LMFDB - Embedded newform 112.9.c.c.97.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,9,Mod(97,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("112.97");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	112.c (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:	$45.6264043268$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.3520512.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 120x^{2} + 3438 $ x^4 + 120*x^2 + 3438
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.3
Root			$6.87547i$ of defining polynomial
Character	$\chi$	$=$	112.97
Dual form			112.9.c.c.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+63.0589i q^{3} -390.317i q^{5} +(687.442 + 2300.48i) q^{7} +2584.58 q^{9} +O(q^{10})$	$q+63.0589i q^{3} -390.317i q^{5} +(687.442 + 2300.48i) q^{7} +2584.58 q^{9} -4009.17 q^{11} -350.590i q^{13} +24613.0 q^{15} +97022.3i q^{17} -209322. i q^{19} +(-145066. + 43349.3i) q^{21} +155907. q^{23} +238277. q^{25} +576710. i q^{27} +845181. q^{29} +873444. i q^{31} -252813. i q^{33} +(897919. - 268321. i) q^{35} -1.13106e6 q^{37} +22107.8 q^{39} +1.80188e6i q^{41} -3.94686e6 q^{43} -1.00881e6i q^{45} +2.02387e6i q^{47} +(-4.81965e6 + 3.16290e6i) q^{49} -6.11812e6 q^{51} -1.13828e7 q^{53} +1.56485e6i q^{55} +1.31996e7 q^{57} +9.30141e6i q^{59} +1.87417e7i q^{61} +(1.77675e6 + 5.94578e6i) q^{63} -136841. q^{65} +3.87489e7 q^{67} +9.83134e6i q^{69} -4.45370e7 q^{71} +4.68614e7i q^{73} +1.50255e7i q^{75} +(-2.75607e6 - 9.22302e6i) q^{77} +3.24052e6 q^{79} -1.94092e7 q^{81} +7.02713e7i q^{83} +3.78695e7 q^{85} +5.32962e7i q^{87} -8.92209e7i q^{89} +(806527. - 241011. i) q^{91} -5.50784e7 q^{93} -8.17021e7 q^{95} +1.27076e8i q^{97} -1.03620e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6076 q^{7} - 13692 q^{9}+O(q^{10}) $ 4 * q + 6076 * q^7 - 13692 * q^9	$ 4 q + 6076 q^{7} - 13692 q^{9} + 13560 q^{11} + 13056 q^{15} - 413952 q^{21} + 894072 q^{23} + 1216900 q^{25} + 317064 q^{29} + 1655808 q^{35} - 2495096 q^{37} - 10228992 q^{39} - 9186568 q^{43} + 931588 q^{49} + 324096 q^{51} - 38727288 q^{53} - 30690816 q^{57} - 40780740 q^{63} - 11891712 q^{65} + 12320248 q^{67} - 62168712 q^{71} + 45208968 q^{77} - 24889736 q^{79} - 70788348 q^{81} + 89943552 q^{85} - 38158848 q^{91} - 408466944 q^{93} - 227967744 q^{95} - 224220168 q^{99}+O(q^{100}) $ 4 * q + 6076 * q^7 - 13692 * q^9 + 13560 * q^11 + 13056 * q^15 - 413952 * q^21 + 894072 * q^23 + 1216900 * q^25 + 317064 * q^29 + 1655808 * q^35 - 2495096 * q^37 - 10228992 * q^39 - 9186568 * q^43 + 931588 * q^49 + 324096 * q^51 - 38727288 * q^53 - 30690816 * q^57 - 40780740 * q^63 - 11891712 * q^65 + 12320248 * q^67 - 62168712 * q^71 + 45208968 * q^77 - 24889736 * q^79 - 70788348 * q^81 + 89943552 * q^85 - 38158848 * q^91 - 408466944 * q^93 - 227967744 * q^95 - 224220168 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			63.0589i			0.778505i	0.921131	+	0.389252i	$0.127267\pi$
$3$			63.0589i			0.778505i	−0.921131	+	0.389252i	$0.872733\pi$
$4$		0			0
$5$		−	390.317i		−	0.624508i	−0.949999	−	0.312254i	$-0.898916\pi$
$5$		−	390.317i		−	0.624508i	0.949999	−	0.312254i	$-0.101084\pi$
$6$		0			0
$7$	687.442	+	2300.48i	0.286315	+	0.958136i
$7$	687.442	+	2300.48i	0.286315	+	0.958136i
$8$		0			0
$9$	2584.58			0.393931
$10$		0			0
$11$	−4009.17			−0.273831			−0.136916	−	0.990583i	$-0.543719\pi$
$11$	−4009.17			−0.273831			−0.136916	+	0.990583i	$0.543719\pi$
$12$		0			0
$13$		−	350.590i		−	0.0122751i	−0.999981	−	0.00613757i	$-0.998046\pi$
$13$		−	350.590i		−	0.0122751i	0.999981	−	0.00613757i	$-0.00195366\pi$
$14$		0			0
$15$	24613.0			0.486182
$16$		0			0
$17$			97022.3i			1.16165i	0.814028	+	0.580826i	$0.197270\pi$
$17$			97022.3i			1.16165i	−0.814028	+	0.580826i	$0.802730\pi$
$18$		0			0
$19$		−	209322.i		−	1.60621i	−0.595840	−	0.803103i	$-0.703181\pi$
$19$		−	209322.i		−	1.60621i	0.595840	−	0.803103i	$-0.296819\pi$
$20$		0			0
$21$	−145066.	+	43349.3i	−0.745913	+	0.222898i
$22$		0			0
$23$	155907.			0.557128			0.278564	−	0.960418i	$-0.410142\pi$
$23$	155907.			0.557128			0.278564	+	0.960418i	$0.410142\pi$
$24$		0			0
$25$	238277.			0.609990
$26$		0			0
$27$			576710.i			1.08518i
$28$		0			0
$29$	845181.			1.19497			0.597486	−	0.801879i	$-0.296166\pi$
$29$	845181.			1.19497			0.597486	+	0.801879i	$0.296166\pi$
$30$		0			0
$31$			873444.i			0.945776i	0.881122	+	0.472888i	$0.156789\pi$
$31$			873444.i			0.945776i	−0.881122	+	0.472888i	$0.843211\pi$
$32$		0			0
$33$		−	252813.i		−	0.213179i
$34$		0			0
$35$	897919.	−	268321.i	0.598363	−	0.178806i
$36$		0			0
$37$	−1.13106e6			−0.603501			−0.301751	−	0.953387i	$-0.597571\pi$
$37$	−1.13106e6			−0.603501			−0.301751	+	0.953387i	$0.597571\pi$
$38$		0			0
$39$	22107.8			0.00955626
$40$		0			0
$41$			1.80188e6i			0.637663i	0.947811	+	0.318831i	$0.103290\pi$
$41$			1.80188e6i			0.637663i	−0.947811	+	0.318831i	$0.896710\pi$
$42$		0			0
$43$	−3.94686e6			−1.15446			−0.577229	−	0.816583i	$-0.695866\pi$
$43$	−3.94686e6			−1.15446			−0.577229	+	0.816583i	$0.695866\pi$
$44$		0			0
$45$		−	1.00881e6i		−	0.246013i
$46$		0			0
$47$			2.02387e6i			0.414755i	0.978261	+	0.207378i	$0.0664928\pi$
$47$			2.02387e6i			0.414755i	−0.978261	+	0.207378i	$0.933507\pi$
$48$		0			0
$49$	−4.81965e6	+	3.16290e6i	−0.836047	+	0.548657i
$50$		0			0
$51$	−6.11812e6			−0.904351
$52$		0			0
$53$	−1.13828e7			−1.44260			−0.721300	−	0.692622i	$-0.756455\pi$
$53$	−1.13828e7			−1.44260			−0.721300	+	0.692622i	$0.756455\pi$
$54$		0			0
$55$			1.56485e6i			0.171010i
$56$		0			0
$57$	1.31996e7			1.25044
$58$		0			0
$59$			9.30141e6i			0.767610i	0.923414	+	0.383805i	$0.125387\pi$
$59$			9.30141e6i			0.767610i	−0.923414	+	0.383805i	$0.874613\pi$
$60$		0			0
$61$			1.87417e7i			1.35360i	0.736168	+	0.676799i	$0.236633\pi$
$61$			1.87417e7i			1.35360i	−0.736168	+	0.676799i	$0.763367\pi$
$62$		0			0
$63$	1.77675e6	+	5.94578e6i	0.112788	+	0.377439i
$64$		0			0
