LMFDB - Embedded newform 16.28.a.d.1.2

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			1.2
Root			$-66.9704$ 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)

$f(q)$	$=$	$q+3.44127e6 q^{3} +5.76560e8 q^{5} +1.96873e11 q^{7} +4.21675e12 q^{9} +O(q^{10})$	$q+3.44127e6 q^{3} +5.76560e8 q^{5} +1.96873e11 q^{7} +4.21675e12 q^{9} -2.06714e14 q^{11} -1.66656e14 q^{13} +1.98410e15 q^{15} -5.47219e15 q^{17} -1.61471e17 q^{19} +6.77495e17 q^{21} -2.80341e18 q^{23} -7.11816e18 q^{25} -1.17308e19 q^{27} -2.99842e18 q^{29} -9.09190e19 q^{31} -7.11360e20 q^{33} +1.13509e20 q^{35} +1.50001e21 q^{37} -5.73510e20 q^{39} +5.47146e21 q^{41} +6.81035e21 q^{43} +2.43121e21 q^{45} +9.41657e21 q^{47} -2.69532e22 q^{49} -1.88313e22 q^{51} -5.35936e22 q^{53} -1.19183e23 q^{55} -5.55667e23 q^{57} -9.16258e23 q^{59} +1.47362e24 q^{61} +8.30166e23 q^{63} -9.60874e22 q^{65} +3.21915e24 q^{67} -9.64729e24 q^{69} -3.38255e24 q^{71} +1.32172e25 q^{73} -2.44955e25 q^{75} -4.06965e25 q^{77} -4.66230e25 q^{79} -7.25240e25 q^{81} -7.14690e25 q^{83} -3.15505e24 q^{85} -1.03184e25 q^{87} -1.40708e26 q^{89} -3.28102e25 q^{91} -3.12877e26 q^{93} -9.30980e25 q^{95} +3.77568e26 q^{97} -8.71662e26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.44127e6	1.24618	0.623092	−	0.782149i	$-0.285876\pi$
$3$	3.44127e6	1.24618	0.623092	+	0.782149i	$0.285876\pi$
$4$	0	0
$5$	5.76560e8	0.211227	0.105614	−	0.994407i	$-0.466319\pi$
$5$	5.76560e8	0.211227	0.105614	+	0.994407i	$0.466319\pi$
$6$	0	0
$7$	1.96873e11	0.768004	0.384002	−	0.923332i	$-0.374546\pi$
$7$	1.96873e11	0.768004	0.384002	+	0.923332i	$0.374546\pi$
$8$	0	0
$9$	4.21675e12	0.552973
$10$	0	0
$11$	−2.06714e14	−1.80538	−0.902691	−	0.430290i	$-0.858411\pi$
$11$	−2.06714e14	−1.80538	−0.902691	+	0.430290i	$0.858411\pi$
$12$	0	0
$13$	−1.66656e14	−0.152611	−0.0763057	−	0.997084i	$-0.524312\pi$
$13$	−1.66656e14	−0.152611	−0.0763057	+	0.997084i	$0.524312\pi$
$14$	0	0
$15$	1.98410e15	0.263228
$16$	0	0
$17$	−5.47219e15	−0.133999	−0.0669994	−	0.997753i	$-0.521343\pi$
$17$	−5.47219e15	−0.133999	−0.0669994	+	0.997753i	$0.521343\pi$
$18$	0	0
$19$	−1.61471e17	−0.880891	−0.440445	−	0.897779i	$-0.645179\pi$
$19$	−1.61471e17	−0.880891	−0.440445	+	0.897779i	$0.645179\pi$
$20$	0	0
$21$	6.77495e17	0.957074
$22$	0	0
$23$	−2.80341e18	−1.15974	−0.579870	−	0.814709i	$-0.696897\pi$
$23$	−2.80341e18	−1.15974	−0.579870	+	0.814709i	$0.696897\pi$
$24$	0	0
$25$	−7.11816e18	−0.955383
$26$	0	0
$27$	−1.17308e19	−0.557078
$28$	0	0
$29$	−2.99842e18	−0.0542650	−0.0271325	−	0.999632i	$-0.508638\pi$
$29$	−2.99842e18	−0.0542650	−0.0271325	+	0.999632i	$0.508638\pi$
$30$	0	0
$31$	−9.09190e19	−0.668763	−0.334381	−	0.942438i	$-0.608527\pi$
$31$	−9.09190e19	−0.668763	−0.334381	+	0.942438i	$0.608527\pi$
$32$	0	0
$33$	−7.11360e20	−2.24984
$34$	0	0
$35$	1.13509e20	0.162223
$36$	0	0
$37$	1.50001e21	1.01244	0.506220	−	0.862404i	$-0.331042\pi$
$37$	1.50001e21	1.01244	0.506220	+	0.862404i	$0.331042\pi$
$38$	0	0
$39$	−5.73510e20	−0.190182
$40$	0	0
$41$	5.47146e21	0.923679	0.461840	−	0.886963i	$-0.347190\pi$
$41$	5.47146e21	0.923679	0.461840	+	0.886963i	$0.347190\pi$
$42$	0	0
$43$	6.81035e21	0.604429	0.302215	−	0.953240i	$-0.402274\pi$
$43$	6.81035e21	0.604429	0.302215	+	0.953240i	$0.402274\pi$
$44$	0	0
$45$	2.43121e21	0.116803
$46$	0	0
$47$	9.41657e21	0.251520	0.125760	−	0.992061i	$-0.459863\pi$
$47$	9.41657e21	0.251520	0.125760	+	0.992061i	$0.459863\pi$
$48$	0	0
$49$	−2.69532e22	−0.410169
$50$	0	0
$51$	−1.88313e22	−0.166987
$52$	0	0
$53$	−5.35936e22	−0.282741	−0.141370	−	0.989957i	$-0.545151\pi$
$53$	−5.35936e22	−0.282741	−0.141370	+	0.989957i	$0.545151\pi$
$54$	0	0
$55$	−1.19183e23	−0.381346
$56$	0	0
$57$	−5.55667e23	−1.09775
$58$	0	0
$59$	−9.16258e23	−1.13636	−0.568180	−	0.822904i	$-0.692352\pi$
$59$	−9.16258e23	−1.13636	−0.568180	+	0.822904i	$0.692352\pi$
$60$	0	0
$61$	1.47362e24	1.16529	0.582643	−	0.812728i	$-0.302019\pi$
$61$	1.47362e24	1.16529	0.582643	+	0.812728i	$0.302019\pi$
$62$	0	0
$63$	8.30166e23	0.424685
$64$	0	0
$65$	−9.60874e22	−0.0322356
$66$	0	0
$67$	3.21915e24	0.717349	0.358675	−	0.933463i	$-0.383229\pi$
$67$	3.21915e24	0.717349	0.358675	+	0.933463i	$0.383229\pi$
$68$	0	0
$69$	−9.64729e24	−1.44525
$70$	0	0
$71$	−3.38255e24	−0.344554	−0.172277	−	0.985049i	$-0.555112\pi$
$71$	−3.38255e24	−0.344554	−0.172277	+	0.985049i	$0.555112\pi$
$72$	0	0
$73$	1.32172e25	0.925295	0.462648	−	0.886542i	$-0.346900\pi$
$73$	1.32172e25	0.925295	0.462648	+	0.886542i	$0.346900\pi$
$74$	0	0
$75$	−2.44955e25	−1.19058
$76$	0	0
$77$	−4.06965e25	−1.38654
$78$	0	0
$79$	−4.66230e25	−1.12366	−0.561830	−	0.827253i	$-0.689902\pi$
$79$	−4.66230e25	−1.12366	−0.561830	+	0.827253i	$0.689902\pi$
$80$	0	0
$81$	−7.25240e25	−1.24719
$82$	0	0
$83$	−7.14690e25	−0.884226	−0.442113	−	0.896959i	$-0.645771\pi$
