LMFDB - Embedded newform 832.2.k.e.447.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,2,Mod(255,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 1]))

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

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

chi := DirichletCharacter("832.255");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.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:	$6.64355344817$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 416)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			447.1
Root			$1.58114 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	832.447
Dual form			832.2.k.e.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.58114 + 1.58114i) q^{5} +(1.58114 - 1.58114i) q^{7} +2.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{3} +(-1.58114 + 1.58114i) q^{5} +(1.58114 - 1.58114i) q^{7} +2.00000 q^{9} +(2.16228 - 2.16228i) q^{11} +(-0.418861 - 3.58114i) q^{13} +(-1.58114 - 1.58114i) q^{15} -5.32456i q^{17} +(5.16228 + 5.16228i) q^{19} +(1.58114 + 1.58114i) q^{21} +0.837722 q^{23} +5.00000i q^{27} -5.16228 q^{29} +(5.16228 + 5.16228i) q^{31} +(2.16228 + 2.16228i) q^{33} +5.00000i q^{35} +(0.418861 + 0.418861i) q^{37} +(3.58114 - 0.418861i) q^{39} +(-1.16228 + 1.16228i) q^{41} +5.00000 q^{43} +(-3.16228 + 3.16228i) q^{45} +(2.74342 - 2.74342i) q^{47} +2.00000i q^{49} +5.32456 q^{51} +9.48683 q^{53} +6.83772i q^{55} +(-5.16228 + 5.16228i) q^{57} +(4.00000 - 4.00000i) q^{59} -2.00000 q^{61} +(3.16228 - 3.16228i) q^{63} +(6.32456 + 5.00000i) q^{65} +(-5.32456 - 5.32456i) q^{67} +0.837722i q^{69} +(1.58114 + 1.58114i) q^{71} +(-6.00000 - 6.00000i) q^{73} -6.83772i q^{77} +15.4868i q^{79} +1.00000 q^{81} +(-12.1623 - 12.1623i) q^{83} +(8.41886 + 8.41886i) q^{85} -5.16228i q^{87} +(-9.16228 - 9.16228i) q^{89} +(-6.32456 - 5.00000i) q^{91} +(-5.16228 + 5.16228i) q^{93} -16.3246 q^{95} +(-10.1623 + 10.1623i) q^{97} +(4.32456 - 4.32456i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{9}+O(q^{10}) $ 4 * q + 8 * q^9	$ 4 q + 8 q^{9} - 4 q^{11} - 8 q^{13} + 8 q^{19} + 16 q^{23} - 8 q^{29} + 8 q^{31} - 4 q^{33} + 8 q^{37} + 8 q^{39} + 8 q^{41} + 20 q^{43} - 8 q^{47} - 4 q^{51} - 8 q^{57} + 16 q^{59} - 8 q^{61} + 4 q^{67} - 24 q^{73} + 4 q^{81} - 36 q^{83} + 40 q^{85} - 24 q^{89} - 8 q^{93} - 40 q^{95} - 28 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q + 8 * q^9 - 4 * q^11 - 8 * q^13 + 8 * q^19 + 16 * q^23 - 8 * q^29 + 8 * q^31 - 4 * q^33 + 8 * q^37 + 8 * q^39 + 8 * q^41 + 20 * q^43 - 8 * q^47 - 4 * q^51 - 8 * q^57 + 16 * q^59 - 8 * q^61 + 4 * q^67 - 24 * q^73 + 4 * q^81 - 36 * q^83 + 40 * q^85 - 24 * q^89 - 8 * q^93 - 40 * q^95 - 28 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$			1.00000i			0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$		0			0
$5$	−1.58114	+	1.58114i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$5$	−1.58114	+	1.58114i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$6$		0			0
$7$	1.58114	−	1.58114i	0.597614	−	0.597614i	−0.342063	−	0.939677i	$-0.611126\pi$
$7$	1.58114	−	1.58114i	0.597614	−	0.597614i	0.939677	+	0.342063i	$0.111126\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$	2.16228	−	2.16228i	0.651951	−	0.651951i	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	2.16228	−	2.16228i	0.651951	−	0.651951i	0.953463	+	0.301511i	$0.0974911\pi$
$12$		0			0
$13$	−0.418861	−	3.58114i	−0.116171	−	0.993229i
$13$	−0.418861	−	3.58114i	−0.116171	−	0.993229i
$14$		0			0
$15$	−1.58114	−	1.58114i	−0.408248	−	0.408248i
$16$		0			0
$17$		−	5.32456i		−	1.29139i	−0.763594	−	0.645697i	$-0.776567\pi$
$17$		−	5.32456i		−	1.29139i	0.763594	−	0.645697i	$-0.223433\pi$
$18$		0			0
$19$	5.16228	+	5.16228i	1.18431	+	1.18431i	0.978617	+	0.205691i	$0.0659441\pi$
$19$	5.16228	+	5.16228i	1.18431	+	1.18431i	0.205691	+	0.978617i	$0.434056\pi$
$20$		0			0
$21$	1.58114	+	1.58114i	0.345033	+	0.345033i
$22$		0			0
$23$	0.837722			0.174677			0.0873386	−	0.996179i	$-0.472164\pi$
$23$	0.837722			0.174677			0.0873386	+	0.996179i	$0.472164\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$	−5.16228			−0.958611			−0.479305	−	0.877648i	$-0.659111\pi$
$29$	−5.16228			−0.958611			−0.479305	+	0.877648i	$0.659111\pi$
$30$		0			0
$31$	5.16228	+	5.16228i	0.927172	+	0.927172i	0.997522	−	0.0703499i	$-0.0224116\pi$
$31$	5.16228	+	5.16228i	0.927172	+	0.927172i	−0.0703499	+	0.997522i	$0.522412\pi$
$32$		0			0
$33$	2.16228	+	2.16228i	0.376404	+	0.376404i
$34$		0			0
$35$			5.00000i			0.845154i
$36$		0			0
$37$	0.418861	+	0.418861i	0.0688604	+	0.0688604i	0.740698	−	0.671838i	$-0.234495\pi$
$37$	0.418861	+	0.418861i	0.0688604	+	0.0688604i	−0.671838	+	0.740698i	$0.734495\pi$
$38$		0			0
$39$	3.58114	−	0.418861i	0.573441	−	0.0670715i
$40$		0			0
$41$	−1.16228	+	1.16228i	−0.181517	+	0.181517i	−0.792017	−	0.610499i	$-0.790969\pi$
$41$	−1.16228	+	1.16228i	−0.181517	+	0.181517i	0.610499	+	0.792017i	$0.290969\pi$
$42$		0			0
$43$	5.00000			0.762493			0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000			0.762493			0.381246	+	0.924473i	$0.375495\pi$
