# Properties

 Label 42.6.a.d.1.1 Level $42$ Weight $6$ Character 42.1 Self dual yes Analytic conductor $6.736$ 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] = [42,6,Mod(1,42)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, 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("42.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 42.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: $$6.73612043215$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 42.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +26.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +26.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -104.000 q^{10} +664.000 q^{11} +144.000 q^{12} +318.000 q^{13} +196.000 q^{14} +234.000 q^{15} +256.000 q^{16} +1582.00 q^{17} -324.000 q^{18} +236.000 q^{19} +416.000 q^{20} -441.000 q^{21} -2656.00 q^{22} +2212.00 q^{23} -576.000 q^{24} -2449.00 q^{25} -1272.00 q^{26} +729.000 q^{27} -784.000 q^{28} -4954.00 q^{29} -936.000 q^{30} -7128.00 q^{31} -1024.00 q^{32} +5976.00 q^{33} -6328.00 q^{34} -1274.00 q^{35} +1296.00 q^{36} +4358.00 q^{37} -944.000 q^{38} +2862.00 q^{39} -1664.00 q^{40} +10542.0 q^{41} +1764.00 q^{42} -8452.00 q^{43} +10624.0 q^{44} +2106.00 q^{45} -8848.00 q^{46} +5352.00 q^{47} +2304.00 q^{48} +2401.00 q^{49} +9796.00 q^{50} +14238.0 q^{51} +5088.00 q^{52} -33354.0 q^{53} -2916.00 q^{54} +17264.0 q^{55} +3136.00 q^{56} +2124.00 q^{57} +19816.0 q^{58} -15436.0 q^{59} +3744.00 q^{60} -36762.0 q^{61} +28512.0 q^{62} -3969.00 q^{63} +4096.00 q^{64} +8268.00 q^{65} -23904.0 q^{66} +40972.0 q^{67} +25312.0 q^{68} +19908.0 q^{69} +5096.00 q^{70} -9092.00 q^{71} -5184.00 q^{72} -73454.0 q^{73} -17432.0 q^{74} -22041.0 q^{75} +3776.00 q^{76} -32536.0 q^{77} -11448.0 q^{78} +89400.0 q^{79} +6656.00 q^{80} +6561.00 q^{81} -42168.0 q^{82} -6428.00 q^{83} -7056.00 q^{84} +41132.0 q^{85} +33808.0 q^{86} -44586.0 q^{87} -42496.0 q^{88} -122658. q^{89} -8424.00 q^{90} -15582.0 q^{91} +35392.0 q^{92} -64152.0 q^{93} -21408.0 q^{94} +6136.00 q^{95} -9216.00 q^{96} +21370.0 q^{97} -9604.00 q^{98} +53784.0 q^{99} +O(q^{100})$$

## 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
$$3$$ 9.00000 0.577350
$$4$$ 16.0000 0.500000
$$5$$ 26.0000 0.465102 0.232551 0.972584i $$-0.425293\pi$$
0.232551 + 0.972584i $$0.425293\pi$$
$$6$$ −36.0000 −0.408248
$$7$$ −49.0000 −0.377964
$$8$$ −64.0000 −0.353553
$$9$$ 81.0000 0.333333
$$10$$ −104.000 −0.328877
$$11$$ 664.000 1.65457 0.827287 0.561779i $$-0.189883\pi$$
0.827287 + 0.561779i $$0.189883\pi$$
$$12$$ 144.000 0.288675
$$13$$ 318.000 0.521878 0.260939 0.965355i $$-0.415968\pi$$
0.260939 + 0.965355i $$0.415968\pi$$
$$14$$ 196.000 0.267261
$$15$$ 234.000 0.268527
$$16$$ 256.000 0.250000
$$17$$ 1582.00 1.32765 0.663826 0.747887i $$-0.268932\pi$$
0.663826 + 0.747887i $$0.268932\pi$$
$$18$$ −324.000 −0.235702
$$19$$ 236.000 0.149978 0.0749891 0.997184i $$-0.476108\pi$$
0.0749891 + 0.997184i $$0.476108\pi$$
$$20$$ 416.000 0.232551
$$21$$ −441.000 −0.218218
$$22$$ −2656.00 −1.16996
$$23$$ 2212.00 0.871898 0.435949 0.899971i $$-0.356413\pi$$
0.435949 + 0.899971i $$0.356413\pi$$
$$24$$ −576.000 −0.204124
$$25$$ −2449.00 −0.783680
$$26$$ −1272.00 −0.369023
$$27$$ 729.000 0.192450
$$28$$ −784.000 −0.188982
$$29$$ −4954.00 −1.09386 −0.546929 0.837179i $$-0.684203\pi$$
−0.546929 + 0.837179i $$0.684203\pi$$
$$30$$ −936.000 −0.189877
$$31$$ −7128.00 −1.33218 −0.666091 0.745871i $$-0.732034\pi$$
−0.666091 + 0.745871i $$0.732034\pi$$
$$32$$ −1024.00 −0.176777
$$33$$ 5976.00 0.955269
$$34$$ −6328.00 −0.938792
$$35$$ −1274.00 −0.175792
$$36$$ 1296.00 0.166667
$$37$$ 4358.00 0.523339 0.261669 0.965158i $$-0.415727\pi$$
0.261669 + 0.965158i $$0.415727\pi$$
$$38$$ −944.000 −0.106051
$$39$$ 2862.00 0.301306
$$40$$ −1664.00 −0.164438
$$41$$ 10542.0 0.979407 0.489704 0.871889i $$-0.337105\pi$$
0.489704 + 0.871889i $$0.337105\pi$$
$$42$$ 1764.00 0.154303
$$43$$ −8452.00 −0.697089 −0.348545 0.937292i $$-0.613324\pi$$
−0.348545 + 0.937292i $$0.613324\pi$$
$$44$$ 10624.0 0.827287
$$45$$ 2106.00 0.155034
$$46$$ −8848.00 −0.616525
$$47$$ 5352.00 0.353404 0.176702 0.984264i $$-0.443457\pi$$
0.176702 + 0.984264i $$0.443457\pi$$
$$48$$ 2304.00 0.144338
$$49$$ 2401.00 0.142857
$$50$$ 9796.00 0.554145
$$51$$ 14238.0 0.766520
$$52$$ 5088.00 0.260939
$$53$$ −33354.0 −1.63102 −0.815508 0.578746i $$-0.803542\pi$$
−0.815508 + 0.578746i $$0.803542\pi$$
$$54$$ −2916.00 −0.136083
$$55$$ 17264.0 0.769546
$$56$$ 3136.00 0.133631
$$57$$ 2124.00 0.0865899
$$58$$ 19816.0 0.773475
$$59$$ −15436.0 −0.577304 −0.288652 0.957434i $$-0.593207\pi$$
−0.288652 + 0.957434i $$0.593207\pi$$
$$60$$ 3744.00 0.134263
$$61$$ −36762.0 −1.26495 −0.632477 0.774579i $$-0.717962\pi$$
−0.632477 + 0.774579i $$0.717962\pi$$
$$62$$ 28512.0 0.941995
$$63$$ −3969.00 −0.125988
$$64$$ 4096.00 0.125000
$$65$$ 8268.00 0.242726
$$66$$ −23904.0 −0.675477
$$67$$ 40972.0 1.11506 0.557532 0.830155i $$-0.311748\pi$$
0.557532 + 0.830155i $$0.311748\pi$$
$$68$$ 25312.0 0.663826
$$69$$ 19908.0 0.503390
$$70$$ 5096.00 0.124304
$$71$$ −9092.00 −0.214049 −0.107025 0.994256i $$-0.534132\pi$$
−0.107025 + 0.994256i $$0.534132\pi$$
$$72$$ −5184.00 −0.117851
$$73$$ −73454.0 −1.61327 −0.806637 0.591047i $$-0.798715\pi$$
−0.806637 + 0.591047i $$0.798715\pi$$
$$74$$ −17432.0 −0.370056
