Properties

Label 475.4.b.c
Level $475$
Weight $4$
Character orbit 475.b
Analytic conductor $28.026$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.0259072527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{2} -5 i q^{3} - q^{4} + 15 q^{6} -11 i q^{7} + 21 i q^{8} + 2 q^{9} +O(q^{10})\) \( q + 3 i q^{2} -5 i q^{3} - q^{4} + 15 q^{6} -11 i q^{7} + 21 i q^{8} + 2 q^{9} -54 q^{11} + 5 i q^{12} + 11 i q^{13} + 33 q^{14} -71 q^{16} + 93 i q^{17} + 6 i q^{18} -19 q^{19} -55 q^{21} -162 i q^{22} + 183 i q^{23} + 105 q^{24} -33 q^{26} -145 i q^{27} + 11 i q^{28} + 249 q^{29} + 56 q^{31} -45 i q^{32} + 270 i q^{33} -279 q^{34} -2 q^{36} + 250 i q^{37} -57 i q^{38} + 55 q^{39} + 240 q^{41} -165 i q^{42} -196 i q^{43} + 54 q^{44} -549 q^{46} + 168 i q^{47} + 355 i q^{48} + 222 q^{49} + 465 q^{51} -11 i q^{52} + 435 i q^{53} + 435 q^{54} + 231 q^{56} + 95 i q^{57} + 747 i q^{58} -195 q^{59} -358 q^{61} + 168 i q^{62} -22 i q^{63} -433 q^{64} -810 q^{66} + 961 i q^{67} -93 i q^{68} + 915 q^{69} -246 q^{71} + 42 i q^{72} + 353 i q^{73} -750 q^{74} + 19 q^{76} + 594 i q^{77} + 165 i q^{78} + 34 q^{79} -671 q^{81} + 720 i q^{82} + 234 i q^{83} + 55 q^{84} + 588 q^{86} -1245 i q^{87} -1134 i q^{88} + 168 q^{89} + 121 q^{91} -183 i q^{92} -280 i q^{93} -504 q^{94} -225 q^{96} -758 i q^{97} + 666 i q^{98} -108 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 30q^{6} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 30q^{6} + 4q^{9} - 108q^{11} + 66q^{14} - 142q^{16} - 38q^{19} - 110q^{21} + 210q^{24} - 66q^{26} + 498q^{29} + 112q^{31} - 558q^{34} - 4q^{36} + 110q^{39} + 480q^{41} + 108q^{44} - 1098q^{46} + 444q^{49} + 930q^{51} + 870q^{54} + 462q^{56} - 390q^{59} - 716q^{61} - 866q^{64} - 1620q^{66} + 1830q^{69} - 492q^{71} - 1500q^{74} + 38q^{76} + 68q^{79} - 1342q^{81} + 110q^{84} + 1176q^{86} + 336q^{89} + 242q^{91} - 1008q^{94} - 450q^{96} - 216q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
1.00000i
1.00000i
3.00000i 5.00000i −1.00000 0 15.0000 11.0000i 21.0000i 2.00000 0
324.2 3.00000i 5.00000i −1.00000 0 15.0000 11.0000i 21.0000i 2.00000 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 475.4.b.c 2
5.b even 2 1 inner 475.4.b.c 2
5.c odd 4 1 19.4.a.a 1
5.c odd 4 1 475.4.a.e 1
15.e even 4 1 171.4.a.d 1
20.e even 4 1 304.4.a.b 1
35.f even 4 1 931.4.a.a 1
40.i odd 4 1 1216.4.a.f 1
40.k even 4 1 1216.4.a.a 1
55.e even 4 1 2299.4.a.b 1
95.g even 4 1 361.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 5.c odd 4 1
171.4.a.d 1 15.e even 4 1
304.4.a.b 1 20.e even 4 1
361.4.a.b 1 95.g even 4 1
475.4.a.e 1 5.c odd 4 1
475.4.b.c 2 1.a even 1 1 trivial
475.4.b.c 2 5.b even 2 1 inner
931.4.a.a 1 35.f even 4 1
1216.4.a.a 1 40.k even 4 1
1216.4.a.f 1 40.i odd 4 1
2299.4.a.b 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(475, [\chi])\):

\( T_{2}^{2} + 9 \)
\( T_{3}^{2} + 25 \)
\( T_{7}^{2} + 121 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + T^{2} \)
$3$ \( 25 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 121 + T^{2} \)
$11$ \( ( 54 + T )^{2} \)
$13$ \( 121 + T^{2} \)
$17$ \( 8649 + T^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( 33489 + T^{2} \)
$29$ \( ( -249 + T )^{2} \)
$31$ \( ( -56 + T )^{2} \)
$37$ \( 62500 + T^{2} \)
$41$ \( ( -240 + T )^{2} \)
$43$ \( 38416 + T^{2} \)
$47$ \( 28224 + T^{2} \)
$53$ \( 189225 + T^{2} \)
$59$ \( ( 195 + T )^{2} \)
$61$ \( ( 358 + T )^{2} \)
$67$ \( 923521 + T^{2} \)
$71$ \( ( 246 + T )^{2} \)
$73$ \( 124609 + T^{2} \)
$79$ \( ( -34 + T )^{2} \)
$83$ \( 54756 + T^{2} \)
$89$ \( ( -168 + T )^{2} \)
$97$ \( 574564 + T^{2} \)
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