$65$	−136841.			−0.00766592
$66$		0			0
$67$	3.87489e7			1.92292			0.961458	−	0.274951i	$-0.0886615\pi$
$67$	3.87489e7			1.92292			0.961458	+	0.274951i	$0.0886615\pi$
$68$		0			0
$69$			9.83134e6i			0.433727i
$70$		0			0
$71$	−4.45370e7			−1.75262			−0.876310	−	0.481747i	$-0.840002\pi$
$71$	−4.45370e7			−1.75262			−0.876310	+	0.481747i	$0.840002\pi$
$72$		0			0
$73$			4.68614e7i			1.65015i	0.565021	+	0.825076i	$0.308868\pi$
$73$			4.68614e7i			1.65015i	−0.565021	+	0.825076i	$0.691132\pi$
$74$		0			0
$75$			1.50255e7i			0.474880i
$76$		0			0
$77$	−2.75607e6	−	9.22302e6i	−0.0784020	−	0.262368i
$78$		0			0
$79$	3.24052e6			0.0831968			0.0415984	−	0.999134i	$-0.486755\pi$
$79$	3.24052e6			0.0831968			0.0415984	+	0.999134i	$0.486755\pi$
$80$		0			0
$81$	−1.94092e7			−0.450888
$82$		0			0
$83$			7.02713e7i			1.48070i	0.672224	+	0.740348i	$0.265339\pi$
$83$			7.02713e7i			1.48070i	−0.672224	+	0.740348i	$0.734661\pi$
$84$		0			0
$85$	3.78695e7			0.725461
$86$		0			0
$87$			5.32962e7i			0.930292i
$88$		0			0
$89$		−	8.92209e7i		−	1.42202i	−0.703181	−	0.711011i	$-0.748237\pi$
$89$		−	8.92209e7i		−	1.42202i	0.703181	−	0.711011i	$-0.251763\pi$
$90$		0			0
$91$	806527.	−	241011.i	0.0117613	−	0.00351456i
$92$		0			0
$93$	−5.50784e7			−0.736291
$94$		0			0
$95$	−8.17021e7			−1.00309
$96$		0			0
$97$			1.27076e8i			1.43542i	0.696344	+	0.717708i	$0.254809\pi$
$97$			1.27076e8i			1.43542i	−0.696344	+	0.717708i	$0.745191\pi$
$98$		0			0
$99$	−1.03620e7			−0.107871
$100$		0			0
$101$			4.94803e7i			0.475496i	0.971327	+	0.237748i	$0.0764092\pi$
$101$			4.94803e7i			0.475496i	−0.971327	+	0.237748i	$0.923591\pi$
$102$		0			0
$103$		−	9.96346e7i		−	0.885240i	−0.896709	−	0.442620i	$-0.854049\pi$
$103$		−	9.96346e7i		−	0.885240i	0.896709	−	0.442620i	$-0.145951\pi$
$104$		0			0
$105$	1.69200e7	+	5.66217e7i	0.139201	+	0.465828i
$106$		0			0
$107$	1.83886e8			1.40286			0.701428	−	0.712741i	$-0.252546\pi$
$107$	1.83886e8			1.40286			0.701428	+	0.712741i	$0.252546\pi$
$108$		0			0
$109$	1.58004e7			0.111934			0.0559671	−	0.998433i	$-0.482176\pi$
$109$	1.58004e7			0.111934			0.0559671	+	0.998433i	$0.482176\pi$
$110$		0			0
$111$		−	7.13232e7i		−	0.469828i
$112$		0			0
$113$	−4.97912e7			−0.305379			−0.152689	−	0.988274i	$-0.548793\pi$
$113$	−4.97912e7			−0.305379			−0.152689	+	0.988274i	$0.548793\pi$
$114$		0			0
$115$		−	6.08533e7i		−	0.347931i
$116$		0			0
$117$		−	906129.i		−	0.00483556i
$118$		0			0
$119$	−2.23198e8	+	6.66973e7i	−1.11302	+	0.332598i
$120$		0			0
$121$	−1.98285e8			−0.925016
$122$		0			0
$123$	−1.13625e8			−0.496423
$124$		0			0
$125$		−	2.45471e8i		−	1.00545i
$126$		0			0
$127$	1.40133e8			0.538674			0.269337	−	0.963046i	$-0.413195\pi$
$127$	1.40133e8			0.538674			0.269337	+	0.963046i	$0.413195\pi$
$128$		0			0
$129$		−	2.48884e8i		−	0.898750i
$130$		0			0
$131$		−	2.20973e8i		−	0.750333i	−0.926957	−	0.375166i	$-0.877586\pi$
$131$		−	2.20973e8i		−	0.750333i	0.926957	−	0.375166i	$-0.122414\pi$
$132$		0			0
$133$	4.81543e8	−	1.43897e8i	1.53896	−	0.459881i
$134$		0			0
$135$	2.25100e8			0.677704
$136$		0			0
$137$	1.67452e8			0.475344			0.237672	−	0.971345i	$-0.423616\pi$
$137$	1.67452e8			0.475344			0.237672	+	0.971345i	$0.423616\pi$
$138$		0			0
$139$			1.60687e8i			0.430448i	0.976565	+	0.215224i	$0.0690482\pi$
$139$			1.60687e8i			0.430448i	−0.976565	+	0.215224i	$0.930952\pi$
$140$		0			0
$141$	−1.27623e8			−0.322889
$142$		0			0
$143$			1.40557e6i			0.00336132i
$144$		0			0
$145$		−	3.29889e8i		−	0.746270i
$146$		0			0
$147$	−1.99449e8	−	3.03921e8i	−0.427132	−	0.650867i
$148$		0			0
$149$	3.03708e7			0.0616186			0.0308093	−	0.999525i	$-0.490192\pi$
$149$	3.03708e7			0.0616186			0.0308093	+	0.999525i	$0.490192\pi$
$150$		0			0
$151$	4.30673e8			0.828400			0.414200	−	0.910186i	$-0.364061\pi$
$151$	4.30673e8			0.828400			0.414200	+	0.910186i	$0.364061\pi$
$152$		0			0
$153$			2.50762e8i			0.457610i
$154$		0			0
$155$	3.40920e8			0.590645
$156$		0			0
$157$		−	2.54276e8i		−	0.418510i	−0.977861	−	0.209255i	$-0.932896\pi$
$157$		−	2.54276e8i		−	0.418510i	0.977861	−	0.209255i	$-0.0671039\pi$
$158$		0			0
$159$		−	7.17787e8i		−	1.12307i
$160$		0			0
$161$	1.07177e8	+	3.58662e8i	0.159514	+	0.533804i
$162$		0			0
$163$	−1.17790e6			−0.00166862			−0.000834312	−	1.00000i	$-0.500266\pi$
$163$	−1.17790e6			−0.00166862			−0.000834312		1.00000i	$0.500266\pi$
$164$		0			0
$165$	−9.86775e7			−0.133132
$166$		0			0
$167$		−	6.76671e7i		−	0.0869985i	−0.999053	−	0.0434993i	$-0.986149\pi$
$167$		−	6.76671e7i		−	0.0869985i	0.999053	−	0.0434993i	$-0.0138506\pi$
$168$		0			0
$169$	8.15608e8			0.999849
$170$		0			0
$171$		−	5.41010e8i		−	0.632734i
$172$		0			0
$173$		−	1.75158e9i		−	1.95544i	−0.209905	−	0.977722i	$-0.567316\pi$
$173$		−	1.75158e9i		−	1.95544i	0.209905	−	0.977722i	$-0.432684\pi$
$174$		0			0
$175$	1.63802e8	+	5.48153e8i	0.174649	+	0.584453i
$176$		0			0
$177$	−5.86536e8			−0.597588
$178$		0			0
$179$	−3.22726e8			−0.314356			−0.157178	−	0.987570i	$-0.550240\pi$
$179$	−3.22726e8			−0.314356			−0.157178	+	0.987570i	$0.550240\pi$
$180$		0			0
$181$			2.77541e8i			0.258590i	0.991606	+	0.129295i	$0.0412715\pi$
$181$			2.77541e8i			0.258590i	−0.991606	+	0.129295i	$0.958729\pi$
$182$		0			0
$183$	−1.18183e9			−1.05378
$184$		0			0
$185$			4.41472e8i			0.376891i
$186$		0			0
$187$		−	3.88979e8i		−	0.318097i
$188$		0			0
$189$	−1.32671e9	+	3.96455e8i	−1.03975	+	0.310704i
$190$		0			0
$191$	6.83273e8			0.513406			0.256703	−	0.966490i	$-0.417364\pi$
$191$	6.83273e8			0.513406			0.256703	+	0.966490i	$0.417364\pi$
$192$		0			0
$193$	−8.28870e8			−0.597389			−0.298695	−	0.954349i	$-0.596551\pi$
$193$	−8.28870e8			−0.597389			−0.298695	+	0.954349i	$0.596551\pi$
$194$		0			0
$195$		−	8.62907e6i		−	0.00596795i
$196$		0			0
$197$	9.93472e8			0.659615			0.329807	−	0.944048i	$-0.393016\pi$
$197$	9.93472e8			0.659615			0.329807	+	0.944048i	$0.393016\pi$
$198$		0			0
