LMFDB - Embedded newform 384.8.d.b.193.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,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, 8, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
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:	$119.955849786$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 277x^{4} + 19236x^{2} + 9216 $ x^6 + 277x^4 + 19236x^2 + 9216
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.6
Root			$11.6046i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.b.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000i q^{3} +349.927i q^{5} +856.119 q^{7} -729.000 q^{9} +O(q^{10})$	$q+27.0000i q^{3} +349.927i q^{5} +856.119 q^{7} -729.000 q^{9} +7467.90i q^{11} +7088.26i q^{13} -9448.04 q^{15} -30555.4 q^{17} +16252.2i q^{19} +23115.2i q^{21} -6640.34 q^{23} -44324.2 q^{25} -19683.0i q^{27} -216311. i q^{29} -161213. q^{31} -201633. q^{33} +299579. i q^{35} +102106. i q^{37} -191383. q^{39} +453079. q^{41} +545313. i q^{43} -255097. i q^{45} +212312. q^{47} -90603.3 q^{49} -824996. i q^{51} +1.22506e6i q^{53} -2.61322e6 q^{55} -438810. q^{57} +1.94063e6i q^{59} +2.81362e6i q^{61} -624111. q^{63} -2.48038e6 q^{65} -2.79519e6i q^{67} -179289. i q^{69} +5.11177e6 q^{71} -1.14982e6 q^{73} -1.19675e6i q^{75} +6.39341e6i q^{77} +2.23303e6 q^{79} +531441. q^{81} -8.74755e6i q^{83} -1.06922e7i q^{85} +5.84040e6 q^{87} +6.73480e6 q^{89} +6.06840e6i q^{91} -4.35275e6i q^{93} -5.68709e6 q^{95} -6.60143e6 q^{97} -5.44410e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 136 q^{7} - 4374 q^{9}+O(q^{10}) $ 6 * q + 136 * q^7 - 4374 * q^9	$ 6 q + 136 q^{7} - 4374 q^{9} - 6048 q^{15} - 25804 q^{17} + 12688 q^{23} - 18418 q^{25} + 220472 q^{31} - 264600 q^{33} - 591192 q^{39} - 619652 q^{41} - 326032 q^{47} - 581482 q^{49} - 4724352 q^{55} - 2429784 q^{57} - 99144 q^{63} - 9518720 q^{65} + 5841776 q^{71} - 11075196 q^{73} + 495800 q^{79} + 3188646 q^{81} + 965520 q^{87} + 9660740 q^{89} - 32750208 q^{95} - 9511564 q^{97}+O(q^{100}) $ 6 * q + 136 * q^7 - 4374 * q^9 - 6048 * q^15 - 25804 * q^17 + 12688 * q^23 - 18418 * q^25 + 220472 * q^31 - 264600 * q^33 - 591192 * q^39 - 619652 * q^41 - 326032 * q^47 - 581482 * q^49 - 4724352 * q^55 - 2429784 * q^57 - 99144 * q^63 - 9518720 * q^65 + 5841776 * q^71 - 11075196 * q^73 + 495800 * q^79 + 3188646 * q^81 + 965520 * q^87 + 9660740 * q^89 - 32750208 * q^95 - 9511564 * 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
$3$	27.0000i	0.577350i
$3$	27.0000i	0.577350i
$4$	0	0
$5$	349.927i	1.25194i	0.779848	+	0.625969i	$0.215296\pi$
$5$	349.927i	1.25194i	−0.779848	+	0.625969i	$0.784704\pi$
$6$	0	0
$7$	856.119	0.943389	0.471695	−	0.881762i	$-0.343642\pi$
$7$	856.119	0.943389	0.471695	+	0.881762i	$0.343642\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	7467.90i	1.69170i	0.533419	+	0.845851i	$0.320907\pi$
$11$	7467.90i	1.69170i	−0.533419	+	0.845851i	$0.679093\pi$
$12$	0	0
$13$	7088.26i	0.894825i	0.894328	+	0.447412i	$0.147654\pi$
$13$	7088.26i	0.894825i	−0.894328	+	0.447412i	$0.852346\pi$
$14$	0	0
$15$	−9448.04	−0.722807
$16$	0	0
$17$	−30555.4	−1.50840	−0.754200	−	0.656644i	$-0.771975\pi$
$17$	−30555.4	−1.50840	−0.754200	+	0.656644i	$0.771975\pi$
$18$	0	0
$19$	16252.2i	0.543594i	0.962355	+	0.271797i	$0.0876179\pi$
$19$	16252.2i	0.543594i	−0.962355	+	0.271797i	$0.912382\pi$
$20$	0	0
$21$	23115.2i	0.544666i
$22$	0	0
$23$	−6640.34	−0.113800	−0.0569001	−	0.998380i	$-0.518122\pi$
$23$	−6640.34	−0.113800	−0.0569001	+	0.998380i	$0.518122\pi$
$24$	0	0
$25$	−44324.2	−0.567349
$26$	0	0
$27$	− 19683.0i	− 0.192450i
$28$	0	0
$29$	− 216311.i	− 1.64697i	−0.567337	−	0.823486i	$-0.692026\pi$
$29$	− 216311.i	− 1.64697i	0.567337	−	0.823486i	$-0.307974\pi$
$30$	0	0
$31$	−161213.	−0.971927	−0.485964	−	0.873979i	$-0.661531\pi$
$31$	−161213.	−0.971927	−0.485964	+	0.873979i	$0.661531\pi$
$32$	0	0
$33$	−201633.	−0.976704
$34$	0	0
$35$	299579.i	1.18107i
$36$	0	0
$37$	102106.i	0.331394i	0.986177	+	0.165697i	$0.0529873\pi$
$37$	102106.i	0.331394i	−0.986177	+	0.165697i	$0.947013\pi$
$38$	0	0
$39$	−191383.	−0.516627
$40$	0	0
$41$	453079.	1.02667	0.513334	−	0.858189i	$-0.328410\pi$
$41$	453079.	1.02667	0.513334	+	0.858189i	$0.328410\pi$
$42$	0	0
$43$	545313.i	1.04594i	0.852352	+	0.522969i	$0.175176\pi$
$43$	545313.i	1.04594i	−0.852352	+	0.522969i	$0.824824\pi$
$44$	0	0
$45$	− 255097.i	− 0.417313i
$46$	0	0
$47$	212312.	0.298285	0.149142	−	0.988816i	$-0.452349\pi$
$47$	212312.	0.298285	0.149142	+	0.988816i	$0.452349\pi$
$48$	0	0
$49$	−90603.3	−0.110016
$50$	0	0
$51$	− 824996.i	− 0.870876i
$52$	0	0
$53$	1.22506e6i	1.13030i	0.824989	+	0.565148i	$0.191181\pi$
$53$	1.22506e6i	1.13030i	−0.824989	+	0.565148i	$0.808819\pi$
$54$	0	0
$55$	−2.61322e6	−2.11791
$56$	0	0
$57$	−438810.	−0.313844
$58$	0	0
$59$	1.94063e6i	1.23015i	0.788467	+	0.615077i	$0.210875\pi$
$59$	1.94063e6i	1.23015i	−0.788467	+	0.615077i	$0.789125\pi$
$60$	0	0
$61$	2.81362e6i	1.58712i	0.608489	+	0.793562i	$0.291776\pi$
$61$	2.81362e6i	1.58712i	−0.608489	+	0.793562i	$0.708224\pi$
