LMFDB - Embedded newform 192.9.g.c.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,9,Mod(127,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	192.g (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:	$78.2166931317$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{1801})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 451x^{2} + 450x + 202500 $ x^4 - x^3 + 451x^2 + 450x + 202500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$-10.3595 - 17.9433i$ of defining polynomial
Character	$\chi$	$=$	192.127
Dual form			192.9.g.c.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} +1084.52 q^{5} +426.979i q^{7} -2187.00 q^{9} +O(q^{10})$	$q-46.7654i q^{3} +1084.52 q^{5} +426.979i q^{7} -2187.00 q^{9} +14232.3i q^{11} -34213.5 q^{13} -50717.8i q^{15} +20078.0 q^{17} -196118. i q^{19} +19967.8 q^{21} -347985. i q^{23} +785551. q^{25} +102276. i q^{27} -1.00247e6 q^{29} -1.63986e6i q^{31} +665580. q^{33} +463066. i q^{35} +791050. q^{37} +1.60001e6i q^{39} +1.36054e6 q^{41} +1.50116e6i q^{43} -2.37184e6 q^{45} -1.49258e6i q^{47} +5.58249e6 q^{49} -938954. i q^{51} +8.94860e6 q^{53} +1.54352e7i q^{55} -9.17155e6 q^{57} +8.50216e6i q^{59} +1.78549e7 q^{61} -933804. i q^{63} -3.71051e7 q^{65} -3.49438e7i q^{67} -1.62736e7 q^{69} -3.84799e7i q^{71} +2.11205e7 q^{73} -3.67366e7i q^{75} -6.07690e6 q^{77} -3.67300e7i q^{79} +4.78297e6 q^{81} -2.94243e7i q^{83} +2.17749e7 q^{85} +4.68808e7i q^{87} -3.70040e7 q^{89} -1.46085e7i q^{91} -7.66885e7 q^{93} -2.12694e8i q^{95} +1.26423e8 q^{97} -3.11261e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 264 q^{5} - 8748 q^{9}+O(q^{10}) $ 4 * q + 264 * q^5 - 8748 * q^9	$ 4 q + 264 q^{5} - 8748 q^{9} - 14632 q^{13} + 332904 q^{17} - 250128 q^{21} + 2604428 q^{25} - 2343576 q^{29} + 2002320 q^{33} - 4315784 q^{37} + 9035496 q^{41} - 577368 q^{45} + 3458884 q^{49} + 42186600 q^{53} + 2253744 q^{57} + 48148408 q^{61} - 125450832 q^{65} - 23514624 q^{69} - 21215480 q^{73} + 32354496 q^{77} + 19131876 q^{81} - 235297584 q^{85} - 12675576 q^{89} - 193564080 q^{93} + 263153800 q^{97}+O(q^{100}) $ 4 * q + 264 * q^5 - 8748 * q^9 - 14632 * q^13 + 332904 * q^17 - 250128 * q^21 + 2604428 * q^25 - 2343576 * q^29 + 2002320 * q^33 - 4315784 * q^37 + 9035496 * q^41 - 577368 * q^45 + 3458884 * q^49 + 42186600 * q^53 + 2253744 * q^57 + 48148408 * q^61 - 125450832 * q^65 - 23514624 * q^69 - 21215480 * q^73 + 32354496 * q^77 + 19131876 * q^81 - 235297584 * q^85 - 12675576 * q^89 - 193564080 * q^93 + 263153800 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 46.7654i	− 0.577350i
$3$	− 46.7654i	− 0.577350i
$4$	0	0
$5$	1084.52	1.73523	0.867613	−	0.497240i	$-0.165653\pi$
$5$	1084.52	1.73523	0.867613	+	0.497240i	$0.165653\pi$
$6$	0	0
$7$	426.979i	0.177834i	0.996039	+	0.0889170i	$0.0283406\pi$
$7$	426.979i	0.177834i	−0.996039	+	0.0889170i	$0.971659\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	14232.3i	0.972087i	0.873935	+	0.486043i	$0.161560\pi$
$11$	14232.3i	0.972087i	−0.873935	+	0.486043i	$0.838440\pi$
$12$	0	0
$13$	−34213.5	−1.19791	−0.598955	−	0.800783i	$-0.704417\pi$
$13$	−34213.5	−1.19791	−0.598955	+	0.800783i	$0.704417\pi$
$14$	0	0
$15$	− 50717.8i	− 1.00183i
$16$	0	0
$17$	20078.0	0.240394	0.120197	−	0.992750i	$-0.461647\pi$
$17$	20078.0	0.240394	0.120197	+	0.992750i	$0.461647\pi$
$18$	0	0
$19$	− 196118.i	− 1.50489i	−0.658657	−	0.752443i	$-0.728875\pi$
$19$	− 196118.i	− 1.50489i	0.658657	−	0.752443i	$-0.271125\pi$
$20$	0	0
$21$	19967.8	0.102672
$22$	0	0
$23$	− 347985.i	− 1.24351i	−0.783212	−	0.621754i	$-0.786420\pi$
$23$	− 347985.i	− 1.24351i	0.783212	−	0.621754i	$-0.213580\pi$
$24$	0	0
$25$	785551.	2.01101
$26$	0	0
$27$	102276.i	0.192450i
$28$	0	0
$29$	−1.00247e6	−1.41735	−0.708677	−	0.705533i	$-0.750708\pi$
$29$	−1.00247e6	−1.41735	−0.708677	+	0.705533i	$0.750708\pi$
$30$	0	0
$31$	− 1.63986e6i	− 1.77566i	−0.460175	−	0.887828i	$-0.652213\pi$
$31$	− 1.63986e6i	− 1.77566i	0.460175	−	0.887828i	$-0.347787\pi$
$32$	0	0
$33$	665580.	0.561234
$34$	0	0
$35$	463066.i	0.308582i
$36$	0	0
$37$	791050.	0.422082	0.211041	−	0.977477i	$-0.432315\pi$
$37$	791050.	0.422082	0.211041	+	0.977477i	$0.432315\pi$
$38$	0	0
$39$	1.60001e6i	0.691613i
$40$	0	0
$41$	1.36054e6	0.481478	0.240739	−	0.970590i	$-0.422610\pi$
$41$	1.36054e6	0.481478	0.240739	+	0.970590i	$0.422610\pi$
$42$	0	0
$43$	1.50116e6i	0.439091i	0.975602	+	0.219545i	$0.0704574\pi$
$43$	1.50116e6i	0.439091i	−0.975602	+	0.219545i	$0.929543\pi$
$44$	0	0
$45$	−2.37184e6	−0.578409
$46$	0	0
$47$	− 1.49258e6i	− 0.305878i	−0.988236	−	0.152939i	$-0.951126\pi$
$47$	− 1.49258e6i	− 0.305878i	0.988236	−	0.152939i	$-0.0488737\pi$
$48$	0	0
$49$	5.58249e6	0.968375
$50$	0	0
$51$	− 938954.i	− 0.138792i
$52$	0	0
$53$	8.94860e6	1.13410	0.567050	−	0.823683i	$-0.308085\pi$
$53$	8.94860e6	1.13410	0.567050	+	0.823683i	$0.308085\pi$
$54$	0	0
$55$	1.54352e7i	1.68679i
$56$	0	0
$57$	−9.17155e6	−0.868847
$58$	0	0
$59$	8.50216e6i	0.701651i	0.936441	+	0.350826i	$0.114099\pi$
$59$	8.50216e6i	0.701651i	−0.936441	+	0.350826i	$0.885901\pi$
$60$	0	0
$61$	1.78549e7	1.28955	0.644774	−	0.764374i	$-0.276952\pi$
