LMFDB - Newform orbit 2175.4.a.u

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,4,Mod(1,2175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2175 = 3 \cdot 5^{2} \cdot 29$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$128.329154262$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 3 x^{15} - 92 x^{14} + 239 x^{13} + 3416 x^{12} - 7461 x^{11} - 65355 x^{10} + 115826 x^{9} + \cdots - 891088$ x^16 - 3x^15 - 92x^14 + 239x^13 + 3416x^12 - 7461x^11 - 65355x^10 + 115826x^9 + 679301x^8 - 932552x^7 - 3681931x^6 + 3646901x^5 + 8804276x^4 - 5805166x^3 - 4956752x^2 + 4507400*x - 891088
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + \beta_1 + 4) q^{4} + 3 \beta_1 q^{6} + (\beta_{7} + 1) q^{7} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 5) q^{8} + 9 q^{9} + ( - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{11}+ \cdots + ( - 9 \beta_{4} + 9 \beta_{2} + \cdots + 9) q^{99}+O(q^{100})$ q - b1 * q^2 - 3 * q^3 + (b2 + b1 + 4) * q^4 + 3b1 q^6 + (b7 + 1) * q^7 + (-b3 - b2 - 4b1 - 5) q^8 + 9 * q^9 + (-b4 + b2 + b1 + 1) * q^11 + (-3b2 - 3b1 - 12) * q^12 + (-b10 + b7 - b2 - 3b1 - 2) q^13 + (b12 + b11 - b7 - b3 + b2 - b1 + 2) * q^14 + (b6 - b5 + 2b3 + 3b2 + 11b1 + 9) q^16 + (b6 + b2 - 3b1 - 9) q^17 - 9b1 q^18 + (b14 - b9 - b7 + b2 - b1 + 8) * q^19 + (-3b7 - 3) q^21 + (-b15 - b13 - b12 - b11 - 2b10 + b9 + b6 - 2b5 - b4 - b3 - 7b1 - 7) q^22 + (-b13 + b12 + 2b11 - b10 + b9 + b5 - 2b3 - 2b2 - 12b1 - 1) * q^23 + (3b3 + 3b2 + 12b1 + 15) q^24 + (2b12 + 2b10 - b9 - b7 + b6 + b5 - b4 + 3b2 + 10b1 + 26) * q^26 - 27 * q^27 + (-b15 - b12 - b11 + b10 - b9 + 5b7 + b6 - 2b5 + b4 + b3 + 3b2 - 8b1 + 10) * q^28 - 29 * q^29 + (b14 - b12 + b10 - b8 + 5b7 - 2b5 + b4 + 4b3 + 7b1 + 21) * q^31 + (b15 - 2b14 + b12 + 2b11 - 2b10 + b9 - b8 + b7 - 3b6 + 2b5 - 3b4 - 7b3 - 12b2 - 14b1 - 71) q^32 + (3b4 - 3b2 - 3b1 - 3) q^33 + (b15 - 2b14 + b11 - 4b10 - 2b7 - b6 + b5 - 2b4 - 9b3 + 6b2 + 7b1 + 46) q^34 + (9b2 + 9b1 + 36) * q^36 + (3b15 + 3b13 + b12 + b10 - 3b9 + b8 + 2b7 - b6 + 3b5 - 7b2 + 10b1 - 40) q^37 + (-b15 + 2b14 + 2b13 + b12 - 3b11 + 4b10 - 2b9 + 4b7 + b6 - b4 + 3b3 + 2b2 - 12b1 + 13) q^38 + (3b10 - 3b7 + 3b2 + 9b1 + 6) * q^39 + (-b15 + b14 - b13 - 3b12 + 2b11 - 3b10 + 2b9 + b8 + 5b7 - 2b6 + 3b5 - b4 - 2b3 - 8b2 - 3b1 + 49) * q^41 + (-3b12 - 3b11 + 3b7 + 3b3 - 3b2 + 3b1 - 6) * q^42 + (3b15 + b14 + b12 + 2b11 + b10 + 2b9 - b7 + 2b4 - 2b3 + 5b2 - 21b1 + 44) q^43 + (-2b14 + 2b12 - 2b10 + 5b9 - 5b7 + b6 - 2b5 - 2b4 - 7b3 + 8b2 + 9b1 + 52) * q^44 + (-b15 - 4b14 + b13 - 2b11 - 2b10 + 2b9 + 4b8 + 14b7 + b6 - 3b5 - 3b4 + 4b3 + 12b2 + 28b1 + 121) q^46 + (-3b15 + 5b14 - b13 + b12 - 2b11 + 3b10 + 3b8 + 4b7 + 3b6 - 3b5 + 2b4 + 2b3 - 10b2 - 5b1 - 69) * q^47 + (-3b6 + 3b5 - 6b3 - 9b2 - 33b1 - 27) q^48 + (-3b15 + 3b14 + b13 - 2b12 - 4b11 + b9 + 3b7 - b6 - b5 - 2b4 - 4b3 + 8b2 - 2b1 + 47) q^49 + (-3b6 - 3b2 + 9b1 + 27) q^51 + (-b15 + 2b14 - 5b12 + b11 - 4b9 - 2b8 + 2b7 - 2b6 + 4b4 - 2b3 - b2 - 27b1 - 66) q^52 + (3b15 - 6b14 + 3b13 + 2b11 - 3b10 - 2b9 + 6b7 - 5b6 + 5b5 - 4b4 - 2b3 + 12b2 - 8b1 - 45) q^53 + 27b1 q^54 + (3b15 + 2b13 + b12 + 3b11 + b10 - 3b8 + b7 - 4b6 + 4b5 - 2b4 - 10b3 + 8b2 - 9b1 + 114) * q^56 + (-3b14 + 3b9 + 3b7 - 3b2 + 3b1 - 24) q^57 + 29b1 q^58 + (2b15 + b13 - 4b12 + 4b11 + 2b10 - 2b9 - b8 - 4b7 + 3b5 + 5b4 - 4b3 + 15b2 + 4b1 + 22) q^59 + (2b15 - b14 - 2b13 + b12 - 5b10 - 2b9 - 3b8 + b7 - 2b6 + 6b5 + b4 - 10b2 - 5b1 + 57) q^61 + (-b15 + 2b13 + 8b12 + 3b11 + 5b10 + 3b9 + b8 - 4b7 - 3b6 - 2b5 - 4b4 - 11b3 - 23b2 - 38b1 - 88) * q^62 + (9b7 + 9) q^63 + (-4b15 - 2b14 - 6b13 - 3b12 - 7b11 + b10 + b9 + 3b8 + 9b7 + 7b6 - 7b5 + 3b4 + 20b3 + 24b2 + 101b1 - 3) q^64 + (3b15 + 3b13 + 