LMFDB - Embedded newform 72.8.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.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:	$22.4917218349$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.78233$ of defining polynomial
Character	$\chi$	$=$	72.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-437.552 q^{5} +1722.21 q^{7} +O(q^{10})$	$q-437.552 q^{5} +1722.21 q^{7} +4226.62 q^{11} -6638.83 q^{13} -34165.6 q^{17} -20358.1 q^{19} -45985.9 q^{23} +113327. q^{25} -51462.3 q^{29} -15184.8 q^{31} -753555. q^{35} -384588. q^{37} -602282. q^{41} -248405. q^{43} +718924. q^{47} +2.14246e6 q^{49} -1.39480e6 q^{53} -1.84937e6 q^{55} -1.19713e6 q^{59} +764413. q^{61} +2.90483e6 q^{65} -2.54127e6 q^{67} +4.66313e6 q^{71} -339311. q^{73} +7.27912e6 q^{77} +328624. q^{79} -1.77607e6 q^{83} +1.49492e7 q^{85} +582340. q^{89} -1.14334e7 q^{91} +8.90771e6 q^{95} -3.28652e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 224 q^{5} + 840 q^{7}+O(q^{10}) $ 2 * q - 224 * q^5 + 840 * q^7	$ 2 q - 224 q^{5} + 840 q^{7} + 640 q^{11} - 2860 q^{13} - 35776 q^{17} - 14672 q^{19} - 118016 q^{23} + 80806 q^{25} - 306720 q^{29} - 95480 q^{31} - 941952 q^{35} + 53820 q^{37} - 1143360 q^{41} - 679120 q^{43} - 567552 q^{47} + 2097202 q^{49} - 1165088 q^{53} - 2615296 q^{55} + 121600 q^{59} + 3143564 q^{61} + 3711808 q^{65} - 5468000 q^{67} + 6763520 q^{71} + 5009420 q^{73} + 10443264 q^{77} - 5450104 q^{79} + 6071168 q^{83} + 14605312 q^{85} + 7063680 q^{89} - 14767152 q^{91} + 10121984 q^{95} + 5219740 q^{97}+O(q^{100}) $ 2 * q - 224 * q^5 + 840 * q^7 + 640 * q^11 - 2860 * q^13 - 35776 * q^17 - 14672 * q^19 - 118016 * q^23 + 80806 * q^25 - 306720 * q^29 - 95480 * q^31 - 941952 * q^35 + 53820 * q^37 - 1143360 * q^41 - 679120 * q^43 - 567552 * q^47 + 2097202 * q^49 - 1165088 * q^53 - 2615296 * q^55 + 121600 * q^59 + 3143564 * q^61 + 3711808 * q^65 - 5468000 * q^67 + 6763520 * q^71 + 5009420 * q^73 + 10443264 * q^77 - 5450104 * q^79 + 6071168 * q^83 + 14605312 * q^85 + 7063680 * q^89 - 14767152 * q^91 + 10121984 * q^95 + 5219740 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−437.552	−1.56543	−0.782717	−	0.622378i	$-0.786167\pi$
$5$	−437.552	−1.56543	−0.782717	+	0.622378i	$0.786167\pi$
$6$	0	0
$7$	1722.21	1.89776	0.948882	−	0.315630i	$-0.102216\pi$
$7$	1722.21	1.89776	0.948882	+	0.315630i	$0.102216\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4226.62	0.957456	0.478728	−	0.877963i	$-0.341098\pi$
$11$	4226.62	0.957456	0.478728	+	0.877963i	$0.341098\pi$
$12$	0	0
$13$	−6638.83	−0.838088	−0.419044	−	0.907966i	$-0.637635\pi$
$13$	−6638.83	−0.838088	−0.419044	+	0.907966i	$0.637635\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−34165.6	−1.68662	−0.843311	−	0.537426i	$-0.819397\pi$
$17$	−34165.6	−1.68662	−0.843311	+	0.537426i	$0.819397\pi$
$18$	0	0
$19$	−20358.1	−0.680925	−0.340462	−	0.940258i	$-0.610584\pi$
$19$	−20358.1	−0.680925	−0.340462	+	0.940258i	$0.610584\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−45985.9	−0.788093	−0.394047	−	0.919090i	$-0.628925\pi$
$23$	−45985.9	−0.788093	−0.394047	+	0.919090i	$0.628925\pi$
$24$	0	0
$25$	113327.	1.45058
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−51462.3	−0.391828	−0.195914	−	0.980621i	$-0.562767\pi$
$29$	−51462.3	−0.391828	−0.195914	+	0.980621i	$0.562767\pi$
$30$	0	0
$31$	−15184.8	−0.0915469	−0.0457734	−	0.998952i	$-0.514575\pi$
$31$	−15184.8	−0.0915469	−0.0457734	+	0.998952i	$0.514575\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−753555.	−2.97082
$36$	0	0
$37$	−384588.	−1.24821	−0.624107	−	0.781339i	$-0.714537\pi$
$37$	−384588.	−1.24821	−0.624107	+	0.781339i	$0.714537\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−602282.	−1.36476	−0.682380	−	0.730998i	$-0.739055\pi$
$41$	−602282.	−1.36476	−0.682380	+	0.730998i	$0.739055\pi$
$42$	0	0
$43$	−248405.	−0.476455	−0.238227	−	0.971209i	$-0.576566\pi$
$43$	−248405.	−0.476455	−0.238227	+	0.971209i	$0.576566\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	718924.	1.01004	0.505022	−	0.863106i	$-0.331484\pi$
$47$	718924.	1.01004	0.505022	+	0.863106i	$0.331484\pi$
$48$	0	0
$49$	2.14246e6	2.60151
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.39480e6	−1.28690	−0.643450	−	0.765488i	$-0.722498\pi$
$53$	−1.39480e6	−1.28690	−0.643450	+	0.765488i	$0.722498\pi$
$54$	0	0
$55$	−1.84937e6	−1.49883
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.19713e6	−0.758857	−0.379429	−	0.925221i	$-0.623879\pi$
$59$	−1.19713e6	−0.758857	−0.379429	+	0.925221i	$0.623879\pi$
$60$	0	0
$61$	764413.	0.431195	0.215598	−	0.976482i	$-0.430830\pi$
$61$	764413.	0.431195	0.215598	+	0.976482i	$0.430830\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.90483e6	1.31197
$66$	0	0
$67$	−2.54127e6	−1.03226	−0.516131	−	0.856510i	$-0.672628\pi$
