Properties

Label 260.2.m.c
Level $260$
Weight $2$
Character orbit 260.m
Analytic conductor $2.076$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.31678304256.2
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{6} + \beta_{3}) q^{5} + (\beta_{5} + \beta_{4}) q^{7} + (\beta_{5} - \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{6} + \beta_{3}) q^{5} + (\beta_{5} + \beta_{4}) q^{7} + (\beta_{5} - \beta_{3} - \beta_1) q^{9} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_1 + 2) q^{11} + ( - \beta_{7} - \beta_1 + 2) q^{13} + ( - \beta_{5} - 2 \beta_{4} + \beta_1 + 1) q^{15} + ( - \beta_{7} - \beta_{6} + 2 \beta_{3}) q^{17} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{19} + (\beta_{7} - \beta_{6} - \beta_{4} - 4 \beta_1 + 1) q^{21} + (\beta_{7} - \beta_{6} + 3 \beta_{4} - \beta_1 - 3) q^{23} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{25} + (\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{27} + ( - 2 \beta_{7} - \beta_{5} - 5 \beta_{4} + \beta_{3} + \beta_1) q^{29} + (\beta_{7} + \beta_{6} + \beta_{3}) q^{31} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{33} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_1 - 1) q^{35} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_1) q^{37} + (\beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{39} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{41} + ( - 3 \beta_{7} + 3 \beta_{6} + \beta_{4} + 3 \beta_1 - 1) q^{43} + (2 \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{45} + (2 \beta_{7} - 3 \beta_{5} - \beta_{4}) q^{47} + (2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{49} + ( - 2 \beta_{5} - 4 \beta_{4} + \beta_{3} + \beta_1) q^{51} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} - 2) q^{53} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_1 - 4) q^{55} + (2 \beta_{6} - 4 \beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{57} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{59} + (4 \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{61} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 - 7) q^{63} + ( - 2 \beta_{6} - \beta_{5} + 4 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 3) q^{65} + (\beta_{3} + \beta_{2} - \beta_1 + 3) q^{67} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{69} + (4 \beta_{7} + 4 \beta_{6} - \beta_{4} - \beta_{3} - 1) q^{71} + ( - 4 \beta_{6} + \beta_{3} - 3 \beta_{2} - \beta_1 - 5) q^{73} + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{75} + ( - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} + \beta_{2}) q^{77} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1) q^{79} + ( - 2 \beta_{6} + 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{81} + (2 \beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{83} + (\beta_{5} + 6 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{85} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + 8 \beta_1) q^{87} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2) q^{89} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_1) q^{91} + ( - \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{93} + ( - 3 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - \beta_1 + 1) q^{95} + ( - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{97} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 2 q^{5} + 14 q^{11} + 14 q^{13} + 10 q^{15} + 2 q^{19} + 4 q^{21} - 22 q^{23} + 12 q^{25} - 16 q^{27} - 6 q^{31} + 16 q^{33} - 4 q^{35} - 34 q^{39} - 8 q^{41} - 14 q^{43} + 22 q^{45} - 24 q^{49} - 8 q^{53} - 30 q^{55} + 16 q^{57} + 2 q^{59} - 12 q^{61} - 64 q^{63} + 32 q^{65} + 20 q^{67} - 20 q^{69} - 22 q^{71} - 28 q^{73} - 14 q^{75} + 8 q^{77} - 20 q^{81} + 12 q^{85} + 12 q^{87} + 4 q^{89} - 6 q^{95} + 12 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 97\nu^{7} - 173\nu^{6} + 220\nu^{5} + 316\nu^{4} + 4010\nu^{3} - 1148\nu^{2} - 1300\nu - 6130 ) / 2462 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 140\nu^{7} - 237\nu^{6} + 216\nu^{5} + 1116\nu^{4} + 5280\nu^{3} - 1530\nu^{2} + 738\nu + 696 ) / 2462 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 174\nu^{7} - 488\nu^{6} + 585\nu^{5} + 1176\nu^{4} + 4452\nu^{3} - 8760\nu^{2} + 2922\nu + 654 ) / 2462 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -382\nu^{7} + 1227\nu^{6} - 1539\nu^{5} - 2412\nu^{4} - 8076\nu^{3} + 22288\nu^{2} - 5566\nu - 1266 ) / 2462 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 452\nu^{7} - 730\nu^{6} + 416\nu^{5} + 4201\nu^{4} + 15640\nu^{3} - 4588\nu^{2} - 5144\nu + 4076 ) / 2462 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 605\nu^{7} - 1244\nu^{6} + 1461\nu^{5} + 