LMFDB - Embedded newform 16.28.a.f.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8494973x^{2} - 3687596342x + 10439241475305 $ x^4 - 8494973x^2 - 3687596342x + 10439241475305
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{42}\cdot 3^{5}\cdot 5 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2061.48$ 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)

$f(q)$	$=$	$q+4.98397e6 q^{3} +3.10348e9 q^{5} +2.36469e11 q^{7} +1.72143e13 q^{9} +2.12352e14 q^{11} -1.46908e15 q^{13} +1.54676e16 q^{15} +2.34431e15 q^{17} +9.03530e16 q^{19} +1.17855e18 q^{21} -1.52631e18 q^{23} +2.18098e18 q^{25} +4.77899e19 q^{27} +1.80158e19 q^{29} -3.08284e19 q^{31} +1.05835e21 q^{33} +7.33876e20 q^{35} -6.43627e20 q^{37} -7.32184e21 q^{39} -7.21227e21 q^{41} -1.26012e22 q^{43} +5.34243e22 q^{45} -6.23837e22 q^{47} -9.79482e21 q^{49} +1.16840e22 q^{51} +1.22507e23 q^{53} +6.59029e23 q^{55} +4.50316e23 q^{57} -1.26232e24 q^{59} -4.90924e23 q^{61} +4.07065e24 q^{63} -4.55925e24 q^{65} +2.60342e24 q^{67} -7.60710e24 q^{69} -7.66670e24 q^{71} +2.19552e25 q^{73} +1.08699e25 q^{75} +5.02146e25 q^{77} +7.76957e25 q^{79} +1.06914e26 q^{81} -3.00643e25 q^{83} +7.27551e24 q^{85} +8.97901e25 q^{87} +2.29131e26 q^{89} -3.47391e26 q^{91} -1.53648e26 q^{93} +2.80408e26 q^{95} -1.71940e26 q^{97} +3.65549e27 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 122512 q^{3} + 3544066168 q^{5} + 211767036576 q^{7} + 19748930504020 q^{9} + 137002338905648 q^{11} - 5580886697000 q^{13} + 25\!\cdots\!12 q^{15} + 44\!\cdots\!44 q^{17} + 17\!\cdots\!72 q^{19}+ \cdots + 53\!\cdots\!16 q^{99}+O(q^{100}) $ 4 * q + 122512 * q^3 + 3544066168 * q^5 + 211767036576 * q^7 + 19748930504020 * q^9 + 137002338905648 * q^11 - 5580886697000 * q^13 + 2557147082175712 * q^15 + 44214930716819144 * q^17 + 177042777372267472 * q^19 + 29921143201836672 * q^21 - 1767092590673925664 * q^23 + 598259983019495324 * q^25 - 5639315117824052576 * q^27 + 6230479481762520216 * q^29 - 211925592023301030272 * q^31 + 840492384365367174848 * q^33 - 295751663414571139392 * q^35 + 3315679506055104172536 * q^37 - 16187282823779225026976 * q^39 + 16742683588694565402600 * q^41 - 28929100862580808746832 * q^43 + 109794415746949511800792 * q^45 - 127204639057684425317952 * q^47 + 154785230493187619859620 * q^49 - 318357332417527091023840 * q^51 + 279552147064338019634872 * q^53 + 518548397855800513776032 * q^55 - 438125799682933882052288 * q^57 + 747757532145939859769840 * q^59 - 2481631547272388497527272 * q^61 + 10528130607790692595312416 * q^63 - 4197418205948650615753904 * q^65 + 14053964155982437041984400 * q^67 - 4086733995329892788833408 * q^69 + 14160923375603434909530016 * q^71 + 25756757389668481118224744 * q^73 + 21854795385788245746680816 * q^75 + 129360200718523083267780480 * q^77 + 32479711484972151883608128 * q^79 + 342342850435863984906534436 * q^81 - 79535495988836946955941680 * q^83 + 439674534481061013744364400 * q^85 + 406870958670348922758232416 * q^87 + 543797853914541667326783528 * q^89 + 743433170089429876643395008 * q^91 + 700751851717865814480816640 * q^93 + 1917664584916076964174787168 * q^95 - 408937442977415638801906552 * q^97 + 5389670612142233536540165616 * q^99

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
$3$	4.98397e6	1.80484	0.902419	−	0.430860i	$-0.141790\pi$
$3$	4.98397e6	1.80484	0.902419	+	0.430860i	$0.141790\pi$
$4$	0	0
$5$	3.10348e9	1.13698	0.568491	−	0.822690i	$-0.307528\pi$
$5$	3.10348e9	1.13698	0.568491	+	0.822690i	$0.307528\pi$
$6$	0	0
$7$	2.36469e11	0.922466	0.461233	−	0.887279i	$-0.347407\pi$
$7$	2.36469e11	0.922466	0.461233	+	0.887279i	$0.347407\pi$
$8$	0	0
$9$	1.72143e13	2.25744
$10$	0	0
$11$	2.12352e14	1.85462	0.927310	−	0.374295i	$-0.122115\pi$
$11$	2.12352e14	1.85462	0.927310	+	0.374295i	$0.122115\pi$
$12$	0	0
$13$	−1.46908e15	−1.34527	−0.672635	−	0.739974i	$-0.734838\pi$
$13$	−1.46908e15	−1.34527	−0.672635	+	0.739974i	$0.734838\pi$
$14$	0	0
$15$	1.54676e16	2.05207
$16$	0	0
$17$	2.34431e15	0.0574056	0.0287028	−	0.999588i	$-0.490862\pi$
$17$	2.34431e15	0.0574056	0.0287028	+	0.999588i	$0.490862\pi$
$18$	0	0
$19$	9.03530e16	0.492911	0.246456	−	0.969154i	$-0.420734\pi$
$19$	9.03530e16	0.492911	0.246456	+	0.969154i	$0.420734\pi$
$20$	0	0
$21$	1.17855e18	1.66490
$22$	0	0
$23$	−1.52631e18	−0.631419	−0.315710	−	0.948856i	$-0.602243\pi$
$23$	−1.52631e18	−0.631419	−0.315710	+	0.948856i	$0.602243\pi$
$24$	0	0
$25$	2.18098e18	0.292726
$26$	0	0
$27$	4.77899e19	2.26948
$28$	0	0
$29$	1.80158e19	0.326047	0.163023	−	0.986622i	$-0.447875\pi$
$29$	1.80158e19	0.326047	0.163023	+	0.986622i	$0.447875\pi$
$30$	0	0
$31$	−3.08284e19	−0.226761	−0.113381	−	0.993552i	$-0.536168\pi$
$31$	−3.08284e19	−0.226761	−0.113381	+	0.993552i	$0.536168\pi$
$32$	0	0
$33$	1.05835e21	3.34729
$34$	0	0
$35$	7.33876e20	1.04883
$36$	0	0
$37$	−6.43627e20	−0.434421	−0.217210	−	0.976125i	$-0.569696\pi$
$37$	−6.43627e20	−0.434421	−0.217210	+	0.976125i	$0.569696\pi$
$38$	0	0
$39$	−7.32184e21	−2.42800
$40$	0	0
$41$	−7.21227e21	−1.21756	−0.608780	−	0.793339i	$-0.708341\pi$
$41$	−7.21227e21	−1.21756	−0.608780	+	0.793339i	$0.708341\pi$
$42$	0	0
$43$	−1.26012e22	−1.11838	−0.559188	−	0.829041i	$-0.688887\pi$
