LMFDB - Embedded newform 48.24.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,24,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 24, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	48.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:	$160.897937926$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 43285816x - 66694276650 $ x^3 - 43285816*x - 66694276650
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1643.31$ of defining polynomial
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-177147. q^{3} +1.22619e8 q^{5} +4.92683e9 q^{7} +3.13811e10 q^{9} +O(q^{10})$	$q-177147. q^{3} +1.22619e8 q^{5} +4.92683e9 q^{7} +3.13811e10 q^{9} -1.05178e11 q^{11} -4.97595e11 q^{13} -2.17216e13 q^{15} -3.26230e13 q^{17} +7.69259e14 q^{19} -8.72773e14 q^{21} +5.13304e15 q^{23} +3.11455e15 q^{25} -5.55906e15 q^{27} +7.47248e16 q^{29} +1.28452e17 q^{31} +1.86320e16 q^{33} +6.04124e17 q^{35} -8.87673e17 q^{37} +8.81474e16 q^{39} +2.40518e18 q^{41} +5.95568e18 q^{43} +3.84792e18 q^{45} -2.36340e19 q^{47} -3.09512e18 q^{49} +5.77906e18 q^{51} +3.46656e19 q^{53} -1.28968e19 q^{55} -1.36272e20 q^{57} -8.87272e19 q^{59} -4.63114e20 q^{61} +1.54609e20 q^{63} -6.10147e19 q^{65} -6.77287e20 q^{67} -9.09302e20 q^{69} -1.80462e19 q^{71} -8.89872e20 q^{73} -5.51734e20 q^{75} -5.18194e20 q^{77} +9.36842e21 q^{79} +9.84771e20 q^{81} +1.60161e22 q^{83} -4.00021e21 q^{85} -1.32373e22 q^{87} -1.20990e22 q^{89} -2.45156e21 q^{91} -2.27549e22 q^{93} +9.43260e22 q^{95} -7.53070e22 q^{97} -3.30060e21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 531441 q^{3} - 45012150 q^{5} - 281292072 q^{7} + 94143178827 q^{9}+O(q^{10}) $ 3 * q - 531441 * q^3 - 45012150 * q^5 - 281292072 * q^7 + 94143178827 * q^9	$ 3 q - 531441 q^{3} - 45012150 q^{5} - 281292072 q^{7} + 94143178827 q^{9} + 1198676057172 q^{11} - 282994829454 q^{13} + 7973767336050 q^{15} + 186934623333462 q^{17} + 118422337604076 q^{19} + 49830046678584 q^{21} + 63\!\cdots\!44 q^{23}+ \cdots + 37\!\cdots\!48 q^{99}+O(q^{100}) $ 3 * q - 531441 * q^3 - 45012150 * q^5 - 281292072 * q^7 + 94143178827 * q^9 + 1198676057172 * q^11 - 282994829454 * q^13 + 7973767336050 * q^15 + 186934623333462 * q^17 + 118422337604076 * q^19 + 49830046678584 * q^21 + 6395792138657544 * q^23 + 2573426290385925 * q^25 - 16677181699666569 * q^27 - 157790889708267966 * q^29 + 231251640669554784 * q^31 - 212341867499848284 * q^33 + 230145243295803600 * q^35 - 830491747621421334 * q^37 + 50131685053287738 * q^39 - 1400039039679261042 * q^41 - 761384577026785356 * q^43 - 1412528962279249350 * q^45 - 20706191933413048944 * q^47 + 26742597511830070875 * q^49 - 33114907719652792914 * q^51 - 64674463906638102342 * q^53 - 78855502334731828200 * q^55 - 20978161839549251172 * q^57 + 42370056643552637028 * q^59 - 516293667927208025118 * q^61 - 8827243278971119848 * q^63 - 1576925490906516336900 * q^65 + 1081135725593602868988 * q^67 - 1132995389986767946968 * q^69 - 22690312119034642536 * q^71 - 4166296087828394097618 * q^73 - 455874747062995455975 * q^75 - 7709458077454199230560 * q^77 - 5410946432474399204880 * q^79 + 2954312706550833698643 * q^81 + 18793771492158984347820 * q^83 - 38369307356296983199500 * q^85 + 27952182739150545373002 * q^87 - 23506358152298677341570 * q^89 + 128231916389440977486864 * q^91 - 40965534389689621321248 * q^93 + 196572725231897227799400 * q^95 - 129282257648368490763546 * q^97 + 37615724801995623965748 * 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
$3$	−177147.	−0.577350
$3$	−177147.	−0.577350
$4$	0	0
$5$	1.22619e8	1.12306	0.561531	−	0.827456i	$-0.310213\pi$
$5$	1.22619e8	1.12306	0.561531	+	0.827456i	$0.310213\pi$
$6$	0	0
$7$	4.92683e9	0.941759	0.470880	−	0.882197i	$-0.343937\pi$
$7$	4.92683e9	0.941759	0.470880	+	0.882197i	$0.343937\pi$
$8$	0	0
$9$	3.13811e10	0.333333
$10$	0	0
$11$	−1.05178e11	−0.111150	−0.0555749	−	0.998455i	$-0.517699\pi$
$11$	−1.05178e11	−0.111150	−0.0555749	+	0.998455i	$0.517699\pi$
$12$	0	0
$13$	−4.97595e11	−0.0770065	−0.0385032	−	0.999258i	$-0.512259\pi$
$13$	−4.97595e11	−0.0770065	−0.0385032	+	0.999258i	$0.512259\pi$
$14$	0	0
$15$	−2.17216e13	−0.648400
$16$	0	0
$17$	−3.26230e13	−0.230867	−0.115433	−	0.993315i	$-0.536826\pi$
$17$	−3.26230e13	−0.230867	−0.115433	+	0.993315i	$0.536826\pi$
$18$	0	0
$19$	7.69259e14	1.51498	0.757489	−	0.652848i	$-0.226426\pi$
$19$	7.69259e14	1.51498	0.757489	+	0.652848i	$0.226426\pi$
$20$	0	0
$21$	−8.72773e14	−0.543725
$22$	0	0
$23$	5.13304e15	1.12332	0.561661	−	0.827367i	$-0.310162\pi$
$23$	5.13304e15	1.12332	0.561661	+	0.827367i	$0.310162\pi$
$24$	0	0
$25$	3.11455e15	0.261268
$26$	0	0
$27$	−5.55906e15	−0.192450
$28$	0	0
$29$	7.47248e16	1.13733	0.568667	−	0.822568i	$-0.307459\pi$
$29$	7.47248e16	1.13733	0.568667	+	0.822568i	$0.307459\pi$
$30$	0	0
$31$	1.28452e17	0.907990	0.453995	−	0.891004i	$-0.349998\pi$
$31$	1.28452e17	0.907990	0.453995	+	0.891004i	$0.349998\pi$
$32$	0	0
$33$	1.86320e16	0.0641724
$34$	0	0
$35$	6.04124e17	1.05765
$36$	0	0
$37$	−8.87673e17	−0.820225	−0.410113	−	0.912035i	$-0.634511\pi$
$37$	−8.87673e17	−0.820225	−0.410113	+	0.912035i	$0.634511\pi$
$38$	0	0
$39$	8.81474e16	0.0444597
$40$	0	0
$41$	2.40518e18	0.682548	0.341274	−	0.939964i	$-0.389142\pi$
