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Properties

Label 175.4.e.g
Level $175$
Weight $4$
Character orbit 175.e
Analytic conductor $10.325$
Analytic rank $0$
Dimension $20$
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] = [175,4,Mod(51,175)]
 
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([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.51");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.e (of order \(3\), 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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 55 x^{18} + 2018 x^{16} + 42095 x^{14} + 639938 x^{12} + 5744691 x^{10} + 35287093 x^{8} + \cdots + 9834496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{2} - \beta_{13} q^{3} + ( - \beta_{4} - 3 \beta_{3} - 3) q^{4} + (\beta_{7} + \beta_{2} - 5) q^{6} + ( - \beta_{19} + \beta_{18} + \cdots - \beta_{12}) q^{7}+ \cdots + (\beta_{11} - \beta_{9} + \cdots + 8 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{2} - \beta_{13} q^{3} + ( - \beta_{4} - 3 \beta_{3} - 3) q^{4} + (\beta_{7} + \beta_{2} - 5) q^{6} + ( - \beta_{19} + \beta_{18} + \cdots - \beta_{12}) q^{7}+ \cdots + ( - 10 \beta_{11} + 62 \beta_{10} + \cdots - 678) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 30 q^{4} - 96 q^{6} - 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 30 q^{4} - 96 q^{6} - 82 q^{9} + 36 q^{11} - 26 q^{14} - 22 q^{16} + 192 q^{19} - 404 q^{21} + 444 q^{24} + 434 q^{26} - 260 q^{29} + 834 q^{31} + 160 q^{34} + 516 q^{36} - 868 q^{39} - 1224 q^{41} - 542 q^{44} - 1274 q^{46} + 326 q^{49} + 986 q^{51} + 2808 q^{54} - 36 q^{56} + 2514 q^{59} + 512 q^{61} - 6900 q^{64} + 1396 q^{66} - 3064 q^{69} - 2944 q^{71} - 1590 q^{74} + 44 q^{76} - 46 q^{79} - 130 q^{81} + 12952 q^{84} - 1592 q^{86} + 5876 q^{89} + 4348 q^{91} + 3314 q^{94} + 3756 q^{96} - 13860 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 55 x^{18} + 2018 x^{16} + 42095 x^{14} + 639938 x^{12} + 5744691 x^{10} + 35287093 x^{8} + \cdots + 9834496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 77\!\cdots\!35 \nu^{18} + \cdots + 31\!\cdots\!44 ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26\!\cdots\!41 \nu^{18} + \cdots + 20\!\cdots\!32 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 94\!\cdots\!97 \nu^{18} + \cdots - 24\!\cdots\!52 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 26\!\cdots\!41 \nu^{19} + \cdots - 20\!\cdots\!32 \nu ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 99\!\cdots\!73 \nu^{18} + \cdots - 33\!\cdots\!52 ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 50\!\cdots\!73 \nu^{18} + \cdots + 27\!\cdots\!00 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 74\!\cdots\!19 \nu^{18} + \cdots + 23\!\cdots\!52 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 61\!\cdots\!51 \nu^{18} + \cdots + 19\!\cdots\!20 ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!05 \nu^{18} + \cdots + 97\!\cdots\!36 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 70\!\cdots\!21 \nu^{18} + \cdots + 58\!\cdots\!32 ) / 59\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\!\cdots\!97 \nu^{19} + \cdots + 84\!\cdots\!36 \nu ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 50\!\cdots\!83 \nu^{19} + \cdots + 39\!\cdots\!68 \nu ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 25\!\cdots\!81 \nu^{19} + \cdots - 79\!\cdots\!36 \nu ) / 63\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 33\!\cdots\!57 \nu^{19} + \cdots + 10\!\cdots\!56 \nu ) / 66\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 24\!\cdots\!27 \nu^{19} + \cdots - 26\!\cdots\!92 \nu ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 75\!\cdots\!49 \nu^{19} + \cdots + 23\!\cdots\!20 \nu ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!13 \nu^{19} + \cdots - 12\!\cdots\!16 \nu ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 29\!\cdots\!27 \nu^{19} + \cdots + 22\!