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Properties

Label 30.7.b.a
Level $30$
Weight $7$
Character orbit 30.b
Analytic conductor $6.902$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [30,7,Mod(29,30)]
 
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, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.29");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 30.b (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: \(6.90162250860\)
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: \( x^{12} + 1376x^{10} + 682902x^{8} + 143683588x^{6} + 11001884985x^{4} + 4485788300x^{2} + 183826562500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{8}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + 32 q^{4} + (\beta_{6} + \beta_{2} - \beta_1) q^{5} + (\beta_{4} - 9) q^{6} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{7} + 32 \beta_1 q^{8} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \cdots + 34) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + 32 q^{4} + (\beta_{6} + \beta_{2} - \beta_1) q^{5} + (\beta_{4} - 9) q^{6} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{7} + 32 \beta_1 q^{8} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \cdots + 34) q^{9}+ \cdots + (1179 \beta_{11} - 1442 \beta_{10} + \cdots - 589792) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 384 q^{4} - 112 q^{6} + 424 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 384 q^{4} - 112 q^{6} + 424 q^{9} - 576 q^{10} + 5168 q^{15} + 12288 q^{16} + 384 q^{19} - 24796 q^{21} - 3584 q^{24} - 39516 q^{25} + 23408 q^{30} + 33936 q^{31} + 35328 q^{34} + 13568 q^{36} - 269760 q^{39} - 18432 q^{40} + 110812 q^{45} - 71712 q^{46} + 790980 q^{49} + 623216 q^{51} - 473360 q^{54} - 553008 q^{55} + 165376 q^{60} - 681072 q^{61} + 393216 q^{64} + 314944 q^{66} - 2583764 q^{69} - 1023456 q^{70} + 2366128 q^{75} + 12288 q^{76} + 3306096 q^{79} + 1443524 q^{81} - 793472 q^{84} - 4091952 q^{85} + 2384768 q^{90} - 1409280 q^{91} + 2706720 q^{94} - 114688 q^{96} - 7084144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1376x^{10} + 682902x^{8} + 143683588x^{6} + 11001884985x^{4} + 4485788300x^{2} + 183826562500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 67357827 \nu^{11} - 91520675077 \nu^{9} - 44637385015329 \nu^{7} + \cdots + 24\!\cdots\!00 \nu ) / 13\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17296654608 \nu^{11} + 54360205375 \nu^{10} - 23783884371108 \nu^{9} + \cdots + 13\!\cdots\!50 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 56739727368 \nu^{11} + 93072545125 \nu^{10} + 77941141718868 \nu^{9} + \cdots + 43\!\cdots\!50 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2387403241 \nu^{11} - 854959500 \nu^{10} + 3295087259866 \nu^{9} - 1000219414500 \nu^{8} + \cdots + 45\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4774806482 \nu^{11} + 47785714625 \nu^{10} + 6590174519732 \nu^{9} + 69262050167750 \nu^{8} + \cdots - 21\!\cdots\!50 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 113924475811 \nu^{11} - 54360205375 \nu^{10} - 157743442333186 \nu^{9} + \cdots - 13\!\cdots\!50 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 148156022803 \nu^{11} + 656839770125 \nu^{10} + 206213645943178 \nu^{9} + \cdots + 10\!\cdots\!50 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23896611797 \nu^{11} - 83833099875 \nu^{10} - 32614763784422 \nu^{9} + \cdots - 15\!\cdots\!50 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18913242456 \nu^{11} - 54360205375 \nu^{10} - 25980380572956 \nu^{9} + \cdots - 13\!\cdots\!50 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 93202201213 \nu^{11} + 277300916375 \nu^{10} - 128997523586338 \nu^{9} + \cdots + 18\!\cdots\!50 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27745942819 \nu^{11} - 11185581625 \nu^{10} + 38272190885194 \nu^{9} + \cdots + 17\!\cdots\!50 ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -5\beta_{11} - 2\beta_{8} - 2\beta_{7} - 7\beta_{6} + 5\beta_{4} - 2\beta_{3} - 6\beta_{2} + 39\beta_1 ) / 180 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 38 \beta_{11} + 48 \beta_{10} - 25 \beta_{9} + 38 \beta_{8} - 22 \beta_{7} + 22 \beta_{6} + \cdots - 41172 ) / 180 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7330 \beta_{11} - 2205 \beta_{10} + 1680 \beta_{9} + 2806 \beta_{8} + 5011 \beta_{7} + \cdots - 110157 \beta_1 ) / 720 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11153 \beta_{11} - 9993 \beta_{10} + 9105 \beta_{9} - 11153 \beta_{8} + 7822 \beta_{7} + \cdots + 7884612 ) / 90 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1444460 \beta_{11} + 690165 \beta_{10} - 723800 \beta_{9} - 671906 \beta_{8} + \cdots + 36505827 \beta_1 ) / 360 