Properties

Label 525.3.l.d
Level $525$
Weight $3$
Character orbit 525.l
Analytic conductor $14.305$
Analytic rank $0$
Dimension $16$
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] = [525,3,Mod(43,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.l (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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 36x^{14} + 424x^{12} + 2232x^{10} + 5580x^{8} + 6288x^{6} + 2704x^{4} + 384x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_1) q^{2} - \beta_{2} q^{3} + ( - 3 \beta_{7} - \beta_{4}) q^{4} + ( - \beta_{5} - 2) q^{6} + (\beta_{12} + \beta_{6}) q^{7} + ( - \beta_{15} + 2 \beta_{9} + 3 \beta_{8} - \beta_{2}) q^{8} + 3 \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_1) q^{2} - \beta_{2} q^{3} + ( - 3 \beta_{7} - \beta_{4}) q^{4} + ( - \beta_{5} - 2) q^{6} + (\beta_{12} + \beta_{6}) q^{7} + ( - \beta_{15} + 2 \beta_{9} + 3 \beta_{8} - \beta_{2}) q^{8} + 3 \beta_{7} q^{9} + (\beta_{10} - 2 \beta_{5} + 2) q^{11} + ( - \beta_{14} - 3 \beta_1) q^{12} + 4 \beta_{9} q^{13} + (\beta_{11} + 4 \beta_{7} + \beta_{4}) q^{14} + (2 \beta_{10} - 4 \beta_{5} - 4 \beta_{3} - 5) q^{16} + ( - \beta_{14} + 4 \beta_{12} - 2 \beta_{6} - 2 \beta_1) q^{17} + ( - 3 \beta_{8} + 3 \beta_{2}) q^{18} + (2 \beta_{11} - 6 \beta_{7} - 3 \beta_{4}) q^{19} + (\beta_{5} + \beta_{3}) q^{21} + ( - 3 \beta_{14} + 2 \beta_{12} - 2 \beta_{6} - 10 \beta_1) q^{22} + ( - \beta_{15} + 10 \beta_{9} + 8 \beta_{8} - 2 \beta_{2}) q^{23} + (3 \beta_{13} + 2 \beta_{11} + \beta_{7} + 3 \beta_{4}) q^{24} + ( - 4 \beta_{5} - 4 \beta_{3} + 8) q^{26} + 3 \beta_1 q^{27} + (\beta_{15} - \beta_{9} - 5 \beta_{8} + 2 \beta_{2}) q^{28} + (6 \beta_{13} + 8 \beta_{11} + 12 \beta_{7} + \beta_{4}) q^{29} + ( - 2 \beta_{10} + 4 \beta_{5} - 4 \beta_{3} - 6) q^{31} + ( - 2 \beta_{14} + 8 \beta_{12} + \beta_{6} - 13 \beta_1) q^{32} + (\beta_{15} - 6 \beta_{8}) q^{33} + (12 \beta_{13} + 6 \beta_{11} - 16 \beta_{7} + \beta_{4}) q^{34} + ( - 3 \beta_{10} + 9) q^{36} + (\beta_{14} + 2 \beta_{12} + 12 \beta_1) q^{37} + ( - 3 \beta_{15} + 16 \beta_{8} - 20 \beta_{2}) q^{38} + (4 \beta_{11} + 8 \beta_{7}) q^{39} + ( - 10 \beta_{10} + 4 \beta_{5} + 4 \beta_{3} - 4) q^{41} + (\beta_{14} - 3 \beta_{12} + 2 \beta_1) q^{42} + ( - 5 \beta_{15} - 2 \beta_{9} - 8 \beta_{8} - 8 \beta_{2}) q^{43} + (18 \beta_{13} + 8 \beta_{11} + 16 \beta_{7} + 3 \beta_{4}) q^{44} + (11 \beta_{10} - 8 \beta_{5} - 12 \beta_{3} - 28) q^{46} + (5 \beta_{14} + 12 \beta_{12} + 10 \beta_{6} + 14 \beta_1) q^{47} + (2 \beta_{15} - 12 \beta_{9} - 12 \beta_{8} + 5 \beta_{2}) q^{48} - 7 \beta_{7} q^{49} + ( - 3 \beta_{10} - 2 \beta_{5} + 4 \beta_{3} + 12) q^{51} + ( - 4 \beta_{14} - 4 \beta_{12} - 8 \beta_{6}) q^{52} + ( - 5 \beta_{15} + 6 \beta_{9} + 4 \beta_{8} + 10 \beta_{2}) q^{53} + ( - 3 \beta_{13} - 3 \beta_{7}) q^{54} + ( - 4 \beta_{10} + 2 \beta_{5} - \beta_{3} + 19) q^{56} + ( - 3 \beta_{14} - 6 \beta_{12} - 10 \beta_1) q^{57} + (7 \beta_{15} - 26 \beta_{9} - 18 \beta_{8} + 18 \beta_{2}) q^{58} + (8 \beta_{11} - 36 \beta_{7} - 2 \beta_{4}) q^{59} + (7 \beta_{10} + 4 \beta_{5} - 6 \beta_{3} + 12) q^{61} + (6 \beta_{14} + 8 \beta_{12} + 14 \beta_{6} + 30 \beta_1) q^{62} + (3 \beta_{9} + 3 \beta_{8}) q^{63} + (12 \beta_{13} - 4 \beta_{11} - 13 \beta_{7} - \beta_{4}) q^{64} + ( - 9 \beta_{10} - 2 \beta_{5} + 2 \beta_{3} + 34) q^{66} + ( - \beta_{14} - 18 \beta_{12} + 16 \beta_{6} - 28 \beta_1) q^{67} + (9 \beta_{15} - 4 \beta_{9} + 10 \beta_{8} + 26 \beta_{2}) q^{68} + (8 \beta_{13} + 10 \beta_{11} + 10 \beta_{7} + 3 \beta_{4}) q^{69} + ( - 6 \beta_{10} - 8 \beta_{5} + 22) q^{71} + (3 \beta_{14} - 6 \beta_{12} - 9 \beta_{6} + 3 \beta_1) q^{72} + (4 \beta_{8} - 40 \beta_{2}) q^{73} + ( - 14 \beta_{13} - 18 \beta_{7} - 3 \beta_{4}) q^{74} + (13 \beta_{10} - 16 \beta_{5} - 14 \beta_{3} - 100) q^{76} + (3 \beta_{14} + 2 \beta_{12} + 4 \beta_{6} + 6 \beta_1) q^{77} + ( - 12 \beta_{9} - 12 \beta_{8} - 8 \beta_{2}) q^{78} + (20 \beta_{13} - 4 \beta_{11} + 38 \beta_{7} - 12 \beta_{4}) q^{79} - 9 q^{81} + (14 \beta_{14} - 32 \beta_{12} - 36 \beta_{6} + 24 \beta_1) q^{82} + (8 \beta_{15} + 12 \beta_{9} - 4 \beta_{8} + 12 \beta_{2}) q^{83} + ( - 5 \beta_{13} - \beta_{11} + 2 \beta_{7} - 3 \beta_{4}) q^{84} + (7 \beta_{10} - 34 \beta_{5} - 8 \beta_{3}) q^{86} + (\beta_{14} - 24 \beta_{12} - 18 \beta_{6} + 8 \beta_1) q^{87} + (9 \beta_{15} - 22 \beta_{9} - 26 \beta_{8} + 58 \beta_{2}) q^{88} + ( - 4 \beta_{13} + 12 \beta_{11} + 40 \beta_{7} + 6 \beta_{4}) q^{89} + (4 \beta_{10} + 4 \beta_{5} + 16) q^{91} + ( - 15 \beta_{14} + 18 \beta_{12} + 48 \beta_{6} - 46 \beta_1) q^{92} + ( - 2 \beta_{15} - 12 \beta_{9} + 12 \beta_{8} - 2 \beta_{2}) q^{93} + ( - 32 \beta_{13} + 2 \beta_{11} + 12 \beta_{7} - 5 \beta_{4}) q^{94} + ( - 6 \beta_{10} + \beta_{5} + 8 \beta_{3} + 46) q^{96} + (10 \beta_{14} - 16 \beta_{12} + 12 \beta_1) q^{97} + (7 \beta_{8} - 7 \beta_{2}) q^{98} + ( - 6 \beta_{13} + 12 \beta_{7} - 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{6} + 48 q^{11} - 80 q^{16} + 128 q^{26} - 160 q^{31} + 144 q^{36} - 64 q^{41} - 480 q^{46} + 240 q^{51} + 280 q^{56} + 112 q^{61} + 576 q^{66} + 416 q^{71} - 1584 q^{76} - 144 q^{81} + 208 q^{86} + 224 q^{91} + 792 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 36x^{14} + 424x^{12} + 2232x^{10} + 5580x^{8} + 6288x^{6} + 2704x^{4} + 384x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 121 \nu^{15} + 560 \nu^{14} - 4114 \nu^{13} + 20200 \nu^{12} - 43076 \nu^{11} + 238710 \nu^{10} - 182710 \nu^{9} + 1261180 \nu^{8} - 273460 \nu^{7} + 3155940 \nu^{6} + \cdots + 106880 ) / 19360 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 121 \nu^{15} + 560 \nu^{14} + 4114 \nu^{13} + 20200 \nu^{12} + 43076 \nu^{11} + 238710 \nu^{10} + 182710 \nu^{9} + 1261180 \nu^{8} + 273460 \nu^{7} + 3155940 \nu^{6} + \cdots + 106880 ) / 19360 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 221 \nu^{14} + 7894 \nu^{12} + 91796 \nu^{10} + 477150 \nu^{8} + 1185760 \nu^{6} + 1361988 \nu^{4} + 630808 \nu^{2} + 65688 ) / 4840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65 \nu^{15} + 2336 \nu^{13} + 27554 \nu^{11} + 147584 \nu^{9} + 388868 \nu^{7} + 497584 \nu^{5} + 285008 \nu^{3} + 64816 \nu ) / 1936 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 401 \nu^{14} - 14214 \nu^{12} - 163036 \nu^{10} - 831450 \nu^{8} - 1997840 \nu^{6} - 2108628 \nu^{4} - 763608 \nu^{2} - 49568 ) / 4840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1873 \nu^{15} - 186 \nu^{14} + 67242 \nu^{13} - 6934 \nu^{12} + 787218 \nu^{11} - 85876 \nu^{10} + 4094660 \nu^{9} - 468960 \nu^{8} + 9982380 \nu^{7} - 1179240 \nu^{6} + \cdots - 10608 ) / 19360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1873 \nu^{15} - 67242 \nu^{13} - 787218 \nu^{11} - 4094660 \nu^{9} - 9982380 \nu^{7} - 10598184 \nu^{5} - 3804224 \nu^{3} - 298704 \nu ) / 19360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1873 \nu^{15} - 186 \nu^{14} - 67242 \nu^{13} - 6934 \nu^{12} - 787218 \nu^{11} - 85876 \nu^{10} - 4094660 \nu^{9} - 468960 \nu^{8} - 9982380 \nu^{7} - 1179240 \nu^{6} + \cdots - 10608 ) / 19360 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1108 \nu^{15} - 512 \nu^{14} + 39777 \nu^{13} - 18313 \nu^{12} + 465693 \nu^{11} - 212977 \nu^{10} + 2423115 \nu^{9} - 1097730 \nu^{8} + 5916360 \nu^{7} - 2639870 \nu^{6} + \cdots - 81816 ) / 9680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 494 \nu^{14} + 17681 \nu^{12} + 205974 \nu^{10} + 1065930 \nu^{8} + 2587460 \nu^{6} + 2738812 \nu^{4} + 978712 \nu^{2} + 74232 ) / 2420 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 503 \nu^{15} - 18118 \nu^{13} - 213650 \nu^{11} - 1127568 \nu^{9} - 2835700 \nu^{7} - 3240784 \nu^{5} - 1419984 \nu^{3} - 150624 \nu ) / 3872 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1108 \nu^{15} - 512 \nu^{14} - 39777 \nu^{13} - 18313 \nu^{12} - 465693 \nu^{11} - 212977 \nu^{10} - 2423115 \nu^{9} - 1097730 \nu^{8} - 5916360 \nu^{7} - 2639870 \nu^{6} + \cdots - 81816 ) / 9680 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1277 \nu^{15} - 45778 \nu^{13} - 534502 \nu^{11} - 2769275 \nu^{9} - 6705890 \nu^{7} - 7011566 \nu^{5} - 2391776 \nu^{3} - 145196 \nu ) / 4840 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3430 \nu^{15} + 221 \nu^{14} - 123120 \nu^{13} + 7894 \nu^{12} - 1441680 \nu^{11} + 91191 \nu^{10} - 7512675 \nu^{9} + 459000 \nu^{8} - 18412650 \nu^{7} + 1039350 \nu^{6} + \cdots - 31112 ) / 4840 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3430 \nu^{15} + 221 \nu^{14} + 123120 \nu^{13} + 7894 \nu^{12} + 1441680 \nu^{11} + 91191 \nu^{10} + 7512675 \nu^{9} + 459000 \nu^{8} + 18412650 \nu^{7} + 1039350 \nu^{6} + \cdots - 31112 ) / 4840 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} - 2\beta_{7} - \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + 2\beta_{8} + 2\beta_{6} + 2\beta_{5} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + \beta_{14} - 6 \beta_{13} + \beta_{12} - \beta_{9} - 13 \beta_{8} + 26 \beta_{7} + 13 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{12} - 8 \beta_{10} - 2 \beta_{9} - 22 \beta_{8} - 22 \beta_{6} - 19 \beta_{5} - \beta_{3} - 6 \beta_{2} - 6 \beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23 \beta_{15} - 23 \beta_{14} + 150 \beta_{13} - 20 \beta_{12} + 10 \beta_{11} + 20 \beta_{9} + 216 \beta_{8} - 482 \beta_{7} - 216 \beta_{6} - 60 \beta_{4} + 70 \beta_{2} - 70 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 49 \beta_{15} + 49 \beta_{14} + 40 \beta_{12} + 138 \beta_{10} + 40 \beta_{9} + 436 \beta_{8} + 436 \beta_{6} + 362 \beta_{5} + 28 \beta_{3} + 150 \beta_{2} + 150 \beta _1 - 778 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 230 \beta_{15} + 