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Properties

Label 349.3.f.a
Level $349$
Weight $3$
Character orbit 349.f
Analytic conductor $9.510$
Analytic rank $0$
Dimension $228$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [349,3,Mod(24,349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.24");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 349.f (of order \(12\), degree \(4\), 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: \(9.50956122617\)
Analytic rank: \(0\)
Dimension: \(228\)
Relative dimension: \(57\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 228 q - 6 q^{2} - 12 q^{3} + 6 q^{4} - 38 q^{6} + 22 q^{7} - 12 q^{8} + 322 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 228 q - 6 q^{2} - 12 q^{3} + 6 q^{4} - 38 q^{6} + 22 q^{7} - 12 q^{8} + 322 q^{9} + 58 q^{10} - 14 q^{11} - 38 q^{12} + 24 q^{13} + 28 q^{14} - 38 q^{15} + 434 q^{16} + 106 q^{18} - 82 q^{19} - 60 q^{20} + 40 q^{21} - 42 q^{22} + 100 q^{23} - 90 q^{24} + 386 q^{25} + 36 q^{26} - 340 q^{28} + 168 q^{29} + 96 q^{30} - 8 q^{31} + 52 q^{32} + 76 q^{33} - 218 q^{34} + 138 q^{35} - 8 q^{38} - 118 q^{39} - 28 q^{40} - 300 q^{41} + 174 q^{42} - 184 q^{43} + 394 q^{44} - 200 q^{46} - 314 q^{47} - 180 q^{49} + 172 q^{50} + 112 q^{51} - 72 q^{52} - 238 q^{53} + 180 q^{54} - 142 q^{55} - 1020 q^{56} - 684 q^{57} - 760 q^{58} + 50 q^{59} - 72 q^{61} + 308 q^{62} + 166 q^{63} - 56 q^{65} + 256 q^{66} - 264 q^{67} - 202 q^{68} - 126 q^{71} - 376 q^{72} - 102 q^{73} + 356 q^{74} + 678 q^{76} + 170 q^{77} - 468 q^{78} + 80 q^{79} - 666 q^{81} + 144 q^{82} - 78 q^{83} + 316 q^{84} + 344 q^{85} + 56 q^{87} - 72 q^{88} - 408 q^{89} + 122 q^{90} + 642 q^{91} - 774 q^{93} - 106 q^{94} + 666 q^{95} - 104 q^{96} - 394 q^{97} + 394 q^{98} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −3.80282 − 1.01896i 1.91378 + 1.10492i 9.95902 + 5.74984i −5.58818 − 3.22634i −6.15188 − 6.15188i 2.85509 − 10.6553i −20.8780 − 20.8780i −2.05830 − 3.56508i 17.9633 + 17.9633i
24.2 −3.56895 − 0.956297i 1.12404 + 0.648965i 8.35880 + 4.82596i 4.17243 + 2.40895i −3.39104 − 3.39104i −2.39015 + 8.92014i −14.7665 − 14.7665i −3.65769 − 6.33530i −12.5875 − 12.5875i
24.3 −3.54709 − 0.950440i −3.26366 − 1.88427i 8.21442 + 4.74260i 3.06852 + 1.77161i 9.78560 + 9.78560i 1.17423 − 4.38227i −14.2431 − 14.2431i 2.60097 + 4.50501i −9.20052 − 9.20052i
24.4 −3.50839 − 0.940071i −2.37014 − 1.36840i 7.96100 + 4.59628i −6.36657 − 3.67574i 7.02899 + 7.02899i −3.26808 + 12.1966i −13.3362 − 13.3362i −0.754959 − 1.30763i 18.8810 + 18.8810i
24.5 −3.42229 − 0.916999i 0.648510 + 0.374417i 7.40705 + 4.27646i 2.22060 + 1.28207i −1.87605 − 1.87605i 1.11319 − 4.15447i −11.4064 − 11.4064i −4.21962 − 7.30860i −6.42389 − 6.42389i
24.6 −3.35096 − 0.897887i −3.87966 − 2.23992i 6.95864 + 4.01757i −1.83026 − 1.05670i 10.9894 + 10.9894i 1.22522 − 4.57258i −9.89850 − 9.89850i 5.53451 + 9.58604i 5.18434 + 5.18434i
