LMFDB - Embedded newform 16.36.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,36,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 36, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 36);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 36 $
Character orbit:	$[\chi]$	$=$	16.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:	$124.152209014$
Analytic rank:	$1$
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} - 12422194x - 2645665785 $ x^3 - 12422194*x - 2645665785
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{3}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-213.765$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.45913e8 q^{3} -2.05014e12 q^{5} +1.25160e14 q^{7} +6.96245e16 q^{9} +O(q^{10})$	$q+3.45913e8 q^{3} -2.05014e12 q^{5} +1.25160e14 q^{7} +6.96245e16 q^{9} +1.70842e18 q^{11} -4.94986e19 q^{13} -7.09169e20 q^{15} +1.32045e21 q^{17} -3.94388e21 q^{19} +4.32945e22 q^{21} +3.48881e23 q^{23} +1.29267e24 q^{25} +6.77748e24 q^{27} +3.21628e25 q^{29} -3.41044e25 q^{31} +5.90965e26 q^{33} -2.56595e26 q^{35} -4.03624e27 q^{37} -1.71222e28 q^{39} +8.65857e27 q^{41} -9.89389e26 q^{43} -1.42740e29 q^{45} -1.95852e29 q^{47} -3.63154e29 q^{49} +4.56760e29 q^{51} -9.96909e29 q^{53} -3.50249e30 q^{55} -1.36424e30 q^{57} -3.91953e30 q^{59} +7.64909e29 q^{61} +8.71420e30 q^{63} +1.01479e32 q^{65} +1.64301e32 q^{67} +1.20682e32 q^{69} -7.65930e31 q^{71} -7.08063e32 q^{73} +4.47152e32 q^{75} +2.13826e32 q^{77} -2.34599e33 q^{79} -1.13900e33 q^{81} -5.18971e33 q^{83} -2.70710e33 q^{85} +1.11255e34 q^{87} +1.44578e34 q^{89} -6.19525e33 q^{91} -1.17972e34 q^{93} +8.08549e33 q^{95} -2.87399e34 q^{97} +1.18948e35 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+O(q^{10}) $ 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9	$ 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+ \cdots - 30\!\cdots\!52 q^{99}+O(q^{100}) $ 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9 + 1157945428549987044 * q^11 - 62139610550998650558 * q^13 - 706062479352366674520 * q^15 - 3932636930193139724394 * q^17 + 32303396242934620550940 * q^19 + 222122458753667398807776 * q^21 + 514822069421375217224808 * q^23 + 646885156836409574920725 * q^25 - 27373531144823515220747400 * q^27 - 38460399143984437598248110 * q^29 - 103731594563724689730029856 * q^31 + 1184565114402567148957069584 * q^33 - 1596564453000564139676248560 * q^35 + 2464613172948888017021063706 * q^37 - 18611981954062877742706387512 * q^39 + 23522986584532768129037087406 * q^41 + 47516297135443858184325274308 * q^43 - 110576300682246572226345828030 * q^45 - 164346768841318947159729902256 * q^47 - 595089038084925731622037450821 * q^49 + 1803044074850794727352259927704 * q^51 - 1670480551111723484744458809558 * q^53 - 3067178595313878889174894808520 * q^55 + 4048712721984947152977307522800 * q^57 - 4370528583092646699845243631180 * q^59 + 23537078276516028947219080488306 * q^61 - 45614247892704690123948892506792 * q^63 + 75762337609865790449417835274620 * q^65 + 188845033384156918977315211921644 * q^67 - 325983212396274250308653315934048 * q^69 - 348774602295358210538621162006856 * q^71 - 285174964854822786615792218987058 * q^73 + 1577819189216049218913461709237300 * q^75 + 691501421964810422051915307575712 * q^77 + 426212862591061661901736630874160 * q^79 + 1867221150796978455595408934723163 * q^81 - 14867187331664065317008883105766692 * q^83 - 8752040931978292430566755304420140 * q^85 + 7833119385476156993771606256673800 * q^87 + 30583590846193751335205992418034270 * q^89 - 1003391702519487362566122156997296 * q^91 - 40914650682166972019906140161446016 * q^93 + 84521392849966457182141665888160200 * q^95 - 106165667044630951063328618701091994 * q^97 - 30288686341597731928441860752034252 * 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
$3$	3.45913e8	1.54648	0.773242	−	0.634111i	$-0.218634\pi$
$3$	3.45913e8	1.54648	0.773242	+	0.634111i	$0.218634\pi$
$4$	0	0
$5$	−2.05014e12	−1.20173	−0.600866	−	0.799350i	$-0.705177\pi$
$5$	−2.05014e12	−1.20173	−0.600866	+	0.799350i	$0.705177\pi$
$6$	0	0
$7$	1.25160e14	0.203353	0.101676	−	0.994818i	$-0.467579\pi$
$7$	1.25160e14	0.203353	0.101676	+	0.994818i	$0.467579\pi$
$8$	0	0
$9$	6.96245e16	1.39161
$10$	0	0
$11$	1.70842e18	1.01911	0.509557	−	0.860437i	$-0.329809\pi$
$11$	1.70842e18	1.01911	0.509557	+	0.860437i	$0.329809\pi$
$12$	0	0
$13$	−4.94986e19	−1.58703	−0.793514	−	0.608552i	$-0.791751\pi$
$13$	−4.94986e19	−1.58703	−0.793514	+	0.608552i	$0.791751\pi$
$14$	0	0
$15$	−7.09169e20	−1.85846
$16$	0	0
$17$	1.32045e21	0.387137	0.193569	−	0.981087i	$-0.437994\pi$
$17$	1.32045e21	0.387137	0.193569	+	0.981087i	$0.437994\pi$
$18$	0	0
$19$	−3.94388e21	−0.165096	−0.0825479	−	0.996587i	$-0.526306\pi$
$19$	−3.94388e21	−0.165096	−0.0825479	+	0.996587i	$0.526306\pi$
$20$	0	0
$21$	4.32945e22	0.314482
$22$	0	0
$23$	3.48881e23	0.515750	0.257875	−	0.966178i	$-0.416978\pi$
$23$	3.48881e23	0.515750	0.257875	+	0.966178i	$0.416978\pi$
$24$	0	0
$25$	1.29267e24	0.444159
$26$	0	0
$27$	6.77748e24	0.605623
$28$	0	0
$29$	3.21628e25	0.822979	0.411489	−	0.911415i	$-0.365009\pi$
$29$	3.21628e25	0.822979	0.411489	+	0.911415i	$0.365009\pi$
$30$	0	0
$31$	−3.41044e25	−0.271632	−0.135816	−	0.990734i	$-0.543366\pi$
$31$	−3.41044e25	−0.271632	−0.135816	+	0.990734i	$0.543366\pi$
$32$	0	0
