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Properties

Label 175.2.k.b
Level $175$
Weight $2$
Character orbit 175.k
Analytic conductor $1.397$
Analytic rank $0$
Dimension $12$
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] = [175,2,Mod(74,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.74");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.k (of order \(6\), degree \(2\), 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: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{11} + \beta_{10}) q^{3} + ( - \beta_{8} + \beta_{4} + \beta_{3} + 1) q^{4} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_{2}) q^{6} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{8} - \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{11} + \beta_{10}) q^{3} + ( - \beta_{8} + \beta_{4} + \beta_{3} + 1) q^{4} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_{2}) q^{6} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_1) q^{7}+ \cdots + (2 \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} - 8 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} - 8 q^{6} + 8 q^{9} + 4 q^{11} - 34 q^{14} + 10 q^{16} - 8 q^{19} - 22 q^{21} - 20 q^{24} - 30 q^{26} + 32 q^{29} - 10 q^{31} + 48 q^{34} + 60 q^{36} + 24 q^{39} - 4 q^{41} + 12 q^{44} + 20 q^{46} - 24 q^{49} - 22 q^{51} - 54 q^{54} - 42 q^{56} - 10 q^{59} - 10 q^{61} + 64 q^{64} - 52 q^{66} - 48 q^{69} + 28 q^{71} - 30 q^{74} + 60 q^{76} - 14 q^{79} + 26 q^{81} + 26 q^{84} + 60 q^{86} - 12 q^{89} - 6 q^{91} - 34 q^{94} - 34 q^{96} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} - 36\nu^{8} + 236\nu^{6} - 987\nu^{4} + 1612\nu^{2} - 792 ) / 795 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -81\nu^{10} + 531\nu^{8} - 3481\nu^{6} + 3627\nu^{4} - 1782\nu^{2} - 54303 ) / 21995 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1298\nu^{10} - 10953\nu^{8} + 71803\nu^{6} - 202311\nu^{4} + 490451\nu^{2} - 240966 ) / 197955 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 704\nu^{10} - 7059\nu^{8} + 38944\nu^{6} - 109728\nu^{4} + 147458\nu^{2} - 23328 ) / 65985 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 81\nu^{11} - 531\nu^{9} + 3481\nu^{7} - 3627\nu^{5} + 1782\nu^{3} + 76298\nu ) / 21995 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -288\nu^{11} + 1888\nu^{9} - 9933\nu^{7} + 12896\nu^{5} - 6336\nu^{3} - 90434\nu ) / 65985 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 211\nu^{10} - 1872\nu^{8} + 12272\nu^{6} - 38286\nu^{4} + 83824\nu^{2} - 41184 ) / 13197 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1298\nu^{11} + 10953\nu^{9} - 71803\nu^{7} + 202311\nu^{5} - 490451\nu^{3} + 43011\nu ) / 197955 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4712\nu^{11} - 47997\nu^{9} + 314647\nu^{7} - 1022364\nu^{5} + 2149199\nu^{3} - 1055934\nu ) / 593865 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4463\nu^{11} - 39033\nu^{9} + 255883\nu^{7} - 776601\nu^{5} + 1747811\nu^{3} - 858726\nu ) / 197955 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + 3\beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - 4\beta_{9} - \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{8} + 12\beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{11} + 3\beta_{10} - 17\beta_{9} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{5} - 23\beta_{3} + 9\beta_{2} - 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} + 32\beta_{6} - 74\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 106\beta_{8} - 59\beta_{5} - 222\beta_{4} - 106\beta_{3} - 222 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 165\beta_{11} - 177\beta_{10} + 328\beta_{9} + 177\beta_{7} + 165\beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 493\beta_{8} - 984\beta_{4} - 342\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 835\beta_{11} - 1026\beta_{10} + 1477\beta_{9} + 1477\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(\beta_{4}\) \(-1\)

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
−1.90412 + 1.09935i
−1.65604 + 0.956115i
−0.617942 + 0.356769i
0.617942 − 0.356769i
1.65604 − 0.956115i
1.90412 − 1.09935i
−1.90412 − 1.09935i
−1.65604 − 0.956115i
−0.617942 − 0.356769i
0.617942 + 0.356769i
1.65604 + 0.956115i
1.90412 + 1.09935i
−1.90412 + 1.09935i 2.45453 + 1.41712i 1.41712 − 2.45453i 0 −6.23163 1.03810 + 2.43359i 1.83424i 2.51647 + 4.35865i 0