4471\nu^{4} + 19300\nu^{3} - 11272\nu^{2} + 12070\nu + 2656 ) / 2462 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 3\beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} + 8\beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 12\beta_{3} - 8\beta_{2} - 12\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} - 8\beta_{6} + 12\beta_{5} + 26\beta_{4} - 12\beta_{2} - 66\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24\beta_{7} + 66\beta_{5} + 158\beta_{4} - 120\beta_{3} - 120\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 66\beta_{7} + 66\beta_{6} + 120\beta_{5} + 270\beta_{4} - 572\beta_{3} + 120\beta_{2} + 270 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(131\) \(157\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
−1.42497 1.42497i
−0.285451 0.285451i
0.575868 + 0.575868i
2.13456 + 2.13456i
−1.42497 + 1.42497i
−0.285451 + 0.285451i
0.575868 0.575868i
2.13456 2.13456i
0 −1.42497 + 1.42497i 0 1.72321 1.42497i 0 4.91106i 0 1.06111i 0
57.2 0 −0.285451 + 0.285451i 0 −2.21777 0.285451i 0 1.26613i 0 2.83704i 0
57.3 0 0.575868 0.575868i 0 2.16064 + 0.575868i 0 2.48849i 0 2.33675i 0
57.4 0 2.13456 2.13456i 0 −0.666078 + 2.13456i 0 2.84356i 0 6.11268i 0
73.1 0 −1.42497 1.42497i 0 1.72321 + 1.42497i 0 4.91106i 0 1.06111i 0
73.2 0 −0.285451 0.285451i 0 −2.21777 + 0.285451i 0 1.26613i 0 2.83704i 0
73.3 0 0.575868 + 0.575868i 0 2.16064 0.575868i 0 2.48849i 0 2.33675i 0
73.4 0 2.13456 + 2.13456i 0 −0.666078 2.13456i 0 2.84356i 0 6.11268i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.m.c 8
3.b odd 2 1 2340.2.u.g 8
4.b odd 2 1 1040.2.bg.m 8
5.b even 2 1 1300.2.m.c 8
5.c odd 4 1 260.2.r.c yes 8
5.c odd 4 1 1300.2.r.c 8
13.d odd 4 1 260.2.r.c yes 8
15.e even 4 1 2340.2.bp.g 8
20.e even 4 1 1040.2.cd.m 8
39.f even 4 1 2340.2.bp.g 8
52.f even 4 1 1040.2.cd.m 8
65.f even 4 1 1300.2.m.c 8
65.g odd 4 1 1300.2.r.c 8
65.k even 4 1 inner 260.2.m.c 8
195.j odd 4 1 2340.2.u.g 8
260.s odd 4 1 1040.2.bg.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.m.c 8 1.a even 1 1 trivial
260.2.m.c 8 65.k even 4 1 inner
260.2.r.c yes 8 5.c odd 4 1
260.2.r.c yes 8 13.d odd 4 1
1040.2.bg.m 8 4.b odd 2 1
1040.2.bg.m 8 260.s odd 4 1
1040.2.cd.m 8 20.e even 4 1
1040.2.cd.m 8 52.f even 4 1
1300.2.m.c 8 5.b even 2 1
1300.2.m.c 8 65.f even 4 1
1300.2.r.c 8 5.c odd 4 1
1300.2.r.c 8 65.g odd 4 1
2340.2.u.g 8 3.b odd 2 1
2340.2.u.g 8 195.j odd 4 1
2340.2.bp.g 8 15.e even 4 1
2340.2.bp.g 8 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} + 8T_{3}^{5} + 32T_{3}^{4} - 20T_{3}^{3} + 8T_{3}^{2} + 8T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + 2 T^{6} + 8 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} - 4 T^{6} + 10 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 40 T^{6} + 456 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( T^{8} - 14 T^{7} + 98 T^{6} - 332 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{7} + 112 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{5} + 360 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 86436 \) Copy content Toggle raw display
$23$ \( T^{8} + 22 T^{7} + 242 T^{6} + \cdots + 6724 \) Copy content Toggle raw display
$29$ \( T^{8} + 156 T^{6} + 8136 T^{4} + \cdots + 810000 \) Copy content Toggle raw display
$31$ \( T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 22500 \) Copy content Toggle raw display
$37$ \( T^{8} + 76 T^{6} + 1848 T^{4} + \cdots + 35344 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + 32 T^{6} - 152 T^{5} + \cdots + 784 \) Copy content Toggle raw display
$43$ \( T^{8} + 14 T^{7} + 98 T^{6} + \cdots + 1085764 \) Copy content Toggle raw display
$47$ \( T^{8} + 400 T^{6} + \cdots + 29637136 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + 32 T^{6} - 928 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 86436 \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} - 168 T^{2} - 1376 T - 1452)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 10 T^{3} + 24 T - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 22 T^{7} + 242 T^{6} + \cdots + 46321636 \) Copy content Toggle raw display
$73$ \( (T^{4} + 14 T^{3} - 168 T^{2} + \cdots - 13212)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 340 T^{6} + 25584 T^{4} + \cdots + 440896 \) Copy content Toggle raw display
$83$ \( T^{8} + 160 T^{6} + 4536 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$89$ \( T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 22391824 \) Copy content Toggle raw display
$97$ \( (T^{4} - 6 T^{3} - 216 T^{2} + 2192 T - 4392)^{2} \) Copy content Toggle raw display
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