# Properties

 Label 2200.2.a.y.1.1 Level $2200$ Weight $2$ Character 2200.1 Self dual yes Analytic conductor $17.567$ 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] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$17.5670884447$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.54764.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 9x^{2} + 3x + 2$$ x^4 - x^3 - 9*x^2 + 3*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.67673$$ of defining polynomial Character $$\chi$$ $$=$$ 2200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.67673 q^{3} +4.38559 q^{7} +4.16490 q^{9} +O(q^{10})$$ $$q-2.67673 q^{3} +4.38559 q^{7} +4.16490 q^{9} +1.00000 q^{11} +4.00000 q^{13} +5.87995 q^{17} -0.526486 q^{19} -11.7391 q^{21} +6.15025 q^{23} -3.11812 q^{27} -0.967873 q^{29} -9.60629 q^{31} -2.67673 q^{33} +1.76466 q^{37} -10.7069 q^{39} +2.50564 q^{41} -3.85911 q^{43} -4.91208 q^{47} +12.2334 q^{49} -15.7391 q^{51} -5.29767 q^{53} +1.40926 q^{57} -2.63841 q^{59} +8.96787 q^{61} +18.2655 q^{63} -7.97440 q^{67} -16.4626 q^{69} +13.8718 q^{71} +12.7712 q^{73} +4.38559 q^{77} -13.2768 q^{79} -4.14832 q^{81} -7.61901 q^{83} +2.59074 q^{87} +2.83510 q^{89} +17.5424 q^{91} +25.7135 q^{93} -10.4800 q^{97} +4.16490 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} + q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q + q^3 + q^7 + 7 * q^9 $$4 q + q^{3} + q^{7} + 7 q^{9} + 4 q^{11} + 16 q^{13} + 7 q^{17} - 9 q^{19} - 7 q^{21} + 6 q^{23} + 13 q^{27} + 3 q^{29} - 15 q^{31} + q^{33} + 5 q^{37} + 4 q^{39} + 10 q^{41} + 8 q^{43} - 10 q^{47} + 9 q^{49} - 23 q^{51} + 5 q^{53} - 15 q^{57} + 6 q^{59} + 29 q^{61} + 40 q^{63} + 6 q^{67} - 30 q^{69} - q^{71} + 18 q^{73} + q^{77} - 20 q^{79} + 44 q^{81} + 26 q^{83} + 31 q^{87} + 21 q^{89} + 4 q^{91} + 25 q^{93} - 4 q^{97} + 7 q^{99}+O(q^{100})$$ 4 * q + q^3 + q^7 + 7 * q^9 + 4 * q^11 + 16 * q^13 + 7 * q^17 - 9 * q^19 - 7 * q^21 + 6 * q^23 + 13 * q^27 + 3 * q^29 - 15 * q^31 + q^33 + 5 * q^37 + 4 * q^39 + 10 * q^41 + 8 * q^43 - 10 * q^47 + 9 * q^49 - 23 * q^51 + 5 * q^53 - 15 * q^57 + 6 * q^59 + 29 * q^61 + 40 * q^63 + 6 * q^67 - 30 * q^69 - q^71 + 18 * q^73 + q^77 - 20 * q^79 + 44 * q^81 + 26 * q^83 + 31 * q^87 + 21 * q^89 + 4 * q^91 + 25 * q^93 - 4 * q^97 + 7 * 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
$$3$$ −2.67673 −1.54541 −0.772706 0.634764i $$-0.781097\pi$$
−0.772706 + 0.634764i $$0.781097\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.38559 1.65760 0.828799 0.559546i $$-0.189025\pi$$
0.828799 + 0.559546i $$0.189025\pi$$
$$8$$ 0 0
$$9$$ 4.16490 1.38830
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.87995 1.42610 0.713049 0.701114i $$-0.247314\pi$$
0.713049 + 0.701114i $$0.247314\pi$$
$$18$$ 0 0
$$19$$ −0.526486 −0.120784 −0.0603920 0.998175i $$-0.519235\pi$$
−0.0603920 + 0.998175i $$0.519235\pi$$
$$20$$ 0 0
$$21$$ −11.7391 −2.56167
$$22$$ 0 0
$$23$$ 6.15025 1.28242 0.641208 0.767367i $$-0.278434\pi$$
0.641208 + 0.767367i $$0.278434\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −3.11812 −0.600083
$$28$$ 0 0
$$29$$ −0.967873 −0.179730 −0.0898648 0.995954i $$-0.528643\pi$$
−0.0898648 + 0.995954i $$0.528643\pi$$
$$30$$ 0 0
$$31$$ −9.60629 −1.72534 −0.862670 0.505767i $$-0.831209\pi$$
−0.862670 + 0.505767i $$0.831209\pi$$
$$32$$ 0 0
$$33$$ −2.67673 −0.465959
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.76466 0.290108 0.145054 0.989424i $$-0.453664\pi$$
0.145054 + 0.989424i $$0.453664\pi$$
$$38$$ 0 0
$$39$$ −10.7069 −1.71448
$$40$$ 0 0
$$41$$ 2.50564 0.391315 0.195658 0.980672i $$-0.437316\pi$$
0.195658 + 0.980672i $$0.437316\pi$$
$$42$$ 0 0
$$43$$ −3.85911 −0.588508 −0.294254 0.955727i $$-0.595071\pi$$
−0.294254 + 0.955727i $$0.595071\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.91208 −0.716500 −0.358250 0.933626i $$-0.616626\pi$$
−0.358250 + 0.933626i $$0.616626\pi$$
$$48$$ 0 0
$$49$$ 12.2334 1.74763
$$50$$ 0 0
$$51$$ −15.7391 −2.20391
$$52$$ 0 0
$$53$$ −5.29767 −0.727691 −0.363845 0.931459i $$-0.618536\pi$$
−0.363845 + 0.931459i $$0.618536\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.40926 0.186661
$$58$$ 0 0
$$59$$ −2.63841 −0.343492 −0.171746 0.985141i $$-0.554941\pi$$
