LMFDB - Embedded newform 48.26.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,26,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, 26, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 26);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 26 $
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:	$190.078454377$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 1647414391x^{2} - 16173027965094x + 235708219135253151 $ x^4 - 1647414391x^2 - 16173027965094x + 235708219135253151
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{42}\cdot 3^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8122.68$ 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)

$f(q)$	$=$	$q+531441. q^{3} -6.59716e8 q^{5} +5.77845e9 q^{7} +2.82430e11 q^{9} +O(q^{10})$	$q+531441. q^{3} -6.59716e8 q^{5} +5.77845e9 q^{7} +2.82430e11 q^{9} +1.69759e13 q^{11} +2.46717e13 q^{13} -3.50600e14 q^{15} -5.10189e14 q^{17} +1.76184e16 q^{19} +3.07091e15 q^{21} +1.34881e17 q^{23} +1.37202e17 q^{25} +1.50095e17 q^{27} -3.40314e18 q^{29} -2.54458e18 q^{31} +9.02169e18 q^{33} -3.81214e18 q^{35} -1.75155e19 q^{37} +1.31115e19 q^{39} -1.90338e20 q^{41} +1.16473e20 q^{43} -1.86323e20 q^{45} -6.74671e20 q^{47} -1.30768e21 q^{49} -2.71135e20 q^{51} +1.40766e21 q^{53} -1.11993e22 q^{55} +9.36314e21 q^{57} +2.44172e22 q^{59} +3.64580e21 q^{61} +1.63201e21 q^{63} -1.62763e22 q^{65} +2.07427e22 q^{67} +7.16812e22 q^{69} +1.15904e23 q^{71} +1.69249e23 q^{73} +7.29148e22 q^{75} +9.80944e22 q^{77} -2.79035e23 q^{79} +7.97664e22 q^{81} +5.68102e23 q^{83} +3.36580e23 q^{85} -1.80857e24 q^{87} -1.67552e23 q^{89} +1.42564e23 q^{91} -1.35229e24 q^{93} -1.16231e25 q^{95} +1.20192e25 q^{97} +4.79449e24 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2125764 q^{3} + 266436088 q^{5} - 484584576 q^{7} + 1129718145924 q^{9}+O(q^{10}) $ 4 * q + 2125764 * q^3 + 266436088 * q^5 - 484584576 * q^7 + 1129718145924 * q^9	$ 4 q + 2125764 q^{3} + 266436088 q^{5} - 484584576 q^{7} + 1129718145924 q^{9} - 4052955557392 q^{11} + 82663381481080 q^{13} + 141595061042808 q^{15} + 10\!\cdots\!28 q^{17}+ \cdots - 11\!\cdots\!52 q^{99}+O(q^{100}) $ 4 * q + 2125764 * q^3 + 266436088 * q^5 - 484584576 * q^7 + 1129718145924 * q^9 - 4052955557392 * q^11 + 82663381481080 * q^13 + 141595061042808 * q^15 + 1091839531710728 * q^17 + 5291818285650448 * q^19 - 257528111654016 * q^21 - 22957555441633504 * q^23 + 329573428119320444 * q^25 + 600378541187996484 * q^27 - 2255995699601091048 * q^29 + 6422134828063294816 * q^31 - 2153906754375961872 * q^33 + 26450723987305847040 * q^35 - 84572383178406307176 * q^37 + 43930710117686636280 * q^39 - 191923587571359535128 * q^41 + 393512700764425221104 * q^43 + 75249420835650926328 * q^45 + 426139906889905851648 * q^47 + 1697461944491957688548 * q^49 + 580248292571880999048 * q^51 + 4165797514523716100344 * q^53 - 1377411551242297580512 * q^55 + 2812289201544359735568 * q^57 - 22289642820361283440912 * q^59 + 31420833466214461354360 * q^61 - 136860997185521917056 * q^63 + 49904798993160775849744 * q^65 + 105958367596085546226448 * q^67 - 12200586221457150999264 * q^69 + 260836089844500944794336 * q^71 + 96585608913237168931624 * q^73 + 175148832213159776079804 * q^75 + 609668306798881251586560 * q^77 + 228314222637752365466528 * q^79 + 319065772307490039453444 * q^81 - 508041934949405113277936 * q^83 + 3527355809877820452506096 * q^85 - 1198928610591703427640168 * q^87 + 9334569288453070817495784 * q^89 - 6414440882259952713889536 * q^91 + 3412985755160785460309856 * q^93 - 9916967691165637498776608 * q^95 + 17898782631688573069195784 * q^97 - 1144674359452315553217552 * 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
$3$	531441.	0.577350
$3$	531441.	0.577350
$4$	0	0
$5$	−6.59716e8	−1.20846	−0.604230	−	0.796810i	$-0.706519\pi$
$5$	−6.59716e8	−1.20846	−0.604230	+	0.796810i	$0.706519\pi$
$6$	0	0
$7$	5.77845e9	0.157792	0.0788962	−	0.996883i	$-0.474860\pi$
$7$	5.77845e9	0.157792	0.0788962	+	0.996883i	$0.474860\pi$
$8$	0	0
$9$	2.82430e11	0.333333
$10$	0	0
$11$	1.69759e13	1.63089	0.815444	−	0.578836i	$-0.196493\pi$
$11$	1.69759e13	1.63089	0.815444	+	0.578836i	$0.196493\pi$
$12$	0	0
$13$	2.46717e13	0.293702	0.146851	−	0.989159i	$-0.453086\pi$
$13$	2.46717e13	0.293702	0.146851	+	0.989159i	$0.453086\pi$
$14$	0	0
$15$	−3.50600e14	−0.697704
$16$	0	0
$17$	−5.10189e14	−0.212383	−0.106191	−	0.994346i	$-0.533866\pi$
$17$	−5.10189e14	−0.212383	−0.106191	+	0.994346i	$0.533866\pi$
$18$	0	0
$19$	1.76184e16	1.82619	0.913096	−	0.407745i	$-0.133685\pi$
$19$	1.76184e16	1.82619	0.913096	+	0.407745i	$0.133685\pi$
$20$	0	0
$21$	3.07091e15	0.0911015
$22$	0	0
$23$	1.34881e17	1.28337	0.641686	−	0.766968i	$-0.278235\pi$
$23$	1.34881e17	1.28337	0.641686	+	0.766968i	$0.278235\pi$
$24$	0	0
$25$	1.37202e17	0.460374
$26$	0	0
$27$	1.50095e17	0.192450
$28$	0	0
$29$	−3.40314e18	−1.78609	−0.893047	−	0.449963i	$-0.851437\pi$
$29$	−3.40314e18	−1.78609	−0.893047	+	0.449963i	$0.851437\pi$
$30$	0	0
$31$	−2.54458e18	−0.580223	−0.290112	−	0.956993i	$-0.593692\pi$
$31$	−2.54458e18	−0.580223	−0.290112	+	0.956993i	$0.593692\pi$
$32$	0	0
$33$	9.02169e18	0.941593
$34$	0	0
$35$	−3.81214e18	−0.190686
$36$	0	0
$37$	−1.75155e19	−0.437422	−0.218711	−	0.975790i	$-0.570185\pi$
