Properties

Label 475.3.c.b
Level $475$
Weight $3$
Character orbit 475.c
Analytic conductor $12.943$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-13}) \)
Defining polynomial: \(x^{2} + 13\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -\beta q^{3} -9 q^{4} + 13 q^{6} + 5 q^{7} -5 \beta q^{8} -4 q^{9} +O(q^{10})\) \( q + \beta q^{2} -\beta q^{3} -9 q^{4} + 13 q^{6} + 5 q^{7} -5 \beta q^{8} -4 q^{9} -10 q^{11} + 9 \beta q^{12} + \beta q^{13} + 5 \beta q^{14} + 29 q^{16} -15 q^{17} -4 \beta q^{18} + ( -6 - 5 \beta ) q^{19} -5 \beta q^{21} -10 \beta q^{22} -35 q^{23} -65 q^{24} -13 q^{26} -5 \beta q^{27} -45 q^{28} -5 \beta q^{29} + 10 \beta q^{31} + 9 \beta q^{32} + 10 \beta q^{33} -15 \beta q^{34} + 36 q^{36} -6 \beta q^{37} + ( 65 - 6 \beta ) q^{38} + 13 q^{39} -10 \beta q^{41} + 65 q^{42} + 20 q^{43} + 90 q^{44} -35 \beta q^{46} -10 q^{47} -29 \beta q^{48} -24 q^{49} + 15 \beta q^{51} -9 \beta q^{52} -21 \beta q^{53} + 65 q^{54} -25 \beta q^{56} + ( -65 + 6 \beta ) q^{57} + 65 q^{58} -5 \beta q^{59} -40 q^{61} -130 q^{62} -20 q^{63} - q^{64} -130 q^{66} + 11 \beta q^{67} + 135 q^{68} + 35 \beta q^{69} -30 \beta q^{71} + 20 \beta q^{72} -105 q^{73} + 78 q^{74} + ( 54 + 45 \beta ) q^{76} -50 q^{77} + 13 \beta q^{78} + 10 \beta q^{79} -101 q^{81} + 130 q^{82} + 40 q^{83} + 45 \beta q^{84} + 20 \beta q^{86} -65 q^{87} + 50 \beta q^{88} + 5 \beta q^{91} + 315 q^{92} + 130 q^{93} -10 \beta q^{94} + 117 q^{96} + 34 \beta q^{97} -24 \beta q^{98} + 40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{4} + 26q^{6} + 10q^{7} - 8q^{9} + O(q^{10}) \) \( 2q - 18q^{4} + 26q^{6} + 10q^{7} - 8q^{9} - 20q^{11} + 58q^{16} - 30q^{17} - 12q^{19} - 70q^{23} - 130q^{24} - 26q^{26} - 90q^{28} + 72q^{36} + 130q^{38} + 26q^{39} + 130q^{42} + 40q^{43} + 180q^{44} - 20q^{47} - 48q^{49} + 130q^{54} - 130q^{57} + 130q^{58} - 80q^{61} - 260q^{62} - 40q^{63} - 2q^{64} - 260q^{66} + 270q^{68} - 210q^{73} + 156q^{74} + 108q^{76} - 100q^{77} - 202q^{81} + 260q^{82} + 80q^{83} - 130q^{87} + 630q^{92} + 260q^{93} + 234q^{96} + 80q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
3.60555i
3.60555i
3.60555i 3.60555i −9.00000 0 13.0000 5.00000 18.0278i −4.00000 0
151.2 3.60555i 3.60555i −9.00000 0 13.0000 5.00000 18.0278i −4.00000 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
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.c.b 2
5.b even 2 1 19.3.b.b 2
5.c odd 4 2 475.3.d.b 4
15.d odd 2 1 171.3.c.b 2
19.b odd 2 1 inner 475.3.c.b 2
20.d odd 2 1 304.3.e.d 2
40.e odd 2 1 1216.3.e.h 2
40.f even 2 1 1216.3.e.g 2
60.h even 2 1 2736.3.o.d 2
95.d odd 2 1 19.3.b.b 2
95.g even 4 2 475.3.d.b 4
95.h odd 6 2 361.3.d.b 4
95.i even 6 2 361.3.d.b 4
95.o odd 18 6 361.3.f.d 12
95.p even 18 6 361.3.f.d 12
285.b even 2 1 171.3.c.b 2
380.d even 2 1 304.3.e.d 2
760.b odd 2 1 1216.3.e.g 2
760.p even 2 1 1216.3.e.h 2
1140.p odd 2 1 2736.3.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 5.b even 2 1
19.3.b.b 2 95.d odd 2 1
171.3.c.b 2 15.d odd 2 1
171.3.c.b 2 285.b even 2 1
304.3.e.d 2 20.d odd 2 1
304.3.e.d 2 380.d even 2 1
361.3.d.b 4 95.h odd 6 2
361.3.d.b 4 95.i even 6 2
361.3.f.d 12 95.o odd 18 6
361.3.f.d 12 95.p even 18 6
475.3.c.b 2 1.a even 1 1 trivial
475.3.c.b 2 19.b odd 2 1 inner
475.3.d.b 4 5.c odd 4 2
475.3.d.b 4 95.g even 4 2
1216.3.e.g 2 40.f even 2 1
1216.3.e.g 2 760.b odd 2 1
1216.3.e.h 2 40.e odd 2 1
1216.3.e.h 2 760.p even 2 1
2736.3.o.d 2 60.h even 2 1
2736.3.o.d 2 1140.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 13 + T^{2} \)
$3$ \( 13 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -5 + T )^{2} \)
$11$ \( ( 10 + T )^{2} \)
$13$ \( 13 + T^{2} \)
$17$ \( ( 15 + T )^{2} \)
$19$ \( 361 + 12 T + T^{2} \)
$23$ \( ( 35 + T )^{2} \)
$29$ \( 325 + T^{2} \)
$31$ \( 1300 + T^{2} \)
$37$ \( 468 + T^{2} \)
$41$ \( 1300 + T^{2} \)
$43$ \( ( -20 + T )^{2} \)
$47$ \( ( 10 + T )^{2} \)
$53$ \( 5733 + T^{2} \)
$59$ \( 325 + T^{2} \)
$61$ \( ( 40 + T )^{2} \)
$67$ \( 1573 + T^{2} \)
$71$ \( 11700 + T^{2} \)
$73$ \( ( 105 + T )^{2} \)
$79$ \( 1300 + T^{2} \)
$83$ \( ( -40 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( 15028 + T^{2} \)
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