Properties

Label 418.2.b.c
Level $418$
Weight $2$
Character orbit 418.b
Analytic conductor $3.338$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14584320320.1
Defining polynomial: \( x^{8} - x^{7} + 4x^{6} + 11x^{5} - 11x^{4} + 32x^{3} + 44x^{2} - 18x + 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{3} q^{3} + q^{4} + \beta_{2} q^{5} + \beta_{3} q^{6} + (\beta_{5} + \beta_{3}) q^{7} - q^{8} + (\beta_{6} - \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{3} q^{3} + q^{4} + \beta_{2} q^{5} + \beta_{3} q^{6} + (\beta_{5} + \beta_{3}) q^{7} - q^{8} + (\beta_{6} - \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{9} - \beta_{2} q^{10} + (\beta_{5} + \beta_{4} - \beta_1 - 1) q^{11} - \beta_{3} q^{12} + ( - \beta_{2} - 2 \beta_1 + 1) q^{13} + ( - \beta_{5} - \beta_{3}) q^{14} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4}) q^{15} + q^{16} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{17} + ( - \beta_{6} + \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{18} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{19} + \beta_{2} q^{20} + ( - \beta_{6} + \beta_{4} + 2 \beta_1 + 3) q^{21} + ( - \beta_{5} - \beta_{4} + \beta_1 + 1) q^{22} + ( - \beta_{2} + \beta_1 + 2) q^{23} + \beta_{3} q^{24} + (\beta_{6} - \beta_{4} - 2 \beta_{2} - \beta_1) q^{25} + (\beta_{2} + 2 \beta_1 - 1) q^{26} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 3 \beta_{3}) q^{27} + (\beta_{5} + \beta_{3}) q^{28} + ( - \beta_1 - 2) q^{29} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{30} + (3 \beta_{7} - \beta_{6} - \beta_{4}) q^{31} - q^{32} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{33} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{34} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4}) q^{35} + (\beta_{6} - \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{36} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{37} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{2} + \beta_1) q^{38} + ( - 3 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3}) q^{39} - \beta_{2} q^{40} + (2 \beta_{2} + 3 \beta_1 + 3) q^{41} + (\beta_{6} - \beta_{4} - 2 \beta_1 - 3) q^{42} + ( - 3 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4}) q^{43} + (\beta_{5} + \beta_{4} - \beta_1 - 1) q^{44} + (\beta_{6} - \beta_{4} - \beta_1 - 1) q^{45} + (\beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{47} - \beta_{3} q^{48} + ( - \beta_{6} + \beta_{4} - \beta_1 + 1) q^{49} + ( - \beta_{6} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{50} + (\beta_{6} - \beta_{4} - \beta_{2} - 5 \beta_1 - 4) q^{51} + ( - \beta_{2} - 2 \beta_1 + 1) q^{52} + (\beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{53} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 3 \beta_{3}) q^{54} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{55} + ( - \beta_{5} - \beta_{3}) q^{56} + (\beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{57} + (\beta_1 + 2) q^{58} + ( - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3}) q^{59} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4}) q^{60} + ( - 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3}) q^{61} + ( - 3 \beta_{7} + \beta_{6} + \beta_{4}) q^{62} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 4 \beta_{3}) q^{63} + q^{64} + ( - \beta_{6} + \beta_{4} + 3 \beta_{2} - \beta_1 - 3) q^{65} + (3 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{66} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + \beta_{3}) q^{67} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{68} + (3 \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{69} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4}) q^{70} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{71} + ( - \beta_{6} + \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{72} + (3 \beta_{7} + \beta_{5} + \beta_{3}) q^{73} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{74} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3}) q^{75} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{76} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{77} + (3 \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{78} + (3 \beta_{6} - 3 \beta_{4} - 2 \beta_1 - 2) q^{79} + \beta_{2} q^{80} + ( - 2 \beta_{6} + 2 \beta_{4} + 5 \beta_{2} + 11 \beta_1 + 6) q^{81} + ( - 2 \beta_{2} - 3 \beta_1 - 3) q^{82} + ( - 5 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{83} + ( - \beta_{6} + \beta_{4} + 2 \beta_1 + 3) q^{84} + (\beta_{6} + \beta_{4}) q^{85} + (3 \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4}) q^{86} + ( - 2 \beta_{7} + \beta_{5} + 3 \beta_{3}) q^{87} + ( - \beta_{5} - \beta_{4} + \beta_1 + 1) q^{88} + (6 \beta_{7} - 2 \beta_{6} - 6 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{89} + ( - \beta_{6} + \beta_{4} + \beta_1 + 1) q^{90} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{91} + ( - \beta_{2} + \beta_1 + 2) q^{92} + ( - \beta_{6} + \beta_{4} - \beta_{2} - 5 \beta_1 - 5) q^{93} + (2 \beta_{2} - 2 \beta_1 - 4) q^{94} + ( - \beta_{7} - 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{95} + \beta_{3} q^{96} + ( - \beta_{6} - \beta_{4}) q^{97} + (\beta_{6} - \beta_{4} + \beta_1 - 1) q^{98} + ( - \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{8} - 14 q^{9} - 2 q^{10} - 6 q^{11} + 10 q^{13} + 8 q^{16} + 14 q^{18} + 2 q^{20} + 20 q^{21} + 6 q^{22} + 12 q^{23} - 2 q^{25} - 10 q^{26} - 14 q^{29} - 8 q^{32} - 8 q^{33} - 14 q^{36} - 2 q^{40} + 22 q^{41} - 20 q^{42} - 6 q^{44} - 6 q^{45} - 12 q^{46} + 24 q^{47} + 10 q^{49} + 2 q^{50} - 24 q^{51} + 10 q^{52} + 10 q^{57} + 14 q^{58} + 8 q^{64} - 16 q^{65} + 8 q^{66} + 14 q^{72} - 16 q^{77} - 12 q^{79} + 2 q^{80} + 36 q^{81} - 22 q^{82} + 20 q^{84} + 6 q^{88} + 6 q^{90} + 12 q^{92} - 32 q^{93} - 24 q^{94} - 12 q^{95} - 10 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 4x^{6} + 11x^{5} - 11x^{4} + 32x^{3} + 44x^{2} - 18x + 46 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 53\nu^{7} + 1084\nu^{6} - 556\nu^{5} + 2891\nu^{4} + 14726\nu^{3} - 4474\nu^{2} + 10896\nu + 54404 ) / 23578 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -355\nu^{7} + 302\nu^{6} - 2504\nu^{5} - 3349\nu^{4} + 1014\nu^{3} - 26086\nu^{2} + 12432\nu - 4506 ) / 23578 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -955\nu^{7} - 848\nu^{6} + 1566\nu^{5} - 23621\nu^{4} + 11362\nu^{3} + 4544\nu^{2} - 107360\nu + 42008 ) / 23578 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1247\nu^{7} - 2522\nu^{6} + 11386\nu^{5} + 1735\nu^{4} + 2150\nu^{3} + 67788\nu^{2} + 5014\nu + 53088 ) / 23578 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 686\nu^{7} - 650\nu^{6} + 2813\nu^{5} + 4944\nu^{4} - 4915\nu^{3} + 8599\nu^{2} + 7126\nu - 1388 ) / 11789 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1501\nu^{7} - 4000\nu^{6} + 10056\nu^{5} - 1315\nu^{4} - 19364\nu^{3} + 58358\nu^{2} - 1490\nu + 5968 ) / 23578 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1727\nu^{7} - 1602\nu^{6} + 8130\nu^{5} + 13237\nu^{4} - 10844\nu^{3} + 43284\nu^{2} + 48976\nu + 1730 ) / 23578 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} - 2\beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{7} - 2\beta_{6} + 9\beta_{5} + 8\beta_{4} - 6\beta_{3} - 3\beta_{2} - 4\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{7} - 21\beta_{6} + 6\beta_{5} + 13\beta_{4} + 6\beta_{3} + 14\beta_{2} - 23\beta _1 + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 58\beta_{7} - 16\beta_{6} - 22\beta_{5} - 20\beta_{4} + 22\beta_{3} - \beta_{2} - 7\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 107\beta_{7} + 72\beta_{6} - 47\beta_{5} - 122\beta_{4} + 46\beta_{3} - 23\beta_{2} + 126\beta _1 - 134 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/418\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(343\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
