LMFDB - Embedded newform 384.8.d.d.193.5

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 449x^{6} + 50632x^{4} + 69129x^{2} + 18225 $ x^8 + 449x^6 + 50632x^4 + 69129*x^2 + 18225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.5
Root			$14.6646i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.d.193.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000i q^{3} -344.306i q^{5} +1669.75 q^{7} -729.000 q^{9} +O(q^{10})$	$q+27.0000i q^{3} -344.306i q^{5} +1669.75 q^{7} -729.000 q^{9} +2282.14i q^{11} +11526.9i q^{13} +9296.26 q^{15} -17243.9 q^{17} -14550.2i q^{19} +45083.1i q^{21} -40886.4 q^{23} -40421.6 q^{25} -19683.0i q^{27} +159495. i q^{29} -171754. q^{31} -61617.9 q^{33} -574903. i q^{35} +9029.61i q^{37} -311227. q^{39} -735509. q^{41} +280634. i q^{43} +250999. i q^{45} -529498. q^{47} +1.96451e6 q^{49} -465585. i q^{51} +540438. i q^{53} +785756. q^{55} +392855. q^{57} +727412. i q^{59} -2.21789e6i q^{61} -1.21724e6 q^{63} +3.96878e6 q^{65} +2.59799e6i q^{67} -1.10393e6i q^{69} -524418. q^{71} +1.37636e6 q^{73} -1.09138e6i q^{75} +3.81060e6i q^{77} +6.61970e6 q^{79} +531441. q^{81} -6.38593e6i q^{83} +5.93718e6i q^{85} -4.30636e6 q^{87} -9.60915e6 q^{89} +1.92470e7i q^{91} -4.63736e6i q^{93} -5.00971e6 q^{95} -9.87790e6 q^{97} -1.66368e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2880 q^{7} - 5832 q^{9}+O(q^{10}) $ 8 * q + 2880 * q^7 - 5832 * q^9	$ 8 q + 2880 q^{7} - 5832 q^{9} - 6048 q^{15} + 22896 q^{17} + 207360 q^{23} - 204696 q^{25} - 17856 q^{31} + 7776 q^{33} - 116640 q^{39} + 687056 q^{41} - 1987200 q^{47} + 4815560 q^{49} + 2077056 q^{55} + 878688 q^{57} - 2099520 q^{63} + 12871808 q^{65} - 6336000 q^{71} + 8920752 q^{73} + 1251648 q^{79} + 4251528 q^{81} - 13385952 q^{87} + 4447408 q^{89} - 31607424 q^{95} - 14157584 q^{97}+O(q^{100}) $ 8 * q + 2880 * q^7 - 5832 * q^9 - 6048 * q^15 + 22896 * q^17 + 207360 * q^23 - 204696 * q^25 - 17856 * q^31 + 7776 * q^33 - 116640 * q^39 + 687056 * q^41 - 1987200 * q^47 + 4815560 * q^49 + 2077056 * q^55 + 878688 * q^57 - 2099520 * q^63 + 12871808 * q^65 - 6336000 * q^71 + 8920752 * q^73 + 1251648 * q^79 + 4251528 * q^81 - 13385952 * q^87 + 4447408 * q^89 - 31607424 * q^95 - 14157584 * 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
$5$	− 344.306i	− 1.23183i	−0.787814	−	0.615913i	$-0.788787\pi$
$5$	− 344.306i	− 1.23183i	0.787814	−	0.615913i	$-0.211213\pi$
$6$	0	0
$7$	1669.75	1.83995	0.919977	−	0.391972i	$-0.128207\pi$
$7$	1669.75	1.83995	0.919977	+	0.391972i	$0.128207\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	2282.14i	0.516974i	0.966015	+	0.258487i	$0.0832239\pi$
$11$	2282.14i	0.516974i	−0.966015	+	0.258487i	$0.916776\pi$
$12$	0	0
$13$	11526.9i	1.45516i	0.686022	+	0.727581i	$0.259355\pi$
$13$	11526.9i	1.45516i	−0.686022	+	0.727581i	$0.740645\pi$
$14$	0	0
$15$	9296.26	0.711195
$16$	0	0
$17$	−17243.9	−0.851264	−0.425632	−	0.904896i	$-0.639948\pi$
$17$	−17243.9	−0.851264	−0.425632	+	0.904896i	$0.639948\pi$
$18$	0	0
$19$	− 14550.2i	− 0.486666i	−0.969943	−	0.243333i	$-0.921759\pi$
$19$	− 14550.2i	− 0.486666i	0.969943	−	0.243333i	$-0.0782407\pi$
$20$	0	0
$21$	45083.1i	1.06230i
$22$	0	0
$23$	−40886.4	−0.700698	−0.350349	−	0.936619i	$-0.613937\pi$
$23$	−40886.4	−0.700698	−0.350349	+	0.936619i	$0.613937\pi$
$24$	0	0
$25$	−40421.6	−0.517397
$26$	0	0
$27$	− 19683.0i	− 0.192450i
$28$	0	0
$29$	159495.i	1.21438i	0.794558	+	0.607188i	$0.207703\pi$
$29$	159495.i	1.21438i	−0.794558	+	0.607188i	$0.792297\pi$
$30$	0	0
$31$	−171754.	−1.03548	−0.517739	−	0.855539i	$-0.673226\pi$
$31$	−171754.	−1.03548	−0.517739	+	0.855539i	$0.673226\pi$
$32$	0	0
$33$	−61617.9	−0.298475
$34$	0	0
$35$	− 574903.i	− 2.26651i
$36$	0	0
$37$	9029.61i	0.0293064i	0.999893	+	0.0146532i	$0.00466443\pi$
$37$	9029.61i	0.0293064i	−0.999893	+	0.0146532i	$0.995336\pi$
$38$	0	0
$39$	−311227.	−0.840138
$40$	0	0
$41$	−735509.	−1.66665	−0.833325	−	0.552784i	$-0.813565\pi$
$41$	−735509.	−1.66665	−0.833325	+	0.552784i	$0.813565\pi$
$42$	0	0
$43$	280634.i	0.538271i	0.963102	+	0.269135i	$0.0867379\pi$
$43$	280634.i	0.538271i	−0.963102	+	0.269135i	$0.913262\pi$
$44$	0	0
$45$	250999.i	0.410609i
$46$	0	0
$47$	−529498.	−0.743912	−0.371956	−	0.928250i	$-0.621313\pi$
$47$	−529498.	−0.743912	−0.371956	+	0.928250i	$0.621313\pi$
$48$	0	0
$49$	1.96451e6	2.38543
$50$	0	0
$51$	− 465585.i	− 0.491477i
$52$	0	0
$53$	540438.i	0.498632i	0.968422	+	0.249316i	$0.0802058\pi$
$53$	540438.i	0.498632i	−0.968422	+	0.249316i	$0.919794\pi$
$54$	0	0
$55$	785756.	0.636822
$56$	0	0
$57$	392855.	0.280977
$58$	0	0
$59$	727412.i	0.461104i	0.973060	+	0.230552i	$0.0740531\pi$
$59$	727412.i	0.461104i	−0.973060	+	0.230552i	$0.925947\pi$
$60$	0	0
$61$	− 2.21789e6i	− 1.25108i	−0.780193	−	0.625539i	$-0.784879\pi$
$61$	− 2.21789e6i	− 1.25108i	0.780193	−	0.625539i	$-0.215121\pi$
$62$	0	0
$63$	−1.21724e6	−0.613318
$64$	0	0
$65$	3.96878e6	1.79251
$66$	0	0
$67$	2.59799e6i	1.05530i	0.849462	+	0.527649i	$0.176926\pi$
$67$	2.59799e6i	1.05530i	−0.849462	+	0.527649i	$0.823074\pi$
