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Properties

Label 315.8.a.h.1.4
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 1.4
Root \(8.09919\) 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)
 
\(f(q)\) \(=\) \(q+14.7424 q^{2} +89.3375 q^{4} +125.000 q^{5} -343.000 q^{7} -569.977 q^{8} +1842.80 q^{10} -91.4856 q^{11} +2264.72 q^{13} -5056.63 q^{14} -19838.0 q^{16} +27072.2 q^{17} +28606.0 q^{19} +11167.2 q^{20} -1348.71 q^{22} -105060. q^{23} +15625.0 q^{25} +33387.3 q^{26} -30642.8 q^{28} +144497. q^{29} +243623. q^{31} -219502. q^{32} +399109. q^{34} -42875.0 q^{35} -379034. q^{37} +421721. q^{38} -71247.1 q^{40} +335219. q^{41} +665150. q^{43} -8173.10 q^{44} -1.54883e6 q^{46} +1.24521e6 q^{47} +117649. q^{49} +230350. q^{50} +202324. q^{52} +949002. q^{53} -11435.7 q^{55} +195502. q^{56} +2.13023e6 q^{58} +2.47644e6 q^{59} -2.23460e6 q^{61} +3.59158e6 q^{62} -696719. q^{64} +283090. q^{65} -5705.57 q^{67} +2.41856e6 q^{68} -632079. q^{70} +2.42993e6 q^{71} -3.78652e6 q^{73} -5.58787e6 q^{74} +2.55559e6 q^{76} +31379.6 q^{77} -4.40140e6 q^{79} -2.47975e6 q^{80} +4.94193e6 q^{82} -6.68955e6 q^{83} +3.38403e6 q^{85} +9.80589e6 q^{86} +52144.7 q^{88} +9.24865e6 q^{89} -776798. q^{91} -9.38579e6 q^{92} +1.83573e7 q^{94} +3.57575e6 q^{95} +1.12707e7 q^{97} +1.73443e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 106 q^{4} + 500 q^{5} - 1372 q^{7} - 12 q^{8} - 500 q^{10} - 3032 q^{11} + 952 q^{13} + 1372 q^{14} - 58494 q^{16} + 50464 q^{17} - 15672 q^{19} + 13250 q^{20} - 40964 q^{22} + 89656 q^{23}+ \cdots - 470596 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.7424 1.30305 0.651527 − 0.758625i \(-0.274129\pi\)
0.651527 + 0.758625i \(0.274129\pi\)
\(3\) 0 0
\(4\) 89.3375 0.697949
\(5\) 125.000 0.447214
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) −569.977 −0.393588
\(9\) 0 0
\(10\) 1842.80 0.582743
\(11\) −91.4856 −0.0207242 −0.0103621 − 0.999946i \(-0.503298\pi\)
−0.0103621 + 0.999946i \(0.503298\pi\)
\(12\) 0 0
\(13\) 2264.72 0.285899 0.142949 − 0.989730i \(-0.454341\pi\)
0.142949 + 0.989730i \(0.454341\pi\)
\(14\) −5056.63 −0.492508
\(15\) 0 0
\(16\) −19838.0 −1.21082
\(17\) 27072.2 1.33645 0.668225 − 0.743959i \(-0.267054\pi\)
0.668225 + 0.743959i \(0.267054\pi\)
\(18\) 0 0
\(19\) 28606.0 0.956797 0.478399 − 0.878143i \(-0.341217\pi\)
0.478399 + 0.878143i \(0.341217\pi\)
\(20\) 11167.2 0.312132
\(21\) 0 0
\(22\) −1348.71 −0.0270048
\(23\) −105060. −1.80049 −0.900243 − 0.435388i \(-0.856611\pi\)
−0.900243 + 0.435388i \(0.856611\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 33387.3 0.372542
\(27\) 0 0
\(28\) −30642.8 −0.263800
\(29\) 144497. 1.10019 0.550094 − 0.835103i \(-0.314592\pi\)
0.550094 + 0.835103i \(0.314592\pi\)
\(30\) 0 0
\(31\) 243623. 1.46876 0.734382 − 0.678737i \(-0.237472\pi\)
