LMFDB - Newform orbit 2100.2.f.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1049,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.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:	$16.7685844245$
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} - 8 x^{15} + 22 x^{14} - 4 x^{13} - 80 x^{12} - 84 x^{11} + 1324 x^{10} - 3800 x^{9} + \cdots + 3204 $ x^16 - 8x^15 + 22x^14 - 4x^13 - 80x^12 - 84x^11 + 1324x^10 - 3800x^9 + 5642x^8 - 4872x^7 + 4136x^6 - 11608x^5 + 30032x^4 - 44288x^3 + 37232x^2 - 16848*x + 3204
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q + \beta_{3} q^{3} + ( - \beta_{8} + \beta_1) q^{7} + ( - \beta_{9} - \beta_{7} - 1) q^{9}+O(q^{10}) $ q + b3 * q^3 + (-b8 + b1) * q^7 + (-b9 - b7 - 1) * q^9	$ q + \beta_{3} q^{3} + ( - \beta_{8} + \beta_1) q^{7} + ( - \beta_{9} - \beta_{7} - 1) q^{9} + (\beta_{9} + \beta_{7} + \beta_{5}) q^{11} + ( - 2 \beta_{2} - \beta_1) q^{13} - \beta_{12} q^{17} + ( - \beta_{13} + \beta_{11}) q^{19} + (\beta_{13} + \beta_{6} - \beta_{5} - 1) q^{21} + ( - \beta_{10} + \beta_{8} + \beta_{4}) q^{23} + ( - \beta_{12} - \beta_{3} + \cdots + \beta_1) q^{27}+ \cdots + ( - 2 \beta_{9} - 2 \beta_{7} + \cdots + 4) q^{99}+O(q^{100}) $ q + b3 * q^3 + (-b8 + b1) * q^7 + (-b9 - b7 - 1) * q^9 + (b9 + b7 + b5) * q^11 + (-2b2 - b1) q^13 - b12 * q^17 + (-b13 + b11) * q^19 + (b13 + b6 - b5 - 1) * q^21 + (-b10 + b8 + b4) * q^23 + (-b12 - b3 - 2b2 + b1) q^27 + (b9 - b7 + 3b5 + 2) q^29 - b13 * q^31 + (-b3 + 3b2) q^33 - b8 * q^37 + (2b9 + 2b7 + 3b5 - 1) q^39 + (b13 + 2b6) q^41 + (-b15 - 2b8) q^43 + (-3b12 - 4b3 + 2b2 - 2b1) * q^47 + (-2b11 - 3) q^49 + (2b9 - b7 + b5) q^51 + (-b15 + 2b10 - b8) q^53 + (b10 - b4) * q^57 + (-b14 + b13 + 2b6) q^59 + (-b13 - 2b11) q^61 + (b12 + 2b8 + b4 - b2 - b1) q^63 - 3b8 q^67 + (-2b13 + 3b11 + b6) * q^69 + (-b9 - 3b7 + b5 + 2) q^71 + b1 * q^73 + (-b12 + b10 - b8 - b4 - 2b3 + b2 - b1) q^77 + (2b9 + 3) q^79 + (5b9 + 2b7 + 2b5 - 2) q^81 + (2b12 + 2b3 - b2 + b1) * q^83 + (-2b12 + b3 - b2 + 8b1) * q^87 + (b14 + b13 + 2b6) q^89 + (-2b13 + 3b11 - 2b9 - 4) q^91 + (b8 - b4) * q^93 + (-3b2 + 5b1) * q^97 + (-2b9 - 2b7 - 3b5 + 4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{9}+O(q^{10}) $ 16 * q - 24 * q^9	$ 16 q - 24 q^{9} - 8 q^{21} - 24 q^{39} - 48 q^{49} - 16 q^{51} + 48 q^{79} - 32 q^{81} - 64 q^{91} + 72 q^{99}+O(q^{100}) $ 16 * q - 24 * q^9 - 8 * q^21 - 24 * q^39 - 48 * q^49 - 16 * q^51 + 48 * q^79 - 32 * q^81 - 64 * q^91 + 72 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} + 22 x^{14} - 4 x^{13} - 80 x^{12} - 84 x^{11} + 1324 x^{10} - 3800 x^{9} + \cdots + 3204 $ x^16 - 8*x^15 + 22*x^14 - 4*x^13 - 80*x^12 - 84*x^11 + 1324*x^10 - 3800*x^9 + 5642*x^8 - 4872*x^7 + 4136*x^6 - 11608*x^5 + 30032*x^4 - 44288*x^3 + 37232*x^2 - 16848*x + 3204 :