$37$	−1.75155e19	−0.437422	−0.218711	+	0.975790i	$0.570185\pi$
$38$	0	0
$39$	1.31115e19	0.169569
$40$	0	0
$41$	−1.90338e20	−1.31743	−0.658715	−	0.752393i	$-0.728900\pi$
$41$	−1.90338e20	−1.31743	−0.658715	+	0.752393i	$0.728900\pi$
$42$	0	0
$43$	1.16473e20	0.444496	0.222248	−	0.974990i	$-0.428661\pi$
$43$	1.16473e20	0.444496	0.222248	+	0.974990i	$0.428661\pi$
$44$	0	0
$45$	−1.86323e20	−0.402820
$46$	0	0
$47$	−6.74671e20	−0.846971	−0.423486	−	0.905903i	$-0.639194\pi$
$47$	−6.74671e20	−0.846971	−0.423486	+	0.905903i	$0.639194\pi$
$48$	0	0
$49$	−1.30768e21	−0.975102
$50$	0	0
$51$	−2.71135e20	−0.122619
$52$	0	0
$53$	1.40766e21	0.393596	0.196798	−	0.980444i	$-0.436946\pi$
$53$	1.40766e21	0.393596	0.196798	+	0.980444i	$0.436946\pi$
$54$	0	0
$55$	−1.11993e22	−1.97086
$56$	0	0
$57$	9.36314e21	1.05435
$58$	0	0
$59$	2.44172e22	1.78667	0.893337	−	0.449387i	$-0.148357\pi$
$59$	2.44172e22	1.78667	0.893337	+	0.449387i	$0.148357\pi$
$60$	0	0
$61$	3.64580e21	0.175861	0.0879305	−	0.996127i	$-0.471975\pi$
$61$	3.64580e21	0.175861	0.0879305	+	0.996127i	$0.471975\pi$
$62$	0	0
$63$	1.63201e21	0.0525975
$64$	0	0
$65$	−1.62763e22	−0.354927
$66$	0	0
$67$	2.07427e22	0.309692	0.154846	−	0.987939i	$-0.450512\pi$
$67$	2.07427e22	0.309692	0.154846	+	0.987939i	$0.450512\pi$
$68$	0	0
$69$	7.16812e22	0.740955
$70$	0	0
$71$	1.15904e23	0.838245	0.419122	−	0.907930i	$-0.362338\pi$
$71$	1.15904e23	0.838245	0.419122	+	0.907930i	$0.362338\pi$
$72$	0	0
$73$	1.69249e23	0.864951	0.432475	−	0.901646i	$-0.357640\pi$
$73$	1.69249e23	0.864951	0.432475	+	0.901646i	$0.357640\pi$
$74$	0	0
$75$	7.29148e22	0.265797
$76$	0	0
$77$	9.80944e22	0.257342
$78$	0	0
$79$	−2.79035e23	−0.531276	−0.265638	−	0.964073i	$-0.585583\pi$
$79$	−2.79035e23	−0.531276	−0.265638	+	0.964073i	$0.585583\pi$
$80$	0	0
$81$	7.97664e22	0.111111
$82$	0	0
$83$	5.68102e23	0.583378	0.291689	−	0.956513i	$-0.405783\pi$
$83$	5.68102e23	0.583378	0.291689	+	0.956513i	$0.405783\pi$
$84$	0	0
$85$	3.36580e23	0.256656
$86$	0	0
$87$	−1.80857e24	−1.03120
$88$	0	0
$89$	−1.67552e23	−0.0719076	−0.0359538	−	0.999353i	$-0.511447\pi$
$89$	−1.67552e23	−0.0719076	−0.0359538	+	0.999353i	$0.511447\pi$
$90$	0	0
$91$	1.42564e23	0.0463440
$92$	0	0
$93$	−1.35229e24	−0.334992
$94$	0	0
$95$	−1.16231e25	−2.20688
$96$	0	0
$97$	1.20192e25	1.75885	0.879423	−	0.476042i	$-0.157929\pi$
$97$	1.20192e25	1.75885	0.879423	+	0.476042i	$0.157929\pi$
$98$	0	0
$99$	4.79449e24	0.543629
$100$	0	0
$101$	1.28681e25	1.13631	0.568154	−	0.822922i	$-0.307658\pi$
$101$	1.28681e25	1.13631	0.568154	+	0.822922i	$0.307658\pi$
$102$	0	0
$103$	1.26127e25	0.871651	0.435826	−	0.900031i	$-0.356456\pi$
$103$	1.26127e25	0.871651	0.435826	+	0.900031i	$0.356456\pi$
$104$	0	0
$105$	−2.02593e24	−0.110092
$106$	0	0
$107$	−2.77748e25	−1.19221	−0.596106	−	0.802906i	$-0.703286\pi$
$107$	−2.77748e25	−1.19221	−0.596106	+	0.802906i	$0.703286\pi$
$108$	0	0
$109$	−7.02114e24	−0.239098	−0.119549	−	0.992828i	$-0.538145\pi$
$109$	−7.02114e24	−0.239098	−0.119549	+	0.992828i	$0.538145\pi$
$110$	0	0
$111$	−9.30845e24	−0.252546
$112$	0	0
$113$	1.77197e25	0.384570	0.192285	−	0.981339i	$-0.438410\pi$
$113$	1.77197e25	0.384570	0.192285	+	0.981339i	$0.438410\pi$
$114$	0	0
$115$	−8.89831e25	−1.55090
$116$	0	0
$117$	6.96802e24	0.0979007
$118$	0	0
$119$	−2.94810e24	−0.0335124
$120$	0	0
$121$	1.79834e26	1.65979
$122$	0	0
$123$	−1.01153e26	−0.760618
$124$	0	0
$125$	1.06096e26	0.652116
$126$	0	0
$127$	−1.86687e26	−0.940953	−0.470476	−	0.882413i	$-0.655918\pi$
$127$	−1.86687e26	−0.940953	−0.470476	+	0.882413i	$0.655918\pi$
$128$	0	0
$129$	6.18983e25	0.256630
$130$	0	0
$131$	−3.98693e26	−1.36379	−0.681894	−	0.731451i	$-0.738844\pi$
$131$	−3.98693e26	−1.36379	−0.681894	+	0.731451i	$0.738844\pi$
$132$	0	0
$133$	1.01807e26	0.288159
$134$	0	0
$135$	−9.90198e25	−0.232568
$136$	0	0
$137$	−6.00786e26	−1.17412	−0.587060	−	0.809543i	$-0.699715\pi$
$137$	−6.00786e26	−1.17412	−0.587060	+	0.809543i	$0.699715\pi$
$138$	0	0
$139$	9.16223e26	1.49388	0.746940	−	0.664891i	$-0.231522\pi$
$139$	9.16223e26	1.49388	0.746940	+	0.664891i	$0.231522\pi$
$140$	0	0
$141$	−3.58548e26	−0.488999
$142$	0	0
$143$	4.18824e26	0.478995
$144$	0	0
$145$	2.24510e27	2.15842
$146$	0	0
$147$	−6.94954e26	−0.562975
$148$	0	0
$149$	8.98500e26	0.614738	0.307369	−	0.951590i	$-0.400551\pi$
$149$	8.98500e26	0.614738	0.307369	+	0.951590i	$0.400551\pi$
$150$	0	0
$151$	2.12872e27	1.23284	0.616420	−	0.787417i	$-0.288582\pi$
$151$	2.12872e27	1.23284	0.616420	+	0.787417i	$0.288582\pi$
$152$	0	0
$153$	−1.44092e26	−0.0707943
$154$	0	0
$155$	1.67870e27	0.701176
$156$	0	0
$157$	−8.57089e26	−0.304986	−0.152493	−	0.988305i	$-0.548730\pi$
$157$	−8.57089e26	−0.304986	−0.152493	+	0.988305i	$0.548730\pi$
$158$	0	0
$159$	7.48090e26	0.227243
$160$	0	0
$161$	7.79403e26	0.202506
$162$	0	0
$163$	3.87803e27	0.863507	0.431754	−	0.901992i	$-0.357895\pi$
$163$	3.87803e27	0.863507	0.431754	+	0.901992i	$0.357895\pi$
$164$	0	0
$165$	−5.95175e27	−1.13788
$166$	0	0
$167$	−5.45143e27	−0.896509	−0.448254	−	0.893906i	$-0.647954\pi$
$167$	−5.45143e27	−0.896509	−0.448254	+	0.893906i	$0.647954\pi$
$168$	0	0
$169$	−6.44772e27	−0.913739
$170$	0	0
$171$	4.97595e27	0.608730
$172$	0	0
$173$	−1.04470e28	−1.10513	−0.552567	−	0.833469i	$-0.686352\pi$
