LMFDB - Newform orbit 3136.2.f.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(3135,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3136.3135");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.f (of order $2$, degree $1$, not 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:	$25.0410860739$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.2353561680715186176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 $ x^16 - 4x^15 + 2x^14 + 41x^12 - 92x^11 + 66x^10 - 104x^9 + 291x^8 - 388x^7 + 366x^6 - 344x^5 + 286x^4 - 184x^3 + 84x^2 - 24x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 224)
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_1 q^{3} + \beta_{13} q^{5} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) $ q + b1 * q^3 + b13 * q^5 + (-b2 + 1) * q^9	$ q + \beta_1 q^{3} + \beta_{13} q^{5} + ( - \beta_{2} + 1) q^{9} + ( - \beta_{11} + \beta_{4}) q^{11} + (\beta_{12} + \beta_{8}) q^{13} + ( - \beta_{15} - \beta_{11} + \beta_{4}) q^{15} - \beta_{12} q^{17} - \beta_{5} q^{19} + ( - \beta_{15} + \beta_{10}) q^{23} + ( - \beta_{9} - 2) q^{25} + ( - \beta_{7} - \beta_{6} + 3 \beta_1) q^{27} + (\beta_{2} - 1) q^{29} + (\beta_{7} - \beta_{5}) q^{31} + (\beta_{14} + 2 \beta_{13} - \beta_{8}) q^{33} + (\beta_{3} - \beta_{2} - 1) q^{37} + ( - \beta_{15} - \beta_{10} + \beta_{4}) q^{39} + ( - \beta_{12} - \beta_{8}) q^{41} + ( - 2 \beta_{11} - 2 \beta_{10}) q^{43} + (\beta_{14} + \beta_{12} + \beta_{8}) q^{45} + ( - \beta_{7} - \beta_{5} + 2 \beta_1) q^{47} + (\beta_{15} - \beta_{4}) q^{51} + (\beta_{9} + \beta_{3} - \beta_{2} - 1) q^{53} + ( - \beta_{6} - 3 \beta_{5} - 2 \beta_1) q^{55} - q^{57} + (2 \beta_{6} + \beta_1) q^{59} + (\beta_{14} - \beta_{13} + \cdots - \beta_{8}) q^{61}+ \cdots + ( - 5 \beta_{15} - 2 \beta_{11} + \cdots + \beta_{4}) q^{99}+O(q^{100}) $ q + b1 * q^3 + b13 * q^5 + (-b2 + 1) * q^9 + (-b11 + b4) * q^11 + (b12 + b8) * q^13 + (-b15 - b11 + b4) * q^15 - b12 * q^17 - b5 * q^19 + (-b15 + b10) * q^23 + (-b9 - 2) * q^25 + (-b7 - b6 + 3b1) q^27 + (b2 - 1) * q^29 + (b7 - b5) * q^31 + (b14 + 2b13 - b8) q^33 + (b3 - b2 - 1) * q^37 + (-b15 - b10 + b4) * q^39 + (-b12 - b8) * q^41 + (-2b11 - 2b10) * q^43 + (b14 + b12 + b8) * q^45 + (-b7 - b5 + 2b1) q^47 + (b15 - b4) * q^51 + (b9 + b3 - b2 - 1) * q^53 + (-b6 - 3b5 - 2b1) * q^55 - q^57 + (2b6 + b1) q^59 + (b14 - b13 - b12 - b8) * q^61 + (-b9 + 2b3 + b2 - 1) q^65 + (b15 - b11 - b10 - 2b4) q^67 + (b14 + b13 - 2b8) q^69 + (-b15 - b11 - 2b4) q^71 + (-b14 - b8) * q^73 + (-b7 - 3b6 + b5 + b1) q^75 + (b15 - 3b10) q^79 + (-b9 + 2b3 - b2 + 6) q^81 + (-b7 + b6 - b5 - 3b1) q^83 + (b9 - b3 - b2 + 1) * q^85 + (b7 + b6 - 6b1) q^87 + (-2b13 + 2b12 + 3b8) q^89 + (-b3 + b2 + 1) * q^93 + (-2b15 + b11 + b10 - 2b4) * q^95 + (-b14 + 2b13 - b12 + 3b8) * q^97 + (-5b15 - 2b11 + 5b10 + b4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^9	$ 16 q + 16 q^{9} - 32 q^{25} - 16 q^{29} - 16 q^{37} - 16 q^{53} - 16 q^{57} - 16 q^{65} + 96 q^{81} + 16 q^{85} + 16 q^{93}+O(q^{100}) $ 16 * q + 16 * q^9 - 32 * q^25 - 16 * q^29 - 16 * q^37 - 16 * q^53 - 16 * q^57 - 16 * q^65 + 96 * q^81 + 16 * q^85 + 16 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 $ x^16 - 4*x^15 + 2*x^14 + 41*x^12 - 92*x^11 + 66*x^10 - 104*x^9 + 291*x^8 - 388*x^7 + 366*x^6 - 344*x^5 + 286*x^4 - 184*x^3 + 84*x^2 - 24*x + 4 :