$67$	−2.54127e6	−1.03226	−0.516131	+	0.856510i	$0.672628\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.66313e6	1.54623	0.773114	−	0.634267i	$-0.218698\pi$
$71$	4.66313e6	1.54623	0.773114	+	0.634267i	$0.218698\pi$
$72$	0	0
$73$	−339311.	−0.102086	−0.0510432	−	0.998696i	$-0.516255\pi$
$73$	−339311.	−0.102086	−0.0510432	+	0.998696i	$0.516255\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.27912e6	1.81703
$78$	0	0
$79$	328624.	0.0749902	0.0374951	−	0.999297i	$-0.488062\pi$
$79$	328624.	0.0749902	0.0374951	+	0.999297i	$0.488062\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.77607e6	−0.340947	−0.170474	−	0.985362i	$-0.554530\pi$
$83$	−1.77607e6	−0.340947	−0.170474	+	0.985362i	$0.554530\pi$
$84$	0	0
$85$	1.49492e7	2.64029
$86$	0	0
$87$	0	0
$88$	0	0
$89$	582340.	0.0875612	0.0437806	−	0.999041i	$-0.486060\pi$
$89$	582340.	0.0875612	0.0437806	+	0.999041i	$0.486060\pi$
$90$	0	0
$91$	−1.14334e7	−1.59049
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.90771e6	1.06594
$96$	0	0
$97$	−3.28652e6	−0.365625	−0.182812	−	0.983148i	$-0.558520\pi$
$97$	−3.28652e6	−0.365625	−0.182812	+	0.983148i	$0.558520\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.54549e6	−0.342414	−0.171207	−	0.985235i	$-0.554767\pi$
$101$	−3.54549e6	−0.342414	−0.171207	+	0.985235i	$0.554767\pi$
$102$	0	0
$103$	−1.24694e7	−1.12438	−0.562192	−	0.827007i	$-0.690042\pi$
$103$	−1.24694e7	−1.12438	−0.562192	+	0.827007i	$0.690042\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.77904e7	1.40392	0.701962	−	0.712214i	$-0.252308\pi$
$107$	1.77904e7	1.40392	0.701962	+	0.712214i	$0.252308\pi$
$108$	0	0
$109$	668188.	0.0494204	0.0247102	−	0.999695i	$-0.492134\pi$
$109$	668188.	0.0494204	0.0247102	+	0.999695i	$0.492134\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.59643e7	−1.04082	−0.520408	−	0.853918i	$-0.674220\pi$
$113$	−1.59643e7	−1.04082	−0.520408	+	0.853918i	$0.674220\pi$
$114$	0	0
$115$	2.01212e7	1.23371
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.88402e7	−3.20081
$120$	0	0
$121$	−1.62284e6	−0.0832772
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.54025e7	−0.705354
$126$	0	0
$127$	3.97232e7	1.72081	0.860403	−	0.509615i	$-0.170212\pi$
$127$	3.97232e7	1.72081	0.860403	+	0.509615i	$0.170212\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.07841e6	−0.236233	−0.118116	−	0.993000i	$-0.537686\pi$
$131$	−6.07841e6	−0.236233	−0.118116	+	0.993000i	$0.537686\pi$
$132$	0	0
$133$	−3.50608e7	−1.29223
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.87491e7	0.955216	0.477608	−	0.878573i	$-0.341504\pi$
$137$	2.87491e7	0.955216	0.477608	+	0.878573i	$0.341504\pi$
$138$	0	0
$139$	−4.65405e7	−1.46987	−0.734935	−	0.678138i	$-0.762787\pi$
$139$	−4.65405e7	−1.46987	−0.734935	+	0.678138i	$0.762787\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.80598e7	−0.802433
$144$	0	0
$145$	2.25174e7	0.613381
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.10645e7	−0.769330	−0.384665	−	0.923056i	$-0.625683\pi$
$149$	−3.10645e7	−0.769330	−0.384665	+	0.923056i	$0.625683\pi$
$150$	0	0
$151$	1.42377e7	0.336528	0.168264	−	0.985742i	$-0.446184\pi$
$151$	1.42377e7	0.336528	0.168264	+	0.985742i	$0.446184\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.64414e6	0.143311
$156$	0	0
$157$	−7.35878e7	−1.51760	−0.758800	−	0.651324i	$-0.774214\pi$
$157$	−7.35878e7	−1.51760	−0.758800	+	0.651324i	$0.774214\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.91973e7	−1.49562
$162$	0	0
$163$	8.10928e7	1.46665	0.733324	−	0.679880i	$-0.237968\pi$
$163$	8.10928e7	1.46665	0.733324	+	0.679880i	$0.237968\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.11778e8	1.85716	0.928581	−	0.371130i	$-0.121029\pi$
$167$	1.11778e8	1.85716	0.928581	+	0.371130i	$0.121029\pi$
$168$	0	0
$169$	−1.86745e7	−0.297608
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.97788e7	1.31829	0.659147	−	0.752014i	$-0.270917\pi$
$173$	8.97788e7	1.31829	0.659147	+	0.752014i	$0.270917\pi$
$174$	0	0
$175$	1.95172e8	2.75286
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.65667e7	0.606862	0.303431	−	0.952853i	$-0.401868\pi$
$179$	4.65667e7	0.606862	0.303431	+	0.952853i	$0.401868\pi$
$180$	0	0
$181$	−6.73507e7	−0.844242	−0.422121	−	0.906539i	$-0.638714\pi$
$181$	−6.73507e7	−0.844242	−0.422121	+	0.906539i	$0.638714\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.68277e8	1.95400
$186$	0	0
$187$	−1.44405e8	−1.61487
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.95557e7	0.306920	0.153460	−	0.988155i	$-0.450958\pi$
$191$	2.95557e7	0.306920	0.153460	+	0.988155i	$0.450958\pi$
$192$	0	0
$193$	−1.23751e8	−1.23908	−0.619540	−	0.784965i	$-0.712681\pi$
$193$	−1.23751e8	−1.23908	−0.619540	+	0.784965i	$0.712681\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.28755e7	0.213177	0.106588	−	0.994303i	$-0.466007\pi$
$197$	2.28755e7	0.213177	0.106588	+	0.994303i	$0.466007\pi$