−0.171746 + 0.985141i $$0.554941\pi$$
$$60$$ 0 0
$$61$$ 8.96787 1.14822 0.574109 0.818779i $$-0.305348\pi$$
0.574109 + 0.818779i $$0.305348\pi$$
$$62$$ 0 0
$$63$$ 18.2655 2.30124
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7.97440 −0.974228 −0.487114 0.873338i $$-0.661950\pi$$
−0.487114 + 0.873338i $$0.661950\pi$$
$$68$$ 0 0
$$69$$ −16.4626 −1.98186
$$70$$ 0 0
$$71$$ 13.8718 1.64628 0.823142 0.567836i $$-0.192219\pi$$
0.823142 + 0.567836i $$0.192219\pi$$
$$72$$ 0 0
$$73$$ 12.7712 1.49475 0.747377 0.664400i $$-0.231313\pi$$
0.747377 + 0.664400i $$0.231313\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.38559 0.499785
$$78$$ 0 0
$$79$$ −13.2768 −1.49376 −0.746880 0.664959i $$-0.768449\pi$$
−0.746880 + 0.664959i $$0.768449\pi$$
$$80$$ 0 0
$$81$$ −4.14832 −0.460924
$$82$$ 0 0
$$83$$ −7.61901 −0.836295 −0.418147 0.908379i $$-0.637320\pi$$
−0.418147 + 0.908379i $$0.637320\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.59074 0.277756
$$88$$ 0 0
$$89$$ 2.83510 0.300520 0.150260 0.988646i $$-0.451989\pi$$
0.150260 + 0.988646i $$0.451989\pi$$
$$90$$ 0 0
$$91$$ 17.5424 1.83894
$$92$$ 0 0
$$93$$ 25.7135 2.66636
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −10.4800 −1.06409 −0.532044 0.846717i $$-0.678576\pi$$
−0.532044 + 0.846717i $$0.678576\pi$$
$$98$$ 0 0
$$99$$ 4.16490 0.418588
$$100$$ 0 0
$$101$$ 8.70693 0.866372 0.433186 0.901305i $$-0.357389\pi$$
0.433186 + 0.901305i $$0.357389\pi$$
$$102$$ 0 0
$$103$$ 15.6190 1.53899 0.769493 0.638655i $$-0.220509\pi$$
0.769493 + 0.638655i $$0.220509\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.08792 0.685215 0.342608 0.939479i $$-0.388690\pi$$
0.342608 + 0.939479i $$0.388690\pi$$
$$108$$ 0 0
$$109$$ −15.5424 −1.48869 −0.744344 0.667796i $$-0.767238\pi$$
−0.744344 + 0.667796i $$0.767238\pi$$
$$110$$ 0 0
$$111$$ −4.72351 −0.448336
$$112$$ 0 0
$$113$$ 8.11530 0.763423 0.381711 0.924282i $$-0.375335\pi$$
0.381711 + 0.924282i $$0.375335\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 16.6596 1.54018
$$118$$ 0 0
$$119$$ 25.7871 2.36390
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −6.70693 −0.604744
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 4.32980 0.384207 0.192104 0.981375i $$-0.438469\pi$$
0.192104 + 0.981375i $$0.438469\pi$$
$$128$$ 0 0
$$129$$ 10.3298 0.909488
$$130$$ 0 0
$$131$$ −17.6748 −1.54425 −0.772127 0.635468i $$-0.780807\pi$$
−0.772127 + 0.635468i $$0.780807\pi$$
$$132$$ 0 0
$$133$$ −2.30895 −0.200211
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.4800 1.57886 0.789428 0.613843i $$-0.210377\pi$$
0.789428 + 0.613843i $$0.210377\pi$$
$$138$$ 0 0
$$139$$ −5.55861 −0.471475 −0.235738 0.971817i $$-0.575751\pi$$
−0.235738 + 0.971817i $$0.575751\pi$$
$$140$$ 0 0
$$141$$ 13.1483 1.10729
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −32.7456 −2.70081
$$148$$ 0 0
$$149$$ 20.9516 1.71642 0.858212 0.513295i $$-0.171575\pi$$
0.858212 + 0.513295i $$0.171575\pi$$
$$150$$ 0 0
$$151$$ −1.67020 −0.135919 −0.0679596 0.997688i $$-0.521649\pi$$
−0.0679596 + 0.997688i $$0.521649\pi$$
$$152$$ 0 0
$$153$$ 24.4894 1.97985
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −8.94228 −0.713671 −0.356836 0.934167i $$-0.616144\pi$$
−0.356836 + 0.934167i $$0.616144\pi$$
$$158$$ 0 0
$$159$$ 14.1804 1.12458
$$160$$ 0 0
$$161$$ 26.9725 2.12573
$$162$$ 0 0
$$163$$ −9.43856 −0.739285 −0.369643 0.929174i $$-0.620520\pi$$
−0.369643 + 0.929174i $$0.620520\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.96787 0.693955 0.346977 0.937873i $$-0.387208\pi$$
0.346977 + 0.937873i $$0.387208\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −2.19276 −0.167684
$$172$$ 0 0
$$173$$ 2.64653 0.201212 0.100606 0.994926i $$-0.467922\pi$$
0.100606 + 0.994926i $$0.467922\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 7.06232 0.530837
$$178$$ 0 0
$$179$$ 20.2225 1.51150 0.755749 0.654861i $$-0.227273\pi$$
0.755749 + 0.654861i $$0.227273\pi$$
$$180$$ 0 0
$$181$$ 15.3453 1.14061 0.570305 0.821433i $$-0.306825\pi$$
0.570305 + 0.821433i $$0.306825\pi$$
$$182$$ 0 0
$$183$$ −24.0046 −1.77447
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 5.87995 0.429985
$$188$$ 0 0
