Properties

Label 147.6.e.m
Level $147$
Weight $6$
Character orbit 147.e
Analytic conductor $23.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.5764215125\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
Defining polynomial: \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + (9 \beta_{2} - 9) q^{3} + (3 \beta_{3} + 17 \beta_{2} + 3 \beta_1 - 20) q^{4} + 36 \beta_{2} q^{5} + (9 \beta_{3} - 18) q^{6} + ( - 9 \beta_{3} - 120) q^{8} - 81 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + (9 \beta_{2} - 9) q^{3} + (3 \beta_{3} + 17 \beta_{2} + 3 \beta_1 - 20) q^{4} + 36 \beta_{2} q^{5} + (9 \beta_{3} - 18) q^{6} + ( - 9 \beta_{3} - 120) q^{8} - 81 \beta_{2} q^{9} + (36 \beta_{3} + 36 \beta_{2} + 36 \beta_1 - 72) q^{10} + (8 \beta_{3} + 236 \beta_{2} + 8 \beta_1 - 244) q^{11} + ( - 153 \beta_{2} - 27 \beta_1) q^{12} + (72 \beta_{3} - 684) q^{13} - 324 q^{15} + (847 \beta_{2} - 15 \beta_1) q^{16} + (216 \beta_{3} - 576 \beta_{2} + 216 \beta_1 + 360) q^{17} + ( - 81 \beta_{3} - 81 \beta_{2} - 81 \beta_1 + 162) q^{18} + (36 \beta_{2} - 288 \beta_1) q^{19} + (108 \beta_{3} - 720) q^{20} + (252 \beta_{3} - 872) q^{22} + (224 \beta_{2} + 56 \beta_1) q^{23} + (81 \beta_{3} - 1161 \beta_{2} + 81 \beta_1 + 1080) q^{24} + ( - 1829 \beta_{2} + 1829) q^{25} + ( - 4068 \beta_{2} - 756 \beta_1) q^{26} + 729 q^{27} + ( - 640 \beta_{3} + 3506) q^{29} + ( - 324 \beta_{2} - 324 \beta_1) q^{30} + (576 \beta_{3} - 5256 \beta_{2} + 576 \beta_1 + 4680) q^{31} + (529 \beta_{3} + 4255 \beta_{2} + 529 \beta_1 - 4784) q^{32} + ( - 2124 \beta_{2} - 72 \beta_1) q^{33} + ( - 144 \beta_{3} - 9648) q^{34} + ( - 243 \beta_{3} + 1620) q^{36} + ( - 6090 \beta_{2} + 1056 \beta_1) q^{37} + ( - 540 \beta_{3} - 13788 \beta_{2} - 540 \beta_1 + 14328) q^{38} + ( - 648 \beta_{3} - 5508 \beta_{2} - 648 \beta_1 + 6156) q^{39} + ( - 4644 \beta_{2} + 324 \beta_1) q^{40} + (216 \beta_{3} - 10584) q^{41} + (2400 \beta_{3} - 4332) q^{43} + ( - 5164 \beta_{2} - 868 \beta_1) q^{44} + ( - 2916 \beta_{2} + 2916) q^{45} + (336 \beta_{3} + 2912 \beta_{2} + 336 \beta_1 - 3248) q^{46} + (2232 \beta_{2} + 3456 \beta_1) q^{47} + ( - 135 \beta_{3} - 7488) q^{48} + ( - 1829 \beta_{3} + 3658) q^{50} + (5184 \beta_{2} - 1944 \beta_1) q^{51} + ( - 3276 \beta_{3} - 20772 \beta_{2} - 3276 \beta_1 + 24048) q^{52} + ( - 1184 \beta_{3} + 1702 \beta_{2} - 1184 \beta_1 - 518) q^{53} + (729 \beta_{2} + 729 \beta_1) q^{54} + (288 \beta_{3} - 8784) q^{55} + ( - 2592 \beta_{3} + 2268) q^{57} + (33586 \beta_{2} + 4146 \beta_1) q^{58} + ( - 864 \beta_{3} - 14436 \beta_{2} - 864 \beta_1 + 15300) q^{59} + ( - 972 \beta_{3} - 5508 \beta_{2} - 972 \beta_1 + 6480) q^{60} + ( - 10980 \beta_{2} + 4680 \beta_1) q^{61} + ( - 4104 \beta_{3} - 18288) q^{62} + (5793 \beta_{3} - 8336) q^{64} + ( - 22032 \beta_{2} - 2592 \beta_1) q^{65} + ( - 2268 \beta_{3} - 5580 \beta_{2} - 2268 \beta_1 + 7848) q^{66} + (6480 \beta_{3} - 13580 \beta_{2} + 6480 \beta_1 + 7100) q^{67} + ( - 21312 \beta_{2} - 2592 \beta_1) q^{68} + (504 \beta_{3} - 2520) q^{69} + ( - 2552 \beta_{3} - 44864) q^{71} + (10449 \beta_{2} - 729 \beta_1) q^{72} + (1872 \beta_{3} - 29232 \beta_{2} + 1872 \beta_1 + 27360) q^{73} + ( - 3978 \beta_{3} + 44598 \beta_{2} - 3978 \beta_1 - 40620) q^{74} + 16461 \beta_{2} q^{75} + ( - 5652 \beta_{3} + 46512) q^{76} + ( - 6804 \beta_{3} + 43416) q^{78} + (24808 \beta_{2} + 6480 \beta_1) q^{79} + ( - 540 \beta_{3} + 30492 \beta_{2} - 540 \beta_1 - 29952) q^{80} + (6561 \beta_{2} - 6561) q^{81} + ( - 20736 \beta_{2} - 10800 \beta_1) q^{82} + (9504 \beta_{3} + 30924) q^{83} + (7776 \beta_{3} + 12960) q^{85} + ( - 117132 \beta_{2} - 6732 \beta_1) q^{86} + (5760 \beta_{3} + 25794 \beta_{2} + 5760 \beta_1 - 31554) q^{87} + (1164 \beta_{3} - 26988 \beta_{2} + 1164 \beta_1 + 25824) q^{88} + (63432 \beta_{2} - 3672 \beta_1) q^{89} + ( - 2916 \beta_{3} + 5832) q^{90} + (1792 \beta_{3} - 13664) q^{92} + (47304 \beta_{2} - 5184 \beta_1) q^{93} + (9144 \beta_{3} + 168120 \beta_{2} + 9144 \beta_1 - 177264) q^{94} + ( - 10368 \beta_{3} + 1296 \beta_{2} - 10368 \beta_1 + 9072) q^{95} + ( - 38295 \beta_{2} - 4761 \beta_1) q^{96} + (19584 \beta_{3} - 27720) q^{97} + ( - 648 \beta_{3} + 19764) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 18 q^{3} - 37 q^{4} + 72 q^{5} - 54 q^{6} - 498 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} - 18 q^{3} - 37 q^{4} + 72 q^{5} - 54 q^{6} - 498 q^{8} - 162 q^{9} - 108 q^{10} - 480 q^{11} - 333 q^{12} - 2592 q^{13} - 1296 q^{15} + 1679 q^{16} + 936 q^{17} + 243 q^{18} - 216 q^{19} - 2664 q^{20} - 2984 q^{22} + 504 q^{23} + 2241 q^{24} + 3658 q^{25} - 8892 q^{26} + 2916 q^{27} + 12744 q^{29} - 972 q^{30} + 9936 q^{31} - 9039 q^{32} - 4320 q^{33} - 38880 q^{34} + 5994 q^{36} - 11124 q^{37} + 28116 q^{38} + 11664 q^{39} - 8964 q^{40} - 41904 q^{41} - 12528 q^{43} - 11196 q^{44} + 5832 q^{45} - 6160 q^{46} + 7920 q^{47} - 30222 q^{48} + 10974 q^{50} + 8424 q^{51} + 44820 q^{52} - 2220 q^{53} + 2187 q^{54} - 34560 q^{55} + 3888 q^{57} + 71318 q^{58} + 29736 q^{59} + 11988 q^{60} - 17280 q^{61} - 81360 q^{62} - 21758 q^{64} - 46656 q^{65} + 13428 q^{66} + 20680 q^{67} - 45216 q^{68} - 9072 q^{69} - 184560 q^{71} + 20169 q^{72} + 56592 q^{73} - 85218 q^{74} + 32922 q^{75} + 174744 q^{76} + 160056 q^{78} + 56096 q^{79} - 60444 q^{80} - 13122 q^{81} - 52272 q^{82} + 142704 q^{83} + 67392 q^{85} - 240996 q^{86} - 57348 q^{87} + 52812 q^{88} + 123192 q^{89} + 17496 q^{90} - 51072 q^{92} + 89424 q^{93} - 345384 q^{94} + 7776 q^{95} - 81351 q^{96} - 71712 q^{97} + 77760 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 49\nu^{2} - 49\nu + 2304 ) / 2352 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 97 ) / 49 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 48\beta_{2} + \beta _1 - 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 49\beta_{3} - 97 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−3.22311 5.58259i
3.72311 + 6.44862i
−3.22311 + 5.58259i
3.72311 6.44862i
