Properties

Label 525.3.s.h
Level $525$
Weight $3$
Character orbit 525.s
Analytic conductor $14.305$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 22x^{14} + 343x^{12} - 2542x^{10} + 13621x^{8} - 35080x^{6} + 64300x^{4} - 28000x^{2} + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + (2 \beta_{13} - \beta_{10}) q^{3} + ( - \beta_{8} - \beta_{6} + \beta_{3} - \beta_{2} + 1) q^{4} + (\beta_{5} + 2 \beta_{3} - 1) q^{6} + ( - \beta_{15} - \beta_{13} + \beta_{12} + 2 \beta_{11} - 3 \beta_{10} + \beta_{9} - \beta_1) q^{7} + (\beta_{14} + 4 \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_1) q^{8} - 3 \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + (2 \beta_{13} - \beta_{10}) q^{3} + ( - \beta_{8} - \beta_{6} + \beta_{3} - \beta_{2} + 1) q^{4} + (\beta_{5} + 2 \beta_{3} - 1) q^{6} + ( - \beta_{15} - \beta_{13} + \beta_{12} + 2 \beta_{11} - 3 \beta_{10} + \beta_{9} - \beta_1) q^{7} + (\beta_{14} + 4 \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_1) q^{8} - 3 \beta_{6} q^{9} + ( - \beta_{8} + \beta_{7} - 7 \beta_{6} + 2 \beta_{4} - \beta_{3} + 6) q^{11} + ( - \beta_{15} + 2 \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - 2 \beta_1) q^{12} + (3 \beta_{15} - \beta_{14} - 2 \beta_{13} + 2 \beta_{11} - 4 \beta_{9} + 2 \beta_1) q^{13} + ( - \beta_{8} + \beta_{7} - 11 \beta_{6} - 5 \beta_{5} + \beta_{3} - \beta_{2} + 10) q^{14} + (3 \beta_{8} - 4 \beta_{7} + 5 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + \cdots - 7) q^{16}+ \cdots + (3 \beta_{7} - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{4} - 24 q^{9} + 40 q^{11} + 32 q^{14} - 4 q^{16} + 96 q^{21} - 96 q^{24} + 240 q^{26} + 200 q^{29} - 252 q^{31} - 72 q^{36} + 24 q^{39} + 36 q^{44} - 164 q^{46} - 76 q^{49} + 36 q^{51} - 36 q^{54} + 392 q^{56} + 108 q^{59} - 792 q^{61} + 8 q^{64} + 48 q^{66} + 328 q^{71} + 280 q^{74} + 412 q^{79} - 72 q^{81} - 264 q^{84} + 356 q^{86} - 564 q^{89} - 228 q^{91} - 60 q^{94} - 216 q^{96} - 240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 22x^{14} + 343x^{12} - 2542x^{10} + 13621x^{8} - 35080x^{6} + 64300x^{4} - 28000x^{2} + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3243 \nu^{14} + 70156 \nu^{12} - 972969 \nu^{10} + 6085386 \nu^{8} - 17251573 \nu^{6} + 31891800 \nu^{4} - 13938000 \nu^{2} + 1162228000 ) / 298803000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2097 \nu^{14} - 57414 \nu^{12} + 952781 \nu^{10} - 8714814 \nu^{8} + 49729797 \nu^{6} - 160507590 \nu^{4} + 245765100 \nu^{2} - 107196000 ) / 117007000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 86391 \nu^{14} + 1089782 \nu^{12} - 13023473 \nu^{10} - 31499288 \nu^{8} + 476316679 \nu^{6} - 5048799890 \nu^{4} + 9068677900 \nu^{2} + \cdots - 14205490000 ) / 2457147000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9374291 \nu^{14} - 190524872 \nu^{12} + 2812486753 \nu^{10} - 17590557882 \nu^{8} + 77297768001 \nu^{6} - 92187176600 \nu^{4} + \cdots + 411726170000 ) / 253086141000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6119178 \nu^{14} - 131308181 \nu^{12} + 2033326684 \nu^{10} - 14560759471 \nu^{8} + 77131205568 \nu^{6} - 183317921405 \nu^{4} + \cdots - 30551903500 ) / 126543070500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3255576 \nu^{14} - 69896422 \nu^{12} + 1071617318 \nu^{10} - 7533395317 \nu^{8} + 37675335571 \nu^{6} - 75410643205 \nu^{4} + \cdots + 62550453500 ) / 18077581500 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23640839 \nu^{14} + 499185228 \nu^{12} - 7729957392 \nu^{10} + 54583447073 \nu^{8} - 293224368384 \nu^{6} + 696907060140 \nu^{4} + \cdots + 116147058000 ) / 126543070500 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4776241 \nu^{15} + 115310592 \nu^{13} - 1863758843 \nu^{11} + 15788353442 \nu^{9} - 93939460491 \nu^{7} + 347196076720 \nu^{5} + \cdots + 1505969628000 \nu ) / 180775815000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 73224919 \nu^{15} - 1471847198 \nu^{13} + 21969033677 \nu^{11} - 137404223538 \nu^{9} + 632989616059 \nu^{7} + \cdots + 3668496441000 \nu ) / 2530861410000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6119178 \nu^{15} + 131308181 \nu^{13} - 2033326684 \nu^{11} + 14560759471 \nu^{9} - 77131205568 \nu^{7} + 183317921405 \nu^{5} + \cdots + 157094974000 \nu ) / 126543070500 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 226365739 \nu^{15} + 4362137888 \nu^{13} - 65222132237 \nu^{11} + 386625669928 \nu^{9} - 1857250226079 \nu^{7} + \cdots - 19091613416000 \nu ) / 2530861410000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 53598 \nu^{15} - 1168671 \nu^{13} + 18097044 \nu^{11} - 131482211 \nu^{9} + 686484288 \nu^{7} - 1631568855 \nu^{5} + 2643813450 \nu^{3} + \cdots - 271918500 \nu ) / 585035000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 54476337 \nu^{15} + 1090797454 \nu^{13} - 16344060171 \nu^{11} + 102223107774 \nu^{9} - 478394080057 \nu^{7} + \cdots - 2845044101000 \nu ) / 506172282000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 25018339 \nu^{15} - 562043572 \nu^{13} + 8812748265 \nu^{11} - 67088899600 \nu^{9} + 362617097947 \nu^{7} - 982088475474 \nu^{5} + \cdots - 750541216000 \nu ) / 101234456400 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 5\beta_{6} - \beta_{5} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - 4\beta_{13} + 2\beta_{12} - 9\beta_{11} + \beta_{10} + \beta_{9} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{8} + 2\beta_{7} + 39\beta_{6} + 4\beta_{4} - 17\beta_{3} + 11\beta_{2} - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{15} + 13\beta_{14} - 59\beta_{13} + 13\beta_{12} - 95\beta_{11} + 72\beta_{10} + 26\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{7} + 243\beta_{5} + 34\beta_{4} + 155\beta_{2} - 684 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 466\beta_{14} + 155\beta_{13} - 155\beta_{12} + 905\beta_{10} + 155\beta_{9} - 1081\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -925\beta_{8} - 932\beta_{7} - 4007\beta_{6} + 3229\beta_{5} - 466\beta_{4} + 3229\beta_{3} + 466\beta_{2} - 3695 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2304 \beta_{15} + 4161 \beta_{14} + 14288 \beta_{13} - 3714 \beta_{12} + 12807 \beta_{11} - 1857 \beta_{10} - 1857 \beta_{9} - 12807 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -10503\beta_{8} - 6018\beta_{7} - 45981\beta_{6} - 12036\beta_{4} + 41435\beta_{3} - 16521\beta_{2} + 51999 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 