3b12 + 3b11 + 6b10 - 3b9 - 3b6 + 6b5 + 3b4 + 3b3 + 21b1 + 21) q^66 + (2b15 + 3b14 - b13 + 4b12 + 2b11 + 5b10 - 3b9 - 7b8 - 3b7 + b6 + b5 + 4b4 - 2b3 - 3b2 - 63) q^67 + (2b15 - 6b14 - 4b13 - 4b11 + 2b10 - b9 + b7 + 6b6 - 6b5 - 3b4 + 3b3 + 40b2 - 43b1 + 12) q^68 + (3b13 - 3b12 - 6b11 + 3b10 - 3b9 - 3b5 + 6b3 + 6b2 + 36b1 + 3) q^69 + (b15 - 5b13 + 2b12 - 8b11 + 5b10 - 2b9 - 10b7 + 7b6 - 5b5 + 2b4 + 4b3 + 10b2 - 32b1 + 81) * q^71 + (-9b3 - 9b2 - 36b1 - 45) q^72 + (-8b15 + 2b14 - 3b13 - 3b12 - 4b11 - 3b10 + 5b9 + 10b7 + 3b6 - 9b5 - 3b4 + 8b3 + 44b2 - 36b1 + 43) * q^73 + (2b15 + 4b14 - 3b13 + 2b11 + 6b10 - 12b9 - 5b8 - 12b7 + 4b6 + 7b5 + 15b4 + 8b3 + 5b2 + 78b1 - 120) * q^74 + (-6b15 + 8b14 - b13 - 3b12 + 2b11 - 3b10 - b9 - 6b8 - 2b7 - b6 - b5 + 11b4 + 2b3 + 6b2 - 12b1 + 131) q^76 + (-6b14 - 6b13 - b12 + 4b11 - 2b10 - 5b9 - b8 - b7 + 2b6 + 4b5 + 4b4 - 8b3 - 5b2 + 2b1 - 147) q^77 + (-6b12 - 6b10 + 3b9 + 3b7 - 3b6 - 3b5 + 3b4 - 9b2 - 30b1 - 78) q^78 + (7b15 - 6b14 + b13 + 2b10 + 2b9 + 16b7 - 5b6 + 5b5 - 7b4 + 26b2 + 56b1 + 91) * q^79 + 81 * q^81 + (-3b15 - 4b14 + 6b12 - 5b11 + 9b9 + 11b8 - 14b7 + 5b6 - 4b5 - 7b4 + 17b3 - b2 + 13b1 - 38) * q^82 + (-4b15 + 8b14 + b13 - b12 + 6b11 - 3b10 + 3b9 - 4b6 + 3b5 - 8b4 + 6b3 + 26b2 - 50b1 - 39) q^83 + (3b15 + 3b12 + 3b11 - 3b10 + 3b9 - 15b7 - 3b6 + 6b5 - 3b4 - 3b3 - 9b2 + 24b1 - 30) * q^84 + (-6b14 + b13 - 3b12 - 3b11 + b10 + 6b9 + 4b8 + 4b7 - 7b5 - b4 - 11b3 + 32b2 - 72b1 + 290) * q^86 + 87 * q^87 + (4b15 - 2b14 - b13 - 7b12 + 3b11 + 4b10 - b8 + 9b7 + 6b4 + 28b2 - 67b1 - 14) q^88 + (-7b15 - b14 + b13 + 2b12 - 5b10 + 5b9 + 8b8 - 4b7 - b6 - b5 - 5b4 - 8b3 + 18b2 + 22b1 + 109) * q^89 + (-3b15 - b14 + 9b13 + 3b12 - 4b11 - 13b10 + 6b9 + 7b8 + 8b7 - 3b6 - 5b5 - 11b4 - 13b2 - 18b1 + 369) q^91 + (6b15 - 2b14 - 3b13 + b12 + 18b11 - 3b10 - b9 - 12b8 - 24b7 - 10b6 + 7b5 + 4b4 - 40b3 - 20b2 - 162b1 - 181) q^92 + (-3b14 + 3b12 - 3b10 + 3b8 - 15b7 + 6b5 - 3b4 - 12b3 - 21b1 - 63) q^93 + (-5b15 + 14b14 + 9b13 + 4b12 + 10b11 + 2b10 + 2b9 - 3b8 + 11b7 - 13b6 + 5b5 - 2b4 + 2b3 - 2b2 + 119b1 + 11) q^94 + (-3b15 + 6b14 - 3b12 - 6b11 + 6b10 - 3b9 + 3b8 - 3b7 + 9b6 - 6b5 + 9b4 + 21b3 + 36b2 + 42b1 + 213) * q^96 + (-3b15 - b14 + 2b13 - 11b12 - 11b10 + 4b8 + 3b7 - 4b6 + 8b5 + 6b4 + 20b3 + 7b2 + 35b1 + 76) * q^97 + (-12b15 + 20b14 - 2b13 - 3b12 - 10b11 + 5b10 - 22b7 + 12b6 - 15b5 + 2b4 + 22b3 + 38b2 - 98b1 + 63) q^98 + (-9b4 + 9b2 + 9b1 + 9) q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 3 q^{2} - 48 q^{3} + 65 q^{4} + 9 q^{6} + 12 q^{7} - 90 q^{8} + 144 q^{9} + 22 q^{11} - 195 q^{12} - 42 q^{13} + 39 q^{14} + 181 q^{16} - 152 q^{17} - 27 q^{18} + 128 q^{19} - 36 q^{21} - 129 q^{22}+ \cdots + 198 q^{99}+O(q^{100})$ 16 * q - 3 * q^2 - 48 * q^3 + 65 * q^4 + 9 * q^6 + 12 * q^7 - 90 * q^8 + 144 * q^9 + 22 * q^11 - 195 * q^12 - 42 * q^13 + 39 * q^14 + 181 * q^16 - 152 * q^17 - 27 * q^18 + 128 * q^19 - 36 * q^21 - 129 * q^22 - 58 * q^23 + 270 * q^24 + 462 * q^26 - 432 * q^27 + 122 * q^28 - 464 * q^29 + 331 * q^31 - 1153 * q^32 - 66 * q^33 + 769 * q^34 + 585 * q^36 - 585 * q^37 + 174 * q^38 + 126 * q^39 + 715 * q^41 - 117 * q^42 + 608 * q^43 + 881 * q^44 + 2004 * q^46 - 1121 * q^47 - 543 * q^48 + 700 * q^49 + 456 * q^51 - 1183 * q^52 - 750 * q^53 + 81 * q^54 + 1760 * q^56 - 384 * q^57 + 87 * q^58 + 301 * q^59 + 867 * q^61 - 1389 * q^62 + 108 * q^63 + 170 * q^64 + 387 * q^66 - 1013 * q^67 + 55 * q^68 + 174 * q^69 + 1234 * q^71 - 810 * q^72 + 464 * q^73 - 1737 * q^74 + 1946 * q^76 - 2343 * q^77 - 1386 * q^78 + 1506 * q^79 + 1296 * q^81 - 418 * q^82 - 864 * q^83 - 366 * q^84 + 4382 * q^86 + 1392 * q^87 - 592 * q^88 + 1844 * q^89 + 5995 * q^91 - 3338 * q^92 - 993 * q^93 + 440 * q^94 + 3459 * q^96 + 1176 * q^97 + 704 * q^98 + 198 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 3 x^{15} - 92 x^{14} + 239 x^{13} + 3416 x^{12} - 7461 x^{11} - 65355 x^{10} + 115826 x^{9} + \cdots - 891088$ x^16 - 3*x^15 - 92*x^14 + 239*x^13 + 3416*x^12 - 7461*x^11 - 65355*x^10 + 115826*x^9 + 679301*x^8 - 932552*x^7 - 3681931*x^6 + 3646901*x^5 + 8804276*x^4 - 5805166*x^3 - 4956752*x^2 + 4507400*x - 891088 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - \nu - 12$ v^2 - v - 12
