LMFDB - Newform orbit 464.3.d.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,3,Mod(175,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("464.175");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$464 = 2^{4} \cdot 29$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	464.d (of order $2$ , degree $1$ , minimal)

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:	no
Analytic conductor:	$12.6430842663$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} + 69 x^{18} + 1795 x^{16} + 24222 x^{14} + 189561 x^{12} + 892623 x^{10} + 2508433 x^{8} + \cdots + 21609$ x^20 + 69x^18 + 1795x^16 + 24222x^14 + 189561x^12 + 892623x^10 + 2508433x^8 + 3989370x^6 + 3148495x^4 + 862449*x^2 + 21609
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$2^{36}\cdot 7^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{11} q^{3} - \beta_{3} q^{5} - \beta_{12} q^{7} + (\beta_{2} - 3) q^{9} + \beta_{16} q^{11} + ( - \beta_{8} + 1) q^{13} + ( - \beta_{18} + \beta_{16} + \cdots - \beta_{4}) q^{15} + ( - \beta_{9} - \beta_{3} - \beta_1 + 2) q^{17}+ \cdots + (3 \beta_{19} - 3 \beta_{18} + \cdots - 4 \beta_{4}) q^{99}+O(q^{100})$ q - b11 * q^3 - b3 * q^5 - b12 * q^7 + (b2 - 3) * q^9 + b16 * q^11 + (-b8 + 1) * q^13 + (-b18 + b16 + b14 + b12 - b4) * q^15 + (-b9 - b3 - b1 + 2) * q^17 + (-b15 + b13 - b12 + b11) * q^19 + (-b10 - b3 - 2b1 - 2) q^21 + (-b18 - b14 - b13 + b12 + b11) * q^23 + (2b6 - b5 + b3 + 9) q^25 + (-b19 - b18 + b16 - b15 + 3b11) q^27 - b1 * q^29 + (b19 + b15 - b14 + b12 - 2b11) q^31 + (b8 + b7 - 2b6 + b5 - b2 - 6b1 - 7) * q^33 + (b19 - 2b18 - b17 - b15 - b14 - b13 + 3b11 - b4) * q^35 + (b9 + b6 - b5 + 3b3 + 2b2 + b1 - 4) * q^37 + (-b19 + b18 + b17 + b16 + 3b15 - b14 - b11 + b4) q^39 + (2b10 + 2b8 - b6 + b5 - 2b1 - 4) q^41 + (b19 - b18 + 2b16 - b13 - b12 - 3b11) * q^43 + (2b10 + 2b9 - b7 - 2b6 + 2b5 + 4b3 - 3b2 + 2) * q^45 + (-b19 - b18 + b17 + b16 + b14 + b13 + b12 + 4b11) q^47 + (-2b10 - 2b9 + b8 - b7 - b3 - b2 - 6b1 - 2) q^49 + (b19 - b18 + b17 + b15 + 2b14 + 2b13 - 4b11 - 3b4) * q^51 + (2b10 - 2b9 + 2b8 - b7 - 2b6 - b3 - b2 + 2b1 + 4) q^53 + (-2b18 - b17 - 2b16 + b15 + 3b14 + b13 + b12 - 6b11 - 3b4) q^55 + (4b9 - 2b8 + 2b7 - b5 + b3 - 5b2 - 8b1 + 4) q^57 + (-b19 - 2b18 - b17 + 4b16 - b15 - b14 - 3b13 + 3b11 + b4) * q^59 + (-b10 + 3b9 - 2b8 + b6 + b5 + 2b3 + b1 - 4) q^61 + (-3b18 - 2b17 - 2b16 + b14 - b13 - b11 - 2b4) * q^63 + (-2b10 + 2b9 - 2b8 + 2b7 + 2b6 - 4b5 + 2b3 + b2 - 8b1 - 4) * q^65 + (-2b19 + b18 + 2b16 + b14 + b13 - b11) * q^67 + (b10 - b9 + 2b8 - 2b7 + 3b6 - b5 + 2b3 - 2b2 + 7b1 + 16) * q^69 + (2b18 + 2b17 - 2b16 + 2b15 - 2b14 + 2b13) * q^71 + (-3b9 - 4b7 - 3b6 + 3b5 + b3 + 4b2 + 3b1 - 14) * q^73 + (2b18 + 2b17 + 5b16 + 3b15 + 4b14 + 3b13 - b12 - 10b11 - 2b4) * q^75 + (-3b10 - b9 - 2b8 + 2b7 + b6 - b5 - 2b3 + 4b2 - b1 - 8) q^77 + (2b18 - 3b17 - 2b16 - 2b15 - 3b14 + 3b11 + 4b4) q^79 + (2b10 + 2b9 + b8 + b7 + 3b5 + 6b3 - 2b2 - 6b1) * q^81 + (-b19 + b18 + b17 - 3b15 - 2b14 - 2b12 + 4b11 + 5b4) q^83 + (-b10 - 2b9 + 2b6 - 7b3 - 2b2 + 8b1 + 18) q^85 + (-b16 + b15 + b12) * q^87 + (-2b10 - 2b9 - 2b8 - 2b7 + b6 + 3b5 - 2b3 + 6b2 - 4) q^89 + (b19 + b18 + b17 - 3b15 - 2b14 - 2b13 - 6b12 - 4b11 + 3b4) * q^91 + (3b8 - b7 - 4b5 - 2b3 + 5b2 + 6b1 - 21) q^93 + (7b19 - 2b18 - b17 - 2b16 - 3b15 - 5b12 - b11 - 4b4) * q^95 + (-3b9 - 4b8 + 2b7 - 2b6 - 2b5 - 11b3 - 3b1 + 6) q^97 + (3b19 - 3b18 - 4b17 - 7b16 + b15 - 4b14 - 4b13 + 6b12 - 2b11 - 4b4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$20 q - 8 q^{5} - 68 q^{9} + 16 q^{13} + 40 q^{17} - 48 q^{21} + 188 q^{25} - 120 q^{33} - 80 q^{37} - 72 q^{41} + 72 q^{45} - 28 q^{49} + 96 q^{53} + 104 q^{57} - 96 q^{61} - 80 q^{65} + 352 q^{69} - 312 q^{73}+ \cdots + 56 q^{97}+O(q^{100})$ 20 * q - 8 * q^5 - 68 * q^9 + 16 * q^13 + 40 * q^17 - 48 * q^21 + 188 * q^25 - 120 * q^33 - 80 * q^37 - 72 * q^41 + 72 * q^45 - 28 * q^49 + 96 * q^53 + 104 * q^57 - 96 * q^61 - 80 * q^65 + 352 * q^69 - 312 * q^73 - 192 * q^77 + 60 * q^81 + 336 * q^85 - 152 * q^89 - 472 * q^93 + 56 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} + 69 x^{18} + 1795 x^{16} + 24222 x^{14} + 189561 x^{12} + 892623 x^{10} + 2508433 x^{8} + \cdots + 21609$ x^20 + 69*x^18 + 1795*x^16 + 24222*x^14 + 189561*x^12 + 892623*x^10 + 2508433*x^8 + 3989370*x^6 + 3148495*x^4 + 862449*x^2 + 21609 :

