Properties

Label 22.10.c.a
Level $22$
Weight $10$
Character orbit 22.c
Analytic conductor $11.331$
Analytic rank $0$
Dimension $16$
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] = [22,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("22.3");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(11.3307883956\)
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: \( x^{16} - 5 x^{15} + 24560 x^{14} + 765588 x^{13} + 435488595 x^{12} + 44737683780 x^{11} + 8370702126799 x^{10} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4}\cdot 11^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - 16 \beta_{3} + 16 \beta_{2} + 16 \beta_1 + 16) q^{2} + ( - \beta_{7} - \beta_{5} + \beta_{4} - 21 \beta_{3} - 39 \beta_{2} + 21 \beta_1) q^{3} + 256 \beta_1 q^{4} + (\beta_{10} - 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 315 \beta_{3} + 272 \beta_{2} + \cdots + 272) q^{5}+ \cdots + (3 \beta_{15} - 7 \beta_{14} + 5 \beta_{12} - \beta_{11} - 3 \beta_{9} + 14 \beta_{7} + \cdots - 400) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 16 \beta_{3} + 16 \beta_{2} + 16 \beta_1 + 16) q^{2} + ( - \beta_{7} - \beta_{5} + \beta_{4} - 21 \beta_{3} - 39 \beta_{2} + 21 \beta_1) q^{3} + 256 \beta_1 q^{4} + (\beta_{10} - 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 315 \beta_{3} + 272 \beta_{2} + \cdots + 272) q^{5}+ \cdots + (96701 \beta_{15} + 10549 \beta_{14} - 175846 \beta_{13} + \cdots + 217483970) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 64 q^{2} - 13 q^{3} - 1024 q^{4} + 1999 q^{5} - 6512 q^{6} + 3779 q^{7} + 16384 q^{8} + 23375 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 64 q^{2} - 13 q^{3} - 1024 q^{4} + 1999 q^{5} - 6512 q^{6} + 3779 q^{7} + 16384 q^{8} + 23375 q^{9} - 45664 q^{10} + 151745 q^{11} - 201728 q^{12} - 12323 q^{13} - 60464 q^{14} + 901537 q^{15} - 262144 q^{16} - 80729 q^{17} + 620080 q^{18} - 1466984 q^{19} + 511744 q^{20} + 1874242 q^{21} - 297620 q^{23} - 1667072 q^{24} + 3069187 q^{25} + 2411328 q^{26} - 4665340 q^{27} - 877056 q^{28} + 6880191 q^{29} - 14424592 q^{30} - 1252991 q^{31} - 16777216 q^{32} + 41049250 q^{33} - 10830016 q^{34} - 1658661 q^{35} - 9921280 q^{36} - 27989673 q^{37} - 28454496 q^{38} + 46356979 q^{39} + 14032896 q^{40} - 24595449 q^{41} - 29575152 q^{42} + 23960050 q^{43} + 9841920 q^{44} + 87671812 q^{45} - 1083200 q^{46} + 7180727 q^{47} - 851968 q^{48} - 265363849 q^{49} + 65767808 q^{50} + 268370090 q^{51} - 38581248 q^{52} - 211091351 q^{53} + 312243200 q^{54} + 213516303 q^{55} + 2891776 q^{56} - 247877128 q^{57} - 110083056 q^{58} - 250912448 q^{59} - 144768768 q^{60} + 188103441 q^{61} - 158631744 q^{62} + 270549028 q^{63} - 67108864 q^{64} + 766524298 q^{65} + 518657920 q^{66} - 394231390 q^{67} - 20666624 q^{68} - 962042050 q^{69} + 18467056 q^{70} - 617600991 q^{71} - 95744000 q^{72} - 802370053 q^{73} + 447834768 q^{74} - 65519420 q^{75} - 159448064 q^{76} + 1876704291 q^{77} + 1615302656 q^{78} - 972901163 q^{79} - 224526336 q^{80} - 1915905826 q^{81} - 196304816 q^{82} - 597454384 q^{83} - 713105408 q^{84} + 2600299681 q^{85} + 329572240 q^{86} + 478576778 q^{87} + 681246720 q^{88} + 383698150 q^{89} - 420438192 q^{90} - 1391700687 q^{91} + 20764160 q^{92} - 1630241747 q^{93} - 1376883392 q^{94} - 249531699 q^{95} + 13631488 q^{96} + 3234138806 q^{97} - 1139014816 q^{98} + 6547100857 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 24560 x^{14} + 765588 x^{13} + 435488595 x^{12} + 44737683780 x^{11} + 8370702126799 x^{10} + \cdots + 74\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20\!