Properties

Label 310.2.b.a
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: 8.0.619810816.2
Defining polynomial: \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2 \) 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
−1.49094 1.49094i
1.18254 + 1.18254i
0.561103 + 0.561103i
−0.252709 0.252709i
−0.252709 + 0.252709i
0.561103 0.561103i
1.18254 1.18254i
−1.49094 + 1.49094i
1.00000i 3.44579i −1.00000 −1.82630 + 1.29021i −3.44579 2.40146i 1.00000i −8.87345 1.29021 + 1.82630i
249.2 1.00000i 1.79682i −1.00000 1.60536 + 1.55654i −1.79682 3.47817i 1.00000i −0.228545 1.55654 1.60536i
249.3 1.00000i 0.370326i −1.00000 1.45220 1.70032i 0.370326 4.27844i 1.00000i 2.86286 −1.70032 1.45220i
249.4 1.00000i 0.872276i −1.00000 −2.23127 0.146426i 0.872276 2.79827i 1.00000i 2.23913 −0.146426 + 2.23127i
249.5 1.00000i 0.872276i −1.00000 −2.23127 + 0.146426i 0.872276 2.79827i 1.00000i 2.23913 −0.146426 2.23127i
249.6 1.00000i 0.370326i −1.00000 1.45220 + 1.70032i 0.370326 4.27844i 1.00000i 2.86286 −1.70032 + 1.45220i
249.7 1.00000i 1.79682i −1.00000 1.60536 1.55654i −1.79682 3.47817i 1.00000i −0.228545 1.55654 + 1.60536i
249.8 1.00000i 3.44579i −1.00000 −1.82630 1.29021i −3.44579 2.40146i 1.00000i −8.87345 1.29021 1.82630i
\(n\): 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.a 8
3.b odd 2 1 2790.2.d.m 8
4.b odd 2 1 2480.2.d.d 8
5.b even 2 1 inner 310.2.b.a 8
5.c odd 4 1 1550.2.a.o 4
5.c odd 4 1 1550.2.a.p 4
15.d odd 2 1 2790.2.d.m 8
20.d odd 2 1 2480.2.d.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.b.a 8 1.a even 1 1 trivial
310.2.b.a 8 5.b even 2 1 inner
1550.2.a.o 4 5.c odd 4 1
1550.2.a.p 4 5.c odd 4 1
2480.2.d.d 8 4.b odd 2 1
2480.2.d.d 8 20.d odd 2 1
2790.2.d.m 8 3.b odd 2 1
2790.2.d.m 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 16T_{3}^{6} + 52T_{3}^{4} + 36T_{3}^{2} + 4 \) 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
$3$ \( T^{8} + 16 T^{6} + 52 T^{4} + 36 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} - 4 T^{6} + 6 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 44 T^{6} + 680 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( (T^{4} + 8 T^{3} + 2 T^{2} - 62 T - 82)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 84 T^{6} + 2044 T^{4} + \cdots + 36100 \) Copy content Toggle raw display
$17$ \( T^{8} + 128 T^{6} + 5256 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} + 4 T^{2} - 16 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 32 T^{6} + 184 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} - 38 T^{2} + 202 T - 2)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 152 T^{6} + 4804 T^{4} + \cdots + 37636 \) Copy content Toggle raw display
$41$ \( (T^{4} - 104 T^{2} - 44 T + 1468)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 172 T^{6} + 8044 T^{4} + \cdots + 6724 \) Copy content Toggle raw display
$47$ \( T^{8} + 172 T^{6} + 8584 T^{4} + \cdots + 795664 \) Copy content Toggle raw display
$53$ \( T^{8} + 148 T^{6} + 6364 T^{4} + \cdots + 3364 \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} - 64 T^{2} - 128 T - 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{3} - 62 T^{2} + 10 T + 574)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 308 T^{6} + \cdots + 16321600 \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} + 32 T^{2} - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 400 T^{6} + 42712 T^{4} + \cdots + 2999824 \) Copy content Toggle raw display
$79$ \( (T^{4} - 16 T^{3} - 80 T^{2} + 1328 T + 2800)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 168 T^{6} + 8308 T^{4} + \cdots + 36100 \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} - 176 T^{2} + \cdots + 10048)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 436 T^{6} + 53528 T^{4} + \cdots + 4080400 \) Copy content Toggle raw display
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