LMFDB - Embedded newform 48.14.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,14,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, 14, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 14 $
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:	$51.4708458969$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q+729.000 q^{3} -30210.0 q^{5} -235088. q^{7} +531441. q^{9} +O(q^{10})$

$q+729.000 q^{3} -30210.0 q^{5} -235088. q^{7} +531441. q^{9} +1.11829e7 q^{11} +8.04961e6 q^{13} -2.20231e7 q^{15} -1.17495e8 q^{17} +2.14061e8 q^{19} -1.71379e8 q^{21} -8.30556e8 q^{23} -3.08059e8 q^{25} +3.87420e8 q^{27} -1.25240e9 q^{29} -6.15935e9 q^{31} +8.15234e9 q^{33} +7.10201e9 q^{35} -5.49819e9 q^{37} +5.86817e9 q^{39} -4.67869e9 q^{41} -7.11501e9 q^{43} -1.60548e10 q^{45} +2.95288e10 q^{47} -4.16226e10 q^{49} -8.56536e10 q^{51} -2.04125e11 q^{53} -3.37836e11 q^{55} +1.56051e11 q^{57} +2.99098e10 q^{59} -1.34392e11 q^{61} -1.24935e11 q^{63} -2.43179e11 q^{65} -3.48519e11 q^{67} -6.05475e11 q^{69} -1.31434e12 q^{71} -1.17888e12 q^{73} -2.24575e11 q^{75} -2.62897e12 q^{77} +1.07242e12 q^{79} +2.82430e11 q^{81} -1.12403e12 q^{83} +3.54951e12 q^{85} -9.13000e11 q^{87} +2.23561e12 q^{89} -1.89237e12 q^{91} -4.49017e12 q^{93} -6.46679e12 q^{95} -1.42153e13 q^{97} +5.94306e12 q^{99} +O(q^{100})$

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
$3$	729.000	0.577350
$3$	729.000	0.577350
$4$	0	0
$5$	−30210.0	−0.864661	−0.432330	−	0.901715i	$-0.642309\pi$
$5$	−30210.0	−0.864661	−0.432330	+	0.901715i	$0.642309\pi$
$6$	0	0
$7$	−235088.	−0.755254	−0.377627	−	0.925958i	$-0.623260\pi$
$7$	−235088.	−0.755254	−0.377627	+	0.925958i	$0.623260\pi$
$8$	0	0
$9$	531441.	0.333333
$10$	0	0
$11$	1.11829e7	1.90328	0.951639	−	0.307218i	$-0.0993981\pi$
$11$	1.11829e7	1.90328	0.951639	+	0.307218i	$0.0993981\pi$
$12$	0	0
$13$	8.04961e6	0.462534	0.231267	−	0.972890i	$-0.425713\pi$
$13$	8.04961e6	0.462534	0.231267	+	0.972890i	$0.425713\pi$
$14$	0	0
$15$	−2.20231e7	−0.499212
$16$	0	0
$17$	−1.17495e8	−1.18059	−0.590296	−	0.807187i	$-0.700989\pi$
$17$	−1.17495e8	−1.18059	−0.590296	+	0.807187i	$0.700989\pi$
$18$	0	0
$19$	2.14061e8	1.04385	0.521927	−	0.852990i	$-0.325213\pi$
$19$	2.14061e8	1.04385	0.521927	+	0.852990i	$0.325213\pi$
$20$	0	0
$21$	−1.71379e8	−0.436046
$22$	0	0
$23$	−8.30556e8	−1.16987	−0.584935	−	0.811080i	$-0.698880\pi$
$23$	−8.30556e8	−1.16987	−0.584935	+	0.811080i	$0.698880\pi$
$24$	0	0
$25$	−3.08059e8	−0.252362
$26$	0	0
$27$	3.87420e8	0.192450
$28$	0	0
$29$	−1.25240e9	−0.390981	−0.195491	−	0.980706i	$-0.562630\pi$
$29$	−1.25240e9	−0.390981	−0.195491	+	0.980706i	$0.562630\pi$
$30$	0	0
$31$	−6.15935e9	−1.24648	−0.623238	−	0.782032i	$-0.714183\pi$
$31$	−6.15935e9	−1.24648	−0.623238	+	0.782032i	$0.714183\pi$
$32$	0	0
$33$	8.15234e9	1.09886
$34$	0	0
$35$	7.10201e9	0.653039
$36$	0	0
$37$	−5.49819e9	−0.352297	−0.176148	−	0.984364i	$-0.556364\pi$
$37$	−5.49819e9	−0.352297	−0.176148	+	0.984364i	$0.556364\pi$
$38$	0	0
$39$	5.86817e9	0.267044
$40$	0	0
$41$	−4.67869e9	−0.153826	−0.0769129	−	0.997038i	$-0.524506\pi$
$41$	−4.67869e9	−0.153826	−0.0769129	+	0.997038i	$0.524506\pi$
$42$	0	0
$43$	−7.11501e9	−0.171645	−0.0858224	−	0.996310i	$-0.527352\pi$
$43$	−7.11501e9	−0.171645	−0.0858224	+	0.996310i	$0.527352\pi$
$44$	0	0
$45$	−1.60548e10	−0.288220
$46$	0	0
$47$	2.95288e10	0.399585	0.199793	−	0.979838i	$-0.435973\pi$
$47$	2.95288e10	0.399585	0.199793	+	0.979838i	$0.435973\pi$
$48$	0	0
$49$	−4.16226e10	−0.429591
$50$	0	0
$51$	−8.56536e10	−0.681615
$52$	0	0
$53$	−2.04125e11	−1.26504	−0.632518	−	0.774545i	$-0.717979\pi$
$53$	−2.04125e11	−1.26504	−0.632518	+	0.774545i	$0.717979\pi$
$54$	0	0
$55$	−3.37836e11	−1.64569
$56$	0	0
$57$	1.56051e11	0.602670
$58$	0	0
$59$	2.99098e10	0.0923157	0.0461579	−	0.998934i	$-0.485302\pi$
$59$	2.99098e10	0.0923157	0.0461579	+	0.998934i	$0.485302\pi$
$60$	0	0
$61$	−1.34392e11	−0.333987	−0.166993	−	0.985958i	$-0.553406\pi$
$61$	−1.34392e11	−0.333987	−0.166993	+	0.985958i	$0.553406\pi$
$62$	0	0
$63$	−1.24935e11	−0.251751
$64$	0	0
$65$	−2.43179e11	−0.399935
$66$	0	0
$67$	−3.48519e11	−0.470695	−0.235348	−	0.971911i	$-0.575623\pi$
$67$	−3.48519e11	−0.470695	−0.235348	+	0.971911i	$0.575623\pi$
$68$	0	0
$69$	−6.05475e11	−0.675425
$70$	0	0
$71$	−1.31434e12	−1.21766	−0.608831	−	0.793300i	$-0.708361\pi$
$71$	−1.31434e12	−1.21766	−0.608831	+	0.793300i	$0.708361\pi$
$72$	0	0
$73$	−1.17888e12	−0.911737	−0.455868	−	0.890047i	$-0.650671\pi$
$73$	−1.17888e12	−0.911737	−0.455868	+	0.890047i	$0.650671\pi$
$74$	0	0
$75$	−2.24575e11	−0.145701
$76$	0	0
$77$	−2.62897e12	−1.43746
$78$	0	0
$79$	1.07242e12	0.496351	0.248176	−	0.968715i	$-0.420169\pi$
