# Properties

 Label 1936.4.a.w.1.2 Level $1936$ Weight $4$ Character 1936.1 Self dual yes Analytic conductor $114.228$ 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] = [1936,4,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, 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("1936.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1936 = 2^{4} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1936.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: $$114.227697771$$ 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 $$\chi$$ $$=$$ 1936.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+7.92820 q^{3} +14.8564 q^{5} +3.07180 q^{7} +35.8564 q^{9} +O(q^{10})$$ $$q+7.92820 q^{3} +14.8564 q^{5} +3.07180 q^{7} +35.8564 q^{9} -5.35898 q^{13} +117.785 q^{15} +41.2154 q^{17} +139.923 q^{19} +24.3538 q^{21} +111.354 q^{23} +95.7128 q^{25} +70.2154 q^{27} +24.9948 q^{29} -31.4974 q^{31} +45.6359 q^{35} +13.1436 q^{37} -42.4871 q^{39} -261.072 q^{41} -57.7128 q^{43} +532.697 q^{45} +343.846 q^{47} -333.564 q^{49} +326.764 q^{51} -342.995 q^{53} +1109.34 q^{57} -88.3693 q^{59} -738.697 q^{61} +110.144 q^{63} -79.6152 q^{65} -342.359 q^{67} +882.836 q^{69} +207.364 q^{71} +1010.60 q^{73} +758.831 q^{75} +1294.23 q^{79} -411.441 q^{81} +441.846 q^{83} +612.313 q^{85} +198.164 q^{87} -1489.11 q^{89} -16.4617 q^{91} -249.718 q^{93} +2078.75 q^{95} +1346.42 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 20 q^{7} + 44 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 20 * q^7 + 44 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 20 q^{7} + 44 q^{9} - 80 q^{13} + 194 q^{15} + 124 q^{17} + 72 q^{19} - 76 q^{21} + 98 q^{23} + 136 q^{25} + 182 q^{27} - 144 q^{29} + 34 q^{31} - 172 q^{35} + 54 q^{37} + 400 q^{39} - 536 q^{41} - 60 q^{43} + 428 q^{45} + 272 q^{47} - 390 q^{49} - 164 q^{51} - 492 q^{53} + 1512 q^{57} - 634 q^{59} - 840 q^{61} + 248 q^{63} + 880 q^{65} - 754 q^{67} + 962 q^{69} + 678 q^{71} + 400 q^{73} + 520 q^{75} + 316 q^{79} - 1294 q^{81} + 468 q^{83} - 452 q^{85} + 1200 q^{87} - 1842 q^{89} - 1280 q^{91} - 638 q^{93} + 2952 q^{95} + 2194 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 20 * q^7 + 44 * q^9 - 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 - 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 + 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 + 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

## 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 $$-0.473572\pi$$
0.0829307 + 0.996555i $$0.473572\pi$$
$$8$$ 0 0
$$9$$ 35.8564 1.32802
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −5.35898 −0.114332 −0.0571659 0.998365i $$-0.518206\pi$$
−0.0571659 + 0.998365i $$0.518206\pi$$
$$14$$ 0 0
$$15$$ 117.785 2.02746
$$16$$ 0 0
$$17$$ 41.2154 0.588012 0.294006 0.955804i $$-0.405011\pi$$
0.294006 + 0.955804i $$0.405011\pi$$
$$18$$ 0 0
$$19$$ 139.923 1.68950 0.844751 0.535159i $$-0.179748\pi$$
0.844751 + 0.535159i $$0.179748\pi$$
$$20$$ 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 $$-0.474500\pi$$
0.0800246 + 0.996793i $$0.474500\pi$$
$$30$$ 0 0
$$31$$ −31.4974 −0.182487 −0.0912436 0.995829i $$-0.529084\pi$$
−0.0912436 + 0.995829i $$0.529084\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 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 $$-0.665648\pi$$
−0.497226 + 0.867621i $$0.665648\pi$$
$$42$$ 0 0
$$43$$ −57.7128 −0.204677 −0.102339 0.994750i $$-0.532633\pi$$
−0.102339 + 0.994750i $$0.532633\pi$$
$$44$$ 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
