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Properties

Label 74.5.d.b
Level $74$
Weight $5$
Character orbit 74.d
Analytic conductor $7.649$
Analytic rank $0$
Dimension $14$
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(31,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.31");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 74.d (of order \(4\), 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: \(7.64937726820\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 727 x^{12} + 207381 x^{10} + 29788577 x^{8} + 2302194203 x^{6} + 92916575085 x^{4} + \cdots + 6531254919424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{5} + 2) q^{2} + ( - \beta_{5} + \beta_1) q^{3} + 8 \beta_{5} q^{4} + ( - \beta_{7} - 3 \beta_{5} + 3) q^{5} + ( - 2 \beta_{5} - 2 \beta_{2} + \cdots + 2) q^{6}+ \cdots + (\beta_{3} + \beta_{2} - 24) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{5} + 2) q^{2} + ( - \beta_{5} + \beta_1) q^{3} + 8 \beta_{5} q^{4} + ( - \beta_{7} - 3 \beta_{5} + 3) q^{5} + ( - 2 \beta_{5} - 2 \beta_{2} + \cdots + 2) q^{6}+ \cdots + (8 \beta_{13} + 8 \beta_{12} + \cdots + 354 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 28 q^{2} + 36 q^{5} + 40 q^{6} - 48 q^{7} - 224 q^{8} - 346 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 28 q^{2} + 36 q^{5} + 40 q^{6} - 48 q^{7} - 224 q^{8} - 346 q^{9} + 144 q^{10} + 160 q^{12} - 104 q^{13} - 96 q^{14} - 378 q^{15} - 896 q^{16} + 516 q^{17} - 692 q^{18} - 328 q^{19} + 288 q^{20} - 320 q^{22} + 154 q^{23} + 320 q^{24} - 416 q^{26} + 1686 q^{29} + 3834 q^{31} - 1792 q^{32} + 2104 q^{33} + 2064 q^{34} - 1502 q^{35} + 2640 q^{37} - 1312 q^{38} - 4526 q^{39} - 5984 q^{42} + 3616 q^{43} - 1280 q^{44} - 2238 q^{45} + 616 q^{46} - 6892 q^{47} + 12854 q^{49} + 7516 q^{50} - 6742 q^{51} - 832 q^{52} + 12572 q^{53} - 1072 q^{54} + 5510 q^{55} + 768 q^{56} - 6302 q^{57} - 8422 q^{59} + 3024 q^{60} - 6386 q^{61} + 22244 q^{63} + 4208 q^{66} + 4128 q^{68} + 1728 q^{69} - 6008 q^{70} + 8680 q^{71} + 5536 q^{72} + 1316 q^{74} - 37980 q^{75} - 2624 q^{76} - 28520 q^{79} - 2304 q^{80} - 33962 q^{81} + 9136 q^{82} - 22688 q^{83} - 23936 q^{84} + 14464 q^{86} + 1828 q^{87} - 2560 q^{88} + 18344 q^{89} - 8952 q^{90} - 4918 q^{91} + 1232 q^{92} + 24 q^{93} - 13784 q^{94} - 2560 q^{96} + 23246 q^{97} + 25708 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 727 x^{12} + 207381 x^{10} + 29788577 x^{8} + 2302194203 x^{6} + 92916575085 x^{4} + \cdots + 6531254919424 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1200481 \nu^{12} + 766025276 \nu^{10} + 180814035481 \nu^{8} + 19656581442877 \nu^{6} + \cdots + 86\!\cdots\!16 ) / 146604462483438 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1200481 \nu^{12} + 766025276 \nu^{10} + 180814035481 \nu^{8} + 19656581442877 \nu^{6} + \cdots + 10\!\cdots\!68 ) / 146604462483438 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18530585749 \nu^{12} - 12189038115456 \nu^{10} + \cdots - 17\!\cdots\!04 ) / 65\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33988179063 \nu^{13} + 21641418519809 \nu^{11} + \cdots + 22\!\cdots\!28 \nu ) / 37\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 58347870445 \nu^{13} + 36628602238213 \nu^{11} + \cdots + 97\!\cdots\!52 \nu ) / 46\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!77 \nu^{13} + \cdots - 12\!\cdots\!92 ) / 50\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 22\!\cdots\!77 \nu^{13} + \cdots + 12\!\cdots\!92 ) / 50\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 55\!\cdots\!25 \nu^{13} + \cdots + 39\!\cdots\!16 ) / 50\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 55\!\cdots\!25 \nu^{13} + \cdots - 39\!\cdots\!16 ) / 50\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 30\!\cdots\!89 \nu^{13} + \cdots - 25\!\cdots\!84 \nu ) / 25\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 54\!\cdots\!17 \nu^{13} + \cdots - 55\!\cdots\!04 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 54\!\cdots\!17 \nu^{13} + \cdots + 55\!