LMFDB - Newform orbit 175.4.k.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(74,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

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

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

chi := DirichletCharacter("175.74");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$175 = 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	175.k (of order $6$ , degree $2$ , 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:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - 75 x^{18} + 3638 x^{16} - 105775 x^{14} + 2246038 x^{12} - 30934571 x^{10} + 307864753 x^{8} + \cdots + 16777216$ x^20 - 75x^18 + 3638x^16 - 105775x^14 + 2246038x^12 - 30934571x^10 + 307864753x^8 - 1719258496x^6 + 6352449536x^4 - 328728576*x^2 + 16777216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{20}$
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{14} + \beta_1) q^{2} + (\beta_{8} + \beta_{6}) q^{3} + ( - \beta_{12} - 7 \beta_{7} + 7) q^{4} + ( - \beta_{10} + \beta_{9} + \beta_{4} + \cdots + 3) q^{6} + (\beta_{17} + 4 \beta_{15} + \cdots - \beta_1) q^{7}+ \cdots + ( - 15 \beta_{10} + 15 \beta_{9} + \cdots + 6) q^{99}+O(q^{100})$ q + (-b14 + b1) * q^2 + (b8 + b6) * q^3 + (-b12 - 7b7 + 7) q^4 + (-b10 + b9 + b4 - b3 + 2b2 + 3) q^6 + (b17 + 4b15 + 2b14 - b13 - b8 - b6 - b1) * q^7 + (2b15 - 5b14 - 4b13 - 4b6) * q^8 + (b12 + 2b11 + b9 + 17b7 + b5 - b3 + b2 - 1) * q^9 + (b12 - b10 - 2b9 + 9b7 - 4b5 - 8) q^11 + (-b19 - b18 - b16 - 13b15 - 11b14 - 9b13 + 13b8 + 11b1) q^12 + (b19 + 3b18 + b17 - 2b16 - b15 - b13 - b6) * q^13 + (-b12 + 5b11 - 2b10 + 2b9 + 27b7 + 6b5 + 3b4 - 6b3 - b2 - 15) q^14 + (-5b12 + 4b11 + 4b10 + 4b9 - 35b7 + 2b5 - 2b3 - 5b2) * q^16 + (-3b19 + 3b18 - 4b8 - 5b6 - b1) * q^17 + (5b19 - 5b18 - 3b17 + 3b16 + 11b8 - 2b6 + 27b1) q^18 + (b12 + 6b10 + 3b9 - 9b7 - 3b5 + 3b3 + b2 + 3) q^19 + (3b12 + 3b11 - 3b10 - 5b9 - 30b7 + 3b5 - b4 + 8b3 - b2 + 29) q^21 + (-3b19 - b18 - 3b17 - 2b16 - 51b15 + 16b14 + 9b13 + 9b6) q^22 + (-3b19 + 3b18 + 6b17 + 3b16 - 14b15 + 4b14 + 6b13 + 14b8 - 4b1) q^23 + (-14b12 + 3b11 - 2b10 + 2b9 - 172b7 - 9b5 + 3b4 + 171) q^24 + (3b12 - 15b11 + 13b10 - b9 - b7 + b5 - b3 + 3b2 + 14) * q^26 + (5b19 + 4b18 + 5b17 + b16 + 54b15 - 11b14 + 6b13 + 6b6) q^27 + (12b19 - 7b18 + 11b16 + 5b15 + 10b14 + 66b8 - b1) * q^28 + (3b10 - 3b9 - b4 - 17b3 + 13b2 - 29) * q^29 + (-8b12 + 8b11 - 4b10 + 8b9 + 78b7 - 3b5 + 8b4 - 82) q^31 + (6b19 - 6b18 - 4b17 + 4b16 - 10b8 - 18b6 - 11b1) q^32 + (8b19 - 2b18 - 10b17 - 2b16 + 21b15 + 18b14 - 2b13 - 21b8 - 18b1) q^33 + (2b10 - 2b9 + 10b4 + 19b3 - 8b2 - 30) q^34 + (-b10 + b9 - 17b4 - 16b3 + 7b2 + 248) q^36 + (-5b19 - 8b18 - 3b17 - 8b16 - 15b15 + 45b14 - 4b13 + 15b8 - 45b1) q^37 + (-9b19 + 9b18 + 3b17 - 3b16 - 95b8 - 26b6 + 9b1) q^38 + (2b12 + 16b11 - 14b10 + 4b9 - 8b7 - 39b5 + 16b4 + 6) q^39 + (13b4 + b3 + 14b2 - 54) * q^41 + (14b19 - 2b18 + 7b17 + 21b16 + 212b15 - 26b14 + 56b13 - 127b8 + 33b6 + 9b1) * q^42 + (2b19 + 8b18 + 2b17 - 6b16 + 61b15 + 54b14 + 9b13 + 9b6) * q^43 + (27b12 - 17b11 + 11b10 - 3b9 + 195b7 - 33b5 + 33b3 + 27b2 + 14) * q^44 + (7b12 + 30b11 - 30b10 + 93b7 + 4b5 + 30b4 - 93) * q^46 + (9b19 - 3b18 - 12b17 - 3b16 - 56b15 + 36b14 - 21b13 + 56b8 - 36b1) q^47 + (b19 + 3b18 + b17 - 2b16 - 9b15 - 153b14 + b13 + b6) * q^48 + (-19b12 - 16b11 + 9b10 - 2b9 - 56b7 + 9b5 - 15b4 + 5b3 - 26b2 - 71) q^49 + (-16b12 - 8b11 - 4b9 - 166b7 - 45b5 + 45b3 - 16b2 + 4) q^51 + (-26b19 + 26b18 + 5b17 - 5b16 - 57b8 + 15b6 - 2b1) q^52 + (8b19 - 8b18 - 3b17 + 3b16 - 24b8 + 23b6 + 2b1) q^53 + (4b12 - 9b11 - b10 - 5b9 - 126b7 + 75b5 - 75b3 + 4b2 + 4) q^54 + (4b12 + 5b11 - 3b10 + 7b9 + 54b7 + 32b5 - 28b4 - 62b3 - b2 - 153) * q^56 + (-5b19 + 10b18 - 5b17 - 15b16 - 111b15 - 85b14 + 17b13 + 17b6) * q^57 + (-b19 - 15b18 - 14b17 - 15b16 - 207b15 - 37b14 - 43b13 + 207b8 + 37b1) * q^58 + (-8b12 - 7b11 + 3b10 - 8b9 - 134b7 + 21b5 - 7b4 + 138) q^59 + (-b12 + 2b11 + 4b10 + 3b9 - 382b7 + 22b5 - 22b3 - b2 + 1) * q^61 + (b19 - 12b18 + b17 + 13b16 - 44b15 + 53b14 - 63b13 - 63b6) * q^62 + (-26b19 + 14b18 + 10b17 - 11b16 + 98b15 + 66b14 - 3b13 - 13b8 + 11b6 - 135b1) * q^63 + (-8b10 + 8b9 - 24b4 + 12b3 - 17b2 + 9) q^64 + (32b12 - 64b11 + 56b10 - 16b9 + 256b7 + 3b5 - 64b4 - 248) q^66 + (7b19 - 7b18 + 12b17 - 12b16 + 116b8 - 52b6 + 47b1) q^67 + (-22b19 + 7b18 + 29b17 + 7b16 + 346b15 + 69b14 + 39b13 - 346b8 - 69b1) q^68 + (b10 - b9 - 30b4 + 100b3 - 23b2 - 120) q^69 + (4b10 - 4b9 + 3b4 + 29b3 - 32b2 + 165) q^71 + (11b19 - 18b18 - 29b17 - 18b16 - 271b15 - 91b14 - 86b13 + 271b8 + 91b1) q^72 + (-3b19 + 3b18 - 2b17 + 2b16 + 178b8 + 41b6 + 79b1) q^73 + (35b12 - b11 + 2b10 + 2b9 + 635b7 - 50b5 - b4 - 636) q^74 + (-7b10 + 7b9 + 25b4 + 104b3 - 39b2 + 96) q^76 + (-23b19 + 33b18 + 21b17 - 17b16 - 63b15 - 30b14 + 42b13 + 44b8 + 12b6 + 99b1) * q^77 + (-25b19 - 30b18 - 25b17 + 5b16 - 562b15 + 65b14 - 33b13 - 33b6) * q^78 + (-22b12 + 51b11 - 39b10 + 6b9 + 80b7 + 7b5 - 7b3 - 22b2 - 45) * q^79 + (-28b12 + 41b11 - 47b10 - 12b9 - 65b7 - 35b5 + 41b4 + 71) q^81 + (-26b19 + 14b18 + 40b17 + 14b16 + 122b15 - 5b14 - 42b13 - 122b8 + 5b1) q^82 + (-7b19 - 36b18 - 7b17 + 29b16 - 130b15 - 15b14 + 32b13 + 32b6) * q^83 + (38b12 + 28b11 - 47b10 - 25b9 + 98b7 + 192b5 - 7b4 - 107b3 + 2b2 + 13) q^84 + (86b12 - 55b11 + 25b10 - 15b9 + 852b7 + 63b5 - 63b3 + 86b2 + 40) * q^86 + (45b19 - 45b18 - 16b17 + 16b16 + 97b8 + 38b6 + 147b1) q^87 + (-62b19 + 62b18 + 39b17 - 39b16 - 231b8 + 13b6 + 236b1) q^88 + (3b12 - 43b11 - 3b10 - 23b9 + 476b7 + 46b5 - 46b3 + 3b2 + 20) * q^89 + (-38b12 + 12b11 - 17b10 + 7b9 + 656b7 - 17b5 - 21b4 + 64b3 + 27b2 - 314) q^91 + (10b19 - 36b18 + 10b17 + 46b16 + 342b15 + 129b14 + 14b13 + 14b6) * q^92 + (18b19 + b18 - 17b17 + b16 + 220b15 - 211b14 + 111b13 - 220b8 + 211b1) q^93 + (15b12 - 57b11 + 66b10 + 18b9 + 447b7 - 80b5 - 57b4 - 456) q^94 + (-34b12 - 41b11 + 19b10 - 11b9 - 912b7 + 65b5 - 65b3 - 34b2 + 30) * q^96 + (8b19 - 8b18 + 8b17 + 16b16 - 51b15 - 48b14 + 8b13 + 8b6) * q^97 + (-2b19 + 31b18 - 4b17 - 31b16 + 105b15 + 83b14 + 18b13 + 26b8 - 56b6 - 287b1) * q^98 + (-15b10 + 15b9 + 49b4 - 38b3 - b2 + 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 70 q^{4} + 64 q^{6} + 162 q^{9} - 94 q^{11} - 14 q^{14} - 342 q^{16} - 42 q^{19} + 356 q^{21} + 1696 q^{24} + 94 q^{26} - 760 q^{29} - 776 q^{31} - 520 q^{34} + 4916 q^{36} + 28 q^{39} - 1124 q^{41}+ \cdots - 140 q^{99}+O(q^{100})$ 20 * q + 70 * q^4 + 64 * q^6 + 162 * q^9 - 94 * q^11 - 14 * q^14 - 342 * q^16 - 42 * q^19 + 356 * q^21 + 1696 * q^24 + 94 * q^26 - 760 * q^29 - 776 * q^31 - 520 * q^34 + 4916 * q^36 + 28 * q^39 - 1124 * q^41 + 2182 * q^44 - 674 * q^46 - 1996 * q^49 - 1464 * q^51 - 1548 * q^54 - 2676 * q^56 + 1416 * q^59 - 3888 * q^61 + 500 * q^64 - 2964 * q^66 - 1496 * q^69 + 3456 * q^71 - 6570 * q^74 + 2764 * q^76 + 436 * q^79 + 910 * q^81 + 1468 * q^84 + 8528 * q^86 + 4644 * q^89 + 1048 * q^91 - 5354 * q^94 - 9184 * q^96 - 140 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} - 75 x^{18} + 3638 x^{16} - 105775 x^{14} + 2246038 x^{12} - 30934571 x^{10} + 307864753 x^{8} + \cdots + 16777216$ x^20 - 75*x^18 + 3638*x^16 - 105775*x^14 + 2246038*x^12 - 30934571*x^10 + 307864753*x^8 - 1719258496*x^6 + 6352449536*x^4 - 328728576*x^2 + 16777216 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 47\!\cdots\!27 \nu^{18} + \cdots - 25\!\cdots\!16 ) / 17\!\cdots\!80$ (-47982427449047527v^18 + 3363516706106142765v^16 - 158522853331880350010v^14 + 4319505709196966456681v^12 - 87482635518242248167130v^10 + 1067199380902491452298445v^8 - 9683701022226946094452023v^6 + 27485717931873584130682880v^4 - 1422376046277104141598720*v^2 - 2575987397196925320917692416) / 172331914394149674471444480
