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Properties

Label 315.8.a.h.1.1
Level $315$
Weight $8$
Character 315.1
Self dual yes
Analytic conductor $98.401$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,8,Mod(1,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(98.4012830275\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 267x^{2} + 1358x + 2744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(12.2539\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-14.5922 q^{2} +84.9312 q^{4} +125.000 q^{5} -343.000 q^{7} +628.467 q^{8} -1824.02 q^{10} +8206.98 q^{11} +2452.76 q^{13} +5005.11 q^{14} -20041.9 q^{16} +22121.0 q^{17} +38239.8 q^{19} +10616.4 q^{20} -119758. q^{22} +48920.4 q^{23} +15625.0 q^{25} -35791.1 q^{26} -29131.4 q^{28} +88120.2 q^{29} -21710.1 q^{31} +212011. q^{32} -322793. q^{34} -42875.0 q^{35} +492908. q^{37} -558001. q^{38} +78558.4 q^{40} +639208. q^{41} -191535. q^{43} +697028. q^{44} -713855. q^{46} -163494. q^{47} +117649. q^{49} -228003. q^{50} +208316. q^{52} -830698. q^{53} +1.02587e6 q^{55} -215564. q^{56} -1.28586e6 q^{58} +995483. q^{59} +1.02098e6 q^{61} +316798. q^{62} -528332. q^{64} +306595. q^{65} -4.40173e6 q^{67} +1.87876e6 q^{68} +625639. q^{70} -4.62049e6 q^{71} -1.34218e6 q^{73} -7.19259e6 q^{74} +3.24775e6 q^{76} -2.81499e6 q^{77} -3.62717e6 q^{79} -2.50524e6 q^{80} -9.32742e6 q^{82} +1.63270e6 q^{83} +2.76512e6 q^{85} +2.79491e6 q^{86} +5.15782e6 q^{88} -3.83917e6 q^{89} -841296. q^{91} +4.15487e6 q^{92} +2.38573e6 q^{94} +4.77998e6 q^{95} -2.79847e6 q^{97} -1.71675e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 106 q^{4} + 500 q^{5} - 1372 q^{7} - 12 q^{8} - 500 q^{10} - 3032 q^{11} + 952 q^{13} + 1372 q^{14} - 58494 q^{16} + 50464 q^{17} - 15672 q^{19} + 13250 q^{20} - 40964 q^{22} + 89656 q^{23}+ \cdots - 470596 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.5922 −1.28978 −0.644889 − 0.764277i \(-0.723096\pi\)
−0.644889 + 0.764277i \(0.723096\pi\)
\(3\) 0 0
\(4\) 84.9312 0.663525
\(5\) 125.000 0.447214
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 628.467 0.433978
\(9\) 0 0
\(10\) −1824.02 −0.576806
\(11\) 8206.98 1.85913 0.929563 − 0.368663i \(-0.120184\pi\)
0.929563 + 0.368663i \(0.120184\pi\)
\(12\) 0 0
\(13\) 2452.76 0.309637 0.154819 − 0.987943i \(-0.450521\pi\)
0.154819 + 0.987943i \(0.450521\pi\)
\(14\) 5005.11 0.487490
\(15\) 0 0
\(16\) −20041.9 −1.22326
\(17\) 22121.0 1.09203 0.546013 − 0.837777i \(-0.316145\pi\)
0.546013 + 0.837777i \(0.316145\pi\)
\(18\) 0 0
\(19\) 38239.8 1.27902 0.639511 − 0.768782i \(-0.279137\pi\)
0.639511 + 0.768782i \(0.279137\pi\)
\(20\) 10616.4 0.296737
\(21\) 0 0
\(22\) −119758. −2.39786
\(23\) 48920.4 0.838384 0.419192 − 0.907898i \(-0.362313\pi\)
0.419192 + 0.907898i \(0.362313\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) −35791.1 −0.399363
\(27\) 0 0
\(28\) −29131.4 −0.250789
\(29\) 88120.2 0.670938 0.335469 − 0.942051i \(-0.391105\pi\)
