# Properties

 Label 539.6.a.e.1.1 Level $539$ Weight $6$ Character 539.1 Self dual yes Analytic conductor $86.447$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,6,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, 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("539.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 539.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: $$86.4468788792$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.54492.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 52x - 38$$ x^3 - x^2 - 52*x - 38 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.749680$$ of defining polynomial Character $$\chi$$ $$=$$ 539.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-10.3963 q^{2} -20.6466 q^{3} +76.0833 q^{4} -8.64919 q^{5} +214.649 q^{6} -458.304 q^{8} +183.283 q^{9} +O(q^{10})$$ $$q-10.3963 q^{2} -20.6466 q^{3} +76.0833 q^{4} -8.64919 q^{5} +214.649 q^{6} -458.304 q^{8} +183.283 q^{9} +89.9197 q^{10} +121.000 q^{11} -1570.86 q^{12} +585.236 q^{13} +178.577 q^{15} +2330.01 q^{16} +945.333 q^{17} -1905.47 q^{18} -1148.76 q^{19} -658.060 q^{20} -1257.95 q^{22} -1346.27 q^{23} +9462.44 q^{24} -3050.19 q^{25} -6084.30 q^{26} +1232.95 q^{27} +899.585 q^{29} -1856.54 q^{30} +390.700 q^{31} -9557.75 q^{32} -2498.24 q^{33} -9827.97 q^{34} +13944.8 q^{36} -4473.41 q^{37} +11942.9 q^{38} -12083.2 q^{39} +3963.96 q^{40} -16018.7 q^{41} -19905.5 q^{43} +9206.08 q^{44} -1585.25 q^{45} +13996.2 q^{46} -1871.38 q^{47} -48106.8 q^{48} +31710.7 q^{50} -19517.9 q^{51} +44526.7 q^{52} +23565.1 q^{53} -12818.1 q^{54} -1046.55 q^{55} +23718.0 q^{57} -9352.37 q^{58} +34709.8 q^{59} +13586.7 q^{60} -25776.2 q^{61} -4061.84 q^{62} +24805.1 q^{64} -5061.82 q^{65} +25972.5 q^{66} +55384.6 q^{67} +71924.1 q^{68} +27795.9 q^{69} +56898.4 q^{71} -83999.6 q^{72} +46871.8 q^{73} +46506.9 q^{74} +62976.2 q^{75} -87401.5 q^{76} +125620. q^{78} -325.479 q^{79} -20152.7 q^{80} -69994.1 q^{81} +166536. q^{82} +92908.3 q^{83} -8176.37 q^{85} +206943. q^{86} -18573.4 q^{87} -55454.8 q^{88} -23058.0 q^{89} +16480.8 q^{90} -102428. q^{92} -8066.65 q^{93} +19455.5 q^{94} +9935.84 q^{95} +197335. q^{96} +5013.44 q^{97} +22177.3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 34 q^{3} + 84 q^{4} - 24 q^{5} + 206 q^{6} - 564 q^{8} - 7 q^{9}+O(q^{10})$$ 3 * q - 34 * q^3 + 84 * q^4 - 24 * q^5 + 206 * q^6 - 564 * q^8 - 7 * q^9 $$3 q - 34 q^{3} + 84 q^{4} - 24 q^{5} + 206 q^{6} - 564 q^{8} - 7 q^{9} + 414 q^{10} + 363 q^{11} - 992 q^{12} - 486 q^{13} + 1654 q^{15} + 1992 q^{16} - 1086 q^{17} - 3706 q^{18} - 1380 q^{19} + 3480 q^{20} - 3066 q^{23} + 11748 q^{24} - 57 q^{25} - 12132 q^{26} + 2990 q^{27} - 3426 q^{29} + 2650 q^{30} + 4098 q^{31} - 12408 q^{32} - 4114 q^{33} - 25320 q^{34} + 4756 q^{36} + 17724 q^{37} + 9240 q^{38} - 6560 q^{39} + 15276 q^{40} - 5994 q^{41} - 26208 q^{43} + 10164 q^{44} - 18458 q^{45} - 16806 q^{46} + 17232 q^{47} - 61064 q^{48} + 41070 q^{50} - 22724 q^{51} + 35304 q^{52} + 50586 q^{53} - 18814 q^{54} - 2904 q^{55} + 20160 q^{57} - 29172 q^{58} + 3738 q^{59} - 13456 q^{60} - 18486 q^{61} + 19974 q^{62} - 20352 q^{64} - 7668 q^{65} + 24926 q^{66} - 47754 q^{67} + 12600 q^{68} - 35042 q^{69} + 39282 q^{71} - 95040 q^{72} - 15426 q^{73} + 153294 q^{74} + 21916 q^{75} - 103920 q^{76} + 124984 q^{78} + 125148 q^{79} - 118680 q^{80} - 86917 q^{81} + 255372 q^{82} + 143928 q^{83} - 104040 q^{85} + 243060 q^{86} + 19368 q^{87} - 68244 q^{88} + 106824 q^{89} - 103424 q^{90} - 336528 q^{92} - 16622 q^{93} + 74928 q^{94} - 22200 q^{95} + 76456 q^{96} - 9684 q^{97} - 847 q^{99}+O(q^{100})$$ 3 * q - 34 * q^3 + 84 * q^4 - 24 * q^5 + 206 * q^6 - 564 * q^8 - 7 * q^9 + 414 * q^10 + 363 * q^11 - 992 * q^12 - 486 * q^13 + 1654 * q^15 + 1992 * q^16 - 1086 * q^17 - 3706 * q^18 - 1380 * q^19 + 3480 * q^20 - 3066 * q^23 + 11748 * q^24 - 57 * q^25 - 12132 * q^26 + 2990 * q^27 - 3426 * q^29 + 2650 * q^30 + 4098 * q^31 - 12408 * q^32 - 4114 * q^33 - 25320 * q^34 + 4756 * q^36 + 17724 * q^37 + 9240 * q^38 - 6560 * q^39 + 15276 * q^40 - 5994 * q^41 - 26208 * q^43 + 10164 * q^44 - 18458 * q^45 - 16806 * q^46 + 17232 * q^47 - 61064 * q^48 + 41070 * q^50 - 22724 * q^51 + 35304 * q^52 + 50586 * q^53 - 18814 * q^54 - 2904 * q^55 + 20160 * q^57 - 29172 * q^58 + 3738 * q^59 - 13456 * q^60 - 18486 * q^61 + 19974 * q^62 - 20352 * q^64 - 7668 * q^65 + 24926 * q^66 - 47754 * q^67 + 12600 * q^68 - 35042 * q^69 + 39282 * q^71 - 95040 * q^72 - 15426 * q^73 + 153294 * q^74 + 21916 * q^75 - 103920 * q^76 + 124984 * q^78 + 125148 * q^79 - 118680 * q^80 - 86917 * q^81 + 255372 * q^82 + 143928 * q^83 - 104040 * q^85 + 243060 * q^86 + 19368 * q^87 - 68244 * q^88 + 106824 * q^89 - 103424 * q^90 - 336528 * q^92 - 16622 * q^93 + 74928 * q^94 - 22200 * q^95 + 76456 * q^96 - 9684 * q^97 - 847 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −10.3963 −1.83783 −0.918913 0.394460i $$-0.870932\pi$$
−0.918913 + 0.394460i $$0.870932\pi$$
$$3$$ −20.6466 −1.32448 −0.662241 0.749291i $$-0.730395\pi$$
−0.662241 + 0.749291i $$0.730395\pi$$
$$4$$ 76.0833 2.37760
$$5$$ −8.64919 −0.154721 −0.0773607 0.997003i $$-0.524649\pi$$
−0.0773607 + 0.997003i $$0.524649\pi$$
$$6$$ 214.649 2.43417
$$7$$ 0 0
$$8$$ −458.304 −2.53180
$$9$$ 183.283 0.754253
$$10$$ 89.9197 0.284351
$$11$$ 121.000 0.301511
$$12$$ −1570.86 −3.14909
$$13$$ 585.236 0.960446 0.480223 0.877147i $$-0.340556\pi$$
