# Properties

 Label 9200.2.a.bu.1.1 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.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: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 9200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.618034 q^{3} +1.61803 q^{7} -2.61803 q^{9} +O(q^{10})$$ $$q-0.618034 q^{3} +1.61803 q^{7} -2.61803 q^{9} -3.85410 q^{11} -4.09017 q^{13} +5.09017 q^{17} +4.85410 q^{19} -1.00000 q^{21} +1.00000 q^{23} +3.47214 q^{27} -4.76393 q^{29} +2.09017 q^{31} +2.38197 q^{33} +2.47214 q^{37} +2.52786 q^{39} -12.3262 q^{41} +9.70820 q^{47} -4.38197 q^{49} -3.14590 q^{51} +8.47214 q^{53} -3.00000 q^{57} +11.7082 q^{59} +6.32624 q^{61} -4.23607 q^{63} +5.52786 q^{67} -0.618034 q^{69} -7.09017 q^{71} +1.23607 q^{73} -6.23607 q^{77} -10.4721 q^{79} +5.70820 q^{81} +10.9443 q^{83} +2.94427 q^{87} -1.52786 q^{89} -6.61803 q^{91} -1.29180 q^{93} -14.6180 q^{97} +10.0902 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + q^7 - 3 * q^9 $$2 q + q^{3} + q^{7} - 3 q^{9} - q^{11} + 3 q^{13} - q^{17} + 3 q^{19} - 2 q^{21} + 2 q^{23} - 2 q^{27} - 14 q^{29} - 7 q^{31} + 7 q^{33} - 4 q^{37} + 14 q^{39} - 9 q^{41} + 6 q^{47} - 11 q^{49} - 13 q^{51} + 8 q^{53} - 6 q^{57} + 10 q^{59} - 3 q^{61} - 4 q^{63} + 20 q^{67} + q^{69} - 3 q^{71} - 2 q^{73} - 8 q^{77} - 12 q^{79} - 2 q^{81} + 4 q^{83} - 12 q^{87} - 12 q^{89} - 11 q^{91} - 16 q^{93} - 27 q^{97} + 9 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^7 - 3 * q^9 - q^11 + 3 * q^13 - q^17 + 3 * q^19 - 2 * q^21 + 2 * q^23 - 2 * q^27 - 14 * q^29 - 7 * q^31 + 7 * q^33 - 4 * q^37 + 14 * q^39 - 9 * q^41 + 6 * q^47 - 11 * q^49 - 13 * q^51 + 8 * q^53 - 6 * q^57 + 10 * q^59 - 3 * q^61 - 4 * q^63 + 20 * q^67 + q^69 - 3 * q^71 - 2 * q^73 - 8 * q^77 - 12 * q^79 - 2 * q^81 + 4 * q^83 - 12 * q^87 - 12 * q^89 - 11 * q^91 - 16 * q^93 - 27 * q^97 + 9 * 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
$$3$$ −0.618034 −0.356822 −0.178411 0.983956i $$-0.557096\pi$$
−0.178411 + 0.983956i $$0.557096\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.61803 0.611559 0.305780 0.952102i $$-0.401083\pi$$
0.305780 + 0.952102i $$0.401083\pi$$
$$8$$ 0 0
$$9$$ −2.61803 −0.872678
$$10$$ 0 0
$$11$$ −3.85410 −1.16206 −0.581028 0.813884i $$-0.697349\pi$$
−0.581028 + 0.813884i $$0.697349\pi$$
$$12$$ 0 0
$$13$$ −4.09017 −1.13441 −0.567205 0.823577i $$-0.691975\pi$$
−0.567205 + 0.823577i $$0.691975\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.09017 1.23455 0.617274 0.786748i $$-0.288237\pi$$
0.617274 + 0.786748i $$0.288237\pi$$
$$18$$ 0 0
$$19$$ 4.85410 1.11361 0.556804 0.830644i $$-0.312028\pi$$
0.556804 + 0.830644i $$0.312028\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 3.47214 0.668213
$$28$$ 0 0
$$29$$ −4.76393 −0.884640 −0.442320 0.896857i $$-0.645844\pi$$
−0.442320 + 0.896857i $$0.645844\pi$$
$$30$$ 0 0
$$31$$ 2.09017 0.375406 0.187703 0.982226i $$-0.439896\pi$$
0.187703 + 0.982226i $$0.439896\pi$$
$$32$$ 0 0
$$33$$ 2.38197 0.414647
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.47214 0.406417 0.203208 0.979136i $$-0.434863\pi$$
0.203208 + 0.979136i $$0.434863\pi$$
$$38$$ 0 0
$$39$$ 2.52786 0.404782
$$40$$ 0 0
$$41$$ −12.3262 −1.92503 −0.962517 0.271220i $$-0.912573\pi$$
−0.962517 + 0.271220i $$0.912573\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 9.70820 1.41609 0.708044 0.706169i $$-0.249578\pi$$
0.708044 + 0.706169i $$0.249578\pi$$
$$48$$ 0 0
$$49$$ −4.38197 −0.625995
$$50$$ 0 0
$$51$$ −3.14590 −0.440514
$$52$$ 0 0
$$53$$ 8.47214 1.16374 0.581869 0.813283i $$-0.302322\pi$$
0.581869 + 0.813283i $$0.302322\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −3.00000 −0.397360
$$58$$ 0 0
$$59$$ 11.7082 1.52428 0.762139 0.647413i $$-0.224149\pi$$
0.762139 + 0.647413i $$0.224149\pi$$
$$60$$ 0 0
$$61$$ 6.32624 0.809992 0.404996 0.914319i $$-0.367273\pi$$
0.404996 + 0.914319i $$0.367273\pi$$
$$62$$ 0 0
$$63$$ −4.23607 −0.533694
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.52786 0.675336 0.337668 0.941265i $$-0.390362\pi$$
