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Properties

Label 931.2.bn.a
Level $931$
Weight $2$
Character orbit 931.bn
Analytic conductor $7.434$
Analytic rank $0$
Dimension $1080$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(64,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([30, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.bn (of order \(21\), degree \(12\), minimal)

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: no
Analytic conductor: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(1080\)
Relative dimension: \(90\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1080 q - 5 q^{2} - 9 q^{3} + 79 q^{4} - 7 q^{5} - 13 q^{6} - 28 q^{7} - 40 q^{8} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1080 q - 5 q^{2} - 9 q^{3} + 79 q^{4} - 7 q^{5} - 13 q^{6} - 28 q^{7} - 40 q^{8} + 77 q^{9} - 11 q^{10} - 38 q^{11} - 16 q^{12} + q^{13} - 3 q^{14} - 11 q^{15} + 71 q^{16} + 10 q^{17} - 68 q^{18} - 8 q^{19} - 96 q^{20} - 3 q^{21} + 3 q^{22} - 13 q^{23} - 11 q^{24} + 67 q^{25} - 88 q^{26} - 12 q^{27} - 7 q^{28} - q^{29} - 88 q^{30} - 224 q^{31} - 25 q^{32} - 7 q^{33} - 13 q^{34} - 23 q^{35} + 55 q^{36} + 82 q^{37} + 14 q^{38} + 52 q^{39} - 7 q^{40} - 9 q^{41} + 119 q^{42} + 9 q^{43} - 21 q^{44} + 2 q^{45} + 4 q^{46} + 35 q^{47} + 138 q^{48} - 48 q^{49} - 4 q^{50} - 17 q^{51} - 99 q^{52} + 7 q^{53} - 30 q^{54} + 22 q^{55} + 172 q^{56} + 52 q^{57} - 16 q^{58} + 9 q^{59} - 15 q^{60} - 45 q^{61} - 55 q^{62} - 80 q^{63} - 200 q^{64} + 8 q^{65} - 95 q^{66} - 20 q^{67} - 92 q^{68} - 36 q^{69} - 19 q^{70} - 20 q^{71} - 136 q^{72} - 23 q^{73} - 23 q^{74} + 92 q^{75} + 103 q^{76} - 20 q^{77} - 109 q^{78} + 2 q^{79} + 48 q^{80} + 75 q^{81} - 24 q^{82} + 66 q^{83} - 206 q^{84} - 95 q^{85} - 12 q^{86} + 24 q^{87} + 80 q^{88} - 48 q^{89} - 87 q^{90} - 69 q^{91} + 63 q^{92} - 60 q^{93} - 284 q^{94} + 67 q^{95} - 148 q^{96} - 2 q^{97} + 37 q^{98} - 82 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −2.58683 + 0.797932i 3.16554 + 0.477128i 4.40252 − 3.00159i −2.42818 − 0.365989i −8.56943 + 1.29163i −1.35019 − 2.27530i −5.61782 + 7.04453i 6.92626 + 2.13647i 6.57332 − 0.990768i
64.2 −2.57920 + 0.795578i 0.624832 + 0.0941782i 4.36685 − 2.97727i −2.11037 − 0.318087i −1.68649 + 0.254198i 1.89976 + 1.84144i −5.52858 + 6.93263i −2.48517 − 0.766574i 5.69613 − 0.858553i
64.3 −2.56616 + 0.791555i 2.63378 + 0.396979i 4.30613 − 2.93587i 4.14847 + 0.625282i −7.07294 + 1.06607i −1.37729 + 2.25900i −5.37759 + 6.74328i 3.91251 + 1.20685i −11.1406 + 1.67917i
64.4 −2.54080 + 0.783733i −2.47902 − 0.373652i 4.18895 − 2.85598i 0.986936 + 0.148757i 6.59154 − 0.993515i −0.0415961 + 2.64542i −5.08933 + 6.38182i 3.13921 + 0.968318i −2.62419 + 0.395533i
64.5 −2.49267 + 0.768886i 0.371286 + 0.0559624i 3.96973 − 2.70652i 2.89912 + 0.436973i −0.968523 + 0.145981i 0.995070 − 2.45150i −4.56140 + 5.71981i −2.73200 − 0.842709i −7.56253 + 1.13987i
64.6 −2.47891 + 0.764642i −1.62665 − 0.245177i 3.90783 − 2.66431i −0.447056 − 0.0673829i 4.21978 − 0.636029i 1.99241 − 1.74078i −4.41504 + 5.53629i −0.280853 − 0.0866317i 1.15974 − 0.174802i