0.480223 + 0.877147i $$0.340556\pi$$
$$14$$ 0 0
$$15$$ 178.577 0.204926
$$16$$ 2330.01 2.27540
$$17$$ 945.333 0.793345 0.396673 0.917960i $$-0.370165\pi$$
0.396673 + 0.917960i $$0.370165\pi$$
$$18$$ −1905.47 −1.38619
$$19$$ −1148.76 −0.730037 −0.365019 0.931000i $$-0.618937\pi$$
−0.365019 + 0.931000i $$0.618937\pi$$
$$20$$ −658.060 −0.367866
$$21$$ 0 0
$$22$$ −1257.95 −0.554125
$$23$$ −1346.27 −0.530654 −0.265327 0.964158i $$-0.585480\pi$$
−0.265327 + 0.964158i $$0.585480\pi$$
$$24$$ 9462.44 3.35332
$$25$$ −3050.19 −0.976061
$$26$$ −6084.30 −1.76513
$$27$$ 1232.95 0.325488
$$28$$ 0 0
$$29$$ 899.585 0.198631 0.0993155 0.995056i $$-0.468335\pi$$
0.0993155 + 0.995056i $$0.468335\pi$$
$$30$$ −1856.54 −0.376618
$$31$$ 390.700 0.0730196 0.0365098 0.999333i $$-0.488376\pi$$
0.0365098 + 0.999333i $$0.488376\pi$$
$$32$$ −9557.75 −1.64999
$$33$$ −2498.24 −0.399346
$$34$$ −9827.97 −1.45803
$$35$$ 0 0
$$36$$ 13944.8 1.79332
$$37$$ −4473.41 −0.537198 −0.268599 0.963252i $$-0.586561\pi$$
−0.268599 + 0.963252i $$0.586561\pi$$
$$38$$ 11942.9 1.34168
$$39$$ −12083.2 −1.27209
$$40$$ 3963.96 0.391723
$$41$$ −16018.7 −1.48822 −0.744111 0.668056i $$-0.767127\pi$$
−0.744111 + 0.668056i $$0.767127\pi$$
$$42$$ 0 0
$$43$$ −19905.5 −1.64173 −0.820864 0.571124i $$-0.806508\pi$$
−0.820864 + 0.571124i $$0.806508\pi$$
$$44$$ 9206.08 0.716875
$$45$$ −1585.25 −0.116699
$$46$$ 13996.2 0.975250
$$47$$ −1871.38 −0.123571 −0.0617856 0.998089i $$-0.519680\pi$$
−0.0617856 + 0.998089i $$0.519680\pi$$
$$48$$ −48106.8 −3.01372
$$49$$ 0 0
$$50$$ 31710.7 1.79383
$$51$$ −19517.9 −1.05077
$$52$$ 44526.7 2.28356
$$53$$ 23565.1 1.15234 0.576169 0.817330i $$-0.304547\pi$$
0.576169 + 0.817330i $$0.304547\pi$$
$$54$$ −12818.1 −0.598189
$$55$$ −1046.55 −0.0466503
$$56$$ 0 0
$$57$$ 23718.0 0.966922
$$58$$ −9352.37 −0.365049
$$59$$ 34709.8 1.29814 0.649071 0.760727i $$-0.275158\pi$$
0.649071 + 0.760727i $$0.275158\pi$$
$$60$$ 13586.7 0.487233
$$61$$ −25776.2 −0.886940 −0.443470 0.896289i $$-0.646253\pi$$
−0.443470 + 0.896289i $$0.646253\pi$$
$$62$$ −4061.84 −0.134197
$$63$$ 0 0
$$64$$ 24805.1 0.756993
$$65$$ −5061.82 −0.148602
$$66$$ 25972.5 0.733929
$$67$$ 55384.6 1.50731 0.753655 0.657271i $$-0.228289\pi$$
0.753655 + 0.657271i $$0.228289\pi$$
$$68$$ 71924.1 1.88626
$$69$$ 27795.9 0.702842
$$70$$ 0 0
$$71$$ 56898.4 1.33954 0.669768 0.742571i $$-0.266394\pi$$
0.669768 + 0.742571i $$0.266394\pi$$
$$72$$ −83999.6 −1.90962
$$73$$ 46871.8 1.02945 0.514724 0.857356i $$-0.327894\pi$$
0.514724 + 0.857356i $$0.327894\pi$$
$$74$$ 46506.9 0.987276
$$75$$ 62976.2 1.29278
$$76$$ −87401.5 −1.73574
$$77$$ 0 0
$$78$$ 125620. 2.33789
$$79$$ −325.479 −0.00586753 −0.00293377 0.999996i $$-0.500934\pi$$
−0.00293377 + 0.999996i $$0.500934\pi$$
$$80$$ −20152.7 −0.352053
$$81$$ −69994.1 −1.18536
$$82$$ 166536. 2.73509
$$83$$ 92908.3 1.48033 0.740166 0.672424i $$-0.234747\pi$$
0.740166 + 0.672424i $$0.234747\pi$$
$$84$$ 0 0
$$85$$ −8176.37 −0.122748
$$86$$ 206943. 3.01721
$$87$$ −18573.4 −0.263083
$$88$$ −55454.8 −0.763365
$$89$$ −23058.0 −0.308565 −0.154283 0.988027i $$-0.549307\pi$$
−0.154283 + 0.988027i $$0.549307\pi$$
$$90$$ 16480.8 0.214473
$$91$$ 0 0
$$92$$ −102428. −1.26169
$$93$$ −8066.65 −0.0967132
$$94$$ 19455.5 0.227103
$$95$$ 9935.84 0.112952
$$96$$ 197335. 2.18538
$$97$$ 5013.44 0.0541011 0.0270506 0.999634i $$-0.491388\pi$$
0.0270506 + 0.999634i $$0.491388\pi$$
$$98$$ 0 0
$$99$$ 22177.3 0.227416
$$100$$ −232069. −2.32069
$$101$$ −37928.6 −0.369968 −0.184984 0.982742i $$-0.559223\pi$$
−0.184984 + 0.982742i $$0.559223\pi$$
$$102$$ 202915. 1.93114
$$103$$ −180296. −1.67453 −0.837265 0.546798i $$-0.815847\pi$$
−0.837265 + 0.546798i $$0.815847\pi$$
$$104$$ −268216. −2.43165
$$105$$ 0 0
$$106$$ −244990. −2.11780
$$107$$ 92860.5 0.784100 0.392050 0.919944i $$-0.371766\pi$$
0.392050 + 0.919944i $$0.371766\pi$$
$$108$$ 93806.6 0.773881
$$109$$ 180736. 1.45707 0.728533 0.685011i $$-0.240203\pi$$
0.728533 + 0.685011i $$0.240203\pi$$
$$110$$ 10880.3 0.0857351
$$111$$ 92360.8 0.711509
$$112$$ 0 0
$$113$$ −68275.4 −0.503000 −0.251500 0.967857i $$-0.580924\pi$$
−0.251500 + 0.967857i $$0.580924\pi$$
$$114$$ −246580. −1.77703
$$115$$ 11644.1 0.0821036
$$116$$ 68443.4 0.472266
$$117$$ 107264. 0.724419
$$118$$ −360854. −2.38576
$$119$$ 0 0
$$120$$ −81842.5 −0.518831
$$121$$ 14641.0 0.0909091
$$122$$ 267977. 1.63004
$$123$$ 330732. 1.97112
$$124$$ 29725.8 0.173612
$$125$$ 53410.4 0.305739
$$126$$ 0 0
$$127$$ 27233.1 0.149826 0.0749130 0.997190i $$-0.476132\pi$$
0.0749130 + 0.997190i $$0.476132\pi$$
$$128$$ 47966.0 0.258767
$$129$$ 410981. 2.17444
$$130$$ 52624.3 0.273104
$$131$$ 11887.1 0.0605199 0.0302600 0.999542i $$-0.490366\pi$$
0.0302600 + 0.999542i $$0.490366\pi$$
$$132$$ −190075. −0.949488
$$133$$ 0 0
$$134$$ −575796. −2.77017
$$135$$ −10664.0 −0.0503599
$$136$$ −433250. −2.00859
$$137$$ 35302.2 0.160694 0.0803471 0.996767i $$-0.474397\pi$$
0.0803471 + 0.996767i $$0.474397\pi$$
$$138$$ −288975. −1.29170
$$139$$ −26248.0 −0.115228 −0.0576141 0.998339i $$-0.518349\pi$$
−0.0576141 + 0.998339i $$0.518349\pi$$
$$140$$ 0 0
$$141$$ 38637.7 0.163668
$$142$$ −591533. −2.46183
$$143$$ 70813.6 0.289585
$$144$$ 427052. 1.71623
$$145$$ −7780.68 −0.0307325
$$146$$ −487294. −1.89195
$$147$$ 0 0
$$148$$ −340352. −1.27724
$$149$$ −226321. −0.835139 −0.417570 0.908645i $$-0.637118\pi$$
