Properties

 Label 338.8.a.a.1.1 Level $338$ Weight $8$ Character 338.1 Self dual yes Analytic conductor $105.586$ 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] = [338,8,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$105.586138614$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-8.00000 q^{2} -87.0000 q^{3} +64.0000 q^{4} -321.000 q^{5} +696.000 q^{6} +181.000 q^{7} -512.000 q^{8} +5382.00 q^{9} +O(q^{10})$$ $$q-8.00000 q^{2} -87.0000 q^{3} +64.0000 q^{4} -321.000 q^{5} +696.000 q^{6} +181.000 q^{7} -512.000 q^{8} +5382.00 q^{9} +2568.00 q^{10} -7782.00 q^{11} -5568.00 q^{12} -1448.00 q^{14} +27927.0 q^{15} +4096.00 q^{16} +9069.00 q^{17} -43056.0 q^{18} +37150.0 q^{19} -20544.0 q^{20} -15747.0 q^{21} +62256.0 q^{22} +19008.0 q^{23} +44544.0 q^{24} +24916.0 q^{25} -277965. q^{27} +11584.0 q^{28} +174750. q^{29} -223416. q^{30} -29012.0 q^{31} -32768.0 q^{32} +677034. q^{33} -72552.0 q^{34} -58101.0 q^{35} +344448. q^{36} -323669. q^{37} -297200. q^{38} +164352. q^{40} -795312. q^{41} +125976. q^{42} -314137. q^{43} -498048. q^{44} -1.72762e6 q^{45} -152064. q^{46} +447441. q^{47} -356352. q^{48} -790782. q^{49} -199328. q^{50} -789003. q^{51} -1.46923e6 q^{53} +2.22372e6 q^{54} +2.49802e6 q^{55} -92672.0 q^{56} -3.23205e6 q^{57} -1.39800e6 q^{58} -1.62777e6 q^{59} +1.78733e6 q^{60} -2.39961e6 q^{61} +232096. q^{62} +974142. q^{63} +262144. q^{64} -5.41627e6 q^{66} +64066.0 q^{67} +580416. q^{68} -1.65370e6 q^{69} +464808. q^{70} +322383. q^{71} -2.75558e6 q^{72} +4.45478e6 q^{73} +2.58935e6 q^{74} -2.16769e6 q^{75} +2.37760e6 q^{76} -1.40854e6 q^{77} +753560. q^{79} -1.31482e6 q^{80} +1.24125e7 q^{81} +6.36250e6 q^{82} +1.21909e6 q^{83} -1.00781e6 q^{84} -2.91115e6 q^{85} +2.51310e6 q^{86} -1.52032e7 q^{87} +3.98438e6 q^{88} -3.39033e6 q^{89} +1.38210e7 q^{90} +1.21651e6 q^{92} +2.52404e6 q^{93} -3.57953e6 q^{94} -1.19252e7 q^{95} +2.85082e6 q^{96} -1.62877e6 q^{97} +6.32626e6 q^{98} -4.18827e7 q^{99} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −8.00000 −0.707107
$$3$$ −87.0000 −1.86035 −0.930175 0.367115i $$-0.880345\pi$$
−0.930175 + 0.367115i $$0.880345\pi$$
$$4$$ 64.0000 0.500000
$$5$$ −321.000 −1.14844 −0.574222 0.818699i $$-0.694695\pi$$
−0.574222 + 0.818699i $$0.694695\pi$$
$$6$$ 696.000 1.31547
$$7$$ 181.000 0.199451 0.0997253 0.995015i $$-0.468204\pi$$
0.0997253 + 0.995015i $$0.468204\pi$$
$$8$$ −512.000 −0.353553
$$9$$ 5382.00 2.46091
$$10$$ 2568.00 0.812073
$$11$$ −7782.00 −1.76286 −0.881428 0.472318i $$-0.843417\pi$$
−0.881428 + 0.472318i $$0.843417\pi$$
$$12$$ −5568.00 −0.930175
$$13$$ 0 0
$$14$$ −1448.00 −0.141033
$$15$$ 27927.0 2.13651
$$16$$ 4096.00 0.250000
$$17$$ 9069.00 0.447701 0.223851 0.974623i $$-0.428137\pi$$
0.223851 + 0.974623i $$0.428137\pi$$
$$18$$ −43056.0 −1.74012
$$19$$ 37150.0 1.24257 0.621286 0.783584i $$-0.286611\pi$$
0.621286 + 0.783584i $$0.286611\pi$$
$$20$$ −20544.0 −0.574222
$$21$$ −15747.0 −0.371048
$$22$$ 62256.0 1.24653
$$23$$ 19008.0 0.325753 0.162877 0.986646i $$-0.447923\pi$$
0.162877 + 0.986646i $$0.447923\pi$$
$$24$$ 44544.0 0.657733
$$25$$ 24916.0 0.318925
$$26$$ 0 0
$$27$$ −277965. −2.71780
$$28$$ 11584.0 0.0997253
$$29$$ 174750. 1.33053 0.665264 0.746608i $$-0.268319\pi$$
0.665264 + 0.746608i $$0.268319\pi$$
$$30$$ −223416. −1.51074
$$31$$ −29012.0 −0.174909 −0.0874544 0.996169i $$-0.527873\pi$$
−0.0874544 + 0.996169i $$0.527873\pi$$
$$32$$ −32768.0 −0.176777
$$33$$ 677034. 3.27953
$$34$$ −72552.0 −0.316572
$$35$$ −58101.0 −0.229058
$$36$$ 344448. 1.23045
$$37$$ −323669. −1.05050 −0.525249 0.850949i $$-0.676028\pi$$
−0.525249 + 0.850949i $$0.676028\pi$$
$$38$$ −297200. −0.878630
$$39$$ 0 0
$$40$$ 164352. 0.406036
$$41$$ −795312. −1.80216 −0.901081 0.433650i $$-0.857225\pi$$
−0.901081 + 0.433650i $$0.857225\pi$$
$$42$$ 125976. 0.262371
$$43$$ −314137. −0.602531 −0.301266 0.953540i $$-0.597409\pi$$
−0.301266 + 0.953540i $$0.597409\pi$$
$$44$$ −498048. −0.881428
$$45$$ −1.72762e6 −2.82621
$$46$$ −152064. −0.230342
$$47$$ 447441. 0.628627 0.314314 0.949319i $$-0.398226\pi$$
0.314314 + 0.949319i $$0.398226\pi$$
$$48$$ −356352. −0.465088
$$49$$ −790782. −0.960219
$$50$$ −199328. −0.225514
$$51$$ −789003. −0.832881
$$52$$ 0 0
$$53$$ −1.46923e6 −1.35558 −0.677790 0.735256i $$-0.737062\pi$$
−0.677790 + 0.735256i $$0.737062\pi$$
$$54$$ 2.22372e6 1.92177
$$55$$ 2.49802e6 2.02454
$$56$$ −92672.0 −0.0705165
$$57$$ −3.23205e6 −2.31162
$$58$$ −1.39800e6 −0.940826
$$59$$ −1.62777e6 −1.03184 −0.515918 0.856638i $$-0.672549\pi$$
−0.515918 + 0.856638i $$0.672549\pi$$
$$60$$ 1.78733e6 1.06825
$$61$$ −2.39961e6 −1.35359 −0.676793 0.736173i $$-0.736631\pi$$
−0.676793 + 0.736173i $$0.736631\pi$$
$$62$$ 232096. 0.123679
$$63$$ 974142. 0.490829
$$64$$ 262144. 0.125000
$$65$$ 0 0
$$66$$ −5.41627e6 −2.31898
$$67$$ 64066.0 0.0260235 0.0130118 0.999915i $$-0.495858\pi$$
0.0130118 + 0.999915i $$0.495858\pi$$
$$68$$ 580416. 0.223851
$$69$$ −1.65370e6 −0.606016
$$70$$ 464808. 0.161968
$$71$$ 322383. 0.106898 0.0534488 0.998571i $$-0.482979\pi$$
0.0534488 + 0.998571i $$0.482979\pi$$
$$72$$ −2.75558e6 −0.870061
$$73$$ 4.45478e6 1.34028 0.670141 0.742233i $$-0.266233\pi$$
