Properties

Label 75.9.d.c
Level $75$
Weight $9$
Character orbit 75.d
Analytic conductor $30.553$
Analytic rank $0$
Dimension $20$
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.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \( x^{20} + 943 x^{18} + 318815 x^{16} + 48938090 x^{14} + 3842259173 x^{12} + 159675554657 x^{10} + 3390679484573 x^{8} + 32981662033730 x^{6} + \cdots + 336685801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{22}\cdot 5^{32} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} - 2 \beta_1) q^{3} + ( - \beta_{2} + 79) q^{4} + ( - \beta_{16} + 2 \beta_{2} - 529) q^{6} + (\beta_{17} + \beta_{15} + 7 \beta_{4} - 16 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{7} + \beta_{5} + 6 \beta_{4} - \beta_{3} + 146 \beta_1) q^{8} + (\beta_{18} + 2 \beta_{16} - \beta_{13} - 3 \beta_{12} - \beta_{10} + \beta_{9} - 392) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} - 2 \beta_1) q^{3} + ( - \beta_{2} + 79) q^{4} + ( - \beta_{16} + 2 \beta_{2} - 529) q^{6} + (\beta_{17} + \beta_{15} + 7 \beta_{4} - 16 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{7} + \beta_{5} + 6 \beta_{4} - \beta_{3} + 146 \beta_1) q^{8} + (\beta_{18} + 2 \beta_{16} - \beta_{13} - 3 \beta_{12} - \beta_{10} + \beta_{9} - 392) q^{9} + ( - \beta_{16} - 2 \beta_{13} - 8 \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{2} + \cdots + 1) q^{11}+ \cdots + (11750 \beta_{18} + 25429 \beta_{16} + 12388 \beta_{13} + \cdots + 3356693) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9} + 560772 q^{16} + 463032 q^{19} + 579144 q^{21} - 2272668 q^{24} + 1763240 q^{31} + 2222552 q^{34} - 1337324 q^{36} - 3653584 q^{39} - 50849208 q^{46} - 18708428 q^{49} - 55465384 q^{51} + 15959596 q^{54} + 44834040 q^{61} + 45870004 q^{64} - 54839600 q^{66} - 67125264 q^{69} + 397844872 q^{76} - 324621848 q^{79} - 187150780 q^{81} + 394693536 q^{84} + 888576928 q^{91} + 184100072 q^{94} - 721614812 q^{96} + 67930400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 943 x^{18} + 318815 x^{16} + 48938090 x^{14} + 3842259173 x^{12} + 159675554657 x^{10} + 3390679484573 x^{8} + 32981662033730 x^{6} + \cdots + 336685801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 47\!\cdots\!45 \nu^{18} + \cdots - 65\!\cdots\!87 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!80 \nu^{18} + \cdots - 30\!\cdots\!43 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28\!\cdots\!88 \nu^{19} + \cdots - 38\!\cdots\!39 \nu ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 37\!\cdots\!62 \nu^{19} + \cdots + 37\!\cdots\!07 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80\!\cdots\!82 \nu^{19} + \cdots + 15\!\cdots\!41 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!48 \nu^{19} + \cdots + 21\!\cdots\!85 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!54 \nu^{19} + \cdots - 44\!\cdots\!29 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!16 \nu^{19} + \cdots - 19\!\cdots\!88 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!24 \nu^{19} + \cdots - 81\!\cdots\!60 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 81\!\cdots\!28 \nu^{19} + \cdots - 84\!\cdots\!21 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26\!\cdots\!52 \nu^{19} + \cdots - 33\!\cdots\!79 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 54\!\cdots\!84 \nu^{19} + \cdots + 72\!\cdots\!95 \nu ) / 96\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 29\!\cdots\!20 \nu^{19} + \cdots + 13\!\cdots\!33 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!94 \nu^{19} + \cdots - 14\!\cdots\!31 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 72\!\cdots\!14 \nu^{19} + \cdots + 37\!\cdots\!07 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 43\!\cdots\!92 \nu^{19} + \cdots - 36\!\cdots\!90 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 58\!\cdots\!92 \nu^{19} + \cdots - 11\!\cdots\!39 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 63\!\cdots\!40 \nu^{19} + \cdots - 25\!\cdots\!63 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 39\!\cdots\!34 \nu^{19} + \cdots - 14\!