Properties

Label 960.2.b.d
Level $960$
Weight $2$
Character orbit 960.b
Analytic conductor $7.666$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + q^{5} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{7} + \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + q^{5} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{7} + \beta_{7} q^{9} - \beta_{11} q^{11} + (\beta_{10} - \beta_{8} + \beta_{7}) q^{13} - \beta_{4} q^{15} + (\beta_{10} - \beta_{2} - \beta_1) q^{17} + \beta_{9} q^{19} + (\beta_{7} + \beta_1 - 2) q^{21} + ( - \beta_{9} + \beta_{5} + \beta_{4}) q^{23} + q^{25} + ( - \beta_{11} + \beta_{5} + \beta_{3}) q^{27} + (\beta_{8} + \beta_{7} - \beta_{2} + \beta_1 + 2) q^{29} + (\beta_{11} - \beta_{6}) q^{31} + ( - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{2} + 1) q^{33} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{35} + (\beta_{10} - \beta_{8} + \beta_{7} + \beta_{2} + \beta_1) q^{37} + (\beta_{9} + 2 \beta_{5} + \beta_{3}) q^{39} + (\beta_{8} - \beta_{7}) q^{41} + ( - 3 \beta_{5} - 3 \beta_{4}) q^{43} + \beta_{7} q^{45} + (\beta_{9} + \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{47} + (\beta_{8} + \beta_{7} - 2 \beta_{2} + 2 \beta_1 + 1) q^{49} + (2 \beta_{9} - 3 \beta_{6} - 2 \beta_{5} - \beta_{3}) q^{51} + (2 \beta_{8} + 2 \beta_{7} + \beta_{2} - \beta_1 + 4) q^{53} - \beta_{11} q^{55} + ( - 2 \beta_{10} + \beta_{8} - \beta_{7} + \beta_{2} - 1) q^{57} + ( - \beta_{11} + 2 \beta_{5} - 2 \beta_{4}) q^{59} + (\beta_{8} - \beta_{7} + \beta_{2} + \beta_1) q^{61} + ( - \beta_{9} + \beta_{5} + 3 \beta_{4} + 2 \beta_{3}) q^{63} + (\beta_{10} - \beta_{8} + \beta_{7}) q^{65} + (2 \beta_{9} - \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{67} + (2 \beta_{10} - \beta_{8} - \beta_{2} - 2) q^{69} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{71} + ( - 2 \beta_{8} - 2 \beta_{7} + \beta_{2} - \beta_1 - 4) q^{73} - \beta_{4} q^{75} + ( - \beta_{2} + \beta_1 + 2) q^{77} + ( - 3 \beta_{11} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4}) q^{79} + ( - 2 \beta_{10} + \beta_{8} - \beta_{7} - 2 \beta_{2} - 1) q^{81} + (6 \beta_{6} + \beta_{5} - \beta_{4}) q^{83} + (\beta_{10} - \beta_{2} - \beta_1) q^{85} + ( - \beta_{9} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{3}) q^{87} + (4 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{2} + 2 \beta_1) q^{89} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{91} + (\beta_{10} + \beta_{8} + \beta_{7} + \beta_{2} - \beta_1 - 2) q^{93} + \beta_{9} q^{95} + ( - \beta_{2} + \beta_1 + 4) q^{97} + ( - \beta_{9} - 3 \beta_{6} + 4 \beta_{5} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{5} - 4 q^{9} - 32 q^{21} + 12 q^{25} + 8 q^{29} + 16 q^{33} - 4 q^{45} - 12 q^{49} + 40 q^{53} - 8 q^{57} - 24 q^{69} - 24 q^{73} + 16 q^{77} - 20 q^{81} - 24 q^{93} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3193 \nu^{11} - 2244 \nu^{10} + 318 \nu^{9} + 2790 \nu^{8} - 55642 \nu^{7} - 38316 \nu^{6} + 52914 \nu^{5} + 250740 \nu^{4} + 773740 \nu^{3} + 629112 \nu^{2} + \cdots + 244884 ) / 51972 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4689 \nu^{11} - 3876 \nu^{10} + 3854 \nu^{9} - 5534 \nu^{8} - 68264 \nu^{7} - 56268 \nu^{6} + 37850 \nu^{5} + 438544 \nu^{4} + 1102444 \nu^{3} + 911136 \nu^{2} + \cdots + 48876 ) / 51972 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 34 \nu^{11} + 4 \nu^{10} - 17 \nu^{9} + 12 \nu^{8} - 546 \nu^{7} - 896 \nu^{6} + 196 \nu^{5} + 3468 \nu^{4} + 10454 \nu^{3} + 13152 \nu^{2} + 7260 \nu + 2736 ) / 366 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9709 \nu^{11} + 6990 \nu^{10} - 6270 \nu^{9} + 4284 \nu^{8} + 155434 \nu^{7} + 