0.337668 + 0.941265i $$0.390362\pi$$
$$68$$ 0 0
$$69$$ −0.618034 −0.0744025
$$70$$ 0 0
$$71$$ −7.09017 −0.841448 −0.420724 0.907189i $$-0.638224\pi$$
−0.420724 + 0.907189i $$0.638224\pi$$
$$72$$ 0 0
$$73$$ 1.23607 0.144671 0.0723354 0.997380i $$-0.476955\pi$$
0.0723354 + 0.997380i $$0.476955\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −6.23607 −0.710666
$$78$$ 0 0
$$79$$ −10.4721 −1.17821 −0.589104 0.808057i $$-0.700519\pi$$
−0.589104 + 0.808057i $$0.700519\pi$$
$$80$$ 0 0
$$81$$ 5.70820 0.634245
$$82$$ 0 0
$$83$$ 10.9443 1.20129 0.600645 0.799516i $$-0.294911\pi$$
0.600645 + 0.799516i $$0.294911\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.94427 0.315659
$$88$$ 0 0
$$89$$ −1.52786 −0.161953 −0.0809766 0.996716i $$-0.525804\pi$$
−0.0809766 + 0.996716i $$0.525804\pi$$
$$90$$ 0 0
$$91$$ −6.61803 −0.693758
$$92$$ 0 0
$$93$$ −1.29180 −0.133953
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −14.6180 −1.48424 −0.742118 0.670269i $$-0.766179\pi$$
−0.742118 + 0.670269i $$0.766179\pi$$
$$98$$ 0 0
$$99$$ 10.0902 1.01410
$$100$$ 0 0
$$101$$ −13.7082 −1.36402 −0.682009 0.731344i $$-0.738893\pi$$
−0.682009 + 0.731344i $$0.738893\pi$$
$$102$$ 0 0
$$103$$ −3.56231 −0.351004 −0.175502 0.984479i $$-0.556155\pi$$
−0.175502 + 0.984479i $$0.556155\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −4.18034 −0.404129 −0.202064 0.979372i $$-0.564765\pi$$
−0.202064 + 0.979372i $$0.564765\pi$$
$$108$$ 0 0
$$109$$ 8.56231 0.820120 0.410060 0.912059i $$-0.365508\pi$$
0.410060 + 0.912059i $$0.365508\pi$$
$$110$$ 0 0
$$111$$ −1.52786 −0.145018
$$112$$ 0 0
$$113$$ −18.9443 −1.78213 −0.891064 0.453878i $$-0.850040\pi$$
−0.891064 + 0.453878i $$0.850040\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 10.7082 0.989974
$$118$$ 0 0
$$119$$ 8.23607 0.754999
$$120$$ 0 0
$$121$$ 3.85410 0.350373
$$122$$ 0 0
$$123$$ 7.61803 0.686895
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −6.18034 −0.548416 −0.274208 0.961670i $$-0.588416\pi$$
−0.274208 + 0.961670i $$0.588416\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.9443 1.30569 0.652844 0.757493i $$-0.273576\pi$$
0.652844 + 0.757493i $$0.273576\pi$$
$$132$$ 0 0
$$133$$ 7.85410 0.681037
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −5.32624 −0.455051 −0.227526 0.973772i $$-0.573064\pi$$
−0.227526 + 0.973772i $$0.573064\pi$$
$$138$$ 0 0
$$139$$ −17.2361 −1.46194 −0.730972 0.682407i $$-0.760933\pi$$
−0.730972 + 0.682407i $$0.760933\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ 15.7639 1.31825
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.70820 0.223369
$$148$$ 0 0
$$149$$ −1.14590 −0.0938756 −0.0469378 0.998898i $$-0.514946\pi$$
−0.0469378 + 0.998898i $$0.514946\pi$$
$$150$$ 0 0
$$151$$ −17.5623 −1.42920 −0.714600 0.699533i $$-0.753391\pi$$
−0.714600 + 0.699533i $$0.753391\pi$$
$$152$$ 0 0
$$153$$ −13.3262 −1.07736
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −9.70820 −0.774799 −0.387400 0.921912i $$-0.626627\pi$$
−0.387400 + 0.921912i $$0.626627\pi$$
$$158$$ 0 0
$$159$$ −5.23607 −0.415247
$$160$$ 0 0
$$161$$ 1.61803 0.127519
$$162$$ 0 0
$$163$$ 3.61803 0.283386 0.141693 0.989911i $$-0.454745\pi$$
0.141693 + 0.989911i $$0.454745\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 3.72949 0.286884
$$170$$ 0 0
$$171$$ −12.7082 −0.971821
$$172$$ 0 0
$$173$$ 21.5623 1.63935 0.819676 0.572828i $$-0.194154\pi$$
0.819676 + 0.572828i $$0.194154\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −7.23607 −0.543896
$$178$$ 0 0
$$179$$ 20.1803 1.50835 0.754175 0.656674i $$-0.228037\pi$$
0.754175 + 0.656674i $$0.228037\pi$$
$$180$$ 0 0
$$181$$ −18.8541 −1.40141 −0.700707 0.713449i $$-0.747132\pi$$
−0.700707 + 0.713449i $$0.747132\pi$$
$$182$$ 0 0
$$183$$ −3.90983 −0.289023
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −19.6180 −1.43461
$$188$$ 0 0
$$189$$ 5.61803 0.408652
$$190$$ 0 0
$$191$$ 0.291796 0.0211136 0.0105568 0.999944i $$-0.496640\pi$$
0.0105568 + 0.999944i $$0.496640\pi$$
$$192$$ 0 0
$$193$$ −5.23607 −0.376900 −0.188450 0.982083i $$-0.560346\pi$$