30932 \beta_{15} - 22539 \beta_{14} + 162097 \beta_{13} - 22539 \beta_{12} + 155033 \beta_{11} - 184636 \beta_{10} - 45078 \beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 76010\beta_{7} - 522621\beta_{5} - 76010\beta_{4} - 276121\beta_{2} + 1221776 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -950762\beta_{14} - 276121\beta_{13} + 276121\beta_{12} - 2060863\beta_{10} - 276121\beta_{9} + 1898119\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1499599 \beta_{8} + 1901524 \beta_{7} + 6638309 \beta_{6} - 6535147 \beta_{5} + 950762 \beta_{4} - 6535147 \beta_{3} - 950762 \beta_{2} + 7485909 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 5035548 \beta_{15} - 8436671 \beta_{14} - 29274612 \beta_{13} + 6802246 \beta_{12} - 23376825 \beta_{11} + 3401123 \beta_{10} + 3401123 \beta_{9} + 23376825 \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
−3.04878 + 1.76021i
−2.18275 + 1.26021i
−1.44926 + 0.836732i
−0.583237 + 0.336732i
0.583237 0.336732i
1.44926 0.836732i
2.18275 1.26021i
3.04878 1.76021i
−3.04878 1.76021i
−2.18275 1.26021i
−1.44926 0.836732i
−0.583237 0.336732i
0.583237 + 0.336732i
1.44926 + 0.836732i
2.18275 + 1.26021i
3.04878 + 1.76021i
−3.04878 1.76021i −0.866025 1.50000i 4.19671 + 7.26891i 0 6.09756i −6.99575 0.244004i 15.4667i −1.50000 + 2.59808i 0
124.2 −2.18275 1.26021i 0.866025 + 1.50000i 1.17628 + 2.03737i 0 4.36551i 3.28656 6.18050i 4.15226i −1.50000 + 2.59808i 0
124.3 −1.44926 0.836732i −0.866025 1.50000i −0.599760 1.03881i 0 2.89852i −5.13152 4.76104i 8.70121i −1.50000 + 2.59808i 0
124.4 −0.583237 0.336732i 0.866025 + 1.50000i −1.77322 3.07131i 0 1.16647i −1.55742 6.82455i 5.08226i −1.50000 + 2.59808i 0
124.5 0.583237 + 0.336732i −0.866025 1.50000i −1.77322 3.07131i 0 1.16647i 1.55742 + 6.82455i 5.08226i −1.50000 + 2.59808i 0
124.6 1.44926 + 0.836732i 0.866025 + 1.50000i −0.599760 1.03881i 0 2.89852i 5.13152 + 4.76104i 8.70121i −1.50000 + 2.59808i 0
124.7 2.18275 + 1.26021i −0.866025 1.50000i 1.17628 + 2.03737i 0 4.36551i −3.28656 + 6.18050i 4.15226i −1.50000 + 2.59808i 0
124.8 3.04878 + 1.76021i 0.866025 + 1.50000i 4.19671 + 7.26891i 0 6.09756i 6.99575 + 0.244004i 15.4667i −1.50000 + 2.59808i 0
199.1 −3.04878 + 1.76021i −0.866025 + 1.50000i 4.19671 7.26891i 0 6.09756i −6.99575 + 0.244004i 15.4667i −1.50000 2.59808i 0
199.2 −2.18275 + 1.26021i 0.866025 1.50000i 1.17628 2.03737i 0 4.36551i 3.28656 + 6.18050i 4.15226i −1.50000 2.59808i 0
199.3 −1.44926 + 0.836732i −0.866025 + 1.50000i −0.599760 + 1.03881i 0 2.89852i −5.13152 + 4.76104i 8.70121i −1.50000 2.59808i 0
199.4 −0.583237 + 0.336732i 0.866025 1.50000i −1.77322 + 3.07131i 0 1.16647i −1.55742 + 6.82455i 5.08226i −1.50000 2.59808i 0
199.5 0.583237 0.336732i −0.866025 + 1.50000i −1.77322 + 3.07131i 0 1.16647i 1.55742 6.82455i 5.08226i −1.50000 2.59808i 0
199.6 1.44926 0.836732i 0.866025 1.50000i −0.599760 + 1.03881i 0 2.89852i 5.13152 4.76104i 8.70121i −1.50000 2.59808i 0
199.7 2.18275 1.26021i −0.866025 + 1.50000i 1.17628 2.03737i 0 4.36551i −3.28656 6.18050i 4.15226i −1.50000 2.59808i 0
199.8 3.04878 1.76021i 0.866025 1.50000i 4.19671 7.26891i 0 6.09756i 6.99575 0.244004i 15.4667i −1.50000 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.s.h 16
