Properties

Label 75.4.a.f
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} + ( - 6 \beta + 6) q^{7} + (\beta + 25) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} + ( - 6 \beta + 6) q^{7} + (\beta + 25) q^{8} + 9 q^{9} + ( - 6 \beta - 18) q^{11} + (9 \beta + 9) q^{12} + ( - 6 \beta + 42) q^{13} + ( - 6 \beta - 54) q^{14} + (3 \beta + 11) q^{16} + (10 \beta + 46) q^{17} + (9 \beta + 9) q^{18} + ( - 24 \beta + 40) q^{19} + ( - 18 \beta + 18) q^{21} + ( - 30 \beta - 78) q^{22} + (8 \beta - 28) q^{23} + (3 \beta + 75) q^{24} + (30 \beta - 18) q^{26} + 27 q^{27} + ( - 18 \beta - 162) q^{28} + (42 \beta - 180) q^{29} + ( - 12 \beta + 32) q^{31} + (9 \beta - 159) q^{32} + ( - 18 \beta - 54) q^{33} + (66 \beta + 146) q^{34} + (27 \beta + 27) q^{36} + (54 \beta + 126) q^{37} + ( - 8 \beta - 200) q^{38} + ( - 18 \beta + 126) q^{39} + ( - 12 \beta - 198) q^{41} + ( - 18 \beta - 162) q^{42} + (96 \beta + 12) q^{43} + ( - 90 \beta - 234) q^{44} + ( - 12 \beta + 52) q^{46} + ( - 92 \beta + 136) q^{47} + (9 \beta + 33) q^{48} + ( - 36 \beta + 53) q^{49} + (30 \beta + 138) q^{51} + (90 \beta - 54) q^{52} + ( - 82 \beta + 242) q^{53} + (27 \beta + 27) q^{54} + ( - 150 \beta + 90) q^{56} + ( - 72 \beta + 120) q^{57} + ( - 96 \beta + 240) q^{58} + ( - 6 \beta - 90) q^{59} + (96 \beta + 122) q^{61} + (8 \beta - 88) q^{62} + ( - 54 \beta + 54) q^{63} + ( - 165 \beta - 157) q^{64} + ( - 90 \beta - 234) q^{66} + (60 \beta + 336) q^{67} + (198 \beta + 438) q^{68} + (24 \beta - 84) q^{69} + (180 \beta - 108) q^{71} + (9 \beta + 225) q^{72} + (108 \beta + 612) q^{73} + (234 \beta + 666) q^{74} + ( - 24 \beta - 600) q^{76} + (108 \beta + 252) q^{77} + (90 \beta - 54) q^{78} + (300 \beta + 40) q^{79} + 81 q^{81} + ( - 222 \beta - 318) q^{82} + ( - 208 \beta - 388) q^{83} + ( - 54 \beta - 486) q^{84} + (204 \beta + 972) q^{86} + (126 \beta - 540) q^{87} + ( - 174 \beta - 510) q^{88} + ( - 144 \beta + 630) q^{89} + ( - 252 \beta + 612) q^{91} + ( - 36 \beta + 156) q^{92} + ( - 36 \beta + 96) q^{93} + ( - 48 \beta - 784) q^{94} + (27 \beta - 477) q^{96} + ( - 240 \beta - 264) q^{97} + ( - 19 \beta - 307) q^{98} + ( - 54 \beta - 162) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 6 q^{3} + 9 q^{4} + 9 q^{6} + 6 q^{7} + 51 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 6 q^{3} + 9 q^{4} + 9 q^{6} + 6 q^{7} + 51 q^{8} + 18 q^{9} - 42 q^{11} + 27 q^{12} + 78 q^{13} - 114 q^{14} + 25 q^{16} + 102 q^{17} + 27 q^{18} + 56 q^{19} + 18 q^{21} - 186 q^{22} - 48 q^{23} + 153 q^{24} - 6 q^{26} + 54 q^{27} - 342 q^{28} - 318 q^{29} + 52 q^{31} - 309 q^{32} - 126 q^{33} + 358 q^{34} + 81 q^{36} + 306 q^{37} - 408 q^{38} + 234 q^{39} - 408 q^{41} - 342 q^{42} + 120 q^{43} - 558 q^{44} + 92 q^{46} + 180 q^{47} + 75 q^{48} + 70 q^{49} + 306 q^{51} - 18 q^{52} + 402 q^{53} + 81 q^{54} + 30 q^{56} + 168 q^{57} + 