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Properties

Label 2075.4.a.l
Level $2075$
Weight $4$
Character orbit 2075.a
Self dual yes
Analytic conductor $122.429$
Analytic rank $0$
Dimension $47$
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] = [2075,4,Mod(1,2075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2075.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2075 = 5^{2} \cdot 83 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2075.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: \(122.428963262\)
Analytic rank: \(0\)
Dimension: \(47\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 47 q + q^{2} - 6 q^{3} + 229 q^{4} + 9 q^{6} - 6 q^{7} + 39 q^{8} + 531 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 47 q + q^{2} - 6 q^{3} + 229 q^{4} + 9 q^{6} - 6 q^{7} + 39 q^{8} + 531 q^{9} + 90 q^{11} + 6 q^{12} + 50 q^{13} + 115 q^{14} + 1149 q^{16} - 68 q^{17} + 8 q^{18} + 344 q^{19} + 625 q^{21} + 402 q^{22} - 469 q^{23} + 262 q^{24} + 1209 q^{26} + 12 q^{27} - 603 q^{28} + 480 q^{29} + 876 q^{31} + 1653 q^{32} - 757 q^{33} + 733 q^{34} + 4318 q^{36} + 634 q^{37} - 1190 q^{38} + 618 q^{39} + 1455 q^{41} + 2747 q^{42} - 652 q^{43} + 595 q^{44} + 1662 q^{46} + 446 q^{47} - 2145 q^{48} + 4099 q^{49} + 1681 q^{51} + 207 q^{52} - 126 q^{53} + 670 q^{54} + 3100 q^{56} + 650 q^{57} - 1082 q^{58} + 1144 q^{59} + 2762 q^{61} + 1916 q^{62} - 885 q^{63} + 6891 q^{64} + 2025 q^{66} - 100 q^{67} - 2052 q^{68} + 2556 q^{69} + 2546 q^{71} + 2940 q^{72} + 14 q^{73} + 1066 q^{74} + 4828 q^{76} - 210 q^{77} - 343 q^{78} + 1636 q^{79} + 8203 q^{81} + 480 q^{82} - 3901 q^{83} + 6209 q^{84} + 2706 q^{86} - 5479 q^{87} + 2178 q^{88} + 2558 q^{89} + 3342 q^{91} - 7501 q^{92} + 3353 q^{93} + 5769 q^{94} - 918 q^{96} - 3648 q^{97} + 8109 q^{98} + 13991 q^{99}+O(q^{100}) \) 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 −5.57387 −1.57034 23.0681 0 8.75287 1.33710 −83.9875 −24.5340 0
1.2 −5.25014 −7.31471 19.5639 0 38.4032 3.44162 −60.7122 26.5050 0
1.3 −5.05617 −8.79982 17.5648 0 44.4934 −26.9237 −48.3615 50.4369 0
1.4 −4.99853 7.70104 16.9853 0 −38.4939 −27.7181 −44.9131 32.3061 0
1.5 −4.95126 −1.24838 16.5150 0 6.18104 −20.7297 −42.1601 −25.4416 0
1.6 −4.75798 9.44125 14.6383 0 −44.9212 −12.8875 −31.5850 62.1371 0
1.7 −4.52319 6.73619 12.4592 0 −30.4691 5.86549 −20.1699 18.3763 0
1.8 −4.47908 1.96805 12.0621 0 −8.81505 27.9268 −18.1946 −23.1268 0
1.9 −4.07887 −9.75064 8.63719 0 39.7716 35.1031 −2.59900 68.0750 0
1.10 −4.02716 1.58965 8.21802 0 −6.40176 −9.91849 −0.877984 −24.4730 0
1.11 −3.55451 9.31621 4.63456 0 −33.1146 22.0806 11.9625 59.7918 0
1.12 −3.44131 −7.83773 3.84262 0 26.9721 −20.1133 14.3068 34.4300 0
1.13 −3.28905 2.26603 2.81787 0 −7.45310 −5.89219 17.0443 −21.8651 0
1.14 −3.02429 −0.588028 1.14634 0 1.77837 28.5702 20.7275 −26.6542 0
1.15 −2.72908 −2.68762 −0.552106 0 7.33473 19.6479 23.3394 −19.7767 0
1.16 −2.59042 −7.16542 −1.28973 0 18.5615 −15.3025 24.0643 24.3433 0
1.17 −2.17523 2.57891 −3.26836 0 −5.60972 −0.561178 24.5113 −20.3492 0
1.18 −1.44203 −2.76719 −5.92054 0 3.99038 −27.6505 20.0739 −19.3427 0
1.19 −1.43393 9.11382 −5.94384 0 −13.0686 2.42127 19.9945 56.0618 0
1.20 −0.938348 −3.05022 −7.11950 0 2.86217 −6.41572 14.1874 −17.6962 0
See all 47 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.47
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(83\) \( +1 \)

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 2075.4.a.l yes 47
5.b even 2 1 2075.4.a.k ✓ 47
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2075.4.a.k ✓ 47 5.b even 2 1
2075.4.a.l yes 47 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{47} - T_{2}^{46} - 302 T_{2}^{45} + 284 T_{2}^{44} + 42455 T_{2}^{43} - 37645 T_{2}^{42} + \cdots - 12\!\cdots\!24 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2075))\). Copy content Toggle raw display