# Properties

 Label 1078.6.a.a.1.1 Level $1078$ Weight $6$ Character 1078.1 Self dual yes Analytic conductor $172.894$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,6,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1078.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1078.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: $$172.893757758$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1078.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} -1.00000 q^{3} +16.0000 q^{4} +51.0000 q^{5} +4.00000 q^{6} -64.0000 q^{8} -242.000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{2} -1.00000 q^{3} +16.0000 q^{4} +51.0000 q^{5} +4.00000 q^{6} -64.0000 q^{8} -242.000 q^{9} -204.000 q^{10} -121.000 q^{11} -16.0000 q^{12} -692.000 q^{13} -51.0000 q^{15} +256.000 q^{16} +738.000 q^{17} +968.000 q^{18} -1424.00 q^{19} +816.000 q^{20} +484.000 q^{22} -1779.00 q^{23} +64.0000 q^{24} -524.000 q^{25} +2768.00 q^{26} +485.000 q^{27} -2064.00 q^{29} +204.000 q^{30} -6245.00 q^{31} -1024.00 q^{32} +121.000 q^{33} -2952.00 q^{34} -3872.00 q^{36} -14785.0 q^{37} +5696.00 q^{38} +692.000 q^{39} -3264.00 q^{40} -5304.00 q^{41} +17798.0 q^{43} -1936.00 q^{44} -12342.0 q^{45} +7116.00 q^{46} +17184.0 q^{47} -256.000 q^{48} +2096.00 q^{50} -738.000 q^{51} -11072.0 q^{52} -30726.0 q^{53} -1940.00 q^{54} -6171.00 q^{55} +1424.00 q^{57} +8256.00 q^{58} +34989.0 q^{59} -816.000 q^{60} +45940.0 q^{61} +24980.0 q^{62} +4096.00 q^{64} -35292.0 q^{65} -484.000 q^{66} +25343.0 q^{67} +11808.0 q^{68} +1779.00 q^{69} +13311.0 q^{71} +15488.0 q^{72} +53260.0 q^{73} +59140.0 q^{74} +524.000 q^{75} -22784.0 q^{76} -2768.00 q^{78} +77234.0 q^{79} +13056.0 q^{80} +58321.0 q^{81} +21216.0 q^{82} -55014.0 q^{83} +37638.0 q^{85} -71192.0 q^{86} +2064.00 q^{87} +7744.00 q^{88} -125415. q^{89} +49368.0 q^{90} -28464.0 q^{92} +6245.00 q^{93} -68736.0 q^{94} -72624.0 q^{95} +1024.00 q^{96} +88807.0 q^{97} +29282.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.00000 −0.707107
$$3$$ −1.00000 −0.0641500 −0.0320750 0.999485i $$-0.510212\pi$$
−0.0320750 + 0.999485i $$0.510212\pi$$
$$4$$ 16.0000 0.500000
$$5$$ 51.0000 0.912316 0.456158 0.889899i $$-0.349225\pi$$
0.456158 + 0.889899i $$0.349225\pi$$
$$6$$ 4.00000 0.0453609
$$7$$ 0 0
$$8$$ −64.0000 −0.353553
$$9$$ −242.000 −0.995885
$$10$$ −204.000 −0.645105
$$11$$ −121.000 −0.301511
$$12$$ −16.0000 −0.0320750
$$13$$ −692.000 −1.13566 −0.567829 0.823146i $$-0.692217\pi$$
−0.567829 + 0.823146i $$0.692217\pi$$
$$14$$ 0 0
$$15$$ −51.0000 −0.0585251
$$16$$ 256.000 0.250000
$$17$$ 738.000 0.619347 0.309674 0.950843i $$-0.399780\pi$$
0.309674 + 0.950843i $$0.399780\pi$$
$$18$$ 968.000 0.704197
$$19$$ −1424.00 −0.904953 −0.452476 0.891776i $$-0.649459\pi$$
−0.452476 + 0.891776i $$0.649459\pi$$
$$20$$ 816.000 0.456158
$$21$$ 0 0
$$22$$ 484.000 0.213201
$$23$$ −1779.00 −0.701223 −0.350612 0.936521i $$-0.614026\pi$$
−0.350612 + 0.936521i $$0.614026\pi$$
$$24$$ 64.0000 0.0226805
$$25$$ −524.000 −0.167680
$$26$$ 2768.00 0.803032
$$27$$ 485.000 0.128036
$$28$$ 0 0
$$29$$ −2064.00 −0.455737 −0.227869 0.973692i $$-0.573176\pi$$
−0.227869 + 0.973692i $$0.573176\pi$$
$$30$$ 204.000 0.0413835
$$31$$ −6245.00 −1.16715 −0.583577 0.812058i $$-0.698347\pi$$
−0.583577 + 0.812058i $$0.698347\pi$$
$$32$$ −1024.00 −0.176777
$$33$$ 121.000 0.0193420
$$34$$ −2952.00 −0.437944
$$35$$ 0 0
$$36$$ −3872.00 −0.497942
$$37$$ −14785.0 −1.77549 −0.887743 0.460340i $$-0.847727\pi$$
−0.887743 + 0.460340i $$0.847727\pi$$
$$38$$ 5696.00 0.639898
$$39$$ 692.000 0.0728525
$$40$$ −3264.00 −0.322552
$$41$$ −5304.00 −0.492770 −0.246385 0.969172i $$-0.579243\pi$$
−0.246385 + 0.969172i $$0.579243\pi$$
$$42$$ 0 0
$$43$$ 17798.0 1.46791 0.733956 0.679197i $$-0.237672\pi$$
0.733956 + 0.679197i $$0.237672\pi$$
$$44$$ −1936.00 −0.150756
$$45$$ −12342.0 −0.908561
$$46$$ 7116.00 0.495840
$$47$$ 17184.0 1.13470 0.567348 0.823478i $$-0.307969\pi$$
0.567348 + 0.823478i $$0.307969\pi$$
$$48$$ −256.000 −0.0160375
$$49$$ 0 0
$$50$$ 2096.00 0.118568
$$51$$ −738.000 −0.0397311
$$52$$ −11072.0 −0.567829
$$53$$ −30726.0 −1.50251 −0.751253 0.660014i $$-0.770550\pi$$
−0.751253 + 0.660014i $$0.770550\pi$$
$$54$$ −1940.00 −0.0905352
$$55$$ −6171.00 −0.275074
$$56$$ 0 0
$$57$$ 1424.00 0.0580528
$$58$$ 8256.00 0.322255
$$59$$ 34989.0 1.30858 0.654292 0.756242i $$-0.272967\pi$$
0.654292 + 0.756242i $$0.272967\pi$$
$$60$$ −816.000 −0.0292625
$$61$$ 45940.0 1.58076 0.790381 0.612616i $$-0.209883\pi$$
0.790381 + 0.612616i $$0.209883\pi$$
$$62$$ 24980.0 0.825303
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ −35292.0 −1.03608
$$66$$ −484.000 −0.0136768
$$67$$ 25343.0 0.689717 0.344859 0.938655i $$-0.387927\pi$$
0.344859 + 0.938655i $$0.387927\pi$$
$$68$$ 11808.0 0.309674
$$69$$ 1779.00 0.0449835
$$70$$ 0 0
$$71$$ 13311.0 0.313375 0.156688 0.987648i $$-0.449918\pi$$
0.156688 + 0.987648i $$0.449918\pi$$
$$72$$ 15488.0 0.352098
$$73$$ 53260.0 1.16975 0.584876 0.811123i $$-0.301143\pi$$
0.584876 + 0.811123i $$0.301143\pi$$
