LMFDB - Embedded newform 384.9.g.a.127.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,9,Mod(127,384)]

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([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("384.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	384.g (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:	$156.433386263$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.24
Character	$\chi$	$=$	384.127
Dual form			384.9.g.a.127.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} +317.614 q^{5} -2582.66i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} +317.614 q^{5} -2582.66i q^{7} -2187.00 q^{9} +24940.0i q^{11} +22690.6 q^{13} +14853.3i q^{15} -83491.9 q^{17} +131385. i q^{19} +120779. q^{21} +6296.48i q^{23} -289746. q^{25} -102276. i q^{27} +448021. q^{29} -848824. i q^{31} -1.16633e6 q^{33} -820288. i q^{35} +612861. q^{37} +1.06113e6i q^{39} +701562. q^{41} +533412. i q^{43} -694622. q^{45} -9.53625e6i q^{47} -905307. q^{49} -3.90453e6i q^{51} +569341. q^{53} +7.92128e6i q^{55} -6.14425e6 q^{57} +8.97760e6i q^{59} -1.79584e7 q^{61} +5.64827e6i q^{63} +7.20684e6 q^{65} +2.97195e7i q^{67} -294457. q^{69} +2.29609e7i q^{71} -5.29487e7 q^{73} -1.35501e7i q^{75} +6.44113e7 q^{77} +3.82620e7i q^{79} +4.78297e6 q^{81} +8.08697e7i q^{83} -2.65182e7 q^{85} +2.09519e7i q^{87} +6.45751e7 q^{89} -5.86019e7i q^{91} +3.96956e7 q^{93} +4.17296e7i q^{95} +2.32734e7 q^{97} -5.45437e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) $ 32 * q - 1344 * q^5 - 69984 * q^9	$ 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) $ 32 * q - 1344 * q^5 - 69984 * q^9 - 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 - 275520 * q^29 + 2421440 * q^37 - 4374720 * q^41 + 2939328 * q^45 - 14219104 * q^49 - 6224448 * q^53 + 3100032 * q^57 - 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 + 154517760 * q^77 + 153055008 * q^81 - 384830848 * q^85 - 182669760 * q^89 + 149817600 * q^93 - 149408192 * 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
$3$	46.7654i	0.577350i
$3$	46.7654i	0.577350i
$4$	0	0
$5$	317.614	0.508182	0.254091	−	0.967180i	$-0.418224\pi$
$5$	317.614	0.508182	0.254091	+	0.967180i	$0.418224\pi$
$6$	0	0
$7$	− 2582.66i	− 1.07566i	−0.843054	−	0.537829i	$-0.819245\pi$
$7$	− 2582.66i	− 1.07566i	0.843054	−	0.537829i	$-0.180755\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	24940.0i	1.70343i	0.524002	+	0.851717i	$0.324438\pi$
$11$	24940.0i	1.70343i	−0.524002	+	0.851717i	$0.675562\pi$
$12$	0	0
$13$	22690.6	0.794459	0.397230	−	0.917719i	$-0.369972\pi$
$13$	22690.6	0.794459	0.397230	+	0.917719i	$0.369972\pi$
$14$	0	0
$15$	14853.3i	0.293399i
$16$	0	0
$17$	−83491.9	−0.999651	−0.499826	−	0.866126i	$-0.666603\pi$
$17$	−83491.9	−0.999651	−0.499826	+	0.866126i	$0.666603\pi$
$18$	0	0
$19$	131385.i	1.00816i	0.863657	+	0.504080i	$0.168169\pi$
$19$	131385.i	1.00816i	−0.863657	+	0.504080i	$0.831831\pi$
$20$	0	0
$21$	120779.	0.621032
$22$	0	0
$23$	6296.48i	0.0225002i	0.999937	+	0.0112501i	$0.00358110\pi$
$23$	6296.48i	0.0225002i	−0.999937	+	0.0112501i	$0.996419\pi$
$24$	0	0
$25$	−289746.	−0.741751
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	448021.	0.633441	0.316720	−	0.948519i	$-0.397418\pi$
$29$	448021.	0.633441	0.316720	+	0.948519i	$0.397418\pi$
$30$	0	0
$31$	− 848824.i	− 0.919117i	−0.888148	−	0.459558i	$-0.848008\pi$
$31$	− 848824.i	− 0.919117i	0.888148	−	0.459558i	$-0.151992\pi$
$32$	0	0
$33$	−1.16633e6	−0.983478
$34$	0	0
$35$	− 820288.i	− 0.546631i
$36$	0	0
$37$	612861.	0.327006	0.163503	−	0.986543i	$-0.447721\pi$
$37$	612861.	0.327006	0.163503	+	0.986543i	$0.447721\pi$
$38$	0	0
$39$	1.06113e6i	0.458681i
$40$	0	0
$41$	701562.	0.248274	0.124137	−	0.992265i	$-0.460384\pi$
$41$	701562.	0.248274	0.124137	+	0.992265i	$0.460384\pi$
$42$	0	0
$43$	533412.i	0.156023i	0.996952	+	0.0780115i	$0.0248571\pi$
$43$	533412.i	0.156023i	−0.996952	+	0.0780115i	$0.975143\pi$
$44$	0	0
$45$	−694622.	−0.169394
$46$	0	0
$47$	− 9.53625e6i	− 1.95428i	−0.212602	−	0.977139i	$-0.568194\pi$
$47$	− 9.53625e6i	− 1.95428i	0.212602	−	0.977139i	$-0.431806\pi$
$48$	0	0
$49$	−905307.	−0.157041
$50$	0	0
$51$	− 3.90453e6i	− 0.577149i
$52$	0	0
$53$	569341.	0.0721555	0.0360777	−	0.999349i	$-0.488514\pi$
$53$	569341.	0.0721555	0.0360777	+	0.999349i	$0.488514\pi$
$54$	0	0
$55$	7.92128e6i	0.865655i
$56$	0	0
$57$	−6.14425e6	−0.582062
$58$	0	0
$59$	8.97760e6i	0.740887i	0.928855	+	0.370444i	$0.120794\pi$
$59$	8.97760e6i	0.740887i	−0.928855	+	0.370444i	$0.879206\pi$
$60$	0	0
$61$	−1.79584e7	−1.29703	−0.648513	−	0.761204i	$-0.724609\pi$
$61$	−1.79584e7	−1.29703	−0.648513	+	0.761204i	$0.724609\pi$
$62$	0	0
$63$	5.64827e6i	0.358553i
$64$	0	0
$65$	7.20684e6	0.403730
$66$	0	0
$67$	2.97195e7i	1.47483i	0.675441	+	0.737414i	$0.263953\pi$
$67$	2.97195e7i	1.47483i	−0.675441	+	0.737414i	$0.736047\pi$
$68$	0	0
$69$	−294457.	−0.0129905
$70$	0	0
$71$	2.29609e7i	0.903556i	0.892131	+	0.451778i	$0.149210\pi$
$71$	2.29609e7i	0.903556i	−0.892131	+	0.451778i	$0.850790\pi$