$173$	−1.04470e28	−1.10513	−0.552567	+	0.833469i	$0.686352\pi$
$174$	0	0
$175$	7.92816e26	0.0726435
$176$	0	0
$177$	1.29763e28	1.03154
$178$	0	0
$179$	1.92963e28	1.33294	0.666470	−	0.745532i	$-0.267805\pi$
$179$	1.92963e28	1.33294	0.666470	+	0.745532i	$0.267805\pi$
$180$	0	0
$181$	−1.45926e28	−0.877304	−0.438652	−	0.898657i	$-0.644544\pi$
$181$	−1.45926e28	−0.877304	−0.438652	+	0.898657i	$0.644544\pi$
$182$	0	0
$183$	1.93753e27	0.101533
$184$	0	0
$185$	1.15552e28	0.528607
$186$	0	0
$187$	−8.66091e27	−0.346372
$188$	0	0
$189$	8.67315e26	0.0303672
$190$	0	0
$191$	5.72468e28	1.75725	0.878626	−	0.477511i	$-0.158461\pi$
$191$	5.72468e28	1.75725	0.878626	+	0.477511i	$0.158461\pi$
$192$	0	0
$193$	3.27780e28	0.883317	0.441659	−	0.897183i	$-0.354390\pi$
$193$	3.27780e28	0.883317	0.441659	+	0.897183i	$0.354390\pi$
$194$	0	0
$195$	−8.64990e27	−0.204917
$196$	0	0
$197$	−3.26544e28	−0.680947	−0.340474	−	0.940254i	$-0.610587\pi$
$197$	−3.26544e28	−0.680947	−0.340474	+	0.940254i	$0.610587\pi$
$198$	0	0
$199$	4.11556e28	0.756425	0.378212	−	0.925719i	$-0.376539\pi$
$199$	4.11556e28	0.756425	0.378212	+	0.925719i	$0.376539\pi$
$200$	0	0
$201$	1.10235e28	0.178801
$202$	0	0
$203$	−1.96649e28	−0.281832
$204$	0	0
$205$	1.25569e29	1.59206
$206$	0	0
$207$	3.80943e28	0.427791
$208$	0	0
$209$	2.99088e29	2.97831
$210$	0	0
$211$	−8.70821e27	−0.0769836	−0.0384918	−	0.999259i	$-0.512255\pi$
$211$	−8.70821e27	−0.0769836	−0.0384918	+	0.999259i	$0.512255\pi$
$212$	0	0
$213$	6.15964e28	0.483961
$214$	0	0
$215$	−7.68388e28	−0.537155
$216$	0	0
$217$	−1.47037e28	−0.0915548
$218$	0	0
$219$	8.99461e28	0.499380
$220$	0	0
$221$	−1.25872e28	−0.0623773
$222$	0	0
$223$	−2.79722e29	−1.23856	−0.619278	−	0.785172i	$-0.712574\pi$
$223$	−2.79722e29	−1.23856	−0.619278	+	0.785172i	$0.712574\pi$
$224$	0	0
$225$	3.87499e28	0.153458
$226$	0	0
$227$	3.38606e29	1.20053	0.600263	−	0.799802i	$-0.295062\pi$
$227$	3.38606e29	1.20053	0.600263	+	0.799802i	$0.295062\pi$
$228$	0	0
$229$	2.27844e29	0.723927	0.361964	−	0.932192i	$-0.382106\pi$
$229$	2.27844e29	0.723927	0.361964	+	0.932192i	$0.382106\pi$
$230$	0	0
$231$	5.21314e28	0.148576
$232$	0	0
$233$	4.78405e29	1.22418	0.612092	−	0.790786i	$-0.290328\pi$
$233$	4.78405e29	1.22418	0.612092	+	0.790786i	$0.290328\pi$
$234$	0	0
$235$	4.45091e29	1.02353
$236$	0	0
$237$	−1.48291e29	−0.306732
$238$	0	0
$239$	5.65226e28	0.105256	0.0526282	−	0.998614i	$-0.483240\pi$
$239$	5.65226e28	0.105256	0.0526282	+	0.998614i	$0.483240\pi$
$240$	0	0
$241$	1.69166e29	0.283858	0.141929	−	0.989877i	$-0.454670\pi$
$241$	1.69166e29	0.283858	0.141929	+	0.989877i	$0.454670\pi$
$242$	0	0
$243$	4.23912e28	0.0641500
$244$	0	0
$245$	8.62696e29	1.17837
$246$	0	0
$247$	4.34676e29	0.536356
$248$	0	0
$249$	3.01913e29	0.336814
$250$	0	0
$251$	8.30590e29	0.838427	0.419214	−	0.907888i	$-0.362306\pi$
$251$	8.30590e29	0.838427	0.419214	+	0.907888i	$0.362306\pi$
$252$	0	0
$253$	2.28972e30	2.09303
$254$	0	0
$255$	1.78872e29	0.148180
$256$	0	0
$257$	−2.30152e30	−1.72922	−0.864611	−	0.502442i	$-0.832435\pi$
$257$	−2.30152e30	−1.72922	−0.864611	+	0.502442i	$0.832435\pi$
$258$	0	0
$259$	−1.01212e29	−0.0690219
$260$	0	0
$261$	−9.61146e29	−0.595365
$262$	0	0
$263$	1.42946e30	0.804869	0.402434	−	0.915449i	$-0.368164\pi$
$263$	1.42946e30	0.804869	0.402434	+	0.915449i	$0.368164\pi$
$264$	0	0
$265$	−9.28658e29	−0.475644
$266$	0	0
$267$	−8.90441e28	−0.0415159
$268$	0	0
$269$	−3.97725e30	−1.68919	−0.844597	−	0.535402i	$-0.820160\pi$
$269$	−3.97725e30	−1.68919	−0.844597	+	0.535402i	$0.820160\pi$
$270$	0	0
$271$	4.54476e30	1.75952	0.879761	−	0.475416i	$-0.157702\pi$
$271$	4.54476e30	1.75952	0.879761	+	0.475416i	$0.157702\pi$
$272$	0	0
$273$	7.57645e28	0.0267567
$274$	0	0
$275$	2.32913e30	0.750818
$276$	0	0
$277$	6.69939e30	1.97260	0.986299	−	0.164969i	$-0.0527526\pi$
$277$	6.69939e30	1.97260	0.986299	+	0.164969i	$0.0527526\pi$
$278$	0	0
$279$	−7.18665e29	−0.193408
$280$	0	0
$281$	8.85736e29	0.218010	0.109005	−	0.994041i	$-0.465234\pi$
$281$	8.85736e29	0.218010	0.109005	+	0.994041i	$0.465234\pi$
$282$	0	0
$283$	1.36031e30	0.306413	0.153207	−	0.988194i	$-0.451040\pi$
$283$	1.36031e30	0.306413	0.153207	+	0.988194i	$0.451040\pi$
$284$	0	0
$285$	−6.17701e30	−1.27414
$286$	0	0
$287$	−1.09986e30	−0.207880
$288$	0	0
$289$	−5.51033e30	−0.954894
$290$	0	0
$291$	6.38748e30	1.01547
$292$	0	0
$293$	1.18165e31	1.72443	0.862213	−	0.506546i	$-0.169078\pi$
$293$	1.18165e31	1.72443	0.862213	+	0.506546i	$0.169078\pi$
$294$	0	0
$295$	−1.61084e31	−2.15912
$296$	0	0
$297$	2.54799e30	0.313864
$298$	0	0
$299$	3.32774e30	0.376929
$300$	0	0
$301$	6.73031e29	0.0701381
$302$	0	0
$303$	6.83862e30	0.656048
$304$	0	0
$305$	−2.40519e30	−0.212521
$306$	0	0
$307$	−1.18664e29	−0.00966246	−0.00483123	−	0.999988i	$-0.501538\pi$
$307$	−1.18664e29	−0.00966246	−0.00483123	+	0.999988i	$0.501538\pi$
$308$	0	0
$309$	6.70291e30	0.503248
$310$	0	0
$311$	−3.54542e30	−0.245563	−0.122782	−	0.992434i	$-0.539181\pi$
$311$	−3.54542e30	−0.245563	−0.122782	+	0.992434i	$0.539181\pi$
$312$	0	0
$313$	2.18186e31	1.39484	0.697419	−	0.716664i	$-0.254332\pi$
$313$	2.18186e31	1.39484	0.697419	+	0.716664i	$0.254332\pi$
$314$	0	0
$315$	−1.07666e30	−0.0635619
$316$	0	0
$317$	−2.90913e31	−1.58680	−0.793400	−	0.608700i	$-0.791691\pi$
