# Properties

 Label 4368.2.a.bs.1.1 Level $4368$ Weight $2$ Character 4368.1 Self dual yes Analytic conductor $34.879$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.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: $$34.8786556029$$ 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 1.1 Root $$1.21773$$ of defining polynomial Character $$\chi$$ $$=$$ 4368.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -3.70948 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -3.70948 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.51714 q^{11} +1.00000 q^{13} -3.70948 q^{15} -4.95259 q^{17} +3.27403 q^{19} +1.00000 q^{21} +8.14494 q^{23} +8.76025 q^{25} +1.00000 q^{27} -1.27403 q^{29} -10.2266 q^{31} +4.51714 q^{33} -3.70948 q^{35} -4.95259 q^{37} +1.00000 q^{39} -0.435456 q^{41} +8.14494 q^{43} -3.70948 q^{45} -5.24311 q^{47} +1.00000 q^{49} -4.95259 q^{51} -1.74039 q^{53} -16.7562 q^{55} +3.27403 q^{57} -11.4697 q^{59} -2.56791 q^{61} +1.00000 q^{63} -3.70948 q^{65} +12.2899 q^{67} +8.14494 q^{69} +11.4697 q^{71} +14.1449 q^{73} +8.76025 q^{75} +4.51714 q^{77} +3.67857 q^{79} +1.00000 q^{81} -10.1758 q^{83} +18.3716 q^{85} -1.27403 q^{87} +3.24311 q^{89} +1.00000 q^{91} -10.2266 q^{93} -12.1449 q^{95} +9.74039 q^{97} +4.51714 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 $$4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 14 * q^25 + 4 * q^27 + 4 * q^29 - 9 * q^31 + 3 * q^33 + 2 * q^35 + 3 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 15 * q^47 + 4 * q^49 + 3 * q^51 + 13 * q^53 - 7 * q^55 + 4 * q^57 - 8 * q^59 + 9 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^69 + 8 * q^71 + 32 * q^73 + 14 * q^75 + 3 * q^77 + q^79 + 4 * q^81 - 13 * q^83 + 17 * q^85 + 4 * q^87 + 7 * q^89 + 4 * q^91 - 9 * q^93 - 24 * q^95 + 19 * q^97 + 3 * q^99

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −3.70948 −1.65893 −0.829465 0.558558i $$-0.811355\pi$$
−0.829465 + 0.558558i $$0.811355\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 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
$$15$$ −3.70948 −0.957784
$$16$$ 0 0
$$17$$ −4.95259 −1.20118 −0.600590 0.799557i $$-0.705068\pi$$
−0.600590 + 0.799557i $$0.705068\pi$$
$$18$$ 0 0
$$19$$ 3.27403 0.751113 0.375556 0.926800i $$-0.377452\pi$$
0.375556 + 0.926800i $$0.377452\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 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
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −1.27403 −0.236581 −0.118290 0.992979i $$-0.537741\pi$$
−0.118290 + 0.992979i $$0.537741\pi$$
$$30$$ 0 0
$$31$$ −10.2266 −1.83676 −0.918378 0.395705i $$-0.870500\pi$$
−0.918378 + 0.395705i $$0.870500\pi$$
$$32$$ 0 0
$$33$$ 4.51714 0.786333
$$34$$ 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
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −0.435456 −0.0680068 −0.0340034 0.999422i $$-0.510826\pi$$
−0.0340034 + 0.999422i $$0.510826\pi$$
$$42$$ 0 0
$$43$$ 8.14494 1.24209 0.621046 0.783774i $$-0.286708\pi$$
0.621046 + 0.783774i $$0.286708\pi$$
$$44$$ 0 0
$$45$$ −3.70948 −0.552977
$$46$$ 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
$$51$$ −4.95259 −0.693502
$$52$$ 0 0
$$53$$ −1.74039 −0.239061 −0.119531 0.992831i $$-0.538139\pi$$
