LMFDB - Embedded newform 1152.2.k.f.289.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(289,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("1152.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.k (of order $4$, degree $2$, not 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:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			289.3
Root			$0.500000 + 0.691860i$ of defining polynomial
Character	$\chi$	$=$	1152.289
Dual form			1152.2.k.f.865.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.27133 - 1.27133i) q^{5} -0.158942i q^{7} +O(q^{10})$	$q+(1.27133 - 1.27133i) q^{5} -0.158942i q^{7} +(3.79793 - 3.79793i) q^{11} +(4.21215 + 4.21215i) q^{13} -3.05320 q^{17} +(-2.15894 - 2.15894i) q^{19} -2.82843i q^{23} +1.76744i q^{25} +(2.09976 + 2.09976i) q^{29} -4.15894 q^{31} +(-0.202067 - 0.202067i) q^{35} +(5.98737 - 5.98737i) q^{37} -2.60365i q^{41} +(5.75481 - 5.75481i) q^{43} -2.82843 q^{47} +6.97474 q^{49} +(3.55710 - 3.55710i) q^{53} -9.65685i q^{55} +(-4.00000 + 4.00000i) q^{59} +(-3.66949 - 3.66949i) q^{61} +10.7101 q^{65} +(0.767438 + 0.767438i) q^{67} -0.317883i q^{71} +1.33897i q^{73} +(-0.603650 - 0.603650i) q^{77} +9.69382 q^{79} +(-0.115816 - 0.115816i) q^{83} +(-3.88163 + 3.88163i) q^{85} +14.3990i q^{89} +(0.669485 - 0.669485i) q^{91} -5.48946 q^{95} -0.571533 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{11} - 8 q^{19} - 16 q^{29} - 24 q^{31} - 24 q^{35} + 16 q^{37} - 8 q^{43} - 8 q^{49} + 16 q^{53} - 32 q^{59} - 16 q^{61} + 16 q^{65} - 16 q^{67} + 16 q^{77} + 24 q^{79} + 40 q^{83} + 16 q^{85} - 8 q^{91} - 48 q^{95}+O(q^{100}) $ 8 * q + 8 * q^11 - 8 * q^19 - 16 * q^29 - 24 * q^31 - 24 * q^35 + 16 * q^37 - 8 * q^43 - 8 * q^49 + 16 * q^53 - 32 * q^59 - 16 * q^61 + 16 * q^65 - 16 * q^67 + 16 * q^77 + 24 * q^79 + 40 * q^83 + 16 * q^85 - 8 * q^91 - 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.27133	−	1.27133i	0.568556	−	0.568556i	−0.363168	−	0.931724i	$-0.618305\pi$
$5$	1.27133	−	1.27133i	0.568556	−	0.568556i	0.931724	+	0.363168i	$0.118305\pi$
$6$		0			0
$7$		−	0.158942i		−	0.0600743i	−0.999549	−	0.0300371i	$-0.990437\pi$
$7$		−	0.158942i		−	0.0600743i	0.999549	−	0.0300371i	$-0.00956256\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.79793	−	3.79793i	1.14512	−	1.14512i	0.157620	−	0.987500i	$-0.449618\pi$
$11$	3.79793	−	3.79793i	1.14512	−	1.14512i	0.987500	−	0.157620i	$-0.0503821\pi$
$12$		0			0
$13$	4.21215	+	4.21215i	1.16824	+	1.16824i	0.982622	+	0.185617i	$0.0594284\pi$
$13$	4.21215	+	4.21215i	1.16824	+	1.16824i	0.185617	+	0.982622i	$0.440572\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.05320			−0.740511			−0.370255	−	0.928930i	$-0.620730\pi$
$17$	−3.05320			−0.740511			−0.370255	+	0.928930i	$0.620730\pi$
$18$		0			0
$19$	−2.15894	−	2.15894i	−0.495295	−	0.495295i	0.414675	−	0.909970i	$-0.363895\pi$
$19$	−2.15894	−	2.15894i	−0.495295	−	0.495295i	−0.909970	+	0.414675i	$0.863895\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$			1.76744i			0.353488i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.09976	+	2.09976i	0.389915	+	0.389915i	0.874657	−	0.484742i	$-0.161087\pi$
$29$	2.09976	+	2.09976i	0.389915	+	0.389915i	−0.484742	+	0.874657i	$0.661087\pi$
$30$		0			0
$31$	−4.15894			−0.746968			−0.373484	−	0.927637i	$-0.621837\pi$
$31$	−4.15894			−0.746968			−0.373484	+	0.927637i	$0.621837\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.202067	−	0.202067i	−0.0341556	−	0.0341556i
$36$		0			0
$37$	5.98737	−	5.98737i	0.984317	−	0.984317i	−0.0155615	−	0.999879i	$-0.504954\pi$
$37$	5.98737	−	5.98737i	0.984317	−	0.984317i	0.999879	+	0.0155615i	$0.00495359\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	2.60365i		−	0.406622i	−0.979114	−	0.203311i	$-0.934830\pi$
$41$		−	2.60365i		−	0.406622i	0.979114	−	0.203311i	$-0.0651702\pi$
$42$		0			0
$43$	5.75481	−	5.75481i	0.877600	−	0.877600i	−0.115686	−	0.993286i	$-0.536907\pi$
$43$	5.75481	−	5.75481i	0.877600	−	0.877600i	0.993286	+	0.115686i	$0.0369066\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.82843			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	6.97474			0.996391
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.55710	−	3.55710i	0.488605	−	0.488605i	−0.419261	−	0.907866i	$-0.637711\pi$
$53$	3.55710	−	3.55710i	0.488605	−	0.488605i	0.907866	+	0.419261i	$0.137711\pi$
$54$		0			0
$55$		−	9.65685i		−	1.30213i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000	+	4.00000i	−0.520756	+	0.520756i	−0.917800	−	0.397044i	$-0.870036\pi$
$59$	−4.00000	+	4.00000i	−0.520756	+	0.520756i	0.397044	+	0.917800i	$0.370036\pi$
$60$		0			0
$61$	−3.66949	−	3.66949i	−0.469829	−	0.469829i	0.432030	−	0.901859i	$-0.357798\pi$
$61$	−3.66949	−	3.66949i	−0.469829	−	0.469829i	−0.901859	+	0.432030i	$0.857798\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.7101			1.32842
$66$		0			0
$67$	0.767438	+	0.767438i	0.0937575	+	0.0937575i	0.752430	−	0.658672i	$-0.228882\pi$
$67$	0.767438	+	0.767438i	0.0937575	+	0.0937575i	−0.658672	+	0.752430i	$0.728882\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.317883i		−	0.0377258i	−0.999822	−	0.0188629i	$-0.993995\pi$
$71$		−	0.317883i		−	0.0377258i	0.999822	−	0.0188629i	$-0.00600460\pi$
$72$		0			0
$73$			1.33897i			0.156715i	0.996925	+	0.0783573i	$0.0249675\pi$
$73$			1.33897i			0.156715i	−0.996925	+	0.0783573i	$0.975032\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.603650	−	0.603650i	−0.0687923	−	0.0687923i
$78$		0			0
$79$	9.69382			1.09064			0.545320	−	0.838228i	$-0.316408\pi$
$79$	9.69382			1.09064			0.545320	+	0.838228i	$0.316408\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.115816	−	0.115816i	−0.0127125	−	0.0127125i	0.700722	−	0.713434i	$-0.252861\pi$