230 \beta_{14} - 1540 \beta_{13} + 178 \beta_{12} - 126 \beta_{11} - 178 \beta_{9} - 1978 \beta_{8} + 4642 \beta_{7} + 1978 \beta_{6} + 574 \beta_{4} - 708 \beta_{2} + 708 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1000 \beta_{15} - 1000 \beta_{14} - 752 \beta_{12} - 2552 \beta_{10} - 752 \beta_{9} - 8464 \beta_{8} - 8464 \beta_{6} - 6946 \beta_{5} - 586 \beta_{3} - 3088 \beta_{2} - 3088 \beta _1 + 14350 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4473 \beta_{15} - 4473 \beta_{14} + 30210 \beta_{13} - 3304 \beta_{12} + 2586 \beta_{11} + 3304 \beta_{9} + 37522 \beta_{8} - 89590 \beta_{7} - 37522 \beta_{6} - 11016 \beta_{4} + 13842 \beta_{2} - 13842 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 19578 \beta_{15} + 19578 \beta_{14} + 14320 \beta_{12} + 48538 \beta_{10} + 14320 \beta_{9} + 163476 \beta_{8} + 163476 \beta_{6} + 133684 \beta_{5} + 11532 \beta_{3} + 60660 \beta_{2} + \cdots - 273136 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 86420 \beta_{15} + 86420 \beta_{14} - 585288 \beta_{13} + 62858 \beta_{12} - 50688 \beta_{11} - 62858 \beta_{9} - 719798 \beta_{8} + 1727980 \beta_{7} + 719798 \beta_{6} + 212014 \beta_{4} + \cdots + 267958 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 379064 \beta_{15} - 379064 \beta_{14} - 274872 \beta_{12} - 931812 \beta_{10} - 274872 \beta_{9} - 3153048 \beta_{8} - 3153048 \beta_{6} - 2575664 \beta_{5} - 223528 \beta_{3} + \cdots + 5245872 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1666896 \beta_{15} - 1666896 \beta_{14} + 11298976 \beta_{13} - 1206684 \beta_{12} + 981656 \beta_{11} + 1206684 \beta_{9} + 13855240 \beta_{8} - 33316440 \beta_{7} + \cdots - 5171820 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 7316384 \beta_{15} + 7316384 \beta_{14} + 5291544 \beta_{12} + 17940100 \beta_{10} + 5291544 \beta_{9} + 60790304 \beta_{8} + 60790304 \beta_{6} + 49642176 \beta_{5} + \cdots - 101014744 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 32137472 \beta_{15} + 32137472 \beta_{14} - 217900032 \beta_{13} + 23231644 \beta_{12} - 18948216 \beta_{11} - 23231644 \beta_{9} - 266967868 \beta_{8} + \cdots + 99732060 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-\beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.373925i
2.39051i
0.705059i
1.72784i
0.272158i
1.29494i
4.39051i
2.37393i
0.373925i
2.39051i
0.705059i
1.72784i
0.272158i
1.29494i
4.39051i
2.37393i
−2.59867 + 2.59867i 1.22474 + 1.22474i 9.50617i 0 −6.36542 1.87083 1.87083i 14.3087 + 14.3087i 3.00000i 0
43.2 −2.16577 + 2.16577i −1.22474 1.22474i 5.38111i 0 5.30503 1.87083 1.87083i 2.99117 + 2.99117i 3.00000i 0
43.3 −1.51969 + 1.51969i 1.22474 + 1.22474i 0.618887i 0 −3.72245 −1.87083 + 1.87083i −5.13823 5.13823i 3.00000i 0
43.4 −0.496903 + 0.496903i 1.22474 + 1.22474i 3.50617i 0 −1.21716 1.87083 1.87083i −3.72984 3.72984i 3.00000i 0
43.5 0.496903 0.496903i −1.22474 1.22474i 3.50617i 0 −1.21716 −1.87083 + 1.87083i 3.72984 + 3.72984i 3.00000i 0
43.6 1.51969 1.51969i −1.22474 1.22474i 0.618887i 0 −3.72245 1.87083 1.87083i 5.13823 + 5.13823i 3.00000i 0
43.7 2.16577 2.16577i 1.22474 + 1.22474i 5.38111i 0 5.30503 −1.87083 + 1.87083i −2.99117 2.99117i 3.00000i 0
