LMFDB - Embedded newform 48.28.a.g.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:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3386644380 $ x^2 - x - 3386644380
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-58194.4$ 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)

$f(q)$	$=$	$q+1.59432e6 q^{3} +4.36927e9 q^{5} -3.11219e11 q^{7} +2.54187e12 q^{9} +O(q^{10})$	$q+1.59432e6 q^{3} +4.36927e9 q^{5} -3.11219e11 q^{7} +2.54187e12 q^{9} +6.06174e13 q^{11} +6.46570e14 q^{13} +6.96603e15 q^{15} +2.78493e16 q^{17} +3.23609e17 q^{19} -4.96184e17 q^{21} -2.77958e18 q^{23} +1.16400e19 q^{25} +4.05256e18 q^{27} -4.40804e18 q^{29} -1.35313e20 q^{31} +9.66437e19 q^{33} -1.35980e21 q^{35} +2.42430e21 q^{37} +1.03084e21 q^{39} -1.00752e22 q^{41} -6.87508e21 q^{43} +1.11061e22 q^{45} +7.32039e22 q^{47} +3.11452e22 q^{49} +4.44008e22 q^{51} +2.25305e23 q^{53} +2.64854e23 q^{55} +5.15938e23 q^{57} +6.81895e22 q^{59} +2.06905e24 q^{61} -7.91078e23 q^{63} +2.82504e24 q^{65} -5.18244e24 q^{67} -4.43155e24 q^{69} -9.12560e24 q^{71} -6.99708e24 q^{73} +1.85579e25 q^{75} -1.88653e25 q^{77} -3.46799e25 q^{79} +6.46108e24 q^{81} +5.24073e25 q^{83} +1.21681e26 q^{85} -7.02784e24 q^{87} +2.05205e26 q^{89} -2.01225e26 q^{91} -2.15733e26 q^{93} +1.41394e27 q^{95} +3.43252e26 q^{97} +1.54081e26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3188646 q^{3} + 291441036 q^{5} - 121646295328 q^{7} + 5083731656658 q^{9}+O(q^{10}) $ 2 * q + 3188646 * q^3 + 291441036 * q^5 - 121646295328 * q^7 + 5083731656658 * q^9	$ 2 q + 3188646 q^{3} + 291441036 q^{5} - 121646295328 q^{7} + 5083731656658 q^{9} + 231807361766376 q^{11} - 11\!\cdots\!64 q^{13}+ \cdots + 58\!\cdots\!04 q^{99}+O(q^{100}) $ 2 * q + 3188646 * q^3 + 291441036 * q^5 - 121646295328 * q^7 + 5083731656658 * q^9 + 231807361766376 * q^11 - 1155759271611764 * q^13 + 464651146838628 * q^15 - 46125175777027452 * q^17 + 268519854753239000 * q^19 - 193943486506222944 * q^21 - 1299561926005929936 * q^23 + 20818087556919436046 * q^25 + 8105110306037952534 * q^27 + 60178362995785587900 * q^29 - 203575085639304966544 * q^31 + 369575808433453883448 * q^33 - 2132849735738606710464 * q^35 + 3022337597514982851388 * q^37 - 1842653589193882415772 * q^39 - 10870375798082518905516 * q^41 + 1299568293415606314344 * q^43 + 740804010381201908844 * q^45 + 99372996476917699213632 * q^47 + 1370785920275118717906 * q^49 - 73538428620357738354996 * q^51 - 35770850049954566775924 * q^53 - 433229958906724722129552 * q^55 + 428107380389748262197000 * q^57 - 299672958231389705557080 * q^59 + 2165599091776519863171724 * q^61 - 309208561237060882746912 * q^63 + 10174637395564415468537928 * q^65 - 13568283837486181949525608 * q^67 - 2071921468555552233353328 * q^69 + 4319907418871433623040336 * q^71 - 20032895180752990538485964 * q^73 + 33190755808010466035166858 * q^75 + 13587711578810985760484736 * q^77 - 5633936436869082596599600 * q^79 + 12922163778453346597864482 * q^81 + 46762236922353070723154904 * q^83 + 423336865613342459682869784 * q^85 + 95943748226529865857491700 * q^87 - 191930414076503127699406380 * q^89 - 542898500680491813749530304 * q^91 - 324564441261713612175329712 * q^93 + 1638584179808499075287277840 * q^95 + 544725669490755670759486468 * q^97 + 589223211629049495820465704 * 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
$5$	4.36927e9	1.60072	0.800358	−	0.599523i	$-0.204643\pi$
$5$	4.36927e9	1.60072	0.800358	+	0.599523i	$0.204643\pi$
$6$	0	0
$7$	−3.11219e11	−1.21407	−0.607034	−	0.794676i	$-0.707641\pi$
$7$	−3.11219e11	−1.21407	−0.607034	+	0.794676i	$0.707641\pi$
$8$	0	0
$9$	2.54187e12	0.333333
$10$	0	0
$11$	6.06174e13	0.529415	0.264707	−	0.964329i	$-0.414725\pi$
$11$	6.06174e13	0.529415	0.264707	+	0.964329i	$0.414725\pi$
$12$	0	0
$13$	6.46570e14	0.592080	0.296040	−	0.955176i	$-0.404334\pi$
$13$	6.46570e14	0.592080	0.296040	+	0.955176i	$0.404334\pi$
$14$	0	0
$15$	6.96603e15	0.924173
$16$	0	0
$17$	2.78493e16	0.681953	0.340976	−	0.940072i	$-0.389242\pi$
$17$	2.78493e16	0.681953	0.340976	+	0.940072i	$0.389242\pi$
$18$	0	0
$19$	3.23609e17	1.76542	0.882709	−	0.469920i	$-0.155717\pi$
$19$	3.23609e17	1.76542	0.882709	+	0.469920i	$0.155717\pi$
$20$	0	0
$21$	−4.96184e17	−0.700943
$22$	0	0
$23$	−2.77958e18	−1.14988	−0.574941	−	0.818195i	$-0.694975\pi$
$23$	−2.77958e18	−1.14988	−0.574941	+	0.818195i	$0.694975\pi$
$24$	0	0
$25$	1.16400e19	1.56229
$26$	0	0
$27$	4.05256e18	0.192450
$28$	0	0
$29$	−4.40804e18	−0.0797761	−0.0398880	−	0.999204i	$-0.512700\pi$
$29$	−4.40804e18	−0.0797761	−0.0398880	+	0.999204i	$0.512700\pi$
$30$	0	0
$31$	−1.35313e20	−0.995309	−0.497654	−	0.867375i	$-0.665805\pi$
$31$	−1.35313e20	−0.995309	−0.497654	+	0.867375i	$0.665805\pi$
$32$	0	0
$33$	9.66437e19	0.305658
$34$	0	0
$35$	−1.35980e21	−1.94338
$36$	0	0
$37$	2.42430e21	1.63630	0.818150	−	0.575005i	$-0.195000\pi$
$37$	2.42430e21	1.63630	0.818150	+	0.575005i	$0.195000\pi$
$38$	0	0
$39$	1.03084e21	0.341838
$40$	0	0
$41$	−1.00752e22	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$41$	−1.00752e22	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$42$	0	0
$43$	−6.87508e21	−0.610174	−0.305087	−	0.952324i	$-0.598686\pi$
$43$	−6.87508e21	−0.610174	−0.305087	+	0.952324i	$0.598686\pi$
