LMFDB - Embedded newform 1680.2.t.i.1009.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(1009,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1680.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.t (of order $2$, degree $1$, 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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.6
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	1680.1009
Dual form			1680.2.t.i.1009.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.48119 + 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(1.48119 + 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} -0.387873 q^{11} +2.96239i q^{13} +(-1.67513 + 1.48119i) q^{15} +3.35026i q^{17} -2.96239 q^{19} +1.00000 q^{21} -0.962389i q^{23} +(-0.612127 + 4.96239i) q^{25} -1.00000i q^{27} -1.22425 q^{29} -2.96239 q^{31} -0.387873i q^{33} +(1.67513 - 1.48119i) q^{35} +5.92478i q^{37} -2.96239 q^{39} +1.03761 q^{41} +10.7005i q^{43} +(-1.48119 - 1.67513i) q^{45} -3.22425i q^{47} -1.00000 q^{49} -3.35026 q^{51} +5.66291i q^{53} +(-0.574515 - 0.649738i) q^{55} -2.96239i q^{57} +3.22425 q^{59} +14.6253 q^{61} +1.00000i q^{63} +(-4.96239 + 4.38787i) q^{65} +0.962389 q^{69} -5.53690 q^{71} -6.18664i q^{73} +(-4.96239 - 0.612127i) q^{75} +0.387873i q^{77} -13.9248 q^{79} +1.00000 q^{81} +3.22425i q^{83} +(-5.61213 + 4.96239i) q^{85} -1.22425i q^{87} -3.73813 q^{89} +2.96239 q^{91} -2.96239i q^{93} +(-4.38787 - 4.96239i) q^{95} -7.73813i q^{97} +0.387873 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^5 - 6 * q^9	$ 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} + 4 q^{31} + 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} + 20 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 12 q^{71} - 8 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} - 4 q^{91} - 28 q^{95} + 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 + 4 * q^31 + 4 * q^39 + 28 * q^41 + 2 * q^45 - 6 * q^49 + 20 * q^55 + 16 * q^59 + 4 * q^61 - 8 * q^65 - 16 * q^69 + 12 * q^71 - 8 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 - 4 * q^89 - 4 * q^91 - 28 * q^95 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$-1$	$1$	$1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−0.387873			−0.116948			−0.0584741	−	0.998289i	$-0.518623\pi$
$11$	−0.387873			−0.116948			−0.0584741	+	0.998289i	$0.518623\pi$
$12$		0			0
$13$			2.96239i			0.821619i	0.911721	+	0.410809i	$0.134754\pi$
$13$			2.96239i			0.821619i	−0.911721	+	0.410809i	$0.865246\pi$
$14$		0			0
$15$	−1.67513	+	1.48119i	−0.432517	+	0.382443i
$16$		0			0
$17$			3.35026i			0.812558i	0.913749	+	0.406279i	$0.133174\pi$
$17$			3.35026i			0.812558i	−0.913749	+	0.406279i	$0.866826\pi$
$18$		0			0
$19$	−2.96239			−0.679619			−0.339809	−	0.940494i	$-0.610363\pi$
$19$	−2.96239			−0.679619			−0.339809	+	0.940494i	$0.610363\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$		−	0.962389i		−	0.200672i	−0.994954	−	0.100336i	$-0.968008\pi$
$23$		−	0.962389i		−	0.200672i	0.994954	−	0.100336i	$-0.0319918\pi$
$24$		0			0
$25$	−0.612127	+	4.96239i	−0.122425	+	0.992478i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−1.22425			−0.227338			−0.113669	−	0.993519i	$-0.536260\pi$
$29$	−1.22425			−0.227338			−0.113669	+	0.993519i	$0.536260\pi$
$30$		0			0
$31$	−2.96239			−0.532061			−0.266030	−	0.963965i	$-0.585712\pi$
$31$	−2.96239			−0.532061			−0.266030	+	0.963965i	$0.585712\pi$
$32$		0			0
$33$		−	0.387873i		−	0.0675200i
$34$		0			0
$35$	1.67513	−	1.48119i	0.283149	−	0.250368i
$36$		0			0
$37$			5.92478i			0.974027i	0.873394	+	0.487014i	$0.161914\pi$
$37$			5.92478i			0.974027i	−0.873394	+	0.487014i	$0.838086\pi$
$38$		0			0
$39$	−2.96239			−0.474362
$40$		0			0
$41$	1.03761			0.162048			0.0810238	−	0.996712i	$-0.474181\pi$
$41$	1.03761			0.162048			0.0810238	+	0.996712i	$0.474181\pi$
$42$		0			0
$43$			10.7005i			1.63181i	0.578183	+	0.815907i	$0.303762\pi$
$43$			10.7005i			1.63181i	−0.578183	+	0.815907i	$0.696238\pi$
$44$		0			0
$45$	−1.48119	−	1.67513i	−0.220803	−	0.249714i
$46$		0			0
$47$		−	3.22425i		−	0.470306i	−0.971958	−	0.235153i	$-0.924441\pi$
$47$		−	3.22425i		−	0.470306i	0.971958	−	0.235153i	$-0.0755591\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−3.35026			−0.469130
$52$		0			0
$53$			5.66291i			0.777861i	0.921267	+	0.388930i	$0.127155\pi$
$53$			5.66291i			0.777861i	−0.921267	+	0.388930i	$0.872845\pi$
$54$		0			0
$55$	−0.574515	−	0.649738i	−0.0774677	−	0.0876107i
$56$		0			0
$57$		−	2.96239i		−	0.392378i
$58$		0			0
$59$	3.22425			0.419762			0.209881	−	0.977727i	$-0.432692\pi$
$59$	3.22425			0.419762			0.209881	+	0.977727i	$0.432692\pi$
$60$		0			0
$61$	14.6253			1.87258			0.936289	−	0.351231i	$-0.114237\pi$
$61$	14.6253			1.87258			0.936289	+	0.351231i	$0.114237\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−4.96239	+	4.38787i	−0.615509	+	0.544249i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	0.962389			0.115858
$70$		0			0
$71$	−5.53690			−0.657110			−0.328555	−	0.944485i	$-0.606562\pi$
$71$	−5.53690			−0.657110			−0.328555	+	0.944485i	$0.606562\pi$
$72$		0			0
$73$		−	6.18664i		−	0.724092i	−0.932160	−	0.362046i	$-0.882078\pi$
$73$		−	6.18664i		−	0.724092i	0.932160	−	0.362046i	$-0.117922\pi$
$74$		0			0
$75$	−4.96239	−	0.612127i	−0.573007	−	0.0706823i
$76$		0			0
$77$			0.387873i			0.0442022i
$78$		0			0
$79$	−13.9248			−1.56666			−0.783330	−	0.621606i	$-0.786480\pi$
