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Properties

Label 22.10.c.b
Level $22$
Weight $10$
Character orbit 22.c
Analytic conductor $11.331$
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] = [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: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} + 70789 x^{18} - 2031667 x^{17} + 3868772403 x^{16} - 175483947004 x^{15} + \cdots + 89\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{4}\cdot 5\cdot 11^{7} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 16 \beta_{5} - 16 \beta_{4} + \cdots - 16) q^{2}+ \cdots + (\beta_{19} - \beta_{18} + \beta_{17} + \cdots - 7160) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 16 \beta_{5} - 16 \beta_{4} + \cdots - 16) q^{2}+ \cdots + (47213 \beta_{19} - 52951 \beta_{18} + \cdots - 74083989) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 80 q^{2} + 15 q^{3} - 1280 q^{4} - 2455 q^{5} - 640 q^{6} + 17413 q^{7} - 20480 q^{8} - 43338 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 80 q^{2} + 15 q^{3} - 1280 q^{4} - 2455 q^{5} - 640 q^{6} + 17413 q^{7} - 20480 q^{8} - 43338 q^{9} + 74400 q^{10} - 50256 q^{11} + 12800 q^{12} - 258759 q^{13} + 278608 q^{14} + 389527 q^{15} - 327680 q^{16} - 1751831 q^{17} - 389808 q^{18} - 897277 q^{19} - 628480 q^{20} + 903862 q^{21} + 1132624 q^{22} + 2888680 q^{23} - 163840 q^{24} - 4106132 q^{25} + 4743456 q^{26} + 3321849 q^{27} - 4820992 q^{28} - 11135159 q^{29} + 6232432 q^{30} + 4877519 q^{31} + 20971520 q^{32} - 13326739 q^{33} + 13076704 q^{34} - 22827609 q^{35} - 6236928 q^{36} + 1464357 q^{37} + 9983488 q^{38} + 39600685 q^{39} + 532480 q^{40} + 114770053 q^{41} + 4643792 q^{42} - 241560680 q^{43} - 14302976 q^{44} + 174080040 q^{45} + 53640960 q^{46} - 126091345 q^{47} + 983040 q^{48} - 56772934 q^{49} - 98934832 q^{50} - 197566653 q^{51} + 75895296 q^{52} + 105233605 q^{53} - 7906816 q^{54} + 302445283 q^{55} + 11624448 q^{56} - 23992995 q^{57} - 178162544 q^{58} - 193231403 q^{59} - 93881088 q^{60} - 240867751 q^{61} - 191953376 q^{62} + 134770852 q^{63} - 83886080 q^{64} + 802354458 q^{65} + 538801536 q^{66} + 1195877592 q^{67} - 448468736 q^{68} - 863518228 q^{69} + 263175056 q^{70} - 1191332935 q^{71} - 177512448 q^{72} - 618031059 q^{73} + 23429712 q^{74} + 1635225515 q^{75} + 139934208 q^{76} + 1295290459 q^{77} - 58986240 q^{78} - 1467134585 q^{79} + 8519680 q^{80} - 133677908 q^{81} - 1642360512 q^{82} + 543921789 q^{83} - 189995008 q^{84} + 605801379 q^{85} + 2067128000 q^{86} + 4108563958 q^{87} + 388296704 q^{88} + 2138363932 q^{89} - 2600690320 q^{90} - 4649231979 q^{91} - 1228006400 q^{92} - 9320860049 q^{93} + 1166884480 q^{94} + 1843101779 q^{95} + 15728640 q^{96} + 2125400435 q^{97} + 4284423136 q^{98} + 1679214700 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} + 70789 x^{18} - 2031667 x^{17} + 3868772403 x^{16} - 175483947004 x^{15} + \cdots + 89\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!12 \nu^{19} + \cdots + 12\!\cdots\!86 ) / 10\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 75\!\cdots\!41 \nu^{19} + \cdots + 72\!\cdots\!80 ) / 90\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 77\!\cdots\!51 \nu^{19} + \cdots - 14\!\cdots\!38 ) / 90\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!42 \nu^{19} + \cdots + 14\!\cdots\!77 ) / 90\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 45\!\cdots\!97 \nu^{19} + \cdots - 31\!\cdots\!25 ) / 18\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28\!\cdots\!54 \nu^{19} + \cdots + 25\!\cdots\!09 ) / 90\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!52 \nu^{19} + \cdots - 56\!\cdots\!23 ) / 60\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 68\!\cdots\!02 \nu^{19} + \cdots + 20\!\cdots\!77 ) / 30\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!57 \nu^{19} + \cdots + 29\!\cdots\!89 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60\!\cdots\!74 \nu^{19} + \cdots - 10\!\cdots\!52 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 92\!\cdots\!79 \nu^{19} + \cdots + 10\!