LMFDB - Embedded newform 48.26.a.k.1.3

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			1.3
Root			$-25346.6$ 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)

$f(q)$	$=$	$q+531441. q^{3} +7.35582e6 q^{5} -7.04630e10 q^{7} +2.82430e11 q^{9} +O(q^{10})$	$q+531441. q^{3} +7.35582e6 q^{5} -7.04630e10 q^{7} +2.82430e11 q^{9} -1.12977e13 q^{11} +8.35429e13 q^{13} +3.90918e12 q^{15} +3.65306e15 q^{17} -1.04438e16 q^{19} -3.74469e16 q^{21} +1.00701e16 q^{23} -2.97969e17 q^{25} +1.50095e17 q^{27} -7.82440e16 q^{29} -5.49407e18 q^{31} -6.00409e18 q^{33} -5.18313e17 q^{35} -5.63793e19 q^{37} +4.43981e19 q^{39} -9.87269e19 q^{41} -2.39407e20 q^{43} +2.07750e18 q^{45} +1.10033e21 q^{47} +3.62396e21 q^{49} +1.94139e21 q^{51} -3.12209e21 q^{53} -8.31041e19 q^{55} -5.55024e21 q^{57} -2.27078e22 q^{59} +1.52728e22 q^{61} -1.99008e22 q^{63} +6.14526e20 q^{65} +7.81198e22 q^{67} +5.35164e21 q^{69} +1.76416e23 q^{71} -1.17253e23 q^{73} -1.58353e23 q^{75} +7.96073e23 q^{77} +4.00068e22 q^{79} +7.97664e22 q^{81} +6.42357e23 q^{83} +2.68713e22 q^{85} -4.15821e22 q^{87} +2.05843e24 q^{89} -5.88668e24 q^{91} -2.91977e24 q^{93} -7.68223e22 q^{95} +6.79625e24 q^{97} -3.19082e24 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	7.35582e6	0.0134743	0.00673714	−	0.999977i	$-0.497855\pi$
$5$	7.35582e6	0.0134743	0.00673714	+	0.999977i	$0.497855\pi$
$6$	0	0
$7$	−7.04630e10	−1.92414	−0.962068	−	0.272811i	$-0.912046\pi$
$7$	−7.04630e10	−1.92414	−0.962068	+	0.272811i	$0.912046\pi$
$8$	0	0
$9$	2.82430e11	0.333333
$10$	0	0
$11$	−1.12977e13	−1.08538	−0.542692	−	0.839932i	$-0.682595\pi$
$11$	−1.12977e13	−1.08538	−0.542692	+	0.839932i	$0.682595\pi$
$12$	0	0
$13$	8.35429e13	0.994529	0.497265	−	0.867599i	$-0.334338\pi$
$13$	8.35429e13	0.994529	0.497265	+	0.867599i	$0.334338\pi$
$14$	0	0
$15$	3.90918e12	0.00777938
$16$	0	0
$17$	3.65306e15	1.52071	0.760354	−	0.649509i	$-0.225025\pi$
$17$	3.65306e15	1.52071	0.760354	+	0.649509i	$0.225025\pi$
$18$	0	0
$19$	−1.04438e16	−1.08252	−0.541261	−	0.840855i	$-0.682053\pi$
$19$	−1.04438e16	−1.08252	−0.541261	+	0.840855i	$0.682053\pi$
$20$	0	0
$21$	−3.74469e16	−1.11090
$22$	0	0
$23$	1.00701e16	0.0958151	0.0479075	−	0.998852i	$-0.484745\pi$
$23$	1.00701e16	0.0958151	0.0479075	+	0.998852i	$0.484745\pi$
$24$	0	0
$25$	−2.97969e17	−0.999818
$26$	0	0
$27$	1.50095e17	0.192450
$28$	0	0
$29$	−7.82440e16	−0.0410654	−0.0205327	−	0.999789i	$-0.506536\pi$
$29$	−7.82440e16	−0.0410654	−0.0205327	+	0.999789i	$0.506536\pi$
$30$	0	0
$31$	−5.49407e18	−1.25277	−0.626387	−	0.779512i	$-0.715467\pi$
$31$	−5.49407e18	−1.25277	−0.626387	+	0.779512i	$0.715467\pi$
$32$	0	0
$33$	−6.00409e18	−0.626647
$34$	0	0
$35$	−5.18313e17	−0.0259263
$36$	0	0
$37$	−5.63793e19	−1.40799	−0.703993	−	0.710207i	$-0.748601\pi$
$37$	−5.63793e19	−1.40799	−0.703993	+	0.710207i	$0.748601\pi$
$38$	0	0
$39$	4.43981e19	0.574192
$40$	0	0
$41$	−9.87269e19	−0.683341	−0.341670	−	0.939820i	$-0.610993\pi$
$41$	−9.87269e19	−0.683341	−0.341670	+	0.939820i	$0.610993\pi$
$42$	0	0
$43$	−2.39407e20	−0.913654	−0.456827	−	0.889556i	$-0.651014\pi$
$43$	−2.39407e20	−0.913654	−0.456827	+	0.889556i	$0.651014\pi$
$44$	0	0
$45$	2.07750e18	0.00449143
$46$	0	0
$47$	1.10033e21	1.38134	0.690671	−	0.723169i	$-0.257315\pi$
$47$	1.10033e21	1.38134	0.690671	+	0.723169i	$0.257315\pi$
$48$	0	0
$49$	3.62396e21	2.70230
$50$	0	0
$51$	1.94139e21	0.877981
$52$	0	0
$53$	−3.12209e21	−0.872964	−0.436482	−	0.899713i	$-0.643776\pi$
$53$	−3.12209e21	−0.872964	−0.436482	+	0.899713i	$0.643776\pi$
$54$	0	0
$55$	−8.31041e19	−0.0146248
$56$	0	0
$57$	−5.55024e21	−0.624994
$58$	0	0
$59$	−2.27078e22	−1.66160	−0.830799	−	0.556573i	$-0.812116\pi$
$59$	−2.27078e22	−1.66160	−0.830799	+	0.556573i	$0.812116\pi$
$60$	0	0
$61$	1.52728e22	0.736708	0.368354	−	0.929686i	$-0.379921\pi$
$61$	1.52728e22	0.736708	0.368354	+	0.929686i	$0.379921\pi$
$62$	0	0
$63$	−1.99008e22	−0.641378
$64$	0	0
$65$	6.14526e20	0.0134006
$66$	0	0
$67$	7.81198e22	1.16634	0.583171	−	0.812349i	$-0.301812\pi$
$67$	7.81198e22	1.16634	0.583171	+	0.812349i	$0.301812\pi$
$68$	0	0
$69$	5.35164e21	0.0553189
$70$	0	0
$71$	1.76416e23	1.27587	0.637937	−	0.770089i	$-0.279788\pi$
$71$	1.76416e23	1.27587	0.637937	+	0.770089i	$0.279788\pi$
$72$	0	0
$73$	−1.17253e23	−0.599225	−0.299612	−	0.954061i	$-0.596857\pi$
$73$	−1.17253e23	−0.599225	−0.299612	+	0.954061i	$0.596857\pi$
$74$	0	0
$75$	−1.58353e23	−0.577245
$76$	0	0
$77$	7.96073e23	2.08842
$78$	0	0
$79$	4.00068e22	0.0761719	0.0380860	−	0.999274i	$-0.487874\pi$
$79$	4.00068e22	0.0761719	0.0380860	+	0.999274i	$0.487874\pi$
$80$	0	0
$81$	7.97664e22	0.111111
$82$	0	0
$83$	6.42357e23	0.659629	0.329815	−	0.944046i	$-0.393014\pi$
$83$	6.42357e23	0.659629	0.329815	+	0.944046i	$0.393014\pi$
