Properties

Label 75.4.a.d
Level $75$
Weight $4$
Character orbit 75.a
Self dual yes
Analytic conductor $4.425$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \(x^{2} - 19\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + 3 q^{3} + ( 12 - 2 \beta ) q^{4} + ( -3 + 3 \beta ) q^{6} + ( 13 + 4 \beta ) q^{7} + ( -42 + 6 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + 3 q^{3} + ( 12 - 2 \beta ) q^{4} + ( -3 + 3 \beta ) q^{6} + ( 13 + 4 \beta ) q^{7} + ( -42 + 6 \beta ) q^{8} + 9 q^{9} + ( 14 + 4 \beta ) q^{11} + ( 36 - 6 \beta ) q^{12} + ( 9 - 16 \beta ) q^{13} + ( 63 + 9 \beta ) q^{14} + ( 60 - 32 \beta ) q^{16} + ( -34 - 20 \beta ) q^{17} + ( -9 + 9 \beta ) q^{18} + ( 3 - 4 \beta ) q^{19} + ( 39 + 12 \beta ) q^{21} + ( 62 + 10 \beta ) q^{22} + ( 66 - 12 \beta ) q^{23} + ( -126 + 18 \beta ) q^{24} + ( -313 + 25 \beta ) q^{26} + 27 q^{27} + ( 4 + 22 \beta ) q^{28} + ( 46 - 28 \beta ) q^{29} + ( 61 + 28 \beta ) q^{31} + ( -332 + 44 \beta ) q^{32} + ( 42 + 12 \beta ) q^{33} + ( -346 - 14 \beta ) q^{34} + ( 108 - 18 \beta ) q^{36} + ( -142 + 24 \beta ) q^{37} + ( -79 + 7 \beta ) q^{38} + ( 27 - 48 \beta ) q^{39} + ( 196 - 52 \beta ) q^{41} + ( 189 + 27 \beta ) q^{42} + ( 345 - 4 \beta ) q^{43} + ( 16 + 20 \beta ) q^{44} + ( -294 + 78 \beta ) q^{46} + ( -310 - 32 \beta ) q^{47} + ( 180 - 96 \beta ) q^{48} + ( 130 + 104 \beta ) q^{49} + ( -102 - 60 \beta ) q^{51} + ( 716 - 210 \beta ) q^{52} + ( -424 + 28 \beta ) q^{53} + ( -27 + 27 \beta ) q^{54} + ( -90 - 90 \beta ) q^{56} + ( 9 - 12 \beta ) q^{57} + ( -578 + 74 \beta ) q^{58} + ( 62 + 64 \beta ) q^{59} + ( 375 + 56 \beta ) q^{61} + ( 471 + 33 \beta ) q^{62} + ( 117 + 36 \beta ) q^{63} + ( 688 - 120 \beta ) q^{64} + ( 186 + 30 \beta ) q^{66} + ( -179 + 100 \beta ) q^{67} + ( 352 - 172 \beta ) q^{68} + ( 198 - 36 \beta ) q^{69} + ( 412 + 20 \beta ) q^{71} + ( -378 + 54 \beta ) q^{72} + ( -54 + 8 \beta ) q^{73} + ( 598 - 166 \beta ) q^{74} + ( 188 - 54 \beta ) q^{76} + ( 486 + 108 \beta ) q^{77} + ( -939 + 75 \beta ) q^{78} + ( -440 + 160 \beta ) q^{79} + 81 q^{81} + ( -1184 + 248 \beta ) q^{82} + ( 78 + 192 \beta ) q^{83} + ( 12 + 66 \beta ) q^{84} + ( -421 + 349 \beta ) q^{86} + ( 138 - 84 \beta ) q^{87} + ( -132 - 84 \beta ) q^{88} + ( -432 - 144 \beta ) q^{89} + ( -1099 - 172 \beta ) q^{91} + ( 1248 - 276 \beta ) q^{92} + ( 183 + 84 \beta ) q^{93} + ( -298 - 278 \beta ) q^{94} + ( -996 + 132 \beta ) q^{96} + 521 q^{97} + ( 1846 + 26 \beta ) q^{98} + ( 126 + 36 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9} + 28 q^{11} + 72 q^{12} + 18 q^{13} + 126 q^{14} + 120 q^{16} - 