$\beta_{3}$	$=$	$\nu^{3} - \nu^{2} - 19\nu + 7$ v^3 - v^2 - 19*v + 7
$\beta_{4}$	$=$	$( 75\!\cdots\!71 \nu^{15} + \cdots + 10\!\cdots\!16 ) / 38\!\cdots\!60$ (7525158985547571v^15 - 18468864160634930v^14 - 701133986851260822v^13 + 1406217529451714573v^12 + 26392938610685868355v^11 - 41040284226631534166v^10 - 512415974799253121243v^9 + 572695110229039619317v^8 + 5409074550840707386022v^7 - 3815619039608166671546v^6 - 29766900555452876344069v^5 + 9648823604698500662844v^4 + 71733397592254137307258v^3 - 520319671267243624892v^2 - 37987844289224340726288*v + 10525191374852782163616) / 3851148466031842960
$\beta_{5}$	$=$	$( - 49\!\cdots\!79 \nu^{15} + \cdots - 95\!\cdots\!04 ) / 15\!\cdots\!40$ (-49723980721369079v^15 + 125040299747916050v^14 + 4642030527363293238v^13 - 9657902670072152187v^12 - 175076174372711814095v^11 + 287937885248299799024v^10 + 3405692242527261740757v^9 - 4158048597137807846933v^8 - 36035940949094115529748v^7 + 29529428913909209358764v^6 + 199165839935209265990841v^5 - 88515897809133255608246v^4 - 486297285152711198779442v^3 + 62204167558916175899528v^2 + 281094755325319310641032*v - 95769897451626676641904) / 15404593864127371840
$\beta_{6}$	$=$	$( - 49\!\cdots\!79 \nu^{15} + \cdots - 94\!\cdots\!24 ) / 15\!\cdots\!40$ (-49723980721369079v^15 + 125040299747916050v^14 + 4642030527363293238v^13 - 9657902670072152187v^12 - 175076174372711814095v^11 + 287937885248299799024v^10 + 3405692242527261740757v^9 - 4158048597137807846933v^8 - 36035940949094115529748v^7 + 29529428913909209358764v^6 + 199165839935209265990841v^5 - 88500493215269128236406v^4 - 486328094340439453523122v^3 + 61819052712312991603528v^2 + 281556893141243131796232*v - 94583743724088869010224) / 15404593864127371840
$\beta_{7}$	$=$	$( 57\!\cdots\!17 \nu^{15} + \cdots + 10\!\cdots\!32 ) / 15\!\cdots\!40$ (57661152840716017v^15 - 144061189341797950v^14 - 5375210866203602794v^13 + 11074121409120960781v^12 + 202390480730967841145v^11 - 327927926446801111232v^10 - 3929234217723994980211v^9 + 4687298399161483836739v^8 + 41471550261716150424284v^7 - 32713031899734386469732v^6 - 228393513818366110358303v^5 + 94199397876051920337658v^4 + 553985791275737931034126v^3 - 53114911856106795770744v^2 - 311721442179437629386456*v + 101006864468857949303632) / 15404593864127371840
$\beta_{8}$	$=$	$( 87\!\cdots\!21 \nu^{15} + \cdots + 17\!\cdots\!56 ) / 15\!\cdots\!40$ (87971212083953921v^15 - 225347675547325030v^14 - 8191993439632895642v^13 + 17422534097777849373v^12 + 308176102731075211505v^11 - 520373363206756084536v^10 - 5979772361503068944643v^9 + 7541238080965151013627v^8 + 63121365032970733061172v^7 - 53969249731469977672836v^6 - 348126691155603392171839v^5 + 165241174143534171398274v^4 + 848940542490429489017838v^3 - 128708450318016921937352v^2 - 493313130200950891613368*v + 172549121831675287415056) / 15404593864127371840
$\beta_{9}$	$=$	$( 24\!\cdots\!40 \nu^{15} + \cdots + 40\!\cdots\!36 ) / 38\!\cdots\!96$ (2474939781126740v^15 - 6011967499074915v^14 - 231358346517863096v^13 + 461121600919030951v^12 + 8733201371706395151v^11 - 13604075081962913506v^10 - 169896947292905713987v^9 + 193148173231065558213v^8 + 1795613053320818664384v^7 - 1329190648502546413705v^6 - 9890206255881311717230v^5 + 3674688626536218396698v^4 + 23922210933156124720594v^3 - 1482881531944752511728v^2 - 13187023724941864188820*v + 4060968989084822253736) / 385114846603184296
