Properties

Label 75.4.b.c
Level $75$
Weight $4$
Character orbit 75.b
Analytic conductor $4.425$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{19})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 25\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{3} ) q^{2} -3 \beta_{1} q^{3} + ( -12 + 2 \beta_{2} ) q^{4} + ( -3 + 3 \beta_{2} ) q^{6} + ( 13 \beta_{1} + 4 \beta_{3} ) q^{7} + ( 42 \beta_{1} - 6 \beta_{3} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{3} ) q^{2} -3 \beta_{1} q^{3} + ( -12 + 2 \beta_{2} ) q^{4} + ( -3 + 3 \beta_{2} ) q^{6} + ( 13 \beta_{1} + 4 \beta_{3} ) q^{7} + ( 42 \beta_{1} - 6 \beta_{3} ) q^{8} -9 q^{9} + ( 14 + 4 \beta_{2} ) q^{11} + ( 36 \beta_{1} - 6 \beta_{3} ) q^{12} + ( -9 \beta_{1} + 16 \beta_{3} ) q^{13} + ( -63 - 9 \beta_{2} ) q^{14} + ( 60 - 32 \beta_{2} ) q^{16} + ( -34 \beta_{1} - 20 \beta_{3} ) q^{17} + ( 9 \beta_{1} - 9 \beta_{3} ) q^{18} + ( -3 + 4 \beta_{2} ) q^{19} + ( 39 + 12 \beta_{2} ) q^{21} + ( 62 \beta_{1} + 10 \beta_{3} ) q^{22} + ( -66 \beta_{1} + 12 \beta_{3} ) q^{23} + ( 126 - 18 \beta_{2} ) q^{24} + ( -313 + 25 \beta_{2} ) q^{26} + 27 \beta_{1} q^{27} + ( -4 \beta_{1} - 22 \beta_{3} ) q^{28} + ( -46 + 28 \beta_{2} ) q^{29} + ( 61 + 28 \beta_{2} ) q^{31} + ( -332 \beta_{1} + 44 \beta_{3} ) q^{32} + ( -42 \beta_{1} - 12 \beta_{3} ) q^{33} + ( 346 + 14 \beta_{2} ) q^{34} + ( 108 - 18 \beta_{2} ) q^{36} + ( -142 \beta_{1} + 24 \beta_{3} ) q^{37} + ( 79 \beta_{1} - 7 \beta_{3} ) q^{38} + ( -27 + 48 \beta_{2} ) q^{39} + ( 196 - 52 \beta_{2} ) q^{41} + ( 189 \beta_{1} + 27 \beta_{3} ) q^{42} + ( -345 \beta_{1} + 4 \beta_{3} ) q^{43} + ( -16 - 20 \beta_{2} ) q^{44} + ( -294 + 78 \beta_{2} ) q^{46} + ( -310 \beta_{1} - 32 \beta_{3} ) q^{47} + ( -180 \beta_{1} + 96 \beta_{3} ) q^{48} + ( -130 - 104 \beta_{2} ) q^{49} + ( -102 - 60 \beta_{2} ) q^{51} + ( 716 \beta_{1} - 210 \beta_{3} ) q^{52} + ( 424 \beta_{1} - 28 \beta_{3} ) q^{53} + ( 27 - 27 \beta_{2} ) q^{54} + ( -90 - 90 \beta_{2} ) q^{56} + ( 9 \beta_{1} - 12 \beta_{3} ) q^{57} + ( 578 \beta_{1} - 74 \beta_{3} ) q^{58} + ( -62 - 64 \beta_{2} ) q^{59} + ( 375 + 56 \beta_{2} ) q^{61} + ( 471 \beta_{1} + 33 \beta_{3} ) q^{62} + ( -117 \beta_{1} - 36 \beta_{3} ) q^{63} + ( -688 + 120 \beta_{2} ) q^{64} + ( 186 + 30 \beta_{2} ) q^{66} + ( -179 \beta_{1} + 100 \beta_{3} ) q^{67} + ( -352 \beta_{1} + 172 \beta_{3} ) q^{68} + ( -198 + 36 \beta_{2} ) q^{69} + ( 412 + 20 \beta_{2} ) q^{71} + ( -378 \beta_{1} + 54 \beta_{3} ) q^{72} + ( 54 \beta_{1} - 8 \beta_{3} ) q^{73} + ( -598 + 166 \beta_{2} ) q^{74} + ( 188 - 54 \beta_{2} ) q^{76} + ( 486 \beta_{1} + 108 \beta_{3} ) q^{77} + ( 939 \beta_{1} - 75 \beta_{3} ) q^{78} + ( 440 - 160 \beta_{2} ) q^{79} + 81 q^{81} + ( -1184 \beta_{1} + 248 \beta_{3} ) q^{82} + ( -78 \beta_{1} - 192 \beta_{3} ) q^{83} + ( -12 - 66 \beta_{2} ) q^{84} + ( -421 + 349 \beta_{2} ) q^{86} + ( 138 \beta_{1} - 84 \beta_{3} ) q^{87} + ( 132 \beta_{1} + 84 \beta_{3} ) q^{88} + ( 432 + 144 \beta_{2} ) q^{89} + ( -1099 - 172 \beta_{2} ) q^{91} + ( 1248 \beta_{1} - 276 \beta_{3} ) q^{92} + ( -183 \beta_{1} - 84 \beta_{3} ) q^{93} + ( 298 + 278 \beta_{2} ) q^{94} + ( -996 + 132 \beta_{2} ) q^{96} + 521 \beta_{1} q^{97} + ( -1846 \beta_{1} - 26 \beta_{3} ) q^{98} + ( -126 - 36 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 48 q^{4} - 12 q^{6} - 36 q^{9} + O(q^{10}) \) \( 4 q - 48 q^{4} - 12 q^{6} - 36 q^{9} + 56 q^{11} - 252 q^{14} + 240 q^{16} - 12 q^{19} + 156 q^{21} + 504 q^{24} - 1252 q^{26} - 184 q^{29} + 244 q^{31} + 1384 q^{34} + 432 q^{36} - 108 q^{39} + 784 q^{41} - 64 q^{44} - 1176 q^{46} - 520 q^{49} - 408 q^{51} + 108 q^{54} - 360 q^{56} - 248 q^{59} + 1500 q^{61} - 2752 q^{64} + 744 q^{66} - 792 q^{69} + 1648 q^{71} - 2392 q^{74} + 752 q^{76} + 1760 q^{79} + 324 q^{81} - 48 q^{84} - 1684 q^{86} + 1728 q^{89} - 4396 q^{91} + 1192 q^{94} - 3984 q^{96} - 504 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 14 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 9\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 7 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−2.17945 + 0.500000i
2.17945 0.500000i
2.17945 + 0.500000i
−2.17945 0.500000i
5.35890i 3.00000i −20.7178 0 −16.0767 4.43560i 68.1534i −9.00000 0
49.2 3.35890i 3.00000i −3.28220 0 10.0767 30.4356i 15.8466i −9.00000 0
49.3 3.35890i 3.00000i −3.28220 0 10.0767 30.4356i 15.8466i −9.00000 0
49.4 5.35890i 3.00000i −20.7178 0 −16.0767 4.43560i 68.1534i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
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 75.4.b.c 4
3.b odd 2 1 225.4.b.h 4
4.b odd 2 1 1200.4.f.v 4
5.b even 2 1 inner 75.4.b.c 4
5.c odd 4 1 75.4.a.d 2
5.c odd 4 1 75.4.a.e yes 2
15.d odd 2 1 225.4.b.h 4
15.e even 4 1 225.4.a.j 2
15.e even 4 1 225.4.a.n 2
20.d odd 2 1 1200.4.f.v 4
20.e even 4 1 1200.4.a.bl 2
20.e even 4 1 1200.4.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 5.c odd 4 1
75.4.a.e yes 2 5.c odd 4 1
75.4.b.c 4 1.a even 1 1 trivial
75.4.b.c 4 5.b even 2 1 inner
225.4.a.j 2 15.e even 4 1
225.4.a.n 2 15.e even 4 1
225.4.b.h 4 3.b odd 2 1
225.4.b.h 4 15.d odd 2 1
1200.4.a.bl 2 20.e even 4 1
1200.4.a.bu 2 20.e even 4 1
1200.4.f.v 4 4.b odd 2 1
1200.4.f.v 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 40 T_{2}^{2} + 324 \) acting on \(S_{4}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 324 + 40 T^{2} + T^{4} \)
$3$ \( ( 9 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 18225 + 946 T^{2} + T^{4} \)
$11$ \( ( -108 - 28 T + T^{2} )^{2} \)
$13$ \( 22877089 + 9890 T^{2} + T^{4} \)
$17$ \( 41525136 + 17512 T^{2} + T^{4} \)
$19$ \( ( -295 + 6 T + T^{2} )^{2} \)
$23$ \( 2624400 + 14184 T^{2} + T^{4} \)
$29$ \( ( -12780 + 92 T + T^{2} )^{2} \)
$31$ \( ( -11175 - 122 T + T^{2} )^{2} \)
$37$ \( 85008400 + 62216 T^{2} + T^{4} \)
$41$ \( ( -12960 - 392 T + T^{2} )^{2} \)
$43$ \( 14094675841 + 238658 T^{2} + T^{4} \)
$47$ \( 5874302736 + 231112 T^{2} + T^{4} \)
$53$ \( 27185414400 + 389344 T^{2} + T^{4} \)
$59$ \( ( -73980 + 124 T + T^{2} )^{2} \)
$61$ \( ( 81041 - 750 T + T^{2} )^{2} \)
$67$ \( 24951045681 + 444082 T^{2} + T^{4} \)
$71$ \( ( 162144 - 824 T + T^{2} )^{2} \)
$73$ \( 2890000 + 8264 T^{2} + T^{4} \)
$79$ \( ( -292800 - 880 T + T^{2} )^{2} \)
$83$ \( 482096926224 + 1413000 T^{2} + T^{4} \)
$89$ \( ( -207360 - 864 T + T^{2} )^{2} \)
$97$ \( ( 271441 + T^{2} )^{2} \)
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