Properties

Label 1875.2.b.h
Level $1875$
Weight $2$
Character orbit 1875.b
Analytic conductor $14.972$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) 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 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \beta_{9} - \beta_1) q^{2} - \beta_{9} q^{3} + ( - \beta_{15} - \beta_{14} - 1) q^{4} + ( - \beta_{15} - 1) q^{6} + ( - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{7} + ( - \beta_{11} + 2 \beta_{9} - 2 \beta_{4}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} - \beta_1) q^{2} - \beta_{9} q^{3} + ( - \beta_{15} - \beta_{14} - 1) q^{4} + ( - \beta_{15} - 1) q^{6} + ( - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{7} + ( - \beta_{11} + 2 \beta_{9} - 2 \beta_{4}) q^{8} - q^{9} + (\beta_{15} - \beta_{13} + \beta_{8} - 2 \beta_{7} + \beta_{2} + 2) q^{11} + (\beta_{9} - \beta_{4} + \beta_1) q^{12} + (\beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_1) q^{13} + ( - \beta_{15} - \beta_{13} - \beta_{8} + \beta_{5} - 1) q^{14} + (\beta_{15} - \beta_{14} - \beta_{13} + 2 \beta_{8} + 2) q^{16} + ( - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{3} + \beta_1) q^{17} + (\beta_{9} + \beta_1) q^{18} + (\beta_{13} + \beta_{7} + \beta_{5} - 2 \beta_{2} + 1) q^{19} + (\beta_{13} - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{2} - 2) q^{21} + (\beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{9} + \beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_1) q^{22} + (\beta_{10} + \beta_{9} - 2 \beta_{3} - \beta_1) q^{23} + (\beta_{14} + \beta_{8} + 2) q^{24} + (3 \beta_{15} - 3 \beta_{13} + 2 \beta_{7} + \beta_{5} + 4) q^{26} + \beta_{9} q^{27} + ( - \beta_{12} + \beta_{11} + 2 \beta_{9} + 3 \beta_{6} - \beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{28} + ( - 2 \beta_{14} - \beta_{13} + \beta_{8} - 2 \beta_{7} + \beta_{5} + 1) q^{29} + (2 \beta_{15} + \beta_{13} - \beta_{7} + \beta_{5} - 2) q^{31} + ( - \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{6} - \beta_1) q^{32} + ( - 2 \beta_{12} + \beta_{11} - \beta_{9} + \beta_{6} + \beta_{4} + \beta_{3}) q^{33} + (2 \beta_{14} - 3 \beta_{13} + 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{2} + 4) q^{34} + (\beta_{15} + \beta_{14} + 1) q^{36} + (\beta_{12} - 3 \beta_{11} + \beta_{10} - 3 \beta_{9} + \beta_{6} - 2 \beta_{4} - \beta_{3} - \beta_1) q^{37} + ( - \beta_{12} - \beta_{10} - 5 \beta_{9} + 2 \beta_{6} + 2 \beta_{4} - \beta_{3} - 3 \beta_1) q^{38} + (\beta_{15} + \beta_{14} + \beta_{13} - \beta_{8} + 2 \beta_{5} + 2) q^{39} + (\beta_{15} + 2 \beta_{14} - 3 \beta_{8} + \beta_{5} - \beta_{2}) q^{41} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{4} + \beta_1) q^{42} + ( - 2 \beta_{11} + 4 \beta_{9} + 2 \beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{43} + ( - 2 \beta_{15} - 2 \beta_{14} - 3 \beta_{13} - 2 \beta_{7} - \beta_{5} + 2 \beta_{2} + \cdots - 2) q^{44}+ \cdots + ( - \beta_{15} + \beta_{13} - \beta_{8} + 2 \beta_{7} - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{14} - 53\nu^{12} - 345\nu^{10} - 1037\nu^{8} - 1518\nu^{6} - 1020\nu^{4} - 245\nu^{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{15} - 69\nu^{13} - 430\nu^{11} - 1183\nu^{9} - 1413\nu^{7} - 483\nu^{5} + 179\nu^{3} + 45\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{15} + 124\nu^{13} + 810\nu^{11} + 2444\nu^{9} + 3583\nu^{7} + 2400\nu^{5} + 595\nu^{3} + 47\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{14} - 158\nu^{12} - 1017\nu^{10} - 2991\nu^{8} - 4185\nu^{6} - 2527\nu^{4} - 435\nu^{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10\nu^{15} - 177\nu^{13} - 1154\nu^{11} - 3465\nu^{9} - 5013\nu^{7} - 3225\nu^{5} - 675\nu^{3} - 19\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{14} + 194\nu^{12} + 1257\nu^{10} + 3731\nu^{8} + 5277\nu^{6} + 3223\nu^{4} + 565\nu^{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -16\nu^{14} - 282\nu^{12} - 