$83$	−7.14690e25	−0.884226	−0.442113	+	0.896959i	$0.645771\pi$
$84$	0	0
$85$	−3.15505e24	−0.0283042
$86$	0	0
$87$	−1.03184e25	−0.0676242
$88$	0	0
$89$	−1.40708e26	−0.678504	−0.339252	−	0.940696i	$-0.610174\pi$
$89$	−1.40708e26	−0.678504	−0.339252	+	0.940696i	$0.610174\pi$
$90$	0	0
$91$	−3.28102e25	−0.117206
$92$	0	0
$93$	−3.12877e26	−0.833401
$94$	0	0
$95$	−9.30980e25	−0.186068
$96$	0	0
$97$	3.77568e26	0.569609	0.284805	−	0.958586i	$-0.408071\pi$
$97$	3.77568e26	0.569609	0.284805	+	0.958586i	$0.408071\pi$
$98$	0	0
$99$	−8.71662e26	−0.998327
$100$	0	0
$101$	−1.82546e27	−1.59600	−0.798002	−	0.602654i	$-0.794110\pi$
$101$	−1.82546e27	−1.59600	−0.798002	+	0.602654i	$0.794110\pi$
$102$	0	0
$103$	−2.87048e27	−1.92598	−0.962991	−	0.269534i	$-0.913130\pi$
$103$	−2.87048e27	−1.92598	−0.962991	+	0.269534i	$0.913130\pi$
$104$	0	0
$105$	3.90617e26	0.202160
$106$	0	0
$107$	1.31331e27	0.526849	0.263425	−	0.964680i	$-0.415148\pi$
$107$	1.31331e27	0.526849	0.263425	+	0.964680i	$0.415148\pi$
$108$	0	0
$109$	−3.16013e27	−0.987296	−0.493648	−	0.869662i	$-0.664337\pi$
$109$	−3.16013e27	−0.987296	−0.493648	+	0.869662i	$0.664337\pi$
$110$	0	0
$111$	5.16192e27	1.26169
$112$	0	0
$113$	3.68033e27	0.706851	0.353425	−	0.935463i	$-0.385017\pi$
$113$	3.68033e27	0.706851	0.353425	+	0.935463i	$0.385017\pi$
$114$	0	0
$115$	−1.61633e27	−0.244969
$116$	0	0
$117$	−7.02748e26	−0.0843899
$118$	0	0
$119$	−1.07733e27	−0.102912
$120$	0	0
$121$	2.96208e28	2.25940
$122$	0	0
$123$	1.88288e28	1.15107
$124$	0	0
$125$	−8.39976e27	−0.413030
$126$	0	0
$127$	2.69227e28	1.06849	0.534243	−	0.845331i	$-0.320597\pi$
$127$	2.69227e28	1.06849	0.534243	+	0.845331i	$0.320597\pi$
$128$	0	0
$129$	2.34363e28	0.753229
$130$	0	0
$131$	−3.17810e28	−0.829860	−0.414930	−	0.909853i	$-0.636194\pi$
$131$	−3.17810e28	−0.829860	−0.414930	+	0.909853i	$0.636194\pi$
$132$	0	0
$133$	−3.17894e28	−0.676528
$134$	0	0
$135$	−6.76350e27	−0.117670
$136$	0	0
$137$	2.13194e28	0.304122	0.152061	−	0.988371i	$-0.451409\pi$
$137$	2.13194e28	0.304122	0.152061	+	0.988371i	$0.451409\pi$
$138$	0	0
$139$	1.15969e29	1.36032	0.680161	−	0.733062i	$-0.261910\pi$
$139$	1.15969e29	1.36032	0.680161	+	0.733062i	$0.261910\pi$
$140$	0	0
$141$	3.24050e28	0.313439
$142$	0	0
$143$	3.44502e28	0.275522
$144$	0	0
$145$	−1.72877e27	−0.0114622
$146$	0	0
$147$	−9.27533e28	−0.511146
$148$	0	0
$149$	2.70481e29	1.24200	0.621000	−	0.783810i	$-0.286727\pi$
$149$	2.70481e29	1.24200	0.621000	+	0.783810i	$0.286727\pi$
$150$	0	0
$151$	−6.43669e28	−0.246873	−0.123437	−	0.992352i	$-0.539392\pi$
$151$	−6.43669e28	−0.246873	−0.123437	+	0.992352i	$0.539392\pi$
$152$	0	0
$153$	−2.30748e28	−0.0740977
$154$	0	0
$155$	−5.24203e28	−0.141261
$156$	0	0
$157$	−7.36476e29	−1.66922	−0.834609	−	0.550842i	$-0.814307\pi$
$157$	−7.36476e29	−1.66922	−0.834609	+	0.550842i	$0.814307\pi$
$158$	0	0
$159$	−1.84430e29	−0.352347
$160$	0	0
$161$	−5.51917e29	−0.890686
$162$	0	0
$163$	−4.91659e29	−0.671632	−0.335816	−	0.941928i	$-0.609012\pi$
$163$	−4.91659e29	−0.671632	−0.335816	+	0.941928i	$0.609012\pi$
$164$	0	0
$165$	−4.10142e29	−0.475226
$166$	0	0
$167$	−6.37427e29	−0.627709	−0.313854	−	0.949471i	$-0.601620\pi$
$167$	−6.37427e29	−0.627709	−0.313854	+	0.949471i	$0.601620\pi$
$168$	0	0
$169$	−1.16476e30	−0.976710
$170$	0	0
$171$	−6.80884e29	−0.487109
$172$	0	0
$173$	2.71145e30	1.65798	0.828989	−	0.559265i	$-0.188917\pi$
$173$	2.71145e30	1.65798	0.828989	+	0.559265i	$0.188917\pi$
$174$	0	0
$175$	−1.40138e30	−0.733738
$176$	0	0
$177$	−3.15309e30	−1.41611
$178$	0	0
$179$	1.65055e30	0.636961	0.318481	−	0.947929i	$-0.396827\pi$
$179$	1.65055e30	0.636961	0.318481	+	0.947929i	$0.396827\pi$
$180$	0	0
$181$	4.30260e30	1.42913	0.714563	−	0.699571i	$-0.246626\pi$
$181$	4.30260e30	1.42913	0.714563	+	0.699571i	$0.246626\pi$
$182$	0	0
$183$	5.07112e30	1.45216
$184$	0	0
$185$	8.64843e29	0.213855
$186$	0	0
$187$	1.13118e30	0.241919
$188$	0	0
$189$	−2.30948e30	−0.427838
$190$	0	0
$191$	−5.35850e30	−0.861177	−0.430589	−	0.902548i	$-0.641694\pi$
$191$	−5.35850e30	−0.861177	−0.430589	+	0.902548i	$0.641694\pi$
$192$	0	0
$193$	2.47114e30	0.345044	0.172522	−	0.985006i	$-0.444808\pi$
$193$	2.47114e30	0.345044	0.172522	+	0.985006i	$0.444808\pi$
$194$	0	0
$195$	−3.30663e29	−0.0401715
$196$	0	0
$197$	−4.41349e30	−0.467185	−0.233592	−	0.972335i	$-0.575048\pi$
$197$	−4.41349e30	−0.467185	−0.233592	+	0.972335i	$0.575048\pi$
$198$	0	0
$199$	1.58034e31	1.45960	0.729802	−	0.683658i	$-0.239612\pi$
$199$	1.58034e31	1.45960	0.729802	+	0.683658i	$0.239612\pi$
$200$	0	0
$201$	1.10780e31	0.893949
$202$	0	0
$203$	−5.90310e29	−0.0416758
$204$	0	0
$205$	3.15463e30	0.195106
$206$	0	0
$207$	−1.18213e31	−0.641305
$208$	0	0
$209$	3.33784e31	1.59034
$210$	0	0
$211$	1.50086e31	0.628820	0.314410	−	0.949287i	$-0.398193\pi$