68 q^{17} - 18 q^{18} + 6 q^{19} + 78 q^{21} + 124 q^{22} + 132 q^{23} - 252 q^{24} - 626 q^{26} + 54 q^{27} + 8 q^{28} + 92 q^{29} + 122 q^{31} - 664 q^{32} + 84 q^{33} - 692 q^{34} + 216 q^{36} - 284 q^{37} - 158 q^{38} + 54 q^{39} + 392 q^{41} + 378 q^{42} + 690 q^{43} + 32 q^{44} - 588 q^{46} - 620 q^{47} + 360 q^{48} + 260 q^{49} - 204 q^{51} + 1432 q^{52} - 848 q^{53} - 54 q^{54} - 180 q^{56} + 18 q^{57} - 1156 q^{58} + 124 q^{59} + 750 q^{61} + 942 q^{62} + 234 q^{63} + 1376 q^{64} + 372 q^{66} - 358 q^{67} + 704 q^{68} + 396 q^{69} + 824 q^{71} - 756 q^{72} - 108 q^{73} + 1196 q^{74} + 376 q^{76} + 972 q^{77} - 1878 q^{78} - 880 q^{79} + 162 q^{81} - 2368 q^{82} + 156 q^{83} + 24 q^{84} - 842 q^{86} + 276 q^{87} - 264 q^{88} - 864 q^{89} - 2198 q^{91} + 2496 q^{92} + 366 q^{93} - 596 q^{94} - 1992 q^{96} + 1042 q^{97} + 3692 q^{98} + 252 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.35890
4.35890
−5.35890 3.00000 20.7178 0 −16.0767 −4.43560 −68.1534 9.00000 0
1.2 3.35890 3.00000 3.28220 0 10.0767 30.4356 −15.8466 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.a.d 2
3.b odd 2 1 225.4.a.n 2
4.b odd 2 1 1200.4.a.bl 2
5.b even 2 1 75.4.a.e yes 2
5.c odd 4 2 75.4.b.c 4
15.d odd 2 1 225.4.a.j 2
15.e even 4 2 225.4.b.h 4
20.d odd 2 1 1200.4.a.bu 2
20.e even 4 2 1200.4.f.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 1.a even 1 1 trivial
75.4.a.e yes 2 5.b even 2 1
75.4.b.c 4 5.c odd 4 2
225.4.a.j 2 15.d odd 2 1
225.4.a.n 2 3.b odd 2 1
225.4.b.h 4 15.e even 4 2
1200.4.a.bl 2 4.b odd 2 1
1200.4.a.bu 2 20.d odd 2 1
1200.4.f.v 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 T_{2} - 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(75))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -18 + 2 T + T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -135 - 26 T + T^{2} \)
$11$ \( -108 - 28 T + T^{2} \)
$13$ \( -4783 - 18 T + T^{2} \)
$17$ \( -6444 + 68 T + T^{2} \)
$19$ \( -295 - 6 T + T^{2} \)
$23$ \( 1620 - 132 T + T^{2} \)
$29$ \( -12780 - 92 T + T^{2} \)
$31$ \( -11175 - 122 T + T^{2} \)
$37$ \( 9220 + 284 T + T^{2} \)
$41$ \( -12960 - 392 T + T^{2} \)
$43$ \( 118721 - 690 T + T^{2} \)
$47$ \( 76644 + 620 T + T^{2} \)
$53$ \( 164880 + 848 T + T^{2} \)
$59$ \( -73980 - 124 T + T^{2} \)
$61$ \( 81041 - 750 T + T^{2} \)
$67$ \( -157959 + 358 T + T^{2} \)
$71$ \( 162144 - 824 T + T^{2} \)
$73$ \( 1700 + 108 T + T^{2} \)
$79$ \( -292800 + 880 T + T^{2} \)
$83$ \( -694332 - 156 T + T^{2} \)
$89$ \( -207360 + 864 T + T^{2} \)
$97$ \( ( -521 + T )^{2} \)
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