$72$	0	0
$73$	−5.29487e7	−1.86451	−0.932253	−	0.361806i	$-0.882160\pi$
$73$	−5.29487e7	−1.86451	−0.932253	+	0.361806i	$0.882160\pi$
$74$	0	0
$75$	− 1.35501e7i	− 0.428250i
$76$	0	0
$77$	6.44113e7	1.83231
$78$	0	0
$79$	3.82620e7i	0.982335i	0.871065	+	0.491168i	$0.163430\pi$
$79$	3.82620e7i	0.982335i	−0.871065	+	0.491168i	$0.836570\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	8.08697e7i	1.70402i	0.523529	+	0.852008i	$0.324615\pi$
$83$	8.08697e7i	1.70402i	−0.523529	+	0.852008i	$0.675385\pi$
$84$	0	0
$85$	−2.65182e7	−0.508005
$86$	0	0
$87$	2.09519e7i	0.365717i
$88$	0	0
$89$	6.45751e7	1.02921	0.514606	−	0.857427i	$-0.327938\pi$
$89$	6.45751e7	1.02921	0.514606	+	0.857427i	$0.327938\pi$
$90$	0	0
$91$	− 5.86019e7i	− 0.854567i
$92$	0	0
$93$	3.96956e7	0.530652
$94$	0	0
$95$	4.17296e7i	0.512330i
$96$	0	0
$97$	2.32734e7	0.262890	0.131445	−	0.991323i	$-0.458038\pi$
$97$	2.32734e7	0.262890	0.131445	+	0.991323i	$0.458038\pi$
$98$	0	0
$99$	− 5.45437e7i	− 0.567811i
$100$	0	0
$101$	484812.	0.00465895	0.00232947	−	0.999997i	$-0.499259\pi$
$101$	484812.	0.00465895	0.00232947	+	0.999997i	$0.499259\pi$
$102$	0	0
$103$	− 1.73266e8i	− 1.53944i	−0.638380	−	0.769721i	$-0.720395\pi$
$103$	− 1.73266e8i	− 1.53944i	0.638380	−	0.769721i	$-0.279605\pi$
$104$	0	0
$105$	3.83611e7	0.315597
$106$	0	0
$107$	− 1.58384e8i	− 1.20830i	−0.796869	−	0.604152i	$-0.793512\pi$
$107$	− 1.58384e8i	− 1.20830i	0.796869	−	0.604152i	$-0.206488\pi$
$108$	0	0
$109$	−1.56847e7	−0.111114	−0.0555570	−	0.998456i	$-0.517693\pi$
$109$	−1.56847e7	−0.111114	−0.0555570	+	0.998456i	$0.517693\pi$
$110$	0	0
$111$	2.86607e7i	0.188797i
$112$	0	0
$113$	−1.58936e8	−0.974785	−0.487393	−	0.873183i	$-0.662052\pi$
$113$	−1.58936e8	−0.974785	−0.487393	+	0.873183i	$0.662052\pi$
$114$	0	0
$115$	1.99985e6i	0.0114342i
$116$	0	0
$117$	−4.96242e7	−0.264820
$118$	0	0
$119$	2.15631e8i	1.07528i
$120$	0	0
$121$	−4.07643e8	−1.90169
$122$	0	0
$123$	3.28088e7i	0.143341i
$124$	0	0
$125$	−2.16095e8	−0.885127
$126$	0	0
$127$	− 3.54231e7i	− 0.136167i	−0.997680	−	0.0680834i	$-0.978312\pi$
$127$	− 3.54231e7i	− 0.136167i	0.997680	−	0.0680834i	$-0.0216884\pi$
$128$	0	0
$129$	−2.49452e7	−0.0900800
$130$	0	0
$131$	− 2.71009e8i	− 0.920235i	−0.887858	−	0.460117i	$-0.847807\pi$
$131$	− 2.71009e8i	− 0.920235i	0.887858	−	0.460117i	$-0.152193\pi$
$132$	0	0
$133$	3.39321e8	1.08444
$134$	0	0
$135$	− 3.24843e7i	− 0.0977998i
$136$	0	0
$137$	3.35164e7	0.0951427	0.0475714	−	0.998868i	$-0.484852\pi$
$137$	3.35164e7	0.0951427	0.0475714	+	0.998868i	$0.484852\pi$
$138$	0	0
$139$	4.03194e8i	1.08008i	0.841640	+	0.540039i	$0.181591\pi$
$139$	4.03194e8i	1.08008i	−0.841640	+	0.540039i	$0.818409\pi$
$140$	0	0
$141$	4.45966e8	1.12830
$142$	0	0
$143$	5.65902e8i	1.35331i
$144$	0	0
$145$	1.42298e8	0.321904
$146$	0	0
$147$	− 4.23370e7i	− 0.0906674i
$148$	0	0
$149$	−7.84189e8	−1.59102	−0.795510	−	0.605941i	$-0.792797\pi$
$149$	−7.84189e8	−1.59102	−0.795510	+	0.605941i	$0.792797\pi$
$150$	0	0
$151$	− 1.61049e8i	− 0.309777i	−0.987932	−	0.154888i	$-0.950498\pi$
$151$	− 1.61049e8i	− 0.309777i	0.987932	−	0.154888i	$-0.0495018\pi$
$152$	0	0
$153$	1.82597e8	0.333217
$154$	0	0
$155$	− 2.69598e8i	− 0.467079i
$156$	0	0
$157$	−8.51750e8	−1.40189	−0.700944	−	0.713216i	$-0.747238\pi$
$157$	−8.51750e8	−1.40189	−0.700944	+	0.713216i	$0.747238\pi$
$158$	0	0
$159$	2.66255e7i	0.0416590i
$160$	0	0
$161$	1.62616e7	0.0242025
$162$	0	0
$163$	− 6.02189e8i	− 0.853065i	−0.904472	−	0.426533i	$-0.859735\pi$
$163$	− 6.02189e8i	− 0.853065i	0.904472	−	0.426533i	$-0.140265\pi$
$164$	0	0
$165$	−3.70442e8	−0.499786
$166$	0	0
$167$	− 1.94512e8i	− 0.250081i	−0.992152	−	0.125040i	$-0.960094\pi$
$167$	− 1.94512e8i	− 0.250081i	0.992152	−	0.125040i	$-0.0399061\pi$
$168$	0	0
$169$	−3.00870e8	−0.368834
$170$	0	0
$171$	− 2.87338e8i	− 0.336054i
$172$	0	0
$173$	−1.51181e9	−1.68776	−0.843882	−	0.536530i	$-0.819735\pi$
$173$	−1.51181e9	−1.68776	−0.843882	+	0.536530i	$0.819735\pi$
$174$	0	0
$175$	7.48315e8i	0.797870i
$176$	0	0
$177$	−4.19841e8	−0.427751
$178$	0	0
$179$	4.46388e8i	0.434811i	0.976081	+	0.217405i	$0.0697594\pi$
$179$	4.46388e8i	0.434811i	−0.976081	+	0.217405i	$0.930241\pi$
$180$	0	0
$181$	−1.51653e9	−1.41298	−0.706490	−	0.707723i	$-0.749723\pi$
$181$	−1.51653e9	−1.41298	−0.706490	+	0.707723i	$0.749723\pi$
$182$	0	0
$183$	− 8.39832e8i	− 0.748838i
$184$	0	0
$185$	1.94653e8	0.166179
$186$	0	0
$187$	− 2.08228e9i	− 1.70284i
$188$	0	0
$189$	−2.64143e8	−0.207011
$190$	0	0
$191$	8.44075e7i	0.0634231i	0.999497	+	0.0317116i	$0.0100958\pi$
$191$	8.44075e7i	0.0634231i	−0.999497	+	0.0317116i	$0.989904\pi$
$192$	0	0
$193$	6.91816e8	0.498610	0.249305	−	0.968425i	$-0.419798\pi$
$193$	6.91816e8	0.498610	0.249305	+	0.968425i	$0.419798\pi$
$194$	0	0
$195$	3.37030e8i	0.233094i
$196$	0	0
$197$	2.13097e9	1.41486	0.707429	−	0.706785i	$-0.249855\pi$
$197$	2.13097e9	1.41486	0.707429	+	0.706785i	$0.249855\pi$
$198$	0	0
$199$	1.37817e9i	0.878802i	0.898291	+	0.439401i	$0.144809\pi$
