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Properties

Label 74.6.c.a
Level $74$
Weight $6$
Character orbit 74.c
Analytic conductor $11.868$
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] = [74,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.47");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(11.8684026662\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 993 x^{12} - 1336 x^{11} + 753233 x^{10} - 540853 x^{9} + 210096237 x^{8} + \cdots + 309997997697600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \beta_{3} - 4) q^{2} + ( - \beta_{3} - \beta_1) q^{3} - 16 \beta_{3} q^{4} + ( - \beta_{8} + 8 \beta_{3}) q^{5} + ( - 4 \beta_{2} + 4 \beta_1 + 4) q^{6} + (\beta_{10} + 27 \beta_{3}) q^{7} + 64 q^{8} + (\beta_{12} - \beta_{8} - \beta_{5} + \cdots - 41) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (4 \beta_{3} - 4) q^{2} + ( - \beta_{3} - \beta_1) q^{3} - 16 \beta_{3} q^{4} + ( - \beta_{8} + 8 \beta_{3}) q^{5} + ( - 4 \beta_{2} + 4 \beta_1 + 4) q^{6} + (\beta_{10} + 27 \beta_{3}) q^{7} + 64 q^{8} + (\beta_{12} - \beta_{8} - \beta_{5} + \cdots - 41) q^{9}+ \cdots + ( - 153 \beta_{13} + 101 \beta_{12} + \cdots - 63298) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 28 q^{2} - 8 q^{3} - 112 q^{4} + 57 q^{5} + 64 q^{6} + 188 q^{7} + 896 q^{8} - 293 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 28 q^{2} - 8 q^{3} - 112 q^{4} + 57 q^{5} + 64 q^{6} + 188 q^{7} + 896 q^{8} - 293 q^{9} - 456 q^{10} + 704 q^{11} - 128 q^{12} - 704 q^{13} - 1504 q^{14} - 364 q^{15} - 1792 q^{16} - 561 q^{17} - 1172 q^{18} - 3742 q^{19} + 912 q^{20} - 506 q^{21} - 1408 q^{22} - 5700 q^{23} - 512 q^{24} - 2340 q^{25} + 5632 q^{26} + 3124 q^{27} + 3008 q^{28} - 17654 q^{29} - 1456 q^{30} + 11888 q^{31} - 7168 q^{32} + 484 q^{33} - 2244 q^{34} + 1460 q^{35} + 9376 q^{36} + 16941 q^{37} + 29936 q^{38} - 36086 q^{39} + 3648 q^{40} - 9813 q^{41} - 2024 q^{42} + 5724 q^{43} - 5632 q^{44} - 63030 q^{45} + 11400 q^{46} + 23320 q^{47} + 4096 q^{48} - 11829 q^{49} - 9360 q^{50} + 110384 q^{51} - 11264 q^{52} + 16856 q^{53} - 6248 q^{54} + 28408 q^{55} + 12032 q^{56} - 26062 q^{57} + 35308 q^{58} + 50886 q^{59} + 11648 q^{60} - 6247 q^{61} - 23776 q^{62} - 10452 q^{63} + 57344 q^{64} - 4852 q^{65} - 3872 q^{66} - 28570 q^{67} + 17952 q^{68} + 43524 q^{69} + 5840 q^{70} - 8580 q^{71} - 18752 q^{72} - 9252 q^{73} - 36336 q^{74} + 112184 q^{75} - 59872 q^{76} - 56112 q^{77} - 144344 q^{78} + 26930 q^{79} - 29184 q^{80} - 9039 q^{81} + 78504 q^{82} - 137412 q^{83} + 16192 q^{84} - 58830 q^{85} - 11448 q^{86} - 70826 q^{87} + 45056 q^{88} - 27961 q^{89} + 126060 q^{90} + 99802 q^{91} + 45600 q^{92} - 181340 q^{93} - 46640 q^{94} + 210526 q^{95} - 8192 q^{96} + 191206 q^{97} - 47316 q^{98} - 446840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 993 x^{12} - 1336 x^{11} + 753233 x^{10} - 540853 x^{9} + 210096237 x^{8} + \cdots + 309997997697600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26\!\cdots\!55 \nu^{13} + \cdots - 22\!\cdots\!40 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!97 \nu^{13} + \cdots - 72\!\cdots\!00 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 86\!\cdots\!68 \nu^{13} + \cdots + 94\!\cdots\!00 ) / 25\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!93 \nu^{13} + \cdots + 21\!\cdots\!50 ) / 25\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\!\cdots\!57 \nu^{13} + \cdots + 39\!\cdots\!25 ) / 12\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\!\cdots\!63 \nu^{13} + \cdots - 54\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17\!\cdots\!27 \nu^{13} + \cdots + 10\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!77 \nu^{13} + \cdots + 10\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 56\!\cdots\!25 \nu^{13} + \cdots + 49\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39\!\cdots\!09 \nu^{13} + \cdots - 93\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15\!\cdots\!71 \nu^{13} + \cdots + 28\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 71\!\cdots\!55 \nu^{13} + \cdots - 14\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{8} - \beta_{5} + 283\beta_{3} - 283 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} + 4\beta_{7} + 11\beta_{6} + 7\beta_{5} - 5\beta_{4} + 524\beta_{2} - 524\beta _1 + 144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{13} - 675 \beta_{12} - 37 \beta_{11} - 155 \beta_{10} + 8 \beta_{9} + 1228 \beta_{8} + \cdots + 1033 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1470 \beta_{13} + 5943 \beta_{12} + 9294 \beta_{11} + 1347 \beta_{10} - 4548 \beta_{8} + \cdots - 435930 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12330 \beta_{9} + 153738 \beta_{7} + 35526 \beta_{6} + 957823 \beta_{5} - 445303 \beta_{4} + \cdots + 