$84$	0	0
$85$	2.68713e22	0.0204905
$86$	0	0
$87$	−4.15821e22	−0.0237091
$88$	0	0
$89$	2.05843e24	0.883408	0.441704	−	0.897161i	$-0.354374\pi$
$89$	2.05843e24	0.883408	0.441704	+	0.897161i	$0.354374\pi$
$90$	0	0
$91$	−5.88668e24	−1.91361
$92$	0	0
$93$	−2.91977e24	−0.723289
$94$	0	0
$95$	−7.68223e22	−0.0145862
$96$	0	0
$97$	6.79625e24	0.994541	0.497270	−	0.867596i	$-0.334336\pi$
$97$	6.79625e24	0.994541	0.497270	+	0.867596i	$0.334336\pi$
$98$	0	0
$99$	−3.19082e24	−0.361795
$100$	0	0
$101$	1.18453e25	1.04599	0.522997	−	0.852335i	$-0.324814\pi$
$101$	1.18453e25	1.04599	0.522997	+	0.852335i	$0.324814\pi$
$102$	0	0
$103$	−1.73424e25	−1.19852	−0.599260	−	0.800555i	$-0.704538\pi$
$103$	−1.73424e25	−1.19852	−0.599260	+	0.800555i	$0.704538\pi$
$104$	0	0
$105$	−2.75453e23	−0.0149686
$106$	0	0
$107$	2.70656e25	1.16177	0.580885	−	0.813986i	$-0.302707\pi$
$107$	2.70656e25	1.16177	0.580885	+	0.813986i	$0.302707\pi$
$108$	0	0
$109$	−1.66933e25	−0.568474	−0.284237	−	0.958754i	$-0.591740\pi$
$109$	−1.66933e25	−0.568474	−0.284237	+	0.958754i	$0.591740\pi$
$110$	0	0
$111$	−2.99623e25	−0.812901
$112$	0	0
$113$	8.52810e25	1.85085	0.925426	−	0.378929i	$-0.123708\pi$
$113$	8.52810e25	1.85085	0.925426	+	0.378929i	$0.123708\pi$
$114$	0	0
$115$	7.40735e22	0.00129104
$116$	0	0
$117$	2.35950e25	0.331510
$118$	0	0
$119$	−2.57406e26	−2.92605
$120$	0	0
$121$	1.92920e25	0.178058
$122$	0	0
$123$	−5.24675e25	−0.394527
$124$	0	0
$125$	−4.38401e24	−0.0269461
$126$	0	0
$127$	−2.77197e26	−1.39715	−0.698573	−	0.715539i	$-0.746181\pi$
$127$	−2.77197e26	−1.39715	−0.698573	+	0.715539i	$0.746181\pi$
$128$	0	0
$129$	−1.27231e26	−0.527498
$130$	0	0
$131$	3.59333e26	1.22915	0.614577	−	0.788857i	$-0.289327\pi$
$131$	3.59333e26	1.22915	0.614577	+	0.788857i	$0.289327\pi$
$132$	0	0
$133$	7.35898e26	2.08292
$134$	0	0
$135$	1.10407e24	0.00259313
$136$	0	0
$137$	1.17345e26	0.229328	0.114664	−	0.993404i	$-0.463421\pi$
$137$	1.17345e26	0.229328	0.114664	+	0.993404i	$0.463421\pi$
$138$	0	0
$139$	2.44153e26	0.398087	0.199043	−	0.979991i	$-0.436217\pi$
$139$	2.44153e26	0.398087	0.199043	+	0.979991i	$0.436217\pi$
$140$	0	0
$141$	5.84763e26	0.797518
$142$	0	0
$143$	−9.43846e26	−1.07945
$144$	0	0
$145$	−5.75549e23	−0.000553327 0
$146$	0	0
$147$	1.92592e27	1.56017
$148$	0	0
$149$	−1.38520e27	−0.947732	−0.473866	−	0.880597i	$-0.657142\pi$
$149$	−1.38520e27	−0.947732	−0.473866	+	0.880597i	$0.657142\pi$
$150$	0	0
$151$	1.52190e27	0.881404	0.440702	−	0.897653i	$-0.354730\pi$
$151$	1.52190e27	0.881404	0.440702	+	0.897653i	$0.354730\pi$
$152$	0	0
$153$	1.03173e27	0.506903
$154$	0	0
$155$	−4.04133e25	−0.0168802
$156$	0	0
$157$	−2.43800e27	−0.867536	−0.433768	−	0.901025i	$-0.642816\pi$
$157$	−2.43800e27	−0.867536	−0.433768	+	0.901025i	$0.642816\pi$
$158$	0	0
$159$	−1.65920e27	−0.504006
$160$	0	0
$161$	−7.09566e26	−0.184361
$162$	0	0
$163$	1.26921e27	0.282610	0.141305	−	0.989966i	$-0.454870\pi$
$163$	1.26921e27	0.282610	0.141305	+	0.989966i	$0.454870\pi$
$164$	0	0
$165$	−4.41650e25	−0.00844362
$166$	0	0
$167$	3.31297e27	0.544832	0.272416	−	0.962180i	$-0.412177\pi$
$167$	3.31297e27	0.544832	0.272416	+	0.962180i	$0.412177\pi$
$168$	0	0
$169$	−7.69992e25	−0.0109120
$170$	0	0
$171$	−2.94962e27	−0.360840
$172$	0	0
$173$	7.62802e26	0.0806929	0.0403465	−	0.999186i	$-0.487154\pi$
$173$	7.62802e26	0.0806929	0.0403465	+	0.999186i	$0.487154\pi$
$174$	0	0
$175$	2.09958e28	1.92379
$176$	0	0
$177$	−1.20679e28	−0.959324
$178$	0	0
$179$	2.76195e28	1.90789	0.953943	−	0.299988i	$-0.0969825\pi$
$179$	2.76195e28	1.90789	0.953943	+	0.299988i	$0.0969825\pi$
$180$	0	0
$181$	1.80447e27	0.108484	0.0542422	−	0.998528i	$-0.482726\pi$
$181$	1.80447e27	0.108484	0.0542422	+	0.998528i	$0.482726\pi$
$182$	0	0
$183$	8.11659e27	0.425339
$184$	0	0
$185$	−4.14716e26	−0.0189716
$186$	0	0
$187$	−4.12714e28	−1.65055
$188$	0	0
$189$	−1.05761e28	−0.370300
$190$	0	0
$191$	1.72125e28	0.528357	0.264179	−	0.964474i	$-0.414899\pi$
$191$	1.72125e28	0.528357	0.264179	+	0.964474i	$0.414899\pi$
$192$	0	0
$193$	1.32446e27	0.0356922	0.0178461	−	0.999841i	$-0.494319\pi$
$193$	1.32446e27	0.0356922	0.0178461	+	0.999841i	$0.494319\pi$
$194$	0	0
$195$	3.26584e26	0.00773682
$196$	0	0
$197$	4.57065e28	0.953126	0.476563	−	0.879140i	$-0.341882\pi$
$197$	4.57065e28	0.953126	0.476563	+	0.879140i	$0.341882\pi$
$198$	0	0
$199$	2.31782e28	0.426007	0.213003	−	0.977051i	$-0.431675\pi$
$199$	2.31782e28	0.426007	0.213003	+	0.977051i	$0.431675\pi$
$200$	0	0
$201$	4.15161e28	0.673388
$202$	0	0
$203$	5.51331e27	0.0790154
$204$	0	0
$205$	−7.26217e26	−0.00920753
$206$	0	0
$207$	2.84408e27	0.0319384
$208$	0	0
$209$	1.17991e29	1.17495
$210$	0	0
$211$	8.05735e28	0.712298	0.356149	−	0.934429i	$-0.384090\pi$
$211$	8.05735e28	0.712298	0.356149	+	0.934429i	$0.384090\pi$
