# Properties

 Label 6552.2.a.bs.1.4 Level $6552$ Weight $2$ Character 6552.1 Self dual yes Analytic conductor $52.318$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6552,2,Mod(1,6552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6552, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("6552.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6552.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: $$52.3179834043$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.138892.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 10x^{2} + 2x + 12$$ x^4 - x^3 - 10*x^2 + 2*x + 12 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2184) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$1.21773$$ of defining polynomial Character $$\chi$$ $$=$$ 6552.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.70948 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q+3.70948 q^{5} -1.00000 q^{7} +4.51714 q^{11} +1.00000 q^{13} +4.95259 q^{17} -3.27403 q^{19} +8.14494 q^{23} +8.76025 q^{25} +1.27403 q^{29} +10.2266 q^{31} -3.70948 q^{35} -4.95259 q^{37} +0.435456 q^{41} -8.14494 q^{43} -5.24311 q^{47} +1.00000 q^{49} +1.74039 q^{53} +16.7562 q^{55} -11.4697 q^{59} -2.56791 q^{61} +3.70948 q^{65} -12.2899 q^{67} +11.4697 q^{71} +14.1449 q^{73} -4.51714 q^{77} -3.67857 q^{79} -10.1758 q^{83} +18.3716 q^{85} -3.24311 q^{89} -1.00000 q^{91} -12.1449 q^{95} +9.74039 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} - 4 q^{7}+O(q^{10})$$ 4 * q - 2 * q^5 - 4 * q^7 $$4 q - 2 q^{5} - 4 q^{7} + 3 q^{11} + 4 q^{13} - 3 q^{17} - 4 q^{19} + 8 q^{23} + 14 q^{25} - 4 q^{29} + 9 q^{31} + 2 q^{35} + 3 q^{37} - 6 q^{41} - 8 q^{43} - 15 q^{47} + 4 q^{49} - 13 q^{53} + 7 q^{55} - 8 q^{59} + 9 q^{61} - 2 q^{65} + 8 q^{71} + 32 q^{73} - 3 q^{77} - q^{79} - 13 q^{83} + 17 q^{85} - 7 q^{89} - 4 q^{91} - 24 q^{95} + 19 q^{97}+O(q^{100})$$ 4 * q - 2 * q^5 - 4 * q^7 + 3 * q^11 + 4 * q^13 - 3 * q^17 - 4 * q^19 + 8 * q^23 + 14 * q^25 - 4 * q^29 + 9 * q^31 + 2 * q^35 + 3 * q^37 - 6 * q^41 - 8 * q^43 - 15 * q^47 + 4 * q^49 - 13 * q^53 + 7 * q^55 - 8 * q^59 + 9 * q^61 - 2 * q^65 + 8 * q^71 + 32 * q^73 - 3 * q^77 - q^79 - 13 * q^83 + 17 * q^85 - 7 * q^89 - 4 * q^91 - 24 * q^95 + 19 * q^97

## 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
$$4$$ 0 0
$$5$$ 3.70948 1.65893 0.829465 0.558558i $$-0.188645\pi$$
0.829465 + 0.558558i $$0.188645\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.51714 1.36197 0.680984 0.732298i $$-0.261552\pi$$
0.680984 + 0.732298i $$0.261552\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.95259 1.20118 0.600590 0.799557i $$-0.294932\pi$$
0.600590 + 0.799557i $$0.294932\pi$$
$$18$$ 0 0
$$19$$ −3.27403 −0.751113 −0.375556 0.926800i $$-0.622548\pi$$
−0.375556 + 0.926800i $$0.622548\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.14494 1.69834 0.849168 0.528122i $$-0.177104\pi$$
0.849168 + 0.528122i $$0.177104\pi$$
$$24$$ 0 0
$$25$$ 8.76025 1.75205
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.27403 0.236581 0.118290 0.992979i $$-0.462259\pi$$
0.118290 + 0.992979i $$0.462259\pi$$
$$30$$ 0 0
$$31$$ 10.2266 1.83676 0.918378 0.395705i $$-0.129500\pi$$
0.918378 + 0.395705i $$0.129500\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.70948 −0.627017
$$36$$ 0 0
$$37$$ −4.95259 −0.814202 −0.407101 0.913383i $$-0.633460\pi$$
−0.407101 + 0.913383i $$0.633460\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0.435456 0.0680068 0.0340034 0.999422i $$-0.489174\pi$$
0.0340034 + 0.999422i $$0.489174\pi$$
$$42$$ 0 0
$$43$$ −8.14494 −1.24209 −0.621046 0.783774i $$-0.713292\pi$$
−0.621046 + 0.783774i $$0.713292\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −5.24311 −0.764787 −0.382393 0.924000i $$-0.624900\pi$$
−0.382393 + 0.924000i $$0.624900\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.74039 0.239061 0.119531 0.992831i $$-0.461861\pi$$
0.119531 + 0.992831i $$0.461861\pi$$
$$54$$ 0 0
$$55$$ 16.7562 2.25941
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −11.4697 −1.49323 −0.746616 0.665255i $$-0.768323\pi$$
