LMFDB - Embedded newform 384.9.g.b.127.17

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			127.17
Character	$\chi$	$=$	384.127
Dual form			384.9.g.b.127.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} +19.7383 q^{5} +759.796i q^{7} -2187.00 q^{9} +O(q^{10})$	$q-46.7654i q^{3} +19.7383 q^{5} +759.796i q^{7} -2187.00 q^{9} +15847.0i q^{11} +47667.3 q^{13} -923.068i q^{15} -117951. q^{17} -196795. i q^{19} +35532.1 q^{21} -143587. i q^{23} -390235. q^{25} +102276. i q^{27} -1.30803e6 q^{29} +1.08127e6i q^{31} +741092. q^{33} +14997.1i q^{35} +2.61456e6 q^{37} -2.22918e6i q^{39} +956143. q^{41} +4.78185e6i q^{43} -43167.6 q^{45} -2.03435e6i q^{47} +5.18751e6 q^{49} +5.51603e6i q^{51} -4.50474e6 q^{53} +312793. i q^{55} -9.20320e6 q^{57} -1.40186e7i q^{59} +2.05420e7 q^{61} -1.66167e6i q^{63} +940872. q^{65} -2.77586e7i q^{67} -6.71488e6 q^{69} -3.22968e7i q^{71} +2.34885e7 q^{73} +1.82495e7i q^{75} -1.20405e7 q^{77} +1.08547e7i q^{79} +4.78297e6 q^{81} +7.99186e7i q^{83} -2.32815e6 q^{85} +6.11706e7i q^{87} +7.46098e7 q^{89} +3.62174e7i q^{91} +5.05661e7 q^{93} -3.88440e6i q^{95} -5.73670e7 q^{97} -3.46575e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	19.7383	0.0315813	0.0157906	−	0.999875i	$-0.494973\pi$
$5$	19.7383	0.0315813	0.0157906	+	0.999875i	$0.494973\pi$
$6$	0	0
$7$	759.796i	0.316450i	0.987403	+	0.158225i	$0.0505771\pi$
$7$	759.796i	0.316450i	−0.987403	+	0.158225i	$0.949423\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	15847.0i	1.08237i	0.840902	+	0.541187i	$0.182025\pi$
$11$	15847.0i	1.08237i	−0.840902	+	0.541187i	$0.817975\pi$
$12$	0	0
$13$	47667.3	1.66897	0.834483	−	0.551034i	$-0.185766\pi$
$13$	47667.3	1.66897	0.834483	+	0.551034i	$0.185766\pi$
$14$	0	0
$15$	− 923.068i	− 0.0182335i
$16$	0	0
$17$	−117951.	−1.41223	−0.706117	−	0.708095i	$-0.749555\pi$
$17$	−117951.	−1.41223	−0.706117	+	0.708095i	$0.749555\pi$
$18$	0	0
$19$	− 196795.i	− 1.51008i	−0.655678	−	0.755041i	$-0.727617\pi$
$19$	− 196795.i	− 1.51008i	0.655678	−	0.755041i	$-0.272383\pi$
$20$	0	0
$21$	35532.1	0.182702
$22$	0	0
$23$	− 143587.i	− 0.513101i	−0.966531	−	0.256550i	$-0.917414\pi$
$23$	− 143587.i	− 0.513101i	0.966531	−	0.256550i	$-0.0825859\pi$
$24$	0	0
$25$	−390235.	−0.999003
$26$	0	0
$27$	102276.i	0.192450i
$28$	0	0
$29$	−1.30803e6	−1.84938	−0.924690	−	0.380720i	$-0.875676\pi$
$29$	−1.30803e6	−1.84938	−0.924690	+	0.380720i	$0.875676\pi$
$30$	0	0
$31$	1.08127e6i	1.17081i	0.810740	+	0.585407i	$0.199065\pi$
$31$	1.08127e6i	1.17081i	−0.810740	+	0.585407i	$0.800935\pi$
$32$	0	0
$33$	741092.	0.624909
$34$	0	0
$35$	14997.1i	0.00999388i
$36$	0	0
$37$	2.61456e6	1.39506	0.697528	−	0.716558i	$-0.254283\pi$
$37$	2.61456e6	1.39506	0.697528	+	0.716558i	$0.254283\pi$
$38$	0	0
$39$	− 2.22918e6i	− 0.963578i
$40$	0	0
$41$	956143.	0.338367	0.169183	−	0.985585i	$-0.445887\pi$
$41$	956143.	0.338367	0.169183	+	0.985585i	$0.445887\pi$
$42$	0	0
$43$	4.78185e6i	1.39869i	0.714783	+	0.699346i	$0.246525\pi$
$43$	4.78185e6i	1.39869i	−0.714783	+	0.699346i	$0.753475\pi$
$44$	0	0
$45$	−43167.6	−0.0105271
$46$	0	0
$47$	− 2.03435e6i	− 0.416903i	−0.978033	−	0.208451i	$-0.933158\pi$
$47$	− 2.03435e6i	− 0.416903i	0.978033	−	0.208451i	$-0.0668423\pi$
$48$	0	0
$49$	5.18751e6	0.899860
$50$	0	0
$51$	5.51603e6i	0.815353i
$52$	0	0
$53$	−4.50474e6	−0.570908	−0.285454	−	0.958392i	$-0.592144\pi$
$53$	−4.50474e6	−0.570908	−0.285454	+	0.958392i	$0.592144\pi$
$54$	0	0
$55$	312793.i	0.0341827i
$56$	0	0
$57$	−9.20320e6	−0.871846
$58$	0	0
$59$	− 1.40186e7i	− 1.15690i	−0.815718	−	0.578449i	$-0.803658\pi$
$59$	− 1.40186e7i	− 1.15690i	0.815718	−	0.578449i	$-0.196342\pi$
$60$	0	0
$61$	2.05420e7	1.48362	0.741812	−	0.670608i	$-0.233967\pi$
$61$	2.05420e7	1.48362	0.741812	+	0.670608i	$0.233967\pi$
$62$	0	0
$63$	− 1.66167e6i	− 0.105483i
$64$	0	0
$65$	940872.	0.0527080
$66$	0	0
$67$	− 2.77586e7i	− 1.37752i	−0.724989	−	0.688760i	$-0.758155\pi$
$67$	− 2.77586e7i	− 1.37752i	0.724989	−	0.688760i	$-0.241845\pi$
$68$	0	0
$69$	−6.71488e6	−0.296239
$70$	0	0
$71$	− 3.22968e7i	− 1.27094i	−0.772125	−	0.635471i	$-0.780806\pi$
$71$	− 3.22968e7i	− 1.27094i	0.772125	−	0.635471i	$-0.219194\pi$
$72$	0	0
$73$	2.34885e7	0.827112	0.413556	−	0.910479i	$-0.364287\pi$
$73$	2.34885e7	0.827112	0.413556	+	0.910479i	$0.364287\pi$
$74$	0	0
$75$	1.82495e7i	0.576774i
$76$	0	0
$77$	−1.20405e7	−0.342517
$78$	0	0
$79$	1.08547e7i	0.278683i	0.990244	+	0.139341i	$0.0444985\pi$
$79$	1.08547e7i	0.278683i	−0.990244	+	0.139341i	$0.955502\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	7.99186e7i	1.68398i	0.539497	+	0.841988i	$0.318614\pi$
$83$	7.99186e7i	1.68398i	−0.539497	+	0.841988i	$0.681386\pi$
$84$	0	0
$85$	−2.32815e6	−0.0446001
$86$	0	0
$87$	6.11706e7i	1.06774i
$88$	0	0
$89$	7.46098e7	1.18915	0.594574	−	0.804041i	$-0.297321\pi$
$89$	7.46098e7	1.18915	0.594574	+	0.804041i	$0.297321\pi$