0.670141 + 0.742233i $$0.266233\pi$$
$$74$$ 2.58935e6 0.742814
$$75$$ −2.16769e6 −0.593312
$$76$$ 2.37760e6 0.621286
$$77$$ −1.40854e6 −0.351603
$$78$$ 0 0
$$79$$ 753560. 0.171958 0.0859791 0.996297i $$-0.472598\pi$$
0.0859791 + 0.996297i $$0.472598\pi$$
$$80$$ −1.31482e6 −0.287111
$$81$$ 1.24125e7 2.59515
$$82$$ 6.36250e6 1.27432
$$83$$ 1.21909e6 0.234025 0.117013 0.993130i $$-0.462668\pi$$
0.117013 + 0.993130i $$0.462668\pi$$
$$84$$ −1.00781e6 −0.185524
$$85$$ −2.91115e6 −0.514160
$$86$$ 2.51310e6 0.426054
$$87$$ −1.52032e7 −2.47525
$$88$$ 3.98438e6 0.623264
$$89$$ −3.39033e6 −0.509773 −0.254887 0.966971i $$-0.582038\pi$$
−0.254887 + 0.966971i $$0.582038\pi$$
$$90$$ 1.38210e7 1.99843
$$91$$ 0 0
$$92$$ 1.21651e6 0.162877
$$93$$ 2.52404e6 0.325392
$$94$$ −3.57953e6 −0.444507
$$95$$ −1.19252e7 −1.42702
$$96$$ 2.85082e6 0.328867
$$97$$ −1.62877e6 −0.181201 −0.0906003 0.995887i $$-0.528879\pi$$
−0.0906003 + 0.995887i $$0.528879\pi$$
$$98$$ 6.32626e6 0.678978
$$99$$ −4.18827e7 −4.33822
$$100$$ 1.59462e6 0.159462
$$101$$ −1.53503e7 −1.48249 −0.741244 0.671236i $$-0.765764\pi$$
−0.741244 + 0.671236i $$0.765764\pi$$
$$102$$ 6.31202e6 0.588936
$$103$$ 6.87643e6 0.620058 0.310029 0.950727i $$-0.399661\pi$$
0.310029 + 0.950727i $$0.399661\pi$$
$$104$$ 0 0
$$105$$ 5.05479e6 0.426128
$$106$$ 1.17539e7 0.958539
$$107$$ −1.52027e7 −1.19971 −0.599857 0.800107i $$-0.704776\pi$$
−0.599857 + 0.800107i $$0.704776\pi$$
$$108$$ −1.77898e7 −1.35890
$$109$$ −6.73260e6 −0.497955 −0.248978 0.968509i $$-0.580095\pi$$
−0.248978 + 0.968509i $$0.580095\pi$$
$$110$$ −1.99842e7 −1.43157
$$111$$ 2.81592e7 1.95429
$$112$$ 741376. 0.0498627
$$113$$ −1.15292e7 −0.751667 −0.375833 0.926687i $$-0.622644\pi$$
−0.375833 + 0.926687i $$0.622644\pi$$
$$114$$ 2.58564e7 1.63456
$$115$$ −6.10157e6 −0.374110
$$116$$ 1.11840e7 0.665264
$$117$$ 0 0
$$118$$ 1.30222e7 0.729619
$$119$$ 1.64149e6 0.0892943
$$120$$ −1.42986e7 −0.755370
$$121$$ 4.10724e7 2.10766
$$122$$ 1.91969e7 0.957130
$$123$$ 6.91921e7 3.35266
$$124$$ −1.85677e6 −0.0874544
$$125$$ 1.70801e7 0.782177
$$126$$ −7.79314e6 −0.347069
$$127$$ 2.06699e7 0.895418 0.447709 0.894179i $$-0.352240\pi$$
0.447709 + 0.894179i $$0.352240\pi$$
$$128$$ −2.09715e6 −0.0883883
$$129$$ 2.73299e7 1.12092
$$130$$ 0 0
$$131$$ −1.90949e7 −0.742107 −0.371054 0.928611i $$-0.621003\pi$$
−0.371054 + 0.928611i $$0.621003\pi$$
$$132$$ 4.33302e7 1.63977
$$133$$ 6.72415e6 0.247832
$$134$$ −512528. −0.0184014
$$135$$ 8.92268e7 3.12124
$$136$$ −4.64333e6 −0.158286
$$137$$ −2.96901e7 −0.986482 −0.493241 0.869893i $$-0.664188\pi$$
−0.493241 + 0.869893i $$0.664188\pi$$
$$138$$ 1.32296e7 0.428518
$$139$$ 1.55652e7 0.491591 0.245795 0.969322i $$-0.420951\pi$$
0.245795 + 0.969322i $$0.420951\pi$$
$$140$$ −3.71846e6 −0.114529
$$141$$ −3.89274e7 −1.16947
$$142$$ −2.57906e6 −0.0755880
$$143$$ 0 0
$$144$$ 2.20447e7 0.615226
$$145$$ −5.60948e7 −1.52804
$$146$$ −3.56383e7 −0.947723
$$147$$ 6.87980e7 1.78635
$$148$$ −2.07148e7 −0.525249
$$149$$ 2.49675e6 0.0618334 0.0309167 0.999522i $$-0.490157\pi$$
0.0309167 + 0.999522i $$0.490157\pi$$
$$150$$ 1.73415e7 0.419535
$$151$$ −2.39802e7 −0.566804 −0.283402 0.959001i $$-0.591463\pi$$
−0.283402 + 0.959001i $$0.591463\pi$$
$$152$$ −1.90208e7 −0.439315
$$153$$ 4.88094e7 1.10175
$$154$$ 1.12683e7 0.248621
$$155$$ 9.31285e6 0.200873
$$156$$ 0 0
$$157$$ 1.70550e7 0.351725 0.175863 0.984415i $$-0.443729\pi$$
0.175863 + 0.984415i $$0.443729\pi$$
$$158$$ −6.02848e6 −0.121593
$$159$$ 1.27823e8 2.52185
$$160$$ 1.05185e7 0.203018
$$161$$ 3.44045e6 0.0649717
$$162$$ −9.93002e7 −1.83505
$$163$$ 7.34586e7 1.32857 0.664287 0.747477i $$-0.268735\pi$$
0.664287 + 0.747477i $$0.268735\pi$$
$$164$$ −5.09000e7 −0.901081
$$165$$ −2.17328e8 −3.76636
$$166$$ −9.75274e6 −0.165481
$$167$$ −4.66860e7 −0.775674 −0.387837 0.921728i $$-0.626778\pi$$
−0.387837 + 0.921728i $$0.626778\pi$$
$$168$$ 8.06246e6 0.131185
$$169$$ 0 0
$$170$$ 2.32892e7 0.363566
$$171$$ 1.99941e8 3.05785
$$172$$ −2.01048e7 −0.301266
$$173$$ −7.80931e7 −1.14670 −0.573352 0.819309i $$-0.694357\pi$$
−0.573352 + 0.819309i $$0.694357\pi$$
$$174$$ 1.21626e8 1.75027
$$175$$ 4.50980e6 0.0636098
$$176$$ −3.18751e7 −0.440714
$$177$$ 1.41616e8 1.91958
$$178$$ 2.71226e7 0.360464
$$179$$ −5.56163e7 −0.724797 −0.362399 0.932023i $$-0.618042\pi$$
−0.362399 + 0.932023i $$0.618042\pi$$
$$180$$ −1.10568e8 −1.41311
$$181$$ −1.19435e8 −1.49711 −0.748557 0.663070i $$-0.769253\pi$$
−0.748557 + 0.663070i $$0.769253\pi$$
$$182$$ 0 0
$$183$$ 2.08766e8 2.51815
$$184$$ −9.73210e6 −0.115171
$$185$$ 1.03898e8 1.20644
$$186$$ −2.01924e7 −0.230087
$$187$$ −7.05750e7 −0.789233
$$188$$ 2.86362e7 0.314314
$$189$$ −5.03117e7 −0.542066
$$190$$ 9.54012e7 1.00906
$$191$$ 1.05485e8 1.09540 0.547700 0.836675i $$-0.315504\pi$$
0.547700 + 0.836675i $$0.315504\pi$$
$$192$$ −2.28065e7 −0.232544
$$193$$ −2.12059e7 −0.212327 −0.106164 0.994349i $$-0.533857\pi$$
−0.106164 + 0.994349i $$0.533857\pi$$
$$194$$ 1.30302e7 0.128128
$$195$$ 0 0
$$196$$ −5.06100e7 −0.480110
$$197$$ 1.66535e8 1.55194 0.775969 0.630771i $$-0.217261\pi$$
0.775969 + 0.630771i $$0.217261\pi$$
