Properties

Label 3.16.a.b
Level $3$
Weight $16$
Character orbit 3.a
Self dual yes
Analytic conductor $4.281$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 72 q^{2} + 2187 q^{3} - 27584 q^{4} - 221490 q^{5} - 157464 q^{6} - 2149000 q^{7} + 4345344 q^{8} + 4782969 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 72 q^{2} + 2187 q^{3} - 27584 q^{4} - 221490 q^{5} - 157464 q^{6} - 2149000 q^{7} + 4345344 q^{8} + 4782969 q^{9} + 15947280 q^{10} + 37169316 q^{11} - 60326208 q^{12} - 279974266 q^{13} + 154728000 q^{14} - 484398630 q^{15} + 591007744 q^{16} + 2492912754 q^{17} - 344373768 q^{18} - 4669782244 q^{19} + 6109580160 q^{20} - 4699863000 q^{21} - 2676190752 q^{22} - 18467933400 q^{23} + 9503267328 q^{24} + 18540241975 q^{25} + 20158147152 q^{26} + 10460353203 q^{27} + 59278016000 q^{28} - 115953449418 q^{29} + 34876701360 q^{30} - 56187023200 q^{31} - 184940789760 q^{32} + 81289294092 q^{33} - 179489718288 q^{34} + 475982010000 q^{35} - 131933416896 q^{36} + 614764926830 q^{37} + 336224321568 q^{38} - 612303719742 q^{39} - 962450242560 q^{40} + 549859792410 q^{41} + 338390136000 q^{42} - 982884444028 q^{43} - 1025278412544 q^{44} - 1059379803810 q^{45} + 1329691204800 q^{46} + 2076144322896 q^{47} + 1292533936128 q^{48} - 129360509943 q^{49} - 1334897422200 q^{50} + 5452000192998 q^{51} + 7722810153344 q^{52} - 12048378188130 q^{53} - 753145430616 q^{54} - 8232631800840 q^{55} - 9338144256000 q^{56} - 10212813767628 q^{57} + 8348648358096 q^{58} + 23087905758324 q^{59} + 13361651809920 q^{60} - 8505809142442 q^{61} + 4045465670400 q^{62} - 10278600381000 q^{63} - 6050404892672 q^{64} + 62011500176340 q^{65} - 5852829174624 q^{66} - 12331010771476 q^{67} - 68764505406336 q^{68} - 40389370345800 q^{69} - 34270704720000 q^{70} + 58989192692472 q^{71} + 20783645646336 q^{72} - 5609828808070 q^{73} - 44263074731760 q^{74} + 40547509199325 q^{75} + 128811273418496 q^{76} - 79876860084000 q^{77} + 44085867821424 q^{78} + 159918683826800 q^{79} - 130902305218560 q^{80} + 22876792454961 q^{81} - 39589905053520 q^{82} + 57675894342876 q^{83} + 129641020992000 q^{84} - 552155245883460 q^{85} + 70767679970016 q^{86} - 253590193877166 q^{87} + 161513464264704 q^{88} - 362287610413974 q^{89} + 76275345874320 q^{90} + 601664697634000 q^{91} + 509419474905600 q^{92} - 122881019738400 q^{93} - 149482391248512 q^{94} + 10\!\cdots\!60 q^{95}+ \cdots + 177779686179204 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
−72.0000 2187.00 −27584.0 −221490. −157464. −2.14900e6 4.34534e6 4.78297e6 1.59473e7
\(n\): 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.b 1
3.b odd 2 1 9.16.a.c 1
4.b odd 2 1 48.16.a.a 1
5.b even 2 1 75.16.a.a 1
5.c odd 4 2 75.16.b.b 2
7.b odd 2 1 147.16.a.b 1
12.b even 2 1 144.16.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.16.a.b 1 1.a even 1 1 trivial
9.16.a.c 1 3.b odd 2 1
48.16.a.a 1 4.b odd 2 1
75.16.a.a 1 5.b even 2 1
75.16.b.b 2 5.c odd 4 2
144.16.a.l 1 12.b even 2 1
147.16.a.b 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 72 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 72 \) Copy content Toggle raw display
$3$ \( T - 2187 \) Copy content Toggle raw display
$5$ \( T + 221490 \) Copy content Toggle raw display
$7$ \( T + 2149000 \) Copy content Toggle raw display
$11$ \( T - 37169316 \) Copy content Toggle raw display
$13$ \( T + 279974266 \) Copy content Toggle raw display
$17$ \( T - 2492912754 \) Copy content Toggle raw display
$19$ \( T + 4669782244 \) Copy content Toggle raw display
$23$ \( T + 18467933400 \) Copy content Toggle raw display
$29$ \( T + 115953449418 \) Copy content Toggle raw display
$31$ \( T + 56187023200 \) Copy content Toggle raw display
$37$ \( T - 614764926830 \) Copy content Toggle raw display
$41$ \( T - 549859792410 \) Copy content Toggle raw display
$43$ \( T + 982884444028 \) Copy content Toggle raw display
$47$ \( T - 2076144322896 \) Copy content Toggle raw display
$53$ \( T + 12048378188130 \) Copy content Toggle raw display
$59$ \( T - 23087905758324 \) Copy content Toggle raw display
$61$ \( T + 8505809142442 \) Copy content Toggle raw display
$67$ \( T + 12331010771476 \) Copy content Toggle raw display
$71$ \( T - 58989192692472 \) Copy content Toggle raw display
$73$ \( T + 5609828808070 \) Copy content Toggle raw display
$79$ \( T - 159918683826800 \) Copy content Toggle raw display
$83$ \( T - 57675894342876 \) Copy content Toggle raw display
$89$ \( T + 362287610413974 \) Copy content Toggle raw display
$97$ \( T + 539786645144926 \) Copy content Toggle raw display
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