LMFDB - Newform orbit 1680.3.bd.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(769,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.769"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.bd (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,48,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$45.7766844125$
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} + 88 x^{14} + 3876 x^{12} + 102922 x^{10} + 1866070 x^{8} + 23190492 x^{6} + 203608845 x^{4} + \cdots + 3839661225 $ x^16 + 88x^14 + 3876x^12 + 102922x^10 + 1866070x^8 + 23190492x^6 + 203608845x^4 + 1131190758*x^2 + 3839661225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 420)
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{6} q^{3} + (\beta_{11} + \beta_{6}) q^{5} - \beta_{10} q^{7} + 3 q^{9} + (\beta_{4} + 2) q^{11} + (\beta_{12} + \beta_{11} + \cdots - \beta_{9}) q^{13} + (\beta_{2} + 2) q^{15} + (\beta_{13} - 2 \beta_{12} + \cdots - 3 \beta_{6}) q^{17}+ \cdots + (3 \beta_{4} + 6) q^{99}+O(q^{100}) $ q + b6 * q^3 + (b11 + b6) * q^5 - b10 * q^7 + 3 * q^9 + (b4 + 2) * q^11 + (b12 + b11 - b10 - b9) * q^13 + (b2 + 2) * q^15 + (b13 - 2b12 - 2b11 + b10 + b9 - 3b6) q^17 - b3 * q^19 + (-b8 - 1) * q^21 + (-b10 + b9 + b5 + b2 + b1 + 1) * q^23 + (-b10 + b9 - b8 - b7 - b5 - b4 + b2 - b1 - 4) * q^25 + 3b6 q^27 + (-b8 - b7 - 3b4 + 2b2 - 2b1 - 4) q^29 + (b15 - b14 + 3b12 - 3b11) * q^31 + (b13 + b12 + b11 - b10 - b9 + 3b6) q^33 + (-b14 - b13 - b12 + b7 + b4 - b2 - 4) * q^35 + (b15 + b14 + b10 - b9 + b5 - 2b1 - 2) q^37 + (-b8 - b7 + b2 - b1 - 5) * q^39 + (b15 - b14 - b12 + b11 + b8 - b7 + 2b3) q^41 + (-b15 - b14 - 2b10 + 2b9 + b5 - 2b2) q^43 + (3b11 + 3b6) * q^45 + (b13 - 3b12 - 3b11 - b10 - b9 + 7b6) q^47 + (-b15 + b14 + b8 + 3b7 + b4 - b3 - 3b2 + 3b1 - 4) q^49 + (2b8 + 2b7 + 3b4 - 3b2 + 3b1 + 1) q^51 + (3b10 - 3b9 + b5 - b2 - b1 - 1) * q^53 + (2b13 - b12 + b11 - 2b10 - 2b9 - 2b8 + 2b7 + 8b6 - b3) * q^55 + (-b15 - b14 - b5 - b2 + b1 + 1) * q^57 + (b15 - b14 + 6b12 - 6b11 + 3b8 - 3b7) * q^59 + (-b15 + b14 + 5b12 - 5b11 + 2b8 - 2b7 - b3) * q^61 - 3b10 q^63 + (-b15 - b14 - b10 + b9 - b5 + b4 - 3b2 - b1 + 9) q^65 + (-b15 - b14 + b5 + 2b1 + 2) q^67 + (b15 - b14 - 2b12 + 2b11 - b8 + b7 + b3) * q^69 + (-b8 - b7 - 2b4 + 6b2 - 6b1 - 12) q^71 + (-2b13 - 9b12 - 9b11 + 2b10 + 2b9 - 22b6) * q^73 + (-b15 + b14 - b13 + b12 + 3b11 - 2b10 - 2b9 - b8 + b7 - b3) q^75 + (-b15 - b14 + b13 + 3b12 + 3b11 - 3b10 + 3b9 + 15b6 - 2b5 + 2b2 + 4b1 + 4) * q^77 + (-2b8 - 2b7 + 4b4 - 6b2 + 6b1) q^79 + 9 * q^81 + (-4b13 - 4b12 - 4b11 + 8b10 + 8b9 - 4b6) * q^83 + (b15 + b14 - 3b10 + 3b9 + b8 + b7 + 2b5 + 5b4 - b2 + 7b1 - 21) q^85 + (-3b13 + 3b12 + 3b11 + b6) q^87 + (b15 - b14 + 5b12 - 5b11 - 5b8 + 5b7 - 2b3) q^89 + (-b12 + b11 + 3b8 + 5b7 + 3b4 - b3 - 3b2 + 3b1 + 39) q^91 + (-b15 - b14 + 2b5 - 3b2 - b1 - 1) * q^93 + (-3b10 + 3b9 + 7b8 + 7b7 - 3b5 + 2b4 - 5b2 + 7b1 - 3) * q^95 + (-4b13 + b12 + b11 - 2b10 - 2b9 + 20b6) * q^97 + (3b4 + 6) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{9} + 24 q^{11} + 24 q^{15} - 12 q^{21} - 48 q^{25} - 32 q^{29} - 76 q^{35} - 72 q^{39} - 88 q^{49} - 24 q^{51} + 152 q^{65} - 168 q^{71} - 16 q^{79} + 144 q^{81} - 416 q^{85} + 568 q^{91} - 136 q^{95}+ \cdots + 72 q^{99}+O(q^{100}) $ 16 * q + 48 * q^9 + 24 * q^11 + 24 * q^15 - 12 * q^21 - 48 * q^25 - 32 * q^29 - 76 * q^35 - 72 * q^39 - 88 * q^49 - 24 * q^51 + 152 * q^65 - 168 * q^71 - 16 * q^79 + 144 * q^81 - 416 * q^85 + 568 * q^91 - 136 * q^95 + 72 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 88 x^{14} + 3876 x^{12} + 102922 x^{10} + 1866070 x^{8} + 23190492 x^{6} + 203608845 x^{4} + \cdots + 3839661225 $ x^16 + 88*x^14 + 3876*x^12 + 102922*x^10 + 1866070*x^8 + 23190492*x^6 + 203608845*x^4 + 1131190758*x^2 + 3839661225 :