$90$	0	0
$91$	3.62174e7i	0.528144i
$92$	0	0
$93$	5.05661e7	0.675970
$94$	0	0
$95$	− 3.88440e6i	− 0.0476903i
$96$	0	0
$97$	−5.73670e7	−0.648001	−0.324000	−	0.946057i	$-0.605028\pi$
$97$	−5.73670e7	−0.648001	−0.324000	+	0.946057i	$0.605028\pi$
$98$	0	0
$99$	− 3.46575e7i	− 0.360791i
$100$	0	0
$101$	9.65501e7	0.927827	0.463914	−	0.885881i	$-0.346445\pi$
$101$	9.65501e7	0.927827	0.463914	+	0.885881i	$0.346445\pi$
$102$	0	0
$103$	− 1.41970e8i	− 1.26138i	−0.776033	−	0.630692i	$-0.782771\pi$
$103$	− 1.41970e8i	− 1.26138i	0.776033	−	0.630692i	$-0.217229\pi$
$104$	0	0
$105$	701343.	0.00576997
$106$	0	0
$107$	− 6.39218e7i	− 0.487656i	−0.969818	−	0.243828i	$-0.921597\pi$
$107$	− 6.39218e7i	− 0.487656i	0.969818	−	0.243828i	$-0.0784033\pi$
$108$	0	0
$109$	−1.31267e8	−0.929932	−0.464966	−	0.885329i	$-0.653933\pi$
$109$	−1.31267e8	−0.929932	−0.464966	+	0.885329i	$0.653933\pi$
$110$	0	0
$111$	− 1.22271e8i	− 0.805436i
$112$	0	0
$113$	1.49881e8	0.919246	0.459623	−	0.888114i	$-0.347985\pi$
$113$	1.49881e8	0.919246	0.459623	+	0.888114i	$0.347985\pi$
$114$	0	0
$115$	− 2.83415e6i	− 0.0162044i
$116$	0	0
$117$	−1.04248e8	−0.556322
$118$	0	0
$119$	− 8.96188e7i	− 0.446901i
$120$	0	0
$121$	−3.67695e7	−0.171533
$122$	0	0
$123$	− 4.47144e7i	− 0.195356i
$124$	0	0
$125$	−1.54128e7	−0.0631310
$126$	0	0
$127$	− 1.99191e8i	− 0.765695i	−0.923812	−	0.382848i	$-0.874943\pi$
$127$	− 1.99191e8i	− 0.765695i	0.923812	−	0.382848i	$-0.125057\pi$
$128$	0	0
$129$	2.23625e8	0.807536
$130$	0	0
$131$	− 3.03185e8i	− 1.02949i	−0.857343	−	0.514746i	$-0.827886\pi$
$131$	− 3.03185e8i	− 1.02949i	0.857343	−	0.514746i	$-0.172114\pi$
$132$	0	0
$133$	1.49524e8	0.477865
$134$	0	0
$135$	2.01875e6i	0.00607782i
$136$	0	0
$137$	−3.76126e8	−1.06770	−0.533852	−	0.845578i	$-0.679256\pi$
$137$	−3.76126e8	−1.06770	−0.533852	+	0.845578i	$0.679256\pi$
$138$	0	0
$139$	2.37171e8i	0.635334i	0.948202	+	0.317667i	$0.102899\pi$
$139$	2.37171e8i	0.635334i	−0.948202	+	0.317667i	$0.897101\pi$
$140$	0	0
$141$	−9.51372e7	−0.240699
$142$	0	0
$143$	7.55386e8i	1.80644i
$144$	0	0
$145$	−2.58183e7	−0.0584058
$146$	0	0
$147$	− 2.42596e8i	− 0.519534i
$148$	0	0
$149$	−8.73173e8	−1.77156	−0.885779	−	0.464108i	$-0.846375\pi$
$149$	−8.73173e8	−1.77156	−0.885779	+	0.464108i	$0.846375\pi$
$150$	0	0
$151$	1.26207e6i	0.00242758i	0.999999	+	0.00121379i	$0.000386362\pi$
$151$	1.26207e6i	0.00242758i	−0.999999	+	0.00121379i	$0.999614\pi$
$152$	0	0
$153$	2.57959e8	0.470745
$154$	0	0
$155$	2.13424e7i	0.0369758i
$156$	0	0
$157$	−5.79730e8	−0.954174	−0.477087	−	0.878856i	$-0.658307\pi$
$157$	−5.79730e8	−0.954174	−0.477087	+	0.878856i	$0.658307\pi$
$158$	0	0
$159$	2.10666e8i	0.329614i
$160$	0	0
$161$	1.09096e8	0.162370
$162$	0	0
$163$	− 1.36798e9i	− 1.93790i	−0.247264	−	0.968948i	$-0.579532\pi$
$163$	− 1.36798e9i	− 1.93790i	0.247264	−	0.968948i	$-0.420468\pi$
$164$	0	0
$165$	1.46279e7	0.0197354
$166$	0	0
$167$	− 9.31645e8i	− 1.19780i	−0.800824	−	0.598900i	$-0.795605\pi$
$167$	− 9.31645e8i	− 1.19780i	0.800824	−	0.598900i	$-0.204395\pi$
$168$	0	0
$169$	1.45644e9	1.78545
$170$	0	0
$171$	4.30391e8i	0.503360i
$172$	0	0
$173$	6.08081e8	0.678856	0.339428	−	0.940632i	$-0.389767\pi$
$173$	6.08081e8	0.678856	0.339428	+	0.940632i	$0.389767\pi$
$174$	0	0
$175$	− 2.96499e8i	− 0.316134i
$176$	0	0
$177$	−6.55583e8	−0.667936
$178$	0	0
$179$	− 1.47872e8i	− 0.144037i	−0.997403	−	0.0720184i	$-0.977056\pi$
$179$	− 1.47872e8i	− 0.144037i	0.997403	−	0.0720184i	$-0.0229440\pi$
$180$	0	0
$181$	−6.89805e8	−0.642705	−0.321353	−	0.946960i	$-0.604137\pi$
$181$	−6.89805e8	−0.642705	−0.321353	+	0.946960i	$0.604137\pi$
$182$	0	0
$183$	− 9.60655e8i	− 0.856571i
$184$	0	0
$185$	5.16069e7	0.0440576
$186$	0	0
$187$	− 1.86918e9i	− 1.52856i
$188$	0	0
$189$	−7.77088e7	−0.0609008
$190$	0	0
$191$	− 1.82722e9i	− 1.37296i	−0.727148	−	0.686481i	$-0.759155\pi$
$191$	− 1.82722e9i	− 1.37296i	0.727148	−	0.686481i	$-0.240845\pi$
$192$	0	0
$193$	4.57534e8	0.329757	0.164879	−	0.986314i	$-0.447277\pi$
$193$	4.57534e8	0.329757	0.164879	+	0.986314i	$0.447277\pi$
$194$	0	0
$195$	− 4.40002e7i	− 0.0304310i
$196$	0	0
$197$	−4.22625e8	−0.280601	−0.140301	−	0.990109i	$-0.544807\pi$
$197$	−4.22625e8	−0.280601	−0.140301	+	0.990109i	$0.544807\pi$
$198$	0	0
$199$	4.66840e8i	0.297684i	0.988861	+	0.148842i	$0.0475546\pi$
$199$	4.66840e8i	0.297684i	−0.988861	+	0.148842i	$0.952445\pi$
$200$	0	0
$201$	−1.29814e9	−0.795312
$202$	0	0
$203$	− 9.93837e8i	− 0.585236i
$204$	0	0
$205$	1.88726e7	0.0106860
$206$	0	0
$207$	3.14024e8i	0.171034i
$208$	0	0
$209$	3.11862e9	1.63447
$210$	0	0
$211$	1.16847e9i	0.589505i	0.955574	+	0.294753i	$0.0952373\pi$
$211$	1.16847e9i	0.589505i	−0.955574	+	0.294753i	$0.904763\pi$
$212$	0	0
$213$	−1.51037e9	−0.733779
$214$	0	0
$215$	9.43856e7i	0.0441725i
$216$	0	0
$217$	−8.21545e8	−0.370504
$218$	0	0
