LMFDB - Embedded newform 112.10.f.b.111.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(111,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.111"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.13
Character	$\chi$	$=$	112.111
Dual form			112.10.f.b.111.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+52.7793 q^{3} -1902.89i q^{5} +(2477.17 + 5849.55i) q^{7} -16897.3 q^{9} -63301.8i q^{11} +186091. i q^{13} -100433. i q^{15} +216886. i q^{17} -184637. q^{19} +(130744. + 308735. i) q^{21} +1.40080e6i q^{23} -1.66787e6 q^{25} -1.93069e6 q^{27} -36135.1 q^{29} +4.89648e6 q^{31} -3.34103e6i q^{33} +(1.11311e7 - 4.71380e6i) q^{35} +1.90500e7 q^{37} +9.82177e6i q^{39} -1.47387e7i q^{41} +6.75137e6i q^{43} +3.21538e7i q^{45} -3.59822e7 q^{47} +(-2.80808e7 + 2.89807e7i) q^{49} +1.14471e7i q^{51} +4.50817e7 q^{53} -1.20456e8 q^{55} -9.74500e6 q^{57} +1.05991e8 q^{59} +4.47002e6i q^{61} +(-4.18577e7 - 9.88418e7i) q^{63} +3.54112e8 q^{65} +3.11283e8i q^{67} +7.39331e7i q^{69} +4.58231e7i q^{71} +1.23135e8i q^{73} -8.80292e7 q^{75} +(3.70287e8 - 1.56810e8i) q^{77} +2.43378e8i q^{79} +2.30690e8 q^{81} +5.66057e8 q^{83} +4.12710e8 q^{85} -1.90719e6 q^{87} +2.22812e8i q^{89} +(-1.08855e9 + 4.60981e8i) q^{91} +2.58433e8 q^{93} +3.51344e8i q^{95} +1.51339e9i q^{97} +1.06963e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 155768 q^{9} + 220672 q^{21} - 9187656 q^{25} + 14881104 q^{29} + 2829456 q^{37} - 214802472 q^{49} + 327087120 q^{53} + 238245440 q^{57} - 495797952 q^{65} + 347010000 q^{77} + 1816013720 q^{81}+ \cdots + 288442240 q^{93}+O(q^{100}) $ 24 * q + 155768 * q^9 + 220672 * q^21 - 9187656 * q^25 + 14881104 * q^29 + 2829456 * q^37 - 214802472 * q^49 + 327087120 * q^53 + 238245440 * q^57 - 495797952 * q^65 + 347010000 * q^77 + 1816013720 * q^81 - 1291803264 * q^85 + 288442240 * q^93

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
$3$	52.7793			0.376199			0.188100	−	0.982150i	$-0.439767\pi$
$3$	52.7793			0.376199			0.188100	+	0.982150i	$0.439767\pi$
$4$		0			0
$5$		−	1902.89i		−	1.36160i	−0.732470	−	0.680799i	$-0.761633\pi$
$5$		−	1902.89i		−	1.36160i	0.732470	−	0.680799i	$-0.238367\pi$
$6$		0			0
$7$	2477.17	+	5849.55i	0.389956	+	0.920834i
$7$	2477.17	+	5849.55i	0.389956	+	0.920834i
$8$		0			0
$9$	−16897.3			−0.858474
$10$		0			0
$11$		−	63301.8i		−	1.30361i	−0.758385	−	0.651807i	$-0.774011\pi$
$11$		−	63301.8i		−	1.30361i	0.758385	−	0.651807i	$-0.225989\pi$
$12$		0			0
$13$			186091.i			1.80709i	0.428488	+	0.903547i	$0.359046\pi$
$13$			186091.i			1.80709i	−0.428488	+	0.903547i	$0.640954\pi$
$14$		0			0
$15$		−	100433.i		−	0.512233i
$16$		0			0
$17$			216886.i			0.629812i	0.949123	+	0.314906i	$0.101973\pi$
$17$			216886.i			0.629812i	−0.949123	+	0.314906i	$0.898027\pi$
$18$		0			0
$19$	−184637.			−0.325032			−0.162516	−	0.986706i	$-0.551961\pi$
$19$	−184637.			−0.325032			−0.162516	+	0.986706i	$0.551961\pi$
$20$		0			0
$21$	130744.	+	308735.i	0.146701	+	0.346417i
$22$		0			0
$23$			1.40080e6i			1.04376i	0.853020	+	0.521879i	$0.174769\pi$
$23$			1.40080e6i			1.04376i	−0.853020	+	0.521879i	$0.825231\pi$
$24$		0			0
$25$	−1.66787e6			−0.853951
$26$		0			0
$27$	−1.93069e6			−0.699157
$28$		0			0
$29$	−36135.1			−0.00948721			−0.00474361	−	0.999989i	$-0.501510\pi$
$29$	−36135.1			−0.00948721			−0.00474361	+	0.999989i	$0.501510\pi$
$30$		0			0
$31$	4.89648e6			0.952262			0.476131	−	0.879374i	$-0.342039\pi$
$31$	4.89648e6			0.952262			0.476131	+	0.879374i	$0.342039\pi$
$32$		0			0
$33$		−	3.34103e6i		−	0.490419i
$34$		0			0
$35$	1.11311e7	−	4.71380e6i	1.25381	−	0.530963i
$36$		0			0
$37$	1.90500e7			1.67104			0.835522	−	0.549458i	$-0.185166\pi$
$37$	1.90500e7			1.67104			0.835522	+	0.549458i	$0.185166\pi$
$38$		0			0
$39$			9.82177e6i			0.679828i
$40$		0			0
$41$		−	1.47387e7i		−	0.814578i	−0.913299	−	0.407289i	$-0.866474\pi$
$41$		−	1.47387e7i		−	0.814578i	0.913299	−	0.407289i	$-0.133526\pi$
$42$		0			0
$43$			6.75137e6i			0.301151i	0.988599	+	0.150575i	$0.0481126\pi$
$43$			6.75137e6i			0.301151i	−0.988599	+	0.150575i	$0.951887\pi$
$44$		0			0
$45$			3.21538e7i			1.16890i
$46$		0			0
$47$	−3.59822e7			−1.07559			−0.537796	−	0.843075i	$-0.680743\pi$
$47$	−3.59822e7			−1.07559			−0.537796	+	0.843075i	$0.680743\pi$
$48$		0			0
$49$	−2.80808e7	+	2.89807e7i	−0.695869	+	0.718169i
$50$		0			0
$51$			1.14471e7i			0.236935i
$52$		0			0
$53$	4.50817e7			0.784800			0.392400	−	0.919795i	$-0.371645\pi$
$53$	4.50817e7			0.784800			0.392400	+	0.919795i	$0.371645\pi$
$54$		0			0
$55$	−1.20456e8			−1.77500
$56$		0			0
$57$	−9.74500e6			−0.122277
$58$		0			0
$59$	1.05991e8			1.13876			0.569382	−	0.822073i	$-0.307183\pi$
$59$	1.05991e8			1.13876			0.569382	+	0.822073i	$0.307183\pi$
$60$		0			0
$61$			4.47002e6i			0.0413357i	0.999786	+	0.0206678i	$0.00657925\pi$
$61$			4.47002e6i			0.0413357i	−0.999786	+	0.0206678i	$0.993421\pi$
$62$		0			0
$63$	−4.18577e7	−	9.88418e7i	−0.334767	−	0.790512i
$64$		0			0
$65$	3.54112e8			2.46054
$66$		0			0
$67$			3.11283e8i			1.88720i	0.331083	+	0.943602i	$0.392586\pi$
$67$			3.11283e8i			1.88720i	−0.331083	+	0.943602i	$0.607414\pi$
$68$		0			0
$69$			7.39331e7i			0.392661i
$70$		0			0
$71$			4.58231e7i			0.214004i	0.994259	+	0.107002i	$0.0341251\pi$
$71$			4.58231e7i			0.214004i	−0.994259	+	0.107002i	$0.965875\pi$
$72$		0			0
$73$			1.23135e8i			0.507492i	0.967271	+	0.253746i	$0.0816628\pi$
$73$			1.23135e8i			0.507492i	−0.967271	+	0.253746i	$0.918337\pi$
$74$		0			0
$75$	−8.80292e7			−0.321256
$76$		0			0
$77$	3.70287e8	−	1.56810e8i	1.20041	−	0.508352i
$78$		0			0
$79$			2.43378e8i			0.703005i	0.936187	+	0.351503i	$0.114329\pi$
