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Properties

Label 66.7.c.a
Level $66$
Weight $7$
Character orbit 66.c
Analytic conductor $15.184$
Analytic rank $0$
Dimension $20$
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] = [66,7,Mod(23,66)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(66, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("66.23");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 66.c (of order \(2\), degree \(1\), 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: \(15.1835695189\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 745 x^{18} + 29254 x^{17} - 384247 x^{16} - 30144404 x^{15} + 1098678422 x^{14} + \cdots + 17\!\cdots\!08 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{29}\cdot 3^{12}\cdot 11^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_1 - 2) q^{3} - 32 q^{4} + (\beta_{9} + 4 \beta_{2}) q^{5} + (\beta_{12} - 2 \beta_{2} + 11) q^{6} + ( - \beta_{14} - \beta_{12} + 20) q^{7} - 32 \beta_{2} q^{8} + ( - \beta_{12} + \beta_{9} - \beta_{8} + \cdots + 46) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_1 - 2) q^{3} - 32 q^{4} + (\beta_{9} + 4 \beta_{2}) q^{5} + (\beta_{12} - 2 \beta_{2} + 11) q^{6} + ( - \beta_{14} - \beta_{12} + 20) q^{7} - 32 \beta_{2} q^{8} + ( - \beta_{12} + \beta_{9} - \beta_{8} + \cdots + 46) q^{9}+ \cdots + ( - 112 \beta_{19} + 21 \beta_{18} + \cdots - 22068) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 32 q^{3} - 640 q^{4} + 224 q^{6} + 400 q^{7} + 912 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 32 q^{3} - 640 q^{4} + 224 q^{6} + 400 q^{7} + 912 q^{9} - 2496 q^{10} + 1024 q^{12} + 8152 q^{13} - 1848 q^{15} + 20480 q^{16} - 2432 q^{18} - 9632 q^{19} + 280 q^{21} - 7168 q^{24} - 13732 q^{25} + 91576 q^{27} - 12800 q^{28} - 134304 q^{30} - 70064 q^{31} + 42592 q^{33} - 34080 q^{34} - 29184 q^{36} - 145592 q^{37} + 107352 q^{39} + 79872 q^{40} - 268224 q^{42} + 23920 q^{43} - 93500 q^{45} - 153024 q^{46} - 32768 q^{48} + 1286340 q^{49} + 499544 q^{51} - 260864 q^{52} + 365888 q^{54} + 383328 q^{55} + 108640 q^{57} - 422112 q^{58} + 59136 q^{60} - 42632 q^{61} - 436744 q^{63} - 655360 q^{64} - 1435520 q^{67} - 335716 q^{69} - 891840 q^{70} + 77824 q^{72} + 2052328 q^{73} - 1582760 q^{75} + 308224 q^{76} + 124832 q^{78} - 2835008 q^{79} - 356280 q^{81} - 612768 q^{82} - 8960 q^{84} + 3673584 q^{85} + 3948784 q^{87} + 40576 q^{90} - 219520 q^{91} - 3376668 q^{93} + 1953792 q^{94} + 229376 q^{96} + 4152808 q^{97} - 425920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} - 745 x^{18} + 29254 x^{17} - 384247 x^{16} - 30144404 x^{15} + 1098678422 x^{14} + \cdots + 17\!\cdots\!08 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 44\!\cdots\!13 \nu^{19} + \cdots - 41\!\cdots\!80 ) / 26\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\!\cdots\!52 \nu^{19} + \cdots - 12\!\cdots\!52 ) / 78\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20\!\cdots\!22 \nu^{19} + \cdots + 60\!\cdots\!20 ) / 36\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 61\!\cdots\!55 \nu^{19} + \cdots + 51\!\cdots\!92 ) / 85\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!91 \nu^{19} + \cdots + 58\!\cdots\!76 ) / 72\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 72\!\cdots\!17 \nu^{19} + \cdots + 33\!\cdots\!84 ) / 36\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\!\cdots\!55 \nu^{19} + \cdots - 40\!\cdots\!56 ) / 72\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!09 \nu^{19} + \cdots + 14\!\cdots\!08 ) / 40\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 99\!\cdots\!47 \nu^{19} + \cdots + 54\!\cdots\!04 ) / 19\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!03 \nu^{19} + \cdots + 20\!\cdots\!20 ) / 36\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!69 \nu^{19} + \cdots + 32\!