$219$	− 1.09845e9i	− 0.477533i
$220$	0	0
$221$	−5.62242e9	−2.35697
$222$	0	0
$223$	5.79628e8i	0.234385i	0.993109	+	0.117192i	$0.0373894\pi$
$223$	5.79628e8i	0.234385i	−0.993109	+	0.117192i	$0.962611\pi$
$224$	0	0
$225$	8.53445e8	0.333001
$226$	0	0
$227$	− 4.62864e9i	− 1.74321i	−0.490207	−	0.871606i	$-0.663079\pi$
$227$	− 4.62864e9i	− 1.74321i	0.490207	−	0.871606i	$-0.336921\pi$
$228$	0	0
$229$	−3.38866e9	−1.23221	−0.616107	−	0.787663i	$-0.711291\pi$
$229$	−3.38866e9	−1.23221	−0.616107	+	0.787663i	$0.711291\pi$
$230$	0	0
$231$	5.63079e8i	0.197752i
$232$	0	0
$233$	2.20266e9	0.747351	0.373675	−	0.927560i	$-0.378097\pi$
$233$	2.20266e9	0.747351	0.373675	+	0.927560i	$0.378097\pi$
$234$	0	0
$235$	− 4.01546e7i	− 0.0131663i
$236$	0	0
$237$	5.07624e8	0.160897
$238$	0	0
$239$	− 4.26194e9i	− 1.30622i	−0.757264	−	0.653108i	$-0.773465\pi$
$239$	− 4.26194e9i	− 1.30622i	0.757264	−	0.653108i	$-0.226535\pi$
$240$	0	0
$241$	4.80122e9	1.42326	0.711628	−	0.702556i	$-0.247958\pi$
$241$	4.80122e9	1.42326	0.711628	+	0.702556i	$0.247958\pi$
$242$	0	0
$243$	− 2.23677e8i	− 0.0641500i
$244$	0	0
$245$	1.02393e8	0.0284187
$246$	0	0
$247$	− 9.38071e9i	− 2.52027i
$248$	0	0
$249$	3.73742e9	0.972244
$250$	0	0
$251$	1.15369e9i	0.290667i	0.989383	+	0.145333i	$0.0464255\pi$
$251$	1.15369e9i	0.290667i	−0.989383	+	0.145333i	$0.953575\pi$
$252$	0	0
$253$	2.27542e9	0.555366
$254$	0	0
$255$	1.08877e8i	0.0257499i
$256$	0	0
$257$	6.22177e8	0.142620	0.0713102	−	0.997454i	$-0.477282\pi$
$257$	6.22177e8	0.142620	0.0713102	+	0.997454i	$0.477282\pi$
$258$	0	0
$259$	1.98653e9i	0.441465i
$260$	0	0
$261$	2.86067e9	0.616460
$262$	0	0
$263$	− 2.82148e9i	− 0.589732i	−0.955539	−	0.294866i	$-0.904725\pi$
$263$	− 2.82148e9i	− 0.589732i	0.955539	−	0.294866i	$-0.0952750\pi$
$264$	0	0
$265$	−8.89159e7	−0.0180300
$266$	0	0
$267$	− 3.48915e9i	− 0.686555i
$268$	0	0
$269$	2.33848e9	0.446605	0.223303	−	0.974749i	$-0.428316\pi$
$269$	2.33848e9	0.446605	0.223303	+	0.974749i	$0.428316\pi$
$270$	0	0
$271$	− 1.92807e9i	− 0.357475i	−0.983897	−	0.178737i	$-0.942799\pi$
$271$	− 1.92807e9i	− 0.357475i	0.983897	−	0.178737i	$-0.0572012\pi$
$272$	0	0
$273$	1.69372e9	0.304924
$274$	0	0
$275$	− 6.18407e9i	− 1.08129i
$276$	0	0
$277$	−3.15707e9	−0.536248	−0.268124	−	0.963384i	$-0.586404\pi$
$277$	−3.15707e9	−0.536248	−0.268124	+	0.963384i	$0.586404\pi$
$278$	0	0
$279$	− 2.36474e9i	− 0.390271i
$280$	0	0
$281$	2.57086e9	0.412338	0.206169	−	0.978516i	$-0.433900\pi$
$281$	2.57086e9	0.412338	0.206169	+	0.978516i	$0.433900\pi$
$282$	0	0
$283$	1.59550e9i	0.248743i	0.992236	+	0.124372i	$0.0396915\pi$
$283$	1.59550e9i	0.248743i	−0.992236	+	0.124372i	$0.960308\pi$
$284$	0	0
$285$	−1.81656e8	−0.0275340
$286$	0	0
$287$	7.26473e8i	0.107076i
$288$	0	0
$289$	6.93672e9	0.994404
$290$	0	0
$291$	2.68279e9i	0.374123i
$292$	0	0
$293$	−4.12015e9	−0.559039	−0.279520	−	0.960140i	$-0.590175\pi$
$293$	−4.12015e9	−0.559039	−0.279520	+	0.960140i	$0.590175\pi$
$294$	0	0
$295$	− 2.76702e8i	− 0.0365363i
$296$	0	0
$297$	−1.62077e9	−0.208303
$298$	0	0
$299$	− 6.84439e9i	− 0.856347i
$300$	0	0
$301$	−3.63323e9	−0.442616
$302$	0	0
$303$	− 4.51520e9i	− 0.535681i
$304$	0	0
$305$	4.05464e8	0.0468547
$306$	0	0
$307$	− 1.53448e10i	− 1.72746i	−0.503954	−	0.863731i	$-0.668122\pi$
$307$	− 1.53448e10i	− 1.72746i	0.503954	−	0.863731i	$-0.331878\pi$
$308$	0	0
$309$	−6.63928e9	−0.728261
$310$	0	0
$311$	8.41711e9i	0.899749i	0.893092	+	0.449875i	$0.148531\pi$
$311$	8.41711e9i	0.899749i	−0.893092	+	0.449875i	$0.851469\pi$
$312$	0	0
$313$	5.18210e9	0.539918	0.269959	−	0.962872i	$-0.412990\pi$
$313$	5.18210e9	0.539918	0.269959	+	0.962872i	$0.412990\pi$
$314$	0	0
$315$	− 3.27986e7i	− 0.00333129i
$316$	0	0
$317$	4.31162e9	0.426976	0.213488	−	0.976946i	$-0.431518\pi$
$317$	4.31162e9	0.426976	0.213488	+	0.976946i	$0.431518\pi$
$318$	0	0
$319$	− 2.07284e10i	− 2.00172i
$320$	0	0
$321$	−2.98933e9	−0.281548
$322$	0	0
$323$	2.32122e10i	2.13259i
$324$	0	0
$325$	−1.86015e10	−1.66730
$326$	0	0
$327$	6.13877e9i	0.536896i
$328$	0	0
$329$	1.54569e9	0.131929
$330$	0	0
$331$	− 7.84383e9i	− 0.653456i	−0.945118	−	0.326728i	$-0.894054\pi$
$331$	− 7.84383e9i	− 0.653456i	0.945118	−	0.326728i	$-0.105946\pi$
$332$	0	0
$333$	−5.71804e9	−0.465018
$334$	0	0
$335$	− 5.47907e8i	− 0.0435039i
$336$	0	0
$337$	1.60451e10	1.24401	0.622005	−	0.783013i	$-0.286318\pi$
$337$	1.60451e10	1.24401	0.622005	+	0.783013i	$0.286318\pi$
$338$	0	0
$339$	− 7.00922e9i	− 0.530727i
$340$	0	0
$341$	−1.71349e10	−1.26726
$342$	0	0
$343$	8.32152e9i	0.601210i
$344$	0	0
$345$	−1.32540e8	−0.00935559
$346$	0	0
$347$	− 1.36821e10i	− 0.943700i	−0.881679	−	0.471850i	$-0.843586\pi$
$347$	− 1.36821e10i	− 0.943700i	0.881679	−	0.471850i	$-0.156414\pi$
$348$	0	0
$349$	2.11356e10	1.42466	0.712332	−	0.701843i	$-0.247639\pi$
$349$	2.11356e10	1.42466	0.712332	+	0.701843i	$0.247639\pi$
