Properties

Label 5225.2.a.y
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: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} - 1609 x^{6} + 2516 x^{5} + 391 x^{4} - 1081 x^{3} - 45 x^{2} + 153 x + 15 \) 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 \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{5} - \beta_{4}) q^{6} + (\beta_{13} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{10} + \beta_{7} + 1) q^{9} + q^{11} + ( - \beta_{14} - \beta_{13} + \beta_{10} - \beta_{9} + 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{12} + ( - \beta_{11} + 1) q^{13} + ( - \beta_{14} + \beta_{10} + \beta_{6} + \beta_{2} + 2 \beta_1) q^{14} + (\beta_{13} + \beta_{8} - \beta_{7} + \beta_{2} + \beta_1 + 1) q^{16} + (\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{6} - \beta_{5} + 2) q^{17} + (\beta_{13} + \beta_{10} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{18} + q^{19} + ( - \beta_{12} - \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2}) q^{21} + \beta_1 q^{22} + ( - \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{3} + \beta_{2} + 2) q^{23} + ( - \beta_{14} - \beta_{12} + \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{24} + (\beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} - \beta_{5} - \beta_{2} + \beta_1) q^{26} + (\beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \cdots - \beta_1) q^{27}+ \cdots + (\beta_{10} + \beta_{7} + 1) q^{99}+O(q^{100}) \) 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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} - 1609 x^{6} + 2516 x^{5} + 391 x^{4} - 1081 x^{3} - 45 x^{2} + 153 x + 15 \) : 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
\(\beta_{4}\)\(=\) \( ( - 16 \nu^{14} - 2151 \nu^{13} + 9337 \nu^{12} + 28321 \nu^{11} - 152439 \nu^{10} - 86367 \nu^{9} + 886590 \nu^{8} - 204807 \nu^{7} - 2159934 \nu^{6} + 1296036 \nu^{5} + \cdots - 3028 ) / 15503 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 896 \nu^{14} - 3568 \nu^{13} - 11273 \nu^{12} + 57342 \nu^{11} + 25437 \nu^{10} - 310444 \nu^{9} + 162099 \nu^{8} + 601589 \nu^{7} - 773252 \nu^{6} - 54982 \nu^{5} + \cdots - 16468 ) / 15503 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2058 \nu^{14} - 12071 \nu^{13} - 16930 \nu^{12} + 207769 \nu^{11} - 106536 \nu^{10} - 1272128 \nu^{9} + 1479090 \nu^{8} + 3265147 \nu^{7} - 5034600 \nu^{6} + \cdots - 21353 ) / 15503 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2771 \nu^{14} + 10204 \nu^{13} + 46404 \nu^{12} - 192737 \nu^{11} - 267316 \nu^{10} + 1369881 \nu^{9} + 572391 \nu^{8} - 4577378 \nu^{7} - 88887 \nu^{6} + \cdots + 37572 ) / 15503 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3508 \nu^{14} - 12862 \nu^{13} - 51126 \nu^{12} + 220490 \nu^{11} + 210949 \nu^{10} - 1345835 \nu^{9} + 30514 \nu^{8} + 3476043 \nu^{7} - 1779705 \nu^{6} + \cdots + 105781 ) / 15503 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3651 \nu^{14} + 15923 \nu^{13} + 48340 \nu^{12} - 278400 \nu^{11} - 140314 \nu^{10} + 1766692 \nu^{9} - 476298 \nu^{8} - 4974160 \nu^{7} + 2844299 \nu^{6} + \cdots + 57068 ) / 15503 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3831 \nu^{14} + 11103 \nu^{13} + 64239 \nu^{12} - 188458 \nu^{11} - 378592 \nu^{10} + 1124502 \nu^{9} + 916929 \nu^{8} - 2731984 \nu^{7} - 766205 \nu^{6} + \cdots + 54009 ) / 15503 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6164 \nu^{14} - 27868 \nu^{13} - 81774 \nu^{12} + 495667 \nu^{11} + 230430 \nu^{10} - 3225051 \nu^{9} + 925727 \nu^{8} + 9429078 \nu^{7} - 5427250 \nu^{6} + \cdots - 166721 ) / 15503 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 