↑

Properties

Label 74.8.f.b
Level $74$
Weight $8$
Character orbit 74.f
Analytic conductor $23.116$
Analytic rank $0$
Dimension $66$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [74,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.7");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(23.1164918858\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 39 q^{3} + 459 q^{5} - 918 q^{7} + 16896 q^{8} + 3237 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 39 q^{3} + 459 q^{5} - 918 q^{7} + 16896 q^{8} + 3237 q^{9} + 4704 q^{10} + 4539 q^{11} - 2496 q^{12} - 10542 q^{13} + 6744 q^{14} - 39894 q^{15} - 78822 q^{17} + 51792 q^{18} + 79461 q^{19} + 29376 q^{20} - 5835 q^{21} + 18168 q^{22} - 199479 q^{23} + 19968 q^{24} + 356301 q^{25} + 186744 q^{26} - 91689 q^{27} - 91584 q^{28} - 38526 q^{29} + 319152 q^{30} + 1917246 q^{31} - 902847 q^{33} - 205776 q^{34} - 1574793 q^{35} + 3079296 q^{36} - 396513 q^{37} - 692880 q^{38} + 2514306 q^{39} + 470016 q^{40} + 2989350 q^{41} + 46680 q^{42} + 6006348 q^{43} - 145344 q^{44} + 1083726 q^{45} - 588744 q^{46} - 894135 q^{47} + 7973742 q^{49} + 2012400 q^{50} - 524880 q^{51} - 2262720 q^{52} + 1457844 q^{53} + 1452312 q^{54} + 2415690 q^{55} - 1202688 q^{56} + 2734338 q^{57} - 3423360 q^{58} + 3099522 q^{59} - 3102912 q^{60} - 10968396 q^{61} - 3067320 q^{62} - 10727970 q^{63} - 8650752 q^{64} - 13497300 q^{65} - 4598712 q^{66} - 1978119 q^{67} + 5026176 q^{68} - 7251108 q^{69} + 435744 q^{70} - 19555002 q^{71} + 3314688 q^{72} - 31332840 q^{73} + 7607472 q^{74} + 44758074 q^{75} + 5085504 q^{76} + 1982007 q^{77} - 22512264 q^{78} + 10522902 q^{79} + 4816896 q^{80} - 7548876 q^{81} - 7150320 q^{82} + 19771926 q^{83} + 3556224 q^{84} - 6684072 q^{85} + 22982496 q^{86} - 67769667 q^{87} - 2323968 q^{88} - 42018963 q^{89} - 25158672 q^{90} + 4847565 q^{91} - 1096704 q^{92} - 14558475 q^{93} + 3175680 q^{94} + 4508415 q^{95} - 2555904 q^{96} - 39286662 q^{97} + 71630904 q^{98} + 66155463 q^{99}+O(q^{100}) \) 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 7.51754 − 2.73616i −83.1185 − 30.2527i 49.0268 − 41.1384i 77.4784 + 439.402i −707.623 −71.3442 − 404.613i 256.000 − 443.405i 4318.12 + 3623.33i 1784.72 + 3091.23i
7.2 7.51754 − 2.73616i −62.7580 − 22.8420i 49.0268 − 41.1384i 1.66323 + 9.43265i −534.285 141.286 + 801.274i 256.000 − 443.405i 1741.46 + 1461.26i 38.3126 + 66.3594i
7.3 7.51754 − 2.73616i −56.1799 − 20.4478i 49.0268 − 41.1384i −76.5259 − 434.000i −478.283 −144.408 − 818.976i 256.000 − 443.405i 1062.73 + 891.735i −1762.78 − 3053.23i
7.4 7.51754 − 2.73616i −36.4042 − 13.2501i 49.0268 − 41.1384i −1.10281 − 6.25436i −309.925 125.532 + 711.929i 256.000 − 443.405i −525.635 − 441.061i −25.4034 − 43.9999i
7.5 7.51754 − 2.73616i −19.5402 − 7.11207i 49.0268 − 41.1384i 69.5532 + 394.456i −166.354 −65.8495 − 373.451i 256.000 − 443.405i −1344.10 − 1127.83i 1602.16 + 2775.03i