$33$	5.90965e26	1.57604
$34$	0	0
$35$	−2.56595e26	−0.244375
$36$	0	0
$37$	−4.03624e27	−1.45361	−0.726804	−	0.686845i	$-0.758995\pi$
$37$	−4.03624e27	−1.45361	−0.726804	+	0.686845i	$0.758995\pi$
$38$	0	0
$39$	−1.71222e28	−2.45431
$40$	0	0
$41$	8.65857e27	0.517283	0.258642	−	0.965973i	$-0.416725\pi$
$41$	8.65857e27	0.517283	0.258642	+	0.965973i	$0.416725\pi$
$42$	0	0
$43$	−9.89389e26	−0.0256844	−0.0128422	−	0.999918i	$-0.504088\pi$
$43$	−9.89389e26	−0.0256844	−0.0128422	+	0.999918i	$0.504088\pi$
$44$	0	0
$45$	−1.42740e29	−1.67234
$46$	0	0
$47$	−1.95852e29	−1.07205	−0.536025	−	0.844202i	$-0.680075\pi$
$47$	−1.95852e29	−1.07205	−0.536025	+	0.844202i	$0.680075\pi$
$48$	0	0
$49$	−3.63154e29	−0.958648
$50$	0	0
$51$	4.56760e29	0.598702
$52$	0	0
$53$	−9.96909e29	−0.666543	−0.333271	−	0.942831i	$-0.608153\pi$
$53$	−9.96909e29	−0.666543	−0.333271	+	0.942831i	$0.608153\pi$
$54$	0	0
$55$	−3.50249e30	−1.22470
$56$	0	0
$57$	−1.36424e30	−0.255318
$58$	0	0
$59$	−3.91953e30	−0.401166	−0.200583	−	0.979677i	$-0.564284\pi$
$59$	−3.91953e30	−0.401166	−0.200583	+	0.979677i	$0.564284\pi$
$60$	0	0
$61$	7.64909e29	0.0436855	0.0218428	−	0.999761i	$-0.493047\pi$
$61$	7.64909e29	0.0436855	0.0218428	+	0.999761i	$0.493047\pi$
$62$	0	0
$63$	8.71420e30	0.282988
$64$	0	0
$65$	1.01479e32	1.90718
$66$	0	0
$67$	1.64301e32	1.81690	0.908451	−	0.417992i	$-0.137266\pi$
$67$	1.64301e32	1.81690	0.908451	+	0.417992i	$0.137266\pi$
$68$	0	0
$69$	1.20682e32	0.797600
$70$	0	0
$71$	−7.65930e31	−0.307021	−0.153510	−	0.988147i	$-0.549058\pi$
$71$	−7.65930e31	−0.307021	−0.153510	+	0.988147i	$0.549058\pi$
$72$	0	0
$73$	−7.08063e32	−1.74551	−0.872754	−	0.488160i	$-0.837668\pi$
$73$	−7.08063e32	−1.74551	−0.872754	+	0.488160i	$0.837668\pi$
$74$	0	0
$75$	4.47152e32	0.686884
$76$	0	0
$77$	2.13826e32	0.207239
$78$	0	0
$79$	−2.34599e33	−1.45162	−0.725809	−	0.687896i	$-0.758534\pi$
$79$	−2.34599e33	−1.45162	−0.725809	+	0.687896i	$0.758534\pi$
$80$	0	0
$81$	−1.13900e33	−0.455027
$82$	0	0
$83$	−5.18971e33	−1.35293	−0.676467	−	0.736473i	$-0.736490\pi$
$83$	−5.18971e33	−1.35293	−0.676467	+	0.736473i	$0.736490\pi$
$84$	0	0
$85$	−2.70710e33	−0.465235
$86$	0	0
$87$	1.11255e34	1.27272
$88$	0	0
$89$	1.44578e34	1.11116	0.555582	−	0.831461i	$-0.312495\pi$
$89$	1.44578e34	1.11116	0.555582	+	0.831461i	$0.312495\pi$
$90$	0	0
$91$	−6.19525e33	−0.322726
$92$	0	0
$93$	−1.17972e34	−0.420075
$94$	0	0
$95$	8.08549e33	0.198401
$96$	0	0
$97$	−2.87399e34	−0.489756	−0.244878	−	0.969554i	$-0.578748\pi$
$97$	−2.87399e34	−0.489756	−0.244878	+	0.969554i	$0.578748\pi$
$98$	0	0
$99$	1.18948e35	1.41821
$100$	0	0
$101$	1.13198e35	0.951079	0.475539	−	0.879694i	$-0.342253\pi$
$101$	1.13198e35	0.951079	0.475539	+	0.879694i	$0.342253\pi$
$102$	0	0
$103$	−2.53635e35	−1.51202	−0.756011	−	0.654559i	$-0.772855\pi$
$103$	−2.53635e35	−1.51202	−0.756011	+	0.654559i	$0.772855\pi$
$104$	0	0
$105$	−8.87596e34	−0.377922
$106$	0	0
$107$	3.63939e35	1.11381	0.556907	−	0.830575i	$-0.311988\pi$
$107$	3.63939e35	1.11381	0.556907	+	0.830575i	$0.311988\pi$
$108$	0	0
$109$	−4.65371e34	−0.103000	−0.0514998	−	0.998673i	$-0.516400\pi$
$109$	−4.65371e34	−0.103000	−0.0514998	+	0.998673i	$0.516400\pi$
$110$	0	0
$111$	−1.39619e36	−2.24798
$112$	0	0
$113$	−2.18346e35	−0.257201	−0.128601	−	0.991696i	$-0.541049\pi$
$113$	−2.18346e35	−0.257201	−0.128601	+	0.991696i	$0.541049\pi$
$114$	0	0
$115$	−7.15252e35	−0.619794
$116$	0	0
$117$	−3.44632e36	−2.20853
$118$	0	0
$119$	1.65267e35	0.0787254
$120$	0	0
$121$	1.08456e35	0.0385932
$122$	0	0
$123$	2.99511e36	0.799970
$124$	0	0
$125$	3.31653e36	0.667972
$126$	0	0
$127$	−4.19558e36	−0.640070	−0.320035	−	0.947406i	$-0.603695\pi$
$127$	−4.19558e36	−0.640070	−0.320035	+	0.947406i	$0.603695\pi$
$128$	0	0
$129$	−3.42243e35	−0.0397205
$130$	0	0
$131$	−2.40631e36	−0.213355	−0.106678	−	0.994294i	$-0.534021\pi$
$131$	−2.40631e36	−0.213355	−0.106678	+	0.994294i	$0.534021\pi$
$132$	0	0
$133$	−4.93616e35	−0.0335727
$134$	0	0
$135$	−1.38947e37	−0.727796
$136$	0	0
$137$	−2.14718e37	−0.869477	−0.434739	−	0.900557i	$-0.643159\pi$
$137$	−2.14718e37	−0.869477	−0.434739	+	0.900557i	$0.643159\pi$
$138$	0	0
$139$	−2.63209e37	−0.827068	−0.413534	−	0.910489i	$-0.635706\pi$
$139$	−2.63209e37	−0.827068	−0.413534	+	0.910489i	$0.635706\pi$
$140$	0	0
$141$	−6.77477e37	−1.65791
$142$	0	0
$143$	−8.45645e37	−1.61736
$144$	0	0
$145$	−6.59381e37	−0.989000
$146$	0	0
$147$	−1.25620e38	−1.48253
$148$	0	0
$149$	−1.74389e38	−1.62465	−0.812324	−	0.583206i	$-0.801798\pi$
$149$	−1.74389e38	−1.62465	−0.812324	+	0.583206i	$0.801798\pi$
$150$	0	0
$151$	1.40947e38	1.03982	0.519912	−	0.854220i	$-0.325965\pi$
$151$	1.40947e38	1.03982	0.519912	+	0.854220i	$0.325965\pi$
$152$	0	0
$153$	9.19355e37	0.538745
$154$	0	0
$155$	6.99187e37	0.326429
$156$	0	0
$157$	1.49924e38	0.559277	0.279639	−	0.960105i	$-0.409785\pi$
$157$	1.49924e38	0.559277	0.279639	+	0.960105i	$0.409785\pi$
$158$	0	0
$159$	−3.44844e38	−1.03080
$160$	0	0
$161$	4.36659e37	0.104879
$162$	0	0
$163$	−8.60787e38	−1.66576	−0.832880	−	0.553453i	$-0.813310\pi$
$163$	−8.60787e38	−1.66576	−0.832880	+	0.553453i	$0.813310\pi$
$164$	0	0
$165$	−1.21156e39	−1.89398
$166$	0	0
$167$	9.23344e38	1.16903	0.584515	−	0.811383i	$-0.301285\pi$
$167$	9.23344e38	1.16903	0.584515	+	0.811383i	$0.301285\pi$
$168$	0	0
$169$	1.47733e39	1.51866
$170$	0	0
$171$	−2.74591e38	−0.229749