$\beta_{1}$	$=$	$ ( - 508009675 \nu^{15} + 17835297016 \nu^{14} - 56517662852 \nu^{13} + 582270848 \nu^{12} + \cdots + 242647661018 ) / 20109322702 $ (-508009675v^15 + 17835297016v^14 - 56517662852v^13 + 582270848v^12 - 7088890831v^11 + 702274874462v^10 - 1170090037720v^9 + 377376368526v^8 - 1364027322083v^7 + 4137630712600v^6 - 4084528421066v^5 + 3083496853086v^4 - 3236291968028v^3 + 2419000098258v^2 - 1055996592624*v + 242647661018) / 20109322702
$\beta_{2}$	$=$	$ ( - 32131571 \nu^{15} + 338443886 \nu^{14} - 801340954 \nu^{13} + 7179161 \nu^{12} + \cdots + 4131081209 ) / 137735087 $ (-32131571v^15 + 338443886v^14 - 801340954v^13 + 7179161v^12 - 1133962371v^11 + 11663672814v^10 - 17217925190v^9 + 7634184353v^8 - 25474337063v^7 + 64850621098v^6 - 63560694880v^5 + 49589483157v^4 - 50404045920v^3 + 36993277864v^2 - 15863090132*v + 4131081209) / 137735087
$\beta_{3}$	$=$	$ ( 35822062 \nu^{15} + 433216928 \nu^{14} - 1956966382 \nu^{13} + 19122975 \nu^{12} + \cdots + 8659988771 ) / 137735087 $ (35822062v^15 + 433216928v^14 - 1956966382v^13 + 19122975v^12 + 2009475974v^11 + 20646707746v^10 - 39231903560v^9 + 7842172647v^8 - 33485592002v^7 + 130013354394v^6 - 129149629850v^5 + 93264868727v^4 - 101442498290v^3 + 77380096762v^2 - 34464244792*v + 8659988771) / 137735087
$\beta_{4}$	$=$	$ ( 105672451 \nu^{15} - 109829046 \nu^{14} - 700918328 \nu^{13} - 436558020 \nu^{12} + \cdots + 1315263230 ) / 275470174 $ (105672451v^15 - 109829046v^14 - 700918328v^13 - 436558020v^12 + 4053203123v^11 + 2949961848v^10 - 7962445032v^9 - 9442457154v^8 + 2682770827v^7 + 19808691902v^6 - 10287010366v^5 + 12468794286v^4 - 15738046848v^3 + 10437391590v^2 - 7030965088*v + 1315263230) / 275470174
$\beta_{5}$	$=$	$ ( 12270144377 \nu^{15} - 2001240182 \nu^{14} - 139220139804 \nu^{13} - 515735242 \nu^{12} + \cdots + 604682780434 ) / 20109322702 $ (12270144377v^15 - 2001240182v^14 - 139220139804v^13 - 515735242v^12 + 536014841613v^11 + 817190268088v^10 - 2517587450708v^9 - 292469239544v^8 - 121704679471v^7 + 6828229304598v^6 - 7029468488722v^5 + 4618553909332v^4 - 5750410276928v^3 + 4656269089466v^2 - 2173060061984*v + 604682780434) / 20109322702
$\beta_{6}$	$=$	$ ( - 6417480703 \nu^{15} + 50370789199 \nu^{14} - 98389585846 \nu^{13} - 2661290691 \nu^{12} + \cdots + 467331960428 ) / 10054661351 $ (-6417480703v^15 + 50370789199v^14 - 98389585846v^13 - 2661290691v^12 - 241515503387v^11 + 1611250432397v^10 - 2157434364858v^9 + 1108879218603v^8 - 3701846687903v^7 + 8528560000327v^6 - 8237445450944v^5 + 6428988204985v^4 - 6329812789436v^3 + 4586407502358v^2 - 1946952787436*v + 467331960428) / 10054661351
$\beta_{7}$	$=$	$ ( 10098445561 \nu^{15} - 28336109146 \nu^{14} - 23195438426 \nu^{13} + 3012760792 \nu^{12} + \cdots + 19165887948 ) / 10054661351 $ (10098445561v^15 - 28336109146v^14 - 23195438426v^13 + 3012760792v^12 + 425903605649v^11 - 425032575208v^10 - 236817229076v^9 - 763509515140v^8 + 1976846170081v^7 - 824549174740v^6 + 571475657248v^5 - 1017241243896v^4 + 346194838574v^3 + 22456060558v^2 - 124781946756*v + 19165887948) / 10054661351
$\beta_{8}$	$=$	$ ( 11175608166 \nu^{15} - 37005033535 \nu^{14} - 2381595084 \nu^{13} - 3936897701 \nu^{12} + \cdots - 77317890647 ) / 10054661351 $ (11175608166v^15 - 37005033535v^14 - 2381595084v^13 - 3936897701v^12 + 454687961278v^11 - 715591444243v^10 + 275560834120v^9 - 1011875313139v^8 + 2563606006170v^7 - 2641228538245v^6 + 2425489864376v^5 - 2311962665959v^4 + 1743124162188v^3 - 985628049642v^2 + 363876573956*v - 77317890647) / 10054661351
$\beta_{9}$	$=$	$ ( 1285644 \nu^{15} - 4488762 \nu^{14} + 148500 \nu^{13} + 447472 \nu^{12} + 53106540 \nu^{11} + \cdots - 3171608 ) / 824761 $ (1285644v^15 - 4488762v^14 + 148500v^13 + 447472v^12 + 53106540v^11 - 90783902v^10 + 33434776v^9 - 111275056v^8 + 313499636v^7 - 323441210v^6 + 284053676v^5 - 273650576v^4 + 195015220v^3 - 111685928v^2 + 35430808*v - 3171608) / 824761
$\beta_{10}$	$=$	$ ( - 645964009 \nu^{15} + 2523811508 \nu^{14} - 818252060 \nu^{13} - 842352170 \nu^{12} + \cdots + 1739736266 ) / 275470174 $ (-645964009v^15 + 2523811508v^14 - 818252060v^13 - 842352170v^12 - 26747246201v^11 + 56978075786v^10 - 27486499588v^9 + 50402059680v^8 - 181048343649v^7 + 214390120040v^6 - 162924491522v^5 + 160618165676v^4 - 132614038408v^3 + 66424993686v^2 - 19780862232*v + 1739736266) / 275470174
$\beta_{11}$	$=$	$ ( 497104862 \nu^{15} - 1666148904 \nu^{14} - 72475566 \nu^{13} - 57955606 \nu^{12} + \cdots - 2175185090 ) / 137735087 $ (497104862v^15 - 1666148904v^14 - 72475566v^13 - 57955606v^12 + 20247549774v^11 - 32672753184v^10 + 12135987260v^9 - 43325950800v^8 + 115494580002v^7 - 119295220946v^6 + 104558981574v^5 - 100106494512v^4 + 76070269750v^3 - 40368861720v^2 + 12765434508*v - 2175185090) / 137735087
$\beta_{12}$	$=$	$ ( - 44354861173 \nu^{15} + 149820773289 \nu^{14} + 4938441286 \nu^{13} - 2404878406 \nu^{12} + \cdots + 151791368023 ) / 10054661351 $ (-44354861173v^15 + 149820773289v^14 + 4938441286v^13 - 2404878406v^12 - 1809165636353v^11 + 2963583334573v^10 - 1062941010642v^9 + 3758985388524v^8 - 10398060352837v^7 + 10748597638435v^6 - 9112489261848v^5 + 8753508886880v^4 - 6727203730596v^3 + 3493170999458v^2 - 1034530257000*v + 151791368023) / 10054661351
$\beta_{13}$	$=$	$ ( - 295233270 \nu^{15} + 1127321091 \nu^{14} - 308779854 \nu^{13} - 310687794 \nu^{12} + \cdots + 1176205899 ) / 60207553 $ (-295233270v^15 + 1127321091v^14 - 308779854v^13 - 310687794v^12 - 12198290034v^11 + 24979819761v^10 - 11785096208v^9 + 23642435496v^8 - 80586302238v^7 + 93939085995v^6 - 73203453810v^5 + 71848340088v^4 - 58800420118v^3 + 30361885068v^2 - 9719175876*v + 1176205899) / 60207553
$\beta_{14}$	$=$	$ ( 56840957306 \nu^{15} - 212571527808 \nu^{14} + 50782639456 \nu^{13} + 39344149404 \nu^{12} + \cdots - 312932226742 ) / 10054661351 $ (56840957306v^15 - 212571527808v^14 + 50782639456v^13 + 39344149404v^12 + 2341132780458v^11 - 4624600178060v^10 + 2234559128732v^9 - 4799822444568v^8 + 15140899480874v^7 - 17511559188156v^6 + 14353045635964v^5 - 13994895119888v^4 + 11226038499996v^3 - 5962265956612v^2 + 2046409277896*v - 312932226742) / 10054661351
$\beta_{15}$	$=$	$ ( - 1074964770 \nu^{15} + 3995246010 \nu^{14} - 848548822 \nu^{13} - 855240542 \nu^{12} + \cdots + 4371717342 ) / 137735087 $ (-1074964770v^15 + 3995246010v^14 - 848548822v^13 - 855240542v^12 - 44262121802v^11 + 86555323910v^10 - 39355648432v^9 + 87685136064v^8 - 283921727750v^7 + 323850307376v^6 - 257745648386v^5 + 251804653440v^4 - 203350422710v^3 + 105509356192v^2 - 33701710916*v + 4371717342) / 137735087

