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Properties

Label 5225.2.a.v
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: \( x^{15} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{6} + (\beta_{14} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{14} + \beta_{12} + \beta_{9} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{6} + (\beta_{14} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{14} + \beta_{12} + \beta_{9} + \cdots + 1) q^{9}+ \cdots + (\beta_{14} - \beta_{12} - \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 17 q^{7} - 3 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 17 q^{7} - 3 q^{8} + 15 q^{9} - 15 q^{11} - 9 q^{12} - 3 q^{13} - 11 q^{14} + 13 q^{16} - q^{17} + 10 q^{18} + 15 q^{19} + 16 q^{21} - q^{22} - 18 q^{23} - 27 q^{24} + 15 q^{26} - 31 q^{27} - 34 q^{28} + 11 q^{29} - 2 q^{31} - 10 q^{32} + 4 q^{33} - 13 q^{34} + 8 q^{36} - 27 q^{37} + q^{38} + 4 q^{39} + 6 q^{41} - 2 q^{42} - 38 q^{43} - 13 q^{44} - 9 q^{46} - 14 q^{47} - 32 q^{48} + 28 q^{49} - 32 q^{51} - 16 q^{52} - 11 q^{53} - 11 q^{54} + 2 q^{56} - 4 q^{57} + 6 q^{58} + 13 q^{59} + 8 q^{61} - 7 q^{62} - 49 q^{63} + 9 q^{64} + q^{66} - 31 q^{67} + 26 q^{68} + 39 q^{69} - 3 q^{71} + 18 q^{72} - 18 q^{73} + 7 q^{74} + 13 q^{76} + 17 q^{77} - 18 q^{78} + 10 q^{79} + 31 q^{81} - 58 q^{82} - 16 q^{83} + 112 q^{84} - 63 q^{86} - 67 q^{87} + 3 q^{88} - 7 q^{89} - 50 q^{91} - 98 q^{92} - 26 q^{93} + 22 q^{94} - 37 q^{96} - 24 q^{97} + 46 q^{98} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5 \) : 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
\(\beta_{4}\)\(=\) \( ( - 130 \nu^{14} + 267 \nu^{13} + 2498 \nu^{12} - 5165 \nu^{11} - 17984 \nu^{10} + 37341 \nu^{9} + \cdots + 1420 ) / 917 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 149 \nu^{14} - 172 \nu^{13} - 3244 \nu^{12} + 3839 \nu^{11} + 26975 \nu^{10} - 32380 \nu^{9} + \cdots - 7976 ) / 917 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 317 \nu^{14} - 249 \nu^{13} - 6317 \nu^{12} + 4518 \nu^{11} + 47352 \nu^{10} - 28621 \nu^{9} + \cdots + 2251 ) / 917 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 284 \nu^{14} + 414 \nu^{13} + 5697 \nu^{12} - 8462 \nu^{11} - 42547 \nu^{10} + 64844 \nu^{9} + \cdots + 986 ) / 917 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 337 \nu^{14} - 149 \nu^{13} - 7054 \nu^{12} + 3267 \nu^{11} + 56044 \nu^{10} - 26536 \nu^{9} + \cdots - 6291 ) / 917 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 55 \nu^{14} + 118 \nu^{13} + 1077 \nu^{12} - 2422 \nu^{11} - 7659 \nu^{10} + 18534 \nu^{9} + \cdots + 500 ) / 131 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 443 \nu^{14} + 536 \nu^{13} + 8851 \nu^{12} - 11217 \nu^{11} - 65615 \nu^{10} + 88387 \nu^{9} + \cdots + 7731 ) / 917 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 508 \nu^{14} + 1128 \nu^{13} + 10100 \nu^{12} - 23428 \nu^{11} - 73690 \nu^{10} + 182710 \nu^{9} + \cdots + 10275 ) / 917 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 705 \nu^{14} + 1060 \nu^{13} + 14746 \nu^{12} - 22614 \nu^{11} - 115657 \nu^{10} + 180742 \nu^{9} + \cdots + 9958 ) / 917 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 901 \nu^{14} - 997 \nu^{13} - 18484 \nu^{12} + 20808 \nu^{11} + 142334 \nu^{10} - 162143 \nu^{9} + \cdots - 4904 ) / 917 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 155 \nu^{14} + 142 \nu^{13} + 3190 \nu^{12} - 2979 \nu^{11} - 24657 \nu^{10} + 23174 \nu^{9} + \cdots + 897 ) / 131 \) 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
\(\nu^{4}\)\(=\) \( - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{9} + 2 \beta_{8} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} - 5 \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - 11 \beta_{13} - 11 \beta_{12} + 12 \beta_{11} - 13 \beta_{9} + 24 \beta_{8} - 11 \beta_{6} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{14} + 24 \beta_{13} + 14 \beta_{12} - 15 \beta_{11} - 12 \beta_{10} + 28 \beta_{9} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12 \beta_{14} - 97 \beta_{13} - 94 \beta_{12} + 108 \beta_{11} + 3 \beta_{10} - 123 \beta_{9} + \cdots + 539 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 31 \beta_{14} + 219 \beta_{13} + 142 \beta_{12} - 157 \beta_{11} - 105 \beta_{10} + 278 \beta_{9} + \cdots - 149 