LMFDB - Newform orbit 1470.2.m.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1470,2,Mod(97,1470)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1470, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1470.97"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1470.m (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,-8,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$11.7380090971$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 32 x^{13} + 2 x^{12} + 352 x^{10} - 288 x^{9} + 2 x^{8} - 1440 x^{7} + 8800 x^{6} + \cdots + 390625$ x^16 - 32x^13 + 2x^12 + 352x^10 - 288x^9 + 2x^8 - 1440x^7 + 8800x^6 + 1250x^4 - 100000*x^3 + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -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.

$f(q)$	$=$	$q - \beta_{6} q^{2} - \beta_{6} q^{3} - \beta_{5} q^{4} + \beta_{3} q^{5} - \beta_{5} q^{6} + \beta_{10} q^{8} - \beta_{5} q^{9} - \beta_1 q^{10} + ( - \beta_{15} - \beta_{11} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{10} - \beta_{9} + \cdots - \beta_1) q^{99}+O(q^{100})$ q - b6 * q^2 - b6 * q^3 - b5 * q^4 + b3 * q^5 - b5 * q^6 + b10 * q^8 - b5 * q^9 - b1 * q^10 + (-b15 - b11 + b10 + b8 - b7 + b6 - b4 - b2) * q^11 + b10 * q^12 + (b14 - b12 - b11 - b7 + b5 - b4) * q^13 - b1 * q^15 - q^16 + (b15 - b14 - b12 + b11 - 3b8 + b7 - b6 + b5 + 2b4 - b1) * q^17 + b10 * q^18 + (b15 + b14 - b10 - b6 + b3 + b2 + 4) * q^19 + (-b12 + b5) * q^20 + (-b13 - b11 + b5 - b4 + b3) * q^22 + (b15 + b13 - b6 - b5 + b3 - b1) * q^23 - q^24 + (-b15 + 2b11 - 2b8 + 4b4 - b3) q^25 + (-b15 - b10 - b8 + b7 + b6 - b4 + b2) * q^26 + b10 * q^27 + (-b15 - 2b10 + b7 + 2b6 - b4 + b2) * q^29 + (-b12 + b5) * q^30 + (b14 - b13 - b12 - b10 - b9 - 2b8 - b7 - 2b5 + b4 - b3 - b1) * q^31 + b6 * q^32 + (-b13 - b11 + b5 - b4 + b3) * q^33 + (b15 + b13 - b12 + 3b11 - b10 - b8 + b7 - b6 + b4 + b2) q^34 - q^36 + (b15 + b13 - 2b11 - b9 - b7 - b6 - b5 - 2b4 - b3 - b2 + b1) * q^37 + (-b15 + b13 - 3b6 - b5 - b3 - b1) q^38 + (-b15 - b10 - b8 + b7 + b6 - b4 + b2) * q^39 + (-b10 + b2) * q^40 + (-b15 - b13 - b12 - 2b10 - b9 - 2b8 + b6 - 2b5 + b2 - b1) q^41 + (-b15 - 3b8 + b6 + 3b4 + b1) * q^43 + (-b10 - b9 - b8 - b1) * q^44 + (-b12 + b5) * q^45 + (b13 - b12 + b10 + b9 - b1) * q^46 + (-b15 + 2b13 + 4b10 + 3b8 + b6 - 2b5 - 3b4 + 2b3 + b1) * q^47 + b6 * q^48 + (-b13 + 2b11 + 2b8 + b1) * q^50 + (b15 + b13 - b12 + 3b11 - b10 - b8 + b7 - b6 + b4 + b2) q^51 + (-b13 + b11 - 2b8 + b7 + b5 + b4 - b3) q^52 + (-2b15 + b13 - b11 - 2b10 + b9 + 4b8 - b7 + 2b6 - 3b4 + b3 - b2 + 2b1 + 1) * q^53 - q^54 + (b15 + 3b11 - b10 - 3b8 + 5b7 - 5b6 + b4 + b2) * q^55 + (-b15 + b13 - 3b6 - b5 - b3 - b1) q^57 + (-b13 - b8 + 2b5 + b4 - b3 + 1) q^58 + (-b14 + 2b13 - 2b12 - 2b11 + b10 + b9 + 