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Properties

Label 5225.2.a.r
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: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) 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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) : 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
\(\beta_{4}\)\(=\) \( ( - 16 \nu^{14} - 2151 \nu^{13} + 9337 \nu^{12} + 28321 \nu^{11} - 152439 \nu^{10} - 86367 \nu^{9} + \cdots - 3028 ) / 15503 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 896 \nu^{14} - 3568 \nu^{13} - 11273 \nu^{12} + 57342 \nu^{11} + 25437 \nu^{10} - 310444 \nu^{9} + \cdots - 16468 ) / 15503 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2058 \nu^{14} - 12071 \nu^{13} - 16930 \nu^{12} + 207769 \nu^{11} - 106536 \nu^{10} - 1272128 \nu^{9} + \cdots - 21353 ) / 15503 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2771 \nu^{14} + 10204 \nu^{13} + 46404 \nu^{12} - 192737 \nu^{11} - 267316 \nu^{10} + \cdots + 37572 ) / 15503 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3508 \nu^{14} - 12862 \nu^{13} - 51126 \nu^{12} + 220490 \nu^{11} + 210949 \nu^{10} - 1345835 \nu^{9} + \cdots + 105781 ) / 15503 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3651 \nu^{14} + 15923 \nu^{13} + 48340 \nu^{12} - 278400 \nu^{11} - 140314 \nu^{10} + \cdots + 57068 ) / 15503 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3831 \nu^{14} + 11103 \nu^{13} + 64239 \nu^{12} - 188458 \nu^{11} - 378592 \nu^{10} + \cdots + 54009 ) / 15503 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6164 \nu^{14} - 27868 \nu^{13} - 81774 \nu^{12} + 495667 \nu^{11} + 230430 \nu^{10} - 3225051 \nu^{9} + \cdots - 166721 ) / 15503 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 6274 \nu^{14} + 26645 \nu^{13} + 82016 \nu^{12} - 457928 \nu^{11} - 210679 \nu^{10} + \cdots - 140902 ) / 15503 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6279 \nu^{14} + 23066 \nu^{13} + 97530 \nu^{12} - 413227 \nu^{11} - 478265 \nu^{10} + \cdots + 24809 ) / 15503 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 6556 \nu^{14} - 29429 \nu^{13} - 85737 \nu^{12} + 522692 \nu^{11} + 221211 \nu^{10} - 3395752 \nu^{9} + \cdots - 108038 ) / 15503 \) 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
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{8} - \beta_{7} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + 9\beta_{3} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} + 11 \beta_{13} + \beta_{11} - \beta_{10} + 2 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{14} + \beta_{12} + 2 \beta_{11} + 13 \beta_{9} + 3 \beta_{8} - 11 \beta_{7} + 8 \beta_{5} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14 \beta_{14} + 92 \beta_{13} - \beta_{12} + 15 \beta_{11} - 14 \beta_{10} + 29 \beta_{9} + 79 \beta_{8} + \cdots + 513 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 15 \beta_{14} + \beta_{13} + 14 \beta_{12} + 31 \beta_{11} - 2 \beta_{10} + 124 \beta_{9} + 45 \beta_{8} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 137 \beta_{14} + 702 \beta_{13} - 15 \beta_{12} + 156 \beta_{11} - 141 \beta_{10} + 295 \beta_{9} + \cdots + 3211 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 147 \beta_{14} + 25 \beta_{13} + 137 \beta_{12} + 329 \beta_{11} - 42 \beta_{10} + 1054 \beta_{9} + \cdots + 358 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1168 \beta_{14} + 5142 \beta_{13} - 147 \beta_{12} + 1393 \beta_{11} - 1243 \beta_{10} + 2609 \beta_{9} + \cdots + 20555 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1180 \beta_{14} + 382 \beta_{13} + 1168 \beta_{12} + 2981 \beta_{11} - 566 \beta_{10} + 8467 \beta_{9} + \cdots + 3428 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9319 \beta_{14} + 36873 \beta_{13} - 1180 \beta_{12} + 11460 \beta_{11} - 10225 \beta_{10} + \cdots + 133544 \) 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
