Properties

Label 925.2.bq.c
Level $925$
Weight $2$
Character orbit 925.bq
Analytic conductor $7.386$
Analytic rank $0$
Dimension $312$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(32,925)] 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("925.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bq (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [312] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(26\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 48 q^{14} - 60 q^{16} - 24 q^{19} - 36 q^{21} - 12 q^{29} - 24 q^{31} + 60 q^{34} + 12 q^{39} + 48 q^{41} - 120 q^{44} - 24 q^{49} + 120 q^{51} + 12 q^{54} + 96 q^{56} - 48 q^{61} + 240 q^{64} - 432 q^{66}+ \cdots + 648 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −0.955333 − 2.62476i −0.864970 − 1.85493i −4.44460 + 3.72946i 0 −4.04242 + 4.04242i 0.831122 − 1.18696i 9.19703 + 5.30991i −0.764247 + 0.910794i 0
32.2 −0.936768 − 2.57375i 0.241236 + 0.517333i −4.21456 + 3.53644i 0 1.10550 − 1.10550i −2.92337 + 4.17501i 8.30602 + 4.79548i 1.71892 − 2.04853i 0
32.3 −0.749603 − 2.05952i 0.583704 + 1.25176i −2.14762 + 1.80206i 0 2.14047 − 2.14047i 2.26020 − 3.22790i 1.52512 + 0.880526i 0.702177 − 0.836821i 0
32.4 −0.740045 − 2.03326i −1.31874 − 2.82805i −2.05438 + 1.72383i 0 −4.77423 + 4.77423i −0.275284 + 0.393147i 1.27761 + 0.737628i −4.33044 + 5.16082i 0
32.5 −0.707431 − 1.94365i 0.235593 + 0.505231i −1.74523 + 1.46442i 0 0.815326 − 0.815326i 1.95836 − 2.79682i 0.498399 + 0.287751i 1.72861 − 2.06008i 0
32.6 −0.701953 − 1.92860i 1.39930 + 3.00081i −1.69467 + 1.42200i 0 4.80512 − 4.80512i 0.479895 − 0.685361i 0.377238 + 0.217798i −5.11845 + 6.09994i 0
32.7 −0.460374 − 1.26487i 0.0815602 + 0.174907i 0.144147 − 0.120954i 0 0.183685 − 0.183685i −0.645139 + 0.921354i −2.55076 − 1.47268i 1.90442 − 2.26960i 0
32.8 −0.450257 − 1.23707i −1.12153 − 2.40513i 0.204473 − 0.171574i 0 −2.47034 + 2.47034i 0.965566 − 1.37897i −2.58450 − 1.49216i −2.59844 + 3.09670i 0
32.9 −0.333971 − 0.917579i −0.583164 − 1.25060i 0.801675 − 0.672685i 0 −0.952763 + 0.952763i −0.401205 + 0.572980i −2.57627 − 1.48741i 0.704445 − 0.839525i 0
32.10 −0.266672 − 0.732675i 0.606667 + 1.30100i 1.06639 − 0.894808i 0 0.791430 − 0.791430i −2.57882 + 3.68294i −2.29045 − 1.32239i 0.603801 − 0.719582i 0
32.11 −0.259728 − 0.713596i 0.803761 + 1.72367i 1.09033 − 0.914894i 0 1.02125 − 1.02125i 1.66052 − 2.37147i −2.25136 − 1.29982i −0.396646 + 0.472705i 0
32.12 −0.193727 − 0.532261i 1.26509 + 2.71300i 1.28632 − 1.07935i 0 1.19894 − 1.19894i −2.68513 + 3.83477i −1.80476 − 1.04198i −3.83154 + 4.56625i 0
32.13 −0.0617499 − 0.169657i 0.589346 + 1.26386i 1.50712 − 1.26462i 0 0.178030 − 0.178030i 1.81568 − 2.59307i −0.620329 − 0.358147i 0.678357 − 0.808434i 0
32.14 0.0617499 + 0.169657i −0.589346 − 1.26386i 1.50712 − 1.26462i 0 0.178030 − 0.178030i −1.81568 + 2.59307i 0.620329 + 0.358147i 0.678357 − 0.808434i 0
32.15 0.193727 + 0.532261i −1.26509 − 2.71300i 1.28632 − 1.07935i 0 1.19894 − 1.19894i 2.68513 − 3.83477i 1.80476 + 1.04198i −3.83154 + 4.56625i 0
32.16 0.259728 + 0.713596i −0.803761 − 1.72367i 1.09033 − 0.914894i 0 1.02125 − 1.02125i −1.66052 + 2.37147i 2.25136 + 1.29982i −0.396646 + 0.472705i 0
32.17 0.266672 + 0.732675i −0.606667 − 1.30100i 1.06639 − 0.894808i 0 0.791430 − 0.791430i 2.57882 − 3.68294i 2.29045 + 1.32239i 0.603801 − 0.719582i 0
32.18 0.333971 + 0.917579i 0.583164 + 1.25060i 0.801675 − 0.672685i 0 −0.952763 + 0.952763i 0.401205 − 0.572980i 2.57627 + 1.48741i 0.704445 − 0.839525i 0
32.19 0.450257 + 1.23707i 1.12153 + 2.40513i 0.204473 − 0.171574i 0 −2.47034 + 2.47034i −0.965566 + 1.37897i 2.58450 + 1.49216i −2.59844 + 3.09670i 0
32.20 0.460374 + 1.26487i −0.0815602 − 0.174907i 0.144147 − 0.120954i 0 0.183685 − 0.183685i 0.645139 − 0.921354i 2.55076 + 1.47268i 1.90442 − 2.26960i 0
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 32.26
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
185.z even 36 1 inner
185.bc even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.bq.c yes 312
5.b even 2 1 inner 925.2.bq.c yes 312
5.c odd 4 2 925.2.bn.c ✓ 312
37.i odd 36 1 925.2.bn.c ✓ 312
185.z even 36 1 inner 925.2.bq.c yes 312
185.ba odd 36 1 925.2.bn.c ✓ 312
185.bc even 36 1 inner 925.2.bq.c yes 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
925.2.bn.c ✓ 312 5.c odd 4 2
925.2.bn.c ✓ 312 37.i odd 36 1
925.2.bn.c ✓ 312 185.ba odd 36 1
925.2.bq.c yes 312 1.a even 1 1 trivial
925.2.bq.c yes 312 5.b even 2 1 inner
925.2.bq.c yes 312 185.z even 36 1 inner
925.2.bq.c yes 312 185.bc even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{312} + 15 T_{2}^{308} - 2408 T_{2}^{306} + 1665 T_{2}^{304} - 48264 T_{2}^{302} + \cdots + 10\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display