# Properties

 Label 126.6.a.f.1.1 Level $126$ Weight $6$ Character 126.1 Self dual yes Analytic conductor $20.208$ 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] = [126,6,Mod(1,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, 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("126.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 126.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: $$20.2083612964$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 126.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} +16.0000 q^{4} -84.0000 q^{5} +49.0000 q^{7} +64.0000 q^{8} +O(q^{10})$$ $$q+4.00000 q^{2} +16.0000 q^{4} -84.0000 q^{5} +49.0000 q^{7} +64.0000 q^{8} -336.000 q^{10} +336.000 q^{11} +584.000 q^{13} +196.000 q^{14} +256.000 q^{16} +1458.00 q^{17} +470.000 q^{19} -1344.00 q^{20} +1344.00 q^{22} +4200.00 q^{23} +3931.00 q^{25} +2336.00 q^{26} +784.000 q^{28} -4866.00 q^{29} -7372.00 q^{31} +1024.00 q^{32} +5832.00 q^{34} -4116.00 q^{35} +14330.0 q^{37} +1880.00 q^{38} -5376.00 q^{40} -6222.00 q^{41} +3704.00 q^{43} +5376.00 q^{44} +16800.0 q^{46} +1812.00 q^{47} +2401.00 q^{49} +15724.0 q^{50} +9344.00 q^{52} +37242.0 q^{53} -28224.0 q^{55} +3136.00 q^{56} -19464.0 q^{58} -34302.0 q^{59} +24476.0 q^{61} -29488.0 q^{62} +4096.00 q^{64} -49056.0 q^{65} -17452.0 q^{67} +23328.0 q^{68} -16464.0 q^{70} -28224.0 q^{71} +3602.00 q^{73} +57320.0 q^{74} +7520.00 q^{76} +16464.0 q^{77} +42872.0 q^{79} -21504.0 q^{80} -24888.0 q^{82} +35202.0 q^{83} -122472. q^{85} +14816.0 q^{86} +21504.0 q^{88} -26730.0 q^{89} +28616.0 q^{91} +67200.0 q^{92} +7248.00 q^{94} -39480.0 q^{95} -16978.0 q^{97} +9604.00 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 16.0000 0.500000
$$5$$ −84.0000 −1.50264 −0.751319 0.659939i $$-0.770582\pi$$
−0.751319 + 0.659939i $$0.770582\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 64.0000 0.353553
$$9$$ 0 0
$$10$$ −336.000 −1.06253
$$11$$ 336.000 0.837255 0.418627 0.908158i $$-0.362511\pi$$
0.418627 + 0.908158i $$0.362511\pi$$
$$12$$ 0 0
$$13$$ 584.000 0.958417 0.479208 0.877701i $$-0.340924\pi$$
0.479208 + 0.877701i $$0.340924\pi$$
$$14$$ 196.000 0.267261
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 1458.00 1.22359 0.611794 0.791017i $$-0.290448\pi$$
0.611794 + 0.791017i $$0.290448\pi$$
$$18$$ 0 0
$$19$$ 470.000 0.298685 0.149343 0.988786i $$-0.452284\pi$$
0.149343 + 0.988786i $$0.452284\pi$$
$$20$$ −1344.00 −0.751319
$$21$$ 0 0
$$22$$ 1344.00 0.592028
$$23$$ 4200.00 1.65550 0.827751 0.561096i $$-0.189620\pi$$
0.827751 + 0.561096i $$0.189620\pi$$
$$24$$ 0 0
$$25$$ 3931.00 1.25792
$$26$$ 2336.00 0.677703
$$27$$ 0 0
$$28$$ 784.000 0.188982
$$29$$ −4866.00 −1.07443 −0.537214 0.843446i $$-0.680523\pi$$
−0.537214 + 0.843446i $$0.680523\pi$$
$$30$$ 0 0
$$31$$ −7372.00 −1.37778 −0.688892 0.724864i $$-0.741903\pi$$
−0.688892 + 0.724864i $$0.741903\pi$$
$$32$$ 1024.00 0.176777
$$33$$ 0 0
$$34$$ 5832.00 0.865207
$$35$$ −4116.00 −0.567944
$$36$$ 0 0
$$37$$ 14330.0 1.72085 0.860423 0.509581i $$-0.170200\pi$$
0.860423 + 0.509581i $$0.170200\pi$$
$$38$$ 1880.00 0.211202
$$39$$ 0 0
$$40$$ −5376.00 −0.531263
$$41$$ −6222.00 −0.578057 −0.289028 0.957321i $$-0.593332\pi$$
−0.289028 + 0.957321i $$0.593332\pi$$
$$42$$ 0 0
$$43$$ 3704.00 0.305492 0.152746 0.988265i $$-0.451188\pi$$
0.152746 + 0.988265i $$0.451188\pi$$
$$44$$ 5376.00 0.418627
$$45$$ 0 0
$$46$$ 16800.0 1.17062
$$47$$ 1812.00 0.119650 0.0598251 0.998209i $$-0.480946\pi$$
0.0598251 + 0.998209i $$0.480946\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 15724.0 0.889484
$$51$$ 0 0
$$52$$ 9344.00 0.479208
$$53$$ 37242.0 1.82114 0.910570 0.413355i $$-0.135643\pi$$
0.910570 + 0.413355i $$0.135643\pi$$
$$54$$ 0 0
$$55$$ −28224.0 −1.25809
$$56$$ 3136.00 0.133631
$$57$$ 0 0
$$58$$ −19464.0 −0.759735
$$59$$ −34302.0 −1.28289 −0.641445 0.767169i $$-0.721665\pi$$
−0.641445 + 0.767169i $$0.721665\pi$$
$$60$$ 0 0
$$61$$ 24476.0 0.842201 0.421101 0.907014i $$-0.361644\pi$$
0.421101 + 0.907014i $$0.361644\pi$$
$$62$$ −29488.0 −0.974240
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ −49056.0 −1.44015
$$66$$ 0 0
$$67$$ −17452.0 −0.474961 −0.237481 0.971392i $$-0.576322\pi$$
−0.237481 + 0.971392i $$0.576322\pi$$
$$68$$ 23328.0 0.611794
$$69$$ 0 0
$$70$$ −16464.0 −0.401597
$$71$$ −28224.0 −0.664466 −0.332233 0.943197i $$-0.607802\pi$$
−0.332233 + 0.943197i $$0.607802\pi$$
$$72$$ 0 0
$$73$$ 3602.00 0.0791109 0.0395555 0.999217i $$-0.487406\pi$$
0.0395555 + 0.999217i $$0.487406\pi$$
$$74$$ 57320.0 1.21682
$$75$$ 0 0
$$76$$ 7520.00 0.149343
$$77$$ 16464.0 0.316453
$$78$$ 0 0
$$79$$ 42872.0 0.772869 0.386435 0.922317i $$-0.373706\pi$$