$199$		−	1.21114e9i		−	0.772293i	−0.922437	−	0.386146i	$-0.873806\pi$
$199$		−	1.21114e9i		−	0.772293i	0.922437	−	0.386146i	$-0.126194\pi$
$200$		0			0
$201$			2.44346e9i			1.49700i
$202$		0			0
$203$	5.81014e8	+	1.94433e9i	0.342139	+	1.14495i
$204$		0			0
$205$	7.03306e8			0.398225
$206$		0			0
$207$	4.02955e8			0.219470
$208$		0			0
$209$			8.39208e8i			0.439830i
$210$		0			0
$211$	−1.23772e9			−0.624444			−0.312222	−	0.950009i	$-0.601073\pi$
$211$	−1.23772e9			−0.624444			−0.312222	+	0.950009i	$0.601073\pi$
$212$		0			0
$213$		−	2.80845e9i		−	1.36442i
$214$		0			0
$215$			1.54053e9i			0.720967i
$216$		0			0
$217$	−2.00934e9	+	6.00443e8i	−0.906182	+	0.270790i
$218$		0			0
$219$	−2.95503e9			−1.28465
$220$		0			0
$221$	3.40151e7			0.0142594
$222$		0			0
$223$			3.39581e9i			1.37317i	0.727049	+	0.686585i	$0.240891\pi$
$223$			3.39581e9i			1.37317i	−0.727049	+	0.686585i	$0.759109\pi$
$224$		0			0
$225$	6.15847e8			0.240294
$226$		0			0
$227$			1.25356e9i			0.472110i	0.971740	+	0.236055i	$0.0758545\pi$
$227$			1.25356e9i			0.472110i	−0.971740	+	0.236055i	$0.924145\pi$
$228$		0			0
$229$		−	7.90350e8i		−	0.287394i	−0.989622	−	0.143697i	$-0.954101\pi$
$229$		−	7.90350e8i		−	0.287394i	0.989622	−	0.143697i	$-0.0458991\pi$
$230$		0			0
$231$	5.81593e8	−	1.73795e8i	0.204254	−	0.0610363i
$232$		0			0
$233$	2.89689e9			0.982898			0.491449	−	0.870906i	$-0.336467\pi$
$233$	2.89689e9			0.982898			0.491449	+	0.870906i	$0.336467\pi$
$234$		0			0
$235$	7.89952e8			0.259018
$236$		0			0
$237$			2.04344e8i			0.0647691i
$238$		0			0
$239$	3.82362e9			1.17188			0.585940	−	0.810354i	$-0.300725\pi$
$239$	3.82362e9			1.17188			0.585940	+	0.810354i	$0.300725\pi$
$240$		0			0
$241$			5.42966e9i			1.60955i	0.593579	+	0.804776i	$0.297714\pi$
$241$			5.42966e9i			1.60955i	−0.593579	+	0.804776i	$0.702286\pi$
$242$		0			0
$243$			2.55987e9i			0.734163i
$244$		0			0
$245$	1.23453e9	+	1.88119e9i	0.342641	+	0.522118i
$246$		0			0
$247$	−7.33864e7			−0.0197164
$248$		0			0
$249$	−4.43123e9			−1.15273
$250$		0			0
$251$		−	2.98297e9i		−	0.751544i	−0.926712	−	0.375772i	$-0.877378\pi$
$251$		−	2.98297e9i		−	0.751544i	0.926712	−	0.375772i	$-0.122622\pi$
$252$		0			0
$253$	−6.25058e8			−0.152559
$254$		0			0
$255$			2.38801e9i			0.564774i
$256$		0			0
$257$			3.75861e9i			0.861579i	0.902452	+	0.430789i	$0.141765\pi$
$257$			3.75861e9i			0.861579i	−0.902452	+	0.430789i	$0.858235\pi$
$258$		0			0
$259$	−7.77537e8	−	2.60198e9i	−0.172791	−	0.578236i
$260$		0			0
$261$	2.18444e9			0.470736
$262$		0			0
$263$	−5.44132e9			−1.13732			−0.568658	−	0.822574i	$-0.692537\pi$
$263$	−5.44132e9			−1.13732			−0.568658	+	0.822574i	$0.692537\pi$
$264$		0			0
$265$			4.44291e9i			0.900915i
$266$		0			0
$267$	5.62617e9			1.10705
$268$		0			0
$269$		−	2.01871e9i		−	0.385536i	−0.981244	−	0.192768i	$-0.938254\pi$
$269$		−	2.01871e9i		−	0.385536i	0.981244	−	0.192768i	$-0.0617465\pi$
$270$		0			0
$271$		−	4.63979e9i		−	0.860243i	−0.902771	−	0.430121i	$-0.858471\pi$
$271$		−	4.63979e9i		−	0.860243i	0.902771	−	0.430121i	$-0.141529\pi$
$272$		0			0
$273$	1.51979e7	+	5.08587e7i	0.00273610	+	0.00915619i
$274$		0			0
$275$	−9.55293e8			−0.167034
$276$		0			0
$277$	3.96862e9			0.674094			0.337047	−	0.941488i	$-0.390572\pi$
$277$	3.96862e9			0.674094			0.337047	+	0.941488i	$0.390572\pi$
$278$		0			0
$279$			2.25749e9i			0.372570i
$280$		0			0
$281$	−1.06501e10			−1.70817			−0.854083	−	0.520136i	$-0.825881\pi$
$281$	−1.06501e10			−1.70817			−0.854083	+	0.520136i	$0.825881\pi$
$282$		0			0
$283$			3.76400e9i			0.586818i	0.955987	+	0.293409i	$0.0947898\pi$
$283$			3.76400e9i			0.586818i	−0.955987	+	0.293409i	$0.905210\pi$
$284$		0			0
$285$		−	5.15204e9i		−	0.780908i
$286$		0			0
$287$	−4.14520e9	+	1.23869e9i	−0.610967	+	0.182572i
$288$		0			0
$289$	−2.43758e9			−0.349435
$290$		0			0
$291$	−8.01329e9			−1.11748
$292$		0			0
$293$		−	1.33762e10i		−	1.81495i	−0.420111	−	0.907473i	$-0.638009\pi$
$293$		−	1.33762e10i		−	1.81495i	0.420111	−	0.907473i	$-0.361991\pi$
$294$		0			0
$295$	3.63050e9			0.479378
$296$		0			0
$297$		−	2.31213e9i		−	0.297157i
$298$		0			0
$299$		−	5.46596e7i		−	0.00683883i
$300$		0			0
$301$	−2.71324e9	−	9.07968e9i	−0.330538	−	1.10613i
$302$		0			0
$303$	−3.12017e9			−0.370176
$304$		0			0
$305$	7.31521e9			0.845332
$306$		0			0
$307$		−	1.63542e9i		−	0.184110i	−0.995754	−	0.0920549i	$-0.970656\pi$
$307$		−	1.63542e9i		−	0.184110i	0.995754	−	0.0920549i	$-0.0293435\pi$
$308$		0			0
$309$	6.28284e9			0.689164
$310$		0			0
$311$		−	1.06927e10i		−	1.14300i	−0.820604	−	0.571498i	$-0.806363\pi$
$311$		−	1.06927e10i		−	1.14300i	0.820604	−	0.571498i	$-0.193637\pi$
$312$		0			0
$313$			2.98022e9i			0.310507i	0.987875	+	0.155254i	$0.0496195\pi$
$313$			2.98022e9i			0.310507i	−0.987875	+	0.155254i	$0.950381\pi$
$314$		0			0
$315$	2.32074e9	−	6.93496e8i	0.235714	−	0.0704372i
$316$		0			0
$317$	2.57274e9			0.254776			0.127388	−	0.991853i	$-0.459341\pi$
$317$	2.57274e9			0.254776			0.127388	+	0.991853i	$0.459341\pi$
$318$		0			0
$319$	−3.38847e9			−0.327221
$320$		0			0
$321$			1.15956e10i			1.09213i
$322$		0			0
$323$	2.03089e10			1.86585
$324$		0			0
$325$		−	8.35378e7i		−	0.00748772i
$326$		0			0
$327$			9.96358e8i			0.0871414i
$328$		0			0
$329$	−4.65588e9	+	1.39130e9i	−0.397392	+	0.118751i
$330$		0			0
$331$	1.53695e10			1.28041			0.640205	−	0.768204i	$-0.278849\pi$
$331$	1.53695e10			1.28041			0.640205	+	0.768204i	$0.278849\pi$
$332$		0			0
$333$	−2.92331e9			−0.237738
$334$		0			0
$335$		−	1.51244e10i		−	1.20088i
$336$		0			0
$337$	2.21413e10			1.71666			0.858329	−	0.513100i	$-0.171503\pi$
$337$	2.21413e10			1.71666			0.858329	+	0.513100i	$0.171503\pi$
$338$		0			0
$339$		−	3.13978e9i		−	0.237739i
$340$		0			0
$341$		−	3.50178e9i		−	0.258983i
$342$		0			0
$343$	−1.05894e10	−	8.91321e9i	−0.765061	−	0.643958i
$344$		0			0
$345$	3.83734e9			0.270866
$346$		0			0