$83$	−0.115816	−	0.115816i	−0.0127125	−	0.0127125i	−0.713434	+	0.700722i	$0.752861\pi$
$84$		0			0
$85$	−3.88163	+	3.88163i	−0.421022	+	0.421022i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			14.3990i			1.52629i	0.646225	+	0.763147i	$0.276347\pi$
$89$			14.3990i			1.52629i	−0.646225	+	0.763147i	$0.723653\pi$
$90$		0			0
$91$	0.669485	−	0.669485i	0.0701811	−	0.0701811i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.48946			−0.563206
$96$		0			0
$97$	−0.571533			−0.0580304			−0.0290152	−	0.999579i	$-0.509237\pi$
$97$	−0.571533			−0.0580304			−0.0290152	+	0.999579i	$0.509237\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.15296	+	7.15296i	−0.711746	+	0.711746i	−0.966900	−	0.255154i	$-0.917874\pi$
$101$	−7.15296	+	7.15296i	−0.711746	+	0.711746i	0.255154	+	0.966900i	$0.417874\pi$
$102$		0			0
$103$		−	11.3507i		−	1.11841i	−0.829028	−	0.559207i	$-0.811106\pi$
$103$		−	11.3507i		−	1.11841i	0.829028	−	0.559207i	$-0.188894\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.722018	−	0.722018i	0.0698001	−	0.0698001i	−0.671345	−	0.741145i	$-0.734283\pi$
$107$	0.722018	−	0.722018i	0.0698001	−	0.0698001i	0.741145	+	0.671345i	$0.234283\pi$
$108$		0			0
$109$	1.44471	+	1.44471i	0.138378	+	0.138378i	0.772903	−	0.634525i	$-0.218804\pi$
$109$	1.44471	+	1.44471i	0.138378	+	0.138378i	−0.634525	+	0.772903i	$0.718804\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.53488			0.332533			0.166267	−	0.986081i	$-0.446829\pi$
$113$	3.53488			0.332533			0.166267	+	0.986081i	$0.446829\pi$
$114$		0			0
$115$	−3.59587	−	3.59587i	−0.335316	−	0.335316i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.485281i			0.0444857i
$120$		0			0
$121$		−	17.8486i		−	1.62260i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.60365	+	8.60365i	0.769534	+	0.769534i
$126$		0			0
$127$	1.49791			0.132918			0.0664591	−	0.997789i	$-0.478830\pi$
$127$	1.49791			0.132918			0.0664591	+	0.997789i	$0.478830\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.4243	−	10.4243i	−0.910775	−	0.910775i	0.0855585	−	0.996333i	$-0.472733\pi$
$131$	−10.4243	−	10.4243i	−0.910775	−	0.910775i	−0.996333	+	0.0855585i	$0.972733\pi$
$132$		0			0
$133$	−0.343146	+	0.343146i	−0.0297545	+	0.0297545i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	13.7954i		−	1.17862i	−0.807907	−	0.589309i	$-0.799400\pi$
$137$		−	13.7954i		−	1.17862i	0.807907	−	0.589309i	$-0.200600\pi$
$138$		0			0
$139$	−2.42429	+	2.42429i	−0.205626	+	0.205626i	−0.802405	−	0.596779i	$-0.796447\pi$
$139$	−2.42429	+	2.42429i	−0.205626	+	0.205626i	0.596779	+	0.802405i	$0.296447\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	31.9949			2.67555
$144$		0			0
$145$	5.33897			0.443377
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.92818	−	2.92818i	0.239886	−	0.239886i	−0.576917	−	0.816803i	$-0.695744\pi$
$149$	2.92818	−	2.92818i	0.239886	−	0.239886i	0.816803	+	0.576917i	$0.195744\pi$
$150$		0			0
$151$			22.6644i			1.84440i	0.386712	+	0.922201i	$0.373611\pi$
$151$			22.6644i			1.84440i	−0.386712	+	0.922201i	$0.626389\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.28739	+	5.28739i	−0.424693	+	0.424693i
$156$		0			0
$157$	2.78007	+	2.78007i	0.221874	+	0.221874i	0.809287	−	0.587413i	$-0.199854\pi$
$157$	2.78007	+	2.78007i	0.221874	+	0.221874i	−0.587413	+	0.809287i	$0.699854\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.449555			−0.0354299
$162$		0			0
$163$	−5.43692	−	5.43692i	−0.425853	−	0.425853i	0.461360	−	0.887213i	$-0.347362\pi$
$163$	−5.43692	−	5.43692i	−0.425853	−	0.425853i	−0.887213	+	0.461360i	$0.847362\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3.95458i			0.306015i	0.988225	+	0.153007i	$0.0488958\pi$
$167$			3.95458i			0.306015i	−0.988225	+	0.153007i	$0.951104\pi$
$168$		0			0
$169$			22.4844i			1.72957i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−15.9814	−	15.9814i	−1.21504	−	1.21504i	−0.969347	−	0.245695i	$-0.920984\pi$
$173$	−15.9814	−	15.9814i	−1.21504	−	1.21504i	−0.245695	−	0.969347i	$-0.579016\pi$
$174$		0			0
$175$	0.280920			0.0212355
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.2316	+	12.2316i	0.914235	+	0.914235i	0.996602	−	0.0823670i	$-0.0262480\pi$
$179$	12.2316	+	12.2316i	0.914235	+	0.914235i	−0.0823670	+	0.996602i	$0.526248\pi$
$180$		0			0
$181$	−5.76259	+	5.76259i	−0.428330	+	0.428330i	−0.888059	−	0.459729i	$-0.847946\pi$
$181$	−5.76259	+	5.76259i	−0.428330	+	0.428330i	0.459729	+	0.888059i	$0.347946\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	15.2238i		−	1.11928i
$186$		0			0
$187$	−11.5959	+	11.5959i	−0.847974	+	0.847974i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.1674			−1.16983			−0.584916	−	0.811094i	$-0.698873\pi$
$191$	−16.1674			−1.16983			−0.584916	+	0.811094i	$0.698873\pi$
$192$		0			0
$193$	−22.1454			−1.59406			−0.797030	−	0.603940i	$-0.793597\pi$
$193$	−22.1454			−1.59406			−0.797030	+	0.603940i	$0.793597\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.2993	+	14.2993i	−1.01878	+	1.01878i	−0.0189608	+	0.999820i	$0.506036\pi$
$197$	−14.2993	+	14.2993i	−1.01878	+	1.01878i	−0.999820	+	0.0189608i	$0.993964\pi$
$198$		0			0
$199$			25.0075i			1.77274i	0.462981	+	0.886368i	$0.346780\pi$
$199$			25.0075i			1.77274i	−0.462981	+	0.886368i	$0.653220\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.333739	−	0.333739i	0.0234239	−	0.0234239i
$204$		0			0
$205$	−3.31010	−	3.31010i	−0.231187	−	0.231187i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.3990			−1.13434
$210$		0			0
$211$	18.4243	+	18.4243i	1.26838	+	1.26838i	0.946924	+	0.321456i	$0.104172\pi$
$211$	18.4243	+	18.4243i	1.26838	+	1.26838i	0.321456	+	0.946924i	$0.395828\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	14.6325i		−	0.997930i
$216$		0			0
