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Properties

Label 175.3.j.a
Level $175$
Weight $3$
Character orbit 175.j
Analytic conductor $4.768$
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] = [175,3,Mod(24,175)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.24");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 175.j (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: \(4.76840462631\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 31 x^{18} + 628 x^{16} - 7327 x^{14} + 61903 x^{12} - 341587 x^{10} + 1372921 x^{8} + \cdots + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{13} q^{2} + (\beta_{15} - \beta_{12}) q^{3} + ( - \beta_{10} - 2 \beta_{4}) q^{4} + ( - \beta_{7} - \beta_{5} - 2 \beta_{4} - 2) q^{6} + (\beta_{17} - \beta_{15} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{9} - \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{2} + (\beta_{15} - \beta_{12}) q^{3} + ( - \beta_{10} - 2 \beta_{4}) q^{4} + ( - \beta_{7} - \beta_{5} - 2 \beta_{4} - 2) q^{6} + (\beta_{17} - \beta_{15} + \cdots + \beta_1) q^{7}+ \cdots + ( - 5 \beta_{9} - 3 \beta_{8} + \cdots - 22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 22 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 22 q^{4} - 4 q^{9} + 20 q^{11} + 70 q^{14} - 6 q^{16} - 132 q^{19} + 150 q^{21} - 252 q^{24} - 210 q^{26} + 160 q^{29} - 186 q^{31} + 36 q^{36} + 156 q^{39} + 72 q^{44} + 268 q^{46} + 316 q^{49} + 54 q^{51} - 30 q^{54} + 730 q^{56} - 414 q^{59} - 426 q^{61} - 288 q^{64} - 1068 q^{66} - 292 q^{71} + 638 q^{74} + 98 q^{79} + 454 q^{81} + 954 q^{84} + 280 q^{86} - 516 q^{89} + 486 q^{91} - 1590 q^{94} + 90 q^{96} - 596 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 31 x^{18} + 628 x^{16} - 7327 x^{14} + 61903 x^{12} - 341587 x^{10} + 1372921 x^{8} + \cdots + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1486641444 \nu^{18} + 42788039140 \nu^{16} - 840830301610 \nu^{14} + \cdots - 21\!\cdots\!69 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15938661121160 \nu^{18} + 553544737389551 \nu^{16} + \cdots - 11\!\cdots\!61 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 680698911163 \nu^{18} - 20740412375161 \nu^{16} + 417081422699344 \nu^{14} + \cdots - 11\!\cdots\!26 ) / 91\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 165448434390830 \nu^{18} + \cdots + 58\!\cdots\!01 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10056146310008 \nu^{18} + 347110089478691 \nu^{16} + \cdots + 34\!\cdots\!01 ) / 62\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 195068506280491 \nu^{18} + \cdots - 68\!\cdots\!87 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13362109765655 \nu^{18} + 511425153377048 \nu^{16} + \cdots + 29\!\cdots\!22 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 496042312459 \nu^{18} + 15087574308422 \nu^{16} - 296486819945303 \nu^{14} + \cdots - 95\!\cdots\!56 ) / 23\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1240979865362 \nu^{18} + 38014993579982 \nu^{16} - 766055590968278 \nu^{14} + \cdots + 21\!\cdots\!91 ) / 30\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 491214135901 \nu^{19} - 13512660338562 \nu^{17} + 265538091868413 \nu^{15} + \cdots + 70\!\cdots\!28 \nu ) / 83\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 660439710640381 \nu^{19} + \cdots - 47\!\cdots\!87 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 680698911163 \nu^{19} - 20740412375161 \nu^{17} + 417081422699344 \nu^{15} + \cdots - 11\!\cdots\!26 \nu ) / 91\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 7379950661 \nu^{19} + 212790368406 \nu^{17} - 4181556183519 \nu^{15} + \cdots - 59\!\cdots\!52 \nu ) / 53\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 24\!\cdots\!66 \nu^{19} + \cdots - 14\!\cdots\!25 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 21514748207 \nu^{19} + 651723132701 \nu^{17} - 13079344753709 \nu^{15} + \cdots + 64\!\cdots\!67 \nu ) / 31\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 24\!\cdots\!09 \nu^{19} + \cdots + 63\!\cdots\!25 \nu ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 13\!