2.66335
2.61792
2.40444
1.89451
1.83402
1.28989
0.776738
0.692644
−0.102671
−0.442944
−0.527123
−1.59223
−1.66124
−2.27492
−2.57239
−2.66335 1.48323 5.09344 0 −3.95037 −2.27906 −8.23893 −0.800022 0
1.2 −2.61792 −2.17802 4.85352 0 5.70188 −4.94832 −7.47030 1.74376 0
1.3 −2.40444 −1.94792 3.78133 0 4.68365 0.665236 −4.28310 0.794388 0
1.4 −1.89451 3.43936 1.58918 0 −6.51591 −4.54180 0.778295 8.82918 0
1.5 −1.83402 0.596609 1.36363 0 −1.09419 2.23486 1.16712 −2.64406 0
1.6 −1.28989 −3.30063 −0.336174 0 4.25747 −0.563541 3.01342 7.89418 0
1.7 −0.776738 −0.738183 −1.39668 0 0.573374 0.981439 2.63833 −2.45509 0
1.8 −0.692644 −1.14700 −1.52024 0 0.794464 −3.60442 2.43828 −1.68438 0
1.9 0.102671 2.39685 −1.98946 0 0.246087 2.47527 −0.409602 2.74488 0
1.10 0.442944 1.62320 −1.80380 0 0.718985 −2.12336 −1.68487 −0.365237 0
1.11 0.527123 −1.24468 −1.72214 0 −0.656097 −3.22241 −1.96203 −1.45078 0
1.12 1.59223 1.35755 0.535199 0 2.16153 −0.175705 −2.33230 −1.15707 0
1.13 1.66124 −2.93460 0.759729 0 −4.87509 −3.57911 −2.06039 5.61190 0
1.14 2.27492 −1.91743 3.17527 0 −4.36201 2.74127 2.67364 0.676544 0
1.15 2.57239 0.511676 4.61720 0 1.31623 −5.06037 6.73245 −2.73819 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.r ✓ 15
5.b even 2 1 5225.2.a.y yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5225.2.a.r ✓ 15 1.a even 1 1 trivial
5225.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{15} + 5 T_{2}^{14} - 11 T_{2}^{13} - 87 T_{2}^{12} - 4 T_{2}^{11} + 545 T_{2}^{10} + 431 T_{2}^{9} + \cdots - 15 \) Copy content Toggle raw display
\( T_{7}^{15} + 21 T_{7}^{14} + 152 T_{7}^{13} + 253 T_{7}^{12} - 2119 T_{7}^{11} - 10036 T_{7}^{10} + \cdots - 22429 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 5 T^{14} + \cdots - 15 \) Copy content Toggle raw display
$3$ \( T^{15} + 4 T^{14} + \cdots - 683 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 21 T^{14} + \cdots - 22429 \) Copy content Toggle raw display
$11$ \( (T - 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} + 13 T^{14} + \cdots + 531655 \) Copy content Toggle raw display
$17$ \( T^{15} + 17 T^{14} + \cdots - 1116225 \) Copy content Toggle raw display
$19$ \( (T - 1)^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + 26 T^{14} + \cdots + 2682024 \) Copy content Toggle raw display
$29$ \( T^{15} - 9 T^{14} + \cdots - 31375800 \) Copy content Toggle raw display
$31$ \( T^{15} - 14 T^{14} + \cdots - 1388125 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 162016351 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 220545465 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 212132375 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 812451225 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 33084572583 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 766016655 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 2601462755 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 995592677 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 765891181005 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 23072143651 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 1326572185771 \) Copy content Toggle raw display
$83$ \( T^{15} + 14 T^{14} + \cdots - 87522675 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 1998434698983 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 1657280631608 \) Copy content Toggle raw display
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