$$75$$ −22041.0 −0.452458
$$76$$ 3776.00 0.0749891
$$77$$ −32536.0 −0.625370
$$78$$ −11448.0 −0.213056
$$79$$ 89400.0 1.61165 0.805823 0.592156i $$-0.201723\pi$$
0.805823 + 0.592156i $$0.201723\pi$$
$$80$$ 6656.00 0.116276
$$81$$ 6561.00 0.111111
$$82$$ −42168.0 −0.692546
$$83$$ −6428.00 −0.102419 −0.0512095 0.998688i $$-0.516308\pi$$
−0.0512095 + 0.998688i $$0.516308\pi$$
$$84$$ −7056.00 −0.109109
$$85$$ 41132.0 0.617494
$$86$$ 33808.0 0.492916
$$87$$ −44586.0 −0.631539
$$88$$ −42496.0 −0.584980
$$89$$ −122658. −1.64142 −0.820712 0.571342i $$-0.806423\pi$$
−0.820712 + 0.571342i $$0.806423\pi$$
$$90$$ −8424.00 −0.109626
$$91$$ −15582.0 −0.197251
$$92$$ 35392.0 0.435949
$$93$$ −64152.0 −0.769135
$$94$$ −21408.0 −0.249894
$$95$$ 6136.00 0.0697552
$$96$$ −9216.00 −0.102062
$$97$$ 21370.0 0.230608 0.115304 0.993330i $$-0.463216\pi$$
0.115304 + 0.993330i $$0.463216\pi$$
$$98$$ −9604.00 −0.101015
$$99$$ 53784.0 0.551525
$$100$$ −39184.0 −0.391840
$$101$$ −36814.0 −0.359095 −0.179548 0.983749i $$-0.557463\pi$$
−0.179548 + 0.983749i $$0.557463\pi$$
$$102$$ −56952.0 −0.542012
$$103$$ 104528. 0.970822 0.485411 0.874286i $$-0.338670\pi$$
0.485411 + 0.874286i $$0.338670\pi$$
$$104$$ −20352.0 −0.184512
$$105$$ −11466.0 −0.101494
$$106$$ 133416. 1.15330
$$107$$ 214440. 1.81070 0.905350 0.424667i $$-0.139609\pi$$
0.905350 + 0.424667i $$0.139609\pi$$
$$108$$ 11664.0 0.0962250
$$109$$ 28798.0 0.232165 0.116082 0.993240i $$-0.462966\pi$$
0.116082 + 0.993240i $$0.462966\pi$$
$$110$$ −69056.0 −0.544151
$$111$$ 39222.0 0.302150
$$112$$ −12544.0 −0.0944911
$$113$$ −56014.0 −0.412668 −0.206334 0.978482i $$-0.566153\pi$$
−0.206334 + 0.978482i $$0.566153\pi$$
$$114$$ −8496.00 −0.0612283
$$115$$ 57512.0 0.405521
$$116$$ −79264.0 −0.546929
$$117$$ 25758.0 0.173959
$$118$$ 61744.0 0.408216
$$119$$ −77518.0 −0.501805
$$120$$ −14976.0 −0.0949386
$$121$$ 279845. 1.73762
$$122$$ 147048. 0.894457
$$123$$ 94878.0 0.565461
$$124$$ −114048. −0.666091
$$125$$ −144924. −0.829593
$$126$$ 15876.0 0.0890871
$$127$$ 185400. 1.02000 0.510000 0.860174i $$-0.329645\pi$$
0.510000 + 0.860174i $$0.329645\pi$$
$$128$$ −16384.0 −0.0883883
$$129$$ −76068.0 −0.402465
$$130$$ −33072.0 −0.171634
$$131$$ 64532.0 0.328547 0.164273 0.986415i $$-0.447472\pi$$
0.164273 + 0.986415i $$0.447472\pi$$
$$132$$ 95616.0 0.477635
$$133$$ −11564.0 −0.0566864
$$134$$ −163888. −0.788470
$$135$$ 18954.0 0.0895089
$$136$$ −101248. −0.469396
$$137$$ 152930. 0.696131 0.348066 0.937470i $$-0.386839\pi$$
0.348066 + 0.937470i $$0.386839\pi$$
$$138$$ −79632.0 −0.355951
$$139$$ −343460. −1.50778 −0.753892 0.656998i $$-0.771826\pi$$
−0.753892 + 0.656998i $$0.771826\pi$$
$$140$$ −20384.0 −0.0878960
$$141$$ 48168.0 0.204038
$$142$$ 36368.0 0.151356
$$143$$ 211152. 0.863486
$$144$$ 20736.0 0.0833333
$$145$$ −128804. −0.508756
$$146$$ 293816. 1.14076
$$147$$ 21609.0 0.0824786
$$148$$ 69728.0 0.261669
$$149$$ −174858. −0.645238 −0.322619 0.946529i $$-0.604563\pi$$
−0.322619 + 0.946529i $$0.604563\pi$$
$$150$$ 88164.0 0.319936
$$151$$ −452552. −1.61520 −0.807600 0.589731i $$-0.799234\pi$$
−0.807600 + 0.589731i $$0.799234\pi$$
$$152$$ −15104.0 −0.0530253
$$153$$ 128142. 0.442551
$$154$$ 130144. 0.442204
$$155$$ −185328. −0.619601
$$156$$ 45792.0 0.150653
$$157$$ −499066. −1.61588 −0.807940 0.589265i $$-0.799417\pi$$
−0.807940 + 0.589265i $$0.799417\pi$$
$$158$$ −357600. −1.13961
$$159$$ −300186. −0.941668
$$160$$ −26624.0 −0.0822192
$$161$$ −108388. −0.329546
$$162$$ −26244.0 −0.0785674
$$163$$ −475588. −1.40204 −0.701022 0.713139i $$-0.747273\pi$$
−0.701022 + 0.713139i $$0.747273\pi$$
$$164$$ 168672. 0.489704
$$165$$ 155376. 0.444298
$$166$$ 25712.0 0.0724212
$$167$$ 120224. 0.333580 0.166790 0.985992i $$-0.446660\pi$$
0.166790 + 0.985992i $$0.446660\pi$$
$$168$$ 28224.0 0.0771517
$$169$$ −270169. −0.727644
$$170$$ −164528. −0.436634
$$171$$ 19116.0 0.0499927
$$172$$ −135232. −0.348545
$$173$$ 508874. 1.29269 0.646346 0.763045i $$-0.276296\pi$$
0.646346 + 0.763045i $$0.276296\pi$$
$$174$$ 178344. 0.446566
$$175$$ 120001. 0.296203
$$176$$ 169984. 0.413644
$$177$$ −138924. −0.333307
$$178$$ 490632. 1.16066
$$179$$ 487560. 1.13735 0.568677 0.822561i $$-0.307456\pi$$
0.568677 + 0.822561i $$0.307456\pi$$
$$180$$ 33696.0 0.0775170
$$181$$ −544410. −1.23518 −0.617589 0.786501i $$-0.711891\pi$$
−0.617589 + 0.786501i $$0.711891\pi$$
$$182$$ 62328.0 0.139478
$$183$$ −330858. −0.730321
$$184$$ −141568. −0.308262
$$185$$ 113308. 0.243406
$$186$$ 256608. 0.543861
$$187$$ 1.05045e6 2.19670
$$188$$ 85632.0 0.176702
$$189$$ −35721.0 −0.0727393
$$190$$ −24544.0 −0.0493243
$$191$$ 376404. 0.746570 0.373285 0.927717i $$-0.378231\pi$$
0.373285 + 0.927717i $$0.378231\pi$$
$$192$$ 36864.0 0.0721688
$$193$$ 844946. 1.63281 0.816405 0.577480i $$-0.195964\pi$$
0.816405 + 0.577480i $$0.195964\pi$$
$$194$$ −85480.0 −0.163065
$$195$$ 74412.0 0.140138
$$196$$ 38416.0 0.0714286
$$197$$ −492794. −0.904690 −0.452345 0.891843i $$-0.649412\pi$$
−0.452345 + 0.891843i $$0.649412\pi$$
$$198$$ −215136. −0.389987
$$199$$ −914776. −1.63750 −0.818751 0.574148i $$-0.805333\pi$$
−0.818751 + 0.574148i $$0.805333\pi$$
$$200$$ 156736. 0.277073
$$201$$ 368748. 0.643783
$$202$$ 147256. 0.253919
$$203$$ 242746. 0.413440
$$204$$ 227808. 0.383260
$$205$$ 274092. 0.455524
$$206$$ −418112. −0.686475