$211$	1.50086e31	0.628820	0.314410	+	0.949287i	$0.398193\pi$
$212$	0	0
$213$	−1.16403e31	−0.429377
$214$	0	0
$215$	3.92658e30	0.127672
$216$	0	0
$217$	−1.78995e31	−0.513613
$218$	0	0
$219$	4.54839e31	1.15309
$220$	0	0
$221$	9.11975e29	0.0204497
$222$	0	0
$223$	1.04534e31	0.207559	0.103779	−	0.994600i	$-0.466906\pi$
$223$	1.04534e31	0.207559	0.103779	+	0.994600i	$0.466906\pi$
$224$	0	0
$225$	−3.00155e31	−0.528301
$226$	0	0
$227$	−9.45028e31	−1.47603	−0.738016	−	0.674783i	$-0.764237\pi$
$227$	−9.45028e31	−1.47603	−0.738016	+	0.674783i	$0.764237\pi$
$228$	0	0
$229$	−6.26689e31	−0.869508	−0.434754	−	0.900549i	$-0.643165\pi$
$229$	−6.26689e31	−0.869508	−0.434754	+	0.900549i	$0.643165\pi$
$230$	0	0
$231$	−1.40048e32	−1.72788
$232$	0	0
$233$	7.87335e31	0.864678	0.432339	−	0.901711i	$-0.357688\pi$
$233$	7.87335e31	0.864678	0.432339	+	0.901711i	$0.357688\pi$
$234$	0	0
$235$	5.42922e30	0.0531277
$236$	0	0
$237$	−1.60442e32	−1.40029
$238$	0	0
$239$	5.65108e30	0.0440311	0.0220156	−	0.999758i	$-0.492992\pi$
$239$	5.65108e30	0.0440311	0.0220156	+	0.999758i	$0.492992\pi$
$240$	0	0
$241$	−4.42939e31	−0.308400	−0.154200	−	0.988040i	$-0.549280\pi$
$241$	−4.42939e31	−0.308400	−0.154200	+	0.988040i	$0.549280\pi$
$242$	0	0
$243$	−1.60121e32	−0.997154
$244$	0	0
$245$	−1.55401e31	−0.0866389
$246$	0	0
$247$	2.69102e31	0.134434
$248$	0	0
$249$	−2.45944e32	−1.10191
$250$	0	0
$251$	3.28116e32	1.31957	0.659785	−	0.751454i	$-0.270647\pi$
$251$	3.28116e32	1.31957	0.659785	+	0.751454i	$0.270647\pi$
$252$	0	0
$253$	5.79505e32	2.09377
$254$	0	0
$255$	−1.08574e31	−0.0352722
$256$	0	0
$257$	3.48288e31	0.101822	0.0509109	−	0.998703i	$-0.483788\pi$
$257$	3.48288e31	0.101822	0.0509109	+	0.998703i	$0.483788\pi$
$258$	0	0
$259$	2.95311e32	0.777559
$260$	0	0
$261$	−1.26436e31	−0.0300071
$262$	0	0
$263$	−3.82694e31	−0.0819311	−0.0409656	−	0.999161i	$-0.513043\pi$
$263$	−3.82694e31	−0.0819311	−0.0409656	+	0.999161i	$0.513043\pi$
$264$	0	0
$265$	−3.08999e31	−0.0597225
$266$	0	0
$267$	−4.84213e32	−0.845541
$268$	0	0
$269$	−4.44965e31	−0.0702539	−0.0351269	−	0.999383i	$-0.511184\pi$
$269$	−4.44965e31	−0.0702539	−0.0351269	+	0.999383i	$0.511184\pi$
$270$	0	0
$271$	−3.62870e32	−0.518401	−0.259200	−	0.965824i	$-0.583459\pi$
$271$	−3.62870e32	−0.518401	−0.259200	+	0.965824i	$0.583459\pi$
$272$	0	0
$273$	−1.12909e32	−0.146060
$274$	0	0
$275$	1.47142e33	1.72483
$276$	0	0
$277$	1.40758e33	1.49622	0.748111	−	0.663574i	$-0.230961\pi$
$277$	1.40758e33	1.49622	0.748111	+	0.663574i	$0.230961\pi$
$278$	0	0
$279$	−3.83383e32	−0.369808
$280$	0	0
$281$	3.59534e32	0.314924	0.157462	−	0.987525i	$-0.449669\pi$
$281$	3.59534e32	0.314924	0.157462	+	0.987525i	$0.449669\pi$
$282$	0	0
$283$	3.59177e32	0.285885	0.142943	−	0.989731i	$-0.454344\pi$
$283$	3.59177e32	0.285885	0.142943	+	0.989731i	$0.454344\pi$
$284$	0	0
$285$	−3.20375e32	−0.231875
$286$	0	0
$287$	1.07719e33	0.709390
$288$	0	0
$289$	−1.63777e33	−0.982044
$290$	0	0
$291$	1.29931e33	0.709838
$292$	0	0
$293$	3.51312e32	0.174976	0.0874882	−	0.996166i	$-0.472116\pi$
$293$	3.51312e32	0.174976	0.0874882	+	0.996166i	$0.472116\pi$
$294$	0	0
$295$	−5.28278e32	−0.240030
$296$	0	0
$297$	2.42492e33	1.00574
$298$	0	0
$299$	4.67206e32	0.176990
$300$	0	0
$301$	1.34078e33	0.464204
$302$	0	0
$303$	−6.28191e33	−1.98891
$304$	0	0
$305$	8.49630e32	0.246140
$306$	0	0
$307$	6.75654e32	0.179208	0.0896038	−	0.995977i	$-0.471440\pi$
$307$	6.75654e32	0.179208	0.0896038	+	0.995977i	$0.471440\pi$
$308$	0	0
$309$	−9.87810e33	−2.40013
$310$	0	0
$311$	−5.16690e33	−1.15071	−0.575354	−	0.817905i	$-0.695136\pi$
$311$	−5.16690e33	−1.15071	−0.575354	+	0.817905i	$0.695136\pi$
$312$	0	0
$313$	−1.10679e33	−0.226056	−0.113028	−	0.993592i	$-0.536055\pi$
$313$	−1.10679e33	−0.226056	−0.113028	+	0.993592i	$0.536055\pi$
$314$	0	0
$315$	4.78641e32	0.0897051
$316$	0	0
$317$	−9.49991e33	−1.63463	−0.817315	−	0.576191i	$-0.804539\pi$
$317$	−9.49991e33	−1.63463	−0.817315	+	0.576191i	$0.804539\pi$
$318$	0	0
$319$	6.19817e32	0.0979691
$320$	0	0
$321$	4.51946e33	0.656551
$322$	0	0
$323$	8.83602e32	0.118038
$324$	0	0
$325$	1.18629e33	0.145802
$326$	0	0
$327$	−1.08749e34	−1.23035
$328$	0	0
$329$	1.85387e33	0.193168
$330$	0	0
$331$	1.25351e34	1.20351	0.601757	−	0.798679i	$-0.294467\pi$
$331$	1.25351e34	1.20351	0.601757	+	0.798679i	$0.294467\pi$
$332$	0	0
$333$	6.32514e33	0.559852
$334$	0	0
$335$	1.85603e33	0.151524
$336$	0	0
$337$	−3.36740e33	−0.253683	−0.126842	−	0.991923i	$-0.540484\pi$
$337$	−3.36740e33	−0.253683	−0.126842	+	0.991923i	$0.540484\pi$
$338$	0	0
$339$	1.26650e34	0.880865
$340$	0	0
$341$	1.87943e34	1.20737
$342$	0	0
$343$	−1.82434e34	−1.08302
$344$	0	0
$345$	−5.56225e33	−0.305276
$346$	0	0
$347$	1.91545e34	0.972353	0.486177	−	0.873861i	$-0.338391\pi$
$347$	1.91545e34	0.972353	0.486177	+	0.873861i	$0.338391\pi$
$348$	0	0