$199$	1.37817e9i	0.878802i	−0.898291	+	0.439401i	$0.855191\pi$
$200$	0	0
$201$	−1.38984e9	−0.851493
$202$	0	0
$203$	− 1.15708e9i	− 0.681366i
$204$	0	0
$205$	2.22826e8	0.126168
$206$	0	0
$207$	− 1.37704e7i	− 0.00750007i
$208$	0	0
$209$	−3.27673e9	−1.71734
$210$	0	0
$211$	− 1.38442e9i	− 0.698452i	−0.937038	−	0.349226i	$-0.886445\pi$
$211$	− 1.38442e9i	− 0.698452i	0.937038	−	0.349226i	$-0.113555\pi$
$212$	0	0
$213$	−1.07377e9	−0.521668
$214$	0	0
$215$	1.69419e8i	0.0792882i
$216$	0	0
$217$	−2.19222e9	−0.988656
$218$	0	0
$219$	− 2.47617e9i	− 1.07647i
$220$	0	0
$221$	−1.89448e9	−0.794182
$222$	0	0
$223$	3.74679e9i	1.51510i	0.652779	+	0.757549i	$0.273603\pi$
$223$	3.74679e9i	1.51510i	−0.652779	+	0.757549i	$0.726397\pi$
$224$	0	0
$225$	6.33675e8	0.247250
$226$	0	0
$227$	4.48761e8i	0.169010i	0.996423	+	0.0845048i	$0.0269308\pi$
$227$	4.48761e8i	0.169010i	−0.996423	+	0.0845048i	$0.973069\pi$
$228$	0	0
$229$	2.10691e9	0.766132	0.383066	−	0.923721i	$-0.374868\pi$
$229$	2.10691e9	0.766132	0.383066	+	0.923721i	$0.374868\pi$
$230$	0	0
$231$	3.01222e9i	1.05789i
$232$	0	0
$233$	4.37823e8	0.148551	0.0742754	−	0.997238i	$-0.476336\pi$
$233$	4.37823e8	0.148551	0.0742754	+	0.997238i	$0.476336\pi$
$234$	0	0
$235$	− 3.02885e9i	− 0.993130i
$236$	0	0
$237$	−1.78934e9	−0.567152
$238$	0	0
$239$	1.98970e9i	0.609813i	0.952382	+	0.304907i	$0.0986252\pi$
$239$	1.98970e9i	0.609813i	−0.952382	+	0.304907i	$0.901375\pi$
$240$	0	0
$241$	−4.13859e9	−1.22683	−0.613415	−	0.789761i	$-0.710205\pi$
$241$	−4.13859e9	−1.22683	−0.613415	+	0.789761i	$0.710205\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	−2.87538e8	−0.0798052
$246$	0	0
$247$	2.98119e9i	0.800943i
$248$	0	0
$249$	−3.78190e9	−0.983814
$250$	0	0
$251$	− 1.51046e9i	− 0.380552i	−0.981731	−	0.190276i	$-0.939062\pi$
$251$	− 1.51046e9i	− 0.380552i	0.981731	−	0.190276i	$-0.0609383\pi$
$252$	0	0
$253$	−1.57034e8	−0.0383276
$254$	0	0
$255$	− 1.24013e9i	− 0.293297i
$256$	0	0
$257$	3.62471e8	0.0830886	0.0415443	−	0.999137i	$-0.486772\pi$
$257$	3.62471e8	0.0830886	0.0415443	+	0.999137i	$0.486772\pi$
$258$	0	0
$259$	− 1.58281e9i	− 0.351746i
$260$	0	0
$261$	−9.79821e8	−0.211147
$262$	0	0
$263$	3.37498e9i	0.705421i	0.935732	+	0.352711i	$0.114740\pi$
$263$	3.37498e9i	0.705421i	−0.935732	+	0.352711i	$0.885260\pi$
$264$	0	0
$265$	1.80831e8	0.0366682
$266$	0	0
$267$	3.01988e9i	0.594216i
$268$	0	0
$269$	−7.44149e9	−1.42119	−0.710593	−	0.703604i	$-0.751573\pi$
$269$	−7.44149e9	−1.42119	−0.710593	+	0.703604i	$0.751573\pi$
$270$	0	0
$271$	1.17692e9i	0.218208i	0.994030	+	0.109104i	$0.0347982\pi$
$271$	1.17692e9i	0.218208i	−0.994030	+	0.109104i	$0.965202\pi$
$272$	0	0
$273$	2.74054e9	0.493384
$274$	0	0
$275$	− 7.22626e9i	− 1.26352i
$276$	0	0
$277$	−7.39130e9	−1.25546	−0.627729	−	0.778432i	$-0.716015\pi$
$277$	−7.39130e9	−1.25546	−0.627729	+	0.778432i	$0.716015\pi$
$278$	0	0
$279$	1.85638e9i	0.306372i
$280$	0	0
$281$	−7.31393e9	−1.17307	−0.586537	−	0.809922i	$-0.699509\pi$
$281$	−7.31393e9	−1.17307	−0.586537	+	0.809922i	$0.699509\pi$
$282$	0	0
$283$	5.77219e9i	0.899902i	0.893053	+	0.449951i	$0.148558\pi$
$283$	5.77219e9i	0.899902i	−0.893053	+	0.449951i	$0.851442\pi$
$284$	0	0
$285$	−1.95150e9	−0.295794
$286$	0	0
$287$	− 1.81189e9i	− 0.267058i
$288$	0	0
$289$	−4.86628e6	−0.000697599 0
$290$	0	0
$291$	1.08839e9i	0.151779i
$292$	0	0
$293$	−4.93503e9	−0.669606	−0.334803	−	0.942288i	$-0.608670\pi$
$293$	−4.93503e9	−0.669606	−0.334803	+	0.942288i	$0.608670\pi$
$294$	0	0
$295$	2.85141e9i	0.376506i
$296$	0	0
$297$	2.55076e9	0.327826
$298$	0	0
$299$	1.42871e8i	0.0178755i
$300$	0	0
$301$	1.37762e9	0.167827
$302$	0	0
$303$	2.26724e7i	0.00268984i
$304$	0	0
$305$	−5.70384e9	−0.659126
$306$	0	0
$307$	− 4.56458e9i	− 0.513863i	−0.966430	−	0.256932i	$-0.917288\pi$
$307$	− 4.56458e9i	− 0.513863i	0.966430	−	0.256932i	$-0.0827116\pi$
$308$	0	0
$309$	8.10283e9	0.888797
$310$	0	0
$311$	1.15164e10i	1.23105i	0.788116	+	0.615526i	$0.211056\pi$
$311$	1.15164e10i	1.23105i	−0.788116	+	0.615526i	$0.788944\pi$
$312$	0	0
$313$	−1.52593e10	−1.58985	−0.794925	−	0.606708i	$-0.792490\pi$
$313$	−1.52593e10	−1.58985	−0.794925	+	0.606708i	$0.792490\pi$
$314$	0	0
$315$	1.79397e9i	0.182210i
$316$	0	0
$317$	−2.49968e9	−0.247541	−0.123771	−	0.992311i	$-0.539499\pi$
$317$	−2.49968e9	−0.247541	−0.123771	+	0.992311i	$0.539499\pi$
$318$	0	0
$319$	1.11736e10i	1.07902i
$320$	0	0
$321$	7.40689e9	0.697615
$322$	0	0
$323$	− 1.09695e10i	− 1.00781i
$324$	0	0
$325$	−6.57450e9	−0.589291
$326$	0	0
$327$	− 7.33499e8i	− 0.0641517i
$328$	0	0
$329$	−2.46288e10	−2.10213
$330$	0	0
$331$	1.02205e10i	0.851456i	0.904851	+	0.425728i	$0.139982\pi$
$331$	1.02205e10i	0.851456i	−0.904851	+	0.425728i	$0.860018\pi$
$332$	0	0
$333$	−1.34033e9	−0.109002
$334$	0	0
$335$	9.43931e9i	0.749482i
$336$	0	0
$337$	−5.85576e9	−0.454008	−0.227004	−	0.973894i	$-0.572893\pi$
$337$	−5.85576e9	−0.454008	−0.227004	+	0.973894i	$0.572893\pi$
$338$	0	0
$339$	− 7.43271e9i	− 0.562793i
$340$	0	0
$341$	2.11696e10	1.56565
$342$	0	0
$343$	− 1.25504e10i	− 0.906736i