$$74$$ 59140.0 1.25546
$$75$$ 524.000 0.0107567
$$76$$ −22784.0 −0.452476
$$77$$ 0 0
$$78$$ −2768.00 −0.0515145
$$79$$ 77234.0 1.39233 0.696163 0.717884i $$-0.254889\pi$$
0.696163 + 0.717884i $$0.254889\pi$$
$$80$$ 13056.0 0.228079
$$81$$ 58321.0 0.987671
$$82$$ 21216.0 0.348441
$$83$$ −55014.0 −0.876553 −0.438276 0.898840i $$-0.644411\pi$$
−0.438276 + 0.898840i $$0.644411\pi$$
$$84$$ 0 0
$$85$$ 37638.0 0.565040
$$86$$ −71192.0 −1.03797
$$87$$ 2064.00 0.0292356
$$88$$ 7744.00 0.106600
$$89$$ −125415. −1.67832 −0.839159 0.543886i $$-0.816953\pi$$
−0.839159 + 0.543886i $$0.816953\pi$$
$$90$$ 49368.0 0.642450
$$91$$ 0 0
$$92$$ −28464.0 −0.350612
$$93$$ 6245.00 0.0748730
$$94$$ −68736.0 −0.802351
$$95$$ −72624.0 −0.825603
$$96$$ 1024.00 0.0113402
$$97$$ 88807.0 0.958336 0.479168 0.877723i $$-0.340938\pi$$
0.479168 + 0.877723i $$0.340938\pi$$
$$98$$ 0 0
$$99$$ 29282.0 0.300271
$$100$$ −8384.00 −0.0838400
$$101$$ −1482.00 −0.0144559 −0.00722794 0.999974i $$-0.502301\pi$$
−0.00722794 + 0.999974i $$0.502301\pi$$
$$102$$ 2952.00 0.0280942
$$103$$ 117496. 1.09126 0.545632 0.838025i $$-0.316290\pi$$
0.545632 + 0.838025i $$0.316290\pi$$
$$104$$ 44288.0 0.401516
$$105$$ 0 0
$$106$$ 122904. 1.06243
$$107$$ −79362.0 −0.670121 −0.335060 0.942197i $$-0.608757\pi$$
−0.335060 + 0.942197i $$0.608757\pi$$
$$108$$ 7760.00 0.0640180
$$109$$ 87842.0 0.708167 0.354084 0.935214i $$-0.384793\pi$$
0.354084 + 0.935214i $$0.384793\pi$$
$$110$$ 24684.0 0.194506
$$111$$ 14785.0 0.113897
$$112$$ 0 0
$$113$$ −47247.0 −0.348079 −0.174040 0.984739i $$-0.555682\pi$$
−0.174040 + 0.984739i $$0.555682\pi$$
$$114$$ −5696.00 −0.0410495
$$115$$ −90729.0 −0.639737
$$116$$ −33024.0 −0.227869
$$117$$ 167464. 1.13098
$$118$$ −139956. −0.925308
$$119$$ 0 0
$$120$$ 3264.00 0.0206917
$$121$$ 14641.0 0.0909091
$$122$$ −183760. −1.11777
$$123$$ 5304.00 0.0316112
$$124$$ −99920.0 −0.583577
$$125$$ −186099. −1.06529
$$126$$ 0 0
$$127$$ −239416. −1.31718 −0.658588 0.752504i $$-0.728846\pi$$
−0.658588 + 0.752504i $$0.728846\pi$$
$$128$$ −16384.0 −0.0883883
$$129$$ −17798.0 −0.0941666
$$130$$ 141168. 0.732618
$$131$$ 98142.0 0.499662 0.249831 0.968289i $$-0.419625\pi$$
0.249831 + 0.968289i $$0.419625\pi$$
$$132$$ 1936.00 0.00967098
$$133$$ 0 0
$$134$$ −101372. −0.487704
$$135$$ 24735.0 0.116809
$$136$$ −47232.0 −0.218972
$$137$$ 400137. 1.82141 0.910704 0.413059i $$-0.135540\pi$$
0.910704 + 0.413059i $$0.135540\pi$$
$$138$$ −7116.00 −0.0318081
$$139$$ −205766. −0.903310 −0.451655 0.892193i $$-0.649166\pi$$
−0.451655 + 0.892193i $$0.649166\pi$$
$$140$$ 0 0
$$141$$ −17184.0 −0.0727908
$$142$$ −53244.0 −0.221590
$$143$$ 83732.0 0.342414
$$144$$ −61952.0 −0.248971
$$145$$ −105264. −0.415776
$$146$$ −213040. −0.827140
$$147$$ 0 0
$$148$$ −236560. −0.887743
$$149$$ 87726.0 0.323715 0.161857 0.986814i $$-0.448252\pi$$
0.161857 + 0.986814i $$0.448252\pi$$
$$150$$ −2096.00 −0.00760612
$$151$$ −432778. −1.54462 −0.772312 0.635243i $$-0.780900\pi$$
−0.772312 + 0.635243i $$0.780900\pi$$
$$152$$ 91136.0 0.319949
$$153$$ −178596. −0.616798
$$154$$ 0 0
$$155$$ −318495. −1.06481
$$156$$ 11072.0 0.0364263
$$157$$ 34075.0 0.110328 0.0551641 0.998477i $$-0.482432\pi$$
0.0551641 + 0.998477i $$0.482432\pi$$
$$158$$ −308936. −0.984523
$$159$$ 30726.0 0.0963858
$$160$$ −52224.0 −0.161276
$$161$$ 0 0
$$162$$ −233284. −0.698389
$$163$$ 45020.0 0.132720 0.0663600 0.997796i $$-0.478861\pi$$
0.0663600 + 0.997796i $$0.478861\pi$$
$$164$$ −84864.0 −0.246385
$$165$$ 6171.00 0.0176460
$$166$$ 220056. 0.619816
$$167$$ −482556. −1.33893 −0.669463 0.742845i $$-0.733476\pi$$
−0.669463 + 0.742845i $$0.733476\pi$$
$$168$$ 0 0
$$169$$ 107571. 0.289720
$$170$$ −150552. −0.399544
$$171$$ 344608. 0.901229
$$172$$ 284768. 0.733956
$$173$$ 766254. 1.94651 0.973257 0.229719i $$-0.0737808\pi$$
0.973257 + 0.229719i $$0.0737808\pi$$
$$174$$ −8256.00 −0.0206727
$$175$$ 0 0
$$176$$ −30976.0 −0.0753778
$$177$$ −34989.0 −0.0839457
$$178$$ 501660. 1.18675
$$179$$ 303399. 0.707753 0.353876 0.935292i $$-0.384863\pi$$
0.353876 + 0.935292i $$0.384863\pi$$
$$180$$ −197472. −0.454281
$$181$$ 285181. 0.647030 0.323515 0.946223i $$-0.395135\pi$$
0.323515 + 0.946223i $$0.395135\pi$$
$$182$$ 0 0
$$183$$ −45940.0 −0.101406
$$184$$ 113856. 0.247920
$$185$$ −754035. −1.61980
$$186$$ −24980.0 −0.0529432
$$187$$ −89298.0 −0.186740
$$188$$ 274944. 0.567348
$$189$$ 0 0
$$190$$ 290496. 0.583789
$$191$$ 767067. 1.52142 0.760711 0.649090i $$-0.224850\pi$$
0.760711 + 0.649090i $$0.224850\pi$$
$$192$$ −4096.00 −0.00801875
$$193$$ 411668. 0.795525 0.397763 0.917488i $$-0.369787\pi$$
0.397763 + 0.917488i $$0.369787\pi$$
$$194$$ −355228. −0.677646
$$195$$ 35292.0 0.0664645
$$196$$ 0 0
$$197$$ −759258. −1.39387 −0.696937 0.717132i $$-0.745455\pi$$
−0.696937 + 0.717132i $$0.745455\pi$$
$$198$$ −117128. −0.212323
$$199$$ 46600.0 0.0834167 0.0417084 0.999130i $$-0.486720\pi$$
0.0417084 + 0.999130i $$0.486720\pi$$
$$200$$ 33536.0 0.0592838
$$201$$ −25343.0 −0.0442454
$$202$$ 5928.00 0.0102219
$$203$$ 0 0