$212$	0	0
$213$	9.37545e28	0.736626
$214$	0	0
$215$	−1.76104e27	−0.0123108
$216$	0	0
$217$	3.87128e29	2.41051
$218$	0	0
$219$	−6.23133e28	−0.345962
$220$	0	0
$221$	3.05188e29	1.51239
$222$	0	0
$223$	1.19175e29	0.527686	0.263843	−	0.964566i	$-0.415010\pi$
$223$	1.19175e29	0.527686	0.263843	+	0.964566i	$0.415010\pi$
$224$	0	0
$225$	−8.41553e28	−0.333273
$226$	0	0
$227$	1.29839e29	0.460342	0.230171	−	0.973150i	$-0.426071\pi$
$227$	1.29839e29	0.460342	0.230171	+	0.973150i	$0.426071\pi$
$228$	0	0
$229$	2.74178e29	0.871143	0.435571	−	0.900154i	$-0.356546\pi$
$229$	2.74178e29	0.871143	0.435571	+	0.900154i	$0.356546\pi$
$230$	0	0
$231$	4.23066e29	1.20575
$232$	0	0
$233$	−5.02468e29	−1.28576	−0.642879	−	0.765968i	$-0.722260\pi$
$233$	−5.02468e29	−1.28576	−0.642879	+	0.765968i	$0.722260\pi$
$234$	0	0
$235$	8.09385e27	0.0186126
$236$	0	0
$237$	2.12612e28	0.0439779
$238$	0	0
$239$	−5.35599e29	−0.997392	−0.498696	−	0.866777i	$-0.666188\pi$
$239$	−5.35599e29	−0.997392	−0.498696	+	0.866777i	$0.666188\pi$
$240$	0	0
$241$	8.02717e29	1.34694	0.673472	−	0.739213i	$-0.264802\pi$
$241$	8.02717e29	1.34694	0.673472	+	0.739213i	$0.264802\pi$
$242$	0	0
$243$	4.23912e28	0.0641500
$244$	0	0
$245$	2.66572e28	0.0364115
$246$	0	0
$247$	−8.72501e29	−1.07660
$248$	0	0
$249$	3.41375e29	0.380837
$250$	0	0
$251$	1.63973e30	1.65520	0.827601	−	0.561316i	$-0.189705\pi$
$251$	1.63973e30	1.65520	0.827601	+	0.561316i	$0.189705\pi$
$252$	0	0
$253$	−1.13769e29	−0.103996
$254$	0	0
$255$	1.42805e28	0.0118302
$256$	0	0
$257$	6.53341e29	0.490881	0.245440	−	0.969412i	$-0.421067\pi$
$257$	6.53341e29	0.490881	0.245440	+	0.969412i	$0.421067\pi$
$258$	0	0
$259$	3.97266e30	2.70915
$260$	0	0
$261$	−2.20984e28	−0.0136885
$262$	0	0
$263$	1.67243e30	0.941674	0.470837	−	0.882220i	$-0.343952\pi$
$263$	1.67243e30	0.941674	0.470837	+	0.882220i	$0.343952\pi$
$264$	0	0
$265$	−2.29655e28	−0.0117626
$266$	0	0
$267$	1.09393e30	0.510036
$268$	0	0
$269$	1.95781e30	0.831510	0.415755	−	0.909477i	$-0.363517\pi$
$269$	1.95781e30	0.831510	0.415755	+	0.909477i	$0.363517\pi$
$270$	0	0
$271$	−1.89742e30	−0.734595	−0.367298	−	0.930103i	$-0.619717\pi$
$271$	−1.89742e30	−0.734595	−0.367298	+	0.930103i	$0.619717\pi$
$272$	0	0
$273$	−3.12842e30	−1.10482
$274$	0	0
$275$	3.36638e30	1.08519
$276$	0	0
$277$	−3.76797e30	−1.10946	−0.554728	−	0.832032i	$-0.687178\pi$
$277$	−3.76797e30	−1.10946	−0.554728	+	0.832032i	$0.687178\pi$
$278$	0	0
$279$	−1.55169e30	−0.417591
$280$	0	0
$281$	−2.78869e30	−0.686390	−0.343195	−	0.939264i	$-0.611509\pi$
$281$	−2.78869e30	−0.686390	−0.343195	+	0.939264i	$0.611509\pi$
$282$	0	0
$283$	−4.53191e30	−1.02082	−0.510412	−	0.859930i	$-0.670507\pi$
$283$	−4.53191e30	−1.02082	−0.510412	+	0.859930i	$0.670507\pi$
$284$	0	0
$285$	−4.08265e28	−0.00842135
$286$	0	0
$287$	6.95659e30	1.31484
$288$	0	0
$289$	7.57425e30	1.31255
$290$	0	0
$291$	3.61181e30	0.574198
$292$	0	0
$293$	−1.99702e30	−0.291432	−0.145716	−	0.989326i	$-0.546549\pi$
$293$	−1.99702e30	−0.291432	−0.145716	+	0.989326i	$0.546549\pi$
$294$	0	0
$295$	−1.67035e29	−0.0223888
$296$	0	0
$297$	−1.69573e30	−0.208882
$298$	0	0
$299$	8.41281e29	0.0952909
$300$	0	0
$301$	1.68693e31	1.75799
$302$	0	0
$303$	6.29508e30	0.603904
$304$	0	0
$305$	1.12344e29	0.00992662
$306$	0	0
$307$	−2.21254e31	−1.80161	−0.900807	−	0.434221i	$-0.857024\pi$
$307$	−2.21254e31	−1.80161	−0.900807	+	0.434221i	$0.857024\pi$
$308$	0	0
$309$	−9.21649e30	−0.691966
$310$	0	0
$311$	−2.37679e31	−1.64621	−0.823106	−	0.567888i	$-0.807761\pi$
$311$	−2.37679e31	−1.64621	−0.823106	+	0.567888i	$0.807761\pi$
$312$	0	0
$313$	1.51520e31	0.968648	0.484324	−	0.874889i	$-0.339066\pi$
$313$	1.51520e31	0.968648	0.484324	+	0.874889i	$0.339066\pi$
$314$	0	0
$315$	−1.46387e29	−0.00864212
$316$	0	0
$317$	−8.66635e30	−0.472711	−0.236355	−	0.971667i	$-0.575953\pi$
$317$	−8.66635e30	−0.472711	−0.236355	+	0.971667i	$0.575953\pi$
$318$	0	0
$319$	8.83981e29	0.0445717
$320$	0	0
$321$	1.43838e31	0.670748
$322$	0	0
$323$	−3.81517e31	−1.64620
$324$	0	0
$325$	−2.48932e31	−0.994348
$326$	0	0
$327$	−8.87150e30	−0.328209
$328$	0	0
$329$	−7.75328e31	−2.65789
$330$	0	0
$331$	−2.48490e31	−0.789695	−0.394848	−	0.918747i	$-0.629203\pi$
$331$	−2.48490e31	−0.789695	−0.394848	+	0.918747i	$0.629203\pi$
$332$	0	0
$333$	−1.59232e31	−0.469329
$334$	0	0
$335$	5.74635e29	0.0157156
$336$	0	0
$337$	2.32202e31	0.589511	0.294755	−	0.955573i	$-0.404762\pi$
$337$	2.32202e31	0.589511	0.294755	+	0.955573i	$0.404762\pi$
$338$	0	0
$339$	4.53218e31	1.06859
$340$	0	0
$341$	6.20706e31	1.35974
$342$	0	0
$343$	−1.60860e32	−3.27545
$344$	0	0
$345$	3.93657e28	0.000745382 0
$346$	0	0
$347$	4.72552e31	0.832399	0.416200	−	0.909273i	$-0.363362\pi$
$347$	4.72552e31	0.832399	0.416200	+	0.909273i	$0.363362\pi$