$\beta_{1}$	$=$	$ ( 672001001684 \nu^{14} + 83638291569392 \nu^{12} + \cdots + 64\!\cdots\!09 ) / 92\!\cdots\!34 $ (672001001684v^14 + 83638291569392v^12 + 4272091016644632v^10 + 123974067663126407v^8 + 2178808884540756332v^6 + 24927134642650563432v^4 + 164573173831945222323*v^2 + 648661629583820097909) / 9275630741049654234
$\beta_{2}$	$=$	$ ( - 1035398609612 \nu^{14} - 111533330833976 \nu^{12} + \cdots - 79\!\cdots\!49 ) / 92\!\cdots\!34 $ (-1035398609612v^14 - 111533330833976v^12 - 5379199494309408v^10 - 149653330607213921v^8 - 2594192598074382740v^6 - 29131419888738608316v^4 - 201602403573332067189*v^2 - 794559645154337067549) / 9275630741049654234
$\beta_{3}$	$=$	$ ( 1601518644256 \nu^{15} + \cdots + 16\!\cdots\!49 \nu ) / 35\!\cdots\!05 $ (1601518644256v^15 - 1243791055410533v^13 - 83130180585664890v^11 - 2518744893744884417v^9 - 38877036874215059939v^7 - 363262564130722362285v^5 - 868324354542205414983v^3 + 16824872697243836939649v) / 3547928758451492744505
$\beta_{4}$	$=$	$ ( 634484072336 \nu^{14} + 36700472608334 \nu^{12} + \cdots - 79\!\cdots\!78 ) / 46\!\cdots\!17 $ (634484072336v^14 + 36700472608334v^12 + 1060452003667296v^10 + 15207683162139494v^8 + 168279695108599388v^6 + 647412056032704486v^4 + 1996888738264010496*v^2 - 79739766948019948578) / 4637815370524827117
$\beta_{5}$	$=$	$ ( - 15533455683352 \nu^{14} + \cdots + 77\!\cdots\!80 ) / 23\!\cdots\!85 $ (-15533455683352v^14 - 1180449492355747v^12 - 43360793116252413v^10 - 876318284088716878v^8 - 11033910504298438969v^6 - 75038762295166027026v^4 - 245160610606893679623*v^2 + 770744626057716884280) / 23189076852624135585
$\beta_{6}$	$=$	$ ( - 19154198883257 \nu^{15} + \cdots - 75\!\cdots\!61 \nu ) / 70\!\cdots\!10 $ (-19154198883257v^15 - 1546569916694156v^13 - 63571822352800752v^11 - 1547919464755800134v^9 - 25920757833965915885v^7 - 285311025541968291384v^5 - 2291825204891570440035v^3 - 7503372377553771177561v) / 7095857516902985489010
$\beta_{7}$	$=$	$ ( - 17963615837495 \nu^{15} - 860461496592840 \nu^{14} + \cdots - 46\!\cdots\!40 ) / 70\!\cdots\!10 $ (-17963615837495v^15 - 860461496592840v^14 + 2955992805288202v^13 - 81389302077257010v^12 + 230861652459815112v^11 - 3655926417831848100v^10 + 8076844319021149333v^9 - 96651721305214940160v^8 + 145437176981878050283v^7 - 1636638493005145024740v^6 + 1893639344756747339244v^5 - 17956389117864022789140v^4 + 14469794400468971075976v^3 - 120358275007032580434300v^2 + 94872323246581515223278*v - 468117452921951691963540) / 7095857516902985489010
$\beta_{8}$	$=$	$ ( 17963615837495 \nu^{15} - 860461496592840 \nu^{14} + \cdots - 46\!\cdots\!40 ) / 70\!\cdots\!10 $ (17963615837495v^15 - 860461496592840v^14 - 2955992805288202v^13 - 81389302077257010v^12 - 230861652459815112v^11 - 3655926417831848100v^10 - 8076844319021149333v^9 - 96651721305214940160v^8 - 145437176981878050283v^7 - 1636638493005145024740v^6 - 1893639344756747339244v^5 - 17956389117864022789140v^4 - 14469794400468971075976v^3 - 120358275007032580434300v^2 - 94872323246581515223278*v - 468117452921951691963540) / 7095857516902985489010
$\beta_{9}$	$=$	$ ( 301208776387831 \nu^{15} + \cdots + 81\!\cdots\!85 ) / 21\!\cdots\!30 $ (301208776387831v^15 + 2692351748830482v^14 + 17964341304450163v^13 + 220158916758918432v^12 + 484549998061732161v^11 + 8678455106638696983v^10 + 3993917644882916062v^9 + 197796238645556772333v^8 - 51003137021880397040v^7 + 2985424552488776600304v^6 - 2046329506496564444148v^5 + 30391105263446609513181v^4 - 20494949904502212421080v^3 + 206681271810133078548108v^2 - 132666874311026007414567*v + 815634568981543791999285) / 21287572550708956467030
$\beta_{10}$	$=$	$ ( 301208776387831 \nu^{15} + \cdots - 81\!\cdots\!85 ) / 21\!\cdots\!30 $ (301208776387831v^15 - 2692351748830482v^14 + 17964341304450163v^13 - 220158916758918432v^12 + 484549998061732161v^11 - 8678455106638696983v^10 + 3993917644882916062v^9 - 197796238645556772333v^8 - 51003137021880397040v^7 - 2985424552488776600304v^6 - 2046329506496564444148v^5 - 30391105263446609513181v^4 - 20494949904502212421080v^3 - 206681271810133078548108v^2 - 132666874311026007414567*v - 815634568981543791999285) / 21287572550708956467030
$\beta_{11}$	$=$	$ ( 543284346539459 \nu^{15} + \cdots - 14\!\cdots\!93 \nu ) / 21\!\cdots\!30 $ (543284346539459v^15 + 36027573534021182v^13 + 1119825456728871999v^11 + 16259996506215883748v^9 + 99792849989394910415v^7 - 927977383114080548367v^5 - 16248590176217896584090v^3 - 146125807930456620557193v) / 21287572550708956467030
$\beta_{12}$	$=$	$ ( 15307198600907 \nu^{15} + \cdots - 13\!\cdots\!69 \nu ) / 49\!\cdots\!10 $ (15307198600907v^15 + 1053651000795026v^13 + 34912939321992477v^11 + 594128216156992664v^9 + 5937613883562567575v^7 + 18229971398551841859v^5 - 58084626671359859160v^3 - 1361172757295722803669v) / 495059826760673406210
$\beta_{13}$	$=$	$ ( 21\!\cdots\!37 \nu^{15} + \cdots + 63\!\cdots\!31 \nu ) / 21\!\cdots\!30 $ (2145495653153737v^15 + 170623655058950176v^13 + 6675329225471977392v^11 + 152661950788010040184v^9 + 2371557898051849791385v^7 + 24782231535664543169664v^5 + 172645003170269350584165v^3 + 633345717488885596886931v) / 21287572550708956467030
$\beta_{14}$	$=$	$ ( 12\!\cdots\!87 \nu^{15} + \cdots + 53\!\cdots\!80 ) / 70\!\cdots\!10 $ (1235084031225187v^15 + 2778962112319995v^14 + 108027505159304863v^13 + 231578769083234895v^12 + 4561024066476113049v^11 + 9461525208149991060v^10 + 111874806596610825592v^9 + 222193009401277106475v^8 + 1767072461067558411118v^7 + 3358421919737873820360v^6 + 17777828841796776235098v^5 + 31603720753251597368025v^4 + 110767274631181060603326v^3 + 191664716523042803725440v^2 + 353101895277362051846949*v + 532329752913369234384780) / 7095857516902985489010
$\beta_{15}$	$=$	$ ( - 12\!\cdots\!87 \nu^{15} + \cdots + 53\!\cdots\!80 ) / 70\!\cdots\!10 $ (-1235084031225187v^15 + 2778962112319995v^14 - 108027505159304863v^13 + 231578769083234895v^12 - 4561024066476113049v^11 + 9461525208149991060v^10 - 111874806596610825592v^9 + 222193009401277106475v^8 - 1767072461067558411118v^7 + 3358421919737873820360v^6 - 17777828841796776235098v^5 + 31603720753251597368025v^4 - 110767274631181060603326v^3 + 191664716523042803725440v^2 - 353101895277362051846949*v + 532329752913369234384780) / 7095857516902985489010