$347$	2.45446e9			0.169293			0.0846465	−	0.996411i	$-0.473024\pi$
$347$	2.45446e9			0.169293			0.0846465	+	0.996411i	$0.473024\pi$
$348$		0			0
$349$		−	2.43917e10i		−	1.64415i	−0.569381	−	0.822073i	$-0.692817\pi$
$349$		−	2.43917e10i		−	1.64415i	0.569381	−	0.822073i	$-0.307183\pi$
$350$		0			0
$351$	2.02189e8			0.0133208
$352$		0			0
$353$			9.23681e9i			0.594872i	0.954742	+	0.297436i	$0.0961314\pi$
$353$			9.23681e9i			0.594872i	−0.954742	+	0.297436i	$0.903869\pi$
$354$		0			0
$355$			1.73836e10i			1.09452i
$356$		0			0
$357$	−4.20585e9	−	1.40746e10i	−0.258929	−	0.866491i
$358$		0			0
$359$	1.47081e9			0.0885481			0.0442740	−	0.999019i	$-0.485903\pi$
$359$	1.47081e9			0.0885481			0.0442740	+	0.999019i	$0.485903\pi$
$360$		0			0
$361$	−2.68323e10			−1.57990
$362$		0			0
$363$		−	1.25037e10i		−	0.720129i
$364$		0			0
$365$	1.82908e10			1.03053
$366$		0			0
$367$			2.03734e10i			1.12305i	0.827460	+	0.561525i	$0.189785\pi$
$367$			2.03734e10i			1.12305i	−0.827460	+	0.561525i	$0.810215\pi$
$368$		0			0
$369$			4.65711e9i			0.251195i
$370$		0			0
$371$	−7.82503e9	−	2.61860e10i	−0.413038	−	1.38221i
$372$		0			0
$373$	5.23051e8			0.0270215			0.0135107	−	0.999909i	$-0.495699\pi$
$373$	5.23051e8			0.0270215			0.0135107	+	0.999909i	$0.495699\pi$
$374$		0			0
$375$	1.54792e10			0.782748
$376$		0			0
$377$		−	2.96312e8i		−	0.0146685i
$378$		0			0
$379$	−2.88196e10			−1.39679			−0.698395	−	0.715712i	$-0.746102\pi$
$379$	−2.88196e10			−1.39679			−0.698395	+	0.715712i	$0.746102\pi$
$380$		0			0
$381$			8.83663e9i			0.419360i
$382$		0			0
$383$		−	1.93044e10i		−	0.897141i	−0.893747	−	0.448570i	$-0.851933\pi$
$383$		−	1.93044e10i		−	0.897141i	0.893747	−	0.448570i	$-0.148067\pi$
$384$		0			0
$385$	−3.59990e9	+	1.07574e9i	−0.163851	+	0.0489627i
$386$		0			0
$387$	−1.02010e10			−0.454776
$388$		0			0
$389$	−2.42405e10			−1.05863			−0.529314	−	0.848426i	$-0.677551\pi$
$389$	−2.42405e10			−1.05863			−0.529314	+	0.848426i	$0.677551\pi$
$390$		0			0
$391$			1.51265e10i			0.647189i
$392$		0			0
$393$	1.39343e10			0.584138
$394$		0			0
$395$		−	1.26483e9i		−	0.0519570i
$396$		0			0
$397$			5.65338e9i			0.227587i	0.993504	+	0.113793i	$0.0363001\pi$
$397$			5.65338e9i			0.227587i	−0.993504	+	0.113793i	$0.963700\pi$
$398$		0			0
$399$	9.07398e9	+	3.03655e10i	0.358019	+	1.19809i
$400$		0			0
$401$	−2.88963e9			−0.111755			−0.0558773	−	0.998438i	$-0.517796\pi$
$401$	−2.88963e9			−0.111755			−0.0558773	+	0.998438i	$0.517796\pi$
$402$		0			0
$403$	3.06221e8			0.0116095
$404$		0			0
$405$			7.57577e9i			0.281583i
$406$		0			0
$407$	4.53460e9			0.165258
$408$		0			0
$409$		−	1.05612e10i		−	0.377416i	−0.982033	−	0.188708i	$-0.939570\pi$
$409$		−	1.05612e10i		−	0.377416i	0.982033	−	0.188708i	$-0.0604300\pi$
$410$		0			0
$411$			1.05593e10i			0.370058i
$412$		0			0
$413$	−2.13977e10	+	6.39418e9i	−0.735474	+	0.219778i
$414$		0			0
$415$	2.74281e10			0.924706
$416$		0			0
$417$	−1.01327e10			−0.335106
$418$		0			0
$419$			2.55995e10i			0.830568i	0.909692	+	0.415284i	$0.136318\pi$
$419$			2.55995e10i			0.830568i	−0.909692	+	0.415284i	$0.863682\pi$
$420$		0			0
$421$	5.42252e10			1.72613			0.863063	−	0.505096i	$-0.168543\pi$
$421$	5.42252e10			1.72613			0.863063	+	0.505096i	$0.168543\pi$
$422$		0			0
$423$			5.23086e9i			0.163385i
$424$		0			0
$425$			2.31182e10i			0.708596i
$426$		0			0
$427$	−4.31150e10	+	1.28838e10i	−1.29693	+	0.387555i
$428$		0			0
$429$	−8.86340e7			−0.00261680
$430$		0			0
$431$	−4.72258e10			−1.36858			−0.684289	−	0.729211i	$-0.739888\pi$
$431$	−4.72258e10			−1.36858			−0.684289	+	0.729211i	$0.739888\pi$
$432$		0			0
$433$			2.47140e10i			0.703058i	0.936177	+	0.351529i	$0.114338\pi$
$433$			2.47140e10i			0.703058i	−0.936177	+	0.351529i	$0.885662\pi$
$434$		0			0
$435$	2.08024e10			0.580974
$436$		0			0
$437$		−	3.26349e10i		−	0.894862i
$438$		0			0
$439$			1.51087e10i			0.406788i	0.979097	+	0.203394i	$0.0651972\pi$
$439$			1.51087e10i			0.406788i	−0.979097	+	0.203394i	$0.934803\pi$
$440$		0			0
$441$	−1.24568e10	+	8.17477e9i	−0.329345	+	0.216133i
$442$		0			0
$443$	2.25425e10			0.585312			0.292656	−	0.956218i	$-0.405461\pi$
$443$	2.25425e10			0.585312			0.292656	+	0.956218i	$0.405461\pi$
$444$		0			0
$445$	−3.48245e10			−0.888064
$446$		0			0
$447$			1.91515e9i			0.0479704i
$448$		0			0
$449$	3.37233e10			0.829744			0.414872	−	0.909880i	$-0.363826\pi$
$449$	3.37233e10			0.829744			0.414872	+	0.909880i	$0.363826\pi$
$450$		0			0
$451$		−	7.22404e9i		−	0.174612i
$452$		0			0
$453$			2.71578e10i			0.644913i
$454$		0			0
$455$	−9.40706e7	−	3.14802e8i	−0.00219487	−	0.00734499i
$456$		0			0
$457$	−6.87894e10			−1.57709			−0.788545	−	0.614977i	$-0.789165\pi$
$457$	−6.87894e10			−1.57709			−0.788545	+	0.614977i	$0.789165\pi$
$458$		0			0
$459$	−5.59537e10			−1.26060
$460$		0			0
$461$			4.88554e9i			0.108170i	0.998536	+	0.0540852i	$0.0172243\pi$
$461$			4.88554e9i			0.108170i	−0.998536	+	0.0540852i	$0.982776\pi$
$462$		0			0
$463$	6.59037e10			1.43412			0.717061	−	0.697011i	$-0.245487\pi$
$463$	6.59037e10			1.43412			0.717061	+	0.697011i	$0.245487\pi$
$464$		0			0
$465$			2.14981e10i			0.459820i
$466$		0			0
$467$			3.75858e10i			0.790234i	0.918631	+	0.395117i	$0.129296\pi$
$467$			3.75858e10i			0.790234i	−0.918631	+	0.395117i	$0.870704\pi$
$468$		0			0
$469$	2.66377e10	+	8.91413e10i	0.550560	+	1.84241i
$470$		0			0
$471$	1.60343e10			0.325812
$472$		0			0
$473$	1.58236e10			0.316127
$474$		0			0
$475$		−	4.98768e10i		−	0.979770i
$476$		0			0
$477$	−2.94198e10			−0.568285
$478$		0			0
$479$		−	3.43929e10i		−	0.653322i	−0.945142	−	0.326661i	$-0.894076\pi$
$479$		−	3.43929e10i		−	0.653322i	0.945142	−	0.326661i	$-0.105924\pi$
$480$		0			0
$481$			3.96538e8i			0.00740806i
$482$		0			0
$483$	−2.26168e10	+	6.75848e9i	−0.415569	+	0.124182i
$484$		0			0
$485$	4.96001e10			0.896428
$486$		0			0
$487$	−2.94678e10			−0.523879			−0.261940	−	0.965084i	$-0.584362\pi$
