LMFDB - Newform orbit 225.6.f.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,Mod(107,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("225.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.f (of order $4$, degree $2$, 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:	$36.0863594579$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} - 468 x^{14} + 88042 x^{12} - 8478288 x^{10} + 442928373 x^{8} - 12512257524 x^{6} + \cdots + 4793095790596 $ x^16 - 468x^14 + 88042x^12 - 8478288x^10 + 442928373x^8 - 12512257524x^6 + 191311236688x^4 - 1487738314296*x^2 + 4793095790596
Coefficient ring:	$\Z[a_1, \ldots, a_{49}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{4} q^{2} + ( - \beta_{5} - 27 \beta_1) q^{4} - \beta_{9} q^{7} + ( - \beta_{6} - 28 \beta_{2}) q^{8}+O(q^{10}) $ q - b4 * q^2 + (-b5 - 27b1) q^4 - b9 * q^7 + (-b6 - 28b2) q^8	\( q - \beta_{4} q^{2} + ( - \beta_{5} - 27 \beta_1) q^{4} - \beta_{9} q^{7} + ( - \beta_{6} - 28 \beta_{2}) q^{8} + (2 \beta_{15} - 3 \beta_{13}) q^{11} + ( - 2 \beta_{10} + \beta_{8}) q^{13} + (7 \beta_{14} - 8 \beta_{12}) q^{14} + ( - 21 \beta_{3} - 787) q^{16} + ( - \beta_{7} + 117 \beta_{4}) q^{17} + ( - 44 \beta_{5} - 902 \beta_1) q^{19} + (11 \beta_{11} + 23 \beta_{9}) q^{22} + (11 \beta_{6} - 253 \beta_{2}) q^{23} + (11 \beta_{15} - 36 \beta_{13}) q^{26} + ( - 31 \beta_{10} + 41 \beta_{8}) q^{28} + 22 \beta_{12} q^{29} + (56 \beta_{3} - 1996) q^{31} + (11 \beta_{7} + 584 \beta_{4}) q^{32} + (142 \beta_{5} + 6902 \beta_1) q^{34} + (22 \beta_{11} - 33 \beta_{9}) q^{37} + ( - 44 \beta_{6} - 2354 \beta_{2}) q^{38} + ( - 88 \beta_{15} - 11 \beta_{13}) q^{41} + ( - 64 \beta_{10} + 32 \beta_{8}) q^{43} + ( - 119 \beta_{14} + 242 \beta_{12}) q^{44} + (22 \beta_{3} - 14938) q^{46} + ( - 44 \beta_{7} + 1012 \beta_{4}) q^{47} + (144 \beta_{5} + 16043 \beta_1) q^{49} + (55 \beta_{11} + 153 \beta_{9}) q^{52} + (55 \beta_{6} + 451 \beta_{2}) q^{53} + (125 \beta_{15} - 506 \beta_{13}) q^{56} + (66 \beta_{10} - 66 \beta_{8}) q^{58} + (154 \beta_{14} + 69 \beta_{12}) q^{59} + (352 \beta_{3} - 21626) q^{61} + (56 \beta_{7} + 148 \beta_{4}) q^{62} + ( - 363 \beta_{5} + 9283 \beta_1) q^{64} + ( - 176 \beta_{11} - 110 \beta_{9}) q^{67} + (110 \beta_{6} + 7844 \beta_{2}) q^{68} + ( - 44 \beta_{15} - 454 \beta_{13}) q^{71} + (132 \beta_{10} + 78 \beta_{8}) q^{73} + (187 \beta_{14} + 44 \beta_{12}) q^{74} + ( - 2046 \beta_{3} - 109978) q^{76} + (99 \beta_{7} - 11727 \beta_{4}) q^{77} + ( - 1408 \beta_{5} + 4664 \beta_1) q^{79} + ( - 55 \beta_{11} - 583 \beta_{9}) q^{82} + ( - 374 \beta_{6} - 4038 \beta_{2}) q^{83} + (352 \beta_{15} - 1152 \beta_{13}) q^{86} + (493 \beta_{10} - 823 \beta_{8}) q^{88} + (136 \beta_{14} + 379 \beta_{12}) q^{89} + (2124 \beta_{3} + 21258) q^{91} + ( - 330 \beta_{7} + 6116 \beta_{4}) q^{92} + (2112 \beta_{5} + 59664 \beta_1) q^{94} + ( - 440 \beta_{11} + 132 \beta_{9}) q^{97} + (144 \beta_{6} + 20795 \beta_{2}) q^{98}+O(q^{100}) \) q - b4 * q^2 + (-b5 - 27b1) q^4 - b9 * q^7 + (-b6 - 28b2) q^8 + (2b15 - 3b13) * q^11 + (-2b10 + b8) q^13 + (7b14 - 8b12) * q^14 + (-21b3 - 787) q^16 + (-b7 + 117b4) q^17 + (-44b5 - 902b1) * q^19 + (11b11 + 23b9) * q^22 + (11b6 - 253b2) * q^23 + (11b15 - 36b13) * q^26 + (-31b10 + 41b8) * q^28 + 22b12 q^29 + (56b3 - 1996) q^31 + (11b7 + 584b4) * q^32 + (142b5 + 6902b1) * q^34 + (22b11 - 33b9) * q^37 + (-44b6 - 2354b2) * q^38 + (-88b15 - 11b13) * q^41 + (-64b10 + 32b8) * q^43 + (-119b14 + 242b12) * q^44 + (22b3 - 14938) q^46 + (-44b7 + 1012b4) * q^47 + (144b5 + 16043b1) * q^49 + (55b11 + 153b9) * q^52 + (55b6 + 451b2) * q^53 + (125b15 - 506b13) * q^56 + (66b10 - 66b8) * q^58 + (154b14 + 69b12) * q^59 + (352b3 - 21626) q^61 + (56b7 + 148b4) * q^62 + (-363b5 + 9283b1) * q^64 + (-176b11 - 110b9) * q^67 + (110b6 + 7844b2) * q^68 + (-44b15 - 454b13) * q^71 + (132b10 + 78b8) * q^73 + (187b14 + 44b12) * q^74 + (-2046b3 - 109978) q^76 + (99b7 - 11727b4) * q^77 + (-1408b5 + 4664b1) * q^79 + (-55b11 - 583b9) * q^82 + (-374b6 - 4038b2) * q^83 + (352b15 - 1152b13) * q^86 + (493b10 - 823b8) * q^88 + (136b14 + 379b12) * q^89 + (2124b3 + 21258) q^91 + (-330b7 + 6116b4) * q^92 + (2112b5 + 59664b1) * q^94 + (-440b11 + 132b9) * q^97 + (144b6 + 20795b2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 12424 q^{16} - 32384 q^{31} - 239184 q^{46} - 348832 q^{61} - 1743280 q^{76} + 323136 q^{91}+O(q^{100}) $ 16 * q - 12424 * q^16 - 32384 * q^31 - 239184 * q^46 - 348832 * q^61 - 1743280 * q^76 + 323136 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 468 x^{14} + 88042 x^{12} - 8478288 x^{10} + 442928373 x^{8} - 12512257524 x^{6} + \cdots + 4793095790596 $ x^16 - 468*x^14 + 88042*x^12 - 8478288*x^10 + 442928373*x^8 - 12512257524*x^6 + 191311236688*x^4 - 1487738314296*x^2 + 4793095790596 :