$198$	0	0
$199$	−8.69164e7	−0.781837	−0.390918	−	0.920425i	$-0.627842\pi$
$199$	−8.69164e7	−0.781837	−0.390918	+	0.920425i	$0.627842\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.86287e7	−0.743598
$204$	0	0
$205$	2.63530e8	2.13644
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.60459e7	−0.651956
$210$	0	0
$211$	−3.82685e7	−0.280448	−0.140224	−	0.990120i	$-0.544782\pi$
$211$	−3.82685e7	−0.280448	−0.140224	+	0.990120i	$0.544782\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.08690e8	0.745858
$216$	0	0
$217$	−2.61514e7	−0.173734
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.26820e8	1.41354
$222$	0	0
$223$	2.07224e8	1.25133	0.625667	−	0.780090i	$-0.284827\pi$
$223$	2.07224e8	1.25133	0.625667	+	0.780090i	$0.284827\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.07779e8	−1.17899	−0.589497	−	0.807770i	$-0.700674\pi$
$227$	−2.07779e8	−1.17899	−0.589497	+	0.807770i	$0.700674\pi$
$228$	0	0
$229$	1.86435e8	1.02589	0.512947	−	0.858420i	$-0.328554\pi$
$229$	1.86435e8	1.02589	0.512947	+	0.858420i	$0.328554\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.55106e7	0.0803309	0.0401654	−	0.999193i	$-0.487212\pi$
$233$	1.55106e7	0.0803309	0.0401654	+	0.999193i	$0.487212\pi$
$234$	0	0
$235$	−3.14566e8	−1.58116
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.74511e8	0.826856	0.413428	−	0.910537i	$-0.364331\pi$
$239$	1.74511e8	0.826856	0.413428	+	0.910537i	$0.364331\pi$
$240$	0	0
$241$	−7.69055e7	−0.353914	−0.176957	−	0.984219i	$-0.556625\pi$
$241$	−7.69055e7	−0.353914	−0.176957	+	0.984219i	$0.556625\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.37435e8	−4.07249
$246$	0	0
$247$	1.35154e8	0.570675
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.99888e8	−0.797865	−0.398933	−	0.916980i	$-0.630619\pi$
$251$	−1.99888e8	−0.797865	−0.398933	+	0.916980i	$0.630619\pi$
$252$	0	0
$253$	−1.94365e8	−0.754565
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.96410e8	−0.721769	−0.360884	−	0.932611i	$-0.617525\pi$
$257$	−1.96410e8	−0.721769	−0.360884	+	0.932611i	$0.617525\pi$
$258$	0	0
$259$	−6.62339e8	−2.36882
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.80764e8	−1.29066	−0.645328	−	0.763905i	$-0.723279\pi$
$263$	−3.80764e8	−1.29066	−0.645328	+	0.763905i	$0.723279\pi$
$264$	0	0
$265$	6.10295e8	2.01456
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.47962e8	1.40316	0.701581	−	0.712590i	$-0.252478\pi$
$269$	4.47962e8	1.40316	0.701581	+	0.712590i	$0.252478\pi$
$270$	0	0
$271$	1.62567e8	0.496180	0.248090	−	0.968737i	$-0.420197\pi$
$271$	1.62567e8	0.496180	0.248090	+	0.968737i	$0.420197\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.78989e8	1.38887
$276$	0	0
$277$	2.52029e8	0.712477	0.356239	−	0.934395i	$-0.384059\pi$
$277$	2.52029e8	0.712477	0.356239	+	0.934395i	$0.384059\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.40349e7	0.145279	0.0726393	−	0.997358i	$-0.476858\pi$
$281$	5.40349e7	0.145279	0.0726393	+	0.997358i	$0.476858\pi$
$282$	0	0
$283$	−5.76782e8	−1.51272	−0.756361	−	0.654155i	$-0.773024\pi$
$283$	−5.76782e8	−1.51272	−0.756361	+	0.654155i	$0.773024\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.03725e9	−2.58999
$288$	0	0
$289$	7.56949e8	1.84469
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.67850e8	1.31885	0.659427	−	0.751769i	$-0.270799\pi$
$293$	5.67850e8	1.31885	0.659427	+	0.751769i	$0.270799\pi$
$294$	0	0
$295$	5.23807e8	1.18794
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.05293e8	0.660492
$300$	0	0
$301$	−4.27806e8	−0.904199
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.34471e8	−0.675007
$306$	0	0
$307$	−2.12785e8	−0.419718	−0.209859	−	0.977732i	$-0.567301\pi$
$307$	−2.12785e8	−0.419718	−0.209859	+	0.977732i	$0.567301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.52178e8	0.852408	0.426204	−	0.904627i	$-0.359851\pi$
$311$	4.52178e8	0.852408	0.426204	+	0.904627i	$0.359851\pi$
$312$	0	0
$313$	−8.42763e8	−1.55346	−0.776730	−	0.629834i	$-0.783123\pi$
$313$	−8.42763e8	−1.55346	−0.776730	+	0.629834i	$0.783123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.02806e8	−0.181264	−0.0906318	−	0.995884i	$-0.528889\pi$
$317$	−1.02806e8	−0.181264	−0.0906318	+	0.995884i	$0.528889\pi$
$318$	0	0
$319$	−2.17512e8	−0.375159
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.95546e8	1.14846
$324$	0	0
$325$	−7.52356e8	−1.21571
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.23814e9	1.91683
$330$	0	0
$331$	4.55210e8	0.689944	0.344972	−	0.938613i	$-0.387888\pi$
$331$	4.55210e8	0.689944	0.344972	+	0.938613i	$0.387888\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.11194e9	1.61594
$336$	0	0
$337$	1.38644e8	0.197331	0.0986655	−	0.995121i	$-0.468543\pi$
$337$	1.38644e8	0.197331	0.0986655	+	0.995121i	$0.468543\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.41805e7	−0.0876522
