LMFDB - Embedded newform 16.28.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,28,Mod(1,16)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("16.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 28 $
Character orbit:	$[\chi]$	$=$	16.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:	$73.8968919741$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{18209}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4552 $ x^2 - x - 4552
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$67.9704$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15499e6 q^{3} +4.86703e9 q^{5} -2.14815e10 q^{7} -2.98161e12 q^{9} +O(q^{10})$	$q-2.15499e6 q^{3} +4.86703e9 q^{5} -2.14815e10 q^{7} -2.98161e12 q^{9} +6.85469e13 q^{11} -5.86777e14 q^{13} -1.04884e16 q^{15} -2.42814e16 q^{17} -2.43094e17 q^{19} +4.62924e16 q^{21} -1.25668e17 q^{23} +1.62374e19 q^{25} +2.28584e19 q^{27} -1.25483e19 q^{29} +6.23745e19 q^{31} -1.47718e20 q^{33} -1.04551e20 q^{35} +3.67692e20 q^{37} +1.26450e21 q^{39} +3.60988e21 q^{41} -1.19560e22 q^{43} -1.45116e22 q^{45} -8.26632e21 q^{47} -6.52509e22 q^{49} +5.23263e22 q^{51} +1.60547e23 q^{53} +3.33619e23 q^{55} +5.23866e23 q^{57} -1.09336e24 q^{59} -1.32576e24 q^{61} +6.40495e22 q^{63} -2.85586e24 q^{65} -6.27072e24 q^{67} +2.70814e23 q^{69} +1.65578e25 q^{71} -7.93292e24 q^{73} -3.49914e25 q^{75} -1.47249e24 q^{77} -1.61913e25 q^{79} -2.65232e25 q^{81} -1.00338e26 q^{83} -1.18178e26 q^{85} +2.70414e25 q^{87} -1.72766e26 q^{89} +1.26049e25 q^{91} -1.34416e26 q^{93} -1.18315e27 q^{95} -1.03077e27 q^{97} -2.04380e26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1286280 q^{3} + 5443587900 q^{5} + 175391963600 q^{7} + 1235136554154 q^{9}+O(q^{10}) $ 2 * q + 1286280 * q^3 + 5443587900 * q^5 + 175391963600 * q^7 + 1235136554154 * q^9	$ 2 q + 1286280 q^{3} + 5443587900 q^{5} + 175391963600 q^{7} + 1235136554154 q^{9} - 138167337691944 q^{11} - 753433801271060 q^{13} - 85\!\cdots\!00 q^{15}+ \cdots - 10\!\cdots\!88 q^{99}+O(q^{100}) $ 2 * q + 1286280 * q^3 + 5443587900 * q^5 + 175391963600 * q^7 + 1235136554154 * q^9 - 138167337691944 * q^11 - 753433801271060 * q^13 - 8504300488438800 * q^15 - 29753620331011740 * q^17 - 404565810372684760 * q^19 + 723787313583184704 * q^21 - 2929078923121218960 * q^23 + 9119218786673228750 * q^25 + 11127665129740313040 * q^27 - 15546679995448558260 * q^29 - 28544554594467385024 * q^31 - 859077391009750054560 * q^33 + 8958395384765013600 * q^35 + 1867697204682824566780 * q^37 + 690990221801025527472 * q^39 + 9081343698046512254964 * q^41 - 5145612605801421773800 * q^43 - 12080376719602732775700 * q^45 + 1150251488862201070560 * q^47 - 92204113217840907690414 * q^49 + 33494974704727214464656 * q^51 + 106953735591470060758620 * q^53 + 214436254091483969701200 * q^55 - 31800538920577362634080 * q^57 - 2009620977624026488631880 * q^59 + 147857426692448940370444 * q^61 + 894215232558851858338320 * q^63 - 2951949350200452960922200 * q^65 - 3051578535738098902157560 * q^67 - 9376480489705868528695872 * q^69 + 13175198820369338598286416 * q^71 + 5284260812951286572116660 * q^73 - 59486914888952006559645000 * q^75 - 42169025756092787413646400 * q^77 - 62814351035720719918179040 * q^79 - 99047155826597708238103278 * q^81 - 171806873410054757883176280 * q^83 - 121333441005595565089302600 * q^85 + 16722988656204557792643120 * q^87 - 313473438761105539763494380 * q^89 - 20205365125040878536694304 * q^91 - 447293490375807514724002560 * q^93 - 1276245244260479145691314000 * q^95 - 653202933397052842883888060 * q^97 - 1076041779280842335610479688 * 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
$3$	−2.15499e6	−0.780384	−0.390192	−	0.920733i	$-0.627591\pi$
$3$	−2.15499e6	−0.780384	−0.390192	+	0.920733i	$0.627591\pi$
$4$	0	0
$5$	4.86703e9	1.78307	0.891536	−	0.452950i	$-0.149629\pi$
$5$	4.86703e9	1.78307	0.891536	+	0.452950i	$0.149629\pi$
$6$	0	0
$7$	−2.14815e10	−0.0837994	−0.0418997	−	0.999122i	$-0.513341\pi$
$7$	−2.14815e10	−0.0837994	−0.0418997	+	0.999122i	$0.513341\pi$
$8$	0	0
$9$	−2.98161e12	−0.391000
$10$	0	0
$11$	6.85469e13	0.598668	0.299334	−	0.954148i	$-0.403235\pi$
$11$	6.85469e13	0.598668	0.299334	+	0.954148i	$0.403235\pi$
$12$	0	0
$13$	−5.86777e14	−0.537326	−0.268663	−	0.963234i	$-0.586582\pi$
$13$	−5.86777e14	−0.537326	−0.268663	+	0.963234i	$0.586582\pi$
$14$	0	0
$15$	−1.04884e16	−1.39148
$16$	0	0
$17$	−2.42814e16	−0.594585	−0.297292	−	0.954786i	$-0.596084\pi$
$17$	−2.42814e16	−0.594585	−0.297292	+	0.954786i	$0.596084\pi$
$18$	0	0
$19$	−2.43094e17	−1.32618	−0.663088	−	0.748541i	$-0.730755\pi$
$19$	−2.43094e17	−1.32618	−0.663088	+	0.748541i	$0.730755\pi$
$20$	0	0
$21$	4.62924e16	0.0653957
$22$	0	0
$23$	−1.25668e17	−0.0519877	−0.0259938	−	0.999662i	$-0.508275\pi$
$23$	−1.25668e17	−0.0519877	−0.0259938	+	0.999662i	$0.508275\pi$
$24$	0	0
$25$	1.62374e19	2.17934
$26$	0	0
$27$	2.28584e19	1.08551
$28$	0	0
$29$	−1.25483e19	−0.227096	−0.113548	−	0.993532i	$-0.536222\pi$
$29$	−1.25483e19	−0.227096	−0.113548	+	0.993532i	$0.536222\pi$
$30$	0	0
$31$	6.23745e19	0.458801	0.229400	−	0.973332i	$-0.426323\pi$
$31$	6.23745e19	0.458801	0.229400	+	0.973332i	$0.426323\pi$
$32$	0	0
$33$	−1.47718e20	−0.467191
$34$	0	0
$35$	−1.04551e20	−0.149420
$36$	0	0
$37$	3.67692e20	0.248177	0.124088	−	0.992271i	$-0.460399\pi$