$172$	0	0
$173$	−1.33161e39	−0.909012	−0.454506	−	0.890744i	$-0.650184\pi$
$173$	−1.33161e39	−0.909012	−0.454506	+	0.890744i	$0.650184\pi$
$174$	0	0
$175$	1.61791e38	0.0903208
$176$	0	0
$177$	−1.35582e39	−0.620396
$178$	0	0
$179$	−6.27202e38	−0.235765	−0.117883	−	0.993028i	$-0.537611\pi$
$179$	−6.27202e38	−0.235765	−0.117883	+	0.993028i	$0.537611\pi$
$180$	0	0
$181$	−2.39965e39	−0.742629	−0.371315	−	0.928507i	$-0.621093\pi$
$181$	−2.39965e39	−0.742629	−0.371315	+	0.928507i	$0.621093\pi$
$182$	0	0
$183$	2.64592e38	0.0675590
$184$	0	0
$185$	8.27484e39	1.74685
$186$	0	0
$187$	2.25588e39	0.394537
$188$	0	0
$189$	8.48269e38	0.123155
$190$	0	0
$191$	−8.14518e39	−0.983596	−0.491798	−	0.870709i	$-0.663660\pi$
$191$	−8.14518e39	−0.983596	−0.491798	+	0.870709i	$0.663660\pi$
$192$	0	0
$193$	1.38549e40	1.39428	0.697139	−	0.716936i	$-0.254456\pi$
$193$	1.38549e40	1.39428	0.697139	+	0.716936i	$0.254456\pi$
$194$	0	0
$195$	3.51029e40	2.94942
$196$	0	0
$197$	−1.15539e40	−0.812026	−0.406013	−	0.913867i	$-0.633081\pi$
$197$	−1.15539e40	−0.812026	−0.406013	+	0.913867i	$0.633081\pi$
$198$	0	0
$199$	4.76691e39	0.280742	0.140371	−	0.990099i	$-0.455170\pi$
$199$	4.76691e39	0.280742	0.140371	+	0.990099i	$0.455170\pi$
$200$	0	0
$201$	5.68340e40	2.80981
$202$	0	0
$203$	4.02549e39	0.167355
$204$	0	0
$205$	−1.77512e40	−0.621636
$206$	0	0
$207$	2.42906e40	0.717725
$208$	0	0
$209$	−6.73781e39	−0.168251
$210$	0	0
$211$	1.87354e40	0.396022	0.198011	−	0.980200i	$-0.436552\pi$
$211$	1.87354e40	0.396022	0.198011	+	0.980200i	$0.436552\pi$
$212$	0	0
$213$	−2.64945e40	−0.474803
$214$	0	0
$215$	2.02838e39	0.0308657
$216$	0	0
$217$	−4.26851e39	−0.0552371
$218$	0	0
$219$	−2.44929e41	−2.69940
$220$	0	0
$221$	−6.53603e40	−0.614398
$222$	0	0
$223$	7.12100e40	0.571750	0.285875	−	0.958267i	$-0.407716\pi$
$223$	7.12100e40	0.571750	0.285875	+	0.958267i	$0.407716\pi$
$224$	0	0
$225$	9.00017e40	0.618097
$226$	0	0
$227$	1.04005e41	0.611787	0.305893	−	0.952066i	$-0.401045\pi$
$227$	1.04005e41	0.611787	0.305893	+	0.952066i	$0.401045\pi$
$228$	0	0
$229$	1.82859e41	0.922564	0.461282	−	0.887254i	$-0.347390\pi$
$229$	1.82859e41	0.922564	0.461282	+	0.887254i	$0.347390\pi$
$230$	0	0
$231$	7.39652e40	0.320493
$232$	0	0
$233$	1.57847e41	0.588177	0.294088	−	0.955778i	$-0.404984\pi$
$233$	1.57847e41	0.588177	0.294088	+	0.955778i	$0.404984\pi$
$234$	0	0
$235$	4.01523e41	1.28832
$236$	0	0
$237$	−8.11510e41	−2.24490
$238$	0	0
$239$	−5.00984e41	−1.19636	−0.598178	−	0.801363i	$-0.704108\pi$
$239$	−5.00984e41	−1.19636	−0.598178	+	0.801363i	$0.704108\pi$
$240$	0	0
$241$	−3.33164e41	−0.687638	−0.343819	−	0.939036i	$-0.611721\pi$
$241$	−3.33164e41	−0.687638	−0.343819	+	0.939036i	$0.611721\pi$
$242$	0	0
$243$	−7.33084e41	−1.30931
$244$	0	0
$245$	7.44514e41	1.15204
$246$	0	0
$247$	1.95217e41	0.262011
$248$	0	0
$249$	−1.79519e42	−2.09229
$250$	0	0
$251$	−1.52760e42	−1.54782	−0.773909	−	0.633297i	$-0.781701\pi$
$251$	−1.52760e42	−1.54782	−0.773909	+	0.633297i	$0.781701\pi$
$252$	0	0
$253$	5.96035e41	0.525608
$254$	0	0
$255$	−9.36421e41	−0.719479
$256$	0	0
$257$	−2.72482e42	−1.82603	−0.913016	−	0.407925i	$-0.866253\pi$
$257$	−2.72482e42	−1.82603	−0.913016	+	0.407925i	$0.866253\pi$
$258$	0	0
$259$	−5.05176e41	−0.295595
$260$	0	0
$261$	2.23932e42	1.14527
$262$	0	0
$263$	−4.22055e40	−0.0188861	−0.00944307	−	0.999955i	$-0.503006\pi$
$263$	−4.22055e40	−0.0188861	−0.00944307	+	0.999955i	$0.503006\pi$
$264$	0	0
$265$	2.04380e42	0.801005
$266$	0	0
$267$	5.00116e42	1.71840
$268$	0	0
$269$	−1.30618e42	−0.393858	−0.196929	−	0.980418i	$-0.563097\pi$
$269$	−1.30618e42	−0.393858	−0.196929	+	0.980418i	$0.563097\pi$
$270$	0	0
$271$	4.11918e41	0.109106	0.0545529	−	0.998511i	$-0.482627\pi$
$271$	4.11918e41	0.109106	0.0545529	+	0.998511i	$0.482627\pi$
$272$	0	0
$273$	−2.14302e42	−0.499091
$274$	0	0
$275$	2.20843e42	0.452648
$276$	0	0
$277$	7.38232e42	1.33290	0.666449	−	0.745551i	$-0.267813\pi$
$277$	7.38232e42	1.33290	0.666449	+	0.745551i	$0.267813\pi$
$278$	0	0
$279$	−2.37451e42	−0.378007
$280$	0	0
$281$	4.08223e42	0.573505	0.286752	−	0.958005i	$-0.407424\pi$
$281$	4.08223e42	0.573505	0.286752	+	0.958005i	$0.407424\pi$
$282$	0	0
$283$	1.06503e43	1.32159	0.660796	−	0.750565i	$-0.270219\pi$
$283$	1.06503e43	1.32159	0.660796	+	0.750565i	$0.270219\pi$
$284$	0	0
$285$	2.79688e42	0.306824
$286$	0	0
$287$	1.08371e42	0.105191
$288$	0	0
$289$	−9.88997e42	−0.850125
$290$	0	0
$291$	−9.94151e42	−0.757399
$292$	0	0
$293$	1.90474e43	1.28722	0.643610	−	0.765353i	$-0.277436\pi$
$293$	1.90474e43	1.28722	0.643610	+	0.765353i	$0.277436\pi$
$294$	0	0
$295$	8.03556e42	0.482093
$296$	0	0
$297$	1.15788e43	0.617199
$298$	0	0
$299$	−1.72691e43	−0.818510
$300$	0	0
$301$	−1.23832e41	−0.00522298
$302$	0	0
$303$	3.91569e43	1.47083
$304$	0	0
$305$	−1.56817e42	−0.0524983
$306$	0	0
$307$	1.39752e43	0.417288	0.208644	−	0.977992i	$-0.433095\pi$
$307$	1.39752e43	0.417288	0.208644	+	0.977992i	$0.433095\pi$
$308$	0	0
$309$	−8.77359e43	−2.33832
$310$	0	0
$311$	−2.22377e43	−0.529399	−0.264700	−	0.964331i	$-0.585273\pi$
$311$	−2.22377e43	−0.529399	−0.264700	+	0.964331i	$0.585273\pi$
$312$	0	0
$313$	3.63171e43	0.772833	0.386417	−	0.922324i	$-0.373713\pi$
$313$	3.63171e43	0.772833	0.386417	+	0.922324i	$0.373713\pi$
$314$	0	0
$315$	−1.78653e43	−0.340076
$316$	0	0
$317$	5.29385e43	0.902058	0.451029	−	0.892509i	$-0.351057\pi$
