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Properties

Label 152.4.v.b
Level $152$
Weight $4$
Character orbit 152.v
Analytic conductor $8.968$
Analytic rank $0$
Dimension $336$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [152,4,Mod(3,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 13]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 152.v (of order \(18\), 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: \(8.96829032087\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(56\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 6 q^{2} - 42 q^{3} + 12 q^{4} + 36 q^{6} - 9 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 6 q^{2} - 42 q^{3} + 12 q^{4} + 36 q^{6} - 9 q^{8} + 126 q^{9} - 105 q^{10} - 6 q^{11} - 9 q^{12} - 129 q^{14} + 192 q^{16} - 12 q^{17} - 12 q^{19} - 582 q^{20} + 1218 q^{22} - 300 q^{24} - 12 q^{25} - 411 q^{26} + 1692 q^{27} - 1152 q^{28} + 516 q^{30} - 51 q^{32} + 120 q^{33} - 1548 q^{34} - 762 q^{35} + 261 q^{36} - 3222 q^{38} + 1056 q^{40} + 1614 q^{41} - 3027 q^{42} - 12 q^{43} - 255 q^{44} + 1566 q^{46} - 7089 q^{48} + 8814 q^{49} + 3204 q^{50} - 498 q^{51} + 207 q^{52} - 999 q^{54} - 12 q^{57} + 4092 q^{58} - 2550 q^{59} + 4476 q^{60} + 3336 q^{62} - 2697 q^{64} - 18 q^{65} + 6228 q^{66} + 198 q^{67} + 2532 q^{68} - 5031 q^{70} - 14034 q^{72} - 3240 q^{73} + 2343 q^{74} - 6558 q^{76} - 6639 q^{78} - 6885 q^{80} + 10632 q^{81} - 2451 q^{82} - 6 q^{83} - 333 q^{84} + 3792 q^{86} + 5823 q^{88} - 12 q^{89} + 1866 q^{90} - 2070 q^{91} + 7656 q^{92} + 18942 q^{96} - 5742 q^{97} - 13749 q^{98} - 5580 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.82600 + 0.117147i 1.03784 + 2.85145i 7.97255 − 0.662116i 12.0023 + 2.11633i −3.26698 − 7.93661i 15.4940 − 8.94547i −22.4529 + 2.80510i 13.6296 − 11.4366i −34.1665 − 4.57472i
3.2 −2.80905 − 0.330514i 2.78543 + 7.65290i 7.78152 + 1.85686i −7.31958 − 1.29064i −5.29502 − 22.4180i −1.31296 + 0.758035i −21.2450 − 7.78791i −30.1251 + 25.2780i 20.1345 + 6.04469i
3.3 −2.80274 + 0.380348i −0.0839815 − 0.230737i 7.71067 − 2.13203i 5.29288 + 0.933278i 0.323139 + 0.614754i −24.4207 + 14.0993i −20.8001 + 8.90826i 20.6370 − 17.3165i −15.1895 − 0.602595i
3.4 −2.78037 − 0.519197i −1.46518 − 4.02554i 7.46087 + 2.88712i −1.25209 − 0.220777i 1.98368 + 11.9532i 9.31103 − 5.37572i −19.2450 − 11.9009i 6.62499 − 5.55903i 3.36663 + 1.26392i
3.5 −2.75470 + 0.641599i −1.70865 − 4.69447i 7.17670 − 3.53482i −16.1264 − 2.84352i 7.71878 + 11.8356i 27.9990 − 16.1652i −17.5017 + 14.3419i 1.56460 − 1.31286i 46.2477 − 2.51366i
3.6 −2.65595 + 0.972576i −3.41377 − 9.37926i 6.10819 − 5.16624i −5.36584 − 0.946143i 18.1889 + 21.5907i −21.7773 + 12.5731i −11.1985 + 19.6620i −55.6335 + 46.6820i 15.1716 − 2.70578i
3.7 −2.59925 − 1.11530i 0.142296 + 0.390954i 5.51222 + 5.79788i −20.2800 − 3.57591i 0.0661683 − 1.17489i −16.1140 + 9.30340i −7.86127 − 21.2179i 20.5506 − 17.2440i 48.7246 + 31.9129i
