Properties

Label 75.9.c.f
Level $75$
Weight $9$
Character orbit 75.c
Analytic conductor $30.553$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \( x^{10} + 1634x^{8} + 776307x^{6} + 148116566x^{4} + 10575941812x^{2} + 105274575720 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{11}\cdot 5^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{4} - \beta_{2} + 2) q^{3} + (\beta_1 - 155) q^{4} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_1 + 224) q^{6} + (\beta_{8} - \beta_{6} - 12 \beta_{4} - 5 \beta_{2} - 5 \beta_1 - 204) q^{7} + (\beta_{9} + \beta_{7} - 11 \beta_{4} - \beta_{3} - 113 \beta_{2} - 5) q^{8} + ( - \beta_{8} + \beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{3} - 55 \beta_{2} + \cdots - 1119) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{4} - \beta_{2} + 2) q^{3} + (\beta_1 - 155) q^{4} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_1 + 224) q^{6} + (\beta_{8} - \beta_{6} - 12 \beta_{4} - 5 \beta_{2} - 5 \beta_1 - 204) q^{7} + (\beta_{9} + \beta_{7} - 11 \beta_{4} - \beta_{3} - 113 \beta_{2} - 5) q^{8} + ( - \beta_{8} + \beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{3} - 55 \beta_{2} + \cdots - 1119) q^{9}+ \cdots + (20700 \beta_{9} + 43198 \beta_{8} + 4208 \beta_{7} + \cdots - 38563731) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 25 q^{3} - 1554 q^{4} + 2257 q^{6} - 1960 q^{7} - 11207 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 25 q^{3} - 1554 q^{4} + 2257 q^{6} - 1960 q^{7} - 11207 q^{9} + 5915 q^{12} + 16920 q^{13} + 44634 q^{16} + 224875 q^{18} - 143934 q^{19} + 673428 q^{21} - 818990 q^{22} - 1016859 q^{24} + 260830 q^{27} - 3810100 q^{28} - 3014060 q^{31} + 4677515 q^{33} + 4977146 q^{34} + 4500527 q^{36} - 3016760 q^{37} - 7513282 q^{39} + 4001760 q^{42} - 11747340 q^{43} - 13938636 q^{46} + 14748755 q^{48} + 8953546 q^{49} + 6209287 q^{51} + 38918320 q^{52} - 8886272 q^{54} - 14759525 q^{57} + 48407900 q^{58} + 1520220 q^{61} - 74748240 q^{63} - 4536998 q^{64} + 10465295 q^{66} + 16269290 q^{67} + 11394978 q^{69} - 172231185 q^{72} + 52090170 q^{73} - 29529046 q^{76} - 198205810 q^{78} + 8549896 q^{79} + 22612945 q^{81} + 295714190 q^{82} - 136883292 q^{84} - 318901610 q^{87} + 310673250 q^{88} - 107224264 q^{91} - 79679130 q^{93} + 356118596 q^{94} + 525424001 q^{96} + 402167800 q^{97} - 382421335 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 1634x^{8} + 776307x^{6} + 148116566x^{4} + 10575941812x^{2} + 105274575720 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 872557 \nu^{8} + 1265751468 \nu^{6} + 444620825535 \nu^{4} + 46859095322612 \nu^{2} + 420291052310412 ) / 157481360352 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8400715 \nu^{9} + 12128076852 \nu^{7} + 4207972253529 \nu^{5} + 435449159046332 \nu^{3} + 32\!\cdots\!04 \nu ) / 153386844982848 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 57764587 \nu^{9} + 16283051488 \nu^{8} - 95509278940 \nu^{7} + 23429801734080 \nu^{6} - 45386771554953 \nu^{5} + \cdots + 85\!\cdots\!12 ) / 818063173241856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 472531299 \nu^{9} - 5511595228 \nu^{8} - 683931862716 \nu^{7} - 7973245106928 \nu^{6} - 239395694736177 \nu^{5} + \cdots - 31\!\cdots\!36 ) / 24\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2093833615 \nu^{9} - 135239021452 \nu^{8} + 3031560854316 \nu^{7} - 195884281212336 \nu^{6} + \cdots - 78\!\cdots\!24 ) / 24\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 227599251 \nu^{9} + 4459205273 \nu^{8} - 326570626108 \nu^{7} + 6426141790740 \nu^{6} - 111468906693249 \nu^{5} + \cdots + 23\!\cdots\!08 ) / 204515793310464 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2367048033 \nu^{9} - 7788738074 \nu^{8} + 3397055634708 \nu^{7} - 11462785669512 \nu^{6} + \cdots - 61\!\cdots\!24 ) / 12\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1932377350 \nu^{9} - 4685079769 \nu^{8} - 2788945929432 \nu^{7} - 6744083301684 \nu^{6} - 968434359217698 \nu^{5} + \cdots - 23\!\cdots\!00 ) / 613547379931392 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3538913459 \nu^{9} + 949770776 \nu^{8} + 5148112932156 \nu^{7} + 1377320091264 \nu^{6} + \cdots + 74\!