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Properties

Label 1305.2.r.d
Level $1305$
Weight $2$
Character orbit 1305.r
Analytic conductor $10.420$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(476,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.476");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.r (of order \(4\), degree \(2\), 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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{5} + 8 q^{7} - 8 q^{11} + 24 q^{14} - 32 q^{16} + 16 q^{19} + 32 q^{25} + 8 q^{26} + 8 q^{29} - 8 q^{31} - 40 q^{32} + 8 q^{35} + 24 q^{37} + 24 q^{38} + 24 q^{41} + 16 q^{43} + 48 q^{44} - 64 q^{46}+ \cdots - 80 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
476.1 −1.88896 + 1.88896i 0 − 5.13635i 1.00000 0 −4.81798 5.92445 + 5.92445i 0 −1.88896 + 1.88896i
476.2 −1.63087 + 1.63087i 0 − 3.31944i 1.00000 0 3.77106 2.15184 + 2.15184i 0 −1.63087 + 1.63087i
476.3 −1.61430 + 1.61430i 0 − 3.21192i 1.00000 0 −0.631105 1.95641 + 1.95641i 0 −1.61430 + 1.61430i
476.4 −1.39641 + 1.39641i 0 − 1.89994i 1.00000 0 4.49373 −0.139719 − 0.139719i 0 −1.39641 + 1.39641i
476.5 −1.26665 + 1.26665i 0 − 1.20879i 1.00000 0 −1.31047 −1.00219 − 1.00219i 0 −1.26665 + 1.26665i
476.6 −1.24864 + 1.24864i 0 − 1.11818i 1.00000 0 −2.67792 −1.10107 − 1.10107i 0 −1.24864 + 1.24864i
476.7 −0.251233 + 0.251233i 0 1.87376i 1.00000 0 −1.71619 −0.973218 − 0.973218i 0 −0.251233 + 0.251233i
476.8 −0.234786 + 0.234786i 0 1.88975i 1.00000 0 3.02559 −0.913258 − 0.913258i 0 −0.234786 + 0.234786i
476.9 0.113859 − 0.113859i 0 1.97407i 1.00000 0 −2.37207 0.452485 + 0.452485i 0 0.113859 − 0.113859i
476.10 0.605142 − 0.605142i 0 1.26761i 1.00000 0 2.21854 1.97737 + 1.97737i 0 0.605142 − 0.605142i
476.11 1.09394 − 1.09394i 0 − 0.393419i 1.00000 0 3.19461 1.75751 + 1.75751i 0 1.09394 − 1.09394i
476.12 1.19445 − 1.19445i 0 − 0.853421i 1.00000 0 −4.00364 1.36953 + 1.36953i 0 1.19445 − 1.19445i
476.13 1.28683 − 1.28683i 0 − 1.31184i 1.00000 0 −1.81123 0.885539 + 0.885539i 0 1.28683 − 1.28683i
476.14 1.47426 − 1.47426i 0 − 2.34690i 1.00000 0 1.04276 −0.511418 − 0.511418i 0 1.47426 − 1.47426i
476.15 1.77908 − 1.77908i 0 − 4.33025i 1.00000 0 2.49153 −4.14571 − 4.14571i 0 1.77908 − 1.77908i
476.16 1.98428 − 1.98428i 0 − 5.87473i 1.00000 0 3.10278 −7.68854 − 7.68854i 0 1.98428 − 1.98428i
1061.1 −1.88896 − 1.88896i 0 5.13635i 1.00000 0 −4.81798 5.92445 − 5.92445i 0 −1.88896 − 1.88896i
1061.2 −1.63087 − 1.63087i 0 3.31944i 1.00000 0 3.77106 2.15184 − 2.15184i 0 −1.63087 − 1.63087i
1061.3 −1.61430 − 1.61430i 0 3.21192i 1.00000 0 −0.631105 1.95641 − 1.95641i 0 −1.61430 − 1.61430i
1061.4 −1.39641 − 1.39641i 0 1.89994i 1.00000 0 4.49373 −0.139719 + 0.139719i 0 −1.39641 − 1.39641i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 476.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
87.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.r.d yes 32
3.b odd 2 1 1305.2.r.c ✓ 32
29.c odd 4 1 1305.2.r.c ✓ 32
87.f even 4 1 inner 1305.2.r.d yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.r.c ✓ 32 3.b odd 2 1
1305.2.r.c ✓ 32 29.c odd 4 1
1305.2.r.d yes 32 1.a even 1 1 trivial
1305.2.r.d yes 32 87.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1305, [\chi])\):

\( T_{2}^{32} + 144 T_{2}^{28} + 8 T_{2}^{27} - 88 T_{2}^{25} + 7720 T_{2}^{24} + 416 T_{2}^{23} + \cdots + 10000 \) Copy content Toggle raw display
\( T_{11}^{32} + 8 T_{11}^{31} + 32 T_{11}^{30} - 48 T_{11}^{29} + 594 T_{11}^{28} + 5720 T_{11}^{27} + \cdots + 770506564 \) Copy content Toggle raw display