$\beta_{1}$	$=$	$ ( 4294 \nu^{14} - 1319877 \nu^{12} + 37859667 \nu^{10} + 29409376230 \nu^{8} + \cdots + 23\!\cdots\!16 ) / 42\!\cdots\!86 $ (4294v^14 - 1319877v^12 + 37859667v^10 + 29409376230v^8 - 4244669562215v^6 + 219605415656679v^4 - 4037762260724186*v^2 + 23732232825921516) / 4209010098945386
$\beta_{2}$	$=$	$ ( - 10555945770082 \nu^{14} + \cdots + 59\!\cdots\!90 ) / 51\!\cdots\!04 $ (-10555945770082v^14 + 4348208197247031v^12 - 686291388708563965v^10 + 51658920344942862120v^8 - 1951187969967889122943v^6 + 42280226983888425873783v^4 - 744690378633220182354868*v^2 + 5970857731550722935015390) / 515464477812676077958804
$\beta_{3}$	$=$	$ ( - 155380 \nu^{14} + 63609390 \nu^{12} - 9889761968 \nu^{10} + 716058156645 \nu^{8} + \cdots - 15\!\cdots\!30 ) / 37\!\cdots\!29 $ (-155380v^14 + 63609390v^12 - 9889761968v^10 + 716058156645v^8 - 24289928058676v^6 + 414097854636960v^4 - 3442726284998328*v^2 - 153508192606073030) / 3701531675221129
$\beta_{4}$	$=$	$ ( - 83163821358150 \nu^{14} + \cdots + 24\!\cdots\!30 ) / 51\!\cdots\!04 $ (-83163821358150v^14 + 36854171605884208v^12 - 6476326358602071582v^10 + 568735726381529841231v^8 - 25878065093092774513458v^6 + 580634211556698343476137v^4 - 6133911956858298804987042*v^2 + 24354217540436013356353530) / 515464477812676077958804
$\beta_{5}$	$=$	$ ( - 8063348422931 \nu^{14} + \cdots - 89\!\cdots\!92 ) / 36\!\cdots\!86 $ (-8063348422931v^14 + 2970202137445506v^12 - 369002276068876527v^10 + 12301212313164612136v^8 + 884870587699129961926v^6 - 72899908313788648718578v^4 + 1471142531140147534486408*v^2 - 8912503245501334010933192) / 36818891272334005568486
$\beta_{6}$	$=$	$ ( 298615658800769 \nu^{14} + \cdots - 42\!\cdots\!82 ) / 51\!\cdots\!04 $ (298615658800769v^14 - 129722146162991617v^12 + 21992756060409097394v^10 - 1827584823049640404860v^8 + 79025123181322317602777v^6 - 2063292954534916186175301v^4 + 47753700971494196391900968*v^2 - 429670789131427531904998282) / 515464477812676077958804
$\beta_{7}$	$=$	$ ( - 56\!\cdots\!86 \nu^{14} + \cdots + 18\!\cdots\!82 ) / 51\!\cdots\!04 $ (-5617294358387486v^14 + 2558368893447016667v^12 - 461539069997264386746v^10 + 41513909243501212594680v^8 - 1925662951353564625753482v^6 + 43715137794192305893368583v^4 - 465162271786812940596031982*v^2 + 1854075598200098698330170282) / 515464477812676077958804
$\beta_{8}$	$=$	$ ( 19\!\cdots\!17 \nu^{15} + \cdots - 21\!\cdots\!20 \nu ) / 11\!\cdots\!56 $ (190720744897470626417v^15 - 97346695663162059615385v^13 + 20172307238035376088937695v^11 - 2159497429588313891178260895v^9 + 125579924625660208640412928698v^7 - 3807629356814680564058021084838v^5 + 51988342208625611034385919418740v^3 - 213799949609107985848305345767320v) / 1128513597777981114940301020456
$\beta_{9}$	$=$	$ ( 21\!\cdots\!67 \nu^{15} + \cdots - 37\!\cdots\!52 \nu ) / 11\!\cdots\!56 $ (215009938407414119167v^15 - 110964545059244483870575v^13 + 23247454766346629097203775v^11 - 2514671585221379331550785225v^9 + 147585815317136355329914817688v^7 - 4522961808379514622219576586770v^5 + 64881196624627589707372506536080v^3 - 370851227268881932364339672094352v) / 1128513597777981114940301020456
$\beta_{10}$	$=$	$ ( 61\!\cdots\!99 \nu^{15} + \cdots - 58\!\cdots\!00 \nu ) / 16\!\cdots\!08 $ (61221650155116341699v^15 - 29911946324964306858811v^13 + 5951744192197790423994831v^11 - 615765866418605561712316515v^9 + 34948622785182469125492050616v^7 - 1045294465549645539924023643276v^5 + 14178926315196638078669230591412v^3 - 58201997502155319310056766037500v) / 161216228253997302134328717208
$\beta_{11}$	$=$	$ ( - 75\!\cdots\!67 \nu^{15} + \cdots + 10\!\cdots\!24 \nu ) / 16\!\cdots\!08 $ (-75489437743353810067v^15 + 36723611126116712975221v^13 - 7265553121183899080738055v^11 + 746383683527752655019966765v^9 - 42063380469622673193311676000v^7 + 1258416787304374045435444358820v^5 - 17826671622310919907589649656900v^3 + 101182430292241025344146188999524v) / 161216228253997302134328717208
$\beta_{12}$	$=$	$ ( 7957705714182 \nu^{15} + \cdots - 15\!\cdots\!10 \nu ) / 11\!\cdots\!36 $ (7957705714182v^15 - 3598119363649968v^13 + 644035157376223023v^11 - 57347815956640217775v^9 + 2613915104655676940037v^7 - 56806316915933494555893v^5 + 533657648661151317221376v^3 - 1547103933501453004623210v) / 11131581817659381517436
$\beta_{13}$	$=$	$ ( 12446636586075 \nu^{15} + \cdots - 48\!\cdots\!78 \nu ) / 13\!\cdots\!08 $ (12446636586075v^15 - 5623748857011024v^13 + 1007487108030208671v^11 - 90219319217181035505v^9 + 4195705656213325089378v^7 - 97411761018323173199583v^5 + 1108282610729611413328482v^3 - 4883021098980463299719178v) / 13590853984273495352108
$\beta_{14}$	$=$	$ ( 12\!\cdots\!41 \nu^{15} + \cdots - 24\!\cdots\!58 \nu ) / 91\!\cdots\!52 $ (124612303971973041v^15 - 56557192104727963941v^13 + 10151637947231585010180v^11 - 905417239479847490800686v^9 + 41288344208434801900415343v^7 - 897367474151544482213399187v^5 + 8430461177369055812214954180v^3 - 24446056079431660996745269458v) / 91913471068413513189469052
$\beta_{15}$	$=$	$ ( 23\!\cdots\!68 \nu^{15} + \cdots - 97\!\cdots\!82 \nu ) / 14\!\cdots\!24 $ (23734424184476868v^15 - 10812037618474507497v^13 + 1953071697173118080376v^11 - 176316437189324754972642v^9 + 8257663444945921701289170v^7 - 192594445223861436617005143v^5 + 2198184045332595498381616374v^3 - 9704006433904861273906606182v) / 14214091716980894210397524