$349$	1.45433e34	0.683158	0.341579	−	0.939853i	$-0.389038\pi$
$349$	1.45433e34	0.683158	0.341579	+	0.939853i	$0.389038\pi$
$350$	0	0
$351$	1.95501e33	0.0850164
$352$	0	0
$353$	−4.39359e33	−0.176954	−0.0884772	−	0.996078i	$-0.528200\pi$
$353$	−4.39359e33	−0.176954	−0.0884772	+	0.996078i	$0.528200\pi$
$354$	0	0
$355$	−1.95025e33	−0.0727791
$356$	0	0
$357$	−3.70738e33	−0.128247
$358$	0	0
$359$	−2.60550e34	−0.835828	−0.417914	−	0.908487i	$-0.637239\pi$
$359$	−2.60550e34	−0.835828	−0.417914	+	0.908487i	$0.637239\pi$
$360$	0	0
$361$	−7.52760e33	−0.224032
$362$	0	0
$363$	1.01933e35	2.81563
$364$	0	0
$365$	7.62050e33	0.195447
$366$	0	0
$367$	−3.50053e34	−0.833953	−0.416977	−	0.908917i	$-0.636910\pi$
$367$	−3.50053e34	−0.833953	−0.416977	+	0.908917i	$0.636910\pi$
$368$	0	0
$369$	2.30718e34	0.510770
$370$	0	0
$371$	−1.05512e34	−0.217146
$372$	0	0
$373$	−2.06100e34	−0.394464	−0.197232	−	0.980357i	$-0.563195\pi$
$373$	−2.06100e34	−0.394464	−0.197232	+	0.980357i	$0.563195\pi$
$374$	0	0
$375$	−2.89058e34	−0.514711
$376$	0	0
$377$	4.99707e32	0.00828146
$378$	0	0
$379$	3.69120e34	0.569559	0.284779	−	0.958593i	$-0.408080\pi$
$379$	3.69120e34	0.569559	0.284779	+	0.958593i	$0.408080\pi$
$380$	0	0
$381$	9.26485e34	1.33153
$382$	0	0
$383$	−6.12833e34	−0.820652	−0.410326	−	0.911939i	$-0.634585\pi$
$383$	−6.12833e34	−0.820652	−0.410326	+	0.911939i	$0.634585\pi$
$384$	0	0
$385$	−2.34640e34	−0.292875
$386$	0	0
$387$	2.87175e34	0.334233
$388$	0	0
$389$	5.65060e34	0.613444	0.306722	−	0.951799i	$-0.400768\pi$
$389$	5.65060e34	0.613444	0.306722	+	0.951799i	$0.400768\pi$
$390$	0	0
$391$	1.53408e34	0.155404
$392$	0	0
$393$	−1.09367e35	−1.03416
$394$	0	0
$395$	−2.68810e34	−0.237347
$396$	0	0
$397$	−2.20649e35	−1.81983	−0.909915	−	0.414794i	$-0.863854\pi$
$397$	−2.20649e35	−1.81983	−0.909915	+	0.414794i	$0.863854\pi$
$398$	0	0
$399$	−1.09396e35	−0.843078
$400$	0	0
$401$	−1.87612e35	−1.35148	−0.675741	−	0.737139i	$-0.736176\pi$
$401$	−1.87612e35	−1.35148	−0.675741	+	0.737139i	$0.736176\pi$
$402$	0	0
$403$	1.51522e34	0.102061
$404$	0	0
$405$	−4.18145e34	−0.263441
$406$	0	0
$407$	−3.10072e35	−1.82784
$408$	0	0
$409$	2.22853e35	1.22957	0.614786	−	0.788694i	$-0.289242\pi$
$409$	2.22853e35	1.22957	0.614786	+	0.788694i	$0.289242\pi$
$410$	0	0
$411$	7.33658e34	0.378991
$412$	0	0
$413$	−1.80387e35	−0.872729
$414$	0	0
$415$	−4.12062e34	−0.186773
$416$	0	0
$417$	3.99081e35	1.69521
$418$	0	0
$419$	5.47879e34	0.218170	0.109085	−	0.994032i	$-0.465208\pi$
$419$	5.47879e34	0.218170	0.109085	+	0.994032i	$0.465208\pi$
$420$	0	0
$421$	−1.97758e35	−0.738458	−0.369229	−	0.929338i	$-0.620378\pi$
$421$	−1.97758e35	−0.738458	−0.369229	+	0.929338i	$0.620378\pi$
$422$	0	0
$423$	3.97073e34	0.139083
$424$	0	0
$425$	3.89519e34	0.128020
$426$	0	0
$427$	2.90116e35	0.894945
$428$	0	0
$429$	1.18553e35	0.343350
$430$	0	0
$431$	4.94928e35	1.34617	0.673083	−	0.739567i	$-0.264970\pi$
$431$	4.94928e35	1.34617	0.673083	+	0.739567i	$0.264970\pi$
$432$	0	0
$433$	3.12882e35	0.799454	0.399727	−	0.916634i	$-0.369105\pi$
$433$	3.12882e35	0.799454	0.399727	+	0.916634i	$0.369105\pi$
$434$	0	0
$435$	−5.94917e33	−0.0142841
$436$	0	0
$437$	4.52671e35	1.02160
$438$	0	0
$439$	−7.22790e35	−1.53370	−0.766852	−	0.641824i	$-0.778178\pi$
$439$	−7.22790e35	−1.53370	−0.766852	+	0.641824i	$0.778178\pi$
$440$	0	0
$441$	−1.13655e35	−0.226813
$442$	0	0
$443$	7.68142e35	1.44209	0.721043	−	0.692891i	$-0.243663\pi$
$443$	7.68142e35	1.44209	0.721043	+	0.692891i	$0.243663\pi$
$444$	0	0
$445$	−8.11264e34	−0.143318
$446$	0	0
$447$	9.30798e35	1.54776
$448$	0	0
$449$	−6.52536e35	−1.02160	−0.510798	−	0.859701i	$-0.670650\pi$
$449$	−6.52536e35	−1.02160	−0.510798	+	0.859701i	$0.670650\pi$
$450$	0	0
$451$	−1.13103e36	−1.66759
$452$	0	0
$453$	−2.21504e35	−0.307649
$454$	0	0
$455$	−1.89171e34	−0.0247571
$456$	0	0
$457$	6.21081e35	0.766090	0.383045	−	0.923730i	$-0.374875\pi$
$457$	6.21081e35	0.766090	0.383045	+	0.923730i	$0.374875\pi$
$458$	0	0
$459$	6.41930e34	0.0746477
$460$	0	0
$461$	1.02161e36	1.12027	0.560133	−	0.828402i	$-0.310750\pi$
$461$	1.02161e36	1.12027	0.560133	+	0.828402i	$0.310750\pi$
$462$	0	0
$463$	1.10341e36	1.14129	0.570643	−	0.821199i	$-0.306694\pi$
$463$	1.10341e36	1.14129	0.570643	+	0.821199i	$0.306694\pi$
$464$	0	0
$465$	−1.80392e35	−0.176037
$466$	0	0
$467$	−8.66440e35	−0.797922	−0.398961	−	0.916968i	$-0.630629\pi$
$467$	−8.66440e35	−0.797922	−0.398961	+	0.916968i	$0.630629\pi$
$468$	0	0
$469$	6.33764e35	0.550927
$470$	0	0
$471$	−2.53441e36	−2.08015
$472$	0	0
$473$	−1.40780e36	−1.09123
$474$	0	0
$475$	1.14938e36	0.841588
$476$	0	0
$477$	−2.25991e35	−0.156348
$478$	0	0
$479$	2.57345e36	1.68262	0.841311	−	0.540552i	$-0.181785\pi$
$479$	2.57345e36	1.68262	0.841311	+	0.540552i	$0.181785\pi$
$480$	0	0
$481$	−2.49985e35	−0.154510
$482$	0	0
$483$	−1.89930e36	−1.10996
