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Properties

Label 22.8.c.b
Level $22$
Weight $8$
Character orbit 22.c
Analytic conductor $6.872$
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] = [22,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("22.3");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(6.87247056065\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 4993 x^{14} - 45580 x^{13} + 36623543 x^{12} - 941636880 x^{11} + 251952145581 x^{10} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 5\cdot 11^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 \beta_{2} q^{2} + (\beta_{7} + 3 \beta_{5} + 3 \beta_{3} + \cdots - 10) q^{3}+ \cdots + (2 \beta_{15} - \beta_{13} - \beta_{11} + \cdots - 61) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 \beta_{2} q^{2} + (\beta_{7} + 3 \beta_{5} + 3 \beta_{3} + \cdots - 10) q^{3}+ \cdots + (1372 \beta_{15} + 4898 \beta_{14} + \cdots - 3253433) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{2} - 96 q^{3} - 256 q^{4} - 82 q^{5} - 352 q^{6} - 700 q^{7} + 2048 q^{8} - 3762 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{2} - 96 q^{3} - 256 q^{4} - 82 q^{5} - 352 q^{6} - 700 q^{7} + 2048 q^{8} - 3762 q^{9} - 1504 q^{10} + 3868 q^{11} + 6656 q^{12} + 30562 q^{13} + 5600 q^{14} - 26612 q^{15} - 16384 q^{16} - 28534 q^{17} + 20496 q^{18} + 34372 q^{19} - 5248 q^{20} - 172652 q^{21} + 8736 q^{22} + 77216 q^{23} - 22528 q^{24} - 233354 q^{25} + 158784 q^{26} + 491916 q^{27} + 53760 q^{28} + 1158 q^{29} + 212896 q^{30} - 2996 q^{31} - 524288 q^{32} - 943264 q^{33} + 108032 q^{34} + 699840 q^{35} - 163968 q^{36} + 640374 q^{37} + 383424 q^{38} - 389492 q^{39} + 6144 q^{40} - 666054 q^{41} - 117424 q^{42} - 1920824 q^{43} - 1363968 q^{44} + 1678872 q^{45} + 1605952 q^{46} + 1692124 q^{47} - 393216 q^{48} + 1316486 q^{49} - 40928 q^{50} + 3869484 q^{51} - 1270272 q^{52} - 2620366 q^{53} - 421888 q^{54} - 7274172 q^{55} + 143360 q^{56} + 2945292 q^{57} - 9264 q^{58} + 1428860 q^{59} + 252672 q^{60} + 10497186 q^{61} + 4703808 q^{62} - 13721936 q^{63} - 1048576 q^{64} - 5436388 q^{65} - 8693568 q^{66} - 1380712 q^{67} - 1826176 q^{68} + 9446132 q^{69} + 9246400 q^{70} - 2404212 q^{71} + 1926144 q^{72} + 12064682 q^{73} - 5122992 q^{74} + 7054316 q^{75} + 1735168 q^{76} - 24265074 q^{77} + 9985536 q^{78} - 10747172 q^{79} - 49152 q^{80} + 18834940 q^{81} - 13256048 q^{82} + 6435508 q^{83} + 4585472 q^{84} - 23787614 q^{85} + 13790432 q^{86} - 3725192 q^{87} - 2021376 q^{88} + 67274636 q^{89} - 12252976 q^{90} - 2787984 q^{91} + 10376704 q^{92} - 49717070 q^{93} + 9942208 q^{94} - 33705972 q^{95} + 3145728 q^{96} + 17674136 q^{97} - 39295328 q^{98} - 47117444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 4993 x^{14} - 45580 x^{13} + 36623543 x^{12} - 941636880 x^{11} + 251952145581 x^{10} + \cdots + 16\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15\!\cdots\!95 \nu^{15} + \cdots - 40\!\cdots\!08 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!85 \nu^{15} + \cdots + 39\!\cdots\!76 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 89\!\cdots\!17 \nu^{15} + \cdots + 10\!\cdots\!20 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!65 \nu^{15} + \cdots + 20\!\cdots\!64 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 60\!\cdots\!53 \nu^{15} + \cdots + 54\!\cdots\!80 ) / 58\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!91 \nu^{15} + \cdots - 17\!