Properties

Label 3850.2.c.ba
Level $3850$
Weight $2$
Character orbit 3850.c
Analytic conductor $30.742$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3850,2,Mod(1849,3850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3850.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (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: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 770)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{5} + \beta_{2}) q^{3} - q^{4} + (\beta_1 + 1) q^{6} + \beta_{2} q^{7} + \beta_{2} q^{8} + ( - \beta_{4} - \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{5} + \beta_{2}) q^{3} - q^{4} + (\beta_1 + 1) q^{6} + \beta_{2} q^{7} + \beta_{2} q^{8} + ( - \beta_{4} - \beta_1 - 3) q^{9} + q^{11} + ( - \beta_{5} - \beta_{2}) q^{12} + ( - \beta_{5} + \beta_{3}) q^{13} + q^{14} + q^{16} + 2 \beta_{5} q^{17} + (\beta_{5} - \beta_{3} + 3 \beta_{2}) q^{18} + (2 \beta_{4} + \beta_1 + 3) q^{19} + ( - \beta_1 - 1) q^{21} - \beta_{2} q^{22} + ( - \beta_{5} + 3 \beta_{2}) q^{23} + ( - \beta_1 - 1) q^{24} + ( - \beta_{4} - \beta_1) q^{26} + ( - 2 \beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{27} - \beta_{2} q^{28} + (3 \beta_1 + 1) q^{29} + ( - \beta_{4} + \beta_1 - 4) q^{31} - \beta_{2} q^{32} + (\beta_{5} + \beta_{2}) q^{33} + 2 \beta_1 q^{34} + (\beta_{4} + \beta_1 + 3) q^{36} + (\beta_{5} - \beta_{2}) q^{37} + ( - \beta_{5} + 2 \beta_{3} - 3 \beta_{2}) q^{38} + (2 \beta_{4} + 2 \beta_1 + 4) q^{39} + (2 \beta_{4} + 3 \beta_1 + 3) q^{41} + (\beta_{5} + \beta_{2}) q^{42} + (\beta_{5} - 3 \beta_{3} + 4 \beta_{2}) q^{43} - q^{44} + ( - \beta_1 + 3) q^{46} + (2 \beta_{5} + 2 \beta_{3}) q^{47} + (\beta_{5} + \beta_{2}) q^{48} - q^{49} + ( - 2 \beta_{4} - 10) q^{51} + (\beta_{5} - \beta_{3}) q^{52} + (3 \beta_{5} + 5 \beta_{2}) q^{53} + ( - 2 \beta_{4} - 2 \beta_1 - 4) q^{54} - q^{56} + (7 \beta_{5} - 3 \beta_{3} + 6 \beta_{2}) q^{57} + ( - 3 \beta_{5} - \beta_{2}) q^{58} + ( - 2 \beta_{4} - 2 \beta_1) q^{59} + (2 \beta_{4} - 2 \beta_1 - 2) q^{61} + ( - \beta_{5} - \beta_{3} + 4 \beta_{2}) q^{62} + ( - \beta_{5} + \beta_{3} - 3 \beta_{2}) q^{63} - q^{64} + (\beta_1 + 1) q^{66} + 4 \beta_{5} q^{67} - 2 \beta_{5} q^{68} + (\beta_{4} - 3 \beta_1 + 2) q^{69} + ( - 2 \beta_{4} - 10) q^{71} + ( - \beta_{5} + \beta_{3} - 3 \beta_{2}) q^{72} + (2 \beta_{5} - 4 \beta_{2}) q^{73} + (\beta_1 - 1) q^{74} + ( - 2 \beta_{4} - \beta_1 - 3) q^{76} + \beta_{2} q^{77} + ( - 2 \beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{78} + ( - 2 \beta_{4} - \beta_1 - 7) q^{79} + (\beta_{4} + 5 \beta_1 + 3) q^{81} + ( - 3 \beta_{5} + 2 \beta_{3} - 3 \beta_{2}) q^{82} + (2 \beta_{5} - 2 \beta_{2}) q^{83} + (\beta_1 + 1) q^{84} + (3 \beta_{4} + \beta_1 + 4) q^{86} + (\beta_{5} - 3 \beta_{3} + 16 \beta_{2}) q^{87} + \beta_{2} q^{88} + (2 \beta_{4} + 4 \beta_1 - 4) q^{89} + (\beta_{4} + \beta_1) q^{91} + (\beta_{5} - 3 \beta_{2}) q^{92} + ( - 6 \beta_{5} + 2 \beta_{2}) q^{93} + ( - 2 \beta_{4} + 2 \beta_1) q^{94} + (\beta_1 + 1) q^{96} + ( - 3 \beta_{5} - 2 \beta_{3} + \beta_{2}) q^{97} + \beta_{2} q^{98} + ( - \beta_{4} - \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 4 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 4 q^{6} - 14 q^{9} + 6 q^{11} + 6 q^{14} + 6 q^{16} + 12 q^{19} - 4 q^{21} - 4 q^{24} + 4 q^{26} - 24 q^{31} - 4 q^{34} + 14 q^{36} + 16 q^{39} + 8 q^{41} - 6 q^{44} + 20 q^{46} - 6 q^{49} - 56 q^{51} - 16 q^{54} - 6 q^{56} + 8 q^{59} - 12 q^{61} - 6 q^{64} + 4 q^{66} + 16 q^{69} - 56 q^{71} - 8 q^{74} - 12 q^{76} - 36 q^{79} + 6 q^{81} + 4 q^{84} + 16 q^{86} - 36 q^{89} - 4 q^{91} + 4 q^{96} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + \nu^{3} + 4\nu^{2} + 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - \nu^{3} + 4\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 7\nu^{3} + 8\nu^{2} - 6\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} + 2\nu^{3} - 3\nu^{2} + 3\nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{5} - \beta_{4} - \beta_{3} + 4\beta_{2} + 3\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 3\beta_{4} + \beta_{3} + 10\beta_{2} + \beta _1 + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{5} + 5\beta_{4} - 3\beta_{3} - 4\beta_{2} + \beta _1 + 10 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