−0.746616 + 0.665255i $$0.768323\pi$$
$$60$$ 0 0
$$61$$ −2.56791 −0.328787 −0.164394 0.986395i $$-0.552567\pi$$
−0.164394 + 0.986395i $$0.552567\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 3.70948 0.460105
$$66$$ 0 0
$$67$$ −12.2899 −1.50145 −0.750724 0.660616i $$-0.770295\pi$$
−0.750724 + 0.660616i $$0.770295\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 11.4697 1.36121 0.680603 0.732652i $$-0.261718\pi$$
0.680603 + 0.732652i $$0.261718\pi$$
$$72$$ 0 0
$$73$$ 14.1449 1.65554 0.827770 0.561068i $$-0.189609\pi$$
0.827770 + 0.561068i $$0.189609\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.51714 −0.514776
$$78$$ 0 0
$$79$$ −3.67857 −0.413871 −0.206936 0.978355i $$-0.566349\pi$$
−0.206936 + 0.978355i $$0.566349\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −10.1758 −1.11694 −0.558472 0.829523i $$-0.688612\pi$$
−0.558472 + 0.829523i $$0.688612\pi$$
$$84$$ 0 0
$$85$$ 18.3716 1.99268
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −3.24311 −0.343769 −0.171885 0.985117i $$-0.554986\pi$$
−0.171885 + 0.985117i $$0.554986\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −12.1449 −1.24604
$$96$$ 0 0
$$97$$ 9.74039 0.988987 0.494494 0.869181i $$-0.335354\pi$$
0.494494 + 0.869181i $$0.335354\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −15.8036 −1.57252 −0.786261 0.617895i $$-0.787986\pi$$
−0.786261 + 0.617895i $$0.787986\pi$$
$$102$$ 0 0
$$103$$ −14.3716 −1.41607 −0.708036 0.706177i $$-0.750418\pi$$
−0.708036 + 0.706177i $$0.750418\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −5.90519 −0.570876 −0.285438 0.958397i $$-0.592139\pi$$
−0.285438 + 0.958397i $$0.592139\pi$$
$$108$$ 0 0
$$109$$ 14.4664 1.38563 0.692813 0.721117i $$-0.256371\pi$$
0.692813 + 0.721117i $$0.256371\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.61131 −0.621939 −0.310970 0.950420i $$-0.600654\pi$$
−0.310970 + 0.950420i $$0.600654\pi$$
$$114$$ 0 0
$$115$$ 30.2135 2.81742
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −4.95259 −0.454004
$$120$$ 0 0
$$121$$ 9.40454 0.854958
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 13.9486 1.24760
$$126$$ 0 0
$$127$$ 18.7761 1.66611 0.833055 0.553191i $$-0.186590\pi$$
0.833055 + 0.553191i $$0.186590\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.3716 0.906167 0.453084 0.891468i $$-0.350324\pi$$
0.453084 + 0.891468i $$0.350324\pi$$
$$132$$ 0 0
$$133$$ 3.27403 0.283894
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.2590 1.04735 0.523677 0.851917i $$-0.324560\pi$$
0.523677 + 0.851917i $$0.324560\pi$$
$$138$$ 0 0
$$139$$ −16.3517 −1.38693 −0.693467 0.720489i $$-0.743917\pi$$
−0.693467 + 0.720489i $$0.743917\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.51714 0.377742
$$144$$ 0 0
$$145$$ 4.72597 0.392471
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −14.5040 −1.18821 −0.594107 0.804386i $$-0.702495\pi$$
−0.594107 + 0.804386i $$0.702495\pi$$
$$150$$ 0 0
$$151$$ 1.62844 0.132521 0.0662604 0.997802i $$-0.478893\pi$$
0.0662604 + 0.997802i $$0.478893\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 37.9355 3.04705
$$156$$ 0 0
$$157$$ 3.04741 0.243209 0.121605 0.992579i $$-0.461196\pi$$
0.121605 + 0.992579i $$0.461196\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.14494 −0.641911
$$162$$ 0 0
$$163$$ −13.0343 −1.02092 −0.510462 0.859900i $$-0.670525\pi$$
−0.510462 + 0.859900i $$0.670525\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −3.22325 −0.249423 −0.124711 0.992193i $$-0.539801\pi$$
−0.124711 + 0.992193i $$0.539801\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 23.3242 1.77330 0.886651 0.462439i $$-0.153025\pi$$
0.886651 + 0.462439i $$0.153025\pi$$
$$174$$ 0 0
$$175$$ −8.76025 −0.662213
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4.61131 0.344665 0.172333 0.985039i $$-0.444870\pi$$
0.172333 + 0.985039i $$0.444870\pi$$
$$180$$ 0 0
$$181$$ 20.8379 1.54887 0.774435 0.632653i $$-0.218034\pi$$
