LMFDB - Embedded newform 1680.2.t.j.1009.3

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(1.48119 + 1.67513i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(1.48119 + 1.67513i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -6.31265 q^{11} -6.96239i q^{13} +(1.67513 - 1.48119i) q^{15} +6.57452i q^{17} -3.73813 q^{19} +1.00000 q^{21} +5.73813i q^{23} +(-0.612127 + 4.96239i) q^{25} +1.00000i q^{27} +2.00000 q^{29} +1.03761 q^{31} +6.31265i q^{33} +(-1.67513 + 1.48119i) q^{35} +10.7005i q^{37} -6.96239 q^{39} -6.96239 q^{41} -5.92478i q^{43} +(-1.48119 - 1.67513i) q^{45} -1.00000 q^{49} +6.57452 q^{51} -1.03761i q^{53} +(-9.35026 - 10.5745i) q^{55} +3.73813i q^{57} -3.22425 q^{59} -13.8496 q^{61} -1.00000i q^{63} +(11.6629 - 10.3127i) q^{65} +4.77575i q^{67} +5.73813 q^{69} -8.23743 q^{71} -4.26187i q^{73} +(4.96239 + 0.612127i) q^{75} -6.31265i q^{77} -5.92478 q^{79} +1.00000 q^{81} +3.22425i q^{83} +(-11.0132 + 9.73813i) q^{85} -2.00000i q^{87} -2.18664 q^{89} +6.96239 q^{91} -1.03761i q^{93} +(-5.53690 - 6.26187i) q^{95} +3.73813i q^{97} +6.31265 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 12 q^{29} + 28 q^{31} - 20 q^{39} - 20 q^{41} + 2 q^{45} - 6 q^{49} + 16 q^{51} - 36 q^{55} - 16 q^{59} + 4 q^{61} + 8 q^{65} + 16 q^{69} + 36 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 16 q^{85} + 12 q^{89} + 20 q^{91} + 12 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 + 12 * q^29 + 28 * q^31 - 20 * q^39 - 20 * q^41 + 2 * q^45 - 6 * q^49 + 16 * q^51 - 36 * q^55 - 16 * q^59 + 4 * q^61 + 8 * q^65 + 16 * q^69 + 36 * q^71 + 8 * q^75 + 8 * q^79 + 6 * q^81 + 16 * q^85 + 12 * q^89 + 20 * q^91 + 12 * 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
$11$	−6.31265			−1.90334			−0.951668	−	0.307129i	$-0.900632\pi$
$11$	−6.31265			−1.90334			−0.951668	+	0.307129i	$0.900632\pi$
$12$		0			0
$13$		−	6.96239i		−	1.93102i	−0.260368	−	0.965510i	$-0.583844\pi$
$13$		−	6.96239i		−	1.93102i	0.260368	−	0.965510i	$-0.416156\pi$
$14$		0			0
$15$	1.67513	−	1.48119i	0.432517	−	0.382443i
$16$		0			0
$17$			6.57452i			1.59455i	0.603613	+	0.797277i	$0.293727\pi$
$17$			6.57452i			1.59455i	−0.603613	+	0.797277i	$0.706273\pi$
$18$		0			0
$19$	−3.73813			−0.857587			−0.428793	−	0.903403i	$-0.641061\pi$
$19$	−3.73813			−0.857587			−0.428793	+	0.903403i	$0.641061\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$			5.73813i			1.19648i	0.801316	+	0.598242i	$0.204134\pi$
$23$			5.73813i			1.19648i	−0.801316	+	0.598242i	$0.795866\pi$
$24$		0			0
$25$	−0.612127	+	4.96239i	−0.122425	+	0.992478i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	1.03761			0.186361			0.0931803	−	0.995649i	$-0.470297\pi$
$31$	1.03761			0.186361			0.0931803	+	0.995649i	$0.470297\pi$
$32$		0			0
$33$			6.31265i			1.09889i
$34$		0			0
$35$	−1.67513	+	1.48119i	−0.283149	+	0.250368i
$36$		0			0
$37$			10.7005i			1.75916i	0.475755	+	0.879578i	$0.342175\pi$
$37$			10.7005i			1.75916i	−0.475755	+	0.879578i	$0.657825\pi$
$38$		0			0
$39$	−6.96239			−1.11487
$40$		0			0
$41$	−6.96239			−1.08734			−0.543671	−	0.839298i	$-0.682966\pi$
$41$	−6.96239			−1.08734			−0.543671	+	0.839298i	$0.682966\pi$
$42$		0			0
$43$		−	5.92478i		−	0.903520i	−0.892139	−	0.451760i	$-0.850796\pi$
$43$		−	5.92478i		−	0.903520i	0.892139	−	0.451760i	$-0.149204\pi$
$44$		0			0
$45$	−1.48119	−	1.67513i	−0.220803	−	0.249714i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	6.57452			0.920616
$52$		0			0
$53$		−	1.03761i		−	0.142527i	−0.997458	−	0.0712634i	$-0.977297\pi$
$53$		−	1.03761i		−	0.142527i	0.997458	−	0.0712634i	$-0.0227031\pi$
$54$		0			0
$55$	−9.35026	−	10.5745i	−1.26079	−	1.42587i
$56$		0			0
$57$			3.73813i			0.495128i
$58$		0			0
$59$	−3.22425			−0.419762			−0.209881	−	0.977727i	$-0.567308\pi$
$59$	−3.22425			−0.419762			−0.209881	+	0.977727i	$0.567308\pi$
$60$		0			0
$61$	−13.8496			−1.77325			−0.886627	−	0.462485i	$-0.846958\pi$
$61$	−13.8496			−1.77325			−0.886627	+	0.462485i	$0.846958\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	11.6629	−	10.3127i	1.44661	−	1.27913i
$66$		0			0
$67$			4.77575i			0.583450i	0.956502	+	0.291725i	$0.0942293\pi$
$67$			4.77575i			0.583450i	−0.956502	+	0.291725i	$0.905771\pi$
$68$		0			0
$69$	5.73813			0.690790
$70$		0			0
$71$	−8.23743			−0.977603			−0.488801	−	0.872395i	$-0.662566\pi$
$71$	−8.23743			−0.977603			−0.488801	+	0.872395i	$0.662566\pi$
$72$		0			0
$73$		−	4.26187i		−	0.498814i	−0.968399	−	0.249407i	$-0.919764\pi$
$73$		−	4.26187i		−	0.498814i	0.968399	−	0.249407i	$-0.0802356\pi$
$74$		0			0
$75$	4.96239	+	0.612127i	0.573007	+	0.0706823i
$76$		0			0
$77$		−	6.31265i		−	0.719393i
$78$		0			0
$79$	−5.92478			−0.666590			−0.333295	−	0.942823i	$-0.608160\pi$
$79$	−5.92478			−0.666590			−0.333295	+	0.942823i	$0.608160\pi$
$80$		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
$85$	−11.0132	+	9.73813i	−1.19455	+	1.05625i
$86$		0			0
$87$		−	2.00000i		−	0.214423i
$88$		0			0
$89$	−2.18664			−0.231784			−0.115892	−	0.993262i	$-0.536973\pi$
$89$	−2.18664			−0.231784			−0.115892	+	0.993262i	$0.536973\pi$
$90$		0			0
$91$	6.96239			0.729857
$92$		0			0
$93$		−	1.03761i		−	0.107595i
$94$		0			0
$95$	−5.53690	−	6.26187i	−0.568074	−	0.642454i
$96$		0			0
$97$			3.73813i			0.379550i	0.981828	+	0.189775i	$0.0607759\pi$
$97$			3.73813i			0.379550i	−0.981828	+	0.189775i	$0.939224\pi$
$98$		0			0