$\beta_{1}$	$=$	$ ( 57403374489 \nu^{15} - 373367002804 \nu^{14} + 701222131854 \nu^{13} + 835306036507 \nu^{12} + \cdots - 119141715042936 ) / 663475361490 $ (57403374489v^15 - 373367002804v^14 + 701222131854v^13 + 835306036507v^12 - 3358106117202v^11 - 9913427677824v^10 + 61257088041072v^9 - 125830353287956v^8 + 133263830063988v^7 - 76905276790596v^6 + 119964804746976v^5 - 485834790073352v^4 + 991542180607608v^3 - 1040225507966284v^2 + 554801595445380*v - 119141715042936) / 663475361490
$\beta_{2}$	$=$	$ ( - 77323022525 \nu^{15} + 626674939546 \nu^{14} - 1652501840140 \nu^{13} + \cdots + 457970972564304 ) / 663475361490 $ (-77323022525v^15 + 626674939546v^14 - 1652501840140v^13 - 183401535319v^12 + 7127194982154v^11 + 8131441630236v^10 - 108330014842670v^9 + 281466093593488v^8 - 359667781317426v^7 + 240749551047996v^6 - 210575615232028v^5 + 871443150537926v^4 - 2212235837321132v^3 + 2826929145871240v^2 - 1800272872764336*v + 457970972564304) / 663475361490
$\beta_{3}$	$=$	$ ( 100088104905 \nu^{15} - 672606326953 \nu^{14} + 1324628699625 \nu^{13} + \cdots - 230259470125842 ) / 663475361490 $ (100088104905v^15 - 672606326953v^14 + 1324628699625v^13 + 1382862362872v^12 - 6334737350562v^11 - 16874919289398v^10 + 111469384632840v^9 - 234153879206074v^8 + 251496180131028v^7 - 145697330406498v^6 + 213655203154554v^5 - 883679203029908v^4 + 1846487578162146v^3 - 1963073630872930v^2 + 1054657479570048*v - 230259470125842) / 663475361490
$\beta_{4}$	$=$	$ ( - 91435174149446 \nu^{15} + 809758733801354 \nu^{14} + \cdots + 78\!\cdots\!66 ) / 477038784911310 $ (-91435174149446v^15 + 809758733801354v^14 - 2380725481593076v^13 + 471516558604255v^12 + 9760817625912960v^11 + 5885969225100684v^10 - 141700307198985008v^9 + 402819089391960698v^8 - 549347824171115100v^7 + 388411383997627392v^6 - 287670523503487168v^5 + 1152719245145689066v^4 - 3160229026930206008v^3 + 4324738922526229916v^2 - 2931975072018951048*v + 789943381126797366) / 477038784911310
$\beta_{5}$	$=$	$ ( - 2586850136 \nu^{15} + 16889035175 \nu^{14} - 31860826666 \nu^{13} - 37727839379 \nu^{12} + \cdots + 5340554258850 ) / 11245345110 $ (-2586850136v^15 + 16889035175v^14 - 31860826666v^13 - 37727839379v^12 + 153099296514v^11 + 446964096000v^10 - 2776835535788v^9 + 5701391057396v^8 - 6015190713816v^7 + 3448029446688v^6 - 5402044811956v^5 + 22006940984932v^4 - 44955950967560v^3 + 46937768878016v^2 - 24837347608704*v + 5340554258850) / 11245345110
$\beta_{6}$	$=$	$ ( 394067700008 \nu^{15} - 2620048717602 \nu^{14} + 5071372130419 \nu^{13} + \cdots - 767042048246670 ) / 1617080626818 $ (394067700008v^15 - 2620048717602v^14 + 5071372130419v^13 + 5644186621575v^12 - 24504385330236v^11 - 67341552918828v^10 + 434039817972950v^9 - 899428368512850v^8 + 947538852220986v^7 - 531807903614832v^6 + 820013525317330v^5 - 3431662430278170v^4 + 7093771930177982v^3 - 7395523643242350v^2 + 3820930691534844*v - 767042048246670) / 1617080626818
$\beta_{7}$	$=$	$ ( - 1508144322 \nu^{15} + 9702247024 \nu^{14} - 17541138297 \nu^{13} - 24143769229 \nu^{12} + \cdots + 2123885484981 ) / 5622672555 $ (-1508144322v^15 + 9702247024v^14 - 17541138297v^13 - 24143769229v^12 + 87076046064v^11 + 271718024049v^10 - 1594178988846v^9 + 3145831094029v^8 - 3140506689666v^7 + 1657608931605v^6 - 3002949942144v^5 + 12564085078403v^4 - 24827023794558v^3 + 24461932213372v^2 - 11723129533632*v + 2123885484981) / 5622672555
$\beta_{8}$	$=$	$ ( 28786349208 \nu^{15} - 203158500162 \nu^{14} + 438648529038 \nu^{13} + 314345478065 \nu^{12} + \cdots - 93251629981978 ) / 94146197930 $ (28786349208v^15 - 203158500162v^14 + 438648529038v^13 + 314345478065v^12 - 2022060235560v^11 - 4393011662922v^10 + 34058987846944v^9 - 76610594463144v^8 + 87791944855140v^7 - 54064370422086v^6 + 65812112235744v^5 - 271315744323048v^4 + 603346777013884v^3 - 687096422851968v^2 + 397058058543384*v - 93251629981978) / 94146197930
$\beta_{9}$	$=$	$ ( - 51174056 \nu^{15} + 369355329 \nu^{14} - 833049106 \nu^{13} - 469586340 \nu^{12} + \cdots + 193580214201 ) / 137138355 $ (-51174056v^15 + 369355329v^14 - 833049106v^13 - 469586340v^12 + 3760534290v^11 + 7314904494v^10 - 62214651188v^9 + 145032796893v^8 - 171605433420v^7 + 108735949992v^6 - 121388663668v^5 + 497098132356v^4 - 1141261635848v^3 + 1345105698096v^2 - 802838700408*v + 193580214201) / 137138355
$\beta_{10}$	$=$	$ ( 237146115290732 \nu^{15} + \cdots - 81\!\cdots\!42 ) / 477038784911310 $ (237146115290732v^15 - 1721592748555913v^14 + 3865712879810812v^13 + 2342293690718270v^12 - 17768161105979250v^11 - 34730436558706908v^10 + 291242148865846016v^9 - 669246871731066416v^8 + 775475237234401080v^7 - 477188249024734434v^6 + 551512082253892276v^5 - 2318111849938045072v^4 + 5271595374806164556v^3 - 6074511014886719072v^2 + 3508792753291626816*v - 814057061299264242) / 477038784911310
$\beta_{11}$	$=$	$ ( 19477599 \nu^{15} - 132952136 \nu^{14} + 273792114 \nu^{13} + 235989785 \nu^{12} + \cdots - 55853497044 ) / 36431730 $ (19477599v^15 - 132952136v^14 + 273792114v^13 + 235989785v^12 - 1271868450v^11 - 3100677486v^10 + 22097309352v^9 - 48356828852v^8 + 54292687260v^7 - 33010227468v^6 + 43220019252v^5 - 175894423504v^4 + 380852817012v^3 - 424709811584v^2 + 241278803772*v - 55853497044) / 36431730
$\beta_{12}$	$=$	$ ( 9185255669 \nu^{15} - 63157440526 \nu^{14} + 129797340514 \nu^{13} + 116532552115 \nu^{12} + \cdots - 23187055339044 ) / 16182325890 $ (9185255669v^15 - 63157440526v^14 + 129797340514v^13 + 116532552115v^12 - 613884121680v^11 - 1488031599246v^10 + 10543806408542v^9 - 22788180451882v^8 + 25037732878050v^7 - 14684014240788v^6 + 19983226662772v^5 - 83617995398834v^4 + 179619253365752v^3 - 195752741941804v^2 + 106776542011392*v - 23187055339044) / 16182325890
$\beta_{13}$	$=$	$ ( - 63212058907 \nu^{15} + 428243944168 \nu^{14} - 860507807102 \nu^{13} + \cdots + 149299677075222 ) / 98602477245 $ (-63212058907v^15 + 428243944168v^14 - 860507807102v^13 - 835204536085v^12 + 4087346785905v^11 + 10424199014343v^10 - 71211400087861v^9 + 151867933915471v^8 - 165055373950875v^7 + 96026202241044v^6 - 135665741954696v^5 + 564422972324267v^4 - 1197020496070456v^3 + 1289895423572662v^2 - 695902825241166*v + 149299677075222) / 98602477245
$\beta_{14}$	$=$	$ ( 9295948386523 \nu^{15} - 67182473206802 \nu^{14} + 151948666431908 \nu^{13} + \cdots - 35\!\cdots\!68 ) / 8085403134090 $ (9295948386523v^15 - 67182473206802v^14 + 151948666431908v^13 + 84256855192265v^12 - 685081077669150v^11 - 1322694341328882v^10 + 11320914231054334v^9 - 26448505828879424v^8 + 31348735924618530v^7 - 19889688242778996v^6 + 22088591186773124v^5 - 90471698713299898v^4 + 208123080439317424v^3 - 245724234499198928v^2 + 146881976453299224*v - 35442084892574268) / 8085403134090
$\beta_{15}$	$=$	$ ( 20502731780 \nu^{15} - 141223574670 \nu^{14} + 291889704550 \nu^{13} + 254773590075 \nu^{12} + \cdots - 54161940862170 ) / 17625670974 $ (20502731780v^15 - 141223574670v^14 + 291889704550v^13 + 254773590075v^12 - 1373424702120v^11 - 3293595733470v^10 + 23573948869640v^9 - 51256745585490v^8 + 56744878249860v^7 - 33632622506310v^6 + 44948659757680v^5 - 187118268731580v^4 + 403945266260300v^3 - 443744604971880v^2 + 244963342068600*v - 54161940862170) / 17625670974