$342$	0	0
$343$	2.27144e9	3.03929
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.99476e8	1.02719	0.513597	−	0.858032i	$-0.328313\pi$
$347$	7.99476e8	1.02719	0.513597	+	0.858032i	$0.328313\pi$
$348$	0	0
$349$	9.15205e8	1.15247	0.576234	−	0.817284i	$-0.304522\pi$
$349$	9.15205e8	1.15247	0.576234	+	0.817284i	$0.304522\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.76398e8	−0.818448	−0.409224	−	0.912434i	$-0.634201\pi$
$353$	−6.76398e8	−0.818448	−0.409224	+	0.912434i	$0.634201\pi$
$354$	0	0
$355$	−2.04036e9	−2.42052
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.99641e8	−0.455869	−0.227934	−	0.973676i	$-0.573197\pi$
$359$	−3.99641e8	−0.455869	−0.227934	+	0.973676i	$0.573197\pi$
$360$	0	0
$361$	−4.79421e8	−0.536342
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.48466e8	0.159809
$366$	0	0
$367$	4.15569e8	0.438846	0.219423	−	0.975630i	$-0.429582\pi$
$367$	4.15569e8	0.438846	0.219423	+	0.975630i	$0.429582\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.40213e9	−2.44223
$372$	0	0
$373$	6.54100e8	0.652624	0.326312	−	0.945262i	$-0.394194\pi$
$373$	6.54100e8	0.652624	0.326312	+	0.945262i	$0.394194\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.41649e8	0.328387
$378$	0	0
$379$	6.68523e8	0.630782	0.315391	−	0.948962i	$-0.397864\pi$
$379$	6.68523e8	0.630782	0.315391	+	0.948962i	$0.397864\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.38983e8	0.672107	0.336054	−	0.941843i	$-0.390908\pi$
$383$	7.38983e8	0.672107	0.336054	+	0.941843i	$0.390908\pi$
$384$	0	0
$385$	−3.18499e9	−2.84443
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.04715e7	−0.0348598	−0.0174299	−	0.999848i	$-0.505548\pi$
$389$	−4.04715e7	−0.0348598	−0.0174299	+	0.999848i	$0.505548\pi$
$390$	0	0
$391$	1.57114e9	1.32922
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.43790e8	−0.117392
$396$	0	0
$397$	−1.87564e9	−1.50447	−0.752233	−	0.658897i	$-0.771023\pi$
$397$	−1.87564e9	−1.50447	−0.752233	+	0.658897i	$0.771023\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.96483e8	0.771728	0.385864	−	0.922556i	$-0.373903\pi$
$401$	9.96483e8	0.771728	0.385864	+	0.922556i	$0.373903\pi$
$402$	0	0
$403$	1.00809e8	0.0767244
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.62551e9	−1.19511
$408$	0	0
$409$	1.01806e8	0.0735772	0.0367886	−	0.999323i	$-0.488287\pi$
$409$	1.01806e8	0.0735772	0.0367886	+	0.999323i	$0.488287\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.06171e9	−1.44013
$414$	0	0
$415$	7.77124e8	0.533730
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.61886e9	−1.07513	−0.537564	−	0.843223i	$-0.680655\pi$
$419$	−1.61886e9	−1.07513	−0.537564	+	0.843223i	$0.680655\pi$
$420$	0	0
$421$	1.76813e9	1.15486	0.577428	−	0.816441i	$-0.304056\pi$
$421$	1.76813e9	1.15486	0.577428	+	0.816441i	$0.304056\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.87187e9	−2.44658
$426$	0	0
$427$	1.31648e9	0.818307
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.37116e8	0.262982	0.131491	−	0.991317i	$-0.458024\pi$
$431$	4.37116e8	0.262982	0.131491	+	0.991317i	$0.458024\pi$
$432$	0	0
$433$	9.69007e8	0.573613	0.286807	−	0.957989i	$-0.407406\pi$
$433$	9.69007e8	0.573613	0.286807	+	0.957989i	$0.407406\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.36185e8	0.536632
$438$	0	0
$439$	−8.86918e8	−0.500331	−0.250166	−	0.968203i	$-0.580485\pi$
$439$	−8.86918e8	−0.500331	−0.250166	+	0.968203i	$0.580485\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.47780e9	−1.35411	−0.677053	−	0.735934i	$-0.736743\pi$
$443$	−2.47780e9	−1.35411	−0.677053	+	0.735934i	$0.736743\pi$
$444$	0	0
$445$	−2.54804e8	−0.137071
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.68169e7	0.0504765	0.0252382	−	0.999681i	$-0.491966\pi$
$449$	9.68169e7	0.0504765	0.0252382	+	0.999681i	$0.491966\pi$
$450$	0	0
$451$	−2.54562e9	−1.30670
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.00272e9	2.48981
$456$	0	0
$457$	3.53320e9	1.73166	0.865828	−	0.500341i	$-0.166792\pi$
$457$	3.53320e9	1.73166	0.865828	+	0.500341i	$0.166792\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.88532e7	0.0279780	0.0139890	−	0.999902i	$-0.495547\pi$
$461$	5.88532e7	0.0279780	0.0139890	+	0.999902i	$0.495547\pi$
$462$	0	0
$463$	−2.28948e9	−1.07202	−0.536011	−	0.844211i	$-0.680069\pi$
$463$	−2.28948e9	−1.07202	−0.536011	+	0.844211i	$0.680069\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.76508e9	1.71067	0.855333	−	0.518079i	$-0.173353\pi$
$467$	3.76508e9	1.71067	0.855333	+	0.518079i	$0.173353\pi$
$468$	0	0
$469$	−4.37660e9	−1.95899
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.04992e9	−0.456185
$474$	0	0
$475$	−2.30711e9	−0.987736
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.15354e9	−0.895320	−0.447660	−	0.894204i	$-0.647743\pi$
$479$	−2.15354e9	−0.895320	−0.447660	+	0.894204i	$0.647743\pi$