$44$		0			0
$45$	−3.16228	+	3.16228i	−0.471405	+	0.471405i
$46$		0			0
$47$	2.74342	−	2.74342i	0.400168	−	0.400168i	−0.478124	−	0.878292i	$-0.658683\pi$
$47$	2.74342	−	2.74342i	0.400168	−	0.400168i	0.878292	+	0.478124i	$0.158683\pi$
$48$		0			0
$49$			2.00000i			0.285714i
$50$		0			0
$51$	5.32456			0.745587
$52$		0			0
$53$	9.48683			1.30312			0.651558	−	0.758599i	$-0.274116\pi$
$53$	9.48683			1.30312			0.651558	+	0.758599i	$0.274116\pi$
$54$		0			0
$55$			6.83772i			0.921998i
$56$		0			0
$57$	−5.16228	+	5.16228i	−0.683760	+	0.683760i
$58$		0			0
$59$	4.00000	−	4.00000i	0.520756	−	0.520756i	−0.397044	−	0.917800i	$-0.629964\pi$
$59$	4.00000	−	4.00000i	0.520756	−	0.520756i	0.917800	+	0.397044i	$0.129964\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$	3.16228	−	3.16228i	0.398410	−	0.398410i
$64$		0			0
$65$	6.32456	+	5.00000i	0.784465	+	0.620174i
$66$		0			0
$67$	−5.32456	−	5.32456i	−0.650498	−	0.650498i	0.302615	−	0.953113i	$-0.402140\pi$
$67$	−5.32456	−	5.32456i	−0.650498	−	0.650498i	−0.953113	+	0.302615i	$0.902140\pi$
$68$		0			0
$69$			0.837722i			0.100850i
$70$		0			0
$71$	1.58114	+	1.58114i	0.187647	+	0.187647i	0.794678	−	0.607031i	$-0.207640\pi$
$71$	1.58114	+	1.58114i	0.187647	+	0.187647i	−0.607031	+	0.794678i	$0.707640\pi$
$72$		0			0
$73$	−6.00000	−	6.00000i	−0.702247	−	0.702247i	0.262646	−	0.964892i	$-0.415405\pi$
$73$	−6.00000	−	6.00000i	−0.702247	−	0.702247i	−0.964892	+	0.262646i	$0.915405\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	6.83772i		−	0.779231i
$78$		0			0
$79$			15.4868i			1.74240i	0.490924	+	0.871202i	$0.336659\pi$
$79$			15.4868i			1.74240i	−0.490924	+	0.871202i	$0.663341\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−12.1623	−	12.1623i	−1.33498	−	1.33498i	−0.900847	−	0.434136i	$-0.857054\pi$
$83$	−12.1623	−	12.1623i	−1.33498	−	1.33498i	−0.434136	−	0.900847i	$-0.642946\pi$
$84$		0			0
$85$	8.41886	+	8.41886i	0.913154	+	0.913154i
$86$		0			0
$87$		−	5.16228i		−	0.553454i
$88$		0			0
$89$	−9.16228	−	9.16228i	−0.971199	−	0.971199i	0.0283972	−	0.999597i	$-0.490960\pi$
$89$	−9.16228	−	9.16228i	−0.971199	−	0.971199i	−0.999597	+	0.0283972i	$0.990960\pi$
$90$		0			0
$91$	−6.32456	−	5.00000i	−0.662994	−	0.524142i
$92$		0			0
$93$	−5.16228	+	5.16228i	−0.535303	+	0.535303i
$94$		0			0
$95$	−16.3246			−1.67486
$96$		0			0
$97$	−10.1623	+	10.1623i	−1.03182	+	1.03182i	−0.0323462	+	0.999477i	$0.510298\pi$
$97$	−10.1623	+	10.1623i	−1.03182	+	1.03182i	−0.999477	+	0.0323462i	$0.989702\pi$
$98$		0			0
$99$	4.32456	−	4.32456i	0.434634	−	0.434634i
$100$		0			0
$101$		−	2.32456i		−	0.231302i	−0.993290	−	0.115651i	$-0.963105\pi$
$101$		−	2.32456i		−	0.231302i	0.993290	−	0.115651i	$-0.0368954\pi$
$102$		0			0
$103$	11.4868			1.13183			0.565916	−	0.824463i	$-0.308523\pi$
$103$	11.4868			1.13183			0.565916	+	0.824463i	$0.308523\pi$
$104$		0			0
$105$	−5.00000			−0.487950
$106$		0			0
$107$			0.324555i			0.0313759i	0.999877	+	0.0156880i	$0.00499384\pi$
$107$			0.324555i			0.0313759i	−0.999877	+	0.0156880i	$0.995006\pi$
$108$		0			0
$109$	10.7434	−	10.7434i	1.02903	−	1.02903i	0.0294669	−	0.999566i	$-0.490619\pi$
$109$	10.7434	−	10.7434i	1.02903	−	1.02903i	0.999566	−	0.0294669i	$-0.00938097\pi$
$110$		0			0
$111$	−0.418861	+	0.418861i	−0.0397565	+	0.0397565i
$112$		0			0
$113$	16.0000			1.50515			0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000			1.50515			0.752577	+	0.658505i	$0.228811\pi$
$114$		0			0
$115$	−1.32456	+	1.32456i	−0.123515	+	0.123515i
$116$		0			0
$117$	−0.837722	−	7.16228i	−0.0774475	−	0.662153i
$118$		0			0
$119$	−8.41886	−	8.41886i	−0.771756	−	0.771756i
$120$		0			0
$121$			1.64911i			0.149919i
$122$		0			0
$123$	−1.16228	−	1.16228i	−0.104799	−	0.104799i
$124$		0			0
$125$	−7.90569	−	7.90569i	−0.707107	−	0.707107i
$126$		0			0
$127$	15.1623			1.34543			0.672717	−	0.739900i	$-0.265127\pi$
$127$	15.1623			1.34543			0.672717	+	0.739900i	$0.265127\pi$
$128$		0			0
$129$			5.00000i			0.440225i
$130$		0			0
$131$			13.6491i			1.19253i	0.802788	+	0.596264i	$0.203349\pi$
$131$			13.6491i			1.19253i	−0.802788	+	0.596264i	$0.796651\pi$
$132$		0			0
$133$	16.3246			1.41552
$134$		0			0
$135$	−7.90569	−	7.90569i	−0.680414	−	0.680414i
$136$		0			0
$137$	−3.00000	−	3.00000i	−0.256307	−	0.256307i	0.567243	−	0.823550i	$-0.308010\pi$
$137$	−3.00000	−	3.00000i	−0.256307	−	0.256307i	−0.823550	+	0.567243i	$0.808010\pi$
$138$		0			0
$139$			2.67544i			0.226928i	0.993542	+	0.113464i	$0.0361947\pi$
$139$			2.67544i			0.226928i	−0.993542	+	0.113464i	$0.963805\pi$
$140$		0			0
$141$	2.74342	+	2.74342i	0.231037	+	0.231037i
$142$		0			0
$143$	−8.64911	−	6.83772i	−0.723275	−	0.571799i
$144$		0			0
$145$	8.16228	−	8.16228i	0.677840	−	0.677840i
$146$		0			0
$147$	−2.00000			−0.164957
$148$		0			0
$149$	−2.00000	+	2.00000i	−0.163846	+	0.163846i	−0.784268	−	0.620422i	$-0.786961\pi$
$149$	−2.00000	+	2.00000i	−0.163846	+	0.163846i	0.620422	+	0.784268i	$0.286961\pi$
$150$		0			0