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

$\nu$	$=$	$ ( \beta_{12} - \beta_{11} + 2\beta_{6} ) / 2 $ (b12 - b11 + 2*b6) / 2
$\nu^{2}$	$=$	$ ( -\beta_{8} - \beta_{7} - \beta_{4} - \beta_{2} - 3\beta _1 - 25 ) / 2 $ (-b8 - b7 - b4 - b2 - 3*b1 - 25) / 2
$\nu^{3}$	$=$	$ ( - 3 \beta_{13} - 3 \beta_{12} + 15 \beta_{11} - 6 \beta_{10} - 6 \beta_{9} - \beta_{8} + \cdots + 2 \beta_{3} ) / 2 $ (-3b13 - 3b12 + 15b11 - 6b10 - 6b9 - b8 + b7 - 69b6 + 2*b3) / 2
$\nu^{4}$	$=$	$ ( 8 \beta_{15} + 8 \beta_{14} - 12 \beta_{10} + 12 \beta_{9} + 16 \beta_{8} + 16 \beta_{7} + 8 \beta_{5} + \cdots + 63 ) / 2 $ (8b15 + 8b14 - 12b10 + 12b9 + 16b8 + 16b7 + 8b5 + 10b4 + 55b2 + 65b1 + 63) / 2
$\nu^{5}$	$=$	$ ( 3 \beta_{15} - 3 \beta_{14} + 110 \beta_{13} - 385 \beta_{12} - 145 \beta_{11} + 310 \beta_{10} + \cdots - 11 \beta_{3} ) / 2 $ (3b15 - 3b14 + 110b13 - 385b12 - 145b11 + 310b10 + 310b9 - 5b8 + 5b7 + 1178b6 - 11*b3) / 2
$\nu^{6}$	$=$	$ ( - 324 \beta_{15} - 324 \beta_{14} + 270 \beta_{10} - 270 \beta_{9} + 461 \beta_{8} + 461 \beta_{7} + \cdots + 7837 ) / 2 $ (-324b15 - 324b14 + 270b10 - 270b9 + 461b8 + 461b7 - 270b5 + 464b4 - 1979b2 - 433b1 + 7837) / 2
$\nu^{7}$	$=$	$ ( 66 \beta_{15} - 66 \beta_{14} - 1876 \beta_{13} + 15596 \beta_{12} - 3682 \beta_{11} + \cdots - 1825 \beta_{3} ) / 2 $ (66b15 - 66b14 - 1876b13 + 15596b12 - 3682b11 - 7595b10 - 7595b9 + 749b8 - 749b7 - 13246b6 - 1825*b3) / 2
$\nu^{8}$	$=$	$ ( 5704 \beta_{15} + 5704 \beta_{14} - 1176 \beta_{10} + 1176 \beta_{9} - 33647 \beta_{8} - 33647 \beta_{7} + \cdots - 341357 ) / 2 $ (5704b15 + 5704b14 - 1176b10 + 1176b9 - 33647b8 - 33647b7 + 4264b5 - 23279b4 + 47961b2 - 21129b1 - 341357) / 2
$\nu^{9}$	$=$	$ ( - 6246 \beta_{15} + 6246 \beta_{14} + 2457 \beta_{13} - 372300 \beta_{12} + 290868 \beta_{11} + \cdots + 88528 \beta_{3} ) / 2 $ (-6246b15 + 6246b14 + 2457b13 - 372300b12 + 290868b11 + 56106b10 + 56106b9 - 18893b8 + 18893b7 - 49707b6 + 88528*b3) / 2
$\nu^{10}$	$=$	$ ( 38668 \beta_{15} + 38668 \beta_{14} - 85110 \beta_{10} + 85110 \beta_{9} + 1072613 \beta_{8} + \cdots + 9538164 ) / 2 $ (38668b15 + 38668b14 - 85110b10 + 85110b9 + 1072613b8 + 1072613b7 + 35320b5 + 616877b4 - 605236b2 + 1104604b1 + 9538164) / 2
$\nu^{11}$	$=$	$ ( 175677 \beta_{15} - 175677 \beta_{14} + 1110769 \beta_{13} + 4509106 \beta_{12} + \cdots - 2309281 \beta_{3} ) / 2 $ (175677b15 - 175677b14 + 1110769b13 + 4509106b12 - 10642508b11 + 3831674b10 + 3831674b9 + 255056b8 - 255056b7 + 10109035b6 - 2309281*b3) / 2
$\nu^{12}$	$=$	$ ( - 6316632 \beta_{15} - 6316632 \beta_{14} + 3252312 \beta_{10} - 3252312 \beta_{9} - 20011151 \beta_{8} + \cdots - 174777046 ) / 2 $ (-6316632b15 - 6316632b14 + 3252312b10 - 3252312b9 - 20011151b8 - 20011151b7 - 5024628b5 - 10041044b4 - 10705114b2 - 31333238b1 - 174777046) / 2
$\nu^{13}$	$=$	$ ( - 2088453 \beta_{15} + 2088453 \beta_{14} - 48906104 \beta_{13} + 84038635 \beta_{12} + \cdots + 28954060 \beta_{3} ) / 2 $ (-2088453b15 + 2088453b14 - 48906104b13 + 84038635b12 + 252111673b11 - 210754765b10 - 210754765b9 - 199082b8 + 199082b7 - 415463150b6 + 28954060*b3) / 2
$\nu^{14}$	$=$	$ ( 241797278 \beta_{15} + 241797278 \beta_{14} - 77052612 \beta_{10} + 77052612 \beta_{9} + \cdots + 536957510 ) / 2 $ (241797278b15 + 241797278b14 - 77052612b10 + 77052612b9 + 35806046b8 + 35806046b7 + 192852308b5 - 7756087b4 + 953129556b2 + 530607876b1 + 536957510) / 2
$\nu^{15}$	$=$	$ ( - 43675362 \beta_{15} + 43675362 \beta_{14} + 1322867439 \beta_{13} - 7067723664 \beta_{12} + \cdots + 566542247 \beta_{3} ) / 2 $ (-43675362b15 + 43675362b14 + 1322867439b13 - 7067723664b12 - 2800880214b11 + 6144114969b10 + 6144114969b9 - 123101224b8 + 123101224b7 + 11121473457b6 + 566542247*b3) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$-1$	$1$	$1$	$1$