−2.72311 4.71657i −4.50000 + 7.79423i 1.16933 2.02534i 18.0000 + 31.1769i 49.0160 0 −187.016 −40.5000 70.1481i 98.0320 169.796i
67.2 4.22311 + 7.31464i −4.50000 + 7.79423i −19.6693 + 34.0683i 18.0000 + 31.1769i −76.0160 0 −61.9840 −40.5000 70.1481i −152.032 + 263.327i
79.1 −2.72311 + 4.71657i −4.50000 7.79423i 1.16933 + 2.02534i 18.0000 31.1769i 49.0160 0 −187.016 −40.5000 + 70.1481i 98.0320 + 169.796i
79.2 4.22311 7.31464i −4.50000 7.79423i −19.6693 34.0683i 18.0000 31.1769i −76.0160 0 −61.9840 −40.5000 + 70.1481i −152.032 263.327i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.e.m 4
7.b odd 2 1 147.6.e.n 4
7.c even 3 1 147.6.a.j yes 2
7.c even 3 1 inner 147.6.e.m 4
7.d odd 6 1 147.6.a.h 2
7.d odd 6 1 147.6.e.n 4
21.g even 6 1 441.6.a.q 2
21.h odd 6 1 441.6.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.h 2 7.d odd 6 1
147.6.a.j yes 2 7.c even 3 1
147.6.e.m 4 1.a even 1 1 trivial
147.6.e.m 4 7.c even 3 1 inner
147.6.e.n 4 7.b odd 2 1
147.6.e.n 4 7.d odd 6 1
441.6.a.q 2 21.g even 6 1
441.6.a.r 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{4} - 3T_{2}^{3} + 55T_{2}^{2} + 138T_{2} + 2116 \) Copy content Toggle raw display
\( T_{5}^{2} - 36T_{5} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + 55 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 36 T + 1296)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 480 T^{3} + \cdots + 2971558144 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1296 T + 169776)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 936 T^{3} + \cdots + 4129544208384 \) Copy content Toggle raw display
$19$ \( T^{4} + 216 T^{3} + \cdots + 15923164467456 \) Copy content Toggle raw display
$23$ \( T^{4} - 504 T^{3} + \cdots + 7710244864 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6372 T - 9612604)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 9936 T^{3} + \cdots + 75218014900224 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 523012566603024 \) Copy content Toggle raw display
$41$ \( (T^{2} + 20952 T + 107495424)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6264 T - 268110576)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 7920 T^{3} + \cdots + 31\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{4} + 2220 T^{3} + \cdots + 44\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{4} - 29736 T^{3} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{4} + 17280 T^{3} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} - 20680 T^{3} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} + 92280 T + 1814661632)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 56592 T^{3} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{4} - 56096 T^{3} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{2} - 71352 T - 3085453296)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 123192 T^{3} + \cdots + 98\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{2} + 35856 T - 18184056768)^{2} \) Copy content Toggle raw display
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