$79$	−13.9248			−1.56666			−0.783330	+	0.621606i	$0.786480\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			3.22425i			0.353908i	0.984219	+	0.176954i	$0.0566244\pi$
$83$			3.22425i			0.353908i	−0.984219	+	0.176954i	$0.943376\pi$
$84$		0			0
$85$	−5.61213	+	4.96239i	−0.608721	+	0.538247i
$86$		0			0
$87$		−	1.22425i		−	0.131254i
$88$		0			0
$89$	−3.73813			−0.396242			−0.198121	−	0.980178i	$-0.563484\pi$
$89$	−3.73813			−0.396242			−0.198121	+	0.980178i	$0.563484\pi$
$90$		0			0
$91$	2.96239			0.310543
$92$		0			0
$93$		−	2.96239i		−	0.307185i
$94$		0			0
$95$	−4.38787	−	4.96239i	−0.450186	−	0.509130i
$96$		0			0
$97$		−	7.73813i		−	0.785689i	−0.919605	−	0.392844i	$-0.871491\pi$
$97$		−	7.73813i		−	0.785689i	0.919605	−	0.392844i	$-0.128509\pi$
$98$		0			0
$99$	0.387873			0.0389827
$100$		0			0
$101$	−7.58769			−0.755003			−0.377502	−	0.926009i	$-0.623217\pi$
$101$	−7.58769			−0.755003			−0.377502	+	0.926009i	$0.623217\pi$
$102$		0			0
$103$			14.7005i			1.44849i	0.689545	+	0.724243i	$0.257811\pi$
$103$			14.7005i			1.44849i	−0.689545	+	0.724243i	$0.742189\pi$
$104$		0			0
$105$	1.48119	+	1.67513i	0.144550	+	0.163476i
$106$		0			0
$107$		−	16.4387i		−	1.58919i	−0.607143	−	0.794593i	$-0.707685\pi$
$107$		−	16.4387i		−	1.58919i	0.607143	−	0.794593i	$-0.292315\pi$
$108$		0			0
$109$	2.77575			0.265868			0.132934	−	0.991125i	$-0.457560\pi$
$109$	2.77575			0.265868			0.132934	+	0.991125i	$0.457560\pi$
$110$		0			0
$111$	−5.92478			−0.562355
$112$		0			0
$113$		−	4.26187i		−	0.400923i	−0.979702	−	0.200461i	$-0.935756\pi$
$113$		−	4.26187i		−	0.400923i	0.979702	−	0.200461i	$-0.0642441\pi$
$114$		0			0
$115$	1.61213	−	1.42548i	0.150332	−	0.132927i
$116$		0			0
$117$		−	2.96239i		−	0.273873i
$118$		0			0
$119$	3.35026			0.307118
$120$		0			0
$121$	−10.8496			−0.986323
$122$		0			0
$123$			1.03761i			0.0935583i
$124$		0			0
$125$	−9.21933	+	6.32487i	−0.824602	+	0.565713i
$126$		0			0
$127$			14.5501i			1.29111i	0.763714	+	0.645555i	$0.223374\pi$
$127$			14.5501i			1.29111i	−0.763714	+	0.645555i	$0.776626\pi$
$128$		0			0
$129$	−10.7005			−0.942129
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$		0			0
$133$			2.96239i			0.256872i
$134$		0			0
$135$	1.67513	−	1.48119i	0.144172	−	0.127481i
$136$		0			0
$137$			10.4387i			0.891835i	0.895074	+	0.445917i	$0.147122\pi$
$137$			10.4387i			0.891835i	−0.895074	+	0.445917i	$0.852878\pi$
$138$		0			0
$139$	5.66291			0.480322			0.240161	−	0.970733i	$-0.422800\pi$
$139$	5.66291			0.480322			0.240161	+	0.970733i	$0.422800\pi$
$140$		0			0
$141$	3.22425			0.271531
$142$		0			0
$143$		−	1.14903i		−	0.0960868i
$144$		0			0
$145$	−1.81336	−	2.05079i	−0.150591	−	0.170308i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	−20.5501			−1.68353			−0.841764	−	0.539846i	$-0.818483\pi$
$149$	−20.5501			−1.68353			−0.841764	+	0.539846i	$0.818483\pi$
$150$		0			0
$151$	22.7005			1.84734			0.923671	−	0.383186i	$-0.125173\pi$
$151$	22.7005			1.84734			0.923671	+	0.383186i	$0.125173\pi$
$152$		0			0
$153$		−	3.35026i		−	0.270853i
$154$		0			0
$155$	−4.38787	−	4.96239i	−0.352442	−	0.398589i
$156$		0			0
$157$		−	2.18664i		−	0.174513i	−0.996186	−	0.0872565i	$-0.972190\pi$
$157$		−	2.18664i		−	0.174513i	0.996186	−	0.0872565i	$-0.0278100\pi$
$158$		0			0
$159$	−5.66291			−0.449098
$160$		0			0
$161$	−0.962389			−0.0758468
$162$		0			0
$163$			7.22425i			0.565847i	0.959142	+	0.282924i	$0.0913043\pi$
$163$			7.22425i			0.565847i	−0.959142	+	0.282924i	$0.908696\pi$
$164$		0			0
$165$	0.649738	−	0.574515i	0.0505821	−	0.0447260i
$166$		0			0
$167$			15.8496i			1.22648i	0.789898	+	0.613238i	$0.210133\pi$
$167$			15.8496i			1.22648i	−0.789898	+	0.613238i	$0.789867\pi$
$168$		0			0
$169$	4.22425			0.324943
$170$		0			0
$171$	2.96239			0.226540
$172$		0			0
$173$		−	23.9756i		−	1.82283i	−0.411490	−	0.911414i	$-0.634992\pi$
$173$		−	23.9756i		−	1.82283i	0.411490	−	0.911414i	$-0.365008\pi$
$174$		0			0
$175$	4.96239	+	0.612127i	0.375121	+	0.0462724i
$176$		0			0
$177$			3.22425i			0.242350i
$178$		0			0
$179$	12.2374			0.914668			0.457334	−	0.889295i	$-0.348804\pi$
$179$	12.2374			0.914668			0.457334	+	0.889295i	$0.348804\pi$
$180$		0			0
$181$	8.70052			0.646705			0.323352	−	0.946279i	$-0.395190\pi$
$181$	8.70052			0.646705			0.323352	+	0.946279i	$0.395190\pi$
$182$		0			0
$183$			14.6253i			1.08113i
$184$		0			0
$185$	−9.92478	+	8.77575i	−0.729684	+	0.645206i
$186$		0			0
$187$		−	1.29948i		−	0.0950271i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	13.5369			0.979496			0.489748	−	0.871864i	$-0.337089\pi$
$191$	13.5369			0.979496			0.489748	+	0.871864i	$0.337089\pi$
$192$		0			0
$193$			1.92478i			0.138548i	0.997598	+	0.0692742i	$0.0220684\pi$
$193$			1.92478i			0.138548i	−0.997598	+	0.0692742i	$0.977932\pi$
$194$		0			0
$195$	−4.38787	−	4.96239i	−0.314222	−	0.355364i
$196$		0			0
$197$			20.1114i			1.43288i	0.697649	+	0.716440i	$0.254229\pi$
$197$			20.1114i			1.43288i	−0.697649	+	0.716440i	$0.745771\pi$
$198$		0			0
$199$	−8.26187			−0.585668			−0.292834	−	0.956163i	$-0.594598\pi$
$199$	−8.26187			−0.585668			−0.292834	+	0.956163i	$0.594598\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1.22425i			0.0859258i
$204$		0			0
$205$	1.53690	+	1.73813i	0.107342	+	0.121397i
$206$		0			0
$207$			0.962389i			0.0668906i
$208$		0			0
$209$	1.14903			0.0794801
$210$		0			0
$211$	18.1768			1.25134			0.625671	−	0.780087i	$-0.284825\pi$
$211$	18.1768			1.25134			0.625671	+	0.780087i	$0.284825\pi$
$212$		0			0
$213$		−	5.53690i		−	0.379382i
