Properties

 Label 2475.4.a.q.1.2 Level $2475$ Weight $4$ Character 2475.1 Self dual yes Analytic conductor $146.030$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$146.029727264$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\chi$$ $$=$$ 2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.73205 q^{2} -0.535898 q^{4} -3.07180 q^{7} -23.3205 q^{8} +O(q^{10})$$ $$q+2.73205 q^{2} -0.535898 q^{4} -3.07180 q^{7} -23.3205 q^{8} +11.0000 q^{11} -5.35898 q^{13} -8.39230 q^{14} -59.4256 q^{16} -41.2154 q^{17} +139.923 q^{19} +30.0526 q^{22} -111.354 q^{23} -14.6410 q^{26} +1.64617 q^{28} +24.9948 q^{29} +31.4974 q^{31} +24.2102 q^{32} -112.603 q^{34} -13.1436 q^{37} +382.277 q^{38} -261.072 q^{41} +57.7128 q^{43} -5.89488 q^{44} -304.224 q^{46} -343.846 q^{47} -333.564 q^{49} +2.87187 q^{52} -342.995 q^{53} +71.6359 q^{56} +68.2872 q^{58} -88.3693 q^{59} +738.697 q^{61} +86.0526 q^{62} +541.549 q^{64} -342.359 q^{67} +22.0873 q^{68} +207.364 q^{71} +1010.60 q^{73} -35.9090 q^{74} -74.9845 q^{76} -33.7898 q^{77} +1294.23 q^{79} -713.261 q^{82} +441.846 q^{83} +157.674 q^{86} -256.526 q^{88} +1489.11 q^{89} +16.4617 q^{91} +59.6743 q^{92} -939.405 q^{94} -1346.42 q^{97} -911.314 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 8 q^{4} - 20 q^{7} - 12 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 8 * q^4 - 20 * q^7 - 12 * q^8 $$2 q + 2 q^{2} - 8 q^{4} - 20 q^{7} - 12 q^{8} + 22 q^{11} - 80 q^{13} + 4 q^{14} - 8 q^{16} - 124 q^{17} + 72 q^{19} + 22 q^{22} - 98 q^{23} + 40 q^{26} + 128 q^{28} - 144 q^{29} - 34 q^{31} - 104 q^{32} - 52 q^{34} - 54 q^{37} + 432 q^{38} - 536 q^{41} + 60 q^{43} - 88 q^{44} - 314 q^{46} - 272 q^{47} - 390 q^{49} + 560 q^{52} - 492 q^{53} - 120 q^{56} + 192 q^{58} - 634 q^{59} + 840 q^{61} + 134 q^{62} + 224 q^{64} - 754 q^{67} + 640 q^{68} + 678 q^{71} + 400 q^{73} - 6 q^{74} + 432 q^{76} - 220 q^{77} + 316 q^{79} - 512 q^{82} + 468 q^{83} + 156 q^{86} - 132 q^{88} + 1842 q^{89} + 1280 q^{91} - 40 q^{92} - 992 q^{94} - 2194 q^{97} - 870 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 8 * q^4 - 20 * q^7 - 12 * q^8 + 22 * q^11 - 80 * q^13 + 4 * q^14 - 8 * q^16 - 124 * q^17 + 72 * q^19 + 22 * q^22 - 98 * q^23 + 40 * q^26 + 128 * q^28 - 144 * q^29 - 34 * q^31 - 104 * q^32 - 52 * q^34 - 54 * q^37 + 432 * q^38 - 536 * q^41 + 60 * q^43 - 88 * q^44 - 314 * q^46 - 272 * q^47 - 390 * q^49 + 560 * q^52 - 492 * q^53 - 120 * q^56 + 192 * q^58 - 634 * q^59 + 840 * q^61 + 134 * q^62 + 224 * q^64 - 754 * q^67 + 640 * q^68 + 678 * q^71 + 400 * q^73 - 6 * q^74 + 432 * q^76 - 220 * q^77 + 316 * q^79 - 512 * q^82 + 468 * q^83 + 156 * q^86 - 132 * q^88 + 1842 * q^89 + 1280 * q^91 - 40 * q^92 - 992 * q^94 - 2194 * q^97 - 870 * q^98

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.73205 0.965926 0.482963 0.875641i $$-0.339561\pi$$
0.482963 + 0.875641i $$0.339561\pi$$
$$3$$ 0 0
$$4$$ −0.535898 −0.0669873
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.07180 −0.165861 −0.0829307 0.996555i $$-0.526428\pi$$
−0.0829307 + 0.996555i $$0.526428\pi$$
$$8$$ −23.3205 −1.03063
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ −5.35898 −0.114332 −0.0571659 0.998365i $$-0.518206\pi$$
−0.0571659 + 0.998365i $$0.518206\pi$$
$$14$$ −8.39230 −0.160210
$$15$$ 0 0
$$16$$ −59.4256 −0.928525
$$17$$ −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 $$-0.179748\pi$$
0.844751 + 0.535159i $$0.179748\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 30.0526 0.291238
$$23$$ −111.354 −1.00952 −0.504758 0.863261i $$-0.668418\pi$$
−0.504758 + 0.863261i $$0.668418\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −14.6410 −0.110436
$$27$$ 0 0
$$28$$ 1.64617 0.0111106
$$29$$ 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 $$-0.470916\pi$$
0.0912436 + 0.995829i $$0.470916\pi$$
$$32$$ 24.2102 0.133744
$$33$$ 0 0
$$34$$ −112.603 −0.567976
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −13.1436 −0.0583998 −0.0291999 0.999574i $$-0.509296\pi$$