$61$	1.78549e7	1.28955	0.644774	+	0.764374i	$0.276952\pi$
$62$	0	0
$63$	− 933804.i	− 0.0592780i
$64$	0	0
$65$	−3.71051e7	−2.07864
$66$	0	0
$67$	− 3.49438e7i	− 1.73409i	−0.498232	−	0.867044i	$-0.666017\pi$
$67$	− 3.49438e7i	− 1.73409i	0.498232	−	0.867044i	$-0.333983\pi$
$68$	0	0
$69$	−1.62736e7	−0.717940
$70$	0	0
$71$	− 3.84799e7i	− 1.51426i	−0.653263	−	0.757131i	$-0.726600\pi$
$71$	− 3.84799e7i	− 1.51426i	0.653263	−	0.757131i	$-0.273400\pi$
$72$	0	0
$73$	2.11205e7	0.743726	0.371863	−	0.928288i	$-0.378719\pi$
$73$	2.11205e7	0.743726	0.371863	+	0.928288i	$0.378719\pi$
$74$	0	0
$75$	− 3.67366e7i	− 1.16106i
$76$	0	0
$77$	−6.07690e6	−0.172870
$78$	0	0
$79$	− 3.67300e7i	− 0.943002i	−0.881865	−	0.471501i	$-0.843712\pi$
$79$	− 3.67300e7i	− 0.943002i	0.881865	−	0.471501i	$-0.156288\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 2.94243e7i	− 0.620003i	−0.950736	−	0.310002i	$-0.899670\pi$
$83$	− 2.94243e7i	− 0.620003i	0.950736	−	0.310002i	$-0.100330\pi$
$84$	0	0
$85$	2.17749e7	0.417139
$86$	0	0
$87$	4.68808e7i	0.818309i
$88$	0	0
$89$	−3.70040e7	−0.589778	−0.294889	−	0.955531i	$-0.595283\pi$
$89$	−3.70040e7	−0.589778	−0.294889	+	0.955531i	$0.595283\pi$
$90$	0	0
$91$	− 1.46085e7i	− 0.213029i
$92$	0	0
$93$	−7.66885e7	−1.02518
$94$	0	0
$95$	− 2.12694e8i	− 2.61132i
$96$	0	0
$97$	1.26423e8	1.42803	0.714017	−	0.700129i	$-0.246874\pi$
$97$	1.26423e8	1.42803	0.714017	+	0.700129i	$0.246874\pi$
$98$	0	0
$99$	− 3.11261e7i	− 0.324029i
$100$	0	0
$101$	−6.91121e7	−0.664154	−0.332077	−	0.943252i	$-0.607749\pi$
$101$	−6.91121e7	−0.664154	−0.332077	+	0.943252i	$0.607749\pi$
$102$	0	0
$103$	− 1.20199e7i	− 0.106795i	−0.998573	−	0.0533975i	$-0.982995\pi$
$103$	− 1.20199e7i	− 0.106795i	0.998573	−	0.0533975i	$-0.0170050\pi$
$104$	0	0
$105$	2.16555e7	0.178160
$106$	0	0
$107$	2.48941e8i	1.89916i	0.313524	+	0.949580i	$0.398490\pi$
$107$	2.48941e8i	1.89916i	−0.313524	+	0.949580i	$0.601510\pi$
$108$	0	0
$109$	−1.61675e8	−1.14534	−0.572672	−	0.819784i	$-0.694093\pi$
$109$	−1.61675e8	−1.14534	−0.572672	+	0.819784i	$0.694093\pi$
$110$	0	0
$111$	− 3.69938e7i	− 0.243689i
$112$	0	0
$113$	−7.05559e7	−0.432733	−0.216366	−	0.976312i	$-0.569421\pi$
$113$	−7.05559e7	−0.432733	−0.216366	+	0.976312i	$0.569421\pi$
$114$	0	0
$115$	− 3.77395e8i	− 2.15777i
$116$	0	0
$117$	7.48249e7	0.399303
$118$	0	0
$119$	8.57288e6i	0.0427503i
$120$	0	0
$121$	1.18000e7	0.0550478
$122$	0	0
$123$	− 6.36263e7i	− 0.277982i
$124$	0	0
$125$	4.28304e8	1.75433
$126$	0	0
$127$	− 9.17381e7i	− 0.352643i	−0.984333	−	0.176321i	$-0.943580\pi$
$127$	− 9.17381e7i	− 0.352643i	0.984333	−	0.176321i	$-0.0564198\pi$
$128$	0	0
$129$	7.02025e7	0.253509
$130$	0	0
$131$	− 2.58852e8i	− 0.878953i	−0.898254	−	0.439477i	$-0.855164\pi$
$131$	− 2.58852e8i	− 0.878953i	0.898254	−	0.439477i	$-0.144836\pi$
$132$	0	0
$133$	8.37384e7	0.267620
$134$	0	0
$135$	1.10920e8i	0.333944i
$136$	0	0
$137$	3.73710e8	1.06085	0.530423	−	0.847733i	$-0.322033\pi$
$137$	3.73710e8	1.06085	0.530423	+	0.847733i	$0.322033\pi$
$138$	0	0
$139$	− 1.61105e7i	− 0.0431569i	−0.999767	−	0.0215785i	$-0.993131\pi$
$139$	− 1.61105e7i	− 0.0431569i	0.999767	−	0.0215785i	$-0.00686917\pi$
$140$	0	0
$141$	−6.98013e7	−0.176598
$142$	0	0
$143$	− 4.86937e8i	− 1.16447i
$144$	0	0
$145$	−1.08719e9	−2.45943
$146$	0	0
$147$	− 2.61067e8i	− 0.559092i
$148$	0	0
$149$	5.78564e8	1.17383	0.586917	−	0.809647i	$-0.300342\pi$
$149$	5.78564e8	1.17383	0.586917	+	0.809647i	$0.300342\pi$
$150$	0	0
$151$	− 6.01470e8i	− 1.15693i	−0.815708	−	0.578463i	$-0.803653\pi$
$151$	− 6.01470e8i	− 1.15693i	0.815708	−	0.578463i	$-0.196347\pi$
$152$	0	0
$153$	−4.39105e7	−0.0801314
$154$	0	0
$155$	− 1.77845e9i	− 3.08117i
$156$	0	0
$157$	−1.28860e8	−0.212090	−0.106045	−	0.994361i	$-0.533819\pi$
$157$	−1.28860e8	−0.212090	−0.106045	+	0.994361i	$0.533819\pi$
$158$	0	0
$159$	− 4.18484e8i	− 0.654773i
$160$	0	0
$161$	1.48582e8	0.221138
$162$	0	0
$163$	− 2.84647e7i	− 0.0403233i	−0.999797	−	0.0201617i	$-0.993582\pi$
$163$	− 2.84647e7i	− 0.0403233i	0.999797	−	0.0201617i	$-0.00641809\pi$
$164$	0	0
$165$	7.21832e8	0.973869
$166$	0	0
$167$	− 178227.i	0 0.000229144i	−1.00000		0.000114572i	$-0.999964\pi$
$167$	− 178227.i	0 0.000229144i	1.00000		0.000114572i	$-3.64693e-5\pi$
$168$	0	0
$169$	3.54833e8	0.434987
$170$	0	0
$171$	4.28911e8i	0.501629i
$172$	0	0
$173$	−1.60561e9	−1.79248	−0.896240	−	0.443569i	$-0.853712\pi$
$173$	−1.60561e9	−1.79248	−0.896240	+	0.443569i	$0.853712\pi$
$174$	0	0
$175$	3.35414e8i	0.357626i
$176$	0	0
$177$	3.97607e8	0.405098
$178$	0	0
$179$	1.44579e9i	1.40829i	0.710054	+	0.704147i	$0.248671\pi$
$179$	1.44579e9i	1.40829i	−0.710054	+	0.704147i	$0.751329\pi$
$180$	0	0
$181$	−1.78411e9	−1.66229	−0.831146	−	0.556054i	$-0.812315\pi$
$181$	−1.78411e9	−1.66229	−0.831146	+	0.556054i	$0.812315\pi$
$182$	0	0
$183$	− 8.34990e8i	− 0.744521i
$184$	0	0
$185$	8.57907e8	0.732409
$186$	0	0
$187$	2.85756e8i	0.233684i
$188$	0	0
$189$	−4.36697e7	−0.0342242
$190$	0	0
$191$	2.30763e9i	1.73394i	0.498364	+	0.866968i	$0.333934\pi$
$191$	2.30763e9i	1.73394i	−0.498364	+	0.866968i	$0.666066\pi$
$192$	0	0
$193$	−5.47360e8	−0.394497	−0.197248	−	0.980354i	$-0.563201\pi$