$79$			2.43378e8i			0.703005i	−0.936187	+	0.351503i	$0.885671\pi$
$80$		0			0
$81$	2.30690e8			0.595451
$82$		0			0
$83$	5.66057e8			1.30921			0.654604	−	0.755972i	$-0.272836\pi$
$83$	5.66057e8			1.30921			0.654604	+	0.755972i	$0.272836\pi$
$84$		0			0
$85$	4.12710e8			0.857551
$86$		0			0
$87$	−1.90719e6			−0.00356908
$88$		0			0
$89$			2.22812e8i			0.376429i	0.982128	+	0.188214i	$0.0602700\pi$
$89$			2.22812e8i			0.376429i	−0.982128	+	0.188214i	$0.939730\pi$
$90$		0			0
$91$	−1.08855e9	+	4.60981e8i	−1.66403	+	0.704687i
$92$		0			0
$93$	2.58433e8			0.358240
$94$		0			0
$95$			3.51344e8i			0.442564i
$96$		0			0
$97$			1.51339e9i			1.73571i	0.496817	+	0.867855i	$0.334502\pi$
$97$			1.51339e9i			1.73571i	−0.496817	+	0.867855i	$0.665498\pi$
$98$		0			0
$99$			1.06963e9i			1.11912i
$100$		0			0
$101$			1.45991e9i			1.39598i	0.716109	+	0.697989i	$0.245921\pi$
$101$			1.45991e9i			1.39598i	−0.716109	+	0.697989i	$0.754079\pi$
$102$		0			0
$103$	−1.39694e9			−1.22295			−0.611477	−	0.791262i	$-0.709424\pi$
$103$	−1.39694e9			−1.22295			−0.611477	+	0.791262i	$0.709424\pi$
$104$		0			0
$105$	5.87490e8	−	2.48791e8i	0.471681	−	0.199748i
$106$		0			0
$107$		−	8.12617e8i		−	0.599320i	−0.954046	−	0.299660i	$-0.903127\pi$
$107$		−	8.12617e8i		−	0.599320i	0.954046	−	0.299660i	$-0.0968733\pi$
$108$		0			0
$109$	1.37236e9			0.931211			0.465605	−	0.884992i	$-0.345837\pi$
$109$	1.37236e9			0.931211			0.465605	+	0.884992i	$0.345837\pi$
$110$		0			0
$111$	1.00545e9			0.628646
$112$		0			0
$113$	2.84484e9			1.64137			0.820683	−	0.571384i	$-0.193593\pi$
$113$	2.84484e9			1.64137			0.820683	+	0.571384i	$0.193593\pi$
$114$		0			0
$115$	2.66556e9			1.42118
$116$		0			0
$117$		−	3.14445e9i		−	1.55134i
$118$		0			0
$119$	−1.26868e9	+	5.37264e8i	−0.579952	+	0.245599i
$120$		0			0
$121$	−1.64917e9			−0.699409
$122$		0			0
$123$		−	7.77900e8i		−	0.306444i
$124$		0			0
$125$		−	5.42804e8i		−	0.198860i
$126$		0			0
$127$			1.97571e8i			0.0673917i	0.999432	+	0.0336959i	$0.0107278\pi$
$127$			1.97571e8i			0.0673917i	−0.999432	+	0.0336959i	$0.989272\pi$
$128$		0			0
$129$			3.56333e8i			0.113293i
$130$		0			0
$131$	−3.98584e9			−1.18250			−0.591248	−	0.806490i	$-0.701364\pi$
$131$	−3.98584e9			−1.18250			−0.591248	+	0.806490i	$0.701364\pi$
$132$		0			0
$133$	−4.57377e8	−	1.08004e9i	−0.126748	−	0.299301i
$134$		0			0
$135$			3.67389e9i			0.951971i
$136$		0			0
$137$	−1.47705e9			−0.358223			−0.179112	−	0.983829i	$-0.557322\pi$
$137$	−1.47705e9			−0.358223			−0.179112	+	0.983829i	$0.557322\pi$
$138$		0			0
$139$	−5.94781e9			−1.35142			−0.675710	−	0.737167i	$-0.736163\pi$
$139$	−5.94781e9			−1.35142			−0.675710	+	0.737167i	$0.736163\pi$
$140$		0			0
$141$	−1.89912e9			−0.404637
$142$		0			0
$143$	1.17799e10			2.35575
$144$		0			0
$145$			6.87612e7i			0.0129178i
$146$		0			0
$147$	−1.48209e9	+	1.52958e9i	−0.261786	+	0.270175i
$148$		0			0
$149$	−1.54686e9			−0.257107			−0.128554	−	0.991703i	$-0.541033\pi$
$149$	−1.54686e9			−0.257107			−0.128554	+	0.991703i	$0.541033\pi$
$150$		0			0
$151$		−	9.78072e9i		−	1.53100i	−0.643437	−	0.765499i	$-0.722492\pi$
$151$		−	9.78072e9i		−	1.53100i	0.643437	−	0.765499i	$-0.277508\pi$
$152$		0			0
$153$		−	3.66479e9i		−	0.540677i
$154$		0			0
$155$		−	9.31747e9i		−	1.29660i
$156$		0			0
$157$			1.20028e10i			1.57665i	0.615259	+	0.788325i	$0.289052\pi$
$157$			1.20028e10i			1.57665i	−0.615259	+	0.788325i	$0.710948\pi$
$158$		0			0
$159$	2.37938e9			0.295241
$160$		0			0
$161$	−8.19402e9	+	3.47002e9i	−0.961127	+	0.407019i
$162$		0			0
$163$		−	2.18950e9i		−	0.242941i	−0.992595	−	0.121470i	$-0.961239\pi$
$163$		−	2.18950e9i		−	0.242941i	0.992595	−	0.121470i	$-0.0387610\pi$
$164$		0			0
$165$	−6.35761e9			−0.667754
$166$		0			0
$167$	4.84297e9			0.481823			0.240911	−	0.970547i	$-0.422554\pi$
$167$	4.84297e9			0.481823			0.240911	+	0.970547i	$0.422554\pi$
$168$		0			0
$169$	−2.40255e10			−2.26559
$170$		0			0
$171$	3.11987e9			0.279032
$172$		0			0
$173$		−	2.30653e10i		−	1.95773i	−0.204517	−	0.978863i	$-0.565562\pi$
$173$		−	2.30653e10i		−	1.95773i	0.204517	−	0.978863i	$-0.434438\pi$
$174$		0			0
$175$	−4.13161e9	−	9.75630e9i	−0.333003	−	0.786347i
$176$		0			0
$177$	5.59412e9			0.428402
$178$		0			0
$179$			2.06483e10i			1.50330i	0.659563	+	0.751649i	$0.270741\pi$
$179$			2.06483e10i			1.50330i	−0.659563	+	0.751649i	$0.729259\pi$
$180$		0			0
$181$			1.04291e10i			0.722257i	0.932516	+	0.361129i	$0.117608\pi$
$181$			1.04291e10i			0.722257i	−0.932516	+	0.361129i	$0.882392\pi$
$182$		0			0
$183$			2.35925e8i			0.0155505i
$184$		0			0
$185$		−	3.62501e10i		−	2.27529i
$186$		0			0
$187$	1.37293e10			0.821031
$188$		0			0
$189$	−4.78265e9	−	1.12936e10i	−0.272640	−	0.643807i
$190$		0			0
$191$		−	1.79251e10i		−	0.974567i	−0.873244	−	0.487283i	$-0.837988\pi$
$191$		−	1.79251e10i		−	0.974567i	0.873244	−	0.487283i	$-0.162012\pi$
$192$		0			0
$193$	−9.88334e8			−0.0512738			−0.0256369	−	0.999671i	$-0.508161\pi$
$193$	−9.88334e8			−0.0512738			−0.0256369	+	0.999671i	$0.508161\pi$
$194$		0			0
$195$	1.86898e10			0.925653
$196$		0			0
$197$	7.92082e9			0.374690			0.187345	−	0.982294i	$-0.440012\pi$
$197$	7.92082e9			0.374690			0.187345	+	0.982294i	$0.440012\pi$
$198$		0			0
$199$	−3.33132e10			−1.50584			−0.752918	−	0.658115i	$-0.771354\pi$
$199$	−3.33132e10			−1.50584			−0.752918	+	0.658115i	$0.771354\pi$
$200$		0			0
$201$			1.64293e10i			0.709965i
$202$		0			0
$203$	−8.95130e7	−	2.11374e8i	−0.00369959	−	0.00873614i
$204$		0			0
$205$	−2.80462e10			−1.10913
$206$		0			0
$207$		−	2.36697e10i		−	0.896039i
$208$		0			0
$209$			1.16878e10i			0.423717i
$210$		0			0