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} $

769.1

−1.73205	+	2.62461i
−1.73205	−	2.62461i
−1.73205	+	3.73386i
−1.73205	−	3.73386i
−1.73205	+	4.99238i
−1.73205	−	4.99238i
−1.73205	+	3.20092i
−1.73205	−	3.20092i
1.73205	+	3.20092i
1.73205	−	3.20092i
1.73205	+	4.99238i
1.73205	−	4.99238i
1.73205	+	3.73386i
1.73205	−	3.73386i
1.73205	+	2.62461i
1.73205	−	2.62461i

−1.73205

−4.25575

−

2.62461i

2.56785

−

6.51200i

3.00000

769.2

−1.73205

−4.25575

2.62461i

2.56785

6.51200i

3.00000

769.3

−1.73205

−3.32540

−

3.73386i

−2.76332

6.43149i

3.00000

769.4

−1.73205

−3.32540

3.73386i

−2.76332

−

6.43149i

3.00000

769.5

−1.73205

0.275940

−

4.99238i

6.91828

1.06651i

3.00000

769.6

−1.73205

0.275940

4.99238i

6.91828

−

1.06651i

3.00000

769.7

−1.73205

3.84111

−

3.20092i

−4.99075

−

4.90840i

3.00000

769.8

−1.73205

3.84111

3.20092i

−4.99075

4.90840i

3.00000

769.9

1.73205

−3.84111

−

3.20092i

4.99075

4.90840i

3.00000

769.10

1.73205

−3.84111

3.20092i

4.99075

−

4.90840i

3.00000

769.11

1.73205

−0.275940

−

4.99238i

−6.91828

−

1.06651i

3.00000

769.12

1.73205

−0.275940

4.99238i

−6.91828

1.06651i

3.00000

769.13

1.73205

3.32540

−

3.73386i

2.76332

−

6.43149i

3.00000

769.14

1.73205

3.32540

3.73386i

2.76332

6.43149i

3.00000

769.15

1.73205

4.25575

−

2.62461i

−2.56785

6.51200i

3.00000

769.16

1.73205

4.25575

2.62461i

−2.56785

−

6.51200i

3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.b	odd	2	1	inner
35.c	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1680.3.bd.b		16
4.b	odd	2	1		420.3.p.a	✓	16
5.b	even	2	1	inner	1680.3.bd.b		16
7.b	odd	2	1	inner	1680.3.bd.b		16
12.b	even	2	1		1260.3.p.e		16
20.d	odd	2	1		420.3.p.a	✓	16
20.e	even	4	2		2100.3.j.g		16
28.d	even	2	1		420.3.p.a	✓	16
35.c	odd	2	1	inner	1680.3.bd.b		16
60.h	even	2	1		1260.3.p.e		16
84.h	odd	2	1		1260.3.p.e		16
140.c	even	2	1		420.3.p.a	✓	16
140.j	odd	4	2		2100.3.j.g		16
420.o	odd	2	1		1260.3.p.e		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
420.3.p.a	✓	16	4.b	odd	2	1
420.3.p.a	✓	16	20.d	odd	2	1
420.3.p.a	✓	16	28.d	even	2	1
420.3.p.a	✓	16	140.c	even	2	1
1260.3.p.e		16	12.b	even	2	1
1260.3.p.e		16	60.h	even	2	1
1260.3.p.e		16	84.h	odd	2	1
1260.3.p.e		16	420.o	odd	2	1
1680.3.bd.b		16	1.a	even	1	1	trivial
1680.3.bd.b		16	5.b	even	2	1	inner
1680.3.bd.b		16	7.b	odd	2	1	inner
1680.3.bd.b		16	35.c	odd	2	1	inner
2100.3.j.g		16	20.e	even	4	2
2100.3.j.g		16	140.j	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{4} - 6T_{11}^{3} - 248T_{11}^{2} - 1296T_{11} - 1904 $ T11^4 - 6*T11^3 - 248*T11^2 - 1296*T11 - 1904 acting on $S_{3}^{\mathrm{new}}(1680, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} - 3)^{8} $ (T^2 - 3)^8
$5$	$ T^{16} + \cdots + 152587890625 $ T^16 + 24T^14 + 1244T^12 + 47208T^10 + 831750T^8 + 29505000T^6 + 485937500T^4 + 5859375000*T^2 + 152587890625
$7$	$ T^{16} + \cdots + 33232930569601 $ T^16 + 44T^14 + 1364T^12 - 121260T^10 - 4243498T^8 - 291145260T^6 + 7863188564T^4 + 609016636844*T^2 + 33232930569601
$11$	$ (T^{4} - 6 T^{3} + \cdots - 1904)^{4} $ (T^4 - 6T^3 - 248T^2 - 1296*T - 1904)^4
$13$	$ (T^{8} - 396 T^{6} + \cdots + 256)^{2} $ (T^8 - 396T^6 + 24736T^4 - 33024*T^2 + 256)^2
$17$	$ (T^{8} - 1500 T^{6} + \cdots + 132342016)^{2} $ (T^8 - 1500T^6 + 565792T^4 - 19526400*T^2 + 132342016)^2
$19$	$ (T^{8} + 2348 T^{6} + \cdots + 63035387136)^{2} $ (T^8 + 2348T^6 + 1857952T^4 + 583855488*T^2 + 63035387136)^2
$23$	$ (T^{8} + 2584 T^{6} + \cdots + 44935513344)^{2} $ (T^8 + 2584T^6 + 1920160T^4 + 514679616*T^2 + 44935513344)^2
$29$	$ (T^{4} + 8 T^{3} + \cdots - 19664)^{4} $ (T^4 + 8T^3 - 1620T^2 - 27568*T - 19664)^4
$31$	$ (T^{8} + 5792 T^{6} + \cdots + 422463239424)^{2} $ (T^8 + 5792T^6 + 10289536T^4 + 5666427840*T^2 + 422463239424)^2
$37$	$ (T^{8} + \cdots + 4504180665600)^{2} $ (T^8 + 8080T^6 + 18083728T^4 + 15453647616*T^2 + 4504180665600)^2
$41$	$ (T^{8} + \cdots + 31643292960000)^{2} $ (T^8 + 9908T^6 + 35844448T^4 + 55914817152*T^2 + 31643292960000)^2
$43$	$ (T^{8} + \cdots + 17480484267264)^{2} $ (T^8 + 11116T^6 + 38638960T^4 + 46584258624*T^2 + 17480484267264)^2