$$189$$ −13.6748 −0.994696
$$190$$ 0 0
$$191$$ −5.62713 −0.407165 −0.203582 0.979058i $$-0.565258\pi$$
−0.203582 + 0.979058i $$0.565258\pi$$
$$192$$ 0 0
$$193$$ −7.23342 −0.520673 −0.260336 0.965518i $$-0.583833\pi$$
−0.260336 + 0.965518i $$0.583833\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.35347 0.381419 0.190709 0.981647i $$-0.438921\pi$$
0.190709 + 0.981647i $$0.438921\pi$$
$$198$$ 0 0
$$199$$ 7.23342 0.512763 0.256382 0.966576i $$-0.417470\pi$$
0.256382 + 0.966576i $$0.417470\pi$$
$$200$$ 0 0
$$201$$ 21.3453 1.50558
$$202$$ 0 0
$$203$$ −4.24470 −0.297919
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 25.6152 1.78038
$$208$$ 0 0
$$209$$ −0.526486 −0.0364178
$$210$$ 0 0
$$211$$ −12.2447 −0.842960 −0.421480 0.906838i $$-0.638489\pi$$
−0.421480 + 0.906838i $$0.638489\pi$$
$$212$$ 0 0
$$213$$ −37.1312 −2.54419
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −42.1292 −2.85992
$$218$$ 0 0
$$219$$ −34.1850 −2.31001
$$220$$ 0 0
$$221$$ 23.5198 1.58211
$$222$$ 0 0
$$223$$ 19.4525 1.30264 0.651318 0.758805i $$-0.274216\pi$$
0.651318 + 0.758805i $$0.274216\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3.90080 0.258905 0.129452 0.991586i $$-0.458678\pi$$
0.129452 + 0.991586i $$0.458678\pi$$
$$228$$ 0 0
$$229$$ 13.4096 0.886131 0.443066 0.896489i $$-0.353891\pi$$
0.443066 + 0.896489i $$0.353891\pi$$
$$230$$ 0 0
$$231$$ −11.7391 −0.772373
$$232$$ 0 0
$$233$$ −3.36192 −0.220247 −0.110123 0.993918i $$-0.535125\pi$$
−0.110123 + 0.993918i $$0.535125\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 35.5385 2.30847
$$238$$ 0 0
$$239$$ −13.2768 −0.858806 −0.429403 0.903113i $$-0.641276\pi$$
−0.429403 + 0.903113i $$0.641276\pi$$
$$240$$ 0 0
$$241$$ −17.7437 −1.14297 −0.571485 0.820613i $$-0.693632\pi$$
−0.571485 + 0.820613i $$0.693632\pi$$
$$242$$ 0 0
$$243$$ 20.4583 1.31240
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.10594 −0.133998
$$248$$ 0 0
$$249$$ 20.3940 1.29242
$$250$$ 0 0
$$251$$ 8.57416 0.541196 0.270598 0.962692i $$-0.412779\pi$$
0.270598 + 0.962692i $$0.412779\pi$$
$$252$$ 0 0
$$253$$ 6.15025 0.386663
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −2.70693 −0.168854 −0.0844269 0.996430i $$-0.526906\pi$$
−0.0844269 + 0.996430i $$0.526906\pi$$
$$258$$ 0 0
$$259$$ 7.73906 0.480882
$$260$$ 0 0
$$261$$ −4.03109 −0.249518
$$262$$ 0 0
$$263$$ 14.3213 0.883092 0.441546 0.897239i $$-0.354430\pi$$
0.441546 + 0.897239i $$0.354430\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −7.58881 −0.464428
$$268$$ 0 0
$$269$$ −11.9357 −0.727735 −0.363868 0.931451i $$-0.618544\pi$$
−0.363868 + 0.931451i $$0.618544\pi$$
$$270$$ 0 0
$$271$$ −24.4195 −1.48338 −0.741689 0.670744i $$-0.765975\pi$$
−0.741689 + 0.670744i $$0.765975\pi$$
$$272$$ 0 0
$$273$$ −46.9562 −2.84192
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −23.0074 −1.38238 −0.691191 0.722672i $$-0.742914\pi$$
−0.691191 + 0.722672i $$0.742914\pi$$
$$278$$ 0 0
$$279$$ −40.0092 −2.39529
$$280$$ 0 0
$$281$$ −10.6596 −0.635898 −0.317949 0.948108i $$-0.602994\pi$$
−0.317949 + 0.948108i $$0.602994\pi$$
$$282$$ 0 0
$$283$$ 10.3771 0.616857 0.308428 0.951248i $$-0.400197\pi$$
0.308428 + 0.951248i $$0.400197\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10.9887 0.648644
$$288$$ 0 0
$$289$$ 17.5738 1.03375
$$290$$ 0 0
$$291$$ 28.0523 1.64445
$$292$$ 0 0
$$293$$ −22.7069 −1.32655 −0.663277 0.748374i $$-0.730835\pi$$
−0.663277 + 0.748374i $$0.730835\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.11812 −0.180932
$$298$$ 0 0
$$299$$ 24.6010 1.42271
$$300$$ 0 0
$$301$$ −16.9245 −0.975510
$$302$$ 0 0
$$303$$ −23.3061 −1.33890
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5.49436 −0.313580 −0.156790 0.987632i $$-0.550115\pi$$
−0.156790 + 0.987632i $$0.550115\pi$$
$$308$$ 0 0
$$309$$ −41.8079 −2.37837
$$310$$ 0 0
$$311$$ −24.2447 −1.37479 −0.687395 0.726283i $$-0.741246\pi$$
−0.687395 + 0.726283i $$0.741246\pi$$
$$312$$ 0 0
$$313$$ −11.7562 −0.664500 −0.332250 0.943191i $$-0.607808\pi$$
−0.332250 + 0.943191i $$0.607808\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −9.05387 −0.508516 −0.254258 0.967136i $$-0.581831\pi$$
−0.254258 + 0.967136i $$0.581831\pi$$
