# Properties

 Label 3800.1.b.d.949.1 Level $3800$ Weight $1$ Character 3800.949 Analytic conductor $1.896$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ CM discriminant -152 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(949,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3800.949");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.b (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: $$1.89644704801$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.419904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 6x^{4} + 9x^{2} + 1$$ x^6 + 6*x^4 + 9*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.8340544000000.1

## Embedding invariants

 Embedding label 949.1 Root $$0.347296i$$ of defining polynomial Character $$\chi$$ $$=$$ 3800.949 Dual form 3800.1.b.d.949.6

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.87939i q^{3} -1.00000 q^{4} -1.87939 q^{6} +1.53209i q^{7} +1.00000i q^{8} -2.53209 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.87939i q^{3} -1.00000 q^{4} -1.87939 q^{6} +1.53209i q^{7} +1.00000i q^{8} -2.53209 q^{9} +1.87939i q^{12} +0.347296i q^{13} +1.53209 q^{14} +1.00000 q^{16} +0.347296i q^{17} +2.53209i q^{18} +1.00000 q^{19} +2.87939 q^{21} +1.87939i q^{23} +1.87939 q^{24} +0.347296 q^{26} +2.87939i q^{27} -1.53209i q^{28} +0.347296 q^{29} -1.00000i q^{32} +0.347296 q^{34} +2.53209 q^{36} +1.00000i q^{37} -1.00000i q^{38} +0.652704 q^{39} -2.87939i q^{42} +1.87939 q^{46} -1.00000i q^{47} -1.87939i q^{48} -1.34730 q^{49} +0.652704 q^{51} -0.347296i q^{52} +1.53209i q^{53} +2.87939 q^{54} -1.53209 q^{56} -1.87939i q^{57} -0.347296i q^{58} -1.87939 q^{59} -3.87939i q^{63} -1.00000 q^{64} -1.53209i q^{67} -0.347296i q^{68} +3.53209 q^{69} -2.53209i q^{72} +1.87939i q^{73} +1.00000 q^{74} -1.00000 q^{76} -0.652704i q^{78} +2.87939 q^{81} -2.87939 q^{84} -0.652704i q^{87} -0.532089 q^{91} -1.87939i q^{92} -1.00000 q^{94} -1.87939 q^{96} +1.34730i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 - 6 * q^9 $$6 q - 6 q^{4} - 6 q^{9} + 6 q^{16} + 6 q^{19} + 6 q^{21} + 6 q^{36} + 6 q^{39} - 6 q^{49} + 6 q^{51} + 6 q^{54} - 6 q^{64} + 12 q^{69} + 6 q^{74} - 6 q^{76} + 6 q^{81} - 6 q^{84} + 6 q^{91} - 6 q^{94}+O(q^{100})$$ 6 * q - 6 * q^4 - 6 * q^9 + 6 * q^16 + 6 * q^19 + 6 * q^21 + 6 * q^36 + 6 * q^39 - 6 * q^49 + 6 * q^51 + 6 * q^54 - 6 * q^64 + 12 * q^69 + 6 * q^74 - 6 * q^76 + 6 * q^81 - 6 * q^84 + 6 * q^91 - 6 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 1.00000i
$$3$$ − 1.87939i − 1.87939i −0.342020 0.939693i $$-0.611111\pi$$
0.342020 0.939693i $$-0.388889\pi$$
$$4$$ −1.00000 −1.00000
$$5$$ 0 0
$$6$$ −1.87939 −1.87939
$$7$$ 1.53209i 1.53209i 0.642788 + 0.766044i $$0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$8$$ 1.00000i 1.00000i
$$9$$ −2.53209 −2.53209
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 1.87939i 1.87939i
$$13$$ 0.347296i 0.347296i 0.984808 + 0.173648i $$0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$14$$ 1.53209 1.53209
$$15$$ 0 0
$$16$$ 1.00000 1.00000
$$17$$ 0.347296i 0.347296i 0.984808 + 0.173648i $$0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$18$$ 2.53209i 2.53209i
$$19$$ 1.00000 1.00000
$$20$$ 0 0
$$21$$ 2.87939 2.87939
$$22$$ 0 0
$$23$$ 1.87939i 1.87939i 0.342020 + 0.939693i $$0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$24$$ 1.87939 1.87939
$$25$$ 0 0
$$26$$ 0.347296 0.347296
$$27$$ 2.87939i 2.87939i
$$28$$ − 1.53209i − 1.53209i
$$29$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ − 1.00000i − 1.00000i
$$33$$ 0 0
$$34$$ 0.347296 0.347296
$$35$$ 0 0
$$36$$ 2.53209 2.53209
$$37$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$38$$ − 1.00000i − 1.00000i
$$39$$ 0.652704 0.652704
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ − 2.87939i − 2.87939i
