LMFDB - Embedded newform 48.28.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,28,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, 28, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 28 $
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:	$221.690675922$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{30001}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7500 $ x^2 - x - 7500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-86.1040$ 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)

$f(q)$	$=$	$q+1.59432e6 q^{3} -6.93920e8 q^{5} -3.81056e11 q^{7} +2.54187e12 q^{9} +O(q^{10})$	$q+1.59432e6 q^{3} -6.93920e8 q^{5} -3.81056e11 q^{7} +2.54187e12 q^{9} -1.26337e14 q^{11} -1.31606e15 q^{13} -1.10633e15 q^{15} -3.53536e16 q^{17} -8.65178e16 q^{19} -6.07526e17 q^{21} -5.10955e17 q^{23} -6.96906e18 q^{25} +4.05256e18 q^{27} -2.74961e19 q^{29} +9.77996e19 q^{31} -2.01422e20 q^{33} +2.64422e20 q^{35} -1.80293e21 q^{37} -2.09822e21 q^{39} -5.17966e21 q^{41} -8.74613e21 q^{43} -1.76385e21 q^{45} -2.51817e22 q^{47} +7.94910e22 q^{49} -5.63650e22 q^{51} +2.11957e23 q^{53} +8.76680e22 q^{55} -1.37937e23 q^{57} +7.00844e23 q^{59} +1.08505e23 q^{61} -9.68592e23 q^{63} +9.13240e23 q^{65} -2.46540e24 q^{67} -8.14627e23 q^{69} -1.29459e25 q^{71} +7.46465e24 q^{73} -1.11109e25 q^{75} +4.81415e25 q^{77} +4.94644e25 q^{79} +6.46108e24 q^{81} +5.54332e25 q^{83} +2.45325e25 q^{85} -4.38376e25 q^{87} -1.32973e26 q^{89} +5.01492e26 q^{91} +1.55924e26 q^{93} +6.00364e25 q^{95} -1.06222e27 q^{97} -3.21132e26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9}+O(q^{10}) $ 2 * q + 3188646 * q^3 - 1771946100 * q^5 - 369665199904 * q^7 + 5083731656658 * q^9	$ 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9} - 75762335668248 q^{11} - 103021079177588 q^{13} - 28\!\cdots\!00 q^{15}+ \cdots - 19\!\cdots\!92 q^{99}+O(q^{100}) $ 2 * q + 3188646 * q^3 - 1771946100 * q^5 - 369665199904 * q^7 + 5083731656658 * q^9 - 75762335668248 * q^11 - 103021079177588 * q^13 - 2825054421990300 * q^15 + 34691611579147908 * q^17 - 111620476558642216 * q^19 - 589365730506544992 * q^21 + 2892743075819387952 * q^23 - 13257495997920171250 * q^25 + 8105110306037952534 * q^27 + 29959552473322806972 * q^29 - 10367463257055494032 * q^31 - 120789634289608156104 * q^33 + 252142979584030920000 * q^35 - 3703660908819524869316 * q^37 - 164248876017649632924 * q^39 + 1481814928479478034772 * q^41 - 9778778623457380249240 * q^43 - 4504049241230841066900 * q^45 - 89494998518809740323520 * q^47 + 13908377820361409575890 * q^49 + 55309634247701830126284 * q^51 + 425671275824601813476748 * q^53 + 33146870181194328159600 * q^55 - 177959093048404133739768 * q^57 - 369418769643380130292824 * q^59 + 925133397428892745249420 * q^61 - 939639339558386331280416 * q^63 - 394447136075147294825400 * q^65 - 4151706674555213948132968 * q^67 + 4611966818869594057796496 * q^69 - 9760455855870867537668784 * q^71 + 28449956347640075157409588 * q^73 - 21136730791892081187813750 * q^75 + 48717589762471785388668288 * q^77 + 49553221898675501744231120 * q^79 + 12922163778453346597864482 * q^81 + 120473408203228735926347736 * q^83 - 50977978054175038499879400 * q^85 + 47765203577925437580019956 * q^87 + 205644052209697133056985556 * q^89 + 515309018103795393250147648 * q^91 - 16529085122378486411580336 * q^93 + 87097784513216888583651600 * q^95 - 1478339006400409594836427772 * q^97 - 192577692109510944264197592 * 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
$3$	1.59432e6	0.577350
$3$	1.59432e6	0.577350
$4$	0	0
$5$	−6.93920e8	−0.254223	−0.127111	−	0.991888i	$-0.540571\pi$
$5$	−6.93920e8	−0.254223	−0.127111	+	0.991888i	$0.540571\pi$
$6$	0	0
$7$	−3.81056e11	−1.48650	−0.743250	−	0.669014i	$-0.766717\pi$
$7$	−3.81056e11	−1.48650	−0.743250	+	0.669014i	$0.766717\pi$
$8$	0	0
$9$	2.54187e12	0.333333
$10$	0	0
$11$	−1.26337e14	−1.10339	−0.551697	−	0.834045i	$-0.686019\pi$
$11$	−1.26337e14	−1.10339	−0.551697	+	0.834045i	$0.686019\pi$
$12$	0	0
$13$	−1.31606e15	−1.20515	−0.602574	−	0.798063i	$-0.705858\pi$
$13$	−1.31606e15	−1.20515	−0.602574	+	0.798063i	$0.705858\pi$
$14$	0	0
$15$	−1.10633e15	−0.146776
$16$	0	0
$17$	−3.53536e16	−0.865711	−0.432855	−	0.901463i	$-0.642494\pi$
$17$	−3.53536e16	−0.865711	−0.432855	+	0.901463i	$0.642494\pi$
$18$	0	0
$19$	−8.65178e16	−0.471989	−0.235994	−	0.971754i	$-0.575835\pi$
$19$	−8.65178e16	−0.471989	−0.235994	+	0.971754i	$0.575835\pi$
$20$	0	0
$21$	−6.07526e17	−0.858231
$22$	0	0
$23$	−5.10955e17	−0.211376	−0.105688	−	0.994399i	$-0.533705\pi$
$23$	−5.10955e17	−0.211376	−0.105688	+	0.994399i	$0.533705\pi$
$24$	0	0
$25$	−6.96906e18	−0.935371
$26$	0	0
$27$	4.05256e18	0.192450
$28$	0	0
$29$	−2.74961e19	−0.497620	−0.248810	−	0.968552i	$-0.580039\pi$
$29$	−2.74961e19	−0.497620	−0.248810	+	0.968552i	$0.580039\pi$
$30$	0	0
$31$	9.77996e19	0.719373	0.359687	−	0.933073i	$-0.382884\pi$
$31$	9.77996e19	0.719373	0.359687	+	0.933073i	$0.382884\pi$
$32$	0	0
$33$	−2.01422e20	−0.637044
$34$	0	0
$35$	2.64422e20	0.377902
$36$	0	0
$37$	−1.80293e21	−1.21690	−0.608450	−	0.793592i	$-0.708208\pi$
$37$	−1.80293e21	−1.21690	−0.608450	+	0.793592i	$0.708208\pi$
$38$	0	0
$39$	−2.09822e21	−0.695793
$40$	0	0
$41$	−5.17966e21	−0.874419	−0.437209	−	0.899360i	$-0.644033\pi$
