LMFDB - Embedded newform 48.20.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,20,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, 20, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 20 $
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:	$109.832014347$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{87481}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 21870 $ x^2 - x - 21870
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$148.386$ 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)

$f(q)$	$=$	$q+19683.0 q^{3} -683163. q^{5} -1.13677e8 q^{7} +3.87420e8 q^{9} +O(q^{10})$	$q+19683.0 q^{3} -683163. q^{5} -1.13677e8 q^{7} +3.87420e8 q^{9} -6.43152e9 q^{11} -5.75132e10 q^{13} -1.34467e10 q^{15} +6.85083e11 q^{17} -1.22275e12 q^{19} -2.23751e12 q^{21} -5.02230e12 q^{23} -1.86068e13 q^{25} +7.62560e12 q^{27} +1.53748e14 q^{29} -3.92872e13 q^{31} -1.26592e14 q^{33} +7.76602e13 q^{35} +1.35709e15 q^{37} -1.13203e15 q^{39} -5.75410e14 q^{41} -3.36667e14 q^{43} -2.64671e14 q^{45} +6.99998e15 q^{47} +1.52367e15 q^{49} +1.34845e16 q^{51} +1.69895e16 q^{53} +4.39377e15 q^{55} -2.40674e16 q^{57} +2.70689e16 q^{59} -5.11533e16 q^{61} -4.40410e16 q^{63} +3.92909e16 q^{65} -2.07437e17 q^{67} -9.88540e16 q^{69} +2.35123e17 q^{71} +7.43858e16 q^{73} -3.66237e17 q^{75} +7.31118e17 q^{77} +6.98697e17 q^{79} +1.50095e17 q^{81} -3.25532e18 q^{83} -4.68023e17 q^{85} +3.02622e18 q^{87} -1.43575e18 q^{89} +6.53796e18 q^{91} -7.73291e17 q^{93} +8.35337e17 q^{95} +1.66911e18 q^{97} -2.49170e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 39366 q^{3} + 6016140 q^{5} - 113892064 q^{7} + 774840978 q^{9}+O(q^{10}) $ 2 * q + 39366 * q^3 + 6016140 * q^5 - 113892064 * q^7 + 774840978 * q^9	$ 2 q + 39366 q^{3} + 6016140 q^{5} - 113892064 q^{7} + 774840978 q^{9} + 6650071272 q^{11} - 44072356148 q^{13} + 118415683620 q^{15} + 336281471748 q^{17} - 602118925096 q^{19} - 2241737495712 q^{21} - 2368252165968 q^{23} + 7200399078350 q^{25} + 15251194969974 q^{27} + 280977251970492 q^{29} - 41610149253712 q^{31} + 130893352846776 q^{33} + 76222408017600 q^{35} + 637994163989884 q^{37} - 867476186061084 q^{39} + 11\!\cdots\!32 q^{41}+ \cdots + 25\!\cdots\!08 q^{99}+O(q^{100}) $ 2 * q + 39366 * q^3 + 6016140 * q^5 - 113892064 * q^7 + 774840978 * q^9 + 6650071272 * q^11 - 44072356148 * q^13 + 118415683620 * q^15 + 336281471748 * q^17 - 602118925096 * q^19 - 2241737495712 * q^21 - 2368252165968 * q^23 + 7200399078350 * q^25 + 15251194969974 * q^27 + 280977251970492 * q^29 - 41610149253712 * q^31 + 130893352846776 * q^33 + 76222408017600 * q^35 + 637994163989884 * q^37 - 867476186061084 * q^39 + 1150038430280532 * q^41 - 2820518953538200 * q^43 + 2330775900692460 * q^45 + 5603091319500480 * q^47 - 9875183230193070 * q^49 + 6619028208415884 * q^51 + 27537868857647628 * q^53 + 92031302621896560 * q^55 - 11851506802664568 * q^57 + 55153474421835816 * q^59 - 87192569066205620 * q^61 - 44124119128099296 * q^63 + 129335487918703560 * q^65 - 499299710877878248 * q^67 - 46614307382748144 * q^69 - 84915048436970544 * q^71 - 241433711161728332 * q^73 + 141725455059163050 * q^75 + 728310910203957888 * q^77 - 1131591466033523440 * q^79 + 300189270593998242 * q^81 - 1607743276003583784 * q^83 - 2804751519619294440 * q^85 + 5530475250535194036 * q^87 + 4322797626369576276 * q^89 + 6535073681312344768 * q^91 - 819012567760813296 * q^93 + 4993123012930175760 * q^95 + 7902201235299458308 * q^97 + 2576373864083092008 * 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
$3$	19683.0	0.577350
$3$	19683.0	0.577350
$4$	0	0
$5$	−683163.	−0.156426	−0.0782131	−	0.996937i	$-0.524921\pi$
$5$	−683163.	−0.156426	−0.0782131	+	0.996937i	$0.524921\pi$
$6$	0	0
$7$	−1.13677e8	−1.06474	−0.532369	−	0.846512i	$-0.678698\pi$
$7$	−1.13677e8	−1.06474	−0.532369	+	0.846512i	$0.678698\pi$
$8$	0	0
$9$	3.87420e8	0.333333
$10$	0	0
$11$	−6.43152e9	−0.822400	−0.411200	−	0.911545i	$-0.634890\pi$
$11$	−6.43152e9	−0.822400	−0.411200	+	0.911545i	$0.634890\pi$
$12$	0	0
$13$	−5.75132e10	−1.50420	−0.752101	−	0.659048i	$-0.770959\pi$
$13$	−5.75132e10	−1.50420	−0.752101	+	0.659048i	$0.770959\pi$
$14$	0	0
$15$	−1.34467e10	−0.0903127
$16$	0	0
$17$	6.85083e11	1.40113	0.700565	−	0.713588i	$-0.252931\pi$
$17$	6.85083e11	1.40113	0.700565	+	0.713588i	$0.252931\pi$
$18$	0	0
$19$	−1.22275e12	−0.869317	−0.434658	−	0.900595i	$-0.643131\pi$
$19$	−1.22275e12	−0.869317	−0.434658	+	0.900595i	$0.643131\pi$
$20$	0	0
$21$	−2.23751e12	−0.614727
$22$	0	0
$23$	−5.02230e12	−0.581418	−0.290709	−	0.956811i	$-0.593891\pi$
$23$	−5.02230e12	−0.581418	−0.290709	+	0.956811i	$0.593891\pi$
$24$	0	0
$25$	−1.86068e13	−0.975531
$26$	0	0
$27$	7.62560e12	0.192450
$28$	0	0
$29$	1.53748e14	1.96801	0.984006	−	0.178134i	$-0.0570060\pi$
$29$	1.53748e14	1.96801	0.984006	+	0.178134i	$0.0570060\pi$
$30$	0	0
$31$	−3.92872e13	−0.266880	−0.133440	−	0.991057i	$-0.542602\pi$
$31$	−3.92872e13	−0.266880	−0.133440	+	0.991057i	$0.542602\pi$
$32$	0	0
$33$	−1.26592e14	−0.474813
$34$	0	0
$35$	7.76602e13	0.166553
$36$	0	0
$37$	1.35709e15	1.71669	0.858346	−	0.513072i	$-0.171493\pi$
$37$	1.35709e15	1.71669	0.858346	+	0.513072i	$0.171493\pi$
$38$	0	0
$39$	−1.13203e15	−0.868451
$40$	0	0
$41$	−5.75410e14	−0.274493	−0.137246	−	0.990537i	$-0.543825\pi$