24.7 −3.32958 − 0.892158i 3.92798 + 2.26782i 6.82604 + 3.94102i 2.76503 + 1.59639i −11.0553 − 11.0553i −1.26026 + 4.70335i −9.46215 − 9.46215i 5.78600 + 10.0217i −7.78215 − 7.78215i
24.8 −2.98222 − 0.799083i 3.28707 + 1.89779i 4.79099 + 2.76608i −6.39629 − 3.69290i −8.28628 − 8.28628i −0.975779 + 3.64166i −3.34490 − 3.34490i 2.70324 + 4.68215i 16.1242 + 16.1242i
24.9 −2.81834 − 0.755171i 4.69373 + 2.70993i 3.90863 + 2.25665i 0.750743 + 0.433442i −11.1820 − 11.1820i 2.54269 − 9.48945i −1.05902 − 1.05902i 10.1874 + 17.6451i −1.78852 − 1.78852i
24.10 −2.68615 − 0.719751i 0.492747 + 0.284488i 3.23325 + 1.86672i 8.08807 + 4.66965i −1.11883 − 1.11883i 2.29792 − 8.57595i 0.524177 + 0.524177i −4.33813 − 7.51387i −18.3648 − 18.3648i
24.11 −2.68157 − 0.718525i −1.47679 − 0.852623i 3.21045 + 1.85355i −0.341681 − 0.197270i 3.34748 + 3.34748i 0.195676 − 0.730274i 0.574961 + 0.574961i −3.04607 − 5.27594i 0.774499 + 0.774499i
24.12 −2.66214 − 0.713317i −4.68585 − 2.70538i 3.11404 + 1.79789i 6.46090 + 3.73020i 10.5446 + 10.5446i −1.45212 + 5.41940i 0.787743 + 0.787743i 10.1382 + 17.5598i −14.5390 − 14.5390i
24.13 −2.38235 − 0.638349i −1.90304 − 1.09872i 1.80400 + 1.04154i 4.02016 + 2.32104i 3.83233 + 3.83233i −2.80885 + 10.4828i 3.34311 + 3.34311i −2.08564 − 3.61243i −8.09579 − 8.09579i
24.14 −2.37492 − 0.636359i 1.14688 + 0.662151i 1.77120 + 1.02260i −5.41072 − 3.12388i −2.30238 − 2.30238i 0.298910 − 1.11555i 3.39853 + 3.39853i −3.62311 − 6.27542i 10.8621 + 10.8621i
24.15 −2.33929 − 0.626810i −1.05010 − 0.606276i 1.61527 + 0.932577i −4.65163 − 2.68562i 2.07647 + 2.07647i 1.28892 − 4.81032i 3.65587 + 3.65587i −3.76486 − 6.52093i 9.19812 + 9.19812i
24.16 −2.08559 − 0.558831i 3.14072 + 1.81330i 0.573276 + 0.330981i −3.41348 − 1.97077i −5.53692 − 5.53692i −2.75230 + 10.2717i 5.09637 + 5.09637i 2.07608 + 3.59588i 6.01777 + 6.01777i
24.17 −1.80049 − 0.482440i −4.68687 − 2.70596i −0.455090 − 0.262746i −5.97349 − 3.44880i 7.13319 + 7.13319i −0.450540 + 1.68144i 5.96482 + 5.96482i 10.1445 + 17.5708i 9.09136 + 9.09136i
24.18 −1.79171 − 0.480087i 2.62204 + 1.51384i −0.484358 − 0.279645i 4.53028 + 2.61556i −3.97117 − 3.97117i −1.61933 + 6.04344i 5.98007 + 5.98007i 0.0834027 + 0.144458i −6.86126 − 6.86126i
24.19 −1.66829 − 0.447016i −4.00726 − 2.31359i −0.880746 − 0.508499i 1.97682 + 1.14132i 5.65105 + 5.65105i 3.18202 − 11.8754i 6.12711 + 6.12711i 6.20543 + 10.7481i −2.78771 − 2.78771i
24.20 −1.34650 − 0.360793i 4.16034 + 2.40197i −1.78122 − 1.02839i 7.46144 + 4.30786i −4.73527 − 4.73527i −0.367833 + 1.37277i 5.97019 + 5.97019i 7.03894 + 12.1918i −8.49256 − 8.49256i
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.57
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
349.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 349.3.f.a ✓ 228
349.f odd 12 1 inner 349.3.f.a ✓ 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.3.f.a ✓ 228 1.a even 1 1 trivial
349.3.f.a ✓ 228 349.f odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(349, [\chi])\).