$41$	2.40518e18	0.682548	0.341274	+	0.939964i	$0.389142\pi$
$42$	0	0
$43$	5.95568e18	0.977336	0.488668	−	0.872470i	$-0.337483\pi$
$43$	5.95568e18	0.977336	0.488668	+	0.872470i	$0.337483\pi$
$44$	0	0
$45$	3.84792e18	0.374354
$46$	0	0
$47$	−2.36340e19	−1.39448	−0.697239	−	0.716839i	$-0.745588\pi$
$47$	−2.36340e19	−1.39448	−0.697239	+	0.716839i	$0.745588\pi$
$48$	0	0
$49$	−3.09512e18	−0.113089
$50$	0	0
$51$	5.77906e18	0.133291
$52$	0	0
$53$	3.46656e19	0.513720	0.256860	−	0.966449i	$-0.417312\pi$
$53$	3.46656e19	0.513720	0.256860	+	0.966449i	$0.417312\pi$
$54$	0	0
$55$	−1.28968e19	−0.124828
$56$	0	0
$57$	−1.36272e20	−0.874673
$58$	0	0
$59$	−8.87272e19	−0.383053	−0.191526	−	0.981487i	$-0.561344\pi$
$59$	−8.87272e19	−0.383053	−0.191526	+	0.981487i	$0.561344\pi$
$60$	0	0
$61$	−4.63114e20	−1.36268	−0.681342	−	0.731965i	$-0.738603\pi$
$61$	−4.63114e20	−1.36268	−0.681342	+	0.731965i	$0.738603\pi$
$62$	0	0
$63$	1.54609e20	0.313920
$64$	0	0
$65$	−6.10147e19	−0.0864830
$66$	0	0
$67$	−6.77287e20	−0.677505	−0.338753	−	0.940875i	$-0.610005\pi$
$67$	−6.77287e20	−0.677505	−0.338753	+	0.940875i	$0.610005\pi$
$68$	0	0
$69$	−9.09302e20	−0.648551
$70$	0	0
$71$	−1.80462e19	−0.00926646	−0.00463323	−	0.999989i	$-0.501475\pi$
$71$	−1.80462e19	−0.00926646	−0.00463323	+	0.999989i	$0.501475\pi$
$72$	0	0
$73$	−8.89872e20	−0.331982	−0.165991	−	0.986127i	$-0.553082\pi$
$73$	−8.89872e20	−0.331982	−0.165991	+	0.986127i	$0.553082\pi$
$74$	0	0
$75$	−5.51734e20	−0.150843
$76$	0	0
$77$	−5.18194e20	−0.104676
$78$	0	0
$79$	9.36842e21	1.40914	0.704571	−	0.709634i	$-0.251139\pi$
$79$	9.36842e21	1.40914	0.704571	+	0.709634i	$0.251139\pi$
$80$	0	0
$81$	9.84771e20	0.111111
$82$	0	0
$83$	1.60161e22	1.36508	0.682540	−	0.730848i	$-0.260875\pi$
$83$	1.60161e22	1.36508	0.682540	+	0.730848i	$0.260875\pi$
$84$	0	0
$85$	−4.00021e21	−0.259277
$86$	0	0
$87$	−1.32373e22	−0.656640
$88$	0	0
$89$	−1.20990e22	−0.462130	−0.231065	−	0.972938i	$-0.574221\pi$
$89$	−1.20990e22	−0.462130	−0.231065	+	0.972938i	$0.574221\pi$
$90$	0	0
$91$	−2.45156e21	−0.0725216
$92$	0	0
$93$	−2.27549e22	−0.524228
$94$	0	0
$95$	9.43260e22	1.70141
$96$	0	0
$97$	−7.53070e22	−1.06896	−0.534478	−	0.845182i	$-0.679492\pi$
$97$	−7.53070e22	−1.06896	−0.534478	+	0.845182i	$0.679492\pi$
$98$	0	0
$99$	−3.30060e21	−0.0370500
$100$	0	0
$101$	−1.92755e23	−1.71913	−0.859566	−	0.511025i	$-0.829266\pi$
$101$	−1.92755e23	−1.71913	−0.859566	+	0.511025i	$0.829266\pi$
$102$	0	0
$103$	2.28334e23	1.62533	0.812666	−	0.582730i	$-0.198016\pi$
$103$	2.28334e23	1.62533	0.812666	+	0.582730i	$0.198016\pi$
$104$	0	0
$105$	−1.07019e23	−0.610637
$106$	0	0
$107$	−2.49043e21	−0.0114383	−0.00571914	−	0.999984i	$-0.501820\pi$
$107$	−2.49043e21	−0.0114383	−0.00571914	+	0.999984i	$0.501820\pi$
$108$	0	0
$109$	3.59463e23	1.33429	0.667144	−	0.744928i	$-0.267516\pi$
$109$	3.59463e23	1.33429	0.667144	+	0.744928i	$0.267516\pi$
$110$	0	0
$111$	1.57249e23	0.473557
$112$	0	0
$113$	3.21111e23	0.787505	0.393753	−	0.919216i	$-0.371177\pi$
$113$	3.21111e23	0.787505	0.393753	+	0.919216i	$0.371177\pi$
$114$	0	0
$115$	6.29409e23	1.26156
$116$	0	0
$117$	−1.56151e22	−0.0256688
$118$	0	0
$119$	−1.60728e23	−0.217421
$120$	0	0
$121$	−8.84368e23	−0.987646
$122$	0	0
$123$	−4.26070e23	−0.394069
$124$	0	0
$125$	−1.07983e24	−0.829642
$126$	0	0
$127$	1.49982e24	0.960053	0.480027	−	0.877254i	$-0.340627\pi$
$127$	1.49982e24	0.960053	0.480027	+	0.877254i	$0.340627\pi$
$128$	0	0
$129$	−1.05503e24	−0.564265
$130$	0	0
$131$	−2.83144e24	−1.26878	−0.634392	−	0.773012i	$-0.718749\pi$
$131$	−2.83144e24	−1.26878	−0.634392	+	0.773012i	$0.718749\pi$
$132$	0	0
$133$	3.79001e24	1.42674
$134$	0	0
$135$	−6.81648e23	−0.216133
$136$	0	0
$137$	−3.38177e24	−0.905436	−0.452718	−	0.891654i	$-0.649546\pi$
$137$	−3.38177e24	−0.905436	−0.452718	+	0.891654i	$0.649546\pi$
$138$	0	0
$139$	−3.64524e24	−0.826145	−0.413072	−	0.910698i	$-0.635544\pi$
$139$	−3.64524e24	−0.826145	−0.413072	+	0.910698i	$0.635544\pi$
$140$	0	0
$141$	4.18669e24	0.805102
$142$	0	0
$143$	5.23360e22	0.00855926
$144$	0	0
$145$	9.16270e24	1.27730
$146$	0	0
$147$	5.48291e23	0.0652923
$148$	0	0
$149$	1.59535e25	1.62635	0.813175	−	0.582019i	$-0.197737\pi$
$149$	1.59535e25	1.62635	0.813175	+	0.582019i	$0.197737\pi$
$150$	0	0
$151$	−4.95271e24	−0.433119	−0.216560	−	0.976269i	$-0.569484\pi$
$151$	−4.95271e24	−0.433119	−0.216560	+	0.976269i	$0.569484\pi$
$152$	0	0
$153$	−1.02374e24	−0.0769555
$154$	0	0
$155$	1.57507e25	1.01973
$156$	0	0
$157$	1.98238e25	1.10749	0.553747	−	0.832685i	$-0.313198\pi$
$157$	1.98238e25	1.10749	0.553747	+	0.832685i	$0.313198\pi$
$158$	0	0
$159$	−6.14091e24	−0.296596
$160$	0	0
$161$	2.52896e25	1.05790
$162$	0	0
$163$	1.26906e25	0.460602	0.230301	−	0.973119i	$-0.426029\pi$
$163$	1.26906e25	0.460602	0.230301	+	0.973119i	$0.426029\pi$
$164$	0	0
$165$	2.28464e24	0.0720696
$166$	0	0
$167$	1.57370e24	0.0432197	0.0216099	−	0.999766i	$-0.493121\pi$
$167$	1.57370e24	0.0432197	0.0216099	+	0.999766i	$0.493121\pi$
$168$	0	0
$169$	−4.15063e25	−0.994070
$170$	0	0
$171$	2.41402e25	0.504993
$172$	0	0
$173$	8.55724e25	1.56604	0.783021	−	0.621996i	$-0.213678\pi$
$173$	8.55724e25	1.56604	0.783021	+	0.621996i	$0.213678\pi$
$174$	0	0
$175$	1.53449e25	0.246051
$176$	0	0
$177$	1.57178e25	0.221156