$$207$$ 179172. 0.290633
$$208$$ 81408.0 0.130469
$$209$$ 156704. 0.248150
$$210$$ 45864.0 0.0717668
$$211$$ 311780. 0.482106 0.241053 0.970512i $$-0.422507\pi$$
0.241053 + 0.970512i $$0.422507\pi$$
$$212$$ −533664. −0.815508
$$213$$ −81828.0 −0.123581
$$214$$ −857760. −1.28036
$$215$$ −219752. −0.324218
$$216$$ −46656.0 −0.0680414
$$217$$ 349272. 0.503517
$$218$$ −115192. −0.164165
$$219$$ −661086. −0.931425
$$220$$ 276224. 0.384773
$$221$$ 503076. 0.692872
$$222$$ −156888. −0.213652
$$223$$ −1.28776e6 −1.73409 −0.867047 0.498226i $$-0.833985\pi$$
−0.867047 + 0.498226i $$0.833985\pi$$
$$224$$ 50176.0 0.0668153
$$225$$ −198369. −0.261227
$$226$$ 224056. 0.291800
$$227$$ 1.28905e6 1.66037 0.830187 0.557485i $$-0.188234\pi$$
0.830187 + 0.557485i $$0.188234\pi$$
$$228$$ 33984.0 0.0432950
$$229$$ 678214. 0.854630 0.427315 0.904103i $$-0.359460\pi$$
0.427315 + 0.904103i $$0.359460\pi$$
$$230$$ −230048. −0.286747
$$231$$ −292824. −0.361058
$$232$$ 317056. 0.386737
$$233$$ −1.11731e6 −1.34829 −0.674146 0.738598i $$-0.735488\pi$$
−0.674146 + 0.738598i $$0.735488\pi$$
$$234$$ −103032. −0.123008
$$235$$ 139152. 0.164369
$$236$$ −246976. −0.288652
$$237$$ 804600. 0.930485
$$238$$ 310072. 0.354830
$$239$$ −1.26196e6 −1.42906 −0.714528 0.699606i $$-0.753359\pi$$
−0.714528 + 0.699606i $$0.753359\pi$$
$$240$$ 59904.0 0.0671317
$$241$$ 948218. 1.05164 0.525818 0.850597i $$-0.323759\pi$$
0.525818 + 0.850597i $$0.323759\pi$$
$$242$$ −1.11938e6 −1.22868
$$243$$ 59049.0 0.0641500
$$244$$ −588192. −0.632477
$$245$$ 62426.0 0.0664432
$$246$$ −379512. −0.399841
$$247$$ 75048.0 0.0782703
$$248$$ 456192. 0.470997
$$249$$ −57852.0 −0.0591317
$$250$$ 579696. 0.586611
$$251$$ −486396. −0.487310 −0.243655 0.969862i $$-0.578347\pi$$
−0.243655 + 0.969862i $$0.578347\pi$$
$$252$$ −63504.0 −0.0629941
$$253$$ 1.46877e6 1.44262
$$254$$ −741600. −0.721249
$$255$$ 370188. 0.356510
$$256$$ 65536.0 0.0625000
$$257$$ −1.03910e6 −0.981349 −0.490675 0.871343i $$-0.663250\pi$$
−0.490675 + 0.871343i $$0.663250\pi$$
$$258$$ 304272. 0.284585
$$259$$ −213542. −0.197803
$$260$$ 132288. 0.121363
$$261$$ −401274. −0.364619
$$262$$ −258128. −0.232317
$$263$$ 1.35104e6 1.20443 0.602213 0.798335i $$-0.294286\pi$$
0.602213 + 0.798335i $$0.294286\pi$$
$$264$$ −382464. −0.337739
$$265$$ −867204. −0.758589
$$266$$ 46256.0 0.0400833
$$267$$ −1.10392e6 −0.947677
$$268$$ 655552. 0.557532
$$269$$ −1.11811e6 −0.942115 −0.471057 0.882103i $$-0.656128\pi$$
−0.471057 + 0.882103i $$0.656128\pi$$
$$270$$ −75816.0 −0.0632924
$$271$$ −190104. −0.157242 −0.0786209 0.996905i $$-0.525052\pi$$
−0.0786209 + 0.996905i $$0.525052\pi$$
$$272$$ 404992. 0.331913
$$273$$ −140238. −0.113883
$$274$$ −611720. −0.492239
$$275$$ −1.62614e6 −1.29666
$$276$$ 318528. 0.251695
$$277$$ −200506. −0.157010 −0.0785051 0.996914i $$-0.525015\pi$$
−0.0785051 + 0.996914i $$0.525015\pi$$
$$278$$ 1.37384e6 1.06616
$$279$$ −577368. −0.444061
$$280$$ 81536.0 0.0621519
$$281$$ 1.09237e6 0.825285 0.412643 0.910893i $$-0.364606\pi$$
0.412643 + 0.910893i $$0.364606\pi$$
$$282$$ −192672. −0.144277
$$283$$ 1.81258e6 1.34534 0.672669 0.739944i $$-0.265148\pi$$
0.672669 + 0.739944i $$0.265148\pi$$
$$284$$ −145472. −0.107025
$$285$$ 55224.0 0.0402732
$$286$$ −844608. −0.610577
$$287$$ −516558. −0.370181
$$288$$ −82944.0 −0.0589256
$$289$$ 1.08287e6 0.762659
$$290$$ 515216. 0.359745
$$291$$ 192330. 0.133142
$$292$$ −1.17526e6 −0.806637
$$293$$ 2.10031e6 1.42927 0.714634 0.699499i $$-0.246593\pi$$
0.714634 + 0.699499i $$0.246593\pi$$
$$294$$ −86436.0 −0.0583212
$$295$$ −401336. −0.268505
$$296$$ −278912. −0.185028
$$297$$ 484056. 0.318423
$$298$$ 699432. 0.456252
$$299$$ 703416. 0.455024
$$300$$ −352656. −0.226229
$$301$$ 414148. 0.263475
$$302$$ 1.81021e6 1.14212
$$303$$ −331326. −0.207324
$$304$$ 60416.0 0.0374945
$$305$$ −955812. −0.588333
$$306$$ −512568. −0.312931
$$307$$ −1.64104e6 −0.993743 −0.496872 0.867824i $$-0.665518\pi$$
−0.496872 + 0.867824i $$0.665518\pi$$
$$308$$ −520576. −0.312685
$$309$$ 940752. 0.560504
$$310$$ 741312. 0.438124
$$311$$ −945232. −0.554163 −0.277081 0.960846i $$-0.589367\pi$$
−0.277081 + 0.960846i $$0.589367\pi$$
$$312$$ −183168. −0.106528
$$313$$ 415354. 0.239639 0.119820 0.992796i $$-0.461768\pi$$
0.119820 + 0.992796i $$0.461768\pi$$
$$314$$ 1.99626e6 1.14260
$$315$$ −103194. −0.0585974
$$316$$ 1.43040e6 0.805823
$$317$$ 1.18481e6 0.662220 0.331110 0.943592i $$-0.392577\pi$$
0.331110 + 0.943592i $$0.392577\pi$$
$$318$$ 1.20074e6 0.665860
$$319$$ −3.28946e6 −1.80987
$$320$$ 106496. 0.0581378
$$321$$ 1.92996e6 1.04541
$$322$$ 433552. 0.233024
$$323$$ 373352. 0.199119
$$324$$ 104976. 0.0555556
$$325$$ −778782. −0.408985
$$326$$ 1.90235e6 0.991395
$$327$$ 259182. 0.134040
$$328$$ −674688. −0.346273
$$329$$ −262248. −0.133574
$$330$$ −621504. −0.314166
$$331$$ 1.37155e6 0.688083 0.344042 0.938954i $$-0.388204\pi$$
0.344042 + 0.938954i $$0.388204\pi$$
$$332$$ −102848. −0.0512095
$$333$$ 352998. 0.174446
$$334$$ −480896. −0.235877
$$335$$ 1.06527e6 0.518619
$$336$$ −112896. −0.0545545
$$337$$ 963522. 0.462154 0.231077 0.972935i $$-0.425775\pi$$
0.231077 + 0.972935i $$0.425775\pi$$
$$338$$ 1.08068e6 0.514522
$$339$$ −504126. −0.238254
$$340$$ 658112. 0.308747
$$341$$ −4.73299e6 −2.20419
$$342$$ −76464.0 −0.0353502
$$343$$ −117649. −0.0539949