−0.119531 + 0.992831i $$0.538139\pi$$
$$54$$ 0 0
$$55$$ −16.7562 −2.25941
$$56$$ 0 0
$$57$$ 3.27403 0.433655
$$58$$ 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
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −3.70948 −0.460105
$$66$$ 0 0
$$67$$ 12.2899 1.50145 0.750724 0.660616i $$-0.229705\pi$$
0.750724 + 0.660616i $$0.229705\pi$$
$$68$$ 0 0
$$69$$ 8.14494 0.980535
$$70$$ 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
$$75$$ 8.76025 1.01155
$$76$$ 0 0
$$77$$ 4.51714 0.514776
$$78$$ 0 0
$$79$$ 3.67857 0.413871 0.206936 0.978355i $$-0.433651\pi$$
0.206936 + 0.978355i $$0.433651\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 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
$$87$$ −1.27403 −0.136590
$$88$$ 0 0
$$89$$ 3.24311 0.343769 0.171885 0.985117i $$-0.445014\pi$$
0.171885 + 0.985117i $$0.445014\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 0 0
$$93$$ −10.2266 −1.06045
$$94$$ 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
$$99$$ 4.51714 0.453990
$$100$$ 0 0
$$101$$ 15.8036 1.57252 0.786261 0.617895i $$-0.212014\pi$$
0.786261 + 0.617895i $$0.212014\pi$$
$$102$$ 0 0
$$103$$ 14.3716 1.41607 0.708036 0.706177i $$-0.249582\pi$$
0.708036 + 0.706177i $$0.249582\pi$$
$$104$$ 0 0
$$105$$ −3.70948 −0.362008
$$106$$ 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
$$111$$ −4.95259 −0.470079
$$112$$ 0 0
$$113$$ 6.61131 0.621939 0.310970 0.950420i $$-0.399346\pi$$
0.310970 + 0.950420i $$0.399346\pi$$
$$114$$ 0 0
$$115$$ −30.2135 −2.81742
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −4.95259 −0.454004
$$120$$ 0 0
$$121$$ 9.40454 0.854958
$$122$$ 0 0
$$123$$ −0.435456 −0.0392637
$$124$$ 0 0
$$125$$ −13.9486 −1.24760
$$126$$ 0 0
$$127$$ −18.7761 −1.66611 −0.833055 0.553191i $$-0.813410\pi$$
−0.833055 + 0.553191i $$0.813410\pi$$
$$128$$ 0 0
$$129$$ 8.14494 0.717122
$$130$$ 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
$$135$$ −3.70948 −0.319261
$$136$$ 0 0
$$137$$ −12.2590 −1.04735 −0.523677 0.851917i $$-0.675440\pi$$
−0.523677 + 0.851917i $$0.675440\pi$$
$$138$$ 0 0
$$139$$ 16.3517 1.38693 0.693467 0.720489i $$-0.256083\pi$$
0.693467 + 0.720489i $$0.256083\pi$$
$$140$$ 0 0
$$141$$ −5.24311 −0.441550
$$142$$ 0 0
$$143$$ 4.51714 0.377742
$$144$$ 0 0
$$145$$ 4.72597 0.392471
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ 14.5040 1.18821 0.594107 0.804386i $$-0.297505\pi$$
0.594107 + 0.804386i $$0.297505\pi$$
$$150$$ 0 0
$$151$$ −1.62844 −0.132521 −0.0662604 0.997802i $$-0.521107\pi$$
−0.0662604 + 0.997802i $$0.521107\pi$$
$$152$$ 0 0
$$153$$ −4.95259 −0.400394
$$154$$ 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
$$159$$ −1.74039 −0.138022
$$160$$ 0 0
$$161$$ 8.14494 0.641911
$$162$$ 0 0
$$163$$ 13.0343 1.02092 0.510462 0.859900i $$-0.329475\pi$$
0.510462 + 0.859900i $$0.329475\pi$$
$$164$$ 0 0
$$165$$ −16.7562 −1.30447
$$166$$ 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
$$171$$ 3.27403 0.250371
$$172$$ 0 0
$$173$$ −23.3242 −1.77330 −0.886651 0.462439i $$-0.846975\pi$$
−0.886651 + 0.462439i $$0.846975\pi$$
$$174$$ 0 0
$$175$$ 8.76025 0.662213
$$176$$ 0 0
$$177$$ −11.4697 −0.862118
$$178$$ 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
$$183$$ −2.56791 −0.189825
$$184$$ 0 0
