Properties

Label 1216.4.a.bh
Level $1216$
Weight $4$
Character orbit 1216.a
Self dual yes
Analytic conductor $71.746$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.7463225670\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 130 x^{5} + 212 x^{4} + 4589 x^{3} - 4178 x^{2} - 46788 x + 7848\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{3} + ( 1 - \beta_{5} ) q^{5} + ( -4 - \beta_{3} ) q^{7} + ( 12 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{3} + ( 1 - \beta_{5} ) q^{5} + ( -4 - \beta_{3} ) q^{7} + ( 12 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{9} + ( 11 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{11} + ( -6 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{13} + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{15} + ( -3 + 6 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{17} + 19 q^{19} + ( -14 - 12 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} + \beta_{4} + 4 \beta_{5} + 3 \beta_{6} ) q^{21} + ( 25 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 11 \beta_{5} + 2 \beta_{6} ) q^{23} + ( 2 + 13 \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{25} + ( 67 + 10 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} + \beta_{4} - 7 \beta_{5} - 4 \beta_{6} ) q^{27} + ( 11 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{29} + ( -14 + 13 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} + \beta_{4} - 7 \beta_{5} - 4 \beta_{6} ) q^{31} + ( -14 + 33 \beta_{1} - 6 \beta_{2} - 5 \beta_{3} - \beta_{4} + \beta_{5} ) q^{33} + ( 44 - 8 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} - 5 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{35} + ( -27 + 12 \beta_{1} - 8 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{37} + ( -46 - 11 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 4 \beta_{4} - \beta_{5} + 7 \beta_{6} ) q^{39} + ( 23 + 26 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{6} ) q^{41} + ( 172 - 8 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 15 \beta_{5} + 9 \beta_{6} ) q^{43} + ( 30 + 5 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} - 6 \beta_{4} - 7 \beta_{5} - 5 \beta_{6} ) q^{45} + ( -131 - 4 \beta_{1} - 4 \beta_{3} - 6 \beta_{4} + 9 \beta_{5} - 2 \beta_{6} ) q^{47} + ( 82 + 26 \beta_{1} + 5 \beta_{2} + 13 \beta_{3} - 2 \beta_{4} - 14 \beta_{5} - \beta_{6} ) q^{49} + ( 282 - 14 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 24 \beta_{5} + 5 \beta_{6} ) q^{51} + ( -2 + 40 \beta_{1} - 18 \beta_{2} - 2 \beta_{3} + \beta_{4} - 12 \beta_{5} + 9 \beta_{6} ) q^{53} + ( -126 - 11 \beta_{1} + 18 \beta_{2} + 4 \beta_{3} - 10 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{55} + ( 19 + 19 \beta_{1} ) q^{57} + ( 232 + 19 \beta_{1} - 4 \beta_{2} + \beta_{3} - 5 \beta_{4} + 7 \beta_{5} - 13 \beta_{6} ) q^{59} + ( 62 - 7 \beta_{1} + 18 \beta_{2} - 16 \beta_{3} + 12 \beta_{4} + 13 \beta_{5} + 17 \beta_{6} ) q^{61} + ( -382 - 39 \beta_{1} + 12 \beta_{2} - 40 \beta_{3} - 18 \beta_{4} + 57 \beta_{5} + 9 \beta_{6} ) q^{63} + ( -29 - 7 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} + 10 \beta_{5} + 5 \beta_{6} ) q^{65} + ( 318 + 8 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} - 20 \beta_{4} - 12 \beta_{5} - 15 \beta_{6} ) q^{67} + ( 65 + 86 \beta_{1} - 6 \beta_{2} + 19 \beta_{3} - 10 \beta_{4} - 11 \beta_{5} - 24 \beta_{6} ) q^{69} + ( -57 + 31 \beta_{1} - 36 \beta_{2} + 16 \beta_{3} + 12 \beta_{4} - 32 \beta_{5} - 3 \beta_{6} ) q^{71} + ( -34 - 51 \beta_{1} + 17 \beta_{2} + \beta_{3} - 8 \beta_{4} + 14 \beta_{5} + 4 \beta_{6} ) q^{73} + ( 419 - 12 \beta_{1} + 10 \beta_{2} - 19 \beta_{3} + 15 \beta_{4} + 31 \beta_{5} + 8 \beta_{6} ) q^{75} + ( -102 + 53 \beta_{1} + 10 \beta_{2} - 14 \beta_{3} - 20 \beta_{4} + 39 \beta_{5} - 7 \beta_{6} ) q^{77} + ( -87 - 4 \beta_{1} + 6 \beta_{2} + 11 \beta_{3} + 5 \beta_{4} + 17 \beta_{5} - 13 \beta_{6} ) q^{79} + ( 119 + 64 \beta_{1} - 18 \beta_{2} + 48 \beta_{3} - 6 \beta_{4} - 36 \beta_{5} - 10 \beta_{6} ) q^{81} + ( 329 + 15 \beta_{1} - 30 \beta_{2} - 14 \beta_{3} - 8 \beta_{4} + 32 \beta_{5} + \beta_{6} ) q^{83} + ( 115 + 66 \beta_{1} + 40 \beta_{2} - 20 \beta_{3} + 4 \beta_{4} + 13 \beta_{5} ) q^{85} + ( -9 + 50 \beta_{1} - 16 \beta_{2} + 31 \beta_{3} + 4 \beta_{4} - 27 \beta_{5} - 6 \beta_{6} ) q^{87} + ( -19 - 34 \beta_{1} - 18 \beta_{2} + 5 \beta_{3} + 15 \beta_{4} + 21 \beta_{5} - 17 \beta_{6} ) q^{89} + ( 586 + \beta_{1} + 10 \beta_{2} + 21 \beta_{3} - 11 \beta_{4} + 13 \beta_{5} - 7 \beta_{6} ) q^{91} + ( 456 + 24 \beta_{1} - 10 \beta_{2} + 66 \beta_{3} + 26 \beta_{4} - 58 \beta_{5} - 4 \beta_{6} ) q^{93} + ( 19 - 19 \beta_{5} ) q^{95} + ( 213 - 17 \beta_{1} - 22 \beta_{2} + 42 \beta_{3} + 26 \beta_{4} - 6 \beta_{5} - \beta_{6} ) q^{97} + ( 795 - 45 \beta_{1} - 5 \beta_{2} + 10 \beta_{3} + 23 \beta_{4} - 9 \beta_{5} + 22 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 9q^{3} + 5q^{5} - 28q^{7} + 86q^{9} + O(q^{10}) \) \( 7q + 9q^{3} + 5q^{5} - 28q^{7} + 86q^{9} + 77q^{11} - 43q^{13} - 6q^{17} + 133q^{19} - 111q^{21} + 159q^{23} + 44q^{25} + 471q^{27} + 69q^{29} - 90q^{31} - 30q^{33} + 297q^{35} - 164q^{37} - 339q^{39} + 214q^{41} + 1167q^{43} + 201q^{45} - 909q^{47} + 597q^{49} + 1903q^{51} + 51q^{53} - 901q^{55} + 171q^{57} + 1663q^{59} + 463q^{61} - 2629q^{63} - 192q^{65} + 2203q^{67} + 581q^{69} - 404q^{71} - 308q^{73} + 2979q^{75} - 537q^{77} - 596q^{79} + 879q^{81} + 2398q^{83} + 963q^{85} - 23q^{87} - 176q^{89} + 4123q^{91} + 3120q^{93} + 95q^{95} + 1444q^{97} + 5479q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 130 x^{5} + 212 x^{4} + 4589 x^{3} - 4178 x^{2} - 46788 x + 7848\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 92 \nu^{5} - 146 \nu^{4} - 10320 \nu^{3} + 8069 \nu^{2} + 233100 \nu - 70812 \)\()/10944\)