$99$	6.31265			0.634445
$100$		0			0
$101$	9.66291			0.961496			0.480748	−	0.876859i	$-0.340365\pi$
$101$	9.66291			0.961496			0.480748	+	0.876859i	$0.340365\pi$
$102$		0			0
$103$			1.29948i			0.128041i	0.997949	+	0.0640206i	$0.0203923\pi$
$103$			1.29948i			0.128041i	−0.997949	+	0.0640206i	$0.979608\pi$
$104$		0			0
$105$	1.48119	+	1.67513i	0.144550	+	0.163476i
$106$		0			0
$107$			3.66291i			0.354107i	0.984201	+	0.177054i	$0.0566566\pi$
$107$			3.66291i			0.354107i	−0.984201	+	0.177054i	$0.943343\pi$
$108$		0			0
$109$	−6.77575			−0.648999			−0.324499	−	0.945886i	$-0.605196\pi$
$109$	−6.77575			−0.648999			−0.324499	+	0.945886i	$0.605196\pi$
$110$		0			0
$111$	10.7005			1.01565
$112$		0			0
$113$			8.88717i			0.836034i	0.908439	+	0.418017i	$0.137275\pi$
$113$			8.88717i			0.836034i	−0.908439	+	0.418017i	$0.862725\pi$
$114$		0			0
$115$	−9.61213	+	8.49929i	−0.896335	+	0.792563i
$116$		0			0
$117$			6.96239i			0.643673i
$118$		0			0
$119$	−6.57452			−0.602685
$120$		0			0
$121$	28.8496			2.62269
$122$		0			0
$123$			6.96239i			0.627777i
$124$		0			0
$125$	−9.21933	+	6.32487i	−0.824602	+	0.565713i
$126$		0			0
$127$			2.70052i			0.239633i	0.992796	+	0.119816i	$0.0382306\pi$
$127$			2.70052i			0.239633i	−0.992796	+	0.119816i	$0.961769\pi$
$128$		0			0
$129$	−5.92478			−0.521648
$130$		0			0
$131$	9.40105			0.821373			0.410687	−	0.911777i	$-0.365289\pi$
$131$	9.40105			0.821373			0.410687	+	0.911777i	$0.365289\pi$
$132$		0			0
$133$		−	3.73813i		−	0.324137i
$134$		0			0
$135$	−1.67513	+	1.48119i	−0.144172	+	0.127481i
$136$		0			0
$137$		−	9.66291i		−	0.825558i	−0.910831	−	0.412779i	$-0.864558\pi$
$137$		−	9.66291i		−	0.825558i	0.910831	−	0.412779i	$-0.135442\pi$
$138$		0			0
$139$	20.8872			1.77163			0.885813	−	0.464042i	$-0.153601\pi$
$139$	20.8872			1.77163			0.885813	+	0.464042i	$0.153601\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			43.9511i			3.67538i
$144$		0			0
$145$	2.96239	+	3.35026i	0.246013	+	0.278224i
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	−0.0752228			−0.00616249			−0.00308125	−	0.999995i	$-0.500981\pi$
$149$	−0.0752228			−0.00616249			−0.00308125	+	0.999995i	$0.500981\pi$
$150$		0			0
$151$	−6.70052			−0.545281			−0.272640	−	0.962116i	$-0.587897\pi$
$151$	−6.70052			−0.545281			−0.272640	+	0.962116i	$0.587897\pi$
$152$		0			0
$153$		−	6.57452i		−	0.531518i
$154$		0			0
$155$	1.53690	+	1.73813i	0.123447	+	0.139610i
$156$		0			0
$157$		−	5.66291i		−	0.451950i	−0.974133	−	0.225975i	$-0.927443\pi$
$157$		−	5.66291i		−	0.451950i	0.974133	−	0.225975i	$-0.0725567\pi$
$158$		0			0
$159$	−1.03761			−0.0822879
$160$		0			0
$161$	−5.73813			−0.452228
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$	−10.5745	+	9.35026i	−0.823225	+	0.727917i
$166$		0			0
$167$			20.6253i			1.59603i	0.602635	+	0.798017i	$0.294117\pi$
$167$			20.6253i			1.59603i	−0.602635	+	0.798017i	$0.705883\pi$
$168$		0			0
$169$	−35.4749			−2.72884
$170$		0			0
$171$	3.73813			0.285862
$172$		0			0
$173$		−	10.5745i		−	0.803966i	−0.915647	−	0.401983i	$-0.868321\pi$
$173$		−	10.5745i		−	0.803966i	0.915647	−	0.401983i	$-0.131679\pi$
$174$		0			0
$175$	−4.96239	−	0.612127i	−0.375121	−	0.0462724i
$176$		0			0
$177$			3.22425i			0.242350i
$178$		0			0
$179$	0.911603			0.0681364			0.0340682	−	0.999420i	$-0.489154\pi$
$179$	0.911603			0.0681364			0.0340682	+	0.999420i	$0.489154\pi$
$180$		0			0
$181$	−13.3258			−0.990501			−0.495250	−	0.868750i	$-0.664924\pi$
$181$	−13.3258			−0.990501			−0.495250	+	0.868750i	$0.664924\pi$
$182$		0			0
$183$			13.8496i			1.02379i
$184$		0			0
$185$	−17.9248	+	15.8496i	−1.31786	+	1.16528i
$186$		0			0
$187$		−	41.5026i		−	3.03497i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	25.7889			1.86602			0.933010	−	0.359849i	$-0.117172\pi$
$191$	25.7889			1.86602			0.933010	+	0.359849i	$0.117172\pi$
$192$		0			0
$193$		−	15.3258i		−	1.10318i	−0.834116	−	0.551588i	$-0.814022\pi$
$193$		−	15.3258i		−	1.10318i	0.834116	−	0.551588i	$-0.185978\pi$
$194$		0			0
$195$	−10.3127	−	11.6629i	−0.738504	−	0.835198i
$196$		0			0
$197$			0.513881i			0.0366125i	0.999832	+	0.0183063i	$0.00582739\pi$
$197$			0.513881i			0.0366125i	−0.999832	+	0.0183063i	$0.994173\pi$
$198$		0			0
$199$	3.73813			0.264989			0.132495	−	0.991184i	$-0.457701\pi$
$199$	3.73813			0.264989			0.132495	+	0.991184i	$0.457701\pi$
$200$		0			0
$201$	4.77575			0.336855
$202$		0			0
$203$			2.00000i			0.140372i
$204$		0			0
$205$	−10.3127	−	11.6629i	−0.720267	−	0.814573i
$206$		0			0
$207$		−	5.73813i		−	0.398828i
$208$		0			0
$209$	23.5975			1.63228
$210$		0			0
$211$	3.22425			0.221967			0.110983	−	0.993822i	$-0.464600\pi$
$211$	3.22425			0.221967			0.110983	+	0.993822i	$0.464600\pi$
$212$		0			0
$213$			8.23743i			0.564419i
$214$		0			0
$215$	9.92478	−	8.77575i	0.676864	−	0.598501i
$216$		0			0
$217$			1.03761i			0.0704377i
$218$		0			0
$219$	−4.26187			−0.287990
$220$		0			0
$221$	45.7743			3.07911
$222$		0			0
$223$		−	11.8496i		−	0.793505i	−0.917926	−	0.396752i	$-0.870137\pi$
$223$		−	11.8496i		−	0.793505i	0.917926	−	0.396752i	$-0.129863\pi$
$224$		0			0
$225$	0.612127	−	4.96239i	0.0408085	−	0.330826i
$226$		0			0
$227$		−	14.7005i		−	0.975708i	−0.872925	−	0.487854i	$-0.837780\pi$
$227$		−	14.7005i		−	0.975708i	0.872925	−	0.487854i	$-0.162220\pi$
$228$		0			0
$229$	−9.47627			−0.626210			−0.313105	−	0.949719i	$-0.601369\pi$