$41$	−5.75410e14	−0.274493	−0.137246	+	0.990537i	$0.543825\pi$
$42$	0	0
$43$	−3.36667e14	−0.102153	−0.0510763	−	0.998695i	$-0.516265\pi$
$43$	−3.36667e14	−0.102153	−0.0510763	+	0.998695i	$0.516265\pi$
$44$	0	0
$45$	−2.64671e14	−0.0521420
$46$	0	0
$47$	6.99998e15	0.912362	0.456181	−	0.889887i	$-0.349217\pi$
$47$	6.99998e15	0.912362	0.456181	+	0.889887i	$0.349217\pi$
$48$	0	0
$49$	1.52367e15	0.133668
$50$	0	0
$51$	1.34845e16	0.808943
$52$	0	0
$53$	1.69895e16	0.707231	0.353615	−	0.935391i	$-0.384952\pi$
$53$	1.69895e16	0.707231	0.353615	+	0.935391i	$0.384952\pi$
$54$	0	0
$55$	4.39377e15	0.128645
$56$	0	0
$57$	−2.40674e16	−0.501900
$58$	0	0
$59$	2.70689e16	0.406796	0.203398	−	0.979096i	$-0.434802\pi$
$59$	2.70689e16	0.406796	0.203398	+	0.979096i	$0.434802\pi$
$60$	0	0
$61$	−5.11533e16	−0.560068	−0.280034	−	0.959990i	$-0.590346\pi$
$61$	−5.11533e16	−0.560068	−0.280034	+	0.959990i	$0.590346\pi$
$62$	0	0
$63$	−4.40410e16	−0.354913
$64$	0	0
$65$	3.92909e16	0.235296
$66$	0	0
$67$	−2.07437e17	−0.931487	−0.465743	−	0.884920i	$-0.654213\pi$
$67$	−2.07437e17	−0.931487	−0.465743	+	0.884920i	$0.654213\pi$
$68$	0	0
$69$	−9.88540e16	−0.335682
$70$	0	0
$71$	2.35123e17	0.608613	0.304306	−	0.952574i	$-0.401575\pi$
$71$	2.35123e17	0.608613	0.304306	+	0.952574i	$0.401575\pi$
$72$	0	0
$73$	7.43858e16	0.147885	0.0739423	−	0.997263i	$-0.476442\pi$
$73$	7.43858e16	0.147885	0.0739423	+	0.997263i	$0.476442\pi$
$74$	0	0
$75$	−3.66237e17	−0.563223
$76$	0	0
$77$	7.31118e17	0.875641
$78$	0	0
$79$	6.98697e17	0.655890	0.327945	−	0.944697i	$-0.393644\pi$
$79$	6.98697e17	0.655890	0.327945	+	0.944697i	$0.393644\pi$
$80$	0	0
$81$	1.50095e17	0.111111
$82$	0	0
$83$	−3.25532e18	−1.91140	−0.955701	−	0.294338i	$-0.904901\pi$
$83$	−3.25532e18	−1.91140	−0.955701	+	0.294338i	$0.904901\pi$
$84$	0	0
$85$	−4.68023e17	−0.219173
$86$	0	0
$87$	3.02622e18	1.13623
$88$	0	0
$89$	−1.43575e18	−0.434385	−0.217193	−	0.976129i	$-0.569690\pi$
$89$	−1.43575e18	−0.434385	−0.217193	+	0.976129i	$0.569690\pi$
$90$	0	0
$91$	6.53796e18	1.60158
$92$	0	0
$93$	−7.73291e17	−0.154083
$94$	0	0
$95$	8.35337e17	0.135984
$96$	0	0
$97$	1.66911e18	0.222922	0.111461	−	0.993769i	$-0.464447\pi$
$97$	1.66911e18	0.222922	0.111461	+	0.993769i	$0.464447\pi$
$98$	0	0
$99$	−2.49170e18	−0.274133
$100$	0	0
$101$	9.99068e18	0.908955	0.454477	−	0.890758i	$-0.349826\pi$
$101$	9.99068e18	0.908955	0.454477	+	0.890758i	$0.349826\pi$
$102$	0	0
$103$	−1.17501e18	−0.0887333	−0.0443666	−	0.999015i	$-0.514127\pi$
$103$	−1.17501e18	−0.0887333	−0.0443666	+	0.999015i	$0.514127\pi$
$104$	0	0
$105$	1.52859e18	0.0961594
$106$	0	0
$107$	1.73164e19	0.910568	0.455284	−	0.890346i	$-0.349538\pi$
$107$	1.73164e19	0.910568	0.455284	+	0.890346i	$0.349538\pi$
$108$	0	0
$109$	1.57748e19	0.695685	0.347842	−	0.937553i	$-0.386914\pi$
$109$	1.57748e19	0.695685	0.347842	+	0.937553i	$0.386914\pi$
$110$	0	0
$111$	2.67116e19	0.991132
$112$	0	0
$113$	4.78817e19	1.49942	0.749712	−	0.661764i	$-0.230192\pi$
$113$	4.78817e19	1.49942	0.749712	+	0.661764i	$0.230192\pi$
$114$	0	0
$115$	3.43105e18	0.0909490
$116$	0	0
$117$	−2.22818e19	−0.501400
$118$	0	0
$119$	−7.78785e19	−1.49184
$120$	0	0
$121$	−1.97947e19	−0.323659
$122$	0	0
$123$	−1.13258e19	−0.158478
$124$	0	0
$125$	2.57418e19	0.309025
$126$	0	0
$127$	1.25232e20	1.29294	0.646472	−	0.762938i	$-0.276244\pi$
$127$	1.25232e20	1.29294	0.646472	+	0.762938i	$0.276244\pi$
$128$	0	0
$129$	−6.62661e18	−0.0589779
$130$	0	0
$131$	2.09329e20	1.60972	0.804862	−	0.593462i	$-0.202239\pi$
$131$	2.09329e20	1.60972	0.804862	+	0.593462i	$0.202239\pi$
$132$	0	0
$133$	1.38999e20	0.925595
$134$	0	0
$135$	−5.20953e18	−0.0301042
$136$	0	0
$137$	1.53205e20	0.769888	0.384944	−	0.922940i	$-0.374221\pi$
$137$	1.53205e20	0.769888	0.384944	+	0.922940i	$0.374221\pi$
$138$	0	0
$139$	2.54374e19	0.111386	0.0556932	−	0.998448i	$-0.482263\pi$
$139$	2.54374e19	0.111386	0.0556932	+	0.998448i	$0.482263\pi$
$140$	0	0
$141$	1.37781e20	0.526752
$142$	0	0
$143$	3.69897e20	1.23705
$144$	0	0
$145$	−1.05035e20	−0.307849
$146$	0	0
$147$	2.99903e19	0.0771732
$148$	0	0
$149$	1.67659e20	0.379453	0.189727	−	0.981837i	$-0.439240\pi$
$149$	1.67659e20	0.379453	0.189727	+	0.981837i	$0.439240\pi$
$150$	0	0
$151$	4.29868e20	0.857144	0.428572	−	0.903508i	$-0.359017\pi$
$151$	4.29868e20	0.857144	0.428572	+	0.903508i	$0.359017\pi$
$152$	0	0
$153$	2.65415e20	0.467043
$154$	0	0
$155$	2.68396e19	0.0417470
$156$	0	0
$157$	9.16306e20	1.26181	0.630905	−	0.775860i	$-0.282684\pi$
$157$	9.16306e20	1.26181	0.630905	+	0.775860i	$0.282684\pi$
$158$	0	0
$159$	3.34405e20	0.408320
$160$	0	0
$161$	5.70923e20	0.619058
$162$	0	0
$163$	−4.50070e20	−0.434008	−0.217004	−	0.976171i	$-0.569628\pi$
$163$	−4.50070e20	−0.434008	−0.217004	+	0.976171i	$0.569628\pi$
$164$	0	0
$165$	8.64827e19	0.0742731
$166$	0	0
$167$	1.62414e20	0.124399	0.0621993	−	0.998064i	$-0.480189\pi$
$167$	1.62414e20	0.124399	0.0621993	+	0.998064i	$0.480189\pi$
$168$	0	0
$169$	1.84585e21	1.26262
$170$	0	0
$171$	−4.73718e20	−0.289772
$172$	0	0
$173$	−3.29538e21	−1.80496	−0.902481	−	0.430731i	$-0.858256\pi$
$173$	−3.29538e21	−1.80496	−0.902481	+	0.430731i	$0.858256\pi$
$174$	0	0
$175$	2.11517e21	1.03869
$176$	0	0
$177$	5.32797e20	0.234864
$178$	0	0
$179$	9.81720e20	0.388941	0.194471	−	0.980908i	$-0.437701\pi$
