Properties

Label 3584.2.b.i
Level $3584$
Weight $2$
Character orbit 3584.b
Analytic conductor $28.618$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6183840844\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + 108 x^{3} + 68 x^{2} + 32 x + 8 \) 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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{7} q^{5} - q^{7} + (\beta_{5} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_{7} q^{5} - q^{7} + (\beta_{5} - 2) q^{9} + (\beta_{7} - \beta_{3}) q^{11} + \beta_{2} q^{13} + ( - \beta_{10} + \beta_1 - 1) q^{15} - \beta_{10} q^{17} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{19} + \beta_{6} q^{21} + (\beta_{9} + \beta_{8}) q^{23} + (\beta_{8} + \beta_1) q^{25} + ( - \beta_{11} - \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2}) q^{27} + (\beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{29} + (\beta_{9} + \beta_1 - 1) q^{31} + ( - \beta_{10} + \beta_{9} - \beta_{8} + \beta_1 + 1) q^{33} - \beta_{7} q^{35} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2}) q^{37} + \beta_{9} q^{39} + (\beta_{10} + \beta_{5} - \beta_1) q^{41} + (\beta_{11} - \beta_{7} + \beta_{4}) q^{43} + (\beta_{11} - 2 \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2}) q^{45} + (\beta_{10} + \beta_{8} - \beta_{5} + \beta_1 + 2) q^{47} + q^{49} + ( - 2 \beta_{7} - \beta_{6} - \beta_{4}) q^{51} + (\beta_{11} - 3 \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{53} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} - 3) q^{55} + ( - 2 \beta_{9} + \beta_{8} + \beta_1 + 1) q^{57} + ( - \beta_{11} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2}) q^{59} + (\beta_{11} + 3 \beta_{7} - \beta_{6} - \beta_{3} - 2 \beta_{2}) q^{61} + ( - \beta_{5} + 2) q^{63} + ( - 2 \beta_{8} - \beta_{5} - 1) q^{65} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{67} + (2 \beta_{11} + \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 3 \beta_{2}) q^{69} + (\beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{5} + \beta_1 + 1) q^{71} + ( - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - \beta_{5} + \beta_1 - 2) q^{73} + (2 \beta_{11} - 3 \beta_{7} + \beta_{4} + \beta_{2}) q^{75} + ( - \beta_{7} + \beta_{3}) q^{77} + (\beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{5} - \beta_1 - 5) q^{79} + (2 \beta_{9} + 3 \beta_{8} + \beta_1 + 2) q^{81} + ( - \beta_{11} + 2 \beta_{7} + \beta_{3} + 2 \beta_{2}) q^{83} + (\beta_{11} + 2 \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3}) q^{85} + ( - \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{5} + \beta_1 + 2) q^{87} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_1 - 3) q^{89} - \beta_{2} q^{91} + (2 \beta_{11} - 2 \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{93} + ( - \beta_{9} + 2 \beta_{8} + 2 \beta_1 - 2) q^{95} + ( - \beta_{9} - 3 \beta_{8} - \beta_{5} + 3) q^{97} + (\beta_{11} - \beta_{7} - \beta_{6} - 2 \beta_{4} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + 108 x^{3} + 68 x^{2} + 32 x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 419637583 \nu^{11} - 1440544428 \nu^{10} + 635340644 \nu^{9} + 4866667572 \nu^{8} - 9783161750 \nu^{7} + 6711732869 \nu^{6} + \cdots + 11421355507 ) / 9640197931 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 638917040 \nu^{11} - 2376776737 \nu^{10} + 1975208857 \nu^{9} + 5237100284 \nu^{8} - 12962071145 \nu^{7} + 10962211405 \nu^{6} + \cdots + 36460193404 ) / 9640197931 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8615178688 \nu^{11} - 52103353005 \nu^{10} + 115591508302 \nu^{9} - 43547709868 \nu^{8} - 308923880124 \nu^{7} + 722748596326 \nu^{6} + \cdots - 39713864336 ) / 96401979310 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 894988184 \nu^{11} + 5710664665 \nu^{10} - 13387713576 \nu^{9} + 6452256724 \nu^{8} + 