$211$		−	1.55776e10i		−	0.541041i	−0.962714	−	0.270520i	$-0.912804\pi$
$211$		−	1.55776e10i		−	0.541041i	0.962714	−	0.270520i	$-0.0871958\pi$
$212$		0			0
$213$			2.41851e9i			0.0805082i
$214$		0			0
$215$	1.28471e10			0.410046
$216$		0			0
$217$	1.21294e10	+	2.86422e10i	0.371340	+	0.876875i
$218$		0			0
$219$			6.49900e9i			0.190918i
$220$		0			0
$221$	−4.03605e10			−1.13813
$222$		0			0
$223$	−1.85063e10			−0.501126			−0.250563	−	0.968100i	$-0.580616\pi$
$223$	−1.85063e10			−0.501126			−0.250563	+	0.968100i	$0.580616\pi$
$224$		0			0
$225$	2.81826e10			0.733095
$226$		0			0
$227$	−4.94686e10			−1.23655			−0.618277	−	0.785960i	$-0.712169\pi$
$227$	−4.94686e10			−1.23655			−0.618277	+	0.785960i	$0.712169\pi$
$228$		0			0
$229$		−	8.82275e9i		−	0.212004i	−0.994366	−	0.106002i	$-0.966195\pi$
$229$		−	8.82275e9i		−	0.212004i	0.994366	−	0.106002i	$-0.0338050\pi$
$230$		0			0
$231$	1.95435e10	−	8.27630e9i	0.451594	−	0.191242i
$232$		0			0
$233$	3.60411e10			0.801117			0.400559	−	0.916271i	$-0.368816\pi$
$233$	3.60411e10			0.801117			0.400559	+	0.916271i	$0.368816\pi$
$234$		0			0
$235$			6.84702e10i			1.46452i
$236$		0			0
$237$			1.28453e10i			0.264470i
$238$		0			0
$239$			5.42652e10i			1.07580i	0.843009	+	0.537899i	$0.180782\pi$
$239$			5.42652e10i			1.07580i	−0.843009	+	0.537899i	$0.819218\pi$
$240$		0			0
$241$		−	3.57024e10i		−	0.681744i	−0.940110	−	0.340872i	$-0.889278\pi$
$241$		−	3.57024e10i		−	0.681744i	0.940110	−	0.340872i	$-0.110722\pi$
$242$		0			0
$243$	5.01774e10			0.923165
$244$		0			0
$245$	5.51471e10	+	5.34348e10i	0.977858	+	0.947494i
$246$		0			0
$247$		−	3.43593e10i		−	0.587365i
$248$		0			0
$249$	2.98761e10			0.492523
$250$		0			0
$251$	−9.24262e10			−1.46982			−0.734909	−	0.678166i	$-0.762775\pi$
$251$	−9.24262e10			−1.46982			−0.734909	+	0.678166i	$0.762775\pi$
$252$		0			0
$253$	8.86729e10			1.36066
$254$		0			0
$255$	2.17826e10			0.322610
$256$		0			0
$257$			3.93789e10i			0.563072i	0.959551	+	0.281536i	$0.0908439\pi$
$257$			3.93789e10i			0.563072i	−0.959551	+	0.281536i	$0.909156\pi$
$258$		0			0
$259$	4.71902e10	+	1.11434e11i	0.651633	+	1.53875i
$260$		0			0
$261$	6.10588e8			0.00814452
$262$		0			0
$263$		−	4.65368e10i		−	0.599785i	−0.953973	−	0.299892i	$-0.903049\pi$
$263$		−	4.65368e10i		−	0.599785i	0.953973	−	0.299892i	$-0.0969508\pi$
$264$		0			0
$265$		−	8.57856e10i		−	1.06858i
$266$		0			0
$267$			1.17599e10i			0.141612i
$268$		0			0
$269$			1.52982e10i			0.178137i	0.996025	+	0.0890686i	$0.0283890\pi$
$269$			1.52982e10i			0.178137i	−0.996025	+	0.0890686i	$0.971611\pi$
$270$		0			0
$271$	1.13154e11			1.27441			0.637203	−	0.770696i	$-0.280091\pi$
$271$	1.13154e11			1.27441			0.637203	+	0.770696i	$0.280091\pi$
$272$		0			0
$273$	−5.74529e10	+	2.43302e10i	−0.626009	+	0.265103i
$274$		0			0
$275$			1.05579e11i			1.11322i
$276$		0			0
$277$	1.76178e11			1.79801			0.899004	−	0.437941i	$-0.144292\pi$
$277$	1.76178e11			1.79801			0.899004	+	0.437941i	$0.144292\pi$
$278$		0			0
$279$	−8.27375e10			−0.817492
$280$		0			0
$281$	2.98340e10			0.285452			0.142726	−	0.989762i	$-0.454413\pi$
$281$	2.98340e10			0.285452			0.142726	+	0.989762i	$0.454413\pi$
$282$		0			0
$283$	−8.07301e10			−0.748164			−0.374082	−	0.927396i	$-0.622042\pi$
$283$	−8.07301e10			−0.748164			−0.374082	+	0.927396i	$0.622042\pi$
$284$		0			0
$285$			1.85437e10i			0.166492i
$286$		0			0
$287$	8.62149e10	−	3.65104e10i	0.750091	−	0.317650i
$288$		0			0
$289$	7.15485e10			0.603337
$290$		0			0
$291$			7.98756e10i			0.652973i
$292$		0			0
$293$			1.65409e11i			1.31116i	0.755127	+	0.655578i	$0.227575\pi$
$293$			1.65409e11i			1.31116i	−0.755127	+	0.655578i	$0.772425\pi$
$294$		0			0
$295$		−	2.01689e11i		−	1.55054i
$296$		0			0
$297$			1.22216e11i			0.911431i
$298$		0			0
$299$	−2.60676e11			−1.88617
$300$		0			0
$301$	−3.94924e10	+	1.67243e10i	−0.277309	+	0.117435i
$302$		0			0
$303$			7.70528e10i			0.525166i
$304$		0			0
$305$	8.50596e9			0.0562826
$306$		0			0
$307$	3.33460e10			0.214250			0.107125	−	0.994246i	$-0.465836\pi$
$307$	3.33460e10			0.214250			0.107125	+	0.994246i	$0.465836\pi$
$308$		0			0
$309$	−7.37295e10			−0.460075
$310$		0			0
$311$	−3.57628e10			−0.216775			−0.108388	−	0.994109i	$-0.534569\pi$
$311$	−3.57628e10			−0.216775			−0.108388	+	0.994109i	$0.534569\pi$
$312$		0			0
$313$			2.97467e10i			0.175182i	0.996157	+	0.0875911i	$0.0279169\pi$
$313$			2.97467e10i			0.175182i	−0.996157	+	0.0875911i	$0.972083\pi$
$314$		0			0
$315$	−1.88085e11	+	7.96506e10i	−1.07636	+	0.455818i
$316$		0			0
$317$	7.86686e10			0.437557			0.218779	−	0.975775i	$-0.429793\pi$
$317$	7.86686e10			0.437557			0.218779	+	0.975775i	$0.429793\pi$
$318$		0			0
$319$			2.28742e9i			0.0123677i
$320$		0			0
$321$		−	4.28894e10i		−	0.225464i
$322$		0			0
$323$		−	4.00451e10i		−	0.204709i
$324$		0			0
$325$		−	3.10377e11i		−	1.54317i
$326$		0			0
$327$	7.24321e10			0.350321
$328$		0			0
$329$	−8.91342e10	−	2.10480e11i	−0.419433	−	0.990440i
$330$		0			0
$331$			2.37508e10i			0.108756i	0.998520	+	0.0543778i	$0.0173175\pi$
$331$			2.37508e10i			0.108756i	−0.998520	+	0.0543778i	$0.982682\pi$
$332$		0			0
$333$	−3.21895e11			−1.43455
$334$		0			0
$335$	5.92338e11			2.56961
$336$		0			0
$337$	−6.52304e10			−0.275496			−0.137748	−	0.990467i	$-0.543986\pi$
$337$	−6.52304e10			−0.275496			−0.137748	+	0.990467i	$0.543986\pi$
$338$		0			0
$339$	1.50149e11			0.617481
$340$		0			0
$341$		−	3.09956e11i		−	1.24138i
$342$		0			0
$343$	−2.39085e11	−	9.24698e10i	−0.932672	−	0.360725i
$344$		0			0
$345$	1.40687e11			0.534647
$346$		0			0
$347$			8.54588e10i			0.316427i	0.987405	+	0.158214i	$0.0505735\pi$
