Properties

Label 3.16.a.a
Level $3$
Weight $16$
Character orbit 3.a
Self dual yes
Analytic conductor $4.281$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.28080515300\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 234 q^{2} - 2187 q^{3} + 21988 q^{4} + 280710 q^{5} + 511758 q^{6} - 1373344 q^{7} + 2522520 q^{8} + 4782969 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 234 q^{2} - 2187 q^{3} + 21988 q^{4} + 280710 q^{5} + 511758 q^{6} - 1373344 q^{7} + 2522520 q^{8} + 4782969 q^{9} - 65686140 q^{10} + 34031052 q^{11} - 48087756 q^{12} + 384022262 q^{13} + 321362496 q^{14} - 613912770 q^{15} - 1310772464 q^{16} + 1259207586 q^{17} - 1119214746 q^{18} - 2499071020 q^{19} + 6172251480 q^{20} + 3003503328 q^{21} - 7963266168 q^{22} + 11284833672 q^{23} - 5516751240 q^{24} + 48280525975 q^{25} - 89861209308 q^{26} - 10460353203 q^{27} - 30197087872 q^{28} - 48413458530 q^{29} + 143655588180 q^{30} + 130547265752 q^{31} + 224062821216 q^{32} - 74425910724 q^{33} - 294654575124 q^{34} - 385511394240 q^{35} + 105167922372 q^{36} - 200223317554 q^{37} + 584782618680 q^{38} - 839856686994 q^{39} + 708096589200 q^{40} + 679141724202 q^{41} - 702819778752 q^{42} + 279482194892 q^{43} + 748274771376 q^{44} + 1342627227990 q^{45} - 2640651079248 q^{46} + 1520672832576 q^{47} + 2866659378768 q^{48} - 2861487767607 q^{49} - 11297643078150 q^{50} - 2753886990582 q^{51} + 8443881496856 q^{52} + 2646053822502 q^{53} + 2447722649502 q^{54} + 9552856606920 q^{55} - 3464287706880 q^{56} + 5465468320740 q^{57} + 11328749296020 q^{58} + 7399371294540 q^{59} - 13498713986760 q^{60} - 42659617819498 q^{61} - 30548060185968 q^{62} - 6568661778336 q^{63} - 9479308064192 q^{64} + 107798889166020 q^{65} + 17415663109416 q^{66} - 56408026065964 q^{67} + 27687456400968 q^{68} - 24679931240664 q^{69} + 90209666252160 q^{70} - 133149677299848 q^{71} + 12065134961880 q^{72} + 105603350884922 q^{73} + 46852256307636 q^{74} - 105589510307325 q^{75} - 54949573587760 q^{76} - 46736341077888 q^{77} + 196526464756596 q^{78} - 55665674361880 q^{79} - 367946938369440 q^{80} + 22876792454961 q^{81} - 158919163463268 q^{82} + 378077412997332 q^{83} + 66041031176064 q^{84} + 353472161466060 q^{85} - 65398833604728 q^{86} + 105880233805110 q^{87} + 85844009291040 q^{88} + 219315065897610 q^{89} - 314174771349660 q^{90} - 527394669384128 q^{91} + 248130922779936 q^{92} - 285506870199624 q^{93} - 355837442822784 q^{94} - 701514226024200 q^{95} - 490025389999392 q^{96} + 703322682162626 q^{97} + 669588137620038 q^{98} + 162769466753388 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
0
−234.000 −2187.00 21988.0 280710. 511758. −1.37334e6 2.52252e6 4.78297e6 −6.56861e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

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 3.16.a.a 1
3.b odd 2 1 9.16.a.d 1
4.b odd 2 1 48.16.a.g 1
5.b even 2 1 75.16.a.b 1
5.c odd 4 2 75.16.b.a 2
7.b odd 2 1 147.16.a.a 1
12.b even 2 1 144.16.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.16.a.a 1 1.a even 1 1 trivial
9.16.a.d 1 3.b odd 2 1
48.16.a.g 1 4.b odd 2 1
75.16.a.b 1 5.b even 2 1
75.16.b.a 2 5.c odd 4 2
144.16.a.b 1 12.b even 2 1
147.16.a.a 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 234 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 234 \) Copy content Toggle raw display
$3$ \( T + 2187 \) Copy content Toggle raw display
$5$ \( T - 280710 \) Copy content Toggle raw display
$7$ \( T + 1373344 \) Copy content Toggle raw display
$11$ \( T - 34031052 \) Copy content Toggle raw display
$13$ \( T - 384022262 \) Copy content Toggle raw display
$17$ \( T - 1259207586 \) Copy content Toggle raw display
$19$ \( T + 2499071020 \) Copy content Toggle raw display
$23$ \( T - 11284833672 \) Copy content Toggle raw display
$29$ \( T + 48413458530 \) Copy content Toggle raw display
$31$ \( T - 130547265752 \) Copy content Toggle raw display
$37$ \( T + 200223317554 \) Copy content Toggle raw display
$41$ \( T - 679141724202 \) Copy content Toggle raw display
$43$ \( T - 279482194892 \) Copy content Toggle raw display
$47$ \( T - 1520672832576 \) Copy content Toggle raw display
$53$ \( T - 2646053822502 \) Copy content Toggle raw display
$59$ \( T - 7399371294540 \) Copy content Toggle raw display
$61$ \( T + 42659617819498 \) Copy content Toggle raw display
$67$ \( T + 56408026065964 \) Copy content Toggle raw display
$71$ \( T + 133149677299848 \) Copy content Toggle raw display
$73$ \( T - 105603350884922 \) Copy content Toggle raw display
$79$ \( T + 55665674361880 \) Copy content Toggle raw display
$83$ \( T - 378077412997332 \) Copy content Toggle raw display
$89$ \( T - 219315065897610 \) Copy content Toggle raw display
$97$ \( T - 703322682162626 \) Copy content Toggle raw display
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