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Properties

Label 2401.4.a.f
Level $2401$
Weight $4$
Character orbit 2401.a
Self dual yes
Analytic conductor $141.664$
Analytic rank $1$
Dimension $78$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2401,4,Mod(1,2401)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2401, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2401.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2401 = 7^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2401.a (trivial)

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: yes
Analytic conductor: \(141.663585924\)
Analytic rank: \(1\)
Dimension: \(78\)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 78 q - q^{2} - 35 q^{3} + 287 q^{4} - 63 q^{5} - 70 q^{6} + 15 q^{8} + 579 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 78 q - q^{2} - 35 q^{3} + 287 q^{4} - 63 q^{5} - 70 q^{6} + 15 q^{8} + 579 q^{9} - 126 q^{10} - q^{11} - 532 q^{12} - 196 q^{13} + 164 q^{15} + 979 q^{16} - 497 q^{17} + 16 q^{18} - 490 q^{19} - 868 q^{20} + 64 q^{22} + 69 q^{23} - 1365 q^{24} + 1419 q^{25} - 1176 q^{26} - 1547 q^{27} - 54 q^{29} + 39 q^{30} - 1162 q^{31} + 1624 q^{32} + 161 q^{33} - 154 q^{34} + 1813 q^{36} - 323 q^{37} - 882 q^{38} - 126 q^{39} - 2723 q^{40} - 1610 q^{41} + 2 q^{43} + 140 q^{44} - 1736 q^{45} + 1478 q^{46} - 2765 q^{47} - 4802 q^{48} - 3092 q^{50} + 2125 q^{51} + 581 q^{52} - 2213 q^{53} - 3514 q^{54} - 3115 q^{55} + 73 q^{57} - 2403 q^{58} - 5061 q^{59} + 1918 q^{60} - 1925 q^{61} - 3178 q^{62} + 2101 q^{64} - 1876 q^{65} - 5670 q^{66} + 202 q^{67} - 5635 q^{68} - 7595 q^{69} + 1262 q^{71} + 4072 q^{72} - 4837 q^{73} - 3821 q^{74} - 7882 q^{75} - 5817 q^{76} + 6692 q^{78} + 608 q^{79} - 12348 q^{80} + 2850 q^{81} + 5208 q^{82} - 3668 q^{83} - 3091 q^{85} + 3610 q^{86} - 938 q^{87} + 3702 q^{88} - 8925 q^{89} + 2198 q^{90} + 2940 q^{92} + 4435 q^{93} - 4102 q^{94} + 3665 q^{95} - 9800 q^{96} - 5558 q^{97} - 5402 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.46905 0.224209 21.9105 −12.5599 −1.22621 0 −76.0772 −26.9497 68.6907
1.2 −5.31700 −0.616628 20.2705 8.49931 3.27861 0 −65.2422 −26.6198 −45.1908
1.3 −5.15954 −8.81575 18.6209 8.56425 45.4852 0 −54.7988 50.7174 −44.1876
1.4 −5.12460 4.83795 18.2615 8.28082 −24.7926 0 −52.5862 −3.59419 −42.4359
1.5 −4.98616 −8.34998 16.8618 −14.5745 41.6343 0 −44.1863 42.7221 72.6708
1.6 −4.83119 −0.633781 15.3404 −16.7962 3.06192 0 −35.4631 −26.5983 81.1455
1.7 −4.78238 −6.76655 14.8712 16.8191 32.3602 0 −32.8605 18.7862 −80.4352
1.8 −4.77561 −0.473082 14.8065 8.05686 2.25925 0 −32.5051 −26.7762 −38.4764
1.9 −4.73293 5.61021 14.4007 11.0770 −26.5527 0 −30.2939 4.47442 −52.4268
1.10 −4.57640 9.32194 12.9435 −11.3149 −42.6610 0 −22.6232 59.8986 51.7816
1.11 −4.49469 2.24598 12.2023 −20.8662 −10.0950 0 −18.8879 −21.9556 93.7872
1.12 −4.22285 −6.99721 9.83247 −20.6393 29.5482 0 −7.73827 21.9610 87.1567
1.13 −4.04817 7.81932 8.38765 16.1362 −31.6539 0 −1.56928 34.1417 −65.3219
1.14 −3.91942 9.45679 7.36189 −0.884980 −37.0652 0 2.50103 62.4308 3.46861
1.15 −3.83595 5.76089 6.71455 3.24218 −22.0985 0 4.93094 6.18786 −12.4368
1.16 −3.71457 −2.52784 5.79800 −3.01785 9.38982 0 8.17948 −20.6100 11.2100
1.17 −3.59774 −4.60658 4.94373 20.3107 16.5733 0 10.9957 −5.77945 −73.0725
1.18 −3.50186 −4.46382 4.26304 10.2875 15.6317 0 13.0863 −7.07429 −36.0253
1.19 −3.38031 2.69306 3.42648 −6.88836 −9.10339 0 15.4599 −19.7474 23.2848
1.20 −3.30386 0.0736277 2.91549 −0.557509 −0.243256 0 16.7985 −26.9946 1.84193
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.78
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2401.4.a.f 78
7.b odd 2 1 2401.4.a.g 78
49.h odd 42 2 49.4.g.a ✓ 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.g.a ✓ 156 49.h odd 42 2
2401.4.a.f 78 1.a even 1 1 trivial
2401.4.a.g 78 7.b odd 2 1