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Properties

Label 225.2.h.e
Level $225$
Weight $2$
Character orbit 225.h
Analytic conductor $1.797$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: 16.0.1130304400000000000000.3
Defining polynomial: \( x^{16} - 5x^{14} + 5x^{12} - 10x^{10} + 205x^{8} - 700x^{6} + 1250x^{4} - 1250x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{14} + \beta_{12} - \beta_{10} + \beta_{3}) q^{2} + ( - \beta_{11} - \beta_{8}) q^{4} + (\beta_{12} + \beta_{5}) q^{5} + (\beta_{13} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{2} - 1) q^{7} - \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{14} + \beta_{12} - \beta_{10} + \beta_{3}) q^{2} + ( - \beta_{11} - \beta_{8}) q^{4} + (\beta_{12} + \beta_{5}) q^{5} + (\beta_{13} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{2} - 1) q^{7} - \beta_{4} q^{8} + ( - \beta_{15} - \beta_{8} - \beta_{7} - 2 \beta_{6} - 1) q^{10} + (\beta_{14} - \beta_{12} + \beta_{9} + \beta_{5} + \beta_1) q^{11} + (\beta_{15} - \beta_{8} - \beta_{6} - 2 \beta_{2}) q^{13} + (\beta_{14} - \beta_{12} + 2 \beta_{10} - \beta_{5} - 2 \beta_{3}) q^{14} + (\beta_{11} - \beta_{7} - \beta_{6} + \beta_{2}) q^{16} + ( - \beta_{14} - 2 \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3}) q^{17} + (\beta_{15} - \beta_{13} - 3 \beta_{11} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{2} + 3) q^{19} + ( - \beta_{14} + 2 \beta_{10} + \beta_{4} - \beta_{3} - \beta_1) q^{20} + (2 \beta_{13} + 2 \beta_{11} - \beta_{8} - 4 \beta_{7} - 4) q^{22} + (\beta_{14} - \beta_{12} - 2 \beta_{9} - 2 \beta_{5} - 2 \beta_1) q^{23} + ( - 2 \beta_{15} - \beta_{11} + 2 \beta_{8} + \beta_{7} + \beta_{2} + 1) q^{25} + (\beta_{14} - 4 \beta_{12} + \beta_{10} + 2 \beta_{9} - \beta_{4} + \beta_1) q^{26} + ( - \beta_{13} + 3 \beta_{11} + \beta_{8} + 2 \beta_{7} + 2) q^{28} + (2 \beta_{14} - 3 \beta_{5} + 3 \beta_{4} - \beta_1) q^{29} + (\beta_{15} - \beta_{13} - 2 \beta_{11} + 3 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} + 3 \beta_{2} + \cdots + 4) q^{31}+ \cdots + (4 \beta_{14} - 4 \beta_{12} + 2 \beta_{9} + \beta_{5} + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{4} + 14 q^{16} + 14 q^{19} - 30 q^{22} + 10 q^{25} + 30 q^{28} + 18 q^{31} - 20 q^{34} + 10 q^{37} - 10 q^{40} - 80 q^{43} - 32 q^{49} - 40 q^{52} - 70 q^{55} - 10 q^{58} + 32 q^{61} - 8 q^{64} - 40 q^{67} + 50 q^{70} + 60 q^{73} - 88 q^{76} + 36 q^{79} + 120 q^{82} + 20 q^{88} + 30 q^{91} + 30 q^{94} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5x^{14} + 5x^{12} - 10x^{10} + 205x^{8} - 700x^{6} + 1250x^{4} - 1250x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 9094 \nu^{15} + 529550 \nu^{13} - 1775155 \nu^{11} - 1261485 \nu^{9} - 4566370 \nu^{7} + 105496925 \nu^{5} - 206804175 \nu^{3} + 136237250 \nu ) / 54753875 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 18559 \nu^{14} + 486170 \nu^{12} - 1233230 \nu^{10} - 2183265 \nu^{8} - 1879805 \nu^{6} + 88524275 \nu^{4} - 87914875 \nu^{2} + 104474000 ) / 54753875 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 85587 \nu^{15} - 683920 \nu^{13} + 729395 \nu^{11} + 734880 \nu^{9} + 21524610 \nu^{7} - 103007550 \nu^{5} + 105654425 \nu^{3} - 113406125 \nu ) / 54753875 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 191309 \nu^{15} - 