LMFDB - Embedded newform 912.2.bn.j.65.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(65,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

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

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

chi := DirichletCharacter("912.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.bn (of order $6$, 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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			65.2
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	912.65
Dual form			912.2.bn.j.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.18614 - 1.26217i) q^{3} +(0.686141 + 0.396143i) q^{5} +2.37228 q^{7} +(-0.186141 - 2.99422i) q^{9} +O(q^{10})$	\(q+(1.18614 - 1.26217i) q^{3} +(0.686141 + 0.396143i) q^{5} +2.37228 q^{7} +(-0.186141 - 2.99422i) q^{9} -3.46410i q^{11} +(-2.87228 + 1.65831i) q^{13} +(1.31386 - 0.396143i) q^{15} +(0.686141 + 0.396143i) q^{17} +(4.00000 + 1.73205i) q^{19} +(2.81386 - 2.99422i) q^{21} +(6.43070 - 3.71277i) q^{23} +(-2.18614 - 3.78651i) q^{25} +(-4.00000 - 3.31662i) q^{27} +(-2.68614 - 4.65253i) q^{29} +4.40387i q^{31} +(-4.37228 - 4.10891i) q^{33} +(1.62772 + 0.939764i) q^{35} +7.86797i q^{37} +(-1.31386 + 5.59230i) q^{39} +(-0.313859 + 0.543620i) q^{41} +(0.127719 - 0.221215i) q^{43} +(1.05842 - 2.12819i) q^{45} +(3.68614 - 2.12819i) q^{47} -1.37228 q^{49} +(1.31386 - 0.396143i) q^{51} +(5.68614 + 9.84868i) q^{53} +(1.37228 - 2.37686i) q^{55} +(6.93070 - 2.99422i) q^{57} +(3.68614 - 6.38458i) q^{59} +(0.500000 + 0.866025i) q^{61} +(-0.441578 - 7.10313i) q^{63} -2.62772 q^{65} +(-10.2446 + 5.91470i) q^{67} +(2.94158 - 12.5205i) q^{69} +(3.31386 - 5.73977i) q^{71} +(2.87228 - 4.97494i) q^{73} +(-7.37228 - 1.73205i) q^{75} -8.21782i q^{77} +(-9.98913 - 5.76722i) q^{79} +(-8.93070 + 1.11469i) q^{81} +3.46410i q^{83} +(0.313859 + 0.543620i) q^{85} +(-9.05842 - 2.12819i) q^{87} +(3.68614 + 6.38458i) q^{89} +(-6.81386 + 3.93398i) q^{91} +(5.55842 + 5.22360i) q^{93} +(2.05842 + 2.77300i) q^{95} +(11.0584 + 6.38458i) q^{97} +(-10.3723 + 0.644810i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) $ 4 * q - q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9	$ 4 q - q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9} + 11 q^{15} - 3 q^{17} + 16 q^{19} + 17 q^{21} - 3 q^{23} - 3 q^{25} - 16 q^{27} - 5 q^{29} - 6 q^{33} + 18 q^{35} - 11 q^{39} - 7 q^{41} + 12 q^{43} - 13 q^{45} + 9 q^{47} + 6 q^{49} + 11 q^{51} + 17 q^{53} - 6 q^{55} - q^{57} + 9 q^{59} + 2 q^{61} - 19 q^{63} - 22 q^{65} - 18 q^{67} + 29 q^{69} + 19 q^{71} - 18 q^{75} + 6 q^{79} - 7 q^{81} + 7 q^{85} - 19 q^{87} + 9 q^{89} - 33 q^{91} + 5 q^{93} - 9 q^{95} + 27 q^{97} - 30 q^{99}+O(q^{100}) $ 4 * q - q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9 + 11 * q^15 - 3 * q^17 + 16 * q^19 + 17 * q^21 - 3 * q^23 - 3 * q^25 - 16 * q^27 - 5 * q^29 - 6 * q^33 + 18 * q^35 - 11 * q^39 - 7 * q^41 + 12 * q^43 - 13 * q^45 + 9 * q^47 + 6 * q^49 + 11 * q^51 + 17 * q^53 - 6 * q^55 - q^57 + 9 * q^59 + 2 * q^61 - 19 * q^63 - 22 * q^65 - 18 * q^67 + 29 * q^69 + 19 * q^71 - 18 * q^75 + 6 * q^79 - 7 * q^81 + 7 * q^85 - 19 * q^87 + 9 * q^89 - 33 * q^91 + 5 * q^93 - 9 * q^95 + 27 * q^97 - 30 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.18614	−	1.26217i	0.684819	−	0.728714i
$3$	1.18614	−	1.26217i	0.684819	−	0.728714i
$4$		0			0
$5$	0.686141	+	0.396143i	0.306851	+	0.177161i	0.645517	−	0.763746i	$-0.276642\pi$
$5$	0.686141	+	0.396143i	0.306851	+	0.177161i	−0.338665	+	0.940907i	$0.609975\pi$
$6$		0			0
$7$	2.37228			0.896638			0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228			0.896638			0.448319	+	0.893874i	$0.352023\pi$
$8$		0			0
$9$	−0.186141	−	2.99422i	−0.0620469	−	0.998073i
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$	−2.87228	+	1.65831i	−0.796628	+	0.459933i	−0.842291	−	0.539024i	$-0.818793\pi$
$13$	−2.87228	+	1.65831i	−0.796628	+	0.459933i	0.0456630	+	0.998957i	$0.485460\pi$
$14$		0			0
$15$	1.31386	−	0.396143i	0.339237	−	0.102284i
$16$		0			0
$17$	0.686141	+	0.396143i	0.166414	+	0.0960789i	0.580894	−	0.813979i	$-0.302703\pi$
$17$	0.686141	+	0.396143i	0.166414	+	0.0960789i	−0.414480	+	0.910058i	$0.636037\pi$
$18$		0			0
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$20$		0			0
$21$	2.81386	−	2.99422i	0.614034	−	0.653392i
$22$		0			0
$23$	6.43070	−	3.71277i	1.34089	−	0.774166i	0.353956	−	0.935262i	$-0.384836\pi$
$23$	6.43070	−	3.71277i	1.34089	−	0.774166i	0.986939	+	0.161096i	$0.0515030\pi$
$24$		0			0
$25$	−2.18614	−	3.78651i	−0.437228	−	0.757301i
$26$		0			0
$27$	−4.00000	−	3.31662i	−0.769800	−	0.638285i
$28$		0			0
$29$	−2.68614	−	4.65253i	−0.498804	−	0.863954i	0.501195	−	0.865334i	$-0.332894\pi$
$29$	−2.68614	−	4.65253i	−0.498804	−	0.863954i	−0.999999	+	0.00138070i	$0.999561\pi$
$30$		0			0
$31$			4.40387i			0.790958i	0.918475	+	0.395479i	$0.129421\pi$
$31$			4.40387i			0.790958i	−0.918475	+	0.395479i	$0.870579\pi$
$32$		0			0
$33$	−4.37228	−	4.10891i	−0.761116	−	0.715270i
$34$		0			0
$35$	1.62772	+	0.939764i	0.275135	+	0.158849i
$36$		0			0
$37$			7.86797i			1.29349i	0.762708	+	0.646743i	$0.223869\pi$
$37$			7.86797i			1.29349i	−0.762708	+	0.646743i	$0.776131\pi$
$38$		0			0
$39$	−1.31386	+	5.59230i	−0.210386	+	0.895484i
$40$		0			0
$41$	−0.313859	+	0.543620i	−0.0490166	+	0.0848992i	−0.889493	−	0.456949i	$-0.848942\pi$
$41$	−0.313859	+	0.543620i	−0.0490166	+	0.0848992i	0.840476	+	0.541849i	$0.182275\pi$
