# Properties

 Label 175.6.a.b.1.1 Level $175$ Weight $6$ Character 175.1 Self dual yes Analytic conductor $28.067$ 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] = [175,6,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, 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("175.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$28.0671684673$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+10.0000 q^{2} +14.0000 q^{3} +68.0000 q^{4} +140.000 q^{6} +49.0000 q^{7} +360.000 q^{8} -47.0000 q^{9} +O(q^{10})$$ $$q+10.0000 q^{2} +14.0000 q^{3} +68.0000 q^{4} +140.000 q^{6} +49.0000 q^{7} +360.000 q^{8} -47.0000 q^{9} +232.000 q^{11} +952.000 q^{12} +140.000 q^{13} +490.000 q^{14} +1424.00 q^{16} +1722.00 q^{17} -470.000 q^{18} -98.0000 q^{19} +686.000 q^{21} +2320.00 q^{22} -1824.00 q^{23} +5040.00 q^{24} +1400.00 q^{26} -4060.00 q^{27} +3332.00 q^{28} +3418.00 q^{29} -7644.00 q^{31} +2720.00 q^{32} +3248.00 q^{33} +17220.0 q^{34} -3196.00 q^{36} +10398.0 q^{37} -980.000 q^{38} +1960.00 q^{39} -17962.0 q^{41} +6860.00 q^{42} -10880.0 q^{43} +15776.0 q^{44} -18240.0 q^{46} -9324.00 q^{47} +19936.0 q^{48} +2401.00 q^{49} +24108.0 q^{51} +9520.00 q^{52} -2262.00 q^{53} -40600.0 q^{54} +17640.0 q^{56} -1372.00 q^{57} +34180.0 q^{58} -2730.00 q^{59} +25648.0 q^{61} -76440.0 q^{62} -2303.00 q^{63} -18368.0 q^{64} +32480.0 q^{66} +48404.0 q^{67} +117096. q^{68} -25536.0 q^{69} -58560.0 q^{71} -16920.0 q^{72} -68082.0 q^{73} +103980. q^{74} -6664.00 q^{76} +11368.0 q^{77} +19600.0 q^{78} +31784.0 q^{79} -45419.0 q^{81} -179620. q^{82} +20538.0 q^{83} +46648.0 q^{84} -108800. q^{86} +47852.0 q^{87} +83520.0 q^{88} -50582.0 q^{89} +6860.00 q^{91} -124032. q^{92} -107016. q^{93} -93240.0 q^{94} +38080.0 q^{96} +58506.0 q^{97} +24010.0 q^{98} -10904.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 10.0000 1.76777 0.883883 0.467707i $$-0.154920\pi$$
0.883883 + 0.467707i $$0.154920\pi$$
$$3$$ 14.0000 0.898100 0.449050 0.893507i $$-0.351762\pi$$
0.449050 + 0.893507i $$0.351762\pi$$
$$4$$ 68.0000 2.12500
$$5$$ 0 0
$$6$$ 140.000 1.58763
$$7$$ 49.0000 0.377964
$$8$$ 360.000 1.98874
$$9$$ −47.0000 −0.193416
$$10$$ 0 0
$$11$$ 232.000 0.578104 0.289052 0.957313i $$-0.406660\pi$$
0.289052 + 0.957313i $$0.406660\pi$$
$$12$$ 952.000 1.90846
$$13$$ 140.000 0.229757 0.114879 0.993380i $$-0.463352\pi$$
0.114879 + 0.993380i $$0.463352\pi$$
$$14$$ 490.000 0.668153
$$15$$ 0 0
$$16$$ 1424.00 1.39062
$$17$$ 1722.00 1.44514 0.722572 0.691296i $$-0.242960\pi$$
0.722572 + 0.691296i $$0.242960\pi$$
$$18$$ −470.000 −0.341914
$$19$$ −98.0000 −0.0622791 −0.0311395 0.999515i $$-0.509914\pi$$
−0.0311395 + 0.999515i $$0.509914\pi$$
$$20$$ 0 0
$$21$$ 686.000 0.339450
$$22$$ 2320.00 1.02195
$$23$$ −1824.00 −0.718961 −0.359480 0.933153i $$-0.617046\pi$$
−0.359480 + 0.933153i $$0.617046\pi$$
$$24$$ 5040.00 1.78609
$$25$$ 0 0
$$26$$ 1400.00 0.406158
$$27$$ −4060.00 −1.07181
$$28$$ 3332.00 0.803175
$$29$$ 3418.00 0.754705 0.377352 0.926070i $$-0.376835\pi$$
0.377352 + 0.926070i $$0.376835\pi$$
$$30$$ 0 0
$$31$$ −7644.00 −1.42862 −0.714310 0.699830i $$-0.753259\pi$$
−0.714310 + 0.699830i $$0.753259\pi$$
$$32$$ 2720.00 0.469563
$$33$$ 3248.00 0.519196
$$34$$ 17220.0 2.55468
$$35$$ 0 0
$$36$$ −3196.00 −0.411008
$$37$$ 10398.0 1.24866 0.624332 0.781159i $$-0.285371\pi$$
0.624332 + 0.781159i $$0.285371\pi$$
$$38$$ −980.000 −0.110095
$$39$$ 1960.00 0.206345
$$40$$ 0 0
$$41$$ −17962.0 −1.66876 −0.834382 0.551186i $$-0.814175\pi$$
−0.834382 + 0.551186i $$0.814175\pi$$
$$42$$ 6860.00 0.600069
$$43$$ −10880.0 −0.897342 −0.448671 0.893697i $$-0.648102\pi$$
−0.448671 + 0.893697i $$0.648102\pi$$
$$44$$ 15776.0 1.22847
$$45$$ 0 0
$$46$$ −18240.0 −1.27096
$$47$$ −9324.00 −0.615684 −0.307842 0.951438i $$-0.599607\pi$$
−0.307842 + 0.951438i $$0.599607\pi$$
$$48$$ 19936.0 1.24892
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ 24108.0 1.29788
$$52$$ 9520.00 0.488235
$$53$$ −2262.00 −0.110612 −0.0553061 0.998469i $$-0.517613\pi$$
−0.0553061 + 0.998469i $$0.517613\pi$$
$$54$$ −40600.0 −1.89471
$$55$$ 0 0
$$56$$ 17640.0 0.751672
$$57$$ −1372.00 −0.0559329
$$58$$ 34180.0 1.33414
$$59$$ −2730.00 −0.102102 −0.0510508 0.998696i $$-0.516257\pi$$
−0.0510508 + 0.998696i $$0.516257\pi$$
$$60$$ 0 0
$$61$$ 25648.0 0.882529 0.441264 0.897377i $$-0.354530\pi$$
0.441264 + 0.897377i $$0.354530\pi$$
$$62$$ −76440.0 −2.52547
$$63$$ −2303.00 −0.0731042
$$64$$ −18368.0 −0.560547
$$65$$ 0 0
$$66$$ 32480.0 0.917817
$$67$$ 48404.0 1.31733 0.658664 0.752437i $$-0.271122\pi$$
0.658664 + 0.752437i $$0.271122\pi$$
$$68$$ 117096. 3.07093
$$69$$ −25536.0 −0.645699
$$70$$ 0 0
$$71$$ −58560.0 −1.37865 −0.689327 0.724450i $$-0.742094\pi$$
−0.689327 + 0.724450i $$0.742094\pi$$
$$72$$ −16920.0 −0.384653
$$73$$ −68082.0 −1.49529 −0.747645 0.664099i $$-0.768815\pi$$
−0.747645 + 0.664099i $$0.768815\pi$$
$$74$$ 103980. 2.20735
$$75$$ 0 0
$$76$$ −6664.00 −0.132343
