Properties

Label 2805.2.a.m
Level $2805$
Weight $2$
Character orbit 2805.a
Self dual yes
Analytic conductor $22.398$
Analytic rank $1$
Dimension $6$
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] = [2805,2,Mod(1,2805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2805, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2805.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2805 = 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2805.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: \(22.3980377670\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.225947669.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} + 14x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} - \beta_1 q^{6} + ( - \beta_{4} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} - \beta_1 q^{6} + ( - \beta_{4} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9} + \beta_1 q^{10} - q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 - 1) q^{14} - q^{15} + \beta_{4} q^{16} + q^{17} + \beta_1 q^{18} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{19} + (\beta_{2} + 1) q^{20} + (\beta_{4} + 1) q^{21} - \beta_1 q^{22} + (2 \beta_{5} + \beta_{4} - 2 \beta_{2} - \beta_1 - 2) q^{23} + ( - \beta_{3} - \beta_1) q^{24} + q^{25} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 - 3) q^{26} - q^{27} + ( - 3 \beta_{2} - \beta_1 - 3) q^{28} + ( - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{29} - \beta_1 q^{30} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{31} + (\beta_{5} - \beta_{3} - \beta_1 + 1) q^{32} + q^{33} + \beta_1 q^{34} + ( - \beta_{4} - 1) q^{35} + (\beta_{2} + 1) q^{36} + ( - \beta_{5} - \beta_{3} - 1) q^{37} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - 1) q^{38} + (\beta_{2} + \beta_1 + 1) q^{39} + (\beta_{3} + \beta_1) q^{40} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{41} + (\beta_{5} + \beta_{3} + 2 \beta_1 + 1) q^{42} + ( - 2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 5) q^{43} + ( - \beta_{2} - 1) q^{44} + q^{45} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - 5 \beta_1 - 2) q^{46} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{47} - \beta_{4} q^{48} + (\beta_{3} + \beta_1 + 2) q^{49} + \beta_1 q^{50} - q^{51} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 6) q^{52} + ( - \beta_{5} - \beta_1 - 2) q^{53} - \beta_1 q^{54} - q^{55} + (2 \beta_{5} - \beta_{3} - \beta_{2} - 5 \beta_1 - 1) q^{56} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{57} + (\beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 4 \beta_1 - 2) q^{58} + ( - \beta_{5} - \beta_{4} - 2 \beta_1 - 2) q^{59} + ( - \beta_{2} - 1) q^{60} + (\beta_{5} + 2 \beta_{4} + \beta_{2} + \beta_1 + 2) q^{61} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 3) q^{62} + ( - \beta_{4} - 1) q^{63} + ( - 2 \beta_{4} - 2 \beta_{2} + \beta_1 - 2) q^{64} + ( - \beta_{2} - \beta_1 - 1) q^{65} + \beta_1 q^{66} + ( - \beta_{5} - \beta_{2} + 4 \beta_1 - 5) q^{67} + (\beta_{2} + 1) q^{68} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1 + 2) q^{69} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 - 1) q^{70} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{71} + (\beta_{3} + \beta_1) q^{72} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{73} + ( - 2 \beta_{4} - \beta_{2} - \beta_1 + 1) q^{74} - q^{75} + ( - 2 \beta_{5} + \beta_{2} - 3 \beta_1 + 1) q^{76} + (\beta_{4} + 1) q^{77} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 3) q^{78} + (2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{79} + \beta_{4} q^{80} + q^{81} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_1 + 3) q^{82} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} + 1) q^{83} + (3 \beta_{2} + \beta_1 + 3) q^{84} + q^{85} + ( - \beta_{4} - \beta_{3} - 7 \beta_1 - 4) q^{86} + (\beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{87} + ( - \beta_{3} - \beta_1) q^{88} + (\beta_{5} + \beta_{4} + \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{89} + \beta_1 q^{90} + (\beta_{5} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 4) q^{91} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 8) q^{92} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{93} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{94} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{95} + ( - \beta_{5} + \beta_{3} + \beta_1 - 1) q^{96} + (\beta_{5} - \beta_{3} + \beta_{2} + \beta_1) q^{97} + (\beta_{4} + 2 \beta_{2} + 2 \beta_1 + 2) q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{7} + 6 q^{9} - 6 q^{11} - 4 q^{12} - 4 q^{13} - 5 q^{14} - 6 q^{15} + 6 q^{17} - 7 q^{19} + 4 q^{20} + 6 q^{21} - 10 q^{23} + 6 q^{25} - 16 q^{26} - 6 q^{27} - 12 q^{28} + 5 q^{29} - 5 q^{31} + 5 q^{32} + 6 q^{33} - 6 q^{35} + 4 q^{36} - 5 q^{37} - 3 q^{38} + 4 q^{39} + 5 q^{41} + 5 q^{42} - 26 q^{43} - 4 q^{44} + 6 q^{45} - 11 q^{46} - 26 q^{47} + 12 q^{49} - 6 q^{51} - 32 q^{52} - 11 q^{53} - 6 q^{55} - 6 q^{56} + 7 q^{57} - 11 q^{58} - 11 q^{59} - 4 q^{60} + 9 q^{61} + 13 q^{62} - 6 q^{63} - 8 q^{64} - 4 q^{65} - 27 q^{67} + 4 q^{68} + 10 q^{69} - 5 q^{70} + 3 q^{71} - 7 q^{73} + 8 q^{74} - 6 q^{75} + 6 q^{76} + 6 q^{77} + 16 q^{78} - 6 q^{79} + 6 q^{81} + 19 q^{82} + 3 q^{83} + 12 q^{84} + 6 q^{85} - 24 q^{86} - 5 q^{87} + 7 q^{89} + 17 q^{91} - 42 q^{92} + 5 q^{93} - 24 q^{94} - 7 q^{95} - 5 q^{96} - 3 q^{97} + 8 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 8x^{4} + 14x^{2} - x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 7\nu^{3} + 8\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 7\beta_{3} + 27\beta _1 + 1 \) 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
−2.34869
−1.49434
−0.542909
0.659660
1.32552
2.40076
−2.34869 −1.00000 3.51636 1.00000 2.34869 −2.33207 −3.56146 1.00000 −2.34869
1.2 −1.49434 −1.00000 0.233061 1.00000 1.49434 3.41180 2.64041 1.00000 −1.49434
1.3 −0.542909 −1.00000 −1.70525 1.00000 0.542909 −3.31838 2.01161 1.00000 −0.542909
1.4 0.659660 −1.00000 −1.56485 1.00000 −0.659660 −2.57845 −2.35159 1.00000 0.659660
1.5 1.32552 −1.00000 −0.242993 1.00000 −1.32552 2.45497 −2.97314 1.00000 1.32552
1.6 2.40076 −1.00000 3.76367 1.00000 −2.40076 −3.63788 4.23416 1.00000 2.40076
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)
\(17\) \(-1\)