$193$	−5.47360e8	−0.394497	−0.197248	+	0.980354i	$0.563201\pi$
$194$	0	0
$195$	1.73523e9i	1.20011i
$196$	0	0
$197$	3.43899e8	0.228332	0.114166	−	0.993462i	$-0.463580\pi$
$197$	3.43899e8	0.228332	0.114166	+	0.993462i	$0.463580\pi$
$198$	0	0
$199$	− 1.12814e9i	− 0.719369i	−0.933074	−	0.359684i	$-0.882884\pi$
$199$	− 1.12814e9i	− 0.719369i	0.933074	−	0.359684i	$-0.117116\pi$
$200$	0	0
$201$	−1.63416e9	−1.00118
$202$	0	0
$203$	− 4.28033e8i	− 0.252054i
$204$	0	0
$205$	1.47553e9	0.835474
$206$	0	0
$207$	7.61043e8i	0.414503i
$208$	0	0
$209$	2.79122e9	1.46288
$210$	0	0
$211$	3.25742e9i	1.64340i	0.569919	+	0.821701i	$0.306975\pi$
$211$	3.25742e9i	1.64340i	−0.569919	+	0.821701i	$0.693025\pi$
$212$	0	0
$213$	−1.79953e9	−0.874259
$214$	0	0
$215$	1.62804e9i	0.761922i
$216$	0	0
$217$	7.00184e8	0.315772
$218$	0	0
$219$	− 9.87709e8i	− 0.429391i
$220$	0	0
$221$	−6.86938e8	−0.287971
$222$	0	0
$223$	− 7.96939e8i	− 0.322259i	−0.986933	−	0.161130i	$-0.948486\pi$
$223$	− 7.96939e8i	− 0.322259i	0.986933	−	0.161130i	$-0.0515137\pi$
$224$	0	0
$225$	−1.71800e9	−0.670337
$226$	0	0
$227$	− 2.32327e9i	− 0.874977i	−0.899224	−	0.437488i	$-0.855868\pi$
$227$	− 2.32327e9i	− 0.874977i	0.899224	−	0.437488i	$-0.144132\pi$
$228$	0	0
$229$	2.17161e9	0.789659	0.394830	−	0.918754i	$-0.370804\pi$
$229$	2.17161e9	0.789659	0.394830	+	0.918754i	$0.370804\pi$
$230$	0	0
$231$	2.84189e8i	0.0998065i
$232$	0	0
$233$	−1.99024e9	−0.675275	−0.337638	−	0.941276i	$-0.609628\pi$
$233$	−1.99024e9	−0.675275	−0.337638	+	0.941276i	$0.609628\pi$
$234$	0	0
$235$	− 1.61873e9i	− 0.530767i
$236$	0	0
$237$	−1.71769e9	−0.544443
$238$	0	0
$239$	1.47132e9i	0.450937i	0.974250	+	0.225469i	$0.0723914\pi$
$239$	1.47132e9i	0.450937i	−0.974250	+	0.225469i	$0.927609\pi$
$240$	0	0
$241$	−2.94516e9	−0.873052	−0.436526	−	0.899692i	$-0.643791\pi$
$241$	−2.94516e9	−0.873052	−0.436526	+	0.899692i	$0.643791\pi$
$242$	0	0
$243$	− 2.23677e8i	− 0.0641500i
$244$	0	0
$245$	6.05430e9	1.68035
$246$	0	0
$247$	6.70989e9i	1.80272i
$248$	0	0
$249$	−1.37604e9	−0.357959
$250$	0	0
$251$	2.41458e9i	0.608341i	0.952618	+	0.304170i	$0.0983792\pi$
$251$	2.41458e9i	0.608341i	−0.952618	+	0.304170i	$0.901621\pi$
$252$	0	0
$253$	4.95263e9	1.20880
$254$	0	0
$255$	− 1.01831e9i	− 0.240835i
$256$	0	0
$257$	5.48578e8	0.125749	0.0628747	−	0.998021i	$-0.479973\pi$
$257$	5.48578e8	0.125749	0.0628747	+	0.998021i	$0.479973\pi$
$258$	0	0
$259$	3.37762e8i	0.0750606i
$260$	0	0
$261$	2.19240e9	0.472451
$262$	0	0
$263$	− 2.76986e9i	− 0.578942i	−0.957187	−	0.289471i	$-0.906521\pi$
$263$	− 2.76986e9i	− 0.578942i	0.957187	−	0.289471i	$-0.0934794\pi$
$264$	0	0
$265$	9.70490e9	1.96792
$266$	0	0
$267$	1.73051e9i	0.340509i
$268$	0	0
$269$	2.47401e9	0.472490	0.236245	−	0.971694i	$-0.424083\pi$
$269$	2.47401e9	0.472490	0.236245	+	0.971694i	$0.424083\pi$
$270$	0	0
$271$	2.51223e9i	0.465781i	0.972503	+	0.232890i	$0.0748183\pi$
$271$	2.51223e9i	0.465781i	−0.972503	+	0.232890i	$0.925182\pi$
$272$	0	0
$273$	−6.83170e8	−0.122992
$274$	0	0
$275$	1.11802e10i	1.95488i
$276$	0	0
$277$	−1.06906e10	−1.81585	−0.907927	−	0.419127i	$-0.862336\pi$
$277$	−1.06906e10	−1.81585	−0.907927	+	0.419127i	$0.862336\pi$
$278$	0	0
$279$	3.58636e9i	0.591885i
$280$	0	0
$281$	−2.60700e9	−0.418134	−0.209067	−	0.977901i	$-0.567043\pi$
$281$	−2.60700e9	−0.418134	−0.209067	+	0.977901i	$0.567043\pi$
$282$	0	0
$283$	1.75548e8i	0.0273684i	0.999906	+	0.0136842i	$0.00435595\pi$
$283$	1.75548e8i	0.0273684i	−0.999906	+	0.0136842i	$0.995644\pi$
$284$	0	0
$285$	−9.94669e9	−1.50765
$286$	0	0
$287$	5.80923e8i	0.0856232i
$288$	0	0
$289$	−6.57263e9	−0.942211
$290$	0	0
$291$	− 5.91221e9i	− 0.824476i
$292$	0	0
$293$	1.83363e9	0.248795	0.124397	−	0.992232i	$-0.460300\pi$
$293$	1.83363e9	0.248795	0.124397	+	0.992232i	$0.460300\pi$
$294$	0	0
$295$	9.22073e9i	1.21752i
$296$	0	0
$297$	−1.45562e9	−0.187078
$298$	0	0
$299$	1.19058e10i	1.48961i
$300$	0	0
$301$	−6.40966e8	−0.0780852
$302$	0	0
$303$	3.23205e9i	0.383450i
$304$	0	0
$305$	1.93639e10	2.23766
$306$	0	0
$307$	1.39249e10i	1.56761i	0.621007	+	0.783805i	$0.286724\pi$
$307$	1.39249e10i	1.56761i	−0.621007	+	0.783805i	$0.713276\pi$
$308$	0	0
$309$	−5.62113e8	−0.0616581
$310$	0	0
$311$	3.88787e9i	0.415595i	0.978172	+	0.207798i	$0.0666295\pi$
$311$	3.88787e9i	0.415595i	−0.978172	+	0.207798i	$0.933370\pi$
$312$	0	0
$313$	−3.59882e9	−0.374958	−0.187479	−	0.982269i	$-0.560032\pi$
$313$	−3.59882e9	−0.374958	−0.187479	+	0.982269i	$0.560032\pi$
$314$	0	0
$315$	− 1.01273e9i	− 0.102861i
$316$	0	0
$317$	8.64268e9	0.855878	0.427939	−	0.903808i	$-0.359240\pi$
$317$	8.64268e9	0.855878	0.427939	+	0.903808i	$0.359240\pi$
$318$	0	0
$319$	− 1.42674e10i	− 1.37779i
$320$	0	0
$321$	1.16418e10	1.09648
$322$	0	0
$323$	− 3.93766e9i	− 0.361766i
$324$	0	0
$325$	−2.68765e10	−2.40901
$326$	0	0
$327$	7.56078e9i	0.661265i
$328$	0	0
$329$	6.37303e8	0.0543954
$330$	0	0
$331$	− 9.59670e9i	− 0.799484i	−0.916628	−	0.399742i	$-0.869100\pi$
$331$	− 9.59670e9i	− 0.799484i	0.916628	−	0.399742i	$-0.130900\pi$
$332$	0	0
$333$	−1.73003e9	−0.140694
$334$	0	0
$335$	− 3.78971e10i	− 3.00904i
$336$	0	0
$337$	−1.75447e10	−1.36027	−0.680137	−	0.733085i	$-0.738080\pi$
