# Properties

 Label 176.4.a.i.1.2 Level $176$ Weight $4$ Character 176.1 Self dual yes Analytic conductor $10.384$ Analytic rank $0$ 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] = [176,4,Mod(1,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("176.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 176.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: $$10.3843361610$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 176.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+7.92820 q^{3} +14.8564 q^{5} -3.07180 q^{7} +35.8564 q^{9} +O(q^{10})$$ $$q+7.92820 q^{3} +14.8564 q^{5} -3.07180 q^{7} +35.8564 q^{9} +11.0000 q^{11} +5.35898 q^{13} +117.785 q^{15} -41.2154 q^{17} -139.923 q^{19} -24.3538 q^{21} +111.354 q^{23} +95.7128 q^{25} +70.2154 q^{27} -24.9948 q^{29} -31.4974 q^{31} +87.2102 q^{33} -45.6359 q^{35} +13.1436 q^{37} +42.4871 q^{39} +261.072 q^{41} +57.7128 q^{43} +532.697 q^{45} +343.846 q^{47} -333.564 q^{49} -326.764 q^{51} -342.995 q^{53} +163.420 q^{55} -1109.34 q^{57} -88.3693 q^{59} +738.697 q^{61} -110.144 q^{63} +79.6152 q^{65} -342.359 q^{67} +882.836 q^{69} +207.364 q^{71} -1010.60 q^{73} +758.831 q^{75} -33.7898 q^{77} -1294.23 q^{79} -411.441 q^{81} -441.846 q^{83} -612.313 q^{85} -198.164 q^{87} -1489.11 q^{89} -16.4617 q^{91} -249.718 q^{93} -2078.75 q^{95} +1346.42 q^{97} +394.420 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - 20 q^{7} + 44 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 - 20 * q^7 + 44 * q^9 $$2 q + 2 q^{3} + 2 q^{5} - 20 q^{7} + 44 q^{9} + 22 q^{11} + 80 q^{13} + 194 q^{15} - 124 q^{17} - 72 q^{19} + 76 q^{21} + 98 q^{23} + 136 q^{25} + 182 q^{27} + 144 q^{29} + 34 q^{31} + 22 q^{33} + 172 q^{35} + 54 q^{37} - 400 q^{39} + 536 q^{41} + 60 q^{43} + 428 q^{45} + 272 q^{47} - 390 q^{49} + 164 q^{51} - 492 q^{53} + 22 q^{55} - 1512 q^{57} - 634 q^{59} + 840 q^{61} - 248 q^{63} - 880 q^{65} - 754 q^{67} + 962 q^{69} + 678 q^{71} - 400 q^{73} + 520 q^{75} - 220 q^{77} - 316 q^{79} - 1294 q^{81} - 468 q^{83} + 452 q^{85} - 1200 q^{87} - 1842 q^{89} - 1280 q^{91} - 638 q^{93} - 2952 q^{95} + 2194 q^{97} + 484 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 - 20 * q^7 + 44 * q^9 + 22 * q^11 + 80 * q^13 + 194 * q^15 - 124 * q^17 - 72 * q^19 + 76 * q^21 + 98 * q^23 + 136 * q^25 + 182 * q^27 + 144 * q^29 + 34 * q^31 + 22 * q^33 + 172 * q^35 + 54 * q^37 - 400 * q^39 + 536 * q^41 + 60 * q^43 + 428 * q^45 + 272 * q^47 - 390 * q^49 + 164 * q^51 - 492 * q^53 + 22 * q^55 - 1512 * q^57 - 634 * q^59 + 840 * q^61 - 248 * q^63 - 880 * q^65 - 754 * q^67 + 962 * q^69 + 678 * q^71 - 400 * q^73 + 520 * q^75 - 220 * q^77 - 316 * q^79 - 1294 * q^81 - 468 * q^83 + 452 * q^85 - 1200 * q^87 - 1842 * q^89 - 1280 * q^91 - 638 * q^93 - 2952 * q^95 + 2194 * q^97 + 484 * 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
$$3$$ 7.92820 1.52578 0.762892 0.646526i $$-0.223779\pi$$
0.762892 + 0.646526i $$0.223779\pi$$
$$4$$ 0 0
$$5$$ 14.8564 1.32880 0.664399 0.747378i $$-0.268688\pi$$
0.664399 + 0.747378i $$0.268688\pi$$
$$6$$ 0 0
$$7$$ −3.07180 −0.165861 −0.0829307 0.996555i $$-0.526428\pi$$
−0.0829307 + 0.996555i $$0.526428\pi$$
$$8$$ 0 0
$$9$$ 35.8564 1.32802
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ 5.35898 0.114332 0.0571659 0.998365i $$-0.481794\pi$$
0.0571659 + 0.998365i $$0.481794\pi$$
$$14$$ 0 0
$$15$$ 117.785 2.02746
$$16$$ 0 0
$$17$$ −41.2154 −0.588012 −0.294006 0.955804i $$-0.594989\pi$$
−0.294006 + 0.955804i $$0.594989\pi$$
$$18$$ 0 0
$$19$$ −139.923 −1.68950 −0.844751 0.535159i $$-0.820252\pi$$
−0.844751 + 0.535159i $$0.820252\pi$$
$$20$$ 0 0
$$21$$ −24.3538 −0.253069
$$22$$ 0 0
$$23$$ 111.354 1.00952 0.504758 0.863261i $$-0.331582\pi$$
0.504758 + 0.863261i $$0.331582\pi$$
$$24$$ 0 0
$$25$$ 95.7128 0.765703
$$26$$ 0 0
$$27$$ 70.2154 0.500480
$$28$$ 0 0
$$29$$ −24.9948 −0.160049 −0.0800246 0.996793i $$-0.525500\pi$$
−0.0800246 + 0.996793i $$0.525500\pi$$
$$30$$ 0 0
$$31$$ −31.4974 −0.182487 −0.0912436 0.995829i $$-0.529084\pi$$
−0.0912436 + 0.995829i $$0.529084\pi$$
$$32$$ 0 0
$$33$$ 87.2102 0.460041
$$34$$ 0 0
$$35$$ −45.6359 −0.220396
$$36$$ 0 0
$$37$$ 13.1436 0.0583998 0.0291999 0.999574i $$-0.490704\pi$$
0.0291999 + 0.999574i $$0.490704\pi$$
$$38$$ 0 0
$$39$$ 42.4871 0.174446
$$40$$ 0 0
$$41$$ 261.072 0.994453 0.497226 0.867621i $$-0.334352\pi$$
0.497226 + 0.867621i $$0.334352\pi$$
$$42$$ 0 0
$$43$$ 57.7128 0.204677 0.102339 0.994750i $$-0.467367\pi$$
