# Properties

 Label 441.6.a.e.1.1 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $1$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## 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: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-32.0000 q^{4} +O(q^{10})$$ $$q-32.0000 q^{4} -427.000 q^{13} +1024.00 q^{16} +3143.00 q^{19} -3125.00 q^{25} +2723.00 q^{31} -6661.00 q^{37} +22475.0 q^{43} +13664.0 q^{52} -38626.0 q^{61} -32768.0 q^{64} -37939.0 q^{67} -78127.0 q^{73} -100576. q^{76} +90857.0 q^{79} -134386. q^{97} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 0 0
$$4$$ −32.0000 −1.00000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −427.000 −0.700760 −0.350380 0.936608i $$-0.613948\pi$$
−0.350380 + 0.936608i $$0.613948\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1024.00 1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 3143.00 1.99738 0.998689 0.0511835i $$-0.0162993\pi$$
0.998689 + 0.0511835i $$0.0162993\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −3125.00 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 2723.00 0.508913 0.254456 0.967084i $$-0.418103\pi$$
0.254456 + 0.967084i $$0.418103\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6661.00 −0.799899 −0.399949 0.916537i $$-0.630972\pi$$
−0.399949 + 0.916537i $$0.630972\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 22475.0 1.85365 0.926827 0.375489i $$-0.122525\pi$$
0.926827 + 0.375489i $$0.122525\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 13664.0 0.700760
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −38626.0 −1.32909 −0.664546 0.747247i $$-0.731375\pi$$
−0.664546 + 0.747247i $$0.731375\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −32768.0 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −37939.0 −1.03252 −0.516260 0.856432i $$-0.672676\pi$$
−0.516260 + 0.856432i $$0.672676\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −78127.0 −1.71591 −0.857954 0.513727i $$-0.828265\pi$$
−0.857954 + 0.513727i $$0.828265\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −100576. −1.99738
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 90857.0 1.63791 0.818956 0.573856i $$-0.194553\pi$$
0.818956 + 0.573856i $$0.194553\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −134386. −1.45019 −0.725095 0.688649i $$-0.758204\pi$$
−0.725095 + 0.688649i $$0.758204\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 100000. 1.00000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ −211477. −1.96413 −0.982065 0.188544i $$-0.939623\pi$$
−0.982065 + 0.188544i $$0.939623\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −247843. −1.99807 −0.999034 0.0439362i $$-0.986010\pi$$
−0.999034 + 0.0439362i $$0.986010\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −161051. −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −87136.0 −0.508913
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 347111. 1.90967 0.954837 0.297131i $$-0.0960299\pi$$
0.954837 + 0.297131i $$0.0960299\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ −454657. −1.99594 −0.997969 0.0637074i $$-0.979708\pi$$
−0.997969 + 0.0637074i $$0.979708\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 213152. 0.799899
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ −408724. −1.45877 −0.729387 0.684102i $$-0.760194\pi$$
−0.729387 + 0.684102i $$0.760194\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 109214. 0.353614 0.176807 0.984246i $$-0.443423\pi$$
0.176807 + 0.984246i $$0.443423\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 678248. 1.99949 0.999746 0.0225538i $$-0.00717969\pi$$
0.999746 + 0.0225538i $$0.00717969\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −188964. −0.508935
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −719200. −1.85365
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −853027. −1.93538 −0.967690 0.252142i $$-0.918865\pi$$
−0.967690 + 0.252142i $$0.918865\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 656375. 1.26841 0.634204 0.773166i $$-0.281328\pi$$
0.634204 + 0.773166i $$0.281328\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −912268. −1.63301 −0.816507 0.577336i $$-0.804092\pi$$
−0.816507 + 0.577336i $$0.804092\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −437248. −0.700760
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −288976. −0.446844 −0.223422 0.974722i $$-0.571723\pi$$
−0.223422 + 0.974722i $$0.571723\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 304052. 0.409436 0.204718 0.978821i $$-0.434372\pi$$
0.204718 + 0.978821i $$0.434372\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −1.45951e6 −1.83915 −0.919576 0.392913i $$-0.871467\pi$$
−0.919576 + 0.392913i $$0.871467\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 1.29697e6 1.43843 0.719215 0.694788i $$-0.244502\pi$$
0.719215 + 0.694788i $$0.244502\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 1.23603e6 1.32909
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.34206e6 −1.39968
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.04858e6 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 1.21405e6 1.03252
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 2.25285e6 1.86341 0.931707 0.363210i $$-0.118319\pi$$
0.931707 + 0.363210i $$0.118319\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 738389. 0.578210 0.289105 0.957297i $$-0.406642\pi$$
0.289105 + 0.957297i $$0.406642\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ −2.33458e6 −1.73277 −0.866387 0.499373i $$-0.833564\pi$$
−0.866387 + 0.499373i $$0.833564\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.41986e6 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.50006e6 1.71591