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + \beta_{8} + \beta_{6} - 2 \beta_{4} + \cdots + 5 ) / 10 $ (b15 - b14 + b13 + b11 + b10 + b8 + b6 - 2b4 - 5b3 + 5*b2 + 5) / 10
$\nu^{2}$	$=$	$ ( - 6 \beta_{13} + 8 \beta_{11} - 2 \beta_{10} + 10 \beta_{9} + 3 \beta_{8} + 5 \beta_{7} + 10 \beta_{5} + \cdots + 15 ) / 10 $ (-6b13 + 8b11 - 2b10 + 10b9 + 3b8 + 5b7 + 10b5 - b4 - 10b1 + 15) / 10
$\nu^{3}$	$=$	$ ( 11 \beta_{15} - 3 \beta_{14} + 6 \beta_{13} - 10 \beta_{12} - 6 \beta_{11} - 5 \beta_{10} - 15 \beta_{9} + \cdots - 10 ) / 10 $ (11b15 - 3b14 + 6b13 - 10b12 - 6b11 - 5b10 - 15b9 + 20b8 + 8b6 - 15b5 - 40b3 + 15b2 - 40*b1 - 10) / 10
$\nu^{4}$	$=$	$ ( - 15 \beta_{15} - 16 \beta_{14} - 8 \beta_{13} + 20 \beta_{12} + 28 \beta_{11} - 4 \beta_{10} + \cdots - 75 ) / 10 $ (-15b15 - 16b14 - 8b13 + 20b12 + 28b11 - 4b10 - 15b9 - 29b8 + 30b7 + 36b6 + 30b5 - 2b4 + 60b3 - 40b2 + 20*b1 - 75) / 10
$\nu^{5}$	$=$	$ ( 5 \beta_{15} + 9 \beta_{14} + 30 \beta_{13} + 10 \beta_{12} - 101 \beta_{11} - 76 \beta_{10} + \cdots + 130 ) / 10 $ (5b15 + 9b14 + 30b13 + 10b12 - 101b11 - 76b10 - 75b9 + 74b8 - 75b7 - 44b6 - 150b5 + 47b4 - 20b3 - 165b2 - 145*b1 + 130) / 10
$\nu^{6}$	$=$	$ ( - 117 \beta_{15} - 78 \beta_{14} + 204 \beta_{13} + 120 \beta_{12} - 170 \beta_{11} + 190 \beta_{10} + \cdots - 635 ) / 10 $ (-117b15 - 78b14 + 204b13 + 120b12 - 170b11 + 190b10 - 455b9 - 285b8 - 120b7 + 228b6 - 260b5 - 130b4 + 240b3 + 330b2 + 370*b1 - 635) / 10
$\nu^{7}$	$=$	$ ( - 290 \beta_{15} + 277 \beta_{14} - 625 \beta_{13} + 435 \beta_{12} + 187 \beta_{11} - 220 \beta_{10} + \cdots + 2350 ) / 10 $ (-290b15 + 277b14 - 625b13 + 435b12 + 187b11 - 220b10 + 1260b9 - 440b8 + 210b7 - 892b6 + 210b5 + 120b4 + 1220b3 - 1085b2 + 230*b1 + 2350) / 10
$\nu^{8}$	$=$	$ ( 357 \beta_{15} - 4 \beta_{14} + 544 \beta_{13} - 640 \beta_{12} - 956 \beta_{11} + 658 \beta_{10} + \cdots - 350 ) / 5 $ (357b15 - 4b14 + 544b13 - 640b12 - 956b11 + 658b10 - 1030b9 + 643b8 - 700b7 + 24b6 - 1330b5 - 406b4 - 2280b3 + 2660b2 - 220*b1 - 350) / 5
$\nu^{9}$	$=$	$ ( - 2255 \beta_{15} + 317 \beta_{14} - 6377 \beta_{13} + 2685 \beta_{12} + 9103 \beta_{11} + 1078 \beta_{10} + \cdots + 4685 ) / 10 $ (-2255b15 + 317b14 - 6377b13 + 2685b12 + 9103b11 + 1078b10 + 11145b9 - 4907b8 + 7800b7 - 1022b6 + 12000b5 - 626b4 + 7930b3 - 1675b2 + 5700*b1 + 4685) / 10
$\nu^{10}$	$=$	$ ( 5586 \beta_{15} + 1140 \beta_{14} + 888 \beta_{13} - 6600 \beta_{12} - 4604 \beta_{11} - 3589 \beta_{10} + \cdots + 7785 ) / 5 $ (5586b15 + 1140b14 + 888b13 - 6600b12 - 4604b11 - 3589b10 - 700b9 + 14511b8 - 3560b7 - 3540b6 - 6265b5 + 2308b4 - 22200b3 + 2550b2 - 16070*b1 + 7785) / 5
$\nu^{11}$	$=$	$ ( - 5570 \beta_{15} - 8349 \beta_{14} + 2994 \beta_{13} + 4915 \beta_{12} + 16869 \beta_{11} + \cdots - 67565 ) / 5 $ (-5570b15 - 8349b14 + 2994b13 + 4915b12 + 16869b11 + 9605b10 - 13035b9 - 15020b8 + 14355b7 + 27454b6 + 22275b5 - 5775b4 + 14320b3 + 14550b2 + 21445*b1 - 67565) / 5
$\nu^{12}$	$=$	$ ( 5799 \beta_{15} + 14338 \beta_{14} - 21214 \beta_{13} + 7090 \beta_{12} - 3310 \beta_{11} + \cdots + 114765 ) / 5 $ (5799b15 + 14338b14 - 21214b13 + 7090b12 - 3310b11 - 59480b10 + 50145b9 + 37775b8 - 1920b7 - 45228b6 - 4710b5 + 37190b4 + 20820b3 - 149450b2 - 73520*b1 + 114765) / 5
$\nu^{13}$	$=$	$ ( 9313 \beta_{15} - 54120 \beta_{14} + 196806 \beta_{13} - 36820 \beta_{12} - 176263 \beta_{11} + \cdots - 507940 ) / 5 $ (9313b15 - 54120b14 + 196806b13 - 36820b12 - 176263b11 + 93127b10 - 393315b9 - 11168b8 - 143910b7 + 177650b6 - 235755b5 - 56444b4 - 122560b3 + 273750b2 + 75055*b1 - 507940) / 5
$\nu^{14}$	$=$	$ ( - 285081 \beta_{15} + 87528 \beta_{14} - 390576 \beta_{13} + 411810 \beta_{12} + 404846 \beta_{11} + \cdots + 817745 ) / 5 $ (-285081b15 + 87528b14 - 390576b13 + 411810b12 + 404846b11 - 129964b10 + 764435b9 - 581109b8 + 335460b7 - 278208b6 + 539420b5 + 82738b4 + 1328040b3 - 972510b2 + 485270*b1 + 817745) / 5
$\nu^{15}$	$=$	$ ( 661913 \beta_{15} + 132176 \beta_{14} + 973606 \beta_{13} - 926670 \beta_{12} - 1913782 \beta_{11} + \cdots + 580685 ) / 5 $ (661913b15 + 132176b14 + 973606b13 - 926670b12 - 1913782b11 + 157210b10 - 1573125b9 + 1431485b8 - 1576980b7 - 416366b6 - 2554170b5 - 92070b4 - 3004010b3 + 1851230b2 - 1263050*b1 + 580685) / 5