−0.0291999 + 0.999574i $$0.509296\pi$$
$$38$$ 382.277 1.63193
$$39$$ 0 0
$$40$$ 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 $$-0.467367\pi$$
0.102339 + 0.994750i $$0.467367\pi$$
$$44$$ −5.89488 −0.0201974
$$45$$ 0 0
$$46$$ −304.224 −0.975118
$$47$$ −343.846 −1.06713 −0.533565 0.845759i $$-0.679148\pi$$
−0.533565 + 0.845759i $$0.679148\pi$$
$$48$$ 0 0
$$49$$ −333.564 −0.972490
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.87187 0.00765879
$$53$$ −342.995 −0.888943 −0.444471 0.895793i $$-0.646608\pi$$
−0.444471 + 0.895793i $$0.646608\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 71.6359 0.170942
$$57$$ 0 0
$$58$$ 68.2872 0.154596
$$59$$ −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 $$0.217624\pi$$
$$62$$ 86.0526 0.176269
$$63$$ 0 0
$$64$$ 541.549 1.05771
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −342.359 −0.624266 −0.312133 0.950038i $$-0.601043\pi$$
−0.312133 + 0.950038i $$0.601043\pi$$
$$68$$ 22.0873 0.0393893
$$69$$ 0 0
$$70$$ 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 $$0.199386\pi$$
$$74$$ −35.9090 −0.0564099
$$75$$ 0 0
$$76$$ −74.9845 −0.113175
$$77$$ −33.7898 −0.0500091
$$78$$ 0 0
$$79$$ 1294.23 1.84319 0.921593 0.388157i $$-0.126888\pi$$
0.921593 + 0.388157i $$0.126888\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −713.261 −0.960568
$$83$$ 441.846 0.584324 0.292162 0.956369i $$-0.405625\pi$$
0.292162 + 0.956369i $$0.405625\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 157.674 0.197703
$$87$$ 0 0
$$88$$ −256.526 −0.310747
$$89$$ 1489.11 1.77355 0.886773 0.462205i $$-0.152942\pi$$
0.886773 + 0.462205i $$0.152942\pi$$
$$90$$ 0 0
$$91$$ 16.4617 0.0189633
$$92$$ 59.6743 0.0676248
$$93$$ 0 0
$$94$$ −939.405 −1.03077
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1346.42 −1.40936 −0.704679 0.709526i $$-0.748909\pi$$
−0.704679 + 0.709526i $$0.748909\pi$$
$$98$$ −911.314 −0.939353
$$99$$ 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 $$0.494708\pi$$
$$104$$ 124.974 0.117834
$$105$$ 0 0
$$106$$ −937.079 −0.858653
$$107$$ 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 $$-0.348274\pi$$
0.458815 + 0.888532i $$0.348274\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 182.543 0.154007
$$113$$ 295.082 0.245654 0.122827 0.992428i $$-0.460804\pi$$
0.122827 + 0.992428i $$0.460804\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −13.3947 −0.0107213
$$117$$ 0 0
$$118$$ −241.429 −0.188351
$$119$$ 126.605 0.0975285
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 2018.16 1.49767
$$123$$ 0 0
$$124$$ −16.8794 −0.0122243
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1317.60 0.920618 0.460309 0.887759i $$-0.347739\pi$$
0.460309 + 0.887759i $$0.347739\pi$$
$$128$$ 1285.86 0.887928
$$129$$ 0 0
$$130$$ 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
$$134$$ −935.342 −0.602994
$$135$$ 0 0
$$136$$ 961.164 0.606023
$$137$$ 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
$$141$$ 0 0
$$142$$ 566.529 0.334803
$$143$$ −58.9488 −0.0344724
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2761.01 1.56509
$$147$$ 0 0
$$148$$ 7.04363 0.00391205
$$149$$ 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 $$0.744298\pi$$
$$152$$ −3263.08 −1.74125
$$153$$ 0 0
$$154$$ −92.3154 −0.0483051
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −2475.94 −1.25861 −0.629305 0.777158i $$-0.716660\pi$$
−0.629305 + 0.777158i $$0.716660\pi$$
$$158$$ 3535.89 1.78038
$$159$$ 0 0
$$160$$ 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 $$0.272776\pi$$
$$164$$ 139.908 0.0666157
$$165$$ 0 0
$$166$$ 1207.15 0.564414
$$167$$ 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
$$171$$ 0 0
$$172$$ −30.9282 −0.0137108
$$173$$ 2307.42 1.01404 0.507022 0.861933i $$-0.330746\pi$$
0.507022 + 0.861933i $$0.330746\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −653.682 −0.279961
$$177$$ 0 0
$$178$$ 4068.33 1.71311
$$179$$ 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 $$0.552735\pi$$