$347$			8.54588e10i			0.316427i	−0.987405	+	0.158214i	$0.949426\pi$
$348$		0			0
$349$		−	5.31212e10i		−	0.191670i	−0.995397	−	0.0958349i	$-0.969448\pi$
$349$		−	5.31212e10i		−	0.191670i	0.995397	−	0.0958349i	$-0.0305521\pi$
$350$		0			0
$351$		−	3.59284e11i		−	1.26344i
$352$		0			0
$353$			3.59942e11i			1.23380i	0.787040	+	0.616902i	$0.211612\pi$
$353$			3.59942e11i			1.23380i	−0.787040	+	0.616902i	$0.788388\pi$
$354$		0			0
$355$	8.71965e10			0.291388
$356$		0			0
$357$	−6.69603e10	+	2.83564e10i	−0.218178	+	0.0923941i
$358$		0			0
$359$		−	5.55243e11i		−	1.76424i	−0.471024	−	0.882121i	$-0.656115\pi$
$359$		−	5.55243e11i		−	1.76424i	0.471024	−	0.882121i	$-0.343885\pi$
$360$		0			0
$361$	−2.88597e11			−0.894354
$362$		0			0
$363$	−8.70420e10			−0.263117
$364$		0			0
$365$	2.34313e11			0.691001
$366$		0			0
$367$	4.62691e11			1.33135			0.665677	−	0.746240i	$-0.268143\pi$
$367$	4.62691e11			1.33135			0.665677	+	0.746240i	$0.268143\pi$
$368$		0			0
$369$			2.49045e11i			0.699294i
$370$		0			0
$371$	1.11675e11	+	2.63708e11i	0.306037	+	0.722670i
$372$		0			0
$373$	−7.18936e11			−1.92310			−0.961548	−	0.274639i	$-0.911442\pi$
$373$	−7.18936e11			−1.92310			−0.961548	+	0.274639i	$0.911442\pi$
$374$		0			0
$375$		−	2.86488e10i		−	0.0748112i
$376$		0			0
$377$		−	6.72443e9i		−	0.0171443i
$378$		0			0
$379$			2.29284e11i			0.570817i	0.958406	+	0.285409i	$0.0921293\pi$
$379$			2.29284e11i			0.570817i	−0.958406	+	0.285409i	$0.907871\pi$
$380$		0			0
$381$			1.04277e10i			0.0253527i
$382$		0			0
$383$	−7.80947e11			−1.85450			−0.927250	−	0.374442i	$-0.877834\pi$
$383$	−7.80947e11			−1.85450			−0.927250	+	0.374442i	$0.877834\pi$
$384$		0			0
$385$	−2.98392e11	−	7.04616e11i	−0.692171	−	1.63448i
$386$		0			0
$387$		−	1.14080e11i		−	0.258530i
$388$		0			0
$389$	−4.75140e11			−1.05208			−0.526040	−	0.850460i	$-0.676324\pi$
$389$	−4.75140e11			−1.05208			−0.526040	+	0.850460i	$0.676324\pi$
$390$		0			0
$391$	−3.03813e11			−0.657371
$392$		0			0
$393$	−2.10370e11			−0.444854
$394$		0			0
$395$	4.63121e11			0.957211
$396$		0			0
$397$			4.45947e10i			0.0901002i	0.998985	+	0.0450501i	$0.0143447\pi$
$397$			4.45947e10i			0.0901002i	−0.998985	+	0.0450501i	$0.985655\pi$
$398$		0			0
$399$	−2.41401e10	−	5.70038e10i	−0.0476827	−	0.112597i
$400$		0			0
$401$	−3.48907e11			−0.673844			−0.336922	−	0.941533i	$-0.609386\pi$
$401$	−3.48907e11			−0.673844			−0.336922	+	0.941533i	$0.609386\pi$
$402$		0			0
$403$			9.11192e11i			1.72083i
$404$		0			0
$405$		−	4.38978e11i		−	0.810766i
$406$		0			0
$407$		−	1.20590e12i		−	2.17839i
$408$		0			0
$409$			7.23232e11i			1.27798i	0.769216	+	0.638988i	$0.220647\pi$
$409$			7.23232e11i			1.27798i	−0.769216	+	0.638988i	$0.779353\pi$
$410$		0			0
$411$	−7.79579e10			−0.134763
$412$		0			0
$413$	2.62557e11	+	6.19998e11i	0.444068	+	1.04861i
$414$		0			0
$415$		−	1.07714e12i		−	1.78261i
$416$		0			0
$417$	−3.13921e11			−0.508404
$418$		0			0
$419$	−1.09706e11			−0.173887			−0.0869436	−	0.996213i	$-0.527710\pi$
$419$	−1.09706e11			−0.173887			−0.0869436	+	0.996213i	$0.527710\pi$
$420$		0			0
$421$	7.45513e11			1.15661			0.578303	−	0.815822i	$-0.303715\pi$
$421$	7.45513e11			1.15661			0.578303	+	0.815822i	$0.303715\pi$
$422$		0			0
$423$	6.08003e11			0.923367
$424$		0			0
$425$		−	3.61738e11i		−	0.537828i
$426$		0			0
$427$	−2.61476e10	+	1.10730e10i	−0.0380633	+	0.0161191i
$428$		0			0
$429$	6.21736e11			0.886233
$430$		0			0
$431$			3.52945e11i			0.492673i	0.969184	+	0.246337i	$0.0792268\pi$
$431$			3.52945e11i			0.492673i	−0.969184	+	0.246337i	$0.920773\pi$
$432$		0			0
$433$		−	8.08524e11i		−	1.10534i	−0.833399	−	0.552672i	$-0.813608\pi$
$433$		−	8.08524e11i		−	1.10534i	0.833399	−	0.552672i	$-0.186392\pi$
$434$		0			0
$435$			3.62917e9i			0.00485966i
$436$		0			0
$437$		−	2.58638e11i		−	0.339255i
$438$		0			0
$439$	4.06391e11			0.522220			0.261110	−	0.965309i	$-0.415912\pi$
$439$	4.06391e11			0.522220			0.261110	+	0.965309i	$0.415912\pi$
$440$		0			0
$441$	4.74491e11	−	4.89697e11i	0.597385	−	0.616529i
$442$		0			0
$443$			1.33247e12i			1.64377i	0.569657	+	0.821883i	$0.307076\pi$
$443$			1.33247e12i			1.64377i	−0.569657	+	0.821883i	$0.692924\pi$
$444$		0			0
$445$	4.23987e11			0.512545
$446$		0			0
$447$	−8.16425e10			−0.0967236
$448$		0			0
$449$	1.06551e12			1.23723			0.618615	−	0.785694i	$-0.287694\pi$
$449$	1.06551e12			1.23723			0.618615	+	0.785694i	$0.287694\pi$
$450$		0			0
$451$	−9.32988e11			−1.06190
$452$		0			0
$453$		−	5.16220e11i		−	0.575961i
$454$		0			0
$455$	8.77196e11	+	2.07139e12i	0.959501	+	2.26575i
$456$		0			0
$457$	1.47301e11			0.157973			0.0789867	−	0.996876i	$-0.474832\pi$
$457$	1.47301e11			0.157973			0.0789867	+	0.996876i	$0.474832\pi$
$458$		0			0
$459$		−	4.18738e11i		−	0.440337i
$460$		0			0
$461$			1.28370e12i			1.32376i	0.749609	+	0.661881i	$0.230241\pi$
$461$			1.28370e12i			1.32376i	−0.749609	+	0.661881i	$0.769759\pi$
$462$		0			0
$463$		−	1.55484e12i		−	1.57243i	−0.617954	−	0.786214i	$-0.712038\pi$
$463$		−	1.55484e12i		−	1.57243i	0.617954	−	0.786214i	$-0.287962\pi$
$464$		0			0
$465$		−	4.91770e11i		−	0.487780i
$466$		0			0
$467$	−1.39737e12			−1.35952			−0.679760	−	0.733435i	$-0.737916\pi$
$467$	−1.39737e12			−1.35952			−0.679760	+	0.733435i	$0.737916\pi$
$468$		0			0
$469$	−1.82086e12	+	7.71102e11i	−1.73780	+	0.735926i
$470$		0			0
$471$			6.33502e11i			0.593135i
$472$		0			0
$473$	4.27374e11			0.392584
$474$		0			0
$475$	3.07950e11			0.277562
$476$		0			0
$477$	−7.61761e11			−0.673730
$478$		0			0
$479$	−1.87728e12			−1.62937			−0.814685	−	0.579904i	$-0.803090\pi$
$479$	−1.87728e12			−1.62937			−0.814685	+	0.579904i	$0.803090\pi$