$$77$$ 11368.0 0.218503
$$78$$ 19600.0 0.364770
$$79$$ 31784.0 0.572982 0.286491 0.958083i $$-0.407511\pi$$
0.286491 + 0.958083i $$0.407511\pi$$
$$80$$ 0 0
$$81$$ −45419.0 −0.769175
$$82$$ −179620. −2.94999
$$83$$ 20538.0 0.327237 0.163619 0.986524i $$-0.447683\pi$$
0.163619 + 0.986524i $$0.447683\pi$$
$$84$$ 46648.0 0.721331
$$85$$ 0 0
$$86$$ −108800. −1.58629
$$87$$ 47852.0 0.677801
$$88$$ 83520.0 1.14970
$$89$$ −50582.0 −0.676894 −0.338447 0.940985i $$-0.609902\pi$$
−0.338447 + 0.940985i $$0.609902\pi$$
$$90$$ 0 0
$$91$$ 6860.00 0.0868402
$$92$$ −124032. −1.52779
$$93$$ −107016. −1.28304
$$94$$ −93240.0 −1.08839
$$95$$ 0 0
$$96$$ 38080.0 0.421715
$$97$$ 58506.0 0.631351 0.315676 0.948867i $$-0.397769\pi$$
0.315676 + 0.948867i $$0.397769\pi$$
$$98$$ 24010.0 0.252538
$$99$$ −10904.0 −0.111814
$$100$$ 0 0
$$101$$ 38696.0 0.377453 0.188726 0.982030i $$-0.439564\pi$$
0.188726 + 0.982030i $$0.439564\pi$$
$$102$$ 241080. 2.29436
$$103$$ −53060.0 −0.492804 −0.246402 0.969168i $$-0.579248\pi$$
−0.246402 + 0.969168i $$0.579248\pi$$
$$104$$ 50400.0 0.456927
$$105$$ 0 0
$$106$$ −22620.0 −0.195537
$$107$$ 146324. 1.23554 0.617769 0.786360i $$-0.288037\pi$$
0.617769 + 0.786360i $$0.288037\pi$$
$$108$$ −276080. −2.27759
$$109$$ 92898.0 0.748928 0.374464 0.927241i $$-0.377827\pi$$
0.374464 + 0.927241i $$0.377827\pi$$
$$110$$ 0 0
$$111$$ 145572. 1.12143
$$112$$ 69776.0 0.525607
$$113$$ 83354.0 0.614088 0.307044 0.951695i $$-0.400660\pi$$
0.307044 + 0.951695i $$0.400660\pi$$
$$114$$ −13720.0 −0.0988762
$$115$$ 0 0
$$116$$ 232424. 1.60375
$$117$$ −6580.00 −0.0444387
$$118$$ −27300.0 −0.180492
$$119$$ 84378.0 0.546213
$$120$$ 0 0
$$121$$ −107227. −0.665795
$$122$$ 256480. 1.56011
$$123$$ −251468. −1.49872
$$124$$ −519792. −3.03582
$$125$$ 0 0
$$126$$ −23030.0 −0.129231
$$127$$ −60384.0 −0.332210 −0.166105 0.986108i $$-0.553119\pi$$
−0.166105 + 0.986108i $$0.553119\pi$$
$$128$$ −270720. −1.46048
$$129$$ −152320. −0.805903
$$130$$ 0 0
$$131$$ −61586.0 −0.313548 −0.156774 0.987635i $$-0.550109\pi$$
−0.156774 + 0.987635i $$0.550109\pi$$
$$132$$ 220864. 1.10329
$$133$$ −4802.00 −0.0235393
$$134$$ 484040. 2.32873
$$135$$ 0 0
$$136$$ 619920. 2.87401
$$137$$ 204462. 0.930703 0.465352 0.885126i $$-0.345928\pi$$
0.465352 + 0.885126i $$0.345928\pi$$
$$138$$ −255360. −1.14145
$$139$$ −35406.0 −0.155432 −0.0777159 0.996976i $$-0.524763\pi$$
−0.0777159 + 0.996976i $$0.524763\pi$$
$$140$$ 0 0
$$141$$ −130536. −0.552946
$$142$$ −585600. −2.43714
$$143$$ 32480.0 0.132824
$$144$$ −66928.0 −0.268969
$$145$$ 0 0
$$146$$ −680820. −2.64332
$$147$$ 33614.0 0.128300
$$148$$ 707064. 2.65341
$$149$$ −20226.0 −0.0746353 −0.0373177 0.999303i $$-0.511881\pi$$
−0.0373177 + 0.999303i $$0.511881\pi$$
$$150$$ 0 0
$$151$$ 70904.0 0.253063 0.126531 0.991963i $$-0.459616\pi$$
0.126531 + 0.991963i $$0.459616\pi$$
$$152$$ −35280.0 −0.123857
$$153$$ −80934.0 −0.279513
$$154$$ 113680. 0.386262
$$155$$ 0 0
$$156$$ 133280. 0.438484
$$157$$ −293524. −0.950374 −0.475187 0.879885i $$-0.657620\pi$$
−0.475187 + 0.879885i $$0.657620\pi$$
$$158$$ 317840. 1.01290
$$159$$ −31668.0 −0.0993408
$$160$$ 0 0
$$161$$ −89376.0 −0.271742
$$162$$ −454190. −1.35972
$$163$$ −13192.0 −0.0388903 −0.0194452 0.999811i $$-0.506190\pi$$
−0.0194452 + 0.999811i $$0.506190\pi$$
$$164$$ −1.22142e6 −3.54612
$$165$$ 0 0
$$166$$ 205380. 0.578479
$$167$$ −493612. −1.36960 −0.684801 0.728730i $$-0.740111\pi$$
−0.684801 + 0.728730i $$0.740111\pi$$
$$168$$ 246960. 0.675077
$$169$$ −351693. −0.947212
$$170$$ 0 0
$$171$$ 4606.00 0.0120457
$$172$$ −739840. −1.90685
$$173$$ −240716. −0.611490 −0.305745 0.952113i $$-0.598906\pi$$
−0.305745 + 0.952113i $$0.598906\pi$$
$$174$$ 478520. 1.19819
$$175$$ 0 0
$$176$$ 330368. 0.803926
$$177$$ −38220.0 −0.0916975
$$178$$ −505820. −1.19659
$$179$$ 294932. 0.688001 0.344001 0.938969i $$-0.388218\pi$$
0.344001 + 0.938969i $$0.388218\pi$$
$$180$$ 0 0
$$181$$ −336980. −0.764553 −0.382277 0.924048i $$-0.624860\pi$$
−0.382277 + 0.924048i $$0.624860\pi$$
$$182$$ 68600.0 0.153513
$$183$$ 359072. 0.792600
$$184$$ −656640. −1.42982
$$185$$ 0 0
$$186$$ −1.07016e6 −2.26812
$$187$$ 399504. 0.835444
$$188$$ −634032. −1.30833
$$189$$ −198940. −0.405105
$$190$$ 0 0
$$191$$ 358264. 0.710591 0.355296 0.934754i $$-0.384380\pi$$
0.355296 + 0.934754i $$0.384380\pi$$
$$192$$ −257152. −0.503427
$$193$$ 989554. 1.91226 0.956128 0.292948i $$-0.0946362\pi$$
0.956128 + 0.292948i $$0.0946362\pi$$
$$194$$ 585060. 1.11608
$$195$$ 0 0
$$196$$ 163268. 0.303571
$$197$$ 990050. 1.81757 0.908786 0.417263i $$-0.137011\pi$$
0.908786 + 0.417263i $$0.137011\pi$$
$$198$$ −109040. −0.197662
$$199$$ −840756. −1.50500 −0.752501 0.658591i $$-0.771153\pi$$
−0.752501 + 0.658591i $$0.771153\pi$$
$$200$$ 0 0
$$201$$ 677656. 1.18309
$$202$$ 386960. 0.667249
$$203$$ 167482. 0.285252
$$204$$ 1.63934e6 2.75800
$$205$$ 0 0
$$206$$ −530600. −0.871163
$$207$$ 85728.0 0.139058
$$208$$ 199360. 0.319506