−0.417570 + 0.908645i $$0.637118\pi$$
$$150$$ −654720. −2.37590
$$151$$ −301067. −1.07453 −0.537267 0.843412i $$-0.680543\pi$$
−0.537267 + 0.843412i $$0.680543\pi$$
$$152$$ 526481. 1.84831
$$153$$ 173264. 0.598383
$$154$$ 0 0
$$155$$ −3379.24 −0.0112977
$$156$$ −919327. −3.02453
$$157$$ −341482. −1.10565 −0.552827 0.833296i $$-0.686451\pi$$
−0.552827 + 0.833296i $$0.686451\pi$$
$$158$$ 3383.78 0.0107835
$$159$$ −486541. −1.52625
$$160$$ 82666.8 0.255289
$$161$$ 0 0
$$162$$ 727680. 2.17848
$$163$$ 604612. 1.78241 0.891205 0.453600i $$-0.149861\pi$$
0.891205 + 0.453600i $$0.149861\pi$$
$$164$$ −1.21876e6 −3.53840
$$165$$ 21607.8 0.0617875
$$166$$ −965904. −2.72059
$$167$$ 159824. 0.443455 0.221728 0.975109i $$-0.428830\pi$$
0.221728 + 0.975109i $$0.428830\pi$$
$$168$$ 0 0
$$169$$ −28791.6 −0.0775442
$$170$$ 85004.1 0.225589
$$171$$ −210549. −0.550633
$$172$$ −1.51447e6 −3.90338
$$173$$ 499771. 1.26957 0.634783 0.772690i $$-0.281089\pi$$
0.634783 + 0.772690i $$0.281089\pi$$
$$174$$ 193095. 0.483501
$$175$$ 0 0
$$176$$ 281931. 0.686058
$$177$$ −716641. −1.71937
$$178$$ 239719. 0.567090
$$179$$ −626569. −1.46163 −0.730813 0.682578i $$-0.760859\pi$$
−0.730813 + 0.682578i $$0.760859\pi$$
$$180$$ −120611. −0.277464
$$181$$ −393700. −0.893243 −0.446621 0.894723i $$-0.647373\pi$$
−0.446621 + 0.894723i $$0.647373\pi$$
$$182$$ 0 0
$$183$$ 532192. 1.17474
$$184$$ 617000. 1.34351
$$185$$ 38691.4 0.0831160
$$186$$ 83863.4 0.177742
$$187$$ 114385. 0.239203
$$188$$ −142381. −0.293804
$$189$$ 0 0
$$190$$ −103296. −0.207587
$$191$$ 205468. 0.407531 0.203766 0.979020i $$-0.434682\pi$$
0.203766 + 0.979020i $$0.434682\pi$$
$$192$$ −512143. −1.00262
$$193$$ −349786. −0.675941 −0.337971 0.941157i $$-0.609740\pi$$
−0.337971 + 0.941157i $$0.609740\pi$$
$$194$$ −52121.3 −0.0994285
$$195$$ 104510. 0.196820
$$196$$ 0 0
$$197$$ 863902. 1.58598 0.792992 0.609232i $$-0.208522\pi$$
0.792992 + 0.609232i $$0.208522\pi$$
$$198$$ −230562. −0.417951
$$199$$ 610140. 1.09219 0.546093 0.837725i $$-0.316115\pi$$
0.546093 + 0.837725i $$0.316115\pi$$
$$200$$ 1.39792e6 2.47119
$$201$$ −1.14351e6 −1.99640
$$202$$ 394318. 0.679936
$$203$$ 0 0
$$204$$ −1.48499e6 −2.49832
$$205$$ 138549. 0.230260
$$206$$ 1.87441e6 3.07749
$$207$$ −246749. −0.400248
$$208$$ 1.36360e6 2.18540
$$209$$ −139000. −0.220115
$$210$$ 0 0
$$211$$ 166602. 0.257616 0.128808 0.991670i $$-0.458885\pi$$
0.128808 + 0.991670i $$0.458885\pi$$
$$212$$ 1.79291e6 2.73981
$$213$$ −1.17476e6 −1.77419
$$214$$ −965407. −1.44104
$$215$$ 172166. 0.254011
$$216$$ −565064. −0.824068
$$217$$ 0 0
$$218$$ −1.87899e6 −2.67783
$$219$$ −967746. −1.36349
$$220$$ −79625.2 −0.110916
$$221$$ 553243. 0.761965
$$222$$ −960212. −1.30763
$$223$$ 1.05575e6 1.42167 0.710836 0.703358i $$-0.248317\pi$$
0.710836 + 0.703358i $$0.248317\pi$$
$$224$$ 0 0
$$225$$ −559050. −0.736197
$$226$$ 709812. 0.924427
$$227$$ −526562. −0.678242 −0.339121 0.940743i $$-0.610130\pi$$
−0.339121 + 0.940743i $$0.610130\pi$$
$$228$$ 1.80455e6 2.29896
$$229$$ −1.11694e6 −1.40748 −0.703740 0.710458i $$-0.748488\pi$$
−0.703740 + 0.710458i $$0.748488\pi$$
$$230$$ −121056. −0.150892
$$231$$ 0 0
$$232$$ −412283. −0.502893
$$233$$ −29262.0 −0.0353113 −0.0176557 0.999844i $$-0.505620\pi$$
−0.0176557 + 0.999844i $$0.505620\pi$$
$$234$$ −1.11515e6 −1.33136
$$235$$ 16185.9 0.0191191
$$236$$ 2.64084e6 3.08647
$$237$$ 6720.05 0.00777144
$$238$$ 0 0
$$239$$ 822476. 0.931384 0.465692 0.884947i $$-0.345806\pi$$
0.465692 + 0.884947i $$0.345806\pi$$
$$240$$ 416085. 0.466288
$$241$$ −762439. −0.845595 −0.422797 0.906224i $$-0.638952\pi$$
−0.422797 + 0.906224i $$0.638952\pi$$
$$242$$ −152212. −0.167075
$$243$$ 1.14554e6 1.24449
$$244$$ −1.96114e6 −2.10879
$$245$$ 0 0
$$246$$ −3.43840e6 −3.62258
$$247$$ −672296. −0.701161
$$248$$ −179060. −0.184871
$$249$$ −1.91824e6 −1.96067
$$250$$ −555272. −0.561895
$$251$$ −561364. −0.562419 −0.281209 0.959646i $$-0.590736\pi$$
−0.281209 + 0.959646i $$0.590736\pi$$
$$252$$ 0 0
$$253$$ −162898. −0.159998
$$254$$ −283123. −0.275354
$$255$$ 168814. 0.162577
$$256$$ −1.29243e6 −1.23256
$$257$$ 764965. 0.722451 0.361226 0.932478i $$-0.382358\pi$$
0.361226 + 0.932478i $$0.382358\pi$$
$$258$$ −4.27269e6 −3.99624
$$259$$ 0 0
$$260$$ −385120. −0.353316
$$261$$ 164879. 0.149818
$$262$$ −123582. −0.111225
$$263$$ −763627. −0.680756 −0.340378 0.940289i $$-0.610555\pi$$
−0.340378 + 0.940289i $$0.610555\pi$$
$$264$$ 1.14495e6 1.01106
$$265$$ −203819. −0.178292
$$266$$ 0 0
$$267$$ 476071. 0.408689
$$268$$ 4.21385e6 3.58378
$$269$$ −800885. −0.674823 −0.337411 0.941357i $$-0.609551\pi$$
−0.337411 + 0.941357i $$0.609551\pi$$
$$270$$ 110866. 0.0925528
$$271$$ −98139.7 −0.0811749 −0.0405874 0.999176i $$-0.512923\pi$$
−0.0405874 + 0.999176i $$0.512923\pi$$
$$272$$ 2.20263e6 1.80518
$$273$$ 0 0
$$274$$ −367013. −0.295328
$$275$$ −369073. −0.294294
$$276$$ 2.11480e6 1.67108
$$277$$ −620993. −0.486281 −0.243140 0.969991i $$-0.578178\pi$$
−0.243140 + 0.969991i $$0.578178\pi$$
$$278$$ 272882. 0.211769
$$279$$ 71608.9 0.0550753
$$280$$ 0 0
$$281$$ 1.31191e6 0.991149 0.495575 0.868565i $$-0.334957\pi$$
0.495575 + 0.868565i $$0.334957\pi$$
$$282$$ −401690. −0.300793
$$283$$ 26897.8 0.0199641 0.00998205 0.999950i $$-0.496823\pi$$
0.00998205 + 0.999950i $$0.496823\pi$$
$$284$$ 4.32902e6 3.18488
$$285$$ −205142. −0.149604
$$286$$ −736200. −0.532207
$$287$$ 0 0
$$288$$ −1.75178e6 −1.24451