$317$	−2.90913e31	−1.58680	−0.793400	+	0.608700i	$0.791691\pi$
$318$	0	0
$319$	−5.77713e31	−2.91292
$320$	0	0
$321$	−1.47607e31	−0.688324
$322$	0	0
$323$	−8.98870e30	−0.387852
$324$	0	0
$325$	3.38501e30	0.135213
$326$	0	0
$327$	−3.73132e30	−0.138044
$328$	0	0
$329$	−3.89855e30	−0.133646
$330$	0	0
$331$	−5.89110e31	−1.87217	−0.936087	−	0.351769i	$-0.885580\pi$
$331$	−5.89110e31	−1.87217	−0.936087	+	0.351769i	$0.885580\pi$
$332$	0	0
$333$	−4.94689e30	−0.145807
$334$	0	0
$335$	−1.36843e31	−0.374250
$336$	0	0
$337$	−8.89828e30	−0.225908	−0.112954	−	0.993600i	$-0.536031\pi$
$337$	−8.89828e30	−0.225908	−0.112954	+	0.993600i	$0.536031\pi$
$338$	0	0
$339$	9.41696e30	0.222031
$340$	0	0
$341$	−4.31965e31	−0.946279
$342$	0	0
$343$	−1.53057e31	−0.311656
$344$	0	0
$345$	−4.72893e31	−0.895414
$346$	0	0
$347$	4.53339e31	0.798555	0.399277	−	0.916830i	$-0.369261\pi$
$347$	4.53339e31	0.798555	0.399277	+	0.916830i	$0.369261\pi$
$348$	0	0
$349$	−1.40922e31	−0.231025	−0.115513	−	0.993306i	$-0.536851\pi$
$349$	−1.40922e31	−0.231025	−0.115513	+	0.993306i	$0.536851\pi$
$350$	0	0
$351$	3.70309e30	0.0565230
$352$	0	0
$353$	−2.48459e31	−0.353241	−0.176621	−	0.984279i	$-0.556517\pi$
$353$	−2.48459e31	−0.353241	−0.176621	+	0.984279i	$0.556517\pi$
$354$	0	0
$355$	−7.64640e31	−1.01298
$356$	0	0
$357$	−1.56674e30	−0.0193484
$358$	0	0
$359$	−2.22458e31	−0.256193	−0.128097	−	0.991762i	$-0.540887\pi$
$359$	−2.22458e31	−0.256193	−0.128097	+	0.991762i	$0.540887\pi$
$360$	0	0
$361$	2.17331e32	2.33497
$362$	0	0
$363$	9.55711e31	0.958283
$364$	0	0
$365$	−1.11657e32	−1.04526
$366$	0	0
$367$	5.11985e31	0.447643	0.223821	−	0.974630i	$-0.428147\pi$
$367$	5.11985e31	0.447643	0.223821	+	0.974630i	$0.428147\pi$
$368$	0	0
$369$	−5.37571e31	−0.439143
$370$	0	0
$371$	8.13412e30	0.0621064
$372$	0	0
$373$	−1.66177e32	−1.18634	−0.593171	−	0.805076i	$-0.702124\pi$
$373$	−1.66177e32	−1.18634	−0.593171	+	0.805076i	$0.702124\pi$
$374$	0	0
$375$	5.63839e31	0.376499
$376$	0	0
$377$	−8.39611e31	−0.524580
$378$	0	0
$379$	−1.10695e29	−0.000647346 0	−0.000323673	−	1.00000i	$-0.500103\pi$
$379$	−1.10695e29	−0.000647346 0	−0.000323673		1.00000i	$0.500103\pi$
$380$	0	0
$381$	−9.92132e31	−0.543259
$382$	0	0
$383$	1.29713e32	0.665271	0.332635	−	0.943056i	$-0.392062\pi$
$383$	1.29713e32	0.665271	0.332635	+	0.943056i	$0.392062\pi$
$384$	0	0
$385$	−6.47145e31	−0.310987
$386$	0	0
$387$	3.28953e31	0.148165
$388$	0	0
$389$	−2.00775e31	−0.0847890	−0.0423945	−	0.999101i	$-0.513499\pi$
$389$	−2.00775e31	−0.0847890	−0.0423945	+	0.999101i	$0.513499\pi$
$390$	0	0
$391$	−6.88147e31	−0.272566
$392$	0	0
$393$	−2.11882e32	−0.787384
$394$	0	0
$395$	1.84084e32	0.642026
$396$	0	0
$397$	3.90449e32	1.27845	0.639223	−	0.769021i	$-0.279256\pi$
$397$	3.90449e32	1.27845	0.639223	+	0.769021i	$0.279256\pi$
$398$	0	0
$399$	5.41044e31	0.166369
$400$	0	0
$401$	−4.05146e32	−1.17033	−0.585163	−	0.810916i	$-0.698969\pi$
$401$	−4.05146e32	−1.17033	−0.585163	+	0.810916i	$0.698969\pi$
$402$	0	0
$403$	−6.27791e31	−0.170413
$404$	0	0
$405$	−5.26232e31	−0.134273
$406$	0	0
$407$	−2.97341e32	−0.713386
$408$	0	0
$409$	7.12404e32	1.60763	0.803813	−	0.594882i	$-0.202801\pi$
$409$	7.12404e32	1.60763	0.803813	+	0.594882i	$0.202801\pi$
$410$	0	0
$411$	−3.19282e32	−0.677879
$412$	0	0
$413$	1.41094e32	0.281924
$414$	0	0
$415$	−3.74786e32	−0.704989
$416$	0	0
$417$	4.86918e32	0.862493
$418$	0	0
$419$	9.98252e31	0.166558	0.0832789	−	0.996526i	$-0.473461\pi$
$419$	9.98252e31	0.166558	0.0832789	+	0.996526i	$0.473461\pi$
$420$	0	0
$421$	7.21900e32	1.13488	0.567441	−	0.823414i	$-0.307934\pi$
$421$	7.21900e32	1.13488	0.567441	+	0.823414i	$0.307934\pi$
$422$	0	0
$423$	−1.90547e32	−0.282324
$424$	0	0
$425$	−6.99990e31	−0.0977755
$426$	0	0
$427$	2.10671e31	0.0277495
$428$	0	0
$429$	2.22580e32	0.276548
$430$	0	0
$431$	7.60853e32	0.891938	0.445969	−	0.895048i	$-0.352859\pi$
$431$	7.60853e32	0.891938	0.445969	+	0.895048i	$0.352859\pi$
$432$	0	0
$433$	−8.46517e31	−0.0936563	−0.0468281	−	0.998903i	$-0.514911\pi$
$433$	−8.46517e31	−0.0936563	−0.0468281	+	0.998903i	$0.514911\pi$
$434$	0	0
$435$	1.19314e33	1.24617
$436$	0	0
$437$	2.37638e33	2.34368
$438$	0	0
$439$	1.65733e33	1.54384	0.771921	−	0.635719i	$-0.219296\pi$
$439$	1.65733e33	1.54384	0.771921	+	0.635719i	$0.219296\pi$
$440$	0	0
$441$	−3.69327e32	−0.325034
$442$	0	0
$443$	−2.53888e32	−0.211152	−0.105576	−	0.994411i	$-0.533669\pi$
$443$	−2.53888e32	−0.211152	−0.105576	+	0.994411i	$0.533669\pi$
$444$	0	0
$445$	1.10537e32	0.0868974
$446$	0	0
$447$	4.77500e32	0.354919
$448$	0	0
$449$	1.87816e33	1.32024	0.660122	−	0.751159i	$-0.270505\pi$
$449$	1.87816e33	1.32024	0.660122	+	0.751159i	$0.270505\pi$
$450$	0	0
$451$	−3.23116e33	−2.14858
$452$	0	0
$453$	1.13129e33	0.711781
$454$	0	0
$455$	−9.40519e31	−0.0560048
$456$	0	0
$457$	−2.65546e33	−1.49688	−0.748440	−	0.663203i	$-0.769197\pi$
$457$	−2.65546e33	−1.49688	−0.748440	+	0.663203i	$0.769197\pi$
$458$	0	0
$459$	−7.65766e31	−0.0408731
$460$	0	0
$461$	−1.59288e33	−0.805232	−0.402616	−	0.915369i	$-0.631899\pi$
$461$	−1.59288e33	−0.805232	−0.402616	+	0.915369i	$0.631899\pi$
$462$	0	0
$463$	2.74292e33	1.31356	0.656782	−	0.754080i	$-0.271917\pi$
$463$	2.74292e33	1.31356	0.656782	+	0.754080i	$0.271917\pi$
$464$	0	0
$465$	8.92131e32	0.404824
$466$	0	0