$$198$$ 3.35062e8 3.06759
$$199$$ −1.26351e8 −1.13656 −0.568279 0.822836i $$-0.692391\pi$$
−0.568279 + 0.822836i $$0.692391\pi$$
$$200$$ −1.27570e7 −0.112757
$$201$$ −5.57374e6 −0.0484129
$$202$$ 1.22802e8 1.04828
$$203$$ 3.16298e7 0.265375
$$204$$ −5.04962e7 −0.416441
$$205$$ 2.55295e8 2.06968
$$206$$ −5.50114e7 −0.438448
$$207$$ 1.02301e8 0.801648
$$208$$ 0 0
$$209$$ −2.89101e8 −2.19047
$$210$$ −4.04383e7 −0.301318
$$211$$ 1.08571e8 0.795655 0.397828 0.917460i $$-0.369764\pi$$
0.397828 + 0.917460i $$0.369764\pi$$
$$212$$ −9.40308e7 −0.677790
$$213$$ −2.80473e7 −0.198867
$$214$$ 1.21622e8 0.848326
$$215$$ 1.00838e8 0.691974
$$216$$ 1.42318e8 0.960886
$$217$$ −5.25117e6 −0.0348857
$$218$$ 5.38608e7 0.352108
$$219$$ −3.87566e8 −2.49340
$$220$$ 1.59873e8 1.01227
$$221$$ 0 0
$$222$$ −2.25274e8 −1.38189
$$223$$ 1.25603e8 0.758459 0.379229 0.925303i $$-0.376189\pi$$
0.379229 + 0.925303i $$0.376189\pi$$
$$224$$ −5.93101e6 −0.0352582
$$225$$ 1.34098e8 0.784844
$$226$$ 9.22338e7 0.531509
$$227$$ −1.90774e8 −1.08250 −0.541252 0.840861i $$-0.682049\pi$$
−0.541252 + 0.840861i $$0.682049\pi$$
$$228$$ −2.06851e8 −1.15581
$$229$$ 5.28911e7 0.291044 0.145522 0.989355i $$-0.453514\pi$$
0.145522 + 0.989355i $$0.453514\pi$$
$$230$$ 4.88125e7 0.264536
$$231$$ 1.22543e8 0.654104
$$232$$ −8.94720e7 −0.470413
$$233$$ 1.51254e8 0.783359 0.391680 0.920102i $$-0.371894\pi$$
0.391680 + 0.920102i $$0.371894\pi$$
$$234$$ 0 0
$$235$$ −1.43629e8 −0.721944
$$236$$ −1.04177e8 −0.515918
$$237$$ −6.55597e7 −0.319903
$$238$$ −1.31319e7 −0.0631406
$$239$$ −2.61917e8 −1.24100 −0.620498 0.784208i $$-0.713070\pi$$
−0.620498 + 0.784208i $$0.713070\pi$$
$$240$$ 1.14389e8 0.534127
$$241$$ 1.31752e8 0.606312 0.303156 0.952941i $$-0.401960\pi$$
0.303156 + 0.952941i $$0.401960\pi$$
$$242$$ −3.28579e8 −1.49034
$$243$$ −4.71980e8 −2.11009
$$244$$ −1.53575e8 −0.676793
$$245$$ 2.53841e8 1.10276
$$246$$ −5.53537e8 −2.37069
$$247$$ 0 0
$$248$$ 1.48541e7 0.0618396
$$249$$ −1.06061e8 −0.435370
$$250$$ −1.36641e8 −0.553083
$$251$$ 2.47061e8 0.986159 0.493080 0.869984i $$-0.335871\pi$$
0.493080 + 0.869984i $$0.335871\pi$$
$$252$$ 6.23451e7 0.245415
$$253$$ −1.47920e8 −0.574256
$$254$$ −1.65359e8 −0.633156
$$255$$ 2.53270e8 0.956518
$$256$$ 1.67772e7 0.0625000
$$257$$ 2.27286e8 0.835231 0.417616 0.908624i $$-0.362866\pi$$
0.417616 + 0.908624i $$0.362866\pi$$
$$258$$ −2.18639e8 −0.792610
$$259$$ −5.85841e7 −0.209522
$$260$$ 0 0
$$261$$ 9.40504e8 3.27430
$$262$$ 1.52759e8 0.524749
$$263$$ −4.25872e8 −1.44356 −0.721779 0.692124i $$-0.756675\pi$$
−0.721779 + 0.692124i $$0.756675\pi$$
$$264$$ −3.46641e8 −1.15949
$$265$$ 4.71623e8 1.55681
$$266$$ −5.37932e7 −0.175243
$$267$$ 2.94959e8 0.948357
$$268$$ 4.10022e6 0.0130118
$$269$$ −5.14154e8 −1.61050 −0.805250 0.592936i $$-0.797969\pi$$
−0.805250 + 0.592936i $$0.797969\pi$$
$$270$$ −7.13814e8 −2.20705
$$271$$ −4.57096e7 −0.139513 −0.0697565 0.997564i $$-0.522222\pi$$
−0.0697565 + 0.997564i $$0.522222\pi$$
$$272$$ 3.71466e7 0.111925
$$273$$ 0 0
$$274$$ 2.37521e8 0.697548
$$275$$ −1.93896e8 −0.562218
$$276$$ −1.05837e8 −0.303008
$$277$$ −2.73964e8 −0.774487 −0.387244 0.921977i $$-0.626573\pi$$
−0.387244 + 0.921977i $$0.626573\pi$$
$$278$$ −1.24522e8 −0.347607
$$279$$ −1.56143e8 −0.430434
$$280$$ 2.97477e7 0.0809842
$$281$$ −4.21707e8 −1.13381 −0.566903 0.823784i $$-0.691859\pi$$
−0.566903 + 0.823784i $$0.691859\pi$$
$$282$$ 3.11419e8 0.826938
$$283$$ 3.81957e8 1.00176 0.500878 0.865518i $$-0.333010\pi$$
0.500878 + 0.865518i $$0.333010\pi$$
$$284$$ 2.06325e7 0.0534488
$$285$$ 1.03749e9 2.65477
$$286$$ 0 0
$$287$$ −1.43951e8 −0.359443
$$288$$ −1.76357e8 −0.435031
$$289$$ −3.28092e8 −0.799564
$$290$$ 4.48758e8 1.08049
$$291$$ 1.41703e8 0.337097
$$292$$ 2.85106e8 0.670141
$$293$$ 4.04833e8 0.940240 0.470120 0.882602i $$-0.344211\pi$$
0.470120 + 0.882602i $$0.344211\pi$$
$$294$$ −5.50384e8 −1.26314
$$295$$ 5.22514e8 1.18501
$$296$$ 1.65719e8 0.371407
$$297$$ 2.16312e9 4.79108
$$298$$ −1.99740e7 −0.0437228
$$299$$ 0 0
$$300$$ −1.38732e8 −0.296656
$$301$$ −5.68588e7 −0.120175
$$302$$ 1.91841e8 0.400791
$$303$$ 1.33547e9 2.75795
$$304$$ 1.52166e8 0.310643
$$305$$ 7.70274e8 1.55452
$$306$$ −3.90475e8 −0.779055
$$307$$ 4.75520e7 0.0937960 0.0468980 0.998900i $$-0.485066\pi$$
0.0468980 + 0.998900i $$0.485066\pi$$
$$308$$ −9.01467e7 −0.175801
$$309$$ −5.98249e8 −1.15353
$$310$$ −7.45028e7 −0.142039
$$311$$ −3.02841e8 −0.570892 −0.285446 0.958395i $$-0.592142\pi$$
−0.285446 + 0.958395i $$0.592142\pi$$
$$312$$ 0 0
$$313$$ −6.31685e8 −1.16438 −0.582191 0.813052i $$-0.697804\pi$$
−0.582191 + 0.813052i $$0.697804\pi$$
$$314$$ −1.36440e8 −0.248707
$$315$$ −3.12700e8 −0.563690
$$316$$ 4.82278e7 0.0859791
$$317$$ −7.93332e8 −1.39877 −0.699387 0.714743i $$-0.746544\pi$$
−0.699387 + 0.714743i $$0.746544\pi$$
$$318$$ −1.02259e9 −1.78322
$$319$$ −1.35990e9 −2.34553
$$320$$ −8.41482e7 −0.143556
$$321$$ 1.32264e9 2.23189
$$322$$ −2.75236e7 −0.0459420
$$323$$ 3.36913e8 0.556300
$$324$$ 7.94401e8 1.29757
$$325$$ 0 0
$$326$$ −5.87669e8 −0.939444
$$327$$ 5.85737e8 0.926372
$$328$$ 4.07200e8 0.637161
$$329$$ 8.09868e7 0.125380