$344$	0	0
$345$	−9.35238e7	−0.00660155
$346$	0	0
$347$	− 2.13783e10i	− 1.47454i	−0.675599	−	0.737269i	$-0.736115\pi$
$347$	− 2.13783e10i	− 1.47454i	0.675599	−	0.737269i	$-0.263885\pi$
$348$	0	0
$349$	−1.57760e10	−1.06340	−0.531698	−	0.846934i	$-0.678446\pi$
$349$	−1.57760e10	−1.06340	−0.531698	+	0.846934i	$0.678446\pi$
$350$	0	0
$351$	− 2.32070e9i	− 0.152894i
$352$	0	0
$353$	2.47946e10	1.59683	0.798416	−	0.602107i	$-0.205672\pi$
$353$	2.47946e10	1.59683	0.798416	+	0.602107i	$0.205672\pi$
$354$	0	0
$355$	7.29269e9i	0.459171i
$356$	0	0
$357$	−1.00840e10	−0.620815
$358$	0	0
$359$	− 1.31718e10i	− 0.792989i	−0.918037	−	0.396494i	$-0.870227\pi$
$359$	− 1.31718e10i	− 0.792989i	0.918037	−	0.396494i	$-0.129773\pi$
$360$	0	0
$361$	−2.78336e8	−0.0163886
$362$	0	0
$363$	− 1.90636e10i	− 1.09794i
$364$	0	0
$365$	−1.68173e10	−0.947510
$366$	0	0
$367$	− 1.08235e10i	− 0.596627i	−0.954468	−	0.298313i	$-0.903576\pi$
$367$	− 1.08235e10i	− 0.596627i	0.954468	−	0.298313i	$-0.0964240\pi$
$368$	0	0
$369$	−1.53432e9	−0.0827579
$370$	0	0
$371$	− 1.47041e9i	− 0.0776146i
$372$	0	0
$373$	2.38473e10	1.23198	0.615989	−	0.787754i	$-0.288756\pi$
$373$	2.38473e10	1.23198	0.615989	+	0.787754i	$0.288756\pi$
$374$	0	0
$375$	− 1.01058e10i	− 0.511028i
$376$	0	0
$377$	1.01658e10	0.503243
$378$	0	0
$379$	3.67617e8i	0.0178172i	0.999960	+	0.00890859i	$0.00283573\pi$
$379$	3.67617e8i	0.0178172i	−0.999960	+	0.00890859i	$0.997164\pi$
$380$	0	0
$381$	1.65657e9	0.0786160
$382$	0	0
$383$	− 3.47247e10i	− 1.61378i	−0.590703	−	0.806889i	$-0.701149\pi$
$383$	− 3.47247e10i	− 1.61378i	0.590703	−	0.806889i	$-0.298851\pi$
$384$	0	0
$385$	2.04579e10	0.931149
$386$	0	0
$387$	− 1.16657e9i	− 0.0520077i
$388$	0	0
$389$	−8.85561e9	−0.386741	−0.193371	−	0.981126i	$-0.561942\pi$
$389$	−8.85561e9	−0.386741	−0.193371	+	0.981126i	$0.561942\pi$
$390$	0	0
$391$	− 5.25705e8i	− 0.0224924i
$392$	0	0
$393$	1.26738e10	0.531298
$394$	0	0
$395$	1.21526e10i	0.499206i
$396$	0	0
$397$	−3.13858e10	−1.26349	−0.631744	−	0.775177i	$-0.717661\pi$
$397$	−3.13858e10	−1.26349	−0.631744	+	0.775177i	$0.717661\pi$
$398$	0	0
$399$	1.58685e10i	0.626100i
$400$	0	0
$401$	2.99111e10	1.15679	0.578396	−	0.815756i	$-0.303679\pi$
$401$	2.99111e10	1.15679	0.578396	+	0.815756i	$0.303679\pi$
$402$	0	0
$403$	− 1.92603e10i	− 0.730201i
$404$	0	0
$405$	1.51914e9	0.0564647
$406$	0	0
$407$	1.52847e10i	0.557032i
$408$	0	0
$409$	−1.37201e10	−0.490301	−0.245151	−	0.969485i	$-0.578837\pi$
$409$	−1.37201e10	−0.490301	−0.245151	+	0.969485i	$0.578837\pi$
$410$	0	0
$411$	1.56741e9i	0.0549307i
$412$	0	0
$413$	2.31860e10	0.796941
$414$	0	0
$415$	2.56854e10i	0.865951i
$416$	0	0
$417$	−1.88555e10	−0.623583
$418$	0	0
$419$	2.72163e10i	0.883025i	0.897255	+	0.441512i	$0.145558\pi$
$419$	2.72163e10i	0.883025i	−0.897255	+	0.441512i	$0.854442\pi$
$420$	0	0
$421$	5.98486e10	1.90513	0.952567	−	0.304329i	$-0.0984323\pi$
$421$	5.98486e10	1.90513	0.952567	+	0.304329i	$0.0984323\pi$
$422$	0	0
$423$	2.08558e10i	0.651426i
$424$	0	0
$425$	2.41915e10	0.741492
$426$	0	0
$427$	4.63804e10i	1.39516i
$428$	0	0
$429$	−2.64646e10	−0.781333
$430$	0	0
$431$	− 2.20986e10i	− 0.640405i	−0.947349	−	0.320203i	$-0.896249\pi$
$431$	− 2.20986e10i	− 0.640405i	0.947349	−	0.320203i	$-0.103751\pi$
$432$	0	0
$433$	2.27219e10	0.646388	0.323194	−	0.946333i	$-0.395243\pi$
$433$	2.27219e10	0.646388	0.323194	+	0.946333i	$0.395243\pi$
$434$	0	0
$435$	6.65460e9i	0.185851i
$436$	0	0
$437$	−8.27261e8	−0.0226838
$438$	0	0
$439$	7.11736e10i	1.91629i	0.286285	+	0.958144i	$0.407579\pi$
$439$	7.11736e10i	1.91629i	−0.286285	+	0.958144i	$0.592421\pi$
$440$	0	0
$441$	1.97991e9	0.0523468
$442$	0	0
$443$	1.03544e10i	0.268850i	0.990924	+	0.134425i	$0.0429187\pi$
$443$	1.03544e10i	0.268850i	−0.990924	+	0.134425i	$0.957081\pi$
$444$	0	0
$445$	2.05100e10	0.523028
$446$	0	0
$447$	− 3.66729e10i	− 0.918576i
$448$	0	0
$449$	4.46154e10	1.09774	0.548869	−	0.835908i	$-0.315058\pi$
$449$	4.46154e10	1.09774	0.548869	+	0.835908i	$0.315058\pi$
$450$	0	0
$451$	1.74969e10i	0.422918i
$452$	0	0
$453$	7.53149e9	0.178850
$454$	0	0
$455$	− 1.86128e10i	− 0.434276i
$456$	0	0
$457$	−8.17052e10	−1.87320	−0.936601	−	0.350398i	$-0.886046\pi$
$457$	−8.17052e10	−1.87320	−0.936601	+	0.350398i	$0.886046\pi$
$458$	0	0
$459$	8.53920e9i	0.192383i
$460$	0	0
$461$	−3.43599e9	−0.0760761	−0.0380381	−	0.999276i	$-0.512111\pi$
$461$	−3.43599e9	−0.0760761	−0.0380381	+	0.999276i	$0.512111\pi$
$462$	0	0
$463$	− 8.08434e10i	− 1.75922i	−0.475693	−	0.879611i	$-0.657803\pi$
$463$	− 8.08434e10i	− 1.75922i	0.475693	−	0.879611i	$-0.342197\pi$
$464$	0	0
$465$	1.26079e10	0.269668
$466$	0	0
$467$	4.72355e10i	0.993119i	0.868003	+	0.496559i	$0.165404\pi$
$467$	4.72355e10i	0.993119i	−0.868003	+	0.496559i	$0.834596\pi$
$468$	0	0
$469$	7.67551e10	1.58641
$470$	0	0
$471$	− 3.98324e10i	− 0.809380i
$472$	0	0
$473$	−1.33033e10	−0.265775
$474$	0	0
$475$	− 3.80682e10i	− 0.747804i
$476$	0	0
$477$	−1.24515e9	−0.0240518
$478$	0	0
$479$	− 1.57036e10i	− 0.298303i	−0.988814	−	0.149152i	$-0.952346\pi$