3b8 - 3b7 - 3b4 - b3 - b1 - 4) * q^59 + (-b10 + b2) * q^60 + (b15 - b14 - 2b10 + b9 + 2b8 + 4b7 + 3b6 - 4b4 + b3 - b2 + b1) q^61 + (-b15 - b14 - b12 + 2b11 + 2b10 - b9 - b8 + 2b7 + b6 + b5 - b4 + b2 + b1) q^62 + b5 * q^64 + (b15 - b14 + b13 + b12 - b11 + 2b8 + b7 - b5 + b4 + 4) q^65 + (-b10 - b9 - b8 - b1) * q^66 + (-2b15 - b14 + b13 + b12 - b11 + b9 + 2b7 + 2b6 + 2b5 - b4 - b3 + b2 - 2b1 - 4) q^67 + (b13 + b11 + b9 + 2b7 - b5 + b4 - b3 + b2) q^68 + (b13 - b12 + b10 + b9 - b1) * q^69 + (-2b15 + b13 - b12 + b10 - b9 + 2b8 - 2b7 + 2b6 - 2b4 - 2b2 + b1) * q^71 + b6 * q^72 + (-b15 + 2b14 - 2b12 - b11 + b7 - 3b6 + 2b5 - b4 - b1) * q^73 + (-b14 + b13 + b12 + b10 + b9 - 2b8 + b7 - 2b5 - b4 + b3 + b1) * q^74 + (-b13 + 2b11 + 2b8 + b1) * q^75 + (-b13 - b12 + b10 + b9 - 2b5 + b1) q^76 + (-b13 + b11 - 2b8 + b7 + b5 + b4 - b3) q^78 + (b15 - b14 + b13 + b12 - 2b10 + b9 + 2b8 - 4b7 + 3b6 - 2b5 + 4b4 + b3 - b2 + b1) * q^79 - b3 * q^80 - q^81 + (-b14 - b13 - b12 + 2b11 + 2b10 - b9 - 2b8 + 2b7 + 2b5 - b3 + b2) q^82 + (-b15 - b14 - b13 + b12 + 2b11 + b9 + 2b7 + 5b6 + 2b4 + b3 + b2 - b1) * q^83 + (-2b15 + b14 + b13 + b12 - b11 + 7b8 - b7 + 5b6 - b5 - 2b4 + b1 - 4) * q^85 + (-b13 + b12 + 3b11) q^86 + (-b13 - b8 + 2b5 + b4 - b3 + 1) q^87 + (-b14 - b12 + b11 - b8 + b7 + b5) * q^88 + (-2b15 + b13 - b12 - 4b11 - 2b9 + 3b8 - 3b7 + 2b6 - 3b4 - 2b2 + 2b1 + 4) q^89 + (-b10 + b2) * q^90 + (b14 - b12 + b9 + b5 + b2) * q^92 + (-b15 - b14 - b12 + 2b11 + 2b10 - b9 - b8 + 2b7 + b6 + b5 - b4 + b2 + b1) q^93 + (-b13 + b12 - 3b11 + 2b10 + 2b9 - 2b1 - 4) * q^94 + (-b15 - b14 + b10 - 4b7 + 5b6 + 4b4 + 3b3 - b2 + 4) * q^95 + b5 * q^96 + (2b15 - b14 + b13 - b12 - b11 + 2b10 - b9 - b7 - 2b6 + 4b5 + b4 + b3 + b2 - 2b1 + 4) q^97 + (-b10 - b9 - b8 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 8 q^{5} - 8 q^{13} - 16 q^{16} + 8 q^{17} + 48 q^{19} - 8 q^{22} - 8 q^{23} - 16 q^{24} + 8 q^{25} - 8 q^{33} - 16 q^{36} + 8 q^{37} + 8 q^{38} - 16 q^{47} + 8 q^{52} + 8 q^{53} - 16 q^{54} + 8 q^{57}+ \cdots + 64 q^{97}+O(q^{100})$ 16 * q - 8 * q^5 - 8 * q^13 - 16 * q^16 + 8 * q^17 + 48 * q^19 - 8 * q^22 - 8 * q^23 - 16 * q^24 + 8 * q^25 - 8 * q^33 - 16 * q^36 + 8 * q^37 + 8 * q^38 - 16 * q^47 + 8 * q^52 + 8 * q^53 - 16 * q^54 + 8 * q^57 + 24 * q^58 - 48 * q^59 + 8 * q^62 + 72 * q^65 - 48 * q^67 + 8 * q^68 - 16 * q^73 + 8 * q^78 + 8 * q^80 - 16 * q^81 + 16 * q^82 - 72 * q^85 + 24 * q^87 + 8 * q^88 + 64 * q^89 - 8 * q^92 + 8 * q^93 - 64 * q^94 + 48 * q^95 + 64 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 32 x^{13} + 2 x^{12} + 352 x^{10} - 288 x^{9} + 2 x^{8} - 1440 x^{7} + 8800 x^{6} + \cdots + 390625$ x^16 - 32*x^13 + 2*x^12 + 352*x^10 - 288*x^9 + 2*x^8 - 1440*x^7 + 8800*x^6 + 1250*x^4 - 100000*x^3 + 390625 :