$179$	9.81720e20	0.388941	0.194471	+	0.980908i	$0.437701\pi$
$180$	0	0
$181$	1.51701e20	0.0540807	0.0270403	−	0.999634i	$-0.491392\pi$
$181$	1.51701e20	0.0540807	0.0270403	+	0.999634i	$0.491392\pi$
$182$	0	0
$183$	−1.00685e21	−0.323355
$184$	0	0
$185$	−9.27114e20	−0.268535
$186$	0	0
$187$	−4.40612e21	−1.15229
$188$	0	0
$189$	−8.66858e20	−0.204909
$190$	0	0
$191$	5.13044e21	1.09733	0.548666	−	0.836042i	$-0.315136\pi$
$191$	5.13044e21	1.09733	0.548666	+	0.836042i	$0.315136\pi$
$192$	0	0
$193$	−9.72873e21	−1.88478	−0.942392	−	0.334510i	$-0.891429\pi$
$193$	−9.72873e21	−1.88478	−0.942392	+	0.334510i	$0.891429\pi$
$194$	0	0
$195$	7.73363e20	0.135848
$196$	0	0
$197$	7.68506e21	1.22523	0.612616	−	0.790381i	$-0.290117\pi$
$197$	7.68506e21	1.22523	0.612616	+	0.790381i	$0.290117\pi$
$198$	0	0
$199$	−4.55786e21	−0.660172	−0.330086	−	0.943951i	$-0.607078\pi$
$199$	−4.55786e21	−0.660172	−0.330086	+	0.943951i	$0.607078\pi$
$200$	0	0
$201$	−4.08299e21	−0.537794
$202$	0	0
$203$	−1.74777e22	−2.09542
$204$	0	0
$205$	3.93099e20	0.0429378
$206$	0	0
$207$	−1.94574e21	−0.193806
$208$	0	0
$209$	7.86413e21	0.714926
$210$	0	0
$211$	1.71334e22	1.42285	0.711426	−	0.702761i	$-0.248050\pi$
$211$	1.71334e22	1.42285	0.711426	+	0.702761i	$0.248050\pi$
$212$	0	0
$213$	4.62793e21	0.351383
$214$	0	0
$215$	2.29998e20	0.0159794
$216$	0	0
$217$	4.46607e21	0.284157
$218$	0	0
$219$	1.46414e21	0.0853812
$220$	0	0
$221$	−3.94014e22	−2.10758
$222$	0	0
$223$	−3.17895e22	−1.56095	−0.780473	−	0.625189i	$-0.785022\pi$
$223$	−3.17895e22	−1.56095	−0.780473	+	0.625189i	$0.785022\pi$
$224$	0	0
$225$	−7.20865e21	−0.325177
$226$	0	0
$227$	−2.55774e22	−1.06074	−0.530372	−	0.847765i	$-0.677948\pi$
$227$	−2.55774e22	−1.06074	−0.530372	+	0.847765i	$0.677948\pi$
$228$	0	0
$229$	−2.23445e22	−0.852576	−0.426288	−	0.904588i	$-0.640179\pi$
$229$	−2.23445e22	−0.852576	−0.426288	+	0.904588i	$0.640179\pi$
$230$	0	0
$231$	1.43906e22	0.505551
$232$	0	0
$233$	−2.52369e21	−0.0816874	−0.0408437	−	0.999166i	$-0.513005\pi$
$233$	−2.52369e21	−0.0816874	−0.0408437	+	0.999166i	$0.513005\pi$
$234$	0	0
$235$	−4.78213e21	−0.142717
$236$	0	0
$237$	1.37524e22	0.378678
$238$	0	0
$239$	−3.93020e22	−0.999158	−0.499579	−	0.866268i	$-0.666512\pi$
$239$	−3.93020e22	−0.999158	−0.499579	+	0.866268i	$0.666512\pi$
$240$	0	0
$241$	−1.99943e22	−0.469618	−0.234809	−	0.972042i	$-0.575447\pi$
$241$	−1.99943e22	−0.469618	−0.234809	+	0.972042i	$0.575447\pi$
$242$	0	0
$243$	2.95431e21	0.0641500
$244$	0	0
$245$	−1.04091e21	−0.0209091
$246$	0	0
$247$	7.03242e22	1.30763
$248$	0	0
$249$	−6.40745e22	−1.10355
$250$	0	0
$251$	4.94152e22	0.788788	0.394394	−	0.918942i	$-0.370955\pi$
$251$	4.94152e22	0.788788	0.394394	+	0.918942i	$0.370955\pi$
$252$	0	0
$253$	3.23010e22	0.478158
$254$	0	0
$255$	−9.21211e21	−0.126540
$256$	0	0
$257$	1.79354e22	0.228742	0.114371	−	0.993438i	$-0.463515\pi$
$257$	1.79354e22	0.228742	0.114371	+	0.993438i	$0.463515\pi$
$258$	0	0
$259$	−1.54271e23	−1.82783
$260$	0	0
$261$	5.95650e22	0.656004
$262$	0	0
$263$	−2.64384e22	−0.270804	−0.135402	−	0.990791i	$-0.543233\pi$
$263$	−2.64384e22	−0.270804	−0.135402	+	0.990791i	$0.543233\pi$
$264$	0	0
$265$	−1.16066e22	−0.110629
$266$	0	0
$267$	−2.82599e22	−0.250792
$268$	0	0
$269$	−6.26512e22	−0.517944	−0.258972	−	0.965885i	$-0.583384\pi$
$269$	−6.26512e22	−0.517944	−0.258972	+	0.965885i	$0.583384\pi$
$270$	0	0
$271$	8.23488e22	0.634525	0.317263	−	0.948338i	$-0.397236\pi$
$271$	8.23488e22	0.634525	0.317263	+	0.948338i	$0.397236\pi$
$272$	0	0
$273$	1.28687e23	0.924673
$274$	0	0
$275$	1.19670e23	0.802276
$276$	0	0
$277$	1.42704e23	0.893057	0.446528	−	0.894769i	$-0.352660\pi$
$277$	1.42704e23	0.893057	0.446528	+	0.894769i	$0.352660\pi$
$278$	0	0
$279$	−1.52207e22	−0.0889599
$280$	0	0
$281$	4.36986e22	0.238648	0.119324	−	0.992855i	$-0.461927\pi$
$281$	4.36986e22	0.238648	0.119324	+	0.992855i	$0.461927\pi$
$282$	0	0
$283$	2.42461e23	1.23786	0.618929	−	0.785447i	$-0.287567\pi$
$283$	2.42461e23	1.23786	0.618929	+	0.785447i	$0.287567\pi$
$284$	0	0
$285$	1.64419e22	0.0785103
$286$	0	0
$287$	6.54112e22	0.292263
$288$	0	0
$289$	2.30266e23	0.963166
$290$	0	0
$291$	3.28531e22	0.128704
$292$	0	0
$293$	1.65262e22	0.0606638	0.0303319	−	0.999540i	$-0.490344\pi$
$293$	1.65262e22	0.0606638	0.0303319	+	0.999540i	$0.490344\pi$
$294$	0	0
$295$	−1.84925e22	−0.0636335
$296$	0	0
$297$	−4.90442e22	−0.158271
$298$	0	0
$299$	2.88849e23	0.874570
$300$	0	0
$301$	3.82714e22	0.108766
$302$	0	0
$303$	1.96647e23	0.524785
$304$	0	0
$305$	3.49461e22	0.0876092
$306$	0	0
$307$	7.14354e23	1.68306	0.841528	−	0.540213i	$-0.181656\pi$
$307$	7.14354e23	1.68306	0.841528	+	0.540213i	$0.181656\pi$
$308$	0	0
$309$	−2.31276e22	−0.0512302
$310$	0	0
$311$	−3.74882e23	−0.781037	−0.390518	−	0.920595i	$-0.627704\pi$
$311$	−3.74882e23	−0.781037	−0.390518	+	0.920595i	$0.627704\pi$
$312$	0	0
$313$	−2.04132e23	−0.400166	−0.200083	−	0.979779i	$-0.564121\pi$
$313$	−2.04132e23	−0.400166	−0.200083	+	0.979779i	$0.564121\pi$
$314$	0	0
$315$	3.00872e22	0.0555176
$316$	0	0
$317$	−2.74437e23	−0.476848	−0.238424	−	0.971161i	$-0.576631\pi$
$317$	−2.74437e23	−0.476848	−0.238424	+	0.971161i	$0.576631\pi$
$318$	0	0
$319$	−9.88832e23	−1.61849
$320$	0	0
$321$	3.40839e23	0.525717
$322$	0	0
$323$	−8.37684e23	−1.21803
$324$	0	0
