# Properties

 Label 22.6.a.b.1.1 Level $22$ Weight $6$ Character 22.1 Self dual yes Analytic conductor $3.528$ Analytic rank $1$ 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] = [22,6,Mod(1,22)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(22, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("22.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$22 = 2 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 22.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: $$3.52844403589$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 22.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} +1.00000 q^{3} +16.0000 q^{4} -51.0000 q^{5} -4.00000 q^{6} -166.000 q^{7} -64.0000 q^{8} -242.000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{2} +1.00000 q^{3} +16.0000 q^{4} -51.0000 q^{5} -4.00000 q^{6} -166.000 q^{7} -64.0000 q^{8} -242.000 q^{9} +204.000 q^{10} -121.000 q^{11} +16.0000 q^{12} +692.000 q^{13} +664.000 q^{14} -51.0000 q^{15} +256.000 q^{16} -738.000 q^{17} +968.000 q^{18} +1424.00 q^{19} -816.000 q^{20} -166.000 q^{21} +484.000 q^{22} -1779.00 q^{23} -64.0000 q^{24} -524.000 q^{25} -2768.00 q^{26} -485.000 q^{27} -2656.00 q^{28} -2064.00 q^{29} +204.000 q^{30} +6245.00 q^{31} -1024.00 q^{32} -121.000 q^{33} +2952.00 q^{34} +8466.00 q^{35} -3872.00 q^{36} -14785.0 q^{37} -5696.00 q^{38} +692.000 q^{39} +3264.00 q^{40} +5304.00 q^{41} +664.000 q^{42} +17798.0 q^{43} -1936.00 q^{44} +12342.0 q^{45} +7116.00 q^{46} -17184.0 q^{47} +256.000 q^{48} +10749.0 q^{49} +2096.00 q^{50} -738.000 q^{51} +11072.0 q^{52} -30726.0 q^{53} +1940.00 q^{54} +6171.00 q^{55} +10624.0 q^{56} +1424.00 q^{57} +8256.00 q^{58} -34989.0 q^{59} -816.000 q^{60} -45940.0 q^{61} -24980.0 q^{62} +40172.0 q^{63} +4096.00 q^{64} -35292.0 q^{65} +484.000 q^{66} +25343.0 q^{67} -11808.0 q^{68} -1779.00 q^{69} -33864.0 q^{70} +13311.0 q^{71} +15488.0 q^{72} -53260.0 q^{73} +59140.0 q^{74} -524.000 q^{75} +22784.0 q^{76} +20086.0 q^{77} -2768.00 q^{78} +77234.0 q^{79} -13056.0 q^{80} +58321.0 q^{81} -21216.0 q^{82} +55014.0 q^{83} -2656.00 q^{84} +37638.0 q^{85} -71192.0 q^{86} -2064.00 q^{87} +7744.00 q^{88} +125415. q^{89} -49368.0 q^{90} -114872. q^{91} -28464.0 q^{92} +6245.00 q^{93} +68736.0 q^{94} -72624.0 q^{95} -1024.00 q^{96} -88807.0 q^{97} -42996.0 q^{98} +29282.0 q^{99} +O(q^{100})$$

## 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 $$-0.489788\pi$$
0.0320750 + 0.999485i $$0.489788\pi$$
$$4$$ 16.0000 0.500000
$$5$$ −51.0000 −0.912316 −0.456158 0.889899i $$-0.650775\pi$$
−0.456158 + 0.889899i $$0.650775\pi$$
$$6$$ −4.00000 −0.0453609
$$7$$ −166.000 −1.28045 −0.640226 0.768187i $$-0.721159\pi$$
−0.640226 + 0.768187i $$0.721159\pi$$
$$8$$ −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 $$-0.307783\pi$$
0.567829 + 0.823146i $$0.307783\pi$$
$$14$$ 664.000 0.905416
$$15$$ −51.0000 −0.0585251
$$16$$ 256.000 0.250000
$$17$$ −738.000 −0.619347 −0.309674 0.950843i $$-0.600220\pi$$
−0.309674 + 0.950843i $$0.600220\pi$$
$$18$$ 968.000 0.704197
$$19$$ 1424.00 0.904953 0.452476 0.891776i $$-0.350541\pi$$
0.452476 + 0.891776i $$0.350541\pi$$
$$20$$ −816.000 −0.456158
$$21$$ −166.000 −0.0821410
$$22$$ 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
$$28$$ −2656.00 −0.640226
$$29$$ −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 $$-0.301653\pi$$
0.583577 + 0.812058i $$0.301653\pi$$
$$32$$ −1024.00 −0.176777
$$33$$ −121.000 −0.0193420
$$34$$ 2952.00 0.437944
$$35$$ 8466.00 1.16818
$$36$$ −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 $$-0.420757\pi$$
