Properties

Label 230.6.a.f
Level $230$
Weight $6$
Character orbit 230.a
Self dual yes
Analytic conductor $36.888$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.8882785570\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 774 x^{3} - 197 x^{2} + 66287 x + 154128\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 q^{2} + \beta_{1} q^{3} + 16 q^{4} + 25 q^{5} -4 \beta_{1} q^{6} + ( 20 + 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{7} -64 q^{8} + ( 67 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q -4 q^{2} + \beta_{1} q^{3} + 16 q^{4} + 25 q^{5} -4 \beta_{1} q^{6} + ( 20 + 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{7} -64 q^{8} + ( 67 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} -100 q^{10} + ( 49 + 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{11} + 16 \beta_{1} q^{12} + ( 353 - 11 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{13} + ( -80 - 16 \beta_{1} - 8 \beta_{2} - 4 \beta_{4} ) q^{14} + 25 \beta_{1} q^{15} + 256 q^{16} + ( 385 + 5 \beta_{3} + 7 \beta_{4} ) q^{17} + ( -268 - 16 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{18} + ( -161 - 27 \beta_{1} - 11 \beta_{2} - 23 \beta_{3} - 6 \beta_{4} ) q^{19} + 400 q^{20} + ( 897 + 194 \beta_{1} + 6 \beta_{2} + 19 \beta_{3} - 2 \beta_{4} ) q^{21} + ( -196 - 16 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} - 12 \beta_{4} ) q^{22} -529 q^{23} -64 \beta_{1} q^{24} + 625 q^{25} + ( -1412 + 44 \beta_{1} - 8 \beta_{2} + 12 \beta_{3} + 8 \beta_{4} ) q^{26} + ( 456 + 139 \beta_{1} + 11 \beta_{2} + 45 \beta_{3} - 7 \beta_{4} ) q^{27} + ( 320 + 64 \beta_{1} + 32 \beta_{2} + 16 \beta_{4} ) q^{28} + ( -839 + 169 \beta_{1} - 20 \beta_{2} - \beta_{3} - 17 \beta_{4} ) q^{29} -100 \beta_{1} q^{30} + ( -3154 + 30 \beta_{1} - 5 \beta_{2} - 72 \beta_{3} + 25 \beta_{4} ) q^{31} -1024 q^{32} + ( 533 + 303 \beta_{1} + 17 \beta_{2} + 17 \beta_{3} - 10 \beta_{4} ) q^{33} + ( -1540 - 20 \beta_{3} - 28 \beta_{4} ) q^{34} + ( 500 + 100 \beta_{1} + 50 \beta_{2} + 25 \beta_{4} ) q^{35} + ( 1072 + 64 \beta_{1} + 64 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} ) q^{36} + ( -640 - 330 \beta_{1} - 31 \beta_{2} - 27 \beta_{3} + 20 \beta_{4} ) q^{37} + ( 644 + 108 \beta_{1} + 44 \beta_{2} + 92 \beta_{3} + 24 \beta_{4} ) q^{38} + ( -3465 + 458 \beta_{1} - 75 \beta_{2} + 25 \beta_{3} - 8 \beta_{4} ) q^{39} -1600 q^{40} + ( -2303 - 420 \beta_{1} - 35 \beta_{2} - 43 \beta_{3} + 5 \beta_{4} ) q^{41} + ( -3588 - 776 \beta_{1} - 24 \beta_{2} - 76 \beta_{3} + 8 \beta_{4} ) q^{42} + ( 1445 - 211 \beta_{1} - 10 \beta_{2} - 12 \beta_{3} - 27 \beta_{4} ) q^{43} + ( 784 + 64 \beta_{1} + 48 \beta_{2} + 32 \beta_{3} + 48 \beta_{4} ) q^{44} + ( 1675 + 100 \beta_{1} + 100 \beta_{2} + 25 \beta_{3} + 25 \beta_{4} ) q^{45} + 2116 q^{46} + ( 2766 - 584 \beta_{1} - 133 \beta_{2} + 12 \beta_{3} - 56 \beta_{4} ) q^{47} + 256 \beta_{1} q^{48} + ( 18172 + 618 \beta_{1} + 160 \beta_{2} + 161 \beta_{3} - 139 \beta_{4} ) q^{49} -2500 q^{50} + ( -478 + 420 \beta_{1} + 27 \beta_{2} - 53 \beta_{3} - 19 \beta_{4} ) q^{51} + ( 5648 - 176 \beta_{1} + 32 \beta_{2} - 48 \beta_{3} - 32 \beta_{4} ) q^{52} + ( 10851 - 1333 \beta_{1} - 137 \beta_{2} - 153 \beta_{3} - 7 \beta_{4} ) q^{53} + ( -1824 - 556 \beta_{1} - 44 \beta_{2} - 180 \beta_{3} + 28 \beta_{4} ) q^{54} + ( 1225 + 100 \beta_{1} + 75 \beta_{2} + 50 \beta_{3} + 75 \beta_{4} ) q^{55} + ( -1280 - 256 \beta_{1} - 128 \beta_{2} - 64 \beta_{4} ) q^{56} + ( -4253 - 982 \beta_{1} - 304 \beta_{2} - 151 \beta_{3} + 30 \beta_{4} ) q^{57} + ( 3356 - 676 \beta_{1} + 80 \beta_{2} + 4 \beta_{3} + 68 \beta_{4} ) q^{58} + ( -6421 - 13 \beta_{1} - 291 \beta_{2} - 77 \beta_{3} + 195 \beta_{4} ) q^{59} + 400 \beta_{1} q^{60} + ( 14327 + 26 \beta_{1} + 153 \beta_{2} + 192 \beta_{3} - 35 \beta_{4} ) q^{61} + ( 12616 - 120 \beta_{1} + 20 \beta_{2} + 288 \beta_{3} - 100 \beta_{4} ) q^{62} + ( 52403 + 992 \beta_{1} + 489 \beta_{2} + 322 \beta_{3} - 76 \beta_{4} ) q^{63} + 4096 q^{64} + ( 8825 - 275 \beta_{1} + 50 \beta_{2} - 75 \beta_{3} - 50 \beta_{4} ) q^{65} + ( -2132 - 1212 \beta_{1} - 68 \beta_{2} - 68 \beta_{3} + 40 \beta_{4} ) q^{66} + ( 14990 + 588 \beta_{1} - 89 \beta_{2} + 99 \beta_{3} + 94 \beta_{4} ) q^{67} + ( 6160 + 80 \beta_{3} + 112 \beta_{4} ) q^{68} -529 \beta_{1} q^{69} + ( -2000 - 400 \beta_{1} - 200 \beta_{2} - 100 \beta_{4} ) q^{70} + ( 11010 - 319 \beta_{1} + 347 \beta_{2} + 81 \beta_{3} + 150 \beta_{4} ) q^{71} + ( -4288 - 256 \beta_{1} - 256 \beta_{2} - 64 \beta_{3} - 64 \beta_{4} ) q^{72} + ( 14113 - 1395 \beta_{1} - 251 \beta_{2} - 440 \beta_{3} + 93 \beta_{4} ) q^{73} + ( 2560 + 1320 \beta_{1} + 124 \beta_{2} + 108 \beta_{3} - 80 \beta_{4} ) q^{74} + 625 \beta_{1} q^{75} + ( -2576 - 432 \beta_{1} - 176 \beta_{2} - 368 \beta_{3} - 96 \beta_{4} ) q^{76} + ( 56792 + 1015 \beta_{1} + 152 \beta_{2} + 37 \beta_{3} - 555 \beta_{4} ) q^{77} + ( 13860 - 1832 \beta_{1} + 300 \beta_{2} - 100 \beta_{3} + 32 \beta_{4} ) q^{78} + ( 20391 + 2467 \beta_{1} - 164 \beta_{2} - 4 \beta_{3} + 139 \beta_{4} ) q^{79} + 6400 q^{80} + ( 20545 + 469 \beta_{1} + 74 \beta_{2} + 181 \beta_{3} - 157 \beta_{4} ) q^{81} + ( 9212 + 1680 \beta_{1} + 140 \beta_{2} + 172 \beta_{3} - 20 \beta_{4} ) q^{82} + ( 17427 + 2359 \beta_{1} - 341 \beta_{2} - 195 \beta_{3} - 291 \beta_{4} ) q^{83} + ( 14352 + 3104 \beta_{1} + 96 \beta_{2} + 304 \beta_{3} - 32 \beta_{4} ) q^{84} + ( 9625 + 125 \beta_{3} + 175 \beta_{4} ) q^{85} + ( -5780 + 844 \beta_{1} + 40 \beta_{2} + 48 \beta_{3} + 108 \beta_{4} ) q^{86} + ( 55910 - 1806 \beta_{1} + 793 \beta_{2} + 80 \beta_{3} + 244 \beta_{4} ) q^{87} + ( -3136 - 256 \beta_{1} - 192 \beta_{2} - 128 \beta_{3} - 192 \beta_{4} ) q^{88} + ( 18765 + 1569 \beta_{1} - 510 \beta_{2} - 72 \beta_{3} - 51 \beta_{4} ) q^{89} + ( -6700 - 400 \beta_{1} - 400 \beta_{2} - 100 \beta_{3} - 100 \beta_{4} ) q^{90} + ( 12131 - 1464 \beta_{1} + 991 \beta_{2} + 194 \beta_{3} + 929 \beta_{4} ) q^{91} -8464 q^{92} + ( 17169 - 2650 \beta_{1} - 757 \beta_{2} - 399 \beta_{3} + 