$\beta_{1}$	$=$	$( 23 \nu^{15} + 151945 \nu^{14} - 513400 \nu^{13} - 30736 \nu^{12} - 3356569 \nu^{11} + \cdots + 17286250000 ) / 263500000$ (23v^15 + 151945v^14 - 513400v^13 - 30736v^12 - 3356569v^11 + 10867065v^10 - 183704v^9 + 19988016v^8 - 99781289v^7 + 133354345v^6 - 9130200v^5 + 988282000v^4 - 2959145625v^3 - 176434375v^2 - 5904875000*v + 17286250000) / 263500000
$\beta_{2}$	$=$	$( - 12176 \nu^{15} - 711160 \nu^{14} + 2464225 \nu^{13} + 334507 \nu^{12} + 13822768 \nu^{11} + \cdots - 88433203125 ) / 1317500000$ (-12176v^15 - 711160v^14 + 2464225v^13 + 334507v^12 + 13822768v^11 - 53602520v^10 + 2515873v^9 - 71108757v^8 + 489526928v^7 - 655014680v^6 - 20484575v^5 - 4050419125v^4 + 14571370000v^3 + 743925000v^2 + 21019390625*v - 88433203125) / 1317500000
$\beta_{3}$	$=$	$( - 7153 \nu^{15} + 234160 \nu^{14} - 230600 \nu^{13} + 200771 \nu^{12} - 4866801 \nu^{11} + \cdots + 7283671875 ) / 263500000$ (-7153v^15 + 234160v^14 - 230600v^13 + 200771v^12 - 4866801v^11 + 4897520v^10 - 1954056v^9 + 29256259v^8 - 86582961v^7 + 61131440v^6 - 80373000v^5 + 1382043875v^4 - 1294480625v^3 + 436350000v^2 - 7983125000*v + 7283671875) / 263500000
$\beta_{4}$	$=$	$( - 52781 \nu^{15} - 277595 \nu^{14} + 417475 \nu^{13} + 1210867 \nu^{12} + 5695603 \nu^{11} + \cdots - 6183828125 ) / 1317500000$ (-52781v^15 - 277595v^14 + 417475v^13 + 1210867v^12 + 5695603v^11 - 6617515v^10 - 6615837v^9 - 21020637v^8 + 97051123v^7 - 100536475v^6 - 290692925v^5 - 1656298125v^4 + 1662561875v^3 + 2208303125v^2 + 9931171875*v - 6183828125) / 1317500000
$\beta_{5}$	$=$	$( 10532 \nu^{15} - 94195 \nu^{14} + 91100 \nu^{13} - 196524 \nu^{12} + 2002804 \nu^{11} + \cdots - 3498281250 ) / 164687500$ (10532v^15 - 94195v^14 + 91100v^13 - 196524v^12 + 2002804v^11 - 2187965v^10 + 1205964v^9 - 14275106v^8 + 36913924v^7 - 25413395v^6 + 64163100v^5 - 565159000v^4 + 555072500v^3 - 278928125v^2 + 3362187500*v - 3498281250) / 164687500
$\beta_{6}$	$=$	$( 5499 \nu^{15} - 36198 \nu^{14} + 69810 \nu^{13} - 110818 \nu^{12} + 799959 \nu^{11} + \cdots - 2325781250 ) / 65875000$ (5499v^15 - 36198v^14 + 69810v^13 - 110818v^12 + 799959v^11 - 1500066v^10 + 759218v^9 - 6093858v^8 + 17642267v^7 - 19619686v^6 + 35441490v^5 - 226977250v^4 + 401156375v^3 - 182746250v^2 + 1367131250*v - 2325781250) / 65875000
$\beta_{7}$	$=$	$( - 124784 \nu^{15} + 289765 \nu^{14} + 543850 \nu^{13} + 2971213 \nu^{12} - 6082048 \nu^{11} + \cdots - 16529921875 ) / 1317500000$ (-124784v^15 + 289765v^14 + 543850v^13 + 2971213v^12 - 6082048v^11 - 9726795v^10 - 17542518v^9 + 63233197v^8 - 18736688v^7 - 125413435v^6 - 834999350v^5 + 1529546125v^4 + 2922380000v^3 + 5544903125v^2 - 8980093750*v - 16529921875) / 1317500000
$\beta_{8}$	$=$	$( 128033 \nu^{15} - 35880 \nu^{14} + 411075 \nu^{13} - 2683056 \nu^{12} + 1413601 \nu^{11} + \cdots - 10391250000 ) / 1317500000$ (128033v^15 - 35880v^14 + 411075v^13 - 2683056v^12 + 1413601v^11 - 7551160v^10 + 15219891v^9 - 45725264v^8 + 46597281v^7 - 118375880v^6 + 765575875v^5 - 356214000v^4 + 2128850625v^3 - 4479975000v^2 + 3063921875*v - 10391250000) / 1317500000
