LMFDB - Embedded newform 1305.2.f.h.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(289,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1305.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1305 = 3^{2} \cdot 5 \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1305.f (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:	$10.4204774638$
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:	$2^{4}$
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	1305.289
Dual form			1305.2.f.h.289.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{2} +3.00000 q^{4} +(1.00000 + 2.00000i) q^{5} +2.00000i q^{7} -2.23607 q^{8} +(-2.23607 - 4.47214i) q^{10} -4.47214i q^{11} -4.00000i q^{13} -4.47214i q^{14} -1.00000 q^{16} -4.47214 q^{17} +4.47214i q^{19} +(3.00000 + 6.00000i) q^{20} +10.0000i q^{22} +6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +8.94427i q^{26} +6.00000i q^{28} +(-3.00000 + 4.47214i) q^{29} -4.47214i q^{31} +6.70820 q^{32} +10.0000 q^{34} +(-4.00000 + 2.00000i) q^{35} -4.47214 q^{37} -10.0000i q^{38} +(-2.23607 - 4.47214i) q^{40} -13.4164i q^{44} -13.4164i q^{46} -8.94427 q^{47} +3.00000 q^{49} +(6.70820 - 8.94427i) q^{50} -12.0000i q^{52} -4.00000i q^{53} +(8.94427 - 4.47214i) q^{55} -4.47214i q^{56} +(6.70820 - 10.0000i) q^{58} -4.00000 q^{59} +8.94427i q^{61} +10.0000i q^{62} -13.0000 q^{64} +(8.00000 - 4.00000i) q^{65} -2.00000i q^{67} -13.4164 q^{68} +(8.94427 - 4.47214i) q^{70} -13.4164 q^{73} +10.0000 q^{74} +13.4164i q^{76} +8.94427 q^{77} +13.4164i q^{79} +(-1.00000 - 2.00000i) q^{80} -6.00000i q^{83} +(-4.47214 - 8.94427i) q^{85} +10.0000i q^{88} +17.8885i q^{89} +8.00000 q^{91} +18.0000i q^{92} +20.0000 q^{94} +(-8.94427 + 4.47214i) q^{95} +4.47214 q^{97} -6.70820 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 12 q^{4} + 4 q^{5} - 4 q^{16} + 12 q^{20} - 12 q^{25} - 12 q^{29} + 40 q^{34} - 16 q^{35} + 12 q^{49} - 16 q^{59} - 52 q^{64} + 32 q^{65} + 40 q^{74} - 4 q^{80} + 32 q^{91} + 80 q^{94}+O(q^{100})$ 4 * q + 12 * q^4 + 4 * q^5 - 4 * q^16 + 12 * q^20 - 12 * q^25 - 12 * q^29 + 40 * q^34 - 16 * q^35 + 12 * q^49 - 16 * q^59 - 52 * q^64 + 32 * q^65 + 40 * q^74 - 4 * q^80 + 32 * q^91 + 80 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$ .

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$-1$	$-1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	−2.23607			−1.58114			−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607			−1.58114			−0.790569	+	0.612372i	$0.790215\pi$
$3$		0			0
$3$		0			0
$4$	3.00000			1.50000
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	−2.23607			−0.790569
$9$		0			0
$10$	−2.23607	−	4.47214i	−0.707107	−	1.41421i
$11$		−	4.47214i		−	1.34840i	−0.738549	−	0.674200i	$-0.764489\pi$
$11$		−	4.47214i		−	1.34840i	0.738549	−	0.674200i	$-0.235511\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		−	4.47214i		−	1.19523i
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−4.47214			−1.08465			−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214			−1.08465			−0.542326	+	0.840168i	$0.682456\pi$
$18$		0			0
$19$			4.47214i			1.02598i	0.858395	+	0.512989i	$0.171462\pi$
$19$			4.47214i			1.02598i	−0.858395	+	0.512989i	$0.828538\pi$
$20$	3.00000	+	6.00000i	0.670820	+	1.34164i
$21$		0			0
$22$			10.0000i			2.13201i
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$			8.94427i			1.75412i
$27$		0			0
$28$			6.00000i			1.13389i
$29$	−3.00000	+	4.47214i	−0.557086	+	0.830455i
$29$	−3.00000	+	4.47214i	−0.557086	+	0.830455i
$30$		0			0
$31$		−	4.47214i		−	0.803219i	−0.915811	−	0.401610i	$-0.868451\pi$
$31$		−	4.47214i		−	0.803219i	0.915811	−	0.401610i	$-0.131549\pi$
$32$	6.70820			1.18585
$33$		0			0
$34$	10.0000			1.71499
$35$	−4.00000	+	2.00000i	−0.676123	+	0.338062i
$36$		0			0
$37$	−4.47214			−0.735215			−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214			−0.735215			−0.367607	+	0.929981i	$0.619823\pi$
$38$		−	10.0000i		−	1.62221i
$39$		0			0
$40$	−2.23607	−	4.47214i	−0.353553	−	0.707107i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		−	13.4164i		−	2.02260i
$45$		0			0
$46$		−	13.4164i		−	1.97814i
$47$	−8.94427			−1.30466			−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427			−1.30466			−0.652328	+	0.757937i	$0.726208\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	6.70820	−	8.94427i	0.948683	−	1.26491i
$51$		0			0
$52$		−	12.0000i		−	1.66410i
$53$		−	4.00000i		−	0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$		−	4.00000i		−	0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$		0			0
$55$	8.94427	−	4.47214i	1.20605	−	0.603023i
$56$		−	4.47214i		−	0.597614i
$57$		0			0
$58$	6.70820	−	10.0000i	0.880830	−	1.31306i
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$			8.94427i			1.14520i	0.819836	+	0.572598i	$0.194065\pi$
$61$			8.94427i			1.14520i	−0.819836	+	0.572598i	$0.805935\pi$
$62$			10.0000i			1.27000i
$63$		0			0
$64$	−13.0000			−1.62500
$65$	8.00000	−	4.00000i	0.992278	−	0.496139i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	−13.4164			−1.62698
$69$		0			0
$70$	8.94427	−	4.47214i	1.06904	−	0.534522i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−13.4164			−1.57027			−0.785136	−	0.619324i	$-0.787407\pi$