$178$	0	0
$179$	6.75207e25	0.834885	0.417443	−	0.908703i	$-0.362926\pi$
$179$	6.75207e25	0.834885	0.417443	+	0.908703i	$0.362926\pi$
$180$	0	0
$181$	−1.36830e25	−0.148894	−0.0744470	−	0.997225i	$-0.523719\pi$
$181$	−1.36830e25	−0.148894	−0.0744470	+	0.997225i	$0.523719\pi$
$182$	0	0
$183$	8.20393e25	0.786746
$184$	0	0
$185$	−1.08846e26	−0.921163
$186$	0	0
$187$	3.43122e24	0.0256608
$188$	0	0
$189$	−2.73885e25	−0.181242
$190$	0	0
$191$	2.15561e26	1.26383	0.631913	−	0.775039i	$-0.282270\pi$
$191$	2.15561e26	1.26383	0.631913	+	0.775039i	$0.282270\pi$
$192$	0	0
$193$	1.03535e26	0.538493	0.269247	−	0.963071i	$-0.413225\pi$
$193$	1.03535e26	0.538493	0.269247	+	0.963071i	$0.413225\pi$
$194$	0	0
$195$	1.08086e25	0.0499310
$196$	0	0
$197$	4.07258e24	0.0167305	0.00836523	−	0.999965i	$-0.497337\pi$
$197$	4.07258e24	0.0167305	0.00836523	+	0.999965i	$0.497337\pi$
$198$	0	0
$199$	1.51661e26	0.554707	0.277353	−	0.960768i	$-0.410543\pi$
$199$	1.51661e26	0.554707	0.277353	+	0.960768i	$0.410543\pi$
$200$	0	0
$201$	1.19979e26	0.391158
$202$	0	0
$203$	3.68156e26	1.07109
$204$	0	0
$205$	2.94921e26	0.766544
$206$	0	0
$207$	1.61080e26	0.374441
$208$	0	0
$209$	−8.09092e25	−0.168390
$210$	0	0
$211$	5.11863e26	0.954784	0.477392	−	0.878690i	$-0.341582\pi$
$211$	5.11863e26	0.954784	0.477392	+	0.878690i	$0.341582\pi$
$212$	0	0
$213$	3.19682e24	0.00534999
$214$	0	0
$215$	7.30280e26	1.09761
$216$	0	0
$217$	6.32860e26	0.855108
$218$	0	0
$219$	1.57638e26	0.191670
$220$	0	0
$221$	1.62330e25	0.0177782
$222$	0	0
$223$	8.73476e26	0.862472	0.431236	−	0.902239i	$-0.358078\pi$
$223$	8.73476e26	0.862472	0.431236	+	0.902239i	$0.358078\pi$
$224$	0	0
$225$	9.77379e25	0.0870892
$226$	0	0
$227$	−1.78719e27	−1.43838	−0.719191	−	0.694812i	$-0.755487\pi$
$227$	−1.78719e27	−1.43838	−0.719191	+	0.694812i	$0.755487\pi$
$228$	0	0
$229$	1.76502e27	1.28423	0.642114	−	0.766609i	$-0.278058\pi$
$229$	1.76502e27	1.28423	0.642114	+	0.766609i	$0.278058\pi$
$230$	0	0
$231$	9.17965e25	0.0604350
$232$	0	0
$233$	2.81361e27	1.67753	0.838767	−	0.544490i	$-0.183277\pi$
$233$	2.81361e27	1.67753	0.838767	+	0.544490i	$0.183277\pi$
$234$	0	0
$235$	−2.89798e27	−1.56609
$236$	0	0
$237$	−1.65959e27	−0.813569
$238$	0	0
$239$	5.03925e26	0.224280	0.112140	−	0.993692i	$-0.464230\pi$
$239$	5.03925e26	0.224280	0.112140	+	0.993692i	$0.464230\pi$
$240$	0	0
$241$	4.72946e27	1.91256	0.956280	−	0.292452i	$-0.0944711\pi$
$241$	4.72946e27	1.91256	0.956280	+	0.292452i	$0.0944711\pi$
$242$	0	0
$243$	−1.74449e26	−0.0641500
$244$	0	0
$245$	−3.79521e26	−0.127006
$246$	0	0
$247$	−3.82779e26	−0.116663
$248$	0	0
$249$	−2.83720e27	−0.788130
$250$	0	0
$251$	3.22757e27	0.817765	0.408882	−	0.912587i	$-0.365919\pi$
$251$	3.22757e27	0.817765	0.408882	+	0.912587i	$0.365919\pi$
$252$	0	0
$253$	−5.39883e26	−0.124857
$254$	0	0
$255$	7.08624e26	0.149694
$256$	0	0
$257$	−1.29904e27	−0.250837	−0.125419	−	0.992104i	$-0.540027\pi$
$257$	−1.29904e27	−0.250837	−0.125419	+	0.992104i	$0.540027\pi$
$258$	0	0
$259$	−4.37341e27	−0.772455
$260$	0	0
$261$	2.34494e27	0.379111
$262$	0	0
$263$	1.14557e27	0.169640	0.0848201	−	0.996396i	$-0.472968\pi$
$263$	1.14557e27	0.169640	0.0848201	+	0.996396i	$0.472968\pi$
$264$	0	0
$265$	4.25067e27	0.576939
$266$	0	0
$267$	2.14330e27	0.266811
$268$	0	0
$269$	−9.43029e27	−1.07739	−0.538696	−	0.842500i	$-0.681083\pi$
$269$	−9.43029e27	−1.07739	−0.538696	+	0.842500i	$0.681083\pi$
$270$	0	0
$271$	−1.27487e28	−1.33757	−0.668786	−	0.743455i	$-0.733186\pi$
$271$	−1.27487e28	−1.33757	−0.668786	+	0.743455i	$0.733186\pi$
$272$	0	0
$273$	4.34287e26	0.0418703
$274$	0	0
$275$	−3.27582e26	−0.0290399
$276$	0	0
$277$	−6.50417e27	−0.530487	−0.265244	−	0.964181i	$-0.585452\pi$
$277$	−6.50417e27	−0.530487	−0.265244	+	0.964181i	$0.585452\pi$
$278$	0	0
$279$	4.03096e27	0.302663
$280$	0	0
$281$	−2.82907e28	−1.95669	−0.978344	−	0.206986i	$-0.933635\pi$
$281$	−2.82907e28	−1.95669	−0.978344	+	0.206986i	$0.933635\pi$
$282$	0	0
$283$	−1.68494e28	−1.07409	−0.537043	−	0.843555i	$-0.680459\pi$
$283$	−1.68494e28	−1.07409	−0.537043	+	0.843555i	$0.680459\pi$
$284$	0	0
$285$	−1.67096e28	−0.982312
$286$	0	0
$287$	1.18499e28	0.642796
$288$	0	0
$289$	−1.89033e28	−0.946701
$290$	0	0
$291$	1.33404e28	0.617163
$292$	0	0
$293$	5.49498e27	0.234957	0.117478	−	0.993075i	$-0.462519\pi$
$293$	5.49498e27	0.234957	0.117478	+	0.993075i	$0.462519\pi$
$294$	0	0
$295$	−1.08797e28	−0.430192
$296$	0	0
$297$	5.84691e26	0.0213908
$298$	0	0
$299$	−2.55417e27	−0.0865031
$300$	0	0
$301$	2.93426e28	0.920415
$302$	0	0
$303$	3.41459e28	0.992541
$304$	0	0
$305$	−5.67867e28	−1.53038
$306$	0	0
$307$	−4.79364e28	−1.19832	−0.599162	−	0.800628i	$-0.704499\pi$
$307$	−4.79364e28	−1.19832	−0.599162	+	0.800628i	$0.704499\pi$
$308$	0	0
$309$	−4.04486e28	−0.938386
$310$	0	0
$311$	1.90432e28	0.410201	0.205100	−	0.978741i	$-0.434248\pi$
$311$	1.90432e28	0.410201	0.205100	+	0.978741i	$0.434248\pi$
$312$	0	0
$313$	2.96861e28	0.594010	0.297005	−	0.954876i	$-0.404012\pi$
$313$	2.96861e28	0.594010	0.297005	+	0.954876i	$0.404012\pi$
$314$	0	0
$315$	1.89580e28	0.352551
$316$	0	0
$317$	6.68653e28	1.15616	0.578082	−	0.815979i	$-0.303801\pi$
$317$	6.68653e28	1.15616	0.578082	+	0.815979i	$0.303801\pi$
$318$	0	0
$319$	−7.85941e27	−0.126415
$320$	0	0
