Properties

Label 245.6.a.m
Level $245$
Weight $6$
Character orbit 245.a
Self dual yes
Analytic conductor $39.294$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 246 x^{8} - 192 x^{7} + 20336 x^{6} + 25380 x^{5} - 639206 x^{4} - 722920 x^{3} + 7583055 x^{2} + 5935300 x - 22888100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 7^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{3} + 6) q^{3} + (\beta_{4} - \beta_{3} - \beta_1 + 18) q^{4} + 25 q^{5} + ( - \beta_{6} - \beta_{5} - 2 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 15) q^{6} + ( - \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 4 \beta_{2} + \cdots + 27) q^{8}+ \cdots + (2 \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{5} - 3 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} + \cdots + 73) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{3} + 6) q^{3} + (\beta_{4} - \beta_{3} - \beta_1 + 18) q^{4} + 25 q^{5} + ( - \beta_{6} - \beta_{5} - 2 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 15) q^{6} + ( - \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 4 \beta_{2} + \cdots + 27) q^{8}+ \cdots + (394 \beta_{9} - 572 \beta_{8} - 232 \beta_{7} - 500 \beta_{6} + \cdots + 12314) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 58 q^{3} + 182 q^{4} + 250 q^{5} + 144 q^{6} + 270 q^{8} + 700 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 58 q^{3} + 182 q^{4} + 250 q^{5} + 144 q^{6} + 270 q^{8} + 700 q^{9} + 250 q^{10} + 794 q^{11} + 2560 q^{12} + 474 q^{13} + 1450 q^{15} + 2394 q^{16} + 802 q^{17} + 3702 q^{18} + 7292 q^{19} + 4550 q^{20} + 3948 q^{22} + 3708 q^{23} + 2092 q^{24} + 6250 q^{25} + 6576 q^{26} + 11818 q^{27} - 8866 q^{29} + 3600 q^{30} + 13292 q^{31} + 2590 q^{32} + 9854 q^{33} + 44468 q^{34} - 10690 q^{36} + 16124 q^{37} + 2180 q^{38} - 24982 q^{39} + 6750 q^{40} + 34836 q^{41} - 28604 q^{43} - 31120 q^{44} + 17500 q^{45} - 39732 q^{46} + 18106 q^{47} + 101788 q^{48} + 6250 q^{50} + 31602 q^{51} - 22480 q^{52} + 36440 q^{53} + 80836 q^{54} + 19850 q^{55} + 126988 q^{57} - 100356 q^{58} + 18644 q^{59} + 64000 q^{60} + 68120 q^{61} + 181052 q^{62} - 59358 q^{64} + 11850 q^{65} + 157780 q^{66} + 92328 q^{67} + 288540 q^{68} + 170888 q^{69} + 5044 q^{71} - 61654 q^{72} + 170160 q^{73} - 216584 q^{74} + 36250 q^{75} + 505180 q^{76} - 158008 q^{78} + 26442 q^{79} + 59850 q^{80} - 56314 q^{81} + 353948 q^{82} + 353360 q^{83} + 20050 q^{85} - 52940 q^{86} - 3190 q^{87} - 114916 q^{88} + 90704 q^{89} + 92550 q^{90} + 183520 q^{92} + 188560 q^{93} - 121388 q^{94} + 182300 q^{95} + 442220 q^{96} + 236382 q^{97} + 109024 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 246 x^{8} - 192 x^{7} + 20336 x^{6} + 25380 x^{5} - 639206 x^{4} - 722920 x^{3} + 7583055 x^{2} + 5935300 x - 22888100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 433 \nu^{9} - 1947 \nu^{8} + 98973 \nu^{7} + 554383 \nu^{6} - 6721179 \nu^{5} - 43954077 \nu^{4} + 111936823 \nu^{3} + 837690357 \nu^{2} + \cdots - 2613859980 ) / 69843200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 83137 \nu^{9} + 200027 \nu^{8} - 19567597 \nu^{7} - 57068271 \nu^{6} + 1395513643 \nu^{5} + 4236045917 \nu^{4} - 27571140327 \nu^{3} + \cdots - 107486394420 ) / 9847891200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83137 \nu^{9} + 200027 \nu^{8} - 19567597 \nu^{7} - 57068271 \nu^{6} + 1395513643 \nu^{5} + 4236045917 \nu^{4} - 27571140327 \nu^{3} + \cdots - 590033063220 ) / 9847891200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 76219 \nu^{9} + 438897 \nu^{8} - 21012719 \nu^{7} - 98631325 \nu^{6} + 1760844265 \nu^{5} + 6994915287 \nu^{4} - 45551323789 \nu^{3} + \cdots + 621408182180 ) / 4923945600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 95695 \nu^{9} - 431909 \nu^{8} + 23439115 \nu^{7} + 118409649 \nu^{6} - 1626243037 \nu^{5} - 9503152979 \nu^{4} + 23203382625 \nu^{3} + \cdots - 737664945060 ) / 4923945600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 271659 \nu^{9} - 2579273 \nu^{8} + 61777919 \nu^{7} + 579871301 \nu^{6} - 3997103913 \nu^{5} - 38975931263 \nu^{4} + \cdots - 1980608051620 ) / 9847891200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 91631 \nu^{9} - 419483 \nu^{8} - 20161043 \nu^{7} + 77430063 \nu^{6} + 1507316885 \nu^{5} - 4622163965 \nu^{4} - 44108638233 \nu^{3} + \cdots - 489826503660 ) / 1969578240 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25091 \nu^{9} + 50621 \nu^{8} - 5627811 \nu^{7} - 16286853 \nu^{6} + 381410769 \nu^{5} + 1347458171 \nu^{4} - 6897894681 \nu^{3} + \cdots + 64943739940 ) / 205164400 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta_{3} + 4\beta_{2} + 80\beta _1 + 57 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} - 