$467$	−4.35039e32	−0.187097	−0.0935487	−	0.995615i	$-0.529821\pi$
$467$	−4.35039e32	−0.187097	−0.0935487	+	0.995615i	$0.529821\pi$
$468$	0	0
$469$	1.19861e32	0.0488670
$470$	0	0
$471$	−4.55492e32	−0.176084
$472$	0	0
$473$	1.97722e33	0.724923
$474$	0	0
$475$	2.41728e33	0.840731
$476$	0	0
$477$	3.97566e32	0.131199
$478$	0	0
$479$	−1.40824e32	−0.0441045	−0.0220522	−	0.999757i	$-0.507020\pi$
$479$	−1.40824e32	−0.0441045	−0.0220522	+	0.999757i	$0.507020\pi$
$480$	0	0
$481$	−4.32137e32	−0.128472
$482$	0	0
$483$	4.14207e32	0.116917
$484$	0	0
$485$	−7.92924e33	−2.12549
$486$	0	0
$487$	5.46270e33	1.39090	0.695449	−	0.718576i	$-0.255206\pi$
$487$	5.46270e33	1.39090	0.695449	+	0.718576i	$0.255206\pi$
$488$	0	0
$489$	2.06094e33	0.498546
$490$	0	0
$491$	3.09492e33	0.711429	0.355714	−	0.934595i	$-0.384238\pi$
$491$	3.09492e33	0.711429	0.355714	+	0.934595i	$0.384238\pi$
$492$	0	0
$493$	1.73624e33	0.379336
$494$	0	0
$495$	−3.16300e33	−0.656954
$496$	0	0
$497$	6.69749e32	0.132269
$498$	0	0
$499$	8.30005e33	1.55892	0.779460	−	0.626451i	$-0.215493\pi$
$499$	8.30005e33	1.55892	0.779460	+	0.626451i	$0.215493\pi$
$500$	0	0
$501$	−2.89711e33	−0.517600
$502$	0	0
$503$	−7.45393e33	−1.26703	−0.633513	−	0.773732i	$-0.718388\pi$
$503$	−7.45393e33	−1.26703	−0.633513	+	0.773732i	$0.718388\pi$
$504$	0	0
$505$	−8.48928e33	−1.37318
$506$	0	0
$507$	−3.42658e33	−0.527547
$508$	0	0
$509$	−1.01652e33	−0.148986	−0.0744928	−	0.997222i	$-0.523734\pi$
$509$	−1.01652e33	−0.148986	−0.0744928	+	0.997222i	$0.523734\pi$
$510$	0	0
$511$	9.78000e32	0.136483
$512$	0	0
$513$	2.64443e33	0.351451
$514$	0	0
$515$	−8.32080e33	−1.05335
$516$	0	0
$517$	−1.14531e34	−1.38132
$518$	0	0
$519$	−5.55196e33	−0.638049
$520$	0	0
$521$	−6.40558e33	−0.701595	−0.350798	−	0.936451i	$-0.614090\pi$
$521$	−6.40558e33	−0.701595	−0.350798	+	0.936451i	$0.614090\pi$
$522$	0	0
$523$	−1.21503e34	−1.26857	−0.634287	−	0.773098i	$-0.718706\pi$
$523$	−1.21503e34	−1.26857	−0.634287	+	0.773098i	$0.718706\pi$
$524$	0	0
$525$	4.21335e32	0.0419407
$526$	0	0
$527$	1.29822e33	0.123229
$528$	0	0
$529$	7.14708e33	0.647042
$530$	0	0
$531$	6.89614e33	0.595558
$532$	0	0
$533$	−4.69596e33	−0.386932
$534$	0	0
$535$	1.83235e34	1.44074
$536$	0	0
$537$	1.02548e34	0.769573
$538$	0	0
$539$	−2.21990e34	−1.59028
$540$	0	0
$541$	−7.26293e33	−0.496759	−0.248380	−	0.968663i	$-0.579898\pi$
$541$	−7.26293e33	−0.496759	−0.248380	+	0.968663i	$0.579898\pi$
$542$	0	0
$543$	−7.75510e33	−0.506512
$544$	0	0
$545$	4.63196e33	0.288941
$546$	0	0
$547$	−1.10164e34	−0.656447	−0.328224	−	0.944600i	$-0.606450\pi$
$547$	−1.10164e34	−0.656447	−0.328224	+	0.944600i	$0.606450\pi$
$548$	0	0
$549$	1.02968e33	0.0586204
$550$	0	0
$551$	−5.99578e34	−3.26175
$552$	0	0
$553$	−1.61239e33	−0.0838314
$554$	0	0
$555$	6.14093e33	0.305191
$556$	0	0
$557$	3.12718e34	1.48581	0.742904	−	0.669398i	$-0.233448\pi$
$557$	3.12718e34	1.48581	0.742904	+	0.669398i	$0.233448\pi$
$558$	0	0
$559$	2.87357e33	0.130549
$560$	0	0
$561$	−4.60276e33	−0.199978
$562$	0	0
$563$	1.77967e34	0.739580	0.369790	−	0.929115i	$-0.379430\pi$
$563$	1.77967e34	0.739580	0.369790	+	0.929115i	$0.379430\pi$
$564$	0	0
$565$	−1.16899e34	−0.464737
$566$	0	0
$567$	4.60927e32	0.0175325
$568$	0	0
$569$	−2.73469e34	−0.995417	−0.497709	−	0.867344i	$-0.665825\pi$
$569$	−2.73469e34	−0.995417	−0.497709	+	0.867344i	$0.665825\pi$
$570$	0	0
$571$	−1.46598e34	−0.510711	−0.255356	−	0.966847i	$-0.582193\pi$
$571$	−1.46598e34	−0.510711	−0.255356	+	0.966847i	$0.582193\pi$
$572$	0	0
$573$	3.04233e34	1.01455
$574$	0	0
$575$	1.85059e34	0.590831
$576$	0	0
$577$	4.08942e34	1.25015	0.625077	−	0.780563i	$-0.285067\pi$
$577$	4.08942e34	1.25015	0.625077	+	0.780563i	$0.285067\pi$
$578$	0	0
$579$	1.74196e34	0.509983
$580$	0	0
$581$	3.28275e33	0.0920527
$582$	0	0
$583$	2.38963e34	0.641911
$584$	0	0
$585$	−4.59691e33	−0.118309
$586$	0	0
$587$	4.33835e34	1.06991	0.534956	−	0.844880i	$-0.320328\pi$
$587$	4.33835e34	1.06991	0.534956	+	0.844880i	$0.320328\pi$
$588$	0	0
$589$	−4.48314e34	−1.05960
$590$	0	0
$591$	−1.73539e34	−0.393145
$592$	0	0
$593$	−7.66669e34	−1.66504	−0.832519	−	0.553996i	$-0.813102\pi$
$593$	−7.66669e34	−1.66504	−0.832519	+	0.553996i	$0.813102\pi$
$594$	0	0
$595$	1.94491e33	0.0404984
$596$	0	0
$597$	2.18718e34	0.436722
$598$	0	0
$599$	−6.19192e34	−1.18574	−0.592871	−	0.805298i	$-0.702005\pi$
$599$	−6.19192e34	−1.18574	−0.592871	+	0.805298i	$0.702005\pi$
$600$	0	0
$601$	1.72697e34	0.317215	0.158607	−	0.987342i	$-0.449300\pi$
$601$	1.72697e34	0.317215	0.158607	+	0.987342i	$0.449300\pi$
$602$	0	0
$603$	5.85834e33	0.103231
$604$	0	0
$605$	−1.18639e35	−2.00579
$606$	0	0
$607$	−3.27364e33	−0.0531096	−0.0265548	−	0.999647i	$-0.508454\pi$
$607$	−3.27364e33	−0.0531096	−0.0265548	+	0.999647i	$0.508454\pi$
$608$	0	0
$609$	−1.04507e34	−0.162716
$610$	0	0
$611$	−1.66453e34	−0.248757
$612$	0	0
$613$	1.33981e35	1.92215	0.961075	−	0.276287i	$-0.0891041\pi$
$613$	1.33981e35	1.92215	0.961075	+	0.276287i	$0.0891041\pi$
$614$	0	0
$615$	6.67326e34	0.919176
$616$	0	0
$617$	4.87480e34	0.644751	0.322376	−	0.946612i	$-0.395519\pi$
$617$	4.87480e34	0.644751	0.322376	+	0.946612i	$0.395519\pi$
$618$	0	0
$619$	−2.87493e34	−0.365170	−0.182585	−	0.983190i	$-0.558446\pi$
$619$	−2.87493e34	−0.365170	−0.182585	+	0.983190i	$0.558446\pi$