$214$		0			0
$215$	−17.9248	+	15.8496i	−1.22246	+	1.08093i
$216$		0			0
$217$			2.96239i			0.201100i
$218$		0			0
$219$	6.18664			0.418055
$220$		0			0
$221$	−9.92478			−0.667613
$222$		0			0
$223$			21.4010i			1.43312i	0.697525	+	0.716560i	$0.254284\pi$
$223$			21.4010i			1.43312i	−0.697525	+	0.716560i	$0.745716\pi$
$224$		0			0
$225$	0.612127	−	4.96239i	0.0408085	−	0.330826i
$226$		0			0
$227$		−	18.5501i		−	1.23121i	−0.788054	−	0.615606i	$-0.788911\pi$
$227$		−	18.5501i		−	1.23121i	0.788054	−	0.615606i	$-0.211089\pi$
$228$		0			0
$229$	28.5501			1.88664			0.943321	−	0.331881i	$-0.107683\pi$
$229$	28.5501			1.88664			0.943321	+	0.331881i	$0.107683\pi$
$230$		0			0
$231$	−0.387873			−0.0255202
$232$		0			0
$233$		−	18.9624i		−	1.24227i	−0.783705	−	0.621134i	$-0.786672\pi$
$233$		−	18.9624i		−	1.24227i	0.783705	−	0.621134i	$-0.213328\pi$
$234$		0			0
$235$	5.40105	−	4.77575i	0.352325	−	0.311535i
$236$		0			0
$237$		−	13.9248i		−	0.904511i
$238$		0			0
$239$	−18.1622			−1.17482			−0.587408	−	0.809291i	$-0.699851\pi$
$239$	−18.1622			−1.17482			−0.587408	+	0.809291i	$0.699851\pi$
$240$		0			0
$241$	14.3733			0.925865			0.462932	−	0.886394i	$-0.346797\pi$
$241$	14.3733			0.925865			0.462932	+	0.886394i	$0.346797\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−1.48119	−	1.67513i	−0.0946300	−	0.107020i
$246$		0			0
$247$		−	8.77575i		−	0.558387i
$248$		0			0
$249$	−3.22425			−0.204329
$250$		0			0
$251$	8.62530			0.544424			0.272212	−	0.962237i	$-0.412245\pi$
$251$	8.62530			0.544424			0.272212	+	0.962237i	$0.412245\pi$
$252$		0			0
$253$			0.373285i			0.0234682i
$254$		0			0
$255$	−4.96239	−	5.61213i	−0.310757	−	0.351445i
$256$		0			0
$257$		−	27.8251i		−	1.73568i	−0.496841	−	0.867842i	$-0.665507\pi$
$257$		−	27.8251i		−	1.73568i	0.496841	−	0.867842i	$-0.334493\pi$
$258$		0			0
$259$	5.92478			0.368148
$260$		0			0
$261$	1.22425			0.0757794
$262$		0			0
$263$		−	0.589104i		−	0.0363257i	−0.999835	−	0.0181629i	$-0.994218\pi$
$263$		−	0.589104i		−	0.0363257i	0.999835	−	0.0181629i	$-0.00578173\pi$
$264$		0			0
$265$	−9.48612	+	8.38787i	−0.582728	+	0.515263i
$266$		0			0
$267$		−	3.73813i		−	0.228770i
$268$		0			0
$269$	−15.7381			−0.959571			−0.479786	−	0.877386i	$-0.659286\pi$
$269$	−15.7381			−0.959571			−0.479786	+	0.877386i	$0.659286\pi$
$270$		0			0
$271$	7.11283			0.432074			0.216037	−	0.976385i	$-0.430687\pi$
$271$	7.11283			0.432074			0.216037	+	0.976385i	$0.430687\pi$
$272$		0			0
$273$			2.96239i			0.179292i
$274$		0			0
$275$	0.237428	−	1.92478i	0.0143174	−	0.116068i
$276$		0			0
$277$			5.92478i			0.355985i	0.984032	+	0.177993i	$0.0569603\pi$
$277$			5.92478i			0.355985i	−0.984032	+	0.177993i	$0.943040\pi$
$278$		0			0
$279$	2.96239			0.177354
$280$		0			0
$281$	29.3258			1.74943			0.874716	−	0.484636i	$-0.161048\pi$
$281$	29.3258			1.74943			0.874716	+	0.484636i	$0.161048\pi$
$282$		0			0
$283$		−	16.1016i		−	0.957139i	−0.878050	−	0.478570i	$-0.841155\pi$
$283$		−	16.1016i		−	0.957139i	0.878050	−	0.478570i	$-0.158845\pi$
$284$		0			0
$285$	4.96239	−	4.38787i	0.293947	−	0.259915i
$286$		0			0
$287$		−	1.03761i		−	0.0612483i
$288$		0			0
$289$	5.77575			0.339750
$290$		0			0
$291$	7.73813			0.453617
$292$		0			0
$293$		−	9.27504i		−	0.541854i	−0.962600	−	0.270927i	$-0.912670\pi$
$293$		−	9.27504i		−	0.541854i	0.962600	−	0.270927i	$-0.0873301\pi$
$294$		0			0
$295$	4.77575	+	5.40105i	0.278055	+	0.314461i
$296$		0			0
$297$			0.387873i			0.0225067i
$298$		0			0
$299$	2.85097			0.164876
$300$		0			0
$301$	10.7005			0.616768
$302$		0			0
$303$		−	7.58769i		−	0.435901i
$304$		0			0
$305$	21.6629	+	24.4993i	1.24041	+	1.40283i
$306$		0			0
$307$		−	10.7005i		−	0.610711i	−0.952238	−	0.305356i	$-0.901225\pi$
$307$		−	10.7005i		−	0.610711i	0.952238	−	0.305356i	$-0.0987753\pi$
$308$		0			0
$309$	−14.7005			−0.836284
$310$		0			0
$311$	−12.1016			−0.686217			−0.343109	−	0.939296i	$-0.611480\pi$
$311$	−12.1016			−0.686217			−0.343109	+	0.939296i	$0.611480\pi$
$312$		0			0
$313$		−	28.3634i		−	1.60320i	−0.597863	−	0.801598i	$-0.703983\pi$
$313$		−	28.3634i		−	1.60320i	0.597863	−	0.801598i	$-0.296017\pi$
$314$		0			0
$315$	−1.67513	+	1.48119i	−0.0943829	+	0.0834558i
$316$		0			0
$317$		−	17.5125i		−	0.983598i	−0.870709	−	0.491799i	$-0.836339\pi$
$317$		−	17.5125i		−	0.983598i	0.870709	−	0.491799i	$-0.163661\pi$
$318$		0			0
$319$	0.474855			0.0265868
$320$		0			0
$321$	16.4387			0.917516
$322$		0			0
$323$		−	9.92478i		−	0.552229i
$324$		0			0
$325$	−14.7005	−	1.81336i	−0.815438	−	0.100587i
$326$		0			0
$327$			2.77575i			0.153499i
$328$		0			0
$329$	−3.22425			−0.177759
$330$		0			0
$331$	19.2243			1.05666			0.528330	−	0.849039i	$-0.322818\pi$
$331$	19.2243			1.05666			0.528330	+	0.849039i	$0.322818\pi$
$332$		0			0
$333$		−	5.92478i		−	0.324676i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$		0			0
$339$	4.26187			0.231473
$340$		0			0
$341$	1.14903			0.0622235
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	1.42548	+	1.61213i	0.0767455	+	0.0867940i
$346$		0			0
$347$			1.11283i			0.0597401i	0.999554	+	0.0298700i	$0.00950934\pi$
$347$			1.11283i			0.0597401i	−0.999554	+	0.0298700i	$0.990491\pi$
$348$		0			0
$349$	32.0263			1.71433			0.857166	−	0.515041i	$-0.172223\pi$
$349$	32.0263			1.71433			0.857166	+	0.515041i	$0.172223\pi$
$350$		0			0
$351$	2.96239			0.158121
$352$		0			0
$353$		−	6.20123i		−	0.330058i	−0.986289	−	0.165029i	$-0.947228\pi$