0.102339 + 0.994750i $$0.467367\pi$$
$$44$$ 0 0
$$45$$ 532.697 1.76466
$$46$$ 0 0
$$47$$ 343.846 1.06713 0.533565 0.845759i $$-0.320852\pi$$
0.533565 + 0.845759i $$0.320852\pi$$
$$48$$ 0 0
$$49$$ −333.564 −0.972490
$$50$$ 0 0
$$51$$ −326.764 −0.897179
$$52$$ 0 0
$$53$$ −342.995 −0.888943 −0.444471 0.895793i $$-0.646608\pi$$
−0.444471 + 0.895793i $$0.646608\pi$$
$$54$$ 0 0
$$55$$ 163.420 0.400647
$$56$$ 0 0
$$57$$ −1109.34 −2.57782
$$58$$ 0 0
$$59$$ −88.3693 −0.194995 −0.0974975 0.995236i $$-0.531084\pi$$
−0.0974975 + 0.995236i $$0.531084\pi$$
$$60$$ 0 0
$$61$$ 738.697 1.55050 0.775250 0.631654i $$-0.217624\pi$$
0.775250 + 0.631654i $$0.217624\pi$$
$$62$$ 0 0
$$63$$ −110.144 −0.220266
$$64$$ 0 0
$$65$$ 79.6152 0.151924
$$66$$ 0 0
$$67$$ −342.359 −0.624266 −0.312133 0.950038i $$-0.601043\pi$$
−0.312133 + 0.950038i $$0.601043\pi$$
$$68$$ 0 0
$$69$$ 882.836 1.54030
$$70$$ 0 0
$$71$$ 207.364 0.346614 0.173307 0.984868i $$-0.444555\pi$$
0.173307 + 0.984868i $$0.444555\pi$$
$$72$$ 0 0
$$73$$ −1010.60 −1.62030 −0.810149 0.586224i $$-0.800614\pi$$
−0.810149 + 0.586224i $$0.800614\pi$$
$$74$$ 0 0
$$75$$ 758.831 1.16830
$$76$$ 0 0
$$77$$ −33.7898 −0.0500091
$$78$$ 0 0
$$79$$ −1294.23 −1.84319 −0.921593 0.388157i $$-0.873112\pi$$
−0.921593 + 0.388157i $$0.873112\pi$$
$$80$$ 0 0
$$81$$ −411.441 −0.564391
$$82$$ 0 0
$$83$$ −441.846 −0.584324 −0.292162 0.956369i $$-0.594375\pi$$
−0.292162 + 0.956369i $$0.594375\pi$$
$$84$$ 0 0
$$85$$ −612.313 −0.781349
$$86$$ 0 0
$$87$$ −198.164 −0.244200
$$88$$ 0 0
$$89$$ −1489.11 −1.77355 −0.886773 0.462205i $$-0.847058\pi$$
−0.886773 + 0.462205i $$0.847058\pi$$
$$90$$ 0 0
$$91$$ −16.4617 −0.0189633
$$92$$ 0 0
$$93$$ −249.718 −0.278436
$$94$$ 0 0
$$95$$ −2078.75 −2.24501
$$96$$ 0 0
$$97$$ 1346.42 1.40936 0.704679 0.709526i $$-0.251091\pi$$
0.704679 + 0.709526i $$0.251091\pi$$
$$98$$ 0 0
$$99$$ 394.420 0.400412
$$100$$ 0 0
$$101$$ −161.461 −0.159069 −0.0795347 0.996832i $$-0.525343\pi$$
−0.0795347 + 0.996832i $$0.525343\pi$$
$$102$$ 0 0
$$103$$ 34.7592 0.0332517 0.0166259 0.999862i $$-0.494708\pi$$
0.0166259 + 0.999862i $$0.494708\pi$$
$$104$$ 0 0
$$105$$ −361.810 −0.336277
$$106$$ 0 0
$$107$$ −832.179 −0.751867 −0.375934 0.926647i $$-0.622678\pi$$
−0.375934 + 0.926647i $$0.622678\pi$$
$$108$$ 0 0
$$109$$ 1044.26 0.917629 0.458815 0.888532i $$-0.348274\pi$$
0.458815 + 0.888532i $$0.348274\pi$$
$$110$$ 0 0
$$111$$ 104.205 0.0891055
$$112$$ 0 0
$$113$$ 295.082 0.245654 0.122827 0.992428i $$-0.460804\pi$$
0.122827 + 0.992428i $$0.460804\pi$$
$$114$$ 0 0
$$115$$ 1654.32 1.34144
$$116$$ 0 0
$$117$$ 192.154 0.151834
$$118$$ 0 0
$$119$$ 126.605 0.0975285
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ 2069.83 1.51732
$$124$$ 0 0
$$125$$ −435.102 −0.311334
$$126$$ 0 0
$$127$$ 1317.60 0.920618 0.460309 0.887759i $$-0.347739\pi$$
0.460309 + 0.887759i $$0.347739\pi$$
$$128$$ 0 0
$$129$$ 457.559 0.312293
$$130$$ 0 0
$$131$$ 1600.71 1.06759 0.533797 0.845612i $$-0.320765\pi$$
0.533797 + 0.845612i $$0.320765\pi$$
$$132$$ 0 0
$$133$$ 429.815 0.280223
$$134$$ 0 0
$$135$$ 1043.15 0.665036
$$136$$ 0 0
$$137$$ 1611.68 1.00507 0.502536 0.864556i $$-0.332400\pi$$
0.502536 + 0.864556i $$0.332400\pi$$
$$138$$ 0 0
$$139$$ 31.8619 0.0194424 0.00972120 0.999953i $$-0.496906\pi$$
0.00972120 + 0.999953i $$0.496906\pi$$
$$140$$ 0 0
$$141$$ 2726.08 1.62821
$$142$$ 0 0
$$143$$ 58.9488 0.0344724
$$144$$ 0 0
$$145$$ −371.334 −0.212673
$$146$$ 0 0
$$147$$ −2644.56 −1.48381
$$148$$ 0 0
$$149$$ −2428.34 −1.33515 −0.667576 0.744542i $$-0.732668\pi$$
−0.667576 + 0.744542i $$0.732668\pi$$
$$150$$ 0 0
$$151$$ 2576.68 1.38866 0.694328 0.719659i $$-0.255702\pi$$
0.694328 + 0.719659i $$0.255702\pi$$
$$152$$ 0 0
$$153$$ −1477.84 −0.780889
$$154$$ 0 0
$$155$$ −467.939 −0.242489
$$156$$ 0 0
$$157$$ 2475.94 1.25861 0.629305 0.777158i $$-0.283340\pi$$
0.629305 + 0.777158i $$0.283340\pi$$
$$158$$ 0 0
$$159$$ −2719.33 −1.35633
$$160$$ 0 0
$$161$$ −342.056 −0.167440
$$162$$ 0 0
$$163$$ 2725.11 1.30949 0.654745 0.755850i $$-0.272776\pi$$
0.654745 + 0.755850i $$0.272776\pi$$
$$164$$ 0 0
$$165$$ 1295.63 0.611301
$$166$$ 0 0
$$167$$ −2737.30 −1.26837 −0.634187 0.773180i $$-0.718665\pi$$
−0.634187 + 0.773180i $$0.718665\pi$$
$$168$$ 0 0
$$169$$ −2168.28 −0.986928
$$170$$ 0 0
$$171$$ −5017.14 −2.24368
$$172$$ 0 0
$$173$$ 2307.42 1.01404 0.507022 0.861933i $$-0.330746\pi$$
0.507022 + 0.861933i $$0.330746\pi$$
$$174$$ 0 0
$$175$$ −294.010 −0.127001
$$176$$ 0 0