$484$	0	0
$485$	2.17691e35	0.120317
$486$	0	0
$487$	2.28927e36	1.19690	0.598448	−	0.801162i	$-0.295784\pi$
$487$	2.28927e36	1.19690	0.598448	+	0.801162i	$0.295784\pi$
$488$	0	0
$489$	−1.69193e36	−0.836977
$490$	0	0
$491$	−1.74114e36	−0.815142	−0.407571	−	0.913174i	$-0.633624\pi$
$491$	−1.74114e36	−0.815142	−0.407571	+	0.913174i	$0.633624\pi$
$492$	0	0
$493$	1.64079e34	0.00727145
$494$	0	0
$495$	−5.02565e35	−0.210874
$496$	0	0
$497$	−6.65935e35	−0.264619
$498$	0	0
$499$	−2.80174e36	−1.05456	−0.527278	−	0.849693i	$-0.676787\pi$
$499$	−2.80174e36	−1.05456	−0.527278	+	0.849693i	$0.676787\pi$
$500$	0	0
$501$	−2.19356e36	−0.782240
$502$	0	0
$503$	−3.44032e36	−1.16260	−0.581301	−	0.813688i	$-0.697456\pi$
$503$	−3.44032e36	−1.16260	−0.581301	+	0.813688i	$0.697456\pi$
$504$	0	0
$505$	−1.05249e36	−0.337119
$506$	0	0
$507$	−4.00825e36	−1.21716
$508$	0	0
$509$	1.50719e35	0.0433989	0.0216995	−	0.999765i	$-0.493092\pi$
$509$	1.50719e35	0.0433989	0.0216995	+	0.999765i	$0.493092\pi$
$510$	0	0
$511$	2.60211e36	0.710631
$512$	0	0
$513$	1.89418e36	0.490725
$514$	0	0
$515$	−1.65501e36	−0.406820
$516$	0	0
$517$	−1.94654e36	−0.454089
$518$	0	0
$519$	9.33082e36	2.06614
$520$	0	0
$521$	8.14778e36	1.71289	0.856446	−	0.516236i	$-0.172667\pi$
$521$	8.14778e36	1.71289	0.856446	+	0.516236i	$0.172667\pi$
$522$	0	0
$523$	4.14380e36	0.827229	0.413615	−	0.910452i	$-0.364266\pi$
$523$	4.14380e36	0.827229	0.413615	+	0.910452i	$0.364266\pi$
$524$	0	0
$525$	−4.82252e36	−0.914372
$526$	0	0
$527$	4.97526e35	0.0896133
$528$	0	0
$529$	2.01590e36	0.344999
$530$	0	0
$531$	−3.86363e36	−0.628376
$532$	0	0
$533$	−9.11854e35	−0.140964
$534$	0	0
$535$	7.57203e35	0.111285
$536$	0	0
$537$	5.67999e36	0.793770
$538$	0	0
$539$	5.57161e36	0.740512
$540$	0	0
$541$	1.06017e37	1.34033	0.670163	−	0.742214i	$-0.266224\pi$
$541$	1.06017e37	1.34033	0.670163	+	0.742214i	$0.266224\pi$
$542$	0	0
$543$	1.48064e37	1.78095
$544$	0	0
$545$	−1.82201e36	−0.208544
$546$	0	0
$547$	−8.42305e36	−0.917573	−0.458786	−	0.888547i	$-0.651716\pi$
$547$	−8.42305e36	−0.917573	−0.458786	+	0.888547i	$0.651716\pi$
$548$	0	0
$549$	6.21388e36	0.644371
$550$	0	0
$551$	4.84160e35	0.0478016
$552$	0	0
$553$	−9.17883e36	−0.862975
$554$	0	0
$555$	2.97616e36	0.266502
$556$	0	0
$557$	−5.59363e36	−0.477143	−0.238571	−	0.971125i	$-0.576679\pi$
$557$	−5.59363e36	−0.477143	−0.238571	+	0.971125i	$0.576679\pi$
$558$	0	0
$559$	−1.13499e36	−0.0922427
$560$	0	0
$561$	3.89269e36	0.301475
$562$	0	0
$563$	−2.06136e37	−1.52157	−0.760783	−	0.649006i	$-0.775185\pi$
$563$	−2.06136e37	−1.52157	−0.760783	+	0.649006i	$0.775185\pi$
$564$	0	0
$565$	2.12193e36	0.149306
$566$	0	0
$567$	−1.42780e37	−0.957850
$568$	0	0
$569$	4.96854e36	0.317844	0.158922	−	0.987291i	$-0.449198\pi$
$569$	4.96854e36	0.317844	0.158922	+	0.987291i	$0.449198\pi$
$570$	0	0
$571$	−1.02940e36	−0.0628055	−0.0314028	−	0.999507i	$-0.509997\pi$
$571$	−1.02940e36	−0.0628055	−0.0314028	+	0.999507i	$0.509997\pi$
$572$	0	0
$573$	−1.84400e37	−1.07318
$574$	0	0
$575$	1.99551e37	1.10800
$576$	0	0
$577$	−4.63258e36	−0.245442	−0.122721	−	0.992441i	$-0.539162\pi$
$577$	−4.63258e36	−0.245442	−0.122721	+	0.992441i	$0.539162\pi$
$578$	0	0
$579$	8.50387e36	0.429987
$580$	0	0
$581$	−1.40703e37	−0.679090
$582$	0	0
$583$	1.10786e37	0.510455
$584$	0	0
$585$	−4.05177e35	−0.0178254
$586$	0	0
$587$	−2.64137e36	−0.110972	−0.0554862	−	0.998459i	$-0.517671\pi$
$587$	−2.64137e36	−0.110972	−0.0554862	+	0.998459i	$0.517671\pi$
$588$	0	0
$589$	1.46808e37	0.589107
$590$	0	0
$591$	−1.51880e37	−0.582198
$592$	0	0
$593$	−3.52090e37	−1.28948	−0.644741	−	0.764401i	$-0.723035\pi$
$593$	−3.52090e37	−1.28948	−0.644741	+	0.764401i	$0.723035\pi$
$594$	0	0
$595$	−6.21145e35	−0.0217377
$596$	0	0
$597$	5.43839e37	1.81893
$598$	0	0
$599$	−4.76915e37	−1.52468	−0.762341	−	0.647176i	$-0.775950\pi$
$599$	−4.76915e37	−1.52468	−0.762341	+	0.647176i	$0.775950\pi$
$600$	0	0
$601$	−2.39124e37	−0.730831	−0.365416	−	0.930844i	$-0.619073\pi$
$601$	−2.39124e37	−0.730831	−0.365416	+	0.930844i	$0.619073\pi$
$602$	0	0
$603$	1.35743e37	0.396675
$604$	0	0
$605$	1.70782e37	0.477247
$606$	0	0
$607$	−3.35424e37	−0.896493	−0.448246	−	0.893910i	$-0.647951\pi$
$607$	−3.35424e37	−0.896493	−0.448246	+	0.893910i	$0.647951\pi$
$608$	0	0
$609$	−2.03142e36	−0.0519357
$610$	0	0
$611$	−1.56933e36	−0.0383847
$612$	0	0
$613$	−6.25036e37	−1.46281	−0.731407	−	0.681942i	$-0.761136\pi$
$613$	−6.25036e37	−1.46281	−0.731407	+	0.681942i	$0.761136\pi$
$614$	0	0
$615$	1.08559e37	0.243138
$616$	0	0
$617$	−6.57455e37	−1.40934	−0.704671	−	0.709534i	$-0.748906\pi$
$617$	−6.57455e37	−1.40934	−0.704671	+	0.709534i	$0.748906\pi$
$618$	0	0
$619$	−2.39694e37	−0.491851	−0.245926	−	0.969289i	$-0.579092\pi$
$619$	−2.39694e37	−0.491851	−0.245926	+	0.969289i	$0.579092\pi$
$620$	0	0