0.774435 + 0.632653i $$0.218034\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −18.3716 −1.35070
$$186$$ 0 0
$$187$$ 22.3716 1.63597
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −24.2767 −1.75660 −0.878302 0.478107i $$-0.841323\pi$$
−0.878302 + 0.478107i $$0.841323\pi$$
$$192$$ 0 0
$$193$$ 23.9052 1.72073 0.860367 0.509676i $$-0.170235\pi$$
0.860367 + 0.509676i $$0.170235\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.40247 −0.456157 −0.228079 0.973643i $$-0.573244\pi$$
−0.228079 + 0.973643i $$0.573244\pi$$
$$198$$ 0 0
$$199$$ −23.4058 −1.65920 −0.829598 0.558361i $$-0.811430\pi$$
−0.829598 + 0.558361i $$0.811430\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −1.27403 −0.0894191
$$204$$ 0 0
$$205$$ 1.61531 0.112818
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −14.7892 −1.02299
$$210$$ 0 0
$$211$$ −17.1792 −1.18267 −0.591333 0.806428i $$-0.701398\pi$$
−0.591333 + 0.806428i $$0.701398\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −30.2135 −2.06054
$$216$$ 0 0
$$217$$ −10.2266 −0.694228
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.95259 0.333148
$$222$$ 0 0
$$223$$ 3.67857 0.246335 0.123168 0.992386i $$-0.460695\pi$$
0.123168 + 0.992386i $$0.460695\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 20.5040 1.36090 0.680450 0.732795i $$-0.261785\pi$$
0.680450 + 0.732795i $$0.261785\pi$$
$$228$$ 0 0
$$229$$ −16.8379 −1.11268 −0.556341 0.830954i $$-0.687795\pi$$
−0.556341 + 0.830954i $$0.687795\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 17.6786 1.15816 0.579081 0.815270i $$-0.303412\pi$$
0.579081 + 0.815270i $$0.303412\pi$$
$$234$$ 0 0
$$235$$ −19.4492 −1.26873
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −11.3682 −0.735347 −0.367674 0.929955i $$-0.619846\pi$$
−0.367674 + 0.929955i $$0.619846\pi$$
$$240$$ 0 0
$$241$$ −19.9355 −1.28416 −0.642078 0.766639i $$-0.721927\pi$$
−0.642078 + 0.766639i $$0.721927\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 3.70948 0.236990
$$246$$ 0 0
$$247$$ −3.27403 −0.208321
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.7562 0.805167 0.402583 0.915383i $$-0.368112\pi$$
0.402583 + 0.915383i $$0.368112\pi$$
$$252$$ 0 0
$$253$$ 36.7918 2.31308
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 9.89846 0.617449 0.308724 0.951152i $$-0.400098\pi$$
0.308724 + 0.951152i $$0.400098\pi$$
$$258$$ 0 0
$$259$$ 4.95259 0.307739
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −3.74039 −0.230643 −0.115321 0.993328i $$-0.536790\pi$$
−0.115321 + 0.993328i $$0.536790\pi$$
$$264$$ 0 0
$$265$$ 6.45596 0.396586
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −14.3517 −0.875039 −0.437519 0.899209i $$-0.644143\pi$$
−0.437519 + 0.899209i $$0.644143\pi$$
$$270$$ 0 0
$$271$$ 27.0013 1.64021 0.820106 0.572212i $$-0.193915\pi$$
0.820106 + 0.572212i $$0.193915\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 39.5713 2.38624
$$276$$ 0 0
$$277$$ 0.644291 0.0387117 0.0193559 0.999813i $$-0.493838\pi$$
0.0193559 + 0.999813i $$0.493838\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8.49728 0.506905 0.253453 0.967348i $$-0.418434\pi$$
0.253453 + 0.967348i $$0.418434\pi$$
$$282$$ 0 0
$$283$$ −31.8104 −1.89093 −0.945465 0.325723i $$-0.894392\pi$$
−0.945465 + 0.325723i $$0.894392\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −0.435456 −0.0257041
$$288$$ 0 0
$$289$$ 7.52819 0.442835
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −21.3366 −1.24650 −0.623250 0.782023i $$-0.714188\pi$$
−0.623250 + 0.782023i $$0.714188\pi$$
$$294$$ 0 0
$$295$$ −42.5468 −2.47717
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 8.14494 0.471034
$$300$$ 0 0
$$301$$ 8.14494 0.469466
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −9.52561 −0.545435
$$306$$ 0 0
$$307$$ −6.32816 −0.361167 −0.180584 0.983560i $$-0.557799\pi$$
−0.180584 + 0.983560i $$0.557799\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 24.5151 1.39012 0.695061 0.718951i $$-0.255377\pi$$
0.695061 + 0.718951i $$0.255377\pi$$
$$312$$ 0 0
$$313$$ 14.1634 0.800561 0.400280 0.916393i $$-0.368913\pi$$
