LMFDB - Newform orbit 110.4.g.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [110,4,Mod(31,110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(110, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 6]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("110.31");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$110 = 2 \cdot 5 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	110.g (of order $5$ , degree $4$ , 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:	$6.49021010063$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$3$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - x^{11} + 80 x^{10} - 71 x^{9} + 3603 x^{8} - 191 x^{7} + 110280 x^{6} + 142285 x^{5} + \cdots + 2085136$ x^12 - x^11 + 80x^10 - 71x^9 + 3603x^8 - 191x^7 + 110280x^6 + 142285x^5 + 9710517x^4 - 1952612x^3 + 6445016x^2 - 5490088x + 2085136
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - 2 \beta_{8} - 2 \beta_{5} + \cdots + 2) q^{2} - \beta_{4} q^{3} - 4 \beta_{2} q^{4} + 5 \beta_{5} q^{5} - 2 \beta_{3} q^{6} + (\beta_{11} + \beta_{9} + 2 \beta_{7} + \cdots + 5) q^{7} + 8 \beta_{8} q^{8}+ \cdots + (9 \beta_{11} + 15 \beta_{10} + \cdots + 220) q^{99}+O(q^{100})$ q + (-2b8 - 2b5 - 2b2 + 2) q^2 - b4 * q^3 - 4b2 q^4 + 5b5 q^5 - 2b3 q^6 + (b11 + b9 + 2b7 - b6 - 5b5 - 2b4 + 5) q^7 + 8b8 q^8 + (3b11 + 7b8 + 7b5 + 13b2 - 13) * q^9 + 10 * q^10 + (b11 - 3b9 + 8b8 - b7 + 2b6 + 8b5 + b4 + b3 + 5b2 - 2b1) * q^11 + (-4b7 + 4b4 - 4b3 + 4b1) * q^12 + (b11 - 3b10 - 10b8 - 3b7 - 3b6 - 10b5 - 4b4 + 4b3 - 13b2 + 13) * q^13 + (-2b10 - 10b8 - 10b5 - 4b3 - 10b2 + 4b1) * q^14 + 5b1 q^15 - 16b5 q^16 + (b11 + 2b10 - 2b9 + 22b8 - 4b7 + b6 + 10b5 - 7b3 + 4b1 - 22) q^17 + (6b6 + 12b5 + 26b2 - 12) q^18 + (-5b10 + 4b9 + 14b8 + 45b5 + 6b4 - b3 + 45b2 + b1) * q^19 + (-20b8 - 20b5 - 20b2 + 20) q^20 + (6b11 + 6b10 - 7b9 + 80b8 + b7 + 7b6 + b3 + 80b2 - 88) * q^21 + (6b10 - 2b9 - 10b8 + 2b7 + 8b6 - 16b5 + 2b4 + 4b3 - 4b1 + 16) q^22 + (-8b11 - 8b10 - 29b8 - 3b7 + 4b4 - 3b3 - 29b2 + 4b1 + 50) * q^23 - 8b7 q^24 - 25b8 q^25 + (-6b11 - 6b9 + 8b7 + 8b6 - 6b5 - 8b4 - 26b2 - 14b1 + 6) * q^26 + (3b10 - 3b9 - 12b8 + 9b7 + 3b6 + 7b3 - 9b1 + 12) q^27 + (-4b11 + 20b8 - 8b7 + 4b6 + 20b5 - 8b3 + 8b1 - 20) q^28 + (5b11 + 5b9 + 9b7 - 20b6 - 86b5 - 9b4 - 80b2 - 9b1 + 86) * q^29 - 10b4 q^30 + (-6b11 - 17b10 - 76b8 + 14b7 - 17b6 - 76b5 + 8b4 - 8b3 - 44b2 + 44) q^31 - 32 * q^32 + (-10b11 - 6b10 + 2b9 - 64b8 - 9b7 - 5b6 - 6b5 - 14b4 + 23b3 - 88b2 - 10b1 + 132) q^33 + (4b11 + 4b10 - 2b9 + 44b8 - 14b7 + 2b6 + 6b4 - 14b3 + 44b2 + 6b1 - 24) * q^34 + (5b10 + 25b8 + 10b7 + 5b6 + 25b5 - 10b4 + 10b3) q^35 + (12b9 - 28b8 + 24b5 + 24b2) * q^36 + (-3b11 - 3b9 + 18b7 + 27b5 - 18b4 - 31b2 - 24b1 - 27) q^37 + (-10b11 - 8b10 + 8b9 - 90b8 - 2b7 + 2b6 - 28b5 + 10b3 + 2b1 + 90) q^38 + (9b11 - 8b10 + 8b9 + 114b8 - 20b7 - 17b6 + 202b5 - 9b3 + 20b1 - 114) q^39 - 40b2 q^40 + (14b10 - 5b9 + 25b8 - 41b5 + 25b4 + 5b3 - 41b2 - 5b1) * q^41 + (12b11 + 14b10 + 16b8 + 2b7 + 14b6 + 16b5 - 2b4 + 2b3 + 176b2 - 176) q^42 + (-20b11 - 20b10 + 13b9 + 24b8 + 6b7 - 13b6 + 18b4 + 6b3 + 24b2 + 18b1 + 66) * q^43 + (12b11 + 4b10 + 12b9 - 32b8 + 8b7 - 8b6 - 12b5 + 12b3 - 32b2 - 4b1) * q^44 + (15b11 + 15b10 + 30b8 + 30b2 - 65) * q^45 + (-16b11 - 42b8 - 6b7 - 42b5 - 2b4 + 2b3 - 100b2 + 100) q^46 + (22b10 - 8b9 - 114b8 - 89b5 + 22b4 - 9b3 - 89b2 + 9b1) * q^47 - 16b1 q^48 + (26b11 + 13b10 - 13b9 + 97b8 + b7 - 13b6 + 17b5 + 10b3 - b1 - 97) q^49 + 50b5 q^50 + (11b11 + 11b9 - 30b7 + 8b6 + 186b5 + 30b4 + 316b2 + 3b1 - 186) * q^51 + (12b10 + 4b9 + 40b8 - 12b5 + 28b4 - 16b3 - 12b2 + 16b1) * q^52 + (-21b11 + 23b10 + 71b8 + 28b7 + 23b6 + 71b5 + 20b4 - 20b3 + 310b2 - 