$621$	3.28862e37	0.646066
$622$	0	0
$623$	−2.77016e37	−0.521094
$624$	0	0
$625$	4.81915e37	0.868140
$626$	0	0
$627$	1.14864e38	1.98186
$628$	0	0
$629$	−8.20831e36	−0.135666
$630$	0	0
$631$	2.72032e37	0.430748	0.215374	−	0.976532i	$-0.430903\pi$
$631$	2.72032e37	0.430748	0.215374	+	0.976532i	$0.430903\pi$
$632$	0	0
$633$	5.16486e37	0.783625
$634$	0	0
$635$	1.55226e37	0.225693
$636$	0	0
$637$	4.49192e36	0.0625965
$638$	0	0
$639$	−1.42634e37	−0.190529
$640$	0	0
$641$	6.03099e37	0.772334	0.386167	−	0.922429i	$-0.373799\pi$
$641$	6.03099e37	0.772334	0.386167	+	0.922429i	$0.373799\pi$
$642$	0	0
$643$	1.01937e38	1.25165	0.625823	−	0.779965i	$-0.284763\pi$
$643$	1.01937e38	1.25165	0.625823	+	0.779965i	$0.284763\pi$
$644$	0	0
$645$	1.35124e37	0.159102
$646$	0	0
$647$	−1.38190e36	−0.0156052	−0.00780260	−	0.999970i	$-0.502484\pi$
$647$	−1.38190e36	−0.0156052	−0.00780260	+	0.999970i	$0.502484\pi$
$648$	0	0
$649$	1.89404e38	2.05156
$650$	0	0
$651$	−6.15972e37	−0.640055
$652$	0	0
$653$	−1.67383e38	−1.66872	−0.834358	−	0.551224i	$-0.814161\pi$
$653$	−1.67383e38	−1.66872	−0.834358	+	0.551224i	$0.814161\pi$
$654$	0	0
$655$	−1.83236e37	−0.175289
$656$	0	0
$657$	5.57335e37	0.511663
$658$	0	0
$659$	1.83021e38	1.61268	0.806339	−	0.591454i	$-0.201446\pi$
$659$	1.83021e38	1.61268	0.806339	+	0.591454i	$0.201446\pi$
$660$	0	0
$661$	6.09366e37	0.515417	0.257708	−	0.966223i	$-0.417033\pi$
$661$	6.09366e37	0.515417	0.257708	+	0.966223i	$0.417033\pi$
$662$	0	0
$663$	3.13835e36	0.0254841
$664$	0	0
$665$	−1.83285e37	−0.142901
$666$	0	0
$667$	8.40581e36	0.0629334
$668$	0	0
$669$	3.59729e37	0.258656
$670$	0	0
$671$	−3.04618e38	−2.10379
$672$	0	0
$673$	−6.36340e37	−0.422168	−0.211084	−	0.977468i	$-0.567699\pi$
$673$	−6.36340e37	−0.422168	−0.211084	+	0.977468i	$0.567699\pi$
$674$	0	0
$675$	8.35015e37	0.532223
$676$	0	0
$677$	−5.37104e37	−0.328936	−0.164468	−	0.986382i	$-0.552591\pi$
$677$	−5.37104e37	−0.328936	−0.164468	+	0.986382i	$0.552591\pi$
$678$	0	0
$679$	7.43332e37	0.437462
$680$	0	0
$681$	−3.25210e38	−1.83941
$682$	0	0
$683$	−1.59190e38	−0.865437	−0.432718	−	0.901529i	$-0.642446\pi$
$683$	−1.59190e38	−0.865437	−0.432718	+	0.901529i	$0.642446\pi$
$684$	0	0
$685$	1.22919e37	0.0642387
$686$	0	0
$687$	−2.15661e38	−1.08357
$688$	0	0
$689$	8.93171e36	0.0431495
$690$	0	0
$691$	1.11503e38	0.518006	0.259003	−	0.965877i	$-0.416606\pi$
$691$	1.11503e38	0.518006	0.259003	+	0.965877i	$0.416606\pi$
$692$	0	0
$693$	−1.71607e38	−0.766719
$694$	0	0
$695$	6.68631e37	0.287337
$696$	0	0
$697$	−2.99409e37	−0.123772
$698$	0	0
$699$	2.70943e38	1.07755
$700$	0	0
$701$	9.04824e37	0.346235	0.173117	−	0.984901i	$-0.444616\pi$
$701$	9.04824e37	0.346235	0.173117	+	0.984901i	$0.444616\pi$
$702$	0	0
$703$	−2.42208e38	−0.891849
$704$	0	0
$705$	1.86834e37	0.0662069
$706$	0	0
$707$	−3.59385e38	−1.22574
$708$	0	0
$709$	2.52263e38	0.828190	0.414095	−	0.910234i	$-0.364098\pi$
$709$	2.52263e38	0.828190	0.414095	+	0.910234i	$0.364098\pi$
$710$	0	0
$711$	−1.96598e38	−0.621353
$712$	0	0
$713$	2.54883e38	0.775591
$714$	0	0
$715$	1.98626e37	0.0581976
$716$	0	0
$717$	1.94469e37	0.0548709
$718$	0	0
$719$	2.15368e38	0.585249	0.292624	−	0.956227i	$-0.405471\pi$
$719$	2.15368e38	0.585249	0.292624	+	0.956227i	$0.405471\pi$
$720$	0	0
$721$	−5.65122e38	−1.47916
$722$	0	0
$723$	−1.52427e38	−0.384323
$724$	0	0
$725$	2.13433e37	0.0518439
$726$	0	0
$727$	6.18554e38	1.44765	0.723824	−	0.689985i	$-0.242383\pi$
$727$	6.18554e38	1.44765	0.723824	+	0.689985i	$0.242383\pi$
$728$	0	0
$729$	2.02058e36	0.00455675
$730$	0	0
$731$	−3.72675e37	−0.0809927
$732$	0	0
$733$	−5.13013e36	−0.0107455	−0.00537273	−	0.999986i	$-0.501710\pi$
$733$	−5.13013e36	−0.0107455	−0.00537273	+	0.999986i	$0.501710\pi$
$734$	0	0
$735$	−5.34779e37	−0.107968
$736$	0	0
$737$	−6.65443e38	−1.29509
$738$	0	0
$739$	5.22466e38	0.980298	0.490149	−	0.871639i	$-0.336942\pi$
$739$	5.22466e38	0.980298	0.490149	+	0.871639i	$0.336942\pi$
$740$	0	0
$741$	9.26054e37	0.167529
$742$	0	0
$743$	−1.00984e39	−1.76158	−0.880789	−	0.473509i	$-0.842987\pi$
$743$	−1.00984e39	−1.76158	−0.880789	+	0.473509i	$0.842987\pi$
$744$	0	0
$745$	1.55948e38	0.262344
$746$	0	0
$747$	−3.01367e38	−0.488953
$748$	0	0
$749$	2.58556e38	0.404622
$750$	0	0
$751$	4.44629e38	0.671210	0.335605	−	0.942003i	$-0.391059\pi$
$751$	4.44629e38	0.671210	0.335605	+	0.942003i	$0.391059\pi$
$752$	0	0
$753$	1.12914e39	1.64443
$754$	0	0
$755$	−3.71114e37	−0.0521463
$756$	0	0
$757$	8.24934e38	1.11847	0.559235	−	0.829009i	$-0.311095\pi$
$757$	8.24934e38	1.11847	0.559235	+	0.829009i	$0.311095\pi$
$758$	0	0
$759$	1.99423e39	2.60923
$760$	0	0
$761$	−5.69669e38	−0.719333	−0.359666	−	0.933081i	$-0.617109\pi$
$761$	−5.69669e38	−0.719333	−0.359666	+	0.933081i	$0.617109\pi$
$762$	0	0
$763$	−6.22146e38	−0.758248
$764$	0	0
$765$	−1.33040e37	−0.0156514
