LMFDB - Embedded newform 192.8.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,8,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.a (trivial)

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:	yes
Analytic conductor:	$59.9779248930$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.78233$ of defining polynomial
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} -423.552 q^{5} +1228.66 q^{7} +729.000 q^{9} +O(q^{10})$	$q-27.0000 q^{3} -423.552 q^{5} +1228.66 q^{7} +729.000 q^{9} -2781.31 q^{11} -3729.10 q^{13} +11435.9 q^{15} -11630.2 q^{17} +45178.2 q^{19} -33173.7 q^{21} +27729.3 q^{23} +101271. q^{25} -19683.0 q^{27} +210915. q^{29} +172577. q^{31} +75095.4 q^{33} -520399. q^{35} +269522. q^{37} +100686. q^{39} -379673. q^{41} -1.03330e6 q^{43} -308769. q^{45} -624148. q^{47} +686051. q^{49} +314016. q^{51} -1.84130e6 q^{53} +1.17803e6 q^{55} -1.21981e6 q^{57} -2.74738e6 q^{59} +2.70188e6 q^{61} +895690. q^{63} +1.57947e6 q^{65} -989057. q^{67} -748691. q^{69} +3.30668e6 q^{71} -1.10275e6 q^{73} -2.73432e6 q^{75} -3.41727e6 q^{77} -8.02881e6 q^{79} +531441. q^{81} +3.55645e6 q^{83} +4.92600e6 q^{85} -5.69471e6 q^{87} -5.21527e6 q^{89} -4.58178e6 q^{91} -4.65959e6 q^{93} -1.91353e7 q^{95} -7.07452e6 q^{97} -2.02758e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9	$ 2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9} - 1656 q^{11} - 6156 q^{13} + 5292 q^{15} + 17108 q^{17} + 504 q^{19} - 13608 q^{21} + 51552 q^{23} + 74926 q^{25} - 39366 q^{27} + 199804 q^{29} + 257256 q^{31} + 44712 q^{33} - 685296 q^{35} + 468724 q^{37} + 166212 q^{39} - 106940 q^{41} - 1617336 q^{43} - 142884 q^{45} + 646416 q^{47} + 387634 q^{49} - 461916 q^{51} - 1469492 q^{53} + 1434096 q^{55} - 13608 q^{57} - 4541544 q^{59} + 481412 q^{61} + 367416 q^{63} + 1027224 q^{65} - 4775256 q^{67} - 1391904 q^{69} - 1094400 q^{71} - 5731884 q^{73} - 2023002 q^{75} - 4232736 q^{77} - 10402776 q^{79} + 1062882 q^{81} - 2212200 q^{83} + 11465432 q^{85} - 5394708 q^{87} - 3604364 q^{89} - 2823120 q^{91} - 6945912 q^{93} - 29300976 q^{95} - 7156188 q^{97} - 1207224 q^{99}+O(q^{100}) $ 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9 - 1656 * q^11 - 6156 * q^13 + 5292 * q^15 + 17108 * q^17 + 504 * q^19 - 13608 * q^21 + 51552 * q^23 + 74926 * q^25 - 39366 * q^27 + 199804 * q^29 + 257256 * q^31 + 44712 * q^33 - 685296 * q^35 + 468724 * q^37 + 166212 * q^39 - 106940 * q^41 - 1617336 * q^43 - 142884 * q^45 + 646416 * q^47 + 387634 * q^49 - 461916 * q^51 - 1469492 * q^53 + 1434096 * q^55 - 13608 * q^57 - 4541544 * q^59 + 481412 * q^61 + 367416 * q^63 + 1027224 * q^65 - 4775256 * q^67 - 1391904 * q^69 - 1094400 * q^71 - 5731884 * q^73 - 2023002 * q^75 - 4232736 * q^77 - 10402776 * q^79 + 1062882 * q^81 - 2212200 * q^83 + 11465432 * q^85 - 5394708 * q^87 - 3604364 * q^89 - 2823120 * q^91 - 6945912 * q^93 - 29300976 * q^95 - 7156188 * q^97 - 1207224 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−423.552	−1.51535	−0.757673	−	0.652635i	$-0.773664\pi$
$5$	−423.552	−1.51535	−0.757673	+	0.652635i	$0.773664\pi$
$6$	0	0
$7$	1228.66	1.35390	0.676951	−	0.736028i	$-0.263301\pi$
$7$	1228.66	1.35390	0.676951	+	0.736028i	$0.263301\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−2781.31	−0.630050	−0.315025	−	0.949083i	$-0.602013\pi$
$11$	−2781.31	−0.630050	−0.315025	+	0.949083i	$0.602013\pi$
$12$	0	0
$13$	−3729.10	−0.470763	−0.235382	−	0.971903i	$-0.575634\pi$
$13$	−3729.10	−0.470763	−0.235382	+	0.971903i	$0.575634\pi$
$14$	0	0
$15$	11435.9	0.874885
$16$	0	0
$17$	−11630.2	−0.574138	−0.287069	−	0.957910i	$-0.592681\pi$
$17$	−11630.2	−0.574138	−0.287069	+	0.957910i	$0.592681\pi$
$18$	0	0
$19$	45178.2	1.51109	0.755546	−	0.655096i	$-0.227372\pi$
$19$	45178.2	1.51109	0.755546	+	0.655096i	$0.227372\pi$
$20$	0	0
$21$	−33173.7	−0.781675
$22$	0	0
$23$	27729.3	0.475217	0.237608	−	0.971361i	$-0.423637\pi$
$23$	27729.3	0.475217	0.237608	+	0.971361i	$0.423637\pi$
$24$	0	0
$25$	101271.	1.29627
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	210915.	1.60589	0.802943	−	0.596056i	$-0.203266\pi$
$29$	210915.	1.60589	0.802943	+	0.596056i	$0.203266\pi$
$30$	0	0
$31$	172577.	1.04044	0.520221	−	0.854031i	$-0.325849\pi$
$31$	172577.	1.04044	0.520221	+	0.854031i	$0.325849\pi$
$32$	0	0
$33$	75095.4	0.363760
$34$	0	0
$35$	−520399.	−2.05163
$36$	0	0
$37$	269522.	0.874757	0.437379	−	0.899277i	$-0.355907\pi$
$37$	269522.	0.874757	0.437379	+	0.899277i	$0.355907\pi$
$38$	0	0
$39$	100686.	0.271795
$40$	0	0
$41$	−379673.	−0.860332	−0.430166	−	0.902750i	$-0.641545\pi$
$41$	−379673.	−0.860332	−0.430166	+	0.902750i	$0.641545\pi$
$42$	0	0
$43$	−1.03330e6	−1.98192	−0.990960	−	0.134154i	$-0.957168\pi$
$43$	−1.03330e6	−1.98192	−0.990960	+	0.134154i	$0.957168\pi$
$44$	0	0
$45$	−308769.	−0.505115
$46$	0	0
$47$	−624148.	−0.876890	−0.438445	−	0.898758i	$-0.644471\pi$
$47$	−624148.	−0.876890	−0.438445	+	0.898758i	$0.644471\pi$
$48$	0	0
$49$	686051.	0.833049
$50$	0	0
$51$	314016.	0.331479
$52$	0	0
$53$	−1.84130e6	−1.69886	−0.849431	−	0.527699i	$-0.823055\pi$