$\beta_{3}$	$=$	$( 29\!\cdots\!47 \nu^{18} + \cdots - 37\!\cdots\!64 ) / 82\!\cdots\!40$ (2939380168937507947v^18 - 234135343333599631065v^16 + 11034820378237300072210v^14 - 313219389060614078641061v^12 + 6089690848155231324295730v^10 - 74288048874380904164996345v^8 + 562254852092151028568115323v^6 - 1913288550958252882144276480v^4 + 99011996384616253025157120*v^2 - 3768730556964256490253451264) / 8271931890919184374629335040
$\beta_{4}$	$=$	$( - 32\!\cdots\!31 \nu^{18} + \cdots - 20\!\cdots\!84 ) / 33\!\cdots\!60$ (-32402339935084445831v^18 + 2345338987859150921997v^16 - 110536043335532726568698v^14 + 3069544066670525760001225v^12 - 61000569870554557652459674v^10 + 744145052499934325324972461v^8 - 5883385318439568777391317463v^6 + 19165454346605594770369409024v^4 - 991805389482543356201926656*v^2 - 20370095855727967772183363584) / 33087727563676737498517340160
$\beta_{5}$	$=$	$( 3217726829 \nu^{18} - 235003656367 \nu^{16} + 11151370466878 \nu^{14} + \cdots - 82\!\cdots\!52 ) / 17\!\cdots\!20$ (3217726829v^18 - 235003656367v^16 + 11151370466878v^14 - 314211062845059v^12 + 6470384095377694v^10 - 85111140113036751v^8 + 814619436970368925v^6 - 4254197494692015424v^4 + 15907432735462977536*v^2 - 823176880234954752) / 1784405262918942720
$\beta_{6}$	$=$	$( 62\!\cdots\!59 \nu^{19} + \cdots - 52\!\cdots\!24 \nu ) / 62\!\cdots\!60$ (6268178813259515959v^19 - 446271515617923389853v^17 + 21032817790994357874202v^15 - 578238027612570603159065v^13 + 11607199177010516892228026v^11 - 141596051631692115744406589v^9 + 1183818392933118450688727687v^7 - 3646806028058680645735346176v^5 + 188720904165082938245382144v^3 - 52414574457933824811537793024v) / 62282781296332682350150287360
$\beta_{7}$	$=$	$( 17\!\cdots\!93 \nu^{18} + \cdots - 77\!\cdots\!68 ) / 55\!\cdots\!60$ (17149579458846657093v^18 - 1284683021735129761111v^16 + 62282537536688741935854v^14 - 1808924035952884982811755v^12 + 38380382965894725077233742v^10 - 527715439053249740014713943v^8 + 5245600663962820024338592789v^6 - 29174681754737935293316447392v^4 + 108062295102125623253397106688*v^2 - 77419540411855272924807168) / 5514621260612789583086223360
$\beta_{8}$	$=$	$( 18\!\cdots\!75 \nu^{19} + \cdots + 46\!\cdots\!48 \nu ) / 52\!\cdots\!60$ (180260048387148772775v^19 - 12752977426083732914541v^17 + 601049005164672301476794v^15 - 16464807238740240731166121v^13 + 331695714165198880352588122v^11 - 4046351126804992838385323533v^9 + 34999077450003602862786678007v^7 - 104213765220355396202229788672v^5 + 5393024977888041911667130368v^3 + 4653879799366085671062265397248v) / 529403641018827799976277442560
$\beta_{9}$	$=$	$( - 10\!\cdots\!37 \nu^{18} + \cdots + 74\!\cdots\!44 ) / 41\!\cdots\!20$ (-103829433873288867137v^18 + 7815402514063440679555v^16 - 379636020619219489151710v^14 + 11071882358743119104948031v^12 - 235662400995836723476314910v^10 + 3262941657800146168547608635v^8 - 32634030564632991739621406233v^6 + 184801893682032891437844697720v^4 - 688039488325081133989458813440*v^2 + 74855451021913926806803513344) / 4135965945459592187314667520
$\beta_{10}$	$=$	$( - 91\!\cdots\!37 \nu^{18} + \cdots - 30\!\cdots\!88 ) / 33\!\cdots\!60$ (-912662652669118280137v^18 + 68312517217667009270819v^16 - 3309938272767941772688166v^14 + 96057105552197472808520295v^12 - 2035874636249255020868273798v^10 + 27940400692038314404223421507v^8 - 276670425004566070379654440121v^6 + 1525723666451987664918586401728v^4 - 5506764105389794724053180813312*v^2 - 309955317916268638480925196288) / 33087727563676737498517340160
$\beta_{11}$	$=$	$( - 94\!\cdots\!07 \nu^{18} + \cdots - 27\!\cdots\!92 ) / 33\!\cdots\!60$ (-944702278843436327407v^18 + 70515560449191456155397v^16 - 3414088042005471989087338v^14 + 98872899770326675946340833v^12 - 2093954189258386493243217674v^10 + 28675951406489492703246778661v^8 - 284358747558708793673119499327v^6 + 1568164746197265113013287714944v^4 - 5753711982427451827763997966336*v^2 - 276754974328921367910480084992) / 33087727563676737498517340160