$37$	3.67692e20	0.248177	0.124088	+	0.992271i	$0.460399\pi$
$38$	0	0
$39$	1.26450e21	0.419321
$40$	0	0
$41$	3.60988e21	0.609412	0.304706	−	0.952446i	$-0.401442\pi$
$41$	3.60988e21	0.609412	0.304706	+	0.952446i	$0.401442\pi$
$42$	0	0
$43$	−1.19560e22	−1.06111	−0.530555	−	0.847651i	$-0.678016\pi$
$43$	−1.19560e22	−1.06111	−0.530555	+	0.847651i	$0.678016\pi$
$44$	0	0
$45$	−1.45116e22	−0.697182
$46$	0	0
$47$	−8.26632e21	−0.220796	−0.110398	−	0.993887i	$-0.535213\pi$
$47$	−8.26632e21	−0.220796	−0.110398	+	0.993887i	$0.535213\pi$
$48$	0	0
$49$	−6.52509e22	−0.992978
$50$	0	0
$51$	5.23263e22	0.464005
$52$	0	0
$53$	1.60547e23	0.846991	0.423496	−	0.905898i	$-0.360803\pi$
$53$	1.60547e23	0.846991	0.423496	+	0.905898i	$0.360803\pi$
$54$	0	0
$55$	3.33619e23	1.06747
$56$	0	0
$57$	5.23866e23	1.03493
$58$	0	0
$59$	−1.09336e24	−1.35601	−0.678004	−	0.735058i	$-0.737155\pi$
$59$	−1.09336e24	−1.35601	−0.678004	+	0.735058i	$0.737155\pi$
$60$	0	0
$61$	−1.32576e24	−1.04837	−0.524183	−	0.851606i	$-0.675629\pi$
$61$	−1.32576e24	−1.04837	−0.524183	+	0.851606i	$0.675629\pi$
$62$	0	0
$63$	6.40495e22	0.0327656
$64$	0	0
$65$	−2.85586e24	−0.958091
$66$	0	0
$67$	−6.27072e24	−1.39736	−0.698679	−	0.715435i	$-0.746229\pi$
$67$	−6.27072e24	−1.39736	−0.698679	+	0.715435i	$0.746229\pi$
$68$	0	0
$69$	2.70814e23	0.0405704
$70$	0	0
$71$	1.65578e25	1.68661	0.843303	−	0.537438i	$-0.180608\pi$
$71$	1.65578e25	1.68661	0.843303	+	0.537438i	$0.180608\pi$
$72$	0	0
$73$	−7.93292e24	−0.555360	−0.277680	−	0.960674i	$-0.589565\pi$
$73$	−7.93292e24	−0.555360	−0.277680	+	0.960674i	$0.589565\pi$
$74$	0	0
$75$	−3.49914e25	−1.70073
$76$	0	0
$77$	−1.47249e24	−0.0501680
$78$	0	0
$79$	−1.61913e25	−0.390227	−0.195113	−	0.980781i	$-0.562507\pi$
$79$	−1.61913e25	−0.390227	−0.195113	+	0.980781i	$0.562507\pi$
$80$	0	0
$81$	−2.65232e25	−0.456118
$82$	0	0
$83$	−1.00338e26	−1.24140	−0.620698	−	0.784049i	$-0.713151\pi$
$83$	−1.00338e26	−1.24140	−0.620698	+	0.784049i	$0.713151\pi$
$84$	0	0
$85$	−1.18178e26	−1.06019
$86$	0	0
$87$	2.70414e25	0.177223
$88$	0	0
$89$	−1.72766e26	−0.833092	−0.416546	−	0.909115i	$-0.636759\pi$
$89$	−1.72766e26	−0.833092	−0.416546	+	0.909115i	$0.636759\pi$
$90$	0	0
$91$	1.26049e25	0.0450276
$92$	0	0
$93$	−1.34416e26	−0.358041
$94$	0	0
$95$	−1.18315e27	−2.36467
$96$	0	0
$97$	−1.03077e27	−1.55505	−0.777524	−	0.628853i	$-0.783525\pi$
$97$	−1.03077e27	−1.55505	−0.777524	+	0.628853i	$0.783525\pi$
$98$	0	0
$99$	−2.04380e26	−0.234080
$100$	0	0
$101$	4.65230e26	0.406751	0.203376	−	0.979101i	$-0.434809\pi$
$101$	4.65230e26	0.406751	0.203376	+	0.979101i	$0.434809\pi$
$102$	0	0
$103$	−7.02209e26	−0.471155	−0.235577	−	0.971856i	$-0.575698\pi$
$103$	−7.02209e26	−0.471155	−0.235577	+	0.971856i	$0.575698\pi$
$104$	0	0
$105$	2.25306e26	0.116605
$106$	0	0
$107$	3.21175e27	1.28843	0.644215	−	0.764845i	$-0.277184\pi$
$107$	3.21175e27	1.28843	0.644215	+	0.764845i	$0.277184\pi$
$108$	0	0
$109$	1.11613e27	0.348705	0.174353	−	0.984683i	$-0.444217\pi$
$109$	1.11613e27	0.348705	0.174353	+	0.984683i	$0.444217\pi$
$110$	0	0
$111$	−7.92373e26	−0.193673
$112$	0	0
$113$	−3.24502e27	−0.623244	−0.311622	−	0.950206i	$-0.600872\pi$
$113$	−3.24502e27	−0.623244	−0.311622	+	0.950206i	$0.600872\pi$
$114$	0	0
$115$	−6.11632e26	−0.0926978
$116$	0	0
$117$	1.74954e27	0.210095
$118$	0	0
$119$	5.21601e26	0.0498258
$120$	0	0
$121$	−8.41132e27	−0.641596
$122$	0	0
$123$	−7.77927e27	−0.475576
$124$	0	0
$125$	4.27656e28	2.10285
$126$	0	0
$127$	3.05563e28	1.21269	0.606346	−	0.795201i	$-0.292635\pi$
$127$	3.05563e28	1.21269	0.606346	+	0.795201i	$0.292635\pi$
$128$	0	0
$129$	2.57650e28	0.828074
$130$	0	0
$131$	3.92050e28	1.02371	0.511857	−	0.859071i	$-0.328958\pi$
$131$	3.92050e28	1.02371	0.511857	+	0.859071i	$0.328958\pi$
$132$	0	0
$133$	5.22203e27	0.111133
$134$	0	0
$135$	1.11253e29	1.93555
$136$	0	0
$137$	−1.05475e28	−0.150461	−0.0752305	−	0.997166i	$-0.523969\pi$
$137$	−1.05475e28	−0.150461	−0.0752305	+	0.997166i	$0.523969\pi$
$138$	0	0
$139$	−1.42615e29	−1.67289	−0.836443	−	0.548054i	$-0.815369\pi$
$139$	−1.42615e29	−1.67289	−0.836443	+	0.548054i	$0.815369\pi$
$140$	0	0
$141$	1.78138e28	0.172306
$142$	0	0
$143$	−4.02217e28	−0.321680
$144$	0	0
$145$	−6.10727e28	−0.404929
$146$	0	0
$147$	1.40615e29	0.774904
$148$	0	0
$149$	1.04309e29	0.478968	0.239484	−	0.970900i	$-0.423022\pi$
$149$	1.04309e29	0.478968	0.239484	+	0.970900i	$0.423022\pi$
$150$	0	0
$151$	−3.58958e29	−1.37675	−0.688374	−	0.725356i	$-0.741675\pi$
$151$	−3.58958e29	−1.37675	−0.688374	+	0.725356i	$0.741675\pi$
$152$	0	0
$153$	7.23978e28	0.232483
$154$	0	0
$155$	3.03578e29	0.818074
$156$	0	0
$157$	−6.56383e29	−1.48769	−0.743844	−	0.668353i	$-0.767000\pi$
$157$	−6.56383e29	−1.48769	−0.743844	+	0.668353i	$0.767000\pi$
$158$	0	0
$159$	−3.45978e29	−0.660979
$160$	0	0
$161$	2.69955e27	0.00435654
$162$	0	0
$163$	−3.98777e29	−0.544750	−0.272375	−	0.962191i	$-0.587809\pi$
$163$	−3.98777e29	−0.544750	−0.272375	+	0.962191i	$0.587809\pi$
$164$	0	0
$165$	−7.18947e29	−0.833036
$166$	0	0
$167$	−6.97711e29	−0.687074	−0.343537	−	0.939139i	$-0.611625\pi$
$167$	−6.97711e29	−0.687074	−0.343537	+	0.939139i	$0.611625\pi$
$168$	0	0
$169$	−8.48226e29	−0.711280
$170$	0	0
$171$	7.24813e29	0.518536
$172$	0	0
$173$	1.20460e30	0.736580	0.368290	−	0.929711i	$-0.379943\pi$
$173$	1.20460e30	0.736580	0.368290	+	0.929711i	$0.379943\pi$
$174$	0	0
