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Properties

Label 150.6.e.b
Level $150$
Weight $6$
Character orbit 150.e
Analytic conductor $24.058$
Analytic rank $0$
Dimension $20$
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] = [150,6,Mod(107,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.107");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.e (of order \(4\), 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: \(24.0575729719\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 1126 x^{18} + 420245 x^{16} + 66878446 x^{14} + 5652274660 x^{12} + 280235806770 x^{10} + \cdots + 87\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{14}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{7} q^{3} - 16 \beta_{3} q^{4} + ( - \beta_{6} + 4) q^{6} + (\beta_{18} - \beta_{12} + \beta_{6} + \cdots - 4) q^{7}+ \cdots + (\beta_{19} - \beta_{18} + \cdots - 8 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{7} q^{3} - 16 \beta_{3} q^{4} + ( - \beta_{6} + 4) q^{6} + (\beta_{18} - \beta_{12} + \beta_{6} + \cdots - 4) q^{7}+ \cdots + (98 \beta_{19} - 23 \beta_{18} + \cdots + 14799 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{3} + 80 q^{6} - 76 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{3} + 80 q^{6} - 76 q^{7} - 64 q^{12} - 2640 q^{13} - 5120 q^{16} + 2272 q^{18} - 760 q^{21} + 464 q^{22} - 16556 q^{27} - 1216 q^{28} + 23960 q^{31} + 7684 q^{33} - 13120 q^{36} + 60696 q^{37} - 46864 q^{42} - 48024 q^{43} + 58880 q^{46} - 1024 q^{48} - 155240 q^{51} + 42240 q^{52} - 167744 q^{57} - 38000 q^{58} + 160680 q^{61} + 219364 q^{63} - 158560 q^{66} + 118736 q^{67} - 36352 q^{72} - 379100 q^{73} + 78080 q^{76} + 25440 q^{78} - 282260 q^{81} + 262976 q^{82} - 249380 q^{87} + 7424 q^{88} + 499200 q^{91} + 304072 q^{93} - 20480 q^{96} + 288228 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 1126 x^{18} + 420245 x^{16} + 66878446 x^{14} + 5652274660 x^{12} + 280235806770 x^{10} + \cdots + 87\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 27\!\cdots\!34 \nu^{19} + \cdots + 31\!\cdots\!76 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27\!\cdots\!34 \nu^{19} + \cdots - 31\!\cdots\!76 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!95 \nu^{19} + \cdots + 37\!\cdots\!36 \nu ) / 83\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!03 \nu^{19} + \cdots + 25\!\cdots\!40 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 72\!\cdots\!93 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 65\!\cdots\!41 \nu^{19} + \cdots - 40\!\cdots\!80 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!85 \nu^{19} + \cdots - 21\!\cdots\!48 ) / 70\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14\!\cdots\!61 \nu^{19} + \cdots + 83\!\cdots\!28 ) / 70\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 58\!\cdots\!56 \nu^{19} + \cdots - 18\!\cdots\!16 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 61\!\cdots\!97 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 40\!\cdots\!45 \nu^{19} + \cdots + 14\!\cdots\!52 ) / 53\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\!\cdots\!23 \nu^{19} + \cdots - 64\!\cdots\!24 ) / 35\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!28 \nu^{19} + \cdots - 27\!\cdots\!16 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 39\!\cdots\!81 \nu^{19} + \cdots + 70\!\cdots\!88 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 41\!\cdots\!25 \nu^{19} + \cdots - 82\!\cdots\!08 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!34 \nu^{19} + \cdots + 13\!\cdots\!16 ) / 53\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 82\!\cdots\!71 \nu^{19} + \cdots - 68\!\cdots\!20 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 95\!\cdots\!11 \nu^{19} + \cdots - 60\!