$79$	1.07242e12	0.496351	0.248176	+	0.968715i	$0.420169\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	0	0
$83$	−1.12403e12	−0.377371	−0.188685	−	0.982038i	$-0.560423\pi$
$83$	−1.12403e12	−0.377371	−0.188685	+	0.982038i	$0.560423\pi$
$84$	0	0
$85$	3.54951e12	1.02081
$86$	0	0
$87$	−9.13000e11	−0.225733
$88$	0	0
$89$	2.23561e12	0.476827	0.238414	−	0.971164i	$-0.423373\pi$
$89$	2.23561e12	0.476827	0.238414	+	0.971164i	$0.423373\pi$
$90$	0	0
$91$	−1.89237e12	−0.349330
$92$	0	0
$93$	−4.49017e12	−0.719653
$94$	0	0
$95$	−6.46679e12	−0.902580
$96$	0	0
$97$	−1.42153e13	−1.73276	−0.866380	−	0.499385i	$-0.833559\pi$
$97$	−1.42153e13	−1.73276	−0.866380	+	0.499385i	$0.833559\pi$
$98$	0	0
$99$	5.94306e12	0.634426
$100$	0	0
$101$	1.70194e13	1.59535	0.797675	−	0.603088i	$-0.206063\pi$
$101$	1.70194e13	1.59535	0.797675	+	0.603088i	$0.206063\pi$
$102$	0	0
$103$	−1.09904e13	−0.906928	−0.453464	−	0.891275i	$-0.649812\pi$
$103$	−1.09904e13	−0.906928	−0.453464	+	0.891275i	$0.649812\pi$
$104$	0	0
$105$	5.17736e12	0.377032
$106$	0	0
$107$	1.96403e13	1.26519	0.632593	−	0.774485i	$-0.281991\pi$
$107$	1.96403e13	1.26519	0.632593	+	0.774485i	$0.281991\pi$
$108$	0	0
$109$	−9.82099e12	−0.560897	−0.280448	−	0.959869i	$-0.590483\pi$
$109$	−9.82099e12	−0.560897	−0.280448	+	0.959869i	$0.590483\pi$
$110$	0	0
$111$	−4.00818e12	−0.203399
$112$	0	0
$113$	−1.70267e13	−0.769344	−0.384672	−	0.923053i	$-0.625685\pi$
$113$	−1.70267e13	−0.769344	−0.384672	+	0.923053i	$0.625685\pi$
$114$	0	0
$115$	2.50911e13	1.01154
$116$	0	0
$117$	4.27789e12	0.154178
$118$	0	0
$119$	2.76216e13	0.891648
$120$	0	0
$121$	9.05347e13	2.62247
$122$	0	0
$123$	−3.41076e12	−0.0888113
$124$	0	0
$125$	4.61839e13	1.08287
$126$	0	0
$127$	4.49347e13	0.950292	0.475146	−	0.879907i	$-0.342395\pi$
$127$	4.49347e13	0.950292	0.475146	+	0.879907i	$0.342395\pi$
$128$	0	0
$129$	−5.18685e12	−0.0990992
$130$	0	0
$131$	1.20182e12	0.0207768	0.0103884	−	0.999946i	$-0.496693\pi$
$131$	1.20182e12	0.0207768	0.0103884	+	0.999946i	$0.496693\pi$
$132$	0	0
$133$	−5.03233e13	−0.788375
$134$	0	0
$135$	−1.17040e13	−0.166404
$136$	0	0
$137$	1.71562e13	0.221685	0.110842	−	0.993838i	$-0.464645\pi$
$137$	1.71562e13	0.221685	0.110842	+	0.993838i	$0.464645\pi$
$138$	0	0
$139$	−1.05644e14	−1.24236	−0.621182	−	0.783666i	$-0.713347\pi$
$139$	−1.05644e14	−1.24236	−0.621182	+	0.783666i	$0.713347\pi$
$140$	0	0
$141$	2.15265e13	0.230701
$142$	0	0
$143$	9.00181e13	0.880330
$144$	0	0
$145$	3.78350e13	0.338066
$146$	0	0
$147$	−3.03429e13	−0.248024
$148$	0	0
$149$	−8.53533e13	−0.639012	−0.319506	−	0.947584i	$-0.603517\pi$
$149$	−8.53533e13	−0.639012	−0.319506	+	0.947584i	$0.603517\pi$
$150$	0	0
$151$	6.16414e13	0.423177	0.211589	−	0.977359i	$-0.432136\pi$
$151$	6.16414e13	0.423177	0.211589	+	0.977359i	$0.432136\pi$
$152$	0	0
$153$	−6.24415e13	−0.393531
$154$	0	0
$155$	1.86074e14	1.07778
$156$	0	0
$157$	−1.18021e14	−0.628942	−0.314471	−	0.949267i	$-0.601827\pi$
$157$	−1.18021e14	−0.628942	−0.314471	+	0.949267i	$0.601827\pi$
$158$	0	0
$159$	−1.48807e14	−0.730369
$160$	0	0
$161$	1.95254e14	0.883550
$162$	0	0
$163$	−1.54710e14	−0.646099	−0.323050	−	0.946382i	$-0.604708\pi$
$163$	−1.54710e14	−0.646099	−0.323050	+	0.946382i	$0.604708\pi$
$164$	0	0
$165$	−2.46282e14	−0.950139
$166$	0	0
$167$	−3.76012e14	−1.34136	−0.670679	−	0.741748i	$-0.733997\pi$
$167$	−3.76012e14	−1.34136	−0.670679	+	0.741748i	$0.733997\pi$
$168$	0	0
$169$	−2.38079e14	−0.786063
$170$	0	0
$171$	1.13761e14	0.347951
$172$	0	0
$173$	3.73562e14	1.05941	0.529704	−	0.848182i	$-0.322303\pi$
$173$	3.73562e14	1.05941	0.529704	+	0.848182i	$0.322303\pi$
$174$	0	0
$175$	7.24210e13	0.190597
$176$	0	0
$177$	2.18043e13	0.0532985
$178$	0	0
$179$	−4.23349e13	−0.0961952	−0.0480976	−	0.998843i	$-0.515316\pi$
$179$	−4.23349e13	−0.0961952	−0.0480976	+	0.998843i	$0.515316\pi$
$180$	0	0
$181$	−3.10447e14	−0.656261	−0.328130	−	0.944632i	$-0.606419\pi$
$181$	−3.10447e14	−0.656261	−0.328130	+	0.944632i	$0.606419\pi$
$182$	0	0
$183$	−9.79718e13	−0.192827
$184$	0	0
$185$	1.66100e14	0.304617
$186$	0	0
$187$	−1.31393e15	−2.24700
$188$	0	0
$189$	−9.10779e13	−0.145349
$190$	0	0
$191$	8.62273e14	1.28507	0.642537	−	0.766255i	$-0.277882\pi$
$191$	8.62273e14	1.28507	0.642537	+	0.766255i	$0.277882\pi$
$192$	0	0
$193$	−9.37837e14	−1.30618	−0.653092	−	0.757278i	$-0.726529\pi$
$193$	−9.37837e14	−1.30618	−0.653092	+	0.757278i	$0.726529\pi$
$194$	0	0
$195$	−1.77277e14	−0.230902
$196$	0	0
$197$	−6.71715e14	−0.818756	−0.409378	−	0.912365i	$-0.634254\pi$
$197$	−6.71715e14	−0.818756	−0.409378	+	0.912365i	$0.634254\pi$
$198$	0	0
$199$	4.36451e13	0.0498185	0.0249093	−	0.999690i	$-0.492070\pi$