$350$	0	0
$351$	4.87522e9i	0.321193i
$352$	0	0
$353$	1.47373e10	0.949112	0.474556	−	0.880225i	$-0.342609\pi$
$353$	1.47373e10	0.949112	0.474556	+	0.880225i	$0.342609\pi$
$354$	0	0
$355$	− 6.37483e8i	− 0.0401379i
$356$	0	0
$357$	−4.19106e9	−0.258018
$358$	0	0
$359$	− 2.49699e10i	− 1.50328i	−0.659575	−	0.751638i	$-0.729264\pi$
$359$	− 2.49699e10i	− 1.50328i	0.659575	−	0.751638i	$-0.270736\pi$
$360$	0	0
$361$	−2.17448e10	−1.28034
$362$	0	0
$363$	1.71954e9i	0.0990344i
$364$	0	0
$365$	4.63623e8	0.0261212
$366$	0	0
$367$	1.11559e10i	0.614951i	0.951556	+	0.307475i	$0.0994842\pi$
$367$	1.11559e10i	0.614951i	−0.951556	+	0.307475i	$0.900516\pi$
$368$	0	0
$369$	−2.09108e9	−0.112789
$370$	0	0
$371$	− 3.42268e9i	− 0.180664i
$372$	0	0
$373$	3.65556e10	1.88851	0.944253	−	0.329221i	$-0.106786\pi$
$373$	3.65556e10	1.88851	0.944253	+	0.329221i	$0.106786\pi$
$374$	0	0
$375$	7.20788e8i	0.0364487i
$376$	0	0
$377$	−6.23504e10	−3.08655
$378$	0	0
$379$	− 2.16949e9i	− 0.105148i	−0.998617	−	0.0525740i	$-0.983257\pi$
$379$	− 2.16949e9i	− 0.105148i	0.998617	−	0.0525740i	$-0.0167425\pi$
$380$	0	0
$381$	−9.31526e9	−0.442074
$382$	0	0
$383$	− 8.10758e9i	− 0.376787i	−0.982094	−	0.188394i	$-0.939672\pi$
$383$	− 8.10758e9i	− 0.376787i	0.982094	−	0.188394i	$-0.0603280\pi$
$384$	0	0
$385$	−2.37659e8	−0.0108171
$386$	0	0
$387$	− 1.04579e10i	− 0.466231i
$388$	0	0
$389$	−1.73636e10	−0.758300	−0.379150	−	0.925335i	$-0.623783\pi$
$389$	−1.73636e10	−0.758300	−0.379150	+	0.925335i	$0.623783\pi$
$390$	0	0
$391$	1.69362e10i	0.724618i
$392$	0	0
$393$	−1.41786e10	−0.594377
$394$	0	0
$395$	2.14253e8i	0.00880115i
$396$	0	0
$397$	7.49508e9	0.301727	0.150864	−	0.988555i	$-0.451795\pi$
$397$	7.49508e9	0.301727	0.150864	+	0.988555i	$0.451795\pi$
$398$	0	0
$399$	− 6.99255e9i	− 0.275895i
$400$	0	0
$401$	3.17286e10	1.22708	0.613541	−	0.789663i	$-0.289745\pi$
$401$	3.17286e10	1.22708	0.613541	+	0.789663i	$0.289745\pi$
$402$	0	0
$403$	5.15413e10i	1.95405i
$404$	0	0
$405$	9.44076e7	0.00350903
$406$	0	0
$407$	4.14330e10i	1.50997i
$408$	0	0
$409$	3.02548e10	1.08119	0.540595	−	0.841283i	$-0.318199\pi$
$409$	3.02548e10	1.08119	0.540595	+	0.841283i	$0.318199\pi$
$410$	0	0
$411$	1.75897e10i	0.616440i
$412$	0	0
$413$	1.06512e10	0.366100
$414$	0	0
$415$	1.57746e9i	0.0531821i
$416$	0	0
$417$	1.10914e10	0.366810
$418$	0	0
$419$	− 3.95924e10i	− 1.28456i	−0.766468	−	0.642282i	$-0.777988\pi$
$419$	− 3.95924e10i	− 1.28456i	0.766468	−	0.642282i	$-0.222012\pi$
$420$	0	0
$421$	−3.41632e10	−1.08750	−0.543751	−	0.839247i	$-0.682996\pi$
$421$	−3.41632e10	−1.08750	−0.543751	+	0.839247i	$0.682996\pi$
$422$	0	0
$423$	4.44913e9i	0.138968i
$424$	0	0
$425$	4.60287e10	1.41083
$426$	0	0
$427$	1.56077e10i	0.469492i
$428$	0	0
$429$	3.53259e10	1.04295
$430$	0	0
$431$	4.74352e10i	1.37465i	0.726351	+	0.687324i	$0.241215\pi$
$431$	4.74352e10i	1.37465i	−0.726351	+	0.687324i	$0.758785\pi$
$432$	0	0
$433$	−3.46150e10	−0.984720	−0.492360	−	0.870392i	$-0.663866\pi$
$433$	−3.46150e10	−0.984720	−0.492360	+	0.870392i	$0.663866\pi$
$434$	0	0
$435$	1.20740e9i	0.0337206i
$436$	0	0
$437$	−2.82572e10	−0.774823
$438$	0	0
$439$	− 3.47839e10i	− 0.936525i	−0.883589	−	0.468263i	$-0.844880\pi$
$439$	− 3.47839e10i	− 0.936525i	0.883589	−	0.468263i	$-0.155120\pi$
$440$	0	0
$441$	−1.13451e10	−0.299953
$442$	0	0
$443$	− 2.71982e10i	− 0.706196i	−0.935586	−	0.353098i	$-0.885128\pi$
$443$	− 2.71982e10i	− 0.706196i	0.935586	−	0.353098i	$-0.114872\pi$
$444$	0	0
$445$	1.47267e9	0.0375548
$446$	0	0
$447$	4.08343e10i	1.02281i
$448$	0	0
$449$	−6.44839e10	−1.58659	−0.793297	−	0.608834i	$-0.791637\pi$
$449$	−6.44839e10	−1.58659	−0.793297	+	0.608834i	$0.791637\pi$
$450$	0	0
$451$	1.51520e10i	0.366239i
$452$	0	0
$453$	5.90210e7	0.00140157
$454$	0	0
$455$	7.14870e8i	0.0166794i
$456$	0	0
$457$	−7.83314e10	−1.79586	−0.897928	−	0.440143i	$-0.854928\pi$
$457$	−7.83314e10	−1.79586	−0.897928	+	0.440143i	$0.854928\pi$
$458$	0	0
$459$	− 1.20636e10i	− 0.271784i
$460$	0	0
$461$	7.86144e10	1.74060	0.870299	−	0.492525i	$-0.163926\pi$
$461$	7.86144e10	1.74060	0.870299	+	0.492525i	$0.163926\pi$
$462$	0	0
$463$	− 3.70967e10i	− 0.807257i	−0.914923	−	0.403628i	$-0.867749\pi$
$463$	− 3.70967e10i	− 0.807257i	0.914923	−	0.403628i	$-0.132251\pi$
$464$	0	0
$465$	9.98088e8	0.0213480
$466$	0	0
$467$	4.80694e10i	1.01065i	0.862929	+	0.505325i	$0.168627\pi$
$467$	4.80694e10i	1.01065i	−0.862929	+	0.505325i	$0.831373\pi$
$468$	0	0
$469$	2.10909e10	0.435916
$470$	0	0
$471$	2.71113e10i	0.550892i
$472$	0	0
$473$	−7.57782e10	−1.51391
$474$	0	0
$475$	7.67965e10i	1.50857i
$476$	0	0
$477$	9.85187e9	0.190303
$478$	0	0
$479$	6.39132e10i	1.21408i	0.794670	+	0.607042i	$0.207644\pi$
$479$	6.39132e10i	1.21408i	−0.794670	+	0.607042i	$0.792356\pi$
$480$	0	0
$481$	1.24629e11	2.32830
$482$	0	0
$483$	− 5.10194e9i	− 0.0937447i
$484$	0	0
