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Properties

Label 30.11.d.a
Level $30$
Weight $11$
Character orbit 30.d
Analytic conductor $19.061$
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] = [30,11,Mod(11,30)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(30, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.11");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 30.d (of order \(2\), degree \(1\), 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: \(19.0607175802\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 830x^{10} + 197097x^{8} + 14857520x^{6} + 438897856x^{4} + 4804439040x^{2} + 12615782400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{15}\cdot 5^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} + \beta_{2} - 4) q^{3} - 512 q^{4} + 5 \beta_{8} q^{5} + ( - 4 \beta_{8} + \beta_{6} + \cdots - 389) q^{6}+ \cdots + (2 \beta_{11} + 5 \beta_{10} + \cdots - 11252) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{3} + \beta_{2} - 4) q^{3} - 512 q^{4} + 5 \beta_{8} q^{5} + ( - 4 \beta_{8} + \beta_{6} + \cdots - 389) q^{6}+ \cdots + ( - 1047519 \beta_{11} + \cdots - 2382910569) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 44 q^{3} - 6144 q^{4} - 4672 q^{6} + 71832 q^{7} - 134864 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 44 q^{3} - 6144 q^{4} - 4672 q^{6} + 71832 q^{7} - 134864 q^{9} + 22528 q^{12} - 980520 q^{13} + 687500 q^{15} + 3145728 q^{16} + 1852928 q^{18} - 2003184 q^{19} - 7389076 q^{21} - 13893888 q^{22} + 2392064 q^{24} - 23437500 q^{25} - 71000468 q^{27} - 36777984 q^{28} + 19000000 q^{30} - 53848944 q^{31} - 2364984 q^{33} - 197196288 q^{34} + 69050368 q^{36} + 379238568 q^{37} + 49018520 q^{39} - 146895872 q^{42} + 278964984 q^{43} - 218687500 q^{45} - 333048192 q^{46} - 11534336 q^{48} - 135396660 q^{49} + 1331155896 q^{51} + 502026240 q^{52} + 262727360 q^{54} + 498750000 q^{55} - 852857032 q^{57} + 584970240 q^{58} - 352000000 q^{60} + 1854585168 q^{61} + 3426768176 q^{63} - 1610612736 q^{64} - 4430336256 q^{66} + 2885038008 q^{67} + 974962764 q^{69} + 366000000 q^{70} - 948699136 q^{72} - 9318808200 q^{73} + 85937500 q^{75} + 1025630208 q^{76} + 152033920 q^{78} + 319339104 q^{79} - 6732881756 q^{81} - 12203523072 q^{82} + 3783206912 q^{84} - 4624125000 q^{85} + 6816951840 q^{87} + 7113670656 q^{88} + 3916000000 q^{90} - 5157182160 q^{91} - 24337692352 q^{93} - 2762682240 q^{94} - 1224736768 q^{96} + 16147526904 q^{97} - 28587620736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 830x^{10} + 197097x^{8} + 14857520x^{6} + 438897856x^{4} + 4804439040x^{2} + 12615782400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 184489 \nu^{10} - 148847886 \nu^{8} - 32914254417 \nu^{6} - 1978884460112 \nu^{4} + \cdots - 114086197246464 ) / 27309436416 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10723 \nu^{11} + 8683380 \nu^{9} + 1940065431 \nu^{7} + 121761000890 \nu^{5} + \cdots + 14588784236160 \nu ) / 554722927200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6719027 \nu^{11} - 24842480 \nu^{10} + 5966320770 \nu^{9} - 21341268000 \nu^{8} + \cdots - 10\!\cdots\!00 ) / 284018138726400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2148439 \nu^{11} + 107232060 \nu^{10} + 1921420890 \nu^{9} + 86946280200 \nu^{8} + \cdots + 79\!\cdots\!00 ) / 23668178227200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36341709 \nu^{11} - 49834720 \nu^{10} + 28175768190 \nu^{9} - 30120667200 \nu^{8} + \cdots - 76\!\cdots\!00 ) / 284018138726400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25505793 \nu^{11} - 62106200 \nu^{10} + 16935639030 \nu^{9} - 53353170000 \nu^{8} + \cdots - 25\!\cdots\!00 ) / 142009069363200 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3441444 \nu^{11} + 78600145 \nu^{10} + 2774080440 \nu^{9} + 63596397150 \nu^{8} + \cdots + 14\!\cdots\!00 ) / 17751133670400 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 252025 \nu^{11} - 205527750 \nu^{9} - 46706485425 \nu^{7} - 3077145473000 \nu^{5} + \cdots - 397531995264000 \nu ) / 1262302838784 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 57062101 \nu^{11} + 352446640 \nu^{10} - 42430019310 \nu^{9} + 320433328800 \nu^{8} + \cdots + 24\!\cdots\!00 ) / 284018138726400 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12384077 \nu^{11} + 370480240 \nu^{10} + 9934191870 \nu^{9} + 297491968800 \nu^{8} + \cdots + 24\!\cdots\!00 ) / 31557570969600 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11437393 \nu^{11} - 69973680 \nu^{10} + 9806134230 \nu^{9} - 56255882400 \nu^{8} + \cdots - 29\!