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Properties

Label 5225.2.a.ba
Level $5225$
Weight $2$
Character orbit 5225.a
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $2$

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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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 26 x^{18} + 281 x^{16} - 1640 x^{14} + 5623 x^{12} - 11551 x^{10} + 13894 x^{8} - 9095 x^{6} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 1045)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{14} - \beta_{2} - 1) q^{6} - \beta_{3} q^{7} + (\beta_{7} + \beta_{6}) q^{8} + ( - \beta_{11} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{14} - \beta_{2} - 1) q^{6} - \beta_{3} q^{7} + (\beta_{7} + \beta_{6}) q^{8} + ( - \beta_{11} + 1) q^{9} + q^{11} + (\beta_{4} - \beta_1) q^{12} + ( - \beta_{18} + \beta_{6} + \beta_{3}) q^{13} + ( - \beta_{17} + \beta_{15} + \beta_{9} + \cdots - 1) q^{14}+ \cdots + ( - \beta_{11} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 12 q^{4} - 8 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 12 q^{4} - 8 q^{6} + 10 q^{9} + 20 q^{11} - 24 q^{14} - 4 q^{16} - 20 q^{19} - 30 q^{21} - 38 q^{24} + 8 q^{26} - 50 q^{29} - 50 q^{31} - 28 q^{34} - 12 q^{36} - 48 q^{39} - 34 q^{41} + 12 q^{44} - 36 q^{46} + 6 q^{49} - 40 q^{51} + 6 q^{54} - 40 q^{56} - 30 q^{59} - 14 q^{61} - 36 q^{64} - 8 q^{66} + 12 q^{69} - 40 q^{71} - 50 q^{74} - 12 q^{76} - 106 q^{79} + 30 q^{84} + 56 q^{86} - 36 q^{89} - 56 q^{91} - 28 q^{94} + 66 q^{96} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 26 x^{18} + 281 x^{16} - 1640 x^{14} + 5623 x^{12} - 11551 x^{10} + 13894 x^{8} - 9095 x^{6} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61 \nu^{19} - 1299 \nu^{17} + 10262 \nu^{15} - 33228 \nu^{13} + 5459 \nu^{11} + 240718 \nu^{9} + \cdots - 1604 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57 \nu^{19} + 1205 \nu^{17} - 9104 \nu^{15} + 23376 \nu^{13} + 49969 \nu^{11} - 428112 \nu^{9} + \cdots - 27024 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 127 \nu^{18} - 3119 \nu^{16} + 31252 \nu^{14} - 164582 \nu^{12} + 489083 \nu^{10} - 819526 \nu^{8} + \cdots - 1064 ) / 538 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 424 \nu^{19} + 10233 \nu^{17} - 99883 \nu^{15} + 504698 \nu^{13} - 1398132 \nu^{11} + \cdots - 13202 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 424 \nu^{19} - 10233 \nu^{17} + 99883 \nu^{15} - 504698 \nu^{13} + 1398132 \nu^{11} - 2051391 \nu^{9} + \cdots + 8898 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 117 \nu^{19} - 3153 \nu^{17} + 35620 \nu^{15} - 219576 \nu^{13} + 805123 \nu^{11} - 1793688 \nu^{9} + \cdots - 62380 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 365 \nu^{18} + 8981 \nu^{16} - 90200 \nu^{14} + 476396 \nu^{12} - 1420387 \nu^{10} + 2386344 \nu^{8} + \cdots + 8116 ) / 1076 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 563 \nu^{19} - 13634 \nu^{17} + 133533 \nu^{15} - 676778 \nu^{13} + 1878649 \nu^{11} + \cdots - 2730 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 119 \nu^{18} - 2931 \nu^{16} + 29474 \nu^{14} - 155907 \nu^{12} + 465652 \nu^{10} - 783140 \nu^{8} + \cdots - 29 ) / 269 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 284 \nu^{18} - 6943 \nu^{16} + 69037 \nu^{14} - 358938 \nu^{12} + 1043638 \nu^{10} - 1682291 \nu^{8} + \cdots - 1100 ) / 538 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 615 \nu^{19} - 15125 \nu^{17} + 151546 \nu^{15} - 795708 \nu^{13} + 2343125 \nu^{11} + \cdots - 11744 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 791 \nu^{18} - 19261 \nu^{16} + 190662 \nu^{14} - 986020 \nu^{12} + 2846233 \nu^{10} - 