$43$	−1.26012e22	−1.11838	−0.559188	+	0.829041i	$0.688887\pi$
$44$	0	0
$45$	5.34243e22	2.56667
$46$	0	0
$47$	−6.23837e22	−1.66629	−0.833144	−	0.553055i	$-0.813462\pi$
$47$	−6.23837e22	−1.66629	−0.833144	+	0.553055i	$0.813462\pi$
$48$	0	0
$49$	−9.79482e21	−0.149056
$50$	0	0
$51$	1.16840e22	0.103608
$52$	0	0
$53$	1.22507e23	0.646302	0.323151	−	0.946347i	$-0.395258\pi$
$53$	1.22507e23	0.646302	0.323151	+	0.946347i	$0.395258\pi$
$54$	0	0
$55$	6.59029e23	2.10867
$56$	0	0
$57$	4.50316e23	0.889625
$58$	0	0
$59$	−1.26232e24	−1.56555	−0.782773	−	0.622307i	$-0.786195\pi$
$59$	−1.26232e24	−1.56555	−0.782773	+	0.622307i	$0.786195\pi$
$60$	0	0
$61$	−4.90924e23	−0.388206	−0.194103	−	0.980981i	$-0.562180\pi$
$61$	−4.90924e23	−0.388206	−0.194103	+	0.980981i	$0.562180\pi$
$62$	0	0
$63$	4.07065e24	2.08241
$64$	0	0
$65$	−4.55925e24	−1.52955
$66$	0	0
$67$	2.60342e24	0.580141	0.290071	−	0.957005i	$-0.406321\pi$
$67$	2.60342e24	0.580141	0.290071	+	0.957005i	$0.406321\pi$
$68$	0	0
$69$	−7.60710e24	−1.13961
$70$	0	0
$71$	−7.66670e24	−0.780946	−0.390473	−	0.920614i	$-0.627689\pi$
$71$	−7.66670e24	−0.780946	−0.390473	+	0.920614i	$0.627689\pi$
$72$	0	0
$73$	2.19552e25	1.53702	0.768509	−	0.639839i	$-0.220999\pi$
$73$	2.19552e25	1.53702	0.768509	+	0.639839i	$0.220999\pi$
$74$	0	0
$75$	1.08699e25	0.528324
$76$	0	0
$77$	5.02146e25	1.71082
$78$	0	0
$79$	7.76957e25	1.87254	0.936270	−	0.351280i	$-0.114253\pi$
$79$	7.76957e25	1.87254	0.936270	+	0.351280i	$0.114253\pi$
$80$	0	0
$81$	1.06914e26	1.83860
$82$	0	0
$83$	−3.00643e25	−0.371961	−0.185980	−	0.982553i	$-0.559546\pi$
$83$	−3.00643e25	−0.371961	−0.185980	+	0.982553i	$0.559546\pi$
$84$	0	0
$85$	7.27551e24	0.0652691
$86$	0	0
$87$	8.97901e25	0.588462
$88$	0	0
$89$	2.29131e26	1.10489	0.552444	−	0.833550i	$-0.313695\pi$
$89$	2.29131e26	1.10489	0.552444	+	0.833550i	$0.313695\pi$
$90$	0	0
$91$	−3.47391e26	−1.24097
$92$	0	0
$93$	−1.53648e26	−0.409267
$94$	0	0
$95$	2.80408e26	0.560431
$96$	0	0
$97$	−1.71940e26	−0.259393	−0.129697	−	0.991554i	$-0.541400\pi$
$97$	−1.71940e26	−0.259393	−0.129697	+	0.991554i	$0.541400\pi$
$98$	0	0
$99$	3.65549e27	4.18669
$100$	0	0
$101$	4.98011e25	0.0435412	0.0217706	−	0.999763i	$-0.493070\pi$
$101$	4.98011e25	0.0435412	0.0217706	+	0.999763i	$0.493070\pi$
$102$	0	0
$103$	−3.83420e26	−0.257260	−0.128630	−	0.991693i	$-0.541058\pi$
$103$	−3.83420e26	−0.257260	−0.128630	+	0.991693i	$0.541058\pi$
$104$	0	0
$105$	3.65761e27	1.89296
$106$	0	0
$107$	7.51558e25	0.0301496	0.0150748	−	0.999886i	$-0.495201\pi$
$107$	7.51558e25	0.0301496	0.0150748	+	0.999886i	$0.495201\pi$
$108$	0	0
$109$	−3.10968e27	−0.971533	−0.485767	−	0.874089i	$-0.661460\pi$
$109$	−3.10968e27	−0.971533	−0.485767	+	0.874089i	$0.661460\pi$
$110$	0	0
$111$	−3.20781e27	−0.784059
$112$	0	0
$113$	7.25589e27	1.39358	0.696790	−	0.717276i	$-0.254611\pi$
$113$	7.25589e27	1.39358	0.696790	+	0.717276i	$0.254611\pi$
$114$	0	0
$115$	−4.73688e27	−0.717912
$116$	0	0
$117$	−2.52892e28	−3.03687
$118$	0	0
$119$	5.54356e26	0.0529548
$120$	0	0
$121$	3.19833e28	2.43961
$122$	0	0
$123$	−3.59457e28	−2.19750
$124$	0	0
$125$	−1.63541e28	−0.804157
$126$	0	0
$127$	−2.34263e28	−0.929722	−0.464861	−	0.885384i	$-0.653896\pi$
$127$	−2.34263e28	−0.929722	−0.464861	+	0.885384i	$0.653896\pi$
$128$	0	0
$129$	−6.28040e28	−2.01849
$130$	0	0
$131$	4.25742e27	0.111169	0.0555845	−	0.998454i	$-0.482298\pi$
$131$	4.25742e27	0.111169	0.0555845	+	0.998454i	$0.482298\pi$
$132$	0	0
$133$	2.13657e28	0.454694
$134$	0	0
$135$	1.48315e29	2.58035
$136$	0	0
$137$	5.93659e28	0.846856	0.423428	−	0.905930i	$-0.360827\pi$
$137$	5.93659e28	0.846856	0.423428	+	0.905930i	$0.360827\pi$
$138$	0	0
$139$	5.87133e28	0.688711	0.344355	−	0.938839i	$-0.388097\pi$
$139$	5.87133e28	0.688711	0.344355	+	0.938839i	$0.388097\pi$
$140$	0	0
$141$	−3.10918e29	−3.00738
$142$	0	0
$143$	−3.11961e29	−2.49496
$144$	0	0
$145$	5.59115e28	0.370709
$146$	0	0
$147$	−4.88170e28	−0.269022
$148$	0	0
$149$	2.14214e29	0.983635	0.491818	−	0.870698i	$-0.336333\pi$
$149$	2.14214e29	0.983635	0.491818	+	0.870698i	$0.336333\pi$
$150$	0	0
$151$	−3.76796e27	−0.0144517	−0.00722583	−	0.999974i	$-0.502300\pi$
$151$	−3.76796e27	−0.0144517	−0.00722583	+	0.999974i	$0.502300\pi$
$152$	0	0
$153$	4.03557e28	0.129590
$154$	0	0
$155$	−9.56753e28	−0.257823
$156$	0	0
$157$	3.45050e29	0.782053	0.391027	−	0.920379i	$-0.372120\pi$
$157$	3.45050e29	0.782053	0.391027	+	0.920379i	$0.372120\pi$
$158$	0	0
$159$	6.10570e29	1.16647
$160$	0	0
$161$	−3.60926e29	−0.582463
$162$	0	0
$163$	5.68411e29	0.776479	0.388240	−	0.921558i	$-0.373083\pi$
$163$	5.68411e29	0.776479	0.388240	+	0.921558i	$0.373083\pi$
$164$	0	0
$165$	3.28458e30	3.80580
$166$	0	0
$167$	−1.38467e29	−0.136356	−0.0681781	−	0.997673i	$-0.521719\pi$
$167$	−1.38467e29	−0.136356	−0.0681781	+	0.997673i	$0.521719\pi$
$168$	0	0
$169$	9.65658e29	0.809753
$170$	0	0
$171$	1.55537e30	1.11272
$172$	0	0
$173$	−2.35976e30	−1.44293	−0.721464	−	0.692452i	$-0.756530\pi$
$173$	−2.35976e30	−1.44293	−0.721464	+	0.692452i	$0.756530\pi$
$174$	0	0
$175$	5.15734e29	0.270030
$176$	0	0
$177$	−6.29134e30	−2.82556
$178$	0	0
$179$	1.46454e29	0.0565178	0.0282589	−	0.999601i	$-0.491004\pi$
$179$	1.46454e29	0.0565178	0.0282589	+	0.999601i	$0.491004\pi$