$62$	0	0
$63$	−624111.	−0.314463
$64$	0	0
$65$	−2.48038e6	−1.12027
$66$	0	0
$67$	− 2.79519e6i	− 1.13540i	−0.823235	−	0.567701i	$-0.807833\pi$
$67$	− 2.79519e6i	− 1.13540i	0.823235	−	0.567701i	$-0.192167\pi$
$68$	0	0
$69$	− 179289.i	− 0.0657025i
$70$	0	0
$71$	5.11177e6	1.69499	0.847495	−	0.530804i	$-0.178110\pi$
$71$	5.11177e6	1.69499	0.847495	+	0.530804i	$0.178110\pi$
$72$	0	0
$73$	−1.14982e6	−0.345938	−0.172969	−	0.984927i	$-0.555336\pi$
$73$	−1.14982e6	−0.345938	−0.172969	+	0.984927i	$0.555336\pi$
$74$	0	0
$75$	− 1.19675e6i	− 0.327559i
$76$	0	0
$77$	6.39341e6i	1.59593i
$78$	0	0
$79$	2.23303e6	0.509566	0.254783	−	0.966998i	$-0.417996\pi$
$79$	2.23303e6	0.509566	0.254783	+	0.966998i	$0.417996\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	− 8.74755e6i	− 1.67924i	−0.543174	−	0.839620i	$-0.682777\pi$
$83$	− 8.74755e6i	− 1.67924i	0.543174	−	0.839620i	$-0.317223\pi$
$84$	0	0
$85$	− 1.06922e7i	− 1.88842i
$86$	0	0
$87$	5.84040e6	0.950880
$88$	0	0
$89$	6.73480e6	1.01265	0.506326	−	0.862342i	$-0.331003\pi$
$89$	6.73480e6	1.01265	0.506326	+	0.862342i	$0.331003\pi$
$90$	0	0
$91$	6.06840e6i	0.844168i
$92$	0	0
$93$	− 4.35275e6i	− 0.561142i
$94$	0	0
$95$	−5.68709e6	−0.680546
$96$	0	0
$97$	−6.60143e6	−0.734407	−0.367204	−	0.930141i	$-0.619685\pi$
$97$	−6.60143e6	−0.734407	−0.367204	+	0.930141i	$0.619685\pi$
$98$	0	0
$99$	− 5.44410e6i	− 0.563901i
$100$	0	0
$101$	4.93817e6i	0.476915i	0.971153	+	0.238458i	$0.0766418\pi$
$101$	4.93817e6i	0.476915i	−0.971153	+	0.238458i	$0.923358\pi$
$102$	0	0
$103$	−1.50259e7	−1.35491	−0.677454	−	0.735565i	$-0.736917\pi$
$103$	−1.50259e7	−1.35491	−0.677454	+	0.735565i	$0.736917\pi$
$104$	0	0
$105$	−8.08865e6	−0.681888
$106$	0	0
$107$	− 2.17099e7i	− 1.71322i	−0.515961	−	0.856612i	$-0.672565\pi$
$107$	− 2.17099e7i	− 1.71322i	0.515961	−	0.856612i	$-0.327435\pi$
$108$	0	0
$109$	− 6.16840e6i	− 0.456226i	−0.973635	−	0.228113i	$-0.926744\pi$
$109$	− 6.16840e6i	− 0.456226i	0.973635	−	0.228113i	$-0.0732556\pi$
$110$	0	0
$111$	−2.75686e6	−0.191330
$112$	0	0
$113$	2.01906e7	1.31636	0.658180	−	0.752861i	$-0.271327\pi$
$113$	2.01906e7	1.31636	0.658180	+	0.752861i	$0.271327\pi$
$114$	0	0
$115$	− 2.32364e6i	− 0.142471i
$116$	0	0
$117$	− 5.16734e6i	− 0.298275i
$118$	0	0
$119$	−2.61591e7	−1.42301
$120$	0	0
$121$	−3.62823e7	−1.86185
$122$	0	0
$123$	1.22331e7i	0.592747i
$124$	0	0
$125$	1.18278e7i	0.541652i
$126$	0	0
$127$	6.65226e6	0.288175	0.144088	−	0.989565i	$-0.453975\pi$
$127$	6.65226e6	0.288175	0.144088	+	0.989565i	$0.453975\pi$
$128$	0	0
$129$	−1.47234e7	−0.603873
$130$	0	0
$131$	6.09598e6i	0.236916i	0.992959	+	0.118458i	$0.0377951\pi$
$131$	6.09598e6i	0.236916i	−0.992959	+	0.118458i	$0.962205\pi$
$132$	0	0
$133$	1.39138e7i	0.512821i
$134$	0	0
$135$	6.88762e6	0.240936
$136$	0	0
$137$	4.57811e7	1.52112	0.760561	−	0.649266i	$-0.224924\pi$
$137$	4.57811e7	1.52112	0.760561	+	0.649266i	$0.224924\pi$
$138$	0	0
$139$	− 5.69460e6i	− 0.179850i	−0.995949	−	0.0899251i	$-0.971337\pi$
$139$	− 5.69460e6i	− 0.179850i	0.995949	−	0.0899251i	$-0.0286628\pi$
$140$	0	0
$141$	5.73241e6i	0.172215i
$142$	0	0
$143$	−5.29344e7	−1.51378
$144$	0	0
$145$	7.56932e7	2.06191
$146$	0	0
$147$	− 2.44629e6i	− 0.0635180i
$148$	0	0
$149$	− 2.05029e7i	− 0.507766i	−0.967235	−	0.253883i	$-0.918292\pi$
$149$	− 2.05029e7i	− 0.507766i	0.967235	−	0.253883i	$-0.0817078\pi$
$150$	0	0
$151$	−6.71513e7	−1.58721	−0.793606	−	0.608432i	$-0.791799\pi$
$151$	−6.71513e7	−1.58721	−0.793606	+	0.608432i	$0.791799\pi$
$152$	0	0
$153$	2.22749e7	0.502800
$154$	0	0
$155$	− 5.64128e7i	− 1.21679i
$156$	0	0
$157$	− 2.43450e7i	− 0.502066i	−0.967978	−	0.251033i	$-0.919230\pi$
$157$	− 2.43450e7i	− 0.502066i	0.967978	−	0.251033i	$-0.0807703\pi$
$158$	0	0
$159$	−3.30767e7	−0.652577
$160$	0	0
$161$	−5.68492e6	−0.107358
$162$	0	0
$163$	− 6.19753e7i	− 1.12089i	−0.828192	−	0.560444i	$-0.810630\pi$
$163$	− 6.19753e7i	− 1.12089i	0.828192	−	0.560444i	$-0.189370\pi$
$164$	0	0
$165$	− 7.05570e7i	− 1.22277i
$166$	0	0
$167$	−3.69791e7	−0.614397	−0.307198	−	0.951646i	$-0.599392\pi$
$167$	−3.69791e7	−0.614397	−0.307198	+	0.951646i	$0.599392\pi$
$168$	0	0
$169$	1.25050e7	0.199288
$170$	0	0
$171$	− 1.18479e7i	− 0.181198i
$172$	0	0
$173$	7.27428e7i	1.06814i	0.845440	+	0.534071i	$0.179338\pi$
$173$	7.27428e7i	1.06814i	−0.845440	+	0.534071i	$0.820662\pi$
$174$	0	0
$175$	−3.79468e7	−0.535231
$176$	0	0
$177$	−5.23969e7	−0.710230
$178$	0	0
$179$	8.54503e7i	1.11360i	0.830648	+	0.556799i	$0.187970\pi$
$179$	8.54503e7i	1.11360i	−0.830648	+	0.556799i	$0.812030\pi$
$180$	0	0
$181$	− 6.78832e7i	− 0.850917i	−0.904978	−	0.425458i	$-0.860113\pi$
$181$	− 6.78832e7i	− 0.850917i	0.904978	−	0.425458i	$-0.139887\pi$
$182$	0	0
$183$	−7.59677e7	−0.916327
$184$	0	0
$185$	−3.57296e7	−0.414884
$186$	0	0
$187$	− 2.28185e8i	− 2.55176i
$188$	0	0
$189$	− 1.68510e7i	− 0.181555i
$190$	0	0
$191$	1.06405e8	1.10495	0.552476	−	0.833529i	$-0.313683\pi$
$191$	1.06405e8	1.10495	0.552476	+	0.833529i	$0.313683\pi$
$192$	0	0
$193$	−1.19393e8	−1.19544	−0.597720	−	0.801705i	$-0.703927\pi$