$$330$$ 1.73862e9 2.66322
$$331$$ 1.21628e9 1.84346 0.921731 0.387829i $$-0.126775\pi$$
0.921731 + 0.387829i $$0.126775\pi$$
$$332$$ 7.80219e7 0.117013
$$333$$ −1.74199e9 −2.58518
$$334$$ 3.73488e8 0.548484
$$335$$ −2.05652e7 −0.0298866
$$336$$ −6.44997e7 −0.0927620
$$337$$ −1.51221e8 −0.215232 −0.107616 0.994193i $$-0.534322\pi$$
−0.107616 + 0.994193i $$0.534322\pi$$
$$338$$ 0 0
$$339$$ 1.00304e9 1.39836
$$340$$ −1.86314e8 −0.257080
$$341$$ 2.25771e8 0.308339
$$342$$ −1.59953e9 −2.16223
$$343$$ −2.92193e8 −0.390967
$$344$$ 1.60838e8 0.213027
$$345$$ 5.30836e8 0.695975
$$346$$ 6.24745e8 0.810842
$$347$$ −5.97234e8 −0.767347 −0.383673 0.923469i $$-0.625341\pi$$
−0.383673 + 0.923469i $$0.625341\pi$$
$$348$$ −9.73008e8 −1.23762
$$349$$ −1.19600e8 −0.150606 −0.0753029 0.997161i $$-0.523992\pi$$
−0.0753029 + 0.997161i $$0.523992\pi$$
$$350$$ −3.60784e7 −0.0449789
$$351$$ 0 0
$$352$$ 2.55001e8 0.311632
$$353$$ 4.66414e8 0.564366 0.282183 0.959361i $$-0.408942\pi$$
0.282183 + 0.959361i $$0.408942\pi$$
$$354$$ −1.13293e9 −1.35735
$$355$$ −1.03485e8 −0.122766
$$356$$ −2.16981e8 −0.254887
$$357$$ −1.42810e8 −0.166119
$$358$$ 4.44931e8 0.512509
$$359$$ 7.70102e8 0.878451 0.439225 0.898377i $$-0.355253\pi$$
0.439225 + 0.898377i $$0.355253\pi$$
$$360$$ 8.84542e8 0.999217
$$361$$ 4.86251e8 0.543983
$$362$$ 9.55477e8 1.05862
$$363$$ −3.57329e9 −3.92099
$$364$$ 0 0
$$365$$ −1.42999e9 −1.53924
$$366$$ −1.67013e9 −1.78060
$$367$$ −8.55319e8 −0.903227 −0.451613 0.892214i $$-0.649151\pi$$
−0.451613 + 0.892214i $$0.649151\pi$$
$$368$$ 7.78568e7 0.0814384
$$369$$ −4.28037e9 −4.43495
$$370$$ −8.31182e8 −0.853081
$$371$$ −2.65931e8 −0.270371
$$372$$ 1.61539e8 0.162696
$$373$$ −5.29609e8 −0.528414 −0.264207 0.964466i $$-0.585110\pi$$
−0.264207 + 0.964466i $$0.585110\pi$$
$$374$$ 5.64600e8 0.558072
$$375$$ −1.48597e9 −1.45512
$$376$$ −2.29090e8 −0.222253
$$377$$ 0 0
$$378$$ 4.02493e8 0.383299
$$379$$ −1.98358e9 −1.87159 −0.935797 0.352540i $$-0.885318\pi$$
−0.935797 + 0.352540i $$0.885318\pi$$
$$380$$ −7.63210e8 −0.713512
$$381$$ −1.79828e9 −1.66579
$$382$$ −8.43877e8 −0.774564
$$383$$ −8.98756e8 −0.817422 −0.408711 0.912664i $$-0.634022\pi$$
−0.408711 + 0.912664i $$0.634022\pi$$
$$384$$ 1.82452e8 0.164433
$$385$$ 4.52142e8 0.403796
$$386$$ 1.69647e8 0.150138
$$387$$ −1.69069e9 −1.48277
$$388$$ −1.04242e8 −0.0906003
$$389$$ 1.82475e9 1.57174 0.785868 0.618395i $$-0.212217\pi$$
0.785868 + 0.618395i $$0.212217\pi$$
$$390$$ 0 0
$$391$$ 1.72384e8 0.145840
$$392$$ 4.04880e8 0.339489
$$393$$ 1.66125e9 1.38058
$$394$$ −1.33228e9 −1.09739
$$395$$ −2.41893e8 −0.197485
$$396$$ −2.68049e9 −2.16911
$$397$$ −4.93083e8 −0.395506 −0.197753 0.980252i $$-0.563364\pi$$
−0.197753 + 0.980252i $$0.563364\pi$$
$$398$$ 1.01081e9 0.803668
$$399$$ −5.85001e8 −0.461054
$$400$$ 1.02056e8 0.0797312
$$401$$ 5.68280e8 0.440105 0.220053 0.975488i $$-0.429377\pi$$
0.220053 + 0.975488i $$0.429377\pi$$
$$402$$ 4.45899e7 0.0342331
$$403$$ 0 0
$$404$$ −9.82417e8 −0.741244
$$405$$ −3.98442e9 −2.98039
$$406$$ −2.53038e8 −0.187648
$$407$$ 2.51879e9 1.85188
$$408$$ 4.03970e8 0.294468
$$409$$ 1.28472e9 0.928489 0.464245 0.885707i $$-0.346326\pi$$
0.464245 + 0.885707i $$0.346326\pi$$
$$410$$ −2.04236e9 −1.46349
$$411$$ 2.58304e9 1.83520
$$412$$ 4.40091e8 0.310029
$$413$$ −2.94626e8 −0.205801
$$414$$ −8.18408e8 −0.566851
$$415$$ −3.91329e8 −0.268765
$$416$$ 0 0
$$417$$ −1.35418e9 −0.914532
$$418$$ 2.31281e9 1.54890
$$419$$ 2.74847e8 0.182533 0.0912667 0.995826i $$-0.470908\pi$$
0.0912667 + 0.995826i $$0.470908\pi$$
$$420$$ 3.23506e8 0.213064
$$421$$ −7.51368e8 −0.490756 −0.245378 0.969428i $$-0.578912\pi$$
−0.245378 + 0.969428i $$0.578912\pi$$
$$422$$ −8.68568e8 −0.562613
$$423$$ 2.40813e9 1.54699
$$424$$ 7.52247e8 0.479270
$$425$$ 2.25963e8 0.142783
$$426$$ 2.24379e8 0.140620
$$427$$ −4.34329e8 −0.269974
$$428$$ −9.72974e8 −0.599857
$$429$$ 0 0
$$430$$ −8.06704e8 −0.489299
$$431$$ 1.30756e8 0.0786668 0.0393334 0.999226i $$-0.487477\pi$$
0.0393334 + 0.999226i $$0.487477\pi$$
$$432$$ −1.13854e9 −0.679449
$$433$$ 1.66736e9 0.987010 0.493505 0.869743i $$-0.335716\pi$$
0.493505 + 0.869743i $$0.335716\pi$$
$$434$$ 4.20094e7 0.0246679
$$435$$ 4.88024e9 2.84269
$$436$$ −4.30887e8 −0.248978
$$437$$ 7.06147e8 0.404772
$$438$$ 3.10053e9 1.76310
$$439$$ 2.31478e9 1.30582 0.652910 0.757436i $$-0.273548\pi$$
0.652910 + 0.757436i $$0.273548\pi$$
$$440$$ −1.27899e9 −0.715784
$$441$$ −4.25599e9 −2.36301
$$442$$ 0 0
$$443$$ −6.90047e8 −0.377108 −0.188554 0.982063i $$-0.560380\pi$$
−0.188554 + 0.982063i $$0.560380\pi$$
$$444$$ 1.80219e9 0.977147
$$445$$ 1.08830e9 0.585446
$$446$$ −1.00482e9 −0.536311
$$447$$ −2.17217e8 −0.115032
$$448$$ 4.74481e7 0.0249313
$$449$$ −2.63806e9 −1.37538 −0.687690 0.726004i $$-0.741375\pi$$
−0.687690 + 0.726004i $$0.741375\pi$$
$$450$$ −1.07278e9 −0.554968
$$451$$ 6.18912e9 3.17695
$$452$$ −7.37870e8 −0.375833
$$453$$ 2.08627e9 1.05445
$$454$$ 1.52619e9 0.765445
$$455$$ 0 0
$$456$$ 1.65481e9 0.817280
$$457$$ 6.16222e8 0.302016 0.151008 0.988533i $$-0.451748\pi$$
0.151008 + 0.988533i $$0.451748\pi$$
$$458$$ −4.23129e8 −0.205799
$$459$$ −2.52086e9 −1.21676
