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Properties

Label 147.3.l.a
Level $147$
Weight $3$
Character orbit 147.l
Analytic conductor $4.005$
Analytic rank $0$
Dimension $216$
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] = [147,3,Mod(8,147)]
 
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, 12]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.8");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 147.l (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: \(4.00545988610\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9} - 42 q^{10} - 20 q^{12} + 22 q^{13} - 17 q^{15} - 170 q^{16} - 86 q^{18} - 40 q^{19} - 21 q^{21} - 118 q^{22} + 119 q^{24} + 174 q^{25} + 88 q^{27} - 168 q^{28} + 36 q^{30} - 164 q^{31} - 35 q^{33} - 294 q^{34} + 307 q^{36} + 8 q^{37} - 61 q^{39} - 42 q^{40} - 133 q^{42} + 138 q^{43} - 336 q^{45} - 46 q^{46} - 52 q^{48} - 14 q^{49} + 111 q^{51} + 550 q^{52} + 147 q^{54} + 126 q^{55} - 363 q^{57} + 630 q^{58} + 353 q^{60} + 86 q^{61} + 21 q^{63} + 146 q^{64} + 105 q^{66} + 100 q^{67} - 7 q^{69} - 532 q^{70} - 167 q^{72} + 18 q^{73} + 1107 q^{75} - 762 q^{76} - 699 q^{78} - 272 q^{79} - 265 q^{81} + 504 q^{82} - 1834 q^{84} - 650 q^{85} - 595 q^{87} - 242 q^{88} - 1323 q^{90} + 126 q^{91} + 233 q^{93} + 1358 q^{94} - 882 q^{96} - 20 q^{97} - 332 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −3.00834 + 2.39907i −2.98897 + 0.256973i 2.40448 − 10.5347i 3.25362 + 6.75620i 8.37534 − 7.94382i 3.12235 + 6.26506i 11.3620 + 23.5934i 8.86793 − 1.53617i −25.9966 − 12.5193i
8.2 −2.90182 + 2.31413i −1.61498 − 2.52821i 2.17531 − 9.53066i −3.38013 − 7.01891i 10.5370 + 3.59914i −4.23095 − 5.57665i 9.30122 + 19.3142i −3.78367 + 8.16602i 26.0512 + 12.5456i
8.3 −2.88499 + 2.30070i 1.88790 + 2.33149i 2.13985 − 9.37531i −2.07438 − 4.30749i −10.8106 − 2.38284i 6.99908 + 0.113215i 8.99215 + 18.6724i −1.87169 + 8.80323i 15.8948 + 7.65454i
8.4 −2.59560 + 2.06992i 2.01782 − 2.22000i 1.56247 − 6.84563i −0.342344 − 0.710886i −0.642229 + 9.93894i −1.82701 + 6.75737i 4.35256 + 9.03819i −0.856797 − 8.95912i 2.36006 + 1.13655i
8.5 −2.46298 + 1.96416i −0.664950 + 2.92538i 1.31826 − 5.77568i 1.01622 + 2.11020i −4.10816 − 8.51123i −5.04359 − 4.85409i 2.63011 + 5.46149i −8.11568 − 3.89046i −6.64769 − 3.20136i
8.6 −2.29766 + 1.83232i 2.90685 + 0.741768i 1.03175 − 4.52040i 0.779870 + 1.61942i −8.03811 + 3.62196i −6.58273 + 2.38068i 0.811795 + 1.68571i 7.89956 + 4.31242i −4.75917 − 2.29189i
8.7 −2.25505 + 1.79834i 0.366861 − 2.97748i 0.961132 − 4.21100i 3.58165 + 7.43737i 4.52725 + 7.37412i −0.00666399 − 7.00000i 0.399578 + 0.829731i −8.73083 − 2.18464i −21.4518 − 10.3306i
8.8 −1.89369 + 1.51017i −2.37508 + 1.83275i 0.415378 − 1.81989i −2.09584 − 4.35206i 1.72992 − 7.05745i 6.79676 − 1.67455i −2.24194 − 4.65543i 2.28205 − 8.70587i 10.5412 + 5.07639i