$766$	0	0
$767$	1.52700e38	0.173421
$768$	0	0
$769$	1.47938e39	1.62209	0.811043	−	0.584986i	$-0.198900\pi$
$769$	1.47938e39	1.62209	0.811043	+	0.584986i	$0.198900\pi$
$770$	0	0
$771$	1.19855e38	0.126889
$772$	0	0
$773$	−1.83758e39	−1.87855	−0.939277	−	0.343160i	$-0.888503\pi$
$773$	−1.83758e39	−1.87855	−0.939277	+	0.343160i	$0.888503\pi$
$774$	0	0
$775$	6.47176e38	0.638924
$776$	0	0
$777$	1.01625e39	0.968981
$778$	0	0
$779$	−8.83484e38	−0.813660
$780$	0	0
$781$	6.99222e38	0.622051
$782$	0	0
$783$	3.51738e37	0.0302298
$784$	0	0
$785$	−4.24623e38	−0.352584
$786$	0	0
$787$	3.93218e38	0.315482	0.157741	−	0.987481i	$-0.449579\pi$
$787$	3.93218e38	0.315482	0.157741	+	0.987481i	$0.449579\pi$
$788$	0	0
$789$	−1.31695e38	−0.102101
$790$	0	0
$791$	7.24559e38	0.542864
$792$	0	0
$793$	−2.45588e38	−0.177836
$794$	0	0
$795$	−1.06335e38	−0.0744252
$796$	0	0
$797$	−1.32458e39	−0.896168	−0.448084	−	0.893991i	$-0.647893\pi$
$797$	−1.32458e39	−0.896168	−0.448084	+	0.893991i	$0.647893\pi$
$798$	0	0
$799$	−5.15293e37	−0.0337033
$800$	0	0
$801$	−5.93329e38	−0.375194
$802$	0	0
$803$	−2.73218e39	−1.67051
$804$	0	0
$805$	−3.18213e38	−0.188137
$806$	0	0
$807$	−1.53125e38	−0.0875492
$808$	0	0
$809$	−2.19258e39	−1.21241	−0.606206	−	0.795308i	$-0.707309\pi$
$809$	−2.19258e39	−1.21241	−0.606206	+	0.795308i	$0.707309\pi$
$810$	0	0
$811$	−2.76373e39	−1.47813	−0.739066	−	0.673633i	$-0.764733\pi$
$811$	−2.76373e39	−1.47813	−0.739066	+	0.673633i	$0.764733\pi$
$812$	0	0
$813$	−1.24873e39	−0.646022
$814$	0	0
$815$	−2.83471e38	−0.141867
$816$	0	0
$817$	−1.09968e39	−0.532436
$818$	0	0
$819$	−1.38352e38	−0.0648118
$820$	0	0
$821$	−9.42212e38	−0.427087	−0.213543	−	0.976934i	$-0.568500\pi$
$821$	−9.42212e38	−0.427087	−0.213543	+	0.976934i	$0.568500\pi$
$822$	0	0
$823$	−2.17900e39	−0.955782	−0.477891	−	0.878419i	$-0.658599\pi$
$823$	−2.17900e39	−0.955782	−0.477891	+	0.878419i	$0.658599\pi$
$824$	0	0
$825$	5.06357e39	2.14946
$826$	0	0
$827$	8.15484e38	0.335036	0.167518	−	0.985869i	$-0.446425\pi$
$827$	8.15484e38	0.335036	0.167518	+	0.985869i	$0.446425\pi$
$828$	0	0
$829$	1.51577e39	0.602763	0.301382	−	0.953504i	$-0.402552\pi$
$829$	1.51577e39	0.602763	0.301382	+	0.953504i	$0.402552\pi$
$830$	0	0
$831$	4.84386e39	1.86457
$832$	0	0
$833$	1.47493e38	0.0549622
$834$	0	0
$835$	−3.67515e38	−0.132589
$836$	0	0
$837$	1.06655e39	0.372553
$838$	0	0
$839$	−3.04177e39	−1.02882	−0.514410	−	0.857544i	$-0.671989\pi$
$839$	−3.04177e39	−1.02882	−0.514410	+	0.857544i	$0.671989\pi$
$840$	0	0
$841$	−3.04414e39	−0.997055
$842$	0	0
$843$	1.23726e39	0.392453
$844$	0	0
$845$	−6.71554e38	−0.206308
$846$	0	0
$847$	5.83154e39	1.73523
$848$	0	0
$849$	1.23603e39	0.356266
$850$	0	0
$851$	−4.20513e39	−1.17417
$852$	0	0
$853$	−9.78044e38	−0.264574	−0.132287	−	0.991211i	$-0.542232\pi$
$853$	−9.78044e38	−0.264574	−0.132287	+	0.991211i	$0.542232\pi$
$854$	0	0
$855$	−3.92571e38	−0.102891
$856$	0	0
$857$	−6.93851e39	−1.76208	−0.881039	−	0.473044i	$-0.843155\pi$
$857$	−6.93851e39	−1.76208	−0.881039	+	0.473044i	$0.843155\pi$
$858$	0	0
$859$	6.79141e39	1.67129	0.835646	−	0.549268i	$-0.185093\pi$
$859$	6.79141e39	1.67129	0.835646	+	0.549268i	$0.185093\pi$
$860$	0	0
$861$	3.70689e39	0.884030
$862$	0	0
$863$	5.27048e38	0.121816	0.0609080	−	0.998143i	$-0.480600\pi$
$863$	5.27048e38	0.121816	0.0609080	+	0.998143i	$0.480600\pi$
$864$	0	0
$865$	1.56331e39	0.350210
$866$	0	0
$867$	−5.63600e39	−1.22381
$868$	0	0
$869$	9.63764e39	2.02863
$870$	0	0
$871$	−5.36491e38	−0.109476
$872$	0	0
$873$	1.59211e39	0.314978
$874$	0	0
$875$	−1.65369e39	−0.317209
$876$	0	0
$877$	3.42049e39	0.636200	0.318100	−	0.948057i	$-0.396955\pi$
$877$	3.42049e39	0.636200	0.318100	+	0.948057i	$0.396955\pi$
$878$	0	0
$879$	1.20896e39	0.218053
$880$	0	0
$881$	−7.31752e39	−1.27994	−0.639969	−	0.768401i	$-0.721053\pi$
$881$	−7.31752e39	−1.27994	−0.639969	+	0.768401i	$0.721053\pi$
$882$	0	0
$883$	2.06213e39	0.349821	0.174910	−	0.984584i	$-0.444036\pi$
$883$	2.06213e39	0.349821	0.174910	+	0.984584i	$0.444036\pi$
$884$	0	0
$885$	−1.81795e39	−0.299121
$886$	0	0
$887$	−6.98969e39	−1.11555	−0.557775	−	0.829992i	$-0.688345\pi$
$887$	−6.98969e39	−1.11555	−0.557775	+	0.829992i	$0.688345\pi$
$888$	0	0
$889$	5.30037e39	0.820602
$890$	0	0
$891$	1.49917e40	2.25166
$892$	0	0
$893$	−1.52051e39	−0.221561
$894$	0	0
$895$	9.51642e38	0.134543
$896$	0	0
$897$	1.60778e39	0.220561
$898$	0	0
$899$	2.72614e38	0.0362904
$900$	0	0
$901$	2.93274e38	0.0378869
$902$	0	0
$903$	4.61398e39	0.578483
$904$	0	0
$905$	2.48071e39	0.301870
$906$	0	0
$907$	2.28488e39	0.269876	0.134938	−	0.990854i	$-0.456916\pi$
$907$	2.28488e39	0.269876	0.134938	+	0.990854i	$0.456916\pi$
$908$	0	0
$909$	−7.69751e39	−0.882547
$910$	0	0
$911$	−7.51662e39	−0.836612	−0.418306	−	0.908306i	$-0.637376\pi$
$911$	−7.51662e39	−0.836612	−0.418306	+	0.908306i	$0.637376\pi$
