Properties

Label 75.4.a.c
Level $75$
Weight $4$
Character orbit 75.a
Self dual yes
Analytic conductor $4.425$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
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.

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

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
3.70156
−2.70156
−4.70156 −3.00000 14.1047 0 14.1047 16.2094 −28.7016 9.00000 0
1.2 1.70156 −3.00000 −5.10469 0 −5.10469 −22.2094 −22.2984 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.c 2
3.b odd 2 1 225.4.a.o 2
4.b odd 2 1 1200.4.a.bt 2
5.b even 2 1 75.4.a.f 2
5.c odd 4 2 15.4.b.a 4
15.d odd 2 1 225.4.a.i 2
15.e even 4 2 45.4.b.b 4
20.d odd 2 1 1200.4.a.bn 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 1.a even 1 1 trivial
75.4.a.f 2 5.b even 2 1
225.4.a.i 2 15.d odd 2 1
225.4.a.o 2 3.b 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 20.d odd 2 1
1200.4.a.bt 2 4.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

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