$180$	0	0
$181$	−2.42478e29	−0.0805400	−0.0402700	−	0.999189i	$-0.512822\pi$
$181$	−2.42478e29	−0.0805400	−0.0402700	+	0.999189i	$0.512822\pi$
$182$	0	0
$183$	−2.44675e30	−0.700648
$184$	0	0
$185$	−1.99748e30	−0.493928
$186$	0	0
$187$	4.97818e29	0.106466
$188$	0	0
$189$	1.13008e31	2.09351
$190$	0	0
$191$	3.20304e30	0.514768	0.257384	−	0.966309i	$-0.417139\pi$
$191$	3.20304e30	0.514768	0.257384	+	0.966309i	$0.417139\pi$
$192$	0	0
$193$	−1.12177e30	−0.156632	−0.0783162	−	0.996929i	$-0.524954\pi$
$193$	−1.12177e30	−0.156632	−0.0783162	+	0.996929i	$0.524954\pi$
$194$	0	0
$195$	−2.27231e31	−2.76059
$196$	0	0
$197$	−9.10597e30	−0.963900	−0.481950	−	0.876199i	$-0.660071\pi$
$197$	−9.10597e30	−0.963900	−0.481950	+	0.876199i	$0.660071\pi$
$198$	0	0
$199$	1.05803e31	0.977192	0.488596	−	0.872510i	$-0.337509\pi$
$199$	1.05803e31	0.977192	0.488596	+	0.872510i	$0.337509\pi$
$200$	0	0
$201$	1.29753e31	1.04706
$202$	0	0
$203$	4.26017e30	0.300767
$204$	0	0
$205$	−2.23831e31	−1.38434
$206$	0	0
$207$	−2.62745e31	−1.42539
$208$	0	0
$209$	1.91866e31	0.914163
$210$	0	0
$211$	−1.17863e31	−0.493814	−0.246907	−	0.969039i	$-0.579414\pi$
$211$	−1.17863e31	−0.493814	−0.246907	+	0.969039i	$0.579414\pi$
$212$	0	0
$213$	−3.82106e31	−1.40948
$214$	0	0
$215$	−3.91076e31	−1.27157
$216$	0	0
$217$	−7.28997e30	−0.209180
$218$	0	0
$219$	1.09424e32	2.77407
$220$	0	0
$221$	−3.44397e30	−0.0772261
$222$	0	0
$223$	−5.14026e31	−1.02063	−0.510316	−	0.859987i	$-0.670472\pi$
$223$	−5.14026e31	−1.02063	−0.510316	+	0.859987i	$0.670472\pi$
$224$	0	0
$225$	3.75441e31	0.660812
$226$	0	0
$227$	6.85208e31	1.07022	0.535110	−	0.844782i	$-0.320270\pi$
$227$	6.85208e31	1.07022	0.535110	+	0.844782i	$0.320270\pi$
$228$	0	0
$229$	−1.20413e32	−1.67069	−0.835343	−	0.549730i	$-0.814731\pi$
$229$	−1.20413e32	−1.67069	−0.835343	+	0.549730i	$0.814731\pi$
$230$	0	0
$231$	2.50268e32	3.08776
$232$	0	0
$233$	2.77909e31	0.305210	0.152605	−	0.988287i	$-0.451234\pi$
$233$	2.77909e31	0.305210	0.152605	+	0.988287i	$0.451234\pi$
$234$	0	0
$235$	−1.93606e32	−1.89454
$236$	0	0
$237$	3.87233e32	3.37963
$238$	0	0
$239$	−3.12220e31	−0.243271	−0.121635	−	0.992575i	$-0.538814\pi$
$239$	−3.12220e31	−0.243271	−0.121635	+	0.992575i	$0.538814\pi$
$240$	0	0
$241$	2.33705e32	1.62719	0.813596	−	0.581431i	$-0.197507\pi$
$241$	2.33705e32	1.62719	0.813596	+	0.581431i	$0.197507\pi$
$242$	0	0
$243$	1.68428e32	1.04889
$244$	0	0
$245$	−3.03980e31	−0.169474
$246$	0	0
$247$	−1.32736e32	−0.663099
$248$	0	0
$249$	−1.49840e32	−0.671329
$250$	0	0
$251$	−2.44628e32	−0.983808	−0.491904	−	0.870650i	$-0.663699\pi$
$251$	−2.44628e32	−0.983808	−0.491904	+	0.870650i	$0.663699\pi$
$252$	0	0
$253$	−3.24115e32	−1.17104
$254$	0	0
$255$	3.62609e31	0.117800
$256$	0	0
$257$	−5.77863e32	−1.68938	−0.844691	−	0.535254i	$-0.820216\pi$
$257$	−5.77863e32	−1.68938	−0.844691	+	0.535254i	$0.820216\pi$
$258$	0	0
$259$	−1.52198e32	−0.400739
$260$	0	0
$261$	3.10130e32	0.736031
$262$	0	0
$263$	−7.32407e32	−1.56801	−0.784007	−	0.620752i	$-0.786827\pi$
$263$	−7.32407e32	−1.56801	−0.784007	+	0.620752i	$0.786827\pi$
$264$	0	0
$265$	3.80197e32	0.734834
$266$	0	0
$267$	1.14198e33	1.99414
$268$	0	0
$269$	5.19389e32	0.820043	0.410022	−	0.912076i	$-0.365521\pi$
$269$	5.19389e32	0.820043	0.410022	+	0.912076i	$0.365521\pi$
$270$	0	0
$271$	4.92441e32	0.703508	0.351754	−	0.936092i	$-0.385585\pi$
$271$	4.92441e32	0.703508	0.351754	+	0.936092i	$0.385585\pi$
$272$	0	0
$273$	−1.73139e33	−2.23974
$274$	0	0
$275$	4.63135e32	0.542896
$276$	0	0
$277$	−1.77095e31	−0.0188248	−0.00941241	−	0.999956i	$-0.502996\pi$
$277$	−1.77095e31	−0.0188248	−0.00941241	+	0.999956i	$0.502996\pi$
$278$	0	0
$279$	−5.30691e32	−0.511900
$280$	0	0
$281$	−1.51610e33	−1.32798	−0.663991	−	0.747741i	$-0.731139\pi$
$281$	−1.51610e33	−1.32798	−0.663991	+	0.747741i	$0.731139\pi$
$282$	0	0
$283$	−6.97399e32	−0.555091	−0.277546	−	0.960712i	$-0.589521\pi$
$283$	−6.97399e32	−0.555091	−0.277546	+	0.960712i	$0.589521\pi$
$284$	0	0
$285$	1.39755e33	1.01149
$286$	0	0
$287$	−1.70548e33	−1.12316
$288$	0	0
$289$	−1.66222e33	−0.996705
$290$	0	0
$291$	−8.56944e32	−0.468163
$292$	0	0
$293$	−1.70891e33	−0.851149	−0.425575	−	0.904923i	$-0.639928\pi$
$293$	−1.70891e33	−0.851149	−0.425575	+	0.904923i	$0.639928\pi$
$294$	0	0
$295$	−3.91757e33	−1.78000
$296$	0	0
$297$	1.01483e34	4.20901
$298$	0	0
$299$	2.24227e33	0.849430
$300$	0	0
$301$	−2.97979e33	−1.03166
$302$	0	0
$303$	2.48207e32	0.0785849
$304$	0	0
$305$	−1.52357e33	−0.441383
$306$	0	0
$307$	3.85932e33	1.02363	0.511814	−	0.859096i	$-0.328974\pi$
$307$	3.85932e33	1.02363	0.511814	+	0.859096i	$0.328974\pi$
$308$	0	0
$309$	−1.91095e33	−0.464313
$310$	0	0
$311$	3.26279e33	0.726649	0.363324	−	0.931663i	$-0.381642\pi$
$311$	3.26279e33	0.726649	0.363324	+	0.931663i	$0.381642\pi$
$312$	0	0
$313$	8.82413e33	1.80228	0.901142	−	0.433525i	$-0.142730\pi$
$313$	8.82413e33	1.80228	0.901142	+	0.433525i	$0.142730\pi$
$314$	0	0
$315$	1.26332e34	2.36766
$316$	0	0
$317$	3.96580e33	0.682387	0.341194	−	0.939993i	$-0.389169\pi$
$317$	3.96580e33	0.682387	0.341194	+	0.939993i	$0.389169\pi$
$318$	0	0
$319$	3.82568e33	0.604693
$320$	0	0
$321$	3.74574e32	0.0544151
$322$	0	0