$53$	−1.84130e6	−1.69886	−0.849431	+	0.527699i	$0.823055\pi$
$54$	0	0
$55$	1.17803e6	0.954744
$56$	0	0
$57$	−1.21981e6	−0.872429
$58$	0	0
$59$	−2.74738e6	−1.74155	−0.870776	−	0.491679i	$-0.836383\pi$
$59$	−2.74738e6	−1.74155	−0.870776	+	0.491679i	$0.836383\pi$
$60$	0	0
$61$	2.70188e6	1.52409	0.762046	−	0.647523i	$-0.224195\pi$
$61$	2.70188e6	1.52409	0.762046	+	0.647523i	$0.224195\pi$
$62$	0	0
$63$	895690.	0.451300
$64$	0	0
$65$	1.57947e6	0.713369
$66$	0	0
$67$	−989057.	−0.401753	−0.200877	−	0.979617i	$-0.564379\pi$
$67$	−989057.	−0.401753	−0.200877	+	0.979617i	$0.564379\pi$
$68$	0	0
$69$	−748691.	−0.274366
$70$	0	0
$71$	3.30668e6	1.09645	0.548224	−	0.836331i	$-0.315304\pi$
$71$	3.30668e6	1.09645	0.548224	+	0.836331i	$0.315304\pi$
$72$	0	0
$73$	−1.10275e6	−0.331779	−0.165889	−	0.986144i	$-0.553049\pi$
$73$	−1.10275e6	−0.331779	−0.165889	+	0.986144i	$0.553049\pi$
$74$	0	0
$75$	−2.73432e6	−0.748402
$76$	0	0
$77$	−3.41727e6	−0.853026
$78$	0	0
$79$	−8.02881e6	−1.83213	−0.916065	−	0.401031i	$-0.868652\pi$
$79$	−8.02881e6	−1.83213	−0.916065	+	0.401031i	$0.868652\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	3.55645e6	0.682722	0.341361	−	0.939932i	$-0.389112\pi$
$83$	3.55645e6	0.682722	0.341361	+	0.939932i	$0.389112\pi$
$84$	0	0
$85$	4.92600e6	0.870018
$86$	0	0
$87$	−5.69471e6	−0.927159
$88$	0	0
$89$	−5.21527e6	−0.784172	−0.392086	−	0.919928i	$-0.628247\pi$
$89$	−5.21527e6	−0.784172	−0.392086	+	0.919928i	$0.628247\pi$
$90$	0	0
$91$	−4.58178e6	−0.637367
$92$	0	0
$93$	−4.65959e6	−0.600700
$94$	0	0
$95$	−1.91353e7	−2.28983
$96$	0	0
$97$	−7.07452e6	−0.787038	−0.393519	−	0.919316i	$-0.628743\pi$
$97$	−7.07452e6	−0.787038	−0.393519	+	0.919316i	$0.628743\pi$
$98$	0	0
$99$	−2.02758e6	−0.210017
$100$	0	0
$101$	6.27236e6	0.605768	0.302884	−	0.953027i	$-0.402051\pi$
$101$	6.27236e6	0.605768	0.302884	+	0.953027i	$0.402051\pi$
$102$	0	0
$103$	−1.22232e7	−1.10219	−0.551093	−	0.834444i	$-0.685789\pi$
$103$	−1.22232e7	−1.10219	−0.551093	+	0.834444i	$0.685789\pi$
$104$	0	0
$105$	1.40508e7	1.18451
$106$	0	0
$107$	−8.74065e6	−0.689764	−0.344882	−	0.938646i	$-0.612081\pi$
$107$	−8.74065e6	−0.689764	−0.344882	+	0.938646i	$0.612081\pi$
$108$	0	0
$109$	−5.57044e6	−0.412000	−0.206000	−	0.978552i	$-0.566045\pi$
$109$	−5.57044e6	−0.412000	−0.206000	+	0.978552i	$0.566045\pi$
$110$	0	0
$111$	−7.27708e6	−0.505041
$112$	0	0
$113$	1.25206e7	0.816302	0.408151	−	0.912914i	$-0.366174\pi$
$113$	1.25206e7	0.816302	0.408151	+	0.912914i	$0.366174\pi$
$114$	0	0
$115$	−1.17448e7	−0.720117
$116$	0	0
$117$	−2.71852e6	−0.156921
$118$	0	0
$119$	−1.42895e7	−0.777327
$120$	0	0
$121$	−1.17515e7	−0.603037
$122$	0	0
$123$	1.02512e7	0.496713
$124$	0	0
$125$	−9.80360e6	−0.448953
$126$	0	0
$127$	−1.60536e7	−0.695438	−0.347719	−	0.937599i	$-0.613044\pi$
$127$	−1.60536e7	−0.695438	−0.347719	+	0.937599i	$0.613044\pi$
$128$	0	0
$129$	2.78991e7	1.14426
$130$	0	0
$131$	1.53592e7	0.596923	0.298462	−	0.954422i	$-0.403526\pi$
$131$	1.53592e7	0.596923	0.298462	+	0.954422i	$0.403526\pi$
$132$	0	0
$133$	5.55084e7	2.04587
$134$	0	0
$135$	8.33677e6	0.291628
$136$	0	0
$137$	2.09949e7	0.697576	0.348788	−	0.937202i	$-0.386593\pi$
$137$	2.09949e7	0.697576	0.348788	+	0.937202i	$0.386593\pi$
$138$	0	0
$139$	3.99151e7	1.26062	0.630312	−	0.776342i	$-0.282927\pi$
$139$	3.99151e7	1.26062	0.630312	+	0.776342i	$0.282927\pi$
$140$	0	0
$141$	1.68520e7	0.506273
$142$	0	0
$143$	1.03718e7	0.296605
$144$	0	0
$145$	−8.93335e7	−2.43347
$146$	0	0
$147$	−1.85234e7	−0.480961
$148$	0	0
$149$	−5.79531e7	−1.43524	−0.717621	−	0.696434i	$-0.754769\pi$
$149$	−5.79531e7	−1.43524	−0.717621	+	0.696434i	$0.754769\pi$
$150$	0	0
$151$	1.00182e7	0.236793	0.118397	−	0.992966i	$-0.462225\pi$
$151$	1.00182e7	0.236793	0.118397	+	0.992966i	$0.462225\pi$
$152$	0	0
$153$	−8.47843e6	−0.191379
$154$	0	0
$155$	−7.30955e7	−1.57663
$156$	0	0
$157$	−1.88494e7	−0.388732	−0.194366	−	0.980929i	$-0.562265\pi$
$157$	−1.88494e7	−0.388732	−0.194366	+	0.980929i	$0.562265\pi$
$158$	0	0
$159$	4.97150e7	0.980839
$160$	0	0
$161$	3.40698e7	0.643396
$162$	0	0
$163$	−7.98770e7	−1.44466	−0.722329	−	0.691550i	$-0.756928\pi$
$163$	−7.98770e7	−1.44466	−0.722329	+	0.691550i	$0.756928\pi$
$164$	0	0
$165$	−3.18068e7	−0.551221
$166$	0	0
$167$	2.20039e7	0.365587	0.182794	−	0.983151i	$-0.441486\pi$
$167$	2.20039e7	0.365587	0.182794	+	0.983151i	$0.441486\pi$
$168$	0	0
$169$	−4.88423e7	−0.778382
$170$	0	0
$171$	3.29349e7	0.503697
$172$	0	0
$173$	−6.83727e7	−1.00397	−0.501986	−	0.864876i	$-0.667397\pi$
$173$	−6.83727e7	−1.00397	−0.501986	+	0.864876i	$0.667397\pi$
$174$	0	0
$175$	1.24427e8	1.75502
$176$	0	0
$177$	7.41793e7	1.00549
$178$	0	0
$179$	−5.95976e7	−0.776682	−0.388341	−	0.921516i	$-0.626952\pi$
$179$	−5.95976e7	−0.776682	−0.388341	+	0.921516i	$0.626952\pi$
$180$	0	0
$181$	1.51283e8	1.89634	0.948168	−	0.317769i	$-0.102934\pi$
$181$	1.51283e8	1.89634	0.948168	+	0.317769i	$0.102934\pi$
$182$	0	0
$183$	−7.29507e7	−0.879935
$184$	0	0
$185$	−1.14156e8	−1.32556
$186$	0	0
$187$	3.23472e7	0.361736
$188$	0	0
$189$	−2.41836e7	−0.260558
$190$	0	0