$$204$$ −11808.0 −0.0198656
$$205$$ −270504. −0.449561
$$206$$ −469984. −0.771641
$$207$$ 430518. 0.698338
$$208$$ −177152. −0.283915
$$209$$ 172304. 0.272854
$$210$$ 0 0
$$211$$ −932428. −1.44181 −0.720907 0.693032i $$-0.756274\pi$$
−0.720907 + 0.693032i $$0.756274\pi$$
$$212$$ −491616. −0.751253
$$213$$ −13311.0 −0.0201030
$$214$$ 317448. 0.473847
$$215$$ 907698. 1.33920
$$216$$ −31040.0 −0.0452676
$$217$$ 0 0
$$218$$ −351368. −0.500750
$$219$$ −53260.0 −0.0750397
$$220$$ −98736.0 −0.137537
$$221$$ −510696. −0.703367
$$222$$ −59140.0 −0.0805376
$$223$$ −169745. −0.228578 −0.114289 0.993448i $$-0.536459\pi$$
−0.114289 + 0.993448i $$0.536459\pi$$
$$224$$ 0 0
$$225$$ 126808. 0.166990
$$226$$ 188988. 0.246129
$$227$$ −198078. −0.255136 −0.127568 0.991830i $$-0.540717\pi$$
−0.127568 + 0.991830i $$0.540717\pi$$
$$228$$ 22784.0 0.0290264
$$229$$ 849997. 1.07110 0.535548 0.844505i $$-0.320105\pi$$
0.535548 + 0.844505i $$0.320105\pi$$
$$230$$ 362916. 0.452362
$$231$$ 0 0
$$232$$ 132096. 0.161128
$$233$$ −401832. −0.484903 −0.242451 0.970164i $$-0.577952\pi$$
−0.242451 + 0.970164i $$0.577952\pi$$
$$234$$ −669856. −0.799727
$$235$$ 876384. 1.03520
$$236$$ 559824. 0.654292
$$237$$ −77234.0 −0.0893177
$$238$$ 0 0
$$239$$ 855174. 0.968411 0.484206 0.874954i $$-0.339109\pi$$
0.484206 + 0.874954i $$0.339109\pi$$
$$240$$ −13056.0 −0.0146313
$$241$$ −1.12546e6 −1.24821 −0.624107 0.781339i $$-0.714537\pi$$
−0.624107 + 0.781339i $$0.714537\pi$$
$$242$$ −58564.0 −0.0642824
$$243$$ −176176. −0.191395
$$244$$ 735040. 0.790381
$$245$$ 0 0
$$246$$ −21216.0 −0.0223525
$$247$$ 985408. 1.02772
$$248$$ 399680. 0.412651
$$249$$ 55014.0 0.0562309
$$250$$ 744396. 0.753276
$$251$$ 1.19751e6 1.19976 0.599882 0.800088i $$-0.295214\pi$$
0.599882 + 0.800088i $$0.295214\pi$$
$$252$$ 0 0
$$253$$ 215259. 0.211427
$$254$$ 957664. 0.931384
$$255$$ −37638.0 −0.0362473
$$256$$ 65536.0 0.0625000
$$257$$ −37758.0 −0.0356596 −0.0178298 0.999841i $$-0.505676\pi$$
−0.0178298 + 0.999841i $$0.505676\pi$$
$$258$$ 71192.0 0.0665858
$$259$$ 0 0
$$260$$ −564672. −0.518040
$$261$$ 499488. 0.453862
$$262$$ −392568. −0.353315
$$263$$ −631254. −0.562749 −0.281375 0.959598i $$-0.590790\pi$$
−0.281375 + 0.959598i $$0.590790\pi$$
$$264$$ −7744.00 −0.00683842
$$265$$ −1.56703e6 −1.37076
$$266$$ 0 0
$$267$$ 125415. 0.107664
$$268$$ 405488. 0.344859
$$269$$ 1.08034e6 0.910292 0.455146 0.890417i $$-0.349587\pi$$
0.455146 + 0.890417i $$0.349587\pi$$
$$270$$ −98940.0 −0.0825967
$$271$$ 816100. 0.675025 0.337513 0.941321i $$-0.390414\pi$$
0.337513 + 0.941321i $$0.390414\pi$$
$$272$$ 188928. 0.154837
$$273$$ 0 0
$$274$$ −1.60055e6 −1.28793
$$275$$ 63404.0 0.0505574
$$276$$ 28464.0 0.0224917
$$277$$ 1.68820e6 1.32198 0.660989 0.750396i $$-0.270137\pi$$
0.660989 + 0.750396i $$0.270137\pi$$
$$278$$ 823064. 0.638736
$$279$$ 1.51129e6 1.16235
$$280$$ 0 0
$$281$$ −879042. −0.664116 −0.332058 0.943259i $$-0.607743\pi$$
−0.332058 + 0.943259i $$0.607743\pi$$
$$282$$ 68736.0 0.0514709
$$283$$ −1.54027e6 −1.14322 −0.571611 0.820525i $$-0.693681\pi$$
−0.571611 + 0.820525i $$0.693681\pi$$
$$284$$ 212976. 0.156688
$$285$$ 72624.0 0.0529624
$$286$$ −334928. −0.242123
$$287$$ 0 0
$$288$$ 247808. 0.176049
$$289$$ −875213. −0.616409
$$290$$ 421056. 0.293998
$$291$$ −88807.0 −0.0614773
$$292$$ 852160. 0.584876
$$293$$ −720840. −0.490535 −0.245267 0.969455i $$-0.578876\pi$$
−0.245267 + 0.969455i $$0.578876\pi$$
$$294$$ 0 0
$$295$$ 1.78444e6 1.19384
$$296$$ 946240. 0.627729
$$297$$ −58685.0 −0.0386043
$$298$$ −350904. −0.228901
$$299$$ 1.23107e6 0.796350
$$300$$ 8384.00 0.00537834
$$301$$ 0 0
$$302$$ 1.73111e6 1.09221
$$303$$ 1482.00 0.000927346 0
$$304$$ −364544. −0.226238
$$305$$ 2.34294e6 1.44215
$$306$$ 714384. 0.436142
$$307$$ 1.03905e6 0.629201 0.314601 0.949224i $$-0.398129\pi$$
0.314601 + 0.949224i $$0.398129\pi$$
$$308$$ 0 0
$$309$$ −117496. −0.0700046
$$310$$ 1.27398e6 0.752937
$$311$$ 1.25135e6 0.733630 0.366815 0.930294i $$-0.380448\pi$$
0.366815 + 0.930294i $$0.380448\pi$$
$$312$$ −44288.0 −0.0257573
$$313$$ 1.44336e6 0.832749 0.416375 0.909193i $$-0.363301\pi$$
0.416375 + 0.909193i $$0.363301\pi$$
$$314$$ −136300. −0.0780139
$$315$$ 0 0
$$316$$ 1.23574e6 0.696163
$$317$$ −2.01208e6 −1.12460 −0.562298 0.826934i $$-0.690083\pi$$
−0.562298 + 0.826934i $$0.690083\pi$$
$$318$$ −122904. −0.0681551
$$319$$ 249744. 0.137410
$$320$$ 208896. 0.114039
$$321$$ 79362.0 0.0429883
$$322$$ 0 0
$$323$$ −1.05091e6 −0.560480
$$324$$ 933136. 0.493836
$$325$$ 362608. 0.190427
$$326$$ −180080. −0.0938472
$$327$$ −87842.0 −0.0454290
$$328$$ 339456. 0.174220
$$329$$ 0 0
$$330$$ −24684.0 −0.0124776
$$331$$ 2.01734e6 1.01207 0.506033 0.862514i $$-0.331112\pi$$
0.506033 + 0.862514i $$0.331112\pi$$
$$332$$ −880224. −0.438276
$$333$$ 3.57797e6 1.76818
$$334$$ 1.93022e6 0.946764
$$335$$ 1.29249e6 0.629240
$$336$$ 0 0
$$337$$ 264122. 0.126686 0.0633432 0.997992i $$-0.479824\pi$$
0.0633432 + 0.997992i $$0.479824\pi$$
$$338$$ −430284. −0.204863
$$339$$ 47247.0 0.0223293