0.400280 + 0.916393i $$0.368913\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −3.69105 −0.207310 −0.103655 0.994613i $$-0.533054\pi$$
−0.103655 + 0.994613i $$0.533054\pi$$
$$318$$ 0 0
$$319$$ 5.75495 0.322215
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −16.2149 −0.902222
$$324$$ 0 0
$$325$$ 8.76025 0.485931
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 5.24311 0.289062
$$330$$ 0 0
$$331$$ −0.479496 −0.0263555 −0.0131777 0.999913i $$-0.504195\pi$$
−0.0131777 + 0.999913i $$0.504195\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −45.5891 −2.49080
$$336$$ 0 0
$$337$$ −11.5969 −0.631723 −0.315861 0.948805i $$-0.602293\pi$$
−0.315861 + 0.948805i $$0.602293\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 46.1951 2.50160
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 11.1291 0.597441 0.298720 0.954341i $$-0.403440\pi$$
0.298720 + 0.954341i $$0.403440\pi$$
$$348$$ 0 0
$$349$$ −12.2266 −0.654476 −0.327238 0.944942i $$-0.606118\pi$$
−0.327238 + 0.944942i $$0.606118\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −5.46973 −0.291125 −0.145562 0.989349i $$-0.546499\pi$$
−0.145562 + 0.989349i $$0.546499\pi$$
$$354$$ 0 0
$$355$$ 42.5468 2.25815
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −11.3682 −0.599990 −0.299995 0.953941i $$-0.596985\pi$$
−0.299995 + 0.953941i $$0.596985\pi$$
$$360$$ 0 0
$$361$$ −8.28076 −0.435829
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 52.4704 2.74643
$$366$$ 0 0
$$367$$ 32.1621 1.67885 0.839423 0.543478i $$-0.182893\pi$$
0.839423 + 0.543478i $$0.182893\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1.74039 −0.0903567
$$372$$ 0 0
$$373$$ −0.864181 −0.0447456 −0.0223728 0.999750i $$-0.507122\pi$$
−0.0223728 + 0.999750i $$0.507122\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.27403 0.0656157
$$378$$ 0 0
$$379$$ 20.1621 1.03566 0.517828 0.855485i $$-0.326741\pi$$
0.517828 + 0.855485i $$0.326741\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.77274 0.192778 0.0963889 0.995344i $$-0.469271\pi$$
0.0963889 + 0.995344i $$0.469271\pi$$
$$384$$ 0 0
$$385$$ −16.7562 −0.853977
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ 40.3386 2.04001
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −13.6456 −0.686584
$$396$$ 0 0
$$397$$ 15.5838 0.782126 0.391063 0.920364i $$-0.372107\pi$$
0.391063 + 0.920364i $$0.372107\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 31.9560 1.59580 0.797902 0.602787i $$-0.205943\pi$$
0.797902 + 0.602787i $$0.205943\pi$$
$$402$$ 0 0
$$403$$ 10.2266 0.509424
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −22.3716 −1.10892
$$408$$ 0 0
$$409$$ −20.8564 −1.03128 −0.515640 0.856805i $$-0.672446\pi$$
−0.515640 + 0.856805i $$0.672446\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 11.4697 0.564389
$$414$$ 0 0
$$415$$ −37.7471 −1.85293
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −3.01442 −0.147264 −0.0736320 0.997285i $$-0.523459\pi$$
−0.0736320 + 0.997285i $$0.523459\pi$$
$$420$$ 0 0
$$421$$ 18.3517 0.894407 0.447204 0.894432i $$-0.352420\pi$$
0.447204 + 0.894432i $$0.352420\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 43.3860 2.10453
$$426$$ 0 0
$$427$$ 2.56791 0.124270
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −16.8557 −0.811911 −0.405955 0.913893i $$-0.633061\pi$$
−0.405955 + 0.913893i $$0.633061\pi$$
$$432$$ 0 0
$$433$$ 3.77868 0.181592 0.0907959 0.995870i $$-0.471059\pi$$
0.0907959 + 0.995870i $$0.471059\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −26.6667 −1.27564
$$438$$ 0 0
$$439$$ 25.8921 1.23576 0.617880 0.786272i $$-0.287992\pi$$
0.617880 + 0.786272i $$0.287992\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −26.0988 −1.23999 −0.619996 0.784605i $$-0.712866\pi$$
−0.619996 + 0.784605i $$0.712866\pi$$
$$444$$ 0 0
$$445$$ −12.0303 −0.570289
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 15.3881 0.726207 0.363103 0.931749i $$-0.381717\pi$$
0.363103 + 0.931749i $$0.381717\pi$$
$$450$$ 0 0