$199$	4.36451e13	0.0498185	0.0249093	+	0.999690i	$0.492070\pi$
$200$	0	0
$201$	−2.54070e14	−0.271756
$202$	0	0
$203$	2.94424e14	0.295290
$204$	0	0
$205$	1.41343e14	0.133007
$206$	0	0
$207$	−4.41391e14	−0.389957
$208$	0	0
$209$	2.39383e15	1.98675
$210$	0	0
$211$	1.62162e15	1.26507	0.632534	−	0.774533i	$-0.282015\pi$
$211$	1.62162e15	1.26507	0.632534	+	0.774533i	$0.282015\pi$
$212$	0	0
$213$	−9.58151e14	−0.703018
$214$	0	0
$215$	2.14945e14	0.148415
$216$	0	0
$217$	1.44799e15	0.941406
$218$	0	0
$219$	−8.59401e14	−0.526392
$220$	0	0
$221$	−9.45786e14	−0.546064
$222$	0	0
$223$	−1.47333e15	−0.802266	−0.401133	−	0.916020i	$-0.631383\pi$
$223$	−1.47333e15	−0.802266	−0.401133	+	0.916020i	$0.631383\pi$
$224$	0	0
$225$	−1.63715e14	−0.0841207
$226$	0	0
$227$	3.74889e15	1.81859	0.909294	−	0.416153i	$-0.136622\pi$
$227$	3.74889e15	1.81859	0.909294	+	0.416153i	$0.136622\pi$
$228$	0	0
$229$	−1.47993e13	−0.00678126	−0.00339063	−	0.999994i	$-0.501079\pi$
$229$	−1.47993e13	−0.00678126	−0.00339063	+	0.999994i	$0.501079\pi$
$230$	0	0
$231$	−1.91652e15	−0.829917
$232$	0	0
$233$	3.63053e15	1.48647	0.743236	−	0.669030i	$-0.233290\pi$
$233$	3.63053e15	1.48647	0.743236	+	0.669030i	$0.233290\pi$
$234$	0	0
$235$	−8.92064e14	−0.345506
$236$	0	0
$237$	7.81795e14	0.286569
$238$	0	0
$239$	4.33900e15	1.50592	0.752962	−	0.658063i	$-0.228624\pi$
$239$	4.33900e15	1.50592	0.752962	+	0.658063i	$0.228624\pi$
$240$	0	0
$241$	3.02372e15	0.994103	0.497051	−	0.867721i	$-0.334416\pi$
$241$	3.02372e15	0.994103	0.497051	+	0.867721i	$0.334416\pi$
$242$	0	0
$243$	2.05891e14	0.0641500
$244$	0	0
$245$	1.25742e15	0.371450
$246$	0	0
$247$	1.72311e15	0.482818
$248$	0	0
$249$	−8.19414e14	−0.217875
$250$	0	0
$251$	1.75146e15	0.442099	0.221050	−	0.975263i	$-0.429052\pi$
$251$	1.75146e15	0.442099	0.221050	+	0.975263i	$0.429052\pi$
$252$	0	0
$253$	−9.28803e15	−2.22659
$254$	0	0
$255$	2.58759e15	0.589366
$256$	0	0
$257$	4.87604e15	1.05561	0.527803	−	0.849367i	$-0.323016\pi$
$257$	4.87604e15	1.05561	0.527803	+	0.849367i	$0.323016\pi$
$258$	0	0
$259$	1.29256e15	0.266074
$260$	0	0
$261$	−6.65577e14	−0.130327
$262$	0	0
$263$	−4.67882e15	−0.871815	−0.435907	−	0.899992i	$-0.643572\pi$
$263$	−4.67882e15	−0.871815	−0.435907	+	0.899992i	$0.643572\pi$
$264$	0	0
$265$	6.16662e15	1.09383
$266$	0	0
$267$	1.62976e15	0.275296
$268$	0	0
$269$	−1.80262e15	−0.290078	−0.145039	−	0.989426i	$-0.546331\pi$
$269$	−1.80262e15	−0.290078	−0.145039	+	0.989426i	$0.546331\pi$
$270$	0	0
$271$	−6.10016e15	−0.935494	−0.467747	−	0.883862i	$-0.654934\pi$
$271$	−6.10016e15	−0.935494	−0.467747	+	0.883862i	$0.654934\pi$
$272$	0	0
$273$	−1.37954e15	−0.201686
$274$	0	0
$275$	−3.44500e15	−0.480315
$276$	0	0
$277$	−1.07023e16	−1.42351	−0.711754	−	0.702428i	$-0.752099\pi$
$277$	−1.07023e16	−1.42351	−0.711754	+	0.702428i	$0.752099\pi$
$278$	0	0
$279$	−3.27333e15	−0.415492
$280$	0	0
$281$	−2.45460e15	−0.297433	−0.148717	−	0.988880i	$-0.547514\pi$
$281$	−2.45460e15	−0.297433	−0.148717	+	0.988880i	$0.547514\pi$
$282$	0	0
$283$	−4.01155e15	−0.464195	−0.232098	−	0.972692i	$-0.574559\pi$
$283$	−4.01155e15	−0.464195	−0.232098	+	0.972692i	$0.574559\pi$
$284$	0	0
$285$	−4.71429e15	−0.521105
$286$	0	0
$287$	1.09990e15	0.116178
$288$	0	0
$289$	3.90041e15	0.393799
$290$	0	0
$291$	−1.03629e16	−1.00041
$292$	0	0
$293$	2.08187e15	0.192227	0.0961133	−	0.995370i	$-0.469359\pi$
$293$	2.08187e15	0.192227	0.0961133	+	0.995370i	$0.469359\pi$
$294$	0	0
$295$	−9.03576e14	−0.0798218
$296$	0	0
$297$	4.33249e15	0.366286
$298$	0	0
$299$	−6.68565e15	−0.541104
$300$	0	0
$301$	1.67265e15	0.129636
$302$	0	0
$303$	1.24072e16	0.921075
$304$	0	0
$305$	4.05998e15	0.288785
$306$	0	0
$307$	1.32352e16	0.902260	0.451130	−	0.892458i	$-0.351021\pi$
$307$	1.32352e16	0.902260	0.451130	+	0.892458i	$0.351021\pi$
$308$	0	0
$309$	−8.01202e15	−0.523615
$310$	0	0
$311$	8.09301e15	0.507187	0.253593	−	0.967311i	$-0.418388\pi$
$311$	8.09301e15	0.507187	0.253593	+	0.967311i	$0.418388\pi$
$312$	0	0
$313$	−1.48181e16	−0.890748	−0.445374	−	0.895345i	$-0.646929\pi$
$313$	−1.48181e16	−0.890748	−0.445374	+	0.895345i	$0.646929\pi$
$314$	0	0
$315$	3.77430e15	0.217680
$316$	0	0
$317$	−2.43171e16	−1.34594	−0.672970	−	0.739670i	$-0.734982\pi$
$317$	−2.43171e16	−1.34594	−0.672970	+	0.739670i	$0.734982\pi$
$318$	0	0
$319$	−1.40055e16	−0.744147
$320$	0	0
$321$	1.43178e16	0.730455
$322$	0	0
$323$	−2.51511e16	−1.23237
$324$	0	0
$325$	−2.47976e15	−0.116726
$326$	0	0
$327$	−7.15950e15	−0.323834
$328$	0	0
$329$	−6.94186e15	−0.301788
$330$	0	0
$331$	−1.16232e16	−0.485783	−0.242892	−	0.970053i	$-0.578096\pi$
$331$	−1.16232e16	−0.485783	−0.242892	+	0.970053i	$0.578096\pi$
$332$	0	0
$333$	−2.92196e15	−0.117432