$68$	0	0
$69$	− 1.10393e6i	− 0.404548i
$70$	0	0
$71$	−524418.	−0.173889	−0.0869447	−	0.996213i	$-0.527710\pi$
$71$	−524418.	−0.173889	−0.0869447	+	0.996213i	$0.527710\pi$
$72$	0	0
$73$	1.37636e6	0.414097	0.207049	−	0.978331i	$-0.433614\pi$
$73$	1.37636e6	0.414097	0.207049	+	0.978331i	$0.433614\pi$
$74$	0	0
$75$	− 1.09138e6i	− 0.298719i
$76$	0	0
$77$	3.81060e6i	0.951208i
$78$	0	0
$79$	6.61970e6	1.51058	0.755290	−	0.655391i	$-0.227496\pi$
$79$	6.61970e6	1.51058	0.755290	+	0.655391i	$0.227496\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	− 6.38593e6i	− 1.22589i	−0.790126	−	0.612944i	$-0.789985\pi$
$83$	− 6.38593e6i	− 1.22589i	0.790126	−	0.612944i	$-0.210015\pi$
$84$	0	0
$85$	5.93718e6i	1.04861i
$86$	0	0
$87$	−4.30636e6	−0.701120
$88$	0	0
$89$	−9.60915e6	−1.44484	−0.722420	−	0.691454i	$-0.756970\pi$
$89$	−9.60915e6	−1.44484	−0.722420	+	0.691454i	$0.756970\pi$
$90$	0	0
$91$	1.92470e7i	2.67743i
$92$	0	0
$93$	− 4.63736e6i	− 0.597834i
$94$	0	0
$95$	−5.00971e6	−0.599488
$96$	0	0
$97$	−9.87790e6	−1.09891	−0.549457	−	0.835522i	$-0.685165\pi$
$97$	−9.87790e6	−1.09891	−0.549457	+	0.835522i	$0.685165\pi$
$98$	0	0
$99$	− 1.66368e6i	− 0.172325i
$100$	0	0
$101$	1.59806e7i	1.54336i	0.636008	+	0.771682i	$0.280585\pi$
$101$	1.59806e7i	1.54336i	−0.636008	+	0.771682i	$0.719415\pi$
$102$	0	0
$103$	1.40912e7	1.27062	0.635312	−	0.772255i	$-0.280871\pi$
$103$	1.40912e7	1.27062	0.635312	+	0.772255i	$0.280871\pi$
$104$	0	0
$105$	1.55224e7	1.30857
$106$	0	0
$107$	2.31609e7i	1.82773i	0.406018	+	0.913865i	$0.366917\pi$
$107$	2.31609e7i	1.82773i	−0.406018	+	0.913865i	$0.633083\pi$
$108$	0	0
$109$	2.10486e6i	0.155679i	0.996966	+	0.0778395i	$0.0248022\pi$
$109$	2.10486e6i	0.155679i	−0.996966	+	0.0778395i	$0.975198\pi$
$110$	0	0
$111$	−243799.	−0.0169201
$112$	0	0
$113$	−2.30926e7	−1.50556	−0.752780	−	0.658272i	$-0.771288\pi$
$113$	−2.30926e7	−1.50556	−0.752780	+	0.658272i	$0.771288\pi$
$114$	0	0
$115$	1.40774e7i	0.863139i
$116$	0	0
$117$	− 8.40312e6i	− 0.485054i
$118$	0	0
$119$	−2.87929e7	−1.56629
$120$	0	0
$121$	1.42790e7	0.732738
$122$	0	0
$123$	− 1.98587e7i	− 0.962240i
$124$	0	0
$125$	− 1.29815e7i	− 0.594483i
$126$	0	0
$127$	−4.56266e7	−1.97654	−0.988269	−	0.152724i	$-0.951195\pi$
$127$	−4.56266e7	−1.97654	−0.988269	+	0.152724i	$0.951195\pi$
$128$	0	0
$129$	−7.57712e6	−0.310771
$130$	0	0
$131$	− 1.78031e7i	− 0.691904i	−0.938252	−	0.345952i	$-0.887556\pi$
$131$	− 1.78031e7i	− 0.691904i	0.938252	−	0.345952i	$-0.112444\pi$
$132$	0	0
$133$	− 2.42951e7i	− 0.895443i
$134$	0	0
$135$	−6.77698e6	−0.237065
$136$	0	0
$137$	3.05756e6	0.101591	0.0507953	−	0.998709i	$-0.483824\pi$
$137$	3.05756e6	0.101591	0.0507953	+	0.998709i	$0.483824\pi$
$138$	0	0
$139$	− 5.22305e7i	− 1.64958i	−0.565441	−	0.824789i	$-0.691294\pi$
$139$	− 5.22305e7i	− 1.64958i	0.565441	−	0.824789i	$-0.308706\pi$
$140$	0	0
$141$	− 1.42964e7i	− 0.429498i
$142$	0	0
$143$	−2.63061e7	−0.752280
$144$	0	0
$145$	5.49150e7	1.49590
$146$	0	0
$147$	5.30417e7i	1.37723i
$148$	0	0
$149$	2.81604e6i	0.0697407i	0.999392	+	0.0348703i	$0.0111018\pi$
$149$	2.81604e6i	0.0697407i	−0.999392	+	0.0348703i	$0.988898\pi$
$150$	0	0
$151$	−5.76673e7	−1.36305	−0.681523	−	0.731797i	$-0.738682\pi$
$151$	−5.76673e7	−1.36305	−0.681523	+	0.731797i	$0.738682\pi$
$152$	0	0
$153$	1.25708e7	0.283755
$154$	0	0
$155$	5.91360e7i	1.27553i
$156$	0	0
$157$	− 9.17674e7i	− 1.89252i	−0.323412	−	0.946258i	$-0.604830\pi$
$157$	− 9.17674e7i	− 1.89252i	0.323412	−	0.946258i	$-0.395170\pi$
$158$	0	0
$159$	−1.45918e7	−0.287885
$160$	0	0
$161$	−6.82698e7	−1.28925
$162$	0	0
$163$	9.20744e7i	1.66526i	0.553829	+	0.832630i	$0.313166\pi$
$163$	9.20744e7i	1.66526i	−0.553829	+	0.832630i	$0.686834\pi$
$164$	0	0
$165$	2.12154e7i	0.367670i
$166$	0	0
$167$	1.62432e7	0.269875	0.134938	−	0.990854i	$-0.456917\pi$
$167$	1.62432e7	0.269875	0.134938	+	0.990854i	$0.456917\pi$
$168$	0	0
$169$	−7.01211e7	−1.11749
$170$	0	0
$171$	1.06071e7i	0.162222i
$172$	0	0
$173$	4.74152e7i	0.696236i	0.937451	+	0.348118i	$0.113179\pi$
$173$	4.74152e7i	0.696236i	−0.937451	+	0.348118i	$0.886821\pi$
$174$	0	0
$175$	−6.74939e7	−0.951987
$176$	0	0
$177$	−1.96401e7	−0.266218
$178$	0	0
$179$	8.00081e7i	1.04267i	0.853351	+	0.521337i	$0.174567\pi$
$179$	8.00081e7i	1.04267i	−0.853351	+	0.521337i	$0.825433\pi$
$180$	0	0
$181$	1.96745e7i	0.246620i	0.992368	+	0.123310i	$0.0393509\pi$
$181$	1.96745e7i	0.246620i	−0.992368	+	0.123310i	$0.960649\pi$
$182$	0	0
$183$	5.98829e7	0.722311
$184$	0	0
$185$	3.10895e6	0.0361004
$186$	0	0
$187$	− 3.93530e7i	− 0.440081i
$188$	0	0
$189$	− 3.28656e7i	− 0.354099i
$190$	0	0
$191$	−8.78405e7	−0.912174	−0.456087	−	0.889935i	$-0.650749\pi$
$191$	−8.78405e7	−0.912174	−0.456087	+	0.889935i	$0.650749\pi$
$192$	0	0
$193$	1.29628e8	1.29792	0.648959	−	0.760824i	$-0.275205\pi$
$193$	1.29628e8	1.29792	0.648959	+	0.760824i	$0.275205\pi$
$194$	0	0
$195$	1.07157e8i	1.03490i
$196$	0	0
$197$	1.21765e8i	1.13472i	0.823469	+	0.567362i	$0.192036\pi$
$197$	1.21765e8i	1.13472i	−0.823469	+	0.567362i	$0.807964\pi$