6274 \nu^{14} + 26645 \nu^{13} + 82016 \nu^{12} - 457928 \nu^{11} - 210679 \nu^{10} + 2799873 \nu^{9} - 1120763 \nu^{8} - 7195859 \nu^{7} + 6154343 \nu^{6} + \cdots - 140902 ) / 15503 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6279 \nu^{14} + 23066 \nu^{13} + 97530 \nu^{12} - 413227 \nu^{11} - 478265 \nu^{10} + 2715716 \nu^{9} + 541877 \nu^{8} - 8053421 \nu^{7} + 1690818 \nu^{6} + \cdots + 24809 ) / 15503 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 6556 \nu^{14} - 29429 \nu^{13} - 85737 \nu^{12} + 522692 \nu^{11} + 221211 \nu^{10} - 3395752 \nu^{9} + 1156520 \nu^{8} + 9912222 \nu^{7} - 6350786 \nu^{6} + \cdots - 108038 ) / 15503 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) 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
\(\nu^{6}\)\(=\) \( \beta_{14} + 11 \beta_{13} + \beta_{11} - \beta_{10} + 2 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 45 \beta_{2} + 12 \beta _1 + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{14} + \beta_{12} + 2 \beta_{11} + 13 \beta_{9} + 3 \beta_{8} - 11 \beta_{7} + 8 \beta_{5} + 13 \beta_{4} + 66 \beta_{3} + 3 \beta_{2} + 192 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14 \beta_{14} + 92 \beta_{13} - \beta_{12} + 15 \beta_{11} - 14 \beta_{10} + 29 \beta_{9} + 79 \beta_{8} - 77 \beta_{7} - 16 \beta_{6} - 20 \beta_{5} + 16 \beta_{4} + 16 \beta_{3} + 287 \beta_{2} + 108 \beta _1 + 513 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 15 \beta_{14} + \beta_{13} + 14 \beta_{12} + 31 \beta_{11} - 2 \beta_{10} + 124 \beta_{9} + 45 \beta_{8} - 93 \beta_{7} - \beta_{6} + 39 \beta_{5} + 126 \beta_{4} + 457 \beta_{3} + 48 \beta_{2} + 1267 \beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 137 \beta_{14} + 702 \beta_{13} - 15 \beta_{12} + 156 \beta_{11} - 141 \beta_{10} + 295 \beta_{9} + 581 \beta_{8} - 550 \beta_{7} - 170 \beta_{6} - 244 \beta_{5} + 178 \beta_{4} + 171 \beta_{3} + 1841 \beta_{2} + 884 \beta _1 + 3211 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 147 \beta_{14} + 25 \beta_{13} + 137 \beta_{12} + 329 \beta_{11} - 42 \beta_{10} + 1054 \beta_{9} + 466 \beta_{8} - 721 \beta_{7} - 23 \beta_{6} + 80 \beta_{5} + 1089 \beta_{4} + 3109 \beta_{3} + 513 \beta_{2} + 8500 \beta _1 + 358 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1168 \beta_{14} + 5142 \beta_{13} - 147 \beta_{12} + 1393 \beta_{11} - 1243 \beta_{10} + 2609 \beta_{9} + 4163 \beta_{8} - 3830 \beta_{7} - 1524 \beta_{6} - 2421 \beta_{5} + 1695 \beta_{4} + 1553 \beta_{3} + \cdots + 20555 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1180 \beta_{14} + 382 \beta_{13} + 1168 \beta_{12} + 2981 \beta_{11} - 566 \beta_{10} + 8467 \beta_{9} + 4162 \beta_{8} - 5383 \beta_{7} - 328 \beta_{6} - 968 \beta_{5} + 8860 \beta_{4} + 21073 \beta_{3} + \cdots + 3428 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9319 \beta_{14} + 36873 \beta_{13} - 1180 \beta_{12} + 11460 \beta_{11} - 10225 \beta_{10} + 21489 \beta_{9} + 29540 \beta_{8} - 26456 \beta_{7} - 12497 \beta_{6} - 21532 \beta_{5} + 14818 \beta_{4} + \cdots + 133544 \) Copy content Toggle raw display

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
−2.57239
−2.27492
−1.66124
−1.59223
−0.527123
−0.442944
−0.102671
0.692644
0.776738
1.28989
1.83402
1.89451
2.40444
2.61792
2.66335
−2.57239 −0.511676 4.61720 0 1.31623 5.06037 −6.73245 −2.73819 0
1.2 −2.27492 1.91743 3.17527 0 −4.36201 −2.74127 −2.67364 0.676544 0
1.3 −1.66124 2.93460 0.759729 0 −4.87509 3.57911 2.06039 5.61190 0
1.4 −1.59223 −1.35755 0.535199 0 2.16153 0.175705 2.33230 −1.15707 0
1.5 −0.527123 1.24468 −1.72214 0 −0.656097 3.22241 1.96203 −1.45078 0
