			−0.613076	+	0.790024i	$0.710068\pi$
$174$		0			0
$175$		−	28725.9i		−	0.937989i
$176$	−107787.			−3.47969
$177$		0			0
$178$	31898.5			1.00677
$179$		−	50619.1i		−	1.57982i	−0.613220	−	0.789912i	$-0.710126\pi$
$179$		−	50619.1i		−	1.57982i	0.613220	−	0.789912i	$-0.289874\pi$
$180$		0			0
$181$	−31719.0			−0.968194			−0.484097	−	0.875014i	$-0.660852\pi$
$181$	−31719.0			−0.968194			−0.484097	+	0.875014i	$0.660852\pi$
$182$	128158.			3.86903
$183$		0			0
$184$			36054.4i			1.06494i
$185$	−11061.2			−0.323191
$186$		0			0
$187$	4875.73			0.139430
$188$	−124473.			−3.52175
$189$		0			0
$190$		−	21424.9i		−	0.593486i
$191$			49921.7i			1.36843i	0.729280	+	0.684215i	$0.239855\pi$
$191$			49921.7i			1.36843i	−0.729280	+	0.684215i	$0.760145\pi$
$192$		0			0
$193$	541.363			0.0145336			0.00726681	−	0.999974i	$-0.497687\pi$
$193$	541.363			0.0145336			0.00726681	+	0.999974i	$0.497687\pi$
$194$			26864.8i			0.713805i
$195$		0			0
$196$	18655.6			0.485621
$197$	31142.6			0.802458			0.401229	−	0.915978i	$-0.368583\pi$
$197$	31142.6			0.802458			0.401229	+	0.915978i	$0.368583\pi$
$198$		0			0
$199$			59618.5i			1.50548i	0.658319	+	0.752739i	$0.271268\pi$
$199$			59618.5i			1.50548i	−0.658319	+	0.752739i	$0.728732\pi$
$200$			97069.1i			2.42673i
$201$		0			0
$202$			23518.8i			0.576384i
$203$	14382.7			0.349018
$204$		0			0
$205$		−	1997.77i		−	0.0475377i
$206$			54139.9i			1.27580i
$207$		0			0
$208$	−227852.			−5.26654
$209$			45968.1i			1.05236i
$210$		0			0
$211$			45278.4i			1.01701i	0.861059	+	0.508506i	$0.169802\pi$
$211$			45278.4i			1.01701i	−0.861059	+	0.508506i	$0.830198\pi$
$212$	106888.			2.37825
$213$		0			0
$214$		−	85209.8i		−	1.86064i
$215$	−11853.6	+	12715.8i	−0.256434	+	0.275086i
$216$		0			0
$217$		−	35622.7i		−	0.756496i
$218$		−	51177.7i		−	1.07688i
$219$		0			0
$220$			57044.0i			1.17860i
$221$	10306.8			0.211029
$222$		0			0
$223$		−	26029.7i		−	0.523431i	−0.965145	−	0.261716i	$-0.915712\pi$
$223$		−	26029.7i		−	0.523431i	0.965145	−	0.261716i	$-0.0842883\pi$
$224$	−131105.			−2.61290
$225$		0			0
$226$	112018.			2.19316
$227$			57788.9i			1.12148i	0.827991	+	0.560741i	$0.189484\pi$
$227$			57788.9i			1.12148i	−0.827991	+	0.560741i	$0.810516\pi$
$228$		0			0
$229$	−15622.4			−0.297905			−0.148953	−	0.988844i	$-0.547590\pi$
$229$	−15622.4			−0.297905			−0.148953	+	0.988844i	$0.547590\pi$
$230$	14040.7			0.265420
$231$		0			0
$232$	−48601.3			−0.902967
$233$		−	60252.0i		−	1.10984i	−0.831904	−	0.554919i	$-0.812749\pi$
$233$		−	60252.0i		−	1.10984i	0.831904	−	0.554919i	$-0.187251\pi$
$234$		0			0
$235$			29153.0i			0.527894i
$236$	110330.			1.98094
$237$		0			0
$238$	12939.2			0.228430
$239$	−91262.4			−1.59770			−0.798851	−	0.601528i	$-0.794559\pi$
$239$	−91262.4			−1.59770			−0.798851	+	0.601528i	$0.794559\pi$
$240$		0			0
$241$		−	11694.8i		−	0.201354i	−0.994919	−	0.100677i	$-0.967899\pi$
$241$		−	11694.8i		−	0.201354i	0.994919	−	0.100677i	$-0.0321008\pi$
$242$		−	61471.1i		−	1.04964i
$243$		0			0
$244$			236066.i			3.96510i
$245$		−	4369.36i		−	0.0727923i
$246$		0			0
$247$			97172.4i			1.59275i
$248$			120374.i			1.95718i
$249$		0			0
$250$	81830.6			1.30929
$251$	−95330.3			−1.51316			−0.756578	−	0.653904i	$-0.773130\pi$
$251$	−95330.3			−1.51316			−0.756578	+	0.653904i	$0.773130\pi$
$252$		0			0
$253$	−30125.1			−0.470639
$254$			83122.6i			1.28840i
$255$		0			0
$256$	−15007.1			−0.228991
$257$			98138.1i			1.48584i	0.669382	+	0.742919i	$0.266559\pi$
$257$			98138.1i			1.48584i	−0.669382	+	0.742919i	$0.733441\pi$
$258$		0			0
$259$	62981.1			0.938881
$260$			120586.i			1.78381i
$261$		0			0
$262$	56356.3			0.820994
$263$		−	7996.23i		−	0.115604i	−0.998328	−	0.0578022i	$-0.981591\pi$
$263$		−	7996.23i		−	0.115604i	0.998328	−	0.0578022i	$-0.0184093\pi$
$264$		0			0
$265$		−	25034.4i		−	0.356488i
$266$			121990.i			1.72410i
$267$		0			0
$268$	43890.2			0.611080
$269$	42975.4			0.593903			0.296951	−	0.954893i	$-0.404030\pi$
$269$	42975.4			0.593903			0.296951	+	0.954893i	$0.404030\pi$
$270$		0			0
$271$	22222.2			0.302585			0.151293	−	0.988489i	$-0.451656\pi$
$271$	22222.2			0.302585			0.151293	+	0.988489i	$0.451656\pi$
$272$	−23004.6			−0.310940
$273$		0			0
$274$	−116744.			−1.55502
$275$	−81105.7			−1.07247
$276$		0			0
$277$			67337.1i			0.877596i	0.898586	+	0.438798i	$0.144596\pi$
$277$			67337.1i			0.877596i	−0.898586	+	0.438798i	$0.855404\pi$
$278$		−	70389.9i		−	0.910795i
$279$		0			0