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 105 \beta_{14} - 788 \beta_{13} - 737 \beta_{12} + 873 \beta_{11} + 52 \beta_{10} - 1031 \beta_{9} + \cdots + 3461 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 328 \beta_{14} + 1811 \beta_{13} + 1263 \beta_{12} - 1420 \beta_{11} - 821 \beta_{10} + 2413 \beta_{9} + \cdots - 1419 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 823 \beta_{14} - 6135 \beta_{13} - 5557 \beta_{12} + 6706 \beta_{11} + 599 \beta_{10} - 8148 \beta_{9} + \cdots + 22751 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2969 \beta_{14} + 14281 \beta_{13} + 10487 \beta_{12} - 11909 \beta_{11} - 6107 \beta_{10} + \cdots - 12688 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6162 \beta_{14} - 46595 \beta_{13} - 41077 \beta_{12} + 50153 \beta_{11} + 5802 \beta_{10} + \cdots + 152169 \) 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
−2.70405
−2.21115
−2.20121
−1.31913
−1.26824
−0.451441
−0.414796
−0.0619135
0.814490
0.916988
1.38157
1.52359
1.98766
2.47156
2.53606
−2.70405 0.205258 5.31188 0 −0.555028 −2.55265 −8.95548 −2.95787 0
1.2 −2.21115 −2.94549 2.88918 0 6.51291 −2.83613 −1.96611 5.67589 0
1.3 −2.20121 2.15466 2.84534 0 −4.74286 2.43464 −1.86078 1.64255 0
1.4 −1.31913 0.487250 −0.259889 0 −0.642748 0.817715 2.98109 −2.76259 0
1.5 −1.26824 −1.37964 −0.391576 0 1.74971 −2.74560 3.03308 −1.09659 0
1.6 −0.451441 −2.92844 −1.79620 0 1.32202 3.27317 1.71376 5.57574 0
1.7 −0.414796 0.335749 −1.82794 0 −0.139267 −3.31574 1.58782 −2.88727 0
1.8 −0.0619135 2.42872 −1.99617 0 −0.150370 −0.484101 0.247417 2.89868 0
1.9 0.814490 −2.50015 −1.33661 0 −2.03634 −4.83914 −2.71763 3.25073 0
1.10 0.916988 −1.46608 −1.15913 0 −1.34437 3.11204 −2.89689 −0.850619 0
1.11 1.38157 2.80596 −0.0912568 0 3.87664 −4.96734 −2.88922 4.87342 0
1.12 1.52359 1.71830 0.321328 0 2.61798 1.47987 −2.55761 −0.0474472 0
1.13 1.98766 0.104882 1.95080 0 0.208469 −1.24528 −0.0977875 −2.98900 0
1.14 2.47156 0.237530 4.10863 0 0.587070 −0.606053 5.21162 −2.94358 0
1.15 2.53606 −3.25852 4.43161 0 −8.26381 −4.52539 6.16671 7.61796 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.v yes 15
5.b even 2 1 5225.2.a.u ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5225.2.a.u ✓ 15 5.b even 2 1
5225.2.a.v yes 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{15} - T_{2}^{14} - 21 T_{2}^{13} + 21 T_{2}^{12} + 168 T_{2}^{11} - 165 T_{2}^{10} - 645 T_{2}^{9} + \cdots - 5 \) Copy content Toggle raw display
\( T_{7}^{15} + 17 T_{7}^{14} + 78 T_{7}^{13} - 197 T_{7}^{12} - 2497 T_{7}^{11} - 3534 T_{7}^{10} + \cdots - 78609 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - T^{14} + \cdots - 5 \) Copy content Toggle raw display
$3$ \( T^{15} + 4 T^{14} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 17 T^{14} + \cdots - 78609 \) Copy content Toggle raw display
$11$ \( (T + 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} + 3 T^{14} + \cdots + 6684389 \) Copy content Toggle raw display
$17$ \( T^{15} + T^{14} + \cdots - 13089 \) Copy content Toggle raw display
$19$ \( (T - 1)^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + 18 T^{14} + \cdots + 132200 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 1033217016 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 1751332969 \) Copy content Toggle raw display
$37$ \( T^{15} + 27 T^{14} + \cdots - 11533069 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 2328385177 \) Copy content Toggle raw display
$43$ \( T^{15} + 38 T^{14} + \cdots - 38811601 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 558835437 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 665077574267 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 51376062089 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 4019482787 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 1253487348767 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 8569180307 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 40852177291 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 6539792293 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 7176867851 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 464591789865437 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 799389137752 \) Copy content Toggle raw display
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