$217$			0.661029i			0.0448736i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.8605	−	12.8605i	−0.865094	−	0.865094i
$222$		0			0
$223$	−18.3465			−1.22857			−0.614286	−	0.789083i	$-0.710556\pi$
$223$	−18.3465			−1.22857			−0.614286	+	0.789083i	$0.710556\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.115816	+	0.115816i	0.00768697	+	0.00768697i	0.710940	−	0.703253i	$-0.248270\pi$
$227$	0.115816	+	0.115816i	0.00768697	+	0.00768697i	−0.703253	+	0.710940i	$0.748270\pi$
$228$		0			0
$229$	−2.84791	+	2.84791i	−0.188195	+	0.188195i	−0.794916	−	0.606720i	$-0.792485\pi$
$229$	−2.84791	+	2.84791i	−0.188195	+	0.188195i	0.606720	+	0.794916i	$0.292485\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			11.7211i			0.767874i	0.923359	+	0.383937i	$0.125432\pi$
$233$			11.7211i			0.767874i	−0.923359	+	0.383937i	$0.874568\pi$
$234$		0			0
$235$	−3.59587	+	3.59587i	−0.234568	+	0.234568i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.6517			−0.883058			−0.441529	−	0.897247i	$-0.645564\pi$
$239$	−13.6517			−0.883058			−0.441529	+	0.897247i	$0.645564\pi$
$240$		0			0
$241$	2.13167			0.137313			0.0686565	−	0.997640i	$-0.478129\pi$
$241$	2.13167			0.137313			0.0686565	+	0.997640i	$0.478129\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.86720	−	8.86720i	0.566504	−	0.566504i
$246$		0			0
$247$		−	18.1876i		−	1.15725i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.43370	−	4.43370i	0.279853	−	0.279853i	−0.553198	−	0.833050i	$-0.686593\pi$
$251$	4.43370	−	4.43370i	0.279853	−	0.279853i	0.833050	+	0.553198i	$0.186593\pi$
$252$		0			0
$253$	−10.7422	−	10.7422i	−0.675355	−	0.675355i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.0853			−0.940997			−0.470498	−	0.882401i	$-0.655926\pi$
$257$	−15.0853			−0.940997			−0.470498	+	0.882401i	$0.655926\pi$
$258$		0			0
$259$	−0.951642	−	0.951642i	−0.0591322	−	0.0591322i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			26.1706i			1.61375i	0.590722	+	0.806875i	$0.298843\pi$
$263$			26.1706i			1.61375i	−0.590722	+	0.806875i	$0.701157\pi$
$264$		0			0
$265$		−	9.04449i		−	0.555599i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.59700	+	8.59700i	0.524168	+	0.524168i	0.918828	−	0.394659i	$-0.129137\pi$
$269$	8.59700	+	8.59700i	0.524168	+	0.524168i	−0.394659	+	0.918828i	$0.629137\pi$
$270$		0			0
$271$	−10.6644			−0.647815			−0.323907	−	0.946089i	$-0.604997\pi$
$271$	−10.6644			−0.647815			−0.323907	+	0.946089i	$0.604997\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.71261	+	6.71261i	0.404786	+	0.404786i
$276$		0			0
$277$	2.66170	−	2.66170i	0.159926	−	0.159926i	−0.622608	−	0.782534i	$-0.713927\pi$
$277$	2.66170	−	2.66170i	0.159926	−	0.159926i	0.782534	+	0.622608i	$0.213927\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			10.4496i			0.623368i	0.950186	+	0.311684i	$0.100893\pi$
$281$			10.4496i			0.623368i	−0.950186	+	0.311684i	$0.899107\pi$
$282$		0			0
$283$	−12.4853	+	12.4853i	−0.742173	+	0.742173i	−0.972996	−	0.230823i	$-0.925858\pi$
$283$	−12.4853	+	12.4853i	−0.742173	+	0.742173i	0.230823	+	0.972996i	$0.425858\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.413828			−0.0244275
$288$		0			0
$289$	−7.67794			−0.451644
$290$		0			0
$291$		0			0
$292$		0			0
$293$	21.7410	−	21.7410i	1.27013	−	1.27013i	0.324104	−	0.946022i	$-0.394937\pi$
$293$	21.7410	−	21.7410i	1.27013	−	1.27013i	0.946022	−	0.324104i	$-0.105063\pi$
$294$		0			0
$295$			10.1706i			0.592158i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.9137	−	11.9137i	0.688990	−	0.688990i
$300$		0			0
$301$	−0.914679	−	0.914679i	−0.0527212	−	0.0527212i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.33026			−0.534249
$306$		0			0
$307$	15.0601	+	15.0601i	0.859523	+	0.859523i	0.991282	−	0.131759i	$-0.0420624\pi$
$307$	15.0601	+	15.0601i	0.859523	+	0.859523i	−0.131759	+	0.991282i	$0.542062\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.77883i			0.100868i	0.998727	+	0.0504342i	$0.0160605\pi$
$311$			1.77883i			0.100868i	−0.998727	+	0.0504342i	$0.983939\pi$
$312$		0			0
$313$			2.70320i			0.152794i	0.997077	+	0.0763971i	$0.0243417\pi$
$313$			2.70320i			0.152794i	−0.997077	+	0.0763971i	$0.975658\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.6025	+	15.6025i	0.876325	+	0.876325i	0.993152	−	0.116828i	$-0.0372725\pi$
$317$	15.6025	+	15.6025i	0.876325	+	0.876325i	−0.116828	+	0.993152i	$0.537272\pi$
$318$		0			0
$319$	15.9495			0.892999
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.59169	+	6.59169i	0.366771	+	0.366771i
$324$		0			0
$325$	−7.44471	+	7.44471i	−0.412958	+	0.412958i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			0.449555i			0.0247848i
$330$		0			0
$331$	15.4454	−	15.4454i	0.848955	−	0.848955i	−0.141048	−	0.990003i	$-0.545047\pi$
$331$	15.4454	−	15.4454i	0.848955	−	0.848955i	0.990003	+	0.141048i	$0.0450472\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.95133			0.106613
$336$		0			0
$337$	−18.8738			−1.02812			−0.514062	−	0.857753i	$-0.671860\pi$
$337$	−18.8738			−1.02812			−0.514062	+	0.857753i	$0.671860\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−15.7954	+	15.7954i	−0.855368	+	0.855368i
$342$		0			0
$343$		−	2.22117i		−	0.119932i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.8337	+	19.8337i	−1.06473	+	1.06473i	−0.0669717	+	0.997755i	$0.521334\pi$
$347$	−19.8337	+	19.8337i	−1.06473	+	1.06473i	−0.997755	+	0.0669717i	$0.978666\pi$
$348$		0			0
$349$	−11.9718	−	11.9718i	−0.640836	−	0.640836i	0.309925	−	0.950761i	$-0.399696\pi$
$349$	−11.9718	−	11.9718i	−0.640836	−	0.640836i	−0.950761	+	0.309925i	$0.899696\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.6202			0.671705			0.335853	−	0.941915i	$-0.390976\pi$
$353$	12.6202			0.671705			0.335853	+	0.941915i	$0.390976\pi$