$47$	$ (T^{8} - 4076 T^{6} + \cdots + 462030153984)^{2} $ (T^8 - 4076T^6 + 5402080T^4 - 2799649536*T^2 + 462030153984)^2
$53$	$ (T^{8} + 5512 T^{6} + \cdots + 45361440000)^{2} $ (T^8 + 5512T^6 + 6399136T^4 + 1746117696*T^2 + 45361440000)^2
$59$	$ (T^{8} + 20708 T^{6} + \cdots + 985548324096)^{2} $ (T^8 + 20708T^6 + 133818112T^4 + 262762411008*T^2 + 985548324096)^2
$61$	$ (T^{8} + \cdots + 136350926228736)^{2} $ (T^8 + 16956T^6 + 100066752T^4 + 228597417216*T^2 + 136350926228736)^2
$67$	$ (T^{8} + \cdots + 4662962908416)^{2} $ (T^8 + 7756T^6 + 17273008T^4 + 15174606144*T^2 + 4662962908416)^2
$71$	$ (T^{4} + 42 T^{3} + \cdots + 3482224)^{4} $ (T^4 + 42T^3 - 6104T^2 - 204480*T + 3482224)^4
$73$	$ (T^{8} + \cdots + 8315148960000)^{2} $ (T^8 - 20240T^6 + 83250016T^4 - 77997297600*T^2 + 8315148960000)^2
$79$	$ (T^{4} + 4 T^{3} + \cdots + 27896112)^{4} $ (T^4 + 4T^3 - 16400T^2 + 84720*T + 27896112)^4
$83$	$ (T^{8} + \cdots + 4449737113600)^{2} $ (T^8 - 40320T^6 + 464711680T^4 - 1651376652288*T^2 + 4449737113600)^2
$89$	$ (T^{8} + \cdots + 62\!\cdots\!00)^{2} $ (T^8 + 53156T^6 + 945510304T^4 + 5961378813312*T^2 + 6272136975801600)^2
$97$	$ (T^{8} + \cdots + 95144885842176)^{2} $ (T^8 - 21056T^6 + 127426912T^4 - 215549244096*T^2 + 95144885842176)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.bd (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + (\beta_{11} + \beta_{6}) q^{5} - \beta_{10} q^{7} + 3 q^{9} + (\beta_{4} + 2) q^{11} + (\beta_{12} + \beta_{11} + \cdots - \beta_{9}) q^{13} + (\beta_{2} + 2) q^{15} + (\beta_{13} - 2 \beta_{12} + \cdots - 3 \beta_{6}) q^{17}+ \cdots + (3 \beta_{4} + 6) q^{99}+O(q^{100}) \) q + b6 * q^3 + (b11 + b6) * q^5 - b10 * q^7 + 3 * q^9 + (b4 + 2) * q^11 + (b12 + b11 - b10 - b9) * q^13 + (b2 + 2) * q^15 + (b13 - 2b12 - 2b11 + b10 + b9 - 3b6) q^17 - b3 * q^19 + (-b8 - 1) * q^21 + (-b10 + b9 + b5 + b2 + b1 + 1) * q^23 + (-b10 + b9 - b8 - b7 - b5 - b4 + b2 - b1 - 4) * q^25 + 3b6 q^27 + (-b8 - b7 - 3b4 + 2b2 - 2b1 - 4) q^29 + (b15 - b14 + 3b12 - 3b11) * q^31 + (b13 + b12 + b11 - b10 - b9 + 3b6) q^33 + (-b14 - b13 - b12 + b7 + b4 - b2 - 4) * q^35 + (b15 + b14 + b10 - b9 + b5 - 2b1 - 2) q^37 + (-b8 - b7 + b2 - b1 - 5) * q^39 + (b15 - b14 - b12 + b11 + b8 - b7 + 2b3) q^41 + (-b15 - b14 - 2b10 + 2b9 + b5 - 2b2) q^43 + (3b11 + 3b6) * q^45 + (b13 - 3b12 - 3b11 - b10 - b9 + 7b6) q^47 + (-b15 + b14 + b8 + 3b7 + b4 - b3 - 3b2 + 3b1 - 4) q^49 + (2b8 + 2b7 + 3b4 - 3b2 + 3b1 + 1) q^51 + (3b10 - 3b9 + b5 - b2 - b1 - 1) * q^53 + (2b13 - b12 + b11 - 2b10 - 2b9 - 2b8 + 2b7 + 8b6 - b3) * q^55 + (-b15 - b14 - b5 - b2 + b1 + 1) * q^57 + (b15 - b14 + 6b12 - 6b11 + 3b8 - 3b7) * q^59 + (-b15 + b14 + 5b12 - 5b11 + 2b8 - 2b7 - b3) * q^61 - 3b10 q^63 + (-b15 - b14 - b10 + b9 - b5 + b4 - 3b2 - b1 + 9) q^65 + (-b15 - b14 + b5 + 2b1 + 2) q^67 + (b15 - b14 - 2b12 + 2b11 - b8 + b7 + b3) * q^69 + (-b8 - b7 - 2b4 + 6b2 - 6b1 - 12) q^71 + (-2b13 - 9b12 - 9b11 + 2b10 + 2b9 - 22b6) * q^73 + (-b15 + b14 - b13 + b12 + 3b11 - 2b10 - 2b9 - b8 + b7 - b3) q^75 + (-b15 - b14 + b13 + 3b12 + 3b11 - 3b10 + 3b9 + 15b6 - 2b5 + 2b2 + 4b1 + 4) * q^77 + (-2b8 - 2b7 + 4b4 - 6b2 + 6b1) q^79 + 9 * q^81 + (-4b13 - 4b12 - 4b11 + 8b10 + 8b9 - 4b6) * q^83 + (b15 + b14 - 3b10 + 3b9 + b8 + b7 + 2b5 + 5b4 - b2 + 7b1 - 21) q^85 + (-3b13 + 3b12 + 3b11 + b6) q^87 + (b15 - b14 + 5b12 - 5b11 - 5b8 + 5b7 - 2b3) q^89 + (-b12 + b11 + 3b8 + 5b7 + 3b4 - b3 - 3b2 + 3b1 + 39) q^91 + (-b15 - b14 + 2b5 - 3b2 - b1 - 1) * q^93 + (-3b10 + 3b9 + 7b8 + 7b7 - 3b5 + 2b4 - 5b2 + 7b1 - 3) * q^95 + (-4b13 + b12 + b11 - 2b10 - 2b9 + 20b6) * q^97 + (3b4 + 6) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{9} + 24 q^{11} + 24 q^{15} - 12 q^{21} - 48 q^{25} - 32 q^{29} - 76 q^{35} - 72 q^{39} - 88 q^{49} - 24 q^{51} + 152 q^{65} - 168 q^{71} - 16 q^{79} + 144 q^{81} - 416 q^{85} + 568 q^{91} - 136 q^{95}+ \cdots + 72 q^{99}+O(q^{100}) \) 16 * q + 48 * q^9 + 24 * q^11 + 24 * q^15 - 12 * q^21 - 48 * q^25 - 32 * q^29 - 76 * q^35 - 72 * q^39 - 88 * q^49 - 24 * q^51 + 152 * q^65 - 168 * q^71 - 16 * q^79 + 144 * q^81 - 416 * q^85 + 568 * q^91 - 136 * q^95 + 72 * q^99