$325$	1.07014e24	1.46739
$326$	0	0
$327$	3.10496e23	0.401654
$328$	0	0
$329$	−7.95740e23	−0.971427
$330$	0	0
$331$	2.31592e23	0.266906	0.133453	−	0.991055i	$-0.457393\pi$
$331$	2.31592e23	0.266906	0.133453	+	0.991055i	$0.457393\pi$
$332$	0	0
$333$	5.25764e23	0.572230
$334$	0	0
$335$	1.41714e23	0.145709
$336$	0	0
$337$	−1.11237e23	−0.108085	−0.0540425	−	0.998539i	$-0.517211\pi$
$337$	−1.11237e23	−0.108085	−0.0540425	+	0.998539i	$0.517211\pi$
$338$	0	0
$339$	9.42456e23	0.865693
$340$	0	0
$341$	2.52677e23	0.219482
$342$	0	0
$343$	1.12259e24	0.922417
$344$	0	0
$345$	6.75334e22	0.0525094
$346$	0	0
$347$	−2.05609e24	−1.51326	−0.756629	−	0.653844i	$-0.773155\pi$
$347$	−2.05609e24	−1.51326	−0.756629	+	0.653844i	$0.773155\pi$
$348$	0	0
$349$	−1.75492e24	−1.22297	−0.611487	−	0.791255i	$-0.709428\pi$
$349$	−1.75492e24	−1.22297	−0.611487	+	0.791255i	$0.709428\pi$
$350$	0	0
$351$	−4.38573e23	−0.289484
$352$	0	0
$353$	8.21430e23	0.513702	0.256851	−	0.966451i	$-0.417315\pi$
$353$	8.21430e23	0.513702	0.256851	+	0.966451i	$0.417315\pi$
$354$	0	0
$355$	−1.60627e23	−0.0952029
$356$	0	0
$357$	−1.53288e24	−0.861313
$358$	0	0
$359$	−5.29683e23	−0.282240	−0.141120	−	0.989992i	$-0.545070\pi$
$359$	−5.29683e23	−0.282240	−0.141120	+	0.989992i	$0.545070\pi$
$360$	0	0
$361$	−4.83305e23	−0.244289
$362$	0	0
$363$	−3.89618e23	−0.186864
$364$	0	0
$365$	−5.08176e22	−0.0231330
$366$	0	0
$367$	2.30180e24	0.994809	0.497404	−	0.867519i	$-0.334287\pi$
$367$	2.30180e24	0.994809	0.497404	+	0.867519i	$0.334287\pi$
$368$	0	0
$369$	−2.22926e23	−0.0914976
$370$	0	0
$371$	−1.93133e24	−0.753016
$372$	0	0
$373$	1.64953e24	0.611120	0.305560	−	0.952173i	$-0.401156\pi$
$373$	1.64953e24	0.611120	0.305560	+	0.952173i	$0.401156\pi$
$374$	0	0
$375$	5.06675e23	0.178415
$376$	0	0
$377$	−8.84253e24	−2.96029
$378$	0	0
$379$	−3.36454e24	−1.07116	−0.535579	−	0.844485i	$-0.679906\pi$
$379$	−3.36454e24	−1.07116	−0.535579	+	0.844485i	$0.679906\pi$
$380$	0	0
$381$	2.46494e24	0.746482
$382$	0	0
$383$	−1.55219e24	−0.447257	−0.223628	−	0.974674i	$-0.571790\pi$
$383$	−1.55219e24	−0.447257	−0.223628	+	0.974674i	$0.571790\pi$
$384$	0	0
$385$	−4.99473e23	−0.136973
$386$	0	0
$387$	−1.30432e23	−0.0340509
$388$	0	0
$389$	−1.19996e23	−0.0298295	−0.0149148	−	0.999889i	$-0.504748\pi$
$389$	−1.19996e23	−0.0298295	−0.0149148	+	0.999889i	$0.504748\pi$
$390$	0	0
$391$	−3.44070e24	−0.814643
$392$	0	0
$393$	4.12022e24	0.929375
$394$	0	0
$395$	−4.77324e23	−0.102598
$396$	0	0
$397$	−1.23812e24	−0.253661	−0.126831	−	0.991924i	$-0.540480\pi$
$397$	−1.23812e24	−0.253661	−0.126831	+	0.991924i	$0.540480\pi$
$398$	0	0
$399$	2.73592e24	0.534392
$400$	0	0
$401$	−6.55287e24	−1.22056	−0.610281	−	0.792185i	$-0.708943\pi$
$401$	−6.55287e24	−1.22056	−0.610281	+	0.792185i	$0.708943\pi$
$402$	0	0
$403$	2.25954e24	0.401441
$404$	0	0
$405$	−1.02539e23	−0.0173807
$406$	0	0
$407$	−8.72815e24	−1.41181
$408$	0	0
$409$	8.56848e24	1.32292	0.661458	−	0.749982i	$-0.269938\pi$
$409$	8.56848e24	1.32292	0.661458	+	0.749982i	$0.269938\pi$
$410$	0	0
$411$	3.01553e24	0.444495
$412$	0	0
$413$	−3.07712e24	−0.433131
$414$	0	0
$415$	2.22392e24	0.298993
$416$	0	0
$417$	5.00685e23	0.0643090
$418$	0	0
$419$	−8.09987e24	−0.994134	−0.497067	−	0.867712i	$-0.665590\pi$
$419$	−8.09987e24	−0.994134	−0.497067	+	0.867712i	$0.665590\pi$
$420$	0	0
$421$	9.99748e24	1.17276	0.586382	−	0.810035i	$-0.300552\pi$
$421$	9.99748e24	1.17276	0.586382	+	0.810035i	$0.300552\pi$
$422$	0	0
$423$	2.71194e24	0.304121
$424$	0	0
$425$	−1.27472e25	−1.36685
$426$	0	0
$427$	5.81498e24	0.596326
$428$	0	0
$429$	7.28069e24	0.714214
$430$	0	0
$431$	4.59733e24	0.431491	0.215745	−	0.976450i	$-0.430782\pi$
$431$	4.59733e24	0.431491	0.215745	+	0.976450i	$0.430782\pi$
$432$	0	0
$433$	−7.51468e24	−0.674956	−0.337478	−	0.941333i	$-0.609574\pi$
$433$	−7.51468e24	−0.674956	−0.337478	+	0.941333i	$0.609574\pi$
$434$	0	0
$435$	−2.06740e24	−0.177736
$436$	0	0
$437$	6.14102e24	0.505436
$438$	0	0
$439$	2.04197e25	1.60930	0.804649	−	0.593751i	$-0.202353\pi$
$439$	2.04197e25	1.60930	0.804649	+	0.593751i	$0.202353\pi$
$440$	0	0
$441$	5.90299e23	0.0445560
$442$	0	0
$443$	1.84340e25	1.33286	0.666428	−	0.745569i	$-0.267822\pi$
$443$	1.84340e25	1.33286	0.666428	+	0.745569i	$0.267822\pi$
$444$	0	0
$445$	9.80854e23	0.0679492
$446$	0	0
$447$	3.30004e24	0.219077
$448$	0	0
$449$	7.82616e24	0.497976	0.248988	−	0.968507i	$-0.419902\pi$
$449$	7.82616e24	0.497976	0.248988	+	0.968507i	$0.419902\pi$
$450$	0	0
$451$	3.70076e24	0.225743
$452$	0	0
$453$	8.46108e24	0.494872
$454$	0	0
$455$	−4.46649e24	−0.250529
$456$	0	0
$457$	2.15612e25	1.16003	0.580015	−	0.814606i	$-0.303047\pi$
$457$	2.15612e25	1.16003	0.580015	+	0.814606i	$0.303047\pi$
$458$	0	0
$459$	5.22417e24	0.269648
$460$	0	0
$461$	4.52754e24	0.224235	0.112118	−	0.993695i	$-0.464237\pi$
$461$	4.52754e24	0.224235	0.112118	+	0.993695i	$0.464237\pi$
$462$	0	0
$463$	−1.12552e25	−0.534974	−0.267487	−	0.963562i	$-0.586193\pi$
$463$	−1.12552e25	−0.534974	−0.267487	+	0.963562i	$0.586193\pi$
$464$	0	0
$465$	5.28284e23	0.0241026
$466$	0	0
$467$	−1.73792e25	−0.761237	−0.380618	−	0.924732i	$-0.624289\pi$
$467$	−1.73792e25	−0.761237	−0.380618	+	0.924732i	$0.624289\pi$
$468$	0	0
$469$	2.35810e25	0.991790
$470$	0	0
$471$	1.80356e25	0.728506
$472$	0	0