$\beta_{9}$	$=$	$( - 221264 \nu^{15} + 115 \nu^{14} + 759725 \nu^{13} + 4513448 \nu^{12} - 596208 \nu^{11} + \cdots - 29524375000 ) / 1317500000$ (-221264v^15 + 115v^14 + 759725v^13 + 4513448v^12 - 596208v^11 - 16782845v^10 - 23549603v^9 + 62805512v^8 + 99497552v^7 - 180286285v^6 - 1280351475v^5 - 45651000v^4 + 4664830000v^3 + 7330671875v^2 + 435328125*v - 29524375000) / 1317500000
$\beta_{10}$	$=$	$( - 72414 \nu^{15} + 116510 \nu^{14} - 145225 \nu^{13} + 1495498 \nu^{12} - 2394398 \nu^{11} + \cdots + 4653906250 ) / 329375000$ (-72414v^15 + 116510v^14 - 145225v^13 + 1495498v^12 - 2394398v^11 + 3173970v^10 - 8656053v^9 + 31497002v^8 - 46480158v^7 + 46384230v^6 - 430608425v^5 + 654882250v^4 - 823538750v^3 + 2448231250v^2 - 3672203125*v + 4653906250) / 329375000
$\beta_{11}$	$=$	$( - 2029 \nu^{15} - 230 \nu^{14} + 12455 \nu^{13} + 40928 \nu^{12} + 1427 \nu^{11} + \cdots - 448000000 ) / 8500000$ (-2029v^15 - 230v^14 + 12455v^13 + 40928v^12 + 1427v^11 - 265270v^10 - 205673v^9 + 515392v^8 + 1584467v^7 - 3089990v^6 - 11581465v^5 - 2887200v^4 + 73925875v^3 + 65686250v^2 + 8384375*v - 448000000) / 8500000
$\beta_{12}$	$=$	$( 374136 \nu^{15} + 550015 \nu^{14} - 2860075 \nu^{13} - 7558352 \nu^{12} - 10487208 \nu^{11} + \cdots + 107188750000 ) / 1317500000$ (374136v^15 + 550015v^14 - 2860075v^13 - 7558352v^12 - 10487208v^11 + 63213055v^10 + 36024597v^9 - 32747888v^8 - 508697448v^7 + 726564415v^6 + 1997199925v^5 + 3498498000v^4 - 17288645000v^3 - 12093690625v^2 - 19156796875*v + 107188750000) / 1317500000
$\beta_{13}$	$=$	$( - 372809 \nu^{15} + 616665 \nu^{14} + 2116200 \nu^{13} + 7167888 \nu^{12} - 13513273 \nu^{11} + \cdots - 73231250000 ) / 1317500000$ (-372809v^15 + 616665v^14 + 2116200v^13 + 7167888v^12 - 13513273v^11 - 43000695v^10 - 34237368v^9 + 168261072v^8 + 85472887v^7 - 527876935v^6 - 2104753400v^5 + 3333950000v^4 + 12378994375v^3 + 11838115625v^2 - 20797375000*v - 73231250000) / 1317500000
$\beta_{14}$	$=$	$( 377912 \nu^{15} - 1106320 \nu^{14} + 575 \nu^{13} - 8294559 \nu^{12} + 23323064 \nu^{11} + \cdots - 4410859375 ) / 1317500000$ (377912v^15 - 1106320v^14 + 575v^13 - 8294559v^12 + 23323064v^11 - 2981040v^10 + 49110799v^9 - 226586671v^8 + 314783384v^7 - 46705520v^6 + 2424194175v^5 - 6401757375v^4 + 244135000v^3 - 14467050000v^2 + 36653359375*v - 4410859375) / 1317500000
$\beta_{15}$	$=$	$( - 95903 \nu^{15} + 201256 \nu^{14} - 10640 \nu^{13} + 2025921 \nu^{12} - 4520623 \nu^{11} + \cdots + 559140625 ) / 263500000$ (-95903v^15 + 201256v^14 - 10640v^13 + 2025921v^12 - 4520623v^11 + 677992v^10 - 11953936v^9 + 50671601v^8 - 55705439v^7 + 10534952v^6 - 590045520v^5 + 1207441825v^4 - 26643375v^3 + 3471705000v^2 - 7407050000*v + 559140625) / 263500000

