Properties

Label 74.4.c.a
Level $74$
Weight $4$
Character orbit 74.c
Analytic conductor $4.366$
Analytic rank $0$
Dimension $8$
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] = [74,4,Mod(47,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.47");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 74.c (of order \(3\), degree \(2\), 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: \(4.36614134042\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 48x^{6} + 157x^{5} + 1944x^{4} + 3005x^{3} + 12895x^{2} - 11550x + 44100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + 2) q^{2} + ( - \beta_{2} - \beta_1) q^{3} - 4 \beta_{2} q^{4} + (\beta_{7} - 2 \beta_{2}) q^{5} + (2 \beta_{3} - 2 \beta_1 - 2) q^{6} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + \beta_1) q^{7} - 8 q^{8} + (\beta_{6} + 5 \beta_{3} - 3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + 2) q^{2} + ( - \beta_{2} - \beta_1) q^{3} - 4 \beta_{2} q^{4} + (\beta_{7} - 2 \beta_{2}) q^{5} + (2 \beta_{3} - 2 \beta_1 - 2) q^{6} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + \beta_1) q^{7} - 8 q^{8} + (\beta_{6} + 5 \beta_{3} - 3 \beta_{2} + 3) q^{9} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{2} - 6) q^{10} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 10) q^{11} + (4 \beta_{3} + 4 \beta_{2} - 4) q^{12} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 14 \beta_{2} - 3 \beta_1) q^{13} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{14} + (\beta_{6} + 3 \beta_{5} + 5 \beta_{3} + 8 \beta_{2} - 8) q^{15} + (16 \beta_{2} - 16) q^{16} + (2 \beta_{6} - 4 \beta_{5} - 8 \beta_{3} + \beta_{2} - 1) q^{17} + (2 \beta_{6} + 2 \beta_{4} - 6 \beta_{2} + 10 \beta_1) q^{18} + ( - 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 18 \beta_{2} - 5 \beta_1) q^{19} + ( - 4 \beta_{5} + 12 \beta_{2} - 12) q^{20} + ( - 3 \beta_{6} + 4 \beta_{5} - 11 \beta_{3} - 16 \beta_{2} + 16) q^{21} + ( - 4 \beta_{6} + 4 \beta_{5} + 8 \beta_{3} - 20 \beta_{2} + 20) q^{22} + (6 \beta_{7} - 6 \beta_{5} - 7 \beta_{4} - 6 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 11) q^{23} + (8 \beta_{2} + 8 \beta_1) q^{24} + (3 \beta_{6} - 5 \beta_{5} - 18 \beta_{3} - 5 \beta_{2} + 5) q^{25} + (2 \beta_{7} - 2 \beta_{5} - 4 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - 6 \beta_1 - 30) q^{26} + ( - \beta_{7} + \beta_{5} + 8 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 83) q^{27} + ( - 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{3}) q^{28} + ( - 11 \beta_{7} + 11 \beta_{5} + \beta_{4} - 19 \beta_{3} - 11 \beta_{2} + \cdots + 41) q^{29}+ \cdots + ( - 30 \beta_{6} + 38 \beta_{5} + 124 \beta_{3} - 6 \beta_{2} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 5 q^{3} - 16 q^{4} - 10 q^{5} - 20 q^{6} + 3 q^{7} - 64 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 5 q^{3} - 16 q^{4} - 10 q^{5} - 20 q^{6} + 3 q^{7} - 64 q^{8} + 7 q^{9} - 40 q^{10} + 64 q^{11} - 20 q^{12} - 61 q^{13} + 12 q^{14} - 43 q^{15} - 64 q^{16} + 12 q^{17} - 14 q^{18} - 71 q^{19} - 40 q^{20} + 67 q^{21} + 64 q^{22} - 52 q^{23} + 40 q^{24} + 48 q^{25} - 244 q^{26} + 658 q^{27} + 12 q^{28} + 322 q^{29} + 86 q^{30} - 112 q^{31} + 128 q^{32} + 280 q^{33} - 24 q^{34} - 359 q^{35} - 56 q^{36} + 557 q^{37} - 284 q^{38} - 389 q^{39} + 80 q^{40} + 92 q^{41} - 134 q^{42} + 532 q^{43} - 128 q^{44} + 