$354$		0			0
$355$	−0.404135	−	0.404135i	−0.0214492	−	0.0214492i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			27.0867i			1.42958i	0.699339	+	0.714790i	$0.253478\pi$
$359$			27.0867i			1.42958i	−0.699339	+	0.714790i	$0.746522\pi$
$360$		0			0
$361$		−	9.67794i		−	0.509365i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.70227	+	1.70227i	0.0891011	+	0.0891011i
$366$		0			0
$367$	20.4937			1.06976			0.534882	−	0.844927i	$-0.320356\pi$
$367$	20.4937			1.06976			0.534882	+	0.844927i	$0.320356\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.565371	−	0.565371i	−0.0293526	−	0.0293526i
$372$		0			0
$373$	−1.03372	+	1.03372i	−0.0535239	+	0.0535239i	−0.733362	−	0.679838i	$-0.762050\pi$
$373$	−1.03372	+	1.03372i	−0.0535239	+	0.0535239i	0.679838	+	0.733362i	$0.262050\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			17.6890i			0.911028i
$378$		0			0
$379$	17.6686	−	17.6686i	0.907573	−	0.907573i	−0.0885032	−	0.996076i	$-0.528208\pi$
$379$	17.6686	−	17.6686i	0.907573	−	0.907573i	0.996076	+	0.0885032i	$0.0282083\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−31.0958			−1.58892			−0.794460	−	0.607316i	$-0.792246\pi$
$383$	−31.0958			−1.58892			−0.794460	+	0.607316i	$0.792246\pi$
$384$		0			0
$385$	−1.53488			−0.0782245
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.56127	+	2.56127i	−0.129862	+	0.129862i	−0.769050	−	0.639188i	$-0.779270\pi$
$389$	−2.56127	+	2.56127i	−0.129862	+	0.129862i	0.639188	+	0.769050i	$0.279270\pi$
$390$		0			0
$391$			8.63577i			0.436729i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.3240	−	12.3240i	0.620090	−	0.620090i
$396$		0			0
$397$	5.09795	+	5.09795i	0.255859	+	0.255859i	0.823367	−	0.567509i	$-0.192093\pi$
$397$	5.09795	+	5.09795i	0.255859	+	0.255859i	−0.567509	+	0.823367i	$0.692093\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.2660			0.762349			0.381174	−	0.924503i	$-0.375520\pi$
$401$	15.2660			0.762349			0.381174	+	0.924503i	$0.375520\pi$
$402$		0			0
$403$	−17.5181	−	17.5181i	−0.872637	−	0.872637i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	45.4792i		−	2.25432i
$408$		0			0
$409$			11.3779i			0.562603i	0.959619	+	0.281302i	$0.0907661\pi$
$409$			11.3779i			0.562603i	−0.959619	+	0.281302i	$0.909234\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.635767	+	0.635767i	0.0312840	+	0.0312840i
$414$		0			0
$415$	−0.294481			−0.0144555
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−23.3075	−	23.3075i	−1.13865	−	1.13865i	−0.988693	−	0.149955i	$-0.952087\pi$
$419$	−23.3075	−	23.3075i	−1.13865	−	1.13865i	−0.149955	−	0.988693i	$-0.547913\pi$
$420$		0			0
$421$	17.6154	−	17.6154i	0.858520	−	0.858520i	−0.132644	−	0.991164i	$-0.542347\pi$
$421$	17.6154	−	17.6154i	0.858520	−	0.858520i	0.991164	+	0.132644i	$0.0423467\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	5.39635i		−	0.261761i
$426$		0			0
$427$	−0.583234	+	0.583234i	−0.0282247	+	0.0282247i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	10.3211			0.497151			0.248576	−	0.968612i	$-0.420038\pi$
$431$	10.3211			0.497151			0.248576	+	0.968612i	$0.420038\pi$
$432$		0			0
$433$	−15.3137			−0.735930			−0.367965	−	0.929840i	$-0.619945\pi$
$433$	−15.3137			−0.735930			−0.367965	+	0.929840i	$0.619945\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.10641	+	6.10641i	−0.292109	+	0.292109i
$438$		0			0
$439$		−	22.5735i		−	1.07738i	−0.842505	−	0.538688i	$-0.818920\pi$
$439$		−	22.5735i		−	1.07738i	0.842505	−	0.538688i	$-0.181080\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−23.7117	+	23.7117i	−1.12658	+	1.12658i	−0.135846	+	0.990730i	$0.543375\pi$
$443$	−23.7117	+	23.7117i	−1.12658	+	1.12658i	−0.990730	+	0.135846i	$0.956625\pi$
$444$		0			0
$445$	18.3059	+	18.3059i	0.867784	+	0.867784i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.75506			0.0828266			0.0414133	−	0.999142i	$-0.486814\pi$
$449$	1.75506			0.0828266			0.0414133	+	0.999142i	$0.486814\pi$
$450$		0			0
$451$	−9.88849	−	9.88849i	−0.465631	−	0.465631i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1.70227i		−	0.0798039i
$456$		0			0
$457$		−	26.7422i		−	1.25095i	−0.780246	−	0.625473i	$-0.784906\pi$
$457$		−	26.7422i		−	1.25095i	0.780246	−	0.625473i	$-0.215094\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.23921	+	9.23921i	0.430313	+	0.430313i	0.888735	−	0.458422i	$-0.151585\pi$
$461$	9.23921	+	9.23921i	0.430313	+	0.430313i	−0.458422	+	0.888735i	$0.651585\pi$
$462$		0			0
$463$	−29.4474			−1.36854			−0.684268	−	0.729231i	$-0.739878\pi$
$463$	−29.4474			−1.36854			−0.684268	+	0.729231i	$0.739878\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.5897	+	19.5897i	0.906503	+	0.906503i	0.995988	−	0.0894848i	$-0.0285221\pi$
$467$	19.5897	+	19.5897i	0.906503	+	0.906503i	−0.0894848	+	0.995988i	$0.528522\pi$
$468$		0			0
$469$	0.121978	−	0.121978i	0.00563242	−	0.00563242i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	43.7127i		−	2.00991i
$474$		0			0
$475$	3.81580	−	3.81580i	0.175081	−	0.175081i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	35.5499			1.62432			0.812159	−	0.583436i	$-0.198292\pi$
$479$	35.5499			1.62432			0.812159	+	0.583436i	$0.198292\pi$
$480$		0			0
$481$	50.4393			2.29984
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.726607	+	0.726607i	−0.0329935	+	0.0329935i
$486$		0			0
$487$			9.86632i			0.447086i	0.974694	+	0.223543i	$0.0717623\pi$
$487$			9.86632i			0.447086i	−0.974694	+	0.223543i	$0.928238\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.449555	−	0.449555i	0.0202881	−	0.0202881i	−0.696890	−	0.717178i	$-0.745433\pi$
$491$	0.449555	−	0.449555i	0.0202881	−	0.0202881i	0.717178	+	0.696890i	$0.245433\pi$
$492$		0			0
