Properties

Label 225.10.b.m
Level $225$
Weight $10$
Character orbit 225.b
Analytic conductor $115.883$
Analytic rank $0$
Dimension $6$
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] = [225,10,Mod(199,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.199");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 1305x^{4} + 433104x^{2} + 16000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{8} \)
Twist minimal: no (minimal twist has level 25)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_1 - 114) q^{4} + (26 \beta_{4} - 4 \beta_{3} - 140 \beta_{2}) q^{7} + ( - 33 \beta_{4} - 27 \beta_{3} + 279 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_1 - 114) q^{4} + (26 \beta_{4} - 4 \beta_{3} - 140 \beta_{2}) q^{7} + ( - 33 \beta_{4} - 27 \beta_{3} + 279 \beta_{2}) q^{8} + ( - 17 \beta_{5} - 76 \beta_1 + 18264) q^{11} + ( - 460 \beta_{4} + 832 \beta_{3} - 928 \beta_{2}) q^{13} + (30 \beta_{5} + 282 \beta_1 + 90870) q^{14} + ( - 6 \beta_{5} + 95 \beta_1 - 233112) q^{16} + ( - 168 \beta_{4} + 1521 \beta_{3} - 3974 \beta_{2}) q^{17} + (351 \beta_{5} + 92 \beta_1 - 273096) q^{19} + (4310 \beta_{4} - 3558 \beta_{3} + 28107 \beta_{2}) q^{22} + (3426 \beta_{4} - 11160 \beta_{3} - 4764 \beta_{2}) q^{23} + ( - 1292 \beta_{5} - 4588 \beta_1 + 428628) q^{26} + (826 \beta_{4} + 238 \beta_{3} - 15778 \beta_{2}) q^{28} + (508 \beta_{5} - 9832 \beta_1 + 728268) q^{29} + ( - 534 \beta_{5} + 2648 \beta_1 + 1423984) q^{31} + ( - 19395 \beta_{4} - 18369 \beta_{3} - 101287 \beta_{2}) q^{32} + ( - 1689 \beta_{5} + 775 \beta_1 + 2279959) q^{34} + (37572 \beta_{4} + 15866 \beta_{3} + 137524 \beta_{2}) q^{37} + ( - 40242 \beta_{4} + 113346 \beta_{3} - 300541 \beta_{2}) q^{38} + (4904 \beta_{5} - 73088 \beta_1 - 1952709) q^{41} + (2732 \beta_{4} - 83000 \beta_{3} + 34976 \beta_{2}) q^{43} + ( - 836 \beta_{5} + 44675 \beta_1 - 7346514) q^{44} + (14586 \beta_{5} - 2322 \beta_1 + 4736994) q^{46} + ( - 9384 \beta_{4} + 121500 \beta_{3} + 894880 \beta_{2}) q^{47} + ( - 17904 \beta_{5} - 93952 \beta_1 + 2188595) q^{49} + (52836 \beta_{4} + 123500 \beta_{3} + 560188 \beta_{2}) q^{52} + ( - 314856 \beta_{4} - 428022 \beta_{3} + 208484 \beta_{2}) q^{53} + (15948 \beta_{5} + 143838 \beta_1 + 56459448) q^{56} + (270608 \beta_{4} + 433104 \beta_{3} + 1874400 \beta_{2}) q^{58} + (21680 \beta_{5} + 193768 \beta_1 + 1818504) q^{59} + (22980 \beta_{5} - 359960 \beta_1 + 41881302) q^{61} + ( - 30780 \beta_{4} - 247716 \beta_{3} + 1133970 \beta_{2}) q^{62} + ( - 4098 \beta_{5} - 474207 \beta_1 - 55688032) q^{64} + ( - 322747 \beta_{4} - 906256 \beta_{3} + 6018994 \beta_{2}) q^{67} + (67443 \beta_{4} + 200457 \beta_{3} + 232429 \beta_{2}) q^{68} + ( - 48040 \beta_{5} - 786920 \beta_1 - 98905212) q^{71} + (56864 \beta_{4} - 497441 \beta_{3} + 10112822 \beta_{2}) q^{73} + (21706 \beta_{5} + 855578 \beta_1 - 84128214) q^{74} + (26124 \beta_{5} - 370821 \beta_1 + 29919854) q^{76} + ( - 701052 \beta_{4} + 1613286 \beta_{3} + 2388876 \beta_{2}) q^{77} + ( - 75234 \beta_{5} + 1255264 \beta_1 + 102948380) q^{79} + (1892080 \beta_{4} + 3591696 \beta_{3} + 6514275 \beta_{2}) q^{82} + (165897 \beta_{4} + 1204200 \beta_{3} + 10762890 \beta_{2}) q^{83} + (85732 \beta_{5} - 333580 \beta_1 - 11233116) q^{86} + (821061 \beta_{4} - 3303801 \beta_{3} + 1767237 \beta_{2}) q^{88} + ( - 4176 \beta_{5} - 1107576 \beta_1 - 367582761) q^{89} + (447552 \beta_{5} + 796432 \beta_1 + 393981224) q^{91} + (284622 \beta_{4} - 837846 \beta_{3} + 1888602 \beta_{2}) q^{92} + ( - 130884 \beta_{5} + 1342852 \beta_1 - 576367700) q^{94} + ( - 62616 \beta_{4} - 2021122 \beta_{3} + 29036476 \beta_{2}) q^{97} + (4998240 \beta_{4} - 3371616 \beta_{3} + 14210371 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 682 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 682 q^{4} + 109398 q^{11} + 545844 q^{14} - 1398494 q^{16} - 1637690 q^{19} + 2560008 q^{26} + 4350960 q^{29} + 8548132 q^{31} + 13677926 q^{34} - 11852622 q^{41} - 43991406 q^{44} + 28446492 q^{46} + 12907858 q^{49} + 339076260 q^{56} + 11341920 q^{59} + 250613852 q^{61} - 335084802 q^{64} - 595101192 q^{71} - 503014716 q^{74} + 178829730 q^{76} + 620050340 q^{79} - 67894392 q^{86} - 2207720070 q^{89} + 2366375312 q^{91} - 3455782264 