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

$\nu$	$=$	$ ( \beta_{13} - \beta_{12} + 3\beta_{11} + 3\beta_{10} + 3\beta_{9} - 3\beta_{8} ) / 162 $ (b13 - b12 + 3b11 + 3b10 + 3b9 - 3b8) / 162
$\nu^{2}$	$=$	$ \beta_{3} - 2\beta_{2} + \beta _1 + 59 $ b3 - 2*b2 + b1 + 59
$\nu^{3}$	$=$	$ ( - 81 \beta_{15} + 81 \beta_{14} + 199 \beta_{13} - 197 \beta_{12} + 234 \beta_{11} + \cdots - 378 \beta_{8} ) / 162 $ (-81b15 + 81b14 + 199b13 - 197b12 + 234b11 + 216b10 + 396b9 - 378b8) / 162
$\nu^{4}$	$=$	$ -4\beta_{6} + 6\beta_{5} - 4\beta_{4} + 117\beta_{3} - 368\beta_{2} + 354\beta _1 + 5426 $ -4b6 + 6b5 - 4b4 + 117b3 - 368b2 + 354b1 + 5426
$\nu^{5}$	$=$	$ ( - 16065 \beta_{15} + 15525 \beta_{14} + 31889 \beta_{13} - 30569 \beta_{12} + 24132 \beta_{11} + \cdots - 36966 \beta_{8} ) / 162 $ (-16065b15 + 15525b14 + 31889b13 - 30569b12 + 24132b11 + 19632b10 + 44706b9 - 36966b8) / 162
$\nu^{6}$	$=$	$ 20 \beta_{7} - 702 \beta_{6} + 1755 \beta_{5} - 1840 \beta_{4} + 12198 \beta_{3} - 55722 \beta_{2} + \cdots + 546990 $ 20b7 - 702b6 + 1755b5 - 1840b4 + 12198b3 - 55722b2 + 81404*b1 + 546990
$\nu^{7}$	$=$	$ ( - 2418255 \beta_{15} + 2197125 \beta_{14} + 4650785 \beta_{13} - 4213567 \beta_{12} + \cdots - 3336960 \beta_{8} ) / 162 $ (-2418255b15 + 2197125b14 + 4650785b13 - 4213567b12 + 2681790b11 + 1762158b10 + 5052660b9 - 3336960b8) / 162
$\nu^{8}$	$=$	$ 6552 \beta_{7} - 97648 \beta_{6} + 341936 \beta_{5} - 520120 \beta_{4} + 1248039 \beta_{3} + \cdots + 55711234 $ 6552b7 - 97648b6 + 341936b5 - 520120b4 + 1248039b3 - 7602080b2 + 15338848*b1 + 55711234
$\nu^{9}$	$=$	$ ( - 332563725 \beta_{15} + 277167501 \beta_{14} + 636375907 \beta_{13} - 529975171 \beta_{12} + \cdots - 275465988 \beta_{8} ) / 162 $ (-332563725b15 + 277167501b14 + 636375907b13 - 529975171b12 + 306069030b11 + 145557702b10 + 578402532b9 - 275465988b8) / 162
$\nu^{10}$	$=$	$ 1465440 \beta_{7} - 12532806 \beta_{6} + 56505735 \beta_{5} - 114097440 \beta_{4} + 126387720 \beta_{3} + \cdots + 5638946796 $ 1465440b7 - 12532806b6 + 56505735b5 - 114097440b4 + 126387720b3 - 973126234b2 + 2522960866*b1 + 5638946796
$\nu^{11}$	$=$	$ ( - 43741978533 \beta_{15} + 32551063083 \beta_{14} + 83634619135 \beta_{13} + \cdots - 19207698312 \beta_{8} ) / 162 $ (-43741978533b15 + 32551063083b14 + 83634619135b13 - 62226781913b12 + 35277259542b11 + 10154532594b10 + 66698912544b9 - 19207698312b8) / 162
$\nu^{12}$	$=$	$ 276277716 \beta_{7} - 1537756384 \beta_{6} + 8499573984 \beta_{5} - 21452913516 \beta_{4} + \cdots + 562664836178 $ 276277716b7 - 1537756384b6 + 8499573984b5 - 21452913516b4 + 12611461941b3 - 119360031344b2 + 379257628200*b1 + 562664836178
$\nu^{13}$	$=$	$ ( - 5596420498695 \beta_{15} + 3606011257095 \beta_{14} + 10698996010013 \beta_{13} + \cdots - 789921475224 \beta_{8} ) / 162 $ (-5596420498695b15 + 3606011257095b14 + 10698996010013b13 - 6893557488557b12 + 4068483058950b11 + 417669150390b10 + 7692830423016b9 - 789921475224b8) / 162
$\nu^{14}$	$=$	$ 46896760768 \beta_{7} - 182750587110 \beta_{6} + 1200452263101 \beta_{5} - 3640168052144 \beta_{4} + \cdots + 54828981404032 $ 46896760768b7 - 182750587110b6 + 1200452263101b5 - 3640168052144b4 + 1228910341940b3 - 14184467391010b2 + 53560444058666*b1 + 54828981404032
$\nu^{15}$	$=$	$ ( - 701869907118915 \beta_{15} + 377165783058885 \beta_{14} + \cdots + 69182953351632 \beta_{8} ) / 162 $ (-701869907118915b15 + 377165783058885b14 + 1341779541476921b13 - 721030066045903b12 + 466482034303218b11 - 36590291767314b10 + 882047096187264b9 + 69182953351632b8) / 162