$$182$$ 44.9742 0.0183171
$$183$$ 0 0
$$184$$ 2596.83 1.04044
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −453.369 −0.177292
$$188$$ 184.267 0.0714842
$$189$$ 0 0
$$190$$ 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 $$0.580404\pi$$
$$194$$ −3678.48 −1.36134
$$195$$ 0 0
$$196$$ 178.756 0.0651445
$$197$$ −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 $$-0.453132\pi$$
0.146709 + 0.989180i $$0.453132\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 441.121 0.153649
$$203$$ −76.7791 −0.0265460
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 94.9639 0.0321187
$$207$$ 0 0
$$208$$ 318.461 0.106160
$$209$$ 1539.15 0.509404
$$210$$ 0 0
$$211$$ −107.343 −0.0350228 −0.0175114 0.999847i $$-0.505574\pi$$
−0.0175114 + 0.999847i $$0.505574\pi$$
$$212$$ 183.810 0.0595479
$$213$$ 0 0
$$214$$ 2273.56 0.726248
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −96.7537 −0.0302676
$$218$$ 2852.96 0.886362
$$219$$ 0 0
$$220$$ 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 $$0.298881\pi$$
$$224$$ −74.3689 −0.0221830
$$225$$ 0 0
$$226$$ 806.178 0.237284
$$227$$ −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
$$232$$ −582.892 −0.164952
$$233$$ 4396.32 1.23610 0.618052 0.786137i $$-0.287922\pi$$
0.618052 + 0.786137i $$0.287922\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 47.3570 0.0130622
$$237$$ 0 0
$$238$$ 345.892 0.0942053
$$239$$ 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 $$0.325054\pi$$
$$242$$ 330.578 0.0878114
$$243$$ 0 0
$$244$$ −395.867 −0.103864
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −749.845 −0.193164
$$248$$ −734.536 −0.188077
$$249$$ 0 0
$$250$$ 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
$$254$$ 3599.76 0.889249
$$255$$ 0 0
$$256$$ −819.364 −0.200040
$$257$$ 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
$$261$$ 0 0
$$262$$ 4373.23 1.03122
$$263$$ 6180.06 1.44897 0.724484 0.689292i $$-0.242078\pi$$
0.724484 + 0.689292i $$0.242078\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −1174.28 −0.270675
$$267$$ 0 0
$$268$$ 183.470 0.0418179
$$269$$ −986.965 −0.223704 −0.111852 0.993725i $$-0.535678\pi$$
−0.111852 + 0.993725i $$0.535678\pi$$
$$270$$ 0 0
$$271$$ 4576.99 1.02595 0.512975 0.858404i $$-0.328543\pi$$
0.512975 + 0.858404i $$0.328543\pi$$
$$272$$ 2449.25 0.545984
$$273$$ 0 0
$$274$$ 4403.18 0.970825
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −567.836 −0.123169 −0.0615847 0.998102i $$-0.519615\pi$$
−0.0615847 + 0.998102i $$0.519615\pi$$
$$278$$ −87.0484 −0.0187799
$$279$$ 0 0
$$280$$ 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 $$-0.334581\pi$$
0.496602 + 0.867978i $$0.334581\pi$$
$$284$$ −111.126 −0.0232187
$$285$$ 0 0
$$286$$ −161.051 −0.0332977
$$287$$ 801.960 0.164941
$$288$$ 0 0
$$289$$ −3214.29 −0.654242
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −541.579 −0.108539
$$293$$ 2328.92 0.464358 0.232179 0.972673i $$-0.425415\pi$$
0.232179 + 0.972673i $$0.425415\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 306.515 0.0601886
$$297$$ 0 0
$$298$$ 6634.36 1.28966
$$299$$ 596.743 0.115420
$$300$$ 0 0
$$301$$ −177.282 −0.0339481
$$302$$ −7039.61 −1.34134
$$303$$ 0 0
$$304$$ −8315.01 −1.56875
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1678.07 0.311962 0.155981 0.987760i $$-0.450146\pi$$
0.155981 + 0.987760i $$0.450146\pi$$
$$308$$ 18.1079 0.00334997
$$309$$ 0 0
$$310$$ 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 $$-0.724686\pi$$
−0.648697 + 0.761047i $$0.724686\pi$$
$$314$$ −6764.40 −1.21572
$$315$$ 0 0
$$316$$ −693.573 −0.123470
$$317$$ −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
$$321$$ 0 0
$$322$$ 934.515 0.161734
$$323$$ −5766.98 −0.993447
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 7445.13 1.26487
$$327$$ 0 0
$$328$$ 6088.33 1.02491
$$329$$ 1056.23 0.176996
$$330$$ 0 0
$$331$$ −1318.95 −0.219022 −0.109511 0.993986i $$-0.534928\pi$$
−0.109511 + 0.993986i $$0.534928\pi$$
$$332$$ −236.785 −0.0391423
$$333$$ 0 0