8.9 −1.86964 + 1.49099i −1.60663 − 2.53352i 0.382431 − 1.67554i 0.269795 + 0.560236i 6.78129 + 2.34131i 6.21651 + 3.21792i −2.36709 − 4.91532i −3.83747 + 8.14087i −1.33973 − 0.645180i
8.10 −1.70385 + 1.35878i −2.98293 − 0.319571i 0.166754 − 0.730599i −1.36590 − 2.83633i 5.51670 − 3.50864i −6.10925 + 3.41717i −3.07367 − 6.38254i 8.79575 + 1.90652i 6.18123 + 2.97672i
8.11 −1.45285 + 1.15861i 2.57601 + 1.53758i −0.121689 + 0.533152i 3.36170 + 6.98065i −5.52401 + 0.750718i 6.92701 − 1.00823i −3.66600 − 7.61252i 4.27169 + 7.92166i −12.9719 − 6.24692i
8.12 −1.35034 + 1.07686i 2.84803 − 0.942709i −0.226298 + 0.991478i −2.39688 − 4.97718i −2.83064 + 4.33990i 1.24252 − 6.88884i −3.75962 − 7.80693i 7.22260 − 5.36974i 8.59630 + 4.13976i
8.13 −1.07234 + 0.855164i 1.25138 + 2.72655i −0.471473 + 2.06566i −3.74354 − 7.77355i −3.67355 − 1.85366i −2.26460 + 6.62356i −3.64131 − 7.56127i −5.86812 + 6.82387i 10.6620 + 5.13455i
8.14 −0.907101 + 0.723389i −1.11078 + 2.78678i −0.590543 + 2.58734i 2.37634 + 4.93452i −1.00834 − 3.33142i −1.20041 + 6.89630i −3.34958 − 6.95548i −6.53233 − 6.19101i −5.72516 − 2.75709i
8.15 −0.603355 + 0.481160i −2.99331 + 0.200270i −0.757561 + 3.31909i 3.33510 + 6.92541i 1.70967 − 1.56109i −3.51861 − 6.05139i −2.47928 − 5.14828i 8.91978 − 1.19894i −5.34448 − 2.57377i
8.16 −0.497631 + 0.396848i 1.93233 − 2.29480i −0.799935 + 3.50474i 1.71979 + 3.57118i −0.0509003 + 1.90880i 3.29295 + 6.17709i −2.09744 − 4.35537i −1.53222 − 8.86861i −2.27303 − 1.09463i
8.17 −0.488839 + 0.389836i 0.192295 − 2.99383i −0.803092 + 3.51858i −1.15070 − 2.38946i 1.07310 + 1.53847i −6.83519 − 1.51004i −2.06423 − 4.28641i −8.92605 − 1.15140i 1.49400 + 0.719474i
8.18 −0.200107 + 0.159580i −2.28574 − 1.94303i −0.875507 + 3.83585i −0.404578 − 0.840116i 0.767464 + 0.0240552i 5.69424 − 4.07132i −0.881135 − 1.82969i 1.44925 + 8.88255i 0.215025 + 0.103551i
8.19 0.200107 − 0.159580i 0.0939851 + 2.99853i −0.875507 + 3.83585i 0.404578 + 0.840116i 0.497313 + 0.585029i 5.69424 − 4.07132i 0.881135 + 1.82969i −8.98233 + 0.563634i 0.215025 + 0.103551i
8.20 0.488839 − 0.389836i 2.46056 + 1.71628i −0.803092 + 3.51858i 1.15070 + 2.38946i 1.87189 − 0.120232i −6.83519 − 1.51004i 2.06423 + 4.28641i 3.10876 + 8.44604i 1.49400 + 0.719474i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.e even 7 1 inner
147.l odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.3.l.a ✓ 216
3.b odd 2 1 inner 147.3.l.a ✓ 216
49.e even 7 1 inner 147.3.l.a ✓ 216
147.l odd 14 1 inner 147.3.l.a ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.3.l.a ✓ 216 1.a even 1 1 trivial
147.3.l.a ✓ 216 3.b odd 2 1 inner
147.3.l.a ✓ 216 49.e even 7 1 inner
147.3.l.a ✓ 216 147.l odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(147, [\chi])\).