$485$	−1.13233e9	−0.0204647
$486$	0	0
$487$	− 2.04932e10i	− 0.364329i	−0.983268	−	0.182164i	$-0.941690\pi$
$487$	− 2.04932e10i	− 0.364329i	0.983268	−	0.182164i	$-0.0583103\pi$
$488$	0	0
$489$	−6.39743e10	−1.11884
$490$	0	0
$491$	9.60306e10i	1.65228i	0.563464	+	0.826140i	$0.309468\pi$
$491$	9.60306e10i	1.65228i	−0.563464	+	0.826140i	$0.690532\pi$
$492$	0	0
$493$	1.54284e11	2.61176
$494$	0	0
$495$	− 6.84079e8i	− 0.0113942i
$496$	0	0
$497$	2.45389e10	0.402189
$498$	0	0
$499$	− 1.03307e9i	− 0.0166620i	−0.999965	−	0.00833099i	$-0.997348\pi$
$499$	− 1.03307e9i	− 0.0166620i	0.999965	−	0.00833099i	$-0.00265187\pi$
$500$	0	0
$501$	−4.35687e10	−0.691550
$502$	0	0
$503$	− 2.88572e10i	− 0.450798i	−0.974267	−	0.225399i	$-0.927632\pi$
$503$	− 2.88572e10i	− 0.450798i	0.974267	−	0.225399i	$-0.0723685\pi$
$504$	0	0
$505$	1.90573e9	0.0293020
$506$	0	0
$507$	− 6.81111e10i	− 1.03083i
$508$	0	0
$509$	4.12139e10	0.614005	0.307003	−	0.951709i	$-0.400674\pi$
$509$	4.12139e10	0.614005	0.307003	+	0.951709i	$0.400674\pi$
$510$	0	0
$511$	1.78465e10i	0.261739i
$512$	0	0
$513$	2.01274e10	0.290615
$514$	0	0
$515$	− 2.80224e9i	− 0.0398361i
$516$	0	0
$517$	3.22384e10	0.451244
$518$	0	0
$519$	− 2.84372e10i	− 0.391937i
$520$	0	0
$521$	−7.02598e10	−0.953577	−0.476789	−	0.879018i	$-0.658199\pi$
$521$	−7.02598e10	−0.953577	−0.476789	+	0.879018i	$0.658199\pi$
$522$	0	0
$523$	1.04627e11i	1.39842i	0.714918	+	0.699208i	$0.246464\pi$
$523$	1.04627e11i	1.39842i	−0.714918	+	0.699208i	$0.753536\pi$
$524$	0	0
$525$	−1.38659e10	−0.182520
$526$	0	0
$527$	− 1.27537e11i	− 1.65346i
$528$	0	0
$529$	5.76939e10	0.736728
$530$	0	0
$531$	3.06586e10i	0.385633i
$532$	0	0
$533$	4.55768e10	0.564722
$534$	0	0
$535$	− 1.26171e9i	− 0.0154008i
$536$	0	0
$537$	−6.91529e9	−0.0831597
$538$	0	0
$539$	8.22067e10i	0.973984i
$540$	0	0
$541$	1.70512e10	0.199052	0.0995260	−	0.995035i	$-0.468267\pi$
$541$	1.70512e10	0.199052	0.0995260	+	0.995035i	$0.468267\pi$
$542$	0	0
$543$	3.22590e10i	0.371066i
$544$	0	0
$545$	−2.59099e9	−0.0293684
$546$	0	0
$547$	8.74356e10i	0.976650i	0.872662	+	0.488325i	$0.162392\pi$
$547$	8.74356e10i	0.976650i	−0.872662	+	0.488325i	$0.837608\pi$
$548$	0	0
$549$	−4.49254e10	−0.494541
$550$	0	0
$551$	2.57415e11i	2.79271i
$552$	0	0
$553$	−8.24736e9	−0.0881890
$554$	0	0
$555$	− 2.41342e9i	− 0.0254367i
$556$	0	0
$557$	3.25190e10	0.337844	0.168922	−	0.985629i	$-0.445971\pi$
$557$	3.25190e10	0.337844	0.168922	+	0.985629i	$0.445971\pi$
$558$	0	0
$559$	2.27938e11i	2.33437i
$560$	0	0
$561$	−8.74127e10	−0.882517
$562$	0	0
$563$	2.12615e10i	0.211621i	0.994386	+	0.105811i	$0.0337438\pi$
$563$	2.12615e10i	0.211621i	−0.994386	+	0.105811i	$0.966256\pi$
$564$	0	0
$565$	2.95839e9	0.0290309
$566$	0	0
$567$	3.63408e9i	0.0351611i
$568$	0	0
$569$	−1.50487e11	−1.43566	−0.717830	−	0.696219i	$-0.754864\pi$
$569$	−1.50487e11	−1.43566	−0.717830	+	0.696219i	$0.754864\pi$
$570$	0	0
$571$	− 5.23675e9i	− 0.0492626i	−0.999697	−	0.0246313i	$-0.992159\pi$
$571$	− 5.23675e9i	− 0.0492626i	0.999697	−	0.0246313i	$-0.00784118\pi$
$572$	0	0
$573$	−8.54508e10	−0.792679
$574$	0	0
$575$	5.60326e10i	0.512589i
$576$	0	0
$577$	7.74601e9	0.0698835	0.0349418	−	0.999389i	$-0.488875\pi$
$577$	7.74601e9	0.0698835	0.0349418	+	0.999389i	$0.488875\pi$
$578$	0	0
$579$	− 2.13968e10i	− 0.190385i
$580$	0	0
$581$	−6.07218e10	−0.532893
$582$	0	0
$583$	− 7.13868e10i	− 0.617936i
$584$	0	0
$585$	−2.05769e9	−0.0175693
$586$	0	0
$587$	− 1.50562e11i	− 1.26813i	−0.773279	−	0.634066i	$-0.781385\pi$
$587$	− 1.50562e11i	− 1.26813i	0.773279	−	0.634066i	$-0.218615\pi$
$588$	0	0
$589$	2.12789e11	1.76802
$590$	0	0
$591$	1.97642e10i	0.162005i
$592$	0	0
$593$	4.15664e10	0.336143	0.168071	−	0.985775i	$-0.446246\pi$
$593$	4.15664e10	0.336143	0.168071	+	0.985775i	$0.446246\pi$
$594$	0	0
$595$	− 1.76892e9i	− 0.0141137i
$596$	0	0
$597$	2.18320e10	0.171868
$598$	0	0
$599$	6.23752e10i	0.484512i	0.970212	+	0.242256i	$0.0778874\pi$
$599$	6.23752e10i	0.484512i	−0.970212	+	0.242256i	$0.922113\pi$
$600$	0	0
$601$	−1.01687e11	−0.779414	−0.389707	−	0.920939i	$-0.627424\pi$
$601$	−1.01687e11	−0.779414	−0.389707	+	0.920939i	$0.627424\pi$
$602$	0	0
$603$	6.07080e10i	0.459174i
$604$	0	0
$605$	−7.25768e8	−0.00541722
$606$	0	0
$607$	− 2.50543e11i	− 1.84555i	−0.385334	−	0.922777i	$-0.625914\pi$
$607$	− 2.50543e11i	− 1.84555i	0.385334	−	0.922777i	$-0.374086\pi$
$608$	0	0
$609$	−4.64772e10	−0.337886
$610$	0	0
$611$	− 9.69721e10i	− 0.695796i
$612$	0	0
$613$	1.28165e11	0.907667	0.453833	−	0.891087i	$-0.350056\pi$
$613$	1.28165e11	0.907667	0.453833	+	0.891087i	$0.350056\pi$
$614$	0	0
$615$	− 8.82585e8i	− 0.00616959i
$616$	0	0
$617$	−1.17897e11	−0.813510	−0.406755	−	0.913537i	$-0.633340\pi$
$617$	−1.17897e11	−0.813510	−0.406755	+	0.913537i	$0.633340\pi$
$618$	0	0
$619$	4.79933e10i	0.326902i	0.986551	+	0.163451i	$0.0522626\pi$