$348$	0	0
$349$	4.81267e31	0.788983	0.394492	−	0.918900i	$-0.370921\pi$
$349$	4.81267e31	0.788983	0.394492	+	0.918900i	$0.370921\pi$
$350$	0	0
$351$	1.25393e31	0.191397
$352$	0	0
$353$	−1.08011e32	−1.53562	−0.767811	−	0.640676i	$-0.778654\pi$
$353$	−1.08011e32	−1.53562	−0.767811	+	0.640676i	$0.778654\pi$
$354$	0	0
$355$	1.29768e30	0.0171915
$356$	0	0
$357$	−1.36796e32	−1.68935
$358$	0	0
$359$	6.79068e31	0.782047	0.391023	−	0.920381i	$-0.372121\pi$
$359$	6.79068e31	0.782047	0.391023	+	0.920381i	$0.372121\pi$
$360$	0	0
$361$	1.59954e31	0.171852
$362$	0	0
$363$	1.02526e31	0.102802
$364$	0	0
$365$	−8.62494e29	−0.00807412
$366$	0	0
$367$	−1.08848e32	−0.951688	−0.475844	−	0.879530i	$-0.657857\pi$
$367$	−1.08848e32	−0.951688	−0.475844	+	0.879530i	$0.657857\pi$
$368$	0	0
$369$	−2.78834e31	−0.227780
$370$	0	0
$371$	2.19991e32	1.67970
$372$	0	0
$373$	−1.87013e31	−0.133509	−0.0667545	−	0.997769i	$-0.521264\pi$
$373$	−1.87013e31	−0.133509	−0.0667545	+	0.997769i	$0.521264\pi$
$374$	0	0
$375$	−2.32984e30	−0.0155574
$376$	0	0
$377$	−6.53673e30	−0.0408408
$378$	0	0
$379$	1.62852e32	0.952365	0.476182	−	0.879347i	$-0.342020\pi$
$379$	1.62852e32	0.952365	0.476182	+	0.879347i	$0.342020\pi$
$380$	0	0
$381$	−1.47314e32	−0.806643
$382$	0	0
$383$	−2.06758e32	−1.06042	−0.530209	−	0.847867i	$-0.677887\pi$
$383$	−2.06758e32	−1.06042	−0.530209	+	0.847867i	$0.677887\pi$
$384$	0	0
$385$	5.85577e30	0.0281400
$386$	0	0
$387$	−6.76157e31	−0.304551
$388$	0	0
$389$	−4.09258e32	−1.72833	−0.864166	−	0.503206i	$-0.832154\pi$
$389$	−4.09258e32	−1.72833	−0.864166	+	0.503206i	$0.832154\pi$
$390$	0	0
$391$	3.67866e31	0.145707
$392$	0	0
$393$	1.90964e32	0.709652
$394$	0	0
$395$	2.94283e29	0.00102636
$396$	0	0
$397$	−1.49338e31	−0.0488976	−0.0244488	−	0.999701i	$-0.507783\pi$
$397$	−1.49338e31	−0.0488976	−0.0244488	+	0.999701i	$0.507783\pi$
$398$	0	0
$399$	3.91086e32	1.20257
$400$	0	0
$401$	−2.82574e32	−0.816257	−0.408129	−	0.912924i	$-0.633818\pi$
$401$	−2.82574e32	−0.816257	−0.408129	+	0.912924i	$0.633818\pi$
$402$	0	0
$403$	−4.58990e32	−1.24592
$404$	0	0
$405$	5.86747e29	0.00149714
$406$	0	0
$407$	6.36959e32	1.52820
$408$	0	0
$409$	5.98603e32	1.35082	0.675410	−	0.737443i	$-0.263967\pi$
$409$	5.98603e32	1.35082	0.675410	+	0.737443i	$0.263967\pi$
$410$	0	0
$411$	6.23620e31	0.132403
$412$	0	0
$413$	1.60006e33	3.19714
$414$	0	0
$415$	4.72506e30	0.00888804
$416$	0	0
$417$	1.29753e32	0.229835
$418$	0	0
$419$	−6.61704e32	−1.10405	−0.552025	−	0.833828i	$-0.686145\pi$
$419$	−6.61704e32	−1.10405	−0.552025	+	0.833828i	$0.686145\pi$
$420$	0	0
$421$	7.21760e32	1.13466	0.567332	−	0.823489i	$-0.307976\pi$
$421$	7.21760e32	1.13466	0.567332	+	0.823489i	$0.307976\pi$
$422$	0	0
$423$	3.10767e32	0.460447
$424$	0	0
$425$	−1.08850e33	−1.52043
$426$	0	0
$427$	−1.07617e33	−1.41753
$428$	0	0
$429$	−5.01599e32	−0.623218
$430$	0	0
$431$	−6.95497e32	−0.815322	−0.407661	−	0.913133i	$-0.633656\pi$
$431$	−6.95497e32	−0.815322	−0.407661	+	0.913133i	$0.633656\pi$
$432$	0	0
$433$	7.81526e32	0.864659	0.432330	−	0.901716i	$-0.357692\pi$
$433$	7.81526e32	0.864659	0.432330	+	0.901716i	$0.357692\pi$
$434$	0	0
$435$	−3.05870e29	−0.000319464 0
$436$	0	0
$437$	−1.05169e32	−0.103722
$438$	0	0
$439$	2.01338e33	1.87551	0.937756	−	0.347294i	$-0.112899\pi$
$439$	2.01338e33	1.87551	0.937756	+	0.347294i	$0.112899\pi$
$440$	0	0
$441$	1.02351e33	0.900765
$442$	0	0
$443$	7.26129e32	0.603903	0.301952	−	0.953323i	$-0.402362\pi$
$443$	7.26129e32	0.603903	0.301952	+	0.953323i	$0.402362\pi$
$444$	0	0
$445$	1.51414e31	0.0119033
$446$	0	0
$447$	−7.36154e32	−0.547173
$448$	0	0
$449$	8.82996e31	0.0620697	0.0310348	−	0.999518i	$-0.490120\pi$
$449$	8.82996e31	0.0620697	0.0310348	+	0.999518i	$0.490120\pi$
$450$	0	0
$451$	1.11539e33	0.741687
$452$	0	0
$453$	8.08802e32	0.508879
$454$	0	0
$455$	−4.33013e31	−0.0257845
$456$	0	0
$457$	2.69756e33	1.52061	0.760305	−	0.649566i	$-0.225049\pi$
$457$	2.69756e33	1.52061	0.760305	+	0.649566i	$0.225049\pi$
$458$	0	0
$459$	5.48305e32	0.292660
$460$	0	0
$461$	3.50635e33	1.77253	0.886265	−	0.463178i	$-0.153291\pi$
$461$	3.50635e33	1.77253	0.886265	+	0.463178i	$0.153291\pi$
$462$	0	0
$463$	2.05859e33	0.985844	0.492922	−	0.870074i	$-0.335929\pi$
$463$	2.05859e33	0.985844	0.492922	+	0.870074i	$0.335929\pi$
$464$	0	0
$465$	−2.14773e31	−0.00974581
$466$	0	0
$467$	−3.73179e32	−0.160493	−0.0802465	−	0.996775i	$-0.525571\pi$
$467$	−3.73179e32	−0.160493	−0.0802465	+	0.996775i	$0.525571\pi$
$468$	0	0
$469$	−5.50455e33	−2.24420
$470$	0	0
$471$	−1.29565e33	−0.500872
$472$	0	0
$473$	2.70476e33	0.991665
$474$	0	0
$475$	3.11191e33	1.08232
$476$	0	0
$477$	−8.81769e32	−0.290988
$478$	0	0
$479$	6.00669e33	1.88123	0.940614	−	0.339477i	$-0.110250\pi$