$73$	−13.4164			−1.57027			−0.785136	+	0.619324i	$0.787407\pi$
$74$	10.0000			1.16248
$75$		0			0
$76$			13.4164i			1.53897i
$77$	8.94427			1.01929
$78$		0			0
$79$			13.4164i			1.50946i	0.656033	+	0.754732i	$0.272233\pi$
$79$			13.4164i			1.50946i	−0.656033	+	0.754732i	$0.727767\pi$
$80$	−1.00000	−	2.00000i	−0.111803	−	0.223607i
$81$		0			0
$82$		0			0
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$	−4.47214	−	8.94427i	−0.485071	−	0.970143i
$86$		0			0
$87$		0			0
$88$			10.0000i			1.06600i
$89$			17.8885i			1.89618i	0.317999	+	0.948091i	$0.396989\pi$
$89$			17.8885i			1.89618i	−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	8.00000			0.838628
$92$			18.0000i			1.87663i
$93$		0			0
$94$	20.0000			2.06284
$95$	−8.94427	+	4.47214i	−0.917663	+	0.458831i
$96$		0			0
$97$	4.47214			0.454077			0.227038	−	0.973886i	$-0.427096\pi$
$97$	4.47214			0.454077			0.227038	+	0.973886i	$0.427096\pi$
$98$	−6.70820			−0.677631
$99$		0			0
$100$	−9.00000	+	12.0000i	−0.900000	+	1.20000i
$101$		−	8.94427i		−	0.889988i	−0.895533	−	0.444994i	$-0.853206\pi$
$101$		−	8.94427i		−	0.889988i	0.895533	−	0.444994i	$-0.146794\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$			8.94427i			0.877058i
$105$		0			0
$106$			8.94427i			0.868744i
$107$			2.00000i			0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$			2.00000i			0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	−20.0000	+	10.0000i	−1.90693	+	0.953463i
$111$		0			0
$112$		−	2.00000i		−	0.188982i
$113$	−4.47214			−0.420703			−0.210352	−	0.977626i	$-0.567461\pi$
$113$	−4.47214			−0.420703			−0.210352	+	0.977626i	$0.567461\pi$
$114$		0			0
$115$	−12.0000	+	6.00000i	−1.11901	+	0.559503i
$116$	−9.00000	+	13.4164i	−0.835629	+	1.24568i
$117$		0			0
$118$	8.94427			0.823387
$119$		−	8.94427i		−	0.819920i
$120$		0			0
$121$	−9.00000			−0.818182
$122$		−	20.0000i		−	1.81071i
$123$		0			0
$124$		−	13.4164i		−	1.20483i
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$	−8.94427			−0.793676			−0.396838	−	0.917889i	$-0.629892\pi$
$127$	−8.94427			−0.793676			−0.396838	+	0.917889i	$0.629892\pi$
$128$	15.6525			1.38350
$129$		0			0
$130$	−17.8885	+	8.94427i	−1.56893	+	0.784465i
$131$		−	4.47214i		−	0.390732i	−0.980730	−	0.195366i	$-0.937410\pi$
$131$		−	4.47214i		−	0.390732i	0.980730	−	0.195366i	$-0.0625895\pi$
$132$		0			0
$133$	−8.94427			−0.775567
$134$			4.47214i			0.386334i
$135$		0			0
$136$	10.0000			0.857493
$137$	13.4164			1.14624			0.573121	−	0.819471i	$-0.305733\pi$
$137$	13.4164			1.14624			0.573121	+	0.819471i	$0.305733\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$	−12.0000	+	6.00000i	−1.01419	+	0.507093i
$141$		0			0
$142$		0			0
$143$	−17.8885			−1.49592
$144$		0			0
$145$	−11.9443	−	1.52786i	−0.991918	−	0.126882i
$146$	30.0000			2.48282
$147$		0			0
$148$	−13.4164			−1.10282
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		−	10.0000i		−	0.811107i
$153$		0			0
$154$	−20.0000			−1.61165
$155$	8.94427	−	4.47214i	0.718421	−	0.359211i
$156$		0			0
$157$	13.4164			1.07075			0.535373	−	0.844616i	$-0.320171\pi$
$157$	13.4164			1.07075			0.535373	+	0.844616i	$0.320171\pi$
$158$		−	30.0000i		−	2.38667i
$159$		0			0
$160$	6.70820	+	13.4164i	0.530330	+	1.06066i
$161$	−12.0000			−0.945732
$162$		0			0
$163$	−17.8885			−1.40114			−0.700569	−	0.713584i	$-0.747071\pi$
$163$	−17.8885			−1.40114			−0.700569	+	0.713584i	$0.747071\pi$
$164$		0			0
$165$		0			0
$166$			13.4164i			1.04132i
$167$			22.0000i			1.70241i	0.524832	+	0.851206i	$0.324128\pi$
$167$			22.0000i			1.70241i	−0.524832	+	0.851206i	$0.675872\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	10.0000	+	20.0000i	0.766965	+	1.53393i
$171$		0			0
$172$		0			0
$173$			4.00000i			0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$			4.00000i			0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$	−8.00000	−	6.00000i	−0.604743	−	0.453557i
$176$			4.47214i			0.337100i
$177$		0			0
$178$		−	40.0000i		−	2.99813i
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	−17.8885			−1.32599
$183$		0			0
$184$		−	13.4164i		−	0.989071i
$185$	−4.47214	−	8.94427i	−0.328798	−	0.657596i
$186$		0			0
$187$			20.0000i			1.46254i
$188$	−26.8328			−1.95698
$189$		0			0
$190$	20.0000	−	10.0000i	1.45095	−	0.725476i
$191$		−	13.4164i		−	0.970777i	−0.874299	−	0.485389i	$-0.838678\pi$
$191$		−	13.4164i		−	0.970777i	0.874299	−	0.485389i	$-0.161322\pi$
$192$		0			0
$193$	4.47214			0.321911			0.160956	−	0.986962i	$-0.448542\pi$
$193$	4.47214			0.321911			0.160956	+	0.986962i	$0.448542\pi$
$194$	−10.0000			−0.717958
$195$		0			0
$196$	9.00000			0.642857
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$	6.70820	−	8.94427i	0.474342	−	0.632456i
$201$		0			0
$202$			20.0000i			1.40720i
$203$	−8.94427	−	6.00000i	−0.627765	−	0.421117i
$204$		0			0
$205$		0			0
$206$			13.4164i			0.934765i
$207$		0			0
$208$			4.00000i			0.277350i
$209$	20.0000			1.38343
$210$		0			0
$211$			22.3607i			1.53937i	0.638422	+	0.769686i	$0.279587\pi$
$211$			22.3607i			1.53937i	−0.638422	+	0.769686i	$0.720413\pi$
$212$		−	12.0000i		−	0.824163i
$213$		0			0
$214$		−	4.47214i		−	0.305709i
$215$		0			0
$216$		0			0