$$289$$ −526203. −0.370603
$$290$$ 80890.4 0.0564810
$$291$$ −103511. −0.0716560
$$292$$ 3.56617e6 2.44762
$$293$$ −638546. −0.434534 −0.217267 0.976112i $$-0.569714\pi$$
−0.217267 + 0.976112i $$0.569714\pi$$
$$294$$ 0 0
$$295$$ −300212. −0.200851
$$296$$ 2.05018e6 1.36008
$$297$$ 149186. 0.0981382
$$298$$ 2.35290e6 1.53484
$$299$$ −787884. −0.509665
$$300$$ 4.79144e6 3.07371
$$301$$ 0 0
$$302$$ 3.12998e6 1.97481
$$303$$ 783099. 0.490016
$$304$$ −2.67662e6 −1.66113
$$305$$ 222943. 0.137229
$$306$$ −1.80131e6 −1.09972
$$307$$ −550428. −0.333315 −0.166658 0.986015i $$-0.553297\pi$$
−0.166658 + 0.986015i $$0.553297\pi$$
$$308$$ 0 0
$$309$$ 3.72250e6 2.21788
$$310$$ 35131.7 0.0207632
$$311$$ −186775. −0.109501 −0.0547504 0.998500i $$-0.517436\pi$$
−0.0547504 + 0.998500i $$0.517436\pi$$
$$312$$ 5.53776e6 3.22068
$$313$$ 934239. 0.539010 0.269505 0.962999i $$-0.413140\pi$$
0.269505 + 0.962999i $$0.413140\pi$$
$$314$$ 3.55016e6 2.03200
$$315$$ 0 0
$$316$$ −24763.5 −0.0139507
$$317$$ −1.88280e6 −1.05234 −0.526170 0.850379i $$-0.676372\pi$$
−0.526170 + 0.850379i $$0.676372\pi$$
$$318$$ 5.05823e6 2.80499
$$319$$ 108850. 0.0598895
$$320$$ −214544. −0.117123
$$321$$ −1.91726e6 −1.03853
$$322$$ 0 0
$$323$$ −1.08596e6 −0.579172
$$324$$ −5.32538e6 −2.81831
$$325$$ −1.78508e6 −0.937454
$$326$$ −6.28574e6 −3.27576
$$327$$ −3.73160e6 −1.92986
$$328$$ 7.34144e6 3.76788
$$329$$ 0 0
$$330$$ −224641. −0.113555
$$331$$ 197056. 0.0988596 0.0494298 0.998778i $$-0.484260\pi$$
0.0494298 + 0.998778i $$0.484260\pi$$
$$332$$ 7.06877e6 3.51964
$$333$$ −819901. −0.405183
$$334$$ −1.66158e6 −0.814993
$$335$$ −479033. −0.233213
$$336$$ 0 0
$$337$$ 387484. 0.185857 0.0929285 0.995673i $$-0.470377\pi$$
0.0929285 + 0.995673i $$0.470377\pi$$
$$338$$ 299327. 0.142513
$$339$$ 1.40966e6 0.666215
$$340$$ −622085. −0.291845
$$341$$ 47274.7 0.0220162
$$342$$ 2.18893e6 1.01197
$$343$$ 0 0
$$344$$ 9.12276e6 4.15652
$$345$$ −240412. −0.108745
$$346$$ −5.19577e6 −2.33324
$$347$$ −2.94793e6 −1.31430 −0.657148 0.753761i $$-0.728238\pi$$
−0.657148 + 0.753761i $$0.728238\pi$$
$$348$$ −1.41313e6 −0.625508
$$349$$ 924908. 0.406476 0.203238 0.979129i $$-0.434854\pi$$
0.203238 + 0.979129i $$0.434854\pi$$
$$350$$ 0 0
$$351$$ 721564. 0.312613
$$352$$ −1.15649e6 −0.497490
$$353$$ 4.46816e6 1.90850 0.954249 0.299012i $$-0.0966570\pi$$
0.954249 + 0.299012i $$0.0966570\pi$$
$$354$$ 7.45043e6 3.15990
$$355$$ −492125. −0.207255
$$356$$ −1.75433e6 −0.733647
$$357$$ 0 0
$$358$$ 6.51401e6 2.68621
$$359$$ −995937. −0.407846 −0.203923 0.978987i $$-0.565369\pi$$
−0.203923 + 0.978987i $$0.565369\pi$$
$$360$$ 726529. 0.295459
$$361$$ −1.15645e6 −0.467045
$$362$$ 4.09303e6 1.64162
$$363$$ −302287. −0.120407
$$364$$ 0 0
$$365$$ −405404. −0.159278
$$366$$ −5.53283e6 −2.15896
$$367$$ 1.21088e6 0.469284 0.234642 0.972082i $$-0.424608\pi$$
0.234642 + 0.972082i $$0.424608\pi$$
$$368$$ −3.13681e6 −1.20745
$$369$$ −2.93596e6 −1.12250
$$370$$ −402248. −0.152753
$$371$$ 0 0
$$372$$ −613737. −0.229946
$$373$$ 1.82235e6 0.678203 0.339102 0.940750i $$-0.389877\pi$$
0.339102 + 0.940750i $$0.389877\pi$$
$$374$$ −1.18918e6 −0.439613
$$375$$ −1.10275e6 −0.404946
$$376$$ 857662. 0.312857
$$377$$ 526470. 0.190774
$$378$$ 0 0
$$379$$ −419357. −0.149964 −0.0749819 0.997185i $$-0.523890\pi$$
−0.0749819 + 0.997185i $$0.523890\pi$$
$$380$$ 755952. 0.268556
$$381$$ −562271. −0.198442
$$382$$ −2.13611e6 −0.748972
$$383$$ −2.95656e6 −1.02989 −0.514943 0.857224i $$-0.672187\pi$$
−0.514943 + 0.857224i $$0.672187\pi$$
$$384$$ −990336. −0.342732
$$385$$ 0 0
$$386$$ 3.63648e6 1.24226
$$387$$ −3.64834e6 −1.23828
$$388$$ 381439. 0.128631
$$389$$ 2.35429e6 0.788834 0.394417 0.918932i $$-0.370947\pi$$
0.394417 + 0.918932i $$0.370947\pi$$
$$390$$ −1.08651e6 −0.361721
$$391$$ −1.27267e6 −0.420992
$$392$$ 0 0
$$393$$ −245429. −0.0801576
$$394$$ −8.98139e6 −2.91476
$$395$$ 2815.13 0.000907833 0
$$396$$ 1.68732e6 0.540705
$$397$$ −3.94809e6 −1.25722 −0.628609 0.777722i $$-0.716375\pi$$
−0.628609 + 0.777722i $$0.716375\pi$$
$$398$$ −6.34320e6 −2.00725
$$399$$ 0 0
$$400$$ −7.10697e6 −2.22093
$$401$$ −5.76535e6 −1.79046 −0.895230 0.445604i $$-0.852989\pi$$
−0.895230 + 0.445604i $$0.852989\pi$$
$$402$$ 1.18883e7 3.66904
$$403$$ 228652. 0.0701314
$$404$$ −2.88574e6 −0.879637
$$405$$ 605392. 0.183400
$$406$$ 0 0
$$407$$ −541282. −0.161971
$$408$$ 8.94515e6 2.66034
$$409$$ −2.39693e6 −0.708512 −0.354256 0.935148i $$-0.615266\pi$$
−0.354256 + 0.935148i $$0.615266\pi$$
$$410$$ −1.44040e6 −0.423178
$$411$$ −728872. −0.212837
$$412$$ −1.37175e7 −3.98137
$$413$$ 0 0
$$414$$ 2.56527e6 0.735585
$$415$$ −803582. −0.229039
$$416$$ −5.59354e6 −1.58472
$$417$$ 541932. 0.152618
$$418$$ 1.44509e6 0.404532
$$419$$ 1.41668e6 0.394220 0.197110 0.980381i $$-0.436844\pi$$
0.197110 + 0.980381i $$0.436844\pi$$
$$420$$ 0 0
$$421$$ −4.80538e6 −1.32136 −0.660682 0.750666i $$-0.729733\pi$$
−0.660682 + 0.750666i $$0.729733\pi$$
$$422$$ −1.73204e6 −0.473454
$$423$$ −342993. −0.0932040
$$424$$ −1.08000e7 −2.91749
$$425$$ −2.88345e6 −0.774354
$$426$$ 1.22132e7 3.26065
$$427$$ 0 0
$$428$$ 7.06514e6 1.86428
$$429$$ −1.46206e6 −0.383551
$$430$$ −1.78989e6 −0.466827
$$431$$ −2.73465e6 −0.709103 −0.354551 0.935037i $$-0.615366\pi$$
−0.354551 + 0.935037i $$0.615366\pi$$
$$432$$ 2.87277e6 0.740613
$$433$$ −2.71922e6 −0.696986 −0.348493 0.937311i $$-0.613307\pi$$