$$185$$ 18.3716 1.35070
$$186$$ 0 0
$$187$$ −22.3716 −1.63597
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 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
$$195$$ −3.70948 −0.265641
$$196$$ 0 0
$$197$$ 6.40247 0.456157 0.228079 0.973643i $$-0.426756\pi$$
0.228079 + 0.973643i $$0.426756\pi$$
$$198$$ 0 0
$$199$$ 23.4058 1.65920 0.829598 0.558361i $$-0.188570\pi$$
0.829598 + 0.558361i $$0.188570\pi$$
$$200$$ 0 0
$$201$$ 12.2899 0.866861
$$202$$ 0 0
$$203$$ −1.27403 −0.0894191
$$204$$ 0 0
$$205$$ 1.61531 0.112818
$$206$$ 0 0
$$207$$ 8.14494 0.566112
$$208$$ 0 0
$$209$$ 14.7892 1.02299
$$210$$ 0 0
$$211$$ 17.1792 1.18267 0.591333 0.806428i $$-0.298602\pi$$
0.591333 + 0.806428i $$0.298602\pi$$
$$212$$ 0 0
$$213$$ 11.4697 0.785893
$$214$$ 0 0
$$215$$ −30.2135 −2.06054
$$216$$ 0 0
$$217$$ −10.2266 −0.694228
$$218$$ 0 0
$$219$$ 14.1449 0.955826
$$220$$ 0 0
$$221$$ −4.95259 −0.333148
$$222$$ 0 0
$$223$$ −3.67857 −0.246335 −0.123168 0.992386i $$-0.539305\pi$$
−0.123168 + 0.992386i $$0.539305\pi$$
$$224$$ 0 0
$$225$$ 8.76025 0.584017
$$226$$ 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
$$231$$ 4.51714 0.297206
$$232$$ 0 0
$$233$$ −17.6786 −1.15816 −0.579081 0.815270i $$-0.696588\pi$$
−0.579081 + 0.815270i $$0.696588\pi$$
$$234$$ 0 0
$$235$$ 19.4492 1.26873
$$236$$ 0 0
$$237$$ 3.67857 0.238949
$$238$$ 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
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −3.70948 −0.236990
$$246$$ 0 0
$$247$$ 3.27403 0.208321
$$248$$ 0 0
$$249$$ −10.1758 −0.644868
$$250$$ 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
$$255$$ 18.3716 1.15047
$$256$$ 0 0
$$257$$ −9.89846 −0.617449 −0.308724 0.951152i $$-0.599902\pi$$
−0.308724 + 0.951152i $$0.599902\pi$$
$$258$$ 0 0
$$259$$ −4.95259 −0.307739
$$260$$ 0 0
$$261$$ −1.27403 −0.0788602
$$262$$ 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
$$267$$ 3.24311 0.198475
$$268$$ 0 0
$$269$$ 14.3517 0.875039 0.437519 0.899209i $$-0.355857\pi$$
0.437519 + 0.899209i $$0.355857\pi$$
$$270$$ 0 0
$$271$$ −27.0013 −1.64021 −0.820106 0.572212i $$-0.806085\pi$$
−0.820106 + 0.572212i $$0.806085\pi$$
$$272$$ 0 0
$$273$$ 1.00000 0.0605228
$$274$$ 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
$$279$$ −10.2266 −0.612252
$$280$$ 0 0
$$281$$ −8.49728 −0.506905 −0.253453 0.967348i $$-0.581566\pi$$
−0.253453 + 0.967348i $$0.581566\pi$$
$$282$$ 0 0
$$283$$ 31.8104 1.89093 0.945465 0.325723i $$-0.105608\pi$$
0.945465 + 0.325723i $$0.105608\pi$$
$$284$$ 0 0
$$285$$ −12.1449 −0.719404
$$286$$ 0 0
$$287$$ −0.435456 −0.0257041
$$288$$ 0 0
$$289$$ 7.52819 0.442835
$$290$$ 0 0
$$291$$ 9.74039 0.570992
$$292$$ 0 0
$$293$$ 21.3366 1.24650 0.623250 0.782023i $$-0.285812\pi$$
0.623250 + 0.782023i $$0.285812\pi$$
$$294$$ 0 0
$$295$$ 42.5468 2.47717
$$296$$ 0 0
$$297$$ 4.51714 0.262111
$$298$$ 0 0
$$299$$ 8.14494 0.471034
$$300$$ 0 0
$$301$$ 8.14494 0.469466
$$302$$ 0 0
$$303$$ 15.8036 0.907896
$$304$$ 0 0
$$305$$ 9.52561 0.545435
$$306$$ 0 0
$$307$$ 6.32816 0.361167 0.180584 0.983560i $$-0.442201\pi$$
0.180584 + 0.983560i $$0.442201\pi$$
$$308$$ 0 0
$$309$$ 14.3716 0.817569
$$310$$ 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
$$315$$ −3.70948 −0.209006
$$316$$ 0 0