$$334$$ 7478.43 1.22515
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 239.183 0.0386621 0.0193310 0.999813i $$-0.493846\pi$$
0.0193310 + 0.999813i $$0.493846\pi$$
$$338$$ −5923.85 −0.953299
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 346.472 0.0550220
$$342$$ 0 0
$$343$$ 2078.27 0.327160
$$344$$ −1345.89 −0.210947
$$345$$ 0 0
$$346$$ 6303.98 0.979491
$$347$$ −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 $$-0.413711\pi$$
0.267776 + 0.963481i $$0.413711\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 266.313 0.0403253
$$353$$ −10916.7 −1.64600 −0.822999 0.568043i $$-0.807701\pi$$
−0.822999 + 0.568043i $$0.807701\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −798.013 −0.118805
$$357$$ 0 0
$$358$$ 3584.87 0.529236
$$359$$ 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
$$362$$ −2194.31 −0.318592
$$363$$ 0 0
$$364$$ −8.82180 −0.00127030
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −6767.01 −0.962493 −0.481246 0.876585i $$-0.659816\pi$$
−0.481246 + 0.876585i $$0.659816\pi$$
$$368$$ 6617.27 0.937362
$$369$$ 0 0
$$370$$ 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 $$0.379848\pi$$
$$374$$ −1238.63 −0.171251
$$375$$ 0 0
$$376$$ 8018.67 1.09982
$$377$$ −133.947 −0.0182987
$$378$$ 0 0
$$379$$ −838.267 −0.113612 −0.0568059 0.998385i $$-0.518092\pi$$
−0.0568059 + 0.998385i $$0.518092\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −4694.34 −0.628752
$$383$$ −2832.16 −0.377851 −0.188925 0.981991i $$-0.560500\pi$$
−0.188925 + 0.981991i $$0.560500\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3661.45 −0.482806
$$387$$ 0 0
$$388$$ 721.542 0.0944091
$$389$$ −3111.25 −0.405519 −0.202759 0.979229i $$-0.564991\pi$$
−0.202759 + 0.979229i $$0.564991\pi$$
$$390$$ 0 0
$$391$$ 4589.49 0.593608
$$392$$ 7778.88 1.00228
$$393$$ 0 0
$$394$$ −9612.25 −1.22908
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −14208.7 −1.79626 −0.898131 0.439728i $$-0.855075\pi$$
−0.898131 + 0.439728i $$0.855075\pi$$
$$398$$ 2250.37 0.283419
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6261.68 0.779784 0.389892 0.920861i $$-0.372512\pi$$
0.389892 + 0.920861i $$0.372512\pi$$
$$402$$ 0 0
$$403$$ −168.794 −0.0208641
$$404$$ −86.5269 −0.0106556
$$405$$ 0 0
$$406$$ −209.764 −0.0256415
$$407$$ −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
$$411$$ 0 0
$$412$$ −18.6274 −0.00222744
$$413$$ 271.453 0.0323421
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −129.742 −0.0152912
$$417$$ 0 0
$$418$$ 4205.05 0.492047
$$419$$ 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 $$0.224746\pi$$
$$422$$ −293.267 −0.0338294
$$423$$ 0 0
$$424$$ 7998.81 0.916172
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −2269.13 −0.257168
$$428$$ −445.964 −0.0503656
$$429$$ 0 0
$$430$$ 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 $$-0.274068\pi$$
0.651671 + 0.758502i $$0.274068\pi$$
$$434$$ −264.336 −0.0292363
$$435$$ 0 0
$$436$$ −559.615 −0.0614695
$$437$$ −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
$$441$$ 0 0
$$442$$ 603.435 0.0649377
$$443$$ 10102.1 1.08344 0.541722 0.840558i $$-0.317772\pi$$
0.541722 + 0.840558i $$0.317772\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 10747.0 1.14100
$$447$$ 0 0
$$448$$ −1663.53 −0.175434
$$449$$ 345.254 0.0362885 0.0181443 0.999835i $$-0.494224\pi$$
0.0181443 + 0.999835i $$0.494224\pi$$
$$450$$ 0 0
$$451$$ −2871.79 −0.299839
$$452$$ −158.134 −0.0164557
$$453$$ 0 0
$$454$$ −4840.93 −0.500431
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10567.1 1.08164 0.540821 0.841138i $$-0.318114\pi$$
0.540821 + 0.841138i $$0.318114\pi$$
$$458$$ 5232.89 0.533879
$$459$$ 0 0
$$460$$ 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 $$0.555089\pi$$
$$464$$ −1485.33 −0.148610
$$465$$ 0 0
$$466$$ 12011.0 1.19399
$$467$$ 5116.96 0.507034 0.253517 0.967331i $$-0.418413\pi$$
0.253517 + 0.967331i $$0.418413\pi$$
$$468$$ 0 0
$$469$$ 1051.66 0.103542
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 2060.82 0.200968