$\beta_{10}$	$=$	$( - 64\!\cdots\!07 \nu^{15} + \cdots - 11\!\cdots\!12 ) / 77\!\cdots\!20$ (-64125927134749907v^15 + 159895030256962110v^14 + 5979978267094794934v^13 - 12287730432291409711v^12 - 225247273751768526895v^11 + 363617500715981114132v^10 + 4374747412141896585121v^9 - 5189317561591878866829v^8 - 46192694536010463145544v^7 + 36078475689612078459172v^6 + 254472817798552868938853v^5 - 102664278421917537107338v^4 - 617018373175601894827626v^3 + 52968611248279540787744v^2 + 344849011248423215055096*v - 110035949619285316680912) / 7702296932063685920
$\beta_{11}$	$=$	$( - 97\!\cdots\!33 \nu^{15} + \cdots - 16\!\cdots\!88 ) / 77\!\cdots\!20$ (-97836377803514133v^15 + 242388137812385290v^14 + 9129448719751065426v^13 - 18624027280728329169v^12 - 344070155989383595425v^11 + 550984168104283120788v^10 + 6685444961785829427959v^9 - 7860317918152478587331v^8 - 70611587449346023081376v^7 + 54611950227177649067868v^6 + 389049287259944491814547v^5 - 155126502413351096237622v^4 - 943387818702785719342334v^3 + 79014274940156342590016v^2 + 527587599325395945281144*v - 167689465759822932212688) / 7702296932063685920
$\beta_{12}$	$=$	$( 11\!\cdots\!91 \nu^{15} + \cdots + 19\!\cdots\!96 ) / 77\!\cdots\!20$ (112205819633697091v^15 - 279226330054419650v^14 - 10463607092947783182v^13 + 21451096671747846023v^12 + 394124428905976970195v^11 - 634553344418183831816v^10 - 7654380925764182143913v^9 + 9052831379800760825617v^8 + 80817866828113589985692v^7 - 62923902337846969717996v^6 - 445203485129173465605869v^5 + 179065657757432016533734v^4 + 1079579590569776127658778v^3 - 92632431705117452502392v^2 - 604139453837199193459848*v + 192633256360833012446896) / 7702296932063685920
$\beta_{13}$	$=$	$( - 11\!\cdots\!03 \nu^{15} + \cdots - 20\!\cdots\!08 ) / 77\!\cdots\!20$ (-113544981007880103v^15 + 284725439383133050v^14 + 10587508998406989066v^13 - 21930863286715794539v^12 - 398750118731754633075v^11 + 651321878239102365388v^10 + 7743554458013892501029v^9 - 9352760911196051875241v^8 - 81761539063879003700096v^7 + 65816040013825264697348v^6 + 450610328255889048357957v^5 - 193381636302394171872942v^4 - 1095239750850298783317874v^3 + 122714855481147967385736v^2 + 623647422887909348640664*v - 206646897257072472488208) / 7702296932063685920
$\beta_{14}$	$=$	$( 33\!\cdots\!91 \nu^{15} + \cdots + 60\!\cdots\!96 ) / 15\!\cdots\!40$ (337364356611789191v^15 - 841403545559897290v^14 - 31472283617399877662v^13 + 64762586829933601923v^12 + 1185827113175794347215v^11 - 1921145190734947043336v^10 - 23036244499893591082973v^9 + 27530704989458854671397v^8 + 243279723002784361380092v^7 - 192942417353100420990276v^6 - 1340627534501736208504369v^5 + 560729167002875335148614v^4 + 3255168055758419710578338v^3 - 333405827236261228526792v^2 - 1840512147691375293043528*v + 602816938810735048758896) / 15404593864127371840
$\beta_{15}$	$=$	$( 55\!\cdots\!81 \nu^{15} + \cdots + 99\!\cdots\!56 ) / 15\!\cdots\!40$ (557049271734611181v^15 - 1391720717654881590v^14 - 51949260887095876562v^13 + 107088424045363262873v^12 + 1956793553577465198645v^11 - 3175478321139451493696v^10 - 38004477053961170368583v^9 + 45482054123730299006807v^8 + 401297530302099991696172v^7 - 318517519633791710334036v^6 - 2211331275016766167933939v^5 + 924546344891495523404994v^4 + 5369732863926323685236278v^3 - 547125713080694448583352v^2 - 3036159125555796270458808*v + 993806077237860058005456) / 15404593864127371840