$\beta_{12}$	$=$	$( - 25\!\cdots\!31 \nu^{18} + \cdots + 83\!\cdots\!32 ) / 55\!\cdots\!60$ (-255708254204330335531v^18 + 19162612791431549848185v^16 - 929165331743710957837490v^14 + 26995636356598971815562533v^12 - 572906300151837124217157970v^10 + 7881581205609866373747158905v^8 - 78374131526731038090056427099v^6 + 436740683347249074707564858720v^4 - 1615374289237790691885339217920*v^2 + 83592889816479439363238264832) / 5514621260612789583086223360
$\beta_{13}$	$=$	$( - 28\!\cdots\!41 \nu^{19} + \cdots + 90\!\cdots\!24 \nu ) / 10\!\cdots\!20$ (-2836508935426760814041v^19 + 211739783289573824354131v^17 - 10250343740613577470581254v^15 + 296785611403190870593124439v^13 - 6282784858764039514380801382v^11 + 85954778211157494834477369843v^9 - 851407752711591739194092130121v^7 + 4684811958963496462979151871552v^5 - 17455946367886637260280862814208v^3 + 903315111221548609382395674624v) / 1058807282037655599952554885120
$\beta_{14}$	$=$	$( 17\!\cdots\!93 \nu^{19} + \cdots - 77\!\cdots\!68 \nu ) / 55\!\cdots\!60$ (17149579458846657093v^19 - 1284683021735129761111v^17 + 62282537536688741935854v^15 - 1808924035952884982811755v^13 + 38380382965894725077233742v^11 - 527715439053249740014713943v^9 + 5245600663962820024338592789v^7 - 29174681754737935293316447392v^5 + 108062295102125623253397106688v^3 - 77419540411855272924807168v) / 5514621260612789583086223360
$\beta_{15}$	$=$	$( 1570085297079 \nu^{19} - 117653430022397 \nu^{17} + \cdots - 70\!\cdots\!52 \nu ) / 57\!\cdots\!40$ (1570085297079v^19 - 117653430022397v^17 + 5704450193769658v^15 - 165718928443591129v^13 + 3516416486469681114v^11 - 48362862807494331901v^9 + 480650365690540780455v^7 - 2673314664464702927584v^5 + 9837753297079771211776v^3 - 7094056372501348352v) / 57100968413406167040
$\beta_{16}$	$=$	$( - 50\!\cdots\!73 \nu^{19} + \cdots + 56\!\cdots\!76 \nu ) / 52\!\cdots\!60$ (-50375822257177986169973v^19 + 3783796197174593326055975v^17 - 183702535610447016317167950v^15 + 5349866698789743934056608699v^13 - 113782042446398902019601707950v^11 + 1572061860294355413409681601255v^9 - 15703335172548814074637732802437v^7 + 88572936082822066681744141580640v^5 - 331323002677617818261582438287360v^3 + 56575432711265409553852967550976v) / 529403641018827799976277442560
$\beta_{17}$	$=$	$( - 10\!\cdots\!19 \nu^{19} + \cdots - 78\!\cdots\!12 \nu ) / 10\!\cdots\!20$ (-109511924017838985226919v^19 + 8193283471736005676066605v^17 - 396893949546247258619186170v^15 + 11510336175842357994335001897v^13 - 243837857034520013137884128410v^11 + 3342647227016139374804080560525v^9 - 33098843607374368581097553673271v^7 + 182258844678705731644657046799040v^5 - 662910599849215307707685128540160v^3 - 78376345446720275293457953062912v) / 1058807282037655599952554885120
$\beta_{18}$	$=$	$( - 12\!\cdots\!23 \nu^{19} + \cdots + 56\!\cdots\!00 \nu ) / 52\!\cdots\!60$ (-120874013512838356724123v^19 + 9069510347726646189339657v^17 - 440029174157397337428682898v^15 + 12799295351929799824801545781v^13 - 271877748972689719206088262194v^11 + 3747216039679682056095207278281v^9 - 37316814147359900246654515051531v^7 + 208789102839972508363101866522144v^5 - 772663638766908376953178676975616v^3 + 56894445132804908997394035507200v) / 529403641018827799976277442560
$\beta_{19}$	$=$	$( - 82\!\cdots\!55 \nu^{19} + \cdots - 10\!\cdots\!72 \nu ) / 35\!\cdots\!40$ (-82666382758504962321855v^19 + 6194890637600059094116149v^17 - 300353977130308488999242506v^15 + 8725128887922819885106722449v^13 - 185115521204412950326712358058v^11 + 2545277106183594468239839784597v^9 - 25281897435622524710078876977103v^7 + 140406647601332014448189018185408v^5 - 515171912041339069512622697910272v^3 - 10901003584996989342234427523072v) / 352935760679218533317518295040

