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Properties

Label 22.12.c.a
Level $22$
Weight $12$
Character orbit 22.c
Analytic conductor $16.904$
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] = [22,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("22.3");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(16.9035499723\)
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: \( x^{20} - 4 x^{19} + 339028 x^{18} - 38195378 x^{17} + 220926638311 x^{16} - 35915431412664 x^{15} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{4}\cdot 5\cdot 11^{8} \)
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.

\(f(q)\) \(=\) \( q + 32 \beta_{2} q^{2} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 7) q^{3}+ \cdots + ( - 4 \beta_{18} + 2 \beta_{16} + \cdots - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 32 \beta_{2} q^{2} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 7) q^{3}+ \cdots + (263902 \beta_{19} + \cdots - 18625348631) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 160 q^{2} + 56 q^{3} - 5120 q^{4} - 362 q^{5} - 3488 q^{6} - 167178 q^{7} - 163840 q^{8} + 206411 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 160 q^{2} + 56 q^{3} - 5120 q^{4} - 362 q^{5} - 3488 q^{6} - 167178 q^{7} - 163840 q^{8} + 206411 q^{9} - 354944 q^{10} - 449801 q^{11} + 108544 q^{12} + 3369596 q^{13} - 5349696 q^{14} + 12264628 q^{15} - 5242880 q^{16} + 1661262 q^{17} - 19177088 q^{18} + 49176291 q^{19} - 370688 q^{20} - 74218796 q^{21} - 5153632 q^{22} - 127653292 q^{23} - 3571712 q^{24} + 33758041 q^{25} - 99025728 q^{26} - 135657199 q^{27} + 156930048 q^{28} + 555960754 q^{29} + 392468096 q^{30} - 13977916 q^{31} + 671088640 q^{32} - 1768265081 q^{33} - 357119936 q^{34} + 584997860 q^{35} - 613666816 q^{36} - 161000626 q^{37} - 1456805888 q^{38} + 1145816470 q^{39} + 193593344 q^{40} + 772614122 q^{41} - 545038272 q^{42} + 3749722062 q^{43} - 50755584 q^{44} - 1321716548 q^{45} - 1394938624 q^{46} + 1924889212 q^{47} + 58720256 q^{48} + 822161735 q^{49} - 805012448 q^{50} + 5262939047 q^{51} - 3168823296 q^{52} + 1531782992 q^{53} + 295824512 q^{54} + 3851857218 q^{55} + 912654336 q^{56} - 17257259731 q^{57} + 17790744128 q^{58} - 16014164779 q^{59} - 3503935488 q^{60} + 7822116636 q^{61} + 6939975808 q^{62} - 79975953122 q^{63} - 5368709120 q^{64} + 73754374272 q^{65} - 3094015232 q^{66} - 22092583538 q^{67} + 1701132288 q^{68} + 43044165458 q^{69} - 5227997760 q^{70} + 23107180542 q^{71} + 6763675648 q^{72} + 55731754682 q^{73} - 5152020032 q^{74} - 37848703409 q^{75} - 7477467136 q^{76} - 52140560296 q^{77} - 110695592320 q^{78} + 43369591752 q^{79} + 6194987008 q^{80} - 19607689666 q^{81} + 6103740704 q^{82} + 105219370951 q^{83} + 55441248256 q^{84} - 14375063614 q^{85} - 53785859936 q^{86} - 216242949124 q^{87} + 14186872832 q^{88} - 227930374154 q^{89} + 138289668864 q^{90} + 236440118278 q^{91} + 109996521472 q^{92} + 215529449050 q^{93} + 24423998144 q^{94} + 242007717988 q^{95} + 1879048192 q^{96} - 200605898215 q^{97} + 128896464000 q^{98} - 634415317547 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 339028 x^{18} - 38195378 x^{17} + 220926638311 x^{16} - 35915431412664 x^{15} + \cdots + 11\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!87 \nu^{19} + \cdots - 37\!\cdots\!20 ) / 55\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\!\cdots\!50 \nu^{19} + \cdots - 66\!\cdots\!00 ) / 69\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27\!\cdots\!05 \nu^{19} + \cdots + 29\!\cdots\!60 ) / 55\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!85 \nu^{19} + \cdots + 72\!\cdots\!00 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22\!\cdots\!67 \nu^{19} + \cdots - 48\!\cdots\!00 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 46\!\cdots\!89 \nu^{19} + \cdots - 58\!\cdots\!00 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\!\cdots\!99 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!11 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!03 \nu^{19} + \cdots - 84\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!93 \nu^{19} + \cdots - 42\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24\!\cdots\!60 \nu^{19} + \cdots - 75\!\cdots\!00 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 20\!