$479$	− 1.57036e10i	− 0.298303i	0.988814	−	0.149152i	$-0.0476542\pi$
$480$	0	0
$481$	1.39062e10	0.259793
$482$	0	0
$483$	7.60482e8i	0.0139733i
$484$	0	0
$485$	7.39197e9	0.133596
$486$	0	0
$487$	− 6.08794e10i	− 1.08232i	−0.840921	−	0.541159i	$-0.817986\pi$
$487$	− 6.08794e10i	− 1.08232i	0.840921	−	0.541159i	$-0.182014\pi$
$488$	0	0
$489$	2.81616e10	0.492517
$490$	0	0
$491$	5.83389e10i	1.00376i	0.864936	+	0.501882i	$0.167359\pi$
$491$	5.83389e10i	1.00376i	−0.864936	+	0.501882i	$0.832641\pi$
$492$	0	0
$493$	−3.74061e10	−0.633220
$494$	0	0
$495$	− 1.73239e10i	− 0.288552i
$496$	0	0
$497$	5.93000e10	0.971917
$498$	0	0
$499$	1.06843e11i	1.72323i	0.507563	+	0.861614i	$0.330546\pi$
$499$	1.06843e11i	1.72323i	−0.507563	+	0.861614i	$0.669454\pi$
$500$	0	0
$501$	9.09643e9	0.144384
$502$	0	0
$503$	− 1.14502e11i	− 1.78871i	−0.447357	−	0.894356i	$-0.647635\pi$
$503$	− 1.14502e11i	− 1.78871i	0.447357	−	0.894356i	$-0.352365\pi$
$504$	0	0
$505$	1.53983e8	0.00236760
$506$	0	0
$507$	− 1.40703e10i	− 0.212947i
$508$	0	0
$509$	−4.92670e10	−0.733981	−0.366990	−	0.930225i	$-0.619612\pi$
$509$	−4.92670e10	−0.733981	−0.366990	+	0.930225i	$0.619612\pi$
$510$	0	0
$511$	1.36748e11i	2.00557i
$512$	0	0
$513$	1.34375e10	0.194021
$514$	0	0
$515$	− 5.50316e10i	− 0.782318i
$516$	0	0
$517$	2.37834e11	3.32898
$518$	0	0
$519$	− 7.07001e10i	− 0.974430i
$520$	0	0
$521$	−1.15586e11	−1.56876	−0.784379	−	0.620281i	$-0.787018\pi$
$521$	−1.15586e11	−1.56876	−0.784379	+	0.620281i	$0.787018\pi$
$522$	0	0
$523$	− 2.07087e10i	− 0.276788i	−0.990377	−	0.138394i	$-0.955806\pi$
$523$	− 2.07087e10i	− 0.276788i	0.990377	−	0.138394i	$-0.0441940\pi$
$524$	0	0
$525$	−3.49952e10	−0.460650
$526$	0	0
$527$	7.08699e10i	0.918796i
$528$	0	0
$529$	7.82713e10	0.999494
$530$	0	0
$531$	− 1.96340e10i	− 0.246962i
$532$	0	0
$533$	1.59188e10	0.197243
$534$	0	0
$535$	− 5.03050e10i	− 0.614039i
$536$	0	0
$537$	−2.08755e10	−0.251038
$538$	0	0
$539$	− 2.25783e10i	− 0.267508i
$540$	0	0
$541$	1.07693e11	1.25718	0.628592	−	0.777735i	$-0.283632\pi$
$541$	1.07693e11	1.25718	0.628592	+	0.777735i	$0.283632\pi$
$542$	0	0
$543$	− 7.09210e10i	− 0.815785i
$544$	0	0
$545$	−4.98167e9	−0.0564662
$546$	0	0
$547$	1.53744e11i	1.71731i	0.512555	+	0.858654i	$0.328699\pi$
$547$	1.53744e11i	1.71731i	−0.512555	+	0.858654i	$0.671301\pi$
$548$	0	0
$549$	3.92750e10	0.432342
$550$	0	0
$551$	5.88630e10i	0.638610i
$552$	0	0
$553$	9.88177e10	1.05666
$554$	0	0
$555$	9.10304e9i	0.0959432i
$556$	0	0
$557$	−4.42783e10	−0.460013	−0.230007	−	0.973189i	$-0.573875\pi$
$557$	−4.42783e10	−0.460013	−0.230007	+	0.973189i	$0.573875\pi$
$558$	0	0
$559$	1.21034e10i	0.123954i
$560$	0	0
$561$	9.73788e10	0.983135
$562$	0	0
$563$	1.34868e10i	0.134237i	0.997745	+	0.0671187i	$0.0213806\pi$
$563$	1.34868e10i	0.134237i	−0.997745	+	0.0671187i	$0.978619\pi$
$564$	0	0
$565$	−5.04804e10	−0.495369
$566$	0	0
$567$	− 1.23528e10i	− 0.119518i
$568$	0	0
$569$	1.00150e11	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$569$	1.00150e11	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$570$	0	0
$571$	1.93335e11i	1.81872i	0.416006	+	0.909362i	$0.363430\pi$
$571$	1.93335e11i	1.81872i	−0.416006	+	0.909362i	$0.636570\pi$
$572$	0	0
$573$	−3.94735e9	−0.0366174
$574$	0	0
$575$	− 1.82438e9i	− 0.0166896i
$576$	0	0
$577$	−3.63612e10	−0.328046	−0.164023	−	0.986456i	$-0.552447\pi$
$577$	−3.63612e10	−0.328046	−0.164023	+	0.986456i	$0.552447\pi$
$578$	0	0
$579$	3.23530e10i	0.287873i
$580$	0	0
$581$	2.08859e11	1.83294
$582$	0	0
$583$	1.41994e10i	0.122912i
$584$	0	0
$585$	−1.57614e10	−0.134577
$586$	0	0
$587$	5.69180e10i	0.479399i	0.970847	+	0.239700i	$0.0770489\pi$
$587$	5.69180e10i	0.479399i	−0.970847	+	0.239700i	$0.922951\pi$
$588$	0	0
$589$	1.11522e11	0.926618
$590$	0	0
$591$	9.96557e10i	0.816868i
$592$	0	0
$593$	1.98584e11	1.60592	0.802961	−	0.596031i	$-0.203256\pi$
$593$	1.98584e11	1.60592	0.802961	+	0.596031i	$0.203256\pi$
$594$	0	0
$595$	6.84873e10i	0.546440i
$596$	0	0
$597$	−6.44507e10	−0.507376
$598$	0	0
$599$	− 1.50549e11i	− 1.16942i	−0.811243	−	0.584709i	$-0.801209\pi$
$599$	− 1.50549e11i	− 1.16942i	0.811243	−	0.584709i	$-0.198791\pi$
$600$	0	0
$601$	−2.29830e11	−1.76161	−0.880805	−	0.473479i	$-0.842998\pi$
$601$	−2.29830e11	−1.76161	−0.880805	+	0.473479i	$0.842998\pi$
$602$	0	0
$603$	− 6.49964e10i	− 0.491610i
$604$	0	0
$605$	−1.29473e11	−0.966403
$606$	0	0
$607$	4.25474e10i	0.313414i	0.987645	+	0.156707i	$0.0500879\pi$
$607$	4.25474e10i	0.313414i	−0.987645	+	0.156707i	$0.949912\pi$
$608$	0	0
$609$	5.41114e10	0.393387
$610$	0	0
$611$	− 2.16383e11i	− 1.55259i
$612$	0	0
$613$	−1.47420e11	−1.04403	−0.522017	−	0.852935i	$-0.674820\pi$
$613$	−1.47420e11	−1.04403	−0.522017	+	0.852935i	$0.674820\pi$
$614$	0	0
$615$	1.04205e10i	0.0728433i
$616$	0	0
$617$	−3.59496e10	−0.248058	−0.124029	−	0.992279i	$-0.539582\pi$
$617$	−3.59496e10	−0.248058	−0.124029	+	0.992279i	$0.539582\pi$
$618$	0	0
$619$	− 1.67371e10i	− 0.114004i	−0.998374	−	0.0570018i	$-0.981846\pi$