0.264658 1.38923i
−0.671462 + 1.24464i
1.40680 + 0.144584i
1.40680 0.144584i
−0.671462 1.24464i
0.264658 + 1.38923i
1.00000i 2.24914i −1.00000 0 −2.24914 1.00000i 1.00000i −2.05863 0
1849.2 1.00000i 1.14637i −1.00000 0 1.14637 1.00000i 1.00000i 1.68585 0
1849.3 1.00000i 3.10278i −1.00000 0 3.10278 1.00000i 1.00000i −6.62721 0
1849.4 1.00000i 3.10278i −1.00000 0 3.10278 1.00000i 1.00000i −6.62721 0
1849.5 1.00000i 1.14637i −1.00000 0 1.14637 1.00000i 1.00000i 1.68585 0
1849.6 1.00000i 2.24914i −1.00000 0 −2.24914 1.00000i 1.00000i −2.05863 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.6
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 3850.2.c.ba 6
5.b even 2 1 inner 3850.2.c.ba 6
5.c odd 4 1 770.2.a.m 3
5.c odd 4 1 3850.2.a.bt 3
15.e even 4 1 6930.2.a.ce 3
20.e even 4 1 6160.2.a.bf 3
35.f even 4 1 5390.2.a.ca 3
55.e even 4 1 8470.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.m 3 5.c odd 4 1
3850.2.a.bt 3 5.c odd 4 1
3850.2.c.ba 6 1.a even 1 1 trivial
3850.2.c.ba 6 5.b even 2 1 inner
5390.2.a.ca 3 35.f even 4 1
6160.2.a.bf 3 20.e even 4 1
6930.2.a.ce 3 15.e even 4 1
8470.2.a.ci 3 55.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 \(S_{2}^{\mathrm{new}}(3850, [\chi])\):

\( T_{3}^{6} + 16T_{3}^{4} + 68T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{6} + 36T_{13}^{4} + 320T_{13}^{2} + 256 \) Copy content Toggle raw display
\( T_{17}^{6} + 60T_{17}^{4} + 752T_{17}^{2} + 64 \) Copy content Toggle raw display
\( T_{19}^{3} - 6T_{19}^{2} - 46T_{19} + 232 \) Copy content Toggle raw display
\( T_{37}^{6} + 20T_{37}^{4} + 36T_{37}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 16 T^{4} + 68 T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 36 T^{4} + 320 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{6} + 60 T^{4} + 752 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{3} - 6 T^{2} - 46 T + 232)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 48 T^{4} + 356 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{3} - 66 T + 92)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 12 T^{2} + 20 T - 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 20 T^{4} + 36 T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T^{3} - 4 T^{2} - 90 T - 164)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 280 T^{4} + 25232 T^{2} + \cdots + 719104 \) Copy content Toggle raw display
$47$ \( T^{6} + 224 T^{4} + 12544 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$53$ \( T^{6} + 180 T^{4} + 7332 T^{2} + \cdots + 85264 \) Copy content Toggle raw display
$59$ \( (T^{3} - 4 T^{2} - 64 T + 128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 6 T^{2} - 100 T - 344)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 240 T^{4} + 12032 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{3} + 28 T^{2} + 200 T + 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 124 T^{4} + 1520 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$79$ \( (T^{3} + 18 T^{2} + 50 T - 256)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 80 T^{4} + 576 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} - 28 T - 1208)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 324 T^{4} + 21188 T^{2} + \cdots + 99856 \) Copy content Toggle raw display
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