\cdots\!17 \nu^{19} + \cdots - 10\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24\!\cdots\!35 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 33\!\cdots\!98 \nu^{19} + \cdots + 62\!\cdots\!00 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14\!\cdots\!35 \nu^{19} + \cdots + 29\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 94\!\cdots\!59 \nu^{19} + \cdots + 23\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 48\!\cdots\!79 \nu^{19} + \cdots + 94\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 62\!\cdots\!85 \nu^{19} + \cdots - 36\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3 \beta_{19} + 3 \beta_{18} - 4 \beta_{17} + \beta_{16} - 9 \beta_{15} - 2 \beta_{13} - 3 \beta_{12} + \cdots + 32289 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 539 \beta_{18} + 969 \beta_{17} - 653 \beta_{16} + 4946 \beta_{15} - 539 \beta_{14} + 1192 \beta_{13} + \cdots - 539 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 1166911 \beta_{19} + 56777 \beta_{18} + 1119846 \beta_{17} - 368741 \beta_{16} + 1295084 \beta_{15} + \cdots - 19787485707 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 445949710 \beta_{19} + 94942210 \beta_{18} - 467319368 \beta_{17} + 167554622 \beta_{16} + \cdots + 11392098641624 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 460811864498 \beta_{19} - 489983550107 \beta_{18} + 834215364107 \beta_{17} + 576162688578 \beta_{16} + \cdots - 40\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 41251840195469 \beta_{19} + 196083153299729 \beta_{18} - 235538432806061 \beta_{17} + \cdots + 83\!\cdots\!41 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 43\!\cdots\!23 \beta_{18} + \cdots - 43\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18\!\cdots\!12 \beta_{19} + \cdots - 51\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 73\!\cdots\!14 \beta_{19} + \cdots + 93\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 45\!\cdots\!74 \beta_{19} + \cdots - 39\!\cdots\!29 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 24\!\cdots\!27 \beta_{19} + \cdots + 21\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 49\!\cdots\!86 \beta_{18} + \cdots - 49\!\cdots\!86 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 10\!\cdots\!67 \beta_{19} + \cdots - 12\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 87\!\cdots\!88 \beta_{19} + \cdots + 13\!\cdots\!94 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 38\!\cdots\!77 \beta_{19} + \cdots - 12\!\cdots\!99 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 49\!\cdots\!36 \beta_{19} + \cdots + 51\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 14\!\cdots\!32 \beta_{18} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 14\!\cdots\!15 \beta_{19} + \cdots - 27\!\cdots\!17 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(13\)
\(\chi(n)\) \(-1 - \beta_{2} - \beta_{3} - \beta_{4}\)

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
180.810 − 556.477i
92.5251 − 284.763i
1.61693 − 4.97641i
−76.6553 + 235.921i
−197.297 + 607.218i
509.799 − 370.390i
396.589 − 288.139i
−104.347 + 75.8125i
−203.932 + 148.166i
−597.108 + 433.824i
509.799 + 370.390i
396.589 + 288.139i
−104.347 − 75.8125i
−203.932 − 148.166i
−597.108 − 433.824i
180.810 + 556.477i
92.5251 + 284.763i
1.61693 + 4.97641i
−76.6553 − 235.921i
−197.297 − 607.218i
−25.8885 + 18.8091i −176.692 − 543.803i 316.433 − 973.882i 3174.43 + 2306.36i 14802.8 + 10754.8i −18336.8 + 56434.7i 10125.9 + 31164.2i −121186. + 88047.0i −125562.
3.2 −25.8885 + 18.8091i −88.4071 − 272.089i 316.433 − 973.882i −354.805 − 257.781i 7406.49 + 5381.13i 12304.0 − 37867.7i 10125.9 + 31164.2i 77098.3 − 56015.2i 14034.0
3.3 −25.8885 + 18.8091i 2.50110 + 7.69759i 316.433 − 973.882i −10011.6 − 7273.85i −209.535 − 152.236i −8899.33 + 27389.3i 10125.9 + 31164.2i 143262. − 104086.i 396000.
3.4 −25.8885 + 18.8091i 80.7733 + 248.595i 316.433 − 973.882i 7368.51 + 5353.53i −6766.95 − 4916.48i 11301.1 − 34781.1i 10125.9 + 31164.2i 88039.9 − 63964.8i −291455.
3.5 −25.8885 + 18.8091i 201.415 + 619.892i 316.433 − 973.882i 932.620 + 677.588i −16874.0 − 12259.7i −16358.4 + 50346.1i 10125.9 + 31164.2i −200383. + 145587.i −36889.0
5.1 9.88854 + 30.4338i −507.917 − 369.023i −828.433 + 601.892i −2319.38 + 7138.31i 6208.22 − 19106.9i −52039.9 + 37809.2i −26509.9 − 19260.5i 67059.9 + 206389.i −240181.