$334$	0	0
$335$	1.05288e16	0.406992
$336$	0	0
$337$	4.62652e16	1.72052	0.860262	−	0.509853i	$-0.170300\pi$
$337$	4.62652e16	1.72052	0.860262	+	0.509853i	$0.170300\pi$
$338$	0	0
$339$	−1.24125e16	−0.444181
$340$	0	0
$341$	−6.88795e16	−2.37239
$342$	0	0
$343$	3.25624e16	1.07970
$344$	0	0
$345$	1.82914e16	0.584013
$346$	0	0
$347$	−4.79404e15	−0.147421	−0.0737106	−	0.997280i	$-0.523484\pi$
$347$	−4.79404e15	−0.147421	−0.0737106	+	0.997280i	$0.523484\pi$
$348$	0	0
$349$	3.76900e16	1.11651	0.558253	−	0.829671i	$-0.311472\pi$
$349$	3.76900e16	1.11651	0.558253	+	0.829671i	$0.311472\pi$
$350$	0	0
$351$	3.11859e15	0.0890146
$352$	0	0
$353$	4.80179e16	1.32089	0.660446	−	0.750873i	$-0.270367\pi$
$353$	4.80179e16	1.32089	0.660446	+	0.750873i	$0.270367\pi$
$354$	0	0
$355$	3.97061e16	1.05286
$356$	0	0
$357$	2.01361e16	0.514793
$358$	0	0
$359$	−4.06616e16	−1.00247	−0.501234	−	0.865312i	$-0.667120\pi$
$359$	−4.06616e16	−1.00247	−0.501234	+	0.865312i	$0.667120\pi$
$360$	0	0
$361$	3.76929e15	0.0896320
$362$	0	0
$363$	6.59998e16	1.51408
$364$	0	0
$365$	3.56138e16	0.788343
$366$	0	0
$367$	−2.96733e16	−0.633923	−0.316961	−	0.948438i	$-0.602663\pi$
$367$	−2.96733e16	−0.633923	−0.316961	+	0.948438i	$0.602663\pi$
$368$	0	0
$369$	−2.48645e15	−0.0512752
$370$	0	0
$371$	4.79873e16	0.955424
$372$	0	0
$373$	−9.01346e16	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$373$	−9.01346e16	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$374$	0	0
$375$	3.36681e16	0.625194
$376$	0	0
$377$	−1.00813e16	−0.180842
$378$	0	0
$379$	1.54841e16	0.268369	0.134184	−	0.990956i	$-0.457159\pi$
$379$	1.54841e16	0.268369	0.134184	+	0.990956i	$0.457159\pi$
$380$	0	0
$381$	3.27574e16	0.548652
$382$	0	0
$383$	−9.37088e15	−0.151701	−0.0758505	−	0.997119i	$-0.524167\pi$
$383$	−9.37088e15	−0.151701	−0.0758505	+	0.997119i	$0.524167\pi$
$384$	0	0
$385$	7.94211e16	1.24291
$386$	0	0
$387$	−3.78121e15	−0.0572150
$388$	0	0
$389$	2.95806e16	0.432847	0.216423	−	0.976300i	$-0.430561\pi$
$389$	2.95806e16	0.432847	0.216423	+	0.976300i	$0.430561\pi$
$390$	0	0
$391$	9.75858e16	1.38114
$392$	0	0
$393$	8.76130e14	0.0119955
$394$	0	0
$395$	−3.23978e16	−0.429175
$396$	0	0
$397$	1.80617e16	0.231538	0.115769	−	0.993276i	$-0.463067\pi$
$397$	1.80617e16	0.231538	0.115769	+	0.993276i	$0.463067\pi$
$398$	0	0
$399$	−3.66857e16	−0.455169
$400$	0	0
$401$	−1.20412e17	−1.44621	−0.723107	−	0.690736i	$-0.757287\pi$
$401$	−1.20412e17	−1.44621	−0.723107	+	0.690736i	$0.757287\pi$
$402$	0	0
$403$	−4.95804e16	−0.576537
$404$	0	0
$405$	−8.53220e15	−0.0960734
$406$	0	0
$407$	−6.14858e16	−0.670519
$408$	0	0
$409$	−1.77522e16	−0.187521	−0.0937606	−	0.995595i	$-0.529889\pi$
$409$	−1.77522e16	−0.187521	−0.0937606	+	0.995595i	$0.529889\pi$
$410$	0	0
$411$	1.25068e16	0.127990
$412$	0	0
$413$	−7.03144e15	−0.0697219
$414$	0	0
$415$	3.39568e16	0.326298
$416$	0	0
$417$	−7.70145e16	−0.717279
$418$	0	0
$419$	−1.75670e17	−1.58602	−0.793008	−	0.609212i	$-0.791486\pi$
$419$	−1.75670e17	−1.58602	−0.793008	+	0.609212i	$0.791486\pi$
$420$	0	0
$421$	1.84473e17	1.61473	0.807365	−	0.590052i	$-0.200893\pi$
$421$	1.84473e17	1.61473	0.807365	+	0.590052i	$0.200893\pi$
$422$	0	0
$423$	1.56928e16	0.133195
$424$	0	0
$425$	3.61953e16	0.297937
$426$	0	0
$427$	3.15939e16	0.252245
$428$	0	0
$429$	6.56232e16	0.508259
$430$	0	0
$431$	−8.05532e16	−0.605314	−0.302657	−	0.953100i	$-0.597874\pi$
$431$	−8.05532e16	−0.605314	−0.302657	+	0.953100i	$0.597874\pi$
$432$	0	0
$433$	−1.97092e17	−1.43714	−0.718568	−	0.695457i	$-0.755202\pi$
$433$	−1.97092e17	−1.43714	−0.718568	+	0.695457i	$0.755202\pi$
$434$	0	0
$435$	2.75817e16	0.195183
$436$	0	0
$437$	−1.77790e17	−1.22117
$438$	0	0
$439$	−9.89007e16	−0.659447	−0.329724	−	0.944078i	$-0.606956\pi$
$439$	−9.89007e16	−0.659447	−0.329724	+	0.944078i	$0.606956\pi$
$440$	0	0
$441$	−2.21200e16	−0.143197
$442$	0	0
$443$	1.25104e17	0.786404	0.393202	−	0.919452i	$-0.371367\pi$
$443$	1.25104e17	0.786404	0.393202	+	0.919452i	$0.371367\pi$
$444$	0	0
$445$	−6.75378e16	−0.412294
$446$	0	0
$447$	−6.22225e16	−0.368934
$448$	0	0
$449$	−1.80095e17	−1.03729	−0.518645	−	0.854990i	$-0.673563\pi$
$449$	−1.80095e17	−1.03729	−0.518645	+	0.854990i	$0.673563\pi$
$450$	0	0
$451$	−5.23213e16	−0.292773
$452$	0	0
$453$	4.49366e16	0.244322
$454$	0	0
$455$	5.71684e16	0.302052
$456$	0	0
$457$	−9.43597e16	−0.484542	−0.242271	−	0.970209i	$-0.577892\pi$
$457$	−9.43597e16	−0.484542	−0.242271	+	0.970209i	$0.577892\pi$
$458$	0	0
$459$	−4.55198e16	−0.227205
$460$	0	0
$461$	8.00500e16	0.388423	0.194212	−	0.980960i	$-0.437785\pi$
$461$	8.00500e16	0.388423	0.194212	+	0.980960i	$0.437785\pi$
$462$	0	0
$463$	−2.14174e17	−1.01039	−0.505196	−	0.863004i	$-0.668580\pi$