$323$	2.11815e32	0.0282959
$324$	0	0
$325$	−3.20403e33	−0.393796
$326$	0	0
$327$	−1.54985e34	−1.75346
$328$	0	0
$329$	−1.47518e34	−1.53710
$330$	0	0
$331$	−1.17780e34	−1.13082	−0.565410	−	0.824810i	$-0.691282\pi$
$331$	−1.17780e34	−1.13082	−0.565410	+	0.824810i	$0.691282\pi$
$332$	0	0
$333$	−1.10796e34	−0.980679
$334$	0	0
$335$	8.07964e33	0.659610
$336$	0	0
$337$	1.94526e34	1.46546	0.732731	−	0.680518i	$-0.238245\pi$
$337$	1.94526e34	1.46546	0.732731	+	0.680518i	$0.238245\pi$
$338$	0	0
$339$	3.61631e34	2.51518
$340$	0	0
$341$	−6.54648e33	−0.420556
$342$	0	0
$343$	−1.78551e34	−1.05997
$344$	0	0
$345$	−2.36084e34	−1.29571
$346$	0	0
$347$	−2.25323e34	−1.14382	−0.571910	−	0.820316i	$-0.693797\pi$
$347$	−2.25323e34	−1.14382	−0.571910	+	0.820316i	$0.693797\pi$
$348$	0	0
$349$	2.87978e34	1.35275	0.676373	−	0.736559i	$-0.263551\pi$
$349$	2.87978e34	1.35275	0.676373	+	0.736559i	$0.263551\pi$
$350$	0	0
$351$	−7.02071e34	−3.05306
$352$	0	0
$353$	1.55573e34	0.626577	0.313289	−	0.949658i	$-0.398569\pi$
$353$	1.55573e34	0.626577	0.313289	+	0.949658i	$0.398569\pi$
$354$	0	0
$355$	−2.37934e34	−0.887921
$356$	0	0
$357$	2.76289e33	0.0955748
$358$	0	0
$359$	−2.82527e33	−0.0906326	−0.0453163	−	0.998973i	$-0.514430\pi$
$359$	−2.82527e33	−0.0906326	−0.0453163	+	0.998973i	$0.514430\pi$
$360$	0	0
$361$	−2.54370e34	−0.757038
$362$	0	0
$363$	1.59404e35	4.40310
$364$	0	0
$365$	6.81375e34	1.74756
$366$	0	0
$367$	−6.62769e34	−1.57896	−0.789478	−	0.613778i	$-0.789649\pi$
$367$	−6.62769e34	−1.57896	−0.789478	+	0.613778i	$0.789649\pi$
$368$	0	0
$369$	−1.24154e35	−2.74857
$370$	0	0
$371$	2.89690e34	0.596192
$372$	0	0
$373$	−3.28983e33	−0.0629657	−0.0314829	−	0.999504i	$-0.510023\pi$
$373$	−3.28983e33	−0.0629657	−0.0314829	+	0.999504i	$0.510023\pi$
$374$	0	0
$375$	−8.15082e34	−1.45137
$376$	0	0
$377$	−2.64666e34	−0.438621
$378$	0	0
$379$	2.68710e34	0.414624	0.207312	−	0.978275i	$-0.433529\pi$
$379$	2.68710e34	0.414624	0.207312	+	0.978275i	$0.433529\pi$
$380$	0	0
$381$	−1.16756e35	−1.67800
$382$	0	0
$383$	1.34101e35	1.79577	0.897883	−	0.440234i	$-0.145104\pi$
$383$	1.34101e35	1.79577	0.897883	+	0.440234i	$0.145104\pi$
$384$	0	0
$385$	1.55840e35	1.94517
$386$	0	0
$387$	−2.16921e35	−2.52467
$388$	0	0
$389$	−4.33242e34	−0.470339	−0.235170	−	0.971954i	$-0.575565\pi$
$389$	−4.33242e34	−0.470339	−0.235170	+	0.971954i	$0.575565\pi$
$390$	0	0
$391$	−3.57815e33	−0.0362470
$392$	0	0
$393$	2.12188e34	0.200642
$394$	0	0
$395$	2.41127e35	2.12904
$396$	0	0
$397$	1.60466e35	1.32346	0.661731	−	0.749742i	$-0.269822\pi$
$397$	1.60466e35	1.32346	0.661731	+	0.749742i	$0.269822\pi$
$398$	0	0
$399$	1.06486e35	0.820649
$400$	0	0
$401$	−1.76431e34	−0.127094	−0.0635471	−	0.997979i	$-0.520241\pi$
$401$	−1.76431e34	−0.127094	−0.0635471	+	0.997979i	$0.520241\pi$
$402$	0	0
$403$	4.52894e34	0.305055
$404$	0	0
$405$	3.31805e35	2.09045
$406$	0	0
$407$	−1.36675e35	−0.805685
$408$	0	0
$409$	2.15166e35	1.18716	0.593580	−	0.804775i	$-0.297714\pi$
$409$	2.15166e35	1.18716	0.593580	+	0.804775i	$0.297714\pi$
$410$	0	0
$411$	2.95878e35	1.52844
$412$	0	0
$413$	−2.98498e35	−1.44416
$414$	0	0
$415$	−9.33038e34	−0.422912
$416$	0	0
$417$	2.92625e35	1.24301
$418$	0	0
$419$	3.84206e34	0.152994	0.0764970	−	0.997070i	$-0.475626\pi$
$419$	3.84206e34	0.152994	0.0764970	+	0.997070i	$0.475626\pi$
$420$	0	0
$421$	1.44796e35	0.540690	0.270345	−	0.962763i	$-0.412862\pi$
$421$	1.44796e35	0.540690	0.270345	+	0.962763i	$0.412862\pi$
$422$	0	0
$423$	−1.07389e36	−3.76155
$424$	0	0
$425$	5.11289e33	0.0168041
$426$	0	0
$427$	−1.16088e35	−0.358107
$428$	0	0
$429$	−1.55481e36	−4.50301
$430$	0	0
$431$	4.31104e35	1.17257	0.586286	−	0.810104i	$-0.300590\pi$
$431$	4.31104e35	1.17257	0.586286	+	0.810104i	$0.300590\pi$
$432$	0	0
$433$	−1.65756e35	−0.423528	−0.211764	−	0.977321i	$-0.567921\pi$
$433$	−1.65756e35	−0.423528	−0.211764	+	0.977321i	$0.567921\pi$
$434$	0	0
$435$	2.78661e35	0.669070
$436$	0	0
$437$	−1.37907e35	−0.311234
$438$	0	0
$439$	2.34003e35	0.496537	0.248268	−	0.968691i	$-0.420139\pi$
$439$	2.34003e35	0.496537	0.248268	+	0.968691i	$0.420139\pi$
$440$	0	0
$441$	−1.68611e35	−0.336485
$442$	0	0
$443$	2.30890e35	0.433466	0.216733	−	0.976231i	$-0.430460\pi$
$443$	2.30890e35	0.433466	0.216733	+	0.976231i	$0.430460\pi$
$444$	0	0
$445$	7.11102e35	1.25624
$446$	0	0
$447$	1.06764e36	1.77530
$448$	0	0
$449$	−1.25981e34	−0.0197233	−0.00986163	−	0.999951i	$-0.503139\pi$
$449$	−1.25981e34	−0.0197233	−0.00986163	+	0.999951i	$0.503139\pi$
$450$	0	0
$451$	−1.53154e36	−2.25811
$452$	0	0
$453$	−1.87794e34	−0.0260829
$454$	0	0
$455$	−1.07812e36	−1.41096
$456$	0	0
$457$	−4.05132e35	−0.499722	−0.249861	−	0.968282i	$-0.580385\pi$
$457$	−4.05132e35	−0.499722	−0.249861	+	0.968282i	$0.580385\pi$
$458$	0	0
$459$	1.12034e35	0.130281
$460$	0	0
$461$	1.47179e36	1.61393	0.806964	−	0.590601i	$-0.201109\pi$
$461$	1.47179e36	1.61393	0.806964	+	0.590601i	$0.201109\pi$
$462$	0	0
$463$	−1.22965e35	−0.127185	−0.0635927	−	0.997976i	$-0.520256\pi$
$463$	−1.22965e35	−0.127185	−0.0635927	+	0.997976i	$0.520256\pi$
$464$	0	0
$465$	−4.76843e35	−0.465329
$466$	0	0
$467$	−4.73907e35	−0.436430	−0.218215	−	0.975901i	$-0.570023\pi$
$467$	−4.73907e35	−0.436430	−0.218215	+	0.975901i	$0.570023\pi$