$321$	4.41172e26	0.00660390
$322$	0	0
$323$	−2.50955e28	−0.349758
$324$	0	0
$325$	−1.54978e27	−0.0201193
$326$	0	0
$327$	−6.36779e28	−0.770352
$328$	0	0
$329$	−1.16441e29	−1.31326
$330$	0	0
$331$	−1.73581e29	−1.82592	−0.912958	−	0.408054i	$-0.866208\pi$
$331$	−1.73581e29	−1.82592	−0.912958	+	0.408054i	$0.866208\pi$
$332$	0	0
$333$	−2.78561e28	−0.273408
$334$	0	0
$335$	−8.30485e28	−0.760880
$336$	0	0
$337$	1.36628e29	1.16895	0.584476	−	0.811411i	$-0.301300\pi$
$337$	1.36628e29	1.16895	0.584476	+	0.811411i	$0.301300\pi$
$338$	0	0
$339$	−5.68839e28	−0.454666
$340$	0	0
$341$	−1.35103e28	−0.100923
$342$	0	0
$343$	−1.50090e29	−1.04826
$344$	0	0
$345$	−1.11498e29	−0.728362
$346$	0	0
$347$	8.77740e28	0.536509	0.268254	−	0.963348i	$-0.413553\pi$
$347$	8.77740e28	0.536509	0.268254	+	0.963348i	$0.413553\pi$
$348$	0	0
$349$	1.17201e29	0.670563	0.335282	−	0.942118i	$-0.391169\pi$
$349$	1.17201e29	0.670563	0.335282	+	0.942118i	$0.391169\pi$
$350$	0	0
$351$	2.76616e27	0.0148199
$352$	0	0
$353$	2.02723e29	1.01740	0.508702	−	0.860943i	$-0.330126\pi$
$353$	2.02723e29	1.01740	0.508702	+	0.860943i	$0.330126\pi$
$354$	0	0
$355$	−2.21281e27	−0.0104068
$356$	0	0
$357$	2.84725e28	0.125528
$358$	0	0
$359$	2.38809e29	0.987337	0.493669	−	0.869650i	$-0.335656\pi$
$359$	2.38809e29	0.987337	0.493669	+	0.869650i	$0.335656\pi$
$360$	0	0
$361$	3.33930e29	1.29516
$362$	0	0
$363$	1.56663e29	0.570218
$364$	0	0
$365$	−1.09115e29	−0.372836
$366$	0	0
$367$	3.86709e29	1.24086	0.620432	−	0.784260i	$-0.286957\pi$
$367$	3.86709e29	1.24086	0.620432	+	0.784260i	$0.286957\pi$
$368$	0	0
$369$	7.54771e28	0.227516
$370$	0	0
$371$	1.70791e29	0.483800
$372$	0	0
$373$	2.97586e28	0.0792429	0.0396215	−	0.999215i	$-0.487385\pi$
$373$	2.97586e28	0.0792429	0.0396215	+	0.999215i	$0.487385\pi$
$374$	0	0
$375$	1.91289e29	0.478994
$376$	0	0
$377$	−3.71827e28	−0.0875821
$378$	0	0
$379$	7.12311e29	1.57877	0.789385	−	0.613899i	$-0.210400\pi$
$379$	7.12311e29	1.57877	0.789385	+	0.613899i	$0.210400\pi$
$380$	0	0
$381$	−2.65688e29	−0.554287
$382$	0	0
$383$	−3.80004e29	−0.746452	−0.373226	−	0.927740i	$-0.621748\pi$
$383$	−3.80004e29	−0.746452	−0.373226	+	0.927740i	$0.621748\pi$
$384$	0	0
$385$	−6.35405e28	−0.117558
$386$	0	0
$387$	1.86895e29	0.325779
$388$	0	0
$389$	7.80080e29	1.28150	0.640750	−	0.767749i	$-0.278623\pi$
$389$	7.80080e29	1.28150	0.640750	+	0.767749i	$0.278623\pi$
$390$	0	0
$391$	−1.67455e29	−0.259338
$392$	0	0
$393$	5.01581e29	0.732532
$394$	0	0
$395$	1.14875e30	1.58255
$396$	0	0
$397$	−1.25264e30	−1.62831	−0.814154	−	0.580648i	$-0.802799\pi$
$397$	−1.25264e30	−1.62831	−0.814154	+	0.580648i	$0.802799\pi$
$398$	0	0
$399$	−6.71389e29	−0.823732
$400$	0	0
$401$	4.44805e29	0.515240	0.257620	−	0.966246i	$-0.417062\pi$
$401$	4.44805e29	0.515240	0.257620	+	0.966246i	$0.417062\pi$
$402$	0	0
$403$	−6.39170e28	−0.0699211
$404$	0	0
$405$	1.20752e29	0.124785
$406$	0	0
$407$	9.33636e28	0.0911679
$408$	0	0
$409$	−8.66461e29	−0.799707	−0.399854	−	0.916579i	$-0.630939\pi$
$409$	−8.66461e29	−0.799707	−0.399854	+	0.916579i	$0.630939\pi$
$410$	0	0
$411$	5.99071e29	0.522754
$412$	0	0
$413$	−4.37144e29	−0.360744
$414$	0	0
$415$	1.96388e30	1.53307
$416$	0	0
$417$	6.45743e29	0.476975
$418$	0	0
$419$	1.68295e30	1.17655	0.588274	−	0.808662i	$-0.299808\pi$
$419$	1.68295e30	1.17655	0.588274	+	0.808662i	$0.299808\pi$
$420$	0	0
$421$	3.76812e29	0.249391	0.124695	−	0.992195i	$-0.460205\pi$
$421$	3.76812e29	0.249391	0.124695	+	0.992195i	$0.460205\pi$
$422$	0	0
$423$	−7.41660e29	−0.464826
$424$	0	0
$425$	−1.01606e29	−0.0603180
$426$	0	0
$427$	−2.28168e30	−1.28332
$428$	0	0
$429$	−9.27117e27	−0.00494169
$430$	0	0
$431$	−7.90836e29	−0.399574	−0.199787	−	0.979839i	$-0.564025\pi$
$431$	−7.90836e29	−0.399574	−0.199787	+	0.979839i	$0.564025\pi$
$432$	0	0
$433$	−3.73979e28	−0.0179158	−0.00895791	−	0.999960i	$-0.502851\pi$
$433$	−3.73979e28	−0.0179158	−0.00895791	+	0.999960i	$0.502851\pi$
$434$	0	0
$435$	−1.62315e30	−0.737447
$436$	0	0
$437$	3.94864e30	1.70181
$438$	0	0
$439$	1.22519e29	0.0501027	0.0250513	−	0.999686i	$-0.492025\pi$
$439$	1.22519e29	0.0501027	0.0250513	+	0.999686i	$0.492025\pi$
$440$	0	0
$441$	−9.71281e28	−0.0376965
$442$	0	0
$443$	−3.48534e30	−1.28411	−0.642055	−	0.766658i	$-0.721918\pi$
$443$	−3.48534e30	−1.28411	−0.642055	+	0.766658i	$0.721918\pi$
$444$	0	0
$445$	−1.48357e30	−0.519001
$446$	0	0
$447$	−2.82612e30	−0.938974
$448$	0	0
$449$	3.51619e30	1.10979	0.554893	−	0.831922i	$-0.312759\pi$
$449$	3.51619e30	1.10979	0.554893	+	0.831922i	$0.312759\pi$
$450$	0	0
$451$	−2.52972e29	−0.0758651
$452$	0	0
$453$	8.77357e29	0.250062
$454$	0	0
$455$	−3.00609e29	−0.0814462
$456$	0	0
$457$	−4.89657e30	−1.26141	−0.630705	−	0.776023i	$-0.717234\pi$
$457$	−4.89657e30	−1.26141	−0.630705	+	0.776023i	$0.717234\pi$
$458$	0	0
$459$	1.81353e29	0.0444303
$460$	0	0
$461$	2.36643e30	0.551485	0.275742	−	0.961232i	$-0.411076\pi$
$461$	2.36643e30	0.551485	0.275742	+	0.961232i	$0.411076\pi$
$462$	0	0
$463$	2.72149e30	0.603428	0.301714	−	0.953399i	$-0.402441\pi$
$463$	2.72149e30	0.603428	0.301714	+	0.953399i	$0.402441\pi$
$464$	0	0
$465$	−2.79018e30	−0.588741
$466$	0	0
$467$	−2.92477e30	−0.587418	−0.293709	−	0.955895i	$-0.594890\pi$
$467$	−2.92477e30	−0.587418	−0.293709	+	0.955895i	$0.594890\pi$