$$177$$ −700.610 −0.297520
$$178$$ 0 0
$$179$$ 1312.15 0.547905 0.273953 0.961743i $$-0.411669\pi$$
0.273953 + 0.961743i $$0.411669\pi$$
$$180$$ 0 0
$$181$$ −803.174 −0.329831 −0.164916 0.986308i $$-0.552735\pi$$
−0.164916 + 0.986308i $$0.552735\pi$$
$$182$$ 0 0
$$183$$ 5856.54 2.36573
$$184$$ 0 0
$$185$$ 195.267 0.0776015
$$186$$ 0 0
$$187$$ −453.369 −0.177292
$$188$$ 0 0
$$189$$ −215.687 −0.0830103
$$190$$ 0 0
$$191$$ −1718.25 −0.650932 −0.325466 0.945554i $$-0.605521\pi$$
−0.325466 + 0.945554i $$0.605521\pi$$
$$192$$ 0 0
$$193$$ 1340.18 0.499837 0.249919 0.968267i $$-0.419596\pi$$
0.249919 + 0.968267i $$0.419596\pi$$
$$194$$ 0 0
$$195$$ 631.206 0.231803
$$196$$ 0 0
$$197$$ −3518.33 −1.27244 −0.636220 0.771508i $$-0.719503\pi$$
−0.636220 + 0.771508i $$0.719503\pi$$
$$198$$ 0 0
$$199$$ −823.692 −0.293417 −0.146709 0.989180i $$-0.546868\pi$$
−0.146709 + 0.989180i $$0.546868\pi$$
$$200$$ 0 0
$$201$$ −2714.29 −0.952494
$$202$$ 0 0
$$203$$ 76.7791 0.0265460
$$204$$ 0 0
$$205$$ 3878.59 1.32143
$$206$$ 0 0
$$207$$ 3992.75 1.34065
$$208$$ 0 0
$$209$$ −1539.15 −0.509404
$$210$$ 0 0
$$211$$ 107.343 0.0350228 0.0175114 0.999847i $$-0.494426\pi$$
0.0175114 + 0.999847i $$0.494426\pi$$
$$212$$ 0 0
$$213$$ 1644.03 0.528858
$$214$$ 0 0
$$215$$ 857.405 0.271975
$$216$$ 0 0
$$217$$ 96.7537 0.0302676
$$218$$ 0 0
$$219$$ −8012.24 −2.47222
$$220$$ 0 0
$$221$$ −220.873 −0.0672285
$$222$$ 0 0
$$223$$ 3933.68 1.18125 0.590625 0.806946i $$-0.298881\pi$$
0.590625 + 0.806946i $$0.298881\pi$$
$$224$$ 0 0
$$225$$ 3431.92 1.01686
$$226$$ 0 0
$$227$$ 1771.90 0.518085 0.259042 0.965866i $$-0.416593\pi$$
0.259042 + 0.965866i $$0.416593\pi$$
$$228$$ 0 0
$$229$$ 1915.37 0.552713 0.276356 0.961055i $$-0.410873\pi$$
0.276356 + 0.961055i $$0.410873\pi$$
$$230$$ 0 0
$$231$$ −267.892 −0.0763031
$$232$$ 0 0
$$233$$ 4396.32 1.23610 0.618052 0.786137i $$-0.287922\pi$$
0.618052 + 0.786137i $$0.287922\pi$$
$$234$$ 0 0
$$235$$ 5108.32 1.41800
$$236$$ 0 0
$$237$$ −10260.9 −2.81230
$$238$$ 0 0
$$239$$ 4084.49 1.10546 0.552728 0.833362i $$-0.313587\pi$$
0.552728 + 0.833362i $$0.313587\pi$$
$$240$$ 0 0
$$241$$ 3908.58 1.04471 0.522353 0.852730i $$-0.325054\pi$$
0.522353 + 0.852730i $$0.325054\pi$$
$$242$$ 0 0
$$243$$ −5157.80 −1.36162
$$244$$ 0 0
$$245$$ −4955.56 −1.29224
$$246$$ 0 0
$$247$$ −749.845 −0.193164
$$248$$ 0 0
$$249$$ −3503.05 −0.891552
$$250$$ 0 0
$$251$$ −1094.89 −0.275335 −0.137667 0.990479i $$-0.543960\pi$$
−0.137667 + 0.990479i $$0.543960\pi$$
$$252$$ 0 0
$$253$$ 1224.89 0.304381
$$254$$ 0 0
$$255$$ −4854.54 −1.19217
$$256$$ 0 0
$$257$$ 783.179 0.190091 0.0950454 0.995473i $$-0.469700\pi$$
0.0950454 + 0.995473i $$0.469700\pi$$
$$258$$ 0 0
$$259$$ −40.3744 −0.00968628
$$260$$ 0 0
$$261$$ −896.225 −0.212548
$$262$$ 0 0
$$263$$ −6180.06 −1.44897 −0.724484 0.689292i $$-0.757922\pi$$
−0.724484 + 0.689292i $$0.757922\pi$$
$$264$$ 0 0
$$265$$ −5095.67 −1.18122
$$266$$ 0 0
$$267$$ −11806.0 −2.70605
$$268$$ 0 0
$$269$$ 986.965 0.223704 0.111852 0.993725i $$-0.464322\pi$$
0.111852 + 0.993725i $$0.464322\pi$$
$$270$$ 0 0
$$271$$ −4576.99 −1.02595 −0.512975 0.858404i $$-0.671457\pi$$
−0.512975 + 0.858404i $$0.671457\pi$$
$$272$$ 0 0
$$273$$ −130.512 −0.0289338
$$274$$ 0 0
$$275$$ 1052.84 0.230868
$$276$$ 0 0
$$277$$ 567.836 0.123169 0.0615847 0.998102i $$-0.480385\pi$$
0.0615847 + 0.998102i $$0.480385\pi$$
$$278$$ 0 0
$$279$$ −1129.38 −0.242346
$$280$$ 0 0
$$281$$ 5311.01 1.12750 0.563752 0.825944i $$-0.309357\pi$$
0.563752 + 0.825944i $$0.309357\pi$$
$$282$$ 0 0
$$283$$ 4728.44 0.993204 0.496602 0.867978i $$-0.334581\pi$$
0.496602 + 0.867978i $$0.334581\pi$$
$$284$$ 0 0
$$285$$ −16480.8 −3.42539
$$286$$ 0 0
$$287$$ −801.960 −0.164941
$$288$$ 0 0
$$289$$ −3214.29 −0.654242
$$290$$ 0 0
$$291$$ 10674.7 2.15038
$$292$$ 0 0
$$293$$ 2328.92 0.464358 0.232179 0.972673i $$-0.425415\pi$$
0.232179 + 0.972673i $$0.425415\pi$$
$$294$$ 0 0
$$295$$ −1312.85 −0.259109
$$296$$ 0 0
$$297$$ 772.369 0.150900
$$298$$ 0 0
$$299$$ 596.743 0.115420
$$300$$ 0 0
$$301$$ −177.282 −0.0339481
$$302$$ 0 0
$$303$$ −1280.10 −0.242705
$$304$$ 0 0
$$305$$ 10974.4 2.06030
$$306$$ 0 0
$$307$$ 1678.07 0.311962 0.155981 0.987760i $$-0.450146\pi$$
0.155981 + 0.987760i $$0.450146\pi$$
$$308$$ 0 0
$$309$$ 275.578 0.0507349
$$310$$ 0 0
$$311$$ −3572.71 −0.651413 −0.325707 0.945471i $$-0.605602\pi$$
−0.325707 + 0.945471i $$0.605602\pi$$
$$312$$ 0 0
$$313$$ 7184.36 1.29739 0.648697 0.761047i $$-0.275314\pi$$
