Properties

Label 475.2.g.b
Level $475$
Weight $2$
Character orbit 475.g
Analytic conductor $3.793$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 35 x^{8} + 223 x^{4} + 289\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 35 x^{8} + 223 x^{4} + 289\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{8} + 58 \nu^{4} + 417 \)\()/190\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{9} + 58 \nu^{5} + 607 \nu \)\()/190\)
\(\beta_{4}\)\(=\)\((\)\( -7 \nu^{8} - 216 \nu^{4} - 639 \)\()/190\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{9} - 216 \nu^{5} - 829 \nu \)\()/190\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{10} + 332 \nu^{6} + 2993 \nu^{2} \)\()/3230\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{11} + 332 \nu^{7} + 2993 \nu^{3} \)\()/3230\)
\(\beta_{8}\)\(=\)\((\)\( -27 \nu^{10} - 996 \nu^{6} - 5749 \nu^{2} \)\()/3230\)
\(\beta_{9}\)\(=\)\((\)\( -18 \nu^{11} - 664 \nu^{7} - 4371 \nu^{3} \)\()/1615\)
\(\beta_{10}\)\(=\)\((\)\( -32 \nu^{10} - 1001 \nu^{6} - 3464 \nu^{2} \)\()/1615\)
\(\beta_{11}\)\(=\)\((\)\( -73 \nu^{11} - 2334 \nu^{7} - 9921 \nu^{3} \)\()/3230\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + 3 \beta_{6}\)
\(\nu^{3}\)\(=\)\(\beta_{9} + 4 \beta_{7}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 7 \beta_{2} - 12\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 7 \beta_{3} - 18 \beta_{1}\)
\(\nu^{6}\)\(=\)\(9 \beta_{10} - 40 \beta_{8} - 56 \beta_{6}\)
\(\nu^{7}\)\(=\)\(9 \beta_{11} - 40 \beta_{9} - 87 \beta_{7}\)
\(\nu^{8}\)\(=\)\(-58 \beta_{4} - 216 \beta_{2} + 279\)
\(\nu^{9}\)\(=\)\(-58 \beta_{5} - 216 \beta_{3} + 437 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-332 \beta_{10} + 1143 \beta_{8} + 1427 \beta_{6}\)
\(\nu^{11}\)\(=\)\(-332 \beta_{11} + 1143 \beta_{9} + 2238 \beta_{7}\)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(\beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
−1.61467 + 1.61467i
−1.10924 + 1.10924i
−0.813901 + 0.813901i
0.813901 0.813901i
1.10924 1.10924i
1.61467 1.61467i
−1.61467 1.61467i
−1.10924 1.10924i
−0.813901 0.813901i
0.813901 + 0.813901i
1.10924 + 1.10924i
1.61467 + 1.61467i
−1.61467 1.61467i 1.11233 1.11233i 3.21432i 0 −3.59210 −1.21432 + 1.21432i 1.96073 1.96073i 0.525428i 0
18.2 −1.10924 1.10924i −1.29790 + 1.29790i 0.460811i 0 2.87936 1.53919 1.53919i −1.70732 + 1.70732i 0.369102i 0
18.3 −0.813901 0.813901i 2.01945 2.01945i 0.675131i 0 −3.28726 2.67513 2.67513i −2.17729 + 2.17729i 5.15633i 0
18.4 0.813901 + 0.813901i −2.01945 + 2.01945i 0.675131i 0 −3.28726 2.67513 2.67513i 2.17729 2.17729i 5.15633i 0
18.5 1.10924 + 1.10924i 1.29790 1.29790i 0.460811i 0 2.87936 1.53919 1.53919i 1.70732 1.70732i 0.369102i 0
18.6 1.61467 + 1.61467i −1.11233 + 1.11233i 3.21432i 0 −3.59210 −1.21432 + 1.21432i −1.96073 + 1.96073i 0.525428i 0
132.1 −1.61467 + 1.61467i 1.11233 + 1.11233i 3.21432i 0 −3.59210 −1.21432 1.21432i 1.96073 + 1.96073i 0.525428i 0
132.2 −1.10924 + 1.10924i −1.29790 1.29790i 0.460811i 0 2.87936 1.53919 + 1.53919i −1.70732 1.70732i 0.369102i 0