$620$	0	0
$621$	2.02449e34	0.246985
$622$	0	0
$623$	−9.68192e32	−0.0113465
$624$	0	0
$625$	−1.10883e35	−1.24843
$626$	0	0
$627$	1.58948e35	1.71953
$628$	0	0
$629$	8.93620e33	0.0929009
$630$	0	0
$631$	7.21691e33	0.0721081	0.0360540	−	0.999350i	$-0.488521\pi$
$631$	7.21691e33	0.0721081	0.0360540	+	0.999350i	$0.488521\pi$
$632$	0	0
$633$	−4.62790e33	−0.0444465
$634$	0	0
$635$	1.23160e35	1.13710
$636$	0	0
$637$	−3.22626e34	−0.286389
$638$	0	0
$639$	3.27348e34	0.279415
$640$	0	0
$641$	2.51881e33	0.0206761	0.0103381	−	0.999947i	$-0.496709\pi$
$641$	2.51881e33	0.0206761	0.0103381	+	0.999947i	$0.496709\pi$
$642$	0	0
$643$	2.04098e35	1.61139	0.805697	−	0.592328i	$-0.201791\pi$
$643$	2.04098e35	1.61139	0.805697	+	0.592328i	$0.201791\pi$
$644$	0	0
$645$	−4.08353e34	−0.310127
$646$	0	0
$647$	4.69374e34	0.342938	0.171469	−	0.985190i	$-0.445149\pi$
$647$	4.69374e34	0.342938	0.171469	+	0.985190i	$0.445149\pi$
$648$	0	0
$649$	4.14504e35	2.91387
$650$	0	0
$651$	−7.81417e33	−0.0528592
$652$	0	0
$653$	−5.09343e34	−0.331586	−0.165793	−	0.986161i	$-0.553018\pi$
$653$	−5.09343e34	−0.331586	−0.165793	+	0.986161i	$0.553018\pi$
$654$	0	0
$655$	2.63024e35	1.64808
$656$	0	0
$657$	4.78010e34	0.288317
$658$	0	0
$659$	−1.83670e35	−1.06653	−0.533263	−	0.845950i	$-0.679034\pi$
$659$	−1.83670e35	−1.06653	−0.533263	+	0.845950i	$0.679034\pi$
$660$	0	0
$661$	3.11068e35	1.73915	0.869575	−	0.493800i	$-0.164393\pi$
$661$	3.11068e35	1.73915	0.869575	+	0.493800i	$0.164393\pi$
$662$	0	0
$663$	−6.68936e33	−0.0360135
$664$	0	0
$665$	−6.71638e34	−0.348229
$666$	0	0
$667$	−4.59018e35	−2.29222
$668$	0	0
$669$	−1.48656e35	−0.715080
$670$	0	0
$671$	6.18907e34	0.286810
$672$	0	0
$673$	3.12272e35	1.39426	0.697132	−	0.716943i	$-0.254459\pi$
$673$	3.12272e35	1.39426	0.697132	+	0.716943i	$0.254459\pi$
$674$	0	0
$675$	2.05933e34	0.0885990
$676$	0	0
$677$	−3.31383e35	−1.37395	−0.686977	−	0.726679i	$-0.741063\pi$
$677$	−3.31383e35	−1.37395	−0.686977	+	0.726679i	$0.741063\pi$
$678$	0	0
$679$	6.94522e34	0.277532
$680$	0	0
$681$	1.79949e35	0.693124
$682$	0	0
$683$	−3.63892e35	−1.35118	−0.675590	−	0.737277i	$-0.736111\pi$
$683$	−3.63892e35	−1.35118	−0.675590	+	0.737277i	$0.736111\pi$
$684$	0	0
$685$	3.96348e35	1.41888
$686$	0	0
$687$	1.21086e35	0.417960
$688$	0	0
$689$	3.47294e34	0.115600
$690$	0	0
$691$	9.21168e34	0.295708	0.147854	−	0.989009i	$-0.452763\pi$
$691$	9.21168e34	0.295708	0.147854	+	0.989009i	$0.452763\pi$
$692$	0	0
$693$	2.77048e34	0.0857806
$694$	0	0
$695$	−6.04447e35	−1.80529
$696$	0	0
$697$	9.71083e34	0.279799
$698$	0	0
$699$	2.54244e35	0.706784
$700$	0	0
$701$	−5.33586e35	−1.43130	−0.715648	−	0.698462i	$-0.753868\pi$
$701$	−5.33586e35	−1.43130	−0.715648	+	0.698462i	$0.753868\pi$
$702$	0	0
$703$	−3.08595e35	−0.798816
$704$	0	0
$705$	2.36540e35	0.590936
$706$	0	0
$707$	7.43576e34	0.179301
$708$	0	0
$709$	−5.45418e35	−1.26956	−0.634778	−	0.772695i	$-0.718908\pi$
$709$	−5.45418e35	−1.26956	−0.634778	+	0.772695i	$0.718908\pi$
$710$	0	0
$711$	−7.88078e34	−0.177092
$712$	0	0
$713$	−3.43215e35	−0.744642
$714$	0	0
$715$	−2.76305e35	−0.578846
$716$	0	0
$717$	3.00384e34	0.0607698
$718$	0	0
$719$	9.27091e34	0.181139	0.0905693	−	0.995890i	$-0.471131\pi$
$719$	9.27091e34	0.181139	0.0905693	+	0.995890i	$0.471131\pi$
$720$	0	0
$721$	7.28819e34	0.137540
$722$	0	0
$723$	8.99020e34	0.163885
$724$	0	0
$725$	−4.66917e35	−0.822271
$726$	0	0
$727$	−6.94605e35	−1.18184	−0.590919	−	0.806731i	$-0.701235\pi$
$727$	−6.94605e35	−1.18184	−0.590919	+	0.806731i	$0.701235\pi$
$728$	0	0
$729$	2.25284e34	0.0370370
$730$	0	0
$731$	−5.94230e34	−0.0944033
$732$	0	0
$733$	−3.87749e35	−0.595321	−0.297661	−	0.954672i	$-0.596206\pi$
$733$	−3.87749e35	−0.595321	−0.297661	+	0.954672i	$0.596206\pi$
$734$	0	0
$735$	4.58472e35	0.680333
$736$	0	0
$737$	3.52125e35	0.505073
$738$	0	0
$739$	8.52722e35	1.18237	0.591183	−	0.806537i	$-0.298661\pi$
$739$	8.52722e35	1.18237	0.591183	+	0.806537i	$0.298661\pi$
$740$	0	0
$741$	2.31004e35	0.309665
$742$	0	0
$743$	−5.10098e35	−0.661140	−0.330570	−	0.943782i	$-0.607241\pi$
$743$	−5.10098e35	−0.661140	−0.330570	+	0.943782i	$0.607241\pi$
$744$	0	0
$745$	−5.92755e35	−0.742885
$746$	0	0
$747$	1.60449e35	0.194459
$748$	0	0
$749$	−1.60495e35	−0.188122
$750$	0	0
$751$	1.36643e36	1.54913	0.774565	−	0.632494i	$-0.217969\pi$
$751$	1.36643e36	1.54913	0.774565	+	0.632494i	$0.217969\pi$
$752$	0	0
$753$	4.41410e35	0.484066
$754$	0	0
$755$	−1.40435e36	−1.48984
$756$	0	0
$757$	−1.77865e36	−1.82555	−0.912773	−	0.408466i	$-0.866064\pi$
$757$	−1.77865e36	−1.82555	−0.912773	+	0.408466i	$0.866064\pi$
$758$	0	0
$759$	1.21685e36	1.20841
$760$	0	0
$761$	−9.12849e35	−0.877184	−0.438592	−	0.898686i	$-0.644523\pi$
$761$	−9.12849e35	−0.877184	−0.438592	+	0.898686i	$0.644523\pi$
$762$	0	0
$763$	−4.05714e34	−0.0377279
$764$	0	0
$765$	9.50600e34	0.0855520
$766$	0	0
$767$	6.02413e35	0.524750
$768$	0	0
$769$	−4.80131e35	−0.404838	−0.202419	−	0.979299i	$-0.564880\pi$
$769$	−4.80131e35	−0.404838	−0.202419	+	0.979299i	$0.564880\pi$
$770$	0	0
$771$	−1.22312e36	−0.998367
$772$	0	0
$773$	−1.89080e36	−1.49418	−0.747088	−	0.664726i	$-0.768548\pi$
$773$	−1.89080e36	−1.49418	−0.747088	+	0.664726i	$0.768548\pi$
$774$	0	0
$775$	−3.49122e35	−0.267120
$776$	0	0
$777$	−5.37884e34	−0.0398498