$317$	5.29385e43	0.902058	0.451029	+	0.892509i	$0.351057\pi$
$318$	0	0
$319$	5.49476e43	0.838709
$320$	0	0
$321$	1.25891e44	1.72249
$322$	0	0
$323$	−5.20769e42	−0.0639147
$324$	0	0
$325$	−6.39855e43	−0.704892
$326$	0	0
$327$	−1.60978e43	−0.159287
$328$	0	0
$329$	−2.45128e43	−0.218004
$330$	0	0
$331$	−6.25551e43	−0.500349	−0.250174	−	0.968201i	$-0.580488\pi$
$331$	−6.25551e43	−0.500349	−0.250174	+	0.968201i	$0.580488\pi$
$332$	0	0
$333$	−2.81022e44	−2.02286
$334$	0	0
$335$	−3.36840e44	−2.18343
$336$	0	0
$337$	−1.04185e44	−0.608529	−0.304264	−	0.952588i	$-0.598411\pi$
$337$	−1.04185e44	−0.608529	−0.304264	+	0.952588i	$0.598411\pi$
$338$	0	0
$339$	−7.55289e43	−0.397757
$340$	0	0
$341$	−5.82647e43	−0.276824
$342$	0	0
$343$	−9.28652e43	−0.398296
$344$	0	0
$345$	−2.47415e44	−0.958501
$346$	0	0
$347$	3.07657e44	1.07721	0.538606	−	0.842557i	$-0.318951\pi$
$347$	3.07657e44	1.07721	0.538606	+	0.842557i	$0.318951\pi$
$348$	0	0
$349$	−2.62767e44	−0.832005	−0.416003	−	0.909363i	$-0.636569\pi$
$349$	−2.62767e44	−0.832005	−0.416003	+	0.909363i	$0.636569\pi$
$350$	0	0
$351$	−3.35476e44	−0.961140
$352$	0	0
$353$	7.06671e44	1.83299	0.916494	−	0.400048i	$-0.131006\pi$
$353$	7.06671e44	1.83299	0.916494	+	0.400048i	$0.131006\pi$
$354$	0	0
$355$	1.57026e44	0.368957
$356$	0	0
$357$	5.71681e43	0.121748
$358$	0	0
$359$	5.35788e44	1.03476	0.517380	−	0.855756i	$-0.326907\pi$
$359$	5.35788e44	1.03476	0.517380	+	0.855756i	$0.326907\pi$
$360$	0	0
$361$	−5.55104e44	−0.972743
$362$	0	0
$363$	3.75165e43	0.0596838
$364$	0	0
$365$	1.45163e45	2.09763
$366$	0	0
$367$	3.00056e44	0.394045	0.197022	−	0.980399i	$-0.436873\pi$
$367$	3.00056e44	0.394045	0.197022	+	0.980399i	$0.436873\pi$
$368$	0	0
$369$	6.02849e44	0.719858
$370$	0	0
$371$	−1.24773e44	−0.135543
$372$	0	0
$373$	1.08201e45	1.06986	0.534932	−	0.844895i	$-0.320337\pi$
$373$	1.08201e45	1.06986	0.534932	+	0.844895i	$0.320337\pi$
$374$	0	0
$375$	1.14723e45	1.03301
$376$	0	0
$377$	−1.59201e45	−1.30609
$378$	0	0
$379$	−1.21847e45	−0.911229	−0.455614	−	0.890177i	$-0.650580\pi$
$379$	−1.21847e45	−0.911229	−0.455614	+	0.890177i	$0.650580\pi$
$380$	0	0
$381$	−1.45131e45	−0.989858
$382$	0	0
$383$	9.53201e43	0.0593207	0.0296603	−	0.999560i	$-0.490557\pi$
$383$	9.53201e43	0.0593207	0.0296603	+	0.999560i	$0.490557\pi$
$384$	0	0
$385$	−4.38372e44	−0.249046
$386$	0	0
$387$	−6.88857e43	−0.0357427
$388$	0	0
$389$	1.90493e45	0.903152	0.451576	−	0.892233i	$-0.350862\pi$
$389$	1.90493e45	0.903152	0.451576	+	0.892233i	$0.350862\pi$
$390$	0	0
$391$	4.60678e44	0.199666
$392$	0	0
$393$	−8.32373e44	−0.329950
$394$	0	0
$395$	4.80960e45	1.74446
$396$	0	0
$397$	3.96399e45	1.31613	0.658065	−	0.752961i	$-0.271375\pi$
$397$	3.96399e45	1.31613	0.658065	+	0.752961i	$0.271375\pi$
$398$	0	0
$399$	−1.70748e44	−0.0519196
$400$	0	0
$401$	−5.53039e44	−0.154074	−0.0770369	−	0.997028i	$-0.524546\pi$
$401$	−5.53039e44	−0.154074	−0.0770369	+	0.997028i	$0.524546\pi$
$402$	0	0
$403$	1.68812e45	0.431088
$404$	0	0
$405$	2.33511e45	0.546820
$406$	0	0
$407$	−6.89560e45	−1.48139
$408$	0	0
$409$	3.35495e45	0.661498	0.330749	−	0.943719i	$-0.392699\pi$
$409$	3.35495e45	0.661498	0.330749	+	0.943719i	$0.392699\pi$
$410$	0	0
$411$	−7.42738e45	−1.34463
$412$	0	0
$413$	−4.90568e44	−0.0815781
$414$	0	0
$415$	1.06396e46	1.62586
$416$	0	0
$417$	−9.10474e45	−1.27905
$418$	0	0
$419$	7.69138e45	0.993708	0.496854	−	0.867834i	$-0.334488\pi$
$419$	7.69138e45	0.993708	0.496854	+	0.867834i	$0.334488\pi$
$420$	0	0
$421$	−2.77229e45	−0.329535	−0.164767	−	0.986332i	$-0.552687\pi$
$421$	−2.77229e45	−0.329535	−0.164767	+	0.986332i	$0.552687\pi$
$422$	0	0
$423$	−1.36361e46	−1.49188
$424$	0	0
$425$	1.70691e45	0.171950
$426$	0	0
$427$	9.57360e43	0.00888357
$428$	0	0
$429$	−2.92520e46	−2.50122
$430$	0	0
$431$	−6.18484e45	−0.487502	−0.243751	−	0.969838i	$-0.578378\pi$
$431$	−6.18484e45	−0.487502	−0.243751	+	0.969838i	$0.578378\pi$
$432$	0	0
$433$	2.21783e46	1.61209	0.806047	−	0.591852i	$-0.201603\pi$
$433$	2.21783e46	1.61209	0.806047	+	0.591852i	$0.201603\pi$
$434$	0	0
$435$	−2.28089e46	−1.52947
$436$	0	0
$437$	−1.37594e45	−0.0851482
$438$	0	0
$439$	−1.91766e45	−0.109557	−0.0547787	−	0.998499i	$-0.517445\pi$
$439$	−1.91766e45	−0.109557	−0.0547787	+	0.998499i	$0.517445\pi$
$440$	0	0
$441$	−2.52844e46	−1.33407
$442$	0	0
$443$	3.00864e46	1.46658	0.733290	−	0.679916i	$-0.237984\pi$
$443$	3.00864e46	1.46658	0.733290	+	0.679916i	$0.237984\pi$
$444$	0	0
$445$	−2.96405e46	−1.33532
$446$	0	0
$447$	−6.03235e46	−2.51249
$448$	0	0
$449$	5.68788e45	0.219099	0.109549	−	0.993981i	$-0.465059\pi$
$449$	5.68788e45	0.219099	0.109549	+	0.993981i	$0.465059\pi$
$450$	0	0
$451$	1.47925e46	0.527171
$452$	0	0
$453$	4.87555e46	1.60807
$454$	0	0
$455$	1.27011e46	0.387830
$456$	0	0
$457$	4.67626e46	1.32241	0.661204	−	0.750206i	$-0.270046\pi$
$457$	4.67626e46	1.32241	0.661204	+	0.750206i	$0.270046\pi$
$458$	0	0
$459$	8.94930e45	0.234459
$460$	0	0
$461$	−3.72978e46	−0.905560	−0.452780	−	0.891622i	$-0.649568\pi$
$461$	−3.72978e46	−0.905560	−0.452780	+	0.891622i	$0.649568\pi$
$462$	0	0
$463$	3.67649e46	0.827497	0.413748	−	0.910391i	$-0.364219\pi$
$463$	3.67649e46	0.827497	0.413748	+	0.910391i	$0.364219\pi$
$464$	0	0
$465$	2.41858e46	0.504817
$466$	0	0
$467$	2.37743e46	0.460323	0.230161	−	0.973152i	$-0.426075\pi$
$467$	2.37743e46	0.460323	0.230161	+	0.973152i	$0.426075\pi$