$493$	−6.41099	−	6.41099i	−0.288736	−	0.288736i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.0505249			−0.00226635
$498$		0			0
$499$	2.70645	+	2.70645i	0.121157	+	0.121157i	0.765086	−	0.643928i	$-0.222697\pi$
$499$	2.70645	+	2.70645i	0.121157	+	0.121157i	−0.643928	+	0.765086i	$0.722697\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			23.6719i			1.05548i	0.849407	+	0.527739i	$0.176960\pi$
$503$			23.6719i			1.05548i	−0.849407	+	0.527739i	$0.823040\pi$
$504$		0			0
$505$			18.1876i			0.809336i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−24.6052	−	24.6052i	−1.09061	−	1.09061i	−0.995464	−	0.0951425i	$-0.969669\pi$
$509$	−24.6052	−	24.6052i	−1.09061	−	1.09061i	−0.0951425	−	0.995464i	$-0.530331\pi$
$510$		0			0
$511$	0.212818			0.00941453
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−14.4305	−	14.4305i	−0.635882	−	0.635882i
$516$		0			0
$517$	−10.7422	+	10.7422i	−0.472440	+	0.472440i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			14.4889i			0.634770i	0.948297	+	0.317385i	$0.102805\pi$
$521$			14.4889i			0.634770i	−0.948297	+	0.317385i	$0.897195\pi$
$522$		0			0
$523$	−19.4979	+	19.4979i	−0.852584	+	0.852584i	−0.990451	−	0.137867i	$-0.955975\pi$
$523$	−19.4979	+	19.4979i	−0.852584	+	0.852584i	0.137867	+	0.990451i	$0.455975\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.6981			0.553138
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.9670	−	10.9670i	0.475031	−	0.475031i
$534$		0			0
$535$		−	1.83585i		−	0.0793706i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	26.4896	−	26.4896i	1.14099	−	1.14099i
$540$		0			0
$541$	10.0396	+	10.0396i	0.431638	+	0.431638i	0.889185	−	0.457547i	$-0.151272\pi$
$541$	10.0396	+	10.0396i	0.431638	+	0.431638i	−0.457547	+	0.889185i	$0.651272\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.67340			0.157351
$546$		0			0
$547$	−7.19884	−	7.19884i	−0.307800	−	0.307800i	0.536255	−	0.844056i	$-0.319838\pi$
$547$	−7.19884	−	7.19884i	−0.307800	−	0.307800i	−0.844056	+	0.536255i	$0.819838\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	9.06651i		−	0.386246i
$552$		0			0
$553$		−	1.54075i		−	0.0655194i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.02129	+	1.02129i	0.0432735	+	0.0432735i	0.728412	−	0.685139i	$-0.240259\pi$
$557$	1.02129	+	1.02129i	0.0432735	+	0.0432735i	−0.685139	+	0.728412i	$0.740259\pi$
$558$		0			0
$559$	48.4802			2.05049
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.70751	−	6.70751i	−0.282688	−	0.282688i	0.551492	−	0.834180i	$-0.314059\pi$
$563$	−6.70751	−	6.70751i	−0.282688	−	0.282688i	−0.834180	+	0.551492i	$0.814059\pi$
$564$		0			0
$565$	4.49400	−	4.49400i	0.189064	−	0.189064i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			8.98711i			0.376759i	0.982096	+	0.188380i	$0.0603235\pi$
$569$			8.98711i			0.376759i	−0.982096	+	0.188380i	$0.939676\pi$
$570$		0			0
$571$	−9.17157	+	9.17157i	−0.383818	+	0.383818i	−0.872476	−	0.488657i	$-0.837487\pi$
$571$	−9.17157	+	9.17157i	−0.383818	+	0.383818i	0.488657	+	0.872476i	$0.337487\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.99907			0.208476
$576$		0			0
$577$	29.5013			1.22815			0.614077	−	0.789246i	$-0.289528\pi$
$577$	29.5013			1.22815			0.614077	+	0.789246i	$0.289528\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.0184080	+	0.0184080i	−0.000763692	+	0.000763692i
$582$		0			0
$583$		−	27.0192i		−	1.11902i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.82425	+	1.82425i	−0.0752950	+	0.0752950i	−0.743751	−	0.668456i	$-0.766955\pi$
$587$	−1.82425	+	1.82425i	−0.0752950	+	0.0752950i	0.668456	+	0.743751i	$0.266955\pi$
$588$		0			0
$589$	8.97891	+	8.97891i	0.369970	+	0.369970i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	35.4338			1.45509			0.727546	−	0.686058i	$-0.240661\pi$
$593$	35.4338			1.45509			0.727546	+	0.686058i	$0.240661\pi$
$594$		0			0
$595$	0.616953	+	0.616953i	0.0252926	+	0.0252926i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	27.1632i		−	1.10986i	−0.831897	−	0.554930i	$-0.812745\pi$
$599$		−	27.1632i		−	1.10986i	0.831897	−	0.554930i	$-0.187255\pi$
$600$		0			0
$601$			5.33897i			0.217781i	0.994054	+	0.108891i	$0.0347298\pi$
$601$			5.33897i			0.217781i	−0.994054	+	0.108891i	$0.965270\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.6914	−	22.6914i	−0.922539	−	0.922539i
$606$		0			0
$607$	−16.1084			−0.653820			−0.326910	−	0.945055i	$-0.606007\pi$
$607$	−16.1084			−0.653820			−0.326910	+	0.945055i	$0.606007\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−11.9137	−	11.9137i	−0.481979	−	0.481979i
$612$		0			0
$613$	−0.436924	+	0.436924i	−0.0176472	+	0.0176472i	−0.715875	−	0.698228i	$-0.753972\pi$
$613$	−0.436924	+	0.436924i	−0.0176472	+	0.0176472i	0.698228	+	0.715875i	$0.253972\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.80641i		−	0.354533i	−0.984163	−	0.177266i	$-0.943275\pi$
$617$		−	8.80641i		−	0.354533i	0.984163	−	0.177266i	$-0.0567254\pi$
$618$		0			0
$619$	1.92932	−	1.92932i	0.0775458	−	0.0775458i	−0.667270	−	0.744816i	$-0.732537\pi$
$619$	1.92932	−	1.92932i	0.0775458	−	0.0775458i	0.744816	+	0.667270i	$0.232537\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.28861			0.0916910
$624$		0			0
$625$	13.0390			0.521559
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−18.2807	+	18.2807i	−0.728898	+	0.728898i
$630$		0			0
$631$		−	38.7864i		−	1.54406i	−0.635586	−	0.772030i	$-0.719241\pi$
$631$		−	38.7864i		−	1.54406i	0.635586	−	0.772030i	$-0.280759\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.90434	−	1.90434i	0.0755715	−	0.0755715i
$636$		0			0
$637$	29.3786	+	29.3786i	1.16402	+	1.16402i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−33.1091			−1.30773			−0.653865	−	0.756611i	$-0.726854\pi$
$641$	−33.1091			−1.30773			−0.653865	+	0.756611i	$0.726854\pi$