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

$\nu$	$=$	$ ( - \beta_{15} - \beta_{14} + 2 \beta_{12} + \beta_{11} - \beta_{9} + 4 \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 4 ) / 16 $ (-b15 - b14 + 2b12 + b11 - b9 + 4b8 + b7 - b6 - b5 + 4b2 - 3b1 + 4) / 16
$\nu^{2}$	$=$	$ ( - \beta_{15} + \beta_{14} + \beta_{11} + 4 \beta_{10} - \beta_{9} - 6 \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 6 ) / 8 $ (-b15 + b14 + b11 + 4b10 - b9 - 6b8 + b7 - b6 - b5 + 2*b2 + b1 + 6) / 8
$\nu^{3}$	$=$	$ ( - \beta_{15} - 5 \beta_{14} - 4 \beta_{13} + 10 \beta_{12} + 13 \beta_{11} - 3 \beta_{9} + 3 \beta_{7} + \cdots + 40 ) / 16 $ (-b15 - 5b14 - 4b13 + 10b12 + 13b11 - 3b9 + 3b7 - 3b6 - 3b5 + 15*b1 + 40) / 16
$\nu^{4}$	$=$	$ ( - 6 \beta_{15} - \beta_{14} - 2 \beta_{13} + 10 \beta_{12} + 10 \beta_{11} + 12 \beta_{10} - \beta_{9} + \cdots - 14 ) / 8 $ (-6b15 - b14 - 2b13 + 10b12 + 10b11 + 12b10 - b9 - 14b8 + 6b7 + 2b6 - 2b5 + 4b4 - 2b3 + 12b2 - 22*b1 - 14) / 8
$\nu^{5}$	$=$	$ ( - 29 \beta_{15} + 7 \beta_{14} - 10 \beta_{13} + 36 \beta_{12} + 39 \beta_{11} + 80 \beta_{10} + \cdots + 104 ) / 16 $ (-29b15 + 7b14 - 10b13 + 36b12 + 39b11 + 80b10 - 7b9 - 104b8 - 9b7 - 21b6 - b5 + 20b4 + 10b3 - 22b2 + 77b1 + 104) / 16
$\nu^{6}$	$=$	$ ( 5 \beta_{15} - 32 \beta_{14} - 40 \beta_{13} + 64 \beta_{12} + 85 \beta_{11} + 8 \beta_{9} - \beta_{7} + \cdots - 24 ) / 8 $ (5b15 - 32b14 - 40b13 + 64b12 + 85b11 + 8b9 - b7 + 11b6 - 9b5 - 15*b1 - 24) / 8
$\nu^{7}$	$=$	$ ( - 87 \beta_{15} + 34 \beta_{14} + 3 \beta_{13} + 79 \beta_{12} + 52 \beta_{11} + 196 \beta_{10} + \cdots - 236 ) / 8 $ (-87b15 + 34b14 + 3b13 + 79b12 + 52b11 + 196b10 + 34b9 - 236b8 - 11b7 + 46b6 - 24b5 + 70b4 + 3b3 + 11b2 - 128*b1 - 236) / 8
$\nu^{8}$	$=$	$ ( - 24 \beta_{15} - 51 \beta_{14} - 110 \beta_{13} + 182 \beta_{12} + 232 \beta_{11} + 128 \beta_{10} + \cdots + 166 ) / 8 $ (-24b15 - 51b14 - 110b13 + 182b12 + 232b11 + 128b10 + 51b9 - 166b8 - 184b7 - 56b6 - 24b5 + 32b4 + 110b3 - 284b2 + 440*b1 + 166) / 8
$\nu^{9}$	$=$	$ ( 77 \beta_{15} - 329 \beta_{14} - 444 \beta_{13} + 658 \beta_{12} + 883 \beta_{11} + 633 \beta_{9} + \cdots - 3632 ) / 16 $ (77b15 - 329b14 - 444b13 + 658b12 + 883b11 + 633b9 - 231b7 + 867b6 - 405b5 - 1791b1 - 3632) / 16
$\nu^{10}$	$=$	$ ( - 350 \beta_{15} + 256 \beta_{14} + 210 \beta_{13} + 14 \beta_{12} - 190 \beta_{11} + 681 \beta_{10} + \cdots - 812 ) / 4 $ (-350b15 + 256b14 + 210b13 + 14b12 - 190b11 + 681b10 + 256b9 - 812b8 - 403b7 + 118b6 - 137b5 + 255b4 + 210b3 - 498b2 + 243*b1 - 812) / 4
$\nu^{11}$	$=$	$ ( 2485 \beta_{15} - 2221 \beta_{14} - 1982 \beta_{13} + 592 \beta_{12} + 2509 \beta_{11} - 4576 \beta_{10} + \cdots - 5232 ) / 16 $ (2485b15 - 2221b14 - 1982b13 + 592b12 + 2509b11 - 4576b10 + 2221b9 + 5232b8 - 3807b7 + 793b6 - 1187b5 - 1980b4 + 1982b3 - 5034b2 + 3831*b1 - 5232) / 16
$\nu^{12}$	$=$	$ ( - 314 \beta_{15} + 539 \beta_{14} + 962 \beta_{13} - 1078 \beta_{12} - 1534 \beta_{11} + 1742 \beta_{9} + \cdots - 10066 ) / 4 $ (-314b15 + 539b14 + 962b13 - 1078b12 - 1534b11 + 1742b9 - 642b7 + 2386b6 - 1102b5 - 4954b1 - 10066) / 4
$\nu^{13}$	$=$	$ ( - 695 \beta_{15} + 2986 \beta_{14} + 5709 \beta_{13} - 7935 \beta_{12} - 10108 \beta_{11} - 2132 \beta_{10} + \cdots + 2316 ) / 8 $ (-695b15 + 2986b14 + 5709b13 - 7935b12 - 10108b11 - 2132b10 + 2986b9 + 2316b8 - 9575b7 - 2294b6 - 1228b5 - 1066b4 + 5709b3 - 13907b2 + 18988*b1 + 2316) / 8
$\nu^{14}$	$=$	$ ( 18403 \beta_{15} - 9634 \beta_{14} - 1340 \beta_{13} - 15520 \beta_{12} - 6793 \beta_{11} - 42470 \beta_{10} + \cdots - 49232 ) / 8 $ (18403b15 - 9634b14 - 1340b13 - 15520b12 - 6793b11 - 42470b10 + 9634b9 + 49232b8 - 5653b7 + 11621b6 - 5957b5 - 17578b4 + 1340b3 - 3748b2 - 19543*b1 - 49232) / 8
$\nu^{15}$	$=$	$ ( - 29757 \beta_{15} + 61371 \beta_{14} + 102052 \beta_{13} - 122742 \beta_{12} - 171851 \beta_{11} + \cdots - 151360 ) / 16 $ (-29757b15 + 61371b14 + 102052b13 - 122742b12 - 171851b11 + 25973b9 - 9689b7 + 35589b6 - 16211b5 - 74345b1 - 151360) / 16