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{12} + 15\beta_{7} + \beta_{2}$ b12 + 15*b7 + b2
$\nu^{3}$	$=$	$-2\beta_{15} + 21\beta_{14} + 4\beta_{13} + 4\beta_{6}$ -2b15 + 21b14 + 4b13 + 4b6
$\nu^{4}$	$=$	$29\beta_{12} - 4\beta_{11} - 8\beta_{9} + 331\beta_{7} - 2\beta_{5} - 4\beta_{4} - 327$ 29b12 - 4b11 - 8b9 + 331b7 - 2b5 - 4b4 - 327
$\nu^{5}$	$=$	$4 \beta_{19} - 2 \beta_{18} - 6 \beta_{17} - 2 \beta_{16} - 54 \beta_{15} + 491 \beta_{14} + \cdots - 491 \beta_1$ 4b19 - 2b18 - 6b17 - 2b16 - 54b15 + 491b14 + 146b13 + 54b8 - 491*b1
$\nu^{6}$	$=$	$152\beta_{10} - 152\beta_{9} - 184\beta_{4} - 68\beta_{3} - 793\beta_{2} - 7823$ 152b10 - 152b9 - 184b4 - 68b3 - 793*b2 - 7823
$\nu^{7}$	$=$	$316\beta_{19} - 316\beta_{18} - 216\beta_{17} + 216\beta_{16} + 1498\beta_{8} - 4288\beta_{6} - 12129\beta_1$ 316b19 - 316b18 - 216b17 + 216b16 + 1498b8 - 4288b6 - 12129*b1
$\nu^{8}$	$=$	$- 21237 \beta_{12} + 6300 \beta_{11} + 2908 \beta_{10} + 4604 \beta_{9} - 198787 \beta_{7} + \cdots - 1696$ -21237b12 + 6300b11 + 2908b10 + 4604b9 - 198787b7 + 2214b5 - 2214b3 - 21237b2 - 1696
$\nu^{9}$	$=$	$3910 \beta_{19} - 7996 \beta_{18} + 3910 \beta_{17} + 11906 \beta_{16} + 46638 \beta_{15} + \cdots - 117870 \beta_{6}$ 3910b19 - 7996b18 + 3910b17 + 11906b16 + 46638b15 - 308311b14 - 117870b13 - 117870b6
$\nu^{10}$	$=$	$- 563953 \beta_{12} + 193392 \beta_{11} - 63616 \beta_{10} + 259552 \beta_{9} - 5084415 \beta_{7} + \cdots + 4954639$ -563953b12 + 193392b11 - 63616b10 + 259552b9 - 5084415b7 + 74184b5 + 193392*b4 + 4954639
$\nu^{11}$	$=$	$- 257008 \beta_{19} + 137800 \beta_{18} + 394808 \beta_{17} + 137800 \beta_{16} + 1528562 \beta_{15} + \cdots + 7959597 \beta_1$ -257008b19 + 137800b18 + 394808b17 + 137800b16 + 1528562b15 - 7959597b14 - 3156220b13 - 1528562b8 + 7959597*b1
$\nu^{12}$	$=$	$-3551028\beta_{10} + 3551028\beta_{9} + 5644276\beta_{4} + 2474570\beta_{3} + 14923853\beta_{2} + 130147655$ -3551028b10 + 3551028b9 + 5644276b4 + 2474570b3 + 14923853*b2 + 130147655
$\nu^{13}$	$=$	$- 12305342 \beta_{19} + 12305342 \beta_{18} + 7737524 \beta_{17} - 7737524 \beta_{16} + \cdots + 207325763 \beta_1$ -12305342b19 + 12305342b18 + 7737524b17 - 7737524b16 - 50041062b8 + 83535818b6 + 207325763*b1
$\nu^{14}$	$=$	$394440265 \beta_{12} - 160537576 \beta_{11} - 31144744 \beta_{10} - 95841160 \beta_{9} + 3430326263 \beta_{7} + \cdots + 64696416$ 394440265b12 - 160537576b11 - 31144744b10 - 95841160b9 + 3430326263b7 - 80617676b5 + 80617676b3 + 394440265b2 + 64696416
$\nu^{15}$	$=$	$- 145314092 \beta_{19} + 225233992 \beta_{18} - 145314092 \beta_{17} - 370548084 \beta_{16} + \cdots + 2199176248 \beta_{6}$ -145314092b19 + 225233992b18 - 145314092b17 - 370548084b16 - 1604371402b15 + 5429524473b14 + 2199176248b13 + 2199176248b6
$\nu^{16}$	$=$	$10423659045 \beta_{12} - 4502384652 \beta_{11} + 1932660320 \beta_{10} - 5139448664 \beta_{9} + \cdots - 87233905479$ 10423659045b12 - 4502384652b11 + 1932660320b10 - 5139448664b9 + 89803629811b7 - 2556175854b5 - 4502384652*b4 - 87233905479
$\nu^{17}$	$=$	$6435044972 \beta_{19} - 4488836174 \beta_{18} - 10923881146 \beta_{17} - 4488836174 \beta_{16} + \cdots - 142690202287 \beta_1$ 6435044972b19 - 4488836174b18 - 10923881146b17 - 4488836174b16 - 50214574686b15 + 142690202287b14 + 57763594662b13 + 50214574686b8 - 142690202287*b1
$\nu^{18}$	$=$	$68687475808 \beta_{10} - 68687475808 \beta_{9} - 125253090336 \beta_{4} - 79093800528 \beta_{3} + \cdots - 2345819043151$ 68687475808b10 - 68687475808b9 - 125253090336b4 - 79093800528b3 - 275576317729*b2 - 2345819043151
$\nu^{19}$	$=$	$317478119920 \beta_{19} - 317478119920 \beta_{18} - 181818704864 \beta_{17} + 181818704864 \beta_{16} + \cdots - 3759177093573 \beta_1$ 317478119920b19 - 317478119920b18 - 181818704864b17 + 181818704864b16 + 1537859848802b8 - 1516145662324b6 - 3759177093573*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$ .