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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} $

1049.1

0.365728	+	1.53127i
0.904362	−	0.230889i
0.904362	+	0.230889i
0.365728	−	1.53127i
−2.44874	+	1.42852i
1.59751	+	0.247497i
1.59751	−	0.247497i
−2.44874	−	1.42852i
1.77273	+	1.86553i
1.07850	−	0.189519i
1.07850	+	0.189519i
1.77273	−	1.86553i
1.93465	+	0.387145i
−1.20474	+	0.913233i
−1.20474	−	0.913233i
1.93465	−	0.387145i

−1.14412

−

1.30038i

1.41421

−

2.23607i

−0.381966

2.97558i

1049.2

−1.14412

−

1.30038i

1.41421

2.23607i

−0.381966

2.97558i

1049.3

−1.14412

1.30038i

1.41421

−

2.23607i

−0.381966

−

2.97558i

1049.4

−1.14412

1.30038i

1.41421

2.23607i

−0.381966

−

2.97558i

1049.5

−0.437016

−

1.67601i

−1.41421

−

2.23607i

−2.61803

1.46489i

1049.6

−0.437016

−

1.67601i

−1.41421

2.23607i

−2.61803

1.46489i

1049.7

−0.437016

1.67601i

−1.41421

−

2.23607i

−2.61803

−

1.46489i

1049.8

−0.437016

1.67601i

−1.41421

2.23607i

−2.61803

−

1.46489i

1049.9

0.437016

−

1.67601i

1.41421

−

2.23607i

−2.61803

−

1.46489i

1049.10

0.437016

−

1.67601i

1.41421

2.23607i

−2.61803

−

1.46489i

1049.11

0.437016

1.67601i

1.41421

−

2.23607i

−2.61803

1.46489i

1049.12

0.437016

1.67601i

1.41421

2.23607i

−2.61803

1.46489i

1049.13

1.14412

−

1.30038i

−1.41421

−

2.23607i

−0.381966

−

2.97558i

1049.14

1.14412

−

1.30038i

−1.41421

2.23607i

−0.381966

−

2.97558i

1049.15

1.14412

1.30038i

−1.41421

−

2.23607i

−0.381966

2.97558i

1049.16

1.14412

1.30038i

−1.41421

2.23607i

−0.381966

2.97558i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
15.d	odd	2	1	inner
21.c	even	2	1	inner
35.c	odd	2	1	inner
105.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2100.2.f.i		16
3.b	odd	2	1	inner	2100.2.f.i		16
5.b	even	2	1	inner	2100.2.f.i		16
5.c	odd	4	1		2100.2.d.k	✓	8
5.c	odd	4	1		2100.2.d.l	yes	8
7.b	odd	2	1	inner	2100.2.f.i		16
15.d	odd	2	1	inner	2100.2.f.i		16
15.e	even	4	1		2100.2.d.k	✓	8
15.e	even	4	1		2100.2.d.l	yes	8
21.c	even	2	1	inner	2100.2.f.i		16
35.c	odd	2	1	inner	2100.2.f.i		16
35.f	even	4	1		2100.2.d.k	✓	8
35.f	even	4	1		2100.2.d.l	yes	8
105.g	even	2	1	inner	2100.2.f.i		16
105.k	odd	4	1		2100.2.d.k	✓	8
105.k	odd	4	1		2100.2.d.l	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2100.2.d.k	✓	8	5.c	odd	4	1
2100.2.d.k	✓	8	15.e	even	4	1
2100.2.d.k	✓	8	35.f	even	4	1
2100.2.d.k	✓	8	105.k	odd	4	1
2100.2.d.l	yes	8	5.c	odd	4	1
2100.2.d.l	yes	8	15.e	even	4	1
2100.2.d.l	yes	8	35.f	even	4	1
2100.2.d.l	yes	8	105.k	odd	4	1
2100.2.f.i		16	1.a	even	1	1	trivial
2100.2.f.i		16	3.b	odd	2	1	inner
2100.2.f.i		16	5.b	even	2	1	inner
2100.2.f.i		16	7.b	odd	2	1	inner
2100.2.f.i		16	15.d	odd	2	1	inner
2100.2.f.i		16	21.c	even	2	1	inner
2100.2.f.i		16	35.c	odd	2	1	inner
2100.2.f.i		16	105.g	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 $S_{2}^{\mathrm{new}}(2100, [\chi])$:

$ T_{11}^{4} + 16T_{11}^{2} + 19 $

$ T_{13}^{4} - 36T_{13}^{2} + 4 $

$ T_{23}^{4} - 80T_{23}^{2} + 475 $

$ T_{41}^{4} - 90T_{41}^{2} + 1900 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 6 T^{6} + 22 T^{4} + \cdots + 81)^{2} $ (T^8 + 6T^6 + 22T^4 + 54*T^2 + 81)^2
$5$	$ T^{16} $ T^16
$7$	$ (T^{4} + 6 T^{2} + 49)^{4} $ (T^4 + 6*T^2 + 49)^4
$11$	$ (T^{4} + 16 T^{2} + 19)^{4} $ (T^4 + 16*T^2 + 19)^4
$13$	$ (T^{4} - 36 T^{2} + 4)^{4} $ (T^4 - 36*T^2 + 4)^4
$17$	$ (T^{4} + 22 T^{2} + 76)^{4} $ (T^4 + 22*T^2 + 76)^4
$19$	$ (T^{4} + 30 T^{2} + 100)^{4} $ (T^4 + 30*T^2 + 100)^4
$23$	$ (T^{4} - 80 T^{2} + 475)^{4} $ (T^4 - 80*T^2 + 475)^4
$29$	$ (T^{4} + 104 T^{2} + 2299)^{4} $ (T^4 + 104*T^2 + 2299)^4
$31$	$ (T^{4} + 30 T^{2} + 100)^{4} $ (T^4 + 30*T^2 + 100)^4
$37$	$ (T^{2} + 5)^{8} $ (T^2 + 5)^8
$41$	$ (T^{4} - 90 T^{2} + 1900)^{4} $ (T^4 - 90*T^2 + 1900)^4
$43$	$ (T^{4} + 90 T^{2} + 25)^{4} $ (T^4 + 90*T^2 + 25)^4
$47$	$ (T^{4} + 222 T^{2} + 9196)^{4} $ (T^4 + 222*T^2 + 9196)^4
$53$	$ (T^{4} - 180 T^{2} + 7600)^{4} $ (T^4 - 180*T^2 + 7600)^4
$59$	$ (T^{4} - 160 T^{2} + 1900)^{4} $ (T^4 - 160*T^2 + 1900)^4
$61$	$ (T^{4} + 150 T^{2} + 2500)^{4} $ (T^4 + 150*T^2 + 2500)^4
$67$	$ (T^{2} + 45)^{8} $ (T^2 + 45)^8
$71$	$ (T^{4} + 120 T^{2} + 475)^{4} $ (T^4 + 120*T^2 + 475)^4
$73$	$ (T^{2} - 2)^{8} $ (T^2 - 2)^8
$79$	$ (T^{2} - 6 T - 11)^{8} $ (T^2 - 6*T - 11)^8
$83$	$ (T^{4} + 90 T^{2} + 1900)^{4} $ (T^4 + 90*T^2 + 1900)^4
$89$	$ (T^{4} - 240 T^{2} + 1900)^{4} $ (T^4 - 240*T^2 + 1900)^4
$97$	$ (T^{4} - 94 T^{2} + 4)^{4} $ (T^4 - 94*T^2 + 4)^4
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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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{3} + ( - \beta_{8} + \beta_1) q^{7} + ( - \beta_{9} - \beta_{7} - 1) q^{9}+O(q^{10}) \) q + b3 * q^3 + (-b8 + b1) * q^7 + (-b9 - b7 - 1) * q^9	\( q + \beta_{3} q^{3} + ( - \beta_{8} + \beta_1) q^{7} + ( - \beta_{9} - \beta_{7} - 1) q^{9} + (\beta_{9} + \beta_{7} + \beta_{5}) q^{11} + ( - 2 \beta_{2} - \beta_1) q^{13} - \beta_{12} q^{17} + ( - \beta_{13} + \beta_{11}) q^{19} + (\beta_{13} + \beta_{6} - \beta_{5} - 1) q^{21} + ( - \beta_{10} + \beta_{8} + \beta_{4}) q^{23} + ( - \beta_{12} - \beta_{3} + \cdots + \beta_1) q^{27}+ \cdots + ( - 2 \beta_{9} - 2 \beta_{7} + \cdots + 4) q^{99}+O(q^{100}) \) q + b3 * q^3 + (-b8 + b1) * q^7 + (-b9 - b7 - 1) * q^9 + (b9 + b7 + b5) * q^11 + (-2b2 - b1) q^13 - b12 * q^17 + (-b13 + b11) * q^19 + (b13 + b6 - b5 - 1) * q^21 + (-b10 + b8 + b4) * q^23 + (-b12 - b3 - 2b2 + b1) q^27 + (b9 - b7 + 3b5 + 2) q^29 - b13 * q^31 + (-b3 + 3b2) q^33 - b8 * q^37 + (2b9 + 2b7 + 3b5 - 1) q^39 + (b13 + 2b6) q^41 + (-b15 - 2b8) q^43 + (-3b12 - 4b3 + 2b2 - 2b1) * q^47 + (-2b11 - 3) q^49 + (2b9 - b7 + b5) q^51 + (-b15 + 2b10 - b8) q^53 + (b10 - b4) * q^57 + (-b14 + b13 + 2b6) q^59 + (-b13 - 2b11) q^61 + (b12 + 2b8 + b4 - b2 - b1) q^63 - 3b8 q^67 + (-2b13 + 3b11 + b6) * q^69 + (-b9 - 3b7 + b5 + 2) q^71 + b1 * q^73 + (-b12 + b10 - b8 - b4 - 2b3 + b2 - b1) q^77 + (2b9 + 3) q^79 + (5b9 + 2b7 + 2b5 - 2) q^81 + (2b12 + 2b3 - b2 + b1) * q^83 + (-2b12 + b3 - b2 + 8b1) * q^87 + (b14 + b13 + 2b6) q^89 + (-2b13 + 3b11 - 2b9 - 4) q^91 + (b8 - b4) * q^93 + (-3b2 + 5b1) * q^97 + (-2b9 - 2b7 - 3b5 + 4) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{9}+O(q^{10}) \) 16 * q - 24 * q^9	\( 16 q - 24 q^{9} - 8 q^{21} - 24 q^{39} - 48 q^{49} - 16 q^{51} + 48 q^{79} - 32 q^{81} - 64 q^{91} + 72 q^{99}+O(q^{100}) \) 16 * q - 24 * q^9 - 8 * q^21 - 24 * q^39 - 48 * q^49 - 16 * q^51 + 48 * q^79 - 32 * q^81 - 64 * q^91 + 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	2100.2.f.i