$$451$$ 1.96701 0.0926231
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −3.70948 −0.173903
$$456$$ 0 0
$$457$$ −37.3530 −1.74730 −0.873650 0.486556i $$-0.838253\pi$$
−0.873650 + 0.486556i $$0.838253\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −21.1667 −0.985833 −0.492916 0.870077i $$-0.664069\pi$$
−0.492916 + 0.870077i $$0.664069\pi$$
$$462$$ 0 0
$$463$$ −13.6272 −0.633308 −0.316654 0.948541i $$-0.602559\pi$$
−0.316654 + 0.948541i $$0.602559\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.24505 0.288986 0.144493 0.989506i $$-0.453845\pi$$
0.144493 + 0.989506i $$0.453845\pi$$
$$468$$ 0 0
$$469$$ 12.2899 0.567494
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −36.7918 −1.69169
$$474$$ 0 0
$$475$$ −28.6813 −1.31599
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 12.3222 0.563016 0.281508 0.959559i $$-0.409165\pi$$
0.281508 + 0.959559i $$0.409165\pi$$
$$480$$ 0 0
$$481$$ −4.95259 −0.225819
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 36.1318 1.64066
$$486$$ 0 0
$$487$$ −7.25560 −0.328782 −0.164391 0.986395i $$-0.552566\pi$$
−0.164391 + 0.986395i $$0.552566\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 13.9052 0.627532 0.313766 0.949500i $$-0.398409\pi$$
0.313766 + 0.949500i $$0.398409\pi$$
$$492$$ 0 0
$$493$$ 6.30973 0.284176
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −11.4697 −0.514488
$$498$$ 0 0
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −1.35041 −0.0602117 −0.0301058 0.999547i $$-0.509584\pi$$
−0.0301058 + 0.999547i $$0.509584\pi$$
$$504$$ 0 0
$$505$$ −58.6233 −2.60870
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 10.5737 0.468669 0.234335 0.972156i $$-0.424709\pi$$
0.234335 + 0.972156i $$0.424709\pi$$
$$510$$ 0 0
$$511$$ −14.1449 −0.625735
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −53.3110 −2.34916
$$516$$ 0 0
$$517$$ −23.6839 −1.04162
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −25.1778 −1.10306 −0.551529 0.834155i $$-0.685956\pi$$
−0.551529 + 0.834155i $$0.685956\pi$$
$$522$$ 0 0
$$523$$ 20.1621 0.881626 0.440813 0.897599i $$-0.354690\pi$$
0.440813 + 0.897599i $$0.354690\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 50.6483 2.20627
$$528$$ 0 0
$$529$$ 43.3400 1.88435
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0.435456 0.0188617
$$534$$ 0 0
$$535$$ −21.9052 −0.947044
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 4.51714 0.194567
$$540$$ 0 0
$$541$$ −1.01442 −0.0436133 −0.0218066 0.999762i $$-0.506942\pi$$
−0.0218066 + 0.999762i $$0.506942\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 53.6627 2.29866
$$546$$ 0 0
$$547$$ −27.4243 −1.17258 −0.586288 0.810102i $$-0.699411\pi$$
−0.586288 + 0.810102i $$0.699411\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4.17119 −0.177699
$$552$$ 0 0
$$553$$ 3.67857 0.156429
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −7.02322 −0.297584 −0.148792 0.988869i $$-0.547538\pi$$
−0.148792 + 0.988869i $$0.547538\pi$$
$$558$$ 0 0
$$559$$ −8.14494 −0.344494
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 23.3661 0.984764 0.492382 0.870379i $$-0.336126\pi$$
0.492382 + 0.870379i $$0.336126\pi$$
$$564$$ 0 0
$$565$$ −24.5245 −1.03175
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −45.1331 −1.89208 −0.946039 0.324053i $$-0.894954\pi$$
−0.946039 + 0.324053i $$0.894954\pi$$
$$570$$ 0 0
$$571$$ 42.0370 1.75919 0.879597 0.475720i $$-0.157812\pi$$
0.879597 + 0.475720i $$0.157812\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 71.3517 2.97557
$$576$$ 0 0
$$577$$ 16.0330 0.667462 0.333731 0.942668i $$-0.391692\pi$$
0.333731 + 0.942668i $$0.391692\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 10.1758 0.422165
$$582$$ 0 0
$$583$$ 7.86160 0.325594
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −13.6595 −0.563788 −0.281894 0.959446i $$-0.590963\pi$$
−0.281894 + 0.959446i $$0.590963\pi$$
$$588$$ 0 0
$$589$$ −33.4822 −1.37961
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 3.53428 0.145135 0.0725677 0.997363i $$-0.476881\pi$$