$41$	−5.17966e21	−0.874419	−0.437209	+	0.899360i	$0.644033\pi$
$42$	0	0
$43$	−8.74613e21	−0.776233	−0.388116	−	0.921610i	$-0.626874\pi$
$43$	−8.74613e21	−0.776233	−0.388116	+	0.921610i	$0.626874\pi$
$44$	0	0
$45$	−1.76385e21	−0.0847409
$46$	0	0
$47$	−2.51817e22	−0.672611	−0.336306	−	0.941753i	$-0.609177\pi$
$47$	−2.51817e22	−0.672611	−0.336306	+	0.941753i	$0.609177\pi$
$48$	0	0
$49$	7.94910e22	1.20968
$50$	0	0
$51$	−5.63650e22	−0.499818
$52$	0	0
$53$	2.11957e23	1.11821	0.559106	−	0.829096i	$-0.311144\pi$
$53$	2.11957e23	1.11821	0.559106	+	0.829096i	$0.311144\pi$
$54$	0	0
$55$	8.76680e22	0.280508
$56$	0	0
$57$	−1.37937e23	−0.272503
$58$	0	0
$59$	7.00844e23	0.869199	0.434600	−	0.900624i	$-0.356890\pi$
$59$	7.00844e23	0.869199	0.434600	+	0.900624i	$0.356890\pi$
$60$	0	0
$61$	1.08505e23	0.0858020	0.0429010	−	0.999079i	$-0.486340\pi$
$61$	1.08505e23	0.0858020	0.0429010	+	0.999079i	$0.486340\pi$
$62$	0	0
$63$	−9.68592e23	−0.495500
$64$	0	0
$65$	9.13240e23	0.306376
$66$	0	0
$67$	−2.46540e24	−0.549387	−0.274693	−	0.961532i	$-0.588576\pi$
$67$	−2.46540e24	−0.549387	−0.274693	+	0.961532i	$0.588576\pi$
$68$	0	0
$69$	−8.14627e23	−0.122038
$70$	0	0
$71$	−1.29459e25	−1.31870	−0.659349	−	0.751837i	$-0.729168\pi$
$71$	−1.29459e25	−1.31870	−0.659349	+	0.751837i	$0.729168\pi$
$72$	0	0
$73$	7.46465e24	0.522577	0.261289	−	0.965261i	$-0.415853\pi$
$73$	7.46465e24	0.522577	0.261289	+	0.965261i	$0.415853\pi$
$74$	0	0
$75$	−1.11109e25	−0.540037
$76$	0	0
$77$	4.81415e25	1.64019
$78$	0	0
$79$	4.94644e25	1.19214	0.596070	−	0.802933i	$-0.296728\pi$
$79$	4.94644e25	1.19214	0.596070	+	0.802933i	$0.296728\pi$
$80$	0	0
$81$	6.46108e24	0.111111
$82$	0	0
$83$	5.54332e25	0.685828	0.342914	−	0.939367i	$-0.388586\pi$
$83$	5.54332e25	0.685828	0.342914	+	0.939367i	$0.388586\pi$
$84$	0	0
$85$	2.45325e25	0.220083
$86$	0	0
$87$	−4.38376e25	−0.287301
$88$	0	0
$89$	−1.32973e26	−0.641206	−0.320603	−	0.947214i	$-0.603886\pi$
$89$	−1.32973e26	−0.641206	−0.320603	+	0.947214i	$0.603886\pi$
$90$	0	0
$91$	5.01492e26	1.79145
$92$	0	0
$93$	1.55924e26	0.415330
$94$	0	0
$95$	6.00364e25	0.119990
$96$	0	0
$97$	−1.06222e27	−1.60249	−0.801243	−	0.598339i	$-0.795827\pi$
$97$	−1.06222e27	−1.60249	−0.801243	+	0.598339i	$0.795827\pi$
$98$	0	0
$99$	−3.21132e26	−0.367798
$100$	0	0
$101$	1.74011e27	1.52138	0.760691	−	0.649114i	$-0.224860\pi$
$101$	1.74011e27	1.52138	0.760691	+	0.649114i	$0.224860\pi$
$102$	0	0
$103$	−4.99563e26	−0.335187	−0.167594	−	0.985856i	$-0.553600\pi$
$103$	−4.99563e26	−0.335187	−0.167594	+	0.985856i	$0.553600\pi$
$104$	0	0
$105$	4.21574e26	0.218182
$106$	0	0
$107$	1.30395e27	0.523096	0.261548	−	0.965190i	$-0.415767\pi$
$107$	1.30395e27	0.523096	0.261548	+	0.965190i	$0.415767\pi$
$108$	0	0
$109$	−5.74920e27	−1.79618	−0.898089	−	0.439814i	$-0.855044\pi$
$109$	−5.74920e27	−1.79618	−0.898089	+	0.439814i	$0.855044\pi$
$110$	0	0
$111$	−2.87445e27	−0.702578
$112$	0	0
$113$	8.29240e27	1.59265	0.796326	−	0.604867i	$-0.206774\pi$
$113$	8.29240e27	1.59265	0.796326	+	0.604867i	$0.206774\pi$
$114$	0	0
$115$	3.54562e26	0.0537367
$116$	0	0
$117$	−3.34525e27	−0.401716
$118$	0	0
$119$	1.34717e28	1.28688
$120$	0	0
$121$	2.85111e27	0.217476
$122$	0	0
$123$	−8.25805e27	−0.504846
$124$	0	0
$125$	1.00061e28	0.492015
$126$	0	0
$127$	−2.16098e27	−0.0857631	−0.0428815	−	0.999080i	$-0.513654\pi$
$127$	−2.16098e27	−0.0857631	−0.0428815	+	0.999080i	$0.513654\pi$
$128$	0	0
$129$	−1.39442e28	−0.448158
$130$	0	0
$131$	−7.19824e28	−1.87959	−0.939797	−	0.341734i	$-0.888986\pi$
$131$	−7.19824e28	−1.87959	−0.939797	+	0.341734i	$0.888986\pi$
$132$	0	0
$133$	3.29681e28	0.701611
$134$	0	0
$135$	−2.81215e27	−0.0489252
$136$	0	0
$137$	−8.23477e28	−1.17469	−0.587346	−	0.809336i	$-0.699827\pi$
$137$	−8.23477e28	−1.17469	−0.587346	+	0.809336i	$0.699827\pi$
$138$	0	0
$139$	−1.17233e29	−1.37515	−0.687574	−	0.726114i	$-0.741324\pi$
$139$	−1.17233e29	−1.37515	−0.687574	+	0.726114i	$0.741324\pi$
$140$	0	0
$141$	−4.01478e28	−0.388332
$142$	0	0
$143$	1.66267e29	1.32975
$144$	0	0
$145$	1.90801e28	0.126506
$146$	0	0
$147$	1.26734e29	0.698410
$148$	0	0
$149$	−2.66795e28	−0.122508	−0.0612538	−	0.998122i	$-0.519510\pi$
$149$	−2.66795e28	−0.122508	−0.0612538	+	0.998122i	$0.519510\pi$
$150$	0	0
$151$	4.49422e29	1.72372	0.861859	−	0.507149i	$-0.169301\pi$
$151$	4.49422e29	1.72372	0.861859	+	0.507149i	$0.169301\pi$
$152$	0	0
$153$	−8.98640e28	−0.288570
$154$	0	0
$155$	−6.78651e28	−0.182881
$156$	0	0
$157$	2.00427e29	0.454268	0.227134	−	0.973864i	$-0.427065\pi$
$157$	2.00427e29	0.454268	0.227134	+	0.973864i	$0.427065\pi$
$158$	0	0
$159$	3.37928e29	0.645600
$160$	0	0
$161$	1.94702e29	0.314211
$162$	0	0
$163$	−6.62774e29	−0.905384	−0.452692	−	0.891667i	$-0.649536\pi$
$163$	−6.62774e29	−0.905384	−0.452692	+	0.891667i	$0.649536\pi$
$164$	0	0
$165$	1.39771e29	0.161951
$166$	0	0
$167$	−1.07500e30	−1.05861	−0.529307	−	0.848431i	$-0.677548\pi$
$167$	−1.07500e30	−1.05861	−0.529307	+	0.848431i	$0.677548\pi$
$168$	0	0
$169$	5.39481e29	0.452382
$170$	0	0
$171$	−2.19917e29	−0.157330
$172$	0	0
$173$	8.13189e29	0.497244	0.248622	−	0.968601i	$-0.420022\pi$
$173$	8.13189e29	0.497244	0.248622	+	0.968601i	$0.420022\pi$
$174$	0	0
$175$	2.65560e30	1.39043
$176$	0	0
$177$	1.11737e30	0.501832
$178$	0	0