$280$			91044.7i			1.16129i
$281$	14673.1			0.185828			0.0929138	−	0.995674i	$-0.470382\pi$
$281$	14673.1			0.185828			0.0929138	+	0.995674i	$0.470382\pi$
$282$		0			0
$283$	25223.6			0.314944			0.157472	−	0.987523i	$-0.449666\pi$
$283$	25223.6			0.314944			0.157472	+	0.987523i	$0.449666\pi$
$284$		−	234481.i		−	2.90717i
$285$		0			0
$286$		−	361845.i		−	4.42375i
$287$			11375.0i			0.138098i
$288$		0			0
$289$	−82480.4			−0.987541
$290$			18926.9i			0.225052i
$291$		0			0
$292$		−	26634.9i		−	0.312382i
$293$	−112480.			−1.31020			−0.655102	−	0.755540i	$-0.727374\pi$
$293$	−112480.			−1.31020			−0.655102	+	0.755540i	$0.727374\pi$
$294$		0			0
$295$		−	25840.6i		−	0.296933i
$296$	−212823.			−2.42904
$297$		0			0
$298$	−15081.6			−0.169830
$299$	−63681.7			−0.712315
$300$		0			0
$301$	67493.0	−	72402.2i	0.744948	−	0.799132i
$302$	−55801.7			−0.611834
$303$		0			0
$304$		−	216886.i		−	2.34685i
$305$	55289.5			0.594351
$306$		0			0
$307$	136783.			1.45130			0.725648	−	0.688067i	$-0.241540\pi$
$307$	136783.			1.45130			0.725648	+	0.688067i	$0.241540\pi$
$308$		−	324801.i		−	3.42386i
$309$		0			0
$310$	46877.5			0.487800
$311$	46529.6			0.481070			0.240535	−	0.970640i	$-0.422677\pi$
$311$	46529.6			0.481070			0.240535	+	0.970640i	$0.422677\pi$
$312$		0			0
$313$		−	68823.9i		−	0.702507i	−0.936280	−	0.351253i	$-0.885756\pi$
$313$		−	68823.9i		−	0.702507i	0.936280	−	0.351253i	$-0.114244\pi$
$314$	122668.			1.24414
$315$		0			0
$316$	−254053.			−2.54419
$317$	48726.3			0.484892			0.242446	−	0.970165i	$-0.422050\pi$
$317$	48726.3			0.484892			0.242446	+	0.970165i	$0.422050\pi$
$318$		0			0
$319$		−	40608.6i		−	0.399058i
$320$		−	65251.1i		−	0.637218i
$321$		0			0
$322$	−79946.0			−0.771054
$323$			9810.83i			0.0940374i
$324$		0			0
$325$	−171450.			−1.62319
$326$	338019.			3.18058
$327$		0			0
$328$		−	38437.9i		−	0.357283i
$329$		−	165993.i		−	1.53355i
$330$		0			0
$331$			61590.4i			0.562156i	0.959685	+	0.281078i	$0.0906919\pi$
$331$			61590.4i			0.562156i	−0.959685	+	0.281078i	$0.909308\pi$
$332$	335444.			3.04329
$333$		0			0
$334$		−	181413.i		−	1.62620i
$335$		−	10279.6i		−	0.0915980i
$336$		0			0
$337$	53084.4			0.467420			0.233710	−	0.972306i	$-0.424913\pi$
$337$	53084.4			0.467420			0.233710	+	0.972306i	$0.424913\pi$
$338$		−	550904.i		−	4.82217i
$339$		0			0
$340$			12174.7i			0.105318i
$341$	−100578.			−0.864958
$342$		0			0
$343$		−	103653.i		−	0.881038i
$344$	−228069.	+	244658.i	−1.92730	+	2.06748i
$345$		0			0
$346$			274968.i			2.29683i
$347$			80407.5i			0.667787i	0.942611	+	0.333893i	$0.108363\pi$
$347$			80407.5i			0.667787i	−0.942611	+	0.333893i	$0.891637\pi$
$348$		0			0
$349$			173808.i			1.42698i	0.700663	+	0.713492i	$0.252888\pi$
$349$			173808.i			1.42698i	−0.700663	+	0.713492i	$0.747112\pi$
$350$	−215238.			−1.75705
$351$		0			0
$352$			370165.i			2.98752i
$353$	80068.2			0.642556			0.321278	−	0.946985i	$-0.395888\pi$
$353$	80068.2			0.642556			0.321278	+	0.946985i	$0.395888\pi$
$354$		0			0
$355$	−54918.0			−0.435771
$356$		−	170895.i		−	1.34843i
$357$		0			0
$358$	−379280.			−2.95934
$359$	−83187.2			−0.645457			−0.322729	−	0.946492i	$-0.604600\pi$
$359$	−83187.2			−0.645457			−0.322729	+	0.946492i	$0.604600\pi$
$360$		0			0
$361$	37825.1			0.290245
$362$			237665.i			1.81363i
$363$		0			0
$364$		−	686598.i		−	5.18203i
$365$	−6238.20			−0.0468245
$366$		0			0
$367$	248797.			1.84719			0.923597	−	0.383365i	$-0.125235\pi$
$367$	248797.			1.84719			0.923597	+	0.383365i	$0.125235\pi$
$368$	142136.			1.04956
$369$		0			0
$370$			82879.8i			0.605404i
$371$			142542.i			1.03561i
$372$		0			0
$373$		−	209386.i		−	1.50498i	−0.658603	−	0.752490i	$-0.728852\pi$
$373$		−	209386.i		−	1.50498i	0.658603	−	0.752490i	$-0.271148\pi$
$374$		−	36533.0i		−	0.261181i
$375$		0			0
$376$			560915.i			3.96754i
$377$		−	85842.7i		−	0.603978i
$378$		0			0
$379$	−92058.9			−0.640895			−0.320448	−	0.947266i	$-0.603833\pi$
$379$	−92058.9			−0.640895			−0.320448	+	0.947266i	$0.603833\pi$
$380$	−114783.			−0.794893
$381$		0			0
$382$	374055.			2.56335
$383$		−	51836.8i		−	0.353379i	−0.984267	−	0.176690i	$-0.943461\pi$
$383$		−	51836.8i		−	0.353379i	0.984267	−	0.176690i	$-0.0565388\pi$
$384$		0			0
$385$	−76072.0			−0.513220
$386$		−	4056.34i		−	0.0272245i
$387$		0			0
$388$	143926.			0.956043
$389$			170164.i			1.12452i	0.826959	+	0.562262i	$0.190069\pi$
$389$			170164.i			1.12452i	−0.826959	+	0.562262i	$0.809931\pi$
$390$		0			0