$337$	−1.75447e10	−1.36027	−0.680137	+	0.733085i	$0.738080\pi$
$338$	0	0
$339$	3.29957e9i	0.249838i
$340$	0	0
$341$	2.33390e10	1.72609
$342$	0	0
$343$	4.84506e9i	0.350044i
$344$	0	0
$345$	−1.76490e10	−1.24579
$346$	0	0
$347$	− 1.71291e10i	− 1.18146i	−0.806871	−	0.590728i	$-0.798841\pi$
$347$	− 1.71291e10i	− 1.18146i	0.806871	−	0.590728i	$-0.201159\pi$
$348$	0	0
$349$	−1.99032e9	−0.134160	−0.0670799	−	0.997748i	$-0.521368\pi$
$349$	−1.99032e9	−0.134160	−0.0670799	+	0.997748i	$0.521368\pi$
$350$	0	0
$351$	− 3.49922e9i	− 0.230538i
$352$	0	0
$353$	2.53774e8	0.0163436	0.00817181	−	0.999967i	$-0.497399\pi$
$353$	2.53774e8	0.0163436	0.00817181	+	0.999967i	$0.497399\pi$
$354$	0	0
$355$	− 4.17321e10i	− 2.62759i
$356$	0	0
$357$	4.00914e8	0.0246819
$358$	0	0
$359$	2.26677e10i	1.36468i	0.731036	+	0.682339i	$0.239037\pi$
$359$	2.26677e10i	1.36468i	−0.731036	+	0.682339i	$0.760963\pi$
$360$	0	0
$361$	−2.14788e10	−1.26468
$362$	0	0
$363$	− 5.51830e8i	− 0.0317818i
$364$	0	0
$365$	2.29056e10	1.29053
$366$	0	0
$367$	− 7.30278e9i	− 0.402554i	−0.979534	−	0.201277i	$-0.935491\pi$
$367$	− 7.30278e9i	− 0.402554i	0.979534	−	0.201277i	$-0.0645091\pi$
$368$	0	0
$369$	−2.97551e9	−0.160493
$370$	0	0
$371$	3.82087e9i	0.201682i
$372$	0	0
$373$	−1.11279e10	−0.574879	−0.287439	−	0.957799i	$-0.592804\pi$
$373$	−1.11279e10	−0.574879	−0.287439	+	0.957799i	$0.592804\pi$
$374$	0	0
$375$	− 2.00298e10i	− 1.01286i
$376$	0	0
$377$	3.42979e10	1.69786
$378$	0	0
$379$	− 5.68377e9i	− 0.275473i	−0.990469	−	0.137737i	$-0.956017\pi$
$379$	− 5.68377e9i	− 0.275473i	0.990469	−	0.137737i	$-0.0439827\pi$
$380$	0	0
$381$	−4.29017e9	−0.203598
$382$	0	0
$383$	1.92155e9i	0.0893009i	0.999003	+	0.0446504i	$0.0142174\pi$
$383$	1.92155e9i	0.0893009i	−0.999003	+	0.0446504i	$0.985783\pi$
$384$	0	0
$385$	−6.59050e9	−0.299969
$386$	0	0
$387$	− 3.28304e9i	− 0.146364i
$388$	0	0
$389$	1.05878e10	0.462389	0.231194	−	0.972908i	$-0.425737\pi$
$389$	1.05878e10	0.462389	0.231194	+	0.972908i	$0.425737\pi$
$390$	0	0
$391$	− 6.98683e9i	− 0.298932i
$392$	0	0
$393$	−1.21053e10	−0.507464
$394$	0	0
$395$	− 3.98343e10i	− 1.63632i
$396$	0	0
$397$	2.38291e10	0.959282	0.479641	−	0.877465i	$-0.340767\pi$
$397$	2.38291e10	0.959282	0.479641	+	0.877465i	$0.340767\pi$
$398$	0	0
$399$	− 3.91606e9i	− 0.154510i
$400$	0	0
$401$	1.59390e10	0.616428	0.308214	−	0.951317i	$-0.400269\pi$
$401$	1.59390e10	0.616428	0.308214	+	0.951317i	$0.400269\pi$
$402$	0	0
$403$	5.61052e10i	2.12708i
$404$	0	0
$405$	5.18721e9	0.192803
$406$	0	0
$407$	1.12585e10i	0.410301i
$408$	0	0
$409$	8.96615e9	0.320415	0.160208	−	0.987083i	$-0.448784\pi$
$409$	8.96615e9	0.320415	0.160208	+	0.987083i	$0.448784\pi$
$410$	0	0
$411$	− 1.74767e10i	− 0.612480i
$412$	0	0
$413$	−3.63025e9	−0.124777
$414$	0	0
$415$	− 3.19112e10i	− 1.07585i
$416$	0	0
$417$	−7.53415e8	−0.0249167
$418$	0	0
$419$	− 1.06743e10i	− 0.346326i	−0.984893	−	0.173163i	$-0.944601\pi$
$419$	− 1.06743e10i	− 0.346326i	0.984893	−	0.173163i	$-0.0553987\pi$
$420$	0	0
$421$	2.60816e10	0.830245	0.415123	−	0.909765i	$-0.363739\pi$
$421$	2.60816e10	0.830245	0.415123	+	0.909765i	$0.363739\pi$
$422$	0	0
$423$	3.26428e9i	0.101959i
$424$	0	0
$425$	1.57723e10	0.483436
$426$	0	0
$427$	7.62366e9i	0.229325i
$428$	0	0
$429$	−2.27718e10	−0.672308
$430$	0	0
$431$	1.20312e9i	0.0348658i	0.999848	+	0.0174329i	$0.00554935\pi$
$431$	1.20312e9i	0.0348658i	−0.999848	+	0.0174329i	$0.994451\pi$
$432$	0	0
$433$	1.13142e10	0.321865	0.160932	−	0.986965i	$-0.448550\pi$
$433$	1.13142e10	0.321865	0.160932	+	0.986965i	$0.448550\pi$
$434$	0	0
$435$	5.08430e10i	1.41995i
$436$	0	0
$437$	−6.82462e10	−1.87134
$438$	0	0
$439$	− 1.76972e10i	− 0.476482i	−0.971206	−	0.238241i	$-0.923429\pi$
$439$	− 1.76972e10i	− 0.476482i	0.971206	−	0.238241i	$-0.0765708\pi$
$440$	0	0
$441$	−1.22089e10	−0.322792
$442$	0	0
$443$	− 6.47192e10i	− 1.68042i	−0.542260	−	0.840211i	$-0.682431\pi$
$443$	− 6.47192e10i	− 1.68042i	0.542260	−	0.840211i	$-0.317569\pi$
$444$	0	0
$445$	−4.01315e10	−1.02340
$446$	0	0
$447$	− 2.70568e10i	− 0.677713i
$448$	0	0
$449$	6.07525e10	1.49478	0.747392	−	0.664383i	$-0.231306\pi$
$449$	6.07525e10	1.49478	0.747392	+	0.664383i	$0.231306\pi$
$450$	0	0
$451$	1.93637e10i	0.468038i
$452$	0	0
$453$	−2.81279e10	−0.667952
$454$	0	0
$455$	− 1.58431e10i	− 0.369653i
$456$	0	0
$457$	4.77740e10	1.09528	0.547642	−	0.836713i	$-0.315526\pi$
$457$	4.77740e10	1.09528	0.547642	+	0.836713i	$0.315526\pi$
$458$	0	0
$459$	2.05349e9i	0.0462639i
$460$	0	0
$461$	7.17546e9	0.158872	0.0794358	−	0.996840i	$-0.474688\pi$
$461$	7.17546e9	0.158872	0.0794358	+	0.996840i	$0.474688\pi$
$462$	0	0
$463$	5.86354e9i	0.127596i	0.997963	+	0.0637978i	$0.0203213\pi$
$463$	5.86354e9i	0.127596i	−0.997963	+	0.0637978i	$0.979679\pi$
$464$	0	0
$465$	−8.31699e10	−1.77891
$466$	0	0
$467$	7.53384e10i	1.58398i	0.610537	+	0.791988i	$0.290954\pi$
$467$	7.53384e10i	1.58398i	−0.610537	+	0.791988i	$0.709046\pi$
$468$	0	0
$469$	1.49203e10	0.308380
$470$	0	0
$471$	6.02620e9i	0.122450i
$472$	0	0
$473$	−2.13650e10	−0.426834
$474$	0	0
$475$	− 1.54061e11i	− 3.02634i
$476$	0	0
$477$	−1.95706e10	−0.378033
$478$	0	0