$468$	0	0
$469$	2.05639e46	0.369472
$470$	0	0
$471$	5.18607e46	0.864914
$472$	0	0
$473$	−1.69029e45	−0.0261753
$474$	0	0
$475$	−5.09814e45	−0.0733287
$476$	0	0
$477$	−6.94094e46	−0.927569
$478$	0	0
$479$	8.80315e46	1.09337	0.546684	−	0.837339i	$-0.315890\pi$
$479$	8.80315e46	1.09337	0.546684	+	0.837339i	$0.315890\pi$
$480$	0	0
$481$	1.99788e47	2.30691
$482$	0	0
$483$	1.51046e46	0.162194
$484$	0	0
$485$	5.89206e46	0.588555
$486$	0	0
$487$	−7.13443e46	−0.663136	−0.331568	−	0.943431i	$-0.607578\pi$
$487$	−7.13443e46	−0.663136	−0.331568	+	0.943431i	$0.607578\pi$
$488$	0	0
$489$	−2.97758e47	−2.57607
$490$	0	0
$491$	−8.49115e46	−0.683975	−0.341988	−	0.939704i	$-0.611100\pi$
$491$	−8.49115e46	−0.683975	−0.341988	+	0.939704i	$0.611100\pi$
$492$	0	0
$493$	4.24693e46	0.318606
$494$	0	0
$495$	−2.43859e47	−1.70431
$496$	0	0
$497$	−9.58638e45	−0.0624335
$498$	0	0
$499$	1.75876e47	1.06770	0.533848	−	0.845581i	$-0.320745\pi$
$499$	1.75876e47	1.06770	0.533848	+	0.845581i	$0.320745\pi$
$500$	0	0
$501$	3.19397e47	1.80789
$502$	0	0
$503$	−1.24051e47	−0.654880	−0.327440	−	0.944872i	$-0.606186\pi$
$503$	−1.24051e47	−0.654880	−0.327440	+	0.944872i	$0.606186\pi$
$504$	0	0
$505$	−2.32072e47	−1.14294
$506$	0	0
$507$	5.11027e47	2.34858
$508$	0	0
$509$	3.71512e47	1.59372	0.796859	−	0.604166i	$-0.206493\pi$
$509$	3.71512e47	1.59372	0.796859	+	0.604166i	$0.206493\pi$
$510$	0	0
$511$	−8.86212e46	−0.354954
$512$	0	0
$513$	−2.67296e46	−0.0999857
$514$	0	0
$515$	5.19987e47	1.81705
$516$	0	0
$517$	−3.34597e47	−1.09254
$518$	0	0
$519$	−4.60621e47	−1.40577
$520$	0	0
$521$	7.89659e45	0.0225309	0.0112655	−	0.999937i	$-0.496414\pi$
$521$	7.89659e45	0.0225309	0.0112655	+	0.999937i	$0.496414\pi$
$522$	0	0
$523$	−1.18122e47	−0.315175	−0.157588	−	0.987505i	$-0.550372\pi$
$523$	−1.18122e47	−0.315175	−0.157588	+	0.987505i	$0.550372\pi$
$524$	0	0
$525$	5.59656e46	0.139680
$526$	0	0
$527$	−4.50331e46	−0.105159
$528$	0	0
$529$	−3.35870e47	−0.734001
$530$	0	0
$531$	−2.72895e47	−0.558267
$532$	0	0
$533$	−4.28587e47	−0.820943
$534$	0	0
$535$	−7.46124e47	−1.33850
$536$	0	0
$537$	−2.16958e47	−0.364607
$538$	0	0
$539$	−6.20419e47	−0.976971
$540$	0	0
$541$	−6.62658e47	−0.977997	−0.488998	−	0.872285i	$-0.662638\pi$
$541$	−6.62658e47	−0.977997	−0.488998	+	0.872285i	$0.662638\pi$
$542$	0	0
$543$	−8.30070e47	−1.14846
$544$	0	0
$545$	9.54073e46	0.123778
$546$	0	0
$547$	1.12424e47	0.136798	0.0683990	−	0.997658i	$-0.478211\pi$
$547$	1.12424e47	0.136798	0.0683990	+	0.997658i	$0.478211\pi$
$548$	0	0
$549$	5.32564e46	0.0607933
$550$	0	0
$551$	−1.26846e47	−0.135870
$552$	0	0
$553$	−2.93624e47	−0.295190
$554$	0	0
$555$	2.86238e48	2.70147
$556$	0	0
$557$	6.68090e47	0.592064	0.296032	−	0.955178i	$-0.404336\pi$
$557$	6.68090e47	0.592064	0.296032	+	0.955178i	$0.404336\pi$
$558$	0	0
$559$	4.89734e46	0.0407618
$560$	0	0
$561$	7.80339e47	0.610145
$562$	0	0
$563$	−2.47733e48	−1.82007	−0.910033	−	0.414536i	$-0.863944\pi$
$563$	−2.47733e48	−1.82007	−0.910033	+	0.414536i	$0.863944\pi$
$564$	0	0
$565$	4.47639e47	0.309087
$566$	0	0
$567$	−1.42558e47	−0.0925309
$568$	0	0
$569$	8.59661e47	0.524641	0.262321	−	0.964981i	$-0.415512\pi$
$569$	8.59661e47	0.524641	0.262321	+	0.964981i	$0.415512\pi$
$570$	0	0
$571$	3.39185e48	1.94673	0.973364	−	0.229266i	$-0.0736327\pi$
$571$	3.39185e48	1.94673	0.973364	+	0.229266i	$0.0736327\pi$
$572$	0	0
$573$	−2.81753e48	−1.52112
$574$	0	0
$575$	4.50988e47	0.229075
$576$	0	0
$577$	2.12534e48	1.01590	0.507951	−	0.861386i	$-0.330403\pi$
$577$	2.12534e48	1.01590	0.507951	+	0.861386i	$0.330403\pi$
$578$	0	0
$579$	4.79258e48	2.15623
$580$	0	0
$581$	−6.49544e47	−0.275123
$582$	0	0
$583$	−1.70314e48	−0.679283
$584$	0	0
$585$	7.06542e48	2.65406
$586$	0	0
$587$	−1.06269e48	−0.376044	−0.188022	−	0.982165i	$-0.560208\pi$
$587$	−1.06269e48	−0.376044	−0.188022	+	0.982165i	$0.560208\pi$
$588$	0	0
$589$	1.34504e47	0.0448453
$590$	0	0
$591$	−3.99665e48	−1.25578
$592$	0	0
$593$	−3.98260e48	−1.17953	−0.589766	−	0.807574i	$-0.700780\pi$
$593$	−3.98260e48	−1.17953	−0.589766	+	0.807574i	$0.700780\pi$
$594$	0	0
$595$	−3.38820e47	−0.0946068
$596$	0	0
$597$	1.64894e48	0.434163
$598$	0	0
$599$	−5.14535e48	−1.27775	−0.638874	−	0.769311i	$-0.720599\pi$
$599$	−5.14535e48	−1.27775	−0.638874	+	0.769311i	$0.720599\pi$
$600$	0	0
$601$	−1.83878e48	−0.430751	−0.215375	−	0.976531i	$-0.569098\pi$
$601$	−1.83878e48	−0.430751	−0.215375	+	0.976531i	$0.569098\pi$
$602$	0	0
$603$	1.14394e49	2.52842
$604$	0	0
$605$	−2.22350e47	−0.0463787
$606$	0	0
$607$	−2.06846e48	−0.407236	−0.203618	−	0.979050i	$-0.565270\pi$
$607$	−2.06846e48	−0.407236	−0.203618	+	0.979050i	$0.565270\pi$
$608$	0	0
$609$	1.39247e48	0.258812
$610$	0	0
$611$	9.69439e48	1.70137
$612$	0	0
$613$	1.02050e48	0.169143	0.0845717	−	0.996417i	$-0.473048\pi$
$613$	1.02050e48	0.169143	0.0845717	+	0.996417i	$0.473048\pi$
$614$	0	0
$615$	−6.14039e48	−0.961350
$616$	0	0
$617$	−3.46313e48	−0.512246	−0.256123	−	0.966644i	$-0.582445\pi$
$617$	−3.46313e48	−0.512246	−0.256123	+	0.966644i	$0.582445\pi$
$618$	0	0
$619$	−6.56428e48	−0.917488	−0.458744	−	0.888568i	$-0.651701\pi$
$619$	−6.56428e48	−0.917488	−0.458744	+	0.888568i	$0.651701\pi$
$620$	0	0
$621$	2.36453e48	0.312350
$622$	0	0
$623$	1.80954e48	0.225958
$624$	0	0
$625$	−1.05615e49	−1.24688
$626$	0	0
$627$	−2.33070e48	−0.260198
$628$	0	0
