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Properties

Label 312.4.m.a
Level $312$
Weight $4$
Character orbit 312.m
Analytic conductor $18.409$
Analytic rank $0$
Dimension $84$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [312,4,Mod(181,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("312.181");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 312.m (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: \(18.4085959218\)
Analytic rank: \(0\)
Dimension: \(84\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 756 q^{9} - 36 q^{10} + 12 q^{12} - 208 q^{14} - 148 q^{16} - 104 q^{17} + 620 q^{22} + 2188 q^{25} + 444 q^{26} - 204 q^{30} - 40 q^{38} - 1924 q^{40} + 192 q^{42} + 624 q^{48} - 3396 q^{49} - 1292 q^{52}+ \cdots + 2480 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −2.82580 − 0.121986i 3.00000i 7.97024 + 0.689413i 14.7613 0.365957 − 8.47739i 28.6139i −22.4382 − 2.92040i −9.00000 −41.7123 − 1.80066i
181.2 −2.82580 + 0.121986i − 3.00000i 7.97024 − 0.689413i 14.7613 0.365957 + 8.47739i − 28.6139i −22.4382 + 2.92040i −9.00000 −41.7123 + 1.80066i
181.3 −2.81504 − 0.274823i 3.00000i 7.84894 + 1.54728i −8.71484 0.824470 − 8.44513i − 16.6675i −21.6699 − 6.51273i −9.00000 24.5327 + 2.39504i
181.4 −2.81504 + 0.274823i − 3.00000i 7.84894 − 1.54728i −8.71484 0.824470 + 8.44513i 16.6675i −21.6699 + 6.51273i −9.00000 24.5327 − 2.39504i
181.5 −2.80130 − 0.390824i − 3.00000i 7.69451 + 2.18963i 14.7907 −1.17247 + 8.40389i − 0.217747i −20.6988 − 9.14099i −9.00000 −41.4331 − 5.78056i
181.6 −2.80130 + 0.390824i 3.00000i 7.69451 − 2.18963i 14.7907 −1.17247 − 8.40389i 0.217747i −20.6988 + 9.14099i −9.00000 −41.4331 + 5.78056i
181.7 −2.73458 − 0.722559i − 3.00000i 6.95582 + 3.95178i −2.18910 −2.16768 + 8.20373i − 4.02754i −16.1658 − 15.8324i −9.00000 5.98626 + 1.58175i
181.8 −2.73458 + 0.722559i 3.00000i 6.95582 − 3.95178i −2.18910 −2.16768 − 8.20373i 4.02754i −16.1658 + 15.8324i −9.00000 5.98626 − 1.58175i
181.9 −2.56970 − 1.18180i 3.00000i 5.20671 + 6.07373i −8.43796 3.54539 − 7.70910i − 8.36709i −6.20176 − 21.7609i −9.00000 21.6830 + 9.97196i
181.10 −2.56970 + 1.18180i − 3.00000i 5.20671 − 6.07373i −8.43796 3.54539 + 7.70910i 8.36709i −6.20176 + 21.7609i −9.00000 21.6830 − 9.97196i
181.11 −2.48687 − 1.34740i 3.00000i 4.36905 + 6.70160i 8.37088 4.04219 − 7.46061i − 16.5971i −1.83557 − 22.5528i −9.00000 −20.8173 − 11.2789i
181.12 −2.48687 + 1.34740i − 3.00000i 4.36905 − 6.70160i 8.37088 4.04219 + 7.46061i 16.5971i −1.83557 + 22.5528i −9.00000 −20.8173 + 11.2789i
181.13 −2.47275 − 1.37314i − 3.00000i 4.22898 + 6.79086i −19.0411 −4.11942 + 7.41825i 2.83206i −1.13242 − 22.5991i −9.00000 47.0838 + 26.1460i
181.14 −2.47275 + 1.37314i 3.00000i 4.22898 − 6.79086i −19.0411 −4.11942 − 7.41825i − 2.83206i −1.13242 + 22.5991i −9.00000 47.0838 − 26.1460i
181.15 −2.32798 − 1.60640i − 3.00000i 2.83895 + 7.47933i 6.73184 −4.81920 + 6.98393i 30.2729i 5.40578 − 21.9722i −9.00000 −15.6716 − 10.8140i
181.16 −2.32798 + 1.60640i 3.00000i 2.83895 − 7.47933i 6.73184 −4.81920 − 6.98393i − 30.2729i 5.40578 + 21.9722i −9.00000 −15.6716 + 10.8140i
181.17 −2.30493 − 1.63930i − 3.00000i 2.62541 + 7.55693i −3.15976 −4.91789 + 6.91479i − 31.8225i 6.33669 − 21.7220i −9.00000 7.28302 + 5.17979i
181.18 −2.30493 + 1.63930i 3.00000i 2.62541 − 7.55693i −3.15976 −4.91789 − 6.91479i 31.8225i 6.33669 + 21.7220i −9.00000 7.28302 − 5.17979i
181.19 −2.23230 − 1.73690i 3.00000i 1.96633 + 7.75458i −12.2791 5.21071 − 6.69690i 32.4044i 9.07951 − 20.7259i −9.00000 27.4107 + 21.3277i
181.20 −2.23230 + 1.73690i − 3.00000i 1.96633 − 7.75458i −12.2791 5.21071 + 6.69690i − 32.4044i 9.07951 + 20.7259i −9.00000 27.4107 − 21.3277i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.84
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.b even 2 1 inner
104.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.4.m.a ✓ 84
4.b odd 2 1 1248.4.m.a 84
8.b even 2 1 inner 312.4.m.a ✓ 84
8.d odd 2 1 1248.4.m.a 84
13.b even 2 1 inner 312.4.m.a ✓ 84
52.b odd 2 1 1248.4.m.a 84
104.e even 2 1 inner 312.4.m.a ✓ 84
104.h odd 2 1 1248.4.m.a 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.m.a ✓ 84 1.a even 1 1 trivial
312.4.m.a ✓ 84 8.b even 2 1 inner
312.4.m.a ✓ 84 13.b even 2 1 inner
312.4.m.a ✓ 84 104.e even 2 1 inner
1248.4.m.a 84 4.b odd 2 1
1248.4.m.a 84 8.d odd 2 1
1248.4.m.a 84 52.b odd 2 1
1248.4.m.a 84 104.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(312, [\chi])\).