\cdots\!82 \nu^{15} + \cdots - 16\!\cdots\!60 ) / 64\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24\!\cdots\!39 \nu^{15} + \cdots + 41\!\cdots\!20 ) / 42\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 75\!\cdots\!33 \nu^{15} + \cdots - 38\!\cdots\!40 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 48\!\cdots\!37 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!21 \nu^{15} + \cdots - 67\!\cdots\!00 ) / 60\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!39 \nu^{15} + \cdots - 66\!\cdots\!00 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\!\cdots\!01 \nu^{15} + \cdots + 58\!\cdots\!00 ) / 27\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!65 \nu^{15} + \cdots + 30\!\cdots\!16 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 54\!\cdots\!89 \nu^{15} + \cdots - 63\!\cdots\!60 ) / 64\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 58\!\cdots\!97 \nu^{15} + \cdots - 26\!\cdots\!60 ) / 64\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 35\!\cdots\!02 \nu^{15} + \cdots - 98\!\cdots\!00 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 56\!\cdots\!55 \nu^{15} + \cdots - 15\!\cdots\!00 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 53\!\cdots\!83 \nu^{15} + \cdots - 67\!\cdots\!40 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15\!\cdots\!66 \nu^{15} + \cdots - 36\!\cdots\!00 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 29\!\cdots\!83 \nu^{15} + \cdots + 38\!\cdots\!40 ) / 37\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{8} + 12 \beta_{7} + 7 \beta_{6} + 12 \beta_{5} - 19 \beta_{4} + 24 \beta_{3} - 9 \beta_{2} - 24 \beta_1 ) / 44 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 21 \beta_{15} + 3 \beta_{14} - 132 \beta_{12} - 203 \beta_{11} + 21 \beta_{9} + 1045 \beta_{7} + 717 \beta_{6} - 717 \beta_{5} - 1045 \beta_{4} + 390560 \beta_{3} - 390560 \beta_{2} - 440701 \beta _1 - 440701 ) / 44 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 10896 \beta_{15} - 2068 \beta_{14} - 10896 \beta_{13} - 13701 \beta_{12} - 12816 \beta_{10} + 8828 \beta_{9} + 13701 \beta_{8} - 91713 \beta_{6} - 52325 \beta_{5} + 52325 \beta_{4} + \cdots - 8558375 ) / 22 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 885111 \beta_{14} - 885111 \beta_{13} - 4614339 \beta_{12} + 4614339 \beta_{11} - 3571032 \beta_{10} + 1039060 \beta_{9} + 8185371 \beta_{8} - 27563867 \beta_{7} - 27563867 \beta_{6} + \cdots + 365240717 ) / 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 104206915 \beta_{15} + 104206915 \beta_{14} + 18163364 \beta_{13} + 171226255 \beta_{11} - 171226255 \beta_{10} - 18163364 \beta_{9} + 297149117 \beta_{8} + \cdots - 25542001007 ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23832471249 \beta_{15} - 2853205866 \beta_{13} + 98826828169 \beta_{12} + 90679584309 \beta_{11} - 98826828169 \beta_{10} + 90679584309 \beta_{8} + \cdots + 6643497212717 \beta_1 ) / 44 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 788263462008 \beta_{15} - 7972179392893 \beta_{14} + 16513758531975 \beta_{12} + 10994368632700 \beta_{11} + 788263462008 \beta_{9} + \cdots + 22\!