−0.188450 + 0.982083i $$0.560346\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.43769 0.173679 0.0868393 0.996222i $$-0.472323\pi$$
0.0868393 + 0.996222i $$0.472323\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 0 0
$$201$$ −3.41641 −0.240975
$$202$$ 0 0
$$203$$ −7.70820 −0.541010
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −2.61803 −0.181966
$$208$$ 0 0
$$209$$ −18.7082 −1.29407
$$210$$ 0 0
$$211$$ −14.0000 −0.963800 −0.481900 0.876226i $$-0.660053\pi$$
−0.481900 + 0.876226i $$0.660053\pi$$
$$212$$ 0 0
$$213$$ 4.38197 0.300247
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.38197 0.229583
$$218$$ 0 0
$$219$$ −0.763932 −0.0516217
$$220$$ 0 0
$$221$$ −20.8197 −1.40048
$$222$$ 0 0
$$223$$ −3.05573 −0.204627 −0.102313 0.994752i $$-0.532624\pi$$
−0.102313 + 0.994752i $$0.532624\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −23.2361 −1.54223 −0.771116 0.636695i $$-0.780301\pi$$
−0.771116 + 0.636695i $$0.780301\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 3.85410 0.253581
$$232$$ 0 0
$$233$$ −19.7082 −1.29113 −0.645564 0.763706i $$-0.723378\pi$$
−0.645564 + 0.763706i $$0.723378\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 6.47214 0.420410
$$238$$ 0 0
$$239$$ −24.3607 −1.57576 −0.787881 0.615828i $$-0.788822\pi$$
−0.787881 + 0.615828i $$0.788822\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ −13.9443 −0.894525
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −19.8541 −1.26329
$$248$$ 0 0
$$249$$ −6.76393 −0.428647
$$250$$ 0 0
$$251$$ −12.8541 −0.811344 −0.405672 0.914019i $$-0.632962\pi$$
−0.405672 + 0.914019i $$0.632962\pi$$
$$252$$ 0 0
$$253$$ −3.85410 −0.242305
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 30.1803 1.88260 0.941299 0.337574i $$-0.109606\pi$$
0.941299 + 0.337574i $$0.109606\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 12.4721 0.772006
$$262$$ 0 0
$$263$$ −21.7426 −1.34071 −0.670354 0.742041i $$-0.733858\pi$$
−0.670354 + 0.742041i $$0.733858\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0.944272 0.0577885
$$268$$ 0 0
$$269$$ 8.18034 0.498764 0.249382 0.968405i $$-0.419773\pi$$
0.249382 + 0.968405i $$0.419773\pi$$
$$270$$ 0 0
$$271$$ 14.6738 0.891368 0.445684 0.895190i $$-0.352961\pi$$
0.445684 + 0.895190i $$0.352961\pi$$
$$272$$ 0 0
$$273$$ 4.09017 0.247548
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.58359 −0.155233 −0.0776165 0.996983i $$-0.524731\pi$$
−0.0776165 + 0.996983i $$0.524731\pi$$
$$278$$ 0 0
$$279$$ −5.47214 −0.327608
$$280$$ 0 0
$$281$$ 27.2361 1.62477 0.812384 0.583123i $$-0.198169\pi$$
0.812384 + 0.583123i $$0.198169\pi$$
$$282$$ 0 0
$$283$$ −9.05573 −0.538307 −0.269154 0.963097i $$-0.586744\pi$$
−0.269154 + 0.963097i $$0.586744\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −19.9443 −1.17727
$$288$$ 0 0
$$289$$ 8.90983 0.524108
$$290$$ 0 0
$$291$$ 9.03444 0.529608
$$292$$ 0 0
$$293$$ −15.8885 −0.928219 −0.464109 0.885778i $$-0.653626\pi$$
−0.464109 + 0.885778i $$0.653626\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −13.3820 −0.776500
$$298$$ 0 0
$$299$$ −4.09017 −0.236541
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 8.47214 0.486711
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 27.4508 1.56670 0.783351 0.621579i $$-0.213509\pi$$
0.783351 + 0.621579i $$0.213509\pi$$
$$308$$ 0 0
$$309$$ 2.20163 0.125246
$$310$$ 0 0
$$311$$ −4.00000 −0.226819 −0.113410 0.993548i $$-0.536177\pi$$
−0.113410 + 0.993548i $$0.536177\pi$$
$$312$$ 0 0
$$313$$ 11.7984 0.666884 0.333442 0.942771i $$-0.391790\pi$$
0.333442 + 0.942771i $$0.391790\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −0.0901699 −0.00506445 −0.00253222 0.999997i $$-0.500806\pi$$
−0.00253222 + 0.999997i $$0.500806\pi$$
$$318$$ 0 0
$$319$$ 18.3607 1.02800
$$320$$ 0 0
$$321$$ 2.58359 0.144202
$$322$$ 0 0
$$323$$ 24.7082 1.37480
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −5.29180 −0.292637
$$328$$ 0 0
$$329$$ 15.7082 0.866021
$$330$$ 0 0
$$331$$ −14.7639 −0.811499 −0.405750 0.913984i $$-0.632989\pi$$