$$340$$ 602208. 0.282520
$$341$$ 755645. 0.351910
$$342$$ −1.37843e6 −0.637265
$$343$$ 0 0
$$344$$ −1.13907e6 −0.518985
$$345$$ 90729.0 0.0410392
$$346$$ −3.06502e6 −1.37639
$$347$$ −1.71049e6 −0.762601 −0.381300 0.924451i $$-0.624524\pi$$
−0.381300 + 0.924451i $$0.624524\pi$$
$$348$$ 33024.0 0.0146178
$$349$$ −218822. −0.0961673 −0.0480836 0.998843i $$-0.515311\pi$$
−0.0480836 + 0.998843i $$0.515311\pi$$
$$350$$ 0 0
$$351$$ −335620. −0.145405
$$352$$ 123904. 0.0533002
$$353$$ −3.68192e6 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$354$$ 139956. 0.0593586
$$355$$ 678861. 0.285897
$$356$$ −2.00664e6 −0.839159
$$357$$ 0 0
$$358$$ −1.21360e6 −0.500457
$$359$$ 1.88528e6 0.772042 0.386021 0.922490i $$-0.373849\pi$$
0.386021 + 0.922490i $$0.373849\pi$$
$$360$$ 789888. 0.321225
$$361$$ −448323. −0.181060
$$362$$ −1.14072e6 −0.457519
$$363$$ −14641.0 −0.00583182
$$364$$ 0 0
$$365$$ 2.71626e6 1.06718
$$366$$ 183760. 0.0717048
$$367$$ 3.11666e6 1.20788 0.603940 0.797029i $$-0.293596\pi$$
0.603940 + 0.797029i $$0.293596\pi$$
$$368$$ −455424. −0.175306
$$369$$ 1.28357e6 0.490742
$$370$$ 3.01614e6 1.14537
$$371$$ 0 0
$$372$$ 99920.0 0.0374365
$$373$$ 1.39441e6 0.518943 0.259471 0.965751i $$-0.416452\pi$$
0.259471 + 0.965751i $$0.416452\pi$$
$$374$$ 357192. 0.132045
$$375$$ 186099. 0.0683386
$$376$$ −1.09978e6 −0.401176
$$377$$ 1.42829e6 0.517562
$$378$$ 0 0
$$379$$ −4.26036e6 −1.52352 −0.761759 0.647860i $$-0.775664\pi$$
−0.761759 + 0.647860i $$0.775664\pi$$
$$380$$ −1.16198e6 −0.412801
$$381$$ 239416. 0.0844969
$$382$$ −3.06827e6 −1.07581
$$383$$ −201765. −0.0702828 −0.0351414 0.999382i $$-0.511188\pi$$
−0.0351414 + 0.999382i $$0.511188\pi$$
$$384$$ 16384.0 0.00567012
$$385$$ 0 0
$$386$$ −1.64667e6 −0.562521
$$387$$ −4.30712e6 −1.46187
$$388$$ 1.42091e6 0.479168
$$389$$ 1.94882e6 0.652977 0.326489 0.945201i $$-0.394135\pi$$
0.326489 + 0.945201i $$0.394135\pi$$
$$390$$ −141168. −0.0469975
$$391$$ −1.31290e6 −0.434301
$$392$$ 0 0
$$393$$ −98142.0 −0.0320534
$$394$$ 3.03703e6 0.985618
$$395$$ 3.93893e6 1.27024
$$396$$ 468512. 0.150135
$$397$$ 1.46826e6 0.467548 0.233774 0.972291i $$-0.424892\pi$$
0.233774 + 0.972291i $$0.424892\pi$$
$$398$$ −186400. −0.0589845
$$399$$ 0 0
$$400$$ −134144. −0.0419200
$$401$$ 2.24618e6 0.697563 0.348781 0.937204i $$-0.386596\pi$$
0.348781 + 0.937204i $$0.386596\pi$$
$$402$$ 101372. 0.0312862
$$403$$ 4.32154e6 1.32549
$$404$$ −23712.0 −0.00722794
$$405$$ 2.97437e6 0.901068
$$406$$ 0 0
$$407$$ 1.78898e6 0.535329
$$408$$ 47232.0 0.0140471
$$409$$ 3.61488e6 1.06853 0.534263 0.845318i $$-0.320589\pi$$
0.534263 + 0.845318i $$0.320589\pi$$
$$410$$ 1.08202e6 0.317888
$$411$$ −400137. −0.116843
$$412$$ 1.87994e6 0.545632
$$413$$ 0 0
$$414$$ −1.72207e6 −0.493799
$$415$$ −2.80571e6 −0.799693
$$416$$ 708608. 0.200758
$$417$$ 205766. 0.0579473
$$418$$ −689216. −0.192937
$$419$$ 3.81239e6 1.06087 0.530435 0.847726i $$-0.322029\pi$$
0.530435 + 0.847726i $$0.322029\pi$$
$$420$$ 0 0
$$421$$ 1.97346e6 0.542655 0.271327 0.962487i $$-0.412537\pi$$
0.271327 + 0.962487i $$0.412537\pi$$
$$422$$ 3.72971e6 1.01952
$$423$$ −4.15853e6 −1.13003
$$424$$ 1.96646e6 0.531216
$$425$$ −386712. −0.103852
$$426$$ 53244.0 0.0142150
$$427$$ 0 0
$$428$$ −1.26979e6 −0.335060
$$429$$ −83732.0 −0.0219659
$$430$$ −3.63079e6 −0.946957
$$431$$ −2.08359e6 −0.540280 −0.270140 0.962821i $$-0.587070\pi$$
−0.270140 + 0.962821i $$0.587070\pi$$
$$432$$ 124160. 0.0320090
$$433$$ 72691.0 0.0186321 0.00931603 0.999957i $$-0.497035\pi$$
0.00931603 + 0.999957i $$0.497035\pi$$
$$434$$ 0 0
$$435$$ 105264. 0.0266721
$$436$$ 1.40547e6 0.354084
$$437$$ 2.53330e6 0.634574
$$438$$ 213040. 0.0530611
$$439$$ −594392. −0.147201 −0.0736007 0.997288i $$-0.523449\pi$$
−0.0736007 + 0.997288i $$0.523449\pi$$
$$440$$ 394944. 0.0972532
$$441$$ 0 0
$$442$$ 2.04278e6 0.497355
$$443$$ 4.56651e6 1.10554 0.552770 0.833334i $$-0.313571\pi$$
0.552770 + 0.833334i $$0.313571\pi$$
$$444$$ 236560. 0.0569487
$$445$$ −6.39616e6 −1.53116
$$446$$ 678980. 0.161629
$$447$$ −87726.0 −0.0207663
$$448$$ 0 0
$$449$$ −5.44382e6 −1.27435 −0.637174 0.770720i $$-0.719897\pi$$
−0.637174 + 0.770720i $$0.719897\pi$$
$$450$$ −507232. −0.118080
$$451$$ 641784. 0.148576
$$452$$ −755952. −0.174040
$$453$$ 432778. 0.0990877
$$454$$ 792312. 0.180408
$$455$$ 0 0
$$456$$ −91136.0 −0.0205247
$$457$$ 6.70312e6 1.50137 0.750683 0.660662i $$-0.229724\pi$$
0.750683 + 0.660662i $$0.229724\pi$$
$$458$$ −3.39999e6 −0.757380
$$459$$ 357930. 0.0792988
$$460$$ −1.45166e6 −0.319869
$$461$$ 1.25994e6 0.276120 0.138060 0.990424i $$-0.455913\pi$$
0.138060 + 0.990424i $$0.455913\pi$$
$$462$$ 0 0
$$463$$ −5.02308e6 −1.08897 −0.544487 0.838769i $$-0.683276\pi$$
−0.544487 + 0.838769i $$0.683276\pi$$
$$464$$ −528384. −0.113934
$$465$$ 318495. 0.0683078
$$466$$ 1.60733e6 0.342878
$$467$$ 2.35660e6 0.500028 0.250014 0.968242i $$-0.419565\pi$$
0.250014 + 0.968242i $$0.419565\pi$$
$$468$$ 2.67942e6 0.565492
$$469$$ 0 0
$$470$$ −3.50554e6 −0.731998
$$471$$ −34075.0 −0.00707756