$179$	−1.37936e30	−0.532305	−0.266152	−	0.963931i	$-0.585752\pi$
$179$	−1.37936e30	−0.532305	−0.266152	+	0.963931i	$0.585752\pi$
$180$	0	0
$181$	−1.97778e30	−0.656927	−0.328464	−	0.944517i	$-0.606531\pi$
$181$	−1.97778e30	−0.656927	−0.328464	+	0.944517i	$0.606531\pi$
$182$	0	0
$183$	1.72992e29	0.0495378
$184$	0	0
$185$	1.25109e30	0.309364
$186$	0	0
$187$	4.46647e30	0.955219
$188$	0	0
$189$	−1.54425e30	−0.286077
$190$	0	0
$191$	−8.23146e30	−1.32290	−0.661449	−	0.749990i	$-0.730058\pi$
$191$	−8.23146e30	−1.32290	−0.661449	+	0.749990i	$0.730058\pi$
$192$	0	0
$193$	−2.63632e30	−0.368107	−0.184054	−	0.982916i	$-0.558922\pi$
$193$	−2.63632e30	−0.368107	−0.184054	+	0.982916i	$0.558922\pi$
$194$	0	0
$195$	1.45600e30	0.176886
$196$	0	0
$197$	9.66937e30	1.02354	0.511769	−	0.859123i	$-0.328990\pi$
$197$	9.66937e30	1.02354	0.511769	+	0.859123i	$0.328990\pi$
$198$	0	0
$199$	8.33709e30	0.770012	0.385006	−	0.922914i	$-0.374199\pi$
$199$	8.33709e30	0.770012	0.385006	+	0.922914i	$0.374199\pi$
$200$	0	0
$201$	−3.93065e30	−0.317189
$202$	0	0
$203$	1.04775e31	0.739712
$204$	0	0
$205$	3.59427e30	0.222297
$206$	0	0
$207$	−1.29878e30	−0.0704588
$208$	0	0
$209$	1.09304e31	0.520789
$210$	0	0
$211$	−1.34579e31	−0.563850	−0.281925	−	0.959437i	$-0.590973\pi$
$211$	−1.34579e31	−0.563850	−0.281925	+	0.959437i	$0.590973\pi$
$212$	0	0
$213$	−2.06400e31	−0.761351
$214$	0	0
$215$	6.06912e30	0.197336
$216$	0	0
$217$	−3.72671e31	−1.06935
$218$	0	0
$219$	1.19011e31	0.301710
$220$	0	0
$221$	4.65274e31	1.04331
$222$	0	0
$223$	−7.20166e30	−0.142994	−0.0714969	−	0.997441i	$-0.522778\pi$
$223$	−7.20166e30	−0.142994	−0.0714969	+	0.997441i	$0.522778\pi$
$224$	0	0
$225$	−1.77144e31	−0.311790
$226$	0	0
$227$	1.24397e32	1.94294	0.971472	−	0.237153i	$-0.0762144\pi$
$227$	1.24397e32	1.94294	0.971472	+	0.237153i	$0.0762144\pi$
$228$	0	0
$229$	1.14944e32	1.59481	0.797403	−	0.603447i	$-0.206206\pi$
$229$	1.14944e32	1.59481	0.797403	+	0.603447i	$0.206206\pi$
$230$	0	0
$231$	7.67531e31	0.946966
$232$	0	0
$233$	1.39267e32	1.52948	0.764741	−	0.644337i	$-0.222867\pi$
$233$	1.39267e32	1.52948	0.764741	+	0.644337i	$0.222867\pi$
$234$	0	0
$235$	1.74741e31	0.170993
$236$	0	0
$237$	7.88622e31	0.688282
$238$	0	0
$239$	−1.18142e32	−0.920517	−0.460259	−	0.887785i	$-0.652243\pi$
$239$	−1.18142e32	−0.920517	−0.460259	+	0.887785i	$0.652243\pi$
$240$	0	0
$241$	1.39602e32	0.971991	0.485996	−	0.873961i	$-0.338457\pi$
$241$	1.39602e32	0.971991	0.485996	+	0.873961i	$0.338457\pi$
$242$	0	0
$243$	1.03011e31	0.0641500
$244$	0	0
$245$	−5.51604e31	−0.307528
$246$	0	0
$247$	1.13863e32	0.568817
$248$	0	0
$249$	8.83784e31	0.395963
$250$	0	0
$251$	−9.88006e31	−0.397342	−0.198671	−	0.980066i	$-0.563662\pi$
$251$	−9.88006e31	−0.397342	−0.198671	+	0.980066i	$0.563662\pi$
$252$	0	0
$253$	6.45526e31	0.233231
$254$	0	0
$255$	3.91128e31	0.127065
$256$	0	0
$257$	−8.60649e31	−0.251611	−0.125805	−	0.992055i	$-0.540151\pi$
$257$	−8.60649e31	−0.251611	−0.125805	+	0.992055i	$0.540151\pi$
$258$	0	0
$259$	6.87016e32	1.80892
$260$	0	0
$261$	−6.98913e31	−0.165873
$262$	0	0
$263$	2.80912e32	0.601405	0.300703	−	0.953718i	$-0.402779\pi$
$263$	2.80912e32	0.601405	0.300703	+	0.953718i	$0.402779\pi$
$264$	0	0
$265$	−1.47081e32	−0.284275
$266$	0	0
$267$	−2.12002e32	−0.370200
$268$	0	0
$269$	−1.02703e32	−0.162153	−0.0810767	−	0.996708i	$-0.525836\pi$
$269$	−1.02703e32	−0.162153	−0.0810767	+	0.996708i	$0.525836\pi$
$270$	0	0
$271$	1.27505e33	1.82155	0.910775	−	0.412903i	$-0.135485\pi$
$271$	1.27505e33	1.82155	0.910775	+	0.412903i	$0.135485\pi$
$272$	0	0
$273$	7.99540e32	1.03430
$274$	0	0
$275$	8.80451e32	1.03208
$276$	0	0
$277$	1.09712e33	1.16621	0.583107	−	0.812396i	$-0.301837\pi$
$277$	1.09712e33	1.16621	0.583107	+	0.812396i	$0.301837\pi$
$278$	0	0
$279$	2.48594e32	0.239791
$280$	0	0
$281$	1.35684e33	1.18849	0.594244	−	0.804285i	$-0.297451\pi$
$281$	1.35684e33	1.18849	0.594244	+	0.804285i	$0.297451\pi$
$282$	0	0
$283$	−2.13864e33	−1.70224	−0.851120	−	0.524972i	$-0.824076\pi$
$283$	−2.13864e33	−1.70224	−0.851120	+	0.524972i	$0.824076\pi$
$284$	0	0
$285$	9.57174e31	0.0692764
$286$	0	0
$287$	1.97374e33	1.29982
$288$	0	0
$289$	−4.17837e32	−0.250545
$290$	0	0
$291$	−1.69351e33	−0.925195
$292$	0	0
$293$	4.30088e32	0.214212	0.107106	−	0.994248i	$-0.465842\pi$
$293$	4.30088e32	0.214212	0.107106	+	0.994248i	$0.465842\pi$
$294$	0	0
$295$	−4.86330e32	−0.220970
$296$	0	0
$297$	−5.11989e32	−0.212348
$298$	0	0
$299$	6.72447e32	0.254740
$300$	0	0
$301$	3.33276e33	1.15387
$302$	0	0
$303$	2.77430e33	0.878370
$304$	0	0
$305$	−7.52938e31	−0.0218128
$306$	0	0
$307$	5.70033e32	0.151193	0.0755966	−	0.997138i	$-0.475914\pi$
$307$	5.70033e32	0.151193	0.0755966	+	0.997138i	$0.475914\pi$
$308$	0	0
$309$	−7.96464e32	−0.193520
$310$	0	0
$311$	4.65284e33	1.03622	0.518111	−	0.855313i	$-0.326635\pi$
$311$	4.65284e33	1.03622	0.518111	+	0.855313i	$0.326635\pi$
$312$	0	0
$313$	4.33109e33	0.884604	0.442302	−	0.896866i	$-0.354162\pi$
$313$	4.33109e33	0.884604	0.442302	+	0.896866i	$0.354162\pi$
$314$	0	0
$315$	6.72126e32	0.125967
$316$	0	0
$317$	−2.47386e33	−0.425671	−0.212836	−	0.977088i	$-0.568270\pi$
$317$	−2.47386e33	−0.425671	−0.212836	+	0.977088i	$0.568270\pi$
$318$	0	0
$319$	3.47378e33	0.549070
$320$	0	0
$321$	2.07892e33	0.302010
$322$	0	0