310) q^53 + (6b11 + 6b10 - 24b8 + 14b7 + 4b4 + 14b3 - 24b2 + 4b1 + 24) * q^54 + (15b11 + 5b10 - 15b8 - 15b7 - 15b6 + 25b5 + 10b4 - 5b3 - 15b2 - 25) q^55 + (8b9 + 40b8 - 16b7 - 8b6 - 16b3 + 40b2) * q^56 + (-11b11 + 8b10 - 218b8 + 64b7 + 8b6 - 218b5 - 79b4 + 79b3 - 230b2 + 230) q^57 + (-10b10 - 30b9 - 12b8 - 172b5 + 18b4 - 18b3 - 172b2 + 18b1) * q^58 + (34b11 + 34b9 + 15b7 - 38b6 + 71b5 - 15b4 + 57b2 - 17b1 - 71) * q^59 - 20b3 q^60 + (-11b11 + 9b10 - 9b9 + 246b8 - 15b7 + 20b6 + 300b5 + 19b3 + 15b1 - 246) q^61 + (-34b11 - 34b9 - 16b7 + 22b6 + 64b5 + 16b4 - 88b2 + 44b1 - 64) * q^62 + (-17b10 - 9b9 - 157b8 - 151b5 + 9b4 + 74b3 - 151b2 - 74b1) * q^63 + (64b8 + 64b5 + 64b2 - 64) q^64 + (5b11 + 5b10 + 15b9 - 15b8 - 15b7 - 15b6 + 35b4 - 15b3 - 15b2 + 35b1 + 65) * q^65 + (-12b11 - 4b10 - 6b9 - 88b8 + 46b7 - 12b6 - 136b5 - 26b4 + 18b3 - 264b2 - 64b1 + 252) q^66 + (-8b11 - 8b10 + 10b9 + 162b8 - 20b7 - 10b6 - 9b4 - 20b3 + 162b2 - 9b1 + 380) * q^67 + (8b11 + 4b10 - 40b8 - 28b7 + 4b6 - 40b5 + 16b4 - 16b3 + 48b2 - 48) q^68 + (3b10 + 17b9 - 130b8 - 48b5 - 69b4 - 37b3 - 48b2 + 37b1) * q^69 + (10b11 + 10b9 + 20b7 - 10b6 - 50b5 - 20b4 + 50) * q^70 + (-28b11 - 32b10 + 32b9 - 170b8 - 65b7 - 4b6 - 180b5 - 4b3 + 65b1 + 170) q^71 + (-24b10 + 24b9 - 48b8 - 24b6 + 56b5 + 48) q^72 + (-42b11 - 42b9 - 9b7 + 30b6 + 102b5 + 9b4 + 212b2 - 113b1 - 102) * q^73 + (6b10 - 6b9 + 116b8 + 54b5 + 48b4 - 36b3 + 54b2 + 36b1) * q^74 + 25b7 q^75 + (-16b11 - 16b10 + 20b9 - 180b8 + 20b7 - 20b6 - 24b4 + 20b3 - 180b2 - 24b1 + 124) * q^76 + (-40b11 - 33b10 + 6b9 - 317b8 - 7b7 + 25b6 + 149b5 - 19b4 + 6b3 - 296b2 - 58b1 + 261) q^77 + (-16b11 - 16b10 - 18b9 + 228b8 - 18b7 + 18b6 - 22b4 - 18b3 + 228b2 - 22b1 + 176) * q^78 + (53b11 - 23b10 - 240b8 + 78b7 - 23b6 - 240b5 - 49b4 + 49b3 - 264b2 + 264) q^79 + 80b8 q^80 + (-27b11 - 27b9 + 9b7 + 84b6 - 174b5 - 9b4 + 29b2 - 18b1 + 174) * q^81 + (28b11 + 10b10 - 10b9 + 82b8 + 10b7 - 18b6 - 50b5 + 60b3 - 10b1 - 82) q^82 + (-42b11 + 15b10 - 15b9 + 648b8 + 97b7 + 57b6 + 142b5 + 24b3 - 97b1 - 648) q^83 + (28b11 + 28b9 + 4b7 - 4b6 + 320b5 - 4b4 + 352b2 - 320) q^84 + (5b10 - 10b9 - 50b8 - 110b5 - 15b4 - 20b3 - 110b2 + 20b1) * q^85 + (-40b11 - 26b10 - 180b8 + 12b7 - 26b6 - 180b5 - 48b4 + 48b3 - 132b2 + 132) q^86 + (-15b11 - 15b10 - 32b9 + 366b8 - 180b7 + 32b6 + 109b4 - 180b3 + 366b2 + 109b1 - 94) * q^87 + (8b11 - 24b10 + 8b9 + 64b8 + 24b7 - 8b6 + 64b5 - 16b4 + 24b3 - 8b1 - 24) * q^88 + (31b11 + 31b10 - 4b9 - 83b8 - 101b7 + 4b6 + 67b4 - 101b3 - 83b2 + 67b1 + 442) * q^89 + (30b11 + 70b8 + 70b5 + 130b2 - 130) * q^90 + (-70b10 - 32b9 - 173b8 - 66b5 - 54b4 - 10b3 - 66b2 + 10b1) * q^91 + (4b7 - 32b6 - 116b5 - 4b4 - 200b2 - 16b1 + 116) * q^92 + (-41b11 + 36b10 - 36b9 - 164b8 - 76b7 + 77b6 - 768b5 - 103b3 + 76b1 + 164) q^93 + (44b11 + 16b10 - 16b9 + 178b8 - 18b7 - 28b6 + 228b5 + 26b3 + 18b1 - 178) q^94 + (25b11 + 25b9 + 5b7 - 5b6 + 225b5 - 5b4 + 155b2 - 30b1 - 225) * q^95 + 32b4 q^96 + (66b11 - 17b10 + 186b8 + 90b7 - 17b6 + 186b5 - 15b4 + 15b3 + 360b2 - 360) q^97 + (26b11 + 26b10 - 52b9 + 194b8 + 20b7 + 52b6 - 18b4 + 20b3 + 194b2 - 18b1 - 160) * q^98 + (9b11 + 15b10 + 10b9 + 263b8 - 22b7 + 8b6 + 404b5 - 25b4 - 101b3 - 432b2 + 68b1 + 220) q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 6 q^{2} - q^{3} - 12 q^{4} + 15 q^{5} + 2 q^{6} + 38 q^{7} + 24 q^{8} - 78 q^{9} + 120 q^{10} + 66 q^{11} + 16 q^{12} + 57 q^{13} - 76 q^{14} + 5 q^{15} - 48 q^{16} - 157 q^{17} - 24 q^{18} + 331 q^{19}+ \cdots + 3455 q^{99}+O(q^{100})$ 12 * q + 6 * q^2 - q^3 - 12 * q^4 + 15 * q^5 + 2 * q^6 + 38 * q^7 + 24 * q^8 - 78 * q^9 + 120 * q^10 + 66 * q^11 + 16 * q^12 + 57 * q^13 - 76 * q^14 + 5 * q^15 - 48 * q^16 - 