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$-1$

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} $

3135.1

−0.349168	+	0.778942i
−0.349168	−	0.778942i
0.224274	−	0.447866i
0.224274	+	0.447866i
0.849168	−	0.0870829i
0.849168	+	0.0870829i
1.50047	−	0.288947i
1.50047	+	0.288947i
0.275726	+	0.418160i
0.275726	−	0.418160i
2.07391	+	0.620024i
2.07391	−	0.620024i
−1.00047	−	1.15497i
−1.00047	+	1.15497i
−1.57391	+	1.48605i
−1.57391	−	1.48605i

−3.20318

−

1.15894i

7.26038

3135.2

−3.20318

1.15894i

7.26038

3135.3

−2.26676

−

3.55765i

2.13818

3135.4

−2.26676

3.55765i

2.13818

3135.5

−0.753692

−

1.15894i

−2.43195

3135.6

−0.753692

1.15894i

−2.43195

3135.7

−0.182734

−

3.55765i

−2.96661

3135.8

−0.182734

3.55765i

−2.96661

3135.9

0.182734

−

3.55765i

−2.96661

3135.10

0.182734

3.55765i

−2.96661

3135.11

0.753692

−

1.15894i

−2.43195

3135.12

0.753692

1.15894i

−2.43195

3135.13

2.26676

−

3.55765i

2.13818

3135.14

2.26676

3.55765i

2.13818

3135.15

3.20318

−

1.15894i

7.26038

3135.16

3.20318

1.15894i

7.26038

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3136.2.f.j		16
4.b	odd	2	1	inner	3136.2.f.j		16
7.b	odd	2	1	inner	3136.2.f.j		16
7.c	even	3	1		448.2.p.e		16
7.d	odd	6	1		448.2.p.e		16
8.b	even	2	1		1568.2.f.b		16
8.d	odd	2	1		1568.2.f.b		16
28.d	even	2	1	inner	3136.2.f.j		16
28.f	even	6	1		448.2.p.e		16
28.g	odd	6	1		448.2.p.e		16
56.e	even	2	1		1568.2.f.b		16
56.h	odd	2	1		1568.2.f.b		16
56.j	odd	6	1		224.2.p.a	✓	16
56.j	odd	6	1		1568.2.p.b		16
56.k	odd	6	1		224.2.p.a	✓	16
56.k	odd	6	1		1568.2.p.b		16
56.m	even	6	1		224.2.p.a	✓	16
56.m	even	6	1		1568.2.p.b		16
56.p	even	6	1		224.2.p.a	✓	16
56.p	even	6	1		1568.2.p.b		16
168.s	odd	6	1		2016.2.cs.b		16
168.v	even	6	1		2016.2.cs.b		16
168.ba	even	6	1		2016.2.cs.b		16
168.be	odd	6	1		2016.2.cs.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.2.p.a	✓	16	56.j	odd	6	1
224.2.p.a	✓	16	56.k	odd	6	1
224.2.p.a	✓	16	56.m	even	6	1
224.2.p.a	✓	16	56.p	even	6	1
448.2.p.e		16	7.c	even	3	1
448.2.p.e		16	7.d	odd	6	1
448.2.p.e		16	28.f	even	6	1
448.2.p.e		16	28.g	odd	6	1
1568.2.f.b		16	8.b	even	2	1
1568.2.f.b		16	8.d	odd	2	1
1568.2.f.b		16	56.e	even	2	1
1568.2.f.b		16	56.h	odd	2	1
1568.2.p.b		16	56.j	odd	6	1
1568.2.p.b		16	56.k	odd	6	1
1568.2.p.b		16	56.m	even	6	1
1568.2.p.b		16	56.p	even	6	1
2016.2.cs.b		16	168.s	odd	6	1
2016.2.cs.b		16	168.v	even	6	1
2016.2.cs.b		16	168.ba	even	6	1
2016.2.cs.b		16	168.be	odd	6	1
3136.2.f.j		16	1.a	even	1	1	trivial
3136.2.f.j		16	4.b	odd	2	1	inner
3136.2.f.j		16	7.b	odd	2	1	inner
3136.2.f.j		16	28.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3136, [\chi])$:

$ T_{3}^{8} - 16T_{3}^{6} + 62T_{3}^{4} - 32T_{3}^{2} + 1 $

$ T_{5}^{4} + 14T_{5}^{2} + 17 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 16 T^{6} + 62 T^{4} + \cdots + 1)^{2} $ (T^8 - 16T^6 + 62T^4 - 32*T^2 + 1)^2
$5$	$ (T^{4} + 14 T^{2} + 17)^{4} $ (T^4 + 14*T^2 + 17)^4
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} + 64 T^{6} + \cdots + 18769)^{2} $ (T^8 + 64T^6 + 1326T^4 + 9808*T^2 + 18769)^2
$13$	$ (T^{8} + 56 T^{6} + \cdots + 12544)^{2} $ (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$17$	$ (T^{8} + 44 T^{6} + \cdots + 49)^{2} $ (T^8 + 44T^6 + 470T^4 + 460*T^2 + 49)^2
$19$	$ (T^{8} - 32 T^{6} + 62 T^{4} + \cdots + 1)^{2} $ (T^8 - 32T^6 + 62T^4 - 16*T^2 + 1)^2
$23$	$ (T^{8} + 48 T^{6} + \cdots + 1681)^{2} $ (T^8 + 48T^6 + 686T^4 + 2592*T^2 + 1681)^2
$29$	$ (T^{4} + 4 T^{3} + \cdots + 112)^{4} $ (T^4 + 4T^3 - 28T^2 - 16*T + 112)^4
$31$	$ (T^{8} - 128 T^{6} + \cdots + 113569)^{2} $ (T^8 - 128T^6 + 4574T^4 - 42352*T^2 + 113569)^2
$37$	$ (T^{4} + 4 T^{3} + \cdots + 337)^{4} $ (T^4 + 4T^3 - 58T^2 - 220*T + 337)^4
$41$	$ (T^{8} + 56 T^{6} + \cdots + 12544)^{2} $ (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$43$	$ (T^{8} + 192 T^{6} + \cdots + 12544)^{2} $ (T^8 + 192T^6 + 6368T^4 + 18432*T^2 + 12544)^2
$47$	$ (T^{8} - 176 T^{6} + \cdots + 97969)^{2} $ (T^8 - 176T^6 + 8846T^4 - 107776*T^2 + 97969)^2
$53$	$ (T^{4} + 4 T^{3} - 90 T^{2} + \cdots + 49)^{4} $ (T^4 + 4T^3 - 90T^2 + 100*T + 49)^4
$59$	$ (T^{8} - 256 T^{6} + \cdots + 208849)^{2} $ (T^8 - 256T^6 + 14510T^4 - 240752*T^2 + 208849)^2
$61$	$ (T^{8} + 268 T^{6} + \cdots + 4844401)^{2} $ (T^8 + 268T^6 + 23126T^4 + 712748*T^2 + 4844401)^2
$67$	$ (T^{8} + 288 T^{6} + \cdots + 124609)^{2} $ (T^8 + 288T^6 + 21374T^4 + 276144*T^2 + 124609)^2
$71$	$ (T^{8} + 320 T^{6} + \cdots + 6635776)^{2} $ (T^8 + 320T^6 + 29664T^4 + 806912*T^2 + 6635776)^2
$73$	$ (T^{8} + 188 T^{6} + \cdots + 564001)^{2} $ (T^8 + 188T^6 + 10406T^4 + 150652*T^2 + 564001)^2
$79$	$ (T^{8} + 336 T^{6} + \cdots + 4092529)^{2} $ (T^8 + 336T^6 + 23630T^4 + 554880*T^2 + 4092529)^2
$83$	$ (T^{8} - 416 T^{6} + \cdots + 3211264)^{2} $ (T^8 - 416T^6 + 39680T^4 - 827392*T^2 + 3211264)^2
$89$	$ (T^{8} + 364 T^{6} + \cdots + 27952369)^{2} $ (T^8 + 364T^6 + 40982T^4 + 1829516*T^2 + 27952369)^2
$97$	$ (T^{8} + 472 T^{6} + \cdots + 118026496)^{2} $ (T^8 + 472T^6 + 78896T^4 + 5378432*T^2 + 118026496)^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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_{13} q^{5} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) q + b1 * q^3 + b13 * q^5 + (-b2 + 1) * q^9	\( q + \beta_1 q^{3} + \beta_{13} q^{5} + ( - \beta_{2} + 1) q^{9} + ( - \beta_{11} + \beta_{4}) q^{11} + (\beta_{12} + \beta_{8}) q^{13} + ( - \beta_{15} - \beta_{11} + \beta_{4}) q^{15} - \beta_{12} q^{17} - \beta_{5} q^{19} + ( - \beta_{15} + \beta_{10}) q^{23} + ( - \beta_{9} - 2) q^{25} + ( - \beta_{7} - \beta_{6} + 3 \beta_1) q^{27} + (\beta_{2} - 1) q^{29} + (\beta_{7} - \beta_{5}) q^{31} + (\beta_{14} + 2 \beta_{13} - \beta_{8}) q^{33} + (\beta_{3} - \beta_{2} - 1) q^{37} + ( - \beta_{15} - \beta_{10} + \beta_{4}) q^{39} + ( - \beta_{12} - \beta_{8}) q^{41} + ( - 2 \beta_{11} - 2 \beta_{10}) q^{43} + (\beta_{14} + \beta_{12} + \beta_{8}) q^{45} + ( - \beta_{7} - \beta_{5} + 2 \beta_1) q^{47} + (\beta_{15} - \beta_{4}) q^{51} + (\beta_{9} + \beta_{3} - \beta_{2} - 1) q^{53} + ( - \beta_{6} - 3 \beta_{5} - 2 \beta_1) q^{55} - q^{57} + (2 \beta_{6} + \beta_1) q^{59} + (\beta_{14} - \beta_{13} + \cdots - \beta_{8}) q^{61}+ \cdots + ( - 5 \beta_{15} - 2 \beta_{11} + \cdots + \beta_{4}) q^{99}+O(q^{100}) \) q + b1 * q^3 + b13 * q^5 + (-b2 + 1) * q^9 + (-b11 + b4) * q^11 + (b12 + b8) * q^13 + (-b15 - b11 + b4) * q^15 - b12 * q^17 - b5 * q^19 + (-b15 + b10) * q^23 + (-b9 - 2) * q^25 + (-b7 - b6 + 3b1) q^27 + (b2 - 1) * q^29 + (b7 - b5) * q^31 + (b14 + 2b13 - b8) q^33 + (b3 - b2 - 1) * q^37 + (-b15 - b10 + b4) * q^39 + (-b12 - b8) * q^41 + (-2b11 - 2b10) * q^43 + (b14 + b12 + b8) * q^45 + (-b7 - b5 + 2b1) q^47 + (b15 - b4) * q^51 + (b9 + b3 - b2 - 1) * q^53 + (-b6 - 3b5 - 2b1) * q^55 - q^57 + (2b6 + b1) q^59 + (b14 - b13 - b12 - b8) * q^61 + (-b9 + 2b3 + b2 - 1) q^65 + (b15 - b11 - b10 - 2b4) q^67 + (b14 + b13 - 2b8) q^69 + (-b15 - b11 - 2b4) q^71 + (-b14 - b8) * q^73 + (-b7 - 3b6 + b5 + b1) q^75 + (b15 - 3b10) q^79 + (-b9 + 2b3 - b2 + 6) q^81 + (-b7 + b6 - b5 - 3b1) q^83 + (b9 - b3 - b2 + 1) * q^85 + (b7 + b6 - 6b1) q^87 + (-2b13 + 2b12 + 3b8) q^89 + (-b3 + b2 + 1) * q^93 + (-2b15 + b11 + b10 - 2b4) * q^95 + (-b14 + 2b13 - b12 + 3b8) * q^97 + (-5b15 - 2b11 + 5b10 + b4) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^9	\( 16 q + 16 q^{9} - 32 q^{25} - 16 q^{29} - 16 q^{37} - 16 q^{53} - 16 q^{57} - 16 q^{65} + 96 q^{81} + 16 q^{85} + 16 q^{93}+O(q^{100}) \) 16 * q + 16 * q^9 - 32 * q^25 - 16 * q^29 - 16 * q^37 - 16 * q^53 - 16 * q^57 - 16 * q^65 + 96 * q^81 + 16 * q^85 + 16 * q^93