\cdots\!96 \nu ) / 22\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 11\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - \beta_{16} + \beta_{14} + \beta_{13} - \beta_{12} - 18\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 2\beta_{8} - 2\beta_{6} - 23\beta_{4} - 203\beta_{3} - 203 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{17} - 30\beta_{15} - 28\beta_{14} + 52\beta_{12} + 345\beta_{5} - 345\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{11} + 72\beta_{10} - 84\beta_{7} - 521\beta_{2} + 4063 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -811\beta_{19} - 296\beta_{18} + 695\beta_{16} + 811\beta_{15} - 1695\beta_{13} + 6952\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 875 \beta_{11} - 1982 \beta_{10} - 875 \beta_{9} + 1982 \beta_{8} + 2622 \beta_{7} + \cdots + 11955 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20928 \beta_{19} + 8336 \beta_{18} + 8336 \beta_{17} - 16562 \beta_{16} + 16562 \beta_{14} + \cdots - 146143 \beta_{5} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20532\beta_{9} - 49796\beta_{8} - 72912\beta_{6} - 277077\beta_{4} - 1853999\beta_{3} - 1853999 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -213568\beta_{17} - 525077\beta_{15} - 388469\beta_{14} + 1249029\beta_{12} + 3177414\beta_{5} - 3177414\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -465429\beta_{11} + 1204074\beta_{10} - 1910714\beta_{7} - 6467103\beta_{2} + 41410267 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 12937890 \beta_{19} - 5249720 \beta_{18} + 9067464 \beta_{16} + 12937890 \beta_{15} + \cdots + 70864837 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 10446758 \beta_{11} - 28634368 \beta_{10} - 10446758 \beta_{9} + 28634368 \beta_{8} + \cdots + 151687873 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 314988255 \beta_{19} + 126533176 \beta_{18} + 126533176 \beta_{17} - 211662515 \beta_{16} + \cdots - 1610074820 \beta_{5} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 234869951 \beta_{9} - 676391382 \beta_{8} - 1198431894 \beta_{6} - 3570027435 \beta_{4} + \cdots - 21737945691 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3020954560 \beta_{17} - 7606844548 \beta_{15} - 4951028670 \beta_{14} + 19016822366 \beta_{12} + \cdots - 37066850795 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( -5316167352\beta_{11} + 15943966460\beta_{10} - 29279482792\beta_{7} - 84220374157\beta_{2} + 505840425135 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 182686621641 \beta_{19} - 71799750416 \beta_{18} + 116112842673 \beta_{16} + \cdots + 861246448994 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
2.43500 + 4.21755i
2.03314 + 3.52149i
1.81568 + 3.14484i
0.457951 + 0.793194i
0.425124 + 0.736336i
−0.425124 − 0.736336i
−0.457951 − 0.793194i
−1.81568 − 3.14484i
−2.03314 − 3.52149i
−2.43500 − 4.21755i
2.43500 − 4.21755i
2.03314 − 3.52149i
1.81568 − 3.14484i
0.457951 − 0.793194i
0.425124 − 0.736336i
−0.425124 + 0.736336i
−0.457951 + 0.793194i
−1.81568 + 3.14484i
−2.03314 + 3.52149i
−2.43500 + 4.21755i
−2.43500 + 4.21755i 4.00765 + 6.94145i −7.85846 − 13.6113i 0 −39.0345 −5.65324 + 17.6364i 37.5814 −18.6225 + 32.2551i 0
51.2 −2.03314 + 3.52149i −2.27280 − 3.93660i −4.26728 − 7.39115i 0 18.4836 −7.19331 − 17.0662i 2.17369 3.16878 − 5.48848i 0
51.3 −1.81568 + 3.14484i −0.194267 − 0.336480i −2.59336 − 4.49183i 0 1.41090 18.2971 + 2.86629i −10.2160 13.4245 − 23.2520i 0
51.4 −0.457951 + 0.793194i 4.38709 + 7.59866i 3.58056 + 6.20172i 0 −8.03628 −12.2610 − 13.8805i −13.8861 −24.9931 + 43.2892i 0
51.5 −0.425124 + 0.736336i −1.86785 − 3.23521i 3.63854 + 6.30214i 0 3.17627 −18.1502 + 3.68367i −12.9893 6.52228 − 11.2969i 0
51.6 0.425124 − 0.736336i 1.86785 + 3.23521i 3.63854 + 6.30214i 0 3.17627 18.1502 − 3.68367i 12.9893 6.52228 − 11.2969i 0
51.7 0.457951 − 0.793194i −4.38709 − 7.59866i 3.58056 + 6.20172i 0 −8.03628 12.2610 + 13.8805i 13.8861 −24.9931 + 43.2892i 0
