↑

Properties

Label 3900.2.j.k
Level $3900$
Weight $2$
Character orbit 3900.j
Analytic conductor $31.142$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3900,2,Mod(649,3900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3900.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.j (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: \(31.1416567883\)
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: \( x^{12} + 37 x^{10} - 186 x^{9} + 646 x^{8} - 3588 x^{7} + 14267 x^{6} - 30978 x^{5} + 45388 x^{4} + \cdots + 3205 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_1 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + \beta_1 q^{7} - q^{9} + \beta_{11} q^{11} + ( - \beta_{7} + \beta_{4}) q^{13} + ( - \beta_{8} - \beta_{7} + \beta_{4}) q^{17} + ( - \beta_{11} - \beta_{10}) q^{19} - \beta_{10} q^{21} + (\beta_{9} - \beta_{4}) q^{23} + \beta_{4} q^{27} + (\beta_{6} - \beta_{5}) q^{29} + ( - \beta_{11} + \beta_{6} + \beta_{5} - 1) q^{31} - \beta_{2} q^{33} + ( - \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{37}+ \cdots - \beta_{11} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} + 6 q^{39} + 40 q^{49} + 12 q^{61} - 16 q^{69} + 8 q^{79} + 12 q^{81} - 18 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 37 x^{10} - 186 x^{9} + 646 x^{8} - 3588 x^{7} + 14267 x^{6} - 30978 x^{5} + 45388 x^{4} + \cdots + 3205 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 304451257918 \nu^{11} - 328670641086 \nu^{10} - 11510736145753 \nu^{9} + \cdots + 17\!\cdots\!67 ) / 25\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15229039502257 \nu^{11} - 18798187246899 \nu^{10} - 581372549607564 \nu^{9} + \cdots - 50\!\cdots\!74 ) / 46\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17014033080 \nu^{11} - 16877418165 \nu^{10} - 638864378562 \nu^{9} + \cdots - 305587799445484 ) / 35365944130441 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 189516957 \nu^{11} - 358531426 \nu^{10} - 7662373332 \nu^{9} + 20808540621 \nu^{8} + \cdots + 278946392075 ) / 56991798810 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 688872593243906 \nu^{11} + \cdots - 92\!\cdots\!35 ) / 77\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 711213127560086 \nu^{11} + \cdots - 95\!\cdots\!40 ) / 77\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26\!\cdots\!61 \nu^{11} + \cdots + 54\!\cdots\!35 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26\!\cdots\!26 \nu^{11} + \cdots + 37\!\cdots\!15 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39694630196973 \nu^{11} + 63900224389634 \nu^{10} + \cdots - 10\!\cdots\!75 ) / 31\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 971441720198553 \nu^{11} + \cdots + 16\!\cdots\!35 ) / 77\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!43 \nu^{11} + \cdots - 23\!\cdots\!10 ) / 77\!\cdots\!90 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{8} + \beta_{7} - 2\beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{11} - 2\beta_{10} - 3\beta_{8} - 3\beta_{7} - \beta_{6} - \beta_{5} + 3\beta_{4} - \beta_{3} + 6\beta _1 - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 13 \beta_{11} - 6 \beta_{10} - 12 \beta_{8} - 8 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 11 \beta_{4} + \cdots + 93 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 142 \beta_{11} + 66 \beta_{10} + 15 \beta_{9} + 141 \beta_{8} + 141 \beta_{7} + 31 \beta_{6} + 43 \beta_{5} + \cdots - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 248 \beta_{11} - 180 \beta_{10} - 37 \beta_{9} - 161 \beta_{8} - 305 \beta_{7} - 159 \beta_{6} + \cdots - 2586 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3124 \beta_{11} - 1840 \beta_{10} - 387 \beta_{9} - 3366 \beta_{8} - 2646 \beta_{7} - 268 \beta_{6} + \cdots + 10555 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19550 \beta_{11} + 14691 \beta_{10} + 3187 \beta_{9} + 17552 \beta_{8} + 19826 \beta_{7} + 6789 \beta_{6} + \cdots + 31494 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 11798 \beta_{11} + 702 \beta_{10} - 207 \beta_{9} + 28617 \beta_{8} - 6183 \beta_{7} - 30169 \beta_{6} + \cdots - 437808 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 612167 \beta_{11} - 511212 \beta_{10} - 113643 \beta_{9} - 624088 \beta_{8} - 549348 \beta_{7} + \cdots + 852072 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2138052 \beta_{11} + 2213506 \beta_{10} + 503541 \beta_{9} + 1610031 \beta_{8} + 2472087 \beta_{7} + \cdots + 9421920 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9242243 \beta_{11} + 7717647 \beta_{10} + 1711923 \beta_{9} + 11975858 \beta_{8} + 5723564 \beta_{7} + \cdots - 63761439 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
−2.10271 − 5.18463i
−1.81651 + 4.66885i
−0.170050 − 1.45157i
0.170050 + 0.257784i
1.81651 − 0.00671896i
2.10271 − 1.28371i
−2.10271 + 5.18463i
−1.81651 − 4.66885i
−0.170050 + 1.45157i
0.170050 − 0.257784i
1.81651 + 0.00671896i
2.10271 + 1.28371i
0 − 1.00000i 0 0 0 −4.20542 0 −1.00000 0
649.2 0 − 1.00000i 0 0 0 −3.63301 0 −1.00000 0
649.3 0 − 1.00000i 0 0 0 −0.340099 0 −1.00000 0
649.4 0 − 1.00000i 0 0 0 0.340099 0 −1.00000 0
649.5 0 − 1.00000i 0 0 0 3.63301 0 −1.00000 0
649.6 0 − 1.00000i 0 0 0 4.20542 0 −1.00000 0
649.7 0 1.00000i 0 0 0 −4.20542 0 −1.00000 0
649.8 0 1.00000i 0 0 0 −3.63301 0 −1.00000 0
649.9 0 1.00000i 0 0 0 −0.340099 0 −1.00000 0
649.10 0 1.00000i 0 0 0 0.340099 0 −1.00000 0
649.11 0 1.00000i 0 0 0 3.63301 0 −1.00000 0
649.12 0 1.00000i 0 0 0 4.20542 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.b even 2 1 inner
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3900.2.j.k 12
5.b even 2 1 inner 3900.2.j.k 12
5.c odd 4 1 3900.2.c.i ✓ 6
5.c odd 4 1 3900.2.c.j yes 6
13.b even 2 1 inner 3900.2.j.k 12
65.d even 2 1 inner 3900.2.j.k 12
65.h odd 4 1 3900.2.c.i ✓ 6
65.h odd 4 1 3900.2.c.j yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3900.2.c.i ✓ 6 5.c odd 4 1
3900.2.c.i ✓ 6 65.h odd 4 1
3900.2.c.j yes 6 5.c odd 4 1
3900.2.c.j yes 6 65.h odd 4 1
3900.2.j.k 12 1.a even 1 1 trivial
3900.2.j.k 12 5.b even 2 1 inner
3900.2.j.k 12 13.b even 2 1 inner
3900.2.j.k 12 65.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\):

