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Properties

Label 354.6.c.b
Level $354$
Weight $6$
Character orbit 354.c
Analytic conductor $56.776$
Analytic rank $0$
Dimension $50$
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] = [354,6,Mod(353,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.353");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 354.c (of order \(2\), degree \(1\), 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: \(56.7758722138\)
Analytic rank: \(0\)
Dimension: \(50\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 200 q^{2} - 13 q^{3} + 800 q^{4} - 52 q^{6} + 38 q^{7} + 3200 q^{8} + 51 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 200 q^{2} - 13 q^{3} + 800 q^{4} - 52 q^{6} + 38 q^{7} + 3200 q^{8} + 51 q^{9} + 652 q^{11} - 208 q^{12} + 152 q^{14} - 2107 q^{15} + 12800 q^{16} + 204 q^{18} - 894 q^{19} - 3801 q^{21} + 2608 q^{22} - 2456 q^{23} - 832 q^{24} - 23956 q^{25} + 9890 q^{27} + 608 q^{28} - 8428 q^{30} + 51200 q^{32} + 9744 q^{33} + 816 q^{36} - 3576 q^{38} - 1388 q^{39} - 15204 q^{42} + 10432 q^{44} + 33067 q^{45} - 9824 q^{46} + 27144 q^{47} - 3328 q^{48} + 85768 q^{49} - 95824 q^{50} + 3338 q^{51} + 39560 q^{54} + 2432 q^{56} - 63969 q^{57} + 23840 q^{59} - 33712 q^{60} + 94781 q^{63} + 204800 q^{64} + 9400 q^{65} + 38976 q^{66} + 115930 q^{69} + 3264 q^{72} + 24248 q^{75} - 14304 q^{76} + 150240 q^{77} - 5552 q^{78} - 68658 q^{79} + 47095 q^{81} + 175140 q^{83} - 60816 q^{84} - 138524 q^{85} + 138261 q^{87} + 41728 q^{88} - 104908 q^{89} + 132268 q^{90} - 39296 q^{92} + 91204 q^{93} + 108576 q^{94} - 13312 q^{96} + 343072 q^{98} + 111406 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1 4.00000 −15.5824 − 0.434789i 16.0000 − 38.7472i −62.3296 − 1.73916i −41.9790 64.0000 242.622 + 13.5501i − 154.989i
353.2 4.00000 −15.5824 + 0.434789i 16.0000 38.7472i −62.3296 + 1.73916i −41.9790 64.0000 242.622 − 13.5501i 154.989i
353.3 4.00000 −15.5391 − 1.23947i 16.0000 14.2214i −62.1564 − 4.95787i 129.028 64.0000 239.927 + 38.5204i 56.8857i
353.4 4.00000 −15.5391 + 1.23947i 16.0000 − 14.2214i −62.1564 + 4.95787i 129.028 64.0000 239.927 − 38.5204i − 56.8857i
353.5 4.00000 −14.5480 − 5.59965i 16.0000 83.1853i −58.1920 − 22.3986i 242.629 64.0000 180.288 + 162.927i 332.741i
353.6 4.00000 −14.5480 + 5.59965i 16.0000 − 83.1853i −58.1920 + 22.3986i 242.629 64.0000 180.288 − 162.927i − 332.741i
353.7 4.00000 −13.3335 − 8.07581i 16.0000 83.4718i −53.3339 − 32.3032i −123.882 64.0000 112.563 + 215.357i 333.887i
353.8 4.00000 −13.3335 + 8.07581i 16.0000 − 83.4718i −53.3339 + 32.3032i −123.882 64.0000 112.563 − 215.357i − 333.887i
353.9 4.00000 −12.9095 − 8.73750i 16.0000 − 27.1818i −51.6381 − 34.9500i −249.363 64.0000 90.3121 + 225.594i − 108.727i
353.10 4.00000 −12.9095 + 8.73750i 16.0000 27.1818i −51.6381 + 34.9500i −249.363 64.0000 90.3121 − 225.594i 108.727i
353.11 4.00000 −11.6908 − 10.3115i 16.0000 − 106.082i −46.7630 − 41.2458i 71.9340 64.0000 30.3476 + 241.098i − 424.326i
353.12 4.00000 −11.6908 + 10.3115i 16.0000 106.082i −46.7630 + 41.2458i 71.9340 64.0000 30.3476 − 241.098i 424.326i
353.13 4.00000 −11.5231 − 10.4984i 16.0000 − 67.9871i −46.0925 − 41.9938i −61.6810 64.0000 22.5654 + 241.950i − 271.948i
353.14 4.00000 −11.5231 + 10.4984i 16.0000 67.9871i −46.0925 + 41.9938i −61.6810 64.0000 22.5654 − 241.950i 271.948i
353.15 4.00000 −9.75722 − 12.1572i 16.0000 43.2791i −39.0289 − 48.6287i −5.13563 64.0000 −52.5934 + 237.240i 173.116i
353.16 4.00000 −9.75722 + 12.1572i 16.0000 − 43.2791i −39.0289 + 48.6287i −5.13563 64.0000 −52.5934 − 237.240i − 173.116i
353.17 4.00000 −9.68192 − 12.2172i 16.0000 − 48.6180i −38.7277 − 48.8689i 149.149 64.0000 −55.5209 + 236.572i − 194.472i
353.18 4.00000 −9.68192 + 12.2172i 16.0000 48.6180i −38.7277 + 48.8689i 149.149 64.0000 −55.5209 − 236.572i 194.472i
353.19 4.00000 −4.24050 − 15.0006i 16.0000 2.81364i −16.9620 − 60.0024i −187.063 64.0000 −207.036 + 127.220i 11.2546i
353.20 4.00000 −4.24050 + 15.0006i 16.0000 − 2.81364i −16.9620 + 60.0024i −187.063 64.0000 −207.036 − 127.220i − 11.2546i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
177.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.6.c.b yes 50
3.b odd 2 1 354.6.c.a ✓ 50
59.b odd 2 1 354.6.c.a ✓ 50
177.d even 2 1 inner 354.6.c.b yes 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.6.c.a ✓ 50 3.b odd 2 1
354.6.c.a ✓ 50 59.b odd 2 1
354.6.c.b yes 50 1.a even 1 1 trivial
354.6.c.b yes 50 177.d even 2 1 inner