$619$	4.79933e10i	0.326902i	−0.986551	+	0.163451i	$0.947737\pi$
$620$	0	0
$621$	1.46854e10	0.0987462
$622$	0	0
$623$	5.66882e10i	0.376305i
$624$	0	0
$625$	1.52131e11	0.997009
$626$	0	0
$627$	− 1.45843e11i	− 0.943663i
$628$	0	0
$629$	−3.08390e11	−1.97014
$630$	0	0
$631$	− 1.90637e11i	− 1.20251i	−0.799057	−	0.601256i	$-0.794667\pi$
$631$	− 1.90637e11i	− 1.20251i	0.799057	−	0.601256i	$-0.205333\pi$
$632$	0	0
$633$	5.46439e10	0.340351
$634$	0	0
$635$	− 3.93170e9i	− 0.0241816i
$636$	0	0
$637$	2.47275e11	1.50183
$638$	0	0
$639$	7.06330e10i	0.423647i
$640$	0	0
$641$	2.61699e10	0.155014	0.0775069	−	0.996992i	$-0.475304\pi$
$641$	2.61699e10	0.155014	0.0775069	+	0.996992i	$0.475304\pi$
$642$	0	0
$643$	6.78239e10i	0.396770i	0.980124	+	0.198385i	$0.0635697\pi$
$643$	6.78239e10i	0.396770i	−0.980124	+	0.198385i	$0.936430\pi$
$644$	0	0
$645$	4.41398e9	0.0255030
$646$	0	0
$647$	− 2.04734e11i	− 1.16835i	−0.811628	−	0.584174i	$-0.801419\pi$
$647$	− 2.04734e11i	− 1.16835i	0.811628	−	0.584174i	$-0.198581\pi$
$648$	0	0
$649$	2.22153e11	1.25220
$650$	0	0
$651$	3.84199e10i	0.213910i
$652$	0	0
$653$	−1.51233e10	−0.0831753	−0.0415876	−	0.999135i	$-0.513242\pi$
$653$	−1.51233e10	−0.0831753	−0.0415876	+	0.999135i	$0.513242\pi$
$654$	0	0
$655$	− 5.98436e9i	− 0.0325126i
$656$	0	0
$657$	−5.13694e10	−0.275704
$658$	0	0
$659$	− 4.16908e10i	− 0.221054i	−0.993873	−	0.110527i	$-0.964746\pi$
$659$	− 4.16908e10i	− 0.221054i	0.993873	−	0.110527i	$-0.0352539\pi$
$660$	0	0
$661$	−3.51774e11	−1.84271	−0.921357	−	0.388719i	$-0.872918\pi$
$661$	−3.51774e11	−1.84271	−0.921357	+	0.388719i	$0.872918\pi$
$662$	0	0
$663$	2.62934e11i	1.36080i
$664$	0	0
$665$	2.95135e9	0.0150916
$666$	0	0
$667$	1.87816e11i	0.948918i
$668$	0	0
$669$	2.71065e10	0.135322
$670$	0	0
$671$	3.25530e11i	1.60584i
$672$	0	0
$673$	1.26006e11	0.614229	0.307115	−	0.951673i	$-0.400637\pi$
$673$	1.26006e11	0.614229	0.307115	+	0.951673i	$0.400637\pi$
$674$	0	0
$675$	− 3.99117e10i	− 0.192258i
$676$	0	0
$677$	−1.43623e11	−0.683708	−0.341854	−	0.939753i	$-0.611055\pi$
$677$	−1.43623e11	−0.683708	−0.341854	+	0.939753i	$0.611055\pi$
$678$	0	0
$679$	− 4.35872e10i	− 0.205060i
$680$	0	0
$681$	−2.16460e11	−1.00644
$682$	0	0
$683$	− 2.24502e11i	− 1.03166i	−0.856690	−	0.515831i	$-0.827483\pi$
$683$	− 2.24502e11i	− 1.03166i	0.856690	−	0.515831i	$-0.172517\pi$
$684$	0	0
$685$	−7.42409e9	−0.0337195
$686$	0	0
$687$	1.58472e11i	0.711419i
$688$	0	0
$689$	−2.14729e11	−0.952826
$690$	0	0
$691$	1.36709e11i	0.599634i	0.953997	+	0.299817i	$0.0969256\pi$
$691$	1.36709e11i	0.599634i	−0.953997	+	0.299817i	$0.903074\pi$
$692$	0	0
$693$	2.63326e10	0.114172
$694$	0	0
$695$	4.68135e9i	0.0200647i
$696$	0	0
$697$	−1.12778e11	−0.477853
$698$	0	0
$699$	− 1.03008e11i	− 0.431483i
$700$	0	0
$701$	−3.04497e11	−1.26099	−0.630493	−	0.776195i	$-0.717147\pi$
$701$	−3.04497e11	−1.26099	−0.630493	+	0.776195i	$0.717147\pi$
$702$	0	0
$703$	− 5.14533e11i	− 2.10665i
$704$	0	0
$705$	−1.87785e9	−0.00760157
$706$	0	0
$707$	7.33583e10i	0.293611i
$708$	0	0
$709$	1.06434e11	0.421206	0.210603	−	0.977572i	$-0.432457\pi$
$709$	1.06434e11	0.421206	0.210603	+	0.977572i	$0.432457\pi$
$710$	0	0
$711$	− 2.37392e10i	− 0.0928942i
$712$	0	0
$713$	1.55256e11	0.600745
$714$	0	0
$715$	1.49100e10i	0.0570498i
$716$	0	0
$717$	−1.99311e11	−0.754145
$718$	0	0
$719$	− 6.97035e10i	− 0.260819i	−0.991460	−	0.130410i	$-0.958371\pi$
$719$	− 6.97035e10i	− 0.260819i	0.991460	−	0.130410i	$-0.0416292\pi$
$720$	0	0
$721$	1.07868e11	0.399165
$722$	0	0
$723$	− 2.24531e11i	− 0.821717i
$724$	0	0
$725$	5.10440e11	1.84754
$726$	0	0
$727$	2.36941e11i	0.848207i	0.905614	+	0.424104i	$0.139411\pi$
$727$	2.36941e11i	0.848207i	−0.905614	+	0.424104i	$0.860589\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 5.64025e11i	− 1.97528i
$732$	0	0
$733$	−3.77173e11	−1.30655	−0.653274	−	0.757122i	$-0.726605\pi$
$733$	−3.77173e11	−1.30655	−0.653274	+	0.757122i	$0.726605\pi$
$734$	0	0
$735$	− 4.78843e9i	− 0.0164075i
$736$	0	0
$737$	4.39891e11	1.49099
$738$	0	0
$739$	2.53030e11i	0.848387i	0.905572	+	0.424193i	$0.139442\pi$
$739$	2.53030e11i	0.848387i	−0.905572	+	0.424193i	$0.860558\pi$
$740$	0	0
$741$	−4.38692e11	−1.45508
$742$	0	0
$743$	1.74065e10i	0.0571159i	0.999592	+	0.0285580i	$0.00909152\pi$
$743$	1.74065e10i	0.0571159i	−0.999592	+	0.0285580i	$0.990908\pi$
$744$	0	0
$745$	−1.72349e10	−0.0559480
$746$	0	0
$747$	− 1.74782e11i	− 0.561325i
$748$	0	0
$749$	4.85675e10	0.154319
$750$	0	0
$751$	− 4.54672e11i	− 1.42935i	−0.699457	−	0.714675i	$-0.746575\pi$
$751$	− 4.54672e11i	− 1.42935i	0.699457	−	0.714675i	$-0.253425\pi$
$752$	0	0
$753$	5.39529e10	0.167817
$754$	0	0
$755$	2.49110e7i	0 7.66661e-5i
$756$	0	0
$757$	5.54282e11	1.68790	0.843952	−	0.536419i	$-0.180223\pi$
$757$	5.54282e11	1.68790	0.843952	+	0.536419i	$0.180223\pi$
$758$	0	0
$759$	− 1.06411e11i	− 0.320641i