$191$	−8.66555e6	−0.0899869	−0.0449935	−	0.998987i	$-0.514327\pi$
$191$	−8.66555e6	−0.0899869	−0.0449935	+	0.998987i	$0.514327\pi$
$192$	0	0
$193$	1.51312e8	1.51504	0.757520	−	0.652812i	$-0.226411\pi$
$193$	1.51312e8	1.51504	0.757520	+	0.652812i	$0.226411\pi$
$194$	0	0
$195$	−4.26457e7	−0.411864
$196$	0	0
$197$	−9.77091e7	−0.910549	−0.455274	−	0.890351i	$-0.650459\pi$
$197$	−9.77091e7	−0.910549	−0.455274	+	0.890351i	$0.650459\pi$
$198$	0	0
$199$	3.19522e7	0.287419	0.143709	−	0.989620i	$-0.454097\pi$
$199$	3.19522e7	0.287419	0.143709	+	0.989620i	$0.454097\pi$
$200$	0	0
$201$	2.67045e7	0.231952
$202$	0	0
$203$	2.59142e8	2.17421
$204$	0	0
$205$	1.60811e8	1.30370
$206$	0	0
$207$	2.02147e7	0.158406
$208$	0	0
$209$	−1.25654e8	−0.952064
$210$	0	0
$211$	3.47789e7	0.254875	0.127437	−	0.991847i	$-0.459325\pi$
$211$	3.47789e7	0.254875	0.127437	+	0.991847i	$0.459325\pi$
$212$	0	0
$213$	−8.92804e7	−0.633035
$214$	0	0
$215$	4.37656e8	3.00329
$216$	0	0
$217$	2.12038e8	1.40866
$218$	0	0
$219$	2.97743e7	0.191552
$220$	0	0
$221$	4.33703e7	0.270283
$222$	0	0
$223$	2.06077e7	0.124441	0.0622204	−	0.998062i	$-0.480182\pi$
$223$	2.06077e7	0.124441	0.0622204	+	0.998062i	$0.480182\pi$
$224$	0	0
$225$	7.38267e7	0.432090
$226$	0	0
$227$	2.38134e8	1.35124	0.675619	−	0.737251i	$-0.263877\pi$
$227$	2.38134e8	1.35124	0.675619	+	0.737251i	$0.263877\pi$
$228$	0	0
$229$	5.34105e7	0.293902	0.146951	−	0.989144i	$-0.453054\pi$
$229$	5.34105e7	0.293902	0.146951	+	0.989144i	$0.453054\pi$
$230$	0	0
$231$	9.22664e7	0.492495
$232$	0	0
$233$	−2.32484e8	−1.20406	−0.602028	−	0.798475i	$-0.705640\pi$
$233$	−2.32484e8	−1.20406	−0.602028	+	0.798475i	$0.705640\pi$
$234$	0	0
$235$	2.64359e8	1.32879
$236$	0	0
$237$	2.16778e8	1.05778
$238$	0	0
$239$	−2.35251e8	−1.11465	−0.557326	−	0.830294i	$-0.688173\pi$
$239$	−2.35251e8	−1.11465	−0.557326	+	0.830294i	$0.688173\pi$
$240$	0	0
$241$	−3.44508e8	−1.58540	−0.792701	−	0.609610i	$-0.791326\pi$
$241$	−3.44508e8	−1.58540	−0.792701	+	0.609610i	$0.791326\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−2.90578e8	−1.26236
$246$	0	0
$247$	−1.68474e8	−0.711367
$248$	0	0
$249$	−9.60242e7	−0.394170
$250$	0	0
$251$	3.57037e7	0.142513	0.0712567	−	0.997458i	$-0.477299\pi$
$251$	3.57037e7	0.142513	0.0712567	+	0.997458i	$0.477299\pi$
$252$	0	0
$253$	−7.71238e7	−0.299410
$254$	0	0
$255$	−1.33002e8	−0.502305
$256$	0	0
$257$	1.77388e8	0.651866	0.325933	−	0.945393i	$-0.394322\pi$
$257$	1.77388e8	0.651866	0.325933	+	0.945393i	$0.394322\pi$
$258$	0	0
$259$	3.31149e8	1.18433
$260$	0	0
$261$	1.53757e8	0.535295
$262$	0	0
$263$	−2.51237e8	−0.851604	−0.425802	−	0.904816i	$-0.640008\pi$
$263$	−2.51237e8	−0.851604	−0.425802	+	0.904816i	$0.640008\pi$
$264$	0	0
$265$	7.79885e8	2.57436
$266$	0	0
$267$	1.40812e8	0.452742
$268$	0	0
$269$	−3.42878e8	−1.07400	−0.537002	−	0.843581i	$-0.680443\pi$
$269$	−3.42878e8	−1.07400	−0.537002	+	0.843581i	$0.680443\pi$
$270$	0	0
$271$	−4.49518e8	−1.37200	−0.686000	−	0.727601i	$-0.740635\pi$
$271$	−4.49518e8	−1.37200	−0.686000	+	0.727601i	$0.740635\pi$
$272$	0	0
$273$	1.23708e8	0.367984
$274$	0	0
$275$	−2.81667e8	−0.816716
$276$	0	0
$277$	2.29030e8	0.647461	0.323730	−	0.946149i	$-0.395063\pi$
$277$	2.29030e8	0.647461	0.323730	+	0.946149i	$0.395063\pi$
$278$	0	0
$279$	1.25809e8	0.346814
$280$	0	0
$281$	−1.49209e8	−0.401165	−0.200582	−	0.979677i	$-0.564283\pi$
$281$	−1.49209e8	−0.401165	−0.200582	+	0.979677i	$0.564283\pi$
$282$	0	0
$283$	−4.05047e8	−1.06231	−0.531157	−	0.847273i	$-0.678243\pi$
$283$	−4.05047e8	−1.06231	−0.531157	+	0.847273i	$0.678243\pi$
$284$	0	0
$285$	5.16653e8	1.32203
$286$	0	0
$287$	−4.66487e8	−1.16480
$288$	0	0
$289$	−2.75077e8	−0.670365
$290$	0	0
$291$	1.91012e8	0.454397
$292$	0	0
$293$	−5.59019e7	−0.129834	−0.0649172	−	0.997891i	$-0.520678\pi$
$293$	−5.59019e7	−0.129834	−0.0649172	+	0.997891i	$0.520678\pi$
$294$	0	0
$295$	1.16366e9	2.63905
$296$	0	0
$297$	5.47445e7	0.121253
$298$	0	0
$299$	−1.03405e8	−0.223715
$300$	0	0
$301$	−1.26957e9	−2.68333
$302$	0	0
$303$	−1.69354e8	−0.349740
$304$	0	0
$305$	−1.14439e9	−2.30953
$306$	0	0
$307$	−7.13287e8	−1.40695	−0.703477	−	0.710718i	$-0.748370\pi$
$307$	−7.13287e8	−1.40695	−0.703477	+	0.710718i	$0.748370\pi$
$308$	0	0
$309$	3.30027e8	0.636347
$310$	0	0
$311$	2.32785e8	0.438827	0.219414	−	0.975632i	$-0.429586\pi$
$311$	2.32785e8	0.438827	0.219414	+	0.975632i	$0.429586\pi$
$312$	0	0
$313$	1.35376e8	0.249538	0.124769	−	0.992186i	$-0.460181\pi$
$313$	1.35376e8	0.249538	0.124769	+	0.992186i	$0.460181\pi$
$314$	0	0
$315$	−3.79371e8	−0.683876
$316$	0	0
$317$	5.29613e8	0.933794	0.466897	−	0.884312i	$-0.345372\pi$
$317$	5.29613e8	0.933794	0.466897	+	0.884312i	$0.345372\pi$
$318$	0	0
$319$	−5.86621e8	−1.01179
$320$	0	0
$321$	2.35997e8	0.398235
$322$	0	0
$323$	−5.25432e8	−0.867576
$324$	0	0
$325$	−3.77651e8	−0.610237
$326$	0	0
$327$	1.50402e8	0.237868
$328$	0	0
$329$	−7.66863e8	−1.18722
$330$	0	0
$331$	9.24935e8	1.40189	0.700944	−	0.713217i	$-0.252762\pi$
$331$	9.24935e8	1.40189	0.700944	+	0.713217i	$0.252762\pi$
$332$	0	0
$333$	1.96481e8	0.291586
$334$	0	0
$335$	4.18917e8	0.608795
$336$	0	0