$$55$$ 0 0
$$56$$ 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 $$-0.782376\pi$$
−0.775250 + 0.631654i $$0.782376\pi$$
$$62$$ 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 $$-0.199386\pi$$
0.810149 + 0.586224i $$0.199386\pi$$
$$74$$ 0 0
$$75$$ 758.831 1.16830
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1294.23 1.84319 0.921593 0.388157i $$-0.126888\pi$$
0.921593 + 0.388157i $$0.126888\pi$$
$$80$$ 0 0
$$81$$ −411.441 −0.564391
$$82$$ 0 0
$$83$$ 441.846 0.584324 0.292162 0.956369i $$-0.405625\pi$$
0.292162 + 0.956369i $$0.405625\pi$$
$$84$$ 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
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 161.461 0.159069 0.0795347 0.996832i $$-0.474657\pi$$
0.0795347 + 0.996832i $$0.474657\pi$$
$$102$$ 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 $$-0.377322\pi$$
0.375934 + 0.926647i $$0.377322\pi$$
$$108$$ 0 0
$$109$$ −1044.26 −0.917629 −0.458815 0.888532i $$-0.651726\pi$$
−0.458815 + 0.888532i $$0.651726\pi$$
$$110$$ 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
$$121$$ 0 0
$$122$$ 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 $$-0.652261\pi$$
−0.460309 + 0.887759i $$0.652261\pi$$
$$128$$ 0 0
$$129$$ −457.559 −0.312293
$$130$$ 0 0
$$131$$ −1600.71 −1.06759 −0.533797 0.845612i $$-0.679235\pi$$
−0.533797 + 0.845612i $$0.679235\pi$$
$$132$$ 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 $$-0.503094\pi$$
−0.00972120 + 0.999953i $$0.503094\pi$$
$$140$$ 0 0
$$141$$ 2726.08 1.62821
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 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 $$-0.267332\pi$$
0.667576 + 0.744542i $$0.267332\pi$$
$$150$$ 0 0
$$151$$ −2576.68 −1.38866 −0.694328 0.719659i $$-0.744298\pi$$
−0.694328 + 0.719659i $$0.744298\pi$$
$$152$$ 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
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2737.30 1.26837 0.634187 0.773180i $$-0.281335\pi$$
0.634187 + 0.773180i $$0.281335\pi$$
$$168$$ 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 $$-0.669254\pi$$
−0.507022 + 0.861933i $$0.669254\pi$$
$$174$$ 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
$$187$$ 0 0
$$188$$ 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 $$-0.580404\pi$$
−0.249919 + 0.968267i $$0.580404\pi$$
$$194$$ 0 0
$$195$$ −631.206 −0.231803
$$196$$ 0 0
$$197$$ 3518.33 1.27244 0.636220 0.771508i $$-0.280497\pi$$
0.636220 + 0.771508i $$0.280497\pi$$
$$198$$ 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
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −107.343 −0.0350228 −0.0175114 0.999847i $$-0.505574\pi$$
−0.0175114 + 0.999847i $$0.505574\pi$$
$$212$$ 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 $$-0.583407\pi$$
−0.259042 + 0.965866i $$0.583407\pi$$
$$228$$ 0 0
$$229$$ 1915.37 0.552713 0.276356 0.961055i $$-0.410873\pi$$
0.276356 + 0.961055i $$0.410873\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −4396.32 −1.23610 −0.618052 0.786137i $$-0.712078\pi$$
−0.618052 + 0.786137i $$0.712078\pi$$
$$234$$ 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 $$-0.686413\pi$$
−0.552728 + 0.833362i $$0.686413\pi$$
$$240$$ 0 0
$$241$$ −3908.58 −1.04471 −0.522353 0.852730i $$-0.674946\pi$$
−0.522353 + 0.852730i $$0.674946\pi$$
$$242$$ 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
$$253$$ 0 0
$$254$$ 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 $$-0.242078\pi$$
0.724484 + 0.689292i $$0.242078\pi$$
$$264$$ 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 $$-0.328543\pi$$