$217$	8.94427			0.607177
$218$	22.3607			1.51446
$219$		0			0
$220$	26.8328	−	13.4164i	1.80907	−	0.904534i
$221$			17.8885i			1.20331i
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$			13.4164i			0.896421i
$225$		0			0
$226$	10.0000			0.665190
$227$			18.0000i			1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$			18.0000i			1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$		0			0
$229$		−	26.8328i		−	1.77316i	−0.462573	−	0.886581i	$-0.653074\pi$
$229$		−	26.8328i		−	1.77316i	0.462573	−	0.886581i	$-0.346926\pi$
$230$	26.8328	−	13.4164i	1.76930	−	0.884652i
$231$		0			0
$232$	6.70820	−	10.0000i	0.440415	−	0.656532i
$233$		−	24.0000i		−	1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$		−	24.0000i		−	1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$		0			0
$235$	−8.94427	−	17.8885i	−0.583460	−	1.16692i
$236$	−12.0000			−0.781133
$237$		0			0
$238$			20.0000i			1.29641i
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$	20.1246			1.29366
$243$		0			0
$244$			26.8328i			1.71780i
$245$	3.00000	+	6.00000i	0.191663	+	0.383326i
$246$		0			0
$247$	17.8885			1.13822
$248$			10.0000i			0.635001i
$249$		0			0
$250$	24.5967	+	4.47214i	1.55563	+	0.282843i
$251$		−	4.47214i		−	0.282279i	−0.989990	−	0.141139i	$-0.954923\pi$
$251$		−	4.47214i		−	0.282279i	0.989990	−	0.141139i	$-0.0450766\pi$
$252$		0			0
$253$	26.8328			1.68696
$254$	20.0000			1.25491
$255$		0			0
$256$	−9.00000			−0.562500
$257$			8.00000i			0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$			8.00000i			0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$		0			0
$259$		−	8.94427i		−	0.555770i
$260$	24.0000	−	12.0000i	1.48842	−	0.744208i
$261$		0			0
$262$			10.0000i			0.617802i
$263$	−26.8328			−1.65458			−0.827291	−	0.561773i	$-0.810119\pi$
$263$	−26.8328			−1.65458			−0.827291	+	0.561773i	$0.810119\pi$
$264$		0			0
$265$	8.00000	−	4.00000i	0.491436	−	0.245718i
$266$	20.0000			1.22628
$267$		0			0
$268$		−	6.00000i		−	0.366508i
$269$		−	8.94427i		−	0.545342i	−0.962107	−	0.272671i	$-0.912093\pi$
$269$		−	8.94427i		−	0.545342i	0.962107	−	0.272671i	$-0.0879070\pi$
$270$		0			0
$271$			13.4164i			0.814989i	0.913208	+	0.407494i	$0.133597\pi$
$271$			13.4164i			0.814989i	−0.913208	+	0.407494i	$0.866403\pi$
$272$	4.47214			0.271163
$273$		0			0
$274$	−30.0000			−1.81237
$275$	17.8885	+	13.4164i	1.07872	+	0.809040i
$276$		0			0
$277$		−	28.0000i		−	1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$		−	28.0000i		−	1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	44.7214			2.68221
$279$		0			0
$280$	8.94427	−	4.47214i	0.534522	−	0.267261i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$			6.00000i			0.356663i	0.983970	+	0.178331i	$0.0570699\pi$
$283$			6.00000i			0.356663i	−0.983970	+	0.178331i	$0.942930\pi$
$284$		0			0
$285$		0			0
$286$	40.0000			2.36525
$287$		0			0
$288$		0			0
$289$	3.00000			0.176471
$290$	26.7082	+	3.41641i	1.56836	+	0.200618i
$291$		0			0
$292$	−40.2492			−2.35541
$293$	4.47214			0.261265			0.130632	−	0.991431i	$-0.458299\pi$
$293$	4.47214			0.261265			0.130632	+	0.991431i	$0.458299\pi$
$294$		0			0
$295$	−4.00000	−	8.00000i	−0.232889	−	0.465778i
$296$	10.0000			0.581238
$297$		0			0
$298$	−22.3607			−1.29532
$299$	24.0000			1.38796
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	4.47214i		−	0.256495i
$305$	−17.8885	+	8.94427i	−1.02430	+	0.512148i
$306$		0			0
$307$	−17.8885			−1.02095			−0.510477	−	0.859892i	$-0.670531\pi$
$307$	−17.8885			−1.02095			−0.510477	+	0.859892i	$0.670531\pi$
$308$	26.8328			1.52894
$309$		0			0
$310$	−20.0000	+	10.0000i	−1.13592	+	0.567962i
$311$			22.3607i			1.26796i	0.773350	+	0.633979i	$0.218579\pi$
$311$			22.3607i			1.26796i	−0.773350	+	0.633979i	$0.781421\pi$
$312$		0			0
$313$			16.0000i			0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$			16.0000i			0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$	−30.0000			−1.69300
$315$		0			0
$316$			40.2492i			2.26420i
$317$	22.3607			1.25590			0.627950	−	0.778253i	$-0.283894\pi$
$317$	22.3607			1.25590			0.627950	+	0.778253i	$0.283894\pi$
$318$		0			0
$319$	20.0000	+	13.4164i	1.11979	+	0.751175i
$320$	−13.0000	−	26.0000i	−0.726722	−	1.45344i
$321$		0			0
$322$	26.8328			1.49533
$323$		−	20.0000i		−	1.11283i
$324$		0			0
$325$	16.0000	+	12.0000i	0.887520	+	0.665640i
$326$	40.0000			2.21540
$327$		0			0
$328$		0			0
$329$		−	17.8885i		−	0.986227i
$330$		0			0
$331$		−	13.4164i		−	0.737432i	−0.929542	−	0.368716i	$-0.879797\pi$
$331$		−	13.4164i		−	0.737432i	0.929542	−	0.368716i	$-0.120203\pi$
$332$		−	18.0000i		−	0.987878i
$333$		0			0
$334$		−	49.1935i		−	2.69175i
$335$	4.00000	−	2.00000i	0.218543	−	0.109272i
$336$		0			0
$337$	4.47214			0.243613			0.121806	−	0.992554i	$-0.461131\pi$
$337$	4.47214			0.243613			0.121806	+	0.992554i	$0.461131\pi$
$338$	6.70820			0.364878
$339$		0			0
$340$	−13.4164	−	26.8328i	−0.727607	−	1.45521i
$341$	−20.0000			−1.08306
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$		−	8.94427i		−	0.480847i
$347$			2.00000i			0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$			2.00000i			0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$		0			0
