Properties

Label 75.4.b.b
Level $75$
Weight $4$
Character orbit 75.b
Analytic conductor $4.425$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{2} -3 i q^{3} + 7 q^{4} + 3 q^{6} -24 i q^{7} + 15 i q^{8} -9 q^{9} +O(q^{10})\) \( q + i q^{2} -3 i q^{3} + 7 q^{4} + 3 q^{6} -24 i q^{7} + 15 i q^{8} -9 q^{9} + 52 q^{11} -21 i q^{12} -22 i q^{13} + 24 q^{14} + 41 q^{16} -14 i q^{17} -9 i q^{18} + 20 q^{19} -72 q^{21} + 52 i q^{22} + 168 i q^{23} + 45 q^{24} + 22 q^{26} + 27 i q^{27} -168 i q^{28} -230 q^{29} -288 q^{31} + 161 i q^{32} -156 i q^{33} + 14 q^{34} -63 q^{36} -34 i q^{37} + 20 i q^{38} -66 q^{39} + 122 q^{41} -72 i q^{42} + 188 i q^{43} + 364 q^{44} -168 q^{46} + 256 i q^{47} -123 i q^{48} -233 q^{49} -42 q^{51} -154 i q^{52} + 338 i q^{53} -27 q^{54} + 360 q^{56} -60 i q^{57} -230 i q^{58} -100 q^{59} + 742 q^{61} -288 i q^{62} + 216 i q^{63} + 167 q^{64} + 156 q^{66} -84 i q^{67} -98 i q^{68} + 504 q^{69} -328 q^{71} -135 i q^{72} + 38 i q^{73} + 34 q^{74} + 140 q^{76} -1248 i q^{77} -66 i q^{78} + 240 q^{79} + 81 q^{81} + 122 i q^{82} -1212 i q^{83} -504 q^{84} -188 q^{86} + 690 i q^{87} + 780 i q^{88} -330 q^{89} -528 q^{91} + 1176 i q^{92} + 864 i q^{93} -256 q^{94} + 483 q^{96} + 866 i q^{97} -233 i q^{98} -468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{4} + 6q^{6} - 18q^{9} + O(q^{10}) \) \( 2q + 14q^{4} + 6q^{6} - 18q^{9} + 104q^{11} + 48q^{14} + 82q^{16} + 40q^{19} - 144q^{21} + 90q^{24} + 44q^{26} - 460q^{29} - 576q^{31} + 28q^{34} - 126q^{36} - 132q^{39} + 244q^{41} + 728q^{44} - 336q^{46} - 466q^{49} - 84q^{51} - 54q^{54} + 720q^{56} - 200q^{59} + 1484q^{61} + 334q^{64} + 312q^{66} + 1008q^{69} - 656q^{71} + 68q^{74} + 280q^{76} + 480q^{79} + 162q^{81} - 1008q^{84} - 376q^{86} - 660q^{89} - 1056q^{91} - 512q^{94} + 966q^{96} - 936q^{99} + O(q^{100}) \)

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
1.00000i
1.00000i
1.00000i 3.00000i 7.00000 0 3.00000 24.0000i 15.0000i −9.00000 0
49.2 1.00000i 3.00000i 7.00000 0 3.00000 24.0000i 15.0000i −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.b 2
3.b odd 2 1 225.4.b.e 2
4.b odd 2 1 1200.4.f.b 2
5.b even 2 1 inner 75.4.b.b 2
5.c odd 4 1 15.4.a.a 1
5.c odd 4 1 75.4.a.b 1
15.d odd 2 1 225.4.b.e 2
15.e even 4 1 45.4.a.c 1
15.e even 4 1 225.4.a.f 1
20.d odd 2 1 1200.4.f.b 2
20.e even 4 1 240.4.a.e 1
20.e even 4 1 1200.4.a.t 1
35.f even 4 1 735.4.a.e 1
40.i odd 4 1 960.4.a.b 1
40.k even 4 1 960.4.a.ba 1
45.k odd 12 2 405.4.e.g 2
45.l even 12 2 405.4.e.i 2
55.e even 4 1 1815.4.a.e 1
60.l odd 4 1 720.4.a.n 1
105.k odd 4 1 2205.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 5.c odd 4 1
45.4.a.c 1 15.e even 4 1
75.4.a.b 1 5.c odd 4 1
75.4.b.b 2 1.a even 1 1 trivial
75.4.b.b 2 5.b even 2 1 inner
225.4.a.f 1 15.e even 4 1
225.4.b.e 2 3.b odd 2 1
225.4.b.e 2 15.d odd 2 1
240.4.a.e 1 20.e even 4 1
405.4.e.g 2 45.k odd 12 2
405.4.e.i 2 45.l even 12 2
720.4.a.n 1 60.l odd 4 1
735.4.a.e 1 35.f even 4 1
960.4.a.b 1 40.i odd 4 1
960.4.a.ba 1 40.k even 4 1
1200.4.a.t 1 20.e even 4 1
1200.4.f.b 2 4.b odd 2 1
1200.4.f.b 2 20.d odd 2 1
1815.4.a.e 1 55.e even 4 1
2205.4.a.l 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 576 + T^{2} \)
$11$ \( ( -52 + T )^{2} \)
$13$ \( 484 + T^{2} \)
$17$ \( 196 + T^{2} \)
$19$ \( ( -20 + T )^{2} \)
$23$ \( 28224 + T^{2} \)
$29$ \( ( 230 + T )^{2} \)
$31$ \( ( 288 + T )^{2} \)
$37$ \( 1156 + T^{2} \)
$41$ \( ( -122 + T )^{2} \)
$43$ \( 35344 + T^{2} \)
$47$ \( 65536 + T^{2} \)
$53$ \( 114244 + T^{2} \)
$59$ \( ( 100 + T )^{2} \)
$61$ \( ( -742 + T )^{2} \)
$67$ \( 7056 + T^{2} \)
$71$ \( ( 328 + T )^{2} \)
$73$ \( 1444 + T^{2} \)
$79$ \( ( -240 + T )^{2} \)
$83$ \( 1468944 + T^{2} \)
$89$ \( ( 330 + T )^{2} \)
$97$ \( 749956 + T^{2} \)
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