Properties

Label 475.2.e.g
Level $475$
Weight $2$
Character orbit 475.e
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 6x^{10} + 29x^{8} + 40x^{6} + 43x^{4} + 7x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{5} + \beta_{4}) q^{3} + ( - \beta_{9} + \beta_{8} - \beta_{3} + \beta_{2}) q^{4} + (\beta_{8} - 2 \beta_{6} + \beta_{2}) q^{6} + ( - 2 \beta_{11} + \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{7} + ( - 2 \beta_{11} + \beta_{10} + 2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{8} + (2 \beta_{9} + \beta_{8} - 2 \beta_{6} + 2 \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{5} + \beta_{4}) q^{3} + ( - \beta_{9} + \beta_{8} - \beta_{3} + \beta_{2}) q^{4} + (\beta_{8} - 2 \beta_{6} + \beta_{2}) q^{6} + ( - 2 \beta_{11} + \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{7} + ( - 2 \beta_{11} + \beta_{10} + 2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{8} + (2 \beta_{9} + \beta_{8} - 2 \beta_{6} + 2 \beta_{3} + \beta_{2}) q^{9} + \beta_{3} q^{11} + ( - 2 \beta_{11} + 2 \beta_{10} + \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{12} + ( - \beta_{11} + \beta_{10} + 2 \beta_{7} - 2 \beta_1) q^{13} + (2 \beta_{9} - 2 \beta_{8} + 3 \beta_{6} - 3) q^{14} + (\beta_{9} + 2 \beta_{6} - 2) q^{16} + ( - \beta_{5} + 2 \beta_{4}) q^{17} + ( - 2 \beta_{10} + \beta_{7} - 2 \beta_{4}) q^{18} + (\beta_{9} + 2 \beta_{8} - \beta_{6} + 2 \beta_{3} + \beta_{2} + 1) q^{19} + (\beta_{9} - \beta_{8} + 3 \beta_{6} - 3) q^{21} + ( - 2 \beta_{5} - \beta_{4} + \beta_1) q^{22} + (\beta_{10} - \beta_{7} + \beta_1) q^{23} + (4 \beta_{9} - \beta_{6} + 1) q^{24} + ( - 2 \beta_{3} + \beta_{2} - 3) q^{26} + (2 \beta_{11} - 3 \beta_{10} + 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{27} + (7 \beta_{11} - 2 \beta_{10} - 5 \beta_{7} + 5 \beta_1) q^{28} + ( - 3 \beta_{9} + 3 \beta_{6} - 3 \beta_{3}) q^{29} + ( - \beta_{2} + 5) q^{31} + (\beta_{10} - 3 \beta_{7} + 3 \beta_1) q^{32} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{33} + (\beta_{9} + \beta_{8} - 2 \beta_{6} + \beta_{3} + \beta_{2}) q^{34} + (2 \beta_{9} + 2 \beta_{8} - 3 \beta_{6} + 3) q^{36} + ( - 2 \beta_{10} - 3 \beta_{7} - 2 \beta_{4}) q^{37} + ( - 3 \beta_{11} - \beta_{10} + 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - \beta_1) q^{38} + ( - \beta_{3} + 2 \beta_{2}) q^{39} + ( - 3 \beta_{9} - 2 \beta_{8} + 3 \beta_{6} - 3) q^{41} + (7 \beta_{11} - \beta_{10} - 5 \beta_{7} + 5 \beta_1) q^{42} + ( - \beta_{5} + 5 \beta_1) q^{43} + ( - \beta_{9} + 2 \beta_{8} - 3 \beta_{6} - \beta_{3} + 2 \beta_{2}) q^{44} + ( - \beta_{3} + 1) q^{46} + (4 \beta_{11} + 4 \beta_{10} - \beta_{7} + \beta_1) q^{47} + (3 \beta_{11} - \beta_{7} + \beta_1) q^{48} + (4 \beta_{3} - 3 \beta_{2} - 1) q^{49} + (4 \beta_{9} + \beta_{8} - 8 \beta_{6} + 4 \beta_{3} + \beta_{2}) q^{51} + (7 \beta_{5} - 5 \beta_1) q^{52} + (\beta_{11} - 2 \beta_{10} + 3 \beta_{7} - 3 \beta_1) q^{53} + ( - 5 \beta_{9} + 2 \beta_{8} + \beta_{6} - 1) q^{54} + (5 \beta_{3} - 3 \beta_{2} + 6) q^{56} + ( - 2 \beta_{11} - 3 \beta_{10} + 6 \beta_{7} - \beta_{5} - 4 \beta_{4} - \beta_1) q^{57} + ( - 3 \beta_{11} + 3 \beta_{10} + 3 \beta_{5} + 3 \beta_{4} - 3 \beta_1) q^{58} + (\beta_{9} - 4 \beta_{8} + 3 \beta_{6} - 3) q^{59} + ( - 2 \beta_{9} + 2 \beta_{8} + \beta_{6} - 2 \beta_{3} + 2 \beta_{2}) q^{61} + ( - 7 \beta_{5} + \beta_1) q^{62} + ( - \beta_{11} + \beta_{10} - \beta_{7} + \beta_1) q^{63} + ( - 3 \beta_{3} - 1) q^{64} + ( - 3 \beta_{9} + \beta_{8} - 3 \beta_{3} + \beta_{2}) q^{66} + (2 \beta_{11} - 4 \beta_{7} + 4 \beta_1) q^{67} + (3 \beta_{10} + 3 \beta_{4}) q^{68} + ( - 3 \beta_{3} - \beta_{2} + 4) q^{69} + ( - \beta_{9} - 5 \beta_{8}) q^{71} + ( - 3 \beta_{11} - 6 \beta_{10} + 5 \beta_{7} - 5 \beta_1) q^{72} + (2 \beta_{5} + \beta_{4} - 7 \beta_1) q^{73} + ( - 2 \beta_{9} - 3 \beta_{6} + 3) q^{74} + (4 \beta_{8} - 3 \beta_{6} - 4 \beta_{3} + 3 \beta_{2} - 3) q^{76} + ( - 4 \beta_{11} + \beta_{10} + 2 \beta_{7} + 4 \beta_{5} + \beta_{4} - 4 \beta_1) q^{77} + (6 \beta_{5} + \beta_{4} - 3 \beta_1) q^{78} + ( - 4 \beta_{8} + 4 \beta_{6} - 4) q^{79} + ( - 5 \beta_{9} + 4 \beta_{6} - 4) q^{81} + (\beta_{11} + 3 \beta_{10} - 2 \beta_{7} + 2 \beta_1) q^{82} + (5 \beta_{11} - 2 \beta_{10} - 9 \beta_{7} - 5 \beta_{5} - 2 \beta_{4} + 5 \beta_1) q^{83} + (6 \beta_{3} - 5 \beta_{2} + 6) q^{84} + ( - \beta_{9} + \beta_{8} + 3 \beta_{6} - \beta_{3} + \beta_{2}) q^{86} + (9 \beta_{10} - 6 \beta_{7} + 9 \beta_{4}) q^{87} + ( - 5 \beta_{11} - \beta_{10} + 4 \beta_{7} + 5 \beta_{5} - \beta_{4} - 5 \beta_1) q^{88} + ( - 4 \beta_{8} - 6 \beta_{6} - 4 \beta_{2}) q^{89} + (3 \beta_{9} - 4 \beta_{8} + 3 \beta_{6} + 3 \beta_{3} - 4 \beta_{2}) q^{91} + (\beta_{5} - \beta_{4} + \beta_1) q^{92} + ( - 6 \beta_{5} + 5 \beta_{4} - \beta_1) q^{93} + ( - 4 \beta_{2} + 5) q^{94} + ( - 5 \beta_{3} - 3 \beta_{2} + 6) q^{96} + ( - 5 \beta_{5} + 7 \beta_{4} + 4 \beta_1) q^{97} + ( - 13 \beta_{5} - 4 \beta_{4} + 7 \beta_1) q^{98} + ( - 3 \beta_{9} - \beta_{8} + 6 \beta_{6} - 3 \beta_{3} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{4} - 12 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{4} - 12 q^{6} - 8 q^{9} + 4 q^{11} - 22 q^{14} - 14 q^{16} + 12 q^{19} - 20 q^{21} - 2 q^{24} - 44 q^{26} + 12 q^{29} + 60 q^{31} - 10 q^{34} + 14 q^{36} - 4 q^{39} - 12 q^{41} - 20 q^{44} + 8 q^{46} + 4 q^{49} - 40 q^{51} + 4 q^{54} + 92 q^{56} - 20 q^{59} + 2 q^{61} - 24 q^{64} - 6 q^{66} + 36 q^{69} + 2 q^{71} + 22 q^{74} - 70 q^{76} - 24 q^{79} - 14 q^{81} + 96 q^{84} + 16 q^{86} - 36 q^{89} + 24 q^{91} + 60 q^{94} + 52 q^{96} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 6x^{10} + 29x^{8} + 40x^{6} + 43x^{4} + 7x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36\nu^{10} + 174\nu^{8} + 841\nu^{6} + 258\nu^{4} + 42\nu^{2} - 2207 ) / 1205 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -138\nu^{10} - 667\nu^{8} - 3023\nu^{6} - 989\nu^{4} - 161\nu^{2} + 2636 ) / 1205 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -138\nu^{11} - 667\nu^{9} - 3023\nu^{7} - 989\nu^{5} - 161\nu^{3} + 2636\nu ) / 1205 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -174\nu^{11} - 841\nu^{9} - 3864\nu^{7} - 1247\nu^{5} - 203\nu^{3} + 7253\nu ) / 1205 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -203\nu^{10} - 1182\nu^{8} - 5713\nu^{6} - 7279\nu^{4} - 8471\nu^{2} - 174 ) / 1205 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -203\nu^{11} - 1182\nu^{9} - 5713\nu^{7} - 7279\nu^{5} - 8471\nu^{3} - 174\nu ) / 1205 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 