$463$	−2.14174e17	−1.01039	−0.505196	+	0.863004i	$0.668580\pi$
$464$	0	0
$465$	1.35648e17	0.622256
$466$	0	0
$467$	1.80681e17	0.806031	0.403015	−	0.915193i	$-0.367962\pi$
$467$	1.80681e17	0.806031	0.403015	+	0.915193i	$0.367962\pi$
$468$	0	0
$469$	8.19326e16	0.355495
$470$	0	0
$471$	−8.60372e16	−0.363120
$472$	0	0
$473$	−7.95665e16	−0.326688
$474$	0	0
$475$	−6.59435e16	−0.263429
$476$	0	0
$477$	−1.08480e17	−0.421679
$478$	0	0
$479$	2.66712e17	1.00893	0.504466	−	0.863431i	$-0.331689\pi$
$479$	2.66712e17	1.00893	0.504466	+	0.863431i	$0.331689\pi$
$480$	0	0
$481$	−4.42583e16	−0.162949
$482$	0	0
$483$	1.42340e17	0.510118
$484$	0	0
$485$	4.29443e17	1.49825
$486$	0	0
$487$	2.63552e17	0.895216	0.447608	−	0.894230i	$-0.352276\pi$
$487$	2.63552e17	0.895216	0.447608	+	0.894230i	$0.352276\pi$
$488$	0	0
$489$	−1.12784e17	−0.373026
$490$	0	0
$491$	−4.11733e17	−1.32613	−0.663065	−	0.748562i	$-0.730745\pi$
$491$	−4.11733e17	−1.32613	−0.663065	+	0.748562i	$0.730745\pi$
$492$	0	0
$493$	1.47150e17	0.461590
$494$	0	0
$495$	−1.79540e17	−0.548563
$496$	0	0
$497$	3.08984e17	0.919645
$498$	0	0
$499$	−3.99658e17	−1.15887	−0.579435	−	0.815018i	$-0.696727\pi$
$499$	−3.99658e17	−1.15887	−0.579435	+	0.815018i	$0.696727\pi$
$500$	0	0
$501$	−2.74113e17	−0.774433
$502$	0	0
$503$	2.83581e17	0.780702	0.390351	−	0.920666i	$-0.372354\pi$
$503$	2.83581e17	0.780702	0.390351	+	0.920666i	$0.372354\pi$
$504$	0	0
$505$	−5.14157e17	−1.37944
$506$	0	0
$507$	−1.73559e17	−0.453834
$508$	0	0
$509$	6.40327e17	1.63206	0.816030	−	0.578009i	$-0.196170\pi$
$509$	6.40327e17	1.63206	0.816030	+	0.578009i	$0.196170\pi$
$510$	0	0
$511$	2.77140e17	0.688593
$512$	0	0
$513$	8.29318e16	0.200890
$514$	0	0
$515$	3.32021e17	0.784185
$516$	0	0
$517$	3.30218e17	0.760522
$518$	0	0
$519$	2.72327e17	0.611650
$520$	0	0
$521$	−4.01348e17	−0.879175	−0.439588	−	0.898200i	$-0.644875\pi$
$521$	−4.01348e17	−0.879175	−0.439588	+	0.898200i	$0.644875\pi$
$522$	0	0
$523$	5.05985e17	1.08113	0.540564	−	0.841303i	$-0.318211\pi$
$523$	5.05985e17	1.08113	0.540564	+	0.841303i	$0.318211\pi$
$524$	0	0
$525$	5.27949e16	0.110041
$526$	0	0
$527$	7.23691e17	1.47158
$528$	0	0
$529$	1.85786e17	0.368597
$530$	0	0
$531$	1.58953e16	0.0307719
$532$	0	0
$533$	−3.76616e16	−0.0711496
$534$	0	0
$535$	−5.93334e17	−1.09396
$536$	0	0
$537$	−3.08621e16	−0.0555383
$538$	0	0
$539$	−4.65462e17	−0.817631
$540$	0	0
$541$	−1.69124e17	−0.290017	−0.145009	−	0.989430i	$-0.546321\pi$
$541$	−1.69124e17	−0.290017	−0.145009	+	0.989430i	$0.546321\pi$
$542$	0	0
$543$	−2.26316e17	−0.378892
$544$	0	0
$545$	2.96692e17	0.484986
$546$	0	0
$547$	4.32104e17	0.689717	0.344858	−	0.938655i	$-0.387927\pi$
$547$	4.32104e17	0.689717	0.344858	+	0.938655i	$0.387927\pi$
$548$	0	0
$549$	−7.14214e16	−0.111329
$550$	0	0
$551$	−2.68091e17	−0.408128
$552$	0	0
$553$	−2.52113e17	−0.374871
$554$	0	0
$555$	1.21087e17	0.175871
$556$	0	0
$557$	1.36804e18	1.94107	0.970534	−	0.240966i	$-0.0774642\pi$
$557$	1.36804e18	1.94107	0.970534	+	0.240966i	$0.0774642\pi$
$558$	0	0
$559$	−5.72731e16	−0.0793915
$560$	0	0
$561$	−9.57856e17	−1.29730
$562$	0	0
$563$	9.52405e17	1.26043	0.630213	−	0.776422i	$-0.282968\pi$
$563$	9.52405e17	1.26043	0.630213	+	0.776422i	$0.282968\pi$
$564$	0	0
$565$	5.14377e17	0.665221
$566$	0	0
$567$	−6.63958e16	−0.0839171
$568$	0	0
$569$	1.53632e17	0.189780	0.0948902	−	0.995488i	$-0.469750\pi$
$569$	1.53632e17	0.189780	0.0948902	+	0.995488i	$0.469750\pi$
$570$	0	0
$571$	1.27956e18	1.54500	0.772498	−	0.635017i	$-0.219007\pi$
$571$	1.27956e18	1.54500	0.772498	+	0.635017i	$0.219007\pi$
$572$	0	0
$573$	6.28597e17	0.741938
$574$	0	0
$575$	2.55860e17	0.295231
$576$	0	0
$577$	3.56770e17	0.402481	0.201241	−	0.979542i	$-0.435503\pi$
$577$	3.56770e17	0.402481	0.201241	+	0.979542i	$0.435503\pi$
$578$	0	0
$579$	−6.83683e17	−0.754126
$580$	0	0
$581$	2.64245e17	0.285011
$582$	0	0
$583$	−2.28271e18	−2.40772
$584$	0	0
$585$	−1.29235e17	−0.133312
$586$	0	0
$587$	−1.28968e18	−1.30118	−0.650588	−	0.759431i	$-0.725477\pi$
$587$	−1.28968e18	−1.30118	−0.650588	+	0.759431i	$0.725477\pi$
$588$	0	0
$589$	−1.31848e18	−1.30114
$590$	0	0
$591$	−4.89680e17	−0.472709
$592$	0	0
$593$	−1.88640e18	−1.78147	−0.890735	−	0.454523i	$-0.849810\pi$
$593$	−1.88640e18	−1.78147	−0.890735	+	0.454523i	$0.849810\pi$
$594$	0	0
$595$	−8.34448e17	−0.770973
$596$	0	0
$597$	3.18173e16	0.0287627
$598$	0	0
$599$	1.44668e18	1.27967	0.639834	−	0.768513i	$-0.279003\pi$
$599$	1.44668e18	1.27967	0.639834	+	0.768513i	$0.279003\pi$
$600$	0	0
$601$	−4.44358e16	−0.0384635	−0.0192317	−	0.999815i	$-0.506122\pi$
$601$	−4.44358e16	−0.0384635	−0.0192317	+	0.999815i	$0.506122\pi$
$602$	0	0