−0.348493 + 0.937311i $$0.613307\pi$$
$$434$$ 0 0
$$435$$ 160645. 0.0407046
$$436$$ 1.37510e7 3.46433
$$437$$ 1.54654e6 0.387397
$$438$$ 1.00610e7 2.50585
$$439$$ 4.17101e6 1.03295 0.516476 0.856301i $$-0.327243\pi$$
0.516476 + 0.856301i $$0.327243\pi$$
$$440$$ 479639. 0.118109
$$441$$ 0 0
$$442$$ −5.75169e6 −1.40036
$$443$$ 6.86870e6 1.66290 0.831448 0.555603i $$-0.187513\pi$$
0.831448 + 0.555603i $$0.187513\pi$$
$$444$$ 7.02712e6 1.69169
$$445$$ 199433. 0.0477417
$$446$$ −1.09759e7 −2.61278
$$447$$ 4.67276e6 1.10613
$$448$$ 0 0
$$449$$ −693812. −0.162415 −0.0812075 0.996697i $$-0.525878\pi$$
−0.0812075 + 0.996697i $$0.525878\pi$$
$$450$$ 5.81206e6 1.35300
$$451$$ −1.93826e6 −0.448716
$$452$$ −5.19462e6 −1.19594
$$453$$ 6.21601e6 1.42320
$$454$$ 5.47431e6 1.24649
$$455$$ 0 0
$$456$$ −1.08701e7 −2.44805
$$457$$ 8.12461e6 1.81975 0.909876 0.414880i $$-0.136176\pi$$
0.909876 + 0.414880i $$0.136176\pi$$
$$458$$ 1.16121e7 2.58670
$$459$$ 1.16554e6 0.258224
$$460$$ 885924. 0.195210
$$461$$ 4.48975e6 0.983944 0.491972 0.870611i $$-0.336276\pi$$
0.491972 + 0.870611i $$0.336276\pi$$
$$462$$ 0 0
$$463$$ −9.04494e6 −1.96089 −0.980445 0.196793i $$-0.936947\pi$$
−0.980445 + 0.196793i $$0.936947\pi$$
$$464$$ 2.09604e6 0.451965
$$465$$ 69770.0 0.0149636
$$466$$ 304217. 0.0648961
$$467$$ −7.17275e6 −1.52192 −0.760962 0.648796i $$-0.775273\pi$$
−0.760962 + 0.648796i $$0.775273\pi$$
$$468$$ 8.16101e6 1.72238
$$469$$ 0 0
$$470$$ −168274. −0.0351376
$$471$$ 7.05046e6 1.46442
$$472$$ −1.59077e7 −3.28663
$$473$$ −2.40856e6 −0.495000
$$474$$ −69863.7 −0.0142826
$$475$$ 3.50394e6 0.712561
$$476$$ 0 0
$$477$$ 4.31910e6 0.869155
$$478$$ −8.55072e6 −1.71172
$$479$$ 1.51089e6 0.300881 0.150440 0.988619i $$-0.451931\pi$$
0.150440 + 0.988619i $$0.451931\pi$$
$$480$$ −1.70679e6 −0.338125
$$481$$ −2.61800e6 −0.515949
$$482$$ 7.92655e6 1.55406
$$483$$ 0 0
$$484$$ 1.11394e6 0.216146
$$485$$ −43362.2 −0.00837061
$$486$$ −1.19093e7 −2.28716
$$487$$ 1.63265e6 0.311940 0.155970 0.987762i $$-0.450150\pi$$
0.155970 + 0.987762i $$0.450150\pi$$
$$488$$ 1.18133e7 2.24555
$$489$$ −1.24832e7 −2.36077
$$490$$ 0 0
$$491$$ 1.24459e6 0.232982 0.116491 0.993192i $$-0.462835\pi$$
0.116491 + 0.993192i $$0.462835\pi$$
$$492$$ 2.51632e7 4.68655
$$493$$ 850407. 0.157583
$$494$$ 6.98940e6 1.28861
$$495$$ −191816. −0.0351861
$$496$$ 910334. 0.166149
$$497$$ 0 0
$$498$$ 1.99427e7 3.60338
$$499$$ −7.66211e6 −1.37752 −0.688759 0.724991i $$-0.741844\pi$$
−0.688759 + 0.724991i $$0.741844\pi$$
$$500$$ 4.06364e6 0.726927
$$501$$ −3.29982e6 −0.587348
$$502$$ 5.83611e6 1.03363
$$503$$ 1.07030e7 1.88619 0.943097 0.332518i $$-0.107898\pi$$
0.943097 + 0.332518i $$0.107898\pi$$
$$504$$ 0 0
$$505$$ 328052. 0.0572420
$$506$$ 1.69354e6 0.294049
$$507$$ 594450. 0.102706
$$508$$ 2.07198e6 0.356227
$$509$$ −6.27494e6 −1.07353 −0.536766 0.843731i $$-0.680354\pi$$
−0.536766 + 0.843731i $$0.680354\pi$$
$$510$$ −1.75505e6 −0.298788
$$511$$ 0 0
$$512$$ 1.19016e7 2.00647
$$513$$ −1.41636e6 −0.237618
$$514$$ −7.95281e6 −1.32774
$$515$$ 1.55941e6 0.259086
$$516$$ 3.12688e7 5.16996
$$517$$ −226437. −0.0372581
$$518$$ 0 0
$$519$$ −1.03186e7 −1.68152
$$520$$ 2.31985e6 0.376229
$$521$$ −3.60326e6 −0.581570 −0.290785 0.956788i $$-0.593916\pi$$
−0.290785 + 0.956788i $$0.593916\pi$$
$$522$$ −1.71413e6 −0.275340
$$523$$ −3.56925e6 −0.570589 −0.285294 0.958440i $$-0.592091\pi$$
−0.285294 + 0.958440i $$0.592091\pi$$
$$524$$ 904412. 0.143892
$$525$$ 0 0
$$526$$ 7.93890e6 1.25111
$$527$$ 369342. 0.0579298
$$528$$ −5.82092e6 −0.908672
$$529$$ −4.62391e6 −0.718406
$$530$$ 2.11897e6 0.327669
$$531$$ 6.36174e6 0.979128
$$532$$ 0 0
$$533$$ −9.37473e6 −1.42936
$$534$$ −4.94938e6 −0.751100
$$535$$ −803169. −0.121317
$$536$$ −2.53830e7 −3.81620
$$537$$ 1.29365e7 1.93590
$$538$$ 8.32625e6 1.24021
$$539$$ 0 0
$$540$$ −811351. −0.119736
$$541$$ 1.16842e7 1.71635 0.858176 0.513355i $$-0.171598\pi$$
0.858176 + 0.513355i $$0.171598\pi$$
$$542$$ 1.02029e6 0.149185
$$543$$ 8.12859e6 1.18308
$$544$$ −9.03525e6 −1.30901
$$545$$ −1.56322e6 −0.225439
$$546$$ 0 0
$$547$$ −4.05598e6 −0.579600 −0.289800 0.957087i $$-0.593589\pi$$
−0.289800 + 0.957087i $$0.593589\pi$$
$$548$$ 2.68591e6 0.382067
$$549$$ −4.72435e6 −0.668977
$$550$$ 3.83700e6 0.540860
$$551$$ −1.03341e6 −0.145008
$$552$$ −1.27390e7 −1.77945
$$553$$ 0 0
$$554$$ 6.45603e6 0.893699
$$555$$ −798846. −0.110086
$$556$$ −1.99703e6 −0.273967
$$557$$ −284385. −0.0388390 −0.0194195 0.999811i $$-0.506182\pi$$
−0.0194195 + 0.999811i $$0.506182\pi$$
$$558$$ −744469. −0.101219
$$559$$ −1.16494e7 −1.57679
$$560$$ 0 0
$$561$$ −2.36167e6 −0.316820
$$562$$ −1.36391e7 −1.82156
$$563$$ −1.20582e6 −0.160329 −0.0801646 0.996782i $$-0.525545\pi$$
−0.0801646 + 0.996782i $$0.525545\pi$$
$$564$$ 2.93969e6 0.389138
$$565$$ 590527. 0.0778249
$$566$$ −279637. −0.0366906
$$567$$ 0 0
$$568$$ −2.60768e7 −3.39143
$$569$$ 3.94580e6 0.510922 0.255461 0.966819i $$-0.417773\pi$$
0.255461 + 0.966819i $$0.417773\pi$$
$$570$$ 2.13272e6 0.274945
$$571$$ 5.88346e6 0.755166 0.377583 0.925976i $$-0.376755\pi$$
0.377583 + 0.925976i $$0.376755\pi$$
$$572$$ 5.38773e6 0.688519
$$573$$ −4.24222e6 −0.539768
$$574$$ 0 0
$$575$$ 4.10637e6 0.517951
$$576$$ 4.54637e6 0.570964
$$577$$ −3.61005e6 −0.451413 −0.225706 0.974195i $$-0.572469\pi$$
−0.225706 + 0.974195i $$0.572469\pi$$
$$578$$ 5.47057e6 0.681104
$$579$$ 7.22190e6 0.895272
