Properties

Label 1225.4.a.bs
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
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] = [1225,4,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

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: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
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: \( x^{12} - 82x^{10} + 2261x^{8} - 25924x^{6} + 124444x^{4} - 217392x^{2} + 51984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_{2} q^{3} + (\beta_{3} + 4) q^{4} + ( - \beta_{11} + \beta_{8}) q^{6} + ( - \beta_{9} - \beta_{6} - 3 \beta_{5}) q^{8} + (\beta_{7} - \beta_{3} - 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_{2} q^{3} + (\beta_{3} + 4) q^{4} + ( - \beta_{11} + \beta_{8}) q^{6} + ( - \beta_{9} - \beta_{6} - 3 \beta_{5}) q^{8} + (\beta_{7} - \beta_{3} - 12) q^{9} + (\beta_{7} - 3 \beta_{3} - 3) q^{11} + ( - \beta_{4} + 2 \beta_{2} - 4 \beta_1) q^{12} + ( - \beta_{4} - 2 \beta_{2} + 9 \beta_1) q^{13} + ( - 5 \beta_{7} + 3 \beta_{3} + 8) q^{16} + (5 \beta_{4} - 8 \beta_{2} - 9 \beta_1) q^{17} + (3 \beta_{9} + \beta_{6} + 15 \beta_{5}) q^{18} + (6 \beta_{11} + 2 \beta_{10} - 6 \beta_{8}) q^{19} + (5 \beta_{9} + 3 \beta_{6} + 20 \beta_{5}) q^{22} + (2 \beta_{9} + 9 \beta_{6} + 19 \beta_{5}) q^{23} + (2 \beta_{11} + 2 \beta_{10} - 6 \beta_{8}) q^{24} + (11 \beta_{11} - 11 \beta_{10} - 2 \beta_{8}) q^{26} + ( - 4 \beta_{4} - 27 \beta_{2}) q^{27} + ( - 13 \beta_{7} + 5 \beta_{3} - 23) q^{29} + (7 \beta_{11} - 11 \beta_{10} - 13 \beta_{8}) q^{31} + ( - 5 \beta_{9} + 5 \beta_{6} + 15 \beta_{5}) q^{32} + ( - 2 \beta_{4} + 13 \beta_{2} + 8 \beta_1) q^{33} + ( - \beta_{11} + 19 \beta_{10} - 8 \beta_{8}) q^{34} + (5 \beta_{7} - 25 \beta_{3} - 100) q^{36} + ( - 11 \beta_{6} + 59 \beta_{5}) q^{37} + (14 \beta_{4} - 60 \beta_{2} + 4 \beta_1) q^{38} + (\beta_{7} - 7 \beta_{3} + 13) q^{39} + ( - 21 \beta_{11} + 18 \beta_{10} + 42 \beta_{8}) q^{41} + (24 \beta_{9} - 12 \beta_{6}) q^{43} + (15 \beta_{7} - 30 \beta_{3} - 240) q^{44} + (17 \beta_{7} - 56 \beta_{3} - 222) q^{46} + ( - 12 \beta_{4} + 69 \beta_{2} - 96 \beta_1) q^{47} + (22 \beta_{4} - 48 \beta_{2} + 8 \beta_1) q^{48} + ( - 23 \beta_{7} + 17 \beta_{3} - 155) q^{51} + ( - 34 \beta_{4} - 67 \beta_{2} + 109 \beta_1) q^{52} + ( - 30 \beta_{9} - 5 \beta_{6} - 23 \beta_{5}) q^{53} + (27 \beta_{11} - 8 \beta_{10} - 27 \beta_{8}) q^{54} + ( - 24 \beta_{9} - 6 \beta_{6} + 82 \beta_{5}) q^{57} + ( - 31 \beta_{9} - 5 \beta_{6} + 40 \beta_{5}) q^{58} + ( - 23 \beta_{11} - 29 \beta_{10} - 15 \beta_{8}) q^{59} + ( - 4 \beta_{11} - 5 \beta_{10} - 15 \beta_{8}) q^{61} + ( - 31 \beta_{4} - 88 \beta_{2} + 120 \beta_1) q^{62} + (25 \beta_{7} - 29 \beta_{3} - 204) q^{64} + ( - 5 \beta_{11} - 12 \beta_{10} + 13 \beta_{8}) q^{66} + (18 \beta_{9} + \beta_{6} + 47 \beta_{5}) q^{67} + (44 \beta_{4} + 47 \beta_{2} - 149 \beta_1) q^{68} + (9 \beta_{11} + 3 \beta_{10} + 41 \beta_{8}) q^{69} + (26 \beta_{7} - 38 \beta_{3} - 166) q^{71} + (11 \beta_{9} + 17 \beta_{6} + 135 \beta_{5}) q^{72} + ( - 53 \beta_{4} - 111 \beta_{2} + 31 \beta_1) q^{73} + ( - 11 \beta_{7} - 26 \beta_{3} - 730) q^{74} + (16 \beta_{11} + 8 \beta_{10} - 12 \beta_{8}) q^{76} + (9 \beta_{9} + 7 \beta_{6} + 32 \beta_{5}) q^{78} + ( - 3 \beta_{7} + 49 \beta_{3} - 403) q^{79} + ( - 42 \beta_{7} + 54 \beta_{3} - 89) q^{81} + (30 \beta_{4} + 273 \beta_{2} - 201 \beta_1) q^{82} + ( - 12 \beta_{4} - 156 \beta_{2} + 12 \beta_1) q^{83} + (84 \beta_{7} - 84 \beta_{3} - 168) q^{86} + (60 \beta_{4} - 163 \beta_{2} + 32 \beta_1) q^{87} + (20 \beta_{9} + 6 \beta_{6} + 230 \beta_{5}) q^{88} + ( - 26 \beta_{11} + 25 \beta_{10} + 99 \beta_{8}) q^{89} + (74 \beta_{9} - 16 \beta_{6} + 394 \beta_{5}) q^{92} + (6 \beta_{9} - 19 \beta_{6} + 59 \beta_{5}) q^{93} + ( - 165 \beta_{11} + 72 \beta_{10} + 69 \beta_{8}) q^{94} + (40 \beta_{11} + 