$$473$$ 634.841 0.0617125
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −67.8476 −0.00653317
$$477$$ 0 0
$$478$$ 11159.0 1.06779
$$479$$ −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
$$482$$ 10678.4 1.00911
$$483$$ 0 0
$$484$$ −64.8437 −0.00608975
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 18326.5 1.70525 0.852623 0.522527i $$-0.175010\pi$$
0.852623 + 0.522527i $$0.175010\pi$$
$$488$$ −17226.8 −1.59799
$$489$$ 0 0
$$490$$ 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
$$494$$ −2048.62 −0.186582
$$495$$ 0 0
$$496$$ −1871.75 −0.169444
$$497$$ −636.980 −0.0574899
$$498$$ 0 0
$$499$$ 12909.1 1.15810 0.579050 0.815292i $$-0.303424\pi$$
0.579050 + 0.815292i $$0.303424\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −2991.30 −0.265953
$$503$$ 10165.7 0.901121 0.450561 0.892746i $$-0.351224\pi$$
0.450561 + 0.892746i $$0.351224\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −3346.47 −0.294009
$$507$$ 0 0
$$508$$ −706.102 −0.0616697
$$509$$ −6449.93 −0.561666 −0.280833 0.959757i $$-0.590611\pi$$
−0.280833 + 0.959757i $$0.590611\pi$$
$$510$$ 0 0
$$511$$ −3104.36 −0.268745
$$512$$ −12525.4 −1.08115
$$513$$ 0 0
$$514$$ 2139.68 0.183614
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3782.31 −0.321752
$$518$$ 110.305 0.00935623
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 19327.4 1.62524 0.812620 0.582794i $$-0.198041\pi$$
0.812620 + 0.582794i $$0.198041\pi$$
$$522$$ 0 0
$$523$$ −6259.09 −0.523310 −0.261655 0.965161i $$-0.584268\pi$$
−0.261655 + 0.965161i $$0.584268\pi$$
$$524$$ −857.819 −0.0715153
$$525$$ 0 0
$$526$$ 16884.2 1.39960
$$527$$ −1298.18 −0.107305
$$528$$ 0 0
$$529$$ 232.675 0.0191235
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 230.337 0.0187714
$$533$$ 1399.08 0.113698
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 7983.99 0.643387
$$537$$ 0 0
$$538$$ −2696.44 −0.216081
$$539$$ −3669.20 −0.293217
$$540$$ 0 0
$$541$$ −14008.2 −1.11323 −0.556616 0.830770i $$-0.687900\pi$$
−0.556616 + 0.830770i $$0.687900\pi$$
$$542$$ 12504.6 0.990991
$$543$$ 0 0
$$544$$ −997.834 −0.0786430
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4949.45 0.386879 0.193440 0.981112i $$-0.438036\pi$$
0.193440 + 0.981112i $$0.438036\pi$$
$$548$$ −863.695 −0.0673270
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3497.35 0.270404
$$552$$ 0 0
$$553$$ −3975.60 −0.305714
$$554$$ −1551.36 −0.118973
$$555$$ 0 0
$$556$$ 17.0748 0.00130239
$$557$$ −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
$$561$$ 0 0
$$562$$ −14510.0 −1.08908
$$563$$ −9900.11 −0.741101 −0.370551 0.928812i $$-0.620831\pi$$
−0.370551 + 0.928812i $$0.620831\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 12918.3 0.959361
$$567$$ 0 0
$$568$$ −4835.84 −0.357231
$$569$$ −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 $$0.713516\pi$$
$$572$$ 31.5906 0.00230921
$$573$$ 0 0
$$574$$ 2190.99 0.159321
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 15487.0 1.11738 0.558692 0.829375i $$-0.311303\pi$$
0.558692 + 0.829375i $$0.311303\pi$$
$$578$$ −8781.61 −0.631949
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1357.26 −0.0969169
$$582$$ 0 0
$$583$$ −3772.94 −0.268026
$$584$$ −23567.7 −1.66993
$$585$$ 0 0
$$586$$ 6362.72 0.448535
$$587$$ 11084.2 0.779373 0.389686 0.920948i $$-0.372583\pi$$
0.389686 + 0.920948i $$0.372583\pi$$
$$588$$ 0 0
$$589$$ 4407.22 0.308313
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 781.066 0.0542257
$$593$$ 4349.68 0.301214 0.150607 0.988594i $$-0.451877\pi$$
0.150607 + 0.988594i $$0.451877\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1301.34 −0.0894382
$$597$$ 0 0
$$598$$ 1630.33 0.111487
$$599$$ −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 $$0.719745\pi$$
$$602$$ −484.344 −0.0327913
$$603$$ 0 0
$$604$$ 1380.84 0.0930223
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −21871.4 −1.46249 −0.731244 0.682116i $$-0.761060\pi$$
−0.731244 + 0.682116i $$0.761060\pi$$