$479$	1.97647e10i	0.375446i	0.982222	+	0.187723i	$0.0601107\pi$
$479$	1.97647e10i	0.375446i	−0.982222	+	0.187723i	$0.939889\pi$
$480$	0	0
$481$	−2.70646e10	−0.505617
$482$	0	0
$483$	− 6.94850e9i	− 0.127674i
$484$	0	0
$485$	1.37108e11	2.47796
$486$	0	0
$487$	− 5.47310e10i	− 0.973011i	−0.873677	−	0.486506i	$-0.838271\pi$
$487$	− 5.47310e10i	− 0.973011i	0.873677	−	0.486506i	$-0.161729\pi$
$488$	0	0
$489$	−1.33116e9	−0.0232807
$490$	0	0
$491$	− 2.44118e10i	− 0.420024i	−0.977699	−	0.210012i	$-0.932650\pi$
$491$	− 2.44118e10i	− 0.420024i	0.977699	−	0.210012i	$-0.0673502\pi$
$492$	0	0
$493$	−2.01275e10	−0.340724
$494$	0	0
$495$	− 3.37568e10i	− 0.562263i
$496$	0	0
$497$	1.64301e10	0.269287
$498$	0	0
$499$	3.29856e10i	0.532014i	0.963971	+	0.266007i	$0.0857044\pi$
$499$	3.29856e10i	0.532014i	−0.963971	+	0.266007i	$0.914296\pi$
$500$	0	0
$501$	−8.33485e6	−0.000132296 0
$502$	0	0
$503$	7.46160e9i	0.116563i	0.998300	+	0.0582814i	$0.0185621\pi$
$503$	7.46160e9i	0.116563i	−0.998300	+	0.0582814i	$0.981438\pi$
$504$	0	0
$505$	−7.49533e10	−1.15246
$506$	0	0
$507$	− 1.65939e10i	− 0.251140i
$508$	0	0
$509$	−2.46374e10	−0.367049	−0.183524	−	0.983015i	$-0.558751\pi$
$509$	−2.46374e10	−0.367049	−0.183524	+	0.983015i	$0.558751\pi$
$510$	0	0
$511$	9.01802e9i	0.132260i
$512$	0	0
$513$	2.00582e10	0.289616
$514$	0	0
$515$	− 1.30357e10i	− 0.185313i
$516$	0	0
$517$	2.12429e10	0.297339
$518$	0	0
$519$	7.50867e10i	1.03489i
$520$	0	0
$521$	−2.99951e10	−0.407099	−0.203549	−	0.979065i	$-0.565248\pi$
$521$	−2.99951e10	−0.407099	−0.203549	+	0.979065i	$0.565248\pi$
$522$	0	0
$523$	1.04737e11i	1.39988i	0.714199	+	0.699942i	$0.246791\pi$
$523$	1.04737e11i	1.39988i	−0.714199	+	0.699942i	$0.753209\pi$
$524$	0	0
$525$	1.56858e10	0.206475
$526$	0	0
$527$	− 3.29250e10i	− 0.426858i
$528$	0	0
$529$	−4.27824e10	−0.546314
$530$	0	0
$531$	− 1.85942e10i	− 0.233884i
$532$	0	0
$533$	−4.65489e10	−0.576767
$534$	0	0
$535$	2.69981e11i	3.29547i
$536$	0	0
$537$	6.76130e10	0.813079
$538$	0	0
$539$	7.94518e10i	0.941344i
$540$	0	0
$541$	5.69348e10	0.664644	0.332322	−	0.943166i	$-0.392168\pi$
$541$	5.69348e10	0.664644	0.332322	+	0.943166i	$0.392168\pi$
$542$	0	0
$543$	8.34346e10i	0.959725i
$544$	0	0
$545$	−1.75339e11	−1.98743
$546$	0	0
$547$	1.13814e11i	1.27130i	0.771979	+	0.635648i	$0.219267\pi$
$547$	1.13814e11i	1.27130i	−0.771979	+	0.635648i	$0.780733\pi$
$548$	0	0
$549$	−3.90486e10	−0.429849
$550$	0	0
$551$	1.96602e11i	2.13296i
$552$	0	0
$553$	1.56830e10	0.167698
$554$	0	0
$555$	− 4.01204e10i	− 0.422856i
$556$	0	0
$557$	1.32489e11	1.37645	0.688225	−	0.725497i	$-0.258390\pi$
$557$	1.32489e11	1.37645	0.688225	+	0.725497i	$0.258390\pi$
$558$	0	0
$559$	− 5.13600e10i	− 0.525991i
$560$	0	0
$561$	1.33635e10	0.134918
$562$	0	0
$563$	1.68664e11i	1.67876i	0.543545	+	0.839380i	$0.317082\pi$
$563$	1.68664e11i	1.67876i	−0.543545	+	0.839380i	$0.682918\pi$
$564$	0	0
$565$	−7.65191e10	−0.750889
$566$	0	0
$567$	2.04223e9i	0.0197593i
$568$	0	0
$569$	1.36419e11	1.30145	0.650725	−	0.759314i	$-0.274465\pi$
$569$	1.36419e11	1.30145	0.650725	+	0.759314i	$0.274465\pi$
$570$	0	0
$571$	− 5.62677e10i	− 0.529315i	−0.964342	−	0.264658i	$-0.914741\pi$
$571$	− 5.62677e10i	− 0.529315i	0.964342	−	0.264658i	$-0.0852590\pi$
$572$	0	0
$573$	1.07917e11	1.00109
$574$	0	0
$575$	− 2.73360e11i	− 2.50071i
$576$	0	0
$577$	1.35733e11	1.22457	0.612285	−	0.790637i	$-0.290250\pi$
$577$	1.35733e11	1.22457	0.612285	+	0.790637i	$0.290250\pi$
$578$	0	0
$579$	2.55975e10i	0.227763i
$580$	0	0
$581$	1.25636e10	0.110258
$582$	0	0
$583$	1.27359e11i	1.10244i
$584$	0	0
$585$	8.11489e10	0.692882
$586$	0	0
$587$	1.54607e11i	1.30220i	0.758993	+	0.651098i	$0.225691\pi$
$587$	1.54607e11i	1.30220i	−0.758993	+	0.651098i	$0.774309\pi$
$588$	0	0
$589$	−3.21606e11	−2.67216
$590$	0	0
$591$	− 1.60826e10i	− 0.131827i
$592$	0	0
$593$	1.44789e11	1.17089	0.585447	−	0.810711i	$-0.300919\pi$
$593$	1.44789e11	1.17089	0.585447	+	0.810711i	$0.300919\pi$
$594$	0	0
$595$	9.29743e9i	0.0741814i
$596$	0	0
$597$	−5.27580e10	−0.415328
$598$	0	0
$599$	− 2.20988e10i	− 0.171657i	−0.996310	−	0.0858284i	$-0.972646\pi$
$599$	− 2.20988e10i	− 0.171657i	0.996310	−	0.0858284i	$-0.0273537\pi$
$600$	0	0
$601$	−1.29762e11	−0.994602	−0.497301	−	0.867578i	$-0.665676\pi$
$601$	−1.29762e11	−0.994602	−0.497301	+	0.867578i	$0.665676\pi$
$602$	0	0
$603$	7.64221e10i	0.578029i
$604$	0	0
$605$	1.27973e10	0.0955204
$606$	0	0
$607$	− 1.15676e10i	− 0.0852099i	−0.999092	−	0.0426050i	$-0.986434\pi$
$607$	− 1.15676e10i	− 0.0852099i	0.999092	−	0.0426050i	$-0.0135657\pi$
$608$	0	0
$609$	−2.00171e10	−0.145523
$610$	0	0
$611$	5.10665e10i	0.366414i
$612$	0	0
$613$	1.57261e11	1.11373	0.556865	−	0.830603i	$-0.312004\pi$
$613$	1.57261e11	1.11373	0.556865	+	0.830603i	$0.312004\pi$
$614$	0	0
$615$	− 6.90037e10i	− 0.482361i
$616$	0	0
$617$	−3.94877e10	−0.272471	−0.136236	−	0.990676i	$-0.543500\pi$
$617$	−3.94877e10	−0.272471	−0.136236	+	0.990676i	$0.543500\pi$
$618$	0	0
$619$	− 1.02727e11i	− 0.699718i	−0.936802	−	0.349859i	$-0.886230\pi$
$619$	− 1.02727e11i	− 0.699718i	0.936802	−	0.349859i	$-0.113770\pi$
$620$	0	0
$621$	3.55904e10	0.239313
$622$	0	0