$603$	−1.85217e17	−0.156898
$604$	0	0
$605$	−2.73505e18	−2.26755
$606$	0	0
$607$	−2.98050e16	−0.0241860	−0.0120930	−	0.999927i	$-0.503849\pi$
$607$	−2.98050e16	−0.0241860	−0.0120930	+	0.999927i	$0.503849\pi$
$608$	0	0
$609$	2.14635e17	0.170486
$610$	0	0
$611$	2.37695e17	0.184822
$612$	0	0
$613$	−8.84082e17	−0.672976	−0.336488	−	0.941688i	$-0.609239\pi$
$613$	−8.84082e17	−0.672976	−0.336488	+	0.941688i	$0.609239\pi$
$614$	0	0
$615$	1.03039e17	0.0767917
$616$	0	0
$617$	1.43684e18	1.04846	0.524232	−	0.851575i	$-0.324352\pi$
$617$	1.43684e18	1.04846	0.524232	+	0.851575i	$0.324352\pi$
$618$	0	0
$619$	−1.68862e18	−1.20654	−0.603272	−	0.797535i	$-0.706137\pi$
$619$	−1.68862e18	−1.20654	−0.603272	+	0.797535i	$0.706137\pi$
$620$	0	0
$621$	−3.21774e17	−0.225142
$622$	0	0
$623$	−5.25565e17	−0.360126
$624$	0	0
$625$	−1.01917e18	−0.683951
$626$	0	0
$627$	1.74510e18	1.14705
$628$	0	0
$629$	6.46008e17	0.415919
$630$	0	0
$631$	3.53490e17	0.222939	0.111470	−	0.993768i	$-0.464444\pi$
$631$	3.53490e17	0.222939	0.111470	+	0.993768i	$0.464444\pi$
$632$	0	0
$633$	1.18216e18	0.730387
$634$	0	0
$635$	−1.35748e18	−0.821680
$636$	0	0
$637$	−3.35046e17	−0.198700
$638$	0	0
$639$	−6.98492e17	−0.405888
$640$	0	0
$641$	1.61802e18	0.921313	0.460656	−	0.887579i	$-0.347614\pi$
$641$	1.61802e18	0.921313	0.460656	+	0.887579i	$0.347614\pi$
$642$	0	0
$643$	1.96065e18	1.09403	0.547015	−	0.837123i	$-0.315764\pi$
$643$	1.96065e18	1.09403	0.547015	+	0.837123i	$0.315764\pi$
$644$	0	0
$645$	1.56695e17	0.0856872
$646$	0	0
$647$	5.96114e17	0.319486	0.159743	−	0.987159i	$-0.448933\pi$
$647$	5.96114e17	0.319486	0.159743	+	0.987159i	$0.448933\pi$
$648$	0	0
$649$	3.34479e17	0.175703
$650$	0	0
$651$	1.05558e18	0.543521
$652$	0	0
$653$	−2.58318e18	−1.30382	−0.651912	−	0.758295i	$-0.726033\pi$
$653$	−2.58318e18	−1.30382	−0.651912	+	0.758295i	$0.726033\pi$
$654$	0	0
$655$	−3.63071e16	−0.0179648
$656$	0	0
$657$	−6.26503e17	−0.303912
$658$	0	0
$659$	−2.64137e18	−1.25624	−0.628121	−	0.778116i	$-0.716176\pi$
$659$	−2.64137e18	−1.25624	−0.628121	+	0.778116i	$0.716176\pi$
$660$	0	0
$661$	4.12451e18	1.92337	0.961685	−	0.274156i	$-0.0883983\pi$
$661$	4.12451e18	1.92337	0.961685	+	0.274156i	$0.0883983\pi$
$662$	0	0
$663$	−6.89478e17	−0.315270
$664$	0	0
$665$	1.52027e18	0.681677
$666$	0	0
$667$	1.04019e18	0.457398
$668$	0	0
$669$	−1.07406e18	−0.463189
$670$	0	0
$671$	−1.50289e18	−0.635670
$672$	0	0
$673$	2.79726e18	1.16047	0.580236	−	0.814449i	$-0.302961\pi$
$673$	2.79726e18	1.16047	0.580236	+	0.814449i	$0.302961\pi$
$674$	0	0
$675$	−1.19348e17	−0.0485671
$676$	0	0
$677$	−4.25553e18	−1.69874	−0.849372	−	0.527795i	$-0.823019\pi$
$677$	−4.25553e18	−1.69874	−0.849372	+	0.527795i	$0.823019\pi$
$678$	0	0
$679$	3.34184e18	1.30867
$680$	0	0
$681$	2.73294e18	1.04996
$682$	0	0
$683$	−1.60893e18	−0.606461	−0.303230	−	0.952917i	$-0.598065\pi$
$683$	−1.60893e18	−0.606461	−0.303230	+	0.952917i	$0.598065\pi$
$684$	0	0
$685$	−5.18287e17	−0.191682
$686$	0	0
$687$	−1.07887e16	−0.00391516
$688$	0	0
$689$	−1.64313e18	−0.585122
$690$	0	0
$691$	3.06331e18	1.07049	0.535247	−	0.844696i	$-0.320218\pi$
$691$	3.06331e18	1.07049	0.535247	+	0.844696i	$0.320218\pi$
$692$	0	0
$693$	−1.39714e18	−0.479153
$694$	0	0
$695$	3.19151e18	1.07422
$696$	0	0
$697$	5.49721e17	0.181606
$698$	0	0
$699$	2.64666e18	0.858214
$700$	0	0
$701$	2.99144e18	0.952166	0.476083	−	0.879400i	$-0.342056\pi$
$701$	2.99144e18	0.952166	0.476083	+	0.879400i	$0.342056\pi$
$702$	0	0
$703$	−1.17695e18	−0.367747
$704$	0	0
$705$	−6.50315e17	−0.199478
$706$	0	0
$707$	−4.00106e18	−1.20489
$708$	0	0
$709$	−5.31694e18	−1.57203	−0.786015	−	0.618207i	$-0.787859\pi$
$709$	−5.31694e18	−1.57203	−0.786015	+	0.618207i	$0.787859\pi$
$710$	0	0
$711$	5.69928e17	0.165450
$712$	0	0
$713$	5.11568e18	1.45822
$714$	0	0
$715$	−2.71945e18	−0.761187
$716$	0	0
$717$	3.16313e18	0.869446
$718$	0	0
$719$	4.03153e18	1.08826	0.544129	−	0.839001i	$-0.316860\pi$
$719$	4.03153e18	1.08826	0.544129	+	0.839001i	$0.316860\pi$
$720$	0	0
$721$	2.58372e18	0.684961
$722$	0	0
$723$	2.20429e18	0.573945
$724$	0	0
$725$	3.85813e17	0.0986688
$726$	0	0
$727$	4.77643e18	1.19986	0.599928	−	0.800054i	$-0.295196\pi$
$727$	4.77643e18	1.19986	0.599928	+	0.800054i	$0.295196\pi$
$728$	0	0
$729$	1.50095e17	0.0370370
$730$	0	0
$731$	8.35976e17	0.202643
$732$	0	0
$733$	1.71668e18	0.408803	0.204401	−	0.978887i	$-0.434475\pi$
$733$	1.71668e18	0.408803	0.204401	+	0.978887i	$0.434475\pi$
$734$	0	0
$735$	9.16659e17	0.214457
$736$	0	0
$737$	−3.89745e18	−0.895864
$738$	0	0
$739$	8.69723e17	0.196423	0.0982114	−	0.995166i	$-0.468688\pi$
$739$	8.69723e17	0.196423	0.0982114	+	0.995166i	$0.468688\pi$
$740$	0	0
$741$	1.25615e18	0.278755
