Properties

Label 75.9.f.d
Level $75$
Weight $9$
Character orbit 75.f
Analytic conductor $30.553$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 192 x^{9} + 27713 x^{8} - 24384 x^{7} + 18432 x^{6} - 2072064 x^{5} + 128589064 x^{4} - 223046400 x^{3} + 184320000 x^{2} + 558720000 x + 846810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{18}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{9} q^{3} + ( - \beta_{5} + 193 \beta_1) q^{4} + (\beta_{4} - \beta_{3} + 189) q^{6} + ( - 2 \beta_{8} + 8 \beta_{6} + 64 \beta_{2}) q^{7} + (7 \beta_{10} - 47 \beta_{9} - 179 \beta_{7}) q^{8} - 2187 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{9} q^{3} + ( - \beta_{5} + 193 \beta_1) q^{4} + (\beta_{4} - \beta_{3} + 189) q^{6} + ( - 2 \beta_{8} + 8 \beta_{6} + 64 \beta_{2}) q^{7} + (7 \beta_{10} - 47 \beta_{9} - 179 \beta_{7}) q^{8} - 2187 \beta_1 q^{9} + ( - 10 \beta_{4} + 16 \beta_{3} - 7410) q^{11} + ( - 27 \beta_{8} + 155 \beta_{6} - 351 \beta_{2}) q^{12} + (56 \beta_{10} - 374 \beta_{9} - 584 \beta_{7}) q^{13} + ( - 22 \beta_{11} + 26 \beta_{5} - 27042 \beta_1) q^{14} + ( - 96 \beta_{4} + 75 \beta_{3} - 40483) q^{16} + ( - 34 \beta_{8} - 106 \beta_{6} - 3024 \beta_{2}) q^{17} + 2187 \beta_{7} q^{18} + ( - 132 \beta_{11} - 248 \beta_{5} - 4866 \beta_1) q^{19} + ( - 90 \beta_{4} - 72 \beta_{3} + 6642) q^{21} + (312 \beta_{8} - 4392 \beta_{6} + 10482 \beta_{2}) q^{22} + ( - 518 \beta_{10} + 958 \beta_{9} - 2528 \beta_{7}) q^{23} + ( - 88 \beta_{11} - 655 \beta_{5} + 140967 \beta_1) q^{24} + ( - 766 \beta_{4} + 1798 \beta_{3} - 337998) q^{26} - 2187 \beta_{6} q^{27} + ( - 254 \beta_{10} - 4738 \beta_{9} + 18710 \beta_{7}) q^{28} + ( - 660 \beta_{11} + 2256 \beta_{5} - 242352 \beta_1) q^{29} + ( - 696 \beta_{4} - 760 \beta_{3} + 471558) q^{31} + (653 \beta_{8} - 26437 \beta_{6} + 5129 \beta_{2}) q^{32} + (162 \beta_{10} - 5292 \beta_{9} - 23976 \beta_{7}) q^{33} + ( - 132 \beta_{11} - 3428 \beta_{5} + 1340836 \beta_1) q^{34} + ( - 2187 \beta_{3} + 422091) q^{36} + ( - 3404 \beta_{8} - 26422 \beta_{6} - 53288 \beta_{2}) q^{37} + (4376 \beta_{10} - 59704 \beta_{9} - 44590 \beta_{7}) q^{38} + (144 \beta_{11} - 4392 \beta_{5} + 963090 \beta_1) q^{39} + (3188 \beta_{4} + 3568 \beta_{3} + 149838) q^{41} + (1296 \beta_{8} - 29376 \beta_{6} - 31266 \beta_{2}) q^{42} + ( - 80 \beta_{10} - 59128 \beta_{9} + 115808 \beta_{7}) q^{43} + (4016 \beta_{11} + 15458 \beta_{5} - 3667938 \beta_1) q^{44} + (4584 \beta_{4} - 6200 \beta_{3} - 906872) q^{46} + ( - 3782 \beta_{8} - 120638 \beta_{6} - 27680 \beta_{2}) q^{47} + ( - 567 \beta_{10} - 23137 \beta_{9} - 202581 \beta_{7}) q^{48} + (2808 \beta_{11} + 576 \beta_{5} - 2860213 \beta_1) q^{49} + (2582 \beta_{4} - 5336 \beta_{3} + 360828) q^{51} + (13570 \beta_{8} - 267586 \beta_{6} + 562330 \beta_{2}) q^{52} + ( - 14236 \beta_{10} - 24352 \beta_{9} - 64720 \beta_{7}) q^{53} + (2187 \beta_{11} + 2187 \beta_{5} - 413343 \beta_1) q^{54} + (2672 \beta_{4} - 11126 \beta_{3} + 605670) q^{56} + ( - 3132 \beta_{8} - 34222 \beta_{6} - 329400 \beta_{2}) q^{57} + ( - 2592 \beta_{10} - 134208 \beta_{9} + 841104 \beta_{7}) q^{58} + ( - 4594 \beta_{11} - 28000 \beta_{5} + 556086 \beta_1) q^{59} + (6384 \beta_{4} - 4448 \beta_{3} + 2804154) q^{61} + (8600 \beta_{8} - 217624 \beta_{6} - 711158 \beta_{2}) q^{62} + ( - 4374 \beta_{10} + 17496 \beta_{9} - 139968 \beta_{7}) q^{63} + (6432 \beta_{11} + 22161 \beta_{5} + 3004711 \beta_1) q^{64} + ( - 6426 \beta_{4} + 31698 \beta_{3} - 11780154) q^{66} + ( - 15708 \beta_{8} - 57936 \beta_{6} + 735072 \beta_{2}) q^{67} + (17932 \beta_{10} - 236300 \beta_{9} - 1385708 \beta_{7}) q^{68} + ( - 9262 \beta_{11} + 32696 \beta_{5} - 1939032 \beta_1) q^{69} + ( - 11852 \beta_{4} - 20464 \beta_{3} + 27958872) q^{71} + ( - 15309 \beta_{8} + 102789 \beta_{6} - 391473 \beta_{2}) q^{72} + (35760 \beta_{10} - 70932 \beta_{9} + 1674384 \beta_{7}) q^{73} + (2594 \beta_{11} - 77926 \beta_{5} + 19242318 \beta_1) q^{74} + ( - 56544 \beta_{4} + 106446 \beta_{3} - 32948878) q^{76} + ( - 5760 \beta_{8} + 230436 \beta_{6} - 78768 \beta_{2}) q^{77} + (27864 \beta_{10} - 154008 \beta_{9} - 2037474 \beta_{7}) q^{78} + ( - 60624 \beta_{11} + 9608 \beta_{5} + 9251190 \beta_1) q^{79} - 4782969 q^{81} + ( - 38784 \beta_{8} + 992736 \beta_{6} + 968658 \beta_{2}) q^{82} + (5660 \beta_{10} + 607040 \beta_{9} - 1432704 \beta_{7}) q^{83} + (15408 \beta_{11} + 35982 \beta_{5} + 6668082 \beta_1) q^{84} + ( - 58568 \beta_{4} - 57880 \beta_{3} + 40829880) q^{86} + (78732 \beta_{8} - 256284 \beta_{6} - 419904 \beta_{2}) q^{87} + ( - 108654 \beta_{10} + 1063998 \beta_{9} + \cdots + 4404102 \beta_{7}) q^{88}+ \cdots + ( - 21870 \beta_{11} - 34992 \beta_{5} + 16205670 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2268 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2268 q^{6} - 88920 q^{11} - 485796 q^{16} + 79704 q^{21} - 4055976 q^{26} + 5658696 q^{31} + 5065092 q^{36} + 1798056 q^{41} - 10882464 q^{46} + 4329936 q^{51} + 7268040 q^{56} + 33649848 q^{61} - 141361848 q^{66} + 335506464 q^{71} - 395386536 q^{76} - 57395628 q^{81} + 489958560 q^{86} - 216875664 q^{91} - 710311356 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 192 x^{9} + 27713 x^{8} - 24384 x^{7} + 18432 x^{6} - 2072064 x^{5} + 128589064 x^{4} - 223046400 x^{3} + 184320000 x^{2} + 558720000 x + 846810000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 15\!\cdots\!13 \nu^{11} + \cdots - 44\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27\!\cdots\!31 \nu^{11} + \cdots + 28\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 86\!\cdots\!87 \nu^{11} + \cdots - 31\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 76\!\cdots\!71 \nu^{11} + \cdots - 13\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!15 \nu^{11} + \cdots + 12\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!09 \nu^{11} + \cdots + 24\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 67\!\cdots\!41 \nu^{11} + \cdots - 29\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!79 \nu^{11} + \cdots - 19\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39\!\cdots\!91 \nu^{11} + \cdots + 16\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 94\!\cdots\!63 \nu^{11} + \cdots - 41\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!51 \nu^{11} + \cdots - 42\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} - 60\beta_{6} + 28\beta_{5} - \beta_{4} + 28\beta_{3} ) / 1620 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 16 \beta_{11} + 9 \beta_{10} + 15 \beta_{9} - 9 \beta_{8} - 144 \beta_{7} - 15 \beta_{6} + 2 \beta_{5} - 144 \beta_{2} + 20430 \beta_1 ) / 270 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 59 \beta_{11} + 162 \beta_{10} + 5850 \beta_{9} + 9558 \beta_{7} + 1652 \beta_{5} + 59 \beta_{4} - 1652 \beta_{3} - 38880 \beta _1 + 38880 ) / 810 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1008 \beta_{10} + 720 \beta_{9} + 1008 \beta_{8} - 16128 \beta_{7} + 720 \beta_{6} - 1024 \beta_{4} + 97 \beta_{3} + 16128 \beta_{2} - 1247085 ) / 135 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4951 \beta_{11} + 12285 \beta_{8} + 546735 \beta_{6} - 106228 \beta_{5} + 4951 \beta_{4} - 106228 \beta_{3} - 967815 \beta_{2} + 4114800 \beta _1 + 4114800 ) / 405 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 45824 \beta_{11} - 57546 \beta_{10} - 9510 \beta_{9} + 57546 \beta_{8} + 985536 \beta_{7} + 9510 \beta_{6} + 13997 \beta_{5} + 985536 \beta_{2} - 55540245 \beta_1 ) / 45 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 844378 \beta_{11} - 1502739 \beta_{10} - 89731725 \beta_{9} - 167928201 \beta_{7} - 14538184 \beta_{5} - 844378 \beta_{4} + 14538184 \beta_{3} + 767536560 \beta _1 - 767536560 ) / 405 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 27182592 \beta_{10} + 10014720 \beta_{9} - 27182592 \beta_{8} + 496870272 \beta_{7} + 10014720 \beta_{6} + 19373216 \beta_{4} - 9859973 \beta_{3} + \cdots + 23408806365 ) / 135 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 143676484 \beta_{11} - 155810385 \beta_{8} - 13968737415 \beta_{6} + 2068541152 \beta_{5} - 143676484 \beta_{4} + 2068541152 \beta_{3} + 27508062915 \beta_{2} + \cdots - 137868188400 ) / 405 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2812855648 \beta_{11} + 4135232232 \beta_{10} - 3713987880 \beta_{9} - 4135232232 \beta_{8} - 80328282912 \beta_{7} + 3713987880 \beta_{6} + \cdots + 3388757125815 \beta_1 ) / 135 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 24324075352 \beta_{11} + 12118610691 \beta_{10} + 2125699807125 \beta_{9} + 4388701336569 \beta_{7} + 301159712656 \beta_{5} + 24324075352 \beta_{4} + \cdots + 24219053155440 ) / 405 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−6.36871 6.36871i
−0.782794 0.782794i
8.37625 + 8.37625i
5.92676 + 5.92676i
1.66670 + 1.66670i
−8.81820 8.81820i
−6.36871 + 6.36871i
−0.782794 + 0.782794i
8.37625 8.37625i
5.92676 5.92676i
1.66670 1.66670i
−8.81820 + 8.81820i
−20.7460 20.7460i −33.0681 + 33.0681i 604.790i 0 1372.06 500.835 + 500.835i 7236.00 7236.00i 2187.00i 0
7.2 −15.2792 15.2792i 33.0681 33.0681i 210.909i 0 −1010.51 1776.62 + 1776.62i −688.952 + 688.952i 2187.00i 0
7.3 −3.10647 3.10647i −33.0681 + 33.0681i 236.700i 0 205.450 974.502 + 974.502i −1530.56 + 1530.56i 2187.00i 0
7.4 3.10647 + 3.10647i 33.0681 33.0681i 236.700i 0 205.450 −974.502 974.502i 1530.56 1530.56i 2187.00i 0
7.5 15.2792 + 15.2792i −33.0681 + 33.0681i 210.909i 0 −1010.51 −1776.62 1776.62i 688.952 688.952i 2187.00i 0
7.6 20.7460 + 20.7460i 33.0681 33.0681i 604.790i 0 1372.06 −500.835 500.835i −7236.00 + 7236.00i 2187.00i 0
43.1 −20.7460 + 20.7460i −33.0681 33.0681i 604.790i 0 1372.06 500.835 500.835i 7236.00 + 7236.00i 2187.00i 0
43.2 −15.2792 + 15.2792i 33.0681 + 33.0681i 210.909i 0 −1010.51 1776.62 1776.62i −688.952 688.952i 2187.00i 0
43.3 −3.10647 + 3.10647i −33.0681 33.0681i 236.700i 0 205.450 974.502 974.502i −1530.56 1530.56i 2187.00i 0
43.4 3.10647 3.10647i 33.0681 + 33.0681i 236.700i 0 205.450 −974.502 + 974.502i 1530.56 + 1530.56i 2187.00i 0
43.5 15.2792 15.2792i −33.0681 33.0681i 210.909i 0 −1010.51 −1776.62 + 1776.62i 688.952 + 688.952i 2187.00i 0
43.6 20.7460 20.7460i 33.0681 + 33.0681i 604.790i 0 1372.06 −500.835 + 500.835i −7236.00 7236.00i 2187.00i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.6
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 75.9.f.d 12
5.b even 2 1 inner 75.9.f.d 12
5.c odd 4 2 inner 75.9.f.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.f.d 12 1.a even 1 1 trivial
75.9.f.d 12 5.b even 2 1 inner
75.9.f.d 12 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 959337T_{2}^{8} + 161889694848T_{2}^{4} + 60170924888064 \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 959337 T^{8} + \cdots + 60170924888064 \) Copy content Toggle raw display
$3$ \( (T^{4} + 4782969)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 43710302371248 T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{3} + 22230 T^{2} + \cdots - 485718860664)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{6} + 68348445324 T^{4} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + 2662083864576 T^{4} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 1414674 T^{2} + \cdots + 54\!\cdots\!00)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{3} - 449514 T^{2} + \cdots + 13\!\cdots\!20)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 42\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + 353695390563564 T^{4} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 8412462 T^{2} + \cdots + 17\!\cdots\!08)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{3} - 83876616 T^{2} + \cdots - 16\!\cdots\!48)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
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