LMFDB - Embedded newform 416.2.k.d.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(31,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("416.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.k (of order $4$, degree $2$, 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:	$3.32177672409$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.1
Root			$-1.58114 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	416.31
Dual form			416.2.k.d.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-1.58114 + 1.58114i) q^{5} +(-1.58114 + 1.58114i) q^{7} +2.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{3} +(-1.58114 + 1.58114i) q^{5} +(-1.58114 + 1.58114i) q^{7} +2.00000 q^{9} +(4.16228 - 4.16228i) q^{11} +(3.58114 + 0.418861i) q^{13} +(1.58114 + 1.58114i) q^{15} +7.32456i q^{17} +(1.16228 + 1.16228i) q^{19} +(1.58114 + 1.58114i) q^{21} +7.16228 q^{23} -5.00000i q^{27} -1.16228 q^{29} +(-1.16228 - 1.16228i) q^{31} +(-4.16228 - 4.16228i) q^{33} -5.00000i q^{35} +(-3.58114 - 3.58114i) q^{37} +(0.418861 - 3.58114i) q^{39} +(5.16228 - 5.16228i) q^{41} -5.00000 q^{43} +(-3.16228 + 3.16228i) q^{45} +(-6.74342 + 6.74342i) q^{47} +2.00000i q^{49} +7.32456 q^{51} +9.48683 q^{53} +13.1623i q^{55} +(1.16228 - 1.16228i) q^{57} +(-4.00000 + 4.00000i) q^{59} +2.00000 q^{61} +(-3.16228 + 3.16228i) q^{63} +(-6.32456 + 5.00000i) q^{65} +(-7.32456 - 7.32456i) q^{67} -7.16228i q^{69} +(-1.58114 - 1.58114i) q^{71} +(-6.00000 - 6.00000i) q^{73} +13.1623i q^{77} -3.48683i q^{79} +1.00000 q^{81} +(5.83772 + 5.83772i) q^{83} +(-11.5811 - 11.5811i) q^{85} +1.16228i q^{87} +(-2.83772 - 2.83772i) q^{89} +(-6.32456 + 5.00000i) q^{91} +(-1.16228 + 1.16228i) q^{93} -3.67544 q^{95} +(-3.83772 + 3.83772i) q^{97} +(8.32456 - 8.32456i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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	$-0.906785\pi$
$3$		−	1.00000i		−	0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$		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	−0.939677	−	0.342063i	$-0.888874\pi$
$7$	−1.58114	+	1.58114i	−0.597614	+	0.597614i	0.342063	+	0.939677i	$0.388874\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$	4.16228	−	4.16228i	1.25497	−	1.25497i	0.301511	−	0.953463i	$-0.402509\pi$
$11$	4.16228	−	4.16228i	1.25497	−	1.25497i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$		0			0
$13$	3.58114	+	0.418861i	0.993229	+	0.116171i
$13$	3.58114	+	0.418861i	0.993229	+	0.116171i
$14$		0			0
$15$	1.58114	+	1.58114i	0.408248	+	0.408248i
$16$		0			0
$17$			7.32456i			1.77647i	0.459394	+	0.888233i	$0.348067\pi$
$17$			7.32456i			1.77647i	−0.459394	+	0.888233i	$0.651933\pi$
$18$		0			0
$19$	1.16228	+	1.16228i	0.266645	+	0.266645i	0.827747	−	0.561102i	$-0.189622\pi$
$19$	1.16228	+	1.16228i	0.266645	+	0.266645i	−0.561102	+	0.827747i	$0.689622\pi$
$20$		0			0
$21$	1.58114	+	1.58114i	0.345033	+	0.345033i
$22$		0			0
$23$	7.16228			1.49344			0.746719	−	0.665140i	$-0.231628\pi$
$23$	7.16228			1.49344			0.746719	+	0.665140i	$0.231628\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	5.00000i		−	0.962250i
$28$		0			0
$29$	−1.16228			−0.215830			−0.107915	−	0.994160i	$-0.534417\pi$
$29$	−1.16228			−0.215830			−0.107915	+	0.994160i	$0.534417\pi$
$30$		0			0
$31$	−1.16228	−	1.16228i	−0.208751	−	0.208751i	0.594985	−	0.803737i	$-0.297158\pi$
$31$	−1.16228	−	1.16228i	−0.208751	−	0.208751i	−0.803737	+	0.594985i	$0.797158\pi$
$32$		0			0
$33$	−4.16228	−	4.16228i	−0.724560	−	0.724560i
$34$		0			0
$35$		−	5.00000i		−	0.845154i
$36$		0			0
$37$	−3.58114	−	3.58114i	−0.588736	−	0.588736i	0.348553	−	0.937289i	$-0.386673\pi$
$37$	−3.58114	−	3.58114i	−0.588736	−	0.588736i	−0.937289	+	0.348553i	$0.886673\pi$
$38$		0			0
$39$	0.418861	−	3.58114i	0.0670715	−	0.573441i
$40$		0			0
$41$	5.16228	−	5.16228i	0.806212	−	0.806212i	−0.177846	−	0.984058i	$-0.556913\pi$
$41$	5.16228	−	5.16228i	0.806212	−	0.806212i	0.984058	+	0.177846i	$0.0569129\pi$
$42$		0			0
$43$	−5.00000			−0.762493			−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000			−0.762493			−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$	−3.16228	+	3.16228i	−0.471405	+	0.471405i
$46$		0			0
$47$	−6.74342	+	6.74342i	−0.983628	+	0.983628i	−0.999868	−	0.0162397i	$-0.994831\pi$
$47$	−6.74342	+	6.74342i	−0.983628	+	0.983628i	0.0162397	+	0.999868i	$0.494831\pi$
$48$		0			0
$49$			2.00000i			0.285714i
$50$		0			0
$51$	7.32456			1.02564
$52$		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
$55$			13.1623i			1.77480i
$56$		0			0
$57$	1.16228	−	1.16228i	0.153947	−	0.153947i
$58$		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
