Properties

Label 224.3.g.b
Level 224
Weight 3
Character orbit 224.g
Analytic conductor 6.104
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.292213762624.3
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + \beta_{2} q^{7} + ( 6 + \beta_{3} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + \beta_{2} q^{7} + ( 6 + \beta_{3} - \beta_{7} ) q^{9} + ( 4 + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{1} - \beta_{2} + 3 \beta_{4} ) q^{13} + ( -\beta_{1} - \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{15} + ( -10 + 4 \beta_{3} ) q^{17} + ( -7 + \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{2} + \beta_{5} ) q^{21} + ( 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{5} ) q^{23} + ( -2 - \beta_{3} + \beta_{7} ) q^{25} + ( 4 + 4 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{27} + ( 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{29} + ( 6 \beta_{1} + 6 \beta_{2} ) q^{31} + ( 4 + 4 \beta_{6} + 4 \beta_{7} ) q^{33} + ( -7 + \beta_{3} + \beta_{6} + \beta_{7} ) q^{35} + ( -2 \beta_{1} - 14 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{37} + ( -11 \beta_{1} + 5 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{39} + ( 16 + 2 \beta_{3} - 4 \beta_{6} - 2 \beta_{7} ) q^{41} + ( 4 \beta_{3} + \beta_{6} - 3 \beta_{7} ) q^{43} + ( -13 \beta_{1} + 13 \beta_{2} + 5 \beta_{4} + 4 \beta_{5} ) q^{45} + ( 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{4} + 4 \beta_{5} ) q^{47} -7 q^{49} + ( 46 - 10 \beta_{3} - 4 \beta_{7} ) q^{51} + ( 4 \beta_{1} - 4 \beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -6 \beta_{1} - 14 \beta_{2} + 8 \beta_{4} - 4 \beta_{5} ) q^{55} + ( 7 - 7 \beta_{3} - \beta_{7} ) q^{57} + ( -13 - 13 \beta_{3} - 2 \beta_{6} + 6 \beta_{7} ) q^{59} + ( -3 \beta_{1} + 19 \beta_{2} - 5 \beta_{4} ) q^{61} + ( -5 \beta_{1} + 6 \beta_{2} + 2 \beta_{5} ) q^{63} + ( -9 - 11 \beta_{3} - 5 \beta_{7} ) q^{65} + ( -38 + 6 \beta_{3} - \beta_{6} + 3 \beta_{7} ) q^{67} + ( 2 \beta_{1} - 18 \beta_{2} - 10 \beta_{4} - 12 \beta_{5} ) q^{69} + ( -8 \beta_{1} - 16 \beta_{4} ) q^{71} + ( -14 + 4 \beta_{6} + 4 \beta_{7} ) q^{73} + ( -9 - 9 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{75} + ( 6 \beta_{1} + 4 \beta_{2} - 7 \beta_{4} - \beta_{5} ) q^{77} + ( 16 \beta_{1} - 8 \beta_{2} - 12 \beta_{4} + 4 \beta_{5} ) q^{79} + ( 6 + 3 \beta_{3} - 8 \beta_{6} - 3 \beta_{7} ) q^{81} + ( -9 - \beta_{3} - 6 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 10 \beta_{1} - 18 \beta_{2} + 2 \beta_{4} + 8 \beta_{5} ) q^{85} + ( 18 \beta_{1} - 6 \beta_{2} - 24 \beta_{4} - 8 \beta_{5} ) q^{87} + ( -64 - 6 \beta_{3} - 4 \beta_{6} - 6 \beta_{7} ) q^{89} + ( 7 - \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{91} + ( 24 \beta_{2} - 12 \beta_{4} ) q^{93} + ( 7 \beta_{1} - 9 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} ) q^{95} + ( 8 - 6 \beta_{3} - 8 \beta_{6} + 2 \beta_{7} ) q^{97} + ( -32 - 12 \beta_{3} + 7 \beta_{6} + 7 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + 48q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + 48q^{9} + 32q^{11} - 80q^{17} - 56q^{19} - 16q^{25} + 32q^{27} + 32q^{33} - 56q^{35} + 128q^{41} - 56q^{49} + 368q^{51} + 56q^{57} - 104q^{59} - 72q^{65} - 304q^{67} - 112q^{73} - 72q^{75} + 48q^{81} - 72q^{83} - 512q^{89} + 56q^{91} + 64q^{97} - 256q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 2 x^{6} - 2 x^{5} + 24 x^{4} - 8 x^{3} - 32 x^{2} - 64 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} - 5 \nu^{6} + 4 \nu^{5} - 6 \nu^{4} + 4 \nu^{3} - 160 \nu^{2} + 96 \nu \)\()/128\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} - 12 \nu^{5} + 6 \nu^{4} + 12 \nu^{3} - 96 \nu \)\()/128\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 2 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} + 8 \nu - 16 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} - 14 \nu^{4} - 4 \nu^{3} - 128 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} - \nu^{6} - 4 \nu^{5} - 22 \nu^{4} + 84 \nu^{3} + 48 \nu^{2} + 128 \nu - 384 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{5} - 4 \nu^{4} + 22 \nu^{3} - 88 \nu - 64 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} + \nu^{6} + 18 \nu^{4} - 20 \nu^{3} - 8 \nu^{2} + 16 \nu + 192 \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 7 \beta_{1} + 5\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{7} + 3 \beta_{6} + 5 \beta_{5} - \beta_{4} - 3 \beta_{3} - 7 \beta_{2} + \beta_{1} + 13\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{7} - \beta_{6} + \beta_{5} - 21 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} - 3 \beta_{1} - 71\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-4 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - \beta_{4} - 35 \beta_{3} - 71 \beta_{2} - 15 \beta_{1} - 19\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(12 \beta_{7} + 23 \beta_{6} - 7 \beta_{5} - 61 \beta_{4} + 89 \beta_{3} - 91 \beta_{2} + 29 \beta_{1} - 55\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(-84 \beta_{7} - 21 \beta_{6} - 27 \beta_{5} - 153 \beta_{4} + 21 \beta_{3} + 49 \beta_{2} + 9 \beta_{1} - 27\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−1.67467 + 1.09337i
−1.67467 1.09337i
1.85837 + 0.739226i
1.85837 0.739226i
−1.05468 + 1.69931i
−1.05468 1.69931i
1.37098 1.45617i
1.37098 + 1.45617i
0 −4.56747 0 5.73252i 0 2.64575i 0 11.8618 0
15.2 0 −4.56747 0 5.73252i 0 2.64575i 0 11.8618 0
15.3 0 −0.0974366 0 3.46547i 0 2.64575i 0 −8.99051 0
15.4 0 −0.0974366 0 3.46547i 0 2.64575i 0 −8.99051 0
15.5 0 3.44128 0 4.88287i 0 2.64575i 0 2.84239 0
15.6 0 3.44128 0 4.88287i 0 2.64575i 0 2.84239 0
15.7 0 5.22363 0 6.26788i 0 2.64575i 0 18.2863 0
15.8 0 5.22363 0 6.26788i 0 2.64575i 0 18.2863 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.g.b 8
3.b odd 2 1 2016.3.g.b 8
4.b odd 2 1 56.3.g.b 8
7.b odd 2 1 1568.3.g.m 8
8.b even 2 1 56.3.g.b 8
8.d odd 2 1 inner 224.3.g.b 8
12.b even 2 1 504.3.g.b 8
16.e even 4 2 1792.3.d.j 16
16.f odd 4 2 1792.3.d.j 16
24.f even 2 1 2016.3.g.b 8
24.h odd 2 1 504.3.g.b 8
28.d even 2 1 392.3.g.m 8
28.f even 6 2 392.3.k.n 16
28.g odd 6 2 392.3.k.o 16
56.e even 2 1 1568.3.g.m 8
56.h odd 2 1 392.3.g.m 8
56.j odd 6 2 392.3.k.n 16
56.p even 6 2 392.3.k.o 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.b 8 4.b odd 2 1
56.3.g.b 8 8.b even 2 1
224.3.g.b 8 1.a even 1 1 trivial
224.3.g.b 8 8.d odd 2 1 inner
392.3.g.m 8 28.d even 2 1
392.3.g.m 8 56.h odd 2 1
392.3.k.n 16 28.f even 6 2
392.3.k.n 16 56.j odd 6 2
392.3.k.o 16 28.g odd 6 2
392.3.k.o 16 56.p even 6 2
504.3.g.b 8 12.b even 2 1
504.3.g.b 8 24.h odd 2 1
1568.3.g.m 8 7.b odd 2 1
1568.3.g.m 8 56.e even 2 1
1792.3.d.j 16 16.e even 4 2
1792.3.d.j 16 16.f odd 4 2
2016.3.g.b 8 3.b odd 2 1
2016.3.g.b 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4 T_{3}^{3} - 22 T_{3}^{2} + 80 T_{3} + 8 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 4 T + 14 T^{2} - 28 T^{3} + 98 T^{4} - 252 T^{5} + 1134 T^{6} - 2916 T^{7} + 6561 T^{8} )^{2} \)
$5$ \( 1 - 92 T^{2} + 5464 T^{4} - 211956 T^{6} + 6231214 T^{8} - 132472500 T^{10} + 2134375000 T^{12} - 22460937500 T^{14} + 152587890625 T^{16} \)
$7$ \( ( 1 + 7 T^{2} )^{4} \)
$11$ \( ( 1 - 16 T + 328 T^{2} - 4944 T^{3} + 49230 T^{4} - 598224 T^{5} + 4802248 T^{6} - 28344976 T^{7} + 214358881 T^{8} )^{2} \)