Level	$2100$
Weight	$2$
Character orbit	2100.f
Analytic conductor	$16.769$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(16.7685844245\)
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:	\( x^{16} - 8 x^{15} + 22 x^{14} - 4 x^{13} - 80 x^{12} - 84 x^{11} + 1324 x^{10} - 3800 x^{9} + \cdots + 3204 \) x^16 - 8x^15 + 22x^14 - 4x^13 - 80x^12 - 84x^11 + 1324x^10 - 3800x^9 + 5642x^8 - 4872x^7 + 4136x^6 - 11608x^5 + 30032x^4 - 44288x^3 + 37232x^2 - 16848*x + 3204
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{10}\cdot 5^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 57403374489 \nu^{15} - 373367002804 \nu^{14} + 701222131854 \nu^{13} + 835306036507 \nu^{12} + \cdots - 119141715042936 ) / 663475361490 \) (57403374489v^15 - 373367002804v^14 + 701222131854v^13 + 835306036507v^12 - 3358106117202v^11 - 9913427677824v^10 + 61257088041072v^9 - 125830353287956v^8 + 133263830063988v^7 - 76905276790596v^6 + 119964804746976v^5 - 485834790073352v^4 + 991542180607608v^3 - 1040225507966284v^2 + 554801595445380*v - 119141715042936) / 663475361490
\(\beta_{2}\)	\(=\)	\( ( - 77323022525 \nu^{15} + 626674939546 \nu^{14} - 1652501840140 \nu^{13} + \cdots + 457970972564304 ) / 663475361490 \) (-77323022525v^15 + 626674939546v^14 - 1652501840140v^13 - 183401535319v^12 + 7127194982154v^11 + 8131441630236v^10 - 108330014842670v^9 + 281466093593488v^8 - 359667781317426v^7 + 240749551047996v^6 - 210575615232028v^5 + 871443150537926v^4 - 2212235837321132v^3 + 2826929145871240v^2 - 1800272872764336*v + 457970972564304) / 663475361490
\(\beta_{3}\)	\(=\)	\( ( 100088104905 \nu^{15} - 672606326953 \nu^{14} + 1324628699625 \nu^{13} + \cdots - 230259470125842 ) / 663475361490 \) (100088104905v^15 - 672606326953v^14 + 1324628699625v^13 + 1382862362872v^12 - 6334737350562v^11 - 16874919289398v^10 + 111469384632840v^9 - 234153879206074v^8 + 251496180131028v^7 - 145697330406498v^6 + 213655203154554v^5 - 883679203029908v^4 + 1846487578162146v^3 - 1963073630872930v^2 + 1054657479570048*v - 230259470125842) / 663475361490
\(\beta_{4}\)	\(=\)	\( ( - 91435174149446 \nu^{15} + 809758733801354 \nu^{14} + \cdots + 78\!\cdots\!66 ) / 477038784911310 \) (-91435174149446v^15 + 809758733801354v^14 - 2380725481593076v^13 + 471516558604255v^12 + 9760817625912960v^11 + 5885969225100684v^10 - 141700307198985008v^9 + 402819089391960698v^8 - 549347824171115100v^7 + 388411383997627392v^6 - 287670523503487168v^5 + 1152719245145689066v^4 - 3160229026930206008v^3 + 4324738922526229916v^2 - 2931975072018951048*v + 789943381126797366) / 477038784911310
\(\beta_{5}\)	\(=\)	\( ( - 2586850136 \nu^{15} + 16889035175 \nu^{14} - 31860826666 \nu^{13} - 37727839379 \nu^{12} + \cdots + 5340554258850 ) / 11245345110 \) (-2586850136v^15 + 16889035175v^14 - 31860826666v^13 - 37727839379v^12 + 153099296514v^11 + 446964096000v^10 - 2776835535788v^9 + 5701391057396v^8 - 6015190713816v^7 + 3448029446688v^6 - 5402044811956v^5 + 22006940984932v^4 - 44955950967560v^3 + 46937768878016v^2 - 24837347608704*v + 5340554258850) / 11245345110
\(\beta_{6}\)	\(=\)	\( ( 394067700008 \nu^{15} - 2620048717602 \nu^{14} + 5071372130419 \nu^{13} + \cdots - 767042048246670 ) / 1617080626818 \) (394067700008v^15 - 2620048717602v^14 + 5071372130419v^13 + 5644186621575v^12 - 24504385330236v^11 - 67341552918828v^10 + 434039817972950v^9 - 899428368512850v^8 + 947538852220986v^7 - 531807903614832v^6 + 820013525317330v^5 - 3431662430278170v^4 + 7093771930177982v^3 - 7395523643242350v^2 + 3820930691534844*v - 767042048246670) / 1617080626818
\(\beta_{7}\)	\(=\)	\( ( - 1508144322 \nu^{15} + 9702247024 \nu^{14} - 17541138297 \nu^{13} - 24143769229 \nu^{12} + \cdots + 2123885484981 ) / 5622672555 \) (-1508144322v^15 + 9702247024v^14 - 17541138297v^13 - 24143769229v^12 + 87076046064v^11 + 271718024049v^10 - 1594178988846v^9 + 3145831094029v^8 - 3140506689666v^7 + 1657608931605v^6 - 3002949942144v^5 + 12564085078403v^4 - 24827023794558v^3 + 24461932213372v^2 - 11723129533632*v + 2123885484981) / 5622672555
\(\beta_{8}\)	\(=\)	\( ( 28786349208 \nu^{15} - 203158500162 \nu^{14} + 438648529038 \nu^{13} + 314345478065 \nu^{12} + \cdots - 93251629981978 ) / 94146197930 \) (28786349208v^15 - 203158500162v^14 + 438648529038v^13 + 314345478065v^12 - 2022060235560v^11 - 4393011662922v^10 + 34058987846944v^9 - 76610594463144v^8 + 87791944855140v^7 - 54064370422086v^6 + 65812112235744v^5 - 271315744323048v^4 + 603346777013884v^3 - 687096422851968v^2 + 397058058543384*v - 93251629981978) / 94146197930
\(\beta_{9}\)	\(=\)	\( ( - 51174056 \nu^{15} + 369355329 \nu^{14} - 833049106 \nu^{13} - 469586340 \nu^{12} + \cdots + 193580214201 ) / 137138355 \) (-51174056v^15 + 369355329v^14 - 833049106v^13 - 469586340v^12 + 3760534290v^11 + 7314904494v^10 - 62214651188v^9 + 145032796893v^8 - 171605433420v^7 + 108735949992v^6 - 121388663668v^5 + 497098132356v^4 - 1141261635848v^3 + 1345105698096v^2 - 802838700408*v + 193580214201) / 137138355
\(\beta_{10}\)	\(=\)	\( ( 237146115290732 \nu^{15} + \cdots - 81\!\cdots\!42 ) / 477038784911310 \) (237146115290732v^15 - 1721592748555913v^14 + 3865712879810812v^13 + 2342293690718270v^12 - 17768161105979250v^11 - 34730436558706908v^10 + 291242148865846016v^9 - 669246871731066416v^8 + 775475237234401080v^7 - 477188249024734434v^6 + 551512082253892276v^5 - 2318111849938045072v^4 + 5271595374806164556v^3 - 6074511014886719072v^2 + 3508792753291626816*v - 814057061299264242) / 477038784911310
\(\beta_{11}\)	\(=\)	\( ( 19477599 \nu^{15} - 132952136 \nu^{14} + 273792114 \nu^{13} + 235989785 \nu^{12} + \cdots - 55853497044 ) / 36431730 \) (19477599v^15 - 132952136v^14 + 273792114v^13 + 235989785v^12 - 1271868450v^11 - 3100677486v^10 + 22097309352v^9 - 48356828852v^8 + 54292687260v^7 - 33010227468v^6 + 43220019252v^5 - 175894423504v^4 + 380852817012v^3 - 424709811584v^2 + 241278803772*v - 55853497044) / 36431730
\(\beta_{12}\)	\(=\)	\( ( 9185255669 \nu^{15} - 63157440526 \nu^{14} + 129797340514 \nu^{13} + 116532552115 \nu^{12} + \cdots - 23187055339044 ) / 16182325890 \) (9185255669v^15 - 63157440526v^14 + 129797340514v^13 + 116532552115v^12 - 613884121680v^11 - 1488031599246v^10 + 10543806408542v^9 - 22788180451882v^8 + 25037732878050v^7 - 14684014240788v^6 + 19983226662772v^5 - 83617995398834v^4 + 179619253365752v^3 - 195752741941804v^2 + 106776542011392*v - 23187055339044) / 16182325890
\(\beta_{13}\)	\(=\)	\( ( - 63212058907 \nu^{15} + 428243944168 \nu^{14} - 860507807102 \nu^{13} + \cdots + 149299677075222 ) / 98602477245 \) (-63212058907v^15 + 428243944168v^14 - 860507807102v^13 - 835204536085v^12 + 4087346785905v^11 + 10424199014343v^10 - 71211400087861v^9 + 151867933915471v^8 - 165055373950875v^7 + 96026202241044v^6 - 135665741954696v^5 + 564422972324267v^4 - 1197020496070456v^3 + 1289895423572662v^2 - 695902825241166*v + 149299677075222) / 98602477245
\(\beta_{14}\)	\(=\)	\( ( 9295948386523 \nu^{15} - 67182473206802 \nu^{14} + 151948666431908 \nu^{13} + \cdots - 35\!\cdots\!68 ) / 8085403134090 \) (9295948386523v^15 - 67182473206802v^14 + 151948666431908v^13 + 84256855192265v^12 - 685081077669150v^11 - 1322694341328882v^10 + 11320914231054334v^9 - 26448505828879424v^8 + 31348735924618530v^7 - 19889688242778996v^6 + 22088591186773124v^5 - 90471698713299898v^4 + 208123080439317424v^3 - 245724234499198928v^2 + 146881976453299224*v - 35442084892574268) / 8085403134090
\(\beta_{15}\)	\(=\)	\( ( 20502731780 \nu^{15} - 141223574670 \nu^{14} + 291889704550 \nu^{13} + 254773590075 \nu^{12} + \cdots - 54161940862170 ) / 17625670974 \) (20502731780v^15 - 141223574670v^14 + 291889704550v^13 + 254773590075v^12 - 1373424702120v^11 - 3293595733470v^10 + 23573948869640v^9 - 51256745585490v^8 + 56744878249860v^7 - 33632622506310v^6 + 44948659757680v^5 - 187118268731580v^4 + 403945266260300v^3 - 443744604971880v^2 + 244963342068600*v - 54161940862170) / 17625670974