$$318$$ 0 0
$$319$$ −0.967873 −0.0541905
$$320$$ 0 0
$$321$$ −18.9725 −1.05894
$$322$$ 0 0
$$323$$ −3.09571 −0.172250
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 41.6028 2.30064
$$328$$ 0 0
$$329$$ −21.5424 −1.18767
$$330$$ 0 0
$$331$$ −3.29733 −0.181238 −0.0906190 0.995886i $$-0.528885\pi$$
−0.0906190 + 0.995886i $$0.528885\pi$$
$$332$$ 0 0
$$333$$ 7.34961 0.402756
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −3.17302 −0.172845 −0.0864227 0.996259i $$-0.527544\pi$$
−0.0864227 + 0.996259i $$0.527544\pi$$
$$338$$ 0 0
$$339$$ −21.7225 −1.17980
$$340$$ 0 0
$$341$$ −9.60629 −0.520210
$$342$$ 0 0
$$343$$ 22.9516 1.23927
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −4.44139 −0.238426 −0.119213 0.992869i $$-0.538037\pi$$
−0.119213 + 0.992869i $$0.538037\pi$$
$$348$$ 0 0
$$349$$ 29.2380 1.56508 0.782538 0.622603i $$-0.213925\pi$$
0.782538 + 0.622603i $$0.213925\pi$$
$$350$$ 0 0
$$351$$ −12.4725 −0.665732
$$352$$ 0 0
$$353$$ 1.04927 0.0558469 0.0279234 0.999610i $$-0.491111\pi$$
0.0279234 + 0.999610i $$0.491111\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −69.0251 −3.65319
$$358$$ 0 0
$$359$$ 9.69565 0.511717 0.255858 0.966714i $$-0.417642\pi$$
0.255858 + 0.966714i $$0.417642\pi$$
$$360$$ 0 0
$$361$$ −18.7228 −0.985411
$$362$$ 0 0
$$363$$ −2.67673 −0.140492
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 11.3921 0.594664 0.297332 0.954774i $$-0.403903\pi$$
0.297332 + 0.954774i $$0.403903\pi$$
$$368$$ 0 0
$$369$$ 10.4357 0.543263
$$370$$ 0 0
$$371$$ −23.2334 −1.20622
$$372$$ 0 0
$$373$$ −6.47634 −0.335332 −0.167666 0.985844i $$-0.553623\pi$$
−0.167666 + 0.985844i $$0.553623\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −3.87149 −0.199392
$$378$$ 0 0
$$379$$ −33.4993 −1.72074 −0.860372 0.509667i $$-0.829768\pi$$
−0.860372 + 0.509667i $$0.829768\pi$$
$$380$$ 0 0
$$381$$ −11.5897 −0.593759
$$382$$ 0 0
$$383$$ 30.8702 1.57740 0.788698 0.614781i $$-0.210756\pi$$
0.788698 + 0.614781i $$0.210756\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −16.0728 −0.817026
$$388$$ 0 0
$$389$$ −23.3453 −1.18366 −0.591828 0.806064i $$-0.701594\pi$$
−0.591828 + 0.806064i $$0.701594\pi$$
$$390$$ 0 0
$$391$$ 36.1632 1.82885
$$392$$ 0 0
$$393$$ 47.3107 2.38651
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 19.4781 0.977579 0.488789 0.872402i $$-0.337439\pi$$
0.488789 + 0.872402i $$0.337439\pi$$
$$398$$ 0 0
$$399$$ 6.18045 0.309409
$$400$$ 0 0
$$401$$ 2.24470 0.112095 0.0560474 0.998428i $$-0.482150\pi$$
0.0560474 + 0.998428i $$0.482150\pi$$
$$402$$ 0 0
$$403$$ −38.4251 −1.91409
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.76466 0.0874707
$$408$$ 0 0
$$409$$ 29.4781 1.45760 0.728799 0.684727i $$-0.240079\pi$$
0.728799 + 0.684727i $$0.240079\pi$$
$$410$$ 0 0
$$411$$ −49.4661 −2.43998
$$412$$ 0 0
$$413$$ −11.5710 −0.569372
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 14.8789 0.728624
$$418$$ 0 0
$$419$$ 7.64831 0.373644 0.186822 0.982394i $$-0.440181\pi$$
0.186822 + 0.982394i $$0.440181\pi$$
$$420$$ 0 0
$$421$$ 29.4781 1.43668 0.718338 0.695695i $$-0.244903\pi$$
0.718338 + 0.695695i $$0.244903\pi$$
$$422$$ 0 0
$$423$$ −20.4583 −0.994717
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 39.3294 1.90328
$$428$$ 0 0
$$429$$ −10.7069 −0.516935
$$430$$ 0 0
$$431$$ −2.08970 −0.100657 −0.0503286 0.998733i $$-0.516027\pi$$
−0.0503286 + 0.998733i $$0.516027\pi$$
$$432$$ 0 0
$$433$$ −5.17777 −0.248828 −0.124414 0.992230i $$-0.539705\pi$$
−0.124414 + 0.992230i $$0.539705\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −3.23802 −0.154895
$$438$$ 0 0
$$439$$ 27.6066 1.31759 0.658796 0.752322i $$-0.271066\pi$$
0.658796 + 0.752322i $$0.271066\pi$$
$$440$$ 0 0
$$441$$ 50.9509 2.42623
$$442$$ 0 0
$$443$$ −14.1502 −0.672299 −0.336149 0.941809i $$-0.609125\pi$$
−0.336149 + 0.941809i $$0.609125\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −56.0819 −2.65258
$$448$$ 0 0
$$449$$ −6.46257 −0.304987 −0.152494 0.988304i $$-0.548730\pi$$
−0.152494 + 0.988304i $$0.548730\pi$$
$$450$$ 0 0
$$451$$ 2.50564 0.117986
$$452$$ 0 0
$$453$$ 4.47069 0.210051
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −38.5582 −1.80368 −0.901839 0.432071i $$-0.857783\pi$$