$$317$$ 3.69105 0.207310 0.103655 0.994613i $$-0.466946\pi$$
0.103655 + 0.994613i $$0.466946\pi$$
$$318$$ 0 0
$$319$$ −5.75495 −0.322215
$$320$$ 0 0
$$321$$ −5.90519 −0.329596
$$322$$ 0 0
$$323$$ −16.2149 −0.902222
$$324$$ 0 0
$$325$$ 8.76025 0.485931
$$326$$ 0 0
$$327$$ 14.4664 0.799992
$$328$$ 0 0
$$329$$ −5.24311 −0.289062
$$330$$ 0 0
$$331$$ 0.479496 0.0263555 0.0131777 0.999913i $$-0.495805\pi$$
0.0131777 + 0.999913i $$0.495805\pi$$
$$332$$ 0 0
$$333$$ −4.95259 −0.271401
$$334$$ 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
$$339$$ 6.61131 0.359077
$$340$$ 0 0
$$341$$ −46.1951 −2.50160
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −30.2135 −1.62664
$$346$$ 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
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ 5.46973 0.291125 0.145562 0.989349i $$-0.453501\pi$$
0.145562 + 0.989349i $$0.453501\pi$$
$$354$$ 0 0
$$355$$ −42.5468 −2.25815
$$356$$ 0 0
$$357$$ −4.95259 −0.262119
$$358$$ 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
$$363$$ 9.40454 0.493611
$$364$$ 0 0
$$365$$ −52.4704 −2.74643
$$366$$ 0 0
$$367$$ −32.1621 −1.67885 −0.839423 0.543478i $$-0.817107\pi$$
−0.839423 + 0.543478i $$0.817107\pi$$
$$368$$ 0 0
$$369$$ −0.435456 −0.0226689
$$370$$ 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
$$375$$ −13.9486 −0.720302
$$376$$ 0 0
$$377$$ −1.27403 −0.0656157
$$378$$ 0 0
$$379$$ −20.1621 −1.03566 −0.517828 0.855485i $$-0.673259\pi$$
−0.517828 + 0.855485i $$0.673259\pi$$
$$380$$ 0 0
$$381$$ −18.7761 −0.961929
$$382$$ 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
$$387$$ 8.14494 0.414030
$$388$$ 0 0
$$389$$ −2.00000 −0.101404 −0.0507020 0.998714i $$-0.516146\pi$$
−0.0507020 + 0.998714i $$0.516146\pi$$
$$390$$ 0 0
$$391$$ −40.3386 −2.04001
$$392$$ 0 0
$$393$$ 10.3716 0.523176
$$394$$ 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
$$399$$ 3.27403 0.163906
$$400$$ 0 0
$$401$$ −31.9560 −1.59580 −0.797902 0.602787i $$-0.794057\pi$$
−0.797902 + 0.602787i $$0.794057\pi$$
$$402$$ 0 0
$$403$$ −10.2266 −0.509424
$$404$$ 0 0
$$405$$ −3.70948 −0.184326
$$406$$ 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
$$411$$ −12.2590 −0.604690
$$412$$ 0 0
$$413$$ −11.4697 −0.564389
$$414$$ 0 0
$$415$$ 37.7471 1.85293
$$416$$ 0 0
$$417$$ 16.3517 0.800746
$$418$$ 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
$$423$$ −5.24311 −0.254929
$$424$$ 0 0
$$425$$ −43.3860 −2.10453
$$426$$ 0 0
$$427$$ −2.56791 −0.124270
$$428$$ 0 0
$$429$$ 4.51714 0.218090
$$430$$ 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
$$435$$ 4.72597 0.226593
$$436$$ 0 0
$$437$$ 26.6667 1.27564
$$438$$ 0 0
$$439$$ −25.8921 −1.23576 −0.617880 0.786272i $$-0.712008\pi$$
−0.617880 + 0.786272i $$0.712008\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 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
$$447$$ 14.5040 0.686016
$$448$$ 0 0
$$449$$ −15.3881 −0.726207 −0.363103 0.931749i $$-0.618283\pi$$
−0.363103 + 0.931749i $$0.618283\pi$$
$$450$$ 0 0
$$451$$ −1.96701 −0.0926231
$$452$$ 0 0
$$453$$ −1.62844 −0.0765109
$$454$$ 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
$$459$$ −4.95259 −0.231167
$$460$$ 0 0
$$461$$ 21.1667 0.985833 0.492916 0.870077i $$-0.335931\pi$$
0.492916 + 0.870077i $$0.335931\pi$$
