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Properties

Label 361.10.a.j
Level $361$
Weight $10$
Character orbit 361.a
Self dual yes
Analytic conductor $185.928$
Analytic rank $0$
Dimension $42$
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] = [361,10,Mod(1,361)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(361, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("361.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 361.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: \(185.927936855\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: no (minimal twist has level 19)
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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 48 q^{2} + 486 q^{3} + 9984 q^{4} + 852 q^{5} + 1536 q^{6} - 5715 q^{7} + 36861 q^{8} + 236196 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 48 q^{2} + 486 q^{3} + 9984 q^{4} + 852 q^{5} + 1536 q^{6} - 5715 q^{7} + 36861 q^{8} + 236196 q^{9} + 58464 q^{10} - 11295 q^{11} + 373245 q^{12} + 117150 q^{13} + 642912 q^{14} + 548451 q^{15} + 2162688 q^{16} + 997809 q^{17} + 1359318 q^{18} + 2160357 q^{20} + 1367568 q^{21} + 5031171 q^{22} + 2213706 q^{23} - 6545730 q^{24} + 13853862 q^{25} + 9164091 q^{26} + 9159258 q^{27} - 14025501 q^{28} + 9949038 q^{29} + 31527234 q^{30} + 14008185 q^{31} + 10978065 q^{32} + 14330319 q^{33} - 585930 q^{34} - 48129252 q^{35} + 7475208 q^{36} + 34869216 q^{37} + 15720432 q^{39} + 103478568 q^{40} + 86266776 q^{41} + 69259017 q^{42} - 12988605 q^{43} - 203224311 q^{44} + 61014765 q^{45} - 63830328 q^{46} + 17328918 q^{47} + 213121044 q^{48} + 163636377 q^{49} + 314983974 q^{50} + 96577575 q^{51} + 75305472 q^{52} + 103187364 q^{53} - 272434584 q^{54} - 56373354 q^{55} + 372695601 q^{56} + 287431734 q^{58} + 660784728 q^{59} + 368982060 q^{60} + 169972728 q^{61} + 300679941 q^{62} - 1173015477 q^{63} + 866999625 q^{64} - 153878493 q^{65} + 853102866 q^{66} + 189790482 q^{67} + 821601474 q^{68} + 1193576865 q^{69} - 629548281 q^{70} + 920008254 q^{71} - 118366770 q^{72} - 976066764 q^{73} - 782973429 q^{74} + 2015364294 q^{75} - 479229111 q^{77} + 2285947218 q^{78} + 2003560392 q^{79} + 3171353712 q^{80} + 1356085014 q^{81} - 1540624044 q^{82} + 1226988585 q^{83} + 1236853875 q^{84} - 372049179 q^{85} + 1472174664 q^{86} + 601158705 q^{87} + 2100827301 q^{88} + 853591443 q^{89} + 5492348259 q^{90} + 3495862842 q^{91} - 536774244 q^{92} - 1750089960 q^{93} + 7410646467 q^{94} + 4632812853 q^{96} + 1807268862 q^{97} + 5836044465 q^{98} + 777469992 q^{99}+O(q^{100}) \) 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 −44.8174 43.2874 1496.60 −242.785 −1940.03 −3029.65 −44127.1 −17809.2 10881.0
1.2 −40.9230 143.903 1162.70 1518.26 −5888.95 −7227.14 −26628.4 1025.10 −62131.6
1.3 −39.5082 −137.184 1048.89 −78.6288 5419.87 −2950.00 −21211.7 −863.640 3106.48
1.4 −39.1701 248.674 1022.30 −1256.48 −9740.57 −166.160 −19988.3 42155.6 49216.5
1.5 −36.5487 −78.6519 823.810 −1263.64 2874.63 3799.68 −11396.3 −13496.9 46184.3
1.6 −29.9877 −210.186 387.262 279.550 6303.00 −12302.0 3740.60 24495.3 −8383.07
1.7 −29.5161 −115.902 359.198 574.021 3420.98 10961.6 4510.12 −6249.64 −16942.8
1.8 −29.1328 17.8284 336.719 −75.8140 −519.391 10206.3 5106.42 −19365.1 2208.67
1.9 −27.8138 202.682 261.606 1212.37 −5637.35 132.925 6964.40 21397.0 −33720.6
1.10 −25.8270 112.150 155.035 2021.69 −2896.51 −9700.37 9219.35 −7105.29 −52214.2
1.11 −25.1217 −213.575 119.099 1408.64 5365.38 −3039.06 9870.34 25931.5 −35387.4
1.12 −24.1995 64.7558 73.6175 −2657.37 −1567.06 9.52165 10608.7 −15489.7 64307.0
1.13 −22.5036 233.999 −5.58674 −2258.01 −5265.82 −3766.80 11647.6 35072.4 50813.5
1.14 −19.8022 103.109 −119.872 1238.11 −2041.78 7532.98 12512.5 −9051.64 −24517.3
1.15 −14.0969 −182.238 −313.277 −1892.02 2569.00 −4502.76 11633.9 13527.7 26671.6
1.16 −10.7299 −190.529 −396.870 2270.76 2044.35 2159.10 9752.06 16618.2 −24365.0
1.17 −7.44560 −50.8856 −456.563 −406.582 378.874 −9956.34 7211.54 −17093.7 3027.25
1.18 −6.78857 113.043 −465.915 −1369.74 −767.398 −1512.12 6638.65 −6904.38 9298.60
1.19 −4.46004 101.331 −492.108 29.6030 −451.939 −3904.84 4478.37 −9415.12 −132.031
1.20 −2.39002 −57.0853 −506.288 2362.94 136.435 −2159.97 2433.73 −16424.3 −5647.49
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.42
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(19\) \( -1 \)

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 361.10.a.j 42
19.b odd 2 1 361.10.a.i 42
19.f odd 18 2 19.10.e.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.10.e.a ✓ 84 19.f odd 18 2
361.10.a.i 42 19.b odd 2 1
361.10.a.j 42 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} - 48 T_{2}^{41} - 14592 T_{2}^{40} + 708609 T_{2}^{39} + 97517520 T_{2}^{38} + \cdots + 11\!\cdots\!24 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(361))\). Copy content Toggle raw display