$619$	− 1.67371e10i	− 0.114004i	0.998374	−	0.0570018i	$-0.0181541\pi$
$620$	0	0
$621$	6.43978e8	0.00433017
$622$	0	0
$623$	− 1.66775e11i	− 1.10708i
$624$	0	0
$625$	4.45472e10	0.291944
$626$	0	0
$627$	− 1.53237e11i	− 0.991504i
$628$	0	0
$629$	−5.11689e10	−0.326892
$630$	0	0
$631$	2.71103e11i	1.71008i	0.518558	+	0.855042i	$0.326469\pi$
$631$	2.71103e11i	1.71008i	−0.518558	+	0.855042i	$0.673531\pi$
$632$	0	0
$633$	6.47427e10	0.403252
$634$	0	0
$635$	− 1.12509e10i	− 0.0691976i
$636$	0	0
$637$	−2.05419e10	−0.124762
$638$	0	0
$639$	− 5.02154e10i	− 0.301185i
$640$	0	0
$641$	2.85187e11	1.68927	0.844633	−	0.535346i	$-0.179819\pi$
$641$	2.85187e11	1.68927	0.844633	+	0.535346i	$0.179819\pi$
$642$	0	0
$643$	6.46097e10i	0.377967i	0.981980	+	0.188983i	$0.0605192\pi$
$643$	6.46097e10i	0.377967i	−0.981980	+	0.188983i	$0.939481\pi$
$644$	0	0
$645$	−7.92295e9	−0.0457771
$646$	0	0
$647$	− 2.55775e9i	− 0.0145963i	−0.999973	−	0.00729813i	$-0.997677\pi$
$647$	− 2.55775e9i	− 0.0145963i	0.999973	−	0.00729813i	$-0.00232309\pi$
$648$	0	0
$649$	−2.23901e11	−1.26205
$650$	0	0
$651$	− 1.02520e11i	− 0.570801i
$652$	0	0
$653$	−2.38378e11	−1.31103	−0.655515	−	0.755182i	$-0.727548\pi$
$653$	−2.38378e11	−1.31103	−0.655515	+	0.755182i	$0.727548\pi$
$654$	0	0
$655$	− 8.60763e10i	− 0.467647i
$656$	0	0
$657$	1.15799e11	0.621502
$658$	0	0
$659$	2.02450e10i	0.107344i	0.998559	+	0.0536718i	$0.0170925\pi$
$659$	2.02450e10i	0.107344i	−0.998559	+	0.0536718i	$0.982908\pi$
$660$	0	0
$661$	8.31440e10	0.435537	0.217769	−	0.976000i	$-0.430122\pi$
$661$	8.31440e10	0.435537	0.217769	+	0.976000i	$0.430122\pi$
$662$	0	0
$663$	− 8.85959e10i	− 0.458521i
$664$	0	0
$665$	1.07773e11	0.551092
$666$	0	0
$667$	2.82096e9i	0.0142526i
$668$	0	0
$669$	−1.75220e11	−0.874742
$670$	0	0
$671$	− 4.47882e11i	− 2.20940i
$672$	0	0
$673$	−3.55384e11	−1.73236	−0.866180	−	0.499732i	$-0.833432\pi$
$673$	−3.55384e11	−1.73236	−0.866180	+	0.499732i	$0.833432\pi$
$674$	0	0
$675$	2.96341e10i	0.142750i
$676$	0	0
$677$	2.28161e11	1.08614	0.543071	−	0.839687i	$-0.317262\pi$
$677$	2.28161e11	1.08614	0.543071	+	0.839687i	$0.317262\pi$
$678$	0	0
$679$	− 6.01072e10i	− 0.282779i
$680$	0	0
$681$	−2.09865e10	−0.0975777
$682$	0	0
$683$	− 3.11677e11i	− 1.43226i	−0.697967	−	0.716130i	$-0.745912\pi$
$683$	− 3.11677e11i	− 1.43226i	0.697967	−	0.716130i	$-0.254088\pi$
$684$	0	0
$685$	1.06453e10	0.0483499
$686$	0	0
$687$	9.85303e10i	0.442327i
$688$	0	0
$689$	1.29187e10	0.0573246
$690$	0	0
$691$	− 1.42781e11i	− 0.626265i	−0.949709	−	0.313133i	$-0.898622\pi$
$691$	− 1.42781e11i	− 0.626265i	0.949709	−	0.313133i	$-0.101378\pi$
$692$	0	0
$693$	−1.40868e11	−0.610771
$694$	0	0
$695$	1.28060e11i	0.548877i
$696$	0	0
$697$	−5.85747e10	−0.248187
$698$	0	0
$699$	2.04750e10i	0.0857658i
$700$	0	0
$701$	2.40399e11	0.995543	0.497772	−	0.867308i	$-0.334152\pi$
$701$	2.40399e11	0.995543	0.497772	+	0.867308i	$0.334152\pi$
$702$	0	0
$703$	8.05205e10i	0.329674i
$704$	0	0
$705$	1.41645e11	0.573384
$706$	0	0
$707$	− 1.25210e9i	− 0.00501144i
$708$	0	0
$709$	2.25436e10	0.0892153	0.0446076	−	0.999005i	$-0.485796\pi$
$709$	2.25436e10	0.0892153	0.0446076	+	0.999005i	$0.485796\pi$
$710$	0	0
$711$	− 8.36791e10i	− 0.327445i
$712$	0	0
$713$	5.34461e9	0.0206803
$714$	0	0
$715$	1.79738e11i	0.687728i
$716$	0	0
$717$	−9.30492e10	−0.352076
$718$	0	0
$719$	2.41719e11i	0.904471i	0.891899	+	0.452235i	$0.149373\pi$
$719$	2.41719e11i	0.904471i	−0.891899	+	0.452235i	$0.850627\pi$
$720$	0	0
$721$	−4.47485e11	−1.65591
$722$	0	0
$723$	− 1.93543e11i	− 0.708310i
$724$	0	0
$725$	−1.29812e11	−0.469855
$726$	0	0
$727$	4.62533e11i	1.65579i	0.560883	+	0.827895i	$0.310462\pi$
$727$	4.62533e11i	1.65579i	−0.560883	+	0.827895i	$0.689538\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 4.45355e10i	− 0.155969i
$732$	0	0
$733$	2.10934e11	0.730687	0.365344	−	0.930873i	$-0.380952\pi$
$733$	2.10934e11	0.730687	0.365344	+	0.930873i	$0.380952\pi$
$734$	0	0
$735$	− 1.34468e10i	− 0.0460756i
$736$	0	0
$737$	−7.41202e11	−2.51227
$738$	0	0
$739$	4.27731e11i	1.43415i	0.696998	+	0.717073i	$0.254519\pi$
$739$	4.27731e11i	1.43415i	−0.696998	+	0.717073i	$0.745481\pi$
$740$	0	0
$741$	−1.39416e11	−0.462425
$742$	0	0
$743$	− 2.50623e10i	− 0.0822366i	−0.999154	−	0.0411183i	$-0.986908\pi$
$743$	− 2.50623e10i	− 0.0822366i	0.999154	−	0.0411183i	$-0.0130921\pi$
$744$	0	0
$745$	−2.49069e11	−0.808528
$746$	0	0
$747$	− 1.76862e11i	− 0.568005i
$748$	0	0
$749$	−4.09052e11	−1.29972
$750$	0	0
$751$	1.62048e11i	0.509429i	0.967016	+	0.254715i	$0.0819816\pi$
$751$	1.62048e11i	0.509429i	−0.967016	+	0.254715i	$0.918018\pi$
$752$	0	0
$753$	7.06372e10	0.219712
$754$	0	0
$755$	− 5.11513e10i	− 0.157423i
$756$	0	0
$757$	3.25684e11	0.991776	0.495888	−	0.868387i	$-0.334843\pi$
$757$	3.25684e11	0.991776	0.495888	+	0.868387i	$0.334843\pi$
$758$	0	0
$759$	− 7.34376e9i	− 0.0221285i
$760$	0	0
$761$	−1.60973e11	−0.479971	−0.239985	−	0.970776i	$-0.577143\pi$
$761$	−1.60973e11	−0.479971	−0.239985	+	0.970776i	$0.577143\pi$
$762$	0	0