$323$	3.05871e33	0.408606
$324$	0	0
$325$	9.17170e33	1.12726
$326$	0	0
$327$	−9.16607e33	−1.03702
$328$	0	0
$329$	9.59563e33	0.999836
$330$	0	0
$331$	2.93911e33	0.282188	0.141094	−	0.989996i	$-0.454938\pi$
$331$	2.93911e33	0.282188	0.141094	+	0.989996i	$0.454938\pi$
$332$	0	0
$333$	−4.58280e33	−0.405633
$334$	0	0
$335$	1.71079e33	0.139667
$336$	0	0
$337$	−1.57199e34	−1.18426	−0.592128	−	0.805844i	$-0.701712\pi$
$337$	−1.57199e34	−1.18426	−0.592128	+	0.805844i	$0.701712\pi$
$338$	0	0
$339$	1.32208e34	0.919518
$340$	0	0
$341$	−1.23557e34	−0.793752
$342$	0	0
$343$	−5.25043e33	−0.311691
$344$	0	0
$345$	5.65286e32	0.0310249
$346$	0	0
$347$	−2.55505e34	−1.29703	−0.648516	−	0.761201i	$-0.724610\pi$
$347$	−2.55505e34	−1.29703	−0.648516	+	0.761201i	$0.724610\pi$
$348$	0	0
$349$	−3.50116e34	−1.64463	−0.822316	−	0.569031i	$-0.807319\pi$
$349$	−3.50116e34	−1.64463	−0.822316	+	0.569031i	$0.807319\pi$
$350$	0	0
$351$	−5.33341e33	−0.231931
$352$	0	0
$353$	−6.64200e33	−0.267510	−0.133755	−	0.991014i	$-0.542704\pi$
$353$	−6.64200e33	−0.267510	−0.133755	+	0.991014i	$0.542704\pi$
$354$	0	0
$355$	8.98343e33	0.335243
$356$	0	0
$357$	2.14782e34	0.742980
$358$	0	0
$359$	−4.11061e34	−1.31865	−0.659327	−	0.751856i	$-0.729159\pi$
$359$	−4.11061e34	−1.31865	−0.659327	+	0.751856i	$0.729159\pi$
$360$	0	0
$361$	−2.61153e34	−0.777227
$362$	0	0
$363$	4.54559e33	0.125560
$364$	0	0
$365$	−5.17987e33	−0.132851
$366$	0	0
$367$	3.04370e33	0.0725121	0.0362560	−	0.999343i	$-0.488457\pi$
$367$	3.04370e33	0.0725121	0.0362560	+	0.999343i	$0.488457\pi$
$368$	0	0
$369$	−1.31660e34	−0.291473
$370$	0	0
$371$	−8.07675e34	−1.66222
$372$	0	0
$373$	4.11966e34	0.788483	0.394242	−	0.919007i	$-0.371007\pi$
$373$	4.11966e34	0.788483	0.394242	+	0.919007i	$0.371007\pi$
$374$	0	0
$375$	1.59529e34	0.284065
$376$	0	0
$377$	3.61865e34	0.599706
$378$	0	0
$379$	−8.35010e34	−1.28843	−0.644217	−	0.764842i	$-0.722817\pi$
$379$	−8.35010e34	−1.28843	−0.644217	+	0.764842i	$0.722817\pi$
$380$	0	0
$381$	−3.44530e33	−0.0495153
$382$	0	0
$383$	5.20579e34	0.697113	0.348557	−	0.937288i	$-0.386672\pi$
$383$	5.20579e34	0.697113	0.348557	+	0.937288i	$0.386672\pi$
$384$	0	0
$385$	−3.34064e34	−0.416974
$386$	0	0
$387$	−2.22315e34	−0.258744
$388$	0	0
$389$	−3.57703e34	−0.388332	−0.194166	−	0.980969i	$-0.562200\pi$
$389$	−3.57703e34	−0.388332	−0.194166	+	0.980969i	$0.562200\pi$
$390$	0	0
$391$	1.80641e34	0.182991
$392$	0	0
$393$	−1.14763e35	−1.08518
$394$	0	0
$395$	−3.43243e34	−0.303069
$396$	0	0
$397$	−7.79828e34	−0.643172	−0.321586	−	0.946880i	$-0.604216\pi$
$397$	−7.79828e34	−0.643172	−0.321586	+	0.946880i	$0.604216\pi$
$398$	0	0
$399$	5.25618e34	0.405075
$400$	0	0
$401$	−2.28761e34	−0.164791	−0.0823955	−	0.996600i	$-0.526257\pi$
$401$	−2.28761e34	−0.164791	−0.0823955	+	0.996600i	$0.526257\pi$
$402$	0	0
$403$	−1.28710e35	−0.866952
$404$	0	0
$405$	−4.48347e33	−0.0282470
$406$	0	0
$407$	2.27777e35	1.34272
$408$	0	0
$409$	−1.15859e34	−0.0639243	−0.0319621	−	0.999489i	$-0.510176\pi$
$409$	−1.15859e34	−0.0639243	−0.0319621	+	0.999489i	$0.510176\pi$
$410$	0	0
$411$	−1.31289e35	−0.678209
$412$	0	0
$413$	−2.67061e35	−1.29206
$414$	0	0
$415$	−3.84662e34	−0.174353
$416$	0	0
$417$	−1.86907e35	−0.793942
$418$	0	0
$419$	−6.19060e34	−0.246515	−0.123258	−	0.992375i	$-0.539334\pi$
$419$	−6.19060e34	−0.246515	−0.123258	+	0.992375i	$0.539334\pi$
$420$	0	0
$421$	−8.53227e33	−0.0318608	−0.0159304	−	0.999873i	$-0.505071\pi$
$421$	−8.53227e33	−0.0318608	−0.0159304	+	0.999873i	$0.505071\pi$
$422$	0	0
$423$	−6.40085e34	−0.224204
$424$	0	0
$425$	2.46381e35	0.809761
$426$	0	0
$427$	−4.13465e34	−0.127545
$428$	0	0
$429$	2.65084e35	0.767733
$430$	0	0
$431$	−4.08719e35	−1.11169	−0.555843	−	0.831288i	$-0.687604\pi$
$431$	−4.08719e35	−1.11169	−0.555843	+	0.831288i	$0.687604\pi$
$432$	0	0
$433$	1.44666e35	0.369642	0.184821	−	0.982772i	$-0.440830\pi$
$433$	1.44666e35	0.369642	0.184821	+	0.982772i	$0.440830\pi$
$434$	0	0
$435$	3.04198e34	0.0730385
$436$	0	0
$437$	4.42067e34	0.0997673
$438$	0	0
$439$	−2.97115e34	−0.0630454	−0.0315227	−	0.999503i	$-0.510036\pi$
$439$	−2.97115e34	−0.0630454	−0.0315227	+	0.999503i	$0.510036\pi$
$440$	0	0
$441$	2.02055e35	0.403227
$442$	0	0
$443$	1.94735e35	0.365589	0.182795	−	0.983151i	$-0.441486\pi$
$443$	1.94735e35	0.365589	0.182795	+	0.983151i	$0.441486\pi$
$444$	0	0
$445$	9.22725e34	0.163009
$446$	0	0
$447$	−4.25357e34	−0.0707298
$448$	0	0
$449$	−3.89051e35	−0.609089	−0.304544	−	0.952498i	$-0.598504\pi$
$449$	−3.89051e35	−0.609089	−0.304544	+	0.952498i	$0.598504\pi$
$450$	0	0
$451$	6.54384e35	0.964827
$452$	0	0
$453$	7.16525e35	0.995188
$454$	0	0
$455$	−3.47995e35	−0.455428
$456$	0	0
$457$	−7.79952e35	−0.962053	−0.481027	−	0.876706i	$-0.659736\pi$
$457$	−7.79952e35	−0.962053	−0.481027	+	0.876706i	$0.659736\pi$
$458$	0	0
$459$	−1.43272e35	−0.166606
$460$	0	0
$461$	−1.33828e36	−1.46752	−0.733760	−	0.679408i	$-0.762236\pi$
$461$	−1.33828e36	−1.46752	−0.733760	+	0.679408i	$0.762236\pi$
$462$	0	0
$463$	−3.79932e35	−0.392973	−0.196486	−	0.980507i	$-0.562953\pi$
$463$	−3.79932e35	−0.392973	−0.196486	+	0.980507i	$0.562953\pi$
$464$	0	0
$465$	−1.08199e35	−0.105586
$466$	0	0
$467$	7.20473e35	0.663499	0.331749	−	0.943368i	$-0.392361\pi$
$467$	7.20473e35	0.663499	0.331749	+	0.943368i	$0.392361\pi$