\cdots\!04 ) / 16\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 104 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -9\beta_{11} + 12\beta_{8} + 12\beta_{7} + 4\beta_{6} - 94\beta_{5} - 152\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{13} - 3 \beta_{12} + 30 \beta_{10} - 30 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - 72 \beta_{4} + \cdots + 16308 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 51 \beta_{13} - 51 \beta_{12} + 2628 \beta_{11} - 60 \beta_{10} - 60 \beta_{9} - 3234 \beta_{8} + \cdots + 28155 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 648 \beta_{13} + 648 \beta_{12} - 8757 \beta_{10} + 8757 \beta_{9} - 360 \beta_{8} + 360 \beta_{7} + \cdots - 3063460 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16686 \beta_{13} + 16686 \beta_{12} - 636552 \beta_{11} + 12141 \beta_{10} + 12141 \beta_{9} + \cdots - 5735587 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 115002 \beta_{13} - 115002 \beta_{12} + 2038356 \beta_{10} - 2038356 \beta_{9} - 307134 \beta_{8} + \cdots + 626407172 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4076208 \beta_{13} - 4076208 \beta_{12} + 145919295 \beta_{11} - 1232964 \beta_{10} + \cdots + 1217832374 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 19321059 \beta_{13} + 19321059 \beta_{12} - 446561616 \beta_{10} + 446561616 \beta_{9} + \cdots - 132932680284 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 906403263 \beta_{13} + 906403263 \beta_{12} - 32835563334 \beta_{11} - 121553616 \beta_{10} + \cdots - 263349429657 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3095029710 \beta_{13} - 3095029710 \beta_{12} + 96097274469 \beta_{10} - 96097274469 \beta_{9} + \cdots + 28687394074840 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 194754750246 \beta_{13} - 194754750246 \beta_{12} + 7346159931312 \beta_{11} + \cdots + 57465660764815 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
− 14.0933i
− 9.07204i
− 8.86388i
− 2.31020i
6.93683i
9.34569i
15.0569i
− 15.0569i
− 9.34569i
− 6.93683i
2.31020i
8.86388i
9.07204i
14.0933i
2.00000 + 2.00000i − 15.0933i 8.00000i 30.6150 − 30.6150i 30.1866 − 30.1866i −89.3132 −16.0000 + 16.0000i −146.808 122.460
31.2 2.00000 + 2.00000i − 10.0720i 8.00000i −28.1665 + 28.1665i 20.1441 − 20.1441i −38.8317 −16.0000 + 16.0000i −20.4460 −112.666
31.3 2.00000 + 2.00000i − 9.86388i 8.00000i 0.419185 − 0.419185i 19.7278 − 19.7278i 41.6300 −16.0000 + 16.0000i −16.2962 1.67674
31.4 2.00000 + 2.00000i − 3.31020i 8.00000i −1.68780 + 1.68780i 6.62039 − 6.62039i 49.9045 −16.0000 + 16.0000i 70.0426 −6.75118
31.5 2.00000 + 2.00000i 5.93683i 8.00000i 34.8311 − 34.8311i −11.8737 + 11.8737i 33.1642 −16.0000 + 16.0000i 45.7541 139.324
31.6 2.00000 + 2.00000i 8.34569i 8.00000i −6.01044 + 6.01044i −16.6914 + 16.6914i −74.3653 −16.0000 + 16.0000i 11.3495 −24.0418
31.7 2.00000 + 2.00000i 14.0569i 8.00000i −12.0005 + 12.0005i −28.1138 + 28.1138i 53.8114 −16.0000 + 16.0000i −116.596 −48.0021
43.1 2.00000 − 2.00000i − 14.0569i − 8.00000i −12.0005 − 12.0005i −28.1138 − 28.1138i 53.8114 −16.0000 − 16.0000i −116.596 −48.0021
43.2 2.00000 − 2.00000i − 8.34569i − 8.00000i −6.01044 − 6.01044i −16.6914 − 16.6914i −74.3653 −16.0000 − 16.0000i 11.3495 −24.0418
43.3 2.00000 − 2.00000i − 5.93683i − 8.00000i 34.8311 + 34.8311i −11.8737 − 11.8737i 33.1642 −16.0000 − 16.0000i 45.7541 139.324
43.4 2.00000 − 2.00000i 3.31020i − 8.00000i −1.68780 − 1.68780i 6.62039 + 6.62039i 49.9045 −16.0000 − 16.0000i 70.0426 −6.75118
43.5 2.00000 − 2.00000i 9.86388i − 8.00000i 0.419185 + 0.419185i 19.7278 + 19.7278i 41.6300 −16.0000 − 16.0000i −16.2962 1.67674
43.6 2.00000 − 2.00000i 10.0720i − 8.00000i −28.1665 − 28.1665i 20.1441 + 20.1441i −38.8317 −16.0000 − 16.0000i −20.4460 −112.666
43.7 2.00000 − 2.00000i 15.0933i − 8.00000i 30.6150 + 30.6150i 30.1866 + 30.1866i −89.3132 −16.0000 − 16.0000i −146.808 122.460
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.5.d.b ✓ 14
37.d odd 4 1 inner 74.5.d.b ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.5.d.b ✓ 14 1.a even 1 1 trivial
74.5.d.b ✓ 14 37.d odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 740 T_{3}^{12} + 215344 T_{3}^{10} + 31575042 T_{3}^{8} + 2486540160 T_{3}^{6} + \cdots + 11951402126400 \) acting on \(S_{5}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 4 T + 8)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 11951402126400 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 300707794649088 \) Copy content Toggle raw display
$7$ \( (T^{7} + 24 T^{6} + \cdots + 956231061528)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 78\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 35\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 29\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 46\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 21\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 81\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 89\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 14\!\cdots\!12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots + 29\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 15\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 10\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 62\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 28\!\cdots\!48 \) Copy content Toggle raw display
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