\cdots\!20 ) / 24\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 57\!\cdots\!31 \nu^{19} + \cdots + 35\!\cdots\!12 ) / 67\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 52\!\cdots\!51 \nu^{19} + \cdots + 37\!\cdots\!20 ) / 19\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10\!\cdots\!75 \nu^{19} + \cdots - 60\!\cdots\!08 ) / 36\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 29\!\cdots\!45 \nu^{19} + \cdots + 27\!\cdots\!64 ) / 72\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 21\!\cdots\!98 \nu^{19} + \cdots - 17\!\cdots\!40 ) / 36\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 45\!\cdots\!83 \nu^{19} + \cdots + 34\!\cdots\!04 ) / 72\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 25\!\cdots\!97 \nu^{19} + \cdots + 15\!\cdots\!48 ) / 36\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 17\!\cdots\!87 \nu^{19} + \cdots + 13\!\cdots\!60 ) / 24\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6\beta_{9} - 6\beta_{7} - 115\beta_{2} + 466\beta _1 + 236 ) / 484 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 48 \beta_{19} - 33 \beta_{18} + 21 \beta_{17} + 51 \beta_{16} - 72 \beta_{15} - 189 \beta_{14} + \cdots + 35613 ) / 484 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9728 \beta_{19} - 12021 \beta_{18} + 5883 \beta_{17} - 6953 \beta_{16} - 7776 \beta_{15} + \cdots - 4016965 ) / 968 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 27212 \beta_{19} - 5670 \beta_{18} - 2517 \beta_{17} - 62352 \beta_{16} + 10798 \beta_{15} + \cdots + 29087456 ) / 242 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4073992 \beta_{19} + 2054409 \beta_{18} - 4464729 \beta_{17} - 950749 \beta_{16} - 985944 \beta_{15} + \cdots + 2502638967 ) / 968 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 118046360 \beta_{19} - 43027617 \beta_{18} + 9302844 \beta_{17} + 32860107 \beta_{16} + \cdots - 38479530835 ) / 484 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1720221144 \beta_{19} + 8820990 \beta_{18} + 1779777063 \beta_{17} + 700888209 \beta_{16} + \cdots + 1553070319175 ) / 484 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 44245558980 \beta_{19} + 33929018754 \beta_{18} - 19262483958 \beta_{17} - 50563224204 \beta_{16} + \cdots + 24316599467004 ) / 242 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6182329421696 \beta_{19} - 3711990857163 \beta_{18} - 337248764517 \beta_{17} + 4527222630505 \beta_{16} + \cdots - 656070684316111 ) / 968 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 37950749464208 \beta_{19} - 83348769224718 \beta_{18} + 8436928494741 \beta_{17} + \cdots - 90\!\cdots\!58 ) / 484 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 45066196745768 \beta_{19} - 229733747026377 \beta_{18} + 300769732916433 \beta_{17} + \cdots - 18\!\cdots\!23 ) / 88 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 36\!\cdots\!24 \beta_{19} + \cdots + 68\!\cdots\!52 ) / 242 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 13\!\cdots\!28 \beta_{19} + \cdots - 29\!\cdots\!16 ) / 484 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 54\!\cdots\!92 \beta_{19} + \cdots - 71\!\cdots\!17 ) / 484 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 16\!\cdots\!92 \beta_{19} + \cdots - 35\!\cdots\!37 ) / 968 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 13\!\cdots\!76 \beta_{19} + \cdots + 57\!\cdots\!96 ) / 242 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 12\!\cdots\!00 \beta_{19} + \cdots + 89\!\cdots\!83 ) / 968 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 17\!\cdots\!92 \beta_{19} + \cdots - 11\!\cdots\!01 ) / 484 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 41\!\cdots\!88 \beta_{19} + \cdots - 23\!\cdots\!05 ) / 484 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(1\) \(-1\)