157 * q^17 - 24 * q^18 + 331 * q^19 + 60 * q^20 - 588 * q^21 + 98 * q^22 + 472 * q^23 + 8 * q^24 - 75 * q^25 - 34 * q^26 + 80 * q^27 - 88 * q^28 + 477 * q^29 - 10 * q^30 - 18 * q^31 - 384 * q^32 + 1093 * q^33 + 4 * q^34 + 110 * q^35 + 48 * q^36 - 390 * q^37 + 748 * q^38 - 381 * q^39 - 120 * q^40 - 193 * q^41 - 1534 * q^42 + 1014 * q^43 - 296 * q^44 - 660 * q^45 + 666 * q^46 - 894 * q^47 - 16 * q^48 - 899 * q^49 + 150 * q^50 - 677 * q^51 + 68 * q^52 - 2377 * q^53 + 100 * q^54 - 330 * q^55 + 256 * q^56 + 535 * q^57 - 954 * q^58 - 621 * q^59 + 20 * q^60 - 1290 * q^61 - 674 * q^62 - 1456 * q^63 - 192 * q^64 + 740 * q^65 + 1424 * q^66 + 5566 * q^67 - 628 * q^68 - 699 * q^69 + 380 * q^70 + 1212 * q^71 + 624 * q^72 - 263 * q^73 + 780 * q^74 - 25 * q^75 + 344 * q^76 + 1822 * q^77 + 3572 * q^78 + 753 * q^79 + 240 * q^80 + 1755 * q^81 - 1034 * q^82 - 5555 * q^83 - 1892 * q^84 - 790 * q^85 + 92 * q^86 + 1770 * q^87 + 72 * q^88 + 5026 * q^89 - 780 * q^90 - 707 * q^91 + 388 * q^92 - 527 * q^93 - 1012 * q^94 - 1655 * q^95 + 32 * q^96 - 2276 * q^97 - 832 * q^98 + 3455 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - x^{11} + 80 x^{10} - 71 x^{9} + 3603 x^{8} - 191 x^{7} + 110280 x^{6} + 142285 x^{5} + \cdots + 2085136$ x^12 - x^11 + 80*x^10 - 71*x^9 + 3603*x^8 - 191*x^7 + 110280*x^6 + 142285*x^5 + 9710517*x^4 - 1952612*x^3 + 6445016*x^2 - 5490088*x + 2085136 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 48\!\cdots\!35 \nu^{11} + \cdots + 23\!\cdots\!92 ) / 12\!\cdots\!12$ (4893067357854173202868698435v^11 + 1447584511374736914219223599v^10 + 403031863434207540562076439702v^9 + 141209664614055337153260218427v^8 + 18608259917888128272349918079523v^7 + 20096219870515693737209990802769v^6 + 602409544796683532727350461825002v^5 + 1393651144655773148282012284103631v^4 + 50350109801185778302834905160799145v^3 + 55410786360192643073009781653285262v^2 + 190755825117658282634664030094344576*v + 23031914998526911984666490361106992) / 124650303471709735516271473368664712
$\beta_{3}$	$=$	$( 87\!\cdots\!97 \nu^{11} + \cdots - 14\!\cdots\!80 ) / 17\!\cdots\!96$ (8782066300871066644166097v^11 + 16047749038606210986953691v^10 + 676755466789060435674429096v^9 + 1355316104625404740185592269v^8 + 29128525949952688114900155407v^7 + 86983485557514282430734623841v^6 + 965984841611484922592575522548v^5 + 3927833851243415367637064044125v^4 + 89979291412668969416910854074581v^3 + 220526215660956247472858757315228v^2 + 69107043466861238276533929660276*v - 14131178529482852209884801080280) / 172645849683808497944974339845796
$\beta_{4}$	$=$	$( - 52\!\cdots\!83 \nu^{11} + \cdots + 32\!\cdots\!98 ) / 43\!\cdots\!49$ (-5258707512245618984613383v^11 + 7780652763541742852675078v^10 - 421607964479885254499129465v^9 + 576158863253287133482176946v^8 - 19248984110185970293895482311v^7 + 10287778876056335165837471056v^6 - 592316317606284799561891175883v^5 - 457075079338429013367119662280v^4 - 51273681923177914082633517394176v^3 + 34838155397149678616787985485676v^2 - 50810308593568288621349644469857*v + 32763496385027800289216470516298) / 43161462420952124486243584961449
$\beta_{5}$	$=$	$( - 20\!\cdots\!11 \nu^{11} + \cdots + 32\!\cdots\!32 ) / 12\!\cdots\!12$ (-20303260024890186207201180911v^11 - 6428333163892834285078060751v^10 - 1605705152890006376347067017798v^9 - 678780699213120750685395534603v^8 - 71922452731918436173230528834619v^7 - 91090191696710154566536354823733v^6 - 2264179566543735985083091295048194v^5 - 5807438239333500091263936217248687v^4 - 201853723890759585723130773365034241v^3 - 222506031046804060164492275680995038v^2 - 156150948106130518777257170951685524*v + 32211601403323012542414447497542232) / 124650303471709735516271473368664712