132.3 −0.813901 + 0.813901i 2.01945 + 2.01945i 0.675131i 0 −3.28726 2.67513 + 2.67513i −2.17729 2.17729i 5.15633i 0
132.4 0.813901 0.813901i −2.01945 2.01945i 0.675131i 0 −3.28726 2.67513 + 2.67513i 2.17729 + 2.17729i 5.15633i 0
132.5 1.10924 1.10924i 1.29790 + 1.29790i 0.460811i 0 2.87936 1.53919 + 1.53919i 1.70732 + 1.70732i 0.369102i 0
132.6 1.61467 1.61467i −1.11233 1.11233i 3.21432i 0 −3.59210 −1.21432 1.21432i −1.96073 1.96073i 0.525428i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 132.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.g.b 12
5.b even 2 1 95.2.g.b 12
5.c odd 4 1 95.2.g.b 12
5.c odd 4 1 inner 475.2.g.b 12
15.d odd 2 1 855.2.p.f 12
15.e even 4 1 855.2.p.f 12
19.b odd 2 1 inner 475.2.g.b 12
95.d odd 2 1 95.2.g.b 12
95.g even 4 1 95.2.g.b 12
95.g even 4 1 inner 475.2.g.b 12
285.b even 2 1 855.2.p.f 12
285.j odd 4 1 855.2.p.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.g.b 12 5.b even 2 1
95.2.g.b 12 5.c odd 4 1
95.2.g.b 12 95.d odd 2 1
95.2.g.b 12 95.g even 4 1
475.2.g.b 12 1.a even 1 1 trivial
475.2.g.b 12 5.c odd 4 1 inner
475.2.g.b 12 19.b odd 2 1 inner
475.2.g.b 12 95.g even 4 1 inner
855.2.p.f 12 15.d odd 2 1
855.2.p.f 12 15.e even 4 1
855.2.p.f 12 285.b even 2 1
855.2.p.f 12 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 35 T_{2}^{8} + 223 T_{2}^{4} + 289 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 289 + 223 T^{4} + 35 T^{8} + T^{12} \)
$3$ \( 4624 + 1232 T^{4} + 84 T^{8} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( ( 200 - 40 T + 4 T^{2} - 8 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$11$ \( ( -20 - 4 T + 4 T^{2} + T^{3} )^{4} \)
$13$ \( 4624 + 1232 T^{4} + 84 T^{8} + T^{12} \)
$17$ \( ( 200 - 120 T + 36 T^{2} + 80 T^{3} + 50 T^{4} + 10 T^{5} + T^{6} )^{2} \)
$19$ \( 47045881 - 9122470 T^{2} + 903583 T^{4} - 58228 T^{6} + 2503 T^{8} - 70 T^{10} + T^{12} \)
$23$ \( ( 200 - 40 T + 4 T^{2} - 8 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$29$ \( ( -13600 + 4448 T^{2} - 132 T^{4} + T^{6} )^{2} \)
$31$ \( ( 13600 + 3728 T^{2} + 148 T^{4} + T^{6} )^{2} \)
$37$ \( 4624 + 49200 T^{4} + 692 T^{8} + T^{12} \)
$41$ \( ( 54400 + 6432 T^{2} + 176 T^{4} + T^{6} )^{2} \)
$43$ \( ( 57800 - 33320 T + 9604 T^{2} - 536 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$47$ \( ( 33800 + 9880 T + 1444 T^{2} - 1096 T^{3} + 242 T^{4} - 22 T^{5} + T^{6} )^{2} \)
$53$ \( 4624 + 8604592 T^{4} + 7124 T^{8} + T^{12} \)
$59$ \( ( -54400 + 6432 T^{2} - 176 T^{4} + T^{6} )^{2} \)
$61$ \( ( -460 - 88 T + 6 T^{2} + T^{3} )^{4} \)
$67$ \( 3270467344 + 104485936 T^{4} + 21428 T^{8} + T^{12} \)
$71$ \( ( 13600 + 2832 T^{2} + 140 T^{4} + T^{6} )^{2} \)
$73$ \( ( 897800 + 163480 T + 14884 T^{2} - 368 T^{3} + 98 T^{4} + 14 T^{5} + T^{6} )^{2} \)
$79$ \( ( -54400 + 6688 T^{2} - 200 T^{4} + T^{6} )^{2} \)
$83$ \( ( 33800 - 1560 T + 36 T^{2} - 152 T^{3} + 162 T^{4} - 18 T^{5} + T^{6} )^{2} \)
$89$ \( ( -340000 + 20768 T^{2} - 348 T^{4} + T^{6} )^{2} \)
$97$ \( 1806250000 + 110070000 T^{4} + 31956 T^{8} + T^{12} \)
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