$198$	0	0
$199$	−1.43807e8	−1.29359	−0.646793	−	0.762665i	$-0.723890\pi$
$199$	−1.43807e8	−1.29359	−0.646793	+	0.762665i	$0.723890\pi$
$200$	0	0
$201$	−7.01457e7	−0.609277
$202$	0	0
$203$	2.66315e8i	2.23440i
$204$	0	0
$205$	2.53240e8i	2.05302i
$206$	0	0
$207$	2.98062e7	0.233566
$208$	0	0
$209$	3.32056e7	0.251593
$210$	0	0
$211$	1.17120e8i	0.858303i	0.903233	+	0.429152i	$0.141187\pi$
$211$	1.17120e8i	0.858303i	−0.903233	+	0.429152i	$0.858813\pi$
$212$	0	0
$213$	− 1.41593e7i	− 0.100395i
$214$	0	0
$215$	9.66240e7	0.663056
$216$	0	0
$217$	−2.86785e8	−1.90523
$218$	0	0
$219$	3.71617e7i	0.239079i
$220$	0	0
$221$	− 1.98769e8i	− 1.23873i
$222$	0	0
$223$	−3.15918e8	−1.90769	−0.953844	−	0.300301i	$-0.902913\pi$
$223$	−3.15918e8	−1.90769	−0.953844	+	0.300301i	$0.902913\pi$
$224$	0	0
$225$	2.94674e7	0.172466
$226$	0	0
$227$	− 6.46937e7i	− 0.367089i	−0.983011	−	0.183545i	$-0.941243\pi$
$227$	− 6.46937e7i	− 0.367089i	0.983011	−	0.183545i	$-0.0587572\pi$
$228$	0	0
$229$	2.06223e8i	1.13478i	0.823448	+	0.567392i	$0.192048\pi$
$229$	2.06223e8i	1.13478i	−0.823448	+	0.567392i	$0.807952\pi$
$230$	0	0
$231$	−1.02886e8	−0.549180
$232$	0	0
$233$	3.20225e8	1.65848	0.829238	−	0.558895i	$-0.188775\pi$
$233$	3.20225e8	1.65848	0.829238	+	0.558895i	$0.188775\pi$
$234$	0	0
$235$	1.82309e8i	0.916371i
$236$	0	0
$237$	1.78732e8i	0.872134i
$238$	0	0
$239$	−1.24636e8	−0.590540	−0.295270	−	0.955414i	$-0.595410\pi$
$239$	−1.24636e8	−0.590540	−0.295270	+	0.955414i	$0.595410\pi$
$240$	0	0
$241$	2.47831e8	1.14050	0.570251	−	0.821470i	$-0.306846\pi$
$241$	2.47831e8	1.14050	0.570251	+	0.821470i	$0.306846\pi$
$242$	0	0
$243$	1.43489e7i	0.0641500i
$244$	0	0
$245$	− 6.76391e8i	− 2.93844i
$246$	0	0
$247$	1.67719e8	0.708177
$248$	0	0
$249$	1.72420e8	0.707767
$250$	0	0
$251$	1.73374e8i	0.692031i	0.938229	+	0.346016i	$0.112466\pi$
$251$	1.73374e8i	0.692031i	−0.938229	+	0.346016i	$0.887534\pi$
$252$	0	0
$253$	− 9.33086e7i	− 0.362243i
$254$	0	0
$255$	−1.60304e8	−0.605415
$256$	0	0
$257$	9.68797e7	0.356014	0.178007	−	0.984029i	$-0.443035\pi$
$257$	9.68797e7	0.356014	0.178007	+	0.984029i	$0.443035\pi$
$258$	0	0
$259$	1.50771e7i	0.0539225i
$260$	0	0
$261$	− 1.16272e8i	− 0.404792i
$262$	0	0
$263$	−3.23068e8	−1.09509	−0.547544	−	0.836777i	$-0.684437\pi$
$263$	−3.23068e8	−1.09509	−0.547544	+	0.836777i	$0.684437\pi$
$264$	0	0
$265$	1.86076e8	0.614229
$266$	0	0
$267$	− 2.59447e8i	− 0.834179i
$268$	0	0
$269$	1.84244e8i	0.577112i	0.957463	+	0.288556i	$0.0931752\pi$
$269$	1.84244e8i	0.577112i	−0.957463	+	0.288556i	$0.906825\pi$
$270$	0	0
$271$	1.62257e8	0.495235	0.247618	−	0.968858i	$-0.420352\pi$
$271$	1.62257e8	0.495235	0.247618	+	0.968858i	$0.420352\pi$
$272$	0	0
$273$	−5.19669e8	−1.54582
$274$	0	0
$275$	− 9.22480e7i	− 0.267481i
$276$	0	0
$277$	9.60497e7i	0.271529i	0.990741	+	0.135765i	$0.0433491\pi$
$277$	9.60497e7i	0.271529i	−0.990741	+	0.135765i	$0.956651\pi$
$278$	0	0
$279$	1.25209e8	0.345159
$280$	0	0
$281$	3.61577e8	0.972139	0.486070	−	0.873920i	$-0.338430\pi$
$281$	3.61577e8	0.972139	0.486070	+	0.873920i	$0.338430\pi$
$282$	0	0
$283$	3.43417e8i	0.900677i	0.892858	+	0.450338i	$0.148697\pi$
$283$	3.43417e8i	0.900677i	−0.892858	+	0.450338i	$0.851303\pi$
$284$	0	0
$285$	− 1.35262e8i	− 0.346114i
$286$	0	0
$287$	−1.22811e9	−3.06656
$288$	0	0
$289$	−1.12987e8	−0.275350
$290$	0	0
$291$	− 2.66703e8i	− 0.634458i
$292$	0	0
$293$	2.82677e8i	0.656528i	0.944586	+	0.328264i	$0.106464\pi$
$293$	2.82677e8i	0.656528i	−0.944586	+	0.328264i	$0.893536\pi$
$294$	0	0
$295$	2.50452e8	0.568000
$296$	0	0
$297$	4.49194e7	0.0994917
$298$	0	0
$299$	− 4.71293e8i	− 1.01963i
$300$	0	0
$301$	4.68587e8i	0.990394i
$302$	0	0
$303$	−4.31477e8	−0.891062
$304$	0	0
$305$	−7.63631e8	−1.54111
$306$	0	0
$307$	− 1.57819e8i	− 0.311296i	−0.987813	−	0.155648i	$-0.950253\pi$
$307$	− 1.57819e8i	− 0.311296i	0.987813	−	0.155648i	$-0.0497466\pi$
$308$	0	0
$309$	3.80462e8i	0.733595i
$310$	0	0
$311$	5.79231e8	1.09192	0.545960	−	0.837811i	$-0.316165\pi$
$311$	5.79231e8	1.09192	0.545960	+	0.837811i	$0.316165\pi$
$312$	0	0
$313$	−8.81032e7	−0.162400	−0.0812001	−	0.996698i	$-0.525875\pi$
$313$	−8.81032e7	−0.162400	−0.0812001	+	0.996698i	$0.525875\pi$
$314$	0	0
$315$	4.19105e8i	0.755502i
$316$	0	0
$317$	− 1.19161e8i	− 0.210100i	−0.994467	−	0.105050i	$-0.966500\pi$
$317$	− 1.19161e8i	− 0.210100i	0.994467	−	0.105050i	$-0.0335002\pi$
$318$	0	0
$319$	−3.63990e8	−0.627801
$320$	0	0
$321$	−6.25344e8	−1.05524
$322$	0	0
$323$	2.50902e8i	0.414281i
$324$	0	0
$325$	− 4.65937e8i	− 0.752896i
$326$	0	0
$327$	−5.68312e7	−0.0898813
$328$	0	0
$329$	−8.84126e8	−1.36876
$330$	0	0
$331$	9.04996e8i	1.37167i	0.727758	+	0.685834i	$0.240562\pi$
$331$	9.04996e8i	1.37167i	−0.727758	+	0.685834i	$0.759438\pi$
$332$	0	0
$333$	− 6.58258e6i	− 0.00976881i
$334$	0	0
$335$	8.94503e8	1.29994
$336$	0	0
$337$	3.49042e8	0.496790	0.248395	−	0.968659i	$-0.420097\pi$
$337$	3.49042e8	0.496790	0.248395	+	0.968659i	$0.420097\pi$
$338$	0	0
$339$	− 6.23500e8i	− 0.869236i
$340$	0	0
$341$	− 3.91967e8i	− 0.535315i
