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Properties

Label 3525.2.a.bg
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $1$
Dimension $10$
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] = [3525,2,Mod(1,3525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.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: \(28.1472667125\)
Analytic rank: \(1\)
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: \( x^{10} - 3x^{9} - 9x^{8} + 29x^{7} + 25x^{6} - 91x^{5} - 21x^{4} + 101x^{3} + 6x^{2} - 30x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 705)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_1 q^{6} - \beta_{4} q^{7} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_1 q^{6} - \beta_{4} q^{7} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{8}+ \cdots + (\beta_{9} - \beta_{4} - \beta_{3} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{2} - 10 q^{3} + 7 q^{4} - 3 q^{6} + 9 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{2} - 10 q^{3} + 7 q^{4} - 3 q^{6} + 9 q^{8} + 10 q^{9} - 16 q^{11} - 7 q^{12} + q^{13} - 12 q^{14} - 3 q^{16} + 14 q^{17} + 3 q^{18} - 26 q^{19} + 7 q^{23} - 9 q^{24} - 10 q^{26} - 10 q^{27} - 24 q^{28} - 14 q^{29} - 22 q^{31} + 11 q^{32} + 16 q^{33} - 12 q^{34} + 7 q^{36} + 2 q^{37} - 2 q^{38} - q^{39} - 22 q^{41} + 12 q^{42} - 11 q^{43} - 36 q^{44} - 14 q^{46} + 10 q^{47} + 3 q^{48} + 2 q^{49} - 14 q^{51} + 14 q^{52} + 22 q^{53} - 3 q^{54} - 48 q^{56} + 26 q^{57} - 20 q^{58} - 37 q^{59} - 25 q^{61} - 2 q^{62} - 7 q^{64} - 4 q^{67} + 8 q^{68} - 7 q^{69} - 27 q^{71} + 9 q^{72} + q^{73} + 4 q^{74} - 42 q^{76} + 34 q^{77} + 10 q^{78} + 5 q^{79} + 10 q^{81} - 32 q^{82} + 2 q^{83} + 24 q^{84} - 6 q^{86} + 14 q^{87} - 58 q^{88} + 9 q^{89} - 64 q^{91} + 34 q^{92} + 22 q^{93} + 3 q^{94} - 11 q^{96} - 40 q^{97} + 29 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 9x^{8} + 29x^{7} + 25x^{6} - 91x^{5} - 21x^{4} + 101x^{3} + 6x^{2} - 30x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{9} + 3\nu^{8} + 22\nu^{7} - 23\nu^{6} - 82\nu^{5} + 43\nu^{4} + 110\nu^{3} - 3\nu^{2} - 40\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{9} - 3\nu^{8} - 22\nu^{7} + 23\nu^{6} + 84\nu^{5} - 47\nu^{4} - 118\nu^{3} + 19\nu^{2} + 40\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{9} - 3\nu^{8} - 22\nu^{7} + 23\nu^{6} + 84\nu^{5} - 45\nu^{4} - 122\nu^{3} + 11\nu^{2} + 54\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{9} - \nu^{8} - 12\nu^{7} + 7\nu^{6} + 49\nu^{5} - 9\nu^{4} - 73\nu^{3} - 9\nu^{2} + 29\nu + 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} - 5\nu^{8} - 33\nu^{7} + 43\nu^{6} + 121\nu^{5} - 105\nu^{4} - 155\nu^{3} + 63\nu^{2} + 48\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{9} + 4\nu^{8} + 35\nu^{7} - 32\nu^{6} - 139\nu^{5} + 66\nu^{4} + 201\nu^{3} - 20\nu^{2} - 78\nu - 10 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} + 2\beta_{3} + 6\beta_{2} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{6} + 7\beta_{5} + \beta_{4} + 8\beta_{3} + 8\beta_{2} + 18\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{7} - 8\beta_{6} + 9\beta_{5} + 2\beta_{4} + 17\beta_{3} + 34\beta_{2} + 11\beta _1 + 65 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} - 32 \beta_{6} + 42 \beta_{5} + 10 \beta_{4} + 52 \beta_{3} + \cdots + 78 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 17 \beta_{9} + 13 \beta_{8} + 17 \beta_{7} - 51 \beta_{6} + 64 \beta_{5} + 24 \beta_{4} + 115 \beta_{3} + \cdots + 351 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 58 \beta_{9} + 30 \beta_{8} + 47 \beta_{7} - 167 \beta_{6} + 244 \beta_{5} + 81 \beta_{4} + 319 \beta_{3} + \cdots + 551 \) Copy content Toggle raw display

