Properties

Label 41.4.f.a
Level $41$
Weight $4$
Character orbit 41.f
Analytic conductor $2.419$
Analytic rank $0$
Dimension $40$
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] = [41,4,Mod(4,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 41.f (of order \(10\), 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: \(2.41907831024\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + q^{2} - 43 q^{4} - q^{5} - 15 q^{6} - 5 q^{7} + 112 q^{8} - 370 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + q^{2} - 43 q^{4} - q^{5} - 15 q^{6} - 5 q^{7} + 112 q^{8} - 370 q^{9} + 96 q^{10} + 120 q^{11} + 130 q^{12} - 5 q^{13} + 190 q^{15} - 219 q^{16} + 35 q^{17} - 368 q^{18} + 220 q^{19} + 156 q^{20} + 454 q^{21} - 365 q^{22} - 477 q^{23} + 490 q^{24} + 349 q^{25} - 95 q^{26} + 510 q^{28} - 495 q^{29} - 570 q^{30} - 487 q^{31} - 1588 q^{32} + 551 q^{33} - 405 q^{34} - 985 q^{35} + 770 q^{36} - 395 q^{37} + 1376 q^{39} + 4238 q^{40} - 1159 q^{41} + 984 q^{42} + 976 q^{43} - 1355 q^{45} + 3176 q^{46} + 985 q^{47} - 2725 q^{48} - 31 q^{49} - 4464 q^{50} + 248 q^{51} - 2535 q^{52} + 95 q^{53} - 980 q^{54} + 3845 q^{56} - 826 q^{57} - 1490 q^{58} + 1345 q^{59} + 6540 q^{60} + 941 q^{61} + 328 q^{62} - 3945 q^{63} - 4262 q^{64} + 1175 q^{65} - 2396 q^{66} - 3800 q^{67} + 2660 q^{69} - 6085 q^{70} - 1915 q^{71} - 4653 q^{72} + 1046 q^{73} + 3519 q^{74} + 2255 q^{75} - 2590 q^{76} + 275 q^{77} + 4096 q^{78} + 7081 q^{80} + 6620 q^{81} - 439 q^{82} + 4986 q^{83} + 8110 q^{84} + 5587 q^{86} + 10346 q^{87} + 2140 q^{88} - 4015 q^{89} - 4332 q^{90} - 8454 q^{91} - 3849 q^{92} - 4030 q^{93} - 12450 q^{94} - 1685 q^{95} + 705 q^{97} - 642 q^{98} - 7000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −4.41144 + 3.20510i 1.48791i 6.71601 20.6698i 0.515160 1.58550i 4.76888 + 6.56381i 3.18727 4.38691i 23.1412 + 71.2212i 24.7861 2.80908 + 8.64547i
4.2 −2.78864 + 2.02606i 7.54001i 1.19943 3.69147i 0.380962 1.17248i −15.2765 21.0264i −1.94929 + 2.68297i −4.38694 13.5016i −29.8517 1.31316 + 4.04148i
4.3 −2.43695 + 1.77055i 6.59197i 0.331746 1.02101i −4.60057 + 14.1591i 11.6714 + 16.0643i −15.8644 + 21.8355i −6.44735 19.8429i −16.4541 −13.8580 42.6505i
4.4 −1.73189 + 1.25830i 4.21774i −1.05598 + 3.24998i 4.47618 13.7763i 5.30716 + 7.30468i 9.95533 13.7023i −7.55278 23.2451i 9.21068 9.58234 + 29.4914i
4.5 0.355319 0.258154i 4.28641i −2.41253 + 7.42500i 1.11581 3.43412i 1.10655 + 1.52304i −14.0326 + 19.3142i 2.14533 + 6.60266i 8.62667 −0.490063 1.50826i
