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Properties

Label 361.6.a.k
Level $361$
Weight $6$
Character orbit 361.a
Self dual yes
Analytic conductor $57.899$
Analytic rank $1$
Dimension $21$
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] = [361,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("361.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(57.8985589525\)
Analytic rank: \(1\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 12 q^{2} - 27 q^{3} + 240 q^{4} - 33 q^{5} + 96 q^{6} + 180 q^{7} - 573 q^{8} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q - 12 q^{2} - 27 q^{3} + 240 q^{4} - 33 q^{5} + 96 q^{6} + 180 q^{7} - 573 q^{8} + 972 q^{9} - 504 q^{10} + 240 q^{11} - 861 q^{12} - 228 q^{13} - 2088 q^{14} - 1971 q^{15} + 768 q^{16} - 2832 q^{17} - 6330 q^{18} + 2997 q^{20} - 9873 q^{21} - 3339 q^{22} + 5442 q^{23} - 1218 q^{24} - 1458 q^{25} - 2637 q^{26} - 858 q^{27} + 13059 q^{28} - 18201 q^{29} - 5886 q^{30} - 23841 q^{31} - 7473 q^{32} + 1821 q^{33} - 4818 q^{34} + 3723 q^{35} + 29688 q^{36} - 10032 q^{37} - 2388 q^{39} - 44328 q^{40} - 32484 q^{41} + 31089 q^{42} - 47508 q^{43} + 52425 q^{44} - 55725 q^{45} - 83112 q^{46} + 22770 q^{47} - 99540 q^{48} + 14349 q^{49} - 39594 q^{50} - 32307 q^{51} + 4704 q^{52} - 80922 q^{53} + 144996 q^{54} + 42651 q^{55} - 49809 q^{56} + 82662 q^{58} - 84957 q^{59} + 4980 q^{60} + 20019 q^{61} - 248211 q^{62} + 223299 q^{63} - 237975 q^{64} - 271482 q^{65} + 182622 q^{66} - 156318 q^{67} - 186078 q^{68} - 162690 q^{69} + 133641 q^{70} - 161262 q^{71} - 511278 q^{72} + 223008 q^{73} + 76131 q^{74} - 22329 q^{75} - 116997 q^{77} - 108138 q^{78} - 16269 q^{79} + 200832 q^{80} - 452847 q^{81} + 213204 q^{82} - 297720 q^{83} - 892659 q^{84} + 273651 q^{85} - 479484 q^{86} - 217299 q^{87} - 549381 q^{88} + 44205 q^{89} - 35259 q^{90} - 552945 q^{91} + 319356 q^{92} + 38088 q^{93} - 606699 q^{94} + 188277 q^{96} - 557115 q^{97} - 510045 q^{98} + 142458 q^{99}+O(q^{100}) \) 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 −10.2544 −26.4937 73.1517 −5.12360 271.676 173.333 −421.984 458.915 52.5392
1.2 −10.2129 14.3749 72.3025 65.3174 −146.808 −186.125 −411.604 −36.3634 −667.078
1.3 −8.91400 −14.5915 47.4593 31.5762 130.069 63.2843 −137.805 −30.0878 −281.470
1.4 −8.17963 25.4306 34.9063 −46.5123 −208.013 196.394 −23.7729 403.715 380.453
1.5 −7.57463 −23.2378 25.3750 17.6617 176.018 172.038 50.1821 296.997 −133.781
1.6 −7.53738 12.9787 24.8122 −1.21278 −97.8252 −58.6562 54.1775 −74.5540 9.14118
1.7 −5.51879 5.94065 −1.54295 −91.0666 −32.7852 −35.5326 185.117 −207.709 502.578
1.8 −3.47543 7.94464 −19.9214 54.4454 −27.6110 −183.423 180.449 −179.883 −189.221
1.9 −1.81905 −14.9069 −28.6911 −100.455 27.1163 −10.7711 110.400 −20.7857 182.733
1.10 −1.46769 −9.62996 −29.8459 95.1614 14.1338 97.8706 90.7706 −150.264 −139.667
1.11 −1.00142 −17.8641 −30.9972 −68.0171 17.8895 −208.515 63.0867 76.1263 68.1138
1.12 −0.337551 10.7337 −31.8861 60.5407 −3.62316 81.4322 21.5648 −127.789 −20.4356
1.13 2.05166 17.8017 −27.7907 −51.3282 36.5230 168.186 −122.670 73.9011 −105.308
1.14 2.70634 −19.6473 −24.6757 −9.58097 −53.1722 −121.611 −153.384 143.016 −25.9293
1.15 3.84404 −22.1689 −17.2234 84.1324 −85.2179 119.889 −189.216 248.458 323.408
1.16 4.52405 17.1735 −11.5329 −8.41910 77.6939 41.7123 −196.945 51.9299 −38.0884
1.17 6.25174 26.8960 7.08421 −64.4365 168.147 −100.365 −155.767 480.393 −402.840
1.18 7.69522 11.6782 27.2164 46.6786 89.8666 −192.771 −36.8109 −106.619 359.202
1.19 8.39336 −18.4914 38.4484 −28.6904 −155.205 129.010 54.1238 98.9323 −240.809
1.20 9.15366 −5.32766 51.7896 10.2169 −48.7676 82.9588 181.147 −214.616 93.5224
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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 361.6.a.k 21
19.b odd 2 1 361.6.a.l 21
19.e even 9 2 19.6.e.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.6.e.a ✓ 42 19.e even 9 2
361.6.a.k 21 1.a even 1 1 trivial
361.6.a.l 21 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} + 12 T_{2}^{20} - 384 T_{2}^{19} - 4737 T_{2}^{18} + 60660 T_{2}^{17} + 774717 T_{2}^{16} + \cdots - 26051659333632 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(361))\). Copy content Toggle raw display