\cdots\!83 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 33 \beta_{19} + 9 \beta_{18} + 144 \beta_{17} + 18 \beta_{16} + 56 \beta_{15} + 80 \beta_{14} + 18 \beta_{13} - 9 \beta_{12} - 9 \beta_{11} + 9 \beta_{8} + 3326 \beta_{4} - 627 \beta_{3} + 9 \beta_{2} + 1340 \beta_1 ) / 67500 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 486 \beta_{18} - 1432 \beta_{16} + 83 \beta_{15} + 83 \beta_{14} + 1638 \beta_{13} + 496 \beta_{12} + 506 \beta_{11} + 646 \beta_{10} + 130 \beta_{9} - 1422 \beta_{8} + 333 \beta_{7} - 84 \beta_{6} + \cdots - 6373730 ) / 67500 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15795 \beta_{19} - 7587 \beta_{18} - 110160 \beta_{17} - 27024 \beta_{16} - 26190 \beta_{15} - 34450 \beta_{14} - 5049 \beta_{13} + 64392 \beta_{12} + 2262 \beta_{11} + 150 \beta_{10} + 4650 \beta_{9} + \cdots - 2325 ) / 135000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 71016 \beta_{18} + 556192 \beta_{16} - 5507 \beta_{15} - 4657 \beta_{14} - 452328 \beta_{13} - 108976 \beta_{12} - 146936 \beta_{11} - 234376 \beta_{10} - 49480 \beta_{9} + \cdots + 1701715130 ) / 67500 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5029449 \beta_{19} + 4014405 \beta_{18} + 40684932 \beta_{17} + 16973060 \beta_{16} + 9106118 \beta_{15} + 9883490 \beta_{14} + 800685 \beta_{13} - 35813930 \beta_{12} + \cdots + 2009125 ) / 135000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7755015 \beta_{18} - 100229780 \beta_{16} - 205668 \beta_{15} - 2078543 \beta_{14} + 74741295 \beta_{13} + 16468640 \beta_{12} + 25182265 \beta_{11} + \cdots - 282224428450 ) / 33750 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 868816092 \beta_{19} - 938121672 \beta_{18} - 7275464406 \beta_{17} - 4154201894 \beta_{16} - 1632278194 \beta_{15} - 1656683170 \beta_{14} - 102186969 \beta_{13} + \cdots - 517288975 ) / 67500 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4659901944 \beta_{18} + 71594191728 \beta_{16} + 436401407 \beta_{15} + 2695206757 \beta_{14} - 52328928552 \beta_{13} - 11256561984 \beta_{12} + \cdots + 197865004894170 ) / 67500 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 617117792445 \beta_{19} + 824255098209 \beta_{18} + 5210028399060 \beta_{17} + 3694221992568 \beta_{16} + 1172610194490 \beta_{15} + \cdots + 464602407675 ) / 135000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1595993368704 \beta_{18} - 25733393246248 \beta_{16} - 207929591855 \beta_{15} - 1247591848605 \beta_{14} + 18720829739832 \beta_{13} + \cdots - 70\!\cdots\!70 ) / 67500 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 222380240304261 \beta_{19} - 347804003712909 \beta_{18} + \cdots - 197038392273525 ) / 135000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 142923037562205 \beta_{18} + \cdots + 64\!\cdots\!25 ) / 16875 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 40\!\cdots\!53 \beta_{19} + \cdots + 40\!\cdots\!00 ) / 67500 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 20\!\cdots\!26 \beta_{18} + \cdots - 93\!\cdots\!30 ) / 67500 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 59\!\cdots\!75 \beta_{19} + \cdots - 65\!\cdots\!25 ) / 27000 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 76\!\cdots\!96 \beta_{18} + \cdots + 34\!\cdots\!30 ) / 67500 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 11\!\cdots\!89 \beta_{19} + \cdots + 12\!\cdots\!25 ) / 135000 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 14\!\cdots\!25 \beta_{18} + \cdots - 64\!\cdots\!50 ) / 33750 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 20\!\cdots\!32 \beta_{19} + \cdots - 25\!\cdots\!75 ) / 67500 \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
17.2930i
17.2930i
0.00158591i
0.00158591i
9.57665i
9.57665i
1.32388i
1.32388i
8.01205i
8.01205i
5.77599i
5.77599i
3.55995i
3.55995i
7.34058i
7.34058i
2.23448i
2.23448i
19.5290i
19.5290i
−29.5009 69.5124 41.5815i 614.301 0 −2050.68 + 1226.69i 3174.27i −10570.2 3102.96 5780.86i 0
74.2 −29.5009 69.5124 + 41.5815i 614.301 0 −2050.68 1226.69i 3174.27i −10570.2 3102.96 + 5780.86i 0