90329323 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 991232 \beta_{13} - 5225723 \beta_{12} - 6675245 \beta_{11} - 434926 \beta_{10} + 991232 \beta_{9} + \cdots + 206445998 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 11394938 \beta_{13} + 298358877 \beta_{12} + 32449243 \beta_{11} + 118281233 \beta_{10} + \cdots - 58263415553 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 667972980 \beta_{9} + 245940189 \beta_{7} + 4598640288 \beta_{6} + 3138939966 \beta_{5} + \cdots + 511372232370 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9270777060 \beta_{13} - 202799771065 \beta_{12} - 29104102932 \beta_{11} - 83871052416 \beta_{10} + \cdots + 1344300516492 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 454487146862 \beta_{13} + 3238971487481 \beta_{12} + 3128000798399 \beta_{11} + \cdots - 431757371330064 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 7183329309068 \beta_{9} + 57553771176551 \beta_{7} + 25114662814129 \beta_{6} + 316732471584232 \beta_{5} + \cdots + 26\!\cdots\!13 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 312096586783290 \beta_{13} + \cdots + 62\!\cdots\!53 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(-\beta_{3}\)

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
12.2489 + 21.2157i
9.50448 + 16.4622i
2.87888 + 4.98636i
0.902874 + 1.56382i
−5.62948 − 9.75055i
−6.04251 − 10.4659i
−13.3631 − 23.1456i
12.2489 − 21.2157i
9.50448 − 16.4622i
2.87888 − 4.98636i
0.902874 − 1.56382i
−5.62948 + 9.75055i
−6.04251 + 10.4659i
−13.3631 + 23.1456i
−2.00000 + 3.46410i −12.7489 − 22.0817i −8.00000 − 13.8564i 32.6269 + 56.5115i 101.991 27.0670 + 46.8814i 64.0000 −203.569 + 352.591i −261.016
47.2 −2.00000 + 3.46410i −10.0045 − 17.3283i −8.00000 − 13.8564i −32.4177 − 56.1491i 80.0358 −55.2452 − 95.6875i 64.0000 −78.6791 + 136.276i 259.342
47.3 −2.00000 + 3.46410i −3.37888 − 5.85239i −8.00000 − 13.8564i −12.8799 − 22.3087i 27.0310 127.514 + 220.861i 64.0000 98.6664 − 170.895i 103.039
47.4 −2.00000 + 3.46410i −1.40287 − 2.42985i −8.00000 − 13.8564i 41.4911 + 71.8648i 11.2230 −4.94404 − 8.56332i 64.0000 117.564 − 203.627i −331.929
47.5 −2.00000 + 3.46410i 5.12948 + 8.88453i −8.00000 − 13.8564i 7.23785 + 12.5363i −41.0359 −90.6154 − 156.951i 64.0000 68.8768 − 119.298i −57.9028
47.6 −2.00000 + 3.46410i 5.54251 + 9.59990i −8.00000 − 13.8564i −35.1636 − 60.9052i −44.3400 48.2136 + 83.5084i 64.0000 60.0613 − 104.029i 281.309
47.7 −2.00000 + 3.46410i 12.8631 + 22.2796i −8.00000 − 13.8564i 27.6053 + 47.8138i −102.905 42.0100 + 72.7634i 64.0000 −209.420 + 362.727i −220.842
63.1 −2.00000 − 3.46410i −12.7489 + 22.0817i −8.00000 + 13.8564i 32.6269 − 56.5115i 101.991 27.0670 − 46.8814i 64.0000 −203.569 − 352.591i −261.016
63.2 −2.00000 − 3.46410i −10.0045 + 17.3283i −8.00000 + 13.8564i −32.4177 + 56.1491i 80.0358 −55.2452 + 95.6875i 64.0000 −78.6791 − 136.276i 259.342
63.3 −2.00000 − 3.46410i −3.37888 + 5.85239i −8.00000 + 13.8564i −12.8799 + 22.3087i 27.0310 127.514 − 220.861i 64.0000 98.6664 + 170.895i 103.039
63.4 −2.00000 − 3.46410i −1.40287 + 2.42985i −8.00000 + 13.8564i 41.4911 − 71.8648i 11.2230 −4.94404 + 8.56332i 64.0000 117.564 + 203.627i −331.929
63.5 −2.00000 − 3.46410i 5.12948 − 8.88453i −8.00000 + 13.8564i 7.23785 − 12.5363i −41.0359 −90.6154 + 156.951i 64.0000 68.8768 + 119.298i −57.9028
63.6 −2.00000 − 3.46410i 5.54251 − 9.59990i −8.00000 + 13.8564i −35.1636 + 60.9052i −44.3400 48.2136 − 83.5084i 64.0000 60.0613 + 104.029i 281.309
63.7 −2.00000 − 3.46410i 12.8631 − 22.2796i −8.00000 + 13.8564i 27.6053 − 47.8138i −102.905 42.0100 − 72.7634i 64.0000 −209.420 − 362.727i −220.842
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.7
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 74.6.c.a ✓ 14
37.c even 3 1 inner 74.6.c.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.c.a ✓ 14 1.a even 1 1 trivial
74.6.c.a ✓ 14 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 8 T_{3}^{13} + 1029 T_{3}^{12} + 4428 T_{3}^{11} + 762343 T_{3}^{10} + \cdots + 800920337887296 \) acting on \(S_{6}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 16)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 800920337887296 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 25\!\cdots\!69 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots - 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 77\!\cdots\!93 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 75\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 32\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 20\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 37\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 43\!\cdots\!89 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
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