$175$	−3.48803e29	−0.182628
$176$	0	0
$177$	2.35619e30	1.05821
$178$	0	0
$179$	−3.25033e30	−1.25433	−0.627164	−	0.778887i	$-0.715785\pi$
$179$	−3.25033e30	−1.25433	−0.627164	+	0.778887i	$0.715785\pi$
$180$	0	0
$181$	−5.16248e30	−1.71474	−0.857369	−	0.514702i	$-0.827903\pi$
$181$	−5.16248e30	−1.71474	−0.857369	+	0.514702i	$0.827903\pi$
$182$	0	0
$183$	2.85700e30	0.818128
$184$	0	0
$185$	1.78957e30	0.442517
$186$	0	0
$187$	−1.66442e30	−0.355959
$188$	0	0
$189$	−4.91033e29	−0.0909655
$190$	0	0
$191$	5.77134e30	0.927526	0.463763	−	0.885959i	$-0.346499\pi$
$191$	5.77134e30	0.927526	0.463763	+	0.885959i	$0.346499\pi$
$192$	0	0
$193$	−2.85746e30	−0.398985	−0.199492	−	0.979899i	$-0.563929\pi$
$193$	−2.85746e30	−0.398985	−0.199492	+	0.979899i	$0.563929\pi$
$194$	0	0
$195$	6.15436e30	0.747679
$196$	0	0
$197$	1.69381e31	1.79296	0.896482	−	0.443081i	$-0.146115\pi$
$197$	1.69381e31	1.79296	0.896482	+	0.443081i	$0.146115\pi$
$198$	0	0
$199$	8.37737e30	0.773733	0.386866	−	0.922136i	$-0.373557\pi$
$199$	8.37737e30	0.773733	0.386866	+	0.922136i	$0.373557\pi$
$200$	0	0
$201$	1.35134e31	1.09048
$202$	0	0
$203$	2.69555e29	0.0190305
$204$	0	0
$205$	1.75694e31	1.08663
$206$	0	0
$207$	3.74695e29	0.0203272
$208$	0	0
$209$	−1.66634e31	−0.793940
$210$	0	0
$211$	1.95424e31	0.818776	0.409388	−	0.912360i	$-0.365742\pi$
$211$	1.95424e31	0.818776	0.409388	+	0.912360i	$0.365742\pi$
$212$	0	0
$213$	−3.56818e31	−1.31620
$214$	0	0
$215$	−5.81900e31	−1.89204
$216$	0	0
$217$	−1.33990e30	−0.0384472
$218$	0	0
$219$	1.70954e31	0.433394
$220$	0	0
$221$	1.42478e31	0.319486
$222$	0	0
$223$	5.21969e30	0.103640	0.0518202	−	0.998656i	$-0.483498\pi$
$223$	5.21969e30	0.103640	0.0518202	+	0.998656i	$0.483498\pi$
$224$	0	0
$225$	−4.84136e31	−0.852124
$226$	0	0
$227$	1.42268e31	0.222208	0.111104	−	0.993809i	$-0.464561\pi$
$227$	1.42268e31	0.222208	0.111104	+	0.993809i	$0.464561\pi$
$228$	0	0
$229$	4.79201e31	0.664874	0.332437	−	0.943126i	$-0.392129\pi$
$229$	4.79201e31	0.664874	0.332437	+	0.943126i	$0.392129\pi$
$230$	0	0
$231$	3.17320e30	0.0391504
$232$	0	0
$233$	−1.09089e31	−0.119805	−0.0599025	−	0.998204i	$-0.519079\pi$
$233$	−1.09089e31	−0.119805	−0.0599025	+	0.998204i	$0.519079\pi$
$234$	0	0
$235$	−4.02324e31	−0.393695
$236$	0	0
$237$	3.48922e31	0.304527
$238$	0	0
$239$	−1.93175e32	−1.50515	−0.752573	−	0.658509i	$-0.771187\pi$
$239$	−1.93175e32	−1.50515	−0.752573	+	0.658509i	$0.771187\pi$
$240$	0	0
$241$	−1.37107e32	−0.954619	−0.477310	−	0.878735i	$-0.658388\pi$
$241$	−1.37107e32	−0.954619	−0.477310	+	0.878735i	$0.658388\pi$
$242$	0	0
$243$	−1.17152e32	−0.729567
$244$	0	0
$245$	−3.17578e32	−1.77055
$246$	0	0
$247$	1.42642e32	0.712590
$248$	0	0
$249$	2.16227e32	0.968767
$250$	0	0
$251$	−6.44860e31	−0.259341	−0.129670	−	0.991557i	$-0.541392\pi$
$251$	−6.44860e31	−0.259341	−0.129670	+	0.991557i	$0.541392\pi$
$252$	0	0
$253$	−8.61418e30	−0.0311234
$254$	0	0
$255$	2.54673e32	0.827353
$256$	0	0
$257$	−1.37557e32	−0.402147	−0.201073	−	0.979576i	$-0.564443\pi$
$257$	−1.37557e32	−0.402147	−0.201073	+	0.979576i	$0.564443\pi$
$258$	0	0
$259$	−7.89857e30	−0.0207971
$260$	0	0
$261$	3.74140e31	0.0887948
$262$	0	0
$263$	6.29327e31	0.134733	0.0673665	−	0.997728i	$-0.478540\pi$
$263$	6.29327e31	0.134733	0.0673665	+	0.997728i	$0.478540\pi$
$264$	0	0
$265$	7.81388e32	1.51025
$266$	0	0
$267$	3.72309e32	0.650132
$268$	0	0
$269$	7.34021e32	1.15892	0.579459	−	0.815001i	$-0.303264\pi$
$269$	7.34021e32	1.15892	0.579459	+	0.815001i	$0.303264\pi$
$270$	0	0
$271$	−6.75590e32	−0.965156	−0.482578	−	0.875853i	$-0.660300\pi$
$271$	−6.75590e32	−0.965156	−0.482578	+	0.875853i	$0.660300\pi$
$272$	0	0
$273$	−2.71633e31	−0.0351388
$274$	0	0
$275$	1.11302e33	1.30470
$276$	0	0
$277$	−1.48766e33	−1.58134	−0.790671	−	0.612241i	$-0.790268\pi$
$277$	−1.48766e33	−1.58134	−0.790671	+	0.612241i	$0.790268\pi$
$278$	0	0
$279$	−1.85976e32	−0.179391
$280$	0	0
$281$	9.53617e32	0.835293	0.417646	−	0.908610i	$-0.362855\pi$
$281$	9.53617e32	0.835293	0.417646	+	0.908610i	$0.362855\pi$
$282$	0	0
$283$	1.55041e33	1.23404	0.617022	−	0.786946i	$-0.288339\pi$
$283$	1.55041e33	1.23404	0.617022	+	0.786946i	$0.288339\pi$
$284$	0	0
$285$	2.54967e33	1.84535
$286$	0	0
$287$	−7.75457e31	−0.0510684
$288$	0	0
$289$	−1.07812e33	−0.646469
$290$	0	0
$291$	2.22130e33	1.21354
$292$	0	0
$293$	−2.76230e33	−1.37581	−0.687903	−	0.725802i	$-0.741469\pi$
$293$	−2.76230e33	−1.37581	−0.687903	+	0.725802i	$0.741469\pi$
$294$	0	0
$295$	−5.32143e33	−2.41786
$296$	0	0
$297$	1.56687e33	0.649863
$298$	0	0
$299$	7.37394e31	0.0279344
$300$	0	0
$301$	2.56832e32	0.0889204
$302$	0	0
$303$	−1.00257e33	−0.317422
$304$	0	0
$305$	−6.45252e33	−1.86931
$306$	0	0
$307$	−4.04426e33	−1.07268	−0.536341	−	0.844001i	$-0.680194\pi$
$307$	−4.04426e33	−1.07268	−0.536341	+	0.844001i	$0.680194\pi$
$308$	0	0
$309$	1.51325e33	0.367682
$310$	0	0
$311$	2.20209e33	0.490423	0.245211	−	0.969470i	$-0.421143\pi$
$311$	2.20209e33	0.490423	0.245211	+	0.969470i	$0.421143\pi$
$312$	0	0
$313$	1.52934e33	0.312361	0.156180	−	0.987729i	$-0.450082\pi$
$313$	1.52934e33	0.312361	0.156180	+	0.987729i	$0.450082\pi$
$314$	0	0
$315$	3.11731e32	0.0584234
$316$	0	0
$317$	5.64949e33	0.972096	0.486048	−	0.873932i	$-0.338438\pi$
$317$	5.64949e33	0.972096	0.486048	+	0.873932i	$0.338438\pi$
$318$	0	0
$319$	−8.60143e32	−0.135955
$320$	0	0
$321$	−6.92129e33	−1.00547
