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Properties

Label 315.8.b.a
Level $315$
Weight $8$
Character orbit 315.b
Analytic conductor $98.401$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,8,Mod(251,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.251");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 315.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(98.4012830275\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2048 q^{4} - 4500 q^{5} - 892 q^{7} - 18132 q^{14} + 87748 q^{16} + 256000 q^{20} + 223780 q^{22} + 562500 q^{25} + 149616 q^{26} + 384860 q^{28} + 111500 q^{35} - 2046344 q^{37} - 8352 q^{38} + 30960 q^{41}+ \cdots - 31624212 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 − 21.8020i 0 −347.326 −125.000 0 −420.595 − 804.141i 4781.73i 0 2725.25i
251.2 − 20.8373i 0 −306.192 −125.000 0 −776.826 + 469.131i 3713.03i 0 2604.66i
251.3 − 19.6445i 0 −257.906 −125.000 0 176.401 − 890.183i 2551.94i 0 2455.56i
251.4 − 18.6170i 0 −218.591 −125.000 0 906.053 + 51.1032i 1686.53i 0 2327.12i
251.5 − 17.2579i 0 −169.836 −125.000 0 678.561 + 602.576i 722.000i 0 2157.24i
251.6 − 16.6252i 0 −148.397 −125.000 0 −749.494 + 511.666i 339.111i 0 2078.15i
251.7 − 16.2847i 0 −137.191 −125.000 0 −907.402 + 12.8416i 149.667i 0 2035.58i
251.8 − 13.4946i 0 −54.1036 −125.000 0 −238.879 + 875.488i − 997.200i 0 1686.82i
251.9 − 13.1741i 0 −45.5578 −125.000 0 599.562 − 681.225i − 1086.10i 0 1646.77i
251.10 − 11.1415i 0 3.86589 −125.000 0 40.2068 − 906.602i − 1469.19i 0 1392.69i
251.11 − 10.1889i 0 24.1857 −125.000 0 817.883 − 393.204i − 1550.61i 0 1273.62i
251.12 − 9.96079i 0 28.7827 −125.000 0 766.665 + 485.559i − 1561.68i 0 1245.10i
251.13 − 8.97766i 0 47.4015 −125.000 0 −783.882 − 457.244i − 1574.70i 0 1122.21i
251.14 − 5.28772i 0 100.040 −125.000 0 −322.098 + 848.408i − 1205.81i 0 660.965i
251.15 − 5.21139i 0 100.841 −125.000 0 −839.402 + 344.886i − 1192.58i 0 651.424i
251.16 − 5.04326i 0 102.566 −125.000 0 787.697 + 450.639i − 1162.80i 0 630.407i
251.17 − 1.17691i 0 126.615 −125.000 0 −415.490 − 806.790i − 299.658i 0 147.113i
251.18 − 1.09451i 0 126.802 −125.000 0 235.038 − 876.527i − 278.884i 0 136.814i
251.19 1.09451i 0 126.802 −125.000 0 235.038 + 876.527i 278.884i 0 − 136.814i
251.20 1.17691i 0 126.615 −125.000 0 −415.490 + 806.790i 299.658i 0 − 147.113i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.8.b.a ✓ 36
3.b odd 2 1 315.8.b.b yes 36
7.b odd 2 1 315.8.b.b yes 36
21.c even 2 1 inner 315.8.b.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.8.b.a ✓ 36 1.a even 1 1 trivial
315.8.b.a ✓ 36 21.c even 2 1 inner
315.8.b.b yes 36 3.b odd 2 1
315.8.b.b yes 36 7.b odd 2 1