$$344$$ 540928. 0.246458
$$345$$ 517608. 0.234128
$$346$$ −2.03550e6 −0.914071
$$347$$ 2.57731e6 1.14906 0.574531 0.818483i $$-0.305185\pi$$
0.574531 + 0.818483i $$0.305185\pi$$
$$348$$ −713376. −0.315770
$$349$$ −3.06751e6 −1.34810 −0.674051 0.738684i $$-0.735447\pi$$
−0.674051 + 0.738684i $$0.735447\pi$$
$$350$$ −480004. −0.209447
$$351$$ 231822. 0.100435
$$352$$ −679936. −0.292490
$$353$$ −3.10144e6 −1.32473 −0.662364 0.749182i $$-0.730447\pi$$
−0.662364 + 0.749182i $$0.730447\pi$$
$$354$$ 555696. 0.235683
$$355$$ −236392. −0.0995547
$$356$$ −1.96253e6 −0.820712
$$357$$ −697662. −0.289717
$$358$$ −1.95024e6 −0.804230
$$359$$ −327508. −0.134118 −0.0670588 0.997749i $$-0.521362\pi$$
−0.0670588 + 0.997749i $$0.521362\pi$$
$$360$$ −134784. −0.0548128
$$361$$ −2.42040e6 −0.977507
$$362$$ 2.17764e6 0.873403
$$363$$ 2.51860e6 1.00321
$$364$$ −249312. −0.0986256
$$365$$ −1.90980e6 −0.750337
$$366$$ 1.32343e6 0.516415
$$367$$ −2.86739e6 −1.11128 −0.555638 0.831424i $$-0.687526\pi$$
−0.555638 + 0.831424i $$0.687526\pi$$
$$368$$ 566272. 0.217974
$$369$$ 853902. 0.326469
$$370$$ −453232. −0.172114
$$371$$ 1.63435e6 0.616466
$$372$$ −1.02643e6 −0.384568
$$373$$ 3.58029e6 1.33244 0.666218 0.745757i $$-0.267912\pi$$
0.666218 + 0.745757i $$0.267912\pi$$
$$374$$ −4.20179e6 −1.55330
$$375$$ −1.30432e6 −0.478966
$$376$$ −342528. −0.124947
$$377$$ −1.57537e6 −0.570860
$$378$$ 142884. 0.0514344
$$379$$ 1.64235e6 0.587310 0.293655 0.955912i $$-0.405128\pi$$
0.293655 + 0.955912i $$0.405128\pi$$
$$380$$ 98176.0 0.0348776
$$381$$ 1.66860e6 0.588898
$$382$$ −1.50562e6 −0.527905
$$383$$ −2.05698e6 −0.716527 −0.358263 0.933621i $$-0.616631\pi$$
−0.358263 + 0.933621i $$0.616631\pi$$
$$384$$ −147456. −0.0510310
$$385$$ −845936. −0.290861
$$386$$ −3.37978e6 −1.15457
$$387$$ −684612. −0.232363
$$388$$ 341920. 0.115304
$$389$$ 616142. 0.206446 0.103223 0.994658i $$-0.467084\pi$$
0.103223 + 0.994658i $$0.467084\pi$$
$$390$$ −297648. −0.0990927
$$391$$ 3.49938e6 1.15758
$$392$$ −153664. −0.0505076
$$393$$ 580788. 0.189686
$$394$$ 1.97118e6 0.639713
$$395$$ 2.32440e6 0.749580
$$396$$ 860544. 0.275762
$$397$$ 2.19212e6 0.698052 0.349026 0.937113i $$-0.386513\pi$$
0.349026 + 0.937113i $$0.386513\pi$$
$$398$$ 3.65910e6 1.15789
$$399$$ −104076. −0.0327279
$$400$$ −626944. −0.195920
$$401$$ 3.28454e6 1.02003 0.510015 0.860165i $$-0.329640\pi$$
0.510015 + 0.860165i $$0.329640\pi$$
$$402$$ −1.47499e6 −0.455223
$$403$$ −2.26670e6 −0.695236
$$404$$ −589024. −0.179548
$$405$$ 170586. 0.0516780
$$406$$ −970984. −0.292346
$$407$$ 2.89371e6 0.865903
$$408$$ −911232. −0.271006
$$409$$ −3.61219e6 −1.06773 −0.533866 0.845569i $$-0.679261\pi$$
−0.533866 + 0.845569i $$0.679261\pi$$
$$410$$ −1.09637e6 −0.322104
$$411$$ 1.37637e6 0.401912
$$412$$ 1.67245e6 0.485411
$$413$$ 756364. 0.218200
$$414$$ −716688. −0.205508
$$415$$ −167128. −0.0476353
$$416$$ −325632. −0.0922558
$$417$$ −3.09114e6 −0.870520
$$418$$ −626816. −0.175469
$$419$$ 5.41489e6 1.50680 0.753398 0.657564i $$-0.228413\pi$$
0.753398 + 0.657564i $$0.228413\pi$$
$$420$$ −183456. −0.0507468
$$421$$ 3.60629e6 0.991644 0.495822 0.868424i $$-0.334867\pi$$
0.495822 + 0.868424i $$0.334867\pi$$
$$422$$ −1.24712e6 −0.340900
$$423$$ 433512. 0.117801
$$424$$ 2.13466e6 0.576651
$$425$$ −3.87432e6 −1.04045
$$426$$ 327312. 0.0873852
$$427$$ 1.80134e6 0.478107
$$428$$ 3.43104e6 0.905350
$$429$$ 1.90037e6 0.498534
$$430$$ 879008. 0.229257
$$431$$ −2.78214e6 −0.721416 −0.360708 0.932679i $$-0.617465\pi$$
−0.360708 + 0.932679i $$0.617465\pi$$
$$432$$ 186624. 0.0481125
$$433$$ 6.27619e6 1.60871 0.804353 0.594152i $$-0.202512\pi$$
0.804353 + 0.594152i $$0.202512\pi$$
$$434$$ −1.39709e6 −0.356041
$$435$$ −1.15924e6 −0.293730
$$436$$ 460768. 0.116082
$$437$$ 522032. 0.130766
$$438$$ 2.64434e6 0.658617
$$439$$ 641592. 0.158890 0.0794452 0.996839i $$-0.474685\pi$$
0.0794452 + 0.996839i $$0.474685\pi$$
$$440$$ −1.10490e6 −0.272076
$$441$$ 194481. 0.0476190
$$442$$ −2.01230e6 −0.489934
$$443$$ 6.05546e6 1.46601 0.733006 0.680222i $$-0.238117\pi$$
0.733006 + 0.680222i $$0.238117\pi$$
$$444$$ 627552. 0.151075
$$445$$ −3.18911e6 −0.763430
$$446$$ 5.15104e6 1.22619
$$447$$ −1.57372e6 −0.372528
$$448$$ −200704. −0.0472456
$$449$$ −5.16681e6 −1.20950 −0.604752 0.796414i $$-0.706728\pi$$
−0.604752 + 0.796414i $$0.706728\pi$$
$$450$$ 793476. 0.184715
$$451$$ 6.99989e6 1.62050
$$452$$ −896224. −0.206334
$$453$$ −4.07297e6 −0.932536
$$454$$ −5.15621e6 −1.17406
$$455$$ −405132. −0.0917420
$$456$$ −135936. −0.0306142
$$457$$ −227798. −0.0510222 −0.0255111 0.999675i $$-0.508121\pi$$
−0.0255111 + 0.999675i $$0.508121\pi$$
$$458$$ −2.71286e6 −0.604315
$$459$$ 1.15328e6 0.255507
$$460$$ 920192. 0.202761
$$461$$ 585146. 0.128237 0.0641183 0.997942i $$-0.479577\pi$$
0.0641183 + 0.997942i $$0.479577\pi$$
$$462$$ 1.17130e6 0.255306
$$463$$ −3.41454e6 −0.740251 −0.370126 0.928982i $$-0.620685\pi$$
−0.370126 + 0.928982i $$0.620685\pi$$
$$464$$ −1.26822e6 −0.273465
$$465$$ −1.66795e6 −0.357727
$$466$$ 4.46924e6 0.953386
$$467$$ 716300. 0.151986 0.0759929 0.997108i $$-0.475787\pi$$
0.0759929 + 0.997108i $$0.475787\pi$$
$$468$$ 412128. 0.0869796
$$469$$ −2.00763e6 −0.421455
$$470$$ −556608. −0.116226
$$471$$ −4.49159e6 −0.932928
$$472$$ 987904. 0.204108
$$473$$ −5.61213e6 −1.15339
$$474$$ −3.21840e6 −0.657952