\( T_{7}^{6} - 31T_{7}^{4} + 237T_{7}^{2} - 27 \) Copy content Toggle raw display
\( T_{11}^{6} + 40T_{11}^{4} + 441T_{11}^{2} + 972 \) Copy content Toggle raw display
\( T_{37}^{6} - 92T_{37}^{4} + 1540T_{37}^{2} - 6912 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 31 T^{4} + \cdots - 27)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 40 T^{4} + \cdots + 972)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 13 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} + 62 T^{4} + \cdots + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 71 T^{4} + \cdots + 300)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 172 T^{4} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 31 T + 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 79 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 92 T^{4} + \cdots - 6912)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 92 T^{4} + \cdots + 6912)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 185 T^{4} + \cdots + 39204)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 160 T^{4} + \cdots - 972)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 190 T^{4} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 104 T^{4} + \cdots + 8748)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 3 T^{2} + \cdots + 171)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 383 T^{4} + \cdots - 363)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 196 T^{4} + \cdots + 139968)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 196 T^{4} + \cdots - 139968)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 2 T^{2} + \cdots - 160)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 160 T^{4} + \cdots - 972)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 416 T^{4} + \cdots + 559872)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 179 T^{4} + \cdots - 17328)^{2} \) Copy content Toggle raw display
show more
show less