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Properties

Label 975.2.bp.h
Level $975$
Weight $2$
Character orbit 975.bp
Analytic conductor $7.785$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(149,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bp (of order \(12\), degree \(4\), not 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{6} - 24 q^{12} - 4 q^{13} + 24 q^{16} + 40 q^{18} + 12 q^{19} - 36 q^{21} + 72 q^{22} + 56 q^{24} + 64 q^{28} - 28 q^{31} + 4 q^{33} - 40 q^{34} + 12 q^{36} + 4 q^{37} - 32 q^{39} - 48 q^{42} + 16 q^{43} + 64 q^{46} + 84 q^{49} + 88 q^{52} - 32 q^{54} + 36 q^{57} + 24 q^{58} - 8 q^{61} + 88 q^{63} + 120 q^{66} - 68 q^{67} - 72 q^{69} - 12 q^{72} + 60 q^{73} - 8 q^{76} - 36 q^{78} + 16 q^{79} + 12 q^{81} + 88 q^{82} - 44 q^{84} + 12 q^{87} + 40 q^{88} - 12 q^{91} - 72 q^{93} - 8 q^{94} + 4 q^{96} - 32 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −0.659148 − 2.45997i 1.68663 + 0.394058i −3.88494 + 2.24297i 0 −0.142367 − 4.40880i −3.19491 − 0.856074i 4.47674 + 4.47674i 2.68944 + 1.32926i 0
149.2 −0.614171 − 2.29212i −0.762566 − 1.55515i −3.14455 + 1.81551i 0 −3.09624 + 2.70302i −4.37590 − 1.17252i 2.73676 + 2.73676i −1.83698 + 2.37181i 0
149.3 −0.572069 − 2.13499i 0.480688 + 1.66401i −2.49887 + 1.44273i 0 3.27767 − 1.97820i 1.63647 + 0.438491i 1.38389 + 1.38389i −2.53788 + 1.59974i 0
149.4 −0.553380 − 2.06524i 0.240414 − 1.71528i −2.22694 + 1.28572i 0 −3.67552 + 0.452691i 3.20573 + 0.858972i 0.863953 + 0.863953i −2.88440 − 0.824757i 0
149.5 −0.396555 − 1.47996i −1.65035 + 0.525689i −0.300984 + 0.173773i 0 1.43245 + 2.23399i 0.816332 + 0.218736i −1.79028 − 1.79028i 2.44730 − 1.73514i 0
149.6 −0.297051 − 1.10861i −0.769034 + 1.55196i 0.591272 − 0.341371i 0 1.94897 + 0.391547i −0.219935 − 0.0589315i −2.17721 − 2.17721i −1.81717 − 2.38702i 0
149.7 −0.156721 − 0.584891i −0.645804 − 1.60715i 1.41451 − 0.816670i 0 −0.838798 + 0.629600i 4.30957 + 1.15475i −1.55569 − 1.55569i −2.16587 + 2.07581i 0
149.8 −0.116204 − 0.433679i 0.853659 − 1.50707i 1.55748 − 0.899210i 0 −0.752784 − 0.195086i −3.58664 − 0.961038i −1.20590 − 1.20590i −1.54253 − 2.57305i 0
149.9 −0.113734 − 0.424461i 1.08557 + 1.34964i 1.56482 − 0.903449i 0 0.449405 − 0.614282i 0.543270 + 0.145569i −1.18291 − 1.18291i −0.643078 + 2.93026i 0
149.10 0.113734 + 0.424461i 1.71161 + 0.265309i 1.56482 − 0.903449i 0 0.0820551 + 0.756687i 0.543270 + 0.145569i 1.18291 + 1.18291i 2.85922 + 0.908210i 0
149.11 0.116204 + 0.433679i −0.878333 + 1.49283i 1.55748 − 0.899210i 0 −0.749473 − 0.207442i −3.58664 − 0.961038i 1.20590 + 1.20590i −1.45706 − 2.62240i 0
149.12 0.156721 + 0.584891i −1.71474 + 0.244293i 1.41451 − 0.816670i 0 −0.411620 − 0.964648i 4.30957 + 1.15475i 1.55569 + 1.55569i 2.88064 − 0.837797i 0