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$	17.8885	+	13.4164i	0.956183	+	0.717137i
$351$		0			0
$352$		−	30.0000i		−	1.59901i
$353$			24.0000i			1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$			24.0000i			1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$		0			0
$355$		0			0
$356$			53.6656i			2.84427i
$357$		0			0
$358$	44.7214			2.36360
$359$		−	13.4164i		−	0.708091i	−0.935228	−	0.354045i	$-0.884806\pi$
$359$		−	13.4164i		−	0.708091i	0.935228	−	0.354045i	$-0.115194\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	22.3607			1.17525
$363$		0			0
$364$	24.0000			1.25794
$365$	−13.4164	−	26.8328i	−0.702247	−	1.40449i
$366$		0			0
$367$	8.94427			0.466887			0.233444	−	0.972370i	$-0.425001\pi$
$367$	8.94427			0.466887			0.233444	+	0.972370i	$0.425001\pi$
$368$		−	6.00000i		−	0.312772i
$369$		0			0
$370$	10.0000	+	20.0000i	0.519875	+	1.03975i
$371$	8.00000			0.415339
$372$		0			0
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		−	44.7214i		−	2.31249i
$375$		0			0
$376$	20.0000			1.03142
$377$	17.8885	+	12.0000i	0.921307	+	0.618031i
$378$		0			0
$379$			4.47214i			0.229718i	0.993382	+	0.114859i	$0.0366417\pi$
$379$			4.47214i			0.229718i	−0.993382	+	0.114859i	$0.963358\pi$
$380$	−26.8328	+	13.4164i	−1.37649	+	0.688247i
$381$		0			0
$382$			30.0000i			1.53493i
$383$			6.00000i			0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$			6.00000i			0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$		0			0
$385$	8.94427	+	17.8885i	0.455842	+	0.911685i
$386$	−10.0000			−0.508987
$387$		0			0
$388$	13.4164			0.681115
$389$			8.94427i			0.453493i	0.973954	+	0.226746i	$0.0728088\pi$
$389$			8.94427i			0.453493i	−0.973954	+	0.226746i	$0.927191\pi$
$390$		0			0
$391$		−	26.8328i		−	1.35699i
$392$	−6.70820			−0.338815
$393$		0			0
$394$			26.8328i			1.35182i
$395$	−26.8328	+	13.4164i	−1.35011	+	0.675053i
$396$		0			0
$397$		−	12.0000i		−	0.602263i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$		−	12.0000i		−	0.602263i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$	−35.7771			−1.79334
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$	−17.8885			−0.891092
$404$		−	26.8328i		−	1.33498i
$405$		0			0
$406$	20.0000	+	13.4164i	0.992583	+	0.665845i
$407$			20.0000i			0.991363i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		−	18.0000i		−	0.886796i
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	12.0000	−	6.00000i	0.589057	−	0.294528i
$416$		−	26.8328i		−	1.31559i
$417$		0			0
$418$	−44.7214			−2.18739
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$		−	8.94427i		−	0.435917i	−0.975958	−	0.217959i	$-0.930060\pi$
$421$		−	8.94427i		−	0.435917i	0.975958	−	0.217959i	$-0.0699398\pi$
$422$		−	50.0000i		−	2.43396i
$423$		0			0
$424$			8.94427i			0.434372i
$425$	13.4164	−	17.8885i	0.650791	−	0.867722i
$426$		0			0
$427$	−17.8885			−0.865687
$428$			6.00000i			0.290021i
$429$		0			0
$430$		0			0
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$	40.2492			1.93425			0.967127	−	0.254293i	$-0.0818429\pi$
$433$	40.2492			1.93425			0.967127	+	0.254293i	$0.0818429\pi$
$434$	−20.0000			−0.960031
$435$		0			0
$436$	−30.0000			−1.43674
$437$	−26.8328			−1.28359
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$	−20.0000	+	10.0000i	−0.953463	+	0.476731i
$441$		0			0
$442$		−	40.0000i		−	1.90261i
$443$	17.8885			0.849910			0.424955	−	0.905214i	$-0.360290\pi$
$443$	17.8885			0.849910			0.424955	+	0.905214i	$0.360290\pi$
$444$		0			0
$445$	−35.7771	+	17.8885i	−1.69600	+	0.847998i
$446$			31.3050i			1.48233i
$447$		0			0
$448$		−	26.0000i		−	1.22838i
$449$			17.8885i			0.844213i	0.906546	+	0.422106i	$0.138709\pi$
$449$			17.8885i			0.844213i	−0.906546	+	0.422106i	$0.861291\pi$
$450$		0			0
$451$		0			0
$452$	−13.4164			−0.631055
$453$		0			0
$454$		−	40.2492i		−	1.88899i
$455$	8.00000	+	16.0000i	0.375046	+	0.750092i
$456$		0			0
$457$			32.0000i			1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$			32.0000i			1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$			60.0000i			2.80362i
$459$		0			0
$460$	−36.0000	+	18.0000i	−1.67851	+	0.839254i
$461$		−	26.8328i		−	1.24973i	−0.780733	−	0.624864i	$-0.785154\pi$
$461$		−	26.8328i		−	1.24973i	0.780733	−	0.624864i	$-0.214846\pi$
$462$		0			0
$463$			34.0000i			1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$			34.0000i			1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$	3.00000	−	4.47214i	0.139272	−	0.207614i
$465$		0			0
$466$			53.6656i			2.48601i
$467$	−17.8885			−0.827783			−0.413892	−	0.910326i	$-0.635831\pi$
$467$	−17.8885			−0.827783			−0.413892	+	0.910326i	$0.635831\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$	20.0000	+	40.0000i	0.922531	+	1.84506i
$471$		0			0
$472$	8.94427			0.411693
$473$		0			0
$474$		0			0
$475$	−17.8885	−	13.4164i	−0.820783	−	0.615587i
$476$		−	26.8328i		−	1.22988i
$477$		0			0
$478$	53.6656			2.45461
$479$		−	13.4164i		−	0.613011i	−0.951869	−	0.306506i	$-0.900840\pi$
$479$		−	13.4164i		−	0.613011i	0.951869	−	0.306506i	$-0.0991598\pi$
$480$		0			0
$481$			17.8885i			0.815647i
$482$	−22.3607			−1.01850
$483$		0			0
$484$	−27.0000			−1.22727
$485$	4.47214	+	8.94427i	0.203069	+	0.406138i
$486$		0			0