$\beta_{6}$	$=$	$( - 38\!\cdots\!65 \nu^{11} + \cdots - 18\!\cdots\!32 ) / 18\!\cdots\!68$ (-38185469433305349663497688565v^11 - 10772995433009884411137232161v^10 - 3331179406267907460775936665486v^9 - 989359871527172028668767082361v^8 - 159878337922511412986461672457241v^7 - 160321871913173123185924003088895v^6 - 5354541415710371483620860116881446v^5 - 11162984135086596304125728840740113v^4 - 413016140286201808884847246306165023v^3 - 453831937091432660424220360420383138v^2 - 3301134699439945136701982015726699948*v - 188747502893588500772027622282670032) / 186975455207564603274407210052997068
$\beta_{7}$	$=$	$( - 37\!\cdots\!71 \nu^{11} + \cdots + 58\!\cdots\!68 ) / 17\!\cdots\!96$ (-37024367297483407884043271v^11 + 25700345015524266245190381v^10 - 2936720444571085832959389722v^9 + 1703868611857208769134800537v^8 - 131534784434160914310404127947v^7 - 34814475067654086084411446237v^6 - 4042366879074792170044759267766v^5 - 6507717816680068455593052228107v^4 - 363089224668317086474941534545685v^3 - 35036166750073227740811973353734v^2 - 109771582853471329738928350980288*v + 58635814951882857766397010470968) / 172645849683808497944974339845796
$\beta_{8}$	$=$	$( - 29\!\cdots\!33 \nu^{11} + \cdots + 17\!\cdots\!64 ) / 12\!\cdots\!12$ (-29968611190343211772975648433v^11 + 35172405214531974520603517957v^10 - 2405160493953430593041134269360v^9 + 2541862307055081757727526715795v^8 - 108521690687031545730264057889983v^7 + 24070257252736810553022064110415v^6 - 3312893362250677760126861227011600v^5 - 3753545537416426555592105189192113v^4 - 290468065036302113217180192810499497v^3 + 107624048250038564071706079903813796v^2 - 226459401146230313769470497698626308*v + 171800242363385986849934804086998464) / 124650303471709735516271473368664712
$\beta_{9}$	$=$	$( - 45\!\cdots\!43 \nu^{11} + \cdots + 27\!\cdots\!00 ) / 18\!\cdots\!68$ (-459147840190939310995962682643v^11 + 603765487543439024970006339887v^10 - 36945833539519578712526060759688v^9 + 44151464374873064771179196451817v^8 - 1669805270588493450178361952194145v^7 + 607490683691613385331234892360793v^6 - 50886343329631855887992302455119832v^5 - 49282497795164467299479094898818095v^4 - 4444699992331870071612007373475027615v^3 + 2284379322816994196728670180662119532v^2 - 3901412580224818639602680180435820712*v + 2727004067324368866759054271481958200) / 186975455207564603274407210052997068
$\beta_{10}$	$=$	$( - 33\!\cdots\!33 \nu^{11} + \cdots + 16\!\cdots\!60 ) / 93\!\cdots\!34$ (-338105978804868745767979062133v^11 + 199869801773337401486502992705v^10 - 26848080125260743935263430294344v^9 + 12987982719305969382506829649271v^8 - 1203521778523750407473013548127739v^7 - 426465737498748275386512664442509v^6 - 37042087229206481435201705336511442v^5 - 62792663558249046440031983091864661v^4 - 3295896481539837440184421347971473099v^3 - 652469130450751495050230324260761004v^2 - 1287732897575200690402829288895947874*v + 1660949488699709913375568754533922460) / 93487727603782301637203605026498534
$\beta_{11}$	$=$	$( 37\!\cdots\!40 \nu^{11} + \cdots - 71\!\cdots\!60 ) / 93\!\cdots\!34$ (373898450082166502190398943340v^11 - 254141781915027267441634633389v^10 + 29704858236263440123997136523165v^9 - 16794647075555741348163249503240v^8 + 1331689334612306555019366258483463v^7 + 369158752192384510142925691692979v^6 + 40897595349439553902155065505945589v^5 + 66483340497637173181567384304706514v^4 + 3638333917709444355135056334783983467v^3 + 420131569511714390891468855998478755v^2 + 1338477711008198102952703340253245604*v - 714611506916199136337442748043447060) / 93487727603782301637203605026498534