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

$\nu$	$=$	$( -\beta_{11} + 2\beta_{9} + \beta_{8} ) / 2$ (-b11 + 2*b9 + b8) / 2
$\nu^{2}$	$=$	$\beta_{13} - 2\beta_{11} - 2\beta_{8} - \beta_1$ b13 - 2b11 - 2b8 - b1
$\nu^{3}$	$=$	$( 2\beta_{15} - 2\beta_{14} - 3\beta_{11} + 3\beta_{8} + 2\beta_{7} - 8\beta_{4} - 6\beta_{3} + 8 ) / 2$ (2b15 - 2b14 - 3b11 + 3b8 + 2b7 - 8b4 - 6*b3 + 8) / 2
$\nu^{4}$	$=$	$-\beta_{14} + \beta_{12} - 8\beta_{11} + 8\beta_{10} + 7\beta_{9} + 8\beta_{8} - 4\beta_{7} - \beta_{5} + 7\beta_{2} - 1$ -b14 + b12 - 8b11 + 8b10 + 7b9 + 8b8 - 4b7 - b5 + 7b2 - 1
$\nu^{5}$	$=$	$( - 2 \beta_{15} - 2 \beta_{13} - 22 \beta_{12} - 41 \beta_{11} - 2 \beta_{9} - 31 \beta_{8} + \cdots - 22 \beta_1 ) / 2$ (-2b15 - 2b13 - 22b12 - 41b11 - 2b9 - 31b8 + 10b6 - 48b5 - 2b4 - 22b1) / 2
$\nu^{6}$	$=$	$- 25 \beta_{15} - 15 \beta_{14} - \beta_{13} + 2 \beta_{11} - \beta_{10} + 2 \beta_{8} + 20 \beta_{7} + \cdots + 40$ -25b15 - 15b14 - b13 + 2b11 - b10 + 2b8 + 20b7 - 55b6 - 20b4 - 25b3 + b2 - 15*b1 + 40
$\nu^{7}$	$=$	$( - 2 \beta_{15} + 130 \beta_{14} + 2 \beta_{13} + 38 \beta_{12} - 75 \beta_{11} + 160 \beta_{10} + \cdots + 280 ) / 2$ (-2b15 + 130b14 + 2b13 + 38b12 - 75b11 + 160b10 + 130b9 + 77b8 + 120b7 - 48b5 + 8b4 + 38b3 + 90*b2 + 280) / 2
$\nu^{8}$	$=$	$- 23 \beta_{15} + \beta_{14} - 15 \beta_{13} + 15 \beta_{12} - 80 \beta_{11} - 72 \beta_{10} + 121 \beta_{9} + \cdots + 1$ -23b15 + b14 - 15b13 + 15b12 - 80b11 - 72b10 + 121b9 - 260b8 + 4b7 + 95b6 - 240b5 - 28b4 + b3 - 23b2 - 121*b1 + 1
$\nu^{9}$	$=$	$( - 510 \beta_{15} - 386 \beta_{14} + 386 \beta_{13} + 54 \beta_{12} - 279 \beta_{11} - 64 \beta_{10} + \cdots + 824 ) / 2$ (-510b15 - 386b14 + 386b13 + 54b12 - 279b11 - 64b10 - 545b8 + 616b7 + 150b6 + 176b5 - 360b4 - 438b3 + 54b2 + 438b1 + 824) / 2
$\nu^{10}$	$=$	$281 \beta_{15} + 399 \beta_{14} + 32 \beta_{13} + 281 \beta_{12} - 686 \beta_{11} + 696 \beta_{10} + \cdots - 696$ 281b15 + 399b14 + 32b13 + 281b12 - 686b11 + 696b10 + 631b9 + 718b8 + 876b7 + 135b6 - 416b5 - 684b4 - 631b3 + 399b2 + 32*b1 - 696
$\nu^{11}$	$=$	$( - 768 \beta_{15} - 768 \beta_{14} - 130 \beta_{13} + 954 \beta_{12} - 2725 \beta_{11} + 3680 \beta_{10} + \cdots - 1088 ) / 2$ (-768b15 - 768b14 - 130b13 + 954b12 - 2725b11 + 3680b10 - 130b9 - 3715b8 - 2042b7 + 2810b6 - 4944b5 - 1088b4 + 2630b2 - 2630b1 - 1088) / 2
$\nu^{12}$	$=$	$- 2537 \beta_{15} - 2336 \beta_{14} - 1905 \beta_{13} - 2336 \beta_{12} - 456 \beta_{11} - 544 \beta_{10} + \cdots + 4239$ -2537b15 - 2336b14 - 1905b13 - 2336b12 - 456b11 - 544b10 - 544b9 - 1124b8 + 4256b7 + 2385b6 - 4239b5 - 2212b4 - 1905b3 + 544b2 + 2537*b1 + 4239
$\nu^{13}$	$=$	$( - 12288 \beta_{15} + 5378 \beta_{14} + 7424 \beta_{12} - 1727 \beta_{11} + 10816 \beta_{10} + \cdots - 19992 ) / 2$ (-12288b15 + 5378b14 + 7424b12 - 1727b11 + 10816b10 + 12288b9 + 7167b8 + 19992b7 - 23360b6 - 12864b5 + 6762b4 - 8234b3 + 8234b2 - 7424b1 - 19992) / 2
$\nu^{14}$	$=$	$- 6432 \beta_{15} + 14112 \beta_{14} + 736 \beta_{13} + 5879 \beta_{12} - 242 \beta_{11} + 19849 \beta_{10} + \cdots + 32672$ -6432b15 + 14112b14 + 736b13 + 5879b12 - 242b11 + 19849b10 - 5879b9 - 29742b8 + 16608b7 + 18560b6 - 26464b5 - 6432b4 + 6432b3 + 736b2 - 14112*b1 + 32672
$\nu^{15}$	$=$	$( - 51456 \beta_{15} - 52480 \beta_{14} - 51456 \beta_{13} + 31744 \beta_{12} + 76979 \beta_{11} + \cdots + 88640 ) / 2$ (-51456b15 - 52480b14 - 51456b13 + 31744b12 + 76979b11 - 64320b10 + 52480b9 - 69645b8 + 116800b7 + 58790b6 - 39104b5 - 58816b4 - 31744b3 + 50406b1 + 88640) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$ .