2\beta_{6} + 3\beta_{5} + 106\beta_{4} - 137\beta_{3} + 11\beta_{2} + 209\beta _1 + 3956 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 114 \beta_{9} - 112 \beta_{8} + 116 \beta_{7} + 264 \beta_{6} + 134 \beta_{5} + 336 \beta_{4} - 378 \beta_{3} + 342 \beta_{2} + 7421 \beta _1 + 10834 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 366 \beta_{9} - 76 \beta_{8} - 116 \beta_{6} + 480 \beta_{5} + 10577 \beta_{4} - 14513 \beta_{3} + 2852 \beta_{2} + 30111 \beta _1 + 369337 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11979 \beta_{9} - 11273 \beta_{8} + 11413 \beta_{7} + 28830 \beta_{6} + 13957 \beta_{5} + 45098 \beta_{4} - 55466 \beta_{3} + 31280 \beta_{2} + 728286 \beta _1 + 1524905 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 50612 \beta_{9} - 17680 \beta_{8} - 1948 \beta_{7} + 11354 \beta_{6} + 64475 \beta_{5} + 1060034 \beta_{4} - 1476853 \beta_{3} + 432195 \beta_{2} + 3821255 \beta _1 + 36321428 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1265068 \beta_{9} - 1114544 \beta_{8} + 1075408 \beta_{7} + 3012396 \beta_{6} + 1388854 \beta_{5} + 5559868 \beta_{4} - 7296106 \beta_{3} + 3305446 \beta_{2} + 73448747 \beta _1 + 191858062 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
10.4745
10.2693
4.72637
4.48290
1.63335
−2.66104
−4.37629
−6.46320
−8.75161
−9.33424
−9.47445 20.3001 57.7653 25.0000 −192.333 0 −244.112 169.095 −236.861
1.2 −9.26930 1.27429 53.9199 25.0000 −11.8118 0 −203.182 −241.376 −231.732
1.3 −3.72637 −21.0476 −18.1142 25.0000 78.4311 0 186.744 200.001 −93.1592
1.4 −3.48290 0.931256 −19.8694 25.0000 −3.24347 0 180.656 −242.133 −87.0725
1.5 −0.633352 19.5171 −31.5989 25.0000 −12.3612 0 40.2804 137.918 −15.8338
1.6 3.66104 29.0676 −18.5968 25.0000 106.418 0 −185.237 601.928 91.5259
1.7 5.37629 −16.5213 −3.09546 25.0000 −88.8236 0 −188.684 29.9545 134.407
1.8 7.46320 −9.93685 23.6994 25.0000 −74.1607 0 −61.9494 −144.259 186.580
1.9 9.75161 23.6367 63.0938 25.0000 230.496 0 303.214 315.692 243.790
1.10 10.3342 10.7786 74.7964 25.0000 111.389 0 442.268 −126.821 258.356
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.6.a.m yes 10
7.b odd 2 1 245.6.a.l 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.6.a.l 10 7.b odd 2 1
245.6.a.m yes 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{10} - 10 T_{2}^{9} - 201 T_{2}^{8} + 2040 T_{2}^{7} + 12314 T_{2}^{6} - 129840 T_{2}^{5} - 230996 T_{2}^{4} + 2639600 T_{2}^{3} + 2127024 T_{2}^{2} - 16621440 T_{2} - 10686592 \) Copy content Toggle raw display
\( T_{3}^{10} - 58 T_{3}^{9} + 117 T_{3}^{8} + 44916 T_{3}^{7} - 571489 T_{3}^{6} - 9277842 T_{3}^{5} + 168931223 T_{3}^{4} + 285902184 T_{3}^{3} - 11335365576 T_{3}^{2} + \cdots - 12031198596 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 10 T^{9} - 201 T^{8} + \cdots - 10686592 \) Copy content Toggle raw display
$3$ \( T^{10} - 58 T^{9} + \cdots - 12031198596 \) Copy content Toggle raw display
$5$ \( (T - 25)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 794 T^{9} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} - 474 T^{9} + \cdots - 10\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{10} - 802 T^{9} + \cdots + 31\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{10} - 7292 T^{9} + \cdots - 53\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{10} - 3708 T^{9} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{10} + 8866 T^{9} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} - 13292 T^{9} + \cdots + 52\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{10} - 16124 T^{9} + \cdots - 14\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{10} - 34836 T^{9} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{10} + 28604 T^{9} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} - 18106 T^{9} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} - 36440 T^{9} + \cdots - 52\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{10} - 18644 T^{9} + \cdots - 26\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{10} - 68120 T^{9} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{10} - 92328 T^{9} + \cdots - 35\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{10} - 5044 T^{9} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{10} - 170160 T^{9} + \cdots - 37\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} - 26442 T^{9} + \cdots + 56\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{10} - 353360 T^{9} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} - 90704 T^{9} + \cdots - 31\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{10} - 236382 T^{9} + \cdots + 30\!\cdots\!88 \) Copy content Toggle raw display
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