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Properties

Label 74.5.i.a
Level $74$
Weight $5$
Character orbit 74.i
Analytic conductor $7.649$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [74,5,Mod(5,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([23]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 74.i (of order \(36\), 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: \(7.64937726820\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 78 q^{5} + 576 q^{8} - 210 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 78 q^{5} + 576 q^{8} - 210 q^{9} + 36 q^{10} + 720 q^{11} - 240 q^{12} - 144 q^{13} + 288 q^{14} - 66 q^{15} + 264 q^{17} + 840 q^{18} + 648 q^{19} - 624 q^{20} + 3450 q^{21} + 432 q^{22} - 4518 q^{23} - 480 q^{24} + 162 q^{25} - 300 q^{26} - 5832 q^{27} - 1968 q^{28} - 684 q^{29} + 2532 q^{30} + 10680 q^{31} + 408 q^{33} + 1116 q^{34} + 5808 q^{35} - 12 q^{37} - 6672 q^{38} + 2346 q^{39} - 2112 q^{40} - 7236 q^{41} - 7320 q^{42} - 5208 q^{43} - 25668 q^{45} + 6960 q^{46} + 4518 q^{47} + 3456 q^{48} + 5550 q^{49} + 13608 q^{50} - 6480 q^{51} - 1152 q^{52} + 4146 q^{53} + 21708 q^{54} - 18054 q^{55} - 2112 q^{56} + 5808 q^{57} + 2076 q^{58} - 14262 q^{59} - 19008 q^{60} + 8208 q^{61} - 8340 q^{62} + 1296 q^{63} + 2094 q^{65} - 22140 q^{66} + 6258 q^{67} + 1584 q^{68} + 13530 q^{69} + 18732 q^{70} + 40176 q^{71} - 6720 q^{72} + 1104 q^{74} - 36672 q^{75} + 5184 q^{76} + 28752 q^{77} + 7992 q^{78} + 14712 q^{79} + 1152 q^{80} - 1212 q^{81} - 28776 q^{82} + 41484 q^{83} + 7020 q^{85} - 18456 q^{86} - 37380 q^{87} + 7680 q^{88} - 17484 q^{89} + 40752 q^{90} + 68682 q^{91} - 39648 q^{92} - 25092 q^{93} + 540 q^{94} + 3762 q^{95} - 7680 q^{96} - 78294 q^{97} + 6720 q^{98} - 30378 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.19534 + 2.56343i −3.97829 + 10.9303i −5.14230 − 6.12836i −26.2346 + 37.4669i −23.2635 − 23.2635i −4.31069 − 24.4472i 21.8564 − 5.85641i −41.5942 − 34.9017i −64.6843 − 112.036i
5.2 −1.19534 + 2.56343i −3.83349 + 10.5324i −5.14230 − 6.12836i 18.1658 − 25.9435i −22.4167 − 22.4167i −12.7637 − 72.3868i 21.8564 − 5.85641i −34.1868 − 28.6861i 44.7897 + 77.5781i
5.3 −1.19534 + 2.56343i −0.787669 + 2.16410i −5.14230 − 6.12836i 11.8497 − 16.9232i −4.60598 − 4.60598i 6.69926 + 37.9934i 21.8564 − 5.85641i 57.9867 + 48.6566i 29.2168 + 50.6049i
5.4 −1.19534 + 2.56343i 1.78189 − 4.89570i −5.14230 − 6.12836i −6.16941 + 8.81083i 10.4198 + 10.4198i 0.427138 + 2.42242i 21.8564 − 5.85641i 41.2568 + 34.6186i −15.2113 − 26.3468i
5.5 −1.19534 + 2.56343i 3.21518 − 8.83363i −5.14230 − 6.12836i −8.81494 + 12.5890i 18.8011 + 18.8011i −6.42919 − 36.4617i 21.8564 − 5.85641i −5.64597 − 4.73753i −21.7342 − 37.6447i
5.6 −1.19534 + 2.56343i 5.97348 − 16.4120i −5.14230 − 6.12836i 22.9095 − 32.7181i 34.9306 + 34.9306i 10.5058 + 59.5815i 21.8564 − 5.85641i −171.622 − 144.008i 56.4858 + 97.8362i