$342$	0	0
$343$	1.90512e9	2.54913
$344$	0	0
$345$	−3.80090e8	−0.498334
$346$	0	0
$347$	− 6.41085e8i	− 0.823688i	−0.911254	−	0.411844i	$-0.864885\pi$
$347$	− 6.41085e8i	− 0.823688i	0.911254	−	0.411844i	$-0.135115\pi$
$348$	0	0
$349$	4.06717e8i	0.512158i	0.966656	+	0.256079i	$0.0824307\pi$
$349$	4.06717e8i	0.512158i	−0.966656	+	0.256079i	$0.917569\pi$
$350$	0	0
$351$	2.26884e8	0.280046
$352$	0	0
$353$	9.01273e8	1.09055	0.545274	−	0.838258i	$-0.316426\pi$
$353$	9.01273e8	1.09055	0.545274	+	0.838258i	$0.316426\pi$
$354$	0	0
$355$	1.80560e8i	0.214202i
$356$	0	0
$357$	− 7.77409e8i	− 0.904296i
$358$	0	0
$359$	1.69288e9	1.93106	0.965532	−	0.260285i	$-0.0838164\pi$
$359$	1.69288e9	1.93106	0.965532	+	0.260285i	$0.0838164\pi$
$360$	0	0
$361$	6.82164e8	0.763156
$362$	0	0
$363$	3.85533e8i	0.423046i
$364$	0	0
$365$	− 4.73889e8i	− 0.510096i
$366$	0	0
$367$	−4.61417e8	−0.487262	−0.243631	−	0.969868i	$-0.578339\pi$
$367$	−4.61417e8	−0.487262	−0.243631	+	0.969868i	$0.578339\pi$
$368$	0	0
$369$	5.36186e8	0.555550
$370$	0	0
$371$	9.02394e8i	0.917461i
$372$	0	0
$373$	6.51827e8i	0.650356i	0.945653	+	0.325178i	$0.105424\pi$
$373$	6.51827e8i	0.650356i	−0.945653	+	0.325178i	$0.894576\pi$
$374$	0	0
$375$	3.50500e8	0.343225
$376$	0	0
$377$	−1.83848e9	−1.76711
$378$	0	0
$379$	− 1.31212e9i	− 1.23804i	−0.785374	−	0.619021i	$-0.787529\pi$
$379$	− 1.31212e9i	− 1.23804i	0.785374	−	0.619021i	$-0.212471\pi$
$380$	0	0
$381$	− 1.23192e9i	− 1.14115i
$382$	0	0
$383$	−2.86348e8	−0.260434	−0.130217	−	0.991486i	$-0.541567\pi$
$383$	−2.86348e8	−0.260434	−0.130217	+	0.991486i	$0.541567\pi$
$384$	0	0
$385$	1.31201e9	1.17172
$386$	0	0
$387$	− 2.04582e8i	− 0.179424i
$388$	0	0
$389$	− 5.07035e8i	− 0.436731i	−0.975867	−	0.218366i	$-0.929927\pi$
$389$	− 5.07035e8i	− 0.436731i	0.975867	−	0.218366i	$-0.0700725\pi$
$390$	0	0
$391$	7.05040e8	0.596479
$392$	0	0
$393$	4.80684e8	0.399471
$394$	0	0
$395$	− 2.27920e9i	− 1.86077i
$396$	0	0
$397$	− 7.17590e8i	− 0.575585i	−0.957693	−	0.287793i	$-0.907079\pi$
$397$	− 7.17590e8i	− 0.575585i	0.957693	−	0.287793i	$-0.0929213\pi$
$398$	0	0
$399$	6.55968e8	0.516984
$400$	0	0
$401$	−2.14391e8	−0.166036	−0.0830178	−	0.996548i	$-0.526456\pi$
$401$	−2.14391e8	−0.166036	−0.0830178	+	0.996548i	$0.526456\pi$
$402$	0	0
$403$	− 1.97979e9i	− 1.50679i
$404$	0	0
$405$	− 1.82978e8i	− 0.136870i
$406$	0	0
$407$	−2.06069e7	−0.0151507
$408$	0	0
$409$	6.86290e8	0.495994	0.247997	−	0.968761i	$-0.420228\pi$
$409$	6.86290e8	0.495994	0.247997	+	0.968761i	$0.420228\pi$
$410$	0	0
$411$	8.25542e7i	0.0586534i
$412$	0	0
$413$	1.21459e9i	0.848410i
$414$	0	0
$415$	−2.19871e9	−1.51008
$416$	0	0
$417$	1.41022e9	0.952384
$418$	0	0
$419$	1.08929e9i	0.723428i	0.932289	+	0.361714i	$0.117808\pi$
$419$	1.08929e9i	0.723428i	−0.932289	+	0.361714i	$0.882192\pi$
$420$	0	0
$421$	1.47613e9i	0.964136i	0.876134	+	0.482068i	$0.160114\pi$
$421$	1.47613e9i	0.964136i	−0.876134	+	0.482068i	$0.839886\pi$
$422$	0	0
$423$	3.86004e8	0.247971
$424$	0	0
$425$	6.97027e8	0.440441
$426$	0	0
$427$	− 3.70330e9i	− 2.30193i
$428$	0	0
$429$	− 7.10264e8i	− 0.434329i
$430$	0	0
$431$	1.47143e8	0.0885256	0.0442628	−	0.999020i	$-0.485906\pi$
$431$	1.47143e8	0.0885256	0.0442628	+	0.999020i	$0.485906\pi$
$432$	0	0
$433$	1.31693e9	0.779572	0.389786	−	0.920905i	$-0.372549\pi$
$433$	1.31693e9	0.779572	0.389786	+	0.920905i	$0.372549\pi$
$434$	0	0
$435$	1.48270e9i	0.863659i
$436$	0	0
$437$	5.94904e8i	0.341006i
$438$	0	0
$439$	6.94899e8	0.392009	0.196004	−	0.980603i	$-0.437203\pi$
$439$	6.94899e8	0.392009	0.196004	+	0.980603i	$0.437203\pi$
$440$	0	0
$441$	−1.43212e9	−0.795144
$442$	0	0
$443$	− 2.53160e9i	− 1.38351i	−0.722133	−	0.691754i	$-0.756838\pi$
$443$	− 2.53160e9i	− 1.38351i	0.722133	−	0.691754i	$-0.243162\pi$
$444$	0	0
$445$	3.30849e9i	1.77979i
$446$	0	0
$447$	−7.60330e7	−0.0402648
$448$	0	0
$449$	−3.62399e9	−1.88940	−0.944701	−	0.327934i	$-0.893648\pi$
$449$	−3.62399e9	−1.88940	−0.944701	+	0.327934i	$0.893648\pi$
$450$	0	0
$451$	− 1.67854e9i	− 0.861614i
$452$	0	0
$453$	− 1.55702e9i	− 0.786954i
$454$	0	0
$455$	6.62686e9	3.29813
$456$	0	0
$457$	2.06640e9	1.01277	0.506383	−	0.862309i	$-0.330982\pi$
$457$	2.06640e9	1.01277	0.506383	+	0.862309i	$0.330982\pi$
$458$	0	0
$459$	3.39412e8i	0.163826i
$460$	0	0
$461$	− 1.53131e9i	− 0.727966i	−0.931406	−	0.363983i	$-0.881417\pi$
$461$	− 1.53131e9i	− 0.727966i	0.931406	−	0.363983i	$-0.118583\pi$
$462$	0	0
$463$	−2.09022e8	−0.0978721	−0.0489360	−	0.998802i	$-0.515583\pi$
$463$	−2.09022e8	−0.0978721	−0.0489360	+	0.998802i	$0.515583\pi$
$464$	0	0
$465$	−1.59667e9	−0.736428
$466$	0	0
$467$	− 1.40416e9i	− 0.637979i	−0.947758	−	0.318989i	$-0.896657\pi$
$467$	− 1.40416e9i	− 0.637979i	0.947758	−	0.318989i	$-0.103343\pi$
$468$	0	0
$469$	4.33798e9i	1.94170i
$470$	0	0
$471$	2.47772e9	1.09264
$472$	0	0
$473$	−6.40447e8	−0.278272
$474$	0	0
$475$	5.88142e8i	0.251799i
$476$	0	0
$477$	− 3.93979e8i	− 0.166211i
$478$	0	0
$479$	−3.95376e9	−1.64375	−0.821876	−	0.569666i	$-0.807073\pi$