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	1680.3.bd.b

Level	$1680$
Weight	$3$
Character orbit	1680.bd
Analytic conductor	$45.777$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(45.7766844125\)
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:	\( x^{16} + 88 x^{14} + 3876 x^{12} + 102922 x^{10} + 1866070 x^{8} + 23190492 x^{6} + 203608845 x^{4} + \cdots + 3839661225 \) x^16 + 88x^14 + 3876x^12 + 102922x^10 + 1866070x^8 + 23190492x^6 + 203608845x^4 + 1131190758*x^2 + 3839661225
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{13}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 672001001684 \nu^{14} + 83638291569392 \nu^{12} + \cdots + 64\!\cdots\!09 ) / 92\!\cdots\!34 \) (672001001684v^14 + 83638291569392v^12 + 4272091016644632v^10 + 123974067663126407v^8 + 2178808884540756332v^6 + 24927134642650563432v^4 + 164573173831945222323*v^2 + 648661629583820097909) / 9275630741049654234
\(\beta_{2}\)	\(=\)	\( ( - 1035398609612 \nu^{14} - 111533330833976 \nu^{12} + \cdots - 79\!\cdots\!49 ) / 92\!\cdots\!34 \) (-1035398609612v^14 - 111533330833976v^12 - 5379199494309408v^10 - 149653330607213921v^8 - 2594192598074382740v^6 - 29131419888738608316v^4 - 201602403573332067189*v^2 - 794559645154337067549) / 9275630741049654234
\(\beta_{3}\)	\(=\)	\( ( 1601518644256 \nu^{15} + \cdots + 16\!\cdots\!49 \nu ) / 35\!\cdots\!05 \) (1601518644256v^15 - 1243791055410533v^13 - 83130180585664890v^11 - 2518744893744884417v^9 - 38877036874215059939v^7 - 363262564130722362285v^5 - 868324354542205414983v^3 + 16824872697243836939649v) / 3547928758451492744505
\(\beta_{4}\)	\(=\)	\( ( 634484072336 \nu^{14} + 36700472608334 \nu^{12} + \cdots - 79\!\cdots\!78 ) / 46\!\cdots\!17 \) (634484072336v^14 + 36700472608334v^12 + 1060452003667296v^10 + 15207683162139494v^8 + 168279695108599388v^6 + 647412056032704486v^4 + 1996888738264010496*v^2 - 79739766948019948578) / 4637815370524827117
\(\beta_{5}\)	\(=\)	\( ( - 15533455683352 \nu^{14} + \cdots + 77\!\cdots\!80 ) / 23\!\cdots\!85 \) (-15533455683352v^14 - 1180449492355747v^12 - 43360793116252413v^10 - 876318284088716878v^8 - 11033910504298438969v^6 - 75038762295166027026v^4 - 245160610606893679623*v^2 + 770744626057716884280) / 23189076852624135585
\(\beta_{6}\)	\(=\)	\( ( - 19154198883257 \nu^{15} + \cdots - 75\!\cdots\!61 \nu ) / 70\!\cdots\!10 \) (-19154198883257v^15 - 1546569916694156v^13 - 63571822352800752v^11 - 1547919464755800134v^9 - 25920757833965915885v^7 - 285311025541968291384v^5 - 2291825204891570440035v^3 - 7503372377553771177561v) / 7095857516902985489010
\(\beta_{7}\)	\(=\)	\( ( - 17963615837495 \nu^{15} - 860461496592840 \nu^{14} + \cdots - 46\!\cdots\!40 ) / 70\!\cdots\!10 \) (-17963615837495v^15 - 860461496592840v^14 + 2955992805288202v^13 - 81389302077257010v^12 + 230861652459815112v^11 - 3655926417831848100v^10 + 8076844319021149333v^9 - 96651721305214940160v^8 + 145437176981878050283v^7 - 1636638493005145024740v^6 + 1893639344756747339244v^5 - 17956389117864022789140v^4 + 14469794400468971075976v^3 - 120358275007032580434300v^2 + 94872323246581515223278*v - 468117452921951691963540) / 7095857516902985489010
\(\beta_{8}\)	\(=\)	\( ( 17963615837495 \nu^{15} - 860461496592840 \nu^{14} + \cdots - 46\!\cdots\!40 ) / 70\!\cdots\!10 \) (17963615837495v^15 - 860461496592840v^14 - 2955992805288202v^13 - 81389302077257010v^12 - 230861652459815112v^11 - 3655926417831848100v^10 - 8076844319021149333v^9 - 96651721305214940160v^8 - 145437176981878050283v^7 - 1636638493005145024740v^6 - 1893639344756747339244v^5 - 17956389117864022789140v^4 - 14469794400468971075976v^3 - 120358275007032580434300v^2 - 94872323246581515223278*v - 468117452921951691963540) / 7095857516902985489010
\(\beta_{9}\)	\(=\)	\( ( 301208776387831 \nu^{15} + \cdots + 81\!\cdots\!85 ) / 21\!\cdots\!30 \) (301208776387831v^15 + 2692351748830482v^14 + 17964341304450163v^13 + 220158916758918432v^12 + 484549998061732161v^11 + 8678455106638696983v^10 + 3993917644882916062v^9 + 197796238645556772333v^8 - 51003137021880397040v^7 + 2985424552488776600304v^6 - 2046329506496564444148v^5 + 30391105263446609513181v^4 - 20494949904502212421080v^3 + 206681271810133078548108v^2 - 132666874311026007414567*v + 815634568981543791999285) / 21287572550708956467030
\(\beta_{10}\)	\(=\)	\( ( 301208776387831 \nu^{15} + \cdots - 81\!\cdots\!85 ) / 21\!\cdots\!30 \) (301208776387831v^15 - 2692351748830482v^14 + 17964341304450163v^13 - 220158916758918432v^12 + 484549998061732161v^11 - 8678455106638696983v^10 + 3993917644882916062v^9 - 197796238645556772333v^8 - 51003137021880397040v^7 - 2985424552488776600304v^6 - 2046329506496564444148v^5 - 30391105263446609513181v^4 - 20494949904502212421080v^3 - 206681271810133078548108v^2 - 132666874311026007414567*v - 815634568981543791999285) / 21287572550708956467030
\(\beta_{11}\)	\(=\)	\( ( 543284346539459 \nu^{15} + \cdots - 14\!\cdots\!93 \nu ) / 21\!\cdots\!30 \) (543284346539459v^15 + 36027573534021182v^13 + 1119825456728871999v^11 + 16259996506215883748v^9 + 99792849989394910415v^7 - 927977383114080548367v^5 - 16248590176217896584090v^3 - 146125807930456620557193v) / 21287572550708956467030
\(\beta_{12}\)	\(=\)	\( ( 15307198600907 \nu^{15} + \cdots - 13\!\cdots\!69 \nu ) / 49\!\cdots\!10 \) (15307198600907v^15 + 1053651000795026v^13 + 34912939321992477v^11 + 594128216156992664v^9 + 5937613883562567575v^7 + 18229971398551841859v^5 - 58084626671359859160v^3 - 1361172757295722803669v) / 495059826760673406210
\(\beta_{13}\)	\(=\)	\( ( 21\!\cdots\!37 \nu^{15} + \cdots + 63\!\cdots\!31 \nu ) / 21\!\cdots\!30 \) (2145495653153737v^15 + 170623655058950176v^13 + 6675329225471977392v^11 + 152661950788010040184v^9 + 2371557898051849791385v^7 + 24782231535664543169664v^5 + 172645003170269350584165v^3 + 633345717488885596886931v) / 21287572550708956467030
\(\beta_{14}\)	\(=\)	\( ( 12\!\cdots\!87 \nu^{15} + \cdots + 53\!\cdots\!80 ) / 70\!\cdots\!10 \) (1235084031225187v^15 + 2778962112319995v^14 + 108027505159304863v^13 + 231578769083234895v^12 + 4561024066476113049v^11 + 9461525208149991060v^10 + 111874806596610825592v^9 + 222193009401277106475v^8 + 1767072461067558411118v^7 + 3358421919737873820360v^6 + 17777828841796776235098v^5 + 31603720753251597368025v^4 + 110767274631181060603326v^3 + 191664716523042803725440v^2 + 353101895277362051846949*v + 532329752913369234384780) / 7095857516902985489010
\(\beta_{15}\)	\(=\)	\( ( - 12\!\cdots\!87 \nu^{15} + \cdots + 53\!\cdots\!80 ) / 70\!\cdots\!10 \) (-1235084031225187v^15 + 2778962112319995v^14 - 108027505159304863v^13 + 231578769083234895v^12 - 4561024066476113049v^11 + 9461525208149991060v^10 - 111874806596610825592v^9 + 222193009401277106475v^8 - 1767072461067558411118v^7 + 3358421919737873820360v^6 - 17777828841796776235098v^5 + 31603720753251597368025v^4 - 110767274631181060603326v^3 + 191664716523042803725440v^2 - 353101895277362051846949*v + 532329752913369234384780) / 7095857516902985489010

