Newform orbit 308.4.i.a

• Label: 308.4.i.a
• Level: $308$
• Weight: $4$
• Character orbit: 308.i
• Analytic conductor: $18.173$
• Analytic rank: $0$
• Dimension: $20$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$308 = 2^{2} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	308.i (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.1725882818$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
Defining polynomial:	x^20 - 4*x^19 + 194*x^18 - 432*x^17 + 24205*x^16 - 47156*x^15 + 1632616*x^14 - 1838210*x^13 + 75876043*x^12 - 82333872*x^11 + 1801672853*x^10 - 471307576*x^9 + 27041915377*x^8 - 14255551602*x^7 + 176709560868*x^6 - 669217680*x^5 + 678547482048*x^4 - 436737562560*x^3 + 226025971776*x^2 - 48231013248*x + 7996651776
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{14}\cdot 3^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (-b3 - b2 - b1) * q^3 + (b9 - b5 + b3 - 1) * q^5 + (-b14 - b8 - b3 + b2 + b1 - 1) * q^7 + (b10 + 11*b3 + b1 - 11) * q^9 - 11*b3 * q^11 + (b16 + b15 + b14 - b13 + b12 - b8 - b6 - 3*b2 - 1) * q^13 + (b17 + b16 + b15 + 2*b14 - 2*b13 + b12 - b8 - b6 - 2*b2 + 5) * q^15 + (b18 + b13 + b11 + b10 + b9 - 2*b7 - b6 - b4 - 17*b3) * q^17 + (b19 - b17 + b16 + b14 - 2*b13 + b10 + 2*b9 + 2*b8 - b7 - b6 - 2*b5 + 35*b3 - b2 - 3*b1 - 33) * q^19 + (-2*b16 - 2*b15 + 2*b13 - 2*b11 - b9 - 2*b7 + 3*b5 - 3*b4 - 21*b3 - b2 - 2*b1 + 10) * q^21 + (-b19 - 2*b18 + b17 + 2*b16 - 3*b13 - 2*b12 - b10 + b9 + 3*b8 + 2*b7 + 2*b6 - b5 + 6*b1 + 5) * q^23 + (-2*b19 + b18 + b13 + b10 - 4*b9 - 3*b7 - b6 - b4 - 20*b3) * q^25 + (b17 + b16 - 2*b15 + 2*b14 - 5*b13 - 2*b12 - 2*b11 - 2*b10 - b8 - b6 - 5*b5 + 3*b3 + 16*b2 + 3*b1 + 39) * q^27 + (-3*b17 + 4*b16 - b15 - 5*b14 - 3*b12 + 3*b11 + 3*b10 - 4*b8 - b6 + 5*b5 - 2*b4 + 5*b3 - 5*b2 + 5*b1 - 16) * q^29 + (-b18 + b16 + b15 - 2*b14 - 5*b13 + 3*b11 - 2*b10 + 2*b9 - b8 + 2*b7 + 2*b6 + 3*b4 - 41*b3 + 10*b2 + 13*b1 - 3) * q^31 + (11*b3 + 11*b1 - 11) * q^33 + (3*b19 - 6*b18 - 2*b17 + 4*b16 - 3*b15 - 5*b14 - 6*b13 - 4*b12 - 3*b11 - 6*b10 - 2*b9 + b8 + 5*b7 + 5*b6 + 6*b5 + b4 - 5*b3 - 6*b2 + b1 + 20) * q^35 + (2*b19 + b18 - 2*b17 - 4*b15 + b14 - 2*b13 + b12 - 2*b11 + 4*b10 + 3*b9 + 4*b8 - 3*b7 - b6 - 3*b5 - 2*b4 + 31*b3 + b2 + 9*b1 - 26) * q^37 + (b19 + 4*b18 + 5*b16 + 4*b15 - 8*b14 - 12*b13 + 2*b11 - 12*b9 - 3*b8 + 5*b7 + 4*b4 + 53*b3 - 4*b2 + 8*b1 - 12) * q^39 + (-2*b16 + b15 + 5*b14 - 2*b13 + 2*b12 - 4*b11 - 4*b10 + 2*b8 + 7*b6 - 5*b5 + b4 - 3*b3 + 10*b2 - 3*b1 + 75) * q^41 + (-b17 - 5*b16 + 3*b15 + 13*b14 - 5*b13 + 7*b12 - 2*b11 - 2*b10 + 5*b8 + 6*b6 + 11*b5 + 4*b4 - 8*b3 - 13*b2 - 8*b1 - 21) * q^43 + (-b19 + 5*b18 + 8*b16 + 5*b15 - 11*b14 - 16*b13 + 9*b11 + b9 - 3*b8 + 10*b7 + 5*b4 + 5*b3 + 10*b2 + 26*b1 - 16) * q^45 + (-2*b19 + b18 + 2*b17 + 2*b16 + 2*b15 + 3*b14 - 11*b13 + b12 + b11 - b10 + 9*b9 + 10*b8 + 6*b7 + 5*b6 - 9*b5 + b4 + 30*b3 - 4*b2 - 10*b1 - 22) * q^47 + (-4*b19 + 5*b18 + 5*b17 + 8*b16 + b15 + 6*b14 - 11*b13 + b12 - 4*b9 - 6*b7 - 11*b6 + 5*b5 + b4 + 41*b3 - 16*b2 - 11*b1 + 7) * q^49 + (-2*b19 + 5*b18 + 2*b17 - b16 + 4*b15 + 4*b14 - 7*b13 + 5*b12 + 2*b11 + b10 - 3*b9 + 5*b8 + b7 - b6 + 3*b5 + 2*b4 - 4*b3 - 6*b2 + 43*b1 + 2) * q^51 + (5*b19 - 7*b18 + 6*b16 - 5*b15 - 11*b14 - 8*b13 - 5*b11 - 2*b10 - 12*b9 - 5*b8 + 4*b7 + 2*b6 - 3*b4 - 60*b3 - 16*b2 - 10*b1 - 6) * q^53 + (11*b5 + 11) * q^55 + (6*b17 + 4*b16 - 5*b15 + 3*b14 - 12*b13 + 2*b11 + 2*b10 - 4*b8 - 2*b6 - 3*b5 + 5*b4 + 9*b3 + 52*b2 + 9*b1 - 55) * q^57 + (5*b19 - 14*b18 - b16 - 7*b15 - 3*b14 - 3*b13 - 6*b11 - 7*b10 + 7*b9 - 4*b8 - b7 + 7*b6 - 74*b3 + 36*b2 + 32*b1 + 4) * q^59 + (-6*b19 + 2*b18 + 6*b17 + 6*b16 + 8*b14 - b13 + 2*b12 - b10 + 18*b9 + b8 - 6*b7 - 6*b6 - 18*b5 - 32*b3 - 8*b2 - 21*b1 + 31) * q^61 + (-4*b19 + 9*b18 - 2*b17 - 2*b16 + 19*b15 + 12*b14 - 16*b13 + 13*b12 + 10*b11 - b10 - 13*b9 + 10*b8 + 11*b7 + 8*b6 + 25*b5 + 15*b4 + 61*b3 - 63*b2 - 18*b1 - 119) * q^63 + (6*b19 - b18 - 6*b17 + 5*b16 - 2*b15 + 4*b14 + 2*b13 - b12 - b11 + 5*b10 - b8 - 10*b7 - 9*b6 - b4 + 79*b3 - 3*b2 + 52*b1 - 78) * q^65 + (b19 - 8*b18 - 4*b16 + b15 + 5*b14 - 5*b13 - 9*b11 - 9*b10 - 18*b9 + b8 - 3*b7 + 9*b6 + 10*b4 - 50*b3 - 20*b2 - 24*b1 + 4) * q^67 + (-2*b17 + 20*b16 - 6*b15 - 23*b14 - 3*b13 - 8*b12 + 5*b11 + 5*b10 - 20*b8 - 11*b6 - 23*b5 - 2*b4 + 26*b3 + 50*b2 + 26*b1 + 145) * q^69 + (b17 + 5*b16 - 4*b15 - 9*b13 - 11*b12 - 2*b11 - 2*b10 - 5*b8 - 11*b6 + 15*b5 - 7*b4 + 9*b3 + 2*b2 + 9*b1 + 15) * q^71 + (-11*b19 + 5*b18 - 3*b16 - 7*b15 + 5*b14 + 24*b13 - 9*b11 + 12*b10 + 11*b9 + 2*b8 - 25*b7 - 12*b6 - 19*b4 + 15*b3 - 13*b2 - 25*b1 + 12) * q^73 + (5*b19 - 5*b18 - 5*b17 - b16 - 12*b15 - 6*b14 - 5*b12 - 6*b11 + 2*b10 + 21*b9 + 6*b8 - 5*b7 + b6 - 21*b5 - 6*b4 + 24*b3 + 12*b2 - 16*b1 - 7) * q^75 + (11*b14 + 11*b3 - 11*b2 - 11*b1 - 11) * q^77 + (-8*b19 + b18 + 8*b17 + 6*b16 + 14*b15 + 7*b14 - 16*b13 + b12 + 7*b11 - 10*b10 + 23*b9 + 9*b8 + 12*b7 + 5*b6 - 23*b5 + 7*b4 - 46*b3 - 14*b2 + 32*b1 + 47) * q^79 + (2*b19 - 13*b18 + 2*b16 - 16*b15 - 8*b14 + 11*b13 - 4*b11 + 3*b10 - 10*b9 - 6*b8 + b7 - 3*b6 - 19*b4 - 438*b3 - 52*b2 - 60*b1 + 8) * q^81 + (-15*b17 - 15*b16 - 2*b15 + b14 + 12*b13 - 9*b12 + 8*b11 + 8*b10 + 15*b8 + 15*b6 - 9*b5 - 7*b4 - 13*b3 + 84*b2 - 13*b1 + 38) * q^83 + (-b17 - 9*b16 - 18*b14 + 27*b13 - 10*b12 + b11 + b10 + 9*b8 + 15*b6 + 25*b5 - 10*b4 - 9*b3 - 9*b1 - 45) * q^85 + (3*b19 + 12*b18 - 5*b16 - 3*b15 - 14*b14 + 4*b13 - 5*b11 + 15*b10 + 14*b9 - 19*b8 - 12*b7 - 15*b6 - 18*b4 + 221*b3 + 24*b2 + 35*b1 - 11) * q^87 + (5*b19 - 6*b18 - 5*b17 + 11*b16 + 22*b15 + 5*b14 - 21*b13 - 6*b12 + 11*b11 - 13*b10 + 17*b9 + 10*b8 + 27*b7 + 16*b6 - 17*b5 + 11*b4 + 184*b3 - 16*b2 - 46*b1 - 179) * q^89 + (9*b19 + 5*b18 - 6*b17 - 2*b16 - 2*b15 - 13*b14 + 27*b13 - 6*b12 + 8*b11 + 20*b10 - 8*b9 - 15*b8 - 17*b7 - 14*b6 + 17*b5 - 14*b4 + 183*b3 - 94*b2 - 40*b1 - 143) * q^91 + (b19 - 4*b18 - b17 - 8*b16 - 10*b15 - 12*b14 + 28*b13 - 4*b12 - 5*b11 - 7*b10 - 7*b9 - 23*b8 - 4*b7 + b6 + 7*b5 - 5*b4 - 264*b3 + 17*b2 + 120*b1 + 250) * q^93 + (-9*b19 + 38*b18 - 26*b16 + 12*b15 + 13*b14 + 27*b13 + 18*b11 + 26*b10 - 34*b9 - 13*b8 - 27*b7 - 26*b6 - 14*b4 - 277*b3 - 19*b2 - 20*b1 + 1) * q^95 + (11*b17 + b16 + 21*b15 + 25*b14 - 5*b13 + 22*b12 + 10*b11 + 10*b10 - b8 - 14*b6 - 17*b5 + b4 - 20*b3 + 23*b2 - 20*b1 + 149) * q^97 + (-11*b11 - 11*b10 + 11*b2 + 121) * q^99
$\operatorname{Tr}(f)(q)$	$=$	20 * q - 6 * q^3 - 10 * q^5 - 20 * q^7 - 104 * q^9 - 110 * q^11 + 16 * q^13 + 108 * q^15 - 166 * q^17 - 342 * q^19 - 42 * q^21 + 54 * q^23 - 198 * q^25 + 612 * q^27 - 160 * q^29 - 492 * q^31 - 66 * q^33 + 310 * q^35 - 258 * q^37 + 418 * q^39 + 1268 * q^41 - 548 * q^43 - 138 * q^45 - 312 * q^47 + 572 * q^49 + 196 * q^51 - 656 * q^53 + 220 * q^55 - 1476 * q^57 - 934 * q^59 + 228 * q^61 - 1270 * q^63 - 524 * q^65 - 478 * q^67 + 3144 * q^69 + 276 * q^71 + 364 * q^73 - 234 * q^75 - 154 * q^77 + 616 * q^79 - 4082 * q^81 - 264 * q^83 - 992 * q^85 + 2120 * q^87 - 1906 * q^89 - 94 * q^91 + 2984 * q^93 - 2322 * q^95 + 2700 * q^97 + 2288 * q^99