$391$	−6429.50			−0.0420556
$392$		−	84068.2i		−	0.547091i
$393$		0			0
$394$		−	233346.i		−	1.50317i
$395$			59502.1i			0.381362i
$396$		0			0
$397$	−201719.			−1.27987			−0.639936	−	0.768428i	$-0.721039\pi$
$397$	−201719.			−1.27987			−0.639936	+	0.768428i	$0.721039\pi$
$398$	446711.			2.82007
$399$		0			0
$400$	382671.			2.39170
$401$	−38112.2			−0.237015			−0.118507	−	0.992953i	$-0.537811\pi$
$401$	−38112.2			−0.237015			−0.118507	+	0.992953i	$0.537811\pi$
$402$		0			0
$403$	−212613.			−1.30912
$404$	126001.			0.771987
$405$		0			0
$406$		−	107767.i		−	0.653783i
$407$		−	177823.i		−	1.07349i
$408$		0			0
$409$			301746.i			1.80383i	0.431918	+	0.901913i	$0.357837\pi$
$409$			301746.i			1.80383i	−0.431918	+	0.901913i	$0.642163\pi$
$410$	−14968.9			−0.0890478
$411$		0			0
$412$	290051.			1.70876
$413$			147133.i			0.862601i
$414$		0			0
$415$		−	78564.7i		−	0.456175i
$416$			782495.i			4.52163i
$417$		0			0
$418$	344431.			1.97129
$419$			15464.5i			0.0880859i	0.999030	+	0.0440430i	$0.0140238\pi$
$419$			15464.5i			0.0880859i	−0.999030	+	0.0440430i	$0.985976\pi$
$420$		0			0
$421$			218060.i			1.23030i	0.788409	+	0.615151i	$0.210905\pi$
$421$			218060.i			1.23030i	−0.788409	+	0.615151i	$0.789095\pi$
$422$	339263.			1.90507
$423$		0			0
$424$		−	481672.i		−	2.67929i
$425$	−17310.1			−0.0958345
$426$		0			0
$427$	−314811.			−1.72661
$428$	−456507.			−2.49207
$429$		0			0
$430$	95277.5	+	88817.3i	0.515292	+	0.480353i
$431$	173738.			0.935277			0.467639	−	0.883920i	$-0.345105\pi$
$431$	173738.			0.935277			0.467639	+	0.883920i	$0.345105\pi$
$432$		0			0
$433$		−	252649.i		−	1.34754i	−0.738942	−	0.673769i	$-0.764674\pi$
$433$		−	252649.i		−	1.34754i	0.738942	−	0.673769i	$-0.235326\pi$
$434$	−266914.			−1.41707
$435$		0			0
$436$	−274181.			−1.44233
$437$		−	60617.0i		−	0.317418i
$438$		0			0
$439$	−197576.			−1.02519			−0.512597	−	0.858630i	$-0.671316\pi$
$439$	−197576.			−1.02519			−0.512597	+	0.858630i	$0.671316\pi$
$440$	257059.			1.32778
$441$		0			0
$442$		−	77227.3i		−	0.395300i
$443$	−116118.			−0.591685			−0.295842	−	0.955237i	$-0.595600\pi$
$443$	−116118.			−0.591685			−0.295842	+	0.955237i	$0.595600\pi$
$444$		0			0
$445$	−40025.4			−0.202123
$446$	−195036.			−0.980495
$447$		0			0
$448$			371531.i			1.85114i
$449$			221303.i			1.09773i	0.835911	+	0.548864i	$0.184940\pi$
$449$			221303.i			1.09773i	−0.835911	+	0.548864i	$0.815060\pi$
$450$		0			0
$451$	32116.6			0.157898
$452$		−	600130.i		−	2.93744i
$453$		0			0
$454$	433002.			2.10077
$455$	−160809.			−0.776762
$456$		0			0
$457$			79170.3i			0.379079i	0.981873	+	0.189540i	$0.0606996\pi$
$457$			79170.3i			0.379079i	−0.981873	+	0.189540i	$0.939300\pi$
$458$			117056.i			0.558038i
$459$		0			0
$460$		−	75222.5i		−	0.355494i
$461$	−272584.			−1.28262			−0.641311	−	0.767281i	$-0.721609\pi$
$461$	−272584.			−1.28262			−0.641311	+	0.767281i	$0.721609\pi$
$462$		0			0
$463$			324723.i			1.51478i	0.652960	+	0.757392i	$0.273527\pi$
$463$			324723.i			1.51478i	−0.652960	+	0.757392i	$0.726473\pi$
$464$			191599.i			0.889932i
$465$		0			0
$466$	−451458.			−2.07896
$467$		−	48808.8i		−	0.223802i	−0.993719	−	0.111901i	$-0.964306\pi$
$467$		−	48808.8i		−	0.223802i	0.993719	−	0.111901i	$-0.0356940\pi$
$468$		0			0
$469$			58530.5i			0.266095i
$470$	218438.			0.988855
$471$		0			0
$472$		−	497184.i		−	2.23169i
$473$	−204423.	−	190562.i	−0.913707	−	0.851754i
$474$		0			0
$475$		−	163199.i		−	0.723319i
$476$		−	69321.1i		−	0.305951i
$477$		0			0
$478$			683813.i			2.99283i
$479$	−383151.			−1.66993			−0.834966	−	0.550301i	$-0.814513\pi$
$479$	−383151.			−1.66993			−0.834966	+	0.550301i	$0.814513\pi$
$480$		0			0
$481$		−	375901.i		−	1.62474i
$482$	−87627.3			−0.377177
$483$		0			0
$484$	−329328.			−1.40585
$485$		−	33709.2i		−	0.143306i
$486$		0			0
$487$	296539.			1.25033			0.625164	−	0.780494i	$-0.285032\pi$
$487$	296539.			1.25033			0.625164	+	0.780494i	$0.285032\pi$
$488$	1.06379e6			4.46701
$489$		0			0
$490$	−32738.8			−0.136355
$491$		−	234793.i		−	0.973916i	−0.873425	−	0.486958i	$-0.838106\pi$
$491$		−	234793.i		−	0.973916i	0.873425	−	0.486958i	$-0.161894\pi$
$492$		0			0
$493$		−	8666.95i		−	0.0356593i
$494$	728095.			2.98356
$495$		0			0
$496$	474546.			1.92893
$497$	312696.			1.26593
$498$		0			0
$499$			223696.i			0.898372i	0.893438	+	0.449186i	$0.148286\pi$
$499$			223696.i			0.898372i	−0.893438	+	0.449186i	$0.851714\pi$
$500$		−	438403.i		−	1.75361i
$501$		0			0