−0.901839 + 0.432071i $$0.857783\pi$$
$$458$$ 0 0
$$459$$ −18.3344 −0.855776
$$460$$ 0 0
$$461$$ −18.4234 −0.858064 −0.429032 0.903289i $$-0.641145\pi$$
−0.429032 + 0.903289i $$0.641145\pi$$
$$462$$ 0 0
$$463$$ 7.09163 0.329576 0.164788 0.986329i $$-0.447306\pi$$
0.164788 + 0.986329i $$0.447306\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 40.4328 1.87101 0.935503 0.353319i $$-0.114947\pi$$
0.935503 + 0.353319i $$0.114947\pi$$
$$468$$ 0 0
$$469$$ −34.9725 −1.61488
$$470$$ 0 0
$$471$$ 23.9361 1.10292
$$472$$ 0 0
$$473$$ −3.85911 −0.177442
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −22.0643 −1.01025
$$478$$ 0 0
$$479$$ −10.7486 −0.491117 −0.245559 0.969382i $$-0.578971\pi$$
−0.245559 + 0.969382i $$0.578971\pi$$
$$480$$ 0 0
$$481$$ 7.05862 0.321845
$$482$$ 0 0
$$483$$ −72.1981 −3.28513
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −38.6396 −1.75093 −0.875465 0.483282i $$-0.839445\pi$$
−0.875465 + 0.483282i $$0.839445\pi$$
$$488$$ 0 0
$$489$$ 25.2645 1.14250
$$490$$ 0 0
$$491$$ 32.9460 1.48683 0.743416 0.668830i $$-0.233204\pi$$
0.743416 + 0.668830i $$0.233204\pi$$
$$492$$ 0 0
$$493$$ −5.69105 −0.256312
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 60.8362 2.72888
$$498$$ 0 0
$$499$$ 26.9562 1.20673 0.603363 0.797466i $$-0.293827\pi$$
0.603363 + 0.797466i $$0.293827\pi$$
$$500$$ 0 0
$$501$$ −24.0046 −1.07245
$$502$$ 0 0
$$503$$ 14.0766 0.627646 0.313823 0.949481i $$-0.398390\pi$$
0.313823 + 0.949481i $$0.398390\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −8.03020 −0.356634
$$508$$ 0 0
$$509$$ 1.28109 0.0567833 0.0283917 0.999597i $$-0.490961\pi$$
0.0283917 + 0.999597i $$0.490961\pi$$
$$510$$ 0 0
$$511$$ 56.0092 2.47770
$$512$$ 0 0
$$513$$ 1.64165 0.0724804
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −4.91208 −0.216033
$$518$$ 0 0
$$519$$ −7.08407 −0.310956
$$520$$ 0 0
$$521$$ −29.2394 −1.28100 −0.640501 0.767958i $$-0.721273\pi$$
−0.640501 + 0.767958i $$0.721273\pi$$
$$522$$ 0 0
$$523$$ −2.50564 −0.109564 −0.0547820 0.998498i $$-0.517446\pi$$
−0.0547820 + 0.998498i $$0.517446\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −56.4845 −2.46050
$$528$$ 0 0
$$529$$ 14.8255 0.644589
$$530$$ 0 0
$$531$$ −10.9887 −0.476870
$$532$$ 0 0
$$533$$ 10.0226 0.434125
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −54.1301 −2.33589
$$538$$ 0 0
$$539$$ 12.2334 0.526931
$$540$$ 0 0
$$541$$ −3.27222 −0.140684 −0.0703420 0.997523i $$-0.522409\pi$$
−0.0703420 + 0.997523i $$0.522409\pi$$
$$542$$ 0 0
$$543$$ −41.0754 −1.76271
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 3.03673 0.129841 0.0649205 0.997890i $$-0.479321\pi$$
0.0649205 + 0.997890i $$0.479321\pi$$
$$548$$ 0 0
$$549$$ 37.3503 1.59407
$$550$$ 0 0
$$551$$ 0.509572 0.0217085
$$552$$ 0 0
$$553$$ −58.2267 −2.47605
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1.52366 −0.0645596 −0.0322798 0.999479i $$-0.510277\pi$$
−0.0322798 + 0.999479i $$0.510277\pi$$
$$558$$ 0 0
$$559$$ −15.4364 −0.652891
$$560$$ 0 0
$$561$$ −15.7391 −0.664504
$$562$$ 0 0
$$563$$ 27.9195 1.17667 0.588333 0.808618i $$-0.299784\pi$$
0.588333 + 0.808618i $$0.299784\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −18.1928 −0.764027
$$568$$ 0 0
$$569$$ −13.7437 −0.576164 −0.288082 0.957606i $$-0.593018\pi$$
−0.288082 + 0.957606i $$0.593018\pi$$
$$570$$ 0 0
$$571$$ −1.42618 −0.0596836 −0.0298418 0.999555i $$-0.509500\pi$$
−0.0298418 + 0.999555i $$0.509500\pi$$
$$572$$ 0 0
$$573$$ 15.0623 0.629238
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6.35154 0.264418 0.132209 0.991222i $$-0.457793\pi$$
0.132209 + 0.991222i $$0.457793\pi$$
$$578$$ 0 0
$$579$$ 19.3619 0.804654
$$580$$ 0 0
$$581$$ −33.4139 −1.38624
$$582$$ 0 0
$$583$$ −5.29767 −0.219407
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 22.1681 0.914974 0.457487 0.889216i $$-0.348750\pi$$
0.457487 + 0.889216i $$0.348750\pi$$
$$588$$ 0 0
$$589$$ 5.05757 0.208394
$$590$$ 0 0
$$591$$ −14.3298 −0.589449
$$592$$ 0 0
$$593$$ 1.05297 0.0432403 0.0216202 0.999766i $$-0.493118\pi$$
0.0216202 + 0.999766i $$0.493118\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −19.3619 −0.792431
$$598$$ 0 0