Basis of coefficient ring in terms of a root $\nu$ of x^20 - 4*x^19 + 194*x^18 - 432*x^17 + 24205*x^16 - 47156*x^15 + 1632616*x^14 - 1838210*x^13 + 75876043*x^12 - 82333872*x^11 + 1801672853*x^10 - 471307576*x^9 + 27041915377*x^8 - 14255551602*x^7 + 176709560868*x^6 - 669217680*x^5 + 678547482048*x^4 - 436737562560*x^3 + 226025971776*x^2 - 48231013248*x + 7996651776

$\nu$	$=$	$\beta_1$
$\nu^{2}$	$=$	$\beta_{11} - 37\beta_{3} + \beta_{2} + \beta_1$
$\nu^{3}$	$=$	b17 + b16 - 2*b15 + 2*b14 - 5*b13 - 2*b12 + b11 + b10 - b8 - b6 - 5*b5 + 3*b3 + 70*b2 + 3*b1 - 19
$\nu^{4}$	$=$	-6*b19 + 5*b18 + 6*b17 + 26*b16 + 16*b15 + 31*b14 - 22*b13 + 5*b12 + 8*b11 + 88*b10 + 30*b9 + 14*b8 - 13*b7 - 21*b6 - 30*b5 + 8*b4 + 2652*b3 - 39*b2 - 149*b1 - 2651
$\nu^{5}$	$=$	-136*b19 - 210*b18 + 336*b16 - 198*b15 + 280*b14 + 466*b13 - 388*b11 - 12*b10 + 734*b9 + 616*b8 - 397*b7 + 12*b6 - 186*b4 + 3484*b3 - 5868*b2 - 6346*b1 + 478
$\nu^{6}$	$=$	-987*b17 - 2355*b16 - 1159*b15 - 2435*b14 + 3631*b13 - 355*b12 - 8200*b11 - 8200*b10 + 2355*b8 + 3339*b6 + 4521*b5 + 804*b4 - 1196*b3 - 12393*b2 - 1196*b1 + 217751
$\nu^{7}$	$=$	14269*b19 + 21739*b18 - 14269*b17 - 66597*b16 + 34448*b15 - 44858*b14 + 8482*b13 + 21739*b12 + 17224*b11 - 22447*b10 - 84872*b9 - 25706*b8 + 41258*b7 + 24034*b6 + 84872*b5 + 17224*b4 - 525067*b3 + 27634*b2 + 513769*b1 + 460398
$\nu^{8}$	$=$	121599*b19 - 4518*b18 - 233769*b16 - 127371*b15 - 332337*b14 - 82113*b13 + 630571*b11 + 122853*b10 - 543663*b9 - 566106*b8 + 136260*b7 - 122853*b6 - 250224*b4 - 18624736*b3 + 2103778*b2 + 2308744*b1 - 204966
$\nu^{9}$	$=$	1443637*b17 + 4976029*b16 - 1437284*b15 - 179278*b14 - 6234035*b13 - 2226458*b12 + 3338263*b11 + 3338263*b10 - 4976029*b8 - 3386800*b6 - 9081185*b5 - 789174*b4 + 6413313*b3 + 43259218*b2 + 6413313*b1 - 64005193
$\nu^{10}$	$=$	-13573437*b19 - 2707906*b18 + 13573437*b17 + 65049551*b16 + 25343746*b15 + 62341645*b14 - 36697981*b13 - 2707906*b12 + 12671873*b11 + 56289256*b10 + 61441887*b9 + 24026108*b8 - 14117233*b7 - 26789106*b6 - 61441887*b5 + 12671873*b4 + 1693685589*b3 - 75013518*b2 - 232721663*b1 - 1679623448
$\nu^{11}$	$=$	-146326591*b19 - 224501367*b18 + 175070226*b16 - 116731782*b15 + 559345177*b14 + 568307374*b13 - 502151266*b11 - 107769585*b10 + 941664524*b9 + 734415403*b8 - 387927235*b7 + 107769585*b6 - 8962197*b4 + 6565273318*b3 - 4344524253*b2 - 5020601212*b1 + 676076959
$\nu^{12}$	$=$	-1451952870*b17 - 4727356764*b16 - 1202407774*b15 - 1833479240*b14 + 5358428230*b13 + 552080849*b12 - 6896244124*b11 - 6896244124*b10 + 4727356764*b8 + 4475880945*b6 + 6752506080*b5 + 1754488623*b4 - 3524948990*b3 - 17436578220*b2 - 3524948990*b1 + 158340131897