$642$		0			0
$643$	−19.2897	−	19.2897i	−0.760711	−	0.760711i	0.215740	−	0.976451i	$-0.430784\pi$
$643$	−19.2897	−	19.2897i	−0.760711	−	0.760711i	−0.976451	+	0.215740i	$0.930784\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	41.8477i		−	1.64520i	−0.568620	−	0.822601i	$-0.692522\pi$
$647$		−	41.8477i		−	1.64520i	0.568620	−	0.822601i	$-0.307478\pi$
$648$		0			0
$649$			30.3835i			1.19266i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.7741	+	14.7741i	0.578155	+	0.578155i	0.934395	−	0.356240i	$-0.115941\pi$
$653$	14.7741	+	14.7741i	0.578155	+	0.578155i	−0.356240	+	0.934395i	$0.615941\pi$
$654$		0			0
$655$	−26.5054			−1.03565
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.22839	+	2.22839i	0.0868056	+	0.0868056i	0.749176	−	0.662371i	$-0.230450\pi$
$659$	2.22839	+	2.22839i	0.0868056	+	0.0868056i	−0.662371	+	0.749176i	$0.730450\pi$
$660$		0			0
$661$	18.0685	−	18.0685i	0.702784	−	0.702784i	−0.262223	−	0.965007i	$-0.584456\pi$
$661$	18.0685	−	18.0685i	0.702784	−	0.702784i	0.965007	+	0.262223i	$0.0844557\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			0.872503i			0.0338342i
$666$		0			0
$667$	5.93901	−	5.93901i	0.229959	−	0.229959i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−27.8729			−1.07602
$672$		0			0
$673$	20.7981			0.801706			0.400853	−	0.916142i	$-0.368714\pi$
$673$	20.7981			0.801706			0.400853	+	0.916142i	$0.368714\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.0213	−	29.0213i	1.11538	−	1.11538i	0.122968	−	0.992411i	$-0.460759\pi$
$677$	29.0213	−	29.0213i	1.11538	−	1.11538i	0.992411	−	0.122968i	$-0.0392413\pi$
$678$		0			0
$679$			0.0908404i			0.00348613i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	18.7938	−	18.7938i	0.719123	−	0.719123i	−0.249303	−	0.968426i	$-0.580201\pi$
$683$	18.7938	−	18.7938i	0.719123	−	0.719123i	0.968426	+	0.249303i	$0.0802013\pi$
$684$		0			0
$685$	−17.5385	−	17.5385i	−0.670111	−	0.670111i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	29.9660			1.14161
$690$		0			0
$691$	10.4580	+	10.4580i	0.397841	+	0.397841i	0.877471	−	0.479630i	$-0.159229\pi$
$691$	10.4580	+	10.4580i	0.397841	+	0.397841i	−0.479630	+	0.877471i	$0.659229\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			6.16415i			0.233820i
$696$		0			0
$697$			7.94948i			0.301108i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.3314	+	18.3314i	0.692367	+	0.692367i	0.962752	−	0.270385i	$-0.0871511\pi$
$701$	18.3314	+	18.3314i	0.692367	+	0.692367i	−0.270385	+	0.962752i	$0.587151\pi$
$702$		0			0
$703$	−25.8528			−0.975055
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.13690	+	1.13690i	0.0427577	+	0.0427577i
$708$		0			0
$709$	−14.5722	+	14.5722i	−0.547271	+	0.547271i	−0.925650	−	0.378380i	$-0.876481\pi$
$709$	−14.5722	+	14.5722i	−0.547271	+	0.547271i	0.378380	+	0.925650i	$0.376481\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			11.7633i			0.440538i
$714$		0			0
$715$	40.6761	−	40.6761i	1.52120	−	1.52120i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	44.0949			1.64446			0.822230	−	0.569155i	$-0.192730\pi$
$719$	44.0949			1.64446			0.822230	+	0.569155i	$0.192730\pi$
$720$		0			0
$721$	−1.80409			−0.0671880
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.71119	+	3.71119i	−0.137830	+	0.137830i
$726$		0			0
$727$			9.23457i			0.342491i	0.985228	+	0.171246i	$0.0547792\pi$
$727$			9.23457i			0.342491i	−0.985228	+	0.171246i	$0.945221\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−17.5706	+	17.5706i	−0.649872	+	0.649872i
$732$		0			0
$733$	−18.2764	−	18.2764i	−0.675053	−	0.675053i	0.283823	−	0.958877i	$-0.408397\pi$
$733$	−18.2764	−	18.2764i	−0.675053	−	0.675053i	−0.958877	+	0.283823i	$0.908397\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.82936			0.214727
$738$		0			0
$739$	16.9991	+	16.9991i	0.625321	+	0.625321i	0.946887	−	0.321566i	$-0.104209\pi$
$739$	16.9991	+	16.9991i	0.625321	+	0.625321i	−0.321566	+	0.946887i	$0.604209\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			17.8748i			0.655762i	0.944719	+	0.327881i	$0.106335\pi$
$743$			17.8748i			0.655762i	−0.944719	+	0.327881i	$0.893665\pi$
$744$		0			0
$745$		−	7.44538i		−	0.272778i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.114759	−	0.114759i	−0.00419319	−	0.00419319i
$750$		0			0
$751$	35.0731			1.27984			0.639918	−	0.768443i	$-0.278968\pi$
$751$	35.0731			1.27984			0.639918	+	0.768443i	$0.278968\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	28.8139	+	28.8139i	1.04865	+	1.04865i
$756$		0			0
$757$	32.8071	−	32.8071i	1.19239	−	1.19239i	0.216000	−	0.976393i	$-0.430699\pi$
$757$	32.8071	−	32.8071i	1.19239	−	1.19239i	0.976393	−	0.216000i	$-0.0693012\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	10.5531i		−	0.382550i	−0.981536	−	0.191275i	$-0.938738\pi$
$761$		−	10.5531i		−	0.382550i	0.981536	−	0.191275i	$-0.0612623\pi$
$762$		0			0
$763$	0.229624	−	0.229624i	0.00831296	−	0.00831296i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−33.6972			−1.21673
$768$		0			0
$769$	−35.2068			−1.26959			−0.634795	−	0.772681i	$-0.718915\pi$
$769$	−35.2068			−1.26959			−0.634795	+	0.772681i	$0.718915\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.3897	+	19.3897i	−0.697399	+	0.697399i	−0.963849	−	0.266450i	$-0.914149\pi$
$773$	−19.3897	+	19.3897i	−0.697399	+	0.697399i	0.266450	+	0.963849i	$0.414149\pi$
$774$		0			0
$775$		−	7.35067i		−	0.264044i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.62113	+	5.62113i	−0.201398	+	0.201398i
$780$		0			0
$781$	−1.20730	−	1.20730i	−0.0432006	−	0.0432006i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.06877			0.252295
$786$		0			0
$787$	−6.68964	−	6.68964i	−0.238460	−	0.238460i	0.577752	−	0.816212i	$-0.303930\pi$