0.386435 + 0.922317i $$0.373706\pi$$
$$80$$ −21504.0 −0.375659
$$81$$ 0 0
$$82$$ −24888.0 −0.408748
$$83$$ 35202.0 0.560883 0.280441 0.959871i $$-0.409519\pi$$
0.280441 + 0.959871i $$0.409519\pi$$
$$84$$ 0 0
$$85$$ −122472. −1.83861
$$86$$ 14816.0 0.216015
$$87$$ 0 0
$$88$$ 21504.0 0.296014
$$89$$ −26730.0 −0.357704 −0.178852 0.983876i $$-0.557238\pi$$
−0.178852 + 0.983876i $$0.557238\pi$$
$$90$$ 0 0
$$91$$ 28616.0 0.362248
$$92$$ 67200.0 0.827751
$$93$$ 0 0
$$94$$ 7248.00 0.0846055
$$95$$ −39480.0 −0.448816
$$96$$ 0 0
$$97$$ −16978.0 −0.183213 −0.0916067 0.995795i $$-0.529200\pi$$
−0.0916067 + 0.995795i $$0.529200\pi$$
$$98$$ 9604.00 0.101015
$$99$$ 0 0
$$100$$ 62896.0 0.628960
$$101$$ −99204.0 −0.967667 −0.483833 0.875160i $$-0.660756\pi$$
−0.483833 + 0.875160i $$0.660756\pi$$
$$102$$ 0 0
$$103$$ −131644. −1.22267 −0.611333 0.791373i $$-0.709366\pi$$
−0.611333 + 0.791373i $$0.709366\pi$$
$$104$$ 37376.0 0.338852
$$105$$ 0 0
$$106$$ 148968. 1.28774
$$107$$ −48852.0 −0.412499 −0.206250 0.978499i $$-0.566126\pi$$
−0.206250 + 0.978499i $$0.566126\pi$$
$$108$$ 0 0
$$109$$ −56374.0 −0.454478 −0.227239 0.973839i $$-0.572970\pi$$
−0.227239 + 0.973839i $$0.572970\pi$$
$$110$$ −112896. −0.889604
$$111$$ 0 0
$$112$$ 12544.0 0.0944911
$$113$$ −8742.00 −0.0644043 −0.0322021 0.999481i $$-0.510252\pi$$
−0.0322021 + 0.999481i $$0.510252\pi$$
$$114$$ 0 0
$$115$$ −352800. −2.48762
$$116$$ −77856.0 −0.537214
$$117$$ 0 0
$$118$$ −137208. −0.907140
$$119$$ 71442.0 0.462473
$$120$$ 0 0
$$121$$ −48155.0 −0.299005
$$122$$ 97904.0 0.595526
$$123$$ 0 0
$$124$$ −117952. −0.688892
$$125$$ −67704.0 −0.387560
$$126$$ 0 0
$$127$$ 315992. 1.73847 0.869234 0.494401i $$-0.164612\pi$$
0.869234 + 0.494401i $$0.164612\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 0 0
$$130$$ −196224. −1.01834
$$131$$ 24666.0 0.125580 0.0627900 0.998027i $$-0.480000\pi$$
0.0627900 + 0.998027i $$0.480000\pi$$
$$132$$ 0 0
$$133$$ 23030.0 0.112892
$$134$$ −69808.0 −0.335848
$$135$$ 0 0
$$136$$ 93312.0 0.432604
$$137$$ −303234. −1.38031 −0.690155 0.723662i $$-0.742458\pi$$
−0.690155 + 0.723662i $$0.742458\pi$$
$$138$$ 0 0
$$139$$ 250586. 1.10007 0.550034 0.835142i $$-0.314615\pi$$
0.550034 + 0.835142i $$0.314615\pi$$
$$140$$ −65856.0 −0.283972
$$141$$ 0 0
$$142$$ −112896. −0.469848
$$143$$ 196224. 0.802439
$$144$$ 0 0
$$145$$ 408744. 1.61448
$$146$$ 14408.0 0.0559399
$$147$$ 0 0
$$148$$ 229280. 0.860423
$$149$$ 60594.0 0.223596 0.111798 0.993731i $$-0.464339\pi$$
0.111798 + 0.993731i $$0.464339\pi$$
$$150$$ 0 0
$$151$$ 124448. 0.444166 0.222083 0.975028i $$-0.428714\pi$$
0.222083 + 0.975028i $$0.428714\pi$$
$$152$$ 30080.0 0.105601
$$153$$ 0 0
$$154$$ 65856.0 0.223766
$$155$$ 619248. 2.07031
$$156$$ 0 0
$$157$$ 76040.0 0.246203 0.123101 0.992394i $$-0.460716\pi$$
0.123101 + 0.992394i $$0.460716\pi$$
$$158$$ 171488. 0.546501
$$159$$ 0 0
$$160$$ −86016.0 −0.265631
$$161$$ 205800. 0.625721
$$162$$ 0 0
$$163$$ 124256. 0.366310 0.183155 0.983084i $$-0.441369\pi$$
0.183155 + 0.983084i $$0.441369\pi$$
$$164$$ −99552.0 −0.289028
$$165$$ 0 0
$$166$$ 140808. 0.396604
$$167$$ 72420.0 0.200940 0.100470 0.994940i $$-0.467965\pi$$
0.100470 + 0.994940i $$0.467965\pi$$
$$168$$ 0 0
$$169$$ −30237.0 −0.0814370
$$170$$ −489888. −1.30009
$$171$$ 0 0
$$172$$ 59264.0 0.152746
$$173$$ 441552. 1.12167 0.560837 0.827926i $$-0.310479\pi$$
0.560837 + 0.827926i $$0.310479\pi$$
$$174$$ 0 0
$$175$$ 192619. 0.475449
$$176$$ 86016.0 0.209314
$$177$$ 0 0
$$178$$ −106920. −0.252935
$$179$$ 10692.0 0.0249417 0.0124709 0.999922i $$-0.496030\pi$$
0.0124709 + 0.999922i $$0.496030\pi$$
$$180$$ 0 0
$$181$$ −546064. −1.23893 −0.619465 0.785024i $$-0.712651\pi$$
−0.619465 + 0.785024i $$0.712651\pi$$
$$182$$ 114464. 0.256148
$$183$$ 0 0
$$184$$ 268800. 0.585308
$$185$$ −1.20372e6 −2.58581
$$186$$ 0 0
$$187$$ 489888. 1.02445
$$188$$ 28992.0 0.0598251
$$189$$ 0 0
$$190$$ −157920. −0.317361
$$191$$ 575976. 1.14241 0.571204 0.820808i $$-0.306477\pi$$
0.571204 + 0.820808i $$0.306477\pi$$
$$192$$ 0 0
$$193$$ −413938. −0.799912 −0.399956 0.916534i $$-0.630975\pi$$
−0.399956 + 0.916534i $$0.630975\pi$$
$$194$$ −67912.0 −0.129551
$$195$$ 0 0
$$196$$ 38416.0 0.0714286
$$197$$ 494946. 0.908641 0.454320 0.890838i $$-0.349882\pi$$
0.454320 + 0.890838i $$0.349882\pi$$
$$198$$ 0 0
$$199$$ 520364. 0.931482 0.465741 0.884921i $$-0.345788\pi$$
0.465741 + 0.884921i $$0.345788\pi$$
$$200$$ 251584. 0.444742
$$201$$ 0 0
$$202$$ −396816. −0.684244
$$203$$ −238434. −0.406095
$$204$$ 0 0
$$205$$ 522648. 0.868610
$$206$$ −526576. −0.864556
$$207$$ 0 0
$$208$$ 149504. 0.239604
$$209$$ 157920. 0.250076
$$210$$ 0 0