0.246385 + 0.969172i $$0.420757\pi$$
$$42$$ 664.000 0.0580824
$$43$$ 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 $$-0.692031\pi$$
−0.567348 + 0.823478i $$0.692031\pi$$
$$48$$ 256.000 0.0160375
$$49$$ 10749.0 0.639555
$$50$$ 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
$$56$$ 10624.0 0.452708
$$57$$ 1424.00 0.0580528
$$58$$ 8256.00 0.322255
$$59$$ −34989.0 −1.30858 −0.654292 0.756242i $$-0.727033\pi$$
−0.654292 + 0.756242i $$0.727033\pi$$
$$60$$ −816.000 −0.0292625
$$61$$ −45940.0 −1.58076 −0.790381 0.612616i $$-0.790117\pi$$
−0.790381 + 0.612616i $$0.790117\pi$$
$$62$$ −24980.0 −0.825303
$$63$$ 40172.0 1.27518
$$64$$ 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
$$70$$ −33864.0 −0.826025
$$71$$ 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 $$-0.698857\pi$$
−0.584876 + 0.811123i $$0.698857\pi$$
$$74$$ 59140.0 1.25546
$$75$$ −524.000 −0.0107567
$$76$$ 22784.0 0.452476
$$77$$ 20086.0 0.386071
$$78$$ −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 $$-0.355589\pi$$
0.438276 + 0.898840i $$0.355589\pi$$
$$84$$ −2656.00 −0.0410705
$$85$$ 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 $$-0.183047\pi$$
0.839159 + 0.543886i $$0.183047\pi$$
$$90$$ −49368.0 −0.642450
$$91$$ −114872. −1.45416
$$92$$ −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 $$-0.659062\pi$$
−0.479168 + 0.877723i $$0.659062\pi$$
$$98$$ −42996.0 −0.452234
$$99$$ 29282.0 0.300271
$$100$$ −8384.00 −0.0838400
$$101$$ 1482.00 0.0144559 0.00722794 0.999974i $$-0.497699\pi$$
0.00722794 + 0.999974i $$0.497699\pi$$
$$102$$ 2952.00 0.0280942
$$103$$ −117496. −1.09126 −0.545632 0.838025i $$-0.683710\pi$$
−0.545632 + 0.838025i $$0.683710\pi$$
$$104$$ −44288.0 −0.401516
$$105$$ 8466.00 0.0749385
$$106$$ 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
$$112$$ −42496.0 −0.320113
$$113$$ −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
$$119$$ 122508. 0.793044
$$120$$ 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
$$126$$ −160688. −0.901690
$$127$$ −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 $$-0.580375\pi$$
−0.249831 + 0.968289i $$0.580375\pi$$
$$132$$ −1936.00 −0.00967098
$$133$$ −236384. −1.15875
$$134$$ −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 $$-0.350834\pi$$
0.451655 + 0.892193i $$0.350834\pi$$
$$140$$ 135456. 0.584088
$$141$$ −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
$$147$$ 10749.0 0.0410275
$$148$$ −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
$$154$$ −80344.0 −0.272993
$$155$$ −318495. −1.06481
$$156$$ 11072.0 0.0364263
$$157$$ −34075.0 −0.110328 −0.0551641 0.998477i $$-0.517568\pi$$
−0.0551641 + 0.998477i $$0.517568\pi$$
$$158$$ −308936. −0.984523
$$159$$ −30726.0 −0.0963858
$$160$$ 52224.0 0.161276
$$161$$ 295314. 0.897882
$$162$$ −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 $$-0.266524\pi$$
0.669463 + 0.742845i $$0.266524\pi$$
$$168$$ 10624.0 0.0290412
$$169$$ 107571. 0.289720
$$170$$ −150552. −0.399544
$$171$$ −344608. −0.901229
$$172$$ 284768. 0.733956
$$173$$ −766254. −1.94651 −0.973257 0.229719i $$-0.926219\pi$$
−0.973257 + 0.229719i $$0.926219\pi$$
$$174$$ 8256.00 0.0206727
$$175$$ 86984.0 0.214706
$$176$$ −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 $$-0.604865\pi$$
−0.323515 + 0.946223i $$0.604865\pi$$
$$182$$ 459488. 1.02824
$$183$$ −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
$$189$$ 80510.0 0.163944
$$190$$ 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
$$196$$ 171984. 0.319777