0.274776 0.839339i
0.682410 + 2.29682i
−1.66113 + 0.0964267i
1.20394 + 1.50360i
1.20394 1.50360i
−1.66113 0.0964267i
0.682410 2.29682i
0.274776 + 0.839339i
−1.00000 3.37171i 1.00000 0.549551 3.37171i 2.61555i −1.00000 −8.36845 −0.549551
417.2 −1.00000 2.00783i 1.00000 1.36482 2.00783i 0.331974i −1.00000 −1.03137 −1.36482
417.3 −1.00000 1.58627i 1.00000 −3.32225 1.58627i 2.41983i −1.00000 0.483751 3.32225
417.4 −1.00000 1.04112i 1.00000 2.40788 1.04112i 3.19266i −1.00000 1.91607 −2.40788
417.5 −1.00000 1.04112i 1.00000 2.40788 1.04112i 3.19266i −1.00000 1.91607 −2.40788
417.6 −1.00000 1.58627i 1.00000 −3.32225 1.58627i 2.41983i −1.00000 0.483751 3.32225
417.7 −1.00000 2.00783i 1.00000 1.36482 2.00783i 0.331974i −1.00000 −1.03137 −1.36482
417.8 −1.00000 3.37171i 1.00000 0.549551 3.37171i 2.61555i −1.00000 −8.36845 −0.549551
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.b.c 8
3.b odd 2 1 3762.2.g.h 8
11.b odd 2 1 418.2.b.d yes 8
19.b odd 2 1 418.2.b.d yes 8
33.d even 2 1 3762.2.g.g 8
57.d even 2 1 3762.2.g.g 8
209.d even 2 1 inner 418.2.b.c 8
627.b odd 2 1 3762.2.g.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.b.c 8 1.a even 1 1 trivial
418.2.b.c 8 209.d even 2 1 inner
418.2.b.d yes 8 11.b odd 2 1
418.2.b.d yes 8 19.b odd 2 1
3762.2.g.g 8 33.d even 2 1
3762.2.g.g 8 57.d even 2 1
3762.2.g.h 8 3.b odd 2 1
3762.2.g.h 8 627.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\):

\( T_{3}^{8} + 19T_{3}^{6} + 104T_{3}^{4} + 207T_{3}^{2} + 125 \) Copy content Toggle raw display
\( T_{13}^{4} - 5T_{13}^{3} - 16T_{13}^{2} + 79T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 19 T^{6} + 104 T^{4} + \cdots + 125 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} - 9 T^{2} + 16 T - 6)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 23 T^{6} + 172 T^{4} + \cdots + 45 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + 10 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} - 16 T^{2} + 79 T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 38 T^{6} + 365 T^{4} + \cdots + 1280 \) Copy content Toggle raw display
$19$ \( T^{8} + 64 T^{5} + 302 T^{4} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} - 7 T^{2} + 44 T + 48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 7 T^{3} + 12 T^{2} - T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 121 T^{6} + 4229 T^{4} + \cdots + 95220 \) Copy content Toggle raw display
$37$ \( T^{8} + 152 T^{6} + 7120 T^{4} + \cdots + 184320 \) Copy content Toggle raw display
$41$ \( (T^{4} - 11 T^{3} - 21 T^{2} + 212 T + 450)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 145 T^{6} + 4673 T^{4} + \cdots + 162000 \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{3} - 28 T^{2} + 352 T + 768)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 266 T^{6} + 19229 T^{4} + \cdots + 2366720 \) Copy content Toggle raw display
$59$ \( T^{8} + 290 T^{6} + 23949 T^{4} + \cdots + 1458000 \) Copy content Toggle raw display
$61$ \( T^{8} + 288 T^{6} + 25652 T^{4} + \cdots + 3732480 \) Copy content Toggle raw display
$67$ \( T^{8} + 259 T^{6} + 16856 T^{4} + \cdots + 595125 \) Copy content Toggle raw display
$71$ \( T^{8} + 313 T^{6} + 31417 T^{4} + \cdots + 4762880 \) Copy content Toggle raw display
$73$ \( T^{8} + 182 T^{6} + 11341 T^{4} + \cdots + 2592000 \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} - 162 T^{2} - 1504 T - 3200)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 265 T^{6} + 16109 T^{4} + \cdots + 36980 \) Copy content Toggle raw display
$89$ \( T^{8} + 568 T^{6} + \cdots + 184832000 \) Copy content Toggle raw display
$97$ \( T^{8} + 64 T^{6} + 1460 T^{4} + \cdots + 46080 \) Copy content Toggle raw display
show more
show less