$763$	4.05081e10i	0.119521i
$764$	0	0
$765$	5.79953e10	0.169335
$766$	0	0
$767$	2.03707e11i	0.588605i
$768$	0	0
$769$	5.65984e11	1.61845	0.809224	−	0.587501i	$-0.199888\pi$
$769$	5.65984e11	1.61845	0.809224	+	0.587501i	$0.199888\pi$
$770$	0	0
$771$	1.69511e10i	0.0479712i
$772$	0	0
$773$	−4.84358e11	−1.35659	−0.678295	−	0.734790i	$-0.737281\pi$
$773$	−4.84358e11	−1.35659	−0.678295	+	0.734790i	$0.737281\pi$
$774$	0	0
$775$	2.45944e11i	0.681755i
$776$	0	0
$777$	7.40207e10	0.203081
$778$	0	0
$779$	9.21744e10i	0.250300i
$780$	0	0
$781$	−5.72643e11	−1.53915
$782$	0	0
$783$	− 4.58217e10i	− 0.121906i
$784$	0	0
$785$	−2.70528e11	−0.712415
$786$	0	0
$787$	3.78123e10i	0.0985675i	0.998785	+	0.0492838i	$0.0156939\pi$
$787$	3.78123e10i	0.0985675i	−0.998785	+	0.0492838i	$0.984306\pi$
$788$	0	0
$789$	−1.57832e11	−0.407275
$790$	0	0
$791$	4.10477e11i	1.04854i
$792$	0	0
$793$	−4.07486e11	−1.03043
$794$	0	0
$795$	8.45662e9i	0.0211704i
$796$	0	0
$797$	−5.04890e11	−1.25131	−0.625653	−	0.780102i	$-0.715167\pi$
$797$	−5.04890e11	−1.25131	−0.625653	+	0.780102i	$0.715167\pi$
$798$	0	0
$799$	7.96199e11i	1.95360i
$800$	0	0
$801$	−1.41226e11	−0.343071
$802$	0	0
$803$	− 1.32054e12i	− 3.17606i
$804$	0	0
$805$	5.16493e9	0.0122993
$806$	0	0
$807$	− 3.48004e11i	− 0.820522i
$808$	0	0
$809$	6.91435e11	1.61420	0.807100	−	0.590415i	$-0.201036\pi$
$809$	6.91435e11	1.61420	0.807100	+	0.590415i	$0.201036\pi$
$810$	0	0
$811$	− 8.32734e10i	− 0.192497i	−0.995357	−	0.0962483i	$-0.969316\pi$
$811$	− 8.32734e10i	− 0.192497i	0.995357	−	0.0962483i	$-0.0306843\pi$
$812$	0	0
$813$	−5.50393e10	−0.125983
$814$	0	0
$815$	− 1.91264e11i	− 0.433513i
$816$	0	0
$817$	−7.00821e10	−0.157296
$818$	0	0
$819$	1.28162e11i	0.284856i
$820$	0	0
$821$	−2.54437e11	−0.560026	−0.280013	−	0.959996i	$-0.590339\pi$
$821$	−2.54437e11	−0.560026	−0.280013	+	0.959996i	$0.590339\pi$
$822$	0	0
$823$	7.96956e9i	0.0173714i	0.999962	+	0.00868570i	$0.00276478\pi$
$823$	7.96956e9i	0.0173714i	−0.999962	+	0.00868570i	$0.997235\pi$
$824$	0	0
$825$	3.37939e11	0.729495
$826$	0	0
$827$	− 2.38385e11i	− 0.509633i	−0.966989	−	0.254816i	$-0.917985\pi$
$827$	− 2.38385e11i	− 0.509633i	0.966989	−	0.254816i	$-0.0820150\pi$
$828$	0	0
$829$	8.34982e11	1.76790	0.883952	−	0.467577i	$-0.154873\pi$
$829$	8.34982e11	1.76790	0.883952	+	0.467577i	$0.154873\pi$
$830$	0	0
$831$	− 3.45657e11i	− 0.724839i
$832$	0	0
$833$	7.55858e10	0.156986
$834$	0	0
$835$	− 6.17798e10i	− 0.127087i
$836$	0	0
$837$	−8.68142e10	−0.176884
$838$	0	0
$839$	5.10577e11i	1.03042i	0.857065	+	0.515209i	$0.172286\pi$
$839$	5.10577e11i	1.03042i	−0.857065	+	0.515209i	$0.827714\pi$
$840$	0	0
$841$	−2.99524e11	−0.598753
$842$	0	0
$843$	− 3.42039e11i	− 0.677275i
$844$	0	0
$845$	−9.55604e10	−0.187435
$846$	0	0
$847$	1.05280e12i	2.04556i
$848$	0	0
$849$	−2.69939e11	−0.519558
$850$	0	0
$851$	3.85887e9i	0.00735770i
$852$	0	0
$853$	8.34513e11	1.57629	0.788147	−	0.615488i	$-0.211041\pi$
$853$	8.34513e11	1.57629	0.788147	+	0.615488i	$0.211041\pi$
$854$	0	0
$855$	− 9.12626e10i	− 0.170777i
$856$	0	0
$857$	−2.88552e11	−0.534935	−0.267468	−	0.963567i	$-0.586187\pi$
$857$	−2.88552e11	−0.534935	−0.267468	+	0.963567i	$0.586187\pi$
$858$	0	0
$859$	4.49879e11i	0.826272i	0.910669	+	0.413136i	$0.135567\pi$
$859$	4.49879e11i	0.826272i	−0.910669	+	0.413136i	$0.864433\pi$
$860$	0	0
$861$	8.47338e10	0.154186
$862$	0	0
$863$	− 2.80389e11i	− 0.505496i	−0.967532	−	0.252748i	$-0.918666\pi$
$863$	− 2.80389e11i	− 0.505496i	0.967532	−	0.252748i	$-0.0813344\pi$
$864$	0	0
$865$	−4.80171e11	−0.857692
$866$	0	0
$867$	− 2.27573e8i	0 0.000402759i
$868$	0	0
$869$	−9.54254e11	−1.67334
$870$	0	0
$871$	6.74351e11i	1.17169i
$872$	0	0
$873$	−5.08990e10	−0.0876299
$874$	0	0
$875$	5.58100e11i	0.952094i
$876$	0	0
$877$	2.88997e11	0.488534	0.244267	−	0.969708i	$-0.421453\pi$
$877$	2.88997e11	0.488534	0.244267	+	0.969708i	$0.421453\pi$
$878$	0	0
$879$	− 2.30788e11i	− 0.386597i
$880$	0	0
$881$	−3.81429e11	−0.633155	−0.316577	−	0.948567i	$-0.602534\pi$
$881$	−3.81429e11	−0.633155	−0.316577	+	0.948567i	$0.602534\pi$
$882$	0	0
$883$	− 2.51251e11i	− 0.413299i	−0.978415	−	0.206650i	$-0.933744\pi$
$883$	− 2.51251e11i	− 0.413299i	0.978415	−	0.206650i	$-0.0662560\pi$
$884$	0	0
$885$	−1.33347e11	−0.217376
$886$	0	0
$887$	− 1.23122e11i	− 0.198903i	−0.995042	−	0.0994516i	$-0.968291\pi$
$887$	− 1.23122e11i	− 0.198903i	0.995042	−	0.0994516i	$-0.0317088\pi$
$888$	0	0
$889$	−9.14856e10	−0.146469
$890$	0	0
$891$	1.19287e11i	0.189270i
$892$	0	0
$893$	1.25292e12	1.97023
$894$	0	0
$895$	1.41779e11i	0.220963i
$896$	0	0
$897$	−6.68140e9	−0.0103204
$898$	0	0
$899$	− 3.80291e11i	− 0.582206i
$900$	0	0
$901$	−4.75354e10	−0.0721303
$902$	0	0
$903$	6.44249e10i	0.0968952i
$904$	0	0
$905$	−4.81671e11	−0.718052
$906$	0	0
$907$	− 5.83116e11i	− 0.861640i	−0.902438	−	0.430820i	$-0.858224\pi$
$907$	− 5.83116e11i	− 0.861640i	0.902438	−	0.430820i	$-0.141776\pi$
$908$	0	0
$909$	−1.06028e9	−0.00155298
$910$	0	0
$911$	− 6.35868e11i	− 0.923196i	−0.887089	−	0.461598i	$-0.847276\pi$
