# Properties

 Label 768.2.c.g.767.2 Level $768$ Weight $2$ Character 768.767 Analytic conductor $6.133$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(767,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.767");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 192) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 767.2 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 768.767 Dual form 768.2.c.g.767.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205i q^{3} +4.00000i q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{3} +4.00000i q^{7} -3.00000 q^{9} +6.92820 q^{13} +3.46410i q^{19} +6.92820 q^{21} +5.00000 q^{25} +5.19615i q^{27} +4.00000i q^{31} +6.92820 q^{37} -12.0000i q^{39} -10.3923i q^{43} -9.00000 q^{49} +6.00000 q^{57} +6.92820 q^{61} -12.0000i q^{63} -3.46410i q^{67} -10.0000 q^{73} -8.66025i q^{75} +4.00000i q^{79} +9.00000 q^{81} +27.7128i q^{91} +6.92820 q^{93} +14.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^9 $$4 q - 12 q^{9} + 20 q^{25} - 36 q^{49} + 24 q^{57} - 40 q^{73} + 36 q^{81} + 56 q^{97}+O(q^{100})$$ 4 * q - 12 * q^9 + 20 * q^25 - 36 * q^49 + 24 * q^57 - 40 * q^73 + 36 * q^81 + 56 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ − 1.73205i − 1.00000i
$$4$$ 0 0
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 6.92820 1.92154 0.960769 0.277350i $$-0.0894562\pi$$
0.960769 + 0.277350i $$0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 3.46410i 0.794719i 0.917663 + 0.397360i $$0.130073\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 6.92820 1.51186
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ 5.19615i 1.00000i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 4.00000i 0.718421i 0.933257 + 0.359211i $$0.116954\pi$$
−0.933257 + 0.359211i $$0.883046\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.92820 1.13899 0.569495 0.821995i $$-0.307139\pi$$
0.569495 + 0.821995i $$0.307139\pi$$
$$38$$ 0 0
$$39$$ − 12.0000i − 1.92154i
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ − 10.3923i − 1.58481i −0.609994 0.792406i $$-0.708828\pi$$
0.609994 0.792406i $$-0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.00000 0.794719
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 6.92820 0.887066 0.443533 0.896258i $$-0.353725\pi$$
0.443533 + 0.896258i $$0.353725\pi$$
$$62$$ 0 0
$$63$$ − 12.0000i − 1.51186i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 3.46410i − 0.423207i −0.977356 0.211604i $$-0.932131\pi$$
0.977356 0.211604i $$-0.0678686\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ − 8.66025i − 1.00000i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000i 0.450035i 0.974355 + 0.225018i $$0.0722440\pi$$
−0.974355 + 0.225018i $$0.927756\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 27.7128i 2.90509i
$$92$$ 0 0
$$93$$ 6.92820 0.718421
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 20.0000i 1.97066i 0.170664 + 0.985329i $$0.445409\pi$$
−0.170664 + 0.985329i $$0.554591\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −20.7846 −1.99080 −0.995402 0.0957826i $$-0.969465\pi$$
−0.995402 + 0.0957826i $$0.969465\pi$$
$$110$$ 0 0
$$111$$ − 12.0000i − 1.13899i
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −20.7846 −1.92154
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 20.0000i 1.77471i 0.461084 + 0.887357i $$0.347461\pi$$
−0.461084 + 0.887357i $$0.652539\pi$$
$$128$$ 0 0
$$129$$ −18.0000 −1.58481
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −13.8564 −1.20150
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ − 17.3205i − 1.46911i −0.678551 0.734553i $$-0.737392\pi$$
0.678551 0.734553i $$-0.262608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 15.5885i 1.28571i
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 4.00000i 0.325515i 0.986666 + 0.162758i $$0.0520389\pi$$
−0.986666 + 0.162758i $$0.947961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −20.7846 −1.65879 −0.829396 0.558661i $$-0.811315\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 24.2487i − 1.89931i −0.313304 0.949653i $$-0.601436\pi$$
0.313304 0.949653i $$-0.398564\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 35.0000 2.69231
$$170$$ 0 0
$$171$$ − 10.3923i − 0.794719i
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 20.0000i 1.51186i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 6.92820 0.514969 0.257485 0.966282i $$-0.417106\pi$$
0.257485 + 0.966282i $$0.417106\pi$$
$$182$$ 0 0
$$183$$ − 12.0000i − 0.887066i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −20.7846 −1.51186
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ − 28.0000i − 1.98487i −0.122782 0.992434i $$-0.539182\pi$$
0.122782 0.992434i $$-0.460818\pi$$
$$200$$ 0 0
$$201$$ −6.00000 −0.423207