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 1305x^{4} + 433104x^{2} + 16000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -35\nu^{4} - 17275\nu^{2} + 2252704 ) / 13392 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 77\nu^{5} + 71485\nu^{3} + 13296008\nu ) / 3348000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -491\nu^{5} - 908755\nu^{3} - 378730064\nu ) / 13392000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1177\nu^{5} - 1291985\nu^{3} - 198655408\nu ) / 13392000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -185\nu^{4} - 171025\nu^{2} - 22789928 ) / 6696 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 25\beta_{4} - 11\beta_{3} + 78\beta_{2} ) / 250 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -21\beta_{5} + 222\beta _1 - 108817 ) / 250 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -16225\beta_{4} - 253\beta_{3} - 62406\beta_{2} ) / 250 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2073\beta_{5} - 41046\beta _1 + 13959941 ) / 50 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10746025\beta_{4} + 2134309\beta_{3} + 55337718\beta_{2} ) / 250 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
27.7229i
22.2334i
6.48955i
6.48955i
22.2334i
27.7229i
31.7828i 0 −498.143 0 0 637.237i 440.406i 0 0
199.2 21.4187i 0 53.2406 0 0 9905.49i 12106.7i 0 0
199.3 20.2014i 0 103.903 0 0 4010.25i 12442.1i 0 0
199.4 20.2014i 0 103.903 0 0 4010.25i 12442.1i 0 0
199.5 21.4187i 0 53.2406 0 0 9905.49i 12106.7i 0 0
199.6 31.7828i 0 −498.143 0 0 637.237i 440.406i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
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 225.10.b.m 6
3.b odd 2 1 25.10.b.c 6
5.b even 2 1 inner 225.10.b.m 6
5.c odd 4 1 225.10.a.m 3
5.c odd 4 1 225.10.a.p 3
12.b even 2 1 400.10.c.q 6
15.d odd 2 1 25.10.b.c 6
15.e even 4 1 25.10.a.c 3
15.e even 4 1 25.10.a.d yes 3
60.h even 2 1 400.10.c.q 6
60.l odd 4 1 400.10.a.u 3
60.l odd 4 1 400.10.a.y 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.a.c 3 15.e even 4 1
25.10.a.d yes 3 15.e even 4 1
25.10.b.c 6 3.b odd 2 1
25.10.b.c 6 15.d odd 2 1
225.10.a.m 3 5.c odd 4 1
225.10.a.p 3 5.c odd 4 1
225.10.b.m 6 1.a even 1 1 trivial
225.10.b.m 6 5.b even 2 1 inner
400.10.a.u 3 60.l odd 4 1
400.10.a.y 3 60.l odd 4 1
400.10.c.q 6 12.b even 2 1
400.10.c.q 6 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{6} + 1877T_{2}^{4} + 1062868T_{2}^{2} + 189117504 \) Copy content Toggle raw display
\( T_{11}^{3} - 54699T_{11}^{2} - 1257655633T_{11} + 75283351667163 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 1877 T^{4} + \cdots + 189117504 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 114606892 T^{4} + \cdots + 64\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( (T^{3} - 54699 T^{2} + \cdots + 75283351667163)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 47782585008 T^{4} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{6} + 113679545147 T^{4} + \cdots + 31\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( (T^{3} + 818845 T^{2} + \cdots - 22\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 5844421656108 T^{4} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2175480 T^{2} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4274066 T^{2} + \cdots - 81\!\cdots\!48)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 204279936810732 T^{4} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{3} + 5926311 T^{2} + \cdots - 15\!\cdots\!47)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 260277885634608 T^{4} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{3} - 5670960 T^{2} + \cdots - 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 125306926 T^{2} + \cdots + 62\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 77\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{3} + 297550596 T^{2} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 19\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( (T^{3} - 310025170 T^{2} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 70\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{3} + 1103860035 T^{2} + \cdots - 19\!\cdots\!75)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
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