$322$	0	0
$323$	5.90268e33	0.788525
$324$	0	0
$325$	−9.52773e33	−1.17102
$326$	0	0
$327$	−2.40526e33	−0.272124
$328$	0	0
$329$	1.77573e32	0.0185026
$330$	0	0
$331$	1.79482e33	0.172323	0.0861616	−	0.996281i	$-0.472540\pi$
$331$	1.79482e33	0.172323	0.0861616	+	0.996281i	$0.472540\pi$
$332$	0	0
$333$	−1.09631e33	−0.0970372
$334$	0	0
$335$	−3.05198e34	−2.49159
$336$	0	0
$337$	3.56328e33	0.268439	0.134220	−	0.990952i	$-0.457147\pi$
$337$	3.56328e33	0.268439	0.134220	+	0.990952i	$0.457147\pi$
$338$	0	0
$339$	6.99298e33	0.486370
$340$	0	0
$341$	4.27557e33	0.274669
$342$	0	0
$343$	2.81329e33	0.167010
$344$	0	0
$345$	1.31806e33	0.0723399
$346$	0	0
$347$	2.88728e34	1.46569	0.732843	−	0.680398i	$-0.238193\pi$
$347$	2.88728e34	1.46569	0.732843	+	0.680398i	$0.238193\pi$
$348$	0	0
$349$	1.78587e34	0.838893	0.419447	−	0.907780i	$-0.362224\pi$
$349$	1.78587e34	0.838893	0.419447	+	0.907780i	$0.362224\pi$
$350$	0	0
$351$	−1.34128e34	−0.583276
$352$	0	0
$353$	2.71254e34	1.09249	0.546245	−	0.837626i	$-0.316057\pi$
$353$	2.71254e34	1.09249	0.546245	+	0.837626i	$0.316057\pi$
$354$	0	0
$355$	8.05870e34	3.00734
$356$	0	0
$357$	−1.12405e33	−0.0388833
$358$	0	0
$359$	−2.08159e34	−0.667760	−0.333880	−	0.942616i	$-0.608358\pi$
$359$	−2.08159e34	−0.667760	−0.333880	+	0.942616i	$0.608358\pi$
$360$	0	0
$361$	2.54943e34	0.758744
$362$	0	0
$363$	1.81263e34	0.500692
$364$	0	0
$365$	−3.86097e34	−0.990246
$366$	0	0
$367$	−1.50702e34	−0.359028	−0.179514	−	0.983755i	$-0.557453\pi$
$367$	−1.50702e34	−0.359028	−0.179514	+	0.983755i	$0.557453\pi$
$368$	0	0
$369$	−1.07633e34	−0.238280
$370$	0	0
$371$	−3.44880e33	−0.0709773
$372$	0	0
$373$	3.06644e34	0.586901	0.293451	−	0.955974i	$-0.405196\pi$
$373$	3.06644e34	0.586901	0.293451	+	0.955974i	$0.405196\pi$
$374$	0	0
$375$	−9.21594e34	−1.64103
$376$	0	0
$377$	7.36303e33	0.122025
$378$	0	0
$379$	2.54352e34	0.392470	0.196235	−	0.980557i	$-0.437128\pi$
$379$	2.54352e34	0.392470	0.196235	+	0.980557i	$0.437128\pi$
$380$	0	0
$381$	−6.58485e34	−0.946365
$382$	0	0
$383$	−2.05837e34	−0.275638	−0.137819	−	0.990457i	$-0.544009\pi$
$383$	−2.05837e34	−0.275638	−0.137819	+	0.990457i	$0.544009\pi$
$384$	0	0
$385$	−7.16664e33	−0.0894532
$386$	0	0
$387$	3.56480e34	0.414894
$388$	0	0
$389$	6.64448e34	0.721342	0.360671	−	0.932693i	$-0.382548\pi$
$389$	6.64448e34	0.721342	0.360671	+	0.932693i	$0.382548\pi$
$390$	0	0
$391$	3.05141e33	0.0309111
$392$	0	0
$393$	−8.44864e34	−0.798891
$394$	0	0
$395$	−7.88037e34	−0.695802
$396$	0	0
$397$	−7.11637e34	−0.586931	−0.293465	−	0.955970i	$-0.594809\pi$
$397$	−7.11637e34	−0.586931	−0.293465	+	0.955970i	$0.594809\pi$
$398$	0	0
$399$	−1.12534e34	−0.0867263
$400$	0	0
$401$	7.78771e34	0.560997	0.280499	−	0.959854i	$-0.409500\pi$
$401$	7.78771e34	0.560997	0.280499	+	0.959854i	$0.409500\pi$
$402$	0	0
$403$	−3.65999e34	−0.246526
$404$	0	0
$405$	−1.29089e35	−0.813292
$406$	0	0
$407$	2.52041e34	0.148575
$408$	0	0
$409$	−3.21718e35	−1.77505	−0.887526	−	0.460758i	$-0.847578\pi$
$409$	−3.21718e35	−1.77505	−0.887526	+	0.460758i	$0.847578\pi$
$410$	0	0
$411$	2.27299e34	0.117417
$412$	0	0
$413$	2.34871e34	0.113633
$414$	0	0
$415$	−4.88347e35	−2.21350
$416$	0	0
$417$	3.07335e35	1.30549
$418$	0	0
$419$	4.31477e35	1.71818	0.859089	−	0.511827i	$-0.171031\pi$
$419$	4.31477e35	1.71818	0.859089	+	0.511827i	$0.171031\pi$
$420$	0	0
$421$	5.01390e35	1.87227	0.936133	−	0.351647i	$-0.114378\pi$
$421$	5.01390e35	1.87227	0.936133	+	0.351647i	$0.114378\pi$
$422$	0	0
$423$	2.46470e34	0.0863313
$424$	0	0
$425$	−3.94267e35	−1.29580
$426$	0	0
$427$	2.84793e34	0.0878524
$428$	0	0
$429$	8.66775e34	0.251034
$430$	0	0
$431$	−1.64027e35	−0.446141	−0.223070	−	0.974802i	$-0.571608\pi$
$431$	−1.64027e35	−0.446141	−0.223070	+	0.974802i	$0.571608\pi$
$432$	0	0
$433$	−1.75800e33	−0.00449193	−0.00224596	−	0.999997i	$-0.500715\pi$
$433$	−1.75800e33	−0.00449193	−0.00224596	+	0.999997i	$0.500715\pi$
$434$	0	0
$435$	1.31611e35	0.316000
$436$	0	0
$437$	3.05493e34	0.0689449
$438$	0	0
$439$	3.72911e35	0.791287	0.395644	−	0.918404i	$-0.370522\pi$
$439$	3.72911e35	0.791287	0.395644	+	0.918404i	$0.370522\pi$
$440$	0	0
$441$	1.94553e35	0.388255
$442$	0	0
$443$	5.23878e35	0.983513	0.491756	−	0.870733i	$-0.336355\pi$
$443$	5.23878e35	0.983513	0.491756	+	0.870733i	$0.336355\pi$
$444$	0	0
$445$	−8.40856e35	−1.48546
$446$	0	0
$447$	−2.24784e35	−0.373779
$448$	0	0
$449$	6.50462e35	1.01835	0.509174	−	0.860664i	$-0.329951\pi$
$449$	6.50462e35	1.01835	0.509174	+	0.860664i	$0.329951\pi$
$450$	0	0
$451$	2.47446e35	0.364836
$452$	0	0
$453$	7.73550e35	1.07439
$454$	0	0
$455$	6.13482e34	0.0802875
$456$	0	0
$457$	9.00061e35	1.11020	0.555102	−	0.831782i	$-0.312679\pi$
$457$	9.00061e35	1.11020	0.555102	+	0.831782i	$0.312679\pi$
$458$	0	0
$459$	−5.55036e35	−0.645431
$460$	0	0
$461$	3.94034e35	0.432087	0.216043	−	0.976384i	$-0.430685\pi$
$461$	3.94034e35	0.432087	0.216043	+	0.976384i	$0.430685\pi$
$462$	0	0
$463$	1.39468e36	1.44255	0.721276	−	0.692648i	$-0.243556\pi$
$463$	1.39468e36	1.44255	0.721276	+	0.692648i	$0.243556\pi$
$464$	0	0
$465$	−6.54209e35	−0.638412
$466$	0	0
$467$	−2.87062e35	−0.264361	−0.132180	−	0.991226i	$-0.542198\pi$
$467$	−2.87062e35	−0.264361	−0.132180	+	0.991226i	$0.542198\pi$
$468$	0	0
$469$	1.34704e35	0.117098
$470$	0	0
$471$	1.41450e36	1.16097
$472$	0	0
$473$	−8.19544e35	−0.635253
$474$	0	0
