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Properties

Label 1150.4.b.s
Level $1150$
Weight $4$
Character orbit 1150.b
Analytic conductor $67.852$
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] = [1150,4,Mod(599,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.599");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not 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: \(67.8521965066\)
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: \( x^{12} + 219x^{10} + 17685x^{8} + 640366x^{6} + 10000368x^{4} + 54897345x^{2} + 95531076 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + 2 \beta_{7} q^{2} + (\beta_{7} + \beta_1) q^{3} - 4 q^{4} + (2 \beta_{2} - 2) q^{6} + (\beta_{11} - \beta_{10} + \cdots - 8 \beta_{7}) q^{7}+ \cdots + ( - \beta_{6} - 2 \beta_{5} + \cdots - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{7} q^{2} + (\beta_{7} + \beta_1) q^{3} - 4 q^{4} + (2 \beta_{2} - 2) q^{6} + (\beta_{11} - \beta_{10} + \cdots - 8 \beta_{7}) q^{7}+ \cdots + (15 \beta_{6} + 18 \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 48 q^{4} - 20 q^{6} - 122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 48 q^{4} - 20 q^{6} - 122 q^{9} - 98 q^{11} + 168 q^{14} + 192 q^{16} + 458 q^{19} + 184 q^{21} + 80 q^{24} - 64 q^{26} + 364 q^{29} + 228 q^{31} + 700 q^{34} + 488 q^{36} + 286 q^{39} + 486 q^{41} + 392 q^{44} - 552 q^{46} - 1296 q^{49} - 2062 q^{51} + 1376 q^{54} - 672 q^{56} + 1118 q^{59} - 1376 q^{61} - 768 q^{64} - 2564 q^{66} - 230 q^{69} + 1168 q^{71} + 256 q^{74} - 1832 q^{76} + 2864 q^{79} + 68 q^{81} - 736 q^{84} - 728 q^{86} + 1182 q^{89} - 5728 q^{91} + 1992 q^{94} - 320 q^{96} + 284 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 219x^{10} + 17685x^{8} + 640366x^{6} + 10000368x^{4} + 54897345x^{2} + 95531076 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 493\nu^{10} + 104304\nu^{8} + 8300763\nu^{6} + 298655455\nu^{4} + 4331050533\nu^{2} + 13383850968 ) / 528060438 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8181965 \nu^{10} - 2120230500 \nu^{8} - 196265438991 \nu^{6} - 7730751216542 \nu^{4} + \cdots - 401803012045467 ) / 3466980805689 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3064139 \nu^{10} + 611504826 \nu^{8} + 44178469263 \nu^{6} + 1401237732521 \nu^{4} + \cdots + 62678814427368 ) / 770440179042 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 46159615 \nu^{10} + 9929512596 \nu^{8} + 777751057467 \nu^{6} + 26525551369165 \nu^{4} + \cdots + 10\!\cdots\!02 ) / 6933961611378 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48378049 \nu^{10} - 10115020758 \nu^{8} - 765365013585 \nu^{6} - 24978460712557 \nu^{4} + \cdots - 900007863283566 ) / 6933961611378 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25358 \nu^{11} - 5464169 \nu^{9} - 429577206 \nu^{7} - 14735962925 \nu^{5} + \cdots - 608166728037 \nu ) / 95578939278 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5792972380 \nu^{11} + 1238291606847 \nu^{9} + 95680321371936 \nu^{7} + \cdots + 99\!\cdots\!91 \nu ) / 37\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9750615365 \nu^{11} + 2140956371733 \nu^{9} + 171005797212273 \nu^{7} + \cdots + 26\!\cdots\!15 \nu ) / 37\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9852184303 \nu^{11} + 2125628714784 \nu^{9} + 166557230007255 \nu^{7} + \cdots + 20\!\cdots\!66 \nu ) / 37\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 599004734 \nu^{11} - 126770931720 \nu^{9} - 9745735846671 \nu^{7} + \cdots - 13\!