$629$	−5.32965e48	−0.562746
$630$	0	0
$631$	−5.18692e48	−0.518079	−0.259039	−	0.965867i	$-0.583406\pi$
$631$	−5.18692e48	−0.518079	−0.259039	+	0.965867i	$0.583406\pi$
$632$	0	0
$633$	6.48082e48	0.612442
$634$	0	0
$635$	8.60152e48	0.769192
$636$	0	0
$637$	1.79756e49	1.52140
$638$	0	0
$639$	−5.33275e48	−0.427254
$640$	0	0
$641$	−6.15569e48	−0.466940	−0.233470	−	0.972364i	$-0.575008\pi$
$641$	−6.15569e48	−0.466940	−0.233470	+	0.972364i	$0.575008\pi$
$642$	0	0
$643$	6.50853e48	0.467510	0.233755	−	0.972296i	$-0.424899\pi$
$643$	6.50853e48	0.467510	0.233755	+	0.972296i	$0.424899\pi$
$644$	0	0
$645$	7.01644e47	0.0477333
$646$	0	0
$647$	−2.43533e49	−1.56940	−0.784698	−	0.619878i	$-0.787182\pi$
$647$	−2.43533e49	−1.56940	−0.784698	+	0.619878i	$0.787182\pi$
$648$	0	0
$649$	−6.69620e48	−0.408833
$650$	0	0
$651$	−1.47654e48	−0.0854233
$652$	0	0
$653$	4.33343e48	0.237602	0.118801	−	0.992918i	$-0.462095\pi$
$653$	4.33343e48	0.237602	0.118801	+	0.992918i	$0.462095\pi$
$654$	0	0
$655$	4.93325e48	0.256396
$656$	0	0
$657$	−4.92986e49	−2.42907
$658$	0	0
$659$	−2.26798e49	−1.05960	−0.529802	−	0.848121i	$-0.677734\pi$
$659$	−2.26798e49	−1.05960	−0.529802	+	0.848121i	$0.677734\pi$
$660$	0	0
$661$	−2.84364e49	−1.25994	−0.629969	−	0.776620i	$-0.716932\pi$
$661$	−2.84364e49	−1.25994	−0.629969	+	0.776620i	$0.716932\pi$
$662$	0	0
$663$	−2.26090e49	−0.950156
$664$	0	0
$665$	1.01198e48	0.0403453
$666$	0	0
$667$	1.12210e49	0.424452
$668$	0	0
$669$	2.46325e49	0.884202
$670$	0	0
$671$	1.30679e48	0.0445205
$672$	0	0
$673$	1.14434e49	0.370077	0.185039	−	0.982731i	$-0.440759\pi$
$673$	1.14434e49	0.370077	0.185039	+	0.982731i	$0.440759\pi$
$674$	0	0
$675$	8.76105e48	0.268993
$676$	0	0
$677$	−3.00594e49	−0.876351	−0.438175	−	0.898890i	$-0.644375\pi$
$677$	−3.00594e49	−0.876351	−0.438175	+	0.898890i	$0.644375\pi$
$678$	0	0
$679$	−3.59708e48	−0.0995931
$680$	0	0
$681$	3.59766e49	0.946119
$682$	0	0
$683$	7.78736e49	1.94549	0.972743	−	0.231885i	$-0.0744894\pi$
$683$	7.78736e49	1.94549	0.972743	+	0.231885i	$0.0744894\pi$
$684$	0	0
$685$	4.40201e49	1.04488
$686$	0	0
$687$	6.32535e49	1.42673
$688$	0	0
$689$	4.93456e49	1.05782
$690$	0	0
$691$	−9.61614e49	−1.95945	−0.979726	−	0.200340i	$-0.935795\pi$
$691$	−9.61614e49	−1.95945	−0.979726	+	0.200340i	$0.935795\pi$
$692$	0	0
$693$	1.48875e49	0.288397
$694$	0	0
$695$	5.39613e49	0.993913
$696$	0	0
$697$	1.14332e49	0.200260
$698$	0	0
$699$	5.46014e49	0.909606
$700$	0	0
$701$	1.19741e49	0.189748	0.0948742	−	0.995489i	$-0.469755\pi$
$701$	1.19741e49	0.189748	0.0948742	+	0.995489i	$0.469755\pi$
$702$	0	0
$703$	1.59185e49	0.239984
$704$	0	0
$705$	1.38892e50	1.99236
$706$	0	0
$707$	1.41679e49	0.193404
$708$	0	0
$709$	−4.80032e49	−0.623680	−0.311840	−	0.950135i	$-0.600945\pi$
$709$	−4.80032e49	−0.623680	−0.311840	+	0.950135i	$0.600945\pi$
$710$	0	0
$711$	−1.63339e50	−2.02009
$712$	0	0
$713$	−1.18984e49	−0.140094
$714$	0	0
$715$	1.73369e50	1.94363
$716$	0	0
$717$	−1.73297e50	−1.85015
$718$	0	0
$719$	1.45387e49	0.147832	0.0739162	−	0.997264i	$-0.476450\pi$
$719$	1.45387e49	0.147832	0.0739162	+	0.997264i	$0.476450\pi$
$720$	0	0
$721$	−3.17450e49	−0.307474
$722$	0	0
$723$	−1.15246e50	−1.06342
$724$	0	0
$725$	4.15759e49	0.365533
$726$	0	0
$727$	−1.25503e50	−1.05148	−0.525742	−	0.850644i	$-0.676212\pi$
$727$	−1.25503e50	−1.05148	−0.525742	+	0.850644i	$0.676212\pi$
$728$	0	0
$729$	−1.96598e50	−1.56981
$730$	0	0
$731$	−1.30644e48	−0.00994338
$732$	0	0
$733$	−1.40440e50	−1.01899	−0.509495	−	0.860473i	$-0.670168\pi$
$733$	−1.40440e50	−1.01899	−0.509495	+	0.860473i	$0.670168\pi$
$734$	0	0
$735$	2.57537e50	1.78161
$736$	0	0
$737$	2.80695e50	1.85163
$738$	0	0
$739$	−2.93572e49	−0.184687	−0.0923435	−	0.995727i	$-0.529436\pi$
$739$	−2.93572e49	−0.184687	−0.0923435	+	0.995727i	$0.529436\pi$
$740$	0	0
$741$	6.75281e49	0.405197
$742$	0	0
$743$	−1.10795e50	−0.634185	−0.317093	−	0.948395i	$-0.602707\pi$
$743$	−1.10795e50	−0.634185	−0.317093	+	0.948395i	$0.602707\pi$
$744$	0	0
$745$	3.57521e50	1.95239
$746$	0	0
$747$	−3.61331e50	−1.88276
$748$	0	0
$749$	4.55506e49	0.226497
$750$	0	0
$751$	4.13601e50	1.96283	0.981416	−	0.191892i	$-0.0614622\pi$
$751$	4.13601e50	1.96283	0.981416	+	0.191892i	$0.0614622\pi$
$752$	0	0
$753$	−5.28418e50	−2.39368
$754$	0	0
$755$	−2.88961e50	−1.24959
$756$	0	0
$757$	3.15399e50	1.30221	0.651106	−	0.758987i	$-0.274305\pi$
$757$	3.15399e50	1.30221	0.651106	+	0.758987i	$0.274305\pi$
$758$	0	0
$759$	2.06176e50	0.812845
$760$	0	0
$761$	−3.65841e50	−1.37740	−0.688702	−	0.725045i	$-0.741819\pi$
$761$	−3.65841e50	−1.37740	−0.688702	+	0.725045i	$0.741819\pi$
$762$	0	0
$763$	−5.82458e48	−0.0209452
$764$	0	0
$765$	−1.88480e50	−0.647427
$766$	0	0
$767$	1.94011e50	0.636661
$768$	0	0
$769$	−2.07264e50	−0.649849	−0.324925	−	0.945740i	$-0.605339\pi$
$769$	−2.07264e50	−0.649849	−0.324925	+	0.945740i	$0.605339\pi$
$770$	0	0
$771$	−9.42552e50	−2.82393
$772$	0	0
$773$	2.04692e50	0.586084	0.293042	−	0.956100i	$-0.405332\pi$
$773$	2.04692e50	0.586084	0.293042	+	0.956100i	$0.405332\pi$
$774$	0	0
$775$	−4.40859e49	−0.120648
$776$	0	0
$777$	−1.74747e50	−0.457133
$778$	0	0
$779$	−3.41484e49	−0.0854013
$780$	0	0
$781$	−1.30853e50	−0.312889
$782$	0	0
$783$	2.17983e50	0.498415
$784$	0	0
$785$	−3.07365e50	−0.672101
$786$	0	0
$787$	2.74398e50	0.573882	0.286941	−	0.957948i	$-0.407362\pi$