$623$	− 1.57999e10i	− 0.104883i
$624$	0	0
$625$	1.57647e11	1.03315
$626$	0	0
$627$	− 1.30532e11i	− 0.844594i
$628$	0	0
$629$	1.58827e10	0.101466
$630$	0	0
$631$	− 2.04152e10i	− 0.128777i	−0.997925	−	0.0643883i	$-0.979490\pi$
$631$	− 2.04152e10i	− 0.128777i	0.997925	−	0.0643883i	$-0.0205096\pi$
$632$	0	0
$633$	1.52334e11	0.948819
$634$	0	0
$635$	− 9.94915e10i	− 0.611915i
$636$	0	0
$637$	−1.90996e11	−1.16003
$638$	0	0
$639$	8.41556e10i	0.504754i
$640$	0	0
$641$	−2.17056e11	−1.28570	−0.642851	−	0.765992i	$-0.722248\pi$
$641$	−2.17056e11	−1.28570	−0.642851	+	0.765992i	$0.722248\pi$
$642$	0	0
$643$	− 1.02152e11i	− 0.597590i	−0.954317	−	0.298795i	$-0.903415\pi$
$643$	− 1.02152e11i	− 0.597590i	0.954317	−	0.298795i	$-0.0965848\pi$
$644$	0	0
$645$	7.61357e10	0.439896
$646$	0	0
$647$	9.09290e10i	0.518902i	0.965756	+	0.259451i	$0.0835416\pi$
$647$	9.09290e10i	0.518902i	−0.965756	+	0.259451i	$0.916458\pi$
$648$	0	0
$649$	−1.21005e11	−0.682065
$650$	0	0
$651$	− 3.27444e10i	− 0.182311i
$652$	0	0
$653$	−1.71409e11	−0.942717	−0.471359	−	0.881942i	$-0.656236\pi$
$653$	−1.71409e11	−0.942717	−0.471359	+	0.881942i	$0.656236\pi$
$654$	0	0
$655$	− 2.80729e11i	− 1.52518i
$656$	0	0
$657$	−4.61906e10	−0.247909
$658$	0	0
$659$	2.27365e9i	0.0120554i	0.999982	+	0.00602769i	$0.00191869\pi$
$659$	2.27365e9i	0.0120554i	−0.999982	+	0.00602769i	$0.998081\pi$
$660$	0	0
$661$	2.92200e11	1.53065	0.765323	−	0.643647i	$-0.222579\pi$
$661$	2.92200e11	1.53065	0.765323	+	0.643647i	$0.222579\pi$
$662$	0	0
$663$	3.21249e10i	0.166260i
$664$	0	0
$665$	9.08157e10	0.464381
$666$	0	0
$667$	3.48843e11i	1.76249i
$668$	0	0
$669$	−3.72691e10	−0.186056
$670$	0	0
$671$	2.54116e11i	1.25355i
$672$	0	0
$673$	−1.51294e10	−0.0737501	−0.0368751	−	0.999320i	$-0.511740\pi$
$673$	−1.51294e10	−0.0737501	−0.0368751	+	0.999320i	$0.511740\pi$
$674$	0	0
$675$	8.03429e10i	0.387019i
$676$	0	0
$677$	−3.80775e11	−1.81265	−0.906324	−	0.422584i	$-0.861123\pi$
$677$	−3.80775e11	−1.81265	−0.906324	+	0.422584i	$0.861123\pi$
$678$	0	0
$679$	5.39799e10i	0.253953i
$680$	0	0
$681$	−1.08649e11	−0.505168
$682$	0	0
$683$	− 4.18285e11i	− 1.92216i	−0.276269	−	0.961080i	$-0.589098\pi$
$683$	− 4.18285e11i	− 1.92216i	0.276269	−	0.961080i	$-0.410902\pi$
$684$	0	0
$685$	4.05295e11	1.84081
$686$	0	0
$687$	− 1.01556e11i	− 0.455910i
$688$	0	0
$689$	−3.06163e11	−1.35855
$690$	0	0
$691$	− 8.10077e10i	− 0.355315i	−0.984092	−	0.177658i	$-0.943148\pi$
$691$	− 8.10077e10i	− 0.355315i	0.984092	−	0.177658i	$-0.0568520\pi$
$692$	0	0
$693$	1.32902e10	0.0576233
$694$	0	0
$695$	− 1.74721e10i	− 0.0748870i
$696$	0	0
$697$	2.73169e10	0.115745
$698$	0	0
$699$	9.30741e10i	0.389870i
$700$	0	0
$701$	3.22597e11	1.33594	0.667971	−	0.744187i	$-0.267163\pi$
$701$	3.22597e11	1.33594	0.667971	+	0.744187i	$0.267163\pi$
$702$	0	0
$703$	− 1.55139e11i	− 0.635186i
$704$	0	0
$705$	−7.57006e10	−0.306438
$706$	0	0
$707$	− 2.95094e10i	− 0.118109i
$708$	0	0
$709$	2.99903e11	1.18685	0.593424	−	0.804890i	$-0.297776\pi$
$709$	2.99903e11	1.18685	0.593424	+	0.804890i	$0.297776\pi$
$710$	0	0
$711$	8.03286e10i	0.314334i
$712$	0	0
$713$	−5.70645e11	−2.20804
$714$	0	0
$715$	− 5.28092e11i	− 2.02062i
$716$	0	0
$717$	6.88070e10	0.260349
$718$	0	0
$719$	1.62338e11i	0.607442i	0.952761	+	0.303721i	$0.0982290\pi$
$719$	1.62338e11i	0.607442i	−0.952761	+	0.303721i	$0.901771\pi$
$720$	0	0
$721$	5.13223e9	0.0189918
$722$	0	0
$723$	1.37731e11i	0.504057i
$724$	0	0
$725$	−7.87489e11	−2.85031
$726$	0	0
$727$	2.20900e11i	0.790786i	0.918512	+	0.395393i	$0.129392\pi$
$727$	2.20900e11i	0.790786i	−0.918512	+	0.395393i	$0.870608\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	3.01403e10i	0.105555i
$732$	0	0
$733$	−1.15228e11	−0.399154	−0.199577	−	0.979882i	$-0.563957\pi$
$733$	−1.15228e11	−0.399154	−0.199577	+	0.979882i	$0.563957\pi$
$734$	0	0
$735$	− 2.83132e11i	− 0.970151i
$736$	0	0
$737$	4.97332e11	1.68568
$738$	0	0
$739$	5.64364e11i	1.89226i	0.323781	+	0.946132i	$0.395046\pi$
$739$	5.64364e11i	1.89226i	−0.323781	+	0.946132i	$0.604954\pi$
$740$	0	0
$741$	3.13791e11	1.04080
$742$	0	0
$743$	− 1.89167e11i	− 0.620713i	−0.950620	−	0.310357i	$-0.899552\pi$
$743$	− 1.89167e11i	− 0.620713i	0.950620	−	0.310357i	$-0.100448\pi$
$744$	0	0
$745$	6.27462e11	2.03687
$746$	0	0
$747$	6.43510e10i	0.206668i
$748$	0	0
$749$	−1.06293e11	−0.337735
$750$	0	0
$751$	3.66355e11i	1.15171i	0.817553	+	0.575853i	$0.195330\pi$
$751$	3.66355e11i	1.15171i	−0.817553	+	0.575853i	$0.804670\pi$
$752$	0	0
$753$	1.12919e11	0.351226
$754$	0	0
$755$	− 6.52304e11i	− 2.00753i
$756$	0	0
$757$	−1.30666e11	−0.397904	−0.198952	−	0.980009i	$-0.563754\pi$
$757$	−1.30666e11	−0.397904	−0.198952	+	0.980009i	$0.563754\pi$
$758$	0	0
$759$	− 2.31612e11i	− 0.697900i
$760$	0	0
$761$	−3.92845e11	−1.17134	−0.585670	−	0.810550i	$-0.699168\pi$
$761$	−3.92845e11	−1.17134	−0.585670	+	0.810550i	$0.699168\pi$
$762$	0	0
$763$	− 6.90318e10i	− 0.203681i
$764$	0	0
$765$	−4.76217e10	−0.139046
$766$	0	0
$767$	− 2.90889e11i	− 0.840515i
$768$	0	0
$769$	5.36177e11	1.53322	0.766608	−	0.642116i	$-0.221943\pi$
$769$	5.36177e11	1.53322	0.766608	+	0.642116i	$0.221943\pi$
$770$	0	0
$771$	− 2.56545e10i	− 0.0726015i