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$3\beta_{10} - 3\beta_{9} + 6\beta_{8} + 3\beta_{6} - 34\beta_{5} - 6$ 3b10 - 3b9 + 6b8 + 3b6 - 34*b5 - 6
$\nu^{3}$	$=$	$-3\beta_{11} - 61\beta_{7} + 9\beta_{4} - 9\beta_{3} + 12\beta_{2} - 12$ -3b11 - 61b7 + 9b4 - 9b3 + 12*b2 - 12
$\nu^{4}$	$=$	$-27\beta_{10} + 186\beta_{9} - 1822\beta_{8} + 660\beta_{5} - 18\beta_{4} + 9\beta_{3} + 660\beta_{2} - 9\beta_1$ -27b10 + 186b9 - 1822b8 + 660b5 - 18b4 + 9b3 + 660b2 - 9b1
$\nu^{5}$	$=$	$213 \beta_{11} + 213 \beta_{10} - 996 \beta_{8} + 3811 \beta_{7} - 2803 \beta_{4} + 3811 \beta_{3} + \cdots + 546$ 213b11 + 213b10 - 996b8 + 3811b7 - 2803b4 + 3811b3 - 996b2 - 2803b1 + 546
$\nu^{6}$	$=$	$- 3024 \beta_{11} - 3024 \beta_{9} + 783 \beta_{7} - 8622 \beta_{6} - 56286 \beta_{5} - 783 \beta_{4} + \cdots + 56286$ -3024b11 - 3024b9 + 783b7 - 8622b6 - 56286b5 - 783b4 - 157636b2 - 1728b1 + 56286
$\nu^{7}$	$=$	$- 675 \beta_{11} - 14481 \beta_{10} + 14481 \beta_{9} + 67536 \beta_{8} - 86076 \beta_{7} + \cdots - 67536$ -675b11 - 14481b10 + 14481b9 + 67536b8 - 86076b7 - 13806b6 + 41958b5 - 242182b3 + 86076*b1 - 67536
$\nu^{8}$	$=$	$741027 \beta_{11} + 258903 \beta_{10} + 5821360 \beta_{8} + 76464 \beta_{7} + 258903 \beta_{6} + \cdots - 10143112$ 741027b11 + 258903b10 + 5821360b8 + 76464b7 + 258903b6 + 5821360b5 + 49005b4 - 49005b3 + 10143112*b2 - 10143112
$\nu^{9}$	$=$	$111888 \beta_{10} - 970419 \beta_{9} + 2936304 \beta_{8} + 4171494 \beta_{5} + 8973007 \beta_{4} + \cdots - 6616197 \beta_1$ 111888b10 - 970419b9 + 2936304b8 + 4171494b5 + 8973007b4 + 6616197b3 + 4171494b2 - 6616197b1
$\nu^{10}$	$=$	$- 47738031 \beta_{11} - 47738031 \beta_{10} + 19960479 \beta_{9} - 314604246 \beta_{8} + \cdots + 658936114$ -47738031b11 - 47738031b10 + 19960479b9 - 314604246b8 - 5669631b7 - 19960479b6 + 8199378b4 - 5669631b3 - 314604246b2 + 8199378b1 + 658936114
$\nu^{11}$	$=$	$12371238 \beta_{11} + 12371238 \beta_{9} + 482105151 \beta_{7} + 52375686 \beta_{6} - 242945736 \beta_{5} + \cdots + 242945736$ 12371238b11 + 12371238b9 + 482105151b7 + 52375686b6 - 242945736b5 - 482105151b4 - 59187282b2 + 530957659b1 + 242945736