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
24.0911 + 5.04721i
23.6808 − 3.35403i
15.3461 − 24.6607i
13.7980 − 15.8588i
11.5596 + 20.0707i
0.321554 + 33.3276i
−9.37785 − 19.4360i
−23.8065 + 11.6518i
−26.6903 − 13.2792i
−27.9224 + 0.834486i
24.0911 − 5.04721i
23.6808 + 3.35403i
15.3461 + 24.6607i
13.7980 + 15.8588i
11.5596 − 20.0707i
0.321554 − 33.3276i
−9.37785 + 19.4360i
−23.8065 − 11.6518i
−26.6903 + 13.2792i
−27.9224 − 0.834486i
− 5.65685i −25.5911 − 8.60793i −32.0000 25.2203i −48.6938 + 144.765i −360.262 181.019i 580.807 + 440.573i 142.667
23.2 − 5.65685i −25.1808 + 9.74318i −32.0000 53.7098i 55.1158 + 142.444i 454.241 181.019i 539.141 − 490.681i 303.828
23.3 − 5.65685i −16.8461 + 21.1000i −32.0000 − 128.417i 119.360 + 95.2958i −107.520 181.019i −161.420 − 710.904i −726.437
23.4 − 5.65685i −15.2980 + 22.2480i −32.0000 207.822i 125.853 + 86.5385i −486.434 181.019i −260.943 − 680.698i 1175.62
23.5 − 5.65685i −13.0596 − 23.6315i −32.0000 105.154i −133.680 + 73.8763i 625.179 181.019i −387.893 + 617.236i 594.839
23.6 − 5.65685i −1.82155 − 26.9385i −32.0000 − 70.4133i −152.387 + 10.3043i −200.521 181.019i −722.364 + 98.1398i −398.318
23.7 − 5.65685i 7.87785 + 25.8252i −32.0000 − 121.263i 146.089 − 44.5638i 246.804 181.019i −604.879 + 406.894i −685.969
23.8 − 5.65685i 22.3065 − 15.2126i −32.0000 − 220.995i −86.0552 − 126.184i 616.013 181.019i 266.157 − 678.676i −1250.14
23.9 − 5.65685i 25.1903 + 9.71846i −32.0000 − 130.067i 54.9759 − 142.498i −603.054 181.019i 540.103 + 489.622i −735.771
23.10 − 5.65685i 26.4224 + 5.55466i −32.0000 58.6335i 31.4219 − 149.468i 15.5539 181.019i 667.291 + 293.536i 331.681
23.11 5.65685i −25.5911 + 8.60793i −32.0000 − 25.2203i −48.6938 − 144.765i −360.262 − 181.019i 580.807 − 440.573i 142.667
23.12 5.65685i −25.1808 − 9.74318i −32.0000 − 53.7098i 55.1158 − 142.444i 454.241 − 181.019i 539.141 + 490.681i 303.828
23.13 5.65685i −16.8461 − 21.1000i −32.0000 128.417i 119.360 − 95.2958i −107.520 − 181.019i −161.420 + 710.904i −726.437
23.14 5.65685i −15.2980 − 22.2480i −32.0000 − 207.822i 125.853 − 86.5385i −486.434 − 181.019i −260.943 + 680.698i 1175.62
23.15 5.65685i −13.0596 + 23.6315i −32.0000 − 105.154i −133.680 − 73.8763i 625.179 − 181.019i −387.893 − 617.236i 594.839
23.16 5.65685i −1.82155 + 26.9385i −32.0000 70.4133i −152.387 − 10.3043i −200.521 − 181.019i −722.364 − 98.1398i −398.318
23.17 5.65685i 7.87785 − 25.8252i −32.0000 121.263i 146.089 + 44.5638i 246.804 − 181.019i −604.879 − 406.894i −685.969
23.18 5.65685i 22.3065 + 15.2126i −32.0000 220.995i −86.0552 + 126.184i 616.013 − 181.019i 266.157 + 678.676i −1250.14
23.19 5.65685i 25.1903 − 9.71846i −32.0000 130.067i 54.9759 + 142.498i −603.054 − 181.019i 540.103 − 489.622i −735.771
23.20 5.65685i 26.4224 − 5.55466i −32.0000 − 58.6335i 31.4219 + 149.468i 15.5539 − 181.019i 667.291 − 293.536i 331.681
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 66.7.c.a ✓ 20
3.b odd 2 1 inner 66.7.c.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.7.c.a ✓ 20 1.a even 1 1 trivial
66.7.c.a ✓ 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(66, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots - 15\!\cdots\!04)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 161051)^{10} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 41\!\cdots\!68)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 40\!\cdots\!68)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 45\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 17\!\cdots\!72)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 38\!\cdots\!92)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 94\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 39\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
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