$$211$$ 183284. 0.283412 0.141706 0.989909i $$-0.454741\pi$$
0.141706 + 0.989909i $$0.454741\pi$$
$$212$$ 595872. 0.910570
$$213$$ 0 0
$$214$$ −195408. −0.291681
$$215$$ −311136. −0.459044
$$216$$ 0 0
$$217$$ −361228. −0.520753
$$218$$ −225496. −0.321364
$$219$$ 0 0
$$220$$ −451584. −0.629045
$$221$$ 851472. 1.17271
$$222$$ 0 0
$$223$$ −1.27746e6 −1.72023 −0.860115 0.510100i $$-0.829608\pi$$
−0.860115 + 0.510100i $$0.829608\pi$$
$$224$$ 50176.0 0.0668153
$$225$$ 0 0
$$226$$ −34968.0 −0.0455407
$$227$$ 1.28764e6 1.65856 0.829279 0.558835i $$-0.188752\pi$$
0.829279 + 0.558835i $$0.188752\pi$$
$$228$$ 0 0
$$229$$ 350936. 0.442221 0.221110 0.975249i $$-0.429032\pi$$
0.221110 + 0.975249i $$0.429032\pi$$
$$230$$ −1.41120e6 −1.75901
$$231$$ 0 0
$$232$$ −311424. −0.379867
$$233$$ −836154. −1.00901 −0.504506 0.863408i $$-0.668325\pi$$
−0.504506 + 0.863408i $$0.668325\pi$$
$$234$$ 0 0
$$235$$ −152208. −0.179791
$$236$$ −548832. −0.641445
$$237$$ 0 0
$$238$$ 285768. 0.327018
$$239$$ −774336. −0.876869 −0.438434 0.898763i $$-0.644467\pi$$
−0.438434 + 0.898763i $$0.644467\pi$$
$$240$$ 0 0
$$241$$ −1.15285e6 −1.27859 −0.639293 0.768963i $$-0.720773\pi$$
−0.639293 + 0.768963i $$0.720773\pi$$
$$242$$ −192620. −0.211428
$$243$$ 0 0
$$244$$ 391616. 0.421101
$$245$$ −201684. −0.214663
$$246$$ 0 0
$$247$$ 274480. 0.286265
$$248$$ −471808. −0.487120
$$249$$ 0 0
$$250$$ −270816. −0.274047
$$251$$ −1.35801e6 −1.36056 −0.680282 0.732951i $$-0.738142\pi$$
−0.680282 + 0.732951i $$0.738142\pi$$
$$252$$ 0 0
$$253$$ 1.41120e6 1.38608
$$254$$ 1.26397e6 1.22928
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 317742. 0.300083 0.150042 0.988680i $$-0.452059\pi$$
0.150042 + 0.988680i $$0.452059\pi$$
$$258$$ 0 0
$$259$$ 702170. 0.650418
$$260$$ −784896. −0.720077
$$261$$ 0 0
$$262$$ 98664.0 0.0887985
$$263$$ −1.05101e6 −0.936951 −0.468475 0.883477i $$-0.655196\pi$$
−0.468475 + 0.883477i $$0.655196\pi$$
$$264$$ 0 0
$$265$$ −3.12833e6 −2.73651
$$266$$ 92120.0 0.0798270
$$267$$ 0 0
$$268$$ −279232. −0.237481
$$269$$ −1.18958e6 −1.00234 −0.501169 0.865349i $$-0.667097\pi$$
−0.501169 + 0.865349i $$0.667097\pi$$
$$270$$ 0 0
$$271$$ −1.43008e6 −1.18287 −0.591435 0.806353i $$-0.701438\pi$$
−0.591435 + 0.806353i $$0.701438\pi$$
$$272$$ 373248. 0.305897
$$273$$ 0 0
$$274$$ −1.21294e6 −0.976026
$$275$$ 1.32082e6 1.05320
$$276$$ 0 0
$$277$$ 63302.0 0.0495699 0.0247849 0.999693i $$-0.492110\pi$$
0.0247849 + 0.999693i $$0.492110\pi$$
$$278$$ 1.00234e6 0.777866
$$279$$ 0 0
$$280$$ −263424. −0.200798
$$281$$ 496614. 0.375192 0.187596 0.982246i $$-0.439930\pi$$
0.187596 + 0.982246i $$0.439930\pi$$
$$282$$ 0 0
$$283$$ −1.15842e6 −0.859803 −0.429902 0.902876i $$-0.641452\pi$$
−0.429902 + 0.902876i $$0.641452\pi$$
$$284$$ −451584. −0.332233
$$285$$ 0 0
$$286$$ 784896. 0.567410
$$287$$ −304878. −0.218485
$$288$$ 0 0
$$289$$ 705907. 0.497168
$$290$$ 1.63498e6 1.14161
$$291$$ 0 0
$$292$$ 57632.0 0.0395555
$$293$$ −1.43886e6 −0.979151 −0.489575 0.871961i $$-0.662848\pi$$
−0.489575 + 0.871961i $$0.662848\pi$$
$$294$$ 0 0
$$295$$ 2.88137e6 1.92772
$$296$$ 917120. 0.608411
$$297$$ 0 0
$$298$$ 242376. 0.158106
$$299$$ 2.45280e6 1.58666
$$300$$ 0 0
$$301$$ 181496. 0.115465
$$302$$ 497792. 0.314073
$$303$$ 0 0
$$304$$ 120320. 0.0746713
$$305$$ −2.05598e6 −1.26552
$$306$$ 0 0
$$307$$ −989098. −0.598954 −0.299477 0.954104i $$-0.596812\pi$$
−0.299477 + 0.954104i $$0.596812\pi$$
$$308$$ 263424. 0.158226
$$309$$ 0 0
$$310$$ 2.47699e6 1.46393
$$311$$ 2.22050e6 1.30182 0.650909 0.759155i $$-0.274388\pi$$
0.650909 + 0.759155i $$0.274388\pi$$
$$312$$ 0 0
$$313$$ 2.33008e6 1.34434 0.672171 0.740396i $$-0.265362\pi$$
0.672171 + 0.740396i $$0.265362\pi$$
$$314$$ 304160. 0.174092
$$315$$ 0 0
$$316$$ 685952. 0.386435
$$317$$ −427542. −0.238963 −0.119481 0.992836i $$-0.538123\pi$$
−0.119481 + 0.992836i $$0.538123\pi$$
$$318$$ 0 0
$$319$$ −1.63498e6 −0.899569
$$320$$ −344064. −0.187830
$$321$$ 0 0
$$322$$ 823200. 0.442452
$$323$$ 685260. 0.365468
$$324$$ 0 0
$$325$$ 2.29570e6 1.20561
$$326$$ 497024. 0.259020
$$327$$ 0 0
$$328$$ −398208. −0.204374
$$329$$ 88788.0 0.0452235
$$330$$ 0 0
$$331$$ −396616. −0.198976 −0.0994879 0.995039i $$-0.531720\pi$$
−0.0994879 + 0.995039i $$0.531720\pi$$
$$332$$ 563232. 0.280441
$$333$$ 0 0
$$334$$ 289680. 0.142086
$$335$$ 1.46597e6 0.713695
$$336$$ 0 0
$$337$$ −3.21819e6 −1.54361 −0.771805 0.635860i $$-0.780646\pi$$
−0.771805 + 0.635860i $$0.780646\pi$$
$$338$$ −120948. −0.0575847
$$339$$ 0 0
$$340$$ −1.95955e6 −0.919305
$$341$$ −2.47699e6 −1.15356
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 237056. 0.108008
$$345$$ 0 0
$$346$$ 1.76621e6 0.793143
