Properties

Label 36.7.g.a
Level $36$
Weight $7$
Character orbit 36.g
Analytic conductor $8.282$
Analytic rank $0$
Dimension $12$
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] = [36,7,Mod(5,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.5");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 36.g (of order \(6\), degree \(2\), 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: \(8.28194701031\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 59 x^{10} + 602 x^{9} + 26655 x^{8} + 692184 x^{7} - 4209870 x^{6} - 34154736 x^{5} + 2261179815 x^{4} - 21421205158 x^{3} + \cdots + 166668145981081 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{15} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 4 \beta_1 - 2) q^{3} + ( - \beta_{6} + \beta_{2} + 12 \beta_1 - 24) q^{5} + (\beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 21 \beta_1 - 19) q^{7} + ( - \beta_{11} - \beta_{9} + 2 \beta_{6} - \beta_{5} + 2 \beta_{3} + 4 \beta_{2} - 94 \beta_1 + 64) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 4 \beta_1 - 2) q^{3} + ( - \beta_{6} + \beta_{2} + 12 \beta_1 - 24) q^{5} + (\beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 21 \beta_1 - 19) q^{7} + ( - \beta_{11} - \beta_{9} + 2 \beta_{6} - \beta_{5} + 2 \beta_{3} + 4 \beta_{2} - 94 \beta_1 + 64) q^{9} + ( - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots - 122) q^{11}+ \cdots + ( - 165 \beta_{11} + 18 \beta_{10} + 519 \beta_{9} + 1395 \beta_{8} + \cdots - 615507) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} - 216 q^{5} - 120 q^{7} + 174 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} - 216 q^{5} - 120 q^{7} + 174 q^{9} - 2214 q^{11} - 840 q^{13} - 5040 q^{15} - 12900 q^{19} + 5892 q^{21} - 8208 q^{23} + 24078 q^{25} + 40176 q^{27} - 29268 q^{29} + 3120 q^{31} + 2718 q^{33} + 12768 q^{37} + 223812 q^{39} - 16578 q^{41} + 71430 q^{43} - 61524 q^{45} - 329508 q^{47} - 238914 q^{49} - 153198 q^{51} + 5400 q^{55} + 652854 q^{57} + 428058 q^{59} - 93576 q^{61} - 493392 q^{63} - 1426464 q^{65} + 104334 q^{67} - 860256 q^{69} + 221820 q^{73} + 2598978 q^{75} + 3461184 q^{77} + 123468 q^{79} - 1125882 q^{81} - 2901420 q^{83} + 377568 q^{85} - 4083948 q^{87} + 91488 q^{91} + 6111144 q^{93} + 7249716 q^{95} - 1033482 q^{97} - 3503484 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 59 x^{10} + 602 x^{9} + 26655 x^{8} + 692184 x^{7} - 4209870 x^{6} - 34154736 x^{5} + 2261179815 x^{4} - 21421205158 x^{3} + \cdots + 166668145981081 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 173906459 \nu^{11} - 3983241235 \nu^{10} - 40768097000 \nu^{9} - 75429664038 \nu^{8} + 289754667033 \nu^{7} + \cdots - 54\!\cdots\!37 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 202671109 \nu^{11} + 32090850959 \nu^{10} + 838755316840 \nu^{9} + 8246596605498 \nu^{8} + 16787157567837 \nu^{7} + \cdots + 22\!\cdots\!13 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1112185019 \nu^{11} - 37350275471 \nu^{10} - 750659956264 \nu^{9} - 7631243937594 \nu^{8} + 17029627188195 \nu^{7} + \cdots - 46\!\cdots\!57 ) / 37\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4343008675 \nu^{11} + 813456210967 \nu^{10} + 12980013229928 \nu^{9} + 134444070700938 \nu^{8} + 198710181513525 \nu^{7} + \cdots + 61\!\cdots\!89 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6929181461 \nu^{11} + 257492325313 \nu^{10} + 2751723174248 \nu^{9} + 7820632027530 \nu^{8} - 527349043103487 \nu^{7} + \cdots + 15\!\cdots\!59 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11949198187 \nu^{11} - 187112610218 \nu^{10} - 1940155302616 \nu^{9} - 2744719897806 \nu^{8} - 15388945404333 \nu^{7} + \cdots + 14\!