33852806542 \nu^{7} - 84141470418 \nu^{6} + \cdots + 8956064308 ) / 7415536870 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 495332004 \nu^{11} - 2247729805 \nu^{10} + 2867331746 \nu^{9} + 3802764346 \nu^{8} - 15503170202 \nu^{7} + 17719026533 \nu^{6} + \cdots + 6862964757 ) / 3707768435 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1410256356 \nu^{11} + 5921534715 \nu^{10} - 6862437384 \nu^{9} - 9554539644 \nu^{8} + 34683435378 \nu^{7} - 39753339342 \nu^{6} + \cdots - 25484705788 ) / 7415536870 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13383971527 \nu^{11} + 62676653525 \nu^{10} - 94344403828 \nu^{9} - 51680040438 \nu^{8} + 369964045326 \nu^{7} + \cdots - 86421877996 ) / 48200989655 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6278889 \nu^{11} + 25797695 \nu^{10} - 26888646 \nu^{9} - 52239876 \nu^{8} + 164047612 \nu^{7} - 164419198 \nu^{6} - 1118786356 \nu^{5} + \cdots - 96252252 ) / 22141015 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2182781366 \nu^{11} - 9596353535 \nu^{10} + 11698649024 \nu^{9} + 16394832844 \nu^{8} - 62798712888 \nu^{7} + 69749861327 \nu^{6} + \cdots - 8063859872 ) / 6885855665 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3571797907 \nu^{11} + 15225607627 \nu^{10} - 17186575048 \nu^{9} - 29155049066 \nu^{8} + 100637456110 \nu^{7} + \cdots - 63122961648 ) / 9640197931 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3742088406 \nu^{11} + 17147484255 \nu^{10} - 24666155409 \nu^{9} - 16843522124 \nu^{8} + 101347688343 \nu^{7} + \cdots - 34789559518 ) / 6885855665 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 3\beta_{6} + \beta_{5} + \beta_{3} + 2\beta _1 + 3 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{11} + 4\beta_{10} - 4\beta_{8} - 4\beta_{7} - \beta_{6} + 4\beta_{5} + 4\beta_{4} + 7\beta_{3} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + 7 \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} + 15 \beta_{6} + 11 \beta_{5} + 22 \beta_{4} + 29 \beta_{3} + 4 \beta_{2} - 9 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{11} - 14\beta_{7} + 42\beta_{6} + 43\beta_{4} + 51\beta_{3} + 14\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 11 \beta_{11} - 65 \beta_{10} + 9 \beta_{9} + 53 \beta_{8} - 68 \beta_{7} + 143 \beta_{6} - 85 \beta_{5} + 222 \beta_{4} + 293 \beta_{3} + 44 \beta_{2} - 12 \beta _1 - 13 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 111 \beta_{11} - 340 \beta_{10} - 72 \beta_{9} + 188 \beta_{8} - 236 \beta_{7} + 67 \beta_{6} - 316 \beta_{5} + 468 \beta_{4} + 699 \beta_{3} + 24 \beta_{2} - 176 \beta _1 - 700 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 315 \beta_{11} - 1259 \beta_{10} - 615 \beta_{9} + 393 \beta_{8} - 420 \beta_{7} - 209 \beta_{6} - 839 \beta_{5} + 716 \beta_{4} + 1183 \beta_{3} - 68 \beta_{2} - 1102 \beta _1 - 4397 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1064\beta_{10} - 594\beta_{9} + 279\beta_{8} - 625\beta_{5} - 995\beta _1 - 4114 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2147 \beta_{11} - 12873 \beta_{10} - 5921 \beta_{9} + 4367 \beta_{8} + 3980 \beta_{7} - 383 \beta_{6} - 8893 \beta_{5} - 7892 \beta_{4} - 12071 \beta_{3} - 124 \beta_{2} - 10710 \beta _1 - 43091 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5767 \beta_{11} - 32820 \beta_{10} - 10864 \beta_{9} + 14660 \beta_{8} + 21420 \beta_{7} - 18435 \beta_{6} - 26876 \beta_{5} - 50636 \beta_{4} - 71715 \beta_{3} - 5944 \beta_{2} - 22264 \beta _1 - 87764 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5365 \beta_{11} - 57879 \beta_{10} - 15021 \beta_{9} + 29199 \beta_{8} + 82804 \beta_{7} - 119319 \beta_{6} - 51627 \beta_{5} - 221478 \beta_{4} - 299877 \beta_{3} - 38276 \beta_{2} + \cdots - 133047 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1793.1
0.960396 2.31860i
