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Properties

Label 3800.2.d.p
Level $3800$
Weight $2$
Character orbit 3800.d
Analytic conductor $30.343$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 28x^{10} + 274x^{8} + 1078x^{6} + 1385x^{4} + 478x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{9} q^{7} + (\beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{9} q^{7} + (\beta_{3} + \beta_{2} - 1) q^{9} + (\beta_{10} - \beta_{7} + \beta_{2} + 1) q^{11} + \beta_{8} q^{13} + ( - \beta_{11} - \beta_{8} + \beta_{5}) q^{17} - q^{19} + ( - \beta_{7} + \beta_{4} + 2) q^{21} + ( - \beta_{11} - \beta_{9}) q^{23} + ( - 2 \beta_{9} + 3 \beta_{6} - 3 \beta_1) q^{27} + ( - \beta_{10} + 2 \beta_{2} - 1) q^{29} + (\beta_{7} - \beta_{4} + \beta_{2} + 1) q^{31} + (\beta_{11} - \beta_{9} - 2 \beta_{8} + \cdots + \beta_1) q^{33}+ \cdots + (\beta_{10} + 6 \beta_{7} - 3 \beta_{4} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 20 q^{9} + 6 q^{11} - 12 q^{19} + 22 q^{21} - 14 q^{29} + 10 q^{31} - 16 q^{39} + 22 q^{41} + 4 q^{49} + 26 q^{51} + 8 q^{59} + 26 q^{61} - 14 q^{69} + 58 q^{71} - 56 q^{79} + 76 q^{81} + 24 q^{89} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 28x^{10} + 274x^{8} + 1078x^{6} + 1385x^{4} + 478x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 27\nu^{8} + 252\nu^{6} + 926\nu^{4} + 1024\nu^{2} + 189 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} - 27\nu^{8} - 252\nu^{6} - 926\nu^{4} - 1014\nu^{2} - 149 ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 28\nu^{8} + 272\nu^{6} + 1037\nu^{4} + 1153\nu^{2} + 203 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -37\nu^{11} - 1029\nu^{9} - 9914\nu^{7} - 37422\nu^{5} - 40878\nu^{3} - 6003\nu ) / 140 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\nu^{11} + 749\nu^{9} + 7209\nu^{7} + 27342\nu^{5} + 30913\nu^{3} + 5738\nu ) / 70 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{10} + 221\nu^{8} + 2116\nu^{6} + 7973\nu^{4} + 8927\nu^{2} + 1622 ) / 20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -79\nu^{11} - 2198\nu^{9} - 21233\nu^{7} - 80899\nu^{5} - 92006\nu^{3} - 16706\nu ) / 140 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 81\nu^{11} + 2247\nu^{9} + 21627\nu^{7} + 82026\nu^{5} + 92669\nu^{3} + 16584\nu ) / 140 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19\nu^{10} + 528\nu^{8} + 5093\nu^{6} + 19369\nu^{4} + 21976\nu^{2} + 4046 ) / 20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -101\nu^{11} - 2807\nu^{9} - 27072\nu^{7} - 102886\nu^{5} - 116484\nu^{3} - 21559\nu ) / 70 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} + 3\beta_{6} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 2\beta_{4} - 9\beta_{3} - 12\beta_{2} + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} + 26\beta_{9} + 4\beta_{8} - 43\beta_{6} + 2\beta_{5} + 84\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{10} - 44\beta_{7} + 32\beta_{4} + 81\beta_{3} + 139\beta_{2} - 285 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 58\beta_{11} - 302\beta_{9} - 80\beta_{8} + 531\beta_{6} - 32\beta_{5} - 808\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -80\beta_{10} + 658\beta_{7} - 414\beta_{4} - 750\beta_{3} - 1587\beta_{2} + 2650 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -837\beta_{11} + 3400\beta_{9} + 1152\beta_{8} - 6262\beta_{6} + 414\beta_{5} + 7985\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1152\beta_{10} - 8530\beta_{7} + 4966\beta_{4} + 7148\beta_{3} + 17919\beta_{2} - 25455 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10771\beta_{11} - 37724\beta_{9} - 14648\beta_{8} + 72048\beta_{6} - 4966\beta_{5} - 80746\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3649.1