$468$	0	0
$469$	−3.33688e30	−0.638047
$470$	0	0
$471$	−3.51173e30	−0.639412
$472$	0	0
$473$	−6.26406e29	−0.108631
$474$	0	0
$475$	2.39590e30	0.395815
$476$	0	0
$477$	1.08784e30	0.171240
$478$	0	0
$479$	−9.30674e30	−1.39617	−0.698087	−	0.716013i	$-0.745965\pi$
$479$	−9.30674e30	−1.39617	−0.698087	+	0.716013i	$0.745965\pi$
$480$	0	0
$481$	4.41701e29	0.0631626
$482$	0	0
$483$	−4.47998e30	−0.610778
$484$	0	0
$485$	−9.23408e30	−1.20050
$486$	0	0
$487$	1.13258e31	1.40438	0.702192	−	0.711988i	$-0.252205\pi$
$487$	1.13258e31	1.40438	0.702192	+	0.711988i	$0.252205\pi$
$488$	0	0
$489$	−2.24811e30	−0.265929
$490$	0	0
$491$	5.29107e30	0.597181	0.298591	−	0.954381i	$-0.403483\pi$
$491$	5.29107e30	0.597181	0.298591	+	0.954381i	$0.403483\pi$
$492$	0	0
$493$	−2.43775e30	−0.262572
$494$	0	0
$495$	−4.04717e29	−0.0416094
$496$	0	0
$497$	−8.89103e28	−0.00872678
$498$	0	0
$499$	−2.03153e31	−1.90400	−0.951999	−	0.306100i	$-0.900976\pi$
$499$	−2.03153e31	−1.90400	−0.951999	+	0.306100i	$0.900976\pi$
$500$	0	0
$501$	−2.78776e29	−0.0249529
$502$	0	0
$503$	1.39093e31	1.18925	0.594624	−	0.804004i	$-0.297301\pi$
$503$	1.39093e31	1.18925	0.594624	+	0.804004i	$0.297301\pi$
$504$	0	0
$505$	−2.36354e31	−1.93069
$506$	0	0
$507$	7.35272e30	0.573927
$508$	0	0
$509$	−1.23503e31	−0.921345	−0.460672	−	0.887570i	$-0.652392\pi$
$509$	−1.23503e31	−0.921345	−0.460672	+	0.887570i	$0.652392\pi$
$510$	0	0
$511$	−4.38424e30	−0.312647
$512$	0	0
$513$	−4.27636e30	−0.291558
$514$	0	0
$515$	2.79981e31	1.82535
$516$	0	0
$517$	2.48578e30	0.154996
$518$	0	0
$519$	−1.51589e31	−0.904155
$520$	0	0
$521$	2.91599e31	1.66400	0.831999	−	0.554777i	$-0.187196\pi$
$521$	2.91599e31	1.66400	0.831999	+	0.554777i	$0.187196\pi$
$522$	0	0
$523$	−5.50484e30	−0.300591	−0.150295	−	0.988641i	$-0.548022\pi$
$523$	−5.50484e30	−0.300591	−0.150295	+	0.988641i	$0.548022\pi$
$524$	0	0
$525$	−2.71830e30	−0.142058
$526$	0	0
$527$	−4.19048e30	−0.209625
$528$	0	0
$529$	5.46762e30	0.261853
$530$	0	0
$531$	−2.78435e30	−0.127684
$532$	0	0
$533$	−1.19681e30	−0.0525606
$534$	0	0
$535$	−3.05375e29	−0.0128459
$536$	0	0
$537$	−1.19611e31	−0.482021
$538$	0	0
$539$	3.25538e29	0.0125699
$540$	0	0
$541$	−4.49931e31	−1.66486	−0.832430	−	0.554131i	$-0.813051\pi$
$541$	−4.49931e31	−1.66486	−0.832430	+	0.554131i	$0.813051\pi$
$542$	0	0
$543$	2.42390e30	0.0859640
$544$	0	0
$545$	4.40771e31	1.49849
$546$	0	0
$547$	−4.16665e31	−1.35810	−0.679051	−	0.734091i	$-0.737608\pi$
$547$	−4.16665e31	−1.35810	−0.679051	+	0.734091i	$0.737608\pi$
$548$	0	0
$549$	−1.45330e31	−0.454228
$550$	0	0
$551$	5.74828e31	1.72304
$552$	0	0
$553$	4.61566e31	1.32707
$554$	0	0
$555$	1.92817e31	0.531834
$556$	0	0
$557$	3.47269e31	0.919032	0.459516	−	0.888169i	$-0.348023\pi$
$557$	3.47269e31	0.919032	0.459516	+	0.888169i	$0.348023\pi$
$558$	0	0
$559$	−2.96351e30	−0.0752612
$560$	0	0
$561$	−6.07830e29	−0.0148153
$562$	0	0
$563$	2.02906e31	0.474732	0.237366	−	0.971420i	$-0.423716\pi$
$563$	2.02906e31	0.474732	0.237366	+	0.971420i	$0.423716\pi$
$564$	0	0
$565$	3.93744e31	0.884417
$566$	0	0
$567$	4.85180e30	0.104640
$568$	0	0
$569$	5.13990e31	1.06454	0.532272	−	0.846573i	$-0.321338\pi$
$569$	5.13990e31	1.06454	0.532272	+	0.846573i	$0.321338\pi$
$570$	0	0
$571$	−5.88151e31	−1.16997	−0.584983	−	0.811045i	$-0.698899\pi$
$571$	−5.88151e31	−1.16997	−0.584983	+	0.811045i	$0.698899\pi$
$572$	0	0
$573$	−3.81861e31	−0.729670
$574$	0	0
$575$	1.59871e31	0.293488
$576$	0	0
$577$	7.75000e31	1.36704	0.683518	−	0.729933i	$-0.260449\pi$
$577$	7.75000e31	1.36704	0.683518	+	0.729933i	$0.260449\pi$
$578$	0	0
$579$	−1.83410e31	−0.310899
$580$	0	0
$581$	7.89085e31	1.28558
$582$	0	0
$583$	−3.64606e30	−0.0570999
$584$	0	0
$585$	−1.91471e30	−0.0288277
$586$	0	0
$587$	1.07681e32	1.55884	0.779418	−	0.626505i	$-0.215515\pi$
$587$	1.07681e32	1.55884	0.779418	+	0.626505i	$0.215515\pi$
$588$	0	0
$589$	9.88128e31	1.37559
$590$	0	0
$591$	−7.21445e29	−0.00965934
$592$	0	0
$593$	2.56069e30	0.0329783	0.0164892	−	0.999864i	$-0.494751\pi$
$593$	2.56069e30	0.0329783	0.0164892	+	0.999864i	$0.494751\pi$
$594$	0	0
$595$	−1.97083e31	−0.244177
$596$	0	0
$597$	−2.68663e31	−0.320260
$598$	0	0
$599$	9.62872e31	1.10449	0.552243	−	0.833683i	$-0.313772\pi$
$599$	9.62872e31	1.10449	0.552243	+	0.833683i	$0.313772\pi$
$600$	0	0
$601$	7.68681e31	0.848574	0.424287	−	0.905528i	$-0.360525\pi$
$601$	7.68681e31	0.848574	0.424287	+	0.905528i	$0.360525\pi$
$602$	0	0
$603$	−2.12540e31	−0.225835
$604$	0	0
$605$	−1.08441e32	−1.10919
$606$	0	0
$607$	−1.10560e32	−1.08875	−0.544375	−	0.838842i	$-0.683233\pi$
$607$	−1.10560e32	−1.08875	−0.544375	+	0.838842i	$0.683233\pi$
$608$	0	0
$609$	−6.52178e31	−0.618397
$610$	0	0
$611$	1.17602e31	0.107384
$612$	0	0
$613$	−1.57475e32	−1.38489	−0.692445	−	0.721471i	$-0.743467\pi$
$613$	−1.57475e32	−1.38489	−0.692445	+	0.721471i	$0.743467\pi$
$614$	0	0
$615$	−5.22444e31	−0.442564
$616$	0	0
$617$	−1.11995e32	−0.913945	−0.456972	−	0.889481i	$-0.651066\pi$
$617$	−1.11995e32	−0.913945	−0.456972	+	0.889481i	$0.651066\pi$
$618$	0	0
$619$	1.68378e32	1.32386	0.661929	−	0.749566i	$-0.269738\pi$
$619$	1.68378e32	1.32386	0.661929	+	0.749566i	$0.269738\pi$
$620$	0	0
$621$	−2.85349e31	−0.216184
$622$	0	0
$623$	−5.96097e31	−0.435215
$624$	0	0