$44$	0	0
$45$	1.11061e22	0.533572
$46$	0	0
$47$	7.32039e22	1.95530	0.977650	−	0.210241i	$-0.0674249\pi$
$47$	7.32039e22	1.95530	0.977650	+	0.210241i	$0.0674249\pi$
$48$	0	0
$49$	3.11452e22	0.473962
$50$	0	0
$51$	4.44008e22	0.393726
$52$	0	0
$53$	2.25305e23	1.18863	0.594316	−	0.804232i	$-0.297423\pi$
$53$	2.25305e23	1.18863	0.594316	+	0.804232i	$0.297423\pi$
$54$	0	0
$55$	2.64854e23	0.847442
$56$	0	0
$57$	5.15938e23	1.01926
$58$	0	0
$59$	6.81895e22	0.0845698	0.0422849	−	0.999106i	$-0.486536\pi$
$59$	6.81895e22	0.0845698	0.0422849	+	0.999106i	$0.486536\pi$
$60$	0	0
$61$	2.06905e24	1.63613	0.818066	−	0.575124i	$-0.195046\pi$
$61$	2.06905e24	1.63613	0.818066	+	0.575124i	$0.195046\pi$
$62$	0	0
$63$	−7.91078e23	−0.404689
$64$	0	0
$65$	2.82504e24	0.947752
$66$	0	0
$67$	−5.18244e24	−1.15485	−0.577424	−	0.816444i	$-0.695942\pi$
$67$	−5.18244e24	−1.15485	−0.577424	+	0.816444i	$0.695942\pi$
$68$	0	0
$69$	−4.43155e24	−0.663885
$70$	0	0
$71$	−9.12560e24	−0.929552	−0.464776	−	0.885428i	$-0.653865\pi$
$71$	−9.12560e24	−0.929552	−0.464776	+	0.885428i	$0.653865\pi$
$72$	0	0
$73$	−6.99708e24	−0.489845	−0.244922	−	0.969543i	$-0.578762\pi$
$73$	−6.99708e24	−0.489845	−0.244922	+	0.969543i	$0.578762\pi$
$74$	0	0
$75$	1.85579e25	0.901988
$76$	0	0
$77$	−1.88653e25	−0.642746
$78$	0	0
$79$	−3.46799e25	−0.835818	−0.417909	−	0.908489i	$-0.637237\pi$
$79$	−3.46799e25	−0.835818	−0.417909	+	0.908489i	$0.637237\pi$
$80$	0	0
$81$	6.46108e24	0.111111
$82$	0	0
$83$	5.24073e25	0.648392	0.324196	−	0.945990i	$-0.394906\pi$
$83$	5.24073e25	0.648392	0.324196	+	0.945990i	$0.394906\pi$
$84$	0	0
$85$	1.21681e26	1.09161
$86$	0	0
$87$	−7.02784e24	−0.0460587
$88$	0	0
$89$	2.05205e26	0.989516	0.494758	−	0.869031i	$-0.335257\pi$
$89$	2.05205e26	0.989516	0.494758	+	0.869031i	$0.335257\pi$
$90$	0	0
$91$	−2.01225e26	−0.718826
$92$	0	0
$93$	−2.15733e26	−0.574642
$94$	0	0
$95$	1.41394e27	2.82593
$96$	0	0
$97$	3.43252e26	0.517839	0.258919	−	0.965899i	$-0.416634\pi$
$97$	3.43252e26	0.517839	0.258919	+	0.965899i	$0.416634\pi$
$98$	0	0
$99$	1.54081e26	0.176472
$100$	0	0
$101$	−5.43435e26	−0.475126	−0.237563	−	0.971372i	$-0.576349\pi$
$101$	−5.43435e26	−0.475126	−0.237563	+	0.971372i	$0.576349\pi$
$102$	0	0
$103$	−5.07992e25	−0.0340843	−0.0170421	−	0.999855i	$-0.505425\pi$
$103$	−5.07992e25	−0.0340843	−0.0170421	+	0.999855i	$0.505425\pi$
$104$	0	0
$105$	−2.16796e27	−1.12201
$106$	0	0
$107$	1.31347e27	0.526913	0.263456	−	0.964671i	$-0.415138\pi$
$107$	1.31347e27	0.526913	0.263456	+	0.964671i	$0.415138\pi$
$108$	0	0
$109$	3.20537e27	1.00143	0.500715	−	0.865612i	$-0.333071\pi$
$109$	3.20537e27	1.00143	0.500715	+	0.865612i	$0.333071\pi$
$110$	0	0
$111$	3.86512e27	0.944718
$112$	0	0
$113$	1.64220e27	0.315403	0.157702	−	0.987487i	$-0.449592\pi$
$113$	1.64220e27	0.315403	0.157702	+	0.987487i	$0.449592\pi$
$114$	0	0
$115$	−1.21447e28	−1.84063
$116$	0	0
$117$	1.64349e27	0.197360
$118$	0	0
$119$	−8.66725e27	−0.827937
$120$	0	0
$121$	−9.43553e27	−0.719720
$122$	0	0
$123$	−1.60632e28	−0.982003
$124$	0	0
$125$	1.83045e28	0.900065
$126$	0	0
$127$	−1.82085e28	−0.722644	−0.361322	−	0.932441i	$-0.617675\pi$
$127$	−1.82085e28	−0.722644	−0.361322	+	0.932441i	$0.617675\pi$
$128$	0	0
$129$	−1.09611e28	−0.352284
$130$	0	0
$131$	4.91416e28	1.28318	0.641589	−	0.767049i	$-0.278275\pi$
$131$	4.91416e28	1.28318	0.641589	+	0.767049i	$0.278275\pi$
$132$	0	0
$133$	−1.00714e29	−2.14334
$134$	0	0
$135$	1.77067e28	0.308058
$136$	0	0
$137$	5.37814e28	0.767193	0.383597	−	0.923501i	$-0.374685\pi$
$137$	5.37814e28	0.767193	0.383597	+	0.923501i	$0.374685\pi$
$138$	0	0
$139$	−4.60592e28	−0.540277	−0.270138	−	0.962821i	$-0.587069\pi$
$139$	−4.60592e28	−0.540277	−0.270138	+	0.962821i	$0.587069\pi$
$140$	0	0
$141$	1.16711e29	1.12889
$142$	0	0
$143$	3.91934e28	0.313456
$144$	0	0
$145$	−1.92599e28	−0.127699
$146$	0	0
$147$	4.96555e28	0.273642
$148$	0	0
$149$	6.39934e28	0.293846	0.146923	−	0.989148i	$-0.453063\pi$
$149$	6.39934e28	0.293846	0.146923	+	0.989148i	$0.453063\pi$
$150$	0	0
$151$	4.57912e29	1.75628	0.878139	−	0.478406i	$-0.158785\pi$
$151$	4.57912e29	1.75628	0.878139	+	0.478406i	$0.158785\pi$
$152$	0	0
$153$	7.07893e28	0.227318
$154$	0	0
$155$	−5.91221e29	−1.59321
$156$	0	0
$157$	−7.23684e29	−1.64023	−0.820113	−	0.572202i	$-0.806089\pi$
$157$	−7.23684e29	−1.64023	−0.820113	+	0.572202i	$0.806089\pi$
$158$	0	0
$159$	3.59209e29	0.686256
$160$	0	0
$161$	8.65059e29	1.39604
$162$	0	0
$163$	−4.01419e29	−0.548360	−0.274180	−	0.961678i	$-0.588406\pi$
$163$	−4.01419e29	−0.548360	−0.274180	+	0.961678i	$0.588406\pi$
$164$	0	0
$165$	4.22263e29	0.489271
$166$	0	0
$167$	1.03017e30	1.01447	0.507233	−	0.861809i	$-0.330668\pi$
$167$	1.03017e30	1.01447	0.507233	+	0.861809i	$0.330668\pi$
$168$	0	0
$169$	−7.74480e29	−0.649441
$170$	0	0
$171$	8.22572e29	0.588473
$172$	0	0
$173$	−2.68290e30	−1.64052	−0.820261	−	0.571989i	$-0.806172\pi$
$173$	−2.68290e30	−1.64052	−0.820261	+	0.571989i	$0.806172\pi$
$174$	0	0
$175$	−3.62258e30	−1.89673
$176$	0	0
$177$	1.08716e29	0.0488264
$178$	0	0
$179$	−2.21883e30	−0.856264	−0.428132	−	0.903716i	$-0.640828\pi$
$179$	−2.21883e30	−0.856264	−0.428132	+	0.903716i	$0.640828\pi$
$180$	0	0
$181$	4.61780e30	1.53382	0.766911	−	0.641754i	$-0.221793\pi$
