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Properties

Label 75.13.d.d
Level $75$
Weight $13$
Character orbit 75.d
Analytic conductor $68.550$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.74");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 13 \)
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: \(68.5495362957\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64612 q^{4} - 64798 q^{6} + 331480 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64612 q^{4} - 64798 q^{6} + 331480 q^{9} + 70468916 q^{16} - 163784080 q^{19} + 216668112 q^{21} + 112707894 q^{24} + 1367213824 q^{31} - 1335293556 q^{34} - 1210176866 q^{36} + 6504423728 q^{39} - 78043620696 q^{46} - 83957215152 q^{49} + 66113769512 q^{51} + 160688077792 q^{54} - 194449308176 q^{61} - 442384961204 q^{64} + 669829542910 q^{66} + 642007476432 q^{69} - 2056182630572 q^{76} - 902580024640 q^{79} + 1889883392072 q^{81} + 3337990221528 q^{84} - 1634112577696 q^{91} + 1038700364664 q^{94} + 225774532546 q^{96} - 183110396920 q^{99}+O(q^{100}) \) 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 −116.687 217.113 − 695.919i 9519.86 0 −25334.3 + 81204.7i 159101.i −632894. −437165. − 302187.i 0
74.2 −116.687 217.113 + 695.919i 9519.86 0 −25334.3 − 81204.7i − 159101.i −632894. −437165. + 302187.i 0
74.3 −107.354 −724.083 + 84.5279i 7428.92 0 77733.3 − 9074.43i − 76606.4i −357803. 517151. − 122410.i 0
74.4 −107.354 −724.083 − 84.5279i 7428.92 0 77733.3 + 9074.43i 76606.4i −357803. 517151. + 122410.i 0
74.5 −97.7240 706.715 + 178.872i 5453.97 0 −69063.0 − 17480.1i − 195864.i −132707. 467451. + 252823.i 0
74.6 −97.7240 706.715 − 178.872i 5453.97 0 −69063.0 + 17480.1i 195864.i −132707. 467451. − 252823.i 0
74.7 −88.3498 67.1399 − 725.902i 3709.68 0 −5931.79 + 64133.3i − 144393.i 34131.0 −522425. − 97473.9i 0
74.8 −88.3498 67.1399 + 725.902i 3709.68 0 −5931.79 − 64133.3i 144393.i 34131.0 −522425. + 97473.9i 0
74.9 −54.6723 −528.380 − 502.251i −1106.94 0 28887.8 + 27459.2i 26114.9i 284457. 26929.8 + 530758.i 0
74.10 −54.6723 −528.380 + 502.251i −1106.94 0 28887.8 − 27459.2i − 26114.9i 284457. 26929.8 − 530758.i 0
74.11 −45.6637 315.050 + 657.408i −2010.83 0 −14386.3 − 30019.7i 82178.2i 278860. −332928. + 414232.i 0
74.12 −45.6637 315.050 − 657.408i −2010.83 0 −14386.3 + 30019.7i − 82178.2i 278860. −332928. − 414232.i 0
74.13 −26.2870 727.487 + 46.9464i −3404.99 0 −19123.5 − 1234.08i 158582.i 197179. 527033. + 68305.8i 0
74.14 −26.2870 727.487 − 46.9464i −3404.99 0 −19123.5 + 1234.08i − 158582.i 197179. 527033. − 68305.8i 0
74.15 −25.6772 −429.106 + 589.329i −3436.68 0 11018.3 − 15132.3i − 93502.2i 193418. −163176. − 505770.i 0
74.16 −25.6772 −429.106 − 589.329i −3436.68 0 11018.3 + 15132.3i 93502.2i 193418. −163176. + 505770.i 0
74.17 25.6772 429.106 + 589.329i −3436.68 0 11018.3 + 15132.3i − 93502.2i −193418. −163176. + 505770.i 0
74.18 25.6772 429.106 − 589.329i −3436.68 0 11018.3 − 15132.3i 93502.2i −193418. −163176. − 505770.i 0
74.19 26.2870 −727.487 + 46.9464i −3404.99 0 −19123.5 + 1234.08i 158582.i −197179. 527033. − 68305.8i 0
74.20 26.2870 −727.487 − 46.9464i −3404.99 0 −19123.5 − 1234.08i − 158582.i −197179. 527033. + 68305.8i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.32
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 75.13.d.d 32
3.b odd 2 1 inner 75.13.d.d 32
5.b even 2 1 inner 75.13.d.d 32
5.c odd 4 1 75.13.c.e ✓ 16
5.c odd 4 1 75.13.c.f yes 16
15.d odd 2 1 inner 75.13.d.d 32
15.e even 4 1 75.13.c.e ✓ 16
15.e even 4 1 75.13.c.f yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.13.c.e ✓ 16 5.c odd 4 1
75.13.c.e ✓ 16 15.e even 4 1
75.13.c.f yes 16 5.c odd 4 1
75.13.c.f yes 16 15.e even 4 1
75.13.d.d 32 1.a even 1 1 trivial
75.13.d.d 32 3.b odd 2 1 inner
75.13.d.d 32 5.b even 2 1 inner
75.13.d.d 32 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 48921 T_{2}^{14} + 954361746 T_{2}^{12} - 9474039334576 T_{2}^{10} + \cdots + 33\!\cdots\!00 \) acting on \(S_{13}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display