116508 \nu^{6} - 57894 \nu^{5} - 856824 \nu^{4} - 2383876 \nu^{3} - 2154324 \nu^{2} + \cdots - 596232 ) / 103944 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12807 \nu^{11} - 7336 \nu^{10} + 6002 \nu^{9} - 4224 \nu^{8} - 204280 \nu^{7} - 188332 \nu^{6} + 79370 \nu^{5} + 1157448 \nu^{4} + 3257084 \nu^{3} + 3239028 \nu^{2} + \cdots + 820728 ) / 103944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1688 \nu^{11} + 1054 \nu^{10} - 840 \nu^{9} + 684 \nu^{8} + 26378 \nu^{7} + 24587 \nu^{6} - 12648 \nu^{5} - 151968 \nu^{4} - 420176 \nu^{3} - 400856 \nu^{2} + \cdots - 105120 ) / 12993 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2365 \nu^{11} + 1728 \nu^{10} - 620 \nu^{9} - 408 \nu^{8} + 38369 \nu^{7} + 28380 \nu^{6} - 29398 \nu^{5} - 210938 \nu^{4} - 548176 \nu^{3} - 448716 \nu^{2} + \cdots - 128100 ) / 17324 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7568 \nu^{11} - 5700 \nu^{10} + 2978 \nu^{9} - 1790 \nu^{8} - 119671 \nu^{7} - 90816 \nu^{6} + 85724 \nu^{5} + 699454 \nu^{4} + 1757628 \nu^{3} + 1442196 \nu^{2} + \cdots + 138132 ) / 51972 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8153 \nu^{11} + 4002 \nu^{10} - 8316 \nu^{9} + 6516 \nu^{8} - 135194 \nu^{7} - 253752 \nu^{6} + 29934 \nu^{5} + 882480 \nu^{4} + 2784512 \nu^{3} + 3554004 \nu^{2} + \cdots + 743400 ) / 51972 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} - 10408 \nu^{4} - 27758 \nu^{3} - 22776 \nu^{2} - 12468 \nu - 4320 ) / 426 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17023 \nu^{11} - 10696 \nu^{10} + 8738 \nu^{9} - 6744 \nu^{8} - 267980 \nu^{7} - 238924 \nu^{6} + 119690 \nu^{5} + 1529352 \nu^{4} + 4246428 \nu^{3} + 4054116 \nu^{2} + \cdots + 1063800 ) / 51972 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + \beta_{8} + \beta_{7} - 2\beta_{5} - \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{9} + 3\beta_{6} - 2\beta_{5} - 2\beta_{4} + 3\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{10} - 4\beta_{5} - 4\beta_{4} + \beta_{3} + 2\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{10} + 5\beta_{8} + \beta_{7} + 3\beta_{2} + 4\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19 \beta_{11} + 6 \beta_{10} + \beta_{9} + 20 \beta_{8} + 18 \beta_{7} + 21 \beta_{6} + 2 \beta_{5} + 34 \beta_{4} - 7 \beta_{3} + \beta_{2} + 17 \beta _1 + 58 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{11} + 27\beta_{6} - 8\beta_{5} + 8\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 47 \beta_{11} + 16 \beta_{10} - 4 \beta_{9} - 43 \beta_{8} - 51 \beta_{7} + 60 \beta_{6} - 78 \beta_{5} - 8 \beta_{4} + 20 \beta_{3} + 39 \beta_{2} + 4 \beta _1 - 155 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 94\beta_{10} - 48\beta_{8} - 118\beta_{7} + 131\beta_{2} + 83\beta _1 - 306 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 83 \beta_{10} + 24 \beta_{9} + 24 \beta_{8} - 24 \beta_{7} + 212 \beta_{5} + 212 \beta_{4} - 107 \beta_{3} + 106 \beta_{2} + 106 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 213\beta_{11} + 82\beta_{9} + 318\beta_{6} + 426\beta_{5} + 688\beta_{4} - 318\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1193 \beta_{11} - 426 \beta_{10} + 131 \beta_{9} - 1324 \beta_{8} - 1062 \beta_{7} + 1671 \beta_{6} + 262 \beta_{5} + 1862 \beta_{4} - 557 \beta_{3} - 131 \beta_{2} - 931 \beta _1 - 4142 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
671.1
0.583700 + 2.17840i
0.583700 2.17840i
−1.50511 + 0.403293i
−1.50511 0.403293i
−0.673288 0.180407i
−0.673288 + 0.180407i
−0.180407 0.673288i
−0.180407 + 0.673288i
−0.403293 + 1.50511i
−0.403293 1.50511i
2.17840 + 0.583700i
2.17840 0.583700i
0 −1.59470 0.675970i 0 1.00000 0 0.648061i 0 2.08613 + 2.15594i 0