$502$			714293.i			2.83445i
$503$		−	171252.i		−	0.676861i	−0.940991	−	0.338430i	$-0.890104\pi$
$503$		−	171252.i		−	0.676861i	0.940991	−	0.338430i	$-0.109896\pi$
$504$		0			0
$505$		−	29510.8i		−	0.115717i
$506$			225722.i			0.881603i
$507$		0			0
$508$	445325.			1.72564
$509$	489934.			1.89105			0.945524	−	0.325554i	$-0.105551\pi$
$509$	489934.			1.89105			0.945524	+	0.325554i	$0.105551\pi$
$510$		0			0
$511$	35519.4			0.136027
$512$			317522.i			1.21125i
$513$		0			0
$514$	735331.			2.78328
$515$		−	67933.4i		−	0.256135i
$516$		0			0
$517$	−468670.			−1.75342
$518$		−	471906.i		−	1.75872i
$519$		0			0
$520$	543398.			2.00961
$521$			286304.i			1.05476i	0.849630	+	0.527378i	$0.176825\pi$
$521$			286304.i			1.05476i	−0.849630	+	0.527378i	$0.823175\pi$
$522$		0			0
$523$		−	132186.i		−	0.483263i	−0.970368	−	0.241631i	$-0.922318\pi$
$523$		−	132186.i		−	0.483263i	0.970368	−	0.241631i	$-0.0776825\pi$
$524$		−	301926.i		−	1.09961i
$525$		0			0
$526$	−59914.4			−0.216551
$527$	−21466.1			−0.0772914
$528$		0			0
$529$	−240116.			−0.858044
$530$	−187578.			−0.667775
$531$		0			0
$532$	653556.			2.30919
$533$	67891.6			0.238980
$534$		0			0
$535$			106919.i			0.373549i
$536$		−	197784.i		−	0.688431i
$537$		0			0
$538$		−	322007.i		−	1.11250i
$539$	70242.8			0.241782
$540$		0			0
$541$	114352.			0.390704			0.195352	−	0.980733i	$-0.437415\pi$
$541$	114352.			0.390704			0.195352	+	0.980733i	$0.437415\pi$
$542$		−	166507.i		−	0.566805i
$543$		0			0
$544$			79003.2i			0.266960i
$545$			64216.4i			0.216199i
$546$		0			0
$547$	48485.5			0.162046			0.0810228	−	0.996712i	$-0.474181\pi$
$547$	48485.5			0.162046			0.0810228	+	0.996712i	$0.474181\pi$
$548$			625452.i			2.08273i
$549$		0			0
$550$			607710.i			2.00896i
$551$	81711.5			0.269141
$552$		0			0
$553$		−	338797.i		−	1.10787i
$554$	504545.			1.64392
$555$		0			0
$556$	−377110.			−1.21988
$557$	96856.1			0.312188			0.156094	−	0.987742i	$-0.450110\pi$
$557$	96856.1			0.312188			0.156094	+	0.987742i	$0.450110\pi$
$558$		0			0
$559$	−432131.	−	402830.i	−1.38290	−	1.28914i
$560$	358922.			1.14452
$561$		0			0
$562$		−	109943.i		−	0.348093i
$563$	26855.0			0.0847244			0.0423622	−	0.999102i	$-0.486512\pi$
$563$	26855.0			0.0847244			0.0423622	+	0.999102i	$0.486512\pi$
$564$		0			0
$565$	−140557.			−0.440308
$566$		−	188996.i		−	0.589955i
$567$		0			0
$568$	−1.05665e6			−3.27516
$569$	−228140.			−0.704654			−0.352327	−	0.935877i	$-0.614610\pi$
$569$	−228140.			−0.704654			−0.352327	+	0.935877i	$0.614610\pi$
$570$		0			0
$571$		−	319649.i		−	0.980396i	−0.871611	−	0.490198i	$-0.836924\pi$
$571$		−	319649.i		−	0.980396i	0.871611	−	0.490198i	$-0.163076\pi$
$572$	−1.93856e6			−5.92500
$573$		0			0
$574$	85231.1			0.258687
$575$	106952.			0.323484
$576$		0			0
$577$		−	527850.i		−	1.58547i	−0.609565	−	0.792736i	$-0.708656\pi$
$577$		−	527850.i		−	1.58547i	0.609565	−	0.792736i	$-0.291344\pi$
$578$			618011.i			1.84987i
$579$		0			0
$580$	101400.			0.301426
$581$			447336.i			1.32520i
$582$		0			0
$583$	402458.			1.18409
$584$	−120025.			−0.351923
$585$		0			0
$586$			842791.i			2.45428i
$587$		−	571037.i		−	1.65725i	−0.559804	−	0.828625i	$-0.689124\pi$
$587$		−	571037.i		−	1.65725i	0.559804	−	0.828625i	$-0.310876\pi$
$588$		0			0
$589$		−	202381.i		−	0.583363i
$590$	−193619.			−0.556217
$591$		0			0
$592$			839001.i			2.39397i
$593$			564829.i			1.60623i	0.595824	+	0.803115i	$0.296825\pi$
$593$			564829.i			1.60623i	−0.595824	+	0.803115i	$0.703175\pi$
$594$		0			0
$595$	−16235.8			−0.0458606
$596$			80799.0i			0.227464i
$597$		0			0
$598$			477156.i			1.33431i
$599$	−323806.			−0.902467			−0.451233	−	0.892406i	$-0.649016\pi$
$599$	−323806.			−0.902467			−0.451233	+	0.892406i	$0.649016\pi$
$600$		0			0
$601$		−	239522.i		−	0.663126i	−0.943433	−	0.331563i	$-0.892424\pi$
$601$		−	239522.i		−	0.663126i	0.943433	−	0.331563i	$-0.107576\pi$
$602$	−542497.	−	505713.i	−1.49694	−	1.39544i
$603$		0			0
$604$			298954.i			0.819466i
$605$			77132.4i			0.210730i
$606$		0			0
$607$			344432.i			0.934816i	0.884042	+	0.467408i	$0.154812\pi$
$607$			344432.i			0.934816i	−0.884042	+	0.467408i	$0.845188\pi$
$608$	−744838.			−2.01490
$609$		0			0
$610$		−	414274.i		−	1.11334i
$611$	−990725.			−2.65382
$612$		0			0
$613$	446348.			1.18783			0.593913	−	0.804529i	$-0.297582\pi$
$613$	446348.			1.18783			0.593913	+	0.804529i	$0.297582\pi$
$614$		−	1.02489e6i		−	2.71857i
$615$		0			0
$616$	−1.46366e6			−3.85725