$193$	−1.19393e8	−1.19544	−0.597720	+	0.801705i	$0.703927\pi$
$194$	0	0
$195$	− 6.69702e7i	− 0.646786i
$196$	0	0
$197$	− 6.78895e6i	− 0.0632660i	−0.999500	−	0.0316330i	$-0.989929\pi$
$197$	− 6.78895e6i	− 0.0632660i	0.999500	−	0.0316330i	$-0.0100708\pi$
$198$	0	0
$199$	1.59438e7	0.143418	0.0717092	−	0.997426i	$-0.477155\pi$
$199$	1.59438e7	0.143418	0.0717092	+	0.997426i	$0.477155\pi$
$200$	0	0
$201$	7.54702e7	0.655525
$202$	0	0
$203$	− 1.85188e8i	− 1.55374i
$204$	0	0
$205$	1.58545e8i	1.28533i
$206$	0	0
$207$	4.84081e6	0.0379334
$208$	0	0
$209$	−1.21370e8	−0.919599
$210$	0	0
$211$	− 2.09686e8i	− 1.53667i	−0.640048	−	0.768335i	$-0.721086\pi$
$211$	− 2.09686e8i	− 1.53667i	0.640048	−	0.768335i	$-0.278914\pi$
$212$	0	0
$213$	1.38018e8i	0.978602i
$214$	0	0
$215$	−1.90820e8	−1.30945
$216$	0	0
$217$	−1.38017e8	−0.916906
$218$	0	0
$219$	− 3.10450e7i	− 0.199727i
$220$	0	0
$221$	− 2.16585e8i	− 1.34975i
$222$	0	0
$223$	−5.34173e7	−0.322563	−0.161281	−	0.986908i	$-0.551563\pi$
$223$	−5.34173e7	−0.322563	−0.161281	+	0.986908i	$0.551563\pi$
$224$	0	0
$225$	3.23123e7	0.189116
$226$	0	0
$227$	− 1.79538e8i	− 1.01874i	−0.860546	−	0.509372i	$-0.829878\pi$
$227$	− 1.79538e8i	− 1.01874i	0.860546	−	0.509372i	$-0.170122\pi$
$228$	0	0
$229$	1.76192e8i	0.969532i	0.874644	+	0.484766i	$0.161095\pi$
$229$	1.76192e8i	0.969532i	−0.874644	+	0.484766i	$0.838905\pi$
$230$	0	0
$231$	−1.72622e8	−0.921413
$232$	0	0
$233$	1.43237e7	0.0741841	0.0370920	−	0.999312i	$-0.488191\pi$
$233$	1.43237e7	0.0741841	0.0370920	+	0.999312i	$0.488191\pi$
$234$	0	0
$235$	7.42936e7i	0.373434i
$236$	0	0
$237$	6.02919e7i	0.294198i
$238$	0	0
$239$	−1.97318e8	−0.934917	−0.467458	−	0.884015i	$-0.654830\pi$
$239$	−1.97318e8	−0.934917	−0.467458	+	0.884015i	$0.654830\pi$
$240$	0	0
$241$	2.79237e8	1.28503	0.642515	−	0.766273i	$-0.277891\pi$
$241$	2.79237e8	1.28503	0.642515	+	0.766273i	$0.277891\pi$
$242$	0	0
$243$	1.43489e7i	0.0641500i
$244$	0	0
$245$	− 3.17046e7i	− 0.137734i
$246$	0	0
$247$	−1.15200e8	−0.486422
$248$	0	0
$249$	2.36184e8	0.969510
$250$	0	0
$251$	2.11424e7i	0.0843909i	0.999109	+	0.0421954i	$0.0134352\pi$
$251$	2.11424e7i	0.0843909i	−0.999109	+	0.0421954i	$0.986565\pi$
$252$	0	0
$253$	− 4.95894e7i	− 0.192516i
$254$	0	0
$255$	2.88689e8	1.09028
$256$	0	0
$257$	3.27572e7	0.120376	0.0601882	−	0.998187i	$-0.480830\pi$
$257$	3.27572e7	0.120376	0.0601882	+	0.998187i	$0.480830\pi$
$258$	0	0
$259$	8.74147e7i	0.312633i
$260$	0	0
$261$	1.57691e8i	0.548991i
$262$	0	0
$263$	−4.61730e8	−1.56510	−0.782552	−	0.622586i	$-0.786082\pi$
$263$	−4.61730e8	−1.56510	−0.782552	+	0.622586i	$0.786082\pi$
$264$	0	0
$265$	−4.28683e8	−1.41506
$266$	0	0
$267$	1.81840e8i	0.584655i
$268$	0	0
$269$	− 3.27505e8i	− 1.02585i	−0.858433	−	0.512926i	$-0.828562\pi$
$269$	− 3.27505e8i	− 1.02585i	0.858433	−	0.512926i	$-0.171438\pi$
$270$	0	0
$271$	−4.81942e8	−1.47097	−0.735483	−	0.677543i	$-0.763045\pi$
$271$	−4.81942e8	−1.47097	−0.735483	+	0.677543i	$0.763045\pi$
$272$	0	0
$273$	−1.63847e8	−0.487381
$274$	0	0
$275$	− 3.31008e8i	− 0.959786i
$276$	0	0
$277$	3.40859e8i	0.963597i	0.876282	+	0.481798i	$0.160016\pi$
$277$	3.40859e8i	0.963597i	−0.876282	+	0.481798i	$0.839984\pi$
$278$	0	0
$279$	1.17524e8	0.323976
$280$	0	0
$281$	5.49375e7	0.147705	0.0738527	−	0.997269i	$-0.476471\pi$
$281$	5.49375e7	0.147705	0.0738527	+	0.997269i	$0.476471\pi$
$282$	0	0
$283$	5.31794e8i	1.39473i	0.716716	+	0.697366i	$0.245645\pi$
$283$	5.31794e8i	1.39473i	−0.716716	+	0.697366i	$0.754355\pi$
$284$	0	0
$285$	− 1.53551e8i	− 0.392914i
$286$	0	0
$287$	3.87889e8	0.968548
$288$	0	0
$289$	5.23294e8	1.27527
$290$	0	0
$291$	− 1.78239e8i	− 0.424010i
$292$	0	0
$293$	1.62432e8i	0.377256i	0.982049	+	0.188628i	$0.0604040\pi$
$293$	1.62432e8i	0.377256i	−0.982049	+	0.188628i	$0.939596\pi$
$294$	0	0
$295$	−6.79078e8	−1.54008
$296$	0	0
$297$	1.46991e8	0.325568
$298$	0	0
$299$	− 4.70685e7i	− 0.101831i
$300$	0	0
$301$	4.66853e8i	0.986727i
$302$	0	0
$303$	−1.33331e8	−0.275347
$304$	0	0
$305$	−9.84563e8	−1.98698
$306$	0	0
$307$	− 3.26529e8i	− 0.644077i	−0.946727	−	0.322039i	$-0.895632\pi$
$307$	− 3.26529e8i	− 0.644077i	0.946727	−	0.322039i	$-0.104368\pi$
$308$	0	0
$309$	− 4.05699e8i	− 0.782257i
$310$	0	0
$311$	−7.48794e8	−1.41156	−0.705782	−	0.708429i	$-0.749404\pi$
$311$	−7.48794e8	−1.41156	−0.705782	+	0.708429i	$0.749404\pi$
$312$	0	0
$313$	−5.11542e8	−0.942923	−0.471461	−	0.881887i	$-0.656273\pi$
$313$	−5.11542e8	−0.942923	−0.471461	+	0.881887i	$0.656273\pi$
$314$	0	0
$315$	− 2.18393e8i	− 0.393688i
$316$	0	0
$317$	− 1.49935e8i	− 0.264360i	−0.991226	−	0.132180i	$-0.957802\pi$
$317$	− 1.49935e8i	− 0.264360i	0.991226	−	0.132180i	$-0.0421977\pi$
$318$	0	0
$319$	1.61539e9	2.78619
$320$	0	0
$321$	5.86167e8	0.989130
$322$	0	0
$323$	− 4.96593e8i	− 0.819958i
$324$	0	0
$325$	− 3.14181e8i	− 0.507678i
$326$	0	0
$327$	1.66547e8	0.263402
$328$	0	0
$329$	1.81764e8	0.281399
$330$	0	0
$331$	− 3.62025e8i	− 0.548707i	−0.961629	−	0.274353i	$-0.911536\pi$
$331$	− 3.62025e8i	− 0.548707i	0.961629	−	0.274353i	$-0.0884638\pi$
$332$	0	0