$$462$$ 0 0
$$463$$ 13.6272 0.633308 0.316654 0.948541i $$-0.397441\pi$$
0.316654 + 0.948541i $$0.397441\pi$$
$$464$$ 0 0
$$465$$ 37.9355 1.75921
$$466$$ 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
$$471$$ 3.04741 0.140417
$$472$$ 0 0
$$473$$ 36.7918 1.69169
$$474$$ 0 0
$$475$$ 28.6813 1.31599
$$476$$ 0 0
$$477$$ −1.74039 −0.0796872
$$478$$ 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
$$483$$ 8.14494 0.370607
$$484$$ 0 0
$$485$$ −36.1318 −1.64066
$$486$$ 0 0
$$487$$ 7.25560 0.328782 0.164391 0.986395i $$-0.447434\pi$$
0.164391 + 0.986395i $$0.447434\pi$$
$$488$$ 0 0
$$489$$ 13.0343 0.589430
$$490$$ 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
$$495$$ −16.7562 −0.753137
$$496$$ 0 0
$$497$$ 11.4697 0.514488
$$498$$ 0 0
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 0 0
$$501$$ −3.22325 −0.144004
$$502$$ 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
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ −10.5737 −0.468669 −0.234335 0.972156i $$-0.575291\pi$$
−0.234335 + 0.972156i $$0.575291\pi$$
$$510$$ 0 0
$$511$$ 14.1449 0.625735
$$512$$ 0 0
$$513$$ 3.27403 0.144552
$$514$$ 0 0
$$515$$ −53.3110 −2.34916
$$516$$ 0 0
$$517$$ −23.6839 −1.04162
$$518$$ 0 0
$$519$$ −23.3242 −1.02382
$$520$$ 0 0
$$521$$ 25.1778 1.10306 0.551529 0.834155i $$-0.314044\pi$$
0.551529 + 0.834155i $$0.314044\pi$$
$$522$$ 0 0
$$523$$ −20.1621 −0.881626 −0.440813 0.897599i $$-0.645310\pi$$
−0.440813 + 0.897599i $$0.645310\pi$$
$$524$$ 0 0
$$525$$ 8.76025 0.382329
$$526$$ 0 0
$$527$$ 50.6483 2.20627
$$528$$ 0 0
$$529$$ 43.3400 1.88435
$$530$$ 0 0
$$531$$ −11.4697 −0.497744
$$532$$ 0 0
$$533$$ −0.435456 −0.0188617
$$534$$ 0 0
$$535$$ 21.9052 0.947044
$$536$$ 0 0
$$537$$ 4.61131 0.198993
$$538$$ 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
$$543$$ 20.8379 0.894241
$$544$$ 0 0
$$545$$ −53.6627 −2.29866
$$546$$ 0 0
$$547$$ 27.4243 1.17258 0.586288 0.810102i $$-0.300589\pi$$
0.586288 + 0.810102i $$0.300589\pi$$
$$548$$ 0 0
$$549$$ −2.56791 −0.109596
$$550$$ 0 0
$$551$$ −4.17119 −0.177699
$$552$$ 0 0
$$553$$ 3.67857 0.156429
$$554$$ 0 0
$$555$$ 18.3716 0.779829
$$556$$ 0 0
$$557$$ 7.02322 0.297584 0.148792 0.988869i $$-0.452462\pi$$
0.148792 + 0.988869i $$0.452462\pi$$
$$558$$ 0 0
$$559$$ 8.14494 0.344494
$$560$$ 0 0
$$561$$ −22.3716 −0.944528
$$562$$ 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
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 45.1331 1.89208 0.946039 0.324053i $$-0.105046\pi$$
0.946039 + 0.324053i $$0.105046\pi$$
$$570$$ 0 0
$$571$$ −42.0370 −1.75919 −0.879597 0.475720i $$-0.842188\pi$$
−0.879597 + 0.475720i $$0.842188\pi$$
$$572$$ 0 0
$$573$$ −24.2767 −1.01418
$$574$$ 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
$$579$$ 23.9052 0.993466
$$580$$ 0 0
$$581$$ −10.1758 −0.422165
$$582$$ 0 0
$$583$$ −7.86160 −0.325594
$$584$$ 0 0
$$585$$ −3.70948 −0.153368
$$586$$ 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
$$591$$ 6.40247 0.263362
$$592$$ 0 0
$$593$$ −3.53428 −0.145135 −0.0725677 0.997363i $$-0.523119\pi$$
−0.0725677 + 0.997363i $$0.523119\pi$$
$$594$$ 0 0
$$595$$ 18.3716 0.753160
$$596$$ 0 0
$$597$$ 23.4058 0.957937
$$598$$ 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
$$603$$ 12.2899 0.500482