$42$		0			0
$43$	0.127719	−	0.221215i	0.0194769	−	0.0337350i	−0.856123	−	0.516773i	$-0.827133\pi$
$43$	0.127719	−	0.221215i	0.0194769	−	0.0337350i	0.875600	+	0.483038i	$0.160467\pi$
$44$		0			0
$45$	1.05842	−	2.12819i	0.157780	−	0.317252i
$46$		0			0
$47$	3.68614	−	2.12819i	0.537679	−	0.310429i	−0.206459	−	0.978455i	$-0.566194\pi$
$47$	3.68614	−	2.12819i	0.537679	−	0.310429i	0.744138	+	0.668026i	$0.232861\pi$
$48$		0			0
$49$	−1.37228			−0.196040
$50$		0			0
$51$	1.31386	−	0.396143i	0.183977	−	0.0554712i
$52$		0			0
$53$	5.68614	+	9.84868i	0.781051	+	1.35282i	0.931330	+	0.364177i	$0.118650\pi$
$53$	5.68614	+	9.84868i	0.781051	+	1.35282i	−0.150278	+	0.988644i	$0.548017\pi$
$54$		0			0
$55$	1.37228	−	2.37686i	0.185038	−	0.320496i
$56$		0			0
$57$	6.93070	−	2.99422i	0.917994	−	0.396594i
$58$		0			0
$59$	3.68614	−	6.38458i	0.479895	−	0.831202i	−0.519839	−	0.854264i	$-0.674008\pi$
$59$	3.68614	−	6.38458i	0.479895	−	0.831202i	0.999734	+	0.0230621i	$0.00734155\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$	−0.441578	−	7.10313i	−0.0556336	−	0.894910i
$64$		0			0
$65$	−2.62772			−0.325928
$66$		0			0
$67$	−10.2446	+	5.91470i	−1.25157	+	0.722596i	−0.971422	−	0.237360i	$-0.923718\pi$
$67$	−10.2446	+	5.91470i	−1.25157	+	0.722596i	−0.280151	+	0.959956i	$0.590384\pi$
$68$		0			0
$69$	2.94158	−	12.5205i	0.354124	−	1.50729i
$70$		0			0
$71$	3.31386	−	5.73977i	0.393283	−	0.681186i	−0.599598	−	0.800302i	$-0.704673\pi$
$71$	3.31386	−	5.73977i	0.393283	−	0.681186i	0.992880	+	0.119116i	$0.0380060\pi$
$72$		0			0
$73$	2.87228	−	4.97494i	0.336175	−	0.582272i	−0.647535	−	0.762036i	$-0.724200\pi$
$73$	2.87228	−	4.97494i	0.336175	−	0.582272i	0.983710	+	0.179764i	$0.0575333\pi$
$74$		0			0
$75$	−7.37228	−	1.73205i	−0.851278	−	0.200000i
$76$		0			0
$77$		−	8.21782i		−	0.936508i
$78$		0			0
$79$	−9.98913	−	5.76722i	−1.12386	−	0.648863i	−0.181480	−	0.983395i	$-0.558089\pi$
$79$	−9.98913	−	5.76722i	−1.12386	−	0.648863i	−0.942385	+	0.334531i	$0.891422\pi$
$80$		0			0
$81$	−8.93070	+	1.11469i	−0.992300	+	0.123855i
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$	0.313859	+	0.543620i	0.0340428	+	0.0589639i
$86$		0			0
$87$	−9.05842	−	2.12819i	−0.971165	−	0.228166i
$88$		0			0
$89$	3.68614	+	6.38458i	0.390730	+	0.676764i	0.992546	−	0.121870i	$-0.0388892\pi$
$89$	3.68614	+	6.38458i	0.390730	+	0.676764i	−0.601816	+	0.798635i	$0.705556\pi$
$90$		0			0
$91$	−6.81386	+	3.93398i	−0.714287	+	0.412394i
$92$		0			0
$93$	5.55842	+	5.22360i	0.576382	+	0.541662i
$94$		0			0
$95$	2.05842	+	2.77300i	0.211190	+	0.284504i
$96$		0			0
$97$	11.0584	+	6.38458i	1.12281	+	0.648256i	0.942117	−	0.335283i	$-0.108832\pi$
$97$	11.0584	+	6.38458i	1.12281	+	0.648256i	0.180695	+	0.983539i	$0.442165\pi$
$98$		0			0
$99$	−10.3723	+	0.644810i	−1.04245	+	0.0648059i
$100$		0			0
$101$	0.686141	−	0.396143i	0.0682735	−	0.0394178i	−0.465475	−	0.885061i	$-0.654116\pi$
$101$	0.686141	−	0.396143i	0.0682735	−	0.0394178i	0.533748	+	0.845643i	$0.320783\pi$
$102$		0			0
$103$		−	0.644810i		−	0.0635350i	−0.999495	−	0.0317675i	$-0.989886\pi$
$103$		−	0.644810i		−	0.0635350i	0.999495	−	0.0317675i	$-0.0101136\pi$
$104$		0			0
$105$	3.11684	−	0.939764i	0.304173	−	0.0917116i
$106$		0			0
$107$	−16.7446			−1.61876			−0.809379	−	0.587287i	$-0.800196\pi$
$107$	−16.7446			−1.61876			−0.809379	+	0.587287i	$0.800196\pi$
$108$		0			0
$109$	5.05842	+	2.92048i	0.484509	+	0.279731i	0.722294	−	0.691587i	$-0.243088\pi$
$109$	5.05842	+	2.92048i	0.484509	+	0.279731i	−0.237785	+	0.971318i	$0.576421\pi$
$110$		0			0
$111$	9.93070	+	9.33252i	0.942581	+	0.885803i
$112$		0			0
$113$	−14.7446			−1.38705			−0.693526	−	0.720432i	$-0.743944\pi$
$113$	−14.7446			−1.38705			−0.693526	+	0.720432i	$0.743944\pi$
$114$		0			0
$115$	5.88316			0.548607
$116$		0			0
$117$	5.50000	+	8.29156i	0.508475	+	0.766555i
$118$		0			0
$119$	1.62772	+	0.939764i	0.149213	+	0.0861480i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	0.313859	+	1.04095i	0.0282997	+	0.0938596i
$124$		0			0
$125$		−	7.42554i		−	0.664160i
$126$		0			0
$127$	16.8030	−	9.70121i	1.49102	−	0.860843i	0.491077	−	0.871116i	$-0.336604\pi$
$127$	16.8030	−	9.70121i	1.49102	−	0.860843i	0.999947	+	0.0102734i	$0.00327020\pi$
$128$		0			0
$129$	−0.127719	−	0.423595i	−0.0112450	−	0.0372955i
$130$		0			0
$131$	−4.80298	−	2.77300i	−0.419639	−	0.242279i	0.275284	−	0.961363i	$-0.411228\pi$
$131$	−4.80298	−	2.77300i	−0.419639	−	0.242279i	−0.694923	+	0.719084i	$0.744561\pi$
$132$		0			0
$133$	9.48913	+	4.10891i	0.822812	+	0.356288i
$134$		0			0
$135$	−1.43070	−	3.86025i	−0.123135	−	0.332237i
$136$		0			0
$137$	−11.3139	+	6.53206i	−0.966608	+	0.558072i	−0.898201	−	0.439586i	$-0.855125\pi$
$137$	−11.3139	+	6.53206i	−0.966608	+	0.558072i	−0.0684077	+	0.997657i	$0.521792\pi$
$138$		0			0
$139$	8.24456	+	14.2800i	0.699295	+	1.21121i	0.968711	+	0.248190i	$0.0798358\pi$
$139$	8.24456	+	14.2800i	0.699295	+	1.21121i	−0.269417	+	0.963024i	$0.586831\pi$
$140$		0			0
$141$	1.68614	−	7.17687i	0.141999	−	0.604401i
$142$		0			0
$143$	5.74456	+	9.94987i	0.480384	+	0.832050i
$144$		0			0
$145$		−	4.25639i		−	0.353474i
$146$		0			0
$147$	−1.62772	+	1.73205i	−0.134252	+	0.142857i
$148$		0			0
$149$	−2.05842	−	1.18843i	−0.168632	−	0.0973600i	0.413308	−	0.910591i	$-0.364373\pi$
$149$	−2.05842	−	1.18843i	−0.168632	−	0.0973600i	−0.581941	+	0.813231i	$0.697706\pi$
$150$		0			0
$151$			13.5615i			1.10362i	0.833971	+	0.551808i	$0.186062\pi$