$$202$$ 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
$$211$$ 24.2487i 1.66935i 0.550743 + 0.834675i $$0.314345\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 0 0
$$219$$ 17.3205i 1.17041i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 28.0000i − 1.87502i −0.347960 0.937509i $$-0.613126\pi$$
0.347960 0.937509i $$-0.386874\pi$$
$$224$$ 0 0
$$225$$ −15.0000 −1.00000
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −20.7846 −1.37349 −0.686743 0.726900i $$-0.740960\pi$$
−0.686743 + 0.726900i $$0.740960\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 6.92820 0.450035
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ − 15.5885i − 1.00000i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 24.0000i 1.52708i
$$248$$ 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
$$256$$ 0 0
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 27.7128i 1.72199i
$$260$$ 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
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ − 28.0000i − 1.70088i −0.526073 0.850439i $$-0.676336\pi$$
0.526073 0.850439i $$-0.323664\pi$$
$$272$$ 0 0
$$273$$ 48.0000 2.90509
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −20.7846 −1.24883 −0.624413 0.781094i $$-0.714662\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ 0 0
$$279$$ − 12.0000i − 0.718421i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 10.3923i 0.617758i 0.951101 + 0.308879i $$0.0999539\pi$$
−0.951101 + 0.308879i $$0.900046\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ − 24.2487i − 1.42148i
$$292$$ 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
$$301$$ 41.5692 2.39601
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 31.1769i − 1.77936i −0.456584 0.889680i $$-0.650927\pi$$
0.456584 0.889680i $$-0.349073\pi$$
$$308$$ 0 0
$$309$$ 34.6410 1.97066
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 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
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 34.6410 1.92154
$$326$$ 0 0
$$327$$ 36.0000i 1.99080i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 17.3205i − 0.952021i −0.879440 0.476011i $$-0.842082\pi$$
0.879440 0.476011i $$-0.157918\pi$$
$$332$$ 0 0
$$333$$ −20.7846 −1.13899
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 8.00000i − 0.431959i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 34.6410 1.85429 0.927146 0.374701i $$-0.122255\pi$$
0.927146 + 0.374701i $$0.122255\pi$$
$$350$$ 0 0
$$351$$ 36.0000i 1.92154i
$$352$$ 0 0
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 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
$$361$$ 7.00000 0.368421
$$362$$ 0 0
$$363$$ 19.0526i 1.00000i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 4.00000i 0.208798i 0.994535 + 0.104399i $$0.0332919\pi$$
−0.994535 + 0.104399i $$0.966708\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6.92820 0.358729 0.179364 0.983783i $$-0.442596\pi$$
0.179364 + 0.983783i $$0.442596\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 38.1051i − 1.95733i −0.205466 0.978664i $$-0.565871\pi$$
0.205466 0.978664i $$-0.434129\pi$$
$$380$$ 0 0
$$381$$ 34.6410 1.77471
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 31.1769i 1.58481i
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −20.7846 −1.04315 −0.521575 0.853206i $$-0.674655\pi$$
−0.521575 + 0.853206i $$0.674655\pi$$
$$398$$ 0 0
$$399$$ 24.0000i 1.20150i
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 27.7128i 1.38047i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −30.0000 −1.46911
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 34.6410 1.68830 0.844150 0.536107i $$-0.180106\pi$$
0.844150 + 0.536107i $$0.180106\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 27.7128i 1.34112i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ − 28.0000i − 1.33637i −0.743996 0.668184i $$-0.767072\pi$$
0.743996 0.668184i $$-0.232928\pi$$
$$440$$ 0 0
$$441$$ 27.0000 1.28571
$$442$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 6.92820 0.325515
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 20.0000i 0.929479i 0.885448 + 0.464739i $$0.153852\pi$$
−0.885448 + 0.464739i $$0.846148\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 13.8564 0.639829
$$470$$ 0 0
$$471$$ 36.0000i 1.65879i
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 17.3205i 0.794719i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 48.0000 2.18861
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 44.0000i − 1.99383i −0.0784867 0.996915i $$-0.525009\pi$$
0.0784867 0.996915i $$-0.474991\pi$$
$$488$$ 0 0
$$489$$ −42.0000 −1.89931
$$490$$ 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
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 31.1769i − 1.39567i −0.716258 0.697835i $$-0.754147\pi$$
