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Properties

Label 207.4.i.a
Level $207$
Weight $4$
Character orbit 207.i
Analytic conductor $12.213$
Analytic rank $0$
Dimension $50$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,4,Mod(55,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 207.i (of order \(11\), degree \(10\), 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: \(12.2133953712\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(5\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 23)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 11 q^{2} - 27 q^{4} + 19 q^{5} - 19 q^{7} - 28 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 11 q^{2} - 27 q^{4} + 19 q^{5} - 19 q^{7} - 28 q^{8} + 47 q^{10} + 53 q^{11} - 65 q^{13} - 117 q^{14} - 499 q^{16} + 117 q^{17} + 73 q^{19} - 529 q^{20} + 310 q^{22} - 542 q^{23} + 246 q^{25} - 324 q^{26} - 677 q^{28} + 497 q^{29} - 471 q^{31} + 915 q^{32} - 2751 q^{34} + 737 q^{35} - 1071 q^{37} + 1504 q^{38} + 1479 q^{40} - 569 q^{41} + 1615 q^{43} - 2518 q^{44} + 4041 q^{46} - 2904 q^{47} + 1226 q^{49} - 1322 q^{50} - 2156 q^{52} - 391 q^{53} - 3323 q^{55} + 7028 q^{56} - 5639 q^{58} + 2445 q^{59} - 1059 q^{61} - 1468 q^{62} + 4570 q^{64} - 2641 q^{65} + 27 q^{67} - 8350 q^{68} + 9702 q^{70} - 3465 q^{71} + 435 q^{73} + 994 q^{74} - 3598 q^{76} + 5931 q^{77} - 2559 q^{79} + 14052 q^{80} - 3822 q^{82} + 3967 q^{83} + 299 q^{85} - 721 q^{86} + 5825 q^{88} - 3717 q^{89} + 7238 q^{91} - 9550 q^{92} + 6035 q^{94} - 4551 q^{95} - 2419 q^{97} + 5687 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −3.51370 + 1.03172i 0 4.55162 − 2.92515i −0.606340 − 4.21719i 0 −6.05808 + 13.2653i 6.20988 − 7.16659i 0 6.48144 + 14.1924i
55.2 −1.42264 + 0.417725i 0 −4.88062 + 3.13658i 1.63503 + 11.3719i 0 8.78092 − 19.2275i 13.4008 − 15.4654i 0 −7.07638 − 15.4951i
55.3 1.12632 − 0.330717i 0 −5.57081 + 3.58014i 0.0835881 + 0.581368i 0 10.1613 − 22.2502i −11.2403 + 12.9719i 0 0.286415 + 0.627161i
55.4 2.49786 − 0.733439i 0 −1.02864 + 0.661064i −2.23195 − 15.5235i 0 −2.46994 + 5.40842i −15.7230 + 18.1453i 0 −16.9607 − 37.1386i
55.5 4.76776 − 1.39994i 0 14.0417 − 9.02406i 2.93015 + 20.3796i 0 −4.70707 + 10.3070i 28.2822 − 32.6393i 0 42.5006 + 93.0633i
64.1 −3.51370 − 1.03172i 0 4.55162 + 2.92515i −0.606340 + 4.21719i 0 −6.05808 − 13.2653i 6.20988 + 7.16659i 0 6.48144 − 14.1924i
64.2 −1.42264 − 0.417725i 0 −4.88062 − 3.13658i 1.63503 − 11.3719i 0 8.78092 + 19.2275i 13.4008 + 15.4654i 0 −7.07638 + 15.4951i
64.3 1.12632 + 0.330717i 0 −5.57081 − 3.58014i 0.0835881 − 0.581368i 0 10.1613 + 22.2502i −11.2403 − 12.9719i 0 0.286415 − 0.627161i
64.4 2.49786 + 0.733439i 0 −1.02864 − 0.661064i −2.23195 + 15.5235i 0 −2.46994 − 5.40842i −15.7230 − 18.1453i 0 −16.9607 + 37.1386i
64.5 4.76776 + 1.39994i 0 14.0417 + 9.02406i 2.93015 − 20.3796i 0 −4.70707 − 10.3070i 28.2822 + 32.6393i 0 42.5006 − 93.0633i
73.1 −1.61009 − 3.52560i 0 −4.59861 + 5.30707i 10.7624 + 6.91659i 0 −0.249078 − 1.73238i −3.63606 − 1.06764i 0 7.05669 − 49.0804i
73.2 −1.49850 − 3.28126i 0 −3.28227 + 3.78794i −15.0482 − 9.67090i 0 −2.56161 − 17.8164i −10.3412 − 3.03646i 0 −9.18297 + 63.8690i
73.3 0.308069 + 0.674576i 0 4.87874 − 5.63037i −3.36936 − 2.16536i 0 0.387311 + 2.69380i 10.9935 + 3.22799i 0 0.422704 − 2.93997i
73.4 1.58245 + 3.46509i 0 −4.26380 + 4.92069i 4.88780 + 3.14120i 0 −2.25200 − 15.6630i 5.44232 + 1.59801i 0 −3.14982 + 21.9075i
73.5 2.10729 + 4.61431i 0 −11.6123 + 13.4013i 5.02341 + 3.22835i 0 3.12135 + 21.7095i −47.3705 − 13.9092i 0 −4.31085 + 29.9826i
82.1 −3.50231 − 4.04189i 0 −2.93212 + 20.3933i −6.92690 + 15.1678i 0 7.38228 + 2.16763i 56.7033 − 36.4410i 0 85.5667 − 25.1246i
82.2 −1.61249 − 1.86092i 0 0.275643 − 1.91714i 3.07518 − 6.73370i 0 −11.6919 − 3.43306i −20.5838 + 13.2284i 0 −17.4896 + 5.13540i
82.3 0.639172 + 0.737644i 0 1.00294 − 6.97561i −0.787834 + 1.72512i 0 10.9823 + 3.22469i 12.3554 − 7.94031i 0 −1.77608 + 0.521505i
82.4 1.60666 + 1.85419i 0 0.281873 − 1.96047i −2.81378 + 6.16132i 0 8.67641 + 2.54762i 20.5997 − 13.2386i 0 −15.9450 + 4.68188i
82.5 3.54027 + 4.08569i 0 −3.02082 + 21.0103i −1.32774 + 2.90735i 0 −29.0289 − 8.52364i −60.1524 + 38.6576i 0 −16.5791 + 4.86806i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.4.i.a 50
3.b odd 2 1 23.4.c.a ✓ 50
23.c even 11 1 inner 207.4.i.a 50
69.g even 22 1 529.4.a.m 25
69.h odd 22 1 23.4.c.a ✓ 50
69.h odd 22 1 529.4.a.n 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.4.c.a ✓ 50 3.b odd 2 1
23.4.c.a ✓ 50 69.h odd 22 1
207.4.i.a 50 1.a even 1 1 trivial
207.4.i.a 50 23.c even 11 1 inner
529.4.a.m 25 69.g even 22 1
529.4.a.n 25 69.h odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} - 11 T_{2}^{49} + 94 T_{2}^{48} - 581 T_{2}^{47} + 3806 T_{2}^{46} - 23017 T_{2}^{45} + \cdots + 17\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display