−0.405750 + 0.913984i $$0.632989\pi$$
$$332$$ 0 0
$$333$$ −6.47214 −0.354671
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −29.3262 −1.59750 −0.798751 0.601662i $$-0.794506\pi$$
−0.798751 + 0.601662i $$0.794506\pi$$
$$338$$ 0 0
$$339$$ 11.7082 0.635902
$$340$$ 0 0
$$341$$ −8.05573 −0.436242
$$342$$ 0 0
$$343$$ −18.4164 −0.994393
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −8.61803 −0.462640 −0.231320 0.972878i $$-0.574304\pi$$
−0.231320 + 0.972878i $$0.574304\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ −14.2016 −0.758027
$$352$$ 0 0
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −5.09017 −0.269400
$$358$$ 0 0
$$359$$ 18.3607 0.969040 0.484520 0.874780i $$-0.338994\pi$$
0.484520 + 0.874780i $$0.338994\pi$$
$$360$$ 0 0
$$361$$ 4.56231 0.240121
$$362$$ 0 0
$$363$$ −2.38197 −0.125021
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −2.47214 −0.129044 −0.0645222 0.997916i $$-0.520552\pi$$
−0.0645222 + 0.997916i $$0.520552\pi$$
$$368$$ 0 0
$$369$$ 32.2705 1.67994
$$370$$ 0 0
$$371$$ 13.7082 0.711694
$$372$$ 0 0
$$373$$ 2.18034 0.112894 0.0564469 0.998406i $$-0.482023\pi$$
0.0564469 + 0.998406i $$0.482023\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 19.4853 1.00354
$$378$$ 0 0
$$379$$ −33.4508 −1.71825 −0.859127 0.511762i $$-0.828993\pi$$
−0.859127 + 0.511762i $$0.828993\pi$$
$$380$$ 0 0
$$381$$ 3.81966 0.195687
$$382$$ 0 0
$$383$$ 17.8885 0.914062 0.457031 0.889451i $$-0.348913\pi$$
0.457031 + 0.889451i $$0.348913\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 5.67376 0.287671 0.143836 0.989602i $$-0.454056\pi$$
0.143836 + 0.989602i $$0.454056\pi$$
$$390$$ 0 0
$$391$$ 5.09017 0.257421
$$392$$ 0 0
$$393$$ −9.23607 −0.465898
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 8.32624 0.417882 0.208941 0.977928i $$-0.432998\pi$$
0.208941 + 0.977928i $$0.432998\pi$$
$$398$$ 0 0
$$399$$ −4.85410 −0.243009
$$400$$ 0 0
$$401$$ 11.7082 0.584680 0.292340 0.956314i $$-0.405566\pi$$
0.292340 + 0.956314i $$0.405566\pi$$
$$402$$ 0 0
$$403$$ −8.54915 −0.425864
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −9.52786 −0.472279
$$408$$ 0 0
$$409$$ 21.2148 1.04900 0.524502 0.851409i $$-0.324252\pi$$
0.524502 + 0.851409i $$0.324252\pi$$
$$410$$ 0 0
$$411$$ 3.29180 0.162372
$$412$$ 0 0
$$413$$ 18.9443 0.932187
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 10.6525 0.521654
$$418$$ 0 0
$$419$$ −5.52786 −0.270054 −0.135027 0.990842i $$-0.543112\pi$$
−0.135027 + 0.990842i $$0.543112\pi$$
$$420$$ 0 0
$$421$$ −28.7426 −1.40083 −0.700415 0.713735i $$-0.747002\pi$$
−0.700415 + 0.713735i $$0.747002\pi$$
$$422$$ 0 0
$$423$$ −25.4164 −1.23579
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 10.2361 0.495358
$$428$$ 0 0
$$429$$ −9.74265 −0.470379
$$430$$ 0 0
$$431$$ −34.6525 −1.66915 −0.834576 0.550894i $$-0.814287\pi$$
−0.834576 + 0.550894i $$0.814287\pi$$
$$432$$ 0 0
$$433$$ 29.5066 1.41800 0.708998 0.705211i $$-0.249148\pi$$
0.708998 + 0.705211i $$0.249148\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.85410 0.232203
$$438$$ 0 0
$$439$$ 15.6180 0.745408 0.372704 0.927950i $$-0.378431\pi$$
0.372704 + 0.927950i $$0.378431\pi$$
$$440$$ 0 0
$$441$$ 11.4721 0.546292
$$442$$ 0 0
$$443$$ 13.9098 0.660876 0.330438 0.943828i $$-0.392804\pi$$
0.330438 + 0.943828i $$0.392804\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0.708204 0.0334969
$$448$$ 0 0
$$449$$ 18.5623 0.876009 0.438005 0.898973i $$-0.355685\pi$$
0.438005 + 0.898973i $$0.355685\pi$$
$$450$$ 0 0
$$451$$ 47.5066 2.23700
$$452$$ 0 0
$$453$$ 10.8541 0.509970
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −33.7771 −1.58003 −0.790013 0.613090i $$-0.789926\pi$$
−0.790013 + 0.613090i $$0.789926\pi$$
$$458$$ 0 0
$$459$$ 17.6738 0.824941
$$460$$ 0 0
$$461$$ −34.7639 −1.61912 −0.809559 0.587039i $$-0.800294\pi$$
−0.809559 + 0.587039i $$0.800294\pi$$
$$462$$ 0 0
$$463$$ 2.00000 0.0929479 0.0464739 0.998920i $$-0.485202\pi$$
0.0464739 + 0.998920i $$0.485202\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 23.1246 1.07008 0.535040 0.844827i $$-0.320297\pi$$