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 1.87939 1.87939
$$47$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$48$$ − 1.87939i − 1.87939i
$$49$$ −1.34730 −1.34730
$$50$$ 0 0
$$51$$ 0.652704 0.652704
$$52$$ − 0.347296i − 0.347296i
$$53$$ 1.53209i 1.53209i 0.642788 + 0.766044i $$0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$54$$ 2.87939 2.87939
$$55$$ 0 0
$$56$$ −1.53209 −1.53209
$$57$$ − 1.87939i − 1.87939i
$$58$$ − 0.347296i − 0.347296i
$$59$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ − 3.87939i − 3.87939i
$$64$$ −1.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 1.53209i − 1.53209i −0.642788 0.766044i $$-0.722222\pi$$
0.642788 0.766044i $$-0.277778\pi$$
$$68$$ − 0.347296i − 0.347296i
$$69$$ 3.53209 3.53209
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ − 2.53209i − 2.53209i
$$73$$ 1.87939i 1.87939i 0.342020 + 0.939693i $$0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$74$$ 1.00000 1.00000
$$75$$ 0 0
$$76$$ −1.00000 −1.00000
$$77$$ 0 0
$$78$$ − 0.652704i − 0.652704i
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 2.87939 2.87939
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ −2.87939 −2.87939
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 0.652704i − 0.652704i
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ −0.532089 −0.532089
$$92$$ − 1.87939i − 1.87939i
$$93$$ 0 0
$$94$$ −1.00000 −1.00000
$$95$$ 0 0
$$96$$ −1.87939 −1.87939
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 1.34730i 1.34730i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ − 0.652704i − 0.652704i
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −0.347296 −0.347296
$$105$$ 0 0
$$106$$ 1.53209 1.53209
$$107$$ − 0.347296i − 0.347296i −0.984808 0.173648i $$-0.944444\pi$$
0.984808 0.173648i $$-0.0555556\pi$$
$$108$$ − 2.87939i − 2.87939i
$$109$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$110$$ 0 0
$$111$$ 1.87939 1.87939
$$112$$ 1.53209i 1.53209i
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ −1.87939 −1.87939
$$115$$ 0 0
$$116$$ −0.347296 −0.347296
$$117$$ − 0.879385i − 0.879385i
$$118$$ 1.87939i 1.87939i
$$119$$ −0.532089 −0.532089
$$120$$ 0 0
$$121$$ 1.00000 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −3.87939 −3.87939
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 1.00000i 1.00000i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 1.53209i 1.53209i
$$134$$ −1.53209 −1.53209
$$135$$ 0 0
$$136$$ −0.347296 −0.347296
$$137$$ 1.53209i 1.53209i 0.642788 + 0.766044i $$0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$138$$ − 3.53209i − 3.53209i
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ −1.87939 −1.87939
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −2.53209 −2.53209
$$145$$ 0 0
$$146$$ 1.87939 1.87939
$$147$$ 2.53209i 2.53209i
$$148$$ − 1.00000i − 1.00000i
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 1.00000i 1.00000i
$$153$$ − 0.879385i − 0.879385i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −0.652704 −0.652704
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ 0 0
$$159$$ 2.87939 2.87939
$$160$$ 0 0
$$161$$ −2.87939 −2.87939
$$162$$ − 2.87939i − 2.87939i
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 2.87939i 2.87939i
$$169$$ 0.879385 0.879385
$$170$$ 0 0
$$171$$ −2.53209 −2.53209
$$172$$ 0 0
$$173$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$174$$ −0.652704 −0.652704
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3.53209i 3.53209i
$$178$$ 0 0
$$179$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$180$$ 0 0
$$181$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$182$$ 0.532089i 0.532089i
$$183$$ 0 0
$$184$$ −1.87939 −1.87939
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 1.00000i 1.00000i
$$189$$ −4.41147 −4.41147
$$190$$ 0 0