$$209$$ −22736.0 −0.0360038
$$210$$ 0 0
$$211$$ 1.15073e6 1.77938 0.889689 0.456568i $$-0.150921\pi$$
0.889689 + 0.456568i $$0.150921\pi$$
$$212$$ −153816. −0.235051
$$213$$ −819840. −1.23817
$$214$$ 1.46324e6 2.18414
$$215$$ 0 0
$$216$$ −1.46160e6 −2.13154
$$217$$ −374556. −0.539967
$$218$$ 928980. 1.32393
$$219$$ −953148. −1.34292
$$220$$ 0 0
$$221$$ 241080. 0.332032
$$222$$ 1.45572e6 1.98242
$$223$$ 824264. 1.10995 0.554976 0.831866i $$-0.312727\pi$$
0.554976 + 0.831866i $$0.312727\pi$$
$$224$$ 133280. 0.177478
$$225$$ 0 0
$$226$$ 833540. 1.08556
$$227$$ −74382.0 −0.0958083 −0.0479042 0.998852i $$-0.515254\pi$$
−0.0479042 + 0.998852i $$0.515254\pi$$
$$228$$ −93296.0 −0.118857
$$229$$ 1.13196e6 1.42640 0.713199 0.700961i $$-0.247245\pi$$
0.713199 + 0.700961i $$0.247245\pi$$
$$230$$ 0 0
$$231$$ 159152. 0.196238
$$232$$ 1.23048e6 1.50091
$$233$$ 198726. 0.239809 0.119904 0.992785i $$-0.461741\pi$$
0.119904 + 0.992785i $$0.461741\pi$$
$$234$$ −65800.0 −0.0785572
$$235$$ 0 0
$$236$$ −185640. −0.216966
$$237$$ 444976. 0.514595
$$238$$ 843780. 0.965577
$$239$$ 482904. 0.546847 0.273424 0.961894i $$-0.411844\pi$$
0.273424 + 0.961894i $$0.411844\pi$$
$$240$$ 0 0
$$241$$ 805910. 0.893807 0.446904 0.894582i $$-0.352527\pi$$
0.446904 + 0.894582i $$0.352527\pi$$
$$242$$ −1.07227e6 −1.17697
$$243$$ 350714. 0.381011
$$244$$ 1.74406e6 1.87537
$$245$$ 0 0
$$246$$ −2.51468e6 −2.64938
$$247$$ −13720.0 −0.0143091
$$248$$ −2.75184e6 −2.84115
$$249$$ 287532. 0.293892
$$250$$ 0 0
$$251$$ 430738. 0.431548 0.215774 0.976443i $$-0.430773\pi$$
0.215774 + 0.976443i $$0.430773\pi$$
$$252$$ −156604. −0.155347
$$253$$ −423168. −0.415634
$$254$$ −603840. −0.587270
$$255$$ 0 0
$$256$$ −2.11942e6 −2.02124
$$257$$ 1.17691e6 1.11150 0.555751 0.831349i $$-0.312431\pi$$
0.555751 + 0.831349i $$0.312431\pi$$
$$258$$ −1.52320e6 −1.42465
$$259$$ 509502. 0.471951
$$260$$ 0 0
$$261$$ −160646. −0.145972
$$262$$ −615860. −0.554279
$$263$$ −1.29098e6 −1.15088 −0.575438 0.817845i $$-0.695169\pi$$
−0.575438 + 0.817845i $$0.695169\pi$$
$$264$$ 1.16928e6 1.03254
$$265$$ 0 0
$$266$$ −48020.0 −0.0416119
$$267$$ −708148. −0.607919
$$268$$ 3.29147e6 2.79932
$$269$$ −1.27756e6 −1.07646 −0.538232 0.842797i $$-0.680907\pi$$
−0.538232 + 0.842797i $$0.680907\pi$$
$$270$$ 0 0
$$271$$ 1.65054e6 1.36522 0.682612 0.730781i $$-0.260844\pi$$
0.682612 + 0.730781i $$0.260844\pi$$
$$272$$ 2.45213e6 2.00965
$$273$$ 96040.0 0.0779912
$$274$$ 2.04462e6 1.64527
$$275$$ 0 0
$$276$$ −1.73645e6 −1.37211
$$277$$ 1.06409e6 0.833257 0.416628 0.909077i $$-0.363212\pi$$
0.416628 + 0.909077i $$0.363212\pi$$
$$278$$ −354060. −0.274767
$$279$$ 359268. 0.276317
$$280$$ 0 0
$$281$$ −22342.0 −0.0168794 −0.00843969 0.999964i $$-0.502686\pi$$
−0.00843969 + 0.999964i $$0.502686\pi$$
$$282$$ −1.30536e6 −0.977479
$$283$$ 2.49574e6 1.85239 0.926196 0.377042i $$-0.123059\pi$$
0.926196 + 0.377042i $$0.123059\pi$$
$$284$$ −3.98208e6 −2.92964
$$285$$ 0 0
$$286$$ 324800. 0.234802
$$287$$ −880138. −0.630734
$$288$$ −127840. −0.0908208
$$289$$ 1.54543e6 1.08844
$$290$$ 0 0
$$291$$ 819084. 0.567017
$$292$$ −4.62958e6 −3.17749
$$293$$ 1.93178e6 1.31458 0.657291 0.753637i $$-0.271702\pi$$
0.657291 + 0.753637i $$0.271702\pi$$
$$294$$ 336140. 0.226805
$$295$$ 0 0
$$296$$ 3.74328e6 2.48326
$$297$$ −941920. −0.619616
$$298$$ −202260. −0.131938
$$299$$ −255360. −0.165187
$$300$$ 0 0
$$301$$ −533120. −0.339163
$$302$$ 709040. 0.447356
$$303$$ 541744. 0.338991
$$304$$ −139552. −0.0866068
$$305$$ 0 0
$$306$$ −809340. −0.494114
$$307$$ 459074. 0.277995 0.138997 0.990293i $$-0.455612\pi$$
0.138997 + 0.990293i $$0.455612\pi$$
$$308$$ 773024. 0.464319
$$309$$ −742840. −0.442587
$$310$$ 0 0
$$311$$ 667128. 0.391118 0.195559 0.980692i $$-0.437348\pi$$
0.195559 + 0.980692i $$0.437348\pi$$
$$312$$ 705600. 0.410367
$$313$$ 111034. 0.0640612 0.0320306 0.999487i $$-0.489803\pi$$
0.0320306 + 0.999487i $$0.489803\pi$$
$$314$$ −2.93524e6 −1.68004
$$315$$ 0 0
$$316$$ 2.16131e6 1.21759
$$317$$ 68778.0 0.0384416 0.0192208 0.999815i $$-0.493881\pi$$
0.0192208 + 0.999815i $$0.493881\pi$$
$$318$$ −316680. −0.175611
$$319$$ 792976. 0.436298
$$320$$ 0 0
$$321$$ 2.04854e6 1.10964
$$322$$ −893760. −0.480376
$$323$$ −168756. −0.0900022
$$324$$ −3.08849e6 −1.63450
$$325$$ 0 0
$$326$$ −131920. −0.0687490
$$327$$ 1.30057e6 0.672613
$$328$$ −6.46632e6 −3.31874
$$329$$ −456876. −0.232707
$$330$$ 0 0
$$331$$ −564448. −0.283174 −0.141587 0.989926i $$-0.545221\pi$$
−0.141587 + 0.989926i $$0.545221\pi$$
$$332$$ 1.39658e6 0.695379
$$333$$ −488706. −0.241511
$$334$$ −4.93612e6 −2.42114
$$335$$ 0 0
$$336$$ 976864. 0.472048
$$337$$ −2.07729e6 −0.996376 −0.498188 0.867069i $$-0.666001\pi$$
−0.498188 + 0.867069i $$0.666001\pi$$
$$338$$ −3.51693e6 −1.67445
$$339$$ 1.16696e6 0.551512
$$340$$ 0 0
$$341$$ −1.77341e6 −0.825891
$$342$$ 46060.0 0.0212941
$$343$$ 117649. 0.0539949
$$344$$ −3.91680e6 −1.78458
$$345$$ 0 0
$$346$$ −2.40716e6 −1.08097