$181$	4.61780e30	1.53382	0.766911	+	0.641754i	$0.221793\pi$
$182$	0	0
$183$	3.29874e30	0.944622
$184$	0	0
$185$	1.05924e31	2.61925
$186$	0	0
$187$	1.68815e30	0.361036
$188$	0	0
$189$	−1.26123e30	−0.233648
$190$	0	0
$191$	4.74656e30	0.762831	0.381415	−	0.924404i	$-0.375437\pi$
$191$	4.74656e30	0.762831	0.381415	+	0.924404i	$0.375437\pi$
$192$	0	0
$193$	2.52769e30	0.352940	0.176470	−	0.984306i	$-0.443532\pi$
$193$	2.52769e30	0.352940	0.176470	+	0.984306i	$0.443532\pi$
$194$	0	0
$195$	4.50403e30	0.547185
$196$	0	0
$197$	−2.31410e30	−0.244956	−0.122478	−	0.992471i	$-0.539084\pi$
$197$	−2.31410e30	−0.244956	−0.122478	+	0.992471i	$0.539084\pi$
$198$	0	0
$199$	3.03480e30	0.280294	0.140147	−	0.990131i	$-0.455242\pi$
$199$	3.03480e30	0.280294	0.140147	+	0.990131i	$0.455242\pi$
$200$	0	0
$201$	−8.26249e30	−0.666752
$202$	0	0
$203$	1.37187e30	0.0968536
$204$	0	0
$205$	−4.40215e31	−2.72262
$206$	0	0
$207$	−7.06532e30	−0.383294
$208$	0	0
$209$	1.96164e31	0.934638
$210$	0	0
$211$	1.95873e31	0.820655	0.410328	−	0.911938i	$-0.365414\pi$
$211$	1.95873e31	0.820655	0.410328	+	0.911938i	$0.365414\pi$
$212$	0	0
$213$	−1.45492e31	−0.536677
$214$	0	0
$215$	−3.00391e31	−0.976715
$216$	0	0
$217$	4.21122e31	1.20837
$218$	0	0
$219$	−1.11556e31	−0.282812
$220$	0	0
$221$	1.80065e31	0.403771
$222$	0	0
$223$	2.74869e31	0.545771	0.272886	−	0.962047i	$-0.412022\pi$
$223$	2.74869e31	0.545771	0.272886	+	0.962047i	$0.412022\pi$
$224$	0	0
$225$	2.95872e31	0.520763
$226$	0	0
$227$	1.10085e30	0.0171940	0.00859701	−	0.999963i	$-0.497263\pi$
$227$	1.10085e30	0.0171940	0.00859701	+	0.999963i	$0.497263\pi$
$228$	0	0
$229$	7.85209e31	1.08945	0.544724	−	0.838615i	$-0.316634\pi$
$229$	7.85209e31	1.08945	0.544724	+	0.838615i	$0.316634\pi$
$230$	0	0
$231$	−3.00774e31	−0.371089
$232$	0	0
$233$	1.08140e32	1.18763	0.593817	−	0.804600i	$-0.297620\pi$
$233$	1.08140e32	1.18763	0.593817	+	0.804600i	$0.297620\pi$
$234$	0	0
$235$	3.19848e32	3.12988
$236$	0	0
$237$	−5.52909e31	−0.482560
$238$	0	0
$239$	3.85813e31	0.300612	0.150306	−	0.988640i	$-0.451974\pi$
$239$	3.85813e31	0.300612	0.150306	+	0.988640i	$0.451974\pi$
$240$	0	0
$241$	2.43286e32	1.69389	0.846947	−	0.531677i	$-0.178438\pi$
$241$	2.43286e32	1.69389	0.846947	+	0.531677i	$0.178438\pi$
$242$	0	0
$243$	1.03011e31	0.0641500
$244$	0	0
$245$	1.36082e32	0.758678
$246$	0	0
$247$	2.09236e32	1.04527
$248$	0	0
$249$	8.35542e31	0.374349
$250$	0	0
$251$	−1.35106e31	−0.0543350	−0.0271675	−	0.999631i	$-0.508649\pi$
$251$	−1.35106e31	−0.0543350	−0.0271675	+	0.999631i	$0.508649\pi$
$252$	0	0
$253$	−1.68491e32	−0.608765
$254$	0	0
$255$	1.93999e32	0.630243
$256$	0	0
$257$	−2.50576e32	−0.732559	−0.366280	−	0.930505i	$-0.619369\pi$
$257$	−2.50576e32	−0.732559	−0.366280	+	0.930505i	$0.619369\pi$
$258$	0	0
$259$	−7.54489e32	−1.98658
$260$	0	0
$261$	−1.12046e31	−0.0265920
$262$	0	0
$263$	4.15757e32	0.890096	0.445048	−	0.895507i	$-0.353187\pi$
$263$	4.15757e32	0.890096	0.445048	+	0.895507i	$0.353187\pi$
$264$	0	0
$265$	9.84420e32	1.90266
$266$	0	0
$267$	3.27163e32	0.571297
$268$	0	0
$269$	−1.81575e32	−0.286682	−0.143341	−	0.989673i	$-0.545785\pi$
$269$	−1.81575e32	−0.286682	−0.143341	+	0.989673i	$0.545785\pi$
$270$	0	0
$271$	4.40915e32	0.629897	0.314949	−	0.949109i	$-0.398013\pi$
$271$	4.40915e32	0.629897	0.314949	+	0.949109i	$0.398013\pi$
$272$	0	0
$273$	−3.20818e32	−0.415014
$274$	0	0
$275$	7.05584e32	0.827099
$276$	0	0
$277$	4.47660e32	0.475852	0.237926	−	0.971283i	$-0.423532\pi$
$277$	4.47660e32	0.475852	0.237926	+	0.971283i	$0.423532\pi$
$278$	0	0
$279$	−3.43948e32	−0.331770
$280$	0	0
$281$	−8.16751e32	−0.715409	−0.357704	−	0.933835i	$-0.616440\pi$
$281$	−8.16751e32	−0.715409	−0.357704	+	0.933835i	$0.616440\pi$
$282$	0	0
$283$	3.19801e32	0.254544	0.127272	−	0.991868i	$-0.459378\pi$
$283$	3.19801e32	0.254544	0.127272	+	0.991868i	$0.459378\pi$
$284$	0	0
$285$	2.25427e33	1.63155
$286$	0	0
$287$	3.13561e33	2.06498
$288$	0	0
$289$	−8.92126e32	−0.534940
$290$	0	0
$291$	5.47255e32	0.298974
$292$	0	0
$293$	2.58899e33	1.28949	0.644744	−	0.764399i	$-0.276964\pi$
$293$	2.58899e33	1.28949	0.644744	+	0.764399i	$0.276964\pi$
$294$	0	0
$295$	2.97939e32	0.135372
$296$	0	0
$297$	2.45655e32	0.101886
$298$	0	0
$299$	−1.79719e33	−0.680823
$300$	0	0
$301$	2.13966e33	0.740793
$302$	0	0
$303$	−8.66410e32	−0.274314
$304$	0	0
$305$	9.04025e33	2.61898
$306$	0	0
$307$	−1.64496e33	−0.436303	−0.218152	−	0.975915i	$-0.570003\pi$
$307$	−1.64496e33	−0.436303	−0.218152	+	0.975915i	$0.570003\pi$
$308$	0	0
$309$	−8.09903e31	−0.0196786
$310$	0	0
$311$	6.49387e33	1.44623	0.723117	−	0.690725i	$-0.242709\pi$
$311$	6.49387e33	1.44623	0.723117	+	0.690725i	$0.242709\pi$
$312$	0	0
$313$	−8.69247e33	−1.77539	−0.887697	−	0.460428i	$-0.847696\pi$
$313$	−8.69247e33	−1.77539	−0.887697	+	0.460428i	$0.847696\pi$
$314$	0	0
$315$	−3.45644e33	−0.647793
$316$	0	0
$317$	1.93111e33	0.332282	0.166141	−	0.986102i	$-0.446869\pi$
$317$	1.93111e33	0.332282	0.166141	+	0.986102i	$0.446869\pi$
$318$	0	0
$319$	−2.67204e32	−0.0422346
$320$	0	0
$321$	2.09409e33	0.304213
$322$	0	0
$323$	9.01231e33	1.20393
$324$	0	0
$325$	7.52605e33	0.925000
$326$	0	0
$327$	5.11039e33	0.578176
$328$	0	0
$329$	−2.27825e34	−2.37387
$330$	0	0