$$604$$ 0 0
$$605$$ −34.8860 −1.41832
$$606$$ 0 0
$$607$$ 16.9593 0.688358 0.344179 0.938904i $$-0.388157\pi$$
0.344179 + 0.938904i $$0.388157\pi$$
$$608$$ 0 0
$$609$$ −1.27403 −0.0516261
$$610$$ 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
$$615$$ 1.61531 0.0651358
$$616$$ 0 0
$$617$$ −3.76600 −0.151614 −0.0758068 0.997123i $$-0.524153\pi$$
−0.0758068 + 0.997123i $$0.524153\pi$$
$$618$$ 0 0
$$619$$ −24.1752 −0.971684 −0.485842 0.874047i $$-0.661487\pi$$
−0.485842 + 0.874047i $$0.661487\pi$$
$$620$$ 0 0
$$621$$ 8.14494 0.326845
$$622$$ 0 0
$$623$$ 3.24311 0.129933
$$624$$ 0 0
$$625$$ 7.94076 0.317630
$$626$$ 0 0
$$627$$ 14.7892 0.590625
$$628$$ 0 0
$$629$$ 24.5282 0.978003
$$630$$ 0 0
$$631$$ −7.24376 −0.288369 −0.144185 0.989551i $$-0.546056\pi$$
−0.144185 + 0.989551i $$0.546056\pi$$
$$632$$ 0 0
$$633$$ 17.1792 0.682812
$$634$$ 0 0
$$635$$ 69.6496 2.76396
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ 11.4697 0.453736
$$640$$ 0 0
$$641$$ 36.3887 1.43727 0.718634 0.695389i $$-0.244768\pi$$
0.718634 + 0.695389i $$0.244768\pi$$
$$642$$ 0 0
$$643$$ −18.4731 −0.728508 −0.364254 0.931300i $$-0.618676\pi$$
−0.364254 + 0.931300i $$0.618676\pi$$
$$644$$ 0 0
$$645$$ −30.2135 −1.18966
$$646$$ 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
$$651$$ −10.2266 −0.400813
$$652$$ 0 0
$$653$$ −18.7931 −0.735431 −0.367715 0.929938i $$-0.619860\pi$$
−0.367715 + 0.929938i $$0.619860\pi$$
$$654$$ 0 0
$$655$$ −38.4731 −1.50327
$$656$$ 0 0
$$657$$ 14.1449 0.551847
$$658$$ 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
$$663$$ −4.95259 −0.192343
$$664$$ 0 0
$$665$$ −12.1449 −0.470960
$$666$$ 0 0
$$667$$ −10.3769 −0.401794
$$668$$ 0 0
$$669$$ −3.67857 −0.142222
$$670$$ 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
$$675$$ 8.76025 0.337182
$$676$$ 0 0
$$677$$ −2.97245 −0.114241 −0.0571203 0.998367i $$-0.518192\pi$$
−0.0571203 + 0.998367i $$0.518192\pi$$
$$678$$ 0 0
$$679$$ 9.74039 0.373802
$$680$$ 0 0
$$681$$ 20.5040 0.785715
$$682$$ 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
$$687$$ −16.8379 −0.642407
$$688$$ 0 0
$$689$$ −1.74039 −0.0663037
$$690$$ 0 0
$$691$$ −5.22920 −0.198928 −0.0994641 0.995041i $$-0.531713\pi$$
−0.0994641 + 0.995041i $$0.531713\pi$$
$$692$$ 0 0
$$693$$ 4.51714 0.171592
$$694$$ 0 0
$$695$$ −60.6563 −2.30083
$$696$$ 0 0
$$697$$ 2.15664 0.0816884
$$698$$ 0 0
$$699$$ −17.6786 −0.668665
$$700$$ 0 0
$$701$$ 5.41366 0.204471 0.102236 0.994760i $$-0.467400\pi$$
0.102236 + 0.994760i $$0.467400\pi$$
$$702$$ 0 0
$$703$$ −16.2149 −0.611557
$$704$$ 0 0
$$705$$ 19.4492 0.732500
$$706$$ 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
$$711$$ 3.67857 0.137957
$$712$$ 0 0
$$713$$ −83.2952 −3.11943
$$714$$ 0 0
$$715$$ −16.7562 −0.626648
$$716$$ 0 0
$$717$$ −11.3682 −0.424553
$$718$$ 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
$$723$$ −19.9355 −0.741408
$$724$$ 0 0
$$725$$ −11.1608 −0.414501
$$726$$ 0 0
$$727$$ 44.4651 1.64912 0.824559 0.565776i $$-0.191423\pi$$
0.824559 + 0.565776i $$0.191423\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 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
$$735$$ −3.70948 −0.136826
$$736$$ 0 0
$$737$$ 55.5151 2.04492
$$738$$ 0 0
$$739$$ −15.3174 −0.563460 −0.281730 0.959494i $$-0.590908\pi$$