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + \beta _1 + 12$ b2 + b1 + 12
$\nu^{3}$	$=$	$\beta_{3} + \beta_{2} + 20\beta _1 + 5$ b3 + b2 + 20*b1 + 5
$\nu^{4}$	$=$	$\beta_{6} - \beta_{5} + 2\beta_{3} + 27\beta_{2} + 35\beta _1 + 233$ b6 - b5 + 2b3 + 27b2 + 35*b1 + 233
$\nu^{5}$	$=$	$- \beta_{15} + 2 \beta_{14} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} + \cdots + 231$ -b15 + 2b14 - b12 - 2b11 + 2b10 - b9 + b8 - b7 + 3b6 - 2b5 + 3b4 + 39b3 + 44b2 + 462*b1 + 231
$\nu^{6}$	$=$	$- 4 \beta_{15} - 2 \beta_{14} - 6 \beta_{13} - 3 \beta_{12} - 7 \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 5221$ -4b15 - 2b14 - 6b13 - 3b12 - 7b11 + b10 + b9 + 3b8 + 9b7 + 47b6 - 47b5 + 3b4 + 100b3 + 720b2 + 1117*b1 + 5221
$\nu^{7}$	$=$	$- 61 \beta_{15} + 94 \beta_{14} - 12 \beta_{13} - 58 \beta_{12} - 127 \beta_{11} + 105 \beta_{10} + \cdots + 8341$ -61b15 + 94b14 - 12b13 - 58b12 - 127b11 + 105b10 - 57b9 + 49b8 + 4b7 + 171b6 - 129b5 + 145b4 + 1281b3 + 1590b2 + 11616*b1 + 8341
$\nu^{8}$	$=$	$- 243 \beta_{15} - 82 \beta_{14} - 320 \beta_{13} - 230 \beta_{12} - 488 \beta_{11} + 114 \beta_{10} + \cdots + 127456$ -243b15 - 82b14 - 320b13 - 230b12 - 488b11 + 114b10 + 24b9 + 198b8 + 605b7 + 1730b6 - 1720b5 + 230b4 + 3915b3 + 19632b2 + 34739*b1 + 127456
$\nu^{9}$	$=$	$- 2605 \beta_{15} + 3262 \beta_{14} - 764 \beta_{13} - 2455 \beta_{12} - 5535 \beta_{11} + \cdots + 277339$ -2605b15 + 3262b14 - 764b13 - 2455b12 - 5535b11 + 4010b10 - 2219b9 + 1914b8 + 2051b7 + 7107b6 - 5789b5 + 5245b4 + 40251b3 + 53524b2 + 308689*b1 + 277339
$\nu^{10}$	$=$	$- 10575 \beta_{15} - 2152 \beta_{14} - 12254 \beta_{13} - 11197 \beta_{12} - 23141 \beta_{11} + \cdots + 3296347$ -10575b15 - 2152b14 - 12254b13 - 11197b12 - 23141b11 + 6637b10 - 328b9 + 8976b8 + 27954b7 + 58707b6 - 58004b5 + 11158b4 + 140483b3 + 548334b2 + 1070745*b1 + 3296347
$\nu^{11}$	$=$	$- 96349 \beta_{15} + 101188 \beta_{14} - 33842 \beta_{13} - 91445 \beta_{12} - 208426 \beta_{11} + \cdots + 8897540$ -96349b15 + 101188b14 - 33842b13 - 91445b12 - 208426b11 + 136877b10 - 74989b9 + 69150b8 + 124326b7 + 261597b6 - 224148b5 + 171935b4 + 1247032b3 + 1746695b2 + 8526359*b1 + 8897540
$\nu^{12}$	$=$	$- 404233 \beta_{15} - 38158 \beta_{14} - 415796 \beta_{13} - 450371 \beta_{12} - 931600 \beta_{11} + \cdots + 88851805$ -404233b15 - 38158b14 - 415796b13 - 450371b12 - 931600b11 + 300909b10 - 54061b9 + 350238b8 + 1103898b7 + 1922237b6 - 1888794b5 + 450209b4 + 4820029b3 + 15643231b2 + 32943770*b1 + 88851805
$\nu^{13}$	$=$	$- 3314995 \beta_{15} + 2981458 \beta_{14} - 1297704 \beta_{13} - 3184681 \beta_{12} - 7301092 \beta_{11} + \cdots + 280930931$ -3314995b15 + 2981458b14 - 1297704b13 - 3184681b12 - 7301092b11 + 4447121b10 - 2378111b9 + 2396684b8 + 5471124b7 + 9063610b6 - 8035529b5 + 5406051b4 + 38465410b3 + 56123337b2 + 242245629*b1 + 280930931
$\nu^{14}$	$=$	$- 14448930 \beta_{15} - 103100 \beta_{14} - 13361028 \beta_{13} - 16403098 \beta_{12} + \cdots + 2470535187$ -14448930b15 - 103100b14 - 13361028b13 - 16403098b12 - 34344928b11 + 12028769b10 - 3065744b9 + 12647573b8 + 40111031b7 + 61789510b6 - 60440626b5 + 16540738b4 + 160896881b3 + 454236430b2 + 1014860450*b1 + 2470535187
$\nu^{15}$	$=$	$- 109564961 \beta_{15} + 85725756 \beta_{14} - 46225438 \beta_{13} - 106614312 \beta_{12} + \cdots + 8809490893$ -109564961b15 + 85725756b14 - 46225438b13 - 106614312b12 - 245589465b11 + 141091688b10 - 73179136b9 + 80837845b8 + 209671193b7 + 303376200b6 - 275284835b5 + 166794888b4 + 1185512812b3 + 1787739680b2 + 7030132055*b1 + 8809490893