$479$	6.00669e33	1.88123	0.940614	+	0.339477i	$0.110250\pi$
$480$	0	0
$481$	−4.71009e33	−1.40028
$482$	0	0
$483$	−3.77092e32	−0.106441
$484$	0	0
$485$	4.99920e31	0.0134007
$486$	0	0
$487$	−4.60284e33	−1.17196	−0.585981	−	0.810325i	$-0.699291\pi$
$487$	−4.60284e33	−1.17196	−0.585981	+	0.810325i	$0.699291\pi$
$488$	0	0
$489$	6.74509e32	0.163165
$490$	0	0
$491$	−2.42260e33	−0.556882	−0.278441	−	0.960453i	$-0.589818\pi$
$491$	−2.42260e33	−0.556882	−0.278441	+	0.960453i	$0.589818\pi$
$492$	0	0
$493$	−2.85831e32	−0.0624485
$494$	0	0
$495$	−2.34711e31	−0.00487492
$496$	0	0
$497$	−1.24308e34	−2.45495
$498$	0	0
$499$	4.61633e32	0.0867043	0.0433521	−	0.999060i	$-0.486196\pi$
$499$	4.61633e32	0.0867043	0.0433521	+	0.999060i	$0.486196\pi$
$500$	0	0
$501$	1.76065e33	0.314559
$502$	0	0
$503$	−7.49779e33	−1.27448	−0.637241	−	0.770665i	$-0.719924\pi$
$503$	−7.49779e33	−1.27448	−0.637241	+	0.770665i	$0.719924\pi$
$504$	0	0
$505$	8.71318e31	0.0140940
$506$	0	0
$507$	−4.09205e31	−0.00630002
$508$	0	0
$509$	−5.38724e33	−0.789576	−0.394788	−	0.918772i	$-0.629182\pi$
$509$	−5.38724e33	−0.789576	−0.394788	+	0.918772i	$0.629182\pi$
$510$	0	0
$511$	8.26202e33	1.15299
$512$	0	0
$513$	−1.56755e33	−0.208331
$514$	0	0
$515$	−1.27568e32	−0.0161492
$516$	0	0
$517$	−1.24313e34	−1.49929
$518$	0	0
$519$	4.05384e32	0.0465881
$520$	0	0
$521$	1.02938e34	1.12746	0.563732	−	0.825958i	$-0.309365\pi$
$521$	1.02938e34	1.12746	0.563732	+	0.825958i	$0.309365\pi$
$522$	0	0
$523$	1.59277e34	1.66297	0.831483	−	0.555551i	$-0.187492\pi$
$523$	1.59277e34	1.66297	0.831483	+	0.555551i	$0.187492\pi$
$524$	0	0
$525$	1.11580e34	1.11070
$526$	0	0
$527$	−2.00702e34	−1.90510
$528$	0	0
$529$	−1.09444e34	−0.990819
$530$	0	0
$531$	−6.41337e33	−0.553866
$532$	0	0
$533$	−8.24793e33	−0.679602
$534$	0	0
$535$	1.99089e32	0.0156540
$536$	0	0
$537$	1.46781e34	1.10152
$538$	0	0
$539$	−4.09426e34	−2.93303
$540$	0	0
$541$	1.94327e34	1.32913	0.664564	−	0.747231i	$-0.268617\pi$
$541$	1.94327e34	1.32913	0.664564	+	0.747231i	$0.268617\pi$
$542$	0	0
$543$	9.58969e32	0.0626335
$544$	0	0
$545$	−1.22793e32	−0.00765979
$546$	0	0
$547$	−5.73576e33	−0.341782	−0.170891	−	0.985290i	$-0.554665\pi$
$547$	−5.73576e33	−0.341782	−0.170891	+	0.985290i	$0.554665\pi$
$548$	0	0
$549$	4.31349e33	0.245569
$550$	0	0
$551$	8.17161e32	0.0444542
$552$	0	0
$553$	−2.81900e33	−0.146565
$554$	0	0
$555$	−2.20397e32	−0.0109533
$556$	0	0
$557$	−3.12346e34	−1.48404	−0.742020	−	0.670377i	$-0.766132\pi$
$557$	−3.12346e34	−1.48404	−0.742020	+	0.670377i	$0.766132\pi$
$558$	0	0
$559$	−2.00008e34	−0.908655
$560$	0	0
$561$	−2.19333e34	−0.952947
$562$	0	0
$563$	8.06088e33	0.334987	0.167493	−	0.985873i	$-0.446433\pi$
$563$	8.06088e33	0.334987	0.167493	+	0.985873i	$0.446433\pi$
$564$	0	0
$565$	6.27311e32	0.0249389
$566$	0	0
$567$	−5.62058e33	−0.213793
$568$	0	0
$569$	5.27095e34	1.91860	0.959302	−	0.282382i	$-0.0911247\pi$
$569$	5.27095e34	1.91860	0.959302	+	0.282382i	$0.0911247\pi$
$570$	0	0
$571$	−1.86409e34	−0.649405	−0.324702	−	0.945816i	$-0.605264\pi$
$571$	−1.86409e34	−0.649405	−0.324702	+	0.945816i	$0.605264\pi$
$572$	0	0
$573$	9.14744e33	0.305047
$574$	0	0
$575$	−3.00056e33	−0.0957977
$576$	0	0
$577$	−2.14156e34	−0.654685	−0.327343	−	0.944906i	$-0.606153\pi$
$577$	−2.14156e34	−0.654685	−0.327343	+	0.944906i	$0.606153\pi$
$578$	0	0
$579$	7.03874e32	0.0206069
$580$	0	0
$581$	−4.52624e34	−1.26922
$582$	0	0
$583$	3.52725e34	0.947501
$584$	0	0
$585$	1.73560e32	0.00446686
$586$	0	0
$587$	1.38546e34	0.341679	0.170840	−	0.985299i	$-0.445352\pi$
$587$	1.38546e34	0.341679	0.170840	+	0.985299i	$0.445352\pi$
$588$	0	0
$589$	5.73787e34	1.35615
$590$	0	0
$591$	2.42903e34	0.550288
$592$	0	0
$593$	7.47136e33	0.162262	0.0811309	−	0.996703i	$-0.474147\pi$
$593$	7.47136e33	0.162262	0.0811309	+	0.996703i	$0.474147\pi$
$594$	0	0
$595$	−1.89343e33	−0.0394264
$596$	0	0
$597$	1.23178e34	0.245955
$598$	0	0
$599$	−2.63388e34	−0.504384	−0.252192	−	0.967677i	$-0.581151\pi$
$599$	−2.63388e34	−0.504384	−0.252192	+	0.967677i	$0.581151\pi$
$600$	0	0
$601$	−3.39470e34	−0.623549	−0.311775	−	0.950156i	$-0.600923\pi$
$601$	−3.39470e34	−0.623549	−0.311775	+	0.950156i	$0.600923\pi$
$602$	0	0
$603$	2.20633e34	0.388781
$604$	0	0
$605$	1.41909e32	0.00239920
$606$	0	0
$607$	2.88827e34	0.468574	0.234287	−	0.972167i	$-0.424724\pi$
$607$	2.88827e34	0.468574	0.234287	+	0.972167i	$0.424724\pi$
$608$	0	0
$609$	2.93000e33	0.0456196
$610$	0	0
$611$	9.19251e34	1.37379
$612$	0	0
$613$	−7.43917e34	−1.06726	−0.533628	−	0.845719i	$-0.679172\pi$
$613$	−7.43917e34	−1.06726	−0.533628	+	0.845719i	$0.679172\pi$
$614$	0	0
$615$	−3.85942e32	−0.00531597
$616$	0	0
$617$	−6.95737e34	−0.920196	−0.460098	−	0.887868i	$-0.652186\pi$
$617$	−6.95737e34	−0.920196	−0.460098	+	0.887868i	$0.652186\pi$