$479$	−3.95376e9	−1.64375	−0.821876	+	0.569666i	$0.807073\pi$
$480$	0	0
$481$	−1.04083e8	−0.0426456
$482$	0	0
$483$	− 1.84329e9i	− 0.744351i
$484$	0	0
$485$	3.40102e9i	1.35367i
$486$	0	0
$487$	−7.40238e7	−0.0290416	−0.0145208	−	0.999895i	$-0.504622\pi$
$487$	−7.40238e7	−0.0290416	−0.0145208	+	0.999895i	$0.504622\pi$
$488$	0	0
$489$	−2.48601e9	−0.961439
$490$	0	0
$491$	4.08459e9i	1.55727i	0.627479	+	0.778634i	$0.284087\pi$
$491$	4.08459e9i	1.55727i	−0.627479	+	0.778634i	$0.715913\pi$
$492$	0	0
$493$	− 2.75031e9i	− 1.03375i
$494$	0	0
$495$	−5.72816e8	−0.212274
$496$	0	0
$497$	−8.75644e8	−0.319949
$498$	0	0
$499$	− 3.35811e9i	− 1.20988i	−0.796270	−	0.604941i	$-0.793197\pi$
$499$	− 3.35811e9i	− 1.20988i	0.796270	−	0.604941i	$-0.206803\pi$
$500$	0	0
$501$	4.38565e8i	0.155812i
$502$	0	0
$503$	−2.31504e9	−0.811093	−0.405546	−	0.914074i	$-0.632919\pi$
$503$	−2.31504e9	−0.811093	−0.405546	+	0.914074i	$0.632919\pi$
$504$	0	0
$505$	5.50222e9	1.90116
$506$	0	0
$507$	− 1.89327e9i	− 0.645186i
$508$	0	0
$509$	2.61529e9i	0.879039i	0.898233	+	0.439520i	$0.144851\pi$
$509$	2.61529e9i	0.879039i	−0.898233	+	0.439520i	$0.855149\pi$
$510$	0	0
$511$	2.29817e9	0.761920
$512$	0	0
$513$	−2.86391e8	−0.0936589
$514$	0	0
$515$	− 4.85168e9i	− 1.56519i
$516$	0	0
$517$	− 1.20839e9i	− 0.384583i
$518$	0	0
$519$	−1.28021e9	−0.401972
$520$	0	0
$521$	9.75395e8	0.302168	0.151084	−	0.988521i	$-0.451724\pi$
$521$	9.75395e8	0.302168	0.151084	+	0.988521i	$0.451724\pi$
$522$	0	0
$523$	2.45790e9i	0.751291i	0.926763	+	0.375646i	$0.122579\pi$
$523$	2.45790e9i	0.751291i	−0.926763	+	0.375646i	$0.877421\pi$
$524$	0	0
$525$	− 1.82233e9i	− 0.549630i
$526$	0	0
$527$	2.96171e9	0.881465
$528$	0	0
$529$	−1.73313e9	−0.509022
$530$	0	0
$531$	− 5.30284e8i	− 0.153701i
$532$	0	0
$533$	− 8.47814e9i	− 2.42524i
$534$	0	0
$535$	7.97444e9	2.25145
$536$	0	0
$537$	−2.16022e9	−0.601988
$538$	0	0
$539$	4.48329e9i	1.23321i
$540$	0	0
$541$	1.30734e9i	0.354977i	0.984123	+	0.177488i	$0.0567972\pi$
$541$	1.30734e9i	0.354977i	−0.984123	+	0.177488i	$0.943203\pi$
$542$	0	0
$543$	−5.31211e8	−0.142386
$544$	0	0
$545$	7.24715e8	0.191770
$546$	0	0
$547$	5.91915e9i	1.54634i	0.634201	+	0.773168i	$0.281329\pi$
$547$	5.91915e9i	1.54634i	−0.634201	+	0.773168i	$0.718671\pi$
$548$	0	0
$549$	1.61684e9i	0.417026i
$550$	0	0
$551$	2.32068e9	0.590995
$552$	0	0
$553$	1.10532e10	2.77940
$554$	0	0
$555$	8.39416e7i	0.0208426i
$556$	0	0
$557$	− 4.93049e9i	− 1.20892i	−0.796636	−	0.604459i	$-0.793389\pi$
$557$	− 4.93049e9i	− 1.20892i	0.796636	−	0.604459i	$-0.206611\pi$
$558$	0	0
$559$	−3.23484e9	−0.783271
$560$	0	0
$561$	1.06253e9	0.254081
$562$	0	0
$563$	− 6.84747e8i	− 0.161715i	−0.996726	−	0.0808576i	$-0.974234\pi$
$563$	− 6.84747e8i	− 0.161715i	0.996726	−	0.0808576i	$-0.0257659\pi$
$564$	0	0
$565$	7.95092e9i	1.85459i
$566$	0	0
$567$	8.87371e8	0.204439
$568$	0	0
$569$	−5.26374e9	−1.19785	−0.598924	−	0.800806i	$-0.704405\pi$
$569$	−5.26374e9	−1.19785	−0.598924	+	0.800806i	$0.704405\pi$
$570$	0	0
$571$	3.59028e9i	0.807053i	0.914968	+	0.403527i	$0.132216\pi$
$571$	3.59028e9i	0.807053i	−0.914968	+	0.403527i	$0.867784\pi$
$572$	0	0
$573$	− 2.37169e9i	− 0.526644i
$574$	0	0
$575$	1.65269e9	0.362539
$576$	0	0
$577$	3.91183e9	0.847744	0.423872	−	0.905722i	$-0.360671\pi$
$577$	3.91183e9	0.847744	0.423872	+	0.905722i	$0.360671\pi$
$578$	0	0
$579$	3.49995e9i	0.749353i
$580$	0	0
$581$	− 1.06629e10i	− 2.25558i
$582$	0	0
$583$	−1.23336e9	−0.257780
$584$	0	0
$585$	−2.89324e9	−0.597502
$586$	0	0
$587$	− 3.88957e9i	− 0.793721i	−0.917879	−	0.396860i	$-0.870100\pi$
$587$	− 3.88957e9i	− 0.793721i	0.917879	−	0.396860i	$-0.129900\pi$
$588$	0	0
$589$	2.49905e9i	0.503932i
$590$	0	0
$591$	−3.28765e9	−0.655133
$592$	0	0
$593$	4.76138e9	0.937652	0.468826	−	0.883291i	$-0.344677\pi$
$593$	4.76138e9	0.937652	0.468826	+	0.883291i	$0.344677\pi$
$594$	0	0
$595$	9.91357e9i	1.92939i
$596$	0	0
$597$	− 3.88280e9i	− 0.746852i
$598$	0	0
$599$	−5.22390e9	−0.993118	−0.496559	−	0.868003i	$-0.665403\pi$
$599$	−5.22390e9	−0.993118	−0.496559	+	0.868003i	$0.665403\pi$
$600$	0	0
$601$	−8.36008e9	−1.57091	−0.785453	−	0.618922i	$-0.787570\pi$
$601$	−8.36008e9	−1.57091	−0.785453	+	0.618922i	$0.787570\pi$
$602$	0	0
$603$	− 1.89393e9i	− 0.351766i
$604$	0	0
$605$	− 4.91634e9i	− 0.902606i
$606$	0	0
$607$	5.78582e9	1.05004	0.525018	−	0.851091i	$-0.324058\pi$
$607$	5.78582e9	1.05004	0.525018	+	0.851091i	$0.324058\pi$
$608$	0	0
$609$	−7.19052e9	−1.29003
$610$	0	0
$611$	− 6.10347e9i	− 1.08251i
$612$	0	0
$613$	1.73833e9i	0.304803i	0.988319	+	0.152402i	$0.0487007\pi$
$613$	1.73833e9i	0.304803i	−0.988319	+	0.152402i	$0.951299\pi$
$614$	0	0
$615$	−6.83748e9	−1.18531
$616$	0	0
$617$	−4.44016e9	−0.761028	−0.380514	−	0.924775i	$-0.624253\pi$
$617$	−4.44016e9	−0.761028	−0.380514	+	0.924775i	$0.624253\pi$
$618$	0	0
$619$	− 1.64970e9i	− 0.279568i	−0.990182	−	0.139784i	$-0.955359\pi$
$619$	− 1.64970e9i	− 0.279568i	0.990182	−	0.139784i	$-0.0446408\pi$
$620$	0	0
$621$	8.04766e8i	0.134849i
$622$	0	0
$623$	−1.60448e10	−2.65844
$624$	0	0
$625$	−7.62755e9	−1.24970