$331$	8.64145e33	0.829676	0.414838	−	0.909895i	$-0.363838\pi$
$331$	8.64145e33	0.829676	0.414838	+	0.909895i	$0.363838\pi$
$332$	0	0
$333$	6.16224e33	0.545433
$334$	0	0
$335$	−2.26435e34	−1.84858
$336$	0	0
$337$	−2.03843e33	−0.153565	−0.0767826	−	0.997048i	$-0.524465\pi$
$337$	−2.03843e33	−0.153565	−0.0767826	+	0.997048i	$0.524465\pi$
$338$	0	0
$339$	2.61819e33	0.182098
$340$	0	0
$341$	−8.20234e33	−0.526931
$342$	0	0
$343$	1.07580e34	0.638646
$344$	0	0
$345$	−1.93626e34	−1.06269
$346$	0	0
$347$	−8.12227e33	−0.412315	−0.206158	−	0.978519i	$-0.566096\pi$
$347$	−8.12227e33	−0.412315	−0.206158	+	0.978519i	$0.566096\pi$
$348$	0	0
$349$	4.94258e33	0.232172	0.116086	−	0.993239i	$-0.462965\pi$
$349$	4.94258e33	0.232172	0.116086	+	0.993239i	$0.462965\pi$
$350$	0	0
$351$	2.62026e33	0.113946
$352$	0	0
$353$	1.97509e34	0.795478	0.397739	−	0.917499i	$-0.369795\pi$
$353$	1.97509e34	0.795478	0.397739	+	0.917499i	$0.369795\pi$
$354$	0	0
$355$	−3.98722e34	−1.48795
$356$	0	0
$357$	−1.38184e34	−0.478010
$358$	0	0
$359$	−1.80645e34	−0.579495	−0.289748	−	0.957103i	$-0.593571\pi$
$359$	−1.80645e34	−0.579495	−0.289748	+	0.957103i	$0.593571\pi$
$360$	0	0
$361$	7.11225e34	2.11670
$362$	0	0
$363$	−1.50433e34	−0.415531
$364$	0	0
$365$	−3.05722e34	−0.784102
$366$	0	0
$367$	2.27625e34	0.542286	0.271143	−	0.962539i	$-0.412598\pi$
$367$	2.27625e34	0.542286	0.271143	+	0.962539i	$0.412598\pi$
$368$	0	0
$369$	−2.56099e34	−0.566960
$370$	0	0
$371$	−7.01194e34	−1.44308
$372$	0	0
$373$	−6.70992e34	−1.28425	−0.642123	−	0.766601i	$-0.721946\pi$
$373$	−6.70992e34	−1.28425	−0.642123	+	0.766601i	$0.721946\pi$
$374$	0	0
$375$	2.91833e34	0.519652
$376$	0	0
$377$	−2.85011e33	−0.0472338
$378$	0	0
$379$	3.50560e34	0.540921	0.270460	−	0.962731i	$-0.412824\pi$
$379$	3.50560e34	0.540921	0.270460	+	0.962731i	$0.412824\pi$
$380$	0	0
$381$	−2.90303e34	−0.417219
$382$	0	0
$383$	−3.70361e33	−0.0495955	−0.0247978	−	0.999692i	$-0.507894\pi$
$383$	−3.70361e33	−0.0495955	−0.0247978	+	0.999692i	$0.507894\pi$
$384$	0	0
$385$	−8.24277e34	−1.02885
$386$	0	0
$387$	−1.74755e34	−0.203391
$388$	0	0
$389$	−7.02105e34	−0.762224	−0.381112	−	0.924529i	$-0.624459\pi$
$389$	−7.02105e34	−0.762224	−0.381112	+	0.924529i	$0.624459\pi$
$390$	0	0
$391$	−7.74095e34	−0.784166
$392$	0	0
$393$	7.83476e34	0.740843
$394$	0	0
$395$	−1.51526e35	−1.33791
$396$	0	0
$397$	−1.30976e35	−1.08024	−0.540118	−	0.841589i	$-0.681620\pi$
$397$	−1.30976e35	−1.08024	−0.540118	+	0.841589i	$0.681620\pi$
$398$	0	0
$399$	−1.60570e35	−1.23746
$400$	0	0
$401$	1.32379e35	0.953612	0.476806	−	0.879009i	$-0.341794\pi$
$401$	1.32379e35	0.953612	0.476806	+	0.879009i	$0.341794\pi$
$402$	0	0
$403$	−8.74896e34	−0.589303
$404$	0	0
$405$	2.82302e34	0.177857
$406$	0	0
$407$	1.46955e35	0.866281
$408$	0	0
$409$	−2.54536e35	−1.40438	−0.702190	−	0.711989i	$-0.747794\pi$
$409$	−2.54536e35	−1.40438	−0.702190	+	0.711989i	$0.747794\pi$
$410$	0	0
$411$	8.57450e34	0.442939
$412$	0	0
$413$	−2.12219e34	−0.102674
$414$	0	0
$415$	2.28982e35	1.03789
$416$	0	0
$417$	−7.34332e34	−0.311929
$418$	0	0
$419$	2.64873e35	1.05475	0.527373	−	0.849634i	$-0.323177\pi$
$419$	2.64873e35	1.05475	0.527373	+	0.849634i	$0.323177\pi$
$420$	0	0
$421$	8.67841e34	0.324065	0.162032	−	0.986785i	$-0.448195\pi$
$421$	8.67841e34	0.324065	0.162032	+	0.986785i	$0.448195\pi$
$422$	0	0
$423$	1.86075e35	0.651766
$424$	0	0
$425$	3.24165e35	1.06541
$426$	0	0
$427$	−6.43929e35	−1.98638
$428$	0	0
$429$	6.24869e34	0.180974
$430$	0	0
$431$	8.57490e34	0.233231	0.116615	−	0.993177i	$-0.462796\pi$
$431$	8.57490e34	0.233231	0.116615	+	0.993177i	$0.462796\pi$
$432$	0	0
$433$	−1.22839e35	−0.313870	−0.156935	−	0.987609i	$-0.550161\pi$
$433$	−1.22839e35	−0.313870	−0.156935	+	0.987609i	$0.550161\pi$
$434$	0	0
$435$	−3.07065e34	−0.0737269
$436$	0	0
$437$	−8.99499e35	−2.03002
$438$	0	0
$439$	2.95742e34	0.0627542	0.0313771	−	0.999508i	$-0.490011\pi$
$439$	2.95742e34	0.0627542	0.0313771	+	0.999508i	$0.490011\pi$
$440$	0	0
$441$	7.91669e34	0.157987
$442$	0	0
$443$	−2.15913e35	−0.405348	−0.202674	−	0.979246i	$-0.564963\pi$
$443$	−2.15913e35	−0.405348	−0.202674	+	0.979246i	$0.564963\pi$
$444$	0	0
$445$	8.96596e35	1.58393
$446$	0	0
$447$	1.02026e35	0.169652
$448$	0	0
$449$	2.22802e35	0.348813	0.174407	−	0.984674i	$-0.444199\pi$
$449$	2.22802e35	0.348813	0.174407	+	0.984674i	$0.444199\pi$
$450$	0	0
$451$	−6.10735e35	−0.900470
$452$	0	0
$453$	7.30060e35	1.01399
$454$	0	0
$455$	−8.79208e35	−1.15064
$456$	0	0
$457$	−5.04203e35	−0.621923	−0.310961	−	0.950423i	$-0.600651\pi$
$457$	−5.04203e35	−0.621923	−0.310961	+	0.950423i	$0.600651\pi$
$458$	0	0
$459$	1.12861e35	0.131242
$460$	0	0
$461$	−1.02767e36	−1.12691	−0.563456	−	0.826146i	$-0.690529\pi$
$461$	−1.02767e36	−1.12691	−0.563456	+	0.826146i	$0.690529\pi$
$462$	0	0
$463$	−1.28910e36	−1.33335	−0.666673	−	0.745350i	$-0.732282\pi$
$463$	−1.28910e36	−1.33335	−0.666673	+	0.745350i	$0.732282\pi$
$464$	0	0
$465$	−9.42597e35	−0.919838
$466$	0	0
$467$	1.89841e36	1.74829	0.874143	−	0.485668i	$-0.161424\pi$
$467$	1.89841e36	1.74829	0.874143	+	0.485668i	$0.161424\pi$
$468$	0	0
$469$	1.61288e36	1.40206
$470$	0	0
$471$	−1.15379e36	−0.946985
$472$	0	0
$473$	−4.16750e35	−0.323035
$474$	0	0
$475$	3.76680e36	2.75809
$476$	0	0
