Properties

 Label 441.2.a.h.1.2 Level $441$ Weight $2$ Character 441.1 Self dual yes Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,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, 2, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

 Embedding label 1.2 Root $$2.64575$$ of defining polynomial 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)

 $$f(q)$$ $$=$$ $$q+2.64575 q^{2} +5.00000 q^{4} +7.93725 q^{8} +O(q^{10})$$ $$q+2.64575 q^{2} +5.00000 q^{4} +7.93725 q^{8} -5.29150 q^{11} +11.0000 q^{16} -14.0000 q^{22} +5.29150 q^{23} -5.00000 q^{25} -10.5830 q^{29} +13.2288 q^{32} +6.00000 q^{37} +12.0000 q^{43} -26.4575 q^{44} +14.0000 q^{46} -13.2288 q^{50} -10.5830 q^{53} -28.0000 q^{58} +13.0000 q^{64} +4.00000 q^{67} -5.29150 q^{71} +15.8745 q^{74} +8.00000 q^{79} +31.7490 q^{86} -42.0000 q^{88} +26.4575 q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{4}+O(q^{10})$$ 2 * q + 10 * q^4 $$2 q + 10 q^{4} + 22 q^{16} - 28 q^{22} - 10 q^{25} + 12 q^{37} + 24 q^{43} + 28 q^{46} - 56 q^{58} + 26 q^{64} + 8 q^{67} + 16 q^{79} - 84 q^{88}+O(q^{100})$$ 2 * q + 10 * q^4 + 22 * q^16 - 28 * q^22 - 10 * q^25 + 12 * q^37 + 24 * q^43 + 28 * q^46 - 56 * q^58 + 26 * q^64 + 8 * q^67 + 16 * q^79 - 84 * q^88

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.64575 1.87083 0.935414 0.353553i $$-0.115027\pi$$
0.935414 + 0.353553i $$0.115027\pi$$
$$3$$ 0 0
$$4$$ 5.00000 2.50000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 7.93725 2.80624
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −5.29150 −1.59545 −0.797724 0.603023i $$-0.793963\pi$$
−0.797724 + 0.603023i $$0.793963\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 11.0000 2.75000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −14.0000 −2.98481
$$23$$ 5.29150 1.10335 0.551677 0.834058i $$-0.313988\pi$$
0.551677 + 0.834058i $$0.313988\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −10.5830 −1.96521 −0.982607 0.185695i $$-0.940546\pi$$
−0.982607 + 0.185695i $$0.940546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 13.2288 2.33854
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ −26.4575 −3.98862
$$45$$ 0 0
$$46$$ 14.0000 2.06419
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −13.2288 −1.87083
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −10.5830 −1.45369 −0.726844 0.686803i $$-0.759014\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −28.0000 −3.67658
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 13.0000 1.62500
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −5.29150 −0.627986 −0.313993 0.949425i $$-0.601667\pi$$
−0.313993 + 0.949425i $$0.601667\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 15.8745 1.84537
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 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
$$86$$ 31.7490 3.42358
$$87$$ 0 0
$$88$$ −42.0000 −4.47722
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 26.4575 2.75839
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −25.0000 −2.50000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −28.0000 −2.71960
$$107$$ −5.29150 −0.511549 −0.255774 0.966736i $$-0.582330\pi$$
−0.255774 + 0.966736i $$0.582330\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 21.1660 1.99113 0.995565 0.0940721i $$-0.0299884\pi$$
0.995565 + 0.0940721i $$0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −52.9150 −4.91304
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 17.0000 1.54545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 7.93725 0.701561
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 10.5830 0.914232
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 21.1660 1.80833 0.904167 0.427179i $$-0.140493\pi$$
0.904167 + 0.427179i $$0.140493\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −14.0000 −1.17485
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 30.0000 2.46598
$$149$$ 10.5830 0.866994 0.433497 0.901155i $$-0.357280\pi$$
0.433497 + 0.901155i $$0.357280\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ 21.1660 1.68388
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 60.0000 4.57496
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −58.2065 −4.38748
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 26.4575 1.97753 0.988764 0.149487i $$-0.0477622\pi$$
0.988764 + 0.149487i $$0.0477622\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 42.0000 3.09628
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −26.4575 −1.91440 −0.957199 0.289430i $$-0.906534\pi$$
−0.957199 + 0.289430i $$0.906534\pi$$
$$192$$ 0 0
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 10.5830 0.754008 0.377004 0.926212i $$-0.376954\pi$$