\cdots\!60 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 34\!\cdots\!95 \nu^{19} + \cdots - 11\!\cdots\!60 ) / 10\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6 \beta_{19} + 9 \beta_{18} + 9 \beta_{17} + 15 \beta_{16} + 9 \beta_{15} + 9 \beta_{14} + \beta_{13} + \cdots + 11 ) / 360 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 63 \beta_{18} - 63 \beta_{17} - 231 \beta_{16} - 165 \beta_{15} + 27 \beta_{14} - 233 \beta_{13} + \cdots - 40449 ) / 360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 732 \beta_{19} - 2502 \beta_{18} - 2502 \beta_{17} - 2905 \beta_{16} - 396 \beta_{15} - 396 \beta_{14} + \cdots - 3647 ) / 180 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 54873 \beta_{18} + 54873 \beta_{17} + 112635 \beta_{16} + 85059 \beta_{15} - 13941 \beta_{14} + \cdots + 15390959 ) / 360 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1145274 \beta_{19} + 2475639 \beta_{18} + 2475639 \beta_{17} + 2801085 \beta_{16} + 261567 \beta_{15} + \cdots + 4463819 ) / 360 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9260118 \beta_{18} - 9260118 \beta_{17} - 14854185 \beta_{16} - 11049009 \beta_{15} + 1879641 \beta_{14} + \cdots - 1883102449 ) / 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 697600074 \beta_{19} - 1263538809 \beta_{18} - 1263538809 \beta_{17} - 1423087915 \beta_{16} + \cdots - 2578550875 ) / 360 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7598227335 \beta_{18} + 7598227335 \beta_{17} + 10760303659 \beta_{16} + 7846484949 \beta_{15} + \cdots + 1302305379385 ) / 120 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 202999449048 \beta_{19} + 332802078228 \beta_{18} + 332802078228 \beta_{17} + 373994823345 \beta_{16} + \cdots + 731679665371 ) / 180 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 13412302492293 \beta_{18} - 13412302492293 \beta_{17} - 17741171442459 \beta_{16} + \cdots - 20\!\cdots\!39 ) / 360 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 231816298143534 \beta_{19} - 358348701148929 \beta_{18} - 358348701148929 \beta_{17} + \cdots - 824018883699505 ) / 360 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 19\!\cdots\!71 \beta_{18} + \cdots + 28\!\cdots\!18 ) / 90 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 13\!\cdots\!10 \beta_{19} + \cdots + 46\!\cdots\!03 ) / 360 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 43\!\cdots\!39 \beta_{18} + \cdots - 62\!\cdots\!57 ) / 360 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 36\!\cdots\!52 \beta_{19} + \cdots - 12\!\cdots\!23 ) / 180 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 82\!\cdots\!75 \beta_{18} + \cdots + 11\!\cdots\!05 ) / 120 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 41\!\cdots\!90 \beta_{19} + \cdots + 14\!\cdots\!67 ) / 360 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 34\!\cdots\!46 \beta_{18} + \cdots - 47\!\cdots\!43 ) / 90 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 23\!\cdots\!82 \beta_{19} + \cdots - 80\!\cdots\!91 ) / 360 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
5.20628i
− 3.30354i
6.04149i
− 23.6117i
18.0817i
− 1.34618i
− 6.17029i
7.90019i
5.27572i
− 6.07365i
− 5.20628i
3.30354i
− 6.04149i
23.6117i
− 18.0817i
1.34618i
6.17029i
− 7.90019i
− 5.27572i
6.07365i
−2.82843 + 2.82843i −14.4107 − 5.94397i − 16.0000i 0 57.5718 − 23.9476i −93.4247 − 93.4247i 45.2548 + 45.2548i 172.338 + 171.314i 0
107.2 −2.82843 + 2.82843i −9.83224 + 12.0966i − 16.0000i 0 −6.40449 − 62.0240i 72.4506 + 72.4506i 45.2548 + 45.2548i −49.6540 − 237.873i 0
107.3 −2.82843 + 2.82843i 2.86199 − 15.3235i − 16.0000i 0 35.2464 + 51.4363i −137.897 − 137.897i 45.2548 + 45.2548i −226.618 − 87.7113i 0
107.4 −2.82843 + 2.82843i 6.50878 + 14.1646i − 16.0000i 0 −58.4731 − 21.6539i −1.93288 − 1.93288i 45.2548 + 45.2548i −158.272 + 184.389i 0