$n$	$491$	$1081$	$1177$
$\chi(n)$	$1$	$-1$	$-\beta_{5}$

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}

97.1

2.23272	+	0.122339i
−1.22045	−	1.87363i
−0.410538	+	2.19806i
−2.01595	+	0.967451i
−1.07534	+	1.96052i
−1.12594	−	1.93191i
2.21573	+	0.300921i
1.39977	−	1.74375i
2.23272	−	0.122339i
−1.22045	+	1.87363i
−0.410538	−	2.19806i
−2.01595	−	0.967451i
−1.07534	−	1.96052i
−1.12594	+	1.93191i
2.21573	−	0.300921i
1.39977	+	1.74375i

−0.707107

−

0.707107i

−0.707107

−

0.707107i

1.00000i

−2.10958

0.741398i

1.00000i

0.707107

−

0.707107i

1.00000i

2.01595

0.967451i

97.2

−0.707107

−

0.707107i

−0.707107

−

0.707107i

1.00000i

−1.84456

−

1.26397i

1.00000i

0.707107

−

0.707107i

1.00000i

0.410538

2.19806i

97.3

−0.707107

−

0.707107i

−0.707107

−

0.707107i

1.00000i

0.461873

2.18785i

1.00000i

0.707107

−

0.707107i

1.00000i

1.22045

−

1.87363i

97.4

−0.707107

−

0.707107i

−0.707107

−

0.707107i

1.00000i

1.49226

−

1.66528i

1.00000i

0.707107

−

0.707107i

1.00000i

−2.23272

0.122339i

97.5

0.707107

0.707107i

0.707107

0.707107i

1.00000i

−2.22280

−

0.243230i

1.00000i

−0.707107

0.707107i

1.00000i

−1.39977

−

1.74375i

97.6

0.707107

0.707107i

0.707107

0.707107i

1.00000i

−1.35397

1.77954i

1.00000i

−0.707107

0.707107i

1.00000i

−2.21573

0.300921i

97.7

0.707107

0.707107i

0.707107

0.707107i

1.00000i

−0.569907

−

2.16222i

1.00000i

−0.707107

0.707107i

1.00000i

1.12594

−

1.93191i

97.8

0.707107

0.707107i

0.707107

0.707107i

1.00000i

2.14668

0.625913i

1.00000i

−0.707107

0.707107i

1.00000i

1.07534

1.96052i

1273.1

−0.707107

0.707107i

−0.707107

0.707107i

−

1.00000i

−2.10958

−

0.741398i

−

1.00000i

0.707107

0.707107i

−

1.00000i

2.01595

−

0.967451i

1273.2

−0.707107

0.707107i

−0.707107

0.707107i

−

1.00000i

−1.84456

1.26397i

−

1.00000i

0.707107

0.707107i

−

1.00000i

0.410538

−

2.19806i

1273.3

−0.707107

0.707107i

−0.707107

0.707107i

−

1.00000i

0.461873

−

2.18785i

−

1.00000i

0.707107

0.707107i

−

1.00000i

1.22045

1.87363i

1273.4

−0.707107

0.707107i

−0.707107

0.707107i

−

1.00000i

1.49226

1.66528i

−

1.00000i

0.707107

0.707107i

−

1.00000i

−2.23272

−

0.122339i

1273.5

0.707107

−

0.707107i

0.707107

−

0.707107i

−

1.00000i

−2.22280

0.243230i

−

1.00000i

−0.707107

−

0.707107i

−

1.00000i

−1.39977

1.74375i

1273.6

0.707107

−

0.707107i

0.707107

−

0.707107i

−

1.00000i

−1.35397

−

1.77954i

−

1.00000i

−0.707107

−

0.707107i

−

1.00000i

−2.21573

−

0.300921i

1273.7

0.707107

−

0.707107i

0.707107

−

0.707107i

−

1.00000i

−0.569907

2.16222i

−

1.00000i

−0.707107

−

0.707107i

−

1.00000i

1.12594

1.93191i

1273.8

0.707107

−

0.707107i

0.707107

−

0.707107i

−

1.00000i

2.14668

−

0.625913i

−

1.00000i

−0.707107

−

0.707107i

−

1.00000i

1.07534

−

1.96052i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
35.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1470.2.m.c	✓	16
5.c	odd	4	1		1470.2.m.f	yes	16
7.b	odd	2	1		1470.2.m.f	yes	16
35.f	even	4	1	inner	1470.2.m.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1470.2.m.c	✓	16	1.a	even	1	1	trivial
1470.2.m.c	✓	16	35.f	even	4	1	inner
1470.2.m.f	yes	16	5.c	odd	4	1
1470.2.m.f	yes	16	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1470, [\chi])$ :

T_{11}^{8} - 40T_{11}^{6} + 8T_{11}^{5} + 382T_{11}^{4} - 160T_{11}^{3} - 1088T_{11}^{2} + 512T_{11} + 512

T11^8 - 40*T11^6 + 8*T11^5 + 382*T11^4 - 160*T11^3 - 1088*T11^2 + 512*T11 + 512

T_{13}^{16} + 8 T_{13}^{15} + 32 T_{13}^{14} + 16 T_{13}^{13} + 740 T_{13}^{12} + 5632 T_{13}^{11} + \cdots + 18496

T13^16 + 8*T13^15 + 32*T13^14 + 16*T13^13 + 740*T13^12 + 5632*T13^11 + 21504*T13^10 + 12480*T13^9 + 107444*T13^8 + 648992*T13^7 + 1900160*T13^6 + 2597824*T13^5 + 1910048*T13^4 + 394624*T13^3 + 4608*T13^2 - 13056*T13 + 18496