$487$			2.00000i			0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$			2.00000i			0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$		−	20.0000i		−	0.905357i
$489$		0			0
$490$	−6.70820	−	13.4164i	−0.303046	−	0.606092i
$491$			13.4164i			0.605474i	0.953074	+	0.302737i	$0.0979004\pi$
$491$			13.4164i			0.605474i	−0.953074	+	0.302737i	$0.902100\pi$
$492$		0			0
$493$	13.4164	−	20.0000i	0.604245	−	0.900755i
$494$	−40.0000			−1.79969
$495$		0			0
$496$			4.47214i			0.200805i
$497$		0			0
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$	−33.0000	−	6.00000i	−1.47580	−	0.268328i
$501$		0			0
$502$			10.0000i			0.446322i
$503$	8.94427			0.398805			0.199403	−	0.979918i	$-0.436100\pi$
$503$	8.94427			0.398805			0.199403	+	0.979918i	$0.436100\pi$
$504$		0			0
$505$	17.8885	−	8.94427i	0.796030	−	0.398015i
$506$	−60.0000			−2.66733
$507$		0			0
$508$	−26.8328			−1.19051
$509$	10.0000			0.443242			0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000			0.443242			0.221621	+	0.975133i	$0.428865\pi$
$510$		0			0
$511$		−	26.8328i		−	1.18701i
$512$	−11.1803			−0.494106
$513$		0			0
$514$		−	17.8885i		−	0.789030i
$515$	12.0000	−	6.00000i	0.528783	−	0.264392i
$516$		0			0
$517$			40.0000i			1.75920i
$518$			20.0000i			0.878750i
$519$		0			0
$520$	−17.8885	+	8.94427i	−0.784465	+	0.392232i
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$		−	26.0000i		−	1.13690i	−0.822718	−	0.568450i	$-0.807543\pi$
$523$		−	26.0000i		−	1.13690i	0.822718	−	0.568450i	$-0.192457\pi$
$524$		−	13.4164i		−	0.586098i
$525$		0			0
$526$	60.0000			2.61612
$527$			20.0000i			0.871214i
$528$		0			0
$529$	−13.0000			−0.565217
$530$	−17.8885	+	8.94427i	−0.777029	+	0.388514i
$531$		0			0
$532$	−26.8328			−1.16335
$533$		0			0
$534$		0			0
$535$	−4.00000	+	2.00000i	−0.172935	+	0.0864675i
$536$			4.47214i			0.193167i
$537$		0			0
$538$			20.0000i			0.862261i
$539$		−	13.4164i		−	0.577886i
$540$		0			0
$541$		−	8.94427i		−	0.384544i	−0.981342	−	0.192272i	$-0.938414\pi$
$541$		−	8.94427i		−	0.384544i	0.981342	−	0.192272i	$-0.0615856\pi$
$542$		−	30.0000i		−	1.28861i
$543$		0			0
$544$	−30.0000			−1.28624
$545$	−10.0000	−	20.0000i	−0.428353	−	0.856706i
$546$		0			0
$547$			38.0000i			1.62476i	0.583127	+	0.812381i	$0.301829\pi$
$547$			38.0000i			1.62476i	−0.583127	+	0.812381i	$0.698171\pi$
$548$	40.2492			1.71936
$549$		0			0
$550$	−40.0000	−	30.0000i	−1.70561	−	1.27920i
$551$	−20.0000	−	13.4164i	−0.852029	−	0.571558i
$552$		0			0
$553$	−26.8328			−1.14105
$554$			62.6099i			2.66004i
$555$		0			0
$556$	−60.0000			−2.54457
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$		0			0
$559$		0			0
$560$	4.00000	−	2.00000i	0.169031	−	0.0845154i
$561$		0			0
$562$	40.2492			1.69781
$563$	17.8885			0.753912			0.376956	−	0.926231i	$-0.376971\pi$
$563$	17.8885			0.753912			0.376956	+	0.926231i	$0.376971\pi$
$564$		0			0
$565$	−4.47214	−	8.94427i	−0.188144	−	0.376288i
$566$		−	13.4164i		−	0.563934i
$567$		0			0
$568$		0			0
$569$			35.7771i			1.49985i	0.661521	+	0.749927i	$0.269911\pi$
$569$			35.7771i			1.49985i	−0.661521	+	0.749927i	$0.730089\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$	−53.6656			−2.24387
$573$		0			0
$574$		0			0
$575$	−24.0000	−	18.0000i	−1.00087	−	0.750652i
$576$		0			0
$577$	4.47214			0.186177			0.0930887	−	0.995658i	$-0.470326\pi$
$577$	4.47214			0.186177			0.0930887	+	0.995658i	$0.470326\pi$
$578$	−6.70820			−0.279024
$579$		0			0
$580$	−35.8328	−	4.58359i	−1.48788	−	0.190323i
$581$	12.0000			0.497844
$582$		0			0
$583$	−17.8885			−0.740868
$584$	30.0000			1.24141
$585$		0			0
$586$	−10.0000			−0.413096
$587$		−	38.0000i		−	1.56843i	−0.620491	−	0.784214i	$-0.713066\pi$
$587$		−	38.0000i		−	1.56843i	0.620491	−	0.784214i	$-0.286934\pi$
$588$		0			0
$589$	20.0000			0.824086
$590$	8.94427	+	17.8885i	0.368230	+	0.736460i
$591$		0			0
$592$	4.47214			0.183804
$593$			24.0000i			0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$			24.0000i			0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$		0			0
$595$	17.8885	−	8.94427i	0.733359	−	0.366679i
$596$	30.0000			1.22885
$597$		0			0
$598$	−53.6656			−2.19455
$599$		−	31.3050i		−	1.27909i	−0.768755	−	0.639543i	$-0.779124\pi$
$599$		−	31.3050i		−	1.27909i	0.768755	−	0.639543i	$-0.220876\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.00000	−	18.0000i	−0.365902	−	0.731804i
$606$		0			0
$607$	−26.8328			−1.08911			−0.544555	−	0.838725i	$-0.683302\pi$
$607$	−26.8328			−1.08911			−0.544555	+	0.838725i	$0.683302\pi$
$608$			30.0000i			1.21666i
$609$		0			0
$610$	40.0000	−	20.0000i	1.61955	−	0.809776i
$611$			35.7771i			1.44739i
$612$		0			0
$613$		−	4.00000i		−	0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$		−	4.00000i		−	0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	40.0000			1.61427
$615$		0			0
$616$	−20.0000			−0.805823
$617$	13.4164			0.540124			0.270062	−	0.962843i	$-0.412956\pi$
$617$	13.4164			0.540124			0.270062	+	0.962843i	$0.412956\pi$
$618$		0			0
$619$			4.47214i			0.179750i	0.995953	+	0.0898752i	$0.0286468\pi$
$619$			4.47214i			0.179750i	−0.995953	+	0.0898752i	$0.971353\pi$
$620$	26.8328	−	13.4164i	1.07763	−	0.538816i
$621$		0			0
$622$		−	50.0000i		−	2.00482i