$487$	−2.94678e10			−0.523879			−0.261940	+	0.965084i	$0.584362\pi$
$488$		0			0
$489$		−	7.42771e7i		−	0.00129903i
$490$		0			0
$491$	7.21979e10			1.24222			0.621110	−	0.783723i	$-0.286682\pi$
$491$	7.21979e10			1.24222			0.621110	+	0.783723i	$0.286682\pi$
$492$		0			0
$493$			8.20015e10i			1.38814i
$494$		0			0
$495$			4.04447e9i			0.0673660i
$496$		0			0
$497$	−3.06166e10	−	1.02457e11i	−0.501802	−	1.67925i
$498$		0			0
$499$	3.79818e10			0.612594			0.306297	−	0.951936i	$-0.400910\pi$
$499$	3.79818e10			0.612594			0.306297	+	0.951936i	$0.400910\pi$
$500$		0			0
$501$	4.26701e9			0.0677288
$502$		0			0
$503$			6.53658e10i			1.02113i	0.859841	+	0.510563i	$0.170563\pi$
$503$			6.53658e10i			1.02113i	−0.859841	+	0.510563i	$0.829437\pi$
$504$		0			0
$505$	1.93130e10			0.296951
$506$		0			0
$507$			5.14313e10i			0.778387i
$508$		0			0
$509$		−	1.00823e11i		−	1.50207i	−0.660264	−	0.751034i	$-0.729555\pi$
$509$		−	1.00823e11i		−	1.50207i	0.660264	−	0.751034i	$-0.270445\pi$
$510$		0			0
$511$	−1.07804e11	+	3.22145e10i	−1.58107	+	0.472464i
$512$		0			0
$513$	1.20718e11			1.74302
$514$		0			0
$515$	−3.88891e10			−0.552839
$516$		0			0
$517$		−	8.11404e9i		−	0.113573i
$518$		0			0
$519$	1.10453e11			1.52232
$520$		0			0
$521$		−	1.25916e11i		−	1.70895i	−0.519491	−	0.854476i	$-0.673878\pi$
$521$		−	1.25916e11i		−	1.70895i	0.519491	−	0.854476i	$-0.326122\pi$
$522$		0			0
$523$		−	4.72666e10i		−	0.631753i	−0.948800	−	0.315877i	$-0.897701\pi$
$523$		−	4.72666e10i		−	0.631753i	0.948800	−	0.315877i	$-0.102299\pi$
$524$		0			0
$525$	−3.45659e10	+	1.03292e10i	−0.454999	+	0.135965i
$526$		0			0
$527$	−8.47436e10			−1.09866
$528$		0			0
$529$	−5.40039e10			−0.689608
$530$		0			0
$531$			2.40402e10i			0.302385i
$532$		0			0
$533$	6.31723e8			0.00782740
$534$		0			0
$535$		−	7.17738e10i		−	0.876094i
$536$		0			0
$537$		−	2.03507e10i		−	0.244727i
$538$		0			0
$539$	1.93228e10	−	1.26806e10i	0.228936	−	0.150240i
$540$		0			0
$541$	−1.02877e11			−1.20097			−0.600483	−	0.799637i	$-0.705025\pi$
$541$	−1.02877e11			−1.20097			−0.600483	+	0.799637i	$0.705025\pi$
$542$		0			0
$543$	−1.75014e10			−0.201314
$544$		0			0
$545$		−	6.16719e9i		−	0.0699038i
$546$		0			0
$547$	4.11293e10			0.459411			0.229706	−	0.973260i	$-0.426224\pi$
$547$	4.11293e10			0.459411			0.229706	+	0.973260i	$0.426224\pi$
$548$		0			0
$549$			4.84394e10i			0.533224i
$550$		0			0
$551$		−	1.76915e11i		−	1.91937i
$552$		0			0
$553$	2.22767e9	+	7.45477e9i	0.0238205	+	0.0797138i
$554$		0			0
$555$	−2.78387e10			−0.293411
$556$		0			0
$557$	−4.40191e10			−0.457320			−0.228660	−	0.973506i	$-0.573434\pi$
$557$	−4.40191e10			−0.457320			−0.228660	+	0.973506i	$0.573434\pi$
$558$		0			0
$559$			1.38373e9i			0.0141711i
$560$		0			0
$561$	2.45286e10			0.247640
$562$		0			0
$563$		−	7.42103e10i		−	0.738636i	−0.929303	−	0.369318i	$-0.879591\pi$
$563$		−	7.42103e10i		−	0.738636i	0.929303	−	0.369318i	$-0.120409\pi$
$564$		0			0
$565$			1.94344e10i			0.190711i
$566$		0			0
$567$	−1.33427e10	−	4.46507e10i	−0.129096	−	0.432012i
$568$		0			0
$569$	1.35895e11			1.29645			0.648223	−	0.761450i	$-0.275512\pi$
$569$	1.35895e11			1.29645			0.648223	+	0.761450i	$0.275512\pi$
$570$		0			0
$571$	5.09298e9			0.0479101			0.0239551	−	0.999713i	$-0.492374\pi$
$571$	5.09298e9			0.0479101			0.0239551	+	0.999713i	$0.492374\pi$
$572$		0			0
$573$			4.30865e10i			0.399689i
$574$		0			0
$575$	3.71492e10			0.339843
$576$		0			0
$577$		−	1.48143e11i		−	1.33652i	−0.743926	−	0.668262i	$-0.767038\pi$
$577$		−	1.48143e11i		−	1.33652i	0.743926	−	0.668262i	$-0.232962\pi$
$578$		0			0
$579$		−	5.22676e10i		−	0.465070i
$580$		0			0
$581$	−1.61658e11	+	4.83075e10i	−1.41871	+	0.423945i
$582$		0			0
$583$	4.56356e10			0.395029
$584$		0			0
$585$	−3.53678e8			−0.00301984
$586$		0			0
$587$		−	1.42185e11i		−	1.19757i	−0.800908	−	0.598787i	$-0.795650\pi$
$587$		−	1.42185e11i		−	1.19757i	0.800908	−	0.598787i	$-0.204350\pi$
$588$		0			0
$589$	1.82831e11			1.51911
$590$		0			0
$591$			6.26472e10i			0.513513i
$592$		0			0
$593$		−	2.13933e11i		−	1.73005i	−0.501729	−	0.865025i	$-0.667303\pi$
$593$		−	2.13933e11i		−	1.73005i	0.501729	−	0.865025i	$-0.332697\pi$
$594$		0			0
$595$	2.60331e10	+	8.71182e10i	0.207710	+	0.695090i
$596$		0			0
$597$	7.63731e10			0.601234
$598$		0			0
$599$	−3.18702e9			−0.0247558			−0.0123779	−	0.999923i	$-0.503940\pi$
$599$	−3.18702e9			−0.0247558			−0.0123779	+	0.999923i	$0.503940\pi$
$600$		0			0
$601$			1.64772e10i			0.126295i	0.998004	+	0.0631475i	$0.0201139\pi$
$601$			1.64772e10i			0.126295i	−0.998004	+	0.0631475i	$0.979886\pi$
$602$		0			0
$603$	1.00150e11			0.757496
$604$		0			0
$605$			7.73943e10i			0.577680i
$606$		0			0
$607$			8.85174e10i			0.652040i	0.945363	+	0.326020i	$0.105708\pi$
$607$			8.85174e10i			0.652040i	−0.945363	+	0.326020i	$0.894292\pi$
$608$		0			0
$609$	−1.22607e11	+	3.66381e10i	−0.891345	+	0.266356i
$610$		0			0
$611$	7.09550e8			0.00509118
$612$		0			0
$613$	−2.18305e10			−0.154604			−0.0773021	−	0.997008i	$-0.524631\pi$
$613$	−2.18305e10			−0.154604			−0.0773021	+	0.997008i	$0.524631\pi$
$614$		0			0
$615$			4.43497e10i			0.310020i
$616$		0			0
$617$	2.42237e11			1.67148			0.835739	−	0.549127i	$-0.185040\pi$
$617$	2.42237e11			1.67148			0.835739	+	0.549127i	$0.185040\pi$
$618$		0			0
$619$			6.65872e10i			0.453553i	0.973947	+	0.226777i	$0.0728187\pi$
$619$			6.65872e10i			0.453553i	−0.973947	+	0.226777i	$0.927181\pi$
$620$		0			0
$621$			8.99133e10i			0.604585i
$622$		0			0
$623$	2.05251e11	−	6.13342e10i	1.36249	−	0.407146i
$624$		0			0
$625$	−2.73467e9			−0.0179219
$626$		0			0
$627$	−5.29195e10			−0.342409
$628$		0			0
$629$		−	1.09738e11i		−	0.701058i
$630$		0			0
$631$	−1.31642e11			−0.830379			−0.415189	−	0.909735i	$-0.636285\pi$
$631$	−1.31642e11			−0.830379			−0.415189	+	0.909735i	$0.636285\pi$
$632$		0			0
$633$		−	7.80494e10i		−	0.486133i
$634$		0			0
$635$		−	5.46964e10i		−	0.336406i
$636$		0			0
