Properties

Label 1875.2.b.f
Level $1875$
Weight $2$
Character orbit 1875.b
Analytic conductor $14.972$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 22x^{10} + 179x^{8} + 641x^{6} + 869x^{4} + 67x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{9} q^{3} + (\beta_{6} + \beta_{4} - 1) q^{4} + \beta_{2} q^{6} + (\beta_{9} - \beta_1) q^{7} + ( - \beta_{7} + \beta_{5} - 2 \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{9} q^{3} + (\beta_{6} + \beta_{4} - 1) q^{4} + \beta_{2} q^{6} + (\beta_{9} - \beta_1) q^{7} + ( - \beta_{7} + \beta_{5} - 2 \beta_1) q^{8} - q^{9} + ( - \beta_{8} + \beta_{2}) q^{11} + (\beta_{11} + \beta_{9} + \beta_{7}) q^{12} + ( - \beta_{10} - \beta_{7} + \beta_1) q^{13} + ( - \beta_{6} - \beta_{4} - \beta_{2} + 3) q^{14} + (\beta_{8} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{16} + (\beta_{11} - \beta_{10} + 3 \beta_{9} - \beta_{7} + \beta_{5}) q^{17} - \beta_1 q^{18} + ( - \beta_{6} - 2 \beta_{4} - \beta_{3} - 3) q^{19} + ( - \beta_{2} + 1) q^{21} + (\beta_{11} + 2 \beta_{9} + \beta_{7} + 2 \beta_{5}) q^{22} + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{7} - \beta_1) q^{23} + (\beta_{4} - \beta_{3} - 2 \beta_{2}) q^{24} + (\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 4) q^{26} + \beta_{9} q^{27} + ( - \beta_{11} - \beta_{9} - \beta_{5} + 4 \beta_1) q^{28} + ( - \beta_{8} - \beta_{3} - \beta_{2}) q^{29} + (\beta_{8} - \beta_{6} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{31} + (\beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 3 \beta_{5} + \beta_1) q^{32} + ( - \beta_{10} - \beta_1) q^{33} + (\beta_{8} - 2 \beta_{4} - \beta_{3} - 4 \beta_{2} - 3) q^{34} + ( - \beta_{6} - \beta_{4} + 1) q^{36} + ( - \beta_{10} + 4 \beta_{9} - \beta_{5} - \beta_1) q^{37} + (\beta_{10} + 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{5} + \beta_1) q^{38} + (\beta_{8} + \beta_{4} + \beta_{2}) q^{39} + ( - \beta_{8} + \beta_{6} - \beta_{4} - 3 \beta_{3} - \beta_{2} - 1) q^{41} + ( - \beta_{11} - 3 \beta_{9} - \beta_{7} + \beta_1) q^{42} + (\beta_{11} - \beta_{9} + \beta_{7} + 2 \beta_{5} + \beta_1) q^{43} + ( - 5 \beta_{4} - \beta_{3} - 3 \beta_{2} - 4) q^{44} + ( - \beta_{6} - 2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 2) q^{46} + ( - \beta_{11} + 4 \beta_{9} - 3 \beta_{7} + 2 \beta_1) q^{47} + (\beta_{10} - 2 \beta_{9} - 3 \beta_{7} + \beta_{5} - \beta_1) q^{48} + (\beta_{6} + \beta_{4} + 2 \beta_{2} + 3) q^{49} + (\beta_{8} - \beta_{6} + \beta_{4} - \beta_{3} + 3) q^{51} + (\beta_{11} - \beta_{10} + 5 \beta_{9} - 5 \beta_{7} + \beta_{5} - 5 \beta_1) q^{52} + (\beta_{11} - 2 \beta_{9} - 4 \beta_{7} - 2 \beta_{5}) q^{53} - \beta_{2} q^{54} + ( - \beta_{8} + 2 \beta_{6} + 4 \beta_{4} + \beta_{2} - 4) q^{56} + ( - \beta_{11} + 3 \beta_{9} - 2 \beta_{7} - \beta_{5}) q^{57} + ( - \beta_{11} + \beta_{10} - 2 \beta_{9} - 4 \beta_{7} + 2 \beta_{5}) q^{58} + ( - 2 \beta_{8} - 3 \beta_{4} - \beta_{3} + \beta_{2} - 4) q^{59} + (\beta_{8} + \beta_{6} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 3) q^{61} + ( - 2 \beta_{11} + 2 \beta_{10} - \beta_{9} - 7 \beta_{7} - \beta_{5}) q^{62} + ( - \beta_{9} + \beta_1) q^{63} + ( - \beta_{8} + \beta_{6} + 5 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 6) q^{64} + ( - \beta_{6} - \beta_{4} - 2 \beta_{3} + 2) q^{66} + ( - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{7} + 2 \beta_{5} - \beta_1) q^{67} + ( - 2 \beta_{11} - \beta_{10} - 3 \beta_{9} - 9 \beta_{7} - 2 \beta_{5} - \beta_1) q^{68} + (\beta_{8} + \beta_{6} - \beta_{4} - \beta_{2} - 2) q^{69} + ( - \beta_{8} + 2 \beta_{6} - 3 \beta_{4} + \beta_{3} - \beta_{2} - 3) q^{71} + (\beta_{7} - \beta_{5} + 2 \beta_1) q^{72} + (2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{7} + \beta_{5}) q^{73} + ( - \beta_{8} - \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4) q^{74} + ( - 2 \beta_{8} - \beta_{6} + 3 \beta_{4} + 2 \beta_{3} - 4) q^{76} + ( - \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{7} - 2 \beta_{5} + \beta_1) q^{77} + (\beta_{11} + 4 \beta_{9} + \beta_{7} - \beta_{5} - \beta_1) q^{78} + (2 \beta_{8} - 2 \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 1) q^{79} + q^{81} + ( - \beta_{11} + 3 \beta_{10} + 2 \beta_{9} - 11 \beta_{7} + \beta_{5} - 2 \beta_1) q^{82} + (2 \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{5}) q^{83} + (\beta_{6} + \beta_{3} + 4 \beta_{2} - 1) q^{84} + (2 \beta_{8} + \beta_{6} - 4 \beta_{4} - \beta_{3} - 2 \beta_{2} - 7) q^{86} + ( - \beta_{10} - \beta_{5} + \beta_1) q^{87} + ( - \beta_{11} + \beta_{10} - 3 \beta_{9} - 4 \beta_{7} - \beta_{5} + \beta_1) q^{88} + (2 \beta_{6} + 8 \beta_{4} - 3 \beta_{3} + \beta_{2} + 5) q^{89} + ( - \beta_{8} - \beta_{6} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 4) q^{91} + (\beta_{11} + \beta_{10} + 11 \beta_{9} - 3 \beta_{7} - 2 \beta_{5} + 4 \beta_1) q^{92} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - 2 \beta_{5} + 2 \beta_1) q^{93} + (2 \beta_{6} + \beta_{4} + 3 \beta_{3} + \beta_{2} - 6) q^{94} + (\beta_{8} - \beta_{6} - 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 2) q^{96} + ( - \beta_{11} - 2 \beta_{10} + 2 \beta_{5} - 3 \beta_1) q^{97} + (2 \beta_{11} + 6 \beta_{9} + \beta_{7} + \beta_{5}) q^{98} + (\beta_{8} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 22x^{10} + 179x^{8} + 641x^{6} + 869x^{4} + 67x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 19\nu^{8} + 122\nu^{6} + 289\nu^{4} + 156\nu^{2} + 5 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{10} - 157\nu^{8} - 902\nu^{6} - 1747\nu^{4} - 242\nu^{2} - 17 ) / 56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} + 71\nu^{8} + 618\nu^{6} + 2337\nu^{4} + 3254\nu^{2} + 99 ) / 56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{11} + 71\nu^{9} + 618\nu^{7} + 2337\nu^{5} + 3254\nu^{3} + 155\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{10} - 71\nu^{8} - 618\nu^{6} - 2337\nu^{4} - 3198\nu^{2} + 69 ) / 56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{11} + 71\nu^{9} + 618\nu^{7} + 2337\nu^{5} + 3198\nu^{3} - 181\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{10} + 21\nu^{8} + 162\nu^{6} + 547\nu^{4} + 694\nu^{2} + 29 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{11} + 109\nu^{9} + 876\nu^{7} + 3083\nu^{5} + 4056\nu^{3} + 179\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -11\nu^{11} - 251\nu^{9} - 2126\nu^{7} - 7953\nu^{5} - 11306\nu^{3} - 979\nu ) / 28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -59\nu^{11} - 1303\nu^{9} - 10642\nu^{7} - 38177\nu^{5} - 51246\nu^{3} - 1947\nu ) / 56 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{5} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - 6\beta_{6} - 9\beta_{4} + \beta_{3} + \beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{10} + 2\beta_{9} + 10\beta_{7} - 11\beta_{5} + 37\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{8} + 37\beta_{6} + 71\beta_{4} - 12\beta_{3} - 14\beta_{2} - 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14\beta_{11} + 12\beta_{10} - 29\beta_{9} - 87\beta_{7} + 93\beta_{5} - 235\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 93\beta_{8} - 235\beta_{6} - 528\beta_{4} + 111\beta_{3} + 144\beta_{2} + 531 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 144\beta_{11} - 111\beta_{10} + 303\beta_{9} + 712\beta_{7} - 714\beta_{5} + 1529\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -714\beta_{8} + 1529\beta_{6} + 3815\beta_{4} - 934\beta_{3} - 1303\beta_{2} - 3270 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1303\beta_{11} + 934\beta_{10} - 2755\beta_{9} - 5634\beta_{7} + 5243\beta_{5} - 10143\beta_1 \) 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
2.68704i
2.44028i
2.16056i
2.01887i
0.246759i
0.141689i
0.141689i
0.246759i
2.01887i
2.16056i
2.44028i