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		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
$67$	−7.32456	−	7.32456i	−0.894837	−	0.894837i	0.100137	−	0.994974i	$-0.468072\pi$
$67$	−7.32456	−	7.32456i	−0.894837	−	0.894837i	−0.994974	+	0.100137i	$0.968072\pi$
$68$		0			0
$69$		−	7.16228i		−	0.862237i
$70$		0			0
$71$	−1.58114	−	1.58114i	−0.187647	−	0.187647i	0.607031	−	0.794678i	$-0.292360\pi$
$71$	−1.58114	−	1.58114i	−0.187647	−	0.187647i	−0.794678	+	0.607031i	$0.792360\pi$
$72$		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
$77$			13.1623i			1.49998i
$78$		0			0
$79$		−	3.48683i		−	0.392299i	−0.980574	−	0.196150i	$-0.937156\pi$
$79$		−	3.48683i		−	0.392299i	0.980574	−	0.196150i	$-0.0628438\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	5.83772	+	5.83772i	0.640773	+	0.640773i	0.950746	−	0.309972i	$-0.100320\pi$
$83$	5.83772	+	5.83772i	0.640773	+	0.640773i	−0.309972	+	0.950746i	$0.600320\pi$
$84$		0			0
$85$	−11.5811	−	11.5811i	−1.25615	−	1.25615i
$86$		0			0
$87$			1.16228i			0.124609i
$88$		0			0
$89$	−2.83772	−	2.83772i	−0.300798	−	0.300798i	0.540528	−	0.841326i	$-0.318224\pi$
$89$	−2.83772	−	2.83772i	−0.300798	−	0.300798i	−0.841326	+	0.540528i	$0.818224\pi$
$90$		0			0
$91$	−6.32456	+	5.00000i	−0.662994	+	0.524142i
$92$		0			0
$93$	−1.16228	+	1.16228i	−0.120523	+	0.120523i
$94$		0			0
$95$	−3.67544			−0.377093
$96$		0			0
$97$	−3.83772	+	3.83772i	−0.389662	+	0.389662i	−0.874567	−	0.484905i	$-0.838854\pi$
$97$	−3.83772	+	3.83772i	−0.389662	+	0.389662i	0.484905	+	0.874567i	$0.338854\pi$
$98$		0			0
$99$	8.32456	−	8.32456i	0.836649	−	0.836649i
$100$		0			0
$101$		−	10.3246i		−	1.02733i	−0.857990	−	0.513666i	$-0.828287\pi$
$101$		−	10.3246i		−	1.02733i	0.857990	−	0.513666i	$-0.171713\pi$
$102$		0			0
$103$	−7.48683			−0.737700			−0.368850	−	0.929489i	$-0.620248\pi$
$103$	−7.48683			−0.737700			−0.368850	+	0.929489i	$0.620248\pi$
$104$		0			0
$105$	−5.00000			−0.487950
$106$		0			0
$107$			12.3246i			1.19146i	0.803185	+	0.595730i	$0.203137\pi$
$107$			12.3246i			1.19146i	−0.803185	+	0.595730i	$0.796863\pi$
$108$		0			0
$109$	−1.25658	+	1.25658i	−0.120359	+	0.120359i	−0.764721	−	0.644362i	$-0.777123\pi$
$109$	−1.25658	+	1.25658i	−0.120359	+	0.120359i	0.644362	+	0.764721i	$0.277123\pi$
$110$		0			0
$111$	−3.58114	+	3.58114i	−0.339907	+	0.339907i
$112$		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
$115$	−11.3246	+	11.3246i	−1.05602	+	1.05602i
$116$		0			0
$117$	7.16228	+	0.837722i	0.662153	+	0.0774475i
$118$		0			0
$119$	−11.5811	−	11.5811i	−1.06164	−	1.06164i
$120$		0			0
$121$		−	23.6491i		−	2.14992i
$122$		0			0
$123$	−5.16228	−	5.16228i	−0.465467	−	0.465467i
$124$		0			0
$125$	−7.90569	−	7.90569i	−0.707107	−	0.707107i
$126$		0			0
$127$	8.83772			0.784221			0.392111	−	0.919918i	$-0.371745\pi$
$127$	8.83772			0.784221			0.392111	+	0.919918i	$0.371745\pi$
$128$		0			0
$129$			5.00000i			0.440225i
$130$		0			0
$131$			11.6491i			1.01779i	0.860829	+	0.508894i	$0.169945\pi$
$131$			11.6491i			1.01779i	−0.860829	+	0.508894i	$0.830055\pi$
$132$		0			0
$133$	−3.67544			−0.318701
$134$		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
$139$		−	15.3246i		−	1.29981i	−0.760015	−	0.649906i	$-0.774808\pi$
$139$		−	15.3246i		−	1.29981i	0.760015	−	0.649906i	$-0.225192\pi$
$140$		0			0
$141$	6.74342	+	6.74342i	0.567898	+	0.567898i
$142$		0			0
$143$	16.6491	−	13.1623i	1.39227	−	1.10068i
$144$		0			0
$145$	1.83772	−	1.83772i	0.152615	−	0.152615i
$146$		0			0
$147$	2.00000			0.164957
$148$		0			0
$149$	2.00000	−	2.00000i	0.163846	−	0.163846i	−0.620422	−	0.784268i	$-0.713039\pi$
$149$	2.00000	−	2.00000i	0.163846	−	0.163846i	0.784268	+	0.620422i	$0.213039\pi$
$150$		0			0
$151$	−1.25658	+	1.25658i	−0.102259	+	0.102259i	−0.756385	−	0.654126i	$-0.773037\pi$
$151$	−1.25658	+	1.25658i	−0.102259	+	0.102259i	0.654126	+	0.756385i	$0.273037\pi$
$152$		0			0
$153$			14.6491i			1.18431i
$154$		0			0
$155$	3.67544			0.295219
$156$		0			0
$157$	−22.9737			−1.83350			−0.916749	−	0.399464i	$-0.869196\pi$
$157$	−22.9737			−1.83350			−0.916749	+	0.399464i	$0.869196\pi$
$158$		0			0
$159$		−	9.48683i		−	0.752355i
$160$		0			0
$161$	−11.3246	+	11.3246i	−0.892500	+	0.892500i
$162$		0			0
$163$	−9.00000	+	9.00000i	−0.704934	+	0.704934i	−0.965465	−	0.260531i	$-0.916102\pi$
$163$	−9.00000	+	9.00000i	−0.704934	+	0.704934i	0.260531	+	0.965465i	$0.416102\pi$
$164$		0			0
$165$	13.1623			1.02468
$166$		0			0
$167$	17.4868	−	17.4868i	1.35317	−	1.35317i	0.471083	−	0.882089i	$-0.343863\pi$
$167$	17.4868	−	17.4868i	1.35317	−	1.35317i	0.882089	−	0.471083i	$-0.156137\pi$