$477$	5.72696e35	0.396210
$478$	0	0
$479$	1.33288e36	0.871491	0.435746	−	0.900070i	$-0.356485\pi$
$479$	1.33288e36	0.871491	0.435746	+	0.900070i	$0.356485\pi$
$480$	0	0
$481$	1.56748e36	0.968821
$482$	0	0
$483$	1.37918e36	0.806002
$484$	0	0
$485$	1.49976e36	0.828913
$486$	0	0
$487$	3.24682e36	1.69753	0.848764	−	0.528772i	$-0.177347\pi$
$487$	3.24682e36	1.69753	0.848764	+	0.528772i	$0.177347\pi$
$488$	0	0
$489$	−6.39992e35	−0.316596
$490$	0	0
$491$	−5.80339e35	−0.271695	−0.135847	−	0.990730i	$-0.543376\pi$
$491$	−5.80339e35	−0.271695	−0.135847	+	0.990730i	$0.543376\pi$
$492$	0	0
$493$	−1.22761e35	−0.0544035
$494$	0	0
$495$	6.73223e35	0.282481
$496$	0	0
$497$	2.84006e36	1.12854
$498$	0	0
$499$	2.77354e36	1.04394	0.521972	−	0.852963i	$-0.325197\pi$
$499$	2.77354e36	1.04394	0.521972	+	0.852963i	$0.325197\pi$
$500$	0	0
$501$	1.64243e36	0.585702
$502$	0	0
$503$	−1.25022e36	−0.422491	−0.211245	−	0.977433i	$-0.567752\pi$
$503$	−1.25022e36	−0.422491	−0.211245	+	0.977433i	$0.567752\pi$
$504$	0	0
$505$	−2.37441e36	−0.760541
$506$	0	0
$507$	−1.23477e36	−0.374955
$508$	0	0
$509$	6.18070e36	1.77970	0.889852	−	0.456250i	$-0.150808\pi$
$509$	6.18070e36	1.77970	0.889852	+	0.456250i	$0.150808\pi$
$510$	0	0
$511$	2.17763e36	0.594705
$512$	0	0
$513$	1.31145e36	0.339755
$514$	0	0
$515$	−2.21955e35	−0.0545592
$516$	0	0
$517$	4.43743e36	1.03516
$518$	0	0
$519$	−4.27741e36	−0.947156
$520$	0	0
$521$	−4.44502e36	−0.934468	−0.467234	−	0.884134i	$-0.654749\pi$
$521$	−4.44502e36	−0.934468	−0.467234	+	0.884134i	$0.654749\pi$
$522$	0	0
$523$	−7.64607e36	−1.52639	−0.763195	−	0.646169i	$-0.776370\pi$
$523$	−7.64607e36	−1.52639	−0.763195	+	0.646169i	$0.776370\pi$
$524$	0	0
$525$	−5.77557e36	−1.09508
$526$	0	0
$527$	−3.76839e36	−0.678754
$528$	0	0
$529$	1.88286e36	0.322230
$530$	0	0
$531$	1.73329e35	0.0281899
$532$	0	0
$533$	−6.51435e36	−1.00706
$534$	0	0
$535$	5.73890e36	0.843437
$536$	0	0
$537$	−3.53753e36	−0.494364
$538$	0	0
$539$	1.88794e36	0.250923
$540$	0	0
$541$	−1.30472e37	−1.64950	−0.824751	−	0.565495i	$-0.808685\pi$
$541$	−1.30472e37	−1.64950	−0.824751	+	0.565495i	$0.808685\pi$
$542$	0	0
$543$	7.36227e36	0.885552
$544$	0	0
$545$	1.40051e37	1.60300
$546$	0	0
$547$	−1.03586e37	−1.12843	−0.564213	−	0.825629i	$-0.690820\pi$
$547$	−1.03586e37	−1.12843	−0.564213	+	0.825629i	$0.690820\pi$
$548$	0	0
$549$	5.25925e36	0.545378
$550$	0	0
$551$	−1.42648e36	−0.140838
$552$	0	0
$553$	1.07931e37	1.01474
$554$	0	0
$555$	1.68877e37	1.51222
$556$	0	0
$557$	−1.07139e37	−0.913904	−0.456952	−	0.889491i	$-0.651059\pi$
$557$	−1.07139e37	−0.913904	−0.456952	+	0.889491i	$0.651059\pi$
$558$	0	0
$559$	−4.44522e36	−0.361272
$560$	0	0
$561$	2.69146e36	0.208444
$562$	0	0
$563$	1.24198e37	0.916749	0.458374	−	0.888759i	$-0.348432\pi$
$563$	1.24198e37	0.916749	0.458374	+	0.888759i	$0.348432\pi$
$564$	0	0
$565$	7.17521e36	0.504871
$566$	0	0
$567$	−2.01081e36	−0.134896
$568$	0	0
$569$	−6.72192e36	−0.430009	−0.215005	−	0.976613i	$-0.568977\pi$
$569$	−6.72192e36	−0.430009	−0.215005	+	0.976613i	$0.568977\pi$
$570$	0	0
$571$	−6.75688e36	−0.412248	−0.206124	−	0.978526i	$-0.566085\pi$
$571$	−6.75688e36	−0.412248	−0.206124	+	0.978526i	$0.566085\pi$
$572$	0	0
$573$	7.56754e36	0.440420
$574$	0	0
$575$	−3.23542e37	−1.79645
$576$	0	0
$577$	−1.83494e37	−0.972184	−0.486092	−	0.873908i	$-0.661578\pi$
$577$	−1.83494e37	−0.972184	−0.486092	+	0.873908i	$0.661578\pi$
$578$	0	0
$579$	4.02996e36	0.203770
$580$	0	0
$581$	−1.63102e37	−0.787193
$582$	0	0
$583$	1.36574e37	0.629279
$584$	0	0
$585$	7.18088e36	0.315917
$586$	0	0
$587$	−1.38416e37	−0.581528	−0.290764	−	0.956795i	$-0.593910\pi$
$587$	−1.38416e37	−0.581528	−0.290764	+	0.956795i	$0.593910\pi$
$588$	0	0
$589$	−4.37887e37	−1.75714
$590$	0	0
$591$	−3.68943e36	−0.141426
$592$	0	0
$593$	−5.74168e36	−0.210281	−0.105141	−	0.994457i	$-0.533529\pi$
$593$	−5.74168e36	−0.210281	−0.105141	+	0.994457i	$0.533529\pi$
$594$	0	0
$595$	−3.78696e37	−1.32529
$596$	0	0
$597$	4.83845e36	0.161828
$598$	0	0
$599$	−9.06518e36	−0.289811	−0.144905	−	0.989446i	$-0.546288\pi$
$599$	−9.06518e36	−0.289811	−0.144905	+	0.989446i	$0.546288\pi$
$600$	0	0
$601$	3.75922e36	0.114893	0.0574464	−	0.998349i	$-0.481704\pi$
$601$	3.75922e36	0.114893	0.0574464	+	0.998349i	$0.481704\pi$
$602$	0	0
$603$	−1.31731e37	−0.384949
$604$	0	0
$605$	−4.12264e37	−1.15207
$606$	0	0
$607$	4.26047e37	1.13870	0.569351	−	0.822094i	$-0.307194\pi$
$607$	4.26047e37	1.13870	0.569351	+	0.822094i	$0.307194\pi$
$608$	0	0
$609$	2.18720e36	0.0559184
$610$	0	0
$611$	4.73315e37	1.15769
$612$	0	0
$613$	5.38870e37	1.26115	0.630576	−	0.776127i	$-0.282819\pi$
$613$	5.38870e37	1.26115	0.630576	+	0.776127i	$0.282819\pi$
$614$	0	0
$615$	−7.01844e37	−1.57191
$616$	0	0
$617$	−3.49858e37	−0.749967	−0.374983	−	0.927031i	$-0.622352\pi$
$617$	−3.49858e37	−0.749967	−0.374983	+	0.927031i	$0.622352\pi$
$618$	0	0
$619$	−3.09916e37	−0.635946	−0.317973	−	0.948100i	$-0.603002\pi$
$619$	−3.09916e37	−0.635946	−0.317973	+	0.948100i	$0.603002\pi$
$620$	0	0
$621$	−1.12644e37	−0.221295
$622$	0	0
$623$	−6.38638e37	−1.20134
$624$	0	0
$625$	−6.74694e36	−0.121542
$626$	0	0
$627$	3.12748e37	0.539614
$628$	0	0
$629$	6.75151e37	1.11588
$630$	0	0
$631$	7.75162e36	0.122743	0.0613714	−	0.998115i	$-0.480453\pi$
