↑

Properties

Label 22.6.c.a
Level $22$
Weight $6$
Character orbit 22.c
Analytic conductor $3.528$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 121x^{6} - 241x^{5} + 7111x^{4} + 2910x^{3} + 163440x^{2} + 626400x + 27248400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{3} q^{2} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + 3) q^{3} - 16 \beta_1 q^{4} + (2 \beta_{6} - 27 \beta_{4} + \cdots + 27) q^{5}+ \cdots + (6 \beta_{7} - 6 \beta_{5} + 119 \beta_{3} + \cdots + 87) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{3} q^{2} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + 3) q^{3} - 16 \beta_1 q^{4} + (2 \beta_{6} - 27 \beta_{4} + \cdots + 27) q^{5}+ \cdots + ( - 2200 \beta_{7} - 3146 \beta_{6} + \cdots + 27907) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 20 q^{3} - 32 q^{4} - 30 q^{5} + 40 q^{6} + 48 q^{7} + 128 q^{8} + 284 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 20 q^{3} - 32 q^{4} - 30 q^{5} + 40 q^{6} + 48 q^{7} + 128 q^{8} + 284 q^{9} + 1200 q^{10} - 320 q^{12} - 1746 q^{13} - 192 q^{14} - 3578 q^{15} - 512 q^{16} - 90 q^{17} + 2344 q^{18} - 1824 q^{19} - 480 q^{20} + 10324 q^{21} + 1640 q^{23} + 640 q^{24} - 4572 q^{25} - 7536 q^{26} - 5500 q^{27} - 1312 q^{28} - 6666 q^{29} + 14312 q^{30} + 6878 q^{31} - 8192 q^{32} + 38962 q^{33} - 2960 q^{34} + 2456 q^{35} - 9376 q^{36} + 9062 q^{37} + 2136 q^{38} - 27728 q^{39} - 11520 q^{40} + 8014 q^{41} - 18216 q^{42} - 10792 q^{43} + 12320 q^{44} - 67880 q^{45} - 3120 q^{46} - 1304 q^{47} + 5120 q^{48} + 56724 q^{49} - 13832 q^{50} - 29014 q^{51} + 30144 q^{52} - 27598 q^{53} + 62480 q^{54} + 84238 q^{55} - 4352 q^{56} + 42926 q^{57} + 26664 q^{58} - 146416 q^{59} + 53952 q^{60} - 1942 q^{61} - 40872 q^{62} + 110074 q^{63} - 8192 q^{64} - 32972 q^{65} - 57728 q^{66} - 335736 q^{67} - 1440 q^{68} + 220148 q^{69} + 46776 q^{70} + 144862 q^{71} - 18176 q^{72} + 139906 q^{73} - 36248 q^{74} - 142640 q^{75} + 75456 q^{76} + 36586 q^{77} - 123568 q^{78} + 124046 q^{79} + 46080 q^{80} - 3952 q^{81} + 169104 q^{82} - 60618 q^{83} - 155456 q^{84} - 105966 q^{85} - 296352 q^{86} - 173656 q^{87} - 49280 q^{88} - 534800 q^{89} + 67560 q^{90} + 64510 q^{91} - 25600 q^{92} + 369082 q^{93} + 30376 q^{94} + 295574 q^{95} - 20480 q^{96} + 36414 q^{97} + 180304 q^{98} + 379390 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 121x^{6} - 241x^{5} + 7111x^{4} + 2910x^{3} + 163440x^{2} + 626400x + 27248400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 77850234 \nu^{7} - 6531055843 \nu^{6} + 53545129558 \nu^{5} - 1589244563098 \nu^{4} + \cdots - 50\!\cdots\!50 ) / 24\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 836002071 \nu^{7} + 44675133197 \nu^{6} + 111512733143 \nu^{5} - 1007500139903 \nu^{4} + \cdots - 18\!\cdots\!60 ) / 33\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 836002071 \nu^{7} + 44675133197 \nu^{6} + 111512733143 \nu^{5} - 1007500139903 \nu^{4} + \cdots - 18\!\cdots\!60 ) / 33\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10871935891 \nu^{7} + 762637084705 \nu^{6} - 1591900836565 \nu^{5} + \cdots + 13\!\cdots\!60 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 169485681739 \nu^{7} + 2273797273633 \nu^{6} - 40184056104373 \nu^{5} + \cdots + 18\!\cdots\!60 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 695201798695 \nu^{7} - 1501846958149 \nu^{6} + 174165923774089 \nu^{5} + \cdots + 11\!\cdots\!60 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 761430227483 \nu^{7} + 417490701662 \nu^{6} - 80009285641712 \nu^{5} + \cdots - 32\!