$$347$$ 53248.0 0.0237399 0.0118700 0.999930i $$-0.496222\pi$$
0.0118700 + 0.999930i $$0.496222\pi$$
$$348$$ 3.25394e6 1.44033
$$349$$ −2.27200e6 −0.998494 −0.499247 0.866460i $$-0.666390\pi$$
−0.499247 + 0.866460i $$0.666390\pi$$
$$350$$ 0 0
$$351$$ −568400. −0.246256
$$352$$ 631040. 0.271456
$$353$$ −4.00645e6 −1.71129 −0.855644 0.517565i $$-0.826838\pi$$
−0.855644 + 0.517565i $$0.826838\pi$$
$$354$$ −382200. −0.162100
$$355$$ 0 0
$$356$$ −3.43958e6 −1.43840
$$357$$ 1.18129e6 0.490554
$$358$$ 2.94932e6 1.21623
$$359$$ 73784.0 0.0302152 0.0151076 0.999886i $$-0.495191\pi$$
0.0151076 + 0.999886i $$0.495191\pi$$
$$360$$ 0 0
$$361$$ −2.46650e6 −0.996121
$$362$$ −3.36980e6 −1.35155
$$363$$ −1.50118e6 −0.597951
$$364$$ 466480. 0.184535
$$365$$ 0 0
$$366$$ 3.59072e6 1.40113
$$367$$ −1.40431e6 −0.544250 −0.272125 0.962262i $$-0.587726\pi$$
−0.272125 + 0.962262i $$0.587726\pi$$
$$368$$ −2.59738e6 −0.999805
$$369$$ 844214. 0.322765
$$370$$ 0 0
$$371$$ −110838. −0.0418075
$$372$$ −7.27709e6 −2.72647
$$373$$ 1.60323e6 0.596657 0.298329 0.954463i $$-0.403571\pi$$
0.298329 + 0.954463i $$0.403571\pi$$
$$374$$ 3.99504e6 1.47687
$$375$$ 0 0
$$376$$ −3.35664e6 −1.22443
$$377$$ 478520. 0.173399
$$378$$ −1.98940e6 −0.716131
$$379$$ −4.77012e6 −1.70581 −0.852906 0.522064i $$-0.825162\pi$$
−0.852906 + 0.522064i $$0.825162\pi$$
$$380$$ 0 0
$$381$$ −845376. −0.298358
$$382$$ 3.58264e6 1.25616
$$383$$ 2.23079e6 0.777072 0.388536 0.921434i $$-0.372981\pi$$
0.388536 + 0.921434i $$0.372981\pi$$
$$384$$ −3.79008e6 −1.31166
$$385$$ 0 0
$$386$$ 9.89554e6 3.38042
$$387$$ 511360. 0.173560
$$388$$ 3.97841e6 1.34162
$$389$$ 4.84024e6 1.62178 0.810892 0.585196i $$-0.198982\pi$$
0.810892 + 0.585196i $$0.198982\pi$$
$$390$$ 0 0
$$391$$ −3.14093e6 −1.03900
$$392$$ 864360. 0.284105
$$393$$ −862204. −0.281597
$$394$$ 9.90050e6 3.21304
$$395$$ 0 0
$$396$$ −741472. −0.237606
$$397$$ −995820. −0.317106 −0.158553 0.987350i $$-0.550683\pi$$
−0.158553 + 0.987350i $$0.550683\pi$$
$$398$$ −8.40756e6 −2.66049
$$399$$ −67228.0 −0.0211406
$$400$$ 0 0
$$401$$ −3.31605e6 −1.02982 −0.514909 0.857245i $$-0.672174\pi$$
−0.514909 + 0.857245i $$0.672174\pi$$
$$402$$ 6.77656e6 2.09143
$$403$$ −1.07016e6 −0.328236
$$404$$ 2.63133e6 0.802087
$$405$$ 0 0
$$406$$ 1.67482e6 0.504258
$$407$$ 2.41234e6 0.721858
$$408$$ 8.67888e6 2.58115
$$409$$ 3.07273e6 0.908274 0.454137 0.890932i $$-0.349948\pi$$
0.454137 + 0.890932i $$0.349948\pi$$
$$410$$ 0 0
$$411$$ 2.86247e6 0.835865
$$412$$ −3.60808e6 −1.04721
$$413$$ −133770. −0.0385908
$$414$$ 857280. 0.245823
$$415$$ 0 0
$$416$$ 380800. 0.107886
$$417$$ −495684. −0.139593
$$418$$ −227360. −0.0636463
$$419$$ 2.81438e6 0.783154 0.391577 0.920145i $$-0.371930\pi$$
0.391577 + 0.920145i $$0.371930\pi$$
$$420$$ 0 0
$$421$$ 3.05802e6 0.840883 0.420441 0.907320i $$-0.361875\pi$$
0.420441 + 0.907320i $$0.361875\pi$$
$$422$$ 1.15073e7 3.14552
$$423$$ 438228. 0.119083
$$424$$ −814320. −0.219979
$$425$$ 0 0
$$426$$ −8.19840e6 −2.18880
$$427$$ 1.25675e6 0.333565
$$428$$ 9.95003e6 2.62552
$$429$$ 454720. 0.119289
$$430$$ 0 0
$$431$$ 1.93750e6 0.502398 0.251199 0.967936i $$-0.419175\pi$$
0.251199 + 0.967936i $$0.419175\pi$$
$$432$$ −5.78144e6 −1.49048
$$433$$ −3.94790e6 −1.01192 −0.505961 0.862557i $$-0.668862\pi$$
−0.505961 + 0.862557i $$0.668862\pi$$
$$434$$ −3.74556e6 −0.954536
$$435$$ 0 0
$$436$$ 6.31706e6 1.59147
$$437$$ 178752. 0.0447762
$$438$$ −9.53148e6 −2.37397
$$439$$ −7.41770e6 −1.83700 −0.918498 0.395426i $$-0.870597\pi$$
−0.918498 + 0.395426i $$0.870597\pi$$
$$440$$ 0 0
$$441$$ −112847. −0.0276308
$$442$$ 2.41080e6 0.586956
$$443$$ −1.40269e6 −0.339589 −0.169794 0.985480i $$-0.554310\pi$$
−0.169794 + 0.985480i $$0.554310\pi$$
$$444$$ 9.89890e6 2.38303
$$445$$ 0 0
$$446$$ 8.24264e6 1.96214
$$447$$ −283164. −0.0670300
$$448$$ −900032. −0.211867
$$449$$ −590574. −0.138248 −0.0691239 0.997608i $$-0.522020\pi$$
−0.0691239 + 0.997608i $$0.522020\pi$$
$$450$$ 0 0
$$451$$ −4.16718e6 −0.964720
$$452$$ 5.66807e6 1.30494
$$453$$ 992656. 0.227276
$$454$$ −743820. −0.169367
$$455$$ 0 0
$$456$$ −493920. −0.111236
$$457$$ 2.90484e6 0.650627 0.325313 0.945606i $$-0.394530\pi$$
0.325313 + 0.945606i $$0.394530\pi$$
$$458$$ 1.13196e7 2.52154
$$459$$ −6.99132e6 −1.54891
$$460$$ 0 0
$$461$$ −922684. −0.202209 −0.101105 0.994876i $$-0.532238\pi$$
−0.101105 + 0.994876i $$0.532238\pi$$
$$462$$ 1.59152e6 0.346902
$$463$$ −7.18235e6 −1.55709 −0.778546 0.627588i $$-0.784042\pi$$
−0.778546 + 0.627588i $$0.784042\pi$$
$$464$$ 4.86723e6 1.04951
$$465$$ 0 0
$$466$$ 1.98726e6 0.423926
$$467$$ 612570. 0.129976 0.0649881 0.997886i $$-0.479299\pi$$
0.0649881 + 0.997886i $$0.479299\pi$$
$$468$$ −447440. −0.0944322
$$469$$ 2.37180e6 0.497904
$$470$$ 0 0
$$471$$ −4.10934e6 −0.853531
$$472$$ −982800. −0.203053
$$473$$ −2.52416e6 −0.518757
$$474$$ 4.44976e6 0.909684
$$475$$ 0 0
$$476$$ 5.73770e6 1.16070
$$477$$ 106314. 0.0213941
$$478$$ 4.82904e6 0.966699