$617$	663045.			1.74170			0.870848	−	0.491552i	$-0.163570\pi$
$617$	663045.			1.74170			0.870848	+	0.491552i	$0.163570\pi$
$618$		0			0
$619$	−298590.			−0.779281			−0.389641	−	0.920967i	$-0.627401\pi$
$619$	−298590.			−0.779281			−0.389641	+	0.920967i	$0.627401\pi$
$620$		−	251144.i		−	0.653340i
$621$		0			0
$622$		−	348638.i		−	0.901143i
$623$	227899.			0.587174
$624$		0			0
$625$	232700.			0.595712
$626$	−515685.			−1.31594
$627$		0			0
$628$		−	657185.i		−	1.66636i
$629$		−	37952.1i		−	0.0959257i
$630$		0			0
$631$		−	204687.i		−	0.514082i	−0.966400	−	0.257041i	$-0.917252\pi$
$631$		−	204687.i		−	0.514082i	0.966400	−	0.257041i	$-0.0827475\pi$
$632$			1.14484e6i			2.86624i
$633$		0			0
$634$		−	365098.i		−	0.908303i
$635$		−	104300.i		−	0.258665i
$636$		0			0
$637$	148487.			0.365939
$638$	−304273.			−0.747518
$639$		0			0
$640$	−120505.			−0.294202
$641$		−	293809.i		−	0.715070i	−0.933900	−	0.357535i	$-0.883617\pi$
$641$		−	293809.i		−	0.715070i	0.933900	−	0.357535i	$-0.116383\pi$
$642$		0			0
$643$	−376164.			−0.909820			−0.454910	−	0.890537i	$-0.650329\pi$
$643$	−376164.			−0.909820			−0.454910	+	0.890537i	$0.650329\pi$
$644$			428306.i			1.03272i
$645$		0			0
$646$	73510.8			0.176151
$647$			72724.1i			0.173728i	0.996220	+	0.0868640i	$0.0276846\pi$
$647$			72724.1i			0.173728i	−0.996220	+	0.0868640i	$0.972315\pi$
$648$		0			0
$649$	415420.			0.986275
$650$			1.28464e6i			3.04058i
$651$		0			0
$652$		−	1.81092e6i		−	4.25995i
$653$		−	338825.i		−	0.794602i	−0.917688	−	0.397301i	$-0.869947\pi$
$653$		−	338825.i		−	0.794602i	0.917688	−	0.397301i	$-0.130053\pi$
$654$		0			0
$655$	−70714.5			−0.164826
$656$	−151532.			−0.352125
$657$		0			0
$658$	−1.24376e6			−2.87265
$659$	−548017.			−1.26189			−0.630947	−	0.775826i	$-0.717334\pi$
$659$	−548017.			−1.26189			−0.630947	+	0.775826i	$0.717334\pi$
$660$		0			0
$661$	487460.			1.11567			0.557836	−	0.829951i	$-0.311632\pi$
$661$	487460.			1.11567			0.557836	+	0.829951i	$0.311632\pi$
$662$	461486.			1.05303
$663$		0			0
$664$		−	1.51162e6i		−	3.42851i
$665$		−	153070.i		−	0.346136i
$666$		0			0
$667$			53549.5i			0.120366i
$668$	−971908.			−2.17807
$669$		0			0
$670$	−77023.1			−0.171582
$671$			888846.i			1.97416i
$672$		0			0
$673$			738888.i			1.63136i	0.578507	+	0.815678i	$0.303636\pi$
$673$			738888.i			1.63136i	−0.578507	+	0.815678i	$0.696364\pi$
$674$		−	397752.i		−	0.875573i
$675$		0			0
$676$	−2.95144e6			−6.45863
$677$		−	227328.i		−	0.495994i	−0.968761	−	0.247997i	$-0.920228\pi$
$677$		−	227328.i		−	0.495994i	0.968761	−	0.247997i	$-0.0797723\pi$
$678$		0			0
$679$			191936.i			0.416309i
$680$	54863.2			0.118649
$681$		0			0
$682$			753615.i			1.62024i
$683$	−161434.			−0.346061			−0.173031	−	0.984916i	$-0.555356\pi$
$683$	−161434.			−0.346061			−0.173031	+	0.984916i	$0.555356\pi$
$684$		0			0
$685$	146488.			0.312191
$686$	−776655.			−1.65036
$687$		0			0
$688$	964504.	+	899107.i	2.03764	+	1.89948i
$689$	850760.			1.79213
$690$		0			0
$691$			327344.i			0.685565i	0.939415	+	0.342783i	$0.111369\pi$
$691$			327344.i			0.685565i	−0.939415	+	0.342783i	$0.888631\pi$
$692$	1.47312e6			3.07629
$693$		0			0
$694$	602479.			1.25090
$695$			88323.4i			0.182855i
$696$		0			0
$697$	6854.54			0.0141095
$698$	1.30231e6			2.67303
$699$		0			0
$700$			1.15313e6i			2.35332i
$701$	−436072.			−0.887406			−0.443703	−	0.896174i	$-0.646335\pi$
$701$	−436072.			−0.887406			−0.443703	+	0.896174i	$0.646335\pi$
$702$		0			0
$703$	357811.			0.724007
$704$	1.04899e6			2.11654
$705$		0			0
$706$		−	599937.i		−	1.20364i
$707$			168030.i			0.336162i
$708$		0			0
$709$	−453704.			−0.902569			−0.451285	−	0.892380i	$-0.649034\pi$
$709$	−453704.			−0.902569			−0.451285	+	0.892380i	$0.649034\pi$
$710$			411491.i			0.816289i
$711$		0			0
$712$	−770106.			−1.51911
$713$	132630.			0.260893
$714$		0			0
$715$			454034.i			0.888129i
$716$			2.03197e6i			3.96362i
$717$		0			0
$718$			623307.i			1.20907i
$719$	677749.			1.31103			0.655513	−	0.755184i	$-0.272453\pi$
$719$	677749.			1.31103			0.655513	+	0.755184i	$0.272453\pi$
$720$		0			0
$721$			386803.i			0.744080i
$722$		−	283417.i		−	0.543689i
$723$		0			0
$724$	1.27328e6			2.42910
$725$			144171.i			0.274285i
$726$		0			0
$727$			421634.i			0.797751i	0.917005	+	0.398875i	$0.130599\pi$
$727$			421634.i			0.797751i	−0.917005	+	0.398875i	$0.869401\pi$
$728$	−3.09403e6			−5.83798
$729$		0			0
$730$			46741.7i			0.0877120i
$731$	−43629.2	−	40671.0i	−0.0816475	−	0.0761115i