$$347$$ −2.78018e6 −1.23951 −0.619755 0.784796i $$-0.712768\pi$$
−0.619755 + 0.784796i $$0.712768\pi$$
$$348$$ 0 0
$$349$$ −338800. −0.148895 −0.0744475 0.997225i $$-0.523719\pi$$
−0.0744475 + 0.997225i $$0.523719\pi$$
$$350$$ 770476. 0.336193
$$351$$ 0 0
$$352$$ 344064. 0.148007
$$353$$ 362046. 0.154642 0.0773209 0.997006i $$-0.475363\pi$$
0.0773209 + 0.997006i $$0.475363\pi$$
$$354$$ 0 0
$$355$$ 2.37082e6 0.998451
$$356$$ −427680. −0.178852
$$357$$ 0 0
$$358$$ 42768.0 0.0176365
$$359$$ −876528. −0.358946 −0.179473 0.983763i $$-0.557439\pi$$
−0.179473 + 0.983763i $$0.557439\pi$$
$$360$$ 0 0
$$361$$ −2.25520e6 −0.910787
$$362$$ −2.18426e6 −0.876056
$$363$$ 0 0
$$364$$ 457856. 0.181124
$$365$$ −302568. −0.118875
$$366$$ 0 0
$$367$$ 2.98062e6 1.15516 0.577578 0.816335i $$-0.303998\pi$$
0.577578 + 0.816335i $$0.303998\pi$$
$$368$$ 1.07520e6 0.413875
$$369$$ 0 0
$$370$$ −4.81488e6 −1.82844
$$371$$ 1.82486e6 0.688326
$$372$$ 0 0
$$373$$ 3.91441e6 1.45678 0.728391 0.685162i $$-0.240268\pi$$
0.728391 + 0.685162i $$0.240268\pi$$
$$374$$ 1.95955e6 0.724399
$$375$$ 0 0
$$376$$ 115968. 0.0423027
$$377$$ −2.84174e6 −1.02975
$$378$$ 0 0
$$379$$ 3.60661e6 1.28974 0.644868 0.764294i $$-0.276912\pi$$
0.644868 + 0.764294i $$0.276912\pi$$
$$380$$ −631680. −0.224408
$$381$$ 0 0
$$382$$ 2.30390e6 0.807804
$$383$$ 2.66644e6 0.928826 0.464413 0.885619i $$-0.346265\pi$$
0.464413 + 0.885619i $$0.346265\pi$$
$$384$$ 0 0
$$385$$ −1.38298e6 −0.475513
$$386$$ −1.65575e6 −0.565623
$$387$$ 0 0
$$388$$ −271648. −0.0916067
$$389$$ 213366. 0.0714910 0.0357455 0.999361i $$-0.488619\pi$$
0.0357455 + 0.999361i $$0.488619\pi$$
$$390$$ 0 0
$$391$$ 6.12360e6 2.02565
$$392$$ 153664. 0.0505076
$$393$$ 0 0
$$394$$ 1.97978e6 0.642506
$$395$$ −3.60125e6 −1.16134
$$396$$ 0 0
$$397$$ −4.09408e6 −1.30371 −0.651854 0.758345i $$-0.726008\pi$$
−0.651854 + 0.758345i $$0.726008\pi$$
$$398$$ 2.08146e6 0.658657
$$399$$ 0 0
$$400$$ 1.00634e6 0.314480
$$401$$ −942366. −0.292657 −0.146328 0.989236i $$-0.546746\pi$$
−0.146328 + 0.989236i $$0.546746\pi$$
$$402$$ 0 0
$$403$$ −4.30525e6 −1.32049
$$404$$ −1.58726e6 −0.483833
$$405$$ 0 0
$$406$$ −953736. −0.287153
$$407$$ 4.81488e6 1.44079
$$408$$ 0 0
$$409$$ −4.84561e6 −1.43232 −0.716160 0.697936i $$-0.754102\pi$$
−0.716160 + 0.697936i $$0.754102\pi$$
$$410$$ 2.09059e6 0.614200
$$411$$ 0 0
$$412$$ −2.10630e6 −0.611333
$$413$$ −1.68080e6 −0.484887
$$414$$ 0 0
$$415$$ −2.95697e6 −0.842804
$$416$$ 598016. 0.169426
$$417$$ 0 0
$$418$$ 631680. 0.176830
$$419$$ 1.73485e6 0.482754 0.241377 0.970431i $$-0.422401\pi$$
0.241377 + 0.970431i $$0.422401\pi$$
$$420$$ 0 0
$$421$$ −1.65145e6 −0.454109 −0.227055 0.973882i $$-0.572910\pi$$
−0.227055 + 0.973882i $$0.572910\pi$$
$$422$$ 733136. 0.200403
$$423$$ 0 0
$$424$$ 2.38349e6 0.643870
$$425$$ 5.73140e6 1.53918
$$426$$ 0 0
$$427$$ 1.19932e6 0.318322
$$428$$ −781632. −0.206250
$$429$$ 0 0
$$430$$ −1.24454e6 −0.324593
$$431$$ −4.14360e6 −1.07445 −0.537223 0.843440i $$-0.680527\pi$$
−0.537223 + 0.843440i $$0.680527\pi$$
$$432$$ 0 0
$$433$$ −3.03966e6 −0.779121 −0.389561 0.921001i $$-0.627373\pi$$
−0.389561 + 0.921001i $$0.627373\pi$$
$$434$$ −1.44491e6 −0.368228
$$435$$ 0 0
$$436$$ −901984. −0.227239
$$437$$ 1.97400e6 0.494474
$$438$$ 0 0
$$439$$ 2.54271e6 0.629703 0.314852 0.949141i $$-0.398045\pi$$
0.314852 + 0.949141i $$0.398045\pi$$
$$440$$ −1.80634e6 −0.444802
$$441$$ 0 0
$$442$$ 3.40589e6 0.829229
$$443$$ 2.43210e6 0.588806 0.294403 0.955681i $$-0.404879\pi$$
0.294403 + 0.955681i $$0.404879\pi$$
$$444$$ 0 0
$$445$$ 2.24532e6 0.537500
$$446$$ −5.10986e6 −1.21639
$$447$$ 0 0
$$448$$ 200704. 0.0472456
$$449$$ −1.82853e6 −0.428042 −0.214021 0.976829i $$-0.568656\pi$$
−0.214021 + 0.976829i $$0.568656\pi$$
$$450$$ 0 0
$$451$$ −2.09059e6 −0.483981
$$452$$ −139872. −0.0322021
$$453$$ 0 0
$$454$$ 5.15057e6 1.17278
$$455$$ −2.40374e6 −0.544327
$$456$$ 0 0
$$457$$ 1.58063e6 0.354030 0.177015 0.984208i $$-0.443356\pi$$
0.177015 + 0.984208i $$0.443356\pi$$
$$458$$ 1.40374e6 0.312697
$$459$$ 0 0
$$460$$ −5.64480e6 −1.24381
$$461$$ −5.09604e6 −1.11681 −0.558407 0.829567i $$-0.688587\pi$$
−0.558407 + 0.829567i $$0.688587\pi$$
$$462$$ 0 0
$$463$$ −7.02338e6 −1.52263 −0.761313 0.648384i $$-0.775445\pi$$
−0.761313 + 0.648384i $$0.775445\pi$$
$$464$$ −1.24570e6 −0.268607
$$465$$ 0 0
$$466$$ −3.34462e6 −0.713479
$$467$$ 4.24845e6 0.901443 0.450722 0.892665i $$-0.351167\pi$$
0.450722 + 0.892665i $$0.351167\pi$$
$$468$$ 0 0
$$469$$ −855148. −0.179518
$$470$$ −608832. −0.127131
$$471$$ 0 0
$$472$$ −2.19533e6 −0.453570
$$473$$ 1.24454e6 0.255775
$$474$$ 0 0
$$475$$ 1.84757e6 0.375722
$$476$$ 1.14307e6 0.231236
$$477$$ 0 0