$333$	− 7.44351e7i	− 0.110465i
$334$	0	0
$335$	9.78114e8	1.42145
$336$	0	0
$337$	−3.58096e6	−0.00509676	−0.00254838	−	0.999997i	$-0.500811\pi$
$337$	−3.58096e6	−0.00509676	−0.00254838	+	0.999997i	$0.500811\pi$
$338$	0	0
$339$	5.45146e8i	0.760001i
$340$	0	0
$341$	− 1.20392e9i	− 1.64421i
$342$	0	0
$343$	−7.82618e8	−1.04718
$344$	0	0
$345$	6.27382e7	0.0822555
$346$	0	0
$347$	9.42728e8i	1.21125i	0.795751	+	0.605624i	$0.207076\pi$
$347$	9.42728e8i	1.21125i	−0.795751	+	0.605624i	$0.792924\pi$
$348$	0	0
$349$	− 4.64187e8i	− 0.584526i	−0.956338	−	0.292263i	$-0.905592\pi$
$349$	− 4.64187e8i	− 0.584526i	0.956338	−	0.292263i	$-0.0944083\pi$
$350$	0	0
$351$	1.39518e8	0.172209
$352$	0	0
$353$	1.35352e9	1.63777	0.818887	−	0.573955i	$-0.194592\pi$
$353$	1.35352e9	1.63777	0.818887	+	0.573955i	$0.194592\pi$
$354$	0	0
$355$	1.78875e9i	2.12202i
$356$	0	0
$357$	− 7.06295e8i	− 0.821575i
$358$	0	0
$359$	1.63713e9	1.86746	0.933731	−	0.357977i	$-0.116533\pi$
$359$	1.63713e9	1.86746	0.933731	+	0.357977i	$0.116533\pi$
$360$	0	0
$361$	6.29738e8	0.704505
$362$	0	0
$363$	− 9.79622e8i	− 1.07494i
$364$	0	0
$365$	− 4.02352e8i	− 0.433093i
$366$	0	0
$367$	7.86689e8	0.830753	0.415376	−	0.909650i	$-0.363650\pi$
$367$	7.86689e8	0.830753	0.415376	+	0.909650i	$0.363650\pi$
$368$	0	0
$369$	−3.30294e8	−0.342223
$370$	0	0
$371$	1.04880e9i	1.06631i
$372$	0	0
$373$	1.69017e9i	1.68636i	0.537631	+	0.843180i	$0.319319\pi$
$373$	1.69017e9i	1.68636i	−0.537631	+	0.843180i	$0.680681\pi$
$374$	0	0
$375$	−3.19352e8	−0.312723
$376$	0	0
$377$	1.53327e9	1.47375
$378$	0	0
$379$	− 1.35599e9i	− 1.27944i	−0.768610	−	0.639718i	$-0.779051\pi$
$379$	− 1.35599e9i	− 1.27944i	0.768610	−	0.639718i	$-0.220949\pi$
$380$	0	0
$381$	1.79611e8i	0.166378i
$382$	0	0
$383$	1.62361e9	1.47668	0.738338	−	0.674431i	$-0.235611\pi$
$383$	1.62361e9	1.47668	0.738338	+	0.674431i	$0.235611\pi$
$384$	0	0
$385$	−2.23723e9	−1.99801
$386$	0	0
$387$	− 3.97533e8i	− 0.348646i
$388$	0	0
$389$	1.86513e9i	1.60652i	0.595630	+	0.803259i	$0.296902\pi$
$389$	1.86513e9i	1.60652i	−0.595630	+	0.803259i	$0.703098\pi$
$390$	0	0
$391$	2.02898e8	0.171656
$392$	0	0
$393$	−1.64592e8	−0.136783
$394$	0	0
$395$	7.81400e8i	0.637945i
$396$	0	0
$397$	6.77670e7i	0.0543565i	0.999631	+	0.0271782i	$0.00865217\pi$
$397$	6.77670e7i	0.0543565i	−0.999631	+	0.0271782i	$0.991348\pi$
$398$	0	0
$399$	−3.75673e8	−0.296077
$400$	0	0
$401$	1.52433e9	1.18052	0.590259	−	0.807214i	$-0.299026\pi$
$401$	1.52433e9	1.18052	0.590259	+	0.807214i	$0.299026\pi$
$402$	0	0
$403$	− 1.14272e9i	− 0.869705i
$404$	0	0
$405$	1.85966e8i	0.139104i
$406$	0	0
$407$	−7.62515e8	−0.560619
$408$	0	0
$409$	−2.22045e9	−1.60476	−0.802379	−	0.596815i	$-0.796433\pi$
$409$	−2.22045e9	−1.60476	−0.802379	+	0.596815i	$0.796433\pi$
$410$	0	0
$411$	1.23609e9i	0.878220i
$412$	0	0
$413$	1.66141e9i	1.16052i
$414$	0	0
$415$	3.06101e9	2.10231
$416$	0	0
$417$	1.53754e8	0.103837
$418$	0	0
$419$	1.85555e9i	1.23232i	0.787620	+	0.616162i	$0.211313\pi$
$419$	1.85555e9i	1.23232i	−0.787620	+	0.616162i	$0.788687\pi$
$420$	0	0
$421$	3.25527e7i	0.0212618i	0.999943	+	0.0106309i	$0.00338398\pi$
$421$	3.25527e7i	0.0212618i	−0.999943	+	0.0106309i	$0.996616\pi$
$422$	0	0
$423$	−1.54775e8	−0.0994282
$424$	0	0
$425$	1.35434e9	0.855790
$426$	0	0
$427$	2.40879e9i	1.49728i
$428$	0	0
$429$	− 1.42923e9i	− 0.873980i
$430$	0	0
$431$	−1.61783e8	−0.0973335	−0.0486667	−	0.998815i	$-0.515497\pi$
$431$	−1.61783e8	−0.0973335	−0.0486667	+	0.998815i	$0.515497\pi$
$432$	0	0
$433$	−3.15622e9	−1.86836	−0.934179	−	0.356806i	$-0.883866\pi$
$433$	−3.15622e9	−1.86836	−0.934179	+	0.356806i	$0.883866\pi$
$434$	0	0
$435$	2.04372e9i	1.19044i
$436$	0	0
$437$	− 1.07920e8i	− 0.0618611i
$438$	0	0
$439$	−2.07384e9	−1.16990	−0.584950	−	0.811070i	$-0.698886\pi$
$439$	−2.07384e9	−1.16990	−0.584950	+	0.811070i	$0.698886\pi$
$440$	0	0
$441$	6.60498e7	0.0366721
$442$	0	0
$443$	8.34953e8i	0.456299i	0.973626	+	0.228149i	$0.0732674\pi$
$443$	8.34953e8i	0.456299i	−0.973626	+	0.228149i	$0.926733\pi$
$444$	0	0
$445$	2.35669e9i	1.26778i
$446$	0	0
$447$	5.53579e8	0.293159
$448$	0	0
$449$	−3.78596e8	−0.197385	−0.0986923	−	0.995118i	$-0.531466\pi$
$449$	−3.78596e8	−0.197385	−0.0986923	+	0.995118i	$0.531466\pi$
$450$	0	0
$451$	3.38354e9i	1.73682i
$452$	0	0
$453$	− 1.81308e9i	− 0.916377i
$454$	0	0
$455$	−2.12350e9	−1.05685
$456$	0	0
$457$	3.86778e8	0.189564	0.0947819	−	0.995498i	$-0.469785\pi$
$457$	3.86778e8	0.189564	0.0947819	+	0.995498i	$0.469785\pi$
$458$	0	0
$459$	6.01422e8i	0.290292i
$460$	0	0
$461$	− 1.81651e9i	− 0.863543i	−0.901983	−	0.431771i	$-0.857889\pi$
$461$	− 1.81651e9i	− 0.863543i	0.901983	−	0.431771i	$-0.142111\pi$
$462$	0	0
$463$	3.73240e9	1.74765	0.873825	−	0.486241i	$-0.161632\pi$
$463$	3.73240e9	1.74765	0.873825	+	0.486241i	$0.161632\pi$
$464$	0	0
$465$	1.52315e9	0.702516
$466$	0	0
$467$	1.07804e9i	0.489808i	0.969547	+	0.244904i	$0.0787564\pi$
$467$	1.07804e9i	0.489808i	−0.969547	+	0.244904i	$0.921244\pi$
$468$	0	0
$469$	− 2.39302e9i	− 1.07113i
$470$	0	0
$471$	6.57315e8	0.289868
$472$	0	0
$473$	−4.07234e9	−1.76942