671.2 0 −1.59470 + 0.675970i 0 1.00000 0 0.648061i 0 2.08613 2.15594i 0
671.3 0 −1.10182 1.33641i 0 1.00000 0 4.67282i 0 −0.571993 + 2.94497i 0
671.4 0 −1.10182 + 1.33641i 0 1.00000 0 4.67282i 0 −0.571993 2.94497i 0
671.5 0 −0.492881 1.66044i 0 1.00000 0 1.32088i 0 −2.51414 + 1.63680i 0
671.6 0 −0.492881 + 1.66044i 0 1.00000 0 1.32088i 0 −2.51414 1.63680i 0
671.7 0 0.492881 1.66044i 0 1.00000 0 1.32088i 0 −2.51414 1.63680i 0
671.8 0 0.492881 + 1.66044i 0 1.00000 0 1.32088i 0 −2.51414 + 1.63680i 0
671.9 0 1.10182 1.33641i 0 1.00000 0 4.67282i 0 −0.571993 2.94497i 0
671.10 0 1.10182 + 1.33641i 0 1.00000 0 4.67282i 0 −0.571993 + 2.94497i 0
671.11 0 1.59470 0.675970i 0 1.00000 0 0.648061i 0 2.08613 2.15594i 0
671.12 0 1.59470 + 0.675970i 0 1.00000 0 0.648061i 0 2.08613 + 2.15594i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 671.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.b.d yes 12
3.b odd 2 1 960.2.b.c 12
4.b odd 2 1 inner 960.2.b.d yes 12
8.b even 2 1 960.2.b.c 12
8.d odd 2 1 960.2.b.c 12
12.b even 2 1 960.2.b.c 12
24.f even 2 1 inner 960.2.b.d yes 12
24.h odd 2 1 inner 960.2.b.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.b.c 12 3.b odd 2 1
960.2.b.c 12 8.b even 2 1
960.2.b.c 12 8.d odd 2 1
960.2.b.c 12 12.b even 2 1
960.2.b.d yes 12 1.a even 1 1 trivial
960.2.b.d yes 12 4.b odd 2 1 inner
960.2.b.d yes 12 24.f even 2 1 inner
960.2.b.d yes 12 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{6} + 24T_{7}^{4} + 48T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{29}^{3} - 2T_{29}^{2} - 44T_{29} + 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{10} + 7 T^{8} + 12 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 24 T^{4} + 48 T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 44 T^{4} + 496 T^{2} + 576)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 52 T^{4} + 304 T^{2} + 192)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 100 T^{4} + 2608 T^{2} + \cdots + 15552)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 52 T^{4} + 304 T^{2} - 192)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 60 T^{4} + 1008 T^{2} + \cdots - 3888)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 44 T + 72)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 64 T^{4} + 768 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 100 T^{4} + 3184 T^{2} + \cdots + 32448)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 64 T^{4} + 1216 T^{2} + \cdots + 6912)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 144 T^{4} + 5184 T^{2} + \cdots - 34992)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 172 T^{4} + 6256 T^{2} + \cdots - 3888)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 10 T^{2} - 116 T + 1128)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} + 92 T^{4} + 2224 T^{2} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 144 T^{4} + 3456 T^{2} + \cdots + 6912)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 256 T^{4} + 12736 T^{2} + \cdots - 178608)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 208 T^{4} + 5632 T^{2} + \cdots - 27648)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 6 T^{2} - 84 T + 104)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 384 T^{4} + 27840 T^{2} + \cdots + 565504)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 404 T^{4} + 49168 T^{2} + \cdots + 1838736)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 640 T^{4} + 114688 T^{2} + \cdots + 3981312)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 10 T^{2} - 4 T + 136)^{4} \) Copy content Toggle raw display
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