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

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

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} $

107.1

10.8371	+	0.707107i
−10.8371	−	0.707107i
−4.49969	−	0.707107i
4.49969	+	0.707107i
−3.08547	+	0.707107i
3.08547	−	0.707107i
9.42286	−	0.707107i
−9.42286	+	0.707107i
10.8371	−	0.707107i
−10.8371	+	0.707107i
−4.49969	+	0.707107i
4.49969	−	0.707107i
−3.08547	−	0.707107i
3.08547	+	0.707107i
9.42286	+	0.707107i
−9.42286	−	0.707107i

−7.16297

7.16297i

−

70.6163i

−139.876

−

139.876i

276.608

276.608i

107.2

−7.16297

7.16297i

−

70.6163i

139.876

139.876i

276.608

276.608i

107.3

−2.68176

2.68176i

17.6163i

−114.946

−

114.946i

−133.059

−

133.059i

107.4

−2.68176

2.68176i

17.6163i

114.946

114.946i

−133.059

−

133.059i

107.5

2.68176

−

2.68176i

17.6163i

−114.946

−

114.946i

133.059

133.059i

107.6

2.68176

−

2.68176i

17.6163i

114.946

114.946i

133.059

133.059i

107.7

7.16297

−

7.16297i

−

70.6163i

−139.876

−

139.876i

−276.608

−

276.608i

107.8

7.16297

−

7.16297i

−

70.6163i

139.876

139.876i

−276.608

−

276.608i

143.1

−7.16297

−

7.16297i

70.6163i

−139.876

139.876i

276.608

−

276.608i

143.2

−7.16297

−

7.16297i

70.6163i

139.876

−

139.876i

276.608

−

276.608i

143.3

−2.68176

−

2.68176i

−

17.6163i

−114.946

114.946i

−133.059

133.059i

143.4

−2.68176

−

2.68176i

−

17.6163i

114.946

−

114.946i

−133.059

133.059i

143.5

2.68176

2.68176i

−

17.6163i

−114.946

114.946i

133.059

−

133.059i

143.6

2.68176

2.68176i

−

17.6163i

114.946

−

114.946i

133.059

−

133.059i

143.7

7.16297

7.16297i

70.6163i

−139.876

139.876i

−276.608

276.608i

143.8

7.16297

7.16297i

70.6163i

139.876

−

139.876i

−276.608

276.608i

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
5.c	odd	4	2	inner
15.d	odd	2	1	inner
15.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.6.f.a	✓	16
3.b	odd	2	1	inner	225.6.f.a	✓	16
5.b	even	2	1	inner	225.6.f.a	✓	16
5.c	odd	4	2	inner	225.6.f.a	✓	16
15.d	odd	2	1	inner	225.6.f.a	✓	16
15.e	even	4	2	inner	225.6.f.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.6.f.a	✓	16	1.a	even	1	1	trivial
225.6.f.a	✓	16	3.b	odd	2	1	inner
225.6.f.a	✓	16	5.b	even	2	1	inner
225.6.f.a	✓	16	5.c	odd	4	2	inner
225.6.f.a	✓	16	15.d	odd	2	1	inner
225.6.f.a	✓	16	15.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 10737T_{2}^{4} + 2178576 $ T2^8 + 10737*T2^4 + 2178576 acting on $S_{6}^{\mathrm{new}}(225, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 10737 T^{4} + 2178576)^{2} $ (T^8 + 10737*T^4 + 2178576)^2
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} + \cdots + 10\!\cdots\!36)^{2} $ (T^8 + 2229509448*T^4 + 1069238398979544336)^2
$11$	$ (T^{4} + 437076 T^{2} + 3809605284)^{4} $ (T^4 + 437076*T^2 + 3809605284)^4
$13$	$ (T^{8} + \cdots + 45\!\cdots\!16)^{2} $ (T^8 + 151721632008*T^4 + 4595392991233898624016)^2
$17$	$ (T^{8} + \cdots + 13\!\cdots\!76)^{2} $ (T^8 + 2793413378592*T^4 + 1390153162414992466176)^2
$19$	$ (T^{4} + \cdots + 8961281731600)^{4} $ (T^4 + 9084680*T^2 + 8961281731600)^4
$23$	$ (T^{8} + \cdots + 11\!\cdots\!36)^{2} $ (T^8 + 314070542878752*T^4 + 1125605912162123511258644736)^2
$29$	$ (T^{2} - 6351048)^{8} $ (T^2 - 6351048)^8
$31$	$ (T^{2} + 4048 T - 2006864)^{8} $ (T^2 + 4048*T - 2006864)^8
$37$	$ (T^{8} + \cdots + 68\!\cdots\!76)^{2} $ (T^8 + 21656893118999688*T^4 + 682513399683881255513986204176)^2
$41$	$ (T^{4} + \cdots + 64\!\cdots\!04)^{4} $ (T^4 + 575553924*T^2 + 64926300461734404)^4
$43$	$ (T^{8} + \cdots + 50\!\cdots\!16)^{2} $ (T^8 + 159091662004420608*T^4 + 5052688028062005574842398766268416)^2
$47$	$ (T^{8} + \cdots + 34\!\cdots\!56)^{2} $ (T^8 + 57301477970153472*T^4 + 347764169628016589340491913363456)^2
$53$	$ (T^{8} + \cdots + 53\!\cdots\!36)^{2} $ (T^8 + 56036677833034272*T^4 + 530957487924379409292628908155136)^2
$59$	$ (T^{4} + \cdots + 44\!\cdots\!44)^{4} $ (T^4 - 1989561204*T^2 + 444730185467759844)^4
$61$	$ (T^{2} + 43604 T + 234179044)^{8} $ (T^2 + 43604*T + 234179044)^8
$67$	$ (T^{8} + \cdots + 15\!\cdots\!96)^{2} $ (T^8 + 10885021764383657088*T^4 + 15766438961720227228174621883359236096)^2
$71$	$ (T^{4} + \cdots + 77\!\cdots\!84)^{4} $ (T^4 + 5840124624*T^2 + 7752975089544926784)^4
$73$	$ (T^{8} + \cdots + 65\!\cdots\!36)^{2} $ (T^8 + 8050345089135308928*T^4 + 65960687965937161482420373221543936)^2