$168$		0			0
$169$	12.6491	+	3.00000i	0.973009	+	0.230769i
$170$		0			0
$171$	2.32456	+	2.32456i	0.177763	+	0.177763i
$172$		0			0
$173$			3.48683i			0.265099i	0.991176	+	0.132550i	$0.0423164\pi$
$173$			3.48683i			0.265099i	−0.991176	+	0.132550i	$0.957684\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.00000	+	4.00000i	0.300658	+	0.300658i
$178$		0			0
$179$	−15.6491			−1.16967			−0.584835	−	0.811152i	$-0.698841\pi$
$179$	−15.6491			−1.16967			−0.584835	+	0.811152i	$0.698841\pi$
$180$		0			0
$181$			2.83772i			0.210926i	0.994423	+	0.105463i	$0.0336325\pi$
$181$			2.83772i			0.210926i	−0.994423	+	0.105463i	$0.966368\pi$
$182$		0			0
$183$		−	2.00000i		−	0.147844i
$184$		0			0
$185$	11.3246			0.832598
$186$		0			0
$187$	30.4868	+	30.4868i	2.22942	+	2.22942i
$188$		0			0
$189$	7.90569	+	7.90569i	0.575055	+	0.575055i
$190$		0			0
$191$		−	18.8377i		−	1.36305i	−0.731795	−	0.681525i	$-0.761317\pi$
$191$		−	18.8377i		−	1.36305i	0.731795	−	0.681525i	$-0.238683\pi$
$192$		0			0
$193$	−2.16228	−	2.16228i	−0.155644	−	0.155644i	0.624989	−	0.780633i	$-0.285103\pi$
$193$	−2.16228	−	2.16228i	−0.155644	−	0.155644i	−0.780633	+	0.624989i	$0.785103\pi$
$194$		0			0
$195$	5.00000	+	6.32456i	0.358057	+	0.452911i
$196$		0			0
$197$	7.58114	−	7.58114i	0.540134	−	0.540134i	−0.383434	−	0.923568i	$-0.625259\pi$
$197$	7.58114	−	7.58114i	0.540134	−	0.540134i	0.923568	+	0.383434i	$0.125259\pi$
$198$		0			0
$199$	−27.1623			−1.92548			−0.962741	−	0.270424i	$-0.912836\pi$
$199$	−27.1623			−1.92548			−0.962741	+	0.270424i	$0.912836\pi$
$200$		0			0
$201$	−7.32456	+	7.32456i	−0.516634	+	0.516634i
$202$		0			0
$203$	1.83772	−	1.83772i	0.128983	−	0.128983i
$204$		0			0
$205$			16.3246i			1.14016i
$206$		0			0
$207$	14.3246			0.995625
$208$		0			0
$209$	9.67544			0.669265
$210$		0			0
$211$		−	9.64911i		−	0.664272i	−0.943231	−	0.332136i	$-0.892231\pi$
$211$		−	9.64911i		−	0.664272i	0.943231	−	0.332136i	$-0.107769\pi$
$212$		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
$217$	3.67544			0.249505
$218$		0			0
$219$	−6.00000	+	6.00000i	−0.405442	+	0.405442i
$220$		0			0
$221$	−3.06797	+	26.2302i	−0.206374	+	1.76444i
$222$		0			0
$223$	9.06797	+	9.06797i	0.607236	+	0.607236i	0.942223	−	0.334987i	$-0.108732\pi$
$223$	9.06797	+	9.06797i	0.607236	+	0.607236i	−0.334987	+	0.942223i	$0.608732\pi$
$224$		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
$229$	5.90569	+	5.90569i	0.390259	+	0.390259i	0.874780	−	0.484521i	$-0.161006\pi$
$229$	5.90569	+	5.90569i	0.390259	+	0.390259i	−0.484521	+	0.874780i	$0.661006\pi$
$230$		0			0
$231$	13.1623			0.866014
$232$		0			0
$233$		−	7.32456i		−	0.479848i	−0.970792	−	0.239924i	$-0.922878\pi$
$233$		−	7.32456i		−	0.479848i	0.970792	−	0.239924i	$-0.0771225\pi$
$234$		0			0
$235$		−	21.3246i		−	1.39106i
$236$		0			0
$237$	−3.48683			−0.226494
$238$		0			0
$239$	−12.4189	−	12.4189i	−0.803309	−	0.803309i	0.180302	−	0.983611i	$-0.442293\pi$
$239$	−12.4189	−	12.4189i	−0.803309	−	0.803309i	−0.983611	+	0.180302i	$0.942293\pi$
$240$		0			0
$241$	3.48683	+	3.48683i	0.224607	+	0.224607i	0.810435	−	0.585828i	$-0.199231\pi$
$241$	3.48683	+	3.48683i	0.224607	+	0.224607i	−0.585828	+	0.810435i	$0.699231\pi$
$242$		0			0
$243$		−	16.0000i		−	1.02640i
$244$		0			0
$245$	−3.16228	−	3.16228i	−0.202031	−	0.202031i
$246$		0			0
$247$	3.67544	+	4.64911i	0.233863	+	0.295816i
$248$		0			0
$249$	5.83772	−	5.83772i	0.369951	−	0.369951i
$250$		0			0
$251$	−6.64911			−0.419688			−0.209844	−	0.977735i	$-0.567296\pi$
$251$	−6.64911			−0.419688			−0.209844	+	0.977735i	$0.567296\pi$
$252$		0			0
$253$	29.8114	−	29.8114i	1.87423	−	1.87423i
$254$		0			0
$255$	−11.5811	+	11.5811i	−0.725239	+	0.725239i
$256$		0			0
$257$			4.35089i			0.271401i	0.990750	+	0.135701i	$0.0433285\pi$
$257$			4.35089i			0.271401i	−0.990750	+	0.135701i	$0.956672\pi$
$258$		0			0
$259$	11.3246			0.703674
$260$		0			0
$261$	−2.32456			−0.143886
$262$		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
$267$	−2.83772	+	2.83772i	−0.173666	+	0.173666i
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$	−9.06797	+	9.06797i	−0.550840	+	0.550840i	−0.926683	−	0.375843i	$-0.877353\pi$
$271$	−9.06797	+	9.06797i	−0.550840	+	0.550840i	0.375843	+	0.926683i	$0.377353\pi$
$272$		0			0
$273$	5.00000	+	6.32456i	0.302614	+	0.382780i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	12.3246i		−	0.740511i	−0.928930	−	0.370255i	$-0.879270\pi$
$277$		−	12.3246i		−	0.740511i	0.928930	−	0.370255i	$-0.120730\pi$
$278$		0			0
$279$	−2.32456	−	2.32456i	−0.139167	−	0.139167i
$280$		0			0