$$478$$ −3.09734e6 −0.620040
$$479$$ −559284. −0.111377 −0.0556883 0.998448i $$-0.517735\pi$$
−0.0556883 + 0.998448i $$0.517735\pi$$
$$480$$ 0 0
$$481$$ 8.36872e6 1.64929
$$482$$ −4.61140e6 −0.904097
$$483$$ 0 0
$$484$$ −770480. −0.149502
$$485$$ 1.42615e6 0.275303
$$486$$ 0 0
$$487$$ −1.32057e6 −0.252312 −0.126156 0.992010i $$-0.540264\pi$$
−0.126156 + 0.992010i $$0.540264\pi$$
$$488$$ 1.56646e6 0.297763
$$489$$ 0 0
$$490$$ −806736. −0.151789
$$491$$ −6.27193e6 −1.17408 −0.587040 0.809558i $$-0.699707\pi$$
−0.587040 + 0.809558i $$0.699707\pi$$
$$492$$ 0 0
$$493$$ −7.09463e6 −1.31466
$$494$$ 1.09792e6 0.202420
$$495$$ 0 0
$$496$$ −1.88723e6 −0.344446
$$497$$ −1.38298e6 −0.251144
$$498$$ 0 0
$$499$$ −3.93785e6 −0.707959 −0.353979 0.935253i $$-0.615172\pi$$
−0.353979 + 0.935253i $$0.615172\pi$$
$$500$$ −1.08326e6 −0.193780
$$501$$ 0 0
$$502$$ −5.43204e6 −0.962063
$$503$$ 7.59830e6 1.33905 0.669525 0.742790i $$-0.266498\pi$$
0.669525 + 0.742790i $$0.266498\pi$$
$$504$$ 0 0
$$505$$ 8.33314e6 1.45405
$$506$$ 5.64480e6 0.980104
$$507$$ 0 0
$$508$$ 5.05587e6 0.869234
$$509$$ 7.82664e6 1.33900 0.669501 0.742812i $$-0.266508\pi$$
0.669501 + 0.742812i $$0.266508\pi$$
$$510$$ 0 0
$$511$$ 176498. 0.0299011
$$512$$ 262144. 0.0441942
$$513$$ 0 0
$$514$$ 1.27097e6 0.212191
$$515$$ 1.10581e7 1.83722
$$516$$ 0 0
$$517$$ 608832. 0.100178
$$518$$ 2.80868e6 0.459915
$$519$$ 0 0
$$520$$ −3.13958e6 −0.509171
$$521$$ −8.94454e6 −1.44366 −0.721828 0.692072i $$-0.756698\pi$$
−0.721828 + 0.692072i $$0.756698\pi$$
$$522$$ 0 0
$$523$$ 4.07481e6 0.651407 0.325704 0.945472i $$-0.394399\pi$$
0.325704 + 0.945472i $$0.394399\pi$$
$$524$$ 394656. 0.0627900
$$525$$ 0 0
$$526$$ −4.20403e6 −0.662524
$$527$$ −1.07484e7 −1.68584
$$528$$ 0 0
$$529$$ 1.12037e7 1.74069
$$530$$ −1.25133e7 −1.93501
$$531$$ 0 0
$$532$$ 368480. 0.0564462
$$533$$ −3.63365e6 −0.554019
$$534$$ 0 0
$$535$$ 4.10357e6 0.619837
$$536$$ −1.11693e6 −0.167924
$$537$$ 0 0
$$538$$ −4.75834e6 −0.708760
$$539$$ 806736. 0.119608
$$540$$ 0 0
$$541$$ −1.18676e7 −1.74329 −0.871644 0.490140i $$-0.836946\pi$$
−0.871644 + 0.490140i $$0.836946\pi$$
$$542$$ −5.72032e6 −0.836416
$$543$$ 0 0
$$544$$ 1.49299e6 0.216302
$$545$$ 4.73542e6 0.682915
$$546$$ 0 0
$$547$$ −5.37801e6 −0.768516 −0.384258 0.923226i $$-0.625543\pi$$
−0.384258 + 0.923226i $$0.625543\pi$$
$$548$$ −4.85174e6 −0.690155
$$549$$ 0 0
$$550$$ 5.28326e6 0.744724
$$551$$ −2.28702e6 −0.320916
$$552$$ 0 0
$$553$$ 2.10073e6 0.292117
$$554$$ 253208. 0.0350512
$$555$$ 0 0
$$556$$ 4.00938e6 0.550034
$$557$$ 5.64878e6 0.771466 0.385733 0.922611i $$-0.373949\pi$$
0.385733 + 0.922611i $$0.373949\pi$$
$$558$$ 0 0
$$559$$ 2.16314e6 0.292789
$$560$$ −1.05370e6 −0.141986
$$561$$ 0 0
$$562$$ 1.98646e6 0.265301
$$563$$ −4.56407e6 −0.606850 −0.303425 0.952855i $$-0.598130\pi$$
−0.303425 + 0.952855i $$0.598130\pi$$
$$564$$ 0 0
$$565$$ 734328. 0.0967763
$$566$$ −4.63367e6 −0.607973
$$567$$ 0 0
$$568$$ −1.80634e6 −0.234924
$$569$$ −8.00165e6 −1.03609 −0.518047 0.855352i $$-0.673341\pi$$
−0.518047 + 0.855352i $$0.673341\pi$$
$$570$$ 0 0
$$571$$ −1.37164e7 −1.76055 −0.880275 0.474464i $$-0.842642\pi$$
−0.880275 + 0.474464i $$0.842642\pi$$
$$572$$ 3.13958e6 0.401220
$$573$$ 0 0
$$574$$ −1.21951e6 −0.154492
$$575$$ 1.65102e7 2.08249
$$576$$ 0 0
$$577$$ 6.09797e6 0.762510 0.381255 0.924470i $$-0.375492\pi$$
0.381255 + 0.924470i $$0.375492\pi$$
$$578$$ 2.82363e6 0.351551
$$579$$ 0 0
$$580$$ 6.53990e6 0.807238
$$581$$ 1.72490e6 0.211994
$$582$$ 0 0
$$583$$ 1.25133e7 1.52476
$$584$$ 230528. 0.0279699
$$585$$ 0 0
$$586$$ −5.75544e6 −0.692364
$$587$$ 8.08462e6 0.968422 0.484211 0.874951i $$-0.339107\pi$$
0.484211 + 0.874951i $$0.339107\pi$$
$$588$$ 0 0
$$589$$ −3.46484e6 −0.411524
$$590$$ 1.15255e7 1.36310
$$591$$ 0 0
$$592$$ 3.66848e6 0.430211
$$593$$ −1.41575e6 −0.165330 −0.0826649 0.996577i $$-0.526343\pi$$
−0.0826649 + 0.996577i $$0.526343\pi$$
$$594$$ 0 0
$$595$$ −6.00113e6 −0.694929
$$596$$ 969504. 0.111798
$$597$$ 0 0
$$598$$ 9.81120e6 1.12194
$$599$$ −8.75460e6 −0.996941 −0.498470 0.866907i $$-0.666105\pi$$
−0.498470 + 0.866907i $$0.666105\pi$$
$$600$$ 0 0
$$601$$ 8.70276e6 0.982813 0.491407 0.870930i $$-0.336483\pi$$
0.491407 + 0.870930i $$0.336483\pi$$
$$602$$ 725984. 0.0816462
$$603$$ 0 0
$$604$$ 1.99117e6 0.222083
$$605$$ 4.04502e6 0.449296
$$606$$ 0 0
$$607$$ −1.69578e7 −1.86809 −0.934045 0.357157i $$-0.883746\pi$$
−0.934045 + 0.357157i $$0.883746\pi$$
$$608$$ 481280. 0.0528006
$$609$$ 0 0
$$610$$ −8.22394e6 −0.894860
$$611$$ 1.05821e6 0.114675
$$612$$ 0 0
$$613$$ 1.76743e7 1.89973 0.949866 0.312658i $$-0.101220\pi$$
0.949866 + 0.312658i $$0.101220\pi$$