$473$	2.16528e24	0.0840104
$474$	0	0
$475$	2.27514e25	0.848045
$476$	0	0
$477$	6.58210e24	0.235744
$478$	0	0
$479$	−4.26798e25	−1.46904	−0.734522	−	0.678585i	$-0.762593\pi$
$479$	−4.26798e25	−1.46904	−0.734522	+	0.678585i	$0.762593\pi$
$480$	0	0
$481$	−7.80506e25	−2.58225
$482$	0	0
$483$	1.12375e25	0.357413
$484$	0	0
$485$	−1.14027e24	−0.0348708
$486$	0	0
$487$	6.57071e24	0.193236	0.0966178	−	0.995322i	$-0.469198\pi$
$487$	6.57071e24	0.193236	0.0966178	+	0.995322i	$0.469198\pi$
$488$	0	0
$489$	−8.85872e24	−0.250574
$490$	0	0
$491$	−5.24168e25	−1.42625	−0.713126	−	0.701035i	$-0.752721\pi$
$491$	−5.24168e25	−1.42625	−0.713126	+	0.701035i	$0.752721\pi$
$492$	0	0
$493$	1.05330e26	2.75744
$494$	0	0
$495$	1.70224e24	0.0428816
$496$	0	0
$497$	−2.67282e25	−0.648013
$498$	0	0
$499$	2.55146e25	0.595434	0.297717	−	0.954654i	$-0.403775\pi$
$499$	2.55146e25	0.595434	0.297717	+	0.954654i	$0.403775\pi$
$500$	0	0
$501$	3.19679e24	0.0718216
$502$	0	0
$503$	5.69466e25	1.23189	0.615945	−	0.787789i	$-0.288775\pi$
$503$	5.69466e25	1.23189	0.615945	+	0.787789i	$0.288775\pi$
$504$	0	0
$505$	−6.82527e24	−0.142184
$506$	0	0
$507$	3.63319e25	0.728975
$508$	0	0
$509$	−3.26878e25	−0.631781	−0.315891	−	0.948796i	$-0.602303\pi$
$509$	−3.26878e25	−0.631781	−0.315891	+	0.948796i	$0.602303\pi$
$510$	0	0
$511$	−8.45598e24	−0.157458
$512$	0	0
$513$	−9.32419e24	−0.167300
$514$	0	0
$515$	8.02721e23	0.0138802
$516$	0	0
$517$	−4.50205e25	−0.750326
$518$	0	0
$519$	−6.48630e25	−1.04209
$520$	0	0
$521$	−9.30404e25	−1.44116	−0.720581	−	0.693370i	$-0.756125\pi$
$521$	−9.30404e25	−1.44116	−0.720581	+	0.693370i	$0.756125\pi$
$522$	0	0
$523$	−4.88002e25	−0.728879	−0.364440	−	0.931227i	$-0.618739\pi$
$523$	−4.88002e25	−0.728879	−0.364440	+	0.931227i	$0.618739\pi$
$524$	0	0
$525$	4.16329e25	0.599685
$526$	0	0
$527$	−2.69150e25	−0.373933
$528$	0	0
$529$	−4.93919e25	−0.661953
$530$	0	0
$531$	1.04870e25	0.135599
$532$	0	0
$533$	3.30937e25	0.412892
$534$	0	0
$535$	−1.18299e25	−0.142437
$536$	0	0
$537$	1.93232e25	0.224555
$538$	0	0
$539$	−9.79948e24	−0.109928
$540$	0	0
$541$	8.45858e25	0.916059	0.458030	−	0.888937i	$-0.348555\pi$
$541$	8.45858e25	0.916059	0.458030	+	0.888937i	$0.348555\pi$
$542$	0	0
$543$	2.98593e24	0.0312235
$544$	0	0
$545$	−1.07768e25	−0.108823
$546$	0	0
$547$	2.08591e25	0.203430	0.101715	−	0.994814i	$-0.467567\pi$
$547$	2.08591e25	0.203430	0.101715	+	0.994814i	$0.467567\pi$
$548$	0	0
$549$	−1.98179e25	−0.186689
$550$	0	0
$551$	−1.87995e26	−1.71083
$552$	0	0
$553$	−7.94260e25	−0.698352
$554$	0	0
$555$	−1.82484e25	−0.155039
$556$	0	0
$557$	−1.19573e25	−0.0981768	−0.0490884	−	0.998794i	$-0.515632\pi$
$557$	−1.19573e25	−0.0981768	−0.0490884	+	0.998794i	$0.515632\pi$
$558$	0	0
$559$	1.93628e25	0.153658
$560$	0	0
$561$	−8.67257e25	−0.665274
$562$	0	0
$563$	1.51109e26	1.12063	0.560313	−	0.828281i	$-0.310681\pi$
$563$	1.51109e26	1.12063	0.560313	+	0.828281i	$0.310681\pi$
$564$	0	0
$565$	−3.27110e25	−0.234549
$566$	0	0
$567$	−1.70624e25	−0.118304
$568$	0	0
$569$	−1.31930e26	−0.884661	−0.442331	−	0.896852i	$-0.645848\pi$
$569$	−1.31930e26	−0.884661	−0.442331	+	0.896852i	$0.645848\pi$
$570$	0	0
$571$	−2.65239e26	−1.72026	−0.860130	−	0.510076i	$-0.829617\pi$
$571$	−2.65239e26	−1.72026	−0.860130	+	0.510076i	$0.829617\pi$
$572$	0	0
$573$	1.00982e26	0.633544
$574$	0	0
$575$	9.34489e25	0.567191
$576$	0	0
$577$	−9.99355e25	−0.586880	−0.293440	−	0.955977i	$-0.594800\pi$
$577$	−9.99355e25	−0.586880	−0.293440	+	0.955977i	$0.594800\pi$
$578$	0	0
$579$	−1.91491e26	−1.08818
$580$	0	0
$581$	3.70057e26	2.03514
$582$	0	0
$583$	−1.09269e26	−0.581626
$584$	0	0
$585$	1.52221e25	0.0784321
$586$	0	0
$587$	1.90494e26	0.950209	0.475104	−	0.879930i	$-0.342410\pi$
$587$	1.90494e26	0.950209	0.475104	+	0.879930i	$0.342410\pi$
$588$	0	0
$589$	4.80384e25	0.232003
$590$	0	0
$591$	1.51265e26	0.707388
$592$	0	0
$593$	−2.37669e26	−1.07635	−0.538173	−	0.842834i	$-0.680885\pi$
$593$	−2.37669e26	−1.07635	−0.538173	+	0.842834i	$0.680885\pi$
$594$	0	0
$595$	5.32037e25	0.233362
$596$	0	0
$597$	−8.97124e25	−0.381150
$598$	0	0
$599$	3.08975e26	1.27165	0.635826	−	0.771832i	$-0.280659\pi$
$599$	3.08975e26	1.27165	0.635826	+	0.771832i	$0.280659\pi$
$600$	0	0
$601$	−2.61533e26	−1.04284	−0.521421	−	0.853299i	$-0.674598\pi$
$601$	−2.61533e26	−1.04284	−0.521421	+	0.853299i	$0.674598\pi$
$602$	0	0
$603$	−8.03655e25	−0.310496
$604$	0	0
$605$	1.35230e25	0.0506287
$606$	0	0
$607$	−5.37620e25	−0.195067	−0.0975334	−	0.995232i	$-0.531095\pi$
$607$	−5.37620e25	−0.195067	−0.0975334	+	0.995232i	$0.531095\pi$
$608$	0	0
$609$	−3.44013e26	−1.20979
$610$	0	0
$611$	−4.02591e26	−1.37238
$612$	0	0
$613$	3.16087e26	1.04456	0.522278	−	0.852775i	$-0.325082\pi$
$613$	3.16087e26	1.04456	0.522278	+	0.852775i	$0.325082\pi$
$614$	0	0
$615$	7.73737e24	0.0247902
$616$	0	0
$617$	1.98562e26	0.616862	0.308431	−	0.951247i	$-0.400196\pi$
$617$	1.98562e26	0.616862	0.308431	+	0.951247i	$0.400196\pi$
$618$	0	0
$619$	4.82992e26	1.45505	0.727527	−	0.686080i	$-0.240670\pi$
$619$	4.82992e26	1.45505	0.727527	+	0.686080i	$0.240670\pi$
$620$	0	0
$621$	−3.82981e25	−0.111894
$622$	0	0
$623$	1.63213e26	0.462507
$624$	0	0
$625$	3.37310e26	0.927191
$626$	0	0
$627$	1.54790e26	0.412763
$628$	0	0
$629$	9.29719e26	2.40531
$630$	0	0
$631$	6.27693e26	1.57568	0.787841	−	0.615879i	$-0.211199\pi$