0.535040 + 0.844827i $$0.320297\pi$$
$$468$$ 0 0
$$469$$ 8.94427 0.413008
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −22.1803 −1.01557
$$478$$ 0 0
$$479$$ −3.88854 −0.177672 −0.0888361 0.996046i $$-0.528315\pi$$
−0.0888361 + 0.996046i $$0.528315\pi$$
$$480$$ 0 0
$$481$$ −10.1115 −0.461043
$$482$$ 0 0
$$483$$ −1.00000 −0.0455016
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −42.1803 −1.91137 −0.955687 0.294385i $$-0.904885\pi$$
−0.955687 + 0.294385i $$0.904885\pi$$
$$488$$ 0 0
$$489$$ −2.23607 −0.101118
$$490$$ 0 0
$$491$$ 16.1803 0.730209 0.365104 0.930967i $$-0.381033\pi$$
0.365104 + 0.930967i $$0.381033\pi$$
$$492$$ 0 0
$$493$$ −24.2492 −1.09213
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −11.4721 −0.514596
$$498$$ 0 0
$$499$$ 32.3607 1.44866 0.724331 0.689452i $$-0.242149\pi$$
0.724331 + 0.689452i $$0.242149\pi$$
$$500$$ 0 0
$$501$$ 4.94427 0.220894
$$502$$ 0 0
$$503$$ 20.6738 0.921797 0.460899 0.887453i $$-0.347527\pi$$
0.460899 + 0.887453i $$0.347527\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −2.30495 −0.102366
$$508$$ 0 0
$$509$$ 5.34752 0.237025 0.118512 0.992953i $$-0.462187\pi$$
0.118512 + 0.992953i $$0.462187\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ 0 0
$$513$$ 16.8541 0.744127
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −37.4164 −1.64557
$$518$$ 0 0
$$519$$ −13.3262 −0.584957
$$520$$ 0 0
$$521$$ −24.4721 −1.07214 −0.536072 0.844172i $$-0.680092\pi$$
−0.536072 + 0.844172i $$0.680092\pi$$
$$522$$ 0 0
$$523$$ −26.0000 −1.13690 −0.568450 0.822718i $$-0.692457\pi$$
−0.568450 + 0.822718i $$0.692457\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 10.6393 0.463456
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −30.6525 −1.33020
$$532$$ 0 0
$$533$$ 50.4164 2.18378
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −12.4721 −0.538212
$$538$$ 0 0
$$539$$ 16.8885 0.727441
$$540$$ 0 0
$$541$$ −30.8328 −1.32561 −0.662803 0.748794i $$-0.730633\pi$$
−0.662803 + 0.748794i $$0.730633\pi$$
$$542$$ 0 0
$$543$$ 11.6525 0.500056
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −36.9230 −1.57871 −0.789356 0.613935i $$-0.789586\pi$$
−0.789356 + 0.613935i $$0.789586\pi$$
$$548$$ 0 0
$$549$$ −16.5623 −0.706862
$$550$$ 0 0
$$551$$ −23.1246 −0.985142
$$552$$ 0 0
$$553$$ −16.9443 −0.720544
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −30.8328 −1.30643 −0.653214 0.757173i $$-0.726580\pi$$
−0.653214 + 0.757173i $$0.726580\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 12.1246 0.511902
$$562$$ 0 0
$$563$$ 21.8885 0.922492 0.461246 0.887272i $$-0.347403\pi$$
0.461246 + 0.887272i $$0.347403\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 9.23607 0.387878
$$568$$ 0 0
$$569$$ −2.00000 −0.0838444 −0.0419222 0.999121i $$-0.513348\pi$$
−0.0419222 + 0.999121i $$0.513348\pi$$
$$570$$ 0 0
$$571$$ 30.9787 1.29642 0.648209 0.761462i $$-0.275518\pi$$
0.648209 + 0.761462i $$0.275518\pi$$
$$572$$ 0 0
$$573$$ −0.180340 −0.00753381
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 12.4721 0.519222 0.259611 0.965713i $$-0.416406\pi$$
0.259611 + 0.965713i $$0.416406\pi$$
$$578$$ 0 0
$$579$$ 3.23607 0.134486
$$580$$ 0 0
$$581$$ 17.7082 0.734660
$$582$$ 0 0
$$583$$ −32.6525 −1.35233
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 11.3820 0.469784 0.234892 0.972021i $$-0.424526\pi$$
0.234892 + 0.972021i $$0.424526\pi$$
$$588$$ 0 0
$$589$$ 10.1459 0.418054
$$590$$ 0 0
$$591$$ −1.50658 −0.0619723
$$592$$ 0 0
$$593$$ −34.7639 −1.42758 −0.713792 0.700358i $$-0.753024\pi$$
−0.713792 + 0.700358i $$0.753024\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1.23607 0.0505889
$$598$$ 0 0
$$599$$ 20.6180 0.842430 0.421215 0.906961i $$-0.361604\pi$$
0.421215 + 0.906961i $$0.361604\pi$$
$$600$$ 0 0
$$601$$ 0.270510 0.0110343 0.00551716 0.999985i $$-0.498244\pi$$
0.00551716 + 0.999985i $$0.498244\pi$$
$$602$$ 0 0
$$603$$ −14.4721 −0.589351
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 17.5279 0.711434 0.355717 0.934594i $$-0.384237\pi$$