7.6 7.51754 − 2.73616i 6.89307 + 2.50887i 49.0268 − 41.1384i −27.8528 − 157.961i 58.6836 117.132 + 664.288i 256.000 − 443.405i −1634.12 − 1371.19i −641.591 − 1111.27i
7.7 7.51754 − 2.73616i 25.6856 + 9.34879i 49.0268 − 41.1384i 46.1726 + 261.858i 218.672 −206.971 − 1173.79i 256.000 − 443.405i −1102.99 − 925.518i 1063.59 + 1842.19i
7.8 7.51754 − 2.73616i 28.9563 + 10.5392i 49.0268 − 41.1384i −48.7600 − 276.532i 246.517 −219.425 − 1244.42i 256.000 − 443.405i −947.948 − 795.422i −1123.19 − 1945.43i
7.9 7.51754 − 2.73616i 46.7880 + 17.0295i 49.0268 − 41.1384i 58.4927 + 331.728i 398.326 201.162 + 1140.85i 256.000 − 443.405i 223.779 + 187.773i 1347.38 + 2333.74i
7.10 7.51754 − 2.73616i 65.8840 + 23.9798i 49.0268 − 41.1384i 42.5352 + 241.229i 560.898 44.8916 + 254.593i 256.000 − 443.405i 2090.33 + 1754.00i 979.802 + 1697.07i
7.11 7.51754 − 2.73616i 79.5512 + 28.9543i 49.0268 − 41.1384i −57.6869 − 327.158i 677.253 −6.84167 − 38.8010i 256.000 − 443.405i 3814.71 + 3200.92i −1328.82 − 2301.59i
9.1 −1.38919 − 7.87846i −14.0898 + 79.9071i −60.1403 + 21.8893i −46.9471 − 39.3933i 649.118 −514.898 − 432.050i 256.000 + 443.405i −4131.51 − 1503.75i −245.140 + 424.595i
9.2 −1.38919 − 7.87846i −11.0757 + 62.8133i −60.1403 + 21.8893i 48.5860 + 40.7685i 510.258 745.438 + 625.497i 256.000 + 443.405i −1767.73 − 643.401i 253.698 − 439.418i
9.3 −1.38919 − 7.87846i −8.32711 + 47.2254i −60.1403 + 21.8893i −300.074 − 251.792i 383.631 −303.488 − 254.657i 256.000 + 443.405i −105.789 − 38.5041i −1566.88 + 2713.91i
9.4 −1.38919 − 7.87846i −6.48463 + 36.7762i −60.1403 + 21.8893i 381.143 + 319.817i 298.748 975.665 + 818.680i 256.000 + 443.405i 744.672 + 271.038i 1990.19 − 3447.11i
9.5 −1.38919 − 7.87846i −2.39984 + 13.6102i −60.1403 + 21.8893i −34.7022 − 29.1186i 110.561 −140.013 − 117.485i 256.000 + 443.405i 1875.63 + 682.674i −181.202 + 313.852i
9.6 −1.38919 − 7.87846i −0.307798 + 1.74561i −60.1403 + 21.8893i 323.926 + 271.806i 14.1803 −1225.86 − 1028.62i 256.000 + 443.405i 2052.16 + 746.923i 1691.42 − 2929.63i
9.7 −1.38919 − 7.87846i 2.48533 − 14.0950i −60.1403 + 21.8893i −356.271 − 298.947i −114.500 1364.62 + 1145.05i 256.000 + 443.405i 1862.62 + 677.936i −1860.32 + 3222.16i
9.8 −1.38919 − 7.87846i 6.70779 − 38.0417i −60.1403 + 21.8893i 124.780 + 104.703i −309.029 25.1771 + 21.1261i 256.000 + 443.405i 652.928 + 237.646i 651.556 − 1128.53i
9.9 −1.38919 − 7.87846i 9.76159 − 55.3608i −60.1403 + 21.8893i −282.513 − 237.057i −449.718 −1337.20 − 1122.04i 256.000 + 443.405i −914.416 − 332.820i −1475.18 + 2555.09i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.11
Significant digits:
Format:

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 74.8.f.b ✓ 66
37.f even 9 1 inner 74.8.f.b ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.f.b ✓ 66 1.a even 1 1 trivial
74.8.f.b ✓ 66 37.f even 9 1 inner