$151$	−10.7434	+	10.7434i	−0.874287	+	0.874287i	−0.992936	−	0.118649i	$-0.962144\pi$
$151$	−10.7434	+	10.7434i	−0.874287	+	0.874287i	0.118649	+	0.992936i	$0.462144\pi$
$152$		0			0
$153$		−	10.6491i		−	0.860930i
$154$		0			0
$155$	−16.3246			−1.31122
$156$		0			0
$157$	−14.9737			−1.19503			−0.597514	−	0.801858i	$-0.703845\pi$
$157$	−14.9737			−1.19503			−0.597514	+	0.801858i	$0.703845\pi$
$158$		0			0
$159$			9.48683i			0.752355i
$160$		0			0
$161$	1.32456	−	1.32456i	0.104390	−	0.104390i
$162$		0			0
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	−0.260531	−	0.965465i	$-0.583898\pi$
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	0.965465	+	0.260531i	$0.0838976\pi$
$164$		0			0
$165$	−6.83772			−0.532316
$166$		0			0
$167$	−1.48683	+	1.48683i	−0.115055	+	0.115055i	−0.762290	−	0.647236i	$-0.775925\pi$
$167$	−1.48683	+	1.48683i	−0.115055	+	0.115055i	0.647236	+	0.762290i	$0.275925\pi$
$168$		0			0
$169$	−12.6491	+	3.00000i	−0.973009	+	0.230769i
$170$		0			0
$171$	10.3246	+	10.3246i	0.789538	+	0.789538i
$172$		0			0
$173$			15.4868i			1.17744i	0.808336	+	0.588721i	$0.200368\pi$
$173$			15.4868i			1.17744i	−0.808336	+	0.588721i	$0.799632\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.00000	+	4.00000i	0.300658	+	0.300658i
$178$		0			0
$179$	−9.64911			−0.721208			−0.360604	−	0.932719i	$-0.617429\pi$
$179$	−9.64911			−0.721208			−0.360604	+	0.932719i	$0.617429\pi$
$180$		0			0
$181$		−	9.16228i		−	0.681027i	−0.940240	−	0.340513i	$-0.889399\pi$
$181$		−	9.16228i		−	0.681027i	0.940240	−	0.340513i	$-0.110601\pi$
$182$		0			0
$183$		−	2.00000i		−	0.147844i
$184$		0			0
$185$	−1.32456			−0.0973832
$186$		0			0
$187$	−11.5132	−	11.5132i	−0.841926	−	0.841926i
$188$		0			0
$189$	7.90569	+	7.90569i	0.575055	+	0.575055i
$190$		0			0
$191$		−	25.1623i		−	1.82068i	−0.413863	−	0.910339i	$-0.635821\pi$
$191$		−	25.1623i		−	1.82068i	0.413863	−	0.910339i	$-0.364179\pi$
$192$		0			0
$193$	4.16228	+	4.16228i	0.299607	+	0.299607i	0.840860	−	0.541253i	$-0.182050\pi$
$193$	4.16228	+	4.16228i	0.299607	+	0.299607i	−0.541253	+	0.840860i	$0.682050\pi$
$194$		0			0
$195$	−5.00000	+	6.32456i	−0.358057	+	0.452911i
$196$		0			0
$197$	−4.41886	+	4.41886i	−0.314831	+	0.314831i	−0.846778	−	0.531947i	$-0.821461\pi$
$197$	−4.41886	+	4.41886i	−0.314831	+	0.314831i	0.531947	+	0.846778i	$0.321461\pi$
$198$		0			0
$199$	−20.8377			−1.47715			−0.738573	−	0.674173i	$-0.764500\pi$
$199$	−20.8377			−1.47715			−0.738573	+	0.674173i	$0.764500\pi$
$200$		0			0
$201$	5.32456	−	5.32456i	0.375565	−	0.375565i
$202$		0			0
$203$	−8.16228	+	8.16228i	−0.572880	+	0.572880i
$204$		0			0
$205$		−	3.67544i		−	0.256704i
$206$		0			0
$207$	1.67544			0.116451
$208$		0			0
$209$	22.3246			1.54422
$210$		0			0
$211$		−	15.6491i		−	1.07733i	−0.842520	−	0.538665i	$-0.818929\pi$
$211$		−	15.6491i		−	1.07733i	0.842520	−	0.538665i	$-0.181071\pi$
$212$		0			0
$213$	−1.58114	+	1.58114i	−0.108338	+	0.108338i
$214$		0			0
$215$	−7.90569	+	7.90569i	−0.539164	+	0.539164i
$216$		0			0
$217$	16.3246			1.10818
$218$		0			0
$219$	6.00000	−	6.00000i	0.405442	−	0.405442i
$220$		0			0
$221$	−19.0680	+	2.23025i	−1.28265	+	0.150023i
$222$		0			0
$223$	−13.0680	−	13.0680i	−0.875096	−	0.875096i	0.117926	−	0.993022i	$-0.462375\pi$
$223$	−13.0680	−	13.0680i	−0.875096	−	0.875096i	−0.993022	+	0.117926i	$0.962375\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.16228	+	3.16228i	0.209888	+	0.209888i	0.804220	−	0.594332i	$-0.202583\pi$
$227$	3.16228	+	3.16228i	0.209888	+	0.209888i	−0.594332	+	0.804220i	$0.702583\pi$
$228$		0			0
$229$	9.90569	+	9.90569i	0.654587	+	0.654587i	0.954094	−	0.299507i	$-0.0968223\pi$
$229$	9.90569	+	9.90569i	0.654587	+	0.654587i	−0.299507	+	0.954094i	$0.596822\pi$
$230$		0			0
$231$	6.83772			0.449889
$232$		0			0
$233$			5.32456i			0.348823i	0.984673	+	0.174412i	$0.0558023\pi$
$233$			5.32456i			0.348823i	−0.984673	+	0.174412i	$0.944198\pi$
$234$		0			0
$235$			8.67544i			0.565924i
$236$		0			0
$237$	−15.4868			−1.00598
$238$		0			0
$239$	−15.5811	−	15.5811i	−1.00786	−	1.00786i	−0.999969	−	0.00789122i	$-0.997488\pi$
$239$	−15.5811	−	15.5811i	−1.00786	−	1.00786i	−0.00789122	−	0.999969i	$-0.502512\pi$
$240$		0			0
$241$	−15.4868	−	15.4868i	−0.997595	−	0.997595i	0.00240251	−	0.999997i	$-0.499235\pi$
$241$	−15.4868	−	15.4868i	−0.997595	−	0.997595i	−0.999997	+	0.00240251i	$0.999235\pi$
$242$		0			0
$243$			16.0000i			1.02640i
$244$		0			0
$245$	−3.16228	−	3.16228i	−0.202031	−	0.202031i
$246$		0			0
$247$	16.3246	−	20.6491i	1.03871	−	1.31387i
$248$		0			0
$249$	12.1623	−	12.1623i	0.770753	−	0.770753i
$250$		0			0
$251$	−18.6491			−1.17712			−0.588561	−	0.808453i	$-0.700305\pi$
$251$	−18.6491			−1.17712			−0.588561	+	0.808453i	$0.700305\pi$
$252$		0			0
$253$	1.81139	−	1.81139i	0.113881	−	0.113881i
$254$		0			0
$255$	−8.41886	+	8.41886i	−0.527210	+	0.527210i
$256$		0			0
$257$			29.6491i			1.84946i	0.380623	+	0.924730i	$0.375710\pi$
$257$			29.6491i			1.84946i	−0.380623	+	0.924730i	$0.624290\pi$
$258$		0			0
$259$	1.32456			0.0823039