$474$	0	0
$475$	− 7.20366e8i	− 0.308408i
$476$	0	0
$477$	− 8.93070e8i	− 0.376766i
$478$	0	0
$479$	2.03431e9	0.845753	0.422876	−	0.906187i	$-0.361020\pi$
$479$	2.03431e9	0.845753	0.422876	+	0.906187i	$0.361020\pi$
$480$	0	0
$481$	−7.23753e8	−0.296539
$482$	0	0
$483$	− 1.53493e8i	− 0.0619831i
$484$	0	0
$485$	− 2.31002e9i	− 0.919433i
$486$	0	0
$487$	5.11664e8	0.200740	0.100370	−	0.994950i	$-0.467997\pi$
$487$	5.11664e8	0.200740	0.100370	+	0.994950i	$0.467997\pi$
$488$	0	0
$489$	1.67333e9	0.647145
$490$	0	0
$491$	3.49186e8i	0.133129i	0.997782	+	0.0665644i	$0.0212038\pi$
$491$	3.49186e8i	0.133129i	−0.997782	+	0.0665644i	$0.978796\pi$
$492$	0	0
$493$	6.60948e9i	2.48429i
$494$	0	0
$495$	1.90504e9	0.705969
$496$	0	0
$497$	4.37628e9	1.59903
$498$	0	0
$499$	8.31193e7i	0.0299468i	0.999888	+	0.0149734i	$0.00476635\pi$
$499$	8.31193e7i	0.0299468i	−0.999888	+	0.0149734i	$0.995234\pi$
$500$	0	0
$501$	− 9.98436e8i	− 0.354722i
$502$	0	0
$503$	−2.63771e9	−0.924144	−0.462072	−	0.886842i	$-0.652894\pi$
$503$	−2.63771e9	−0.924144	−0.462072	+	0.886842i	$0.652894\pi$
$504$	0	0
$505$	−1.72800e9	−0.597069
$506$	0	0
$507$	3.37636e8i	0.115059i
$508$	0	0
$509$	7.45908e8i	0.250711i	0.992112	+	0.125355i	$0.0400071\pi$
$509$	7.45908e8i	0.250711i	−0.992112	+	0.125355i	$0.959993\pi$
$510$	0	0
$511$	−9.84379e8	−0.326354
$512$	0	0
$513$	3.19892e8	0.104615
$514$	0	0
$515$	− 5.25797e9i	− 1.69626i
$516$	0	0
$517$	1.58552e9i	0.504609i
$518$	0	0
$519$	−1.96406e9	−0.616692
$520$	0	0
$521$	1.31492e8	0.0407350	0.0203675	−	0.999793i	$-0.493516\pi$
$521$	1.31492e8	0.0407350	0.0203675	+	0.999793i	$0.493516\pi$
$522$	0	0
$523$	2.76999e9i	0.846684i	0.905970	+	0.423342i	$0.139143\pi$
$523$	2.76999e9i	0.846684i	−0.905970	+	0.423342i	$0.860857\pi$
$524$	0	0
$525$	− 1.02456e9i	− 0.309016i
$526$	0	0
$527$	4.92592e9	1.46606
$528$	0	0
$529$	−3.36073e9	−0.987050
$530$	0	0
$531$	− 1.41472e9i	− 0.410052i
$532$	0	0
$533$	3.21154e9i	0.918689i
$534$	0	0
$535$	7.59688e9	2.14485
$536$	0	0
$537$	−2.30716e9	−0.642936
$538$	0	0
$539$	− 6.76616e8i	− 0.186115i
$540$	0	0
$541$	3.26395e9i	0.886244i	0.896461	+	0.443122i	$0.146129\pi$
$541$	3.26395e9i	0.886244i	−0.896461	+	0.443122i	$0.853871\pi$
$542$	0	0
$543$	1.83285e9	0.491277
$544$	0	0
$545$	2.15849e9	0.571167
$546$	0	0
$547$	− 3.11944e9i	− 0.814931i	−0.913221	−	0.407466i	$-0.866413\pi$
$547$	− 3.11944e9i	− 0.814931i	0.913221	−	0.407466i	$-0.133587\pi$
$548$	0	0
$549$	− 2.05113e9i	− 0.529041i
$550$	0	0
$551$	3.51554e9	0.895284
$552$	0	0
$553$	1.91174e9	0.480719
$554$	0	0
$555$	− 9.64699e8i	− 0.239534i
$556$	0	0
$557$	− 7.07578e9i	− 1.73493i	−0.497500	−	0.867464i	$-0.665749\pi$
$557$	− 7.07578e9i	− 1.73493i	0.497500	−	0.867464i	$-0.334251\pi$
$558$	0	0
$559$	−3.86532e9	−0.935932
$560$	0	0
$561$	6.16098e9	1.47326
$562$	0	0
$563$	3.22997e9i	0.762815i	0.924407	+	0.381408i	$0.124561\pi$
$563$	3.22997e9i	0.762815i	−0.924407	+	0.381408i	$0.875439\pi$
$564$	0	0
$565$	7.06524e9i	1.64800i
$566$	0	0
$567$	4.54977e8	0.104821
$568$	0	0
$569$	6.84349e7	0.0155734	0.00778672	−	0.999970i	$-0.497521\pi$
$569$	6.84349e7	0.0155734	0.00778672	+	0.999970i	$0.497521\pi$
$570$	0	0
$571$	4.35493e9i	0.978937i	0.872021	+	0.489469i	$0.162809\pi$
$571$	4.35493e9i	0.978937i	−0.872021	+	0.489469i	$0.837191\pi$
$572$	0	0
$573$	2.87292e9i	0.637945i
$574$	0	0
$575$	2.94328e8	0.0645644
$576$	0	0
$577$	−6.05562e9	−1.31233	−0.656165	−	0.754617i	$-0.727823\pi$
$577$	−6.05562e9	−1.31233	−0.656165	+	0.754617i	$0.727823\pi$
$578$	0	0
$579$	− 3.22361e9i	− 0.690188i
$580$	0	0
$581$	− 7.48894e9i	− 1.58418i
$582$	0	0
$583$	−9.14863e9	−1.91213
$584$	0	0
$585$	1.80820e9	0.373422
$586$	0	0
$587$	− 3.45962e9i	− 0.705984i	−0.935626	−	0.352992i	$-0.885164\pi$
$587$	− 3.45962e9i	− 0.705984i	0.935626	−	0.352992i	$-0.114836\pi$
$588$	0	0
$589$	− 2.62006e9i	− 0.528334i
$590$	0	0
$591$	1.83302e8	0.0365267
$592$	0	0
$593$	1.46699e8	0.0288893	0.0144446	−	0.999896i	$-0.495402\pi$
$593$	1.46699e8	0.0288893	0.0144446	+	0.999896i	$0.495402\pi$
$594$	0	0
$595$	− 9.15377e9i	− 1.78152i
$596$	0	0
$597$	4.30482e8i	0.0828027i
$598$	0	0
$599$	−7.05880e9	−1.34195	−0.670976	−	0.741479i	$-0.734125\pi$
$599$	−7.05880e9	−1.34195	−0.670976	+	0.741479i	$0.734125\pi$
$600$	0	0
$601$	4.96971e9	0.933835	0.466918	−	0.884301i	$-0.345364\pi$
$601$	4.96971e9	0.933835	0.466918	+	0.884301i	$0.345364\pi$
$602$	0	0
$603$	2.03769e9i	0.378467i
$604$	0	0
$605$	− 1.26962e10i	− 2.33093i
$606$	0	0
$607$	−7.27371e9	−1.32006	−0.660032	−	0.751237i	$-0.729457\pi$
$607$	−7.27371e9	−1.32006	−0.660032	+	0.751237i	$0.729457\pi$
$608$	0	0
$609$	5.00008e9	0.897050
$610$	0	0
$611$	1.50492e9i	0.266913i
$612$	0	0
$613$	1.02845e10i	1.80331i	0.432453	+	0.901656i	$0.357648\pi$
$613$	1.02845e10i	1.80331i	−0.432453	+	0.901656i	$0.642352\pi$
$614$	0	0
$615$	−4.28071e9	−0.742083
$616$	0	0
$617$	9.60238e9	1.64581	0.822907	−	0.568175i	$-0.192351\pi$
$617$	9.60238e9	1.64581	0.822907	+	0.568175i	$0.192351\pi$
$618$	0	0
$619$	− 6.07813e9i	− 1.03004i	−0.857179	−	0.515018i	$-0.827785\pi$