$151$			13.5615i			1.10362i	−0.833971	+	0.551808i	$0.813938\pi$
$152$		0			0
$153$	1.05842	−	2.12819i	0.0855683	−	0.172054i
$154$		0			0
$155$	−1.74456	+	3.02167i	−0.140127	+	0.242706i
$156$		0			0
$157$	−10.2446	+	17.7441i	−0.817605	+	1.41613i	0.0898370	+	0.995956i	$0.471365\pi$
$157$	−10.2446	+	17.7441i	−0.817605	+	1.41613i	−0.907442	+	0.420177i	$0.861968\pi$
$158$		0			0
$159$	19.1753	+	4.50506i	1.52070	+	0.357274i
$160$		0			0
$161$	15.2554	−	8.80773i	1.20230	−	0.694146i
$162$		0			0
$163$	−9.62772			−0.754101			−0.377051	−	0.926193i	$-0.623062\pi$
$163$	−9.62772			−0.754101			−0.377051	+	0.926193i	$0.623062\pi$
$164$		0			0
$165$	−1.37228	−	4.55134i	−0.106832	−	0.354322i
$166$		0			0
$167$	9.05842	+	15.6896i	0.700962	+	1.21410i	0.968129	+	0.250451i	$0.0805790\pi$
$167$	9.05842	+	15.6896i	0.700962	+	1.21410i	−0.267167	+	0.963650i	$0.586088\pi$
$168$		0			0
$169$	−1.00000	+	1.73205i	−0.0769231	+	0.133235i
$170$		0			0
$171$	4.44158	−	12.2993i	0.339656	−	0.940550i
$172$		0			0
$173$	−7.43070	+	12.8704i	−0.564946	+	0.978515i	0.432109	+	0.901821i	$0.357770\pi$
$173$	−7.43070	+	12.8704i	−0.564946	+	0.978515i	−0.997055	+	0.0766935i	$0.975564\pi$
$174$		0			0
$175$	−5.18614	−	8.98266i	−0.392035	−	0.679025i
$176$		0			0
$177$	−3.68614	−	12.2255i	−0.277067	−	0.918928i
$178$		0			0
$179$	−21.4891			−1.60617			−0.803086	−	0.595863i	$-0.796810\pi$
$179$	−21.4891			−1.60617			−0.803086	+	0.595863i	$0.796810\pi$
$180$		0			0
$181$	1.80298	−	1.04095i	0.134015	−	0.0773735i	−0.431494	−	0.902116i	$-0.642013\pi$
$181$	1.80298	−	1.04095i	0.134015	−	0.0773735i	0.565508	+	0.824742i	$0.308680\pi$
$182$		0			0
$183$	1.68614	+	0.396143i	0.124643	+	0.0292838i
$184$		0			0
$185$	−3.11684	+	5.39853i	−0.229155	+	0.396908i
$186$		0			0
$187$	1.37228	−	2.37686i	0.100351	−	0.173813i
$188$		0			0
$189$	−9.48913	−	7.86797i	−0.690232	−	0.572310i
$190$		0			0
$191$		−	6.63325i		−	0.479965i	−0.970777	−	0.239983i	$-0.922858\pi$
$191$		−	6.63325i		−	0.479965i	0.970777	−	0.239983i	$-0.0771417\pi$
$192$		0			0
$193$	13.5000	+	7.79423i	0.971751	+	0.561041i	0.899770	−	0.436365i	$-0.143734\pi$
$193$	13.5000	+	7.79423i	0.971751	+	0.561041i	0.0719816	+	0.997406i	$0.477068\pi$
$194$		0			0
$195$	−3.11684	+	3.31662i	−0.223202	+	0.237508i
$196$		0			0
$197$			16.4356i			1.17099i	0.810676	+	0.585496i	$0.199100\pi$
$197$			16.4356i			1.17099i	−0.810676	+	0.585496i	$0.800900\pi$
$198$		0			0
$199$	10.2446	+	17.7441i	0.726218	+	1.25785i	0.958471	+	0.285191i	$0.0920569\pi$
$199$	10.2446	+	17.7441i	0.726218	+	1.25785i	−0.232253	+	0.972655i	$0.574610\pi$
$200$		0			0
$201$	−4.68614	+	19.9460i	−0.330535	+	1.40688i
$202$		0			0
$203$	−6.37228	−	11.0371i	−0.447246	−	0.774654i
$204$		0			0
$205$	−0.430703	+	0.248667i	−0.0300816	+	0.0173676i
$206$		0			0
$207$	−12.3139	−	18.5638i	−0.855872	−	1.29028i
$208$		0			0
$209$	6.00000	−	13.8564i	0.415029	−	0.958468i
$210$		0			0
$211$	12.9891	+	7.49927i	0.894208	+	0.516271i	0.875317	−	0.483550i	$-0.160653\pi$
$211$	12.9891	+	7.49927i	0.894208	+	0.516271i	0.0188916	+	0.999822i	$0.493986\pi$
$212$		0			0
$213$	−3.31386	−	10.9908i	−0.227062	−	0.753079i
$214$		0			0
$215$	0.175266	−	0.101190i	0.0119530	−	0.00690109i
$216$		0			0
$217$			10.4472i			0.709203i
$218$		0			0
$219$	−2.87228	−	9.52628i	−0.194091	−	0.643726i
$220$		0			0
$221$	−2.62772			−0.176759
$222$		0			0
$223$	15.7337	+	9.08385i	1.05361	+	0.608300i	0.923657	−	0.383221i	$-0.125185\pi$
$223$	15.7337	+	9.08385i	1.05361	+	0.608300i	0.129949	+	0.991521i	$0.458519\pi$
$224$		0			0
$225$	−10.9307	+	7.25061i	−0.728714	+	0.483374i
$226$		0			0
$227$	7.25544			0.481560			0.240780	−	0.970580i	$-0.422597\pi$
$227$	7.25544			0.481560			0.240780	+	0.970580i	$0.422597\pi$
$228$		0			0
$229$	−21.1168			−1.39544			−0.697720	−	0.716370i	$-0.745802\pi$
$229$	−21.1168			−1.39544			−0.697720	+	0.716370i	$0.745802\pi$
$230$		0			0
$231$	−10.3723	−	9.74749i	−0.682446	−	0.641338i
$232$		0			0
$233$	3.94158	+	2.27567i	0.258221	+	0.149084i	0.623523	−	0.781805i	$-0.285701\pi$
$233$	3.94158	+	2.27567i	0.258221	+	0.149084i	−0.365302	+	0.930889i	$0.619034\pi$
$234$		0			0
$235$	3.37228			0.219983
$236$		0			0
$237$	−19.1277	+	5.76722i	−1.24248	+	0.374621i
$238$		0			0
$239$			10.3923i			0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$			10.3923i			0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$		0			0
$241$	0.383156	−	0.221215i	0.0246812	−	0.0142497i	−0.487609	−	0.873062i	$-0.662131\pi$
$241$	0.383156	−	0.221215i	0.0246812	−	0.0142497i	0.512290	+	0.858813i	$0.328797\pi$
$242$		0			0
$243$	−9.18614	+	12.5942i	−0.589291	+	0.807921i
$244$		0			0
$245$	−0.941578	−	0.543620i	−0.0601552	−	0.0347306i
$246$		0			0
$247$	−14.3614	+	1.65831i	−0.913794	+	0.105516i
$248$		0			0
$249$	4.37228	+	4.10891i	0.277082	+	0.260392i
$250$		0			0
$251$	−5.56930	+	3.21543i	−0.351531	+	0.202956i	−0.665359	−	0.746523i	$-0.731722\pi$
$251$	−5.56930	+	3.21543i	−0.351531	+	0.202956i	0.313828	+	0.949480i	$0.398388\pi$
$252$		0			0
$253$	−12.8614	−	22.2766i	−0.808590	−	1.40052i
$254$		0			0
$255$	1.05842	+	0.248667i	0.0662810	+	0.0155721i
$256$		0			0
$257$	0.941578	+	1.63086i	0.0587340	+	0.101730i	0.893897	−	0.448272i	$-0.147960\pi$
$257$	0.941578	+	1.63086i	0.0587340	+	0.101730i	−0.835163	+	0.550002i	$0.814627\pi$
$258$		0			0
$259$			18.6650i			1.15979i
$260$		0			0
$261$	−13.4307	+	8.90892i	−0.831340	+	0.551448i
$262$		0			0
$263$	3.43070	+	1.98072i	0.211546	+	0.122136i	0.602030	−	0.798474i	$-0.294359\pi$