Display $\beta_i$ in terms of $\nu^j$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

5.58145
4.87062
4.67314
3.58918
3.08631
2.66029
0.515365
0.501041
0.450617
−1.18577
−1.86231
−2.99493
−3.73424
−3.83202
−4.35146
−4.96729

−5.58145

−3.00000

23.1526

16.7443

10.0377

−84.5733

9.00000

1.2

−4.87062

−3.00000

15.7230

14.6119

−16.1001

−37.6156

9.00000

1.3

−4.67314

−3.00000

13.8382

14.0194

−23.4063

−27.2828

9.00000

1.4

−3.58918

−3.00000

4.88219

10.7675

30.7036

11.1904

9.00000

1.5

−3.08631

−3.00000

1.52531

9.25893

25.9780

19.9829

9.00000

1.6

−2.66029

−3.00000

−0.922855

7.98087

−18.8651

23.7374

9.00000

1.7

−0.515365

−3.00000

−7.73440

1.54609

15.3973

8.10895

9.00000

1.8

−0.501041

−3.00000

−7.74896

1.50312

−17.4547

7.89087

9.00000

1.9

−0.450617

−3.00000

−7.79694

1.35185

−0.788922

7.11838

9.00000

1.10

1.18577

−3.00000

−6.59395

−3.55731

−7.22246

−17.3051

9.00000

1.11

1.86231

−3.00000

−4.53181

−5.58692

13.3897

−23.3381

9.00000

1.12

2.99493

−3.00000

0.969579

−8.98478

4.43075

−21.0556

9.00000

1.13

3.73424

−3.00000

5.94451

−11.2027

−11.3963

−7.67567

9.00000

1.14

3.83202

−3.00000

6.68436

−11.4961

−36.2405

−5.04155

9.00000

1.15

4.35146

−3.00000

10.9352

−13.0544

29.5878

12.7724

9.00000

1.16

4.96729

−3.00000

16.6740

−14.9019

13.9493

43.0864

9.00000

Atkin-Lehner signs

$p$	Sign
$3$	$+1$
$5$	$+1$
$29$	$+1$

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	2175.4.a.u	✓	16
5.b	even	2	1		2175.4.a.v	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2175.4.a.u	✓	16	1.a	even	1	1	trivial
2175.4.a.v	yes	16	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2175))$ :

T_{2}^{16} + 3 T_{2}^{15} - 92 T_{2}^{14} - 239 T_{2}^{13} + 3416 T_{2}^{12} + 7461 T_{2}^{11} + \cdots - 891088