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 3.21843e6 1.99738
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 3.20232e6 1.93919 0.969593 0.244723i $$-0.0786971\pi$$
0.969593 + 0.244723i $$0.0786971\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 2.56708e6 1.48108 0.740539 0.672014i $$-0.234570\pi$$
0.740539 + 0.672014i $$0.234570\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −2.90742e6 −1.63791
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 1.33438e6 0.700760
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −546151. −0.273995 −0.136998 0.990571i $$-0.543745\pi$$
−0.136998 + 0.990571i $$0.543745\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.63172e6 1.26231 0.631155 0.775657i $$-0.282581\pi$$
0.631155 + 0.775657i $$0.282581\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 4.27561e6 1.87904 0.939518 0.342501i $$-0.111274\pi$$
0.939518 + 0.342501i $$0.111274\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 7.40235e6 2.98952
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −2.58318e6 −1.00113 −0.500563 0.865700i $$-0.666874\pi$$
−0.500563 + 0.865700i $$0.666874\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −5.31727e6 −1.97887 −0.989434 0.144983i $$-0.953687\pi$$
−0.989434 + 0.144983i $$0.953687\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 69893.0 0.0249940 0.0124970 0.999922i $$-0.496022\pi$$
0.0124970 + 0.999922i $$0.496022\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 4.30035e6 1.45019
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −5.00342e6 −1.59328 −0.796638 0.604456i $$-0.793390\pi$$
−0.796638 + 0.604456i $$0.793390\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −3.20000e6 −1.00000
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ −1.16272e6 −0.356626
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −5.17021e6 −1.52827 −0.764134 0.645057i $$-0.776834\pi$$
−0.764134 + 0.645057i $$0.776834\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 6.76726e6 1.96413
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −4.87580e6 −1.34073 −0.670364 0.742033i $$-0.733862\pi$$
−0.670364 + 0.742033i $$0.733862\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −7.49192e6 −1.92032 −0.960160 0.279450i $$-0.909848\pi$$
−0.960160 + 0.279450i $$0.909848\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 7.93098e6 1.99807
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 3.99333e6 0.988950 0.494475 0.869192i $$-0.335360\pi$$
0.494475 + 0.869192i $$0.335360\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.18576e6 0.265587 0.132793 0.991144i $$-0.457605\pi$$
0.132793 + 0.991144i $$0.457605\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 2.92217e6 0.633510 0.316755 0.948507i $$-0.397407\pi$$
0.316755 + 0.948507i $$0.397407\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −9.82188e6 −1.99738
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 2.84425e6 0.560537
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 5.15363e6 1.00000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −1.36487e6 −0.260778 −0.130389 0.991463i $$-0.541623\pi$$
−0.130389 + 0.991463i $$0.541623\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 2.78835e6 0.508913
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −2.17369e6 −0.390793 −0.195397 0.980724i $$-0.562599\pi$$
−0.195397 + 0.980724i $$0.562599\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −1.11076e7 −1.90967
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ −1.17287e7 −1.87497 −0.937486 0.348023i $$-0.886853\pi$$
−0.937486 + 0.348023i $$0.886853\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −6.43634e6 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −3.02235e6 −0.443968 −0.221984 0.975050i $$-0.571253\pi$$
−0.221984 + 0.975050i $$0.571253\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1.27982e7 1.82886 0.914430 0.404744i $$-0.132639\pi$$
0.914430 + 0.404744i $$0.132639\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 1.45490e7 1.99594
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ −9.59682e6 −1.29897
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 5.88780e6 0.755723 0.377862 0.925862i $$-0.376660\pi$$
0.377862 + 0.925862i $$0.376660\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −1.29099e7 −1.61430 −0.807150 0.590347i $$-0.798991\pi$$
−0.807150 + 0.590347i $$0.798991\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 8.55839e6 1.01649
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −6.82086e6 −0.799899
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −1.74342e7 −1.96887 −0.984435 0.175749i $$-0.943765\pi$$
−0.984435 + 0.175749i $$0.943765\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 1.30792e7 1.45877
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −1.56805e7 −1.72739 −0.863693 0.504019i $$-0.831854\pi$$
−0.863693 + 0.504019i $$0.831854\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −1.58234e7 −1.70079 −0.850394 0.526147i $$-0.823636\pi$$
−0.850394 + 0.526147i $$0.823636\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ −1.22260e7 −1.28251 −0.641253 0.767330i $$-0.721585\pi$$
−0.641253 + 0.767330i $$0.721585\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 9.76562e6 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −3.49485e6 −0.353614
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −1.62439e7 −1.62411 −0.812057 0.583579i $$-0.801652\pi$$
−0.812057 + 0.583579i $$0.801652\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ −1.33848e7 −1.27668 −0.638342 0.769753i $$-0.720380\pi$$