$480$		0			0
$481$			3.54504e12i			3.01973i
$482$		0			0
$483$	−4.32475e11	+	1.83145e11i	−0.361575	+	0.153120i
$484$		0			0
$485$	2.87981e12			2.36334
$486$		0			0
$487$			1.50389e12i			1.21153i	0.795642	+	0.605767i	$0.207134\pi$
$487$			1.50389e12i			1.21153i	−0.795642	+	0.605767i	$0.792866\pi$
$488$		0			0
$489$		−	1.15560e11i		−	0.0913942i
$490$		0			0
$491$		−	8.23056e11i		−	0.639091i	−0.947571	−	0.319545i	$-0.896470\pi$
$491$		−	8.23056e11i		−	0.639091i	0.947571	−	0.319545i	$-0.103530\pi$
$492$		0			0
$493$		−	7.83719e9i		−	0.00597516i
$494$		0			0
$495$	2.03539e12			1.52379
$496$		0			0
$497$	−2.68045e11	+	1.13512e11i	−0.197062	+	0.0834522i
$498$		0			0
$499$		−	5.33140e11i		−	0.384936i	−0.981303	−	0.192468i	$-0.938351\pi$
$499$		−	5.33140e11i		−	0.384936i	0.981303	−	0.192468i	$-0.0616492\pi$
$500$		0			0
$501$	2.55609e11			0.181262
$502$		0			0
$503$	1.52196e12			1.06010			0.530049	−	0.847967i	$-0.322173\pi$
$503$	1.52196e12			1.06010			0.530049	+	0.847967i	$0.322173\pi$
$504$		0			0
$505$	2.77804e12			1.90076
$506$		0			0
$507$	−1.26805e12			−0.852315
$508$		0			0
$509$		−	6.36533e11i		−	0.420330i	−0.977666	−	0.210165i	$-0.932600\pi$
$509$		−	6.36533e11i		−	0.420330i	0.977666	−	0.210165i	$-0.0674002\pi$
$510$		0			0
$511$	−7.20286e11	+	3.05028e11i	−0.467316	+	0.197900i
$512$		0			0
$513$	3.56475e11			0.227249
$514$		0			0
$515$			2.65822e12i			1.66517i
$516$		0			0
$517$			2.27774e12i			1.40216i
$518$		0			0
$519$		−	1.21737e12i		−	0.736496i
$520$		0			0
$521$		−	1.60777e12i		−	0.955991i	−0.878362	−	0.477996i	$-0.841363\pi$
$521$		−	1.60777e12i		−	0.955991i	0.878362	−	0.477996i	$-0.158637\pi$
$522$		0			0
$523$	2.47269e12			1.44515			0.722575	−	0.691293i	$-0.242959\pi$
$523$	2.47269e12			1.44515			0.722575	+	0.691293i	$0.242959\pi$
$524$		0			0
$525$	−2.18064e11	−	5.14931e11i	−0.125276	−	0.295823i
$526$		0			0
$527$			1.06198e12i			0.599746i
$528$		0			0
$529$	−1.61076e11			−0.0894296
$530$		0			0
$531$	−1.79096e12			−0.977599
$532$		0			0
$533$	2.74275e12			1.47202
$534$		0			0
$535$	−1.54632e12			−0.816034
$536$		0			0
$537$			1.08980e12i			0.565540i
$538$		0			0
$539$	1.83453e12	+	1.77757e12i	0.936215	+	0.907144i
$540$		0			0
$541$	−1.79418e12			−0.900491			−0.450246	−	0.892905i	$-0.648664\pi$
$541$	−1.79418e12			−0.900491			−0.450246	+	0.892905i	$0.648664\pi$
$542$		0			0
$543$			5.50439e11i			0.271713i
$544$		0			0
$545$		−	2.61145e12i		−	1.26794i
$546$		0			0
$547$		−	3.16569e12i		−	1.51191i	−0.654623	−	0.755955i	$-0.727173\pi$
$547$		−	3.16569e12i		−	1.51191i	0.654623	−	0.755955i	$-0.272827\pi$
$548$		0			0
$549$		−	7.55314e10i		−	0.0354856i
$550$		0			0
$551$	6.67187e9			0.00308365
$552$		0			0
$553$	−1.42365e12	+	6.02889e11i	−0.647351	+	0.274141i
$554$		0			0
$555$		−	1.91326e12i		−	0.855963i
$556$		0			0
$557$	−1.44434e12			−0.635800			−0.317900	−	0.948124i	$-0.602978\pi$
$557$	−1.44434e12			−0.635800			−0.317900	+	0.948124i	$0.602978\pi$
$558$		0			0
$559$	−1.25637e12			−0.544208
$560$		0			0
$561$	7.24621e11			0.308872
$562$		0			0
$563$	3.33261e11			0.139797			0.0698984	−	0.997554i	$-0.477732\pi$
$563$	3.33261e11			0.139797			0.0698984	+	0.997554i	$0.477732\pi$
$564$		0			0
$565$		−	5.41343e12i		−	2.23488i
$566$		0			0
$567$	5.71460e11	+	1.34943e12i	0.232200	+	0.548312i
$568$		0			0
$569$	6.79467e11			0.271746			0.135873	−	0.990726i	$-0.456616\pi$
$569$	6.79467e11			0.271746			0.135873	+	0.990726i	$0.456616\pi$
$570$		0			0
$571$			1.53580e12i			0.604607i	0.953212	+	0.302303i	$0.0977556\pi$
$571$			1.53580e12i			0.604607i	−0.953212	+	0.302303i	$0.902244\pi$
$572$		0			0
$573$		−	9.46075e11i		−	0.366632i
$574$		0			0
$575$		−	2.33635e12i		−	0.891318i
$576$		0			0
$577$		−	1.06430e12i		−	0.399735i	−0.979823	−	0.199868i	$-0.935949\pi$
$577$		−	1.06430e12i		−	0.399735i	0.979823	−	0.199868i	$-0.0640512\pi$
$578$		0			0
$579$	−5.21636e10			−0.0192892
$580$		0			0
$581$	1.40222e12	+	3.31117e12i	0.510533	+	1.20556i
$582$		0			0
$583$		−	2.85375e12i		−	1.02308i
$584$		0			0
$585$	−5.98355e12			−2.11231
$586$		0			0
$587$	−2.36920e12			−0.823626			−0.411813	−	0.911268i	$-0.635104\pi$
$587$	−2.36920e12			−0.823626			−0.411813	+	0.911268i	$0.635104\pi$
$588$		0			0
$589$	−9.04070e11			−0.309516
$590$		0			0
$591$	4.18055e11			0.140958
$592$		0			0
$593$			3.86798e12i			1.28451i	0.766490	+	0.642256i	$0.222001\pi$
$593$			3.86798e12i			1.28451i	−0.766490	+	0.642256i	$0.777999\pi$
$594$		0			0
$595$	1.02235e12	+	2.41417e12i	0.334407	+	0.789662i
$596$		0			0
$597$	−1.75825e12			−0.566495
$598$		0			0
$599$			3.83243e12i			1.21634i	0.793808	+	0.608168i	$0.208095\pi$
$599$			3.83243e12i			1.21634i	−0.793808	+	0.608168i	$0.791905\pi$
$600$		0			0
$601$		−	1.13398e12i		−	0.354545i	−0.984162	−	0.177272i	$-0.943273\pi$
$601$		−	1.13398e12i		−	0.354545i	0.984162	−	0.177272i	$-0.0567273\pi$
$602$		0			0
$603$		−	5.25985e12i		−	1.62011i
$604$		0			0
$605$			3.13819e12i			0.952314i
$606$		0			0
$607$	−2.01649e12			−0.602903			−0.301452	−	0.953481i	$-0.597471\pi$
$607$	−2.01649e12			−0.602903			−0.301452	+	0.953481i	$0.597471\pi$
$608$		0			0
$609$	−4.72444e9	−	1.11562e10i	−0.00139178	−	0.00328653i
$610$		0			0
$611$		−	6.69597e12i		−	1.94370i
$612$		0			0
$613$	−4.03272e12			−1.15352			−0.576761	−	0.816913i	$-0.695683\pi$
$613$	−4.03272e12			−1.15352			−0.576761	+	0.816913i	$0.695683\pi$
$614$		0			0
$615$	−1.48026e12			−0.417254
$616$		0			0
$617$	1.73007e12			0.480596			0.240298	−	0.970699i	$-0.422755\pi$
$617$	1.73007e12			0.480596			0.240298	+	0.970699i	$0.422755\pi$
$618$		0			0
$619$	6.04924e12			1.65612			0.828062	−	0.560637i	$-0.189444\pi$