0.355717 + 0.934594i $$0.384237\pi$$
$$608$$ 0 0
$$609$$ 4.76393 0.193044
$$610$$ 0 0
$$611$$ −39.7082 −1.60642
$$612$$ 0 0
$$613$$ 43.3050 1.74907 0.874535 0.484962i $$-0.161167\pi$$
0.874535 + 0.484962i $$0.161167\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −22.9098 −0.922315 −0.461158 0.887318i $$-0.652566\pi$$
−0.461158 + 0.887318i $$0.652566\pi$$
$$618$$ 0 0
$$619$$ −21.7984 −0.876151 −0.438075 0.898938i $$-0.644340\pi$$
−0.438075 + 0.898938i $$0.644340\pi$$
$$620$$ 0 0
$$621$$ 3.47214 0.139332
$$622$$ 0 0
$$623$$ −2.47214 −0.0990440
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 11.5623 0.461754
$$628$$ 0 0
$$629$$ 12.5836 0.501741
$$630$$ 0 0
$$631$$ 16.0689 0.639692 0.319846 0.947470i $$-0.396369\pi$$
0.319846 + 0.947470i $$0.396369\pi$$
$$632$$ 0 0
$$633$$ 8.65248 0.343905
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 17.9230 0.710135
$$638$$ 0 0
$$639$$ 18.5623 0.734313
$$640$$ 0 0
$$641$$ −44.3607 −1.75214 −0.876071 0.482183i $$-0.839844\pi$$
−0.876071 + 0.482183i $$0.839844\pi$$
$$642$$ 0 0
$$643$$ −21.7082 −0.856088 −0.428044 0.903758i $$-0.640797\pi$$
−0.428044 + 0.903758i $$0.640797\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 44.2492 1.73962 0.869808 0.493390i $$-0.164242\pi$$
0.869808 + 0.493390i $$0.164242\pi$$
$$648$$ 0 0
$$649$$ −45.1246 −1.77130
$$650$$ 0 0
$$651$$ −2.09017 −0.0819202
$$652$$ 0 0
$$653$$ −21.0344 −0.823141 −0.411571 0.911378i $$-0.635020\pi$$
−0.411571 + 0.911378i $$0.635020\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −3.23607 −0.126251
$$658$$ 0 0
$$659$$ −34.2492 −1.33416 −0.667080 0.744986i $$-0.732456\pi$$
−0.667080 + 0.744986i $$0.732456\pi$$
$$660$$ 0 0
$$661$$ 34.3262 1.33514 0.667568 0.744549i $$-0.267335\pi$$
0.667568 + 0.744549i $$0.267335\pi$$
$$662$$ 0 0
$$663$$ 12.8673 0.499723
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4.76393 −0.184460
$$668$$ 0 0
$$669$$ 1.88854 0.0730153
$$670$$ 0 0
$$671$$ −24.3820 −0.941255
$$672$$ 0 0
$$673$$ −6.94427 −0.267682 −0.133841 0.991003i $$-0.542731\pi$$
−0.133841 + 0.991003i $$0.542731\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 33.0557 1.27043 0.635217 0.772333i $$-0.280910\pi$$
0.635217 + 0.772333i $$0.280910\pi$$
$$678$$ 0 0
$$679$$ −23.6525 −0.907699
$$680$$ 0 0
$$681$$ 14.3607 0.550302
$$682$$ 0 0
$$683$$ −11.4377 −0.437651 −0.218826 0.975764i $$-0.570223\pi$$
−0.218826 + 0.975764i $$0.570223\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −6.18034 −0.235795
$$688$$ 0 0
$$689$$ −34.6525 −1.32015
$$690$$ 0 0
$$691$$ 24.7639 0.942064 0.471032 0.882116i $$-0.343882\pi$$
0.471032 + 0.882116i $$0.343882\pi$$
$$692$$ 0 0
$$693$$ 16.3262 0.620182
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −62.7426 −2.37655
$$698$$ 0 0
$$699$$ 12.1803 0.460703
$$700$$ 0 0
$$701$$ −48.3394 −1.82575 −0.912877 0.408235i $$-0.866144\pi$$
−0.912877 + 0.408235i $$0.866144\pi$$
$$702$$ 0 0
$$703$$ 12.0000 0.452589
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −22.1803 −0.834178
$$708$$ 0 0
$$709$$ −14.9098 −0.559950 −0.279975 0.960007i $$-0.590326\pi$$
−0.279975 + 0.960007i $$0.590326\pi$$
$$710$$ 0 0
$$711$$ 27.4164 1.02820
$$712$$ 0 0
$$713$$ 2.09017 0.0782775
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 15.0557 0.562266
$$718$$ 0 0
$$719$$ −1.72949 −0.0644991 −0.0322495 0.999480i $$-0.510267\pi$$
−0.0322495 + 0.999480i $$0.510267\pi$$
$$720$$ 0 0
$$721$$ −5.76393 −0.214660
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 52.7984 1.95818 0.979092 0.203420i $$-0.0652056\pi$$
0.979092 + 0.203420i $$0.0652056\pi$$
$$728$$ 0 0
$$729$$ −8.50658 −0.315058
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 2.58359 0.0954272 0.0477136 0.998861i $$-0.484807\pi$$
0.0477136 + 0.998861i $$0.484807\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −21.3050 −0.784778
$$738$$ 0 0
$$739$$ −21.8885 −0.805183 −0.402592 0.915380i $$-0.631890\pi$$
−0.402592 + 0.915380i $$0.631890\pi$$
$$740$$ 0 0
$$741$$ 12.2705 0.450768
$$742$$ 0 0