$$599$$ −5.12747 −0.209503 −0.104751 0.994498i $$-0.533405\pi$$
−0.104751 + 0.994498i $$0.533405\pi$$
$$600$$ 0 0
$$601$$ −1.71821 −0.0700874 −0.0350437 0.999386i $$-0.511157\pi$$
−0.0350437 + 0.999386i $$0.511157\pi$$
$$602$$ 0 0
$$603$$ −33.2126 −1.35252
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 4.42728 0.179698 0.0898489 0.995955i $$-0.471362\pi$$
0.0898489 + 0.995955i $$0.471362\pi$$
$$608$$ 0 0
$$609$$ 11.3619 0.460408
$$610$$ 0 0
$$611$$ −19.6483 −0.794886
$$612$$ 0 0
$$613$$ −33.8334 −1.36652 −0.683258 0.730177i $$-0.739438\pi$$
−0.683258 + 0.730177i $$0.739438\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 40.3778 1.62555 0.812775 0.582578i $$-0.197956\pi$$
0.812775 + 0.582578i $$0.197956\pi$$
$$618$$ 0 0
$$619$$ −1.62713 −0.0653999 −0.0327000 0.999465i $$-0.510411\pi$$
−0.0327000 + 0.999465i $$0.510411\pi$$
$$620$$ 0 0
$$621$$ −19.1772 −0.769555
$$622$$ 0 0
$$623$$ 12.4336 0.498142
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 1.40926 0.0562805
$$628$$ 0 0
$$629$$ 10.3761 0.413722
$$630$$ 0 0
$$631$$ 3.71223 0.147781 0.0738907 0.997266i $$-0.476458\pi$$
0.0738907 + 0.997266i $$0.476458\pi$$
$$632$$ 0 0
$$633$$ 32.7758 1.30272
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 48.9337 1.93882
$$638$$ 0 0
$$639$$ 57.7748 2.28553
$$640$$ 0 0
$$641$$ 32.5562 1.28589 0.642946 0.765911i $$-0.277712\pi$$
0.642946 + 0.765911i $$0.277712\pi$$
$$642$$ 0 0
$$643$$ 27.9017 1.10034 0.550168 0.835054i $$-0.314564\pi$$
0.550168 + 0.835054i $$0.314564\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.7873 1.13174 0.565872 0.824493i $$-0.308540\pi$$
0.565872 + 0.824493i $$0.308540\pi$$
$$648$$ 0 0
$$649$$ −2.63841 −0.103567
$$650$$ 0 0
$$651$$ 112.769 4.41976
$$652$$ 0 0
$$653$$ 34.7152 1.35851 0.679256 0.733901i $$-0.262303\pi$$
0.679256 + 0.733901i $$0.262303\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 53.1907 2.07517
$$658$$ 0 0
$$659$$ −25.5632 −0.995801 −0.497901 0.867234i $$-0.665896\pi$$
−0.497901 + 0.867234i $$0.665896\pi$$
$$660$$ 0 0
$$661$$ 1.40960 0.0548269 0.0274135 0.999624i $$-0.491273\pi$$
0.0274135 + 0.999624i $$0.491273\pi$$
$$662$$ 0 0
$$663$$ −62.9562 −2.44502
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −5.95266 −0.230488
$$668$$ 0 0
$$669$$ −52.0692 −2.01311
$$670$$ 0 0
$$671$$ 8.96787 0.346201
$$672$$ 0 0
$$673$$ 0.0557959 0.00215078 0.00107539 0.999999i $$-0.499658\pi$$
0.00107539 + 0.999999i $$0.499658\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 20.8316 0.800623 0.400311 0.916379i $$-0.368902\pi$$
0.400311 + 0.916379i $$0.368902\pi$$
$$678$$ 0 0
$$679$$ −45.9612 −1.76383
$$680$$ 0 0
$$681$$ −10.4414 −0.400115
$$682$$ 0 0
$$683$$ 21.5085 0.822999 0.411499 0.911410i $$-0.365005\pi$$
0.411499 + 0.911410i $$0.365005\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −35.8939 −1.36944
$$688$$ 0 0
$$689$$ −21.1907 −0.807301
$$690$$ 0 0
$$691$$ 5.49862 0.209178 0.104589 0.994516i $$-0.466647\pi$$
0.104589 + 0.994516i $$0.466647\pi$$
$$692$$ 0 0
$$693$$ 18.2655 0.693851
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 14.7330 0.558054
$$698$$ 0 0
$$699$$ 8.99897 0.340372
$$700$$ 0 0
$$701$$ −6.00460 −0.226791 −0.113395 0.993550i $$-0.536173\pi$$
−0.113395 + 0.993550i $$0.536173\pi$$
$$702$$ 0 0
$$703$$ −0.929066 −0.0350404
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 38.1850 1.43610
$$708$$ 0 0
$$709$$ 17.7168 0.665369 0.332685 0.943038i $$-0.392046\pi$$
0.332685 + 0.943038i $$0.392046\pi$$
$$710$$ 0 0
$$711$$ −55.2966 −2.07379
$$712$$ 0 0
$$713$$ −59.0810 −2.21260
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 35.5385 1.32721
$$718$$ 0 0
$$719$$ 30.5308 1.13860 0.569302 0.822128i $$-0.307213\pi$$
0.569302 + 0.822128i $$0.307213\pi$$
$$720$$ 0 0
$$721$$ 68.4986 2.55102
$$722$$ 0 0
$$723$$ 47.4950 1.76636
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −43.9063 −1.62839 −0.814197 0.580589i $$-0.802823\pi$$
−0.814197 + 0.580589i $$0.802823\pi$$
$$728$$ 0 0
$$729$$ −42.3165 −1.56728
$$730$$ 0 0
$$731$$ −22.6914 −0.839270
$$732$$ 0 0
$$733$$ −20.6427 −0.762455 −0.381227 0.924481i $$-0.624498\pi$$
−0.381227 + 0.924481i $$0.624498\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −7.97440 −0.293741