\cdots\!31 \nu ) / 209174508609903 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - 2\beta_{5} + \beta_{4} - \beta_{3} - 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} - 10\beta_{10} + 4\beta_{9} + 5\beta_{8} - 9\beta_{7} - 60\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 110\beta_{6} + 172\beta_{5} - 42\beta_{4} + 71\beta_{3} - 46\beta_{2} + 2059 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 58\beta_{11} + 971\beta_{10} - 287\beta_{9} - 646\beta_{8} + 2403\beta_{7} + 3848\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9006\beta_{6} - 13101\beta_{5} + 1640\beta_{4} - 4842\beta_{3} + 6887\beta_{2} - 131591 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1683\beta_{11} - 80988\beta_{10} + 18138\beta_{9} + 57789\beta_{8} - 304755\beta_{7} - 258163\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 685888\beta_{6} + 975677\beta_{5} - 63301\beta_{4} + 329020\beta_{3} - 729924\beta_{2} + 8824371 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 421024 \beta_{11} + 6468613 \beta_{10} - 1078363 \beta_{9} - 4703489 \beta_{8} + \cdots + 17842203 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -51329087\beta_{6} - 72465223\beta_{5} + 2437677\beta_{4} - 22310942\beta_{3} + 67409431\beta_{2} - 609551932 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 44244317 \beta_{11} - 507464663 \beta_{10} + 60406061 \beta_{9} + 370596664 \beta_{8} + \cdots - 1259250047 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
− 8.71212i
− 4.90184i
− 2.16288i
1.92662i
6.96868i
7.88153i
− 7.88153i
− 6.96868i
− 1.92662i
2.16288i
4.90184i
8.71212i
− 2.00000i − 9.71212i −4.00000 0 −19.4242 33.1288i 8.00000i −67.3252 0
599.2 − 2.00000i − 5.90184i −4.00000 0 −11.8037 1.63515i 8.00000i −7.83172 0
599.3 − 2.00000i − 3.16288i −4.00000 0 −6.32576 − 3.52475i 8.00000i 16.9962 0
599.4 − 2.00000i 0.926620i −4.00000 0 1.85324 − 28.2323i 8.00000i 26.1414 0
599.5 − 2.00000i 5.96868i −4.00000 0 11.9374 15.2484i 8.00000i −8.62516 0
599.6 − 2.00000i 6.88153i −4.00000 0 13.7631 23.7447i 8.00000i −20.3555 0
599.7 2.00000i − 6.88153i −4.00000 0 13.7631 − 23.7447i − 8.00000i −20.3555 0
599.8 2.00000i − 5.96868i −4.00000 0 11.9374 − 15.2484i − 8.00000i −8.62516 0
599.9 2.00000i − 0.926620i −4.00000 0 1.85324 28.2323i − 8.00000i 26.1414 0
599.10 2.00000i 3.16288i −4.00000 0 −6.32576 3.52475i − 8.00000i 16.9962 0
599.11 2.00000i 5.90184i −4.00000 0 −11.8037 − 1.63515i − 8.00000i −7.83172 0
599.12 2.00000i 9.71212i −4.00000 0 −19.4242 − 33.1288i − 8.00000i −67.3252 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 599.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1150.4.b.s 12
5.b even 2 1 inner 1150.4.b.s 12
5.c odd 4 1 1150.4.a.w ✓ 6
5.c odd 4 1 1150.4.a.x yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1150.4.a.w ✓ 6 5.c odd 4 1
1150.4.a.x yes 6 5.c odd 4 1
1150.4.b.s 12 1.a even 1 1 trivial
1150.4.b.s 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{12} + 223T_{3}^{10} + 18003T_{3}^{8} + 662782T_{3}^{6} + 11005897T_{3}^{4} + 64421736T_{3}^{2} + 47610000 \) Copy content Toggle raw display
\( T_{7}^{12} + 2706 T_{7}^{10} + 2555241 T_{7}^{8} + 983037160 T_{7}^{6} + 129030059664 T_{7}^{4} + \cdots + 3809429554176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 223 T^{10} + \cdots + 47610000 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 3809429554176 \) Copy content Toggle raw display
$11$ \( (T^{6} + 49 T^{5} + \cdots - 17113032)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} - 229 T^{5} + \cdots + 315942692928)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{6} \) Copy content Toggle raw display
$29$ \( (T^{6} - 182 T^{5} + \cdots + 194358059568)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 1219003806240)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 190890178116489)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 95\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 98\!\cdots\!52)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 388743615564288)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 10\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 86\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 31\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 13\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
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