$625$	−1.69536e32	−1.19301
$626$	0	0
$627$	1.43328e31	0.0972198
$628$	0	0
$629$	2.89585e31	0.189363
$630$	0	0
$631$	5.39770e31	0.340307	0.170154	−	0.985418i	$-0.445574\pi$
$631$	5.39770e31	0.340307	0.170154	+	0.985418i	$0.445574\pi$
$632$	0	0
$633$	−9.06749e31	−0.551245
$634$	0	0
$635$	1.83906e32	1.07820
$636$	0	0
$637$	1.54011e30	0.00870862
$638$	0	0
$639$	−5.66308e29	−0.00308882
$640$	0	0
$641$	1.80943e32	0.952082	0.476041	−	0.879423i	$-0.342071\pi$
$641$	1.80943e32	0.952082	0.476041	+	0.879423i	$0.342071\pi$
$642$	0	0
$643$	−2.10276e32	−1.06749	−0.533744	−	0.845646i	$-0.679216\pi$
$643$	−2.10276e32	−1.06749	−0.533744	+	0.845646i	$0.679216\pi$
$644$	0	0
$645$	−1.29367e32	−0.633704
$646$	0	0
$647$	2.02830e32	0.958809	0.479404	−	0.877594i	$-0.340853\pi$
$647$	2.02830e32	0.958809	0.479404	+	0.877594i	$0.340853\pi$
$648$	0	0
$649$	9.33215e30	0.0425763
$650$	0	0
$651$	−1.12109e32	−0.493697
$652$	0	0
$653$	2.20269e32	0.936381	0.468190	−	0.883628i	$-0.344906\pi$
$653$	2.20269e32	0.936381	0.468190	+	0.883628i	$0.344906\pi$
$654$	0	0
$655$	−3.47189e32	−1.42492
$656$	0	0
$657$	−2.79251e31	−0.110661
$658$	0	0
$659$	5.42710e31	0.207676	0.103838	−	0.994594i	$-0.466888\pi$
$659$	5.42710e31	0.207676	0.103838	+	0.994594i	$0.466888\pi$
$660$	0	0
$661$	−3.83005e32	−1.41543	−0.707713	−	0.706500i	$-0.750273\pi$
$661$	−3.83005e32	−1.41543	−0.707713	+	0.706500i	$0.750273\pi$
$662$	0	0
$663$	−2.87563e30	−0.0102643
$664$	0	0
$665$	4.64728e32	1.60232
$666$	0	0
$667$	3.83565e32	1.27759
$668$	0	0
$669$	−1.54734e32	−0.497949
$670$	0	0
$671$	4.87094e31	0.151462
$672$	0	0
$673$	−1.49416e32	−0.448978	−0.224489	−	0.974477i	$-0.572071\pi$
$673$	−1.49416e32	−0.448978	−0.224489	+	0.974477i	$0.572071\pi$
$674$	0	0
$675$	−1.73140e31	−0.0502810
$676$	0	0
$677$	−2.42529e32	−0.680760	−0.340380	−	0.940288i	$-0.610556\pi$
$677$	−2.42529e32	−0.680760	−0.340380	+	0.940288i	$0.610556\pi$
$678$	0	0
$679$	−3.71024e32	−1.00670
$680$	0	0
$681$	3.16596e32	0.830450
$682$	0	0
$683$	4.63262e31	0.117487	0.0587433	−	0.998273i	$-0.481291\pi$
$683$	4.63262e31	0.117487	0.0587433	+	0.998273i	$0.481291\pi$
$684$	0	0
$685$	−4.14671e32	−1.01686
$686$	0	0
$687$	−3.12668e32	−0.741449
$688$	0	0
$689$	−1.72494e31	−0.0395597
$690$	0	0
$691$	−2.22770e32	−0.494151	−0.247076	−	0.968996i	$-0.579470\pi$
$691$	−2.22770e32	−0.494151	−0.247076	+	0.968996i	$0.579470\pi$
$692$	0	0
$693$	−1.62615e31	−0.0348921
$694$	0	0
$695$	−4.46977e32	−0.927811
$696$	0	0
$697$	−7.84642e31	−0.157578
$698$	0	0
$699$	−4.98423e32	−0.968525
$700$	0	0
$701$	−9.31121e30	−0.0175085	−0.00875425	−	0.999962i	$-0.502787\pi$
$701$	−9.31121e30	−0.0175085	−0.00875425	+	0.999962i	$0.502787\pi$
$702$	0	0
$703$	−6.82851e32	−1.24262
$704$	0	0
$705$	5.13369e32	0.904180
$706$	0	0
$707$	−9.49669e32	−1.61901
$708$	0	0
$709$	−1.09895e33	−1.81362	−0.906811	−	0.421536i	$-0.861491\pi$
$709$	−1.09895e33	−1.81362	−0.906811	+	0.421536i	$0.861491\pi$
$710$	0	0
$711$	2.93991e32	0.469714
$712$	0	0
$713$	6.59349e32	1.01997
$714$	0	0
$715$	6.41740e30	0.00961258
$716$	0	0
$717$	−8.92687e31	−0.129488
$718$	0	0
$719$	6.61949e32	0.929913	0.464957	−	0.885334i	$-0.346070\pi$
$719$	6.61949e32	0.929913	0.464957	+	0.885334i	$0.346070\pi$
$720$	0	0
$721$	1.12496e33	1.53067
$722$	0	0
$723$	−8.37809e32	−1.10422
$724$	0	0
$725$	2.32734e32	0.297148
$726$	0	0
$727$	1.41564e33	1.75109	0.875543	−	0.483139i	$-0.160504\pi$
$727$	1.41564e33	1.75109	0.875543	+	0.483139i	$0.160504\pi$
$728$	0	0
$729$	3.09032e31	0.0370370
$730$	0	0
$731$	−1.94292e32	−0.225634
$732$	0	0
$733$	−1.60767e33	−1.80926	−0.904630	−	0.426198i	$-0.859853\pi$
$733$	−1.60767e33	−1.80926	−0.904630	+	0.426198i	$0.859853\pi$
$734$	0	0
$735$	6.72310e31	0.0733272
$736$	0	0
$737$	7.12357e31	0.0753046
$738$	0	0
$739$	7.34540e32	0.752669	0.376335	−	0.926484i	$-0.377184\pi$
$739$	7.34540e32	0.752669	0.376335	+	0.926484i	$0.377184\pi$
$740$	0	0
$741$	6.78082e31	0.0673555
$742$	0	0
$743$	2.97134e32	0.286142	0.143071	−	0.989712i	$-0.454302\pi$
$743$	2.97134e32	0.286142	0.143071	+	0.989712i	$0.454302\pi$
$744$	0	0
$745$	1.95621e33	1.82649
$746$	0	0
$747$	5.02602e32	0.455027
$748$	0	0
$749$	−1.22699e31	−0.0107721
$750$	0	0
$751$	6.73550e32	0.573470	0.286735	−	0.958010i	$-0.407430\pi$
$751$	6.73550e32	0.573470	0.286735	+	0.958010i	$0.407430\pi$
$752$	0	0
$753$	−5.71755e32	−0.472137
$754$	0	0
$755$	−6.07297e32	−0.486420
$756$	0	0
$757$	2.73159e32	0.212233	0.106117	−	0.994354i	$-0.466158\pi$
$757$	2.73159e32	0.212233	0.106117	+	0.994354i	$0.466158\pi$
$758$	0	0
$759$	9.56386e31	0.0720863
$760$	0	0
$761$	1.14628e33	0.838236	0.419118	−	0.907932i	$-0.362339\pi$
$761$	1.14628e33	0.838236	0.419118	+	0.907932i	$0.362339\pi$
$762$	0	0
$763$	1.77101e33	1.25658
$764$	0	0
$765$	−1.25531e32	−0.0864258
$766$	0	0
$767$	4.41502e31	0.0294975
$768$	0	0
$769$	−2.18756e33	−1.41843	−0.709215	−	0.704992i	$-0.750951\pi$
$769$	−2.18756e33	−1.41843	−0.709215	+	0.704992i	$0.750951\pi$
$770$	0	0
$771$	2.30121e32	0.144821
$772$	0	0
$773$	−2.45768e33	−1.50128	−0.750640	−	0.660711i	$-0.770255\pi$
$773$	−2.45768e33	−1.50128	−0.750640	+	0.660711i	$0.770255\pi$
$774$	0	0
$775$	4.00070e32	0.237228
$776$	0	0
$777$	7.74736e32	0.445977
$778$	0	0
$779$	1.85021e33	1.03405
$780$	0	0
$781$	1.89806e30	0.00102997
