LMFDB - Embedded newform 384.10.d.f.193.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, 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:	$197.773761087$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 $ x^20 - 8x^19 - 300700x^18 + 6140664x^17 + 35387063979x^16 - 1130222504088x^15 - 2027709461599244x^14 + 84850591004671480x^13 + 57031067836759165823x^12 - 2761881829653049004184x^11 - 715774112122919877685260x^10 + 33804686703728196561023832x^9 + 4440127592352179578492608037x^8 - 186062650255909170075184345736x^7 - 13450754766452318411315895909412x^6 + 457952106473696272472169073091112x^5 + 15298236539262291502286172763910964x^4 - 382483105264444074951018570546548352x^3 + 3006361776222337276289989101322653304x^2 - 10187939951443559461324435287012809264*x + 12842550816412602186535854197192532452
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{175}\cdot 3^{32} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.5
Root			$-67.7122 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.10.d.f.193.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000i q^{3} -471.295i q^{5} -3820.46 q^{7} -6561.00 q^{9} +O(q^{10})$	$q-81.0000i q^{3} -471.295i q^{5} -3820.46 q^{7} -6561.00 q^{9} -88862.6i q^{11} +9364.95i q^{13} -38174.9 q^{15} -160780. q^{17} +686701. i q^{19} +309458. i q^{21} +1.96343e6 q^{23} +1.73101e6 q^{25} +531441. i q^{27} +2.90860e6i q^{29} +9.54671e6 q^{31} -7.19787e6 q^{33} +1.80057e6i q^{35} -1.01166e7i q^{37} +758561. q^{39} +2.63848e7 q^{41} +8.49707e6i q^{43} +3.09217e6i q^{45} -4.66576e7 q^{47} -2.57577e7 q^{49} +1.30232e7i q^{51} +9.42340e7i q^{53} -4.18805e7 q^{55} +5.56228e7 q^{57} +6.64602e7i q^{59} -1.72970e8i q^{61} +2.50661e7 q^{63} +4.41365e6 q^{65} -6.28736e7i q^{67} -1.59038e8i q^{69} +3.01482e8 q^{71} +9.43629e7 q^{73} -1.40211e8i q^{75} +3.39496e8i q^{77} +3.92555e8 q^{79} +4.30467e7 q^{81} -3.32735e8i q^{83} +7.57749e7i q^{85} +2.35597e8 q^{87} -3.51490e8 q^{89} -3.57784e7i q^{91} -7.73283e8i q^{93} +3.23639e8 q^{95} +5.72032e7 q^{97} +5.83027e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 131220 q^{9}+O(q^{10}) $ 20 * q - 131220 * q^9	$ 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) $ 20 * q - 131220 * q^9 - 905768 * q^17 - 9682620 * q^25 + 12054096 * q^33 + 74264008 * q^41 + 252775700 * q^49 - 5335632 * q^57 + 245588672 * q^65 - 895193896 * q^73 + 860934420 * q^81 + 882422136 * q^89 + 433683736 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 81.0000i	− 0.577350i
$3$	− 81.0000i	− 0.577350i
$4$	0	0
$5$	− 471.295i	− 0.337231i	−0.985682	−	0.168616i	$-0.946070\pi$
$5$	− 471.295i	− 0.337231i	0.985682	−	0.168616i	$-0.0539297\pi$
$6$	0	0
$7$	−3820.46	−0.601416	−0.300708	−	0.953716i	$-0.597223\pi$
$7$	−3820.46	−0.601416	−0.300708	+	0.953716i	$0.597223\pi$
$8$	0	0
$9$	−6561.00	−0.333333
$10$	0	0
$11$	− 88862.6i	− 1.83000i	−0.403450	−	0.915002i	$-0.632189\pi$
$11$	− 88862.6i	− 1.83000i	0.403450	−	0.915002i	$-0.367811\pi$
$12$	0	0
$13$	9364.95i	0.0909411i	0.998966	+	0.0454706i	$0.0144787\pi$
$13$	9364.95i	0.0909411i	−0.998966	+	0.0454706i	$0.985521\pi$
$14$	0	0
$15$	−38174.9	−0.194701
$16$	0	0
$17$	−160780.	−0.466888	−0.233444	−	0.972370i	$-0.574999\pi$
$17$	−160780.	−0.466888	−0.233444	+	0.972370i	$0.574999\pi$
$18$	0	0
$19$	686701.i	1.20886i	0.796658	+	0.604430i	$0.206599\pi$
$19$	686701.i	1.20886i	−0.796658	+	0.604430i	$0.793401\pi$
$20$	0	0
$21$	309458.i	0.347228i
$22$	0	0
$23$	1.96343e6	1.46299	0.731494	−	0.681848i	$-0.238824\pi$
$23$	1.96343e6	1.46299	0.731494	+	0.681848i	$0.238824\pi$
$24$	0	0
$25$	1.73101e6	0.886275
$26$	0	0
$27$	531441.i	0.192450i
$28$	0	0
$29$	2.90860e6i	0.763649i	0.924235	+	0.381824i	$0.124704\pi$
$29$	2.90860e6i	0.763649i	−0.924235	+	0.381824i	$0.875296\pi$
$30$	0	0
$31$	9.54671e6	1.85663	0.928316	−	0.371791i	$-0.121256\pi$
$31$	9.54671e6	1.85663	0.928316	+	0.371791i	$0.121256\pi$
$32$	0	0
$33$	−7.19787e6	−1.05655
$34$	0	0
$35$	1.80057e6i	0.202816i
$36$	0	0
$37$	− 1.01166e7i	− 0.887413i	−0.896172	−	0.443706i	$-0.853663\pi$
$37$	− 1.01166e7i	− 0.887413i	0.896172	−	0.443706i	$-0.146337\pi$
$38$	0	0
$39$	758561.	0.0525049
$40$	0	0
$41$	2.63848e7	1.45823	0.729116	−	0.684391i	$-0.239932\pi$
$41$	2.63848e7	1.45823	0.729116	+	0.684391i	$0.239932\pi$
$42$	0	0
$43$	8.49707e6i	0.379019i	0.981879	+	0.189510i	$0.0606898\pi$
$43$	8.49707e6i	0.379019i	−0.981879	+	0.189510i	$0.939310\pi$
$44$	0	0
$45$	3.09217e6i	0.112410i
$46$	0	0
$47$	−4.66576e7	−1.39470	−0.697352	−	0.716729i	$-0.745639\pi$
$47$	−4.66576e7	−1.39470	−0.697352	+	0.716729i	$0.745639\pi$
$48$	0	0
$49$	−2.57577e7	−0.638299
$50$	0	0
$51$	1.30232e7i	0.269558i
$52$	0	0
$53$	9.42340e7i	1.64046i	0.572033	+	0.820231i	$0.306155\pi$
$53$	9.42340e7i	1.64046i	−0.572033	+	0.820231i	$0.693845\pi$
$54$	0	0
$55$	−4.18805e7	−0.617134
$56$	0	0
$57$	5.56228e7	0.697936
$58$	0	0
$59$	6.64602e7i	0.714048i	0.934095	+	0.357024i	$0.116209\pi$
$59$	6.64602e7i	0.714048i	−0.934095	+	0.357024i	$0.883791\pi$
$60$	0	0
$61$	− 1.72970e8i	− 1.59951i	−0.600329	−	0.799753i	$-0.704964\pi$