$$479$$ 2.60330e6 0.518424 0.259212 0.965820i $$-0.416537\pi$$
0.259212 + 0.965820i $$0.416537\pi$$
$$480$$ 0 0
$$481$$ 1.45572e6 0.286890
$$482$$ 8.05910e6 1.58004
$$483$$ −1.25126e6 −0.244051
$$484$$ −7.29144e6 −1.41482
$$485$$ 0 0
$$486$$ 3.50714e6 0.673539
$$487$$ −5.46309e6 −1.04380 −0.521898 0.853008i $$-0.674776\pi$$
−0.521898 + 0.853008i $$0.674776\pi$$
$$488$$ 9.23328e6 1.75512
$$489$$ −184688. −0.0349274
$$490$$ 0 0
$$491$$ 1.64090e6 0.307170 0.153585 0.988135i $$-0.450918\pi$$
0.153585 + 0.988135i $$0.450918\pi$$
$$492$$ −1.70998e7 −3.18478
$$493$$ 5.88580e6 1.09066
$$494$$ −137200. −0.0252951
$$495$$ 0 0
$$496$$ −1.08851e7 −1.98667
$$497$$ −2.86944e6 −0.521082
$$498$$ 2.87532e6 0.519533
$$499$$ 2.99796e6 0.538983 0.269491 0.963003i $$-0.413144\pi$$
0.269491 + 0.963003i $$0.413144\pi$$
$$500$$ 0 0
$$501$$ −6.91057e6 −1.23004
$$502$$ 4.30738e6 0.762876
$$503$$ 6.89405e6 1.21494 0.607469 0.794343i $$-0.292185\pi$$
0.607469 + 0.794343i $$0.292185\pi$$
$$504$$ −829080. −0.145385
$$505$$ 0 0
$$506$$ −4.23168e6 −0.734745
$$507$$ −4.92370e6 −0.850691
$$508$$ −4.10611e6 −0.705946
$$509$$ 2.30476e6 0.394305 0.197152 0.980373i $$-0.436831\pi$$
0.197152 + 0.980373i $$0.436831\pi$$
$$510$$ 0 0
$$511$$ −3.33602e6 −0.565166
$$512$$ −1.25312e7 −2.11260
$$513$$ 397880. 0.0667511
$$514$$ 1.17691e7 1.96488
$$515$$ 0 0
$$516$$ −1.03578e7 −1.71254
$$517$$ −2.16317e6 −0.355929
$$518$$ 5.09502e6 0.834299
$$519$$ −3.37002e6 −0.549180
$$520$$ 0 0
$$521$$ −1.20960e7 −1.95231 −0.976155 0.217073i $$-0.930349\pi$$
−0.976155 + 0.217073i $$0.930349\pi$$
$$522$$ −1.60646e6 −0.258044
$$523$$ −5.48443e6 −0.876753 −0.438377 0.898791i $$-0.644446\pi$$
−0.438377 + 0.898791i $$0.644446\pi$$
$$524$$ −4.18785e6 −0.666289
$$525$$ 0 0
$$526$$ −1.29098e7 −2.03448
$$527$$ −1.31630e7 −2.06456
$$528$$ 4.62515e6 0.722007
$$529$$ −3.10937e6 −0.483095
$$530$$ 0 0
$$531$$ 128310. 0.0197480
$$532$$ −326536. −0.0500210
$$533$$ −2.51468e6 −0.383411
$$534$$ −7.08148e6 −1.07466
$$535$$ 0 0
$$536$$ 1.74254e7 2.61982
$$537$$ 4.12905e6 0.617894
$$538$$ −1.27756e7 −1.90294
$$539$$ 557032. 0.0825863
$$540$$ 0 0
$$541$$ −6.71799e6 −0.986839 −0.493420 0.869791i $$-0.664253\pi$$
−0.493420 + 0.869791i $$0.664253\pi$$
$$542$$ 1.65054e7 2.41340
$$543$$ −4.71772e6 −0.686646
$$544$$ 4.68384e6 0.678586
$$545$$ 0 0
$$546$$ 960400. 0.137870
$$547$$ 5.00235e6 0.714835 0.357418 0.933945i $$-0.383657\pi$$
0.357418 + 0.933945i $$0.383657\pi$$
$$548$$ 1.39034e7 1.97774
$$549$$ −1.20546e6 −0.170695
$$550$$ 0 0
$$551$$ −334964. −0.0470023
$$552$$ −9.19296e6 −1.28413
$$553$$ 1.55742e6 0.216567
$$554$$ 1.06409e7 1.47300
$$555$$ 0 0
$$556$$ −2.40761e6 −0.330293
$$557$$ −9.01961e6 −1.23183 −0.615913 0.787814i $$-0.711213\pi$$
−0.615913 + 0.787814i $$0.711213\pi$$
$$558$$ 3.59268e6 0.488465
$$559$$ −1.52320e6 −0.206171
$$560$$ 0 0
$$561$$ 5.59306e6 0.750312
$$562$$ −223420. −0.0298388
$$563$$ −1.24051e7 −1.64941 −0.824707 0.565561i $$-0.808660\pi$$
−0.824707 + 0.565561i $$0.808660\pi$$
$$564$$ −8.87645e6 −1.17501
$$565$$ 0 0
$$566$$ 2.49574e7 3.27460
$$567$$ −2.22553e6 −0.290721
$$568$$ −2.10816e7 −2.74178
$$569$$ 6.48804e6 0.840103 0.420052 0.907500i $$-0.362012\pi$$
0.420052 + 0.907500i $$0.362012\pi$$
$$570$$ 0 0
$$571$$ −1.02285e7 −1.31287 −0.656435 0.754382i $$-0.727936\pi$$
−0.656435 + 0.754382i $$0.727936\pi$$
$$572$$ 2.20864e6 0.282251
$$573$$ 5.01570e6 0.638182
$$574$$ −8.80138e6 −1.11499
$$575$$ 0 0
$$576$$ 863296. 0.108419
$$577$$ −2.65338e6 −0.331787 −0.165894 0.986144i $$-0.553051\pi$$
−0.165894 + 0.986144i $$0.553051\pi$$
$$578$$ 1.54543e7 1.92411
$$579$$ 1.38538e7 1.71740
$$580$$ 0 0
$$581$$ 1.00636e6 0.123684
$$582$$ 8.19084e6 1.00235
$$583$$ −524784. −0.0639454
$$584$$ −2.45095e7 −2.97374
$$585$$ 0 0
$$586$$ 1.93178e7 2.32387
$$587$$ 1.43044e7 1.71346 0.856729 0.515766i $$-0.172493\pi$$
0.856729 + 0.515766i $$0.172493\pi$$
$$588$$ 2.28575e6 0.272638
$$589$$ 749112. 0.0889731
$$590$$ 0 0
$$591$$ 1.38607e7 1.63236
$$592$$ 1.48068e7 1.73642
$$593$$ 1.00265e7 1.17088 0.585442 0.810714i $$-0.300921\pi$$
0.585442 + 0.810714i $$0.300921\pi$$
$$594$$ −9.41920e6 −1.09534
$$595$$ 0 0
$$596$$ −1.37537e6 −0.158600
$$597$$ −1.17706e7 −1.35164
$$598$$ −2.55360e6 −0.292011
$$599$$ −7.52292e6 −0.856681 −0.428341 0.903617i $$-0.640902\pi$$
−0.428341 + 0.903617i $$0.640902\pi$$
$$600$$ 0 0
$$601$$ 3.38625e6 0.382413 0.191207 0.981550i $$-0.438760\pi$$
0.191207 + 0.981550i $$0.438760\pi$$
$$602$$ −5.33120e6 −0.599562
$$603$$ −2.27499e6 −0.254792
$$604$$ 4.82147e6 0.537759
$$605$$ 0 0
$$606$$ 5.41744e6 0.599256
$$607$$ 6.90861e6 0.761060 0.380530 0.924769i $$-0.375742\pi$$
0.380530 + 0.924769i $$0.375742\pi$$
$$608$$ −266560. −0.0292439
$$609$$ 2.34475e6 0.256185
$$610$$ 0 0
$$611$$ −1.30536e6 −0.141458
$$612$$ −5.50351e6 −0.593966
$$613$$ 9.68896e6 1.04142 0.520710 0.853734i $$-0.325667\pi$$
0.520710 + 0.853734i $$0.325667\pi$$
$$614$$ 4.59074e6 0.491430