$732$		0			0
$733$		−	545085.i		−	1.01451i	−0.861796	−	0.507255i	$-0.830660\pi$
$733$		−	545085.i		−	1.01451i	0.861796	−	0.507255i	$-0.169340\pi$
$734$		−	1.86419e6i		−	3.46017i
$735$		0			0
$736$		−	488128.i		−	0.901110i
$737$	165257.			0.304246
$738$		0			0
$739$		−	308914.i		−	0.565651i	−0.959171	−	0.282826i	$-0.908728\pi$
$739$		−	308914.i		−	0.565651i	0.959171	−	0.282826i	$-0.0912718\pi$
$740$	444024.			0.810855
$741$		0			0
$742$	1.06804e6			1.93991
$743$			1.03791e6i			1.88011i	0.341026	+	0.940054i	$0.389225\pi$
$743$			1.03791e6i			1.88011i	−0.341026	+	0.940054i	$0.610775\pi$
$744$		0			0
$745$	18924.0			0.0340958
$746$	−1.56890e6			−2.81914
$747$		0			0
$748$	−195724.			−0.349816
$749$		−	608782.i		−	1.08517i
$750$		0			0
$751$		−	162809.i		−	0.288669i	−0.989529	−	0.144334i	$-0.953896\pi$
$751$		−	162809.i		−	0.288669i	0.989529	−	0.144334i	$-0.0461041\pi$
$752$	2.21127e6			3.91027
$753$		0			0
$754$	−643204.			−1.13137
$755$	70018.5			0.122834
$756$		0			0
$757$			24344.0i			0.0424816i	0.999774	+	0.0212408i	$0.00676166\pi$
$757$			24344.0i			0.0424816i	−0.999774	+	0.0212408i	$0.993238\pi$
$758$			689781.i			1.20053i
$759$		0			0
$760$			517247.i			0.895511i
$761$		−	935074.i		−	1.61464i	−0.590111	−	0.807322i	$-0.700916\pi$
$761$		−	935074.i		−	1.61464i	0.590111	−	0.807322i	$-0.299084\pi$
$762$		0			0
$763$		−	365639.i		−	0.628064i
$764$		−	2.00398e6i		−	3.43325i
$765$		0			0
$766$	−388404.			−0.661952
$767$	878159.			1.49273
$768$		0			0
$769$	314204.			0.531323			0.265662	−	0.964066i	$-0.414410\pi$
$769$	314204.			0.531323			0.265662	+	0.964066i	$0.414410\pi$
$770$			569994.i			0.961366i
$771$		0			0
$772$	−21731.6			−0.0364634
$773$		−	786232.i		−	1.31581i	−0.753103	−	0.657903i	$-0.771444\pi$
$773$		−	786232.i		−	1.31581i	0.753103	−	0.657903i	$-0.228556\pi$
$774$		0			0
$775$	357079.			0.594512
$776$		−	648579.i		−	1.07706i
$777$		0			0
$778$	1.27501e6			2.10646
$779$			64624.3i			0.106493i
$780$		0			0
$781$		−	882875.i		−	1.44743i
$782$			48175.1i			0.0787788i
$783$		0			0
$784$	−331418.			−0.539193
$785$	−153920.			−0.249779
$786$		0			0
$787$	229414.			0.370400			0.185200	−	0.982701i	$-0.440707\pi$
$787$	229414.			0.370400			0.185200	+	0.982701i	$0.440707\pi$
$788$	−1.25014e6			−2.01329
$789$		0			0
$790$	445839.			0.714370
$791$	800313.			1.27911
$792$		0			0
$793$			1.87894e6i			2.98790i
$794$			1.51145e6i			2.39746i
$795$		0			0
$796$		−	2.39323e6i		−	3.77709i
$797$	1.10312e6			1.73663			0.868316	−	0.496011i	$-0.165202\pi$
$797$	1.10312e6			1.73663			0.868316	+	0.496011i	$0.165202\pi$
$798$		0			0
$799$	−100027.			−0.156683
$800$		−	1.31418e6i		−	2.05341i
$801$		0			0
$802$			285568.i			0.443977i
$803$		−	100287.i		−	0.155529i
$804$		0			0
$805$	100314.			0.154800
$806$			1.59307e6i			2.45225i
$807$		0			0
$808$		−	567800.i		−	0.869706i
$809$	−602078.			−0.919932			−0.459966	−	0.887937i	$-0.652138\pi$
$809$	−602078.			−0.919932			−0.459966	+	0.887937i	$0.652138\pi$
$810$		0			0
$811$			930391.i			1.41457i	0.706930	+	0.707284i	$0.250080\pi$
$811$			930391.i			1.41457i	−0.706930	+	0.707284i	$0.749920\pi$
$812$	−577356.			−0.875651
$813$		0			0
$814$	−1.33240e6			−2.01087
$815$	−424138.			−0.638546
$816$		0			0
$817$	383444.	−	411334.i	0.574458	−	0.616241i
$818$	2.26093e6			3.37894
$819$		0			0
$820$			80195.3i			0.119267i
$821$	857913.			1.27279			0.636395	−	0.771363i	$-0.280425\pi$
$821$	857913.			1.27279			0.636395	+	0.771363i	$0.280425\pi$
$822$		0			0
$823$	−1.18057e6			−1.74298			−0.871488	−	0.490417i	$-0.836845\pi$
$823$	−1.18057e6			−1.74298			−0.871488	+	0.490417i	$0.836845\pi$
$824$		−	1.30707e6i		−	1.92505i
$825$		0			0
$826$	1.10244e6			1.61583
$827$	−48059.2			−0.0702693			−0.0351347	−	0.999383i	$-0.511186\pi$
$827$	−48059.2			−0.0702693			−0.0351347	+	0.999383i	$0.511186\pi$
$828$		0			0
$829$			640828.i			0.932465i	0.884662	+	0.466233i	$0.154389\pi$
$829$			640828.i			0.932465i	−0.884662	+	0.466233i	$0.845611\pi$
$830$	−588671.			−0.854509
$831$		0			0
$832$	2.21747e6			3.20340
$833$	14991.7			0.0216053
$834$		0			0
$835$			227632.i			0.326483i
$836$		−	1.84527e6i		−	2.64027i
$837$		0			0
$838$	115872.			0.165003
$839$		−	380215.i		−	0.540138i	−0.962841	−	0.270069i	$-0.912953\pi$
$839$		−	380215.i		−	0.540138i	0.962841	−	0.270069i	$-0.0870465\pi$
$840$		0			0
$841$	635096.			0.897941
$842$	1.63388e6			2.30461
$843$		0			0
$844$		−	1.81758e6i		−	2.55158i
$845$			691260.i			0.968118i