3.8 −2.59333 − 1.12900i −2.83786 − 7.79695i 5.45072 + 5.85574i 1.42267 + 0.250855i −1.44326 + 23.4240i 4.55142 − 2.62776i −7.52437 − 21.3397i −32.0558 + 26.8980i −3.40624 − 2.25675i
3.9 −2.46139 + 1.39340i 3.38339 + 9.29578i 4.11687 − 6.85940i 15.3871 + 2.71316i −21.2806 − 18.1661i −12.1479 + 7.01358i −0.575331 + 22.6201i −54.2810 + 45.5472i −41.6541 + 14.7622i
3.10 −2.40608 + 1.48687i −1.85247 − 5.08962i 3.57842 − 7.15506i 17.6265 + 3.10803i 12.0248 + 9.49162i 8.07536 − 4.66231i 2.02872 + 22.5363i −1.78935 + 1.50144i −47.0320 + 18.7302i
3.11 −2.37580 + 1.53478i 1.49181 + 4.09871i 3.28887 − 7.29269i −9.95185 − 1.75478i −9.83487 − 7.44811i 20.6799 − 11.9396i 3.37899 + 22.3737i 6.10930 − 5.12631i 26.3368 − 11.1049i
3.12 −2.34056 − 1.58802i 2.24927 + 6.17982i 2.95641 + 7.43368i 1.00572 + 0.177336i 4.54912 − 18.0361i 16.9939 − 9.81142i 4.88518 − 22.0938i −12.4478 + 10.4449i −2.07234 − 2.01217i
3.13 −2.33650 − 1.59398i −1.53541 − 4.21851i 2.91843 + 7.44867i 9.26153 + 1.63306i −3.13675 + 12.3039i −18.2291 + 10.5246i 5.05415 − 22.0557i 5.24490 − 4.40099i −19.0365 − 18.5784i
3.14 −2.33394 + 1.59773i 1.26287 + 3.46971i 2.89454 − 7.45799i −10.3094 − 1.81783i −8.49112 − 6.08037i −13.9249 + 8.03952i 5.16016 + 22.0312i 10.2391 − 8.59165i 26.9660 − 12.2290i
3.15 −2.20806 − 1.76762i 1.06368 + 2.92245i 1.75101 + 7.80602i 20.8129 + 3.66988i 2.81712 − 8.33312i −3.59923 + 2.07802i 9.93177 − 20.3313i 13.2739 − 11.1381i −39.4691 − 44.8927i
3.16 −1.66083 − 2.28947i 0.347584 + 0.954979i −2.48332 + 7.60481i −8.45010 − 1.48998i 1.60912 − 2.38184i 14.6333 − 8.44854i 21.5353 − 6.94479i 19.8920 − 16.6914i 10.6229 + 21.8208i
3.17 −1.54216 − 2.37102i 2.73860 + 7.52425i −3.24350 + 7.31298i −2.41626 − 0.426052i 13.6168 − 18.0969i −24.4488 + 14.1155i 22.3412 − 3.58735i −28.4311 + 23.8566i 2.71608 + 6.38605i
3.18 −1.39959 − 2.45788i −2.30554 − 6.33443i −4.08232 + 6.88002i 11.5991 + 2.04524i −12.3424 + 14.5323i 28.6907 − 16.5646i 22.6238 + 0.404658i −14.1263 + 11.8533i −11.2070 − 31.3717i
3.19 −1.38290 − 2.46730i −2.76797 − 7.60493i −4.17515 + 6.82408i −15.4492 − 2.72410i −14.9358 + 17.3463i −5.72949 + 3.30792i 22.6109 + 0.864300i −29.4902 + 24.7452i 14.6435 + 41.8849i
3.20 −1.16817 + 2.57592i 1.26287 + 3.46971i −5.27076 − 6.01823i 10.3094 + 1.81783i −10.4130 − 0.800152i 13.9249 − 8.03952i 21.6596 − 6.54676i 10.2391 − 8.59165i −16.7258 + 24.4328i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.f odd 18 1 inner
152.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.4.v.b ✓ 336
8.d odd 2 1 inner 152.4.v.b ✓ 336
19.f odd 18 1 inner 152.4.v.b ✓ 336
152.v even 18 1 inner 152.4.v.b ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.v.b ✓ 336 1.a even 1 1 trivial
152.4.v.b ✓ 336 8.d odd 2 1 inner
152.4.v.b ✓ 336 19.f odd 18 1 inner
152.4.v.b ✓ 336 152.v even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{168} + 21 T_{3}^{167} + 189 T_{3}^{166} + 600 T_{3}^{165} - 8331 T_{3}^{164} + \cdots + 17\!\cdots\!25 \) acting on \(S_{4}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display