149.13 0.297051 + 1.10861i 0.959522 − 1.44198i 0.591272 − 0.341371i 0 1.88363 + 0.635393i −0.219935 − 0.0589315i 2.17721 + 2.17721i −1.15864 − 2.76723i 0
149.14 0.396555 + 1.47996i −0.369915 − 1.69209i −0.300984 + 0.173773i 0 2.35754 − 1.21847i 0.816332 + 0.218736i 1.79028 + 1.79028i −2.72633 + 1.25186i 0
149.15 0.553380 + 2.06524i −1.36527 + 1.06585i −2.22694 + 1.28572i 0 −2.95675 − 2.22980i 3.20573 + 0.858972i −0.863953 − 0.863953i 0.727941 − 2.91034i 0
149.16 0.572069 + 2.13499i 1.68142 − 0.415718i −2.49887 + 1.44273i 0 1.84944 + 3.35200i 1.63647 + 0.438491i −1.38389 − 1.38389i 2.65436 − 1.39800i 0
149.17 0.614171 + 2.29212i −1.72808 + 0.117173i −3.14455 + 1.81551i 0 −1.32991 − 3.88901i −4.37590 − 1.17252i −2.73676 − 2.73676i 2.97254 − 0.404970i 0
149.18 0.659148 + 2.45997i 1.18458 + 1.26364i −3.88494 + 2.24297i 0 −2.32769 + 3.74695i −3.19491 − 0.856074i −4.47674 − 4.47674i −0.193547 + 2.99375i 0
449.1 −2.60640 + 0.698381i −1.17399 − 1.27348i 4.57351 − 2.64052i 0 3.94925 + 2.49931i 0.724221 − 2.70283i −6.26025 + 6.26025i −0.243511 + 2.99010i 0
449.2 −2.42430 + 0.649589i 1.40950 − 1.00663i 3.72321 − 2.14960i 0 −2.76316 + 3.35597i −0.743154 + 2.77349i −4.08041 + 4.08041i 0.973391 − 2.83769i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
65.s odd 12 1 inner
195.bh even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bp.h 72
3.b odd 2 1 inner 975.2.bp.h 72
5.b even 2 1 975.2.bp.i 72
5.c odd 4 1 195.2.bg.a ✓ 72
5.c odd 4 1 975.2.bo.e 72
13.f odd 12 1 975.2.bp.i 72
15.d odd 2 1 975.2.bp.i 72
15.e even 4 1 195.2.bg.a ✓ 72
15.e even 4 1 975.2.bo.e 72
39.k even 12 1 975.2.bp.i 72
65.o even 12 1 195.2.bg.a ✓ 72
65.s odd 12 1 inner 975.2.bp.h 72
65.t even 12 1 975.2.bo.e 72
195.bc odd 12 1 975.2.bo.e 72
195.bh even 12 1 inner 975.2.bp.h 72
195.bn odd 12 1 195.2.bg.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bg.a ✓ 72 5.c odd 4 1
195.2.bg.a ✓ 72 15.e even 4 1
195.2.bg.a ✓ 72 65.o even 12 1
195.2.bg.a ✓ 72 195.bn odd 12 1
975.2.bo.e 72 5.c odd 4 1
975.2.bo.e 72 15.e even 4 1
975.2.bo.e 72 65.t even 12 1
975.2.bo.e 72 195.bc odd 12 1
975.2.bp.h 72 1.a even 1 1 trivial
975.2.bp.h 72 3.b odd 2 1 inner
975.2.bp.h 72 65.s odd 12 1 inner
975.2.bp.h 72 195.bh even 12 1 inner
975.2.bp.i 72 5.b even 2 1
975.2.bp.i 72 13.f odd 12 1
975.2.bp.i 72 15.d odd 2 1
975.2.bp.i 72 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{72} - 126 T_{2}^{68} + 10081 T_{2}^{64} + 3396 T_{2}^{62} - 487974 T_{2}^{60} - 247176 T_{2}^{58} + \cdots + 1679616 \) Copy content Toggle raw display
\( T_{7}^{36} - 21 T_{7}^{34} + 50 T_{7}^{33} - 235 T_{7}^{32} - 1360 T_{7}^{31} + 9272 T_{7}^{30} + \cdots + 3833714889 \) Copy content Toggle raw display