$760$	0	0
$761$	8.46497e10	0.252399	0.126199	−	0.992005i	$-0.459722\pi$
$761$	8.46497e10	0.252399	0.126199	+	0.992005i	$0.459722\pi$
$762$	0	0
$763$	− 9.97364e10i	− 0.294277i
$764$	0	0
$765$	5.09167e9	0.0148667
$766$	0	0
$767$	− 6.68227e11i	− 1.93082i
$768$	0	0
$769$	1.61130e11	0.460756	0.230378	−	0.973101i	$-0.426004\pi$
$769$	1.61130e11	0.460756	0.230378	+	0.973101i	$0.426004\pi$
$770$	0	0
$771$	− 2.90963e10i	− 0.0823419i
$772$	0	0
$773$	−5.45092e11	−1.52669	−0.763346	−	0.645990i	$-0.776445\pi$
$773$	−5.45092e11	−1.52669	−0.763346	+	0.645990i	$0.776445\pi$
$774$	0	0
$775$	− 4.21950e11i	− 1.16965i
$776$	0	0
$777$	9.29008e10	0.254880
$778$	0	0
$779$	− 1.88164e11i	− 0.510961i
$780$	0	0
$781$	5.11808e11	1.37563
$782$	0	0
$783$	− 1.33780e11i	− 0.355914i
$784$	0	0
$785$	−1.14429e10	−0.0301340
$786$	0	0
$787$	− 4.18854e11i	− 1.09185i	−0.837833	−	0.545926i	$-0.816178\pi$
$787$	− 4.18854e11i	− 1.09185i	0.837833	−	0.545926i	$-0.183822\pi$
$788$	0	0
$789$	−1.31948e11	−0.340482
$790$	0	0
$791$	1.13879e11i	0.290895i
$792$	0	0
$793$	9.79183e11	2.47612
$794$	0	0
$795$	4.15818e9i	0.0104096i
$796$	0	0
$797$	−5.31554e11	−1.31739	−0.658694	−	0.752411i	$-0.728891\pi$
$797$	−5.31554e11	−1.31739	−0.658694	+	0.752411i	$0.728891\pi$
$798$	0	0
$799$	2.39954e11i	0.588764i
$800$	0	0
$801$	−1.63172e11	−0.396382
$802$	0	0
$803$	3.72223e11i	0.895244i
$804$	0	0
$805$	2.15338e9	0.00512787
$806$	0	0
$807$	− 1.09360e11i	− 0.257848i
$808$	0	0
$809$	−3.98989e11	−0.931466	−0.465733	−	0.884925i	$-0.654209\pi$
$809$	−3.98989e11	−0.931466	−0.465733	+	0.884925i	$0.654209\pi$
$810$	0	0
$811$	1.83588e10i	0.0424385i	0.999775	+	0.0212192i	$0.00675480\pi$
$811$	1.83588e10i	0.0424385i	−0.999775	+	0.0212192i	$0.993245\pi$
$812$	0	0
$813$	−9.01668e10	−0.206388
$814$	0	0
$815$	− 2.70017e10i	− 0.0612012i
$816$	0	0
$817$	9.41046e11	2.11214
$818$	0	0
$819$	− 7.92075e10i	− 0.176048i
$820$	0	0
$821$	−1.50902e11	−0.332141	−0.166070	−	0.986114i	$-0.553108\pi$
$821$	−1.50902e11	−0.332141	−0.166070	+	0.986114i	$0.553108\pi$
$822$	0	0
$823$	− 5.25782e11i	− 1.14606i	−0.819535	−	0.573029i	$-0.805768\pi$
$823$	− 5.25782e11i	− 1.14606i	0.819535	−	0.573029i	$-0.194232\pi$
$824$	0	0
$825$	−2.89200e11	−0.624285
$826$	0	0
$827$	1.65875e11i	0.354617i	0.984155	+	0.177309i	$0.0567390\pi$
$827$	1.65875e11i	0.354617i	−0.984155	+	0.177309i	$0.943261\pi$
$828$	0	0
$829$	1.36343e11	0.288678	0.144339	−	0.989528i	$-0.453894\pi$
$829$	1.36343e11	0.288678	0.144339	+	0.989528i	$0.453894\pi$
$830$	0	0
$831$	1.47642e11i	0.309603i
$832$	0	0
$833$	−6.11873e11	−1.27081
$834$	0	0
$835$	− 1.83891e10i	− 0.0378281i
$836$	0	0
$837$	−1.10588e11	−0.225323
$838$	0	0
$839$	6.41586e11i	1.29481i	0.762145	+	0.647406i	$0.224146\pi$
$839$	6.41586e11i	1.29481i	−0.762145	+	0.647406i	$0.775854\pi$
$840$	0	0
$841$	1.21070e12	2.42021
$842$	0	0
$843$	− 1.20227e11i	− 0.238063i
$844$	0	0
$845$	2.87477e10	0.0563867
$846$	0	0
$847$	− 2.79373e10i	− 0.0542814i
$848$	0	0
$849$	7.46142e10	0.143612
$850$	0	0
$851$	− 3.75415e11i	− 0.715804i
$852$	0	0
$853$	7.03535e9	0.0132889	0.00664446	−	0.999978i	$-0.497885\pi$
$853$	7.03535e9	0.0132889	0.00664446	+	0.999978i	$0.497885\pi$
$854$	0	0
$855$	8.49519e9i	0.0158968i
$856$	0	0
$857$	4.30808e11	0.798657	0.399328	−	0.916808i	$-0.369243\pi$
$857$	4.30808e11	0.798657	0.399328	+	0.916808i	$0.369243\pi$
$858$	0	0
$859$	7.05962e11i	1.29661i	0.761382	+	0.648304i	$0.224521\pi$
$859$	7.05962e11i	1.29661i	−0.761382	+	0.648304i	$0.775479\pi$
$860$	0	0
$861$	3.39738e10	0.0618203
$862$	0	0
$863$	− 3.49784e11i	− 0.630604i	−0.948991	−	0.315302i	$-0.897894\pi$
$863$	− 3.49784e11i	− 0.630604i	0.948991	−	0.315302i	$-0.102106\pi$
$864$	0	0
$865$	1.20025e10	0.0214391
$866$	0	0
$867$	− 3.24398e11i	− 0.574119i
$868$	0	0
$869$	−1.72015e11	−0.301639
$870$	0	0
$871$	− 1.32318e12i	− 2.29904i
$872$	0	0
$873$	1.25462e11	0.216000
$874$	0	0
$875$	− 1.17106e10i	− 0.0199778i
$876$	0	0
$877$	−5.02015e11	−0.848631	−0.424315	−	0.905514i	$-0.639485\pi$
$877$	−5.02015e11	−0.848631	−0.424315	+	0.905514i	$0.639485\pi$
$878$	0	0
$879$	1.92680e11i	0.322761i
$880$	0	0
$881$	−1.95986e11	−0.325328	−0.162664	−	0.986682i	$-0.552009\pi$
$881$	−1.95986e11	−0.325328	−0.162664	+	0.986682i	$0.552009\pi$
$882$	0	0
$883$	6.69240e11i	1.10088i	0.834876	+	0.550439i	$0.185540\pi$
$883$	6.69240e11i	1.10088i	−0.834876	+	0.550439i	$0.814460\pi$
$884$	0	0
$885$	−1.29401e10	−0.0210943
$886$	0	0
$887$	5.04570e11i	0.815130i	0.913176	+	0.407565i	$0.133622\pi$
$887$	5.04570e11i	0.815130i	−0.913176	+	0.407565i	$0.866378\pi$
$888$	0	0
$889$	1.51345e11	0.242304
$890$	0	0
$891$	7.57959e10i	0.120264i
$892$	0	0
$893$	−4.00351e11	−0.629557
$894$	0	0
$895$	− 2.91874e9i	− 0.00454887i
$896$	0	0
$897$	−3.20080e11	−0.494412
$898$	0	0
$899$	− 1.41434e12i	− 2.16528i
$900$	0	0
$901$	5.31340e11	0.806256
$902$	0	0