$337$	1.02943e9	1.46518	0.732592	−	0.680668i	$-0.238311\pi$
$337$	1.02943e9	1.46518	0.732592	+	0.680668i	$0.238311\pi$
$338$	0	0
$339$	−3.38057e8	−0.471292
$340$	0	0
$341$	−4.79992e8	−0.655531
$342$	0	0
$343$	−1.68930e8	−0.226036
$344$	0	0
$345$	3.17110e8	0.415760
$346$	0	0
$347$	−3.00011e8	−0.385464	−0.192732	−	0.981251i	$-0.561735\pi$
$347$	−3.00011e8	−0.385464	−0.192732	+	0.981251i	$0.561735\pi$
$348$	0	0
$349$	5.43096e8	0.683893	0.341946	−	0.939719i	$-0.388914\pi$
$349$	5.43096e8	0.683893	0.341946	+	0.939719i	$0.388914\pi$
$350$	0	0
$351$	7.33999e7	0.0905985
$352$	0	0
$353$	−3.10616e8	−0.375849	−0.187924	−	0.982184i	$-0.560176\pi$
$353$	−3.10616e8	−0.375849	−0.187924	+	0.982184i	$0.560176\pi$
$354$	0	0
$355$	−1.40055e9	−1.66150
$356$	0	0
$357$	3.85817e8	0.448790
$358$	0	0
$359$	−2.00225e8	−0.228395	−0.114198	−	0.993458i	$-0.536430\pi$
$359$	−2.00225e8	−0.228395	−0.114198	+	0.993458i	$0.536430\pi$
$360$	0	0
$361$	1.14719e9	1.28340
$362$	0	0
$363$	3.17290e8	0.348163
$364$	0	0
$365$	4.67073e8	0.502759
$366$	0	0
$367$	−7.09899e8	−0.749662	−0.374831	−	0.927093i	$-0.622299\pi$
$367$	−7.09899e8	−0.749662	−0.374831	+	0.927093i	$0.622299\pi$
$368$	0	0
$369$	−2.76782e8	−0.286777
$370$	0	0
$371$	−2.26232e9	−2.30009
$372$	0	0
$373$	−1.90313e8	−0.189883	−0.0949416	−	0.995483i	$-0.530266\pi$
$373$	−1.90313e8	−0.189883	−0.0949416	+	0.995483i	$0.530266\pi$
$374$	0	0
$375$	2.64697e8	0.259203
$376$	0	0
$377$	−7.86525e8	−0.755993
$378$	0	0
$379$	−1.52757e9	−1.44133	−0.720665	−	0.693284i	$-0.756163\pi$
$379$	−1.52757e9	−1.44133	−0.720665	+	0.693284i	$0.756163\pi$
$380$	0	0
$381$	4.33446e8	0.401511
$382$	0	0
$383$	4.22576e8	0.384334	0.192167	−	0.981362i	$-0.438448\pi$
$383$	4.22576e8	0.384334	0.192167	+	0.981362i	$0.438448\pi$
$384$	0	0
$385$	1.44739e9	1.29263
$386$	0	0
$387$	−7.53275e8	−0.660640
$388$	0	0
$389$	−8.37610e8	−0.721470	−0.360735	−	0.932668i	$-0.617474\pi$
$389$	−8.37610e8	−0.721470	−0.360735	+	0.932668i	$0.617474\pi$
$390$	0	0
$391$	−3.22498e8	−0.272840
$392$	0	0
$393$	−4.14698e8	−0.344634
$394$	0	0
$395$	3.40062e9	2.77631
$396$	0	0
$397$	1.46607e9	1.17595	0.587973	−	0.808881i	$-0.299926\pi$
$397$	1.46607e9	1.17595	0.587973	+	0.808881i	$0.299926\pi$
$398$	0	0
$399$	−1.49873e9	−1.18118
$400$	0	0
$401$	1.61531e9	1.25098	0.625489	−	0.780233i	$-0.284899\pi$
$401$	1.61531e9	1.25098	0.625489	+	0.780233i	$0.284899\pi$
$402$	0	0
$403$	−6.43559e8	−0.489802
$404$	0	0
$405$	−2.25093e8	−0.168372
$406$	0	0
$407$	−7.49623e8	−0.551141
$408$	0	0
$409$	−1.00624e9	−0.727230	−0.363615	−	0.931549i	$-0.618458\pi$
$409$	−1.00624e9	−0.727230	−0.363615	+	0.931549i	$0.618458\pi$
$410$	0	0
$411$	−5.66862e8	−0.402746
$412$	0	0
$413$	−3.37558e9	−2.35789
$414$	0	0
$415$	−1.50634e9	−1.03456
$416$	0	0
$417$	−1.07771e9	−0.727822
$418$	0	0
$419$	−3.49577e8	−0.232164	−0.116082	−	0.993240i	$-0.537033\pi$
$419$	−3.49577e8	−0.232164	−0.116082	+	0.993240i	$0.537033\pi$
$420$	0	0
$421$	7.71712e8	0.504043	0.252022	−	0.967722i	$-0.418905\pi$
$421$	7.71712e8	0.504043	0.252022	+	0.967722i	$0.418905\pi$
$422$	0	0
$423$	−4.55004e8	−0.292297
$424$	0	0
$425$	−1.17781e9	−0.744239
$426$	0	0
$427$	3.31968e9	2.06347
$428$	0	0
$429$	−2.80039e8	−0.171245
$430$	0	0
$431$	−2.24587e9	−1.35118	−0.675592	−	0.737276i	$-0.736112\pi$
$431$	−2.24587e9	−1.35118	−0.675592	+	0.737276i	$0.736112\pi$
$432$	0	0
$433$	−1.16044e8	−0.0686937	−0.0343468	−	0.999410i	$-0.510935\pi$
$433$	−1.16044e8	−0.0686937	−0.0343468	+	0.999410i	$0.510935\pi$
$434$	0	0
$435$	2.41200e9	1.40497
$436$	0	0
$437$	1.25276e9	0.718096
$438$	0	0
$439$	2.46503e9	1.39058	0.695290	−	0.718729i	$-0.255276\pi$
$439$	2.46503e9	1.39058	0.695290	+	0.718729i	$0.255276\pi$
$440$	0	0
$441$	5.00131e8	0.277683
$442$	0	0
$443$	2.71781e9	1.48527	0.742635	−	0.669696i	$-0.233576\pi$
$443$	2.71781e9	1.48527	0.742635	+	0.669696i	$0.233576\pi$
$444$	0	0
$445$	2.20894e9	1.18829
$446$	0	0
$447$	1.56474e9	0.828637
$448$	0	0
$449$	−3.31427e9	−1.72793	−0.863964	−	0.503553i	$-0.832026\pi$
$449$	−3.31427e9	−1.72793	−0.863964	+	0.503553i	$0.832026\pi$
$450$	0	0
$451$	1.05599e9	0.542052
$452$	0	0
$453$	−2.70491e8	−0.136713
$454$	0	0
$455$	1.94062e9	0.965831
$456$	0	0
$457$	−1.13520e9	−0.556374	−0.278187	−	0.960527i	$-0.589733\pi$
$457$	−1.13520e9	−0.556374	−0.278187	+	0.960527i	$0.589733\pi$
$458$	0	0
$459$	2.28918e8	0.110493
$460$	0	0
$461$	−1.89698e9	−0.901799	−0.450900	−	0.892575i	$-0.648897\pi$
$461$	−1.89698e9	−0.901799	−0.450900	+	0.892575i	$0.648897\pi$
$462$	0	0
$463$	1.13818e9	0.532940	0.266470	−	0.963843i	$-0.414143\pi$
$463$	1.13818e9	0.532940	0.266470	+	0.963843i	$0.414143\pi$
$464$	0	0
$465$	1.97358e9	0.910268
$466$	0	0
$467$	−2.10626e9	−0.956978	−0.478489	−	0.878094i	$-0.658815\pi$
$467$	−2.10626e9	−0.956978	−0.478489	+	0.878094i	$0.658815\pi$
$468$	0	0
$469$	−1.21521e9	−0.543935
$470$	0	0
$471$	5.08935e8	0.224434
$472$	0	0
$473$	2.87393e9	1.24871
$474$	0	0
$475$	4.57524e9	1.95878
$476$	0	0
$477$	−1.34231e9	−0.566288
$478$	0	0
$479$	−1.63874e8	−0.0681295	−0.0340648	−	0.999420i	$-0.510845\pi$
$479$	−1.63874e8	−0.0681295	−0.0340648	+	0.999420i	$0.510845\pi$