$229$	−9.47627			−0.626210			−0.313105	+	0.949719i	$0.601369\pi$
$230$		0			0
$231$	−6.31265			−0.415342
$232$		0			0
$233$		−	29.5125i		−	1.93343i	−0.255863	−	0.966713i	$-0.582360\pi$
$233$		−	29.5125i		−	1.93343i	0.255863	−	0.966713i	$-0.417640\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			5.92478i			0.384856i
$238$		0			0
$239$	−25.0132			−1.61797			−0.808984	−	0.587831i	$-0.799982\pi$
$239$	−25.0132			−1.61797			−0.808984	+	0.587831i	$0.799982\pi$
$240$		0			0
$241$	7.92478			0.510480			0.255240	−	0.966878i	$-0.417846\pi$
$241$	7.92478			0.510480			0.255240	+	0.966878i	$0.417846\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−1.48119	−	1.67513i	−0.0946300	−	0.107020i
$246$		0			0
$247$			26.0263i			1.65602i
$248$		0			0
$249$	3.22425			0.204329
$250$		0			0
$251$	15.0738			0.951450			0.475725	−	0.879594i	$-0.342186\pi$
$251$	15.0738			0.951450			0.475725	+	0.879594i	$0.342186\pi$
$252$		0			0
$253$		−	36.2228i		−	2.27731i
$254$		0			0
$255$	9.73813	+	11.0132i	0.609826	+	0.689672i
$256$		0			0
$257$			22.0508i			1.37549i	0.725952	+	0.687745i	$0.241399\pi$
$257$			22.0508i			1.37549i	−0.725952	+	0.687745i	$0.758601\pi$
$258$		0			0
$259$	−10.7005			−0.664898
$260$		0			0
$261$	−2.00000			−0.123797
$262$		0			0
$263$			11.8134i			0.728443i	0.931312	+	0.364221i	$0.118665\pi$
$263$			11.8134i			0.728443i	−0.931312	+	0.364221i	$0.881335\pi$
$264$		0			0
$265$	1.73813	−	1.53690i	0.106773	−	0.0944113i
$266$		0			0
$267$			2.18664i			0.133820i
$268$		0			0
$269$	27.0640			1.65012			0.825059	−	0.565046i	$-0.191142\pi$
$269$	27.0640			1.65012			0.825059	+	0.565046i	$0.191142\pi$
$270$		0			0
$271$	10.8119			0.656779			0.328389	−	0.944542i	$-0.393494\pi$
$271$	10.8119			0.656779			0.328389	+	0.944542i	$0.393494\pi$
$272$		0			0
$273$		−	6.96239i		−	0.421383i
$274$		0			0
$275$	3.86414	−	31.3258i	0.233017	−	1.88902i
$276$		0			0
$277$		−	22.5501i		−	1.35490i	−0.735568	−	0.677451i	$-0.763084\pi$
$277$		−	22.5501i		−	1.35490i	0.735568	−	0.677451i	$-0.236916\pi$
$278$		0			0
$279$	−1.03761			−0.0621202
$280$		0			0
$281$	−10.3733			−0.618818			−0.309409	−	0.950929i	$-0.600131\pi$
$281$	−10.3733			−0.618818			−0.309409	+	0.950929i	$0.600131\pi$
$282$		0			0
$283$		−	10.7005i		−	0.636080i	−0.948077	−	0.318040i	$-0.896975\pi$
$283$		−	10.7005i		−	0.636080i	0.948077	−	0.318040i	$-0.103025\pi$
$284$		0			0
$285$	−6.26187	+	5.53690i	−0.370921	+	0.327978i
$286$		0			0
$287$		−	6.96239i		−	0.410977i
$288$		0			0
$289$	−26.2243			−1.54260
$290$		0			0
$291$	3.73813			0.219133
$292$		0			0
$293$			10.5745i			0.617770i	0.951099	+	0.308885i	$0.0999558\pi$
$293$			10.5745i			0.617770i	−0.951099	+	0.308885i	$0.900044\pi$
$294$		0			0
$295$	−4.77575	−	5.40105i	−0.278055	−	0.314461i
$296$		0			0
$297$		−	6.31265i		−	0.366297i
$298$		0			0
$299$	39.9511			2.31043
$300$		0			0
$301$	5.92478			0.341498
$302$		0			0
$303$		−	9.66291i		−	0.555120i
$304$		0			0
$305$	−20.5139	−	23.1998i	−1.17462	−	1.32842i
$306$		0			0
$307$		−	6.55008i		−	0.373833i	−0.982376	−	0.186916i	$-0.940151\pi$
$307$		−	6.55008i		−	0.373833i	0.982376	−	0.186916i	$-0.0598493\pi$
$308$		0			0
$309$	1.29948			0.0739246
$310$		0			0
$311$	17.2995			0.980963			0.490482	−	0.871452i	$-0.336821\pi$
$311$	17.2995			0.980963			0.490482	+	0.871452i	$0.336821\pi$
$312$		0			0
$313$			20.2130i			1.14251i	0.820774	+	0.571253i	$0.193542\pi$
$313$			20.2130i			1.14251i	−0.820774	+	0.571253i	$0.806458\pi$
$314$		0			0
$315$	1.67513	−	1.48119i	0.0943829	−	0.0834558i
$316$		0			0
$317$			19.3357i			1.08600i	0.839733	+	0.543000i	$0.182712\pi$
$317$			19.3357i			1.08600i	−0.839733	+	0.543000i	$0.817288\pi$
$318$		0			0
$319$	−12.6253			−0.706881
$320$		0			0
$321$	3.66291			0.204444
$322$		0			0
$323$		−	24.5764i		−	1.36747i
$324$		0			0
$325$	34.5501	+	4.26187i	1.91649	+	0.236406i
$326$		0			0
$327$			6.77575i			0.374700i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−24.6253			−1.35353			−0.676764	−	0.736200i	$-0.736618\pi$
$331$	−24.6253			−1.35353			−0.676764	+	0.736200i	$0.736618\pi$
$332$		0			0
$333$		−	10.7005i		−	0.586385i
$334$		0			0
$335$	−8.00000	+	7.07381i	−0.437087	+	0.386483i
$336$		0			0
$337$			6.44851i			0.351273i	0.984455	+	0.175636i	$0.0561983\pi$
$337$			6.44851i			0.351273i	−0.984455	+	0.175636i	$0.943802\pi$
$338$		0			0
$339$	8.88717			0.482685
$340$		0			0
$341$	−6.55008			−0.354707
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	8.49929	+	9.61213i	0.457587	+	0.517500i
$346$		0			0
$347$			11.6629i			0.626098i	0.949737	+	0.313049i	$0.101350\pi$
$347$			11.6629i			0.626098i	−0.949737	+	0.313049i	$0.898650\pi$
$348$		0			0
$349$	−12.4485			−0.666353			−0.333177	−	0.942864i	$-0.608121\pi$
$349$	−12.4485			−0.666353			−0.333177	+	0.942864i	$0.608121\pi$
$350$		0			0
$351$	6.96239			0.371625
$352$		0			0
$353$			13.0230i			0.693146i	0.938023	+	0.346573i	$0.112655\pi$
$353$			13.0230i			0.693146i	−0.938023	+	0.346573i	$0.887345\pi$
$354$		0			0
$355$	−12.2012	−	13.7988i	−0.647574	−	0.732363i
$356$		0			0
$357$			6.57452i			0.347960i
$358$		0			0
$359$	−20.3879			−1.07603			−0.538015	−	0.842935i	$-0.680826\pi$
$359$	−20.3879			−1.07603			−0.538015	+	0.842935i	$0.680826\pi$
$360$		0			0
$361$	−5.02635			−0.264545
$362$		0			0
$363$		−	28.8496i		−	1.51421i
$364$		0			0
$365$	7.13918	−	6.31265i	0.373682	−	0.330419i
$366$		0			0