$637$	1.10888e9	+	1.68972e9i	0.00673485	+	0.0102626i
$638$		0			0
$639$	−1.15109e11			−0.690411
$640$		0			0
$641$	−1.05575e11			−0.625360			−0.312680	−	0.949859i	$-0.601227\pi$
$641$	−1.05575e11			−0.625360			−0.312680	+	0.949859i	$0.601227\pi$
$642$		0			0
$643$			2.06835e11i			1.20998i	0.796232	+	0.604991i	$0.206823\pi$
$643$			2.06835e11i			1.20998i	−0.796232	+	0.604991i	$0.793177\pi$
$644$		0			0
$645$	−9.71439e10			−0.561276
$646$		0			0
$647$		−	2.50568e10i		−	0.142991i	−0.997441	−	0.0714955i	$-0.977223\pi$
$647$		−	2.50568e10i		−	0.142991i	0.997441	−	0.0714955i	$-0.0227772\pi$
$648$		0			0
$649$		−	3.72909e10i		−	0.210196i
$650$		0			0
$651$	−3.78632e10	−	1.26707e11i	−0.210811	−	0.705467i
$652$		0			0
$653$	1.61819e11			0.889973			0.444986	−	0.895537i	$-0.353209\pi$
$653$	1.61819e11			0.889973			0.444986	+	0.895537i	$0.353209\pi$
$654$		0			0
$655$	−8.62496e10			−0.468589
$656$		0			0
$657$			1.21117e11i			0.650046i
$658$		0			0
$659$	1.02652e11			0.544283			0.272142	−	0.962257i	$-0.412268\pi$
$659$	1.02652e11			0.544283			0.272142	+	0.962257i	$0.412268\pi$
$660$		0			0
$661$			3.00191e11i			1.57250i	0.617907	+	0.786252i	$0.287981\pi$
$661$			3.00191e11i			1.57250i	−0.617907	+	0.786252i	$0.712019\pi$
$662$		0			0
$663$			2.14495e9i			0.0111010i
$664$		0			0
$665$	−5.61655e10	−	1.87954e11i	−0.287199	−	0.961094i
$666$		0			0
$667$	1.31770e11			0.665753
$668$		0			0
$669$	−2.14136e11			−1.06902
$670$		0			0
$671$		−	7.51386e10i		−	0.370658i
$672$		0			0
$673$	1.07564e11			0.524333			0.262167	−	0.965023i	$-0.415563\pi$
$673$	1.07564e11			0.524333			0.262167	+	0.965023i	$0.415563\pi$
$674$		0			0
$675$			1.37417e11i			0.661950i
$676$		0			0
$677$			2.08464e11i			0.992376i	0.868215	+	0.496188i	$0.165267\pi$
$677$			2.08464e11i			0.992376i	−0.868215	+	0.496188i	$0.834733\pi$
$678$		0			0
$679$	−2.92337e11	+	8.73577e10i	−1.37532	+	0.410981i
$680$		0			0
$681$	−7.90483e10			−0.367540
$682$		0			0
$683$	−8.80075e10			−0.404424			−0.202212	−	0.979342i	$-0.564813\pi$
$683$	−8.80075e10			−0.404424			−0.202212	+	0.979342i	$0.564813\pi$
$684$		0			0
$685$		−	6.53595e10i		−	0.296856i
$686$		0			0
$687$	4.98386e10			0.223738
$688$		0			0
$689$			3.99071e9i			0.0177081i
$690$		0			0
$691$			1.24749e11i			0.547175i	0.961847	+	0.273587i	$0.0882102\pi$
$691$			1.24749e11i			0.547175i	−0.961847	+	0.273587i	$0.911790\pi$
$692$		0			0
$693$	−7.12328e9	−	2.38376e10i	−0.0308850	−	0.103355i
$694$		0			0
$695$	6.27188e10			0.268818
$696$		0			0
$697$	−1.74823e11			−0.740742
$698$		0			0
$699$			1.82675e11i			0.765191i
$700$		0			0
$701$	6.16985e10			0.255507			0.127753	−	0.991806i	$-0.459223\pi$
$701$	6.16985e10			0.255507			0.127753	+	0.991806i	$0.459223\pi$
$702$		0			0
$703$			2.36756e11i			0.969347i
$704$		0			0
$705$			4.98135e10i			0.201646i
$706$		0			0
$707$	−1.13829e11	+	3.40148e10i	−0.455589	+	0.136142i
$708$		0			0
$709$	−2.62101e11			−1.03725			−0.518626	−	0.855002i	$-0.673556\pi$
$709$	−2.62101e11			−1.03725			−0.518626	+	0.855002i	$0.673556\pi$
$710$		0			0
$711$	8.37538e9			0.0327738
$712$		0			0
$713$			1.36176e11i			0.526919i
$714$		0			0
$715$	5.48620e8			0.00209917
$716$		0			0
$717$			2.41113e11i			0.912314i
$718$		0			0
$719$			2.34993e11i			0.879305i	0.898168	+	0.439652i	$0.144898\pi$
$719$			2.34993e11i			0.879305i	−0.898168	+	0.439652i	$0.855102\pi$
$720$		0			0
$721$	2.29208e11	−	6.84930e10i	0.848180	−	0.253458i
$722$		0			0
$723$	−3.42389e11			−1.25304
$724$		0			0
$725$	2.01388e11			0.728922
$726$		0			0
$727$		−	4.00180e11i		−	1.43258i	−0.697804	−	0.716288i	$-0.745839\pi$
$727$		−	4.00180e11i		−	1.43258i	0.697804	−	0.716288i	$-0.254161\pi$
$728$		0			0
$729$	−2.88766e11			−1.02244
$730$		0			0
$731$		−	3.82934e11i		−	1.34108i
$732$		0			0
$733$			1.72841e11i			0.598729i	0.954139	+	0.299364i	$0.0967746\pi$
$733$			1.72841e11i			0.598729i	−0.954139	+	0.299364i	$0.903225\pi$
$734$		0			0
$735$	−1.18626e11	+	7.78484e10i	−0.406471	+	0.266747i
$736$		0			0
$737$	−1.55351e11			−0.526555
$738$		0			0
$739$	−8.57210e9			−0.0287415			−0.0143708	−	0.999897i	$-0.504575\pi$
$739$	−8.57210e9			−0.0287415			−0.0143708	+	0.999897i	$0.504575\pi$
$740$		0			0
$741$		−	4.62766e9i		−	0.0153493i
$742$		0			0
$743$	3.54812e11			1.16424			0.582121	−	0.813102i	$-0.302223\pi$
$743$	3.54812e11			1.16424			0.582121	+	0.813102i	$0.302223\pi$
$744$		0			0
$745$		−	1.18543e10i		−	0.0384813i
$746$		0			0
$747$			1.81622e11i			0.583291i
$748$		0			0
$749$	1.26411e11	+	4.23026e11i	0.401658	+	1.34413i
$750$		0			0
$751$	4.80478e11			1.51047			0.755237	−	0.655451i	$-0.227522\pi$
$751$	4.80478e11			1.51047			0.755237	+	0.655451i	$0.227522\pi$
$752$		0			0
$753$	1.88103e11			0.585080
$754$		0			0
$755$		−	1.68099e11i		−	0.517342i
$756$		0			0
$757$	−1.55185e11			−0.472570			−0.236285	−	0.971684i	$-0.575930\pi$
$757$	−1.55185e11			−0.472570			−0.236285	+	0.971684i	$0.575930\pi$
$758$		0			0
$759$		−	3.94155e10i		−	0.118768i
$760$		0			0
$761$			1.40466e11i			0.418826i	0.977827	+	0.209413i	$0.0671553\pi$
$761$			1.40466e11i			0.418826i	−0.977827	+	0.209413i	$0.932845\pi$
$762$		0			0
$763$	1.08619e10	+	3.63486e10i	0.0320485	+	0.107248i
$764$		0			0
$765$	9.78767e10			0.285781
$766$		0			0
$767$	3.26098e9			0.00942252
$768$		0			0
$769$			1.78759e11i			0.511168i	0.966787	+	0.255584i	$0.0822677\pi$
$769$			1.78759e11i			0.511168i	−0.966787	+	0.255584i	$0.917732\pi$
$770$		0			0
$771$	−2.37014e11			−0.670743
$772$		0			0
$773$		−	5.28570e11i		−	1.48042i	−0.672376	−	0.740210i	$-0.734726\pi$
$773$		−	5.28570e11i		−	1.48042i	0.672376	−	0.740210i	$-0.265274\pi$
$774$		0			0
$775$			2.08122e11i			0.576914i
$776$		0			0
$777$	1.64078e11	−	4.90306e10i	0.450159	−	0.134519i
$778$		0			0
$779$	3.77174e11			1.02422
$780$		0			0
$781$	1.78556e11			0.479922
$782$		0			0
$783$			4.87424e11i			1.29676i
$784$		0			0
$785$	−9.92482e10			−0.261363
$786$		0			0