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11244362 \beta_{11} + 8463342 \beta_{10} - 11242865 \beta_{9} + 11244362 \beta_{8} - 8423248 \beta_{7} + \cdots - 6491369568 ) / 180 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1205319410 \beta_{11} - 698025825 \beta_{10} + 986307280 \beta_{9} + 670310654 \beta_{8} + \cdots - 43824632073 \beta_1 ) / 720 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 889475009 \beta_{11} - 584161059 \beta_{10} + 1090894970 \beta_{9} - 889475009 \beta_{8} + \cdots + 460075911216 ) / 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 128329815815 \beta_{11} + 80788066695 \beta_{10} - 150056293380 \beta_{9} + \cdots + 6297603986874 \beta_1 ) / 180 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2448287942978 \beta_{11} + 1391955674808 \beta_{10} - 3676278270445 \beta_{9} + 2448287942978 \beta_{8} + \cdots - 11\!\cdots\!52 ) / 180 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 217759064763850 \beta_{11} - 142093167464745 \beta_{10} + 341829802350960 \beta_{9} + \cdots - 14\!\cdots\!57 \beta_1 ) / 720 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−1.41421 + 1.44735i
−1.41421 − 1.44735i
−1.41421 + 14.7657i
−1.41421 − 14.7657i
−1.41421 + 21.7688i
−1.41421 − 21.7688i
1.41421 − 21.7688i
1.41421 + 21.7688i
1.41421 − 14.7657i
1.41421 + 14.7657i
1.41421 − 1.44735i
1.41421 + 1.44735i
−5.65685 −21.5067 − 16.3237i 32.0000 −74.4540 − 100.407i 121.660 + 92.3408i − 41.6060i −181.019 196.074 + 702.137i 421.175 + 567.989i
29.2 −5.65685 −21.5067 + 16.3237i 32.0000 −74.4540 + 100.407i 121.660 − 92.3408i 41.6060i −181.019 196.074 − 702.137i 421.175 − 567.989i
29.3 −5.65685 0.305514 − 26.9983i 32.0000 113.056 + 53.3219i −1.72825 + 152.725i − 274.097i −181.019 −728.813 − 16.4967i −639.544 − 301.634i
29.4 −5.65685 0.305514 + 26.9983i 32.0000 113.056 − 53.3219i −1.72825 − 152.725i 274.097i −181.019 −728.813 + 16.4967i −639.544 + 301.634i
29.5 −5.65685 26.1509 − 6.71790i 32.0000 −13.1467 − 124.307i −147.932 + 38.0022i 279.896i −181.019 638.740 − 351.358i 74.3688 + 703.185i
29.6 −5.65685 26.1509 + 6.71790i 32.0000 −13.1467 + 124.307i −147.932 − 38.0022i − 279.896i −181.019 638.740 + 351.358i 74.3688 − 703.185i
29.7 5.65685 −26.1509 − 6.71790i 32.0000 13.1467 + 124.307i −147.932 − 38.0022i 279.896i 181.019 638.740 + 351.358i 74.3688 + 703.185i
29.8 5.65685 −26.1509 + 6.71790i 32.0000 13.1467 − 124.307i −147.932 + 38.0022i − 279.896i 181.019 638.740 − 351.358i 74.3688 − 703.185i
29.9 5.65685 −0.305514 − 26.9983i 32.0000 −113.056 − 53.3219i −1.72825 − 152.725i − 274.097i 181.019 −728.813 + 16.4967i −639.544 − 301.634i
29.10 5.65685 −0.305514 + 26.9983i 32.0000 −113.056 + 53.3219i −1.72825 + 152.725i 274.097i 181.019 −728.813 − 16.4967i −639.544 + 301.634i
29.11 5.65685 21.5067 − 16.3237i 32.0000 74.4540 + 100.407i 121.660 − 92.3408i − 41.6060i 181.019 196.074 − 702.137i 421.175 + 567.989i
29.12 5.65685 21.5067 + 16.3237i 32.0000 74.4540 − 100.407i 121.660 + 92.3408i 41.6060i 181.019 196.074 + 702.137i 421.175 − 567.989i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.7.b.a ✓ 12
3.b odd 2 1 inner 30.7.b.a ✓ 12
4.b odd 2 1 240.7.c.d 12
5.b even 2 1 inner 30.7.b.a ✓ 12
5.c odd 4 2 150.7.d.e 12
12.b even 2 1 240.7.c.d 12
15.d odd 2 1 inner 30.7.b.a ✓ 12
15.e even 4 2 150.7.d.e 12
20.d odd 2 1 240.7.c.d 12
60.h even 2 1 240.7.c.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.7.b.a ✓ 12 1.a even 1 1 trivial
30.7.b.a ✓ 12 3.b odd 2 1 inner
30.7.b.a ✓ 12 5.b even 2 1 inner
30.7.b.a ✓ 12 15.d odd 2 1 inner
150.7.d.e 12 5.c odd 4 2
150.7.d.e 12 15.e even 4 2
240.7.c.d 12 4.b odd 2 1
240.7.c.d 12 12.b even 2 1
240.7.c.d 12 20.d odd 2 1
240.7.c.d 12 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 10188575924224)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 46\!\cdots\!08)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 20\!\cdots\!72)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 96 T^{2} + \cdots + 6543430992)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 52\!\cdots\!12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 42680587422848)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 89\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 80\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 96\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 93\!\cdots\!04)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 16\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 20\!\cdots\!28)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 16\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 42\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 42\!\cdots\!56)^{2} \) Copy content Toggle raw display
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