$353$		−	6.20123i		−	0.330058i	0.986289	−	0.165029i	$-0.0527718\pi$
$354$		0			0
$355$	−8.20123	−	9.27504i	−0.435276	−	0.492268i
$356$		0			0
$357$			3.35026i			0.177315i
$358$		0			0
$359$	−1.68735			−0.0890549			−0.0445275	−	0.999008i	$-0.514178\pi$
$359$	−1.68735			−0.0890549			−0.0445275	+	0.999008i	$0.514178\pi$
$360$		0			0
$361$	−10.2243			−0.538119
$362$		0			0
$363$		−	10.8496i		−	0.569454i
$364$		0			0
$365$	10.3634	−	9.16362i	0.542447	−	0.479646i
$366$		0			0
$367$		−	3.84955i		−	0.200945i	−0.994940	−	0.100473i	$-0.967965\pi$
$367$		−	3.84955i		−	0.200945i	0.994940	−	0.100473i	$-0.0320355\pi$
$368$		0			0
$369$	−1.03761			−0.0540159
$370$		0			0
$371$	5.66291			0.294004
$372$		0			0
$373$			24.8773i			1.28810i	0.764984	+	0.644049i	$0.222747\pi$
$373$			24.8773i			1.28810i	−0.764984	+	0.644049i	$0.777253\pi$
$374$		0			0
$375$	−6.32487	−	9.21933i	−0.326615	−	0.476084i
$376$		0			0
$377$		−	3.62672i		−	0.186785i
$378$		0			0
$379$	2.70052			0.138717			0.0693583	−	0.997592i	$-0.477905\pi$
$379$	2.70052			0.138717			0.0693583	+	0.997592i	$0.477905\pi$
$380$		0			0
$381$	−14.5501			−0.745423
$382$		0			0
$383$		−	0.523730i		−	0.0267614i	−0.999910	−	0.0133807i	$-0.995741\pi$
$383$		−	0.523730i		−	0.0267614i	0.999910	−	0.0133807i	$-0.00425933\pi$
$384$		0			0
$385$	−0.649738	+	0.574515i	−0.0331137	+	0.0292800i
$386$		0			0
$387$		−	10.7005i		−	0.543938i
$388$		0			0
$389$	−6.77575			−0.343544			−0.171772	−	0.985137i	$-0.554949\pi$
$389$	−6.77575			−0.343544			−0.171772	+	0.985137i	$0.554949\pi$
$390$		0			0
$391$	3.22425			0.163058
$392$		0			0
$393$			4.00000i			0.201773i
$394$		0			0
$395$	−20.6253	−	23.3258i	−1.03777	−	1.17365i
$396$		0			0
$397$			0.635150i			0.0318773i	0.999873	+	0.0159386i	$0.00507364\pi$
$397$			0.635150i			0.0318773i	−0.999873	+	0.0159386i	$0.994926\pi$
$398$		0			0
$399$	−2.96239			−0.148305
$400$		0			0
$401$	10.3733			0.518017			0.259009	−	0.965875i	$-0.416604\pi$
$401$	10.3733			0.518017			0.259009	+	0.965875i	$0.416604\pi$
$402$		0			0
$403$		−	8.77575i		−	0.437151i
$404$		0			0
$405$	1.48119	+	1.67513i	0.0736011	+	0.0832379i
$406$		0			0
$407$		−	2.29806i		−	0.113911i
$408$		0			0
$409$	−15.7743			−0.779991			−0.389995	−	0.920817i	$-0.627523\pi$
$409$	−15.7743			−0.779991			−0.389995	+	0.920817i	$0.627523\pi$
$410$		0			0
$411$	−10.4387			−0.514901
$412$		0			0
$413$		−	3.22425i		−	0.158655i
$414$		0			0
$415$	−5.40105	+	4.77575i	−0.265127	+	0.234432i
$416$		0			0
$417$			5.66291i			0.277314i
$418$		0			0
$419$	25.1490			1.22861			0.614305	−	0.789068i	$-0.289436\pi$
$419$	25.1490			1.22861			0.614305	+	0.789068i	$0.289436\pi$
$420$		0			0
$421$	20.1768			0.983357			0.491678	−	0.870777i	$-0.336384\pi$
$421$	20.1768			0.983357			0.491678	+	0.870777i	$0.336384\pi$
$422$		0			0
$423$			3.22425i			0.156769i
$424$		0			0
$425$	−16.6253	−	2.05079i	−0.806446	−	0.0994777i
$426$		0			0
$427$		−	14.6253i		−	0.707768i
$428$		0			0
$429$	1.14903			0.0554757
$430$		0			0
$431$	38.3127			1.84546			0.922728	−	0.385452i	$-0.125955\pi$
$431$	38.3127			1.84546			0.922728	+	0.385452i	$0.125955\pi$
$432$		0			0
$433$			12.8872i			0.619318i	0.950848	+	0.309659i	$0.100215\pi$
$433$			12.8872i			0.619318i	−0.950848	+	0.309659i	$0.899785\pi$
$434$		0			0
$435$	2.05079	−	1.81336i	0.0983276	−	0.0869438i
$436$		0			0
$437$			2.85097i			0.136380i
$438$		0			0
$439$	−29.6629			−1.41573			−0.707867	−	0.706346i	$-0.750342\pi$
$439$	−29.6629			−1.41573			−0.707867	+	0.706346i	$0.750342\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	24.8119i		−	1.17885i	−0.807823	−	0.589425i	$-0.799354\pi$
$443$		−	24.8119i		−	1.17885i	0.807823	−	0.589425i	$-0.200646\pi$
$444$		0			0
$445$	−5.53690	−	6.26187i	−0.262474	−	0.296841i
$446$		0			0
$447$		−	20.5501i		−	0.971985i
$448$		0			0
$449$	36.6516			1.72970			0.864849	−	0.502032i	$-0.167414\pi$
$449$	36.6516			1.72970			0.864849	+	0.502032i	$0.167414\pi$
$450$		0			0
$451$	−0.402462			−0.0189512
$452$		0			0
$453$			22.7005i			1.06656i
$454$		0			0
$455$	4.38787	+	4.96239i	0.205707	+	0.232640i
$456$		0			0
$457$			15.4763i			0.723949i	0.932188	+	0.361975i	$0.117897\pi$
$457$			15.4763i			0.723949i	−0.932188	+	0.361975i	$0.882103\pi$
$458$		0			0
$459$	3.35026			0.156377
$460$		0			0
$461$	41.2605			1.92169			0.960845	−	0.277085i	$-0.0893684\pi$
$461$	41.2605			1.92169			0.960845	+	0.277085i	$0.0893684\pi$
$462$		0			0
$463$			15.5975i			0.724879i	0.932007	+	0.362440i	$0.118056\pi$
$463$			15.5975i			0.724879i	−0.932007	+	0.362440i	$0.881944\pi$
$464$		0			0
$465$	4.96239	−	4.38787i	0.230125	−	0.203483i
$466$		0			0
$467$			18.5501i			0.858395i	0.903211	+	0.429198i	$0.141204\pi$
$467$			18.5501i			0.858395i	−0.903211	+	0.429198i	$0.858796\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	2.18664			0.100755
$472$		0			0
$473$		−	4.15045i		−	0.190838i
$474$		0			0
$475$	1.81336	−	14.7005i	0.0832026	−	0.674506i
$476$		0			0
$477$		−	5.66291i		−	0.259287i
$478$		0			0
$479$	−41.5026			−1.89630			−0.948151	−	0.317819i	$-0.897050\pi$
$479$	−41.5026			−1.89630			−0.948151	+	0.317819i	$0.897050\pi$
$480$		0			0
$481$	−17.5515			−0.800279
$482$		0			0
$483$		−	0.962389i		−	0.0437902i
$484$		0			0
$485$	12.9624	−	11.4617i	0.588592	−	0.520448i
$486$		0			0
$487$			13.6728i			0.619572i	0.950806	+	0.309786i	$0.100257\pi$
$487$			13.6728i			0.619572i	−0.950806	+	0.309786i	$0.899743\pi$
$488$		0			0
$489$	−7.22425			−0.326692
$490$		0			0
$491$	−23.3404			−1.05334			−0.526669	−	0.850070i	$-0.676559\pi$