$631$	7.75162e36	0.122743	0.0613714	+	0.998115i	$0.480453\pi$
$632$	0	0
$633$	3.12284e37	0.473806
$634$	0	0
$635$	−7.95580e37	−1.15675
$636$	0	0
$637$	2.01375e37	0.280624
$638$	0	0
$639$	−2.31961e37	−0.309851
$640$	0	0
$641$	−7.99511e37	−1.02386	−0.511931	−	0.859027i	$-0.671070\pi$
$641$	−7.99511e37	−1.02386	−0.511931	+	0.859027i	$0.671070\pi$
$642$	0	0
$643$	−2.47951e37	−0.304451	−0.152226	−	0.988346i	$-0.548644\pi$
$643$	−2.47951e37	−0.304451	−0.152226	+	0.988346i	$0.548644\pi$
$644$	0	0
$645$	−4.78920e37	−0.563907
$646$	0	0
$647$	3.52397e37	0.397946	0.198973	−	0.980005i	$-0.436239\pi$
$647$	3.52397e37	0.397946	0.198973	+	0.980005i	$0.436239\pi$
$648$	0	0
$649$	4.13347e36	0.0447725
$650$	0	0
$651$	6.71404e37	0.697655
$652$	0	0
$653$	1.58081e37	0.157599	0.0787993	−	0.996891i	$-0.474891\pi$
$653$	1.58081e37	0.157599	0.0787993	+	0.996891i	$0.474891\pi$
$654$	0	0
$655$	2.14713e38	2.05400
$656$	0	0
$657$	−1.77856e37	−0.163282
$658$	0	0
$659$	−2.71928e37	−0.239608	−0.119804	−	0.992798i	$-0.538227\pi$
$659$	−2.71928e37	−0.239608	−0.119804	+	0.992798i	$0.538227\pi$
$660$	0	0
$661$	8.48665e36	0.0717821	0.0358911	−	0.999356i	$-0.488573\pi$
$661$	8.48665e36	0.0717821	0.0358911	+	0.999356i	$0.488573\pi$
$662$	0	0
$663$	2.87083e37	0.233117
$664$	0	0
$665$	−4.40045e38	−3.43087
$666$	0	0
$667$	1.22525e37	0.0917331
$668$	0	0
$669$	4.38230e37	0.315101
$670$	0	0
$671$	1.25421e38	0.866193
$672$	0	0
$673$	2.46452e38	1.63504	0.817522	−	0.575898i	$-0.195347\pi$
$673$	2.46452e38	1.63504	0.817522	+	0.575898i	$0.195347\pi$
$674$	0	0
$675$	4.71716e37	0.300663
$676$	0	0
$677$	2.19397e38	1.34364	0.671822	−	0.740713i	$-0.265512\pi$
$677$	2.19397e38	1.34364	0.671822	+	0.740713i	$0.265512\pi$
$678$	0	0
$679$	−1.06827e38	−0.628692
$680$	0	0
$681$	1.75510e36	0.00992698
$682$	0	0
$683$	−1.45078e38	−0.788717	−0.394358	−	0.918957i	$-0.629033\pi$
$683$	−1.45078e38	−0.788717	−0.394358	+	0.918957i	$0.629033\pi$
$684$	0	0
$685$	2.34986e38	1.22806
$686$	0	0
$687$	1.25188e38	0.628993
$688$	0	0
$689$	1.45676e38	0.703765
$690$	0	0
$691$	−4.04073e38	−1.87718	−0.938591	−	0.345031i	$-0.887869\pi$
$691$	−4.04073e38	−1.87718	−0.938591	+	0.345031i	$0.887869\pi$
$692$	0	0
$693$	−4.79531e37	−0.214249
$694$	0	0
$695$	−2.01245e38	−0.864829
$696$	0	0
$697$	−2.80589e38	−1.15992
$698$	0	0
$699$	1.72411e38	0.685681
$700$	0	0
$701$	6.40419e37	0.245059	0.122529	−	0.992465i	$-0.460899\pi$
$701$	6.40419e37	0.245059	0.122529	+	0.992465i	$0.460899\pi$
$702$	0	0
$703$	7.84526e38	2.88875
$704$	0	0
$705$	5.09941e38	1.80704
$706$	0	0
$707$	1.69127e38	0.576835
$708$	0	0
$709$	7.71389e37	0.253250	0.126625	−	0.991951i	$-0.459586\pi$
$709$	7.71389e37	0.253250	0.126625	+	0.991951i	$0.459586\pi$
$710$	0	0
$711$	−8.81516e37	−0.278606
$712$	0	0
$713$	3.76114e38	1.14449
$714$	0	0
$715$	1.71247e38	0.501754
$716$	0	0
$717$	6.15111e37	0.173558
$718$	0	0
$719$	−6.01226e38	−1.63380	−0.816898	−	0.576783i	$-0.804308\pi$
$719$	−6.01226e38	−1.63380	−0.816898	+	0.576783i	$0.804308\pi$
$720$	0	0
$721$	1.58097e37	0.0413806
$722$	0	0
$723$	3.87876e38	0.977971
$724$	0	0
$725$	−5.13094e37	−0.124633
$726$	0	0
$727$	5.96046e38	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$727$	5.96046e38	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$728$	0	0
$729$	1.64232e37	0.0370370
$730$	0	0
$731$	−1.91466e38	−0.416110
$732$	0	0
$733$	−3.15157e38	−0.660120	−0.330060	−	0.943960i	$-0.607069\pi$
$733$	−3.15157e38	−0.660120	−0.330060	+	0.943960i	$0.607069\pi$
$734$	0	0
$735$	2.16958e38	0.438023
$736$	0	0
$737$	−3.14146e38	−0.611393
$738$	0	0
$739$	3.24282e38	0.608447	0.304223	−	0.952601i	$-0.401603\pi$
$739$	3.24282e38	0.608447	0.304223	+	0.952601i	$0.401603\pi$
$740$	0	0
$741$	3.33590e38	0.603486
$742$	0	0
$743$	−5.80289e38	−1.01227	−0.506134	−	0.862455i	$-0.668926\pi$
$743$	−5.80289e38	−1.01227	−0.506134	+	0.862455i	$0.668926\pi$
$744$	0	0
$745$	2.79604e38	0.470364
$746$	0	0
$747$	1.33212e38	0.216131
$748$	0	0
$749$	−4.08777e38	−0.639708
$750$	0	0
$751$	−2.21124e38	−0.333808	−0.166904	−	0.985973i	$-0.553377\pi$
$751$	−2.21124e38	−0.333808	−0.166904	+	0.985973i	$0.553377\pi$
$752$	0	0
$753$	−2.15403e37	−0.0313703
$754$	0	0
$755$	2.00074e39	2.81130
$756$	0	0
$757$	−2.01995e38	−0.273870	−0.136935	−	0.990580i	$-0.543725\pi$
$757$	−2.01995e38	−0.273870	−0.136935	+	0.990580i	$0.543725\pi$
$758$	0	0
$759$	−2.68629e38	−0.351470
$760$	0	0
$761$	6.78605e38	0.856888	0.428444	−	0.903568i	$-0.359062\pi$
$761$	6.78605e38	0.856888	0.428444	+	0.903568i	$0.359062\pi$
$762$	0	0
$763$	−9.97573e38	−1.21580
$764$	0	0
$765$	3.09298e38	0.363871
$766$	0	0
$767$	4.40893e37	0.0500721
$768$	0	0
$769$	9.79619e38	1.07412	0.537059	−	0.843545i	$-0.319535\pi$
$769$	9.79619e38	1.07412	0.537059	+	0.843545i	$0.319535\pi$
$770$	0	0
$771$	−3.99500e38	−0.422943
$772$	0	0
$773$	−1.58678e39	−1.62216	−0.811081	−	0.584934i	$-0.801120\pi$
$773$	−1.58678e39	−1.62216	−0.811081	+	0.584934i	$0.801120\pi$
$774$	0	0
$775$	−1.57504e39	−1.55496
$776$	0	0
$777$	−1.20290e39	−1.14695
$778$	0	0
$779$	−3.26044e39	−3.00276
$780$	0	0
$781$	−5.53170e38	−0.492119
$782$	0	0
$783$	−1.78638e37	−0.0153529
$784$	0	0
$785$	−3.16197e39	−2.62553
$786$	0	0
$787$	1.82547e39	1.46459	0.732296	−	0.680987i	$-0.238449\pi$
