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Properties

Label 5225.2.a.s
Level $5225$
Weight $2$
Character orbit 5225.a
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
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: \(1\)
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: \( x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 \) 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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 191 \nu^{14} - 593 \nu^{13} - 3078 \nu^{12} + 10145 \nu^{11} + 17645 \nu^{10} - 64855 \nu^{9} + \cdots + 2586 ) / 199 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 175 \nu^{14} + 784 \nu^{13} + 2269 \nu^{12} - 13520 \nu^{11} - 6966 \nu^{10} + 87304 \nu^{9} + \cdots - 8952 ) / 199 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 232 \nu^{14} - 912 \nu^{13} - 3273 \nu^{12} + 15638 \nu^{11} + 13655 \nu^{10} - 100250 \nu^{9} + \cdots + 8984 ) / 199 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 182 \nu^{14} - 1086 \nu^{13} - 1715 \nu^{12} + 18797 \nu^{11} - 3573 \nu^{10} - 122071 \nu^{9} + \cdots + 18082 ) / 199 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 251 \nu^{14} - 1220 \nu^{13} - 2878 \nu^{12} + 20921 \nu^{11} + 3414 \nu^{10} - 134349 \nu^{9} + \cdots + 18215 ) / 199 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 517 \nu^{14} + 2348 \nu^{13} + 6502 \nu^{12} - 40357 \nu^{11} - 17051 \nu^{10} + 259704 \nu^{9} + \cdots - 29641 ) / 199 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 556 \nu^{14} + 2467 \nu^{13} + 6969 \nu^{12} - 42267 \nu^{11} - 17522 \nu^{10} + 271065 \nu^{9} + \cdots - 33601 ) / 199 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 556 \nu^{14} - 2467 \nu^{13} - 6969 \nu^{12} + 42267 \nu^{11} + 17522 \nu^{10} - 271065 \nu^{9} + \cdots + 34795 ) / 199 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 829 \nu^{14} - 3698 \nu^{13} - 10437 \nu^{12} + 63398 \nu^{11} + 27187 \nu^{10} - 406710 \nu^{9} + \cdots + 47988 ) / 199 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 881 \nu^{14} + 4719 \nu^{13} + 9335 \nu^{12} - 81334 \nu^{11} + 443 \nu^{10} + 525338 \nu^{9} + \cdots - 72770 ) / 199 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1163 \nu^{14} - 5656 \nu^{13} - 13768 \nu^{12} + 97395 \nu^{11} + 23873 \nu^{10} - 628244 \nu^{9} + \cdots + 78228 ) / 199 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{14} - \beta_{13} - \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} + 8\beta_{3} + 8\beta_{2} + 29\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{14} - \beta_{13} + \beta_{12} + 9 \beta_{11} + 10 \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14 \beta_{14} - 13 \beta_{13} + \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 14 \beta_{9} - 14 \beta_{8} + \cdots + 79 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{14} - 14 \beta_{13} + 13 \beta_{12} + 67 \beta_{11} + 79 \beta_{10} - 16 \beta_{9} + \cdots + 525 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 135 \beta_{14} - 121 \beta_{13} + 15 \beta_{12} + 34 \beta_{11} + 34 \beta_{10} - 138 \beta_{9} + \cdots + 582 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 145 \beta_{14} - 142 \beta_{13} + 119 \beta_{12} + 479 \beta_{11} + 580 \beta_{10} - 185 \beta_{9} + \cdots + 3308 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1126 \beta_{14} - 990 \beta_{13} + 150 \beta_{12} + 387 \beta_{11} + 383 \beta_{10} - 1197 \beta_{9} + \cdots + 4167 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1335 \beta_{14} - 1273 \beta_{13} + 949 \beta_{12} + 3404 \beta_{11} + 4137 \beta_{10} - 1849 \beta_{9} + \cdots + 21219 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8760 \beta_{14} - 7610 \beta_{13} + 1278 \beta_{12} + 3722 \beta_{11} + 3634 \beta_{10} + \cdots + 29419 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11521 \beta_{14} - 10707 \beta_{13} + 7065 \beta_{12} + 24266 \beta_{11} + 29151 \beta_{10} + \cdots + 137637 \) 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