$$608$$ 3387.57 0.225961
$$609$$ 0 0
$$610$$ 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 $$0.462921\pi$$
$$614$$ 4584.56 0.301332
$$615$$ 0 0
$$616$$ 787.994 0.0515409
$$617$$ −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 $$-0.745121\pi$$
−0.696187 + 0.717861i $$0.745121\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −9760.81 −0.629217
$$623$$ −4574.25 −0.294163
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −19628.0 −1.25319
$$627$$ 0 0
$$628$$ 1326.85 0.0843109
$$629$$ 541.718 0.0343398
$$630$$ 0 0
$$631$$ 21532.0 1.35844 0.679219 0.733936i $$-0.262319\pi$$
0.679219 + 0.733936i $$0.262319\pi$$
$$632$$ −30182.0 −1.89964
$$633$$ 0 0
$$634$$ −42.9141 −0.00268823
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1787.56 0.111187
$$638$$ 751.159 0.0466123
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −20148.3 −1.24151 −0.620756 0.784004i $$-0.713174\pi$$
−0.620756 + 0.784004i $$0.713174\pi$$
$$642$$ 0 0
$$643$$ −28869.7 −1.77062 −0.885310 0.465000i $$-0.846054\pi$$
−0.885310 + 0.465000i $$0.846054\pi$$
$$644$$ −183.307 −0.0112163
$$645$$ 0 0
$$646$$ −15755.7 −0.959597
$$647$$ −1590.02 −0.0966155 −0.0483077 0.998833i $$-0.515383\pi$$
−0.0483077 + 0.998833i $$0.515383\pi$$
$$648$$ 0 0
$$649$$ −972.062 −0.0587932
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −1460.38 −0.0877192
$$653$$ 20028.1 1.20024 0.600122 0.799909i $$-0.295119\pi$$
0.600122 + 0.799909i $$0.295119\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 15514.4 0.923375
$$657$$ 0 0
$$658$$ 2885.66 0.170965
$$659$$ 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 $$0.469085\pi$$
$$662$$ −3603.45 −0.211559
$$663$$ 0 0
$$664$$ −10304.1 −0.602222
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2783.27 −0.161572
$$668$$ −1466.91 −0.0849649
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 8125.67 0.467493
$$672$$ 0 0
$$673$$ 1187.64 0.0680239 0.0340119 0.999421i $$-0.489172\pi$$
0.0340119 + 0.999421i $$0.489172\pi$$
$$674$$ 653.460 0.0373447
$$675$$ 0 0
$$676$$ 1161.98 0.0661117
$$677$$ 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
$$681$$ 0 0
$$682$$ 946.578 0.0531471
$$683$$ −13831.4 −0.774882 −0.387441 0.921894i $$-0.626641\pi$$
−0.387441 + 0.921894i $$0.626641\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 5677.93 0.316012
$$687$$ 0 0
$$688$$ −3429.62 −0.190048
$$689$$ 1838.10 0.101635
$$690$$ 0 0
$$691$$ −9817.07 −0.540462 −0.270231 0.962796i $$-0.587100\pi$$
−0.270231 + 0.962796i $$0.587100\pi$$
$$692$$ −1236.54 −0.0679280
$$693$$ 0 0
$$694$$ −16017.4 −0.876101
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 10760.2 0.584750
$$698$$ 9539.58 0.517304
$$699$$ 0 0
$$700$$ 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
$$704$$ 5957.03 0.318912
$$705$$ 0 0
$$706$$ −29825.0 −1.58991
$$707$$ −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
$$711$$ 0 0
$$712$$ −34726.9 −1.82787
$$713$$ −3507.36 −0.184224
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −703.181 −0.0367027
$$717$$ 0 0
$$718$$ 31420.6 1.63316
$$719$$ 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
$$722$$ 34750.2 1.79123
$$723$$ 0 0
$$724$$ 430.420 0.0220945
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 502.545 0.0256373 0.0128187 0.999918i $$-0.495920\pi$$
0.0128187 + 0.999918i $$0.495920\pi$$
$$728$$ −383.895 −0.0195441
$$729$$ 0 0
$$730$$ 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 $$0.569780\pi$$
$$734$$ −18487.8 −0.929697
$$735$$ 0 0
$$736$$ −2695.90 −0.135017
$$737$$ −3765.95 −0.188223
$$738$$ 0 0
$$739$$ −18357.5 −0.913792 −0.456896 0.889520i $$-0.651039\pi$$
−0.456896 + 0.889520i $$0.651039\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 2878.52 0.142417
$$743$$ 11182.6 0.552155 0.276078 0.961135i $$-0.410965\pi$$
0.276078 + 0.961135i $$0.410965\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 14507.8 0.712021
$$747$$ 0 0
$$748$$ 242.960 0.0118763
$$749$$ −2556.29 −0.124706
$$750$$ 0 0