$787$			2.40815e11i			0.627746i	0.949465	+	0.313873i	$0.101627\pi$
$787$			2.40815e11i			0.627746i	−0.949465	+	0.313873i	$0.898373\pi$
$788$		0			0
$789$		−	3.43123e11i		−	0.885406i
$790$		0			0
$791$	−3.42286e10	−	1.14544e11i	−0.0874345	−	0.292594i
$792$		0			0
$793$	6.57066e9			0.0166156
$794$		0			0
$795$	−2.80165e11			−0.701367
$796$		0			0
$797$		−	6.80241e11i		−	1.68589i	−0.537999	−	0.842946i	$-0.680820\pi$
$797$		−	6.80241e11i		−	1.68589i	0.537999	−	0.842946i	$-0.319180\pi$
$798$		0			0
$799$	−1.96361e11			−0.481801
$800$		0			0
$801$		−	2.30598e11i		−	0.560178i
$802$		0			0
$803$		−	1.87875e11i		−	0.451864i
$804$		0			0
$805$	1.39992e11	−	4.18331e10i	0.333365	−	0.0996178i
$806$		0			0
$807$	1.27298e11			0.300142
$808$		0			0
$809$	6.57997e11			1.53614			0.768068	−	0.640369i	$-0.221218\pi$
$809$	6.57997e11			1.53614			0.768068	+	0.640369i	$0.221218\pi$
$810$		0			0
$811$			6.43078e10i			0.148655i	0.997234	+	0.0743276i	$0.0236811\pi$
$811$			6.43078e10i			0.148655i	−0.997234	+	0.0743276i	$0.976319\pi$
$812$		0			0
$813$	2.92580e11			0.669703
$814$		0			0
$815$			4.59755e8i			0.00104207i
$816$		0			0
$817$			8.26166e11i			1.85430i
$818$		0			0
$819$	2.08453e9	−	6.22911e8i	0.00463312	−	0.00138449i
$820$		0			0
$821$	−5.94524e11			−1.30857			−0.654285	−	0.756248i	$-0.727030\pi$
$821$	−5.94524e11			−1.30857			−0.654285	+	0.756248i	$0.727030\pi$
$822$		0			0
$823$	4.42327e11			0.964149			0.482075	−	0.876130i	$-0.339883\pi$
$823$	4.42327e11			0.964149			0.482075	+	0.876130i	$0.339883\pi$
$824$		0			0
$825$		−	6.02397e10i		−	0.130037i
$826$		0			0
$827$	−1.25503e11			−0.268307			−0.134153	−	0.990961i	$-0.542831\pi$
$827$	−1.25503e11			−0.268307			−0.134153	+	0.990961i	$0.542831\pi$
$828$		0			0
$829$			8.34051e10i			0.176593i	0.996094	+	0.0882967i	$0.0281424\pi$
$829$			8.34051e10i			0.176593i	−0.996094	+	0.0882967i	$0.971858\pi$
$830$		0			0
$831$			2.50257e11i			0.524785i
$832$		0			0
$833$	−3.06872e11	−	4.67613e11i	−0.637349	−	0.971196i
$834$		0			0
$835$	−2.64117e10			−0.0543313
$836$		0			0
$837$	−5.03724e11			−1.02634
$838$		0			0
$839$		−	9.34541e11i		−	1.88604i	−0.332738	−	0.943019i	$-0.607972\pi$
$839$		−	9.34541e11i		−	1.88604i	0.332738	−	0.943019i	$-0.392028\pi$
$840$		0			0
$841$	2.14085e11			0.427960
$842$		0			0
$843$		−	6.71586e11i		−	1.32982i
$844$		0			0
$845$		−	3.18346e11i		−	0.624414i
$846$		0			0
$847$	−1.36310e11	−	4.56152e11i	−0.264846	−	0.886291i
$848$		0			0
$849$	−2.37353e11			−0.456841
$850$		0			0
$851$	−1.76340e11			−0.336227
$852$		0			0
$853$		−	4.03969e11i		−	0.763049i	−0.924359	−	0.381524i	$-0.875399\pi$
$853$		−	4.03969e11i		−	0.763049i	0.924359	−	0.381524i	$-0.124601\pi$
$854$		0			0
$855$	−2.11166e11			−0.395147
$856$		0			0
$857$			1.51089e11i			0.280097i	0.990145	+	0.140049i	$0.0447259\pi$
$857$			1.51089e11i			0.280097i	−0.990145	+	0.140049i	$0.955274\pi$
$858$		0			0
$859$		−	6.87985e11i		−	1.26359i	−0.775135	−	0.631795i	$-0.782318\pi$
$859$		−	6.87985e11i		−	1.26359i	0.775135	−	0.631795i	$-0.217682\pi$
$860$		0			0
$861$	−7.81104e10	−	2.61392e11i	−0.142133	−	0.475641i
$862$		0			0
$863$	−5.26400e11			−0.949015			−0.474507	−	0.880252i	$-0.657374\pi$
$863$	−5.26400e11			−0.949015			−0.474507	+	0.880252i	$0.657374\pi$
$864$		0			0
$865$	−6.83672e11			−1.22119
$866$		0			0
$867$		−	1.53711e11i		−	0.272037i
$868$		0			0
$869$	−1.29918e10			−0.0227819
$870$		0			0
$871$		−	1.35850e10i		−	0.0236041i
$872$		0			0
$873$			3.28439e11i			0.565454i
$874$		0			0
$875$	5.64703e11	−	1.68748e11i	0.963359	−	0.287876i
$876$		0			0
$877$	6.90759e11			1.16769			0.583846	−	0.811865i	$-0.301547\pi$
$877$	6.90759e11			1.16769			0.583846	+	0.811865i	$0.301547\pi$
$878$		0			0
$879$	8.43491e11			1.41294
$880$		0			0
$881$			6.90270e10i			0.114582i	0.998358	+	0.0572909i	$0.0182462\pi$
$881$			6.90270e10i			0.114582i	−0.998358	+	0.0572909i	$0.981754\pi$
$882$		0			0
$883$	−3.66101e11			−0.602224			−0.301112	−	0.953589i	$-0.597358\pi$
$883$	−3.66101e11			−0.602224			−0.301112	+	0.953589i	$0.597358\pi$
$884$		0			0
$885$			2.28935e11i			0.373198i
$886$		0			0
$887$		−	2.07121e11i		−	0.334603i	−0.985906	−	0.167301i	$-0.946495\pi$
$887$		−	2.07121e11i		−	0.334603i	0.985906	−	0.167301i	$-0.0535053\pi$
$888$		0			0
$889$	9.63334e10	+	3.22374e11i	0.154230	+	0.516122i
$890$		0			0
$891$	7.78149e10			0.123467
$892$		0			0
$893$	4.23642e11			0.666182
$894$		0			0
$895$			1.25965e11i			0.196318i
$896$		0			0
$897$	3.44677e9			0.00532406
$898$		0			0
$899$			7.38219e11i			1.13018i
$900$		0			0
$901$		−	1.10439e12i		−	1.67580i
$902$		0			0
$903$	5.72555e11	−	1.71094e11i	0.861124	−	0.257326i
$904$		0			0
$905$	1.08329e11			0.161492
$906$		0			0
$907$	9.59162e11			1.41730			0.708652	−	0.705558i	$-0.249304\pi$
$907$	9.59162e11			1.41730			0.708652	+	0.705558i	$0.249304\pi$
$908$		0			0
$909$			1.27886e11i			0.187312i
$910$		0			0
$911$	−9.11205e11			−1.32295			−0.661474	−	0.749968i	$-0.730069\pi$
$911$	−9.11205e11			−1.32295			−0.661474	+	0.749968i	$0.730069\pi$
$912$		0			0
$913$		−	2.81729e11i		−	0.405461i
$914$		0			0
$915$			4.61289e11i			0.658095i
$916$		0			0
$917$	5.08345e11	−	1.51906e11i	0.718921	−	0.214832i
$918$		0			0
$919$	−1.32026e12			−1.85096			−0.925478	−	0.378802i	$-0.876336\pi$
$919$	−1.32026e12			−1.85096			−0.925478	+	0.378802i	$0.876336\pi$
$920$		0			0
$921$	1.03128e11			0.143330
$922$		0			0
$923$			1.56143e10i			0.0215137i
$924$		0			0
$925$	−2.69506e11			−0.368130
$926$		0			0
$927$		−	2.57513e11i		−	0.348723i
$928$		0			0
$929$		−	1.91730e11i		−	0.257411i	−0.991683	−	0.128706i	$-0.958918\pi$
$929$		−	1.91730e11i		−	0.257411i	0.991683	−	0.128706i	$-0.0410823\pi$
$930$		0			0
$931$	6.62066e11	+	1.00886e12i	0.881256	+	1.34286i
$932$		0			0
$933$	6.74268e11			0.889827
$934$		0			0
$935$	−1.51825e11			−0.198654
$936$		0			0
$937$			4.83837e11i			0.627684i	0.949475	+	0.313842i	$0.101616\pi$
$937$			4.83837e11i			0.627684i	−0.949475	+	0.313842i	$0.898384\pi$
