Properties

Label 1216.4.a.bf
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: \(x^{7} - 3 x^{6} - 121 x^{5} + 402 x^{4} + 4234 x^{3} - 14542 x^{2} - 40996 x + 141664\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 2 + \beta_{3} ) q^{5} + ( 6 - \beta_{2} + \beta_{5} ) q^{7} + ( 12 - 2 \beta_{2} + \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 2 + \beta_{3} ) q^{5} + ( 6 - \beta_{2} + \beta_{5} ) q^{7} + ( 12 - 2 \beta_{2} + \beta_{5} + \beta_{6} ) q^{9} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{11} + ( 6 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{13} + ( 17 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{15} + ( 7 + 7 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{17} + 19 q^{19} + ( 3 - 17 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{21} + ( 60 + \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{23} + ( -1 + 11 \beta_{1} - 4 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{25} + ( 5 - 8 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} ) q^{27} + ( -35 - 11 \beta_{1} - 5 \beta_{2} + 7 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} ) q^{29} + ( 13 + 9 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} ) q^{31} + ( -33 - 7 \beta_{1} + 2 \beta_{3} + 10 \beta_{4} - 5 \beta_{5} + 3 \beta_{6} ) q^{33} + ( -22 - \beta_{1} + 2 \beta_{2} + 13 \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{35} + ( 89 - 33 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} - 10 \beta_{4} + 9 \beta_{5} + 2 \beta_{6} ) q^{37} + ( 110 - 20 \beta_{1} + 7 \beta_{2} + 23 \beta_{3} + 16 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} ) q^{39} + ( -1 - 7 \beta_{1} + 14 \beta_{2} - 10 \beta_{3} - 2 \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{41} + ( -104 + 21 \beta_{1} - 18 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - 5 \beta_{6} ) q^{43} + ( -44 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} + 7 \beta_{6} ) q^{45} + ( 210 + 14 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 16 \beta_{4} - 6 \beta_{6} ) q^{47} + ( 44 - 29 \beta_{1} + 14 \beta_{2} - 23 \beta_{3} - 11 \beta_{4} - \beta_{5} ) q^{49} + ( -242 - 7 \beta_{1} + 32 \beta_{2} - 4 \beta_{3} + 14 \beta_{4} - 4 \beta_{5} - 19 \beta_{6} ) q^{51} + ( -1 + 5 \beta_{1} - 21 \beta_{2} - 15 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 12 \beta_{6} ) q^{53} + ( 266 + 10 \beta_{1} + 6 \beta_{2} + 33 \beta_{3} + 4 \beta_{4} + 10 \beta_{5} - 7 \beta_{6} ) q^{55} -19 \beta_{1} q^{57} + ( -9 - 20 \beta_{1} + 18 \beta_{2} - 22 \beta_{3} - 22 \beta_{4} + 3 \beta_{5} + 12 \beta_{6} ) q^{59} + ( -14 + 6 \beta_{1} + 8 \beta_{2} - 15 \beta_{3} - 4 \beta_{4} - 30 \beta_{5} + 7 \beta_{6} ) q^{61} + ( 440 - 14 \beta_{1} - 12 \beta_{2} - 11 \beta_{3} - 14 \beta_{4} - 6 \beta_{5} + 15 \beta_{6} ) q^{63} + ( -82 - 90 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} + 26 \beta_{4} - 