$911$	− 6.35868e11i	− 0.923196i	0.887089	−	0.461598i	$-0.152724\pi$
$912$	0	0
$913$	−2.01689e12	−2.90268
$914$	0	0
$915$	− 2.66742e11i	− 0.380546i
$916$	0	0
$917$	−6.99923e11	−0.989858
$918$	0	0
$919$	− 2.80127e11i	− 0.392729i	−0.980531	−	0.196365i	$-0.937086\pi$
$919$	− 2.80127e11i	− 0.392729i	0.980531	−	0.196365i	$-0.0629136\pi$
$920$	0	0
$921$	2.13464e11	0.296679
$922$	0	0
$923$	5.20995e11i	0.717838i
$924$	0	0
$925$	−1.77574e11	−0.242557
$926$	0	0
$927$	3.78932e11i	0.513147i
$928$	0	0
$929$	9.35745e11	1.25630	0.628152	−	0.778091i	$-0.283812\pi$
$929$	9.35745e11	1.25630	0.628152	+	0.778091i	$0.283812\pi$
$930$	0	0
$931$	− 1.18943e11i	− 0.158322i
$932$	0	0
$933$	−5.38570e11	−0.710748
$934$	0	0
$935$	− 6.61363e11i	− 0.865353i
$936$	0	0
$937$	3.35693e11	0.435496	0.217748	−	0.976005i	$-0.430129\pi$
$937$	3.35693e11	0.435496	0.217748	+	0.976005i	$0.430129\pi$
$938$	0	0
$939$	− 7.13605e11i	− 0.917900i
$940$	0	0
$941$	−1.40491e11	−0.179180	−0.0895899	−	0.995979i	$-0.528556\pi$
$941$	−1.40491e11	−0.179180	−0.0895899	+	0.995979i	$0.528556\pi$
$942$	0	0
$943$	4.41737e9i	0.00558621i
$944$	0	0
$945$	−8.38956e10	−0.105199
$946$	0	0
$947$	1.97092e10i	0.0245058i	0.999925	+	0.0122529i	$0.00390031\pi$
$947$	1.97092e10i	0.0245058i	−0.999925	+	0.0122529i	$0.996100\pi$
$948$	0	0
$949$	−1.20144e12	−1.48127
$950$	0	0
$951$	− 1.16898e11i	− 0.142918i
$952$	0	0
$953$	9.61755e11	1.16598	0.582992	−	0.812478i	$-0.301882\pi$
$953$	9.61755e11	1.16598	0.582992	+	0.812478i	$0.301882\pi$
$954$	0	0
$955$	2.68090e10i	0.0322305i
$956$	0	0
$957$	−5.22539e11	−0.622975
$958$	0	0
$959$	− 8.65614e10i	− 0.102341i
$960$	0	0
$961$	1.32389e11	0.155224
$962$	0	0
$963$	3.46386e11i	0.402768i
$964$	0	0
$965$	2.19730e11	0.253385
$966$	0	0
$967$	− 1.05599e12i	− 1.20769i	−0.797103	−	0.603844i	$-0.793635\pi$
$967$	− 1.05599e12i	− 1.20769i	0.797103	−	0.603844i	$-0.206365\pi$
$968$	0	0
$969$	5.12995e11	0.581859
$970$	0	0
$971$	1.64566e11i	0.185124i	0.995707	+	0.0925619i	$0.0295056\pi$
$971$	1.64566e11i	0.185124i	−0.995707	+	0.0925619i	$0.970494\pi$
$972$	0	0
$973$	1.04131e12	1.16179
$974$	0	0
$975$	− 3.07459e11i	− 0.340227i
$976$	0	0
$977$	8.97096e11	0.984602	0.492301	−	0.870425i	$-0.336156\pi$
$977$	8.97096e11	0.984602	0.492301	+	0.870425i	$0.336156\pi$
$978$	0	0
$979$	1.61050e12i	1.75320i
$980$	0	0
$981$	3.43024e10	0.0370380
$982$	0	0
$983$	1.08369e12i	1.16062i	0.814395	+	0.580311i	$0.197069\pi$
$983$	1.08369e12i	1.16062i	−0.814395	+	0.580311i	$0.802931\pi$
$984$	0	0
$985$	6.76826e11	0.719006
$986$	0	0
$987$	− 1.15178e12i	− 1.21367i
$988$	0	0
$989$	−3.35862e9	−0.00351055
$990$	0	0
$991$	− 1.19310e12i	− 1.23703i	−0.785772	−	0.618516i	$-0.787734\pi$
$991$	− 1.19310e12i	− 1.23703i	0.785772	−	0.618516i	$-0.212266\pi$
$992$	0	0
$993$	−4.77968e11	−0.491588
$994$	0	0
$995$	4.37727e11i	0.446592i
$996$	0	0
$997$	−4.65045e11	−0.470667	−0.235334	−	0.971915i	$-0.575618\pi$
$997$	−4.65045e11	−0.470667	−0.235334	+	0.971915i	$0.575618\pi$
$998$	0	0
$999$	− 6.26809e10i	− 0.0629323i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.a.127.24	yes	32
4.3	odd	2	inner	384.9.g.a.127.23	✓	32
8.3	odd	2		384.9.g.b.127.10	yes	32
8.5	even	2		384.9.g.b.127.9	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.23	✓	32	4.3	odd	2	inner
384.9.g.a.127.24	yes	32	1.1	even	1	trivial
384.9.g.b.127.9	yes	32	8.5	even	2
384.9.g.b.127.10	yes	32	8.3	odd	2

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} +317.614 q^{5} -2582.66i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} +317.614 q^{5} -2582.66i q^{7} -2187.00 q^{9} +24940.0i q^{11} +22690.6 q^{13} +14853.3i q^{15} -83491.9 q^{17} +131385. i q^{19} +120779. q^{21} +6296.48i q^{23} -289746. q^{25} -102276. i q^{27} +448021. q^{29} -848824. i q^{31} -1.16633e6 q^{33} -820288. i q^{35} +612861. q^{37} +1.06113e6i q^{39} +701562. q^{41} +533412. i q^{43} -694622. q^{45} -9.53625e6i q^{47} -905307. q^{49} -3.90453e6i q^{51} +569341. q^{53} +7.92128e6i q^{55} -6.14425e6 q^{57} +8.97760e6i q^{59} -1.79584e7 q^{61} +5.64827e6i q^{63} +7.20684e6 q^{65} +2.97195e7i q^{67} -294457. q^{69} +2.29609e7i q^{71} -5.29487e7 q^{73} -1.35501e7i q^{75} +6.44113e7 q^{77} +3.82620e7i q^{79} +4.78297e6 q^{81} +8.08697e7i q^{83} -2.65182e7 q^{85} +2.09519e7i q^{87} +6.45751e7 q^{89} -5.86019e7i q^{91} +3.96956e7 q^{93} +4.17296e7i q^{95} +2.32734e7 q^{97} -5.45437e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 32 * q - 1344 * q^5 - 69984 * q^9	\( 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) \) 32 * q - 1344 * q^5 - 69984 * q^9 - 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 - 275520 * q^29 + 2421440 * q^37 - 4374720 * q^41 + 2939328 * q^45 - 14219104 * q^49 - 6224448 * q^53 + 3100032 * q^57 - 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 + 154517760 * q^77 + 153055008 * q^81 - 384830848 * q^85 - 182669760 * q^89 + 149817600 * q^93 - 149408192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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