0.377004 + 0.926212i $$0.376954\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ −39.6863 −2.80624
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −52.9150 −3.63422
$$213$$ 0 0
$$214$$ −14.0000 −0.957020
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −47.6235 −3.22547
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 56.0000 3.72506
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −84.0000 −5.51487
$$233$$ −21.1660 −1.38663 −0.693316 0.720634i $$-0.743851\pi$$
−0.693316 + 0.720634i $$0.743851\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 26.4575 1.71139 0.855697 0.517477i $$-0.173129\pi$$
0.855697 + 0.517477i $$0.173129\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 44.9778 2.89128
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ −28.0000 −1.76034
$$254$$ 42.3320 2.65615
$$255$$ 0 0
$$256$$ −5.00000 −0.312500
$$257$$ 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
$$263$$ 5.29150 0.326288 0.163144 0.986602i $$-0.447836\pi$$
0.163144 + 0.986602i $$0.447836\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 20.0000 1.22169
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 56.0000 3.38308
$$275$$ 26.4575 1.59545
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −21.1660 −1.26266 −0.631329 0.775515i $$-0.717490\pi$$
−0.631329 + 0.775515i $$0.717490\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ −26.4575 −1.56996
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 47.6235 2.76806
$$297$$ 0 0
$$298$$ 28.0000 1.62200
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 63.4980 3.65390
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 40.0000 2.25018
$$317$$ −10.5830 −0.594401 −0.297200 0.954815i $$-0.596053\pi$$
−0.297200 + 0.954815i $$0.596053\pi$$
$$318$$ 0 0
$$319$$ 56.0000 3.13540
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −52.9150 −2.93069
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −36.0000 −1.97874 −0.989369 0.145424i $$-0.953545\pi$$
−0.989369 + 0.145424i $$0.953545\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −30.0000 −1.63420 −0.817102 0.576493i $$-0.804421\pi$$
−0.817102 + 0.576493i $$0.804421\pi$$
$$338$$ −34.3948 −1.87083
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 95.2470 5.13538
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −37.0405 −1.98844 −0.994220 0.107366i $$-0.965758\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −70.0000 −3.73101
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 70.0000 3.69961
$$359$$ 37.0405 1.95492 0.977462 0.211112i $$-0.0677085\pi$$
0.977462 + 0.211112i $$0.0677085\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 58.2065 3.03423
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −70.0000 −3.58151
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −47.6235 −2.42397
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 10.5830 0.536580 0.268290 0.963338i $$-0.413542\pi$$
0.268290 + 0.963338i $$0.413542\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 28.0000 1.41062
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −55.0000 −2.75000
$$401$$ 21.1660 1.05698 0.528490 0.848939i $$-0.322758\pi$$
0.528490 + 0.848939i $$0.322758\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −31.7490 −1.57374
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 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
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 31.7490 1.54552
$$423$$ 0 0
$$424$$ −84.0000 −4.07940
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −26.4575 −1.27887
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −26.4575 −1.27441 −0.637207 0.770693i $$-0.719910\pi$$
−0.637207 + 0.770693i $$0.719910\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −90.0000 −4.31022
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −37.0405 −1.75985 −0.879924 0.475114i $$-0.842407\pi$$
−0.879924 + 0.475114i $$0.842407\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −42.3320 −1.99777 −0.998886 0.0471929i $$-0.984972\pi$$
−0.998886 + 0.0471929i $$0.984972\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 105.830 4.97783
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ −40.0000 −1.85896 −0.929479 0.368875i $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ −116.413 −5.40434
$$465$$ 0 0
$$466$$ −56.0000 −2.59415
$$467$$ 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
$$473$$ −63.4980 −2.91964
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 70.0000 3.20173
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 85.0000 3.86364
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 24.0000 1.08754 0.543772 0.839233i $$-0.316996\pi$$