\cdots\!29 \nu^{19} + \cdots + 42\!\cdots\!45 \nu ) / 15\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 870848149084408 \nu^{19} + \cdots - 14\!\cdots\!37 \nu ) / 67\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 6\beta_{4} + \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{18} - \beta_{16} - 2\beta_{15} + 8\beta_{13} + \beta_{12} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{7} - 5\beta_{6} + \beta_{5} + 59\beta_{4} - \beta_{3} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} + 37 \beta_{18} - 2 \beta_{17} - 23 \beta_{16} - 16 \beta_{15} - 36 \beta_{14} + 78 \beta_{13} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 39\beta_{9} + 22\beta_{8} + 16\beta_{7} - 44\beta_{6} - 16\beta_{5} + 22\beta_{4} - 44\beta_{3} - 184\beta_{2} - 664 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -23\beta_{19} - 23\beta_{17} + 209\beta_{15} - 524\beta_{14} + 209\beta_{12} - 384\beta_{11} - 917\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2011 \beta_{10} + 977 \beta_{9} - 361 \beta_{8} - 209 \beta_{7} + 977 \beta_{6} - 418 \beta_{5} + \cdots - 8340 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 814 \beta_{19} - 8291 \beta_{18} + 407 \beta_{17} + 5770 \beta_{16} + 5298 \beta_{15} + \cdots - 10978 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 27153 \beta_{10} + 5363 \beta_{9} - 10726 \beta_{8} - 5298 \beta_{7} + 25194 \beta_{6} - 2649 \beta_{5} + \cdots + 65 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6456 \beta_{19} - 117541 \beta_{18} + 12912 \beta_{17} + 82904 \beta_{16} + 33942 \beta_{15} + \cdots + 6456 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 130453 \beta_{9} - 76448 \beta_{8} - 33942 \beta_{7} + 152896 \beta_{6} + 33942 \beta_{5} + \cdots + 1440589 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 96511 \beta_{19} + 96511 \beta_{17} - 443128 \beta_{15} + 1450888 \beta_{14} - 443128 \beta_{12} + \cdots + 1964982 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5059780 \beta_{10} - 2906392 \beta_{9} + 1069460 \beta_{8} + 443128 \beta_{7} - 2906392 \beta_{6} + \cdots + 19274061 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2787608 \beta_{19} + 22814588 \beta_{18} - 1393804 \beta_{17} - 16219864 \beta_{16} + \cdots + 25093425 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 69328797 \beta_{10} - 14826060 \beta_{9} + 29652120 \beta_{8} + 11763232 \beta_{7} - 70080376 \beta_{6} + \cdots - 3062828 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 19720580 \beta_{19} + 315196678 \beta_{18} - 39441160 \beta_{17} - 224315609 \beta_{16} + \cdots - 19720580 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 354637838 \beta_{9} + 204595029 \beta_{8} + 79016297 \beta_{7} - 409190058 \beta_{6} - 79016297 \beta_{5} + \cdots - 3612256474 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 275621541 \beta_{19} - 275621541 \beta_{17} + 1070179728 \beta_{15} - 3791722983 \beta_{14} + \cdots - 4907900740 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
−3.20834 − 1.85234i
−2.46987 − 1.42598i
−1.99886 − 1.15404i
−1.61094 − 0.930078i
−0.515466 − 0.297605i
0.515466 + 0.297605i
1.61094 + 0.930078i
1.99886 + 1.15404i
2.46987 + 1.42598i
3.20834 + 1.85234i
−3.20834 + 1.85234i
−2.46987 + 1.42598i
−1.99886 + 1.15404i
−1.61094 + 0.930078i
−0.515466 + 0.297605i
0.515466 − 0.297605i
1.61094 − 0.930078i
1.99886 − 1.15404i
2.46987 − 1.42598i
3.20834 − 1.85234i
−3.20834 + 1.85234i −0.796349 + 1.37932i 4.86231 − 8.42177i 0 − 5.90043i −2.34938 − 6.59397i 21.2079i 3.23166 + 5.59739i 0
24.2 −2.46987 + 1.42598i −2.20797 + 3.82432i 2.06684 − 3.57988i 0 − 12.5941i 5.20242 − 4.68346i 0.381273i −5.25031 − 9.09380i 0
24.3 −1.99886 + 1.15404i 1.30322 − 2.25725i 0.663615 − 1.14942i 0 6.01589i −6.99491 + 0.266827i − 6.16896i 1.10321 + 1.91082i 0
24.4 −1.61094 + 0.930078i 1.83746 − 3.18257i −0.269910 + 0.467497i 0 6.83591i 5.72890 + 4.02240i − 8.44477i −2.25249 − 3.90143i 0
24.5 −0.515466 + 0.297605i −1.07983 + 1.87032i −1.82286 + 3.15729i 0 − 1.28545i −6.90407 + 1.15494i − 4.55081i 2.16793 + 3.75496i 0