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/110\mathbb{Z}\right)^\times$ .

$n$	$67$	$101$
$\chi(n)$	$1$	$-\beta_{2}$

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}

31.1

5.73804	+	4.16893i
0.396392	+	0.287995i
−5.32541	−	3.86914i
5.73804	−	4.16893i
0.396392	−	0.287995i
−5.32541	+	3.86914i
2.51400	+	7.73730i
−0.292839	−	0.901266i
−2.53018	−	7.78709i
2.51400	−	7.73730i
−0.292839	+	0.901266i
−2.53018	+	7.78709i

−0.618034

1.90211i

−5.73804

4.16893i

−3.23607

−

2.35114i

−1.54508

−

4.75528i

−4.38347

−

13.4909i

−9.28009

−

6.74238i

6.47214

−

4.70228i

7.20166

−

22.1644i

10.0000

31.2

−0.618034

1.90211i

−0.396392

0.287995i

−3.23607

−

2.35114i

−1.54508

−

4.75528i

−0.302816

−

0.931973i

19.1679

13.9263i

6.47214

−

4.70228i

−8.26927

25.4502i

10.0000

31.3

−0.618034

1.90211i

5.32541

−

3.86914i

−3.23607

−

2.35114i

−1.54508

−

4.75528i

4.06825

12.5208i

7.43847

5.40436i

6.47214

−

4.70228i

5.04633

−

15.5310i

10.0000

71.1

−0.618034

−

1.90211i

−5.73804

−

4.16893i

−3.23607

2.35114i

−1.54508

4.75528i

−4.38347

13.4909i

−9.28009

6.74238i

6.47214

4.70228i

7.20166

22.1644i

10.0000

71.2

−0.618034

−

1.90211i

−0.396392

−

0.287995i

−3.23607

2.35114i

−1.54508

4.75528i

−0.302816

0.931973i

19.1679

−

13.9263i

6.47214

4.70228i

−8.26927

−

25.4502i

10.0000

71.3

−0.618034

−

1.90211i

5.32541

3.86914i

−3.23607

2.35114i

−1.54508

4.75528i

4.06825

−

12.5208i

7.43847

−

5.40436i

6.47214

4.70228i

5.04633

15.5310i

10.0000

81.1

1.61803

1.17557i

−2.51400

7.73730i

1.23607

3.80423i

4.04508

−

2.93893i

−13.1635

9.56383i

7.65943

23.5733i

−2.47214

7.60845i

−31.7021

−

23.0330i

10.0000

81.2

1.61803

1.17557i

0.292839

−

0.901266i

1.23607

3.80423i

4.04508

−

2.93893i

1.53333

−

1.11403i

2.73261

8.41010i

−2.47214

7.60845i

21.1169

15.3423i

10.0000

81.3

1.61803

1.17557i

2.53018

−

7.78709i

1.23607

3.80423i

4.04508

−

2.93893i

13.2482

−

9.62537i

−8.71828

−

26.8321i

−2.47214

7.60845i

−32.3935

−

23.5353i

10.0000

91.1

1.61803

−

1.17557i

−2.51400

−

7.73730i

1.23607

−

3.80423i

4.04508

2.93893i

−13.1635

−

9.56383i

7.65943

−

23.5733i

−2.47214

−

7.60845i

−31.7021

23.0330i

10.0000

91.2

1.61803

−

1.17557i

0.292839

0.901266i

1.23607

−

3.80423i

4.04508

2.93893i

1.53333

1.11403i

2.73261

−

8.41010i

−2.47214

−

7.60845i

21.1169

−

15.3423i

10.0000

91.3

1.61803

−

1.17557i

2.53018

7.78709i

1.23607

−

3.80423i

4.04508

2.93893i

13.2482

9.62537i

−8.71828

26.8321i

−2.47214

−

7.60845i

−32.3935

23.5353i

10.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	110.4.g.c	✓	12
11.c	even	5	1	inner	110.4.g.c	✓	12
11.c	even	5	1		1210.4.a.bd		6
11.d	odd	10	1		1210.4.a.bg		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
110.4.g.c	✓	12	1.a	even	1	1	trivial
110.4.g.c	✓	12	11.c	even	5	1	inner
1210.4.a.bd		6	11.c	even	5	1
1210.4.a.bg		6	11.d	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{12} + T_{3}^{11} + 80 T_{3}^{10} + 71 T_{3}^{9} + 3603 T_{3}^{8} + 191 T_{3}^{7} + 110280 T_{3}^{6} + \cdots + 2085136$ T3^12 + T3^11 + 80*T3^10 + 71*T3^9 + 3603*T3^8 + 191*T3^7 + 110280*T3^6 - 142285*T3^5 + 9710517*T3^4 + 1952612*T3^3 + 6445016*T3^2 + 5490088*T3 + 2085136 acting on $S_{4}^{\mathrm{new}}(110, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{3}$ (T^4 - 2T^3 + 4T^2 - 8*T + 16)^3