$903$	1.69909e11i	0.255544i
$904$	0	0
$905$	−1.36156e10	−0.0202974
$906$	0	0
$907$	1.07760e11i	0.159232i	0.996826	+	0.0796158i	$0.0253693\pi$
$907$	1.07760e11i	0.159232i	−0.996826	+	0.0796158i	$0.974631\pi$
$908$	0	0
$909$	−2.11155e11	−0.309276
$910$	0	0
$911$	− 3.44029e11i	− 0.499485i	−0.968312	−	0.249742i	$-0.919654\pi$
$911$	− 3.44029e11i	− 0.499485i	0.968312	−	0.249742i	$-0.0803459\pi$
$912$	0	0
$913$	−1.26647e12	−1.82269
$914$	0	0
$915$	− 1.89617e10i	− 0.0270516i
$916$	0	0
$917$	2.30359e11	0.325782
$918$	0	0
$919$	− 6.25848e11i	− 0.877419i	−0.898629	−	0.438709i	$-0.855436\pi$
$919$	− 6.25848e11i	− 0.877419i	0.898629	−	0.438709i	$-0.144564\pi$
$920$	0	0
$921$	−7.17606e11	−0.997350
$922$	0	0
$923$	− 1.53950e12i	− 2.12116i
$924$	0	0
$925$	−1.02029e12	−1.39366
$926$	0	0
$927$	3.10488e11i	0.420461i
$928$	0	0
$929$	−1.19959e11	−0.161054	−0.0805269	−	0.996752i	$-0.525660\pi$
$929$	−1.19959e11	−0.161054	−0.0805269	+	0.996752i	$0.525660\pi$
$930$	0	0
$931$	− 1.02088e12i	− 1.35886i
$932$	0	0
$933$	3.93629e11	0.519470
$934$	0	0
$935$	− 3.68943e10i	− 0.0482740i
$936$	0	0
$937$	−1.81236e11	−0.235118	−0.117559	−	0.993066i	$-0.537507\pi$
$937$	−1.81236e11	−0.235118	−0.117559	+	0.993066i	$0.537507\pi$
$938$	0	0
$939$	− 2.42343e11i	− 0.311722i
$940$	0	0
$941$	−3.59055e11	−0.457934	−0.228967	−	0.973434i	$-0.573535\pi$
$941$	−3.59055e11	−0.457934	−0.228967	+	0.973434i	$0.573535\pi$
$942$	0	0
$943$	− 1.37289e11i	− 0.173616i
$944$	0	0
$945$	−1.53384e9	−0.00192332
$946$	0	0
$947$	1.42565e12i	1.77260i	0.463109	+	0.886302i	$0.346734\pi$
$947$	1.42565e12i	1.77260i	−0.463109	+	0.886302i	$0.653266\pi$
$948$	0	0
$949$	1.11963e12	1.38042
$950$	0	0
$951$	− 2.01635e11i	− 0.246515i
$952$	0	0
$953$	1.01220e12	1.22715	0.613573	−	0.789638i	$-0.289732\pi$
$953$	1.01220e12	1.22715	0.613573	+	0.789638i	$0.289732\pi$
$954$	0	0
$955$	− 3.60663e10i	− 0.0433598i
$956$	0	0
$957$	−9.69372e11	−1.15569
$958$	0	0
$959$	− 2.85779e11i	− 0.337875i
$960$	0	0
$961$	−3.16257e11	−0.370805
$962$	0	0
$963$	1.39797e11i	0.162552i
$964$	0	0
$965$	9.03094e9	0.0104141
$966$	0	0
$967$	1.29366e12i	1.47950i	0.672881	+	0.739750i	$0.265056\pi$
$967$	1.29366e12i	1.47950i	−0.672881	+	0.739750i	$0.734944\pi$
$968$	0	0
$969$	1.08553e12	1.23125
$970$	0	0
$971$	3.23711e11i	0.364150i	0.983285	+	0.182075i	$0.0582813\pi$
$971$	3.23711e11i	0.364150i	−0.983285	+	0.182075i	$0.941719\pi$
$972$	0	0
$973$	−1.80201e11	−0.201051
$974$	0	0
$975$	8.69905e11i	0.962617i
$976$	0	0
$977$	3.69052e11	0.405050	0.202525	−	0.979277i	$-0.435085\pi$
$977$	3.69052e11	0.405050	0.202525	+	0.979277i	$0.435085\pi$
$978$	0	0
$979$	1.18234e12i	1.28710i
$980$	0	0
$981$	2.87082e11	0.309977
$982$	0	0
$983$	5.92111e11i	0.634146i	0.948401	+	0.317073i	$0.102700\pi$
$983$	5.92111e11i	0.634146i	−0.948401	+	0.317073i	$0.897300\pi$
$984$	0	0
$985$	−8.34189e9	−0.00886175
$986$	0	0
$987$	− 7.22849e10i	− 0.0761691i
$988$	0	0
$989$	6.86610e11	0.717670
$990$	0	0
$991$	2.22630e11i	0.230828i	0.993317	+	0.115414i	$0.0368195\pi$
$991$	2.22630e11i	0.230828i	−0.993317	+	0.115414i	$0.963181\pi$
$992$	0	0
$993$	−3.66820e11	−0.377273
$994$	0	0
$995$	9.21463e9i	0.00940125i
$996$	0	0
$997$	2.32972e11	0.235788	0.117894	−	0.993026i	$-0.462386\pi$
$997$	2.32972e11	0.235788	0.117894	+	0.993026i	$0.462386\pi$
$998$	0	0
$999$	2.67406e11i	0.268479i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.b.127.17	yes	32
4.3	odd	2	inner	384.9.g.b.127.18	yes	32
8.3	odd	2		384.9.g.a.127.15	✓	32
8.5	even	2		384.9.g.a.127.16	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.15	✓	32	8.3	odd	2
384.9.g.a.127.16	yes	32	8.5	even	2
384.9.g.b.127.17	yes	32	1.1	even	1	trivial
384.9.g.b.127.18	yes	32	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} +19.7383 q^{5} +759.796i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q-46.7654i q^{3} +19.7383 q^{5} +759.796i q^{7} -2187.00 q^{9} +15847.0i q^{11} +47667.3 q^{13} -923.068i q^{15} -117951. q^{17} -196795. i q^{19} +35532.1 q^{21} -143587. i q^{23} -390235. q^{25} +102276. i q^{27} -1.30803e6 q^{29} +1.08127e6i q^{31} +741092. q^{33} +14997.1i q^{35} +2.61456e6 q^{37} -2.22918e6i q^{39} +956143. q^{41} +4.78185e6i q^{43} -43167.6 q^{45} -2.03435e6i q^{47} +5.18751e6 q^{49} +5.51603e6i q^{51} -4.50474e6 q^{53} +312793. i q^{55} -9.20320e6 q^{57} -1.40186e7i q^{59} +2.05420e7 q^{61} -1.66167e6i q^{63} +940872. q^{65} -2.77586e7i q^{67} -6.71488e6 q^{69} -3.22968e7i q^{71} +2.34885e7 q^{73} +1.82495e7i q^{75} -1.20405e7 q^{77} +1.08547e7i q^{79} +4.78297e6 q^{81} +7.99186e7i q^{83} -2.32815e6 q^{85} +6.11706e7i q^{87} +7.46098e7 q^{89} +3.62174e7i q^{91} +5.05661e7 q^{93} -3.88440e6i q^{95} -5.73670e7 q^{97} -3.46575e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 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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