$$472$$ −2.23930e6 −0.462654
$$473$$ −2.15356e6 −0.442592
$$474$$ 308936. 0.0631572
$$475$$ 746176. 0.151743
$$476$$ 0 0
$$477$$ 7.43569e6 1.49632
$$478$$ −3.42070e6 −0.684770
$$479$$ 6.72258e6 1.33874 0.669371 0.742928i $$-0.266563\pi$$
0.669371 + 0.742928i $$0.266563\pi$$
$$480$$ 52224.0 0.0103459
$$481$$ 1.02312e7 2.01634
$$482$$ 4.50186e6 0.882620
$$483$$ 0 0
$$484$$ 234256. 0.0454545
$$485$$ 4.52916e6 0.874305
$$486$$ 704704. 0.135337
$$487$$ 1.96001e6 0.374487 0.187243 0.982314i $$-0.440045\pi$$
0.187243 + 0.982314i $$0.440045\pi$$
$$488$$ −2.94016e6 −0.558884
$$489$$ −45020.0 −0.00851399
$$490$$ 0 0
$$491$$ −579624. −0.108503 −0.0542516 0.998527i $$-0.517277\pi$$
−0.0542516 + 0.998527i $$0.517277\pi$$
$$492$$ 84864.0 0.0158056
$$493$$ −1.52323e6 −0.282260
$$494$$ −3.94163e6 −0.726706
$$495$$ 1.49338e6 0.273942
$$496$$ −1.59872e6 −0.291789
$$497$$ 0 0
$$498$$ −220056. −0.0397612
$$499$$ 1.36905e6 0.246132 0.123066 0.992398i $$-0.460727\pi$$
0.123066 + 0.992398i $$0.460727\pi$$
$$500$$ −2.97758e6 −0.532646
$$501$$ 482556. 0.0858921
$$502$$ −4.79005e6 −0.848361
$$503$$ −1.83343e6 −0.323105 −0.161552 0.986864i $$-0.551650\pi$$
−0.161552 + 0.986864i $$0.551650\pi$$
$$504$$ 0 0
$$505$$ −75582.0 −0.0131883
$$506$$ −861036. −0.149501
$$507$$ −107571. −0.0185855
$$508$$ −3.83066e6 −0.658588
$$509$$ 1.71266e6 0.293006 0.146503 0.989210i $$-0.453198\pi$$
0.146503 + 0.989210i $$0.453198\pi$$
$$510$$ 150552. 0.0256307
$$511$$ 0 0
$$512$$ −262144. −0.0441942
$$513$$ −690640. −0.115867
$$514$$ 151032. 0.0252151
$$515$$ 5.99230e6 0.995578
$$516$$ −284768. −0.0470833
$$517$$ −2.07926e6 −0.342124
$$518$$ 0 0
$$519$$ −766254. −0.124869
$$520$$ 2.25869e6 0.366309
$$521$$ 789435. 0.127415 0.0637077 0.997969i $$-0.479707\pi$$
0.0637077 + 0.997969i $$0.479707\pi$$
$$522$$ −1.99795e6 −0.320929
$$523$$ −627392. −0.100296 −0.0501481 0.998742i $$-0.515969\pi$$
−0.0501481 + 0.998742i $$0.515969\pi$$
$$524$$ 1.57027e6 0.249831
$$525$$ 0 0
$$526$$ 2.52502e6 0.397924
$$527$$ −4.60881e6 −0.722873
$$528$$ 30976.0 0.00483549
$$529$$ −3.27150e6 −0.508286
$$530$$ 6.26810e6 0.969274
$$531$$ −8.46734e6 −1.30320
$$532$$ 0 0
$$533$$ 3.67037e6 0.559618
$$534$$ −501660. −0.0761301
$$535$$ −4.04746e6 −0.611362
$$536$$ −1.62195e6 −0.243852
$$537$$ −303399. −0.0454024
$$538$$ −4.32137e6 −0.643673
$$539$$ 0 0
$$540$$ 395760. 0.0584047
$$541$$ 3.20895e6 0.471379 0.235689 0.971828i $$-0.424265\pi$$
0.235689 + 0.971828i $$0.424265\pi$$
$$542$$ −3.26440e6 −0.477315
$$543$$ −285181. −0.0415070
$$544$$ −755712. −0.109486
$$545$$ 4.47994e6 0.646072
$$546$$ 0 0
$$547$$ 3.42658e6 0.489658 0.244829 0.969566i $$-0.421268\pi$$
0.244829 + 0.969566i $$0.421268\pi$$
$$548$$ 6.40219e6 0.910704
$$549$$ −1.11175e7 −1.57426
$$550$$ −253616. −0.0357495
$$551$$ 2.93914e6 0.412421
$$552$$ −113856. −0.0159041
$$553$$ 0 0
$$554$$ −6.75279e6 −0.934779
$$555$$ 754035. 0.103910
$$556$$ −3.29226e6 −0.451655
$$557$$ 1.05198e7 1.43672 0.718358 0.695674i $$-0.244894\pi$$
0.718358 + 0.695674i $$0.244894\pi$$
$$558$$ −6.04516e6 −0.821906
$$559$$ −1.23162e7 −1.66705
$$560$$ 0 0
$$561$$ 89298.0 0.0119794
$$562$$ 3.51617e6 0.469601
$$563$$ −5.47288e6 −0.727687 −0.363844 0.931460i $$-0.618536\pi$$
−0.363844 + 0.931460i $$0.618536\pi$$
$$564$$ −274944. −0.0363954
$$565$$ −2.40960e6 −0.317558
$$566$$ 6.16107e6 0.808379
$$567$$ 0 0
$$568$$ −851904. −0.110795
$$569$$ −1.17787e7 −1.52516 −0.762580 0.646893i $$-0.776068\pi$$
−0.762580 + 0.646893i $$0.776068\pi$$
$$570$$ −290496. −0.0374501
$$571$$ −8.35628e6 −1.07256 −0.536281 0.844039i $$-0.680171\pi$$
−0.536281 + 0.844039i $$0.680171\pi$$
$$572$$ 1.33971e6 0.171207
$$573$$ −767067. −0.0975993
$$574$$ 0 0
$$575$$ 932196. 0.117581
$$576$$ −991232. −0.124486
$$577$$ 1.37758e7 1.72258 0.861288 0.508117i $$-0.169658\pi$$
0.861288 + 0.508117i $$0.169658\pi$$
$$578$$ 3.50085e6 0.435867
$$579$$ −411668. −0.0510330
$$580$$ −1.68422e6 −0.207888
$$581$$ 0 0
$$582$$ 355228. 0.0434710
$$583$$ 3.71785e6 0.453023
$$584$$ −3.40864e6 −0.413570
$$585$$ 8.54066e6 1.03182
$$586$$ 2.88336e6 0.346860
$$587$$ 1.27093e7 1.52239 0.761196 0.648522i $$-0.224612\pi$$
0.761196 + 0.648522i $$0.224612\pi$$
$$588$$ 0 0
$$589$$ 8.89288e6 1.05622
$$590$$ −7.13776e6 −0.844173
$$591$$ 759258. 0.0894171
$$592$$ −3.78496e6 −0.443871
$$593$$ −1.00825e6 −0.117742 −0.0588711 0.998266i $$-0.518750\pi$$
−0.0588711 + 0.998266i $$0.518750\pi$$
$$594$$ 234740. 0.0272974
$$595$$ 0 0
$$596$$ 1.40362e6 0.161857
$$597$$ −46600.0 −0.00535119
$$598$$ −4.92427e6 −0.563105
$$599$$ 1.05100e7 1.19684 0.598421 0.801182i $$-0.295795\pi$$
0.598421 + 0.801182i $$0.295795\pi$$
$$600$$ −33536.0 −0.00380306
$$601$$ 199390. 0.0225173 0.0112587 0.999937i $$-0.496416\pi$$
0.0112587 + 0.999937i $$0.496416\pi$$
$$602$$ 0 0
$$603$$ −6.13301e6 −0.686879
$$604$$ −6.92445e6 −0.772312
$$605$$ 746691. 0.0829378
$$606$$ −5928.00 −0.000655732 0
$$607$$ −16190.0 −0.00178351 −0.000891754 1.00000i $$-0.500284\pi$$
−0.000891754 1.00000i $$0.500284\pi$$
$$608$$ 1.45818e6 0.159975
$$609$$ 0 0