$61$	− 1.72970e8i	− 1.59951i	0.600329	−	0.799753i	$-0.295036\pi$
$62$	0	0
$63$	2.50661e7	0.200472
$64$	0	0
$65$	4.41365e6	0.0306682
$66$	0	0
$67$	− 6.28736e7i	− 0.381182i	−0.981670	−	0.190591i	$-0.938960\pi$
$67$	− 6.28736e7i	− 0.381182i	0.981670	−	0.190591i	$-0.0610404\pi$
$68$	0	0
$69$	− 1.59038e8i	− 0.844656i
$70$	0	0
$71$	3.01482e8	1.40799	0.703993	−	0.710207i	$-0.251399\pi$
$71$	3.01482e8	1.40799	0.703993	+	0.710207i	$0.251399\pi$
$72$	0	0
$73$	9.43629e7	0.388909	0.194455	−	0.980912i	$-0.437706\pi$
$73$	9.43629e7	0.388909	0.194455	+	0.980912i	$0.437706\pi$
$74$	0	0
$75$	− 1.40211e8i	− 0.511691i
$76$	0	0
$77$	3.39496e8i	1.10059i
$78$	0	0
$79$	3.92555e8	1.13391	0.566955	−	0.823749i	$-0.308121\pi$
$79$	3.92555e8	1.13391	0.566955	+	0.823749i	$0.308121\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	− 3.32735e8i	− 0.769569i	−0.923007	−	0.384784i	$-0.874276\pi$
$83$	− 3.32735e8i	− 0.769569i	0.923007	−	0.384784i	$-0.125724\pi$
$84$	0	0
$85$	7.57749e7i	0.157449i
$86$	0	0
$87$	2.35597e8	0.440893
$88$	0	0
$89$	−3.51490e8	−0.593823	−0.296912	−	0.954905i	$-0.595957\pi$
$89$	−3.51490e8	−0.593823	−0.296912	+	0.954905i	$0.595957\pi$
$90$	0	0
$91$	− 3.57784e7i	− 0.0546934i
$92$	0	0
$93$	− 7.73283e8i	− 1.07193i
$94$	0	0
$95$	3.23639e8	0.407666
$96$	0	0
$97$	5.72032e7	0.0656066	0.0328033	−	0.999462i	$-0.489557\pi$
$97$	5.72032e7	0.0656066	0.0328033	+	0.999462i	$0.489557\pi$
$98$	0	0
$99$	5.83027e8i	0.610001i
$100$	0	0
$101$	1.06377e9i	1.01719i	0.861007	+	0.508593i	$0.169834\pi$
$101$	1.06377e9i	1.01719i	−0.861007	+	0.508593i	$0.830166\pi$
$102$	0	0
$103$	5.55617e8	0.486416	0.243208	−	0.969974i	$-0.421800\pi$
$103$	5.55617e8	0.486416	0.243208	+	0.969974i	$0.421800\pi$
$104$	0	0
$105$	1.45846e8	0.117096
$106$	0	0
$107$	4.33293e8i	0.319562i	0.987153	+	0.159781i	$0.0510787\pi$
$107$	4.33293e8i	0.319562i	−0.987153	+	0.159781i	$0.948921\pi$
$108$	0	0
$109$	− 2.85477e8i	− 0.193710i	−0.995299	−	0.0968548i	$-0.969122\pi$
$109$	− 2.85477e8i	− 0.193710i	0.995299	−	0.0968548i	$-0.0308783\pi$
$110$	0	0
$111$	−8.19443e8	−0.512348
$112$	0	0
$113$	1.79071e9	1.03317	0.516585	−	0.856236i	$-0.327203\pi$
$113$	1.79071e9	1.03317	0.516585	+	0.856236i	$0.327203\pi$
$114$	0	0
$115$	− 9.25356e8i	− 0.493365i
$116$	0	0
$117$	− 6.14434e7i	− 0.0303137i
$118$	0	0
$119$	6.14255e8	0.280794
$120$	0	0
$121$	−5.53861e9	−2.34891
$122$	0	0
$123$	− 2.13717e9i	− 0.841910i
$124$	0	0
$125$	− 1.73631e9i	− 0.636111i
$126$	0	0
$127$	−8.68203e8	−0.296145	−0.148073	−	0.988977i	$-0.547307\pi$
$127$	−8.68203e8	−0.296145	−0.148073	+	0.988977i	$0.547307\pi$
$128$	0	0
$129$	6.88263e8	0.218827
$130$	0	0
$131$	3.66000e9i	1.08583i	0.839789	+	0.542914i	$0.182679\pi$
$131$	3.66000e9i	1.08583i	−0.839789	+	0.542914i	$0.817321\pi$
$132$	0	0
$133$	− 2.62352e9i	− 0.727028i
$134$	0	0
$135$	2.50465e8	0.0649002
$136$	0	0
$137$	2.03358e9	0.493195	0.246598	−	0.969118i	$-0.420687\pi$
$137$	2.03358e9	0.493195	0.246598	+	0.969118i	$0.420687\pi$
$138$	0	0
$139$	5.52975e9i	1.25643i	0.778039	+	0.628216i	$0.216215\pi$
$139$	5.52975e9i	1.25643i	−0.778039	+	0.628216i	$0.783785\pi$
$140$	0	0
$141$	3.77927e9i	0.805233i
$142$	0	0
$143$	8.32193e8	0.166423
$144$	0	0
$145$	1.37081e9	0.257526
$146$	0	0
$147$	2.08637e9i	0.368522i
$148$	0	0
$149$	− 8.86505e9i	− 1.47348i	−0.676178	−	0.736738i	$-0.736365\pi$
$149$	− 8.86505e9i	− 1.47348i	0.676178	−	0.736738i	$-0.263635\pi$
$150$	0	0
$151$	4.97073e9	0.778079	0.389039	−	0.921221i	$-0.372807\pi$
$151$	4.97073e9	0.778079	0.389039	+	0.921221i	$0.372807\pi$
$152$	0	0
$153$	1.05488e9	0.155629
$154$	0	0
$155$	− 4.49932e9i	− 0.626115i
$156$	0	0
$157$	− 1.99628e8i	− 0.0262225i	−0.999914	−	0.0131112i	$-0.995826\pi$
$157$	− 1.99628e8i	− 0.0262225i	0.999914	−	0.0131112i	$-0.00417356\pi$
$158$	0	0
$159$	7.63295e9	0.947121
$160$	0	0
$161$	−7.50122e9	−0.879864
$162$	0	0
$163$	− 1.56968e10i	− 1.74167i	−0.491572	−	0.870837i	$-0.663578\pi$
$163$	− 1.56968e10i	− 1.74167i	0.491572	−	0.870837i	$-0.336422\pi$
$164$	0	0
$165$	3.39232e9i	0.356303i
$166$	0	0
$167$	9.13988e8	0.0909319	0.0454660	−	0.998966i	$-0.485523\pi$
$167$	9.13988e8	0.0909319	0.0454660	+	0.998966i	$0.485523\pi$
$168$	0	0
$169$	1.05168e10	0.991730
$170$	0	0
$171$	− 4.50544e9i	− 0.402954i
$172$	0	0
$173$	− 1.41680e10i	− 1.20254i	−0.799045	−	0.601271i	$-0.794661\pi$
$173$	− 1.41680e10i	− 1.20254i	0.799045	−	0.601271i	$-0.205339\pi$
$174$	0	0
$175$	−6.61325e9	−0.533020
$176$	0	0
$177$	5.38327e9	0.412256
$178$	0	0
$179$	1.10063e10i	0.801316i	0.916228	+	0.400658i	$0.131218\pi$
$179$	1.10063e10i	0.801316i	−0.916228	+	0.400658i	$0.868782\pi$
$180$	0	0
$181$	2.27200e9i	0.157346i	0.996900	+	0.0786728i	$0.0250682\pi$
$181$	2.27200e9i	0.157346i	−0.996900	+	0.0786728i	$0.974932\pi$
$182$	0	0
$183$	−1.40105e10	−0.923475
$184$	0	0
$185$	−4.76789e9	−0.299263
$186$	0	0
$187$	1.42873e10i	0.854406i
$188$	0	0
$189$	− 2.03035e9i	− 0.115743i
$190$	0	0
$191$	−2.98830e10	−1.62470	−0.812352	−	0.583168i	$-0.801813\pi$
$191$	−2.98830e10	−1.62470	−0.812352	+	0.583168i	$0.801813\pi$
$192$	0	0
$193$	−9.03947e9	−0.468959	−0.234480	−	0.972121i	$-0.575339\pi$