$619$	− 6.07813e9i	− 1.03004i	0.857179	−	0.515018i	$-0.172215\pi$
$620$	0	0
$621$	1.30702e8i	0.0219008i
$622$	0	0
$623$	5.76579e9	0.955325
$624$	0	0
$625$	−7.60171e9	−1.24546
$626$	0	0
$627$	− 3.27698e9i	− 0.530931i
$628$	0	0
$629$	− 3.11988e9i	− 0.499875i
$630$	0	0
$631$	−3.61125e9	−0.572210	−0.286105	−	0.958198i	$-0.592361\pi$
$631$	−3.61125e9	−0.572210	−0.286105	+	0.958198i	$0.592361\pi$
$632$	0	0
$633$	5.66152e9	0.887197
$634$	0	0
$635$	2.32781e9i	0.360777i
$636$	0	0
$637$	− 6.42220e8i	− 0.0984455i
$638$	0	0
$639$	−3.72648e9	−0.564996
$640$	0	0
$641$	−6.07091e9	−0.910439	−0.455219	−	0.890379i	$-0.650439\pi$
$641$	−6.07091e9	−0.910439	−0.455219	+	0.890379i	$0.650439\pi$
$642$	0	0
$643$	8.23573e9i	1.22170i	0.791747	+	0.610849i	$0.209172\pi$
$643$	8.23573e9i	1.22170i	−0.791747	+	0.610849i	$0.790828\pi$
$644$	0	0
$645$	− 5.15214e9i	− 0.756011i
$646$	0	0
$647$	3.84864e9	0.558653	0.279326	−	0.960196i	$-0.409889\pi$
$647$	3.84864e9	0.558653	0.279326	+	0.960196i	$0.409889\pi$
$648$	0	0
$649$	−1.44924e10	−2.08106
$650$	0	0
$651$	− 3.72647e9i	− 0.529376i
$652$	0	0
$653$	− 2.49171e9i	− 0.350188i	−0.984552	−	0.175094i	$-0.943977\pi$
$653$	− 2.49171e9i	− 0.350188i	0.984552	−	0.175094i	$-0.0560229\pi$
$654$	0	0
$655$	−2.13315e9	−0.296604
$656$	0	0
$657$	8.38216e8	0.115313
$658$	0	0
$659$	1.48339e9i	0.201909i	0.994891	+	0.100955i	$0.0321897\pi$
$659$	1.48339e9i	0.201909i	−0.994891	+	0.100955i	$0.967810\pi$
$660$	0	0
$661$	− 7.01612e9i	− 0.944914i	−0.881354	−	0.472457i	$-0.843367\pi$
$661$	− 7.01612e9i	− 0.944914i	0.881354	−	0.472457i	$-0.156633\pi$
$662$	0	0
$663$	5.84779e9	0.779281
$664$	0	0
$665$	−4.86883e9	−0.642020
$666$	0	0
$667$	1.43638e9i	0.187426i
$668$	0	0
$669$	− 1.44227e9i	− 0.186232i
$670$	0	0
$671$	−2.10118e10	−2.68494
$672$	0	0
$673$	9.91585e9	1.25394	0.626971	−	0.779043i	$-0.284295\pi$
$673$	9.91585e9	1.25394	0.626971	+	0.779043i	$0.284295\pi$
$674$	0	0
$675$	8.72433e8i	0.109186i
$676$	0	0
$677$	8.67366e9i	1.07434i	0.843474	+	0.537170i	$0.180507\pi$
$677$	8.67366e9i	1.07434i	−0.843474	+	0.537170i	$0.819493\pi$
$678$	0	0
$679$	−5.65161e9	−0.692832
$680$	0	0
$681$	4.84752e9	0.588173
$682$	0	0
$683$	− 1.33330e10i	− 1.60123i	−0.599178	−	0.800616i	$-0.704506\pi$
$683$	− 1.33330e10i	− 1.60123i	0.599178	−	0.800616i	$-0.295494\pi$
$684$	0	0
$685$	1.60201e10i	1.90435i
$686$	0	0
$687$	−4.75719e9	−0.559760
$688$	0	0
$689$	−8.68356e9	−1.01142
$690$	0	0
$691$	1.26732e9i	0.146122i	0.997327	+	0.0730608i	$0.0232767\pi$
$691$	1.26732e9i	0.146122i	−0.997327	+	0.0730608i	$0.976723\pi$
$692$	0	0
$693$	− 4.66079e9i	− 0.531978i
$694$	0	0
$695$	1.99270e9	0.225161
$696$	0	0
$697$	−1.38440e10	−1.54863
$698$	0	0
$699$	3.86741e8i	0.0428302i
$700$	0	0
$701$	8.74112e9i	0.958416i	0.877701	+	0.479208i	$0.159076\pi$
$701$	8.74112e9i	0.958416i	−0.877701	+	0.479208i	$0.840924\pi$
$702$	0	0
$703$	−1.65944e9	−0.180144
$704$	0	0
$705$	−2.00593e9	−0.215602
$706$	0	0
$707$	4.22766e9i	0.449917i
$708$	0	0
$709$	1.12357e10i	1.18397i	0.805951	+	0.591983i	$0.201655\pi$
$709$	1.12357e10i	1.18397i	−0.805951	+	0.591983i	$0.798345\pi$
$710$	0	0
$711$	−1.62788e9	−0.169855
$712$	0	0
$713$	1.07051e9	0.110605
$714$	0	0
$715$	− 1.85232e10i	− 1.89516i
$716$	0	0
$717$	− 5.32757e9i	− 0.539775i
$718$	0	0
$719$	−1.84643e10	−1.85260	−0.926298	−	0.376791i	$-0.877028\pi$
$719$	−1.84643e10	−1.85260	−0.926298	+	0.376791i	$0.877028\pi$
$720$	0	0
$721$	−1.28639e10	−1.27821
$722$	0	0
$723$	7.53940e9i	0.741913i
$724$	0	0
$725$	9.58782e9i	0.934409i
$726$	0	0
$727$	4.87792e9	0.470830	0.235415	−	0.971895i	$-0.424355\pi$
$727$	4.87792e9	0.470830	0.235415	+	0.971895i	$0.424355\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	− 1.66623e10i	− 1.57769i
$732$	0	0
$733$	3.61813e8i	0.0339329i	0.999856	+	0.0169664i	$0.00540084\pi$
$733$	3.61813e8i	0.0339329i	−0.999856	+	0.0169664i	$0.994599\pi$
$734$	0	0
$735$	8.56023e8	0.0795206
$736$	0	0
$737$	2.08742e10	1.92076
$738$	0	0
$739$	− 1.33914e10i	− 1.22059i	−0.792172	−	0.610297i	$-0.791050\pi$
$739$	− 1.33914e10i	− 1.22059i	0.792172	−	0.610297i	$-0.208950\pi$
$740$	0	0
$741$	− 3.11040e9i	− 0.280836i
$742$	0	0
$743$	−1.06383e10	−0.951504	−0.475752	−	0.879580i	$-0.657824\pi$
$743$	−1.06383e10	−0.951504	−0.475752	+	0.879580i	$0.657824\pi$
$744$	0	0
$745$	7.17453e9	0.635692
$746$	0	0
$747$	6.37696e9i	0.559747i
$748$	0	0
$749$	− 1.85862e10i	− 1.61624i
$750$	0	0
$751$	1.45550e10	1.25393	0.626964	−	0.779048i	$-0.284297\pi$
$751$	1.45550e10	1.25393	0.626964	+	0.779048i	$0.284297\pi$
$752$	0	0
$753$	−5.70844e8	−0.0487231
$754$	0	0
$755$	− 2.34981e10i	− 1.98709i
$756$	0	0
$757$	− 2.11332e9i	− 0.177064i	−0.996073	−	0.0885321i	$-0.971782\pi$
$757$	− 2.11332e9i	− 0.177064i	0.996073	−	0.0885321i	$-0.0282176\pi$
$758$	0	0
$759$	1.33891e9	0.111149
$760$	0	0
$761$	1.32240e10	1.08772	0.543859	−	0.839177i	$-0.316963\pi$
$761$	1.32240e10	1.08772	0.543859	+	0.839177i	$0.316963\pi$
$762$	0	0
$763$	− 5.28089e9i	− 0.430399i
$764$	0	0
$765$	7.79459e9i	0.629475i
$766$	0	0
$767$	−1.37557e10	−1.10077
$768$	0	0
$769$	9.85486e9	0.781463	0.390731	−	0.920505i	$-0.372222\pi$