$$191$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$192$$ 1.87939i 1.87939i
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 1.34730 1.34730
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$200$$ 0 0
$$201$$ −2.87939 −2.87939
$$202$$ 0 0
$$203$$ 0.532089i 0.532089i
$$204$$ −0.652704 −0.652704
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 4.75877i − 4.75877i
$$208$$ 0.347296i 0.347296i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$212$$ − 1.53209i − 1.53209i
$$213$$ 0 0
$$214$$ −0.347296 −0.347296
$$215$$ 0 0
$$216$$ −2.87939 −2.87939
$$217$$ 0 0
$$218$$ − 1.53209i − 1.53209i
$$219$$ 3.53209 3.53209
$$220$$ 0 0
$$221$$ −0.120615 −0.120615
$$222$$ − 1.87939i − 1.87939i
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 1.53209 1.53209
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 1.53209i − 1.53209i −0.642788 0.766044i $$-0.722222\pi$$
0.642788 0.766044i $$-0.277778\pi$$
$$228$$ 1.87939i 1.87939i
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0.347296i 0.347296i
$$233$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$234$$ −0.879385 −0.879385
$$235$$ 0 0
$$236$$ 1.87939 1.87939
$$237$$ 0 0
$$238$$ 0.532089i 0.532089i
$$239$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ − 1.00000i − 1.00000i
$$243$$ − 2.53209i − 2.53209i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.347296i 0.347296i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 3.87939i 3.87939i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ −1.53209 −1.53209
$$260$$ 0 0
$$261$$ −0.879385 −0.879385
$$262$$ 0 0
$$263$$ − 2.00000i − 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 1.53209 1.53209
$$267$$ 0 0
$$268$$ 1.53209i 1.53209i
$$269$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$270$$ 0 0
$$271$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$272$$ 0.347296i 0.347296i
$$273$$ 1.00000i 1.00000i
$$274$$ 1.53209 1.53209
$$275$$ 0 0
$$276$$ −3.53209 −3.53209
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 1.87939i 1.87939i
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 2.53209i 2.53209i
$$289$$ 0.879385 0.879385
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 1.87939i − 1.87939i
$$293$$ 0.347296i 0.347296i 0.984808 + 0.173648i $$0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$294$$ 2.53209 2.53209
$$295$$ 0 0
$$296$$ −1.00000 −1.00000
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −0.652704 −0.652704
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 1.00000 1.00000
$$305$$ 0 0
$$306$$ −0.879385 −0.879385
$$307$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$312$$ 0.652704i 0.652704i
$$313$$ − 0.347296i − 0.347296i −0.984808 0.173648i $$-0.944444\pi$$
0.984808 0.173648i $$-0.0555556\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.87939i 1.87939i 0.342020 + 0.939693i $$0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$318$$ − 2.87939i − 2.87939i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −0.652704 −0.652704
$$322$$ 2.87939i 2.87939i
$$323$$ 0.347296i 0.347296i
$$324$$ −2.87939 −2.87939
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 2.87939i − 2.87939i
$$328$$ 0 0
$$329$$ 1.53209 1.53209
$$330$$ 0 0
$$331$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$332$$ 0 0
$$333$$ − 2.53209i − 2.53209i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.87939 2.87939
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ − 0.879385i − 0.879385i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 2.53209i 2.53209i
$$343$$ − 0.532089i − 0.532089i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −1.00000 −1.00000
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0.652704i 0.652704i
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −1.00000
$$352$$ 0 0
$$353$$ − 1.53209i − 1.53209i −0.642788 0.766044i $$-0.722222\pi$$