$367$			19.8496i			1.03614i	0.855339	+	0.518069i	$0.173349\pi$
$367$			19.8496i			1.03614i	−0.855339	+	0.518069i	$0.826651\pi$
$368$		0			0
$369$	6.96239			0.362447
$370$		0			0
$371$	1.03761			0.0538701
$372$		0			0
$373$		−	2.85097i		−	0.147618i	−0.997272	−	0.0738088i	$-0.976485\pi$
$373$		−	2.85097i		−	0.147618i	0.997272	−	0.0738088i	$-0.0235155\pi$
$374$		0			0
$375$	6.32487	+	9.21933i	0.326615	+	0.476084i
$376$		0			0
$377$		−	13.9248i		−	0.717163i
$378$		0			0
$379$	5.29948			0.272216			0.136108	−	0.990694i	$-0.456541\pi$
$379$	5.29948			0.272216			0.136108	+	0.990694i	$0.456541\pi$
$380$		0			0
$381$	2.70052			0.138352
$382$		0			0
$383$		−	12.2520i		−	0.626049i	−0.949745	−	0.313024i	$-0.898658\pi$
$383$		−	12.2520i		−	0.626049i	0.949745	−	0.313024i	$-0.101342\pi$
$384$		0			0
$385$	10.5745	−	9.35026i	0.538927	−	0.476533i
$386$		0			0
$387$			5.92478i			0.301173i
$388$		0			0
$389$	−16.4485			−0.833972			−0.416986	−	0.908913i	$-0.636914\pi$
$389$	−16.4485			−0.833972			−0.416986	+	0.908913i	$0.636914\pi$
$390$		0			0
$391$	−37.7255			−1.90786
$392$		0			0
$393$		−	9.40105i		−	0.474220i
$394$		0			0
$395$	−8.77575	−	9.92478i	−0.441556	−	0.499370i
$396$		0			0
$397$		−	3.58769i		−	0.180061i	−0.995939	−	0.0900305i	$-0.971304\pi$
$397$		−	3.58769i		−	0.180061i	0.995939	−	0.0900305i	$-0.0286964\pi$
$398$		0			0
$399$	−3.73813			−0.187141
$400$		0			0
$401$	19.9248			0.994996			0.497498	−	0.867465i	$-0.334252\pi$
$401$	19.9248			0.994996			0.497498	+	0.867465i	$0.334252\pi$
$402$		0			0
$403$		−	7.22425i		−	0.359866i
$404$		0			0
$405$	1.48119	+	1.67513i	0.0736011	+	0.0832379i
$406$		0			0
$407$		−	67.5487i		−	3.34826i
$408$		0			0
$409$	25.1754			1.24484			0.622421	−	0.782682i	$-0.286149\pi$
$409$	25.1754			1.24484			0.622421	+	0.782682i	$0.286149\pi$
$410$		0			0
$411$	−9.66291			−0.476636
$412$		0			0
$413$		−	3.22425i		−	0.158655i
$414$		0			0
$415$	−5.40105	+	4.77575i	−0.265127	+	0.234432i
$416$		0			0
$417$		−	20.8872i		−	1.02285i
$418$		0			0
$419$	1.14903			0.0561338			0.0280669	−	0.999606i	$-0.491065\pi$
$419$	1.14903			0.0561338			0.0280669	+	0.999606i	$0.491065\pi$
$420$		0			0
$421$	−0.176793			−0.00861637			−0.00430819	−	0.999991i	$-0.501371\pi$
$421$	−0.176793			−0.00861637			−0.00430819	+	0.999991i	$0.501371\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−32.6253	−	4.02444i	−1.58256	−	0.195214i
$426$		0			0
$427$		−	13.8496i		−	0.670227i
$428$		0			0
$429$	43.9511			2.12198
$430$		0			0
$431$	−12.3879			−0.596703			−0.298351	−	0.954456i	$-0.596437\pi$
$431$	−12.3879			−0.596703			−0.298351	+	0.954456i	$0.596437\pi$
$432$		0			0
$433$			1.41090i			0.0678033i	0.999425	+	0.0339017i	$0.0107933\pi$
$433$			1.41090i			0.0678033i	−0.999425	+	0.0339017i	$0.989207\pi$
$434$		0			0
$435$	3.35026	−	2.96239i	0.160633	−	0.142036i
$436$		0			0
$437$		−	21.4499i		−	1.02609i
$438$		0			0
$439$	31.5877			1.50760			0.753799	−	0.657105i	$-0.228219\pi$
$439$	31.5877			1.50760			0.753799	+	0.657105i	$0.228219\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			28.0362i			1.33204i	0.745934	+	0.666020i	$0.232003\pi$
$443$			28.0362i			1.33204i	−0.745934	+	0.666020i	$0.767997\pi$
$444$		0			0
$445$	−3.23884	−	3.66291i	−0.153536	−	0.173639i
$446$		0			0
$447$			0.0752228i			0.00355792i
$448$		0			0
$449$	19.4010			0.915592			0.457796	−	0.889057i	$-0.348639\pi$
$449$	19.4010			0.915592			0.457796	+	0.889057i	$0.348639\pi$
$450$		0			0
$451$	43.9511			2.06958
$452$		0			0
$453$			6.70052i			0.314818i
$454$		0			0
$455$	10.3127	+	11.6629i	0.483465	+	0.546766i
$456$		0			0
$457$			19.3258i			0.904024i	0.892012	+	0.452012i	$0.149294\pi$
$457$			19.3258i			0.904024i	−0.892012	+	0.452012i	$0.850706\pi$
$458$		0			0
$459$	−6.57452			−0.306872
$460$		0			0
$461$	2.81194			0.130965			0.0654826	−	0.997854i	$-0.479141\pi$
$461$	2.81194			0.130965			0.0654826	+	0.997854i	$0.479141\pi$
$462$		0			0
$463$			24.1016i			1.12009i	0.828461	+	0.560047i	$0.189217\pi$
$463$			24.1016i			1.12009i	−0.828461	+	0.560047i	$0.810783\pi$
$464$		0			0
$465$	1.73813	−	1.53690i	0.0806041	−	0.0712722i
$466$		0			0
$467$			21.1490i			0.978660i	0.872099	+	0.489330i	$0.162759\pi$
$467$			21.1490i			0.978660i	−0.872099	+	0.489330i	$0.837241\pi$
$468$		0			0
$469$	−4.77575			−0.220523
$470$		0			0
$471$	−5.66291			−0.260933
$472$		0			0
$473$			37.4010i			1.71970i
$474$		0			0
$475$	2.28821	−	18.5501i	0.104990	−	0.851136i
$476$		0			0
$477$			1.03761i			0.0475090i
$478$		0			0
$479$	−17.2995			−0.790433			−0.395217	−	0.918588i	$-0.629330\pi$
$479$	−17.2995			−0.790433			−0.395217	+	0.918588i	$0.629330\pi$
$480$		0			0
$481$	74.5012			3.39696
$482$		0			0
$483$			5.73813i			0.261094i
$484$		0			0
$485$	−6.26187	+	5.53690i	−0.284337	+	0.251418i
$486$		0			0
$487$		−	40.4749i		−	1.83409i	−0.398783	−	0.917045i	$-0.630567\pi$
$487$		−	40.4749i		−	1.83409i	0.398783	−	0.917045i	$-0.369433\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	19.9854			0.901929			0.450964	−	0.892542i	$-0.351080\pi$
$491$	19.9854			0.901929			0.450964	+	0.892542i	$0.351080\pi$
$492$		0			0
$493$			13.1490i			0.592203i
$494$		0			0
$495$	9.35026	+	10.5745i	0.420263	+	0.475289i
$496$		0			0
$497$		−	8.23743i		−	0.369499i
$498$		0			0
$499$	14.8510			0.664821			0.332410	−	0.943135i	$-0.392138\pi$
$499$	14.8510			0.664821			0.332410	+	0.943135i	$0.392138\pi$
$500$		0			0
$501$	20.6253			0.921470