$\nu^{13}$	$=$	14918189656*b19 + 22390586023*b18 - 14918189656*b17 - 76397571852*b16 + 18512681540*b15 - 54006985829*b14 + 6793048372*b13 + 22390586023*b12 + 9256340770*b11 - 29649150964*b10 - 96276776771*b9 - 16049389142*b8 + 37207984574*b7 + 27951643804*b6 + 96276776771*b5 + 9256340770*b4 - 802564541647*b3 + 44750645059*b2 + 413359535362*b1 + 754868225712
$\nu^{14}$	$=$	152093627136*b19 + 79318393848*b18 - 233867751861*b16 - 111296434638*b15 - 523470114516*b14 - 221558851392*b13 + 561122722579*b11 + 190614828486*b10 - 729254399724*b9 - 757337866377*b8 + 165995676693*b7 - 190614828486*b6 - 301911263124*b4 - 14370241908133*b3 + 2691431285563*b2 + 3103604965441*b1 - 412173679878
$\nu^{15}$	$=$	1526872041136*b17 + 6362779623634*b16 - 715604115908*b15 - 668564953639*b14 - 6409818785903*b13 - 2220636664883*b12 + 4636953019495*b11 + 4636953019495*b10 - 6362779623634*b8 - 4370550385996*b6 - 9779779201199*b5 - 1505032548975*b4 + 7078383739542*b3 + 34989499645765*b2 + 7078383739542*b1 - 89750328272596
$\nu^{16}$	$=$	-15755378016579*b19 - 9980471229007*b18 + 15755378016579*b17 + 79407426350690*b16 + 20342453551738*b15 + 69426955121683*b14 - 33036425182516*b13 - 9980471229007*b12 + 10171226775869*b11 + 46077373057561*b10 + 77774158745319*b9 + 22865198406647*b8 - 18151857531373*b7 - 28323084307242*b6 - 77774158745319*b5 + 10171226775869*b4 + 1419570175581132*b3 - 79598181897552*b2 - 284740839994199*b1 - 1396895732721347
$\nu^{17}$	$=$	-156570936098419*b19 - 219863389279494*b18 + 155429834177496*b16 - 53624415313710*b15 + 661879491863527*b14 + 549264933211453*b13 - 548461050953599*b11 - 166238973965784*b10 + 990734491754447*b9 + 817309326041023*b8 - 346692232604395*b7 + 166238973965784*b6 + 112614558652074*b4 + 8620135149424717*b3 - 3913030144310679*b2 - 4628534051487916*b1 + 715503907177237
$\nu^{18}$	$=$	-1622068008586602*b17 - 6009001647063972*b16 - 924581240022598*b15 - 1067744532580043*b14 + 6152164939621417*b13 + 1172523572115935*b12 - 6574005251741755*b11 - 6574005251741755*b10 + 6009001647063972*b8 + 4994244713352204*b6 + 8216420143275966*b5 + 2097104812138533*b4 - 5084420407041374*b3 - 21915483159785868*b2 - 5084420407041374*b1 + 140741316328785557
$\nu^{19}$	$=$	16065497287644790*b19 + 21785484810745678*b18 - 16065497287644790*b17 - 84182615967189174*b16 + 7692849688436348*b15 - 62397131156443496*b14 + 12034851563306506*b13 + 21785484810745678*b12 + 3846424844218174*b11 - 34313415609943288*b10 - 100291614172371794*b9 - 15881276407524680*b8 + 33788759298049541*b7 + 29942334453831367*b6 + 100291614172371794*b5 + 3846424844218174*b4 - 990047127708359692*b3 + 58550706312225322*b2 + 386799252670076656*b1 + 948533941645871160