$79$	$ (T^{4} + \cdots + 14\!\cdots\!96)^{4} $ (T^4 + 7774371968*T^2 + 14665492539663978496)^4
$83$	$ (T^{8} + \cdots + 34\!\cdots\!36)^{2} $ (T^8 + 124008796527911281152*T^4 + 3495253234441172781399592065630784782336)^2
$89$	$ (T^{4} + \cdots + 26\!\cdots\!64)^{4} $ (T^4 - 5854602564*T^2 + 2661397578676837764)^4
$97$	$ (T^{8} + \cdots + 63\!\cdots\!56)^{2} $ (T^8 + 613561480633573705728*T^4 + 6383804236571419372276116196251329888256)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{2} + ( - \beta_{5} - 27 \beta_1) q^{4} - \beta_{9} q^{7} + ( - \beta_{6} - 28 \beta_{2}) q^{8}+O(q^{10}) \) q - b4 * q^2 + (-b5 - 27b1) q^4 - b9 * q^7 + (-b6 - 28b2) q^8	\( q - \beta_{4} q^{2} + ( - \beta_{5} - 27 \beta_1) q^{4} - \beta_{9} q^{7} + ( - \beta_{6} - 28 \beta_{2}) q^{8} + (2 \beta_{15} - 3 \beta_{13}) q^{11} + ( - 2 \beta_{10} + \beta_{8}) q^{13} + (7 \beta_{14} - 8 \beta_{12}) q^{14} + ( - 21 \beta_{3} - 787) q^{16} + ( - \beta_{7} + 117 \beta_{4}) q^{17} + ( - 44 \beta_{5} - 902 \beta_1) q^{19} + (11 \beta_{11} + 23 \beta_{9}) q^{22} + (11 \beta_{6} - 253 \beta_{2}) q^{23} + (11 \beta_{15} - 36 \beta_{13}) q^{26} + ( - 31 \beta_{10} + 41 \beta_{8}) q^{28} + 22 \beta_{12} q^{29} + (56 \beta_{3} - 1996) q^{31} + (11 \beta_{7} + 584 \beta_{4}) q^{32} + (142 \beta_{5} + 6902 \beta_1) q^{34} + (22 \beta_{11} - 33 \beta_{9}) q^{37} + ( - 44 \beta_{6} - 2354 \beta_{2}) q^{38} + ( - 88 \beta_{15} - 11 \beta_{13}) q^{41} + ( - 64 \beta_{10} + 32 \beta_{8}) q^{43} + ( - 119 \beta_{14} + 242 \beta_{12}) q^{44} + (22 \beta_{3} - 14938) q^{46} + ( - 44 \beta_{7} + 1012 \beta_{4}) q^{47} + (144 \beta_{5} + 16043 \beta_1) q^{49} + (55 \beta_{11} + 153 \beta_{9}) q^{52} + (55 \beta_{6} + 451 \beta_{2}) q^{53} + (125 \beta_{15} - 506 \beta_{13}) q^{56} + (66 \beta_{10} - 66 \beta_{8}) q^{58} + (154 \beta_{14} + 69 \beta_{12}) q^{59} + (352 \beta_{3} - 21626) q^{61} + (56 \beta_{7} + 148 \beta_{4}) q^{62} + ( - 363 \beta_{5} + 9283 \beta_1) q^{64} + ( - 176 \beta_{11} - 110 \beta_{9}) q^{67} + (110 \beta_{6} + 7844 \beta_{2}) q^{68} + ( - 44 \beta_{15} - 454 \beta_{13}) q^{71} + (132 \beta_{10} + 78 \beta_{8}) q^{73} + (187 \beta_{14} + 44 \beta_{12}) q^{74} + ( - 2046 \beta_{3} - 109978) q^{76} + (99 \beta_{7} - 11727 \beta_{4}) q^{77} + ( - 1408 \beta_{5} + 4664 \beta_1) q^{79} + ( - 55 \beta_{11} - 583 \beta_{9}) q^{82} + ( - 374 \beta_{6} - 4038 \beta_{2}) q^{83} + (352 \beta_{15} - 1152 \beta_{13}) q^{86} + (493 \beta_{10} - 823 \beta_{8}) q^{88} + (136 \beta_{14} + 379 \beta_{12}) q^{89} + (2124 \beta_{3} + 21258) q^{91} + ( - 330 \beta_{7} + 6116 \beta_{4}) q^{92} + (2112 \beta_{5} + 59664 \beta_1) q^{94} + ( - 440 \beta_{11} + 132 \beta_{9}) q^{97} + (144 \beta_{6} + 20795 \beta_{2}) q^{98}+O(q^{100}) \) q - b4 * q^2 + (-b5 - 27b1) q^4 - b9 * q^7 + (-b6 - 28b2) q^8 + (2b15 - 3b13) * q^11 + (-2b10 + b8) q^13 + (7b14 - 8b12) * q^14 + (-21b3 - 787) q^16 + (-b7 + 117b4) q^17 + (-44b5 - 902b1) * q^19 + (11b11 + 23b9) * q^22 + (11b6 - 253b2) * q^23 + (11b15 - 36b13) * q^26 + (-31b10 + 41b8) * q^28 + 22b12 q^29 + (56b3 - 1996) q^31 + (11b7 + 584b4) * q^32 + (142b5 + 6902b1) * q^34 + (22b11 - 33b9) * q^37 + (-44b6 - 2354b2) * q^38 + (-88b15 - 11b13) * q^41 + (-64b10 + 32b8) * q^43 + (-119b14 + 242b12) * q^44 + (22b3 - 14938) q^46 + (-44b7 + 1012b4) * q^47 + (144b5 + 16043b1) * q^49 + (55b11 + 153b9) * q^52 + (55b6 + 451b2) * q^53 + (125b15 - 506b13) * q^56 + (66b10 - 66b8) * q^58 + (154b14 + 69b12) * q^59 + (352b3 - 21626) q^61 + (56b7 + 148b4) * q^62 + (-363b5 + 9283b1) * q^64 + (-176b11 - 110b9) * q^67 + (110b6 + 7844b2) * q^68 + (-44b15 - 454b13) * q^71 + (132b10 + 78b8) * q^73 + (187b14 + 44b12) * q^74 + (-2046b3 - 109978) q^76 + (99b7 - 11727b4) * q^77 + (-1408b5 + 4664b1) * q^79 + (-55b11 - 583b9) * q^82 + (-374b6 - 4038b2) * q^83 + (352b15 - 1152b13) * q^86 + (493b10 - 823b8) * q^88 + (136b14 + 379b12) * q^89 + (2124b3 + 21258) q^91 + (-330b7 + 6116b4) * q^92 + (2112b5 + 59664b1) * q^94 + (-440b11 + 132b9) * q^97 + (144b6 + 20795b2) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 12424 q^{16} - 32384 q^{31} - 239184 q^{46} - 348832 q^{61} - 1743280 q^{76} + 323136 q^{91}+O(q^{100}) \) 16 * q - 12424 * q^16 - 32384 * q^31 - 239184 * q^46 - 348832 * q^61 - 1743280 * q^76 + 323136 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	225.6.f.a