$619$	6.04924e12			1.65612			0.828062	+	0.560637i	$0.189444\pi$
$620$		0			0
$621$		−	2.70450e12i		−	0.729750i
$622$		0			0
$623$	−1.30335e12	+	5.51944e11i	−0.346628	+	0.146791i
$624$		0			0
$625$	−4.29046e12			−1.12472
$626$		0			0
$627$			6.16876e11i			0.159402i
$628$		0			0
$629$			4.13168e12i			1.05244i
$630$		0			0
$631$		−	3.02086e12i		−	0.758575i	−0.925279	−	0.379287i	$-0.876169\pi$
$631$		−	3.02086e12i		−	0.758575i	0.925279	−	0.379287i	$-0.123831\pi$
$632$		0			0
$633$		−	8.22176e11i		−	0.203539i
$634$		0			0
$635$	3.75956e11			0.0917605
$636$		0			0
$637$	−5.39306e12	−	5.22560e12i	−1.29780	−	1.25750i
$638$		0			0
$639$		−	7.74289e11i		−	0.183717i
$640$		0			0
$641$	−2.03172e12			−0.475338			−0.237669	−	0.971346i	$-0.576383\pi$
$641$	−2.03172e12			−0.475338			−0.237669	+	0.971346i	$0.576383\pi$
$642$		0			0
$643$	7.02493e12			1.62066			0.810332	−	0.585971i	$-0.199287\pi$
$643$	7.02493e12			1.62066			0.810332	+	0.585971i	$0.199287\pi$
$644$		0			0
$645$	6.78062e11			0.154259
$646$		0			0
$647$	3.02288e12			0.678190			0.339095	−	0.940752i	$-0.389879\pi$
$647$	3.02288e12			0.678190			0.339095	+	0.940752i	$0.389879\pi$
$648$		0			0
$649$		−	6.70940e12i		−	1.48451i
$650$		0			0
$651$	6.40184e11	+	1.51172e12i	0.139698	+	0.329880i
$652$		0			0
$653$	4.82107e12			1.03761			0.518805	−	0.854893i	$-0.326377\pi$
$653$	4.82107e12			1.03761			0.518805	+	0.854893i	$0.326377\pi$
$654$		0			0
$655$			7.58463e12i			1.61008i
$656$		0			0
$657$		−	2.08066e12i		−	0.435669i
$658$		0			0
$659$		−	8.39092e12i		−	1.73311i	−0.499085	−	0.866553i	$-0.666330\pi$
$659$		−	8.39092e12i		−	1.73311i	0.499085	−	0.866553i	$-0.333670\pi$
$660$		0			0
$661$		−	5.46093e12i		−	1.11265i	−0.830964	−	0.556327i	$-0.812210\pi$
$661$		−	5.46093e12i		−	1.11265i	0.830964	−	0.556327i	$-0.187790\pi$
$662$		0			0
$663$	−2.13020e12			−0.428164
$664$		0			0
$665$	−2.05520e12	+	8.70340e11i	−0.407528	+	0.172580i
$666$		0			0
$667$		−	5.06179e10i		−	0.00990235i
$668$		0			0
$669$	−9.76748e11			−0.188523
$670$		0			0
$671$	2.82960e11			0.0538858
$672$		0			0
$673$	1.43102e12			0.268892			0.134446	−	0.990921i	$-0.457075\pi$
$673$	1.43102e12			0.268892			0.134446	+	0.990921i	$0.457075\pi$
$674$		0			0
$675$	3.22014e12			0.597046
$676$		0			0
$677$		−	3.55512e12i		−	0.650436i	−0.945639	−	0.325218i	$-0.894562\pi$
$677$		−	3.55512e12i		−	0.650436i	0.945639	−	0.325218i	$-0.105438\pi$
$678$		0			0
$679$	−8.85263e12	+	3.74892e12i	−1.59830	+	0.676850i
$680$		0			0
$681$	−2.61092e12			−0.465191
$682$		0			0
$683$			8.02521e11i			0.141112i	0.997508	+	0.0705559i	$0.0224773\pi$
$683$			8.02521e11i			0.141112i	−0.997508	+	0.0705559i	$0.977523\pi$
$684$		0			0
$685$			2.81067e12i			0.487756i
$686$		0			0
$687$		−	4.65659e11i		−	0.0797559i
$688$		0			0
$689$			8.38931e12i			1.41821i
$690$		0			0
$691$	1.81053e12			0.302103			0.151051	−	0.988526i	$-0.451734\pi$
$691$	1.81053e12			0.302103			0.151051	+	0.988526i	$0.451734\pi$
$692$		0			0
$693$	−6.25686e12	+	2.64967e12i	−1.03052	+	0.436407i
$694$		0			0
$695$			1.13180e13i			1.84009i
$696$		0			0
$697$	3.19662e12			0.513031
$698$		0			0
$699$	1.90222e12			0.301380
$700$		0			0
$701$	9.88963e12			1.54685			0.773426	−	0.633886i	$-0.218541\pi$
$701$	9.88963e12			1.54685			0.773426	+	0.633886i	$0.218541\pi$
$702$		0			0
$703$	−3.51733e12			−0.543143
$704$		0			0
$705$			3.61381e12i			0.550953i
$706$		0			0
$707$	−8.53978e12	+	3.61644e12i	−1.28546	+	0.544370i
$708$		0			0
$709$	4.62982e12			0.688107			0.344054	−	0.938950i	$-0.388200\pi$
$709$	4.62982e12			0.688107			0.344054	+	0.938950i	$0.388200\pi$
$710$		0			0
$711$		−	4.11243e12i		−	0.603512i
$712$		0			0
$713$			6.85897e12i			0.993930i
$714$		0			0
$715$		−	2.24159e13i		−	3.20759i
$716$		0			0
$717$			2.86408e12i			0.404714i
$718$		0			0
$719$	2.39023e11			0.0333549			0.0166775	−	0.999861i	$-0.494691\pi$
$719$	2.39023e11			0.0333549			0.0166775	+	0.999861i	$0.494691\pi$
$720$		0			0
$721$	−3.46046e12	−	8.17146e12i	−0.476898	−	1.12614i
$722$		0			0
$723$		−	1.88435e12i		−	0.256472i
$724$		0			0
$725$	6.02688e10			0.00810161
$726$		0			0
$727$	9.72932e12			1.29175			0.645874	−	0.763444i	$-0.276493\pi$
$727$	9.72932e12			1.29175			0.645874	+	0.763444i	$0.276493\pi$
$728$		0			0
$729$	−1.89235e12			−0.248157
$730$		0			0
$731$	−1.46427e12			−0.189668
$732$		0			0
$733$			7.61754e12i			0.974645i	0.873222	+	0.487323i	$0.162026\pi$
$733$			7.61754e12i			0.974645i	−0.873222	+	0.487323i	$0.837974\pi$
$734$		0			0
$735$	2.91063e12	+	2.82025e12i	0.367870	+	0.356447i
$736$		0			0
$737$	1.97048e13			2.46018
$738$		0			0
$739$			5.41862e12i			0.668327i	0.942515	+	0.334163i	$0.108454\pi$
$739$			5.41862e12i			0.668327i	−0.942515	+	0.334163i	$0.891546\pi$
$740$		0			0
$741$		−	1.81346e12i		−	0.220966i
$742$		0			0
$743$			9.50797e12i			1.14456i	0.820059	+	0.572280i	$0.193941\pi$
$743$			9.50797e12i			1.14456i	−0.820059	+	0.572280i	$0.806059\pi$
$744$		0			0
$745$			2.94352e12i			0.350077i
$746$		0			0
$747$	−9.56485e12			−1.12392
$748$		0			0
$749$	4.75344e12	−	2.01299e12i	0.551874	−	0.233708i
$750$		0			0
$751$		−	5.78041e12i		−	0.663100i	−0.943438	−	0.331550i	$-0.892428\pi$
$751$		−	5.78041e12i		−	0.663100i	0.943438	−	0.331550i	$-0.107572\pi$
$752$		0			0
$753$	−4.87819e12			−0.552945
$754$		0			0
$755$	−1.86117e13			−2.08461
$756$		0			0
$757$	4.62838e12			0.512268			0.256134	−	0.966641i	$-0.417551\pi$
$757$	4.62838e12			0.512268			0.256134	+	0.966641i	$0.417551\pi$
$758$		0			0
$759$	4.68010e12			0.511878
$760$		0			0
$761$		−	3.61747e12i		−	0.390997i	−0.980704	−	0.195499i	$-0.937367\pi$