$$615$$ 0 0
$$616$$ 4.09248e6 0.434545
$$617$$ 7.84742e6 0.829877 0.414939 0.909849i $$-0.363803\pi$$
0.414939 + 0.909849i $$0.363803\pi$$
$$618$$ −7.42840e6 −0.782391
$$619$$ −1.01972e7 −1.06968 −0.534840 0.844953i $$-0.679628\pi$$
−0.534840 + 0.844953i $$0.679628\pi$$
$$620$$ 0 0
$$621$$ 7.40544e6 0.770587
$$622$$ 6.67128e6 0.691406
$$623$$ −2.47852e6 −0.255842
$$624$$ 2.79104e6 0.286949
$$625$$ 0 0
$$626$$ 1.11034e6 0.113245
$$627$$ −318304. −0.0323350
$$628$$ −1.99596e7 −2.01954
$$629$$ 1.79054e7 1.80450
$$630$$ 0 0
$$631$$ −8.36258e6 −0.836116 −0.418058 0.908420i $$-0.637289\pi$$
−0.418058 + 0.908420i $$0.637289\pi$$
$$632$$ 1.14422e7 1.13951
$$633$$ 1.61102e7 1.59806
$$634$$ 687780. 0.0679558
$$635$$ 0 0
$$636$$ −2.15342e6 −0.211099
$$637$$ 336140. 0.0328225
$$638$$ 7.92976e6 0.771273
$$639$$ 2.75232e6 0.266653
$$640$$ 0 0
$$641$$ 1.10283e6 0.106014 0.0530070 0.998594i $$-0.483119\pi$$
0.0530070 + 0.998594i $$0.483119\pi$$
$$642$$ 2.04854e7 1.96158
$$643$$ −1.71354e7 −1.63443 −0.817217 0.576330i $$-0.804484\pi$$
−0.817217 + 0.576330i $$0.804484\pi$$
$$644$$ −6.07757e6 −0.577451
$$645$$ 0 0
$$646$$ −1.68756e6 −0.159103
$$647$$ 54964.0 0.00516200 0.00258100 0.999997i $$-0.499178\pi$$
0.00258100 + 0.999997i $$0.499178\pi$$
$$648$$ −1.63508e7 −1.52969
$$649$$ −633360. −0.0590254
$$650$$ 0 0
$$651$$ −5.24378e6 −0.484945
$$652$$ −897056. −0.0826420
$$653$$ 485166. 0.0445254 0.0222627 0.999752i $$-0.492913\pi$$
0.0222627 + 0.999752i $$0.492913\pi$$
$$654$$ 1.30057e7 1.18902
$$655$$ 0 0
$$656$$ −2.55779e7 −2.32063
$$657$$ 3.19985e6 0.289212
$$658$$ −4.56876e6 −0.411371
$$659$$ −2.72136e6 −0.244103 −0.122051 0.992524i $$-0.538947\pi$$
−0.122051 + 0.992524i $$0.538947\pi$$
$$660$$ 0 0
$$661$$ −2.14525e6 −0.190974 −0.0954869 0.995431i $$-0.530441\pi$$
−0.0954869 + 0.995431i $$0.530441\pi$$
$$662$$ −5.64448e6 −0.500586
$$663$$ 3.37512e6 0.298198
$$664$$ 7.39368e6 0.650789
$$665$$ 0 0
$$666$$ −4.88706e6 −0.426935
$$667$$ −6.23443e6 −0.542603
$$668$$ −3.35656e7 −2.91041
$$669$$ 1.15397e7 0.996848
$$670$$ 0 0
$$671$$ 5.95034e6 0.510194
$$672$$ 1.86592e6 0.159393
$$673$$ −2.92796e6 −0.249188 −0.124594 0.992208i $$-0.539763\pi$$
−0.124594 + 0.992208i $$0.539763\pi$$
$$674$$ −2.07729e7 −1.76136
$$675$$ 0 0
$$676$$ −2.39151e7 −2.01282
$$677$$ 1.34992e7 1.13198 0.565988 0.824414i $$-0.308495\pi$$
0.565988 + 0.824414i $$0.308495\pi$$
$$678$$ 1.16696e7 0.974945
$$679$$ 2.86679e6 0.238628
$$680$$ 0 0
$$681$$ −1.04135e6 −0.0860455
$$682$$ −1.77341e7 −1.45998
$$683$$ 5.42972e6 0.445375 0.222688 0.974890i $$-0.428517\pi$$
0.222688 + 0.974890i $$0.428517\pi$$
$$684$$ 313208. 0.0255972
$$685$$ 0 0
$$686$$ 1.17649e6 0.0954504
$$687$$ 1.58474e7 1.28105
$$688$$ −1.54931e7 −1.24787
$$689$$ −316680. −0.0254140
$$690$$ 0 0
$$691$$ 2.08280e7 1.65940 0.829702 0.558207i $$-0.188510\pi$$
0.829702 + 0.558207i $$0.188510\pi$$
$$692$$ −1.63687e7 −1.29942
$$693$$ −534296. −0.0422619
$$694$$ 532480. 0.0419667
$$695$$ 0 0
$$696$$ 1.72267e7 1.34797
$$697$$ −3.09306e7 −2.41160
$$698$$ −2.27200e7 −1.76510
$$699$$ 2.78216e6 0.215372
$$700$$ 0 0
$$701$$ 2.35141e7 1.80731 0.903655 0.428261i $$-0.140874\pi$$
0.903655 + 0.428261i $$0.140874\pi$$
$$702$$ −5.68400e6 −0.435323
$$703$$ −1.01900e6 −0.0777656
$$704$$ −4.26138e6 −0.324055
$$705$$ 0 0
$$706$$ −4.00645e7 −3.02516
$$707$$ 1.89610e6 0.142664
$$708$$ −2.59896e6 −0.194857
$$709$$ −1.95747e7 −1.46244 −0.731221 0.682140i $$-0.761049\pi$$
−0.731221 + 0.682140i $$0.761049\pi$$
$$710$$ 0 0
$$711$$ −1.49385e6 −0.110824
$$712$$ −1.82095e7 −1.34617
$$713$$ 1.39427e7 1.02712
$$714$$ 1.18129e7 0.867185
$$715$$ 0 0
$$716$$ 2.00554e7 1.46200
$$717$$ 6.76066e6 0.491124
$$718$$ 737840. 0.0534135
$$719$$ −2.61152e7 −1.88396 −0.941978 0.335674i $$-0.891036\pi$$
−0.941978 + 0.335674i $$0.891036\pi$$
$$720$$ 0 0
$$721$$ −2.59994e6 −0.186262
$$722$$ −2.46650e7 −1.76091
$$723$$ 1.12827e7 0.802729
$$724$$ −2.29146e7 −1.62468
$$725$$ 0 0
$$726$$ −1.50118e7 −1.05704
$$727$$ −1.54126e7 −1.08154 −0.540768 0.841172i $$-0.681866\pi$$
−0.540768 + 0.841172i $$0.681866\pi$$
$$728$$ 2.46960e6 0.172702
$$729$$ 1.59468e7 1.11136
$$730$$ 0 0
$$731$$ −1.87354e7 −1.29679
$$732$$ 2.44169e7 1.68427
$$733$$ 1.69868e7 1.16776 0.583878 0.811841i $$-0.301535\pi$$
0.583878 + 0.811841i $$0.301535\pi$$
$$734$$ −1.40431e7 −0.962107
$$735$$ 0 0
$$736$$ −4.96128e6 −0.337597
$$737$$ 1.12297e7 0.761554
$$738$$ 8.44214e6 0.570574
$$739$$ 2.01511e6 0.135734 0.0678669 0.997694i $$-0.478381\pi$$
0.0678669 + 0.997694i $$0.478381\pi$$
$$740$$ 0 0
$$741$$ −192080. −0.0128510
$$742$$ −1.10838e6 −0.0739059
$$743$$ 1.51381e7 1.00600 0.503001 0.864286i $$-0.332229\pi$$
0.503001 + 0.864286i $$0.332229\pi$$
$$744$$ −3.85258e7 −2.55164
$$745$$ 0 0
$$746$$ 1.60323e7 1.05475
$$747$$ −965286. −0.0632928
$$748$$ 2.71663e7 1.77532
$$749$$ 7.16988e6 0.466989
$$750$$ 0 0
$$751$$ 7.21401e6 0.466742 0.233371 0.972388i $$-0.425024\pi$$