$$610$$ −9.37176e6 −1.01976
$$611$$ −1.18913e7 −1.28863
$$612$$ −2.85754e6 −0.308399
$$613$$ −1.15253e7 −1.23880 −0.619402 0.785074i $$-0.712625\pi$$
−0.619402 + 0.785074i $$0.712625\pi$$
$$614$$ −4.15619e6 −0.444913
$$615$$ 270504. 0.0288394
$$616$$ 0 0
$$617$$ 1.69974e7 1.79750 0.898751 0.438459i $$-0.144476\pi$$
0.898751 + 0.438459i $$0.144476\pi$$
$$618$$ 469984. 0.0495008
$$619$$ 1.84875e7 1.93933 0.969663 0.244445i $$-0.0786058\pi$$
0.969663 + 0.244445i $$0.0786058\pi$$
$$620$$ −5.09592e6 −0.532407
$$621$$ −862815. −0.0897819
$$622$$ −5.00539e6 −0.518755
$$623$$ 0 0
$$624$$ 177152. 0.0182131
$$625$$ −7.85355e6 −0.804203
$$626$$ −5.77344e6 −0.588842
$$627$$ −172304. −0.0175036
$$628$$ 545200. 0.0551641
$$629$$ −1.09113e7 −1.09964
$$630$$ 0 0
$$631$$ −4.54281e6 −0.454204 −0.227102 0.973871i $$-0.572925\pi$$
−0.227102 + 0.973871i $$0.572925\pi$$
$$632$$ −4.94298e6 −0.492261
$$633$$ 932428. 0.0924924
$$634$$ 8.04832e6 0.795210
$$635$$ −1.22102e7 −1.20168
$$636$$ 491616. 0.0481929
$$637$$ 0 0
$$638$$ −998976. −0.0971635
$$639$$ −3.22126e6 −0.312086
$$640$$ −835584. −0.0806381
$$641$$ 1.84286e7 1.77153 0.885764 0.464136i $$-0.153635\pi$$
0.885764 + 0.464136i $$0.153635\pi$$
$$642$$ −317448. −0.0303973
$$643$$ −9.66604e6 −0.921979 −0.460989 0.887406i $$-0.652505\pi$$
−0.460989 + 0.887406i $$0.652505\pi$$
$$644$$ 0 0
$$645$$ −907698. −0.0859097
$$646$$ 4.20365e6 0.396319
$$647$$ 4.51430e6 0.423965 0.211982 0.977273i $$-0.432008\pi$$
0.211982 + 0.977273i $$0.432008\pi$$
$$648$$ −3.73254e6 −0.349195
$$649$$ −4.23367e6 −0.394553
$$650$$ −1.45043e6 −0.134652
$$651$$ 0 0
$$652$$ 720320. 0.0663600
$$653$$ −5.37235e6 −0.493039 −0.246519 0.969138i $$-0.579287\pi$$
−0.246519 + 0.969138i $$0.579287\pi$$
$$654$$ 351368. 0.0321231
$$655$$ 5.00524e6 0.455850
$$656$$ −1.35782e6 −0.123192
$$657$$ −1.28889e7 −1.16494
$$658$$ 0 0
$$659$$ 9.87956e6 0.886184 0.443092 0.896476i $$-0.353881\pi$$
0.443092 + 0.896476i $$0.353881\pi$$
$$660$$ 98736.0 0.00882299
$$661$$ −1.08052e7 −0.961898 −0.480949 0.876748i $$-0.659708\pi$$
−0.480949 + 0.876748i $$0.659708\pi$$
$$662$$ −8.06935e6 −0.715638
$$663$$ 510696. 0.0451210
$$664$$ 3.52090e6 0.309908
$$665$$ 0 0
$$666$$ −1.43119e7 −1.25029
$$667$$ 3.67186e6 0.319574
$$668$$ −7.72090e6 −0.669463
$$669$$ 169745. 0.0146633
$$670$$ −5.16997e6 −0.444940
$$671$$ −5.55874e6 −0.476618
$$672$$ 0 0
$$673$$ 1.13275e7 0.964042 0.482021 0.876160i $$-0.339903\pi$$
0.482021 + 0.876160i $$0.339903\pi$$
$$674$$ −1.05649e6 −0.0895808
$$675$$ −254140. −0.0214691
$$676$$ 1.72114e6 0.144860
$$677$$ 1.20595e7 1.01125 0.505624 0.862754i $$-0.331262\pi$$
0.505624 + 0.862754i $$0.331262\pi$$
$$678$$ −188988. −0.0157892
$$679$$ 0 0
$$680$$ −2.40883e6 −0.199772
$$681$$ 198078. 0.0163670
$$682$$ −3.02258e6 −0.248838
$$683$$ −5.14166e6 −0.421747 −0.210873 0.977513i $$-0.567631\pi$$
−0.210873 + 0.977513i $$0.567631\pi$$
$$684$$ 5.51373e6 0.450614
$$685$$ 2.04070e7 1.66170
$$686$$ 0 0
$$687$$ −849997. −0.0687109
$$688$$ 4.55629e6 0.366978
$$689$$ 2.12624e7 1.70633
$$690$$ −362916. −0.0290191
$$691$$ −1.31243e7 −1.04563 −0.522817 0.852445i $$-0.675119\pi$$
−0.522817 + 0.852445i $$0.675119\pi$$
$$692$$ 1.22601e7 0.973257
$$693$$ 0 0
$$694$$ 6.84197e6 0.539240
$$695$$ −1.04941e7 −0.824104
$$696$$ −132096. −0.0103363
$$697$$ −3.91435e6 −0.305195
$$698$$ 875288. 0.0680005
$$699$$ 401832. 0.0311065
$$700$$ 0 0
$$701$$ 3.65956e6 0.281277 0.140638 0.990061i $$-0.455084\pi$$
0.140638 + 0.990061i $$0.455084\pi$$
$$702$$ 1.34248e6 0.102817
$$703$$ 2.10538e7 1.60673
$$704$$ −495616. −0.0376889
$$705$$ −876384. −0.0664082
$$706$$ 1.47277e7 1.11204
$$707$$ 0 0
$$708$$ −559824. −0.0419728
$$709$$ 1.02252e7 0.763935 0.381968 0.924176i $$-0.375247\pi$$
0.381968 + 0.924176i $$0.375247\pi$$
$$710$$ −2.71544e6 −0.202160
$$711$$ −1.86906e7 −1.38660
$$712$$ 8.02656e6 0.593375
$$713$$ 1.11099e7 0.818436
$$714$$ 0 0
$$715$$ 4.27033e6 0.312390
$$716$$ 4.85438e6 0.353876
$$717$$ −855174. −0.0621236
$$718$$ −7.54114e6 −0.545916
$$719$$ −2.41683e7 −1.74351 −0.871753 0.489945i $$-0.837017\pi$$
−0.871753 + 0.489945i $$0.837017\pi$$
$$720$$ −3.15955e6 −0.227140
$$721$$ 0 0
$$722$$ 1.79329e6 0.128029
$$723$$ 1.12546e6 0.0800730
$$724$$ 4.56290e6 0.323515
$$725$$ 1.08154e6 0.0764181
$$726$$ 58564.0 0.00412372
$$727$$ −1.68246e7 −1.18062 −0.590310 0.807177i $$-0.700994\pi$$
−0.590310 + 0.807177i $$0.700994\pi$$
$$728$$ 0 0
$$729$$ −1.39958e7 −0.975393
$$730$$ −1.08650e7 −0.754613
$$731$$ 1.31349e7 0.909147
$$732$$ −735040. −0.0507030
$$733$$ 5.04168e6 0.346590 0.173295 0.984870i $$-0.444559\pi$$
0.173295 + 0.984870i $$0.444559\pi$$
$$734$$ −1.24666e7 −0.854101
$$735$$ 0 0
$$736$$ 1.82170e6 0.123960
$$737$$ −3.06650e6 −0.207958
$$738$$ −5.13427e6 −0.347007
$$739$$ −6.26375e6 −0.421913 −0.210957 0.977495i $$-0.567658\pi$$
−0.210957 + 0.977495i $$0.567658\pi$$
$$740$$ −1.20646e7 −0.809901
$$741$$ −985408. −0.0659281
$$742$$ 0 0
$$743$$ 3.63976e6 0.241880 0.120940 0.992660i $$-0.461409\pi$$
0.120940 + 0.992660i $$0.461409\pi$$