327040 \nu^{13} - 1605420 \nu^{11} - 1814665 \nu^{9} + 36243345 \nu^{7} - 11269000 \nu^{5} - 84275075 \nu^{3} + 103147500 \nu ) / 54753875 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 216278 \nu^{15} - 1248275 \nu^{13} + 1298755 \nu^{11} - 360830 \nu^{9} + 45929540 \nu^{7} - 186657050 \nu^{5} + 255540700 \nu^{3} - 120294875 \nu ) / 54753875 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 226756 \nu^{14} - 812710 \nu^{12} - 301695 \nu^{10} - 1956335 \nu^{8} + 45336155 \nu^{6} - 87919850 \nu^{4} + 95202125 \nu^{2} - 85476750 ) / 54753875 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 237426 \nu^{14} + 787630 \nu^{12} + 357520 \nu^{10} + 1796010 \nu^{8} - 44976455 \nu^{6} + 87194950 \nu^{4} - 94337250 \nu^{2} - 3771750 ) / 54753875 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26057 \nu^{14} - 128005 \nu^{12} + 108960 \nu^{10} - 210920 \nu^{8} + 5037260 \nu^{6} - 17331500 \nu^{4} + 30950625 \nu^{2} - 22852125 ) / 4977625 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 323013 \nu^{15} + 1471550 \nu^{13} - 371875 \nu^{11} + 1061130 \nu^{9} - 66501065 \nu^{7} + 190202500 \nu^{5} - 199991675 \nu^{3} + \cdots + 54880500 \nu ) / 54753875 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 330256 \nu^{15} + 1021010 \nu^{13} + 321835 \nu^{11} + 4065110 \nu^{9} - 59745955 \nu^{7} + 115300575 \nu^{5} - 188253675 \nu^{3} + \cdots + 81913375 \nu ) / 54753875 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 389298 \nu^{14} - 1338325 \nu^{12} - 665635 \nu^{10} - 3682905 \nu^{8} + 76510440 \nu^{6} - 149141900 \nu^{4} + 161761375 \nu^{2} - 87855125 ) / 54753875 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 443034 \nu^{15} + 2060985 \nu^{13} - 997060 \nu^{11} + 2317165 \nu^{9} - 91265695 \nu^{7} + 274576900 \nu^{5} - 350742825 \nu^{3} + \cdots + 260525500 \nu ) / 54753875 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 40791 \nu^{14} - 196825 \nu^{12} + 194030 \nu^{10} - 383635 \nu^{8} + 8130605 \nu^{6} - 27808800 \nu^{4} + 49697625 \nu^{2} - 36685125 ) / 4977625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 597957 \nu^{15} - 2324530 \nu^{13} + 218325 \nu^{11} - 5390820 \nu^{9} + 116417835 \nu^{7} - 286289900 \nu^{5} + 406932950 \nu^{3} + \cdots - 244363125 \nu ) / 54753875 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 607697 \nu^{14} - 2843530 \nu^{12} + 1509785 \nu^{10} - 3468945 \nu^{8} + 126219210 \nu^{6} - 378889375 \nu^{4} + 538425125 \nu^{2} + \cdots - 393095875 ) / 54753875 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{12} + 2\beta_{10} - \beta_{9} + 3\beta_{5} - \beta_{4} - 3\beta_{3} + 2\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{13} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{14} + \beta_{12} + 2\beta_{10} - \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} - 5\beta_{11} + 2\beta_{8} + \beta_{7} + 9\beta_{6} + \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{12} + 2\beta_{10} - 5\beta_{9} - 5\beta_{5} - 4\beta_{4} - 5\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{15} - 6\beta_{11} - 8\beta_{8} + 11\beta_{7} + 10\beta_{6} + 6\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{14} - 21\beta_{12} + 40\beta_{10} + 14\beta_{9} + 19\beta_{5} + 3\beta_{4} - 46\beta_{3} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 36\beta_{15} - 