$846$		0			0
$847$		−	439181.i		−	0.612177i
$848$	−1.89887e6			−2.64061
$849$		0			0
$850$			129702.i			0.179518i
$851$			234490.i			0.323792i
$852$		0			0
$853$	374276.			0.514392			0.257196	−	0.966359i	$-0.417201\pi$
$853$	374276.			0.514392			0.257196	+	0.966359i	$0.417201\pi$
$854$			2.35882e6i			3.23429i
$855$		0			0
$856$			2.05717e6i			2.80751i
$857$	−581564.			−0.791837			−0.395918	−	0.918286i	$-0.629574\pi$
$857$	−581564.			−0.791837			−0.395918	+	0.918286i	$0.629574\pi$
$858$		0			0
$859$			1.14416e6i			1.55061i	0.631589	+	0.775303i	$0.282403\pi$
$859$			1.14416e6i			1.55061i	−0.631589	+	0.775303i	$0.717597\pi$
$860$	475834.	−	510444.i	0.643366	−	0.690162i
$861$		0			0
$862$		−	1.30179e6i		−	1.75197i
$863$			48167.9i			0.0646750i	0.999477	+	0.0323375i	$0.0102951\pi$
$863$			48167.9i			0.0646750i	−0.999477	+	0.0323375i	$0.989705\pi$
$864$		0			0
$865$		−	345022.i		−	0.461121i
$866$	−1.89305e6			−2.52422
$867$		0			0
$868$			1.42998e6i			1.89797i
$869$	−956570.			−1.26671
$870$		0			0
$871$	349338.			0.460479
$872$			1.23555e6i			1.62490i
$873$		0			0
$874$	−454192.			−0.594589
$875$	584640.			0.763611
$876$		0			0
$877$	−481656.			−0.626235			−0.313118	−	0.949714i	$-0.601373\pi$
$877$	−481656.			−0.626235			−0.313118	+	0.949714i	$0.601373\pi$
$878$			1.48040e6i			1.92040i
$879$		0			0
$880$		−	1.01339e6i		−	1.30862i
$881$	771331.			0.993777			0.496889	−	0.867814i	$-0.334476\pi$
$881$	771331.			0.993777			0.496889	+	0.867814i	$0.334476\pi$
$882$		0			0
$883$	1.36525e6			1.75101			0.875507	−	0.483205i	$-0.160527\pi$
$883$	1.36525e6			1.75101			0.875507	+	0.483205i	$0.160527\pi$
$884$	−413741.			−0.529449
$885$		0			0
$886$			870048.i			1.10835i
$887$		−	395935.i		−	0.503242i	−0.967826	−	0.251621i	$-0.919036\pi$
$887$		−	395935.i		−	0.503242i	0.967826	−	0.251621i	$-0.0809637\pi$
$888$		0			0
$889$			593870.i			0.751429i
$890$			299904.i			0.378618i
$891$		0			0
$892$			1.04490e6i			1.31324i
$893$		−	943046.i		−	1.18258i
$894$		0			0
$895$	475911.			0.594128
$896$	686140.			0.854667
$897$		0			0
$898$	1.65819e6			2.05627
$899$			178785.i			0.221213i
$900$		0			0
$901$	85895.3			0.105808
$902$		−	240644.i		−	0.295776i
$903$		0			0
$904$	−2.70438e6			−3.30926
$905$		−	298216.i		−	0.364111i
$906$		0			0
$907$	1.02709e6			1.24852			0.624260	−	0.781216i	$-0.285400\pi$
$907$	1.02709e6			1.24852			0.624260	+	0.781216i	$0.285400\pi$
$908$		−	2.31978e6i		−	2.81369i
$909$		0			0
$910$			1.20491e6i			1.45503i
$911$			1.39342e6i			1.67898i	0.543374	+	0.839491i	$0.317147\pi$
$911$			1.39342e6i			1.67898i	−0.543374	+	0.839491i	$0.682853\pi$
$912$		0			0
$913$	1.26302e6			1.51520
$914$	593209.			0.710094
$915$		0			0
$916$	627122.			0.747414
$917$	402638.			0.478825
$918$		0			0
$919$	−1.19770e6			−1.41813			−0.709064	−	0.705144i	$-0.750882\pi$
$919$	−1.19770e6			−1.41813			−0.709064	+	0.705144i	$0.750882\pi$
$920$	−338977.			−0.400492
$921$		0			0
$922$			2.04242e6i			2.40261i
$923$		−	1.86632e6i		−	2.19069i
$924$		0			0
$925$			631317.i			0.737843i
$926$	2.43309e6			2.83750
$927$		0			0
$928$	657995.			0.764058
$929$			235931.i			0.273371i	0.990614	+	0.136686i	$0.0436450\pi$
$929$			235931.i			0.273371i	−0.990614	+	0.136686i	$0.956355\pi$
$930$		0			0
$931$			141341.i			0.163068i
$932$			2.41866e6i			2.78447i
$933$		0			0
$934$	−365715.			−0.419227
$935$			45840.7i			0.0524358i
$936$		0			0
$937$		−	1.54482e6i		−	1.75954i	−0.475401	−	0.879769i	$-0.657697\pi$
$937$		−	1.54482e6i		−	1.75954i	0.475401	−	0.879769i	$-0.342303\pi$
$938$	438559.			0.498451
$939$		0			0
$940$		−	1.17027e6i		−	1.32443i
$941$	573364.			0.647517			0.323759	−	0.946140i	$-0.395053\pi$
$941$	573364.			0.647517			0.323759	+	0.946140i	$0.395053\pi$
$942$		0			0
$943$	−42351.4			−0.0476260
$944$	−1.96003e6			−2.19947
$945$		0			0
$946$	−1.42785e6	+	1.53170e6i	−1.59551	+	1.71156i
$947$	−1.66137e6			−1.85253			−0.926265	−	0.376872i	$-0.877000\pi$
$947$	−1.66137e6			−1.85253			−0.926265	+	0.376872i	$0.877000\pi$
$948$		0			0
$949$		−	211997.i		−	0.235395i
$950$	−1.22282e6			−1.35492
$951$		0			0
$952$	−312383.			−0.344678
$953$			1.10535e6i			1.21707i	0.793527	+	0.608535i	$0.208242\pi$
$953$			1.10535e6i			1.21707i	−0.793527	+	0.608535i	$0.791758\pi$
$954$		0			0
$955$	−469354.			−0.514629
$956$	3.66349e6			4.00848
$957$		0			0
$958$			2.87088e6i			3.12813i
$959$	−834083.			−0.906926
$960$		0			0
$961$	−480712.			−0.520521
$962$	−2.81656e6			−3.04347
$963$		0			0
