Properties

Label 575.4.b.i
Level $575$
Weight $4$
Character orbit 575.b
Analytic conductor $33.926$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,4,Mod(24,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.24");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.b (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: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 55x^{8} + 1079x^{6} + 8937x^{4} + 26936x^{2} + 8464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_1) q^{2} + ( - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{3}+ \cdots + ( - 3 \beta_{5} + 6 \beta_{3} - 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_1) q^{2} + ( - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{3}+ \cdots + (35 \beta_{5} + 295 \beta_{4} + \cdots - 433) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 44 q^{4} + 38 q^{6} - 154 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 44 q^{4} + 38 q^{6} - 154 q^{9} + 46 q^{11} - 186 q^{14} + 564 q^{16} + 322 q^{19} - 120 q^{21} - 210 q^{24} - 514 q^{26} - 802 q^{29} + 64 q^{31} + 1326 q^{34} - 1318 q^{36} - 670 q^{39} - 24 q^{41} - 94 q^{44} - 276 q^{46} + 1476 q^{49} - 1986 q^{51} + 16 q^{54} + 686 q^{56} - 2648 q^{59} - 3346 q^{61} - 4932 q^{64} - 5562 q^{66} + 184 q^{69} - 216 q^{71} - 2916 q^{74} - 6954 q^{76} - 1312 q^{79} - 638 q^{81} + 1436 q^{84} + 224 q^{86} - 1140 q^{89} - 3178 q^{91} + 1896 q^{94} - 11982 q^{96} - 4042 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 55x^{8} + 1079x^{6} + 8937x^{4} + 26936x^{2} + 8464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 35\nu^{6} + 491\nu^{4} + 4493\nu^{2} + 14916 ) / 2240 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 63\nu^{6} + 1275\nu^{4} + 8273\nu^{2} + 4500 ) / 1120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 42\nu^{6} + 547\nu^{4} + 2218\nu^{2} + 552 ) / 280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{8} + 427\nu^{6} + 6435\nu^{4} + 32037\nu^{2} + 23140 ) / 2240 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{9} + 119\nu^{7} + 1305\nu^{5} + 1649\nu^{3} - 21220\nu ) / 12880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33\nu^{9} + 2275\nu^{7} + 54283\nu^{5} + 489869\nu^{3} + 1139588\nu ) / 103040 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -29\nu^{9} - 1687\nu^{7} - 31935\nu^{5} - 206457\nu^{3} - 202740\nu ) / 51520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 129\nu^{9} + 6083\nu^{7} + 96043\nu^{5} + 594157\nu^{3} + 1233348\nu ) / 103040 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} - 2\beta_{7} - 8\beta_{6} - 15\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -23\beta_{5} + 21\beta_{4} + 25\beta_{3} - 11\beta_{2} + 169 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -50\beta_{9} + 18\beta_{8} + 70\beta_{7} + 216\beta_{6} + 251\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 509\beta_{5} - 453\beta_{4} - 525\beta_{3} + 93\beta_{2} - 2875 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1090\beta_{9} - 744\beta_{8} - 1810\beta_{7} - 5168\beta_{6} - 4471\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -11015\beta_{5} + 10037\beta_{4} + 10593\beta_{3} - 107\beta_{2} + 52153 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -22586\beta_{9} + 21682\beta_{8} + 42446\beta_{7} + 119728\beta_{6} + 83483\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
4.60878i
3.41740i
3.93900i
2.49214i
0.595043i
0.595043i
2.49214i
3.93900i
3.41740i
4.60878i
5.60878i 1.89520i −23.4584 0 −10.6297 11.4426i 86.7031i 23.4082 0
24.2 4.41740i 7.84147i −11.5134 0 34.6389 8.97260i 15.5200i −34.4886 0
24.3 2.93900i 3.85751i −0.637693 0 11.3372 23.5932i 21.6378i 12.1196 0
24.4 1.49214i 9.02447i 5.77352 0 −13.4658 4.33445i 20.5520i −54.4411 0
24.5 0.404957i 7.11323i 7.83601 0 −2.88055 13.7888i 6.41290i −23.5981 0
24.6 0.404957i 7.11323i 7.83601 0 −2.88055 13.7888i 6.41290i −23.5981 0
24.7 1.49214i 9.02447i 5.77352 0 −13.4658 4.33445i 20.5520i −54.4411 0
24.8 2.93900i 3.85751i −0.637693 0 11.3372 23.5932i 21.6378i 12.1196 0
24.9 4.41740i 7.84147i −11.5134 0 34.6389 8.97260i 15.5200i −34.4886 0
24.10 5.60878i 1.89520i −23.4584 0 −10.6297 11.4426i 86.7031i 23.4082 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.4.b.i 10
5.b even 2 1 inner 575.4.b.i 10
5.c odd 4 1 115.4.a.e 5
5.c odd 4 1 575.4.a.j 5
15.e even 4 1 1035.4.a.k 5
20.e even 4 1 1840.4.a.n 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.e 5 5.c odd 4 1
575.4.a.j 5 5.c odd 4 1
575.4.b.i 10 1.a even 1 1 trivial
575.4.b.i 10 5.b even 2 1 inner
1035.4.a.k 5 15.e even 4 1
1840.4.a.n 5 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(575, [\chi])\):

\( T_{2}^{10} + 62T_{2}^{8} + 1197T_{2}^{6} + 7844T_{2}^{4} + 13060T_{2}^{2} + 1936 \) Copy content Toggle raw display
\( T_{3}^{10} + 212T_{3}^{8} + 15868T_{3}^{6} + 489817T_{3}^{4} + 5334656T_{3}^{2} + 13542400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 62 T^{8} + \cdots + 1936 \) Copy content Toggle raw display
$3$ \( T^{10} + 212 T^{8} + \cdots + 13542400 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 20959511076 \) Copy content Toggle raw display
$11$ \( (T^{5} - 23 T^{4} + \cdots - 74136848)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 2402859613456 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} - 161 T^{4} + \cdots + 801280)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{5} \) Copy content Toggle raw display
$29$ \( (T^{5} + 401 T^{4} + \cdots + 6149898500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 32 T^{4} + \cdots - 438072447)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{5} + 12 T^{4} + \cdots + 114116030755)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 24279649927232)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 34095834816896)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 31\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 15638892903635)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 90481602379776)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 115104799418880)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
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