$772$	0	0
$773$	−3.88295e11	−1.08754	−0.543769	−	0.839235i	$-0.683003\pi$
$773$	−3.88295e11	−1.08754	−0.543769	+	0.839235i	$0.683003\pi$
$774$	0	0
$775$	− 1.28819e12i	− 3.57086i
$776$	0	0
$777$	1.57956e10	0.0433362
$778$	0	0
$779$	− 2.66827e11i	− 0.724570i
$780$	0	0
$781$	5.47659e11	1.47199
$782$	0	0
$783$	− 1.02528e11i	− 0.272770i
$784$	0	0
$785$	−1.39751e11	−0.368025
$786$	0	0
$787$	− 3.56234e11i	− 0.928616i	−0.885674	−	0.464308i	$-0.846303\pi$
$787$	− 3.56234e11i	− 0.928616i	0.885674	−	0.464308i	$-0.153697\pi$
$788$	0	0
$789$	−1.29534e11	−0.334253
$790$	0	0
$791$	− 3.01259e10i	− 0.0769546i
$792$	0	0
$793$	−6.10877e11	−1.54476
$794$	0	0
$795$	− 4.53853e11i	− 1.13618i
$796$	0	0
$797$	−5.81694e11	−1.44165	−0.720827	−	0.693115i	$-0.756238\pi$
$797$	−5.81694e11	−1.44165	−0.720827	+	0.693115i	$0.756238\pi$
$798$	0	0
$799$	− 2.99681e10i	− 0.0735312i
$800$	0	0
$801$	8.09278e10	0.196593
$802$	0	0
$803$	3.00594e11i	0.722966i
$804$	0	0
$805$	1.61140e11	0.383725
$806$	0	0
$807$	− 1.15698e11i	− 0.272792i
$808$	0	0
$809$	5.29788e11	1.23682	0.618412	−	0.785854i	$-0.287776\pi$
$809$	5.29788e11	1.23682	0.618412	+	0.785854i	$0.287776\pi$
$810$	0	0
$811$	1.27857e11i	0.295556i	0.989021	+	0.147778i	$0.0472121\pi$
$811$	1.27857e11i	0.295556i	−0.989021	+	0.147778i	$0.952788\pi$
$812$	0	0
$813$	1.17485e11	0.268919
$814$	0	0
$815$	− 3.08705e10i	− 0.0699701i
$816$	0	0
$817$	2.94406e11	0.660781
$818$	0	0
$819$	3.19487e10i	0.0710097i
$820$	0	0
$821$	5.23718e11	1.15272	0.576361	−	0.817195i	$-0.304472\pi$
$821$	5.23718e11	1.15272	0.576361	+	0.817195i	$0.304472\pi$
$822$	0	0
$823$	− 4.16147e11i	− 0.907083i	−0.891235	−	0.453541i	$-0.850160\pi$
$823$	− 4.16147e11i	− 0.907083i	0.891235	−	0.453541i	$-0.149840\pi$
$824$	0	0
$825$	5.22847e11	1.12865
$826$	0	0
$827$	1.32442e11i	0.283141i	0.989928	+	0.141570i	$0.0452152\pi$
$827$	1.32442e11i	0.283141i	−0.989928	+	0.141570i	$0.954785\pi$
$828$	0	0
$829$	−1.50571e11	−0.318804	−0.159402	−	0.987214i	$-0.550957\pi$
$829$	−1.50571e11	−0.318804	−0.159402	+	0.987214i	$0.550957\pi$
$830$	0	0
$831$	4.99948e11i	1.04838i
$832$	0	0
$833$	1.12085e11	0.232792
$834$	0	0
$835$	− 1.93290e8i	0 0.000397616i
$836$	0	0
$837$	1.67718e11	0.341725
$838$	0	0
$839$	− 5.30040e11i	− 1.06970i	−0.844948	−	0.534848i	$-0.820369\pi$
$839$	− 5.30040e11i	− 1.06970i	0.844948	−	0.534848i	$-0.179631\pi$
$840$	0	0
$841$	5.04694e11	1.00889
$842$	0	0
$843$	1.21917e11i	0.241410i
$844$	0	0
$845$	3.84822e11	0.754802
$846$	0	0
$847$	5.03835e9i	0.00978936i
$848$	0	0
$849$	8.20955e9	0.0158011
$850$	0	0
$851$	− 2.75273e11i	− 0.524863i
$852$	0	0
$853$	−7.37919e11	−1.39384	−0.696919	−	0.717150i	$-0.745446\pi$
$853$	−7.37919e11	−1.39384	−0.696919	+	0.717150i	$0.745446\pi$
$854$	0	0
$855$	4.65161e11i	0.870440i
$856$	0	0
$857$	−2.91969e11	−0.541270	−0.270635	−	0.962682i	$-0.587234\pi$
$857$	−2.91969e11	−0.541270	−0.270635	+	0.962682i	$0.587234\pi$
$858$	0	0
$859$	− 3.42628e11i	− 0.629289i	−0.949210	−	0.314645i	$-0.898115\pi$
$859$	− 3.42628e11i	− 0.629289i	0.949210	−	0.314645i	$-0.101885\pi$
$860$	0	0
$861$	2.71671e10	0.0494346
$862$	0	0
$863$	6.00307e11i	1.08226i	0.840940	+	0.541128i	$0.182003\pi$
$863$	6.00307e11i	1.08226i	−0.840940	+	0.541128i	$0.817997\pi$
$864$	0	0
$865$	−1.74131e12	−3.11036
$866$	0	0
$867$	3.07372e11i	0.543986i
$868$	0	0
$869$	5.22753e11	0.916680
$870$	0	0
$871$	1.19555e12i	2.07728i
$872$	0	0
$873$	−2.76487e11	−0.476011
$874$	0	0
$875$	1.82877e11i	0.311980i
$876$	0	0
$877$	−3.40566e11	−0.575709	−0.287854	−	0.957674i	$-0.592942\pi$
$877$	−3.40566e11	−0.575709	−0.287854	+	0.957674i	$0.592942\pi$
$878$	0	0
$879$	− 8.57504e10i	− 0.143642i
$880$	0	0
$881$	−8.36651e11	−1.38880	−0.694402	−	0.719587i	$-0.744331\pi$
$881$	−8.36651e11	−1.38880	−0.694402	+	0.719587i	$0.744331\pi$
$882$	0	0
$883$	− 6.58705e11i	− 1.08355i	−0.840524	−	0.541774i	$-0.817753\pi$
$883$	− 6.58705e11i	− 1.08355i	0.840524	−	0.541774i	$-0.182247\pi$
$884$	0	0
$885$	4.31211e11	0.702937
$886$	0	0
$887$	− 1.84018e11i	− 0.297281i	−0.988891	−	0.148640i	$-0.952510\pi$
$887$	− 1.84018e11i	− 0.297281i	0.988891	−	0.148640i	$-0.0474897\pi$
$888$	0	0
$889$	3.91703e10	0.0627119
$890$	0	0
$891$	6.80727e10i	0.108010i
$892$	0	0
$893$	−2.92723e11	−0.460311
$894$	0	0
$895$	1.56799e12i	2.44371i
$896$	0	0
$897$	5.56778e11	0.860027
$898$	0	0
$899$	1.64390e12i	2.51673i
$900$	0	0
$901$	1.79670e11	0.272631
$902$	0	0
$903$	2.99750e10i	0.0450825i
$904$	0	0
$905$	−1.93490e12	−2.88445
$906$	0	0
$907$	7.74263e11i	1.14409i	0.820223	+	0.572044i	$0.193849\pi$
$907$	7.74263e11i	1.14409i	−0.820223	+	0.572044i	$0.806151\pi$
$908$	0	0
$909$	1.51148e11	0.221385
$910$	0	0
$911$	− 2.57326e11i	− 0.373602i	−0.982398	−	0.186801i	$-0.940188\pi$
$911$	− 2.57326e11i	− 0.373602i	0.982398	−	0.186801i	$-0.0598120\pi$
$912$	0	0
$913$	4.18776e11	0.602697
$914$	0	0
$915$	− 9.05560e11i	− 1.29191i
$916$	0	0
$917$	1.10524e11	0.156308
$918$	0	0
$919$	− 3.23480e11i	− 0.453508i	−0.973952	−	0.226754i	$-0.927189\pi$
$919$	− 3.23480e11i	− 0.453508i	0.973952	−	0.226754i	$-0.0728114\pi$
$920$	0	0
$921$	6.51202e11	0.905060
$922$	0	0
$923$	1.31653e12i	1.81395i
$924$	0	0