62 \beta_{4} ) q^{93} + ( -11064 + 2336 \beta_{1} + 532 \beta_{2} - 48 \beta_{3} + 224 \beta_{4} ) q^{94} + ( -4025 - 675 \beta_{1} - 275 \beta_{2} - 575 \beta_{3} - 150 \beta_{4} ) q^{95} -1024 \beta_{1} q^{96} + ( 42981 + 2092 \beta_{1} + 209 \beta_{2} - 490 \beta_{3} - \beta_{4} ) q^{97} + ( -72688 - 2472 \beta_{1} - 640 \beta_{2} - 644 \beta_{3} + 556 \beta_{4} ) q^{98} + ( 77440 + 1812 \beta_{1} + 659 \beta_{2} + 145 \beta_{3} - 457 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 20q^{2} + q^{3} + 80q^{4} + 125q^{5} - 4q^{6} + 102q^{7} - 320q^{8} + 334q^{9} + O(q^{10}) \) \( 5q - 20q^{2} + q^{3} + 80q^{4} + 125q^{5} - 4q^{6} + 102q^{7} - 320q^{8} + 334q^{9} - 500q^{10} + 251q^{11} + 16q^{12} + 1743q^{13} - 408q^{14} + 25q^{15} + 1280q^{16} + 1944q^{17} - 1336q^{18} - 845q^{19} + 2000q^{20} + 4682q^{21} - 1004q^{22} - 2645q^{23} - 64q^{24} + 3125q^{25} - 6972q^{26} + 2428q^{27} + 1632q^{28} - 4021q^{29} - 100q^{30} - 15752q^{31} - 5120q^{32} + 2931q^{33} - 7776q^{34} + 2550q^{35} + 5344q^{36} - 3455q^{37} + 3380q^{38} - 16708q^{39} - 8000q^{40} - 11898q^{41} - 18728q^{42} + 6968q^{43} + 4016q^{44} + 8350q^{45} + 10580q^{46} + 13412q^{47} + 256q^{48} + 91041q^{49} - 12500q^{50} - 2115q^{51} + 27888q^{52} + 53029q^{53} - 9712q^{54} + 6275q^{55} - 6528q^{56} - 21730q^{57} + 16084q^{58} - 31223q^{59} + 400q^{60} + 71477q^{61} + 63008q^{62} + 262199q^{63} + 20480q^{64} + 43575q^{65} - 11724q^{66} + 76003q^{67} + 31104q^{68} - 529q^{69} - 10200q^{70} + 54418q^{71} - 21376q^{72} + 69418q^{73} + 13820q^{74} + 625q^{75} - 13520q^{76} + 283598q^{77} + 66832q^{78} + 105024q^{79} + 32000q^{80} + 102913q^{81} + 47592q^{82} + 89399q^{83} + 74912q^{84} + 48600q^{85} - 27872q^{86} + 276726q^{87} - 16064q^{88} + 96240q^{89} - 33400q^{90} + 59261q^{91} - 42320q^{92} + 84434q^{93} - 53648q^{94} - 21125q^{95} - 1024q^{96} + 216087q^{97} - 364164q^{98} + 386925q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 774 x^{3} - 197 x^{2} + 66287 x + 154128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} - 13 \nu^{3} - 1235 \nu^{2} + 6295 \nu + 13806 \)\()/897\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + 18 \nu^{3} + 698 \nu^{2} - 10657 \nu - 37102 \)\()/598\)
\(\beta_{4}\)\(=\)\((\)\( -13 \nu^{4} + 50 \nu^{3} + 9580 \nu^{2} - 25565 \nu - 555282 \)\()/1794\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 4 \beta_{2} + 4 \beta_{1} + 310\)
\(\nu^{3}\)\(=\)\(-7 \beta_{4} + 45 \beta_{3} + 11 \beta_{2} + 625 \beta_{1} + 456\)
\(\nu^{4}\)\(=\)\(572 \beta_{4} + 910 \beta_{3} + 2990 \beta_{2} + 3385 \beta_{1} + 187486\)

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
−25.4526
−8.46823
−2.48374
10.8366
26.5679
−4.00000 −25.4526 16.0000 25.0000 101.810 169.151 −64.0000 404.834 −100.000
1.2 −4.00000 −8.46823 16.0000 25.0000 33.8729 −118.977 −64.0000 −171.289 −100.000
1.3 −4.00000 −2.48374 16.0000 25.0000 9.93494 −252.275 −64.0000 −236.831 −100.000
1.4 −4.00000 10.8366 16.0000 25.0000 −43.3465 46.1561 −64.0000 −125.567 −100.000