$787$	2.74398e50	0.573882	0.286941	+	0.957948i	$0.407362\pi$
$788$	0	0
$789$	−1.45994e49	−0.0292071
$790$	0	0
$791$	−2.73282e49	−0.0523025
$792$	0	0
$793$	−3.78619e49	−0.0693301
$794$	0	0
$795$	7.06978e50	1.23874
$796$	0	0
$797$	3.73868e50	0.626899	0.313450	−	0.949605i	$-0.398515\pi$
$797$	3.73868e50	0.626899	0.313450	+	0.949605i	$0.398515\pi$
$798$	0	0
$799$	−2.58612e50	−0.415030
$800$	0	0
$801$	1.00662e51	1.54631
$802$	0	0
$803$	−1.20967e51	−1.77887
$804$	0	0
$805$	−8.95210e49	−0.126037
$806$	0	0
$807$	−4.51827e50	−0.609095
$808$	0	0
$809$	−3.87407e50	−0.500114	−0.250057	−	0.968231i	$-0.580449\pi$
$809$	−3.87407e50	−0.500114	−0.250057	+	0.968231i	$0.580449\pi$
$810$	0	0
$811$	9.21816e50	1.13967	0.569835	−	0.821759i	$-0.307007\pi$
$811$	9.21816e50	1.13967	0.569835	+	0.821759i	$0.307007\pi$
$812$	0	0
$813$	1.42488e50	0.168730
$814$	0	0
$815$	1.76473e51	2.00180
$816$	0	0
$817$	3.90203e48	0.00424038
$818$	0	0
$819$	−4.31341e50	−0.449110
$820$	0	0
$821$	1.64737e51	1.64357	0.821783	−	0.569801i	$-0.192980\pi$
$821$	1.64737e51	1.64357	0.821783	+	0.569801i	$0.192980\pi$
$822$	0	0
$823$	6.02163e50	0.575727	0.287864	−	0.957671i	$-0.407055\pi$
$823$	6.02163e50	0.575727	0.287864	+	0.957671i	$0.407055\pi$
$824$	0	0
$825$	7.63924e50	0.700013
$826$	0	0
$827$	4.30476e50	0.378097	0.189048	−	0.981968i	$-0.439460\pi$
$827$	4.30476e50	0.378097	0.189048	+	0.981968i	$0.439460\pi$
$828$	0	0
$829$	8.88751e50	0.748300	0.374150	−	0.927368i	$-0.377935\pi$
$829$	8.88751e50	0.748300	0.374150	+	0.927368i	$0.377935\pi$
$830$	0	0
$831$	2.55364e51	2.06131
$832$	0	0
$833$	−4.79525e50	−0.371128
$834$	0	0
$835$	−1.89298e51	−1.40486
$836$	0	0
$837$	−2.31142e50	−0.164507
$838$	0	0
$839$	−3.76912e50	−0.257280	−0.128640	−	0.991691i	$-0.541061\pi$
$839$	−3.76912e50	−0.257280	−0.128640	+	0.991691i	$0.541061\pi$
$840$	0	0
$841$	−4.92875e50	−0.322706
$842$	0	0
$843$	1.41210e51	0.886916
$844$	0	0
$845$	−3.02872e51	−1.82502
$846$	0	0
$847$	1.35744e49	0.00784804
$848$	0	0
$849$	3.68407e51	2.04382
$850$	0	0
$851$	−1.40817e51	−0.749699
$852$	0	0
$853$	−3.00248e51	−1.53417	−0.767083	−	0.641547i	$-0.778293\pi$
$853$	−3.00248e51	−1.53417	−0.767083	+	0.641547i	$0.778293\pi$
$854$	0	0
$855$	5.62949e50	0.276097
$856$	0	0
$857$	−2.46088e50	−0.115858	−0.0579291	−	0.998321i	$-0.518450\pi$
$857$	−2.46088e50	−0.115858	−0.0579291	+	0.998321i	$0.518450\pi$
$858$	0	0
$859$	−2.64506e51	−1.19551	−0.597757	−	0.801677i	$-0.703941\pi$
$859$	−2.64506e51	−1.19551	−0.597757	+	0.801677i	$0.703941\pi$
$860$	0	0
$861$	3.74868e50	0.162676
$862$	0	0
$863$	9.93861e50	0.414130	0.207065	−	0.978327i	$-0.433609\pi$
$863$	9.93861e50	0.414130	0.207065	+	0.978327i	$0.433609\pi$
$864$	0	0
$865$	2.72998e51	1.09239
$866$	0	0
$867$	−3.42107e51	−1.31470
$868$	0	0
$869$	−4.00794e51	−1.47936
$870$	0	0
$871$	−8.13268e51	−2.88347
$872$	0	0
$873$	−2.00100e51	−0.681550
$874$	0	0
$875$	4.15096e50	0.135834
$876$	0	0
$877$	−2.79832e50	−0.0879842	−0.0439921	−	0.999032i	$-0.514008\pi$
$877$	−2.79832e50	−0.0879842	−0.0439921	+	0.999032i	$0.514008\pi$
$878$	0	0
$879$	6.58876e51	1.99067
$880$	0	0
$881$	−5.95050e50	−0.172773	−0.0863863	−	0.996262i	$-0.527532\pi$
$881$	−5.95050e50	−0.172773	−0.0863863	+	0.996262i	$0.527532\pi$
$882$	0	0
$883$	3.85350e51	1.07533	0.537667	−	0.843157i	$-0.319306\pi$
$883$	3.85350e51	1.07533	0.537667	+	0.843157i	$0.319306\pi$
$884$	0	0
$885$	2.77961e51	0.745550
$886$	0	0
$887$	4.56338e50	0.117658	0.0588292	−	0.998268i	$-0.481263\pi$
$887$	4.56338e50	0.117658	0.0588292	+	0.998268i	$0.481263\pi$
$888$	0	0
$889$	−5.25119e50	−0.130160
$890$	0	0
$891$	−1.94590e51	−0.463724
$892$	0	0
$893$	7.72416e50	0.176991
$894$	0	0
$895$	1.28585e51	0.283326
$896$	0	0
$897$	−5.97362e51	−1.26581
$898$	0	0
$899$	−1.09689e51	−0.223548
$900$	0	0
$901$	−1.31637e51	−0.258044
$902$	0	0
$903$	−4.28351e49	−0.00807726
$904$	0	0
$905$	4.91960e51	0.892441
$906$	0	0
$907$	4.12286e51	0.719566	0.359783	−	0.933036i	$-0.382851\pi$
$907$	4.12286e51	0.719566	0.359783	+	0.933036i	$0.382851\pi$
$908$	0	0
$909$	7.88139e51	1.32353
$910$	0	0
$911$	−6.95561e51	−1.12399	−0.561996	−	0.827140i	$-0.689966\pi$
$911$	−6.95561e51	−1.12399	−0.561996	+	0.827140i	$0.689966\pi$
$912$	0	0
$913$	−8.86621e51	−1.37879
$914$	0	0
$915$	−5.42450e50	−0.0811877
$916$	0	0
$917$	−3.01173e50	−0.0433863
$918$	0	0
$919$	−1.23335e51	−0.171027	−0.0855136	−	0.996337i	$-0.527253\pi$
$919$	−1.23335e51	−0.171027	−0.0855136	+	0.996337i	$0.527253\pi$
$920$	0	0
$921$	4.83420e51	0.645329
$922$	0	0
$923$	3.79125e51	0.487251
$924$	0	0
$925$	−5.21754e51	−0.645632
$926$	0	0
$927$	−1.76592e52	−2.10415
$928$	0	0
$929$	−9.97208e51	−1.14422	−0.572112	−	0.820176i	$-0.693876\pi$
$929$	−9.97208e51	−1.14422	−0.572112	+	0.820176i	$0.693876\pi$
$930$	0	0
$931$	1.43224e51	0.158269
$932$	0	0
$933$	−7.69232e51	−0.818707
$934$	0	0
$935$	−4.62486e51	−0.474128
$936$	0	0
$937$	6.01334e51	0.593844	0.296922	−	0.954902i	$-0.404040\pi$
$937$	6.01334e51	0.593844	0.296922	+	0.954902i	$0.404040\pi$
$938$	0	0
$939$	1.25626e52	1.19517
$940$	0	0
$941$	1.38035e52	1.26523	0.632617	−	0.774465i	$-0.281981\pi$
$941$	1.38035e52	1.26523	0.632617	+	0.774465i	$0.281981\pi$
$942$	0	0
$943$	3.02081e51	0.266789
$944$	0	0
$945$	−1.73907e51	−0.147999
$946$	0	0
$947$	9.01648e51	0.739456	0.369728	−	0.929140i	$-0.379451\pi$
$947$	9.01648e51	0.739456	0.369728	+	0.929140i	$0.379451\pi$
$948$	0	0