$468$	0	0
$469$	9.39456e35	0.816664
$470$	0	0
$471$	3.19546e35	0.262271
$472$	0	0
$473$	1.10496e36	0.856490
$474$	0	0
$475$	6.02947e35	0.441485
$476$	0	0
$477$	5.38767e35	0.372737
$478$	0	0
$479$	4.56900e34	0.0298739	0.0149370	−	0.999888i	$-0.495245\pi$
$479$	4.56900e34	0.0298739	0.0149370	+	0.999888i	$0.495245\pi$
$480$	0	0
$481$	2.37276e36	1.46655
$482$	0	0
$483$	3.10418e35	0.181410
$484$	0	0
$485$	7.37092e35	0.407388
$486$	0	0
$487$	2.25055e36	1.17665	0.588326	−	0.808624i	$-0.299787\pi$
$487$	2.25055e36	1.17665	0.588326	+	0.808624i	$0.299787\pi$
$488$	0	0
$489$	−1.05668e36	−0.522723
$490$	0	0
$491$	3.36099e36	1.57350	0.786751	−	0.617271i	$-0.211762\pi$
$491$	3.36099e36	1.57350	0.786751	+	0.617271i	$0.211762\pi$
$492$	0	0
$493$	9.72085e35	0.430795
$494$	0	0
$495$	2.22840e35	0.0935025
$496$	0	0
$497$	4.93312e36	1.96024
$498$	0	0
$499$	−1.11121e35	−0.0418253	−0.0209127	−	0.999781i	$-0.506657\pi$
$499$	−1.11121e35	−0.0418253	−0.0209127	+	0.999781i	$0.506657\pi$
$500$	0	0
$501$	−1.71390e36	−0.611191
$502$	0	0
$503$	1.49384e36	0.504818	0.252409	−	0.967621i	$-0.418777\pi$
$503$	1.49384e36	0.504818	0.252409	+	0.967621i	$0.418777\pi$
$504$	0	0
$505$	−1.20750e36	−0.386770
$506$	0	0
$507$	8.60107e35	0.261183
$508$	0	0
$509$	−5.00593e36	−1.44143	−0.720717	−	0.693230i	$-0.756187\pi$
$509$	−5.00593e36	−1.44143	−0.720717	+	0.693230i	$0.756187\pi$
$510$	0	0
$511$	−2.84445e36	−0.776811
$512$	0	0
$513$	−3.50618e35	−0.0908343
$514$	0	0
$515$	3.46657e35	0.0852122
$516$	0	0
$517$	3.18139e36	0.742154
$518$	0	0
$519$	1.29649e36	0.287084
$520$	0	0
$521$	3.77205e36	0.792990	0.396495	−	0.918037i	$-0.370226\pi$
$521$	3.77205e36	0.792990	0.396495	+	0.918037i	$0.370226\pi$
$522$	0	0
$523$	7.85298e35	0.156770	0.0783848	−	0.996923i	$-0.475024\pi$
$523$	7.85298e35	0.156770	0.0783848	+	0.996923i	$0.475024\pi$
$524$	0	0
$525$	4.23388e36	0.802764
$526$	0	0
$527$	−3.45757e36	−0.622769
$528$	0	0
$529$	−5.58214e36	−0.955320
$530$	0	0
$531$	1.78145e36	0.289733
$532$	0	0
$533$	6.81675e36	1.05380
$534$	0	0
$535$	−9.04840e35	−0.132983
$536$	0	0
$537$	−2.19914e36	−0.307326
$538$	0	0
$539$	−1.00427e37	−1.33475
$540$	0	0
$541$	6.90432e36	0.872887	0.436444	−	0.899732i	$-0.356238\pi$
$541$	6.90432e36	0.872887	0.436444	+	0.899732i	$0.356238\pi$
$542$	0	0
$543$	−3.15322e36	−0.379277
$544$	0	0
$545$	3.98948e36	0.456629
$546$	0	0
$547$	4.90777e36	0.534632	0.267316	−	0.963609i	$-0.413863\pi$
$547$	4.90777e36	0.534632	0.267316	+	0.963609i	$0.413863\pi$
$548$	0	0
$549$	2.75805e35	0.0286007
$550$	0	0
$551$	2.37890e36	0.234871
$552$	0	0
$553$	−1.88487e37	−1.77211
$554$	0	0
$555$	1.99464e36	0.178611
$556$	0	0
$557$	−8.37169e36	−0.714115	−0.357058	−	0.934082i	$-0.616220\pi$
$557$	−8.37169e36	−0.714115	−0.357058	+	0.934082i	$0.616220\pi$
$558$	0	0
$559$	1.15104e37	0.935475
$560$	0	0
$561$	7.12100e36	0.551496
$562$	0	0
$563$	−1.53607e37	−1.13383	−0.566913	−	0.823778i	$-0.691862\pi$
$563$	−1.53607e37	−1.13383	−0.566913	+	0.823778i	$0.691862\pi$
$564$	0	0
$565$	−5.75426e36	−0.404889
$566$	0	0
$567$	−2.46203e36	−0.165167
$568$	0	0
$569$	−2.56048e37	−1.63797	−0.818986	−	0.573814i	$-0.805463\pi$
$569$	−2.56048e37	−1.63797	−0.818986	+	0.573814i	$0.805463\pi$
$570$	0	0
$571$	1.84375e37	1.12490	0.562450	−	0.826831i	$-0.309859\pi$
$571$	1.84375e37	1.12490	0.562450	+	0.826831i	$0.309859\pi$
$572$	0	0
$573$	−1.31236e37	−0.763775
$574$	0	0
$575$	3.56087e36	0.197715
$576$	0	0
$577$	2.60648e37	1.38096	0.690479	−	0.723352i	$-0.257400\pi$
$577$	2.60648e37	1.38096	0.690479	+	0.723352i	$0.257400\pi$
$578$	0	0
$579$	−4.20315e36	−0.212527
$580$	0	0
$581$	−2.11231e37	−1.01948
$582$	0	0
$583$	−2.67781e37	−1.23383
$584$	0	0
$585$	2.32133e36	0.102125
$586$	0	0
$587$	−1.64116e37	−0.689505	−0.344752	−	0.938694i	$-0.612037\pi$
$587$	−1.64116e37	−0.689505	−0.344752	+	0.938694i	$0.612037\pi$
$588$	0	0
$589$	−8.46141e36	−0.339536
$590$	0	0
$591$	1.54161e37	0.590940
$592$	0	0
$593$	6.39232e35	0.0234110	0.0117055	−	0.999931i	$-0.496274\pi$
$593$	6.39232e35	0.0234110	0.0117055	+	0.999931i	$0.496274\pi$
$594$	0	0
$595$	−9.34827e36	−0.327154
$596$	0	0
$597$	1.32920e37	0.444567
$598$	0	0
$599$	−3.67500e37	−1.17488	−0.587442	−	0.809266i	$-0.699865\pi$
$599$	−3.67500e37	−1.17488	−0.587442	+	0.809266i	$0.699865\pi$
$600$	0	0
$601$	4.35901e36	0.133224	0.0666120	−	0.997779i	$-0.478781\pi$
$601$	4.35901e36	0.133224	0.0666120	+	0.997779i	$0.478781\pi$
$602$	0	0
$603$	−6.26673e36	−0.183129
$604$	0	0
$605$	−1.97844e36	−0.0552874
$606$	0	0
$607$	−4.98438e37	−1.33218	−0.666092	−	0.745870i	$-0.732034\pi$
$607$	−4.98438e37	−1.33218	−0.666092	+	0.745870i	$0.732034\pi$
$608$	0	0
$609$	1.67046e37	0.427073
$610$	0	0
$611$	3.31406e37	0.810596
$612$	0	0
$613$	2.09778e37	0.490958	0.245479	−	0.969402i	$-0.421055\pi$
$613$	2.09778e37	0.490958	0.245479	+	0.969402i	$0.421055\pi$
$614$	0	0
$615$	5.73043e36	0.128343
$616$	0	0
$617$	−9.28596e37	−1.99057	−0.995284	−	0.0970039i	$-0.969074\pi$
$617$	−9.28596e37	−1.99057	−0.995284	+	0.0970039i	$0.969074\pi$
$618$	0	0
$619$	−4.21148e37	−0.864194	−0.432097	−	0.901827i	$-0.642226\pi$
$619$	−4.21148e37	−0.864194	−0.432097	+	0.901827i	$0.642226\pi$
$620$	0	0
$621$	−2.07067e36	−0.0406794
$622$	0	0
$623$	5.06700e37	0.953153
$624$	0	0
$625$	4.49801e37	0.810289
$626$	0	0