$623$	−35.7771			−1.43338
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		−	35.7771i		−	1.42994i
$627$		0			0
$628$	40.2492			1.60612
$629$	20.0000			0.797452
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		−	30.0000i		−	1.19334i
$633$		0			0
$634$	−50.0000			−1.98575
$635$	−8.94427	−	17.8885i	−0.354943	−	0.709885i
$636$		0			0
$637$		−	12.0000i		−	0.475457i
$638$	−44.7214	−	30.0000i	−1.77054	−	1.18771i
$639$		0			0
$640$	15.6525	+	31.3050i	0.618718	+	1.23744i
$641$			17.8885i			0.706555i	0.935519	+	0.353278i	$0.114933\pi$
$641$			17.8885i			0.706555i	−0.935519	+	0.353278i	$0.885067\pi$
$642$		0			0
$643$			14.0000i			0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$			14.0000i			0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	−36.0000			−1.41860
$645$		0			0
$646$			44.7214i			1.75954i
$647$		−	2.00000i		−	0.0786281i	−0.999227	−	0.0393141i	$-0.987483\pi$
$647$		−	2.00000i		−	0.0786281i	0.999227	−	0.0393141i	$-0.0125173\pi$
$648$		0			0
$649$			17.8885i			0.702187i
$650$	−35.7771	−	26.8328i	−1.40329	−	1.05247i
$651$		0			0
$652$	−53.6656			−2.10171
$653$	22.3607			0.875041			0.437521	−	0.899208i	$-0.355857\pi$
$653$	22.3607			0.875041			0.437521	+	0.899208i	$0.355857\pi$
$654$		0			0
$655$	8.94427	−	4.47214i	0.349482	−	0.174741i
$656$		0			0
$657$		0			0
$658$			40.0000i			1.55936i
$659$			31.3050i			1.21947i	0.792606	+	0.609734i	$0.208724\pi$
$659$			31.3050i			1.21947i	−0.792606	+	0.609734i	$0.791276\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$			30.0000i			1.16598i
$663$		0			0
$664$			13.4164i			0.520658i
$665$	−8.94427	−	17.8885i	−0.346844	−	0.693688i
$666$		0			0
$667$	−26.8328	−	18.0000i	−1.03897	−	0.696963i
$668$			66.0000i			2.55362i
$669$		0			0
$670$	−8.94427	+	4.47214i	−0.345547	+	0.172774i
$671$	40.0000			1.54418
$672$		0			0
$673$		−	16.0000i		−	0.616755i	−0.951264	−	0.308377i	$-0.900214\pi$
$673$		−	16.0000i		−	0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$	−10.0000			−0.385186
$675$		0			0
$676$	−9.00000			−0.346154
$677$	4.47214			0.171878			0.0859391	−	0.996300i	$-0.472611\pi$
$677$	4.47214			0.171878			0.0859391	+	0.996300i	$0.472611\pi$
$678$		0			0
$679$			8.94427i			0.343250i
$680$	10.0000	+	20.0000i	0.383482	+	0.766965i
$681$		0			0
$682$	44.7214			1.71247
$683$		−	6.00000i		−	0.229584i	−0.993390	−	0.114792i	$-0.963380\pi$
$683$		−	6.00000i		−	0.229584i	0.993390	−	0.114792i	$-0.0366201\pi$
$684$		0			0
$685$	13.4164	+	26.8328i	0.512615	+	1.02523i
$686$		−	44.7214i		−	1.70747i
$687$		0			0
$688$		0			0
$689$	−16.0000			−0.609551
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$			12.0000i			0.456172i
$693$		0			0
$694$		−	4.47214i		−	0.169760i
$695$	−20.0000	−	40.0000i	−0.758643	−	1.51729i
$696$		0			0
$697$		0			0
$698$	−13.4164			−0.507819
$699$		0			0
$700$	−24.0000	−	18.0000i	−0.907115	−	0.680336i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$		−	20.0000i		−	0.754314i
$704$			58.1378i			2.19115i
$705$		0			0
$706$		−	53.6656i		−	2.01973i
$707$	17.8885			0.672768
$708$		0			0
$709$	30.0000			1.12667			0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000			1.12667			0.563337	+	0.826227i	$0.309517\pi$
$710$		0			0
$711$		0			0
$712$		−	40.0000i		−	1.49906i
$713$	26.8328			1.00490
$714$		0			0
$715$	−17.8885	−	35.7771i	−0.668994	−	1.33799i
$716$	−60.0000			−2.24231
$717$		0			0
$718$			30.0000i			1.11959i
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$	2.23607			0.0832178
$723$		0			0
$724$	−30.0000			−1.11494
$725$	−8.88854	−	25.4164i	−0.330112	−	0.943942i
$726$		0			0
$727$	26.8328			0.995174			0.497587	−	0.867414i	$-0.334220\pi$
$727$	26.8328			0.995174			0.497587	+	0.867414i	$0.334220\pi$
$728$	−17.8885			−0.662994
$729$		0			0
$730$	30.0000	+	60.0000i	1.11035	+	2.22070i
$731$		0			0
$732$		0			0
$733$	13.4164			0.495546			0.247773	−	0.968818i	$-0.420301\pi$
$733$	13.4164			0.495546			0.247773	+	0.968818i	$0.420301\pi$
$734$	−20.0000			−0.738213
$735$		0			0
$736$			40.2492i			1.48361i
$737$	−8.94427			−0.329466
$738$		0			0
$739$			40.2492i			1.48059i	0.672281	+	0.740296i	$0.265315\pi$
$739$			40.2492i			1.48059i	−0.672281	+	0.740296i	$0.734685\pi$
$740$	−13.4164	−	26.8328i	−0.493197	−	0.986394i
$741$		0			0
$742$	−17.8885			−0.656709
$743$	8.94427			0.328134			0.164067	−	0.986449i	$-0.447539\pi$
$743$	8.94427			0.328134			0.164067	+	0.986449i	$0.447539\pi$
$744$		0			0
$745$	10.0000	+	20.0000i	0.366372	+	0.732743i
$746$			8.94427i			0.327473i
$747$		0			0
$748$			60.0000i			2.19382i
$749$	−4.00000			−0.146157
$750$		0			0
$751$		−	40.2492i		−	1.46872i	−0.678763	−	0.734358i	$-0.737484\pi$
$751$		−	40.2492i		−	1.46872i	0.678763	−	0.734358i	$-0.262516\pi$
$752$	8.94427			0.326164
$753$		0			0
$754$	−40.0000	−	26.8328i	−1.45671	−	0.977194i
$755$		0			0
$756$		0			0
$757$	31.3050			1.13780			0.568899	−	0.822407i	$-0.307370\pi$
$757$	31.3050			1.13780			0.568899	+	0.822407i	$0.307370\pi$
$758$		−	10.0000i		−	0.363216i
$759$		0			0
$760$	20.0000	−	10.0000i	0.725476	−	0.362738i
$761$	30.0000			1.08750			0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000			1.08750			0.543750	+	0.839248i	$0.317004\pi$
$762$		0			0
$763$		−	20.0000i		−	0.724049i