$$580$$ −591980. −0.0730697
$$581$$ 0 0
$$582$$ 1.07613e6 0.131691
$$583$$ 2.85138e6 0.347443
$$584$$ −2.14816e7 −2.60636
$$585$$ −927748. −0.112083
$$586$$ 6.63853e6 0.798597
$$587$$ −5.01469e6 −0.600688 −0.300344 0.953831i $$-0.597101\pi$$
−0.300344 + 0.953831i $$0.597101\pi$$
$$588$$ 0 0
$$589$$ −448821. −0.0533070
$$590$$ 3.12110e6 0.369128
$$591$$ −1.78367e7 −2.10061
$$592$$ −1.04231e7 −1.22234
$$593$$ −1.64451e7 −1.92044 −0.960220 0.279244i $$-0.909916\pi$$
−0.960220 + 0.279244i $$0.909916\pi$$
$$594$$ −1.55099e6 −0.180361
$$595$$ 0 0
$$596$$ −1.72192e7 −1.98563
$$597$$ −1.25973e7 −1.44658
$$598$$ 8.19109e6 0.936675
$$599$$ 6.45089e6 0.734603 0.367302 0.930102i $$-0.380282\pi$$
0.367302 + 0.930102i $$0.380282\pi$$
$$600$$ −2.88622e7 −3.27305
$$601$$ 8.32443e6 0.940087 0.470044 0.882643i $$-0.344238\pi$$
0.470044 + 0.882643i $$0.344238\pi$$
$$602$$ 0 0
$$603$$ 1.01511e7 1.13689
$$604$$ −2.29062e7 −2.55482
$$605$$ −126633. −0.0140656
$$606$$ −8.14134e6 −0.900563
$$607$$ −1.47290e7 −1.62256 −0.811279 0.584659i $$-0.801228\pi$$
−0.811279 + 0.584659i $$0.801228\pi$$
$$608$$ 1.09796e7 1.20455
$$609$$ 0 0
$$610$$ −2.31779e6 −0.252202
$$611$$ −1.09520e6 −0.118683
$$612$$ 1.31825e7 1.42272
$$613$$ −1.14564e7 −1.23139 −0.615697 0.787983i $$-0.711126\pi$$
−0.615697 + 0.787983i $$0.711126\pi$$
$$614$$ 5.72243e6 0.612575
$$615$$ −2.86057e6 −0.304975
$$616$$ 0 0
$$617$$ −413797. −0.0437597 −0.0218799 0.999761i $$-0.506965\pi$$
−0.0218799 + 0.999761i $$0.506965\pi$$
$$618$$ −3.87003e7 −4.07609
$$619$$ 1.25898e7 1.32067 0.660333 0.750973i $$-0.270415\pi$$
0.660333 + 0.750973i $$0.270415\pi$$
$$620$$ −257104. −0.0268615
$$621$$ −1.65987e6 −0.172721
$$622$$ 1.94177e6 0.201243
$$623$$ 0 0
$$624$$ −2.81538e7 −2.89452
$$625$$ 9.06989e6 0.928757
$$626$$ −9.71264e6 −0.990607
$$627$$ 2.86988e6 0.291538
$$628$$ −2.59811e7 −2.62881
$$629$$ −4.22886e6 −0.426183
$$630$$ 0 0
$$631$$ −1.55648e7 −1.55621 −0.778107 0.628132i $$-0.783820\pi$$
−0.778107 + 0.628132i $$0.783820\pi$$
$$632$$ 149168. 0.0148554
$$633$$ −3.43976e6 −0.341208
$$634$$ 1.95742e7 1.93402
$$635$$ −235544. −0.0231813
$$636$$ −3.70176e7 −3.62882
$$637$$ 0 0
$$638$$ −1.13164e6 −0.110067
$$639$$ 1.04285e7 1.01035
$$640$$ −414867. −0.0400368
$$641$$ −1.23075e7 −1.18311 −0.591556 0.806264i $$-0.701486\pi$$
−0.591556 + 0.806264i $$0.701486\pi$$
$$642$$ 1.99324e7 1.90863
$$643$$ 1.44400e6 0.137733 0.0688667 0.997626i $$-0.478062\pi$$
0.0688667 + 0.997626i $$0.478062\pi$$
$$644$$ 0 0
$$645$$ −3.55465e6 −0.336433
$$646$$ 1.12900e7 1.06442
$$647$$ 7.35610e6 0.690855 0.345427 0.938445i $$-0.387734\pi$$
0.345427 + 0.938445i $$0.387734\pi$$
$$648$$ 3.20786e7 3.00108
$$649$$ 4.19989e6 0.391405
$$650$$ 1.85583e7 1.72288
$$651$$ 0 0
$$652$$ 4.60009e7 4.23787
$$653$$ 3.83734e6 0.352166 0.176083 0.984375i $$-0.443657\pi$$
0.176083 + 0.984375i $$0.443657\pi$$
$$654$$ 3.87948e7 3.54674
$$655$$ −102814. −0.00936373
$$656$$ −3.73237e7 −3.38630
$$657$$ 8.59083e6 0.776465
$$658$$ 0 0
$$659$$ −1.98049e7 −1.77648 −0.888239 0.459382i $$-0.848071\pi$$
−0.888239 + 0.459382i $$0.848071\pi$$
$$660$$ 1.64399e6 0.146906
$$661$$ −1.75724e7 −1.56433 −0.782164 0.623073i $$-0.785884\pi$$
−0.782164 + 0.623073i $$0.785884\pi$$
$$662$$ −2.04865e6 −0.181687
$$663$$ −1.14226e7 −1.00921
$$664$$ −4.25803e7 −3.74790
$$665$$ 0 0
$$666$$ 8.52395e6 0.744656
$$667$$ −1.21108e6 −0.105404
$$668$$ 1.21599e7 1.05436
$$669$$ −2.17977e7 −1.88298
$$670$$ 4.98017e6 0.428605
$$671$$ −3.11892e6 −0.267423
$$672$$ 0 0
$$673$$ 4.88569e6 0.415804 0.207902 0.978150i $$-0.433337\pi$$
0.207902 + 0.978150i $$0.433337\pi$$
$$674$$ −4.02840e6 −0.341573
$$675$$ −3.76072e6 −0.317696
$$676$$ −2.19056e6 −0.184369
$$677$$ 1.56799e7 1.31483 0.657416 0.753528i $$-0.271649\pi$$
0.657416 + 0.753528i $$0.271649\pi$$
$$678$$ −1.46552e7 −1.22439
$$679$$ 0 0
$$680$$ 3.74726e6 0.310772
$$681$$ 1.08717e7 0.898320
$$682$$ −491483. −0.0404620
$$683$$ −1.94703e6 −0.159705 −0.0798527 0.996807i $$-0.525445\pi$$
−0.0798527 + 0.996807i $$0.525445\pi$$
$$684$$ −1.60192e7 −1.30919
$$685$$ −305336. −0.0248629
$$686$$ 0 0
$$687$$ 2.30611e7 1.86418
$$688$$ −4.63799e7 −3.73558
$$689$$ 1.37912e7 1.10676
$$690$$ 2.49940e6 0.199854
$$691$$ 5.39805e6 0.430073 0.215036 0.976606i $$-0.431013\pi$$
0.215036 + 0.976606i $$0.431013\pi$$
$$692$$ 3.80242e7 3.01853
$$693$$ 0 0
$$694$$ 3.06476e7 2.41545
$$695$$ 227024. 0.0178283
$$696$$ 8.51227e6 0.666073
$$697$$ −1.51430e7 −1.18067
$$698$$ −9.61563e6 −0.747032
$$699$$ 604162. 0.0467692
$$700$$ 0 0
$$701$$ 5.48228e6 0.421373 0.210686 0.977554i $$-0.432430\pi$$
0.210686 + 0.977554i $$0.432430\pi$$
$$702$$ −7.50161e6 −0.574528
$$703$$ 5.13887e6 0.392174
$$704$$ 3.00142e6 0.228242
$$705$$ −334185. −0.0253230
$$706$$ −4.64524e7 −3.50749
$$707$$ 0 0
$$708$$ −5.45244e7 −4.08797
$$709$$ −1.30530e7 −0.975200 −0.487600 0.873067i $$-0.662128\pi$$
−0.487600 + 0.873067i $$0.662128\pi$$
$$710$$ 5.11629e6 0.380898
$$711$$ −59654.9 −0.00442560
$$712$$ 1.05676e7 0.781225
$$713$$ −525987. −0.0387482
$$714$$ 0 0
$$715$$ −612480. −0.0448051
$$716$$ −4.76714e7 −3.47517
$$717$$ −1.69814e7 −1.23360
$$718$$ 1.03541e7 0.749550
$$719$$ 2.17045e7 1.56577 0.782885 0.622167i $$-0.213748\pi$$
0.782885 + 0.622167i $$0.213748\pi$$
$$720$$ −3.69365e6 −0.265537
$$721$$ 0 0
$$722$$ 1.20228e7 0.858348
$$723$$ 1.57418e7 1.11998
$$724$$ −2.99540e7 −2.12378
$$725$$ −2.74391e6 −0.193876