1826\nu^{10} - 5419\nu^{8} - 7680\nu^{6} - 4732\nu^{4} - 865\nu^{2} - 13 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -19\nu^{15} - 335\nu^{13} - 2170\nu^{11} - 6440\nu^{9} - 9110\nu^{7} - 5557\nu^{5} - 945\nu^{3} + 15\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -22\nu^{15} - 387\nu^{13} - 2498\nu^{11} - 7373\nu^{9} - 10347\nu^{7} - 6237\nu^{5} - 1041\nu^{3} + 7\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\nu^{15} + 459\nu^{13} + 2980\nu^{11} + 8884\nu^{9} + 12693\nu^{7} + 7957\nu^{5} + 1542\nu^{3} + 40\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 29\nu^{15} + 512\nu^{13} + 3325\nu^{11} + 9921\nu^{9} + 14211\nu^{7} + 8977\nu^{5} + 1785\nu^{3} + 40\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( 16\nu^{14} + 282\nu^{12} + 1826\nu^{10} + 5419\nu^{8} + 7680\nu^{6} + 4731\nu^{4} + 858\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -45\nu^{14} - 794\nu^{12} - 5150\nu^{10} - 15324\nu^{8} - 21803\nu^{6} - 13514\nu^{4} - 2487\nu^{2} - 23 ) / 2 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -45\nu^{14} - 794\nu^{12} - 5150\nu^{10} - 15324\nu^{8} - 21803\nu^{6} - 13514\nu^{4} - 2485\nu^{2} - 19 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{4} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{15} + 7\beta_{14} - \beta_{13} - 2\beta_{8} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{11} + \beta_{10} - 9\beta_{9} + 3\beta_{6} + 9\beta_{4} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 46\beta_{15} - 44\beta_{14} + 12\beta_{13} + 16\beta_{8} - \beta_{7} + 3\beta_{5} - 33 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4\beta_{12} + 44\beta_{11} - 13\beta_{10} + 64\beta_{9} - 34\beta_{6} - 62\beta_{4} + \beta_{3} - 123\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -297\beta_{15} + 273\beta_{14} - 104\beta_{13} - 106\beta_{8} + 18\beta_{7} - 38\beta_{5} - 4\beta_{2} + 181 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 60\beta_{12} - 273\beta_{11} + 122\beta_{10} - 423\beta_{9} + 286\beta_{6} + 399\beta_{4} - 18\beta_{3} + 751\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1906\beta_{15} - 1696\beta_{14} + 803\beta_{13} + 672\beta_{8} - 200\beta_{7} + 346\beta_{5} + 60\beta_{2} - 1061 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 606 \beta_{12} + 1696 \beta_{11} - 1003 \beta_{10} + 2728 \beta_{9} - 2167 \beta_{6} - 2518 \beta_{4} + 200 \beta_{3} - 4663 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 12211 \beta_{15} + 10573 \beta_{14} - 5869 \beta_{13} - 4214 \beta_{8} + 1809 \beta_{7} - 2773 \beta_{5} - 606 \beta_{2} + 6402 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5188 \beta_{12} - 10573 \beta_{11} + 7678 \beta_{10} - 17457 \beta_{9} + 15629 \beta_{6} + 15819 \beta_{4} - 1809 \beta_{3} + 29186 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 78223 \beta_{15} - 66151 \beta_{14} + 41558 \beta_{13} + 26392 \beta_{8} - 14675 \beta_{7} + 20817 \beta_{5} + 5188 \beta_{2} - 39174 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 40680 \beta_{12} + 66151 \beta_{11} - 56233 \beta_{10} + 111499 \beta_{9} - 109584 \beta_{6} - 99427 \beta_{4} + 14675 \beta_{3} - 183548 \beta_1 \) 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
1.53767i
1.35083i
2.53767i
0.536547i
2.35083i
0.0898194i
1.53655i
1.08982i
1.08982i
1.53655i
0.0898194i
2.35083i
0.536547i
2.53767i
1.35083i
1.53767i
2.53767i 1.00000i −4.43979 0 −2.53767 1.04054i 6.19138i −1.00000 0
1249.2 2.35083i 1.00000i −3.52640 0 −2.35083 3.48189i 3.58831i −1.00000 0
1249.3 1.53767i 1.00000i −0.364440 0 1.53767 1.68601i 2.51496i −1.00000 0
1249.4 1.53655i 1.00000i −0.360976 0 −1.53655 1.49550i 2.51844i −1.00000 0
1249.5 1.35083i 1.00000i 0.175259 0 1.35083 1.59580i 2.93840i −1.00000 0
1249.6 1.08982i 1.00000i 0.812294 0 −1.08982 3.08724i 3.06489i −1.00000 0
1249.7 0.536547i 1.00000i 1.71212 0 0.536547 2.57318i 1.99173i −1.00000 0