$949$	3.50482e52	2.77017
$950$	0	0
$951$	1.83121e52	1.39502
$952$	0	0
$953$	4.32450e51	0.317549	0.158774	−	0.987315i	$-0.449246\pi$
$953$	4.32450e51	0.317549	0.158774	+	0.987315i	$0.449246\pi$
$954$	0	0
$955$	1.66987e52	1.18202
$956$	0	0
$957$	1.90071e52	1.29705
$958$	0	0
$959$	−2.68741e51	−0.176810
$960$	0	0
$961$	−1.46006e52	−0.926216
$962$	0	0
$963$	2.53391e52	1.55000
$964$	0	0
$965$	−2.84044e52	−1.67555
$966$	0	0
$967$	1.91085e51	0.108709	0.0543543	−	0.998522i	$-0.482690\pi$
$967$	1.91085e51	0.108709	0.0543543	+	0.998522i	$0.482690\pi$
$968$	0	0
$969$	−1.80141e51	−0.0988431
$970$	0	0
$971$	−2.54698e52	−1.34800	−0.674000	−	0.738732i	$-0.735425\pi$
$971$	−2.54698e52	−1.34800	−0.674000	+	0.738732i	$0.735425\pi$
$972$	0	0
$973$	−3.29432e51	−0.168186
$974$	0	0
$975$	−2.21334e52	−1.09010
$976$	0	0
$977$	2.34465e52	1.11410	0.557048	−	0.830480i	$-0.311934\pi$
$977$	2.34465e52	1.11410	0.557048	+	0.830480i	$0.311934\pi$
$978$	0	0
$979$	2.47001e52	1.13240
$980$	0	0
$981$	−3.24012e51	−0.143335
$982$	0	0
$983$	−1.06892e52	−0.456308	−0.228154	−	0.973625i	$-0.573269\pi$
$983$	−1.06892e52	−0.456308	−0.228154	+	0.973625i	$0.573269\pi$
$984$	0	0
$985$	2.36871e52	0.975837
$986$	0	0
$987$	−8.47931e51	−0.337140
$988$	0	0
$989$	−3.45178e50	−0.0132467
$990$	0	0
$991$	−1.97227e52	−0.730597	−0.365298	−	0.930891i	$-0.619033\pi$
$991$	−1.97227e52	−0.730597	−0.365298	+	0.930891i	$0.619033\pi$
$992$	0	0
$993$	−2.16387e52	−0.773782
$994$	0	0
$995$	−9.77280e51	−0.337377
$996$	0	0
$997$	−1.44359e52	−0.481149	−0.240575	−	0.970631i	$-0.577336\pi$
$997$	−1.44359e52	−0.481149	−0.240575	+	0.970631i	$0.577336\pi$
$998$	0	0
$999$	−2.73555e52	−0.880338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.36.a.d.1.3		3
4.3	odd	2		1.36.a.a.1.1	✓	3
12.11	even	2		9.36.a.b.1.3		3
20.3	even	4		25.36.b.a.24.5		6
20.7	even	4		25.36.b.a.24.2		6
20.19	odd	2		25.36.a.a.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.36.a.a.1.1	✓	3	4.3	odd	2
9.36.a.b.1.3		3	12.11	even	2
16.36.a.d.1.3		3	1.1	even	1	trivial
25.36.a.a.1.3		3	20.19	odd	2
25.36.b.a.24.2		6	20.7	even	4
25.36.b.a.24.5		6	20.3	even	4

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+3.45913e8 q^{3} -2.05014e12 q^{5} +1.25160e14 q^{7} +6.96245e16 q^{9} +O(q^{10})\)	\(q+3.45913e8 q^{3} -2.05014e12 q^{5} +1.25160e14 q^{7} +6.96245e16 q^{9} +1.70842e18 q^{11} -4.94986e19 q^{13} -7.09169e20 q^{15} +1.32045e21 q^{17} -3.94388e21 q^{19} +4.32945e22 q^{21} +3.48881e23 q^{23} +1.29267e24 q^{25} +6.77748e24 q^{27} +3.21628e25 q^{29} -3.41044e25 q^{31} +5.90965e26 q^{33} -2.56595e26 q^{35} -4.03624e27 q^{37} -1.71222e28 q^{39} +8.65857e27 q^{41} -9.89389e26 q^{43} -1.42740e29 q^{45} -1.95852e29 q^{47} -3.63154e29 q^{49} +4.56760e29 q^{51} -9.96909e29 q^{53} -3.50249e30 q^{55} -1.36424e30 q^{57} -3.91953e30 q^{59} +7.64909e29 q^{61} +8.71420e30 q^{63} +1.01479e32 q^{65} +1.64301e32 q^{67} +1.20682e32 q^{69} -7.65930e31 q^{71} -7.08063e32 q^{73} +4.47152e32 q^{75} +2.13826e32 q^{77} -2.34599e33 q^{79} -1.13900e33 q^{81} -5.18971e33 q^{83} -2.70710e33 q^{85} +1.11255e34 q^{87} +1.44578e34 q^{89} -6.19525e33 q^{91} -1.17972e34 q^{93} +8.08549e33 q^{95} -2.87399e34 q^{97} +1.18948e35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+O(q^{10}) \) 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9	\( 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+ \cdots - 30\!\cdots\!52 q^{99}+O(q^{100}) \) 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9 + 1157945428549987044 * q^11 - 62139610550998650558 * q^13 - 706062479352366674520 * q^15 - 3932636930193139724394 * q^17 + 32303396242934620550940 * q^19 + 222122458753667398807776 * q^21 + 514822069421375217224808 * q^23 + 646885156836409574920725 * q^25 - 27373531144823515220747400 * q^27 - 38460399143984437598248110 * q^29 - 103731594563724689730029856 * q^31 + 1184565114402567148957069584 * q^33 - 1596564453000564139676248560 * q^35 + 2464613172948888017021063706 * q^37 - 18611981954062877742706387512 * q^39 + 23522986584532768129037087406 * q^41 + 47516297135443858184325274308 * q^43 - 110576300682246572226345828030 * q^45 - 164346768841318947159729902256 * q^47 - 595089038084925731622037450821 * q^49 + 1803044074850794727352259927704 * q^51 - 1670480551111723484744458809558 * q^53 - 3067178595313878889174894808520 * q^55 + 4048712721984947152977307522800 * q^57 - 4370528583092646699845243631180 * q^59 + 23537078276516028947219080488306 * q^61 - 45614247892704690123948892506792 * q^63 + 75762337609865790449417835274620 * q^65 + 188845033384156918977315211921644 * q^67 - 325983212396274250308653315934048 * q^69 - 348774602295358210538621162006856 * q^71 - 285174964854822786615792218987058 * q^73 + 1577819189216049218913461709237300 * q^75 + 691501421964810422051915307575712 * q^77 + 426212862591061661901736630874160 * q^79 + 1867221150796978455595408934723163 * q^81 - 14867187331664065317008883105766692 * q^83 - 8752040931978292430566755304420140 * q^85 + 7833119385476156993771606256673800 * q^87 + 30583590846193751335205992418034270 * q^89 - 1003391702519487362566122156997296 * q^91 - 40914650682166972019906140161446016 * q^93 + 84521392849966457182141665888160200 * q^95 - 106165667044630951063328618701091994 * q^97 - 30288686341597731928441860752034252 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more