$281$	1.48683	+	1.48683i	0.0886970	+	0.0886970i	0.750063	−	0.661366i	$-0.230023\pi$
$281$	1.48683	+	1.48683i	0.0886970	+	0.0886970i	−0.661366	+	0.750063i	$0.730023\pi$
$282$		0			0
$283$	0.649111			0.0385856			0.0192928	−	0.999814i	$-0.493859\pi$
$283$	0.649111			0.0385856			0.0192928	+	0.999814i	$0.493859\pi$
$284$		0			0
$285$			3.67544i			0.217715i
$286$		0			0
$287$			16.3246i			0.963608i
$288$		0			0
$289$	−36.6491			−2.15583
$290$		0			0
$291$	3.83772	+	3.83772i	0.224971	+	0.224971i
$292$		0			0
$293$	−7.25658	−	7.25658i	−0.423934	−	0.423934i	0.462622	−	0.886556i	$-0.346909\pi$
$293$	−7.25658	−	7.25658i	−0.423934	−	0.423934i	−0.886556	+	0.462622i	$0.846909\pi$
$294$		0			0
$295$		−	12.6491i		−	0.736460i
$296$		0			0
$297$	−20.8114	−	20.8114i	−1.20760	−	1.20760i
$298$		0			0
$299$	25.6491	+	3.00000i	1.48333	+	0.173494i
$300$		0			0
$301$	7.90569	−	7.90569i	0.455677	−	0.455677i
$302$		0			0
$303$	−10.3246			−0.593130
$304$		0			0
$305$	−3.16228	+	3.16228i	−0.181071	+	0.181071i
$306$		0			0
$307$	−0.324555	+	0.324555i	−0.0185234	+	0.0185234i	−0.716308	−	0.697784i	$-0.754169\pi$
$307$	−0.324555	+	0.324555i	−0.0185234	+	0.0185234i	0.697784	+	0.716308i	$0.254169\pi$
$308$		0			0
$309$			7.48683i			0.425911i
$310$		0			0
$311$	24.9737			1.41613			0.708063	−	0.706149i	$-0.249569\pi$
$311$	24.9737			1.41613			0.708063	+	0.706149i	$0.249569\pi$
$312$		0			0
$313$	15.6491			0.884540			0.442270	−	0.896882i	$-0.354173\pi$
$313$	15.6491			0.884540			0.442270	+	0.896882i	$0.354173\pi$
$314$		0			0
$315$		−	10.0000i		−	0.563436i
$316$		0			0
$317$	−20.3246	+	20.3246i	−1.14154	+	1.14154i	−0.153372	+	0.988168i	$0.549013\pi$
$317$	−20.3246	+	20.3246i	−1.14154	+	1.14154i	−0.988168	+	0.153372i	$0.950987\pi$
$318$		0			0
$319$	−4.83772	+	4.83772i	−0.270860	+	0.270860i
$320$		0			0
$321$	12.3246			0.687890
$322$		0			0
$323$	−8.51317	+	8.51317i	−0.473685	+	0.473685i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.25658	+	1.25658i	0.0694892	+	0.0694892i
$328$		0			0
$329$		−	21.3246i		−	1.17566i
$330$		0			0
$331$	−14.4868	−	14.4868i	−0.796268	−	0.796268i	0.186237	−	0.982505i	$-0.440371\pi$
$331$	−14.4868	−	14.4868i	−0.796268	−	0.796268i	−0.982505	+	0.186237i	$0.940371\pi$
$332$		0			0
$333$	−7.16228	−	7.16228i	−0.392490	−	0.392490i
$334$		0			0
$335$	23.1623			1.26549
$336$		0			0
$337$			9.64911i			0.525621i	0.964848	+	0.262810i	$0.0846493\pi$
$337$			9.64911i			0.525621i	−0.964848	+	0.262810i	$0.915351\pi$
$338$		0			0
$339$		−	16.0000i		−	0.869001i
$340$		0			0
$341$	−9.67544			−0.523955
$342$		0			0
$343$	−14.2302	−	14.2302i	−0.768361	−	0.768361i
$344$		0			0
$345$	11.3246	+	11.3246i	0.609694	+	0.609694i
$346$		0			0
$347$			24.2982i			1.30440i	0.758048	+	0.652198i	$0.226153\pi$
$347$			24.2982i			1.30440i	−0.758048	+	0.652198i	$0.773847\pi$
$348$		0			0
$349$	−2.41886	−	2.41886i	−0.129479	−	0.129479i	0.639398	−	0.768876i	$-0.279184\pi$
$349$	−2.41886	−	2.41886i	−0.129479	−	0.129479i	−0.768876	+	0.639398i	$0.779184\pi$
$350$		0			0
$351$	2.09431	−	17.9057i	0.111786	−	0.955735i
$352$		0			0
$353$	0.162278	−	0.162278i	0.00863717	−	0.00863717i	−0.702775	−	0.711412i	$-0.748056\pi$
$353$	0.162278	−	0.162278i	0.00863717	−	0.00863717i	0.711412	+	0.702775i	$0.248056\pi$
$354$		0			0
$355$	5.00000			0.265372
$356$		0			0
$357$	−11.5811	+	11.5811i	−0.612939	+	0.612939i
$358$		0			0
$359$	−14.8377	+	14.8377i	−0.783105	+	0.783105i	−0.980354	−	0.197248i	$-0.936799\pi$
$359$	−14.8377	+	14.8377i	−0.783105	+	0.783105i	0.197248	+	0.980354i	$0.436799\pi$
$360$		0			0
$361$		−	16.2982i		−	0.857801i
$362$		0			0
$363$	−23.6491			−1.24126
$364$		0			0
$365$	18.9737			0.993127
$366$		0			0
$367$			23.1623i			1.20906i	0.796582	+	0.604531i	$0.206639\pi$
$367$			23.1623i			1.20906i	−0.796582	+	0.604531i	$0.793361\pi$
$368$		0			0
$369$	10.3246	−	10.3246i	0.537475	−	0.537475i
$370$		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
$377$	−4.16228	−	0.486833i	−0.214368	−	0.0250732i
$378$		0			0
$379$	−13.0000	−	13.0000i	−0.667765	−	0.667765i	0.289433	−	0.957198i	$-0.406533\pi$
$379$	−13.0000	−	13.0000i	−0.667765	−	0.667765i	−0.957198	+	0.289433i	$0.906533\pi$
$380$		0			0
$381$		−	8.83772i		−	0.452770i
$382$		0			0
$383$	16.7434	+	16.7434i	0.855549	+	0.855549i	0.990810	−	0.135261i	$-0.0431874\pi$
$383$	16.7434	+	16.7434i	0.855549	+	0.855549i	−0.135261	+	0.990810i	$0.543187\pi$
$384$		0			0
$385$	−20.8114	−	20.8114i	−1.06065	−	1.06065i
$386$		0			0
$387$	−10.0000			−0.508329
$388$		0			0
$389$		−	4.00000i		−	0.202808i	−0.994845	−	0.101404i	$-0.967667\pi$
$389$		−	4.00000i		−	0.202808i	0.994845	−	0.101404i	$-0.0323335\pi$
