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Properties

Label 147.4.k.b
Level $147$
Weight $4$
Character orbit 147.k
Analytic conductor $8.673$
Analytic rank $0$
Dimension $312$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,4,Mod(20,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 13]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.20");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.k (of order \(14\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(52\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 7 q^{3} + 214 q^{4} - 35 q^{6} - 42 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 312 q - 7 q^{3} + 214 q^{4} - 35 q^{6} - 42 q^{7} + 11 q^{9} - 14 q^{10} + 378 q^{12} - 14 q^{13} + 133 q^{15} - 682 q^{16} - 210 q^{18} + 77 q^{21} + 126 q^{22} - 7 q^{24} - 1158 q^{25} + 35 q^{27} + 112 q^{28} + 616 q^{30} - 7 q^{33} + 490 q^{34} - 409 q^{36} + 1982 q^{37} - 1959 q^{39} + 490 q^{40} - 2471 q^{42} - 34 q^{43} - 1771 q^{45} - 2982 q^{46} - 378 q^{49} + 1757 q^{51} + 6538 q^{52} - 7 q^{54} + 4648 q^{55} - 2498 q^{57} + 2730 q^{58} - 3759 q^{60} + 2674 q^{61} - 4459 q^{63} + 1026 q^{64} - 5523 q^{66} - 720 q^{67} + 5453 q^{69} + 1876 q^{70} + 1617 q^{72} - 14 q^{73} + 4151 q^{75} - 7686 q^{76} - 2247 q^{78} + 1044 q^{79} + 11239 q^{81} - 6104 q^{82} + 7126 q^{84} - 2702 q^{85} + 721 q^{87} + 686 q^{88} + 2387 q^{90} - 434 q^{91} - 8098 q^{93} + 4942 q^{94} - 5446 q^{96} - 11018 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −2.38006 + 4.94225i 5.19571 − 0.0681019i −13.7732 − 17.2710i 0.369752 + 1.61999i −12.0295 + 25.8405i 17.2266 + 6.80040i 75.3552 − 17.1993i 26.9907 − 0.707675i −8.88641 − 2.02826i
20.2 −2.29550 + 4.76666i −5.15705 + 0.636295i −12.4638 − 15.6291i 3.19104 + 13.9808i 8.80501 − 26.0425i 14.9665 − 10.9089i 61.8455 − 14.1158i 26.1903 − 6.56280i −73.9669 − 16.8825i
20.3 −2.25519 + 4.68295i −4.34587 + 2.84841i −11.8562 − 14.8672i −3.15654 − 13.8297i −3.53820 − 26.7752i −17.6412 + 5.63804i 55.8213 − 12.7409i 10.7731 − 24.7576i 71.8824 + 16.4067i
20.4 −2.22877 + 4.62810i −0.293111 − 5.18788i −11.4639 − 14.3753i 3.32017 + 14.5466i 24.6633 + 10.2061i −17.6411 + 5.63851i 52.0168 − 11.8725i −26.8282 + 3.04125i −74.7230 − 17.0550i
20.5 −2.09179 + 4.34365i 3.01551 + 4.23163i −9.50380 − 11.9174i −1.95252 − 8.55456i −24.6886 + 4.24661i −7.40388 − 16.9759i 34.0432 − 7.77015i −8.81345 + 25.5210i 41.2423 + 9.41329i
20.6 −2.05184 + 4.26070i −1.01818 − 5.09542i −8.95555 − 11.2299i −2.94075 − 12.8843i 23.7992 + 6.11683i 10.2189 + 15.4458i 29.3390 − 6.69644i −24.9266 + 10.3761i 60.9299 + 13.9069i
20.7 −2.04574 + 4.24802i 0.360429 + 5.18364i −8.87271 − 11.1260i 2.25014 + 9.85850i −22.7575 − 9.07326i −1.77366 + 18.4351i 28.6410 − 6.53712i −26.7402 + 3.73667i −46.4823 − 10.6093i
20.8 −1.95192 + 4.05321i −4.01826 − 3.29447i −7.63060 − 9.56847i −2.05577 − 9.00692i 21.1965 − 9.85632i 2.44738 − 18.3578i 18.5899 − 4.24302i 5.29290 + 26.4761i 40.5197 + 9.24835i