$13$ \( 1 - 444 T^{2} + 142936 T^{4} - 35361044 T^{6} + 6575436334 T^{8} - 1009946777684 T^{10} + 116597286336856 T^{12} - 10344349794381564 T^{14} + 665416609183179841 T^{16} \)
$17$ \( ( 1 + 40 T + 1308 T^{2} + 31512 T^{3} + 588230 T^{4} + 9106968 T^{5} + 109245468 T^{6} + 965502760 T^{7} + 6975757441 T^{8} )^{2} \)
$19$ \( ( 1 + 28 T + 1710 T^{2} + 31332 T^{3} + 975266 T^{4} + 11310852 T^{5} + 222848910 T^{6} + 1317284668 T^{7} + 16983563041 T^{8} )^{2} \)
$23$ \( 1 - 1744 T^{2} + 1272156 T^{4} - 412076080 T^{6} + 99307893702 T^{8} - 115315782303280 T^{10} + 99623789791135836 T^{12} - 38219105009443439824 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \)
$29$ \( 1 - 3384 T^{2} + 6555580 T^{4} - 8754768776 T^{6} + 8490907402822 T^{8} - 6192081614658056 T^{10} + 3279405379878872380 T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(25\!\cdots\!21\)\( T^{16} \)
$31$ \( 1 - 3944 T^{2} + 8438620 T^{4} - 12447428312 T^{6} + 13694235978694 T^{8} - 11495461442126552 T^{10} + 7197223366370371420 T^{12} - \)\(31\!\cdots\!84\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( 1 - 3512 T^{2} + 9188668 T^{4} - 18622781448 T^{6} + 27544347275206 T^{8} - 34902090701365128 T^{10} + 32275007558901367228 T^{12} - \)\(23\!\cdots\!72\)\( T^{14} + \)\(12\!\cdots\!41\)\( T^{16} \)
$41$ \( ( 1 - 64 T + 4956 T^{2} - 221760 T^{3} + 11848326 T^{4} - 372778560 T^{5} + 14004471516 T^{6} - 304006671424 T^{7} + 7984925229121 T^{8} )^{2} \)
$43$ \( ( 1 + 4680 T^{2} + 58016 T^{3} + 10251086 T^{4} + 107271584 T^{5} + 15999988680 T^{6} + 11688200277601 T^{8} )^{2} \)
$47$ \( 1 - 8392 T^{2} + 39566748 T^{4} - 127207295352 T^{6} + 316693927920198 T^{8} - 620731022190542712 T^{10} + \)\(94\!\cdots\!28\)\( T^{12} - \)\(97\!\cdots\!72\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( 1 - 18920 T^{2} + 162796828 T^{4} - 840091728600 T^{6} + 2864724835962118 T^{8} - 6628727822775456600 T^{10} + \)\(10\!\cdots\!08\)\( T^{12} - \)\(92\!\cdots\!20\)\( T^{14} + \)\(38\!\cdots\!21\)\( T^{16} \)
$59$ \( ( 1 + 52 T + 2254 T^{2} - 207508 T^{3} - 19795230 T^{4} - 722335348 T^{5} + 27312531694 T^{6} + 2193387749332 T^{7} + 146830437604321 T^{8} )^{2} \)
$61$ \( 1 - 16316 T^{2} + 140172120 T^{4} - 816942037524 T^{6} + 3499102878259502 T^{8} - 11311249557773337684 T^{10} + \)\(26\!\cdots\!20\)\( T^{12} - \)\(43\!\cdots\!36\)\( T^{14} + \)\(36\!\cdots\!61\)\( T^{16} \)
$67$ \( ( 1 + 152 T + 22224 T^{2} + 2037320 T^{3} + 158433022 T^{4} + 9145529480 T^{5} + 447838513104 T^{6} + 13749674089688 T^{7} + 406067677556641 T^{8} )^{2} \)
$71$ \( 1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 7484538771485032376 T^{10} + \)\(33\!\cdots\!24\)\( T^{12} - \)\(16\!\cdots\!24\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} \)
$73$ \( ( 1 + 56 T + 18460 T^{2} + 736008 T^{3} + 138223494 T^{4} + 3922186632 T^{5} + 524231528860 T^{6} + 8474716672184 T^{7} + 806460091894081 T^{8} )^{2} \)
$79$ \( 1 - 24968 T^{2} + 330869788 T^{4} - 3106127956152 T^{6} + 22189846569597766 T^{8} - \)\(12\!\cdots\!12\)\( T^{10} + \)\(50\!\cdots\!68\)\( T^{12} - \)\(14\!\cdots\!88\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \)
$83$ \( ( 1 + 36 T + 16478 T^{2} + 177884 T^{3} + 135298114 T^{4} + 1225442876 T^{5} + 782018213438 T^{6} + 11769853441284 T^{7} + 2252292232139041 T^{8} )^{2} \)
$89$ \( ( 1 + 256 T + 48252 T^{2} + 6269952 T^{3} + 638304966 T^{4} + 49664289792 T^{5} + 3027438612732 T^{6} + 127227210486016 T^{7} + 3936588805702081 T^{8} )^{2} \)
$97$ \( ( 1 - 32 T + 19484 T^{2} - 1437536 T^{3} + 199130566 T^{4} - 13525776224 T^{5} + 1724904511004 T^{6} - 26655104157728 T^{7} + 7837433594376961 T^{8} )^{2} \)
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