29\beta_{13} - 44\beta_{11} + 11\beta_{8} - 37\beta_{7} - 18\beta_{6} + 22\beta_{2} - 48 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -6\beta_{14} - 16\beta_{12} + 3\beta_{10} - 39\beta_{9} - 118\beta_{5} + 56\beta_{4} - 17\beta_{3} - 127\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -85\beta_{15} + 140\beta_{13} - 210\beta_{11} - 140\beta_{8} - 165\beta_{7} + 320\beta_{6} - 55\beta_{2} + 35 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -260\beta_{14} - 340\beta_{12} - 85\beta_{10} - 60\beta_{9} + 80\beta_{5} - 175\beta_{4} - 285\beta_{3} + 225\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 200\beta_{15} + 200\beta_{13} + 185\beta_{11} - 520\beta_{8} - 385\beta_{7} - 1005\beta_{6} + 120\beta_{2} - 1005 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 115 \beta_{14} - 415 \beta_{12} + 225 \beta_{10} + 835 \beta_{9} + 530 \beta_{5} + 820 \beta_{4} - 550 \beta_{3} - 495 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -1105\beta_{15} + 685\beta_{13} - 735\beta_{11} - 4150\beta_{7} - 1180\beta_{6} - 685\beta_{2} - 4885 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 4125 \beta_{14} + 420 \beta_{12} - 7095 \beta_{10} - 2285 \beta_{9} - 4200 \beta_{5} + 495 \beta_{4} + 4810 \beta_{3} - 3095 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{11}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
1.85090 − 0.438393i
−0.733267 − 1.75509i
0.733267 + 1.75509i
−1.85090 + 0.438393i
−1.13937 + 0.289499i
0.994142 + 0.627414i
−0.994142 − 0.627414i
1.13937 − 0.289499i
−1.13937 − 0.289499i
0.994142 − 0.627414i
−0.994142 + 0.627414i
1.13937 + 0.289499i
1.85090 + 0.438393i
−0.733267 + 1.75509i
0.733267 − 1.75509i
−1.85090 − 0.438393i
−0.670386 + 2.06324i 0 −2.18949 − 1.59076i 1.69588 + 1.45739i 0 −3.32440 1.23973 − 0.900718i 0 −4.14384 + 2.52199i
46.2 −0.167451 + 0.515362i 0 1.38048 + 1.00297i 1.16252 − 1.91012i 0 1.08833 −1.62484 + 1.18052i 0 0.789737 + 0.918969i
46.3 0.167451 − 0.515362i 0 1.38048 + 1.00297i −1.16252 + 1.91012i 0 1.08833 1.62484 − 1.18052i 0 0.789737 + 0.918969i
46.4 0.670386 − 2.06324i 0 −2.18949 − 1.59076i −1.69588 − 1.45739i 0 −3.32440 −1.23973 + 0.900718i 0 −4.14384 + 2.52199i
91.1 −1.64259 − 1.19341i 0 0.655837 + 2.01846i −2.23130 − 0.145994i 0 −0.504300 0.0767537 − 0.236224i 0 3.49087 + 2.90266i
91.2 −0.757918 − 0.550660i 0 −0.346820 − 1.06740i 1.42964 + 1.71934i 0 2.74037 −0.903913 + 2.78196i 0 −0.136773 − 2.09036i
91.3 0.757918 + 0.550660i 0 −0.346820 − 1.06740i −1.42964 − 1.71934i 0 2.74037 0.903913 − 2.78196i 0 −0.136773 − 2.09036i
91.4 1.64259 + 1.19341i 0 0.655837 + 2.01846i 2.23130 + 0.145994i 0 −0.504300 −0.0767537 + 0.236224i 0 3.49087 + 2.90266i
136.1 −1.64259 + 1.19341i 0 0.655837 − 2.01846i −2.23130 + 0.145994i 0 −0.504300 0.0767537 + 0.236224i 0 3.49087 − 2.90266i
136.2 −0.757918 + 0.550660i 0 −0.346820 + 1.06740i 1.42964 − 1.71934i 0 2.74037 −0.903913 − 2.78196i 0 −0.136773 + 2.09036i
136.3 0.757918 − 0.550660i 0 −0.346820 + 1.06740i −1.42964 + 1.71934i 0 2.74037 0.903913 + 2.78196i 0 −0.136773 + 2.09036i
136.4 1.64259 − 1.19341i 0 0.655837 − 2.01846i 2.23130 − 0.145994i 0 −0.504300 −0.0767537 − 0.236224i 0 3.49087 − 2.90266i
181.1 −0.670386 − 2.06324i 0 −2.18949 + 1.59076i 1.69588 − 1.45739i 0 −3.32440 1.23973 + 0.900718i 0 −4.14384 − 2.52199i