T_{7}^{16} - 12 T_{7}^{15} - 3022 T_{7}^{14} + 35587 T_{7}^{13} + 3434893 T_{7}^{12} + \cdots + 88\!\cdots\!64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16} + 3 T^{15} + \cdots - 891088$ T^16 + 3T^15 - 92T^14 - 239T^13 + 3416T^12 + 7461T^11 - 65355T^10 - 115826T^9 + 679301T^8 + 932552T^7 - 3681931T^6 - 3646901T^5 + 8804276T^4 + 5805166T^3 - 4956752T^2 - 4507400*T - 891088
$3$	$(T + 3)^{16}$ (T + 3)^16
$5$	$T^{16}$ T^16
$7$	$T^{16} + \cdots + 88\!\cdots\!64$ T^16 - 12T^15 - 3022T^14 + 35587T^13 + 3434893T^12 - 36801489T^11 - 1935115933T^10 + 18219793098T^9 + 583703240023T^8 - 4806817662629T^7 - 94138077858594T^6 + 672574582825666T^5 + 7490069969107083T^4 - 44334982305076294T^3 - 232080656378561439T^2 + 965182129766977689*T + 881456106936347164
$11$	$T^{16} + \cdots + 59\!\cdots\!88$ T^16 - 22T^15 - 14018T^14 + 281289T^13 + 77167983T^12 - 1363874187T^11 - 212517402068T^10 + 3086145689904T^9 + 309322126640378T^8 - 3198857449848958T^7 - 232112144517329001T^6 + 1240509940975355686T^5 + 80742101138944477105T^4 - 68937229938331605675T^3 - 10285370221508402898438T^2 - 30166434073986489587556*T + 59760052487188392933288
$13$	$T^{16} + \cdots + 10\!\cdots\!76$ T^16 + 42T^15 - 20118T^14 - 826663T^13 + 144573201T^12 + 5909031687T^11 - 442481084253T^10 - 18708375241884T^9 + 539004899600907T^8 + 26152699701747923T^7 - 187939090600616766T^6 - 15394508293260596662T^5 - 63518345862032987389T^4 + 2818320618607156949412T^3 + 32823068517853936040245T^2 + 110835913098880415753457*T + 103787974034614539866876
$17$	$T^{16} + \cdots + 19\!\cdots\!16$ T^16 + 152T^15 - 34971T^14 - 5617147T^13 + 428997808T^12 + 74041650136T^11 - 2474670937163T^10 - 448063592539554T^9 + 7879442335870942T^8 + 1305824736886212067T^7 - 16211440837754457312T^6 - 1784222201027905098933T^5 + 19636950242712621121301T^4 + 1051038214088649680957011T^3 - 11408206644877147974155670T^2 - 190781145845449174161498180*T + 1984338126291658007437429016
$19$	$T^{16} + \cdots - 26\!\cdots\!60$ T^16 - 128T^15 - 50878T^14 + 5118700T^13 + 1092297524T^12 - 71758726078T^11 - 12200781795149T^10 + 385956301966716T^9 + 69261339418375352T^8 - 391899933740902064T^7 - 166457603372619022416T^6 - 1379536993460178751200T^5 + 105686629933042278460544T^4 + 595526631791273374124416T^3 - 3650011900518441433985024T^2 - 23811039318376584453137920*T - 26618682354999474936770560
$23$	$T^{16} + \cdots + 19\!\cdots\!76$ T^16 + 58T^15 - 120312T^14 - 6030018T^13 + 5393956471T^12 + 225712341824T^11 - 115569222238840T^10 - 4082504358642496T^9 + 1243501222629903776T^8 + 38657154203709320928T^7 - 6331088798874729544640T^6 - 187953423127057427245568T^5 + 12262165191127510440475904T^4 + 388695660902757429274680320T^3 - 2290032177082526694994075648T^2 - 73865353172184916036373469184*T + 198827406627889122766243352576
$29$	$(T + 29)^{16}$ (T + 29)^16
$31$	$T^{16} + \cdots - 10\!\cdots\!00$ T^16 - 331T^15 - 201948T^14 + 78733790T^13 + 12023939788T^12 - 6718311735902T^11 - 65970007364295T^10 + 251846357369701183T^9 - 13528879586952017654T^8 - 4262604826453470132344T^7 + 385887573118602646086032T^6 + 29406205204474896981756768T^5 - 3724183734804535831254965408T^4 - 46801192767630453657815005888T^3 + 11731089786531297870189510668160T^2 - 12296192095923179134164256281600*T - 10962869436200166023753869519296000
$37$	$T^{16} + \cdots + 10\!\cdots\!28$ T^16 + 585T^15 - 345554T^14 - 282731998T^13 + 16879475877T^12 + 49074598675917T^11 + 6389606502538020T^10 - 3481363938703480168T^9 - 959404482813956141496T^8 + 54941362762830335064176T^7 + 46546949984162789476929216T^6 + 4023864795062468067721886016T^5 - 592806846744879281595349439360T^4 - 130462711698079276725535606181120T^3 - 7383740700873602727114688114549760T^2 + 29184971158081597776884642081685504*T + 10705337621023616579376998756044259328
$41$	$T^{16} + \cdots + 71\!\cdots\!84$ T^16 - 715T^15 - 510484T^14 + 446060302T^13 + 82652123895T^12 - 107179660035457T^11 - 3058762758256346T^10 + 12779067577189477380T^9 - 346932859498505200785T^8 - 825783497769454773072093T^7 + 29102079046705983617644416T^6 + 29307209169644048448583838302T^5 - 440149720833960067820523531655T^4 - 511612245148897178435508244590967T^3 - 8926752032431108245783613879119410T^2 + 2563508807838109772595366521287728432*T + 71042615634369588530681843708707272384
$43$	$T^{16} + \cdots - 27\!\cdots\!16$ T^16 - 608T^15 - 581239T^14 + 367372928T^13 + 107918562751T^12 - 72994335200824T^11 - 7172656645141301T^10 + 5427969179077341304T^9 + 253373630677981221660T^8 - 167825205455144157977440T^7 - 7334414570104447729050064T^6 + 1970667220630215542734283744T^5 + 109804626065985669116915096960T^4 - 3821646374537641331225421885440T^3 - 140985253027739984070535226614784T^2 + 4480851384980231843270437748795904*T - 27477731103898349424252466987554816
$47$	$T^{16} + \cdots + 30\!\cdots\!52$ T^16 + 1121T^15 - 556886T^14 - 982809415T^13 - 7903758764T^12 + 303091454288394T^11 + 46642899719135567T^10 - 41873096469260465258T^9 - 8286040722027631491682T^8 + 2877261114342824277360840T^7 + 502423590154672962196916021T^6 - 111967994486819296343902153737T^5 - 10944512428955028275039533515587T^4 + 2191923706701303144359743471017138T^3 + 34336202903009379391664254961318769T^2 - 10544124669816264674582216893133321692*T + 30668186182395305895751867877756728452
