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Properties

Label 74.6.c.b
Level $74$
Weight $6$
Character orbit 74.c
Analytic conductor $11.868$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1317 x^{14} - 2712 x^{13} + 1206654 x^{12} - 2683704 x^{11} + 572597881 x^{10} + \cdots + 683643854869056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \beta_{3} + 4) q^{2} + (\beta_{3} - \beta_1) q^{3} + 16 \beta_{3} q^{4} + ( - \beta_{9} + \beta_{4} - 9 \beta_{3}) q^{5} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{6} + (\beta_{11} + 4 \beta_{3} + \beta_1) q^{7} - 64 q^{8} + (\beta_{12} - \beta_{9} - 87 \beta_{3} - 87) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (4 \beta_{3} + 4) q^{2} + (\beta_{3} - \beta_1) q^{3} + 16 \beta_{3} q^{4} + ( - \beta_{9} + \beta_{4} - 9 \beta_{3}) q^{5} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{6} + (\beta_{11} + 4 \beta_{3} + \beta_1) q^{7} - 64 q^{8} + (\beta_{12} - \beta_{9} - 87 \beta_{3} - 87) q^{9} + (4 \beta_{4} + 36) q^{10} + (\beta_{8} - \beta_{6} + \beta_{4} + \cdots - 15) q^{11}+ \cdots + (28 \beta_{15} - 167 \beta_{14} + \cdots - 1051) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{2} - 8 q^{3} - 128 q^{4} + 70 q^{5} - 64 q^{6} - 32 q^{7} - 1024 q^{8} - 698 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{2} - 8 q^{3} - 128 q^{4} + 70 q^{5} - 64 q^{6} - 32 q^{7} - 1024 q^{8} - 698 q^{9} + 560 q^{10} - 240 q^{11} - 128 q^{12} + 1614 q^{13} - 256 q^{14} + 728 q^{15} - 2048 q^{16} + 216 q^{17} + 2792 q^{18} - 950 q^{19} + 1120 q^{20} + 1718 q^{21} - 480 q^{22} + 1716 q^{23} + 512 q^{24} - 11760 q^{25} + 12912 q^{26} - 92 q^{27} - 512 q^{28} + 1112 q^{29} - 2912 q^{30} - 16816 q^{31} + 8192 q^{32} + 4744 q^{33} - 864 q^{34} - 5004 q^{35} + 22336 q^{36} + 3556 q^{37} - 7600 q^{38} + 17710 q^{39} - 4480 q^{40} - 8080 q^{41} - 6872 q^{42} - 19012 q^{43} + 1920 q^{44} - 58672 q^{45} + 3432 q^{46} + 29392 q^{47} + 4096 q^{48} + 474 q^{49} + 47040 q^{50} - 17856 q^{51} + 25824 q^{52} - 29250 q^{53} - 184 q^{54} + 15772 q^{55} + 2048 q^{56} + 16182 q^{57} + 2224 q^{58} + 61478 q^{59} - 23296 q^{60} - 132002 q^{61} - 33632 q^{62} - 103324 q^{63} + 65536 q^{64} - 37426 q^{65} + 37952 q^{66} - 45658 q^{67} - 6912 q^{68} - 23816 q^{69} + 20016 q^{70} + 13236 q^{71} + 44672 q^{72} + 270632 q^{73} - 81304 q^{74} + 419904 q^{75} - 15200 q^{76} - 184084 q^{77} - 70840 q^{78} + 123974 q^{79} - 35840 q^{80} + 96448 q^{81} - 64640 q^{82} - 16936 q^{83} - 54976 q^{84} - 251316 q^{85} - 38024 q^{86} + 168290 q^{87} + 15360 q^{88} + 52892 q^{89} - 117344 q^{90} - 432546 q^{91} - 13728 q^{92} + 220328 q^{93} + 58784 q^{94} + 47026 q^{95} + 8192 q^{96} - 546140 q^{97} - 1896 q^{98} - 9800 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1317 x^{14} - 2712 x^{13} + 1206654 x^{12} - 2683704 x^{11} + 572597881 x^{10} + \cdots + 683643854869056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!83 \nu^{15} + \cdots + 57\!\cdots\!64 ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 91\!\cdots\!04 \nu^{15} + \cdots + 21\!\cdots\!08 ) / 40\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 74\!\cdots\!38 \nu^{15} + \cdots + 62\!\cdots\!08 ) / 86\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!77 \nu^{15} + \cdots + 69\!\cdots\!16 ) / 17\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!67 \nu^{15} + \cdots - 14\!\cdots\!24 ) / 86\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 51\!\cdots\!49 \nu^{15} + \cdots - 13\!\cdots\!32 ) / 34\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!91 \nu^{15} + \cdots + 36\!