$$197$$ −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 $$-0.513280\pi$$
−0.0417084 + 0.999130i $$0.513280\pi$$
$$200$$ 33536.0 0.0592838
$$201$$ 25343.0 0.0442454
$$202$$ −5928.00 −0.0102219
$$203$$ 342624. 0.583549
$$204$$ −11808.0 −0.0198656
$$205$$ −270504. −0.449561
$$206$$ 469984. 0.771641
$$207$$ 430518. 0.698338
$$208$$ 177152. 0.283915
$$209$$ −172304. −0.272854
$$210$$ −33864.0 −0.0529895
$$211$$ −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
$$217$$ −1.03667e6 −1.49448
$$218$$ −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 $$-0.463541\pi$$
0.114289 + 0.993448i $$0.463541\pi$$
$$224$$ 169984. 0.226354
$$225$$ 126808. 0.166990
$$226$$ 188988. 0.246129
$$227$$ 198078. 0.255136 0.127568 0.991830i $$-0.459283\pi$$
0.127568 + 0.991830i $$0.459283\pi$$
$$228$$ 22784.0 0.0290264
$$229$$ −849997. −1.07110 −0.535548 0.844505i $$-0.679895\pi$$
−0.535548 + 0.844505i $$0.679895\pi$$
$$230$$ −362916. −0.452362
$$231$$ 20086.0 0.0247664
$$232$$ 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
$$238$$ −490032. −0.560766
$$239$$ 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 $$-0.285463\pi$$
0.624107 + 0.781339i $$0.285463\pi$$
$$242$$ −58564.0 −0.0642824
$$243$$ 176176. 0.191395
$$244$$ −735040. −0.790381
$$245$$ −548199. −0.583476
$$246$$ −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 $$-0.704786\pi$$
−0.599882 + 0.800088i $$0.704786\pi$$
$$252$$ 642752. 0.637591
$$253$$ 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 $$-0.494324\pi$$
0.0178298 + 0.999841i $$0.494324\pi$$
$$258$$ −71192.0 −0.0665858
$$259$$ 2.45431e6 2.27342
$$260$$ −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
$$266$$ 945536. 0.819359
$$267$$ 125415. 0.107664
$$268$$ 405488. 0.344859
$$269$$ −1.08034e6 −0.910292 −0.455146 0.890417i $$-0.650413\pi$$
−0.455146 + 0.890417i $$0.650413\pi$$
$$270$$ −98940.0 −0.0825967
$$271$$ −816100. −0.675025 −0.337513 0.941321i $$-0.609586\pi$$
−0.337513 + 0.941321i $$0.609586\pi$$
$$272$$ −188928. −0.154837
$$273$$ −114872. −0.0932841
$$274$$ −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
$$280$$ −541824. −0.413012
$$281$$ −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 $$-0.306319\pi$$
0.571611 + 0.820525i $$0.306319\pi$$
$$284$$ 212976. 0.156688
$$285$$ −72624.0 −0.0529624
$$286$$ 334928. 0.242123
$$287$$ −880464. −0.630967
$$288$$ 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 $$-0.421124\pi$$
0.245267 + 0.969455i $$0.421124\pi$$
$$294$$ −42996.0 −0.0290108
$$295$$ 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
$$301$$ −2.95447e6 −1.87959
$$302$$ 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 $$-0.601871\pi$$
−0.314601 + 0.949224i $$0.601871\pi$$
$$308$$ 321376. 0.193035
$$309$$ −117496. −0.0700046
$$310$$ 1.27398e6 0.752937
$$311$$ −1.25135e6 −0.733630 −0.366815 0.930294i $$-0.619552\pi$$
−0.366815 + 0.930294i $$0.619552\pi$$
$$312$$ −44288.0 −0.0257573
$$313$$ −1.44336e6 −0.832749 −0.416375 0.909193i $$-0.636699\pi$$
−0.416375 + 0.909193i $$0.636699\pi$$
$$314$$ 136300. 0.0780139
$$315$$ −2.04877e6 −1.16337
$$316$$ 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
$$322$$ −1.18126e6 −0.634899
$$323$$ −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
$$329$$ 2.85254e6 1.45292
$$330$$ −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
$$336$$ −42496.0 −0.0205352
$$337$$ 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
$$343$$ 1.00563e6 0.461532
$$344$$ −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 $$-0.484689\pi$$
