# Properties

 Label 2401.4.a.d Level $2401$ Weight $4$ Character orbit 2401.a Self dual yes Analytic conductor $141.664$ Analytic rank $0$ Dimension $39$ CM no Inner twists $1$

# Related objects

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: $$0$$ Dimension: $$39$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$39 q + q^{2} + q^{3} + 145 q^{4} + 27 q^{5} + 41 q^{6} - 12 q^{8} + 312 q^{9}+O(q^{10})$$ 39 * q + q^2 + q^3 + 145 * q^4 + 27 * q^5 + 41 * q^6 - 12 * q^8 + 312 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$39 q + q^{2} + q^{3} + 145 q^{4} + 27 q^{5} + 41 q^{6} - 12 q^{8} + 312 q^{9} + 78 q^{10} + q^{11} - 91 q^{12} + 77 q^{13} - 161 q^{15} + 461 q^{16} + 211 q^{17} + 8 q^{18} + 314 q^{19} + 476 q^{20} - 61 q^{22} - 69 q^{23} + 330 q^{24} + 606 q^{25} + 504 q^{26} - 50 q^{27} + 57 q^{29} + 42 q^{30} + 638 q^{31} - 1600 q^{32} + 1574 q^{33} + 1343 q^{34} + 782 q^{36} + 71 q^{37} + 1359 q^{38} - 84 q^{39} - 155 q^{40} + 1393 q^{41} - 125 q^{43} + 52 q^{44} + 1129 q^{45} - 1454 q^{46} + 1483 q^{47} - 974 q^{48} + 3074 q^{50} - 2044 q^{51} + 3899 q^{52} + 2213 q^{53} + 1142 q^{54} + 1604 q^{55} + 98 q^{57} + 2403 q^{58} + 2073 q^{59} - 1519 q^{60} + 2575 q^{61} + 1742 q^{62} + 1358 q^{64} + 1876 q^{65} - 48 q^{66} + 176 q^{67} + 3038 q^{68} - 638 q^{69} - 1259 q^{71} - 3799 q^{72} - 307 q^{73} + 3845 q^{74} + 131 q^{75} + 1974 q^{76} - 6041 q^{78} + 22 q^{79} - 804 q^{80} + 795 q^{81} + 8043 q^{82} + 6349 q^{83} + 3094 q^{85} + 1745 q^{86} + 9508 q^{87} + 1299 q^{88} + 2253 q^{89} + 11156 q^{90} + 1284 q^{92} - 3430 q^{93} + 2738 q^{94} - 3290 q^{95} + 3031 q^{96} - 770 q^{97} + 5384 q^{99}+O(q^{100})$$ 39 * q + q^2 + q^3 + 145 * q^4 + 27 * q^5 + 41 * q^6 - 12 * q^8 + 312 * q^9 + 78 * q^10 + q^11 - 91 * q^12 + 77 * q^13 - 161 * q^15 + 461 * q^16 + 211 * q^17 + 8 * q^18 + 314 * q^19 + 476 * q^20 - 61 * q^22 - 69 * q^23 + 330 * q^24 + 606 * q^25 + 504 * q^26 - 50 * q^27 + 57 * q^29 + 42 * q^30 + 638 * q^31 - 1600 * q^32 + 1574 * q^33 + 1343 * q^34 + 782 * q^36 + 71 * q^37 + 1359 * q^38 - 84 * q^39 - 155 * q^40 + 1393 * q^41 - 125 * q^43 + 52 * q^44 + 1129 * q^45 - 1454 * q^46 + 1483 * q^47 - 974 * q^48 + 3074 * q^50 - 2044 * q^51 + 3899 * q^52 + 2213 * q^53 + 1142 * q^54 + 1604 * q^55 + 98 * q^57 + 2403 * q^58 + 2073 * q^59 - 1519 * q^60 + 2575 * q^61 + 1742 * q^62 + 1358 * q^64 + 1876 * q^65 - 48 * q^66 + 176 * q^67 + 3038 * q^68 - 638 * q^69 - 1259 * q^71 - 3799 * q^72 - 307 * q^73 + 3845 * q^74 + 131 * q^75 + 1974 * q^76 - 6041 * q^78 + 22 * q^79 - 804 * q^80 + 795 * q^81 + 8043 * q^82 + 6349 * q^83 + 3094 * q^85 + 1745 * q^86 + 9508 * q^87 + 1299 * q^88 + 2253 * q^89 + 11156 * q^90 + 1284 * q^92 - 3430 * q^93 + 2738 * q^94 - 3290 * q^95 + 3031 * q^96 - 770 * q^97 + 5384 * q^99

## 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 −5.52124 −4.05315 22.4841 14.0923 22.3784 0 −79.9700 −10.5720 −77.8068
1.2 −5.47147 −8.31145 21.9370 −5.94042 45.4758 0 −76.2558 42.0802 32.5028
1.3 −5.14785 6.96013 18.5003 3.24701 −35.8297 0 −54.0542 21.4435 −16.7151
1.4 −4.68909 5.32877 13.9876 −6.47646 −24.9871 0 −28.0763 1.39583 30.3687
1.5 −4.44388 6.83693 11.7481 −2.20264 −30.3825 0 −16.6559 19.7437 9.78829
1.6 −4.34611 −2.32971 10.8887 −2.24811 10.1252 0 −12.5546 −21.5725 9.77055
1.7 −3.93571 −3.09696 7.48982 −18.2977 12.1887 0 2.00794 −17.4089 72.0144
1.8 −3.74761 2.12177 6.04460 10.5286 −7.95158 0 7.32810 −22.4981 −39.4572
1.9 −3.57033 −9.05210 4.74728 15.7342 32.3190 0 11.6133 54.9406 −56.1764
1.10 −2.58597 3.61260 −1.31274 3.92216 −9.34209 0 24.0825 −13.9491 −10.1426
1.11 −2.46450 −5.32585 −1.92624 15.0765 13.1256 0 24.4632 1.36464 −37.1561
1.12 −2.34370 −8.19575 −2.50708 0.508590 19.2083 0 24.6254 40.1702 −1.19198
1.13 −2.11024 −5.68890 −3.54689 −12.6417 12.0050 0 24.3667 5.36363 26.6769
1.14 −2.08878 −2.27918 −3.63702 −20.5308 4.76069 0 24.3071 −21.8054 42.8842
1.15 −2.07454 7.56379 −3.69627 −6.53935 −15.6914 0 24.2644 30.2110 13.5662
1.16 −1.58052 3.81461 −5.50196 14.4403 −6.02907 0 21.3401 −12.4487 −22.8232
1.17 −1.28751 1.89643 −6.34231 17.6887 −2.44167 0 18.4659 −23.4036 −22.7744
1.18 −0.904464 9.08614 −7.18194 −3.14137 −8.21809 0 13.7315 55.5579 2.84125
1.19 −0.275911 3.60983 −7.92387 −3.79895 −0.995991 0 4.39357 −13.9691 1.04817
1.20 0.246994 9.97745 −7.93899 −15.8609 2.46437 0 −3.93684 72.5495 −3.91755
See all 39 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.39 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.d 39
7.b odd 2 1 2401.4.a.c 39
49.e even 7 2 49.4.e.a 78

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.e.a 78 49.e even 7 2
2401.4.a.c 39 7.b odd 2 1
2401.4.a.d 39 1.a even 1 1 trivial