$627$	1.74266e37	0.300678
$628$	0	0
$629$	6.37399e37	1.05348
$630$	0	0
$631$	1.60405e37	0.253993	0.126997	−	0.991903i	$-0.459466\pi$
$631$	1.60405e37	0.253993	0.126997	+	0.991903i	$0.459466\pi$
$632$	0	0
$633$	−2.14562e37	−0.325539
$634$	0	0
$635$	1.49955e36	0.0218029
$636$	0	0
$637$	−1.04615e38	−1.45784
$638$	0	0
$639$	−3.29068e37	−0.439566
$640$	0	0
$641$	−4.27881e37	−0.547948	−0.273974	−	0.961737i	$-0.588338\pi$
$641$	−4.27881e37	−0.547948	−0.273974	+	0.961737i	$0.588338\pi$
$642$	0	0
$643$	−6.44875e36	−0.0791821	−0.0395911	−	0.999216i	$-0.512606\pi$
$643$	−6.44875e36	−0.0791821	−0.0395911	+	0.999216i	$0.512606\pi$
$644$	0	0
$645$	9.67613e36	0.113932
$646$	0	0
$647$	−1.16912e38	−1.32024	−0.660118	−	0.751162i	$-0.729494\pi$
$647$	−1.16912e38	−1.32024	−0.660118	+	0.751162i	$0.729494\pi$
$648$	0	0
$649$	−8.85427e37	−0.959068
$650$	0	0
$651$	−5.94158e37	−0.617389
$652$	0	0
$653$	−7.34409e37	−0.732167	−0.366083	−	0.930582i	$-0.619301\pi$
$653$	−7.34409e37	−0.732167	−0.366083	+	0.930582i	$0.619301\pi$
$654$	0	0
$655$	4.99500e37	0.477835
$656$	0	0
$657$	1.89741e37	0.174192
$658$	0	0
$659$	−3.42254e37	−0.301575	−0.150788	−	0.988566i	$-0.548181\pi$
$659$	−3.42254e37	−0.301575	−0.150788	+	0.988566i	$0.548181\pi$
$660$	0	0
$661$	−1.89312e37	−0.160125	−0.0800625	−	0.996790i	$-0.525512\pi$
$661$	−1.89312e37	−0.160125	−0.0800625	+	0.996790i	$0.525512\pi$
$662$	0	0
$663$	7.41797e37	0.602355
$664$	0	0
$665$	−2.28772e37	−0.178366
$666$	0	0
$667$	1.40493e37	0.105185
$668$	0	0
$669$	−1.14818e37	−0.0825575
$670$	0	0
$671$	−1.37082e37	−0.0946733
$672$	0	0
$673$	2.43532e38	1.61567	0.807835	−	0.589409i	$-0.200639\pi$
$673$	2.43532e38	1.61567	0.807835	+	0.589409i	$0.200639\pi$
$674$	0	0
$675$	−2.82425e37	−0.180012
$676$	0	0
$677$	2.58654e38	1.58406	0.792032	−	0.610479i	$-0.209023\pi$
$677$	2.58654e38	1.58406	0.792032	+	0.610479i	$0.209023\pi$
$678$	0	0
$679$	4.04763e38	2.38209
$680$	0	0
$681$	1.98329e38	1.12176
$682$	0	0
$683$	−5.72592e37	−0.311290	−0.155645	−	0.987813i	$-0.549746\pi$
$683$	−5.72592e37	−0.311290	−0.155645	+	0.987813i	$0.549746\pi$
$684$	0	0
$685$	5.71427e37	0.298633
$686$	0	0
$687$	1.83258e38	0.920762
$688$	0	0
$689$	−2.78949e38	−1.34761
$690$	0	0
$691$	1.45765e38	0.677175	0.338588	−	0.940935i	$-0.390051\pi$
$691$	1.45765e38	0.677175	0.338588	+	0.940935i	$0.390051\pi$
$692$	0	0
$693$	1.22369e38	0.546731
$694$	0	0
$695$	8.13502e37	0.349594
$696$	0	0
$697$	1.83120e38	0.756994
$698$	0	0
$699$	2.22037e38	0.883047
$700$	0	0
$701$	−5.69328e37	−0.217856	−0.108928	−	0.994050i	$-0.534742\pi$
$701$	−5.69328e37	−0.217856	−0.108928	+	0.994050i	$0.534742\pi$
$702$	0	0
$703$	1.55985e38	0.574363
$704$	0	0
$705$	2.78593e37	0.0987229
$706$	0	0
$707$	−6.63079e38	−2.26153
$708$	0	0
$709$	2.15586e38	0.707776	0.353888	−	0.935288i	$-0.384859\pi$
$709$	2.15586e38	0.707776	0.353888	+	0.935288i	$0.384859\pi$
$710$	0	0
$711$	1.25732e38	0.397380
$712$	0	0
$713$	−4.99712e37	−0.152059
$714$	0	0
$715$	−1.15376e38	−0.338053
$716$	0	0
$717$	−1.88356e38	−0.531461
$718$	0	0
$719$	−2.25632e38	−0.613141	−0.306571	−	0.951848i	$-0.599182\pi$
$719$	−2.25632e38	−0.613141	−0.306571	+	0.951848i	$0.599182\pi$
$720$	0	0
$721$	1.90361e38	0.498256
$722$	0	0
$723$	2.22571e38	0.561179
$724$	0	0
$725$	1.91622e38	0.465459
$726$	0	0
$727$	−3.26448e38	−0.764010	−0.382005	−	0.924160i	$-0.624766\pi$
$727$	−3.26448e38	−0.764010	−0.382005	+	0.924160i	$0.624766\pi$
$728$	0	0
$729$	1.64232e37	0.0370370
$730$	0	0
$731$	3.09207e38	0.671993
$732$	0	0
$733$	3.51091e38	0.735387	0.367694	−	0.929947i	$-0.380147\pi$
$733$	3.51091e38	0.735387	0.367694	+	0.929947i	$0.380147\pi$
$734$	0	0
$735$	−8.79435e37	−0.177552
$736$	0	0
$737$	3.11472e38	0.606190
$738$	0	0
$739$	5.12467e38	0.961537	0.480768	−	0.876848i	$-0.340358\pi$
$739$	5.12467e38	0.961537	0.480768	+	0.876848i	$0.340358\pi$
$740$	0	0
$741$	1.81534e38	0.328406
$742$	0	0
$743$	6.94450e38	1.21141	0.605706	−	0.795688i	$-0.292891\pi$
$743$	6.94450e38	1.21141	0.605706	+	0.795688i	$0.292891\pi$
$744$	0	0
$745$	1.85134e37	0.0311442
$746$	0	0
$747$	1.40904e38	0.228609
$748$	0	0
$749$	−4.96879e38	−0.777582
$750$	0	0
$751$	−5.87337e38	−0.886642	−0.443321	−	0.896363i	$-0.646200\pi$
$751$	−5.87337e38	−0.886642	−0.443321	+	0.896363i	$0.646200\pi$
$752$	0	0
$753$	−1.57520e38	−0.229405
$754$	0	0
$755$	−3.11863e38	−0.438208
$756$	0	0
$757$	1.01202e38	0.137212	0.0686062	−	0.997644i	$-0.478145\pi$
$757$	1.01202e38	0.137212	0.0686062	+	0.997644i	$0.478145\pi$
$758$	0	0
$759$	1.02918e38	0.134656
$760$	0	0
$761$	−1.48838e39	−1.87941	−0.939705	−	0.341987i	$-0.888900\pi$
$761$	−1.48838e39	−1.87941	−0.939705	+	0.341987i	$0.888900\pi$
$762$	0	0
$763$	2.19076e39	2.67002
$764$	0	0
$765$	6.23585e37	0.0733611
$766$	0	0
$767$	−9.22353e38	−1.04751
$768$	0	0
$769$	−1.63286e39	−1.79037	−0.895186	−	0.445692i	$-0.852958\pi$
$769$	−1.63286e39	−1.79037	−0.895186	+	0.445692i	$0.852958\pi$
$770$	0	0
$771$	−1.37215e38	−0.145267
$772$	0	0
$773$	9.47969e36	0.00969105	0.00484552	−	0.999988i	$-0.498458\pi$
$773$	9.47969e36	0.00969105	0.00484552	+	0.999988i	$0.498458\pi$
$774$	0	0
$775$	−6.81571e38	−0.672881
$776$	0	0
$777$	1.09532e39	1.04438
$778$	0	0
$779$	4.48133e38	0.412716
$780$	0	0
$781$	1.63555e39	1.45504
$782$	0	0
$783$	−1.11429e38	−0.0957670
$784$	0	0