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/308\mathbb{Z}\right)^\times$.

$n$	$45$	$57$	$155$
$\chi(n)$	$-1 + \beta_{3}$	$1$	$1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

177.1

−4.54700	+	7.87564i
−3.86353	+	6.69182i
−2.05350	+	3.55676i
−1.14233	+	1.97858i
0.138808	−	0.240422i
0.192948	−	0.334196i
1.43296	−	2.48195i
2.62470	−	4.54611i
4.16059	−	7.20635i
5.05636	−	8.75787i
−4.54700	−	7.87564i
−3.86353	−	6.69182i
−2.05350	−	3.55676i
−1.14233	−	1.97858i
0.138808	+	0.240422i
0.192948	+	0.334196i
1.43296	+	2.48195i
2.62470	+	4.54611i
4.16059	+	7.20635i
5.05636	+	8.75787i

0

−5.04700

−

8.74166i

0

3.77667

−

6.54138i

0

17.9318

+

4.63153i

0

−37.4445

+

64.8557i

0

177.2

0

−4.36353

−

7.55785i

0

−6.34995

+

10.9984i

0

−15.6613

+

9.88547i

0

−24.5807

+

42.5751i

0

177.3

0

−2.55350

−

4.42278i

0

3.75702

−

6.50735i

0

−2.07070

−

18.4041i

0

0.459316

−

0.795558i

0

177.4

0

−1.64233

−

2.84460i

0

−10.0843

+

17.4664i

0

15.8994

−

9.49786i

0

8.10549

−

14.0391i

0

177.5

0

−0.361192

−

0.625604i

0

5.92984

−

10.2708i

0

−5.49821

−

17.6853i

0

13.2391

−

22.9308i

0

177.6

0

−0.307052

−

0.531830i

0

−0.388866

+

0.673536i

0

6.69839

+

17.2665i

0

13.3114

−

23.0561i

0

177.7

0

0.932956

+

1.61593i

0

−7.91970

+

13.7173i

0

−18.5160

+

0.398871i

0

11.7592

−

20.3675i

0

177.8

0

2.12470

+

3.68009i

0

7.32246

−

12.6829i

0

−17.0162

+

7.31095i

0

4.47130

−

7.74452i

0

177.9

0

3.66059

+

6.34032i

0

3.91736

−

6.78507i

0

16.2143

+

8.94974i

0

−13.2998

+

23.0359i

0

177.10

0

4.55636

+

7.89185i

0

−4.96059

+

8.59200i

0

−7.98145

−

16.7122i

0

−28.0208

+

48.5335i

0

221.1

0

−5.04700

+

8.74166i

0

3.77667

+

6.54138i

0

17.9318

−

4.63153i

0

−37.4445

−

64.8557i

0

221.2

0

−4.36353

+

7.55785i

0

−6.34995

−

10.9984i

0

−15.6613

−

9.88547i

0

−24.5807

−

42.5751i

0

221.3

0

−2.55350

+

4.42278i

0

3.75702

+

6.50735i

0

−2.07070

+

18.4041i

0

0.459316

+

0.795558i

0

221.4

0

−1.64233

+

2.84460i

0

−10.0843

−

17.4664i

0

15.8994

+

9.49786i

0

8.10549

+

14.0391i

0

221.5

0

−0.361192

+

0.625604i

0

5.92984

+

10.2708i

0

−5.49821

+

17.6853i

0

13.2391

+

22.9308i

0

221.6

0

−0.307052

+

0.531830i

0

−0.388866

−

0.673536i

0

6.69839

−

17.2665i

0

13.3114

+

23.0561i

0

