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Properties

Label 175.5.g.b
Level $175$
Weight $5$
Character orbit 175.g
Analytic conductor $18.090$
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] = [175,5,Mod(43,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.43");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 175.g (of order \(4\), degree \(2\), 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: \(18.0897435397\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 78x^{10} + 2049x^{8} + 22752x^{6} + 104040x^{4} + 162336x^{2} + 15376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4}) q^{2} + (2 \beta_{7} - 3 \beta_{5}) q^{3} + ( - \beta_{9} - \beta_{8} + 8 \beta_1) q^{4} + (2 \beta_{3} - 3 \beta_{2} - 53) q^{6} - 7 \beta_{4} q^{7} + ( - 2 \beta_{10} + 8 \beta_{7} - 24 \beta_{5}) q^{8} + (4 \beta_{9} + 8 \beta_{8} - 42 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{4}) q^{2} + (2 \beta_{7} - 3 \beta_{5}) q^{3} + ( - \beta_{9} - \beta_{8} + 8 \beta_1) q^{4} + (2 \beta_{3} - 3 \beta_{2} - 53) q^{6} - 7 \beta_{4} q^{7} + ( - 2 \beta_{10} + 8 \beta_{7} - 24 \beta_{5}) q^{8} + (4 \beta_{9} + 8 \beta_{8} - 42 \beta_1) q^{9} + ( - 7 \beta_{3} - 11 \beta_{2} + 5) q^{11} + ( - 5 \beta_{11} - 57 \beta_{6} + 91 \beta_{4}) q^{12} + ( - \beta_{10} + 33 \beta_{7} + 24 \beta_{5}) q^{13} + ( - 7 \beta_{8} + 35 \beta_1) q^{14} + (2 \beta_{3} - 26 \beta_{2} - 144) q^{16} + (5 \beta_{11} + 49 \beta_{6} - 20 \beta_{4}) q^{17} + (12 \beta_{10} - 122 \beta_{7} + 258 \beta_{5}) q^{18} + ( - 24 \beta_{9} + 18 \beta_{8} + 142 \beta_1) q^{19} + ( - 14 \beta_{2} - 119) q^{21} + ( - 4 \beta_{11} + \cdots + 167 \beta_{4}) q^{22}+ \cdots + ( - 530 \beta_{9} + 214 \beta_{8} + 5278 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 632 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 632 q^{6} + 132 q^{11} - 1632 q^{16} - 1372 q^{21} - 6296 q^{26} + 4104 q^{31} + 34752 q^{36} - 16040 q^{41} + 10416 q^{46} - 25124 q^{51} - 12544 q^{56} - 1104 q^{61} + 12984 q^{66} + 36184 q^{71} - 52224 q^{76} - 88892 q^{81} + 80768 q^{86} + 20580 q^{91} + 131904 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 78x^{10} + 2049x^{8} + 22752x^{6} + 104040x^{4} + 162336x^{2} + 15376 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{11} - 668\nu^{9} - 44091\nu^{7} - 1164354\nu^{5} - 11868052\nu^{3} - 32195544\nu ) / 9860480 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 293\nu^{10} + 20748\nu^{8} + 460861\nu^{6} + 3932694\nu^{4} + 10999532\nu^{2} + 1039944 ) / 477120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 377\nu^{10} + 28092\nu^{8} + 673009\nu^{6} + 6169326\nu^{4} + 18090908\nu^{2} + 11527656 ) / 477120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2840 \nu^{11} + 279 \nu^{10} - 212716 \nu^{9} + 26908 \nu^{8} - 5202880 \nu^{7} + \cdots + 62524024 ) / 29581440 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2840 \nu^{11} - 279 \nu^{10} - 212716 \nu^{9} - 26908 \nu^{8} - 5202880 \nu^{7} + \cdots - 62524024 ) / 29581440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2831 \nu^{11} + 1581 \nu^{10} - 210712 \nu^{9} + 140740 \nu^{8} - 5070607 \nu^{7} + \cdots + 32804200 ) / 29581440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2831 \nu^{11} - 1581 \nu^{10} - 210712 \nu^{9} - 140740 \nu^{8} - 5070607 \nu^{7} + \cdots - 32804200 ) / 29581440 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6193\nu^{11} + 469228\nu^{9} + 11606441\nu^{7} + 112373774\nu^{5} + 341204572\nu^{3} + 26998504\nu ) / 29581440 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1799\nu^{11} - 148196\nu^{9} - 4218607\nu^{7} - 51711346\nu^{5} - 262949444\nu^{3} - 406660376\nu ) / 5916288 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26797 \nu^{11} - 32178 \nu^{10} + 2081300 \nu^{9} - 2340376 \nu^{8} + 54001109 \nu^{7} + \cdots + 1084611632 ) / 29581440 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26797 \nu^{11} - 32178 \nu^{10} - 2081300 \nu^{9} - 2340376 \nu^{8} - 54001109 \nu^{7} + \cdots + 1084611632 ) / 29581440 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - 2\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} - 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} + \beta_{10} + 3 \beta_{9} + 3 \beta_{8} - 23 \beta_{7} - 23 \beta_{6} + 31 \beta_{5} + \cdots - 74 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{11} + 4 \beta_{10} - 84 \beta_{7} + 84 \beta_{6} + 