$193$	−9.03947e9	−0.468959	−0.234480	+	0.972121i	$0.575339\pi$
$194$	0	0
$195$	− 3.57506e8i	− 0.0177063i
$196$	0	0
$197$	− 2.72855e10i	− 1.29072i	−0.763877	−	0.645362i	$-0.776706\pi$
$197$	− 2.72855e10i	− 1.29072i	0.763877	−	0.645362i	$-0.223294\pi$
$198$	0	0
$199$	−3.07448e10	−1.38974	−0.694868	−	0.719137i	$-0.744537\pi$
$199$	−3.07448e10	−1.38974	−0.694868	+	0.719137i	$0.744537\pi$
$200$	0	0
$201$	−5.09276e9	−0.220075
$202$	0	0
$203$	− 1.11122e10i	− 0.459270i
$204$	0	0
$205$	− 1.24350e10i	− 0.491761i
$206$	0	0
$207$	−1.28821e10	−0.487662
$208$	0	0
$209$	6.10220e10	2.21222
$210$	0	0
$211$	− 2.71487e10i	− 0.942927i	−0.881886	−	0.471463i	$-0.843726\pi$
$211$	− 2.71487e10i	− 0.942927i	0.881886	−	0.471463i	$-0.156274\pi$
$212$	0	0
$213$	− 2.44200e10i	− 0.812901i
$214$	0	0
$215$	4.00463e9	0.127817
$216$	0	0
$217$	−3.64729e10	−1.11661
$218$	0	0
$219$	− 7.64339e9i	− 0.224537i
$220$	0	0
$221$	− 1.50570e9i	− 0.0424593i
$222$	0	0
$223$	4.09594e10	1.10913	0.554564	−	0.832141i	$-0.312885\pi$
$223$	4.09594e10	1.10913	0.554564	+	0.832141i	$0.312885\pi$
$224$	0	0
$225$	−1.13571e10	−0.295425
$226$	0	0
$227$	2.85796e10i	0.714398i	0.934028	+	0.357199i	$0.116268\pi$
$227$	2.85796e10i	0.714398i	−0.934028	+	0.357199i	$0.883732\pi$
$228$	0	0
$229$	− 4.90945e10i	− 1.17970i	−0.807511	−	0.589852i	$-0.799186\pi$
$229$	− 4.90945e10i	− 1.17970i	0.807511	−	0.589852i	$-0.200814\pi$
$230$	0	0
$231$	2.74992e10	0.635428
$232$	0	0
$233$	−6.70534e10	−1.49046	−0.745228	−	0.666810i	$-0.767659\pi$
$233$	−6.70534e10	−1.49046	−0.745228	+	0.666810i	$0.767659\pi$
$234$	0	0
$235$	2.19895e10i	0.470338i
$236$	0	0
$237$	− 3.17970e10i	− 0.654663i
$238$	0	0
$239$	4.32031e10	0.856494	0.428247	−	0.903662i	$-0.359131\pi$
$239$	4.32031e10	0.856494	0.428247	+	0.903662i	$0.359131\pi$
$240$	0	0
$241$	9.43840e10	1.80228	0.901139	−	0.433530i	$-0.142732\pi$
$241$	9.43840e10	1.80228	0.901139	+	0.433530i	$0.142732\pi$
$242$	0	0
$243$	− 3.48678e9i	− 0.0641500i
$244$	0	0
$245$	1.21395e10i	0.215254i
$246$	0	0
$247$	−6.43092e9	−0.109935
$248$	0	0
$249$	−2.69516e10	−0.444311
$250$	0	0
$251$	6.74755e10i	1.07304i	0.843889	+	0.536518i	$0.180261\pi$
$251$	6.74755e10i	1.07304i	−0.843889	+	0.536518i	$0.819739\pi$
$252$	0	0
$253$	− 1.74476e11i	− 2.67727i
$254$	0	0
$255$	6.13777e9	0.0909033
$256$	0	0
$257$	−7.61078e10	−1.08825	−0.544127	−	0.839003i	$-0.683139\pi$
$257$	−7.61078e10	−1.08825	−0.544127	+	0.839003i	$0.683139\pi$
$258$	0	0
$259$	3.86500e10i	0.533704i
$260$	0	0
$261$	− 1.90833e10i	− 0.254550i
$262$	0	0
$263$	1.39533e10	0.179836	0.0899178	−	0.995949i	$-0.471340\pi$
$263$	1.39533e10	0.179836	0.0899178	+	0.995949i	$0.471340\pi$
$264$	0	0
$265$	4.44120e10	0.553215
$266$	0	0
$267$	2.84707e10i	0.342844i
$268$	0	0
$269$	− 5.08365e10i	− 0.591958i	−0.955194	−	0.295979i	$-0.904354\pi$
$269$	− 5.08365e10i	− 0.591958i	0.955194	−	0.295979i	$-0.0956458\pi$
$270$	0	0
$271$	−3.18542e10	−0.358760	−0.179380	−	0.983780i	$-0.557409\pi$
$271$	−3.18542e10	−0.358760	−0.179380	+	0.983780i	$0.557409\pi$
$272$	0	0
$273$	−2.89805e9	−0.0315773
$274$	0	0
$275$	− 1.53822e11i	− 1.62189i
$276$	0	0
$277$	− 1.35604e11i	− 1.38392i	−0.721934	−	0.691962i	$-0.756746\pi$
$277$	− 1.35604e11i	− 1.38392i	0.721934	−	0.691962i	$-0.243254\pi$
$278$	0	0
$279$	−6.26359e10	−0.618878
$280$	0	0
$281$	5.40664e10	0.517307	0.258654	−	0.965970i	$-0.416721\pi$
$281$	5.40664e10	0.517307	0.258654	+	0.965970i	$0.416721\pi$
$282$	0	0
$283$	− 1.20661e11i	− 1.11822i	−0.829094	−	0.559109i	$-0.811143\pi$
$283$	− 1.20661e11i	− 1.11822i	0.829094	−	0.559109i	$-0.188857\pi$
$284$	0	0
$285$	− 2.62147e10i	− 0.235366i
$286$	0	0
$287$	−1.00802e11	−0.877003
$288$	0	0
$289$	−9.27376e10	−0.782016
$290$	0	0
$291$	− 4.63346e9i	− 0.0378780i
$292$	0	0
$293$	− 6.94144e10i	− 0.550231i	−0.961411	−	0.275116i	$-0.911284\pi$
$293$	− 6.94144e10i	− 0.550231i	0.961411	−	0.275116i	$-0.0887161\pi$
$294$	0	0
$295$	3.13223e10	0.240799
$296$	0	0
$297$	4.72252e10	0.352184
$298$	0	0
$299$	1.83874e10i	0.133046i
$300$	0	0
$301$	− 3.24627e10i	− 0.227948i
$302$	0	0
$303$	8.61652e10	0.587273
$304$	0	0
$305$	−8.15198e10	−0.539403
$306$	0	0
$307$	1.55259e11i	0.997550i	0.866732	+	0.498775i	$0.166217\pi$
$307$	1.55259e11i	0.997550i	−0.866732	+	0.498775i	$0.833783\pi$
$308$	0	0
$309$	− 4.50050e10i	− 0.280833i
$310$	0	0
$311$	9.71016e10	0.588579	0.294290	−	0.955716i	$-0.404917\pi$
$311$	9.71016e10	0.588579	0.294290	+	0.955716i	$0.404917\pi$
$312$	0	0
$313$	5.47447e10	0.322398	0.161199	−	0.986922i	$-0.448464\pi$
$313$	5.47447e10	0.322398	0.161199	+	0.986922i	$0.448464\pi$
$314$	0	0
$315$	− 1.18135e10i	− 0.0676054i
$316$	0	0
$317$	− 3.41160e11i	− 1.89754i	−0.315966	−	0.948770i	$-0.602329\pi$
$317$	− 3.41160e11i	− 1.89754i	0.315966	−	0.948770i	$-0.397671\pi$
$318$	0	0
$319$	2.58466e11	1.39748
$320$	0	0
$321$	3.50967e10	0.184499
$322$	0	0
$323$	− 1.10408e11i	− 0.564402i
$324$	0	0
$325$	1.62108e10i	0.0805988i
$326$	0	0
$327$	−2.31236e10	−0.111838
$328$	0	0
$329$	1.78254e11	0.838797
$330$	0	0
$331$	− 4.58666e10i	− 0.210025i	−0.994471	−	0.105012i	$-0.966512\pi$
