Properties

Label 1875.2.b.e
Level $1875$
Weight $2$
Character orbit 1875.b
Analytic conductor $14.972$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{11} q^{6} + (\beta_{9} + \beta_{6} - \beta_{3}) q^{7} + (\beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{11} q^{6} + (\beta_{9} + \beta_{6} - \beta_{3}) q^{7} + (\beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - \beta_1) q^{8} - q^{9} + (\beta_{7} + \beta_{4} - 1) q^{11} + (\beta_{9} + 2 \beta_{6}) q^{12} + ( - \beta_{9} - \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{11} - \beta_{10} + \beta_{7} + \beta_{4} - \beta_{2}) q^{14} + ( - 2 \beta_{11} + 2 \beta_{10} + \beta_{7} + \beta_{4} - \beta_{2} + 1) q^{16} + (\beta_{8} + \beta_{3} + 2 \beta_1) q^{17} - \beta_1 q^{18} + (2 \beta_{10} - \beta_{4} + \beta_{2}) q^{19} + ( - \beta_{4} - \beta_{2} + 1) q^{21} + ( - \beta_{9} - \beta_{6} + 2 \beta_{5}) q^{22} + (\beta_{9} + \beta_{8} + \beta_{6} + 2 \beta_{5} + \beta_{3} + 2 \beta_1) q^{23} + ( - \beta_{11} + \beta_{7} + \beta_{4} - \beta_{2}) q^{24} + (2 \beta_{11} - \beta_{10} - \beta_{7} - \beta_{4} - \beta_{2} + 8) q^{26} + \beta_{6} q^{27} + ( - \beta_{9} - 7 \beta_{6} + 2 \beta_{5} + \beta_{3} + 2 \beta_1) q^{28} + ( - \beta_{2} - 5) q^{29} + ( - 2 \beta_{10} + \beta_{7} - \beta_{2}) q^{31} + ( - 2 \beta_{9} - \beta_{8} - 2 \beta_{6} + 2 \beta_{5} - 7 \beta_{3} + \beta_1) q^{32} + ( - \beta_{8} - \beta_{3}) q^{33} + (2 \beta_{10} + \beta_{2} - 7) q^{34} + ( - \beta_{2} + 2) q^{36} + ( - \beta_{9} - \beta_{8} + 4 \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{37} + (\beta_{9} - \beta_{8} + 7 \beta_{6} - \beta_{5} - 7 \beta_{3} - 2 \beta_1) q^{38} + ( - 2 \beta_{11} - \beta_{4} + \beta_{2}) q^{39} + ( - 2 \beta_{11} - \beta_{7} - \beta_{4} + \beta_{2} + 6) q^{41} + ( - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1) q^{42} + (\beta_{6} + 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{43} + (2 \beta_{11} - \beta_{7} - 7 \beta_{4} + \beta_{2} + 6) q^{44} + ( - 2 \beta_{11} + 2 \beta_{10} - \beta_{7} - 7 \beta_{4} + 1) q^{46} + (2 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{47} + ( - \beta_{9} - \beta_{8} - 2 \beta_{6} + 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{48} + (2 \beta_{11} - 2 \beta_{10} + \beta_{7} - \beta_{2}) q^{49} + (2 \beta_{11} + \beta_{7} + \beta_{4} - 1) q^{51} + (5 \beta_{6} - 2 \beta_{5} + \beta_{3} + 4 \beta_1) q^{52} + ( - 2 \beta_{8} + 2 \beta_{5} + 4 \beta_{3}) q^{53} - \beta_{11} q^{54} + (5 \beta_{11} - \beta_{10} - \beta_{7} - 7 \beta_{4} + \beta_{2}) q^{56} + (\beta_{9} + 2 \beta_{5} + \beta_{3}) q^{57} + ( - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{3} - 4 \beta_1) q^{58} + (2 \beta_{11} + \beta_{7} + \beta_{4} - 2 \beta_{2} + 1) q^{59} + ( - 2 \beta_{11} + 2 \beta_{10} - \beta_{7} - 2 \beta_{4} + \beta_{2} + 6) q^{61} + ( - 2 \beta_{9} + \beta_{8} - 8 \beta_{6} + \beta_{5} + 7 \beta_{3} + \beta_1) q^{62} + ( - \beta_{9} - \beta_{6} + \beta_{3}) q^{63} + (6 \beta_{11} - 4 \beta_{10} - 2 \beta_{7} - 8 \beta_{4} + 2 \beta_{2} + 5) q^{64} + ( - 2 \beta_{10} + \beta_{2} - 1) q^{66} + (\beta_{9} - \beta_{8} + \beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{67} + (\beta_{9} + \beta_{8} + 7 \beta_{6} - 5 \beta_{3} - 4 \beta_1) q^{68} + (2 \beta_{11} - 2 \beta_{10} + \beta_{7} + \beta_{4} - \beta_{2}) q^{69} + ( - 2 \beta_{11} + 2 \beta_{7} + 2 \beta_{4} + \beta_{2} - 3) q^{71} + ( - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{3} + \beta_1) q^{72} + ( - \beta_{9} - 2 \beta_{8} - 7 \beta_{6} - \beta_{3} - 2 \beta_1) q^{73} + ( - 4 \beta_{11} - \beta_{10} - 3 \beta_{7} - 9 \beta_{4} + 4 \beta_{2} - 1) q^{74} + ( - 2 \beta_{11} - 4 \beta_{10} + 2 \beta_{7} + 3 \beta_{4} + 3) q^{76} + ( - 6 \beta_{6} + 6 \beta_{3} - 2 \beta_1) q^{77} + ( - \beta_{9} + \beta_{8} - 7 \beta_{6} - \beta_{5} + \beta_{3} - 2 \beta_1) q^{78} + (2 \beta_{11} + \beta_{4} - 2 \beta_{2} - 4) q^{79} + q^{81} + (\beta_{8} - 6 \beta_{6} - 2 \beta_{5} + \beta_{3} + 4 \beta_1) q^{82} + ( - 2 \beta_{9} + \beta_{8} - 2 \beta_{6} + \beta_{3} + 2 \beta_1) q^{83} + (2 \beta_{11} - 2 \beta_{10} + \beta_{4} + \beta_{2} - 7) q^{84} + ( - \beta_{10} - 2 \beta_{7} - 8 \beta_{4} + 2 \beta_{2}) q^{86} + ( - \beta_{9} + 5 \beta_{6}) q^{87} + (2 \beta_{9} + \beta_{8} + 8 \beta_{6} - 4 \beta_{5} + \beta_{3} - 2 \beta_1) q^{88} + ( - 4 \beta_{11} + 2 \beta_{10} - \beta_{7} - \beta_{4} - 2 \beta_{2} - 3) q^{89} + (4 \beta_{10} - \beta_{7} - \beta_{4} + 2 \beta_{2} + 4) q^{91} + (\beta_{9} + \beta_{6} - 4 \beta_{5} - 6 \beta_{3} - 2 \beta_1) q^{92} + ( - \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5}) q^{93} + ( - 4 \beta_{11} + 4 \beta_{10} - 6 \beta_{4} - 4 \beta_{2} + 10) q^{94} + (\beta_{11} - 2 \beta_{10} - \beta_{7} - 7 \beta_{4} + 2 \beta_{2} - 1) q^{96} + ( - 3 \beta_{8} + 2 \beta_{5} - 4 \beta_{3}) q^{97} + (\beta_{8} + \beta_{5} + 7 \beta_{3} + \beta_1) q^{98} + ( - \beta_{7} - \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9} + 8 q^{14} + 34 q^{16} + 4 q^{19} + 4 q^{21} + 12 q^{24} + 74 q^{26} - 62 q^{29} - 4 q^{31} - 74 q^{34} + 22 q^{36} + 66 q^{41} + 22 q^{44} - 24 q^{46} - 8 q^{49} - 4 q^{51} + 2 q^{54} - 60 q^{56} + 16 q^{59} + 68 q^{61} - 24 q^{64} - 18 q^{66} - 2 q^{69} - 6 q^{71} - 72 q^{74} + 54 q^{76} - 50 q^{79} + 12 q^{81} - 88 q^{84} - 60 q^{86} - 36 q^{89} + 56 q^{91} + 100 q^{94} - 66 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 20\nu^{9} + 139\nu^{7} + 377\nu^{5} + 268\nu^{3} - 93\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{10} + 37\nu^{8} + 227\nu^{6} + 490\nu^{4} + 161\nu^{2} - 9 ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{11} + 37\nu^{9} + 227\nu^{7} + 490\nu^{5} + 161\nu^{3} - 45\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{11} + 97\nu^{9} + 662\nu^{7} + 1873\nu^{5} + 1937\nu^{3} + 522\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + 23\nu^{8} + 190\nu^{6} + 677\nu^{4} + 967\nu^{2} + 342 ) / 36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} - 16\nu^{9} - 71\nu^{7} - \nu^{5} + 472\nu^{3} + 409\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{11} - 55\nu^{9} - 551\nu^{7} - 2434\nu^{5} - 4355\nu^{3} - 1683\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} + 18\nu^{8} + 109\nu^{6} + 249\nu^{4} + 154\nu^{2} + 21 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2\nu^{10} + 37\nu^{8} + 233\nu^{6} + 562\nu^{4} + 377\nu^{2} + 45 ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{11} + 2\beta_{10} + \beta_{7} + \beta_{4} - 7\beta_{2} + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{9} - 9\beta_{8} - 10\beta_{6} + 2\beta_{5} - 15\beta_{3} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26\beta_{11} - 24\beta_{10} - 12\beta_{7} - 18\beta_{4} + 48\beta_{2} - 117 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 86\beta_{9} + 72\beta_{8} + 92\beta_{6} - 30\beta_{5} + 144\beta_{3} - 183\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -250\beta_{11} + 216\beta_{10} + 116\beta_{7} + 206\beta_{4} - 341\beta_{2} + 684 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -707\beta_{9} - 557\beta_{8} - 809\beta_{6} + 322\beta_{5} - 1205\beta_{3} + 1231\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2164\beta_{11} - 1762\beta_{10} - 1029\beta_{7} - 1995\beta_{4} + 2495\beta_{2} - 4193 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5688\beta_{9} + 4257\beta_{8} + 6894\beta_{6} - 3024\beta_{5} + 9543\beta_{3} - 8683\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
2.78712i
2.38719i
2.13324i
2.02791i