$$738$$ 0 0
$$739$$ −7.73446 −0.284517 −0.142258 0.989830i $$-0.545436\pi$$
−0.142258 + 0.989830i $$0.545436\pi$$
$$740$$ 0 0
$$741$$ 5.63705 0.207082
$$742$$ 0 0
$$743$$ −29.1755 −1.07034 −0.535172 0.844743i $$-0.679753\pi$$
−0.535172 + 0.844743i $$0.679753\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −31.7324 −1.16103
$$748$$ 0 0
$$749$$ 31.0847 1.13581
$$750$$ 0 0
$$751$$ −9.38310 −0.342394 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$752$$ 0 0
$$753$$ −22.9507 −0.836371
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 12.8997 0.468847 0.234424 0.972135i $$-0.424680\pi$$
0.234424 + 0.972135i $$0.424680\pi$$
$$758$$ 0 0
$$759$$ −16.4626 −0.597553
$$760$$ 0 0
$$761$$ 15.6963 0.568991 0.284496 0.958677i $$-0.408174\pi$$
0.284496 + 0.958677i $$0.408174\pi$$
$$762$$ 0 0
$$763$$ −68.1625 −2.46765
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −10.5536 −0.381070
$$768$$ 0 0
$$769$$ 34.1123 1.23012 0.615060 0.788480i $$-0.289132\pi$$
0.615060 + 0.788480i $$0.289132\pi$$
$$770$$ 0 0
$$771$$ 7.24573 0.260949
$$772$$ 0 0
$$773$$ −52.2539 −1.87944 −0.939721 0.341942i $$-0.888915\pi$$
−0.939721 + 0.341942i $$0.888915\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −20.7154 −0.743160
$$778$$ 0 0
$$779$$ −1.31918 −0.0472647
$$780$$ 0 0
$$781$$ 13.8718 0.496373
$$782$$ 0 0
$$783$$ 3.01795 0.107853
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 6.89902 0.245924 0.122962 0.992411i $$-0.460761\pi$$
0.122962 + 0.992411i $$0.460761\pi$$
$$788$$ 0 0
$$789$$ −38.3344 −1.36474
$$790$$ 0 0
$$791$$ 35.5904 1.26545
$$792$$ 0 0
$$793$$ 35.8715 1.27383
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −0.544296 −0.0192800 −0.00963998 0.999954i $$-0.503069\pi$$
−0.00963998 + 0.999954i $$0.503069\pi$$
$$798$$ 0 0
$$799$$ −28.8828 −1.02180
$$800$$ 0 0
$$801$$ 11.8079 0.417212
$$802$$ 0 0
$$803$$ 12.7712 0.450685
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 31.9488 1.12465
$$808$$ 0 0
$$809$$ 26.3354 0.925905 0.462952 0.886383i $$-0.346790\pi$$
0.462952 + 0.886383i $$0.346790\pi$$
$$810$$ 0 0
$$811$$ −24.4686 −0.859207 −0.429604 0.903018i $$-0.641347\pi$$
−0.429604 + 0.903018i $$0.641347\pi$$
$$812$$ 0 0
$$813$$ 65.3645 2.29243
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.03176 0.0710824
$$818$$ 0 0
$$819$$ 73.0622 2.55300
$$820$$ 0 0
$$821$$ 42.8503 1.49549 0.747743 0.663989i $$-0.231138\pi$$
0.747743 + 0.663989i $$0.231138\pi$$
$$822$$ 0 0
$$823$$ 18.7417 0.653296 0.326648 0.945146i $$-0.394081\pi$$
0.326648 + 0.945146i $$0.394081\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −30.6681 −1.06644 −0.533218 0.845978i $$-0.679017\pi$$
−0.533218 + 0.845978i $$0.679017\pi$$
$$828$$ 0 0
$$829$$ −19.8758 −0.690314 −0.345157 0.938545i $$-0.612174\pi$$
−0.345157 + 0.938545i $$0.612174\pi$$
$$830$$ 0 0
$$831$$ 61.5847 2.13635
$$832$$ 0 0
$$833$$ 71.9319 2.49229
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 29.9536 1.03535
$$838$$ 0 0
$$839$$ −48.3178 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$840$$ 0 0
$$841$$ −28.0632 −0.967697
$$842$$ 0 0
$$843$$ 28.5329 0.982725
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4.38559 0.150691
$$848$$ 0 0
$$849$$ −27.7768 −0.953298
$$850$$ 0 0
$$851$$ 10.8531 0.372038
$$852$$ 0 0
$$853$$ 39.4686 1.35138 0.675690 0.737186i $$-0.263846\pi$$
0.675690 + 0.737186i $$0.263846\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 16.7666 0.572736 0.286368 0.958120i $$-0.407552\pi$$
0.286368 + 0.958120i $$0.407552\pi$$
$$858$$ 0 0
$$859$$ 0.350976 0.0119751 0.00598757 0.999982i $$-0.498094\pi$$
0.00598757 + 0.999982i $$0.498094\pi$$
$$860$$ 0 0
$$861$$ −29.4139 −1.00242
$$862$$ 0 0
$$863$$ −35.2087 −1.19852 −0.599259 0.800555i $$-0.704538\pi$$
−0.599259 + 0.800555i $$0.704538\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −47.0404 −1.59758
$$868$$ 0 0
$$869$$ −13.2768 −0.450385
$$870$$ 0 0
$$871$$ −31.8976 −1.08081
$$872$$ 0 0
$$873$$ −43.6483 −1.47727
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −8.61790 −0.291006 −0.145503 0.989358i $$-0.546480\pi$$
−0.145503 + 0.989358i $$0.546480\pi$$
$$878$$ 0 0
$$879$$ 60.7804 2.05007
$$880$$ 0 0