0.512975 + 0.858404i $$0.328543\pi$$
$$272$$ 0 0
$$273$$ −130.512 −0.0289338
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −567.836 −0.123169 −0.0615847 0.998102i $$-0.519615\pi$$
−0.0615847 + 0.998102i $$0.519615\pi$$
$$278$$ 0 0
$$279$$ −1129.38 −0.242346
$$280$$ 0 0
$$281$$ −5311.01 −1.12750 −0.563752 0.825944i $$-0.690643\pi$$
−0.563752 + 0.825944i $$0.690643\pi$$
$$282$$ 0 0
$$283$$ −4728.44 −0.993204 −0.496602 0.867978i $$-0.665419\pi$$
−0.496602 + 0.867978i $$0.665419\pi$$
$$284$$ 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 $$-0.574585\pi$$
−0.232179 + 0.972673i $$0.574585\pi$$
$$294$$ 0 0
$$295$$ −1312.85 −0.259109
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 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 $$-0.549854\pi$$
−0.155981 + 0.987760i $$0.549854\pi$$
$$308$$ 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
$$319$$ 0 0
$$320$$ 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 $$-0.493846\pi$$
0.0193310 + 0.999813i $$0.493846\pi$$
$$338$$ 0 0
$$339$$ 2339.47 0.374816
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 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 $$-0.649826\pi$$
−0.453503 + 0.891255i $$0.649826\pi$$
$$348$$ 0 0
$$349$$ −3491.73 −0.535553 −0.267776 0.963481i $$-0.586289\pi$$
−0.267776 + 0.963481i $$0.586289\pi$$
$$350$$ 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 $$-0.820628\pi$$
−0.845384 + 0.534160i $$0.820628\pi$$
$$360$$ 0 0
$$361$$ 12719.5 1.85442
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 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 $$-0.379848\pi$$
0.368569 + 0.929600i $$0.379848\pi$$
$$374$$ 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
$$385$$ 0 0
$$386$$ 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
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 4192.50 0.506860 0.253430 0.967354i $$-0.418441\pi$$
0.253430 + 0.967354i $$0.418441\pi$$
$$410$$ 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
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4909.67 0.548701 0.274351 0.961630i $$-0.411537\pi$$
0.274351 + 0.961630i $$0.411537\pi$$
$$432$$ 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 $$-0.722208\pi$$
−0.642754 + 0.766073i $$0.722208\pi$$
$$440$$ 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
$$451$$ 0 0
$$452$$ 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 $$-0.318114\pi$$
0.540821 + 0.841138i $$0.318114\pi$$
$$458$$ 0 0
$$459$$ 2893.95 0.294288
$$460$$ 0 0
$$461$$ −4733.96 −0.478270 −0.239135 0.970986i $$-0.576864\pi$$
−0.239135 + 0.970986i $$0.576864\pi$$
$$462$$ 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
$$473$$ 0 0
$$474$$ 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 $$-0.313989\pi$$
0.551675 + 0.834059i $$0.313989\pi$$
$$480$$ 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 $$-0.613845\pi$$
−0.350078 + 0.936721i $$0.613845\pi$$
$$492$$ 0 0
$$493$$ 1030.17 0.0941108
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 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 $$-0.351224\pi$$
0.450561 + 0.892746i $$0.351224\pi$$
$$504$$ 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
$$517$$ 0 0
$$518$$ 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 $$-0.415732\pi$$
0.261655 + 0.965161i $$0.415732\pi$$
$$524$$ 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
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14008.2 1.11323 0.556616 0.830770i $$-0.312100\pi$$
0.556616 + 0.830770i $$0.312100\pi$$