$787$	−6.68964	−	6.68964i	−0.238460	−	0.238460i	−0.816212	+	0.577752i	$0.803930\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	0.561839i		−	0.0199767i
$792$		0			0
$793$		−	30.9128i		−	1.09775i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13.5617	+	13.5617i	0.480380	+	0.480380i	0.905253	−	0.424873i	$-0.139681\pi$
$797$	13.5617	+	13.5617i	0.480380	+	0.480380i	−0.424873	+	0.905253i	$0.639681\pi$
$798$		0			0
$799$	8.63577			0.305511
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.08532	+	5.08532i	0.179457	+	0.179457i
$804$		0			0
$805$	−0.571533	+	0.571533i	−0.0201439	+	0.0201439i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			43.1578i			1.51735i	0.651472	+	0.758673i	$0.274152\pi$
$809$			43.1578i			1.51735i	−0.651472	+	0.758673i	$0.725848\pi$
$810$		0			0
$811$	2.74017	−	2.74017i	0.0962203	−	0.0962203i	−0.657358	−	0.753578i	$-0.728326\pi$
$811$	2.74017	−	2.74017i	0.0962203	−	0.0962203i	0.753578	+	0.657358i	$0.228326\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.8243			−0.484242
$816$		0			0
$817$	−24.8486			−0.869342
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.97453	−	3.97453i	0.138712	−	0.138712i	−0.634341	−	0.773053i	$-0.718729\pi$
$821$	3.97453	−	3.97453i	0.138712	−	0.138712i	0.773053	+	0.634341i	$0.218729\pi$
$822$		0			0
$823$			38.5255i			1.34291i	0.741043	+	0.671457i	$0.234331\pi$
$823$			38.5255i			1.34291i	−0.741043	+	0.671457i	$0.765669\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.99583	+	2.99583i	−0.104175	+	0.104175i	−0.757273	−	0.653098i	$-0.773469\pi$
$827$	−2.99583	+	2.99583i	−0.104175	+	0.104175i	0.653098	+	0.757273i	$0.273469\pi$
$828$		0			0
$829$	24.9699	+	24.9699i	0.867240	+	0.867240i	0.992166	−	0.124926i	$-0.0398693\pi$
$829$	24.9699	+	24.9699i	0.867240	+	0.867240i	−0.124926	+	0.992166i	$0.539869\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.2953			−0.737838
$834$		0			0
$835$	5.02758	+	5.02758i	0.173986	+	0.173986i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			39.6005i			1.36716i	0.729876	+	0.683580i	$0.239578\pi$
$839$			39.6005i			1.36716i	−0.729876	+	0.683580i	$0.760422\pi$
$840$		0			0
$841$		−	20.1820i		−	0.695932i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.5850	+	28.5850i	0.983355	+	0.983355i
$846$		0			0
$847$	−2.83688			−0.0974765
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−16.9348	−	16.9348i	−0.580519	−	0.580519i
$852$		0			0
$853$	−7.68505	+	7.68505i	−0.263131	+	0.263131i	−0.826325	−	0.563194i	$-0.809572\pi$
$853$	−7.68505	+	7.68505i	−0.263131	+	0.263131i	0.563194	+	0.826325i	$0.309572\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	34.0082i		−	1.16170i	−0.814011	−	0.580849i	$-0.802721\pi$
$857$		−	34.0082i		−	1.16170i	0.814011	−	0.580849i	$-0.197279\pi$
$858$		0			0
$859$	9.19049	−	9.19049i	0.313576	−	0.313576i	−0.532718	−	0.846293i	$-0.678829\pi$
$859$	9.19049	−	9.19049i	0.313576	−	0.313576i	0.846293	+	0.532718i	$0.178829\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−25.8307			−0.879289			−0.439644	−	0.898172i	$-0.644896\pi$
$863$	−25.8307			−0.879289			−0.439644	+	0.898172i	$0.644896\pi$
$864$		0			0
$865$	−40.6353			−1.38164
$866$		0			0
$867$		0			0
$868$		0			0
$869$	36.8165	−	36.8165i	1.24891	−	1.24891i
$870$		0			0
$871$			6.46512i			0.219062i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.36748	−	1.36748i	0.0462292	−	0.0462292i
$876$		0			0
$877$	−31.9718	−	31.9718i	−1.07961	−	1.07961i	−0.996544	−	0.0830670i	$-0.973528\pi$
$877$	−31.9718	−	31.9718i	−1.07961	−	1.07961i	−0.0830670	−	0.996544i	$-0.526472\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−34.7403			−1.17043			−0.585215	−	0.810878i	$-0.698990\pi$
$881$	−34.7403			−1.17043			−0.585215	+	0.810878i	$0.698990\pi$
$882$		0			0
$883$	−34.6034	−	34.6034i	−1.16450	−	1.16450i	−0.983480	−	0.181017i	$-0.942061\pi$
$883$	−34.6034	−	34.6034i	−1.16450	−	1.16450i	−0.181017	−	0.983480i	$-0.557939\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			22.9284i			0.769860i	0.922946	+	0.384930i	$0.125774\pi$
$887$			22.9284i			0.769860i	−0.922946	+	0.384930i	$0.874226\pi$
$888$		0			0
$889$		−	0.238081i		−	0.00798497i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.10641	+	6.10641i	0.204343	+	0.204343i
$894$		0			0
$895$	31.1009			1.03959
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8.73277	−	8.73277i	−0.291254	−	0.291254i
$900$		0			0
$901$	−10.8605	+	10.8605i	−0.361817	+	0.361817i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			14.6523i			0.487059i
$906$		0			0
$907$	−16.3822	+	16.3822i	−0.543963	+	0.543963i	−0.924688	−	0.380725i	$-0.875674\pi$
$907$	−16.3822	+	16.3822i	−0.543963	+	0.543963i	0.380725	+	0.924688i	$0.375674\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−29.4078			−0.974324			−0.487162	−	0.873312i	$-0.661968\pi$
$911$	−29.4078			−0.974324			−0.487162	+	0.873312i	$0.661968\pi$
$912$		0			0
$913$	−0.879722			−0.0291146
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.65685	+	1.65685i	−0.0547141	+	0.0547141i
$918$		0			0
$919$			6.86029i			0.226300i	0.993578	+	0.113150i	$0.0360941\pi$
$919$			6.86029i			0.226300i	−0.993578	+	0.113150i	$0.963906\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.33897	−	1.33897i	0.0440728	−	0.0440728i
$924$		0			0
$925$	10.5823	+	10.5823i	0.347944	+	0.347944i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−16.1385			−0.529488			−0.264744	−	0.964319i	$-0.585287\pi$
$929$	−16.1385			−0.529488			−0.264744	+	0.964319i	$0.585287\pi$
$930$		0			0
$931$	−15.0581	−	15.0581i	−0.493508	−	0.493508i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			29.4844i			0.964241i
$936$		0			0
$937$		−	34.7669i		−	1.13579i	−0.823102	−	0.567893i	$-0.807759\pi$