107.5 −2.82843 + 2.82843i 12.3367 − 9.52925i − 16.0000i 0 −7.94059 + 61.8462i 141.804 + 141.804i 45.2548 + 45.2548i 61.3868 − 235.118i 0
107.6 2.82843 − 2.82843i −14.1646 − 6.50878i − 16.0000i 0 −58.4731 + 21.6539i −1.93288 − 1.93288i −45.2548 − 45.2548i 158.272 + 184.389i 0
107.7 2.82843 − 2.82843i −12.0966 + 9.83224i − 16.0000i 0 −6.40449 + 62.0240i 72.4506 + 72.4506i −45.2548 − 45.2548i 49.6540 − 237.873i 0
107.8 2.82843 − 2.82843i 5.94397 + 14.4107i − 16.0000i 0 57.5718 + 23.9476i −93.4247 − 93.4247i −45.2548 − 45.2548i −172.338 + 171.314i 0
107.9 2.82843 − 2.82843i 9.52925 − 12.3367i − 16.0000i 0 −7.94059 − 61.8462i 141.804 + 141.804i −45.2548 − 45.2548i −61.3868 − 235.118i 0
107.10 2.82843 − 2.82843i 15.3235 − 2.86199i − 16.0000i 0 35.2464 − 51.4363i −137.897 − 137.897i −45.2548 − 45.2548i 226.618 − 87.7113i 0
143.1 −2.82843 − 2.82843i −14.4107 + 5.94397i 16.0000i 0 57.5718 + 23.9476i −93.4247 + 93.4247i 45.2548 − 45.2548i 172.338 − 171.314i 0
143.2 −2.82843 − 2.82843i −9.83224 − 12.0966i 16.0000i 0 −6.40449 + 62.0240i 72.4506 − 72.4506i 45.2548 − 45.2548i −49.6540 + 237.873i 0
143.3 −2.82843 − 2.82843i 2.86199 + 15.3235i 16.0000i 0 35.2464 − 51.4363i −137.897 + 137.897i 45.2548 − 45.2548i −226.618 + 87.7113i 0
143.4 −2.82843 − 2.82843i 6.50878 − 14.1646i 16.0000i 0 −58.4731 + 21.6539i −1.93288 + 1.93288i 45.2548 − 45.2548i −158.272 − 184.389i 0
143.5 −2.82843 − 2.82843i 12.3367 + 9.52925i 16.0000i 0 −7.94059 − 61.8462i 141.804 − 141.804i 45.2548 − 45.2548i 61.3868 + 235.118i 0
143.6 2.82843 + 2.82843i −14.1646 + 6.50878i 16.0000i 0 −58.4731 − 21.6539i −1.93288 + 1.93288i −45.2548 + 45.2548i 158.272 − 184.389i 0
143.7 2.82843 + 2.82843i −12.0966 − 9.83224i 16.0000i 0 −6.40449 − 62.0240i 72.4506 − 72.4506i −45.2548 + 45.2548i 49.6540 + 237.873i 0
143.8 2.82843 + 2.82843i 5.94397 − 14.4107i 16.0000i 0 57.5718 − 23.9476i −93.4247 + 93.4247i −45.2548 + 45.2548i −172.338 − 171.314i 0
143.9 2.82843 + 2.82843i 9.52925 + 12.3367i 16.0000i 0 −7.94059 + 61.8462i 141.804 − 141.804i −45.2548 + 45.2548i −61.3868 + 235.118i 0
143.10 2.82843 + 2.82843i 15.3235 + 2.86199i 16.0000i 0 35.2464 + 51.4363i −137.897 + 137.897i −45.2548 + 45.2548i 226.618 + 87.7113i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.6.e.b 20
3.b odd 2 1 inner 150.6.e.b 20
5.b even 2 1 30.6.e.a ✓ 20
5.c odd 4 1 30.6.e.a ✓ 20
5.c odd 4 1 inner 150.6.e.b 20
15.d odd 2 1 30.6.e.a ✓ 20
15.e even 4 1 30.6.e.a ✓ 20
15.e even 4 1 inner 150.6.e.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.e.a ✓ 20 5.b even 2 1
30.6.e.a ✓ 20 5.c odd 4 1
30.6.e.a ✓ 20 15.d odd 2 1
30.6.e.a ✓ 20 15.e even 4 1
150.6.e.b 20 1.a even 1 1 trivial
150.6.e.b 20 3.b odd 2 1 inner
150.6.e.b 20 5.c odd 4 1 inner
150.6.e.b 20 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 38 T_{7}^{9} + 722 T_{7}^{8} - 265360 T_{7}^{7} + 1729005288 T_{7}^{6} + \cdots + 20\!\cdots\!32 \) acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 256)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 26\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 96\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 17\!\cdots\!68)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 14\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 27\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 76\!\cdots\!32)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 16\!\cdots\!68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 16\!\cdots\!24)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 24\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 17\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 69\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 20\!\cdots\!68)^{2} \) Copy content Toggle raw display
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