$263$	3.43070	+	1.98072i	0.211546	+	0.122136i	−0.390484	+	0.920610i	$0.627692\pi$
$264$		0			0
$265$			9.01011i			0.553487i
$266$		0			0
$267$	12.4307	+	2.92048i	0.760747	+	0.178731i
$268$		0			0
$269$	6.43070	−	11.1383i	0.392087	−	0.679114i	−0.600638	−	0.799521i	$-0.705087\pi$
$269$	6.43070	−	11.1383i	0.392087	−	0.679114i	0.992725	+	0.120407i	$0.0384199\pi$
$270$		0			0
$271$	4.43070	−	7.67420i	0.269146	−	0.466175i	−0.699495	−	0.714637i	$-0.746592\pi$
$271$	4.43070	−	7.67420i	0.269146	−	0.466175i	0.968642	+	0.248462i	$0.0799252\pi$
$272$		0			0
$273$	−3.11684	+	13.2665i	−0.188640	+	0.802925i
$274$		0			0
$275$	−13.1168	+	7.57301i	−0.790975	+	0.456670i
$276$		0			0
$277$	3.48913			0.209641			0.104821	−	0.994491i	$-0.466573\pi$
$277$	3.48913			0.209641			0.104821	+	0.994491i	$0.466573\pi$
$278$		0			0
$279$	13.1861	−	0.819738i	0.789434	−	0.0490765i
$280$		0			0
$281$	12.8030	+	22.1754i	0.763762	+	1.32287i	0.940899	+	0.338688i	$0.109983\pi$
$281$	12.8030	+	22.1754i	0.763762	+	1.32287i	−0.177137	+	0.984186i	$0.556683\pi$
$282$		0			0
$283$	14.4307	−	24.9947i	0.857816	−	1.48578i	−0.0161912	−	0.999869i	$-0.505154\pi$
$283$	14.4307	−	24.9947i	0.857816	−	1.48578i	0.874007	−	0.485912i	$-0.161513\pi$
$284$		0			0
$285$	5.94158	+	0.691097i	0.351949	+	0.0409371i
$286$		0			0
$287$	−0.744563	+	1.28962i	−0.0439501	+	0.0761239i
$288$		0			0
$289$	−8.18614	−	14.1788i	−0.481538	−	0.834048i
$290$		0			0
$291$	21.1753	−	6.38458i	1.24132	−	0.374271i
$292$		0			0
$293$	−9.25544			−0.540708			−0.270354	−	0.962761i	$-0.587141\pi$
$293$	−9.25544			−0.540708			−0.270354	+	0.962761i	$0.587141\pi$
$294$		0			0
$295$	5.05842	−	2.92048i	0.294513	−	0.170037i
$296$		0			0
$297$	−11.4891	+	13.8564i	−0.666667	+	0.804030i
$298$		0			0
$299$	−12.3139	+	21.3282i	−0.712129	+	1.23344i
$300$		0			0
$301$	0.302985	−	0.524785i	0.0174637	−	0.0302481i
$302$		0			0
$303$	0.313859	−	1.33591i	0.0180307	−	0.0767459i
$304$		0			0
$305$			0.792287i			0.0453662i
$306$		0			0
$307$	−16.2921	−	9.40625i	−0.929840	−	0.536843i	−0.0430789	−	0.999072i	$-0.513717\pi$
$307$	−16.2921	−	9.40625i	−0.929840	−	0.536843i	−0.886761	+	0.462228i	$0.847050\pi$
$308$		0			0
$309$	−0.813859	−	0.764836i	−0.0462988	−	0.0435100i
$310$		0			0
$311$			12.2718i			0.695872i	0.937518	+	0.347936i	$0.113117\pi$
$311$			12.2718i			0.695872i	−0.937518	+	0.347936i	$0.886883\pi$
$312$		0			0
$313$	−16.1753	−	28.0164i	−0.914280	−	1.58358i	−0.807952	−	0.589248i	$-0.799424\pi$
$313$	−16.1753	−	28.0164i	−0.914280	−	1.58358i	−0.106328	−	0.994331i	$-0.533909\pi$
$314$		0			0
$315$	2.51087	−	5.04868i	0.141472	−	0.284461i
$316$		0			0
$317$	−10.6861	−	18.5089i	−0.600193	−	1.03957i	−0.992791	−	0.119855i	$-0.961757\pi$
$317$	−10.6861	−	18.5089i	−0.600193	−	1.03957i	0.392598	−	0.919710i	$-0.371576\pi$
$318$		0			0
$319$	−16.1168	+	9.30506i	−0.902370	+	0.520984i
$320$		0			0
$321$	−19.8614	+	21.1345i	−1.10856	+	1.17961i
$322$		0			0
$323$	2.05842	+	2.77300i	0.114534	+	0.154294i
$324$		0			0
$325$	12.5584	+	7.25061i	0.696616	+	0.402191i
$326$		0			0
$327$	9.68614	−	2.92048i	0.535645	−	0.161503i
$328$		0			0
$329$	8.74456	−	5.04868i	0.482103	−	0.278342i
$330$		0			0
$331$		−	19.5499i		−	1.07456i	−0.843404	−	0.537280i	$-0.819452\pi$
$331$		−	19.5499i		−	1.07456i	0.843404	−	0.537280i	$-0.180548\pi$
$332$		0			0
$333$	23.5584	−	1.46455i	1.29099	−	0.0802568i
$334$		0			0
$335$	−9.37228			−0.512062
$336$		0			0
$337$	9.73369	+	5.61975i	0.530228	+	0.306127i	0.741109	−	0.671385i	$-0.234300\pi$
$337$	9.73369	+	5.61975i	0.530228	+	0.306127i	−0.210881	+	0.977512i	$0.567633\pi$
$338$		0			0
$339$	−17.4891	+	18.6101i	−0.949879	+	1.01076i
$340$		0			0
$341$	15.2554			0.826128
$342$		0			0
$343$	−19.8614			−1.07242
$344$		0			0
$345$	6.97825	−	7.42554i	0.375696	−	0.399777i
$346$		0			0
$347$	−7.54755	−	4.35758i	−0.405174	−	0.233927i	0.283540	−	0.958960i	$-0.408491\pi$
$347$	−7.54755	−	4.35758i	−0.405174	−	0.233927i	−0.688714	+	0.725033i	$0.741824\pi$
$348$		0			0
$349$	−30.6060			−1.63830			−0.819150	−	0.573579i	$-0.805554\pi$
$349$	−30.6060			−1.63830			−0.819150	+	0.573579i	$0.805554\pi$
$350$		0			0
$351$	16.9891	+	2.89303i	0.906812	+	0.154419i
$352$		0			0
$353$		−	35.3407i		−	1.88100i	−0.339798	−	0.940499i	$-0.610359\pi$
$353$		−	35.3407i		−	1.88100i	0.339798	−	0.940499i	$-0.389641\pi$
$354$		0			0
$355$	4.54755	−	2.62553i	0.241359	−	0.139349i
$356$		0			0
$357$	3.11684	−	0.939764i	0.164961	−	0.0497376i
$358$		0			0
$359$	1.80298	+	1.04095i	0.0951579	+	0.0549394i	0.546824	−	0.837248i	$-0.315837\pi$
$359$	1.80298	+	1.04095i	0.0951579	+	0.0549394i	−0.451666	+	0.892187i	$0.649170\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$	−1.18614	+	1.26217i	−0.0622562	+	0.0662467i
$364$		0			0
$365$	3.94158	−	2.27567i	0.206312	−	0.119114i
$366$		0			0
$367$	−9.61684	−	16.6569i	−0.501995	−	0.869481i	−0.999997	−	0.00230536i	$-0.999266\pi$
$367$	−9.61684	−	16.6569i	−0.501995	−	0.869481i	0.498002	−	0.867176i	$-0.334067\pi$
$368$		0			0
$369$	1.68614	+	0.838574i	0.0877770	+	0.0436544i
$370$		0			0
$371$	13.4891	+	23.3639i	0.700320	+	1.21299i
$372$		0			0
$373$		−	6.92820i		−	0.358729i	−0.983783	−	0.179364i	$-0.942596\pi$
$373$		−	6.92820i		−	0.358729i	0.983783	−	0.179364i	$-0.0574041\pi$
$374$		0			0
$375$	−9.37228	−	8.80773i	−0.483983	−	0.454829i
$376$		0			0
$377$	15.4307	+	8.90892i	0.794722	+	0.458833i
$378$		0			0
$379$			10.7422i			0.551788i	0.961188	+	0.275894i	$0.0889738\pi$