10 \beta_{5} + 3 \beta_{6} ) q^{65} + ( -236 + 15 \beta_{1} + 16 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 17 \beta_{6} ) q^{67} + ( -102 - 82 \beta_{1} + 27 \beta_{2} - 9 \beta_{3} + 30 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} ) q^{69} + ( 338 - 12 \beta_{1} - 22 \beta_{2} - 16 \beta_{3} + 10 \beta_{4} - 18 \beta_{5} + 5 \beta_{6} ) q^{71} + ( 21 - \beta_{1} - 46 \beta_{2} + 47 \beta_{3} - 9 \beta_{4} + 13 \beta_{5} - \beta_{6} ) q^{73} + ( -261 - 34 \beta_{1} + 2 \beta_{2} - 28 \beta_{3} + 26 \beta_{4} - 15 \beta_{5} + 7 \beta_{6} ) q^{75} + ( 84 + 42 \beta_{1} + 2 \beta_{2} + 15 \beta_{3} - 20 \beta_{4} - 8 \beta_{5} + 19 \beta_{6} ) q^{77} + ( 383 + 3 \beta_{1} + 48 \beta_{2} - 40 \beta_{3} - 4 \beta_{4} - 15 \beta_{5} - 24 \beta_{6} ) q^{79} + ( 45 - 66 \beta_{1} - 20 \beta_{2} - 18 \beta_{4} - 20 \beta_{5} + 22 \beta_{6} ) q^{81} + ( -108 + 76 \beta_{1} - 34 \beta_{2} + 50 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} - 7 \beta_{6} ) q^{83} + ( -368 + 38 \beta_{1} + 40 \beta_{2} + 39 \beta_{3} + 16 \beta_{4} - 12 \beta_{5} - 20 \beta_{6} ) q^{85} + ( 666 + 48 \beta_{1} - 49 \beta_{2} - 9 \beta_{3} - 16 \beta_{4} + 17 \beta_{5} ) q^{87} + ( -65 - 15 \beta_{1} + 28 \beta_{2} + 18 \beta_{3} + 20 \beta_{4} - \beta_{5} - 32 \beta_{6} ) q^{89} + ( 7 + 12 \beta_{1} - 62 \beta_{2} - 16 \beta_{3} - 16 \beta_{4} + 3 \beta_{5} + 26 \beta_{6} ) q^{91} + ( -148 + 88 \beta_{1} - 14 \beta_{2} - 34 \beta_{3} - 30 \beta_{4} - 18 \beta_{5} - 28 \beta_{6} ) q^{93} + ( 38 + 19 \beta_{3} ) q^{95} + ( -24 - 102 \beta_{1} + 20 \beta_{2} + 14 \beta_{3} - 12 \beta_{4} + 12 \beta_{5} - 7 \beta_{6} ) q^{97} + ( 266 - 35 \beta_{1} - 20 \beta_{2} - 23 \beta_{3} - 39 \beta_{4} + 10 \beta_{5} + 26 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 3q^{3} + 17q^{5} + 42q^{7} + 86q^{9} + O(q^{10}) \) \( 7q - 3q^{3} + 17q^{5} + 42q^{7} + 86q^{9} + 33q^{11} + 35q^{13} + 120q^{15} + 66q^{17} + 133q^{19} - 33q^{21} + 389q^{23} + 44q^{25} + 39q^{27} - 233q^{29} + 158q^{31} - 206q^{33} - 123q^{35} + 436q^{37} + 807q^{39} - 94q^{41} - 645q^{43} - 103q^{45} + 1451q^{47} + 93q^{49} - 1741q^{51} - 3q^{53} + 1971q^{55} - 57q^{57} - 297q^{59} - 93q^{61} + 2999q^{63} - 788q^{65} - 1641q^{67} - 945q^{69} + 2392q^{71} + 324q^{73} - 1909q^{75} + 711q^{77} + 2492q^{79} + 143q^{81} - 310q^{83} - 2353q^{85} + 4795q^{87} - 440q^{89} + 107q^{91} - 900q^{93} + 323q^{95} - 532q^{97} + 1591q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 121 x^{5} + 402 x^{4} + 4234 x^{3} - 14542 x^{2} - 40996 x + 141664\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 31441 \nu^{6} - 53905 \nu^{5} - 3640731 \nu^{4} + 5610384 \nu^{3} + 129105226 \nu^{2} - 93123434 \nu - 1374812288 \)\()/46945020\)
\(\beta_{2}\)\(=\)\((\)\( -39878 \nu^{6} - 170155 \nu^{5} + 4088763 \nu^{4} + 14312913 \nu^{3} - 123786608 \nu^{2} - 269997998 \nu + 1133044114 \)\()/23472510\)