$3$	$T^{12} + T^{11} + \cdots + 2085136$ T^12 + T^11 + 80T^10 + 71T^9 + 3603T^8 + 191T^7 + 110280T^6 - 142285T^5 + 9710517T^4 + 1952612T^3 + 6445016T^2 + 5490088T + 2085136
$5$	$(T^{4} - 5 T^{3} + \cdots + 625)^{3}$ (T^4 - 5T^3 + 25T^2 - 125*T + 625)^3
$7$	$T^{12} + \cdots + 238773729813025$ T^12 - 38T^11 + 1686T^10 - 46714T^9 + 1263334T^8 - 21640742T^7 + 315124289T^6 - 738102031T^5 + 10608951226T^4 - 251682464000T^3 + 5166186177735T^2 - 42056152210875*T + 238773729813025
$11$	$T^{12} + \cdots + 55\!\cdots\!81$ T^12 - 66T^11 - 98T^10 + 35323T^9 + 2441428T^8 + 36753508T^7 - 8410438597T^6 + 48918919148T^5 + 4325138629108T^4 + 83289786289193T^3 - 307565980918658T^2 - 275698379181432966*T + 5559917313492231481
$13$	$T^{12} + \cdots + 81\!\cdots\!61$ T^12 - 57T^11 + 11651T^10 - 547605T^9 + 53576950T^8 - 2462803347T^7 + 91512590024T^6 - 3100651251687T^5 + 89466660867130T^4 - 1555319950337925T^3 + 14780167241231871T^2 - 53305868108181942*T + 81861460759707561
$17$	$T^{12} + \cdots + 10\!\cdots\!56$ T^12 + 157T^11 + 8656T^10 - 322291T^9 + 16122971T^8 - 564818299T^7 + 26526416016T^6 - 436922312673T^5 + 14341956898249T^4 - 262503470340360T^3 + 4451190342231072T^2 - 36344861659444584*T + 1030701122477754256
$19$	$T^{12} + \cdots + 16\!\cdots\!61$ T^12 - 331T^11 + 74839T^10 - 10424578T^9 + 1046034316T^8 - 70424681993T^7 + 4055069582519T^6 - 145574138508969T^5 + 6603816996753274T^4 - 152948543935269690T^3 + 1835349673229799093T^2 - 8443497602608854687*T + 16551813717904304961
$23$	$(T^{6} - 236 T^{5} + \cdots + 35478920461)^{2}$ (T^6 - 236T^5 + 2996T^4 + 2062861T^3 - 58288495T^2 - 4444570893*T + 35478920461)^2
$29$	$T^{12} + \cdots + 38\!\cdots\!00$ T^12 - 477T^11 + 161565T^10 - 24316627T^9 + 3109668814T^8 - 41322314933T^7 + 217892501963165T^6 - 37695045622316668T^5 + 9377586787950600181T^4 - 533777290526821869650T^3 + 21086953466383603456560T^2 - 705916984836471482100400*T + 38786996948247064971216400
$31$	$T^{12} + \cdots + 23\!\cdots\!16$ T^12 + 18T^11 + 112245T^10 - 7890118T^9 + 6175290753T^8 - 284466001582T^7 + 251461580308125T^6 - 17510941913861590T^5 + 15448012923270992697T^4 - 732299129994866062734T^3 + 337116641143021858594504T^2 - 42151177340051443422096224*T + 2389151378982640314359616016
$37$	$T^{12} + \cdots + 99\!\cdots\!61$ T^12 + 390T^11 + 43815T^10 - 7957586T^9 + 1125333375T^8 + 213708878580T^7 + 27727453161421T^6 + 2103910194122250T^5 + 355130515389738120T^4 + 11306618910194838064T^3 + 932707815284288806290T^2 - 4748450519661837236970*T + 99966126282878966309761
$41$	$T^{12} + \cdots + 15\!\cdots\!01$ T^12 + 193T^11 + 178601T^10 - 23807632T^9 - 934118552T^8 + 430813990561T^7 + 303312530642861T^6 - 51422810959133933T^5 + 7166200154940698262T^4 - 520132804045027723836T^3 + 29141674937980160644307T^2 - 936203815527863955150333*T + 15567057865861141138356001
$43$	$(T^{6} + \cdots + 78741334432420)^{2}$ (T^6 - 507T^5 - 107071T^4 + 75167817T^3 - 4692855679T^2 - 925347036790*T + 78741334432420)^2
$47$	$T^{12} + \cdots + 68\!\cdots\!21$ T^12 + 894T^11 + 446689T^10 + 139457767T^9 + 45688682726T^8 + 8908288090707T^7 + 1950080455797134T^6 + 184798334536063231T^5 + 14684602193422610974T^4 + 153067834275589346155T^3 + 329549001170555519547633T^2 + 35115113017110383124414873*T + 6829610863278716135003232721