$785$	−1.39081e38	−0.115485
$786$	0	0
$787$	−1.01258e39	−0.812401	−0.406201	−	0.913784i	$-0.633147\pi$
$787$	−1.01258e39	−0.812401	−0.406201	+	0.913784i	$0.633147\pi$
$788$	0	0
$789$	4.47864e38	0.347222
$790$	0	0
$791$	−3.15986e39	−2.36748
$792$	0	0
$793$	−1.42799e38	−0.103404
$794$	0	0
$795$	−2.34495e38	−0.164126
$796$	0	0
$797$	−1.64542e39	−1.11324	−0.556620	−	0.830767i	$-0.687902\pi$
$797$	−1.64542e39	−1.11324	−0.556620	+	0.830767i	$0.687902\pi$
$798$	0	0
$799$	8.90263e38	0.582287
$800$	0	0
$801$	−3.37999e38	−0.213735
$802$	0	0
$803$	−9.43063e38	−0.576608
$804$	0	0
$805$	−1.35108e38	−0.0798796
$806$	0	0
$807$	−1.63741e38	−0.0936193
$808$	0	0
$809$	−3.76453e38	−0.208164	−0.104082	−	0.994569i	$-0.533190\pi$
$809$	−3.76453e38	−0.208164	−0.104082	+	0.994569i	$0.533190\pi$
$810$	0	0
$811$	−1.84807e39	−0.988408	−0.494204	−	0.869346i	$-0.664540\pi$
$811$	−1.84807e39	−0.988408	−0.494204	+	0.869346i	$0.664540\pi$
$812$	0	0
$813$	2.03284e39	1.05167
$814$	0	0
$815$	4.59912e38	0.230169
$816$	0	0
$817$	7.56696e38	0.366373
$818$	0	0
$819$	1.27473e39	0.597151
$820$	0	0
$821$	−4.29041e39	−1.94476	−0.972380	−	0.233401i	$-0.925014\pi$
$821$	−4.29041e39	−1.94476	−0.972380	+	0.233401i	$0.925014\pi$
$822$	0	0
$823$	1.74258e39	0.764355	0.382177	−	0.924089i	$-0.375174\pi$
$823$	1.74258e39	0.764355	0.382177	+	0.924089i	$0.375174\pi$
$824$	0	0
$825$	1.40372e39	0.595873
$826$	0	0
$827$	−3.90760e39	−1.60541	−0.802704	−	0.596378i	$-0.796606\pi$
$827$	−3.90760e39	−1.60541	−0.802704	+	0.596378i	$0.796606\pi$
$828$	0	0
$829$	1.05175e39	0.418243	0.209121	−	0.977890i	$-0.432940\pi$
$829$	1.05175e39	0.418243	0.209121	+	0.977890i	$0.432940\pi$
$830$	0	0
$831$	1.74917e39	0.673313
$832$	0	0
$833$	−2.81029e39	−1.04723
$834$	0	0
$835$	7.45966e38	0.269124
$836$	0	0
$837$	3.96338e38	0.138443
$838$	0	0
$839$	9.84534e38	0.333000	0.166500	−	0.986041i	$-0.446753\pi$
$839$	9.84534e38	0.333000	0.166500	+	0.986041i	$0.446753\pi$
$840$	0	0
$841$	−2.29710e39	−0.752374
$842$	0	0
$843$	2.16325e39	0.686173
$844$	0	0
$845$	−3.74357e38	−0.115006
$846$	0	0
$847$	−1.08643e39	−0.323278
$848$	0	0
$849$	−3.40968e39	−0.982788
$850$	0	0
$851$	9.21214e38	0.257224
$852$	0	0
$853$	2.30350e39	0.623125	0.311563	−	0.950226i	$-0.399148\pi$
$853$	2.30350e39	0.623125	0.311563	+	0.950226i	$0.399148\pi$
$854$	0	0
$855$	1.52605e38	0.0399968
$856$	0	0
$857$	−1.77038e39	−0.449600	−0.224800	−	0.974405i	$-0.572173\pi$
$857$	−1.77038e39	−0.449600	−0.224800	+	0.974405i	$0.572173\pi$
$858$	0	0
$859$	−5.46881e39	−1.34581	−0.672907	−	0.739727i	$-0.734954\pi$
$859$	−5.46881e39	−1.34581	−0.672907	+	0.739727i	$0.734954\pi$
$860$	0	0
$861$	3.14678e39	0.750453
$862$	0	0
$863$	−5.53394e39	−1.27905	−0.639526	−	0.768769i	$-0.720870\pi$
$863$	−5.53394e39	−1.27905	−0.639526	+	0.768769i	$0.720870\pi$
$864$	0	0
$865$	−5.64288e38	−0.126411
$866$	0	0
$867$	−6.66166e38	−0.144652
$868$	0	0
$869$	−6.24920e39	−1.31540
$870$	0	0
$871$	3.24462e39	0.662093
$872$	0	0
$873$	−2.70001e39	−0.534162
$874$	0	0
$875$	−3.81287e39	−0.731380
$876$	0	0
$877$	7.58578e39	1.41093	0.705466	−	0.708744i	$-0.250738\pi$
$877$	7.58578e39	1.41093	0.705466	+	0.708744i	$0.250738\pi$
$878$	0	0
$879$	6.85699e38	0.123675
$880$	0	0
$881$	−5.97063e38	−0.104435	−0.0522174	−	0.998636i	$-0.516629\pi$
$881$	−5.97063e38	−0.104435	−0.0522174	+	0.998636i	$0.516629\pi$
$882$	0	0
$883$	7.85839e39	1.33310	0.666552	−	0.745459i	$-0.267770\pi$
$883$	7.85839e39	1.33310	0.666552	+	0.745459i	$0.267770\pi$
$884$	0	0
$885$	−7.75367e38	−0.127577
$886$	0	0
$887$	−5.04306e39	−0.804869	−0.402434	−	0.915449i	$-0.631836\pi$
$887$	−5.04306e39	−0.804869	−0.402434	+	0.915449i	$0.631836\pi$
$888$	0	0
$889$	8.23453e38	0.127487
$890$	0	0
$891$	−8.16275e38	−0.122599
$892$	0	0
$893$	2.17866e39	0.317465
$894$	0	0
$895$	9.57163e38	0.135324
$896$	0	0
$897$	1.07210e39	0.147074
$898$	0	0
$899$	−2.68911e39	−0.357975
$900$	0	0
$901$	−7.49345e39	−0.968048
$902$	0	0
$903$	5.31350e39	0.666187
$904$	0	0
$905$	1.37242e39	0.167006
$906$	0	0
$907$	7.33863e39	0.866796	0.433398	−	0.901203i	$-0.357314\pi$
$907$	7.33863e39	0.866796	0.433398	+	0.901203i	$0.357314\pi$
$908$	0	0
$909$	4.42313e39	0.507127
$910$	0	0
$911$	−7.23054e39	−0.804771	−0.402386	−	0.915470i	$-0.631819\pi$
$911$	−7.23054e39	−0.804771	−0.402386	+	0.915470i	$0.631819\pi$
$912$	0	0
$913$	−7.00327e39	−0.756738
$914$	0	0
$915$	−1.20043e38	−0.0125936
$916$	0	0
$917$	2.74293e40	2.79401
$918$	0	0
$919$	1.06263e40	1.05105	0.525523	−	0.850780i	$-0.323870\pi$
$919$	1.06263e40	1.05105	0.525523	+	0.850780i	$0.323870\pi$
$920$	0	0
$921$	9.08817e38	0.0872914
$922$	0	0
$923$	1.70376e40	1.58923
$924$	0	0
$925$	1.25647e40	1.13825
$926$	0	0
$927$	−1.26982e39	−0.111729
$928$	0	0
$929$	−9.62717e39	−0.822785	−0.411392	−	0.911458i	$-0.634957\pi$
$929$	−9.62717e39	−0.822785	−0.411392	+	0.911458i	$0.634957\pi$
$930$	0	0
$931$	−6.87738e39	−0.570956
$932$	0	0
$933$	7.41813e39	0.598263
$934$	0	0
$935$	−3.09938e39	−0.242838
$936$	0	0
$937$	1.21318e40	0.923509	0.461755	−	0.887008i	$-0.347220\pi$
$937$	1.21318e40	0.923509	0.461755	+	0.887008i	$0.347220\pi$
$938$	0	0
$939$	6.90516e39	0.510726
$940$	0	0
$941$	−5.05393e38	−0.0363219	−0.0181610	−	0.999835i	$-0.505781\pi$
$941$	−5.05393e38	−0.0363219	−0.0181610	+	0.999835i	$0.505781\pi$
$942$	0	0