$480$	0	0
$481$	2.55321e9	1.04611
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.43803e9	0.572361
$486$	0	0
$487$	−4.64683e8	−0.182308	−0.0911540	−	0.995837i	$-0.529056\pi$
$487$	−4.64683e8	−0.182308	−0.0911540	+	0.995837i	$0.529056\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.88216e8	−0.300511	−0.150255	−	0.988647i	$-0.548010\pi$
$491$	−7.88216e8	−0.300511	−0.150255	+	0.988647i	$0.548010\pi$
$492$	0	0
$493$	1.75824e9	0.660866
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.03088e9	2.93438
$498$	0	0
$499$	4.30040e9	1.54938	0.774688	−	0.632344i	$-0.217907\pi$
$499$	4.30040e9	1.54938	0.774688	+	0.632344i	$0.217907\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.47920e9	−1.91968	−0.959840	−	0.280547i	$-0.909484\pi$
$503$	−5.47920e9	−1.91968	−0.959840	+	0.280547i	$0.909484\pi$
$504$	0	0
$505$	1.55134e9	0.536027
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.25720e9	0.422562	0.211281	−	0.977425i	$-0.432236\pi$
$509$	1.25720e9	0.422562	0.211281	+	0.977425i	$0.432236\pi$
$510$	0	0
$511$	−5.84364e8	−0.193736
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.45600e9	1.76015
$516$	0	0
$517$	3.03862e9	0.967073
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.69281e8	−0.269295	−0.134648	−	0.990894i	$-0.542990\pi$
$521$	−8.69281e8	−0.269295	−0.134648	+	0.990894i	$0.542990\pi$
$522$	0	0
$523$	8.30165e7	0.0253752	0.0126876	−	0.999920i	$-0.495961\pi$
$523$	8.30165e7	0.0253752	0.0126876	+	0.999920i	$0.495961\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.18798e8	0.154405
$528$	0	0
$529$	−1.29012e9	−0.378909
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.99845e9	1.14379
$534$	0	0
$535$	−7.78424e9	−2.19775
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.05535e9	2.49083
$540$	0	0
$541$	9.05014e8	0.245734	0.122867	−	0.992423i	$-0.460791\pi$
$541$	9.05014e8	0.245734	0.122867	+	0.992423i	$0.460791\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.92367e8	−0.0773643
$546$	0	0
$547$	−2.61326e9	−0.682696	−0.341348	−	0.939937i	$-0.610884\pi$
$547$	−2.61326e9	−0.682696	−0.341348	+	0.939937i	$0.610884\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.04767e9	0.266806
$552$	0	0
$553$	5.65959e8	0.142314
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.06824e9	−1.97827	−0.989136	−	0.147006i	$-0.953036\pi$
$557$	−8.06824e9	−1.97827	−0.989136	+	0.147006i	$0.953036\pi$
$558$	0	0
$559$	1.64912e9	0.399311
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.55851e9	−1.31274	−0.656371	−	0.754439i	$-0.727909\pi$
$563$	−5.55851e9	−1.31274	−0.656371	+	0.754439i	$0.727909\pi$
$564$	0	0
$565$	6.98519e9	1.62933
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.82207e9	−0.642207	−0.321104	−	0.947044i	$-0.604054\pi$
$569$	−2.82207e9	−0.642207	−0.321104	+	0.947044i	$0.604054\pi$
$570$	0	0
$571$	−6.20632e9	−1.39511	−0.697554	−	0.716533i	$-0.745728\pi$
$571$	−6.20632e9	−1.39511	−0.697554	+	0.716533i	$0.745728\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.21143e9	−1.14319
$576$	0	0
$577$	−5.43205e9	−1.17720	−0.588598	−	0.808426i	$-0.700320\pi$
$577$	−5.43205e9	−1.17720	−0.588598	+	0.808426i	$0.700320\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.05876e9	−0.647038
$582$	0	0
$583$	−5.89527e9	−1.23215
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.08447e9	−0.629431	−0.314715	−	0.949186i	$-0.601909\pi$
$587$	−3.08447e9	−0.629431	−0.314715	+	0.949186i	$0.601909\pi$
$588$	0	0
$589$	3.09134e8	0.0623365
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.83257e8	−0.134553	−0.0672764	−	0.997734i	$-0.521431\pi$
$593$	−6.83257e8	−0.134553	−0.0672764	+	0.997734i	$0.521431\pi$
$594$	0	0
$595$	2.57457e10	5.01066
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.24846e9	0.997787	0.498893	−	0.866663i	$-0.333740\pi$
$599$	5.24846e9	0.997787	0.498893	+	0.866663i	$0.333740\pi$
$600$	0	0
$601$	−4.00032e9	−0.751682	−0.375841	−	0.926684i	$-0.622646\pi$
$601$	−4.00032e9	−0.751682	−0.375841	+	0.926684i	$0.622646\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.10075e8	0.130365
$606$	0	0
$607$	−2.52401e9	−0.458069	−0.229034	−	0.973418i	$-0.573557\pi$
$607$	−2.52401e9	−0.458069	−0.229034	+	0.973418i	$0.573557\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.77281e9	−0.846506
$612$	0	0
$613$	−1.17565e9	−0.206142	−0.103071	−	0.994674i	$-0.532867\pi$
$613$	−1.17565e9	−0.206142	−0.103071	+	0.994674i	$0.532867\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.05146e9	−1.03720	−0.518599	−	0.855017i	$-0.673546\pi$
$617$	−6.05146e9	−1.03720	−0.518599	+	0.855017i	$0.673546\pi$
$618$	0	0
$619$	9.38449e9	1.59035	0.795176	−	0.606378i	$-0.207378\pi$
$619$	9.38449e9	1.59035	0.795176	+	0.606378i	$0.207378\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.00291e9	0.166171
$624$	0	0
$625$	−2.11424e9	−0.346396