$475$	−3.94722e36	−2.89020
$476$	0	0
$477$	−4.78690e35	−0.331174
$478$	0	0
$479$	−8.28285e35	−0.541565	−0.270782	−	0.962641i	$-0.587282\pi$
$479$	−8.28285e35	−0.541565	−0.270782	+	0.962641i	$0.587282\pi$
$480$	0	0
$481$	−2.15753e35	−0.133352
$482$	0	0
$483$	−5.81750e33	−0.00339977
$484$	0	0
$485$	−5.01679e36	−2.77276
$486$	0	0
$487$	1.69732e36	0.887405	0.443702	−	0.896174i	$-0.353665\pi$
$487$	1.69732e36	0.887405	0.443702	+	0.896174i	$0.353665\pi$
$488$	0	0
$489$	8.59360e35	0.425114
$490$	0	0
$491$	−2.68983e36	−1.25929	−0.629645	−	0.776883i	$-0.716799\pi$
$491$	−2.68983e36	−1.25929	−0.629645	+	0.776883i	$0.716799\pi$
$492$	0	0
$493$	3.04690e35	0.135028
$494$	0	0
$495$	−9.94724e35	−0.417381
$496$	0	0
$497$	−3.55685e35	−0.141337
$498$	0	0
$499$	4.36140e35	0.164160	0.0820802	−	0.996626i	$-0.473844\pi$
$499$	4.36140e35	0.164160	0.0820802	+	0.996626i	$0.473844\pi$
$500$	0	0
$501$	1.50356e36	0.536182
$502$	0	0
$503$	2.16228e36	0.730708	0.365354	−	0.930869i	$-0.380948\pi$
$503$	2.16228e36	0.730708	0.365354	+	0.930869i	$0.380948\pi$
$504$	0	0
$505$	2.26429e36	0.725267
$506$	0	0
$507$	1.82792e36	0.555072
$508$	0	0
$509$	4.49619e36	1.29466	0.647328	−	0.762211i	$-0.275886\pi$
$509$	4.49619e36	1.29466	0.647328	+	0.762211i	$0.275886\pi$
$510$	0	0
$511$	1.70411e35	0.0465388
$512$	0	0
$513$	−5.55676e36	−1.43958
$514$	0	0
$515$	−3.41767e36	−0.840103
$516$	0	0
$517$	−5.66630e35	−0.132184
$518$	0	0
$519$	−2.59590e36	−0.574815
$520$	0	0
$521$	9.12580e35	0.191850	0.0959250	−	0.995389i	$-0.469419\pi$
$521$	9.12580e35	0.191850	0.0959250	+	0.995389i	$0.469419\pi$
$522$	0	0
$523$	−1.44280e35	−0.0288027	−0.0144013	−	0.999896i	$-0.504584\pi$
$523$	−1.44280e35	−0.0288027	−0.0144013	+	0.999896i	$0.504584\pi$
$524$	0	0
$525$	7.51667e35	0.142520
$526$	0	0
$527$	−1.51454e36	−0.272796
$528$	0	0
$529$	−5.82742e36	−0.997297
$530$	0	0
$531$	3.25998e36	0.530199
$532$	0	0
$533$	−2.11820e36	−0.327453
$534$	0	0
$535$	1.56317e37	2.29736
$536$	0	0
$537$	7.00443e36	0.978858
$538$	0	0
$539$	−4.47274e36	−0.594464
$540$	0	0
$541$	6.50406e36	0.822284	0.411142	−	0.911571i	$-0.365130\pi$
$541$	6.50406e36	0.822284	0.411142	+	0.911571i	$0.365130\pi$
$542$	0	0
$543$	1.11251e37	1.33815
$544$	0	0
$545$	5.43226e36	0.621767
$546$	0	0
$547$	−1.22652e37	−1.33612	−0.668058	−	0.744109i	$-0.732874\pi$
$547$	−1.22652e37	−1.33612	−0.668058	+	0.744109i	$0.732874\pi$
$548$	0	0
$549$	3.95291e36	0.409911
$550$	0	0
$551$	3.05041e36	0.301170
$552$	0	0
$553$	3.47814e35	0.0327008
$554$	0	0
$555$	−3.85650e36	−0.345333
$556$	0	0
$557$	1.32843e37	1.13317	0.566585	−	0.824003i	$-0.308264\pi$
$557$	1.32843e37	1.13317	0.566585	+	0.824003i	$0.308264\pi$
$558$	0	0
$559$	7.01549e36	0.570162
$560$	0	0
$561$	3.58680e36	0.277785
$562$	0	0
$563$	1.08601e37	0.801624	0.400812	−	0.916160i	$-0.368728\pi$
$563$	1.08601e37	0.801624	0.400812	+	0.916160i	$0.368728\pi$
$564$	0	0
$565$	−1.57936e37	−1.11129
$566$	0	0
$567$	5.69757e35	0.0382224
$568$	0	0
$569$	−1.31614e37	−0.841948	−0.420974	−	0.907073i	$-0.638312\pi$
$569$	−1.31614e37	−0.841948	−0.420974	+	0.907073i	$0.638312\pi$
$570$	0	0
$571$	−3.26297e37	−1.99079	−0.995395	−	0.0958616i	$-0.969439\pi$
$571$	−3.26297e37	−1.99079	−0.995395	+	0.0958616i	$0.969439\pi$
$572$	0	0
$573$	−1.24372e37	−0.723827
$574$	0	0
$575$	−2.04053e36	−0.113299
$576$	0	0
$577$	−3.67195e36	−0.194547	−0.0972733	−	0.995258i	$-0.531012\pi$
$577$	−3.67195e36	−0.194547	−0.0972733	+	0.995258i	$0.531012\pi$
$578$	0	0
$579$	6.15780e36	0.311361
$580$	0	0
$581$	2.15541e36	0.104028
$582$	0	0
$583$	1.10050e37	0.507067
$584$	0	0
$585$	8.51507e36	0.374614
$586$	0	0
$587$	9.27744e36	0.389775	0.194888	−	0.980826i	$-0.437566\pi$
$587$	9.27744e36	0.389775	0.194888	+	0.980826i	$0.437566\pi$
$588$	0	0
$589$	−1.51629e37	−0.608451
$590$	0	0
$591$	−3.65015e37	−1.39920
$592$	0	0
$593$	−5.24575e36	−0.192119	−0.0960593	−	0.995376i	$-0.530624\pi$
$593$	−5.24575e36	−0.192119	−0.0960593	+	0.995376i	$0.530624\pi$
$594$	0	0
$595$	2.53865e36	0.0888431
$596$	0	0
$597$	−1.80531e37	−0.603809
$598$	0	0
$599$	−4.76000e37	−1.52176	−0.760878	−	0.648894i	$-0.775232\pi$
$599$	−4.76000e37	−1.52176	−0.760878	+	0.648894i	$0.775232\pi$
$600$	0	0
$601$	1.70669e37	0.521613	0.260807	−	0.965391i	$-0.416012\pi$
$601$	1.70669e37	0.521613	0.260807	+	0.965391i	$0.416012\pi$
$602$	0	0
$603$	1.86969e37	0.546368
$604$	0	0
$605$	−4.09381e37	−1.14401
$606$	0	0
$607$	−1.00572e37	−0.268800	−0.134400	−	0.990927i	$-0.542911\pi$
$607$	−1.00572e37	−0.268800	−0.134400	+	0.990927i	$0.542911\pi$
$608$	0	0
$609$	−5.80889e35	−0.0148511
$610$	0	0
$611$	4.85049e36	0.118640
$612$	0	0
$613$	5.68669e37	1.33089	0.665447	−	0.746445i	$-0.268241\pi$
$613$	5.68669e37	1.33089	0.665447	+	0.746445i	$0.268241\pi$
$614$	0	0
$615$	−3.78619e37	−0.847985
$616$	0	0
$617$	−4.13930e37	−0.887315	−0.443657	−	0.896196i	$-0.646319\pi$
$617$	−4.13930e37	−0.887315	−0.443657	+	0.896196i	$0.646319\pi$
$618$	0	0
$619$	3.55328e37	0.729131	0.364565	−	0.931178i	$-0.381218\pi$
$619$	3.55328e37	0.729131	0.364565	+	0.931178i	$0.381218\pi$
$620$	0	0
$621$	−2.87259e36	−0.0564334
$622$	0	0
$623$	3.71127e36	0.0698126
$624$	0	0
$625$	8.71634e37	1.57020
$626$	0	0
$627$	3.59094e37	0.619578
$628$	0	0
$629$	−8.92809e36	−0.147562
$630$	0	0
$631$	7.29607e37	1.15529	0.577647	−	0.816287i	$-0.303971\pi$
$631$	7.29607e37	1.15529	0.577647	+	0.816287i	$0.303971\pi$