1249.8 0.0898194i 1.00000i 1.99193 0 0.0898194 4.36070i 0.358553i −1.00000 0
1249.9 0.0898194i 1.00000i 1.99193 0 0.0898194 4.36070i 0.358553i −1.00000 0
1249.10 0.536547i 1.00000i 1.71212 0 0.536547 2.57318i 1.99173i −1.00000 0
1249.11 1.08982i 1.00000i 0.812294 0 −1.08982 3.08724i 3.06489i −1.00000 0
1249.12 1.35083i 1.00000i 0.175259 0 1.35083 1.59580i 2.93840i −1.00000 0
1249.13 1.53655i 1.00000i −0.360976 0 −1.53655 1.49550i 2.51844i −1.00000 0
1249.14 1.53767i 1.00000i −0.364440 0 1.53767 1.68601i 2.51496i −1.00000 0
1249.15 2.35083i 1.00000i −3.52640 0 −2.35083 3.48189i 3.58831i −1.00000 0
1249.16 2.53767i 1.00000i −4.43979 0 −2.53767 1.04054i 6.19138i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.16
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.h 16
5.b even 2 1 inner 1875.2.b.h 16
5.c odd 4 1 1875.2.a.m 8
5.c odd 4 1 1875.2.a.p 8
15.e even 4 1 5625.2.a.t 8
15.e even 4 1 5625.2.a.bd 8
25.d even 5 1 75.2.i.a 16
25.d even 5 1 375.2.i.c 16
25.e even 10 1 75.2.i.a 16
25.e even 10 1 375.2.i.c 16
25.f odd 20 2 375.2.g.d 16
25.f odd 20 2 375.2.g.e 16
75.h odd 10 1 225.2.m.b 16
75.j odd 10 1 225.2.m.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.i.a 16 25.d even 5 1
75.2.i.a 16 25.e even 10 1
225.2.m.b 16 75.h odd 10 1
225.2.m.b 16 75.j odd 10 1
375.2.g.d 16 25.f odd 20 2
375.2.g.e 16 25.f odd 20 2
375.2.i.c 16 25.d even 5 1
375.2.i.c 16 25.e even 10 1
1875.2.a.m 8 5.c odd 4 1
1875.2.a.p 8 5.c odd 4 1
1875.2.b.h 16 1.a even 1 1 trivial
1875.2.b.h 16 5.b even 2 1 inner
5625.2.a.t 8 15.e even 4 1
5625.2.a.bd 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 20T_{2}^{14} + 156T_{2}^{12} + 610T_{2}^{10} + 1286T_{2}^{8} + 1440T_{2}^{6} + 761T_{2}^{4} + 130T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 20 T^{14} + 156 T^{12} + 610 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 56 T^{14} + 1236 T^{12} + \cdots + 255025 \) Copy content Toggle raw display
$11$ \( (T^{8} - 2 T^{7} - 43 T^{6} + 90 T^{5} + \cdots + 5281)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 110 T^{14} + 4141 T^{12} + \cdots + 78961 \) Copy content Toggle raw display
$17$ \( T^{16} + 172 T^{14} + \cdots + 53860921 \) Copy content Toggle raw display
$19$ \( (T^{8} - 14 T^{7} + 11 T^{6} + 516 T^{5} + \cdots + 2525)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 174 T^{14} + 10761 T^{12} + \cdots + 4389025 \) Copy content Toggle raw display
$29$ \( (T^{8} + 2 T^{7} - 106 T^{6} - 162 T^{5} + \cdots - 395)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 22 T^{7} + 169 T^{6} + 548 T^{5} + \cdots + 125)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 376 T^{14} + \cdots + 8653650625 \) Copy content Toggle raw display
$41$ \( (T^{8} - 8 T^{7} - 56 T^{6} + 428 T^{5} + \cdots + 4705)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 388 T^{14} + \cdots + 527207521 \) Copy content Toggle raw display
$47$ \( T^{16} + 472 T^{14} + \cdots + 36687479166361 \) Copy content Toggle raw display
$53$ \( T^{16} + 444 T^{14} + \cdots + 40398990025 \) Copy content Toggle raw display
$59$ \( (T^{8} + 14 T^{7} - 29 T^{6} - 886 T^{5} + \cdots - 3595)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 20 T^{7} - 136 T^{6} + \cdots + 16604261)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 532 T^{14} + \cdots + 13980121 \) Copy content Toggle raw display
$71$ \( (T^{8} - 16 T^{7} - 78 T^{6} + \cdots - 159779)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 644 T^{14} + \cdots + 757413387025 \) Copy content Toggle raw display
$79$ \( (T^{8} - 30 T^{7} + 145 T^{6} + \cdots - 1984975)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 520 T^{14} + \cdots + 2356228681 \) Copy content Toggle raw display
$89$ \( (T^{8} + 16 T^{7} - 39 T^{6} - 1454 T^{5} + \cdots + 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 472 T^{14} + \cdots + 216648961 \) Copy content Toggle raw display
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