−0.281730 + 0.959494i $$0.590908\pi$$
$$740$$ 0 0
$$741$$ 3.27403 0.120274
$$742$$ 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
$$747$$ −10.1758 −0.372315
$$748$$ 0 0
$$749$$ −5.90519 −0.215771
$$750$$ 0 0
$$751$$ −5.35571 −0.195433 −0.0977163 0.995214i $$-0.531154\pi$$
−0.0977163 + 0.995214i $$0.531154\pi$$
$$752$$ 0 0
$$753$$ 12.7562 0.464863
$$754$$ 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
$$759$$ 36.7918 1.33546
$$760$$ 0 0
$$761$$ 15.2418 0.552516 0.276258 0.961084i $$-0.410906\pi$$
0.276258 + 0.961084i $$0.410906\pi$$
$$762$$ 0 0
$$763$$ 14.4664 0.523718
$$764$$ 0 0
$$765$$ 18.3716 0.664225
$$766$$ 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
$$771$$ −9.89846 −0.356484
$$772$$ 0 0
$$773$$ −13.0034 −0.467699 −0.233849 0.972273i $$-0.575132\pi$$
−0.233849 + 0.972273i $$0.575132\pi$$
$$774$$ 0 0
$$775$$ −89.5878 −3.21809
$$776$$ 0 0
$$777$$ −4.95259 −0.177673
$$778$$ 0 0
$$779$$ −1.42569 −0.0510808
$$780$$ 0 0
$$781$$ 51.8104 1.85392
$$782$$ 0 0
$$783$$ −1.27403 −0.0455300
$$784$$ 0 0
$$785$$ −11.3043 −0.403468
$$786$$ 0 0
$$787$$ 0.268587 0.00957409 0.00478704 0.999989i $$-0.498476\pi$$
0.00478704 + 0.999989i $$0.498476\pi$$
$$788$$ 0 0
$$789$$ −3.74039 −0.133162
$$790$$ 0 0
$$791$$ 6.61131 0.235071
$$792$$ 0 0
$$793$$ −2.56791 −0.0911891
$$794$$ 0 0
$$795$$ 6.45596 0.228969
$$796$$ 0 0
$$797$$ 29.5810 1.04781 0.523907 0.851775i $$-0.324474\pi$$
0.523907 + 0.851775i $$0.324474\pi$$
$$798$$ 0 0
$$799$$ 25.9670 0.918647
$$800$$ 0 0
$$801$$ 3.24311 0.114590
$$802$$ 0 0
$$803$$ 63.8946 2.25479
$$804$$ 0 0
$$805$$ −30.2135 −1.06489
$$806$$ 0 0
$$807$$ 14.3517 0.505204
$$808$$ 0 0
$$809$$ −26.0921 −0.917349 −0.458675 0.888604i $$-0.651676\pi$$
−0.458675 + 0.888604i $$0.651676\pi$$
$$810$$ 0 0
$$811$$ −37.0461 −1.30087 −0.650433 0.759564i $$-0.725412\pi$$
−0.650433 + 0.759564i $$0.725412\pi$$
$$812$$ 0 0
$$813$$ −27.0013 −0.946977
$$814$$ 0 0
$$815$$ −48.3504 −1.69364
$$816$$ 0 0
$$817$$ 26.6667 0.932951
$$818$$ 0 0
$$819$$ 1.00000 0.0349428
$$820$$ 0 0
$$821$$ −31.3749 −1.09499 −0.547496 0.836808i $$-0.684419\pi$$
−0.547496 + 0.836808i $$0.684419\pi$$
$$822$$ 0 0
$$823$$ −17.8655 −0.622751 −0.311376 0.950287i $$-0.600790\pi$$
−0.311376 + 0.950287i $$0.600790\pi$$
$$824$$ 0 0
$$825$$ 39.5713 1.37769
$$826$$ 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
$$831$$ 0.644291 0.0223502
$$832$$ 0 0
$$833$$ −4.95259 −0.171597
$$834$$ 0 0
$$835$$ 11.9566 0.413775
$$836$$ 0 0
$$837$$ −10.2266 −0.353484
$$838$$ 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
$$843$$ −8.49728 −0.292662
$$844$$ 0 0
$$845$$ −3.70948 −0.127610
$$846$$ 0 0
$$847$$ 9.40454 0.323144
$$848$$ 0 0
$$849$$ 31.8104 1.09173
$$850$$ 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
$$855$$ −12.1449 −0.415348
$$856$$ 0 0
$$857$$ 20.7364 0.708341 0.354171 0.935181i $$-0.384763\pi$$
0.354171 + 0.935181i $$0.384763\pi$$
$$858$$ 0 0
$$859$$ 3.68387 0.125692 0.0628460 0.998023i $$-0.479982\pi$$
0.0628460 + 0.998023i $$0.479982\pi$$
$$860$$ 0 0
$$861$$ −0.435456 −0.0148403
$$862$$ 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
$$867$$ 7.52819 0.255671
$$868$$ 0 0
$$869$$ 16.6166 0.563680
$$870$$ 0 0
$$871$$ 12.2899 0.416427