181.2 −0.167451 − 0.515362i 0 1.38048 − 1.00297i 1.16252 + 1.91012i 0 1.08833 −1.62484 − 1.18052i 0 0.789737 − 0.918969i
181.3 0.167451 + 0.515362i 0 1.38048 − 1.00297i −1.16252 − 1.91012i 0 1.08833 1.62484 + 1.18052i 0 0.789737 − 0.918969i
181.4 0.670386 + 2.06324i 0 −2.18949 + 1.59076i −1.69588 + 1.45739i 0 −3.32440 −1.23973 − 0.900718i 0 −4.14384 − 2.52199i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.h.e ✓ 16
3.b odd 2 1 inner 225.2.h.e ✓ 16
25.d even 5 1 inner 225.2.h.e ✓ 16
25.d even 5 1 5625.2.a.w 8
25.e even 10 1 5625.2.a.v 8
75.h odd 10 1 5625.2.a.v 8
75.j odd 10 1 inner 225.2.h.e ✓ 16
75.j odd 10 1 5625.2.a.w 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.h.e ✓ 16 1.a even 1 1 trivial
225.2.h.e ✓ 16 3.b odd 2 1 inner
225.2.h.e ✓ 16 25.d even 5 1 inner
225.2.h.e ✓ 16 75.j odd 10 1 inner
5625.2.a.v 8 25.e even 10 1
5625.2.a.v 8 75.h odd 10 1
5625.2.a.w 8 25.d even 5 1
5625.2.a.w 8 75.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 5T_{2}^{14} + 20T_{2}^{12} + 75T_{2}^{10} + 385T_{2}^{8} + 25T_{2}^{6} + 250T_{2}^{4} + 125T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 5 T^{14} + 20 T^{12} + 75 T^{10} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 5 T^{14} + 50 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 5 T + 5)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + 30 T^{14} + 475 T^{12} + \cdots + 15625 \) Copy content Toggle raw display
$13$ \( (T^{8} - 5 T^{6} - 5 T^{5} + 310 T^{4} + \cdots + 2025)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 15 T^{14} + 445 T^{12} + \cdots + 17682025 \) Copy content Toggle raw display
$19$ \( (T^{8} - 7 T^{7} + 53 T^{6} - 349 T^{5} + \cdots + 11881)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 75 T^{14} + \cdots + 20393268025 \) Copy content Toggle raw display
$29$ \( T^{16} + 155 T^{14} + 11150 T^{12} + \cdots + 15625 \) Copy content Toggle raw display
$31$ \( (T^{8} - 9 T^{7} + 82 T^{6} - 647 T^{5} + \cdots + 641601)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 5 T^{7} + 155 T^{6} - 415 T^{5} + \cdots + 9025)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 60 T^{14} + \cdots + 16041026265625 \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{3} + 115 T^{2} + 150 T - 45)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} - 25 T^{14} + \cdots + 144120025 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 104248286142025 \) Copy content Toggle raw display
$59$ \( T^{16} + 460 T^{14} + \cdots + 39\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{8} - 16 T^{7} + 137 T^{6} + \cdots + 101761)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 20 T^{7} + 395 T^{6} + \cdots + 40896025)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 425 T^{14} + \cdots + 67\!\cdots\!25 \) Copy content Toggle raw display
$73$ \( (T^{8} - 30 T^{7} + 715 T^{6} + \cdots + 117614025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 18 T^{7} + 353 T^{6} + \cdots + 28185481)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 180 T^{14} + 12225 T^{12} + \cdots + 3258025 \) Copy content Toggle raw display
$89$ \( T^{16} + 450 T^{14} + \cdots + 1272666015625 \) Copy content Toggle raw display
$97$ \( (T^{8} - 20 T^{7} + 485 T^{6} + \cdots + 86397025)^{2} \) Copy content Toggle raw display
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