\cdots\!40 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31\!\cdots\!63 \nu^{15} + \cdots + 36\!\cdots\!76 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32\!\cdots\!01 \nu^{15} + \cdots - 16\!\cdots\!48 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!29 \nu^{15} + \cdots - 96\!\cdots\!88 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!31 \nu^{15} + \cdots - 50\!\cdots\!32 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 31\!\cdots\!61 \nu^{15} + \cdots - 15\!\cdots\!28 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!26 \nu^{15} + \cdots - 91\!\cdots\!88 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!04 \nu^{15} + \cdots + 16\!\cdots\!72 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!31 \nu^{15} + \cdots - 29\!\cdots\!08 ) / 17\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 4 \beta_{15} - \beta_{14} - 3 \beta_{13} - 3 \beta_{11} + 2 \beta_{8} + 8 \beta_{7} - \beta_{6} + \cdots + 154 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 11 \beta_{15} - 29 \beta_{14} + 11 \beta_{12} - 133 \beta_{11} - 18 \beta_{10} + 115 \beta_{9} + \cdots - 10577 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 604 \beta_{15} + 11496 \beta_{14} + 16489 \beta_{13} - 23596 \beta_{12} + 28287 \beta_{11} + \cdots + 604 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 25997 \beta_{15} - 658814 \beta_{14} - 658814 \beta_{13} + 218125 \beta_{12} - 51994 \beta_{11} + \cdots + 110039415 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 79427088 \beta_{15} - 13317183 \beta_{13} - 12183468 \beta_{12} - 11364099 \beta_{11} + \cdots - 35782963178 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3267698244 \beta_{15} + 3681093804 \beta_{14} + 1366239287 \beta_{13} - 635418805 \beta_{12} + \cdots + 1889876797068 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 513175314078 \beta_{15} - 382423803072 \beta_{14} + 513175314078 \beta_{12} - 281804785089 \beta_{11} + \cdots - 302637004503589 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6589234720701 \beta_{15} + 18417221447829 \beta_{14} - 2666386248189 \beta_{13} + \cdots + 6589234720701 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 964204908018060 \beta_{15} - 500545247105645 \beta_{14} - 500545247105645 \beta_{13} + \cdots + 17\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 15\!\cdots\!35 \beta_{15} + \cdots - 10\!\cdots\!75 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 29\!\cdots\!36 \beta_{15} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11\!\cdots\!89 \beta_{15} + \cdots - 46\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 49\!\cdots\!16 \beta_{15} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 31\!\cdots\!33 \beta_{15} + \cdots + 33\!\cdots\!95 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(13\)
\(\chi(n)\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−66.8636 + 48.5792i
−0.884811 + 0.642853i
9.38166 − 6.81617i
58.3667 − 42.4059i
−19.9601 − 61.4310i
−11.0601 − 34.0396i
11.5464 + 35.5361i
19.4739 + 59.9345i
−19.9601 + 61.4310i
−11.0601 + 34.0396i
11.5464 − 35.5361i
19.4739 − 59.9345i
−66.8636 − 48.5792i
−0.884811 − 0.642853i
9.38166 + 6.81617i
58.3667 + 42.4059i
6.47214 − 4.70228i −25.9494 − 79.8642i 19.7771 − 60.8676i 16.5478 + 12.0227i −543.492 − 394.870i −205.912 + 633.733i −158.217 − 486.941i −3935.59 + 2859.38i 163.634
3.2 6.47214 − 4.70228i −0.747798 − 2.30148i 19.7771 − 60.8676i 293.130 + 212.971i −15.6621 − 11.3792i 355.303 − 1093.51i −158.217 − 486.941i 1764.58 − 1282.04i 2898.63