$379$			10.7422i			0.551788i	−0.961188	+	0.275894i	$0.911026\pi$
$380$		0			0
$381$	7.68614	−	32.7152i	0.393773	−	1.67605i
$382$		0			0
$383$	4.80298	−	8.31901i	0.245421	−	0.425082i	−0.716829	−	0.697249i	$-0.754407\pi$
$383$	4.80298	−	8.31901i	0.245421	−	0.425082i	0.962250	+	0.272167i	$0.0877405\pi$
$384$		0			0
$385$	3.25544	−	5.63858i	0.165912	−	0.287369i
$386$		0			0
$387$	−0.686141	−	0.341241i	−0.0348785	−	0.0173462i
$388$		0			0
$389$	−26.6644	+	15.3947i	−1.35194	+	0.780542i	−0.988521	−	0.151084i	$-0.951723\pi$
$389$	−26.6644	+	15.3947i	−1.35194	+	0.780542i	−0.363417	+	0.931626i	$0.618390\pi$
$390$		0			0
$391$	5.88316			0.297524
$392$		0			0
$393$	−9.19702	+	2.77300i	−0.463928	+	0.139880i
$394$		0			0
$395$	−4.56930	−	7.91425i	−0.229906	−	0.398209i
$396$		0			0
$397$	−4.61684	+	7.99661i	−0.231713	+	0.401338i	−0.958312	−	0.285723i	$-0.907766\pi$
$397$	−4.61684	+	7.99661i	−0.231713	+	0.401338i	0.726599	+	0.687061i	$0.241100\pi$
$398$		0			0
$399$	16.4416	−	7.10313i	0.823108	−	0.355601i
$400$		0			0
$401$	9.17527	−	15.8920i	0.458191	−	0.793610i	−0.540675	−	0.841232i	$-0.681831\pi$
$401$	9.17527	−	15.8920i	0.458191	−	0.793610i	0.998865	+	0.0476219i	$0.0151643\pi$
$402$		0			0
$403$	−7.30298	−	12.6491i	−0.363788	−	0.630099i
$404$		0			0
$405$	−6.56930	−	2.77300i	−0.326431	−	0.137792i
$406$		0			0
$407$	27.2554			1.35100
$408$		0			0
$409$	31.2921	−	18.0665i	1.54730	−	0.893331i	0.548948	−	0.835856i	$-0.315028\pi$
$409$	31.2921	−	18.0665i	1.54730	−	0.893331i	0.998347	−	0.0574750i	$-0.0183050\pi$
$410$		0			0
$411$	−5.17527	+	22.0279i	−0.255277	+	1.08656i
$412$		0			0
$413$	8.74456	−	15.1460i	0.430292	−	0.745287i
$414$		0			0
$415$	−1.37228	+	2.37686i	−0.0673626	+	0.116676i
$416$		0			0
$417$	27.8030	+	6.53206i	1.36152	+	0.319876i
$418$		0			0
$419$		−	37.5152i		−	1.83274i	−0.400335	−	0.916369i	$-0.631106\pi$
$419$		−	37.5152i		−	1.83274i	0.400335	−	0.916369i	$-0.368894\pi$
$420$		0			0
$421$	25.2921	+	14.6024i	1.23266	+	0.711678i	0.967584	−	0.252549i	$-0.0812690\pi$
$421$	25.2921	+	14.6024i	1.23266	+	0.711678i	0.265078	+	0.964227i	$0.414602\pi$
$422$		0			0
$423$	−7.05842	−	10.6410i	−0.343192	−	0.517382i
$424$		0			0
$425$		−	3.46410i		−	0.168034i
$426$		0			0
$427$	1.18614	+	2.05446i	0.0574014	+	0.0994221i
$428$		0			0
$429$	19.3723	+	4.55134i	0.935303	+	0.219741i
$430$		0			0
$431$	−18.4307	−	31.9229i	−0.887776	−	1.53767i	−0.842499	−	0.538698i	$-0.818916\pi$
$431$	−18.4307	−	31.9229i	−0.887776	−	1.53767i	−0.0452769	−	0.998974i	$-0.514417\pi$
$432$		0			0
$433$	3.73369	−	2.15565i	0.179430	−	0.103594i	−0.407595	−	0.913163i	$-0.633632\pi$
$433$	3.73369	−	2.15565i	0.179430	−	0.103594i	0.587025	+	0.809569i	$0.300299\pi$
$434$		0			0
$435$	−5.37228	−	5.04868i	−0.257581	−	0.242065i
$436$		0			0
$437$	32.1535	−	3.71277i	1.53811	−	0.177606i
$438$		0			0
$439$	−4.50000	−	2.59808i	−0.214773	−	0.123999i	0.388755	−	0.921341i	$-0.372905\pi$
$439$	−4.50000	−	2.59808i	−0.214773	−	0.123999i	−0.603528	+	0.797342i	$0.706239\pi$
$440$		0			0
$441$	0.255437	+	4.10891i	0.0121637	+	0.195662i
$442$		0			0
$443$	31.0367	−	17.9190i	1.47460	−	0.851359i	0.475007	−	0.879982i	$-0.342446\pi$
$443$	31.0367	−	17.9190i	1.47460	−	0.851359i	0.999590	+	0.0286234i	$0.00911235\pi$
$444$		0			0
$445$			5.84096i			0.276888i
$446$		0			0
$447$	−3.94158	+	1.18843i	−0.186430	+	0.0562108i
$448$		0			0
$449$	−38.4674			−1.81539			−0.907694	−	0.419633i	$-0.862159\pi$
$449$	−38.4674			−1.81539			−0.907694	+	0.419633i	$0.862159\pi$
$450$		0			0
$451$	1.88316	+	1.08724i	0.0886744	+	0.0511962i
$452$		0			0
$453$	17.1168	+	16.0858i	0.804219	+	0.755776i
$454$		0			0
$455$	−6.23369			−0.292240
$456$		0			0
$457$	36.3723			1.70142			0.850712	−	0.525632i	$-0.176171\pi$
$457$	36.3723			1.70142			0.850712	+	0.525632i	$0.176171\pi$
$458$		0			0
$459$	−1.43070	−	3.86025i	−0.0667795	−	0.180181i
$460$		0			0
$461$	11.5693	+	6.67954i	0.538836	+	0.311097i	0.744607	−	0.667503i	$-0.232637\pi$
$461$	11.5693	+	6.67954i	0.538836	+	0.311097i	−0.205771	+	0.978600i	$0.565970\pi$
$462$		0			0
$463$	−17.6277			−0.819230			−0.409615	−	0.912259i	$-0.634337\pi$
$463$	−17.6277			−0.819230			−0.409615	+	0.912259i	$0.634337\pi$
$464$		0			0
$465$	1.74456	+	5.78606i	0.0809022	+	0.268322i
$466$		0			0
$467$		−	4.16381i		−	0.192678i	−0.995349	−	0.0963392i	$-0.969287\pi$
$467$		−	4.16381i		−	0.192678i	0.995349	−	0.0963392i	$-0.0307133\pi$
$468$		0			0
$469$	−24.3030	+	14.0313i	−1.12221	+	0.647907i
$470$		0			0
$471$	10.2446	+	33.9774i	0.472045	+	1.56559i
$472$		0			0
$473$	−0.766312	−	0.442430i	−0.0352351	−	0.0203430i
$474$		0			0
$475$	−2.18614	−	18.9325i	−0.100307	−	0.868684i
$476$		0			0
$477$	28.4307	−	18.8588i	1.30175	−	0.863485i
$478$		0			0
$479$	8.56930	−	4.94749i	0.391541	−	0.226056i	−0.291287	−	0.956636i	$-0.594083\pi$
$479$	8.56930	−	4.94749i	0.391541	−	0.226056i	0.682828	+	0.730579i	$0.260750\pi$
$480$		0			0
$481$	−13.0475	−	22.5990i	−0.594917	−	1.03043i
$482$		0			0
$483$	6.97825	−	29.7021i	0.317521	−	1.35149i
$484$		0			0
$485$	5.05842	+	8.76144i	0.229691	+	0.397837i
$486$		0			0
$487$		−	0.294954i		−	0.0133656i	−0.999978	−	0.00668281i	$-0.997873\pi$
$487$		−	0.294954i		−	0.0133656i	0.999978	−	0.00668281i	$-0.00212722\pi$
$488$		0			0
$489$	−11.4198	+	12.1518i	−0.516423	+	0.549524i
$490$		0			0
$491$	18.1753	+	10.4935i	0.820238	+	0.473565i	0.850499	−	0.525977i	$-0.176300\pi$
$491$	18.1753	+	10.4935i	0.820238	+	0.473565i	−0.0302604	+	0.999542i	$0.509634\pi$