−0.638342 + 0.769753i $$0.720380\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −2.17039e7 −1.99949
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −690949. −0.0615095 −0.0307548 0.999527i $$-0.509791\pi$$
−0.0307548 + 0.999527i $$0.509791\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 1.73115e7 1.47332 0.736661 0.676262i $$-0.236401\pi$$
0.736661 + 0.676262i $$0.236401\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 6.04685e6 0.508935
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2.30144e7 1.85365
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 7.01920e6 0.559233 0.279616 0.960112i $$-0.409793\pi$$
0.279616 + 0.960112i $$0.409793\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ −2.09355e7 −1.59770
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −2.27014e7 −1.69604 −0.848021 0.529962i $$-0.822206\pi$$
−0.848021 + 0.529962i $$0.822206\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 2.72969e7 1.93538
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.56893e7 1.10095 0.550474 0.834853i $$-0.314447\pi$$
0.550474 + 0.834853i $$0.314447\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −1.75648e7 −1.20749 −0.603746 0.797177i $$-0.706326\pi$$
−0.603746 + 0.797177i $$0.706326\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 1.28170e7 0.863330 0.431665 0.902034i $$-0.357926\pi$$
0.431665 + 0.902034i $$0.357926\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −2.73219e7 −1.76771 −0.883857 0.467758i $$-0.845062\pi$$
−0.883857 + 0.467758i $$0.845062\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.98526e7 1.25915 0.629575 0.776940i $$-0.283229\pi$$
0.629575 + 0.776940i $$0.283229\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 2.50017e7 1.52459 0.762296 0.647228i $$-0.224072\pi$$
0.762296 + 0.647228i $$0.224072\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −2.10040e7 −1.26841
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ −8.50938e6 −0.508913
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 3.31061e7 1.90534 0.952668 0.304012i $$-0.0983263\pi$$
0.952668 + 0.304012i $$0.0983263\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.64933e7 0.931375
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 2.91926e7 1.63301
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 2.27708e7 1.21570 0.607849 0.794053i $$-0.292033\pi$$
0.607849 + 0.794053i $$0.292033\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.06389e7 3.70245
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ 2.22923e7 1.14725 0.573623 0.819120i $$-0.305538\pi$$
0.573623 + 0.819120i $$0.305538\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ −2.26485e7 −1.14460 −0.572299 0.820045i $$-0.693948\pi$$
−0.572299 + 0.820045i $$0.693948\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.39919e7 0.700760
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −2.05111e7 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 9.24723e6 0.446844
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −3.00842e6 −0.141568 −0.0707842 0.997492i $$-0.522550\pi$$
−0.0707842 + 0.997492i $$0.522550\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 3.06774e6 0.141852 0.0709259 0.997482i $$-0.477405\pi$$
0.0709259 + 0.997482i $$0.477405\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 1.62000e7 0.723550
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −3.83931e6 −0.168560 −0.0842800 0.996442i $$-0.526859\pi$$
−0.0842800 + 0.996442i $$0.526859\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ −4.58066e7 −1.97709 −0.988545 0.150925i $$-0.951775\pi$$
−0.988545 + 0.150925i $$0.951775\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −9.72966e6 −0.409436
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 2.62120e7 1.05799 0.528995 0.848625i $$-0.322569\pi$$
0.528995 + 0.848625i $$0.322569\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 4.67042e7 1.83915
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −4.21821e6 −0.164755 −0.0823777 0.996601i $$-0.526251\pi$$
−0.0823777 + 0.996601i $$0.526251\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.08156e7 0.799899
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 5.29708e6 0.197100 0.0985501 0.995132i $$-0.468580\pi$$
0.0985501 + 0.995132i $$0.468580\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ 3.33602e7 1.20244
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.12144e7 −0.741008
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −4.15032e7 −1.43843
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −3.23453e7 −1.11236 −0.556180 0.831062i $$-0.687733\pi$$
−0.556180 + 0.831062i $$0.687733\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −3.95530e7 −1.32909
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 4.29460e7 1.39968
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 6.05528e7 1.95862 0.979310 0.202365i $$-0.0648626\pi$$
0.979310 + 0.202365i $$0.0648626\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 6.04728e7 1.92674 0.963368 0.268183i $$-0.0864230\pi$$
0.963368 + 0.268183i $$0.0864230\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.6.a.e.1.1 1
3.2 odd 2 CM 441.6.a.e.1.1 1
7.2 even 3 63.6.e.b.46.1 yes 2
7.4 even 3 63.6.e.b.37.1 2
7.6 odd 2 441.6.a.f.1.1 1
21.2 odd 6 63.6.e.b.46.1 yes 2
21.11 odd 6 63.6.e.b.37.1 2
21.20 even 2 441.6.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
63.6.e.b.37.1 2 7.4 even 3
63.6.e.b.37.1 2 21.11 odd 6
63.6.e.b.46.1 yes 2 7.2 even 3
63.6.e.b.46.1 yes 2 21.2 odd 6
441.6.a.e.1.1 1 1.1 even 1 trivial
441.6.a.e.1.1 1 3.2 odd 2 CM
441.6.a.f.1.1 1 7.6 odd 2
441.6.a.f.1.1 1 21.20 even 2