$912$	0	0
$913$	1.47737e40	1.59637
$914$	0	0
$915$	2.92381e39	0.306736
$916$	0	0
$917$	−6.25683e39	−0.637336
$918$	0	0
$919$	−5.26405e39	−0.520668	−0.260334	−	0.965519i	$-0.583833\pi$
$919$	−5.26405e39	−0.520668	−0.260334	+	0.965519i	$0.583833\pi$
$920$	0	0
$921$	2.32511e39	0.223326
$922$	0	0
$923$	5.63724e38	0.0525828
$924$	0	0
$925$	−1.06773e40	−0.967269
$926$	0	0
$927$	−1.21041e40	−1.06502
$928$	0	0
$929$	−2.48177e39	−0.212104	−0.106052	−	0.994361i	$-0.533821\pi$
$929$	−2.48177e39	−0.212104	−0.106052	+	0.994361i	$0.533821\pi$
$930$	0	0
$931$	4.35217e39	0.361314
$932$	0	0
$933$	−1.77807e40	−1.43399
$934$	0	0
$935$	6.52193e38	0.0510998
$936$	0	0
$937$	−2.96211e39	−0.225485	−0.112742	−	0.993624i	$-0.535963\pi$
$937$	−2.96211e39	−0.225485	−0.112742	+	0.993624i	$0.535963\pi$
$938$	0	0
$939$	−3.80877e39	−0.281708
$940$	0	0
$941$	1.23582e40	0.888168	0.444084	−	0.895985i	$-0.353529\pi$
$941$	1.23582e40	0.888168	0.444084	+	0.895985i	$0.353529\pi$
$942$	0	0
$943$	−1.53388e40	−1.07123
$944$	0	0
$945$	−1.33155e39	−0.0903710
$946$	0	0
$947$	−3.62546e39	−0.239132	−0.119566	−	0.992826i	$-0.538150\pi$
$947$	−3.62546e39	−0.239132	−0.119566	+	0.992826i	$0.538150\pi$
$948$	0	0
$949$	−2.20273e39	−0.141210
$950$	0	0
$951$	−3.26918e40	−2.03705
$952$	0	0
$953$	−2.05559e40	−1.24504	−0.622519	−	0.782605i	$-0.713891\pi$
$953$	−2.05559e40	−1.24504	−0.622519	+	0.782605i	$0.713891\pi$
$954$	0	0
$955$	−3.08950e39	−0.181904
$956$	0	0
$957$	2.13296e39	0.122087
$958$	0	0
$959$	4.19722e39	0.233567
$960$	0	0
$961$	−1.02164e40	−0.552757
$962$	0	0
$963$	5.53790e39	0.291333
$964$	0	0
$965$	1.42476e39	0.0728825
$966$	0	0
$967$	2.81961e40	1.40259	0.701296	−	0.712871i	$-0.252605\pi$
$967$	2.81961e40	1.40259	0.701296	+	0.712871i	$0.252605\pi$
$968$	0	0
$969$	3.04071e39	0.147097
$970$	0	0
$971$	1.81766e40	0.855170	0.427585	−	0.903975i	$-0.359364\pi$
$971$	1.81766e40	0.855170	0.427585	+	0.903975i	$0.359364\pi$
$972$	0	0
$973$	2.28312e40	1.04473
$974$	0	0
$975$	4.08233e39	0.181696
$976$	0	0
$977$	2.03258e40	0.879974	0.439987	−	0.898004i	$-0.354983\pi$
$977$	2.03258e40	0.879974	0.439987	+	0.898004i	$0.354983\pi$
$978$	0	0
$979$	2.90863e40	1.22496
$980$	0	0
$981$	−1.33255e40	−0.545948
$982$	0	0
$983$	−7.72807e39	−0.308034	−0.154017	−	0.988068i	$-0.549221\pi$
$983$	−7.72807e39	−0.308034	−0.154017	+	0.988068i	$0.549221\pi$
$984$	0	0
$985$	−2.54465e39	−0.0986820
$986$	0	0
$987$	6.37968e39	0.240723
$988$	0	0
$989$	−1.90922e40	−0.700981
$990$	0	0
$991$	−1.25885e40	−0.449758	−0.224879	−	0.974387i	$-0.572199\pi$
$991$	−1.25885e40	−0.449758	−0.224879	+	0.974387i	$0.572199\pi$
$992$	0	0
$993$	4.31368e40	1.49980
$994$	0	0
$995$	9.11164e39	0.308308
$996$	0	0
$997$	4.50833e40	1.48467	0.742336	−	0.670028i	$-0.233718\pi$
$997$	4.50833e40	1.48467	0.742336	+	0.670028i	$0.233718\pi$
$998$	0	0
$999$	−1.75962e40	−0.564008

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.28.a.d.1.2		2
4.3	odd	2		1.28.a.a.1.1	✓	2
12.11	even	2		9.28.a.d.1.2		2
20.3	even	4		25.28.b.a.24.4		4
20.7	even	4		25.28.b.a.24.1		4
20.19	odd	2		25.28.a.a.1.2		2
28.27	even	2		49.28.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.28.a.a.1.1	✓	2	4.3	odd	2
9.28.a.d.1.2		2	12.11	even	2
16.28.a.d.1.2		2	1.1	even	1	trivial
25.28.a.a.1.2		2	20.19	odd	2
25.28.b.a.24.1		4	20.7	even	4
25.28.b.a.24.4		4	20.3	even	4
49.28.a.b.1.1		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)

\(f(q)\)	\(=\)	\(q+3.44127e6 q^{3} +5.76560e8 q^{5} +1.96873e11 q^{7} +4.21675e12 q^{9} +O(q^{10})\)	\(q+3.44127e6 q^{3} +5.76560e8 q^{5} +1.96873e11 q^{7} +4.21675e12 q^{9} -2.06714e14 q^{11} -1.66656e14 q^{13} +1.98410e15 q^{15} -5.47219e15 q^{17} -1.61471e17 q^{19} +6.77495e17 q^{21} -2.80341e18 q^{23} -7.11816e18 q^{25} -1.17308e19 q^{27} -2.99842e18 q^{29} -9.09190e19 q^{31} -7.11360e20 q^{33} +1.13509e20 q^{35} +1.50001e21 q^{37} -5.73510e20 q^{39} +5.47146e21 q^{41} +6.81035e21 q^{43} +2.43121e21 q^{45} +9.41657e21 q^{47} -2.69532e22 q^{49} -1.88313e22 q^{51} -5.35936e22 q^{53} -1.19183e23 q^{55} -5.55667e23 q^{57} -9.16258e23 q^{59} +1.47362e24 q^{61} +8.30166e23 q^{63} -9.60874e22 q^{65} +3.21915e24 q^{67} -9.64729e24 q^{69} -3.38255e24 q^{71} +1.32172e25 q^{73} -2.44955e25 q^{75} -4.06965e25 q^{77} -4.66230e25 q^{79} -7.25240e25 q^{81} -7.14690e25 q^{83} -3.15505e24 q^{85} -1.03184e25 q^{87} -1.40708e26 q^{89} -3.28102e25 q^{91} -3.12877e26 q^{93} -9.30980e25 q^{95} +3.77568e26 q^{97} -8.71662e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1286280 q^{3} + 5443587900 q^{5} + 175391963600 q^{7} + 1235136554154 q^{9}+O(q^{10}) \) 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

Varieties

Fields

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Groups

Database

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