$468$	0	0
$469$	6.15627e35	0.535161
$470$	0	0
$471$	1.71972e36	1.41148
$472$	0	0
$473$	−2.67589e36	−2.07416
$474$	0	0
$475$	1.97058e35	0.144288
$476$	0	0
$477$	2.10887e36	1.45899
$478$	0	0
$479$	−8.61638e35	−0.563372	−0.281686	−	0.959507i	$-0.590894\pi$
$479$	−8.61638e35	−0.563372	−0.281686	+	0.959507i	$0.590894\pi$
$480$	0	0
$481$	9.45538e35	0.584414
$482$	0	0
$483$	−1.79884e36	−1.05125
$484$	0	0
$485$	−5.33612e35	−0.294925
$486$	0	0
$487$	−2.71211e36	−1.41797	−0.708983	−	0.705225i	$-0.750846\pi$
$487$	−2.71211e36	−1.41797	−0.708983	+	0.705225i	$0.750846\pi$
$488$	0	0
$489$	2.83294e36	1.40142
$490$	0	0
$491$	1.50407e36	0.704153	0.352077	−	0.935971i	$-0.385476\pi$
$491$	1.50407e36	0.704153	0.352077	+	0.935971i	$0.385476\pi$
$492$	0	0
$493$	4.22346e34	0.0187169
$494$	0	0
$495$	1.13447e37	4.76019
$496$	0	0
$497$	−1.81294e36	−0.720396
$498$	0	0
$499$	−5.13988e36	−1.93462	−0.967309	−	0.253600i	$-0.918385\pi$
$499$	−5.13988e36	−1.93462	−0.967309	+	0.253600i	$0.918385\pi$
$500$	0	0
$501$	−6.90116e35	−0.246101
$502$	0	0
$503$	2.32303e36	0.785032	0.392516	−	0.919745i	$-0.371605\pi$
$503$	2.32303e36	0.785032	0.392516	+	0.919745i	$0.371605\pi$
$504$	0	0
$505$	1.54557e35	0.0495056
$506$	0	0
$507$	4.81281e36	1.46147
$508$	0	0
$509$	2.52922e36	0.728278	0.364139	−	0.931345i	$-0.381363\pi$
$509$	2.52922e36	0.728278	0.364139	+	0.931345i	$0.381363\pi$
$510$	0	0
$511$	5.19173e36	1.41785
$512$	0	0
$513$	4.31796e36	1.11865
$514$	0	0
$515$	−1.18994e36	−0.292500
$516$	0	0
$517$	−1.32473e37	−3.09033
$518$	0	0
$519$	−1.17609e37	−2.60425
$520$	0	0
$521$	3.53956e36	0.744116	0.372058	−	0.928210i	$-0.378652\pi$
$521$	3.53956e36	0.744116	0.372058	+	0.928210i	$0.378652\pi$
$522$	0	0
$523$	2.83783e36	0.566519	0.283259	−	0.959043i	$-0.408584\pi$
$523$	2.83783e36	0.566519	0.283259	+	0.959043i	$0.408584\pi$
$524$	0	0
$525$	2.57040e36	0.487361
$526$	0	0
$527$	−7.22714e34	−0.0130174
$528$	0	0
$529$	−3.51358e36	−0.601310
$530$	0	0
$531$	−2.17299e37	−3.53413
$532$	0	0
$533$	1.05954e37	1.63795
$534$	0	0
$535$	2.33244e35	0.0342795
$536$	0	0
$537$	7.29922e35	0.102005
$538$	0	0
$539$	−2.07995e36	−0.276442
$540$	0	0
$541$	−1.41671e37	−1.79109	−0.895545	−	0.444972i	$-0.853214\pi$
$541$	−1.41671e37	−1.79109	−0.895545	+	0.444972i	$0.853214\pi$
$542$	0	0
$543$	−1.20850e36	−0.145362
$544$	0	0
$545$	−9.65081e36	−1.10462
$546$	0	0
$547$	3.30352e36	0.359872	0.179936	−	0.983678i	$-0.442411\pi$
$547$	3.30352e36	0.359872	0.179936	+	0.983678i	$0.442411\pi$
$548$	0	0
$549$	−8.45094e36	−0.876351
$550$	0	0
$551$	1.62778e36	0.160712
$552$	0	0
$553$	1.83726e37	1.72736
$554$	0	0
$555$	−9.95537e36	−0.891461
$556$	0	0
$557$	−1.48102e37	−1.26333	−0.631664	−	0.775242i	$-0.717628\pi$
$557$	−1.48102e37	−1.26333	−0.631664	+	0.775242i	$0.717628\pi$
$558$	0	0
$559$	1.85122e37	1.50452
$560$	0	0
$561$	2.48111e36	0.192153
$562$	0	0
$563$	−6.27545e36	−0.463214	−0.231607	−	0.972809i	$-0.574398\pi$
$563$	−6.27545e36	−0.463214	−0.231607	+	0.972809i	$0.574398\pi$
$564$	0	0
$565$	2.25185e37	1.58447
$566$	0	0
$567$	2.52818e37	1.69604
$568$	0	0
$569$	1.50622e37	0.963549	0.481774	−	0.876295i	$-0.339992\pi$
$569$	1.50622e37	0.963549	0.481774	+	0.876295i	$0.339992\pi$
$570$	0	0
$571$	1.23412e37	0.752959	0.376480	−	0.926425i	$-0.377134\pi$
$571$	1.23412e37	0.752959	0.376480	+	0.926425i	$0.377134\pi$
$572$	0	0
$573$	1.59638e37	0.929074
$574$	0	0
$575$	−3.32886e36	−0.184833
$576$	0	0
$577$	−2.71273e37	−1.43725	−0.718626	−	0.695397i	$-0.755228\pi$
$577$	−2.71273e37	−1.43725	−0.718626	+	0.695397i	$0.755228\pi$
$578$	0	0
$579$	−5.59088e36	−0.282696
$580$	0	0
$581$	−7.10927e36	−0.343121
$582$	0	0
$583$	2.60145e37	1.19864
$584$	0	0
$585$	−7.84844e37	−3.45286
$586$	0	0
$587$	−2.02495e36	−0.0850744	−0.0425372	−	0.999095i	$-0.513544\pi$
$587$	−2.02495e36	−0.0850744	−0.0425372	+	0.999095i	$0.513544\pi$
$588$	0	0
$589$	−2.78544e36	−0.111773
$590$	0	0
$591$	−4.53838e37	−1.73968
$592$	0	0
$593$	2.96953e37	1.08755	0.543775	−	0.839231i	$-0.316994\pi$
$593$	2.96953e37	1.08755	0.543775	+	0.839231i	$0.316994\pi$
$594$	0	0
$595$	1.72043e36	0.0602086
$596$	0	0
$597$	5.27317e37	1.76367
$598$	0	0
$599$	−3.18999e37	−1.01983	−0.509914	−	0.860225i	$-0.670323\pi$
$599$	−3.18999e37	−1.01983	−0.509914	+	0.860225i	$0.670323\pi$
$600$	0	0
$601$	2.98723e37	0.912985	0.456492	−	0.889727i	$-0.349106\pi$
$601$	2.98723e37	0.912985	0.456492	+	0.889727i	$0.349106\pi$
$602$	0	0
$603$	4.48161e37	1.30963
$604$	0	0
$605$	9.92594e37	2.77379
$606$	0	0
$607$	3.86067e36	0.103185	0.0515924	−	0.998668i	$-0.483570\pi$
$607$	3.86067e36	0.103185	0.0515924	+	0.998668i	$0.483570\pi$
$608$	0	0
$609$	2.12326e37	0.542836
$610$	0	0
$611$	9.16466e37	2.24161
$612$	0	0
$613$	−4.07829e37	−0.954468	−0.477234	−	0.878776i	$-0.658361\pi$
$613$	−4.07829e37	−0.954468	−0.477234	+	0.878776i	$0.658361\pi$
$614$	0	0
$615$	−1.11557e38	−2.49851
$616$	0	0
$617$	4.76142e37	1.02067	0.510336	−	0.859975i	$-0.329521\pi$
$617$	4.76142e37	1.02067	0.510336	+	0.859975i	$0.329521\pi$
$618$	0	0
$619$	−5.61889e37	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$619$	−5.61889e37	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$620$	0	0
$621$	−7.29424e37	−1.43299
$622$	0	0
$623$	5.41823e37	1.01922
