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Properties

Label 5225.2.a.t
Level $5225$
Weight $2$
Character orbit 5225.a
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $15$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 21 x^{13} + 19 x^{12} + 170 x^{11} - 137 x^{10} - 669 x^{9} + 458 x^{8} + 1327 x^{7} + \cdots - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 21 x^{13} + 19 x^{12} + 170 x^{11} - 137 x^{10} - 669 x^{9} + 458 x^{8} + 1327 x^{7} + \cdots - 9 \) : 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
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{14} + 17 \nu^{13} + 102 \nu^{12} - 329 \nu^{11} - 784 \nu^{10} + 2383 \nu^{9} + 2745 \nu^{8} + \cdots + 66 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22 \nu^{14} + 17 \nu^{13} - 411 \nu^{12} - 302 \nu^{11} + 2807 \nu^{10} + 1816 \nu^{9} - 8703 \nu^{8} + \cdots + 12 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22 \nu^{14} + 17 \nu^{13} - 411 \nu^{12} - 302 \nu^{11} + 2807 \nu^{10} + 1816 \nu^{9} - 8703 \nu^{8} + \cdots - 69 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10 \nu^{14} + 47 \nu^{13} - 150 \nu^{12} - 854 \nu^{11} + 623 \nu^{10} + 5548 \nu^{9} - 9 \nu^{8} + \cdots + 192 ) / 27 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20 \nu^{14} + 14 \nu^{13} + 381 \nu^{12} - 290 \nu^{11} - 2677 \nu^{10} + 2350 \nu^{9} + 8523 \nu^{8} + \cdots + 318 ) / 27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{14} - 5 \nu^{13} + 15 \nu^{12} + 92 \nu^{11} - 62 \nu^{10} - 610 \nu^{9} - 3 \nu^{8} + 1810 \nu^{7} + \cdots - 39 ) / 3 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29 \nu^{14} - 4 \nu^{13} + 570 \nu^{12} + 79 \nu^{11} - 4207 \nu^{10} - 494 \nu^{9} + 14625 \nu^{8} + \cdots - 357 ) / 27 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 20 \nu^{14} - 14 \nu^{13} - 408 \nu^{12} + 263 \nu^{11} + 3163 \nu^{10} - 1891 \nu^{9} - 11655 \nu^{8} + \cdots - 183 ) / 27 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26 \nu^{14} + 25 \nu^{13} - 498 \nu^{12} - 460 \nu^{11} + 3553 \nu^{10} + 2966 \nu^{9} - 11952 \nu^{8} + \cdots - 3 ) / 27 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 55 \nu^{14} - 2 \nu^{13} + 1068 \nu^{12} + 26 \nu^{11} - 7733 \nu^{10} + 131 \nu^{9} + 26037 \nu^{8} + \cdots - 138 ) / 27 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 35 \nu^{14} + 70 \nu^{13} - 660 \nu^{12} - 1315 \nu^{11} + 4597 \nu^{10} + 8942 \nu^{9} - 15057 \nu^{8} + \cdots + 483 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) 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
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{8} - 2\beta_{6} + \beta_{4} + 9\beta_{3} + \beta_{2} + 27\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 13 \beta_{6} + 10 \beta_{5} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{14} - \beta_{13} + 13 \beta_{11} - 10 \beta_{9} - 11 \beta_{8} + 3 \beta_{7} - 26 \beta_{6} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{14} - 14 \beta_{13} + \beta_{12} + 17 \beta_{11} + 12 \beta_{10} - \beta_{9} - 13 \beta_{8} + \cdots + 516 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 14 \beta_{14} - 16 \beta_{13} - \beta_{12} + 123 \beta_{11} + \beta_{10} - 77 \beta_{9} - 95 \beta_{8} + \cdots + 202 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 32 \beta_{14} - 138 \beta_{13} + 17 \beta_{12} + 190 \beta_{11} + 106 \beta_{10} - 20 \beta_{9} + \cdots + 3296 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 137 \beta_{14} - 175 \beta_{13} - 17 \beta_{12} + 1037 \beta_{11} + 21 \beta_{10} - 551 \beta_{9} + \cdots + 2061 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 349 \beta_{14} - 1189 \beta_{13} + 190 \beta_{12} + 1798 \beta_{11} + 840 \beta_{10} - 259 \beta_{9} + \cdots + 21900 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1173 \beta_{14} - 1651 \beta_{13} - 186 \beta_{12} + 8296 \beta_{11} + 288 \beta_{10} - 3869 \beta_{9} + \cdots + 19312 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3254 \beta_{14} - 9608 \beta_{13} + 1769 \beta_{12} + 15681 \beta_{11} + 6350 \beta_{10} + \cdots + 150072 \) 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