$761$		−	3.61747e12i		−	0.390997i	0.980704	−	0.195499i	$-0.0626325\pi$
$762$		0			0
$763$	3.39957e12	+	8.02767e12i	0.363131	+	0.857490i
$764$		0			0
$765$	−6.97370e12			−0.736185
$766$		0			0
$767$			1.97239e13i			2.05785i
$768$		0			0
$769$			2.43237e12i			0.250820i	0.992105	+	0.125410i	$0.0400246\pi$
$769$			2.43237e12i			0.250820i	−0.992105	+	0.125410i	$0.959975\pi$
$770$		0			0
$771$			2.07839e12i			0.211828i
$772$		0			0
$773$			1.39891e12i			0.140923i	0.997514	+	0.0704616i	$0.0224472\pi$
$773$			1.39891e12i			0.140923i	−0.997514	+	0.0704616i	$0.977553\pi$
$774$		0			0
$775$	−8.16671e12			−0.813185
$776$		0			0
$777$	2.49067e12	+	5.88141e12i	0.245144	+	0.578878i
$778$		0			0
$779$			2.72131e12i			0.264764i
$780$		0			0
$781$	2.90069e12			0.278979
$782$		0			0
$783$	6.97656e10			0.00663305
$784$		0			0
$785$	2.28401e13			2.14677
$786$		0			0
$787$	4.14624e12			0.385273			0.192636	−	0.981270i	$-0.438296\pi$
$787$	4.14624e12			0.385273			0.192636	+	0.981270i	$0.438296\pi$
$788$		0			0
$789$		−	2.45618e12i		−	0.225639i
$790$		0			0
$791$	7.04717e12	+	1.66411e13i	0.640060	+	1.51142i
$792$		0			0
$793$	−8.31832e11			−0.0746975
$794$		0			0
$795$		−	4.52771e12i		−	0.402000i
$796$		0			0
$797$		−	1.75123e13i		−	1.53738i	−0.639621	−	0.768690i	$-0.720909\pi$
$797$		−	1.75123e13i		−	1.53738i	0.639621	−	0.768690i	$-0.279091\pi$
$798$		0			0
$799$		−	7.80402e12i		−	0.677420i
$800$		0			0
$801$		−	3.76493e12i		−	0.323154i
$802$		0			0
$803$	7.79468e12			0.661574
$804$		0			0
$805$	6.60307e12	+	1.55923e13i	0.554197	+	1.30867i
$806$		0			0
$807$			8.07428e11i			0.0670151i
$808$		0			0
$809$	1.90614e13			1.56454			0.782270	−	0.622940i	$-0.214062\pi$
$809$	1.90614e13			1.56454			0.782270	+	0.622940i	$0.214062\pi$
$810$		0			0
$811$	5.76248e12			0.467752			0.233876	−	0.972266i	$-0.424859\pi$
$811$	5.76248e12			0.467752			0.233876	+	0.972266i	$0.424859\pi$
$812$		0			0
$813$	5.97219e12			0.479431
$814$		0			0
$815$	−4.16638e12			−0.330788
$816$		0			0
$817$		−	1.24655e12i		−	0.0978837i
$818$		0			0
$819$	1.83936e13	−	7.78935e12i	1.42853	−	0.604956i
$820$		0			0
$821$	−9.02105e12			−0.692968			−0.346484	−	0.938056i	$-0.612624\pi$
$821$	−9.02105e12			−0.692968			−0.346484	+	0.938056i	$0.612624\pi$
$822$		0			0
$823$		−	2.95463e11i		−	0.0224493i	−0.999937	−	0.0112247i	$-0.996427\pi$
$823$		−	2.95463e11i		−	0.0224493i	0.999937	−	0.0112247i	$-0.00357300\pi$
$824$		0			0
$825$			5.57241e12i			0.418794i
$826$		0			0
$827$		−	7.67228e12i		−	0.570361i	−0.958474	−	0.285180i	$-0.907946\pi$
$827$		−	7.67228e12i		−	0.570361i	0.958474	−	0.285180i	$-0.0920535\pi$
$828$		0			0
$829$		−	6.85909e12i		−	0.504396i	−0.967676	−	0.252198i	$-0.918847\pi$
$829$		−	6.85909e12i		−	0.504396i	0.967676	−	0.252198i	$-0.0811534\pi$
$830$		0			0
$831$	9.29853e12			0.676410
$832$		0			0
$833$	−6.28550e12	−	6.09033e12i	−0.452311	−	0.438266i
$834$		0			0
$835$		−	9.21564e12i		−	0.656049i
$836$		0			0
$837$	−9.45357e12			−0.665781
$838$		0			0
$839$	−2.36789e13			−1.64981			−0.824904	−	0.565272i	$-0.808771\pi$
$839$	−2.36789e13			−1.64981			−0.824904	+	0.565272i	$0.808771\pi$
$840$		0			0
$841$	−1.45058e13			−0.999910
$842$		0			0
$843$	1.57462e12			0.107387
$844$		0			0
$845$			4.57179e13i			3.08483i
$846$		0			0
$847$	−4.08528e12	−	9.64689e12i	−0.272738	−	0.644039i
$848$		0			0
$849$	−4.26088e12			−0.281459
$850$		0			0
$851$			2.66852e13i			1.74416i
$852$		0			0
$853$			1.80289e12i			0.116600i	0.998299	+	0.0583001i	$0.0185680\pi$
$853$			1.80289e12i			0.116600i	−0.998299	+	0.0583001i	$0.981432\pi$
$854$		0			0
$855$		−	5.93677e12i		−	0.379929i
$856$		0			0
$857$			1.11670e13i			0.707171i	0.935402	+	0.353586i	$0.115038\pi$
$857$			1.11670e13i			0.707171i	−0.935402	+	0.353586i	$0.884962\pi$
$858$		0			0
$859$	2.58811e12			0.162186			0.0810931	−	0.996707i	$-0.474159\pi$
$859$	2.58811e12			0.162186			0.0810931	+	0.996707i	$0.474159\pi$
$860$		0			0
$861$	4.55037e12	−	1.92700e12i	0.282184	−	0.119500i
$862$		0			0
$863$		−	1.36443e13i		−	0.837340i	−0.908139	−	0.418670i	$-0.862496\pi$
$863$		−	1.36443e13i		−	0.837340i	0.908139	−	0.418670i	$-0.137504\pi$
$864$		0			0
$865$	−4.38908e13			−2.66564
$866$		0			0
$867$	3.77628e12			0.226975
$868$		0			0
$869$	1.54062e13			0.916447
$870$		0			0
$871$	−5.79270e13			−3.41036
$872$		0			0
$873$		−	2.55722e13i		−	1.49006i
$874$		0			0
$875$	3.17516e12	−	1.34462e12i	0.183117	−	0.0775468i
$876$		0			0
$877$	−6.75602e12			−0.385649			−0.192825	−	0.981233i	$-0.561765\pi$
$877$	−6.75602e12			−0.385649			−0.192825	+	0.981233i	$0.561765\pi$
$878$		0			0
$879$			8.73017e12i			0.493256i
$880$		0			0
$881$		−	1.17764e13i		−	0.658599i	−0.944225	−	0.329300i	$-0.893187\pi$
$881$		−	1.17764e13i		−	0.658599i	0.944225	−	0.329300i	$-0.106813\pi$
$882$		0			0
$883$			2.52167e13i			1.39594i	0.716129	+	0.697968i	$0.245912\pi$
$883$			2.52167e13i			1.39594i	−0.716129	+	0.697968i	$0.754088\pi$
$884$		0			0
$885$		−	1.06450e13i		−	0.583312i
$886$		0			0
$887$	1.19059e13			0.645810			0.322905	−	0.946431i	$-0.395341\pi$
$887$	1.19059e13			0.645810			0.322905	+	0.946431i	$0.395341\pi$
$888$		0			0
$889$	−1.15570e12	+	4.89418e11i	−0.0620566	+	0.0262798i
$890$		0			0
$891$		−	1.46031e13i		−	0.776239i
$892$		0			0
$893$	6.64363e12			0.349602
$894$		0			0
$895$	3.92915e13			2.04689
$896$		0			0
$897$	−1.37583e13			−0.709576
$898$		0			0
$899$	−1.76935e11			−0.00903431
$900$		0			0
$901$			9.77758e12i			0.494276i
$902$		0			0
$903$	−2.08438e12	+	8.82698e11i	−0.104324	+	0.0441791i
$904$		0			0
$905$	1.98454e13			0.983424
$906$		0			0
$907$		−	3.87772e13i		−	1.90258i	−0.308290	−	0.951292i	$-0.599757\pi$