74\nu^{10} + 438\nu^{8} + 2117\nu^{6} + 2860\nu^{4} + 3139\nu^{2} + 511 ) / 241 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -429\nu^{10} - 2676\nu^{8} - 12934\nu^{6} - 19342\nu^{4} - 19178\nu^{2} - 3122 ) / 1205 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -429\nu^{11} - 2676\nu^{9} - 12934\nu^{7} - 19342\nu^{5} - 19178\nu^{3} - 3122\nu ) / 1205 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1002\nu^{11} + 6048\nu^{9} + 29232\nu^{7} + 40921\nu^{5} + 43344\nu^{3} + 7056\nu ) / 1205 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - 2\beta_{6} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} - 3\beta_{7} + \beta_{5} - \beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 5\beta_{8} + 7\beta_{6} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{11} + 6\beta_{10} + 12\beta_{7} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{3} + 23\beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -23\beta_{5} + 29\beta_{4} + 75\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -29\beta_{9} - 104\beta_{8} - 127\beta_{6} - 29\beta_{3} - 104\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -104\beta_{11} - 133\beta_{10} - 231\beta_{7} + 104\beta_{5} - 133\beta_{4} - 104\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 133\beta_{9} + 468\beta_{8} + 566\beta_{6} - 566 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 468\beta_{11} + 601\beta_{10} + 1034\beta_{7} - 1034\beta_1 \) Copy content Toggle raw display

Character values

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

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

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
0.203566 + 0.352587i
0.579521 + 1.00376i
1.05958 + 1.83525i
−1.05958 1.83525i
−0.579521 1.00376i
−0.203566 0.352587i
0.203566 0.352587i
0.579521 1.00376i
1.05958 1.83525i
−1.05958 + 1.83525i
−0.579521 + 1.00376i
−0.203566 + 0.352587i
−1.22810 2.12713i −0.780522 1.35190i −2.01647 + 3.49262i 0 −1.91712 + 3.32055i 4.50527 4.99330 0.281570 0.487693i 0
26.2 −0.431391 0.747190i −1.53957 2.66661i 0.627804 1.08739i 0 −1.32831 + 2.30070i 0.566520 −2.80888 −3.24054 + 5.61278i 0
26.3 −0.235942 0.408663i 0.520111 + 0.900858i 0.888663 1.53921i 0 0.245432 0.425100i −1.17540 −1.78246 0.958970 1.66098i 0
26.4 0.235942 + 0.408663i −0.520111 0.900858i 0.888663 1.53921i 0 0.245432 0.425100i 1.17540 1.78246 0.958970 1.66098i 0
26.5 0.431391 + 0.747190i 1.53957 + 2.66661i 0.627804 1.08739i 0 −1.32831 + 2.30070i −0.566520 2.80888 −3.24054 + 5.61278i 0
26.6 1.22810 + 2.12713i 0.780522 + 1.35190i −2.01647 + 3.49262i 0 −1.91712 + 3.32055i −4.50527 −4.99330 0.281570 0.487693i 0
201.1 −1.22810 + 2.12713i −0.780522 + 1.35190i −2.01647 3.49262i 0 −1.91712 3.32055i 4.50527 4.99330 0.281570 + 0.487693i 0
201.2 −0.431391 + 0.747190i −1.53957 + 2.66661i 0.627804 + 1.08739i 0 −1.32831 2.30070i 0.566520 −2.80888 −3.24054 5.61278i 0
201.3 −0.235942 + 0.408663i 0.520111 0.900858i 0.888663 + 1.53921i 0 0.245432 + 0.425100i −1.17540 −1.78246 0.958970 + 1.66098i 0
201.4 0.235942 0.408663i −0.520111 + 0.900858i 0.888663 + 1.53921i 0 0.245432 + 0.425100i 1.17540 1.78246 0.958970 + 1.66098i 0
201.5 0.431391 0.747190i 1.53957 2.66661i 0.627804 + 1.08739i 0 −1.32831 2.30070i −0.566520 2.80888 −3.24054 5.61278i 0