$787$	1.82547e39	1.46459	0.732296	+	0.680987i	$0.238449\pi$
$788$	0	0
$789$	6.62851e38	0.513897
$790$	0	0
$791$	−5.11084e38	−0.382921
$792$	0	0
$793$	1.33779e39	0.968722
$794$	0	0
$795$	1.56948e39	1.09850
$796$	0	0
$797$	−6.64299e38	−0.449445	−0.224723	−	0.974423i	$-0.572148\pi$
$797$	−6.64299e38	−0.449445	−0.224723	+	0.974423i	$0.572148\pi$
$798$	0	0
$799$	2.03868e39	1.33342
$800$	0	0
$801$	5.21603e38	0.329839
$802$	0	0
$803$	−4.24145e38	−0.259331
$804$	0	0
$805$	3.77968e39	2.23466
$806$	0	0
$807$	−2.89489e38	−0.165516
$808$	0	0
$809$	−1.75085e39	−0.968150	−0.484075	−	0.875026i	$-0.660844\pi$
$809$	−1.75085e39	−0.968150	−0.484075	+	0.875026i	$0.660844\pi$
$810$	0	0
$811$	−1.01209e39	−0.541301	−0.270651	−	0.962678i	$-0.587239\pi$
$811$	−1.01209e39	−0.541301	−0.270651	+	0.962678i	$0.587239\pi$
$812$	0	0
$813$	7.02961e38	0.363671
$814$	0	0
$815$	−1.75391e39	−0.877768
$816$	0	0
$817$	−2.22484e39	−1.07721
$818$	0	0
$819$	−5.11488e38	−0.239609
$820$	0	0
$821$	−8.40933e38	−0.381179	−0.190589	−	0.981670i	$-0.561040\pi$
$821$	−8.40933e38	−0.381179	−0.190589	+	0.981670i	$0.561040\pi$
$822$	0	0
$823$	−2.05478e39	−0.901297	−0.450649	−	0.892701i	$-0.648807\pi$
$823$	−2.05478e39	−0.901297	−0.450649	+	0.892701i	$0.648807\pi$
$824$	0	0
$825$	1.12493e39	0.477526
$826$	0	0
$827$	9.27685e38	0.381133	0.190566	−	0.981674i	$-0.438968\pi$
$827$	9.27685e38	0.381133	0.190566	+	0.981674i	$0.438968\pi$
$828$	0	0
$829$	3.98790e39	1.58584	0.792918	−	0.609329i	$-0.208561\pi$
$829$	3.98790e39	1.58584	0.792918	+	0.609329i	$0.208561\pi$
$830$	0	0
$831$	7.13715e38	0.274733
$832$	0	0
$833$	8.67372e38	0.323220
$834$	0	0
$835$	4.50110e39	1.62387
$836$	0	0
$837$	−5.48365e38	−0.191547
$838$	0	0
$839$	−4.87871e39	−1.65013	−0.825066	−	0.565036i	$-0.808862\pi$
$839$	−4.87871e39	−1.65013	−0.825066	+	0.565036i	$0.808862\pi$
$840$	0	0
$841$	−3.03370e39	−0.993636
$842$	0	0
$843$	−1.30216e39	−0.413042
$844$	0	0
$845$	−3.38391e39	−1.03957
$846$	0	0
$847$	2.93652e39	0.873789
$848$	0	0
$849$	5.09866e38	0.146961
$850$	0	0
$851$	−6.73853e39	−1.88155
$852$	0	0
$853$	4.52266e38	0.122344	0.0611720	−	0.998127i	$-0.480516\pi$
$853$	4.52266e38	0.122344	0.0611720	+	0.998127i	$0.480516\pi$
$854$	0	0
$855$	3.59404e39	0.941977
$856$	0	0
$857$	−7.11106e39	−1.80590	−0.902949	−	0.429748i	$-0.858602\pi$
$857$	−7.11106e39	−1.80590	−0.902949	+	0.429748i	$0.858602\pi$
$858$	0	0
$859$	−3.37936e39	−0.831624	−0.415812	−	0.909451i	$-0.636503\pi$
$859$	−3.37936e39	−0.831624	−0.415812	+	0.909451i	$0.636503\pi$
$860$	0	0
$861$	4.99918e39	1.19222
$862$	0	0
$863$	4.33269e39	1.00141	0.500705	−	0.865618i	$-0.333074\pi$
$863$	4.33269e39	1.00141	0.500705	+	0.865618i	$0.333074\pi$
$864$	0	0
$865$	−1.17223e40	−2.62601
$866$	0	0
$867$	−1.42234e39	−0.308848
$868$	0	0
$869$	−2.10220e39	−0.442494
$870$	0	0
$871$	−3.35081e39	−0.683763
$872$	0	0
$873$	8.72501e38	0.172613
$874$	0	0
$875$	−5.69673e39	−1.09274
$876$	0	0
$877$	−5.02589e39	−0.934800	−0.467400	−	0.884046i	$-0.654809\pi$
$877$	−5.02589e39	−0.934800	−0.467400	+	0.884046i	$0.654809\pi$
$878$	0	0
$879$	4.12768e39	0.744486
$880$	0	0
$881$	−8.13080e39	−1.42219	−0.711097	−	0.703094i	$-0.751801\pi$
$881$	−8.13080e39	−1.42219	−0.711097	+	0.703094i	$0.751801\pi$
$882$	0	0
$883$	8.35207e39	1.41685	0.708427	−	0.705784i	$-0.249405\pi$
$883$	8.35207e39	1.41685	0.708427	+	0.705784i	$0.249405\pi$
$884$	0	0
$885$	4.75010e38	0.0781572
$886$	0	0
$887$	−3.26135e39	−0.520510	−0.260255	−	0.965540i	$-0.583807\pi$
$887$	−3.26135e39	−0.520510	−0.260255	+	0.965540i	$0.583807\pi$
$888$	0	0
$889$	5.66685e39	0.877339
$890$	0	0
$891$	3.91654e38	0.0588239
$892$	0	0
$893$	2.36895e40	3.45192
$894$	0	0
$895$	−9.69467e39	−1.37064
$896$	0	0
$897$	−2.86531e39	−0.393073
$898$	0	0
$899$	5.96467e38	0.0794018
$900$	0	0
$901$	6.27460e39	0.810590
$902$	0	0
$903$	3.41131e39	0.427697
$904$	0	0
$905$	2.01764e40	2.45521
$906$	0	0
$907$	1.85496e39	0.219096	0.109548	−	0.993981i	$-0.465060\pi$
$907$	1.85496e39	0.219096	0.109548	+	0.993981i	$0.465060\pi$
$908$	0	0
$909$	−1.38134e39	−0.158375
$910$	0	0
$911$	1.23539e40	1.37501	0.687503	−	0.726182i	$-0.258707\pi$
$911$	1.23539e40	1.37501	0.687503	+	0.726182i	$0.258707\pi$
$912$	0	0
$913$	3.17680e39	0.343268
$914$	0	0
$915$	1.44131e40	1.51207
$916$	0	0
$917$	−1.52938e40	−1.55787
$918$	0	0
$919$	−9.73771e39	−0.963157	−0.481579	−	0.876403i	$-0.659936\pi$
$919$	−9.73771e39	−0.963157	−0.481579	+	0.876403i	$0.659936\pi$
$920$	0	0
$921$	−2.62261e39	−0.251900
$922$	0	0
$923$	−5.90034e39	−0.550369
$924$	0	0
$925$	2.82187e40	2.55637
$926$	0	0
$927$	−1.29125e38	−0.0113614
$928$	0	0
$929$	−6.31554e38	−0.0539757	−0.0269879	−	0.999636i	$-0.508592\pi$
$929$	−6.31554e38	−0.0539757	−0.0269879	+	0.999636i	$0.508592\pi$
$930$	0	0
$931$	1.00789e40	0.836741
$932$	0	0
$933$	1.03533e40	0.834984
$934$	0	0
$935$	7.37600e39	0.577916
$936$	0	0
$937$	9.85113e38	0.0749897	0.0374948	−	0.999297i	$-0.488062\pi$
$937$	9.85113e38	0.0749897	0.0374948	+	0.999297i	$0.488062\pi$
$938$	0	0
$939$	−1.38586e40	−1.02502
$940$	0	0
$941$	−1.59948e40	−1.14953	−0.574763	−	0.818320i	$-0.694906\pi$
$941$	−1.59948e40	−1.14953	−0.574763	+	0.818320i	$0.694906\pi$
$942$	0	0
$943$	2.80049e40	1.95581
$944$	0	0
$945$	−5.51067e39	−0.374003
$946$	0	0