$626$	0	0
$627$	8.96551e8i	0.145258i
$628$	0	0
$629$	− 1.55706e8i	− 0.0249475i
$630$	0	0
$631$	−3.28962e9	−0.521246	−0.260623	−	0.965441i	$-0.583928\pi$
$631$	−3.28962e9	−0.521246	−0.260623	+	0.965441i	$0.583928\pi$
$632$	0	0
$633$	−3.16223e9	−0.495541
$634$	0	0
$635$	1.57095e10i	2.43475i
$636$	0	0
$637$	2.26447e10i	3.47119i
$638$	0	0
$639$	3.82300e8	0.0579631
$640$	0	0
$641$	7.66402e9	1.14935	0.574677	−	0.818381i	$-0.305128\pi$
$641$	7.66402e9	1.14935	0.574677	+	0.818381i	$0.305128\pi$
$642$	0	0
$643$	1.16781e7i	0.00173234i	1.00000		0.000866170i	$0.000275710\pi$
$643$	1.16781e7i	0.00173234i	−1.00000		0.000866170i	$0.999724\pi$
$644$	0	0
$645$	2.60885e9i	0.382816i
$646$	0	0
$647$	−1.10751e9	−0.160762	−0.0803811	−	0.996764i	$-0.525614\pi$
$647$	−1.10751e9	−0.160762	−0.0803811	+	0.996764i	$0.525614\pi$
$648$	0	0
$649$	−1.66006e9	−0.238379
$650$	0	0
$651$	− 7.74321e9i	− 1.09999i
$652$	0	0
$653$	5.58523e9i	0.784955i	0.919762	+	0.392477i	$0.128382\pi$
$653$	5.58523e9i	0.784955i	−0.919762	+	0.392477i	$0.871618\pi$
$654$	0	0
$655$	−6.12971e9	−0.852306
$656$	0	0
$657$	−1.00337e9	−0.138032
$658$	0	0
$659$	− 7.10525e9i	− 0.967120i	−0.875311	−	0.483560i	$-0.839343\pi$
$659$	− 7.10525e9i	− 0.967120i	0.875311	−	0.483560i	$-0.160657\pi$
$660$	0	0
$661$	3.08453e9i	0.415416i	0.978191	+	0.207708i	$0.0666005\pi$
$661$	3.08453e9i	0.415416i	−0.978191	+	0.207708i	$0.933400\pi$
$662$	0	0
$663$	5.36676e9	0.715179
$664$	0	0
$665$	−8.36495e9	−1.10303
$666$	0	0
$667$	− 6.52116e9i	− 0.850911i
$668$	0	0
$669$	− 8.52979e9i	− 1.10140i
$670$	0	0
$671$	5.06153e9	0.646775
$672$	0	0
$673$	1.17552e10	1.48655	0.743273	−	0.668988i	$-0.233272\pi$
$673$	1.17552e10	1.48655	0.743273	+	0.668988i	$0.233272\pi$
$674$	0	0
$675$	7.95619e8i	0.0995731i
$676$	0	0
$677$	− 4.23386e9i	− 0.524416i	−0.965011	−	0.262208i	$-0.915549\pi$
$677$	− 4.23386e9i	− 0.524416i	0.965011	−	0.262208i	$-0.0844507\pi$
$678$	0	0
$679$	−1.64936e10	−2.02195
$680$	0	0
$681$	1.74673e9	0.211939
$682$	0	0
$683$	− 3.23179e9i	− 0.388124i	−0.980989	−	0.194062i	$-0.937834\pi$
$683$	− 3.23179e9i	− 0.388124i	0.980989	−	0.194062i	$-0.0621663\pi$
$684$	0	0
$685$	− 1.05274e9i	− 0.125142i
$686$	0	0
$687$	−5.56803e9	−0.655168
$688$	0	0
$689$	−6.22958e9	−0.725590
$690$	0	0
$691$	2.19695e9i	0.253307i	0.991947	+	0.126653i	$0.0404236\pi$
$691$	2.19695e9i	0.253307i	−0.991947	+	0.126653i	$0.959576\pi$
$692$	0	0
$693$	− 2.77793e9i	− 0.317069i
$694$	0	0
$695$	−1.79833e10	−2.03199
$696$	0	0
$697$	1.26830e10	1.41876
$698$	0	0
$699$	8.64607e9i	0.957522i
$700$	0	0
$701$	− 5.54482e9i	− 0.607959i	−0.952679	−	0.303980i	$-0.901684\pi$
$701$	− 5.54482e9i	− 0.607959i	0.952679	−	0.303980i	$-0.0983155\pi$
$702$	0	0
$703$	1.31382e8	0.0142624
$704$	0	0
$705$	−4.92235e9	−0.529067
$706$	0	0
$707$	2.66836e10i	2.83972i
$708$	0	0
$709$	1.08253e10i	1.14071i	0.821397	+	0.570357i	$0.193195\pi$
$709$	1.08253e10i	1.14071i	−0.821397	+	0.570357i	$0.806805\pi$
$710$	0	0
$711$	−4.82576e9	−0.503527
$712$	0	0
$713$	7.02240e9	0.725558
$714$	0	0
$715$	9.05734e9i	0.926679i
$716$	0	0
$717$	− 3.36516e9i	− 0.340948i
$718$	0	0
$719$	1.08562e10	1.08925	0.544623	−	0.838681i	$-0.316673\pi$
$719$	1.08562e10	1.08925	0.544623	+	0.838681i	$0.316673\pi$
$720$	0	0
$721$	2.35287e10	2.33789
$722$	0	0
$723$	6.69144e9i	0.658469i
$724$	0	0
$725$	− 6.44704e9i	− 0.628315i
$726$	0	0
$727$	−1.04714e10	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$727$	−1.04714e10	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	− 4.83922e9i	− 0.458210i
$732$	0	0
$733$	− 1.93042e10i	− 1.81045i	−0.424931	−	0.905226i	$-0.639702\pi$
$733$	− 1.93042e10i	− 1.81045i	0.424931	−	0.905226i	$-0.360298\pi$
$734$	0	0
$735$	1.82626e10	1.69651
$736$	0	0
$737$	−5.92898e9	−0.545562
$738$	0	0
$739$	2.64009e9i	0.240637i	0.992735	+	0.120319i	$0.0383916\pi$
$739$	2.64009e9i	0.240637i	−0.992735	+	0.120319i	$0.961608\pi$
$740$	0	0
$741$	4.52840e9i	0.408866i
$742$	0	0
$743$	2.00716e10	1.79523	0.897617	−	0.440777i	$-0.145297\pi$
$743$	2.00716e10	1.79523	0.897617	+	0.440777i	$0.145297\pi$
$744$	0	0
$745$	9.69578e8	0.0859084
$746$	0	0
$747$	4.65534e9i	0.408629i
$748$	0	0
$749$	3.86728e10i	3.36294i
$750$	0	0
$751$	−1.78345e10	−1.53646	−0.768229	−	0.640175i	$-0.778862\pi$
$751$	−1.78345e10	−1.53646	−0.768229	+	0.640175i	$0.778862\pi$
$752$	0	0
$753$	−4.68110e9	−0.399545
$754$	0	0
$755$	1.98552e10i	1.67904i
$756$	0	0
$757$	− 2.59023e9i	− 0.217021i	−0.994095	−	0.108511i	$-0.965392\pi$
$757$	− 2.59023e9i	− 0.217021i	0.994095	−	0.108511i	$-0.0346082\pi$
$758$	0	0
$759$	2.51933e9	0.209141
$760$	0	0
$761$	1.47456e10	1.21287	0.606436	−	0.795132i	$-0.292599\pi$
$761$	1.47456e10	1.21287	0.606436	+	0.795132i	$0.292599\pi$
$762$	0	0
$763$	3.51458e9i	0.286442i
$764$	0	0
$765$	− 4.32820e9i	− 0.349536i
$766$	0	0
$767$	−8.38482e9	−0.670980
$768$	0	0
$769$	2.16839e9	0.171947	0.0859737	−	0.996297i	$-0.472600\pi$
$769$	2.16839e9	0.171947	0.0859737	+	0.996297i	$0.472600\pi$
$770$	0	0
$771$	2.61575e9i	0.205545i
$772$	0	0
