# Properties

 Label 304.2.h.a.303.1 Level $304$ Weight $2$ Character 304.303 Analytic conductor $2.427$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,2,Mod(303,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("304.303");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 304.h (of order $$2$$, degree $$1$$, 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: $$2.42745222145$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 5$$ x^2 - x + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 303.1 Root $$0.500000 + 2.17945i$$ of defining polynomial Character $$\chi$$ $$=$$ 304.303 Dual form 304.2.h.a.303.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -4.35890i q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{5} -4.35890i q^{7} -3.00000 q^{9} -4.35890i q^{11} +7.00000 q^{17} +4.35890i q^{19} -8.71780i q^{23} -4.00000 q^{25} +4.35890i q^{35} +13.0767i q^{43} +3.00000 q^{45} -4.35890i q^{47} -12.0000 q^{49} +4.35890i q^{55} +15.0000 q^{61} +13.0767i q^{63} +11.0000 q^{73} -19.0000 q^{77} +9.00000 q^{81} +8.71780i q^{83} -7.00000 q^{85} -4.35890i q^{95} +13.0767i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 - 6 * q^9 $$2 q - 2 q^{5} - 6 q^{9} + 14 q^{17} - 8 q^{25} + 6 q^{45} - 24 q^{49} + 30 q^{61} + 22 q^{73} - 38 q^{77} + 18 q^{81} - 14 q^{85}+O(q^{100})$$ 2 * q - 2 * q^5 - 6 * q^9 + 14 * q^17 - 8 * q^25 + 6 * q^45 - 24 * q^49 + 30 * q^61 + 22 * q^73 - 38 * q^77 + 18 * q^81 - 14 * q^85

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ − 4.35890i − 1.64751i −0.566947 0.823754i $$-0.691875\pi$$
0.566947 0.823754i $$-0.308125\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ − 4.35890i − 1.31426i −0.753778 0.657129i $$-0.771771\pi$$
0.753778 0.657129i $$-0.228229\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ 0 0
$$19$$ 4.35890i 1.00000i
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 8.71780i − 1.81779i −0.417029 0.908893i $$-0.636929\pi$$
0.417029 0.908893i $$-0.363071\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.35890i 0.736788i
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 13.0767i 1.99418i 0.0762493 + 0.997089i $$0.475706\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ − 4.35890i − 0.635811i −0.948122 0.317905i $$-0.897021\pi$$
0.948122 0.317905i $$-0.102979\pi$$
$$48$$ 0 0
$$49$$ −12.0000 −1.71429
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 4.35890i 0.587754i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 15.0000 1.92055 0.960277 0.279050i $$-0.0900195\pi$$
0.960277 + 0.279050i $$0.0900195\pi$$
$$62$$ 0 0
$$63$$ 13.0767i 1.64751i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −19.0000 −2.16525
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 8.71780i 0.956903i 0.878114 + 0.478451i $$0.158802\pi$$
−0.878114 + 0.478451i $$0.841198\pi$$
$$84$$ 0 0
$$85$$ −7.00000 −0.759257
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 4.35890i − 0.447214i
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ 13.0767i 1.31426i
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 8.71780i 0.812939i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 30.5123i − 2.79706i
$$120$$ 0 0
$$121$$ −8.00000 −0.727273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 21.7945i − 1.90419i −0.305796 0.952097i $$-0.598923\pi$$
0.305796 0.952097i $$-0.401077\pi$$
$$132$$ 0 0
$$133$$ 19.0000 1.64751
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 23.0000 1.96502 0.982511 0.186203i $$-0.0596182\pi$$
0.982511 + 0.186203i $$0.0596182\pi$$
$$138$$ 0 0
$$139$$ − 21.7945i − 1.84858i −0.381685 0.924292i $$-0.624656\pi$$
0.381685 0.924292i $$-0.375344\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 11.0000 0.901155 0.450578 0.892737i $$-0.351218\pi$$
0.450578 + 0.892737i $$0.351218\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ −21.0000 −1.69775
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −38.0000 −2.99482
$$162$$ 0 0
$$163$$ 8.71780i 0.682831i 0.939913 + 0.341415i $$0.110906\pi$$
−0.939913 + 0.341415i $$0.889094\pi$$
$$164$$ 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
$$171$$ − 13.0767i − 1.00000i
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 17.4356i 1.31801i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 30.5123i − 2.23128i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 21.7945i − 1.57699i −0.615038 0.788497i $$-0.710860\pi$$
0.615038 0.788497i $$-0.289140\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ 0 0
$$199$$ 13.0767i 0.926982i 0.886102 + 0.463491i $$0.153403\pi$$
−0.886102 + 0.463491i $$0.846597\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 26.1534i 1.81779i