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	3136.2.f.j

Level	$3136$
Weight	$2$
Character orbit	3136.f
Analytic conductor	$25.041$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(25.0410860739\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.2353561680715186176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 \) x^16 - 4x^15 + 2x^14 + 41x^12 - 92x^11 + 66x^10 - 104x^9 + 291x^8 - 388x^7 + 366x^6 - 344x^5 + 286x^4 - 184x^3 + 84x^2 - 24x + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{28} \)
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 508009675 \nu^{15} + 17835297016 \nu^{14} - 56517662852 \nu^{13} + 582270848 \nu^{12} + \cdots + 242647661018 ) / 20109322702 \) (-508009675v^15 + 17835297016v^14 - 56517662852v^13 + 582270848v^12 - 7088890831v^11 + 702274874462v^10 - 1170090037720v^9 + 377376368526v^8 - 1364027322083v^7 + 4137630712600v^6 - 4084528421066v^5 + 3083496853086v^4 - 3236291968028v^3 + 2419000098258v^2 - 1055996592624*v + 242647661018) / 20109322702
\(\beta_{2}\)	\(=\)	\( ( - 32131571 \nu^{15} + 338443886 \nu^{14} - 801340954 \nu^{13} + 7179161 \nu^{12} + \cdots + 4131081209 ) / 137735087 \) (-32131571v^15 + 338443886v^14 - 801340954v^13 + 7179161v^12 - 1133962371v^11 + 11663672814v^10 - 17217925190v^9 + 7634184353v^8 - 25474337063v^7 + 64850621098v^6 - 63560694880v^5 + 49589483157v^4 - 50404045920v^3 + 36993277864v^2 - 15863090132*v + 4131081209) / 137735087
\(\beta_{3}\)	\(=\)	\( ( 35822062 \nu^{15} + 433216928 \nu^{14} - 1956966382 \nu^{13} + 19122975 \nu^{12} + \cdots + 8659988771 ) / 137735087 \) (35822062v^15 + 433216928v^14 - 1956966382v^13 + 19122975v^12 + 2009475974v^11 + 20646707746v^10 - 39231903560v^9 + 7842172647v^8 - 33485592002v^7 + 130013354394v^6 - 129149629850v^5 + 93264868727v^4 - 101442498290v^3 + 77380096762v^2 - 34464244792*v + 8659988771) / 137735087
\(\beta_{4}\)	\(=\)	\( ( 105672451 \nu^{15} - 109829046 \nu^{14} - 700918328 \nu^{13} - 436558020 \nu^{12} + \cdots + 1315263230 ) / 275470174 \) (105672451v^15 - 109829046v^14 - 700918328v^13 - 436558020v^12 + 4053203123v^11 + 2949961848v^10 - 7962445032v^9 - 9442457154v^8 + 2682770827v^7 + 19808691902v^6 - 10287010366v^5 + 12468794286v^4 - 15738046848v^3 + 10437391590v^2 - 7030965088*v + 1315263230) / 275470174
\(\beta_{5}\)	\(=\)	\( ( 12270144377 \nu^{15} - 2001240182 \nu^{14} - 139220139804 \nu^{13} - 515735242 \nu^{12} + \cdots + 604682780434 ) / 20109322702 \) (12270144377v^15 - 2001240182v^14 - 139220139804v^13 - 515735242v^12 + 536014841613v^11 + 817190268088v^10 - 2517587450708v^9 - 292469239544v^8 - 121704679471v^7 + 6828229304598v^6 - 7029468488722v^5 + 4618553909332v^4 - 5750410276928v^3 + 4656269089466v^2 - 2173060061984*v + 604682780434) / 20109322702
\(\beta_{6}\)	\(=\)	\( ( - 6417480703 \nu^{15} + 50370789199 \nu^{14} - 98389585846 \nu^{13} - 2661290691 \nu^{12} + \cdots + 467331960428 ) / 10054661351 \) (-6417480703v^15 + 50370789199v^14 - 98389585846v^13 - 2661290691v^12 - 241515503387v^11 + 1611250432397v^10 - 2157434364858v^9 + 1108879218603v^8 - 3701846687903v^7 + 8528560000327v^6 - 8237445450944v^5 + 6428988204985v^4 - 6329812789436v^3 + 4586407502358v^2 - 1946952787436*v + 467331960428) / 10054661351
\(\beta_{7}\)	\(=\)	\( ( 10098445561 \nu^{15} - 28336109146 \nu^{14} - 23195438426 \nu^{13} + 3012760792 \nu^{12} + \cdots + 19165887948 ) / 10054661351 \) (10098445561v^15 - 28336109146v^14 - 23195438426v^13 + 3012760792v^12 + 425903605649v^11 - 425032575208v^10 - 236817229076v^9 - 763509515140v^8 + 1976846170081v^7 - 824549174740v^6 + 571475657248v^5 - 1017241243896v^4 + 346194838574v^3 + 22456060558v^2 - 124781946756*v + 19165887948) / 10054661351
\(\beta_{8}\)	\(=\)	\( ( 11175608166 \nu^{15} - 37005033535 \nu^{14} - 2381595084 \nu^{13} - 3936897701 \nu^{12} + \cdots - 77317890647 ) / 10054661351 \) (11175608166v^15 - 37005033535v^14 - 2381595084v^13 - 3936897701v^12 + 454687961278v^11 - 715591444243v^10 + 275560834120v^9 - 1011875313139v^8 + 2563606006170v^7 - 2641228538245v^6 + 2425489864376v^5 - 2311962665959v^4 + 1743124162188v^3 - 985628049642v^2 + 363876573956*v - 77317890647) / 10054661351
\(\beta_{9}\)	\(=\)	\( ( 1285644 \nu^{15} - 4488762 \nu^{14} + 148500 \nu^{13} + 447472 \nu^{12} + 53106540 \nu^{11} + \cdots - 3171608 ) / 824761 \) (1285644v^15 - 4488762v^14 + 148500v^13 + 447472v^12 + 53106540v^11 - 90783902v^10 + 33434776v^9 - 111275056v^8 + 313499636v^7 - 323441210v^6 + 284053676v^5 - 273650576v^4 + 195015220v^3 - 111685928v^2 + 35430808*v - 3171608) / 824761
\(\beta_{10}\)	\(=\)	\( ( - 645964009 \nu^{15} + 2523811508 \nu^{14} - 818252060 \nu^{13} - 842352170 \nu^{12} + \cdots + 1739736266 ) / 275470174 \) (-645964009v^15 + 2523811508v^14 - 818252060v^13 - 842352170v^12 - 26747246201v^11 + 56978075786v^10 - 27486499588v^9 + 50402059680v^8 - 181048343649v^7 + 214390120040v^6 - 162924491522v^5 + 160618165676v^4 - 132614038408v^3 + 66424993686v^2 - 19780862232*v + 1739736266) / 275470174
\(\beta_{11}\)	\(=\)	\( ( 497104862 \nu^{15} - 1666148904 \nu^{14} - 72475566 \nu^{13} - 57955606 \nu^{12} + \cdots - 2175185090 ) / 137735087 \) (497104862v^15 - 1666148904v^14 - 72475566v^13 - 57955606v^12 + 20247549774v^11 - 32672753184v^10 + 12135987260v^9 - 43325950800v^8 + 115494580002v^7 - 119295220946v^6 + 104558981574v^5 - 100106494512v^4 + 76070269750v^3 - 40368861720v^2 + 12765434508*v - 2175185090) / 137735087
\(\beta_{12}\)	\(=\)	\( ( - 44354861173 \nu^{15} + 149820773289 \nu^{14} + 4938441286 \nu^{13} - 2404878406 \nu^{12} + \cdots + 151791368023 ) / 10054661351 \) (-44354861173v^15 + 149820773289v^14 + 4938441286v^13 - 2404878406v^12 - 1809165636353v^11 + 2963583334573v^10 - 1062941010642v^9 + 3758985388524v^8 - 10398060352837v^7 + 10748597638435v^6 - 9112489261848v^5 + 8753508886880v^4 - 6727203730596v^3 + 3493170999458v^2 - 1034530257000*v + 151791368023) / 10054661351
\(\beta_{13}\)	\(=\)	\( ( - 295233270 \nu^{15} + 1127321091 \nu^{14} - 308779854 \nu^{13} - 310687794 \nu^{12} + \cdots + 1176205899 ) / 60207553 \) (-295233270v^15 + 1127321091v^14 - 308779854v^13 - 310687794v^12 - 12198290034v^11 + 24979819761v^10 - 11785096208v^9 + 23642435496v^8 - 80586302238v^7 + 93939085995v^6 - 73203453810v^5 + 71848340088v^4 - 58800420118v^3 + 30361885068v^2 - 9719175876*v + 1176205899) / 60207553
\(\beta_{14}\)	\(=\)	\( ( 56840957306 \nu^{15} - 212571527808 \nu^{14} + 50782639456 \nu^{13} + 39344149404 \nu^{12} + \cdots - 312932226742 ) / 10054661351 \) (56840957306v^15 - 212571527808v^14 + 50782639456v^13 + 39344149404v^12 + 2341132780458v^11 - 4624600178060v^10 + 2234559128732v^9 - 4799822444568v^8 + 15140899480874v^7 - 17511559188156v^6 + 14353045635964v^5 - 13994895119888v^4 + 11226038499996v^3 - 5962265956612v^2 + 2046409277896*v - 312932226742) / 10054661351
\(\beta_{15}\)	\(=\)	\( ( - 1074964770 \nu^{15} + 3995246010 \nu^{14} - 848548822 \nu^{13} - 855240542 \nu^{12} + \cdots + 4371717342 ) / 137735087 \) (-1074964770v^15 + 3995246010v^14 - 848548822v^13 - 855240542v^12 - 44262121802v^11 + 86555323910v^10 - 39355648432v^9 + 87685136064v^8 - 283921727750v^7 + 323850307376v^6 - 257745648386v^5 + 251804653440v^4 - 203350422710v^3 + 105509356192v^2 - 33701710916*v + 4371717342) / 137735087