330 q^{45} - 52 q^{46} + 280 q^{47} + 160 q^{48} + 87 q^{49} - 96 q^{50} - 1306 q^{51} - 244 q^{52} + 159 q^{53} + 658 q^{54} - 872 q^{55} - 24 q^{56} - 469 q^{57} + 322 q^{58} + 263 q^{59} + 344 q^{60} - 206 q^{61} - 112 q^{62} - 2328 q^{63} + 512 q^{64} - 731 q^{65} + 1120 q^{66} + 245 q^{67} - 96 q^{68} - 360 q^{69} + 718 q^{70} - 957 q^{71} - 56 q^{72} - 272 q^{73} - 178 q^{74} - 3232 q^{75} - 284 q^{76} + 744 q^{77} + 778 q^{78} + 173 q^{79} + 320 q^{80} - 528 q^{81} + 368 q^{82} + 1217 q^{83} - 536 q^{84} + 2988 q^{85} + 532 q^{86} - 2336 q^{87} - 512 q^{88} - 2136 q^{89} + 330 q^{90} + 1575 q^{91} + 104 q^{92} + 2608 q^{93} + 280 q^{94} + 891 q^{95} + 160 q^{96} + 5262 q^{97} - 174 q^{98} - 176 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 48x^{6} + 157x^{5} + 1944x^{4} + 3005x^{3} + 12895x^{2} - 11550x + 44100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1161547 \nu^{7} + 2985827 \nu^{6} + 46810062 \nu^{5} + 367759321 \nu^{4} + 2694757026 \nu^{3} + 11047848851 \nu^{2} + 15096834055 \nu + 39927249180 ) / 50470839930 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 98747 \nu^{7} - 212957 \nu^{6} + 4414201 \nu^{5} + 10397849 \nu^{4} + 179938098 \nu^{3} + 2825845 \nu^{2} + 1270074215 \nu - 1219624350 ) / 1201686665 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 410451 \nu^{7} - 313216 \nu^{6} + 18348033 \nu^{5} + 43219617 \nu^{4} + 833723184 \nu^{3} + 11745885 \nu^{2} + 284259150 \nu - 26942923645 ) / 1201686665 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5600523 \nu^{7} - 60008240 \nu^{6} + 470238386 \nu^{5} - 1552858600 \nu^{4} + 5158399873 \nu^{3} - 36053963593 \nu^{2} + \cdots - 102366278700 ) / 8411806655 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39157703 \nu^{7} - 41841439 \nu^{6} - 1632820752 \nu^{5} - 9768593357 \nu^{4} - 84651611946 \nu^{3} - 203985740113 \nu^{2} + \cdots + 396175255350 ) / 50470839930 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7343719 \nu^{7} - 8871239 \nu^{6} - 139078134 \nu^{5} - 2644841047 \nu^{4} - 8006436282 \nu^{3} - 32824442807 \nu^{2} + \cdots - 118628497260 ) / 7210119990 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 3\beta_{3} + 23\beta_{2} - 23 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} - 5\beta_{4} + 41\beta_{3} + \beta_{2} - 41\beta _1 - 67 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 52\beta_{6} - 52\beta_{4} - 937\beta_{2} - 237\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -342\beta_{6} + 48\beta_{5} - 2119\beta_{3} - 5352\beta_{2} + 5352 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -150\beta_{7} + 150\beta_{5} + 2851\beta_{4} - 14883\beta_{3} - 150\beta_{2} + 14883\beta _1 + 48293 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2251\beta_{7} + 20735\beta_{6} + 20735\beta_{4} + 335106\beta_{2} + 118901\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
3.81550 + 6.60864i
0.810477 + 1.40379i
−1.95521 3.38653i
−2.17076 3.75987i
3.81550 6.60864i
0.810477 1.40379i
−1.95521 + 3.38653i
−2.17076 + 3.75987i
1.00000 1.73205i −4.31550 7.47467i −2.00000 3.46410i −2.90148 5.02551i −17.2620 8.58362 + 14.8673i −8.00000 −23.7471 + 41.1312i −11.6059
47.2 1.00000 1.73205i −1.31048 2.26981i −2.00000 3.46410i 0.0116171 + 0.0201213i −5.24191 −10.2956 17.8325i −8.00000 10.0653 17.4336i 0.0464682