$624$	0	0
$625$	−6.70040e37	−1.20704
$626$	0	0
$627$	9.56255e37	1.64992
$628$	0	0
$629$	−1.50886e36	−0.0249382
$630$	0	0
$631$	8.44313e37	1.33693	0.668463	−	0.743745i	$-0.266952\pi$
$631$	8.44313e37	1.33693	0.668463	+	0.743745i	$0.266952\pi$
$632$	0	0
$633$	−5.87424e37	−0.891254
$634$	0	0
$635$	−7.27029e37	−1.05708
$636$	0	0
$637$	1.43894e37	0.200521
$638$	0	0
$639$	−1.31977e38	−1.76294
$640$	0	0
$641$	6.89743e37	0.883291	0.441646	−	0.897190i	$-0.354395\pi$
$641$	6.89743e37	0.883291	0.441646	+	0.897190i	$0.354395\pi$
$642$	0	0
$643$	−9.85286e37	−1.20980	−0.604901	−	0.796301i	$-0.706787\pi$
$643$	−9.85286e37	−1.20980	−0.604901	+	0.796301i	$0.706787\pi$
$644$	0	0
$645$	−1.94911e38	−2.29498
$646$	0	0
$647$	−4.18480e37	−0.472571	−0.236286	−	0.971684i	$-0.575930\pi$
$647$	−4.18480e37	−0.472571	−0.236286	+	0.971684i	$0.575930\pi$
$648$	0	0
$649$	−2.68055e38	−2.90349
$650$	0	0
$651$	−3.63330e37	−0.377535
$652$	0	0
$653$	−1.30758e38	−1.30359	−0.651796	−	0.758394i	$-0.725984\pi$
$653$	−1.30758e38	−1.30359	−0.651796	+	0.758394i	$0.725984\pi$
$654$	0	0
$655$	1.32128e37	0.126397
$656$	0	0
$657$	3.77944e38	3.46973
$658$	0	0
$659$	2.08098e38	1.83365	0.916823	−	0.399294i	$-0.130745\pi$
$659$	2.08098e38	1.83365	0.916823	+	0.399294i	$0.130745\pi$
$660$	0	0
$661$	1.40577e38	1.18904	0.594519	−	0.804081i	$-0.297342\pi$
$661$	1.40577e38	1.18904	0.594519	+	0.804081i	$0.297342\pi$
$662$	0	0
$663$	−1.71647e37	−0.139381
$664$	0	0
$665$	6.63079e37	0.516979
$666$	0	0
$667$	−2.74977e37	−0.205872
$668$	0	0
$669$	−2.56189e38	−1.84208
$670$	0	0
$671$	−1.04249e38	−0.719974
$672$	0	0
$673$	5.46802e37	0.362766	0.181383	−	0.983413i	$-0.441943\pi$
$673$	5.46802e37	0.362766	0.181383	+	0.983413i	$0.441943\pi$
$674$	0	0
$675$	1.04229e38	0.664335
$676$	0	0
$677$	−9.40445e37	−0.575952	−0.287976	−	0.957638i	$-0.592982\pi$
$677$	−9.40445e37	−0.575952	−0.287976	+	0.957638i	$0.592982\pi$
$678$	0	0
$679$	−4.06585e37	−0.239282
$680$	0	0
$681$	3.41505e38	1.93157
$682$	0	0
$683$	3.05025e38	1.65827	0.829137	−	0.559045i	$-0.188832\pi$
$683$	3.05025e38	1.65827	0.829137	+	0.559045i	$0.188832\pi$
$684$	0	0
$685$	1.84241e38	0.962859
$686$	0	0
$687$	−6.00135e38	−3.01532
$688$	0	0
$689$	−1.79972e38	−0.869452
$690$	0	0
$691$	1.09109e38	0.506883	0.253441	−	0.967351i	$-0.418438\pi$
$691$	1.09109e38	0.506883	0.253441	+	0.967351i	$0.418438\pi$
$692$	0	0
$693$	8.64411e38	3.86208
$694$	0	0
$695$	1.82215e38	0.783051
$696$	0	0
$697$	−1.69078e37	−0.0698948
$698$	0	0
$699$	1.38509e38	0.550854
$700$	0	0
$701$	1.64967e38	0.631253	0.315627	−	0.948883i	$-0.397785\pi$
$701$	1.64967e38	0.631253	0.315627	+	0.948883i	$0.397785\pi$
$702$	0	0
$703$	−5.81536e37	−0.214131
$704$	0	0
$705$	−9.64928e38	−3.41934
$706$	0	0
$707$	1.17764e37	0.0401653
$708$	0	0
$709$	−4.51785e38	−1.48323	−0.741614	−	0.670827i	$-0.765939\pi$
$709$	−4.51785e38	−1.48323	−0.741614	+	0.670827i	$0.765939\pi$
$710$	0	0
$711$	1.33748e39	4.22715
$712$	0	0
$713$	4.70539e37	0.143181
$714$	0	0
$715$	−9.68165e38	−2.83673
$716$	0	0
$717$	−1.55610e38	−0.439064
$718$	0	0
$719$	−5.50353e38	−1.49555	−0.747776	−	0.663951i	$-0.768878\pi$
$719$	−5.50353e38	−1.49555	−0.747776	+	0.663951i	$0.768878\pi$
$720$	0	0
$721$	−9.06669e37	−0.237314
$722$	0	0
$723$	1.16478e39	2.93682
$724$	0	0
$725$	3.92921e37	0.0954425
$726$	0	0
$727$	−4.56014e38	−1.06724	−0.533622	−	0.845723i	$-0.679169\pi$
$727$	−4.56014e38	−1.06724	−0.533622	+	0.845723i	$0.679169\pi$
$728$	0	0
$729$	2.41595e37	0.0544837
$730$	0	0
$731$	−2.95411e37	−0.0642011
$732$	0	0
$733$	8.20814e38	1.71926	0.859629	−	0.510918i	$-0.170694\pi$
$733$	8.20814e38	1.71926	0.859629	+	0.510918i	$0.170694\pi$
$734$	0	0
$735$	−1.51503e38	−0.305873
$736$	0	0
$737$	5.52840e38	1.07594
$738$	0	0
$739$	−2.21038e38	−0.414731	−0.207366	−	0.978264i	$-0.566489\pi$
$739$	−2.21038e38	−0.414731	−0.207366	+	0.978264i	$0.566489\pi$
$740$	0	0
$741$	−6.61550e38	−1.19679
$742$	0	0
$743$	7.17538e38	1.25169	0.625843	−	0.779949i	$-0.284755\pi$
$743$	7.17538e38	1.25169	0.625843	+	0.779949i	$0.284755\pi$
$744$	0	0
$745$	6.64809e38	1.11837
$746$	0	0
$747$	−5.17537e38	−0.839679
$748$	0	0
$749$	1.77720e37	0.0278120
$750$	0	0
$751$	1.48899e37	0.0224778	0.0112389	−	0.999937i	$-0.496422\pi$
$751$	1.48899e37	0.0224778	0.0112389	+	0.999937i	$0.496422\pi$
$752$	0	0
$753$	−1.21922e39	−1.77561
$754$	0	0
$755$	−1.16938e37	−0.0164313
$756$	0	0
$757$	−3.26772e38	−0.443048	−0.221524	−	0.975155i	$-0.571103\pi$
$757$	−3.26772e38	−0.443048	−0.221524	+	0.975155i	$0.571103\pi$
$758$	0	0
$759$	−1.61538e39	−2.11354
$760$	0	0
$761$	−5.38363e38	−0.679802	−0.339901	−	0.940461i	$-0.610394\pi$
$761$	−5.38363e38	−0.679802	−0.339901	+	0.940461i	$0.610394\pi$
$762$	0	0
$763$	−7.35342e38	−0.896207
$764$	0	0
$765$	1.25243e38	0.147341
$766$	0	0
$767$	1.85444e39	2.10608
$768$	0	0
$769$	5.03277e38	0.551825	0.275913	−	0.961183i	$-0.411020\pi$
$769$	5.03277e38	0.551825	0.275913	+	0.961183i	$0.411020\pi$
$770$	0	0
$771$	−2.88005e39	−3.04906
$772$	0	0
$773$	1.25666e39	1.28468	0.642341	−	0.766419i	$-0.277963\pi$
$773$	1.25666e39	1.28468	0.642341	+	0.766419i	$0.277963\pi$
$774$	0	0
$775$	−6.72363e37	−0.0663790
$776$	0	0
$777$	−7.58548e38	−0.723268
$778$	0	0
$779$	−6.51651e38	−0.600149