4530894 \nu^{8} + \cdots - 3848 ) / 1076 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 288 \nu^{18} + 7037 \nu^{16} - 69926 \nu^{14} + 363141 \nu^{12} - 1052798 \nu^{10} + 1682461 \nu^{8} + \cdots - 669 ) / 269 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 784 \nu^{19} - 19769 \nu^{17} + 205179 \nu^{15} - 1134214 \nu^{13} + 3612724 \nu^{11} + \cdots - 56238 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 615 \nu^{18} + 15125 \nu^{16} - 151546 \nu^{14} + 795708 \nu^{12} - 2343125 \nu^{10} + 3838002 \nu^{8} + \cdots + 3674 ) / 538 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 1317 \nu^{19} + 32698 \nu^{17} - 332179 \nu^{15} + 1781218 \nu^{13} - 5425363 \nu^{11} + \cdots + 40090 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 1767 \nu^{18} + 43273 \nu^{16} - 431250 \nu^{14} + 2248272 \nu^{12} - 6554317 \nu^{10} + \cdots + 460 ) / 1076 \) 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
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{19} + \beta_{17} - \beta_{14} + \beta_{5} + 7\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} + \beta_{16} + \beta_{13} - \beta_{10} + 8\beta_{7} + 7\beta_{6} - \beta_{4} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{19} + 9\beta_{17} + 2\beta_{15} - 8\beta_{14} + 8\beta_{5} + 43\beta_{2} + 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{18} + 10 \beta_{16} + 13 \beta_{13} - 12 \beta_{10} + \beta_{8} + 54 \beta_{7} + \cdots + 87 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 77 \beta_{19} + 64 \beta_{17} + 26 \beta_{15} - 51 \beta_{14} + \beta_{11} + 2 \beta_{9} + 52 \beta_{5} + \cdots + 337 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 90 \beta_{18} + 75 \beta_{16} + 116 \beta_{13} - 103 \beta_{10} + 15 \beta_{8} + 350 \beta_{7} + \cdots + 445 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 542 \beta_{19} + 422 \beta_{17} + 239 \beta_{15} - 303 \beta_{14} - \beta_{12} + 14 \beta_{11} + \cdots + 1851 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 662 \beta_{18} + 510 \beta_{16} + 893 \beta_{13} - 772 \beta_{10} + 151 \beta_{8} + 2241 \beta_{7} + \cdots + 2386 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3662 \beta_{19} + 2695 \beta_{17} + 1913 \beta_{15} - 1747 \beta_{14} - 23 \beta_{12} + 133 \beta_{11} + \cdots + 10476 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4631 \beta_{18} + 3326 \beta_{16} + 6393 \beta_{13} - 5403 \beta_{10} + 1282 \beta_{8} + \cdots + 13300 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 24212 \beta_{19} + 16954 \beta_{17} + 14252 \beta_{15} - 9919 \beta_{14} - 309 \beta_{12} + \cdots + 60653 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 31515 \beta_{18} + 21270 \beta_{16} + 43969 \beta_{13} - 36402 \beta_{10} + 9936 \beta_{8} + \cdots + 76488 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 158093 \beta_{19} + 105910 \beta_{17} + 101750 \beta_{15} - 55820 \beta_{14} - 3229 \beta_{12} + \cdots + 357443 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 210889 \beta_{18} + 134747 \beta_{16} + 295289 \beta_{13} - 239877 \beta_{10} + 72913 \beta_{8} + \cdots + 450757 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1024436 \beta_{19} + 659783 \beta_{17} + 706661 \beta_{15} - 312428 \beta_{14} - 29218 \beta_{12} + \cdots + 2136195 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 1395662 \beta_{18} + 849942 \beta_{16} + 1953793 \beta_{13} - 1559922 \beta_{10} + 516502 \beta_{8} + \cdots + 2706467 \beta_1 \) 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
−2.52711
−2.33380
−2.04146
−2.00304
−1.51095
−1.23709
−1.22468
−0.712399
−0.386431
−0.131596
0.131596
0.386431
0.712399
1.22468
1.23709
1.51095
2.00304
2.04146