$331$	− 4.58666e10i	− 0.210025i	0.994471	−	0.105012i	$-0.0334882\pi$
$332$	0	0
$333$	6.63749e10i	0.295804i
$334$	0	0
$335$	−2.96320e10	−0.128546
$336$	0	0
$337$	−2.87436e10	−0.121397	−0.0606983	−	0.998156i	$-0.519333\pi$
$337$	−2.87436e10	−0.121397	−0.0606983	+	0.998156i	$0.519333\pi$
$338$	0	0
$339$	− 1.45047e11i	− 0.596501i
$340$	0	0
$341$	− 8.48345e11i	− 3.39764i
$342$	0	0
$343$	2.52576e11	0.985299
$344$	0	0
$345$	−7.49538e10	−0.284844
$346$	0	0
$347$	3.04671e11i	1.12810i	0.825740	+	0.564051i	$0.190758\pi$
$347$	3.04671e11i	1.12810i	−0.825740	+	0.564051i	$0.809242\pi$
$348$	0	0
$349$	− 2.43522e11i	− 0.878667i	−0.898324	−	0.439334i	$-0.855215\pi$
$349$	− 2.43522e11i	− 0.878667i	0.898324	−	0.439334i	$-0.144785\pi$
$350$	0	0
$351$	−4.97692e9	−0.0175016
$352$	0	0
$353$	2.47800e11	0.849406	0.424703	−	0.905333i	$-0.360378\pi$
$353$	2.47800e11	0.849406	0.424703	+	0.905333i	$0.360378\pi$
$354$	0	0
$355$	− 1.42087e11i	− 0.474817i
$356$	0	0
$357$	− 4.97546e10i	− 0.162116i
$358$	0	0
$359$	2.08035e11	0.661015	0.330508	−	0.943803i	$-0.392780\pi$
$359$	2.08035e11	0.661015	0.330508	+	0.943803i	$0.392780\pi$
$360$	0	0
$361$	−1.48870e11	−0.461345
$362$	0	0
$363$	4.48627e11i	1.35614i
$364$	0	0
$365$	− 4.44727e10i	− 0.131152i
$366$	0	0
$367$	−3.74048e11	−1.07629	−0.538146	−	0.842852i	$-0.680875\pi$
$367$	−3.74048e11	−1.07629	−0.538146	+	0.842852i	$0.680875\pi$
$368$	0	0
$369$	−1.73111e11	−0.486077
$370$	0	0
$371$	− 3.60017e11i	− 0.986600i
$372$	0	0
$373$	2.23482e10i	0.0597795i	0.999553	+	0.0298898i	$0.00951562\pi$
$373$	2.23482e10i	0.0597795i	−0.999553	+	0.0298898i	$0.990484\pi$
$374$	0	0
$375$	−1.40641e11	−0.367259
$376$	0	0
$377$	−2.72389e10	−0.0694471
$378$	0	0
$379$	3.24937e11i	0.808952i	0.914549	+	0.404476i	$0.132546\pi$
$379$	3.24937e11i	0.808952i	−0.914549	+	0.404476i	$0.867454\pi$
$380$	0	0
$381$	7.03244e10i	0.170979i
$382$	0	0
$383$	1.51314e11	0.359322	0.179661	−	0.983729i	$-0.442500\pi$
$383$	1.51314e11	0.359322	0.179661	+	0.983729i	$0.442500\pi$
$384$	0	0
$385$	1.60003e11	0.371154
$386$	0	0
$387$	− 5.57493e10i	− 0.126340i
$388$	0	0
$389$	− 2.59727e10i	− 0.0575100i	−0.999586	−	0.0287550i	$-0.990846\pi$
$389$	− 2.59727e10i	− 0.0575100i	0.999586	−	0.0287550i	$-0.00915427\pi$
$390$	0	0
$391$	−3.15681e11	−0.683051
$392$	0	0
$393$	2.96460e11	0.626903
$394$	0	0
$395$	− 1.85009e11i	− 0.382390i
$396$	0	0
$397$	− 3.92181e11i	− 0.792373i	−0.918170	−	0.396186i	$-0.870333\pi$
$397$	− 3.92181e11i	− 0.792373i	0.918170	−	0.396186i	$-0.129667\pi$
$398$	0	0
$399$	−2.12505e11	−0.419750
$400$	0	0
$401$	−4.49989e11	−0.869066	−0.434533	−	0.900656i	$-0.643087\pi$
$401$	−4.49989e11	−0.869066	−0.434533	+	0.900656i	$0.643087\pi$
$402$	0	0
$403$	8.94044e10i	0.168844i
$404$	0	0
$405$	− 2.02877e10i	− 0.0374701i
$406$	0	0
$407$	−8.98985e11	−1.62397
$408$	0	0
$409$	−1.68203e11	−0.297221	−0.148610	−	0.988896i	$-0.547480\pi$
$409$	−1.68203e11	−0.297221	−0.148610	+	0.988896i	$0.547480\pi$
$410$	0	0
$411$	− 1.64720e11i	− 0.284746i
$412$	0	0
$413$	− 2.53909e11i	− 0.429440i
$414$	0	0
$415$	−1.56816e11	−0.259523
$416$	0	0
$417$	4.47910e11	0.725402
$418$	0	0
$419$	2.11921e11i	0.335900i	0.985795	+	0.167950i	$0.0537148\pi$
$419$	2.11921e11i	0.335900i	−0.985795	+	0.167950i	$0.946285\pi$
$420$	0	0
$421$	− 8.49699e11i	− 1.31824i	−0.752036	−	0.659122i	$-0.770928\pi$
$421$	− 8.49699e11i	− 1.31824i	0.752036	−	0.659122i	$-0.229072\pi$
$422$	0	0
$423$	3.06120e11	0.464901
$424$	0	0
$425$	−2.78311e11	−0.413791
$426$	0	0
$427$	6.60825e11i	0.961968i
$428$	0	0
$429$	− 6.74077e10i	− 0.0960841i
$430$	0	0
$431$	−8.02974e11	−1.12087	−0.560433	−	0.828200i	$-0.689365\pi$
$431$	−8.02974e11	−1.12087	−0.560433	+	0.828200i	$0.689365\pi$
$432$	0	0
$433$	−1.23235e12	−1.68476	−0.842381	−	0.538882i	$-0.818847\pi$
$433$	−1.23235e12	−1.68476	−0.842381	+	0.538882i	$0.818847\pi$
$434$	0	0
$435$	− 1.11036e11i	− 0.148683i
$436$	0	0
$437$	1.34829e12i	1.76855i
$438$	0	0
$439$	4.29854e11	0.552371	0.276185	−	0.961104i	$-0.410930\pi$
$439$	4.29854e11	0.552371	0.276185	+	0.961104i	$0.410930\pi$
$440$	0	0
$441$	1.68996e11	0.212766
$442$	0	0
$443$	2.29365e11i	0.282950i	0.989942	+	0.141475i	$0.0451845\pi$
$443$	2.29365e11i	0.282950i	−0.989942	+	0.141475i	$0.954816\pi$
$444$	0	0
$445$	1.65655e11i	0.200256i
$446$	0	0
$447$	−7.18069e11	−0.850712
$448$	0	0
$449$	5.04155e11	0.585404	0.292702	−	0.956204i	$-0.405446\pi$
$449$	5.04155e11	0.585404	0.292702	+	0.956204i	$0.405446\pi$
$450$	0	0
$451$	− 2.34462e12i	− 2.66857i
$452$	0	0
$453$	− 4.02629e11i	− 0.449224i
$454$	0	0
$455$	−1.68622e10	−0.0184443
$456$	0	0
$457$	9.45992e11	1.01453	0.507264	−	0.861790i	$-0.330657\pi$
$457$	9.45992e11	1.01453	0.507264	+	0.861790i	$0.330657\pi$
$458$	0	0
$459$	− 8.54452e10i	− 0.0898526i
$460$	0	0
$461$	9.98331e11i	1.02949i	0.857345	+	0.514743i	$0.172113\pi$
$461$	9.98331e11i	1.02949i	−0.857345	+	0.514743i	$0.827887\pi$
$462$	0	0
$463$	1.78438e12	1.80457	0.902284	−	0.431142i	$-0.141889\pi$
$463$	1.78438e12	1.80457	0.902284	+	0.431142i	$0.141889\pi$
$464$	0	0
$465$	−3.64445e11	−0.361487
$466$	0	0