$780$	0	0
$781$	−1.62804e39	−1.44836
$782$	0	0
$783$	8.60973e38	0.739956
$784$	0	0
$785$	1.07085e39	0.889180
$786$	0	0
$787$	−1.12469e39	−0.902349	−0.451174	−	0.892436i	$-0.648995\pi$
$787$	−1.12469e39	−0.902349	−0.451174	+	0.892436i	$0.648995\pi$
$788$	0	0
$789$	−3.65029e39	−2.83001
$790$	0	0
$791$	1.71579e39	1.28553
$792$	0	0
$793$	7.21207e38	0.522242
$794$	0	0
$795$	1.89489e39	1.32626
$796$	0	0
$797$	1.13227e39	0.766062	0.383031	−	0.923735i	$-0.374880\pi$
$797$	1.13227e39	0.766062	0.383031	+	0.923735i	$0.374880\pi$
$798$	0	0
$799$	−1.46247e38	−0.0956544
$800$	0	0
$801$	3.94433e39	2.49422
$802$	0	0
$803$	4.66223e39	2.85058
$804$	0	0
$805$	−1.12012e39	−0.662250
$806$	0	0
$807$	2.58862e39	1.48004
$808$	0	0
$809$	−7.83646e38	−0.433326	−0.216663	−	0.976246i	$-0.569517\pi$
$809$	−7.83646e38	−0.433326	−0.216663	+	0.976246i	$0.569517\pi$
$810$	0	0
$811$	2.10581e39	1.12626	0.563130	−	0.826368i	$-0.309597\pi$
$811$	2.10581e39	1.12626	0.563130	+	0.826368i	$0.309597\pi$
$812$	0	0
$813$	2.45431e39	1.26972
$814$	0	0
$815$	1.76405e39	0.882843
$816$	0	0
$817$	−1.13856e39	−0.551261
$818$	0	0
$819$	−5.98011e39	−2.80141
$820$	0	0
$821$	1.97957e39	0.897303	0.448652	−	0.893707i	$-0.351904\pi$
$821$	1.97957e39	0.897303	0.448652	+	0.893707i	$0.351904\pi$
$822$	0	0
$823$	−3.79739e39	−1.66566	−0.832832	−	0.553525i	$-0.813282\pi$
$823$	−3.79739e39	−1.66566	−0.832832	+	0.553525i	$0.813282\pi$
$824$	0	0
$825$	2.30825e39	0.979839
$826$	0	0
$827$	9.06763e38	0.372537	0.186268	−	0.982499i	$-0.440361\pi$
$827$	9.06763e38	0.372537	0.186268	+	0.982499i	$0.440361\pi$
$828$	0	0
$829$	−1.06862e39	−0.424949	−0.212475	−	0.977167i	$-0.568152\pi$
$829$	−1.06862e39	−0.424949	−0.212475	+	0.977167i	$0.568152\pi$
$830$	0	0
$831$	−8.82638e37	−0.0339757
$832$	0	0
$833$	−2.29621e37	−0.00855665
$834$	0	0
$835$	−4.29730e38	−0.155034
$836$	0	0
$837$	−1.47329e39	−0.514629
$838$	0	0
$839$	−4.99062e39	−1.68798	−0.843992	−	0.536356i	$-0.819801\pi$
$839$	−4.99062e39	−1.68798	−0.843992	+	0.536356i	$0.819801\pi$
$840$	0	0
$841$	−2.72857e39	−0.893693
$842$	0	0
$843$	−7.55619e39	−2.39679
$844$	0	0
$845$	2.99690e39	0.920674
$846$	0	0
$847$	7.56305e39	2.25046
$848$	0	0
$849$	−3.47581e39	−1.00185
$850$	0	0
$851$	9.82376e38	0.274302
$852$	0	0
$853$	7.26169e39	1.96438	0.982190	−	0.187891i	$-0.0601650\pi$
$853$	7.26169e39	1.96438	0.982190	+	0.187891i	$0.0601650\pi$
$854$	0	0
$855$	4.82704e39	1.26514
$856$	0	0
$857$	1.14984e38	0.0292008	0.0146004	−	0.999893i	$-0.495352\pi$
$857$	1.14984e38	0.0292008	0.0146004	+	0.999893i	$0.495352\pi$
$858$	0	0
$859$	2.75022e39	0.676798	0.338399	−	0.941003i	$-0.390115\pi$
$859$	2.75022e39	0.676798	0.338399	+	0.941003i	$0.390115\pi$
$860$	0	0
$861$	−8.50005e39	−2.02712
$862$	0	0
$863$	4.61137e39	1.06582	0.532910	−	0.846172i	$-0.321098\pi$
$863$	4.61137e39	1.06582	0.532910	+	0.846172i	$0.321098\pi$
$864$	0	0
$865$	−7.32344e39	−1.64058
$866$	0	0
$867$	−8.28443e39	−1.79889
$868$	0	0
$869$	1.64988e40	3.47285
$870$	0	0
$871$	−3.82462e39	−0.780447
$872$	0	0
$873$	−2.95983e39	−0.585565
$874$	0	0
$875$	−3.86723e39	−0.741808
$876$	0	0
$877$	1.02693e39	0.191006	0.0955028	−	0.995429i	$-0.469554\pi$
$877$	1.02693e39	0.191006	0.0955028	+	0.995429i	$0.469554\pi$
$878$	0	0
$879$	−8.51714e39	−1.53619
$880$	0	0
$881$	−4.80184e38	−0.0839911	−0.0419955	−	0.999118i	$-0.513372\pi$
$881$	−4.80184e38	−0.0839911	−0.0419955	+	0.999118i	$0.513372\pi$
$882$	0	0
$883$	5.06858e39	0.859839	0.429919	−	0.902867i	$-0.358542\pi$
$883$	5.06858e39	0.859839	0.429919	+	0.902867i	$0.358542\pi$
$884$	0	0
$885$	−1.95250e40	−3.21261
$886$	0	0
$887$	9.25025e38	0.147633	0.0738167	−	0.997272i	$-0.476482\pi$
$887$	9.25025e38	0.147633	0.0738167	+	0.997272i	$0.476482\pi$
$888$	0	0
$889$	−5.53959e39	−0.857637
$890$	0	0
$891$	2.27033e40	3.40989
$892$	0	0
$893$	−5.63656e39	−0.821333
$894$	0	0
$895$	4.54516e38	0.0642597
$896$	0	0
$897$	1.11754e40	1.53308
$898$	0	0
$899$	−5.55398e38	−0.0739348
$900$	0	0
$901$	2.87194e38	0.0371014
$902$	0	0
$903$	−1.48512e40	−1.86199
$904$	0	0
$905$	−7.52524e38	−0.0915724
$906$	0	0
$907$	−3.26356e39	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$907$	−3.26356e39	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$908$	0	0
$909$	8.57293e38	0.0982917
$910$	0	0
$911$	−9.43229e38	−0.104983	−0.0524915	−	0.998621i	$-0.516716\pi$
$911$	−9.43229e38	−0.104983	−0.0524915	+	0.998621i	$0.516716\pi$
$912$	0	0
$913$	−6.38421e39	−0.689845
$914$	0	0
$915$	−7.59343e39	−0.796624
$916$	0	0
$917$	1.00675e39	0.102550
$918$	0	0
$919$	−1.63185e40	−1.61407	−0.807033	−	0.590507i	$-0.798928\pi$
$919$	−1.63185e40	−1.61407	−0.807033	+	0.590507i	$0.798928\pi$
$920$	0	0
$921$	1.92347e40	1.84748
$922$	0	0
$923$	1.12630e40	1.05058
$924$	0	0
$925$	−1.40374e39	−0.127166
$926$	0	0
$927$	−6.60032e39	−0.580749
$928$	0	0
$929$	1.93800e40	1.65631	0.828153	−	0.560502i	$-0.189392\pi$
$929$	1.93800e40	1.65631	0.828153	+	0.560502i	$0.189392\pi$
$930$	0	0
$931$	−8.84991e38	−0.0734714
$932$	0	0
$933$	1.62617e40	1.31148
$934$	0	0
$935$	1.54497e39	0.121049
$936$	0	0
$937$	−6.58389e39	−0.501185	−0.250592	−	0.968093i	$-0.580625\pi$
$937$	−6.58389e39	−0.501185	−0.250592	+	0.968093i	$0.580625\pi$
$938$	0	0
$939$	4.39791e40	3.25283