$631$	6.27693e26	1.57568	0.787841	+	0.615879i	$0.211199\pi$
$632$	0	0
$633$	3.37236e26	0.821484
$634$	0	0
$635$	−8.55538e25	−0.202250
$636$	0	0
$637$	−8.76310e25	−0.201063
$638$	0	0
$639$	9.10915e25	0.202871
$640$	0	0
$641$	2.15915e26	0.466800	0.233400	−	0.972381i	$-0.425015\pi$
$641$	2.15915e26	0.466800	0.233400	+	0.972381i	$0.425015\pi$
$642$	0	0
$643$	−4.68779e26	−0.983929	−0.491964	−	0.870615i	$-0.663721\pi$
$643$	−4.68779e26	−0.983929	−0.491964	+	0.870615i	$0.663721\pi$
$644$	0	0
$645$	4.52706e24	0.00922568
$646$	0	0
$647$	2.10829e26	0.417196	0.208598	−	0.978001i	$-0.433110\pi$
$647$	2.10829e26	0.417196	0.208598	+	0.978001i	$0.433110\pi$
$648$	0	0
$649$	−1.74094e26	−0.334549
$650$	0	0
$651$	8.79057e25	0.164058
$652$	0	0
$653$	2.88194e26	0.522408	0.261204	−	0.965284i	$-0.415880\pi$
$653$	2.88194e26	0.522408	0.261204	+	0.965284i	$0.415880\pi$
$654$	0	0
$655$	−1.43006e26	−0.251803
$656$	0	0
$657$	2.88186e25	0.0492948
$658$	0	0
$659$	7.36091e26	1.22326	0.611632	−	0.791143i	$-0.290514\pi$
$659$	7.36091e26	1.22326	0.611632	+	0.791143i	$0.290514\pi$
$660$	0	0
$661$	3.05440e25	0.0493187	0.0246594	−	0.999696i	$-0.492150\pi$
$661$	3.05440e25	0.0493187	0.0246594	+	0.999696i	$0.492150\pi$
$662$	0	0
$663$	−7.75537e26	−1.21681
$664$	0	0
$665$	−9.49589e25	−0.144787
$666$	0	0
$667$	−7.72168e26	−1.14424
$668$	0	0
$669$	−6.25713e26	−0.901213
$670$	0	0
$671$	3.28994e26	0.460600
$672$	0	0
$673$	−9.62645e26	−1.31016	−0.655078	−	0.755561i	$-0.727364\pi$
$673$	−9.62645e26	−1.31016	−0.655078	+	0.755561i	$0.727364\pi$
$674$	0	0
$675$	−1.41888e26	−0.187741
$676$	0	0
$677$	1.18974e27	1.53059	0.765295	−	0.643680i	$-0.222593\pi$
$677$	1.18974e27	1.53059	0.765295	+	0.643680i	$0.222593\pi$
$678$	0	0
$679$	−1.89740e26	−0.237354
$680$	0	0
$681$	−5.03440e26	−0.612421
$682$	0	0
$683$	1.51240e27	1.78925	0.894624	−	0.446819i	$-0.147443\pi$
$683$	1.51240e27	1.78925	0.894624	+	0.446819i	$0.147443\pi$
$684$	0	0
$685$	−1.04664e26	−0.120431
$686$	0	0
$687$	−4.39806e26	−0.492235
$688$	0	0
$689$	−9.77124e26	−1.06382
$690$	0	0
$691$	1.40247e27	1.48543	0.742716	−	0.669606i	$-0.233537\pi$
$691$	1.40247e27	1.48543	0.742716	+	0.669606i	$0.233537\pi$
$692$	0	0
$693$	2.83250e26	0.291880
$694$	0	0
$695$	−1.73779e25	−0.0174238
$696$	0	0
$697$	−3.94204e26	−0.384600
$698$	0	0
$699$	−4.96738e25	−0.0471622
$700$	0	0
$701$	6.64167e26	0.613700	0.306850	−	0.951758i	$-0.400725\pi$
$701$	6.64167e26	0.613700	0.306850	+	0.951758i	$0.400725\pi$
$702$	0	0
$703$	−1.65938e27	−1.49235
$704$	0	0
$705$	−9.41266e25	−0.0823978
$706$	0	0
$707$	−1.13572e27	−0.967799
$708$	0	0
$709$	1.03445e27	0.858163	0.429081	−	0.903266i	$-0.358837\pi$
$709$	1.03445e27	0.858163	0.429081	+	0.903266i	$0.358837\pi$
$710$	0	0
$711$	2.70689e26	0.218630
$712$	0	0
$713$	1.97313e26	0.155169
$714$	0	0
$715$	−2.52700e26	−0.193508
$716$	0	0
$717$	−7.73581e26	−0.576864
$718$	0	0
$719$	1.59383e26	0.115749	0.0578744	−	0.998324i	$-0.481568\pi$
$719$	1.59383e26	0.115749	0.0578744	+	0.998324i	$0.481568\pi$
$720$	0	0
$721$	1.33572e26	0.0944777
$722$	0	0
$723$	−3.93548e26	−0.271134
$724$	0	0
$725$	−2.86075e27	−1.91986
$726$	0	0
$727$	−1.66059e27	−1.08564	−0.542819	−	0.839850i	$-0.682643\pi$
$727$	−1.66059e27	−1.08564	−0.542819	+	0.839850i	$0.682643\pi$
$728$	0	0
$729$	5.81497e25	0.0370370
$730$	0	0
$731$	−2.30645e26	−0.143129
$732$	0	0
$733$	−1.36074e27	−0.822786	−0.411393	−	0.911458i	$-0.634958\pi$
$733$	−1.36074e27	−0.822786	−0.411393	+	0.911458i	$0.634958\pi$
$734$	0	0
$735$	−2.04883e25	−0.0120719
$736$	0	0
$737$	1.33414e27	0.766054
$738$	0	0
$739$	−9.59784e26	−0.537095	−0.268548	−	0.963266i	$-0.586544\pi$
$739$	−9.59784e26	−0.537095	−0.268548	+	0.963266i	$0.586544\pi$
$740$	0	0
$741$	1.38419e27	0.754959
$742$	0	0
$743$	1.18287e27	0.628846	0.314423	−	0.949283i	$-0.398189\pi$
$743$	1.18287e27	0.628846	0.314423	+	0.949283i	$0.398189\pi$
$744$	0	0
$745$	−1.14539e26	−0.0593564
$746$	0	0
$747$	−1.26118e27	−0.637134
$748$	0	0
$749$	−1.96849e27	−0.969516
$750$	0	0
$751$	1.02382e27	0.491637	0.245819	−	0.969316i	$-0.420943\pi$
$751$	1.02382e27	0.491637	0.245819	+	0.969316i	$0.420943\pi$
$752$	0	0
$753$	9.72639e26	0.455407
$754$	0	0
$755$	−2.93670e26	−0.134080
$756$	0	0
$757$	2.74415e25	0.0122179	0.00610897	−	0.999981i	$-0.498055\pi$
$757$	2.74415e25	0.0122179	0.00610897	+	0.999981i	$0.498055\pi$
$758$	0	0
$759$	6.35781e26	0.276065
$760$	0	0
$761$	1.89240e27	0.801416	0.400708	−	0.916206i	$-0.368764\pi$
$761$	1.89240e27	0.801416	0.400708	+	0.916206i	$0.368764\pi$
$762$	0	0
$763$	−1.79324e27	−0.740722
$764$	0	0
$765$	−1.81322e26	−0.0730578
$766$	0	0
$767$	−1.55682e27	−0.611902
$768$	0	0
$769$	−1.93865e27	−0.743359	−0.371679	−	0.928361i	$-0.621218\pi$
$769$	−1.93865e27	−0.743359	−0.371679	+	0.928361i	$0.621218\pi$
$770$	0	0
$771$	3.53022e26	0.132064
$772$	0	0
$773$	−7.63732e24	−0.00278764	−0.00139382	−	0.999999i	$-0.500444\pi$
$773$	−7.63732e24	−0.00278764	−0.00139382	+	0.999999i	$0.500444\pi$
$774$	0	0
$775$	7.31009e26	0.260349
$776$	0	0
$777$	−3.03651e27	−1.05530
$778$	0	0
$779$	7.03582e26	0.238621
$780$	0	0
$781$	−1.51220e27	−0.500523
$782$	0	0
$783$	1.17242e27	0.378744
$784$	0	0
$785$	−6.25986e26	−0.197380
$786$	0	0
$787$	4.61341e27	1.41991	0.709956	−	0.704246i	$-0.248715\pi$
$787$	4.61341e27	1.41991	0.709956	+	0.704246i	$0.248715\pi$