$632$	0	0
$633$	−4.21138e37	−0.638960
$634$	0	0
$635$	1.48718e38	2.16232
$636$	0	0
$637$	3.82878e37	0.533553
$638$	0	0
$639$	−4.93688e37	−0.659464
$640$	0	0
$641$	1.83496e37	0.234987	0.117494	−	0.993074i	$-0.462514\pi$
$641$	1.83496e37	0.234987	0.117494	+	0.993074i	$0.462514\pi$
$642$	0	0
$643$	−5.85846e37	−0.719342	−0.359671	−	0.933079i	$-0.617111\pi$
$643$	−5.85846e37	−0.719342	−0.359671	+	0.933079i	$0.617111\pi$
$644$	0	0
$645$	1.25399e38	1.47651
$646$	0	0
$647$	−1.39594e38	−1.57637	−0.788187	−	0.615435i	$-0.788980\pi$
$647$	−1.39594e38	−1.57637	−0.788187	+	0.615435i	$0.788980\pi$
$648$	0	0
$649$	−7.49466e37	−0.811799
$650$	0	0
$651$	2.88747e36	0.0300036
$652$	0	0
$653$	−8.62301e37	−0.859668	−0.429834	−	0.902908i	$-0.641428\pi$
$653$	−8.62301e37	−0.859668	−0.429834	+	0.902908i	$0.641428\pi$
$654$	0	0
$655$	1.90812e38	1.82536
$656$	0	0
$657$	2.36529e37	0.217146
$658$	0	0
$659$	−1.30986e38	−1.15418	−0.577089	−	0.816681i	$-0.695811\pi$
$659$	−1.30986e38	−1.15418	−0.577089	+	0.816681i	$0.695811\pi$
$660$	0	0
$661$	1.01882e38	0.861740	0.430870	−	0.902414i	$-0.358207\pi$
$661$	1.01882e38	0.861740	0.430870	+	0.902414i	$0.358207\pi$
$662$	0	0
$663$	−3.07039e37	−0.249322
$664$	0	0
$665$	2.54158e37	0.198158
$666$	0	0
$667$	1.57692e36	0.0118062
$668$	0	0
$669$	−1.12484e37	−0.0808794
$670$	0	0
$671$	−9.08768e37	−0.627623
$672$	0	0
$673$	2.29828e36	0.0152476	0.00762378	−	0.999971i	$-0.497573\pi$
$673$	2.29828e36	0.0152476	0.00762378	+	0.999971i	$0.497573\pi$
$674$	0	0
$675$	3.71161e38	2.36571
$676$	0	0
$677$	1.56923e37	0.0961033	0.0480516	−	0.998845i	$-0.484699\pi$
$677$	1.56923e37	0.0961033	0.0480516	+	0.998845i	$0.484699\pi$
$678$	0	0
$679$	2.21425e37	0.130312
$680$	0	0
$681$	−3.06587e37	−0.173408
$682$	0	0
$683$	1.85515e38	1.00856	0.504278	−	0.863542i	$-0.331759\pi$
$683$	1.85515e38	1.00856	0.504278	+	0.863542i	$0.331759\pi$
$684$	0	0
$685$	−5.13352e37	−0.268283
$686$	0	0
$687$	−1.03267e38	−0.518857
$688$	0	0
$689$	−9.42055e37	−0.455111
$690$	0	0
$691$	−1.36613e38	−0.634655	−0.317328	−	0.948316i	$-0.602785\pi$
$691$	−1.36613e38	−0.634655	−0.317328	+	0.948316i	$0.602785\pi$
$692$	0	0
$693$	4.39039e36	0.0196157
$694$	0	0
$695$	−6.94113e38	−2.98288
$696$	0	0
$697$	−8.76531e37	−0.362347
$698$	0	0
$699$	2.35085e37	0.0934940
$700$	0	0
$701$	−2.20458e38	−0.843594	−0.421797	−	0.906690i	$-0.638601\pi$
$701$	−2.20458e38	−0.843594	−0.421797	+	0.906690i	$0.638601\pi$
$702$	0	0
$703$	−8.93839e37	−0.329126
$704$	0	0
$705$	8.67005e37	0.307233
$706$	0	0
$707$	−9.99383e36	−0.0340855
$708$	0	0
$709$	−5.82758e38	−1.91322	−0.956608	−	0.291376i	$-0.905887\pi$
$709$	−5.82758e38	−1.91322	−0.956608	+	0.291376i	$0.905887\pi$
$710$	0	0
$711$	4.82763e37	0.152579
$712$	0	0
$713$	−7.83851e36	−0.0238520
$714$	0	0
$715$	−1.95760e38	−0.573579
$716$	0	0
$717$	4.16290e38	1.17459
$718$	0	0
$719$	−2.83019e38	−0.769089	−0.384544	−	0.923107i	$-0.625641\pi$
$719$	−2.83019e38	−0.769089	−0.384544	+	0.923107i	$0.625641\pi$
$720$	0	0
$721$	1.50845e37	0.0394825
$722$	0	0
$723$	2.95465e38	0.744970
$724$	0	0
$725$	−2.03751e38	−0.494921
$726$	0	0
$727$	2.87575e38	0.673032	0.336516	−	0.941678i	$-0.390751\pi$
$727$	2.87575e38	0.673032	0.336516	+	0.941678i	$0.390751\pi$
$728$	0	0
$729$	4.54717e38	1.02546
$730$	0	0
$731$	2.90308e38	0.630920
$732$	0	0
$733$	6.09181e38	1.27598	0.637989	−	0.770046i	$-0.279767\pi$
$733$	6.09181e38	1.27598	0.637989	+	0.770046i	$0.279767\pi$
$734$	0	0
$735$	6.84378e38	1.38171
$736$	0	0
$737$	−4.29838e38	−0.836554
$738$	0	0
$739$	−5.52340e38	−1.03635	−0.518175	−	0.855275i	$-0.673388\pi$
$739$	−5.52340e38	−1.03635	−0.518175	+	0.855275i	$0.673388\pi$
$740$	0	0
$741$	−3.07393e38	−0.556094
$742$	0	0
$743$	4.68741e38	0.817680	0.408840	−	0.912606i	$-0.365933\pi$
$743$	4.68741e38	0.817680	0.408840	+	0.912606i	$0.365933\pi$
$744$	0	0
$745$	5.07674e38	0.854033
$746$	0	0
$747$	2.99169e38	0.485387
$748$	0	0
$749$	−6.89932e37	−0.107970
$750$	0	0
$751$	7.49018e38	1.13071	0.565357	−	0.824846i	$-0.308738\pi$
$751$	7.49018e38	1.13071	0.565357	+	0.824846i	$0.308738\pi$
$752$	0	0
$753$	1.38967e38	0.202385
$754$	0	0
$755$	−1.74706e39	−2.45484
$756$	0	0
$757$	−2.19892e38	−0.298137	−0.149068	−	0.988827i	$-0.547627\pi$
$757$	−2.19892e38	−0.298137	−0.149068	+	0.988827i	$0.547627\pi$
$758$	0	0
$759$	1.85635e37	0.0242882
$760$	0	0
$761$	4.67111e38	0.589830	0.294915	−	0.955523i	$-0.404709\pi$
$761$	4.67111e38	0.589830	0.294915	+	0.955523i	$0.404709\pi$
$762$	0	0
$763$	−2.39762e37	−0.0292213
$764$	0	0
$765$	3.52362e38	0.414534
$766$	0	0
$767$	6.41560e38	0.728618
$768$	0	0
$769$	9.38023e37	0.102851	0.0514254	−	0.998677i	$-0.483624\pi$
$769$	9.38023e37	0.102851	0.0514254	+	0.998677i	$0.483624\pi$
$770$	0	0
$771$	2.96433e38	0.313829
$772$	0	0
$773$	1.06611e39	1.08988	0.544942	−	0.838474i	$-0.316552\pi$
$773$	1.06611e39	1.08988	0.544942	+	0.838474i	$0.316552\pi$
$774$	0	0
$775$	1.01280e39	0.999885
$776$	0	0
$777$	1.70214e37	0.0162297
$778$	0	0
$779$	−8.77542e38	−0.808188
$780$	0	0
$781$	1.13498e39	1.00972
$782$	0	0
$783$	−2.86834e38	−0.246517
$784$	0	0
$785$	−3.19463e39	−2.65266
$786$	0	0
$787$	1.52283e39	1.22177	0.610887	−	0.791718i	$-0.290813\pi$
$787$	1.52283e39	1.22177	0.610887	+	0.791718i	$0.290813\pi$
$788$	0	0
$789$	−1.35619e38	−0.105143
$790$	0	0
$791$	6.97078e37	0.0522274
$792$	0	0
$793$	7.77927e38	0.563314
$794$	0	0