116 \beta_{5} - 116 \beta_{4} - 31 \beta_{3} + \cdots + 666 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 41 \beta_{11} - 41 \beta_{10} - 135 \beta_{9} - 195 \beta_{8} + 689 \beta_{7} + 689 \beta_{6} + \cdots + 2842 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 206 \beta_{11} - 206 \beta_{10} + 3310 \beta_{7} - 3310 \beta_{6} - 5318 \beta_{5} + 5318 \beta_{4} + \cdots - 21318 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1541 \beta_{11} + 1541 \beta_{10} + 5369 \beta_{9} + 8813 \beta_{8} - 23425 \beta_{7} + \cdots - 110798 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 8632 \beta_{11} + 8632 \beta_{10} - 128664 \beta_{7} + 128664 \beta_{6} + 218360 \beta_{5} + \cdots + 751738 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 57809 \beta_{11} - 57809 \beta_{10} - 208323 \beta_{9} - 357903 \beta_{8} + 847237 \beta_{7} + \cdots + 4288154 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 340922 \beta_{11} - 340922 \beta_{10} + 4957058 \beta_{7} - 4957058 \beta_{6} - 8579754 \beta_{5} + \cdots - 27671302 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2180881 \beta_{11} + 2180881 \beta_{10} + 8006273 \beta_{9} + 13983365 \beta_{8} + \cdots - 164754174 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
4.16946i
1.74581i
0.318006i
− 2.31801i
− 3.74581i
− 6.16946i
− 4.16946i
− 1.74581i
− 0.318006i
2.31801i
3.74581i
6.16946i
−5.16946 + 5.16946i 12.2098 + 12.2098i − 37.4467i 0 −126.236 −13.0958 + 13.0958i 110.868 + 110.868i 217.156i 0
43.2 −2.74581 + 2.74581i 3.62080 + 3.62080i 0.921016i 0 −19.8841 13.0958 − 13.0958i −46.4620 − 46.4620i − 54.7796i 0
43.3 −1.31801 + 1.31801i 4.50684 + 4.50684i 12.5257i 0 −11.8801 −13.0958 + 13.0958i −37.5971 − 37.5971i − 40.3768i 0
43.4 1.31801 − 1.31801i −4.50684 − 4.50684i 12.5257i 0 −11.8801 13.0958 − 13.0958i 37.5971 + 37.5971i − 40.3768i 0
43.5 2.74581 − 2.74581i −3.62080 − 3.62080i 0.921016i 0 −19.8841 −13.0958 + 13.0958i 46.4620 + 46.4620i − 54.7796i 0
43.6 5.16946 − 5.16946i −12.2098 − 12.2098i − 37.4467i 0 −126.236 13.0958 − 13.0958i −110.868 − 110.868i 217.156i 0
57.1 −5.16946 − 5.16946i 12.2098 − 12.2098i 37.4467i 0 −126.236 −13.0958 − 13.0958i 110.868 − 110.868i − 217.156i 0
57.2 −2.74581 − 2.74581i 3.62080 − 3.62080i − 0.921016i 0 −19.8841 13.0958 + 13.0958i −46.4620 + 46.4620i 54.7796i 0
57.3 −1.31801 − 1.31801i 4.50684 − 4.50684i − 12.5257i 0 −11.8801 −13.0958 − 13.0958i −37.5971 + 37.5971i 40.3768i 0
57.4 1.31801 + 1.31801i −4.50684 + 4.50684i − 12.5257i 0 −11.8801 13.0958 + 13.0958i 37.5971 − 37.5971i 40.3768i 0
57.5 2.74581 + 2.74581i −3.62080 + 3.62080i − 0.921016i 0 −19.8841 −13.0958 − 13.0958i 46.4620 − 46.4620i 54.7796i 0
57.6 5.16946 + 5.16946i −12.2098 + 12.2098i 37.4467i 0 −126.236 13.0958 + 13.0958i −110.868 + 110.868i − 217.156i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.5.g.b ✓ 12
5.b even 2 1 inner 175.5.g.b ✓ 12
5.c odd 4 2 inner 175.5.g.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.5.g.b ✓ 12 1.a even 1 1 trivial
175.5.g.b ✓ 12 5.b even 2 1 inner
175.5.g.b ✓ 12 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 3096T_{2}^{8} + 686736T_{2}^{4} + 7840000 \) acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3096 T^{8} + \cdots + 7840000 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 100858961889 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{4} + 117649)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} - 33 T^{2} + \cdots - 1406007)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 43\!\cdots\!69 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 19\!\cdots\!89)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 1026 T^{2} + \cdots + 126947800)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 40\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{3} + 4010 T^{2} + \cdots + 2102148328)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 29\!\cdots\!09 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 35\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 276 T^{2} + \cdots + 52598753312)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{3} - 9046 T^{2} + \cdots + 37011086680)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 10\!\cdots\!49)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 20\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 29\!\cdots\!09 \) Copy content Toggle raw display
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