2.68704i
2.68704i 1.00000i −5.22020 0 2.68704 1.68704i 8.65280i −1.00000 0
1249.2 2.44028i 1.00000i −3.95498 0 −2.44028 3.44028i 4.77071i −1.00000 0
1249.3 2.16056i 1.00000i −2.66802 0 −2.16056 3.16056i 1.44329i −1.00000 0
1249.4 2.01887i 1.00000i −2.07584 0 2.01887 1.01887i 0.153106i −1.00000 0
1249.5 0.246759i 1.00000i 1.93911 0 −0.246759 1.24676i 0.972011i −1.00000 0
1249.6 0.141689i 1.00000i 1.97992 0 0.141689 0.858311i 0.563913i −1.00000 0
1249.7 0.141689i 1.00000i 1.97992 0 0.141689 0.858311i 0.563913i −1.00000 0
1249.8 0.246759i 1.00000i 1.93911 0 −0.246759 1.24676i 0.972011i −1.00000 0
1249.9 2.01887i 1.00000i −2.07584 0 2.01887 1.01887i 0.153106i −1.00000 0
1249.10 2.16056i 1.00000i −2.66802 0 −2.16056 3.16056i 1.44329i −1.00000 0
1249.11 2.44028i 1.00000i −3.95498 0 −2.44028 3.44028i 4.77071i −1.00000 0
1249.12 2.68704i 1.00000i −5.22020 0 2.68704 1.68704i 8.65280i −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.f 12
5.b even 2 1 inner 1875.2.b.f 12
5.c odd 4 1 1875.2.a.j 6
5.c odd 4 1 1875.2.a.k 6
15.e even 4 1 5625.2.a.p 6
15.e even 4 1 5625.2.a.q 6
25.d even 5 2 375.2.i.d 24
25.e even 10 2 375.2.i.d 24
25.f odd 20 2 75.2.g.c 12
25.f odd 20 2 375.2.g.c 12
75.l even 20 2 225.2.h.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.c 12 25.f odd 20 2
225.2.h.d 12 75.l even 20 2
375.2.g.c 12 25.f odd 20 2
375.2.i.d 24 25.d even 5 2
375.2.i.d 24 25.e even 10 2
1875.2.a.j 6 5.c odd 4 1
1875.2.a.k 6 5.c odd 4 1
1875.2.b.f 12 1.a even 1 1 trivial
1875.2.b.f 12 5.b even 2 1 inner
5625.2.a.p 6 15.e even 4 1
5625.2.a.q 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 22T_{2}^{10} + 179T_{2}^{8} + 641T_{2}^{6} + 869T_{2}^{4} + 67T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 22 T^{10} + 179 T^{8} + 641 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 28 T^{10} + 266 T^{8} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} - 24 T^{4} + 66 T^{3} + \cdots + 244)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 88 T^{10} + 2954 T^{8} + \cdots + 10201 \) Copy content Toggle raw display
$17$ \( T^{12} + 139 T^{10} + \cdots + 21520321 \) Copy content Toggle raw display
$19$ \( (T^{6} + 11 T^{5} + 11 T^{4} - 169 T^{3} + \cdots - 380)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 157 T^{10} + 7866 T^{8} + \cdots + 4080400 \) Copy content Toggle raw display
$29$ \( (T^{6} - 3 T^{5} - 41 T^{4} + 173 T^{3} + \cdots + 2105)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 11 T^{5} - 46 T^{4} - 680 T^{3} + \cdots - 2900)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 163 T^{10} + \cdots + 36300625 \) Copy content Toggle raw display
$41$ \( (T^{6} + T^{5} - 181 T^{4} + 110 T^{3} + \cdots - 82655)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 186 T^{10} + \cdots + 39488656 \) Copy content Toggle raw display
$47$ \( T^{12} + 204 T^{10} + 14630 T^{8} + \cdots + 5216656 \) Copy content Toggle raw display
$53$ \( T^{12} + 407 T^{10} + \cdots + 1189905025 \) Copy content Toggle raw display
$59$ \( (T^{6} + 9 T^{5} - 89 T^{4} - 391 T^{3} + \cdots + 3920)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 11 T^{5} - 139 T^{4} + \cdots + 168269)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 374 T^{10} + \cdots + 215619856 \) Copy content Toggle raw display
$71$ \( (T^{6} + 8 T^{5} - 150 T^{4} - 1745 T^{3} + \cdots - 196)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 317 T^{10} + \cdots + 493062025 \) Copy content Toggle raw display
$79$ \( (T^{6} - 5 T^{5} - 160 T^{4} + 530 T^{3} + \cdots - 8000)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 348 T^{10} + \cdots + 1683953296 \) Copy content Toggle raw display
$89$ \( (T^{6} - 4 T^{5} - 464 T^{4} + \cdots - 377055)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 479 T^{10} + 90075 T^{8} + \cdots + 5755201 \) Copy content Toggle raw display
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