\(\beta_{3}\)\(=\)\((\)\( -41606 \nu^{6} - 149275 \nu^{5} + 4108191 \nu^{4} + 9529701 \nu^{3} - 122864216 \nu^{2} - 88940246 \nu + 1155407968 \)\()/23472510\)
\(\beta_{4}\)\(=\)\((\)\( 134983 \nu^{6} + 651005 \nu^{5} - 12792033 \nu^{4} - 43729188 \nu^{3} + 362351638 \nu^{2} + 636664498 \nu - 3246819584 \)\()/46945020\)
\(\beta_{5}\)\(=\)\((\)\( -320173 \nu^{6} - 369335 \nu^{5} + 34597563 \nu^{4} + 19633308 \nu^{3} - 1044937738 \nu^{2} - 252259318 \nu + 8164814204 \)\()/46945020\)
\(\beta_{6}\)\(=\)\((\)\( -169831 \nu^{6} - 229925 \nu^{5} + 21008001 \nu^{4} + 17685276 \nu^{3} - 766255246 \nu^{2} - 318429586 \nu + 7496458508 \)\()/23472510\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + 2 \beta_{3} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - \beta_{3} - \beta_{2} + 5 \beta_{1} + 70\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{6} - 2 \beta_{5} + 54 \beta_{4} + 76 \beta_{3} + 40 \beta_{2} + 18 \beta_{1} - 104\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(39 \beta_{6} + 126 \beta_{5} - 36 \beta_{4} - 182 \beta_{3} - 226 \beta_{2} + 804 \beta_{1} + 6554\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-178 \beta_{6} - 62 \beta_{5} + 1652 \beta_{4} + 1821 \beta_{3} + 1683 \beta_{2} - 558 \beta_{1} - 5331\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(4441 \beta_{6} + 6522 \beta_{5} - 5178 \beta_{4} - 14256 \beta_{3} - 19324 \beta_{2} + 55846 \beta_{1} + 359232\)\()/4\)

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
−8.42108
7.37394
6.59304
−4.00212
4.17107
−6.10834
3.39349
0 −8.54092 0 −13.1578 0 19.9667 0 45.9472 0
1.2 0 −6.88542 0 −4.69330 0 3.62949 0 20.4091 0
1.3 0 −5.59727 0 20.9584 0 13.9085 0 4.32941 0
1.4 0 0.923414 0 0.670967 0 −28.6780 0 −26.1473 0
1.5 0 2.43763 0 9.42708 0 −9.58919 0 −21.0580 0
1.6 0 5.20712 0 −7.79991 0 26.8592 0 0.114065 0
1.7 0 9.45545 0 11.5945 0 15.9032 0 62.4055 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.bf 7
4.b odd 2 1 1216.4.a.bg 7
8.b even 2 1 608.4.a.k yes 7
8.d odd 2 1 608.4.a.j 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.j 7 8.d odd 2 1
608.4.a.k yes 7 8.b even 2 1
1216.4.a.bf 7 1.a even 1 1 trivial
1216.4.a.bg 7 4.b odd 2 1

Hecke kernels

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

\( T_{3}^{7} + 3 T_{3}^{6} - 133 T_{3}^{5} - 367 T_{3}^{4} + 4632 T_{3}^{3} + 6688 T_{3}^{2} - 49248 T_{3} + 36480 \)
\( T_{5}^{7} - 17 T_{5}^{6} - 315 T_{5}^{5} + 4077 T_{5}^{4} + 28950 T_{5}^{3} - 216728 T_{5}^{2} - 972192 T_{5} + 740352 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \)
$3$ \( 36480 - 49248 T + 6688 T^{2} + 4632 T^{3} - 367 T^{4} - 133 T^{5} + 3 T^{6} + T^{7} \)
$5$ \( 740352 - 972192 T - 216728 T^{2} + 28950 T^{3} + 4077 T^{4} - 315 T^{5} - 17 T^{6} + T^{7} \)
$7$ \( -118397740 + 42440813 T - 1626362 T^{2} - 424321 T^{3} + 36816 T^{4} - 365 T^{5} - 42 T^{6} + T^{7} \)