0.642788 0.766044i $$-0.277778\pi$$
$$354$$ 3.53209 3.53209
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1.00000i 1.00000i
$$358$$ 1.00000i 1.00000i
$$359$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$360$$ 0 0
$$361$$ 1.00000 1.00000
$$362$$ − 1.00000i − 1.00000i
$$363$$ − 1.87939i − 1.87939i
$$364$$ 0.532089 0.532089
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 1.87939i 1.87939i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.34730 −2.34730
$$372$$ 0 0
$$373$$ − 1.87939i − 1.87939i −0.342020 0.939693i $$-0.611111\pi$$
0.342020 0.939693i $$-0.388889\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 1.00000 1.00000
$$377$$ 0.120615i 0.120615i
$$378$$ 4.41147i 4.41147i
$$379$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 1.87939i 1.87939i
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 1.87939 1.87939
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ −0.652704 −0.652704
$$392$$ − 1.34730i − 1.34730i
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ 1.53209i 1.53209i
$$399$$ 2.87939 2.87939
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 2.87939i 2.87939i
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0.532089 0.532089
$$407$$ 0 0
$$408$$ 0.652704i 0.652704i
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 2.87939 2.87939
$$412$$ 0 0
$$413$$ − 2.87939i − 2.87939i
$$414$$ −4.75877 −4.75877
$$415$$ 0 0
$$416$$ 0.347296 0.347296
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$422$$ − 1.87939i − 1.87939i
$$423$$ 2.53209i 2.53209i
$$424$$ −1.53209 −1.53209
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0.347296i 0.347296i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 2.87939i 2.87939i
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −1.53209 −1.53209
$$437$$ 1.87939i 1.87939i
$$438$$ − 3.53209i − 3.53209i
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ 3.41147 3.41147
$$442$$ 0.120615i 0.120615i
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ −1.87939 −1.87939
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ − 1.53209i − 1.53209i
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ −1.53209 −1.53209
$$455$$ 0 0
$$456$$ 1.87939 1.87939
$$457$$ 0.347296i 0.347296i 0.984808 + 0.173648i $$0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −1.00000
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$464$$ 0.347296 0.347296
$$465$$ 0 0
$$466$$ 1.00000 1.00000
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0.879385i 0.879385i
$$469$$ 2.34730 2.34730
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 1.87939i − 1.87939i
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0.532089 0.532089
$$477$$ − 3.87939i − 3.87939i
$$478$$ − 1.87939i − 1.87939i
$$479$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$480$$ 0 0
$$481$$ −0.347296 −0.347296
$$482$$ 0 0
$$483$$ 5.41147i 5.41147i
$$484$$ −1.00000 −1.00000
$$485$$ 0 0
$$486$$ −2.53209 −2.53209
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0.120615i 0.120615i
$$494$$ 0.347296 0.347296
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 0.347296i − 0.347296i −0.984808 0.173648i $$-0.944444\pi$$
0.984808 0.173648i $$-0.0555556\pi$$
$$504$$ 3.87939 3.87939
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 1.65270i − 1.65270i
$$508$$ 0 0
$$509$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$510$$ 0 0
$$511$$ −2.87939 −2.87939
$$512$$ − 1.00000i − 1.00000i
$$513$$ 2.87939i 2.87939i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 1.53209i 1.53209i
$$519$$ −1.87939 −1.87939
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0.879385i 0.879385i
$$523$$ 1.53209i 1.53209i 0.642788 + 0.766044i $$0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −2.00000 −2.00000
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −2.53209 −2.53209
$$530$$ 0 0