$$614$$ −3.95639e6 −0.423524
$$615$$ 0 0
$$616$$ 1.05370e6 0.111883
$$617$$ 9.70636e6 1.02646 0.513232 0.858250i $$-0.328448\pi$$
0.513232 + 0.858250i $$0.328448\pi$$
$$618$$ 0 0
$$619$$ 1.48739e7 1.56027 0.780133 0.625613i $$-0.215151\pi$$
0.780133 + 0.625613i $$0.215151\pi$$
$$620$$ 9.90797e6 1.03515
$$621$$ 0 0
$$622$$ 8.88202e6 0.920525
$$623$$ −1.30977e6 −0.135199
$$624$$ 0 0
$$625$$ −6.59724e6 −0.675557
$$626$$ 9.32031e6 0.950593
$$627$$ 0 0
$$628$$ 1.21664e6 0.123101
$$629$$ 2.08931e7 2.10561
$$630$$ 0 0
$$631$$ 1.26353e7 1.26331 0.631656 0.775248i $$-0.282375\pi$$
0.631656 + 0.775248i $$0.282375\pi$$
$$632$$ 2.74381e6 0.273251
$$633$$ 0 0
$$634$$ −1.71017e6 −0.168972
$$635$$ −2.65433e7 −2.61229
$$636$$ 0 0
$$637$$ 1.40218e6 0.136917
$$638$$ −6.53990e6 −0.636092
$$639$$ 0 0
$$640$$ −1.37626e6 −0.132816
$$641$$ −6.23398e6 −0.599267 −0.299634 0.954054i $$-0.596864\pi$$
−0.299634 + 0.954054i $$0.596864\pi$$
$$642$$ 0 0
$$643$$ 1.06874e7 1.01940 0.509701 0.860352i $$-0.329756\pi$$
0.509701 + 0.860352i $$0.329756\pi$$
$$644$$ 3.29280e6 0.312860
$$645$$ 0 0
$$646$$ 2.74104e6 0.258425
$$647$$ −1.83258e7 −1.72109 −0.860544 0.509376i $$-0.829876\pi$$
−0.860544 + 0.509376i $$0.829876\pi$$
$$648$$ 0 0
$$649$$ −1.15255e7 −1.07411
$$650$$ 9.18282e6 0.852496
$$651$$ 0 0
$$652$$ 1.98810e6 0.183155
$$653$$ 7.28857e6 0.668897 0.334448 0.942414i $$-0.391450\pi$$
0.334448 + 0.942414i $$0.391450\pi$$
$$654$$ 0 0
$$655$$ −2.07194e6 −0.188701
$$656$$ −1.59283e6 −0.144514
$$657$$ 0 0
$$658$$ 355152. 0.0319779
$$659$$ −4.54337e6 −0.407534 −0.203767 0.979019i $$-0.565319\pi$$
−0.203767 + 0.979019i $$0.565319\pi$$
$$660$$ 0 0
$$661$$ −2.10021e7 −1.86964 −0.934821 0.355120i $$-0.884440\pi$$
−0.934821 + 0.355120i $$0.884440\pi$$
$$662$$ −1.58646e6 −0.140697
$$663$$ 0 0
$$664$$ 2.25293e6 0.198302
$$665$$ −1.93452e6 −0.169636
$$666$$ 0 0
$$667$$ −2.04372e7 −1.77872
$$668$$ 1.15872e6 0.100470
$$669$$ 0 0
$$670$$ 5.86387e6 0.504658
$$671$$ 8.22394e6 0.705137
$$672$$ 0 0
$$673$$ 3.46923e6 0.295253 0.147627 0.989043i $$-0.452837\pi$$
0.147627 + 0.989043i $$0.452837\pi$$
$$674$$ −1.28728e7 −1.09150
$$675$$ 0 0
$$676$$ −483792. −0.0407185
$$677$$ 1.80916e7 1.51707 0.758536 0.651631i $$-0.225915\pi$$
0.758536 + 0.651631i $$0.225915\pi$$
$$678$$ 0 0
$$679$$ −831922. −0.0692481
$$680$$ −7.83821e6 −0.650047
$$681$$ 0 0
$$682$$ −9.90797e6 −0.815687
$$683$$ −4.67752e6 −0.383675 −0.191838 0.981427i $$-0.561445\pi$$
−0.191838 + 0.981427i $$0.561445\pi$$
$$684$$ 0 0
$$685$$ 2.54717e7 2.07411
$$686$$ 470596. 0.0381802
$$687$$ 0 0
$$688$$ 948224. 0.0763730
$$689$$ 2.17493e7 1.74541
$$690$$ 0 0
$$691$$ 1.68960e7 1.34614 0.673069 0.739579i $$-0.264976\pi$$
0.673069 + 0.739579i $$0.264976\pi$$
$$692$$ 7.06483e6 0.560837
$$693$$ 0 0
$$694$$ −1.11207e7 −0.876466
$$695$$ −2.10492e7 −1.65300
$$696$$ 0 0
$$697$$ −9.07168e6 −0.707303
$$698$$ −1.35520e6 −0.105285
$$699$$ 0 0
$$700$$ 3.08190e6 0.237725
$$701$$ −2.40964e6 −0.185207 −0.0926035 0.995703i $$-0.529519\pi$$
−0.0926035 + 0.995703i $$0.529519\pi$$
$$702$$ 0 0
$$703$$ 6.73510e6 0.513991
$$704$$ 1.37626e6 0.104657
$$705$$ 0 0
$$706$$ 1.44818e6 0.109348
$$707$$ −4.86100e6 −0.365744
$$708$$ 0 0
$$709$$ −5.77010e6 −0.431090 −0.215545 0.976494i $$-0.569153\pi$$
−0.215545 + 0.976494i $$0.569153\pi$$
$$710$$ 9.48326e6 0.706012
$$711$$ 0 0
$$712$$ −1.71072e6 −0.126468
$$713$$ −3.09624e7 −2.28092
$$714$$ 0 0
$$715$$ −1.64828e7 −1.20578
$$716$$ 171072. 0.0124709
$$717$$ 0 0
$$718$$ −3.50611e6 −0.253813
$$719$$ 1.43716e7 1.03677 0.518385 0.855147i $$-0.326533\pi$$
0.518385 + 0.855147i $$0.326533\pi$$
$$720$$ 0 0
$$721$$ −6.45056e6 −0.462124
$$722$$ −9.02080e6 −0.644024
$$723$$ 0 0
$$724$$ −8.73702e6 −0.619465
$$725$$ −1.91282e7 −1.35154
$$726$$ 0 0
$$727$$ −1.40705e7 −0.987353 −0.493676 0.869646i $$-0.664347\pi$$
−0.493676 + 0.869646i $$0.664347\pi$$
$$728$$ 1.83142e6 0.128074
$$729$$ 0 0
$$730$$ −1.21027e6 −0.0840574
$$731$$ 5.40043e6 0.373796
$$732$$ 0 0
$$733$$ −3.75000e6 −0.257793 −0.128897 0.991658i $$-0.541144\pi$$
−0.128897 + 0.991658i $$0.541144\pi$$
$$734$$ 1.19225e7 0.816819
$$735$$ 0 0
$$736$$ 4.30080e6 0.292654
$$737$$ −5.86387e6 −0.397664
$$738$$ 0 0
$$739$$ 2.61318e7 1.76019 0.880093 0.474802i $$-0.157480\pi$$
0.880093 + 0.474802i $$0.157480\pi$$
$$740$$ −1.92595e7 −1.29290
$$741$$ 0 0
$$742$$ 7.29943e6 0.486720
$$743$$ 159072. 0.0105711 0.00528557 0.999986i $$-0.498318\pi$$
0.00528557 + 0.999986i $$0.498318\pi$$
$$744$$ 0 0
$$745$$ −5.08990e6 −0.335984
$$746$$ 1.56577e7 1.03010
$$747$$ 0 0
$$748$$ 7.83821e6 0.512227
$$749$$ −2.39375e6 −0.155910
$$750$$ 0 0
$$751$$ −2.65311e7 −1.71654 −0.858272 0.513196i $$-0.828461\pi$$
