Properties

Label 3675.2.a.cb
Level $3675$
Weight $2$
Character orbit 3675.a
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $4$
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] = [3675,2,Mod(1,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, 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("3675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.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: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{8} + q^{9} + (\beta_{3} - \beta_1 + 1) q^{11} + (\beta_{2} - \beta_1 + 2) q^{12} + ( - 2 \beta_{2} + \beta_1 - 2) q^{13} + ( - 2 \beta_{3} - 2 \beta_1) q^{16} + (2 \beta_{2} - \beta_1 + 1) q^{17} + ( - \beta_1 + 1) q^{18} + (\beta_{2} + \beta_1 + 3) q^{19} + (\beta_{3} + 4) q^{22} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 5) q^{23} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{24} + (2 \beta_{3} - \beta_{2} + 4 \beta_1 - 3) q^{26} + q^{27} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{29} + ( - 2 \beta_{2} + 1) q^{31} + (2 \beta_{2} + 2) q^{32} + (\beta_{3} - \beta_1 + 1) q^{33} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{34} + (\beta_{2} - \beta_1 + 2) q^{36} + ( - \beta_{3} - \beta_1 + 6) q^{37} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{38} + ( - 2 \beta_{2} + \beta_1 - 2) q^{39} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{41} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{43} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{44} + ( - \beta_{2} - 5 \beta_1 + 6) q^{46} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{47} + ( - 2 \beta_{3} - 2 \beta_1) q^{48} + (2 \beta_{2} - \beta_1 + 1) q^{51} + (3 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 10) q^{52} + (2 \beta_{2} + 6) q^{53} + ( - \beta_1 + 1) q^{54} + (\beta_{2} + \beta_1 + 3) q^{57} + ( - 3 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 4) q^{58} + ( - \beta_{2} - 1) q^{59} + ( - 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{61} + (2 \beta_{3} + \beta_1 + 3) q^{62} + 2 \beta_{3} q^{64} + (\beta_{3} + 4) q^{66} + (2 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{67} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1 + 8) q^{68} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 5) q^{69} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 3) q^{71} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{72} + ( - \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 4) q^{73} + ( - \beta_{3} + 2 \beta_{2} - 7 \beta_1 + 9) q^{74} + (3 \beta_{2} - \beta_1 + 6) q^{76} + (2 \beta_{3} - \beta_{2} + 4 \beta_1 - 3) q^{78} + (4 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{79} + q^{81} + (3 \beta_{3} + 4 \beta_1 + 4) q^{82} + (2 \beta_{3} + 3 \beta_1 + 1) q^{83} + (3 \beta_{3} + \beta_1 + 7) q^{86} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{87} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{88} + ( - \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 7) q^{89} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1 + 12) q^{92} + ( - 2 \beta_{2} + 1) q^{93} + (2 \beta_{2} - 2 \beta_1 + 6) q^{94} + (2 \beta_{2} + 2) q^{96} - \beta_{3} q^{97} + (\beta_{3} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{6} + 6 q^{8} + 4 q^{9} + 4 q^{12} - 2 q^{13} - 2 q^{17} + 2 q^{18} + 12 q^{19} + 14 q^{22} + 10 q^{23} + 6 q^{24} - 6 q^{26} + 4 q^{27} - 6 q^{29} + 8 q^{31} + 4 q^{32} + 4 q^{34} + 4 q^{36} + 24 q^{37} - 8 q^{38} - 2 q^{39} - 4 q^{41} + 8 q^{43} + 10 q^{44} + 16 q^{46} + 10 q^{47} - 2 q^{51} - 34 q^{52} + 20 q^{53} + 2 q^{54} + 12 q^{57} + 10 q^{58} - 2 q^{59} + 8 q^{61} + 10 q^{62} - 4 q^{64} + 14 q^{66} + 6 q^{67} + 30 q^{68} + 10 q^{69} - 14 q^{71} + 6 q^{72} - 12 q^{73} + 20 q^{74} + 16 q^{76} - 6 q^{78} - 8 q^{79} + 4 q^{81} + 18 q^{82} + 6 q^{83} + 24 q^{86} - 6 q^{87} - 12 q^{88} - 8 q^{89} + 46 q^{92} + 8 q^{93} + 16 q^{94} + 4 q^{96} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \) 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
