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Properties

Label 315.8.a.h.1.3
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 1.3
Root \(-17.7998\) 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)
 
\(f(q)\) \(=\) \(q+7.38849 q^{2} -73.4103 q^{4} +125.000 q^{5} -343.000 q^{7} -1488.12 q^{8} +923.561 q^{10} -2561.67 q^{11} +3729.61 q^{13} -2534.25 q^{14} -1598.42 q^{16} -17887.7 q^{17} -24060.2 q^{19} -9176.28 q^{20} -18926.9 q^{22} +106178. q^{23} +15625.0 q^{25} +27556.2 q^{26} +25179.7 q^{28} -247987. q^{29} -184899. q^{31} +178669. q^{32} -132163. q^{34} -42875.0 q^{35} +482569. q^{37} -177768. q^{38} -186015. q^{40} +463084. q^{41} -696667. q^{43} +188053. q^{44} +784497. q^{46} +1.25048e6 q^{47} +117649. q^{49} +115445. q^{50} -273792. q^{52} +1.13351e6 q^{53} -320209. q^{55} +510424. q^{56} -1.83225e6 q^{58} +31563.3 q^{59} -2.13264e6 q^{61} -1.36612e6 q^{62} +1.52469e6 q^{64} +466201. q^{65} -3.28284e6 q^{67} +1.31314e6 q^{68} -316781. q^{70} +2.32209e6 q^{71} +2.22542e6 q^{73} +3.56546e6 q^{74} +1.76626e6 q^{76} +878654. q^{77} +2.48636e6 q^{79} -199803. q^{80} +3.42149e6 q^{82} +5.21504e6 q^{83} -2.23597e6 q^{85} -5.14731e6 q^{86} +3.81207e6 q^{88} +271854. q^{89} -1.27926e6 q^{91} -7.79458e6 q^{92} +9.23916e6 q^{94} -3.00752e6 q^{95} +1.48651e7 q^{97} +869248. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 106 q^{4} + 500 q^{5} - 1372 q^{7} - 12 q^{8} - 500 q^{10} - 3032 q^{11} + 952 q^{13} + 1372 q^{14} - 58494 q^{16} + 50464 q^{17} - 15672 q^{19} + 13250 q^{20} - 40964 q^{22} + 89656 q^{23}+ \cdots - 470596 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.38849 0.653056 0.326528 − 0.945187i \(-0.394121\pi\)
0.326528 + 0.945187i \(0.394121\pi\)
\(3\) 0 0
\(4\) −73.4103 −0.573518
\(5\) 125.000 0.447214
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) −1488.12 −1.02760
\(9\) 0 0
\(10\) 923.561 0.292056
\(11\) −2561.67 −0.580296 −0.290148 − 0.956982i \(-0.593705\pi\)
−0.290148 + 0.956982i \(0.593705\pi\)
\(12\) 0 0
\(13\) 3729.61 0.470827 0.235414 − 0.971895i \(-0.424355\pi\)
0.235414 + 0.971895i \(0.424355\pi\)
\(14\) −2534.25 −0.246832
\(15\) 0 0
\(16\) −1598.42 −0.0975600
\(17\) −17887.7 −0.883047 −0.441524 − 0.897250i \(-0.645562\pi\)
−0.441524 + 0.897250i \(0.645562\pi\)
\(18\) 0 0
\(19\) −24060.2 −0.804750 −0.402375 − 0.915475i \(-0.631815\pi\)
−0.402375 + 0.915475i \(0.631815\pi\)
\(20\) −9176.28 −0.256485
\(21\) 0 0
\(22\) −18926.9 −0.378966
\(23\) 106178. 1.81965 0.909826 − 0.414989i \(-0.136215\pi\)
0.909826 + 0.414989i \(0.136215\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 27556.2 0.307477
\(27\) 0 0
\(28\) 25179.7 0.216769
\(29\) −247987. −1.88815 −0.944073 − 0.329736i \(-0.893040\pi\)
−0.944073 + 0.329736i \(0.893040\pi\)
\(30\) 0 0
\(31\) −184899. −1.11473 −0.557363 − 0.830269i \(-0.688187\pi\)