$618$	0	0
$619$	4.41886e34	0.561276	0.280638	−	0.959814i	$-0.409454\pi$
$619$	4.41886e34	0.561276	0.280638	+	0.959814i	$0.409454\pi$
$620$	0	0
$621$	1.51146e33	0.0184396
$622$	0	0
$623$	−1.45043e35	−1.69980
$624$	0	0
$625$	8.87695e34	0.999455
$626$	0	0
$627$	6.27052e34	0.678358
$628$	0	0
$629$	−2.05957e35	−2.14114
$630$	0	0
$631$	3.82744e33	0.0382421	0.0191210	−	0.999817i	$-0.493913\pi$
$631$	3.82744e33	0.0382421	0.0191210	+	0.999817i	$0.493913\pi$
$632$	0	0
$633$	4.28201e34	0.411245
$634$	0	0
$635$	−2.03901e33	−0.0188255
$636$	0	0
$637$	3.02756e35	2.68751
$638$	0	0
$639$	4.98250e34	0.425291
$640$	0	0
$641$	4.59196e33	0.0376940	0.0188470	−	0.999822i	$-0.494000\pi$
$641$	4.59196e33	0.0376940	0.0188470	+	0.999822i	$0.494000\pi$
$642$	0	0
$643$	1.56258e35	1.23369	0.616845	−	0.787085i	$-0.288411\pi$
$643$	1.56258e35	1.23369	0.616845	+	0.787085i	$0.288411\pi$
$644$	0	0
$645$	−9.35886e32	−0.00710766
$646$	0	0
$647$	1.87682e35	1.37126	0.685628	−	0.727952i	$-0.259528\pi$
$647$	1.87682e35	1.37126	0.685628	+	0.727952i	$0.259528\pi$
$648$	0	0
$649$	2.56548e35	1.80347
$650$	0	0
$651$	2.05736e35	1.39171
$652$	0	0
$653$	−1.56012e35	−1.01565	−0.507824	−	0.861461i	$-0.669550\pi$
$653$	−1.56012e35	−1.01565	−0.507824	+	0.861461i	$0.669550\pi$
$654$	0	0
$655$	2.64319e33	0.0165620
$656$	0	0
$657$	−3.31158e34	−0.199742
$658$	0	0
$659$	−1.20697e35	−0.700857	−0.350428	−	0.936590i	$-0.613964\pi$
$659$	−1.20697e35	−0.700857	−0.350428	+	0.936590i	$0.613964\pi$
$660$	0	0
$661$	2.17565e34	0.121639	0.0608193	−	0.998149i	$-0.480629\pi$
$661$	2.17565e34	0.121639	0.0608193	+	0.998149i	$0.480629\pi$
$662$	0	0
$663$	1.62189e35	0.873178
$664$	0	0
$665$	5.41313e33	0.0280658
$666$	0	0
$667$	−7.87922e32	−0.00393469
$668$	0	0
$669$	6.33346e34	0.304659
$670$	0	0
$671$	−1.72548e35	−0.799611
$672$	0	0
$673$	4.39078e35	1.96044	0.980220	−	0.197911i	$-0.0634156\pi$
$673$	4.39078e35	1.96044	0.980220	+	0.197911i	$0.0634156\pi$
$674$	0	0
$675$	−4.47236e34	−0.192415
$676$	0	0
$677$	2.87287e35	1.19113	0.595563	−	0.803309i	$-0.296929\pi$
$677$	2.87287e35	1.19113	0.595563	+	0.803309i	$0.296929\pi$
$678$	0	0
$679$	−4.78884e35	−1.91363
$680$	0	0
$681$	6.90016e34	0.265779
$682$	0	0
$683$	−1.10036e35	−0.408579	−0.204289	−	0.978911i	$-0.565488\pi$
$683$	−1.10036e35	−0.408579	−0.204289	+	0.978911i	$0.565488\pi$
$684$	0	0
$685$	8.63169e32	0.00309004
$686$	0	0
$687$	1.45710e35	0.502955
$688$	0	0
$689$	−2.60828e35	−0.868188
$690$	0	0
$691$	−4.76758e35	−1.53046	−0.765230	−	0.643757i	$-0.777375\pi$
$691$	−4.76758e35	−1.53046	−0.765230	+	0.643757i	$0.777375\pi$
$692$	0	0
$693$	2.24835e35	0.696142
$694$	0	0
$695$	1.79595e33	0.00536393
$696$	0	0
$697$	−3.60656e35	−1.03916
$698$	0	0
$699$	−2.67032e35	−0.742333
$700$	0	0
$701$	−4.35093e35	−1.16710	−0.583548	−	0.812079i	$-0.698336\pi$
$701$	−4.35093e35	−1.16710	−0.583548	+	0.812079i	$0.698336\pi$
$702$	0	0
$703$	5.88812e35	1.52417
$704$	0	0
$705$	4.30141e33	0.0107460
$706$	0	0
$707$	−8.34655e35	−2.01263
$708$	0	0
$709$	−2.34082e35	−0.544865	−0.272433	−	0.962175i	$-0.587828\pi$
$709$	−2.34082e35	−0.544865	−0.272433	+	0.962175i	$0.587828\pi$
$710$	0	0
$711$	1.12991e34	0.0253906
$712$	0	0
$713$	−5.53255e34	−0.120035
$714$	0	0
$715$	−6.94276e33	−0.0145448
$716$	0	0
$717$	−2.84639e35	−0.575844
$718$	0	0
$719$	4.98165e35	0.973335	0.486667	−	0.873587i	$-0.338212\pi$
$719$	4.98165e35	0.973335	0.486667	+	0.873587i	$0.338212\pi$
$720$	0	0
$721$	1.22200e36	2.30611
$722$	0	0
$723$	4.26597e35	0.777658
$724$	0	0
$725$	2.33143e34	0.0410580
$726$	0	0
$727$	5.39232e35	0.917478	0.458739	−	0.888571i	$-0.348301\pi$
$727$	5.39232e35	0.917478	0.458739	+	0.888571i	$0.348301\pi$
$728$	0	0
$729$	2.25284e34	0.0370370
$730$	0	0
$731$	−8.74570e35	−1.38940
$732$	0	0
$733$	−5.94787e35	−0.913192	−0.456596	−	0.889674i	$-0.650931\pi$
$733$	−5.94787e35	−0.913192	−0.456596	+	0.889674i	$0.650931\pi$
$734$	0	0
$735$	1.41667e34	0.0210222
$736$	0	0
$737$	−8.82578e35	−1.26593
$738$	0	0
$739$	−3.20291e34	−0.0444108	−0.0222054	−	0.999753i	$-0.507069\pi$
$739$	−3.20291e34	−0.0444108	−0.0222054	+	0.999753i	$0.507069\pi$
$740$	0	0
$741$	−4.63683e35	−0.621575
$742$	0	0
$743$	4.71096e34	0.0610589	0.0305294	−	0.999534i	$-0.490281\pi$
$743$	4.71096e34	0.0610589	0.0305294	+	0.999534i	$0.490281\pi$
$744$	0	0
$745$	−1.01893e34	−0.0127700
$746$	0	0
$747$	1.81420e35	0.219876
$748$	0	0
$749$	−1.90712e36	−2.23540
$750$	0	0
$751$	−1.19641e36	−1.35638	−0.678191	−	0.734886i	$-0.737236\pi$
$751$	−1.19641e36	−1.35638	−0.678191	+	0.734886i	$0.737236\pi$
$752$	0	0
$753$	8.71420e35	0.955632
$754$	0	0
$755$	1.11948e34	0.0118763
$756$	0	0
$757$	−1.17527e36	−1.20626	−0.603130	−	0.797643i	$-0.706080\pi$
$757$	−1.17527e36	−1.20626	−0.603130	+	0.797643i	$0.706080\pi$