$$531$$ 4.75877 4.75877
$$532$$ − 1.53209i − 1.53209i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 1.53209 1.53209
$$537$$ 1.87939i 1.87939i
$$538$$ 1.00000i 1.00000i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 1.87939i 1.87939i
$$543$$ − 1.87939i − 1.87939i
$$544$$ 0.347296 0.347296
$$545$$ 0 0
$$546$$ 1.00000 1.00000
$$547$$ − 2.00000i − 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$548$$ − 1.53209i − 1.53209i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0.347296 0.347296
$$552$$ 3.53209i 3.53209i
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$564$$ 1.87939 1.87939
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 4.41147i 4.41147i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 3.53209i 3.53209i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 2.53209 2.53209
$$577$$ − 1.87939i − 1.87939i −0.342020 0.939693i $$-0.611111\pi$$
0.342020 0.939693i $$-0.388889\pi$$
$$578$$ − 0.879385i − 0.879385i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −1.87939 −1.87939
$$585$$ 0 0
$$586$$ 0.347296 0.347296
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ − 2.53209i − 2.53209i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.00000i 1.00000i
$$593$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 2.87939i 2.87939i
$$598$$ 0.652704i 0.652704i
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 3.87939i 3.87939i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ − 1.00000i − 1.00000i
$$609$$ 1.00000 1.00000
$$610$$ 0 0
$$611$$ 0.347296 0.347296
$$612$$ 0.879385i 0.879385i
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ 1.00000 1.00000
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ −5.41147 −5.41147
$$622$$ − 1.53209i − 1.53209i
$$623$$ 0 0
$$624$$ 0.652704 0.652704
$$625$$ 0 0
$$626$$ −0.347296 −0.347296
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −0.347296 −0.347296
$$630$$ 0 0
$$631$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$632$$ 0 0
$$633$$ − 3.53209i − 3.53209i
$$634$$ 1.87939 1.87939
$$635$$ 0 0
$$636$$ −2.87939 −2.87939
$$637$$ − 0.467911i − 0.467911i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0.652704i 0.652704i
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 2.87939 2.87939
$$645$$ 0 0
$$646$$ 0.347296 0.347296
$$647$$ 1.53209i 1.53209i 0.642788 + 0.766044i $$0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$648$$ 2.87939i 2.87939i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ −2.87939 −2.87939
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 4.75877i − 4.75877i
$$658$$ − 1.53209i − 1.53209i
$$659$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$660$$ 0 0
$$661$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$662$$ 0.347296i 0.347296i
$$663$$ 0.226682i 0.226682i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −2.53209 −2.53209
$$667$$ 0.652704i 0.652704i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 2.87939i − 2.87939i
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −0.879385 −0.879385
$$677$$ − 1.53209i − 1.53209i −0.642788 0.766044i $$-0.722222\pi$$
0.642788 0.766044i $$-0.277778\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −2.87939 −2.87939
$$682$$ 0 0
$$683$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$684$$ 2.53209 2.53209
$$685$$ 0 0
$$686$$ −0.532089 −0.532089
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −0.532089 −0.532089
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 1.00000i 1.00000i
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0.652704 0.652704
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 1.87939 1.87939
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 1.00000i 1.00000i
$$703$$ 1.00000i 1.00000i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −1.53209 −1.53209
$$707$$ 0 0