$$872$$ 0 0
$$873$$ 9.74039 0.329662
$$874$$ 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
$$879$$ 21.3366 0.719667
$$880$$ 0 0
$$881$$ 51.4126 1.73213 0.866067 0.499929i $$-0.166640\pi$$
0.866067 + 0.499929i $$0.166640\pi$$
$$882$$ 0 0
$$883$$ 12.6957 0.427245 0.213622 0.976916i $$-0.431474\pi$$
0.213622 + 0.976916i $$0.431474\pi$$
$$884$$ 0 0
$$885$$ 42.5468 1.43019
$$886$$ 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
$$891$$ 4.51714 0.151330
$$892$$ 0 0
$$893$$ −17.1661 −0.574441
$$894$$ 0 0
$$895$$ −17.1056 −0.571776
$$896$$ 0 0
$$897$$ 8.14494 0.271952
$$898$$ 0 0
$$899$$ 13.0290 0.434541
$$900$$ 0 0
$$901$$ 8.61946 0.287156
$$902$$ 0 0
$$903$$ 8.14494 0.271047
$$904$$ 0 0
$$905$$ −77.2979 −2.56947
$$906$$ 0 0
$$907$$ −6.02354 −0.200008 −0.100004 0.994987i $$-0.531886\pi$$
−0.100004 + 0.994987i $$0.531886\pi$$
$$908$$ 0 0
$$909$$ 15.8036 0.524174
$$910$$ 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
$$915$$ 9.52561 0.314907
$$916$$ 0 0
$$917$$ 10.3716 0.342499
$$918$$ 0 0
$$919$$ −6.13038 −0.202223 −0.101111 0.994875i $$-0.532240\pi$$
−0.101111 + 0.994875i $$0.532240\pi$$
$$920$$ 0 0
$$921$$ 6.32816 0.208520
$$922$$ 0 0
$$923$$ 11.4697 0.377531
$$924$$ 0 0
$$925$$ −43.3860 −1.42652
$$926$$ 0 0
$$927$$ 14.3716 0.472024
$$928$$ 0 0
$$929$$ −55.2169 −1.81161 −0.905803 0.423699i $$-0.860732\pi$$
−0.905803 + 0.423699i $$0.860732\pi$$
$$930$$ 0 0
$$931$$ 3.27403 0.107302
$$932$$ 0 0
$$933$$ 24.5151 0.802587
$$934$$ 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
$$939$$ 14.1634 0.462204
$$940$$ 0 0
$$941$$ 50.9520 1.66099 0.830493 0.557029i $$-0.188059\pi$$
0.830493 + 0.557029i $$0.188059\pi$$
$$942$$ 0 0
$$943$$ −3.54676 −0.115498
$$944$$ 0 0
$$945$$ −3.70948 −0.120669
$$946$$ 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
$$951$$ 3.69105 0.119691
$$952$$ 0 0
$$953$$ −27.4835 −0.890278 −0.445139 0.895461i $$-0.646846\pi$$
−0.445139 + 0.895461i $$0.646846\pi$$
$$954$$ 0 0
$$955$$ 90.0541 2.91408
$$956$$ 0 0
$$957$$ −5.75495 −0.186031
$$958$$ 0 0
$$959$$ −12.2590 −0.395863
$$960$$ 0 0
$$961$$ 73.5838 2.37367
$$962$$ 0 0
$$963$$ −5.90519 −0.190292
$$964$$ 0 0
$$965$$ −88.6759 −2.85458
$$966$$ 0 0
$$967$$ −42.2450 −1.35851 −0.679255 0.733903i $$-0.737697\pi$$
−0.679255 + 0.733903i $$0.737697\pi$$
$$968$$ 0 0
$$969$$ −16.2149 −0.520898
$$970$$ 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
$$975$$ 8.76025 0.280553
$$976$$ 0 0
$$977$$ −15.0982 −0.483033 −0.241517 0.970397i $$-0.577645\pi$$
−0.241517 + 0.970397i $$0.577645\pi$$
$$978$$ 0 0
$$979$$ 14.6496 0.468203
$$980$$ 0 0
$$981$$ 14.4664 0.461876
$$982$$ 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
$$987$$ −5.24311 −0.166890
$$988$$ 0 0
$$989$$ 66.3400 2.10949
$$990$$ 0 0
$$991$$ −55.9737 −1.77806 −0.889032 0.457844i $$-0.848622\pi$$
−0.889032 + 0.457844i $$0.848622\pi$$
$$992$$ 0 0
$$993$$ 0.479496 0.0152163
$$994$$ 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
$$999$$ −4.95259 −0.156693
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 4368.2.a.bs.1.1 4
4.3 odd 2 2184.2.a.v.1.1 4
12.11 even 2 6552.2.a.bs.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
2184.2.a.v.1.1 4 4.3 odd 2
4368.2.a.bs.1.1 4 1.1 even 1 trivial
6552.2.a.bs.1.4 4 12.11 even 2