0.233371 + 0.972388i $$0.425024\pi$$
$$752$$ −1.32774e7 −0.856185
$$753$$ 6.03033e6 0.387573
$$754$$ 4.78520e6 0.306529
$$755$$ 0 0
$$756$$ −1.35279e7 −0.860848
$$757$$ 1.09697e7 0.695755 0.347877 0.937540i $$-0.386903\pi$$
0.347877 + 0.937540i $$0.386903\pi$$
$$758$$ −4.77012e7 −3.01548
$$759$$ −5.92435e6 −0.373281
$$760$$ 0 0
$$761$$ 1.92442e7 1.20459 0.602293 0.798275i $$-0.294254\pi$$
0.602293 + 0.798275i $$0.294254\pi$$
$$762$$ −8.45376e6 −0.527427
$$763$$ 4.55200e6 0.283068
$$764$$ 2.43620e7 1.51001
$$765$$ 0 0
$$766$$ 2.23079e7 1.37368
$$767$$ −382200. −0.0234586
$$768$$ −2.96719e7 −1.81528
$$769$$ 8.21185e6 0.500755 0.250378 0.968148i $$-0.419445\pi$$
0.250378 + 0.968148i $$0.419445\pi$$
$$770$$ 0 0
$$771$$ 1.64767e7 0.998241
$$772$$ 6.72897e7 4.06355
$$773$$ −1.86187e7 −1.12073 −0.560363 0.828247i $$-0.689338\pi$$
−0.560363 + 0.828247i $$0.689338\pi$$
$$774$$ 5.11360e6 0.306813
$$775$$ 0 0
$$776$$ 2.10622e7 1.25559
$$777$$ 7.13303e6 0.423859
$$778$$ 4.84024e7 2.86694
$$779$$ 1.76028e6 0.103929
$$780$$ 0 0
$$781$$ −1.35859e7 −0.797006
$$782$$ −3.14093e7 −1.83671
$$783$$ −1.38771e7 −0.808898
$$784$$ 3.41902e6 0.198661
$$785$$ 0 0
$$786$$ −8.62204e6 −0.497799
$$787$$ −2.62501e7 −1.51075 −0.755377 0.655291i $$-0.772546\pi$$
−0.755377 + 0.655291i $$0.772546\pi$$
$$788$$ 6.73234e7 3.86234
$$789$$ −1.80737e7 −1.03360
$$790$$ 0 0
$$791$$ 4.08435e6 0.232103
$$792$$ −3.92544e6 −0.222370
$$793$$ 3.59072e6 0.202768
$$794$$ −9.95820e6 −0.560570
$$795$$ 0 0
$$796$$ −5.71714e7 −3.19813
$$797$$ 1.00373e7 0.559720 0.279860 0.960041i $$-0.409712\pi$$
0.279860 + 0.960041i $$0.409712\pi$$
$$798$$ −672280. −0.0373717
$$799$$ −1.60559e7 −0.889751
$$800$$ 0 0
$$801$$ 2.37735e6 0.130922
$$802$$ −3.31605e7 −1.82048
$$803$$ −1.57950e7 −0.864433
$$804$$ 4.60806e7 2.51407
$$805$$ 0 0
$$806$$ −1.07016e7 −0.580245
$$807$$ −1.78858e7 −0.966772
$$808$$ 1.39306e7 0.750655
$$809$$ 1.40884e7 0.756816 0.378408 0.925639i $$-0.376472\pi$$
0.378408 + 0.925639i $$0.376472\pi$$
$$810$$ 0 0
$$811$$ 1.81433e7 0.968646 0.484323 0.874889i $$-0.339066\pi$$
0.484323 + 0.874889i $$0.339066\pi$$
$$812$$ 1.13888e7 0.606160
$$813$$ 2.31076e7 1.22611
$$814$$ 2.41234e7 1.27608
$$815$$ 0 0
$$816$$ 3.43298e7 1.80487
$$817$$ 1.06624e6 0.0558856
$$818$$ 3.07273e7 1.60562
$$819$$ −322420. −0.0167962
$$820$$ 0 0
$$821$$ −2.13669e7 −1.10633 −0.553164 0.833072i $$-0.686580\pi$$
−0.553164 + 0.833072i $$0.686580\pi$$
$$822$$ 2.86247e7 1.47761
$$823$$ −1.78017e7 −0.916142 −0.458071 0.888916i $$-0.651459\pi$$
−0.458071 + 0.888916i $$0.651459\pi$$
$$824$$ −1.91016e7 −0.980058
$$825$$ 0 0
$$826$$ −1.33770e6 −0.0682195
$$827$$ −1.62921e7 −0.828350 −0.414175 0.910197i $$-0.635930\pi$$
−0.414175 + 0.910197i $$0.635930\pi$$
$$828$$ 5.82950e6 0.295499
$$829$$ −2.08499e6 −0.105370 −0.0526851 0.998611i $$-0.516778\pi$$
−0.0526851 + 0.998611i $$0.516778\pi$$
$$830$$ 0 0
$$831$$ 1.48973e7 0.748348
$$832$$ −2.57152e6 −0.128790
$$833$$ 4.13452e6 0.206449
$$834$$ −4.95684e6 −0.246769
$$835$$ 0 0
$$836$$ −1.54605e6 −0.0765081
$$837$$ 3.10346e7 1.53120
$$838$$ 2.81438e7 1.38443
$$839$$ −2.27850e7 −1.11749 −0.558745 0.829340i $$-0.688717\pi$$
−0.558745 + 0.829340i $$0.688717\pi$$
$$840$$ 0 0
$$841$$ −8.82842e6 −0.430421
$$842$$ 3.05802e7 1.48648
$$843$$ −312788. −0.0151594
$$844$$ 7.82498e7 3.78118
$$845$$ 0 0
$$846$$ 4.38228e6 0.210511
$$847$$ −5.25412e6 −0.251647
$$848$$ −3.22109e6 −0.153820
$$849$$ 3.49403e7 1.66363
$$850$$ 0 0
$$851$$ −1.89660e7 −0.897740
$$852$$ −5.57491e7 −2.63111
$$853$$ 2.26975e7 1.06808 0.534042 0.845458i $$-0.320672\pi$$
0.534042 + 0.845458i $$0.320672\pi$$
$$854$$ 1.25675e7 0.589664
$$855$$ 0 0
$$856$$ 5.26766e7 2.45716
$$857$$ −2.52900e7 −1.17624 −0.588120 0.808774i $$-0.700132\pi$$
−0.588120 + 0.808774i $$0.700132\pi$$
$$858$$ 4.54720e6 0.210875
$$859$$ −1.03947e7 −0.480652 −0.240326 0.970692i $$-0.577254\pi$$
−0.240326 + 0.970692i $$0.577254\pi$$
$$860$$ 0 0
$$861$$ −1.23219e7 −0.566462
$$862$$ 1.93750e7 0.888122
$$863$$ −4.33399e7 −1.98089 −0.990447 0.137892i $$-0.955967\pi$$
−0.990447 + 0.137892i $$0.955967\pi$$
$$864$$ −1.10432e7 −0.503281
$$865$$ 0 0
$$866$$ −3.94790e7 −1.78884
$$867$$ 2.16360e7 0.977527
$$868$$ −2.54698e7 −1.14743
$$869$$ 7.37389e6 0.331243
$$870$$ 0 0
$$871$$ 6.77656e6 0.302666
$$872$$ 3.34433e7 1.48942
$$873$$ −2.74978e6 −0.122113
$$874$$ 1.78752e6 0.0791539
$$875$$ 0 0
$$876$$ −6.48141e7 −2.85370
$$877$$ −3.71659e7 −1.63172 −0.815861 0.578248i $$-0.803736\pi$$
−0.815861 + 0.578248i $$0.803736\pi$$
$$878$$ −7.41770e7 −3.24738
$$879$$ 2.70449e7 1.18063
$$880$$ 0 0
$$881$$ 9.04785e6 0.392740 0.196370 0.980530i $$-0.437085\pi$$
0.196370 + 0.980530i $$0.437085\pi$$
$$882$$ −1.12847e6 −0.0488448
$$883$$ −3.29679e7 −1.42295 −0.711474 0.702712i $$-0.751972\pi$$
−0.711474 + 0.702712i $$0.751972\pi$$
$$884$$ 1.63934e7 0.705569
$$885$$ 0 0
$$886$$ −1.40269e7 −0.600313
$$887$$ 1.61099e7 0.687517 0.343758 0.939058i $$-0.388300\pi$$