$778$	0	0
$779$	−3.35345e36	−2.40588
$780$	0	0
$781$	1.96758e36	1.36708
$782$	0	0
$783$	−5.10792e35	−0.343734
$784$	0	0
$785$	5.65435e35	0.368563
$786$	0	0
$787$	2.40170e36	1.51647	0.758237	−	0.651979i	$-0.226061\pi$
$787$	2.40170e36	1.51647	0.758237	+	0.651979i	$0.226061\pi$
$788$	0	0
$789$	7.59674e35	0.464691
$790$	0	0
$791$	1.02392e35	0.0606822
$792$	0	0
$793$	8.99480e34	0.0516508
$794$	0	0
$795$	−4.93527e35	−0.274613
$796$	0	0
$797$	−2.24196e36	−1.20893	−0.604463	−	0.796633i	$-0.706612\pi$
$797$	−2.24196e36	−1.20893	−0.604463	+	0.796633i	$0.706612\pi$
$798$	0	0
$799$	3.44209e35	0.179882
$800$	0	0
$801$	−4.73217e34	−0.0239692
$802$	0	0
$803$	2.87316e36	1.41064
$804$	0	0
$805$	−5.14185e35	−0.244721
$806$	0	0
$807$	−2.11367e36	−0.975257
$808$	0	0
$809$	1.17932e36	0.527566	0.263783	−	0.964582i	$-0.415030\pi$
$809$	1.17932e36	0.527566	0.263783	+	0.964582i	$0.415030\pi$
$810$	0	0
$811$	1.69300e36	0.734339	0.367170	−	0.930154i	$-0.380327\pi$
$811$	1.69300e36	0.734339	0.367170	+	0.930154i	$0.380327\pi$
$812$	0	0
$813$	2.41527e36	1.01586
$814$	0	0
$815$	−2.55840e36	−1.04351
$816$	0	0
$817$	2.05206e36	0.811735
$818$	0	0
$819$	4.02643e34	0.0154480
$820$	0	0
$821$	2.35389e36	0.875984	0.437992	−	0.898979i	$-0.355690\pi$
$821$	2.35389e36	0.875984	0.437992	+	0.898979i	$0.355690\pi$
$822$	0	0
$823$	−1.13654e36	−0.410285	−0.205143	−	0.978732i	$-0.565766\pi$
$823$	−1.13654e36	−0.410285	−0.205143	+	0.978732i	$0.565766\pi$
$824$	0	0
$825$	1.23779e36	0.433485
$826$	0	0
$827$	−1.26535e36	−0.429926	−0.214963	−	0.976622i	$-0.568963\pi$
$827$	−1.26535e36	−0.429926	−0.214963	+	0.976622i	$0.568963\pi$
$828$	0	0
$829$	−4.82159e36	−1.58949	−0.794746	−	0.606942i	$-0.792396\pi$
$829$	−4.82159e36	−1.58949	−0.794746	+	0.606942i	$0.792396\pi$
$830$	0	0
$831$	3.56033e36	1.13888
$832$	0	0
$833$	6.67163e35	0.207095
$834$	0	0
$835$	3.59640e36	1.08339
$836$	0	0
$837$	−3.81928e35	−0.111664
$838$	0	0
$839$	−6.85960e36	−1.94659	−0.973295	−	0.229557i	$-0.926272\pi$
$839$	−6.85960e36	−1.94659	−0.973295	+	0.229557i	$0.926272\pi$
$840$	0	0
$841$	7.95097e36	2.19013
$842$	0	0
$843$	4.70717e35	0.125868
$844$	0	0
$845$	4.25366e36	1.10422
$846$	0	0
$847$	1.03916e36	0.261903
$848$	0	0
$849$	7.22925e35	0.176908
$850$	0	0
$851$	−2.36250e36	−0.561375
$852$	0	0
$853$	−3.50144e36	−0.807948	−0.403974	−	0.914770i	$-0.632371\pi$
$853$	−3.50144e36	−0.807948	−0.403974	+	0.914770i	$0.632371\pi$
$854$	0	0
$855$	−3.28272e36	−0.735626
$856$	0	0
$857$	4.42062e36	0.962106	0.481053	−	0.876691i	$-0.340254\pi$
$857$	4.42062e36	0.962106	0.481053	+	0.876691i	$0.340254\pi$
$858$	0	0
$859$	1.80302e36	0.381141	0.190571	−	0.981673i	$-0.438966\pi$
$859$	1.80302e36	0.381141	0.190571	+	0.981673i	$0.438966\pi$
$860$	0	0
$861$	−5.84511e35	−0.120020
$862$	0	0
$863$	6.23862e36	1.24438	0.622191	−	0.782865i	$-0.286243\pi$
$863$	6.23862e36	1.24438	0.622191	+	0.782865i	$0.286243\pi$
$864$	0	0
$865$	6.89204e36	1.33551
$866$	0	0
$867$	−2.92842e36	−0.551308
$868$	0	0
$869$	−4.73687e36	−0.866452
$870$	0	0
$871$	5.11757e35	0.0909571
$872$	0	0
$873$	3.39457e36	0.586282
$874$	0	0
$875$	6.13072e35	0.102899
$876$	0	0
$877$	−8.04612e36	−1.31248	−0.656238	−	0.754554i	$-0.727853\pi$
$877$	−8.04612e36	−1.31248	−0.656238	+	0.754554i	$0.727853\pi$
$878$	0	0
$879$	6.27979e36	0.995598
$880$	0	0
$881$	3.66662e36	0.565024	0.282512	−	0.959264i	$-0.408832\pi$
$881$	3.66662e36	0.565024	0.282512	+	0.959264i	$0.408832\pi$
$882$	0	0
$883$	9.80248e36	1.46834	0.734171	−	0.678965i	$-0.237571\pi$
$883$	9.80248e36	1.46834	0.734171	+	0.678965i	$0.237571\pi$
$884$	0	0
$885$	−8.56067e36	−1.24657
$886$	0	0
$887$	1.23069e37	1.74222	0.871112	−	0.491085i	$-0.163399\pi$
$887$	1.23069e37	1.74222	0.871112	+	0.491085i	$0.163399\pi$
$888$	0	0
$889$	−1.07876e36	−0.148475
$890$	0	0
$891$	1.35411e36	0.181210
$892$	0	0
$893$	−1.18866e37	−1.54673
$894$	0	0
$895$	−1.27301e37	−1.61080
$896$	0	0
$897$	1.76850e36	0.217620
$898$	0	0
$899$	8.65956e36	1.03633
$900$	0	0
$901$	−7.18174e35	−0.0835930
$902$	0	0
$903$	3.57676e35	0.0404942
$904$	0	0
$905$	9.62696e36	1.06019
$906$	0	0
$907$	1.58042e36	0.169310	0.0846549	−	0.996410i	$-0.473021\pi$
$907$	1.58042e36	0.169310	0.0846549	+	0.996410i	$0.473021\pi$
$908$	0	0
$909$	3.63432e36	0.378770
$910$	0	0
$911$	3.24627e36	0.329159	0.164579	−	0.986364i	$-0.447373\pi$
$911$	3.24627e36	0.329159	0.164579	+	0.986364i	$0.447373\pi$
$912$	0	0
$913$	9.64404e36	0.951424
$914$	0	0
$915$	−1.27822e36	−0.122699
$916$	0	0
$917$	−2.30383e36	−0.215195
$918$	0	0
$919$	1.22143e37	1.11026	0.555131	−	0.831763i	$-0.312668\pi$
$919$	1.22143e37	1.11026	0.555131	+	0.831763i	$0.312668\pi$
$920$	0	0
$921$	−6.30627e34	−0.00557862
$922$	0	0
$923$	2.85956e36	0.246194
$924$	0	0
$925$	−2.40316e36	−0.201378
$926$	0	0
$927$	3.56220e36	0.290550
$928$	0	0
$929$	8.34841e36	0.662838	0.331419	−	0.943484i	$-0.392473\pi$
$929$	8.34841e36	0.662838	0.331419	+	0.943484i	$0.392473\pi$
$930$	0	0
$931$	−2.30392e37	−1.78072
$932$	0	0
$933$	−1.88418e36	−0.141776
$934$	0	0
$935$	5.71374e36	0.418577
$936$	0	0
$937$	−1.25464e37	−0.894899	−0.447450	−	0.894309i	$-0.647668\pi$
$937$	−1.25464e37	−0.894899	−0.447450	+	0.894309i	$0.647668\pi$
$938$	0	0
$939$	1.15953e37	0.805310
$940$	0	0
$941$	−1.71574e37	−1.16033	−0.580164	−	0.814500i	$-0.697012\pi$