$467$	− 8.07502e11i	− 0.785629i	−0.919618	−	0.392814i	$-0.871501\pi$
$467$	− 8.07502e11i	− 0.785629i	0.919618	−	0.392814i	$-0.128499\pi$
$468$	0	0
$469$	2.40206e11i	0.229249i
$470$	0	0
$471$	−1.61699e10	−0.0151396
$472$	0	0
$473$	7.55071e11	0.693606
$474$	0	0
$475$	1.18868e12i	1.07138i
$476$	0	0
$477$	− 6.18269e11i	− 0.546821i
$478$	0	0
$479$	1.08299e12	0.939968	0.469984	−	0.882675i	$-0.344260\pi$
$479$	1.08299e12	0.939968	0.469984	+	0.882675i	$0.344260\pi$
$480$	0	0
$481$	9.47412e10	0.0807023
$482$	0	0
$483$	6.07599e11i	0.507990i
$484$	0	0
$485$	− 2.69596e10i	− 0.0221246i
$486$	0	0
$487$	2.56836e11	0.206907	0.103454	−	0.994634i	$-0.467011\pi$
$487$	2.56836e11	0.206907	0.103454	+	0.994634i	$0.467011\pi$
$488$	0	0
$489$	−1.27144e12	−1.00556
$490$	0	0
$491$	9.27687e11i	0.720336i	0.932888	+	0.360168i	$0.117281\pi$
$491$	9.27687e11i	0.720336i	−0.932888	+	0.360168i	$0.882719\pi$
$492$	0	0
$493$	− 4.67646e11i	− 0.356538i
$494$	0	0
$495$	2.74778e11	0.205711
$496$	0	0
$497$	−1.15180e12	−0.846785
$498$	0	0
$499$	− 9.35334e9i	− 0.00675327i	−0.999994	−	0.00337663i	$-0.998925\pi$
$499$	− 9.35334e9i	− 0.00675327i	0.999994	−	0.00337663i	$-0.00107482\pi$
$500$	0	0
$501$	− 7.40330e10i	− 0.0524996i
$502$	0	0
$503$	−1.60695e11	−0.111930	−0.0559651	−	0.998433i	$-0.517824\pi$
$503$	−1.60695e11	−0.111930	−0.0559651	+	0.998433i	$0.517824\pi$
$504$	0	0
$505$	5.01348e11	0.343027
$506$	0	0
$507$	− 8.51861e11i	− 0.572575i
$508$	0	0
$509$	1.10885e12i	0.732222i	0.930571	+	0.366111i	$0.119311\pi$
$509$	1.10885e12i	0.732222i	−0.930571	+	0.366111i	$0.880689\pi$
$510$	0	0
$511$	−3.60510e11	−0.233896
$512$	0	0
$513$	−3.64941e11	−0.232645
$514$	0	0
$515$	− 2.61860e11i	− 0.164035i
$516$	0	0
$517$	4.14611e12i	2.55231i
$518$	0	0
$519$	−1.14761e12	−0.694288
$520$	0	0
$521$	1.20091e12	0.714069	0.357034	−	0.934091i	$-0.383788\pi$
$521$	1.20091e12	0.714069	0.357034	+	0.934091i	$0.383788\pi$
$522$	0	0
$523$	1.87595e12i	1.09639i	0.836352	+	0.548193i	$0.184684\pi$
$523$	1.87595e12i	1.09639i	−0.836352	+	0.548193i	$0.815316\pi$
$524$	0	0
$525$	5.35673e11i	0.307739i
$526$	0	0
$527$	−1.53492e12	−0.866839
$528$	0	0
$529$	2.05391e12	1.14033
$530$	0	0
$531$	− 4.36045e11i	− 0.238016i
$532$	0	0
$533$	2.47092e11i	0.132613i
$534$	0	0
$535$	2.04209e11	0.107766
$536$	0	0
$537$	8.91513e11	0.462640
$538$	0	0
$539$	2.28889e12i	1.16809i
$540$	0	0
$541$	1.65048e12i	0.828367i	0.910193	+	0.414183i	$0.135933\pi$
$541$	1.65048e12i	0.828367i	−0.910193	+	0.414183i	$0.864067\pi$
$542$	0	0
$543$	1.84032e11	0.0908435
$544$	0	0
$545$	−1.34544e11	−0.0653249
$546$	0	0
$547$	− 3.66906e12i	− 1.75232i	−0.482025	−	0.876158i	$-0.660098\pi$
$547$	− 3.66906e12i	− 1.75232i	0.482025	−	0.876158i	$-0.339902\pi$
$548$	0	0
$549$	1.13485e12i	0.533169i
$550$	0	0
$551$	−1.99734e12	−0.923145
$552$	0	0
$553$	−1.49974e12	−0.681952
$554$	0	0
$555$	3.86199e11i	0.172780i
$556$	0	0
$557$	− 4.06696e11i	− 0.179028i	−0.995986	−	0.0895141i	$-0.971469\pi$
$557$	− 4.06696e11i	− 0.179028i	0.995986	−	0.0895141i	$-0.0285314\pi$
$558$	0	0
$559$	−7.95746e10	−0.0344684
$560$	0	0
$561$	1.15727e12	0.493291
$562$	0	0
$563$	− 2.94016e12i	− 1.23334i	−0.787221	−	0.616671i	$-0.788481\pi$
$563$	− 2.94016e12i	− 1.23334i	0.787221	−	0.616671i	$-0.211519\pi$
$564$	0	0
$565$	− 8.43951e11i	− 0.348417i
$566$	0	0
$567$	−1.64458e11	−0.0668240
$568$	0	0
$569$	−3.53308e12	−1.41302	−0.706510	−	0.707703i	$-0.749732\pi$
$569$	−3.53308e12	−1.41302	−0.706510	+	0.707703i	$0.749732\pi$
$570$	0	0
$571$	3.44664e12i	1.35685i	0.734668	+	0.678427i	$0.237338\pi$
$571$	3.44664e12i	1.35685i	−0.734668	+	0.678427i	$0.762662\pi$
$572$	0	0
$573$	2.42052e12i	0.938023i
$574$	0	0
$575$	3.39871e12	1.29661
$576$	0	0
$577$	2.22018e12	0.833866	0.416933	−	0.908937i	$-0.363105\pi$
$577$	2.22018e12	0.833866	0.416933	+	0.908937i	$0.363105\pi$
$578$	0	0
$579$	7.32197e11i	0.270754i
$580$	0	0
$581$	1.27120e12i	0.462831i
$582$	0	0
$583$	8.37387e12	3.00205
$584$	0	0
$585$	−2.89580e10	−0.0102227
$586$	0	0
$587$	3.73747e12i	1.29929i	0.760238	+	0.649645i	$0.225082\pi$
$587$	3.73747e12i	1.29929i	−0.760238	+	0.649645i	$0.774918\pi$
$588$	0	0
$589$	6.55573e12i	2.24441i
$590$	0	0
$591$	−2.21012e12	−0.745200
$592$	0	0
$593$	2.41973e12	0.803564	0.401782	−	0.915735i	$-0.368391\pi$
$593$	2.41973e12	0.803564	0.401782	+	0.915735i	$0.368391\pi$
$594$	0	0
$595$	− 2.89495e11i	− 0.0946924i
$596$	0	0
$597$	2.49033e12i	0.802365i
$598$	0	0
$599$	1.37627e12	0.436802	0.218401	−	0.975859i	$-0.429916\pi$
$599$	1.37627e12	0.436802	0.218401	+	0.975859i	$0.429916\pi$
$600$	0	0
$601$	−3.92853e12	−1.22827	−0.614137	−	0.789199i	$-0.710496\pi$
$601$	−3.92853e12	−1.22827	−0.614137	+	0.789199i	$0.710496\pi$
$602$	0	0
$603$	4.12514e11i	0.127061i
$604$	0	0
$605$	2.61032e12i	0.792126i
$606$	0	0
$607$	4.52784e12	1.35376	0.676881	−	0.736093i	$-0.263331\pi$
$607$	4.52784e12	1.35376	0.676881	+	0.736093i	$0.263331\pi$
$608$	0	0
$609$	−9.00089e11	−0.265160
$610$	0	0
$611$	− 4.36946e11i	− 0.126836i
$612$	0	0
$613$	− 4.79535e11i	− 0.137167i	−0.997645	−	0.0685833i	$-0.978152\pi$
$613$	− 4.79535e11i	− 0.137167i	0.997645	−	0.0685833i	$-0.0218479\pi$