$758$	0	0
$759$	−6.04615e34	−0.0600422
$760$	0	0
$761$	−1.71081e36	−1.64397	−0.821983	−	0.569512i	$-0.807132\pi$
$761$	−1.71081e36	−1.64397	−0.821983	+	0.569512i	$0.807132\pi$
$762$	0	0
$763$	1.17626e36	1.09382
$764$	0	0
$765$	7.58924e33	0.00683015
$766$	0	0
$767$	−1.89708e36	−1.65251
$768$	0	0
$769$	−5.93052e35	−0.500050	−0.250025	−	0.968239i	$-0.580439\pi$
$769$	−5.93052e35	−0.500050	−0.250025	+	0.968239i	$0.580439\pi$
$770$	0	0
$771$	3.47212e35	0.283410
$772$	0	0
$773$	2.40618e36	1.90145	0.950723	−	0.310041i	$-0.100343\pi$
$773$	2.40618e36	1.90145	0.950723	+	0.310041i	$0.100343\pi$
$774$	0	0
$775$	1.63706e36	1.25255
$776$	0	0
$777$	2.11123e36	1.56413
$778$	0	0
$779$	1.03108e36	0.739731
$780$	0	0
$781$	−1.99310e36	−1.38481
$782$	0	0
$783$	−1.17440e34	−0.00790304
$784$	0	0
$785$	−1.79335e34	−0.0116894
$786$	0	0
$787$	1.04017e36	0.656780	0.328390	−	0.944542i	$-0.393494\pi$
$787$	1.04017e36	0.656780	0.328390	+	0.944542i	$0.393494\pi$
$788$	0	0
$789$	8.88797e35	0.543676
$790$	0	0
$791$	−6.00915e36	−3.56129
$792$	0	0
$793$	1.27593e36	0.732678
$794$	0	0
$795$	−1.22048e34	−0.00679112
$796$	0	0
$797$	−1.54782e36	−0.834623	−0.417312	−	0.908763i	$-0.637028\pi$
$797$	−1.54782e36	−0.834623	−0.417312	+	0.908763i	$0.637028\pi$
$798$	0	0
$799$	4.01959e36	2.10062
$800$	0	0
$801$	5.81362e35	0.294469
$802$	0	0
$803$	1.32470e36	0.650389
$804$	0	0
$805$	−5.21944e33	−0.00248413
$806$	0	0
$807$	1.04046e36	0.480073
$808$	0	0
$809$	−1.50513e36	−0.673315	−0.336658	−	0.941627i	$-0.609296\pi$
$809$	−1.50513e36	−0.673315	−0.336658	+	0.941627i	$0.609296\pi$
$810$	0	0
$811$	−1.93633e35	−0.0839881	−0.0419941	−	0.999118i	$-0.513371\pi$
$811$	−1.93633e35	−0.0839881	−0.0419941	+	0.999118i	$0.513371\pi$
$812$	0	0
$813$	−1.00837e36	−0.424119
$814$	0	0
$815$	9.33606e33	0.00380797
$816$	0	0
$817$	2.50031e36	0.989050
$818$	0	0
$819$	−1.66257e36	−0.637869
$820$	0	0
$821$	−2.59194e36	−0.964574	−0.482287	−	0.876013i	$-0.660194\pi$
$821$	−2.59194e36	−0.964574	−0.482287	+	0.876013i	$0.660194\pi$
$822$	0	0
$823$	−3.39771e36	−1.22656	−0.613280	−	0.789866i	$-0.710150\pi$
$823$	−3.39771e36	−1.22656	−0.613280	+	0.789866i	$0.710150\pi$
$824$	0	0
$825$	1.78903e36	0.626533
$826$	0	0
$827$	−4.21290e36	−1.43140	−0.715702	−	0.698406i	$-0.753893\pi$
$827$	−4.21290e36	−1.43140	−0.715702	+	0.698406i	$0.753893\pi$
$828$	0	0
$829$	8.10835e35	0.267301	0.133651	−	0.991029i	$-0.457330\pi$
$829$	8.10835e35	0.267301	0.133651	+	0.991029i	$0.457330\pi$
$830$	0	0
$831$	−2.00245e36	−0.640544
$832$	0	0
$833$	1.32386e37	4.10940
$834$	0	0
$835$	2.43696e34	0.00734122
$836$	0	0
$837$	−8.24630e35	−0.241096
$838$	0	0
$839$	5.54659e36	1.57399	0.786994	−	0.616961i	$-0.211636\pi$
$839$	5.54659e36	1.57399	0.786994	+	0.616961i	$0.211636\pi$
$840$	0	0
$841$	−3.62424e36	−0.998314
$842$	0	0
$843$	−1.48202e36	−0.396287
$844$	0	0
$845$	−5.66392e32	−0.000147031 0
$846$	0	0
$847$	−1.35938e36	−0.342607
$848$	0	0
$849$	−2.40844e36	−0.589373
$850$	0	0
$851$	−5.67743e35	−0.134906
$852$	0	0
$853$	4.00324e36	0.923737	0.461868	−	0.886948i	$-0.347179\pi$
$853$	4.00324e36	0.923737	0.461868	+	0.886948i	$0.347179\pi$
$854$	0	0
$855$	−2.16969e34	−0.00486207
$856$	0	0
$857$	−2.63034e35	−0.0572469	−0.0286235	−	0.999590i	$-0.509112\pi$
$857$	−2.63034e35	−0.0572469	−0.0286235	+	0.999590i	$0.509112\pi$
$858$	0	0
$859$	9.93120e35	0.209936	0.104968	−	0.994476i	$-0.466526\pi$
$859$	9.93120e35	0.209936	0.104968	+	0.994476i	$0.466526\pi$
$860$	0	0
$861$	3.69702e36	0.759123
$862$	0	0
$863$	5.01460e36	1.00023	0.500117	−	0.865958i	$-0.333290\pi$
$863$	5.01460e36	1.00023	0.500117	+	0.865958i	$0.333290\pi$
$864$	0	0
$865$	5.61103e33	0.00108728
$866$	0	0
$867$	4.02527e36	0.757803
$868$	0	0
$869$	−4.51987e35	−0.0826758
$870$	0	0
$871$	6.52635e36	1.15996
$872$	0	0
$873$	1.91946e36	0.331514
$874$	0	0
$875$	3.08910e35	0.0518480
$876$	0	0
$877$	−1.05998e37	−1.72902	−0.864512	−	0.502612i	$-0.832372\pi$
$877$	−1.05998e37	−1.72902	−0.864512	+	0.502612i	$0.832372\pi$
$878$	0	0
$879$	−1.06130e36	−0.168258
$880$	0	0
$881$	1.08375e37	1.67005	0.835027	−	0.550209i	$-0.185452\pi$
$881$	1.08375e37	1.67005	0.835027	+	0.550209i	$0.185452\pi$
$882$	0	0
$883$	9.53121e36	1.42771	0.713854	−	0.700294i	$-0.246948\pi$
$883$	9.53121e36	1.42771	0.713854	+	0.700294i	$0.246948\pi$
$884$	0	0
$885$	−8.87691e34	−0.0129262
$886$	0	0
$887$	1.76460e36	0.249805	0.124903	−	0.992169i	$-0.460138\pi$
$887$	1.76460e36	0.249805	0.124903	+	0.992169i	$0.460138\pi$
$888$	0	0
$889$	1.95321e37	2.68830
$890$	0	0
$891$	−9.01181e35	−0.120598
$892$	0	0
$893$	−1.14916e37	−1.49533
$894$	0	0
$895$	2.03164e35	0.0257074
$896$	0	0
$897$	4.47091e35	0.0550162
$898$	0	0
$899$	4.29878e35	0.0514457
$900$	0	0
$901$	−1.14052e37	−1.32752
$902$	0	0
$903$	8.96506e36	1.01498