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.31397e10	2.10527
$630$	0	0
$631$	−4.21509e9	−0.667889	−0.333945	−	0.942593i	$-0.608380\pi$
$631$	−4.21509e9	−0.667889	−0.333945	+	0.942593i	$0.608380\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.73810e10	−2.69381
$636$	0	0
$637$	−1.42234e10	−2.18029
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.96210e9	−1.49399	−0.746995	−	0.664829i	$-0.768504\pi$
$641$	−9.96210e9	−1.49399	−0.746995	+	0.664829i	$0.768504\pi$
$642$	0	0
$643$	−6.98296e9	−1.03586	−0.517930	−	0.855423i	$-0.673297\pi$
$643$	−6.98296e9	−1.03586	−0.517930	+	0.855423i	$0.673297\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.65100e9	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$647$	−7.65100e9	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$648$	0	0
$649$	−5.05983e9	−0.726573
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.57145e9	−1.34518	−0.672592	−	0.740014i	$-0.734819\pi$
$653$	−9.57145e9	−1.34518	−0.672592	+	0.740014i	$0.734819\pi$
$654$	0	0
$655$	2.65962e9	0.369807
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.13794e9	1.10768	0.553842	−	0.832622i	$-0.313161\pi$
$659$	8.13794e9	1.10768	0.553842	+	0.832622i	$0.313161\pi$
$660$	0	0
$661$	9.82041e9	1.32259	0.661294	−	0.750127i	$-0.270008\pi$
$661$	9.82041e9	1.32259	0.661294	+	0.750127i	$0.270008\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.53409e10	2.02291
$666$	0	0
$667$	2.36654e9	0.308797
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.23089e9	0.412851
$672$	0	0
$673$	3.93826e9	0.498026	0.249013	−	0.968500i	$-0.419894\pi$
$673$	3.93826e9	0.498026	0.249013	+	0.968500i	$0.419894\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.66884e9	−0.949882	−0.474941	−	0.880018i	$-0.657531\pi$
$677$	−7.66884e9	−0.949882	−0.474941	+	0.880018i	$0.657531\pi$
$678$	0	0
$679$	−5.66008e9	−0.693870
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.65167e9	0.918933	0.459466	−	0.888195i	$-0.348041\pi$
$683$	7.65167e9	0.918933	0.459466	+	0.888195i	$0.348041\pi$
$684$	0	0
$685$	−1.25792e10	−1.49533
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.25981e9	1.07854
$690$	0	0
$691$	1.58561e10	1.82820	0.914098	−	0.405493i	$-0.132900\pi$
$691$	1.58561e10	1.82820	0.914098	+	0.405493i	$0.132900\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.03639e10	2.30098
$696$	0	0
$697$	2.05773e10	2.30183
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.61910e9	0.616104	0.308052	−	0.951370i	$-0.400323\pi$
$701$	5.61910e9	0.616104	0.308052	+	0.951370i	$0.400323\pi$
$702$	0	0
$703$	7.82946e9	0.849940
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.10608e9	−0.649822
$708$	0	0
$709$	−1.34254e10	−1.41471	−0.707353	−	0.706860i	$-0.750111\pi$
$709$	−1.34254e10	−1.41471	−0.707353	+	0.706860i	$0.750111\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.98288e8	0.0721475
$714$	0	0
$715$	1.22776e10	1.25616
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.63313e10	1.63858	0.819292	−	0.573377i	$-0.194367\pi$
$719$	1.63313e10	1.63858	0.819292	+	0.573377i	$0.194367\pi$
$720$	0	0
$721$	−2.14749e10	−2.13382
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.83205e9	−0.568379
$726$	0	0
$727$	−4.33823e9	−0.418738	−0.209369	−	0.977837i	$-0.567141\pi$
$727$	−4.33823e9	−0.418738	−0.209369	+	0.977837i	$0.567141\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.48692e9	0.803599
$732$	0	0
$733$	−7.43119e9	−0.696939	−0.348469	−	0.937320i	$-0.613298\pi$
$733$	−7.43119e9	−0.696939	−0.348469	+	0.937320i	$0.613298\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.07410e10	−0.988345
$738$	0	0
$739$	−5.03186e9	−0.458641	−0.229321	−	0.973351i	$-0.573650\pi$
$739$	−5.03186e9	−0.458641	−0.229321	+	0.973351i	$0.573650\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.63121e10	−1.45898	−0.729491	−	0.683990i	$-0.760243\pi$
$743$	−1.63121e10	−1.45898	−0.729491	+	0.683990i	$0.760243\pi$
$744$	0	0
$745$	1.35923e10	1.20433
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.06388e10	2.66432
$750$	0	0
$751$	−2.07632e10	−1.78877	−0.894385	−	0.447299i	$-0.852386\pi$
$751$	−2.07632e10	−1.78877	−0.894385	+	0.447299i	$0.852386\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.22975e9	−0.526813
$756$	0	0
$757$	3.85677e9	0.323138	0.161569	−	0.986861i	$-0.448345\pi$
$757$	3.85677e9	0.323138	0.161569	+	0.986861i	$0.448345\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.80368e9	0.148358	0.0741792	−	0.997245i	$-0.476366\pi$
$761$	1.80368e9	0.148358	0.0741792	+	0.997245i	$0.476366\pi$
$762$	0	0
$763$	1.15076e9	0.0937882
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.94756e9	0.635989
$768$	0	0
$769$	3.33349e9	0.264336	0.132168	−	0.991227i	$-0.457806\pi$
$769$	3.33349e9	0.264336	0.132168	+	0.991227i	$0.457806\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.11920e10	−0.871526	−0.435763	−	0.900062i	$-0.643521\pi$