$480$	0	0
$481$	−1.00507e9	−0.411804
$482$	0	0
$483$	−9.19884e8	−0.371465
$484$	0	0
$485$	2.99643e9	1.19263
$486$	0	0
$487$	−4.01594e8	−0.157556	−0.0787781	−	0.996892i	$-0.525102\pi$
$487$	−4.01594e8	−0.157556	−0.0787781	+	0.996892i	$0.525102\pi$
$488$	0	0
$489$	2.15668e9	0.834073
$490$	0	0
$491$	−2.05766e9	−0.784490	−0.392245	−	0.919861i	$-0.628301\pi$
$491$	−2.05766e9	−0.784490	−0.392245	+	0.919861i	$0.628301\pi$
$492$	0	0
$493$	−2.45299e9	−0.922001
$494$	0	0
$495$	8.58783e8	0.318248
$496$	0	0
$497$	4.06277e9	1.48448
$498$	0	0
$499$	8.05327e8	0.290148	0.145074	−	0.989421i	$-0.453658\pi$
$499$	8.05327e8	0.290148	0.145074	+	0.989421i	$0.453658\pi$
$500$	0	0
$501$	−5.94104e8	−0.211072
$502$	0	0
$503$	1.32490e9	0.464189	0.232094	−	0.972693i	$-0.425442\pi$
$503$	1.32490e9	0.464189	0.232094	+	0.972693i	$0.425442\pi$
$504$	0	0
$505$	−2.65667e9	−0.917948
$506$	0	0
$507$	1.31874e9	0.449399
$508$	0	0
$509$	2.83357e9	0.952406	0.476203	−	0.879335i	$-0.342013\pi$
$509$	2.83357e9	0.952406	0.476203	+	0.879335i	$0.342013\pi$
$510$	0	0
$511$	−1.35490e9	−0.449195
$512$	0	0
$513$	−8.89242e8	−0.290810
$514$	0	0
$515$	5.17716e9	1.67019
$516$	0	0
$517$	1.73595e9	0.552485
$518$	0	0
$519$	1.84606e9	0.579643
$520$	0	0
$521$	−1.08648e8	−0.0336580	−0.0168290	−	0.999858i	$-0.505357\pi$
$521$	−1.08648e8	−0.0336580	−0.0168290	+	0.999858i	$0.505357\pi$
$522$	0	0
$523$	−5.09356e9	−1.55692	−0.778458	−	0.627696i	$-0.783998\pi$
$523$	−5.09356e9	−1.55692	−0.778458	+	0.627696i	$0.783998\pi$
$524$	0	0
$525$	−3.35954e9	−1.01326
$526$	0	0
$527$	−2.00711e9	−0.597358
$528$	0	0
$529$	−2.63591e9	−0.774169
$530$	0	0
$531$	−2.00284e9	−0.580518
$532$	0	0
$533$	1.41584e9	0.405013
$534$	0	0
$535$	3.70212e9	1.04523
$536$	0	0
$537$	1.60914e9	0.448418
$538$	0	0
$539$	−1.90812e9	−0.524862
$540$	0	0
$541$	3.98616e9	1.08234	0.541171	−	0.840913i	$-0.317981\pi$
$541$	3.98616e9	1.08234	0.541171	+	0.840913i	$0.317981\pi$
$542$	0	0
$543$	−4.08464e9	−1.09485
$544$	0	0
$545$	2.35937e9	0.624322
$546$	0	0
$547$	−4.49694e9	−1.17479	−0.587397	−	0.809299i	$-0.699847\pi$
$547$	−4.49694e9	−1.17479	−0.587397	+	0.809299i	$0.699847\pi$
$548$	0	0
$549$	1.96967e9	0.508031
$550$	0	0
$551$	9.52876e9	2.42664
$552$	0	0
$553$	−9.86464e9	−2.48052
$554$	0	0
$555$	3.08222e9	0.765312
$556$	0	0
$557$	−6.10617e9	−1.49719	−0.748593	−	0.663030i	$-0.769270\pi$
$557$	−6.10617e9	−1.49719	−0.748593	+	0.663030i	$0.769270\pi$
$558$	0	0
$559$	3.85328e9	0.933016
$560$	0	0
$561$	−8.73376e8	−0.208848
$562$	0	0
$563$	1.38158e9	0.326284	0.163142	−	0.986603i	$-0.447837\pi$
$563$	1.38158e9	0.326284	0.163142	+	0.986603i	$0.447837\pi$
$564$	0	0
$565$	−5.30313e9	−1.23698
$566$	0	0
$567$	6.52958e8	0.150433
$568$	0	0
$569$	8.73612e9	1.98804	0.994021	−	0.109188i	$-0.0348251\pi$
$569$	8.73612e9	1.98804	0.994021	+	0.109188i	$0.0348251\pi$
$570$	0	0
$571$	−1.33535e9	−0.300170	−0.150085	−	0.988673i	$-0.547955\pi$
$571$	−1.33535e9	−0.300170	−0.150085	+	0.988673i	$0.547955\pi$
$572$	0	0
$573$	2.33970e8	0.0519540
$574$	0	0
$575$	2.80818e9	0.616009
$576$	0	0
$577$	5.50143e8	0.119223	0.0596115	−	0.998222i	$-0.481014\pi$
$577$	5.50143e8	0.119223	0.0596115	+	0.998222i	$0.481014\pi$
$578$	0	0
$579$	−4.08544e9	−0.874709
$580$	0	0
$581$	4.36966e9	0.924338
$582$	0	0
$583$	5.12122e9	1.07037
$584$	0	0
$585$	1.15143e9	0.237790
$586$	0	0
$587$	−3.96942e9	−0.810017	−0.405008	−	0.914313i	$-0.632731\pi$
$587$	−3.96942e9	−0.810017	−0.405008	+	0.914313i	$0.632731\pi$
$588$	0	0
$589$	7.79673e9	1.57220
$590$	0	0
$591$	2.63815e9	0.525706
$592$	0	0
$593$	2.66508e9	0.524830	0.262415	−	0.964955i	$-0.415481\pi$
$593$	2.66508e9	0.524830	0.262415	+	0.964955i	$0.415481\pi$
$594$	0	0
$595$	6.05236e9	1.17792
$596$	0	0
$597$	−8.62709e8	−0.165941
$598$	0	0
$599$	1.20668e7	0.00229403	0.00114701	−	0.999999i	$-0.499635\pi$
$599$	1.20668e7	0.00229403	0.00114701	+	0.999999i	$0.499635\pi$
$600$	0	0
$601$	1.12678e9	0.211728	0.105864	−	0.994381i	$-0.466239\pi$
$601$	1.12678e9	0.211728	0.105864	+	0.994381i	$0.466239\pi$
$602$	0	0
$603$	−7.21023e8	−0.133918
$604$	0	0
$605$	4.97736e9	0.913809
$606$	0	0
$607$	−3.33232e9	−0.604765	−0.302382	−	0.953187i	$-0.597782\pi$
$607$	−3.33232e9	−0.604765	−0.302382	+	0.953187i	$0.597782\pi$
$608$	0	0
$609$	−6.99684e9	−1.25528
$610$	0	0
$611$	2.32751e9	0.412808
$612$	0	0
$613$	−4.28653e9	−0.751612	−0.375806	−	0.926698i	$-0.622634\pi$
$613$	−4.28653e9	−0.751612	−0.375806	+	0.926698i	$0.622634\pi$
$614$	0	0
$615$	−4.34190e9	−0.752692
$616$	0	0
$617$	3.82726e9	0.655978	0.327989	−	0.944681i	$-0.393629\pi$
$617$	3.82726e9	0.655978	0.327989	+	0.944681i	$0.393629\pi$
$618$	0	0
$619$	1.89529e9	0.321187	0.160594	−	0.987021i	$-0.448659\pi$
$619$	1.89529e9	0.321187	0.160594	+	0.987021i	$0.448659\pi$
$620$	0	0
$621$	−5.45796e8	−0.0914555
$622$	0	0
$623$	−6.40777e9	−1.06169
$624$	0	0
$625$	−3.75948e9	−0.615953
$626$	0	0
$627$	3.39267e9	0.549674
$628$	0	0
$629$	−3.13459e9	−0.502232
$630$	0	0
$631$	−1.20090e10	−1.90285	−0.951424	−	0.307882i	$-0.900380\pi$
$631$	−1.20090e10	−1.90285	−0.951424	+	0.307882i	$0.900380\pi$
$632$	0	0