Level	$225$
Weight	$6$
Character orbit	225.f
Analytic conductor	$36.086$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(36.0863594579\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 468 x^{14} + 88042 x^{12} - 8478288 x^{10} + 442928373 x^{8} - 12512257524 x^{6} + \cdots + 4793095790596 \) x^16 - 468x^14 + 88042x^12 - 8478288x^10 + 442928373x^8 - 12512257524x^6 + 191311236688x^4 - 1487738314296*x^2 + 4793095790596
Coefficient ring:	\(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index:	\( 2^{14}\cdot 3^{24} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 4294 \nu^{14} - 1319877 \nu^{12} + 37859667 \nu^{10} + 29409376230 \nu^{8} + \cdots + 23\!\cdots\!16 ) / 42\!\cdots\!86 \) (4294v^14 - 1319877v^12 + 37859667v^10 + 29409376230v^8 - 4244669562215v^6 + 219605415656679v^4 - 4037762260724186*v^2 + 23732232825921516) / 4209010098945386
\(\beta_{2}\)	\(=\)	\( ( - 10555945770082 \nu^{14} + \cdots + 59\!\cdots\!90 ) / 51\!\cdots\!04 \) (-10555945770082v^14 + 4348208197247031v^12 - 686291388708563965v^10 + 51658920344942862120v^8 - 1951187969967889122943v^6 + 42280226983888425873783v^4 - 744690378633220182354868*v^2 + 5970857731550722935015390) / 515464477812676077958804
\(\beta_{3}\)	\(=\)	\( ( - 155380 \nu^{14} + 63609390 \nu^{12} - 9889761968 \nu^{10} + 716058156645 \nu^{8} + \cdots - 15\!\cdots\!30 ) / 37\!\cdots\!29 \) (-155380v^14 + 63609390v^12 - 9889761968v^10 + 716058156645v^8 - 24289928058676v^6 + 414097854636960v^4 - 3442726284998328*v^2 - 153508192606073030) / 3701531675221129
\(\beta_{4}\)	\(=\)	\( ( - 83163821358150 \nu^{14} + \cdots + 24\!\cdots\!30 ) / 51\!\cdots\!04 \) (-83163821358150v^14 + 36854171605884208v^12 - 6476326358602071582v^10 + 568735726381529841231v^8 - 25878065093092774513458v^6 + 580634211556698343476137v^4 - 6133911956858298804987042*v^2 + 24354217540436013356353530) / 515464477812676077958804
\(\beta_{5}\)	\(=\)	\( ( - 8063348422931 \nu^{14} + \cdots - 89\!\cdots\!92 ) / 36\!\cdots\!86 \) (-8063348422931v^14 + 2970202137445506v^12 - 369002276068876527v^10 + 12301212313164612136v^8 + 884870587699129961926v^6 - 72899908313788648718578v^4 + 1471142531140147534486408*v^2 - 8912503245501334010933192) / 36818891272334005568486
\(\beta_{6}\)	\(=\)	\( ( 298615658800769 \nu^{14} + \cdots - 42\!\cdots\!82 ) / 51\!\cdots\!04 \) (298615658800769v^14 - 129722146162991617v^12 + 21992756060409097394v^10 - 1827584823049640404860v^8 + 79025123181322317602777v^6 - 2063292954534916186175301v^4 + 47753700971494196391900968*v^2 - 429670789131427531904998282) / 515464477812676077958804
\(\beta_{7}\)	\(=\)	\( ( - 56\!\cdots\!86 \nu^{14} + \cdots + 18\!\cdots\!82 ) / 51\!\cdots\!04 \) (-5617294358387486v^14 + 2558368893447016667v^12 - 461539069997264386746v^10 + 41513909243501212594680v^8 - 1925662951353564625753482v^6 + 43715137794192305893368583v^4 - 465162271786812940596031982*v^2 + 1854075598200098698330170282) / 515464477812676077958804
\(\beta_{8}\)	\(=\)	\( ( 19\!\cdots\!17 \nu^{15} + \cdots - 21\!\cdots\!20 \nu ) / 11\!\cdots\!56 \) (190720744897470626417v^15 - 97346695663162059615385v^13 + 20172307238035376088937695v^11 - 2159497429588313891178260895v^9 + 125579924625660208640412928698v^7 - 3807629356814680564058021084838v^5 + 51988342208625611034385919418740v^3 - 213799949609107985848305345767320v) / 1128513597777981114940301020456
\(\beta_{9}\)	\(=\)	\( ( 21\!\cdots\!67 \nu^{15} + \cdots - 37\!\cdots\!52 \nu ) / 11\!\cdots\!56 \) (215009938407414119167v^15 - 110964545059244483870575v^13 + 23247454766346629097203775v^11 - 2514671585221379331550785225v^9 + 147585815317136355329914817688v^7 - 4522961808379514622219576586770v^5 + 64881196624627589707372506536080v^3 - 370851227268881932364339672094352v) / 1128513597777981114940301020456
\(\beta_{10}\)	\(=\)	\( ( 61\!\cdots\!99 \nu^{15} + \cdots - 58\!\cdots\!00 \nu ) / 16\!\cdots\!08 \) (61221650155116341699v^15 - 29911946324964306858811v^13 + 5951744192197790423994831v^11 - 615765866418605561712316515v^9 + 34948622785182469125492050616v^7 - 1045294465549645539924023643276v^5 + 14178926315196638078669230591412v^3 - 58201997502155319310056766037500v) / 161216228253997302134328717208
\(\beta_{11}\)	\(=\)	\( ( - 75\!\cdots\!67 \nu^{15} + \cdots + 10\!\cdots\!24 \nu ) / 16\!\cdots\!08 \) (-75489437743353810067v^15 + 36723611126116712975221v^13 - 7265553121183899080738055v^11 + 746383683527752655019966765v^9 - 42063380469622673193311676000v^7 + 1258416787304374045435444358820v^5 - 17826671622310919907589649656900v^3 + 101182430292241025344146188999524v) / 161216228253997302134328717208
\(\beta_{12}\)	\(=\)	\( ( 7957705714182 \nu^{15} + \cdots - 15\!\cdots\!10 \nu ) / 11\!\cdots\!36 \) (7957705714182v^15 - 3598119363649968v^13 + 644035157376223023v^11 - 57347815956640217775v^9 + 2613915104655676940037v^7 - 56806316915933494555893v^5 + 533657648661151317221376v^3 - 1547103933501453004623210v) / 11131581817659381517436
\(\beta_{13}\)	\(=\)	\( ( 12446636586075 \nu^{15} + \cdots - 48\!\cdots\!78 \nu ) / 13\!\cdots\!08 \) (12446636586075v^15 - 5623748857011024v^13 + 1007487108030208671v^11 - 90219319217181035505v^9 + 4195705656213325089378v^7 - 97411761018323173199583v^5 + 1108282610729611413328482v^3 - 4883021098980463299719178v) / 13590853984273495352108
\(\beta_{14}\)	\(=\)	\( ( 12\!\cdots\!41 \nu^{15} + \cdots - 24\!\cdots\!58 \nu ) / 91\!\cdots\!52 \) (124612303971973041v^15 - 56557192104727963941v^13 + 10151637947231585010180v^11 - 905417239479847490800686v^9 + 41288344208434801900415343v^7 - 897367474151544482213399187v^5 + 8430461177369055812214954180v^3 - 24446056079431660996745269458v) / 91913471068413513189469052
\(\beta_{15}\)	\(=\)	\( ( 23\!\cdots\!68 \nu^{15} + \cdots - 97\!\cdots\!82 \nu ) / 14\!\cdots\!24 \) (23734424184476868v^15 - 10812037618474507497v^13 + 1953071697173118080376v^11 - 176316437189324754972642v^9 + 8257663444945921701289170v^7 - 192594445223861436617005143v^5 + 2198184045332595498381616374v^3 - 9704006433904861273906606182v) / 14214091716980894210397524