$742$	0	0
$743$	−2.40272e18	−0.523933	−0.261966	−	0.965077i	$-0.584371\pi$
$743$	−2.40272e18	−0.523933	−0.261966	+	0.965077i	$0.584371\pi$
$744$	0	0
$745$	2.57852e18	0.552529
$746$	0	0
$747$	−5.97353e17	−0.125790
$748$	0	0
$749$	−4.61721e18	−0.955537
$750$	0	0
$751$	−9.37175e18	−1.90617	−0.953084	−	0.302706i	$-0.902110\pi$
$751$	−9.37175e18	−1.90617	−0.953084	+	0.302706i	$0.902110\pi$
$752$	0	0
$753$	1.27681e18	0.255246
$754$	0	0
$755$	−1.86219e18	−0.365905
$756$	0	0
$757$	3.09120e18	0.597040	0.298520	−	0.954403i	$-0.403507\pi$
$757$	3.09120e18	0.597040	0.298520	+	0.954403i	$0.403507\pi$
$758$	0	0
$759$	−6.77097e18	−1.28552
$760$	0	0
$761$	−7.97787e18	−1.48897	−0.744486	−	0.667638i	$-0.767306\pi$
$761$	−7.97787e18	−1.48897	−0.744486	+	0.667638i	$0.767306\pi$
$762$	0	0
$763$	2.30880e18	0.423620
$764$	0	0
$765$	1.88636e18	0.340271
$766$	0	0
$767$	2.40763e17	0.0426991
$768$	0	0
$769$	7.37344e18	1.28573	0.642863	−	0.765981i	$-0.277746\pi$
$769$	7.37344e18	1.28573	0.642863	+	0.765981i	$0.277746\pi$
$770$	0	0
$771$	3.55463e18	0.609454
$772$	0	0
$773$	−1.67335e18	−0.282111	−0.141056	−	0.990002i	$-0.545050\pi$
$773$	−1.67335e18	−0.282111	−0.141056	+	0.990002i	$0.545050\pi$
$774$	0	0
$775$	1.89744e18	0.314563
$776$	0	0
$777$	9.42275e17	0.153618
$778$	0	0
$779$	−1.00153e18	−0.160572
$780$	0	0
$781$	−1.46981e19	−2.31755
$782$	0	0
$783$	−4.85206e17	−0.0752444
$784$	0	0
$785$	3.56541e18	0.543822
$786$	0	0
$787$	3.75359e18	0.563133	0.281566	−	0.959542i	$-0.409146\pi$
$787$	3.75359e18	0.563133	0.281566	+	0.959542i	$0.409146\pi$
$788$	0	0
$789$	−3.41086e18	−0.503342
$790$	0	0
$791$	4.00277e18	0.581050
$792$	0	0
$793$	−1.08180e18	−0.154480
$794$	0	0
$795$	4.49546e18	0.631522
$796$	0	0
$797$	3.38853e18	0.468309	0.234154	−	0.972199i	$-0.424768\pi$
$797$	3.38853e18	0.468309	0.234154	+	0.972199i	$0.424768\pi$
$798$	0	0
$799$	−3.46947e18	−0.471747
$800$	0	0
$801$	1.18810e18	0.158942
$802$	0	0
$803$	−1.31833e19	−1.73529
$804$	0	0
$805$	−5.89861e18	−0.763971
$806$	0	0
$807$	−1.31411e18	−0.167476
$808$	0	0
$809$	−3.13119e18	−0.392685	−0.196343	−	0.980535i	$-0.562906\pi$
$809$	−3.13119e18	−0.392685	−0.196343	+	0.980535i	$0.562906\pi$
$810$	0	0
$811$	−1.04731e19	−1.29253	−0.646264	−	0.763114i	$-0.723669\pi$
$811$	−1.04731e19	−1.29253	−0.646264	+	0.763114i	$0.723669\pi$
$812$	0	0
$813$	−4.44702e18	−0.540108
$814$	0	0
$815$	4.67379e18	0.558657
$816$	0	0
$817$	−1.52305e18	−0.179172
$818$	0	0
$819$	−1.00568e18	−0.116443
$820$	0	0
$821$	−6.85162e18	−0.780841	−0.390421	−	0.920637i	$-0.627670\pi$
$821$	−6.85162e18	−0.780841	−0.390421	+	0.920637i	$0.627670\pi$
$822$	0	0
$823$	−3.06934e17	−0.0344308	−0.0172154	−	0.999852i	$-0.505480\pi$
$823$	−3.06934e17	−0.0344308	−0.0172154	+	0.999852i	$0.505480\pi$
$824$	0	0
$825$	−2.51140e18	−0.277310
$826$	0	0
$827$	−7.75365e18	−0.842792	−0.421396	−	0.906877i	$-0.638460\pi$
$827$	−7.75365e18	−0.842792	−0.421396	+	0.906877i	$0.638460\pi$
$828$	0	0
$829$	2.34336e18	0.250747	0.125373	−	0.992110i	$-0.459987\pi$
$829$	2.34336e18	0.250747	0.125373	+	0.992110i	$0.459987\pi$
$830$	0	0
$831$	−7.80200e18	−0.821863
$832$	0	0
$833$	4.89044e18	0.507172
$834$	0	0
$835$	1.13593e19	1.15982
$836$	0	0
$837$	−2.38626e18	−0.239884
$838$	0	0
$839$	1.63297e19	1.61632	0.808158	−	0.588965i	$-0.200465\pi$
$839$	1.63297e19	1.61632	0.808158	+	0.588965i	$0.200465\pi$
$840$	0	0
$841$	−8.69212e18	−0.847134
$842$	0	0
$843$	−1.78940e18	−0.171723
$844$	0	0
$845$	7.19236e18	0.679677
$846$	0	0
$847$	−2.12836e19	−1.98063
$848$	0	0
$849$	−2.92442e18	−0.268003
$850$	0	0
$851$	4.56655e18	0.412142
$852$	0	0
$853$	1.93794e19	1.72255	0.861276	−	0.508137i	$-0.169666\pi$
$853$	1.93794e19	1.72255	0.861276	+	0.508137i	$0.169666\pi$
$854$	0	0
$855$	−3.43672e18	−0.300860
$856$	0	0
$857$	1.20537e19	1.03931	0.519656	−	0.854376i	$-0.326060\pi$
$857$	1.20537e19	1.03931	0.519656	+	0.854376i	$0.326060\pi$
$858$	0	0
$859$	1.00612e19	0.854465	0.427232	−	0.904142i	$-0.359489\pi$
$859$	1.00612e19	0.854465	0.427232	+	0.904142i	$0.359489\pi$
$860$	0	0
$861$	8.01830e17	0.0670751
$862$	0	0
$863$	−8.01407e18	−0.660363	−0.330182	−	0.943917i	$-0.607110\pi$
$863$	−8.01407e18	−0.660363	−0.330182	+	0.943917i	$0.607110\pi$
$864$	0	0
$865$	−1.12853e19	−0.916029
$866$	0	0
$867$	2.84340e18	0.227360
$868$	0	0
$869$	1.19928e19	0.944694
$870$	0	0
$871$	−2.80544e18	−0.217712
$872$	0	0
$873$	−7.55457e18	−0.577587
$874$	0	0
$875$	−1.08573e19	−0.817841
$876$	0	0
$877$	8.87791e17	0.0658891	0.0329445	−	0.999457i	$-0.489512\pi$
$877$	8.87791e17	0.0658891	0.0329445	+	0.999457i	$0.489512\pi$
$878$	0	0
$879$	1.51768e18	0.110982
$880$	0	0
$881$	3.78774e18	0.272921	0.136460	−	0.990646i	$-0.456427\pi$