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + \beta_{8} + \beta_{6} - 2 \beta_{4} + \cdots + 5 ) / 10 \) (b15 - b14 + b13 + b11 + b10 + b8 + b6 - 2b4 - 5b3 + 5*b2 + 5) / 10
\(\nu^{2}\)	\(=\)	\( ( - 6 \beta_{13} + 8 \beta_{11} - 2 \beta_{10} + 10 \beta_{9} + 3 \beta_{8} + 5 \beta_{7} + 10 \beta_{5} + \cdots + 15 ) / 10 \) (-6b13 + 8b11 - 2b10 + 10b9 + 3b8 + 5b7 + 10b5 - b4 - 10b1 + 15) / 10
\(\nu^{3}\)	\(=\)	\( ( 11 \beta_{15} - 3 \beta_{14} + 6 \beta_{13} - 10 \beta_{12} - 6 \beta_{11} - 5 \beta_{10} - 15 \beta_{9} + \cdots - 10 ) / 10 \) (11b15 - 3b14 + 6b13 - 10b12 - 6b11 - 5b10 - 15b9 + 20b8 + 8b6 - 15b5 - 40b3 + 15b2 - 40*b1 - 10) / 10
\(\nu^{4}\)	\(=\)	\( ( - 15 \beta_{15} - 16 \beta_{14} - 8 \beta_{13} + 20 \beta_{12} + 28 \beta_{11} - 4 \beta_{10} + \cdots - 75 ) / 10 \) (-15b15 - 16b14 - 8b13 + 20b12 + 28b11 - 4b10 - 15b9 - 29b8 + 30b7 + 36b6 + 30b5 - 2b4 + 60b3 - 40b2 + 20*b1 - 75) / 10
\(\nu^{5}\)	\(=\)	\( ( 5 \beta_{15} + 9 \beta_{14} + 30 \beta_{13} + 10 \beta_{12} - 101 \beta_{11} - 76 \beta_{10} + \cdots + 130 ) / 10 \) (5b15 + 9b14 + 30b13 + 10b12 - 101b11 - 76b10 - 75b9 + 74b8 - 75b7 - 44b6 - 150b5 + 47b4 - 20b3 - 165b2 - 145*b1 + 130) / 10
\(\nu^{6}\)	\(=\)	\( ( - 117 \beta_{15} - 78 \beta_{14} + 204 \beta_{13} + 120 \beta_{12} - 170 \beta_{11} + 190 \beta_{10} + \cdots - 635 ) / 10 \) (-117b15 - 78b14 + 204b13 + 120b12 - 170b11 + 190b10 - 455b9 - 285b8 - 120b7 + 228b6 - 260b5 - 130b4 + 240b3 + 330b2 + 370*b1 - 635) / 10
\(\nu^{7}\)	\(=\)	\( ( - 290 \beta_{15} + 277 \beta_{14} - 625 \beta_{13} + 435 \beta_{12} + 187 \beta_{11} - 220 \beta_{10} + \cdots + 2350 ) / 10 \) (-290b15 + 277b14 - 625b13 + 435b12 + 187b11 - 220b10 + 1260b9 - 440b8 + 210b7 - 892b6 + 210b5 + 120b4 + 1220b3 - 1085b2 + 230*b1 + 2350) / 10
\(\nu^{8}\)	\(=\)	\( ( 357 \beta_{15} - 4 \beta_{14} + 544 \beta_{13} - 640 \beta_{12} - 956 \beta_{11} + 658 \beta_{10} + \cdots - 350 ) / 5 \) (357b15 - 4b14 + 544b13 - 640b12 - 956b11 + 658b10 - 1030b9 + 643b8 - 700b7 + 24b6 - 1330b5 - 406b4 - 2280b3 + 2660b2 - 220*b1 - 350) / 5
\(\nu^{9}\)	\(=\)	\( ( - 2255 \beta_{15} + 317 \beta_{14} - 6377 \beta_{13} + 2685 \beta_{12} + 9103 \beta_{11} + 1078 \beta_{10} + \cdots + 4685 ) / 10 \) (-2255b15 + 317b14 - 6377b13 + 2685b12 + 9103b11 + 1078b10 + 11145b9 - 4907b8 + 7800b7 - 1022b6 + 12000b5 - 626b4 + 7930b3 - 1675b2 + 5700*b1 + 4685) / 10
\(\nu^{10}\)	\(=\)	\( ( 5586 \beta_{15} + 1140 \beta_{14} + 888 \beta_{13} - 6600 \beta_{12} - 4604 \beta_{11} - 3589 \beta_{10} + \cdots + 7785 ) / 5 \) (5586b15 + 1140b14 + 888b13 - 6600b12 - 4604b11 - 3589b10 - 700b9 + 14511b8 - 3560b7 - 3540b6 - 6265b5 + 2308b4 - 22200b3 + 2550b2 - 16070*b1 + 7785) / 5
\(\nu^{11}\)	\(=\)	\( ( - 5570 \beta_{15} - 8349 \beta_{14} + 2994 \beta_{13} + 4915 \beta_{12} + 16869 \beta_{11} + \cdots - 67565 ) / 5 \) (-5570b15 - 8349b14 + 2994b13 + 4915b12 + 16869b11 + 9605b10 - 13035b9 - 15020b8 + 14355b7 + 27454b6 + 22275b5 - 5775b4 + 14320b3 + 14550b2 + 21445*b1 - 67565) / 5
\(\nu^{12}\)	\(=\)	\( ( 5799 \beta_{15} + 14338 \beta_{14} - 21214 \beta_{13} + 7090 \beta_{12} - 3310 \beta_{11} + \cdots + 114765 ) / 5 \) (5799b15 + 14338b14 - 21214b13 + 7090b12 - 3310b11 - 59480b10 + 50145b9 + 37775b8 - 1920b7 - 45228b6 - 4710b5 + 37190b4 + 20820b3 - 149450b2 - 73520*b1 + 114765) / 5
\(\nu^{13}\)	\(=\)	\( ( 9313 \beta_{15} - 54120 \beta_{14} + 196806 \beta_{13} - 36820 \beta_{12} - 176263 \beta_{11} + \cdots - 507940 ) / 5 \) (9313b15 - 54120b14 + 196806b13 - 36820b12 - 176263b11 + 93127b10 - 393315b9 - 11168b8 - 143910b7 + 177650b6 - 235755b5 - 56444b4 - 122560b3 + 273750b2 + 75055*b1 - 507940) / 5
\(\nu^{14}\)	\(=\)	\( ( - 285081 \beta_{15} + 87528 \beta_{14} - 390576 \beta_{13} + 411810 \beta_{12} + 404846 \beta_{11} + \cdots + 817745 ) / 5 \) (-285081b15 + 87528b14 - 390576b13 + 411810b12 + 404846b11 - 129964b10 + 764435b9 - 581109b8 + 335460b7 - 278208b6 + 539420b5 + 82738b4 + 1328040b3 - 972510b2 + 485270*b1 + 817745) / 5
\(\nu^{15}\)	\(=\)	\( ( 661913 \beta_{15} + 132176 \beta_{14} + 973606 \beta_{13} - 926670 \beta_{12} - 1913782 \beta_{11} + \cdots + 580685 ) / 5 \) (661913b15 + 132176b14 + 973606b13 - 926670b12 - 1913782b11 + 157210b10 - 1573125b9 + 1431485b8 - 1576980b7 - 416366b6 - 2554170b5 - 92070b4 - 3004010b3 + 1851230b2 - 1263050*b1 + 580685) / 5