$260$		0			0
$261$	−10.3246			−0.639074
$262$		0			0
$263$		−	6.00000i		−	0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$		−	6.00000i		−	0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$		0			0
$265$	−15.0000	+	15.0000i	−0.921443	+	0.921443i
$266$		0			0
$267$	9.16228	−	9.16228i	0.560722	−	0.560722i
$268$		0			0
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$	13.0680	−	13.0680i	0.793823	−	0.793823i	−0.188291	−	0.982113i	$-0.560295\pi$
$271$	13.0680	−	13.0680i	0.793823	−	0.793823i	0.982113	+	0.188291i	$0.0602947\pi$
$272$		0			0
$273$	5.00000	−	6.32456i	0.302614	−	0.382780i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	0.324555i		−	0.0195006i	−0.999952	−	0.00975032i	$-0.996896\pi$
$277$		−	0.324555i		−	0.0195006i	0.999952	−	0.00975032i	$-0.00310367\pi$
$278$		0			0
$279$	10.3246	+	10.3246i	0.618115	+	0.618115i
$280$		0			0
$281$	−17.4868	−	17.4868i	−1.04318	−	1.04318i	−0.999025	−	0.0441522i	$-0.985941\pi$
$281$	−17.4868	−	17.4868i	−1.04318	−	1.04318i	−0.0441522	−	0.999025i	$-0.514059\pi$
$282$		0			0
$283$	24.6491			1.46524			0.732619	−	0.680639i	$-0.238298\pi$
$283$	24.6491			1.46524			0.732619	+	0.680639i	$0.238298\pi$
$284$		0			0
$285$		−	16.3246i		−	0.966983i
$286$		0			0
$287$			3.67544i			0.216955i
$288$		0			0
$289$	−11.3509			−0.667699
$290$		0			0
$291$	−10.1623	−	10.1623i	−0.595723	−	0.595723i
$292$		0			0
$293$	16.7434	+	16.7434i	0.978161	+	0.978161i	0.999767	−	0.0216057i	$-0.00687785\pi$
$293$	16.7434	+	16.7434i	0.978161	+	0.978161i	−0.0216057	+	0.999767i	$0.506878\pi$
$294$		0			0
$295$			12.6491i			0.736460i
$296$		0			0
$297$	10.8114	+	10.8114i	0.627340	+	0.627340i
$298$		0			0
$299$	−0.350889	−	3.00000i	−0.0202925	−	0.173494i
$300$		0			0
$301$	7.90569	−	7.90569i	0.455677	−	0.455677i
$302$		0			0
$303$	2.32456			0.133542
$304$		0			0
$305$	3.16228	−	3.16228i	0.181071	−	0.181071i
$306$		0			0
$307$	−12.3246	+	12.3246i	−0.703400	+	0.703400i	−0.965139	−	0.261739i	$-0.915704\pi$
$307$	−12.3246	+	12.3246i	−0.703400	+	0.703400i	0.261739	+	0.965139i	$0.415704\pi$
$308$		0			0
$309$			11.4868i			0.653463i
$310$		0			0
$311$	−12.9737			−0.735669			−0.367835	−	0.929891i	$-0.619901\pi$
$311$	−12.9737			−0.735669			−0.367835	+	0.929891i	$0.619901\pi$
$312$		0			0
$313$	−9.64911			−0.545400			−0.272700	−	0.962099i	$-0.587917\pi$
$313$	−9.64911			−0.545400			−0.272700	+	0.962099i	$0.587917\pi$
$314$		0			0
$315$			10.0000i			0.563436i
$316$		0			0
$317$	7.67544	−	7.67544i	0.431096	−	0.431096i	−0.457905	−	0.889001i	$-0.651400\pi$
$317$	7.67544	−	7.67544i	0.431096	−	0.431096i	0.889001	+	0.457905i	$0.151400\pi$
$318$		0			0
$319$	−11.1623	+	11.1623i	−0.624968	+	0.624968i
$320$		0			0
$321$	−0.324555			−0.0181149
$322$		0			0
$323$	27.4868	−	27.4868i	1.52941	−	1.52941i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	10.7434	+	10.7434i	0.594112	+	0.594112i
$328$		0			0
$329$		−	8.67544i		−	0.478293i
$330$		0			0
$331$	−4.48683	−	4.48683i	−0.246619	−	0.246619i	0.572963	−	0.819581i	$-0.305794\pi$
$331$	−4.48683	−	4.48683i	−0.246619	−	0.246619i	−0.819581	+	0.572963i	$0.805794\pi$
$332$		0			0
$333$	0.837722	+	0.837722i	0.0459069	+	0.0459069i
$334$		0			0
$335$	16.8377			0.919943
$336$		0			0
$337$		−	15.6491i		−	0.852461i	−0.904615	−	0.426231i	$-0.859841\pi$
$337$		−	15.6491i		−	0.852461i	0.904615	−	0.426231i	$-0.140159\pi$
$338$		0			0
$339$			16.0000i			0.869001i
$340$		0			0
$341$	22.3246			1.20894
$342$		0			0
$343$	14.2302	+	14.2302i	0.768361	+	0.768361i
$344$		0			0
$345$	−1.32456	−	1.32456i	−0.0713117	−	0.0713117i
$346$		0			0
$347$			26.2982i			1.41176i	0.708330	+	0.705881i	$0.249449\pi$
$347$			26.2982i			1.41176i	−0.708330	+	0.705881i	$0.750551\pi$
$348$		0			0
$349$	5.58114	+	5.58114i	0.298752	+	0.298752i	0.840525	−	0.541773i	$-0.182247\pi$
$349$	5.58114	+	5.58114i	0.298752	+	0.298752i	−0.541773	+	0.840525i	$0.682247\pi$
$350$		0			0
$351$	17.9057	−	2.09431i	0.955735	−	0.111786i
$352$		0			0
$353$	−6.16228	+	6.16228i	−0.327985	+	0.327985i	−0.851820	−	0.523835i	$-0.824501\pi$
$353$	−6.16228	+	6.16228i	−0.327985	+	0.327985i	0.523835	+	0.851820i	$0.324501\pi$
$354$		0			0
$355$	−5.00000			−0.265372
$356$		0			0
$357$	8.41886	−	8.41886i	0.445573	−	0.445573i
$358$		0			0
$359$	−21.1623	+	21.1623i	−1.11690	+	1.11690i	−0.124709	+	0.992193i	$0.539800\pi$
$359$	−21.1623	+	21.1623i	−1.11690	+	1.11690i	−0.992193	+	0.124709i	$0.960200\pi$
$360$		0			0
$361$			34.2982i			1.80517i
$362$		0			0
$363$	−1.64911			−0.0865559
$364$		0			0
$365$	18.9737			0.993127
$366$		0			0
$367$			16.8377i			0.878922i	0.898262	+	0.439461i	$0.144831\pi$
$367$			16.8377i			0.878922i	−0.898262	+	0.439461i	$0.855169\pi$
$368$		0			0
$369$	−2.32456	+	2.32456i	−0.121012	+	0.121012i
$370$		0			0
$371$	15.0000	−	15.0000i	0.778761	−	0.778761i
$372$		0			0
$373$	−12.6491			−0.654946			−0.327473	−	0.944861i	$-0.606197\pi$
$373$	−12.6491			−0.654946			−0.327473	+	0.944861i	$0.606197\pi$
$374$		0			0
$375$	7.90569	−	7.90569i	0.408248	−	0.408248i
$376$		0			0