−0.858272 + 0.513196i $$0.828461\pi$$
$$752$$ 463872. 0.0299126
$$753$$ 0 0
$$754$$ −1.13670e7 −0.728143
$$755$$ −1.04536e7 −0.667421
$$756$$ 0 0
$$757$$ −1.52032e7 −0.964260 −0.482130 0.876100i $$-0.660137\pi$$
−0.482130 + 0.876100i $$0.660137\pi$$
$$758$$ 1.44264e7 0.911981
$$759$$ 0 0
$$760$$ −2.52672e6 −0.158680
$$761$$ −4.71380e6 −0.295059 −0.147530 0.989058i $$-0.547132\pi$$
−0.147530 + 0.989058i $$0.547132\pi$$
$$762$$ 0 0
$$763$$ −2.76233e6 −0.171776
$$764$$ 9.21562e6 0.571204
$$765$$ 0 0
$$766$$ 1.06657e7 0.656779
$$767$$ −2.00324e7 −1.22954
$$768$$ 0 0
$$769$$ −1.58977e6 −0.0969434 −0.0484717 0.998825i $$-0.515435\pi$$
−0.0484717 + 0.998825i $$0.515435\pi$$
$$770$$ −5.53190e6 −0.336239
$$771$$ 0 0
$$772$$ −6.62301e6 −0.399956
$$773$$ 9.69095e6 0.583334 0.291667 0.956520i $$-0.405790\pi$$
0.291667 + 0.956520i $$0.405790\pi$$
$$774$$ 0 0
$$775$$ −2.89793e7 −1.73314
$$776$$ −1.08659e6 −0.0647757
$$777$$ 0 0
$$778$$ 853464. 0.0505518
$$779$$ −2.92434e6 −0.172657
$$780$$ 0 0
$$781$$ −9.48326e6 −0.556327
$$782$$ 2.44944e7 1.43235
$$783$$ 0 0
$$784$$ 614656. 0.0357143
$$785$$ −6.38736e6 −0.369954
$$786$$ 0 0
$$787$$ −1.57170e6 −0.0904549 −0.0452275 0.998977i $$-0.514401\pi$$
−0.0452275 + 0.998977i $$0.514401\pi$$
$$788$$ 7.91914e6 0.454320
$$789$$ 0 0
$$790$$ −1.44050e7 −0.821193
$$791$$ −428358. −0.0243425
$$792$$ 0 0
$$793$$ 1.42940e7 0.807180
$$794$$ −1.63763e7 −0.921860
$$795$$ 0 0
$$796$$ 8.32582e6 0.465741
$$797$$ 2.25298e6 0.125635 0.0628175 0.998025i $$-0.479991\pi$$
0.0628175 + 0.998025i $$0.479991\pi$$
$$798$$ 0 0
$$799$$ 2.64190e6 0.146403
$$800$$ 4.02534e6 0.222371
$$801$$ 0 0
$$802$$ −3.76946e6 −0.206940
$$803$$ 1.21027e6 0.0662360
$$804$$ 0 0
$$805$$ −1.72872e7 −0.940232
$$806$$ −1.72210e7 −0.933728
$$807$$ 0 0
$$808$$ −6.34906e6 −0.342122
$$809$$ 2.37938e7 1.27818 0.639090 0.769132i $$-0.279311\pi$$
0.639090 + 0.769132i $$0.279311\pi$$
$$810$$ 0 0
$$811$$ 5.32300e6 0.284187 0.142093 0.989853i $$-0.454617\pi$$
0.142093 + 0.989853i $$0.454617\pi$$
$$812$$ −3.81494e6 −0.203048
$$813$$ 0 0
$$814$$ 1.92595e7 1.01879
$$815$$ −1.04375e7 −0.550431
$$816$$ 0 0
$$817$$ 1.74088e6 0.0912460
$$818$$ −1.93824e7 −1.01280
$$819$$ 0 0
$$820$$ 8.36237e6 0.434305
$$821$$ −1.48802e7 −0.770464 −0.385232 0.922820i $$-0.625879\pi$$
−0.385232 + 0.922820i $$0.625879\pi$$
$$822$$ 0 0
$$823$$ 2.00601e7 1.03236 0.516182 0.856479i $$-0.327353\pi$$
0.516182 + 0.856479i $$0.327353\pi$$
$$824$$ −8.42522e6 −0.432278
$$825$$ 0 0
$$826$$ −6.72319e6 −0.342867
$$827$$ −1.21539e7 −0.617949 −0.308975 0.951070i $$-0.599986\pi$$
−0.308975 + 0.951070i $$0.599986\pi$$
$$828$$ 0 0
$$829$$ 3.21197e7 1.62325 0.811625 0.584179i $$-0.198583\pi$$
0.811625 + 0.584179i $$0.198583\pi$$
$$830$$ −1.18279e7 −0.595952
$$831$$ 0 0
$$832$$ 2.39206e6 0.119802
$$833$$ 3.50066e6 0.174798
$$834$$ 0 0
$$835$$ −6.08328e6 −0.301941
$$836$$ 2.52672e6 0.125038
$$837$$ 0 0
$$838$$ 6.93938e6 0.341359
$$839$$ 1.01320e6 0.0496922 0.0248461 0.999691i $$-0.492090\pi$$
0.0248461 + 0.999691i $$0.492090\pi$$
$$840$$ 0 0
$$841$$ 3.16681e6 0.154394
$$842$$ −6.60580e6 −0.321104
$$843$$ 0 0
$$844$$ 2.93254e6 0.141706
$$845$$ 2.53991e6 0.122370
$$846$$ 0 0
$$847$$ −2.35960e6 −0.113013
$$848$$ 9.53395e6 0.455285
$$849$$ 0 0
$$850$$ 2.29256e7 1.08836
$$851$$ 6.01860e7 2.84886
$$852$$ 0 0
$$853$$ 234824. 0.0110502 0.00552510 0.999985i $$-0.498241\pi$$
0.00552510 + 0.999985i $$0.498241\pi$$
$$854$$ 4.79730e6 0.225088
$$855$$ 0 0
$$856$$ −3.12653e6 −0.145840
$$857$$ −2.83802e7 −1.31997 −0.659985 0.751279i $$-0.729437\pi$$
−0.659985 + 0.751279i $$0.729437\pi$$
$$858$$ 0 0
$$859$$ 4.00081e7 1.84997 0.924986 0.380001i $$-0.124076\pi$$
0.924986 + 0.380001i $$0.124076\pi$$
$$860$$ −4.97818e6 −0.229522
$$861$$ 0 0
$$862$$ −1.65744e7 −0.759748
$$863$$ 2.08030e7 0.950823 0.475411 0.879764i $$-0.342299\pi$$
0.475411 + 0.879764i $$0.342299\pi$$
$$864$$ 0 0
$$865$$ −3.70904e7 −1.68547
$$866$$ −1.21586e7 −0.550922
$$867$$ 0 0
$$868$$ −5.77965e6 −0.260377
$$869$$ 1.44050e7 0.647088
$$870$$ 0 0
$$871$$ −1.01920e7 −0.455211
$$872$$ −3.60794e6 −0.160682
$$873$$ 0 0
$$874$$ 7.89600e6 0.349646
$$875$$ −3.31750e6 −0.146484
$$876$$ 0 0
$$877$$ 3.03559e7 1.33273 0.666367 0.745624i $$-0.267848\pi$$
0.666367 + 0.745624i $$0.267848\pi$$
$$878$$ 1.01708e7 0.445267
$$879$$ 0 0
$$880$$ −7.22534e6 −0.314523
$$881$$ 2.58936e7 1.12396 0.561981 0.827150i $$-0.310039\pi$$
0.561981 + 0.827150i $$0.310039\pi$$
$$882$$ 0 0
$$883$$ −1.88813e7 −0.814950 −0.407475 0.913216i $$-0.633591\pi$$
−0.407475 + 0.913216i $$0.633591\pi$$
$$884$$ 1.36236e7 0.586354
$$885$$ 0 0
$$886$$ 9.72840e6 0.416349