$$743$$ −44.6312 −1.63736 −0.818680 0.574250i $$-0.805294\pi$$
−0.818680 + 0.574250i $$0.805294\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −28.6525 −1.04834
$$748$$ 0 0
$$749$$ −6.76393 −0.247149
$$750$$ 0 0
$$751$$ −29.0132 −1.05871 −0.529353 0.848402i $$-0.677565\pi$$
−0.529353 + 0.848402i $$0.677565\pi$$
$$752$$ 0 0
$$753$$ 7.94427 0.289505
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −17.8885 −0.650170 −0.325085 0.945685i $$-0.605393\pi$$
−0.325085 + 0.945685i $$0.605393\pi$$
$$758$$ 0 0
$$759$$ 2.38197 0.0864599
$$760$$ 0 0
$$761$$ −35.8673 −1.30019 −0.650094 0.759854i $$-0.725270\pi$$
−0.650094 + 0.759854i $$0.725270\pi$$
$$762$$ 0 0
$$763$$ 13.8541 0.501552
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −47.8885 −1.72916
$$768$$ 0 0
$$769$$ −33.4164 −1.20503 −0.602513 0.798109i $$-0.705834\pi$$
−0.602513 + 0.798109i $$0.705834\pi$$
$$770$$ 0 0
$$771$$ −18.6525 −0.671753
$$772$$ 0 0
$$773$$ −11.0557 −0.397647 −0.198823 0.980035i $$-0.563712\pi$$
−0.198823 + 0.980035i $$0.563712\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −2.47214 −0.0886874
$$778$$ 0 0
$$779$$ −59.8328 −2.14373
$$780$$ 0 0
$$781$$ 27.3262 0.977810
$$782$$ 0 0
$$783$$ −16.5410 −0.591128
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −43.1246 −1.53723 −0.768613 0.639714i $$-0.779053\pi$$
−0.768613 + 0.639714i $$0.779053\pi$$
$$788$$ 0 0
$$789$$ 13.4377 0.478395
$$790$$ 0 0
$$791$$ −30.6525 −1.08988
$$792$$ 0 0
$$793$$ −25.8754 −0.918862
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −0.291796 −0.0103359 −0.00516797 0.999987i $$-0.501645\pi$$
−0.00516797 + 0.999987i $$0.501645\pi$$
$$798$$ 0 0
$$799$$ 49.4164 1.74823
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ 0 0
$$803$$ −4.76393 −0.168116
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −5.05573 −0.177970
$$808$$ 0 0
$$809$$ −4.25735 −0.149681 −0.0748403 0.997196i $$-0.523845\pi$$
−0.0748403 + 0.997196i $$0.523845\pi$$
$$810$$ 0 0
$$811$$ −44.1803 −1.55138 −0.775691 0.631113i $$-0.782598\pi$$
−0.775691 + 0.631113i $$0.782598\pi$$
$$812$$ 0 0
$$813$$ −9.06888 −0.318060
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 17.3262 0.605428
$$820$$ 0 0
$$821$$ −50.9443 −1.77797 −0.888984 0.457939i $$-0.848588\pi$$
−0.888984 + 0.457939i $$0.848588\pi$$
$$822$$ 0 0
$$823$$ −1.41641 −0.0493729 −0.0246864 0.999695i $$-0.507859\pi$$
−0.0246864 + 0.999695i $$0.507859\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 8.29180 0.288334 0.144167 0.989553i $$-0.453950\pi$$
0.144167 + 0.989553i $$0.453950\pi$$
$$828$$ 0 0
$$829$$ 1.05573 0.0366670 0.0183335 0.999832i $$-0.494164\pi$$
0.0183335 + 0.999832i $$0.494164\pi$$
$$830$$ 0 0
$$831$$ 1.59675 0.0553906
$$832$$ 0 0
$$833$$ −22.3050 −0.772821
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 7.25735 0.250851
$$838$$ 0 0
$$839$$ 43.0132 1.48498 0.742490 0.669858i $$-0.233645\pi$$
0.742490 + 0.669858i $$0.233645\pi$$
$$840$$ 0 0
$$841$$ −6.30495 −0.217412
$$842$$ 0 0
$$843$$ −16.8328 −0.579753
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 6.23607 0.214274
$$848$$ 0 0
$$849$$ 5.59675 0.192080
$$850$$ 0 0
$$851$$ 2.47214 0.0847437
$$852$$ 0 0
$$853$$ −13.7984 −0.472447 −0.236224 0.971699i $$-0.575910\pi$$
−0.236224 + 0.971699i $$0.575910\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 33.4164 1.14148 0.570741 0.821130i $$-0.306656\pi$$
0.570741 + 0.821130i $$0.306656\pi$$
$$858$$ 0 0
$$859$$ −34.0689 −1.16242 −0.581208 0.813755i $$-0.697420\pi$$
−0.581208 + 0.813755i $$0.697420\pi$$
$$860$$ 0 0
$$861$$ 12.3262 0.420077
$$862$$ 0 0
$$863$$ 37.2361 1.26753 0.633765 0.773525i $$-0.281509\pi$$
0.633765 + 0.773525i $$0.281509\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −5.50658 −0.187013
$$868$$ 0 0
$$869$$ 40.3607 1.36914
$$870$$ 0 0
$$871$$ −22.6099 −0.766107
$$872$$ 0 0
$$873$$ 38.2705 1.29526
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 23.7426 0.801732 0.400866 0.916137i $$-0.368709\pi$$
0.400866 + 0.916137i $$0.368709\pi$$
$$878$$ 0 0
$$879$$ 9.81966 0.331209
$$880$$ 0 0