$782$	0	0
$783$	−4.15400e32	−0.218880
$784$	0	0
$785$	2.43078e33	1.24378
$786$	0	0
$787$	2.80389e33	1.39332	0.696661	−	0.717401i	$-0.254668\pi$
$787$	2.80389e33	1.39332	0.696661	+	0.717401i	$0.254668\pi$
$788$	0	0
$789$	−2.02933e32	−0.0979418
$790$	0	0
$791$	1.58206e33	0.741640
$792$	0	0
$793$	2.30443e32	0.104936
$794$	0	0
$795$	−7.52993e32	−0.333096
$796$	0	0
$797$	−3.03624e33	−1.30486	−0.652431	−	0.757848i	$-0.726251\pi$
$797$	−3.03624e33	−1.30486	−0.652431	+	0.757848i	$0.726251\pi$
$798$	0	0
$799$	7.71011e32	0.321938
$800$	0	0
$801$	−3.79679e32	−0.154043
$802$	0	0
$803$	9.35949e31	0.0368998
$804$	0	0
$805$	3.10099e33	1.18809
$806$	0	0
$807$	1.67055e33	0.622033
$808$	0	0
$809$	3.80127e33	1.37569	0.687846	−	0.725857i	$-0.258556\pi$
$809$	3.80127e33	1.37569	0.687846	+	0.725857i	$0.258556\pi$
$810$	0	0
$811$	−4.58144e33	−1.61162	−0.805809	−	0.592175i	$-0.798269\pi$
$811$	−4.58144e33	−1.61162	−0.805809	+	0.592175i	$0.798269\pi$
$812$	0	0
$813$	2.25839e33	0.772248
$814$	0	0
$815$	1.55612e33	0.517285
$816$	0	0
$817$	4.58146e33	1.48064
$818$	0	0
$819$	−7.69327e31	−0.0241739
$820$	0	0
$821$	−1.85956e33	−0.568151	−0.284075	−	0.958802i	$-0.591687\pi$
$821$	−1.85956e33	−0.568151	−0.284075	+	0.958802i	$0.591687\pi$
$822$	0	0
$823$	−3.33178e33	−0.989870	−0.494935	−	0.868930i	$-0.664808\pi$
$823$	−3.33178e33	−0.989870	−0.494935	+	0.868930i	$0.664808\pi$
$824$	0	0
$825$	5.80302e31	0.0167662
$826$	0	0
$827$	−8.24438e32	−0.231656	−0.115828	−	0.993269i	$-0.536952\pi$
$827$	−8.24438e32	−0.231656	−0.115828	+	0.993269i	$0.536952\pi$
$828$	0	0
$829$	2.26379e32	0.0618671	0.0309336	−	0.999521i	$-0.490152\pi$
$829$	2.26379e32	0.0618671	0.0309336	+	0.999521i	$0.490152\pi$
$830$	0	0
$831$	1.15219e33	0.306277
$832$	0	0
$833$	1.00972e32	0.0261086
$834$	0	0
$835$	1.92966e32	0.0485384
$836$	0	0
$837$	−7.14072e32	−0.174743
$838$	0	0
$839$	−5.25684e33	−1.25159	−0.625795	−	0.779987i	$-0.715225\pi$
$839$	−5.25684e33	−1.25159	−0.625795	+	0.779987i	$0.715225\pi$
$840$	0	0
$841$	1.26708e33	0.293528
$842$	0	0
$843$	5.01162e33	1.12969
$844$	0	0
$845$	−5.08947e33	−1.11640
$846$	0	0
$847$	−4.35713e33	−0.930124
$848$	0	0
$849$	2.98481e33	0.620124
$850$	0	0
$851$	−4.55646e33	−0.921377
$852$	0	0
$853$	−1.31287e33	−0.258408	−0.129204	−	0.991618i	$-0.541242\pi$
$853$	−1.31287e33	−0.258408	−0.129204	+	0.991618i	$0.541242\pi$
$854$	0	0
$855$	2.96005e33	0.567138
$856$	0	0
$857$	−8.73892e33	−1.62997	−0.814983	−	0.579485i	$-0.803254\pi$
$857$	−8.73892e33	−1.62997	−0.814983	+	0.579485i	$0.803254\pi$
$858$	0	0
$859$	−4.63183e33	−0.841068	−0.420534	−	0.907277i	$-0.638157\pi$
$859$	−4.63183e33	−0.841068	−0.420534	+	0.907277i	$0.638157\pi$
$860$	0	0
$861$	−2.09918e33	−0.371118
$862$	0	0
$863$	−4.55794e33	−0.784593	−0.392296	−	0.919839i	$-0.628319\pi$
$863$	−4.55794e33	−0.784593	−0.392296	+	0.919839i	$0.628319\pi$
$864$	0	0
$865$	1.04928e34	1.75876
$866$	0	0
$867$	3.34866e33	0.546578
$868$	0	0
$869$	−9.85351e32	−0.156626
$870$	0	0
$871$	3.37015e32	0.0521723
$872$	0	0
$873$	−2.36321e33	−0.356319
$874$	0	0
$875$	−5.32014e33	−0.781323
$876$	0	0
$877$	−8.18593e33	−1.17104	−0.585521	−	0.810657i	$-0.699110\pi$
$877$	−8.18593e33	−1.17104	−0.585521	+	0.810657i	$0.699110\pi$
$878$	0	0
$879$	−9.73419e32	−0.135652
$880$	0	0
$881$	7.93018e33	1.07661	0.538307	−	0.842749i	$-0.319064\pi$
$881$	7.93018e33	1.07661	0.538307	+	0.842749i	$0.319064\pi$
$882$	0	0
$883$	−3.70562e33	−0.490132	−0.245066	−	0.969506i	$-0.578810\pi$
$883$	−3.70562e33	−0.490132	−0.245066	+	0.969506i	$0.578810\pi$
$884$	0	0
$885$	1.92730e33	0.248371
$886$	0	0
$887$	−6.24016e33	−0.783563	−0.391781	−	0.920058i	$-0.628141\pi$
$887$	−6.24016e33	−0.783563	−0.391781	+	0.920058i	$0.628141\pi$
$888$	0	0
$889$	7.38934e33	0.904139
$890$	0	0
$891$	−1.03576e32	−0.0123500
$892$	0	0
$893$	−1.81807e34	−2.11260
$894$	0	0
$895$	8.27933e33	0.937628
$896$	0	0
$897$	4.52464e32	0.0499426
$898$	0	0
$899$	9.59855e33	1.03269
$900$	0	0
$901$	−1.13090e33	−0.118601
$902$	0	0
$903$	−5.19795e33	−0.531402
$904$	0	0
$905$	−1.67780e33	−0.167217
$906$	0	0
$907$	−6.25363e33	−0.607643	−0.303821	−	0.952729i	$-0.598263\pi$
$907$	−6.25363e33	−0.607643	−0.303821	+	0.952729i	$0.598263\pi$
$908$	0	0
$909$	−6.04885e33	−0.573044
$910$	0	0
$911$	−3.37934e33	−0.312155	−0.156078	−	0.987745i	$-0.549885\pi$
$911$	−3.37934e33	−0.312155	−0.156078	+	0.987745i	$0.549885\pi$
$912$	0	0
$913$	−1.68454e33	−0.151729
$914$	0	0
$915$	1.00596e34	0.883564
$916$	0	0
$917$	−1.39500e34	−1.19489
$918$	0	0
$919$	8.55820e33	0.714914	0.357457	−	0.933930i	$-0.383644\pi$
$919$	8.55820e33	0.714914	0.357457	+	0.933930i	$0.383644\pi$
$920$	0	0
$921$	8.49179e33	0.691852
$922$	0	0
$923$	8.97968e30	0.000713578 0
$924$	0	0
$925$	−2.76470e33	−0.214298
$926$	0	0
$927$	7.16535e33	0.541777
$928$	0	0
$929$	−2.43124e34	−1.79328	−0.896639	−	0.442762i	$-0.853999\pi$
$929$	−2.43124e34	−1.79328	−0.896639	+	0.442762i	$0.853999\pi$
$930$	0	0
$931$	−2.38095e33	−0.171328
$932$	0	0
$933$	−3.37345e33	−0.236829
$934$	0	0
$935$	4.20734e32	0.0288187
$936$	0	0
$937$	1.52537e34	1.01946	0.509728	−	0.860335i	$-0.329746\pi$
$937$	1.52537e34	1.01946	0.509728	+	0.860335i	$0.329746\pi$
$938$	0	0
$939$	−5.25881e33	−0.342952
$940$	0	0
$941$	−1.74266e34	−1.10900	−0.554501	−	0.832183i	$-0.687091\pi$