0.0480836 + 0.998843i $$0.484689\pi$$
$$350$$ −347936. −0.151820
$$351$$ −335620. −0.145405
$$352$$ 123904. 0.0533002
$$353$$ 3.68192e6 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$354$$ 139956. 0.0593586
$$355$$ −678861. −0.285897
$$356$$ 2.00664e6 0.839159
$$357$$ 122508. 0.0508738
$$358$$ −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
$$364$$ −1.83795e6 −0.727078
$$365$$ 2.71626e6 1.06718
$$366$$ 183760. 0.0717048
$$367$$ −3.11666e6 −1.20788 −0.603940 0.797029i $$-0.706404\pi$$
−0.603940 + 0.797029i $$0.706404\pi$$
$$368$$ −455424. −0.175306
$$369$$ −1.28357e6 −0.490742
$$370$$ −3.01614e6 −1.14537
$$371$$ 5.10052e6 1.92389
$$372$$ 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
$$378$$ −322040. −0.115926
$$379$$ −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 $$-0.488812\pi$$
0.0351414 + 0.999382i $$0.488812\pi$$
$$384$$ −16384.0 −0.00567012
$$385$$ −1.02439e6 −0.352218
$$386$$ −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
$$392$$ −687936. −0.226117
$$393$$ −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 $$-0.575108\pi$$
−0.233774 + 0.972291i $$0.575108\pi$$
$$398$$ 186400. 0.0589845
$$399$$ −236384. −0.0743337
$$400$$ −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
$$406$$ −1.37050e6 −0.412632
$$407$$ 1.78898e6 0.535329
$$408$$ 47232.0 0.0140471
$$409$$ −3.61488e6 −1.06853 −0.534263 0.845318i $$-0.679411\pi$$
−0.534263 + 0.845318i $$0.679411\pi$$
$$410$$ 1.08202e6 0.317888
$$411$$ 400137. 0.116843
$$412$$ −1.87994e6 −0.545632
$$413$$ 5.80817e6 1.67558
$$414$$ −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 $$-0.677971\pi$$
−0.530435 + 0.847726i $$0.677971\pi$$
$$420$$ 135456. 0.0374693
$$421$$ 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
$$427$$ 7.62604e6 2.02409
$$428$$ −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 $$-0.502965\pi$$
−0.00931603 + 0.999957i $$0.502965\pi$$
$$434$$ 4.14668e6 1.05676
$$435$$ 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 $$-0.476551\pi$$
0.0736007 + 0.997288i $$0.476551\pi$$
$$440$$ −394944. −0.0972532
$$441$$ −2.60126e6 −0.636923
$$442$$ 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
$$448$$ −679936. −0.160056
$$449$$ −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
$$455$$ 5.85847e6 1.32665
$$456$$ −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 $$-0.544087\pi$$
−0.138060 + 0.990424i $$0.544087\pi$$
$$462$$ −80344.0 −0.0175125
$$463$$ −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 $$-0.580435\pi$$
−0.250014 + 0.968242i $$0.580435\pi$$
$$468$$ −2.67942e6 −0.565492
$$469$$ −4.20694e6 −0.883149
$$470$$ −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
$$476$$ 1.96013e6 0.396522
$$477$$ 7.43569e6 1.49632
$$478$$ −3.42070e6 −0.684770
$$479$$ −6.72258e6 −1.33874 −0.669371 0.742928i $$-0.733437\pi$$
−0.669371 + 0.742928i $$0.733437\pi$$
$$480$$ 52224.0 0.0103459
$$481$$ −1.02312e7 −2.01634
$$482$$ −4.50186e6 −0.882620
$$483$$ 295314. 0.0575992
$$484$$ 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
$$490$$ 2.19280e6 0.412580
$$491$$ −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
$$497$$ −2.20963e6 −0.401262
$$498$$ −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 $$-0.448350\pi$$
0.161552 + 0.986864i $$0.448350\pi$$
$$504$$ −2.57101e6 −0.450845
$$505$$ −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 $$-0.546802\pi$$
−0.146503 + 0.989210i $$0.546802\pi$$
$$510$$ −150552. −0.0256307
$$511$$ 8.84116e6 1.49781
$$512$$ −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