0.335469 + 0.942051i \(0.391105\pi\)
\(30\) 0 0
\(31\) −21710.1 −0.130887 −0.0654435 − 0.997856i \(-0.520846\pi\)
−0.0654435 + 0.997856i \(0.520846\pi\)
\(32\) 212011. 1.14375
\(33\) 0 0
\(34\) −322793. −1.40847
\(35\) −42875.0 −0.169031
\(36\) 0 0
\(37\) 492908. 1.59978 0.799889 − 0.600148i \(-0.204892\pi\)
0.799889 + 0.600148i \(0.204892\pi\)
\(38\) −558001. −1.64965
\(39\) 0 0
\(40\) 78558.4 0.194081
\(41\) 639208. 1.44843 0.724217 − 0.689573i \(-0.242202\pi\)
0.724217 + 0.689573i \(0.242202\pi\)
\(42\) 0 0
\(43\) −191535. −0.367374 −0.183687 − 0.982985i \(-0.558803\pi\)
−0.183687 + 0.982985i \(0.558803\pi\)
\(44\) 697028. 1.23358
\(45\) 0 0
\(46\) −713855. −1.08133
\(47\) −163494. −0.229699 −0.114850 − 0.993383i \(-0.536639\pi\)
−0.114850 + 0.993383i \(0.536639\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) −228003. −0.257955
\(51\) 0 0
\(52\) 208316. 0.205452
\(53\) −830698. −0.766439 −0.383219 − 0.923657i \(-0.625185\pi\)
−0.383219 + 0.923657i \(0.625185\pi\)
\(54\) 0 0
\(55\) 1.02587e6 0.831426
\(56\) −215564. −0.164028
\(57\) 0 0
\(58\) −1.28586e6 −0.865361
\(59\) 995483. 0.631032 0.315516 − 0.948920i \(-0.397822\pi\)
0.315516 + 0.948920i \(0.397822\pi\)
\(60\) 0 0
\(61\) 1.02098e6 0.575918 0.287959 − 0.957643i \(-0.407023\pi\)
0.287959 + 0.957643i \(0.407023\pi\)
\(62\) 316798. 0.168815
\(63\) 0 0
\(64\) −528332. −0.251928
\(65\) 306595. 0.138474
\(66\) 0 0
\(67\) −4.40173e6 −1.78797 −0.893987 − 0.448092i \(-0.852104\pi\)
−0.893987 + 0.448092i \(0.852104\pi\)
\(68\) 1.87876e6 0.724586
\(69\) 0 0
\(70\) 625639. 0.218012
\(71\) −4.62049e6 −1.53209 −0.766044 − 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(72\) 0 0
\(73\) −1.34218e6 −0.403812 −0.201906 − 0.979405i \(-0.564714\pi\)
−0.201906 + 0.979405i \(0.564714\pi\)
\(74\) −7.19259e6 −2.06336
\(75\) 0 0
\(76\) 3.24775e6 0.848663
\(77\) −2.81499e6 −0.702684
\(78\) 0 0
\(79\) −3.62717e6 −0.827701 −0.413851 − 0.910345i \(-0.635816\pi\)
−0.413851 + 0.910345i \(0.635816\pi\)
\(80\) −2.50524e6 −0.547058
\(81\) 0 0
\(82\) −9.32742e6 −1.86816
\(83\) 1.63270e6 0.313424 0.156712 − 0.987644i \(-0.449911\pi\)
0.156712 + 0.987644i \(0.449911\pi\)
\(84\) 0 0
\(85\) 2.76512e6 0.488369
\(86\) 2.79491e6 0.473830
\(87\) 0 0
\(88\) 5.15782e6 0.806820
\(89\) −3.83917e6 −0.577260 −0.288630 − 0.957441i \(-0.593200\pi\)
−0.288630 + 0.957441i \(0.593200\pi\)
\(90\) 0 0
\(91\) −841296. −0.117032
\(92\) 4.15487e6 0.556288
\(93\) 0 0
\(94\) 2.38573e6 0.296261
\(95\) 4.77998e6 0.571996
\(96\) 0 0
\(97\) −2.79847e6 −0.311329 −0.155664 − 0.987810i \(-0.549752\pi\)
−0.155664 + 0.987810i \(0.549752\pi\)
\(98\) −1.71675e6 −0.184254
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 315.8.a.h.1.1 4
3.2 odd 2 105.8.a.g.1.4 ✓ 4
15.14 odd 2 525.8.a.j.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.8.a.g.1.4 ✓ 4 3.2 odd 2
315.8.a.h.1.1 4 1.1 even 1 trivial
525.8.a.j.1.1 4 15.14 odd 2