$491$	−23.3404			−1.05334			−0.526669	+	0.850070i	$0.676559\pi$
$492$		0			0
$493$		−	4.10157i		−	0.184725i
$494$		0			0
$495$	0.574515	+	0.649738i	0.0258226	+	0.0292036i
$496$		0			0
$497$			5.53690i			0.248364i
$498$		0			0
$499$	−6.85097			−0.306691			−0.153346	−	0.988173i	$-0.549005\pi$
$499$	−6.85097			−0.306691			−0.153346	+	0.988173i	$0.549005\pi$
$500$		0			0
$501$	−15.8496			−0.708106
$502$		0			0
$503$			3.32582i			0.148291i	0.997247	+	0.0741456i	$0.0236230\pi$
$503$			3.32582i			0.148291i	−0.997247	+	0.0741456i	$0.976377\pi$
$504$		0			0
$505$	−11.2388	−	12.7104i	−0.500122	−	0.565604i
$506$		0			0
$507$			4.22425i			0.187606i
$508$		0			0
$509$	6.18664			0.274218			0.137109	−	0.990556i	$-0.456219\pi$
$509$	6.18664			0.274218			0.137109	+	0.990556i	$0.456219\pi$
$510$		0			0
$511$	−6.18664			−0.273681
$512$		0			0
$513$			2.96239i			0.130793i
$514$		0			0
$515$	−24.6253	+	21.7743i	−1.08512	+	0.959492i
$516$		0			0
$517$			1.25060i			0.0550014i
$518$		0			0
$519$	23.9756			1.05241
$520$		0			0
$521$	34.4387			1.50879			0.754393	−	0.656424i	$-0.227932\pi$
$521$	34.4387			1.50879			0.754393	+	0.656424i	$0.227932\pi$
$522$		0			0
$523$			6.59895i			0.288552i	0.989537	+	0.144276i	$0.0460853\pi$
$523$			6.59895i			0.288552i	−0.989537	+	0.144276i	$0.953915\pi$
$524$		0			0
$525$	−0.612127	+	4.96239i	−0.0267154	+	0.216576i
$526$		0			0
$527$		−	9.92478i		−	0.432330i
$528$		0			0
$529$	22.0738			0.959731
$530$		0			0
$531$	−3.22425			−0.139921
$532$		0			0
$533$			3.07381i			0.133141i
$534$		0			0
$535$	27.5369	−	24.3488i	1.19052	−	1.05269i
$536$		0			0
$537$			12.2374i			0.528084i
$538$		0			0
$539$	0.387873			0.0167069
$540$		0			0
$541$	−9.07381			−0.390113			−0.195057	−	0.980792i	$-0.562489\pi$
$541$	−9.07381			−0.390113			−0.195057	+	0.980792i	$0.562489\pi$
$542$		0			0
$543$			8.70052i			0.373375i
$544$		0			0
$545$	4.11142	+	4.64974i	0.176114	+	0.199173i
$546$		0			0
$547$			19.8496i			0.848706i	0.905497	+	0.424353i	$0.139498\pi$
$547$			19.8496i			0.848706i	−0.905497	+	0.424353i	$0.860502\pi$
$548$		0			0
$549$	−14.6253			−0.624193
$550$		0			0
$551$	3.62672			0.154503
$552$		0			0
$553$			13.9248i			0.592142i
$554$		0			0
$555$	−8.77575	−	9.92478i	−0.372510	−	0.421283i
$556$		0			0
$557$			0.785595i			0.0332867i	0.999861	+	0.0166434i	$0.00529799\pi$
$557$			0.785595i			0.0332867i	−0.999861	+	0.0166434i	$0.994702\pi$
$558$		0			0
$559$	−31.6991			−1.34073
$560$		0			0
$561$	1.29948			0.0548639
$562$		0			0
$563$		−	32.2228i		−	1.35803i	−0.734124	−	0.679015i	$-0.762407\pi$
$563$		−	32.2228i		−	1.35803i	0.734124	−	0.679015i	$-0.237593\pi$
$564$		0			0
$565$	7.13918	−	6.31265i	0.300348	−	0.265575i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−16.0752			−0.673908			−0.336954	−	0.941521i	$-0.609397\pi$
$569$	−16.0752			−0.673908			−0.336954	+	0.941521i	$0.609397\pi$
$570$		0			0
$571$	39.8496			1.66765			0.833826	−	0.552027i	$-0.186146\pi$
$571$	39.8496			1.66765			0.833826	+	0.552027i	$0.186146\pi$
$572$		0			0
$573$			13.5369i			0.565512i
$574$		0			0
$575$	4.77575	+	0.589104i	0.199162	+	0.0245673i
$576$		0			0
$577$		−	30.1378i		−	1.25465i	−0.778757	−	0.627326i	$-0.784149\pi$
$577$		−	30.1378i		−	1.25465i	0.778757	−	0.627326i	$-0.215851\pi$
$578$		0			0
$579$	−1.92478			−0.0799910
$580$		0			0
$581$	3.22425			0.133765
$582$		0			0
$583$		−	2.19649i		−	0.0909694i
$584$		0			0
$585$	4.96239	−	4.38787i	0.205170	−	0.181416i
$586$		0			0
$587$			33.7743i			1.39402i	0.717063	+	0.697008i	$0.245486\pi$
$587$			33.7743i			1.39402i	−0.717063	+	0.697008i	$0.754514\pi$
$588$		0			0
$589$	8.77575			0.361598
$590$		0			0
$591$	−20.1114			−0.827273
$592$		0			0
$593$		−	7.19982i		−	0.295661i	−0.989013	−	0.147831i	$-0.952771\pi$
$593$		−	7.19982i		−	0.295661i	0.989013	−	0.147831i	$-0.0472290\pi$
$594$		0			0
$595$	4.96239	+	5.61213i	0.203438	+	0.230075i
$596$		0			0
$597$		−	8.26187i		−	0.338136i
$598$		0			0
$599$	−29.5369			−1.20685			−0.603423	−	0.797422i	$-0.706197\pi$
$599$	−29.5369			−1.20685			−0.603423	+	0.797422i	$0.706197\pi$
$600$		0			0
$601$	8.59895			0.350759			0.175379	−	0.984501i	$-0.443885\pi$
$601$	8.59895			0.350759			0.175379	+	0.984501i	$0.443885\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.0703	−	18.1744i	−0.653351	−	0.738895i
$606$		0			0
$607$		−	37.4010i		−	1.51806i	−0.651055	−	0.759031i	$-0.725673\pi$
$607$		−	37.4010i		−	1.51806i	0.651055	−	0.759031i	$-0.274327\pi$
$608$		0			0
$609$	−1.22425			−0.0496093
$610$		0			0
$611$	9.55149			0.386412
$612$		0			0
$613$			14.5990i			0.589646i	0.955552	+	0.294823i	$0.0952607\pi$
$613$			14.5990i			0.589646i	−0.955552	+	0.294823i	$0.904739\pi$
$614$		0			0
$615$	−1.73813	+	1.53690i	−0.0700884	+	0.0619740i
$616$		0			0
$617$			22.8119i			0.918374i	0.888340	+	0.459187i	$0.151859\pi$
$617$			22.8119i			0.918374i	−0.888340	+	0.459187i	$0.848141\pi$
$618$		0			0
$619$	0.261865			0.0105252			0.00526262	−	0.999986i	$-0.498325\pi$
$619$	0.261865			0.0105252			0.00526262	+	0.999986i	$0.498325\pi$
$620$		0			0
$621$	−0.962389			−0.0386193
$622$		0			0
$623$			3.73813i			0.149765i
$624$		0			0
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$		0			0
$627$			1.14903i			0.0458879i
$628$		0			0
$629$	−19.8496			−0.791454
$630$		0			0
$631$	5.92478			0.235862			0.117931	−	0.993022i	$-0.462374\pi$
$631$	5.92478			0.235862			0.117931	+	0.993022i	$0.462374\pi$
$632$		0			0
$633$			18.1768i			0.722463i
$634$		0			0