$$460$$ −3.90500e8 −0.187055
$$461$$ −1.23621e9 −0.587679 −0.293839 0.955855i $$-0.594933\pi$$
−0.293839 + 0.955855i $$0.594933\pi$$
$$462$$ −9.80345e8 −0.462522
$$463$$ −6.78469e7 −0.0317685 −0.0158843 0.999874i $$-0.505056\pi$$
−0.0158843 + 0.999874i $$0.505056\pi$$
$$464$$ 7.15776e8 0.332632
$$465$$ −8.10218e8 −0.373694
$$466$$ −1.21003e9 −0.553919
$$467$$ −1.17502e9 −0.533869 −0.266934 0.963715i $$-0.586011\pi$$
−0.266934 + 0.963715i $$0.586011\pi$$
$$468$$ 0 0
$$469$$ 1.15959e7 0.00519040
$$470$$ 1.14903e9 0.510491
$$471$$ −1.48379e9 −0.654332
$$472$$ 8.33418e8 0.364809
$$473$$ 2.44461e9 1.06218
$$474$$ 5.24478e8 0.226205
$$475$$ 9.25629e8 0.396287
$$476$$ 1.05055e8 0.0446471
$$477$$ −7.90741e9 −3.33595
$$478$$ 2.09533e9 0.877517
$$479$$ −3.96154e8 −0.164699 −0.0823494 0.996604i $$-0.526242\pi$$
−0.0823494 + 0.996604i $$0.526242\pi$$
$$480$$ −9.15112e8 −0.377685
$$481$$ 0 0
$$482$$ −1.05401e9 −0.428728
$$483$$ −2.99319e8 −0.120870
$$484$$ 2.62863e9 1.05383
$$485$$ 5.22836e8 0.208099
$$486$$ 3.77584e9 1.49206
$$487$$ 3.03665e9 1.19136 0.595680 0.803222i $$-0.296883\pi$$
0.595680 + 0.803222i $$0.296883\pi$$
$$488$$ 1.22860e9 0.478565
$$489$$ −6.39090e9 −2.47162
$$490$$ −2.03073e9 −0.779768
$$491$$ −2.91974e9 −1.11316 −0.556582 0.830793i $$-0.687887\pi$$
−0.556582 + 0.830793i $$0.687887\pi$$
$$492$$ 4.42830e9 1.67633
$$493$$ 1.58481e9 0.595679
$$494$$ 0 0
$$495$$ 1.34444e10 4.98221
$$496$$ −1.18833e8 −0.0437272
$$497$$ 5.83513e7 0.0213208
$$498$$ 8.48488e8 0.307853
$$499$$ 1.62343e9 0.584898 0.292449 0.956281i $$-0.405530\pi$$
0.292449 + 0.956281i $$0.405530\pi$$
$$500$$ 1.09313e9 0.391089
$$501$$ 4.06168e9 1.44303
$$502$$ −1.97649e9 −0.697320
$$503$$ 4.75888e9 1.66731 0.833655 0.552285i $$-0.186244\pi$$
0.833655 + 0.552285i $$0.186244\pi$$
$$504$$ −4.98761e8 −0.173534
$$505$$ 4.92744e9 1.70256
$$506$$ 1.18336e9 0.406061
$$507$$ 0 0
$$508$$ 1.32288e9 0.447709
$$509$$ 9.19375e8 0.309016 0.154508 0.987992i $$-0.450621\pi$$
0.154508 + 0.987992i $$0.450621\pi$$
$$510$$ −2.02616e9 −0.676360
$$511$$ 8.06316e8 0.267320
$$512$$ −1.34218e8 −0.0441942
$$513$$ −1.03264e10 −3.37706
$$514$$ −1.81829e9 −0.590598
$$515$$ −2.20733e9 −0.712103
$$516$$ 1.74911e9 0.560460
$$517$$ −3.48199e9 −1.10818
$$518$$ 4.68673e8 0.148155
$$519$$ 6.79410e9 2.13327
$$520$$ 0 0
$$521$$ −1.46089e9 −0.452569 −0.226284 0.974061i $$-0.572658\pi$$
−0.226284 + 0.974061i $$0.572658\pi$$
$$522$$ −7.52404e9 −2.31528
$$523$$ 2.12856e9 0.650624 0.325312 0.945607i $$-0.394531\pi$$
0.325312 + 0.945607i $$0.394531\pi$$
$$524$$ −1.22207e9 −0.371054
$$525$$ −3.92352e8 −0.118336
$$526$$ 3.40698e9 1.02075
$$527$$ −2.63110e8 −0.0783069
$$528$$ 2.77313e9 0.819883
$$529$$ −3.04352e9 −0.893885
$$530$$ −3.77299e9 −1.10083
$$531$$ −8.76066e9 −2.53925
$$532$$ 4.30346e8 0.123916
$$533$$ 0 0
$$534$$ −2.35967e9 −0.670590
$$535$$ 4.88007e9 1.37781
$$536$$ −3.28018e7 −0.00920070
$$537$$ 4.83862e9 1.34838
$$538$$ 4.11323e9 1.13879
$$539$$ 6.15387e9 1.69273
$$540$$ 5.71051e9 1.56062
$$541$$ 1.72479e7 0.00468324 0.00234162 0.999997i $$-0.499255\pi$$
0.00234162 + 0.999997i $$0.499255\pi$$
$$542$$ 3.65677e8 0.0986507
$$543$$ 1.03908e10 2.78516
$$544$$ −2.97173e8 −0.0791431
$$545$$ 2.16117e9 0.571874
$$546$$ 0 0
$$547$$ 7.51154e8 0.196234 0.0981168 0.995175i $$-0.468718\pi$$
0.0981168 + 0.995175i $$0.468718\pi$$
$$548$$ −1.90016e9 −0.493241
$$549$$ −1.29147e10 −3.33105
$$550$$ 1.55117e9 0.397549
$$551$$ 6.49196e9 1.65328
$$552$$ 8.46692e8 0.214259
$$553$$ 1.36394e8 0.0342972
$$554$$ 2.19171e9 0.547645
$$555$$ −9.03910e9 −2.24440
$$556$$ 9.96175e8 0.245795
$$557$$ −3.00701e9 −0.737295 −0.368647 0.929569i $$-0.620179\pi$$
−0.368647 + 0.929569i $$0.620179\pi$$
$$558$$ 1.24914e9 0.304363
$$559$$ 0 0
$$560$$ −2.37982e8 −0.0572645
$$561$$ 6.14002e9 1.46825
$$562$$ 3.37366e9 0.801722
$$563$$ 2.82880e9 0.668070 0.334035 0.942561i $$-0.391590\pi$$
0.334035 + 0.942561i $$0.391590\pi$$
$$564$$ −2.49135e9 −0.584734
$$565$$ 3.70088e9 0.863248
$$566$$ −3.05566e9 −0.708349
$$567$$ 2.24667e9 0.517604
$$568$$ −1.65060e8 −0.0377940
$$569$$ 7.67290e9 1.74609 0.873045 0.487639i $$-0.162142\pi$$
0.873045 + 0.487639i $$0.162142\pi$$
$$570$$ −8.29990e9 −1.87720
$$571$$ −3.09363e9 −0.695411 −0.347706 0.937604i $$-0.613039\pi$$
−0.347706 + 0.937604i $$0.613039\pi$$
$$572$$ 0 0
$$573$$ −9.17717e9 −2.03783
$$574$$ 1.15161e9 0.254164
$$575$$ 4.73603e8 0.103891
$$576$$ 1.41086e9 0.307613
$$577$$ −3.71815e9 −0.805770 −0.402885 0.915251i $$-0.631993\pi$$
−0.402885 + 0.915251i $$0.631993\pi$$
$$578$$ 2.62474e9 0.565377
$$579$$ 1.84491e9 0.395003
$$580$$ −3.59006e9 −0.764019
$$581$$ 2.20656e8 0.0466765
$$582$$ −1.13363e9 −0.238363
$$583$$ 1.14336e10 2.38969
$$584$$ −2.28085e9 −0.473862
$$585$$ 0 0
$$586$$ −3.23866e9 −0.664850
$$587$$ 2.74853e9 0.560876 0.280438 0.959872i $$-0.409520\pi$$
0.280438 + 0.959872i $$0.409520\pi$$
$$588$$ 4.40307e9 0.893173
$$589$$ −1.07780e9 −0.217337
$$590$$ −4.18011e9 −0.837927
$$591$$ −1.44886e10 −2.88715
$$592$$ −1.32575e9 −0.262624
$$593$$ 9.11262e9 1.79453 0.897267 0.441488i $$-0.145549\pi$$
0.897267 + 0.441488i $$0.145549\pi$$
$$594$$ −1.73050e10 −3.38781
