Properties

Label 224.3.h.d
Level 224
Weight 3
Character orbit 224.h
Analytic conductor 6.104
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.976966189056.51
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} + ( -\beta_{2} + \beta_{3} ) q^{5} + ( 2 + \beta_{1} - \beta_{4} ) q^{7} + ( 1 + 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} + ( -\beta_{2} + \beta_{3} ) q^{5} + ( 2 + \beta_{1} - \beta_{4} ) q^{7} + ( 1 + 2 \beta_{1} ) q^{9} + ( -\beta_{5} - \beta_{6} ) q^{11} + ( -3 \beta_{2} - \beta_{3} ) q^{13} + ( -8 + \beta_{1} ) q^{15} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{17} + ( -4 \beta_{2} - 3 \beta_{3} ) q^{19} + ( -\beta_{2} - 3 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{21} + ( 4 - 6 \beta_{1} ) q^{23} + ( -3 - 6 \beta_{1} ) q^{25} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{27} + ( 4 \beta_{5} - 2 \beta_{6} ) q^{29} + ( \beta_{1} + 2 \beta_{7} ) q^{31} + ( \beta_{1} - 3 \beta_{4} - \beta_{7} ) q^{33} + ( \beta_{3} + 5 \beta_{5} - \beta_{6} ) q^{35} + ( -2 \beta_{5} + 2 \beta_{6} ) q^{37} + ( 16 + 11 \beta_{1} ) q^{39} + ( -2 \beta_{1} + 4 \beta_{4} ) q^{41} + ( -9 \beta_{5} - 5 \beta_{6} ) q^{43} + ( 7 \beta_{2} - 3 \beta_{3} ) q^{45} + ( -\beta_{1} - 2 \beta_{7} ) q^{47} + ( -35 + 5 \beta_{1} - 3 \beta_{4} - \beta_{7} ) q^{49} + ( 2 \beta_{5} + 10 \beta_{6} ) q^{51} + 6 \beta_{6} q^{53} + ( -\beta_{1} + 2 \beta_{4} ) q^{55} + ( 38 + 18 \beta_{1} ) q^{57} + ( 12 \beta_{2} + \beta_{3} ) q^{59} + ( -\beta_{2} + \beta_{3} ) q^{61} + ( 14 + 3 \beta_{1} + \beta_{4} - 2 \beta_{7} ) q^{63} + ( 34 - 14 \beta_{1} ) q^{65} + ( 9 \beta_{5} + 5 \beta_{6} ) q^{67} + ( 12 \beta_{2} + 8 \beta_{3} ) q^{69} + ( -68 - 13 \beta_{1} ) q^{71} + ( -5 \beta_{1} + 11 \beta_{4} + \beta_{7} ) q^{73} + ( 12 \beta_{2} + 15 \beta_{3} ) q^{75} + ( 2 \beta_{2} + 14 \beta_{3} - 3 \beta_{5} - 4 \beta_{6} ) q^{77} + ( 4 - 9 \beta_{1} ) q^{79} + ( -41 - 14 \beta_{1} ) q^{81} + ( 24 \beta_{2} - 17 \beta_{3} ) q^{83} + ( 14 \beta_{5} - 8 \beta_{6} ) q^{85} + ( -4 \beta_{1} + 6 \beta_{4} - 2 \beta_{7} ) q^{87} + ( -7 \beta_{1} + 13 \beta_{4} - \beta_{7} ) q^{89} + ( -4 \beta_{2} - 9 \beta_{3} + 3 \beta_{5} - 11 \beta_{6} ) q^{91} + ( 10 \beta_{5} + 24 \beta_{6} ) q^{93} + ( 32 - 17 \beta_{1} ) q^{95} + ( 9 \beta_{1} - 17 \beta_{4} + \beta_{7} ) q^{97} + ( -5 \beta_{5} - 9 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{7} + 8q^{9} + O(q^{10}) \) \( 8q + 16q^{7} + 8q^{9} - 64q^{15} + 32q^{23} - 24q^{25} + 128q^{39} - 280q^{49} + 304q^{57} + 112q^{63} + 272q^{65} - 544q^{71} + 32q^{79} - 328q^{81} + 256q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 10 x^{6} + 123 x^{4} - 300 x^{3} + 86 x^{2} + 300 x + 526\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -868 \nu^{7} - 9000 \nu^{6} - 26304 \nu^{5} + 91850 \nu^{4} + 320592 \nu^{3} - 425950 \nu^{2} - 816140 \nu + 4767050 \)\()/1582309\)
\(\beta_{2}\)\(=\)\((\)\( 2679764 \nu^{7} + 208186 \nu^{6} - 31814108 \nu^{5} - 10053780 \nu^{4} + 411714584 \nu^{3} - 624881484 \nu^{2} - 1121009602 \nu + 982876622 \)\()/ 685139797 \)