\cdots\!63 ) / 44 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 26904962337429 \beta_{15} - 291615491264431 \beta_{14} + 26904962337429 \beta_{13} + \cdots + 21\!\cdots\!89 ) / 22 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 127036722000550 \beta_{14} + 127036722000550 \beta_{13} + \cdots + 28\!\cdots\!48 ) / 44 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 31\!\cdots\!54 \beta_{15} + \cdots + 13\!\cdots\!53 ) / 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 34\!\cdots\!37 \beta_{15} + \cdots - 65\!\cdots\!93 \beta_1 ) / 44 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 32\!\cdots\!89 \beta_{15} + \cdots - 10\!\cdots\!08 ) / 44 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 30\!\cdots\!16 \beta_{15} + \cdots - 75\!\cdots\!51 ) / 22 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 73\!\cdots\!15 \beta_{14} + \cdots - 12\!\cdots\!85 ) / 44 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 31\!\cdots\!29 \beta_{15} + \cdots - 24\!\cdots\!42 ) / 11 \) Copy content Toggle raw display

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
43.0920 + 132.624i
−39.5513 121.727i
16.7961 + 51.6930i
−19.6458 60.4634i
−35.8410 26.0400i
123.927 + 90.0385i
−20.5281 14.9145i
−65.7493 47.7697i
−35.8410 + 26.0400i
123.927 90.0385i
−20.5281 + 14.9145i
−65.7493 + 47.7697i
43.0920 132.624i
−39.5513 + 121.727i
16.7961 51.6930i
−19.6458 + 60.4634i
12.9443 9.40456i −70.0349 215.545i 79.1084 243.470i 398.827 + 289.765i −2933.66 2131.43i 3341.01 10282.6i −1265.73 3895.53i −25630.9 + 18622.0i 7887.63
3.2 12.9443 9.40456i −30.8657 94.9949i 79.1084 243.470i −1336.79 971.238i −1292.92 939.362i −2863.65 + 8813.41i −1265.73 3895.53i 7852.54 5705.20i −26437.9
3.3 12.9443 9.40456i −16.9465 52.1560i 79.1084 243.470i 2023.50 + 1470.16i −709.865 515.747i −2056.89 + 6330.45i −1265.73 3895.53i 13490.8 9801.65i 40018.9
3.4 12.9443 9.40456i 27.9495 + 86.0196i 79.1084 243.470i −681.370 495.044i 1170.76 + 850.609i 2022.84 6225.65i −1265.73 3895.53i 9305.68 6760.97i −13475.5
5.1 −4.94427 15.2169i −137.617 99.9845i −207.108 + 150.473i −287.658 + 885.320i −841.040 + 2588.45i −3195.55 + 2321.70i 3313.73 + 2407.57i 2859.12 + 8799.47i 14894.1
5.2 −4.94427 15.2169i −23.7921 17.2860i −207.108 + 150.473i 551.542 1697.47i −145.404 + 447.509i −2144.31 + 1557.93i 3313.73 + 2407.57i −5815.12 17897.1i −28557.2
5.3 −4.94427 15.2169i 32.4389 + 23.5682i −207.108 + 150.473i −150.943 + 464.556i 198.249 610.147i 5263.40 3824.09i 3313.73 + 2407.57i −5585.56 17190.6i 7815.41
5.4 −4.94427 15.2169i 212.368 + 154.294i −207.108 + 150.473i 482.401 1484.68i 1297.88 3994.45i 1522.64 1106.26i 3313.73 + 2407.57i 15211.0 + 46814.6i −24977.3
9.1 −4.94427 + 15.2169i −137.617 + 99.9845i −207.108 150.473i −287.658 885.320i −841.040 2588.45i −3195.55 2321.70i 3313.73 2407.57i 2859.12 8799.47i 14894.1
9.2 −4.94427 + 15.2169i −23.7921 + 17.2860i −207.108 150.473i 551.542 + 1697.47i −145.404 447.509i −2144.31 1557.93i 3313.73 2407.57i −5815.12 + 17897.1i −28557.2
9.3 −4.94427 + 15.2169i 32.4389 23.5682i −207.108 150.473i −150.943 464.556i 198.249 + 610.147i 5263.40 + 3824.09i 3313.73 2407.57i −5585.56 + 17190.6i 7815.41
9.4 −4.94427 + 15.2169i 212.368 154.294i −207.108 150.473i 482.401 + 1484.68i 1297.88 + 3994.45i 1522.64 + 1106.26i 3313.73 2407.57i 15211.0 46814.6i −24977.3