$53$	$T^{16} + \cdots + 28\!\cdots\!00$ T^16 + 750T^15 - 1234955T^14 - 1014166668T^13 + 531023785367T^12 + 521503805443600T^11 - 80201436100107200T^10 - 126206487195614133384T^9 - 3555619490238576213264T^8 + 14208794341371886951497600T^7 + 1969124579841700322684326336T^6 - 590387159677643144143386070528T^5 - 139083509369374011880039470293248T^4 + 85644007185586762884256025740800T^3 + 1915231057718221526377094381852119040T^2 + 142350151651603080846228900390035968000*T + 2835291902936377162440666288683485593600
$59$	$T^{16} + \cdots + 82\!\cdots\!40$ T^16 - 301T^15 - 1633380T^14 + 381362970T^13 + 1066094235873T^12 - 168237868828761T^11 - 360661167332881078T^10 + 30073170765117547372T^9 + 67463342935214947513048T^8 - 1380712041651957450583328T^7 - 6855976386023854122017311936T^6 - 147308270593140052875070447232T^5 + 354527347632186947624114468614144T^4 + 12543862803368548686755072950991616T^3 - 8717929158641145440272972800399228416T^2 - 222225992731611443553995552480281175040*T + 82997753585840976448423510357189555722240
$61$	$T^{16} + \cdots - 16\!\cdots\!00$ T^16 - 867T^15 - 1476508T^14 + 1165385410T^13 + 935943872948T^12 - 620713701826710T^11 - 322245671351060123T^10 + 167118210344490037543T^9 + 62713485076208723464034T^8 - 24187671022178124972447808T^7 - 6596678752123027486821649632T^6 + 1854391584650350693844414952544T^5 + 329204604245788822306903161419584T^4 - 66998829610116143069300908156329984T^3 - 5294997335688454440268534061434891008T^2 + 707460796352558194826142570646255246080*T - 16394420984490725499078067911753098380800
$67$	$T^{16} + \cdots - 71\!\cdots\!16$ T^16 + 1013T^15 - 1702914T^14 - 2138458273T^13 + 838634719387T^12 + 1739005794972529T^11 + 102159541470843432T^10 - 649330132308525635588T^9 - 217127936496852338162944T^8 + 89677578525143972171144401T^7 + 64571107311494610946780622207T^6 + 6076234257898746554112420583714T^5 - 4978796825463549467037610176590365T^4 - 1933260924733763512050414174233030136T^3 - 307674249958350786065335825365480270424T^2 - 23626636795081612073116877392189107994496*T - 715000984405213528556801079229964033193616
$71$	$T^{16} + \cdots + 47\!\cdots\!92$ T^16 - 1234T^15 - 2476047T^14 + 3452229684T^13 + 1998968858787T^12 - 3588666765604004T^11 - 464263620520833664T^10 + 1724715031878012463240T^9 - 101623947261033979523296T^8 - 401618213862426684544779616T^7 + 48467140235379290202772029248T^6 + 47620171691869283252734807405056T^5 - 4864014174230949382551635556359168T^4 - 2993203842788145412133486909401082368T^3 + 91236667979584364526222011518131823616T^2 + 81768944935520115606002074017736469970944*T + 4791698251663122085664967368478701561044992
$73$	$T^{16} + \cdots + 18\!\cdots\!12$ T^16 - 464T^15 - 3356011T^14 + 1046318442T^13 + 4412867216987T^12 - 625073973921202T^11 - 2852702095537244768T^10 - 82096704272559698240T^9 + 901045426238427516141280T^8 + 163936888620446074366748320T^7 - 110728880894085208338441953856T^6 - 35949980392169150170107154925696T^5 + 885249620260016909885623810354688T^4 + 1501633690124819589365793934261824512T^3 + 205779958489966490987964611819719618560T^2 + 10710586015637754510080722798541657892864*T + 186112139208443806903469151177937007296512
$79$	$T^{16} + \cdots - 59\!\cdots\!40$ T^16 - 1506T^15 - 2647508T^14 + 5134874678T^13 + 1194827500627T^12 - 5484991584554476T^11 + 1177853533093591624T^10 + 2112595980524129409456T^9 - 902264666444885359117296T^8 - 200181521266020688891707008T^7 + 118030836741514449831270388288T^6 + 8486260380626190965849146428288T^5 - 5621950598775771990287714795730688T^4 - 191827969790792575192351252425727488T^3 + 104307091700196445685396494451889332224T^2 + 1697338397327928482141452320197536235520*T - 594681208743824151985248737921300148387840
$83$	$T^{16} + \cdots - 63\!\cdots\!96$ T^16 + 864T^15 - 5208364T^14 - 5208552534T^13 + 9367864143623T^12 + 11496210589950070T^11 - 6263777797574060484T^10 - 11307288637474934285864T^9 + 187218521059318594994448T^8 + 4892536264188296093295519424T^7 + 1192922326260725252495901218880T^6 - 778697089338072542577479894590592T^5 - 300244218766131274273866885984604672T^4 + 42500301363656565701358558827641365504T^3 + 23936543065697576796188814374189595377664T^2 - 591129839899566191298921488125219327350784*T - 631001962736035729570854865717076050049744896
$89$	$T^{16} + \cdots + 51\!\cdots\!00$ T^16 - 1844T^15 - 2162770T^14 + 5593461447T^13 - 13903281495T^12 - 5276687388213871T^11 + 2065156823816499030T^10 + 1782529083038418885104T^9 - 1231186441525801248402766T^8 - 109338749126399048374908098T^7 + 232025525713246381205513181191T^6 - 27220836439243867406561697691340T^5 - 15970134715422123238363083987758023T^4 + 3545116260406298069266835852467744985T^3 + 241874675880221617856307965931319751804T^2 - 99898919144896462749659407176893984294080*T + 5127898834503382487399789436843932914166400
$97$	$T^{16} + \cdots + 46\!\cdots\!04$ T^16 - 1176T^15 - 8440759T^14 + 9204229760T^13 + 29411615369155T^12 - 29383629020976688T^11 - 55665043399643552085T^10 + 49909507134919339639136T^9 + 62920465457348298040659928T^8 - 48825524745262696744567281976T^7 - 43753169744710488861768530574096T^6 + 27501527908549223343773359876467040T^5 + 18451345068517484250404902579733331392T^4 - 8237051780158716991689186692935881123200T^3 - 4386477892539455716557143887743352055125248T^2 + 1006564582765962113354528071115544821128487936*T + 462396091636444519032353379395096548734025339904
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.16
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2175.4.a.u

Level	$2175$
Weight	$4$
Character orbit	2175.a
Self dual	yes
Analytic conductor	$128.329$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$1$