$11$ \( 47353183200 - 2124226296 T - 141601148 T^{2} + 5879858 T^{3} + 121585 T^{4} - 4463 T^{5} - 33 T^{6} + T^{7} \)
$13$ \( 361563654528 - 16694005472 T - 841106280 T^{2} + 32853942 T^{3} + 332039 T^{4} - 10979 T^{5} - 35 T^{6} + T^{7} \)
$17$ \( 821802019554 + 15242724471 T - 3228972582 T^{2} + 44422771 T^{3} + 1072774 T^{4} - 16363 T^{5} - 66 T^{6} + T^{7} \)
$19$ \( ( -19 + T )^{7} \)
$23$ \( -74757540326400 + 3095668139520 T - 12704460960 T^{2} - 675866924 T^{3} + 5730129 T^{4} + 26319 T^{5} - 389 T^{6} + T^{7} \)
$29$ \( -758502349773120 - 10754165247744 T + 321731362632 T^{2} + 2530188614 T^{3} - 18250053 T^{4} - 95059 T^{5} + 233 T^{6} + T^{7} \)
$31$ \( 542529415987200 - 3085033538560 T - 236662577280 T^{2} + 2134742752 T^{3} + 10706712 T^{4} - 98988 T^{5} - 158 T^{6} + T^{7} \)
$37$ \( 82678995093613056 + 267983927564032 T - 7604616061760 T^{2} + 10555928400 T^{3} + 109616640 T^{4} - 227528 T^{5} - 436 T^{6} + T^{7} \)
$41$ \( -883998869667840 - 14254491666432 T + 285978528000 T^{2} + 3374719808 T^{3} - 9531872 T^{4} - 120544 T^{5} + 94 T^{6} + T^{7} \)
$43$ \( 71381448428213760 + 1941353169222464 T + 9834553228160 T^{2} - 16283660456 T^{3} - 184520873 T^{4} - 177397 T^{5} + 645 T^{6} + T^{7} \)
$47$ \( -600627802971763200 + 3546139636159936 T + 10844867857200 T^{2} - 105167987532 T^{3} + 113245487 T^{4} + 545431 T^{5} - 1451 T^{6} + T^{7} \)
$53$ \( -4321560912460940880 - 21027098460967200 T + 30858153575664 T^{2} + 228513883428 T^{3} - 51556751 T^{4} - 835977 T^{5} + 3 T^{6} + T^{7} \)
$59$ \( 52628953098306720 - 622999119137912 T - 11308157636068 T^{2} + 109566461074 T^{3} - 76071161 T^{4} - 765615 T^{5} + 297 T^{6} + T^{7} \)
$61$ \( 237983714954927600 - 2798008032740272 T - 41948689638248 T^{2} + 252025323152 T^{3} - 6711609 T^{4} - 1042445 T^{5} + 93 T^{6} + T^{7} \)
$67$ \( -2877194950049120 + 405023809237208 T - 1475802521252 T^{2} - 41807403690 T^{3} - 47477153 T^{4} + 599445 T^{5} + 1641 T^{6} + T^{7} \)
$71$ \( 6776730452132536320 - 152541107886041408 T + 910222034503552 T^{2} - 2128919444176 T^{3} + 1660356224 T^{4} + 1098436 T^{5} - 2392 T^{6} + T^{7} \)
$73$ \( 2259972913008529406 - 654677043450823 T - 248683580363216 T^{2} + 531329687407 T^{3} + 570561654 T^{4} - 1465609 T^{5} - 324 T^{6} + T^{7} \)
$79$ \( 11028149872904337920 - 198474186118358976 T + 1231304282483136 T^{2} - 3238110061776 T^{3} + 3168479840 T^{4} + 537708 T^{5} - 2492 T^{6} + T^{7} \)
$83$ \( -13028067932575334400 - 49450580577472512 T + 155159012585472 T^{2} + 496439620448 T^{3} - 487183800 T^{4} - 1445068 T^{5} + 310 T^{6} + T^{7} \)
$89$ \( 3570529620607298560 - 58240077891559936 T - 31758477944832 T^{2} + 1010148984912 T^{3} - 337001472 T^{4} - 1945608 T^{5} + 440 T^{6} + T^{7} \)
$97$ \( 8644623795817792512 + 14560184083842816 T - 545554594114624 T^{2} + 1423699466320 T^{3} - 94699392 T^{4} - 2173064 T^{5} + 532 T^{6} + T^{7} \)
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