0.716258 0.697835i $$-0.245853\pi$$
$$500$$ 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
$$507$$ − 60.6218i − 2.69231i
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ − 40.0000i − 1.76950i
$$512$$ 0 0
$$513$$ −18.0000 −0.794719
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 45.0333i 1.96917i 0.174908 + 0.984585i $$0.444037\pi$$
−0.174908 + 0.984585i $$0.555963\pi$$
$$524$$ 0 0
$$525$$ 34.6410 1.51186
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −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
$$541$$ 6.92820 0.297867 0.148933 0.988847i $$-0.452416\pi$$
0.148933 + 0.988847i $$0.452416\pi$$
$$542$$ 0 0
$$543$$ − 12.0000i − 0.514969i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 24.2487i − 1.03680i −0.855138 0.518400i $$-0.826528\pi$$
0.855138 0.518400i $$-0.173472\pi$$
$$548$$ 0 0
$$549$$ −20.7846 −0.887066
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ − 72.0000i − 3.04528i
$$560$$ 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
$$567$$ 36.0000i 1.51186i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ − 45.0333i − 1.88459i −0.334790 0.942293i $$-0.608665\pi$$
0.334790 0.942293i $$-0.391335\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 46.0000 1.91501 0.957503 0.288425i $$-0.0931316\pi$$
0.957503 + 0.288425i $$0.0931316\pi$$
$$578$$ 0 0
$$579$$ − 3.46410i − 0.143963i
$$580$$ 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
$$589$$ −13.8564 −0.570943
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −48.4974 −1.98487
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 10.3923i 0.423207i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 20.0000i 0.811775i 0.913923 + 0.405887i $$0.133038\pi$$
−0.913923 + 0.405887i $$0.866962\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −48.4974 −1.95879 −0.979396 0.201948i $$-0.935273\pi$$
−0.979396 + 0.201948i $$0.935273\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ 38.1051i 1.53157i 0.643094 + 0.765787i $$0.277650\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ 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
$$631$$ − 44.0000i − 1.75161i −0.482663 0.875806i $$-0.660330\pi$$
0.482663 0.875806i $$-0.339670\pi$$
$$632$$ 0 0
$$633$$ 42.0000 1.66935
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −62.3538 −2.47055
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 31.1769i 1.22950i 0.788723 + 0.614749i $$0.210743\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ 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
$$651$$ 27.7128i 1.08615i
$$652$$ 0 0
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 30.0000 1.17041
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 34.6410 1.34738 0.673690 0.739014i $$-0.264708\pi$$
0.673690 + 0.739014i $$0.264708\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −48.4974 −1.87502
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −50.0000 −1.92736 −0.963679 0.267063i $$-0.913947\pi$$
−0.963679 + 0.267063i $$0.913947\pi$$
$$674$$ 0 0
$$675$$ 25.9808i 1.00000i
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 56.0000i 2.14908i
$$680$$ 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
$$687$$ 36.0000i 1.37349i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 51.9615i − 1.97671i −0.152167 0.988355i $$-0.548625\pi$$
0.152167 0.988355i $$-0.451375\pi$$
$$692$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 24.0000i 0.905177i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −48.4974 −1.82136 −0.910679 0.413114i $$-0.864441\pi$$
−0.910679 + 0.413114i $$0.864441\pi$$
$$710$$ 0 0
$$711$$ − 12.0000i − 0.450035i
$$712$$ 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
$$721$$ −80.0000 −2.97936
$$722$$ 0 0
$$723$$ 24.2487i 0.901819i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 44.0000i − 1.63187i −0.578144 0.815935i $$-0.696223\pi$$
0.578144 0.815935i $$-0.303777\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −20.7846 −0.767697 −0.383849 0.923396i $$-0.625402\pi$$
−0.383849 + 0.923396i $$0.625402\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 51.9615i 1.91144i 0.294285 + 0.955718i $$0.404919\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 41.5692 1.52708
$$742$$ 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
$$751$$ 52.0000i 1.89751i 0.316017 + 0.948753i $$0.397654\pi$$
−0.316017 + 0.948753i $$0.602346\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −48.4974 −1.76267 −0.881334 0.472493i $$-0.843354\pi$$
−0.881334 + 0.472493i $$0.843354\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ − 83.1384i − 3.00981i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 20.0000i 0.718421i
$$776$$ 0 0
$$777$$ 48.0000 1.72199
$$778$$ 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