$53$	$T^{12} + \cdots + 14\!\cdots\!25$ T^12 + 2377T^11 + 3274135T^10 + 3107867502T^9 + 2250782982894T^8 + 1302383197907893T^7 + 633520106476009415T^6 + 251557089031175181078T^5 + 74992033756144462879636T^4 + 14317991170010516894904240T^3 + 1471321244239278519573440790T^2 + 41941300639783582950249333150*T + 1493294873518022698822496658025
$59$	$T^{12} + \cdots + 98\!\cdots\!21$ T^12 + 621T^11 + 84447T^10 - 67222090T^9 + 61924264844T^8 + 29921165081907T^7 + 17523701475133481T^6 + 3889665742611732666T^5 + 807561006319491112216T^4 + 107081511257700809395764T^3 + 13541433112221693166444946T^2 + 1188115614586605428681855612*T + 98568440765183934322642834321
$61$	$T^{12} + \cdots + 58\!\cdots\!00$ T^12 + 1290T^11 + 1526062T^10 + 938240880T^9 + 468312700484T^8 + 135426978042060T^7 + 22896640725220093T^6 + 2111534320715427320T^5 + 115007926818872817001T^4 + 3909199417840588130340T^3 + 124735516143141365796240T^2 + 3426459955979684478465600*T + 58191207035362583630342400
$67$	$(T^{6} + \cdots + 13\!\cdots\!44)^{2}$ (T^6 - 2783T^5 + 2979146T^4 - 1522448254T^3 + 372794937941T^2 - 38623265459970*T + 1363304886646644)^2
$71$	$T^{12} + \cdots + 12\!\cdots\!00$ T^12 - 1212T^11 + 1494591T^10 - 445668331T^9 + 23218377484T^8 + 55930046277107T^7 + 252936188198715789T^6 - 11787679467076469969T^5 + 28608509999353140361001T^4 - 5855900136167053191143920T^3 + 1641628706908755220930686160T^2 - 204392337268549594423464583400*T + 12628961607768390817804937520400
$73$	$T^{12} + \cdots + 21\!\cdots\!76$ T^12 + 263T^11 + 1756336T^10 + 497109221T^9 + 1592638758871T^8 + 825941597440509T^7 + 1074623766469167996T^6 + 1064513187282344477103T^5 + 1193218414282601882229069T^4 + 890749830916782195630675300T^3 + 494778516611317503246869190552T^2 + 148707126888558869843653143105144*T + 21619261534400109623551035622914576
$79$	$T^{12} + \cdots + 17\!\cdots\!76$ T^12 - 753T^11 + 796246T^10 - 1144388951T^9 + 1175409085111T^8 - 196626386505359T^7 + 429375935000240456T^6 - 832679320554744833363T^5 + 737184846041101431978189T^4 - 290488686132166620810897100T^3 + 98351914836607990095882450592T^2 - 12270308041462285546827294105344*T + 1763407918171180562134944564510976
$83$	$T^{12} + \cdots + 51\!\cdots\!56$ T^12 + 5555T^11 + 17082155T^10 + 34874614813T^9 + 52531737091120T^8 + 59460990381018825T^7 + 53383041070298004699T^6 + 38413285717769737339015T^5 + 23270683993972697213377825T^4 + 11133252705726525455300317792T^3 + 4466563479344749212485194808640T^2 + 725249461965426370452770516406440*T + 51321648864300915963312781653479056
$89$	$(T^{6} + \cdots + 27\!\cdots\!19)^{2}$ (T^6 - 2513T^5 + 1245437T^4 + 1152007970T^3 - 1057568078497T^2 + 154188125287647*T + 27633686993982819)^2
$97$	$T^{12} + \cdots + 22\!\cdots\!36$ T^12 + 2276T^11 + 3324989T^10 + 3162949668T^9 + 5326865072541T^8 + 1481699234380868T^7 + 2000486906872632589T^6 - 673079433269485943596T^5 + 76904192965560967477049T^4 + 9539996715372090297978500T^3 + 555399567330694274131589168T^2 + 8285595887844884884860076352*T + 2260379162489721865624488622336
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.3
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	110.4.g.c

Level	$110$
Weight	$4$
Character orbit	110.g
Analytic conductor	$6.490$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$