\cdots\!00 ) / 21847549132800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 50 \beta_{11} - 12 \beta_{10} + 37 \beta_{8} + 44 \beta_{7} - 11 \beta_{6} + 32 \beta_{5} + \cdots - 184 ) / 27000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 149 \beta_{11} - 351 \beta_{10} + 192 \beta_{9} - 129 \beta_{8} + 33 \beta_{7} + 514 \beta_{6} + \cdots - 1867713 ) / 13500 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3175 \beta_{11} + 1059 \beta_{10} - 36998 \beta_{8} - 5158 \beta_{7} + 577 \beta_{6} - 4099 \beta_{5} + \cdots + 21788 ) / 6750 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 31361 \beta_{11} + 85239 \beta_{10} - 17688 \beta_{9} + 35106 \beta_{8} - 26262 \beta_{7} + \cdots + 331570584 ) / 6750 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1521725 \beta_{11} - 343857 \beta_{10} + 22076842 \beta_{8} + 2329034 \beta_{7} - 660971 \beta_{6} + \cdots - 10147924 ) / 6750 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 13436443 \beta_{11} - 40189257 \beta_{10} + 240144 \beta_{9} - 19277478 \beta_{8} + \cdots - 141288114948 ) / 6750 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 755613775 \beta_{11} + 118534371 \beta_{10} - 11504796410 \beta_{8} - 1084586902 \beta_{7} + \cdots + 4805461772 ) / 6750 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6022878149 \beta_{11} + 19065221451 \beta_{10} + 1993174008 \beta_{9} + 9900798654 \beta_{8} + \cdots + 63817483450092 ) / 6750 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 369918368525 \beta_{11} - 45150124593 \beta_{10} + 5738349037762 \beta_{8} + 512669211866 \beta_{7} + \cdots - 2291718647476 ) / 6750 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2784352070227 \beta_{11} - 9092821284873 \beta_{10} - 1479530148384 \beta_{9} - 4913094214542 \beta_{8} + \cdots - 29\!\cdots\!40 ) / 6750 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 179376259665775 \beta_{11} + 18769185047859 \beta_{10} + \cdots + 10\!\cdots\!88 ) / 6750 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
15.6779i
− 1.93969i
− 21.9082i
7.05219i
4.10832i
− 5.81891i
− 15.6779i
1.93969i
21.9082i
− 7.05219i
− 4.10832i
5.81891i
− 22.6274i −230.646 + 76.4935i −512.000 − 1397.54i 1730.85 + 5218.93i 27960.5 11585.2i 47346.5 − 35285.9i −31622.8
11.2 − 22.6274i −185.326 − 157.173i −512.000 1397.54i −3556.41 + 4193.45i −256.600 11585.2i 9642.59 + 58256.4i 31622.8
11.3 − 22.6274i 17.0050 − 242.404i −512.000 − 1397.54i −5484.98 − 384.780i −7765.89 11585.2i −58470.7 − 8244.18i −31622.8
11.4 − 22.6274i 52.4331 + 237.276i −512.000 − 1397.54i 5368.94 − 1186.43i −5130.06 11585.2i −53550.5 + 24882.2i −31622.8
11.5 − 22.6274i 152.578 − 189.127i −512.000 1397.54i −4279.46 − 3452.44i 26923.2 11585.2i −12489.1 − 57713.1i 31622.8
11.6 − 22.6274i 171.957 + 171.697i −512.000 1397.54i 3885.06 − 3890.94i −5815.10 11585.2i 89.2200 + 59048.9i 31622.8
11.7 22.6274i −230.646 − 76.4935i −512.000 1397.54i 1730.85 − 5218.93i 27960.5 − 11585.2i 47346.5 + 35285.9i −31622.8
11.8 22.6274i −185.326 + 157.173i −512.000 − 1397.54i −3556.41 − 4193.45i −256.600 − 11585.2i 9642.59 − 58256.4i 31622.8
11.9 22.6274i 17.0050 + 242.404i −512.000 1397.54i −5484.98 + 384.780i −7765.89 − 11585.2i −58470.7 + 8244.18i −31622.8
11.10 22.6274i 52.4331 − 237.276i −512.000 1397.54i 5368.94 + 1186.43i −5130.06 − 11585.2i −53550.5 − 24882.2i −31622.8
11.11 22.6274i 152.578 + 189.127i −512.000 − 1397.54i −4279.46 + 3452.44i 26923.2 − 11585.2i −12489.1 + 57713.1i 31622.8
11.12 22.6274i 171.957 − 171.697i −512.000 − 1397.54i 3885.06 + 3890.94i −5815.10 − 11585.2i 89.2200 − 59048.9i 31622.8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.11.d.a ✓ 12
3.b odd 2 1 inner 30.11.d.a ✓ 12
4.b odd 2 1 240.11.l.a 12
5.b even 2 1 150.11.d.b 12
5.c odd 4 2 150.11.b.b 24
12.b even 2 1 240.11.l.a 12
15.d odd 2 1 150.11.d.b 12
15.e even 4 2 150.11.b.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.11.d.a ✓ 12 1.a even 1 1 trivial
30.11.d.a ✓ 12 3.b odd 2 1 inner
150.11.b.b 24 5.c odd 4 2
150.11.b.b 24 15.e even 4 2
150.11.d.b 12 5.b even 2 1
150.11.d.b 12 15.d odd 2 1
240.11.l.a 12 4.b odd 2 1
240.11.l.a 12 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 512)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1953125)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 43\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 79\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 97\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 46\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 16\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 21\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 27\!\cdots\!64)^{2} \) Copy content Toggle raw display
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