\(\nu\)	\(=\)	\( ( \beta_{12} - \beta_{11} + 2\beta_{6} ) / 2 \) (b12 - b11 + 2*b6) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{8} - \beta_{7} - \beta_{4} - \beta_{2} - 3\beta _1 - 25 ) / 2 \) (-b8 - b7 - b4 - b2 - 3*b1 - 25) / 2
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{13} - 3 \beta_{12} + 15 \beta_{11} - 6 \beta_{10} - 6 \beta_{9} - \beta_{8} + \cdots + 2 \beta_{3} ) / 2 \) (-3b13 - 3b12 + 15b11 - 6b10 - 6b9 - b8 + b7 - 69b6 + 2*b3) / 2
\(\nu^{4}\)	\(=\)	\( ( 8 \beta_{15} + 8 \beta_{14} - 12 \beta_{10} + 12 \beta_{9} + 16 \beta_{8} + 16 \beta_{7} + 8 \beta_{5} + \cdots + 63 ) / 2 \) (8b15 + 8b14 - 12b10 + 12b9 + 16b8 + 16b7 + 8b5 + 10b4 + 55b2 + 65b1 + 63) / 2
\(\nu^{5}\)	\(=\)	\( ( 3 \beta_{15} - 3 \beta_{14} + 110 \beta_{13} - 385 \beta_{12} - 145 \beta_{11} + 310 \beta_{10} + \cdots - 11 \beta_{3} ) / 2 \) (3b15 - 3b14 + 110b13 - 385b12 - 145b11 + 310b10 + 310b9 - 5b8 + 5b7 + 1178b6 - 11*b3) / 2
\(\nu^{6}\)	\(=\)	\( ( - 324 \beta_{15} - 324 \beta_{14} + 270 \beta_{10} - 270 \beta_{9} + 461 \beta_{8} + 461 \beta_{7} + \cdots + 7837 ) / 2 \) (-324b15 - 324b14 + 270b10 - 270b9 + 461b8 + 461b7 - 270b5 + 464b4 - 1979b2 - 433b1 + 7837) / 2
\(\nu^{7}\)	\(=\)	\( ( 66 \beta_{15} - 66 \beta_{14} - 1876 \beta_{13} + 15596 \beta_{12} - 3682 \beta_{11} + \cdots - 1825 \beta_{3} ) / 2 \) (66b15 - 66b14 - 1876b13 + 15596b12 - 3682b11 - 7595b10 - 7595b9 + 749b8 - 749b7 - 13246b6 - 1825*b3) / 2
\(\nu^{8}\)	\(=\)	\( ( 5704 \beta_{15} + 5704 \beta_{14} - 1176 \beta_{10} + 1176 \beta_{9} - 33647 \beta_{8} - 33647 \beta_{7} + \cdots - 341357 ) / 2 \) (5704b15 + 5704b14 - 1176b10 + 1176b9 - 33647b8 - 33647b7 + 4264b5 - 23279b4 + 47961b2 - 21129b1 - 341357) / 2
\(\nu^{9}\)	\(=\)	\( ( - 6246 \beta_{15} + 6246 \beta_{14} + 2457 \beta_{13} - 372300 \beta_{12} + 290868 \beta_{11} + \cdots + 88528 \beta_{3} ) / 2 \) (-6246b15 + 6246b14 + 2457b13 - 372300b12 + 290868b11 + 56106b10 + 56106b9 - 18893b8 + 18893b7 - 49707b6 + 88528*b3) / 2
\(\nu^{10}\)	\(=\)	\( ( 38668 \beta_{15} + 38668 \beta_{14} - 85110 \beta_{10} + 85110 \beta_{9} + 1072613 \beta_{8} + \cdots + 9538164 ) / 2 \) (38668b15 + 38668b14 - 85110b10 + 85110b9 + 1072613b8 + 1072613b7 + 35320b5 + 616877b4 - 605236b2 + 1104604b1 + 9538164) / 2
\(\nu^{11}\)	\(=\)	\( ( 175677 \beta_{15} - 175677 \beta_{14} + 1110769 \beta_{13} + 4509106 \beta_{12} + \cdots - 2309281 \beta_{3} ) / 2 \) (175677b15 - 175677b14 + 1110769b13 + 4509106b12 - 10642508b11 + 3831674b10 + 3831674b9 + 255056b8 - 255056b7 + 10109035b6 - 2309281*b3) / 2
\(\nu^{12}\)	\(=\)	\( ( - 6316632 \beta_{15} - 6316632 \beta_{14} + 3252312 \beta_{10} - 3252312 \beta_{9} - 20011151 \beta_{8} + \cdots - 174777046 ) / 2 \) (-6316632b15 - 6316632b14 + 3252312b10 - 3252312b9 - 20011151b8 - 20011151b7 - 5024628b5 - 10041044b4 - 10705114b2 - 31333238b1 - 174777046) / 2
\(\nu^{13}\)	\(=\)	\( ( - 2088453 \beta_{15} + 2088453 \beta_{14} - 48906104 \beta_{13} + 84038635 \beta_{12} + \cdots + 28954060 \beta_{3} ) / 2 \) (-2088453b15 + 2088453b14 - 48906104b13 + 84038635b12 + 252111673b11 - 210754765b10 - 210754765b9 - 199082b8 + 199082b7 - 415463150b6 + 28954060*b3) / 2
\(\nu^{14}\)	\(=\)	\( ( 241797278 \beta_{15} + 241797278 \beta_{14} - 77052612 \beta_{10} + 77052612 \beta_{9} + \cdots + 536957510 ) / 2 \) (241797278b15 + 241797278b14 - 77052612b10 + 77052612b9 + 35806046b8 + 35806046b7 + 192852308b5 - 7756087b4 + 953129556b2 + 530607876b1 + 536957510) / 2
\(\nu^{15}\)	\(=\)	\( ( - 43675362 \beta_{15} + 43675362 \beta_{14} + 1322867439 \beta_{13} - 7067723664 \beta_{12} + \cdots + 566542247 \beta_{3} ) / 2 \) (-43675362b15 + 43675362b14 + 1322867439b13 - 7067723664b12 - 2800880214b11 + 6144114969b10 + 6144114969b9 - 123101224b8 + 123101224b7 + 11121473457b6 + 566542247*b3) / 2