0.0725677 + 0.997363i $$0.476881\pi$$
$$594$$ 0 0
$$595$$ −18.3716 −0.753160
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 17.2808 0.706073 0.353036 0.935610i $$-0.385149\pi$$
0.353036 + 0.935610i $$0.385149\pi$$
$$600$$ 0 0
$$601$$ 17.0673 0.696188 0.348094 0.937460i $$-0.386829\pi$$
0.348094 + 0.937460i $$0.386829\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 34.8860 1.41832
$$606$$ 0 0
$$607$$ −16.9593 −0.688358 −0.344179 0.938904i $$-0.611843\pi$$
−0.344179 + 0.938904i $$0.611843\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5.24311 −0.212114
$$612$$ 0 0
$$613$$ −13.7617 −0.555829 −0.277915 0.960606i $$-0.589643\pi$$
−0.277915 + 0.960606i $$0.589643\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 3.76600 0.151614 0.0758068 0.997123i $$-0.475847\pi$$
0.0758068 + 0.997123i $$0.475847\pi$$
$$618$$ 0 0
$$619$$ 24.1752 0.971684 0.485842 0.874047i $$-0.338513\pi$$
0.485842 + 0.874047i $$0.338513\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 3.24311 0.129933
$$624$$ 0 0
$$625$$ 7.94076 0.317630
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −24.5282 −0.978003
$$630$$ 0 0
$$631$$ 7.24376 0.288369 0.144185 0.989551i $$-0.453944\pi$$
0.144185 + 0.989551i $$0.453944\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 69.6496 2.76396
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −36.3887 −1.43727 −0.718634 0.695389i $$-0.755232\pi$$
−0.718634 + 0.695389i $$0.755232\pi$$
$$642$$ 0 0
$$643$$ 18.4731 0.728508 0.364254 0.931300i $$-0.381324\pi$$
0.364254 + 0.931300i $$0.381324\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 11.2912 0.443902 0.221951 0.975058i $$-0.428758\pi$$
0.221951 + 0.975058i $$0.428758\pi$$
$$648$$ 0 0
$$649$$ −51.8104 −2.03374
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18.7931 0.735431 0.367715 0.929938i $$-0.380140\pi$$
0.367715 + 0.929938i $$0.380140\pi$$
$$654$$ 0 0
$$655$$ 38.4731 1.50327
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 7.80508 0.304043 0.152021 0.988377i $$-0.451422\pi$$
0.152021 + 0.988377i $$0.451422\pi$$
$$660$$ 0 0
$$661$$ −25.1991 −0.980130 −0.490065 0.871686i $$-0.663027\pi$$
−0.490065 + 0.871686i $$0.663027\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 12.1449 0.470960
$$666$$ 0 0
$$667$$ 10.3769 0.401794
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −11.5996 −0.447798
$$672$$ 0 0
$$673$$ 17.9553 0.692127 0.346063 0.938211i $$-0.387518\pi$$
0.346063 + 0.938211i $$0.387518\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 2.97245 0.114241 0.0571203 0.998367i $$-0.481808\pi$$
0.0571203 + 0.998367i $$0.481808\pi$$
$$678$$ 0 0
$$679$$ −9.74039 −0.373802
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −8.86884 −0.339357 −0.169678 0.985500i $$-0.554273\pi$$
−0.169678 + 0.985500i $$0.554273\pi$$
$$684$$ 0 0
$$685$$ 45.4744 1.73749
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 1.74039 0.0663037
$$690$$ 0 0
$$691$$ 5.22920 0.198928 0.0994641 0.995041i $$-0.468287\pi$$
0.0994641 + 0.995041i $$0.468287\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −60.6563 −2.30083
$$696$$ 0 0
$$697$$ 2.15664 0.0816884
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −5.41366 −0.204471 −0.102236 0.994760i $$-0.532600\pi$$
−0.102236 + 0.994760i $$0.532600\pi$$
$$702$$ 0 0
$$703$$ 16.2149 0.611557
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 15.8036 0.594357
$$708$$ 0 0
$$709$$ 29.0631 1.09149 0.545744 0.837952i $$-0.316247\pi$$
0.545744 + 0.837952i $$0.316247\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 83.2952 3.11943
$$714$$ 0 0
$$715$$ 16.7562 0.626648
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 27.9987 1.04418 0.522088 0.852892i $$-0.325153\pi$$
0.522088 + 0.852892i $$0.325153\pi$$
$$720$$ 0 0
$$721$$ 14.3716 0.535225
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 11.1608 0.414501
$$726$$ 0 0
$$727$$ −44.4651 −1.64912 −0.824559 0.565776i $$-0.808577\pi$$
−0.824559 + 0.565776i $$0.808577\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −40.3386 −1.49198
