Properties

Label 575.4.a.j
Level $575$
Weight $4$
Character orbit 575.a
Self dual yes
Analytic conductor $33.926$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,4,Mod(1,575)]
 
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([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

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: yes
Analytic conductor: \(33.9260982533\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 4) q^{4}+ \cdots + ( - 3 \beta_{4} + 6 \beta_{3} + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 4) q^{4}+ \cdots + (35 \beta_{4} + 121 \beta_{3} + \cdots + 433) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 6 q^{2} - 4 q^{3} + 22 q^{4} + 19 q^{6} + 3 q^{7} - 138 q^{8} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 6 q^{2} - 4 q^{3} + 22 q^{4} + 19 q^{6} + 3 q^{7} - 138 q^{8} + 77 q^{9} + 23 q^{11} - 47 q^{12} - 132 q^{13} + 93 q^{14} + 282 q^{16} - 23 q^{17} + 15 q^{18} - 161 q^{19} - 60 q^{21} - 193 q^{22} + 115 q^{23} + 105 q^{24} - 257 q^{26} - 577 q^{27} - 17 q^{28} + 401 q^{29} + 32 q^{31} - 670 q^{32} - 189 q^{33} - 663 q^{34} - 659 q^{36} + 38 q^{37} + 875 q^{38} + 335 q^{39} - 12 q^{41} + 798 q^{42} + 566 q^{43} + 47 q^{44} - 138 q^{46} - 919 q^{47} + 773 q^{48} - 738 q^{49} - 993 q^{51} + 305 q^{52} - 1156 q^{53} - 8 q^{54} + 343 q^{56} - 114 q^{57} + 1042 q^{58} + 1324 q^{59} - 1673 q^{61} - 565 q^{62} - 270 q^{63} + 2466 q^{64} - 2781 q^{66} - 558 q^{67} + 2267 q^{68} - 92 q^{69} - 108 q^{71} + 789 q^{72} - 1173 q^{73} + 1458 q^{74} - 3477 q^{76} - 2608 q^{77} - 704 q^{78} + 656 q^{79} - 319 q^{81} - 3505 q^{82} + 82 q^{83} - 718 q^{84} + 112 q^{86} - 2389 q^{87} - 2397 q^{88} + 570 q^{89} - 1589 q^{91} + 506 q^{92} - 911 q^{93} - 948 q^{94} - 5991 q^{96} - 633 q^{97} + 2555 q^{98} + 2021 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + \nu^{3} - 25\nu^{2} - 11\nu + 98 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 3\nu^{3} + 17\nu^{2} - 41\nu - 42 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 7\nu^{3} + 25\nu^{2} - 109\nu - 162 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} + 2\beta_{2} + 15\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 23\beta_{4} - 25\beta_{3} - 11\beta_{2} + 21\beta _1 + 169 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.60878
3.41740
−0.595043
−2.49214
−3.93900
−5.60878 1.89520 23.4584 0 −10.6297 −11.4426 −86.7031 −23.4082 0
1.2 −4.41740 −7.84147 11.5134 0 34.6389 8.97260 −15.5200 34.4886 0
1.3 −0.404957 7.11323 −7.83601 0 −2.88055 −13.7888 6.41290 23.5981 0
1.4 1.49214 −9.02447 −5.77352 0 −13.4658 −4.33445 −20.5520 54.4411 0
1.5 2.93900 3.85751 0.637693 0 11.3372 23.5932 −21.6378 −12.1196 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(23\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(575))\):

\( T_{2}^{5} + 6T_{2}^{4} - 13T_{2}^{3} - 72T_{2}^{2} + 82T_{2} + 44 \) Copy content Toggle raw display
\( T_{3}^{5} + 4T_{3}^{4} - 98T_{3}^{3} - 149T_{3}^{2} + 2536T_{3} - 3680 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 6 T^{4} + \cdots + 44 \) Copy content Toggle raw display
$3$ \( T^{5} + 4 T^{4} + \cdots - 3680 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 3 T^{4} + \cdots + 144774 \) Copy content Toggle raw display
$11$ \( T^{5} - 23 T^{4} + \cdots - 74136848 \) Copy content Toggle raw display
$13$ \( T^{5} + 132 T^{4} + \cdots - 1550116 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots + 1039045340 \) Copy content Toggle raw display
$19$ \( T^{5} + 161 T^{4} + \cdots - 801280 \) Copy content Toggle raw display
$23$ \( (T - 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 6149898500 \) Copy content Toggle raw display
$31$ \( T^{5} - 32 T^{4} + \cdots - 438072447 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 1590700778176 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 114116030755 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 504784881664 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 117787714816 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 5720332226904 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 24279649927232 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 34095834816896 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 5644442112 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 15638892903635 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 100895881632176 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 90481602379776 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 18307318870176 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 115104799418880 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 480989167569272 \) Copy content Toggle raw display
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