\(\beta_{3}\)\(=\)\((\)\( 3872412 \nu^{7} + 6558661 \nu^{6} - 34456727 \nu^{5} - 55216902 \nu^{4} + 394683447 \nu^{3} - 633800883 \nu^{2} - 1116611684 \nu + 1714881404 \)\()/ 685139797 \)
\(\beta_{4}\)\(=\)\((\)\( -12992 \nu^{7} + 18417 \nu^{6} + 167753 \nu^{5} - 3353 \nu^{4} - 1377571 \nu^{3} + 4853780 \nu^{2} - 5682368 \nu + 556407 \)\()/1582309\)
\(\beta_{5}\)\(=\)\((\)\( 5735372 \nu^{7} + 4313372 \nu^{6} - 52238584 \nu^{5} - 59878610 \nu^{4} + 684612832 \nu^{3} - 1065326618 \nu^{2} + 851928604 \nu - 98379406 \)\()/ 685139797 \)
\(\beta_{6}\)\(=\)\((\)\( 8071492 \nu^{7} - 164500 \nu^{6} - 46895176 \nu^{5} + 109451638 \nu^{4} + 845538352 \nu^{3} - 2992899254 \nu^{2} + 2981272772 \nu + 1079942072 \)\()/ 685139797 \)
\(\beta_{7}\)\(=\)\((\)\( -19240 \nu^{7} + 143219 \nu^{6} + 372167 \nu^{5} - 1048471 \nu^{4} - 3401197 \nu^{3} + 23225460 \nu^{2} - 25469724 \nu - 15107251 \)\()/1582309\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - 2 \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_{1} + 10\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} - 4 \beta_{6} + 28 \beta_{5} + 21 \beta_{4} - 16 \beta_{3} + 10 \beta_{2}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(8 \beta_{7} + 21 \beta_{6} - 33 \beta_{5} - 24 \beta_{4} + 16 \beta_{3} - 32 \beta_{2} + 66 \beta_{1} - 146\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-25 \beta_{7} + 6 \beta_{6} + 112 \beta_{5} + 175 \beta_{4} - 376 \beta_{3} + 502 \beta_{2} - 522 \beta_{1} + 1500\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-16 \beta_{7} + 24 \beta_{6} + 93 \beta_{5} + 148 \beta_{4} + 804 \beta_{3} - 1068 \beta_{2} + 743 \beta_{1} - 2948\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(693 \beta_{7} + 1306 \beta_{6} - 2864 \beta_{5} - 2751 \beta_{4} - 3876 \beta_{3} + 5654 \beta_{2} - 4048 \beta_{1} + 18900\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.639946 0.719687i
−0.639946 + 0.719687i
1.52814 1.99551i
1.52814 + 1.99551i
−3.26020 + 1.99551i
−3.26020 1.99551i
2.37200 + 0.719687i
2.37200 0.719687i
0 −4.11439 0 1.10245 0 3.73205 5.92214i 0 7.92820 0
209.2 0 −4.11439 0 1.10245 0 3.73205 + 5.92214i 0 7.92820 0
209.3 0 −1.75265 0 6.54099 0 0.267949 6.99487i 0 −5.92820 0
209.4 0 −1.75265 0 6.54099 0 0.267949 + 6.99487i 0 −5.92820 0
209.5 0 1.75265 0 −6.54099 0 0.267949 6.99487i 0 −5.92820 0
209.6 0 1.75265 0 −6.54099 0 0.267949 + 6.99487i 0 −5.92820 0
209.7 0 4.11439 0 −1.10245 0 3.73205 5.92214i 0 7.92820 0
209.8 0 4.11439 0 −1.10245 0 3.73205 + 5.92214i 0 7.92820 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.b even 2 1 inner
56.h 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.h.d 8
3.b odd 2 1 2016.3.l.f 8
4.b odd 2 1 56.3.h.d 8
7.b odd 2 1 inner 224.3.h.d 8
8.b even 2 1 inner 224.3.h.d 8
8.d odd 2 1 56.3.h.d 8
12.b even 2 1 504.3.l.f 8
21.c even 2 1 2016.3.l.f 8
24.f even 2 1 504.3.l.f 8
24.h odd 2 1 2016.3.l.f 8
28.d even 2 1 56.3.h.d 8
28.f even 6 2 392.3.j.d 16
28.g odd 6 2 392.3.j.d 16