0.648697 + 0.761047i $$0.275314\pi$$
$$314$$ 0 0
$$315$$ −1636.34 −0.292690
$$316$$ 0 0
$$317$$ −15.7077 −0.00278306 −0.00139153 0.999999i $$-0.500443\pi$$
−0.00139153 + 0.999999i $$0.500443\pi$$
$$318$$ 0 0
$$319$$ −274.943 −0.0482566
$$320$$ 0 0
$$321$$ −6597.69 −1.14719
$$322$$ 0 0
$$323$$ 5766.98 0.993447
$$324$$ 0 0
$$325$$ 512.923 0.0875442
$$326$$ 0 0
$$327$$ 8279.08 1.40010
$$328$$ 0 0
$$329$$ −1056.23 −0.176996
$$330$$ 0 0
$$331$$ 1318.95 0.219022 0.109511 0.993986i $$-0.465072\pi$$
0.109511 + 0.993986i $$0.465072\pi$$
$$332$$ 0 0
$$333$$ 471.282 0.0775558
$$334$$ 0 0
$$335$$ −5086.22 −0.829523
$$336$$ 0 0
$$337$$ −239.183 −0.0386621 −0.0193310 0.999813i $$-0.506154\pi$$
−0.0193310 + 0.999813i $$0.506154\pi$$
$$338$$ 0 0
$$339$$ 2339.47 0.374816
$$340$$ 0 0
$$341$$ −346.472 −0.0550220
$$342$$ 0 0
$$343$$ 2078.27 0.327160
$$344$$ 0 0
$$345$$ 13115.8 2.04675
$$346$$ 0 0
$$347$$ 5862.79 0.907006 0.453503 0.891255i $$-0.350174\pi$$
0.453503 + 0.891255i $$0.350174\pi$$
$$348$$ 0 0
$$349$$ 3491.73 0.535553 0.267776 0.963481i $$-0.413711\pi$$
0.267776 + 0.963481i $$0.413711\pi$$
$$350$$ 0 0
$$351$$ 376.283 0.0572208
$$352$$ 0 0
$$353$$ −10916.7 −1.64600 −0.822999 0.568043i $$-0.807701\pi$$
−0.822999 + 0.568043i $$0.807701\pi$$
$$354$$ 0 0
$$355$$ 3080.69 0.460580
$$356$$ 0 0
$$357$$ 1003.75 0.148807
$$358$$ 0 0
$$359$$ 11500.7 1.69077 0.845384 0.534160i $$-0.179372\pi$$
0.845384 + 0.534160i $$0.179372\pi$$
$$360$$ 0 0
$$361$$ 12719.5 1.85442
$$362$$ 0 0
$$363$$ 959.313 0.138708
$$364$$ 0 0
$$365$$ −15013.9 −2.15305
$$366$$ 0 0
$$367$$ −6767.01 −0.962493 −0.481246 0.876585i $$-0.659816\pi$$
−0.481246 + 0.876585i $$0.659816\pi$$
$$368$$ 0 0
$$369$$ 9361.10 1.32065
$$370$$ 0 0
$$371$$ 1053.61 0.147441
$$372$$ 0 0
$$373$$ −5310.22 −0.737139 −0.368569 0.929600i $$-0.620152\pi$$
−0.368569 + 0.929600i $$0.620152\pi$$
$$374$$ 0 0
$$375$$ −3449.58 −0.475028
$$376$$ 0 0
$$377$$ −133.947 −0.0182987
$$378$$ 0 0
$$379$$ 838.267 0.113612 0.0568059 0.998385i $$-0.481908\pi$$
0.0568059 + 0.998385i $$0.481908\pi$$
$$380$$ 0 0
$$381$$ 10446.2 1.40466
$$382$$ 0 0
$$383$$ 2832.16 0.377851 0.188925 0.981991i $$-0.439500\pi$$
0.188925 + 0.981991i $$0.439500\pi$$
$$384$$ 0 0
$$385$$ −501.994 −0.0664520
$$386$$ 0 0
$$387$$ 2069.37 0.271814
$$388$$ 0 0
$$389$$ 3111.25 0.405519 0.202759 0.979229i $$-0.435009\pi$$
0.202759 + 0.979229i $$0.435009\pi$$
$$390$$ 0 0
$$391$$ −4589.49 −0.593608
$$392$$ 0 0
$$393$$ 12690.8 1.62892
$$394$$ 0 0
$$395$$ −19227.5 −2.44922
$$396$$ 0 0
$$397$$ 14208.7 1.79626 0.898131 0.439728i $$-0.144925\pi$$
0.898131 + 0.439728i $$0.144925\pi$$
$$398$$ 0 0
$$399$$ 3407.66 0.427560
$$400$$ 0 0
$$401$$ −6261.68 −0.779784 −0.389892 0.920861i $$-0.627488\pi$$
−0.389892 + 0.920861i $$0.627488\pi$$
$$402$$ 0 0
$$403$$ −168.794 −0.0208641
$$404$$ 0 0
$$405$$ −6112.54 −0.749961
$$406$$ 0 0
$$407$$ 144.580 0.0176082
$$408$$ 0 0
$$409$$ −4192.50 −0.506860 −0.253430 0.967354i $$-0.581559\pi$$
−0.253430 + 0.967354i $$0.581559\pi$$
$$410$$ 0 0
$$411$$ 12777.7 1.53352
$$412$$ 0 0
$$413$$ 271.453 0.0323421
$$414$$ 0 0
$$415$$ −6564.25 −0.776448
$$416$$ 0 0
$$417$$ 252.608 0.0296649
$$418$$ 0 0
$$419$$ 9287.15 1.08283 0.541416 0.840755i $$-0.317888\pi$$
0.541416 + 0.840755i $$0.317888\pi$$
$$420$$ 0 0
$$421$$ 13146.0 1.52185 0.760923 0.648842i $$-0.224746\pi$$
0.760923 + 0.648842i $$0.224746\pi$$
$$422$$ 0 0
$$423$$ 12329.1 1.41716
$$424$$ 0 0
$$425$$ −3944.84 −0.450242
$$426$$ 0 0
$$427$$ −2269.13 −0.257168
$$428$$ 0 0
$$429$$ 467.358 0.0525974
$$430$$ 0 0
$$431$$ −4909.67 −0.548701 −0.274351 0.961630i $$-0.588463\pi$$
−0.274351 + 0.961630i $$0.588463\pi$$
$$432$$ 0 0
$$433$$ −11743.3 −1.30334 −0.651671 0.758502i $$-0.725932\pi$$
−0.651671 + 0.758502i $$0.725932\pi$$
$$434$$ 0 0
$$435$$ −2944.01 −0.324493
$$436$$ 0 0
$$437$$ −15581.0 −1.70558
$$438$$ 0 0
$$439$$ 11824.2 1.28551 0.642754 0.766073i $$-0.277792\pi$$
0.642754 + 0.766073i $$0.277792\pi$$
$$440$$ 0 0
$$441$$ −11960.4 −1.29148
$$442$$ 0 0
$$443$$ −10102.1 −1.08344 −0.541722 0.840558i $$-0.682228\pi$$
−0.541722 + 0.840558i $$0.682228\pi$$
$$444$$ 0 0
$$445$$ −22122.9 −2.35668
$$446$$ 0 0
$$447$$ −19252.4 −2.03715
$$448$$ 0 0
$$449$$ −345.254 −0.0362885 −0.0181443 0.999835i $$-0.505776\pi$$
−0.0181443 + 0.999835i $$0.505776\pi$$
$$450$$ 0 0
$$451$$ 2871.79 0.299839
$$452$$ 0 0
$$453$$ 20428.4 2.11879
$$454$$ 0 0
$$455$$ −244.562 −0.0251983
$$456$$ 0 0
$$457$$ −10567.1 −1.08164 −0.540821 0.841138i $$-0.681886\pi$$