−1.45042 0.600784i
−0.232297 0.560814i
−0.484138 0.200537i
2.93456 1.21553i
0.271901 + 0.656426i
0.271901 0.656426i
2.93456 + 1.21553i
−0.484138 + 0.200537i
−0.232297 + 0.560814i
−1.45042 + 0.600784i
0.960396 + 2.31860i
0 3.22299i 0 0.725063i 0 −1.00000 0 −7.38766 0
1793.2 0 2.61578i 0 3.96111i 0 −1.00000 0 −3.84231 0
1793.3 0 2.53584i 0 1.47134i 0 −1.00000 0 −3.43049 0
1793.4 0 1.81529i 0 2.66965i 0 −1.00000 0 −0.295267 0
1793.5 0 1.01685i 0 1.29145i 0 −1.00000 0 1.96601 0
1793.6 0 0.101362i 0 2.19640i 0 −1.00000 0 2.98973 0
1793.7 0 0.101362i 0 2.19640i 0 −1.00000 0 2.98973 0
1793.8 0 1.01685i 0 1.29145i 0 −1.00000 0 1.96601 0
1793.9 0 1.81529i 0 2.66965i 0 −1.00000 0 −0.295267 0
1793.10 0 2.53584i 0 1.47134i 0 −1.00000 0 −3.43049 0
1793.11 0 2.61578i 0 3.96111i 0 −1.00000 0 −3.84231 0
1793.12 0 3.22299i 0 0.725063i 0 −1.00000 0 −7.38766 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1793.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.b.i 12
4.b odd 2 1 3584.2.b.k 12
8.b even 2 1 inner 3584.2.b.i 12
8.d odd 2 1 3584.2.b.k 12
16.e even 4 1 3584.2.a.f yes 6
16.e even 4 1 3584.2.a.l yes 6
16.f odd 4 1 3584.2.a.e 6
16.f odd 4 1 3584.2.a.k yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.e 6 16.f odd 4 1
3584.2.a.f yes 6 16.e even 4 1
3584.2.a.k yes 6 16.f odd 4 1
3584.2.a.l yes 6 16.e even 4 1
3584.2.b.i 12 1.a even 1 1 trivial
3584.2.b.i 12 8.b even 2 1 inner
3584.2.b.k 12 4.b odd 2 1
3584.2.b.k 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3584, [\chi])\):

\( T_{3}^{12} + 28T_{3}^{10} + 288T_{3}^{8} + 1328T_{3}^{6} + 2612T_{3}^{4} + 1584T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{23}^{6} - 76T_{23}^{4} - 96T_{23}^{3} + 1060T_{23}^{2} + 832T_{23} - 2848 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 28 T^{10} + 288 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} + 32 T^{10} + 348 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 68 T^{10} + 1604 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{12} + 48 T^{10} + 668 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( (T^{6} - 48 T^{4} + 32 T^{3} + 464 T^{2} + \cdots - 896)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 140 T^{10} + 5792 T^{8} + \cdots + 300304 \) Copy content Toggle raw display
$23$ \( (T^{6} - 76 T^{4} - 96 T^{3} + 1060 T^{2} + \cdots - 2848)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 168 T^{10} + 9808 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} - 72 T^{4} - 512 T^{3} + \cdots + 11776)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 168 T^{10} + 9296 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} - 84 T^{4} + 416 T^{3} + \cdots + 5696)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 292 T^{10} + \cdots + 2117472256 \) Copy content Toggle raw display
$47$ \( (T^{6} - 8 T^{5} - 152 T^{4} + \cdots - 179200)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 552 T^{10} + \cdots + 2390818816 \) Copy content Toggle raw display
$59$ \( T^{12} + 364 T^{10} + \cdots + 241740304 \) Copy content Toggle raw display
$61$ \( T^{12} + 448 T^{10} + \cdots + 2466512896 \) Copy content Toggle raw display
$67$ \( T^{12} + 356 T^{10} + \cdots + 18939904 \) Copy content Toggle raw display
$71$ \( (T^{6} - 8 T^{5} - 248 T^{4} + 800 T^{3} + \cdots - 51200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 16 T^{5} - 240 T^{4} + \cdots + 899200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 24 T^{5} + 64 T^{4} - 1696 T^{3} + \cdots - 21632)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 636 T^{10} + \cdots + 72755351824 \) Copy content Toggle raw display
$89$ \( (T^{6} + 16 T^{5} - 32 T^{4} - 1056 T^{3} + \cdots + 6272)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 16 T^{5} - 96 T^{4} + \cdots + 232576)^{2} \) Copy content Toggle raw display
show more
show less