$904$	0	0
$905$	1.32733e34	0.00146175
$906$	0	0
$907$	1.35924e37	1.45614	0.728072	−	0.685501i	$-0.240417\pi$
$907$	1.35924e37	1.45614	0.728072	+	0.685501i	$0.240417\pi$
$908$	0	0
$909$	3.34546e36	0.348664
$910$	0	0
$911$	−7.15807e35	−0.0725799	−0.0362899	−	0.999341i	$-0.511554\pi$
$911$	−7.15807e35	−0.0725799	−0.0362899	+	0.999341i	$0.511554\pi$
$912$	0	0
$913$	−7.25718e36	−0.715951
$914$	0	0
$915$	5.97041e34	0.00573114
$916$	0	0
$917$	−2.53197e37	−2.36506
$918$	0	0
$919$	−1.75818e37	−1.59816	−0.799078	−	0.601227i	$-0.794679\pi$
$919$	−1.75818e37	−1.59816	−0.799078	+	0.601227i	$0.794679\pi$
$920$	0	0
$921$	−1.17583e37	−1.04016
$922$	0	0
$923$	1.47383e37	1.26889
$924$	0	0
$925$	1.67993e37	1.40773
$926$	0	0
$927$	−4.89802e36	−0.399506
$928$	0	0
$929$	3.76047e36	0.298570	0.149285	−	0.988794i	$-0.452303\pi$
$929$	3.76047e36	0.298570	0.149285	+	0.988794i	$0.452303\pi$
$930$	0	0
$931$	−3.78478e37	−2.92529
$932$	0	0
$933$	−1.26312e37	−0.950441
$934$	0	0
$935$	−3.03585e35	−0.0222400
$936$	0	0
$937$	1.04117e37	0.742639	0.371319	−	0.928505i	$-0.378905\pi$
$937$	1.04117e37	0.742639	0.371319	+	0.928505i	$0.378905\pi$
$938$	0	0
$939$	8.05240e36	0.559249
$940$	0	0
$941$	−1.41555e36	−0.0957312	−0.0478656	−	0.998854i	$-0.515242\pi$
$941$	−1.41555e36	−0.0957312	−0.0478656	+	0.998854i	$0.515242\pi$
$942$	0	0
$943$	−9.94185e35	−0.0654744
$944$	0	0
$945$	−7.77960e34	−0.00498953
$946$	0	0
$947$	1.13648e36	0.0709884	0.0354942	−	0.999370i	$-0.488699\pi$
$947$	1.13648e36	0.0709884	0.0354942	+	0.999370i	$0.488699\pi$
$948$	0	0
$949$	−9.79568e36	−0.595946
$950$	0	0
$951$	−4.60565e36	−0.272920
$952$	0	0
$953$	2.29205e37	1.32301	0.661504	−	0.749942i	$-0.269918\pi$
$953$	2.29205e37	1.32301	0.661504	+	0.749942i	$0.269918\pi$
$954$	0	0
$955$	1.26612e35	0.00711923
$956$	0	0
$957$	4.69784e35	0.0257335
$958$	0	0
$959$	−8.26848e36	−0.441259
$960$	0	0
$961$	1.09520e37	0.569442
$962$	0	0
$963$	7.64412e36	0.387257
$964$	0	0
$965$	9.74250e33	0.000480927 0
$966$	0	0
$967$	−1.02269e37	−0.491939	−0.245970	−	0.969278i	$-0.579106\pi$
$967$	−1.02269e37	−0.491939	−0.245970	+	0.969278i	$0.579106\pi$
$968$	0	0
$969$	−2.02754e37	−0.950433
$970$	0	0
$971$	1.06266e37	0.485461	0.242730	−	0.970094i	$-0.421957\pi$
$971$	1.06266e37	0.485461	0.242730	+	0.970094i	$0.421957\pi$
$972$	0	0
$973$	−1.72038e37	−0.765972
$974$	0	0
$975$	−1.32293e37	−0.574087
$976$	0	0
$977$	1.71873e36	0.0726985	0.0363493	−	0.999339i	$-0.488427\pi$
$977$	1.71873e36	0.0726985	0.0363493	+	0.999339i	$0.488427\pi$
$978$	0	0
$979$	−2.32556e37	−0.958837
$980$	0	0
$981$	−4.71468e36	−0.189491
$982$	0	0
$983$	1.01989e37	0.399609	0.199804	−	0.979836i	$-0.435969\pi$
$983$	1.01989e37	0.399609	0.199804	+	0.979836i	$0.435969\pi$
$984$	0	0
$985$	3.36209e35	0.0128427
$986$	0	0
$987$	−4.12041e37	−1.53453
$988$	0	0
$989$	−2.41084e36	−0.0875418
$990$	0	0
$991$	−2.38247e37	−0.843541	−0.421771	−	0.906703i	$-0.638591\pi$
$991$	−2.38247e37	−0.843541	−0.421771	+	0.906703i	$0.638591\pi$
$992$	0	0
$993$	−1.32058e37	−0.455931
$994$	0	0
$995$	1.70495e35	0.00574013
$996$	0	0
$997$	−4.79713e37	−1.57504	−0.787521	−	0.616288i	$-0.788636\pi$
$997$	−4.79713e37	−1.57504	−0.787521	+	0.616288i	$0.788636\pi$
$998$	0	0
$999$	−8.46223e36	−0.270967

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.26.a.k.1.3		4
4.3	odd	2		24.26.a.d.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.26.a.d.1.3	✓	4	4.3	odd	2
48.26.a.k.1.3		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)

\(f(q)\)	\(=\)	\(q+531441. q^{3} +7.35582e6 q^{5} -7.04630e10 q^{7} +2.82430e11 q^{9} +O(q^{10})\)	\(q+531441. q^{3} +7.35582e6 q^{5} -7.04630e10 q^{7} +2.82430e11 q^{9} -1.12977e13 q^{11} +8.35429e13 q^{13} +3.90918e12 q^{15} +3.65306e15 q^{17} -1.04438e16 q^{19} -3.74469e16 q^{21} +1.00701e16 q^{23} -2.97969e17 q^{25} +1.50095e17 q^{27} -7.82440e16 q^{29} -5.49407e18 q^{31} -6.00409e18 q^{33} -5.18313e17 q^{35} -5.63793e19 q^{37} +4.43981e19 q^{39} -9.87269e19 q^{41} -2.39407e20 q^{43} +2.07750e18 q^{45} +1.10033e21 q^{47} +3.62396e21 q^{49} +1.94139e21 q^{51} -3.12209e21 q^{53} -8.31041e19 q^{55} -5.55024e21 q^{57} -2.27078e22 q^{59} +1.52728e22 q^{61} -1.99008e22 q^{63} +6.14526e20 q^{65} +7.81198e22 q^{67} +5.35164e21 q^{69} +1.76416e23 q^{71} -1.17253e23 q^{73} -1.58353e23 q^{75} +7.96073e23 q^{77} +4.00068e22 q^{79} +7.97664e22 q^{81} +6.42357e23 q^{83} +2.68713e22 q^{85} -4.15821e22 q^{87} +2.05843e24 q^{89} -5.88668e24 q^{91} -2.91977e24 q^{93} -7.68223e22 q^{95} +6.79625e24 q^{97} -3.19082e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2125764 q^{3} + 266436088 q^{5} - 484584576 q^{7} + 1129718145924 q^{9}+O(q^{10}) \) 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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