$$708$$ − 3.53209i − 3.53209i
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 1.00000 1.00000
$$715$$ 0 0
$$716$$ 1.00000 1.00000
$$717$$ − 3.53209i − 3.53209i
$$718$$ 0.347296i 0.347296i
$$719$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 1.00000i − 1.00000i
$$723$$ 0 0
$$724$$ −1.00000 −1.00000
$$725$$ 0 0
$$726$$ −1.87939 −1.87939
$$727$$ − 1.87939i − 1.87939i −0.342020 0.939693i $$-0.611111\pi$$
0.342020 0.939693i $$-0.388889\pi$$
$$728$$ − 0.532089i − 0.532089i
$$729$$ −1.87939 −1.87939
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 2.00000 2.00000
$$735$$ 0 0
$$736$$ 1.87939 1.87939
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0.652704 0.652704
$$742$$ 2.34730i 2.34730i
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −1.87939 −1.87939
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0.532089 0.532089
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ − 1.00000i − 1.00000i
$$753$$ 0 0
$$754$$ 0.120615 0.120615
$$755$$ 0 0
$$756$$ 4.41147 4.41147
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ − 1.53209i − 1.53209i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$762$$ 0 0
$$763$$ 2.34730i 2.34730i
$$764$$ 1.87939 1.87939
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 0.652704i − 0.652704i
$$768$$ − 1.87939i − 1.87939i
$$769$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 1.87939i − 1.87939i −0.342020 0.939693i $$-0.611111\pi$$
0.342020 0.939693i $$-0.388889\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 2.87939i 2.87939i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0.652704i 0.652704i
$$783$$ 1.00000i 1.00000i
$$784$$ −1.34730 −1.34730
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 0.347296i − 0.347296i −0.984808 0.173648i $$-0.944444\pi$$
0.984808 0.173648i $$-0.0555556\pi$$
$$788$$ 0 0
$$789$$ −3.75877 −3.75877
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 1.53209 1.53209
$$797$$ − 0.347296i − 0.347296i −0.984808 0.173648i $$-0.944444\pi$$
0.984808 0.173648i $$-0.0555556\pi$$
$$798$$ − 2.87939i − 2.87939i
$$799$$ 0.347296 0.347296
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 2.87939 2.87939
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.87939i 1.87939i
$$808$$ 0 0
$$809$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$810$$ 0 0
$$811$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$812$$ − 0.532089i − 0.532089i
$$813$$ 3.53209i 3.53209i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0.652704 0.652704
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 1.34730 1.34730
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ − 2.87939i − 2.87939i
$$823$$ 1.87939i 1.87939i 0.342020 + 0.939693i $$0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −2.87939 −2.87939
$$827$$ − 0.347296i − 0.347296i −0.984808 0.173648i $$-0.944444\pi$$
0.984808 0.173648i $$-0.0555556\pi$$
$$828$$ 4.75877i 4.75877i
$$829$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 0.347296i − 0.347296i
$$833$$ − 0.467911i − 0.467911i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ −0.879385 −0.879385
$$842$$ − 1.87939i − 1.87939i
$$843$$ 0 0
$$844$$ −1.87939 −1.87939
$$845$$ 0 0
$$846$$ 2.53209 2.53209
$$847$$ 1.53209i 1.53209i
$$848$$ 1.53209i 1.53209i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.87939 −1.87939
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0.347296 0.347296
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 2.87939 2.87939
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 1.65270i − 1.65270i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0.532089 0.532089
$$872$$ 1.53209i 1.53209i
$$873$$ 0 0
$$874$$ 1.87939 1.87939
$$875$$ 0 0
$$876$$ −3.53209 −3.53209
$$877$$ 1.87939i 1.87939i 0.342020 + 0.939693i $$0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$878$$ 0 0