$940$	0	0
$941$	−3.92916e39	−0.282384	−0.141192	−	0.989982i	$-0.545093\pi$
$941$	−3.92916e39	−0.282384	−0.141192	+	0.989982i	$0.545093\pi$
$942$	0	0
$943$	1.10082e40	0.768791
$944$	0	0
$945$	3.50719e40	2.38029
$946$	0	0
$947$	−2.18622e40	−1.44201	−0.721005	−	0.692930i	$-0.756320\pi$
$947$	−2.18622e40	−1.44201	−0.721005	+	0.692930i	$0.756320\pi$
$948$	0	0
$949$	−3.22539e40	−2.06771
$950$	0	0
$951$	1.97654e40	1.23160
$952$	0	0
$953$	−8.73050e39	−0.528792	−0.264396	−	0.964414i	$-0.585173\pi$
$953$	−8.73050e39	−0.528792	−0.264396	+	0.964414i	$0.585173\pi$
$954$	0	0
$955$	9.94056e39	0.585282
$956$	0	0
$957$	1.90671e40	1.09137
$958$	0	0
$959$	1.40382e40	0.781196
$960$	0	0
$961$	−1.75323e40	−0.948579
$962$	0	0
$963$	1.29376e39	0.0680609
$964$	0	0
$965$	−3.48140e39	−0.178088
$966$	0	0
$967$	−1.69507e40	−0.843197	−0.421599	−	0.906783i	$-0.638531\pi$
$967$	−1.69507e40	−0.843197	−0.421599	+	0.906783i	$0.638531\pi$
$968$	0	0
$969$	1.05568e39	0.0510695
$970$	0	0
$971$	−1.20819e40	−0.568430	−0.284215	−	0.958761i	$-0.591733\pi$
$971$	−1.20819e40	−0.568430	−0.284215	+	0.958761i	$0.591733\pi$
$972$	0	0
$973$	1.38839e40	0.635313
$974$	0	0
$975$	−1.59688e40	−0.710738
$976$	0	0
$977$	3.02320e39	0.130885	0.0654426	−	0.997856i	$-0.479154\pi$
$977$	3.02320e39	0.130885	0.0654426	+	0.997856i	$0.479154\pi$
$978$	0	0
$979$	4.86563e40	2.04915
$980$	0	0
$981$	−5.35310e40	−2.19318
$982$	0	0
$983$	−1.75980e39	−0.0701440	−0.0350720	−	0.999385i	$-0.511166\pi$
$983$	−1.75980e39	−0.0701440	−0.0350720	+	0.999385i	$0.511166\pi$
$984$	0	0
$985$	−2.82601e40	−1.09594
$986$	0	0
$987$	−7.35226e40	−2.77421
$988$	0	0
$989$	1.92334e40	0.706165
$990$	0	0
$991$	−4.81245e40	−1.71938	−0.859691	−	0.510815i	$-0.829344\pi$
$991$	−4.81245e40	−1.71938	−0.859691	+	0.510815i	$0.829344\pi$
$992$	0	0
$993$	−5.87011e40	−2.04095
$994$	0	0
$995$	3.28356e40	1.11105
$996$	0	0
$997$	3.20519e40	1.05553	0.527764	−	0.849391i	$-0.323031\pi$
$997$	3.20519e40	1.05553	0.527764	+	0.849391i	$0.323031\pi$
$998$	0	0
$999$	−3.07589e40	−0.985908

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.28.a.f.1.4		4
4.3	odd	2		8.28.a.b.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.28.a.b.1.1	✓	4	4.3	odd	2
16.28.a.f.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+4.98397e6 q^{3} +3.10348e9 q^{5} +2.36469e11 q^{7} +1.72143e13 q^{9} +2.12352e14 q^{11} -1.46908e15 q^{13} +1.54676e16 q^{15} +2.34431e15 q^{17} +9.03530e16 q^{19} +1.17855e18 q^{21} -1.52631e18 q^{23} +2.18098e18 q^{25} +4.77899e19 q^{27} +1.80158e19 q^{29} -3.08284e19 q^{31} +1.05835e21 q^{33} +7.33876e20 q^{35} -6.43627e20 q^{37} -7.32184e21 q^{39} -7.21227e21 q^{41} -1.26012e22 q^{43} +5.34243e22 q^{45} -6.23837e22 q^{47} -9.79482e21 q^{49} +1.16840e22 q^{51} +1.22507e23 q^{53} +6.59029e23 q^{55} +4.50316e23 q^{57} -1.26232e24 q^{59} -4.90924e23 q^{61} +4.07065e24 q^{63} -4.55925e24 q^{65} +2.60342e24 q^{67} -7.60710e24 q^{69} -7.66670e24 q^{71} +2.19552e25 q^{73} +1.08699e25 q^{75} +5.02146e25 q^{77} +7.76957e25 q^{79} +1.06914e26 q^{81} -3.00643e25 q^{83} +7.27551e24 q^{85} +8.97901e25 q^{87} +2.29131e26 q^{89} -3.47391e26 q^{91} -1.53648e26 q^{93} +2.80408e26 q^{95} -1.71940e26 q^{97} +3.65549e27 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 122512 q^{3} + 3544066168 q^{5} + 211767036576 q^{7} + 19748930504020 q^{9} + 137002338905648 q^{11} - 5580886697000 q^{13} + 25\!\cdots\!12 q^{15} + 44\!\cdots\!44 q^{17} + 17\!\cdots\!72 q^{19}+ \cdots + 53\!\cdots\!16 q^{99}+O(q^{100}) \) 4 * q + 122512 * q^3 + 3544066168 * q^5 + 211767036576 * q^7 + 19748930504020 * q^9 + 137002338905648 * q^11 - 5580886697000 * q^13 + 2557147082175712 * q^15 + 44214930716819144 * q^17 + 177042777372267472 * q^19 + 29921143201836672 * q^21 - 1767092590673925664 * q^23 + 598259983019495324 * q^25 - 5639315117824052576 * q^27 + 6230479481762520216 * q^29 - 211925592023301030272 * q^31 + 840492384365367174848 * q^33 - 295751663414571139392 * q^35 + 3315679506055104172536 * q^37 - 16187282823779225026976 * q^39 + 16742683588694565402600 * q^41 - 28929100862580808746832 * q^43 + 109794415746949511800792 * q^45 - 127204639057684425317952 * q^47 + 154785230493187619859620 * q^49 - 318357332417527091023840 * q^51 + 279552147064338019634872 * q^53 + 518548397855800513776032 * q^55 - 438125799682933882052288 * q^57 + 747757532145939859769840 * q^59 - 2481631547272388497527272 * q^61 + 10528130607790692595312416 * q^63 - 4197418205948650615753904 * q^65 + 14053964155982437041984400 * q^67 - 4086733995329892788833408 * q^69 + 14160923375603434909530016 * q^71 + 25756757389668481118224744 * q^73 + 21854795385788245746680816 * q^75 + 129360200718523083267780480 * q^77 + 32479711484972151883608128 * q^79 + 342342850435863984906534436 * q^81 - 79535495988836946955941680 * q^83 + 439674534481061013744364400 * q^85 + 406870958670348922758232416 * q^87 + 543797853914541667326783528 * q^89 + 743433170089429876643395008 * q^91 + 700751851717865814480816640 * q^93 + 1917664584916076964174787168 * q^95 - 408937442977415638801906552 * q^97 + 5389670612142233536540165616 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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