$941$	−1.74266e34	−1.10900	−0.554501	+	0.832183i	$0.687091\pi$
$942$	0	0
$943$	1.23459e34	0.766722
$944$	0	0
$945$	−3.35836e33	−0.203546
$946$	0	0
$947$	9.92620e33	0.587162	0.293581	−	0.955934i	$-0.405153\pi$
$947$	9.92620e33	0.587162	0.293581	+	0.955934i	$0.405153\pi$
$948$	0	0
$949$	4.42796e32	0.0255648
$950$	0	0
$951$	−1.18450e34	−0.667512
$952$	0	0
$953$	2.32496e34	1.27893	0.639465	−	0.768820i	$-0.279156\pi$
$953$	2.32496e34	1.27893	0.639465	+	0.768820i	$0.279156\pi$
$954$	0	0
$955$	2.64320e34	1.41935
$956$	0	0
$957$	1.39227e33	0.0729854
$958$	0	0
$959$	−1.66614e34	−0.852703
$960$	0	0
$961$	−3.51342e33	−0.175554
$962$	0	0
$963$	−7.81524e31	−0.00381276
$964$	0	0
$965$	1.26954e34	0.604761
$966$	0	0
$967$	−2.38294e34	−1.10843	−0.554214	−	0.832374i	$-0.686981\pi$
$967$	−2.38294e34	−1.10843	−0.554214	+	0.832374i	$0.686981\pi$
$968$	0	0
$969$	4.44560e33	0.201933
$970$	0	0
$971$	−3.30809e34	−1.46743	−0.733713	−	0.679460i	$-0.762214\pi$
$971$	−3.30809e34	−1.46743	−0.733713	+	0.679460i	$0.762214\pi$
$972$	0	0
$973$	−1.79595e34	−0.778029
$974$	0	0
$975$	2.74540e32	0.0116159
$976$	0	0
$977$	2.71162e33	0.112058	0.0560288	−	0.998429i	$-0.482156\pi$
$977$	2.71162e33	0.112058	0.0560288	+	0.998429i	$0.482156\pi$
$978$	0	0
$979$	1.27255e33	0.0513657
$980$	0	0
$981$	1.12803e34	0.444763
$982$	0	0
$983$	−4.00190e34	−1.54135	−0.770674	−	0.637229i	$-0.780080\pi$
$983$	−4.00190e34	−1.54135	−0.770674	+	0.637229i	$0.780080\pi$
$984$	0	0
$985$	4.99377e32	0.0187893
$986$	0	0
$987$	2.06271e34	0.758213
$988$	0	0
$989$	3.05707e34	1.09786
$990$	0	0
$991$	2.67938e34	0.940129	0.470064	−	0.882632i	$-0.344231\pi$
$991$	2.67938e34	0.940129	0.470064	+	0.882632i	$0.344231\pi$
$992$	0	0
$993$	3.07494e34	1.05419
$994$	0	0
$995$	1.85966e34	0.622970
$996$	0	0
$997$	3.50538e31	0.00114747	0.000573734	−	1.00000i	$-0.499817\pi$
$997$	3.50538e31	0.00114747	0.000573734		1.00000i	$0.499817\pi$
$998$	0	0
$999$	4.93463e33	0.157852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.24.a.j.1.3		3
4.3	odd	2		24.24.a.c.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.24.a.c.1.3	✓	3	4.3	odd	2
48.24.a.j.1.3		3	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q-177147. q^{3} +1.22619e8 q^{5} +4.92683e9 q^{7} +3.13811e10 q^{9} +O(q^{10})\)	\(q-177147. q^{3} +1.22619e8 q^{5} +4.92683e9 q^{7} +3.13811e10 q^{9} -1.05178e11 q^{11} -4.97595e11 q^{13} -2.17216e13 q^{15} -3.26230e13 q^{17} +7.69259e14 q^{19} -8.72773e14 q^{21} +5.13304e15 q^{23} +3.11455e15 q^{25} -5.55906e15 q^{27} +7.47248e16 q^{29} +1.28452e17 q^{31} +1.86320e16 q^{33} +6.04124e17 q^{35} -8.87673e17 q^{37} +8.81474e16 q^{39} +2.40518e18 q^{41} +5.95568e18 q^{43} +3.84792e18 q^{45} -2.36340e19 q^{47} -3.09512e18 q^{49} +5.77906e18 q^{51} +3.46656e19 q^{53} -1.28968e19 q^{55} -1.36272e20 q^{57} -8.87272e19 q^{59} -4.63114e20 q^{61} +1.54609e20 q^{63} -6.10147e19 q^{65} -6.77287e20 q^{67} -9.09302e20 q^{69} -1.80462e19 q^{71} -8.89872e20 q^{73} -5.51734e20 q^{75} -5.18194e20 q^{77} +9.36842e21 q^{79} +9.84771e20 q^{81} +1.60161e22 q^{83} -4.00021e21 q^{85} -1.32373e22 q^{87} -1.20990e22 q^{89} -2.45156e21 q^{91} -2.27549e22 q^{93} +9.43260e22 q^{95} -7.53070e22 q^{97} -3.30060e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 531441 q^{3} - 45012150 q^{5} - 281292072 q^{7} + 94143178827 q^{9}+O(q^{10}) \) 3 * q - 531441 * q^3 - 45012150 * q^5 - 281292072 * q^7 + 94143178827 * q^9	\( 3 q - 531441 q^{3} - 45012150 q^{5} - 281292072 q^{7} + 94143178827 q^{9} + 1198676057172 q^{11} - 282994829454 q^{13} + 7973767336050 q^{15} + 186934623333462 q^{17} + 118422337604076 q^{19} + 49830046678584 q^{21} + 63\!\cdots\!44 q^{23}+ \cdots + 37\!\cdots\!48 q^{99}+O(q^{100}) \) 3 * q - 531441 * q^3 - 45012150 * q^5 - 281292072 * q^7 + 94143178827 * q^9 + 1198676057172 * q^11 - 282994829454 * q^13 + 7973767336050 * q^15 + 186934623333462 * q^17 + 118422337604076 * q^19 + 49830046678584 * q^21 + 6395792138657544 * q^23 + 2573426290385925 * q^25 - 16677181699666569 * q^27 - 157790889708267966 * q^29 + 231251640669554784 * q^31 - 212341867499848284 * q^33 + 230145243295803600 * q^35 - 830491747621421334 * q^37 + 50131685053287738 * q^39 - 1400039039679261042 * q^41 - 761384577026785356 * q^43 - 1412528962279249350 * q^45 - 20706191933413048944 * q^47 + 26742597511830070875 * q^49 - 33114907719652792914 * q^51 - 64674463906638102342 * q^53 - 78855502334731828200 * q^55 - 20978161839549251172 * q^57 + 42370056643552637028 * q^59 - 516293667927208025118 * q^61 - 8827243278971119848 * q^63 - 1576925490906516336900 * q^65 + 1081135725593602868988 * q^67 - 1132995389986767946968 * q^69 - 22690312119034642536 * q^71 - 4166296087828394097618 * q^73 - 455874747062995455975 * q^75 - 7709458077454199230560 * q^77 - 5410946432474399204880 * q^79 + 2954312706550833698643 * q^81 + 18793771492158984347820 * q^83 - 38369307356296983199500 * q^85 + 27952182739150545373002 * q^87 - 23506358152298677341570 * q^89 + 128231916389440977486864 * q^91 - 40965534389689621321248 * q^93 + 196572725231897227799400 * q^95 - 129282257648368490763546 * q^97 + 37615724801995623965748 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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