\(\nu\)	\(=\)	\( ( - \beta_{15} - \beta_{14} + 2 \beta_{12} + \beta_{11} - \beta_{9} + 4 \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 4 ) / 16 \) (-b15 - b14 + 2b12 + b11 - b9 + 4b8 + b7 - b6 - b5 + 4b2 - 3b1 + 4) / 16
\(\nu^{2}\)	\(=\)	\( ( - \beta_{15} + \beta_{14} + \beta_{11} + 4 \beta_{10} - \beta_{9} - 6 \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 6 ) / 8 \) (-b15 + b14 + b11 + 4b10 - b9 - 6b8 + b7 - b6 - b5 + 2*b2 + b1 + 6) / 8
\(\nu^{3}\)	\(=\)	\( ( - \beta_{15} - 5 \beta_{14} - 4 \beta_{13} + 10 \beta_{12} + 13 \beta_{11} - 3 \beta_{9} + 3 \beta_{7} + \cdots + 40 ) / 16 \) (-b15 - 5b14 - 4b13 + 10b12 + 13b11 - 3b9 + 3b7 - 3b6 - 3b5 + 15*b1 + 40) / 16
\(\nu^{4}\)	\(=\)	\( ( - 6 \beta_{15} - \beta_{14} - 2 \beta_{13} + 10 \beta_{12} + 10 \beta_{11} + 12 \beta_{10} - \beta_{9} + \cdots - 14 ) / 8 \) (-6b15 - b14 - 2b13 + 10b12 + 10b11 + 12b10 - b9 - 14b8 + 6b7 + 2b6 - 2b5 + 4b4 - 2b3 + 12b2 - 22*b1 - 14) / 8
\(\nu^{5}\)	\(=\)	\( ( - 29 \beta_{15} + 7 \beta_{14} - 10 \beta_{13} + 36 \beta_{12} + 39 \beta_{11} + 80 \beta_{10} + \cdots + 104 ) / 16 \) (-29b15 + 7b14 - 10b13 + 36b12 + 39b11 + 80b10 - 7b9 - 104b8 - 9b7 - 21b6 - b5 + 20b4 + 10b3 - 22b2 + 77b1 + 104) / 16
\(\nu^{6}\)	\(=\)	\( ( 5 \beta_{15} - 32 \beta_{14} - 40 \beta_{13} + 64 \beta_{12} + 85 \beta_{11} + 8 \beta_{9} - \beta_{7} + \cdots - 24 ) / 8 \) (5b15 - 32b14 - 40b13 + 64b12 + 85b11 + 8b9 - b7 + 11b6 - 9b5 - 15*b1 - 24) / 8
\(\nu^{7}\)	\(=\)	\( ( - 87 \beta_{15} + 34 \beta_{14} + 3 \beta_{13} + 79 \beta_{12} + 52 \beta_{11} + 196 \beta_{10} + \cdots - 236 ) / 8 \) (-87b15 + 34b14 + 3b13 + 79b12 + 52b11 + 196b10 + 34b9 - 236b8 - 11b7 + 46b6 - 24b5 + 70b4 + 3b3 + 11b2 - 128*b1 - 236) / 8
\(\nu^{8}\)	\(=\)	\( ( - 24 \beta_{15} - 51 \beta_{14} - 110 \beta_{13} + 182 \beta_{12} + 232 \beta_{11} + 128 \beta_{10} + \cdots + 166 ) / 8 \) (-24b15 - 51b14 - 110b13 + 182b12 + 232b11 + 128b10 + 51b9 - 166b8 - 184b7 - 56b6 - 24b5 + 32b4 + 110b3 - 284b2 + 440*b1 + 166) / 8
\(\nu^{9}\)	\(=\)	\( ( 77 \beta_{15} - 329 \beta_{14} - 444 \beta_{13} + 658 \beta_{12} + 883 \beta_{11} + 633 \beta_{9} + \cdots - 3632 ) / 16 \) (77b15 - 329b14 - 444b13 + 658b12 + 883b11 + 633b9 - 231b7 + 867b6 - 405b5 - 1791b1 - 3632) / 16
\(\nu^{10}\)	\(=\)	\( ( - 350 \beta_{15} + 256 \beta_{14} + 210 \beta_{13} + 14 \beta_{12} - 190 \beta_{11} + 681 \beta_{10} + \cdots - 812 ) / 4 \) (-350b15 + 256b14 + 210b13 + 14b12 - 190b11 + 681b10 + 256b9 - 812b8 - 403b7 + 118b6 - 137b5 + 255b4 + 210b3 - 498b2 + 243*b1 - 812) / 4
\(\nu^{11}\)	\(=\)	\( ( 2485 \beta_{15} - 2221 \beta_{14} - 1982 \beta_{13} + 592 \beta_{12} + 2509 \beta_{11} - 4576 \beta_{10} + \cdots - 5232 ) / 16 \) (2485b15 - 2221b14 - 1982b13 + 592b12 + 2509b11 - 4576b10 + 2221b9 + 5232b8 - 3807b7 + 793b6 - 1187b5 - 1980b4 + 1982b3 - 5034b2 + 3831*b1 - 5232) / 16
\(\nu^{12}\)	\(=\)	\( ( - 314 \beta_{15} + 539 \beta_{14} + 962 \beta_{13} - 1078 \beta_{12} - 1534 \beta_{11} + 1742 \beta_{9} + \cdots - 10066 ) / 4 \) (-314b15 + 539b14 + 962b13 - 1078b12 - 1534b11 + 1742b9 - 642b7 + 2386b6 - 1102b5 - 4954b1 - 10066) / 4
\(\nu^{13}\)	\(=\)	\( ( - 695 \beta_{15} + 2986 \beta_{14} + 5709 \beta_{13} - 7935 \beta_{12} - 10108 \beta_{11} - 2132 \beta_{10} + \cdots + 2316 ) / 8 \) (-695b15 + 2986b14 + 5709b13 - 7935b12 - 10108b11 - 2132b10 + 2986b9 + 2316b8 - 9575b7 - 2294b6 - 1228b5 - 1066b4 + 5709b3 - 13907b2 + 18988*b1 + 2316) / 8
\(\nu^{14}\)	\(=\)	\( ( 18403 \beta_{15} - 9634 \beta_{14} - 1340 \beta_{13} - 15520 \beta_{12} - 6793 \beta_{11} - 42470 \beta_{10} + \cdots - 49232 ) / 8 \) (18403b15 - 9634b14 - 1340b13 - 15520b12 - 6793b11 - 42470b10 + 9634b9 + 49232b8 - 5653b7 + 11621b6 - 5957b5 - 17578b4 + 1340b3 - 3748b2 - 19543*b1 - 49232) / 8
\(\nu^{15}\)	\(=\)	\( ( - 29757 \beta_{15} + 61371 \beta_{14} + 102052 \beta_{13} - 122742 \beta_{12} - 171851 \beta_{11} + \cdots - 151360 ) / 16 \) (-29757b15 + 61371b14 + 102052b13 - 122742b12 - 171851b11 + 25973b9 - 9689b7 + 35589b6 - 16211b5 - 74345b1 - 151360) / 16