$773$	− 2.19037e10i	− 1.70565i	−0.522198	−	0.852824i	$-0.674888\pi$
$773$	− 2.19037e10i	− 1.70565i	0.522198	−	0.852824i	$-0.325112\pi$
$774$	0	0
$775$	6.94258e9	0.535753
$776$	0	0
$777$	−4.07083e8	−0.0311322
$778$	0	0
$779$	1.07018e10i	0.811101i
$780$	0	0
$781$	− 1.19680e9i	− 0.0898963i
$782$	0	0
$783$	3.13933e9	0.233707
$784$	0	0
$785$	−3.15961e10	−2.33125
$786$	0	0
$787$	− 9.86968e9i	− 0.721758i	−0.932613	−	0.360879i	$-0.882477\pi$
$787$	− 9.86968e9i	− 0.721758i	0.932613	−	0.360879i	$-0.117523\pi$
$788$	0	0
$789$	− 8.72284e9i	− 0.632249i
$790$	0	0
$791$	−3.85588e10	−2.77016
$792$	0	0
$793$	2.55654e10	1.82052
$794$	0	0
$795$	5.02405e9i	0.354625i
$796$	0	0
$797$	− 1.69316e10i	− 1.18466i	−0.805695	−	0.592331i	$-0.798208\pi$
$797$	− 1.69316e10i	− 1.18466i	0.805695	−	0.592331i	$-0.201792\pi$
$798$	0	0
$799$	9.13060e9	0.633265
$800$	0	0
$801$	7.00507e9	0.481614
$802$	0	0
$803$	3.14105e9i	0.214077i
$804$	0	0
$805$	2.35057e10i	1.58814i
$806$	0	0
$807$	−4.97459e9	−0.333196
$808$	0	0
$809$	−5.74007e8	−0.0381151	−0.0190576	−	0.999818i	$-0.506067\pi$
$809$	−5.74007e8	−0.0381151	−0.0190576	+	0.999818i	$0.506067\pi$
$810$	0	0
$811$	− 2.00117e10i	− 1.31738i	−0.752415	−	0.658689i	$-0.771111\pi$
$811$	− 2.00117e10i	− 1.31738i	0.752415	−	0.658689i	$-0.228889\pi$
$812$	0	0
$813$	4.38095e9i	0.285924i
$814$	0	0
$815$	3.17018e10	2.05131
$816$	0	0
$817$	4.08328e9	0.261958
$818$	0	0
$819$	− 1.40311e10i	− 0.892477i
$820$	0	0
$821$	8.15447e9i	0.514274i	0.966375	+	0.257137i	$0.0827791\pi$
$821$	8.15447e9i	0.514274i	−0.966375	+	0.257137i	$0.917221\pi$
$822$	0	0
$823$	−9.80708e9	−0.613254	−0.306627	−	0.951830i	$-0.599200\pi$
$823$	−9.80708e9	−0.613254	−0.306627	+	0.951830i	$0.599200\pi$
$824$	0	0
$825$	2.49070e9	0.154430
$826$	0	0
$827$	− 1.05065e10i	− 0.645937i	−0.946410	−	0.322969i	$-0.895319\pi$
$827$	− 1.05065e10i	− 0.645937i	0.946410	−	0.322969i	$-0.104681\pi$
$828$	0	0
$829$	1.45134e10i	0.884766i	0.896826	+	0.442383i	$0.145867\pi$
$829$	1.45134e10i	0.884766i	−0.896826	+	0.442383i	$0.854133\pi$
$830$	0	0
$831$	−2.59334e9	−0.156767
$832$	0	0
$833$	−3.38757e10	−2.03063
$834$	0	0
$835$	− 5.59262e9i	− 0.332439i
$836$	0	0
$837$	3.38063e9i	0.199278i
$838$	0	0
$839$	−1.98293e9	−0.115915	−0.0579576	−	0.998319i	$-0.518459\pi$
$839$	−1.98293e9	−0.115915	−0.0579576	+	0.998319i	$0.518459\pi$
$840$	0	0
$841$	−8.18867e9	−0.474709
$842$	0	0
$843$	9.76257e9i	0.561265i
$844$	0	0
$845$	2.41431e10i	1.37656i
$846$	0	0
$847$	2.38423e10	1.34820
$848$	0	0
$849$	−9.27225e9	−0.520006
$850$	0	0
$851$	− 3.69188e8i	− 0.0205350i
$852$	0	0
$853$	7.37240e9i	0.406712i	0.979105	+	0.203356i	$0.0651849\pi$
$853$	7.37240e9i	0.406712i	−0.979105	+	0.203356i	$0.934815\pi$
$854$	0	0
$855$	3.65208e9	0.199829
$856$	0	0
$857$	−2.41520e10	−1.31075	−0.655374	−	0.755304i	$-0.727489\pi$
$857$	−2.41520e10	−1.31075	−0.655374	+	0.755304i	$0.727489\pi$
$858$	0	0
$859$	3.64471e10i	1.96195i	0.194145	+	0.980973i	$0.437807\pi$
$859$	3.64471e10i	1.96195i	−0.194145	+	0.980973i	$0.562193\pi$
$860$	0	0
$861$	− 3.31590e10i	− 1.77048i
$862$	0	0
$863$	4.59737e9	0.243485	0.121742	−	0.992562i	$-0.461152\pi$
$863$	4.59737e9	0.243485	0.121742	+	0.992562i	$0.461152\pi$
$864$	0	0
$865$	1.63253e10	0.857642
$866$	0	0
$867$	− 3.05064e9i	− 0.158973i
$868$	0	0
$869$	1.51071e10i	0.780930i
$870$	0	0
$871$	−2.99468e10	−1.53563
$872$	0	0
$873$	7.20099e9	0.366305
$874$	0	0
$875$	− 2.16758e10i	− 1.09382i
$876$	0	0
$877$	− 1.57473e10i	− 0.788327i	−0.919040	−	0.394164i	$-0.871034\pi$
$877$	− 1.57473e10i	− 0.788327i	0.919040	−	0.394164i	$-0.128966\pi$
$878$	0	0
$879$	−7.63227e9	−0.379047
$880$	0	0
$881$	2.30534e10	1.13584	0.567922	−	0.823082i	$-0.307748\pi$
$881$	2.30534e10	1.13584	0.567922	+	0.823082i	$0.307748\pi$
$882$	0	0
$883$	− 5.63512e9i	− 0.275449i	−0.990471	−	0.137724i	$-0.956021\pi$
$883$	− 5.63512e9i	− 0.275449i	0.990471	−	0.137724i	$-0.0439788\pi$
$884$	0	0
$885$	6.76222e9i	0.327935i
$886$	0	0
$887$	1.11246e10	0.535245	0.267623	−	0.963524i	$-0.413762\pi$
$887$	1.11246e10	0.535245	0.267623	+	0.963524i	$0.413762\pi$
$888$	0	0
$889$	−7.61848e10	−3.63674
$890$	0	0
$891$	1.21282e9i	0.0574415i
$892$	0	0
$893$	7.70429e9i	0.362036i
$894$	0	0
$895$	2.75473e10	1.28439
$896$	0	0
$897$	1.27249e10	0.588683
$898$	0	0
$899$	− 2.73939e10i	− 1.25746i
$900$	0	0
$901$	− 9.31926e9i	− 0.424468i
$902$	0	0
$903$	−1.26519e10	−0.571804
$904$	0	0
$905$	6.77404e9	0.303793
$906$	0	0
$907$	− 7.73698e9i	− 0.344307i	−0.985070	−	0.172154i	$-0.944927\pi$
$907$	− 7.73698e9i	− 0.344307i	0.985070	−	0.172154i	$-0.0550725\pi$
$908$	0	0
$909$	− 1.16499e10i	− 0.514455i
$910$	0	0
$911$	1.03842e10	0.455051	0.227526	−	0.973772i	$-0.426936\pi$
$911$	1.03842e10	0.455051	0.227526	+	0.973772i	$0.426936\pi$
$912$	0	0
$913$	1.45736e10	0.633752
$914$	0	0
$915$	− 2.06180e10i	− 0.889761i
$916$	0	0
$917$	− 2.97266e10i	− 1.27307i
$918$	0	0
$919$	−8.08020e9	−0.343414	−0.171707	−	0.985148i	$-0.554928\pi$
$919$	−8.08020e9	−0.343414	−0.171707	+	0.985148i	$0.554928\pi$
$920$	0	0
$921$	4.26110e9	0.179727
$922$	0	0