$377$	2.16228	+	18.4868i	0.111363	+	0.952120i
$378$		0			0
$379$	13.0000	+	13.0000i	0.667765	+	0.667765i	0.957198	−	0.289433i	$-0.0934668\pi$
$379$	13.0000	+	13.0000i	0.667765	+	0.667765i	−0.289433	+	0.957198i	$0.593467\pi$
$380$		0			0
$381$			15.1623i			0.776787i
$382$		0			0
$383$	7.25658	+	7.25658i	0.370794	+	0.370794i	0.867766	−	0.496972i	$-0.165555\pi$
$383$	7.25658	+	7.25658i	0.370794	+	0.370794i	−0.496972	+	0.867766i	$0.665555\pi$
$384$		0			0
$385$	10.8114	+	10.8114i	0.550999	+	0.550999i
$386$		0			0
$387$	10.0000			0.508329
$388$		0			0
$389$			4.00000i			0.202808i	0.994845	+	0.101404i	$0.0323335\pi$
$389$			4.00000i			0.202808i	−0.994845	+	0.101404i	$0.967667\pi$
$390$		0			0
$391$		−	4.46050i		−	0.225577i
$392$		0			0
$393$	−13.6491			−0.688507
$394$		0			0
$395$	−24.4868	−	24.4868i	−1.23207	−	1.23207i
$396$		0			0
$397$	−27.2982	−	27.2982i	−1.37006	−	1.37006i	−0.860343	−	0.509715i	$-0.829751\pi$
$397$	−27.2982	−	27.2982i	−1.37006	−	1.37006i	−0.509715	−	0.860343i	$-0.670249\pi$
$398$		0			0
$399$			16.3246i			0.817250i
$400$		0			0
$401$	−13.9737	−	13.9737i	−0.697812	−	0.697812i	0.266127	−	0.963938i	$-0.414256\pi$
$401$	−13.9737	−	13.9737i	−0.697812	−	0.697812i	−0.963938	+	0.266127i	$0.914256\pi$
$402$		0			0
$403$	16.3246	−	20.6491i	0.813184	−	1.02861i
$404$		0			0
$405$	−1.58114	+	1.58114i	−0.0785674	+	0.0785674i
$406$		0			0
$407$	1.81139			0.0897872
$408$		0			0
$409$	10.6754	−	10.6754i	0.527867	−	0.527867i	−0.392069	−	0.919936i	$-0.628241\pi$
$409$	10.6754	−	10.6754i	0.527867	−	0.527867i	0.919936	+	0.392069i	$0.128241\pi$
$410$		0			0
$411$	3.00000	−	3.00000i	0.147979	−	0.147979i
$412$		0			0
$413$		−	12.6491i		−	0.622422i
$414$		0			0
$415$	38.4605			1.88795
$416$		0			0
$417$	−2.67544			−0.131017
$418$		0			0
$419$			17.3246i			0.846360i	0.906046	+	0.423180i	$0.139086\pi$
$419$			17.3246i			0.846360i	−0.906046	+	0.423180i	$0.860914\pi$
$420$		0			0
$421$	5.58114	−	5.58114i	0.272008	−	0.272008i	−0.557900	−	0.829908i	$-0.688393\pi$
$421$	5.58114	−	5.58114i	0.272008	−	0.272008i	0.829908	+	0.557900i	$0.188393\pi$
$422$		0			0
$423$	5.48683	−	5.48683i	0.266779	−	0.266779i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.16228	+	3.16228i	−0.153033	+	0.153033i
$428$		0			0
$429$	6.83772	−	8.64911i	0.330128	−	0.417583i
$430$		0			0
$431$	11.3925	+	11.3925i	0.548759	+	0.548759i	0.926082	−	0.377323i	$-0.123155\pi$
$431$	11.3925	+	11.3925i	0.548759	+	0.548759i	−0.377323	+	0.926082i	$0.623155\pi$
$432$		0			0
$433$		−	27.9737i		−	1.34433i	−0.740402	−	0.672164i	$-0.765365\pi$
$433$		−	27.9737i		−	1.34433i	0.740402	−	0.672164i	$-0.234635\pi$
$434$		0			0
$435$	8.16228	+	8.16228i	0.391351	+	0.391351i
$436$		0			0
$437$	4.32456	+	4.32456i	0.206872	+	0.206872i
$438$		0			0
$439$	24.6491			1.17644			0.588219	−	0.808702i	$-0.299829\pi$
$439$	24.6491			1.17644			0.588219	+	0.808702i	$0.299829\pi$
$440$		0			0
$441$			4.00000i			0.190476i
$442$		0			0
$443$		−	20.2982i		−	0.964398i	−0.876062	−	0.482199i	$-0.839838\pi$
$443$		−	20.2982i		−	0.964398i	0.876062	−	0.482199i	$-0.160162\pi$
$444$		0			0
$445$	28.9737			1.37348
$446$		0			0
$447$	−2.00000	−	2.00000i	−0.0945968	−	0.0945968i
$448$		0			0
$449$	−7.51317	−	7.51317i	−0.354568	−	0.354568i	0.507238	−	0.861806i	$-0.330667\pi$
$449$	−7.51317	−	7.51317i	−0.354568	−	0.354568i	−0.861806	+	0.507238i	$0.830667\pi$
$450$		0			0
$451$			5.02633i			0.236681i
$452$		0			0
$453$	−10.7434	−	10.7434i	−0.504770	−	0.504770i
$454$		0			0
$455$	17.9057	−	2.09431i	0.839432	−	0.0981826i
$456$		0			0
$457$	5.81139	−	5.81139i	0.271845	−	0.271845i	−0.557997	−	0.829843i	$-0.688430\pi$
$457$	5.81139	−	5.81139i	0.271845	−	0.271845i	0.829843	+	0.557997i	$0.188430\pi$
$458$		0			0
$459$	26.6228			1.24264
$460$		0			0
$461$	−11.0680	+	11.0680i	−0.515487	+	0.515487i	−0.916202	−	0.400716i	$-0.868762\pi$
$461$	−11.0680	+	11.0680i	−0.515487	+	0.515487i	0.400716	+	0.916202i	$0.368762\pi$
$462$		0			0
$463$	−7.16228	+	7.16228i	−0.332859	+	0.332859i	−0.853671	−	0.520812i	$-0.825629\pi$
$463$	−7.16228	+	7.16228i	−0.332859	+	0.332859i	0.520812	+	0.853671i	$0.325629\pi$
$464$		0			0
$465$		−	16.3246i		−	0.757033i
$466$		0			0
$467$	−34.6491			−1.60337			−0.801685	−	0.597747i	$-0.796063\pi$
$467$	−34.6491			−1.60337			−0.801685	+	0.597747i	$0.796063\pi$
$468$		0			0
$469$	−16.8377			−0.777494
$470$		0			0
$471$		−	14.9737i		−	0.689950i
$472$		0			0
$473$	10.8114	−	10.8114i	0.497108	−	0.497108i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	18.9737			0.868744
$478$		0			0
$479$	−29.3925	+	29.3925i	−1.34298	+	1.34298i	−0.449900	+	0.893079i	$0.648540\pi$
$479$	−29.3925	+	29.3925i	−1.34298	+	1.34298i	−0.893079	+	0.449900i	$0.851460\pi$
$480$		0			0
$481$	1.32456	−	1.67544i	0.0603945	−	0.0763937i
$482$		0			0
$483$	1.32456	+	1.32456i	0.0602694	+	0.0602694i
$484$		0			0
$485$		−	32.1359i		−	1.45922i
$486$		0			0
$487$	−3.81139	−	3.81139i	−0.172710	−	0.172710i	0.615459	−	0.788169i	$-0.288971\pi$
$487$	−3.81139	−	3.81139i	−0.172710	−	0.172710i	−0.788169	+	0.615459i	$0.788971\pi$