$$475$$ −577964. −0.117535
$$476$$ −1.24029e6 −0.250903
$$477$$ −2.70167e6 −0.543672
$$478$$ 5.04782e6 1.01050
$$479$$ 5.24092e6 1.04368 0.521842 0.853042i $$-0.325245\pi$$
0.521842 + 0.853042i $$0.325245\pi$$
$$480$$ −239616. −0.0474693
$$481$$ 1.38584e6 0.273119
$$482$$ −3.79287e6 −0.743619
$$483$$ −975492. −0.190264
$$484$$ 4.47752e6 0.868809
$$485$$ 555620. 0.107256
$$486$$ −236196. −0.0453609
$$487$$ 1.11702e6 0.213421 0.106710 0.994290i $$-0.465968\pi$$
0.106710 + 0.994290i $$0.465968\pi$$
$$488$$ 2.35277e6 0.447229
$$489$$ −4.28029e6 −0.809471
$$490$$ −249704. −0.0469824
$$491$$ 1.34458e6 0.251699 0.125850 0.992049i $$-0.459834\pi$$
0.125850 + 0.992049i $$0.459834\pi$$
$$492$$ 1.51805e6 0.282731
$$493$$ −7.83723e6 −1.45226
$$494$$ −300192. −0.0553454
$$495$$ 1.39838e6 0.256515
$$496$$ −1.82477e6 −0.333045
$$497$$ 445508. 0.0809030
$$498$$ 231408. 0.0418124
$$499$$ −6.54648e6 −1.17695 −0.588473 0.808517i $$-0.700271\pi$$
−0.588473 + 0.808517i $$0.700271\pi$$
$$500$$ −2.31878e6 −0.414797
$$501$$ 1.08202e6 0.192592
$$502$$ 1.94558e6 0.344580
$$503$$ −8.22050e6 −1.44870 −0.724350 0.689432i $$-0.757860\pi$$
−0.724350 + 0.689432i $$0.757860\pi$$
$$504$$ 254016. 0.0445435
$$505$$ −957164. −0.167016
$$506$$ −5.87507e6 −1.02009
$$507$$ −2.43152e6 −0.420105
$$508$$ 2.96640e6 0.510000
$$509$$ −5.11045e6 −0.874308 −0.437154 0.899387i $$-0.644013\pi$$
−0.437154 + 0.899387i $$0.644013\pi$$
$$510$$ −1.48075e6 −0.252091
$$511$$ 3.59925e6 0.609760
$$512$$ −262144. −0.0441942
$$513$$ 172044. 0.0288633
$$514$$ 4.15639e6 0.693919
$$515$$ 2.71773e6 0.451531
$$516$$ −1.21709e6 −0.201232
$$517$$ 3.55373e6 0.584733
$$518$$ 854168. 0.139868
$$519$$ 4.57987e6 0.746336
$$520$$ −529152. −0.0858168
$$521$$ 9.69999e6 1.56559 0.782793 0.622282i $$-0.213794\pi$$
0.782793 + 0.622282i $$0.213794\pi$$
$$522$$ 1.60510e6 0.257825
$$523$$ −3.17295e6 −0.507234 −0.253617 0.967305i $$-0.581620\pi$$
−0.253617 + 0.967305i $$0.581620\pi$$
$$524$$ 1.03251e6 0.164273
$$525$$ 1.08001e6 0.171013
$$526$$ −5.40418e6 −0.851658
$$527$$ −1.12765e7 −1.76867
$$528$$ 1.52986e6 0.238817
$$529$$ −1.54340e6 −0.239794
$$530$$ 3.46882e6 0.536403
$$531$$ −1.25032e6 −0.192435
$$532$$ −185024. −0.0283432
$$533$$ 3.35236e6 0.511131
$$534$$ 4.41569e6 0.670109
$$535$$ 5.57544e6 0.842160
$$536$$ −2.62221e6 −0.394235
$$537$$ 4.38804e6 0.656651
$$538$$ 4.47244e6 0.666176
$$539$$ 1.59426e6 0.236368
$$540$$ 303264. 0.0447545
$$541$$ −6.62575e6 −0.973289 −0.486644 0.873600i $$-0.661779\pi$$
−0.486644 + 0.873600i $$0.661779\pi$$
$$542$$ 760416. 0.111187
$$543$$ −4.89969e6 −0.713131
$$544$$ −1.61997e6 −0.234698
$$545$$ 748748. 0.107980
$$546$$ 560952. 0.0805275
$$547$$ 3.84707e6 0.549745 0.274873 0.961481i $$-0.411364\pi$$
0.274873 + 0.961481i $$0.411364\pi$$
$$548$$ 2.44688e6 0.348066
$$549$$ −2.97772e6 −0.421651
$$550$$ 6.50454e6 0.916875
$$551$$ −1.16914e6 −0.164055
$$552$$ −1.27411e6 −0.177975
$$553$$ −4.38060e6 −0.609145
$$554$$ 802024. 0.111023
$$555$$ 1.01977e6 0.140531
$$556$$ −5.49536e6 −0.753892
$$557$$ 5.00176e6 0.683101 0.341550 0.939863i $$-0.389048\pi$$
0.341550 + 0.939863i $$0.389048\pi$$
$$558$$ 2.30947e6 0.313998
$$559$$ −2.68774e6 −0.363795
$$560$$ −326144. −0.0439480
$$561$$ 9.45403e6 1.26826
$$562$$ −4.36948e6 −0.583565
$$563$$ 2.27772e6 0.302852 0.151426 0.988469i $$-0.451614\pi$$
0.151426 + 0.988469i $$0.451614\pi$$
$$564$$ 770688. 0.102019
$$565$$ −1.45636e6 −0.191933
$$566$$ −7.25032e6 −0.951297
$$567$$ −321489. −0.0419961
$$568$$ 581888. 0.0756778
$$569$$ 8.86979e6 1.14850 0.574252 0.818678i $$-0.305293\pi$$
0.574252 + 0.818678i $$0.305293\pi$$
$$570$$ −220896. −0.0284774
$$571$$ 1.40102e7 1.79826 0.899132 0.437678i $$-0.144199\pi$$
0.899132 + 0.437678i $$0.144199\pi$$
$$572$$ 3.37843e6 0.431743
$$573$$ 3.38764e6 0.431033
$$574$$ 2.06623e6 0.261758
$$575$$ −5.41719e6 −0.683289
$$576$$ 331776. 0.0416667
$$577$$ 8.75327e6 1.09454 0.547269 0.836957i $$-0.315668\pi$$
0.547269 + 0.836957i $$0.315668\pi$$
$$578$$ −4.33147e6 −0.539281
$$579$$ 7.60451e6 0.942703
$$580$$ −2.06086e6 −0.254378
$$581$$ 314972. 0.0387108
$$582$$ −769320. −0.0941455
$$583$$ −2.21471e7 −2.69864
$$584$$ 4.70106e6 0.570379
$$585$$ 669708. 0.0809088
$$586$$ −8.40122e6 −1.01064
$$587$$ −1.06117e7 −1.27113 −0.635564 0.772048i $$-0.719232\pi$$
−0.635564 + 0.772048i $$0.719232\pi$$
$$588$$ 345744. 0.0412393
$$589$$ −1.68221e6 −0.199798
$$590$$ 1.60534e6 0.189862
$$591$$ −4.43515e6 −0.522323
$$592$$ 1.11565e6 0.130835
$$593$$ 1.88552e6 0.220188 0.110094 0.993921i $$-0.464885\pi$$
0.110094 + 0.993921i $$0.464885\pi$$
$$594$$ −1.93622e6 −0.225159
$$595$$ −2.01547e6 −0.233391
$$596$$ −2.79773e6 −0.322619
$$597$$ −8.23298e6 −0.945413
$$598$$ −2.81366e6 −0.321751
$$599$$ 1.27256e7 1.44915 0.724573 0.689198i $$-0.242037\pi$$
0.724573 + 0.689198i $$0.242037\pi$$
$$600$$ 1.41062e6 0.159968
$$601$$ 7.18846e6 0.811801 0.405900 0.913917i $$-0.366958\pi$$
0.405900 + 0.913917i $$0.366958\pi$$
$$602$$ −1.65659e6 −0.186305
$$603$$ 3.31873e6 0.371688
$$604$$ −7.24083e6 −0.807600
$$605$$ 7.27597e6 0.808170
$$606$$ 1.32530e6 0.146600
$$607$$ 1.08494e7 1.19519 0.597593 0.801800i $$-0.296124\pi$$
0.597593 + 0.801800i $$0.296124\pi$$
$$608$$ −241664. −0.0265126
$$609$$ 2.18471e6 0.238699
$$610$$ 3.82325e6 0.416014
$$611$$ 1.70194e6 0.184434
$$612$$ 2.05027e6 0.221275