$633$	−9.39029e8	−0.147152
$634$	0	0
$635$	6.79951e9	1.05383
$636$	0	0
$637$	−2.55836e9	−0.392169
$638$	0	0
$639$	2.41057e9	0.365483
$640$	0	0
$641$	3.33728e9	0.500484	0.250242	−	0.968183i	$-0.419490\pi$
$641$	3.33728e9	0.500484	0.250242	+	0.968183i	$0.419490\pi$
$642$	0	0
$643$	−7.80682e9	−1.15807	−0.579037	−	0.815302i	$-0.696571\pi$
$643$	−7.80682e9	−1.15807	−0.579037	+	0.815302i	$0.696571\pi$
$644$	0	0
$645$	−1.18167e10	−1.73395
$646$	0	0
$647$	−4.81228e9	−0.698531	−0.349265	−	0.937024i	$-0.613569\pi$
$647$	−4.81228e9	−0.698531	−0.349265	+	0.937024i	$0.613569\pi$
$648$	0	0
$649$	7.64132e9	1.09727
$650$	0	0
$651$	−5.72503e9	−0.813288
$652$	0	0
$653$	4.16908e9	0.585927	0.292964	−	0.956124i	$-0.405359\pi$
$653$	4.16908e9	0.585927	0.292964	+	0.956124i	$0.405359\pi$
$654$	0	0
$655$	−6.50541e9	−0.904545
$656$	0	0
$657$	−8.03907e8	−0.110593
$658$	0	0
$659$	−1.02165e10	−1.39060	−0.695299	−	0.718721i	$-0.744728\pi$
$659$	−1.02165e10	−1.39060	−0.695299	+	0.718721i	$0.744728\pi$
$660$	0	0
$661$	−1.41924e9	−0.191139	−0.0955697	−	0.995423i	$-0.530467\pi$
$661$	−1.41924e9	−0.191139	−0.0955697	+	0.995423i	$0.530467\pi$
$662$	0	0
$663$	−1.17100e9	−0.156048
$664$	0	0
$665$	−2.35107e10	−3.10020
$666$	0	0
$667$	5.84853e9	0.763144
$668$	0	0
$669$	−5.56408e8	−0.0718459
$670$	0	0
$671$	−7.51476e9	−0.960255
$672$	0	0
$673$	8.90621e9	1.12626	0.563132	−	0.826367i	$-0.309596\pi$
$673$	8.90621e9	1.12626	0.563132	+	0.826367i	$0.309596\pi$
$674$	0	0
$675$	−1.99332e9	−0.249467
$676$	0	0
$677$	3.21191e9	0.397834	0.198917	−	0.980016i	$-0.436258\pi$
$677$	3.21191e9	0.397834	0.198917	+	0.980016i	$0.436258\pi$
$678$	0	0
$679$	−8.69215e9	−1.06557
$680$	0	0
$681$	−6.42963e9	−0.780137
$682$	0	0
$683$	−1.49063e10	−1.79018	−0.895091	−	0.445883i	$-0.852890\pi$
$683$	−1.49063e10	−1.79018	−0.895091	+	0.445883i	$0.852890\pi$
$684$	0	0
$685$	−8.89242e9	−1.05707
$686$	0	0
$687$	−1.44208e9	−0.169684
$688$	0	0
$689$	6.86639e9	0.799762
$690$	0	0
$691$	−3.38129e9	−0.389860	−0.194930	−	0.980817i	$-0.562448\pi$
$691$	−3.38129e9	−0.389860	−0.194930	+	0.980817i	$0.562448\pi$
$692$	0	0
$693$	−2.49119e9	−0.284342
$694$	0	0
$695$	−1.69061e10	−1.91028
$696$	0	0
$697$	4.41568e9	0.493950
$698$	0	0
$699$	6.27706e9	0.695162
$700$	0	0
$701$	1.19330e10	1.30839	0.654195	−	0.756326i	$-0.273008\pi$
$701$	1.19330e10	1.30839	0.654195	+	0.756326i	$0.273008\pi$
$702$	0	0
$703$	1.21765e10	1.32184
$704$	0	0
$705$	−7.13769e9	−0.767178
$706$	0	0
$707$	7.70657e9	0.820150
$708$	0	0
$709$	−1.92709e9	−0.203068	−0.101534	−	0.994832i	$-0.532375\pi$
$709$	−1.92709e9	−0.203068	−0.101534	+	0.994832i	$0.532375\pi$
$710$	0	0
$711$	−5.85300e9	−0.610710
$712$	0	0
$713$	4.78546e9	0.494436
$714$	0	0
$715$	−4.39299e9	−0.449458
$716$	0	0
$717$	6.35179e9	0.643545
$718$	0	0
$719$	5.16235e9	0.517960	0.258980	−	0.965883i	$-0.416614\pi$
$719$	5.16235e9	0.517960	0.258980	+	0.965883i	$0.416614\pi$
$720$	0	0
$721$	−1.50181e10	−1.49225
$722$	0	0
$723$	9.30171e9	0.915333
$724$	0	0
$725$	2.13596e10	2.08166
$726$	0	0
$727$	−1.38758e10	−1.33933	−0.669667	−	0.742662i	$-0.733563\pi$
$727$	−1.38758e10	−1.33933	−0.669667	+	0.742662i	$0.733563\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.20175e10	1.13790
$732$	0	0
$733$	−3.26388e9	−0.306104	−0.153052	−	0.988218i	$-0.548910\pi$
$733$	−3.26388e9	−0.306104	−0.153052	+	0.988218i	$0.548910\pi$
$734$	0	0
$735$	7.84561e9	0.728822
$736$	0	0
$737$	2.75088e9	0.253125
$738$	0	0
$739$	5.49185e8	0.0500568	0.0250284	−	0.999687i	$-0.492032\pi$
$739$	5.49185e8	0.0500568	0.0250284	+	0.999687i	$0.492032\pi$
$740$	0	0
$741$	4.54880e9	0.410708
$742$	0	0
$743$	−1.11208e10	−0.994663	−0.497332	−	0.867561i	$-0.665687\pi$
$743$	−1.11208e10	−0.994663	−0.497332	+	0.867561i	$0.665687\pi$
$744$	0	0
$745$	2.45462e10	2.17489
$746$	0	0
$747$	2.59265e9	0.227574
$748$	0	0
$749$	−1.07392e10	−0.933872
$750$	0	0
$751$	5.06853e9	0.436659	0.218329	−	0.975875i	$-0.429939\pi$
$751$	5.06853e9	0.436659	0.218329	+	0.975875i	$0.429939\pi$
$752$	0	0
$753$	−9.64001e8	−0.0822801
$754$	0	0
$755$	−4.24322e9	−0.358824
$756$	0	0
$757$	−5.64424e9	−0.472901	−0.236450	−	0.971644i	$-0.575984\pi$
$757$	−5.64424e9	−0.472901	−0.236450	+	0.971644i	$0.575984\pi$
$758$	0	0
$759$	2.08234e9	0.172865
$760$	0	0
$761$	1.57769e10	1.29770	0.648850	−	0.760916i	$-0.275250\pi$
$761$	1.57769e10	1.29770	0.648850	+	0.760916i	$0.275250\pi$
$762$	0	0
$763$	−6.84416e9	−0.557807
$764$	0	0
$765$	3.59105e9	0.290006
$766$	0	0
$767$	1.02453e10	0.819859
$768$	0	0
$769$	1.05126e10	0.833616	0.416808	−	0.908994i	$-0.363149\pi$
$769$	1.05126e10	0.833616	0.416808	+	0.908994i	$0.363149\pi$
$770$	0	0
$771$	−4.78947e9	−0.376355
$772$	0	0
$773$	4.70533e9	0.366405	0.183203	−	0.983075i	$-0.441354\pi$
$773$	4.70533e9	0.366405	0.183203	+	0.983075i	$0.441354\pi$
$774$	0	0
$775$	1.74771e10	1.34870
$776$	0	0
$777$	−8.94103e9	−0.683776
$778$	0	0
$779$	−1.71529e10	−1.30004
$780$	0	0
$781$	−9.19691e9	−0.690818
$782$	0	0
$783$	−4.15144e9	−0.309053
$784$	0	0
$785$	7.98372e9	0.589063
$786$	0	0
$787$	2.07916e10	1.52046	0.760232	−	0.649652i	$-0.225085\pi$
