Properties

Label 304.10.a.e
Level $304$
Weight $10$
Character orbit 304.a
Self dual yes
Analytic conductor $156.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(156.570894194\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 34433 x^{2} - 2723303 x - 48270488\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -21 - \beta_{1} ) q^{3} + ( -350 - 5 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{5} + ( -3075 - 27 \beta_{1} - 2 \beta_{2} + 9 \beta_{3} ) q^{7} + ( 4134 + 135 \beta_{1} - 18 \beta_{2} + 16 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -21 - \beta_{1} ) q^{3} + ( -350 - 5 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{5} + ( -3075 - 27 \beta_{1} - 2 \beta_{2} + 9 \beta_{3} ) q^{7} + ( 4134 + 135 \beta_{1} - 18 \beta_{2} + 16 \beta_{3} ) q^{9} + ( 26056 + 3 \beta_{1} - 25 \beta_{2} ) q^{11} + ( 30071 + 766 \beta_{1} - 33 \beta_{2} - 169 \beta_{3} ) q^{13} + ( 147720 + 1380 \beta_{1} - 264 \beta_{2} + 54 \beta_{3} ) q^{15} + ( -103123 - 276 \beta_{1} - 587 \beta_{2} + 234 \beta_{3} ) q^{17} -130321 q^{19} + ( 609189 + 2976 \beta_{1} - 375 \beta_{2} + 125 \beta_{3} ) q^{21} + ( -752981 + 678 \beta_{1} - 1531 \beta_{2} - 93 \beta_{3} ) q^{23} + ( 2440945 + 6795 \beta_{1} + 1845 \beta_{2} + 1350 \beta_{3} ) q^{25} + ( -3097563 - 12769 \beta_{1} + 1674 \beta_{2} - 3948 \beta_{3} ) q^{27} + ( 1537183 - 1436 \beta_{1} - 1027 \beta_{2} - 3481 \beta_{3} ) q^{29} + ( -3192456 - 9482 \beta_{1} + 2050 \beta_{2} + 2150 \beta_{3} ) q^{31} + ( -815454 - 38998 \beta_{1} - 1596 \beta_{2} - 2198 \beta_{3} ) q^{33} + ( -2453330 + 20875 \beta_{1} - 10051 \beta_{2} + 20156 \beta_{3} ) q^{35} + ( 5128462 - 40262 \beta_{1} - 6770 \beta_{2} + 14570 \beta_{3} ) q^{37} + ( -17471739 - 93298 \beta_{1} + 7047 \beta_{2} - 12559 \beta_{3} ) q^{39} + ( 2893200 - 11750 \beta_{1} - 19020 \beta_{2} - 28480 \beta_{3} ) q^{41} + ( -1937238 + 59233 \beta_{1} - 24965 \beta_{2} - 25660 \beta_{3} ) q^{43} + ( -30988530 - 352695 \beta_{1} + 28557 \beta_{2} + 33138 \beta_{3} ) q^{45} + ( 7894898 - 45347 \beta_{1} + 31111 \beta_{2} - 32602 \beta_{3} ) q^{47} + ( 4720576 + 110482 \beta_{1} + 12239 \beta_{2} - 100318 \beta_{3} ) q^{49} + ( 2126961 - 217655 \beta_{1} - 37392 \beta_{2} - 49576 \beta_{3} ) q^{51} + ( 18108791 + 259234 \beta_{1} - 35357 \beta_{2} - 78901 \beta_{3} ) q^{53} + ( -5350710 - 321855 \beta_{1} - 18501 \beta_{2} - 51834 \beta_{3} ) q^{55} + ( 2736741 + 130321 \beta_{1} ) q^{57} + ( 37327107 + 25343 \beta_{1} + 66072 \beta_{2} + 8236 \beta_{3} ) q^{59} + ( 32233368 - 7739 \beta_{1} - 115309 \beta_{2} + 44408 \beta_{3} ) q^{61} + ( -25788720 - 636137 \beta_{1} + 71559 \beta_{2} - 258888 \beta_{3} ) q^{63} + ( 31222100 - 1113730 \beta_{1} + 309130 \beta_{2} - 112430 \beta_{3} ) q^{65} + ( -33192661 + 873165 \beta_{1} - 156322 \beta_{2} - 19056 \beta_{3} ) q^{67} + ( -11440611 - 73522 \beta_{1} - 91353 \beta_{2} - 141119 \beta_{3} ) q^{69} + ( 11884254 - 50978 \beta_{1} - 8844 \beta_{2} + 407378 \beta_{3} ) q^{71} + ( -9921749 - 1030550 \beta_{1} - 184019 \beta_{2} - 170182 \beta_{3} ) q^{73} + ( -206079195 - 2611045 \beta_{1} + 280530 \beta_{2} + 29700 \beta_{3} ) q^{75} + ( -41488292 - 940039 \beta_{1} - 63511 \beta_{2} + 44062 \beta_{3} ) q^{77} + ( 76710310 + 1674604 \beta_{1} - 293190 \beta_{2} + 123590 \beta_{3} ) q^{79} + ( 326443161 + 3691188 \beta_{1} + 128340 \beta_{2} + 92560 \beta_{3} ) q^{81} + ( 186639194 + 980280 \beta_{1} + 92806 \beta_{2} - 104262 \beta_{3} ) q^{83} + ( -25807650 - 3902385 \beta_{1} + 26637 \beta_{2} + 1748748 \beta_{3} ) q^{85} + ( 20486865 - 1052166 \beta_{1} - 187617 \beta_{2} - 13131 \beta_{3} ) q^{87} + ( 72054508 - 224494 \beta_{1} + 201740 \beta_{2} + 1072810 \beta_{3} ) q^{89} + ( -788716179 - 2598565 \beta_{1} - 102178 \beta_{2} + 1018576 \beta_{3} ) q^{91} + ( 288055008 + 4788454 \beta_{1} + 22674 \beta_{2} + 295762 \beta_{3} ) q^{93} + ( 45612350 + 651605 \beta_{1} + 130321 \beta_{2} + 521284 \beta_{3} ) q^{95} + ( 197503898 - 3887418 \beta_{1} - 963354 \beta_{2} - 2541012 \beta_{3} ) q^{97} + ( 420494730 + 4927511 \beta_{1} - 374571 \beta_{2} + 519682 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 84q^{3} - 1395q^{5} - 12307q^{7} + 16538q^{9} + O(q^{10}) \) \( 4q - 84q^{3} - 1395q^{5} - 12307q^{7} + 16538q^{9} + 104249q^{11} + 120486q^{13} + 591090q^{15} - 412139q^{17} - 521284q^{19} + 2437006q^{21} - 3010300q^{23} + 9760585q^{25} - 12387978q^{27} + 6153240q^{29} - 12774024q^{31} - 3258022q^{33} - 9823425q^{35} + 20506048q^{37} - 69881444q^{39} + 11620300q^{41} - 7698327q^{43} - 124015815q^{45} + 31581083q^{47} + 18970383q^{49} + 8594812q^{51} + 72549422q^{53} - 21332505q^{55} + 10946964q^{57} + 149234120q^{59} + 129004373q^{61} - 102967551q^{63} + 124691700q^{65} - 132595266q^{67} - 45529972q^{69} + 47138482q^{71} - 39332795q^{73} - 824627010q^{75} - 165933719q^{77} + 307010840q^{79} + 1305551744q^{81} + 746568232q^{83} - 105005985q^{85} + 82148208q^{87} + 286943482q^{89} - 3155781114q^{91} + 1151901596q^{93} + 181797795q^{95} + 793519958q^{97} + 1681833809q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 34433 x^{2} - 2723303 x - 48270488\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 9 \nu^{3} - 736 \nu^{2} - 252737 \nu - 5880008 \)\()/20632\)
\(\beta_{2}\)\(=\)\((\)\( -17 \nu^{3} + 244 \nu^{2} + 698613 \nu + 30780440 \)\()/20632\)
\(\beta_{3}\)\(=\)\((\)\( 13 \nu^{3} - 490 \nu^{2} - 413779 \nu - 18350856 \)\()/10316\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 2\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(193 \beta_{3} + 85 \beta_{2} - 397 \beta_{1} + 103370\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(43865 \beta_{3} + 35033 \beta_{2} - 46793 \beta_{1} + 12429538\)\()/6\)

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
−67.1081
−26.2676
219.264
−124.888
0 −265.578 0 −2367.11 0 −5859.40 0 50848.5 0
1.2 0 −25.2570 0 2126.71 0 −11469.0 0 −19045.1 0
1.3 0 66.6053 0 −2418.56 0 1533.95 0 −15246.7 0
