Properties

Label 84.12.a.b
Level $84$
Weight $12$
Character orbit 84.a
Self dual yes
Analytic conductor $64.541$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 84.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.5408271670\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1000465}) \)
Defining polynomial: \( x^{2} - x - 250116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{1000465}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 243 q^{3} + ( - 5 \beta - 4455) q^{5} - 16807 q^{7} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 243 q^{3} + ( - 5 \beta - 4455) q^{5} - 16807 q^{7} + 59049 q^{9} + ( - 115 \beta + 166617) q^{11} + (1458 \beta + 110592) q^{13} + (1215 \beta + 1082565) q^{15} + ( - 2503 \beta + 4120227) q^{17} + (1944 \beta + 4857644) q^{19} + 4084101 q^{21} + (10367 \beta + 8556111) q^{23} + (44550 \beta - 3969475) q^{25} - 14348907 q^{27} + ( - 117800 \beta + 32975262) q^{29} + ( - 34344 \beta + 39729240) q^{31} + (27945 \beta - 40487931) q^{33} + (84035 \beta + 74875185) q^{35} + ( - 483570 \beta + 147033644) q^{37} + ( - 354294 \beta - 26873856) q^{39} + (690591 \beta - 208747935) q^{41} + (1626480 \beta - 149459212) q^{43} + ( - 295245 \beta - 263063295) q^{45} + ( - 2289774 \beta + 241447554) q^{47} + 282475249 q^{49} + (608229 \beta - 1001215161) q^{51} + (3111378 \beta - 2134755648) q^{53} + ( - 320760 \beta - 167011360) q^{55} + ( - 472392 \beta - 1180407492) q^{57} + ( - 4074170 \beta - 2615799582) q^{59} + (8181000 \beta - 924873594) q^{61} - 992436543 q^{63} + ( - 7048350 \beta - 7786077210) q^{65} + ( - 6436746 \beta - 3727394846) q^{67} + ( - 2519181 \beta - 2079134973) q^{69} + (5015153 \beta - 15767760591) q^{71} + ( - 1198638 \beta - 4864591556) q^{73} + ( - 10825650 \beta + 964582425) q^{75} + (1932805 \beta - 2800331919) q^{77} + (13426722 \beta - 25478178222) q^{79} + 3486784401 q^{81} + ( - 86080 \beta - 12122072172) q^{83} + ( - 9450270 \beta - 5834791810) q^{85} + (28625400 \beta - 8012988666) q^{87} + (53744511 \beta + 5742574785) q^{89} + ( - 24504606 \beta - 1858719744) q^{91} + (8345592 \beta - 9654205320) q^{93} + ( - 32948740 \beta - 31365323820) q^{95} + ( - 23050818 \beta - 55400331584) q^{97} + ( - 6790635 \beta + 9838567233) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 486 q^{3} - 8910 q^{5} - 33614 q^{7} + 118098 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 486 q^{3} - 8910 q^{5} - 33614 q^{7} + 118098 q^{9} + 333234 q^{11} + 221184 q^{13} + 2165130 q^{15} + 8240454 q^{17} + 9715288 q^{19} + 8168202 q^{21} + 17112222 q^{23} - 7938950 q^{25} - 28697814 q^{27} + 65950524 q^{29} + 79458480 q^{31} - 80975862 q^{33} + 149750370 q^{35} + 294067288 q^{37} - 53747712 q^{39} - 417495870 q^{41} - 298918424 q^{43} - 526126590 q^{45} + 482895108 q^{47} + 564950498 q^{49} - 2002430322 q^{51} - 4269511296 q^{53} - 334022720 q^{55} - 2360814984 q^{57} - 5231599164 q^{59} - 1849747188 q^{61} - 1984873086 q^{63} - 15572154420 q^{65} - 7454789692 q^{67} - 4158269946 q^{69} - 31535521182 q^{71} - 9729183112 q^{73} + 1929164850 q^{75} - 5600663838 q^{77} - 50956356444 q^{79} + 6973568802 q^{81} - 24244144344 q^{83} - 11669583620 q^{85} - 16025977332 q^{87} + 11485149570 q^{89} - 3717439488 q^{91} - 19308410640 q^{93} - 62730647640 q^{95} - 110800663168 q^{97} + 19677134466 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
500.616
−499.616
0 −243.000 0 −9456.16 0 −16807.0 0 59049.0 0
1.2 0 −243.000 0 546.162 0 −16807.0 0 59049.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

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 84.12.a.b 2
3.b odd 2 1 252.12.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.a.b 2 1.a even 1 1 trivial
252.12.a.d 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8910T_{5} - 5164600 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(84))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 243)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 8910 T - 5164600 \) Copy content Toggle raw display
$7$ \( (T + 16807)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 333234 T + 14530075064 \) Copy content Toggle raw display
$13$ \( T^{2} - 221184 T - 2114521889796 \) Copy content Toggle raw display
$17$ \( T^{2} - 8240454 T + 10708348302344 \) Copy content Toggle raw display
$19$ \( T^{2} - 9715288 T + 19815811932496 \) Copy content Toggle raw display
$23$ \( T^{2} - 17112222 T - 34317629286064 \) Copy content Toggle raw display
$29$ \( T^{2} - 65950524 T - 12\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 398353702671360 \) Copy content Toggle raw display
$37$ \( T^{2} - 294067288 T - 21\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{2} + 417495870 T - 43\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{2} + 298918424 T - 26\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{2} - 482895108 T - 51\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} + 4269511296 T - 51\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{2} + 5231599164 T - 97\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{2} + 1849747188 T - 66\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{2} + 7454789692 T - 27\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{2} + 31535521182 T + 22\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{2} + 9729183112 T + 22\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{2} + 50956356444 T + 46\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{2} + 24244144344 T + 14\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} - 11485149570 T - 28\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{2} + 110800663168 T + 25\!\cdots\!96 \) Copy content Toggle raw display
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