24.6 0.515466 − 0.297605i 1.07983 − 1.87032i −1.82286 + 3.15729i 0 − 1.28545i 6.90407 − 1.15494i 4.55081i 2.16793 + 3.75496i 0
24.7 1.61094 − 0.930078i −1.83746 + 3.18257i −0.269910 + 0.467497i 0 6.83591i −5.72890 − 4.02240i 8.44477i −2.25249 − 3.90143i 0
24.8 1.99886 − 1.15404i −1.30322 + 2.25725i 0.663615 − 1.14942i 0 6.01589i 6.99491 − 0.266827i 6.16896i 1.10321 + 1.91082i 0
24.9 2.46987 − 1.42598i 2.20797 − 3.82432i 2.06684 − 3.57988i 0 − 12.5941i −5.20242 + 4.68346i − 0.381273i −5.25031 − 9.09380i 0
24.10 3.20834 − 1.85234i 0.796349 − 1.37932i 4.86231 − 8.42177i 0 − 5.90043i 2.34938 + 6.59397i − 21.2079i 3.23166 + 5.59739i 0
124.1 −3.20834 − 1.85234i −0.796349 − 1.37932i 4.86231 + 8.42177i 0 5.90043i −2.34938 + 6.59397i − 21.2079i 3.23166 − 5.59739i 0
124.2 −2.46987 − 1.42598i −2.20797 − 3.82432i 2.06684 + 3.57988i 0 12.5941i 5.20242 + 4.68346i − 0.381273i −5.25031 + 9.09380i 0
124.3 −1.99886 − 1.15404i 1.30322 + 2.25725i 0.663615 + 1.14942i 0 − 6.01589i −6.99491 − 0.266827i 6.16896i 1.10321 − 1.91082i 0
124.4 −1.61094 − 0.930078i 1.83746 + 3.18257i −0.269910 − 0.467497i 0 − 6.83591i 5.72890 − 4.02240i 8.44477i −2.25249 + 3.90143i 0
124.5 −0.515466 − 0.297605i −1.07983 − 1.87032i −1.82286 − 3.15729i 0 1.28545i −6.90407 − 1.15494i 4.55081i 2.16793 − 3.75496i 0
124.6 0.515466 + 0.297605i 1.07983 + 1.87032i −1.82286 − 3.15729i 0 1.28545i 6.90407 + 1.15494i − 4.55081i 2.16793 − 3.75496i 0
124.7 1.61094 + 0.930078i −1.83746 − 3.18257i −0.269910 − 0.467497i 0 − 6.83591i −5.72890 + 4.02240i − 8.44477i −2.25249 + 3.90143i 0
124.8 1.99886 + 1.15404i −1.30322 − 2.25725i 0.663615 + 1.14942i 0 − 6.01589i 6.99491 + 0.266827i − 6.16896i 1.10321 − 1.91082i 0
124.9 2.46987 + 1.42598i 2.20797 + 3.82432i 2.06684 + 3.57988i 0 12.5941i −5.20242 − 4.68346i 0.381273i −5.25031 + 9.09380i 0
124.10 3.20834 + 1.85234i 0.796349 + 1.37932i 4.86231 + 8.42177i 0 5.90043i 2.34938 − 6.59397i 21.2079i 3.23166 − 5.59739i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.3.j.a 20
5.b even 2 1 inner 175.3.j.a 20
5.c odd 4 1 175.3.i.a ✓ 10
5.c odd 4 1 175.3.i.b yes 10
7.d odd 6 1 inner 175.3.j.a 20
35.i odd 6 1 inner 175.3.j.a 20
35.k even 12 1 175.3.i.a ✓ 10
35.k even 12 1 175.3.i.b yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.3.i.a ✓ 10 5.c odd 4 1
175.3.i.a ✓ 10 35.k even 12 1
175.3.i.b yes 10 5.c odd 4 1
175.3.i.b yes 10 35.k even 12 1
175.3.j.a 20 1.a even 1 1 trivial
175.3.j.a 20 5.b even 2 1 inner
175.3.j.a 20 7.d odd 6 1 inner
175.3.j.a 20 35.i odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 31 T_{2}^{18} + 628 T_{2}^{16} - 7327 T_{2}^{14} + 61903 T_{2}^{12} - 341587 T_{2}^{10} + \cdots + 531441 \) acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 31 T^{18} + \cdots + 531441 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 448084224 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{10} - 10 T^{9} + \cdots + 5789992464)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 1230 T^{8} + \cdots - 99219178800)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 46\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( (T^{10} + 66 T^{9} + \cdots + 10401976368)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 37\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{5} - 40 T^{4} + \cdots - 14241648)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 6043467398787)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 52457962921875)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 75608242090000)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 95\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 14\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 115474655017008)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{5} + 73 T^{4} + \cdots + 21335037)^{4} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 53\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 36897521394889)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 29\!\cdots\!67)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 34\!\cdots\!75)^{2} \) Copy content Toggle raw display
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