↑

Properties

Label 74.3.g.b
Level $74$
Weight $3$
Character orbit 74.g
Analytic conductor $2.016$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [74,3,Mod(23,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.23");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 74.g (of order \(12\), degree \(4\), 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: \(2.01635395627\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 82x^{10} + 2505x^{8} + 34456x^{6} + 196096x^{4} + 262464x^{2} + 69696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{4}) q^{2} - \beta_{3} q^{3} + 2 \beta_{6} q^{4} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 1) q^{5} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{6} + ( - \beta_{11} - \beta_{7} - \beta_{4} - 1) q^{7} + (2 \beta_{7} - 2 \beta_{6} - 2) q^{8} + ( - \beta_{11} + \beta_{8} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{4}) q^{2} - \beta_{3} q^{3} + 2 \beta_{6} q^{4} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 1) q^{5} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{6} + ( - \beta_{11} - \beta_{7} - \beta_{4} - 1) q^{7} + (2 \beta_{7} - 2 \beta_{6} - 2) q^{8} + ( - \beta_{11} + \beta_{8} + \cdots - \beta_1) q^{9}+ \cdots + ( - 2 \beta_{11} + \beta_{9} + \cdots - 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 4 q^{6} - 8 q^{7} - 24 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} + 4 q^{6} - 8 q^{7} - 24 q^{8} + 28 q^{9} - 36 q^{10} - 4 q^{12} + 16 q^{13} + 16 q^{14} + 24 q^{16} + 40 q^{17} + 28 q^{18} - 26 q^{19} + 66 q^{21} + 4 q^{22} - 80 q^{23} - 4 q^{24} - 54 q^{25} - 124 q^{26} - 12 q^{28} + 16 q^{29} - 6 q^{30} - 32 q^{31} + 24 q^{32} - 20 q^{33} - 10 q^{34} + 12 q^{35} - 148 q^{37} + 92 q^{38} + 216 q^{39} + 36 q^{40} + 66 q^{41} - 46 q^{42} + 152 q^{43} - 16 q^{44} + 84 q^{45} + 80 q^{46} - 112 q^{47} - 160 q^{49} + 168 q^{50} - 446 q^{51} + 32 q^{52} + 74 q^{53} + 230 q^{54} + 28 q^{56} + 50 q^{57} + 84 q^{58} - 114 q^{59} - 12 q^{60} + 448 q^{61} - 204 q^{62} - 784 q^{63} - 138 q^{65} + 40 q^{66} + 468 q^{67} + 20 q^{68} - 278 q^{69} + 18 q^{70} + 116 q^{71} - 56 q^{72} - 2 q^{74} + 76 q^{75} - 52 q^{76} + 60 q^{77} - 366 q^{78} + 114 q^{79} + 72 q^{80} + 14 q^{81} + 128 q^{82} - 20 q^{83} - 80 q^{84} - 152 q^{86} + 770 q^{87} + 32 q^{88} + 340 q^{89} - 84 q^{90} + 792 q^{91} + 68 q^{92} - 498 q^{93} + 20 q^{94} + 60 q^{95} + 8 q^{96} - 356 q^{97} - 160 q^{98} - 348 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 82x^{10} + 2505x^{8} + 34456x^{6} + 196096x^{4} + 262464x^{2} + 69696 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\nu^{10} + 617\nu^{8} + 10275\nu^{6} + 46025\nu^{4} + 86420\nu^{2} + 1280592 ) / 113880 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 62\nu^{8} + 1265\nu^{6} + 9260\nu^{4} + 15160\nu^{2} + 2080\nu + 2112 ) / 4160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} - 62\nu^{8} - 1265\nu^{6} - 9260\nu^{4} - 15160\nu^{2} + 2080\nu - 2112 ) / 4160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 49\nu^{9} + 459\nu^{7} - 7289\nu^{5} - 109484\nu^{3} - 237816\nu - 68640 ) / 137280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -35\nu^{10} - 2222\nu^{8} - 47187\nu^{6} - 368768\nu^{4} - 685976\nu^{2} - 252384 ) / 91104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1195 \nu^{11} - 1716 \nu^{10} - 86440 \nu^{9} - 130416 \nu^{8} - 2260215 \nu^{7} + \cdots - 104113152 ) / 60128640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1195 \nu^{11} - 1716 \nu^{10} + 86440 \nu^{9} - 130416 \nu^{8} + 2260215 \nu^{7} + \cdots - 104113152 ) / 60128640 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26 \nu^{11} - 175 \nu^{10} - 1976 \nu^{9} - 11110 \nu^{8} - 53274 \nu^{7} - 235935 \nu^{6} + \cdots - 1261920 ) / 911040 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2269 \nu^{11} - 8184 \nu^{10} - 260044 \nu^{9} - 390720 \nu^{8} - 10295001 \nu^{7} + \cdots + 954410688 ) / 60128640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2269 \nu^{11} + 8184 \nu^{10} - 260044 \nu^{9} + 390720 \nu^{8} - 10295001 \nu^{7} + \cdots - 1014539328 ) / 60128640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5849 \nu^{11} - 1188 \nu^{10} + 400286 \nu^{9} - 32472 \nu^{8} + 9894465 \nu^{7} + \cdots - 233963136 ) / 60128640 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} + 2\beta_{8} + 7\beta_{7} - 7\beta_{6} - \beta_{5} - 20\beta_{4} - 21\beta_{3} - 21\beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{7} - 10\beta_{6} + 35\beta_{5} - 31\beta_{3} + 31\beta_{2} - 21\beta _1 + 267 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20 \beta_{11} - 35 \beta_{10} - 35 \beta_{9} - 110 \beta_{8} - 309 \beta_{7} + 329 \beta_{6} + \cdots + 305 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 20 \beta_{10} + 20 \beta_{9} + 550 \beta_{7} + 550 \beta_{6} - 1083 \beta_{5} + 879 \beta_{3} + \cdots - 5907 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 860 \beta_{11} + 1023 \beta_{10} + 1023 \beta_{9} + 4286 \beta_{8} + 10345 \beta_{7} - 11205 \beta_{6} + \cdots - 8937 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1120 \beta_{10} - 1120 \beta_{9} - 22210 \beta_{7} - 22210 \beta_{6} + 31819 \beta_{5} - 24183 \beta_{3} + \cdots + 136851 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 28180 \beta_{11} - 28459 \beta_{10} - 28459 \beta_{9} - 145198 \beta_{8} - 314829 \beta_{7} + \cdots + 248921 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 44140 \beta_{10} + 44140 \beta_{9} + 773870 \beta_{7} + 773870 \beta_{6} - 911723 \beta_{5} + \cdots - 3289859 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 840300 \beta_{11} + 779303 \beta_{10} + 779303 \beta_{9} + 4564606 \beta_{8} + 9192353 \beta_{7} + \cdots - 6788617 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
− 3.64005i
− 0.594165i
5.23421i
3.64005i
0.594165i
− 5.23421i
4.36026i
1.17841i
− 4.53867i
− 4.36026i
− 1.17841i
4.53867i
0.366025 + 1.36603i −3.15238 + 1.82002i −1.73205 + 1.00000i −0.866025 + 3.23205i −3.64005 − 3.64005i −5.59738 − 9.69495i −2.00000 − 2.00000i 2.12498 − 3.68058i −4.73205
23.2 0.366025 + 1.36603i −0.514562 + 0.297082i −1.73205 + 1.00000i −0.866025 + 3.23205i −0.594165 − 0.594165i 5.01184 + 8.68076i −2.00000 − 2.00000i −4.32348 + 7.48849i −4.73205
23.3 0.366025 + 1.36603i 4.53296 − 2.61711i −1.73205 + 1.00000i −0.866025 + 3.23205i 5.23421 + 5.23421i −0.548432 − 0.949912i −2.00000 − 2.00000i 9.19850 − 15.9323i −4.73205
29.1 0.366025 − 1.36603i −3.15238 − 1.82002i −1.73205 − 1.00000i −0.866025 − 3.23205i −3.64005 + 3.64005i −5.59738 + 9.69495i −2.00000 + 2.00000i 2.12498 + 3.68058i −4.73205
29.2 0.366025 − 1.36603i −0.514562 − 0.297082i −1.73205 − 1.00000i −0.866025 − 3.23205i −0.594165 + 0.594165i 5.01184 − 8.68076i −2.00000 + 2.00000i −4.32348 − 7.48849i −4.73205
29.3 0.366025 − 1.36603i 4.53296 + 2.61711i −1.73205 − 1.00000i −0.866025 − 3.23205i 5.23421 − 5.23421i −0.548432 + 0.949912i −2.00000 + 2.00000i 9.19850 + 15.9323i −4.73205
45.1 −1.36603 − 0.366025i −3.77609 − 2.18013i 1.73205 + 1.00000i 0.866025 − 0.232051i 4.36026 + 4.36026i −5.10188 + 8.83671i −2.00000 − 2.00000i 5.00591 + 8.67050i −1.26795
45.2 −1.36603 − 0.366025i −1.02054 − 0.589207i 1.73205 + 1.00000i 0.866025 − 0.232051i 1.17841 + 1.17841i 4.87434 − 8.44261i −2.00000 − 2.00000i −3.80567 − 6.59162i −1.26795
45.3 −1.36603 − 0.366025i 3.93060 + 2.26933i 1.73205 + 1.00000i 0.866025 − 0.232051i −4.53867 − 4.53867i −2.63849 + 4.57000i −2.00000 − 2.00000i 5.79976 + 10.0455i −1.26795
51.1 −1.36603 + 0.366025i −3.77609 + 2.18013i 1.73205 − 1.00000i 0.866025 + 0.232051i 4.36026 − 4.36026i −5.10188 − 8.83671i −2.00000 + 2.00000i 5.00591 − 8.67050i −1.26795
51.2 −1.36603 + 0.366025i −1.02054 + 0.589207i 1.73205 − 1.00000i 0.866025 + 0.232051i 1.17841 − 1.17841i 4.87434 + 8.44261i −2.00000 + 2.00000i −3.80567 + 6.59162i −1.26795
51.3 −1.36603 + 0.366025i 3.93060 − 2.26933i 1.73205 − 1.00000i 0.866025 + 0.232051i −4.53867 + 4.53867i −2.63849 − 4.57000i −2.00000 + 2.00000i 5.79976 − 10.0455i −1.26795
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.3.g.b ✓ 12
37.g odd 12 1 inner 74.3.g.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.3.g.b ✓ 12 1.a even 1 1 trivial
74.3.g.b ✓ 12 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 41 T_{3}^{10} + 1269 T_{3}^{8} + 1260 T_{3}^{7} - 16220 T_{3}^{6} - 24720 T_{3}^{5} + \cdots + 69696 \) acting on \(S_{3}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} - 41 T^{10} + \cdots + 69696 \) Copy content Toggle raw display
$5$ \( (T^{4} + 9 T^{2} - 18 T + 9)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 4174193664 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 196795155456 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 57672420423696 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 70631132318289 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 6088062760000 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 1879443581184 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 65\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 77\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 84\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{6} + 56 T^{5} + \cdots + 104298336)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 97\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 58\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
show more
show less