0.858825i
0.364088i
0.364088i
0.858825i
2.02791i
2.13324i
2.38719i
2.78712i
2.78712i 1.00000i −5.76803 0 −2.78712 3.15000i 10.5020i −1.00000 0
1249.2 2.38719i 1.00000i −3.69868 0 2.38719 3.31671i 4.05506i −1.00000 0
1249.3 2.13324i 1.00000i −2.55073 0 −2.13324 2.16876i 1.17484i −1.00000 0
1249.4 2.02791i 1.00000i −2.11242 0 2.02791 0.505614i 0.227977i −1.00000 0
1249.5 0.858825i 1.00000i 1.26242 0 −0.858825 3.88045i 2.80185i −1.00000 0
1249.6 0.364088i 1.00000i 1.86744 0 0.364088 2.24941i 1.40809i −1.00000 0
1249.7 0.364088i 1.00000i 1.86744 0 0.364088 2.24941i 1.40809i −1.00000 0
1249.8 0.858825i 1.00000i 1.26242 0 −0.858825 3.88045i 2.80185i −1.00000 0
1249.9 2.02791i 1.00000i −2.11242 0 2.02791 0.505614i 0.227977i −1.00000 0
1249.10 2.13324i 1.00000i −2.55073 0 −2.13324 2.16876i 1.17484i −1.00000 0
1249.11 2.38719i 1.00000i −3.69868 0 2.38719 3.31671i 4.05506i −1.00000 0
1249.12 2.78712i 1.00000i −5.76803 0 −2.78712 3.15000i 10.5020i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.12
Significant digits:
Format:

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 1875.2.b.e 12
5.b even 2 1 inner 1875.2.b.e 12
5.c odd 4 1 1875.2.a.i 6
5.c odd 4 1 1875.2.a.l yes 6
15.e even 4 1 5625.2.a.o 6
15.e even 4 1 5625.2.a.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.i 6 5.c odd 4 1
1875.2.a.l yes 6 5.c odd 4 1
1875.2.b.e 12 1.a even 1 1 trivial
1875.2.b.e 12 5.b even 2 1 inner
5625.2.a.o 6 15.e even 4 1
5625.2.a.r 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 23T_{2}^{10} + 199T_{2}^{8} + 794T_{2}^{6} + 1399T_{2}^{4} + 783T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 23 T^{10} + 199 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 46 T^{10} + 811 T^{8} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( (T^{6} - 29 T^{4} - 8 T^{3} + 184 T^{2} + \cdots - 144)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 112 T^{10} + 4784 T^{8} + \cdots + 121801 \) Copy content Toggle raw display
$17$ \( T^{12} + 114 T^{10} + 4705 T^{8} + \cdots + 331776 \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} - 74 T^{4} + 120 T^{3} + \cdots - 5725)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 139 T^{10} + 6121 T^{8} + \cdots + 518400 \) Copy content Toggle raw display
$29$ \( (T^{6} + 31 T^{5} + 379 T^{4} + 2320 T^{3} + \cdots + 6480)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 2 T^{5} - 86 T^{4} + 28 T^{3} + \cdots + 3155)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 366 T^{10} + \cdots + 2125210000 \) Copy content Toggle raw display
$41$ \( (T^{6} - 33 T^{5} + 399 T^{4} - 2192 T^{3} + \cdots + 720)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 161 T^{10} + 7790 T^{8} + \cdots + 1661521 \) Copy content Toggle raw display
$47$ \( T^{12} + 404 T^{10} + \cdots + 6410244096 \) Copy content Toggle raw display
$53$ \( T^{12} + 524 T^{10} + \cdots + 207360000 \) Copy content Toggle raw display
$59$ \( (T^{6} - 8 T^{5} - 69 T^{4} + 600 T^{3} + \cdots - 2880)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 34 T^{5} + 406 T^{4} + \cdots - 72001)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 224 T^{10} + 13840 T^{8} + \cdots + 3481 \) Copy content Toggle raw display
$71$ \( (T^{6} + 3 T^{5} - 225 T^{4} + 160 T^{3} + \cdots - 12816)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 434 T^{10} + \cdots + 415344400 \) Copy content Toggle raw display
$79$ \( (T^{6} + 25 T^{5} + 150 T^{4} - 395 T^{3} + \cdots + 2725)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 402 T^{10} + \cdots + 557715456 \) Copy content Toggle raw display
$89$ \( (T^{6} + 18 T^{5} - 219 T^{4} + \cdots - 42480)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 669 T^{10} + \cdots + 1042708681 \) Copy content Toggle raw display
show more
show less