\cdots\!60 ) / 14\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - 36\beta_{4} + 36\beta_{3} + \beta_{2} - 169\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -18\beta_{6} - 85\beta_{5} + 199\beta_{4} + 54\beta_{3} - 18\beta_{2} + 54 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -169\beta_{7} - 36\beta_{6} + 9625\beta_{4} - 15097\beta_{3} + 15097\beta _1 - 9625 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2754\beta_{7} + 7633\beta_{6} + 7633\beta_{5} - 13590\beta_{4} + 2754\beta_{2} - 13590\beta _1 - 23731 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8172\beta_{7} - 8172\beta_{5} + 1403965\beta_{3} - 14305\beta_{2} - 646488\beta _1 + 646488 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 713221 \beta_{7} - 713221 \beta_{6} - 327330 \beta_{5} + 2215206 \beta_{4} - 2215206 \beta_{3} + \cdots + 4783549 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(13\)
\(\chi(n)\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
6.41117 − 4.65799i
−5.60216 + 4.07021i
2.86335 + 8.81248i
−3.17237 − 9.76354i
2.86335 − 8.81248i
−3.17237 + 9.76354i
6.41117 + 4.65799i
−5.60216 − 4.07021i
3.23607 − 2.35114i −4.32475 − 13.3102i 4.94427 − 15.2169i −27.5325 − 20.0035i −45.2894 − 32.9047i 13.8732 − 42.6973i −19.7771 − 60.8676i 38.1327 − 27.7050i −136.128
3.2 3.23607 − 2.35114i 4.85262 + 14.9348i 4.94427 − 15.2169i 50.2194 + 36.4865i 50.8173 + 36.9209i 0.362867 − 1.11679i −19.7771 − 60.8676i −2.90976 + 2.11407i 248.298
5.1 −1.23607 − 3.80423i −11.0656 − 8.03966i −12.9443 + 9.40456i −11.3829 + 35.0330i −16.9068 + 52.0338i −113.191 + 82.2377i 51.7771 + 37.6183i −17.2788 − 53.1788i 147.343
5.2 −1.23607 − 3.80423i 20.5378 + 14.9216i −12.9443 + 9.40456i −26.3040 + 80.9554i 31.3789 − 96.5744i 122.954 − 89.3317i 51.7771 + 37.6183i 124.056 + 381.805i 340.486
9.1 −1.23607 + 3.80423i −11.0656 + 8.03966i −12.9443 − 9.40456i −11.3829 − 35.0330i −16.9068 − 52.0338i −113.191 − 82.2377i 51.7771 − 37.6183i −17.2788 + 53.1788i 147.343
9.2 −1.23607 + 3.80423i 20.5378 − 14.9216i −12.9443 − 9.40456i −26.3040 − 80.9554i 31.3789 + 96.5744i 122.954 + 89.3317i 51.7771 − 37.6183i 124.056 − 381.805i 340.486
15.1 3.23607 + 2.35114i −4.32475 + 13.3102i 4.94427 + 15.2169i −27.5325 + 20.0035i −45.2894 + 32.9047i 13.8732 + 42.6973i −19.7771 + 60.8676i 38.1327 + 27.7050i −136.128
15.2 3.23607 + 2.35114i 4.85262 − 14.9348i 4.94427 + 15.2169i 50.2194 − 36.4865i 50.8173 − 36.9209i 0.362867 + 1.11679i −19.7771 + 60.8676i −2.90976 − 2.11407i 248.298
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.2
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.a ✓ 8
3.b odd 2 1 198.6.f.b 8
11.c even 5 1 inner 22.6.c.a ✓ 8
11.c even 5 1 242.6.a.k 4
11.d odd 10 1 242.6.a.m 4
33.h odd 10 1 198.6.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.c.a ✓ 8 1.a even 1 1 trivial
22.6.c.a ✓ 8 11.c even 5 1 inner
198.6.f.b 8 3.b odd 2 1
198.6.f.b 8 33.h odd 10 1
242.6.a.k 4 11.c even 5 1
242.6.a.m 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 20 T_{3}^{7} + 301 T_{3}^{6} + 100 T_{3}^{5} + 129736 T_{3}^{4} + 1298040 T_{3}^{3} + \cdots + 5823368721 \) acting on \(S_{6}^{\mathrm{new}}(22, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 5823368721 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 43875773006641 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 1256598402361 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 67\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 66\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 54\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 30\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 1617277673216)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 60\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 57\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 37\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 54\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 16\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 54\!\cdots\!21 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 97\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 17\!\cdots\!21 \) Copy content Toggle raw display
show more
show less