$502$		0			0
$503$		−	22.5501i		−	1.00546i	−0.864444	−	0.502729i	$-0.832329\pi$
$503$		−	22.5501i		−	1.00546i	0.864444	−	0.502729i	$-0.167671\pi$
$504$		0			0
$505$	14.3127	+	16.1866i	0.636905	+	0.720296i
$506$		0			0
$507$			35.4749i			1.57549i
$508$		0			0
$509$	−17.5125			−0.776226			−0.388113	−	0.921612i	$-0.626873\pi$
$509$	−17.5125			−0.776226			−0.388113	+	0.921612i	$0.626873\pi$
$510$		0			0
$511$	4.26187			0.188534
$512$		0			0
$513$		−	3.73813i		−	0.165043i
$514$		0			0
$515$	−2.17679	+	1.92478i	−0.0959210	+	0.0848158i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−10.5745			−0.464170
$520$		0			0
$521$	26.4387			1.15830			0.579149	−	0.815221i	$-0.303385\pi$
$521$	26.4387			1.15830			0.579149	+	0.815221i	$0.303385\pi$
$522$		0			0
$523$			31.8496i			1.39268i	0.717710	+	0.696342i	$0.245190\pi$
$523$			31.8496i			1.39268i	−0.717710	+	0.696342i	$0.754810\pi$
$524$		0			0
$525$	−0.612127	+	4.96239i	−0.0267154	+	0.216576i
$526$		0			0
$527$			6.82179i			0.297162i
$528$		0			0
$529$	−9.92619			−0.431574
$530$		0			0
$531$	3.22425			0.139921
$532$		0			0
$533$			48.4749i			2.09968i
$534$		0			0
$535$	−6.13586	+	5.42548i	−0.265276	+	0.234564i
$536$		0			0
$537$		−	0.911603i		−	0.0393386i
$538$		0			0
$539$	6.31265			0.271905
$540$		0			0
$541$	24.1768			1.03944			0.519721	−	0.854336i	$-0.326036\pi$
$541$	24.1768			1.03944			0.519721	+	0.854336i	$0.326036\pi$
$542$		0			0
$543$			13.3258i			0.571866i
$544$		0			0
$545$	−10.0362	−	11.3503i	−0.429903	−	0.486192i
$546$		0			0
$547$		−	34.1768i		−	1.46129i	−0.682755	−	0.730647i	$-0.739219\pi$
$547$		−	34.1768i		−	1.46129i	0.682755	−	0.730647i	$-0.260781\pi$
$548$		0			0
$549$	13.8496			0.591085
$550$		0			0
$551$	−7.47627			−0.318500
$552$		0			0
$553$		−	5.92478i		−	0.251947i
$554$		0			0
$555$	15.8496	+	17.9248i	0.672776	+	0.760864i
$556$		0			0
$557$			24.7367i			1.04813i	0.851679	+	0.524064i	$0.175585\pi$
$557$			24.7367i			1.04813i	−0.851679	+	0.524064i	$0.824415\pi$
$558$		0			0
$559$	−41.2506			−1.74471
$560$		0			0
$561$	−41.5026			−1.75224
$562$		0			0
$563$		−	7.47627i		−	0.315087i	−0.987512	−	0.157544i	$-0.949643\pi$
$563$		−	7.47627i		−	0.315087i	0.987512	−	0.157544i	$-0.0503575\pi$
$564$		0			0
$565$	−14.8872	+	13.1636i	−0.626308	+	0.553798i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	−17.3258			−0.726336			−0.363168	−	0.931724i	$-0.618305\pi$
$569$	−17.3258			−0.726336			−0.363168	+	0.931724i	$0.618305\pi$
$570$		0			0
$571$	−31.8496			−1.33286			−0.666431	−	0.745567i	$-0.732179\pi$
$571$	−31.8496			−1.33286			−0.666431	+	0.745567i	$0.732179\pi$
$572$		0			0
$573$		−	25.7889i		−	1.07735i
$574$		0			0
$575$	−28.4749	−	3.51247i	−1.18748	−	0.146480i
$576$		0			0
$577$		−	20.5139i		−	0.854004i	−0.904251	−	0.427002i	$-0.859570\pi$
$577$		−	20.5139i		−	0.854004i	0.904251	−	0.427002i	$-0.140430\pi$
$578$		0			0
$579$	−15.3258			−0.636920
$580$		0			0
$581$	−3.22425			−0.133765
$582$		0			0
$583$			6.55008i			0.271277i
$584$		0			0
$585$	−11.6629	+	10.3127i	−0.482202	+	0.426376i
$586$		0			0
$587$		−	20.8773i		−	0.861699i	−0.902424	−	0.430850i	$-0.858214\pi$
$587$		−	20.8773i		−	0.861699i	0.902424	−	0.430850i	$-0.141786\pi$
$588$		0			0
$589$	−3.87873			−0.159820
$590$		0			0
$591$	0.513881			0.0211382
$592$		0			0
$593$			5.57593i			0.228976i	0.993425	+	0.114488i	$0.0365227\pi$
$593$			5.57593i			0.228976i	−0.993425	+	0.114488i	$0.963477\pi$
$594$		0			0
$595$	−9.73813	−	11.0132i	−0.399225	−	0.451496i
$596$		0			0
$597$		−	3.73813i		−	0.152992i
$598$		0			0
$599$	1.01317			0.0413972			0.0206986	−	0.999786i	$-0.493411\pi$
$599$	1.01317			0.0413972			0.0206986	+	0.999786i	$0.493411\pi$
$600$		0			0
$601$	−20.2981			−0.827975			−0.413988	−	0.910283i	$-0.635864\pi$
$601$	−20.2981			−0.827975			−0.413988	+	0.910283i	$0.635864\pi$
$602$		0			0
$603$		−	4.77575i		−	0.194483i
$604$		0			0
$605$	42.7318	+	48.3268i	1.73729	+	1.96476i
$606$		0			0
$607$			14.9525i			0.606905i	0.952847	+	0.303452i	$0.0981393\pi$
$607$			14.9525i			0.606905i	−0.952847	+	0.303452i	$0.901861\pi$
$608$		0			0
$609$	2.00000			0.0810441
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	19.0738i		−	0.770384i	−0.922836	−	0.385192i	$-0.874135\pi$
$613$		−	19.0738i		−	0.770384i	0.922836	−	0.385192i	$-0.125865\pi$
$614$		0			0
$615$	−11.6629	+	10.3127i	−0.470294	+	0.415846i
$616$		0			0
$617$		−	15.8886i		−	0.639650i	−0.947477	−	0.319825i	$-0.896376\pi$
$617$		−	15.8886i		−	0.639650i	0.947477	−	0.319825i	$-0.103624\pi$
$618$		0			0
$619$	32.7367			1.31580			0.657900	−	0.753105i	$-0.271445\pi$
$619$	32.7367			1.31580			0.657900	+	0.753105i	$0.271445\pi$
$620$		0			0
$621$	−5.73813			−0.230263
$622$		0			0
$623$		−	2.18664i		−	0.0876060i
$624$		0			0
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$		0			0
$627$		−	23.5975i		−	0.942395i
$628$		0			0
$629$	−70.3508			−2.80507
$630$		0			0
$631$	40.7269			1.62131			0.810656	−	0.585523i	$-0.199111\pi$
$631$	40.7269			1.62131			0.810656	+	0.585523i	$0.199111\pi$
$632$		0			0
$633$		−	3.22425i		−	0.128153i
$634$		0			0
$635$	−4.52373	+	4.00000i	−0.179519	+	0.158735i
$636$		0			0
$637$			6.96239i			0.275860i
$638$		0			0
$639$	8.23743			0.325868
$640$		0			0
$641$	−37.1754			−1.46834			−0.734170	−	0.678966i	$-0.762428\pi$
$641$	−37.1754			−1.46834			−0.734170	+	0.678966i	$0.762428\pi$
$642$		0			0
$643$			18.3996i			0.725611i	0.931865	+	0.362805i	$0.118181\pi$