5.b even 2 1 inner 525.3.s.h 16
5.c odd 4 1 105.3.n.a 8
5.c odd 4 1 525.3.o.l 8
7.d odd 6 1 inner 525.3.s.h 16
15.e even 4 1 315.3.w.a 8
35.i odd 6 1 inner 525.3.s.h 16
35.k even 12 1 105.3.n.a 8
35.k even 12 1 525.3.o.l 8
35.k even 12 1 735.3.h.a 8
35.l odd 12 1 735.3.h.a 8
105.w odd 12 1 315.3.w.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.a 8 5.c odd 4 1
105.3.n.a 8 35.k even 12 1
315.3.w.a 8 15.e even 4 1
315.3.w.a 8 105.w odd 12 1
525.3.o.l 8 5.c odd 4 1
525.3.o.l 8 35.k even 12 1
525.3.s.h 16 1.a even 1 1 trivial
525.3.s.h 16 5.b even 2 1 inner
525.3.s.h 16 7.d odd 6 1 inner
525.3.s.h 16 35.i odd 6 1 inner
735.3.h.a 8 35.k even 12 1
735.3.h.a 8 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{16} - 22 T_{2}^{14} + 343 T_{2}^{12} - 2542 T_{2}^{10} + 13621 T_{2}^{8} - 35080 T_{2}^{6} + 64300 T_{2}^{4} - 28000 T_{2}^{2} + 10000 \) Copy content Toggle raw display
\( T_{11}^{8} - 20T_{11}^{7} + 337T_{11}^{6} - 1880T_{11}^{5} + 10183T_{11}^{4} + 18970T_{11}^{3} + 96982T_{11}^{2} - 4340T_{11} + 196 \) Copy content Toggle raw display
\( T_{13}^{8} - 1164T_{13}^{6} + 420414T_{13}^{4} - 47028060T_{13}^{2} + 1230957225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 22 T^{14} + 343 T^{12} + \cdots + 10000 \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 38 T^{14} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{8} - 20 T^{7} + 337 T^{6} - 1880 T^{5} + \cdots + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 1164 T^{6} + \cdots + 1230957225)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 1104 T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} - 846 T^{6} + 712707 T^{4} + \cdots + 9054081)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 3382 T^{14} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} - 50 T^{3} - 2130 T^{2} + \cdots - 1825400)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 126 T^{7} + \cdots + 385089749136)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 6940 T^{14} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{8} + 3342 T^{6} + \cdots + 13887679716)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 11734 T^{6} + \cdots + 581217165376)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 676520100000000 \) Copy content Toggle raw display
$53$ \( T^{16} - 10558 T^{14} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} - 54 T^{7} + \cdots + 2582886122496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 396 T^{7} + \cdots + 84471609600)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 14518 T^{14} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} - 82 T^{3} - 7998 T^{2} + \cdots + 22760224)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + 27210 T^{14} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{8} - 206 T^{7} + \cdots + 4446784387600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 34440 T^{6} + \cdots + 111959592561216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 282 T^{7} + \cdots + 580473805464576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 30696 T^{6} + \cdots + 2211287961600)^{2} \) Copy content Toggle raw display
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