$635$	−24.3733	+	21.5515i	−0.967224	+	0.855245i
$636$		0			0
$637$		−	2.96239i		−	0.117374i
$638$		0			0
$639$	5.53690			0.219037
$640$		0			0
$641$	6.87732			0.271638			0.135819	−	0.990734i	$-0.456633\pi$
$641$	6.87732			0.271638			0.135819	+	0.990734i	$0.456633\pi$
$642$		0			0
$643$		−	10.7005i		−	0.421987i	−0.977487	−	0.210994i	$-0.932330\pi$
$643$		−	10.7005i		−	0.421987i	0.977487	−	0.210994i	$-0.0676700\pi$
$644$		0			0
$645$	−15.8496	−	17.9248i	−0.624076	−	0.705787i
$646$		0			0
$647$			45.2506i			1.77898i	0.456950	+	0.889492i	$0.348942\pi$
$647$			45.2506i			1.77898i	−0.456950	+	0.889492i	$0.651058\pi$
$648$		0			0
$649$	−1.25060			−0.0490904
$650$		0			0
$651$	−2.96239			−0.116105
$652$		0			0
$653$		−	36.0625i		−	1.41124i	−0.708592	−	0.705618i	$-0.750669\pi$
$653$		−	36.0625i		−	1.41124i	0.708592	−	0.705618i	$-0.249331\pi$
$654$		0			0
$655$	5.92478	+	6.70052i	0.231500	+	0.261811i
$656$		0			0
$657$			6.18664i			0.241364i
$658$		0			0
$659$	43.4617			1.69303			0.846513	−	0.532367i	$-0.178698\pi$
$659$	43.4617			1.69303			0.846513	+	0.532367i	$0.178698\pi$
$660$		0			0
$661$	−22.4749			−0.874171			−0.437085	−	0.899420i	$-0.643989\pi$
$661$	−22.4749			−0.874171			−0.437085	+	0.899420i	$0.643989\pi$
$662$		0			0
$663$		−	9.92478i		−	0.385446i
$664$		0			0
$665$	−4.96239	+	4.38787i	−0.192433	+	0.170154i
$666$		0			0
$667$			1.17821i			0.0456204i
$668$		0			0
$669$	−21.4010			−0.827412
$670$		0			0
$671$	−5.67276			−0.218995
$672$		0			0
$673$		−	47.5487i		−	1.83287i	−0.400188	−	0.916433i	$-0.631055\pi$
$673$		−	47.5487i		−	1.83287i	0.400188	−	0.916433i	$-0.368945\pi$
$674$		0			0
$675$	4.96239	+	0.612127i	0.191002	+	0.0235608i
$676$		0			0
$677$			5.67750i			0.218204i	0.994031	+	0.109102i	$0.0347975\pi$
$677$			5.67750i			0.218204i	−0.994031	+	0.109102i	$0.965202\pi$
$678$		0			0
$679$	−7.73813			−0.296962
$680$		0			0
$681$	18.5501			0.710841
$682$		0			0
$683$			12.2882i			0.470195i	0.971972	+	0.235098i	$0.0755410\pi$
$683$			12.2882i			0.470195i	−0.971972	+	0.235098i	$0.924459\pi$
$684$		0			0
$685$	−17.4861	+	15.4617i	−0.668110	+	0.590760i
$686$		0			0
$687$			28.5501i			1.08925i
$688$		0			0
$689$	−16.7757			−0.639105
$690$		0			0
$691$	1.81336			0.0689834			0.0344917	−	0.999405i	$-0.489019\pi$
$691$	1.81336			0.0689834			0.0344917	+	0.999405i	$0.489019\pi$
$692$		0			0
$693$		−	0.387873i		−	0.0147341i
$694$		0			0
$695$	8.38787	+	9.48612i	0.318170	+	0.359829i
$696$		0			0
$697$			3.47627i			0.131673i
$698$		0			0
$699$	18.9624			0.717223
$700$		0			0
$701$	29.5778			1.11714			0.558570	−	0.829458i	$-0.311350\pi$
$701$	29.5778			1.11714			0.558570	+	0.829458i	$0.311350\pi$
$702$		0			0
$703$		−	17.5515i		−	0.661967i
$704$		0			0
$705$	4.77575	+	5.40105i	0.179865	+	0.203415i
$706$		0			0
$707$			7.58769i			0.285364i
$708$		0			0
$709$	−20.8021			−0.781239			−0.390620	−	0.920552i	$-0.627739\pi$
$709$	−20.8021			−0.781239			−0.390620	+	0.920552i	$0.627739\pi$
$710$		0			0
$711$	13.9248			0.522220
$712$		0			0
$713$			2.85097i			0.106770i
$714$		0			0
$715$	1.92478	−	1.70194i	0.0719826	−	0.0636489i
$716$		0			0
$717$		−	18.1622i		−	0.678280i
$718$		0			0
$719$	19.3258			0.720732			0.360366	−	0.932811i	$-0.382652\pi$
$719$	19.3258			0.720732			0.360366	+	0.932811i	$0.382652\pi$
$720$		0			0
$721$	14.7005			0.547476
$722$		0			0
$723$			14.3733i			0.534548i
$724$		0			0
$725$	0.749399	−	6.07522i	0.0278320	−	0.225628i
$726$		0			0
$727$			21.6531i			0.803068i	0.915844	+	0.401534i	$0.131523\pi$
$727$			21.6531i			0.803068i	−0.915844	+	0.401534i	$0.868477\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−35.8496			−1.32594
$732$		0			0
$733$			48.0625i			1.77523i	0.460586	+	0.887615i	$0.347639\pi$
$733$			48.0625i			1.77523i	−0.460586	+	0.887615i	$0.652361\pi$
$734$		0			0
$735$	1.67513	−	1.48119i	0.0617881	−	0.0546347i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−30.2981			−1.11453			−0.557266	−	0.830334i	$-0.688150\pi$
$739$	−30.2981			−1.11453			−0.557266	+	0.830334i	$0.688150\pi$
$740$		0			0
$741$	8.77575			0.322385
$742$		0			0
$743$		−	40.9624i		−	1.50276i	−0.659867	−	0.751382i	$-0.729388\pi$
$743$		−	40.9624i		−	1.50276i	0.659867	−	0.751382i	$-0.270612\pi$
$744$		0			0
$745$	−30.4387	−	34.4241i	−1.11519	−	1.26120i
$746$		0			0
$747$		−	3.22425i		−	0.117969i
$748$		0			0
$749$	−16.4387			−0.600656
$750$		0			0
$751$	−9.52232			−0.347474			−0.173737	−	0.984792i	$-0.555584\pi$
$751$	−9.52232			−0.347474			−0.173737	+	0.984792i	$0.555584\pi$
$752$		0			0
$753$			8.62530i			0.314323i
$754$		0			0
$755$	33.6239	+	38.0263i	1.22370	+	1.38392i
$756$		0			0
$757$			31.3258i			1.13856i	0.822145	+	0.569278i	$0.192777\pi$
$757$			31.3258i			1.13856i	−0.822145	+	0.569278i	$0.807223\pi$
$758$		0			0
$759$	−0.373285			−0.0135494
$760$		0			0
$761$	−54.2393			−1.96617			−0.983087	−	0.183138i	$-0.941375\pi$
$761$	−54.2393			−1.96617			−0.983087	+	0.183138i	$0.941375\pi$
$762$		0			0
$763$		−	2.77575i		−	0.100489i
$764$		0			0
$765$	5.61213	−	4.96239i	0.202907	−	0.179416i
$766$		0			0
$767$			9.55149i			0.344884i
$768$		0			0
$769$	−35.1002			−1.26574			−0.632872	−	0.774256i	$-0.718124\pi$
$769$	−35.1002			−1.26574			−0.632872	+	0.774256i	$0.718124\pi$
$770$		0			0
$771$	27.8251			1.00210
$772$		0			0
$773$		−	19.8740i		−	0.714818i	−0.933948	−	0.357409i	$-0.883660\pi$
$773$		−	19.8740i		−	0.714818i	0.933948	−	0.357409i	$-0.116340\pi$
$774$		0			0
$775$	1.81336	−	14.7005i	0.0651377	−	0.528058i