$n$	$101$	$127$
$\chi(n)$	$-\beta_{7}$	$-1$

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}

74.1

−4.50647	−	2.60181i
−4.35203	−	2.51265i
−3.33093	−	1.92311i
−2.42233	−	1.39853i
−0.197018	−	0.113749i
0.197018	+	0.113749i
2.42233	+	1.39853i
3.33093	+	1.92311i
4.35203	+	2.51265i
4.50647	+	2.60181i
−4.50647	+	2.60181i
−4.35203	+	2.51265i
−3.33093	+	1.92311i
−2.42233	+	1.39853i
−0.197018	+	0.113749i
0.197018	−	0.113749i
2.42233	−	1.39853i
3.33093	−	1.92311i
4.35203	−	2.51265i
4.50647	−	2.60181i

−4.50647

2.60181i

−4.24917

−

2.45326i

9.53885

−

16.5218i

25.5317

−4.23212

18.0302i

57.6442i

−1.46305

−

2.53407i

74.2

−4.35203

2.51265i

−7.22596

−

4.17191i

8.62677

−

14.9420i

41.9301

7.81984

−

16.7884i

46.5017i

21.3097

36.9094i

74.3

−3.33093

1.92311i

7.76647

4.48397i

3.39672

−

5.88330i

−34.4927

13.8005

−

12.3509i

−

4.64067i

26.7120

46.2666i

74.4

−2.42233

1.39853i

5.38134

3.10692i

−0.0882227

0.152806i

−17.3805

−7.92622

−

16.7384i

−

22.8700i

5.80586

10.0560i

74.5

−0.197018

0.113749i

−1.56628

−

0.904291i

−3.97412

6.88338i

0.411448

16.6030

8.20612i

−

3.62818i

−11.8645

−

20.5499i

74.6

0.197018

−

0.113749i

1.56628

0.904291i

−3.97412

6.88338i

0.411448

−16.6030

−

8.20612i

3.62818i

−11.8645

−

20.5499i

74.7

2.42233

−

1.39853i

−5.38134

−

3.10692i

−0.0882227

0.152806i

−17.3805

7.92622

16.7384i

22.8700i

5.80586

10.0560i

74.8

3.33093

−

1.92311i

−7.76647

−

4.48397i

3.39672

−

5.88330i

−34.4927

−13.8005

12.3509i

4.64067i

26.7120

46.2666i

74.9

4.35203

−

2.51265i

7.22596

4.17191i

8.62677

−

14.9420i

41.9301

−7.81984

16.7884i

−

46.5017i

21.3097

36.9094i

74.10

4.50647

−

2.60181i

4.24917

2.45326i

9.53885

−

16.5218i

25.5317

4.23212

−

18.0302i

−

57.6442i

−1.46305

−

2.53407i

149.1

−4.50647

−

2.60181i

−4.24917

2.45326i

9.53885

16.5218i

25.5317

−4.23212

−

18.0302i

−

57.6442i

−1.46305

2.53407i

149.2

−4.35203

−

2.51265i

−7.22596

4.17191i

8.62677

14.9420i

41.9301

7.81984

16.7884i

−

46.5017i

21.3097

−

36.9094i

149.3

−3.33093

−

1.92311i

7.76647

−

4.48397i

3.39672

5.88330i

−34.4927

13.8005

12.3509i

4.64067i

26.7120

−

46.2666i

149.4

−2.42233

−

1.39853i

5.38134

−

3.10692i

−0.0882227

−

0.152806i

−17.3805

−7.92622

16.7384i

22.8700i

5.80586

−

10.0560i

149.5

−0.197018

−

0.113749i

−1.56628

0.904291i

−3.97412

−

6.88338i

0.411448

16.6030

−

8.20612i

3.62818i

−11.8645

20.5499i

149.6

0.197018

0.113749i

1.56628

−

0.904291i

−3.97412

−

6.88338i

0.411448

−16.6030

8.20612i

−

3.62818i

−11.8645

20.5499i

149.7

2.42233

1.39853i

−5.38134

3.10692i

−0.0882227

−

0.152806i

−17.3805

7.92622

−

16.7384i

−

22.8700i

5.80586

−

10.0560i

149.8

3.33093

1.92311i

−7.76647

4.48397i

3.39672

5.88330i

−34.4927

−13.8005

−

12.3509i

−

4.64067i

26.7120

−

46.2666i

149.9

4.35203

2.51265i

7.22596

−

4.17191i

8.62677

14.9420i

41.9301

−7.81984

−

16.7884i

46.5017i

21.3097

−

36.9094i

149.10

4.50647

2.60181i

4.24917

−

2.45326i

9.53885

16.5218i

25.5317

4.23212

18.0302i

57.6442i

−1.46305

2.53407i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.c	even	3	1	inner
35.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	175.4.k.d		20
5.b	even	2	1	inner	175.4.k.d		20
5.c	odd	4	1		35.4.e.c	✓	10
5.c	odd	4	1		175.4.e.d		10
7.c	even	3	1	inner	175.4.k.d		20
15.e	even	4	1		315.4.j.g		10
20.e	even	4	1		560.4.q.n		10
35.f	even	4	1		245.4.e.o		10
35.j	even	6	1	inner	175.4.k.d		20
35.k	even	12	1		245.4.a.n		5
35.k	even	12	1		245.4.e.o		10
35.k	even	12	1		1225.4.a.bf		5
35.l	odd	12	1		35.4.e.c	✓	10
35.l	odd	12	1		175.4.e.d		10
35.l	odd	12	1		245.4.a.m		5