51.8 1.81568 − 3.14484i 0.194267 + 0.336480i −2.59336 − 4.49183i 0 1.41090 −18.2971 − 2.86629i 10.2160 13.4245 − 23.2520i 0
51.9 2.03314 − 3.52149i 2.27280 + 3.93660i −4.26728 − 7.39115i 0 18.4836 7.19331 + 17.0662i −2.17369 3.16878 − 5.48848i 0
51.10 2.43500 − 4.21755i −4.00765 − 6.94145i −7.85846 − 13.6113i 0 −39.0345 5.65324 − 17.6364i −37.5814 −18.6225 + 32.2551i 0
151.1 −2.43500 − 4.21755i 4.00765 − 6.94145i −7.85846 + 13.6113i 0 −39.0345 −5.65324 − 17.6364i 37.5814 −18.6225 − 32.2551i 0
151.2 −2.03314 − 3.52149i −2.27280 + 3.93660i −4.26728 + 7.39115i 0 18.4836 −7.19331 + 17.0662i 2.17369 3.16878 + 5.48848i 0
151.3 −1.81568 − 3.14484i −0.194267 + 0.336480i −2.59336 + 4.49183i 0 1.41090 18.2971 − 2.86629i −10.2160 13.4245 + 23.2520i 0
151.4 −0.457951 − 0.793194i 4.38709 − 7.59866i 3.58056 − 6.20172i 0 −8.03628 −12.2610 + 13.8805i −13.8861 −24.9931 − 43.2892i 0
151.5 −0.425124 − 0.736336i −1.86785 + 3.23521i 3.63854 − 6.30214i 0 3.17627 −18.1502 − 3.68367i −12.9893 6.52228 + 11.2969i 0
151.6 0.425124 + 0.736336i 1.86785 − 3.23521i 3.63854 − 6.30214i 0 3.17627 18.1502 + 3.68367i 12.9893 6.52228 + 11.2969i 0
151.7 0.457951 + 0.793194i −4.38709 + 7.59866i 3.58056 − 6.20172i 0 −8.03628 12.2610 − 13.8805i 13.8861 −24.9931 − 43.2892i 0
151.8 1.81568 + 3.14484i 0.194267 − 0.336480i −2.59336 + 4.49183i 0 1.41090 −18.2971 + 2.86629i 10.2160 13.4245 + 23.2520i 0
151.9 2.03314 + 3.52149i 2.27280 − 3.93660i −4.26728 + 7.39115i 0 18.4836 7.19331 − 17.0662i −2.17369 3.16878 + 5.48848i 0
151.10 2.43500 + 4.21755i −4.00765 + 6.94145i −7.85846 + 13.6113i 0 −39.0345 5.65324 + 17.6364i −37.5814 −18.6225 − 32.2551i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.10
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 175.4.e.g 20
5.b even 2 1 inner 175.4.e.g 20
5.c odd 4 2 35.4.j.a ✓ 20
7.c even 3 1 inner 175.4.e.g 20
7.c even 3 1 1225.4.a.bp 10
7.d odd 6 1 1225.4.a.bq 10
35.f even 4 2 245.4.j.d 20
35.i odd 6 1 1225.4.a.bq 10
35.j even 6 1 inner 175.4.e.g 20
35.j even 6 1 1225.4.a.bp 10
35.k even 12 2 245.4.b.f 10
35.k even 12 2 245.4.j.d 20
35.l odd 12 2 35.4.j.a ✓ 20
35.l odd 12 2 245.4.b.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.j.a ✓ 20 5.c odd 4 2
35.4.j.a ✓ 20 35.l odd 12 2
175.4.e.g 20 1.a even 1 1 trivial
175.4.e.g 20 5.b even 2 1 inner
175.4.e.g 20 7.c even 3 1 inner
175.4.e.g 20 35.j even 6 1 inner
245.4.b.e 10 35.l odd 12 2
245.4.b.f 10 35.k even 12 2
245.4.j.d 20 35.f even 4 2
245.4.j.d 20 35.k even 12 2
1225.4.a.bp 10 7.c even 3 1
1225.4.a.bp 10 35.j even 6 1
1225.4.a.bq 10 7.d odd 6 1
1225.4.a.bq 10 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 55 T_{2}^{18} + 2018 T_{2}^{16} + 42095 T_{2}^{14} + 639938 T_{2}^{12} + 5744691 T_{2}^{10} + \cdots + 9834496 \) acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 55 T^{18} + \cdots + 9834496 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 46352367616 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 68542503321600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 3445 T^{8} + \cdots - 3071819776)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 80\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 94\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{5} + 65 T^{4} + \cdots - 1210995800)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + 306 T^{4} + \cdots + 30026757530)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 26\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 31\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + 736 T^{4} + \cdots - 445136828288)^{4} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 38\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 22\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 57\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 56\!\cdots\!24)^{2} \) Copy content Toggle raw display
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