\cdots\!82 ) / 37\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35711678585 \nu^{11} - 325369646803 \nu^{10} - 993353112008 \nu^{9} + 29517172094142 \nu^{8} - 178614240947265 \nu^{7} + \cdots + 17\!\cdots\!91 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 39298535645 \nu^{11} - 254427416017 \nu^{10} + 979809244360 \nu^{9} + 52347389598486 \nu^{8} - 193278545752521 \nu^{7} + \cdots + 28\!\cdots\!21 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9488516311 \nu^{11} - 143056562701 \nu^{10} - 3425187928928 \nu^{9} - 53314586786682 \nu^{8} - 140751870439917 \nu^{7} + \cdots - 23\!\cdots\!67 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 76707126133 \nu^{11} + 81415003807 \nu^{10} + 10166518046456 \nu^{9} + 258653426509494 \nu^{8} + 919437592169343 \nu^{7} + \cdots + 10\!\cdots\!93 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 158422959073 \nu^{11} - 159746415521 \nu^{10} + 13476527468456 \nu^{9} + 228158883283086 \nu^{8} + \cdots + 18\!\cdots\!45 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{7} - \beta_{3} + \beta_{2} + 9\beta _1 + 1 ) / 27 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{10} + 3\beta_{9} + \beta_{8} - 2\beta_{7} - 10\beta_{3} + 2\beta_{2} - 282\beta _1 + 58 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 63 \beta_{11} - 90 \beta_{10} - 9 \beta_{9} + 8 \beta_{8} + 22 \beta_{7} + 567 \beta_{6} - 162 \beta_{5} + 99 \beta_{4} - 1300 \beta_{3} + 460 \beta_{2} - 15120 \beta _1 + 3946 ) / 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 270 \beta_{11} + 303 \beta_{10} - 372 \beta_{9} + 75 \beta_{8} - 47 \beta_{7} + 135 \beta_{6} + 1512 \beta_{5} - 918 \beta_{4} + 14684 \beta_{3} + 6816 \beta_{2} - 248307 \beta _1 + 41122 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4599 \beta_{11} - 3879 \beta_{10} + 4707 \beta_{9} - 46417 \beta_{8} - 65087 \beta_{7} - 14985 \beta_{6} - 24705 \beta_{5} - 12006 \beta_{4} - 673651 \beta_{3} - 191624 \beta_{2} + \cdots - 15469244 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 40389 \beta_{11} - 29961 \beta_{10} - 36927 \beta_{9} + 164825 \beta_{8} + 107371 \beta_{7} + 38475 \beta_{6} - 95013 \beta_{5} - 33504 \beta_{4} - 11776 \beta_{3} - 442634 \beta_{2} + \cdots + 21998677 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8220231 \beta_{11} + 1110591 \beta_{10} + 1185921 \beta_{9} - 25277014 \beta_{8} - 15346376 \beta_{7} + 1951047 \beta_{6} - 20601297 \beta_{5} + 6999642 \beta_{4} + \cdots + 975949849 ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 46371465 \beta_{11} + 8960142 \beta_{10} - 58260786 \beta_{9} + 34123966 \beta_{8} + 51337747 \beta_{7} - 46578321 \beta_{6} + 83576853 \beta_{5} - 48000942 \beta_{4} + \cdots - 13496205494 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 983079630 \beta_{11} - 98311005 \beta_{10} + 3027082662 \beta_{9} + 2173653863 \beta_{8} - 554776007 \beta_{7} - 2476888470 \beta_{6} - 3549554379 \beta_{5} + \cdots + 593445455986 ) / 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1949216985 \beta_{11} - 3254274516 \beta_{10} - 7565378205 \beta_{9} - 11040941052 \beta_{8} + 15791195404 \beta_{7} + 22255846794 \beta_{6} + \cdots - 4876287261224 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 186742393902 \beta_{11} + 540934518744 \beta_{10} + 225727678026 \beta_{9} + 331158635420 \beta_{8} - 1785189014252 \beta_{7} - 1154905058664 \beta_{6} + \cdots + 164657189157526 ) / 27 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−9.16225 + 13.8457i
9.14227 + 10.9018i
12.5094 + 5.69894i
−16.7265 5.11172i
8.49212 11.5289i
−3.25500 15.5379i
−9.16225 13.8457i
9.14227 10.9018i
12.5094 5.69894i
−16.7265 + 5.11172i