−0.540821 + 0.841138i $$0.681886\pi$$
$$458$$ 0 0
$$459$$ −2893.95 −0.294288
$$460$$ 0 0
$$461$$ 4733.96 0.478270 0.239135 0.970986i $$-0.423136\pi$$
0.239135 + 0.970986i $$0.423136\pi$$
$$462$$ 0 0
$$463$$ −3431.20 −0.344409 −0.172204 0.985061i $$-0.555089\pi$$
−0.172204 + 0.985061i $$0.555089\pi$$
$$464$$ 0 0
$$465$$ −3709.91 −0.369985
$$466$$ 0 0
$$467$$ −5116.96 −0.507034 −0.253517 0.967331i $$-0.581587\pi$$
−0.253517 + 0.967331i $$0.581587\pi$$
$$468$$ 0 0
$$469$$ 1051.66 0.103542
$$470$$ 0 0
$$471$$ 19629.8 1.92037
$$472$$ 0 0
$$473$$ 634.841 0.0617125
$$474$$ 0 0
$$475$$ −13392.4 −1.29366
$$476$$ 0 0
$$477$$ −12298.6 −1.18053
$$478$$ 0 0
$$479$$ −11566.9 −1.10335 −0.551675 0.834059i $$-0.686011\pi$$
−0.551675 + 0.834059i $$0.686011\pi$$
$$480$$ 0 0
$$481$$ 70.4363 0.00667696
$$482$$ 0 0
$$483$$ −2711.89 −0.255477
$$484$$ 0 0
$$485$$ 20002.9 1.87275
$$486$$ 0 0
$$487$$ 18326.5 1.70525 0.852623 0.522527i $$-0.175010\pi$$
0.852623 + 0.522527i $$0.175010\pi$$
$$488$$ 0 0
$$489$$ 21605.2 1.99800
$$490$$ 0 0
$$491$$ 7617.58 0.700156 0.350078 0.936721i $$-0.386155\pi$$
0.350078 + 0.936721i $$0.386155\pi$$
$$492$$ 0 0
$$493$$ 1030.17 0.0941108
$$494$$ 0 0
$$495$$ 5859.67 0.532066
$$496$$ 0 0
$$497$$ −636.980 −0.0574899
$$498$$ 0 0
$$499$$ −12909.1 −1.15810 −0.579050 0.815292i $$-0.696576\pi$$
−0.579050 + 0.815292i $$0.696576\pi$$
$$500$$ 0 0
$$501$$ −21701.8 −1.93526
$$502$$ 0 0
$$503$$ −10165.7 −0.901121 −0.450561 0.892746i $$-0.648776\pi$$
−0.450561 + 0.892746i $$0.648776\pi$$
$$504$$ 0 0
$$505$$ −2398.74 −0.211371
$$506$$ 0 0
$$507$$ −17190.6 −1.50584
$$508$$ 0 0
$$509$$ 6449.93 0.561666 0.280833 0.959757i $$-0.409389\pi$$
0.280833 + 0.959757i $$0.409389\pi$$
$$510$$ 0 0
$$511$$ 3104.36 0.268745
$$512$$ 0 0
$$513$$ −9824.75 −0.845562
$$514$$ 0 0
$$515$$ 516.397 0.0441848
$$516$$ 0 0
$$517$$ 3782.31 0.321752
$$518$$ 0 0
$$519$$ 18293.7 1.54721
$$520$$ 0 0
$$521$$ −19327.4 −1.62524 −0.812620 0.582794i $$-0.801959\pi$$
−0.812620 + 0.582794i $$0.801959\pi$$
$$522$$ 0 0
$$523$$ −6259.09 −0.523310 −0.261655 0.965161i $$-0.584268\pi$$
−0.261655 + 0.965161i $$0.584268\pi$$
$$524$$ 0 0
$$525$$ −2330.97 −0.193775
$$526$$ 0 0
$$527$$ 1298.18 0.107305
$$528$$ 0 0
$$529$$ 232.675 0.0191235
$$530$$ 0 0
$$531$$ −3168.61 −0.258956
$$532$$ 0 0
$$533$$ 1399.08 0.113698
$$534$$ 0 0
$$535$$ −12363.2 −0.999079
$$536$$ 0 0
$$537$$ 10403.0 0.835985
$$538$$ 0 0
$$539$$ −3669.20 −0.293217
$$540$$ 0 0
$$541$$ −14008.2 −1.11323 −0.556616 0.830770i $$-0.687900\pi$$
−0.556616 + 0.830770i $$0.687900\pi$$
$$542$$ 0 0
$$543$$ −6367.72 −0.503251
$$544$$ 0 0
$$545$$ 15513.9 1.21934
$$546$$ 0 0
$$547$$ 4949.45 0.386879 0.193440 0.981112i $$-0.438036\pi$$
0.193440 + 0.981112i $$0.438036\pi$$
$$548$$ 0 0
$$549$$ 26487.0 2.05909
$$550$$ 0 0
$$551$$ 3497.35 0.270404
$$552$$ 0 0
$$553$$ 3975.60 0.305714
$$554$$ 0 0
$$555$$ 1548.11 0.118403
$$556$$ 0 0
$$557$$ −3801.58 −0.289188 −0.144594 0.989491i $$-0.546188\pi$$
−0.144594 + 0.989491i $$0.546188\pi$$
$$558$$ 0 0
$$559$$ 309.282 0.0234011
$$560$$ 0 0
$$561$$ −3594.40 −0.270510
$$562$$ 0 0
$$563$$ 9900.11 0.741101 0.370551 0.928812i $$-0.379169\pi$$
0.370551 + 0.928812i $$0.379169\pi$$
$$564$$ 0 0
$$565$$ 4383.85 0.326425
$$566$$ 0 0
$$567$$ 1263.86 0.0936107
$$568$$ 0 0
$$569$$ 5329.16 0.392636 0.196318 0.980540i $$-0.437102\pi$$
0.196318 + 0.980540i $$0.437102\pi$$
$$570$$ 0 0
$$571$$ 16962.6 1.24319 0.621597 0.783337i $$-0.286484\pi$$
0.621597 + 0.783337i $$0.286484\pi$$
$$572$$ 0 0
$$573$$ −13622.6 −0.993181
$$574$$ 0 0
$$575$$ 10658.0 0.772989
$$576$$ 0 0
$$577$$ −15487.0 −1.11738 −0.558692 0.829375i $$-0.688697\pi$$
−0.558692 + 0.829375i $$0.688697\pi$$
$$578$$ 0 0
$$579$$ 10625.3 0.762643
$$580$$ 0 0
$$581$$ 1357.26 0.0969169
$$582$$ 0 0
$$583$$ −3772.94 −0.268026
$$584$$ 0 0
$$585$$ 2854.72 0.201757
$$586$$ 0 0
$$587$$ −11084.2 −0.779373 −0.389686 0.920948i $$-0.627417\pi$$
−0.389686 + 0.920948i $$0.627417\pi$$
$$588$$ 0 0
$$589$$ 4407.22 0.308313
$$590$$ 0 0
$$591$$ −27894.0 −1.94147
$$592$$ 0 0
$$593$$ 4349.68 0.301214 0.150607 0.988594i $$-0.451877\pi$$
0.150607 + 0.988594i $$0.451877\pi$$
$$594$$ 0 0
$$595$$ 1880.90 0.129596
$$596$$ 0 0
$$597$$ −6530.40 −0.447691
$$598$$ 0 0
$$599$$ −13183.9 −0.899299 −0.449650 0.893205i $$-0.648451\pi$$
−0.449650 + 0.893205i $$0.648451\pi$$
$$600$$ 0 0
$$601$$ −18765.0 −1.27361 −0.636806 0.771024i $$-0.719745\pi$$
−0.636806 + 0.771024i $$0.719745\pi$$
$$602$$ 0 0