$$595$$ −5.26918e8 −0.102550
$$596$$ 1.59792e8 0.0309167
$$597$$ 1.09925e10 2.11440
$$598$$ 0 0
$$599$$ −5.52493e9 −1.05035 −0.525174 0.850995i $$-0.676000\pi$$
−0.525174 + 0.850995i $$0.676000\pi$$
$$600$$ 1.10986e9 0.209767
$$601$$ −1.78219e9 −0.334883 −0.167441 0.985882i $$-0.553551\pi$$
−0.167441 + 0.985882i $$0.553551\pi$$
$$602$$ 4.54870e8 0.0849767
$$603$$ 3.44803e8 0.0640414
$$604$$ −1.53473e9 −0.283402
$$605$$ −1.31842e10 −2.42053
$$606$$ −1.06838e10 −1.95016
$$607$$ 9.53705e9 1.73083 0.865414 0.501058i $$-0.167056\pi$$
0.865414 + 0.501058i $$0.167056\pi$$
$$608$$ −1.21733e9 −0.219658
$$609$$ −2.75179e9 −0.493690
$$610$$ −6.16219e9 −1.09921
$$611$$ 0 0
$$612$$ 3.12380e9 0.550875
$$613$$ 1.18627e9 0.208004 0.104002 0.994577i $$-0.466835\pi$$
0.104002 + 0.994577i $$0.466835\pi$$
$$614$$ −3.80416e8 −0.0663238
$$615$$ −2.22107e10 −3.85034
$$616$$ 7.21174e8 0.124310
$$617$$ −1.32256e9 −0.226682 −0.113341 0.993556i $$-0.536155\pi$$
−0.113341 + 0.993556i $$0.536155\pi$$
$$618$$ 4.78599e9 0.815666
$$619$$ 3.59450e9 0.609147 0.304573 0.952489i $$-0.401486\pi$$
0.304573 + 0.952489i $$0.401486\pi$$
$$620$$ 5.96023e8 0.100437
$$621$$ −5.28356e9 −0.885332
$$622$$ 2.42273e9 0.403681
$$623$$ −6.13650e8 −0.101675
$$624$$ 0 0
$$625$$ −7.42927e9 −1.21721
$$626$$ 5.05348e9 0.823343
$$627$$ 2.51518e10 4.07505
$$628$$ 1.09152e9 0.175863
$$629$$ −2.93535e9 −0.470309
$$630$$ 2.50160e9 0.398589
$$631$$ −7.49102e6 −0.00118697 −0.000593483 1.00000i $$-0.500189\pi$$
−0.000593483 1.00000i $$0.500189\pi$$
$$632$$ −3.85823e8 −0.0607964
$$633$$ −9.44567e9 −1.48020
$$634$$ 6.34666e9 0.989083
$$635$$ −6.63505e9 −1.02834
$$636$$ 8.18068e9 1.26093
$$637$$ 0 0
$$638$$ 1.08792e10 1.65854
$$639$$ 1.73507e9 0.263065
$$640$$ 6.73186e8 0.101509
$$641$$ 4.06396e9 0.609462 0.304731 0.952438i $$-0.401433\pi$$
0.304731 + 0.952438i $$0.401433\pi$$
$$642$$ −1.05811e10 −1.57818
$$643$$ −1.56544e6 −0.000232219 0 −0.000116109 1.00000i $$-0.500037\pi$$
−0.000116109 1.00000i $$0.500037\pi$$
$$644$$ 2.20189e8 0.0324859
$$645$$ −8.77290e9 −1.28731
$$646$$ −2.69531e9 −0.393364
$$647$$ 1.31025e10 1.90191 0.950956 0.309325i $$-0.100103\pi$$
0.950956 + 0.309325i $$0.100103\pi$$
$$648$$ −6.35521e9 −0.917524
$$649$$ 1.26673e10 1.81898
$$650$$ 0 0
$$651$$ 4.56852e8 0.0648996
$$652$$ 4.70135e9 0.664287
$$653$$ 7.63326e9 1.07279 0.536394 0.843968i $$-0.319786\pi$$
0.536394 + 0.843968i $$0.319786\pi$$
$$654$$ −4.68589e9 −0.655044
$$655$$ 6.12945e9 0.852269
$$656$$ −3.25760e9 −0.450541
$$657$$ 2.39756e10 3.29831
$$658$$ −6.47895e8 −0.0886571
$$659$$ −9.25900e9 −1.26027 −0.630137 0.776484i $$-0.717001\pi$$
−0.630137 + 0.776484i $$0.717001\pi$$
$$660$$ −1.39090e10 −1.88318
$$661$$ −4.79962e9 −0.646401 −0.323201 0.946330i $$-0.604759\pi$$
−0.323201 + 0.946330i $$0.604759\pi$$
$$662$$ −9.73021e9 −1.30352
$$663$$ 0 0
$$664$$ −6.24175e8 −0.0827405
$$665$$ −2.15845e9 −0.284621
$$666$$ 1.39359e10 1.82799
$$667$$ 3.32165e9 0.433424
$$668$$ −2.98791e9 −0.387837
$$669$$ −1.09274e10 −1.41100
$$670$$ 1.64521e8 0.0211330
$$671$$ 1.86737e10 2.38618
$$672$$ 5.15998e8 0.0655927
$$673$$ −1.08997e10 −1.37836 −0.689182 0.724589i $$-0.742030\pi$$
−0.689182 + 0.724589i $$0.742030\pi$$
$$674$$ 1.20977e9 0.152192
$$675$$ −6.92578e9 −0.866773
$$676$$ 0 0
$$677$$ 3.44099e9 0.426210 0.213105 0.977029i $$-0.431642\pi$$
0.213105 + 0.977029i $$0.431642\pi$$
$$678$$ −8.02434e9 −0.988793
$$679$$ −2.94808e8 −0.0361406
$$680$$ 1.49051e9 0.181783
$$681$$ 1.65974e10 2.01384
$$682$$ −1.80617e9 −0.218029
$$683$$ −5.53553e9 −0.664794 −0.332397 0.943140i $$-0.607857\pi$$
−0.332397 + 0.943140i $$0.607857\pi$$
$$684$$ 1.27962e10 1.52892
$$685$$ 9.53051e9 1.13292
$$686$$ 2.33754e9 0.276455
$$687$$ −4.60153e9 −0.541444
$$688$$ −1.28671e9 −0.150633
$$689$$ 0 0
$$690$$ −4.24669e9 −0.492129
$$691$$ −4.21595e8 −0.0486097 −0.0243048 0.999705i $$-0.507737\pi$$
−0.0243048 + 0.999705i $$0.507737\pi$$
$$692$$ −4.99796e9 −0.573352
$$693$$ −7.58077e9 −0.865261
$$694$$ 4.77788e9 0.542596
$$695$$ −4.99644e9 −0.564565
$$696$$ 7.78406e9 0.875133
$$697$$ −7.21268e9 −0.806830
$$698$$ 9.56799e8 0.106494
$$699$$ −1.31591e10 −1.45732
$$700$$ 2.88627e8 0.0318049
$$701$$ −5.14995e9 −0.564663 −0.282332 0.959317i $$-0.591108\pi$$
−0.282332 + 0.959317i $$0.591108\pi$$
$$702$$ 0 0
$$703$$ −1.20243e10 −1.30532
$$704$$ −2.04000e9 −0.220357
$$705$$ 1.24957e10 1.34307
$$706$$ −3.73132e9 −0.399067
$$707$$ −2.77840e9 −0.295683
$$708$$ 9.06342e9 0.959789
$$709$$ −1.05683e10 −1.11363 −0.556817 0.830635i $$-0.687978\pi$$
−0.556817 + 0.830635i $$0.687978\pi$$
$$710$$ 8.27880e8 0.0868086
$$711$$ 4.05566e9 0.423173
$$712$$ 1.73585e9 0.180232
$$713$$ −5.51460e8 −0.0569772
$$714$$ 1.14248e9 0.117464
$$715$$ 0 0
$$716$$ −3.55944e9 −0.362399
$$717$$ 2.27868e10 2.30869
$$718$$ −6.16081e9 −0.621159
$$719$$ 1.53690e10 1.54204 0.771020 0.636811i $$-0.219747\pi$$
0.771020 + 0.636811i $$0.219747\pi$$
$$720$$ −7.07634e9 −0.706553
$$721$$ 1.24463e9 0.123671
$$722$$ −3.89001e9 −0.384654
$$723$$ −1.14624e10 −1.12795
$$724$$ −7.64381e9 −0.748557
$$725$$ 4.35407e9 0.424339
$$726$$ 2.85864e10 2.77256
$$727$$ 4.88599e9 0.471609 0.235804 0.971801i $$-0.424228\pi$$