\cdots\!28 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 94\!\cdots\!82 \nu^{19} + \cdots - 67\!\cdots\!45 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 44\!\cdots\!55 \nu^{19} + \cdots + 13\!\cdots\!60 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15\!\cdots\!12 \nu^{19} + \cdots - 30\!\cdots\!77 ) / 20\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 11\!\cdots\!03 \nu^{19} + \cdots + 50\!\cdots\!06 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 26\!\cdots\!16 \nu^{19} + \cdots + 29\!\cdots\!02 ) / 20\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 14\!\cdots\!50 \nu^{19} + \cdots + 13\!\cdots\!89 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 19\!\cdots\!01 \nu^{19} + \cdots + 93\!\cdots\!35 ) / 11\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - \beta_{14} + \beta_{13} - 5 \beta_{11} + \cdots - 26831 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 171 \beta_{19} - 66 \beta_{18} + 146 \beta_{17} + 129 \beta_{16} + 102 \beta_{15} - 78 \beta_{14} + \cdots + 341615 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 103378 \beta_{19} + 5774 \beta_{18} - 10849 \beta_{17} - 54576 \beta_{16} + 118063 \beta_{15} + \cdots - 13472062 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7729414 \beta_{19} - 3639736 \beta_{18} - 5944540 \beta_{17} - 2061506 \beta_{16} - 3813046 \beta_{15} + \cdots + 1429832747 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2828083519 \beta_{19} + 9249068195 \beta_{18} - 5925272166 \beta_{17} - 7616484687 \beta_{16} + \cdots + 25477481484 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 57687650104 \beta_{19} + 7714795449 \beta_{18} + 220310960327 \beta_{17} + 440621920654 \beta_{16} + \cdots - 998403311717124 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 373440206217663 \beta_{19} + 45582058198325 \beta_{18} + 144688361418167 \beta_{17} + \cdots + 27\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 24\!\cdots\!80 \beta_{19} + \cdots - 22\!\cdots\!13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 37\!\cdots\!28 \beta_{19} + \cdots + 90\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 49\!\cdots\!73 \beta_{19} + \cdots + 57\!\cdots\!21 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 50\!\cdots\!18 \beta_{19} + \cdots - 42\!\cdots\!29 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10\!\cdots\!46 \beta_{19} + \cdots + 66\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 30\!\cdots\!38 \beta_{19} + \cdots - 65\!\cdots\!22 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 22\!\cdots\!83 \beta_{19} + \cdots + 18\!\cdots\!91 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 95\!\cdots\!08 \beta_{19} + \cdots + 72\!\cdots\!87 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 87\!\cdots\!52 \beta_{19} + \cdots - 10\!\cdots\!58 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 11\!\cdots\!73 \beta_{19} + \cdots + 20\!\cdots\!45 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 10\!\cdots\!30 \beta_{19} + \cdots - 15\!\cdots\!78 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(13\)
\(\chi(n)\) \(\beta_{4}\)

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
−61.5537 − 189.443i
−57.6478 − 177.422i
21.2274 + 65.3311i
30.0219 + 92.3979i
67.5252 + 207.821i
−184.431 − 133.997i
−91.1459 − 66.2214i
−0.623625 − 0.453090i
104.104 + 75.6360i
175.023 + 127.162i
−184.431 + 133.997i
−91.1459 + 66.2214i
−0.623625 + 0.453090i
104.104 − 75.6360i
175.023 − 127.162i
−61.5537 + 189.443i
−57.6478 + 177.422i
21.2274 − 65.3311i
30.0219 − 92.3979i
67.5252 − 207.821i
−12.9443 + 9.40456i −59.9357 − 184.463i 79.1084 − 243.470i −2153.81 − 1564.83i 2510.62 + 1824.07i 1122.17 − 3453.68i 1265.73 + 3895.53i −14510.4 + 10542.4i 42596.0
3.2 −12.9443 + 9.40456i −56.0298 − 172.442i 79.1084 − 243.470i 1441.73 + 1047.48i 2347.00 + 1705.20i 807.503 − 2485.24i 1265.73 + 3895.53i −10673.0 + 7754.37i −28513.3
3.3 −12.9443 + 9.40456i 22.8454 + 70.3109i 79.1084 − 243.470i 1005.95 + 730.865i −956.961 − 695.273i 1694.68 − 5215.67i 1265.73 + 3895.53i 11502.2 − 8356.81i −19894.8
