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

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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

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