\(\beta_{3}\)\(=\)\((\)\( -11 \nu^{6} + 14 \nu^{5} + 1492 \nu^{4} - 1050 \nu^{3} - 48517 \nu^{2} + 3180 \nu + 219420 \)\()/10944\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{6} - 4 \nu^{5} + 274 \nu^{4} + 72 \nu^{3} + 18479 \nu^{2} + 3684 \nu - 515988 \)\()/10944\)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{6} + 5 \nu^{5} + 883 \nu^{4} - 489 \nu^{3} - 20491 \nu^{2} + 3432 \nu + 59652 \)\()/5472\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} - 22 \nu^{5} - 32 \nu^{4} + 2106 \nu^{3} - 4813 \nu^{2} - 31380 \nu + 80124 \)\()/1824\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{3} + 38\)
\(\nu^{3}\)\(=\)\(-4 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 6 \beta_{2} + 61 \beta_{1} + 6\)
\(\nu^{4}\)\(=\)\(6 \beta_{6} - 95 \beta_{5} + 77 \beta_{4} + 107 \beta_{3} + 6 \beta_{2} - 22 \beta_{1} + 2296\)
\(\nu^{5}\)\(=\)\(-446 \beta_{6} - 418 \beta_{5} - 254 \beta_{4} + 430 \beta_{3} - 552 \beta_{2} + 4307 \beta_{1} - 20\)
\(\nu^{6}\)\(=\)\(628 \beta_{6} - 8625 \beta_{5} + 5901 \beta_{4} + 9273 \beta_{3} + 684 \beta_{2} - 3036 \beta_{1} + 163166\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.94968
−5.20371
−3.90450
0.165732
4.53243
6.44524
8.91449
0 −7.94968 0 0.0633525 0 −19.8268 0 36.1974 0
1.2 0 −4.20371 0 −3.63602 0 8.86224 0 −9.32886 0
1.3 0 −2.90450 0 12.8220 0 11.9961 0 −18.5639 0
1.4 0 1.16573 0 −9.90212 0 −23.9754 0 −25.6411 0
1.5 0 5.53243 0 15.3317 0 23.3717 0 3.60773 0
1.6 0 7.44524 0 −18.2823 0 6.49029 0 28.4317 0
1.7 0 9.91449 0 8.60335 0 −34.9181 0 71.2970 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.4.a.bh 7
4.b odd 2 1 1216.4.a.be 7
8.b even 2 1 608.4.a.i 7
8.d odd 2 1 608.4.a.l yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.i 7 8.b even 2 1
608.4.a.l yes 7 8.d odd 2 1
1216.4.a.be 7 4.b odd 2 1
1216.4.a.bh 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):

\( T_{3}^{7} - 9 T_{3}^{6} - 97 T_{3}^{5} + 797 T_{3}^{4} + 2516 T_{3}^{3} - 15424 T_{3}^{2} - 26144 T_{3} + 46208 \)
\( T_{5}^{7} - 5 T_{5}^{6} - 447 T_{5}^{5} + 2537 T_{5}^{4} + 46234 T_{5}^{3} - 193600 T_{5}^{2} - 1101184 T_{5} + 70528 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \)
$3$ \( 46208 - 26144 T - 15424 T^{2} + 2516 T^{3} + 797 T^{4} - 97 T^{5} - 9 T^{6} + T^{7} \)
$5$ \( 70528 - 1101184 T - 193600 T^{2} + 46234 T^{3} + 2537 T^{4} - 447 T^{5} - 5 T^{6} + T^{7} \)
$7$ \( 267674508 - 72881673 T + 3182588 T^{2} + 441275 T^{3} - 22084 T^{4} - 1107 T^{5} + 28 T^{6} + T^{7} \)
$11$ \( -8913577312 + 2201195256 T - 121940284 T^{2} - 1702202 T^{3} + 230941 T^{4} - 2035 T^{5} - 77 T^{6} + T^{7} \)