$$603$$ −12275.8 −0.829034
$$604$$ 0 0
$$605$$ 1797.63 0.120800
$$606$$ 0 0
$$607$$ −21871.4 −1.46249 −0.731244 0.682116i $$-0.761060\pi$$
−0.731244 + 0.682116i $$0.761060\pi$$
$$608$$ 0 0
$$609$$ 608.720 0.0405034
$$610$$ 0 0
$$611$$ 1842.67 0.122007
$$612$$ 0 0
$$613$$ −3527.85 −0.232445 −0.116222 0.993223i $$-0.537079\pi$$
−0.116222 + 0.993223i $$0.537079\pi$$
$$614$$ 0 0
$$615$$ 30750.2 2.01621
$$616$$ 0 0
$$617$$ −22728.1 −1.48298 −0.741490 0.670963i $$-0.765881\pi$$
−0.741490 + 0.670963i $$0.765881\pi$$
$$618$$ 0 0
$$619$$ 21443.3 1.39237 0.696187 0.717861i $$-0.254879\pi$$
0.696187 + 0.717861i $$0.254879\pi$$
$$620$$ 0 0
$$621$$ 7818.75 0.505243
$$622$$ 0 0
$$623$$ 4574.25 0.294163
$$624$$ 0 0
$$625$$ −18428.2 −1.17940
$$626$$ 0 0
$$627$$ −12202.7 −0.777240
$$628$$ 0 0
$$629$$ −541.718 −0.0343398
$$630$$ 0 0
$$631$$ −21532.0 −1.35844 −0.679219 0.733936i $$-0.737681\pi$$
−0.679219 + 0.733936i $$0.737681\pi$$
$$632$$ 0 0
$$633$$ 851.038 0.0534372
$$634$$ 0 0
$$635$$ 19574.9 1.22332
$$636$$ 0 0
$$637$$ −1787.56 −0.111187
$$638$$ 0 0
$$639$$ 7435.33 0.460309
$$640$$ 0 0
$$641$$ 20148.3 1.24151 0.620756 0.784004i $$-0.286826\pi$$
0.620756 + 0.784004i $$0.286826\pi$$
$$642$$ 0 0
$$643$$ −28869.7 −1.77062 −0.885310 0.465000i $$-0.846054\pi$$
−0.885310 + 0.465000i $$0.846054\pi$$
$$644$$ 0 0
$$645$$ 6797.68 0.414974
$$646$$ 0 0
$$647$$ 1590.02 0.0966155 0.0483077 0.998833i $$-0.484617\pi$$
0.0483077 + 0.998833i $$0.484617\pi$$
$$648$$ 0 0
$$649$$ −972.062 −0.0587932
$$650$$ 0 0
$$651$$ 767.083 0.0461818
$$652$$ 0 0
$$653$$ 20028.1 1.20024 0.600122 0.799909i $$-0.295119\pi$$
0.600122 + 0.799909i $$0.295119\pi$$
$$654$$ 0 0
$$655$$ 23780.8 1.41862
$$656$$ 0 0
$$657$$ −36236.5 −2.15178
$$658$$ 0 0
$$659$$ 10520.7 0.621897 0.310948 0.950427i $$-0.399353\pi$$
0.310948 + 0.950427i $$0.399353\pi$$
$$660$$ 0 0
$$661$$ 3295.83 0.193938 0.0969690 0.995287i $$-0.469085\pi$$
0.0969690 + 0.995287i $$0.469085\pi$$
$$662$$ 0 0
$$663$$ −1751.12 −0.102576
$$664$$ 0 0
$$665$$ 6385.51 0.372360
$$666$$ 0 0
$$667$$ −2783.27 −0.161572
$$668$$ 0 0
$$669$$ 31187.0 1.80233
$$670$$ 0 0
$$671$$ 8125.67 0.467493
$$672$$ 0 0
$$673$$ −1187.64 −0.0680239 −0.0340119 0.999421i $$-0.510828\pi$$
−0.0340119 + 0.999421i $$0.510828\pi$$
$$674$$ 0 0
$$675$$ 6720.51 0.383219
$$676$$ 0 0
$$677$$ 13221.4 0.750574 0.375287 0.926909i $$-0.377544\pi$$
0.375287 + 0.926909i $$0.377544\pi$$
$$678$$ 0 0
$$679$$ −4135.91 −0.233758
$$680$$ 0 0
$$681$$ 14048.0 0.790485
$$682$$ 0 0
$$683$$ 13831.4 0.774882 0.387441 0.921894i $$-0.373359\pi$$
0.387441 + 0.921894i $$0.373359\pi$$
$$684$$ 0 0
$$685$$ 23943.7 1.33554
$$686$$ 0 0
$$687$$ 15185.4 0.843320
$$688$$ 0 0
$$689$$ −1838.10 −0.101635
$$690$$ 0 0
$$691$$ 9817.07 0.540462 0.270231 0.962796i $$-0.412900\pi$$
0.270231 + 0.962796i $$0.412900\pi$$
$$692$$ 0 0
$$693$$ −1211.58 −0.0664128
$$694$$ 0 0
$$695$$ 473.354 0.0258350
$$696$$ 0 0
$$697$$ −10760.2 −0.584750
$$698$$ 0 0
$$699$$ 34854.9 1.88603
$$700$$ 0 0
$$701$$ 29949.8 1.61368 0.806838 0.590773i $$-0.201177\pi$$
0.806838 + 0.590773i $$0.201177\pi$$
$$702$$ 0 0
$$703$$ −1839.09 −0.0986667
$$704$$ 0 0
$$705$$ 40499.8 2.16356
$$706$$ 0 0
$$707$$ 495.976 0.0263835
$$708$$ 0 0
$$709$$ 11307.5 0.598959 0.299479 0.954103i $$-0.403187\pi$$
0.299479 + 0.954103i $$0.403187\pi$$
$$710$$ 0 0
$$711$$ −46406.3 −2.44778
$$712$$ 0 0
$$713$$ −3507.36 −0.184224
$$714$$ 0 0
$$715$$ 875.768 0.0458068
$$716$$ 0 0
$$717$$ 32382.7 1.68669
$$718$$ 0 0
$$719$$ 32623.4 1.69214 0.846070 0.533071i $$-0.178962\pi$$
0.846070 + 0.533071i $$0.178962\pi$$
$$720$$ 0 0
$$721$$ −106.773 −0.00551518
$$722$$ 0 0
$$723$$ 30988.0 1.59399
$$724$$ 0 0
$$725$$ −2392.33 −0.122550
$$726$$ 0 0
$$727$$ 502.545 0.0256373 0.0128187 0.999918i $$-0.495920\pi$$
0.0128187 + 0.999918i $$0.495920\pi$$
$$728$$ 0 0
$$729$$ −29783.2 −1.51314
$$730$$ 0 0
$$731$$ −2378.66 −0.120353
$$732$$ 0 0
$$733$$ 8631.37 0.434935 0.217467 0.976068i $$-0.430220\pi$$
0.217467 + 0.976068i $$0.430220\pi$$
$$734$$ 0 0
$$735$$ −39288.7 −1.97168
$$736$$ 0 0
$$737$$ −3765.95 −0.188223
$$738$$ 0 0
$$739$$ 18357.5 0.913792 0.456896 0.889520i $$-0.348961\pi$$
0.456896 + 0.889520i $$0.348961\pi$$
$$740$$ 0 0
$$741$$ −5944.93 −0.294726
$$742$$ 0 0
$$743$$ −11182.6 −0.552155 −0.276078 0.961135i $$-0.589035\pi$$
−0.276078 + 0.961135i $$0.589035\pi$$
$$744$$ 0 0
$$745$$ −36076.4 −1.77415
$$746$$ 0 0
$$747$$ −15843.0 −0.775991
$$748$$ 0 0
$$749$$ 2556.29 0.124706