74.3 −23.7888 10.0775 80.3707i 309.906 0 −239.732 + 1911.92i 692.753i −1282.36 −6357.89 1619.88i 0
74.4 −23.7888 10.0775 + 80.3707i 309.906 0 −239.732 1911.92i 692.753i −1282.36 −6357.89 + 1619.88i 0
74.5 −10.2357 −56.7364 57.8099i −151.230 0 580.739 + 591.726i 3448.05i 4168.30 −122.959 + 6559.85i 0
74.6 −10.2357 −56.7364 + 57.8099i −151.230 0 580.739 591.726i 3448.05i 4168.30 −122.959 6559.85i 0
74.7 −8.27106 39.9876 70.4414i −187.590 0 −330.740 + 582.625i 860.291i 3668.96 −3362.98 5633.57i 0
74.8 −8.27106 39.9876 + 70.4414i −187.590 0 −330.740 582.625i 860.291i 3668.96 −3362.98 + 5633.57i 0
74.9 −7.97572 75.3023 29.8424i −192.388 0 −600.590 + 238.014i 3211.86i 3576.22 4779.87 4494.40i 0
74.10 −7.97572 75.3023 + 29.8424i −192.388 0 −600.590 238.014i 3211.86i 3576.22 4779.87 + 4494.40i 0
74.11 7.97572 −75.3023 29.8424i −192.388 0 −600.590 238.014i 3211.86i −3576.22 4779.87 + 4494.40i 0
74.12 7.97572 −75.3023 + 29.8424i −192.388 0 −600.590 + 238.014i 3211.86i −3576.22 4779.87 4494.40i 0
74.13 8.27106 −39.9876 70.4414i −187.590 0 −330.740 582.625i 860.291i −3668.96 −3362.98 + 5633.57i 0
74.14 8.27106 −39.9876 + 70.4414i −187.590 0 −330.740 + 582.625i 860.291i −3668.96 −3362.98 5633.57i 0
74.15 10.2357 56.7364 57.8099i −151.230 0 580.739 591.726i 3448.05i −4168.30 −122.959 6559.85i 0
74.16 10.2357 56.7364 + 57.8099i −151.230 0 580.739 + 591.726i 3448.05i −4168.30 −122.959 + 6559.85i 0
74.17 23.7888 −10.0775 80.3707i 309.906 0 −239.732 1911.92i 692.753i 1282.36 −6357.89 + 1619.88i 0
74.18 23.7888 −10.0775 + 80.3707i 309.906 0 −239.732 + 1911.92i 692.753i 1282.36 −6357.89 1619.88i 0
74.19 29.5009 −69.5124 41.5815i 614.301 0 −2050.68 1226.69i 3174.27i 10570.2 3102.96 + 5780.86i 0
74.20 29.5009 −69.5124 + 41.5815i 614.301 0 −2050.68 + 1226.69i 3174.27i 10570.2 3102.96 5780.86i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.d.c 20
3.b odd 2 1 inner 75.9.d.c 20
5.b even 2 1 inner 75.9.d.c 20
5.c odd 4 1 15.9.c.a 10
5.c odd 4 1 75.9.c.g 10
15.d odd 2 1 inner 75.9.d.c 20
15.e even 4 1 15.9.c.a 10
15.e even 4 1 75.9.c.g 10
20.e even 4 1 240.9.l.b 10
60.l odd 4 1 240.9.l.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.c.a 10 5.c odd 4 1
15.9.c.a 10 15.e even 4 1
75.9.c.g 10 5.c odd 4 1
75.9.c.g 10 15.e even 4 1
75.9.d.c 20 1.a even 1 1 trivial
75.9.d.c 20 3.b odd 2 1 inner
75.9.d.c 20 5.b even 2 1 inner
75.9.d.c 20 15.d odd 2 1 inner
240.9.l.b 10 20.e even 4 1
240.9.l.b 10 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 1673T_{2}^{8} + 850776T_{2}^{6} - 143194100T_{2}^{4} + 9610475200T_{2}^{2} - 224550432000 \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 1673 T^{8} + \cdots - 224550432000)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + 3922 T^{18} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + 33501112 T^{8} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 588896300 T^{8} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 4627542152 T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 43238240228 T^{8} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 115758 T^{4} + \cdots + 11\!\cdots\!32)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} - 440008477848 T^{8} + \cdots - 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 1520697472700 T^{8} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 440810 T^{4} + \cdots + 13\!\cdots\!68)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 24948943182152 T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 63263296371800 T^{8} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 67642272250412 T^{8} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 205484270428088 T^{8} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 513219350099588 T^{8} + \cdots - 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 659758779031100 T^{8} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 11208510 T^{4} + \cdots + 10\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 81155462 T^{4} + \cdots + 23\!\cdots\!32)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
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