1.5 −4.00000 26.5679 16.0000 25.0000 −106.272 257.945 −64.0000 462.854 −100.000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(23\) \(1\)

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 230.6.a.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.a.f 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - T_{3}^{4} - 774 T_{3}^{3} - 197 T_{3}^{2} + 66287 T_{3} + 154128 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{5} \)
$3$ \( 154128 + 66287 T - 197 T^{2} - 774 T^{3} - T^{4} + T^{5} \)
$5$ \( ( -25 + T )^{5} \)
$7$ \( -60445984556 + 1153635228 T + 7298351 T^{2} - 82336 T^{3} - 102 T^{4} + T^{5} \)
$11$ \( 216794838000 - 26175895976 T + 203896930 T^{2} - 408321 T^{3} - 251 T^{4} + T^{5} \)
$13$ \( 15847720477654 - 175955978859 T + 439278623 T^{2} + 463868 T^{3} - 1743 T^{4} + T^{5} \)
$17$ \( 151348597947360 - 1278068818176 T + 3078143043 T^{2} - 1506900 T^{3} - 1944 T^{4} + T^{5} \)
$19$ \( -5767294160903792 + 20631582728744 T - 2061132098 T^{2} - 9048023 T^{3} + 845 T^{4} + T^{5} \)
$23$ \( ( 529 + T )^{5} \)
$29$ \( -75188728129854300 + 144945235918354 T - 8321555417 T^{2} - 31835475 T^{3} + 4021 T^{4} + T^{5} \)
$31$ \( -1523654623422069815 - 3074918384725792 T - 826223314121 T^{2} + 4434631 T^{3} + 15752 T^{4} + T^{5} \)
$37$ \( -3130987952255940608 + 1516383438254720 T + 143867955592 T^{2} - 107053664 T^{3} + 3455 T^{4} + T^{5} \)
$41$ \( 378692981425953 + 11691043765920 T - 170174827371 T^{2} - 91677585 T^{3} + 11898 T^{4} + T^{5} \)
$43$ \( 458308667481725952 + 662829431519040 T + 179644064880 T^{2} - 45378900 T^{3} - 6968 T^{4} + T^{5} \)
$47$ \( -8753317759230157776 + 16198616440057036 T + 6314372401546 T^{2} - 482074695 T^{3} - 13412 T^{4} + T^{5} \)
$53$ \( \)\(76\!\cdots\!48\)\( - 1295592630991409504 T + 67167472285664 T^{2} - 417693180 T^{3} - 53029 T^{4} + T^{5} \)
$59$ \( \)\(87\!\cdots\!32\)\( + 2952424862073508288 T - 105251182924684 T^{2} - 3522871518 T^{3} + 31223 T^{4} + T^{5} \)
$61$ \( \)\(39\!\cdots\!00\)\( - 511464713228530680 T + 8776907919408 T^{2} + 1373935005 T^{3} - 71477 T^{4} + T^{5} \)
$67$ \( -\)\(43\!\cdots\!16\)\( - 335866098731322688 T + 29699188644952 T^{2} + 902777920 T^{3} - 76003 T^{4} + T^{5} \)
$71$ \( -\)\(11\!\cdots\!87\)\( + 117218618523681364 T + 45271165695821 T^{2} - 1032107217 T^{3} - 54418 T^{4} + T^{5} \)
$73$ \( \)\(20\!\cdots\!48\)\( - 1209451035464929768 T + 164609214180772 T^{2} - 2068070543 T^{3} - 69418 T^{4} + T^{5} \)
$79$ \( -\)\(21\!\cdots\!88\)\( - 4164561699752205696 T + 442814035187648 T^{2} - 2401899736 T^{3} - 105024 T^{4} + T^{5} \)
$83$ \( -\)\(59\!\cdots\!24\)\( + 1062463078220218528 T + 622921520176588 T^{2} - 6206153268 T^{3} - 89399 T^{4} + T^{5} \)
$89$ \( \)\(21\!\cdots\!00\)\( - 20206593475449724800 T + 578413154705904 T^{2} - 2898715140 T^{3} - 96240 T^{4} + T^{5} \)
$97$ \( \)\(12\!\cdots\!56\)\( - 69063758418002129604 T + 886628213878754 T^{2} + 8399896175 T^{3} - 216087 T^{4} + T^{5} \)
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