2.77430
2.13794
2.13710
1.67859
1.54588
0.678001
0.643169
−0.151942
−0.367308
−0.722545
−0.761882
−1.76560
−1.93364
−2.41798
−2.47407
−2.77430 1.94173 5.69673 0 −5.38694 2.77587 −10.2558 0.770318 0
1.2 −2.13794 0.537061 2.57079 0 −1.14821 −0.416282 −1.22033 −2.71157 0
1.3 −2.13710 −1.19868 2.56721 0 2.56171 2.23006 −1.21218 −1.56316 0
1.4 −1.67859 −2.69106 0.817650 0 4.51717 2.51534 1.98468 4.24179 0
1.5 −1.54588 3.18552 0.389733 0 −4.92442 −2.77361 2.48927 7.14754 0
1.6 −0.678001 −1.35109 −1.54031 0 0.916040 −1.92484 2.40034 −1.17455 0
1.7 −0.643169 0.586127 −1.58633 0 −0.376979 4.99433 2.30662 −2.65645 0
1.8 0.151942 3.09381 −1.97691 0 0.470080 5.09357 −0.604261 6.57165 0
1.9 0.367308 −0.412171 −1.86509 0 −0.151394 −0.0772451 −1.41968 −2.83011 0
1.10 0.722545 1.71747 −1.47793 0 1.24095 −2.23664 −2.51296 −0.0502999 0
1.11 0.761882 −2.93875 −1.41954 0 −2.23898 −0.140222 −2.60528 5.63627 0
1.12 1.76560 −0.418817 1.11733 0 −0.739462 −4.14636 −1.55843 −2.82459 0
1.13 1.93364 1.29385 1.73897 0 2.50184 3.43011 −0.504738 −1.32596 0
1.14 2.41798 −2.46128 3.84665 0 −5.95133 1.10716 4.46516 3.05789 0
1.15 2.47407 3.11629 4.12105 0 7.70992 0.568753 5.24763 6.71124 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)
\(19\) \(1\)

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 5225.2.a.t ✓ 15
5.b even 2 1 5225.2.a.w yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5225.2.a.t ✓ 15 1.a even 1 1 trivial
5225.2.a.w yes 15 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{15} + T_{2}^{14} - 21 T_{2}^{13} - 19 T_{2}^{12} + 170 T_{2}^{11} + 137 T_{2}^{10} - 669 T_{2}^{9} + \cdots + 9 \) Copy content Toggle raw display
\( T_{7}^{15} - 11 T_{7}^{14} + 2 T_{7}^{13} + 343 T_{7}^{12} - 841 T_{7}^{11} - 3146 T_{7}^{10} + \cdots + 191 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + T^{14} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{15} - 4 T^{14} + \cdots + 227 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} - 11 T^{14} + \cdots + 191 \) Copy content Toggle raw display
$11$ \( (T + 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} - 3 T^{14} + \cdots + 186591 \) Copy content Toggle raw display
$17$ \( T^{15} + 5 T^{14} + \cdots + 261903 \) Copy content Toggle raw display
$19$ \( (T + 1)^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 15324926616 \) Copy content Toggle raw display
$29$ \( T^{15} - 7 T^{14} + \cdots - 5832 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 140385879 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 119957219817 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 4768760805 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 2065651035549 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 632236095 \) Copy content Toggle raw display
$53$ \( T^{15} - 21 T^{14} + \cdots + 16749 \) Copy content Toggle raw display
$59$ \( T^{15} + 11 T^{14} + \cdots + 2749729 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 19802477763 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 50650313397 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 26212478639 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 5780455047 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 28856420091 \) Copy content Toggle raw display
$83$ \( T^{15} + 20 T^{14} + \cdots + 12128217 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 111267524027057 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 3730533486440 \) Copy content Toggle raw display
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