64.7 −2.42848 + 0.749086i 0.0748494 + 0.0112817i 3.68389 − 2.51164i −2.92759 − 0.441264i −0.190221 + 0.0286712i −2.39253 + 1.12952i −3.89577 + 4.88514i −2.86124 − 0.882577i 7.44014 − 1.12142i
64.8 −2.30464 + 0.710888i −2.93369 − 0.442183i 3.15354 − 2.15005i −1.14863 − 0.173128i 7.07546 − 1.06645i −1.17612 − 2.36997i −2.73189 + 3.42568i 5.54431 + 1.71019i 2.77025 − 0.417548i
64.9 −2.27435 + 0.701546i 0.923734 + 0.139231i 3.02805 − 2.06449i 0.528424 + 0.0796471i −2.19858 + 0.331382i −1.11122 + 2.40108i −2.47059 + 3.09802i −2.03282 − 0.627041i −1.25770 + 0.189568i
64.10 −2.20685 + 0.680724i −1.88768 − 0.284522i 2.75433 − 1.87787i −3.04964 − 0.459660i 4.35950 − 0.657090i −2.57827 − 0.593749i −1.92025 + 2.40791i 0.615656 + 0.189905i 7.04301 − 1.06156i
64.11 −2.18993 + 0.675504i 1.50531 + 0.226889i 2.68702 − 1.83198i 0.0974940 + 0.0146949i −3.44979 + 0.519973i −1.42517 − 2.22910i −1.78911 + 2.24348i −0.652236 − 0.201188i −0.223432 + 0.0336769i
64.12 −2.12456 + 0.655341i 2.45870 + 0.370589i 2.43182 − 1.65798i 1.32774 + 0.200124i −5.46651 + 0.823945i 2.63432 + 0.245650i −1.30754 + 1.63961i 3.04113 + 0.938064i −2.95201 + 0.444944i
64.13 −2.05701 + 0.634503i 1.09238 + 0.164649i 2.17622 − 1.48372i −4.38181 − 0.660452i −2.35150 + 0.354432i 2.05423 − 1.66738i −0.850769 + 1.06683i −1.70054 − 0.524547i 9.43249 − 1.42172i
64.14 −2.00440 + 0.618274i −0.995113 − 0.149989i 1.98286 − 1.35189i 3.04433 + 0.458860i 2.08734 − 0.314615i 2.00220 + 1.72950i −0.522954 + 0.655764i −1.89896 − 0.585753i −6.38575 + 0.962497i
64.15 −2.00036 + 0.617031i −2.93329 − 0.442122i 1.96825 − 1.34193i 2.51924 + 0.379715i 6.14045 − 0.925524i 1.86698 + 1.87467i −0.498824 + 0.625505i 5.54200 + 1.70948i −5.27370 + 0.794883i
64.16 −1.93306 + 0.596269i 2.89536 + 0.436405i 1.72870 − 1.17861i −1.67202 − 0.252016i −5.85711 + 0.882817i −1.79685 + 1.94199i −0.116352 + 0.145900i 5.32594 + 1.64283i 3.38238 − 0.509811i
64.17 −1.88300 + 0.580829i −0.847019 − 0.127668i 1.55585 − 1.06076i 3.25191 + 0.490147i 1.66909 − 0.251575i −2.62663 − 0.317545i 0.143681 − 0.180170i −2.16558 − 0.667992i −6.40804 + 0.965857i
64.18 −1.82143 + 0.561836i 0.618881 + 0.0932814i 1.34947 − 0.920050i 0.510303 + 0.0769158i −1.17966 + 0.177805i −0.435599 + 2.60965i 0.435844 − 0.546532i −2.49241 − 0.768805i −0.972695 + 0.146610i
64.19 −1.64980 + 0.508894i −0.505087 − 0.0761296i 0.810374 − 0.552503i −2.06282 − 0.310921i 0.872032 − 0.131438i −0.939428 − 2.47335i 1.09712 − 1.37575i −2.61740 − 0.807361i 3.56146 − 0.536804i
64.20 −1.63928 + 0.505650i 2.91850 + 0.439892i 0.779070 − 0.531161i 0.347665 + 0.0524021i −5.00665 + 0.754632i 1.85704 − 1.88452i 1.13065 − 1.41779i 5.45739 + 1.68338i −0.596417 + 0.0898954i
See next 80 embeddings (of 1080 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner
49.e even 7 1 inner
931.bn even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.bn.a ✓ 1080
19.c even 3 1 inner 931.2.bn.a ✓ 1080
49.e even 7 1 inner 931.2.bn.a ✓ 1080
931.bn even 21 1 inner 931.2.bn.a ✓ 1080
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.bn.a ✓ 1080 1.a even 1 1 trivial
931.2.bn.a ✓ 1080 19.c even 3 1 inner
931.2.bn.a ✓ 1080 49.e even 7 1 inner
931.2.bn.a ✓ 1080 931.bn even 21 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(931, [\chi])\).