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Properties

Label 2075.4.a.h
Level $2075$
Weight $4$
Character orbit 2075.a
Self dual yes
Analytic conductor $122.429$
Analytic rank $0$
Dimension $26$
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: \(26\)
Twist minimal: no (minimal twist has level 415)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 7 q^{2} - 12 q^{3} + 127 q^{4} + 61 q^{6} - 5 q^{7} - 60 q^{8} + 378 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 7 q^{2} - 12 q^{3} + 127 q^{4} + 61 q^{6} - 5 q^{7} - 60 q^{8} + 378 q^{9} + 90 q^{11} - 142 q^{12} - 93 q^{13} + 201 q^{14} + 671 q^{16} - 425 q^{17} - 297 q^{18} + 219 q^{19} + 327 q^{21} + 79 q^{22} - 30 q^{23} + 1028 q^{24} - 525 q^{26} - 525 q^{27} + 565 q^{28} + 876 q^{29} + 483 q^{31} + 28 q^{32} + 21 q^{33} + 805 q^{34} + 1086 q^{36} - 302 q^{37} + 114 q^{38} + 548 q^{39} + 1160 q^{41} + 444 q^{42} + 49 q^{43} + 1766 q^{44} + 1009 q^{46} - 886 q^{47} + 751 q^{48} + 3285 q^{49} + 481 q^{51} - 801 q^{52} - 581 q^{53} + 2139 q^{54} + 2502 q^{56} - 2224 q^{57} + 1305 q^{58} + 1648 q^{59} + 2098 q^{61} - 649 q^{62} + 132 q^{63} + 4666 q^{64} - 994 q^{66} - 983 q^{67} - 3956 q^{68} + 3522 q^{69} + 2034 q^{71} - 1656 q^{72} - 826 q^{73} + 2788 q^{74} + 2126 q^{76} - 2453 q^{77} - 1859 q^{78} + 1948 q^{79} + 7306 q^{81} - 2968 q^{82} - 2158 q^{83} + 3367 q^{84} + 4180 q^{86} + 3489 q^{87} + 100 q^{88} + 5414 q^{89} + 3540 q^{91} - 337 q^{92} - 1614 q^{93} + 2887 q^{94} + 10110 q^{96} - 128 q^{97} - 1676 q^{98} - 5477 q^{99}+O(q^{100}) \) 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 −5.55008 −2.48281 22.8034 0 13.7798 3.39315 −82.1602 −20.8357 0
1.2 −5.09502 −9.88037 17.9592 0 50.3407 20.0408 −50.7426 70.6218 0
1.3 −5.02406 6.78543 17.2412 0 −34.0904 25.0309 −46.4281 19.0420 0
1.4 −4.89264 3.34121 15.9379 0 −16.3473 −27.4693 −38.8374 −15.8363 0
1.5 −4.65647 −6.52969 13.6827 0 30.4053 −29.1290 −26.4612 15.6369 0
1.6 −3.97088 −7.59816 7.76791 0 30.1714 7.31087 0.921600 30.7320 0
1.7 −3.87469 −4.85102 7.01321 0 18.7962 −14.2457 3.82349 −3.46756 0
1.8 −3.19598 7.47907 2.21427 0 −23.9029 −29.6122 18.4911 28.9364 0
1.9 −2.49474 4.88326 −1.77628 0 −12.1825 −4.63960 24.3893 −3.15375 0
1.10 −2.07300 9.65211 −3.70267 0 −20.0088 28.3218 24.2596 66.1632 0
1.11 −1.55036 −10.0582 −5.59637 0 15.5938 −10.0693 21.0793 74.1669 0
1.12 −1.26833 −1.05678 −6.39134 0 1.34035 3.27119 18.2529 −25.8832 0
1.13 −1.20675 1.32080 −6.54376 0 −1.59387 35.8229 17.5507 −25.2555 0
1.14 −0.286911 −6.62100 −7.91768 0 1.89964 −28.6057 4.56695 16.8377 0
1.15 0.975140 −6.63444 −7.04910 0 −6.46951 17.4199 −14.6750 17.0158 0
1.16 1.08429 3.70278 −6.82431 0 4.01489 23.1268 −16.0739 −13.2894 0
1.17 1.13780 8.38350 −6.70541 0 9.53874 −32.7224 −16.7318 43.2831 0
1.18 2.01514 −0.462874 −3.93922 0 −0.932755 −11.6794 −24.0592 −26.7857 0
1.19 2.83850 3.37323 0.0570803 0 9.57492 −27.1568 −22.5460 −15.6213 0
1.20 3.21515 9.60601 2.33720 0 30.8848 2.75722 −18.2068 65.2754 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
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.h 26
5.b even 2 1 415.4.a.d ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
415.4.a.d ✓ 26 5.b even 2 1
2075.4.a.h 26 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} + 7 T_{2}^{25} - 143 T_{2}^{24} - 1058 T_{2}^{23} + 8664 T_{2}^{22} + 69057 T_{2}^{21} + \cdots + 90109228032 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2075))\). Copy content Toggle raw display