\(\nu\)	\(=\)	\( ( \beta_{13} - \beta_{12} + 3\beta_{11} + 3\beta_{10} + 3\beta_{9} - 3\beta_{8} ) / 162 \) (b13 - b12 + 3b11 + 3b10 + 3b9 - 3b8) / 162
\(\nu^{2}\)	\(=\)	\( \beta_{3} - 2\beta_{2} + \beta _1 + 59 \) b3 - 2*b2 + b1 + 59
\(\nu^{3}\)	\(=\)	\( ( - 81 \beta_{15} + 81 \beta_{14} + 199 \beta_{13} - 197 \beta_{12} + 234 \beta_{11} + \cdots - 378 \beta_{8} ) / 162 \) (-81b15 + 81b14 + 199b13 - 197b12 + 234b11 + 216b10 + 396b9 - 378b8) / 162
\(\nu^{4}\)	\(=\)	\( -4\beta_{6} + 6\beta_{5} - 4\beta_{4} + 117\beta_{3} - 368\beta_{2} + 354\beta _1 + 5426 \) -4b6 + 6b5 - 4b4 + 117b3 - 368b2 + 354b1 + 5426
\(\nu^{5}\)	\(=\)	\( ( - 16065 \beta_{15} + 15525 \beta_{14} + 31889 \beta_{13} - 30569 \beta_{12} + 24132 \beta_{11} + \cdots - 36966 \beta_{8} ) / 162 \) (-16065b15 + 15525b14 + 31889b13 - 30569b12 + 24132b11 + 19632b10 + 44706b9 - 36966b8) / 162
\(\nu^{6}\)	\(=\)	\( 20 \beta_{7} - 702 \beta_{6} + 1755 \beta_{5} - 1840 \beta_{4} + 12198 \beta_{3} - 55722 \beta_{2} + \cdots + 546990 \) 20b7 - 702b6 + 1755b5 - 1840b4 + 12198b3 - 55722b2 + 81404*b1 + 546990
\(\nu^{7}\)	\(=\)	\( ( - 2418255 \beta_{15} + 2197125 \beta_{14} + 4650785 \beta_{13} - 4213567 \beta_{12} + \cdots - 3336960 \beta_{8} ) / 162 \) (-2418255b15 + 2197125b14 + 4650785b13 - 4213567b12 + 2681790b11 + 1762158b10 + 5052660b9 - 3336960b8) / 162
\(\nu^{8}\)	\(=\)	\( 6552 \beta_{7} - 97648 \beta_{6} + 341936 \beta_{5} - 520120 \beta_{4} + 1248039 \beta_{3} + \cdots + 55711234 \) 6552b7 - 97648b6 + 341936b5 - 520120b4 + 1248039b3 - 7602080b2 + 15338848*b1 + 55711234
\(\nu^{9}\)	\(=\)	\( ( - 332563725 \beta_{15} + 277167501 \beta_{14} + 636375907 \beta_{13} - 529975171 \beta_{12} + \cdots - 275465988 \beta_{8} ) / 162 \) (-332563725b15 + 277167501b14 + 636375907b13 - 529975171b12 + 306069030b11 + 145557702b10 + 578402532b9 - 275465988b8) / 162
\(\nu^{10}\)	\(=\)	\( 1465440 \beta_{7} - 12532806 \beta_{6} + 56505735 \beta_{5} - 114097440 \beta_{4} + 126387720 \beta_{3} + \cdots + 5638946796 \) 1465440b7 - 12532806b6 + 56505735b5 - 114097440b4 + 126387720b3 - 973126234b2 + 2522960866*b1 + 5638946796
\(\nu^{11}\)	\(=\)	\( ( - 43741978533 \beta_{15} + 32551063083 \beta_{14} + 83634619135 \beta_{13} + \cdots - 19207698312 \beta_{8} ) / 162 \) (-43741978533b15 + 32551063083b14 + 83634619135b13 - 62226781913b12 + 35277259542b11 + 10154532594b10 + 66698912544b9 - 19207698312b8) / 162
\(\nu^{12}\)	\(=\)	\( 276277716 \beta_{7} - 1537756384 \beta_{6} + 8499573984 \beta_{5} - 21452913516 \beta_{4} + \cdots + 562664836178 \) 276277716b7 - 1537756384b6 + 8499573984b5 - 21452913516b4 + 12611461941b3 - 119360031344b2 + 379257628200*b1 + 562664836178
\(\nu^{13}\)	\(=\)	\( ( - 5596420498695 \beta_{15} + 3606011257095 \beta_{14} + 10698996010013 \beta_{13} + \cdots - 789921475224 \beta_{8} ) / 162 \) (-5596420498695b15 + 3606011257095b14 + 10698996010013b13 - 6893557488557b12 + 4068483058950b11 + 417669150390b10 + 7692830423016b9 - 789921475224b8) / 162
\(\nu^{14}\)	\(=\)	\( 46896760768 \beta_{7} - 182750587110 \beta_{6} + 1200452263101 \beta_{5} - 3640168052144 \beta_{4} + \cdots + 54828981404032 \) 46896760768b7 - 182750587110b6 + 1200452263101b5 - 3640168052144b4 + 1228910341940b3 - 14184467391010b2 + 53560444058666*b1 + 54828981404032
\(\nu^{15}\)	\(=\)	\( ( - 701869907118915 \beta_{15} + 377165783058885 \beta_{14} + \cdots + 69182953351632 \beta_{8} ) / 162 \) (-701869907118915b15 + 377165783058885b14 + 1341779541476921b13 - 721030066045903b12 + 466482034303218b11 - 36590291767314b10 + 882047096187264b9 + 69182953351632b8) / 162