$390$		0			0
$391$			52.4605i			2.65304i
$392$		0			0
$393$	11.6491			0.587620
$394$		0			0
$395$	5.51317	+	5.51317i	0.277398	+	0.277398i
$396$		0			0
$397$	−23.2982	−	23.2982i	−1.16930	−	1.16930i	−0.982373	−	0.186931i	$-0.940146\pi$
$397$	−23.2982	−	23.2982i	−1.16930	−	1.16930i	−0.186931	−	0.982373i	$-0.559854\pi$
$398$		0			0
$399$			3.67544i			0.184002i
$400$		0			0
$401$	23.9737	+	23.9737i	1.19719	+	1.19719i	0.975005	+	0.222183i	$0.0713181\pi$
$401$	23.9737	+	23.9737i	1.19719	+	1.19719i	0.222183	+	0.975005i	$0.428682\pi$
$402$		0			0
$403$	−3.67544	−	4.64911i	−0.183087	−	0.231589i
$404$		0			0
$405$	−1.58114	+	1.58114i	−0.0785674	+	0.0785674i
$406$		0			0
$407$	−29.8114			−1.47770
$408$		0			0
$409$	23.3246	−	23.3246i	1.15333	−	1.15333i	0.167443	−	0.985882i	$-0.446449\pi$
$409$	23.3246	−	23.3246i	1.15333	−	1.15333i	0.985882	−	0.167443i	$-0.0535511\pi$
$410$		0			0
$411$	−3.00000	+	3.00000i	−0.147979	+	0.147979i
$412$		0			0
$413$		−	12.6491i		−	0.622422i
$414$		0			0
$415$	−18.4605			−0.906190
$416$		0			0
$417$	−15.3246			−0.750447
$418$		0			0
$419$		−	4.67544i		−	0.228410i	−0.993457	−	0.114205i	$-0.963568\pi$
$419$		−	4.67544i		−	0.228410i	0.993457	−	0.114205i	$-0.0364321\pi$
$420$		0			0
$421$	−2.41886	+	2.41886i	−0.117888	+	0.117888i	−0.763590	−	0.645702i	$-0.776565\pi$
$421$	−2.41886	+	2.41886i	−0.117888	+	0.117888i	0.645702	+	0.763590i	$0.276565\pi$
$422$		0			0
$423$	−13.4868	+	13.4868i	−0.655752	+	0.655752i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.16228	+	3.16228i	−0.153033	+	0.153033i
$428$		0			0
$429$	−13.1623	−	16.6491i	−0.635481	−	0.803827i
$430$		0			0
$431$	−23.3925	−	23.3925i	−1.12678	−	1.12678i	−0.990698	−	0.136081i	$-0.956549\pi$
$431$	−23.3925	−	23.3925i	−1.12678	−	1.12678i	−0.136081	−	0.990698i	$-0.543451\pi$
$432$		0			0
$433$			9.97367i			0.479304i	0.970859	+	0.239652i	$0.0770333\pi$
$433$			9.97367i			0.479304i	−0.970859	+	0.239652i	$0.922967\pi$
$434$		0			0
$435$	−1.83772	−	1.83772i	−0.0881120	−	0.0881120i
$436$		0			0
$437$	8.32456	+	8.32456i	0.398217	+	0.398217i
$438$		0			0
$439$	−0.649111			−0.0309804			−0.0154902	−	0.999880i	$-0.504931\pi$
$439$	−0.649111			−0.0309804			−0.0154902	+	0.999880i	$0.504931\pi$
$440$		0			0
$441$			4.00000i			0.190476i
$442$		0			0
$443$		−	30.2982i		−	1.43951i	−0.694227	−	0.719756i	$-0.744254\pi$
$443$		−	30.2982i		−	1.43951i	0.694227	−	0.719756i	$-0.255746\pi$
$444$		0			0
$445$	8.97367			0.425393
$446$		0			0
$447$	−2.00000	−	2.00000i	−0.0945968	−	0.0945968i
$448$		0			0
$449$	−26.4868	−	26.4868i	−1.24999	−	1.24999i	−0.955723	−	0.294268i	$-0.904924\pi$
$449$	−26.4868	−	26.4868i	−1.24999	−	1.24999i	−0.294268	−	0.955723i	$-0.595076\pi$
$450$		0			0
$451$		−	42.9737i		−	2.02355i
$452$		0			0
$453$	1.25658	+	1.25658i	0.0590394	+	0.0590394i
$454$		0			0
$455$	2.09431	−	17.9057i	0.0981826	−	0.839432i
$456$		0			0
$457$	−25.8114	+	25.8114i	−1.20741	+	1.20741i	−0.235542	+	0.971864i	$0.575686\pi$
$457$	−25.8114	+	25.8114i	−1.20741	+	1.20741i	−0.971864	+	0.235542i	$0.924314\pi$
$458$		0			0
$459$	36.6228			1.70940
$460$		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
$463$	−0.837722	+	0.837722i	−0.0389323	+	0.0389323i	−0.726305	−	0.687373i	$-0.758764\pi$
$463$	−0.837722	+	0.837722i	−0.0389323	+	0.0389323i	0.687373	+	0.726305i	$0.258764\pi$
$464$		0			0
$465$		−	3.67544i		−	0.170445i
$466$		0			0
$467$	9.35089			0.432708			0.216354	−	0.976315i	$-0.430584\pi$
$467$	9.35089			0.432708			0.216354	+	0.976315i	$0.430584\pi$
$468$		0			0
$469$	23.1623			1.06953
$470$		0			0
$471$			22.9737i			1.05857i
$472$		0			0
$473$	−20.8114	+	20.8114i	−0.956909	+	0.956909i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	18.9737			0.868744
$478$		0			0
$479$	5.39253	−	5.39253i	0.246391	−	0.246391i	−0.573097	−	0.819488i	$-0.694258\pi$
$479$	5.39253	−	5.39253i	0.246391	−	0.246391i	0.819488	+	0.573097i	$0.194258\pi$
$480$		0			0
$481$	−11.3246	−	14.3246i	−0.516355	−	0.653144i
$482$		0			0
$483$	11.3246	+	11.3246i	0.515285	+	0.515285i
$484$		0			0
$485$		−	12.1359i		−	0.551065i
$486$		0			0
$487$	27.8114	+	27.8114i	1.26025	+	1.26025i	0.950969	+	0.309285i	$0.100090\pi$
$487$	27.8114	+	27.8114i	1.26025	+	1.26025i	0.309285	+	0.950969i	$0.399910\pi$
$488$		0			0
$489$	9.00000	+	9.00000i	0.406994	+	0.406994i
$490$		0			0
$491$	−28.6754			−1.29410			−0.647052	−	0.762446i	$-0.723998\pi$
$491$	−28.6754			−1.29410			−0.647052	+	0.762446i	$0.723998\pi$
$492$		0			0
$493$		−	8.51317i		−	0.383414i
$494$		0			0
$495$			26.3246i			1.18320i
$496$		0			0
$497$	5.00000			0.224281
$498$		0			0
