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Properties

Label 22.6.c.b
Level $22$
Weight $6$
Character orbit 22.c
Analytic conductor $3.528$
Analytic rank $0$
Dimension $12$
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] = [22,6,Mod(3,22)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(22, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("22.3");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 22.c (of order \(5\), 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: \(3.52844403589\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 329 x^{10} + 1365 x^{9} + 52729 x^{8} + 242031 x^{7} + 7017903 x^{6} + \cdots + 93740681241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 11^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 4 \beta_{3} + 4 \beta_{2} + \cdots - 4) q^{2}+ \cdots + (\beta_{11} + \beta_{10} + 2 \beta_{9} + \cdots - 65) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \beta_{3} + 4 \beta_{2} + \cdots - 4) q^{2}+ \cdots + (751 \beta_{11} - 306 \beta_{10} + \cdots - 40640) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 48 q^{4} + 44 q^{5} - 100 q^{6} - 326 q^{7} - 192 q^{8} - 69 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 48 q^{4} + 44 q^{5} - 100 q^{6} - 326 q^{7} - 192 q^{8} - 69 q^{9} + 96 q^{10} + 1011 q^{11} + 800 q^{12} - 246 q^{13} - 1304 q^{14} + 112 q^{15} - 768 q^{16} + 400 q^{17} - 1416 q^{18} + 7535 q^{19} + 704 q^{20} - 3368 q^{21} + 484 q^{22} + 15732 q^{23} - 1600 q^{24} - 19283 q^{25} - 984 q^{26} - 27411 q^{27} + 3264 q^{28} - 2504 q^{29} + 448 q^{30} + 15336 q^{31} + 12288 q^{32} + 26213 q^{33} + 24600 q^{34} - 33916 q^{35} - 5664 q^{36} - 116 q^{37} - 34880 q^{38} + 4690 q^{39} - 3584 q^{40} - 38836 q^{41} + 19808 q^{42} + 42230 q^{43} - 4704 q^{44} + 139932 q^{45} - 35232 q^{46} - 57840 q^{47} - 125521 q^{49} + 26268 q^{50} + 16467 q^{51} - 3936 q^{52} + 131934 q^{53} + 60176 q^{54} - 53082 q^{55} + 15616 q^{56} - 90021 q^{57} - 10016 q^{58} + 36585 q^{59} - 37248 q^{60} + 43146 q^{61} - 40016 q^{62} - 52922 q^{63} - 12288 q^{64} - 133772 q^{65} - 211008 q^{66} + 325206 q^{67} + 6400 q^{68} - 103894 q^{69} + 68216 q^{70} - 172806 q^{71} - 4416 q^{72} + 49436 q^{73} - 464 q^{74} + 263915 q^{75} + 37920 q^{76} - 207094 q^{77} + 330000 q^{78} + 58364 q^{79} - 14336 q^{80} + 341038 q^{81} + 82836 q^{82} + 82035 q^{83} - 52288 q^{84} - 317604 q^{85} - 104380 q^{86} - 572732 q^{87} - 38336 q^{88} + 97958 q^{89} - 151672 q^{90} - 22162 q^{91} + 15072 q^{92} - 271316 q^{93} + 153720 q^{94} + 549736 q^{95} + 76403 q^{97} + 198656 q^{98} - 199487 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 329 x^{10} + 1365 x^{9} + 52729 x^{8} + 242031 x^{7} + 7017903 x^{6} + \cdots + 93740681241 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 25\!\cdots\!10 \nu^{11} + \cdots + 12\!\cdots\!99 ) / 28\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 87\!\cdots\!90 \nu^{11} + \cdots + 59\!\cdots\!36 ) / 28\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 99\!\cdots\!27 \nu^{11} + \cdots + 14\!\cdots\!39 ) / 28\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25\!\cdots\!34 \nu^{11} + \cdots + 30\!\cdots\!57 ) / 62\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 26\!\cdots\!64 \nu^{11} + \cdots - 15\!\cdots\!61 ) / 61\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 96\!\cdots\!09 \nu^{11} + \cdots - 27\!\cdots\!15 ) / 20\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 55\!\cdots\!37 \nu^{11} + \cdots + 37\!\cdots\!10 ) / 11\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25\!\cdots\!04 \nu^{11} + \cdots + 11\!\cdots\!55 ) / 29\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 62\!\cdots\!54 \nu^{11} + \cdots - 92\!\cdots\!25 ) / 62\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!21 \nu^{11} + \cdots - 22\!\cdots\!76 ) / 20\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18\!\cdots\!83 \nu^{11} + \cdots - 78\!\cdots\!46 ) / 98\!\cdots\!26 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{10} + \beta_{8} - 12\beta_{7} + \beta_{6} - 7\beta_{5} + 7\beta_{4} - 4\beta_{2} - 6\beta _1 + 6 ) / 22 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -23\beta_{11} - 39\beta_{9} + 29\beta_{8} - 39\beta_{7} - 18\beta_{4} - 618\beta_{3} - 2940\beta _1 - 618 ) / 22 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 28 \beta_{11} + 370 \beta_{10} - 668 \beta_{9} + 370 \beta_{8} + 944 \beta_{7} - 28 \beta_{6} + \cdots - 5535 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9263 \beta_{11} + 9263 \beta_{10} - 5747 \beta_{9} + 8898 \beta_{7} - 1522 \beta_{6} + \cdots + 282342 \beta_1 ) / 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 214817 \beta_{11} - 31281 \beta_{10} + 504305 \beta_{9} - 214817 \beta_{8} - 183536 \beta_{6} + \cdots + 5765838 ) / 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1335072 \beta_{10} - 1391448 \beta_{8} - 2966180 \beta_{7} - 1391448 \beta_{6} - 2685288 \beta_{5} + \cdots + 51141116 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 65518455 \beta_{11} - 145596419 \beta_{9} + 19432867 \beta_{8} - 145596419 \beta_{7} + \cdots - 2132821838 ) / 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 836060169 \beta_{11} + 918801931 \beta_{10} - 635348235 \beta_{9} + 918801931 \beta_{8} + \cdots - 75956936510 ) / 22 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4096336324 \beta_{11} + 4096336324 \beta_{10} + 16758126568 \beta_{9} + 38899513584 \beta_{7} + \cdots + 354164978028 \beta_1 ) / 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 312802064249 \beta_{11} - 254920103939 \beta_{10} + 663038391897 \beta_{9} - 312802064249 \beta_{8} + \cdots + 23171526831924 ) / 22 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 595045952311 \beta_{10} - 278771150471 \beta_{8} - 2177065400452 \beta_{7} - 278771150471 \beta_{6} + \cdots + 21120734276566 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
5.60318 − 17.2448i
−0.669902 + 2.06175i
−4.43327 + 13.6442i
−8.87706 + 6.44956i
12.0620 − 8.76353i
−2.68489 + 1.95069i
−8.87706 − 6.44956i
12.0620 + 8.76353i
−2.68489 − 1.95069i
5.60318 + 17.2448i
−0.669902 − 2.06175i
−4.43327 − 13.6442i
−3.23607 + 2.35114i −1.93335 − 5.95024i 4.94427 − 15.2169i 69.5734 + 50.5481i 20.2463 + 14.7098i −63.1076 + 194.225i 19.7771 + 60.8676i 164.924 − 119.824i −343.990
3.2 −3.23607 + 2.35114i −1.32178 − 4.06802i 4.94427 − 15.2169i −21.5825 − 15.6806i 13.8418 + 10.0567i 76.6137 − 235.793i 19.7771 + 60.8676i 181.789 − 132.078i 106.710
3.3 −3.23607 + 2.35114i 8.84530 + 27.2230i 4.94427 − 15.2169i −34.7549 − 25.2509i −92.6292 − 67.2990i −31.2782 + 96.2643i 19.7771 + 60.8676i −466.263 + 338.760i 171.838
5.1 1.23607 + 3.80423i −15.7161 − 11.4184i −12.9443 + 9.40456i 32.8187 − 101.006i 24.0121 − 73.9017i −39.7423 + 28.8745i −51.7771 − 37.6183i 41.5249 + 127.801i 424.814
5.2 1.23607 + 3.80423i −8.33499 − 6.05573i −12.9443 + 9.40456i −30.5739 + 94.0969i 12.7347 − 39.1935i −73.6937 + 53.5416i −51.7771 − 37.6183i −42.2909 − 130.158i −395.757
5.3 1.23607 + 3.80423i 18.4610 + 13.4127i −12.9443 + 9.40456i 6.51915 − 20.0639i −28.2058 + 86.8086i −31.7919 + 23.0982i −51.7771 − 37.6183i 85.8160 + 264.115i 84.3857
9.1 1.23607 − 3.80423i −15.7161 + 11.4184i −12.9443 − 9.40456i 32.8187 + 101.006i 24.0121 + 73.9017i −39.7423 − 28.8745i −51.7771 + 37.6183i 41.5249 − 127.801i 424.814
9.2 1.23607 − 3.80423i −8.33499 + 6.05573i −12.9443 − 9.40456i −30.5739 − 94.0969i 12.7347 + 39.1935i −73.6937 − 53.5416i −51.7771 + 37.6183i −42.2909 + 130.158i −395.757
9.3 1.23607 − 3.80423i 18.4610 − 13.4127i −12.9443 − 9.40456i 6.51915 + 20.0639i −28.2058 − 86.8086i −31.7919 − 23.0982i −51.7771 + 37.6183i 85.8160 − 264.115i 84.3857
15.1 −3.23607 − 2.35114i −1.93335 + 5.95024i 4.94427 + 15.2169i 69.5734 − 50.5481i 20.2463 − 14.7098i −63.1076 − 194.225i 19.7771 − 60.8676i 164.924 + 119.824i −343.990
15.2 −3.23607 − 2.35114i −1.32178 + 4.06802i 4.94427 + 15.2169i −21.5825 + 15.6806i 13.8418 − 10.0567i 76.6137 + 235.793i 19.7771 − 60.8676i 181.789 + 132.078i 106.710
15.3 −3.23607 − 2.35114i 8.84530 − 27.2230i 4.94427 + 15.2169i −34.7549 + 25.2509i −92.6292 + 67.2990i −31.2782 − 96.2643i 19.7771 − 60.8676i −466.263 − 338.760i 171.838
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.6.c.b ✓ 12
3.b odd 2 1 198.6.f.e 12
11.c even 5 1 inner 22.6.c.b ✓ 12
11.c even 5 1 242.6.a.p 6
11.d odd 10 1 242.6.a.o 6
33.h odd 10 1 198.6.f.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.c.b ✓ 12 1.a even 1 1 trivial
22.6.c.b ✓ 12 11.c even 5 1 inner
198.6.f.e 12 3.b odd 2 1
198.6.f.e 12 33.h odd 10 1
242.6.a.o 6 11.d odd 10 1
242.6.a.p 6 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 399 T_{3}^{10} + 13187 T_{3}^{9} + 147431 T_{3}^{8} - 1098409 T_{3}^{7} + \cdots + 12238669621161 \) acting on \(S_{6}^{\mathrm{new}}(22, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 4 T^{3} + \cdots + 256)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 12238669621161 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 33\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 87\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 28\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 93\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 64\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 45\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 26\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 61\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 30\!\cdots\!41 \) Copy content Toggle raw display
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