\(n\)	\(701\)	\(1051\)	\(1177\)	\(1501\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1049.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 6 T^{6} + 22 T^{4} + \cdots + 81)^{2} \) (T^8 + 6T^6 + 22T^4 + 54*T^2 + 81)^2
$5$	\( T^{16} \) T^16
$7$	\( (T^{4} + 6 T^{2} + 49)^{4} \) (T^4 + 6*T^2 + 49)^4
$11$	\( (T^{4} + 16 T^{2} + 19)^{4} \) (T^4 + 16*T^2 + 19)^4
$13$	\( (T^{4} - 36 T^{2} + 4)^{4} \) (T^4 - 36*T^2 + 4)^4
$17$	\( (T^{4} + 22 T^{2} + 76)^{4} \) (T^4 + 22*T^2 + 76)^4
$19$	\( (T^{4} + 30 T^{2} + 100)^{4} \) (T^4 + 30*T^2 + 100)^4
$23$	\( (T^{4} - 80 T^{2} + 475)^{4} \) (T^4 - 80*T^2 + 475)^4
$29$	\( (T^{4} + 104 T^{2} + 2299)^{4} \) (T^4 + 104*T^2 + 2299)^4
$31$	\( (T^{4} + 30 T^{2} + 100)^{4} \) (T^4 + 30*T^2 + 100)^4
$37$	\( (T^{2} + 5)^{8} \) (T^2 + 5)^8
$41$	\( (T^{4} - 90 T^{2} + 1900)^{4} \) (T^4 - 90*T^2 + 1900)^4
$43$	\( (T^{4} + 90 T^{2} + 25)^{4} \) (T^4 + 90*T^2 + 25)^4
$47$	\( (T^{4} + 222 T^{2} + 9196)^{4} \) (T^4 + 222*T^2 + 9196)^4
$53$	\( (T^{4} - 180 T^{2} + 7600)^{4} \) (T^4 - 180*T^2 + 7600)^4
$59$	\( (T^{4} - 160 T^{2} + 1900)^{4} \) (T^4 - 160*T^2 + 1900)^4
$61$	\( (T^{4} + 150 T^{2} + 2500)^{4} \) (T^4 + 150*T^2 + 2500)^4
$67$	\( (T^{2} + 45)^{8} \) (T^2 + 45)^8
$71$	\( (T^{4} + 120 T^{2} + 475)^{4} \) (T^4 + 120*T^2 + 475)^4
$73$	\( (T^{2} - 2)^{8} \) (T^2 - 2)^8
$79$	\( (T^{2} - 6 T - 11)^{8} \) (T^2 - 6*T - 11)^8
$83$	\( (T^{4} + 90 T^{2} + 1900)^{4} \) (T^4 + 90*T^2 + 1900)^4
$89$	\( (T^{4} - 240 T^{2} + 1900)^{4} \) (T^4 - 240*T^2 + 1900)^4
$97$	\( (T^{4} - 94 T^{2} + 4)^{4} \) (T^4 - 94*T^2 + 4)^4
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