$499$	15.4868	+	15.4868i	0.693286	+	0.693286i	0.962953	−	0.269668i	$-0.0869138\pi$
$499$	15.4868	+	15.4868i	0.693286	+	0.693286i	−0.269668	+	0.962953i	$0.586914\pi$
$500$		0			0
$501$	−17.4868	−	17.4868i	−0.781254	−	0.781254i
$502$		0			0
$503$			14.6491i			0.653172i	0.945168	+	0.326586i	$0.105898\pi$
$503$			14.6491i			0.653172i	−0.945168	+	0.326586i	$0.894102\pi$
$504$		0			0
$505$	16.3246	+	16.3246i	0.726433	+	0.726433i
$506$		0			0
$507$	3.00000	−	12.6491i	0.133235	−	0.561767i
$508$		0			0
$509$	2.00000	−	2.00000i	0.0886484	−	0.0886484i	−0.661392	−	0.750040i	$-0.730034\pi$
$509$	2.00000	−	2.00000i	0.0886484	−	0.0886484i	0.750040	+	0.661392i	$0.230034\pi$
$510$		0			0
$511$	18.9737			0.839346
$512$		0			0
$513$	5.81139	−	5.81139i	0.256579	−	0.256579i
$514$		0			0
$515$	11.8377	−	11.8377i	0.521632	−	0.521632i
$516$		0			0
$517$			56.1359i			2.46886i
$518$		0			0
$519$	3.48683			0.153055
$520$		0			0
$521$	−10.6754			−0.467700			−0.233850	−	0.972273i	$-0.575132\pi$
$521$	−10.6754			−0.467700			−0.233850	+	0.972273i	$0.575132\pi$
$522$		0			0
$523$			0.324555i			0.0141918i	0.999975	+	0.00709591i	$0.00225872\pi$
$523$			0.324555i			0.0141918i	−0.999975	+	0.00709591i	$0.997741\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.51317	−	8.51317i	0.370839	−	0.370839i
$528$		0			0
$529$	28.2982			1.23036
$530$		0			0
$531$	−8.00000	+	8.00000i	−0.347170	+	0.347170i
$532$		0			0
$533$	20.6491	−	16.3246i	0.894412	−	0.707095i
$534$		0			0
$535$	−19.4868	−	19.4868i	−0.842489	−	0.842489i
$536$		0			0
$537$			15.6491i			0.675309i
$538$		0			0
$539$	8.32456	+	8.32456i	0.358564	+	0.358564i
$540$		0			0
$541$	15.0680	+	15.0680i	0.647823	+	0.647823i	0.952466	−	0.304644i	$-0.0985374\pi$
$541$	15.0680	+	15.0680i	0.647823	+	0.647823i	−0.304644	+	0.952466i	$0.598537\pi$
$542$		0			0
$543$	2.83772			0.121778
$544$		0			0
$545$		−	3.97367i		−	0.170213i
$546$		0			0
$547$		−	23.3246i		−	0.997286i	−0.866807	−	0.498643i	$-0.833832\pi$
$547$		−	23.3246i		−	0.997286i	0.866807	−	0.498643i	$-0.166168\pi$
$548$		0			0
$549$	4.00000			0.170716
$550$		0			0
$551$	−1.35089	−	1.35089i	−0.0575498	−	0.0575498i
$552$		0			0
$553$	5.51317	+	5.51317i	0.234444	+	0.234444i
$554$		0			0
$555$		−	11.3246i		−	0.480701i
$556$		0			0
$557$	−9.58114	−	9.58114i	−0.405966	−	0.405966i	0.474363	−	0.880329i	$-0.342678\pi$
$557$	−9.58114	−	9.58114i	−0.405966	−	0.405966i	−0.880329	+	0.474363i	$0.842678\pi$
$558$		0			0
$559$	−17.9057	−	2.09431i	−0.757330	−	0.0885797i
$560$		0			0
$561$	30.4868	−	30.4868i	1.28716	−	1.28716i
$562$		0			0
$563$	13.3246			0.561563			0.280782	−	0.959772i	$-0.409406\pi$
$563$	13.3246			0.561563			0.280782	+	0.959772i	$0.409406\pi$
$564$		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
$569$			3.97367i			0.166585i	0.996525	+	0.0832924i	$0.0265435\pi$
$569$			3.97367i			0.166585i	−0.996525	+	0.0832924i	$0.973456\pi$
$570$		0			0
$571$	13.9737			0.584780			0.292390	−	0.956299i	$-0.405550\pi$
$571$	13.9737			0.584780			0.292390	+	0.956299i	$0.405550\pi$
$572$		0			0
$573$	−18.8377			−0.786957
$574$		0			0
$575$		0			0
$576$		0			0
$577$	20.9737	−	20.9737i	0.873145	−	0.873145i	−0.119669	−	0.992814i	$-0.538183\pi$
$577$	20.9737	−	20.9737i	0.873145	−	0.873145i	0.992814	+	0.119669i	$0.0381832\pi$
$578$		0			0
$579$	−2.16228	+	2.16228i	−0.0898612	+	0.0898612i
$580$		0			0
$581$	−18.4605			−0.765871
$582$		0			0
$583$	39.4868	−	39.4868i	1.63538	−	1.63538i
$584$		0			0
$585$	−12.6491	+	10.0000i	−0.522976	+	0.413449i
$586$		0			0
$587$	7.00000	+	7.00000i	0.288921	+	0.288921i	0.836653	−	0.547733i	$-0.184509\pi$
$587$	7.00000	+	7.00000i	0.288921	+	0.288921i	−0.547733	+	0.836653i	$0.684509\pi$
$588$		0			0
$589$		−	2.70178i		−	0.111325i
$590$		0			0
$591$	−7.58114	−	7.58114i	−0.311846	−	0.311846i
$592$		0			0
$593$	−12.4868	−	12.4868i	−0.512773	−	0.512773i	0.402602	−	0.915375i	$-0.368106\pi$
$593$	−12.4868	−	12.4868i	−0.512773	−	0.512773i	−0.915375	+	0.402602i	$0.868106\pi$
$594$		0			0
$595$	36.6228			1.50139
$596$		0			0
$597$			27.1623i			1.11168i
$598$		0			0
$599$		−	27.6228i		−	1.12864i	−0.825557	−	0.564318i	$-0.809139\pi$
$599$		−	27.6228i		−	1.12864i	0.825557	−	0.564318i	$-0.190861\pi$
$600$		0			0
$601$	−30.9473			−1.26237			−0.631184	−	0.775633i	$-0.717431\pi$
$601$	−30.9473			−1.26237			−0.631184	+	0.775633i	$0.717431\pi$
$602$		0			0
$603$	−14.6491	−	14.6491i	−0.596558	−	0.596558i
$604$		0			0
$605$	37.3925	+	37.3925i	1.52022	+	1.52022i
$606$		0			0
$607$			18.9737i			0.770117i	0.922892	+	0.385059i	$0.125819\pi$
$607$			18.9737i			0.770117i	−0.922892	+	0.385059i	$0.874181\pi$