$787$	2.07916e10	1.52046	0.760232	+	0.649652i	$0.225085\pi$
$788$	0	0
$789$	6.78339e9	0.491674
$790$	0	0
$791$	1.53835e10	1.10519
$792$	0	0
$793$	−1.00756e10	−0.717487
$794$	0	0
$795$	−2.10569e10	−1.48631
$796$	0	0
$797$	−1.09035e10	−0.762890	−0.381445	−	0.924392i	$-0.624573\pi$
$797$	−1.09035e10	−0.762890	−0.381445	+	0.924392i	$0.624573\pi$
$798$	0	0
$799$	7.25897e9	0.503456
$800$	0	0
$801$	−3.80193e9	−0.261391
$802$	0	0
$803$	3.06710e9	0.209037
$804$	0	0
$805$	−1.44303e10	−0.974968
$806$	0	0
$807$	9.25769e9	0.620077
$808$	0	0
$809$	−1.20873e10	−0.802620	−0.401310	−	0.915942i	$-0.631445\pi$
$809$	−1.20873e10	−0.802620	−0.401310	+	0.915942i	$0.631445\pi$
$810$	0	0
$811$	−1.39636e10	−0.919229	−0.459615	−	0.888118i	$-0.652012\pi$
$811$	−1.39636e10	−0.919229	−0.459615	+	0.888118i	$0.652012\pi$
$812$	0	0
$813$	1.21370e10	0.792125
$814$	0	0
$815$	3.38320e10	2.18915
$816$	0	0
$817$	−4.66825e10	−2.99486
$818$	0	0
$819$	−3.34012e9	−0.212456
$820$	0	0
$821$	−6.90599e9	−0.435537	−0.217769	−	0.976000i	$-0.569878\pi$
$821$	−6.90599e9	−0.435537	−0.217769	+	0.976000i	$0.569878\pi$
$822$	0	0
$823$	−1.31839e9	−0.0824413	−0.0412206	−	0.999150i	$-0.513125\pi$
$823$	−1.31839e9	−0.0824413	−0.0412206	+	0.999150i	$0.513125\pi$
$824$	0	0
$825$	7.60500e9	0.471531
$826$	0	0
$827$	1.16284e10	0.714907	0.357454	−	0.933931i	$-0.383645\pi$
$827$	1.16284e10	0.714907	0.357454	+	0.933931i	$0.383645\pi$
$828$	0	0
$829$	−1.53849e10	−0.937893	−0.468947	−	0.883226i	$-0.655366\pi$
$829$	−1.53849e10	−0.937893	−0.468947	+	0.883226i	$0.655366\pi$
$830$	0	0
$831$	−6.18382e9	−0.373812
$832$	0	0
$833$	−7.97892e9	−0.478285
$834$	0	0
$835$	−9.31978e9	−0.553991
$836$	0	0
$837$	−3.39684e9	−0.200233
$838$	0	0
$839$	4.92329e9	0.287799	0.143899	−	0.989592i	$-0.454036\pi$
$839$	4.92329e9	0.287799	0.143899	+	0.989592i	$0.454036\pi$
$840$	0	0
$841$	2.72353e10	1.57887
$842$	0	0
$843$	4.02864e9	0.231613
$844$	0	0
$845$	2.06872e10	1.17952
$846$	0	0
$847$	−1.44385e10	−0.816452
$848$	0	0
$849$	1.09363e10	0.613327
$850$	0	0
$851$	7.47365e9	0.415699
$852$	0	0
$853$	7.54820e9	0.416411	0.208205	−	0.978085i	$-0.433238\pi$
$853$	7.54820e9	0.416411	0.208205	+	0.978085i	$0.433238\pi$
$854$	0	0
$855$	−1.39496e10	−0.763275
$856$	0	0
$857$	−1.27814e10	−0.693656	−0.346828	−	0.937929i	$-0.612741\pi$
$857$	−1.27814e10	−0.693656	−0.346828	+	0.937929i	$0.612741\pi$
$858$	0	0
$859$	−1.33691e10	−0.719657	−0.359829	−	0.933018i	$-0.617165\pi$
$859$	−1.33691e10	−0.719657	−0.359829	+	0.933018i	$0.617165\pi$
$860$	0	0
$861$	1.25952e10	0.672500
$862$	0	0
$863$	9.25958e9	0.490404	0.245202	−	0.969472i	$-0.421146\pi$
$863$	9.25958e9	0.490404	0.245202	+	0.969472i	$0.421146\pi$
$864$	0	0
$865$	2.89594e10	1.52136
$866$	0	0
$867$	7.42707e9	0.387036
$868$	0	0
$869$	2.23306e10	1.15433
$870$	0	0
$871$	3.68830e9	0.189131
$872$	0	0
$873$	−5.15733e9	−0.262346
$874$	0	0
$875$	−1.20452e10	−0.607837
$876$	0	0
$877$	−1.40971e10	−0.705717	−0.352859	−	0.935677i	$-0.614790\pi$
$877$	−1.40971e10	−0.705717	−0.352859	+	0.935677i	$0.614790\pi$
$878$	0	0
$879$	1.50935e9	0.0749600
$880$	0	0
$881$	1.97871e10	0.974914	0.487457	−	0.873147i	$-0.337925\pi$
$881$	1.97871e10	0.974914	0.487457	+	0.873147i	$0.337925\pi$
$882$	0	0
$883$	3.39322e10	1.65863	0.829314	−	0.558782i	$-0.188731\pi$
$883$	3.39322e10	1.65863	0.829314	+	0.558782i	$0.188731\pi$
$884$	0	0
$885$	−3.14188e10	−1.52366
$886$	0	0
$887$	−3.05047e10	−1.46769	−0.733846	−	0.679316i	$-0.762276\pi$
$887$	−3.05047e10	−1.46769	−0.733846	+	0.679316i	$0.762276\pi$
$888$	0	0
$889$	−1.97243e10	−0.941554
$890$	0	0
$891$	−1.47810e9	−0.0700056
$892$	0	0
$893$	−2.81978e10	−1.32506
$894$	0	0
$895$	2.52427e10	1.17694
$896$	0	0
$897$	2.79195e9	0.129162
$898$	0	0
$899$	3.63992e10	1.67083
$900$	0	0
$901$	2.14147e10	0.975382
$902$	0	0
$903$	3.42783e10	1.54922
$904$	0	0
$905$	−6.40762e10	−2.87360
$906$	0	0
$907$	−6.76417e9	−0.301016	−0.150508	−	0.988609i	$-0.548091\pi$
$907$	−6.76417e9	−0.301016	−0.150508	+	0.988609i	$0.548091\pi$
$908$	0	0
$909$	4.57255e9	0.201923
$910$	0	0
$911$	1.14881e10	0.503424	0.251712	−	0.967802i	$-0.419006\pi$
$911$	1.14881e10	0.503424	0.251712	+	0.967802i	$0.419006\pi$
$912$	0	0
$913$	−9.89160e9	−0.430149
$914$	0	0
$915$	3.08984e10	1.33341
$916$	0	0
$917$	1.88711e10	0.808175
$918$	0	0
$919$	3.87986e10	1.64897	0.824483	−	0.565887i	$-0.191466\pi$
$919$	3.87986e10	1.64897	0.824483	+	0.565887i	$0.191466\pi$
$920$	0	0
$921$	1.92587e10	0.812305
$922$	0	0
$923$	−1.23310e10	−0.516168
$924$	0	0
$925$	2.72948e10	1.13392
$926$	0	0
$927$	−8.91072e9	−0.367395
$928$	0	0
$929$	4.14694e10	1.69696	0.848482	−	0.529224i	$-0.177517\pi$
$929$	4.14694e10	1.69696	0.848482	+	0.529224i	$0.177517\pi$
$930$	0	0
$931$	3.09945e10	1.25881
$932$	0	0
$933$	−6.28519e9	−0.253357
$934$	0	0
$935$	−1.37007e10	−0.548155
$936$	0	0
$937$	1.25279e10	0.497497	0.248748	−	0.968568i	$-0.419981\pi$
$937$	1.25279e10	0.497497	0.248748	+	0.968568i	$0.419981\pi$
$938$	0	0
$939$	−3.65515e9	−0.144071
$940$	0	0
$941$	−3.90647e10	−1.52834	−0.764171	−	0.645014i	$-0.776852\pi$
$941$	−3.90647e10	−1.52834	−0.764171	+	0.645014i	$0.776852\pi$
$942$	0	0