$$732$$ 0 0
$$733$$ −29.8406 −1.10219 −0.551095 0.834443i $$-0.685790\pi$$
−0.551095 + 0.834443i $$0.685790\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −55.5151 −2.04492
$$738$$ 0 0
$$739$$ 15.3174 0.563460 0.281730 0.959494i $$-0.409092\pi$$
0.281730 + 0.959494i $$0.409092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −1.27338 −0.0467158 −0.0233579 0.999727i $$-0.507436\pi$$
−0.0233579 + 0.999727i $$0.507436\pi$$
$$744$$ 0 0
$$745$$ −53.8024 −1.97117
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 5.90519 0.215771
$$750$$ 0 0
$$751$$ 5.35571 0.195433 0.0977163 0.995214i $$-0.468846\pi$$
0.0977163 + 0.995214i $$0.468846\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 6.04068 0.219843
$$756$$ 0 0
$$757$$ 38.5232 1.40015 0.700075 0.714069i $$-0.253150\pi$$
0.700075 + 0.714069i $$0.253150\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −15.2418 −0.552516 −0.276258 0.961084i $$-0.589094\pi$$
−0.276258 + 0.961084i $$0.589094\pi$$
$$762$$ 0 0
$$763$$ −14.4664 −0.523718
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −11.4697 −0.414148
$$768$$ 0 0
$$769$$ −4.31503 −0.155604 −0.0778020 0.996969i $$-0.524790\pi$$
−0.0778020 + 0.996969i $$0.524790\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 13.0034 0.467699 0.233849 0.972273i $$-0.424868\pi$$
0.233849 + 0.972273i $$0.424868\pi$$
$$774$$ 0 0
$$775$$ 89.5878 3.21809
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.42569 −0.0510808
$$780$$ 0 0
$$781$$ 51.8104 1.85392
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 11.3043 0.403468
$$786$$ 0 0
$$787$$ −0.268587 −0.00957409 −0.00478704 0.999989i $$-0.501524\pi$$
−0.00478704 + 0.999989i $$0.501524\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.61131 0.235071
$$792$$ 0 0
$$793$$ −2.56791 −0.0911891
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −29.5810 −1.04781 −0.523907 0.851775i $$-0.675526\pi$$
−0.523907 + 0.851775i $$0.675526\pi$$
$$798$$ 0 0
$$799$$ −25.9670 −0.918647
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 63.8946 2.25479
$$804$$ 0 0
$$805$$ −30.2135 −1.06489
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 26.0921 0.917349 0.458675 0.888604i $$-0.348324\pi$$
0.458675 + 0.888604i $$0.348324\pi$$
$$810$$ 0 0
$$811$$ 37.0461 1.30087 0.650433 0.759564i $$-0.274588\pi$$
0.650433 + 0.759564i $$0.274588\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −48.3504 −1.69364
$$816$$ 0 0
$$817$$ 26.6667 0.932951
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 31.3749 1.09499 0.547496 0.836808i $$-0.315581\pi$$
0.547496 + 0.836808i $$0.315581\pi$$
$$822$$ 0 0
$$823$$ 17.8655 0.622751 0.311376 0.950287i $$-0.399210\pi$$
0.311376 + 0.950287i $$0.399210\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6.19428 −0.215396 −0.107698 0.994184i $$-0.534348\pi$$
−0.107698 + 0.994184i $$0.534348\pi$$
$$828$$ 0 0
$$829$$ 0.662720 0.0230172 0.0115086 0.999934i $$-0.496337\pi$$
0.0115086 + 0.999934i $$0.496337\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 4.95259 0.171597
$$834$$ 0 0
$$835$$ −11.9566 −0.413775
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −54.3077 −1.87491 −0.937454 0.348108i $$-0.886824\pi$$
−0.937454 + 0.348108i $$0.886824\pi$$
$$840$$ 0 0
$$841$$ −27.3769 −0.944030
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 3.70948 0.127610
$$846$$ 0 0
$$847$$ −9.40454 −0.323144
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −40.3386 −1.38279
$$852$$ 0 0
$$853$$ −30.0053 −1.02736 −0.513681 0.857981i $$-0.671719\pi$$
−0.513681 + 0.857981i $$0.671719\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −20.7364 −0.708341 −0.354171 0.935181i $$-0.615237\pi$$
−0.354171 + 0.935181i $$0.615237\pi$$
$$858$$ 0 0
$$859$$ −3.68387 −0.125692 −0.0628460 0.998023i $$-0.520018\pi$$
−0.0628460 + 0.998023i $$0.520018\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 16.1113 0.548435 0.274218 0.961668i $$-0.411581\pi$$
0.274218 + 0.961668i $$0.411581\pi$$
$$864$$ 0 0
$$865$$ 86.5205 2.94179