4.6 0.454809 0.330438i 1.60812i −2.37447 + 7.30788i −5.11607 + 15.7456i 0.531384 + 0.731388i 13.6716 18.8173i 2.72464 + 8.38558i 24.4139 2.87612 + 8.85179i
4.7 2.08916 1.51786i 9.41969i −0.411459 + 1.26634i −1.61945 + 4.98415i −14.2978 19.6792i 7.26325 9.99701i 7.44643 + 22.9178i −61.7306 4.18197 + 12.8708i
4.8 2.80536 2.03822i 1.54098i 1.24361 3.82743i 3.68903 11.3537i −3.14085 4.32301i −3.25858 + 4.48505i 4.26007 + 13.1112i 24.6254 −12.7922 39.3702i
4.9 3.25753 2.36673i 9.81535i 2.53793 7.81093i 1.19307 3.67189i 23.2303 + 31.9738i 12.9368 17.8060i −0.264905 0.815295i −69.3412 −4.80392 14.7849i
4.10 4.33380 3.14869i 1.16718i 6.39542 19.6831i −5.31529 + 16.3588i −3.67507 5.05830i −12.6004 + 17.3430i −21.0165 64.6822i 25.6377 28.4733 + 87.6317i
23.1 −1.69608 5.22000i 6.95975i −17.8996 + 13.0048i 12.4188 9.02282i −36.3299 + 11.8043i 15.1090 + 4.90923i 62.7209 + 45.5694i −21.4382 −68.1625 49.5229i
23.2 −1.49988 4.61614i 4.75851i −12.5870 + 9.14499i −13.9422 + 10.1296i 21.9660 7.13718i −21.8801 7.10927i 29.6797 + 21.5636i 4.35654 67.6714 + 49.1662i
23.3 −0.925466 2.84829i 6.53368i −0.784140 + 0.569711i 9.17905 6.66897i 18.6098 6.04670i 23.6307 + 7.67807i −17.0348 12.3765i −15.6889 −27.4901 19.9727i
23.4 −0.814530 2.50687i 2.19754i 0.851213 0.618443i 4.32167 3.13988i −5.50894 + 1.78996i −24.2069 7.86529i −19.3034 14.0248i 22.1708 −11.3914 8.27632i
23.5 −0.625259 1.92435i 8.16596i 3.15996 2.29585i −13.3127 + 9.67225i −15.7142 + 5.10585i 17.8022 + 5.78429i −19.4894 14.1599i −39.6830 26.9367 + 19.5707i
23.6 0.239215 + 0.736229i 6.90238i 5.98733 4.35005i −13.1428 + 9.54883i −5.08173 + 1.65116i 5.83421 + 1.89565i 9.64508 + 7.00756i −20.6429 −10.1741 7.39191i
23.7 0.283232 + 0.871700i 1.56069i 5.79250 4.20849i 3.14119 2.28221i 1.36045 0.442038i 0.892763 + 0.290076i 11.2413 + 8.16726i 24.5643 2.87909 + 2.09178i
23.8 1.06811 + 3.28729i 9.65587i −3.19332 + 2.32008i 6.29911 4.57657i 31.7417 10.3135i 1.36476 + 0.443437i 11.3331 + 8.23400i −66.2357 21.7726 + 15.8188i
23.9 1.09430 + 3.36790i 8.13837i −3.67312 + 2.66868i 15.1333 10.9950i −27.4092 + 8.90579i −27.8693 9.05530i 9.91193 + 7.20144i −39.2330 53.5903 + 38.9357i
23.10 1.44931 + 4.46051i 0.304759i −11.3236 + 8.22705i −5.31421 + 3.86100i −1.35938 + 0.441690i 7.51362 + 2.44132i −22.7535 16.5314i 26.9071 −24.9240 18.1083i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.4.f.a 40
41.f even 10 1 inner 41.4.f.a 40
41.g even 20 2 1681.4.a.l 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.4.f.a 40 1.a even 1 1 trivial
41.4.f.a 40 41.f even 10 1 inner
1681.4.a.l 40 41.g even 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(41, [\chi])\).