0.343758 + 0.939058i $$0.388300\pi$$
$$888$$ 5.24059e7 2.23022
$$889$$ −2.95882e6 −0.125564
$$890$$ 0 0
$$891$$ −1.05372e7 −0.444663
$$892$$ 5.60500e7 2.35865
$$893$$ 913752. 0.0383442
$$894$$ −2.83164e6 −0.118493
$$895$$ 0 0
$$896$$ −1.32653e7 −0.552009
$$897$$ −3.57504e6 −0.148354
$$898$$ −5.90574e6 −0.244390
$$899$$ −2.61272e7 −1.07819
$$900$$ 0 0
$$901$$ −3.89516e6 −0.159850
$$902$$ −4.16718e7 −1.70540
$$903$$ −7.46368e6 −0.304603
$$904$$ 3.00074e7 1.22126
$$905$$ 0 0
$$906$$ 9.92656e6 0.401771
$$907$$ 4.47286e7 1.80537 0.902686 0.430300i $$-0.141592\pi$$
0.902686 + 0.430300i $$0.141592\pi$$
$$908$$ −5.05798e6 −0.203593
$$909$$ −1.81871e6 −0.0730053
$$910$$ 0 0
$$911$$ −6.60518e6 −0.263687 −0.131844 0.991271i $$-0.542090\pi$$
−0.131844 + 0.991271i $$0.542090\pi$$
$$912$$ −1.95373e6 −0.0777816
$$913$$ 4.76482e6 0.189177
$$914$$ 2.90484e7 1.15016
$$915$$ 0 0
$$916$$ 7.69730e7 3.03110
$$917$$ −3.01771e6 −0.118510
$$918$$ −6.99132e7 −2.73812
$$919$$ −3.08930e7 −1.20662 −0.603311 0.797506i $$-0.706152\pi$$
−0.603311 + 0.797506i $$0.706152\pi$$
$$920$$ 0 0
$$921$$ 6.42704e6 0.249667
$$922$$ −9.22684e6 −0.357459
$$923$$ −8.19840e6 −0.316756
$$924$$ 1.08223e7 0.417005
$$925$$ 0 0
$$926$$ −7.18235e7 −2.75258
$$927$$ 2.49382e6 0.0953160
$$928$$ 9.29696e6 0.354381
$$929$$ −4.87215e6 −0.185217 −0.0926087 0.995703i $$-0.529521\pi$$
−0.0926087 + 0.995703i $$0.529521\pi$$
$$930$$ 0 0
$$931$$ −235298. −0.00889701
$$932$$ 1.35134e7 0.509593
$$933$$ 9.33979e6 0.351264
$$934$$ 6.12570e6 0.229767
$$935$$ 0 0
$$936$$ −2.36880e6 −0.0883769
$$937$$ 3.25004e7 1.20932 0.604658 0.796485i $$-0.293310\pi$$
0.604658 + 0.796485i $$0.293310\pi$$
$$938$$ 2.37180e7 0.880177
$$939$$ 1.55448e6 0.0575334
$$940$$ 0 0
$$941$$ −2.64040e6 −0.0972066 −0.0486033 0.998818i $$-0.515477\pi$$
−0.0486033 + 0.998818i $$0.515477\pi$$
$$942$$ −4.10934e7 −1.50884
$$943$$ 3.27627e7 1.19978
$$944$$ −3.88752e6 −0.141985
$$945$$ 0 0
$$946$$ −2.52416e7 −0.917042
$$947$$ 4.08179e7 1.47903 0.739513 0.673142i $$-0.235056\pi$$
0.739513 + 0.673142i $$0.235056\pi$$
$$948$$ 3.02584e7 1.09351
$$949$$ −9.53148e6 −0.343554
$$950$$ 0 0
$$951$$ 962892. 0.0345244
$$952$$ 3.03761e7 1.08627
$$953$$ 6.71983e6 0.239677 0.119838 0.992793i $$-0.461762\pi$$
0.119838 + 0.992793i $$0.461762\pi$$
$$954$$ 1.06314e6 0.0378198
$$955$$ 0 0
$$956$$ 3.28375e7 1.16205
$$957$$ 1.11017e7 0.391840
$$958$$ 2.60330e7 0.916454
$$959$$ 1.00186e7 0.351773
$$960$$ 0 0
$$961$$ 2.98016e7 1.04095
$$962$$ 1.45572e7 0.507154
$$963$$ −6.87723e6 −0.238972
$$964$$ 5.48019e7 1.89934
$$965$$ 0 0
$$966$$ −1.25126e7 −0.431426
$$967$$ 2.78979e6 0.0959413 0.0479707 0.998849i $$-0.484725\pi$$
0.0479707 + 0.998849i $$0.484725\pi$$
$$968$$ −3.86017e7 −1.32409
$$969$$ −2.36258e6 −0.0808310
$$970$$ 0 0
$$971$$ 3.33594e7 1.13545 0.567727 0.823217i $$-0.307823\pi$$
0.567727 + 0.823217i $$0.307823\pi$$
$$972$$ 2.38486e7 0.809648
$$973$$ −1.73489e6 −0.0587477
$$974$$ −5.46309e7 −1.84519
$$975$$ 0 0
$$976$$ 3.65228e7 1.22727
$$977$$ 7.60033e6 0.254739 0.127370 0.991855i $$-0.459347\pi$$
0.127370 + 0.991855i $$0.459347\pi$$
$$978$$ −1.84688e6 −0.0617435
$$979$$ −1.17350e7 −0.391316
$$980$$ 0 0
$$981$$ −4.36621e6 −0.144854
$$982$$ 1.64090e7 0.543004
$$983$$ 5.79760e6 0.191366 0.0956829 0.995412i $$-0.469497\pi$$
0.0956829 + 0.995412i $$0.469497\pi$$
$$984$$ −9.05285e7 −2.98056
$$985$$ 0 0
$$986$$ 5.88580e7 1.92803
$$987$$ −6.39626e6 −0.208994
$$988$$ −932960. −0.0304068
$$989$$ 1.98451e7 0.645153
$$990$$ 0 0
$$991$$ 1.26825e7 0.410224 0.205112 0.978739i $$-0.434244\pi$$
0.205112 + 0.978739i $$0.434244\pi$$
$$992$$ −2.07917e7 −0.670827
$$993$$ −7.90227e6 −0.254319
$$994$$ −2.86944e7 −0.921152
$$995$$ 0 0
$$996$$ 1.95522e7 0.624521
$$997$$ −1.44400e7 −0.460077 −0.230039 0.973182i $$-0.573885\pi$$
−0.230039 + 0.973182i $$0.573885\pi$$
$$998$$ 2.99796e7 0.952796
$$999$$ −4.22159e7 −1.33833
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 175.6.a.b.1.1 1
5.2 odd 4 175.6.b.a.99.2 2
5.3 odd 4 175.6.b.a.99.1 2
5.4 even 2 7.6.a.a.1.1 1
15.14 odd 2 63.6.a.e.1.1 1
20.19 odd 2 112.6.a.g.1.1 1
35.4 even 6 49.6.c.c.30.1 2
35.9 even 6 49.6.c.c.18.1 2
35.19 odd 6 49.6.c.b.18.1 2
35.24 odd 6 49.6.c.b.30.1 2
35.34 odd 2 49.6.a.a.1.1 1
40.19 odd 2 448.6.a.c.1.1 1
40.29 even 2 448.6.a.m.1.1 1
55.54 odd 2 847.6.a.b.1.1 1
60.59 even 2 1008.6.a.y.1.1 1
105.104 even 2 441.6.a.k.1.1 1
140.139 even 2 784.6.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 5.4 even 2
49.6.a.a.1.1 1 35.34 odd 2
49.6.c.b.18.1 2 35.19 odd 6
49.6.c.b.30.1 2 35.24 odd 6
49.6.c.c.18.1 2 35.9 even 6
49.6.c.c.30.1 2 35.4 even 6
63.6.a.e.1.1 1 15.14 odd 2
112.6.a.g.1.1 1 20.19 odd 2
175.6.a.b.1.1 1 1.1 even 1 trivial
175.6.b.a.99.1 2 5.3 odd 4
175.6.b.a.99.2 2 5.2 odd 4
441.6.a.k.1.1 1 105.104 even 2
448.6.a.c.1.1 1 40.19 odd 2
448.6.a.m.1.1 1 40.29 even 2
784.6.a.c.1.1 1 140.139 even 2
847.6.a.b.1.1 1 55.54 odd 2
1008.6.a.y.1.1 1 60.59 even 2