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3135.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 16 T^{6} + 62 T^{4} + \cdots + 1)^{2} \) (T^8 - 16T^6 + 62T^4 - 32*T^2 + 1)^2
$5$	\( (T^{4} + 14 T^{2} + 17)^{4} \) (T^4 + 14*T^2 + 17)^4
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} + 64 T^{6} + \cdots + 18769)^{2} \) (T^8 + 64T^6 + 1326T^4 + 9808*T^2 + 18769)^2
$13$	\( (T^{8} + 56 T^{6} + \cdots + 12544)^{2} \) (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$17$	\( (T^{8} + 44 T^{6} + \cdots + 49)^{2} \) (T^8 + 44T^6 + 470T^4 + 460*T^2 + 49)^2
$19$	\( (T^{8} - 32 T^{6} + 62 T^{4} + \cdots + 1)^{2} \) (T^8 - 32T^6 + 62T^4 - 16*T^2 + 1)^2
$23$	\( (T^{8} + 48 T^{6} + \cdots + 1681)^{2} \) (T^8 + 48T^6 + 686T^4 + 2592*T^2 + 1681)^2
$29$	\( (T^{4} + 4 T^{3} + \cdots + 112)^{4} \) (T^4 + 4T^3 - 28T^2 - 16*T + 112)^4
$31$	\( (T^{8} - 128 T^{6} + \cdots + 113569)^{2} \) (T^8 - 128T^6 + 4574T^4 - 42352*T^2 + 113569)^2
$37$	\( (T^{4} + 4 T^{3} + \cdots + 337)^{4} \) (T^4 + 4T^3 - 58T^2 - 220*T + 337)^4
$41$	\( (T^{8} + 56 T^{6} + \cdots + 12544)^{2} \) (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$43$	\( (T^{8} + 192 T^{6} + \cdots + 12544)^{2} \) (T^8 + 192T^6 + 6368T^4 + 18432*T^2 + 12544)^2
$47$	\( (T^{8} - 176 T^{6} + \cdots + 97969)^{2} \) (T^8 - 176T^6 + 8846T^4 - 107776*T^2 + 97969)^2
$53$	\( (T^{4} + 4 T^{3} - 90 T^{2} + \cdots + 49)^{4} \) (T^4 + 4T^3 - 90T^2 + 100*T + 49)^4
$59$	\( (T^{8} - 256 T^{6} + \cdots + 208849)^{2} \) (T^8 - 256T^6 + 14510T^4 - 240752*T^2 + 208849)^2
$61$	\( (T^{8} + 268 T^{6} + \cdots + 4844401)^{2} \) (T^8 + 268T^6 + 23126T^4 + 712748*T^2 + 4844401)^2
$67$	\( (T^{8} + 288 T^{6} + \cdots + 124609)^{2} \) (T^8 + 288T^6 + 21374T^4 + 276144*T^2 + 124609)^2
$71$	\( (T^{8} + 320 T^{6} + \cdots + 6635776)^{2} \) (T^8 + 320T^6 + 29664T^4 + 806912*T^2 + 6635776)^2
$73$	\( (T^{8} + 188 T^{6} + \cdots + 564001)^{2} \) (T^8 + 188T^6 + 10406T^4 + 150652*T^2 + 564001)^2
$79$	\( (T^{8} + 336 T^{6} + \cdots + 4092529)^{2} \) (T^8 + 336T^6 + 23630T^4 + 554880*T^2 + 4092529)^2
$83$	\( (T^{8} - 416 T^{6} + \cdots + 3211264)^{2} \) (T^8 - 416T^6 + 39680T^4 - 827392*T^2 + 3211264)^2
$89$	\( (T^{8} + 364 T^{6} + \cdots + 27952369)^{2} \) (T^8 + 364T^6 + 40982T^4 + 1829516*T^2 + 27952369)^2
$97$	\( (T^{8} + 472 T^{6} + \cdots + 118026496)^{2} \) (T^8 + 472T^6 + 78896T^4 + 5378432*T^2 + 118026496)^2
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\(n\)	\(197\)	\(1471\)	\(1473\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)