$614$	0	0
$615$	−1.00724e12	−0.283918
$616$	0	0
$617$	−1.64760e12	−0.457686	−0.228843	−	0.973463i	$-0.573494\pi$
$617$	−1.64760e12	−0.457686	−0.228843	+	0.973463i	$0.573494\pi$
$618$	0	0
$619$	− 3.27639e12i	− 0.896991i	−0.893785	−	0.448495i	$-0.851960\pi$
$619$	− 3.27639e12i	− 0.896991i	0.893785	−	0.448495i	$-0.148040\pi$
$620$	0	0
$621$	1.04345e12i	0.281552i
$622$	0	0
$623$	1.34285e12	0.357135
$624$	0	0
$625$	2.56256e12	0.671759
$626$	0	0
$627$	− 4.94278e12i	− 1.27723i
$628$	0	0
$629$	1.62654e12i	0.414322i
$630$	0	0
$631$	−1.14441e12	−0.287376	−0.143688	−	0.989623i	$-0.545896\pi$
$631$	−1.14441e12	−0.287376	−0.143688	+	0.989623i	$0.545896\pi$
$632$	0	0
$633$	−2.19904e12	−0.544399
$634$	0	0
$635$	4.09180e11i	0.0998694i
$636$	0	0
$637$	− 2.41219e11i	− 0.0580476i
$638$	0	0
$639$	−1.97802e12	−0.469329
$640$	0	0
$641$	5.76383e11	0.134850	0.0674249	−	0.997724i	$-0.478522\pi$
$641$	5.76383e11	0.134850	0.0674249	+	0.997724i	$0.478522\pi$
$642$	0	0
$643$	− 6.60767e12i	− 1.52440i	−0.647342	−	0.762200i	$-0.724119\pi$
$643$	− 6.60767e12i	− 1.52440i	0.647342	−	0.762200i	$-0.275881\pi$
$644$	0	0
$645$	− 3.24375e11i	− 0.0737952i
$646$	0	0
$647$	−7.08600e12	−1.58976	−0.794880	−	0.606766i	$-0.792466\pi$
$647$	−7.08600e12	−1.58976	−0.794880	+	0.606766i	$0.792466\pi$
$648$	0	0
$649$	5.90582e12	1.30671
$650$	0	0
$651$	2.95430e12i	0.644674i
$652$	0	0
$653$	4.64319e11i	0.0999326i	0.998751	+	0.0499663i	$0.0159114\pi$
$653$	4.64319e11i	0.0999326i	−0.998751	+	0.0499663i	$0.984089\pi$
$654$	0	0
$655$	1.72494e12	0.366175
$656$	0	0
$657$	−6.19115e11	−0.129636
$658$	0	0
$659$	− 6.08969e12i	− 1.25780i	−0.777487	−	0.628899i	$-0.783506\pi$
$659$	− 6.08969e12i	− 1.25780i	0.777487	−	0.628899i	$-0.216494\pi$
$660$	0	0
$661$	8.48828e12i	1.72947i	0.502228	+	0.864735i	$0.332514\pi$
$661$	8.48828e12i	1.72947i	−0.502228	+	0.864735i	$0.667486\pi$
$662$	0	0
$663$	−1.21962e11	−0.0245139
$664$	0	0
$665$	−1.23645e12	−0.245177
$666$	0	0
$667$	5.71085e12i	1.11721i
$668$	0	0
$669$	− 3.31771e12i	− 0.640355i
$670$	0	0
$671$	−1.53705e13	−2.92710
$672$	0	0
$673$	−1.00036e13	−1.87969	−0.939846	−	0.341599i	$-0.889031\pi$
$673$	−1.00036e13	−1.87969	−0.939846	+	0.341599i	$0.889031\pi$
$674$	0	0
$675$	9.19928e11i	0.170564i
$676$	0	0
$677$	4.32214e12i	0.790769i	0.918516	+	0.395385i	$0.129389\pi$
$677$	4.32214e12i	0.790769i	−0.918516	+	0.395385i	$0.870611\pi$
$678$	0	0
$679$	−2.18543e11	−0.0394569
$680$	0	0
$681$	2.31495e12	0.412458
$682$	0	0
$683$	4.77874e12i	0.840273i	0.907461	+	0.420137i	$0.138018\pi$
$683$	4.77874e12i	0.840273i	−0.907461	+	0.420137i	$0.861982\pi$
$684$	0	0
$685$	− 9.58416e11i	− 0.166321i
$686$	0	0
$687$	−3.97665e12	−0.681102
$688$	0	0
$689$	−8.82496e11	−0.149185
$690$	0	0
$691$	1.06361e13i	1.77473i	0.461070	+	0.887364i	$0.347466\pi$
$691$	1.06361e13i	1.77473i	−0.461070	+	0.887364i	$0.652534\pi$
$692$	0	0
$693$	− 2.22743e12i	− 0.366864i
$694$	0	0
$695$	2.60615e12	0.423708
$696$	0	0
$697$	−4.24215e12	−0.680830
$698$	0	0
$699$	5.43132e12i	0.860515i
$700$	0	0
$701$	8.31590e12i	1.30070i	0.759634	+	0.650351i	$0.225378\pi$
$701$	8.31590e12i	1.30070i	−0.759634	+	0.650351i	$0.774622\pi$
$702$	0	0
$703$	6.94706e12	1.07276
$704$	0	0
$705$	1.78115e12	0.271550
$706$	0	0
$707$	− 4.06408e12i	− 0.611752i
$708$	0	0
$709$	− 9.21605e12i	− 1.36974i	−0.728667	−	0.684868i	$-0.759860\pi$
$709$	− 9.21605e12i	− 1.36974i	0.728667	−	0.684868i	$-0.240140\pi$
$710$	0	0
$711$	−2.57555e12	−0.377970
$712$	0	0
$713$	1.87443e13	2.71623
$714$	0	0
$715$	− 3.92209e11i	− 0.0561229i
$716$	0	0
$717$	− 3.49945e12i	− 0.494497i
$718$	0	0
$719$	−4.14703e12	−0.578704	−0.289352	−	0.957223i	$-0.593440\pi$
$719$	−4.14703e12	−0.578704	−0.289352	+	0.957223i	$0.593440\pi$
$720$	0	0
$721$	−2.12272e12	−0.292539
$722$	0	0
$723$	− 7.64511e12i	− 1.04055i
$724$	0	0
$725$	5.03481e12i	0.676803i
$726$	0	0
$727$	9.25541e11	0.122883	0.0614414	−	0.998111i	$-0.480430\pi$
$727$	9.25541e11	0.122883	0.0614414	+	0.998111i	$0.480430\pi$
$728$	0	0
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	− 1.36616e12i	− 0.176959i
$732$	0	0
$733$	1.34522e13i	1.72118i	0.509301	+	0.860589i	$0.329904\pi$
$733$	1.34522e13i	1.72118i	−0.509301	+	0.860589i	$0.670096\pi$
$734$	0	0
$735$	9.83296e11	0.124277
$736$	0	0
$737$	−5.58711e12	−0.697563
$738$	0	0
$739$	− 8.53898e12i	− 1.05319i	−0.850117	−	0.526594i	$-0.823469\pi$
$739$	− 8.53898e12i	− 1.05319i	0.850117	−	0.526594i	$-0.176531\pi$
$740$	0	0
$741$	5.20904e11i	0.0634711i
$742$	0	0
$743$	5.04007e12	0.606719	0.303359	−	0.952876i	$-0.401892\pi$
$743$	5.04007e12	0.606719	0.303359	+	0.952876i	$0.401892\pi$
$744$	0	0
$745$	−4.17805e12	−0.496902
$746$	0	0
$747$	2.18308e12i	0.256523i
$748$	0	0
$749$	− 1.65538e12i	− 0.192189i
$750$	0	0
$751$	8.53326e12	0.978893	0.489446	−	0.872033i	$-0.337199\pi$
$751$	8.53326e12	0.978893	0.489446	+	0.872033i	$0.337199\pi$
$752$	0	0
$753$	5.46552e12	0.619518
$754$	0	0
$755$	− 2.34268e12i	− 0.262392i
$756$	0	0
$757$	− 1.81749e12i	− 0.201160i	−0.994929	−	0.100580i	$-0.967930\pi$