$$744$$ −399680. −0.0264716
$$745$$ 4.47403e6 0.295330
$$746$$ −5.57766e6 −0.366948
$$747$$ 1.33134e7 0.872945
$$748$$ −1.42877e6 −0.0933701
$$749$$ 0 0
$$750$$ −744396. −0.0483227
$$751$$ −1.87370e7 −1.21227 −0.606135 0.795362i $$-0.707281\pi$$
−0.606135 + 0.795362i $$0.707281\pi$$
$$752$$ 4.39910e6 0.283674
$$753$$ −1.19751e6 −0.0769649
$$754$$ −5.71315e6 −0.365972
$$755$$ −2.20717e7 −1.40918
$$756$$ 0 0
$$757$$ 489242. 0.0310302 0.0155151 0.999880i $$-0.495061\pi$$
0.0155151 + 0.999880i $$0.495061\pi$$
$$758$$ 1.70414e7 1.07729
$$759$$ −215259. −0.0135630
$$760$$ 4.64794e6 0.291895
$$761$$ −1.46969e7 −0.919952 −0.459976 0.887931i $$-0.652142\pi$$
−0.459976 + 0.887931i $$0.652142\pi$$
$$762$$ −957664. −0.0597483
$$763$$ 0 0
$$764$$ 1.22731e7 0.760711
$$765$$ −9.10840e6 −0.562715
$$766$$ 807060. 0.0496974
$$767$$ −2.42124e7 −1.48610
$$768$$ −65536.0 −0.00400938
$$769$$ −2.42072e7 −1.47615 −0.738073 0.674721i $$-0.764264\pi$$
−0.738073 + 0.674721i $$0.764264\pi$$
$$770$$ 0 0
$$771$$ 37758.0 0.00228756
$$772$$ 6.58669e6 0.397763
$$773$$ −1.35260e7 −0.814181 −0.407091 0.913388i $$-0.633457\pi$$
−0.407091 + 0.913388i $$0.633457\pi$$
$$774$$ 1.72285e7 1.03370
$$775$$ 3.27238e6 0.195708
$$776$$ −5.68365e6 −0.338823
$$777$$ 0 0
$$778$$ −7.79528e6 −0.461725
$$779$$ 7.55290e6 0.445933
$$780$$ 564672. 0.0332323
$$781$$ −1.61063e6 −0.0944862
$$782$$ 5.25161e6 0.307097
$$783$$ −1.00104e6 −0.0583508
$$784$$ 0 0
$$785$$ 1.73782e6 0.100654
$$786$$ 392568. 0.0226651
$$787$$ −1.42094e7 −0.817786 −0.408893 0.912582i $$-0.634085\pi$$
−0.408893 + 0.912582i $$0.634085\pi$$
$$788$$ −1.21481e7 −0.696937
$$789$$ 631254. 0.0361004
$$790$$ −1.57557e7 −0.898196
$$791$$ 0 0
$$792$$ −1.87405e6 −0.106162
$$793$$ −3.17905e7 −1.79521
$$794$$ −5.87303e6 −0.330606
$$795$$ 1.56703e6 0.0879343
$$796$$ 745600. 0.0417084
$$797$$ 7.93333e6 0.442395 0.221197 0.975229i $$-0.429003\pi$$
0.221197 + 0.975229i $$0.429003\pi$$
$$798$$ 0 0
$$799$$ 1.26818e7 0.702771
$$800$$ 536576. 0.0296419
$$801$$ 3.03504e7 1.67141
$$802$$ −8.98471e6 −0.493251
$$803$$ −6.44446e6 −0.352694
$$804$$ −405488. −0.0221227
$$805$$ 0 0
$$806$$ −1.72862e7 −0.937262
$$807$$ −1.08034e6 −0.0583952
$$808$$ 94848.0 0.00511093
$$809$$ −1.04685e7 −0.562359 −0.281180 0.959655i $$-0.590726\pi$$
−0.281180 + 0.959655i $$0.590726\pi$$
$$810$$ −1.18975e7 −0.637151
$$811$$ −1.19147e7 −0.636110 −0.318055 0.948072i $$-0.603030\pi$$
−0.318055 + 0.948072i $$0.603030\pi$$
$$812$$ 0 0
$$813$$ −816100. −0.0433029
$$814$$ −7.15594e6 −0.378535
$$815$$ 2.29602e6 0.121083
$$816$$ −188928. −0.00993278
$$817$$ −2.53444e7 −1.32839
$$818$$ −1.44595e7 −0.755562
$$819$$ 0 0
$$820$$ −4.32806e6 −0.224781
$$821$$ −1.86112e6 −0.0963645 −0.0481822 0.998839i $$-0.515343\pi$$
−0.0481822 + 0.998839i $$0.515343\pi$$
$$822$$ 1.60055e6 0.0826208
$$823$$ 2.30153e7 1.18445 0.592225 0.805773i $$-0.298250\pi$$
0.592225 + 0.805773i $$0.298250\pi$$
$$824$$ −7.51974e6 −0.385820
$$825$$ −63404.0 −0.00324326
$$826$$ 0 0
$$827$$ −1.68351e7 −0.855959 −0.427980 0.903788i $$-0.640775\pi$$
−0.427980 + 0.903788i $$0.640775\pi$$
$$828$$ 6.88829e6 0.349169
$$829$$ 2.35299e7 1.18914 0.594570 0.804044i $$-0.297322\pi$$
0.594570 + 0.804044i $$0.297322\pi$$
$$830$$ 1.12229e7 0.565468
$$831$$ −1.68820e6 −0.0848049
$$832$$ −2.83443e6 −0.141957
$$833$$ 0 0
$$834$$ −823064. −0.0409750
$$835$$ −2.46104e7 −1.22152
$$836$$ 2.75686e6 0.136427
$$837$$ −3.02882e6 −0.149438
$$838$$ −1.52496e7 −0.750148
$$839$$ −2.91549e7 −1.42990 −0.714952 0.699173i $$-0.753552\pi$$
−0.714952 + 0.699173i $$0.753552\pi$$
$$840$$ 0 0
$$841$$ −1.62511e7 −0.792303
$$842$$ −7.89385e6 −0.383715
$$843$$ 879042. 0.0426030
$$844$$ −1.49188e7 −0.720907
$$845$$ 5.48612e6 0.264316
$$846$$ 1.66341e7 0.799050
$$847$$ 0 0
$$848$$ −7.86586e6 −0.375627
$$849$$ 1.54027e6 0.0733377
$$850$$ 1.54685e6 0.0734345
$$851$$ 2.63025e7 1.24501
$$852$$ −212976. −0.0100515
$$853$$ 9.49052e6 0.446599 0.223299 0.974750i $$-0.428317\pi$$
0.223299 + 0.974750i $$0.428317\pi$$
$$854$$ 0 0
$$855$$ 1.75750e7 0.822205
$$856$$ 5.07917e6 0.236924
$$857$$ 1.81553e6 0.0844405 0.0422203 0.999108i $$-0.486557\pi$$
0.0422203 + 0.999108i $$0.486557\pi$$
$$858$$ 334928. 0.0155322
$$859$$ 1.07812e7 0.498522 0.249261 0.968436i $$-0.419812\pi$$
0.249261 + 0.968436i $$0.419812\pi$$
$$860$$ 1.45232e7 0.669600
$$861$$ 0 0
$$862$$ 8.33436e6 0.382036
$$863$$ −2.83355e7 −1.29510 −0.647550 0.762023i $$-0.724206\pi$$
−0.647550 + 0.762023i $$0.724206\pi$$
$$864$$ −496640. −0.0226338
$$865$$ 3.90790e7 1.77584
$$866$$ −290764. −0.0131749
$$867$$ 875213. 0.0395427
$$868$$ 0 0
$$869$$ −9.34531e6 −0.419802
$$870$$ −421056. −0.0188600
$$871$$ −1.75374e7 −0.783283
$$872$$ −5.62189e6 −0.250375
$$873$$ −2.14913e7 −0.954392
$$874$$ −1.01332e7 −0.448712
$$875$$ 0 0
$$876$$ −852160. −0.0375198
$$877$$ −2.68919e7 −1.18065 −0.590326 0.807165i $$-0.701001\pi$$
−0.590326 + 0.807165i $$0.701001\pi$$
$$878$$ 2.37757e6 0.104087
$$879$$ 720840. 0.0314678
$$880$$ −1.57978e6 −0.0687684