0.543772 + 0.839233i $$0.316996\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 5.29150 0.238802 0.119401 0.992846i $$-0.461903\pi$$
0.119401 + 0.992846i $$0.461903\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 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
$$506$$ −74.0810 −3.29330
$$507$$ 0 0
$$508$$ 80.0000 3.54943
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −29.1033 −1.28619
$$513$$ 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
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 14.0000 0.610429
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 5.00000 0.217391
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 31.7490 1.37135
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ 105.830 4.52084
$$549$$ 0 0
$$550$$ 70.0000 2.98481
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −26.4575 −1.12407
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 10.5830 0.448416 0.224208 0.974541i $$-0.428020\pi$$
0.224208 + 0.974541i $$0.428020\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −56.0000 −2.36222
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ −42.0000 −1.76228
$$569$$ 42.3320 1.77465 0.887325 0.461144i $$-0.152561\pi$$
0.887325 + 0.461144i $$0.152561\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −26.4575 −1.10335
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ −44.9778 −1.87083
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 56.0000 2.31928
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 66.0000 2.71258
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 52.9150 2.16748
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 37.0405 1.51343 0.756717 0.653742i $$-0.226802\pi$$
0.756717 + 0.653742i $$0.226802\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 120.000 4.88273
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −38.0000 −1.53481 −0.767403 0.641165i $$-0.778451\pi$$
−0.767403 + 0.641165i $$0.778451\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.3320 1.70422 0.852111 0.523360i $$-0.175322\pi$$
0.852111 + 0.523360i $$0.175322\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 63.4980 2.52582
$$633$$ 0 0
$$634$$ −28.0000 −1.11202
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 148.162 5.86579
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −21.1660 −0.836007 −0.418004 0.908445i $$-0.637270\pi$$
−0.418004 + 0.908445i $$0.637270\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 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
$$652$$ −100.000 −3.91630
$$653$$ −10.5830 −0.414145 −0.207072 0.978326i $$-0.566394\pi$$
−0.207072 + 0.978326i $$0.566394\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −26.4575 −1.03064 −0.515319 0.856998i $$-0.672327\pi$$
−0.515319 + 0.856998i $$0.672327\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ −95.2470 −3.70188
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −56.0000 −2.16833
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ −79.3725 −3.05732
$$675$$ 0 0
$$676$$ −65.0000 −2.50000
$$677$$ 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
$$683$$ 5.29150 0.202474 0.101237 0.994862i $$-0.467720\pi$$
0.101237 + 0.994862i $$0.467720\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 132.000 5.03245
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −98.0000 −3.72003
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 52.9150 1.99857 0.999286 0.0377695i $$-0.0120253\pi$$
0.999286 + 0.0377695i $$0.0120253\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −68.7895 −2.59260
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 132.288 4.94382
$$717$$ 0 0
$$718$$ 98.0000 3.65733
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −50.2693 −1.87083
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 52.9150 1.96521
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 70.0000 2.58023
$$737$$ −21.1660 −0.779660
$$738$$ 0 0
$$739$$ −52.0000 −1.91285 −0.956425 0.291977i $$-0.905687\pi$$
−0.956425 + 0.291977i $$0.905687\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 37.0405 1.35888 0.679442 0.733729i $$-0.262222\pi$$
0.679442 + 0.733729i $$0.262222\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 58.2065 2.13109
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 54.0000 1.96266 0.981332 0.192323i $$-0.0616021\pi$$
0.981332 + 0.192323i $$0.0616021\pi$$
$$758$$ 31.7490 1.15318
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −132.288 −4.78600
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −90.0000 −3.23917
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 28.0000 1.00385
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 28.0000 1.00192
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 52.9150 1.88502