$$726$$ 3.14267e6 0.221288
$$727$$ 1.07681e7 0.755619 0.377809 0.925883i $$-0.376677\pi$$
0.377809 + 0.925883i $$0.376677\pi$$
$$728$$ 0 0
$$729$$ −6.64290e6 −0.462955
$$730$$ 4.21470e6 0.292725
$$731$$ −1.88173e7 −1.30246
$$732$$ 4.04909e7 2.79306
$$733$$ −7.35742e6 −0.505785 −0.252892 0.967494i $$-0.581382\pi$$
−0.252892 + 0.967494i $$0.581382\pi$$
$$734$$ −1.25887e7 −0.862463
$$735$$ 0 0
$$736$$ 1.28673e7 0.875573
$$737$$ 6.70154e6 0.454471
$$738$$ 3.05232e7 2.06295
$$739$$ −6.81140e6 −0.458802 −0.229401 0.973332i $$-0.573677\pi$$
−0.229401 + 0.973332i $$0.573677\pi$$
$$740$$ 2.94377e6 0.197617
$$741$$ 1.38806e7 0.928676
$$742$$ 0 0
$$743$$ −1.01182e7 −0.672405 −0.336203 0.941790i $$-0.609143\pi$$
−0.336203 + 0.941790i $$0.609143\pi$$
$$744$$ 3.69698e6 0.244858
$$745$$ 1.95749e6 0.129214
$$746$$ −1.89457e7 −1.24642
$$747$$ 1.70286e7 1.11655
$$748$$ 8.70281e6 0.568729
$$749$$ 0 0
$$750$$ 1.14645e7 0.744220
$$751$$ −1.91140e7 −1.23667 −0.618333 0.785916i $$-0.712192\pi$$
−0.618333 + 0.785916i $$0.712192\pi$$
$$752$$ −4.36033e6 −0.281174
$$753$$ 1.15903e7 0.744914
$$754$$ −5.47334e6 −0.350610
$$755$$ 2.60398e6 0.166254
$$756$$ 0 0
$$757$$ 1.48895e7 0.944366 0.472183 0.881501i $$-0.343466\pi$$
0.472183 + 0.881501i $$0.343466\pi$$
$$758$$ 4.35977e6 0.275607
$$759$$ 3.36330e6 0.211915
$$760$$ −4.55364e6 −0.285973
$$761$$ 2.14078e7 1.34002 0.670009 0.742353i $$-0.266290\pi$$
0.670009 + 0.742353i $$0.266290\pi$$
$$762$$ 5.84554e6 0.364702
$$763$$ 0 0
$$764$$ 1.56327e7 0.968948
$$765$$ −1.49859e6 −0.0925827
$$766$$ 3.07373e7 1.89275
$$767$$ 2.03134e7 1.24680
$$768$$ 2.66844e7 1.63251
$$769$$ −1.45027e7 −0.884369 −0.442184 0.896924i $$-0.645796\pi$$
−0.442184 + 0.896924i $$0.645796\pi$$
$$770$$ 0 0
$$771$$ −1.57939e7 −0.956874
$$772$$ −2.66129e7 −1.60712
$$773$$ −3.18546e7 −1.91745 −0.958723 0.284342i $$-0.908225\pi$$
−0.958723 + 0.284342i $$0.908225\pi$$
$$774$$ 3.79293e7 2.27574
$$775$$ −1.19171e6 −0.0712716
$$776$$ −2.29768e6 −0.136973
$$777$$ 0 0
$$778$$ −2.44759e7 −1.44974
$$779$$ 1.84016e7 1.08646
$$780$$ 7.95144e6 0.467960
$$781$$ 6.88470e6 0.403885
$$782$$ 1.32311e7 0.773710
$$783$$ 1.10914e6 0.0646519
$$784$$ 0 0
$$785$$ 2.95355e6 0.171068
$$786$$ 2.55156e6 0.147316
$$787$$ 1.68239e7 0.968253 0.484126 0.874998i $$-0.339138\pi$$
0.484126 + 0.874998i $$0.339138\pi$$
$$788$$ 6.57285e7 3.77084
$$789$$ 1.57663e7 0.901650
$$790$$ −29267.0 −0.00166844
$$791$$ 0 0
$$792$$ −1.01639e7 −0.575771
$$793$$ −1.50852e7 −0.851858
$$794$$ 4.10455e7 2.31055
$$795$$ 4.20818e6 0.236144
$$796$$ 4.64215e7 2.59679
$$797$$ 2.00376e7 1.11738 0.558690 0.829377i $$-0.311304\pi$$
0.558690 + 0.829377i $$0.311304\pi$$
$$798$$ 0 0
$$799$$ −1.76908e6 −0.0980347
$$800$$ 2.91530e7 1.61049
$$801$$ −4.22616e6 −0.232736
$$802$$ 5.99384e7 3.29055
$$803$$ 5.67149e6 0.310391
$$804$$ −8.70018e7 −4.74666
$$805$$ 0 0
$$806$$ −2.37714e6 −0.128889
$$807$$ 1.65356e7 0.893790
$$808$$ 1.73829e7 0.936683
$$809$$ −1.59711e7 −0.857954 −0.428977 0.903315i $$-0.641126\pi$$
−0.428977 + 0.903315i $$0.641126\pi$$
$$810$$ −6.29385e6 −0.337057
$$811$$ −1.37309e7 −0.733074 −0.366537 0.930403i $$-0.619457\pi$$
−0.366537 + 0.930403i $$0.619457\pi$$
$$812$$ 0 0
$$813$$ 2.02625e6 0.107515
$$814$$ 5.62734e6 0.297675
$$815$$ −5.22941e6 −0.275777
$$816$$ −4.54769e7 −2.39092
$$817$$ 2.28666e7 1.19852
$$818$$ 2.49192e7 1.30212
$$819$$ 0 0
$$820$$ 1.05413e7 0.547467
$$821$$ 2.26402e7 1.17226 0.586128 0.810218i $$-0.300652\pi$$
0.586128 + 0.810218i $$0.300652\pi$$
$$822$$ 7.57758e6 0.391157
$$823$$ 1.16510e7 0.599603 0.299802 0.954002i $$-0.403079\pi$$
0.299802 + 0.954002i $$0.403079\pi$$
$$824$$ 8.26304e7 4.23957
$$825$$ 7.62012e6 0.389787
$$826$$ 0 0
$$827$$ 5.48883e6 0.279072 0.139536 0.990217i $$-0.455439\pi$$
0.139536 + 0.990217i $$0.455439\pi$$
$$828$$ −1.87734e7 −0.951630
$$829$$ −1.81680e7 −0.918164 −0.459082 0.888394i $$-0.651822\pi$$
−0.459082 + 0.888394i $$0.651822\pi$$
$$830$$ 8.35429e6 0.420934
$$831$$ 1.28214e7 0.644070
$$832$$ 1.45169e7 0.727050
$$833$$ 0 0
$$834$$ −5.63410e6 −0.280485
$$835$$ −1.38234e6 −0.0686120
$$836$$ −1.05756e7 −0.523345
$$837$$ 481712. 0.0237670
$$838$$ −1.47283e7 −0.724507
$$839$$ 1.83237e7 0.898689 0.449344 0.893359i $$-0.351658\pi$$
0.449344 + 0.893359i $$0.351658\pi$$
$$840$$ 0 0
$$841$$ −1.97019e7 −0.960546
$$842$$ 4.99582e7 2.42844
$$843$$ −2.70866e7 −1.31276
$$844$$ 1.26756e7 0.612509
$$845$$ 249024. 0.0119978
$$846$$ 3.56586e6 0.171293
$$847$$ 0 0
$$848$$ 5.49069e7 2.62203
$$849$$ −555348. −0.0264421
$$850$$ 2.99772e7 1.42313
$$851$$ 6.02240e6 0.285066
$$852$$ −8.93797e7 −4.21832
$$853$$ −3.66987e7 −1.72694 −0.863471 0.504398i $$-0.831715\pi$$
−0.863471 + 0.504398i $$0.831715\pi$$
$$854$$ 0 0
$$855$$ 1.82108e6 0.0851948
$$856$$ −4.25584e7 −1.98518
$$857$$ −3.80021e7 −1.76749 −0.883743 0.467972i $$-0.844985\pi$$
−0.883743 + 0.467972i $$0.844985\pi$$
$$858$$ 1.52001e7 0.704899
$$859$$ 7.40664e6 0.342482 0.171241 0.985229i $$-0.445222\pi$$
0.171241 + 0.985229i $$0.445222\pi$$
$$860$$ 1.30990e7 0.603937
$$861$$ 0 0
$$862$$ 2.84303e7 1.30321
$$863$$ −9.79342e6 −0.447618 −0.223809 0.974633i $$-0.571849\pi$$
−0.223809 + 0.974633i $$0.571849\pi$$
$$864$$ −1.17842e7 −0.537050
$$865$$ −4.32261e6 −0.196429
$$866$$ 2.82698e7 1.28094
$$867$$ 1.08643e7 0.490857
$$868$$ 0 0
$$869$$ −39383.0 −0.00176913
$$870$$ −1.67012e6 −0.0748080
$$871$$ 3.24131e7 1.44769