8.49212 + 11.5289i
−3.25500 + 15.5379i
0 −23.9814 + 12.4054i 0 −19.8824 11.4791i 0 −18.7162 32.4175i 0 421.213 594.996i 0
5.2 0 −18.8825 19.2990i 0 98.8290 + 57.0590i 0 −276.118 478.250i 0 −15.9009 + 728.827i 0
5.3 0 −9.87085 25.1310i 0 −166.078 95.8851i 0 315.624 + 546.677i 0 −534.133 + 496.128i 0
5.4 0 8.85375 + 25.5071i 0 51.3356 + 29.6386i 0 76.8738 + 133.149i 0 −572.222 + 451.667i 0
5.5 0 19.9686 18.1729i 0 114.537 + 66.1282i 0 65.4834 + 113.420i 0 68.4928 725.775i 0
5.6 0 26.9124 + 2.17372i 0 −186.742 107.815i 0 −223.147 386.502i 0 719.550 + 117.000i 0
29.1 0 −23.9814 12.4054i 0 −19.8824 + 11.4791i 0 −18.7162 + 32.4175i 0 421.213 + 594.996i 0
29.2 0 −18.8825 + 19.2990i 0 98.8290 57.0590i 0 −276.118 + 478.250i 0 −15.9009 728.827i 0
29.3 0 −9.87085 + 25.1310i 0 −166.078 + 95.8851i 0 315.624 546.677i 0 −534.133 496.128i 0
29.4 0 8.85375 25.5071i 0 51.3356 29.6386i 0 76.8738 133.149i 0 −572.222 451.667i 0
29.5 0 19.9686 + 18.1729i 0 114.537 66.1282i 0 65.4834 113.420i 0 68.4928 + 725.775i 0
29.6 0 26.9124 2.17372i 0 −186.742 + 107.815i 0 −223.147 + 386.502i 0 719.550 117.000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.7.g.a 12
3.b odd 2 1 108.7.g.a 12
4.b odd 2 1 144.7.q.b 12
9.c even 3 1 108.7.g.a 12
9.c even 3 1 324.7.c.b 12
9.d odd 6 1 inner 36.7.g.a 12
9.d odd 6 1 324.7.c.b 12
12.b even 2 1 432.7.q.c 12
36.f odd 6 1 432.7.q.c 12
36.h even 6 1 144.7.q.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.7.g.a 12 1.a even 1 1 trivial
36.7.g.a 12 9.d odd 6 1 inner
108.7.g.a 12 3.b odd 2 1
108.7.g.a 12 9.c even 3 1
144.7.q.b 12 4.b odd 2 1
144.7.q.b 12 36.h even 6 1
324.7.c.b 12 9.c even 3 1
324.7.c.b 12 9.d odd 6 1
432.7.q.c 12 12.b even 2 1
432.7.q.c 12 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{11} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{12} + 216 T^{11} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + 120 T^{11} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{12} + 2214 T^{11} + \cdots + 79\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{12} + 840 T^{11} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + 143217774 T^{10} + \cdots + 82\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{6} + 6450 T^{5} + \cdots + 80\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 8208 T^{11} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + 29268 T^{11} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} - 3120 T^{11} + \cdots + 89\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( (T^{6} - 6384 T^{5} + \cdots - 71\!\cdots\!52)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 16578 T^{11} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{12} - 71430 T^{11} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{12} + 329508 T^{11} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + 123487951464 T^{10} + \cdots + 61\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} - 428058 T^{11} + \cdots + 60\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{12} + 93576 T^{11} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{12} - 104334 T^{11} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{12} + 396592461360 T^{10} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{6} - 110910 T^{5} + \cdots - 40\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 123468 T^{11} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + 2901420 T^{11} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{12} + 3628807684200 T^{10} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{12} + 1033482 T^{11} + \cdots + 29\!\cdots\!29 \) Copy content Toggle raw display
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