$923$	− 6.04491e9i	− 0.253037i
$924$	0	0
$925$	− 3.64992e8i	− 0.0151631i
$926$	0	0
$927$	−1.02725e10	−0.423541
$928$	0	0
$929$	1.28931e10	0.527598	0.263799	−	0.964578i	$-0.415024\pi$
$929$	1.28931e10	0.527598	0.263799	+	0.964578i	$0.415024\pi$
$930$	0	0
$931$	− 2.85839e10i	− 1.16091i
$932$	0	0
$933$	1.56392e10i	0.630420i
$934$	0	0
$935$	−1.35495e10	−0.542104
$936$	0	0
$937$	−7.17106e9	−0.284770	−0.142385	−	0.989811i	$-0.545477\pi$
$937$	−7.17106e9	−0.284770	−0.142385	+	0.989811i	$0.545477\pi$
$938$	0	0
$939$	− 2.37879e9i	− 0.0937618i
$940$	0	0
$941$	− 2.40976e9i	− 0.0942781i	−0.998888	−	0.0471391i	$-0.984990\pi$
$941$	− 2.40976e9i	− 0.0942781i	0.998888	−	0.0471391i	$-0.0150104\pi$
$942$	0	0
$943$	3.00723e10	1.16782
$944$	0	0
$945$	−1.13158e10	−0.436189
$946$	0	0
$947$	− 4.20590e10i	− 1.60929i	−0.593758	−	0.804644i	$-0.702356\pi$
$947$	− 4.20590e10i	− 1.60929i	0.593758	−	0.804644i	$-0.297644\pi$
$948$	0	0
$949$	1.58652e10i	0.602578i
$950$	0	0
$951$	3.21734e9	0.121301
$952$	0	0
$953$	2.79027e10	1.04429	0.522145	−	0.852857i	$-0.325132\pi$
$953$	2.79027e10	1.04429	0.522145	+	0.852857i	$0.325132\pi$
$954$	0	0
$955$	3.02440e10i	1.12364i
$956$	0	0
$957$	− 9.82772e9i	− 0.362461i
$958$	0	0
$959$	5.10535e9	0.186922
$960$	0	0
$961$	1.98684e9	0.0722155
$962$	0	0
$963$	− 1.68843e10i	− 0.609243i
$964$	0	0
$965$	− 4.46316e10i	− 1.59881i
$966$	0	0
$967$	2.50078e10	0.889371	0.444686	−	0.895687i	$-0.353315\pi$
$967$	2.50078e10	0.889371	0.444686	+	0.895687i	$0.353315\pi$
$968$	0	0
$969$	−6.77435e9	−0.239185
$970$	0	0
$971$	− 3.65828e10i	− 1.28236i	−0.767391	−	0.641179i	$-0.778446\pi$
$971$	− 3.65828e10i	− 1.28236i	0.767391	−	0.641179i	$-0.221554\pi$
$972$	0	0
$973$	− 8.72117e10i	− 3.03515i
$974$	0	0
$975$	1.25803e10	0.434685
$976$	0	0
$977$	2.86412e10	0.982563	0.491281	−	0.871001i	$-0.336529\pi$
$977$	2.86412e10	0.982563	0.491281	+	0.871001i	$0.336529\pi$
$978$	0	0
$979$	− 2.19295e10i	− 0.746945i
$980$	0	0
$981$	− 1.53444e9i	− 0.0518930i
$982$	0	0
$983$	−3.81805e10	−1.28205	−0.641024	−	0.767521i	$-0.721490\pi$
$983$	−3.81805e10	−1.28205	−0.641024	+	0.767521i	$0.721490\pi$
$984$	0	0
$985$	4.19244e10	1.39778
$986$	0	0
$987$	− 2.38714e10i	− 0.790256i
$988$	0	0
$989$	− 1.14741e10i	− 0.377165i
$990$	0	0
$991$	2.06103e10	0.672709	0.336354	−	0.941736i	$-0.390806\pi$
$991$	2.06103e10	0.672709	0.336354	+	0.941736i	$0.390806\pi$
$992$	0	0
$993$	−2.44349e10	−0.791932
$994$	0	0
$995$	4.95137e10i	1.59347i
$996$	0	0
$997$	2.00020e10i	0.639204i	0.947552	+	0.319602i	$0.103549\pi$
$997$	2.00020e10i	0.639204i	−0.947552	+	0.319602i	$0.896451\pi$
$998$	0	0
$999$	1.77730e8	0.00564002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.d.193.5	yes	8
4.3	odd	2		384.8.d.c.193.1	✓	8
8.3	odd	2		384.8.d.c.193.8	yes	8
8.5	even	2	inner	384.8.d.d.193.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.c.193.1	✓	8	4.3	odd	2
384.8.d.c.193.8	yes	8	8.3	odd	2
384.8.d.d.193.4	yes	8	8.5	even	2	inner
384.8.d.d.193.5	yes	8	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)

\(f(q)\)	\(=\)	\(q+27.0000i q^{3} -344.306i q^{5} +1669.75 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q+27.0000i q^{3} -344.306i q^{5} +1669.75 q^{7} -729.000 q^{9} +2282.14i q^{11} +11526.9i q^{13} +9296.26 q^{15} -17243.9 q^{17} -14550.2i q^{19} +45083.1i q^{21} -40886.4 q^{23} -40421.6 q^{25} -19683.0i q^{27} +159495. i q^{29} -171754. q^{31} -61617.9 q^{33} -574903. i q^{35} +9029.61i q^{37} -311227. q^{39} -735509. q^{41} +280634. i q^{43} +250999. i q^{45} -529498. q^{47} +1.96451e6 q^{49} -465585. i q^{51} +540438. i q^{53} +785756. q^{55} +392855. q^{57} +727412. i q^{59} -2.21789e6i q^{61} -1.21724e6 q^{63} +3.96878e6 q^{65} +2.59799e6i q^{67} -1.10393e6i q^{69} -524418. q^{71} +1.37636e6 q^{73} -1.09138e6i q^{75} +3.81060e6i q^{77} +6.61970e6 q^{79} +531441. q^{81} -6.38593e6i q^{83} +5.93718e6i q^{85} -4.30636e6 q^{87} -9.60915e6 q^{89} +1.92470e7i q^{91} -4.63736e6i q^{93} -5.00971e6 q^{95} -9.87790e6 q^{97} -1.66368e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2880 q^{7} - 5832 q^{9}+O(q^{10}) \) 8 * q + 2880 * q^7 - 5832 * q^9	\( 8 q + 2880 q^{7} - 5832 q^{9} - 6048 q^{15} + 22896 q^{17} + 207360 q^{23} - 204696 q^{25} - 17856 q^{31} + 7776 q^{33} - 116640 q^{39} + 687056 q^{41} - 1987200 q^{47} + 4815560 q^{49} + 2077056 q^{55} + 878688 q^{57} - 2099520 q^{63} + 12871808 q^{65} - 6336000 q^{71} + 8920752 q^{73} + 1251648 q^{79} + 4251528 q^{81} - 13385952 q^{87} + 4447408 q^{89} - 31607424 q^{95} - 14157584 q^{97}+O(q^{100}) \) 8 * q + 2880 * q^7 - 5832 * q^9 - 6048 * q^15 + 22896 * q^17 + 207360 * q^23 - 204696 * q^25 - 17856 * q^31 + 7776 * q^33 - 116640 * q^39 + 687056 * q^41 - 1987200 * q^47 + 4815560 * q^49 + 2077056 * q^55 + 878688 * q^57 - 2099520 * q^63 + 12871808 * q^65 - 6336000 * q^71 + 8920752 * q^73 + 1251648 * q^79 + 4251528 * q^81 - 13385952 * q^87 + 4447408 * q^89 - 31607424 * q^95 - 14157584 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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