$$881$$ 35.4164 1.19321 0.596605 0.802535i $$-0.296516\pi$$
0.596605 + 0.802535i $$0.296516\pi$$
$$882$$ 0 0
$$883$$ −4.56231 −0.153534 −0.0767669 0.997049i $$-0.524460\pi$$
−0.0767669 + 0.997049i $$0.524460\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −58.8328 −1.97541 −0.987706 0.156321i $$-0.950037\pi$$
−0.987706 + 0.156321i $$0.950037\pi$$
$$888$$ 0 0
$$889$$ −10.0000 −0.335389
$$890$$ 0 0
$$891$$ −22.0000 −0.737028
$$892$$ 0 0
$$893$$ 47.1246 1.57697
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2.52786 0.0844029
$$898$$ 0 0
$$899$$ −9.95743 −0.332099
$$900$$ 0 0
$$901$$ 43.1246 1.43669
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −33.1246 −1.09988 −0.549942 0.835203i $$-0.685350\pi$$
−0.549942 + 0.835203i $$0.685350\pi$$
$$908$$ 0 0
$$909$$ 35.8885 1.19035
$$910$$ 0 0
$$911$$ 22.0689 0.731175 0.365587 0.930777i $$-0.380868\pi$$
0.365587 + 0.930777i $$0.380868\pi$$
$$912$$ 0 0
$$913$$ −42.1803 −1.39597
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 24.1803 0.798505
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −16.9656 −0.559034
$$922$$ 0 0
$$923$$ 29.0000 0.954547
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 9.32624 0.306314
$$928$$ 0 0
$$929$$ −12.4721 −0.409198 −0.204599 0.978846i $$-0.565589\pi$$
−0.204599 + 0.978846i $$0.565589\pi$$
$$930$$ 0 0
$$931$$ −21.2705 −0.697113
$$932$$ 0 0
$$933$$ 2.47214 0.0809341
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 12.2016 0.398610 0.199305 0.979938i $$-0.436132\pi$$
0.199305 + 0.979938i $$0.436132\pi$$
$$938$$ 0 0
$$939$$ −7.29180 −0.237959
$$940$$ 0 0
$$941$$ −60.5066 −1.97246 −0.986229 0.165385i $$-0.947113\pi$$
−0.986229 + 0.165385i $$0.947113\pi$$
$$942$$ 0 0
$$943$$ −12.3262 −0.401398
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 5.68692 0.184800 0.0924000 0.995722i $$-0.470546\pi$$
0.0924000 + 0.995722i $$0.470546\pi$$
$$948$$ 0 0
$$949$$ −5.05573 −0.164116
$$950$$ 0 0
$$951$$ 0.0557281 0.00180711
$$952$$ 0 0
$$953$$ 20.7984 0.673725 0.336863 0.941554i $$-0.390634\pi$$
0.336863 + 0.941554i $$0.390634\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −11.3475 −0.366813
$$958$$ 0 0
$$959$$ −8.61803 −0.278291
$$960$$ 0 0
$$961$$ −26.6312 −0.859071
$$962$$ 0 0
$$963$$ 10.9443 0.352674
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 50.5410 1.62529 0.812645 0.582759i $$-0.198027\pi$$
0.812645 + 0.582759i $$0.198027\pi$$
$$968$$ 0 0
$$969$$ −15.2705 −0.490559
$$970$$ 0 0
$$971$$ −0.729490 −0.0234105 −0.0117052 0.999931i $$-0.503726\pi$$
−0.0117052 + 0.999931i $$0.503726\pi$$
$$972$$ 0 0
$$973$$ −27.8885 −0.894066
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 3.43769 0.109982 0.0549908 0.998487i $$-0.482487\pi$$
0.0549908 + 0.998487i $$0.482487\pi$$
$$978$$ 0 0
$$979$$ 5.88854 0.188199
$$980$$ 0 0
$$981$$ −22.4164 −0.715701
$$982$$ 0 0
$$983$$ 19.2705 0.614634 0.307317 0.951607i $$-0.400569\pi$$
0.307317 + 0.951607i $$0.400569\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −9.70820 −0.309016
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 10.5066 0.333752 0.166876 0.985978i $$-0.446632\pi$$
0.166876 + 0.985978i $$0.446632\pi$$
$$992$$ 0 0
$$993$$ 9.12461 0.289561
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 41.1935 1.30461 0.652306 0.757956i $$-0.273802\pi$$
0.652306 + 0.757956i $$0.273802\pi$$
$$998$$ 0 0
$$999$$ 8.58359 0.271573
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 9200.2.a.bu.1.1 2
4.3 odd 2 1150.2.a.j.1.2 2
5.4 even 2 1840.2.a.l.1.2 2
20.3 even 4 1150.2.b.i.599.4 4
20.7 even 4 1150.2.b.i.599.1 4
20.19 odd 2 230.2.a.c.1.1 2
40.19 odd 2 7360.2.a.bh.1.2 2
40.29 even 2 7360.2.a.bn.1.1 2
60.59 even 2 2070.2.a.u.1.2 2
460.459 even 2 5290.2.a.o.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.1 2 20.19 odd 2
1150.2.a.j.1.2 2 4.3 odd 2
1150.2.b.i.599.1 4 20.7 even 4
1150.2.b.i.599.4 4 20.3 even 4
1840.2.a.l.1.2 2 5.4 even 2
2070.2.a.u.1.2 2 60.59 even 2
5290.2.a.o.1.1 2 460.459 even 2
7360.2.a.bh.1.2 2 40.19 odd 2
7360.2.a.bn.1.1 2 40.29 even 2
9200.2.a.bu.1.1 2 1.1 even 1 trivial