$492$		0			0
$493$		−	4.25639i		−	0.191698i
$494$		0			0
$495$	−7.37228	−	3.66648i	−0.331359	−	0.164796i
$496$		0			0
$497$	7.86141	−	13.6164i	0.352632	−	0.610777i
$498$		0			0
$499$	−11.8723	+	20.5634i	−0.531476	+	0.920544i	0.467849	+	0.883809i	$0.345029\pi$
$499$	−11.8723	+	20.5634i	−0.531476	+	0.920544i	−0.999325	+	0.0367354i	$0.988304\pi$
$500$		0			0
$501$	30.5475	+	7.17687i	1.36476	+	0.320639i
$502$		0			0
$503$	−36.6861	+	21.1808i	−1.63575	+	0.944403i	−0.653482	+	0.756942i	$0.726693\pi$
$503$	−36.6861	+	21.1808i	−1.63575	+	0.944403i	−0.982272	+	0.187461i	$0.939974\pi$
$504$		0			0
$505$	0.627719			0.0279331
$506$		0			0
$507$	1.00000	+	3.31662i	0.0444116	+	0.147296i
$508$		0			0
$509$	−5.05842	−	8.76144i	−0.224211	−	0.388344i	0.731872	−	0.681442i	$-0.238647\pi$
$509$	−5.05842	−	8.76144i	−0.224211	−	0.388344i	−0.956082	+	0.293098i	$0.905314\pi$
$510$		0			0
$511$	6.81386	−	11.8020i	0.301427	−	0.522088i
$512$		0			0
$513$	−10.2554	−	20.1947i	−0.452789	−	0.891618i
$514$		0			0
$515$	0.255437	−	0.442430i	0.0112559	−	0.0194958i
$516$		0			0
$517$	−7.37228	−	12.7692i	−0.324233	−	0.561587i
$518$		0			0
$519$	7.43070	+	24.6449i	0.326172	+	1.08179i
$520$		0			0
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$	−10.2446	+	5.91470i	−0.447963	+	0.258632i	−0.706970	−	0.707244i	$-0.749938\pi$
$523$	−10.2446	+	5.91470i	−0.447963	+	0.258632i	0.259006	+	0.965876i	$0.416605\pi$
$524$		0			0
$525$	−17.4891	−	4.10891i	−0.763288	−	0.179328i
$526$		0			0
$527$	−1.74456	+	3.02167i	−0.0759943	+	0.131626i
$528$		0			0
$529$	16.0693	−	27.8328i	0.698665	−	1.21012i
$530$		0			0
$531$	−19.8030	−	9.84868i	−0.859376	−	0.427397i
$532$		0			0
$533$		−	2.08191i		−	0.0901774i
$534$		0			0
$535$	−11.4891	−	6.63325i	−0.496718	−	0.286780i
$536$		0			0
$537$	−25.4891	+	27.1229i	−1.09994	+	1.17044i
$538$		0			0
$539$			4.75372i			0.204757i
$540$		0			0
$541$	7.98913	+	13.8376i	0.343479	+	0.594924i	0.985076	−	0.172118i	$-0.0550611\pi$
$541$	7.98913	+	13.8376i	0.343479	+	0.594924i	−0.641597	+	0.767042i	$0.721728\pi$
$542$		0			0
$543$	0.824734	−	3.51039i	0.0353927	−	0.150645i
$544$		0			0
$545$	2.31386	+	4.00772i	0.0991148	+	0.171672i
$546$		0			0
$547$	35.6168	−	20.5634i	1.52287	−	0.879227i	0.523232	−	0.852190i	$-0.324726\pi$
$547$	35.6168	−	20.5634i	1.52287	−	0.879227i	0.999634	−	0.0270369i	$-0.00860716\pi$
$548$		0			0
$549$	2.50000	−	1.65831i	0.106697	−	0.0707750i
$550$		0			0
$551$	−2.68614	−	23.2627i	−0.114433	−	0.991023i
$552$		0			0
$553$	−23.6970	−	13.6815i	−1.00770	−	0.581796i
$554$		0			0
$555$	3.11684	+	10.3374i	0.132303	+	0.438798i
$556$		0			0
$557$	27.4307	−	15.8371i	1.16228	−	0.671040i	0.210428	−	0.977609i	$-0.432514\pi$
$557$	27.4307	−	15.8371i	1.16228	−	0.671040i	0.951848	+	0.306569i	$0.0991810\pi$
$558$		0			0
$559$			0.847190i			0.0358323i
$560$		0			0
$561$	−1.37228	−	4.55134i	−0.0579378	−	0.192158i
$562$		0			0
$563$	43.7228			1.84270			0.921348	−	0.388738i	$-0.127089\pi$
$563$	43.7228			1.84270			0.921348	+	0.388738i	$0.127089\pi$
$564$		0			0
$565$	−10.1168	−	5.84096i	−0.425619	−	0.245731i
$566$		0			0
$567$	−21.1861	+	2.64436i	−0.889734	+	0.111053i
$568$		0			0
$569$	−38.7446			−1.62426			−0.812128	−	0.583479i	$-0.801691\pi$
$569$	−38.7446			−1.62426			−0.812128	+	0.583479i	$0.801691\pi$
$570$		0			0
$571$	−9.62772			−0.402907			−0.201454	−	0.979498i	$-0.564567\pi$
$571$	−9.62772			−0.402907			−0.201454	+	0.979498i	$0.564567\pi$
$572$		0			0
$573$	−8.37228	−	7.86797i	−0.349757	−	0.328689i
$574$		0			0
$575$	−28.1168	−	16.2333i	−1.17255	−	0.676974i
$576$		0			0
$577$	−27.4891			−1.14439			−0.572194	−	0.820119i	$-0.693907\pi$
$577$	−27.4891			−1.14439			−0.572194	+	0.820119i	$0.693907\pi$
$578$		0			0
$579$	25.8505	−	7.79423i	1.07431	−	0.323917i
$580$		0			0
$581$			8.21782i			0.340933i
$582$		0			0
$583$	34.1168	−	19.6974i	1.41298	−	0.815782i
$584$		0			0
$585$	0.489125	+	7.86797i	0.0202228	+	0.325300i
$586$		0			0
$587$	38.3139	+	22.1205i	1.58138	+	0.913011i	0.994658	+	0.103225i	$0.0329160\pi$
$587$	38.3139	+	22.1205i	1.58138	+	0.913011i	0.586724	+	0.809787i	$0.300417\pi$
$588$		0			0
$589$	−7.62772	+	17.6155i	−0.314295	+	0.725832i
$590$		0			0
$591$	20.7446	+	19.4950i	0.853317	+	0.801917i
$592$		0			0
$593$	4.54755	−	2.62553i	0.186745	−	0.107817i	−0.403713	−	0.914886i	$-0.632280\pi$
$593$	4.54755	−	2.62553i	0.186745	−	0.107817i	0.590458	+	0.807068i	$0.298947\pi$
$594$		0			0
$595$	0.744563	+	1.28962i	0.0305241	+	0.0528693i
$596$		0			0
$597$	34.5475	+	8.11663i	1.41394	+	0.332192i
$598$		0			0
$599$	−0.0584220	−	0.101190i	−0.00238706	−	0.00413451i	0.864829	−	0.502066i	$-0.167426\pi$
$599$	−0.0584220	−	0.101190i	−0.00238706	−	0.00413451i	−0.867216	+	0.497931i	$0.834093\pi$
$600$		0			0
$601$		−	27.4728i		−	1.12064i	−0.828277	−	0.560319i	$-0.810679\pi$
$601$		−	27.4728i		−	1.12064i	0.828277	−	0.560319i	$-0.189321\pi$
$602$		0			0
$603$	19.6168	+	29.5735i	0.798860	+	1.20433i
$604$		0			0
$605$	−0.686141	−	0.396143i	−0.0278956	−	0.0161055i
$606$		0			0
$607$		−	2.52434i		−	0.102460i	−0.998687	−	0.0512299i	$-0.983686\pi$
$607$		−	2.52434i		−	0.102460i	0.998687	−	0.0512299i	$-0.0163141\pi$
$608$		0			0
$609$	−21.4891	−	5.04868i	−0.870783	−	0.204583i
$610$		0			0
$611$	−7.05842	+	12.2255i	−0.285553	+	0.494593i
$612$		0			0
$613$	−3.94158	+	6.82701i	−0.159199	+	0.275740i	−0.934580	−	0.355753i	$-0.884224\pi$
$613$	−3.94158	+	6.82701i	−0.159199	+	0.275740i	0.775381	+	0.631494i	$0.217558\pi$