47.3 1.00000 1.73205i 1.45521 + 2.52050i −2.00000 3.46410i −8.20891 14.2183i 5.82085 −6.65276 11.5229i −8.00000 9.26471 16.0469i −32.8357
47.4 1.00000 1.73205i 1.67076 + 2.89385i −2.00000 3.46410i 6.09877 + 10.5634i 6.68306 9.86474 + 17.0862i −8.00000 7.91709 13.7128i 24.3951
63.1 1.00000 + 1.73205i −4.31550 + 7.47467i −2.00000 + 3.46410i −2.90148 + 5.02551i −17.2620 8.58362 14.8673i −8.00000 −23.7471 41.1312i −11.6059
63.2 1.00000 + 1.73205i −1.31048 + 2.26981i −2.00000 + 3.46410i 0.0116171 0.0201213i −5.24191 −10.2956 + 17.8325i −8.00000 10.0653 + 17.4336i 0.0464682
63.3 1.00000 + 1.73205i 1.45521 2.52050i −2.00000 + 3.46410i −8.20891 + 14.2183i 5.82085 −6.65276 + 11.5229i −8.00000 9.26471 + 16.0469i −32.8357
63.4 1.00000 + 1.73205i 1.67076 2.89385i −2.00000 + 3.46410i 6.09877 10.5634i 6.68306 9.86474 17.0862i −8.00000 7.91709 + 13.7128i 24.3951
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.c.a 8
3.b odd 2 1 666.4.f.a 8
37.c even 3 1 inner 74.4.c.a 8
111.i odd 6 1 666.4.f.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.c.a 8 1.a even 1 1 trivial
74.4.c.a 8 37.c even 3 1 inner
666.4.f.a 8 3.b odd 2 1
666.4.f.a 8 111.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 5T_{3}^{7} + 63T_{3}^{6} - 126T_{3}^{5} + 1384T_{3}^{4} - 984T_{3}^{3} + 9384T_{3}^{2} - 7040T_{3} + 48400 \) acting on \(S_{4}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + 63 T^{6} + \cdots + 48400 \) Copy content Toggle raw display
$5$ \( T^{8} + 10 T^{7} + 276 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 8611097616 \) Copy content Toggle raw display
$11$ \( (T^{4} - 32 T^{3} - 2880 T^{2} + \cdots - 460800)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 61 T^{7} + \cdots + 60535681600 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 1531116688689 \) Copy content Toggle raw display
$19$ \( T^{8} + 71 T^{7} + \cdots + 6992428262400 \) Copy content Toggle raw display
$23$ \( (T^{4} + 26 T^{3} - 31000 T^{2} + \cdots - 15115776)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 161 T^{3} - 40343 T^{2} + \cdots - 72087162)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 56 T^{3} - 99216 T^{2} + \cdots + 712674048)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 557 T^{7} + \cdots + 65\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{8} - 92 T^{7} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{4} - 266 T^{3} - 32132 T^{2} + \cdots - 712153104)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 140 T^{3} - 300064 T^{2} + \cdots + 6878004480)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 159 T^{7} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} - 263 T^{7} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + 206 T^{7} + \cdots + 48\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{8} - 245 T^{7} + \cdots + 58\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{8} + 957 T^{7} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{4} + 136 T^{3} - 581912 T^{2} + \cdots - 6461408784)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 173 T^{7} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} - 1217 T^{7} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + 2136 T^{7} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{4} - 2631 T^{3} + \cdots - 492264212614)^{2} \) Copy content Toggle raw display
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