$$872$$ −8.28322e7 −3.68899
$$873$$ 918880. 0.0408059
$$874$$ −1.60783e7 −0.711969
$$875$$ 0 0
$$876$$ −7.36293e7 −3.24183
$$877$$ −3.57898e7 −1.57130 −0.785652 0.618669i $$-0.787672\pi$$
−0.785652 + 0.618669i $$0.787672\pi$$
$$878$$ −4.33632e7 −1.89839
$$879$$ 1.31838e7 0.575532
$$880$$ −2.43847e6 −0.106148
$$881$$ −7.14038e6 −0.309943 −0.154971 0.987919i $$-0.549529\pi$$
−0.154971 + 0.987919i $$0.549529\pi$$
$$882$$ 0 0
$$883$$ −2.77023e7 −1.19568 −0.597838 0.801617i $$-0.703973\pi$$
−0.597838 + 0.801617i $$0.703973\pi$$
$$884$$ 4.20926e7 1.81165
$$885$$ 6.19837e6 0.266023
$$886$$ −7.14091e7 −3.05611
$$887$$ −387870. −0.0165530 −0.00827651 0.999966i $$-0.502635\pi$$
−0.00827651 + 0.999966i $$0.502635\pi$$
$$888$$ −4.23293e7 −1.80140
$$889$$ 0 0
$$890$$ −2.07337e6 −0.0877410
$$891$$ −8.46928e6 −0.357398
$$892$$ 8.03250e7 3.38017
$$893$$ 2.14977e6 0.0902117
$$894$$ −4.85795e7 −2.03287
$$895$$ 5.41932e6 0.226145
$$896$$ 0 0
$$897$$ 1.62672e7 0.675042
$$898$$ 7.21309e6 0.298491
$$899$$ 351468. 0.0145040
$$900$$ −4.25344e7 −1.75039
$$901$$ 2.22769e7 0.914203
$$902$$ 2.01508e7 0.824662
$$903$$ 0 0
$$904$$ 3.12909e7 1.27349
$$905$$ 3.40519e6 0.138204
$$906$$ −6.46236e7 −2.61560
$$907$$ 5.05051e6 0.203853 0.101926 0.994792i $$-0.467499\pi$$
0.101926 + 0.994792i $$0.467499\pi$$
$$908$$ −4.00626e7 −1.61259
$$909$$ −6.95169e6 −0.279049
$$910$$ 0 0
$$911$$ −1.72283e7 −0.687774 −0.343887 0.939011i $$-0.611744\pi$$
−0.343887 + 0.939011i $$0.611744\pi$$
$$912$$ 5.52631e7 2.20013
$$913$$ 1.12419e7 0.446337
$$914$$ −8.44660e7 −3.34439
$$915$$ −4.60303e6 −0.181757
$$916$$ −8.49807e7 −3.34643
$$917$$ 0 0
$$918$$ −1.21174e7 −0.474571
$$919$$ 3.89056e7 1.51958 0.759789 0.650170i $$-0.225302\pi$$
0.759789 + 0.650170i $$0.225302\pi$$
$$920$$ −5.33655e6 −0.207870
$$921$$ 1.13645e7 0.441470
$$922$$ −4.66769e7 −1.80832
$$923$$ 3.32990e7 1.28655
$$924$$ 0 0
$$925$$ 1.36447e7 0.524338
$$926$$ 9.40340e7 3.60377
$$927$$ −3.30453e7 −1.26302
$$928$$ −8.59801e6 −0.327739
$$929$$ −3.18602e7 −1.21118 −0.605590 0.795777i $$-0.707063\pi$$
−0.605590 + 0.795777i $$0.707063\pi$$
$$930$$ −725351. −0.0275005
$$931$$ 0 0
$$932$$ −2.22635e6 −0.0839564
$$933$$ 3.85627e6 0.145032
$$934$$ 7.45701e7 2.79703
$$935$$ −989340. −0.0370098
$$936$$ −4.91596e7 −1.83408
$$937$$ 4.09748e7 1.52464 0.762322 0.647198i $$-0.224059\pi$$
0.762322 + 0.647198i $$0.224059\pi$$
$$938$$ 0 0
$$939$$ −1.92889e7 −0.713909
$$940$$ 1.23148e6 0.0454577
$$941$$ 1.76369e7 0.649304 0.324652 0.945833i $$-0.394753\pi$$
0.324652 + 0.945833i $$0.394753\pi$$
$$942$$ −7.32988e7 −2.69135
$$943$$ 2.15655e7 0.789732
$$944$$ 8.08741e7 2.95379
$$945$$ 0 0
$$946$$ 2.50402e7 0.909723
$$947$$ 4.07232e7 1.47559 0.737797 0.675022i $$-0.235866\pi$$
0.737797 + 0.675022i $$0.235866\pi$$
$$948$$ 511284. 0.0184774
$$949$$ 2.74311e7 0.988730
$$950$$ −3.64280e7 −1.30956
$$951$$ 3.88735e7 1.39381
$$952$$ 0 0
$$953$$ −1.02210e7 −0.364553 −0.182276 0.983247i $$-0.558347\pi$$
−0.182276 + 0.983247i $$0.558347\pi$$
$$954$$ −4.49027e7 −1.59736
$$955$$ −1.77713e6 −0.0630538
$$956$$ 6.25767e7 2.21446
$$957$$ −2.24738e6 −0.0793226
$$958$$ −1.57077e7 −0.552967
$$959$$ 0 0
$$960$$ 4.42962e6 0.155127
$$961$$ −2.84765e7 −0.994668
$$962$$ 2.72175e7 0.948225
$$963$$ 1.70198e7 0.591410
$$964$$ −5.80089e7 −2.01049
$$965$$ 3.02536e6 0.104583
$$966$$ 0 0
$$967$$ −7.46133e6 −0.256596 −0.128298 0.991736i $$-0.540951\pi$$
−0.128298 + 0.991736i $$0.540951\pi$$
$$968$$ −6.71003e6 −0.230163
$$969$$ 2.24214e7 0.767103
$$970$$ 450807. 0.0153837
$$971$$ −4.34819e7 −1.47999 −0.739997 0.672610i $$-0.765173\pi$$
−0.739997 + 0.672610i $$0.765173\pi$$
$$972$$ 8.71562e7 2.95892
$$973$$ 0 0
$$974$$ −1.69736e7 −0.573292
$$975$$ 3.68559e7 1.24164
$$976$$ −6.00587e7 −2.01814
$$977$$ −1.52214e7 −0.510173 −0.255087 0.966918i $$-0.582104\pi$$
−0.255087 + 0.966918i $$0.582104\pi$$
$$978$$ 1.29779e8 4.33869
$$979$$ −2.79002e6 −0.0930360
$$980$$ 0 0
$$981$$ 3.31260e7 1.09900
$$982$$ −1.29392e7 −0.428181
$$983$$ −2.90192e7 −0.957858 −0.478929 0.877854i $$-0.658975\pi$$
−0.478929 + 0.877854i $$0.658975\pi$$
$$984$$ −1.51576e8 −4.99049
$$985$$ −7.47205e6 −0.245386
$$986$$ −8.84110e6 −0.289610
$$987$$ 0 0
$$988$$ −5.11505e7 −1.66708
$$989$$ 2.67981e7 0.871190
$$990$$ 1.99418e6 0.0646660
$$991$$ 3.46744e7 1.12156 0.560782 0.827963i $$-0.310500\pi$$
0.560782 + 0.827963i $$0.310500\pi$$
$$992$$ −3.73422e6 −0.120481
$$993$$ −4.06853e6 −0.130938
$$994$$ 0 0
$$995$$ −5.27722e6 −0.168985
$$996$$ −1.45946e8 −4.66171
$$997$$ −1.05268e7 −0.335397 −0.167699 0.985838i $$-0.553634\pi$$
−0.167699 + 0.985838i $$0.553634\pi$$
$$998$$ 7.96577e7 2.53164
$$999$$ −5.51546e6 −0.174851
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 539.6.a.e.1.1 3
7.6 odd 2 11.6.a.b.1.1 3
21.20 even 2 99.6.a.g.1.3 3
28.27 even 2 176.6.a.i.1.1 3
35.13 even 4 275.6.b.b.199.6 6
35.27 even 4 275.6.b.b.199.1 6
35.34 odd 2 275.6.a.b.1.3 3
56.13 odd 2 704.6.a.q.1.1 3
56.27 even 2 704.6.a.t.1.3 3
77.76 even 2 121.6.a.d.1.3 3
231.230 odd 2 1089.6.a.r.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.1 3 7.6 odd 2
99.6.a.g.1.3 3 21.20 even 2
121.6.a.d.1.3 3 77.76 even 2
176.6.a.i.1.1 3 28.27 even 2
275.6.a.b.1.3 3 35.34 odd 2
275.6.b.b.199.1 6 35.27 even 4
275.6.b.b.199.6 6 35.13 even 4
539.6.a.e.1.1 3 1.1 even 1 trivial
704.6.a.q.1.1 3 56.13 odd 2
704.6.a.t.1.3 3 56.27 even 2
1089.6.a.r.1.1 3 231.230 odd 2