$964$			469458.i			0.505176i
$965$			5089.79i			0.00546569i
$966$		0			0
$967$	352269.			0.376723			0.188361	−	0.982100i	$-0.439682\pi$
$967$	352269.			0.376723			0.188361	+	0.982100i	$0.439682\pi$
$968$			1.48406e6i			1.58380i
$969$		0			0
$970$	−252577.			−0.268442
$971$	697949.			0.740262			0.370131	−	0.928980i	$-0.379313\pi$
$971$	697949.			0.740262			0.370131	+	0.928980i	$0.379313\pi$
$972$		0			0
$973$		−	502901.i		−	0.531199i
$974$		−	2.22191e6i		−	2.34212i
$975$		0			0
$976$		−	4.19374e6i		−	4.40253i
$977$	−84775.3			−0.0888137			−0.0444069	−	0.999014i	$-0.514140\pi$
$977$	−84775.3			−0.0888137			−0.0444069	+	0.999014i	$0.514140\pi$
$978$		0			0
$979$		−	643459.i		−	0.671360i
$980$			175396.i			0.182628i
$981$		0			0
$982$	−1.75926e6			−1.82435
$983$		−	528881.i		−	0.547333i	−0.961825	−	0.273666i	$-0.911764\pi$
$983$		−	528881.i		−	0.547333i	0.961825	−	0.273666i	$-0.0882364\pi$
$984$		0			0
$985$			292796.i			0.301782i
$986$	−64939.9			−0.0667972
$987$		0			0
$988$		−	3.90073e6i		−	3.99606i
$989$	269567.	+	251289.i	0.275597	+	0.256910i
$990$		0			0
$991$			766986.i			0.780980i	0.920607	+	0.390490i	$0.127694\pi$
$991$			766986.i			0.780980i	−0.920607	+	0.390490i	$0.872306\pi$
$992$		−	1.62970e6i		−	1.65609i
$993$		0			0
$994$		−	2.34297e6i		−	2.37134i
$995$	−560521.			−0.566169
$996$		0			0
$997$			932501.i			0.938121i	0.883166	+	0.469060i	$0.155407\pi$
$997$			932501.i			0.938121i	−0.883166	+	0.469060i	$0.844593\pi$
$998$	1.67611e6			1.68284
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.5.b.c.343.1		12
3.2	odd	2		43.5.b.b.42.12	yes	12
12.11	even	2		688.5.b.d.257.11		12
43.42	odd	2	inner	387.5.b.c.343.12		12
129.128	even	2		43.5.b.b.42.1	✓	12
516.515	odd	2		688.5.b.d.257.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.b.b.42.1	✓	12	129.128	even	2
43.5.b.b.42.12	yes	12	3.2	odd	2
387.5.b.c.343.1		12	1.1	even	1	trivial
387.5.b.c.343.12		12	43.42	odd	2	inner
688.5.b.d.257.2		12	516.515	odd	2
688.5.b.d.257.11		12	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	387.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.49282i q^{2} -40.1424 q^{4} +9.40180i q^{5} -53.5326i q^{7} +180.895i q^{8} +O(q^{10})\)	\(q-7.49282i q^{2} -40.1424 q^{4} +9.40180i q^{5} -53.5326i q^{7} +180.895i q^{8} +70.4460 q^{10} -151.146 q^{11} -319.508 q^{13} -401.110 q^{14} +713.133 q^{16} -32.2585 q^{17} -304.131i q^{19} -377.411i q^{20} +1132.51i q^{22} +199.312 q^{23} +536.606 q^{25} +2394.02i q^{26} +2148.92i q^{28} +268.672i q^{29} +665.439 q^{31} -2449.06i q^{32} +241.707i q^{34} +503.303 q^{35} +1176.50i q^{37} -2278.80 q^{38} -1700.74 q^{40} -212.488 q^{41} +(1352.49 + 1260.78i) q^{43} +6067.35 q^{44} -1493.41i q^{46} +3100.78 q^{47} -464.736 q^{49} -4020.69i q^{50} +12825.8 q^{52} -2662.72 q^{53} -1421.04i q^{55} +9683.75 q^{56} +2013.11 q^{58} -2748.48 q^{59} -5880.73i q^{61} -4986.02i q^{62} -6940.28 q^{64} -3003.95i q^{65} -1093.36 q^{67} +1294.93 q^{68} -3771.16i q^{70} +5841.22i q^{71} +663.511i q^{73} +8815.31 q^{74} +12208.6i q^{76} +8091.21i q^{77} +6328.79 q^{79} +6704.74i q^{80} +1592.13i q^{82} -8356.34 q^{83} -303.288i q^{85} +(9446.83 - 10134.0i) q^{86} -27341.4i q^{88} +4257.21i q^{89} +17104.1i q^{91} -8000.85 q^{92} -23233.6i q^{94} +2859.38 q^{95} -3585.40 q^{97} +3482.18i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 92 q^{4}+O(q^{10}) \) 12 * q - 92 * q^4	\( 12 q - 92 q^{4} + 182 q^{10} + 180 q^{11} - 216 q^{13} - 732 q^{14} + 1076 q^{16} - 678 q^{17} - 1566 q^{23} - 174 q^{25} + 5710 q^{31} - 936 q^{35} - 1242 q^{38} - 2618 q^{40} - 4878 q^{41} - 1108 q^{43} + 15168 q^{44} + 5526 q^{47} - 8544 q^{49} + 24084 q^{52} - 1212 q^{53} + 10152 q^{56} - 4666 q^{58} - 14016 q^{59} - 15580 q^{64} - 1088 q^{67} - 15186 q^{68} + 7674 q^{74} + 24302 q^{79} + 7032 q^{83} + 14412 q^{86} - 48354 q^{92} - 606 q^{95} - 5842 q^{97}+O(q^{100}) \) 12 * q - 92 * q^4 + 182 * q^10 + 180 * q^11 - 216 * q^13 - 732 * q^14 + 1076 * q^16 - 678 * q^17 - 1566 * q^23 - 174 * q^25 + 5710 * q^31 - 936 * q^35 - 1242 * q^38 - 2618 * q^40 - 4878 * q^41 - 1108 * q^43 + 15168 * q^44 + 5526 * q^47 - 8544 * q^49 + 24084 * q^52 - 1212 * q^53 + 10152 * q^56 - 4666 * q^58 - 14016 * q^59 - 15580 * q^64 - 1088 * q^67 - 15186 * q^68 + 7674 * q^74 + 24302 * q^79 + 7032 * q^83 + 14412 * q^86 - 48354 * q^92 - 606 * q^95 - 5842 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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