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} + 1)^{4}$ (T^4 + 1)^4
$3$	$(T^{4} + 1)^{4}$ (T^4 + 1)^4
$5$	$T^{16} + 8 T^{15} + \cdots + 390625$ T^16 + 8T^15 + 28T^14 + 56T^13 + 70T^12 + 56T^11 + 28T^10 + 8T^9 + 2T^8 + 40T^7 + 700T^6 + 7000T^5 + 43750T^4 + 175000T^3 + 437500T^2 + 625000*T + 390625
$7$	$T^{16}$ T^16
$11$	$(T^{8} - 40 T^{6} + \cdots + 512)^{2}$ (T^8 - 40T^6 + 8T^5 + 382T^4 - 160T^3 - 1088T^2 + 512T + 512)^2
$13$	$T^{16} + 8 T^{15} + \cdots + 18496$ T^16 + 8T^15 + 32T^14 + 16T^13 + 740T^12 + 5632T^11 + 21504T^10 + 12480T^9 + 107444T^8 + 648992T^7 + 1900160T^6 + 2597824T^5 + 1910048T^4 + 394624T^3 + 4608T^2 - 13056*T + 18496
$17$	$T^{16} - 8 T^{15} + \cdots + 1336336$ T^16 - 8T^15 + 32T^14 + 64T^13 + 3152T^12 - 26096T^11 + 109952T^10 + 242176T^9 - 130504T^8 - 598240T^7 + 18393216T^6 + 75580672T^5 + 151093568T^4 + 153372096T^3 + 85229568T^2 + 15092736*T + 1336336
$19$	$(T^{8} - 24 T^{7} + \cdots + 31744)^{2}$ (T^8 - 24T^7 + 180T^6 - 144T^5 - 3708T^4 + 12352T^3 + 5248T^2 - 54272*T + 31744)^2
$23$	$T^{16} + 8 T^{15} + \cdots + 4194304$ T^16 + 8T^15 + 32T^14 - 48T^13 + 2184T^12 + 14752T^11 + 49280T^10 + 26176T^9 - 8176T^8 + 119808T^7 + 1146880T^6 + 49152T^5 - 475136T^4 - 1572864T^3 + 8388608T^2 - 8388608*T + 4194304
$29$	$T^{16} + 112 T^{14} + \cdots + 64$ T^16 + 112T^14 + 4204T^12 + 63568T^10 + 351348T^8 + 779424T^6 + 620448T^4 + 77184*T^2 + 64
$31$	$T^{16} + \cdots + 303038464$ T^16 + 320T^14 + 39804T^12 + 2468928T^10 + 80125444T^8 + 1273638912T^6 + 7772106752T^4 + 9677307904*T^2 + 303038464
$37$	$T^{16} + \cdots + 1224191770624$ T^16 - 8T^15 + 32T^14 - 272T^13 + 17924T^12 - 162000T^11 + 759424T^10 - 2436896T^9 + 40893956T^8 - 362205056T^7 + 1761953792T^6 - 4308895744T^5 + 14443317248T^4 - 88779096064T^3 + 429404979200T^2 - 1025352663040*T + 1224191770624
$41$	$T^{16} + \cdots + 14484603904$ T^16 + 344T^14 + 44724T^12 + 2790480T^10 + 89708676T^8 + 1468735296T^6 + 10787820672T^4 + 22834207744*T^2 + 14484603904
$43$	$T^{16} + \cdots + 3347316736$ T^16 - 224T^13 + 9164T^12 - 14784T^11 + 25088T^10 - 911680T^9 + 11314820T^8 - 41145216T^7 + 83593216T^6 - 145625088T^5 + 932091904T^4 - 3299573760T^3 + 6576668672T^2 - 6635388928*T + 3347316736
$47$	$T^{16} + \cdots + 1212153856$ T^16 + 16T^15 + 128T^14 + 96T^13 + 26380T^12 + 402592T^11 + 3069440T^10 + 4097984T^9 + 179488004T^8 + 2530418560T^7 + 17973970944T^6 + 40742932480T^5 + 49173127168T^4 + 21445607424T^3 + 2283798528T^2 - 2353004544*T + 1212153856