56.e even 2 1 56.3.h.d 8
56.h odd 2 1 inner 224.3.h.d 8
56.k odd 6 2 392.3.j.d 16
56.m even 6 2 392.3.j.d 16
84.h odd 2 1 504.3.l.f 8
168.e odd 2 1 504.3.l.f 8
168.i even 2 1 2016.3.l.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.d 8 4.b odd 2 1
56.3.h.d 8 8.d odd 2 1
56.3.h.d 8 28.d even 2 1
56.3.h.d 8 56.e even 2 1
224.3.h.d 8 1.a even 1 1 trivial
224.3.h.d 8 7.b odd 2 1 inner
224.3.h.d 8 8.b even 2 1 inner
224.3.h.d 8 56.h odd 2 1 inner
392.3.j.d 16 28.f even 6 2
392.3.j.d 16 28.g odd 6 2
392.3.j.d 16 56.k odd 6 2
392.3.j.d 16 56.m even 6 2
504.3.l.f 8 12.b even 2 1
504.3.l.f 8 24.f even 2 1
504.3.l.f 8 84.h odd 2 1
504.3.l.f 8 168.e odd 2 1
2016.3.l.f 8 3.b odd 2 1
2016.3.l.f 8 21.c even 2 1
2016.3.l.f 8 24.h odd 2 1
2016.3.l.f 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 20 T_{3}^{2} + 52 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 16 T^{2} + 178 T^{4} + 1296 T^{6} + 6561 T^{8} )^{2} \)
$5$ \( ( 1 + 56 T^{2} + 1602 T^{4} + 35000 T^{6} + 390625 T^{8} )^{2} \)
$7$ \( ( 1 - 8 T + 102 T^{2} - 392 T^{3} + 2401 T^{4} )^{2} \)
$11$ \( ( 1 - 364 T^{2} + 59334 T^{4} - 5329324 T^{6} + 214358881 T^{8} )^{2} \)
$13$ \( ( 1 + 344 T^{2} + 86658 T^{4} + 9824984 T^{6} + 815730721 T^{8} )^{2} \)
$17$ \( ( 1 - 148 T^{2} + 165606 T^{4} - 12361108 T^{6} + 6975757441 T^{8} )^{2} \)
$19$ \( ( 1 + 656 T^{2} + 327858 T^{4} + 85490576 T^{6} + 16983563041 T^{8} )^{2} \)
$23$ \( ( 1 - 8 T + 642 T^{2} - 4232 T^{3} + 279841 T^{4} )^{4} \)
$29$ \( ( 1 - 1444 T^{2} + 1048038 T^{4} - 1021313764 T^{6} + 500246412961 T^{8} )^{2} \)
$31$ \( ( 1 + 140 T^{2} + 1206054 T^{4} + 129292940 T^{6} + 852891037441 T^{8} )^{2} \)
$37$ \( ( 1 - 4612 T^{2} + 8955366 T^{4} - 8643630532 T^{6} + 3512479453921 T^{8} )^{2} \)
$41$ \( ( 1 - 5380 T^{2} + 12875334 T^{4} - 15202594180 T^{6} + 7984925229121 T^{8} )^{2} \)
$43$ \( ( 1 - 1324 T^{2} + 5788998 T^{4} - 4526492524 T^{6} + 11688200277601 T^{8} )^{2} \)
$47$ \( ( 1 - 4852 T^{2} + 14998950 T^{4} - 23676212212 T^{6} + 23811286661761 T^{8} )^{2} \)
$53$ \( ( 1 - 7780 T^{2} + 30664230 T^{4} - 61387942180 T^{6} + 62259690411361 T^{8} )^{2} \)
$59$ \( ( 1 + 9200 T^{2} + 44845170 T^{4} + 111479721200 T^{6} + 146830437604321 T^{8} )^{2} \)
$61$ \( ( 1 + 14840 T^{2} + 82747650 T^{4} + 205472280440 T^{6} + 191707312997281 T^{8} )^{2} \)
$67$ \( ( 1 - 11884 T^{2} + 74122758 T^{4} - 239475921964 T^{6} + 406067677556641 T^{8} )^{2} \)
$71$ \( ( 1 + 136 T + 12678 T^{2} + 685576 T^{3} + 25411681 T^{4} )^{4} \)
$73$ \( ( 1 - 9364 T^{2} + 72904614 T^{4} - 265921128724 T^{6} + 806460091894081 T^{8} )^{2} \)
$79$ \( ( 1 - 8 T + 11526 T^{2} - 49928 T^{3} + 38950081 T^{4} )^{4} \)
$83$ \( ( 1 + 6608 T^{2} - 3756750 T^{4} + 313604585168 T^{6} + 2252292232139041 T^{8} )^{2} \)
$89$ \( ( 1 - 17428 T^{2} + 180509478 T^{4} - 1093471776148 T^{6} + 3936588805702081 T^{8} )^{2} \)
$97$ \( ( 1 - 13588 T^{2} + 180804198 T^{4} - 1202935870228 T^{6} + 7837433594376961 T^{8} )^{2} \)
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