$$208$$ 0 0
$$209$$ 19.0000 1.31426
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 13.0767i − 0.891823i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 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
$$225$$ 12.0000 0.800000
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −21.0000 −1.38772 −0.693860 0.720110i $$-0.744091\pi$$
−0.693860 + 0.720110i $$0.744091\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1.00000 −0.0655122 −0.0327561 0.999463i $$-0.510428\pi$$
−0.0327561 + 0.999463i $$0.510428\pi$$
$$234$$ 0 0
$$235$$ 4.35890i 0.284343i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 30.5123i 1.97368i 0.161712 + 0.986838i $$0.448299\pi$$
−0.161712 + 0.986838i $$0.551701\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 12.0000 0.766652
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 21.7945i − 1.37566i −0.725874 0.687828i $$-0.758564\pi$$
0.725874 0.687828i $$-0.241436\pi$$
$$252$$ 0 0
$$253$$ −38.0000 −2.38904
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 30.5123i 1.88147i 0.339145 + 0.940734i $$0.389862\pi$$
−0.339145 + 0.940734i $$0.610138\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 26.1534i 1.58871i 0.607457 + 0.794353i $$0.292190\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 17.4356i 1.05141i
$$276$$ 0 0
$$277$$ −33.0000 −1.98278 −0.991389 0.130950i $$-0.958197\pi$$
−0.991389 + 0.130950i $$0.958197\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 13.0767i 0.777329i 0.921379 + 0.388664i $$0.127063\pi$$
−0.921379 + 0.388664i $$0.872937\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 57.0000 3.28543
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −15.0000 −0.858898
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 4.35890i − 0.247170i −0.992334 0.123585i $$-0.960561\pi$$
0.992334 0.123585i $$-0.0394392\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 0 0
$$315$$ − 13.0767i − 0.736788i
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 30.5123i 1.69775i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −19.0000 −1.04750
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 21.7945i 1.17679i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 4.35890i − 0.233998i −0.993132 0.116999i $$-0.962673\pi$$
0.993132 0.116999i $$-0.0373274\pi$$
$$348$$ 0 0
$$349$$ 35.0000 1.87351 0.936754 0.349990i $$-0.113815\pi$$
0.936754 + 0.349990i $$0.113815\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 21.7945i − 1.15027i −0.818059 0.575135i $$-0.804950\pi$$
0.818059 0.575135i $$-0.195050\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −11.0000 −0.575766
$$366$$ 0 0
$$367$$ 26.1534i 1.36520i 0.730794 + 0.682598i $$0.239150\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 19.0000 0.968330
$$386$$ 0 0
$$387$$ − 39.2301i − 1.99418i
$$388$$ 0 0
$$389$$ −25.0000 −1.26755 −0.633775 0.773517i $$-0.718496\pi$$
−0.633775 + 0.773517i $$0.718496\pi$$
$$390$$ 0 0
$$391$$ − 61.0246i − 3.08615i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 7.00000 0.351320 0.175660 0.984451i $$-0.443794\pi$$
0.175660 + 0.984451i $$0.443794\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −9.00000 −0.447214
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ − 8.71780i − 0.427940i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 8.71780i 0.425892i 0.977064 + 0.212946i $$0.0683059\pi$$
−0.977064 + 0.212946i $$0.931694\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 13.0767i 0.635811i
$$424$$ 0 0
$$425$$ −28.0000 −1.35820
$$426$$ 0 0
$$427$$ − 65.3835i − 3.16413i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 38.0000 1.81779
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 36.0000 1.71429
$$442$$ 0 0
$$443$$ 30.5123i 1.44968i 0.688916 + 0.724841i $$0.258087\pi$$
−0.688916 + 0.724841i $$0.741913\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −17.0000 −0.795226 −0.397613 0.917553i $$-0.630161\pi$$
−0.397613 + 0.917553i $$0.630161\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −37.0000 −1.72326 −0.861631 0.507535i $$-0.830557\pi$$
−0.861631 + 0.507535i $$0.830557\pi$$
$$462$$ 0 0
$$463$$ 13.0767i 0.607726i 0.952716 + 0.303863i $$0.0982765\pi$$
−0.952716 + 0.303863i $$0.901724\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 4.35890i − 0.201706i −0.994901 0.100853i $$-0.967843\pi$$
0.994901 0.100853i $$-0.0321571\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 57.0000 2.62086
$$474$$ 0 0
$$475$$ − 17.4356i − 0.800000i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 43.5890i − 1.99163i −0.0913823 0.995816i $$-0.529129\pi$$