$$881$$ 1.92132e7 0.833989 0.416995 0.908909i $$-0.363083\pi$$
0.416995 + 0.908909i $$0.363083\pi$$
$$882$$ 0 0
$$883$$ 1.15931e7 0.500378 0.250189 0.968197i $$-0.419507\pi$$
0.250189 + 0.968197i $$0.419507\pi$$
$$884$$ −8.17114e6 −0.351683
$$885$$ −1.78444e6 −0.0765850
$$886$$ −1.82660e7 −0.781735
$$887$$ −1.31857e7 −0.562721 −0.281361 0.959602i $$-0.590786\pi$$
−0.281361 + 0.959602i $$0.590786\pi$$
$$888$$ −946240. −0.0402688
$$889$$ 0 0
$$890$$ 2.55847e7 1.08269
$$891$$ −7.05684e6 −0.297794
$$892$$ −2.71592e6 −0.114289
$$893$$ −2.44700e7 −1.02685
$$894$$ 350904. 0.0146840
$$895$$ 1.54733e7 0.645694
$$896$$ 0 0
$$897$$ −1.23107e6 −0.0510859
$$898$$ 2.17753e7 0.901100
$$899$$ 1.28897e7 0.531916
$$900$$ 2.02893e6 0.0834950
$$901$$ −2.26758e7 −0.930573
$$902$$ −2.56714e6 −0.105059
$$903$$ 0 0
$$904$$ 3.02381e6 0.123065
$$905$$ 1.45442e7 0.590295
$$906$$ −1.73111e6 −0.0700656
$$907$$ 2.98195e6 0.120360 0.0601800 0.998188i $$-0.480833\pi$$
0.0601800 + 0.998188i $$0.480833\pi$$
$$908$$ −3.16925e6 −0.127568
$$909$$ 358644. 0.0143964
$$910$$ 0 0
$$911$$ 2.96579e7 1.18398 0.591989 0.805946i $$-0.298343\pi$$
0.591989 + 0.805946i $$0.298343\pi$$
$$912$$ 364544. 0.0145132
$$913$$ 6.65669e6 0.264291
$$914$$ −2.68125e7 −1.06163
$$915$$ −2.34294e6 −0.0925142
$$916$$ 1.36000e7 0.535548
$$917$$ 0 0
$$918$$ −1.43172e6 −0.0560727
$$919$$ 3.18057e7 1.24227 0.621135 0.783704i $$-0.286672\pi$$
0.621135 + 0.783704i $$0.286672\pi$$
$$920$$ 5.80666e6 0.226181
$$921$$ −1.03905e6 −0.0403633
$$922$$ −5.03976e6 −0.195246
$$923$$ −9.21121e6 −0.355887
$$924$$ 0 0
$$925$$ 7.74734e6 0.297713
$$926$$ 2.00923e7 0.770021
$$927$$ −2.84340e7 −1.08677
$$928$$ 2.11354e6 0.0805638
$$929$$ 2.33444e7 0.887451 0.443725 0.896163i $$-0.353657\pi$$
0.443725 + 0.896163i $$0.353657\pi$$
$$930$$ −1.27398e6 −0.0483009
$$931$$ 0 0
$$932$$ −6.42931e6 −0.242451
$$933$$ −1.25135e6 −0.0470624
$$934$$ −9.42642e6 −0.353573
$$935$$ −4.55420e6 −0.170366
$$936$$ −1.07177e7 −0.399864
$$937$$ −2.07372e7 −0.771616 −0.385808 0.922579i $$-0.626077\pi$$
−0.385808 + 0.922579i $$0.626077\pi$$
$$938$$ 0 0
$$939$$ −1.44336e6 −0.0534209
$$940$$ 1.40221e7 0.517601
$$941$$ −2.69193e7 −0.991036 −0.495518 0.868598i $$-0.665022\pi$$
−0.495518 + 0.868598i $$0.665022\pi$$
$$942$$ 136300. 0.00500459
$$943$$ 9.43582e6 0.345542
$$944$$ 8.95718e6 0.327146
$$945$$ 0 0
$$946$$ 8.61423e6 0.312960
$$947$$ 1.01896e7 0.369216 0.184608 0.982812i $$-0.440898\pi$$
0.184608 + 0.982812i $$0.440898\pi$$
$$948$$ −1.23574e6 −0.0446589
$$949$$ −3.68559e7 −1.32844
$$950$$ −2.98470e6 −0.107298
$$951$$ 2.01208e6 0.0721429
$$952$$ 0 0
$$953$$ 1.03924e7 0.370665 0.185333 0.982676i $$-0.440664\pi$$
0.185333 + 0.982676i $$0.440664\pi$$
$$954$$ −2.97428e7 −1.05806
$$955$$ 3.91204e7 1.38802
$$956$$ 1.36828e7 0.484206
$$957$$ −249744. −0.00881486
$$958$$ −2.68903e7 −0.946634
$$959$$ 0 0
$$960$$ −208896. −0.00731564
$$961$$ 1.03709e7 0.362249
$$962$$ −4.09249e7 −1.42577
$$963$$ 1.92056e7 0.667363
$$964$$ −1.80074e7 −0.624107
$$965$$ 2.09951e7 0.725770
$$966$$ 0 0
$$967$$ −8.18877e6 −0.281613 −0.140806 0.990037i $$-0.544970\pi$$
−0.140806 + 0.990037i $$0.544970\pi$$
$$968$$ −937024. −0.0321412
$$969$$ 1.05091e6 0.0359548
$$970$$ −1.81166e7 −0.618227
$$971$$ 1.73274e7 0.589775 0.294887 0.955532i $$-0.404718\pi$$
0.294887 + 0.955532i $$0.404718\pi$$
$$972$$ −2.81882e6 −0.0956976
$$973$$ 0 0
$$974$$ −7.84005e6 −0.264802
$$975$$ −362608. −0.0122159
$$976$$ 1.17606e7 0.395190
$$977$$ −438963. −0.0147127 −0.00735634 0.999973i $$-0.502342\pi$$
−0.00735634 + 0.999973i $$0.502342\pi$$
$$978$$ 180080. 0.00602030
$$979$$ 1.51752e7 0.506032
$$980$$ 0 0
$$981$$ −2.12578e7 −0.705253
$$982$$ 2.31850e6 0.0767234
$$983$$ 2.79124e7 0.921326 0.460663 0.887575i $$-0.347612\pi$$
0.460663 + 0.887575i $$0.347612\pi$$
$$984$$ −339456. −0.0111762
$$985$$ −3.87222e7 −1.27165
$$986$$ 6.09293e6 0.199588
$$987$$ 0 0
$$988$$ 1.57665e7 0.513859
$$989$$ −3.16626e7 −1.02933
$$990$$ −5.97353e6 −0.193706
$$991$$ −4.26846e7 −1.38066 −0.690331 0.723494i $$-0.742535\pi$$
−0.690331 + 0.723494i $$0.742535\pi$$
$$992$$ 6.39488e6 0.206326
$$993$$ −2.01734e6 −0.0649240
$$994$$ 0 0
$$995$$ 2.37660e6 0.0761024
$$996$$ 880224. 0.0281154
$$997$$ 2.21044e7 0.704273 0.352137 0.935949i $$-0.385455\pi$$
0.352137 + 0.935949i $$0.385455\pi$$
$$998$$ −5.47621e6 −0.174042
$$999$$ −7.17072e6 −0.227326
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 1078.6.a.a.1.1 1
7.6 odd 2 22.6.a.b.1.1 1
21.20 even 2 198.6.a.i.1.1 1
28.27 even 2 176.6.a.b.1.1 1
35.13 even 4 550.6.b.f.199.2 2
35.27 even 4 550.6.b.f.199.1 2
35.34 odd 2 550.6.a.f.1.1 1
56.13 odd 2 704.6.a.e.1.1 1
56.27 even 2 704.6.a.f.1.1 1
77.76 even 2 242.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
22.6.a.b.1.1 1 7.6 odd 2
176.6.a.b.1.1 1 28.27 even 2
198.6.a.i.1.1 1 21.20 even 2
242.6.a.d.1.1 1 77.76 even 2
550.6.a.f.1.1 1 35.34 odd 2
550.6.b.f.199.1 2 35.27 even 4
550.6.b.f.199.2 2 35.13 even 4
704.6.a.e.1.1 1 56.13 odd 2
704.6.a.f.1.1 1 56.27 even 2
1078.6.a.a.1.1 1 1.1 even 1 trivial