$488$		0			0
$489$	9.00000	+	9.00000i	0.406994	+	0.406994i
$490$		0			0
$491$	41.3246			1.86495			0.932476	−	0.361233i	$-0.117644\pi$
$491$	41.3246			1.86495			0.932476	+	0.361233i	$0.117644\pi$
$492$		0			0
$493$			27.4868i			1.23794i
$494$		0			0
$495$			13.6754i			0.614666i
$496$		0			0
$497$	5.00000			0.224281
$498$		0			0
$499$	3.48683	+	3.48683i	0.156092	+	0.156092i	0.780833	−	0.624740i	$-0.214795\pi$
$499$	3.48683	+	3.48683i	0.156092	+	0.156092i	−0.624740	+	0.780833i	$0.714795\pi$
$500$		0			0
$501$	−1.48683	−	1.48683i	−0.0664268	−	0.0664268i
$502$		0			0
$503$		−	10.6491i		−	0.474820i	−0.971409	−	0.237410i	$-0.923701\pi$
$503$		−	10.6491i		−	0.474820i	0.971409	−	0.237410i	$-0.0762985\pi$
$504$		0			0
$505$	3.67544	+	3.67544i	0.163555	+	0.163555i
$506$		0			0
$507$	−3.00000	−	12.6491i	−0.133235	−	0.561767i
$508$		0			0
$509$	−2.00000	+	2.00000i	−0.0886484	+	0.0886484i	−0.750040	−	0.661392i	$-0.769966\pi$
$509$	−2.00000	+	2.00000i	−0.0886484	+	0.0886484i	0.661392	+	0.750040i	$0.269966\pi$
$510$		0			0
$511$	−18.9737			−0.839346
$512$		0			0
$513$	−25.8114	+	25.8114i	−1.13960	+	1.13960i
$514$		0			0
$515$	−18.1623	+	18.1623i	−0.800326	+	0.800326i
$516$		0			0
$517$		−	11.8641i		−	0.521781i
$518$		0			0
$519$	−15.4868			−0.679797
$520$		0			0
$521$	−23.3246			−1.02187			−0.510934	−	0.859620i	$-0.670700\pi$
$521$	−23.3246			−1.02187			−0.510934	+	0.859620i	$0.670700\pi$
$522$		0			0
$523$			12.3246i			0.538915i	0.963012	+	0.269458i	$0.0868444\pi$
$523$			12.3246i			0.538915i	−0.963012	+	0.269458i	$0.913156\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	27.4868	−	27.4868i	1.19735	−	1.19735i
$528$		0			0
$529$	−22.2982			−0.969488
$530$		0			0
$531$	8.00000	−	8.00000i	0.347170	−	0.347170i
$532$		0			0
$533$	4.64911	+	3.67544i	0.201375	+	0.159201i
$534$		0			0
$535$	−0.513167	−	0.513167i	−0.0221861	−	0.0221861i
$536$		0			0
$537$		−	9.64911i		−	0.416390i
$538$		0			0
$539$	4.32456	+	4.32456i	0.186272	+	0.186272i
$540$		0			0
$541$	7.06797	+	7.06797i	0.303876	+	0.303876i	0.842528	−	0.538652i	$-0.181066\pi$
$541$	7.06797	+	7.06797i	0.303876	+	0.303876i	−0.538652	+	0.842528i	$0.681066\pi$
$542$		0			0
$543$	9.16228			0.393191
$544$		0			0
$545$			33.9737i			1.45527i
$546$		0			0
$547$			10.6754i			0.456449i	0.973609	+	0.228225i	$0.0732920\pi$
$547$			10.6754i			0.456449i	−0.973609	+	0.228225i	$0.926708\pi$
$548$		0			0
$549$	−4.00000			−0.170716
$550$		0			0
$551$	−26.6491	−	26.6491i	−1.13529	−	1.13529i
$552$		0			0
$553$	24.4868	+	24.4868i	1.04129	+	1.04129i
$554$		0			0
$555$		−	1.32456i		−	0.0562242i
$556$		0			0
$557$	6.41886	+	6.41886i	0.271976	+	0.271976i	0.829895	−	0.557919i	$-0.188400\pi$
$557$	6.41886	+	6.41886i	0.271976	+	0.271976i	−0.557919	+	0.829895i	$0.688400\pi$
$558$		0			0
$559$	−2.09431	−	17.9057i	−0.0885797	−	0.757330i
$560$		0			0
$561$	11.5132	−	11.5132i	0.486086	−	0.486086i
$562$		0			0
$563$	−0.675445			−0.0284666			−0.0142333	−	0.999899i	$-0.504531\pi$
$563$	−0.675445			−0.0284666			−0.0142333	+	0.999899i	$0.504531\pi$
$564$		0			0
$565$	−25.2982	+	25.2982i	−1.06430	+	1.06430i
$566$		0			0
$567$	1.58114	−	1.58114i	0.0664016	−	0.0664016i
$568$		0			0
$569$		−	33.9737i		−	1.42425i	−0.702053	−	0.712125i	$-0.747733\pi$
$569$		−	33.9737i		−	1.42425i	0.702053	−	0.712125i	$-0.252267\pi$
$570$		0			0
$571$	23.9737			1.00327			0.501633	−	0.865080i	$-0.332733\pi$
$571$	23.9737			1.00327			0.501633	+	0.865080i	$0.332733\pi$
$572$		0			0
$573$	25.1623			1.05117
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−16.9737	+	16.9737i	−0.706623	+	0.706623i	−0.965824	−	0.259201i	$-0.916541\pi$
$577$	−16.9737	+	16.9737i	−0.706623	+	0.706623i	0.259201	+	0.965824i	$0.416541\pi$
$578$		0			0
$579$	−4.16228	+	4.16228i	−0.172978	+	0.172978i
$580$		0			0
$581$	−38.4605			−1.59561
$582$		0			0
$583$	20.5132	−	20.5132i	0.849569	−	0.849569i
$584$		0			0
$585$	12.6491	+	10.0000i	0.522976	+	0.413449i
$586$		0			0
$587$	−7.00000	−	7.00000i	−0.288921	−	0.288921i	0.547733	−	0.836653i	$-0.315491\pi$
$587$	−7.00000	−	7.00000i	−0.288921	−	0.288921i	−0.836653	+	0.547733i	$0.815491\pi$
$588$		0			0
$589$			53.2982i			2.19611i
$590$		0			0
$591$	−4.41886	−	4.41886i	−0.181768	−	0.181768i
$592$		0			0
$593$	6.48683	+	6.48683i	0.266382	+	0.266382i	0.827641	−	0.561258i	$-0.189683\pi$
$593$	6.48683	+	6.48683i	0.266382	+	0.266382i	−0.561258	+	0.827641i	$0.689683\pi$
$594$		0			0
$595$	26.6228			1.09143
$596$		0			0
$597$		−	20.8377i		−	0.852831i
$598$		0			0
$599$			35.6228i			1.45551i	0.685839	+	0.727754i	$0.259436\pi$
$599$			35.6228i			1.45551i	−0.685839	+	0.727754i	$0.740564\pi$
$600$		0			0
$601$	44.9473			1.83344			0.916720	−	0.399530i	$-0.130827\pi$
$601$	44.9473			1.83344			0.916720	+	0.399530i	$0.130827\pi$
$602$		0			0
$603$	−10.6491	−	10.6491i	−0.433665	−	0.433665i
$604$		0			0
$605$	−2.60747	−	2.60747i	−0.106009	−	0.106009i
$606$		0			0
$607$		−	18.9737i		−	0.770117i	−0.922892	−	0.385059i	$-0.874181\pi$