$\beta_{1}$	$=$	$( 269119 \nu^{18} + 14955825 \nu^{16} + 264806302 \nu^{14} + 1929025386 \nu^{12} + \cdots - 24957173283 ) / 3750443928$ (269119v^18 + 14955825v^16 + 264806302v^14 + 1929025386v^12 + 4124199909v^10 - 18416701329v^8 - 113611368386v^6 - 221364373482v^4 - 177458469041*v^2 - 24957173283) / 3750443928
$\beta_{2}$	$=$	$( - 538883 \nu^{18} - 36402591 \nu^{16} - 918073114 \nu^{14} - 11927037014 \nu^{12} + \cdots - 29241914583 ) / 4643406768$ (-538883v^18 - 36402591v^16 - 918073114v^14 - 11927037014v^12 - 88899687189v^10 - 389643831661v^8 - 965916438026v^6 - 1187433500870v^4 - 511362980275*v^2 - 29241914583) / 4643406768
$\beta_{3}$	$=$	$( 13214681 \nu^{18} + 897749793 \nu^{16} + 22671980714 \nu^{14} + 290379259542 \nu^{12} + \cdots - 132028188327 ) / 32503847376$ (13214681v^18 + 897749793v^16 + 22671980714v^14 + 290379259542v^12 + 2080514274031v^10 + 8443630548159v^8 + 18342774305998v^6 + 18054214873254v^4 + 4658194352973*v^2 - 132028188327) / 32503847376
$\beta_{4}$	$=$	$( 10936 \nu^{19} + 747381 \nu^{17} + 19101116 \nu^{15} + 250205022 \nu^{13} + \cdots + 10598814513 \nu ) / 236023788$ (10936v^19 + 747381v^17 + 19101116v^15 + 250205022v^13 + 1869567576v^11 + 8253402375v^9 + 21605398480v^7 + 33404128170v^5 + 29241375044v^3 + 10598814513v) / 236023788
$\beta_{5}$	$=$	$( - 757289 \nu^{18} - 50166367 \nu^{16} - 1225295662 \nu^{14} - 15217333392 \nu^{12} + \cdots - 41589924285 ) / 1250147976$ (-757289v^18 - 50166367v^16 - 1225295662v^14 - 15217333392v^12 - 106796641785v^10 - 433226755077v^8 - 979251701432v^6 - 1104754236502v^4 - 482946172219*v^2 - 41589924285) / 1250147976
$\beta_{6}$	$=$	$( - 13082613 \nu^{18} - 855428506 \nu^{16} - 20436920204 \nu^{14} - 245686535916 \nu^{12} + \cdots - 12352318272 ) / 16251923688$ (-13082613v^18 - 855428506v^16 - 20436920204v^14 - 245686535916v^12 - 1645762719753v^10 - 6239923059684v^8 - 12710202390780v^6 - 11905342686538v^4 - 3274322583581*v^2 - 12352318272) / 16251923688
$\beta_{7}$	$=$	$( - 3168803 \nu^{18} - 217019291 \nu^{16} - 5536671654 \nu^{14} - 71565124038 \nu^{12} + \cdots - 75153189783 ) / 2954895216$ (-3168803v^18 - 217019291v^16 - 5536671654v^14 - 71565124038v^12 - 516897110017v^10 - 2117595588861v^8 - 4686723065350v^6 - 4868711941898v^4 - 1599323728387*v^2 - 75153189783) / 2954895216
$\beta_{8}$	$=$	$( - 37977677 \nu^{18} - 2535952577 \nu^{16} - 62516866638 \nu^{14} - 780334603670 \nu^{12} + \cdots - 1132793231829 ) / 32503847376$ (-37977677v^18 - 2535952577v^16 - 62516866638v^14 - 780334603670v^12 - 5458427250335v^10 - 21765200499019v^8 - 47177177916762v^6 - 48320959468642v^4 - 16118928681065*v^2 - 1132793231829) / 32503847376
$\beta_{9}$	$=$	$( 11254547 \nu^{18} + 730259037 \nu^{16} + 17257669802 \nu^{14} + 205149355410 \nu^{12} + \cdots + 183966243909 ) / 8864685648$ (11254547v^18 + 730259037v^16 + 17257669802v^14 + 205149355410v^12 + 1360361904093v^10 + 5121109060023v^8 + 10441706401586v^6 + 10054783986450v^4 + 3186498766283*v^2 + 183966243909) / 8864685648
$\beta_{10}$	$=$	$( 155501111 \nu^{18} + 10347255573 \nu^{16} + 253965882410 \nu^{14} + 3159391510218 \nu^{12} + \cdots + 1264306787709 ) / 97511542128$ (155501111v^18 + 10347255573v^16 + 253965882410v^14 + 3159391510218v^12 + 22078479809865v^10 + 88319024419575v^8 + 193484244030362v^6 + 203067435541530v^4 + 69855863267351*v^2 + 1264306787709) / 97511542128