2.78165
1.28734
−0.552409
−1.51658
−1.78165 1.00000 1.17429 0 −1.78165 0 1.47113 1.00000 0
1.2 −0.287336 1.00000 −1.91744 0 −0.287336 0 1.12562 1.00000 0
1.3 1.55241 1.00000 0.409975 0 1.55241 0 −2.46837 1.00000 0
1.4 2.51658 1.00000 4.33317 0 2.51658 0 5.87162 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 3675.2.a.cb 4
5.b even 2 1 3675.2.a.bn 4
5.c odd 4 2 735.2.d.e 8
7.b odd 2 1 3675.2.a.bz 4
7.d odd 6 2 525.2.i.h 8
15.e even 4 2 2205.2.d.o 8
35.c odd 2 1 3675.2.a.bp 4
35.f even 4 2 735.2.d.d 8
35.i odd 6 2 525.2.i.k 8
35.k even 12 4 105.2.q.a 16
35.l odd 12 4 735.2.q.g 16
105.k odd 4 2 2205.2.d.s 8
105.w odd 12 4 315.2.bf.b 16
140.x odd 12 4 1680.2.di.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.q.a 16 35.k even 12 4
315.2.bf.b 16 105.w odd 12 4
525.2.i.h 8 7.d odd 6 2
525.2.i.k 8 35.i odd 6 2
735.2.d.d 8 35.f even 4 2
735.2.d.e 8 5.c odd 4 2
735.2.q.g 16 35.l odd 12 4
1680.2.di.d 16 140.x odd 12 4
2205.2.d.o 8 15.e even 4 2
2205.2.d.s 8 105.k odd 4 2
3675.2.a.bn 4 5.b even 2 1
3675.2.a.bp 4 35.c odd 2 1
3675.2.a.bz 4 7.b odd 2 1
3675.2.a.cb 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 6T_{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{4} - 18T_{11}^{2} - 14T_{11} + 30 \) Copy content Toggle raw display
\( T_{13}^{4} + 2T_{13}^{3} - 28T_{13}^{2} - 36T_{13} + 127 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 4 T^{2} + 6 T + 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{2} - 14 T + 30 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} - 28 T^{2} - 36 T + 127 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 28 T^{2} - 22 T + 134 \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} + 38 T^{2} - 40 T + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} - 32 T^{2} + \cdots - 1506 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} - 38 T^{2} - 190 T - 22 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} - 6 T^{2} + 96 T + 61 \) Copy content Toggle raw display
$37$ \( T^{4} - 24 T^{3} + 204 T^{2} + \cdots + 907 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} - 50 T^{2} - 146 T - 10 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} - 32 T^{2} + 154 T - 49 \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + 16 T^{2} + \cdots - 120 \) Copy content Toggle raw display
$53$ \( T^{4} - 20 T^{3} + 120 T^{2} + \cdots + 96 \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} - 6 T^{2} - 6 T + 10 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} - 100 T^{2} + \cdots + 500 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} - 72 T^{2} + 544 T - 905 \) Copy content Toggle raw display
$71$ \( T^{4} + 14 T^{3} - 90 T^{2} + \cdots - 3202 \) Copy content Toggle raw display
$73$ \( T^{4} + 12 T^{3} - 64 T^{2} + \cdots - 1389 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} - 154 T^{2} + \cdots + 7081 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} - 56 T^{2} + 238 T + 362 \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} - 194 T^{2} + \cdots - 534 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} - 8 T^{2} + 16 T - 4 \) Copy content Toggle raw display
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