$941$	−1.71574e37	−1.16033	−0.580164	+	0.814500i	$0.697012\pi$
$942$	0	0
$943$	−2.56730e37	−1.69075
$944$	0	0
$945$	−5.72182e35	−0.0366975
$946$	0	0
$947$	1.02793e37	0.642077	0.321038	−	0.947066i	$-0.395968\pi$
$947$	1.02793e37	0.642077	0.321038	+	0.947066i	$0.395968\pi$
$948$	0	0
$949$	4.17567e36	0.254038
$950$	0	0
$951$	−1.54603e37	−0.916140
$952$	0	0
$953$	2.09956e36	0.121190	0.0605951	−	0.998162i	$-0.480700\pi$
$953$	2.09956e36	0.121190	0.0605951	+	0.998162i	$0.480700\pi$
$954$	0	0
$955$	−3.77666e37	−2.12357
$956$	0	0
$957$	−3.07020e37	−1.68177
$958$	0	0
$959$	−3.47161e36	−0.185267
$960$	0	0
$961$	−1.27579e37	−0.663341
$962$	0	0
$963$	−7.84442e36	−0.397404
$964$	0	0
$965$	−2.16242e37	−1.06745
$966$	0	0
$967$	1.19562e36	0.0575123	0.0287561	−	0.999586i	$-0.490845\pi$
$967$	1.19562e36	0.0575123	0.0287561	+	0.999586i	$0.490845\pi$
$968$	0	0
$969$	−4.77697e36	−0.223926
$970$	0	0
$971$	−8.74193e36	−0.399362	−0.199681	−	0.979861i	$-0.563991\pi$
$971$	−8.74193e36	−0.399362	−0.199681	+	0.979861i	$0.563991\pi$
$972$	0	0
$973$	5.29435e36	0.235723
$974$	0	0
$975$	1.79893e36	0.0780651
$976$	0	0
$977$	4.15926e37	1.75928	0.879639	−	0.475642i	$-0.157784\pi$
$977$	4.15926e37	1.75928	0.879639	+	0.475642i	$0.157784\pi$
$978$	0	0
$979$	−2.84435e36	−0.117273
$980$	0	0
$981$	−1.98298e36	−0.0796995
$982$	0	0
$983$	−2.89079e37	−1.13265	−0.566327	−	0.824181i	$-0.691636\pi$
$983$	−2.89079e37	−1.13265	−0.566327	+	0.824181i	$0.691636\pi$
$984$	0	0
$985$	2.15426e37	0.822897
$986$	0	0
$987$	−2.07185e36	−0.0771603
$988$	0	0
$989$	1.57099e37	0.570453
$990$	0	0
$991$	−8.37011e36	−0.296354	−0.148177	−	0.988961i	$-0.547340\pi$
$991$	−8.37011e36	−0.296354	−0.148177	+	0.988961i	$0.547340\pi$
$992$	0	0
$993$	−3.13077e37	−1.08090
$994$	0	0
$995$	−2.71510e37	−0.914109
$996$	0	0
$997$	1.30188e37	0.427448	0.213724	−	0.976894i	$-0.431441\pi$
$997$	1.30188e37	0.427448	0.213724	+	0.976894i	$0.431441\pi$
$998$	0	0
$999$	−2.62898e36	−0.0841819

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.26.a.k.1.1		4
4.3	odd	2		24.26.a.d.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.26.a.d.1.1	✓	4	4.3	odd	2
48.26.a.k.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q+531441. q^{3} -6.59716e8 q^{5} +5.77845e9 q^{7} +2.82430e11 q^{9} +O(q^{10})\)	\(q+531441. q^{3} -6.59716e8 q^{5} +5.77845e9 q^{7} +2.82430e11 q^{9} +1.69759e13 q^{11} +2.46717e13 q^{13} -3.50600e14 q^{15} -5.10189e14 q^{17} +1.76184e16 q^{19} +3.07091e15 q^{21} +1.34881e17 q^{23} +1.37202e17 q^{25} +1.50095e17 q^{27} -3.40314e18 q^{29} -2.54458e18 q^{31} +9.02169e18 q^{33} -3.81214e18 q^{35} -1.75155e19 q^{37} +1.31115e19 q^{39} -1.90338e20 q^{41} +1.16473e20 q^{43} -1.86323e20 q^{45} -6.74671e20 q^{47} -1.30768e21 q^{49} -2.71135e20 q^{51} +1.40766e21 q^{53} -1.11993e22 q^{55} +9.36314e21 q^{57} +2.44172e22 q^{59} +3.64580e21 q^{61} +1.63201e21 q^{63} -1.62763e22 q^{65} +2.07427e22 q^{67} +7.16812e22 q^{69} +1.15904e23 q^{71} +1.69249e23 q^{73} +7.29148e22 q^{75} +9.80944e22 q^{77} -2.79035e23 q^{79} +7.97664e22 q^{81} +5.68102e23 q^{83} +3.36580e23 q^{85} -1.80857e24 q^{87} -1.67552e23 q^{89} +1.42564e23 q^{91} -1.35229e24 q^{93} -1.16231e25 q^{95} +1.20192e25 q^{97} +4.79449e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2125764 q^{3} + 266436088 q^{5} - 484584576 q^{7} + 1129718145924 q^{9}+O(q^{10}) \) 4 * q + 2125764 * q^3 + 266436088 * q^5 - 484584576 * q^7 + 1129718145924 * q^9	\( 4 q + 2125764 q^{3} + 266436088 q^{5} - 484584576 q^{7} + 1129718145924 q^{9} - 4052955557392 q^{11} + 82663381481080 q^{13} + 141595061042808 q^{15} + 10\!\cdots\!28 q^{17}+ \cdots - 11\!\cdots\!52 q^{99}+O(q^{100}) \) 4 * q + 2125764 * q^3 + 266436088 * q^5 - 484584576 * q^7 + 1129718145924 * q^9 - 4052955557392 * q^11 + 82663381481080 * q^13 + 141595061042808 * q^15 + 1091839531710728 * q^17 + 5291818285650448 * q^19 - 257528111654016 * q^21 - 22957555441633504 * q^23 + 329573428119320444 * q^25 + 600378541187996484 * q^27 - 2255995699601091048 * q^29 + 6422134828063294816 * q^31 - 2153906754375961872 * q^33 + 26450723987305847040 * q^35 - 84572383178406307176 * q^37 + 43930710117686636280 * q^39 - 191923587571359535128 * q^41 + 393512700764425221104 * q^43 + 75249420835650926328 * q^45 + 426139906889905851648 * q^47 + 1697461944491957688548 * q^49 + 580248292571880999048 * q^51 + 4165797514523716100344 * q^53 - 1377411551242297580512 * q^55 + 2812289201544359735568 * q^57 - 22289642820361283440912 * q^59 + 31420833466214461354360 * q^61 - 136860997185521917056 * q^63 + 49904798993160775849744 * q^65 + 105958367596085546226448 * q^67 - 12200586221457150999264 * q^69 + 260836089844500944794336 * q^71 + 96585608913237168931624 * q^73 + 175148832213159776079804 * q^75 + 609668306798881251586560 * q^77 + 228314222637752365466528 * q^79 + 319065772307490039453444 * q^81 - 508041934949405113277936 * q^83 + 3527355809877820452506096 * q^85 - 1198928610591703427640168 * q^87 + 9334569288453070817495784 * q^89 - 6414440882259952713889536 * q^91 + 3412985755160785460309856 * q^93 - 9916967691165637498776608 * q^95 + 17898782631688573069195784 * q^97 - 1144674359452315553217552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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