5.2 9.88854 + 30.4338i −394.707 − 286.771i −828.433 + 601.892i 250.688 − 771.537i 4824.47 − 14848.2i 37543.0 − 27276.6i −26509.9 − 19260.5i 18814.2 + 57904.3i 25959.8
5.3 9.88854 + 30.4338i 106.229 + 77.1798i −828.433 + 601.892i 1487.63 − 4578.47i −1298.43 + 3996.15i 9589.00 − 6966.82i −26509.9 − 19260.5i −49413.6 − 152079.i 154051.
5.4 9.88854 + 30.4338i 205.814 + 149.533i −828.433 + 601.892i −2975.41 + 9157.38i −2515.65 + 7742.38i 94.4405 − 68.6150i −26509.9 − 19260.5i −34741.9 − 106925.i −308117.
5.5 9.88854 + 30.4338i 598.990 + 435.192i −828.433 + 601.892i 2266.32 − 6975.01i −7321.40 + 22533.0i −58786.1 + 42710.6i −26509.9 − 19260.5i 114656. + 352874.i 234687.
9.1 9.88854 − 30.4338i −507.917 + 369.023i −828.433 − 601.892i −2319.38 − 7138.31i 6208.22 + 19106.9i −52039.9 − 37809.2i −26509.9 + 19260.5i 67059.9 − 206389.i −240181.
9.2 9.88854 − 30.4338i −394.707 + 286.771i −828.433 − 601.892i 250.688 + 771.537i 4824.47 + 14848.2i 37543.0 + 27276.6i −26509.9 + 19260.5i 18814.2 − 57904.3i 25959.8
9.3 9.88854 − 30.4338i 106.229 − 77.1798i −828.433 − 601.892i 1487.63 + 4578.47i −1298.43 − 3996.15i 9589.00 + 6966.82i −26509.9 + 19260.5i −49413.6 + 152079.i 154051.
9.4 9.88854 − 30.4338i 205.814 − 149.533i −828.433 − 601.892i −2975.41 − 9157.38i −2515.65 − 7742.38i 94.4405 + 68.6150i −26509.9 + 19260.5i −34741.9 + 106925.i −308117.
9.5 9.88854 − 30.4338i 598.990 − 435.192i −828.433 − 601.892i 2266.32 + 6975.01i −7321.40 − 22533.0i −58786.1 − 42710.6i −26509.9 + 19260.5i 114656. − 352874.i 234687.
15.1 −25.8885 − 18.8091i −176.692 + 543.803i 316.433 + 973.882i 3174.43 − 2306.36i 14802.8 − 10754.8i −18336.8 − 56434.7i 10125.9 − 31164.2i −121186. − 88047.0i −125562.
15.2 −25.8885 − 18.8091i −88.4071 + 272.089i 316.433 + 973.882i −354.805 + 257.781i 7406.49 − 5381.13i 12304.0 + 37867.7i 10125.9 − 31164.2i 77098.3 + 56015.2i 14034.0
15.3 −25.8885 − 18.8091i 2.50110 − 7.69759i 316.433 + 973.882i −10011.6 + 7273.85i −209.535 + 152.236i −8899.33 − 27389.3i 10125.9 − 31164.2i 143262. + 104086.i 396000.
15.4 −25.8885 − 18.8091i 80.7733 − 248.595i 316.433 + 973.882i 7368.51 − 5353.53i −6766.95 + 4916.48i 11301.1 + 34781.1i 10125.9 − 31164.2i 88039.9 + 63964.8i −291455.
15.5 −25.8885 − 18.8091i 201.415 − 619.892i 316.433 + 973.882i 932.620 − 677.588i −16874.0 + 12259.7i −16358.4 − 50346.1i 10125.9 − 31164.2i −200383. − 145587.i −36889.0
\(n\): 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 22.12.c.a ✓ 20
11.c even 5 1 inner 22.12.c.a ✓ 20
11.c even 5 1 242.12.a.n 10
11.d odd 10 1 242.12.a.m 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.12.c.a ✓ 20 1.a even 1 1 trivial
22.12.c.a ✓ 20 11.c even 5 1 inner
242.12.a.m 10 11.d odd 10 1
242.12.a.n 10 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 56 T_{3}^{19} + 341230 T_{3}^{18} + 39041407 T_{3}^{17} + 217940592502 T_{3}^{16} + \cdots + 29\!\cdots\!01 \) acting on \(S_{12}^{\mathrm{new}}(22, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 32 T^{3} + \cdots + 1048576)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 29\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 35\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 56\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 67\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 69\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 72\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 38\!\cdots\!61 \) Copy content Toggle raw display
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