$943$	−1.05281e10	−0.408844
$944$	0	0
$945$	1.02430e10	0.394836
$946$	0	0
$947$	1.55171e10	0.593726	0.296863	−	0.954920i	$-0.404059\pi$
$947$	1.55171e10	0.593726	0.296863	+	0.954920i	$0.404059\pi$
$948$	0	0
$949$	4.11228e9	0.156189
$950$	0	0
$951$	−1.42995e10	−0.539126
$952$	0	0
$953$	−3.71024e10	−1.38860	−0.694299	−	0.719687i	$-0.744285\pi$
$953$	−3.71024e10	−1.38860	−0.694299	+	0.719687i	$0.744285\pi$
$954$	0	0
$955$	3.67031e9	0.136361
$956$	0	0
$957$	1.58388e10	0.584157
$958$	0	0
$959$	2.57955e10	0.944449
$960$	0	0
$961$	2.27038e9	0.0825214
$962$	0	0
$963$	−6.37193e9	−0.229921
$964$	0	0
$965$	−6.40887e10	−2.29581
$966$	0	0
$967$	−2.56321e10	−0.911574	−0.455787	−	0.890089i	$-0.650642\pi$
$967$	−2.56321e10	−0.911574	−0.455787	+	0.890089i	$0.650642\pi$
$968$	0	0
$969$	1.41867e10	0.500895
$970$	0	0
$971$	5.08869e10	1.78377	0.891884	−	0.452264i	$-0.149384\pi$
$971$	5.08869e10	1.78377	0.891884	+	0.452264i	$0.149384\pi$
$972$	0	0
$973$	4.90420e10	1.70676
$974$	0	0
$975$	1.01966e10	0.352320
$976$	0	0
$977$	1.68684e10	0.578686	0.289343	−	0.957225i	$-0.406563\pi$
$977$	1.68684e10	0.578686	0.289343	+	0.957225i	$0.406563\pi$
$978$	0	0
$979$	1.45053e10	0.494068
$980$	0	0
$981$	−4.06085e9	−0.137333
$982$	0	0
$983$	−1.61062e10	−0.540825	−0.270412	−	0.962745i	$-0.587160\pi$
$983$	−1.61062e10	−0.540825	−0.270412	+	0.962745i	$0.587160\pi$
$984$	0	0
$985$	4.13849e10	1.37980
$986$	0	0
$987$	2.07053e10	0.685443
$988$	0	0
$989$	−2.86527e10	−0.941842
$990$	0	0
$991$	−3.98303e9	−0.130004	−0.0650019	−	0.997885i	$-0.520705\pi$
$991$	−3.98303e9	−0.130004	−0.0650019	+	0.997885i	$0.520705\pi$
$992$	0	0
$993$	−2.49732e10	−0.809380
$994$	0	0
$995$	−1.35334e10	−0.435538
$996$	0	0
$997$	4.19886e10	1.34183	0.670917	−	0.741533i	$-0.265901\pi$
$997$	4.19886e10	1.34183	0.670917	+	0.741533i	$0.265901\pi$
$998$	0	0
$999$	−5.30499e9	−0.168347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.a.q.1.1		2
3.2	odd	2		576.8.a.bo.1.2		2
4.3	odd	2		192.8.a.t.1.1		2
8.3	odd	2		96.8.a.e.1.2	✓	2
8.5	even	2		96.8.a.h.1.2	yes	2
12.11	even	2		576.8.a.bn.1.2		2
24.5	odd	2		288.8.a.h.1.1		2
24.11	even	2		288.8.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.8.a.e.1.2	✓	2	8.3	odd	2
96.8.a.h.1.2	yes	2	8.5	even	2
192.8.a.q.1.1		2	1.1	even	1	trivial
192.8.a.t.1.1		2	4.3	odd	2
288.8.a.g.1.1		2	24.11	even	2
288.8.a.h.1.1		2	24.5	odd	2
576.8.a.bn.1.2		2	12.11	even	2
576.8.a.bo.1.2		2	3.2	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} -423.552 q^{5} +1228.66 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q-27.0000 q^{3} -423.552 q^{5} +1228.66 q^{7} +729.000 q^{9} -2781.31 q^{11} -3729.10 q^{13} +11435.9 q^{15} -11630.2 q^{17} +45178.2 q^{19} -33173.7 q^{21} +27729.3 q^{23} +101271. q^{25} -19683.0 q^{27} +210915. q^{29} +172577. q^{31} +75095.4 q^{33} -520399. q^{35} +269522. q^{37} +100686. q^{39} -379673. q^{41} -1.03330e6 q^{43} -308769. q^{45} -624148. q^{47} +686051. q^{49} +314016. q^{51} -1.84130e6 q^{53} +1.17803e6 q^{55} -1.21981e6 q^{57} -2.74738e6 q^{59} +2.70188e6 q^{61} +895690. q^{63} +1.57947e6 q^{65} -989057. q^{67} -748691. q^{69} +3.30668e6 q^{71} -1.10275e6 q^{73} -2.73432e6 q^{75} -3.41727e6 q^{77} -8.02881e6 q^{79} +531441. q^{81} +3.55645e6 q^{83} +4.92600e6 q^{85} -5.69471e6 q^{87} -5.21527e6 q^{89} -4.58178e6 q^{91} -4.65959e6 q^{93} -1.91353e7 q^{95} -7.07452e6 q^{97} -2.02758e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9	\( 2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9} - 1656 q^{11} - 6156 q^{13} + 5292 q^{15} + 17108 q^{17} + 504 q^{19} - 13608 q^{21} + 51552 q^{23} + 74926 q^{25} - 39366 q^{27} + 199804 q^{29} + 257256 q^{31} + 44712 q^{33} - 685296 q^{35} + 468724 q^{37} + 166212 q^{39} - 106940 q^{41} - 1617336 q^{43} - 142884 q^{45} + 646416 q^{47} + 387634 q^{49} - 461916 q^{51} - 1469492 q^{53} + 1434096 q^{55} - 13608 q^{57} - 4541544 q^{59} + 481412 q^{61} + 367416 q^{63} + 1027224 q^{65} - 4775256 q^{67} - 1391904 q^{69} - 1094400 q^{71} - 5731884 q^{73} - 2023002 q^{75} - 4232736 q^{77} - 10402776 q^{79} + 1062882 q^{81} - 2212200 q^{83} + 11465432 q^{85} - 5394708 q^{87} - 3604364 q^{89} - 2823120 q^{91} - 6945912 q^{93} - 29300976 q^{95} - 7156188 q^{97} - 1207224 q^{99}+O(q^{100}) \) 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9 - 1656 * q^11 - 6156 * q^13 + 5292 * q^15 + 17108 * q^17 + 504 * q^19 - 13608 * q^21 + 51552 * q^23 + 74926 * q^25 - 39366 * q^27 + 199804 * q^29 + 257256 * q^31 + 44712 * q^33 - 685296 * q^35 + 468724 * q^37 + 166212 * q^39 - 106940 * q^41 - 1617336 * q^43 - 142884 * q^45 + 646416 * q^47 + 387634 * q^49 - 461916 * q^51 - 1469492 * q^53 + 1434096 * q^55 - 13608 * q^57 - 4541544 * q^59 + 481412 * q^61 + 367416 * q^63 + 1027224 * q^65 - 4775256 * q^67 - 1391904 * q^69 - 1094400 * q^71 - 5731884 * q^73 - 2023002 * q^75 - 4232736 * q^77 - 10402776 * q^79 + 1062882 * q^81 - 2212200 * q^83 + 11465432 * q^85 - 5394708 * q^87 - 3604364 * q^89 - 2823120 * q^91 - 6945912 * q^93 - 29300976 * q^95 - 7156188 * q^97 - 1207224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more