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

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
−2.04563
−1.88719
−1.16553
−0.563403
−0.138546
0.801361
1.49126
1.55802
2.38580
2.56385
−2.04563 −1.00000 2.18458 0 2.04563 1.36348 −0.377587 1.00000 0
1.2 −1.88719 −1.00000 1.56149 0 1.88719 1.44822 0.827559 1.00000 0
1.3 −1.16553 −1.00000 −0.641537 0 1.16553 −4.15990 3.07879 1.00000 0
1.4 −0.563403 −1.00000 −1.68258 0 0.563403 1.09545 2.07478 1.00000 0
1.5 −0.138546 −1.00000 −1.98081 0 0.138546 3.39422 0.551524 1.00000 0
1.6 0.801361 −1.00000 −1.35782 0 −0.801361 −0.0381502 −2.69083 1.00000 0
1.7 1.49126 −1.00000 0.223857 0 −1.49126 −1.33606 −2.64869 1.00000 0
1.8 1.55802 −1.00000 0.427425 0 −1.55802 3.87455 −2.45010 1.00000 0
1.9 2.38580 −1.00000 3.69205 0 −2.38580 −1.19234 4.03690 1.00000 0
1.10 2.56385 −1.00000 4.57334 0 −2.56385 −4.44946 6.59766 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(47\) \(-1\)

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 3525.2.a.bg 10
5.b even 2 1 3525.2.a.bf 10
5.c odd 4 2 705.2.c.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.c.b ✓ 20 5.c odd 4 2
3525.2.a.bf 10 5.b even 2 1
3525.2.a.bg 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2}^{10} - 3T_{2}^{9} - 9T_{2}^{8} + 29T_{2}^{7} + 25T_{2}^{6} - 91T_{2}^{5} - 21T_{2}^{4} + 101T_{2}^{3} + 6T_{2}^{2} - 30T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{10} - 36T_{7}^{8} + 20T_{7}^{7} + 382T_{7}^{6} - 388T_{7}^{5} - 980T_{7}^{4} + 1084T_{7}^{3} + 745T_{7}^{2} - 812T_{7} - 32 \) Copy content Toggle raw display
\( T_{11}^{10} + 16 T_{11}^{9} + 65 T_{11}^{8} - 172 T_{11}^{7} - 1652 T_{11}^{6} - 1832 T_{11}^{5} + \cdots + 8192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 3 T^{9} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 36 T^{8} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( T^{10} + 16 T^{9} + \cdots + 8192 \) Copy content Toggle raw display
$13$ \( T^{10} - T^{9} + \cdots - 188 \) Copy content Toggle raw display
$17$ \( T^{10} - 14 T^{9} + \cdots - 13912 \) Copy content Toggle raw display
$19$ \( T^{10} + 26 T^{9} + \cdots - 1782512 \) Copy content Toggle raw display
$23$ \( T^{10} - 7 T^{9} + \cdots - 7036 \) Copy content Toggle raw display
$29$ \( T^{10} + 14 T^{9} + \cdots - 25147124 \) Copy content Toggle raw display
$31$ \( T^{10} + 22 T^{9} + \cdots + 891008 \) Copy content Toggle raw display
$37$ \( T^{10} - 2 T^{9} + \cdots - 33248 \) Copy content Toggle raw display
$41$ \( T^{10} + 22 T^{9} + \cdots - 74416 \) Copy content Toggle raw display
$43$ \( T^{10} + 11 T^{9} + \cdots + 16405888 \) Copy content Toggle raw display
$47$ \( (T - 1)^{10} \) Copy content Toggle raw display
$53$ \( T^{10} - 22 T^{9} + \cdots - 361232 \) Copy content Toggle raw display
$59$ \( T^{10} + 37 T^{9} + \cdots + 7796972 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 507918704 \) Copy content Toggle raw display
$67$ \( T^{10} + 4 T^{9} + \cdots - 1106944 \) Copy content Toggle raw display
$71$ \( T^{10} + 27 T^{9} + \cdots - 17743288 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 102689648 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 182722976 \) Copy content Toggle raw display
$83$ \( T^{10} - 2 T^{9} + \cdots - 1600000 \) Copy content Toggle raw display
$89$ \( T^{10} - 9 T^{9} + \cdots - 11488 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1969378304 \) Copy content Toggle raw display
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