15.1 12.9443 + 9.40456i −70.0349 + 215.545i 79.1084 + 243.470i 398.827 289.765i −2933.66 + 2131.43i 3341.01 + 10282.6i −1265.73 + 3895.53i −25630.9 18622.0i 7887.63
15.2 12.9443 + 9.40456i −30.8657 + 94.9949i 79.1084 + 243.470i −1336.79 + 971.238i −1292.92 + 939.362i −2863.65 8813.41i −1265.73 + 3895.53i 7852.54 + 5705.20i −26437.9
15.3 12.9443 + 9.40456i −16.9465 + 52.1560i 79.1084 + 243.470i 2023.50 1470.16i −709.865 + 515.747i −2056.89 6330.45i −1265.73 + 3895.53i 13490.8 + 9801.65i 40018.9
15.4 12.9443 + 9.40456i 27.9495 86.0196i 79.1084 + 243.470i −681.370 + 495.044i 1170.76 850.609i 2022.84 + 6225.65i −1265.73 + 3895.53i 9305.68 + 6760.97i −13475.5
\(n\): 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 22.10.c.a 16
11.c even 5 1 inner 22.10.c.a 16
11.c even 5 1 242.10.a.m 8
11.d odd 10 1 242.10.a.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.10.c.a 16 1.a even 1 1 trivial
22.10.c.a 16 11.c even 5 1 inner
242.10.a.m 8 11.c even 5 1
242.10.a.o 8 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 13 T_{3}^{15} + 27763 T_{3}^{14} - 3425354 T_{3}^{13} + 2162106109 T_{3}^{12} + 606181677690 T_{3}^{11} + 149579741800233 T_{3}^{10} + \cdots + 34\!\cdots\!01 \) acting on \(S_{10}^{\mathrm{new}}(22, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 16 T^{3} + 256 T^{2} + \cdots + 65536)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 13 T^{15} + \cdots + 34\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{16} - 1999 T^{15} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} - 3779 T^{15} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} - 151745 T^{15} + \cdots + 95\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{16} + 12323 T^{15} + \cdots + 64\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{16} + 80729 T^{15} + \cdots + 15\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{16} + 1466984 T^{15} + \cdots + 23\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( (T^{8} + 148810 T^{7} + \cdots - 27\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 6880191 T^{15} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + 1252991 T^{15} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{16} + 27989673 T^{15} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{16} + 24595449 T^{15} + \cdots + 30\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( (T^{8} - 11980025 T^{7} + \cdots - 23\!\cdots\!80)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 7180727 T^{15} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} + 211091351 T^{15} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + 250912448 T^{15} + \cdots + 47\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{16} - 188103441 T^{15} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + 197115695 T^{7} + \cdots - 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 617600991 T^{15} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + 802370053 T^{15} + \cdots + 57\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{16} + 972901163 T^{15} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + 597454384 T^{15} + \cdots + 68\!\cdots\!61 \) Copy content Toggle raw display
$89$ \( (T^{8} - 191849075 T^{7} + \cdots - 35\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 3234138806 T^{15} + \cdots + 21\!\cdots\!01 \) Copy content Toggle raw display
show more
show less