$607$		−	18.9737i		−	0.770117i	0.922892	−	0.385059i	$-0.125819\pi$
$608$		0			0
$609$	−8.16228	−	8.16228i	−0.330752	−	0.330752i
$610$		0			0
$611$	−10.9737	−	8.67544i	−0.443947	−	0.350971i
$612$		0			0
$613$	−21.4868	+	21.4868i	−0.867845	+	0.867845i	−0.992234	−	0.124389i	$-0.960303\pi$
$613$	−21.4868	+	21.4868i	−0.867845	+	0.867845i	0.124389	+	0.992234i	$0.460303\pi$
$614$		0			0
$615$	3.67544			0.148208
$616$		0			0
$617$	22.9737	−	22.9737i	0.924885	−	0.924885i	−0.0724846	−	0.997370i	$-0.523093\pi$
$617$	22.9737	−	22.9737i	0.924885	−	0.924885i	0.997370	+	0.0724846i	$0.0230928\pi$
$618$		0			0
$619$	−27.4868	+	27.4868i	−1.10479	+	1.10479i	−0.110965	+	0.993824i	$0.535394\pi$
$619$	−27.4868	+	27.4868i	−1.10479	+	1.10479i	−0.993824	+	0.110965i	$0.964606\pi$
$620$		0			0
$621$			4.18861i			0.168083i
$622$		0			0
$623$	−28.9737			−1.16081
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$			22.3246i			0.891557i
$628$		0			0
$629$	2.23025	−	2.23025i	0.0889259	−	0.0889259i
$630$		0			0
$631$	−0.230249	+	0.230249i	−0.00916609	+	0.00916609i	−0.711675	−	0.702509i	$-0.752063\pi$
$631$	−0.230249	+	0.230249i	−0.00916609	+	0.00916609i	0.702509	+	0.711675i	$0.252063\pi$
$632$		0			0
$633$	15.6491			0.621996
$634$		0			0
$635$	−23.9737	+	23.9737i	−0.951366	+	0.951366i
$636$		0			0
$637$	7.16228	−	0.837722i	0.283780	−	0.0331918i
$638$		0			0
$639$	3.16228	+	3.16228i	0.125098	+	0.125098i
$640$		0			0
$641$			34.3246i			1.35574i	0.735183	+	0.677869i	$0.237096\pi$
$641$			34.3246i			1.35574i	−0.735183	+	0.677869i	$0.762904\pi$
$642$		0			0
$643$	6.67544	+	6.67544i	0.263254	+	0.263254i	0.826375	−	0.563121i	$-0.190399\pi$
$643$	6.67544	+	6.67544i	0.263254	+	0.263254i	−0.563121	+	0.826375i	$0.690399\pi$
$644$		0			0
$645$	−7.90569	−	7.90569i	−0.311286	−	0.311286i
$646$		0			0
$647$	14.9737			0.588676			0.294338	−	0.955701i	$-0.404901\pi$
$647$	14.9737			0.588676			0.294338	+	0.955701i	$0.404901\pi$
$648$		0			0
$649$		−	17.2982i		−	0.679015i
$650$		0			0
$651$			16.3246i			0.639810i
$652$		0			0
$653$	−42.9737			−1.68169			−0.840845	−	0.541276i	$-0.817941\pi$
$653$	−42.9737			−1.68169			−0.840845	+	0.541276i	$0.817941\pi$
$654$		0			0
$655$	−21.5811	−	21.5811i	−0.843245	−	0.843245i
$656$		0			0
$657$	−12.0000	−	12.0000i	−0.468165	−	0.468165i
$658$		0			0
$659$			8.97367i			0.349564i	0.984607	+	0.174782i	$0.0559221\pi$
$659$			8.97367i			0.349564i	−0.984607	+	0.174782i	$0.944078\pi$
$660$		0			0
$661$	−16.8377	−	16.8377i	−0.654911	−	0.654911i	0.299260	−	0.954172i	$-0.403260\pi$
$661$	−16.8377	−	16.8377i	−0.654911	−	0.654911i	−0.954172	+	0.299260i	$0.903260\pi$
$662$		0			0
$663$	−2.23025	−	19.0680i	−0.0866157	−	0.740539i
$664$		0			0
$665$	−25.8114	+	25.8114i	−1.00092	+	1.00092i
$666$		0			0
$667$	−4.32456			−0.167447
$668$		0			0
$669$	13.0680	−	13.0680i	0.505237	−	0.505237i
$670$		0			0
$671$	−4.32456	+	4.32456i	−0.166948	+	0.166948i
$672$		0			0
$673$		−	19.9737i		−	0.769928i	−0.922932	−	0.384964i	$-0.874214\pi$
$673$		−	19.9737i		−	0.769928i	0.922932	−	0.384964i	$-0.125786\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10.8377			−0.416528			−0.208264	−	0.978073i	$-0.566781\pi$
$677$	−10.8377			−0.416528			−0.208264	+	0.978073i	$0.566781\pi$
$678$		0			0
$679$			32.1359i			1.23326i
$680$		0			0
$681$	−3.16228	+	3.16228i	−0.121179	+	0.121179i
$682$		0			0
$683$	−0.811388	+	0.811388i	−0.0310469	+	0.0310469i	−0.722460	−	0.691413i	$-0.756989\pi$
$683$	−0.811388	+	0.811388i	−0.0310469	+	0.0310469i	0.691413	+	0.722460i	$0.256989\pi$
$684$		0			0
$685$	9.48683			0.362473
$686$		0			0
$687$	−9.90569	+	9.90569i	−0.377926	+	0.377926i
$688$		0			0
$689$	−3.97367	−	33.9737i	−0.151385	−	1.29429i
$690$		0			0
$691$	2.64911	+	2.64911i	0.100777	+	0.100777i	0.755698	−	0.654921i	$-0.227298\pi$
$691$	2.64911	+	2.64911i	0.100777	+	0.100777i	−0.654921	+	0.755698i	$0.727298\pi$
$692$		0			0
$693$		−	13.6754i		−	0.519487i
$694$		0			0
$695$	−4.23025	−	4.23025i	−0.160463	−	0.160463i
$696$		0			0
$697$	6.18861	+	6.18861i	0.234410	+	0.234410i
$698$		0			0
$699$	−5.32456			−0.201393
$700$		0			0
$701$			31.1623i			1.17698i	0.808503	+	0.588491i	$0.200278\pi$
$701$			31.1623i			1.17698i	−0.808503	+	0.588491i	$0.799722\pi$
$702$		0			0
$703$			4.32456i			0.163104i
$704$		0			0
$705$	−8.67544			−0.326736
$706$		0			0
$707$	−3.67544	−	3.67544i	−0.138229	−	0.138229i
$708$		0			0
$709$	−9.35089	−	9.35089i	−0.351180	−	0.351180i	0.509368	−	0.860549i	$-0.329879\pi$
$709$	−9.35089	−	9.35089i	−0.351180	−	0.351180i	−0.860549	+	0.509368i	$0.829879\pi$
$710$		0			0
$711$			30.9737i			1.16160i
$712$		0			0
$713$	4.32456	+	4.32456i	0.161956	+	0.161956i
$714$		0			0
$715$	24.4868	−	2.86406i	0.915756	−	0.107110i
$716$		0			0
$717$	15.5811	−	15.5811i	0.581888	−	0.581888i
$718$		0			0
$719$	−0.188612			−0.00703403			−0.00351701	−	0.999994i	$-0.501120\pi$
$719$	−0.188612			−0.00703403			−0.00351701	+	0.999994i	$0.501120\pi$
$720$		0			0
$721$	18.1623	−	18.1623i	0.676399	−	0.676399i
$722$		0			0