201.6 1.22810 2.12713i 0.780522 1.35190i −2.01647 3.49262i 0 −1.91712 3.32055i −4.50527 −4.99330 0.281570 + 0.487693i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.g 12
5.b even 2 1 inner 475.2.e.g 12
5.c odd 4 2 95.2.i.b 12
15.e even 4 2 855.2.be.d 12
19.c even 3 1 inner 475.2.e.g 12
19.c even 3 1 9025.2.a.bu 6
19.d odd 6 1 9025.2.a.bt 6
95.h odd 6 1 9025.2.a.bt 6
95.i even 6 1 inner 475.2.e.g 12
95.i even 6 1 9025.2.a.bu 6
95.l even 12 2 1805.2.b.g 6
95.m odd 12 2 95.2.i.b 12
95.m odd 12 2 1805.2.b.f 6
285.v even 12 2 855.2.be.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.b 12 5.c odd 4 2
95.2.i.b 12 95.m odd 12 2
475.2.e.g 12 1.a even 1 1 trivial
475.2.e.g 12 5.b even 2 1 inner
475.2.e.g 12 19.c even 3 1 inner
475.2.e.g 12 95.i even 6 1 inner
855.2.be.d 12 15.e even 4 2
855.2.be.d 12 285.v even 12 2
1805.2.b.f 6 95.m odd 12 2
1805.2.b.g 6 95.l even 12 2
9025.2.a.bt 6 19.d odd 6 1
9025.2.a.bt 6 95.h odd 6 1
9025.2.a.bu 6 19.c even 3 1
9025.2.a.bu 6 95.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 7T_{2}^{10} + 43T_{2}^{8} + 40T_{2}^{6} + 29T_{2}^{4} + 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 7 T^{10} + 43 T^{8} + 40 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 13 T^{10} + 133 T^{8} + \cdots + 625 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 22 T^{4} + 35 T^{2} - 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - T^{2} - 4 T + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} + 31 T^{10} + 722 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{12} + 35 T^{10} + 1027 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( (T^{6} - 6 T^{5} + 30 T^{4} - 115 T^{3} + \cdots + 6859)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{10} + 137 T^{8} + 82 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + 63 T^{4} + 108 T^{3} + \cdots + 729)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 15 T^{2} + 70 T - 97)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 98 T^{4} + 887 T^{2} - 729)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 6 T^{5} + 77 T^{4} - 240 T^{3} + \cdots + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 127 T^{10} + 13613 T^{8} + \cdots + 194481 \) Copy content Toggle raw display
$47$ \( T^{12} + 214 T^{10} + \cdots + 47562811921 \) Copy content Toggle raw display
$53$ \( T^{12} + 99 T^{10} + \cdots + 131079601 \) Copy content Toggle raw display
$59$ \( (T^{6} + 10 T^{5} + 155 T^{4} + \cdots + 84681)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - T^{5} + 42 T^{4} - 185 T^{3} + \cdots + 12769)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 76 T^{10} + \cdots + 207360000 \) Copy content Toggle raw display
$71$ \( (T^{6} - T^{5} + 125 T^{4} + 1078 T^{3} + \cdots + 227529)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 236 T^{10} + \cdots + 9845600625 \) Copy content Toggle raw display
$79$ \( (T^{6} + 12 T^{5} + 176 T^{4} + \cdots + 200704)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 459 T^{4} + 56302 T^{2} + \cdots - 966289)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 18 T^{5} + 296 T^{4} + 648 T^{3} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 529 T^{10} + \cdots + 1534548635361 \) Copy content Toggle raw display
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