$788$	0	0
$789$	−5.20387e26	−0.156349
$790$	0	0
$791$	−5.44307e27	−1.59649
$792$	0	0
$793$	2.94199e27	0.842455
$794$	0	0
$795$	−2.28453e26	−0.0638719
$796$	0	0
$797$	1.01219e27	0.276316	0.138158	−	0.990410i	$-0.455882\pi$
$797$	1.01219e27	0.276316	0.138158	+	0.990410i	$0.455882\pi$
$798$	0	0
$799$	4.79557e27	1.27834
$800$	0	0
$801$	−5.56240e26	−0.144795
$802$	0	0
$803$	−4.78413e26	−0.121620
$804$	0	0
$805$	−3.90033e26	−0.0968369
$806$	0	0
$807$	−1.23316e27	−0.299035
$808$	0	0
$809$	3.00853e27	0.712597	0.356298	−	0.934372i	$-0.384039\pi$
$809$	3.00853e27	0.712597	0.356298	+	0.934372i	$0.384039\pi$
$810$	0	0
$811$	−3.54753e27	−0.820782	−0.410391	−	0.911910i	$-0.634608\pi$
$811$	−3.54753e27	−0.820782	−0.410391	+	0.911910i	$0.634608\pi$
$812$	0	0
$813$	1.62087e27	0.366343
$814$	0	0
$815$	3.07471e26	0.0678901
$816$	0	0
$817$	4.11659e26	0.0888030
$818$	0	0
$819$	2.53294e27	0.533860
$820$	0	0
$821$	−4.28797e27	−0.883063	−0.441532	−	0.897246i	$-0.645565\pi$
$821$	−4.28797e27	−0.883063	−0.441532	+	0.897246i	$0.645565\pi$
$822$	0	0
$823$	−3.20459e27	−0.644874	−0.322437	−	0.946591i	$-0.604502\pi$
$823$	−3.20459e27	−0.644874	−0.322437	+	0.946591i	$0.604502\pi$
$824$	0	0
$825$	2.35546e27	0.463194
$826$	0	0
$827$	4.02192e27	0.772914	0.386457	−	0.922308i	$-0.373699\pi$
$827$	4.02192e27	0.772914	0.386457	+	0.922308i	$0.373699\pi$
$828$	0	0
$829$	−1.58908e27	−0.298455	−0.149227	−	0.988803i	$-0.547679\pi$
$829$	−1.58908e27	−0.298455	−0.149227	+	0.988803i	$0.547679\pi$
$830$	0	0
$831$	2.80885e27	0.515607
$832$	0	0
$833$	1.04384e27	0.187286
$834$	0	0
$835$	−1.10955e26	−0.0194592
$836$	0	0
$837$	−2.99589e26	−0.0513610
$838$	0	0
$839$	5.31813e27	0.891293	0.445646	−	0.895209i	$-0.352974\pi$
$839$	5.31813e27	0.891293	0.445646	+	0.895209i	$0.352974\pi$
$840$	0	0
$841$	1.75351e28	2.87307
$842$	0	0
$843$	8.60120e26	0.137783
$844$	0	0
$845$	−1.26102e27	−0.197507
$846$	0	0
$847$	2.25021e27	0.344612
$848$	0	0
$849$	4.77236e27	0.714677
$850$	0	0
$851$	−6.81572e27	−0.998115
$852$	0	0
$853$	−1.60722e27	−0.230176	−0.115088	−	0.993355i	$-0.536715\pi$
$853$	−1.60722e27	−0.230176	−0.115088	+	0.993355i	$0.536715\pi$
$854$	0	0
$855$	3.23627e26	0.0453279
$856$	0	0
$857$	8.25782e27	1.13122	0.565611	−	0.824672i	$-0.308641\pi$
$857$	8.25782e27	1.13122	0.565611	+	0.824672i	$0.308641\pi$
$858$	0	0
$859$	9.31408e27	1.24797	0.623986	−	0.781436i	$-0.285512\pi$
$859$	9.31408e27	1.24797	0.623986	+	0.781436i	$0.285512\pi$
$860$	0	0
$861$	1.28749e27	0.168738
$862$	0	0
$863$	−6.65457e27	−0.853134	−0.426567	−	0.904456i	$-0.640277\pi$
$863$	−6.65457e27	−0.853134	−0.426567	+	0.904456i	$0.640277\pi$
$864$	0	0
$865$	2.25128e27	0.282343
$866$	0	0
$867$	4.53234e27	0.556084
$868$	0	0
$869$	−4.49368e27	−0.539404
$870$	0	0
$871$	1.19304e28	1.40114
$872$	0	0
$873$	6.46647e26	0.0743074
$874$	0	0
$875$	−2.92626e27	−0.329030
$876$	0	0
$877$	−1.08073e27	−0.118911	−0.0594554	−	0.998231i	$-0.518936\pi$
$877$	−1.08073e27	−0.118911	−0.0594554	+	0.998231i	$0.518936\pi$
$878$	0	0
$879$	3.25285e26	0.0350243
$880$	0	0
$881$	−1.38540e28	−1.45983	−0.729916	−	0.683536i	$-0.760441\pi$
$881$	−1.38540e28	−1.45983	−0.729916	+	0.683536i	$0.760441\pi$
$882$	0	0
$883$	−6.76336e27	−0.697487	−0.348743	−	0.937218i	$-0.613392\pi$
$883$	−6.76336e27	−0.697487	−0.348743	+	0.937218i	$0.613392\pi$
$884$	0	0
$885$	−3.63987e26	−0.0367388
$886$	0	0
$887$	1.60903e28	1.58960	0.794802	−	0.606869i	$-0.207575\pi$
$887$	1.60903e28	1.58960	0.794802	+	0.606869i	$0.207575\pi$
$888$	0	0
$889$	−1.42360e28	−1.37665
$890$	0	0
$891$	−9.65336e26	−0.0913777
$892$	0	0
$893$	−8.55921e27	−0.793131
$894$	0	0
$895$	−6.70675e26	−0.0608406
$896$	0	0
$897$	5.68541e27	0.504933
$898$	0	0
$899$	−6.04033e27	−0.525223
$900$	0	0
$901$	1.16393e28	0.990922
$902$	0	0
$903$	7.53296e26	0.0627960
$904$	0	0
$905$	−1.03636e26	−0.00845963
$906$	0	0
$907$	4.87731e26	0.0389862	0.0194931	−	0.999810i	$-0.493795\pi$
$907$	4.87731e26	0.0389862	0.0194931	+	0.999810i	$0.493795\pi$
$908$	0	0
$909$	3.87060e27	0.302985
$910$	0	0
$911$	−2.04975e28	−1.57136	−0.785680	−	0.618634i	$-0.787687\pi$
$911$	−2.04975e28	−1.57136	−0.785680	+	0.618634i	$0.787687\pi$
$912$	0	0
$913$	2.09367e28	1.57194
$914$	0	0
$915$	6.87844e26	0.0505812
$916$	0	0
$917$	−2.37960e28	−1.71394
$918$	0	0
$919$	−7.36209e27	−0.519402	−0.259701	−	0.965689i	$-0.583624\pi$
$919$	−7.36209e27	−0.519402	−0.259701	+	0.965689i	$0.583624\pi$
$920$	0	0
$921$	1.40606e28	0.971713
$922$	0	0
$923$	−1.35227e28	−0.915476
$924$	0	0
$925$	−2.52511e28	−1.67469
$926$	0	0
$927$	−4.55222e26	−0.0295778
$928$	0	0
$929$	−9.18844e27	−0.584915	−0.292457	−	0.956279i	$-0.594473\pi$
$929$	−9.18844e27	−0.584915	−0.292457	+	0.956279i	$0.594473\pi$
$930$	0	0
$931$	−1.86306e27	−0.116200
$932$	0	0
$933$	−7.37881e27	−0.450932
$934$	0	0
$935$	3.01010e27	0.180248
$936$	0	0
$937$	1.55615e28	0.913113	0.456557	−	0.889694i	$-0.349083\pi$
$937$	1.55615e28	0.913113	0.456557	+	0.889694i	$0.349083\pi$
$938$	0	0
$939$	−4.01793e27	−0.231036
$940$	0	0
$941$	−4.21264e27	−0.237385	−0.118693	−	0.992931i	$-0.537870\pi$
$941$	−4.21264e27	−0.237385	−0.118693	+	0.992931i	$0.537870\pi$
$942$	0	0
$943$	2.88989e27	0.159595
$944$	0	0
$945$	5.92206e26	0.0320531
$946$	0	0
$947$	−1.60906e27	−0.0853584	−0.0426792	−	0.999089i	$-0.513589\pi$
