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Properties

Label 75.17.d.b
Level $75$
Weight $17$
Character orbit 75.d
Analytic conductor $121.743$
Analytic rank $0$
Dimension $8$
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] = [75,17,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, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.74");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 17 \)
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: \(121.743407892\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2145929x^{4} + 555561529600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{16}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{4} + \beta_{2} - 51 \beta_1) q^{3} + ( - \beta_{6} - 2 \beta_{5} + 3116) q^{4} + (81 \beta_{6} + 30 \beta_{5} + \cdots - 100764) q^{6}+ \cdots + (729 \beta_{6} + 297 \beta_{5} + \cdots - 4654665) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{4} + \beta_{2} - 51 \beta_1) q^{3} + ( - \beta_{6} - 2 \beta_{5} + 3116) q^{4} + (81 \beta_{6} + 30 \beta_{5} + \cdots - 100764) q^{6}+ \cdots + (374876825841 \beta_{6} + \cdots - 47\!\cdots\!40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24928 q^{4} - 806112 q^{6} - 37237320 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24928 q^{4} - 806112 q^{6} - 37237320 q^{9} - 29122537216 q^{16} + 112234232720 q^{19} + 248910874128 q^{21} - 201031144704 q^{24} + 4943562312496 q^{31} - 5442523225344 q^{34} - 2439308253024 q^{36} - 14044340454768 q^{39} - 87989159009664 q^{46} - 58956525074328 q^{49} - 165683150445312 q^{51} + 214127321512608 q^{54} + 724539586166416 q^{61} + 13\!\cdots\!84 q^{64}+ \cdots - 37\!\cdots\!20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2145929x^{4} + 555561529600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 2891289\nu^{2} ) / 142140152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} + 8673867\nu^{3} - 4264204560\nu ) / 710700760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} + 8673867\nu^{3} + 4264204560\nu ) / 142140152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3345 \nu^{7} - 125829 \nu^{6} - 2236080 \nu^{5} + 5407157145 \nu^{3} + \cdots - 3087903495600 \nu ) / 15635416720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3411 \nu^{7} - 2236080 \nu^{5} + 98387520 \nu^{4} - 5597982219 \nu^{3} + \cdots + 105566316203040 ) / 7817708360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3411 \nu^{7} + 2236080 \nu^{5} + 49193760 \nu^{4} + 5597982219 \nu^{3} + 3181715995920 \nu + 52783158101520 ) / 3908854180 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10101 \nu^{7} - 377597 \nu^{6} + 6708240 \nu^{5} - 16412296509 \nu^{3} + \cdots + 9357522987120 \nu ) / 15635416720 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} - 5\beta_{2} ) / 60 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} + 15\beta_{4} + 5\beta_{2} + 34322\beta_1 ) / 360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 55\beta_{7} - 55\beta_{6} + 55\beta_{5} - 165\beta_{4} + 3411\beta_{3} + 16780\beta_{2} + 55\beta_1 ) / 180 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1907\beta_{6} + 3814\beta_{5} - 77253444 ) / 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 104885 \beta_{7} + 104885 \beta_{6} - 104885 \beta_{5} - 314655 \beta_{4} - 4268697 \beta_{3} + \cdots + 104885 \beta_1 ) / 180 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4818815\beta_{7} - 14456445\beta_{4} - 4818815\beta_{2} - 16021455446\beta_1 ) / 120 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 53006965 \beta_{7} + 53006965 \beta_{6} - 53006965 \beta_{5} + 159020895 \beta_{4} + \cdots - 53006965 \beta_1 ) / 60 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−26.0598 − 26.0598i
−26.0598 + 26.0598i
−16.5646 + 16.5646i
−16.5646 − 16.5646i
16.5646 − 16.5646i
16.5646 + 16.5646i
26.0598 + 26.0598i
26.0598 − 26.0598i
−312.717 3770.02 − 5369.70i 32256.2 0 −1.17895e6 + 1.67920e6i 7.10881e6i 1.04072e7 −1.46206e7 − 4.04878e7i 0
74.2 −312.717 3770.02 + 5369.70i 32256.2 0 −1.17895e6 − 1.67920e6i − 7.10881e6i 1.04072e7 −1.46206e7 + 4.04878e7i 0
74.3 −198.776 −4917.21 − 4343.70i −26024.2 0 977423. + 863422.i 5.53804e6i 1.81999e7 5.31128e6 + 4.27178e7i 0
74.4 −198.776 −4917.21 + 4343.70i −26024.2 0 977423. − 863422.i − 5.53804e6i 1.81999e7 5.31128e6 − 4.27178e7i 0
74.5 198.776 4917.21 − 4343.70i −26024.2 0 977423. − 863422.i 5.53804e6i −1.81999e7 5.31128e6 − 4.27178e7i 0
74.6 198.776 4917.21 + 4343.70i −26024.2 0 977423. + 863422.i − 5.53804e6i −1.81999e7 5.31128e6 + 4.27178e7i 0
74.7 312.717 −3770.02 − 5369.70i 32256.2 0 −1.17895e6 − 1.67920e6i 7.10881e6i −1.04072e7 −1.46206e7 + 4.04878e7i 0
74.8 312.717 −3770.02 + 5369.70i 32256.2 0 −1.17895e6 + 1.67920e6i − 7.10881e6i −1.04072e7 −1.46206e7 − 4.04878e7i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.8
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 75.17.d.b 8
3.b odd 2 1 inner 75.17.d.b 8
5.b even 2 1 inner 75.17.d.b 8
5.c odd 4 1 3.17.b.a ✓ 4
5.c odd 4 1 75.17.c.d 4
15.d odd 2 1 inner 75.17.d.b 8
15.e even 4 1 3.17.b.a ✓ 4
15.e even 4 1 75.17.c.d 4
20.e even 4 1 48.17.e.b 4
60.l odd 4 1 48.17.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.17.b.a ✓ 4 5.c odd 4 1
3.17.b.a ✓ 4 15.e even 4 1
48.17.e.b 4 20.e even 4 1
48.17.e.b 4 60.l odd 4 1
75.17.c.d 4 5.c odd 4 1
75.17.c.d 4 15.e even 4 1
75.17.d.b 8 1.a even 1 1 trivial
75.17.d.b 8 3.b odd 2 1 inner
75.17.d.b 8 5.b even 2 1 inner
75.17.d.b 8 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 137304T_{2}^{2} + 3863946240 \) acting on \(S_{17}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 137304 T^{2} + 3863946240)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 28\!\cdots\!60)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots - 15\!\cdots\!64)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 32\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots + 38\!\cdots\!44)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 26\!\cdots\!60)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 12\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 77\!\cdots\!04)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 10\!\cdots\!96)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 20\!\cdots\!60)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
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