$$750$$ 0 0
$$751$$ −16733.4 −0.813063 −0.406531 0.913637i $$-0.633262\pi$$
−0.406531 + 0.913637i $$0.633262\pi$$
$$752$$ 0 0
$$753$$ −8680.53 −0.420101
$$754$$ 0 0
$$755$$ 38280.2 1.84524
$$756$$ 0 0
$$757$$ −24402.4 −1.17163 −0.585813 0.810446i $$-0.699225\pi$$
−0.585813 + 0.810446i $$0.699225\pi$$
$$758$$ 0 0
$$759$$ 9711.19 0.464419
$$760$$ 0 0
$$761$$ 8469.33 0.403434 0.201717 0.979444i $$-0.435348\pi$$
0.201717 + 0.979444i $$0.435348\pi$$
$$762$$ 0 0
$$763$$ −3207.74 −0.152199
$$764$$ 0 0
$$765$$ −21955.3 −1.03764
$$766$$ 0 0
$$767$$ −473.570 −0.0222941
$$768$$ 0 0
$$769$$ 32834.7 1.53973 0.769864 0.638208i $$-0.220324\pi$$
0.769864 + 0.638208i $$0.220324\pi$$
$$770$$ 0 0
$$771$$ 6209.20 0.290038
$$772$$ 0 0
$$773$$ −35571.4 −1.65513 −0.827564 0.561371i $$-0.810274\pi$$
−0.827564 + 0.561371i $$0.810274\pi$$
$$774$$ 0 0
$$775$$ −3014.71 −0.139731
$$776$$ 0 0
$$777$$ −320.097 −0.0147792
$$778$$ 0 0
$$779$$ −36530.0 −1.68013
$$780$$ 0 0
$$781$$ 2281.01 0.104508
$$782$$ 0 0
$$783$$ −1755.02 −0.0801014
$$784$$ 0 0
$$785$$ 36783.6 1.67244
$$786$$ 0 0
$$787$$ −15729.6 −0.712452 −0.356226 0.934400i $$-0.615937\pi$$
−0.356226 + 0.934400i $$0.615937\pi$$
$$788$$ 0 0
$$789$$ −48996.7 −2.21081
$$790$$ 0 0
$$791$$ −906.431 −0.0407446
$$792$$ 0 0
$$793$$ 3958.67 0.177272
$$794$$ 0 0
$$795$$ −40399.5 −1.80229
$$796$$ 0 0
$$797$$ 7888.07 0.350577 0.175288 0.984517i $$-0.443914\pi$$
0.175288 + 0.984517i $$0.443914\pi$$
$$798$$ 0 0
$$799$$ −14171.8 −0.627485
$$800$$ 0 0
$$801$$ −53394.2 −2.35530
$$802$$ 0 0
$$803$$ −11116.6 −0.488538
$$804$$ 0 0
$$805$$ −5081.73 −0.222494
$$806$$ 0 0
$$807$$ 7824.86 0.341323
$$808$$ 0 0
$$809$$ 5896.97 0.256275 0.128138 0.991756i $$-0.459100\pi$$
0.128138 + 0.991756i $$0.459100\pi$$
$$810$$ 0 0
$$811$$ −14197.9 −0.614744 −0.307372 0.951589i $$-0.599450\pi$$
−0.307372 + 0.951589i $$0.599450\pi$$
$$812$$ 0 0
$$813$$ −36287.3 −1.56538
$$814$$ 0 0
$$815$$ 40485.3 1.74005
$$816$$ 0 0
$$817$$ −8075.35 −0.345803
$$818$$ 0 0
$$819$$ −590.258 −0.0251835
$$820$$ 0 0
$$821$$ −19841.7 −0.843459 −0.421729 0.906722i $$-0.638577\pi$$
−0.421729 + 0.906722i $$0.638577\pi$$
$$822$$ 0 0
$$823$$ 28202.2 1.19449 0.597246 0.802058i $$-0.296262\pi$$
0.597246 + 0.802058i $$0.296262\pi$$
$$824$$ 0 0
$$825$$ 8347.14 0.352255
$$826$$ 0 0
$$827$$ −34031.0 −1.43092 −0.715462 0.698651i $$-0.753784\pi$$
−0.715462 + 0.698651i $$0.753784\pi$$
$$828$$ 0 0
$$829$$ 4931.55 0.206610 0.103305 0.994650i $$-0.467058\pi$$
0.103305 + 0.994650i $$0.467058\pi$$
$$830$$ 0 0
$$831$$ 4501.92 0.187930
$$832$$ 0 0
$$833$$ 13748.0 0.571836
$$834$$ 0 0
$$835$$ −40666.4 −1.68541
$$836$$ 0 0
$$837$$ −2211.60 −0.0913312
$$838$$ 0 0
$$839$$ 38189.8 1.57146 0.785731 0.618568i $$-0.212287\pi$$
0.785731 + 0.618568i $$0.212287\pi$$
$$840$$ 0 0
$$841$$ −23764.3 −0.974384
$$842$$ 0 0
$$843$$ 42106.8 1.72033
$$844$$ 0 0
$$845$$ −32212.9 −1.31143
$$846$$ 0 0
$$847$$ −371.687 −0.0150783
$$848$$ 0 0
$$849$$ 37488.0 1.51541
$$850$$ 0 0
$$851$$ 1463.59 0.0589556
$$852$$ 0 0
$$853$$ 42966.8 1.72469 0.862343 0.506325i $$-0.168997\pi$$
0.862343 + 0.506325i $$0.168997\pi$$
$$854$$ 0 0
$$855$$ −74536.6 −2.98140
$$856$$ 0 0
$$857$$ −17281.5 −0.688828 −0.344414 0.938818i $$-0.611922\pi$$
−0.344414 + 0.938818i $$0.611922\pi$$
$$858$$ 0 0
$$859$$ −9316.75 −0.370062 −0.185031 0.982733i $$-0.559239\pi$$
−0.185031 + 0.982733i $$0.559239\pi$$
$$860$$ 0 0
$$861$$ −6358.10 −0.251665
$$862$$ 0 0
$$863$$ 9647.65 0.380544 0.190272 0.981731i $$-0.439063\pi$$
0.190272 + 0.981731i $$0.439063\pi$$
$$864$$ 0 0
$$865$$ 34279.9 1.34746
$$866$$ 0 0
$$867$$ −25483.6 −0.998232
$$868$$ 0 0
$$869$$ −14236.5 −0.555742
$$870$$ 0 0
$$871$$ −1834.70 −0.0713735
$$872$$ 0 0
$$873$$ 48277.6 1.87165
$$874$$ 0 0
$$875$$ 1336.55 0.0516383
$$876$$ 0 0
$$877$$ 19728.7 0.759624 0.379812 0.925064i $$-0.375989\pi$$
0.379812 + 0.925064i $$0.375989\pi$$
$$878$$ 0 0
$$879$$ 18464.1 0.708509
$$880$$ 0 0
$$881$$ 19473.9 0.744712 0.372356 0.928090i $$-0.378550\pi$$
0.372356 + 0.928090i $$0.378550\pi$$
$$882$$ 0 0
$$883$$ −49092.4 −1.87100 −0.935499 0.353329i $$-0.885050\pi$$
−0.935499 + 0.353329i $$0.885050\pi$$
$$884$$ 0 0
$$885$$ −10408.5 −0.395344
$$886$$ 0 0
$$887$$ −9292.86 −0.351774 −0.175887 0.984410i $$-0.556279\pi$$
−0.175887 + 0.984410i $$0.556279\pi$$
$$888$$ 0 0
$$889$$ −4047.41 −0.152695
$$890$$ 0 0
$$891$$ −4525.85 −0.170170
$$892$$ 0 0
$$893$$ −48112.0 −1.80292
$$894$$ 0 0
$$895$$ 19493.9 0.728055
$$896$$ 0 0