$795$	−1.68388e39	−1.17857
$796$	0	0
$797$	−2.14824e39	−1.45343	−0.726717	−	0.686937i	$-0.758955\pi$
$797$	−2.14824e39	−1.45343	−0.726717	+	0.686937i	$0.758955\pi$
$798$	0	0
$799$	2.00718e38	0.131282
$800$	0	0
$801$	5.15121e38	0.325739
$802$	0	0
$803$	−5.43777e38	−0.332476
$804$	0	0
$805$	1.31388e37	0.00776802
$806$	0	0
$807$	−1.58181e39	−0.904401
$808$	0	0
$809$	−1.44847e39	−0.800949	−0.400474	−	0.916308i	$-0.631155\pi$
$809$	−1.44847e39	−0.800949	−0.400474	+	0.916308i	$0.631155\pi$
$810$	0	0
$811$	4.08234e38	0.218337	0.109169	−	0.994023i	$-0.465181\pi$
$811$	4.08234e38	0.218337	0.109169	+	0.994023i	$0.465181\pi$
$812$	0	0
$813$	1.45589e39	0.753193
$814$	0	0
$815$	−1.94086e39	−0.971328
$816$	0	0
$817$	2.90643e39	1.40722
$818$	0	0
$819$	−3.75828e37	−0.0176058
$820$	0	0
$821$	−2.94462e39	−1.33474	−0.667371	−	0.744726i	$-0.732580\pi$
$821$	−2.94462e39	−1.33474	−0.667371	+	0.744726i	$0.732580\pi$
$822$	0	0
$823$	1.81936e39	0.798032	0.399016	−	0.916944i	$-0.369352\pi$
$823$	1.81936e39	0.798032	0.399016	+	0.916944i	$0.369352\pi$
$824$	0	0
$825$	−2.39855e39	−1.01817
$826$	0	0
$827$	−2.52233e39	−1.03628	−0.518141	−	0.855295i	$-0.673376\pi$
$827$	−2.52233e39	−1.03628	−0.518141	+	0.855295i	$0.673376\pi$
$828$	0	0
$829$	1.02531e39	0.407726	0.203863	−	0.978999i	$-0.434650\pi$
$829$	1.02531e39	0.407726	0.203863	+	0.978999i	$0.434650\pi$
$830$	0	0
$831$	3.20589e39	1.23405
$832$	0	0
$833$	1.58439e39	0.590409
$834$	0	0
$835$	−3.39578e39	−1.22510
$836$	0	0
$837$	1.42578e39	0.498035
$838$	0	0
$839$	1.22528e39	0.414429	0.207214	−	0.978296i	$-0.433560\pi$
$839$	1.22528e39	0.414429	0.207214	+	0.978296i	$0.433560\pi$
$840$	0	0
$841$	−2.89568e39	−0.948427
$842$	0	0
$843$	−2.05504e39	−0.651849
$844$	0	0
$845$	−4.12834e39	−1.26826
$846$	0	0
$847$	1.80688e38	0.0537654
$848$	0	0
$849$	−3.34112e39	−0.963028
$850$	0	0
$851$	−4.62073e37	−0.0129021
$852$	0	0
$853$	2.17452e39	0.588235	0.294117	−	0.955769i	$-0.404974\pi$
$853$	2.17452e39	0.588235	0.294117	+	0.955769i	$0.404974\pi$
$854$	0	0
$855$	3.52769e39	0.924586
$856$	0	0
$857$	4.29564e39	1.09091	0.545453	−	0.838142i	$-0.316358\pi$
$857$	4.29564e39	1.09091	0.545453	+	0.838142i	$0.316358\pi$
$858$	0	0
$859$	1.62483e39	0.399853	0.199927	−	0.979811i	$-0.435930\pi$
$859$	1.62483e39	0.399853	0.199927	+	0.979811i	$0.435930\pi$
$860$	0	0
$861$	1.67110e38	0.0398529
$862$	0	0
$863$	−3.68820e39	−0.852450	−0.426225	−	0.904617i	$-0.640157\pi$
$863$	−3.68820e39	−0.852450	−0.426225	+	0.904617i	$0.640157\pi$
$864$	0	0
$865$	5.86281e39	1.31337
$866$	0	0
$867$	2.32335e39	0.504494
$868$	0	0
$869$	−1.10987e39	−0.233616
$870$	0	0
$871$	3.67952e39	0.750838
$872$	0	0
$873$	3.07336e39	0.608024
$874$	0	0
$875$	−9.18669e38	−0.176218
$876$	0	0
$877$	−5.53156e39	−1.02885	−0.514426	−	0.857535i	$-0.671995\pi$
$877$	−5.53156e39	−1.02885	−0.514426	+	0.857535i	$0.671995\pi$
$878$	0	0
$879$	5.95272e39	1.07366
$880$	0	0
$881$	9.22430e39	1.61346	0.806731	−	0.590919i	$-0.201235\pi$
$881$	9.22430e39	1.61346	0.806731	+	0.590919i	$0.201235\pi$
$882$	0	0
$883$	7.47767e39	1.26852	0.634259	−	0.773120i	$-0.281305\pi$
$883$	7.47767e39	1.26852	0.634259	+	0.773120i	$0.281305\pi$
$884$	0	0
$885$	1.14676e40	1.88686
$886$	0	0
$887$	3.52542e39	0.562655	0.281328	−	0.959612i	$-0.409225\pi$
$887$	3.52542e39	0.562655	0.281328	+	0.959612i	$0.409225\pi$
$888$	0	0
$889$	−6.56395e38	−0.101623
$890$	0	0
$891$	−1.81808e39	−0.273064
$892$	0	0
$893$	2.00950e39	0.292814
$894$	0	0
$895$	−1.58194e40	−2.23656
$896$	0	0
$897$	−1.58908e38	−0.0217995
$898$	0	0
$899$	−7.82691e38	−0.104192
$900$	0	0
$901$	−3.89832e39	−0.503608
$902$	0	0
$903$	−5.53470e38	−0.0693921
$904$	0	0
$905$	−2.51259e40	−3.05750
$906$	0	0
$907$	1.16020e40	1.37036	0.685180	−	0.728374i	$-0.259724\pi$
$907$	1.16020e40	1.37036	0.685180	+	0.728374i	$0.259724\pi$
$908$	0	0
$909$	−1.38713e39	−0.159040
$910$	0	0
$911$	−1.03198e40	−1.14861	−0.574306	−	0.818640i	$-0.694728\pi$
$911$	−1.03198e40	−1.14861	−0.574306	+	0.818640i	$0.694728\pi$
$912$	0	0
$913$	−6.87785e39	−0.743185
$914$	0	0
$915$	1.39051e40	1.45878
$916$	0	0
$917$	−8.42181e38	−0.0857866
$918$	0	0
$919$	−1.74754e40	−1.72850	−0.864248	−	0.503067i	$-0.832205\pi$
$919$	−1.74754e40	−1.72850	−0.864248	+	0.503067i	$0.832205\pi$
$920$	0	0
$921$	8.71535e39	0.837104
$922$	0	0
$923$	−9.71572e39	−0.906258
$924$	0	0
$925$	5.97035e39	0.540862
$926$	0	0
$927$	2.09371e39	0.184222
$928$	0	0
$929$	8.87284e39	0.758316	0.379158	−	0.925332i	$-0.376214\pi$
$929$	8.87284e39	0.758316	0.379158	+	0.925332i	$0.376214\pi$
$930$	0	0
$931$	1.58621e40	1.31686
$932$	0	0
$933$	−4.74549e39	−0.382718
$934$	0	0
$935$	−8.10076e39	−0.634701
$936$	0	0
$937$	−1.64530e39	−0.125245	−0.0626226	−	0.998037i	$-0.519946\pi$
$937$	−1.64530e39	−0.125245	−0.0626226	+	0.998037i	$0.519946\pi$
$938$	0	0
$939$	−3.29572e39	−0.243761
$940$	0	0
$941$	−7.97642e38	−0.0573255	−0.0286628	−	0.999589i	$-0.509125\pi$
$941$	−7.97642e38	−0.0573255	−0.0286628	+	0.999589i	$0.509125\pi$
$942$	0	0
$943$	−4.53648e38	−0.0316819
$944$	0	0
$945$	−2.38987e39	−0.162198
$946$	0	0
$947$	−1.60271e40	−1.05713	−0.528566	−	0.848892i	$-0.677270\pi$
$947$	−1.60271e40	−1.05713	−0.528566	+	0.848892i	$0.677270\pi$
$948$	0	0
$949$	4.65486e39	0.298410
$950$	0	0
$951$	−1.21746e40	−0.758608
$952$	0	0
$953$	−8.74550e39	−0.529701	−0.264850	−	0.964290i	$-0.585323\pi$