$$879$$ 0.652704 0.652704
$$880$$ 0 0
$$881$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$882$$ − 3.41147i − 3.41147i
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0.120615 0.120615
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 1.87939i 1.87939i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 1.00000i − 1.00000i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.53209 −1.53209
$$897$$ 1.22668i 1.22668i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −0.532089 −0.532089
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 1.53209i − 1.53209i −0.642788 0.766044i $$-0.722222\pi$$
0.642788 0.766044i $$-0.277778\pi$$
$$908$$ 1.53209i 1.53209i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ − 1.87939i − 1.87939i
$$913$$ 0 0
$$914$$ 0.347296 0.347296
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 1.00000i 1.00000i
$$919$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$920$$ 0 0
$$921$$ 1.87939 1.87939
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 1.00000 1.00000
$$927$$ 0 0
$$928$$ − 0.347296i − 0.347296i
$$929$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$930$$ 0 0
$$931$$ −1.34730 −1.34730
$$932$$ − 1.00000i − 1.00000i
$$933$$ − 2.87939i − 2.87939i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0.879385 0.879385
$$937$$ 0.347296i 0.347296i 0.984808 + 0.173648i $$0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$938$$ − 2.34730i − 2.34730i
$$939$$ −0.652704 −0.652704
$$940$$ 0 0
$$941$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −1.87939 −1.87939
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ −0.652704 −0.652704
$$950$$ 0 0
$$951$$ 3.53209 3.53209
$$952$$ − 0.532089i − 0.532089i
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ −3.87939 −3.87939
$$955$$ 0 0
$$956$$ −1.87939 −1.87939
$$957$$ 0 0
$$958$$ − 1.00000i − 1.00000i
$$959$$ −2.34730 −2.34730
$$960$$ 0 0
$$961$$ 1.00000 1.00000
$$962$$ 0.347296i 0.347296i
$$963$$ 0.879385i 0.879385i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 5.41147 5.41147
$$967$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 1.00000i 1.00000i
$$969$$ 0.652704 0.652704
$$970$$ 0 0
$$971$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$972$$ 2.53209i 2.53209i
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −3.87939 −3.87939
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0.120615 0.120615
$$987$$ − 2.87939i − 2.87939i
$$988$$ − 0.347296i − 0.347296i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0.652704i 0.652704i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ 0 0
$$999$$ −2.87939 −2.87939
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 3800.1.b.d.949.1 6
5.2 odd 4 3800.1.o.e.1101.1 yes 3
5.3 odd 4 3800.1.o.c.1101.3 3
5.4 even 2 inner 3800.1.b.d.949.6 6
8.5 even 2 3800.1.b.c.949.6 6
19.18 odd 2 3800.1.b.c.949.6 6
40.13 odd 4 3800.1.o.f.1101.1 yes 3
40.29 even 2 3800.1.b.c.949.1 6
40.37 odd 4 3800.1.o.d.1101.3 yes 3
95.18 even 4 3800.1.o.f.1101.1 yes 3
95.37 even 4 3800.1.o.d.1101.3 yes 3
95.94 odd 2 3800.1.b.c.949.1 6
152.37 odd 2 CM 3800.1.b.d.949.1 6
760.37 even 4 3800.1.o.e.1101.1 yes 3
760.189 odd 2 inner 3800.1.b.d.949.6 6
760.493 even 4 3800.1.o.c.1101.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
3800.1.b.c.949.1 6 40.29 even 2
3800.1.b.c.949.1 6 95.94 odd 2
3800.1.b.c.949.6 6 8.5 even 2
3800.1.b.c.949.6 6 19.18 odd 2
3800.1.b.d.949.1 6 1.1 even 1 trivial
3800.1.b.d.949.1 6 152.37 odd 2 CM
3800.1.b.d.949.6 6 5.4 even 2 inner
3800.1.b.d.949.6 6 760.189 odd 2 inner
3800.1.o.c.1101.3 3 5.3 odd 4
3800.1.o.c.1101.3 3 760.493 even 4
3800.1.o.d.1101.3 yes 3 40.37 odd 4
3800.1.o.d.1101.3 yes 3 95.37 even 4
3800.1.o.e.1101.1 yes 3 5.2 odd 4
3800.1.o.e.1101.1 yes 3 760.37 even 4
3800.1.o.f.1101.1 yes 3 40.13 odd 4
3800.1.o.f.1101.1 yes 3 95.18 even 4