35.l	odd	12	1		1225.4.a.bg		5
105.w	odd	12	1		2205.4.a.bt		5
105.x	even	12	1		315.4.j.g		10
105.x	even	12	1		2205.4.a.bu		5
140.w	even	12	1		560.4.q.n		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.4.e.c	✓	10	5.c	odd	4	1
35.4.e.c	✓	10	35.l	odd	12	1
175.4.e.d		10	5.c	odd	4	1
175.4.e.d		10	35.l	odd	12	1
175.4.k.d		20	1.a	even	1	1	trivial
175.4.k.d		20	5.b	even	2	1	inner
175.4.k.d		20	7.c	even	3	1	inner
175.4.k.d		20	35.j	even	6	1	inner
245.4.a.m		5	35.l	odd	12	1
245.4.a.n		5	35.k	even	12	1
245.4.e.o		10	35.f	even	4	1
245.4.e.o		10	35.k	even	12	1
315.4.j.g		10	15.e	even	4	1
315.4.j.g		10	105.x	even	12	1
560.4.q.n		10	20.e	even	4	1
560.4.q.n		10	140.w	even	12	1
1225.4.a.bf		5	35.k	even	12	1
1225.4.a.bg		5	35.l	odd	12	1
2205.4.a.bt		5	105.w	odd	12	1
2205.4.a.bu		5	105.x	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{20} - 75 T_{2}^{18} + 3638 T_{2}^{16} - 105775 T_{2}^{14} + 2246038 T_{2}^{12} - 30934571 T_{2}^{10} + \cdots + 16777216$ T2^20 - 75*T2^18 + 3638*T2^16 - 105775*T2^14 + 2246038*T2^12 - 30934571*T2^10 + 307864753*T2^8 - 1719258496*T2^6 + 6352449536*T2^4 - 328728576*T2^2 + 16777216 acting on $S_{4}^{\mathrm{new}}(175, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{20} - 75 T^{18} + \cdots + 16777216$ T^20 - 75T^18 + 3638T^16 - 105775T^14 + 2246038T^12 - 30934571T^10 + 307864753T^8 - 1719258496T^6 + 6352449536T^4 - 328728576*T^2 + 16777216
$3$	$T^{20} + \cdots + 289812354063376$ T^20 - 216T^18 + 30026T^16 - 2506936T^14 + 152552563T^12 - 6098601116T^10 + 177477123418T^8 - 3128041979260T^6 + 37122874727153T^4 - 115911911550660*T^2 + 289812354063376
$5$	$T^{20}$ T^20
$7$	$T^{20} + \cdots + 22\!\cdots\!49$ T^20 + 998T^18 + 615963T^16 + 275067592T^14 + 113208276461T^12 + 40819377158070T^10 + 13318840517360189T^8 + 3807289540559489992T^6 + 1003042525009713897387T^4 + 191198068917805281572198*T^2 + 22539340290692258087863249
$11$	$(T^{10} + \cdots + 1044811065600)^{2}$ (T^10 + 47T^9 + 4365T^8 + 118604T^7 + 10867616T^6 + 336886864T^5 + 9779429200T^4 + 119960168832T^3 + 1216551905536T^2 - 1074069373440*T + 1044811065600)^2
$13$	$(T^{10} + \cdots + 94\!\cdots\!16)^{2}$ (T^10 + 15305T^8 + 87447464T^6 + 227597156624T^4 + 258296753002240T^2 + 94444166274748416)^2
$17$	$T^{20} + \cdots + 12\!\cdots\!96$ T^20 - 20604T^18 + 287697056T^16 - 2178474887104T^14 + 11881204800076288T^12 - 33949177989181697024T^10 + 69806007370010729402368T^8 - 76820455599439737370378240T^6 + 57510621178795476733235560448T^4 - 27246740716387752733531176960*T^2 + 12900494833575099457702199296
$19$	$(T^{10} + \cdots + 26\!\cdots\!36)^{2}$ (T^10 + 21T^9 + 14201T^8 - 507400T^7 + 134507884T^6 - 5335779776T^5 + 768980016784T^4 - 39020440485760T^3 + 2937676413456896T^2 - 85444524969652224*T + 2645165718335143936)^2
$23$	$T^{20} + \cdots + 15\!\cdots\!21$ T^20 - 64093T^18 + 2582159055T^16 - 64597606330430T^14 + 1184612573902417965T^12 - 15240835109867008560235T^10 + 146053466147187905040917437T^8 - 944057695194102838253038523646T^6 + 4286252375622269154628992657783135T^4 - 9838460826522772325884534436740198557*T^2 + 15273810899104359477750747792890226459921
$29$	$(T^{5} + 190 T^{4} + \cdots + 13190815450)^{4}$ (T^5 + 190T^4 - 58358T^3 - 7888952T^2 + 35108245T + 13190815450)^4
$31$	$(T^{10} + \cdots + 63\!\cdots\!24)^{2}$ (T^10 + 388T^9 + 126796T^8 + 14113488T^7 + 1682623632T^6 + 74867995008T^5 + 9017518685184T^4 + 292359263680512T^3 + 34462721900371968T^2 - 423777164166365184*T + 6347880411248001024)^2
$37$	$T^{20} + \cdots + 15\!\cdots\!00$ T^20 - 254193T^18 + 42342598593T^16 - 4174257645287200T^14 + 300752394426664299264T^12 - 13381375730080305013530624T^10 + 409906380518984599649129660416T^8 - 4114067944863983966378597862604800T^6 + 30171955634927768495945874721996800000T^4 - 77604885089476390403489936695099392000000*T^2 + 152980295471529412101822705728854425600000000