$937$		−	34.7669i		−	1.13579i	0.823102	−	0.567893i	$-0.192241\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−37.2662	−	37.2662i	−1.21484	−	1.21484i	−0.969414	−	0.245430i	$-0.921071\pi$
$941$	−37.2662	−	37.2662i	−1.21484	−	1.21484i	−0.245430	−	0.969414i	$-0.578929\pi$
$942$		0			0
$943$	−7.36423			−0.239812
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.5243	−	18.5243i	−0.601957	−	0.601957i	0.338874	−	0.940832i	$-0.389954\pi$
$947$	−18.5243	−	18.5243i	−0.601957	−	0.601957i	−0.940832	+	0.338874i	$0.889954\pi$
$948$		0			0
$949$	−5.63994	+	5.63994i	−0.183080	+	0.183080i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	11.1752i		−	0.362000i	−0.983483	−	0.181000i	$-0.942067\pi$
$953$		−	11.1752i		−	0.362000i	0.983483	−	0.181000i	$-0.0579333\pi$
$954$		0			0
$955$	−20.5541	+	20.5541i	−0.665115	+	0.665115i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.19266			−0.0708047
$960$		0			0
$961$	−13.7032			−0.442039
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−28.1541	+	28.1541i	−0.906312	+	0.906312i
$966$		0			0
$967$		−	12.8452i		−	0.413075i	−0.978439	−	0.206537i	$-0.933780\pi$
$967$		−	12.8452i		−	0.413075i	0.978439	−	0.206537i	$-0.0662195\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.94517	+	5.94517i	−0.190790	+	0.190790i	−0.796037	−	0.605248i	$-0.793074\pi$
$971$	−5.94517	+	5.94517i	−0.190790	+	0.190790i	0.605248	+	0.796037i	$0.293074\pi$
$972$		0			0
$973$	0.385321	+	0.385321i	0.0123528	+	0.0123528i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−40.4156			−1.29301			−0.646504	−	0.762910i	$-0.723770\pi$
$977$	−40.4156			−1.29301			−0.646504	+	0.762910i	$0.723770\pi$
$978$		0			0
$979$	54.6865	+	54.6865i	1.74779	+	1.74779i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	13.9202i		−	0.443985i	−0.975048	−	0.221993i	$-0.928744\pi$
$983$		−	13.9202i		−	0.443985i	0.975048	−	0.221993i	$-0.0712561\pi$
$984$		0			0
$985$			36.3582i			1.15847i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.2771	−	16.2771i	−0.517580	−	0.517580i
$990$		0			0
$991$	−41.0309			−1.30339			−0.651695	−	0.758481i	$-0.725942\pi$
$991$	−41.0309			−1.30339			−0.651695	+	0.758481i	$0.725942\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	31.7928	+	31.7928i	1.00790	+	1.00790i
$996$		0			0
$997$	−27.3245	+	27.3245i	−0.865375	+	0.865375i	−0.991956	−	0.126581i	$-0.959600\pi$
$997$	−27.3245	+	27.3245i	−0.865375	+	0.865375i	0.126581	+	0.991956i	$0.459600\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.k.f.289.3		8
3.2	odd	2		384.2.j.a.289.3		8
4.3	odd	2		1152.2.k.c.289.3		8
8.3	odd	2		144.2.k.b.109.4		8
8.5	even	2		576.2.k.b.145.2		8
12.11	even	2		384.2.j.b.289.1		8
16.3	odd	4		144.2.k.b.37.4		8
16.5	even	4	inner	1152.2.k.f.865.3		8
16.11	odd	4		1152.2.k.c.865.3		8
16.13	even	4		576.2.k.b.433.2		8
24.5	odd	2		192.2.j.a.145.2		8
24.11	even	2		48.2.j.a.13.1	✓	8
32.5	even	8		9216.2.a.x.1.4		4
32.11	odd	8		9216.2.a.bo.1.1		4
32.21	even	8		9216.2.a.bn.1.1		4
32.27	odd	8		9216.2.a.y.1.4		4
48.5	odd	4		384.2.j.a.97.3		8
48.11	even	4		384.2.j.b.97.1		8
48.29	odd	4		192.2.j.a.49.2		8
48.35	even	4		48.2.j.a.37.1	yes	8
96.5	odd	8		3072.2.a.n.1.1		4
96.11	even	8		3072.2.a.i.1.4		4
96.29	odd	8		3072.2.d.i.1537.5		8
96.35	even	8		3072.2.d.f.1537.1		8
96.53	odd	8		3072.2.a.o.1.4		4
96.59	even	8		3072.2.a.t.1.1		4
96.77	odd	8		3072.2.d.i.1537.4		8
96.83	even	8		3072.2.d.f.1537.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.1	✓	8	24.11	even	2
48.2.j.a.37.1	yes	8	48.35	even	4
144.2.k.b.37.4		8	16.3	odd	4
144.2.k.b.109.4		8	8.3	odd	2
192.2.j.a.49.2		8	48.29	odd	4
192.2.j.a.145.2		8	24.5	odd	2
384.2.j.a.97.3		8	48.5	odd	4
384.2.j.a.289.3		8	3.2	odd	2
384.2.j.b.97.1		8	48.11	even	4
384.2.j.b.289.1		8	12.11	even	2
576.2.k.b.145.2		8	8.5	even	2
576.2.k.b.433.2		8	16.13	even	4
1152.2.k.c.289.3		8	4.3	odd	2
1152.2.k.c.865.3		8	16.11	odd	4
1152.2.k.f.289.3		8	1.1	even	1	trivial
1152.2.k.f.865.3		8	16.5	even	4	inner
3072.2.a.i.1.4		4	96.11	even	8
3072.2.a.n.1.1		4	96.5	odd	8
3072.2.a.o.1.4		4	96.53	odd	8
3072.2.a.t.1.1		4	96.59	even	8
3072.2.d.f.1537.1		8	96.35	even	8
3072.2.d.f.1537.8		8	96.83	even	8
3072.2.d.i.1537.4		8	96.77	odd	8
3072.2.d.i.1537.5		8	96.29	odd	8
9216.2.a.x.1.4		4	32.5	even	8
9216.2.a.y.1.4		4	32.27	odd	8
9216.2.a.bn.1.1		4	32.21	even	8
9216.2.a.bo.1.1		4	32.11	odd	8

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.27133 - 1.27133i) q^{5} -0.158942i q^{7} +O(q^{10})\)	\(q+(1.27133 - 1.27133i) q^{5} -0.158942i q^{7} +(3.79793 - 3.79793i) q^{11} +(4.21215 + 4.21215i) q^{13} -3.05320 q^{17} +(-2.15894 - 2.15894i) q^{19} -2.82843i q^{23} +1.76744i q^{25} +(2.09976 + 2.09976i) q^{29} -4.15894 q^{31} +(-0.202067 - 0.202067i) q^{35} +(5.98737 - 5.98737i) q^{37} -2.60365i q^{41} +(5.75481 - 5.75481i) q^{43} -2.82843 q^{47} +6.97474 q^{49} +(3.55710 - 3.55710i) q^{53} -9.65685i q^{55} +(-4.00000 + 4.00000i) q^{59} +(-3.66949 - 3.66949i) q^{61} +10.7101 q^{65} +(0.767438 + 0.767438i) q^{67} -0.317883i q^{71} +1.33897i q^{73} +(-0.603650 - 0.603650i) q^{77} +9.69382 q^{79} +(-0.115816 - 0.115816i) q^{83} +(-3.88163 + 3.88163i) q^{85} +14.3990i q^{89} +(0.669485 - 0.669485i) q^{91} -5.48946 q^{95} -0.571533 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{11} - 8 q^{19} - 16 q^{29} - 24 q^{31} - 24 q^{35} + 16 q^{37} - 8 q^{43} - 8 q^{49} + 16 q^{53} - 32 q^{59} - 16 q^{61} + 16 q^{65} - 16 q^{67} + 16 q^{77} + 24 q^{79} + 40 q^{83} + 16 q^{85} - 8 q^{91} - 48 q^{95}+O(q^{100}) \) 8 * q + 8 * q^11 - 8 * q^19 - 16 * q^29 - 24 * q^31 - 24 * q^35 + 16 * q^37 - 8 * q^43 - 8 * q^49 + 16 * q^53 - 32 * q^59 - 16 * q^61 + 16 * q^65 - 16 * q^67 + 16 * q^77 + 24 * q^79 + 40 * q^83 + 16 * q^85 - 8 * q^91 - 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more