$614$		0			0
$615$	−0.197015	+	0.838574i	−0.00794443	+	0.0338146i
$616$		0			0
$617$	3.94158	−	2.27567i	0.158682	−	0.0916151i	−0.418556	−	0.908191i	$-0.637464\pi$
$617$	3.94158	−	2.27567i	0.158682	−	0.0916151i	0.577238	+	0.816576i	$0.304130\pi$
$618$		0			0
$619$	−31.8614			−1.28062			−0.640309	−	0.768117i	$-0.721194\pi$
$619$	−31.8614			−1.28062			−0.640309	+	0.768117i	$0.721194\pi$
$620$		0			0
$621$	−38.0367	−	6.47716i	−1.52636	−	0.259919i
$622$		0			0
$623$	8.74456	+	15.1460i	0.350344	+	0.606813i
$624$		0			0
$625$	−7.98913	+	13.8376i	−0.319565	+	0.553503i
$626$		0			0
$627$	−10.3723	−	24.0087i	−0.414229	−	0.958814i
$628$		0			0
$629$	−3.11684	+	5.39853i	−0.124277	+	0.215254i
$630$		0			0
$631$	4.61684	+	7.99661i	0.183794	+	0.318340i	0.943169	−	0.332312i	$-0.107829\pi$
$631$	4.61684	+	7.99661i	0.183794	+	0.318340i	−0.759376	+	0.650652i	$0.774496\pi$
$632$		0			0
$633$	24.8723	−	7.49927i	0.988584	−	0.298069i
$634$		0			0
$635$	15.3723			0.610030
$636$		0			0
$637$	3.94158	−	2.27567i	0.156171	−	0.0901654i
$638$		0			0
$639$	−17.8030	−	8.85402i	−0.704275	−	0.350260i
$640$		0			0
$641$	−10.3139	+	17.8641i	−0.407373	+	0.705591i	−0.994594	−	0.103836i	$-0.966888\pi$
$641$	−10.3139	+	17.8641i	−0.407373	+	0.705591i	0.587222	+	0.809426i	$0.300222\pi$
$642$		0			0
$643$	−1.50000	+	2.59808i	−0.0591542	+	0.102458i	−0.894086	−	0.447895i	$-0.852174\pi$
$643$	−1.50000	+	2.59808i	−0.0591542	+	0.102458i	0.834932	+	0.550353i	$0.185507\pi$
$644$		0			0
$645$	0.0801714	−	0.341241i	0.00315675	−	0.0134363i
$646$		0			0
$647$		−	44.4434i		−	1.74725i	−0.486599	−	0.873625i	$-0.661763\pi$
$647$		−	44.4434i		−	1.74725i	0.486599	−	0.873625i	$-0.338237\pi$
$648$		0			0
$649$	−22.1168	−	12.7692i	−0.868162	−	0.501234i
$650$		0			0
$651$	13.1861	+	12.3919i	0.516806	+	0.485675i
$652$		0			0
$653$			22.0742i			0.863831i	0.901914	+	0.431916i	$0.142162\pi$
$653$			22.0742i			0.863831i	−0.901914	+	0.431916i	$0.857838\pi$
$654$		0			0
$655$	−2.19702	−	3.80534i	−0.0858445	−	0.148687i
$656$		0			0
$657$	−15.4307	−	7.67420i	−0.602009	−	0.299399i
$658$		0			0
$659$	−19.5475	−	33.8573i	−0.761464	−	1.31889i	−0.942096	−	0.335344i	$-0.891147\pi$
$659$	−19.5475	−	33.8573i	−0.761464	−	1.31889i	0.180631	−	0.983551i	$-0.442186\pi$
$660$		0			0
$661$	−4.19702	+	2.42315i	−0.163245	+	0.0942495i	−0.579397	−	0.815046i	$-0.696712\pi$
$661$	−4.19702	+	2.42315i	−0.163245	+	0.0942495i	0.416152	+	0.909295i	$0.363378\pi$
$662$		0			0
$663$	−3.11684	+	3.31662i	−0.121048	+	0.128807i
$664$		0			0
$665$	4.88316	+	6.57835i	0.189361	+	0.255097i
$666$		0			0
$667$	−34.5475	−	19.9460i	−1.33769	−	0.772314i
$668$		0			0
$669$	30.1277	−	9.08385i	1.16480	−	0.351202i
$670$		0			0
$671$	3.00000	−	1.73205i	0.115814	−	0.0668651i
$672$		0			0
$673$		−	21.1345i		−	0.814674i	−0.913278	−	0.407337i	$-0.866457\pi$
$673$		−	21.1345i		−	0.814674i	0.913278	−	0.407337i	$-0.133543\pi$
$674$		0			0
$675$	−3.81386	+	22.3966i	−0.146796	+	0.862047i
$676$		0			0
$677$	−26.7446			−1.02788			−0.513939	−	0.857827i	$-0.671814\pi$
$677$	−26.7446			−1.02788			−0.513939	+	0.857827i	$0.671814\pi$
$678$		0			0
$679$	26.2337	+	15.1460i	1.00676	+	0.581251i
$680$		0			0
$681$	8.60597	−	9.15759i	0.329781	−	0.350920i
$682$		0			0
$683$	4.74456			0.181546			0.0907728	−	0.995872i	$-0.471066\pi$
$683$	4.74456			0.181546			0.0907728	+	0.995872i	$0.471066\pi$
$684$		0			0
$685$	−10.3505			−0.395473
$686$		0			0
$687$	−25.0475	+	26.6530i	−0.955624	+	1.01688i
$688$		0			0
$689$	−32.6644	−	18.8588i	−1.24441	−	0.718463i
$690$		0			0
$691$	41.9565			1.59610			0.798050	−	0.602591i	$-0.205865\pi$
$691$	41.9565			1.59610			0.798050	+	0.602591i	$0.205865\pi$
$692$		0			0
$693$	−24.6060	+	1.52967i	−0.934703	+	0.0581074i
$694$		0			0
$695$			13.0641i			0.495550i
$696$		0			0
$697$	−0.430703	+	0.248667i	−0.0163141	+	0.00941892i
$698$		0			0
$699$	7.54755	−	2.27567i	0.285474	−	0.0860738i
$700$		0			0
$701$	−14.0584	−	8.11663i	−0.530979	−	0.306561i	0.210436	−	0.977608i	$-0.432512\pi$
$701$	−14.0584	−	8.11663i	−0.530979	−	0.306561i	−0.741415	+	0.671047i	$0.765845\pi$
$702$		0			0
$703$	−13.6277	+	31.4719i	−0.513979	+	1.18698i
$704$		0			0
$705$	4.00000	−	4.25639i	0.150649	−	0.160305i
$706$		0			0
$707$	1.62772	−	0.939764i	0.0612167	−	0.0353435i
$708$		0			0
$709$	−0.244563	−	0.423595i	−0.00918474	−	0.0159084i	0.861397	−	0.507933i	$-0.169590\pi$
$709$	−0.244563	−	0.423595i	−0.00918474	−	0.0159084i	−0.870581	+	0.492025i	$0.836257\pi$
$710$		0			0
$711$	−15.4090	+	30.9832i	−0.577881	+	1.16196i
$712$		0			0
$713$	16.3505	+	28.3200i	0.612332	+	1.06059i
$714$		0			0
$715$			9.10268i			0.340421i
$716$		0			0
$717$	13.1168	+	12.3267i	0.489858	+	0.460350i
$718$		0			0
$719$	−3.68614	−	2.12819i	−0.137470	−	0.0793683i	0.429688	−	0.902978i	$-0.358624\pi$
$719$	−3.68614	−	2.12819i	−0.137470	−	0.0793683i	−0.567158	+	0.823609i	$0.691957\pi$
$720$		0			0
$721$		−	1.52967i		−	0.0569679i
$722$		0			0
$723$	0.175266	−	0.746000i	0.00651821	−	0.0277440i
$724$		0			0
$725$	−11.7446	+	20.3422i	−0.436182	+	0.755490i
$726$		0			0
$727$	16.8723	−	29.2236i	0.625758	−	1.08385i	−0.362635	−	0.931931i	$-0.618123\pi$
$727$	16.8723	−	29.2236i	0.625758	−	1.08385i	0.988394	−	0.151914i	$-0.0485437\pi$
$728$		0			0
$729$	5.00000	+	26.5330i	0.185185	+	0.982704i
$730$		0			0
$731$	0.175266	−	0.101190i	0.00648245	−	0.00374264i
$732$		0			0
$733$	−24.9783			−0.922593			−0.461296	−	0.887246i	$-0.652616\pi$