$$518$$ −9.81724e6 −1.60755
$$519$$ −766254. −0.124869
$$520$$ 2.25869e6 0.366309
$$521$$ −789435. −0.127415 −0.0637077 0.997969i $$-0.520293\pi$$
−0.0637077 + 0.997969i $$0.520293\pi$$
$$522$$ −1.99795e6 −0.320929
$$523$$ 627392. 0.100296 0.0501481 0.998742i $$-0.484031\pi$$
0.0501481 + 0.998742i $$0.484031\pi$$
$$524$$ −1.57027e6 −0.249831
$$525$$ 86984.0 0.0137734
$$526$$ 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
$$532$$ −3.78214e6 −0.579374
$$533$$ 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
$$539$$ −1.30063e6 −0.192833
$$540$$ 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
$$546$$ 459488. 0.0659618
$$547$$ 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
$$553$$ −1.28208e7 −1.78280
$$554$$ −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
$$560$$ 2.16730e6 0.292044
$$561$$ 89298.0 0.0119794
$$562$$ 3.51617e6 0.469601
$$563$$ 5.47288e6 0.727687 0.363844 0.931460i $$-0.381464\pi$$
0.363844 + 0.931460i $$0.381464\pi$$
$$564$$ −274944. −0.0363954
$$565$$ 2.40960e6 0.317558
$$566$$ −6.16107e6 −0.808379
$$567$$ −9.68129e6 −1.26466
$$568$$ −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
$$574$$ 3.52186e6 0.446161
$$575$$ 932196. 0.117581
$$576$$ −991232. −0.124486
$$577$$ −1.37758e7 −1.72258 −0.861288 0.508117i $$-0.830342\pi$$
−0.861288 + 0.508117i $$0.830342\pi$$
$$578$$ 3.50085e6 0.435867
$$579$$ 411668. 0.0510330
$$580$$ 1.68422e6 0.207888
$$581$$ −9.13232e6 −1.12238
$$582$$ 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 $$-0.775388\pi$$
−0.761196 + 0.648522i $$0.775388\pi$$
$$588$$ 171984. 0.0205137
$$589$$ 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 $$-0.481250\pi$$
0.0588711 + 0.998266i $$0.481250\pi$$
$$594$$ −234740. −0.0272974
$$595$$ −6.24791e6 −0.723506
$$596$$ 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 $$-0.503584\pi$$
−0.0112587 + 0.999937i $$0.503584\pi$$
$$602$$ 1.18179e7 1.32907
$$603$$ −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 $$-0.499716\pi$$
0.000891754 1.00000i $$0.499716\pi$$
$$608$$ −1.45818e6 −0.159975
$$609$$ 342624. 0.0374347
$$610$$ −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
$$616$$ −1.28550e6 −0.136497
$$617$$ 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 $$-0.921394\pi$$
−0.969663 + 0.244445i $$0.921394\pi$$
$$620$$ −5.09592e6 −0.532407
$$621$$ 862815. 0.0897819
$$622$$ 5.00539e6 0.518755
$$623$$ −2.08189e7 −2.14901
$$624$$ 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
$$630$$ 8.19509e6 0.822626
$$631$$ −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
$$637$$ 7.43831e6 0.726316
$$638$$ −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 $$-0.347495\pi$$
0.460989 + 0.887406i $$0.347495\pi$$
$$644$$ 4.72502e6 0.448941
$$645$$ −907698. −0.0859097
$$646$$ 4.20365e6 0.396319
$$647$$ −4.51430e6 −0.423965 −0.211982 0.977273i $$-0.567992\pi$$
−0.211982 + 0.977273i $$0.567992\pi$$
$$648$$ −3.73254e6 −0.349195
$$649$$ 4.23367e6 0.394553
$$650$$ 1.45043e6 0.134652
$$651$$ −1.03667e6 −0.0958712
$$652$$ 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
$$658$$ −1.14102e7 −1.02737
$$659$$ 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 $$-0.340292\pi$$
0.480949 + 0.876748i $$0.340292\pi$$
$$662$$ −8.06935e6 −0.715638
$$663$$ −510696. −0.0451210
$$664$$ −3.52090e6 −0.309908
$$665$$ 1.20556e7 1.05714
$$666$$ −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
$$672$$ 169984. 0.0145206
$$673$$ 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 $$-0.668738\pi$$
−0.505624 + 0.862754i $$0.668738\pi$$