20.9 −1.69839 + 3.52674i 3.12415 − 4.15207i −4.56543 − 5.72486i 0.507251 + 2.22241i 9.33723 + 18.0699i 11.2807 − 14.6883i −2.58598 + 0.590233i −7.47936 − 25.9434i −8.69937 − 1.98558i
20.10 −1.67762 + 3.48362i 5.17717 − 0.443709i −4.33327 − 5.43375i 2.43675 + 10.6761i −7.13963 + 18.7797i −18.4226 − 1.89925i −3.95798 + 0.903384i 26.6062 − 4.59432i −41.2793 − 9.42174i
20.11 −1.62625 + 3.37694i 4.91662 − 1.68132i −3.77114 − 4.72886i −4.21623 − 18.4725i −2.31792 + 19.3374i −9.52544 + 15.8829i −7.13132 + 1.62768i 21.3463 − 16.5329i 69.2373 + 15.8030i
20.12 −1.52621 + 3.16921i −3.34207 + 3.97876i −2.72667 − 3.41913i −2.86471 − 12.5511i −7.50883 − 16.6642i 17.7226 + 5.37663i −12.4375 + 2.83878i −4.66110 − 26.5946i 44.1492 + 10.0768i
20.13 −1.50215 + 3.11924i −1.49628 + 4.97606i −2.48529 − 3.11646i 3.00534 + 13.1673i −13.2739 − 12.1420i 0.425776 − 18.5154i −13.5481 + 3.09226i −22.5223 − 14.8912i −45.5863 − 10.4048i
20.14 −1.39113 + 2.88872i −4.37695 − 2.80041i −1.42152 − 1.78253i 0.110890 + 0.485842i 14.1785 − 8.74802i −18.3508 + 2.49934i −17.8800 + 4.08100i 11.3154 + 24.5145i −1.55772 − 0.355540i
20.15 −1.38337 + 2.87259i −5.15946 − 0.616409i −1.35014 − 1.69302i 1.70320 + 7.46219i 8.90811 − 13.9683i 7.53996 + 16.9159i −18.1361 + 4.13944i 26.2401 + 6.36068i −23.7919 − 5.43035i
20.16 −1.22013 + 2.53363i 3.49682 + 3.84347i 0.0573723 + 0.0719426i −3.23518 − 14.1742i −14.0045 + 4.17010i 18.4346 − 1.77883i −22.1851 + 5.06361i −2.54451 + 26.8798i 39.8596 + 9.09768i
20.17 −1.06488 + 2.21125i 4.43379 + 2.70952i 1.23226 + 1.54521i 3.75754 + 16.4628i −10.7129 + 6.91890i 18.1401 + 3.73318i −23.8712 + 5.44845i 12.3170 + 24.0269i −40.4048 − 9.22213i
20.18 −0.902464 + 1.87399i 2.06522 − 4.76811i 2.29054 + 2.87224i 2.14483 + 9.39711i 7.07158 + 8.17324i 4.61331 + 17.9365i −23.6722 + 5.40303i −18.4697 − 19.6944i −19.5457 − 4.46117i
20.19 −0.843275 + 1.75108i −4.28003 + 2.94641i 2.63276 + 3.30137i −0.732320 − 3.20850i −1.55016 − 9.97930i −12.2831 − 13.8610i −23.1597 + 5.28604i 9.63730 − 25.2215i 6.23589 + 1.42330i
20.20 −0.789224 + 1.63884i 1.00679 − 5.09768i 2.92499 + 3.66782i −2.97968 − 13.0548i 7.55972 + 5.67318i −14.2410 − 11.8404i −22.5064 + 5.13695i −24.9728 − 10.2646i 23.7464 + 5.41996i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.f odd 14 1 inner
147.k even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.k.b ✓ 312
3.b odd 2 1 inner 147.4.k.b ✓ 312
49.f odd 14 1 inner 147.4.k.b ✓ 312
147.k even 14 1 inner 147.4.k.b ✓ 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.k.b ✓ 312 1.a even 1 1 trivial
147.4.k.b ✓ 312 3.b odd 2 1 inner
147.4.k.b ✓ 312 49.f odd 14 1 inner
147.4.k.b ✓ 312 147.k even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{312} - 315 T_{2}^{310} + 52815 T_{2}^{308} - 6266799 T_{2}^{306} + 590472659 T_{2}^{304} + \cdots + 28\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display