3.3 6.47214 − 4.70228i 3.17364 + 9.76747i 19.7771 − 60.8676i −396.047 − 287.745i 66.4697 + 48.2930i 102.158 − 314.410i −158.217 − 486.941i 1683.99 − 1223.49i −3916.32
3.4 6.47214 − 4.70228i 21.8843 + 67.3529i 19.7771 − 60.8676i 35.6820 + 25.9245i 458.350 + 333.011i −379.591 + 1168.26i −158.217 − 486.941i −2288.17 + 1662.45i 352.843
5.1 −2.47214 − 7.60845i −63.8465 − 46.3872i −51.7771 + 37.6183i 139.496 − 429.324i −195.098 + 600.449i −760.753 + 552.720i 414.217 + 300.946i 1248.78 + 3843.36i −3611.35
5.2 −2.47214 − 7.60845i −40.5460 − 29.4584i −51.7771 + 37.6183i −135.696 + 417.629i −123.897 + 381.317i 1191.06 − 865.358i 414.217 + 300.946i 100.360 + 308.877i 3512.97
5.3 −2.47214 − 7.60845i 18.6386 + 13.5418i −51.7771 + 37.6183i 65.9538 − 202.985i 56.9546 − 175.288i 538.292 − 391.092i 414.217 + 300.946i −511.801 − 1575.16i −1707.45
5.4 −2.47214 − 7.60845i 39.3932 + 28.6208i −51.7771 + 37.6183i −60.0671 + 184.868i 120.375 − 370.476i −1190.56 + 864.992i 414.217 + 300.946i 56.8510 + 174.969i 1555.05
9.1 −2.47214 + 7.60845i −63.8465 + 46.3872i −51.7771 − 37.6183i 139.496 + 429.324i −195.098 − 600.449i −760.753 − 552.720i 414.217 − 300.946i 1248.78 − 3843.36i −3611.35
9.2 −2.47214 + 7.60845i −40.5460 + 29.4584i −51.7771 − 37.6183i −135.696 − 417.629i −123.897 − 381.317i 1191.06 + 865.358i 414.217 − 300.946i 100.360 − 308.877i 3512.97
9.3 −2.47214 + 7.60845i 18.6386 − 13.5418i −51.7771 − 37.6183i 65.9538 + 202.985i 56.9546 + 175.288i 538.292 + 391.092i 414.217 − 300.946i −511.801 + 1575.16i −1707.45
9.4 −2.47214 + 7.60845i 39.3932 − 28.6208i −51.7771 − 37.6183i −60.0671 − 184.868i 120.375 + 370.476i −1190.56 − 864.992i 414.217 − 300.946i 56.8510 − 174.969i 1555.05
15.1 6.47214 + 4.70228i −25.9494 + 79.8642i 19.7771 + 60.8676i 16.5478 − 12.0227i −543.492 + 394.870i −205.912 − 633.733i −158.217 + 486.941i −3935.59 − 2859.38i 163.634
15.2 6.47214 + 4.70228i −0.747798 + 2.30148i 19.7771 + 60.8676i 293.130 − 212.971i −15.6621 + 11.3792i 355.303 + 1093.51i −158.217 + 486.941i 1764.58 + 1282.04i 2898.63
15.3 6.47214 + 4.70228i 3.17364 − 9.76747i 19.7771 + 60.8676i −396.047 + 287.745i 66.4697 − 48.2930i 102.158 + 314.410i −158.217 + 486.941i 1683.99 + 1223.49i −3916.32
15.4 6.47214 + 4.70228i 21.8843 − 67.3529i 19.7771 + 60.8676i 35.6820 − 25.9245i 458.350 − 333.011i −379.591 − 1168.26i −158.217 + 486.941i −2288.17 − 1662.45i 352.843
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.4
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 22.8.c.b ✓ 16
11.c even 5 1 inner 22.8.c.b ✓ 16
11.c even 5 1 242.8.a.r 8
11.d odd 10 1 242.8.a.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.8.c.b ✓ 16 1.a even 1 1 trivial
22.8.c.b ✓ 16 11.c even 5 1 inner
242.8.a.r 8 11.c even 5 1
242.8.a.s 8 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 96 T_{3}^{15} + 10863 T_{3}^{14} + 519812 T_{3}^{13} + 39084116 T_{3}^{12} + \cdots + 43\!\cdots\!41 \) acting on \(S_{8}^{\mathrm{new}}(22, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 8 T^{3} + \cdots + 4096)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 43\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots - 44\!\cdots\!64)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 24\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 49\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 96\!\cdots\!61 \) Copy content Toggle raw display
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