$$678$$ 188988. 0.0157892
$$679$$ 1.47420e7 1.22710
$$680$$ −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
$$686$$ −4.02251e6 −0.326353
$$687$$ −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 $$-0.324881\pi$$
0.522817 + 0.852445i $$0.324881\pi$$
$$692$$ −1.22601e7 −0.973257
$$693$$ −4.86081e6 −0.384482
$$694$$ 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
$$700$$ 1.39174e6 0.107353
$$701$$ 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
$$707$$ −246012. −0.0185101
$$708$$ −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
$$714$$ −490032. −0.0359732
$$715$$ 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 $$-0.162983\pi$$
0.871753 + 0.489945i $$0.162983\pi$$
$$720$$ 3.15955e6 0.227140
$$721$$ 1.95043e7 1.39731
$$722$$ 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 $$-0.299006\pi$$
0.590310 + 0.807177i $$0.299006\pi$$
$$728$$ 7.35181e6 0.514121
$$729$$ −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 $$-0.555441\pi$$
−0.173295 + 0.984870i $$0.555441\pi$$
$$734$$ 1.24666e7 0.854101
$$735$$ −548199. −0.0374300
$$736$$ 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
$$742$$ −2.04021e7 −1.36039
$$743$$ 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
$$749$$ 1.31741e7 0.858057
$$750$$ −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
$$756$$ 1.28816e6 0.0819720
$$757$$ 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 $$-0.347858\pi$$
0.459976 + 0.887931i $$0.347858\pi$$
$$762$$ 957664. 0.0597483
$$763$$ −1.45818e7 −0.906774
$$764$$ 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 $$-0.235736\pi$$
0.738073 + 0.674721i $$0.235736\pi$$
$$770$$ 4.09754e6 0.249056
$$771$$ 37758.0 0.00228756
$$772$$ 6.58669e6 0.397763
$$773$$ 1.35260e7 0.814181 0.407091 0.913388i $$-0.366543\pi$$
0.407091 + 0.913388i $$0.366543\pi$$
$$774$$ 1.72285e7 1.03370
$$775$$ −3.27238e6 −0.195708
$$776$$ 5.68365e6 0.338823
$$777$$ 2.45431e6 0.145840
$$778$$ −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
$$784$$ 2.75174e6 0.159889
$$785$$ 1.73782e6 0.100654
$$786$$ 392568. 0.0226651
$$787$$ 1.42094e7 0.817786 0.408893 0.912582i $$-0.365915\pi$$
0.408893 + 0.912582i $$0.365915\pi$$
$$788$$ −1.21481e7 −0.696937
$$789$$ −631254. −0.0361004
$$790$$ 1.57557e7 0.898196
$$791$$ 7.84300e6 0.445698
$$792$$ −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 $$-0.570997\pi$$
−0.221197 + 0.975229i $$0.570997\pi$$
$$798$$ 945536. 0.0525619
$$799$$ 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
$$805$$ −1.50610e7 −0.819152
$$806$$ −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 $$-0.396970\pi$$
0.318055 + 0.948072i $$0.396970\pi$$
$$812$$ 5.48198e6 0.291775
$$813$$ −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
$$819$$ 2.77990e7 1.44817
$$820$$ −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
$$826$$ −2.32327e7 −1.18481
$$827$$ −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 $$-0.702678\pi$$
−0.594570 + 0.804044i $$0.702678\pi$$
$$830$$ 1.12229e7 0.565468
$$831$$ 1.68820e6 0.0848049
$$832$$ 2.83443e6 0.141957
$$833$$ −7.93276e6 −0.396106
$$834$$ −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 $$-0.246448\pi$$
0.714952 + 0.699173i $$0.246448\pi$$
$$840$$ −541824. −0.0264948
$$841$$ −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
$$847$$ −2.43041e6 −0.116405
$$848$$ −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 $$-0.571683\pi$$
−0.223299 + 0.974750i $$0.571683\pi$$
$$854$$ −3.05042e7 −1.43125
$$855$$ 1.75750e7 0.822205
$$856$$ 5.07917e6 0.236924