$723$	15.4868	−	15.4868i	0.575962	−	0.575962i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−34.4605			−1.27807			−0.639035	−	0.769178i	$-0.720666\pi$
$727$	−34.4605			−1.27807			−0.639035	+	0.769178i	$0.720666\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		−	26.6228i		−	0.984679i
$732$		0			0
$733$	2.41886	−	2.41886i	0.0893427	−	0.0893427i	−0.661023	−	0.750366i	$-0.729877\pi$
$733$	2.41886	−	2.41886i	0.0893427	−	0.0893427i	0.750366	+	0.661023i	$0.229877\pi$
$734$		0			0
$735$	3.16228	−	3.16228i	0.116642	−	0.116642i
$736$		0			0
$737$	−23.0263			−0.848186
$738$		0			0
$739$	−2.83772	+	2.83772i	−0.104387	+	0.104387i	−0.757372	−	0.652984i	$-0.773517\pi$
$739$	−2.83772	+	2.83772i	−0.104387	+	0.104387i	0.652984	+	0.757372i	$0.273517\pi$
$740$		0			0
$741$	20.6491	+	16.3246i	0.758564	+	0.599698i
$742$		0			0
$743$	−28.2302	−	28.2302i	−1.03567	−	1.03567i	−0.999340	−	0.0363275i	$-0.988434\pi$
$743$	−28.2302	−	28.2302i	−1.03567	−	1.03567i	−0.0363275	−	0.999340i	$-0.511566\pi$
$744$		0			0
$745$		−	6.32456i		−	0.231714i
$746$		0			0
$747$	−24.3246	−	24.3246i	−0.889989	−	0.889989i
$748$		0			0
$749$	0.513167	+	0.513167i	0.0187507	+	0.0187507i
$750$		0			0
$751$	11.0263			0.402357			0.201178	−	0.979555i	$-0.435523\pi$
$751$	11.0263			0.402357			0.201178	+	0.979555i	$0.435523\pi$
$752$		0			0
$753$		−	18.6491i		−	0.679611i
$754$		0			0
$755$		−	33.9737i		−	1.23643i
$756$		0			0
$757$	−28.4605			−1.03441			−0.517207	−	0.855860i	$-0.673028\pi$
$757$	−28.4605			−1.03441			−0.517207	+	0.855860i	$0.673028\pi$
$758$		0			0
$759$	1.81139	+	1.81139i	0.0657492	+	0.0657492i
$760$		0			0
$761$	18.3246	+	18.3246i	0.664265	+	0.664265i	0.956382	−	0.292118i	$-0.0943599\pi$
$761$	18.3246	+	18.3246i	0.664265	+	0.664265i	−0.292118	+	0.956382i	$0.594360\pi$
$762$		0			0
$763$		−	33.9737i		−	1.22993i
$764$		0			0
$765$	16.8377	+	16.8377i	0.608769	+	0.608769i
$766$		0			0
$767$	−16.0000	−	12.6491i	−0.577727	−	0.456733i
$768$		0			0
$769$	−24.6491	+	24.6491i	−0.888870	+	0.888870i	−0.994415	−	0.105545i	$-0.966341\pi$
$769$	−24.6491	+	24.6491i	−0.888870	+	0.888870i	0.105545	+	0.994415i	$0.466341\pi$
$770$		0			0
$771$	−29.6491			−1.06779
$772$		0			0
$773$	30.7434	−	30.7434i	1.10576	−	1.10576i	0.112063	−	0.993701i	$-0.464254\pi$
$773$	30.7434	−	30.7434i	1.10576	−	1.10576i	0.993701	−	0.112063i	$-0.0357457\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			1.32456i			0.0475182i
$778$		0			0
$779$	−12.0000			−0.429945
$780$		0			0
$781$	6.83772			0.244673
$782$		0			0
$783$		−	25.8114i		−	0.922424i
$784$		0			0
$785$	23.6754	−	23.6754i	0.845013	−	0.845013i
$786$		0			0
$787$	−27.4868	+	27.4868i	−0.979800	+	0.979800i	−0.999800	−	0.0200002i	$-0.993633\pi$
$787$	−27.4868	+	27.4868i	−0.979800	+	0.979800i	0.0200002	+	0.999800i	$0.493633\pi$
$788$		0			0
$789$	6.00000			0.213606
$790$		0			0
$791$	25.2982	−	25.2982i	0.899501	−	0.899501i

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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.58114 + 1.58114i) q^{5} +(1.58114 - 1.58114i) q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-1.58114 + 1.58114i) q^{5} +(1.58114 - 1.58114i) q^{7} +2.00000 q^{9} +(2.16228 - 2.16228i) q^{11} +(-0.418861 - 3.58114i) q^{13} +(-1.58114 - 1.58114i) q^{15} -5.32456i q^{17} +(5.16228 + 5.16228i) q^{19} +(1.58114 + 1.58114i) q^{21} +0.837722 q^{23} +5.00000i q^{27} -5.16228 q^{29} +(5.16228 + 5.16228i) q^{31} +(2.16228 + 2.16228i) q^{33} +5.00000i q^{35} +(0.418861 + 0.418861i) q^{37} +(3.58114 - 0.418861i) q^{39} +(-1.16228 + 1.16228i) q^{41} +5.00000 q^{43} +(-3.16228 + 3.16228i) q^{45} +(2.74342 - 2.74342i) q^{47} +2.00000i q^{49} +5.32456 q^{51} +9.48683 q^{53} +6.83772i q^{55} +(-5.16228 + 5.16228i) q^{57} +(4.00000 - 4.00000i) q^{59} -2.00000 q^{61} +(3.16228 - 3.16228i) q^{63} +(6.32456 + 5.00000i) q^{65} +(-5.32456 - 5.32456i) q^{67} +0.837722i q^{69} +(1.58114 + 1.58114i) q^{71} +(-6.00000 - 6.00000i) q^{73} -6.83772i q^{77} +15.4868i q^{79} +1.00000 q^{81} +(-12.1623 - 12.1623i) q^{83} +(8.41886 + 8.41886i) q^{85} -5.16228i q^{87} +(-9.16228 - 9.16228i) q^{89} +(-6.32456 - 5.00000i) q^{91} +(-5.16228 + 5.16228i) q^{93} -16.3246 q^{95} +(-10.1623 + 10.1623i) q^{97} +(4.32456 - 4.32456i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{9}+O(q^{10}) \) 4 * q + 8 * q^9	\( 4 q + 8 q^{9} - 4 q^{11} - 8 q^{13} + 8 q^{19} + 16 q^{23} - 8 q^{29} + 8 q^{31} - 4 q^{33} + 8 q^{37} + 8 q^{39} + 8 q^{41} + 20 q^{43} - 8 q^{47} - 4 q^{51} - 8 q^{57} + 16 q^{59} - 8 q^{61} + 4 q^{67} - 24 q^{73} + 4 q^{81} - 36 q^{83} + 40 q^{85} - 24 q^{89} - 8 q^{93} - 40 q^{95} - 28 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q + 8 * q^9 - 4 * q^11 - 8 * q^13 + 8 * q^19 + 16 * q^23 - 8 * q^29 + 8 * q^31 - 4 * q^33 + 8 * q^37 + 8 * q^39 + 8 * q^41 + 20 * q^43 - 8 * q^47 - 4 * q^51 - 8 * q^57 + 16 * q^59 - 8 * q^61 + 4 * q^67 - 24 * q^73 + 4 * q^81 - 36 * q^83 + 40 * q^85 - 24 * q^89 - 8 * q^93 - 40 * q^95 - 28 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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