$776$		0			0
$777$			5.92478i			0.212550i
$778$		0			0
$779$	−3.07381			−0.110131
$780$		0			0
$781$	2.14762			0.0768478
$782$		0			0
$783$			1.22425i			0.0437513i
$784$		0			0
$785$	3.66291	−	3.23884i	0.130735	−	0.115599i
$786$		0			0
$787$		−	13.2506i		−	0.472333i	−0.971713	−	0.236166i	$-0.924109\pi$
$787$		−	13.2506i		−	0.472333i	0.971713	−	0.236166i	$-0.0758911\pi$
$788$		0			0
$789$	0.589104			0.0209727
$790$		0			0
$791$	−4.26187			−0.151534
$792$		0			0
$793$			43.3258i			1.53855i
$794$		0			0
$795$	−8.38787	−	9.48612i	−0.297487	−	0.336438i
$796$		0			0
$797$		−	15.7235i		−	0.556957i	−0.960443	−	0.278478i	$-0.910170\pi$
$797$		−	15.7235i		−	0.556957i	0.960443	−	0.278478i	$-0.0898300\pi$
$798$		0			0
$799$	10.8021			0.382151
$800$		0			0
$801$	3.73813			0.132081
$802$		0			0
$803$			2.39963i			0.0846812i
$804$		0			0
$805$	−1.42548	−	1.61213i	−0.0502417	−	0.0568200i
$806$		0			0
$807$		−	15.7381i		−	0.554009i
$808$		0			0
$809$	18.5040			0.650567			0.325284	−	0.945617i	$-0.394540\pi$
$809$	18.5040			0.650567			0.325284	+	0.945617i	$0.394540\pi$
$810$		0			0
$811$	−37.6629			−1.32252			−0.661262	−	0.750155i	$-0.729979\pi$
$811$	−37.6629			−1.32252			−0.661262	+	0.750155i	$0.729979\pi$
$812$		0			0
$813$			7.11283i			0.249458i
$814$		0			0
$815$	−12.1016	+	10.7005i	−0.423900	+	0.374823i
$816$		0			0
$817$		−	31.6991i		−	1.10901i
$818$		0			0
$819$	−2.96239			−0.103514
$820$		0			0
$821$	−17.9511			−0.626499			−0.313249	−	0.949671i	$-0.601418\pi$
$821$	−17.9511			−0.626499			−0.313249	+	0.949671i	$0.601418\pi$
$822$		0			0
$823$		−	32.1016i		−	1.11899i	−0.828834	−	0.559495i	$-0.810995\pi$
$823$		−	32.1016i		−	1.11899i	0.828834	−	0.559495i	$-0.189005\pi$
$824$		0			0
$825$	1.92478	+	0.237428i	0.0670121	+	0.00826617i
$826$		0			0
$827$			25.8594i			0.899220i	0.893225	+	0.449610i	$0.148437\pi$
$827$			25.8594i			0.899220i	−0.893225	+	0.449610i	$0.851563\pi$
$828$		0			0
$829$	−11.8035			−0.409953			−0.204976	−	0.978767i	$-0.565712\pi$
$829$	−11.8035			−0.409953			−0.204976	+	0.978767i	$0.565712\pi$
$830$		0			0
$831$	−5.92478			−0.205528
$832$		0			0
$833$		−	3.35026i		−	0.116080i
$834$		0			0
$835$	−26.5501	+	23.4763i	−0.918803	+	0.812430i
$836$		0			0
$837$			2.96239i			0.102395i
$838$		0			0
$839$	19.5778			0.675902			0.337951	−	0.941164i	$-0.390266\pi$
$839$	19.5778			0.675902			0.337951	+	0.941164i	$0.390266\pi$
$840$		0			0
$841$	−27.5012			−0.948317
$842$		0			0
$843$			29.3258i			1.01004i
$844$		0			0
$845$	6.25694	+	7.07618i	0.215245	+	0.243428i
$846$		0			0
$847$			10.8496i			0.372795i
$848$		0			0
$849$	16.1016			0.552604
$850$		0			0
$851$	5.70194			0.195460
$852$		0			0
$853$			40.6155i			1.39065i	0.718697	+	0.695323i	$0.244739\pi$
$853$			40.6155i			1.39065i	−0.718697	+	0.695323i	$0.755261\pi$
$854$		0			0
$855$	4.38787	+	4.96239i	0.150062	+	0.169710i
$856$		0			0
$857$			29.1246i			0.994877i	0.867499	+	0.497439i	$0.165726\pi$
$857$			29.1246i			0.994877i	−0.867499	+	0.497439i	$0.834274\pi$
$858$		0			0
$859$	35.7090			1.21837			0.609187	−	0.793027i	$-0.291496\pi$
$859$	35.7090			1.21837			0.609187	+	0.793027i	$0.291496\pi$
$860$		0			0
$861$	1.03761			0.0353617
$862$		0			0
$863$			21.1852i			0.721154i	0.932730	+	0.360577i	$0.117420\pi$
$863$			21.1852i			0.721154i	−0.932730	+	0.360577i	$0.882580\pi$
$864$		0			0
$865$	40.1622	−	35.5125i	1.36556	−	1.20746i
$866$		0			0
$867$			5.77575i			0.196155i
$868$		0			0
$869$	5.40105			0.183218
$870$		0			0
$871$		0			0
$872$		0			0
$873$			7.73813i			0.261896i
$874$		0			0
$875$	6.32487	+	9.21933i	0.213820	+	0.311670i
$876$		0			0
$877$		−	26.4485i		−	0.893103i	−0.894758	−	0.446551i	$-0.852652\pi$
$877$		−	26.4485i		−	0.893103i	0.894758	−	0.446551i	$-0.147348\pi$
$878$		0			0
$879$	9.27504			0.312839
$880$		0			0
$881$	−26.0362			−0.877182			−0.438591	−	0.898687i	$-0.644522\pi$
$881$	−26.0362			−0.877182			−0.438591	+	0.898687i	$0.644522\pi$
$882$		0			0
$883$			8.87732i			0.298745i	0.988781	+	0.149373i	$0.0477254\pi$
$883$			8.87732i			0.298745i	−0.988781	+	0.149373i	$0.952275\pi$
$884$		0			0
$885$	−5.40105	+	4.77575i	−0.181554	+	0.160535i
$886$		0			0
$887$		−	2.17679i		−	0.0730896i	−0.999332	−	0.0365448i	$-0.988365\pi$
$887$		−	2.17679i		−	0.0730896i	0.999332	−	0.0365448i	$-0.0116352\pi$
$888$		0			0
$889$	14.5501			0.487994
$890$		0			0
$891$	−0.387873			−0.0129942
$892$		0			0
$893$			9.55149i			0.319629i
$894$		0			0
$895$	18.1260	+	20.4993i	0.605886	+	0.685216i
$896$		0			0
$897$			2.85097i			0.0951911i
$898$		0			0
$899$	3.62672			0.120958
$900$		0			0
$901$	−18.9722			−0.632057
$902$		0			0
$903$			10.7005i			0.356091i
$904$		0			0
$905$	12.8872	+	14.5745i	0.428384	+	0.484473i
$906$		0			0
$907$			44.7269i			1.48513i	0.669773	+	0.742566i	$0.266391\pi$
$907$			44.7269i			1.48513i	−0.669773	+	0.742566i	$0.733609\pi$
$908$		0			0
$909$	7.58769			0.251668
$910$		0			0
$911$	−26.4631			−0.876761			−0.438381	−	0.898789i	$-0.644448\pi$
$911$	−26.4631			−0.876761			−0.438381	+	0.898789i	$0.644448\pi$
$912$		0			0
$913$		−	1.25060i		−	0.0413889i
$914$		0			0
$915$	−24.4993	+	21.6629i	−0.809922	+	0.716154i
$916$		0			0
$917$		−	4.00000i		−	0.132092i
$918$		0			0
$919$	−59.2769			−1.95537			−0.977683	−	0.210085i	$-0.932626\pi$
$919$	−59.2769			−1.95537			−0.977683	+	0.210085i	$0.932626\pi$
$920$		0			0
$921$	10.7005			0.352594
$922$		0			0
$923$		−	16.4025i		−	0.539894i
$924$		0			0
$925$	−29.4010	−	3.62672i	−0.966700	−	0.119246i