\cdots\!32 ) / 613547379931392 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 9 \beta_{9} + 17 \beta_{8} + 15 \beta_{7} - 18 \beta_{5} + 358 \beta_{4} + 2 \beta_{3} + 1043 \beta_{2} - 3 \beta _1 + 182 ) / 4320 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 39\beta_{8} - 24\beta_{6} + 2\beta_{5} - 563\beta_{4} + 15\beta_{3} - 236\beta_{2} - 114\beta _1 - 88558 ) / 270 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 375 \beta_{9} - 3967 \beta_{8} - 3153 \beta_{7} - 3696 \beta_{6} + 3198 \beta_{5} - 63242 \beta_{4} + 2882 \beta_{3} - 421117 \beta_{2} + 1389 \beta _1 - 31114 ) / 1440 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8805 \beta_{8} + 4944 \beta_{6} - 1894 \beta_{5} + 123217 \beta_{4} - 3861 \beta_{3} + 51820 \beta_{2} + 13386 \beta _1 + 12132530 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 587295 \beta_{9} + 10471033 \beta_{8} + 8476887 \beta_{7} + 13073904 \beta_{6} - 7623762 \beta_{5} + 144648998 \beta_{4} - 11079758 \beta_{3} + 1244093803 \beta_{2} + \cdots + 69764806 ) / 4320 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4991021 \beta_{8} - 2823976 \beta_{6} + 1189238 \beta_{5} - 69115377 \beta_{4} + 2167045 \beta_{3} - 29077764 \beta_{2} - 5980126 \beta _1 - 6045548182 ) / 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1002570867 \beta_{9} - 10039223749 \beta_{8} - 8231449035 \beta_{7} - 13312393200 \beta_{6} + 7140762666 \beta_{5} - 133013599406 \beta_{4} + \cdots - 63697009294 ) / 4320 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8964366177 \beta_{8} + 5112252240 \beta_{6} - 2161618478 \beta_{5} + 123729099293 \beta_{4} - 3852113937 \beta_{3} + 52054935740 \beta_{2} + \cdots + 10528560319474 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 366131162877 \beta_{9} + 3286296411019 \beta_{8} + 2707361971365 \beta_{7} + 4414995739920 \beta_{6} - 2326880020086 \beta_{5} + \cdots + 20593816306834 ) / 1440 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
3.43232i
13.5890i
31.4704i
13.7835i
16.0371i
16.0371i
13.7835i
31.4704i
13.5890i
3.43232i
29.4751i 29.7353 + 75.3446i −612.784 0 2220.79 876.451i 3164.62 10516.3i −4792.63 + 4480.79i 0
26.2 24.1370i −39.5746 70.6742i −326.594 0 −1705.86 + 955.212i 1042.22 1703.92i −3428.70 + 5593.81i 0
26.3 18.2182i −69.9805 + 40.7889i −75.9021 0 743.099 + 1274.92i −4676.68 3281.06i 3233.54 5708.85i 0
26.4 16.2639i 77.7167 22.8278i −8.51375 0 −371.269 1263.98i −569.895 4025.09i 5518.78 3548.20i 0
26.5 3.03414i 14.6031 + 79.6728i 246.794 0 241.739 44.3080i 59.7348 1525.55i −6134.50 + 2326.94i 0
26.6 3.03414i 14.6031 79.6728i 246.794 0 241.739 + 44.3080i 59.7348 1525.55i −6134.50 2326.94i 0
26.7 16.2639i 77.7167 + 22.8278i −8.51375 0 −371.269 + 1263.98i −569.895 4025.09i 5518.78 + 3548.20i 0
26.8 18.2182i −69.9805 40.7889i −75.9021 0 743.099 1274.92i −4676.68 3281.06i 3233.54 + 5708.85i 0
26.9 24.1370i −39.5746 + 70.6742i −326.594 0 −1705.86 955.212i 1042.22 1703.92i −3428.70 5593.81i 0
26.10 29.4751i 29.7353 75.3446i −612.784 0 2220.79 + 876.451i 3164.62 10516.3i −4792.63 4480.79i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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.c.f yes 10
3.b odd 2 1 inner 75.9.c.f yes 10
5.b even 2 1 75.9.c.e 10
5.c odd 4 2 75.9.d.d 20
15.d odd 2 1 75.9.c.e 10
15.e even 4 2 75.9.d.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.c.e 10 5.b even 2 1
75.9.c.e 10 15.d odd 2 1
75.9.c.f yes 10 1.a even 1 1 trivial
75.9.c.f yes 10 3.b odd 2 1 inner
75.9.d.d 20 5.c odd 4 2
75.9.d.d 20 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{10} + 2057T_{2}^{8} + 1478418T_{2}^{6} + 442732064T_{2}^{4} + 48388216960T_{2}^{2} + 409079808000 \) Copy content Toggle raw display
\( T_{7}^{5} + 980T_{7}^{4} - 16170189T_{7}^{3} + 7054500600T_{7}^{2} + 8426583900000T_{7} - 525098700000000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2057 T^{8} + \cdots + 409079808000 \) Copy content Toggle raw display
$3$ \( T^{10} - 25 T^{9} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} + 980 T^{4} + \cdots - 525098700000000)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 1951248065 T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} - 8460 T^{4} + \cdots - 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 45135371657 T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + 71967 T^{4} + \cdots + 52\!\cdots\!67)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 280193128092 T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + 2261313689660 T^{8} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + 1507030 T^{4} + \cdots + 96\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 1508380 T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 38200056696065 T^{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + 5873670 T^{4} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 143634526671812 T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + 307776345854012 T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + 924675900073040 T^{8} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} - 760110 T^{4} + \cdots + 41\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 8134645 T^{4} + \cdots + 15\!\cdots\!75)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} - 26045085 T^{4} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 4274948 T^{4} + \cdots + 23\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} - 201083900 T^{4} + \cdots - 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
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