20 \beta_{10}) q^{96} + (127 \beta_{4} - 152 \beta_{2} + 89 \beta_1) q^{97} + ( - 8 \beta_{7} + 60 \beta_{3} + 312) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 44 q^{4} - 140 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 44 q^{4} - 140 q^{9} - 24 q^{11} + 84 q^{16} - 296 q^{29} - 1100 q^{36} + 184 q^{39} - 2760 q^{44} - 2440 q^{46} - 1928 q^{51} - 2332 q^{64} - 1840 q^{71} - 8656 q^{74} - 5032 q^{79} - 1284 q^{81} - 1680 q^{86} + 3504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 82x^{10} + 2261x^{8} - 25924x^{6} + 124444x^{4} - 217392x^{2} + 51984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19141\nu^{10} - 1394122\nu^{8} + 30202805\nu^{6} - 198343372\nu^{4} + 69383344\nu^{2} + 858661080 ) / 183642480 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -6733\nu^{10} + 535966\nu^{8} - 14050805\nu^{6} + 147266776\nu^{4} - 567656632\nu^{2} + 359961720 ) / 61214160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 281\nu^{10} - 21443\nu^{8} + 516823\nu^{6} - 4572455\nu^{4} + 13315532\nu^{2} - 10762296 ) / 655866 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2643\nu^{10} - 203761\nu^{8} + 4972335\nu^{6} - 43626671\nu^{4} + 99146962\nu^{2} + 41808980 ) / 5101180 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18437 \nu^{11} - 1447766 \nu^{9} + 36797053 \nu^{7} - 360125144 \nu^{5} + 1251854288 \nu^{3} - 822577560 \nu ) / 299074896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 67514 \nu^{11} + 6523217 \nu^{9} - 226442038 \nu^{7} + 3453258473 \nu^{5} - 21882594914 \nu^{3} + 39937848756 \nu ) / 747687240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3265\nu^{10} - 256153\nu^{8} + 6462551\nu^{6} - 61204135\nu^{4} + 187243276\nu^{2} - 95014992 ) / 3279330 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 273323 \nu^{11} + 21724268 \nu^{9} - 566130859 \nu^{7} + 5888218586 \nu^{5} - 24840474476 \nu^{3} + 43005709248 \nu ) / 1495374480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 471097 \nu^{11} + 36678616 \nu^{9} - 907330769 \nu^{7} + 8066324194 \nu^{5} - 17850992452 \nu^{3} - 18351542112 \nu ) / 1495374480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 512557 \nu^{11} + 40696672 \nu^{9} - 1052514101 \nu^{7} + 10506691354 \nu^{5} - 35838763564 \nu^{3} + 25482790992 \nu ) / 1495374480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 129059 \nu^{11} + 10134362 \nu^{9} - 257579371 \nu^{7} + 2520876008 \nu^{5} - 8762980016 \nu^{3} + 7851567192 \nu ) / 299074896 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + 7\beta_{5} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} - 2\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 44\beta_{11} - 21\beta_{10} + 7\beta_{9} - 21\beta_{8} + 7\beta_{6} + 175\beta_{5} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} + 12\beta_{4} + 39\beta_{3} + 76\beta_{2} - 92\beta _1 + 380 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1614\beta_{11} - 1085\beta_{10} + 399\beta_{9} - 735\beta_{8} + 329\beta_{6} + 5607\beta_{5} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -325\beta_{7} + 666\beta_{4} + 1537\beta_{3} + 2606\beta_{2} - 3774\beta _1 + 12668 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 59102\beta_{11} - 46501\beta_{10} + 18977\beta_{9} - 24451\beta_{8} + 13587\beta_{6} + 199549\beta_{5} ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -15935\beta_{7} + 29672\beta_{4} + 61171\beta_{3} + 91896\beta_{2} - 149752\beta _1 + 458700 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2208838 \beta_{11} - 1891869 \beta_{10} + 826343 \beta_{9} - 848379 \beta_{8} + 548373 \beta_{6} + 7453635 \beta_{5} ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -699605\beta_{7} + 1234598\beta_{4} + 2430601\beta_{3} + 3361426\beta_{2} - 5888530\beta _1 + 17262164 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 84074638 \beta_{11} - 75518289 \beta_{10} + 34332081 \beta_{9} - 30749719 \beta_{8} + 21894691 \beta_{6} + 285095349 \beta_{5} ) / 7 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.43837