$938$		0			0
$939$	−1.87930e11			−0.241731
$940$		0			0
$941$		−	1.03826e12i		−	1.32418i	−0.749424	−	0.662090i	$-0.769669\pi$
$941$		−	1.03826e12i		−	1.32418i	0.749424	−	0.662090i	$-0.230331\pi$
$942$		0			0
$943$			2.80927e11i			0.355260i
$944$		0			0
$945$	1.54743e11	+	5.17838e11i	0.194037	+	0.649332i
$946$		0			0
$947$	6.77497e11			0.842379			0.421190	−	0.906973i	$-0.361613\pi$
$947$	6.77497e11			0.842379			0.421190	+	0.906973i	$0.361613\pi$
$948$		0			0
$949$	1.64292e10			0.0202559
$950$		0			0
$951$			1.62234e11i			0.198345i
$952$		0			0
$953$	−1.22445e12			−1.48446			−0.742231	−	0.670144i	$-0.766232\pi$
$953$	−1.22445e12			−1.48446			−0.742231	+	0.670144i	$0.766232\pi$
$954$		0			0
$955$		−	2.66693e11i		−	0.320626i
$956$		0			0
$957$		−	2.13673e11i		−	0.254743i
$958$		0			0
$959$	1.15114e11	+	3.85221e11i	0.136098	+	0.455444i
$960$		0			0
$961$	8.99861e10			0.105507
$962$		0			0
$963$	4.75267e11			0.552628
$964$		0			0
$965$			3.23522e11i			0.373074i
$966$		0			0
$967$	−8.83520e10			−0.101044			−0.0505220	−	0.998723i	$-0.516089\pi$
$967$	−8.83520e10			−0.101044			−0.0505220	+	0.998723i	$0.516089\pi$
$968$		0			0
$969$			1.28066e12i			1.45257i
$970$		0			0
$971$		−	5.71167e11i		−	0.642519i	−0.946991	−	0.321259i	$-0.895894\pi$
$971$		−	5.71167e11i		−	0.642519i	0.946991	−	0.321259i	$-0.104106\pi$
$972$		0			0
$973$	−3.69657e11	+	1.10463e11i	−0.412428	+	0.123244i
$974$		0			0
$975$	5.26780e9			0.00582922
$976$		0			0
$977$	5.59672e11			0.614265			0.307132	−	0.951667i	$-0.400631\pi$
$977$	5.59672e11			0.614265			0.307132	+	0.951667i	$0.400631\pi$
$978$		0			0
$979$			3.57701e11i			0.389394i
$980$		0			0
$981$	4.08375e10			0.0440944
$982$		0			0
$983$		−	1.08008e12i		−	1.15676i	−0.815767	−	0.578380i	$-0.803685\pi$
$983$		−	1.08008e12i		−	1.15676i	0.815767	−	0.578380i	$-0.196315\pi$
$984$		0			0
$985$		−	3.87769e11i		−	0.411935i
$986$		0			0
$987$	−8.77335e10	−	2.93595e11i	−0.0924479	−	0.309371i
$988$		0			0
$989$	−6.15344e11			−0.643181
$990$		0			0
$991$	−9.22739e11			−0.956719			−0.478360	−	0.878164i	$-0.658768\pi$
$991$	−9.22739e11			−0.956719			−0.478360	+	0.878164i	$0.658768\pi$
$992$		0			0
$993$			9.69186e11i			0.996805i
$994$		0			0
$995$	−4.72729e11			−0.482303
$996$		0			0
$997$			7.27213e11i			0.736006i	0.929825	+	0.368003i	$0.119958\pi$
$997$			7.27213e11i			0.736006i	−0.929825	+	0.368003i	$0.880042\pi$
$998$		0			0
$999$		−	6.52292e11i		−	0.654908i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.9.c.c.97.3		4
4.3	odd	2		14.9.b.a.13.3	✓	4
7.6	odd	2	inner	112.9.c.c.97.2		4
12.11	even	2		126.9.c.a.55.2		4
20.3	even	4		350.9.d.a.349.3		8
20.7	even	4		350.9.d.a.349.6		8
20.19	odd	2		350.9.b.a.251.2		4
28.3	even	6		98.9.d.a.19.2		8
28.11	odd	6		98.9.d.a.19.1		8
28.19	even	6		98.9.d.a.31.1		8
28.23	odd	6		98.9.d.a.31.2		8
28.27	even	2		14.9.b.a.13.4	yes	4
84.83	odd	2		126.9.c.a.55.1		4
140.27	odd	4		350.9.d.a.349.7		8
140.83	odd	4		350.9.d.a.349.2		8
140.139	even	2		350.9.b.a.251.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.9.b.a.13.3	✓	4	4.3	odd	2
14.9.b.a.13.4	yes	4	28.27	even	2
98.9.d.a.19.1		8	28.11	odd	6
98.9.d.a.19.2		8	28.3	even	6
98.9.d.a.31.1		8	28.19	even	6
98.9.d.a.31.2		8	28.23	odd	6
112.9.c.c.97.2		4	7.6	odd	2	inner
112.9.c.c.97.3		4	1.1	even	1	trivial
126.9.c.a.55.1		4	84.83	odd	2
126.9.c.a.55.2		4	12.11	even	2
350.9.b.a.251.1		4	140.139	even	2
350.9.b.a.251.2		4	20.19	odd	2
350.9.d.a.349.2		8	140.83	odd	4
350.9.d.a.349.3		8	20.3	even	4
350.9.d.a.349.6		8	20.7	even	4
350.9.d.a.349.7		8	140.27	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	112.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+63.0589i q^{3} -390.317i q^{5} +(687.442 + 2300.48i) q^{7} +2584.58 q^{9} +O(q^{10})\)	\(q+63.0589i q^{3} -390.317i q^{5} +(687.442 + 2300.48i) q^{7} +2584.58 q^{9} -4009.17 q^{11} -350.590i q^{13} +24613.0 q^{15} +97022.3i q^{17} -209322. i q^{19} +(-145066. + 43349.3i) q^{21} +155907. q^{23} +238277. q^{25} +576710. i q^{27} +845181. q^{29} +873444. i q^{31} -252813. i q^{33} +(897919. - 268321. i) q^{35} -1.13106e6 q^{37} +22107.8 q^{39} +1.80188e6i q^{41} -3.94686e6 q^{43} -1.00881e6i q^{45} +2.02387e6i q^{47} +(-4.81965e6 + 3.16290e6i) q^{49} -6.11812e6 q^{51} -1.13828e7 q^{53} +1.56485e6i q^{55} +1.31996e7 q^{57} +9.30141e6i q^{59} +1.87417e7i q^{61} +(1.77675e6 + 5.94578e6i) q^{63} -136841. q^{65} +3.87489e7 q^{67} +9.83134e6i q^{69} -4.45370e7 q^{71} +4.68614e7i q^{73} +1.50255e7i q^{75} +(-2.75607e6 - 9.22302e6i) q^{77} +3.24052e6 q^{79} -1.94092e7 q^{81} +7.02713e7i q^{83} +3.78695e7 q^{85} +5.32962e7i q^{87} -8.92209e7i q^{89} +(806527. - 241011. i) q^{91} -5.50784e7 q^{93} -8.17021e7 q^{95} +1.27076e8i q^{97} -1.03620e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6076 q^{7} - 13692 q^{9}+O(q^{10}) \) 4 * q + 6076 * q^7 - 13692 * q^9	\( 4 q + 6076 q^{7} - 13692 q^{9} + 13560 q^{11} + 13056 q^{15} - 413952 q^{21} + 894072 q^{23} + 1216900 q^{25} + 317064 q^{29} + 1655808 q^{35} - 2495096 q^{37} - 10228992 q^{39} - 9186568 q^{43} + 931588 q^{49} + 324096 q^{51} - 38727288 q^{53} - 30690816 q^{57} - 40780740 q^{63} - 11891712 q^{65} + 12320248 q^{67} - 62168712 q^{71} + 45208968 q^{77} - 24889736 q^{79} - 70788348 q^{81} + 89943552 q^{85} - 38158848 q^{91} - 408466944 q^{93} - 227967744 q^{95} - 224220168 q^{99}+O(q^{100}) \) 4 * q + 6076 * q^7 - 13692 * q^9 + 13560 * q^11 + 13056 * q^15 - 413952 * q^21 + 894072 * q^23 + 1216900 * q^25 + 317064 * q^29 + 1655808 * q^35 - 2495096 * q^37 - 10228992 * q^39 - 9186568 * q^43 + 931588 * q^49 + 324096 * q^51 - 38727288 * q^53 - 30690816 * q^57 - 40780740 * q^63 - 11891712 * q^65 + 12320248 * q^67 - 62168712 * q^71 + 45208968 * q^77 - 24889736 * q^79 - 70788348 * q^81 + 89943552 * q^85 - 38158848 * q^91 - 408466944 * q^93 - 227967744 * q^95 - 224220168 * q^99

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