\cdots\!72 ) / 86\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!74 \nu^{15} + \cdots + 79\!\cdots\!76 ) / 10\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 95\!\cdots\!14 \nu^{15} + \cdots - 47\!\cdots\!08 ) / 43\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!96 \nu^{15} + \cdots + 60\!\cdots\!36 ) / 62\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!98 \nu^{15} + \cdots + 53\!\cdots\!12 ) / 12\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 50\!\cdots\!03 \nu^{15} + \cdots + 23\!\cdots\!72 ) / 54\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 35\!\cdots\!84 \nu^{15} + \cdots - 84\!\cdots\!44 ) / 31\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 26\!\cdots\!92 \nu^{15} + \cdots + 12\!\cdots\!88 ) / 10\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{9} - 329\beta_{3} + 2\beta_{2} - 329 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - \beta_{8} + 2\beta_{7} - 4\beta_{6} - 3\beta_{5} + 3\beta_{4} - 518\beta_{2} - 518\beta _1 + 507 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{15} + 23 \beta_{14} - 28 \beta_{13} - 667 \beta_{12} + 179 \beta_{11} + 5 \beta_{10} + \cdots + 2170 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 836 \beta_{15} - 524 \beta_{14} + 1972 \beta_{13} + 4125 \beta_{12} - 3383 \beta_{11} - 9150 \beta_{9} + \cdots - 554370 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5836 \beta_{10} - 15922 \beta_{8} + 34928 \beta_{7} - 152512 \beta_{6} - 431725 \beta_{5} + \cdots + 97046633 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 572565 \beta_{15} + 271263 \beta_{14} - 1587324 \beta_{13} - 3986667 \beta_{12} + 2441064 \beta_{11} + \cdots + 183300412 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5282601 \beta_{15} - 8641761 \beta_{14} + 32099124 \beta_{13} + 281615599 \beta_{12} + \cdots - 59048539556 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 378422176 \beta_{10} - 143807242 \beta_{8} + 1196237150 \beta_{7} - 1724105347 \beta_{6} + \cdots + 503292419706 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4390552784 \beta_{15} + 4481982956 \beta_{14} - 26315805496 \beta_{13} - 186754914721 \beta_{12} + \cdots + 1471564841938 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 250995758105 \beta_{15} - 80324939597 \beta_{14} + 877432198318 \beta_{13} + 2773143081867 \beta_{12} + \cdots - 428983578680523 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3504195150013 \beta_{10} - 2376500679691 \beta_{8} + 20434118734244 \beta_{7} - 49731246033331 \beta_{6} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 168695898866700 \beta_{15} + 47904700551432 \beta_{14} - 635992035431184 \beta_{13} + \cdots + 51\!\cdots\!84 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 27\!\cdots\!68 \beta_{15} + \cdots - 16\!\cdots\!77 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 11\!\cdots\!29 \beta_{10} + \cdots + 27\!\cdots\!39 \) 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
11.6855 + 20.2399i
11.5099 + 19.9358i
6.66586 + 11.5456i
1.38002 + 2.39026i
0.0781534 + 0.135366i
−7.78646 − 13.4866i
−10.0937 − 17.4828i
−13.4393 − 23.2776i
11.6855 − 20.2399i
11.5099 − 19.9358i
6.66586 − 11.5456i
1.38002 − 2.39026i
0.0781534 − 0.135366i
−7.78646 + 13.4866i
−10.0937 + 17.4828i
−13.4393 + 23.2776i
2.00000 − 3.46410i −12.1855 − 21.1059i −8.00000 − 13.8564i −5.21821 − 9.03821i −97.4840 −70.8626 − 122.738i −64.0000 −175.473 + 303.928i −41.7457
