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Properties

Label 74.6.f.a
Level $74$
Weight $6$
Character orbit 74.f
Analytic conductor $11.868$
Analytic rank $0$
Dimension $48$
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] = [74,6,Mod(7,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), 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: \(11.8684026662\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{3} - 72 q^{5} + 114 q^{7} - 1536 q^{8} - 93 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{3} - 72 q^{5} + 114 q^{7} - 1536 q^{8} - 93 q^{9} - 648 q^{10} - 9 q^{11} - 48 q^{12} + 288 q^{13} - 660 q^{14} - 3678 q^{15} - 2475 q^{17} + 744 q^{18} + 945 q^{19} - 1152 q^{20} - 2073 q^{21} - 60 q^{22} - 5739 q^{23} - 192 q^{24} - 3834 q^{25} - 5304 q^{26} - 1077 q^{27} - 5232 q^{28} - 11640 q^{29} - 14712 q^{30} - 20394 q^{31} - 62013 q^{33} + 7812 q^{34} + 27999 q^{35} + 62208 q^{36} + 13605 q^{37} + 60696 q^{38} + 25590 q^{39} + 9216 q^{40} - 18267 q^{41} - 8292 q^{42} + 2208 q^{43} - 240 q^{44} - 48132 q^{45} - 40308 q^{46} - 663 q^{47} + 58326 q^{49} + 61776 q^{50} + 8748 q^{51} - 31536 q^{52} + 20940 q^{53} + 37836 q^{54} - 158868 q^{55} + 13632 q^{56} + 164910 q^{57} - 28068 q^{58} + 111258 q^{59} - 31824 q^{60} + 88758 q^{61} + 116100 q^{62} - 202488 q^{63} - 98304 q^{64} + 155997 q^{65} - 53532 q^{66} + 52503 q^{67} + 32544 q^{68} + 298758 q^{69} - 5760 q^{70} + 160434 q^{71} + 11904 q^{72} - 477210 q^{73} + 8844 q^{74} + 808230 q^{75} + 15120 q^{76} - 510765 q^{77} + 26004 q^{78} - 378768 q^{79} + 82944 q^{80} + 55476 q^{81} - 149064 q^{82} - 230082 q^{83} - 42336 q^{84} + 55653 q^{85} - 149904 q^{86} - 453051 q^{87} - 576 q^{88} + 492789 q^{89} + 12276 q^{90} + 220839 q^{91} + 176160 q^{92} - 109767 q^{93} - 6768 q^{94} - 630843 q^{95} + 6144 q^{96} - 632190 q^{97} - 194628 q^{98} - 414765 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.75877 + 1.36808i −23.1101 − 8.41139i 12.2567 − 10.2846i 6.03710 + 34.2381i 98.3730 19.4454 + 110.280i −32.0000 + 55.4256i 277.176 + 232.578i −69.5325 − 120.434i
7.2 −3.75877 + 1.36808i −20.0307 − 7.29057i 12.2567 − 10.2846i −15.8864 − 90.0960i 85.2648 −21.3019 − 120.809i −32.0000 + 55.4256i 161.927 + 135.873i 182.972 + 316.916i
7.3 −3.75877 + 1.36808i −13.8557 − 5.04306i 12.2567 − 10.2846i 11.2687 + 63.9081i 58.9797 −40.1269 − 227.571i −32.0000 + 55.4256i −19.6009 − 16.4471i −129.788 − 224.799i
7.4 −3.75877 + 1.36808i −2.35676 − 0.857791i 12.2567 − 10.2846i −1.87431 − 10.6298i 10.0321 20.8226 + 118.091i −32.0000 + 55.4256i −181.330 − 152.154i 21.5875 + 37.3906i
7.5 −3.75877 + 1.36808i 7.77948 + 2.83150i 12.2567 − 10.2846i −14.8075 − 83.9778i −33.1150 8.29469 + 47.0415i −32.0000 + 55.4256i −133.646 − 112.142i 170.547 + 295.395i
7.6 −3.75877 + 1.36808i 9.99337 + 3.63729i 12.2567 − 10.2846i 0.180282 + 1.02243i −42.5389 −23.4499 − 132.991i −32.0000 + 55.4256i −99.5112 − 83.4998i −2.07641 − 3.59644i