$\beta_{11}$	$=$	$( 108346853 \nu^{19} + 7258096644 \nu^{17} + 179293861268 \nu^{15} + 2227767857166 \nu^{13} + \cdots - 16844008902672 \nu ) / 682580794896$ (108346853v^19 + 7258096644v^17 + 179293861268v^15 + 2227767857166v^13 + 15294758270007v^11 + 58084381410870v^9 + 111157559658890v^7 + 74618847039954v^5 - 25180233298045v^3 - 16844008902672v) / 682580794896
$\beta_{12}$	$=$	$( 24959967 \nu^{19} + 1740535793 \nu^{17} + 45661735278 \nu^{15} + 612950649536 \nu^{13} + \cdots + 4647445493355 \nu ) / 113763465816$ (24959967v^19 + 1740535793v^17 + 45661735278v^15 + 612950649536v^13 + 4658675196655v^11 + 20497839911803v^9 + 50550209017184v^7 + 63496535790058v^5 + 32894333841313v^3 + 4647445493355v) / 113763465816
$\beta_{13}$	$=$	$( - 60588499 \nu^{19} - 4013927296 \nu^{17} - 97597388088 \nu^{15} - 1191456127314 \nu^{13} + \cdots - 4009400719128 \nu ) / 227526931632$ (-60588499v^19 - 4013927296v^17 - 97597388088v^15 - 1191456127314v^13 - 8025223854213v^11 - 29816703645186v^9 - 55470212118058v^7 - 36469643407330v^5 + 4876392032211v^3 - 4009400719128v) / 227526931632
$\beta_{14}$	$=$	$( - 258162071 \nu^{19} - 17021686536 \nu^{17} - 412592166284 \nu^{15} + \cdots + 35240748098808 \nu ) / 682580794896$ (-258162071v^19 - 17021686536v^17 - 412592166284v^15 - 5070395023398v^13 - 35079253355649v^11 - 139499345201478v^9 - 305931614095586v^7 - 323802742801566v^5 - 100416926761109v^3 + 35240748098808v) / 682580794896
$\beta_{15}$	$=$	$( 60980495 \nu^{19} + 4117881696 \nu^{17} + 103406444000 \nu^{15} + 1326071703002 \nu^{13} + \cdots + 19264652092644 \nu ) / 113763465816$ (60980495v^19 + 4117881696v^17 + 103406444000v^15 + 1326071703002v^13 + 9659511066837v^11 + 41127935527246v^9 + 100408982661758v^7 + 132036141139574v^5 + 82774196468957v^3 + 19264652092644v) / 113763465816
$\beta_{16}$	$=$	$( - 389833673 \nu^{19} - 25576963008 \nu^{17} - 613696544504 \nu^{15} + \cdots + 40430927254620 \nu ) / 682580794896$ (-389833673v^19 - 25576963008v^17 - 613696544504v^15 - 7396182653562v^13 - 49338273908355v^11 - 182835724424982v^9 - 344036816132498v^7 - 231870594479958v^5 + 65555112945973v^3 + 40430927254620v) / 682580794896
$\beta_{17}$	$=$	$( - 165753053 \nu^{19} - 11335372240 \nu^{17} - 288959154860 \nu^{15} + \cdots - 13092847367712 \nu ) / 227526931632$ (-165753053v^19 - 11335372240v^17 - 288959154860v^15 - 3743237889602v^13 - 27260928827115v^11 - 113967245421850v^9 - 264159494330678v^7 - 306680573343354v^5 - 139403234684399v^3 - 13092847367712v) / 227526931632
$\beta_{18}$	$=$	$( - 687333589 \nu^{19} - 45371705136 \nu^{17} - 1102288771984 \nu^{15} + \cdots - 63775765633188 \nu ) / 682580794896$ (-687333589v^19 - 45371705136v^17 - 1102288771984v^15 - 13602452080122v^13 - 94812151934631v^11 - 382259266496022v^9 - 861783988272682v^7 - 979529094604326v^5 - 444957631956439v^3 - 63775765633188v) / 682580794896
$\beta_{19}$	$=$	$( - 67516814 \nu^{19} - 4481300403 \nu^{17} - 109749087416 \nu^{15} - 1367135984622 \nu^{13} + \cdots - 8694964918311 \nu ) / 48755771064$ (-67516814v^19 - 4481300403v^17 - 109749087416v^15 - 1367135984622v^13 - 9636005711802v^11 - 39426689228133v^9 - 90997204908572v^7 - 108378279328578v^5 - 55249860475922v^3 - 8694964918311v) / 48755771064