$764$		−	40.2492i		−	1.45617i
$765$		0			0
$766$		−	13.4164i		−	0.484755i
$767$			16.0000i			0.577727i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	−20.0000	−	40.0000i	−0.720750	−	1.44150i
$771$		0			0
$772$	13.4164			0.482867
$773$	40.2492			1.44766			0.723832	−	0.689976i	$-0.242379\pi$
$773$	40.2492			1.44766			0.723832	+	0.689976i	$0.242379\pi$
$774$		0			0
$775$	17.8885	+	13.4164i	0.642575	+	0.481932i
$776$	−10.0000			−0.358979
$777$		0			0
$778$		−	20.0000i		−	0.717035i
$779$		0			0
$780$		0			0
$781$		0			0
$782$			60.0000i			2.14560i
$783$		0			0
$784$	−3.00000			−0.107143
$785$	13.4164	+	26.8328i	0.478852	+	0.957704i
$786$		0			0
$787$			22.0000i			0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$			22.0000i			0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$		−	36.0000i		−	1.28245i
$789$		0			0
$790$	60.0000	−	30.0000i	2.13470	−	1.06735i
$791$		−	8.94427i		−	0.318022i
$792$		0			0
$793$	35.7771			1.27048
$794$			26.8328i			0.952261i
$795$		0			0
$796$	48.0000			1.70131
$797$	−13.4164			−0.475234			−0.237617	−	0.971359i	$-0.576366\pi$
$797$	−13.4164			−0.475234			−0.237617	+	0.971359i	$0.576366\pi$
$798$		0			0
$799$	40.0000			1.41510
$800$	−20.1246	+	26.8328i	−0.711512	+	0.948683i
$801$		0			0
$802$	−67.0820			−2.36875
$803$			60.0000i			2.11735i
$804$		0			0
$805$	−12.0000	−	24.0000i	−0.422944	−	0.845889i
$806$	40.0000			1.40894
$807$		0			0
$808$			20.0000i			0.703598i
$809$			17.8885i			0.628928i	0.949269	+	0.314464i	$0.101825\pi$
$809$			17.8885i			0.628928i	−0.949269	+	0.314464i	$0.898175\pi$
$810$		0			0
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$	−26.8328	−	18.0000i	−0.941647	−	0.631676i
$813$		0			0
$814$		−	44.7214i		−	1.56748i
$815$	−17.8885	−	35.7771i	−0.626608	−	1.25322i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.0000			−0.767805			−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000			−0.767805			−0.383903	+	0.923374i	$0.625420\pi$
$822$		0			0
$823$	26.8328			0.935333			0.467667	−	0.883905i	$-0.345095\pi$
$823$	26.8328			0.935333			0.467667	+	0.883905i	$0.345095\pi$
$824$			13.4164i			0.467383i
$825$		0			0
$826$			17.8885i			0.622422i
$827$	53.6656			1.86614			0.933068	−	0.359699i	$-0.117121\pi$
$827$	53.6656			1.86614			0.933068	+	0.359699i	$0.117121\pi$
$828$		0			0
$829$			8.94427i			0.310647i	0.987864	+	0.155324i	$0.0496421\pi$
$829$			8.94427i			0.310647i	−0.987864	+	0.155324i	$0.950358\pi$
$830$	−26.8328	+	13.4164i	−0.931381	+	0.465690i
$831$		0			0
$832$			52.0000i			1.80278i
$833$	−13.4164			−0.464851
$834$		0			0
$835$	−44.0000	+	22.0000i	−1.52268	+	0.761341i
$836$	60.0000			2.07514
$837$		0			0
$838$	−8.94427			−0.308975
$839$		−	31.3050i		−	1.08077i	−0.841419	−	0.540383i	$-0.818279\pi$
$839$		−	31.3050i		−	1.08077i	0.841419	−	0.540383i	$-0.181721\pi$
$840$		0			0
$841$	−11.0000	−	26.8328i	−0.379310	−	0.925270i
$842$			20.0000i			0.689246i
$843$		0			0
$844$			67.0820i			2.30906i
$845$	−3.00000	−	6.00000i	−0.103203	−	0.206406i
$846$		0			0
$847$		−	18.0000i		−	0.618487i
$848$			4.00000i			0.137361i
$849$		0			0
$850$	−30.0000	+	40.0000i	−1.02899	+	1.37199i
$851$		−	26.8328i		−	0.919817i
$852$		0			0
$853$	−40.2492			−1.37811			−0.689054	−	0.724710i	$-0.741974\pi$
$853$	−40.2492			−1.37811			−0.689054	+	0.724710i	$0.741974\pi$
$854$	40.0000			1.36877
$855$		0			0
$856$		−	4.47214i		−	0.152854i
$857$			48.0000i			1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$857$			48.0000i			1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$858$		0			0
$859$			4.47214i			0.152587i	0.997085	+	0.0762937i	$0.0243086\pi$
$859$			4.47214i			0.152587i	−0.997085	+	0.0762937i	$0.975691\pi$
$860$		0			0
$861$		0			0
$862$	−71.5542			−2.43714
$863$			54.0000i			1.83818i	0.394046	+	0.919091i	$0.371075\pi$
$863$			54.0000i			1.83818i	−0.394046	+	0.919091i	$0.628925\pi$
$864$		0			0
$865$	−8.00000	+	4.00000i	−0.272008	+	0.136004i
$866$	−90.0000			−3.05832
$867$		0			0
$868$	26.8328			0.910765
$869$	60.0000			2.03536
$870$		0			0
$871$	−8.00000			−0.271070
$872$	22.3607			0.757228
$873$		0			0
$874$	60.0000			2.02953
$875$	4.00000	−	22.0000i	0.135225	−	0.743736i
$876$		0			0
$877$			12.0000i			0.405211i	0.979260	+	0.202606i	$0.0649409\pi$
$877$			12.0000i			0.405211i	−0.979260	+	0.202606i	$0.935059\pi$
$878$	35.7771			1.20742
$879$		0			0
$880$	−8.94427	+	4.47214i	−0.301511	+	0.150756i
$881$			17.8885i			0.602680i	0.953517	+	0.301340i	$0.0974340\pi$
$881$			17.8885i			0.602680i	−0.953517	+	0.301340i	$0.902566\pi$
$882$		0			0
$883$			54.0000i			1.81724i	0.417619	+	0.908622i	$0.362865\pi$
$883$			54.0000i			1.81724i	−0.417619	+	0.908622i	$0.637135\pi$
$884$			53.6656i			1.80497i
$885$		0			0
$886$	−40.0000			−1.34383
$887$	−44.7214			−1.50160			−0.750798	−	0.660532i	$-0.770331\pi$
$887$	−44.7214			−1.50160			−0.750798	+	0.660532i	$0.770331\pi$
$888$		0			0
$889$		−	17.8885i		−	0.599963i
$890$	80.0000	−	40.0000i	2.68161	−	1.34080i
$891$		0			0
$892$		−	42.0000i		−	1.40626i
$893$		−	40.0000i		−	1.33855i
$894$		0			0
$895$	−20.0000	−	40.0000i	−0.668526	−	1.33705i
$896$			31.3050i			1.04583i
$897$		0			0
$898$		−	40.0000i		−	1.33482i