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −16.6166 −0.563680
$$870$$ 0 0
$$871$$ −12.2899 −0.416427
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −13.9486 −0.471548
$$876$$ 0 0
$$877$$ −4.38469 −0.148060 −0.0740301 0.997256i $$-0.523586\pi$$
−0.0740301 + 0.997256i $$0.523586\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −51.4126 −1.73213 −0.866067 0.499929i $$-0.833360\pi$$
−0.866067 + 0.499929i $$0.833360\pi$$
$$882$$ 0 0
$$883$$ −12.6957 −0.427245 −0.213622 0.976916i $$-0.568526\pi$$
−0.213622 + 0.976916i $$0.568526\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −32.7828 −1.10074 −0.550370 0.834921i $$-0.685513\pi$$
−0.550370 + 0.834921i $$0.685513\pi$$
$$888$$ 0 0
$$889$$ −18.7761 −0.629730
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 17.1661 0.574441
$$894$$ 0 0
$$895$$ 17.1056 0.571776
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 13.0290 0.434541
$$900$$ 0 0
$$901$$ 8.61946 0.287156
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 77.2979 2.56947
$$906$$ 0 0
$$907$$ 6.02354 0.200008 0.100004 0.994987i $$-0.468114\pi$$
0.100004 + 0.994987i $$0.468114\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −26.1120 −0.865128 −0.432564 0.901603i $$-0.642391\pi$$
−0.432564 + 0.901603i $$0.642391\pi$$
$$912$$ 0 0
$$913$$ −45.9657 −1.52124
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −10.3716 −0.342499
$$918$$ 0 0
$$919$$ 6.13038 0.202223 0.101111 0.994875i $$-0.467760\pi$$
0.101111 + 0.994875i $$0.467760\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 11.4697 0.377531
$$924$$ 0 0
$$925$$ −43.3860 −1.42652
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 55.2169 1.81161 0.905803 0.423699i $$-0.139268\pi$$
0.905803 + 0.423699i $$0.139268\pi$$
$$930$$ 0 0
$$931$$ −3.27403 −0.107302
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 82.9869 2.71396
$$936$$ 0 0
$$937$$ 48.1951 1.57446 0.787232 0.616657i $$-0.211513\pi$$
0.787232 + 0.616657i $$0.211513\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −50.9520 −1.66099 −0.830493 0.557029i $$-0.811941\pi$$
−0.830493 + 0.557029i $$0.811941\pi$$
$$942$$ 0 0
$$943$$ 3.54676 0.115498
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 3.12987 0.101707 0.0508536 0.998706i $$-0.483806\pi$$
0.0508536 + 0.998706i $$0.483806\pi$$
$$948$$ 0 0
$$949$$ 14.1449 0.459164
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 27.4835 0.890278 0.445139 0.895461i $$-0.353154\pi$$
0.445139 + 0.895461i $$0.353154\pi$$
$$954$$ 0 0
$$955$$ −90.0541 −2.91408
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −12.2590 −0.395863
$$960$$ 0 0
$$961$$ 73.5838 2.37367
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 88.6759 2.85458
$$966$$ 0 0
$$967$$ 42.2450 1.35851 0.679255 0.733903i $$-0.262303\pi$$
0.679255 + 0.733903i $$0.262303\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −53.5493 −1.71848 −0.859240 0.511573i $$-0.829063\pi$$
−0.859240 + 0.511573i $$0.829063\pi$$
$$972$$ 0 0
$$973$$ 16.3517 0.524211
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 15.0982 0.483033 0.241517 0.970397i $$-0.422355\pi$$
0.241517 + 0.970397i $$0.422355\pi$$
$$978$$ 0 0
$$979$$ −14.6496 −0.468203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −22.1230 −0.705614 −0.352807 0.935696i $$-0.614773\pi$$
−0.352807 + 0.935696i $$0.614773\pi$$
$$984$$ 0 0
$$985$$ −23.7498 −0.756733
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −66.3400 −2.10949
$$990$$ 0 0
$$991$$ 55.9737 1.77806 0.889032 0.457844i $$-0.151378\pi$$
0.889032 + 0.457844i $$0.151378\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −86.8235 −2.75249
$$996$$ 0 0
$$997$$ 12.3187 0.390138 0.195069 0.980790i $$-0.437507\pi$$
0.195069 + 0.980790i $$0.437507\pi$$
$$998$$ 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 6552.2.a.bs.1.4 4
3.2 odd 2 2184.2.a.v.1.1 4
12.11 even 2 4368.2.a.bs.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
2184.2.a.v.1.1 4 3.2 odd 2
4368.2.a.bs.1.1 4 12.11 even 2
6552.2.a.bs.1.4 4 1.1 even 1 trivial