3.4 −12.9443 + 9.40456i 31.6399 + 97.3777i 79.1084 − 243.470i −185.713 − 134.929i −1325.35 − 962.924i −3615.48 + 11127.3i 1265.73 + 3895.53i 7442.55 − 5407.33i 3672.87
3.5 −12.9443 + 9.40456i 69.1432 + 212.801i 79.1084 − 243.470i −1516.28 − 1101.64i −2896.31 − 2104.29i 2714.85 − 8355.45i 1265.73 + 3895.53i −24579.6 + 17858.1i 29987.6
5.1 4.94427 + 15.2169i −185.049 − 134.446i −207.108 + 150.473i 246.428 − 758.427i 1130.92 − 3480.61i −1418.69 + 1030.74i −3313.73 − 2407.57i 10085.0 + 31038.5i 12759.3
5.2 4.94427 + 15.2169i −91.7640 − 66.6704i −207.108 + 150.473i −750.168 + 2308.78i 560.811 − 1726.00i 6343.08 − 4608.52i −3313.73 − 2407.57i −2106.70 − 6483.76i −38841.5
5.3 4.94427 + 15.2169i −1.24166 − 0.902118i −207.108 + 150.473i 245.194 − 754.629i 7.58835 − 23.3545i −6336.83 + 4603.98i −3313.73 − 2407.57i −6081.65 − 18717.4i 12695.4
5.4 4.94427 + 15.2169i 103.486 + 75.1869i −207.108 + 150.473i 767.043 − 2360.72i −632.450 + 1946.48i 9575.71 − 6957.16i −3313.73 − 2407.57i −1026.12 − 3158.06i 39715.3
5.5 4.94427 + 15.2169i 174.405 + 126.713i −207.108 + 150.473i −327.884 + 1009.12i −1065.87 + 3280.41i −2180.49 + 1584.22i −3313.73 − 2407.57i 8278.70 + 25479.2i −16976.9
9.1 4.94427 − 15.2169i −185.049 + 134.446i −207.108 − 150.473i 246.428 + 758.427i 1130.92 + 3480.61i −1418.69 − 1030.74i −3313.73 + 2407.57i 10085.0 − 31038.5i 12759.3
9.2 4.94427 − 15.2169i −91.7640 + 66.6704i −207.108 − 150.473i −750.168 − 2308.78i 560.811 + 1726.00i 6343.08 + 4608.52i −3313.73 + 2407.57i −2106.70 + 6483.76i −38841.5
9.3 4.94427 − 15.2169i −1.24166 + 0.902118i −207.108 − 150.473i 245.194 + 754.629i 7.58835 + 23.3545i −6336.83 − 4603.98i −3313.73 + 2407.57i −6081.65 + 18717.4i 12695.4
9.4 4.94427 − 15.2169i 103.486 − 75.1869i −207.108 − 150.473i 767.043 + 2360.72i −632.450 − 1946.48i 9575.71 + 6957.16i −3313.73 + 2407.57i −1026.12 + 3158.06i 39715.3
9.5 4.94427 − 15.2169i 174.405 − 126.713i −207.108 − 150.473i −327.884 − 1009.12i −1065.87 − 3280.41i −2180.49 − 1584.22i −3313.73 + 2407.57i 8278.70 − 25479.2i −16976.9
15.1 −12.9443 − 9.40456i −59.9357 + 184.463i 79.1084 + 243.470i −2153.81 + 1564.83i 2510.62 − 1824.07i 1122.17 + 3453.68i 1265.73 − 3895.53i −14510.4 − 10542.4i 42596.0
15.2 −12.9443 − 9.40456i −56.0298 + 172.442i 79.1084 + 243.470i 1441.73 − 1047.48i 2347.00 − 1705.20i 807.503 + 2485.24i 1265.73 − 3895.53i −10673.0 − 7754.37i −28513.3
15.3 −12.9443 − 9.40456i 22.8454 − 70.3109i 79.1084 + 243.470i 1005.95 − 730.865i −956.961 + 695.273i 1694.68 + 5215.67i 1265.73 − 3895.53i 11502.2 + 8356.81i −19894.8
15.4 −12.9443 − 9.40456i 31.6399 − 97.3777i 79.1084 + 243.470i −185.713 + 134.929i −1325.35 + 962.924i −3615.48 − 11127.3i 1265.73 − 3895.53i 7442.55 + 5407.33i 3672.87
15.5 −12.9443 − 9.40456i 69.1432 − 212.801i 79.1084 + 243.470i −1516.28 + 1101.64i −2896.31 + 2104.29i 2714.85 + 8355.45i 1265.73 − 3895.53i −24579.6 − 17858.1i 29987.6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.5
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.b ✓ 20
11.c even 5 1 inner 22.10.c.b ✓ 20
11.c even 5 1 242.10.a.r 10
11.d odd 10 1 242.10.a.q 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.10.c.b ✓ 20 1.a even 1 1 trivial
22.10.c.b ✓ 20 11.c even 5 1 inner
242.10.a.q 10 11.d odd 10 1
242.10.a.r 10 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 15 T_{3}^{19} + 70989 T_{3}^{18} - 1646113 T_{3}^{17} + 3890074283 T_{3}^{16} + \cdots + 42\!\cdots\!81 \) acting on \(S_{10}^{\mathrm{new}}(22, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 16 T^{3} + \cdots + 65536)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 42\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 61\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 14\!\cdots\!80)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 59\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 29\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 93\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 81\!\cdots\!41 \) Copy content Toggle raw display
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