Properties

Label 75.3.c.e
Level $75$
Weight $3$
Character orbit 75.c
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 2 - \beta ) q^{3} - q^{4} + ( 5 + 2 \beta ) q^{6} + 6 q^{7} + 3 \beta q^{8} + ( -1 - 4 \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( 2 - \beta ) q^{3} - q^{4} + ( 5 + 2 \beta ) q^{6} + 6 q^{7} + 3 \beta q^{8} + ( -1 - 4 \beta ) q^{9} + 2 \beta q^{11} + ( -2 + \beta ) q^{12} -16 q^{13} + 6 \beta q^{14} -19 q^{16} -2 \beta q^{17} + ( 20 - \beta ) q^{18} -2 q^{19} + ( 12 - 6 \beta ) q^{21} -10 q^{22} -6 \beta q^{23} + ( 15 + 6 \beta ) q^{24} -16 \beta q^{26} + ( -22 - 7 \beta ) q^{27} -6 q^{28} -14 \beta q^{29} -18 q^{31} -7 \beta q^{32} + ( 10 + 4 \beta ) q^{33} + 10 q^{34} + ( 1 + 4 \beta ) q^{36} + 16 q^{37} -2 \beta q^{38} + ( -32 + 16 \beta ) q^{39} + 28 \beta q^{41} + ( 30 + 12 \beta ) q^{42} -16 q^{43} -2 \beta q^{44} + 30 q^{46} + 22 \beta q^{47} + ( -38 + 19 \beta ) q^{48} -13 q^{49} + ( -10 - 4 \beta ) q^{51} + 16 q^{52} -2 \beta q^{53} + ( 35 - 22 \beta ) q^{54} + 18 \beta q^{56} + ( -4 + 2 \beta ) q^{57} + 70 q^{58} + 2 \beta q^{59} + 82 q^{61} -18 \beta q^{62} + ( -6 - 24 \beta ) q^{63} -41 q^{64} + ( -20 + 10 \beta ) q^{66} -24 q^{67} + 2 \beta q^{68} + ( -30 - 12 \beta ) q^{69} -56 \beta q^{71} + ( 60 - 3 \beta ) q^{72} + 74 q^{73} + 16 \beta q^{74} + 2 q^{76} + 12 \beta q^{77} + ( -80 - 32 \beta ) q^{78} + 138 q^{79} + ( -79 + 8 \beta ) q^{81} -140 q^{82} + 42 \beta q^{83} + ( -12 + 6 \beta ) q^{84} -16 \beta q^{86} + ( -70 - 28 \beta ) q^{87} -30 q^{88} + 48 \beta q^{89} -96 q^{91} + 6 \beta q^{92} + ( -36 + 18 \beta ) q^{93} -110 q^{94} + ( -35 - 14 \beta ) q^{96} + 166 q^{97} -13 \beta q^{98} + ( 40 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} - 2q^{4} + 10q^{6} + 12q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} - 2q^{4} + 10q^{6} + 12q^{7} - 2q^{9} - 4q^{12} - 32q^{13} - 38q^{16} + 40q^{18} - 4q^{19} + 24q^{21} - 20q^{22} + 30q^{24} - 44q^{27} - 12q^{28} - 36q^{31} + 20q^{33} + 20q^{34} + 2q^{36} + 32q^{37} - 64q^{39} + 60q^{42} - 32q^{43} + 60q^{46} - 76q^{48} - 26q^{49} - 20q^{51} + 32q^{52} + 70q^{54} - 8q^{57} + 140q^{58} + 164q^{61} - 12q^{63} - 82q^{64} - 40q^{66} - 48q^{67} - 60q^{69} + 120q^{72} + 148q^{73} + 4q^{76} - 160q^{78} + 276q^{79} - 158q^{81} - 280q^{82} - 24q^{84} - 140q^{87} - 60q^{88} - 192q^{91} - 72q^{93} - 220q^{94} - 70q^{96} + 332q^{97} + 80q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
2.23607i
2.23607i
2.23607i 2.00000 + 2.23607i −1.00000 0 5.00000 4.47214i 6.00000 6.70820i −1.00000 + 8.94427i 0
26.2 2.23607i 2.00000 2.23607i −1.00000 0 5.00000 + 4.47214i 6.00000 6.70820i −1.00000 8.94427i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.c.e 2
3.b odd 2 1 inner 75.3.c.e 2
4.b odd 2 1 1200.3.l.g 2
5.b even 2 1 15.3.c.a 2
5.c odd 4 2 75.3.d.b 4
12.b even 2 1 1200.3.l.g 2
15.d odd 2 1 15.3.c.a 2
15.e even 4 2 75.3.d.b 4
20.d odd 2 1 240.3.l.b 2
20.e even 4 2 1200.3.c.f 4
40.e odd 2 1 960.3.l.b 2
40.f even 2 1 960.3.l.c 2
45.h odd 6 2 405.3.i.b 4
45.j even 6 2 405.3.i.b 4
60.h even 2 1 240.3.l.b 2
60.l odd 4 2 1200.3.c.f 4
120.i odd 2 1 960.3.l.c 2
120.m even 2 1 960.3.l.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.c.a 2 5.b even 2 1
15.3.c.a 2 15.d odd 2 1
75.3.c.e 2 1.a even 1 1 trivial
75.3.c.e 2 3.b odd 2 1 inner
75.3.d.b 4 5.c odd 4 2
75.3.d.b 4 15.e even 4 2
240.3.l.b 2 20.d odd 2 1
240.3.l.b 2 60.h even 2 1
405.3.i.b 4 45.h odd 6 2
405.3.i.b 4 45.j even 6 2
960.3.l.b 2 40.e odd 2 1
960.3.l.b 2 120.m even 2 1
960.3.l.c 2 40.f even 2 1
960.3.l.c 2 120.i odd 2 1
1200.3.c.f 4 20.e even 4 2
1200.3.c.f 4 60.l odd 4 2
1200.3.l.g 2 4.b odd 2 1
1200.3.l.g 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{2} + 5 \)
\( T_{7} - 6 \)
\( T_{13} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5 + T^{2} \)
$3$ \( 9 - 4 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -6 + T )^{2} \)
$11$ \( 20 + T^{2} \)
$13$ \( ( 16 + T )^{2} \)
$17$ \( 20 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 180 + T^{2} \)
$29$ \( 980 + T^{2} \)
$31$ \( ( 18 + T )^{2} \)
$37$ \( ( -16 + T )^{2} \)
$41$ \( 3920 + T^{2} \)
$43$ \( ( 16 + T )^{2} \)
$47$ \( 2420 + T^{2} \)
$53$ \( 20 + T^{2} \)
$59$ \( 20 + T^{2} \)
$61$ \( ( -82 + T )^{2} \)
$67$ \( ( 24 + T )^{2} \)
$71$ \( 15680 + T^{2} \)
$73$ \( ( -74 + T )^{2} \)
$79$ \( ( -138 + T )^{2} \)
$83$ \( 8820 + T^{2} \)
$89$ \( 11520 + T^{2} \)
$97$ \( ( -166 + T )^{2} \)
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