2.66470
2.59653
2.59045
2.12341
1.38376
1.30094
1.10124
0.199845
0.195748
−0.324814
−1.00105
−1.44812
−1.75566
−2.15791
−2.46907
−2.66470 −2.92926 5.10062 0 7.80559 0.523619 −8.26223 5.58055 0
1.2 −2.59653 −1.22746 4.74198 0 3.18713 −3.87200 −7.11963 −1.49335 0
1.3 −2.59045 2.99450 4.71045 0 −7.75712 −1.08607 −7.02129 5.96706 0
1.4 −2.12341 0.300215 2.50886 0 −0.637479 1.66086 −1.08053 −2.90987 0
1.5 −1.38376 0.179641 −0.0852182 0 −0.248579 −4.32482 2.88543 −2.96773 0
1.6 −1.30094 −2.76347 −0.307545 0 3.59512 −4.11821 3.00199 4.63678 0
1.7 −1.10124 1.12617 −0.787279 0 −1.24018 1.81708 3.06945 −1.73174 0
1.8 −0.199845 1.67817 −1.96006 0 −0.335375 −3.54061 0.791399 −0.183735 0
1.9 −0.195748 2.69341 −1.96168 0 −0.527231 0.191992 0.775492 4.25447 0
1.10 0.324814 −3.24319 −1.89450 0 −1.05343 −0.166007 −1.26499 7.51826 0
1.11 1.00105 −1.31958 −0.997895 0 −1.32097 3.51091 −3.00105 −1.25870 0
1.12 1.44812 −2.09272 0.0970385 0 −3.03050 −1.77651 −2.75571 1.37947 0
1.13 1.75566 0.305911 1.08236 0 0.537076 2.74363 −1.61108 −2.90642 0
1.14 2.15791 2.27846 2.65657 0 4.91670 −4.96679 1.41681 2.19136 0
1.15 2.46907 −1.98081 4.09630 0 −4.89075 −1.59709 5.17592 0.923593 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.s ✓ 15
5.b even 2 1 5225.2.a.x yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5225.2.a.s ✓ 15 1.a even 1 1 trivial
5225.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{15} + 5 T_{2}^{14} - 11 T_{2}^{13} - 85 T_{2}^{12} + 6 T_{2}^{11} + 537 T_{2}^{10} + 327 T_{2}^{9} + \cdots + 13 \) Copy content Toggle raw display
\( T_{7}^{15} + 15 T_{7}^{14} + 52 T_{7}^{13} - 241 T_{7}^{12} - 1775 T_{7}^{11} - 576 T_{7}^{10} + \cdots + 1813 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 5 T^{14} + \cdots + 13 \) Copy content Toggle raw display
$3$ \( T^{15} + 4 T^{14} + \cdots + 101 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 15 T^{14} + \cdots + 1813 \) Copy content Toggle raw display
$11$ \( (T - 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} + 13 T^{14} + \cdots + 119259 \) Copy content Toggle raw display
$17$ \( T^{15} + 11 T^{14} + \cdots - 5878079 \) Copy content Toggle raw display
$19$ \( (T + 1)^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + 10 T^{14} + \cdots + 8989816 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 564962440 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 15534356323 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 5312541061 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 1902339571 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 704805109 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 118914861797 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 113540087551 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 17679650455 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 16570478683 \) Copy content Toggle raw display
$67$ \( T^{15} + 35 T^{14} + \cdots - 9019249 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 1830004407127 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 6548500921 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 12035730685 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 1924224829663 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 1557086911875 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 29591370408776 \) Copy content Toggle raw display
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