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Properties

Label 7225.2.a.bt
Level $7225$
Weight $2$
Character orbit 7225.a
Self dual yes
Analytic conductor $57.692$
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] = [7225,2,Mod(1,7225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7225 = 5^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7225.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: \(57.6919154604\)
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: \( x^{15} - 21 x^{13} - 2 x^{12} + 171 x^{11} + 30 x^{10} - 678 x^{9} - 153 x^{8} + 1350 x^{7} + 301 x^{6} + \cdots + 3 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{12} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{5} - \beta_1 - 1) q^{6} + (\beta_{14} + \beta_{12} + \beta_{8} + \cdots - 2) q^{7}+ \cdots + ( - \beta_{14} + \beta_{13} - 2 \beta_{12} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{12} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{5} - \beta_1 - 1) q^{6} + (\beta_{14} + \beta_{12} + \beta_{8} + \cdots - 2) q^{7}+ \cdots + ( - 2 \beta_{14} - \beta_{13} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 9 q^{3} + 12 q^{4} - 9 q^{6} - 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 9 q^{3} + 12 q^{4} - 9 q^{6} - 12 q^{7} + 6 q^{8} + 12 q^{9} - 6 q^{11} - 24 q^{12} + 6 q^{16} + 12 q^{18} + 6 q^{19} + 30 q^{21} - 12 q^{22} - 36 q^{23} - 18 q^{24} + 36 q^{26} - 36 q^{27} - 24 q^{28} + 18 q^{29} + 12 q^{32} + 12 q^{33} - 9 q^{36} - 12 q^{37} - 6 q^{38} - 9 q^{39} + 18 q^{41} + 36 q^{42} - 3 q^{43} + 12 q^{44} - 21 q^{46} - 3 q^{47} + 12 q^{48} + 15 q^{49} - 27 q^{52} - 21 q^{54} + 6 q^{56} - 39 q^{57} - 18 q^{58} - 12 q^{59} + 15 q^{61} - 54 q^{62} - 60 q^{63} - 36 q^{64} + 18 q^{66} - 24 q^{67} + 42 q^{69} - 6 q^{71} + 66 q^{72} + 9 q^{73} + 36 q^{74} - 18 q^{76} + 30 q^{77} - 30 q^{78} + 9 q^{79} + 51 q^{81} + 36 q^{82} + 15 q^{83} + 9 q^{84} - 36 q^{86} - 51 q^{87} - 30 q^{88} - 24 q^{89} - 27 q^{91} - 15 q^{92} - 42 q^{93} - 57 q^{94} - 42 q^{96} - 48 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 21 x^{13} - 2 x^{12} + 171 x^{11} + 30 x^{10} - 678 x^{9} - 153 x^{8} + 1350 x^{7} + 301 x^{6} + \cdots + 3 \) : 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
\(\beta_{4}\)\(=\) \( ( 556 \nu^{14} + 3088 \nu^{13} - 12608 \nu^{12} - 60751 \nu^{11} + 104414 \nu^{10} + 451319 \nu^{9} + \cdots - 5844 ) / 16983 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2816 \nu^{14} - 6395 \nu^{13} + 57055 \nu^{12} + 136148 \nu^{11} - 437155 \nu^{10} - 1097170 \nu^{9} + \cdots + 70773 ) / 33966 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1948 \nu^{14} - 556 \nu^{13} + 37820 \nu^{12} + 16504 \nu^{11} - 272357 \nu^{10} - 162854 \nu^{9} + \cdots - 29130 ) / 16983 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6223 \nu^{14} + 5024 \nu^{13} + 123887 \nu^{12} - 88895 \nu^{11} - 940355 \nu^{10} + 603334 \nu^{9} + \cdots + 9939 ) / 33966 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3131 \nu^{14} - 5090 \nu^{13} + 65542 \nu^{12} + 108896 \nu^{11} - 516409 \nu^{10} - 880180 \nu^{9} + \cdots + 22524 ) / 16983 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6881 \nu^{14} - 1685 \nu^{13} + 137464 \nu^{12} + 48917 \nu^{11} - 1035334 \nu^{10} + \cdots - 186330 ) / 33966 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9334 \nu^{14} + 6349 \nu^{13} - 190238 \nu^{12} - 147082 \nu^{11} + 1464941 \nu^{10} + 1254341 \nu^{9} + \cdots - 205626 ) / 33966 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5560 \nu^{14} + 2575 \nu^{13} - 114758 \nu^{12} - 64054 \nu^{11} + 902615 \nu^{10} + 578795 \nu^{9} + \cdots - 24474 ) / 16983 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12269 \nu^{14} - 2816 \nu^{13} + 251254 \nu^{12} + 81593 \nu^{11} - 1961851 \nu^{10} + \cdots + 162312 ) / 33966 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15718 \nu^{14} + 13867 \nu^{13} - 321482 \nu^{12} - 314854 \nu^{11} + 2480147 \nu^{10} + \cdots - 79194 ) / 33966 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20198 \nu^{14} + 21725 \nu^{13} - 417655 \nu^{12} - 482048 \nu^{11} + 3275203 \nu^{10} + \cdots - 266667 ) / 33966 \) 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
\(\nu^{4}\)\(=\) \( -\beta_{14} + \beta_{11} - \beta_{8} - \beta_{5} + 6\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - 2\beta_{13} - \beta_{8} - 2\beta_{6} + \beta_{5} + 7\beta_{3} + \beta_{2} + 26\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{14} + 11 \beta_{11} + 2 \beta_{9} - 9 \beta_{8} - 2 \beta_{6} - 10 \beta_{5} + \beta_{4} + \cdots + 83 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{14} - 22 \beta_{13} + \beta_{12} - \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - 11 \beta_{8} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 75 \beta_{14} - \beta_{13} + 2 \beta_{12} + 93 \beta_{11} - 3 \beta_{10} + 26 \beta_{9} - 61 \beta_{8} + \cdots + 476 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 116 \beta_{14} - 178 \beta_{13} + 11 \beta_{12} - 14 \beta_{11} - 45 \beta_{10} + 31 \beta_{9} + \cdots + 153 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 506 \beta_{14} - 21 \beta_{13} + 32 \beta_{12} + 707 \beta_{11} - 49 \beta_{10} + 242 \beta_{9} + \cdots + 2792 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 896 \beta_{14} - 1292 \beta_{13} + 84 \beta_{12} - 131 \beta_{11} - 454 \beta_{10} + 321 \beta_{9} + \cdots + 1069 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3254 \beta_{14} - 261 \beta_{13} + 332 \beta_{12} + 5073 \beta_{11} - 529 \beta_{10} + 1967 \beta_{9} + \cdots + 16661 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6446 \beta_{14} - 8928 \beta_{13} + 555 \beta_{12} - 1025 \beta_{11} - 3875 \beta_{10} + 2799 \beta_{9} + \cdots + 7186 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 20437 \beta_{14} - 2579 \beta_{13} + 2846 \beta_{12} + 35128 \beta_{11} - 4781 \beta_{10} + \cdots + 100806 \) 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
−2.50743
−2.19490
−1.93105
−1.76581
−1.10615
−0.970758
−0.525320
0.0377149
0.567962
0.718828
0.788473
1.71131
2.26761
2.35122
2.55831
−2.50743 0.598298 4.28722 0 −1.50019 −3.52603 −5.73504 −2.64204 0
1.2 −2.19490 −1.59866 2.81758 0 3.50890 −3.21427 −1.79451 −0.444285 0
1.3 −1.93105 0.539243 1.72896 0 −1.04130 0.979713 0.523397 −2.70922 0
1.4 −1.76581 −2.30878 1.11810 0 4.07688 3.96414 1.55728 2.33047 0
1.5 −1.10615 −3.09602 −0.776422 0 3.42468 −2.90465 3.07115 6.58534 0
1.6 −0.970758 1.98754 −1.05763 0 −1.92942 0.240703 2.96822 0.950332 0
1.7 −0.525320 1.36683 −1.72404 0 −0.718021 −0.426746 1.95631 −1.13179 0
1.8 0.0377149 2.71683 −1.99858 0 0.102465 0.366049 −0.150806 4.38117 0