$$887$$ 2.34431e7 1.00048 0.500238 0.865888i $$-0.333246\pi$$
0.500238 + 0.865888i $$0.333246\pi$$
$$888$$ 0 0
$$889$$ 1.54836e7 0.657079
$$890$$ 8.98128e6 0.380070
$$891$$ 0 0
$$892$$ −2.04394e7 −0.860115
$$893$$ 851640. 0.0357378
$$894$$ 0 0
$$895$$ −898128. −0.0374784
$$896$$ 802816. 0.0334077
$$897$$ 0 0
$$898$$ −7.31412e6 −0.302671
$$899$$ 3.58722e7 1.48033
$$900$$ 0 0
$$901$$ 5.42988e7 2.22833
$$902$$ −8.36237e6 −0.342226
$$903$$ 0 0
$$904$$ −559488. −0.0227703
$$905$$ 4.58694e7 1.86166
$$906$$ 0 0
$$907$$ −5.60873e6 −0.226384 −0.113192 0.993573i $$-0.536108\pi$$
−0.113192 + 0.993573i $$0.536108\pi$$
$$908$$ 2.06023e7 0.829279
$$909$$ 0 0
$$910$$ −9.61498e6 −0.384897
$$911$$ −2.16215e7 −0.863156 −0.431578 0.902076i $$-0.642043\pi$$
−0.431578 + 0.902076i $$0.642043\pi$$
$$912$$ 0 0
$$913$$ 1.18279e7 0.469602
$$914$$ 6.32252e6 0.250337
$$915$$ 0 0
$$916$$ 5.61498e6 0.221110
$$917$$ 1.20863e6 0.0474648
$$918$$ 0 0
$$919$$ 4.51695e7 1.76424 0.882119 0.471028i $$-0.156117\pi$$
0.882119 + 0.471028i $$0.156117\pi$$
$$920$$ −2.25792e7 −0.879506
$$921$$ 0 0
$$922$$ −2.03842e7 −0.789706
$$923$$ −1.64828e7 −0.636835
$$924$$ 0 0
$$925$$ 5.63312e7 2.16469
$$926$$ −2.80935e7 −1.07666
$$927$$ 0 0
$$928$$ −4.98278e6 −0.189934
$$929$$ 2.28729e7 0.869524 0.434762 0.900545i $$-0.356832\pi$$
0.434762 + 0.900545i $$0.356832\pi$$
$$930$$ 0 0
$$931$$ 1.12847e6 0.0426693
$$932$$ −1.33785e7 −0.504506
$$933$$ 0 0
$$934$$ 1.69938e7 0.637417
$$935$$ −4.11506e7 −1.53938
$$936$$ 0 0
$$937$$ −1.79616e7 −0.668336 −0.334168 0.942514i $$-0.608455\pi$$
−0.334168 + 0.942514i $$0.608455\pi$$
$$938$$ −3.42059e6 −0.126939
$$939$$ 0 0
$$940$$ −2.43533e6 −0.0898955
$$941$$ 1.79697e7 0.661558 0.330779 0.943708i $$-0.392689\pi$$
0.330779 + 0.943708i $$0.392689\pi$$
$$942$$ 0 0
$$943$$ −2.61324e7 −0.956974
$$944$$ −8.78131e6 −0.320722
$$945$$ 0 0
$$946$$ 4.97818e6 0.180860
$$947$$ −4.32115e7 −1.56576 −0.782879 0.622174i $$-0.786250\pi$$
−0.782879 + 0.622174i $$0.786250\pi$$
$$948$$ 0 0
$$949$$ 2.10357e6 0.0758213
$$950$$ 7.39028e6 0.265676
$$951$$ 0 0
$$952$$ 4.57229e6 0.163509
$$953$$ 7.50965e6 0.267848 0.133924 0.990992i $$-0.457242\pi$$
0.133924 + 0.990992i $$0.457242\pi$$
$$954$$ 0 0
$$955$$ −4.83820e7 −1.71662
$$956$$ −1.23894e7 −0.438434
$$957$$ 0 0
$$958$$ −2.23714e6 −0.0787551
$$959$$ −1.48585e7 −0.521708
$$960$$ 0 0
$$961$$ 2.57172e7 0.898288
$$962$$ 3.34749e7 1.16622
$$963$$ 0 0
$$964$$ −1.84456e7 −0.639293
$$965$$ 3.47708e7 1.20198
$$966$$ 0 0
$$967$$ −1.69305e7 −0.582242 −0.291121 0.956686i $$-0.594028\pi$$
−0.291121 + 0.956686i $$0.594028\pi$$
$$968$$ −3.08192e6 −0.105714
$$969$$ 0 0
$$970$$ 5.70461e6 0.194669
$$971$$ −2.86144e7 −0.973949 −0.486974 0.873416i $$-0.661899\pi$$
−0.486974 + 0.873416i $$0.661899\pi$$
$$972$$ 0 0
$$973$$ 1.22787e7 0.415787
$$974$$ −5.28227e6 −0.178412
$$975$$ 0 0
$$976$$ 6.26586e6 0.210550
$$977$$ −3.69445e7 −1.23826 −0.619132 0.785287i $$-0.712515\pi$$
−0.619132 + 0.785287i $$0.712515\pi$$
$$978$$ 0 0
$$979$$ −8.98128e6 −0.299489
$$980$$ −3.22694e6 −0.107331
$$981$$ 0 0
$$982$$ −2.50877e7 −0.830200
$$983$$ 3.88787e7 1.28330 0.641650 0.766998i $$-0.278250\pi$$
0.641650 + 0.766998i $$0.278250\pi$$
$$984$$ 0 0
$$985$$ −4.15755e7 −1.36536
$$986$$ −2.83785e7 −0.929603
$$987$$ 0 0
$$988$$ 4.39168e6 0.143133
$$989$$ 1.55568e7 0.505743
$$990$$ 0 0
$$991$$ 2.49212e7 0.806092 0.403046 0.915180i $$-0.367951\pi$$
0.403046 + 0.915180i $$0.367951\pi$$
$$992$$ −7.54893e6 −0.243560
$$993$$ 0 0
$$994$$ −5.53190e6 −0.177586
$$995$$ −4.37106e7 −1.39968
$$996$$ 0 0
$$997$$ 1.01956e7 0.324845 0.162422 0.986721i $$-0.448069\pi$$
0.162422 + 0.986721i $$0.448069\pi$$
$$998$$ −1.57514e7 −0.500603
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 126.6.a.f.1.1 1
3.2 odd 2 14.6.a.a.1.1 1
4.3 odd 2 1008.6.a.b.1.1 1
7.6 odd 2 882.6.a.x.1.1 1
12.11 even 2 112.6.a.c.1.1 1
15.2 even 4 350.6.c.d.99.1 2
15.8 even 4 350.6.c.d.99.2 2
15.14 odd 2 350.6.a.i.1.1 1
21.2 odd 6 98.6.c.c.67.1 2
21.5 even 6 98.6.c.d.67.1 2
21.11 odd 6 98.6.c.c.79.1 2
21.17 even 6 98.6.c.d.79.1 2
21.20 even 2 98.6.a.a.1.1 1
24.5 odd 2 448.6.a.e.1.1 1
24.11 even 2 448.6.a.l.1.1 1
84.83 odd 2 784.6.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.6.a.a.1.1 1 3.2 odd 2
98.6.a.a.1.1 1 21.20 even 2
98.6.c.c.67.1 2 21.2 odd 6
98.6.c.c.79.1 2 21.11 odd 6
98.6.c.d.67.1 2 21.5 even 6
98.6.c.d.79.1 2 21.17 even 6
112.6.a.c.1.1 1 12.11 even 2
126.6.a.f.1.1 1 1.1 even 1 trivial
350.6.a.i.1.1 1 15.14 odd 2
350.6.c.d.99.1 2 15.2 even 4
350.6.c.d.99.2 2 15.8 even 4
448.6.a.e.1.1 1 24.5 odd 2
448.6.a.l.1.1 1 24.11 even 2
784.6.a.i.1.1 1 84.83 odd 2
882.6.a.x.1.1 1 7.6 odd 2
1008.6.a.b.1.1 1 4.3 odd 2