$643$			18.3996i			0.725611i	−0.931865	+	0.362805i	$0.881819\pi$
$644$		0			0
$645$	−8.77575	−	9.92478i	−0.345545	−	0.390788i
$646$		0			0
$647$		−	15.2243i		−	0.598527i	−0.954170	−	0.299264i	$-0.903259\pi$
$647$		−	15.2243i		−	0.598527i	0.954170	−	0.299264i	$-0.0967410\pi$
$648$		0			0
$649$	20.3536			0.798948
$650$		0			0
$651$	1.03761			0.0406672
$652$		0			0
$653$			0.785595i			0.0307427i	0.999882	+	0.0153714i	$0.00489305\pi$
$653$			0.785595i			0.0307427i	−0.999882	+	0.0153714i	$0.995107\pi$
$654$		0			0
$655$	13.9248	+	15.7480i	0.544086	+	0.615325i
$656$		0			0
$657$			4.26187i			0.166271i
$658$		0			0
$659$	−21.2652			−0.828374			−0.414187	−	0.910192i	$-0.635934\pi$
$659$	−21.2652			−0.828374			−0.414187	+	0.910192i	$0.635934\pi$
$660$		0			0
$661$	−32.4485			−1.26210			−0.631050	−	0.775742i	$-0.717376\pi$
$661$	−32.4485			−1.26210			−0.631050	+	0.775742i	$0.717376\pi$
$662$		0			0
$663$		−	45.7743i		−	1.77773i
$664$		0			0
$665$	6.26187	−	5.53690i	0.242825	−	0.214712i
$666$		0			0
$667$			11.4763i			0.444363i
$668$		0			0
$669$	−11.8496			−0.458130
$670$		0			0
$671$	87.4274			3.37510
$672$		0			0
$673$			5.55149i			0.213994i	0.994259	+	0.106997i	$0.0341236\pi$
$673$			5.55149i			0.213994i	−0.994259	+	0.106997i	$0.965876\pi$
$674$		0			0
$675$	−4.96239	−	0.612127i	−0.191002	−	0.0235608i
$676$		0			0
$677$		−	7.72355i		−	0.296840i	−0.988924	−	0.148420i	$-0.952581\pi$
$677$		−	7.72355i		−	0.296840i	0.988924	−	0.148420i	$-0.0474187\pi$
$678$		0			0
$679$	−3.73813			−0.143456
$680$		0			0
$681$	−14.7005			−0.563325
$682$		0			0
$683$		−	15.5125i		−	0.593568i	−0.954945	−	0.296784i	$-0.904086\pi$
$683$		−	15.5125i		−	0.593568i	0.954945	−	0.296784i	$-0.0959141\pi$
$684$		0			0
$685$	16.1866	−	14.3127i	0.618460	−	0.546858i
$686$		0			0
$687$			9.47627i			0.361542i
$688$		0			0
$689$	−7.22425			−0.275222
$690$		0			0
$691$	24.7367			0.941029			0.470515	−	0.882392i	$-0.344068\pi$
$691$	24.7367			0.941029			0.470515	+	0.882392i	$0.344068\pi$
$692$		0			0
$693$			6.31265i			0.239798i
$694$		0			0
$695$	30.9380	+	34.9887i	1.17354	+	1.32720i
$696$		0			0
$697$		−	45.7743i		−	1.73383i
$698$		0			0
$699$	−29.5125			−1.11626
$700$		0			0
$701$	−3.55149			−0.134138			−0.0670690	−	0.997748i	$-0.521365\pi$
$701$	−3.55149			−0.134138			−0.0670690	+	0.997748i	$0.521365\pi$
$702$		0			0
$703$		−	40.0000i		−	1.50863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			9.66291i			0.363411i
$708$		0			0
$709$	−17.6991			−0.664704			−0.332352	−	0.943155i	$-0.607842\pi$
$709$	−17.6991			−0.664704			−0.332352	+	0.943155i	$0.607842\pi$
$710$		0			0
$711$	5.92478			0.222197
$712$		0			0
$713$			5.95395i			0.222977i
$714$		0			0
$715$	−73.6239	+	65.1002i	−2.75338	+	2.43461i
$716$		0			0
$717$			25.0132i			0.934134i
$718$		0			0
$719$	−12.3733			−0.461446			−0.230723	−	0.973020i	$-0.574109\pi$
$719$	−12.3733			−0.461446			−0.230723	+	0.973020i	$0.574109\pi$
$720$		0			0
$721$	−1.29948			−0.0483950
$722$		0			0
$723$		−	7.92478i		−	0.294726i
$724$		0			0
$725$	−1.22425	+	9.92478i	−0.0454676	+	0.368597i
$726$		0			0
$727$			47.9511i			1.77841i	0.457510	+	0.889204i	$0.348741\pi$
$727$			47.9511i			1.77841i	−0.457510	+	0.889204i	$0.651259\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	38.9525			1.44071
$732$		0			0
$733$		−	37.9149i		−	1.40042i	−0.713937	−	0.700210i	$-0.753090\pi$
$733$		−	37.9149i		−	1.40042i	0.713937	−	0.700210i	$-0.246910\pi$
$734$		0			0
$735$	−1.67513	+	1.48119i	−0.0617881	+	0.0546347i
$736$		0			0
$737$		−	30.1476i		−	1.11050i
$738$		0			0
$739$	41.4010			1.52296			0.761481	−	0.648187i	$-0.224473\pi$
$739$	41.4010			1.52296			0.761481	+	0.648187i	$0.224473\pi$
$740$		0			0
$741$	26.0263			0.956102
$742$		0			0
$743$			7.28963i			0.267430i	0.991020	+	0.133715i	$0.0426908\pi$
$743$			7.28963i			0.267430i	−0.991020	+	0.133715i	$0.957309\pi$
$744$		0			0
$745$	−0.111420	−	0.126008i	−0.00408210	−	0.00461658i
$746$		0			0
$747$		−	3.22425i		−	0.117969i
$748$		0			0
$749$	−3.66291			−0.133840
$750$		0			0
$751$	−4.12127			−0.150387			−0.0751936	−	0.997169i	$-0.523957\pi$
$751$	−4.12127			−0.150387			−0.0751936	+	0.997169i	$0.523957\pi$
$752$		0			0
$753$		−	15.0738i		−	0.549320i
$754$		0			0
$755$	−9.92478	−	11.2243i	−0.361200	−	0.408492i
$756$		0			0
$757$		−	23.4471i		−	0.852199i	−0.904676	−	0.426100i	$-0.859887\pi$
$757$		−	23.4471i		−	0.852199i	0.904676	−	0.426100i	$-0.140113\pi$
$758$		0			0
$759$	−36.2228			−1.31481
$760$		0			0
$761$	10.7104			0.388251			0.194125	−	0.980977i	$-0.437813\pi$
$761$	10.7104			0.388251			0.194125	+	0.980977i	$0.437813\pi$
$762$		0			0
$763$		−	6.77575i		−	0.245298i
$764$		0			0
$765$	11.0132	−	9.73813i	0.398182	−	0.352083i
$766$		0			0
$767$			22.4485i			0.810569i
$768$		0			0
$769$	21.8496			0.787915			0.393958	−	0.919129i	$-0.371106\pi$
$769$	21.8496			0.787915			0.393958	+	0.919129i	$0.371106\pi$
$770$		0			0
$771$	22.0508			0.794140
$772$		0			0
$773$			31.9756i			1.15008i	0.818125	+	0.575041i	$0.195014\pi$
$773$			31.9756i			1.15008i	−0.818125	+	0.575041i	$0.804986\pi$
$774$		0			0
$775$	−0.635150	+	5.14903i	−0.0228153	+	0.184959i
$776$		0			0
$777$			10.7005i			0.383879i
$778$		0			0
$779$	26.0263			0.932491
$780$		0			0
$781$	52.0000			1.86071
$782$		0			0
$783$			2.00000i			0.0714742i
$784$		0			0
$785$	9.48612	−	8.38787i	0.338574	−	0.299376i