$926$		0			0
$927$		−	14.7005i		−	0.482829i
$928$		0			0
$929$	−25.3620			−0.832101			−0.416050	−	0.909342i	$-0.636586\pi$
$929$	−25.3620			−0.832101			−0.416050	+	0.909342i	$0.636586\pi$
$930$		0			0
$931$	2.96239			0.0970884
$932$		0			0
$933$		−	12.1016i		−	0.396188i
$934$		0			0
$935$	2.17679	−	1.92478i	0.0711888	−	0.0629469i
$936$		0			0
$937$		−	3.85940i		−	0.126081i	−0.998011	−	0.0630406i	$-0.979920\pi$
$937$		−	3.85940i		−	0.126081i	0.998011	−	0.0630406i	$-0.0200798\pi$
$938$		0			0
$939$	28.3634			0.925606
$940$		0			0
$941$	27.4861			0.896022			0.448011	−	0.894028i	$-0.352133\pi$
$941$	27.4861			0.896022			0.448011	+	0.894028i	$0.352133\pi$
$942$		0			0
$943$		−	0.998585i		−	0.0325184i
$944$		0			0
$945$	−1.48119	−	1.67513i	−0.0481833	−	0.0544920i
$946$		0			0
$947$			13.8397i			0.449730i	0.974390	+	0.224865i	$0.0721941\pi$
$947$			13.8397i			0.449730i	−0.974390	+	0.224865i	$0.927806\pi$
$948$		0			0
$949$	18.3272			0.594927
$950$		0			0
$951$	17.5125			0.567881
$952$		0			0
$953$			39.2868i			1.27262i	0.771432	+	0.636312i	$0.219541\pi$
$953$			39.2868i			1.27262i	−0.771432	+	0.636312i	$0.780459\pi$
$954$		0			0
$955$	20.0508	+	22.6761i	0.648828	+	0.733781i
$956$		0			0
$957$			0.474855i			0.0153499i
$958$		0			0
$959$	10.4387			0.337082
$960$		0			0
$961$	−22.2243			−0.716911
$962$		0			0
$963$			16.4387i			0.529728i
$964$		0			0
$965$	−3.22425	+	2.85097i	−0.103792	+	0.0917759i
$966$		0			0
$967$			8.67418i			0.278943i	0.990226	+	0.139471i	$0.0445403\pi$
$967$			8.67418i			0.278943i	−0.990226	+	0.139471i	$0.955460\pi$
$968$		0			0
$969$	9.92478			0.318830
$970$		0			0
$971$	25.9248			0.831966			0.415983	−	0.909372i	$-0.363438\pi$
$971$	25.9248			0.831966			0.415983	+	0.909372i	$0.363438\pi$
$972$		0			0
$973$		−	5.66291i		−	0.181545i
$974$		0			0
$975$	1.81336	−	14.7005i	0.0580739	−	0.470794i
$976$		0			0
$977$			7.85940i			0.251445i	0.992065	+	0.125722i	$0.0401248\pi$
$977$			7.85940i			0.251445i	−0.992065	+	0.125722i	$0.959875\pi$
$978$		0			0
$979$	1.44992			0.0463397
$980$		0			0
$981$	−2.77575			−0.0886228
$982$		0			0
$983$		−	18.8510i		−	0.601253i	−0.953742	−	0.300626i	$-0.902804\pi$
$983$		−	18.8510i		−	0.601253i	0.953742	−	0.300626i	$-0.0971957\pi$
$984$		0			0
$985$	−33.6893	+	29.7889i	−1.07343	+	0.949154i
$986$		0			0
$987$		−	3.22425i		−	0.102629i
$988$		0			0
$989$	10.2981			0.327459
$990$		0			0
$991$	43.0738			1.36828			0.684142	−	0.729349i	$-0.260177\pi$
$991$	43.0738			1.36828			0.684142	+	0.729349i	$0.260177\pi$
$992$		0			0
$993$			19.2243i			0.610063i
$994$		0			0
$995$	−12.2374	−	13.8397i	−0.387953	−	0.438748i
$996$		0			0
$997$		−	24.6155i		−	0.779579i	−0.920904	−	0.389790i	$-0.872548\pi$
$997$		−	24.6155i		−	0.779579i	0.920904	−	0.389790i	$-0.127452\pi$
$998$		0			0
$999$	5.92478			0.187452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.t.i.1009.6		6
3.2	odd	2		5040.2.t.ba.1009.1		6
4.3	odd	2		840.2.t.e.169.3	✓	6
5.2	odd	4		8400.2.a.dk.1.2		3
5.3	odd	4		8400.2.a.dh.1.2		3
5.4	even	2	inner	1680.2.t.i.1009.3		6
12.11	even	2		2520.2.t.j.1009.1		6
15.14	odd	2		5040.2.t.ba.1009.2		6
20.3	even	4		4200.2.a.bq.1.2		3
20.7	even	4		4200.2.a.bo.1.2		3
20.19	odd	2		840.2.t.e.169.6	yes	6
60.59	even	2		2520.2.t.j.1009.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.e.169.3	✓	6	4.3	odd	2
840.2.t.e.169.6	yes	6	20.19	odd	2
1680.2.t.i.1009.3		6	5.4	even	2	inner
1680.2.t.i.1009.6		6	1.1	even	1	trivial
2520.2.t.j.1009.1		6	12.11	even	2
2520.2.t.j.1009.2		6	60.59	even	2
4200.2.a.bo.1.2		3	20.7	even	4
4200.2.a.bq.1.2		3	20.3	even	4
5040.2.t.ba.1009.1		6	3.2	odd	2
5040.2.t.ba.1009.2		6	15.14	odd	2
8400.2.a.dh.1.2		3	5.3	odd	4
8400.2.a.dk.1.2		3	5.2	odd	4

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.48119 + 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(1.48119 + 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} -0.387873 q^{11} +2.96239i q^{13} +(-1.67513 + 1.48119i) q^{15} +3.35026i q^{17} -2.96239 q^{19} +1.00000 q^{21} -0.962389i q^{23} +(-0.612127 + 4.96239i) q^{25} -1.00000i q^{27} -1.22425 q^{29} -2.96239 q^{31} -0.387873i q^{33} +(1.67513 - 1.48119i) q^{35} +5.92478i q^{37} -2.96239 q^{39} +1.03761 q^{41} +10.7005i q^{43} +(-1.48119 - 1.67513i) q^{45} -3.22425i q^{47} -1.00000 q^{49} -3.35026 q^{51} +5.66291i q^{53} +(-0.574515 - 0.649738i) q^{55} -2.96239i q^{57} +3.22425 q^{59} +14.6253 q^{61} +1.00000i q^{63} +(-4.96239 + 4.38787i) q^{65} +0.962389 q^{69} -5.53690 q^{71} -6.18664i q^{73} +(-4.96239 - 0.612127i) q^{75} +0.387873i q^{77} -13.9248 q^{79} +1.00000 q^{81} +3.22425i q^{83} +(-5.61213 + 4.96239i) q^{85} -1.22425i q^{87} -3.73813 q^{89} +2.96239 q^{91} -2.96239i q^{93} +(-4.38787 - 4.96239i) q^{95} -7.73813i q^{97} +0.387873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^5 - 6 * q^9	\( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} + 4 q^{31} + 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} + 20 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 12 q^{71} - 8 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} - 4 q^{91} - 28 q^{95} + 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 + 4 * q^31 + 4 * q^39 + 28 * q^41 + 2 * q^45 - 6 * q^49 + 20 * q^55 + 16 * q^59 + 4 * q^61 - 8 * q^65 - 16 * q^69 + 12 * q^71 - 8 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 - 4 * q^89 - 4 * q^91 - 28 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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