0.734382 + 0.678737i \(0.237472\pi\)
\(32\) −219502. −1.18417
\(33\) 0 0
\(34\) 399109. 1.74147
\(35\) −42875.0 −0.169031
\(36\) 0 0
\(37\) −379034. −1.23019 −0.615096 − 0.788452i \(-0.710883\pi\)
−0.615096 + 0.788452i \(0.710883\pi\)
\(38\) 421721. 1.24676
\(39\) 0 0
\(40\) −71247.1 −0.176018
\(41\) 335219. 0.759601 0.379801 − 0.925068i \(-0.375993\pi\)
0.379801 + 0.925068i \(0.375993\pi\)
\(42\) 0 0
\(43\) 665150. 1.27579 0.637897 − 0.770122i \(-0.279805\pi\)
0.637897 + 0.770122i \(0.279805\pi\)
\(44\) −8173.10 −0.0144645
\(45\) 0 0
\(46\) −1.54883e6 −2.34613
\(47\) 1.24521e6 1.74944 0.874721 − 0.484628i \(-0.161045\pi\)
0.874721 + 0.484628i \(0.161045\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 230350. 0.260611
\(51\) 0 0
\(52\) 202324. 0.199543
\(53\) 949002. 0.875591 0.437796 − 0.899075i \(-0.355759\pi\)
0.437796 + 0.899075i \(0.355759\pi\)
\(54\) 0 0
\(55\) −11435.7 −0.00926816
\(56\) 195502. 0.148762
\(57\) 0 0
\(58\) 2.13023e6 1.43360
\(59\) 2.47644e6 1.56980 0.784901 − 0.619621i \(-0.212714\pi\)
0.784901 + 0.619621i \(0.212714\pi\)
\(60\) 0 0
\(61\) −2.23460e6 −1.26051 −0.630254 − 0.776389i \(-0.717049\pi\)
−0.630254 + 0.776389i \(0.717049\pi\)
\(62\) 3.59158e6 1.91388
\(63\) 0 0
\(64\) −696719. −0.332222
\(65\) 283090. 0.127858
\(66\) 0 0
\(67\) −5705.57 −0.00231759 −0.00115880 − 0.999999i \(-0.500369\pi\)
−0.00115880 + 0.999999i \(0.500369\pi\)
\(68\) 2.41856e6 0.932774
\(69\) 0 0
\(70\) −632079. −0.220256
\(71\) 2.42993e6 0.805729 0.402865 − 0.915260i \(-0.368015\pi\)
0.402865 + 0.915260i \(0.368015\pi\)
\(72\) 0 0
\(73\) −3.78652e6 −1.13923 −0.569613 − 0.821913i \(-0.692907\pi\)
−0.569613 + 0.821913i \(0.692907\pi\)
\(74\) −5.58787e6 −1.60301
\(75\) 0 0
\(76\) 2.55559e6 0.667796
\(77\) 31379.6 0.00783302
\(78\) 0 0
\(79\) −4.40140e6 −1.00437 −0.502187 − 0.864759i \(-0.667471\pi\)
−0.502187 + 0.864759i \(0.667471\pi\)
\(80\) −2.47975e6 −0.541493
\(81\) 0 0
\(82\) 4.94193e6 0.989801
\(83\) −6.68955e6 −1.28417 −0.642087 − 0.766632i \(-0.721931\pi\)
−0.642087 + 0.766632i \(0.721931\pi\)
\(84\) 0 0
\(85\) 3.38403e6 0.597678
\(86\) 9.80589e6 1.66243
\(87\) 0 0
\(88\) 52144.7 0.00815681
\(89\) 9.24865e6 1.39064 0.695318 − 0.718702i \(-0.255264\pi\)
0.695318 + 0.718702i \(0.255264\pi\)
\(90\) 0 0
\(91\) −776798. −0.108060
\(92\) −9.38579e6 −1.25665
\(93\) 0 0
\(94\) 1.83573e7 2.27962
\(95\) 3.57575e6 0.427893
\(96\) 0 0
\(97\) 1.12707e7 1.25387 0.626933 − 0.779073i \(-0.284310\pi\)
0.626933 + 0.779073i \(0.284310\pi\)
\(98\) 1.73443e6 0.186151
\(99\) 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.1.4 4
3.2 odd 2 105.8.a.g.1.1 ✓ 4
15.14 odd 2 525.8.a.j.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.8.a.g.1.1 ✓ 4 3.2 odd 2
315.8.a.h.1.4 4 1.1 even 1 trivial
525.8.a.j.1.4 4 15.14 odd 2