$769$	9.85486e9	0.781463	0.390731	+	0.920505i	$0.372222\pi$
$770$	0	0
$771$	8.84445e8i	0.0694993i
$772$	0	0
$773$	1.66051e9i	0.129305i	0.997908	+	0.0646523i	$0.0205938\pi$
$773$	1.66051e9i	0.129305i	−0.997908	+	0.0646523i	$0.979406\pi$
$774$	0	0
$775$	7.14563e9	0.551422
$776$	0	0
$777$	−2.36020e9	−0.180499
$778$	0	0
$779$	7.36353e9i	0.558091i
$780$	0	0
$781$	3.81741e10i	2.86742i
$782$	0	0
$783$	−4.25765e9	−0.316960
$784$	0	0
$785$	8.51898e9	0.628556
$786$	0	0
$787$	3.94514e9i	0.288503i	0.989541	+	0.144252i	$0.0460775\pi$
$787$	3.94514e9i	0.288503i	−0.989541	+	0.144252i	$0.953922\pi$
$788$	0	0
$789$	− 1.24667e10i	− 0.903613i
$790$	0	0
$791$	1.72856e10	1.24184
$792$	0	0
$793$	−1.99437e10	−1.42020
$794$	0	0
$795$	− 1.15744e10i	− 0.816986i
$796$	0	0
$797$	− 2.25092e10i	− 1.57491i	−0.616370	−	0.787457i	$-0.711397\pi$
$797$	− 2.25092e10i	− 1.57491i	0.616370	−	0.787457i	$-0.288603\pi$
$798$	0	0
$799$	−6.48726e9	−0.449933
$800$	0	0
$801$	−4.90967e9	−0.337550
$802$	0	0
$803$	− 8.58671e9i	− 0.585224i
$804$	0	0
$805$	− 1.98931e9i	− 0.134405i
$806$	0	0
$807$	8.84262e9	0.592275
$808$	0	0
$809$	−2.75203e9	−0.182740	−0.0913698	−	0.995817i	$-0.529125\pi$
$809$	−2.75203e9	−0.182740	−0.0913698	+	0.995817i	$0.529125\pi$
$810$	0	0
$811$	− 3.00116e9i	− 0.197568i	−0.995109	−	0.0987840i	$-0.968505\pi$
$811$	− 3.00116e9i	− 0.197568i	0.995109	−	0.0987840i	$-0.0314953\pi$
$812$	0	0
$813$	− 1.30124e10i	− 0.849263i
$814$	0	0
$815$	2.16869e10	1.40328
$816$	0	0
$817$	−8.86254e9	−0.568566
$818$	0	0
$819$	− 4.42386e9i	− 0.281389i
$820$	0	0
$821$	2.30714e10i	1.45504i	0.686089	+	0.727518i	$0.259326\pi$
$821$	2.30714e10i	1.45504i	−0.686089	+	0.727518i	$0.740674\pi$
$822$	0	0
$823$	3.89647e9	0.243653	0.121826	−	0.992551i	$-0.461125\pi$
$823$	3.89647e9	0.243653	0.121826	+	0.992551i	$0.461125\pi$
$824$	0	0
$825$	8.93722e9	0.554133
$826$	0	0
$827$	2.68162e10i	1.64865i	0.566117	+	0.824325i	$0.308445\pi$
$827$	2.68162e10i	1.64865i	−0.566117	+	0.824325i	$0.691555\pi$
$828$	0	0
$829$	− 2.91105e10i	− 1.77463i	−0.461160	−	0.887317i	$-0.652567\pi$
$829$	− 2.91105e10i	− 1.77463i	0.461160	−	0.887317i	$-0.347433\pi$
$830$	0	0
$831$	−9.20319e9	−0.556333
$832$	0	0
$833$	2.76842e9	0.165949
$834$	0	0
$835$	− 1.29400e10i	− 0.769187i
$836$	0	0
$837$	3.17315e9i	0.187047i
$838$	0	0
$839$	2.24547e10	1.31262	0.656311	−	0.754491i	$-0.272116\pi$
$839$	2.24547e10	1.31262	0.656311	+	0.754491i	$0.272116\pi$
$840$	0	0
$841$	−2.95407e10	−1.71252
$842$	0	0
$843$	1.48331e9i	0.0852778i
$844$	0	0
$845$	4.37586e9i	0.249497i
$846$	0	0
$847$	−3.10620e10	−1.75645
$848$	0	0
$849$	−1.43584e10	−0.805249
$850$	0	0
$851$	− 6.78017e8i	− 0.0377126i
$852$	0	0
$853$	− 1.46232e10i	− 0.806715i	−0.915043	−	0.403357i	$-0.867843\pi$
$853$	− 1.46232e10i	− 0.806715i	0.915043	−	0.403357i	$-0.132157\pi$
$854$	0	0
$855$	4.14589e9	0.226849
$856$	0	0
$857$	7.23536e9	0.392670	0.196335	−	0.980537i	$-0.437096\pi$
$857$	7.23536e9	0.392670	0.196335	+	0.980537i	$0.437096\pi$
$858$	0	0
$859$	7.63577e9i	0.411033i	0.978654	+	0.205516i	$0.0658874\pi$
$859$	7.63577e9i	0.411033i	−0.978654	+	0.205516i	$0.934113\pi$
$860$	0	0
$861$	1.04730e10i	0.559192i
$862$	0	0
$863$	−2.72344e10	−1.44238	−0.721190	−	0.692737i	$-0.756405\pi$
$863$	−2.72344e10	−1.44238	−0.721190	+	0.692737i	$0.756405\pi$
$864$	0	0
$865$	−2.54547e10	−1.33725
$866$	0	0
$867$	1.41289e10i	0.736279i
$868$	0	0
$869$	1.66761e10i	0.862034i
$870$	0	0
$871$	1.98131e10	1.01599
$872$	0	0
$873$	4.81244e9	0.244802
$874$	0	0
$875$	1.01260e10i	0.510989i
$876$	0	0
$877$	2.14283e10i	1.07273i	0.843987	+	0.536363i	$0.180202\pi$
$877$	2.14283e10i	1.07273i	−0.843987	+	0.536363i	$0.819798\pi$
$878$	0	0
$879$	−4.38568e9	−0.217809
$880$	0	0
$881$	9.87394e9	0.486491	0.243245	−	0.969965i	$-0.421788\pi$
$881$	9.87394e9	0.486491	0.243245	+	0.969965i	$0.421788\pi$
$882$	0	0
$883$	1.75511e10i	0.857909i	0.903326	+	0.428954i	$0.141118\pi$
$883$	1.75511e10i	0.857909i	−0.903326	+	0.428954i	$0.858882\pi$
$884$	0	0
$885$	− 1.83351e10i	− 0.889164i
$886$	0	0
$887$	−2.19189e10	−1.05460	−0.527298	−	0.849680i	$-0.676795\pi$
$887$	−2.19189e10	−1.05460	−0.527298	+	0.849680i	$0.676795\pi$
$888$	0	0
$889$	5.69513e9	0.271861
$890$	0	0
$891$	3.96875e9i	0.187967i
$892$	0	0
$893$	3.45053e9i	0.162146i
$894$	0	0
$895$	−2.99014e10	−1.39415
$896$	0	0
$897$	1.27085e9	0.0587923
$898$	0	0
$899$	3.48722e10i	1.60074i
$900$	0	0
$901$	− 3.74323e10i	− 1.70494i
$902$	0	0
$903$	−1.26050e10	−0.569687
$904$	0	0
$905$	2.37542e10	1.06530
$906$	0	0
$907$	2.90151e10i	1.29122i	0.763669	+	0.645608i	$0.223396\pi$
$907$	2.90151e10i	1.29122i	−0.763669	+	0.645608i	$0.776604\pi$
$908$	0	0
$909$	− 3.59993e9i	− 0.158972i
$910$	0	0
$911$	−8.49994e8	−0.0372479	−0.0186239	−	0.999827i	$-0.505929\pi$
$911$	−8.49994e8	−0.0372479	−0.0186239	+	0.999827i	$0.505929\pi$
$912$	0	0
$913$	6.53258e10	2.84077
$914$	0	0
$915$	− 2.65832e10i	− 1.14718i
$916$	0	0
$917$	5.21889e9i	0.223504i
$918$	0	0
$919$	2.37664e10	1.01009	0.505043	−	0.863094i	$-0.331477\pi$
$919$	2.37664e10	1.01009	0.505043	+	0.863094i	$0.331477\pi$