2.33380
2.52711
−2.52711 0.473551 4.38628 0 −1.19672 1.17543 −6.03039 −2.77575 0
1.2 −2.33380 1.97214 3.44661 0 −4.60257 4.79282 −3.37609 0.889329 0
1.3 −2.04146 −1.84076 2.16755 0 3.75784 2.14151 −0.342047 0.388411 0
1.4 −2.00304 2.62978 2.01218 0 −5.26757 −4.24477 −0.0243959 3.91575 0
1.5 −1.51095 −0.900785 0.282982 0 1.36104 0.603482 2.59434 −2.18859 0
1.6 −1.23709 2.07277 −0.469598 0 −2.56421 −0.314194 3.05513 1.29637 0
1.7 −1.22468 −1.51450 −0.500160 0 1.85478 0.966830 3.06189 −0.706295 0
1.8 −0.712399 −3.32227 −1.49249 0 2.36678 2.70450 2.48804 8.03747 0
1.9 −0.386431 −0.261531 −1.85067 0 0.101064 −4.13791 1.48802 −2.93160 0
1.10 −0.131596 −1.44045 −1.98268 0 0.189557 −0.456875 0.524103 −0.925103 0
1.11 0.131596 1.44045 −1.98268 0 0.189557 0.456875 −0.524103 −0.925103 0
1.12 0.386431 0.261531 −1.85067 0 0.101064 4.13791 −1.48802 −2.93160 0
1.13 0.712399 3.32227 −1.49249 0 2.36678 −2.70450 −2.48804 8.03747 0
1.14 1.22468 1.51450 −0.500160 0 1.85478 −0.966830 −3.06189 −0.706295 0
1.15 1.23709 −2.07277 −0.469598 0 −2.56421 0.314194 −3.05513 1.29637 0
1.16 1.51095 0.900785 0.282982 0 1.36104 −0.603482 −2.59434 −2.18859 0
1.17 2.00304 −2.62978 2.01218 0 −5.26757 4.24477 0.0243959 3.91575 0
1.18 2.04146 1.84076 2.16755 0 3.75784 −2.14151 0.342047 0.388411 0
1.19 2.33380 −1.97214 3.44661 0 −4.60257 −4.79282 3.37609 0.889329 0
1.20 2.52711 −0.473551 4.38628 0 −1.19672 −1.17543 6.03039 −2.77575 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)
\(19\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5225.2.a.ba 20
5.b even 2 1 inner 5225.2.a.ba 20
5.c odd 4 2 1045.2.b.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.c ✓ 20 5.c odd 4 2
5225.2.a.ba 20 1.a even 1 1 trivial
5225.2.a.ba 20 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{20} - 26 T_{2}^{18} + 281 T_{2}^{16} - 1640 T_{2}^{14} + 5623 T_{2}^{12} - 11551 T_{2}^{10} + \cdots + 4 \) Copy content Toggle raw display
\( T_{7}^{20} - 73 T_{7}^{18} + 2053 T_{7}^{16} - 28024 T_{7}^{14} + 194005 T_{7}^{12} - 670233 T_{7}^{10} + \cdots + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 26 T^{18} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{20} - 35 T^{18} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 73 T^{18} + \cdots + 2304 \) Copy content Toggle raw display
$11$ \( (T - 1)^{20} \) Copy content Toggle raw display
$13$ \( T^{20} - 137 T^{18} + \cdots + 33732864 \) Copy content Toggle raw display
$17$ \( T^{20} - 157 T^{18} + \cdots + 7054336 \) Copy content Toggle raw display
$19$ \( (T + 1)^{20} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 172764736 \) Copy content Toggle raw display
$29$ \( (T^{10} + 25 T^{9} + \cdots + 4194192)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 25 T^{9} + \cdots - 7456)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 61309721664 \) Copy content Toggle raw display
$41$ \( (T^{10} + 17 T^{9} + \cdots + 166848)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 7783474176 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{10} + 15 T^{9} + \cdots + 86423136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 7 T^{9} + \cdots - 26701648)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 345472702914816 \) Copy content Toggle raw display
$71$ \( (T^{10} + 20 T^{9} + \cdots - 400032)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{10} + 53 T^{9} + \cdots - 204893824)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 5917659525376 \) Copy content Toggle raw display
$89$ \( (T^{10} + 18 T^{9} + \cdots + 14189256)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 19285658703936 \) Copy content Toggle raw display
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