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

$\nu$	$=$	$( - 3 \beta_{18} - \beta_{17} + 5 \beta_{16} - 3 \beta_{15} + \beta_{14} - 2 \beta_{13} + \cdots + \beta_{4} ) / 28$ (-3b18 - b17 + 5b16 - 3b15 + b14 - 2b13 - 2b12 + 6b11 + b4) / 28
$\nu^{2}$	$=$	$( - \beta_{10} - \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} - 3 \beta_{5} - 3 \beta_{3} + \cdots - 91 ) / 14$ (-b10 - b9 - 2b8 - 3b7 + 4b6 - 3b5 - 3b3 + 8b2 + 12*b1 - 91) / 14
$\nu^{3}$	$=$	$( 28 \beta_{19} + 27 \beta_{18} - 5 \beta_{17} - 101 \beta_{16} + 55 \beta_{15} - 23 \beta_{14} + \cdots - 44 \beta_{4} ) / 28$ (28b19 + 27b18 - 5b17 - 101b16 + 55b15 - 23b14 + 4b13 + 32b12 - 236b11 - 44b4) / 28
$\nu^{4}$	$=$	$( 8 \beta_{10} + 50 \beta_{9} + 16 \beta_{8} + 94 \beta_{7} - 88 \beta_{6} + 108 \beta_{5} + 80 \beta_{3} + \cdots + 1477 ) / 14$ (8b10 + 50b9 + 16b8 + 94b7 - 88b6 + 108b5 + 80b3 - 274b2 - 369*b1 + 1477) / 14
$\nu^{5}$	$=$	$( - 952 \beta_{19} - 514 \beta_{18} + 370 \beta_{17} + 2700 \beta_{16} - 1368 \beta_{15} + \cdots + 1359 \beta_{4} ) / 28$ (-952b19 - 514b18 + 370b17 + 2700b16 - 1368b15 + 764b14 + 320b13 - 814b12 + 7552b11 + 1359b4) / 28
$\nu^{6}$	$=$	$( - 19 \beta_{10} - 240 \beta_{9} - 23 \beta_{8} - 407 \beta_{7} + 331 \beta_{6} - 493 \beta_{5} + \cdots - 5235 ) / 2$ (-19b10 - 240b9 - 23b8 - 407b7 + 331b6 - 493b5 - 330b3 + 1209b2 + 1588*b1 - 5235) / 2
$\nu^{7}$	$=$	$( 29246 \beta_{19} + 13773 \beta_{18} - 12755 \beta_{17} - 79249 \beta_{16} + 38987 \beta_{15} + \cdots - 41019 \beta_{4} ) / 28$ (29246b19 + 13773b18 - 12755b17 - 79249b16 + 38987b15 - 23715b14 - 13078b13 + 23728b12 - 232086b11 - 41019b4) / 28
$\nu^{8}$	$=$	$( 3478 \beta_{10} + 52240 \beta_{9} + 2336 \beta_{8} + 86398 \beta_{7} - 67028 \beta_{6} + 106040 \beta_{5} + \cdots + 1051337 ) / 14$ (3478b10 + 52240b9 + 2336b8 + 86398b7 - 67028b6 + 106040b5 + 69080b3 - 257018b2 - 336177*b1 + 1051337) / 14
$\nu^{9}$	$=$	$( - 886956 \beta_{19} - 405119 \beta_{18} + 398267 \beta_{17} + 2381551 \beta_{16} + \cdots + 1238732 \beta_{4} ) / 28$ (-886956b19 - 405119b18 + 398267b17 + 2381551b16 - 1159761b15 + 721915b14 + 421922b13 - 712596b12 + 7056562b11 + 1238732b4) / 28
$\nu^{10}$	$=$	$( - 102570 \beta_{10} - 1592555 \beta_{9} - 49551 \beta_{8} - 2618396 \beta_{7} + 2002311 \beta_{6} + \cdots - 31386026 ) / 14$ (-102570b10 - 1592555b9 - 49551b8 - 2618396b7 + 2002311b6 - 3223168b5 - 2085507b3 + 7783663b2 + 10176028*b1 - 31386026) / 14
$\nu^{11}$	$=$	$( 26853946 \beta_{19} + 12180984 \beta_{18} - 12155844 \beta_{17} - 71971874 \beta_{16} + \cdots - 37456957 \beta_{4} ) / 28$ (26853946b19 + 12180984b18 - 12155844b17 - 71971874b16 + 34934050b15 - 21880186b14 - 12973634b13 + 21535484b12 - 213896698b11 - 37456957b4) / 28