$786$		0			0
$787$			29.2506i			1.04267i	0.853352	+	0.521336i	$0.174566\pi$
$787$			29.2506i			1.04267i	−0.853352	+	0.521336i	$0.825434\pi$
$788$		0			0
$789$	11.8134			0.420567
$790$		0			0
$791$	−8.88717			−0.315991
$792$		0			0
$793$			96.4260i			3.42419i
$794$		0			0
$795$	−1.53690	−	1.73813i	−0.0545084	−	0.0616453i
$796$		0			0
$797$		−	29.1246i		−	1.03165i	−0.856695	−	0.515823i	$-0.827486\pi$
$797$		−	29.1246i		−	1.03165i	0.856695	−	0.515823i	$-0.172514\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	2.18664			0.0772612
$802$		0			0
$803$			26.9037i			0.949410i
$804$		0			0
$805$	−8.49929	−	9.61213i	−0.299561	−	0.338783i
$806$		0			0
$807$		−	27.0640i		−	0.952696i
$808$		0			0
$809$	−4.95254			−0.174122			−0.0870610	−	0.996203i	$-0.527748\pi$
$809$	−4.95254			−0.174122			−0.0870610	+	0.996203i	$0.527748\pi$
$810$		0			0
$811$	−31.6893			−1.11276			−0.556380	−	0.830928i	$-0.687810\pi$
$811$	−31.6893			−1.11276			−0.556380	+	0.830928i	$0.687810\pi$
$812$		0			0
$813$		−	10.8119i		−	0.379191i
$814$		0			0
$815$	−6.70052	+	5.92478i	−0.234709	+	0.207536i
$816$		0			0
$817$			22.1476i			0.774847i
$818$		0			0
$819$	−6.96239			−0.243286
$820$		0			0
$821$	−16.8218			−0.587085			−0.293542	−	0.955946i	$-0.594834\pi$
$821$	−16.8218			−0.587085			−0.293542	+	0.955946i	$0.594834\pi$
$822$		0			0
$823$		−	26.7005i		−	0.930722i	−0.885121	−	0.465361i	$-0.845925\pi$
$823$		−	26.7005i		−	0.930722i	0.885121	−	0.465361i	$-0.154075\pi$
$824$		0			0
$825$	−31.3258	−	3.86414i	−1.09063	−	0.134532i
$826$		0			0
$827$		−	32.1866i		−	1.11924i	−0.828750	−	0.559620i	$-0.810947\pi$
$827$		−	32.1866i		−	1.11924i	0.828750	−	0.559620i	$-0.189053\pi$
$828$		0			0
$829$	−46.7269			−1.62289			−0.811446	−	0.584428i	$-0.801319\pi$
$829$	−46.7269			−1.62289			−0.811446	+	0.584428i	$0.801319\pi$
$830$		0			0
$831$	−22.5501			−0.782254
$832$		0			0
$833$		−	6.57452i		−	0.227793i
$834$		0			0
$835$	−34.5501	+	30.5501i	−1.19565	+	1.05723i
$836$		0			0
$837$			1.03761i			0.0358651i
$838$		0			0
$839$	−44.6253			−1.54064			−0.770318	−	0.637660i	$-0.779903\pi$
$839$	−44.6253			−1.54064			−0.770318	+	0.637660i	$0.779903\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			10.3733i			0.357275i
$844$		0			0
$845$	−52.5452	−	59.4250i	−1.80761	−	2.04428i
$846$		0			0
$847$			28.8496i			0.991282i
$848$		0			0
$849$	−10.7005			−0.367241
$850$		0			0
$851$	−61.4010			−2.10480
$852$		0			0
$853$			24.2422i			0.830036i	0.909813	+	0.415018i	$0.136225\pi$
$853$			24.2422i			0.830036i	−0.909813	+	0.415018i	$0.863775\pi$
$854$		0			0
$855$	5.53690	+	6.26187i	0.189358	+	0.214151i
$856$		0			0
$857$			15.6023i			0.532964i	0.963840	+	0.266482i	$0.0858612\pi$
$857$			15.6023i			0.532964i	−0.963840	+	0.266482i	$0.914139\pi$
$858$		0			0
$859$	38.3371			1.30804			0.654022	−	0.756475i	$-0.273080\pi$
$859$	38.3371			1.30804			0.654022	+	0.756475i	$0.273080\pi$
$860$		0			0
$861$	−6.96239			−0.237278
$862$		0			0
$863$		−	29.0640i		−	0.989349i	−0.869078	−	0.494674i	$-0.835287\pi$
$863$		−	29.0640i		−	0.989349i	0.869078	−	0.494674i	$-0.164713\pi$
$864$		0			0
$865$	17.7137	−	15.6629i	0.602284	−	0.532555i
$866$		0			0
$867$			26.2243i			0.890622i
$868$		0			0
$869$	37.4010			1.26874
$870$		0			0
$871$	33.2506			1.12665
$872$		0			0
$873$		−	3.73813i		−	0.126517i
$874$		0			0
$875$	−6.32487	−	9.21933i	−0.213820	−	0.311670i
$876$		0			0
$877$			16.7757i			0.566477i	0.959050	+	0.283238i	$0.0914088\pi$
$877$			16.7757i			0.566477i	−0.959050	+	0.283238i	$0.908591\pi$
$878$		0			0
$879$	10.5745			0.356670
$880$		0			0
$881$	−30.6907			−1.03400			−0.516998	−	0.855987i	$-0.672950\pi$
$881$	−30.6907			−1.03400			−0.516998	+	0.855987i	$0.672950\pi$
$882$		0			0
$883$		−	32.9986i		−	1.11049i	−0.831687	−	0.555245i	$-0.812624\pi$
$883$		−	32.9986i		−	1.11049i	0.831687	−	0.555245i	$-0.187376\pi$
$884$		0			0
$885$	−5.40105	+	4.77575i	−0.181554	+	0.160535i
$886$		0			0
$887$			45.1002i			1.51432i	0.653232	+	0.757158i	$0.273412\pi$
$887$			45.1002i			1.51432i	−0.653232	+	0.757158i	$0.726588\pi$
$888$		0			0
$889$	−2.70052			−0.0905727
$890$		0			0
$891$	−6.31265			−0.211482
$892$		0			0
$893$		0			0
$894$		0			0
$895$	1.35026	+	1.52705i	0.0451343	+	0.0510438i
$896$		0			0
$897$		−	39.9511i		−	1.33393i
$898$		0			0
$899$	2.07522			0.0692126
$900$		0			0
$901$	6.82179			0.227267
$902$		0			0
$903$		−	5.92478i		−	0.197164i
$904$		0			0
$905$	−19.7381	−	22.3225i	−0.656118	−	0.742025i
$906$		0			0
$907$			32.9986i			1.09570i	0.836577	+	0.547850i	$0.184554\pi$
$907$			32.9986i			1.09570i	−0.836577	+	0.547850i	$0.815446\pi$
$908$		0			0
$909$	−9.66291			−0.320499
$910$		0			0
$911$	−35.8153			−1.18661			−0.593306	−	0.804977i	$-0.702178\pi$
$911$	−35.8153			−1.18661			−0.593306	+	0.804977i	$0.702178\pi$
$912$		0			0
$913$		−	20.3536i		−	0.673605i
$914$		0			0
$915$	−23.1998	+	20.5139i	−0.766962	+	0.678168i
$916$		0			0
$917$			9.40105i			0.310450i
$918$		0			0
$919$	−27.0738			−0.893083			−0.446541	−	0.894763i	$-0.647344\pi$
$919$	−27.0738			−0.893083			−0.446541	+	0.894763i	$0.647344\pi$
$920$		0			0
$921$	−6.55008			−0.215832
$922$		0			0
$923$			57.3522i			1.88777i
$924$		0			0
$925$	−53.1002	−	6.55008i	−1.74592	−	0.215365i
$926$		0			0
$927$		−	1.29948i		−	0.0426804i
$928$		0			0
$929$	23.5877			0.773887			0.386943	−	0.922103i	$-0.373531\pi$
$929$	23.5877			0.773887			0.386943	+	0.922103i	$0.373531\pi$