$947$	−1.60906e27	−0.0853584	−0.0426792	+	0.999089i	$0.513589\pi$
$948$	0	0
$949$	−4.27817e27	−0.222448
$950$	0	0
$951$	−5.40175e27	−0.275308
$952$	0	0
$953$	2.00721e28	1.00279	0.501396	−	0.865218i	$-0.332820\pi$
$953$	2.00721e28	1.00279	0.501396	+	0.865218i	$0.332820\pi$
$954$	0	0
$955$	−3.50493e27	−0.171651
$956$	0	0
$957$	−1.94632e28	−0.934437
$958$	0	0
$959$	−1.74160e28	−0.819729
$960$	0	0
$961$	−2.01272e28	−0.928775
$962$	0	0
$963$	6.70874e27	0.303523
$964$	0	0
$965$	6.64631e27	0.294830
$966$	0	0
$967$	−3.69606e28	−1.60764	−0.803818	−	0.594876i	$-0.797201\pi$
$967$	−3.69606e28	−1.60764	−0.803818	+	0.594876i	$0.797201\pi$
$968$	0	0
$969$	−1.64881e28	−0.703228
$970$	0	0
$971$	2.54073e28	1.06261	0.531306	−	0.847180i	$-0.321701\pi$
$971$	2.54073e28	1.06261	0.531306	+	0.847180i	$0.321701\pi$
$972$	0	0
$973$	−2.89166e27	−0.118597
$974$	0	0
$975$	2.10635e28	0.847201
$976$	0	0
$977$	8.52403e27	0.336238	0.168119	−	0.985767i	$-0.446231\pi$
$977$	8.52403e27	0.336238	0.168119	+	0.985767i	$0.446231\pi$
$978$	0	0
$979$	9.23408e27	0.357238
$980$	0	0
$981$	6.11149e27	0.231895
$982$	0	0
$983$	−3.09592e28	−1.15221	−0.576105	−	0.817376i	$-0.695428\pi$
$983$	−3.09592e28	−1.15221	−0.576105	+	0.817376i	$0.695428\pi$
$984$	0	0
$985$	−5.25015e27	−0.191658
$986$	0	0
$987$	−1.56625e28	−0.560853
$988$	0	0
$989$	1.69084e27	0.0593934
$990$	0	0
$991$	2.94856e28	1.01604	0.508020	−	0.861345i	$-0.330378\pi$
$991$	2.94856e28	1.01604	0.508020	+	0.861345i	$0.330378\pi$
$992$	0	0
$993$	4.55843e27	0.154098
$994$	0	0
$995$	3.11376e27	0.103268
$996$	0	0
$997$	−3.29038e28	−1.07064	−0.535318	−	0.844650i	$-0.679808\pi$
$997$	−3.29038e28	−1.07064	−0.535318	+	0.844650i	$0.679808\pi$
$998$	0	0
$999$	1.03486e28	0.330377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.20.a.j.1.1		2
4.3	odd	2		3.20.a.b.1.1	✓	2
12.11	even	2		9.20.a.c.1.2		2
20.3	even	4		75.20.b.b.49.3		4
20.7	even	4		75.20.b.b.49.2		4
20.19	odd	2		75.20.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.20.a.b.1.1	✓	2	4.3	odd	2
9.20.a.c.1.2		2	12.11	even	2
48.20.a.j.1.1		2	1.1	even	1	trivial
75.20.a.b.1.2		2	20.19	odd	2
75.20.b.b.49.2		4	20.7	even	4
75.20.b.b.49.3		4	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q+19683.0 q^{3} -683163. q^{5} -1.13677e8 q^{7} +3.87420e8 q^{9} +O(q^{10})\)	\(q+19683.0 q^{3} -683163. q^{5} -1.13677e8 q^{7} +3.87420e8 q^{9} -6.43152e9 q^{11} -5.75132e10 q^{13} -1.34467e10 q^{15} +6.85083e11 q^{17} -1.22275e12 q^{19} -2.23751e12 q^{21} -5.02230e12 q^{23} -1.86068e13 q^{25} +7.62560e12 q^{27} +1.53748e14 q^{29} -3.92872e13 q^{31} -1.26592e14 q^{33} +7.76602e13 q^{35} +1.35709e15 q^{37} -1.13203e15 q^{39} -5.75410e14 q^{41} -3.36667e14 q^{43} -2.64671e14 q^{45} +6.99998e15 q^{47} +1.52367e15 q^{49} +1.34845e16 q^{51} +1.69895e16 q^{53} +4.39377e15 q^{55} -2.40674e16 q^{57} +2.70689e16 q^{59} -5.11533e16 q^{61} -4.40410e16 q^{63} +3.92909e16 q^{65} -2.07437e17 q^{67} -9.88540e16 q^{69} +2.35123e17 q^{71} +7.43858e16 q^{73} -3.66237e17 q^{75} +7.31118e17 q^{77} +6.98697e17 q^{79} +1.50095e17 q^{81} -3.25532e18 q^{83} -4.68023e17 q^{85} +3.02622e18 q^{87} -1.43575e18 q^{89} +6.53796e18 q^{91} -7.73291e17 q^{93} +8.35337e17 q^{95} +1.66911e18 q^{97} -2.49170e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 39366 q^{3} + 6016140 q^{5} - 113892064 q^{7} + 774840978 q^{9}+O(q^{10}) \) 2 * q + 39366 * q^3 + 6016140 * q^5 - 113892064 * q^7 + 774840978 * q^9	\( 2 q + 39366 q^{3} + 6016140 q^{5} - 113892064 q^{7} + 774840978 q^{9} + 6650071272 q^{11} - 44072356148 q^{13} + 118415683620 q^{15} + 336281471748 q^{17} - 602118925096 q^{19} - 2241737495712 q^{21} - 2368252165968 q^{23} + 7200399078350 q^{25} + 15251194969974 q^{27} + 280977251970492 q^{29} - 41610149253712 q^{31} + 130893352846776 q^{33} + 76222408017600 q^{35} + 637994163989884 q^{37} - 867476186061084 q^{39} + 11\!\cdots\!32 q^{41}+ \cdots + 25\!\cdots\!08 q^{99}+O(q^{100}) \) 2 * q + 39366 * q^3 + 6016140 * q^5 - 113892064 * q^7 + 774840978 * q^9 + 6650071272 * q^11 - 44072356148 * q^13 + 118415683620 * q^15 + 336281471748 * q^17 - 602118925096 * q^19 - 2241737495712 * q^21 - 2368252165968 * q^23 + 7200399078350 * q^25 + 15251194969974 * q^27 + 280977251970492 * q^29 - 41610149253712 * q^31 + 130893352846776 * q^33 + 76222408017600 * q^35 + 637994163989884 * q^37 - 867476186061084 * q^39 + 1150038430280532 * q^41 - 2820518953538200 * q^43 + 2330775900692460 * q^45 + 5603091319500480 * q^47 - 9875183230193070 * q^49 + 6619028208415884 * q^51 + 27537868857647628 * q^53 + 92031302621896560 * q^55 - 11851506802664568 * q^57 + 55153474421835816 * q^59 - 87192569066205620 * q^61 - 44124119128099296 * q^63 + 129335487918703560 * q^65 - 499299710877878248 * q^67 - 46614307382748144 * q^69 - 84915048436970544 * q^71 - 241433711161728332 * q^73 + 141725455059163050 * q^75 + 728310910203957888 * q^77 - 1131591466033523440 * q^79 + 300189270593998242 * q^81 - 1607743276003583784 * q^83 - 2804751519619294440 * q^85 + 5530475250535194036 * q^87 + 4322797626369576276 * q^89 + 6535073681312344768 * q^91 - 819012567760813296 * q^93 + 4993123012930175760 * q^95 + 7902201235299458308 * q^97 + 2576373864083092008 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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