1.4 0 140.229 0 1263.95 0 3487.42 0 −18.7042 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

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 304.10.a.e 4
4.b odd 2 1 38.10.a.d 4
12.b even 2 1 342.10.a.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.d 4 4.b odd 2 1
304.10.a.e 4 1.a even 1 1 trivial
342.10.a.l 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 84 T_{3}^{3} - 44107 T_{3}^{2} + 1329018 T_{3} + 62650008 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(304))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 62650008 + 1329018 T - 44107 T^{2} + 84 T^{3} + T^{4} \)
$5$ \( 15389064288000 - 6547343400 T - 7813530 T^{2} + 1395 T^{3} + T^{4} \)
$7$ \( 359494671206216 - 244743771643 T - 14461281 T^{2} + 12307 T^{3} + T^{4} \)
$11$ \( 115765319454159024 - 36343268701740 T + 3398840664 T^{2} - 104249 T^{3} + T^{4} \)
$13$ \( \)\(23\!\cdots\!00\)\( + 2295095998230370 T - 30806569293 T^{2} - 120486 T^{3} + T^{4} \)
$17$ \( \)\(27\!\cdots\!62\)\( - 99501926037315363 T - 374372926323 T^{2} + 412139 T^{3} + T^{4} \)
$19$ \( ( 130321 + T )^{4} \)
$23$ \( \)\(22\!\cdots\!08\)\( - 2126564557696334952 T + 862408556595 T^{2} + 3010300 T^{3} + T^{4} \)
$29$ \( -\)\(34\!\cdots\!76\)\( + 2981737641494670876 T + 8624111888511 T^{2} - 6153240 T^{3} + T^{4} \)
$31$ \( -\)\(14\!\cdots\!96\)\( + 63865994000171580800 T + 51883371775344 T^{2} + 12774024 T^{3} + T^{4} \)
$37$ \( -\)\(58\!\cdots\!72\)\( + \)\(22\!\cdots\!56\)\( T - 68206760068968 T^{2} - 20506048 T^{3} + T^{4} \)
$41$ \( -\)\(36\!\cdots\!00\)\( + \)\(67\!\cdots\!00\)\( T - 531906213139500 T^{2} - 11620300 T^{3} + T^{4} \)
$43$ \( -\)\(19\!\cdots\!12\)\( + \)\(93\!\cdots\!96\)\( T - 893995448146728 T^{2} + 7698327 T^{3} + T^{4} \)
$47$ \( \)\(52\!\cdots\!00\)\( + \)\(19\!\cdots\!60\)\( T - 1669221829649832 T^{2} - 31581083 T^{3} + T^{4} \)
$53$ \( -\)\(20\!\cdots\!64\)\( + \)\(31\!\cdots\!82\)\( T - 4017930015425685 T^{2} - 72549422 T^{3} + T^{4} \)
$59$ \( -\)\(27\!\cdots\!04\)\( + \)\(16\!\cdots\!70\)\( T + 3811888983551253 T^{2} - 149234120 T^{3} + T^{4} \)
$61$ \( \)\(43\!\cdots\!00\)\( + \)\(91\!\cdots\!60\)\( T - 10642541015571546 T^{2} - 129004373 T^{3} + T^{4} \)
$67$ \( -\)\(18\!\cdots\!00\)\( - \)\(84\!\cdots\!00\)\( T - 58349865773044743 T^{2} + 132595266 T^{3} + T^{4} \)
$71$ \( -\)\(32\!\cdots\!32\)\( + \)\(10\!\cdots\!44\)\( T - 74119300967107956 T^{2} - 47138482 T^{3} + T^{4} \)
$73$ \( \)\(35\!\cdots\!34\)\( - \)\(10\!\cdots\!87\)\( T - 87654713869556991 T^{2} + 39332795 T^{3} + T^{4} \)
$79$ \( \)\(29\!\cdots\!00\)\( - \)\(10\!\cdots\!40\)\( T - 229308708527002548 T^{2} - 307010840 T^{3} + T^{4} \)
$83$ \( -\)\(94\!\cdots\!68\)\( - \)\(21\!\cdots\!04\)\( T + 152802359319519228 T^{2} - 746568232 T^{3} + T^{4} \)
$89$ \( \)\(54\!\cdots\!84\)\( + \)\(99\!\cdots\!80\)\( T - 451932251304917424 T^{2} - 286943482 T^{3} + T^{4} \)
$97$ \( \)\(37\!\cdots\!00\)\( + \)\(15\!\cdots\!80\)\( T - 3736940299059182196 T^{2} - 793519958 T^{3} + T^{4} \)
show more
show less