$$613$$ −4.90511e6 −0.527227 −0.263614 0.964628i $$-0.584914\pi$$
−0.263614 + 0.964628i $$0.584914\pi$$
$$614$$ 6.56418e6 0.702683
$$615$$ 2.46683e6 0.262997
$$616$$ 2.08230e6 0.221102
$$617$$ 2.58445e6 0.273310 0.136655 0.990619i $$-0.456365\pi$$
0.136655 + 0.990619i $$0.456365\pi$$
$$618$$ −3.76301e6 −0.396336
$$619$$ −4.99336e6 −0.523801 −0.261901 0.965095i $$-0.584349\pi$$
−0.261901 + 0.965095i $$0.584349\pi$$
$$620$$ −2.96525e6 −0.309800
$$621$$ 1.61255e6 0.167797
$$622$$ 3.78093e6 0.391852
$$623$$ 6.01024e6 0.620400
$$624$$ 732672. 0.0753266
$$625$$ 3.88510e6 0.397834
$$626$$ −1.66142e6 −0.169450
$$627$$ 1.41034e6 0.143269
$$628$$ −7.98506e6 −0.807940
$$629$$ 6.89436e6 0.694812
$$630$$ 412776. 0.0414346
$$631$$ −1.18219e7 −1.18199 −0.590997 0.806674i $$-0.701265\pi$$
−0.590997 + 0.806674i $$0.701265\pi$$
$$632$$ −5.72160e6 −0.569803
$$633$$ 2.80602e6 0.278344
$$634$$ −4.73926e6 −0.468260
$$635$$ 4.82040e6 0.474404
$$636$$ −4.80298e6 −0.470834
$$637$$ 763518. 0.0745540
$$638$$ 1.31578e7 1.27977
$$639$$ −736452. −0.0713497
$$640$$ −425984. −0.0411096
$$641$$ −5.47007e6 −0.525833 −0.262916 0.964819i $$-0.584684\pi$$
−0.262916 + 0.964819i $$0.584684\pi$$
$$642$$ −7.71984e6 −0.739215
$$643$$ 9.64934e6 0.920386 0.460193 0.887819i $$-0.347780\pi$$
0.460193 + 0.887819i $$0.347780\pi$$
$$644$$ −1.73421e6 −0.164773
$$645$$ −1.97777e6 −0.187187
$$646$$ −1.49341e6 −0.140798
$$647$$ 292368. 0.0274580 0.0137290 0.999906i $$-0.495630\pi$$
0.0137290 + 0.999906i $$0.495630\pi$$
$$648$$ −419904. −0.0392837
$$649$$ −1.02495e7 −0.955193
$$650$$ 3.11513e6 0.289196
$$651$$ 3.14345e6 0.290706
$$652$$ −7.60941e6 −0.701022
$$653$$ 6.94081e6 0.636982 0.318491 0.947926i $$-0.396824\pi$$
0.318491 + 0.947926i $$0.396824\pi$$
$$654$$ −1.03673e6 −0.0947808
$$655$$ 1.67783e6 0.152808
$$656$$ 2.69875e6 0.244852
$$657$$ −5.94977e6 −0.537758
$$658$$ 1.04899e6 0.0944512
$$659$$ −1.32912e7 −1.19221 −0.596104 0.802908i $$-0.703285\pi$$
−0.596104 + 0.802908i $$0.703285\pi$$
$$660$$ 2.48602e6 0.222149
$$661$$ 2.05219e6 0.182690 0.0913448 0.995819i $$-0.470883\pi$$
0.0913448 + 0.995819i $$0.470883\pi$$
$$662$$ −5.48619e6 −0.486548
$$663$$ 4.52768e6 0.400030
$$664$$ 411392. 0.0362106
$$665$$ −300664. −0.0263650
$$666$$ −1.41199e6 −0.123352
$$667$$ −1.09582e7 −0.953732
$$668$$ 1.92358e6 0.166790
$$669$$ −1.15898e7 −1.00118
$$670$$ −4.26109e6 −0.366719
$$671$$ −2.44100e7 −2.09296
$$672$$ 451584. 0.0385758
$$673$$ −1.57039e7 −1.33650 −0.668252 0.743935i $$-0.732957\pi$$
−0.668252 + 0.743935i $$0.732957\pi$$
$$674$$ −3.85409e6 −0.326792
$$675$$ −1.78532e6 −0.150819
$$676$$ −4.32270e6 −0.363822
$$677$$ −969534. −0.0813002 −0.0406501 0.999173i $$-0.512943\pi$$
−0.0406501 + 0.999173i $$0.512943\pi$$
$$678$$ 2.01650e6 0.168471
$$679$$ −1.04713e6 −0.0871618
$$680$$ −2.63245e6 −0.218317
$$681$$ 1.16015e7 0.958617
$$682$$ 1.89320e7 1.55860
$$683$$ −1.49908e7 −1.22962 −0.614812 0.788673i $$-0.710768\pi$$
−0.614812 + 0.788673i $$0.710768\pi$$
$$684$$ 305856. 0.0249964
$$685$$ 3.97618e6 0.323772
$$686$$ 470596. 0.0381802
$$687$$ 6.10393e6 0.493421
$$688$$ −2.16371e6 −0.174272
$$689$$ −1.06066e7 −0.851191
$$690$$ −2.07043e6 −0.165553
$$691$$ −7.16038e6 −0.570481 −0.285240 0.958456i $$-0.592073\pi$$
−0.285240 + 0.958456i $$0.592073\pi$$
$$692$$ 8.14198e6 0.646346
$$693$$ −2.63542e6 −0.208457
$$694$$ −1.03092e7 −0.812509
$$695$$ −8.92996e6 −0.701274
$$696$$ 2.85350e6 0.223283
$$697$$ 1.66774e7 1.30031
$$698$$ 1.22701e7 0.953253
$$699$$ −1.00558e7 −0.778437
$$700$$ 1.92002e6 0.148102
$$701$$ −91834.0 −0.00705844 −0.00352922 0.999994i $$-0.501123\pi$$
−0.00352922 + 0.999994i $$0.501123\pi$$
$$702$$ −927288. −0.0710186
$$703$$ 1.02849e6 0.0784894
$$704$$ 2.71974e6 0.206822
$$705$$ 1.25237e6 0.0948985
$$706$$ 1.24058e7 0.936725
$$707$$ 1.80389e6 0.135725
$$708$$ −2.22278e6 −0.166653
$$709$$ 2.20981e7 1.65097 0.825487 0.564422i $$-0.190901\pi$$
0.825487 + 0.564422i $$0.190901\pi$$
$$710$$ 945568. 0.0703958
$$711$$ 7.24140e6 0.537216
$$712$$ 7.85011e6 0.580331
$$713$$ −1.57671e7 −1.16153
$$714$$ 2.79065e6 0.204861
$$715$$ 5.48995e6 0.401609
$$716$$ 7.80096e6 0.568677
$$717$$ −1.13576e7 −0.825066
$$718$$ 1.31003e6 0.0948355
$$719$$ 1.58388e7 1.14262 0.571308 0.820736i $$-0.306436\pi$$
0.571308 + 0.820736i $$0.306436\pi$$
$$720$$ 539136. 0.0387585
$$721$$ −5.12187e6 −0.366936
$$722$$ 9.68161e6 0.691202
$$723$$ 8.53396e6 0.607163
$$724$$ −8.71056e6 −0.617589
$$725$$ 1.21323e7 0.857235
$$726$$ −1.00744e7 −0.709379
$$727$$ 6.31418e6 0.443078 0.221539 0.975151i $$-0.428892\pi$$
0.221539 + 0.975151i $$0.428892\pi$$
$$728$$ 997248. 0.0697388
$$729$$ 531441. 0.0370370
$$730$$ 7.63922e6 0.530569
$$731$$ −1.33711e7 −0.925492
$$732$$ −5.29373e6 −0.365161
$$733$$ 6.93003e6 0.476404 0.238202 0.971216i $$-0.423442\pi$$
0.238202 + 0.971216i $$0.423442\pi$$
$$734$$ 1.14696e7 0.785791
$$735$$ 561834. 0.0383610
$$736$$ −2.26509e6 −0.154131
$$737$$ 2.72054e7 1.84496
$$738$$ −3.41561e6 −0.230849
$$739$$ 1.42331e7 0.958714 0.479357 0.877620i $$-0.340870\pi$$
0.479357 + 0.877620i $$0.340870\pi$$
$$740$$ 1.81293e6 0.121703
$$741$$ 675432. 0.0451894
$$742$$ −6.53738e6 −0.435907
$$743$$ −5.94460e6 −0.395048 −0.197524 0.980298i $$-0.563290\pi$$
−0.197524 + 0.980298i $$0.563290\pi$$
$$744$$ 4.10573e6 0.271930
$$745$$ −4.54631e6 −0.300102
$$746$$ −1.43212e7 −0.942175