43.8 2.59867 2.59867i −1.22474 1.22474i 9.50617i 0 −6.36542 −1.87083 + 1.87083i −14.3087 14.3087i 3.00000i 0
232.1 −2.59867 2.59867i 1.22474 1.22474i 9.50617i 0 −6.36542 1.87083 + 1.87083i 14.3087 14.3087i 3.00000i 0
232.2 −2.16577 2.16577i −1.22474 + 1.22474i 5.38111i 0 5.30503 1.87083 + 1.87083i 2.99117 2.99117i 3.00000i 0
232.3 −1.51969 1.51969i 1.22474 1.22474i 0.618887i 0 −3.72245 −1.87083 1.87083i −5.13823 + 5.13823i 3.00000i 0
232.4 −0.496903 0.496903i 1.22474 1.22474i 3.50617i 0 −1.21716 1.87083 + 1.87083i −3.72984 + 3.72984i 3.00000i 0
232.5 0.496903 + 0.496903i −1.22474 + 1.22474i 3.50617i 0 −1.21716 −1.87083 1.87083i 3.72984 3.72984i 3.00000i 0
232.6 1.51969 + 1.51969i −1.22474 + 1.22474i 0.618887i 0 −3.72245 1.87083 + 1.87083i 5.13823 5.13823i 3.00000i 0
232.7 2.16577 + 2.16577i 1.22474 1.22474i 5.38111i 0 5.30503 −1.87083 1.87083i −2.99117 + 2.99117i 3.00000i 0
232.8 2.59867 + 2.59867i −1.22474 + 1.22474i 9.50617i 0 −6.36542 −1.87083 1.87083i −14.3087 + 14.3087i 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.l.d 16
5.b even 2 1 inner 525.3.l.d 16
5.c odd 4 2 inner 525.3.l.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.l.d 16 1.a even 1 1 trivial
525.3.l.d 16 5.b even 2 1 inner
525.3.l.d 16 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 292T_{2}^{12} + 21894T_{2}^{8} + 347812T_{2}^{4} + 83521 \) acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 292 T^{12} + 21894 T^{8} + \cdots + 83521 \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 12 T^{3} - 168 T^{2} + 2064 T - 1776)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 358719963529216 \) Copy content Toggle raw display
$17$ \( T^{16} + 518272 T^{12} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + 1504 T^{6} + \cdots + 6194319616)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 2738496 T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + 4864 T^{6} + \cdots + 339133851904)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 40 T^{3} - 1512 T^{2} + \cdots + 908560)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} + 1332352 T^{12} + \cdots + 94\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{4} + 16 T^{3} - 5376 T^{2} + \cdots + 275200)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} + 33104512 T^{12} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{16} + 48501376 T^{12} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + 25340224 T^{12} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{8} + 11968 T^{6} + \cdots + 2548544045056)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 28 T^{3} - 3960 T^{2} + \cdots + 1423696)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} + 184288384 T^{12} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} - 104 T^{3} + 504 T^{2} + \cdots + 161680)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + 84926464 T^{12} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{8} + 53392 T^{6} + \cdots + 18\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 339533824 T^{12} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + 20416 T^{6} + \cdots + 340380655550464)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 549517312 T^{12} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
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