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 2805.2.a.m 6
3.b odd 2 1 8415.2.a.bi 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2805.2.a.m 6 1.a even 1 1 trivial
8415.2.a.bi 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2805))\):

\( T_{2}^{6} - 8T_{2}^{4} + 14T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 6T_{7}^{5} - 9T_{7}^{4} - 106T_{7}^{3} - 83T_{7}^{2} + 421T_{7} + 608 \) Copy content Toggle raw display
\( T_{19}^{6} + 7T_{19}^{5} - 30T_{19}^{4} - 278T_{19}^{3} - 49T_{19}^{2} + 2254T_{19} + 2744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 8 T^{4} + 14 T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{5} - 9 T^{4} - 106 T^{3} + \cdots + 608 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} - 16 T^{4} - 11 T^{3} + \cdots - 20 \) Copy content Toggle raw display
$17$ \( (T - 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 7 T^{5} - 30 T^{4} + \cdots + 2744 \) Copy content Toggle raw display
$23$ \( T^{6} + 10 T^{5} - 63 T^{4} + \cdots + 10976 \) Copy content Toggle raw display
$29$ \( T^{6} - 5 T^{5} - 60 T^{4} + \cdots - 5062 \) Copy content Toggle raw display
$31$ \( T^{6} + 5 T^{5} - 68 T^{4} + \cdots + 11104 \) Copy content Toggle raw display
$37$ \( T^{6} + 5 T^{5} - 46 T^{4} - 144 T^{3} + \cdots - 40 \) Copy content Toggle raw display
$41$ \( T^{6} - 5 T^{5} - 68 T^{4} - 89 T^{3} + \cdots - 230 \) Copy content Toggle raw display
$43$ \( T^{6} + 26 T^{5} + 147 T^{4} + \cdots - 105632 \) Copy content Toggle raw display
$47$ \( T^{6} + 26 T^{5} + 228 T^{4} + \cdots - 29504 \) Copy content Toggle raw display
$53$ \( T^{6} + 11 T^{5} + 22 T^{4} + \cdots + 178 \) Copy content Toggle raw display
$59$ \( T^{6} + 11 T^{5} - 28 T^{4} + \cdots + 532 \) Copy content Toggle raw display
$61$ \( T^{6} - 9 T^{5} - 123 T^{4} + \cdots - 13100 \) Copy content Toggle raw display
$67$ \( T^{6} + 27 T^{5} + 133 T^{4} + \cdots + 284 \) Copy content Toggle raw display
$71$ \( T^{6} - 3 T^{5} - 200 T^{4} + \cdots - 6592 \) Copy content Toggle raw display
$73$ \( T^{6} + 7 T^{5} - 152 T^{4} + \cdots + 22558 \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} - 144 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$83$ \( T^{6} - 3 T^{5} - 86 T^{4} + \cdots - 10976 \) Copy content Toggle raw display
$89$ \( T^{6} - 7 T^{5} - 243 T^{4} + \cdots - 520070 \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} - 70 T^{4} + 90 T^{3} + \cdots + 776 \) Copy content Toggle raw display
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