$$542$$ 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 $$-0.561964\pi$$
−0.193440 + 0.981112i $$0.561964\pi$$
$$548$$ 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 $$-0.453812\pi$$
0.144594 + 0.989491i $$0.453812\pi$$
$$558$$ 0 0
$$559$$ 309.282 0.0234011
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −9900.11 −0.741101 −0.370551 0.928812i $$-0.620831\pi$$
−0.370551 + 0.928812i $$0.620831\pi$$
$$564$$ 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 $$-0.562898\pi$$
−0.196318 + 0.980540i $$0.562898\pi$$
$$570$$ 0 0
$$571$$ −16962.6 −1.24319 −0.621597 0.783337i $$-0.713516\pi$$
−0.621597 + 0.783337i $$0.713516\pi$$
$$572$$ 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
$$583$$ 0 0
$$584$$ 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 $$-0.548123\pi$$
−0.150607 + 0.988594i $$0.548123\pi$$
$$594$$ 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 $$-0.280255\pi$$
0.636806 + 0.771024i $$0.280255\pi$$
$$602$$ 0 0
$$603$$ −12275.8 −0.829034
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 21871.4 1.46249 0.731244 0.682116i $$-0.238940\pi$$
0.731244 + 0.682116i $$0.238940\pi$$
$$608$$ 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 $$-0.462921\pi$$
0.116222 + 0.993223i $$0.462921\pi$$
$$614$$ 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
$$627$$ 0 0
$$628$$ 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
$$649$$ 0 0
$$650$$ 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 $$-0.600647\pi$$
−0.310948 + 0.950427i $$0.600647\pi$$
$$660$$ 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
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 1187.64 0.0680239 0.0340119 0.999421i $$-0.489172\pi$$
0.0340119 + 0.999421i $$0.489172\pi$$
$$674$$ 0 0
$$675$$ 6720.51 0.383219
$$676$$ 0 0
$$677$$ −13221.4 −0.750574 −0.375287 0.926909i $$-0.622456\pi$$
−0.375287 + 0.926909i $$0.622456\pi$$
$$678$$ 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
$$693$$ 0 0
$$694$$ 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 $$-0.798823\pi$$
−0.806838 + 0.590773i $$0.798823\pi$$
$$702$$ 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
$$715$$ 0 0
$$716$$ 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 $$-0.569780\pi$$
−0.217467 + 0.976068i $$0.569780\pi$$
$$734$$ 0 0
$$735$$ −39288.7 −1.97168
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −18357.5 −0.913792 −0.456896 0.889520i $$-0.651039\pi$$
−0.456896 + 0.889520i $$0.651039\pi$$
$$740$$ 0 0
$$741$$ −5944.93 −0.294726
$$742$$ 0 0
$$743$$ 11182.6 0.552155 0.276078 0.961135i $$-0.410965\pi$$
0.276078 + 0.961135i $$0.410965\pi$$
$$744$$ 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
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −8469.33 −0.403434 −0.201717 0.979444i $$-0.564652\pi$$
−0.201717 + 0.979444i $$0.564652\pi$$
$$762$$ 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 $$-0.779676\pi$$
−0.769864 + 0.638208i $$0.779676\pi$$
$$770$$ 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
$$781$$ 0 0
$$782$$ 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 $$-0.384063\pi$$
0.356226 + 0.934400i $$0.384063\pi$$
$$788$$ 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
$$803$$ 0 0
$$804$$ 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 $$-0.540900\pi$$
−0.128138 + 0.991756i $$0.540900\pi$$
$$810$$ 0 0
$$811$$ 14197.9 0.614744 0.307372 0.951589i $$-0.400550\pi$$
0.307372 + 0.951589i $$0.400550\pi$$
$$812$$ 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 $$-0.361423\pi$$