$\nu^{12}$	$=$	$220990 \beta_{10} + 3449872 \beta_{9} + 94163 \beta_{8} + 5664594 \beta_{7} - 4313480 \beta_{6} + \cdots + 67621484$ 220990b10 + 3449872b9 + 94163b8 + 5664594b7 - 4313480b6 + 6977335b5 + 4506852b3 - 16830953b2 - 22004460*b1 + 67621484
$\nu^{13}$	$=$	$( - 812917196 \beta_{19} - 368149435 \beta_{18} + 368819275 \beta_{17} + 2177913421 \beta_{16} + \cdots + 1133409677 \beta_{4} ) / 28$ (-812917196b19 - 368149435b18 + 368819275b17 + 2177913421b16 - 1056047751b15 + 662540285b14 + 394314114b13 - 651706694b12 + 6477730818b11 + 1133409677b4) / 28
$\nu^{14}$	$=$	$( - 93646799 \beta_{10} - 1462797155 \beta_{9} - 38326360 \beta_{8} - 2401197039 \beta_{7} + \cdots - 28632047717 ) / 14$ (-93646799b10 - 1462797155b9 - 38326360b8 - 2401197039b7 + 1826238572b6 - 2958045093b5 - 1909825515b3 + 7133127646b2 + 9326079894*b1 - 28632047717) / 14
$\nu^{15}$	$=$	$( 24608798564 \beta_{19} + 11140467351 \beta_{18} - 11172193765 \beta_{17} - 65925563581 \beta_{16} + \cdots - 34305851260 \beta_{4} ) / 28$ (24608798564b19 + 11140467351b18 - 11172193765b17 - 65925563581b16 + 31956654239b15 - 20058072211b14 - 11949345544b13 + 19727657968b12 - 196121849320b11 - 34305851260b4) / 28
$\nu^{16}$	$=$	$( 2835454462 \beta_{10} + 44288251036 \beta_{9} + 1146402476 \beta_{8} + 72695641994 \beta_{7} + \cdots + 866556678029 ) / 14$ (2835454462b10 + 44288251036b9 + 1146402476b8 + 72695641994b7 - 55269719516b6 + 89556385740b5 + 57813897592b3 - 215937964418b2 - 282329601195*b1 + 866556678029) / 14
$\nu^{17}$	$=$	$( - 744973789700 \beta_{19} - 337220701814 \beta_{18} + 338273969270 \beta_{17} + \cdots + 1038478608483 \beta_{4} ) / 28$ (-744973789700b19 - 337220701814b18 + 338273969270b17 + 1995716950764b16 - 967309572876b15 + 607223498428b14 + 361839479272b13 - 597206890058b12 + 5937381591056b11 + 1038478608483b4) / 28
$\nu^{18}$	$=$	$( - 12263760671 \beta_{10} - 191539148568 \beta_{9} - 4940489665 \beta_{8} - 314393408887 \beta_{7} + \cdots - 3747349638765 ) / 2$ (-12263760671b10 - 191539148568b9 - 4940489665b8 - 314393408887b7 + 239006454275b6 - 387314502395b5 - 250025528646b3 + 933863779023b2 + 1220995599668*b1 - 3747349638765) / 2
$\nu^{19}$	$=$	$( 22552482932770 \beta_{19} + 10208403611445 \beta_{18} - 10241049289423 \beta_{17} + \cdots - 31437203065743 \beta_{4} ) / 28$ (22552482932770b19 + 10208403611445b18 - 10241049289423b17 - 60415935412457b16 + 29282341329343b15 - 18382489344687b14 - 10954729729442b13 + 18079188879932b12 - 179743801500138b11 - 31437203065743b4) / 28