$757$	− 1.81749e12i	− 0.201160i	0.994929	−	0.100580i	$-0.0320698\pi$
$758$	0	0
$759$	−1.41325e13	−1.54572
$760$	0	0
$761$	−1.20280e13	−1.30005	−0.650026	−	0.759912i	$-0.725242\pi$
$761$	−1.20280e13	−1.30005	−0.650026	+	0.759912i	$0.725242\pi$
$762$	0	0
$763$	1.09065e12i	0.116500i
$764$	0	0
$765$	− 4.97159e11i	− 0.0524830i
$766$	0	0
$767$	−6.22396e11	−0.0649363
$768$	0	0
$769$	1.63110e13	1.68194	0.840972	−	0.541079i	$-0.181984\pi$
$769$	1.63110e13	1.68194	0.840972	+	0.541079i	$0.181984\pi$
$770$	0	0
$771$	6.16473e12i	0.628304i
$772$	0	0
$773$	5.02183e12i	0.505888i	0.967481	+	0.252944i	$0.0813988\pi$
$773$	5.02183e12i	0.505888i	−0.967481	+	0.252944i	$0.918601\pi$
$774$	0	0
$775$	1.65254e13	1.64549
$776$	0	0
$777$	3.13065e12	0.308134
$778$	0	0
$779$	1.81185e13i	1.76280i
$780$	0	0
$781$	− 2.67904e13i	− 2.57662i
$782$	0	0
$783$	−1.54575e12	−0.146964
$784$	0	0
$785$	−9.40839e10	−0.00884304
$786$	0	0
$787$	8.19023e11i	0.0761044i	0.999276	+	0.0380522i	$0.0121153\pi$
$787$	8.19023e11i	0.0761044i	−0.999276	+	0.0380522i	$0.987885\pi$
$788$	0	0
$789$	− 1.13022e12i	− 0.103828i
$790$	0	0
$791$	−6.84133e12	−0.621365
$792$	0	0
$793$	1.61985e12	0.145461
$794$	0	0
$795$	− 3.59737e12i	− 0.319399i
$796$	0	0
$797$	− 1.56728e13i	− 1.37589i	−0.725763	−	0.687945i	$-0.758513\pi$
$797$	− 1.56728e13i	− 1.37589i	0.725763	−	0.687945i	$-0.241487\pi$
$798$	0	0
$799$	7.50162e12	0.651170
$800$	0	0
$801$	2.30612e12	0.197941
$802$	0	0
$803$	− 8.38533e12i	− 0.711705i
$804$	0	0
$805$	3.53529e12i	0.296718i
$806$	0	0
$807$	−4.11776e12	−0.341767
$808$	0	0
$809$	−7.52727e12	−0.617830	−0.308915	−	0.951090i	$-0.599966\pi$
$809$	−7.52727e12	−0.617830	−0.308915	+	0.951090i	$0.599966\pi$
$810$	0	0
$811$	2.82861e12i	0.229604i	0.993388	+	0.114802i	$0.0366233\pi$
$811$	2.82861e12i	0.229604i	−0.993388	+	0.114802i	$0.963377\pi$
$812$	0	0
$813$	2.58019e12i	0.207130i
$814$	0	0
$815$	−7.39782e12	−0.587347
$816$	0	0
$817$	−5.83494e12	−0.458181
$818$	0	0
$819$	2.34742e11i	0.0182311i
$820$	0	0
$821$	− 1.50858e13i	− 1.15885i	−0.815027	−	0.579423i	$-0.803278\pi$
$821$	− 1.50858e13i	− 1.15885i	0.815027	−	0.579423i	$-0.196722\pi$
$822$	0	0
$823$	2.01780e13	1.53313	0.766566	−	0.642166i	$-0.221964\pi$
$823$	2.01780e13	1.53313	0.766566	+	0.642166i	$0.221964\pi$
$824$	0	0
$825$	−1.24596e13	−0.936396
$826$	0	0
$827$	− 1.54218e13i	− 1.14646i	−0.819393	−	0.573232i	$-0.805689\pi$
$827$	− 1.54218e13i	− 1.14646i	0.819393	−	0.573232i	$-0.194311\pi$
$828$	0	0
$829$	− 9.26680e12i	− 0.681451i	−0.940163	−	0.340725i	$-0.889327\pi$
$829$	− 9.26680e12i	− 0.681451i	0.940163	−	0.340725i	$-0.110673\pi$
$830$	0	0
$831$	−1.09839e13	−0.799009
$832$	0	0
$833$	4.14132e12	0.298014
$834$	0	0
$835$	− 4.30758e11i	− 0.0306651i
$836$	0	0
$837$	5.07351e12i	0.357309i
$838$	0	0
$839$	−2.62921e13	−1.83188	−0.915940	−	0.401315i	$-0.868553\pi$
$839$	−2.62921e13	−1.83188	−0.915940	+	0.401315i	$0.868553\pi$
$840$	0	0
$841$	6.04717e12	0.416841
$842$	0	0
$843$	− 4.37937e12i	− 0.298668i
$844$	0	0
$845$	− 4.95651e12i	− 0.334442i
$846$	0	0
$847$	2.11601e13	1.41267
$848$	0	0
$849$	−9.77351e12	−0.645603
$850$	0	0
$851$	− 1.98632e13i	− 1.29827i
$852$	0	0
$853$	− 1.80363e13i	− 1.16648i	−0.812300	−	0.583239i	$-0.801785\pi$
$853$	− 1.80363e13i	− 1.16648i	0.812300	−	0.583239i	$-0.198215\pi$
$854$	0	0
$855$	−2.12339e12	−0.135889
$856$	0	0
$857$	2.70275e13	1.71156	0.855781	−	0.517338i	$-0.173077\pi$
$857$	2.70275e13	1.71156	0.855781	+	0.517338i	$0.173077\pi$
$858$	0	0
$859$	1.11739e13i	0.700221i	0.936708	+	0.350110i	$0.113856\pi$
$859$	1.11739e13i	0.700221i	−0.936708	+	0.350110i	$0.886144\pi$
$860$	0	0
$861$	8.16497e12i	0.506338i
$862$	0	0
$863$	3.64282e12	0.223557	0.111779	−	0.993733i	$-0.464345\pi$
$863$	3.64282e12	0.223557	0.111779	+	0.993733i	$0.464345\pi$
$864$	0	0
$865$	−6.67730e12	−0.405535
$866$	0	0
$867$	7.51175e12i	0.451497i
$868$	0	0
$869$	− 3.48835e13i	− 2.07506i
$870$	0	0
$871$	5.88808e11	0.0346651
$872$	0	0
$873$	−3.75310e11	−0.0218689
$874$	0	0
$875$	6.63352e12i	0.382567i
$876$	0	0
$877$	2.93178e13i	1.67353i	0.547561	+	0.836766i	$0.315556\pi$
$877$	2.93178e13i	1.67353i	−0.547561	+	0.836766i	$0.684444\pi$
$878$	0	0
$879$	−5.62256e12	−0.317676
$880$	0	0
$881$	−1.41446e13	−0.791039	−0.395519	−	0.918458i	$-0.629435\pi$
$881$	−1.41446e13	−0.791039	−0.395519	+	0.918458i	$0.629435\pi$
$882$	0	0
$883$	− 2.11725e13i	− 1.17206i	−0.810289	−	0.586030i	$-0.800690\pi$
$883$	− 2.11725e13i	− 1.17206i	0.810289	−	0.586030i	$-0.199310\pi$
$884$	0	0
$885$	− 2.53711e12i	− 0.139026i
$886$	0	0
$887$	8.64246e12	0.468793	0.234397	−	0.972141i	$-0.424689\pi$
$887$	8.64246e12	0.468793	0.234397	+	0.972141i	$0.424689\pi$
$888$	0	0
$889$	3.31694e12	0.178106
$890$	0	0
$891$	− 3.82524e12i	− 0.203334i
$892$	0	0
$893$	− 3.20398e13i	− 1.68600i
$894$	0	0
$895$	5.18723e12	0.270229
$896$	0	0
$897$	1.48938e12	0.0768140
$898$	0	0
$899$	2.77676e13i	1.41781i
$900$	0	0
$901$	− 1.51510e13i	− 0.765911i
$902$	0	0
$903$	−2.62948e12	−0.131606
$904$	0	0
$905$	1.07078e12	0.0530618
$906$	0	0
$907$	5.58474e12i	0.274012i	0.990570	+	0.137006i	$0.0437480\pi$