384 q^{58} - 186 q^{59} + 340 q^{61} - 168 q^{62} + 54 q^{63} - 479 q^{64} - 558 q^{66} + 732 q^{67} + 1074 q^{68} - 144 q^{69} - 36 q^{71} + 459 q^{72} + 1332 q^{73} + 1566 q^{74} - 1224 q^{76} + 612 q^{77} - 18 q^{78} + 380 q^{79} + 162 q^{81} - 858 q^{82} - 984 q^{83} - 1026 q^{84} + 2148 q^{86} - 954 q^{87} - 1194 q^{88} + 1116 q^{89} + 972 q^{91} + 276 q^{92} + 156 q^{93} - 1616 q^{94} - 927 q^{96} - 768 q^{97} - 633 q^{98} - 378 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−2.70156
3.70156
−1.70156 3.00000 −5.10469 0 −5.10469 22.2094 22.2984 9.00000 0
1.2 4.70156 3.00000 14.1047 0 14.1047 −16.2094 28.7016 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.f 2
3.b odd 2 1 225.4.a.i 2
4.b odd 2 1 1200.4.a.bn 2
5.b even 2 1 75.4.a.c 2
5.c odd 4 2 15.4.b.a 4
15.d odd 2 1 225.4.a.o 2
15.e even 4 2 45.4.b.b 4
20.d odd 2 1 1200.4.a.bt 2
20.e even 4 2 240.4.f.f 4
40.i odd 4 2 960.4.f.q 4
40.k even 4 2 960.4.f.p 4
60.l odd 4 2 720.4.f.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 5.c odd 4 2
45.4.b.b 4 15.e even 4 2
75.4.a.c 2 5.b even 2 1
75.4.a.f 2 1.a even 1 1 trivial
225.4.a.i 2 3.b odd 2 1
225.4.a.o 2 15.d odd 2 1
240.4.f.f 4 20.e even 4 2
720.4.f.j 4 60.l odd 4 2
960.4.f.p 4 40.k even 4 2
960.4.f.q 4 40.i odd 4 2
1200.4.a.bn 2 4.b odd 2 1
1200.4.a.bt 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T - 8 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T - 360 \) Copy content Toggle raw display
$11$ \( T^{2} + 42T + 72 \) Copy content Toggle raw display
$13$ \( T^{2} - 78T + 1152 \) Copy content Toggle raw display
$17$ \( T^{2} - 102T + 1576 \) Copy content Toggle raw display
$19$ \( T^{2} - 56T - 5120 \) Copy content Toggle raw display
$23$ \( T^{2} + 48T - 80 \) Copy content Toggle raw display
$29$ \( T^{2} + 318T + 7200 \) Copy content Toggle raw display
$31$ \( T^{2} - 52T - 800 \) Copy content Toggle raw display
$37$ \( T^{2} - 306T - 6480 \) Copy content Toggle raw display
$41$ \( T^{2} + 408T + 40140 \) Copy content Toggle raw display
$43$ \( T^{2} - 120T - 90864 \) Copy content Toggle raw display
$47$ \( T^{2} - 180T - 78656 \) Copy content Toggle raw display
$53$ \( T^{2} - 402T - 28520 \) Copy content Toggle raw display
$59$ \( T^{2} + 186T + 8280 \) Copy content Toggle raw display
$61$ \( T^{2} - 340T - 65564 \) Copy content Toggle raw display
$67$ \( T^{2} - 732T + 97056 \) Copy content Toggle raw display
$71$ \( T^{2} + 36T - 331776 \) Copy content Toggle raw display
$73$ \( T^{2} - 1332 T + 324000 \) Copy content Toggle raw display
$79$ \( T^{2} - 380T - 886400 \) Copy content Toggle raw display
$83$ \( T^{2} + 984T - 201392 \) Copy content Toggle raw display
$89$ \( T^{2} - 1116T + 98820 \) Copy content Toggle raw display
$97$ \( T^{2} + 768T - 442944 \) Copy content Toggle raw display
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