74.2 −1.65604 + 0.956115i −1.43468 − 0.828310i 0.828310 − 1.43468i 0 3.16784 2.52206 + 0.799494i − 0.656620i −0.127804 − 0.221364i 0
74.3 −0.617942 + 0.356769i −1.29113 − 0.745432i −0.745432 + 1.29113i 0 1.06379 −0.248083 − 2.63409i − 2.49086i −0.388663 − 0.673184i 0
74.4 0.617942 − 0.356769i 1.29113 + 0.745432i −0.745432 + 1.29113i 0 1.06379 0.248083 + 2.63409i 2.49086i −0.388663 − 0.673184i 0
74.5 1.65604 − 0.956115i 1.43468 + 0.828310i 0.828310 − 1.43468i 0 3.16784 −2.52206 − 0.799494i 0.656620i −0.127804 − 0.221364i 0
74.6 1.90412 − 1.09935i −2.45453 − 1.41712i 1.41712 − 2.45453i 0 −6.23163 −1.03810 − 2.43359i − 1.83424i 2.51647 + 4.35865i 0
149.1 −1.90412 − 1.09935i 2.45453 − 1.41712i 1.41712 + 2.45453i 0 −6.23163 1.03810 − 2.43359i − 1.83424i 2.51647 − 4.35865i 0
149.2 −1.65604 − 0.956115i −1.43468 + 0.828310i 0.828310 + 1.43468i 0 3.16784 2.52206 − 0.799494i 0.656620i −0.127804 + 0.221364i 0
149.3 −0.617942 − 0.356769i −1.29113 + 0.745432i −0.745432 − 1.29113i 0 1.06379 −0.248083 + 2.63409i 2.49086i −0.388663 + 0.673184i 0
149.4 0.617942 + 0.356769i 1.29113 − 0.745432i −0.745432 − 1.29113i 0 1.06379 0.248083 − 2.63409i − 2.49086i −0.388663 + 0.673184i 0
149.5 1.65604 + 0.956115i 1.43468 − 0.828310i 0.828310 + 1.43468i 0 3.16784 −2.52206 + 0.799494i − 0.656620i −0.127804 + 0.221364i 0
149.6 1.90412 + 1.09935i −2.45453 + 1.41712i 1.41712 + 2.45453i 0 −6.23163 −1.03810 + 2.43359i 1.83424i 2.51647 − 4.35865i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.k.b 12
5.b even 2 1 inner 175.2.k.b 12
5.c odd 4 1 175.2.e.d ✓ 6
5.c odd 4 1 175.2.e.e yes 6
7.c even 3 1 inner 175.2.k.b 12
7.c even 3 1 1225.2.b.l 6
7.d odd 6 1 1225.2.b.m 6
35.i odd 6 1 1225.2.b.m 6
35.j even 6 1 inner 175.2.k.b 12
35.j even 6 1 1225.2.b.l 6
35.k even 12 1 1225.2.a.w 3
35.k even 12 1 1225.2.a.z 3
35.l odd 12 1 175.2.e.d ✓ 6
35.l odd 12 1 175.2.e.e yes 6
35.l odd 12 1 1225.2.a.x 3
35.l odd 12 1 1225.2.a.y 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.e.d ✓ 6 5.c odd 4 1
175.2.e.d ✓ 6 35.l odd 12 1
175.2.e.e yes 6 5.c odd 4 1
175.2.e.e yes 6 35.l odd 12 1
175.2.k.b 12 1.a even 1 1 trivial
175.2.k.b 12 5.b even 2 1 inner
175.2.k.b 12 7.c even 3 1 inner
175.2.k.b 12 35.j even 6 1 inner
1225.2.a.w 3 35.k even 12 1
1225.2.a.x 3 35.l odd 12 1
1225.2.a.y 3 35.l odd 12 1
1225.2.a.z 3 35.k even 12 1
1225.2.b.l 6 7.c even 3 1
1225.2.b.l 6 35.j even 6 1
1225.2.b.m 6 7.d odd 6 1
1225.2.b.m 6 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 9T_{2}^{10} + 59T_{2}^{8} - 180T_{2}^{6} + 403T_{2}^{4} - 198T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 9 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{12} - 13 T^{10} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 12 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} - 2 T^{5} + \cdots + 2025)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 46 T^{4} + \cdots + 49)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 42 T^{10} + \cdots + 194481 \) Copy content Toggle raw display
$19$ \( (T^{6} + 4 T^{5} + 33 T^{4} + \cdots + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 21 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$29$ \( (T^{3} - 8 T^{2} - 5 T + 75)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 5 T^{5} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 82 T^{10} + \cdots + 43046721 \) Copy content Toggle raw display
$41$ \( (T^{3} + T^{2} - 46 T + 105)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 121 T^{4} + \cdots + 2809)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 291843050625 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 228886641 \) Copy content Toggle raw display
$59$ \( (T^{6} + 5 T^{5} + \cdots + 11025)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 5 T^{5} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 88 T^{10} + \cdots + 2560000 \) Copy content Toggle raw display
$71$ \( (T^{3} - 7 T^{2} + \cdots + 423)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 40282095616 \) Copy content Toggle raw display
$79$ \( (T^{6} + 7 T^{5} + \cdots + 22801)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 278 T^{4} + \cdots + 159201)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 6 T^{5} + \cdots + 603729)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 269 T^{4} + \cdots + 67081)^{2} \) Copy content Toggle raw display
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