$953$	−8.74550e39	−0.529701	−0.264850	+	0.964290i	$0.585323\pi$
$954$	0	0
$955$	2.80893e40	1.65384
$956$	0	0
$957$	1.85360e39	0.106098
$958$	0	0
$959$	2.26577e38	0.0126085
$960$	0	0
$961$	−1.45921e40	−0.789502
$962$	0	0
$963$	−9.57619e39	−0.503776
$964$	0	0
$965$	−1.39073e40	−0.711419
$966$	0	0
$967$	−1.60838e40	−0.800075	−0.400038	−	0.916499i	$-0.631003\pi$
$967$	−1.60838e40	−0.800075	−0.400038	+	0.916499i	$0.631003\pi$
$968$	0	0
$969$	−1.27202e40	−0.615352
$970$	0	0
$971$	−1.28013e40	−0.602276	−0.301138	−	0.953581i	$-0.597367\pi$
$971$	−1.28013e40	−0.602276	−0.301138	+	0.953581i	$0.597367\pi$
$972$	0	0
$973$	3.06359e39	0.140187
$974$	0	0
$975$	2.05322e40	0.913845
$976$	0	0
$977$	−1.68476e40	−0.729392	−0.364696	−	0.931127i	$-0.618827\pi$
$977$	−1.68476e40	−0.729392	−0.364696	+	0.931127i	$0.618827\pi$
$978$	0	0
$979$	−1.18426e40	−0.498746
$980$	0	0
$981$	−3.32788e39	−0.136344
$982$	0	0
$983$	3.82928e40	1.52632	0.763159	−	0.646211i	$-0.223647\pi$
$983$	3.82928e40	1.52632	0.763159	+	0.646211i	$0.223647\pi$
$984$	0	0
$985$	8.24383e40	3.19698
$986$	0	0
$987$	−3.82668e38	−0.0144391
$988$	0	0
$989$	1.50249e39	0.0551647
$990$	0	0
$991$	−1.66034e40	−0.593201	−0.296600	−	0.955002i	$-0.595853\pi$
$991$	−1.66034e40	−0.593201	−0.296600	+	0.955002i	$0.595853\pi$
$992$	0	0
$993$	−3.86783e39	−0.134478
$994$	0	0
$995$	4.07729e40	1.37962
$996$	0	0
$997$	−4.81089e39	−0.158431	−0.0792156	−	0.996858i	$-0.525242\pi$
$997$	−4.81089e39	−0.158431	−0.0792156	+	0.996858i	$0.525242\pi$
$998$	0	0
$999$	8.40486e39	0.269399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.28.a.d.1.1		2
4.3	odd	2		1.28.a.a.1.2	✓	2
12.11	even	2		9.28.a.d.1.1		2
20.3	even	4		25.28.b.a.24.2		4
20.7	even	4		25.28.b.a.24.3		4
20.19	odd	2		25.28.a.a.1.1		2
28.27	even	2		49.28.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.28.a.a.1.2	✓	2	4.3	odd	2
9.28.a.d.1.1		2	12.11	even	2
16.28.a.d.1.1		2	1.1	even	1	trivial
25.28.a.a.1.1		2	20.19	odd	2
25.28.b.a.24.2		4	20.3	even	4
25.28.b.a.24.3		4	20.7	even	4
49.28.a.b.1.2		2	28.27	even	2

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 28 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15499e6 q^{3} +4.86703e9 q^{5} -2.14815e10 q^{7} -2.98161e12 q^{9} +O(q^{10})\)	\(q-2.15499e6 q^{3} +4.86703e9 q^{5} -2.14815e10 q^{7} -2.98161e12 q^{9} +6.85469e13 q^{11} -5.86777e14 q^{13} -1.04884e16 q^{15} -2.42814e16 q^{17} -2.43094e17 q^{19} +4.62924e16 q^{21} -1.25668e17 q^{23} +1.62374e19 q^{25} +2.28584e19 q^{27} -1.25483e19 q^{29} +6.23745e19 q^{31} -1.47718e20 q^{33} -1.04551e20 q^{35} +3.67692e20 q^{37} +1.26450e21 q^{39} +3.60988e21 q^{41} -1.19560e22 q^{43} -1.45116e22 q^{45} -8.26632e21 q^{47} -6.52509e22 q^{49} +5.23263e22 q^{51} +1.60547e23 q^{53} +3.33619e23 q^{55} +5.23866e23 q^{57} -1.09336e24 q^{59} -1.32576e24 q^{61} +6.40495e22 q^{63} -2.85586e24 q^{65} -6.27072e24 q^{67} +2.70814e23 q^{69} +1.65578e25 q^{71} -7.93292e24 q^{73} -3.49914e25 q^{75} -1.47249e24 q^{77} -1.61913e25 q^{79} -2.65232e25 q^{81} -1.00338e26 q^{83} -1.18178e26 q^{85} +2.70414e25 q^{87} -1.72766e26 q^{89} +1.26049e25 q^{91} -1.34416e26 q^{93} -1.18315e27 q^{95} -1.03077e27 q^{97} -2.04380e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1286280 q^{3} + 5443587900 q^{5} + 175391963600 q^{7} + 1235136554154 q^{9}+O(q^{10}) \) 2 * q + 1286280 * q^3 + 5443587900 * q^5 + 175391963600 * q^7 + 1235136554154 * q^9	\( 2 q + 1286280 q^{3} + 5443587900 q^{5} + 175391963600 q^{7} + 1235136554154 q^{9} - 138167337691944 q^{11} - 753433801271060 q^{13} - 85\!\cdots\!00 q^{15}+ \cdots - 10\!\cdots\!88 q^{99}+O(q^{100}) \) 2 * q + 1286280 * q^3 + 5443587900 * q^5 + 175391963600 * q^7 + 1235136554154 * q^9 - 138167337691944 * q^11 - 753433801271060 * q^13 - 8504300488438800 * q^15 - 29753620331011740 * q^17 - 404565810372684760 * q^19 + 723787313583184704 * q^21 - 2929078923121218960 * q^23 + 9119218786673228750 * q^25 + 11127665129740313040 * q^27 - 15546679995448558260 * q^29 - 28544554594467385024 * q^31 - 859077391009750054560 * q^33 + 8958395384765013600 * q^35 + 1867697204682824566780 * q^37 + 690990221801025527472 * q^39 + 9081343698046512254964 * q^41 - 5145612605801421773800 * q^43 - 12080376719602732775700 * q^45 + 1150251488862201070560 * q^47 - 92204113217840907690414 * q^49 + 33494974704727214464656 * q^51 + 106953735591470060758620 * q^53 + 214436254091483969701200 * q^55 - 31800538920577362634080 * q^57 - 2009620977624026488631880 * q^59 + 147857426692448940370444 * q^61 + 894215232558851858338320 * q^63 - 2951949350200452960922200 * q^65 - 3051578535738098902157560 * q^67 - 9376480489705868528695872 * q^69 + 13175198820369338598286416 * q^71 + 5284260812951286572116660 * q^73 - 59486914888952006559645000 * q^75 - 42169025756092787413646400 * q^77 - 62814351035720719918179040 * q^79 - 99047155826597708238103278 * q^81 - 171806873410054757883176280 * q^83 - 121333441005595565089302600 * q^85 + 16722988656204557792643120 * q^87 - 313473438761105539763494380 * q^89 - 20205365125040878536694304 * q^91 - 447293490375807514724002560 * q^93 - 1276245244260479145691314000 * q^95 - 653202933397052842883888060 * q^97 - 1076041779280842335610479688 * q^99

Introduction

L-functions

Modular forms

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Database

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