7.7 −3.75877 + 1.36808i 13.8016 + 5.02336i 12.2567 − 10.2846i 19.0297 + 107.923i −58.7493 6.51125 + 36.9271i −32.0000 + 55.4256i −20.8996 − 17.5369i −219.175 − 379.623i
7.8 −3.75877 + 1.36808i 27.4525 + 9.99188i 12.2567 − 10.2846i −2.40304 − 13.6283i −116.857 22.7293 + 128.904i −32.0000 + 55.4256i 467.651 + 392.406i 27.6772 + 47.9382i
9.1 0.694593 + 3.93923i −4.94687 + 28.0551i −15.0351 + 5.47232i 9.88188 + 8.29188i −113.952 −125.671 − 105.450i −32.0000 − 55.4256i −534.272 − 194.459i −25.7998 + 44.6865i
9.2 0.694593 + 3.93923i −2.94557 + 16.7052i −15.0351 + 5.47232i 46.4233 + 38.9537i −67.8514 193.520 + 162.382i −32.0000 − 55.4256i −42.0404 − 15.3014i −121.203 + 209.929i
9.3 0.694593 + 3.93923i −2.84323 + 16.1248i −15.0351 + 5.47232i −37.4537 − 31.4274i −65.4941 −21.7905 − 18.2844i −32.0000 − 55.4256i −23.5794 − 8.58219i 97.7847 − 169.368i
9.4 0.694593 + 3.93923i 0.515998 − 2.92637i −15.0351 + 5.47232i 33.8915 + 28.4384i 11.8861 −11.2213 − 9.41575i −32.0000 − 55.4256i 220.048 + 80.0909i −88.4845 + 153.260i
9.5 0.694593 + 3.93923i 0.677795 − 3.84397i −15.0351 + 5.47232i −48.1268 − 40.3832i 15.6131 36.5328 + 30.6547i −32.0000 − 55.4256i 214.029 + 77.9001i 125.650 − 217.633i
9.6 0.694593 + 3.93923i 1.57867 − 8.95310i −15.0351 + 5.47232i 22.3419 + 18.7471i 36.3648 −145.430 − 122.031i −32.0000 − 55.4256i 150.680 + 54.8429i −58.3305 + 101.031i
9.7 0.694593 + 3.93923i 3.82058 − 21.6676i −15.0351 + 5.47232i −48.1841 − 40.4312i 88.0075 134.280 + 112.674i −32.0000 − 55.4256i −226.543 − 82.4550i 125.800 − 217.891i
9.8 0.694593 + 3.93923i 4.40867 − 25.0028i −15.0351 + 5.47232i 68.9775 + 57.8790i 101.554 55.3841 + 46.4727i −32.0000 − 55.4256i −377.359 − 137.347i −180.088 + 311.921i
33.1 0.694593 − 3.93923i −4.94687 − 28.0551i −15.0351 − 5.47232i 9.88188 − 8.29188i −113.952 −125.671 + 105.450i −32.0000 + 55.4256i −534.272 + 194.459i −25.7998 − 44.6865i
33.2 0.694593 − 3.93923i −2.94557 − 16.7052i −15.0351 − 5.47232i 46.4233 − 38.9537i −67.8514 193.520 − 162.382i −32.0000 + 55.4256i −42.0404 + 15.3014i −121.203 − 209.929i
33.3 0.694593 − 3.93923i −2.84323 − 16.1248i −15.0351 − 5.47232i −37.4537 + 31.4274i −65.4941 −21.7905 + 18.2844i −32.0000 + 55.4256i −23.5794 + 8.58219i 97.7847 + 169.368i
33.4 0.694593 − 3.93923i 0.515998 + 2.92637i −15.0351 − 5.47232i 33.8915 − 28.4384i 11.8861 −11.2213 + 9.41575i −32.0000 + 55.4256i 220.048 − 80.0909i −88.4845 − 153.260i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.6.f.a ✓ 48
37.f even 9 1 inner 74.6.f.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.f.a ✓ 48 1.a even 1 1 trivial
74.6.f.a ✓ 48 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 3 T_{3}^{47} + 51 T_{3}^{46} + 503 T_{3}^{45} + 5451 T_{3}^{44} + 7013421 T_{3}^{43} + \cdots + 10\!\cdots\!24 \) acting on \(S_{6}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display