221.7

0

0.932956

−

1.61593i

0

−7.91970

−

13.7173i

0

−18.5160

−

0.398871i

0

11.7592

+

20.3675i

0

221.8

0

2.12470

−

3.68009i

0

7.32246

+

12.6829i

0

−17.0162

−

7.31095i

0

4.47130

+

7.74452i

0

221.9

0

3.66059

−

6.34032i

0

3.91736

+

6.78507i

0

16.2143

−

8.94974i

0

−13.2998

−

23.0359i

0

221.10

0

4.55636

−

7.89185i

0

−4.96059

−

8.59200i

0

−7.98145

+

16.7122i

0

−28.0208

−

48.5335i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	308.4.i.a	✓	20
7.c	even	3	1	inner	308.4.i.a	✓	20
7.c	even	3	1		2156.4.a.m		10
7.d	odd	6	1		2156.4.a.j		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
308.4.i.a	✓	20	1.a	even	1	1	trivial
308.4.i.a	✓	20	7.c	even	3	1	inner
2156.4.a.j		10	7.d	odd	6	1
2156.4.a.m		10	7.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^20 + 6*T3^19 + 205*T3^18 + 738*T3^17 + 25156*T3^16 + 81030*T3^15 + 1743169*T3^14 + 3638202*T3^13 + 79006747*T3^12 + 163524492*T3^11 + 1981604563*T3^10 + 2439791418*T3^9 + 29541111199*T3^8 + 39561045522*T3^7 + 229096545946*T3^6 + 86531904882*T3^5 + 623840847004*T3^4 + 680180357640*T3^3 + 747603272850*T3^2 + 325763411022*T3 + 120251513529

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{20}$
$3$	T^20 + 6*T^19 + 205*T^18 + 738*T^17 + 25156*T^16 + 81030*T^15 + 1743169*T^14 + 3638202*T^13 + 79006747*T^12 + 163524492*T^11 + 1981604563*T^10 + 2439791418*T^9 + 29541111199*T^8 + 39561045522*T^7 + 229096545946*T^6 + 86531904882*T^5 + 623840847004*T^4 + 680180357640*T^3 + 747603272850*T^2 + 325763411022*T + 120251513529
$5$	T^20 + 10*T^19 + 774*T^18 + 1368*T^17 + 323343*T^16 - 326448*T^15 + 89692080*T^14 - 369783390*T^13 + 17456215347*T^12 - 92525029544*T^11 + 2453532574615*T^10 - 15181521340356*T^9 + 244661982886377*T^8 - 1385216236311066*T^7 + 16171277683615668*T^6 - 81929282470873200*T^5 + 663495732633788928*T^4 - 1941501279704058432*T^3 + 8198184128983550784*T^2 + 6034238828770739328*T + 5845284198667489536
$7$	T^20 + 20*T^19 - 86*T^18 - 7376*T^17 - 128323*T^16 + 1614262*T^15 + 89761511*T^14 + 851673508*T^13 + 161886053*T^12 - 119162345498*T^11 - 5272744024051*T^10 - 40872684505814*T^9 + 19045732249397*T^8 + 34368098034143356*T^7 + 1242414853346720711*T^6 + 7663808138163607066*T^5 - 208962918124662547027*T^4 - 4119834293478302835632*T^3 - 16475985898728711638486*T^2 + 1314247247270685602790860*T + 22539340290692258087863249
$11$	$(T^{2} + 11 T + 121)^{10}$