↑

Properties

Label 2645.2.a.u
Level $2645$
Weight $2$
Character orbit 2645.a
Self dual yes
Analytic conductor $21.120$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2645,2,Mod(1,2645)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2645, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2645.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2645 = 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2645.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: \(21.1204313346\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5x^{9} - 12x^{8} + 78x^{7} + 60x^{6} - 474x^{5} - 231x^{4} + 1353x^{3} + 770x^{2} - 1540x - 1199 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5x^{9} - 12x^{8} + 78x^{7} + 60x^{6} - 474x^{5} - 231x^{4} + 1353x^{3} + 770x^{2} - 1540x - 1199 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{8} - 4\nu^{7} - 12\nu^{6} + 50\nu^{5} + 62\nu^{4} - 212\nu^{3} - 194\nu^{2} + 309\nu + 295 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{9} - 4\nu^{8} - 12\nu^{7} + 50\nu^{6} + 62\nu^{5} - 212\nu^{4} - 194\nu^{3} + 309\nu^{2} + 295\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{8} + 4\nu^{7} + 12\nu^{6} - 50\nu^{5} - 62\nu^{4} + 212\nu^{3} + 195\nu^{2} - 310\nu - 299 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{9} - 4\nu^{8} - 12\nu^{7} + 50\nu^{6} + 62\nu^{5} - 212\nu^{4} - 195\nu^{3} + 310\nu^{2} + 299\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\nu^{8} + 8\nu^{7} + 25\nu^{6} - 103\nu^{5} - 135\nu^{4} + 451\nu^{3} + 435\nu^{2} - 679\nu - 673 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\nu^{8} + 8\nu^{7} + 25\nu^{6} - 103\nu^{5} - 136\nu^{4} + 453\nu^{3} + 444\nu^{2} - 689\nu - 695 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -2\nu^{9} + 10\nu^{8} + 17\nu^{7} - 128\nu^{6} - 32\nu^{5} + 586\nu^{4} - 16\nu^{3} - 1114\nu^{2} + 6\nu + 673 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -4\nu^{9} + 20\nu^{8} + 34\nu^{7} - 256\nu^{6} - 65\nu^{5} + 1175\nu^{4} - 24\nu^{3} - 2249\nu^{2} - 4\nu + 1373 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} - 2\beta_{5} + 11\beta_{4} + 2\beta_{3} + 11\beta_{2} + 9\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 14 \beta_{5} + 20 \beta_{4} + 14 \beta_{3} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{9} + 6 \beta_{8} - 20 \beta_{7} + 21 \beta_{6} - 37 \beta_{5} + 107 \beta_{4} + 37 \beta_{3} + \cdots + 152 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 20 \beta_{9} + 41 \beta_{8} - 66 \beta_{7} + 70 \beta_{6} - 164 \beta_{5} + 258 \beta_{4} + \cdots + 369 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 66 \beta_{9} + 136 \beta_{8} - 292 \beta_{7} + 320 \beta_{6} - 488 \beta_{5} + 1040 \beta_{4} + \cdots + 1215 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 292 \beta_{9} + 612 \beta_{8} - 986 \beta_{7} + 1096 \beta_{6} - 1820 \beta_{5} + 2883 \beta_{4} + \cdots + 3350 \) Copy content Toggle raw display

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

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
−2.23472
3.23472
−1.46906
2.46906
−1.42453
2.42453
−1.81044
2.81044
−1.18882
2.18882
−1.91899 −3.23472 1.68251 1.00000 6.20739 −3.68750 0.609264 7.46343 −1.91899
1.2 −1.91899 2.23472 1.68251 1.00000 −4.28840 0.856675 0.609264 1.99398 −1.91899
1.3 −1.30972 −2.46906 −0.284630 1.00000 3.23379 3.73810 2.99223 3.09628 −1.30972
1.4 −1.30972 1.46906 −0.284630 1.00000 −1.92407 −3.81911 2.99223 −0.841850 −1.30972
1.5 −0.284630 −2.42453 −1.91899 1.00000 0.690092 −5.07928 1.11546 2.87833 −0.284630
1.6 −0.284630 1.42453 −1.91899 1.00000 −0.405462 1.39677 1.11546 −0.970726 −0.284630
1.7 0.830830 −2.81044 −1.30972 1.00000 −2.33500 −0.200064 −2.74982 4.89860 0.830830
1.8 0.830830 1.81044 −1.30972 1.00000 1.50417 −1.51531 −2.74982 0.277711 0.830830
1.9 1.68251 −2.18882 0.830830 1.00000 −3.68271 1.86675 −1.96714 1.79095 1.68251
1.10 1.68251 1.18882 0.830830 1.00000 2.00020 −2.55703 −1.96714 −1.58670 1.68251
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(23\) \(-1\)

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 2645.2.a.u 10
23.b odd 2 1 2645.2.a.t 10
23.d odd 22 2 115.2.g.b ✓ 20
115.i odd 22 2 575.2.k.c 20
115.l even 44 4 575.2.p.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.g.b ✓ 20 23.d odd 22 2
575.2.k.c 20 115.i odd 22 2
575.2.p.c 40 115.l even 44 4
2645.2.a.t 10 23.b odd 2 1
2645.2.a.u 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2645))\):

\( T_{2}^{5} + T_{2}^{4} - 4T_{2}^{3} - 3T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{10} + 9 T_{7}^{9} - T_{7}^{8} - 187 T_{7}^{7} - 292 T_{7}^{6} + 1005 T_{7}^{5} + 1731 T_{7}^{4} + \cdots + 463 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} + 5 T^{9} + \cdots - 1199 \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 9 T^{9} + \cdots + 463 \) Copy content Toggle raw display
$11$ \( T^{10} - 4 T^{9} + \cdots + 67 \) Copy content Toggle raw display
$13$ \( T^{10} + 17 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$17$ \( T^{10} + 12 T^{9} + \cdots - 881 \) Copy content Toggle raw display
$19$ \( T^{10} + T^{9} + \cdots + 14641 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + 16 T^{9} + \cdots + 857 \) Copy content Toggle raw display
$31$ \( T^{10} + 28 T^{9} + \cdots - 248357 \) Copy content Toggle raw display
$37$ \( T^{10} + 2 T^{9} + \cdots - 302368 \) Copy content Toggle raw display
$41$ \( T^{10} + 6 T^{9} + \cdots - 126313 \) Copy content Toggle raw display
$43$ \( T^{10} - 21 T^{9} + \cdots + 11640211 \) Copy content Toggle raw display
$47$ \( T^{10} + T^{9} + \cdots - 3222967 \) Copy content Toggle raw display
$53$ \( T^{10} + 3 T^{9} + \cdots + 12749297 \) Copy content Toggle raw display
$59$ \( T^{10} + 40 T^{9} + \cdots - 12943457 \) Copy content Toggle raw display
$61$ \( T^{10} - T^{9} + \cdots + 6028219 \) Copy content Toggle raw display
$67$ \( T^{10} + 34 T^{9} + \cdots + 367951 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 609853157 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 1152904939 \) Copy content Toggle raw display
$79$ \( T^{10} + 8 T^{9} + \cdots - 4749029 \) Copy content Toggle raw display
$83$ \( T^{10} + 17 T^{9} + \cdots - 79651043 \) Copy content Toggle raw display
$89$ \( T^{10} + 3 T^{9} + \cdots - 546457 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 125429833 \) Copy content Toggle raw display
show more
show less