0.0913823 0.995816i $$-0.470871\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 43.5890i 1.96714i 0.180517 + 0.983572i $$0.442223\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ − 13.0767i − 0.587754i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 21.7945i − 0.975656i −0.872940 0.487828i $$-0.837789\pi$$
0.872940 0.487828i $$-0.162211\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 8.71780i − 0.388707i −0.980932 0.194354i $$-0.937739\pi$$
0.980932 0.194354i $$-0.0622609\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ − 47.9479i − 2.12109i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −19.0000 −0.835619
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −53.0000 −2.30435
$$530$$ 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
$$539$$ 52.3068i 2.25301i
$$540$$ 0 0
$$541$$ −25.0000 −1.07483 −0.537417 0.843317i $$-0.680600\pi$$
−0.537417 + 0.843317i $$0.680600\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ −45.0000 −1.92055
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 47.0000 1.99145 0.995727 0.0923462i $$-0.0294367\pi$$
0.995727 + 0.0923462i $$0.0294367\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 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
$$567$$ − 39.2301i − 1.64751i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ − 26.1534i − 1.09449i −0.836974 0.547243i $$-0.815677\pi$$
0.836974 0.547243i $$-0.184323\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 34.8712i 1.45423i
$$576$$ 0 0
$$577$$ 3.00000 0.124892 0.0624458 0.998048i $$-0.480110\pi$$
0.0624458 + 0.998048i $$0.480110\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 38.0000 1.57651
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 47.9479i 1.97902i 0.144460 + 0.989511i $$0.453855\pi$$
−0.144460 + 0.989511i $$0.546145\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ 30.5123i 1.25088i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 8.00000 0.325246
$$606$$ 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
$$613$$ 39.0000 1.57520 0.787598 0.616190i $$-0.211325\pi$$
0.787598 + 0.616190i $$0.211325\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −13.0000 −0.523360 −0.261680 0.965155i $$-0.584277\pi$$
−0.261680 + 0.965155i $$0.584277\pi$$
$$618$$ 0 0
$$619$$ 43.5890i 1.75199i 0.482321 + 0.875995i $$0.339794\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 47.9479i 1.90878i 0.298570 + 0.954388i $$0.403490\pi$$
−0.298570 + 0.954388i $$0.596510\pi$$
$$632$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 13.0767i 0.515695i 0.966186 + 0.257847i $$0.0830131\pi$$
−0.966186 + 0.257847i $$0.916987\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 47.9479i 1.88503i 0.334169 + 0.942513i $$0.391544\pi$$
−0.334169 + 0.942513i $$0.608456\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −41.0000 −1.60445 −0.802227 0.597019i $$-0.796352\pi$$
−0.802227 + 0.597019i $$0.796352\pi$$
$$654$$ 0 0
$$655$$ 21.7945i 0.851581i
$$656$$ 0 0
$$657$$ −33.0000 −1.28745
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −19.0000 −0.736788
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 65.3835i − 2.52410i
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 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
$$685$$ −23.0000 −0.878785
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 39.2301i − 1.49238i −0.665731 0.746191i $$-0.731880\pi$$
0.665731 0.746191i $$-0.268120\pi$$
$$692$$ 0 0
$$693$$ 57.0000 2.16525
$$694$$ 0 0
$$695$$ 21.7945i 0.826712i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −50.0000 −1.88847 −0.944237 0.329267i $$-0.893198\pi$$
−0.944237 + 0.329267i $$0.893198\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 43.5890i 1.63933i
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 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
$$719$$ − 21.7945i − 0.812798i −0.913696 0.406399i $$-0.866784\pi$$
0.913696 0.406399i $$-0.133216\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 39.2301i − 1.45496i −0.686127 0.727482i $$-0.740691\pi$$
0.686127 0.727482i $$-0.259309\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 91.5369i 3.38561i
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 30.5123i 1.12241i 0.827676 + 0.561206i $$0.189663\pi$$
−0.827676 + 0.561206i $$0.810337\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −11.0000 −0.403009
$$746$$ 0 0
$$747$$ − 26.1534i − 0.956903i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 27.0000 0.981332 0.490666 0.871348i $$-0.336754\pi$$
0.490666 + 0.871348i $$0.336754\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 55.0000 1.99375 0.996874 0.0790050i $$-0.0251743\pi$$