$$857$$ −1.81553e6 −0.0844405 −0.0422203 0.999108i $$-0.513443\pi$$
−0.0422203 + 0.999108i $$0.513443\pi$$
$$858$$ 334928. 0.0155322
$$859$$ −1.07812e7 −0.498522 −0.249261 0.968436i $$-0.580188\pi$$
−0.249261 + 0.968436i $$0.580188\pi$$
$$860$$ −1.45232e7 −0.669600
$$861$$ −880464. −0.0404766
$$862$$ 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
$$868$$ −1.65867e7 −0.747242
$$869$$ −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
$$875$$ −3.08924e7 −1.36406
$$876$$ −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 $$-0.636917\pi$$
−0.416995 + 0.908909i $$0.636917\pi$$
$$882$$ 1.04050e7 0.450373
$$883$$ 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 $$-0.409214\pi$$
0.281361 + 0.959602i $$0.409214\pi$$
$$888$$ 946240. 0.0402688
$$889$$ 3.97431e7 1.68658
$$890$$ 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
$$896$$ 2.71974e6 0.113177
$$897$$ −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
$$903$$ −2.95447e6 −0.120576
$$904$$ 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
$$910$$ −2.34339e7 −0.938082
$$911$$ 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
$$917$$ 1.62916e7 0.639793
$$918$$ −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
$$924$$ 321376. 0.0123832
$$925$$ 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 $$-0.646343\pi$$
−0.443725 + 0.896163i $$0.646343\pi$$
$$930$$ 1.27398e6 0.0483009
$$931$$ 1.53066e7 0.578767
$$932$$ −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 $$-0.373923\pi$$
0.385808 + 0.922579i $$0.373923\pi$$
$$938$$ 1.68278e7 0.624481
$$939$$ −1.44336e6 −0.0534209
$$940$$ 1.40221e7 0.517601
$$941$$ 2.69193e7 0.991036 0.495518 0.868598i $$-0.334978\pi$$
0.495518 + 0.868598i $$0.334978\pi$$
$$942$$ 136300. 0.00500459
$$943$$ −9.43582e6 −0.345542
$$944$$ −8.95718e6 −0.327146
$$945$$ −4.10601e6 −0.149569
$$946$$ 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
$$952$$ −7.84051e6 −0.280383
$$953$$ 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
$$959$$ −6.64227e7 −2.33222
$$960$$ −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
$$966$$ −1.18126e6 −0.0407288
$$967$$ −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 $$-0.595282\pi$$
−0.294887 + 0.955532i $$0.595282\pi$$
$$972$$ 2.81882e6 0.0956976
$$973$$ −3.41572e7 −1.15664
$$974$$ −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
$$980$$ −8.77118e6 −0.291738
$$981$$ −2.12578e7 −0.705253
$$982$$ 2.31850e6 0.0767234
$$983$$ −2.79124e7 −0.921326 −0.460663 0.887575i $$-0.652388\pi$$
−0.460663 + 0.887575i $$0.652388\pi$$
$$984$$ −339456. −0.0111762
$$985$$ 3.87222e7 1.27165
$$986$$ −6.09293e6 −0.199588
$$987$$ 2.85254e6 0.0932051
$$988$$ 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
$$994$$ 8.83850e6 0.283735
$$995$$ 2.37660e6 0.0761024
$$996$$ 880224. 0.0281154
$$997$$ −2.21044e7 −0.704273 −0.352137 0.935949i $$-0.614545\pi$$
−0.352137 + 0.935949i $$0.614545\pi$$
$$998$$ −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 22.6.a.b.1.1 1
3.2 odd 2 198.6.a.i.1.1 1
4.3 odd 2 176.6.a.b.1.1 1
5.2 odd 4 550.6.b.f.199.1 2
5.3 odd 4 550.6.b.f.199.2 2
5.4 even 2 550.6.a.f.1.1 1
7.6 odd 2 1078.6.a.a.1.1 1
8.3 odd 2 704.6.a.f.1.1 1
8.5 even 2 704.6.a.e.1.1 1
11.10 odd 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 1.1 even 1 trivial
176.6.a.b.1.1 1 4.3 odd 2
198.6.a.i.1.1 1 3.2 odd 2
242.6.a.d.1.1 1 11.10 odd 2
550.6.a.f.1.1 1 5.4 even 2
550.6.b.f.199.1 2 5.2 odd 4
550.6.b.f.199.2 2 5.3 odd 4
704.6.a.e.1.1 1 8.5 even 2
704.6.a.f.1.1 1 8.3 odd 2
1078.6.a.a.1.1 1 7.6 odd 2