$$751$$ 16733.4 0.813063 0.406531 0.913637i $$-0.366738\pi$$
0.406531 + 0.913637i $$0.366738\pi$$
$$752$$ 20433.3 0.990857
$$753$$ 0 0
$$754$$ −365.950 −0.0176752
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 24402.4 1.17163 0.585813 0.810446i $$-0.300775\pi$$
0.585813 + 0.810446i $$0.300775\pi$$
$$758$$ −2290.19 −0.109741
$$759$$ 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
$$764$$ 920.805 0.0436042
$$765$$ 0 0
$$766$$ −7737.62 −0.364976
$$767$$ 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
$$771$$ 0 0
$$772$$ 718.202 0.0334827
$$773$$ −35571.4 −1.65513 −0.827564 0.561371i $$-0.810274\pi$$
−0.827564 + 0.561371i $$0.810274\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 31399.1 1.45253
$$777$$ 0 0
$$778$$ −8500.11 −0.391701
$$779$$ −36530.0 −1.68013
$$780$$ 0 0
$$781$$ 2281.01 0.104508
$$782$$ 12538.7 0.573381
$$783$$ 0 0
$$784$$ 19822.3 0.902982
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −15729.6 −0.712452 −0.356226 0.934400i $$-0.615937\pi$$
−0.356226 + 0.934400i $$0.615937\pi$$
$$788$$ 1885.47 0.0852373
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −906.431 −0.0407446
$$792$$ 0 0
$$793$$ −3958.67 −0.177272
$$794$$ −38819.0 −1.73506
$$795$$ 0 0
$$796$$ −441.415 −0.0196552
$$797$$ 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
$$801$$ 0 0
$$802$$ 17107.2 0.753214
$$803$$ 11116.6 0.488538
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −461.154 −0.0201532
$$807$$ 0 0
$$808$$ −3765.36 −0.163942
$$809$$ −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 $$0.400550\pi$$
$$812$$ 41.1458 0.00177824
$$813$$ 0 0
$$814$$ −394.999 −0.0170082
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8075.35 0.345803
$$818$$ −11454.1 −0.489589
$$819$$ 0 0
$$820$$ 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 $$0.296262\pi$$
$$824$$ −810.602 −0.0342702
$$825$$ 0 0
$$826$$ 741.622 0.0312401
$$827$$ 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
$$831$$ 0 0
$$832$$ −2902.15 −0.120930
$$833$$ 13748.0 0.571836
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −824.830 −0.0341236
$$837$$ 0 0
$$838$$ 25373.0 1.04594
$$839$$ 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
$$842$$ 35915.6 1.46999
$$843$$ 0 0
$$844$$ 57.5250 0.00234608
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −371.687 −0.0150783
$$848$$ 20382.7 0.825406
$$849$$ 0 0
$$850$$ 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 $$0.831003\pi$$
$$854$$ −6199.37 −0.248405
$$855$$ 0 0
$$856$$ −19406.8 −0.774898
$$857$$ −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 $$-0.440761\pi$$
0.185031 + 0.982733i $$0.440761\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −13413.5 −0.530005
$$863$$ −9647.65 −0.380544 −0.190272 0.981731i $$-0.560937\pi$$
−0.190272 + 0.981731i $$0.560937\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 32083.3 1.25893
$$867$$ 0 0
$$868$$ 51.8501 0.00202754
$$869$$ 14236.5 0.555742
$$870$$ 0 0
$$871$$ 1834.70 0.0713735
$$872$$ −24352.6 −0.945737
$$873$$ 0 0
$$874$$ −42568.0 −1.64746
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −19728.7 −0.759624 −0.379812 0.925064i $$-0.624011\pi$$
−0.379812 + 0.925064i $$0.624011\pi$$
$$878$$ −32304.3 −1.24171
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −19473.9 −0.744712 −0.372356 0.928090i $$-0.621450\pi$$
−0.372356 + 0.928090i $$0.621450\pi$$
$$882$$ 0 0
$$883$$ −49092.4 −1.87100 −0.935499 0.353329i $$-0.885050\pi$$
−0.935499 + 0.353329i $$0.885050\pi$$
$$884$$ −118.365 −0.00450346
$$885$$ 0 0
$$886$$ 27599.5 1.04653
$$887$$ 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
$$892$$ −2108.05 −0.0791288
$$893$$ −48112.0 −1.80292
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −3949.89 −0.147273
$$897$$ 0 0
$$898$$ 943.252 0.0350520
$$899$$ 787.273 0.0292069
$$900$$ 0 0
$$901$$ 14136.7 0.522709
$$902$$ −7845.88 −0.289622
$$903$$ 0 0
$$904$$ −6881.46 −0.253179
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −37688.7 −1.37975 −0.689875 0.723928i $$-0.742335\pi$$