47.2 2.00000 − 3.46410i −12.0099 − 20.8018i −8.00000 − 13.8564i 51.5512 + 89.2893i −96.0793 102.698 + 177.878i −64.0000 −166.976 + 289.211i 412.410
47.3 2.00000 − 3.46410i −7.16586 − 12.4116i −8.00000 − 13.8564i −51.9940 − 90.0563i −57.3269 36.2078 + 62.7137i −64.0000 18.8008 − 32.5639i −415.952
47.4 2.00000 − 3.46410i −1.88002 − 3.25629i −8.00000 − 13.8564i 7.78936 + 13.4916i −15.0401 −28.3238 − 49.0583i −64.0000 114.431 − 198.200i 62.3149
47.5 2.00000 − 3.46410i −0.578153 − 1.00139i −8.00000 − 13.8564i 40.4037 + 69.9813i −4.62523 −85.9598 − 148.887i −64.0000 120.831 − 209.286i 323.230
47.6 2.00000 − 3.46410i 7.28646 + 12.6205i −8.00000 − 13.8564i −3.95824 − 6.85587i 58.2917 79.3358 + 137.414i −64.0000 15.3149 − 26.5262i −31.6659
47.7 2.00000 − 3.46410i 9.59367 + 16.6167i −8.00000 − 13.8564i −34.0877 − 59.0415i 76.7494 −46.0584 − 79.7755i −64.0000 −62.5771 + 108.387i −272.701
47.8 2.00000 − 3.46410i 12.9393 + 22.4116i −8.00000 − 13.8564i 30.5138 + 52.8515i 103.515 −3.03701 − 5.26025i −64.0000 −213.352 + 369.536i 244.111
63.1 2.00000 + 3.46410i −12.1855 + 21.1059i −8.00000 + 13.8564i −5.21821 + 9.03821i −97.4840 −70.8626 + 122.738i −64.0000 −175.473 − 303.928i −41.7457
63.2 2.00000 + 3.46410i −12.0099 + 20.8018i −8.00000 + 13.8564i 51.5512 − 89.2893i −96.0793 102.698 − 177.878i −64.0000 −166.976 − 289.211i 412.410
63.3 2.00000 + 3.46410i −7.16586 + 12.4116i −8.00000 + 13.8564i −51.9940 + 90.0563i −57.3269 36.2078 − 62.7137i −64.0000 18.8008 + 32.5639i −415.952
63.4 2.00000 + 3.46410i −1.88002 + 3.25629i −8.00000 + 13.8564i 7.78936 − 13.4916i −15.0401 −28.3238 + 49.0583i −64.0000 114.431 + 198.200i 62.3149
63.5 2.00000 + 3.46410i −0.578153 + 1.00139i −8.00000 + 13.8564i 40.4037 − 69.9813i −4.62523 −85.9598 + 148.887i −64.0000 120.831 + 209.286i 323.230
63.6 2.00000 + 3.46410i 7.28646 − 12.6205i −8.00000 + 13.8564i −3.95824 + 6.85587i 58.2917 79.3358 − 137.414i −64.0000 15.3149 + 26.5262i −31.6659
63.7 2.00000 + 3.46410i 9.59367 − 16.6167i −8.00000 + 13.8564i −34.0877 + 59.0415i 76.7494 −46.0584 + 79.7755i −64.0000 −62.5771 − 108.387i −272.701
63.8 2.00000 + 3.46410i 12.9393 − 22.4116i −8.00000 + 13.8564i 30.5138 − 52.8515i 103.515 −3.03701 + 5.26025i −64.0000 −213.352 − 369.536i 244.111
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
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.b ✓ 16
37.c even 3 1 inner 74.6.c.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.c.b ✓ 16 1.a even 1 1 trivial
74.6.c.b ✓ 16 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 8 T_{3}^{15} + 1353 T_{3}^{14} + 8092 T_{3}^{13} + 1233767 T_{3}^{12} + \cdots + 69\!\cdots\!84 \) acting on \(S_{6}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 4 T + 16)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 69\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 38\!\cdots\!72)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 59\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 71\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 41\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 32\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 52\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 37\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 17\!\cdots\!88)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 33\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 78\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 87\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 49\!\cdots\!16)^{2} \) Copy content Toggle raw display
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