$13$ \( -3972040064 + 152415840 T + 71517800 T^{2} - 1579550 T^{3} - 272503 T^{4} - 4711 T^{5} + 43 T^{6} + T^{7} \)
$17$ \( 1272145656338 - 75541045137 T - 1268487102 T^{2} + 93109251 T^{3} - 115034 T^{4} - 20339 T^{5} + 6 T^{6} + T^{7} \)
$19$ \( ( -19 + T )^{7} \)
$23$ \( 43343115459584 - 821129754880 T - 68533307776 T^{2} + 533591556 T^{3} + 6588251 T^{4} - 44585 T^{5} - 159 T^{6} + T^{7} \)
$29$ \( -53253690048 - 90613135392 T - 1103658640 T^{2} + 109357342 T^{3} + 903825 T^{4} - 23731 T^{5} - 69 T^{6} + T^{7} \)
$31$ \( -312246972080128 - 6641247429120 T + 173718134016 T^{2} + 1262836736 T^{3} - 14953640 T^{4} - 103732 T^{5} + 90 T^{6} + T^{7} \)
$37$ \( -1907389311488 - 387224985088 T + 13187680768 T^{2} + 1756840368 T^{3} - 5902384 T^{4} - 88432 T^{5} + 164 T^{6} + T^{7} \)
$41$ \( -460009274261504 + 41438759360512 T - 911184009984 T^{2} + 5267505344 T^{3} + 22177632 T^{4} - 187768 T^{5} - 214 T^{6} + T^{7} \)
$43$ \( -372474759986432 - 230566855966400 T + 5143112607232 T^{2} - 34862907324 T^{3} + 41777451 T^{4} + 351711 T^{5} - 1167 T^{6} + T^{7} \)
$47$ \( 829054396295168 + 4488147445056 T - 308921128048 T^{2} - 2560125732 T^{3} + 18717303 T^{4} + 262671 T^{5} + 909 T^{6} + T^{7} \)
$53$ \( 270594069263939248 - 3015919346013840 T - 11473461597896 T^{2} + 114360583520 T^{3} + 42184719 T^{4} - 691053 T^{5} - 51 T^{6} + T^{7} \)
$59$ \( 56606865830630176 - 2197870298503064 T + 25839910031804 T^{2} - 113262408182 T^{3} + 73466095 T^{4} + 718681 T^{5} - 1663 T^{6} + T^{7} \)
$61$ \( -2497687378177519664 - 70663920067968256 T - 70802380974800 T^{2} + 501411539076 T^{3} + 378769739 T^{4} - 1213825 T^{5} - 463 T^{6} + T^{7} \)
$67$ \( -42011282160203174176 + 190651083744984680 T + 10228334076316 T^{2} - 1122784083566 T^{3} + 1396540435 T^{4} + 773881 T^{5} - 2203 T^{6} + T^{7} \)
$71$ \( -6850823452091877376 - 29358568595232192 T + 168365680834624 T^{2} + 495388901232 T^{3} - 499437024 T^{4} - 1375620 T^{5} + 404 T^{6} + T^{7} \)
$73$ \( -10102144904316754 + 2788371917028561 T + 49574962547176 T^{2} + 112158713695 T^{3} - 249420266 T^{4} - 691729 T^{5} + 308 T^{6} + T^{7} \)
$79$ \( 348341971512695296 - 6882613605194944 T + 25124671608256 T^{2} + 118324132016 T^{3} - 277282784 T^{4} - 668580 T^{5} + 596 T^{6} + T^{7} \)
$83$ \( \)\(16\!\cdots\!92\)\( + 177754003292639232 T - 1164310694307840 T^{2} - 658682891360 T^{3} + 2809042520 T^{4} + 77220 T^{5} - 2398 T^{6} + T^{7} \)
$89$ \( -2521724011831719936 - 24037861271417088 T + 145478084394176 T^{2} + 1138172306096 T^{3} - 528625808 T^{4} - 2310480 T^{5} + 176 T^{6} + T^{7} \)
$97$ \( 2635261381995846656 - 10401539490879744 T - 515042016095680 T^{2} - 870362050224 T^{3} + 4323007040 T^{4} - 2385608 T^{5} - 1444 T^{6} + T^{7} \)
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