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$ .

$n$	$117$	$175$	$321$
$\chi(n)$	$1$	$-1$	$1$

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.

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}

175.1

		5.50209i
		1.44310i
	−	1.53049i
	−	2.95207i
	−	0.166656i
		2.79820i
	−	2.72620i
		0.682100i
		2.50448i
		1.88675i
	−	1.88675i
	−	2.50448i
	−	0.682100i
		2.72620i
	−	2.79820i
		0.166656i
		2.95207i
		1.53049i
	−	1.44310i
	−	5.50209i

−

5.45563i

−3.94765

2.61861i

−20.7639

175.2

−

5.12416i

−8.13026

−

11.0093i

−17.2570

175.3

−

4.55005i

8.26335

−

0.295091i

−11.7029

175.4

−

3.94782i

−0.459342

4.40863i

−6.58528

175.5

−

3.92832i

7.62576

5.54000i

−6.43170

175.6

−

2.34966i

−1.82181

−

0.690645i

3.47908

175.7

−

2.28467i

1.85233

−

13.1301i

3.78028

175.8

−

2.23726i

−3.52607

6.82457i

3.99465

175.9

−

0.707016i

−9.31226

−

6.08820i

8.50013

175.10

−

0.115341i

5.45596

8.31894i

8.98670

175.11

0.115341i

5.45596

−

8.31894i

8.98670

175.12

0.707016i

−9.31226

6.08820i

8.50013

175.13

2.23726i

−3.52607

−

6.82457i

3.99465

175.14

2.28467i

1.85233

13.1301i

3.78028

175.15

2.34966i

−1.82181

0.690645i

3.47908

175.16

3.92832i

7.62576

−

5.54000i

−6.43170

175.17

3.94782i

−0.459342

−

4.40863i

−6.58528

175.18

4.55005i

8.26335

0.295091i

−11.7029

175.19

5.12416i

−8.13026

11.0093i

−17.2570

175.20

5.45563i

−3.94765

−

2.61861i

−20.7639

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	464.3.d.b	✓	20
4.b	odd	2	1	inner	464.3.d.b	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
464.3.d.b	✓	20	1.a	even	1	1	trivial
464.3.d.b	✓	20	4.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{20} + 124 T_{3}^{18} + 6404 T_{3}^{16} + 178968 T_{3}^{14} + 2944622 T_{3}^{12} + 29101248 T_{3}^{10} + \cdots + 3732624$ T3^20 + 124*T3^18 + 6404*T3^16 + 178968*T3^14 + 2944622*T3^12 + 29101248*T3^10 + 169151012*T3^8 + 540771784*T3^6 + 798932953*T3^4 + 291107556*T3^2 + 3732624 acting on $S_{3}^{\mathrm{new}}(464, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{20}$ T^20
$3$	$T^{20} + 124 T^{18} + \cdots + 3732624$ T^20 + 124T^18 + 6404T^16 + 178968T^14 + 2944622T^12 + 29101248T^10 + 169151012T^8 + 540771784T^6 + 798932953T^4 + 291107556*T^2 + 3732624
$5$	$(T^{10} + 4 T^{9} + \cdots + 561636)^{2}$ (T^10 + 4T^9 - 164T^8 - 584T^7 + 8490T^6 + 28376T^5 - 137288T^4 - 504232T^3 + 216541T^2 + 1414164*T + 561636)^2
$7$	$T^{20} + \cdots + 424144797696$ T^20 + 504T^18 + 99952T^16 + 10176000T^14 + 583086336T^12 + 19259154432T^10 + 356166676480T^8 + 3328733970432T^6 + 11993180864512T^4 + 5890158624768*T^2 + 424144797696
$11$	$T^{20} + \cdots + 10\!\cdots\!56$ T^20 + 1340T^18 + 761140T^16 + 240178232T^14 + 46290643486T^12 + 5632535448992T^10 + 430651721130292T^8 + 19816369048941992T^6 + 494261582432190697T^4 + 5210063976242606148*T^2 + 10769590883715674256
$13$	$(T^{10} - 8 T^{9} + \cdots + 17020989468)^{2}$ (T^10 - 8T^9 - 1068T^8 + 9296T^7 + 353034T^6 - 3342152T^5 - 40792184T^4 + 436264896T^3 + 994365309T^2 - 15240254400*T + 17020989468)^2
$17$	$(T^{10} - 20 T^{9} + \cdots - 2764274688)^{2}$ (T^10 - 20T^9 - 1024T^8 + 9488T^7 + 368944T^6 + 769728T^5 - 22741632T^4 - 73933056T^3 + 407409408T^2 + 894716928*T - 2764274688)^2
$19$	$T^{20} + \cdots + 13\!\cdots\!76$ T^20 + 5380T^18 + 12084144T^16 + 14763791488T^14 + 10689643902208T^12 + 4676454627609600T^10 + 1204161193950105600T^8 + 167890712454809026560T^6 + 10116912694136430329856T^4 + 76584732081086542381056*T^2 + 13140586161345502642176
$23$	$T^{20} + \cdots + 36\!\cdots\!76$ T^20 + 7064T^18 + 19909520T^16 + 28696846080T^14 + 22565476218368T^12 + 9642851326826496T^10 + 2170999948277547008T^8 + 256709423699035553792T^6 + 15578935548494717452288T^4 + 432488969813762811887616*T^2 + 3648007297684604951986176
$29$	$(T^{2} - 29)^{10}$ (T^2 - 29)^10
$31$	$T^{20} + \cdots + 76\!\cdots\!24$ T^20 + 9508T^18 + 35592092T^16 + 68022876720T^14 + 72472124901062T^12 + 44294877945958632T^10 + 15249302365933278764T^8 + 2738868177411117267472T^6 + 218962522802422072072417T^4 + 7250557426642469486739636*T^2 + 76347303618832310481108624