−0.557363 + 0.830269i \(0.688187\pi\)
\(32\) 178669. 0.963883
\(33\) 0 0
\(34\) −132163. −0.576679
\(35\) −42875.0 −0.169031
\(36\) 0 0
\(37\) 482569. 1.56622 0.783112 − 0.621881i \(-0.213631\pi\)
0.783112 + 0.621881i \(0.213631\pi\)
\(38\) −177768. −0.525547
\(39\) 0 0
\(40\) −186015. −0.459555
\(41\) 463084. 1.04934 0.524670 − 0.851306i \(-0.324189\pi\)
0.524670 + 0.851306i \(0.324189\pi\)
\(42\) 0 0
\(43\) −696667. −1.33624 −0.668122 − 0.744052i \(-0.732901\pi\)
−0.668122 + 0.744052i \(0.732901\pi\)
\(44\) 188053. 0.332810
\(45\) 0 0
\(46\) 784497. 1.18834
\(47\) 1.25048e6 1.75685 0.878424 − 0.477882i \(-0.158595\pi\)
0.878424 + 0.477882i \(0.158595\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 115445. 0.130611
\(51\) 0 0
\(52\) −273792. −0.270028
\(53\) 1.13351e6 1.04583 0.522915 − 0.852385i \(-0.324845\pi\)
0.522915 + 0.852385i \(0.324845\pi\)
\(54\) 0 0
\(55\) −320209. −0.259516
\(56\) 510424. 0.388395
\(57\) 0 0
\(58\) −1.83225e6 −1.23307
\(59\) 31563.3 0.0200078 0.0100039 − 0.999950i \(-0.496816\pi\)
0.0100039 + 0.999950i \(0.496816\pi\)
\(60\) 0 0
\(61\) −2.13264e6 −1.20299 −0.601497 − 0.798875i \(-0.705429\pi\)
−0.601497 + 0.798875i \(0.705429\pi\)
\(62\) −1.36612e6 −0.727979
\(63\) 0 0
\(64\) 1.52469e6 0.727030
\(65\) 466201. 0.210560
\(66\) 0 0
\(67\) −3.28284e6 −1.33349 −0.666743 − 0.745288i \(-0.732312\pi\)
−0.666743 + 0.745288i \(0.732312\pi\)
\(68\) 1.31314e6 0.506443
\(69\) 0 0
\(70\) −316781. −0.110387
\(71\) 2.32209e6 0.769971 0.384986 − 0.922923i \(-0.374206\pi\)
0.384986 + 0.922923i \(0.374206\pi\)
\(72\) 0 0
\(73\) 2.22542e6 0.669547 0.334774 − 0.942299i \(-0.391340\pi\)
0.334774 + 0.942299i \(0.391340\pi\)
\(74\) 3.56546e6 1.02283
\(75\) 0 0
\(76\) 1.76626e6 0.461539
\(77\) 878654. 0.219331
\(78\) 0 0
\(79\) 2.48636e6 0.567374 0.283687 − 0.958917i \(-0.408442\pi\)
0.283687 + 0.958917i \(0.408442\pi\)
\(80\) −199803. −0.0436302
\(81\) 0 0
\(82\) 3.42149e6 0.685278
\(83\) 5.21504e6 1.00112 0.500558 − 0.865703i \(-0.333128\pi\)
0.500558 + 0.865703i \(0.333128\pi\)
\(84\) 0 0
\(85\) −2.23597e6 −0.394911
\(86\) −5.14731e6 −0.872642
\(87\) 0 0
\(88\) 3.81207e6 0.596309
\(89\) 271854. 0.0408761 0.0204381 − 0.999791i \(-0.493494\pi\)
0.0204381 + 0.999791i \(0.493494\pi\)
\(90\) 0 0
\(91\) −1.27926e6 −0.177956
\(92\) −7.79458e6 −1.04360
\(93\) 0 0
\(94\) 9.23916e6 1.14732
\(95\) −3.00752e6 −0.359895
\(96\) 0 0
\(97\) 1.48651e7 1.65374 0.826868 − 0.562396i \(-0.190120\pi\)
0.826868 + 0.562396i \(0.190120\pi\)
\(98\) 869248. 0.0932937
\(99\) 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.1.3 4
3.2 odd 2 105.8.a.g.1.2 ✓ 4
15.14 odd 2 525.8.a.j.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.8.a.g.1.2 ✓ 4 3.2 odd 2
315.8.a.h.1.3 4 1.1 even 1 trivial
525.8.a.j.1.3 4 15.14 odd 2