$41$	$(T^{5} + 281 T^{4} + \cdots + 77000100765)^{4}$ (T^5 + 281T^4 - 75374T^3 - 25090518T^2 - 783155827T + 77000100765)^4
$43$	$(T^{10} + \cdots + 53\!\cdots\!64)^{2}$ (T^10 + 459720T^8 + 74328956902T^6 + 4763909064451996T^4 + 87470237964549168417T^2 + 5318532104577956652964)^2
$47$	$T^{20} + \cdots + 20\!\cdots\!00$ T^20 - 416129T^18 + 118539699297T^16 - 17898936102266016T^14 + 1944933609354510788352T^12 - 104190122585636994723422208T^10 + 3944995718653652710578590121984T^8 - 65492051946389097641652342038200320T^6 + 781392925943756723777897265254363037696T^4 - 4764936680255632674279931194892671883673600*T^2 + 20188349518695965218115719473215246706933760000
$53$	$T^{20} + \cdots + 60\!\cdots\!56$ T^20 - 195609T^18 + 26875249337T^16 - 1801210335636184T^14 + 87477089964219541936T^12 - 2229736087101045173352320T^10 + 39628487282629785670536806656T^8 - 102218836245137900622055906729984T^6 + 203518104572819399514781699582656512T^4 - 123991248387730372423971508916651556864*T^2 + 60106418349130223770588987278813200121856
$59$	$(T^{10} + \cdots + 85\!\cdots\!00)^{2}$ (T^10 - 708T^9 + 431932T^8 - 88869968T^7 + 18489060880T^6 + 1937728820224T^5 + 361335796367872T^4 + 9255906865242112T^3 + 344303419114799104T^2 - 3699187914711367680*T + 85092046704187801600)^2
$61$	$(T^{10} + \cdots + 28\!\cdots\!56)^{2}$ (T^10 + 1944T^9 + 2336862T^8 + 1788785176T^7 + 1009354276359T^6 + 409778641639428T^5 + 125790963193923286T^4 + 27477673338949799148T^3 + 4407897755356708506489T^2 + 445744633227602973886188*T + 28008916584290292624631056)^2
$67$	$T^{20} + \cdots + 86\!\cdots\!00$ T^20 - 1578196T^18 + 1755895771098T^16 - 878649550737590192T^14 + 307026191550148917992547T^12 - 68307017628966391879099220416T^10 + 11076889080982724389162997463682042T^8 - 1136796044986138843127600349030198662532T^6 + 83334882295499020431569599147284269969568801T^4 - 3279044545976355356506047898228314705460017795600*T^2 + 86286883946876861918041643762679949045854982899360000
$71$	$(T^{5} + \cdots + 6439908260352)^{4}$ (T^5 - 864T^4 - 98636T^3 + 277412208T^2 - 81579114752T + 6439908260352)^4
$73$	$T^{20} + \cdots + 68\!\cdots\!36$ T^20 - 1535208T^18 + 1652302946480T^16 - 817909508877625984T^14 + 283650338987466975768832T^12 - 61517283578739494739510886400T^10 + 9724308398694193049946259117834240T^8 - 990908678337320833637206240691287490560T^6 + 71790409509337182790898616186835985519083520T^4 - 2696485343727309023109804907669430751081825042432*T^2 + 68412776268323750585802230463536337452428016993959936
$79$	$(T^{10} + \cdots + 10\!\cdots\!76)^{2}$ (T^10 - 218T^9 + 1011528T^8 - 320484136T^7 + 859781378784T^6 - 232806411906848T^5 + 186071328075072064T^4 - 29000014385007508992T^3 + 24861924228952434539520T^2 - 4147923537424126645346304*T + 1060677440103157129974398976)^2
$83$	$(T^{10} + \cdots + 31\!\cdots\!44)^{2}$ (T^10 + 2884596T^8 + 2047810289622T^6 + 338362640808839908T^4 + 19436917570303180515729T^2 + 318002197969033695171072144)^2
$89$	$(T^{10} + \cdots + 85\!\cdots\!56)^{2}$ (T^10 - 2322T^9 + 4410398T^8 - 4883559276T^7 + 5243447211091T^6 - 4262390483025378T^5 + 3609068489134031118T^4 - 2210781098969073195480T^3 + 1196388311202585575273313T^2 - 367822896700179035887274214*T + 85748364562135510769162872356)^2
$97$	$(T^{10} + \cdots + 74\!\cdots\!44)^{2}$ (T^10 + 634260T^8 + 130867421344T^6 + 9717340708500096T^4 + 228610675244372788480T^2 + 745444475248831064146944)^2
show more
show less

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 74.10
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	175.4.k.d

Level	$175$
Weight	$4$
Character orbit	175.k
Analytic conductor	$10.325$
Analytic rank	$0$
Dimension	$20$
Inner twists	$4$