Properties

Label 310.2.b.b
Level $310$
Weight $2$
Character orbit 310.b
Analytic conductor $2.475$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.47536246266\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2058981376.2
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -97\nu^{7} + 102\nu^{6} - 30\nu^{5} - 432\nu^{4} - 1769\nu^{3} + 1637\nu^{2} - 408\nu - 4773 ) / 1631 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 145\nu^{7} + 167\nu^{6} - 241\nu^{5} + 444\nu^{4} + 3132\nu^{3} + 2984\nu^{2} - 3930\nu + 4226 ) / 1631 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 485\nu^{7} - 510\nu^{6} + 150\nu^{5} + 529\nu^{4} + 8845\nu^{3} - 8185\nu^{2} + 2040\nu + 4293 ) / 1631 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -552\nu^{7} + 984\nu^{6} - 961\nu^{5} - 138\nu^{4} - 9966\nu^{3} + 16176\nu^{2} - 15353\nu + 2213 ) / 1631 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -137\nu^{7} + 305\nu^{6} - 309\nu^{5} + 24\nu^{4} - 2400\nu^{3} + 5281\nu^{2} - 4948\nu + 1236 ) / 233 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -1084\nu^{7} + 2216\nu^{6} - 1899\nu^{5} - 271\nu^{4} - 19500\nu^{3} + 38219\nu^{2} - 30067\nu + 4334 ) / 1631 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1661\nu^{7} + 2890\nu^{6} - 2481\nu^{5} - 823\nu^{4} - 30006\nu^{3} + 49100\nu^{2} - 37656\nu + 1769 ) / 1631 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{6} + 5\beta_{5} - 3\beta_{4} - 5\beta_{3} + 3\beta_{2} + 3\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{3} - 5\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -34\beta_{7} + 23\beta_{6} + 11\beta_{5} + 13\beta_{4} - 23\beta_{3} - 11\beta_{2} + 11\beta _1 + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + 7\beta_{6} - 23\beta_{5} + 35\beta_{4} + 22\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 103\beta_{6} - 105\beta_{5} + 61\beta_{4} + 103\beta_{3} - 43\beta_{2} - 43\beta _1 - 85 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(251\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
0.148421 + 0.148421i
1.52153 + 1.52153i
−1.43917 1.43917i
0.769222 + 0.769222i
0.769222 0.769222i
−1.43917 + 1.43917i
1.52153 1.52153i
0.148421 0.148421i
1.00000i 1.78165i −1.00000 −0.264435 2.22038i −1.78165 1.70316i 1.00000i −0.174289 −2.22038 + 0.264435i
249.2 1.00000i 0.287336i −1.00000 −0.437190 + 2.19291i −0.287336 1.04306i 1.00000i 2.91744 2.19291 + 0.437190i
249.3 1.00000i 1.55241i −1.00000 −2.23418 0.0917505i 1.55241 4.87834i 1.00000i 0.590025 −0.0917505 + 2.23418i
249.4 1.00000i 2.51658i −1.00000 1.93581 + 1.11922i 2.51658 0.461555i 1.00000i −3.33317 1.11922 1.93581i
249.5 1.00000i 2.51658i −1.00000 1.93581 1.11922i 2.51658 0.461555i 1.00000i −3.33317 1.11922 + 1.93581i
249.6 1.00000i 1.55241i −1.00000 −2.23418 + 0.0917505i 1.55241 4.87834i 1.00000i 0.590025 −0.0917505 2.23418i
249.7 1.00000i 0.287336i −1.00000 −0.437190 2.19291i −0.287336 1.04306i 1.00000i 2.91744 2.19291 0.437190i
249.8 1.00000i 1.78165i −1.00000 −0.264435 + 2.22038i −1.78165 1.70316i 1.00000i −0.174289 −2.22038 0.264435i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.8
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 310.2.b.b 8
3.b odd 2 1 2790.2.d.l 8
4.b odd 2 1 2480.2.d.e 8
5.b even 2 1 inner 310.2.b.b 8
5.c odd 4 1 1550.2.a.n 4
5.c odd 4 1 1550.2.a.q 4
15.d odd 2 1 2790.2.d.l 8
20.d odd 2 1 2480.2.d.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.b.b 8 1.a even 1 1 trivial
310.2.b.b 8 5.b even 2 1 inner
1550.2.a.n 4 5.c odd 4 1
1550.2.a.q 4 5.c odd 4 1
2480.2.d.e 8 4.b odd 2 1
2480.2.d.e 8 20.d odd 2 1
2790.2.d.l 8 3.b odd 2 1
2790.2.d.l 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 12T_{3}^{6} + 44T_{3}^{4} + 52T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(310, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 12 T^{6} + 44 T^{4} + 52 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + 4 T^{6} + 6 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + 104 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} - 6 T^{3} + 6 T^{2} + 10 T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 40 T^{6} + 548 T^{4} + \cdots + 3844 \) Copy content Toggle raw display
$17$ \( T^{8} + 24 T^{6} + 152 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} - 20 T^{2} - 64 T - 40)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 104 T^{6} + 2632 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} - 34 T^{2} + 198 T - 250)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 92 T^{6} + 2748 T^{4} + \cdots + 88804 \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{3} - 56 T^{2} + 116 T - 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 168 T^{6} + 8148 T^{4} + \cdots + 252004 \) Copy content Toggle raw display
$47$ \( T^{8} + 188 T^{6} + 7272 T^{4} + \cdots + 138384 \) Copy content Toggle raw display
$53$ \( T^{8} + 296 T^{6} + 22116 T^{4} + \cdots + 11236 \) Copy content Toggle raw display
$59$ \( (T^{4} - 176 T^{2} - 608 T + 320)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 138 T^{2} - 338 T + 1942)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 292 T^{6} + 17552 T^{4} + \cdots + 341056 \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} - 96 T^{2} + 640 T + 3504)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 392 T^{6} + \cdots + 28005264 \) Copy content Toggle raw display
$79$ \( (T^{4} - 12 T^{3} + 32 T^{2} + 32 T - 80)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 308 T^{6} + 28588 T^{4} + \cdots + 5008644 \) Copy content Toggle raw display
$89$ \( (T^{4} - 12 T^{3} - 240 T^{2} + 3392 T - 5760)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 788 T^{6} + \cdots + 1177038864 \) Copy content Toggle raw display
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