$899$	20.0000	+	13.4164i	0.667037	+	0.447462i
$900$		0			0
$901$			17.8885i			0.595954i
$902$		0			0
$903$		0			0
$904$	10.0000			0.332595
$905$	−10.0000	−	20.0000i	−0.332411	−	0.664822i
$906$		0			0
$907$	−17.8885			−0.593979			−0.296990	−	0.954881i	$-0.595983\pi$
$907$	−17.8885			−0.593979			−0.296990	+	0.954881i	$0.595983\pi$
$908$			54.0000i			1.79205i
$909$		0			0
$910$	−17.8885	−	35.7771i	−0.592999	−	1.18600i
$911$			4.47214i			0.148168i	0.997252	+	0.0740842i	$0.0236034\pi$
$911$			4.47214i			0.148168i	−0.997252	+	0.0740842i	$0.976397\pi$
$912$		0			0
$913$	−26.8328			−0.888037
$914$		−	71.5542i		−	2.36680i
$915$		0			0
$916$		−	80.4984i		−	2.65974i
$917$	8.94427			0.295366
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$	26.8328	−	13.4164i	0.884652	−	0.442326i
$921$		0			0
$922$			60.0000i			1.97599i
$923$		0			0
$924$		0			0
$925$	13.4164	−	17.8885i	0.441129	−	0.588172i
$926$		−	76.0263i		−	2.49838i
$927$		0			0
$928$	−20.1246	+	30.0000i	−0.660623	+	0.984798i
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$			13.4164i			0.439705i
$932$		−	72.0000i		−	2.35844i
$933$		0			0
$934$	40.0000			1.30884
$935$	−40.0000	+	20.0000i	−1.30814	+	0.654070i
$936$		0			0
$937$		−	32.0000i		−	1.04539i	−0.852518	−	0.522697i	$-0.824926\pi$
$937$		−	32.0000i		−	1.04539i	0.852518	−	0.522697i	$-0.175074\pi$
$938$	−8.94427			−0.292041
$939$		0			0
$940$	−26.8328	−	53.6656i	−0.875190	−	1.75038i
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$		0			0
$943$		0			0
$944$	4.00000			0.130189
$945$		0			0
$946$		0			0
$947$	−17.8885			−0.581300			−0.290650	−	0.956830i	$-0.593871\pi$
$947$	−17.8885			−0.581300			−0.290650	+	0.956830i	$0.593871\pi$
$948$		0			0
$949$			53.6656i			1.74206i
$950$	40.0000	+	30.0000i	1.29777	+	0.973329i
$951$		0			0
$952$			20.0000i			0.648204i
$953$		−	24.0000i		−	0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$		−	24.0000i		−	0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$		0			0
$955$	26.8328	−	13.4164i	0.868290	−	0.434145i
$956$	−72.0000			−2.32865
$957$		0			0
$958$			30.0000i			0.969256i
$959$			26.8328i			0.866477i
$960$		0			0
$961$	11.0000			0.354839
$962$		−	40.0000i		−	1.28965i
$963$		0			0
$964$	30.0000			0.966235
$965$	4.47214	+	8.94427i	0.143963	+	0.287926i
$966$		0			0
$967$	44.7214			1.43814			0.719071	−	0.694937i	$-0.244568\pi$
$967$	44.7214			1.43814			0.719071	+	0.694937i	$0.244568\pi$
$968$	20.1246			0.646830
$969$		0			0
$970$	−10.0000	−	20.0000i	−0.321081	−	0.642161i
$971$		−	22.3607i		−	0.717588i	−0.933417	−	0.358794i	$-0.883188\pi$
$971$		−	22.3607i		−	0.717588i	0.933417	−	0.358794i	$-0.116812\pi$
$972$		0			0
$973$		−	40.0000i		−	1.28234i
$974$		−	4.47214i		−	0.143296i
$975$		0			0
$976$		−	8.94427i		−	0.286299i
$977$		−	8.00000i		−	0.255943i	−0.991778	−	0.127971i	$-0.959153\pi$
$977$		−	8.00000i		−	0.255943i	0.991778	−	0.127971i	$-0.0408466\pi$
$978$		0			0
$979$	80.0000			2.55681
$980$	9.00000	+	18.0000i	0.287494	+	0.574989i
$981$		0			0
$982$		−	30.0000i		−	0.957338i
$983$	−26.8328			−0.855834			−0.427917	−	0.903818i	$-0.640752\pi$
$983$	−26.8328			−0.855834			−0.427917	+	0.903818i	$0.640752\pi$
$984$		0			0
$985$	24.0000	−	12.0000i	0.764704	−	0.382352i
$986$	−30.0000	+	44.7214i	−0.955395	+	1.42422i
$987$		0			0
$988$	53.6656			1.70733
$989$		0			0
$990$		0			0
$991$	−40.0000			−1.27064			−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000			−1.27064			−0.635321	+	0.772248i	$0.719132\pi$
$992$		−	30.0000i		−	0.952501i
$993$		0			0
$994$		0			0
$995$	16.0000	+	32.0000i	0.507234	+	1.01447i
$996$		0			0
$997$	−40.2492			−1.27471			−0.637353	−	0.770572i	$-0.719971\pi$
$997$	−40.2492			−1.27471			−0.637353	+	0.770572i	$0.719971\pi$
$998$	−44.7214			−1.41563
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.f.h.289.2		4
3.2	odd	2		145.2.d.b.144.3	yes	4
5.4	even	2	inner	1305.2.f.h.289.3		4
12.11	even	2		2320.2.j.b.289.1		4
15.2	even	4		725.2.c.a.376.2		2
15.8	even	4		725.2.c.b.376.1		2
15.14	odd	2		145.2.d.b.144.2	yes	4
29.28	even	2	inner	1305.2.f.h.289.4		4
60.59	even	2		2320.2.j.b.289.3		4
87.86	odd	2		145.2.d.b.144.1	✓	4
145.144	even	2	inner	1305.2.f.h.289.1		4
348.347	even	2		2320.2.j.b.289.2		4
435.173	even	4		725.2.c.b.376.2		2
435.347	even	4		725.2.c.a.376.1		2
435.434	odd	2		145.2.d.b.144.4	yes	4
1740.1739	even	2		2320.2.j.b.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.d.b.144.1	✓	4	87.86	odd	2
145.2.d.b.144.2	yes	4	15.14	odd	2
145.2.d.b.144.3	yes	4	3.2	odd	2
145.2.d.b.144.4	yes	4	435.434	odd	2
725.2.c.a.376.1		2	435.347	even	4
725.2.c.a.376.2		2	15.2	even	4
725.2.c.b.376.1		2	15.8	even	4
725.2.c.b.376.2		2	435.173	even	4
1305.2.f.h.289.1		4	145.144	even	2	inner
1305.2.f.h.289.2		4	1.1	even	1	trivial
1305.2.f.h.289.3		4	5.4	even	2	inner
1305.2.f.h.289.4		4	29.28	even	2	inner
2320.2.j.b.289.1		4	12.11	even	2
2320.2.j.b.289.2		4	348.347	even	2
2320.2.j.b.289.3		4	60.59	even	2
2320.2.j.b.289.4		4	1740.1739	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 $\F_{q}$

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	1305.2.f.h.289.2

Level	$1305$
Weight	$2$
Character	1305.289
Analytic conductor	$10.420$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$