1.6 −0.442944 −1.62320 −1.80380 0 0.718985 2.12336 1.68487 −0.365237 0
1.7 −0.102671 −2.39685 −1.98946 0 0.246087 −2.47527 0.409602 2.74488 0
1.8 0.692644 1.14700 −1.52024 0 0.794464 3.60442 −2.43828 −1.68438 0
1.9 0.776738 0.738183 −1.39668 0 0.573374 −0.981439 −2.63833 −2.45509 0
1.10 1.28989 3.30063 −0.336174 0 4.25747 0.563541 −3.01342 7.89418 0
1.11 1.83402 −0.596609 1.36363 0 −1.09419 −2.23486 −1.16712 −2.64406 0
1.12 1.89451 −3.43936 1.58918 0 −6.51591 4.54180 −0.778295 8.82918 0
1.13 2.40444 1.94792 3.78133 0 4.68365 −0.665236 4.28310 0.794388 0
1.14 2.61792 2.17802 4.85352 0 5.70188 4.94832 7.47030 1.74376 0
1.15 2.66335 −1.48323 5.09344 0 −3.95037 2.27906 8.23893 −0.800022 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.y yes 15
5.b even 2 1 5225.2.a.r 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5225.2.a.r 15 5.b even 2 1
5225.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{15} - 5 T_{2}^{14} - 11 T_{2}^{13} + 87 T_{2}^{12} - 4 T_{2}^{11} - 545 T_{2}^{10} + 431 T_{2}^{9} + 1480 T_{2}^{8} - 1763 T_{2}^{7} - 1609 T_{2}^{6} + 2516 T_{2}^{5} + 391 T_{2}^{4} - 1081 T_{2}^{3} - 45 T_{2}^{2} + 153 T_{2} + 15 \) Copy content Toggle raw display
\( T_{7}^{15} - 21 T_{7}^{14} + 152 T_{7}^{13} - 253 T_{7}^{12} - 2119 T_{7}^{11} + 10036 T_{7}^{10} - 324 T_{7}^{9} - 76610 T_{7}^{8} + 105408 T_{7}^{7} + 195473 T_{7}^{6} - 448684 T_{7}^{5} - 87941 T_{7}^{4} + \cdots + 22429 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 5 T^{14} - 11 T^{13} + 87 T^{12} + \cdots + 15 \) Copy content Toggle raw display
$3$ \( T^{15} - 4 T^{14} - 22 T^{13} + 101 T^{12} + \cdots + 683 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} - 21 T^{14} + 152 T^{13} + \cdots + 22429 \) Copy content Toggle raw display
$11$ \( (T - 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} - 13 T^{14} - 4 T^{13} + \cdots - 531655 \) Copy content Toggle raw display
$17$ \( T^{15} - 17 T^{14} - 23 T^{13} + \cdots + 1116225 \) Copy content Toggle raw display
$19$ \( (T - 1)^{15} \) Copy content Toggle raw display
$23$ \( T^{15} - 26 T^{14} + 178 T^{13} + \cdots - 2682024 \) Copy content Toggle raw display
$29$ \( T^{15} - 9 T^{14} - 212 T^{13} + \cdots - 31375800 \) Copy content Toggle raw display
$31$ \( T^{15} - 14 T^{14} - 52 T^{13} + \cdots - 1388125 \) Copy content Toggle raw display
$37$ \( T^{15} - 9 T^{14} - 207 T^{13} + \cdots - 162016351 \) Copy content Toggle raw display
$41$ \( T^{15} - 4 T^{14} - 261 T^{13} + \cdots - 220545465 \) Copy content Toggle raw display
$43$ \( T^{15} - 28 T^{14} + \cdots - 212132375 \) Copy content Toggle raw display
$47$ \( T^{15} - 14 T^{14} + \cdots - 812451225 \) Copy content Toggle raw display
$53$ \( T^{15} - 3 T^{14} + \cdots - 33084572583 \) Copy content Toggle raw display
$59$ \( T^{15} - T^{14} - 435 T^{13} + \cdots - 766016655 \) Copy content Toggle raw display
$61$ \( T^{15} - 2 T^{14} + \cdots + 2601462755 \) Copy content Toggle raw display
$67$ \( T^{15} - 37 T^{14} + \cdots + 995592677 \) Copy content Toggle raw display
$71$ \( T^{15} + 7 T^{14} + \cdots - 765891181005 \) Copy content Toggle raw display
$73$ \( T^{15} - 42 T^{14} + \cdots - 23072143651 \) Copy content Toggle raw display
$79$ \( T^{15} + 10 T^{14} + \cdots + 1326572185771 \) Copy content Toggle raw display
$83$ \( T^{15} - 14 T^{14} - 441 T^{13} + \cdots + 87522675 \) Copy content Toggle raw display
$89$ \( T^{15} - 15 T^{14} + \cdots - 1998434698983 \) Copy content Toggle raw display
$97$ \( T^{15} - 8 T^{14} + \cdots + 1657280631608 \) Copy content Toggle raw display
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