$773$	−1.11920e10	−0.871526	−0.435763	+	0.900062i	$0.643521\pi$
$774$	0	0
$775$	−1.72084e9	−0.132796
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.22613e10	0.929299
$780$	0	0
$781$	1.97093e10	1.48045
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.21985e10	2.37570
$786$	0	0
$787$	−2.19525e9	−0.160536	−0.0802679	−	0.996773i	$-0.525578\pi$
$787$	−2.19525e9	−0.160536	−0.0802679	+	0.996773i	$0.525578\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.74938e10	−1.97522
$792$	0	0
$793$	−5.07481e9	−0.361380
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.43541e9	0.380302	0.190151	−	0.981755i	$-0.439102\pi$
$797$	5.43541e9	0.380302	0.190151	+	0.981755i	$0.439102\pi$
$798$	0	0
$799$	−2.45625e10	−1.70356
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.43414e9	−0.0977433
$804$	0	0
$805$	3.46529e10	2.34129
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9.89544e9	−0.657075	−0.328538	−	0.944491i	$-0.606556\pi$
$809$	−9.89544e9	−0.657075	−0.328538	+	0.944491i	$0.606556\pi$
$810$	0	0
$811$	1.00426e10	0.661111	0.330556	−	0.943787i	$-0.392764\pi$
$811$	1.00426e10	0.661111	0.330556	+	0.943787i	$0.392764\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.54823e10	−2.29594
$816$	0	0
$817$	5.05706e9	0.324430
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.07209e10	−1.93746	−0.968731	−	0.248112i	$-0.920190\pi$
$821$	−3.07209e10	−1.93746	−0.968731	+	0.248112i	$0.920190\pi$
$822$	0	0
$823$	2.01255e10	1.25848	0.629241	−	0.777210i	$-0.283366\pi$
$823$	2.01255e10	1.25848	0.629241	+	0.777210i	$0.283366\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.55472e10	0.955832	0.477916	−	0.878405i	$-0.341392\pi$
$827$	1.55472e10	0.955832	0.477916	+	0.878405i	$0.341392\pi$
$828$	0	0
$829$	1.42396e10	0.868077	0.434038	−	0.900894i	$-0.357088\pi$
$829$	1.42396e10	0.868077	0.434038	+	0.900894i	$0.357088\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.31982e10	−4.38776
$834$	0	0
$835$	−4.89088e10	−2.90726
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.17959e9	0.185868	0.0929338	−	0.995672i	$-0.470375\pi$
$839$	3.17959e9	0.185868	0.0929338	+	0.995672i	$0.470375\pi$
$840$	0	0
$841$	−1.46015e10	−0.846470
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.17104e9	0.465885
$846$	0	0
$847$	−2.79486e9	−0.158040
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.76856e10	0.983709
$852$	0	0
$853$	−4.62326e9	−0.255051	−0.127525	−	0.991835i	$-0.540703\pi$
$853$	−4.62326e9	−0.255051	−0.127525	+	0.991835i	$0.540703\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.13062e10	1.15631	0.578153	−	0.815929i	$-0.303774\pi$
$857$	2.13062e10	1.15631	0.578153	+	0.815929i	$0.303774\pi$
$858$	0	0
$859$	−7.86988e9	−0.423635	−0.211818	−	0.977309i	$-0.567938\pi$
$859$	−7.86988e9	−0.423635	−0.211818	+	0.977309i	$0.567938\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.55483e9	−0.0823465	−0.0411732	−	0.999152i	$-0.513110\pi$
$863$	−1.55483e9	−0.0823465	−0.0411732	+	0.999152i	$0.513110\pi$
$864$	0	0
$865$	−3.92829e10	−2.06370
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.38897e9	0.0717999
$870$	0	0
$871$	1.68711e10	0.865126
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.65264e10	−1.33860
$876$	0	0
$877$	−2.34828e10	−1.17558	−0.587790	−	0.809014i	$-0.700002\pi$
$877$	−2.34828e10	−1.17558	−0.587790	+	0.809014i	$0.700002\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.10488e9	0.0544378	0.0272189	−	0.999629i	$-0.491335\pi$
$881$	1.10488e9	0.0544378	0.0272189	+	0.999629i	$0.491335\pi$
$882$	0	0
$883$	1.52780e10	0.746800	0.373400	−	0.927670i	$-0.378192\pi$
$883$	1.52780e10	0.746800	0.373400	+	0.927670i	$0.378192\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.13521e10	0.546189	0.273094	−	0.961987i	$-0.411953\pi$
$887$	1.13521e10	0.546189	0.273094	+	0.961987i	$0.411953\pi$
$888$	0	0
$889$	6.84117e10	3.26568
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.46359e10	−0.687764
$894$	0	0
$895$	−2.03754e10	−0.950002
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.81445e8	0.0358707
$900$	0	0
$901$	4.76540e10	2.17052
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.94694e10	1.32160
$906$	0	0
$907$	−2.41214e9	−0.107344	−0.0536718	−	0.998559i	$-0.517092\pi$
$907$	−2.41214e9	−0.107344	−0.0536718	+	0.998559i	$0.517092\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.63158e10	−1.15319	−0.576597	−	0.817029i	$-0.695620\pi$
$911$	−2.63158e10	−1.15319	−0.576597	+	0.817029i	$0.695620\pi$
$912$	0	0
$913$	−7.50679e9	−0.326442
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.04683e10	−0.448314
$918$	0	0
$919$	2.35629e10	1.00144	0.500719	−	0.865610i	$-0.333069\pi$
$919$	2.35629e10	1.00144	0.500719	+	0.865610i	$0.333069\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.09577e10	−1.29588
$924$	0	0
$925$	−4.35840e10	−1.81064