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −66.1438 −2.33854
$$801$$ 0 0
$$802$$ 56.0000 1.97743
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 42.3320 1.48831 0.744157 0.668004i $$-0.232851\pi$$
0.744157 + 0.668004i $$0.232851\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −84.0000 −2.94420
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −52.9150 −1.84675 −0.923374 0.383903i $$-0.874580\pi$$
−0.923374 + 0.383903i $$0.874580\pi$$
$$822$$ 0 0
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 37.0405 1.28803 0.644013 0.765015i $$-0.277268\pi$$
0.644013 + 0.765015i $$0.277268\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 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
$$841$$ 83.0000 2.86207
$$842$$ −68.7895 −2.37064
$$843$$ 0 0
$$844$$ 60.0000 2.06529
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −116.413 −3.99764
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 31.7490 1.08834
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −42.0000 −1.43553
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −70.0000 −2.38421
$$863$$ −58.2065 −1.98137 −0.990687 0.136162i $$-0.956523\pi$$
−0.990687 + 0.136162i $$0.956523\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −42.3320 −1.43602
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −142.871 −4.83821
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 50.0000 1.68838 0.844190 0.536044i $$-0.180082\pi$$
0.844190 + 0.536044i $$0.180082\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 12.0000 0.403832 0.201916 0.979403i $$-0.435283\pi$$
0.201916 + 0.979403i $$0.435283\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −98.0000 −3.29237
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −112.000 −3.73749
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 168.000 5.58760
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −60.0000 −1.99227 −0.996134 0.0878507i $$-0.972000\pi$$
−0.996134 + 0.0878507i $$0.972000\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 58.2065 1.92847 0.964234 0.265052i $$-0.0853891\pi$$
0.964234 + 0.265052i $$0.0853891\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 15.8745 0.525082
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 48.0000 1.58337 0.791687 0.610927i $$-0.209203\pi$$
0.791687 + 0.610927i $$0.209203\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −30.0000 −0.986394
$$926$$ −105.830 −3.47779
$$927$$ 0 0
$$928$$ −140.000 −4.59573
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −105.830 −3.46658
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$938$$ 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
$$946$$ −168.000 −5.46215
$$947$$ 58.2065 1.89146 0.945729 0.324956i $$-0.105350\pi$$
0.945729 + 0.324956i $$0.105350\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −21.1660 −0.685634 −0.342817 0.939402i $$-0.611381\pi$$
−0.342817 + 0.939402i $$0.611381\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 132.288 4.27849
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 134.933 4.33692
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 63.4980 2.03461
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 42.3320 1.35432 0.677161 0.735835i $$-0.263210\pi$$
0.677161 + 0.735835i $$0.263210\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 14.0000 0.446758
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 63.4980 2.01912
$$990$$ 0 0
$$991$$ 24.0000 0.762385 0.381193 0.924496i $$-0.375513\pi$$
0.381193 + 0.924496i $$0.375513\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ −95.2470 −3.01499
$$999$$ 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 441.2.a.h.1.2 yes 2
3.2 odd 2 inner 441.2.a.h.1.1 2
4.3 odd 2 7056.2.a.co.1.2 2
7.2 even 3 441.2.e.h.361.1 4
7.3 odd 6 441.2.e.h.226.1 4
7.4 even 3 441.2.e.h.226.1 4
7.5 odd 6 441.2.e.h.361.1 4
7.6 odd 2 CM 441.2.a.h.1.2 yes 2
12.11 even 2 7056.2.a.co.1.1 2
21.2 odd 6 441.2.e.h.361.2 4
21.5 even 6 441.2.e.h.361.2 4
21.11 odd 6 441.2.e.h.226.2 4
21.17 even 6 441.2.e.h.226.2 4
21.20 even 2 inner 441.2.a.h.1.1 2
28.27 even 2 7056.2.a.co.1.2 2
84.83 odd 2 7056.2.a.co.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.a.h.1.1 2 3.2 odd 2 inner
441.2.a.h.1.1 2 21.20 even 2 inner
441.2.a.h.1.2 yes 2 1.1 even 1 trivial
441.2.a.h.1.2 yes 2 7.6 odd 2 CM
441.2.e.h.226.1 4 7.3 odd 6
441.2.e.h.226.1 4 7.4 even 3
441.2.e.h.226.2 4 21.11 odd 6
441.2.e.h.226.2 4 21.17 even 6
441.2.e.h.361.1 4 7.2 even 3
441.2.e.h.361.1 4 7.5 odd 6
441.2.e.h.361.2 4 21.2 odd 6
441.2.e.h.361.2 4 21.5 even 6
7056.2.a.co.1.1 2 12.11 even 2
7056.2.a.co.1.1 2 84.83 odd 2
7056.2.a.co.1.2 2 4.3 odd 2
7056.2.a.co.1.2 2 28.27 even 2