\(n\)	\(241\)	\(337\)	\(421\)	\(1121\)	\(1471\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 769.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} - 3)^{8} \) (T^2 - 3)^8
$5$	\( T^{16} + \cdots + 152587890625 \) T^16 + 24T^14 + 1244T^12 + 47208T^10 + 831750T^8 + 29505000T^6 + 485937500T^4 + 5859375000*T^2 + 152587890625
$7$	\( T^{16} + \cdots + 33232930569601 \) T^16 + 44T^14 + 1364T^12 - 121260T^10 - 4243498T^8 - 291145260T^6 + 7863188564T^4 + 609016636844*T^2 + 33232930569601
$11$	\( (T^{4} - 6 T^{3} + \cdots - 1904)^{4} \) (T^4 - 6T^3 - 248T^2 - 1296*T - 1904)^4
$13$	\( (T^{8} - 396 T^{6} + \cdots + 256)^{2} \) (T^8 - 396T^6 + 24736T^4 - 33024*T^2 + 256)^2
$17$	\( (T^{8} - 1500 T^{6} + \cdots + 132342016)^{2} \) (T^8 - 1500T^6 + 565792T^4 - 19526400*T^2 + 132342016)^2
$19$	\( (T^{8} + 2348 T^{6} + \cdots + 63035387136)^{2} \) (T^8 + 2348T^6 + 1857952T^4 + 583855488*T^2 + 63035387136)^2
$23$	\( (T^{8} + 2584 T^{6} + \cdots + 44935513344)^{2} \) (T^8 + 2584T^6 + 1920160T^4 + 514679616*T^2 + 44935513344)^2
$29$	\( (T^{4} + 8 T^{3} + \cdots - 19664)^{4} \) (T^4 + 8T^3 - 1620T^2 - 27568*T - 19664)^4
$31$	\( (T^{8} + 5792 T^{6} + \cdots + 422463239424)^{2} \) (T^8 + 5792T^6 + 10289536T^4 + 5666427840*T^2 + 422463239424)^2
$37$	\( (T^{8} + \cdots + 4504180665600)^{2} \) (T^8 + 8080T^6 + 18083728T^4 + 15453647616*T^2 + 4504180665600)^2
$41$	\( (T^{8} + \cdots + 31643292960000)^{2} \) (T^8 + 9908T^6 + 35844448T^4 + 55914817152*T^2 + 31643292960000)^2
$43$	\( (T^{8} + \cdots + 17480484267264)^{2} \) (T^8 + 11116T^6 + 38638960T^4 + 46584258624*T^2 + 17480484267264)^2
$47$	\( (T^{8} - 4076 T^{6} + \cdots + 462030153984)^{2} \) (T^8 - 4076T^6 + 5402080T^4 - 2799649536*T^2 + 462030153984)^2
$53$	\( (T^{8} + 5512 T^{6} + \cdots + 45361440000)^{2} \) (T^8 + 5512T^6 + 6399136T^4 + 1746117696*T^2 + 45361440000)^2
$59$	\( (T^{8} + 20708 T^{6} + \cdots + 985548324096)^{2} \) (T^8 + 20708T^6 + 133818112T^4 + 262762411008*T^2 + 985548324096)^2
$61$	\( (T^{8} + \cdots + 136350926228736)^{2} \) (T^8 + 16956T^6 + 100066752T^4 + 228597417216*T^2 + 136350926228736)^2
$67$	\( (T^{8} + \cdots + 4662962908416)^{2} \) (T^8 + 7756T^6 + 17273008T^4 + 15174606144*T^2 + 4662962908416)^2
$71$	\( (T^{4} + 42 T^{3} + \cdots + 3482224)^{4} \) (T^4 + 42T^3 - 6104T^2 - 204480*T + 3482224)^4
$73$	\( (T^{8} + \cdots + 8315148960000)^{2} \) (T^8 - 20240T^6 + 83250016T^4 - 77997297600*T^2 + 8315148960000)^2
$79$	\( (T^{4} + 4 T^{3} + \cdots + 27896112)^{4} \) (T^4 + 4T^3 - 16400T^2 + 84720*T + 27896112)^4
$83$	\( (T^{8} + \cdots + 4449737113600)^{2} \) (T^8 - 40320T^6 + 464711680T^4 - 1651376652288*T^2 + 4449737113600)^2
$89$	\( (T^{8} + \cdots + 62\!\cdots\!00)^{2} \) (T^8 + 53156T^6 + 945510304T^4 + 5961378813312*T^2 + 6272136975801600)^2
$97$	\( (T^{8} + \cdots + 95144885842176)^{2} \) (T^8 - 21056T^6 + 127426912T^4 - 215549244096*T^2 + 95144885842176)^2
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