$920$	0	0
$921$	8.81629e9	0.371858
$922$	0	0
$923$	3.62336e10i	1.51672i
$924$	0	0
$925$	− 4.52575e9i	− 0.188016i
$926$	0	0
$927$	1.09539e10	0.451636
$928$	0	0
$929$	−2.77346e10	−1.13493	−0.567463	−	0.823399i	$-0.692075\pi$
$929$	−2.77346e10	−1.13493	−0.567463	+	0.823399i	$0.692075\pi$
$930$	0	0
$931$	− 1.47250e9i	− 0.0598043i
$932$	0	0
$933$	− 2.02174e10i	− 0.814967i
$934$	0	0
$935$	7.98480e10	3.19465
$936$	0	0
$937$	1.10536e10	0.438952	0.219476	−	0.975618i	$-0.429565\pi$
$937$	1.10536e10	0.438952	0.219476	+	0.975618i	$0.429565\pi$
$938$	0	0
$939$	− 1.38116e10i	− 0.544397i
$940$	0	0
$941$	− 2.17481e8i	− 0.00850859i	−0.999991	−	0.00425429i	$-0.998646\pi$
$941$	− 2.17481e8i	− 0.00850859i	0.999991	−	0.00425429i	$-0.00135419\pi$
$942$	0	0
$943$	−3.00860e9	−0.116835
$944$	0	0
$945$	5.89662e9	0.227296
$946$	0	0
$947$	− 2.31911e10i	− 0.887351i	−0.896187	−	0.443676i	$-0.853674\pi$
$947$	− 2.31911e10i	− 0.887351i	0.896187	−	0.443676i	$-0.146326\pi$
$948$	0	0
$949$	− 8.15020e9i	− 0.309554i
$950$	0	0
$951$	4.04825e9	0.152628
$952$	0	0
$953$	2.16450e10	0.810087	0.405044	−	0.914297i	$-0.367256\pi$
$953$	2.16450e10	0.810087	0.405044	+	0.914297i	$0.367256\pi$
$954$	0	0
$955$	3.72339e10i	1.38333i
$956$	0	0
$957$	4.36155e10i	1.60860i
$958$	0	0
$959$	3.91941e10	1.43501
$960$	0	0
$961$	−1.52303e9	−0.0553577
$962$	0	0
$963$	1.58265e10i	0.571075i
$964$	0	0
$965$	− 4.17788e10i	− 1.49662i
$966$	0	0
$967$	−5.17872e9	−0.184175	−0.0920873	−	0.995751i	$-0.529354\pi$
$967$	−5.17872e9	−0.184175	−0.0920873	+	0.995751i	$0.529354\pi$
$968$	0	0
$969$	1.34080e10	0.473403
$970$	0	0
$971$	− 1.74490e10i	− 0.611651i	−0.952088	−	0.305825i	$-0.901068\pi$
$971$	− 1.74490e10i	− 0.611651i	0.952088	−	0.305825i	$-0.0989324\pi$
$972$	0	0
$973$	− 4.87525e9i	− 0.169669i
$974$	0	0
$975$	8.48290e9	0.293108
$976$	0	0
$977$	−2.65151e10	−0.909625	−0.454812	−	0.890587i	$-0.650294\pi$
$977$	−2.65151e10	−0.909625	−0.454812	+	0.890587i	$0.650294\pi$
$978$	0	0
$979$	5.02948e10i	1.71310i
$980$	0	0
$981$	4.49677e9i	0.152075i
$982$	0	0
$983$	−7.58942e9	−0.254842	−0.127421	−	0.991849i	$-0.540670\pi$
$983$	−7.58942e9	−0.254842	−0.127421	+	0.991849i	$0.540670\pi$
$984$	0	0
$985$	2.37564e9	0.0792052
$986$	0	0
$987$	4.90763e9i	0.162466i
$988$	0	0
$989$	− 3.62106e9i	− 0.119028i
$990$	0	0
$991$	2.16676e10	0.707218	0.353609	−	0.935393i	$-0.384954\pi$
$991$	2.16676e10	0.707218	0.353609	+	0.935393i	$0.384954\pi$
$992$	0	0
$993$	9.77467e9	0.316796
$994$	0	0
$995$	5.57916e9i	0.179551i
$996$	0	0
$997$	5.20679e10i	1.66394i	0.554823	+	0.831968i	$0.312786\pi$
$997$	5.20679e10i	1.66394i	−0.554823	+	0.831968i	$0.687214\pi$
$998$	0	0
$999$	2.00975e9	0.0637767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.b.193.6	yes	6
4.3	odd	2		384.8.d.a.193.3	✓	6
8.3	odd	2		384.8.d.a.193.4	yes	6
8.5	even	2	inner	384.8.d.b.193.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.a.193.3	✓	6	4.3	odd	2
384.8.d.a.193.4	yes	6	8.3	odd	2
384.8.d.b.193.1	yes	6	8.5	even	2	inner
384.8.d.b.193.6	yes	6	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+27.0000i q^{3} +349.927i q^{5} +856.119 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q+27.0000i q^{3} +349.927i q^{5} +856.119 q^{7} -729.000 q^{9} +7467.90i q^{11} +7088.26i q^{13} -9448.04 q^{15} -30555.4 q^{17} +16252.2i q^{19} +23115.2i q^{21} -6640.34 q^{23} -44324.2 q^{25} -19683.0i q^{27} -216311. i q^{29} -161213. q^{31} -201633. q^{33} +299579. i q^{35} +102106. i q^{37} -191383. q^{39} +453079. q^{41} +545313. i q^{43} -255097. i q^{45} +212312. q^{47} -90603.3 q^{49} -824996. i q^{51} +1.22506e6i q^{53} -2.61322e6 q^{55} -438810. q^{57} +1.94063e6i q^{59} +2.81362e6i q^{61} -624111. q^{63} -2.48038e6 q^{65} -2.79519e6i q^{67} -179289. i q^{69} +5.11177e6 q^{71} -1.14982e6 q^{73} -1.19675e6i q^{75} +6.39341e6i q^{77} +2.23303e6 q^{79} +531441. q^{81} -8.74755e6i q^{83} -1.06922e7i q^{85} +5.84040e6 q^{87} +6.73480e6 q^{89} +6.06840e6i q^{91} -4.35275e6i q^{93} -5.68709e6 q^{95} -6.60143e6 q^{97} -5.44410e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 136 q^{7} - 4374 q^{9}+O(q^{10}) \) 6 * q + 136 * q^7 - 4374 * q^9	\( 6 q + 136 q^{7} - 4374 q^{9} - 6048 q^{15} - 25804 q^{17} + 12688 q^{23} - 18418 q^{25} + 220472 q^{31} - 264600 q^{33} - 591192 q^{39} - 619652 q^{41} - 326032 q^{47} - 581482 q^{49} - 4724352 q^{55} - 2429784 q^{57} - 99144 q^{63} - 9518720 q^{65} + 5841776 q^{71} - 11075196 q^{73} + 495800 q^{79} + 3188646 q^{81} + 965520 q^{87} + 9660740 q^{89} - 32750208 q^{95} - 9511564 q^{97}+O(q^{100}) \) 6 * q + 136 * q^7 - 4374 * q^9 - 6048 * q^15 - 25804 * q^17 + 12688 * q^23 - 18418 * q^25 + 220472 * q^31 - 264600 * q^33 - 591192 * q^39 - 619652 * q^41 - 326032 * q^47 - 581482 * q^49 - 4724352 * q^55 - 2429784 * q^57 - 99144 * q^63 - 9518720 * q^65 + 5841776 * q^71 - 11075196 * q^73 + 495800 * q^79 + 3188646 * q^81 + 965520 * q^87 + 9660740 * q^89 - 32750208 * q^95 - 9511564 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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