$53$	$T^{16} + \cdots + 1176027464704$ T^16 - 8T^15 + 32T^14 + 208T^13 + 23960T^12 - 178336T^11 + 681600T^10 + 4147648T^9 + 176869584T^8 - 1148941824T^7 + 3965147136T^6 + 20328189440T^5 + 373704430336T^4 - 1667752409088T^3 + 3551688163328T^2 - 2890288160768*T + 1176027464704
$59$	$(T^{8} + 24 T^{7} + \cdots + 15023104)^{2}$ (T^8 + 24T^7 - 52T^6 - 5440T^5 - 37060T^4 + 174912T^3 + 2958208T^2 + 11758592*T + 15023104)^2
$61$	$T^{16} + \cdots + 98867482624$ T^16 + 648T^14 + 160788T^12 + 19485456T^10 + 1183933316T^8 + 31395442176T^6 + 189494849280T^4 + 266839572480*T^2 + 98867482624
$67$	$T^{16} + \cdots + 4228120576$ T^16 + 48T^15 + 1152T^14 + 17248T^13 + 222668T^12 + 3494880T^11 + 59987456T^10 + 825235648T^9 + 8252020868T^8 + 58689541248T^7 + 294240651264T^6 + 1006404622336T^5 + 2237420570624T^4 + 2849732984832T^3 + 1870827880448T^2 + 125778264064*T + 4228120576
$71$	$(T^{8} - 228 T^{6} + \cdots + 591872)^{2}$ (T^8 - 228T^6 - 896T^5 + 8004T^4 + 28544T^3 - 112384T^2 - 208896T + 591872)^2
$73$	$T^{16} + \cdots + 36991502500096$ T^16 + 16T^15 + 128T^14 + 416T^13 + 24668T^12 + 370112T^11 + 2850816T^10 + 11231360T^9 + 123435140T^8 + 1580872000T^7 + 12334017024T^6 + 48415579776T^5 + 134410557248T^4 + 615420529152T^3 + 4808075692032T^2 + 18860431807488*T + 36991502500096
$79$	$T^{16} + \cdots + 148243480576$ T^16 + 720T^14 + 193056T^12 + 24132352T^10 + 1442709760T^8 + 37532991488T^6 + 296029519872T^4 + 739137028096*T^2 + 148243480576
$83$	$T^{16} + \cdots + 1401249857536$ T^16 + 640T^13 + 45600T^12 + 97280T^11 + 204800T^10 + 13629440T^9 + 509747456T^8 + 1911521280T^7 + 4115660800T^6 + 30152458240T^5 + 540572778496T^4 + 2513080156160T^3 + 6060769280000T^2 + 4121323110400*T + 1401249857536
$89$	$(T^{8} - 32 T^{7} + \cdots + 15376)^{2}$ (T^8 - 32T^7 + 52T^6 + 5920T^5 - 32566T^4 - 173952T^3 + 196112T^2 + 781696*T + 15376)^2
$97$	$T^{16} + \cdots + 244234884096256$ T^16 - 64T^15 + 2048T^14 - 38880T^13 + 480604T^12 - 4021248T^11 + 28910080T^10 - 269183232T^9 + 2926317188T^8 - 22710734080T^7 + 117151090688T^6 - 512357261440T^5 + 4080338085696T^4 - 29994906685440T^3 + 135545346394112T^2 - 257312657933312*T + 244234884096256
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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 $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 97.8
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

Character values

Embeddings

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	1470.2.m.c

Level	$1470$
Weight	$2$
Character orbit	1470.m
Analytic conductor	$11.738$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$