6.26680
1.85266
4.68109
2.29734
−0.531089
−2.29734
0.531089
−1.85266
−4.68109
−3.43837
−6.26680
−4.85259 −1.14521 15.5476 0 5.55723 0 −36.6254 −25.6885 0
1.2 −4.85259 1.14521 15.5476 0 −5.55723 0 −36.6254 −25.6885 0
1.3 −3.26688 −5.72526 2.67248 0 18.7037 0 17.4043 5.77855 0
1.4 −3.26688 5.72526 2.67248 0 −18.7037 0 17.4043 5.77855 0
1.5 −0.883124 −3.45108 −7.22009 0 3.04773 0 13.4412 −15.0901 0
1.6 −0.883124 3.45108 −7.22009 0 −3.04773 0 13.4412 −15.0901 0
1.7 0.883124 −3.45108 −7.22009 0 −3.04773 0 −13.4412 −15.0901 0
1.8 0.883124 3.45108 −7.22009 0 3.04773 0 −13.4412 −15.0901 0
1.9 3.26688 −5.72526 2.67248 0 −18.7037 0 −17.4043 5.77855 0
1.10 3.26688 5.72526 2.67248 0 18.7037 0 −17.4043 5.77855 0
1.11 4.85259 −1.14521 15.5476 0 −5.55723 0 36.6254 −25.6885 0
1.12 4.85259 1.14521 15.5476 0 5.55723 0 36.6254 −25.6885 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.4.a.bs 12
5.b even 2 1 inner 1225.4.a.bs 12
5.c odd 4 2 245.4.b.g 12
7.b odd 2 1 inner 1225.4.a.bs 12
35.c odd 2 1 inner 1225.4.a.bs 12
35.f even 4 2 245.4.b.g 12
35.k even 12 4 245.4.j.g 24
35.l odd 12 4 245.4.j.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.b.g 12 5.c odd 4 2
245.4.b.g 12 35.f even 4 2
245.4.j.g 24 35.k even 12 4
245.4.j.g 24 35.l odd 12 4
1225.4.a.bs 12 1.a even 1 1 trivial
1225.4.a.bs 12 5.b even 2 1 inner
1225.4.a.bs 12 7.b odd 2 1 inner
1225.4.a.bs 12 35.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{6} - 35T_{2}^{4} + 278T_{2}^{2} - 196 \) Copy content Toggle raw display
\( T_{3}^{6} - 46T_{3}^{4} + 449T_{3}^{2} - 512 \) Copy content Toggle raw display
\( T_{19}^{6} - 19440T_{19}^{4} + 1000512T_{19}^{2} - 4917248 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 35 T^{4} + 278 T^{2} - 196)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 46 T^{4} + 449 T^{2} - 512)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{3} + 6 T^{2} - 1059 T + 11340)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} - 4280 T^{4} + 2063605 T^{2} + \cdots - 85621698)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 15704 T^{4} + \cdots - 1365240258)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 19440 T^{4} + 1000512 T^{2} + \cdots - 4917248)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 70508 T^{4} + \cdots - 1936238854144)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 74 T^{2} - 34583 T + 1629180)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} - 71300 T^{4} + \cdots - 1421945497728)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 216160 T^{4} + \cdots - 313315913335824)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 425142 T^{4} + \cdots - 25\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 472752 T^{4} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 578286 T^{4} + \cdots - 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 697660 T^{4} + \cdots - 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 634516 T^{4} + \cdots - 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 69794 T^{4} + \cdots - 10292107392)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 328108 T^{4} + \cdots - 323613907089984)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 460 T^{2} - 167276 T - 23030496)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} - 1368434 T^{4} + \cdots - 10\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 1258 T^{2} + 229189 T - 79678656)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 1129248 T^{4} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 2065426 T^{4} + \cdots - 63\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 5777544 T^{4} + \cdots - 50\!\cdots\!58)^{2} \) Copy content Toggle raw display
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