$930$		0			0
$931$	3.73813			0.122512
$932$		0			0
$933$		−	17.2995i		−	0.566359i
$934$		0			0
$935$	69.5223	−	61.4734i	2.27362	−	2.01040i
$936$		0			0
$937$			48.0625i			1.57013i	0.619410	+	0.785067i	$0.287372\pi$
$937$			48.0625i			1.57013i	−0.619410	+	0.785067i	$0.712628\pi$
$938$		0			0
$939$	20.2130			0.659626
$940$		0			0
$941$	−17.4109			−0.567579			−0.283789	−	0.958887i	$-0.591592\pi$
$941$	−17.4109			−0.567579			−0.283789	+	0.958887i	$0.591592\pi$
$942$		0			0
$943$		−	39.9511i		−	1.30099i
$944$		0			0
$945$	−1.48119	−	1.67513i	−0.0481833	−	0.0544920i
$946$		0			0
$947$		−	13.9610i		−	0.453671i	−0.973933	−	0.226835i	$-0.927162\pi$
$947$		−	13.9610i		−	0.453671i	0.973933	−	0.226835i	$-0.0728379\pi$
$948$		0			0
$949$	−29.6728			−0.963219
$950$		0			0
$951$	19.3357			0.627002
$952$		0			0
$953$			27.4861i			0.890363i	0.895440	+	0.445181i	$0.146861\pi$
$953$			27.4861i			0.890363i	−0.895440	+	0.445181i	$0.853139\pi$
$954$		0			0
$955$	38.1984	+	43.1998i	1.23607	+	1.39791i
$956$		0			0
$957$			12.6253i			0.408118i
$958$		0			0
$959$	9.66291			0.312032
$960$		0			0
$961$	−29.9234			−0.965270
$962$		0			0
$963$		−	3.66291i		−	0.118036i
$964$		0			0
$965$	25.6728	−	22.7005i	0.826435	−	0.730756i
$966$		0			0
$967$			26.6713i			0.857693i	0.903377	+	0.428846i	$0.141080\pi$
$967$			26.6713i			0.857693i	−0.903377	+	0.428846i	$0.858920\pi$
$968$		0			0
$969$	−24.5764			−0.789509
$970$		0			0
$971$	−49.6239			−1.59251			−0.796253	−	0.604964i	$-0.793188\pi$
$971$	−49.6239			−1.59251			−0.796253	+	0.604964i	$0.793188\pi$
$972$		0			0
$973$			20.8872i			0.669612i
$974$		0			0
$975$	4.26187	−	34.5501i	0.136489	−	1.10649i
$976$		0			0
$977$			37.5125i			1.20013i	0.799951	+	0.600065i	$0.204859\pi$
$977$			37.5125i			1.20013i	−0.799951	+	0.600065i	$0.795141\pi$
$978$		0			0
$979$	13.8035			0.441162
$980$		0			0
$981$	6.77575			0.216333
$982$		0			0
$983$			52.2228i			1.66565i	0.553536	+	0.832825i	$0.313278\pi$
$983$			52.2228i			1.66565i	−0.553536	+	0.832825i	$0.686722\pi$
$984$		0			0
$985$	−0.860818	+	0.761158i	−0.0274279	+	0.0242525i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	33.9972			1.08105
$990$		0			0
$991$	−26.3272			−0.836312			−0.418156	−	0.908375i	$-0.637324\pi$
$991$	−26.3272			−0.836312			−0.418156	+	0.908375i	$0.637324\pi$
$992$		0			0
$993$			24.6253i			0.781460i
$994$		0			0
$995$	5.53690	+	6.26187i	0.175532	+	0.198514i
$996$		0			0
$997$			29.3620i			0.929905i	0.885336	+	0.464952i	$0.153929\pi$
$997$			29.3620i			0.929905i	−0.885336	+	0.464952i	$0.846071\pi$
$998$		0			0
$999$	−10.7005			−0.338550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.t.j.1009.3		6
3.2	odd	2		5040.2.t.z.1009.1		6
4.3	odd	2		840.2.t.d.169.6	yes	6
5.2	odd	4		8400.2.a.di.1.1		3
5.3	odd	4		8400.2.a.dl.1.1		3
5.4	even	2	inner	1680.2.t.j.1009.6		6
12.11	even	2		2520.2.t.k.1009.1		6
15.14	odd	2		5040.2.t.z.1009.2		6
20.3	even	4		4200.2.a.bn.1.3		3
20.7	even	4		4200.2.a.bp.1.3		3
20.19	odd	2		840.2.t.d.169.3	✓	6
60.59	even	2		2520.2.t.k.1009.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.d.169.3	✓	6	20.19	odd	2
840.2.t.d.169.6	yes	6	4.3	odd	2
1680.2.t.j.1009.3		6	1.1	even	1	trivial
1680.2.t.j.1009.6		6	5.4	even	2	inner
2520.2.t.k.1009.1		6	12.11	even	2
2520.2.t.k.1009.2		6	60.59	even	2
4200.2.a.bn.1.3		3	20.3	even	4
4200.2.a.bp.1.3		3	20.7	even	4
5040.2.t.z.1009.1		6	3.2	odd	2
5040.2.t.z.1009.2		6	15.14	odd	2
8400.2.a.di.1.1		3	5.2	odd	4
8400.2.a.dl.1.1		3	5.3	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(1.48119 + 1.67513i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(1.48119 + 1.67513i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -6.31265 q^{11} -6.96239i q^{13} +(1.67513 - 1.48119i) q^{15} +6.57452i q^{17} -3.73813 q^{19} +1.00000 q^{21} +5.73813i q^{23} +(-0.612127 + 4.96239i) q^{25} +1.00000i q^{27} +2.00000 q^{29} +1.03761 q^{31} +6.31265i q^{33} +(-1.67513 + 1.48119i) q^{35} +10.7005i q^{37} -6.96239 q^{39} -6.96239 q^{41} -5.92478i q^{43} +(-1.48119 - 1.67513i) q^{45} -1.00000 q^{49} +6.57452 q^{51} -1.03761i q^{53} +(-9.35026 - 10.5745i) q^{55} +3.73813i q^{57} -3.22425 q^{59} -13.8496 q^{61} -1.00000i q^{63} +(11.6629 - 10.3127i) q^{65} +4.77575i q^{67} +5.73813 q^{69} -8.23743 q^{71} -4.26187i q^{73} +(4.96239 + 0.612127i) q^{75} -6.31265i q^{77} -5.92478 q^{79} +1.00000 q^{81} +3.22425i q^{83} +(-11.0132 + 9.73813i) q^{85} -2.00000i q^{87} -2.18664 q^{89} +6.96239 q^{91} -1.03761i q^{93} +(-5.53690 - 6.26187i) q^{95} +3.73813i q^{97} +6.31265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^5 - 6 * q^9	\( 6 q - 2 q^{5} - 6 q^{9} + 4 q^{11} - 4 q^{19} + 6 q^{21} - 2 q^{25} + 12 q^{29} + 28 q^{31} - 20 q^{39} - 20 q^{41} + 2 q^{45} - 6 q^{49} + 16 q^{51} - 36 q^{55} - 16 q^{59} + 4 q^{61} + 8 q^{65} + 16 q^{69} + 36 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 16 q^{85} + 12 q^{89} + 20 q^{91} + 12 q^{95} - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^5 - 6 * q^9 + 4 * q^11 - 4 * q^19 + 6 * q^21 - 2 * q^25 + 12 * q^29 + 28 * q^31 - 20 * q^39 - 20 * q^41 + 2 * q^45 - 6 * q^49 + 16 * q^51 - 36 * q^55 - 16 * q^59 + 4 * q^61 + 8 * q^65 + 16 * q^69 + 36 * q^71 + 8 * q^75 + 8 * q^79 + 6 * q^81 + 16 * q^85 + 12 * q^89 + 20 * q^91 + 12 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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