$$747$$ −520668. −0.0341397
$$748$$ 1.68072e7 1.09835
$$749$$ −1.05076e7 −0.684380
$$750$$ 5.21726e6 0.338680
$$751$$ −682752. −0.0441736 −0.0220868 0.999756i $$-0.507031\pi$$
−0.0220868 + 0.999756i $$0.507031\pi$$
$$752$$ 1.37011e6 0.0883510
$$753$$ −4.37756e6 −0.281349
$$754$$ 6.30149e6 0.403659
$$755$$ −1.17664e7 −0.751233
$$756$$ −571536. −0.0363696
$$757$$ 1.46333e7 0.928116 0.464058 0.885805i $$-0.346393\pi$$
0.464058 + 0.885805i $$0.346393\pi$$
$$758$$ −6.56939e6 −0.415291
$$759$$ 1.32189e7 0.832897
$$760$$ −392704. −0.0246622
$$761$$ −1.16367e7 −0.728399 −0.364200 0.931321i $$-0.618657\pi$$
−0.364200 + 0.931321i $$0.618657\pi$$
$$762$$ −6.67440e6 −0.416414
$$763$$ −1.41110e6 −0.0877500
$$764$$ 6.02246e6 0.373285
$$765$$ 3.33169e6 0.205831
$$766$$ 8.22790e6 0.506661
$$767$$ −4.90865e6 −0.301282
$$768$$ 589824. 0.0360844
$$769$$ 1.91472e7 1.16759 0.583793 0.811902i $$-0.301568\pi$$
0.583793 + 0.811902i $$0.301568\pi$$
$$770$$ 3.38374e6 0.205670
$$771$$ −9.35188e6 −0.566582
$$772$$ 1.35191e7 0.816405
$$773$$ −5.39261e6 −0.324601 −0.162301 0.986741i $$-0.551891\pi$$
−0.162301 + 0.986741i $$0.551891\pi$$
$$774$$ 2.73845e6 0.164305
$$775$$ 1.74565e7 1.04400
$$776$$ −1.36768e6 −0.0815324
$$777$$ −1.92188e6 −0.114202
$$778$$ −2.46457e6 −0.145979
$$779$$ 2.48791e6 0.146890
$$780$$ 1.19059e6 0.0700691
$$781$$ −6.03709e6 −0.354160
$$782$$ −1.39975e7 −0.818530
$$783$$ −3.61147e6 −0.210513
$$784$$ 614656. 0.0357143
$$785$$ −1.29757e7 −0.751549
$$786$$ −2.32315e6 −0.134129
$$787$$ 3.04348e6 0.175159 0.0875796 0.996158i $$-0.472087\pi$$
0.0875796 + 0.996158i $$0.472087\pi$$
$$788$$ −7.88470e6 −0.452345
$$789$$ 1.21594e7 0.695376
$$790$$ −9.29760e6 −0.530033
$$791$$ 2.74469e6 0.155974
$$792$$ −3.44218e6 −0.194993
$$793$$ −1.16903e7 −0.660151
$$794$$ −8.76847e6 −0.493597
$$795$$ −7.80484e6 −0.437972
$$796$$ −1.46364e7 −0.818751
$$797$$ 2.29652e7 1.28063 0.640316 0.768111i $$-0.278803\pi$$
0.640316 + 0.768111i $$0.278803\pi$$
$$798$$ 416304. 0.0231421
$$799$$ 8.46686e6 0.469197
$$800$$ 2.50778e6 0.138536
$$801$$ −9.93530e6 −0.547141
$$802$$ −1.31382e7 −0.721271
$$803$$ −4.87735e7 −2.66928
$$804$$ 5.89997e6 0.321892
$$805$$ −2.81809e6 −0.153273
$$806$$ 9.06682e6 0.491606
$$807$$ −1.00630e7 −0.543930
$$808$$ 2.35610e6 0.126959
$$809$$ 1.90787e7 1.02489 0.512445 0.858720i $$-0.328740\pi$$
0.512445 + 0.858720i $$0.328740\pi$$
$$810$$ −682344. −0.0365419
$$811$$ 1.09414e7 0.584147 0.292074 0.956396i $$-0.405655\pi$$
0.292074 + 0.956396i $$0.405655\pi$$
$$812$$ 3.88394e6 0.206720
$$813$$ −1.71094e6 −0.0907836
$$814$$ −1.15748e7 −0.612286
$$815$$ −1.23653e7 −0.652094
$$816$$ 3.64493e6 0.191630
$$817$$ −1.99467e6 −0.104548
$$818$$ 1.44488e7 0.755001
$$819$$ −1.26214e6 −0.0657504
$$820$$ 4.38547e6 0.227762
$$821$$ 2.12594e7 1.10076 0.550380 0.834914i $$-0.314483\pi$$
0.550380 + 0.834914i $$0.314483\pi$$
$$822$$ −5.50548e6 −0.284194
$$823$$ −1.42256e7 −0.732103 −0.366052 0.930595i $$-0.619291\pi$$
−0.366052 + 0.930595i $$0.619291\pi$$
$$824$$ −6.68979e6 −0.343237
$$825$$ −1.46352e7 −0.748625
$$826$$ −3.02546e6 −0.154291
$$827$$ 2.76103e6 0.140381 0.0701904 0.997534i $$-0.477639\pi$$
0.0701904 + 0.997534i $$0.477639\pi$$
$$828$$ 2.86675e6 0.145316
$$829$$ −3.82147e7 −1.93127 −0.965637 0.259895i $$-0.916312\pi$$
−0.965637 + 0.259895i $$0.916312\pi$$
$$830$$ 668512. 0.0336832
$$831$$ −1.80455e6 −0.0906499
$$832$$ 1.30253e6 0.0652347
$$833$$ 3.79838e6 0.189665
$$834$$ 1.23646e7 0.615550
$$835$$ 3.12582e6 0.155149
$$836$$ 2.50726e6 0.124075
$$837$$ −5.19631e6 −0.256378
$$838$$ −2.16596e7 −1.06547
$$839$$ 1.06044e7 0.520094 0.260047 0.965596i $$-0.416262\pi$$
0.260047 + 0.965596i $$0.416262\pi$$
$$840$$ 733824. 0.0358834
$$841$$ 4.03097e6 0.196526
$$842$$ −1.44252e7 −0.701198
$$843$$ 9.83133e6 0.476479
$$844$$ 4.98848e6 0.241053
$$845$$ −7.02439e6 −0.338429
$$846$$ −1.73405e6 −0.0832981
$$847$$ −1.37124e7 −0.656758
$$848$$ −8.53862e6 −0.407754
$$849$$ 1.63132e7 0.776731
$$850$$ 1.54973e7 0.735712
$$851$$ 9.63990e6 0.456298
$$852$$ −1.30925e6 −0.0617907
$$853$$ −4.07009e7 −1.91527 −0.957637 0.287977i $$-0.907017\pi$$
−0.957637 + 0.287977i $$0.907017\pi$$
$$854$$ −7.20535e6 −0.338073
$$855$$ 497016. 0.0232517
$$856$$ −1.37242e7 −0.640179
$$857$$ −3.10120e7 −1.44237 −0.721187 0.692741i $$-0.756403\pi$$
−0.721187 + 0.692741i $$0.756403\pi$$
$$858$$ −7.60147e6 −0.352517
$$859$$ 1.09104e7 0.504495 0.252247 0.967663i $$-0.418830\pi$$
0.252247 + 0.967663i $$0.418830\pi$$
$$860$$ −3.51603e6 −0.162109
$$861$$ −4.64902e6 −0.213724
$$862$$ 1.11286e7 0.510118
$$863$$ 1.04089e7 0.475751 0.237875 0.971296i $$-0.423549\pi$$
0.237875 + 0.971296i $$0.423549\pi$$
$$864$$ −746496. −0.0340207
$$865$$ 1.32307e7 0.601234
$$866$$ −2.51048e7 −1.13753
$$867$$ 9.74580e6 0.440321
$$868$$ 5.58835e6 0.251759
$$869$$ 5.93616e7 2.66659
$$870$$ 4.63694e6 0.207699
$$871$$ 1.30291e7 0.581928
$$872$$ −1.84307e6 −0.0820826
$$873$$ 1.73097e6 0.0768695
$$874$$ −2.08813e6 −0.0924652
$$875$$ 7.10128e6 0.313557
$$876$$ −1.05774e7 −0.465712
$$877$$ 1.64064e7 0.720299 0.360150 0.932895i $$-0.382726\pi$$
0.360150 + 0.932895i $$0.382726\pi$$
$$878$$ −2.56637e6 −0.112352
$$879$$ 1.89028e7 0.825188
$$880$$ 4.41958e6 0.192387
$$881$$ 1.48577e7 0.644927 0.322464 0.946582i $$-0.395489\pi$$
0.322464 + 0.946582i $$0.395489\pi$$