$881$	3.78774e18	0.272921	0.136460	+	0.990646i	$0.456427\pi$
$882$	0	0
$883$	2.75428e19	1.95553	0.977763	−	0.209712i	$-0.0672526\pi$
$883$	2.75428e19	1.95553	0.977763	+	0.209712i	$0.0672526\pi$
$884$	0	0
$885$	−6.58707e17	−0.0460851
$886$	0	0
$887$	2.84165e19	1.95914	0.979572	−	0.201092i	$-0.0644489\pi$
$887$	2.84165e19	1.95914	0.979572	+	0.201092i	$0.0644489\pi$
$888$	0	0
$889$	−1.05636e19	−0.717712
$890$	0	0
$891$	3.15838e18	0.211475
$892$	0	0
$893$	6.32097e18	0.417109
$894$	0	0
$895$	1.27894e18	0.0831762
$896$	0	0
$897$	−4.87384e18	−0.312407
$898$	0	0
$899$	7.71397e18	0.487349
$900$	0	0
$901$	2.39836e19	1.49349
$902$	0	0
$903$	1.21937e18	0.0748451
$904$	0	0
$905$	9.37860e18	0.567443
$906$	0	0
$907$	−7.51023e18	−0.447926	−0.223963	−	0.974598i	$-0.571899\pi$
$907$	−7.51023e18	−0.447926	−0.223963	+	0.974598i	$0.571899\pi$
$908$	0	0
$909$	9.04482e18	0.531783
$910$	0	0
$911$	2.34028e19	1.35643	0.678216	−	0.734863i	$-0.262753\pi$
$911$	2.34028e19	1.35643	0.678216	+	0.734863i	$0.262753\pi$
$912$	0	0
$913$	−1.25699e19	−0.718242
$914$	0	0
$915$	2.95973e18	0.166730
$916$	0	0
$917$	−2.82535e17	−0.0156917
$918$	0	0
$919$	−2.20881e19	−1.20950	−0.604752	−	0.796414i	$-0.706728\pi$
$919$	−2.20881e19	−1.20950	−0.604752	+	0.796414i	$0.706728\pi$
$920$	0	0
$921$	9.64848e18	0.520920
$922$	0	0
$923$	−1.05799e19	−0.563210
$924$	0	0
$925$	1.69377e18	0.0889063
$926$	0	0
$927$	−5.84077e18	−0.302309
$928$	0	0
$929$	−1.05632e19	−0.539132	−0.269566	−	0.962982i	$-0.586880\pi$
$929$	−1.05632e19	−0.539132	−0.269566	+	0.962982i	$0.586880\pi$
$930$	0	0
$931$	−8.90980e18	−0.448430
$932$	0	0
$933$	5.89981e18	0.292824
$934$	0	0
$935$	3.96939e19	1.94289
$936$	0	0
$937$	−1.72833e19	−0.834294	−0.417147	−	0.908839i	$-0.636970\pi$
$937$	−1.72833e19	−0.834294	−0.417147	+	0.908839i	$0.636970\pi$
$938$	0	0
$939$	−1.08024e19	−0.514273
$940$	0	0
$941$	2.14038e19	1.00498	0.502490	−	0.864583i	$-0.332417\pi$
$941$	2.14038e19	1.00498	0.502490	+	0.864583i	$0.332417\pi$
$942$	0	0
$943$	3.88591e18	0.179956
$944$	0	0
$945$	2.75146e18	0.125677
$946$	0	0
$947$	5.91308e17	0.0266403	0.0133202	−	0.999911i	$-0.495760\pi$
$947$	5.91308e17	0.0266403	0.0133202	+	0.999911i	$0.495760\pi$
$948$	0	0
$949$	−9.48950e18	−0.421709
$950$	0	0
$951$	−1.77271e19	−0.777079
$952$	0	0
$953$	−2.39396e19	−1.03517	−0.517587	−	0.855631i	$-0.673170\pi$
$953$	−2.39396e19	−1.03517	−0.517587	+	0.855631i	$0.673170\pi$
$954$	0	0
$955$	−2.60493e19	−1.11115
$956$	0	0
$957$	−1.02100e19	−0.429633
$958$	0	0
$959$	−4.03321e18	−0.167429
$960$	0	0
$961$	1.35201e19	0.553702
$962$	0	0
$963$	1.04377e19	0.421729
$964$	0	0
$965$	2.83320e19	1.12941
$966$	0	0
$967$	−1.99045e18	−0.0782853	−0.0391427	−	0.999234i	$-0.512463\pi$
$967$	−1.99045e18	−0.0782853	−0.0391427	+	0.999234i	$0.512463\pi$
$968$	0	0
$969$	−1.83351e19	−0.711507
$970$	0	0
$971$	−4.12385e19	−1.57898	−0.789492	−	0.613761i	$-0.789656\pi$
$971$	−4.12385e19	−1.57898	−0.789492	+	0.613761i	$0.789656\pi$
$972$	0	0
$973$	2.48357e19	0.938301
$974$	0	0
$975$	−1.80774e18	−0.0673917
$976$	0	0
$977$	−1.25374e19	−0.461202	−0.230601	−	0.973048i	$-0.574069\pi$
$977$	−1.25374e19	−0.461202	−0.230601	+	0.973048i	$0.574069\pi$
$978$	0	0
$979$	2.50006e19	0.907535
$980$	0	0
$981$	−5.21928e18	−0.186966
$982$	0	0
$983$	1.83966e19	0.650340	0.325170	−	0.945656i	$-0.394578\pi$
$983$	1.83966e19	0.650340	0.325170	+	0.945656i	$0.394578\pi$
$984$	0	0
$985$	2.02925e19	0.707946
$986$	0	0
$987$	−5.06062e18	−0.174238
$988$	0	0
$989$	5.90941e18	0.200802
$990$	0	0
$991$	3.43064e19	1.15053	0.575263	−	0.817969i	$-0.304900\pi$
$991$	3.43064e19	1.15053	0.575263	+	0.817969i	$0.304900\pi$
$992$	0	0
$993$	−8.47329e18	−0.280467
$994$	0	0
$995$	−1.31852e18	−0.0430761
$996$	0	0
$997$	−5.72458e19	−1.84597	−0.922985	−	0.384835i	$-0.874258\pi$
$997$	−5.72458e19	−1.84597	−0.922985	+	0.384835i	$0.874258\pi$
$998$	0	0
$999$	−2.13011e18	−0.0677995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.14.a.c.1.1		1
3.2	odd	2		144.14.a.k.1.1		1
4.3	odd	2		3.14.a.a.1.1	✓	1
8.3	odd	2		192.14.a.j.1.1		1
8.5	even	2		192.14.a.e.1.1		1
12.11	even	2		9.14.a.a.1.1		1
20.3	even	4		75.14.b.b.49.2		2
20.7	even	4		75.14.b.b.49.1		2
20.19	odd	2		75.14.a.a.1.1		1
28.27	even	2		147.14.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.14.a.a.1.1	✓	1	4.3	odd	2
9.14.a.a.1.1		1	12.11	even	2
48.14.a.c.1.1		1	1.1	even	1	trivial
75.14.a.a.1.1		1	20.19	odd	2
75.14.b.b.49.1		2	20.7	even	4
75.14.b.b.49.2		2	20.3	even	4
144.14.a.k.1.1		1	3.2	odd	2
147.14.a.a.1.1		1	28.27	even	2
192.14.a.e.1.1		1	8.5	even	2
192.14.a.j.1.1		1	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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