$$897$$ 4731.10 0.176106
$$898$$ 0 0
$$899$$ 787.273 0.0292069
$$900$$ 0 0
$$901$$ 14136.7 0.522709
$$902$$ 0 0
$$903$$ −1405.53 −0.0517974
$$904$$ 0 0
$$905$$ −11932.3 −0.438279
$$906$$ 0 0
$$907$$ −37688.7 −1.37975 −0.689875 0.723928i $$-0.742335\pi$$
−0.689875 + 0.723928i $$0.742335\pi$$
$$908$$ 0 0
$$909$$ −5789.42 −0.211246
$$910$$ 0 0
$$911$$ −33049.6 −1.20196 −0.600979 0.799265i $$-0.705222\pi$$
−0.600979 + 0.799265i $$0.705222\pi$$
$$912$$ 0 0
$$913$$ −4860.31 −0.176180
$$914$$ 0 0
$$915$$ 87007.2 3.14357
$$916$$ 0 0
$$917$$ −4917.06 −0.177073
$$918$$ 0 0
$$919$$ 23148.0 0.830883 0.415442 0.909620i $$-0.363627\pi$$
0.415442 + 0.909620i $$0.363627\pi$$
$$920$$ 0 0
$$921$$ 13304.1 0.475986
$$922$$ 0 0
$$923$$ 1111.26 0.0396290
$$924$$ 0 0
$$925$$ 1258.01 0.0447169
$$926$$ 0 0
$$927$$ 1246.34 0.0441588
$$928$$ 0 0
$$929$$ −23177.9 −0.818561 −0.409280 0.912409i $$-0.634220\pi$$
−0.409280 + 0.912409i $$0.634220\pi$$
$$930$$ 0 0
$$931$$ 46673.3 1.64302
$$932$$ 0 0
$$933$$ −28325.1 −0.993916
$$934$$ 0 0
$$935$$ −6735.44 −0.235585
$$936$$ 0 0
$$937$$ −34574.7 −1.20545 −0.602724 0.797950i $$-0.705918\pi$$
−0.602724 + 0.797950i $$0.705918\pi$$
$$938$$ 0 0
$$939$$ 56959.1 1.97954
$$940$$ 0 0
$$941$$ 41831.2 1.44916 0.724578 0.689192i $$-0.242034\pi$$
0.724578 + 0.689192i $$0.242034\pi$$
$$942$$ 0 0
$$943$$ 29071.3 1.00392
$$944$$ 0 0
$$945$$ −3204.34 −0.110304
$$946$$ 0 0
$$947$$ −27231.2 −0.934419 −0.467209 0.884147i $$-0.654741\pi$$
−0.467209 + 0.884147i $$0.654741\pi$$
$$948$$ 0 0
$$949$$ −5415.79 −0.185252
$$950$$ 0 0
$$951$$ −124.534 −0.00424635
$$952$$ 0 0
$$953$$ 40939.4 1.39156 0.695781 0.718254i $$-0.255058\pi$$
0.695781 + 0.718254i $$0.255058\pi$$
$$954$$ 0 0
$$955$$ −25527.0 −0.864956
$$956$$ 0 0
$$957$$ −2179.81 −0.0736292
$$958$$ 0 0
$$959$$ −4950.74 −0.166703
$$960$$ 0 0
$$961$$ −28798.9 −0.966698
$$962$$ 0 0
$$963$$ −29839.0 −0.998491
$$964$$ 0 0
$$965$$ 19910.3 0.664182
$$966$$ 0 0
$$967$$ 46173.1 1.53550 0.767750 0.640750i $$-0.221376\pi$$
0.767750 + 0.640750i $$0.221376\pi$$
$$968$$ 0 0
$$969$$ 45721.8 1.51579
$$970$$ 0 0
$$971$$ 5153.91 0.170337 0.0851683 0.996367i $$-0.472857\pi$$
0.0851683 + 0.996367i $$0.472857\pi$$
$$972$$ 0 0
$$973$$ −97.8734 −0.00322474
$$974$$ 0 0
$$975$$ 4066.56 0.133574
$$976$$ 0 0
$$977$$ 9692.13 0.317378 0.158689 0.987329i $$-0.449273\pi$$
0.158689 + 0.987329i $$0.449273\pi$$
$$978$$ 0 0
$$979$$ −16380.2 −0.534744
$$980$$ 0 0
$$981$$ 37443.3 1.21863
$$982$$ 0 0
$$983$$ −32915.7 −1.06800 −0.534002 0.845483i $$-0.679313\pi$$
−0.534002 + 0.845483i $$0.679313\pi$$
$$984$$ 0 0
$$985$$ −52269.7 −1.69081
$$986$$ 0 0
$$987$$ −8373.97 −0.270057
$$988$$ 0 0
$$989$$ 6426.54 0.206625
$$990$$ 0 0
$$991$$ −29477.9 −0.944901 −0.472451 0.881357i $$-0.656630\pi$$
−0.472451 + 0.881357i $$0.656630\pi$$
$$992$$ 0 0
$$993$$ 10456.9 0.334180
$$994$$ 0 0
$$995$$ −12237.1 −0.389892
$$996$$ 0 0
$$997$$ −31944.4 −1.01473 −0.507366 0.861731i $$-0.669381\pi$$
−0.507366 + 0.861731i $$0.669381\pi$$
$$998$$ 0 0
$$999$$ 922.883 0.0292279
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 176.4.a.i.1.2 2
3.2 odd 2 1584.4.a.bc.1.1 2
4.3 odd 2 11.4.a.a.1.2 2
8.3 odd 2 704.4.a.p.1.2 2
8.5 even 2 704.4.a.n.1.1 2
11.10 odd 2 1936.4.a.w.1.2 2
12.11 even 2 99.4.a.c.1.1 2
20.3 even 4 275.4.b.c.199.1 4
20.7 even 4 275.4.b.c.199.4 4
20.19 odd 2 275.4.a.b.1.1 2
28.27 even 2 539.4.a.e.1.2 2
44.3 odd 10 121.4.c.c.9.2 8
44.7 even 10 121.4.c.f.27.1 8
44.15 odd 10 121.4.c.c.27.2 8
44.19 even 10 121.4.c.f.9.1 8
44.27 odd 10 121.4.c.c.3.1 8
44.31 odd 10 121.4.c.c.81.1 8
44.35 even 10 121.4.c.f.81.2 8
44.39 even 10 121.4.c.f.3.2 8
44.43 even 2 121.4.a.c.1.1 2
52.51 odd 2 1859.4.a.a.1.1 2
60.59 even 2 2475.4.a.q.1.2 2
132.131 odd 2 1089.4.a.v.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 4.3 odd 2
99.4.a.c.1.1 2 12.11 even 2
121.4.a.c.1.1 2 44.43 even 2
121.4.c.c.3.1 8 44.27 odd 10
121.4.c.c.9.2 8 44.3 odd 10
121.4.c.c.27.2 8 44.15 odd 10
121.4.c.c.81.1 8 44.31 odd 10
121.4.c.f.3.2 8 44.39 even 10
121.4.c.f.9.1 8 44.19 even 10
121.4.c.f.27.1 8 44.7 even 10
121.4.c.f.81.2 8 44.35 even 10
176.4.a.i.1.2 2 1.1 even 1 trivial
275.4.a.b.1.1 2 20.19 odd 2
275.4.b.c.199.1 4 20.3 even 4
275.4.b.c.199.4 4 20.7 even 4
539.4.a.e.1.2 2 28.27 even 2
704.4.a.n.1.1 2 8.5 even 2
704.4.a.p.1.2 2 8.3 odd 2
1089.4.a.v.1.2 2 132.131 odd 2
1584.4.a.bc.1.1 2 3.2 odd 2
1859.4.a.a.1.1 2 52.51 odd 2
1936.4.a.w.1.2 2 11.10 odd 2
2475.4.a.q.1.2 2 60.59 even 2