$$881$$ −42.5162 −1.43241 −0.716204 0.697891i $$-0.754122\pi$$
−0.716204 + 0.697891i $$0.754122\pi$$
$$882$$ 0 0
$$883$$ 25.1568 0.846593 0.423296 0.905991i $$-0.360873\pi$$
0.423296 + 0.905991i $$0.360873\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 6.27505 0.210696 0.105348 0.994435i $$-0.466404\pi$$
0.105348 + 0.994435i $$0.466404\pi$$
$$888$$ 0 0
$$889$$ 18.9887 0.636861
$$890$$ 0 0
$$891$$ −4.14832 −0.138974
$$892$$ 0 0
$$893$$ 2.58614 0.0865418
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −65.8503 −2.19868
$$898$$ 0 0
$$899$$ 9.29767 0.310095
$$900$$ 0 0
$$901$$ −31.1500 −1.03776
$$902$$ 0 0
$$903$$ 45.3023 1.50757
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 11.0925 0.368321 0.184161 0.982896i $$-0.441043\pi$$
0.184161 + 0.982896i $$0.441043\pi$$
$$908$$ 0 0
$$909$$ 36.2635 1.20278
$$910$$ 0 0
$$911$$ −32.8175 −1.08729 −0.543646 0.839315i $$-0.682956\pi$$
−0.543646 + 0.839315i $$0.682956\pi$$
$$912$$ 0 0
$$913$$ −7.61901 −0.252152
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −77.5145 −2.55975
$$918$$ 0 0
$$919$$ 0.658922 0.0217358 0.0108679 0.999941i $$-0.496541\pi$$
0.0108679 + 0.999941i $$0.496541\pi$$
$$920$$ 0 0
$$921$$ 14.7069 0.484610
$$922$$ 0 0
$$923$$ 55.4873 1.82639
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 65.0516 2.13657
$$928$$ 0 0
$$929$$ −28.0519 −0.920354 −0.460177 0.887827i $$-0.652214\pi$$
−0.460177 + 0.887827i $$0.652214\pi$$
$$930$$ 0 0
$$931$$ −6.44072 −0.211086
$$932$$ 0 0
$$933$$ 64.8966 2.12462
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.6652 0.871115 0.435558 0.900161i $$-0.356551\pi$$
0.435558 + 0.900161i $$0.356551\pi$$
$$938$$ 0 0
$$939$$ 31.4682 1.02693
$$940$$ 0 0
$$941$$ −0.222135 −0.00724139 −0.00362069 0.999993i $$-0.501153\pi$$
−0.00362069 + 0.999993i $$0.501153\pi$$
$$942$$ 0 0
$$943$$ 15.4103 0.501829
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −4.99028 −0.162162 −0.0810812 0.996707i $$-0.525837\pi$$
−0.0810812 + 0.996707i $$0.525837\pi$$
$$948$$ 0 0
$$949$$ 51.0847 1.65828
$$950$$ 0 0
$$951$$ 24.2348 0.785867
$$952$$ 0 0
$$953$$ −9.66807 −0.313179 −0.156590 0.987664i $$-0.550050\pi$$
−0.156590 + 0.987664i $$0.550050\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 2.59074 0.0837467
$$958$$ 0 0
$$959$$ 81.0459 2.61711
$$960$$ 0 0
$$961$$ 61.2807 1.97680
$$962$$ 0 0
$$963$$ 29.5205 0.951284
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −19.8033 −0.636832 −0.318416 0.947951i $$-0.603151\pi$$
−0.318416 + 0.947951i $$0.603151\pi$$
$$968$$ 0 0
$$969$$ 8.28639 0.266197
$$970$$ 0 0
$$971$$ 39.0410 1.25289 0.626443 0.779468i $$-0.284510\pi$$
0.626443 + 0.779468i $$0.284510\pi$$
$$972$$ 0 0
$$973$$ −24.3778 −0.781517
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −50.8314 −1.62624 −0.813121 0.582095i $$-0.802233\pi$$
−0.813121 + 0.582095i $$0.802233\pi$$
$$978$$ 0 0
$$979$$ 2.83510 0.0906103
$$980$$ 0 0
$$981$$ −64.7324 −2.06675
$$982$$ 0 0
$$983$$ −21.7569 −0.693936 −0.346968 0.937877i $$-0.612789\pi$$
−0.346968 + 0.937877i $$0.612789\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 57.6632 1.83544
$$988$$ 0 0
$$989$$ −23.7345 −0.754712
$$990$$ 0 0
$$991$$ −33.0622 −1.05025 −0.525127 0.851024i $$-0.675982\pi$$
−0.525127 + 0.851024i $$0.675982\pi$$
$$992$$ 0 0
$$993$$ 8.82608 0.280087
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −19.4008 −0.614430 −0.307215 0.951640i $$-0.599397\pi$$
−0.307215 + 0.951640i $$0.599397\pi$$
$$998$$ 0 0
$$999$$ −5.50241 −0.174088
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 2200.2.a.y.1.1 4
4.3 odd 2 4400.2.a.cb.1.4 4
5.2 odd 4 440.2.b.d.89.7 yes 8
5.3 odd 4 440.2.b.d.89.2 8
5.4 even 2 2200.2.a.x.1.4 4
15.2 even 4 3960.2.d.f.3169.2 8
15.8 even 4 3960.2.d.f.3169.1 8
20.3 even 4 880.2.b.j.529.7 8
20.7 even 4 880.2.b.j.529.2 8
20.19 odd 2 4400.2.a.ce.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.b.d.89.2 8 5.3 odd 4
440.2.b.d.89.7 yes 8 5.2 odd 4
880.2.b.j.529.2 8 20.7 even 4
880.2.b.j.529.7 8 20.3 even 4
2200.2.a.x.1.4 4 5.4 even 2
2200.2.a.y.1.1 4 1.1 even 1 trivial
3960.2.d.f.3169.1 8 15.8 even 4
3960.2.d.f.3169.2 8 15.2 even 4
4400.2.a.cb.1.4 4 4.3 odd 2
4400.2.a.ce.1.1 4 20.19 odd 2