0.421729 + 0.906722i $$0.361423\pi$$
$$822$$ 0 0
$$823$$ 28202.2 1.19449 0.597246 0.802058i $$-0.296262\pi$$
0.597246 + 0.802058i $$0.296262\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 34031.0 1.43092 0.715462 0.698651i $$-0.246216\pi$$
0.715462 + 0.698651i $$0.246216\pi$$
$$828$$ 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
$$847$$ 0 0
$$848$$ 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 $$-0.831003\pi$$
−0.862343 + 0.506325i $$0.831003\pi$$
$$854$$ 0 0
$$855$$ 74536.6 2.98140
$$856$$ 0 0
$$857$$ 17281.5 0.688828 0.344414 0.938818i $$-0.388078\pi$$
0.344414 + 0.938818i $$0.388078\pi$$
$$858$$ 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
$$869$$ 0 0
$$870$$ 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 $$-0.624011\pi$$
−0.379812 + 0.925064i $$0.624011\pi$$
$$878$$ 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 $$-0.443721\pi$$
0.175887 + 0.984410i $$0.443721\pi$$
$$888$$ 0 0
$$889$$ −4047.41 −0.152695
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 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
$$913$$ 0 0
$$914$$ 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 $$-0.636373\pi$$
−0.415442 + 0.909620i $$0.636373\pi$$
$$920$$ 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
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 34574.7 1.20545 0.602724 0.797950i $$-0.294082\pi$$
0.602724 + 0.797950i $$0.294082\pi$$
$$938$$ 0 0
$$939$$ 56959.1 1.97954
$$940$$ 0 0
$$941$$ −41831.2 −1.44916 −0.724578 0.689192i $$-0.757966\pi$$
−0.724578 + 0.689192i $$0.757966\pi$$
$$942$$ 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 $$-0.744942\pi$$
−0.695781 + 0.718254i $$0.744942\pi$$
$$954$$ 0 0
$$955$$ −25527.0 −0.864956
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 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 $$-0.778624\pi$$
−0.767750 + 0.640750i $$0.778624\pi$$
$$968$$ 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
$$979$$ 0 0
$$980$$ 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 $$-0.330619\pi$$
0.507366 + 0.861731i $$0.330619\pi$$
$$998$$ 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 1936.4.a.w.1.2 2
4.3 odd 2 121.4.a.c.1.1 2
11.10 odd 2 176.4.a.i.1.2 2
12.11 even 2 1089.4.a.v.1.2 2
33.32 even 2 1584.4.a.bc.1.1 2
44.3 odd 10 121.4.c.f.9.1 8
44.7 even 10 121.4.c.c.27.2 8
44.15 odd 10 121.4.c.f.27.1 8
44.19 even 10 121.4.c.c.9.2 8
44.27 odd 10 121.4.c.f.3.2 8
44.31 odd 10 121.4.c.f.81.2 8
44.35 even 10 121.4.c.c.81.1 8
44.39 even 10 121.4.c.c.3.1 8
44.43 even 2 11.4.a.a.1.2 2
88.21 odd 2 704.4.a.n.1.1 2
88.43 even 2 704.4.a.p.1.2 2
132.131 odd 2 99.4.a.c.1.1 2
220.43 odd 4 275.4.b.c.199.1 4
220.87 odd 4 275.4.b.c.199.4 4
220.219 even 2 275.4.a.b.1.1 2
308.307 odd 2 539.4.a.e.1.2 2
572.571 even 2 1859.4.a.a.1.1 2
660.659 odd 2 2475.4.a.q.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 44.43 even 2
99.4.a.c.1.1 2 132.131 odd 2
121.4.a.c.1.1 2 4.3 odd 2
121.4.c.c.3.1 8 44.39 even 10
121.4.c.c.9.2 8 44.19 even 10
121.4.c.c.27.2 8 44.7 even 10
121.4.c.c.81.1 8 44.35 even 10
121.4.c.f.3.2 8 44.27 odd 10
121.4.c.f.9.1 8 44.3 odd 10
121.4.c.f.27.1 8 44.15 odd 10
121.4.c.f.81.2 8 44.31 odd 10
176.4.a.i.1.2 2 11.10 odd 2
275.4.a.b.1.1 2 220.219 even 2
275.4.b.c.199.1 4 220.43 odd 4
275.4.b.c.199.4 4 220.87 odd 4
539.4.a.e.1.2 2 308.307 odd 2
704.4.a.n.1.1 2 88.21 odd 2
704.4.a.p.1.2 2 88.43 even 2
1089.4.a.v.1.2 2 12.11 even 2
1584.4.a.bc.1.1 2 33.32 even 2
1859.4.a.a.1.1 2 572.571 even 2
1936.4.a.w.1.2 2 1.1 even 1 trivial
2475.4.a.q.1.2 2 660.659 odd 2