$907$		−	3.87772e13i		−	1.90258i	0.308290	−	0.951292i	$-0.400243\pi$
$908$		0			0
$909$		−	2.46685e13i		−	1.19841i
$910$		0			0
$911$		−	2.82572e13i		−	1.35924i	−0.733563	−	0.679621i	$-0.762144\pi$
$911$		−	2.82572e13i		−	1.35924i	0.733563	−	0.679621i	$-0.237856\pi$
$912$		0			0
$913$		−	3.58324e13i		−	1.70670i
$914$		0			0
$915$	4.48939e11			0.0211735
$916$		0			0
$917$	−9.87363e12	−	2.33154e13i	−0.461121	−	1.08888i
$918$		0			0
$919$			1.46263e13i			0.676415i	0.941072	+	0.338207i	$0.109821\pi$
$919$			1.46263e13i			0.676415i	−0.941072	+	0.338207i	$0.890179\pi$
$920$		0			0
$921$	1.75998e12			0.0806007
$922$		0			0
$923$	−8.52728e12			−0.386726
$924$		0			0
$925$	−3.17730e13			−1.42699
$926$		0			0
$927$	2.36046e13			1.04987
$928$		0			0
$929$			4.65419e12i			0.205009i	0.994733	+	0.102505i	$0.0326856\pi$
$929$			4.65419e12i			0.205009i	−0.994733	+	0.102505i	$0.967314\pi$
$930$		0			0
$931$	5.18475e12	−	5.35090e12i	0.226180	−	0.233428i
$932$		0			0
$933$	−1.88753e12			−0.0815507
$934$		0			0
$935$		−	2.61253e13i		−	1.11792i
$936$		0			0
$937$			2.81507e13i			1.19305i	0.802593	+	0.596527i	$0.203453\pi$
$937$			2.81507e13i			1.19305i	−0.802593	+	0.596527i	$0.796547\pi$
$938$		0			0
$939$			1.57001e12i			0.0659034i
$940$		0			0
$941$		−	1.13137e13i		−	0.470381i	−0.971949	−	0.235191i	$-0.924429\pi$
$941$		−	1.13137e13i		−	0.470381i	0.971949	−	0.235191i	$-0.0755714\pi$
$942$		0			0
$943$	2.06460e13			0.850222
$944$		0			0
$945$	−2.14906e13	+	9.10086e12i	−0.876607	+	0.371227i
$946$		0			0
$947$			2.50689e13i			1.01289i	0.862273	+	0.506443i	$0.169040\pi$
$947$			2.50689e13i			1.01289i	−0.862273	+	0.506443i	$0.830960\pi$
$948$		0			0
$949$	−2.29144e13			−0.917087
$950$		0			0
$951$	4.15208e12			0.164609
$952$		0			0
$953$	−1.46333e13			−0.574679			−0.287340	−	0.957829i	$-0.592771\pi$
$953$	−1.46333e13			−0.574679			−0.287340	+	0.957829i	$0.592771\pi$
$954$		0			0
$955$	−3.41095e13			−1.32697
$956$		0			0
$957$			1.20728e11i			0.00465271i
$958$		0			0
$959$	−3.65892e12	−	8.64009e12i	−0.139691	−	0.329864i
$960$		0			0
$961$	−2.46410e12			−0.0931973
$962$		0			0
$963$			1.37311e13i			0.514501i
$964$		0			0
$965$			1.88069e12i			0.0698144i
$966$		0			0
$967$		−	2.57361e13i		−	0.946506i	−0.880926	−	0.473253i	$-0.843080\pi$
$967$		−	2.57361e13i		−	0.946506i	0.880926	−	0.473253i	$-0.156920\pi$
$968$		0			0
$969$		−	2.11355e12i		−	0.0770115i
$970$		0			0
$971$	9.33583e12			0.337028			0.168514	−	0.985699i	$-0.446103\pi$
$971$	9.33583e12			0.337028			0.168514	+	0.985699i	$0.446103\pi$
$972$		0			0
$973$	−1.47338e13	−	3.47920e13i	−0.526994	−	1.24443i
$974$		0			0
$975$		−	1.63815e13i		−	0.580540i
$976$		0			0
$977$	4.20622e13			1.47695			0.738477	−	0.674279i	$-0.235546\pi$
$977$	4.20622e13			1.47695			0.738477	+	0.674279i	$0.235546\pi$
$978$		0			0
$979$	1.41044e13			0.490718
$980$		0			0
$981$	−2.31892e13			−0.799420
$982$		0			0
$983$	3.78927e13			1.29439			0.647194	−	0.762325i	$-0.275942\pi$
$983$	3.78927e13			1.29439			0.647194	+	0.762325i	$0.275942\pi$
$984$		0			0
$985$		−	1.50725e13i		−	0.510177i
$986$		0			0
$987$	−4.70444e12	−	1.11090e13i	−0.157791	−	0.372603i
$988$		0			0
$989$	−9.45729e12			−0.314328
$990$		0			0
$991$		−	2.00517e13i		−	0.660421i	−0.943907	−	0.330210i	$-0.892880\pi$
$991$		−	2.00517e13i		−	0.660421i	0.943907	−	0.330210i	$-0.107120\pi$
$992$		0			0
$993$			1.25355e12i			0.0409138i
$994$		0			0
$995$			6.33914e13i			2.05034i
$996$		0			0
$997$			4.29461e13i			1.37656i	0.725444	+	0.688281i	$0.241634\pi$
$997$			4.29461e13i			1.37656i	−0.725444	+	0.688281i	$0.758366\pi$
$998$		0			0
$999$	−3.67796e13			−1.16832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.f.b.111.13	yes	24
4.3	odd	2	inner	112.10.f.b.111.11	✓	24
7.6	odd	2	inner	112.10.f.b.111.12	yes	24
28.27	even	2	inner	112.10.f.b.111.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.f.b.111.11	✓	24	4.3	odd	2	inner
112.10.f.b.111.12	yes	24	7.6	odd	2	inner
112.10.f.b.111.13	yes	24	1.1	even	1	trivial
112.10.f.b.111.14	yes	24	28.27	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+52.7793 q^{3} -1902.89i q^{5} +(2477.17 + 5849.55i) q^{7} -16897.3 q^{9} -63301.8i q^{11} +186091. i q^{13} -100433. i q^{15} +216886. i q^{17} -184637. q^{19} +(130744. + 308735. i) q^{21} +1.40080e6i q^{23} -1.66787e6 q^{25} -1.93069e6 q^{27} -36135.1 q^{29} +4.89648e6 q^{31} -3.34103e6i q^{33} +(1.11311e7 - 4.71380e6i) q^{35} +1.90500e7 q^{37} +9.82177e6i q^{39} -1.47387e7i q^{41} +6.75137e6i q^{43} +3.21538e7i q^{45} -3.59822e7 q^{47} +(-2.80808e7 + 2.89807e7i) q^{49} +1.14471e7i q^{51} +4.50817e7 q^{53} -1.20456e8 q^{55} -9.74500e6 q^{57} +1.05991e8 q^{59} +4.47002e6i q^{61} +(-4.18577e7 - 9.88418e7i) q^{63} +3.54112e8 q^{65} +3.11283e8i q^{67} +7.39331e7i q^{69} +4.58231e7i q^{71} +1.23135e8i q^{73} -8.80292e7 q^{75} +(3.70287e8 - 1.56810e8i) q^{77} +2.43378e8i q^{79} +2.30690e8 q^{81} +5.66057e8 q^{83} +4.12710e8 q^{85} -1.90719e6 q^{87} +2.22812e8i q^{89} +(-1.08855e9 + 4.60981e8i) q^{91} +2.58433e8 q^{93} +3.51344e8i q^{95} +1.51339e9i q^{97} +1.06963e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 155768 q^{9} + 220672 q^{21} - 9187656 q^{25} + 14881104 q^{29} + 2829456 q^{37} - 214802472 q^{49} + 327087120 q^{53} + 238245440 q^{57} - 495797952 q^{65} + 347010000 q^{77} + 1816013720 q^{81}+ \cdots + 288442240 q^{93}+O(q^{100}) \) 24 * q + 155768 * q^9 + 220672 * q^21 - 9187656 * q^25 + 14881104 * q^29 + 2829456 * q^37 - 214802472 * q^49 + 327087120 * q^53 + 238245440 * q^57 - 495797952 * q^65 + 347010000 * q^77 + 1816013720 * q^81 - 1291803264 * q^85 + 288442240 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more