0.235804 + 0.971801i $$0.424228\pi$$
$$728$$ 0 0
$$729$$ 1.39161e10 1.33036
$$730$$ 1.14399e10 1.08841
$$731$$ −2.84891e9 −0.269754
$$732$$ 1.33610e10 1.25907
$$733$$ −3.59889e9 −0.337524 −0.168762 0.985657i $$-0.553977\pi$$
−0.168762 + 0.985657i $$0.553977\pi$$
$$734$$ 6.84255e9 0.638678
$$735$$ −2.20842e10 −2.05152
$$736$$ −6.22854e8 −0.0575856
$$737$$ −4.98562e8 −0.0458757
$$738$$ 3.42430e10 3.13598
$$739$$ −2.78886e9 −0.254198 −0.127099 0.991890i $$-0.540567\pi$$
−0.127099 + 0.991890i $$0.540567\pi$$
$$740$$ 6.64946e9 0.603219
$$741$$ 0 0
$$742$$ 2.12745e9 0.191181
$$743$$ 3.08130e9 0.275597 0.137798 0.990460i $$-0.455997\pi$$
0.137798 + 0.990460i $$0.455997\pi$$
$$744$$ −1.29231e9 −0.115043
$$745$$ −8.01457e8 −0.0710122
$$746$$ 4.23687e9 0.373645
$$747$$ 6.56115e9 0.575915
$$748$$ −4.51680e9 −0.394616
$$749$$ −2.75169e9 −0.239284
$$750$$ 1.18877e10 1.02893
$$751$$ −6.41281e8 −0.0552470 −0.0276235 0.999618i $$-0.508794\pi$$
−0.0276235 + 0.999618i $$0.508794\pi$$
$$752$$ 1.83272e9 0.157157
$$753$$ −2.14943e10 −1.83460
$$754$$ 0 0
$$755$$ 7.69763e9 0.650943
$$756$$ −3.21995e9 −0.271033
$$757$$ −1.60219e10 −1.34239 −0.671195 0.741280i $$-0.734219\pi$$
−0.671195 + 0.741280i $$0.734219\pi$$
$$758$$ 1.58686e10 1.32342
$$759$$ 1.28691e10 1.06832
$$760$$ 6.10568e9 0.504529
$$761$$ 5.73623e9 0.471824 0.235912 0.971774i $$-0.424192\pi$$
0.235912 + 0.971774i $$0.424192\pi$$
$$762$$ 1.43863e10 1.17789
$$763$$ −1.21860e9 −0.0993175
$$764$$ 6.75102e9 0.547700
$$765$$ −1.56678e10 −1.26530
$$766$$ 7.19005e9 0.578005
$$767$$ 0 0
$$768$$ −1.45962e9 −0.116272
$$769$$ −2.45874e10 −1.94971 −0.974857 0.222832i $$-0.928470\pi$$
−0.974857 + 0.222832i $$0.928470\pi$$
$$770$$ −3.61714e9 −0.285527
$$771$$ −1.97739e10 −1.55382
$$772$$ −1.35718e9 −0.106164
$$773$$ 1.31517e10 1.02413 0.512065 0.858947i $$-0.328881\pi$$
0.512065 + 0.858947i $$0.328881\pi$$
$$774$$ 1.35255e10 1.04848
$$775$$ −7.22863e8 −0.0557828
$$776$$ 8.33932e8 0.0640641
$$777$$ 5.09682e9 0.389785
$$778$$ −1.45980e10 −1.11138
$$779$$ −2.95458e10 −2.23932
$$780$$ 0 0
$$781$$ −2.50878e9 −0.188445
$$782$$ −1.37907e9 −0.103125
$$783$$ −4.85744e10 −3.61611
$$784$$ −3.23904e9 −0.240055
$$785$$ −5.47466e9 −0.403937
$$786$$ −1.32900e10 −0.976218
$$787$$ 7.38863e9 0.540322 0.270161 0.962815i $$-0.412923\pi$$
0.270161 + 0.962815i $$0.412923\pi$$
$$788$$ 1.06583e10 0.775969
$$789$$ 3.70509e10 2.68552
$$790$$ 1.93514e9 0.139643
$$791$$ −2.08679e9 −0.149920
$$792$$ 2.14440e10 1.53379
$$793$$ 0 0
$$794$$ 3.94467e9 0.279665
$$795$$ −4.10312e10 −2.89621
$$796$$ −8.08645e9 −0.568279
$$797$$ −5.22399e9 −0.365509 −0.182754 0.983159i $$-0.558501\pi$$
−0.182754 + 0.983159i $$0.558501\pi$$
$$798$$ 4.68001e9 0.326014
$$799$$ 4.05784e9 0.281437
$$800$$ −8.16447e8 −0.0563785
$$801$$ −1.82468e10 −1.25450
$$802$$ −4.54624e9 −0.311202
$$803$$ −3.46671e10 −2.36273
$$804$$ −3.56719e8 −0.0242064
$$805$$ −1.10438e9 −0.0746164
$$806$$ 0 0
$$807$$ 4.47314e10 2.99609
$$808$$ 7.85934e9 0.524139
$$809$$ −7.92102e9 −0.525970 −0.262985 0.964800i $$-0.584707\pi$$
−0.262985 + 0.964800i $$0.584707\pi$$
$$810$$ 3.18754e10 2.10745
$$811$$ 8.16607e9 0.537576 0.268788 0.963199i $$-0.413377\pi$$
0.268788 + 0.963199i $$0.413377\pi$$
$$812$$ 2.02430e9 0.132687
$$813$$ 3.97674e9 0.259543
$$814$$ −2.01503e10 −1.30947
$$815$$ −2.35802e10 −1.52579
$$816$$ −3.23176e9 −0.208220
$$817$$ −1.16702e10 −0.748688
$$818$$ −1.02778e10 −0.656541
$$819$$ 0 0
$$820$$ 1.63389e10 1.03484
$$821$$ −2.63749e10 −1.66338 −0.831688 0.555244i $$-0.812625\pi$$
−0.831688 + 0.555244i $$0.812625\pi$$
$$822$$ −2.06643e10 −1.29768
$$823$$ 2.04085e10 1.27618 0.638090 0.769962i $$-0.279725\pi$$
0.638090 + 0.769962i $$0.279725\pi$$
$$824$$ −3.52073e9 −0.219224
$$825$$ 1.68690e10 1.04592
$$826$$ 2.35701e9 0.145523
$$827$$ 2.55307e10 1.56962 0.784809 0.619738i $$-0.212761\pi$$
0.784809 + 0.619738i $$0.212761\pi$$
$$828$$ 6.54727e9 0.400824
$$829$$ 8.48208e9 0.517085 0.258542 0.966000i $$-0.416758\pi$$
0.258542 + 0.966000i $$0.416758\pi$$
$$830$$ 3.13063e9 0.190046
$$831$$ 2.38349e10 1.44082
$$832$$ 0 0
$$833$$ −7.17160e9 −0.429891
$$834$$ 1.08334e10 0.646672
$$835$$ 1.49862e10 0.890819
$$836$$ −1.85025e10 −1.09524
$$837$$ 8.06432e9 0.475367
$$838$$ −2.19878e9 −0.129071
$$839$$ 2.29323e10 1.34055 0.670273 0.742115i $$-0.266177\pi$$
0.670273 + 0.742115i $$0.266177\pi$$
$$840$$ −2.58805e9 −0.150659
$$841$$ 1.32877e10 0.770306
$$842$$ 6.01094e9 0.347017
$$843$$ 3.66885e10 2.10928
$$844$$ 6.94854e9 0.397828
$$845$$ 0 0
$$846$$ −1.92650e10 −1.09389
$$847$$ 7.43410e9 0.420374
$$848$$ −6.01797e9 −0.338895
$$849$$ −3.32303e10 −1.86362
$$850$$ −1.80771e9 −0.100963
$$851$$ −6.15230e9 −0.342203
$$852$$ −1.79503e9 −0.0994335
$$853$$ −2.47175e10 −1.36358 −0.681792 0.731546i $$-0.738799\pi$$
−0.681792 + 0.731546i $$0.738799\pi$$
$$854$$ 3.47463e9 0.190900
$$855$$ −6.41812e10 −3.51177
$$856$$ 7.78379e9 0.424163
$$857$$ 1.19081e10 0.646265 0.323133 0.946354i $$-0.395264\pi$$
0.323133 + 0.946354i $$0.395264\pi$$
$$858$$ 0 0
$$859$$ −4.94214e9 −0.266035 −0.133018 0.991114i $$-0.542467\pi$$
−0.133018 + 0.991114i $$0.542467\pi$$
$$860$$ 6.45363e9 0.345987