$907$	5.58474e12i	0.274012i	−0.990570	+	0.137006i	$0.956252\pi$
$908$	0	0
$909$	− 6.97938e12i	− 0.339062i
$910$	0	0
$911$	−4.45153e12	−0.214130	−0.107065	−	0.994252i	$-0.534145\pi$
$911$	−4.45153e12	−0.214130	−0.107065	+	0.994252i	$0.534145\pi$
$912$	0	0
$913$	−2.95677e13	−1.40831
$914$	0	0
$915$	6.60310e12i	0.311425i
$916$	0	0
$917$	− 1.39829e13i	− 0.653034i
$918$	0	0
$919$	3.68639e13	1.70483	0.852415	−	0.522866i	$-0.175137\pi$
$919$	3.68639e13	1.70483	0.852415	+	0.522866i	$0.175137\pi$
$920$	0	0
$921$	1.25760e13	0.575936
$922$	0	0
$923$	2.82336e12i	0.128044i
$924$	0	0
$925$	− 1.75119e13i	− 0.786492i
$926$	0	0
$927$	−3.64540e12	−0.162139
$928$	0	0
$929$	−1.89511e13	−0.834762	−0.417381	−	0.908732i	$-0.637052\pi$
$929$	−1.89511e13	−0.834762	−0.417381	+	0.908732i	$0.637052\pi$
$930$	0	0
$931$	− 1.76878e13i	− 0.771615i
$932$	0	0
$933$	− 7.86523e12i	− 0.339816i
$934$	0	0
$935$	6.73355e12	0.288132
$936$	0	0
$937$	9.55698e12	0.405035	0.202518	−	0.979279i	$-0.435088\pi$
$937$	9.55698e12	0.405035	0.202518	+	0.979279i	$0.435088\pi$
$938$	0	0
$939$	− 4.43432e12i	− 0.186137i
$940$	0	0
$941$	− 6.56465e12i	− 0.272934i	−0.990645	−	0.136467i	$-0.956425\pi$
$941$	− 6.56465e12i	− 0.272934i	0.990645	−	0.136467i	$-0.0435748\pi$
$942$	0	0
$943$	5.18047e13	2.13337
$944$	0	0
$945$	−9.56894e11	−0.0390320
$946$	0	0
$947$	3.52133e13i	1.42276i	0.702808	+	0.711380i	$0.251929\pi$
$947$	3.52133e13i	1.42276i	−0.702808	+	0.711380i	$0.748071\pi$
$948$	0	0
$949$	8.83703e11i	0.0353678i
$950$	0	0
$951$	−2.76339e13	−1.09555
$952$	0	0
$953$	7.55997e12	0.296895	0.148447	−	0.988920i	$-0.452572\pi$
$953$	7.55997e12	0.296895	0.148447	+	0.988920i	$0.452572\pi$
$954$	0	0
$955$	1.40837e13i	0.547901i
$956$	0	0
$957$	− 2.09357e13i	− 0.806835i
$958$	0	0
$959$	−7.76922e12	−0.296615
$960$	0	0
$961$	6.47000e13	2.44708
$962$	0	0
$963$	− 2.84283e12i	− 0.106521i
$964$	0	0
$965$	4.26026e12i	0.158148i
$966$	0	0
$967$	2.47875e13	0.911621	0.455811	−	0.890077i	$-0.349349\pi$
$967$	2.47875e13	0.911621	0.455811	+	0.890077i	$0.349349\pi$
$968$	0	0
$969$	−8.94304e12	−0.325858
$970$	0	0
$971$	− 4.55193e13i	− 1.64327i	−0.570015	−	0.821634i	$-0.693063\pi$
$971$	− 4.55193e13i	− 1.64327i	0.570015	−	0.821634i	$-0.306937\pi$
$972$	0	0
$973$	− 2.11262e13i	− 0.755639i
$974$	0	0
$975$	1.31307e12	0.0465338
$976$	0	0
$977$	−4.28527e13	−1.50471	−0.752355	−	0.658757i	$-0.771082\pi$
$977$	−4.28527e13	−1.50471	−0.752355	+	0.658757i	$0.771082\pi$
$978$	0	0
$979$	3.12343e13i	1.08670i
$980$	0	0
$981$	1.87301e12i	0.0645699i
$982$	0	0
$983$	−2.65533e13	−0.907042	−0.453521	−	0.891246i	$-0.649832\pi$
$983$	−2.65533e13	−0.907042	−0.453521	+	0.891246i	$0.649832\pi$
$984$	0	0
$985$	−1.28595e13	−0.435273
$986$	0	0
$987$	− 1.44385e13i	− 0.484280i
$988$	0	0
$989$	1.66834e13i	0.554500i
$990$	0	0
$991$	2.92076e13	0.961978	0.480989	−	0.876727i	$-0.340278\pi$
$991$	2.92076e13	0.961978	0.480989	+	0.876727i	$0.340278\pi$
$992$	0	0
$993$	−3.71519e12	−0.121258
$994$	0	0
$995$	1.44899e13i	0.468663i
$996$	0	0
$997$	− 1.07195e13i	− 0.343594i	−0.985132	−	0.171797i	$-0.945043\pi$
$997$	− 1.07195e13i	− 0.343594i	0.985132	−	0.171797i	$-0.0549574\pi$
$998$	0	0
$999$	5.37636e12	0.170783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.d.f.193.5	✓	20
4.3	odd	2	inner	384.10.d.f.193.15	yes	20
8.3	odd	2	inner	384.10.d.f.193.6	yes	20
8.5	even	2	inner	384.10.d.f.193.16	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.d.f.193.5	✓	20	1.1	even	1	trivial
384.10.d.f.193.6	yes	20	8.3	odd	2	inner
384.10.d.f.193.15	yes	20	4.3	odd	2	inner
384.10.d.f.193.16	yes	20	8.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-81.0000i q^{3} -471.295i q^{5} -3820.46 q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q-81.0000i q^{3} -471.295i q^{5} -3820.46 q^{7} -6561.00 q^{9} -88862.6i q^{11} +9364.95i q^{13} -38174.9 q^{15} -160780. q^{17} +686701. i q^{19} +309458. i q^{21} +1.96343e6 q^{23} +1.73101e6 q^{25} +531441. i q^{27} +2.90860e6i q^{29} +9.54671e6 q^{31} -7.19787e6 q^{33} +1.80057e6i q^{35} -1.01166e7i q^{37} +758561. q^{39} +2.63848e7 q^{41} +8.49707e6i q^{43} +3.09217e6i q^{45} -4.66576e7 q^{47} -2.57577e7 q^{49} +1.30232e7i q^{51} +9.42340e7i q^{53} -4.18805e7 q^{55} +5.56228e7 q^{57} +6.64602e7i q^{59} -1.72970e8i q^{61} +2.50661e7 q^{63} +4.41365e6 q^{65} -6.28736e7i q^{67} -1.59038e8i q^{69} +3.01482e8 q^{71} +9.43629e7 q^{73} -1.40211e8i q^{75} +3.39496e8i q^{77} +3.92555e8 q^{79} +4.30467e7 q^{81} -3.32735e8i q^{83} +7.57749e7i q^{85} +2.35597e8 q^{87} -3.51490e8 q^{89} -3.57784e7i q^{91} -7.73283e8i q^{93} +3.23639e8 q^{95} +5.72032e7 q^{97} +5.83027e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 131220 q^{9}+O(q^{10}) \) 20 * q - 131220 * q^9	\( 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) \) 20 * q - 131220 * q^9 - 905768 * q^17 - 9682620 * q^25 + 12054096 * q^33 + 74264008 * q^41 + 252775700 * q^49 - 5335632 * q^57 + 245588672 * q^65 - 895193896 * q^73 + 860934420 * q^81 + 882422136 * q^89 + 433683736 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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