0.996874 + 0.0790050i $$0.0251743\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 21.0000 0.759257
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 51.0000 1.83911 0.919554 0.392965i $$-0.128551\pi$$
0.919554 + 0.392965i $$0.128551\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 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
$$785$$ 18.0000 0.642448
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ − 30.5123i − 1.07945i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 47.9479i − 1.69204i
$$804$$ 0 0
$$805$$ 38.0000 1.33932
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −5.00000 −0.175791 −0.0878953 0.996130i $$-0.528014\pi$$
−0.0878953 + 0.996130i $$0.528014\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 8.71780i − 0.305371i
$$816$$ 0 0
$$817$$ −57.0000 −1.99418
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −53.0000 −1.84971 −0.924856 0.380317i $$-0.875815\pi$$
−0.924856 + 0.380317i $$0.875815\pi$$
$$822$$ 0 0
$$823$$ − 56.6657i − 1.97524i −0.156860 0.987621i $$-0.550137\pi$$
0.156860 0.987621i $$-0.449863\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −84.0000 −2.91043
$$834$$ 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
$$841$$ 29.0000 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −13.0000 −0.447214
$$846$$ 0 0
$$847$$ 34.8712i 1.19819i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ 0 0
$$855$$ 13.0767i 0.447214i
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ − 56.6657i − 1.93341i −0.255897 0.966704i $$-0.582371\pi$$
0.255897 0.966704i $$-0.417629\pi$$
$$860$$ 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
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 39.2301i − 1.32622i
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 35.0000 1.17918 0.589590 0.807703i $$-0.299289\pi$$
0.589590 + 0.807703i $$0.299289\pi$$
$$882$$ 0 0
$$883$$ 30.5123i 1.02682i 0.858143 + 0.513410i $$0.171618\pi$$
−0.858143 + 0.513410i $$0.828382\pi$$
$$884$$ 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
$$891$$ − 39.2301i − 1.31426i
$$892$$ 0 0
$$893$$ 19.0000 0.635811
$$894$$ 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
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ 0 0
$$909$$ 30.0000 0.995037
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 38.0000 1.25762
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −95.0000 −3.13718
$$918$$ 0 0
$$919$$ − 8.71780i − 0.287574i −0.989609 0.143787i $$-0.954072\pi$$
0.989609 0.143787i $$-0.0459280\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 50.0000 1.64045 0.820223 0.572043i $$-0.193849\pi$$
0.820223 + 0.572043i $$0.193849\pi$$
$$930$$ 0 0
$$931$$ − 52.3068i − 1.71429i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 30.5123i 0.997859i
$$936$$ 0 0
$$937$$ 47.0000 1.53542 0.767712 0.640796i $$-0.221395\pi$$
0.767712 + 0.640796i $$0.221395\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 61.0246i − 1.98303i −0.129983 0.991516i $$-0.541492\pi$$
0.129983 0.991516i $$-0.458508\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 21.7945i 0.705253i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 100.255i − 3.23739i
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 61.0246i 1.96242i 0.192947 + 0.981209i $$0.438195\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ −95.0000 −3.04556
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 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
$$985$$ −22.0000 −0.700978
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 114.000 3.62499
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 13.0767i − 0.414559i
$$996$$ 0 0
$$997$$ 63.0000 1.99523 0.997615 0.0690239i $$-0.0219885\pi$$
0.997615 + 0.0690239i $$0.0219885\pi$$
$$998$$ 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 304.2.h.a.303.1 2
3.2 odd 2 2736.2.k.g.2431.1 2
4.3 odd 2 inner 304.2.h.a.303.2 yes 2
8.3 odd 2 1216.2.h.a.1215.2 2
8.5 even 2 1216.2.h.a.1215.1 2
12.11 even 2 2736.2.k.g.2431.2 2
19.18 odd 2 CM 304.2.h.a.303.1 2
57.56 even 2 2736.2.k.g.2431.1 2
76.75 even 2 inner 304.2.h.a.303.2 yes 2
152.37 odd 2 1216.2.h.a.1215.1 2
152.75 even 2 1216.2.h.a.1215.2 2
228.227 odd 2 2736.2.k.g.2431.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
304.2.h.a.303.1 2 1.1 even 1 trivial
304.2.h.a.303.1 2 19.18 odd 2 CM
304.2.h.a.303.2 yes 2 4.3 odd 2 inner
304.2.h.a.303.2 yes 2 76.75 even 2 inner
1216.2.h.a.1215.1 2 8.5 even 2
1216.2.h.a.1215.1 2 152.37 odd 2
1216.2.h.a.1215.2 2 8.3 odd 2
1216.2.h.a.1215.2 2 152.75 even 2
2736.2.k.g.2431.1 2 3.2 odd 2
2736.2.k.g.2431.1 2 57.56 even 2
2736.2.k.g.2431.2 2 12.11 even 2
2736.2.k.g.2431.2 2 228.227 odd 2