$608$		0			0
$609$	−1.83772	−	1.83772i	−0.0744683	−	0.0744683i
$610$		0			0
$611$	−26.9737	+	21.3246i	−1.09124	+	0.862699i
$612$		0			0
$613$	2.51317	−	2.51317i	0.101506	−	0.101506i	−0.654530	−	0.756036i	$-0.727133\pi$
$613$	2.51317	−	2.51317i	0.101506	−	0.101506i	0.756036	+	0.654530i	$0.227133\pi$
$614$		0			0
$615$	16.3246			0.658270
$616$		0			0
$617$	−14.9737	+	14.9737i	−0.602817	+	0.602817i	−0.941059	−	0.338242i	$-0.890168\pi$
$617$	−14.9737	+	14.9737i	−0.602817	+	0.602817i	0.338242	+	0.941059i	$0.390168\pi$
$618$		0			0
$619$	8.51317	−	8.51317i	0.342173	−	0.342173i	−0.515011	−	0.857184i	$-0.672212\pi$
$619$	8.51317	−	8.51317i	0.342173	−	0.342173i	0.857184	+	0.515011i	$0.172212\pi$
$620$		0			0
$621$		−	35.8114i		−	1.43706i
$622$		0			0
$623$	8.97367			0.359522
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		−	9.67544i		−	0.386400i
$628$		0			0
$629$	26.2302	−	26.2302i	1.04587	−	1.04587i
$630$		0			0
$631$	28.2302	−	28.2302i	1.12383	−	1.12383i	0.132668	−	0.991161i	$-0.457646\pi$
$631$	28.2302	−	28.2302i	1.12383	−	1.12383i	0.991161	−	0.132668i	$-0.0423544\pi$
$632$		0			0
$633$	−9.64911			−0.383518
$634$		0			0
$635$	−13.9737	+	13.9737i	−0.554528	+	0.554528i
$636$		0			0
$637$	−0.837722	+	7.16228i	−0.0331918	+	0.283780i
$638$		0			0
$639$	−3.16228	−	3.16228i	−0.125098	−	0.125098i
$640$		0			0
$641$			21.6754i			0.856129i	0.903748	+	0.428064i	$0.140804\pi$
$641$			21.6754i			0.856129i	−0.903748	+	0.428064i	$0.859196\pi$
$642$		0			0
$643$	−19.3246	−	19.3246i	−0.762086	−	0.762086i	0.214613	−	0.976699i	$-0.431151\pi$
$643$	−19.3246	−	19.3246i	−0.762086	−	0.762086i	−0.976699	+	0.214613i	$0.931151\pi$
$644$		0			0
$645$	−7.90569	−	7.90569i	−0.311286	−	0.311286i
$646$		0			0
$647$	−22.9737			−0.903188			−0.451594	−	0.892224i	$-0.649144\pi$
$647$	−22.9737			−0.903188			−0.451594	+	0.892224i	$0.649144\pi$
$648$		0			0
$649$			33.2982i			1.30707i
$650$		0			0
$651$		−	3.67544i		−	0.144052i
$652$		0			0
$653$	5.02633			0.196696			0.0983478	−	0.995152i	$-0.468644\pi$
$653$	5.02633			0.196696			0.0983478	+	0.995152i	$0.468644\pi$
$654$		0			0
$655$	−18.4189	−	18.4189i	−0.719684	−	0.719684i
$656$		0			0
$657$	−12.0000	−	12.0000i	−0.468165	−	0.468165i
$658$		0			0
$659$			28.9737i			1.12865i	0.825551	+	0.564327i	$0.190864\pi$
$659$			28.9737i			1.12865i	−0.825551	+	0.564327i	$0.809136\pi$
$660$		0			0
$661$	23.1623	+	23.1623i	0.900908	+	0.900908i	0.995515	−	0.0946066i	$-0.0301593\pi$
$661$	23.1623	+	23.1623i	0.900908	+	0.900908i	−0.0946066	+	0.995515i	$0.530159\pi$
$662$		0			0
$663$	26.2302	+	3.06797i	1.01870	+	0.119150i
$664$		0			0
$665$	5.81139	−	5.81139i	0.225356	−	0.225356i
$666$		0			0
$667$	−8.32456			−0.322328
$668$		0			0
$669$	9.06797	−	9.06797i	0.350588	−	0.350588i
$670$		0			0
$671$	8.32456	−	8.32456i	0.321366	−	0.321366i
$672$		0			0
$673$			17.9737i			0.692834i	0.938081	+	0.346417i	$0.112602\pi$
$673$			17.9737i			0.692834i	−0.938081	+	0.346417i	$0.887398\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.1623			0.659600			0.329800	−	0.944051i	$-0.393019\pi$
$677$	17.1623			0.659600			0.329800	+	0.944051i	$0.393019\pi$
$678$		0			0
$679$		−	12.1359i		−	0.465735i
$680$		0			0
$681$	3.16228	−	3.16228i	0.121179	−	0.121179i
$682$		0			0
$683$	−30.8114	+	30.8114i	−1.17897	+	1.17897i	−0.198957	+	0.980008i	$0.563755\pi$
$683$	−30.8114	+	30.8114i	−1.17897	+	1.17897i	−0.980008	+	0.198957i	$0.936245\pi$
$684$		0			0
$685$	9.48683			0.362473
$686$		0			0
$687$	5.90569	−	5.90569i	0.225316	−	0.225316i
$688$		0			0
$689$	33.9737	+	3.97367i	1.29429	+	0.151385i
$690$		0			0
$691$	22.6491	+	22.6491i	0.861613	+	0.861613i	0.991525	−	0.129913i	$-0.0414697\pi$
$691$	22.6491	+	22.6491i	0.861613	+	0.861613i	−0.129913	+	0.991525i	$0.541470\pi$
$692$		0			0
$693$			26.3246i			0.999987i
$694$		0			0
$695$	24.2302	+	24.2302i	0.919106	+	0.919106i
$696$		0			0
$697$	37.8114	+	37.8114i	1.43221	+	1.43221i
$698$		0			0
$699$	−7.32456			−0.277040
$700$		0			0
$701$		−	24.8377i		−	0.938108i	−0.883170	−	0.469054i	$-0.844595\pi$
$701$		−	24.8377i		−	0.938108i	0.883170	−	0.469054i	$-0.155405\pi$
$702$		0			0
$703$		−	8.32456i		−	0.313967i
$704$		0			0
$705$	−21.3246			−0.803129
$706$		0			0
$707$	16.3246	+	16.3246i	0.613948	+	0.613948i
$708$		0			0
$709$	34.6491	+	34.6491i	1.30127	+	1.30127i	0.927533	+	0.373742i	$0.121925\pi$
$709$	34.6491	+	34.6491i	1.30127	+	1.30127i	0.373742	+	0.927533i	$0.378075\pi$
$710$		0			0
$711$		−	6.97367i		−	0.261533i
$712$		0			0
$713$	−8.32456	−	8.32456i	−0.311757	−	0.311757i
$714$		0			0
$715$	−5.51317	+	47.1359i	−0.206181	+	1.76278i
$716$		0			0