$37$	$(T^{10} + 40 T^{9} + \cdots + 428081152)^{2}$ (T^10 + 40T^9 - 4308T^8 - 152736T^7 + 3517568T^6 + 46907904T^5 - 1391259648T^4 + 10436214784T^3 - 30137925632T^2 + 26341146624*T + 428081152)^2
$41$	$(T^{10} + \cdots - 3072101280768)^{2}$ (T^10 + 36T^9 - 7088T^8 - 70144T^7 + 14897504T^6 - 282855872T^5 - 1645367040T^4 + 102770484736T^3 - 1126299264512T^2 + 4415152825344*T - 3072101280768)^2
$43$	$T^{20} + \cdots + 98\!\cdots\!76$ T^20 + 12396T^18 + 57485284T^16 + 133503113448T^14 + 170690144732238T^12 + 122964679404322992T^10 + 48993119740047890500T^8 + 10007575248039721517496T^6 + 850124550586328307009145T^4 + 9628458996253213218180420*T^2 + 9895917234029934805776
$47$	$T^{20} + \cdots + 82\!\cdots\!36$ T^20 + 18692T^18 + 128050892T^16 + 409906806000T^14 + 665743864617014T^12 + 545328602864556648T^10 + 202327661481592677308T^8 + 25599573670527784830800T^6 + 1049518999770473380055473T^4 + 8882820152049272536750932*T^2 + 8251643163946859421288336
$53$	$(T^{10} + \cdots - 25\!\cdots\!00)^{2}$ (T^10 - 48T^9 - 13716T^8 + 626424T^7 + 62135042T^6 - 2618008416T^5 - 107887912464T^4 + 3802119335064T^3 + 82644573940069T^2 - 1785078916757760*T - 25213277168597700)^2
$59$	$T^{20} + \cdots + 25\!\cdots\!64$ T^20 + 38328T^18 + 629914384T^16 + 5823989046528T^14 + 33333766596521472T^12 + 122236536361728276480T^10 + 286918003743703675666432T^8 + 417869206640796211176013824T^6 + 353389307631618915655225507840T^4 + 153766294439323538007065814368256*T^2 + 25615179252897442867474944453771264
$61$	$(T^{10} + \cdots - 52\!\cdots\!32)^{2}$ (T^10 + 48T^9 - 13384T^8 - 508736T^7 + 55389712T^6 + 1363723456T^5 - 87922283648T^4 - 1425571346688T^3 + 50157942236928T^2 + 570219691474944*T - 5285057598968832)^2
$67$	$T^{20} + \cdots + 11\!\cdots\!36$ T^20 + 22176T^18 + 188220160T^16 + 786684641280T^14 + 1711215999516672T^12 + 1859921544172535808T^10 + 863655415085252214784T^8 + 125581766588215030972416T^6 + 5377961299939731045351424T^4 + 23028051874867348393426944*T^2 + 11449398835240092655681536
$71$	$T^{20} + \cdots + 27\!\cdots\!16$ T^20 + 49376T^18 + 921581888T^16 + 8360669274624T^14 + 41183327844683264T^12 + 118751269520903577600T^10 + 207302358135660521603072T^8 + 217510334184564350405967872T^6 + 129142339722795682214922747904T^4 + 36643033630407549716416274890752*T^2 + 2761878950530717144396744114569216
$73$	$(T^{10} + \cdots + 22\!\cdots\!72)^{2}$ (T^10 + 156T^9 - 27292T^8 - 4972960T^7 + 163377088T^6 + 46270799360T^5 + 111336174592T^4 - 149138899279872T^3 - 1739516682190848T^2 + 143248894480613376*T + 2235755302725353472)^2
$79$	$T^{20} + \cdots + 33\!\cdots\!96$ T^20 + 78388T^18 + 2641661148T^16 + 49974481459072T^14 + 580965250927451974T^12 + 4265777229417942651480T^10 + 19546145498700242225308908T^8 + 53059275172802531769847888896T^6 + 76327211442810057252438121984737T^4 + 45867958110774775917667307760013044*T^2 + 3306213416000165009101900598951367696
$83$	$T^{20} + \cdots + 43\!\cdots\!96$ T^20 + 45752T^18 + 892760848T^16 + 9688852045568T^14 + 63903206593474048T^12 + 262701570954120009728T^10 + 664110027252280137318400T^8 + 984121987866881443135619072T^6 + 787120114687847763720764194816T^4 + 303947399636063053908550861455360*T^2 + 43082717757818941693664682307485696
$89$	$(T^{10} + \cdots + 75\!\cdots\!08)^{2}$ (T^10 + 76T^9 - 34384T^8 - 2957312T^7 + 286258912T^6 + 25841478464T^5 - 690710395648T^4 - 73496718972928T^3 + 86065250761216T^2 + 60509205905820672*T + 756818116524868608)^2
$97$	$(T^{10} + \cdots + 81\!\cdots\!24)^{2}$ (T^10 - 28T^9 - 58576T^8 - 1394176T^7 + 1202621856T^6 + 85136947520T^5 - 6863516723200T^4 - 962564496921600T^3 - 32632066730234368T^2 - 90809736302922752*T + 8121313541007668224)^2
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Overview	Random
Universe	Knowledge

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 175.20
Significant digits:
Format:

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