$925$	6.21411e11	0.848812
$926$	0	0
$927$	2.62874e10i	0.0355983i
$928$	0	0
$929$	1.13798e12	1.52782	0.763910	−	0.645322i	$-0.223277\pi$
$929$	1.13798e12	1.52782	0.763910	+	0.645322i	$0.223277\pi$
$930$	0	0
$931$	− 1.09483e12i	− 1.45729i
$932$	0	0
$933$	1.81818e11	0.239944
$934$	0	0
$935$	3.09907e11i	0.405495i
$936$	0	0
$937$	1.34507e12	1.74497	0.872483	−	0.488645i	$-0.162509\pi$
$937$	1.34507e12	1.74497	0.872483	+	0.488645i	$0.162509\pi$
$938$	0	0
$939$	1.68300e11i	0.216482i
$940$	0	0
$941$	1.54350e10	0.0196856	0.00984279	−	0.999952i	$-0.496867\pi$
$941$	1.54350e10	0.0196856	0.00984279	+	0.999952i	$0.496867\pi$
$942$	0	0
$943$	− 4.73448e11i	− 0.598722i
$944$	0	0
$945$	−4.73605e10	−0.0593867
$946$	0	0
$947$	− 1.10486e12i	− 1.37374i	−0.726779	−	0.686872i	$-0.758983\pi$
$947$	− 1.10486e12i	− 1.37374i	0.726779	−	0.686872i	$-0.241017\pi$
$948$	0	0
$949$	−7.22607e11	−0.890917
$950$	0	0
$951$	− 4.04178e11i	− 0.494141i
$952$	0	0
$953$	−3.94710e11	−0.478527	−0.239264	−	0.970955i	$-0.576906\pi$
$953$	−3.94710e11	−0.478527	−0.239264	+	0.970955i	$0.576906\pi$
$954$	0	0
$955$	2.50267e12i	3.00877i
$956$	0	0
$957$	−6.67222e11	−0.795468
$958$	0	0
$959$	1.59566e11i	0.188654i
$960$	0	0
$961$	−1.83624e12	−2.15295
$962$	0	0
$963$	− 5.44434e11i	− 0.633053i
$964$	0	0
$965$	−5.93621e11	−0.684541
$966$	0	0
$967$	6.43683e11i	0.736150i	0.929796	+	0.368075i	$0.119983\pi$
$967$	6.43683e11i	0.736150i	−0.929796	+	0.368075i	$0.880017\pi$
$968$	0	0
$969$	−1.84146e11	−0.208866
$970$	0	0
$971$	9.15239e11i	1.02957i	0.857318	+	0.514787i	$0.172129\pi$
$971$	9.15239e11i	1.02957i	−0.857318	+	0.514787i	$0.827871\pi$
$972$	0	0
$973$	6.87886e9	0.00767477
$974$	0	0
$975$	1.25689e12i	1.39084i
$976$	0	0
$977$	1.39678e12	1.53303	0.766515	−	0.642226i	$-0.221989\pi$
$977$	1.39678e12	1.53303	0.766515	+	0.642226i	$0.221989\pi$
$978$	0	0
$979$	− 5.26653e11i	− 0.573316i
$980$	0	0
$981$	3.53583e11	0.381782
$982$	0	0
$983$	4.88577e11i	0.523261i	0.965168	+	0.261631i	$0.0842602\pi$
$983$	4.88577e11i	0.523261i	−0.965168	+	0.261631i	$0.915740\pi$
$984$	0	0
$985$	3.72964e11	0.396207
$986$	0	0
$987$	− 2.98037e10i	− 0.0314052i
$988$	0	0
$989$	5.22382e11	0.546013
$990$	0	0
$991$	− 7.71826e11i	− 0.800248i	−0.916461	−	0.400124i	$-0.868967\pi$
$991$	− 7.71826e11i	− 0.800248i	0.916461	−	0.400124i	$-0.131033\pi$
$992$	0	0
$993$	−4.48793e11	−0.461582
$994$	0	0
$995$	− 1.22349e12i	− 1.24827i
$996$	0	0
$997$	−1.24478e12	−1.25983	−0.629916	−	0.776663i	$-0.716911\pi$
$997$	−1.24478e12	−1.25983	−0.629916	+	0.776663i	$0.716911\pi$
$998$	0	0
$999$	8.09054e10i	0.0812298i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.9.g.c.127.2		4
4.3	odd	2	inner	192.9.g.c.127.4		4
8.3	odd	2		48.9.g.c.31.1	✓	4
8.5	even	2		48.9.g.c.31.3	yes	4
24.5	odd	2		144.9.g.i.127.4		4
24.11	even	2		144.9.g.i.127.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.9.g.c.31.1	✓	4	8.3	odd	2
48.9.g.c.31.3	yes	4	8.5	even	2
144.9.g.i.127.3		4	24.11	even	2
144.9.g.i.127.4		4	24.5	odd	2
192.9.g.c.127.2		4	1.1	even	1	trivial
192.9.g.c.127.4		4	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	192.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} +1084.52 q^{5} +426.979i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q-46.7654i q^{3} +1084.52 q^{5} +426.979i q^{7} -2187.00 q^{9} +14232.3i q^{11} -34213.5 q^{13} -50717.8i q^{15} +20078.0 q^{17} -196118. i q^{19} +19967.8 q^{21} -347985. i q^{23} +785551. q^{25} +102276. i q^{27} -1.00247e6 q^{29} -1.63986e6i q^{31} +665580. q^{33} +463066. i q^{35} +791050. q^{37} +1.60001e6i q^{39} +1.36054e6 q^{41} +1.50116e6i q^{43} -2.37184e6 q^{45} -1.49258e6i q^{47} +5.58249e6 q^{49} -938954. i q^{51} +8.94860e6 q^{53} +1.54352e7i q^{55} -9.17155e6 q^{57} +8.50216e6i q^{59} +1.78549e7 q^{61} -933804. i q^{63} -3.71051e7 q^{65} -3.49438e7i q^{67} -1.62736e7 q^{69} -3.84799e7i q^{71} +2.11205e7 q^{73} -3.67366e7i q^{75} -6.07690e6 q^{77} -3.67300e7i q^{79} +4.78297e6 q^{81} -2.94243e7i q^{83} +2.17749e7 q^{85} +4.68808e7i q^{87} -3.70040e7 q^{89} -1.46085e7i q^{91} -7.66885e7 q^{93} -2.12694e8i q^{95} +1.26423e8 q^{97} -3.11261e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 264 q^{5} - 8748 q^{9}+O(q^{10}) \) 4 * q + 264 * q^5 - 8748 * q^9	\( 4 q + 264 q^{5} - 8748 q^{9} - 14632 q^{13} + 332904 q^{17} - 250128 q^{21} + 2604428 q^{25} - 2343576 q^{29} + 2002320 q^{33} - 4315784 q^{37} + 9035496 q^{41} - 577368 q^{45} + 3458884 q^{49} + 42186600 q^{53} + 2253744 q^{57} + 48148408 q^{61} - 125450832 q^{65} - 23514624 q^{69} - 21215480 q^{73} + 32354496 q^{77} + 19131876 q^{81} - 235297584 q^{85} - 12675576 q^{89} - 193564080 q^{93} + 263153800 q^{97}+O(q^{100}) \) 4 * q + 264 * q^5 - 8748 * q^9 - 14632 * q^13 + 332904 * q^17 - 250128 * q^21 + 2604428 * q^25 - 2343576 * q^29 + 2002320 * q^33 - 4315784 * q^37 + 9035496 * q^41 - 577368 * q^45 + 3458884 * q^49 + 42186600 * q^53 + 2253744 * q^57 + 48148408 * q^61 - 125450832 * q^65 - 23514624 * q^69 - 21215480 * q^73 + 32354496 * q^77 + 19131876 * q^81 - 235297584 * q^85 - 12675576 * q^89 - 193564080 * q^93 + 263153800 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more