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.08846e9	0.0854618	0.0427309	−	0.999087i	$-0.486394\pi$
$929$	2.08846e9	0.0854618	0.0427309	+	0.999087i	$0.486394\pi$
$930$	0	0
$931$	−4.36163e10	−1.77143
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.31847e10	2.52797
$936$	0	0
$937$	3.67845e10	1.46075	0.730376	−	0.683045i	$-0.239345\pi$
$937$	3.67845e10	1.46075	0.730376	+	0.683045i	$0.239345\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.19271e9	0.359650	0.179825	−	0.983699i	$-0.442447\pi$
$941$	9.19271e9	0.359650	0.179825	+	0.983699i	$0.442447\pi$
$942$	0	0
$943$	2.76965e10	1.07556
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.30566e10	−0.499581	−0.249791	−	0.968300i	$-0.580362\pi$
$947$	−1.30566e10	−0.499581	−0.249791	+	0.968300i	$0.580362\pi$
$948$	0	0
$949$	2.25263e9	0.0855574
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.42599e10	0.907954	0.453977	−	0.891013i	$-0.350005\pi$
$953$	2.42599e10	0.907954	0.453977	+	0.891013i	$0.350005\pi$
$954$	0	0
$955$	−1.29322e10	−0.480462
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.95118e10	1.81278
$960$	0	0
$961$	−2.72820e10	−0.991619
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.41476e10	1.93970
$966$	0	0
$967$	1.37434e10	0.488766	0.244383	−	0.969679i	$-0.421415\pi$
$967$	1.37434e10	0.488766	0.244383	+	0.969679i	$0.421415\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.37656e9	−0.223521	−0.111761	−	0.993735i	$-0.535649\pi$
$971$	−6.37656e9	−0.223521	−0.111761	+	0.993735i	$0.535649\pi$
$972$	0	0
$973$	−8.01523e10	−2.78947
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.35347e10	1.49350	0.746748	−	0.665107i	$-0.231614\pi$
$977$	4.35347e10	1.49350	0.746748	+	0.665107i	$0.231614\pi$
$978$	0	0
$979$	2.46133e9	0.0838361
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.03387e10	−0.682946	−0.341473	−	0.939892i	$-0.610926\pi$
$983$	−2.03387e10	−0.682946	−0.341473	+	0.939892i	$0.610926\pi$
$984$	0	0
$985$	−1.00092e10	−0.333714
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.14232e10	0.375491
$990$	0	0
$991$	5.03245e10	1.64256	0.821280	−	0.570525i	$-0.193260\pi$
$991$	5.03245e10	1.64256	0.821280	+	0.570525i	$0.193260\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.80304e10	1.22391
$996$	0	0
$997$	−2.65066e10	−0.847073	−0.423537	−	0.905879i	$-0.639211\pi$
$997$	−2.65066e10	−0.847073	−0.423537	+	0.905879i	$0.639211\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.a.f.1.1	✓	2
3.2	odd	2		72.8.a.g.1.2	yes	2
4.3	odd	2		144.8.a.l.1.1		2
8.3	odd	2		576.8.a.bp.1.2		2
8.5	even	2		576.8.a.bq.1.2		2
12.11	even	2		144.8.a.n.1.2		2
24.5	odd	2		576.8.a.bd.1.1		2
24.11	even	2		576.8.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.a.f.1.1	✓	2	1.1	even	1	trivial
72.8.a.g.1.2	yes	2	3.2	odd	2
144.8.a.l.1.1		2	4.3	odd	2
144.8.a.n.1.2		2	12.11	even	2
576.8.a.bc.1.1		2	24.11	even	2
576.8.a.bd.1.1		2	24.5	odd	2
576.8.a.bp.1.2		2	8.3	odd	2
576.8.a.bq.1.2		2	8.5	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

\(f(q)\)	\(=\)	\(q-437.552 q^{5} +1722.21 q^{7} +O(q^{10})\)	\(q-437.552 q^{5} +1722.21 q^{7} +4226.62 q^{11} -6638.83 q^{13} -34165.6 q^{17} -20358.1 q^{19} -45985.9 q^{23} +113327. q^{25} -51462.3 q^{29} -15184.8 q^{31} -753555. q^{35} -384588. q^{37} -602282. q^{41} -248405. q^{43} +718924. q^{47} +2.14246e6 q^{49} -1.39480e6 q^{53} -1.84937e6 q^{55} -1.19713e6 q^{59} +764413. q^{61} +2.90483e6 q^{65} -2.54127e6 q^{67} +4.66313e6 q^{71} -339311. q^{73} +7.27912e6 q^{77} +328624. q^{79} -1.77607e6 q^{83} +1.49492e7 q^{85} +582340. q^{89} -1.14334e7 q^{91} +8.90771e6 q^{95} -3.28652e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 224 q^{5} + 840 q^{7}+O(q^{10}) \) 2 * q - 224 * q^5 + 840 * q^7	\( 2 q - 224 q^{5} + 840 q^{7} + 640 q^{11} - 2860 q^{13} - 35776 q^{17} - 14672 q^{19} - 118016 q^{23} + 80806 q^{25} - 306720 q^{29} - 95480 q^{31} - 941952 q^{35} + 53820 q^{37} - 1143360 q^{41} - 679120 q^{43} - 567552 q^{47} + 2097202 q^{49} - 1165088 q^{53} - 2615296 q^{55} + 121600 q^{59} + 3143564 q^{61} + 3711808 q^{65} - 5468000 q^{67} + 6763520 q^{71} + 5009420 q^{73} + 10443264 q^{77} - 5450104 q^{79} + 6071168 q^{83} + 14605312 q^{85} + 7063680 q^{89} - 14767152 q^{91} + 10121984 q^{95} + 5219740 q^{97}+O(q^{100}) \) 2 * q - 224 * q^5 + 840 * q^7 + 640 * q^11 - 2860 * q^13 - 35776 * q^17 - 14672 * q^19 - 118016 * q^23 + 80806 * q^25 - 306720 * q^29 - 95480 * q^31 - 941952 * q^35 + 53820 * q^37 - 1143360 * q^41 - 679120 * q^43 - 567552 * q^47 + 2097202 * q^49 - 1165088 * q^53 - 2615296 * q^55 + 121600 * q^59 + 3143564 * q^61 + 3711808 * q^65 - 5468000 * q^67 + 6763520 * q^71 + 5009420 * q^73 + 10443264 * q^77 - 5450104 * q^79 + 6071168 * q^83 + 14605312 * q^85 + 7063680 * q^89 - 14767152 * q^91 + 10121984 * q^95 + 5219740 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more