$947$	7.63606e38	0.0503669	0.0251834	−	0.999683i	$-0.491983\pi$
$947$	7.63606e38	0.0503669	0.0251834	+	0.999683i	$0.491983\pi$
$948$	0	0
$949$	−4.52411e39	−0.290027
$950$	0	0
$951$	3.07881e39	0.191843
$952$	0	0
$953$	−2.87976e40	−1.74422	−0.872112	−	0.489307i	$-0.837250\pi$
$953$	−2.87976e40	−1.74422	−0.872112	+	0.489307i	$0.837250\pi$
$954$	0	0
$955$	2.07390e40	1.22107
$956$	0	0
$957$	−4.26009e38	−0.0243842
$958$	0	0
$959$	−1.67378e40	−0.931425
$960$	0	0
$961$	−1.73001e38	−0.00936016
$962$	0	0
$963$	3.33866e39	0.175638
$964$	0	0
$965$	1.10442e40	0.564956
$966$	0	0
$967$	−2.00353e39	−0.0996639	−0.0498319	−	0.998758i	$-0.515869\pi$
$967$	−2.00353e39	−0.0996639	−0.0498319	+	0.998758i	$0.515869\pi$
$968$	0	0
$969$	1.43685e40	0.695090
$970$	0	0
$971$	−1.39930e40	−0.658343	−0.329172	−	0.944270i	$-0.606769\pi$
$971$	−1.39930e40	−0.658343	−0.329172	+	0.944270i	$0.606769\pi$
$972$	0	0
$973$	1.43345e40	0.655933
$974$	0	0
$975$	1.19990e40	0.534049
$976$	0	0
$977$	−1.69499e39	−0.0733824	−0.0366912	−	0.999327i	$-0.511682\pi$
$977$	−1.69499e39	−0.0733824	−0.0366912	+	0.999327i	$0.511682\pi$
$978$	0	0
$979$	1.24390e40	0.523864
$980$	0	0
$981$	8.14762e39	0.333810
$982$	0	0
$983$	−1.61199e40	−0.642526	−0.321263	−	0.946990i	$-0.604107\pi$
$983$	−1.61199e40	−0.642526	−0.321263	+	0.946990i	$0.604107\pi$
$984$	0	0
$985$	−1.01110e40	−0.392106
$986$	0	0
$987$	−3.63226e40	−1.37055
$988$	0	0
$989$	1.91098e40	0.701629
$990$	0	0
$991$	−6.12985e39	−0.219006	−0.109503	−	0.993986i	$-0.534926\pi$
$991$	−6.12985e39	−0.219006	−0.109503	+	0.993986i	$0.534926\pi$
$992$	0	0
$993$	1.37773e40	0.479014
$994$	0	0
$995$	1.32599e40	0.448671
$996$	0	0
$997$	1.07857e40	0.355191	0.177596	−	0.984104i	$-0.443168\pi$
$997$	1.07857e40	0.355191	0.177596	+	0.984104i	$0.443168\pi$
$998$	0	0
$999$	9.82461e39	0.314906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.28.a.g.1.2		2
4.3	odd	2		6.28.a.d.1.2	✓	2
12.11	even	2		18.28.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.28.a.d.1.2	✓	2	4.3	odd	2
18.28.a.g.1.1		2	12.11	even	2
48.28.a.g.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)

\(f(q)\)	\(=\)	\(q+1.59432e6 q^{3} +4.36927e9 q^{5} -3.11219e11 q^{7} +2.54187e12 q^{9} +O(q^{10})\)	\(q+1.59432e6 q^{3} +4.36927e9 q^{5} -3.11219e11 q^{7} +2.54187e12 q^{9} +6.06174e13 q^{11} +6.46570e14 q^{13} +6.96603e15 q^{15} +2.78493e16 q^{17} +3.23609e17 q^{19} -4.96184e17 q^{21} -2.77958e18 q^{23} +1.16400e19 q^{25} +4.05256e18 q^{27} -4.40804e18 q^{29} -1.35313e20 q^{31} +9.66437e19 q^{33} -1.35980e21 q^{35} +2.42430e21 q^{37} +1.03084e21 q^{39} -1.00752e22 q^{41} -6.87508e21 q^{43} +1.11061e22 q^{45} +7.32039e22 q^{47} +3.11452e22 q^{49} +4.44008e22 q^{51} +2.25305e23 q^{53} +2.64854e23 q^{55} +5.15938e23 q^{57} +6.81895e22 q^{59} +2.06905e24 q^{61} -7.91078e23 q^{63} +2.82504e24 q^{65} -5.18244e24 q^{67} -4.43155e24 q^{69} -9.12560e24 q^{71} -6.99708e24 q^{73} +1.85579e25 q^{75} -1.88653e25 q^{77} -3.46799e25 q^{79} +6.46108e24 q^{81} +5.24073e25 q^{83} +1.21681e26 q^{85} -7.02784e24 q^{87} +2.05205e26 q^{89} -2.01225e26 q^{91} -2.15733e26 q^{93} +1.41394e27 q^{95} +3.43252e26 q^{97} +1.54081e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3188646 q^{3} + 291441036 q^{5} - 121646295328 q^{7} + 5083731656658 q^{9}+O(q^{10}) \) 2 * q + 3188646 * q^3 + 291441036 * q^5 - 121646295328 * q^7 + 5083731656658 * q^9	\( 2 q + 3188646 q^{3} + 291441036 q^{5} - 121646295328 q^{7} + 5083731656658 q^{9} + 231807361766376 q^{11} - 11\!\cdots\!64 q^{13}+ \cdots + 58\!\cdots\!04 q^{99}+O(q^{100}) \) 2 * q + 3188646 * q^3 + 291441036 * q^5 - 121646295328 * q^7 + 5083731656658 * q^9 + 231807361766376 * q^11 - 1155759271611764 * q^13 + 464651146838628 * q^15 - 46125175777027452 * q^17 + 268519854753239000 * q^19 - 193943486506222944 * q^21 - 1299561926005929936 * q^23 + 20818087556919436046 * q^25 + 8105110306037952534 * q^27 + 60178362995785587900 * q^29 - 203575085639304966544 * q^31 + 369575808433453883448 * q^33 - 2132849735738606710464 * q^35 + 3022337597514982851388 * q^37 - 1842653589193882415772 * q^39 - 10870375798082518905516 * q^41 + 1299568293415606314344 * q^43 + 740804010381201908844 * q^45 + 99372996476917699213632 * q^47 + 1370785920275118717906 * q^49 - 73538428620357738354996 * q^51 - 35770850049954566775924 * q^53 - 433229958906724722129552 * q^55 + 428107380389748262197000 * q^57 - 299672958231389705557080 * q^59 + 2165599091776519863171724 * q^61 - 309208561237060882746912 * q^63 + 10174637395564415468537928 * q^65 - 13568283837486181949525608 * q^67 - 2071921468555552233353328 * q^69 + 4319907418871433623040336 * q^71 - 20032895180752990538485964 * q^73 + 33190755808010466035166858 * q^75 + 13587711578810985760484736 * q^77 - 5633936436869082596599600 * q^79 + 12922163778453346597864482 * q^81 + 46762236922353070723154904 * q^83 + 423336865613342459682869784 * q^85 + 95943748226529865857491700 * q^87 - 191930414076503127699406380 * q^89 - 542898500680491813749530304 * q^91 - 324564441261713612175329712 * q^93 + 1638584179808499075287277840 * q^95 + 544725669490755670759486468 * q^97 + 589223211629049495820465704 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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