$717$	−12.4189	+	12.4189i	−0.463791	+	0.463791i
$718$		0			0
$719$	−31.8114			−1.18636			−0.593182	−	0.805068i	$-0.702129\pi$
$719$	−31.8114			−1.18636			−0.593182	+	0.805068i	$0.702129\pi$
$720$		0			0
$721$	11.8377	−	11.8377i	0.440860	−	0.440860i
$722$		0			0
$723$	3.48683	−	3.48683i	0.129677	−	0.129677i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	22.4605			0.833014			0.416507	−	0.909133i	$-0.363254\pi$
$727$	22.4605			0.833014			0.416507	+	0.909133i	$0.363254\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		−	36.6228i		−	1.35454i
$732$		0			0
$733$	−5.58114	+	5.58114i	−0.206144	+	0.206144i	−0.802626	−	0.596482i	$-0.796565\pi$
$733$	−5.58114	+	5.58114i	−0.206144	+	0.206144i	0.596482	+	0.802626i	$0.296565\pi$
$734$		0			0
$735$	−3.16228	+	3.16228i	−0.116642	+	0.116642i
$736$		0			0
$737$	−60.9737			−2.24599
$738$		0			0
$739$	9.16228	−	9.16228i	0.337040	−	0.337040i	−0.518212	−	0.855252i	$-0.673402\pi$
$739$	9.16228	−	9.16228i	0.337040	−	0.337040i	0.855252	+	0.518212i	$0.173402\pi$
$740$		0			0
$741$	4.64911	−	3.67544i	0.170789	−	0.135021i
$742$		0			0
$743$	0.230249	+	0.230249i	0.00844703	+	0.00844703i	0.711318	−	0.702871i	$-0.248099\pi$
$743$	0.230249	+	0.230249i	0.00844703	+	0.00844703i	−0.702871	+	0.711318i	$0.748099\pi$
$744$		0			0
$745$			6.32456i			0.231714i
$746$		0			0
$747$	11.6754	+	11.6754i	0.427182	+	0.427182i
$748$		0			0
$749$	−19.4868	−	19.4868i	−0.712033	−	0.712033i
$750$		0			0
$751$	48.9737			1.78707			0.893537	−	0.448989i	$-0.148216\pi$
$751$	48.9737			1.78707			0.893537	+	0.448989i	$0.148216\pi$
$752$		0			0
$753$			6.64911i			0.242307i
$754$		0			0
$755$		−	3.97367i		−	0.144617i
$756$		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
$759$	−29.8114	−	29.8114i	−1.08208	−	1.08208i
$760$		0			0
$761$	5.67544	+	5.67544i	0.205735	+	0.205735i	0.802452	−	0.596717i	$-0.203529\pi$
$761$	5.67544	+	5.67544i	0.205735	+	0.205735i	−0.596717	+	0.802452i	$0.703529\pi$
$762$		0			0
$763$		−	3.97367i		−	0.143856i
$764$		0			0
$765$	−23.1623	−	23.1623i	−0.837434	−	0.837434i
$766$		0			0
$767$	−16.0000	+	12.6491i	−0.577727	+	0.456733i
$768$		0			0
$769$	0.649111	−	0.649111i	0.0234075	−	0.0234075i	−0.695306	−	0.718714i	$-0.744731\pi$
$769$	0.649111	−	0.649111i	0.0234075	−	0.0234075i	0.718714	+	0.695306i	$0.244731\pi$
$770$		0			0
$771$	4.35089			0.156693
$772$		0			0
$773$	−21.2566	+	21.2566i	−0.764546	+	0.764546i	−0.977141	−	0.212594i	$-0.931809\pi$
$773$	−21.2566	+	21.2566i	−0.764546	+	0.764546i	0.212594	+	0.977141i	$0.431809\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	11.3246i		−	0.406266i
$778$		0			0
$779$	12.0000			0.429945
$780$		0			0
$781$	−13.1623			−0.470983
$782$		0			0
$783$			5.81139i			0.207682i
$784$		0			0
$785$	36.3246	−	36.3246i	1.29648	−	1.29648i
$786$		0			0
$787$	8.51317	−	8.51317i	0.303462	−	0.303462i	−0.538905	−	0.842367i	$-0.681162\pi$
$787$	8.51317	−	8.51317i	0.303462	−	0.303462i	0.842367	+	0.538905i	$0.181162\pi$
$788$		0			0
$789$	−6.00000			−0.213606
$790$		0			0
$791$	−25.2982	+	25.2982i	−0.899501	+	0.899501i
$792$

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-1.58114 + 1.58114i) q^{5} +(-1.58114 + 1.58114i) q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-1.58114 + 1.58114i) q^{5} +(-1.58114 + 1.58114i) q^{7} +2.00000 q^{9} +(4.16228 - 4.16228i) q^{11} +(3.58114 + 0.418861i) q^{13} +(1.58114 + 1.58114i) q^{15} +7.32456i q^{17} +(1.16228 + 1.16228i) q^{19} +(1.58114 + 1.58114i) q^{21} +7.16228 q^{23} -5.00000i q^{27} -1.16228 q^{29} +(-1.16228 - 1.16228i) q^{31} +(-4.16228 - 4.16228i) q^{33} -5.00000i q^{35} +(-3.58114 - 3.58114i) q^{37} +(0.418861 - 3.58114i) q^{39} +(5.16228 - 5.16228i) q^{41} -5.00000 q^{43} +(-3.16228 + 3.16228i) q^{45} +(-6.74342 + 6.74342i) q^{47} +2.00000i q^{49} +7.32456 q^{51} +9.48683 q^{53} +13.1623i q^{55} +(1.16228 - 1.16228i) q^{57} +(-4.00000 + 4.00000i) q^{59} +2.00000 q^{61} +(-3.16228 + 3.16228i) q^{63} +(-6.32456 + 5.00000i) q^{65} +(-7.32456 - 7.32456i) q^{67} -7.16228i q^{69} +(-1.58114 - 1.58114i) q^{71} +(-6.00000 - 6.00000i) q^{73} +13.1623i q^{77} -3.48683i q^{79} +1.00000 q^{81} +(5.83772 + 5.83772i) q^{83} +(-11.5811 - 11.5811i) q^{85} +1.16228i q^{87} +(-2.83772 - 2.83772i) q^{89} +(-6.32456 + 5.00000i) q^{91} +(-1.16228 + 1.16228i) q^{93} -3.67544 q^{95} +(-3.83772 + 3.83772i) q^{97} +(8.32456 - 8.32456i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{9}+O(q^{10}) \) 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

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Database

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