1.9 0.567962 −2.88559 −1.67742 0 −1.63891 −5.00907 −2.08863 5.32662 0
1.10 0.718828 −0.0658528 −1.48329 0 −0.0473368 −2.64277 −2.50388 −2.99566 0
1.11 0.788473 −1.07579 −1.37831 0 −0.848234 2.07026 −2.66371 −1.84267 0
1.12 1.71131 0.109608 0.928572 0 0.187572 4.34210 −1.83354 −2.98799 0
1.13 2.26761 −3.43264 3.14206 0 −7.78389 −3.42036 2.58974 8.78301 0
1.14 2.35122 0.229247 3.52823 0 0.539011 −2.06647 3.59321 −2.94745 0
1.15 2.55831 −2.08426 4.54497 0 −5.33220 −0.752596 6.51082 1.34415 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
\(5\) \(-1\)
\(17\) \(-1\)

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 7225.2.a.bt ✓ 15
5.b even 2 1 7225.2.a.bv yes 15
17.b even 2 1 7225.2.a.bw yes 15
85.c even 2 1 7225.2.a.bu yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7225.2.a.bt ✓ 15 1.a even 1 1 trivial
7225.2.a.bu yes 15 85.c even 2 1
7225.2.a.bv yes 15 5.b even 2 1
7225.2.a.bw yes 15 17.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7225))\):

\( T_{2}^{15} - 21 T_{2}^{13} - 2 T_{2}^{12} + 171 T_{2}^{11} + 30 T_{2}^{10} - 678 T_{2}^{9} - 153 T_{2}^{8} + \cdots + 3 \) Copy content Toggle raw display
\( T_{3}^{15} + 9 T_{3}^{14} + 12 T_{3}^{13} - 105 T_{3}^{12} - 312 T_{3}^{11} + 258 T_{3}^{10} + 1623 T_{3}^{9} + \cdots - 1 \) Copy content Toggle raw display
\( T_{7}^{15} + 12 T_{7}^{14} + 12 T_{7}^{13} - 385 T_{7}^{12} - 1539 T_{7}^{11} + 2058 T_{7}^{10} + \cdots + 3043 \) Copy content Toggle raw display
\( T_{11}^{15} + 6 T_{11}^{14} - 48 T_{11}^{13} - 297 T_{11}^{12} + 666 T_{11}^{11} + 4698 T_{11}^{10} + \cdots + 2049 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 21 T^{13} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{15} + 9 T^{14} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 12 T^{14} + \cdots + 3043 \) Copy content Toggle raw display
$11$ \( T^{15} + 6 T^{14} + \cdots + 2049 \) Copy content Toggle raw display
$13$ \( T^{15} - 126 T^{13} + \cdots + 78319 \) Copy content Toggle raw display
$17$ \( T^{15} \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 2943215727 \) Copy content Toggle raw display
$23$ \( T^{15} + 36 T^{14} + \cdots + 52855497 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 2047004811 \) Copy content Toggle raw display
$31$ \( T^{15} - 210 T^{13} + \cdots + 839331 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 153234181 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 28750724823 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 2893157909 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 5651025984 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 409635139149 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 3599003937 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 291499289 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 329291105993 \) Copy content Toggle raw display
$71$ \( T^{15} + 6 T^{14} + \cdots + 10215693 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 359237523797 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 236431363913 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 228495657729 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 369173604663 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 2376999318151 \) Copy content Toggle raw display
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