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 107.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 10737 T^{4} + 2178576)^{2} \) (T^8 + 10737*T^4 + 2178576)^2
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} + \cdots + 10\!\cdots\!36)^{2} \) (T^8 + 2229509448*T^4 + 1069238398979544336)^2
$11$	\( (T^{4} + 437076 T^{2} + 3809605284)^{4} \) (T^4 + 437076*T^2 + 3809605284)^4
$13$	\( (T^{8} + \cdots + 45\!\cdots\!16)^{2} \) (T^8 + 151721632008*T^4 + 4595392991233898624016)^2
$17$	\( (T^{8} + \cdots + 13\!\cdots\!76)^{2} \) (T^8 + 2793413378592*T^4 + 1390153162414992466176)^2
$19$	\( (T^{4} + \cdots + 8961281731600)^{4} \) (T^4 + 9084680*T^2 + 8961281731600)^4
$23$	\( (T^{8} + \cdots + 11\!\cdots\!36)^{2} \) (T^8 + 314070542878752*T^4 + 1125605912162123511258644736)^2
$29$	\( (T^{2} - 6351048)^{8} \) (T^2 - 6351048)^8
$31$	\( (T^{2} + 4048 T - 2006864)^{8} \) (T^2 + 4048*T - 2006864)^8
$37$	\( (T^{8} + \cdots + 68\!\cdots\!76)^{2} \) (T^8 + 21656893118999688*T^4 + 682513399683881255513986204176)^2
$41$	\( (T^{4} + \cdots + 64\!\cdots\!04)^{4} \) (T^4 + 575553924*T^2 + 64926300461734404)^4
$43$	\( (T^{8} + \cdots + 50\!\cdots\!16)^{2} \) (T^8 + 159091662004420608*T^4 + 5052688028062005574842398766268416)^2
$47$	\( (T^{8} + \cdots + 34\!\cdots\!56)^{2} \) (T^8 + 57301477970153472*T^4 + 347764169628016589340491913363456)^2
$53$	\( (T^{8} + \cdots + 53\!\cdots\!36)^{2} \) (T^8 + 56036677833034272*T^4 + 530957487924379409292628908155136)^2
$59$	\( (T^{4} + \cdots + 44\!\cdots\!44)^{4} \) (T^4 - 1989561204*T^2 + 444730185467759844)^4
$61$	\( (T^{2} + 43604 T + 234179044)^{8} \) (T^2 + 43604*T + 234179044)^8
$67$	\( (T^{8} + \cdots + 15\!\cdots\!96)^{2} \) (T^8 + 10885021764383657088*T^4 + 15766438961720227228174621883359236096)^2
$71$	\( (T^{4} + \cdots + 77\!\cdots\!84)^{4} \) (T^4 + 5840124624*T^2 + 7752975089544926784)^4
$73$	\( (T^{8} + \cdots + 65\!\cdots\!36)^{2} \) (T^8 + 8050345089135308928*T^4 + 65960687965937161482420373221543936)^2
$79$	\( (T^{4} + \cdots + 14\!\cdots\!96)^{4} \) (T^4 + 7774371968*T^2 + 14665492539663978496)^4
$83$	\( (T^{8} + \cdots + 34\!\cdots\!36)^{2} \) (T^8 + 124008796527911281152*T^4 + 3495253234441172781399592065630784782336)^2
$89$	\( (T^{4} + \cdots + 26\!\cdots\!64)^{4} \) (T^4 - 5854602564*T^2 + 2661397578676837764)^4
$97$	\( (T^{8} + \cdots + 63\!\cdots\!56)^{2} \) (T^8 + 613561480633573705728*T^4 + 6383804236571419372276116196251329888256)^2
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\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(\beta_{1}\)