13.1 −1.62232 − 2.31691i −16.6369 + 2.93353i −2.73616 + 7.51754i −17.5967 − 1.53951i 33.7871 + 33.7871i −63.9596 + 53.6684i 21.8564 − 5.85641i 192.065 − 69.9059i 24.9806 + 43.2677i
13.2 −1.62232 − 2.31691i −7.71289 + 1.35999i −2.73616 + 7.51754i 20.5848 + 1.80093i 15.6637 + 15.6637i 27.9792 − 23.4774i 21.8564 − 5.85641i −18.4760 + 6.72472i −29.2225 − 50.6148i
13.3 −1.62232 − 2.31691i −6.74492 + 1.18931i −2.73616 + 7.51754i −42.7538 − 3.74047i 13.6979 + 13.6979i 40.9477 − 34.3592i 21.8564 − 5.85641i −32.0356 + 11.6600i 60.6940 + 105.125i
13.4 −1.62232 − 2.31691i 0.577959 − 0.101910i −2.73616 + 7.51754i 31.2527 + 2.73426i −1.17375 − 1.17375i −47.1074 + 39.5278i 21.8564 − 5.85641i −75.7915 + 27.5858i −44.3668 − 76.8455i
13.5 −1.62232 − 2.31691i 3.97925 − 0.701649i −2.73616 + 7.51754i −12.1726 − 1.06496i −8.08127 − 8.08127i −6.73788 + 5.65375i 21.8564 − 5.85641i −60.7730 + 22.1196i 17.2803 + 29.9304i
13.6 −1.62232 − 2.31691i 13.8762 − 2.44675i −2.73616 + 7.51754i 3.32003 + 0.290465i −28.1805 − 28.1805i 30.1596 − 25.3069i 21.8564 − 5.85641i 110.447 − 40.1995i −4.71317 − 8.16344i
15.1 −1.19534 − 2.56343i −3.97829 − 10.9303i −5.14230 + 6.12836i −26.2346 − 37.4669i −23.2635 + 23.2635i −4.31069 + 24.4472i 21.8564 + 5.85641i −41.5942 + 34.9017i −64.6843 + 112.036i
15.2 −1.19534 − 2.56343i −3.83349 − 10.5324i −5.14230 + 6.12836i 18.1658 + 25.9435i −22.4167 + 22.4167i −12.7637 + 72.3868i 21.8564 + 5.85641i −34.1868 + 28.6861i 44.7897 − 77.5781i
15.3 −1.19534 − 2.56343i −0.787669 − 2.16410i −5.14230 + 6.12836i 11.8497 + 16.9232i −4.60598 + 4.60598i 6.69926 − 37.9934i 21.8564 + 5.85641i 57.9867 − 48.6566i 29.2168 − 50.6049i
15.4 −1.19534 − 2.56343i 1.78189 + 4.89570i −5.14230 + 6.12836i −6.16941 − 8.81083i 10.4198 − 10.4198i 0.427138 − 2.42242i 21.8564 + 5.85641i 41.2568 − 34.6186i −15.2113 + 26.3468i
15.5 −1.19534 − 2.56343i 3.21518 + 8.83363i −5.14230 + 6.12836i −8.81494 − 12.5890i 18.8011 − 18.8011i −6.42919 + 36.4617i 21.8564 + 5.85641i −5.64597 + 4.73753i −21.7342 + 37.6447i
15.6 −1.19534 − 2.56343i 5.97348 + 16.4120i −5.14230 + 6.12836i 22.9095 + 32.7181i 34.9306 − 34.9306i 10.5058 − 59.5815i 21.8564 + 5.85641i −171.622 + 144.008i 56.4858 − 97.8362i
17.1 2.31691 + 1.62232i −12.5684 − 2.21615i 2.73616 + 7.51754i −1.85059 − 21.1523i −25.5246 − 25.5246i 54.5439 + 45.7677i −5.85641 + 21.8564i 76.9391 + 28.0035i 30.0282 − 52.0103i
17.2 2.31691 + 1.62232i −7.41481 − 1.30743i 2.73616 + 7.51754i 0.755709 + 8.63779i −15.0584 − 15.0584i 13.1872 + 11.0654i −5.85641 + 21.8564i −22.8451 − 8.31492i −12.2623 + 21.2390i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.i odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.5.i.a ✓ 72
37.i odd 36 1 inner 74.5.i.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.5.i.a ✓ 72 1.a even 1 1 trivial
74.5.i.a ✓ 72 37.i odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} + 105 T_{3}^{70} + 4860 T_{3}^{69} - 6942 T_{3}^{68} - 445392 T_{3}^{67} + \cdots + 75\!\cdots\!24 \) acting on \(S_{5}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display