$733$	−24.9783			−0.922593			−0.461296	+	0.887246i	$0.652616\pi$
$734$		0			0
$735$	−1.80298	+	0.543620i	−0.0665041	+	0.0200517i
$736$		0			0
$737$	20.4891	+	35.4882i	0.754727	+	1.30722i
$738$		0			0
$739$	15.2446	−	26.4044i	0.560780	−	0.971300i	−0.436648	−	0.899632i	$-0.643835\pi$
$739$	15.2446	−	26.4044i	0.560780	−	0.971300i	0.997429	−	0.0716677i	$-0.0228321\pi$
$740$		0			0
$741$	−14.9416	+	20.0935i	−0.548893	+	0.738154i
$742$		0			0
$743$	5.31386	−	9.20387i	0.194947	−	0.337657i	−0.751936	−	0.659236i	$-0.770880\pi$
$743$	5.31386	−	9.20387i	0.194947	−	0.337657i	0.946883	+	0.321578i	$0.104213\pi$
$744$		0			0
$745$	−0.941578	−	1.63086i	−0.0344967	−	0.0597501i
$746$		0			0
$747$	10.3723	−	0.644810i	0.379502	−	0.0235924i
$748$		0			0
$749$	−39.7228			−1.45144
$750$		0			0
$751$	−24.9891	+	14.4275i	−0.911866	+	0.526466i	−0.881031	−	0.473058i	$-0.843150\pi$
$751$	−24.9891	+	14.4275i	−0.911866	+	0.526466i	−0.0308350	+	0.999524i	$0.509817\pi$
$752$		0			0
$753$	−2.54755	+	10.8434i	−0.0928378	+	0.395154i
$754$		0			0
$755$	−5.37228	+	9.30506i	−0.195517	+	0.338646i
$756$		0			0
$757$	12.8723	−	22.2954i	0.467851	−	0.810342i	−0.531474	−	0.847075i	$-0.678362\pi$
$757$	12.8723	−	22.2954i	0.467851	−	0.810342i	0.999325	+	0.0367328i	$0.0116950\pi$
$758$		0			0
$759$	−43.3723	−	10.1899i	−1.57431	−	0.369871i
$760$		0			0
$761$			33.4612i			1.21297i	0.795096	+	0.606484i	$0.207421\pi$
$761$			33.4612i			1.21297i	−0.795096	+	0.606484i	$0.792579\pi$
$762$		0			0
$763$	12.0000	+	6.92820i	0.434429	+	0.250818i
$764$		0			0
$765$	1.56930	−	1.04095i	0.0567380	−	0.0376358i
$766$		0			0
$767$			24.4511i			0.882878i
$768$		0			0
$769$	10.1277	+	17.5417i	0.365215	+	0.632571i	0.988811	−	0.149176i	$-0.0476622\pi$
$769$	10.1277	+	17.5417i	0.365215	+	0.632571i	−0.623596	+	0.781747i	$0.714329\pi$
$770$		0			0
$771$	3.17527	+	0.746000i	0.114354	+	0.0268665i
$772$		0			0
$773$	21.6861	+	37.5615i	0.779996	+	1.35099i	0.931943	+	0.362604i	$0.118112\pi$
$773$	21.6861	+	37.5615i	0.779996	+	1.35099i	−0.151947	+	0.988389i	$0.548554\pi$
$774$		0			0
$775$	16.6753	−	9.62747i	0.598993	−	0.345829i
$776$		0			0
$777$	23.5584	+	22.1394i	0.845154	+	0.794245i
$778$		0			0
$779$	−2.19702	+	1.63086i	−0.0787162	+	0.0584317i
$780$		0			0
$781$	−19.8832	−	11.4795i	−0.711475	−	0.410770i
$782$		0			0
$783$	−4.68614	+	27.5190i	−0.167469	+	0.983451i
$784$		0			0
$785$	−14.0584	+	8.11663i	−0.501767	+	0.289695i
$786$		0			0
$787$		−	7.57301i		−	0.269949i	−0.990849	−	0.134974i	$-0.956905\pi$
$787$		−	7.57301i		−	0.269949i	0.990849	−	0.134974i	$-0.0430952\pi$
$788$

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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.bn (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.18614 - 1.26217i) q^{3} +(0.686141 + 0.396143i) q^{5} +2.37228 q^{7} +(-0.186141 - 2.99422i) q^{9} +O(q^{10})\)	\(q+(1.18614 - 1.26217i) q^{3} +(0.686141 + 0.396143i) q^{5} +2.37228 q^{7} +(-0.186141 - 2.99422i) q^{9} -3.46410i q^{11} +(-2.87228 + 1.65831i) q^{13} +(1.31386 - 0.396143i) q^{15} +(0.686141 + 0.396143i) q^{17} +(4.00000 + 1.73205i) q^{19} +(2.81386 - 2.99422i) q^{21} +(6.43070 - 3.71277i) q^{23} +(-2.18614 - 3.78651i) q^{25} +(-4.00000 - 3.31662i) q^{27} +(-2.68614 - 4.65253i) q^{29} +4.40387i q^{31} +(-4.37228 - 4.10891i) q^{33} +(1.62772 + 0.939764i) q^{35} +7.86797i q^{37} +(-1.31386 + 5.59230i) q^{39} +(-0.313859 + 0.543620i) q^{41} +(0.127719 - 0.221215i) q^{43} +(1.05842 - 2.12819i) q^{45} +(3.68614 - 2.12819i) q^{47} -1.37228 q^{49} +(1.31386 - 0.396143i) q^{51} +(5.68614 + 9.84868i) q^{53} +(1.37228 - 2.37686i) q^{55} +(6.93070 - 2.99422i) q^{57} +(3.68614 - 6.38458i) q^{59} +(0.500000 + 0.866025i) q^{61} +(-0.441578 - 7.10313i) q^{63} -2.62772 q^{65} +(-10.2446 + 5.91470i) q^{67} +(2.94158 - 12.5205i) q^{69} +(3.31386 - 5.73977i) q^{71} +(2.87228 - 4.97494i) q^{73} +(-7.37228 - 1.73205i) q^{75} -8.21782i q^{77} +(-9.98913 - 5.76722i) q^{79} +(-8.93070 + 1.11469i) q^{81} +3.46410i q^{83} +(0.313859 + 0.543620i) q^{85} +(-9.05842 - 2.12819i) q^{87} +(3.68614 + 6.38458i) q^{89} +(-6.81386 + 3.93398i) q^{91} +(5.55842 + 5.22360i) q^{93} +(2.05842 + 2.77300i) q^{95} +(11.0584 + 6.38458i) q^{97} +(-10.3723 + 0.644810i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) 4 * q - q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9	\( 4 q - q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9} + 11 q^{15} - 3 q^{17} + 16 q^{19} + 17 q^{21} - 3 q^{23} - 3 q^{25} - 16 q^{27} - 5 q^{29} - 6 q^{33} + 18 q^{35} - 11 q^{39} - 7 q^{41} + 12 q^{43} - 13 q^{45} + 9 q^{47} + 6 q^{49} + 11 q^{51} + 17 q^{53} - 6 q^{55} - q^{57} + 9 q^{59} + 2 q^{61} - 19 q^{63} - 22 q^{65} - 18 q^{67} + 29 q^{69} + 19 q^{71} - 18 q^{75} + 6 q^{79} - 7 q^{81} + 7 q^{85} - 19 q^{87} + 9 q^{89} - 33 q^{91} + 5 q^{93} - 9 q^{95} + 27 q^{97} - 30 q^{99}+O(q^{100}) \) 4 * q - q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9 + 11 * q^15 - 3 * q^17 + 16 * q^19 + 17 * q^21 - 3 * q^23 - 3 * q^25 - 16 * q^27 - 5 * q^29 - 6 * q^33 + 18 * q^35 - 11 * q^39 - 7 * q^41 + 12 * q^43 - 13 * q^45 + 9 * q^47 + 6 * q^49 + 11 * q^51 + 17 * q^53 - 6 * q^55 - q^57 + 9 * q^59 + 2 * q^61 - 19 * q^63 - 22 * q^65 - 18 * q^67 + 29 * q^69 + 19 * q^71 - 18 * q^75 + 6 * q^79 - 7 * q^81 + 7 * q^85 - 19 * q^87 + 9 * q^89 - 33 * q^91 + 5 * q^93 - 9 * q^95 + 27 * q^97 - 30 * q^99

Introduction

L-functions

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