− 3.30105i
− 2.79951i
− 2.70452i
− 1.22174i
− 0.486697i
− 0.471016i
0.471016i
0.486697i
1.22174i
2.70452i
2.79951i
3.30105i
0 − 3.30105i 0 0 0 − 1.63094i 0 −7.89694 0
3649.2 0 − 2.79951i 0 0 0 0.127550i 0 −4.83726 0
3649.3 0 − 2.70452i 0 0 0 3.77934i 0 −4.31442 0
3649.4 0 − 1.22174i 0 0 0 3.08602i 0 1.50735 0
3649.5 0 − 0.486697i 0 0 0 3.63249i 0 2.76313 0
3649.6 0 − 0.471016i 0 0 0 0.567324i 0 2.77814 0
3649.7 0 0.471016i 0 0 0 − 0.567324i 0 2.77814 0
3649.8 0 0.486697i 0 0 0 − 3.63249i 0 2.76313 0
3649.9 0 1.22174i 0 0 0 − 3.08602i 0 1.50735 0
3649.10 0 2.70452i 0 0 0 − 3.77934i 0 −4.31442 0
3649.11 0 2.79951i 0 0 0 − 0.127550i 0 −4.83726 0
3649.12 0 3.30105i 0 0 0 1.63094i 0 −7.89694 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3649.12
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 3800.2.d.p 12
5.b even 2 1 inner 3800.2.d.p 12
5.c odd 4 1 3800.2.a.bb ✓ 6
5.c odd 4 1 3800.2.a.bd yes 6
20.e even 4 1 7600.2.a.ci 6
20.e even 4 1 7600.2.a.cm 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.bb ✓ 6 5.c odd 4 1
3800.2.a.bd yes 6 5.c odd 4 1
3800.2.d.p 12 1.a even 1 1 trivial
3800.2.d.p 12 5.b even 2 1 inner
7600.2.a.ci 6 20.e even 4 1
7600.2.a.cm 6 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 \(S_{2}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{12} + 28T_{3}^{10} + 274T_{3}^{8} + 1078T_{3}^{6} + 1385T_{3}^{4} + 478T_{3}^{2} + 49 \) Copy content Toggle raw display
\( T_{7}^{12} + 40T_{7}^{10} + 562T_{7}^{8} + 3178T_{7}^{6} + 5789T_{7}^{4} + 1630T_{7}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{6} - 3T_{11}^{5} - 52T_{11}^{4} + 115T_{11}^{3} + 906T_{11}^{2} - 1080T_{11} - 5400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 28 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 40 T^{10} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} + \cdots - 5400)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 95 T^{10} + \cdots + 1369 \) Copy content Toggle raw display
$17$ \( T^{12} + 128 T^{10} + \cdots + 555025 \) Copy content Toggle raw display
$19$ \( (T + 1)^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 100 T^{10} + \cdots + 344569 \) Copy content Toggle raw display
$29$ \( (T^{6} + 7 T^{5} + \cdots - 14717)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 5 T^{5} + \cdots - 296)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12355878649 \) Copy content Toggle raw display
$41$ \( (T^{6} - 11 T^{5} + \cdots - 3584)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 6332339776 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 8103060289 \) Copy content Toggle raw display
$53$ \( T^{12} + 113 T^{10} + \cdots + 1408969 \) Copy content Toggle raw display
$59$ \( (T^{6} - 4 T^{5} + \cdots - 11944)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 13 T^{5} + \cdots - 3880)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 201838849 \) Copy content Toggle raw display
$71$ \( (T^{6} - 29 T^{5} + \cdots - 39960)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 227 T^{10} + \cdots + 2968729 \) Copy content Toggle raw display
$79$ \( (T^{6} + 28 T^{5} + \cdots + 15040)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 229 T^{10} + \cdots + 40000 \) Copy content Toggle raw display
$89$ \( (T^{6} - 12 T^{5} + \cdots - 274808)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 437 T^{10} + \cdots + 3182656 \) Copy content Toggle raw display
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