$943$	2.64657e39	0.184831
$944$	0	0
$945$	1.07159e39	0.0727273
$946$	0	0
$947$	−8.58094e39	−0.565992	−0.282996	−	0.959121i	$-0.591328\pi$
$947$	−8.58094e39	−0.565992	−0.282996	+	0.959121i	$0.591328\pi$
$948$	0	0
$949$	−9.82392e39	−0.629783
$950$	0	0
$951$	−3.94413e39	−0.245761
$952$	0	0
$953$	−2.53482e39	−0.153530	−0.0767649	−	0.997049i	$-0.524459\pi$
$953$	−2.53482e39	−0.153530	−0.0767649	+	0.997049i	$0.524459\pi$
$954$	0	0
$955$	5.71197e39	0.336311
$956$	0	0
$957$	5.53833e39	0.317006
$958$	0	0
$959$	3.13791e40	1.74618
$960$	0	0
$961$	−8.91794e39	−0.482502
$962$	0	0
$963$	3.31448e39	0.174365
$964$	0	0
$965$	1.82940e39	0.0935813
$966$	0	0
$967$	2.17064e39	0.107977	0.0539884	−	0.998542i	$-0.482807\pi$
$967$	2.17064e39	0.107977	0.0539884	+	0.998542i	$0.482807\pi$
$968$	0	0
$969$	4.87658e39	0.235909
$970$	0	0
$971$	−2.71062e40	−1.27529	−0.637645	−	0.770330i	$-0.720091\pi$
$971$	−2.71062e40	−1.27529	−0.637645	+	0.770330i	$0.720091\pi$
$972$	0	0
$973$	4.46722e40	2.04416
$974$	0	0
$975$	1.46226e40	0.650824
$976$	0	0
$977$	−4.41710e40	−1.91232	−0.956161	−	0.292841i	$-0.905399\pi$
$977$	−4.41710e40	−1.91232	−0.956161	+	0.292841i	$0.905399\pi$
$978$	0	0
$979$	1.67994e40	0.707502
$980$	0	0
$981$	−1.46137e40	−0.598726
$982$	0	0
$983$	2.40915e40	0.960266	0.480133	−	0.877196i	$-0.340588\pi$
$983$	2.40915e40	0.960266	0.480133	+	0.877196i	$0.340588\pi$
$984$	0	0
$985$	−6.70977e39	−0.260207
$986$	0	0
$987$	1.52985e40	0.577256
$988$	0	0
$989$	4.46888e39	0.164077
$990$	0	0
$991$	1.16541e39	0.0416376	0.0208188	−	0.999783i	$-0.493373\pi$
$991$	1.16541e39	0.0416376	0.0208188	+	0.999783i	$0.493373\pi$
$992$	0	0
$993$	4.68589e39	0.162921
$994$	0	0
$995$	−5.78527e39	−0.195755
$996$	0	0
$997$	5.76017e40	1.89693	0.948463	−	0.316888i	$-0.102638\pi$
$997$	5.76017e40	1.89693	0.948463	+	0.316888i	$0.102638\pi$
$998$	0	0
$999$	−7.30646e39	−0.234193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.28.a.e.1.2		2
4.3	odd	2		3.28.a.b.1.2	✓	2
12.11	even	2		9.28.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.28.a.b.1.2	✓	2	4.3	odd	2
9.28.a.b.1.1		2	12.11	even	2
48.28.a.e.1.2		2	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:	\( k \)	\(=\)	\( 28 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q+1.59432e6 q^{3} -6.93920e8 q^{5} -3.81056e11 q^{7} +2.54187e12 q^{9} +O(q^{10})\)	\(q+1.59432e6 q^{3} -6.93920e8 q^{5} -3.81056e11 q^{7} +2.54187e12 q^{9} -1.26337e14 q^{11} -1.31606e15 q^{13} -1.10633e15 q^{15} -3.53536e16 q^{17} -8.65178e16 q^{19} -6.07526e17 q^{21} -5.10955e17 q^{23} -6.96906e18 q^{25} +4.05256e18 q^{27} -2.74961e19 q^{29} +9.77996e19 q^{31} -2.01422e20 q^{33} +2.64422e20 q^{35} -1.80293e21 q^{37} -2.09822e21 q^{39} -5.17966e21 q^{41} -8.74613e21 q^{43} -1.76385e21 q^{45} -2.51817e22 q^{47} +7.94910e22 q^{49} -5.63650e22 q^{51} +2.11957e23 q^{53} +8.76680e22 q^{55} -1.37937e23 q^{57} +7.00844e23 q^{59} +1.08505e23 q^{61} -9.68592e23 q^{63} +9.13240e23 q^{65} -2.46540e24 q^{67} -8.14627e23 q^{69} -1.29459e25 q^{71} +7.46465e24 q^{73} -1.11109e25 q^{75} +4.81415e25 q^{77} +4.94644e25 q^{79} +6.46108e24 q^{81} +5.54332e25 q^{83} +2.45325e25 q^{85} -4.38376e25 q^{87} -1.32973e26 q^{89} +5.01492e26 q^{91} +1.55924e26 q^{93} +6.00364e25 q^{95} -1.06222e27 q^{97} -3.21132e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9}+O(q^{10}) \) 2 * q + 3188646 * q^3 - 1771946100 * q^5 - 369665199904 * q^7 + 5083731656658 * q^9	\( 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9} - 75762335668248 q^{11} - 103021079177588 q^{13} - 28\!\cdots\!00 q^{15}+ \cdots - 19\!\cdots\!92 q^{99}+O(q^{100}) \) 2 * q + 3188646 * q^3 - 1771946100 * q^5 - 369665199904 * q^7 + 5083731656658 * q^9 - 75762335668248 * q^11 - 103021079177588 * q^13 - 2825054421990300 * q^15 + 34691611579147908 * q^17 - 111620476558642216 * q^19 - 589365730506544992 * q^21 + 2892743075819387952 * q^23 - 13257495997920171250 * q^25 + 8105110306037952534 * q^27 + 29959552473322806972 * q^29 - 10367463257055494032 * q^31 - 120789634289608156104 * q^33 + 252142979584030920000 * q^35 - 3703660908819524869316 * q^37 - 164248876017649632924 * q^39 + 1481814928479478034772 * q^41 - 9778778623457380249240 * q^43 - 4504049241230841066900 * q^45 - 89494998518809740323520 * q^47 + 13908377820361409575890 * q^49 + 55309634247701830126284 * q^51 + 425671275824601813476748 * q^53 + 33146870181194328159600 * q^55 - 177959093048404133739768 * q^57 - 369418769643380130292824 * q^59 + 925133397428892745249420 * q^61 - 939639339558386331280416 * q^63 - 394447136075147294825400 * q^65 - 4151706674555213948132968 * q^67 + 4611966818869594057796496 * q^69 - 9760455855870867537668784 * q^71 + 28449956347640075157409588 * q^73 - 21136730791892081187813750 * q^75 + 48717589762471785388668288 * q^77 + 49553221898675501744231120 * q^79 + 12922163778453346597864482 * q^81 + 120473408203228735926347736 * q^83 - 50977978054175038499879400 * q^85 + 47765203577925437580019956 * q^87 + 205644052209697133056985556 * q^89 + 515309018103795393250147648 * q^91 - 16529085122378486411580336 * q^93 + 87097784513216888583651600 * q^95 - 1478339006400409594836427772 * q^97 - 192577692109510944264197592 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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