$$787$$ 3.46410i 0.123482i 0.998092 + 0.0617409i $$0.0196653\pi$$
−0.998092 + 0.0617409i $$0.980335\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 48.0000 1.70453
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ − 10.3923i − 0.364923i −0.983213 0.182462i $$-0.941593\pi$$
0.983213 0.182462i $$-0.0584065\pi$$
$$812$$ 0 0
$$813$$ −48.4974 −1.70088
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 36.0000 1.25948
$$818$$ 0 0
$$819$$ − 83.1384i − 2.90509i
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ 52.0000i 1.81261i 0.422628 + 0.906303i $$0.361108\pi$$
−0.422628 + 0.906303i $$0.638892\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ 34.6410 1.20313 0.601566 0.798823i $$-0.294544\pi$$
0.601566 + 0.798823i $$0.294544\pi$$
$$830$$ 0 0
$$831$$ 36.0000i 1.24883i
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −20.7846 −0.718421
$$838$$ 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
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 44.0000i − 1.51186i
$$848$$ 0 0
$$849$$ 18.0000 0.617758
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 6.92820 0.237217 0.118609 0.992941i $$-0.462157\pi$$
0.118609 + 0.992941i $$0.462157\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 17.3205i 0.590968i 0.955348 + 0.295484i $$0.0954809\pi$$
−0.955348 + 0.295484i $$0.904519\pi$$
$$860$$ 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
$$867$$ − 29.4449i − 1.00000i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ − 24.0000i − 0.813209i
$$872$$ 0 0
$$873$$ −42.0000 −1.42148
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −48.4974 −1.63764 −0.818821 0.574049i $$-0.805372\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 58.8897i 1.98180i 0.134611 + 0.990899i $$0.457022\pi$$
−0.134611 + 0.990899i $$0.542978\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ −80.0000 −2.68311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ − 72.0000i − 2.39601i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 45.0333i 1.49531i 0.664089 + 0.747653i $$0.268820\pi$$
−0.664089 + 0.747653i $$0.731180\pi$$
$$908$$ 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
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 52.0000i 1.71532i 0.514216 + 0.857661i $$0.328083\pi$$
−0.514216 + 0.857661i $$0.671917\pi$$
$$920$$ 0 0
$$921$$ −54.0000 −1.77936
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 34.6410 1.13899
$$926$$ 0 0
$$927$$ − 60.0000i − 1.97066i
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ − 31.1769i − 1.02178i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 0 0
$$939$$ 38.1051i 1.24351i
$$940$$ 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
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ −69.2820 −2.24899
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 15.0000 0.483871
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 20.0000i 0.643157i 0.946883 + 0.321578i $$0.104213\pi$$
−0.946883 + 0.321578i $$0.895787\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 69.2820 2.22108
$$974$$ 0 0
$$975$$ − 60.0000i − 1.92154i
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 62.3538 1.99080
$$982$$ 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
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ − 44.0000i − 1.39771i −0.715265 0.698853i $$-0.753694\pi$$
0.715265 0.698853i $$-0.246306\pi$$
$$992$$ 0 0
$$993$$ −30.0000 −0.952021
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 62.3538 1.97477 0.987383 0.158352i $$-0.0506179\pi$$
0.987383 + 0.158352i $$0.0506179\pi$$
$$998$$ 0 0
$$999$$ 36.0000i 1.13899i
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 768.2.c.g.767.2 4
3.2 odd 2 CM 768.2.c.g.767.2 4
4.3 odd 2 inner 768.2.c.g.767.3 4
8.3 odd 2 inner 768.2.c.g.767.1 4
8.5 even 2 inner 768.2.c.g.767.4 4
12.11 even 2 inner 768.2.c.g.767.3 4
16.3 odd 4 192.2.f.b.95.2 yes 4
16.5 even 4 192.2.f.b.95.1 4
16.11 odd 4 192.2.f.b.95.4 yes 4
16.13 even 4 192.2.f.b.95.3 yes 4
24.5 odd 2 inner 768.2.c.g.767.4 4
24.11 even 2 inner 768.2.c.g.767.1 4
48.5 odd 4 192.2.f.b.95.1 4
48.11 even 4 192.2.f.b.95.4 yes 4
48.29 odd 4 192.2.f.b.95.3 yes 4
48.35 even 4 192.2.f.b.95.2 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.f.b.95.1 4 16.5 even 4
192.2.f.b.95.1 4 48.5 odd 4
192.2.f.b.95.2 yes 4 16.3 odd 4
192.2.f.b.95.2 yes 4 48.35 even 4
192.2.f.b.95.3 yes 4 16.13 even 4
192.2.f.b.95.3 yes 4 48.29 odd 4
192.2.f.b.95.4 yes 4 16.11 odd 4
192.2.f.b.95.4 yes 4 48.11 even 4
768.2.c.g.767.1 4 8.3 odd 2 inner
768.2.c.g.767.1 4 24.11 even 2 inner
768.2.c.g.767.2 4 1.1 even 1 trivial
768.2.c.g.767.2 4 3.2 odd 2 CM
768.2.c.g.767.3 4 4.3 odd 2 inner
768.2.c.g.767.3 4 12.11 even 2 inner
768.2.c.g.767.4 4 8.5 even 2 inner
768.2.c.g.767.4 4 24.5 odd 2 inner