$$882$$ −777924. −0.0336718
$$883$$ −2.72018e7 −1.17407 −0.587037 0.809560i $$-0.699706\pi$$
−0.587037 + 0.809560i $$0.699706\pi$$
$$884$$ 8.04922e6 0.346436
$$885$$ −3.61202e6 −0.155022
$$886$$ −2.42218e7 −1.03663
$$887$$ 2.71242e7 1.15757 0.578785 0.815480i $$-0.303527\pi$$
0.578785 + 0.815480i $$0.303527\pi$$
$$888$$ −2.51021e6 −0.106826
$$889$$ −9.08460e6 −0.385524
$$890$$ 1.27564e7 0.539827
$$891$$ 4.35650e6 0.183842
$$892$$ −2.06042e7 −0.867047
$$893$$ 1.26307e6 0.0530029
$$894$$ 6.29489e6 0.263417
$$895$$ 1.26766e7 0.528986
$$896$$ 802816. 0.0334077
$$897$$ 6.33074e6 0.262708
$$898$$ 2.06673e7 0.855248
$$899$$ 3.53121e7 1.45722
$$900$$ −3.17390e6 −0.130613
$$901$$ −5.27660e7 −2.16542
$$902$$ −2.79996e7 −1.14587
$$903$$ 3.72733e6 0.152117
$$904$$ 3.58490e6 0.145900
$$905$$ −1.41547e7 −0.574484
$$906$$ 1.62919e7 0.659402
$$907$$ −8.42269e6 −0.339964 −0.169982 0.985447i $$-0.554371\pi$$
−0.169982 + 0.985447i $$0.554371\pi$$
$$908$$ 2.06248e7 0.830187
$$909$$ −2.98193e6 −0.119698
$$910$$ 1.62053e6 0.0648714
$$911$$ 3.08637e7 1.23212 0.616060 0.787700i $$-0.288728\pi$$
0.616060 + 0.787700i $$0.288728\pi$$
$$912$$ 543744. 0.0216475
$$913$$ −4.26819e6 −0.169460
$$914$$ 911192. 0.0360782
$$915$$ −8.60231e6 −0.339674
$$916$$ 1.08514e7 0.427315
$$917$$ −3.16207e6 −0.124179
$$918$$ −4.61311e6 −0.180671
$$919$$ 4.93895e6 0.192906 0.0964531 0.995338i $$-0.469250\pi$$
0.0964531 + 0.995338i $$0.469250\pi$$
$$920$$ −3.68077e6 −0.143373
$$921$$ −1.47694e7 −0.573738
$$922$$ −2.34058e6 −0.0906770
$$923$$ −2.89126e6 −0.111707
$$924$$ −4.68518e6 −0.180529
$$925$$ −1.06727e7 −0.410130
$$926$$ 1.36581e7 0.523437
$$927$$ 8.46677e6 0.323607
$$928$$ 5.07290e6 0.193369
$$929$$ 5.62575e6 0.213866 0.106933 0.994266i $$-0.465897\pi$$
0.106933 + 0.994266i $$0.465897\pi$$
$$930$$ 6.67181e6 0.252951
$$931$$ 566636. 0.0214255
$$932$$ −1.78770e7 −0.674146
$$933$$ −8.50709e6 −0.319946
$$934$$ −2.86520e6 −0.107470
$$935$$ 2.73116e7 1.02169
$$936$$ −1.64851e6 −0.0615039
$$937$$ 2.60073e7 0.967714 0.483857 0.875147i $$-0.339236\pi$$
0.483857 + 0.875147i $$0.339236\pi$$
$$938$$ 8.03051e6 0.298014
$$939$$ 3.73819e6 0.138356
$$940$$ 2.22643e6 0.0821845
$$941$$ 3.02160e6 0.111241 0.0556203 0.998452i $$-0.482286\pi$$
0.0556203 + 0.998452i $$0.482286\pi$$
$$942$$ 1.79664e7 0.659680
$$943$$ 2.33189e7 0.853943
$$944$$ −3.95162e6 −0.144326
$$945$$ −928746. −0.0338312
$$946$$ 2.24485e7 0.815567
$$947$$ −3.48282e7 −1.26199 −0.630995 0.775787i $$-0.717353\pi$$
−0.630995 + 0.775787i $$0.717353\pi$$
$$948$$ 1.28736e7 0.465242
$$949$$ −2.33584e7 −0.841932
$$950$$ 2.31186e6 0.0831097
$$951$$ 1.06633e7 0.382333
$$952$$ 4.96115e6 0.177415
$$953$$ −9.39009e6 −0.334917 −0.167459 0.985879i $$-0.553556\pi$$
−0.167459 + 0.985879i $$0.553556\pi$$
$$954$$ 1.08067e7 0.384434
$$955$$ 9.78650e6 0.347232
$$956$$ −2.01913e7 −0.714528
$$957$$ −2.96051e7 −1.04493
$$958$$ −2.09637e7 −0.737996
$$959$$ −7.49357e6 −0.263113
$$960$$ 958464. 0.0335659
$$961$$ 2.21792e7 0.774708
$$962$$ −5.54338e6 −0.193124
$$963$$ 1.73696e7 0.603566
$$964$$ 1.51715e7 0.525818
$$965$$ 2.19686e7 0.759423
$$966$$ 3.90197e6 0.134537
$$967$$ 1.44768e7 0.497860 0.248930 0.968521i $$-0.419921\pi$$
0.248930 + 0.968521i $$0.419921\pi$$
$$968$$ −1.79101e7 −0.614340
$$969$$ 3.36017e6 0.114961
$$970$$ −2.22248e6 −0.0758418
$$971$$ 9.24976e6 0.314834 0.157417 0.987532i $$-0.449683\pi$$
0.157417 + 0.987532i $$0.449683\pi$$
$$972$$ 944784. 0.0320750
$$973$$ 1.68295e7 0.569889
$$974$$ −4.46806e6 −0.150911
$$975$$ −7.00904e6 −0.236128
$$976$$ −9.41107e6 −0.316238
$$977$$ −4.97780e7 −1.66840 −0.834202 0.551459i $$-0.814071\pi$$
−0.834202 + 0.551459i $$0.814071\pi$$
$$978$$ 1.71212e7 0.572382
$$979$$ −8.14449e7 −2.71586
$$980$$ 998816. 0.0332216
$$981$$ 2.33264e6 0.0773882
$$982$$ −5.37830e6 −0.177978
$$983$$ −8.95601e6 −0.295618 −0.147809 0.989016i $$-0.547222\pi$$
−0.147809 + 0.989016i $$0.547222\pi$$
$$984$$ −6.07219e6 −0.199921
$$985$$ −1.28126e7 −0.420773
$$986$$ 3.13489e7 1.02690
$$987$$ −2.36023e6 −0.0771191
$$988$$ 1.20077e6 0.0391351
$$989$$ −1.86958e7 −0.607790
$$990$$ −5.59354e6 −0.181384
$$991$$ 2.62400e7 0.848751 0.424376 0.905486i $$-0.360494\pi$$
0.424376 + 0.905486i $$0.360494\pi$$
$$992$$ 7.29907e6 0.235499
$$993$$ 1.23439e7 0.397265
$$994$$ −1.78203e6 −0.0572070
$$995$$ −2.37842e7 −0.761606
$$996$$ −925632. −0.0295658
$$997$$ 2.80506e7 0.893727 0.446863 0.894602i $$-0.352541\pi$$
0.446863 + 0.894602i $$0.352541\pi$$
$$998$$ 2.61859e7 0.832226
$$999$$ 3.17698e6 0.100717
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 42.6.a.d.1.1 1
3.2 odd 2 126.6.a.i.1.1 1
4.3 odd 2 336.6.a.h.1.1 1
5.2 odd 4 1050.6.g.i.799.1 2
5.3 odd 4 1050.6.g.i.799.2 2
5.4 even 2 1050.6.a.k.1.1 1
7.2 even 3 294.6.e.i.67.1 2
7.3 odd 6 294.6.e.p.79.1 2
7.4 even 3 294.6.e.i.79.1 2
7.5 odd 6 294.6.e.p.67.1 2
7.6 odd 2 294.6.a.b.1.1 1
12.11 even 2 1008.6.a.j.1.1 1
21.20 even 2 882.6.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.a.d.1.1 1 1.1 even 1 trivial
126.6.a.i.1.1 1 3.2 odd 2
294.6.a.b.1.1 1 7.6 odd 2
294.6.e.i.67.1 2 7.2 even 3
294.6.e.i.79.1 2 7.4 even 3
294.6.e.p.67.1 2 7.5 odd 6
294.6.e.p.79.1 2 7.3 odd 6
336.6.a.h.1.1 1 4.3 odd 2
882.6.a.s.1.1 1 21.20 even 2
1008.6.a.j.1.1 1 12.11 even 2
1050.6.a.k.1.1 1 5.4 even 2
1050.6.g.i.799.1 2 5.2 odd 4
1050.6.g.i.799.2 2 5.3 odd 4