↑

Properties

Label 75.11.d.d
Level $75$
Weight $11$
Character orbit 75.d
Analytic conductor $47.652$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,11,Mod(74,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.74");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\), degree \(1\), 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: \(47.6517939505\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 17604 q^{4} + 43772 q^{6} - 232724 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 17604 q^{4} + 43772 q^{6} - 232724 q^{9} + 5743812 q^{16} - 7629288 q^{19} - 4382016 q^{21} - 18909084 q^{24} + 210888616 q^{31} - 169921544 q^{34} + 161936980 q^{36} + 525991904 q^{39} + 605632104 q^{46} - 2679858100 q^{49} - 1039546648 q^{51} + 6343557388 q^{54} - 4745815464 q^{61} - 11326231660 q^{64} + 1831573840 q^{66} + 2064761208 q^{69} - 9855869080 q^{76} + 16667838152 q^{79} - 8569270852 q^{81} - 27675191136 q^{84} + 8026443968 q^{91} + 95003033416 q^{94} + 86264478916 q^{96} - 72516625120 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1 −57.7709 −235.318 − 60.6159i 2313.48 0 13594.6 + 3501.84i − 22792.7i −74494.6 51700.4 + 28528.1i 0
74.2 −57.7709 −235.318 + 60.6159i 2313.48 0 13594.6 − 3501.84i 22792.7i −74494.6 51700.4 − 28528.1i 0
74.3 −52.7178 121.125 − 210.660i 1755.17 0 −6385.47 + 11105.5i 8585.72i −38545.6 −29706.2 − 51032.6i 0
74.4 −52.7178 121.125 + 210.660i 1755.17 0 −6385.47 − 11105.5i − 8585.72i −38545.6 −29706.2 + 51032.6i 0
74.5 −52.0937 142.800 − 196.615i 1689.75 0 −7438.96 + 10242.4i − 32323.0i −34681.6 −18265.6 − 56152.9i 0
74.6 −52.0937 142.800 + 196.615i 1689.75 0 −7438.96 − 10242.4i 32323.0i −34681.6 −18265.6 + 56152.9i 0
74.7 −40.7012 −76.4761 − 230.652i 632.586 0 3112.67 + 9387.81i 19744.7i 15931.0 −47351.8 + 35278.8i 0
74.8 −40.7012 −76.4761 + 230.652i 632.586 0 3112.67 − 9387.81i − 19744.7i 15931.0 −47351.8 − 35278.8i 0
74.9 −27.5253 −229.405 − 80.1400i −266.360 0 6314.43 + 2205.87i 24115.7i 35517.5 46204.2 + 36769.0i 0
74.10 −27.5253 −229.405 + 80.1400i −266.360 0 6314.43 − 2205.87i − 24115.7i 35517.5 46204.2 − 36769.0i 0
74.11 −17.3194 −55.1313 − 236.663i −724.039 0 954.839 + 4098.86i − 2728.90i 30275.0 −52970.1 + 26095.1i 0
74.12 −17.3194 −55.1313 + 236.663i −724.039 0 954.839 − 4098.86i 2728.90i 30275.0 −52970.1 − 26095.1i 0
74.13 −4.94055 −160.089 − 182.813i −999.591 0 790.929 + 903.195i − 5515.83i 9997.66 −7791.87 + 58532.7i 0
74.14 −4.94055 −160.089 + 182.813i −999.591 0 790.929 − 903.195i 5515.83i 9997.66 −7791.87 − 58532.7i 0
74.15 4.94055 160.089 − 182.813i −999.591 0 790.929 − 903.195i − 5515.83i −9997.66 −7791.87 − 58532.7i 0
74.16 4.94055 160.089 + 182.813i −999.591 0 790.929 + 903.195i 5515.83i −9997.66 −7791.87 + 58532.7i 0
74.17 17.3194 55.1313 − 236.663i −724.039 0 954.839 − 4098.86i − 2728.90i −30275.0 −52970.1 − 26095.1i 0
74.18 17.3194 55.1313 + 236.663i −724.039 0 954.839 + 4098.86i 2728.90i −30275.0 −52970.1 + 26095.1i 0
74.19 27.5253 229.405 − 80.1400i −266.360 0 6314.43 − 2205.87i 24115.7i −35517.5 46204.2 − 36769.0i 0
74.20 27.5253 229.405 + 80.1400i −266.360 0 6314.43 + 2205.87i − 24115.7i −35517.5 46204.2 + 36769.0i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.11.d.d 28
3.b odd 2 1 inner 75.11.d.d 28
5.b even 2 1 inner 75.11.d.d 28
5.c odd 4 1 15.11.c.a ✓ 14
5.c odd 4 1 75.11.c.g 14
15.d odd 2 1 inner 75.11.d.d 28
15.e even 4 1 15.11.c.a ✓ 14
15.e even 4 1 75.11.c.g 14
20.e even 4 1 240.11.l.b 14
60.l odd 4 1 240.11.l.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.11.c.a ✓ 14 5.c odd 4 1
15.11.c.a ✓ 14 15.e even 4 1
75.11.c.g 14 5.c odd 4 1
75.11.c.g 14 15.e even 4 1
75.11.d.d 28 1.a even 1 1 trivial
75.11.d.d 28 3.b odd 2 1 inner
75.11.d.d 28 5.b even 2 1 inner
75.11.d.d 28 15.d odd 2 1 inner
240.11.l.b 14 20.e even 4 1
240.11.l.b 14 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 11569 T_{2}^{12} + 52102936 T_{2}^{10} - 114518599604 T_{2}^{8} + 125620895405696 T_{2}^{6} + \cdots - 23\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display