−0.689875 + 0.723928i $$0.742335\pi$$
$$908$$ 949.559 0.0347051
$$909$$ 0 0
$$910$$ 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
$$914$$ 28870.0 1.04479
$$915$$ 0 0
$$916$$ −1026.44 −0.0370247
$$917$$ −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
$$921$$ 0 0
$$922$$ −12933.4 −0.461973
$$923$$ −1111.26 −0.0396290
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −9374.21 −0.332673
$$927$$ 0 0
$$928$$ 605.131 0.0214056
$$929$$ 23177.9 0.818561 0.409280 0.912409i $$-0.365780\pi$$
0.409280 + 0.912409i $$0.365780\pi$$
$$930$$ 0 0
$$931$$ −46673.3 −1.64302
$$932$$ −2355.98 −0.0828033
$$933$$ 0 0
$$934$$ 13979.8 0.489757
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 34574.7 1.20545 0.602724 0.797950i $$-0.294082\pi$$
0.602724 + 0.797950i $$0.294082\pi$$
$$938$$ 2873.18 0.100014
$$939$$ 0 0
$$940$$ 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
$$944$$ 5251.40 0.181058
$$945$$ 0 0
$$946$$ 1734.42 0.0596097
$$947$$ 27231.2 0.934419 0.467209 0.884147i $$-0.345259\pi$$
0.467209 + 0.884147i $$0.345259\pi$$
$$948$$ 0 0
$$949$$ −5415.79 −0.185252
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −2952.50 −0.100516
$$953$$ 40939.4 1.39156 0.695781 0.718254i $$-0.255058\pi$$
0.695781 + 0.718254i $$0.255058\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −2188.87 −0.0740515
$$957$$ 0 0
$$958$$ −31601.3 −1.06575
$$959$$ −4950.74 −0.166703
$$960$$ 0 0
$$961$$ −28798.9 −0.966698
$$962$$ 192.436 0.00644945
$$963$$ 0 0
$$964$$ −2094.60 −0.0699820
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 46173.1 1.53550 0.767750 0.640750i $$-0.221376\pi$$
0.767750 + 0.640750i $$0.221376\pi$$
$$968$$ −2821.78 −0.0936937
$$969$$ 0 0
$$970$$ 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
$$974$$ 50069.0 1.64714
$$975$$ 0 0
$$976$$ −43897.6 −1.43968
$$977$$ 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
$$981$$ 0 0
$$982$$ 20811.6 0.676299
$$983$$ 32915.7 1.06800 0.534002 0.845483i $$-0.320687\pi$$
0.534002 + 0.845483i $$0.320687\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2814.48 −0.0909041
$$987$$ 0 0
$$988$$ 401.841 0.0129395
$$989$$ −6426.54 −0.206625
$$990$$ 0 0
$$991$$ 29477.9 0.944901 0.472451 0.881357i $$-0.343370\pi$$
0.472451 + 0.881357i $$0.343370\pi$$
$$992$$ 762.560 0.0244066
$$993$$ 0 0
$$994$$ −1740.26 −0.0555310
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 31944.4 1.01473 0.507366 0.861731i $$-0.330619\pi$$
0.507366 + 0.861731i $$0.330619\pi$$
$$998$$ 35268.4 1.11864
$$999$$ 0 0
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 2475.4.a.q.1.2 2
3.2 odd 2 275.4.a.b.1.1 2
5.4 even 2 99.4.a.c.1.1 2
15.2 even 4 275.4.b.c.199.1 4
15.8 even 4 275.4.b.c.199.4 4
15.14 odd 2 11.4.a.a.1.2 2
20.19 odd 2 1584.4.a.bc.1.1 2
55.54 odd 2 1089.4.a.v.1.2 2
60.59 even 2 176.4.a.i.1.2 2
105.104 even 2 539.4.a.e.1.2 2
120.29 odd 2 704.4.a.p.1.2 2
120.59 even 2 704.4.a.n.1.1 2
165.14 odd 10 121.4.c.c.9.2 8
165.29 even 10 121.4.c.f.27.1 8
165.59 odd 10 121.4.c.c.27.2 8
165.74 even 10 121.4.c.f.9.1 8
165.104 odd 10 121.4.c.c.3.1 8
165.119 odd 10 121.4.c.c.81.1 8
165.134 even 10 121.4.c.f.81.2 8
165.149 even 10 121.4.c.f.3.2 8
165.164 even 2 121.4.a.c.1.1 2
195.194 odd 2 1859.4.a.a.1.1 2
660.659 odd 2 1936.4.a.w.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 15.14 odd 2
99.4.a.c.1.1 2 5.4 even 2
121.4.a.c.1.1 2 165.164 even 2
121.4.c.c.3.1 8 165.104 odd 10
121.4.c.c.9.2 8 165.14 odd 10
121.4.c.c.27.2 8 165.59 odd 10
121.4.c.c.81.1 8 165.119 odd 10
121.4.c.f.3.2 8 165.149 even 10
121.4.c.f.9.1 8 165.74 even 10
121.4.c.f.27.1 8 165.29 even 10
121.4.c.f.81.2 8 165.134 even 10
176.4.a.i.1.2 2 60.59 even 2
275.4.a.b.1.1 2 3.2 odd 2
275.4.b.c.199.1 4 15.2 even 4
275.4.b.c.199.4 4 15.8 even 4
539.4.a.e.1.2 2 105.104 even 2
704.4.a.n.1.1 2 120.59 even 2
704.4.a.p.1.2 2 120.29 odd 2
1089.4.a.v.1.2 2 55.54 odd 2
1584.4.a.bc.1.1 2 20.19 odd 2
1859.4.a.a.1.1 2 195.194 odd 2
1936.4.a.w.1.2 2 660.659 odd 2
2475.4.a.q.1.2 2 1.1 even 1 trivial