$$861$$ 1.25238e10 0.668689
$$862$$ −1.04605e9 −0.0556259
$$863$$ 2.05387e10 1.08776 0.543881 0.839162i $$-0.316954\pi$$
0.543881 + 0.839162i $$0.316954\pi$$
$$864$$ 9.10836e9 0.480443
$$865$$ 2.50679e10 1.31693
$$866$$ −1.33389e10 −0.697921
$$867$$ 2.85440e10 1.48747
$$868$$ −3.36075e8 −0.0174428
$$869$$ −5.86420e9 −0.303138
$$870$$ −3.90419e10 −2.01008
$$871$$ 0 0
$$872$$ 3.44709e9 0.176054
$$873$$ −8.76606e9 −0.445918
$$874$$ −5.64918e9 −0.286217
$$875$$ 3.09150e9 0.156006
$$876$$ −2.48042e10 −1.24670
$$877$$ 1.42584e10 0.713791 0.356895 0.934144i $$-0.383835\pi$$
0.356895 + 0.934144i $$0.383835\pi$$
$$878$$ −1.85182e10 −0.923354
$$879$$ −3.52204e10 −1.74918
$$880$$ 1.02319e10 0.506136
$$881$$ 1.78398e10 0.878971 0.439486 0.898250i $$-0.355161\pi$$
0.439486 + 0.898250i $$0.355161\pi$$
$$882$$ 3.40479e10 1.67090
$$883$$ 3.79954e10 1.85724 0.928622 0.371028i $$-0.120995\pi$$
0.928622 + 0.371028i $$0.120995\pi$$
$$884$$ 0 0
$$885$$ −4.54587e10 −2.20453
$$886$$ 5.52038e9 0.266656
$$887$$ 8.45195e7 0.00406653 0.00203327 0.999998i $$-0.499353\pi$$
0.00203327 + 0.999998i $$0.499353\pi$$
$$888$$ −1.44175e10 −0.690947
$$889$$ 3.74126e9 0.178592
$$890$$ −8.70637e9 −0.413973
$$891$$ −9.65942e10 −4.57488
$$892$$ 8.03857e9 0.379229
$$893$$ 1.66224e10 0.781114
$$894$$ 1.73774e9 0.0813398
$$895$$ 1.78528e10 0.832390
$$896$$ −3.79585e8 −0.0176291
$$897$$ 0 0
$$898$$ 2.11045e10 0.972541
$$899$$ −5.06985e9 −0.232721
$$900$$ 8.58227e9 0.392422
$$901$$ −1.33245e10 −0.606894
$$902$$ −4.95129e10 −2.24645
$$903$$ 4.94672e9 0.223568
$$904$$ 5.90296e9 0.265754
$$905$$ 3.83385e10 1.71935
$$906$$ −1.66902e10 −0.745611
$$907$$ 1.82024e10 0.810033 0.405017 0.914309i $$-0.367266\pi$$
0.405017 + 0.914309i $$0.367266\pi$$
$$908$$ −1.22096e10 −0.541252
$$909$$ −8.26151e10 −3.64826
$$910$$ 0 0
$$911$$ −3.66963e10 −1.60808 −0.804040 0.594575i $$-0.797320\pi$$
−0.804040 + 0.594575i $$0.797320\pi$$
$$912$$ −1.32385e10 −0.577905
$$913$$ −9.48697e9 −0.412553
$$914$$ −4.92978e9 −0.213558
$$915$$ −6.70139e10 −2.89195
$$916$$ 3.38503e9 0.145522
$$917$$ −3.45617e9 −0.148014
$$918$$ 2.01669e10 0.860380
$$919$$ 1.33474e10 0.567275 0.283638 0.958932i $$-0.408459\pi$$
0.283638 + 0.958932i $$0.408459\pi$$
$$920$$ 3.12400e9 0.132268
$$921$$ −4.13702e9 −0.174493
$$922$$ 9.88970e9 0.415552
$$923$$ 0 0
$$924$$ 7.84276e9 0.327052
$$925$$ −8.06454e9 −0.335030
$$926$$ 5.42775e8 0.0224637
$$927$$ 3.70089e10 1.52591
$$928$$ −5.72621e9 −0.235206
$$929$$ −2.71771e10 −1.11211 −0.556055 0.831146i $$-0.687686\pi$$
−0.556055 + 0.831146i $$0.687686\pi$$
$$930$$ 6.48174e9 0.264242
$$931$$ −2.93776e10 −1.19314
$$932$$ 9.68025e9 0.391680
$$933$$ 2.63472e10 1.06206
$$934$$ 9.40012e9 0.377502
$$935$$ 2.26546e10 0.906390
$$936$$ 0 0
$$937$$ 4.04333e10 1.60565 0.802825 0.596214i $$-0.203329\pi$$
0.802825 + 0.596214i $$0.203329\pi$$
$$938$$ −9.27676e7 −0.00367017
$$939$$ 5.49566e10 2.16616
$$940$$ −9.19223e9 −0.360972
$$941$$ 8.49843e9 0.332487 0.166244 0.986085i $$-0.446836\pi$$
0.166244 + 0.986085i $$0.446836\pi$$
$$942$$ 1.18703e10 0.462683
$$943$$ −1.51173e10 −0.587061
$$944$$ −6.66735e9 −0.257959
$$945$$ 1.61500e10 0.622533
$$946$$ −1.95569e10 −0.751072
$$947$$ 4.40082e9 0.168387 0.0841935 0.996449i $$-0.473169\pi$$
0.0841935 + 0.996449i $$0.473169\pi$$
$$948$$ −4.19582e9 −0.159951
$$949$$ 0 0
$$950$$ −7.40504e9 −0.280217
$$951$$ 6.90199e10 2.60221
$$952$$ −8.40442e8 −0.0315703
$$953$$ −1.73133e10 −0.647970 −0.323985 0.946062i $$-0.605023\pi$$
−0.323985 + 0.946062i $$0.605023\pi$$
$$954$$ 6.32593e10 2.35887
$$955$$ −3.38606e10 −1.25801
$$956$$ −1.67627e10 −0.620498
$$957$$ 1.18312e11 4.36351
$$958$$ 3.16924e9 0.116460
$$959$$ −5.37390e9 −0.196754
$$960$$ 7.32090e9 0.267064
$$961$$ −2.66709e10 −0.969407
$$962$$ 0 0
$$963$$ −8.18210e10 −2.95238
$$964$$ 8.43211e9 0.303156
$$965$$ 6.80708e9 0.243846
$$966$$ 2.39455e9 0.0854681
$$967$$ −1.40918e10 −0.501158 −0.250579 0.968096i $$-0.580621\pi$$
−0.250579 + 0.968096i $$0.580621\pi$$
$$968$$ −2.10290e10 −0.745171
$$969$$ −2.93115e10 −1.03491
$$970$$ −4.18269e9 −0.147148
$$971$$ −7.27843e9 −0.255135 −0.127568 0.991830i $$-0.540717\pi$$
−0.127568 + 0.991830i $$0.540717\pi$$
$$972$$ −3.02067e10 −1.05505
$$973$$ 2.81731e9 0.0980481
$$974$$ −2.42932e10 −0.842419
$$975$$ 0 0
$$976$$ −9.82879e9 −0.338397
$$977$$ −2.43791e10 −0.836348 −0.418174 0.908367i $$-0.637330\pi$$
−0.418174 + 0.908367i $$0.637330\pi$$
$$978$$ 5.11272e10 1.74770
$$979$$ 2.63835e10 0.898657
$$980$$ 1.62458e10 0.551379
$$981$$ −3.62349e10 −1.22542
$$982$$ 2.33579e10 0.787126
$$983$$ −4.06556e10 −1.36516 −0.682579 0.730811i $$-0.739142\pi$$
−0.682579 + 0.730811i $$0.739142\pi$$
$$984$$ −3.54264e10 −1.18534
$$985$$ −5.34578e10 −1.78231
$$986$$ −1.26785e10 −0.421209
$$987$$ −7.04585e9 −0.233251
$$988$$ 0 0
$$989$$ −5.97112e9 −0.196277
$$990$$ −1.07555e11 −3.52295
$$991$$ −4.86636e10 −1.58835 −0.794175 0.607689i $$-0.792097\pi$$
−0.794175 + 0.607689i $$0.792097\pi$$
$$992$$ 9.50665e8 0.0309198
$$993$$ −1.05816e11 −3.42949
$$994$$ −4.66811e8 −0.0150761
$$995$$ 4.05586e10 1.30527
$$996$$ −6.78790e9 −0.217685
$$997$$ −1.76682e10 −0.564622 −0.282311 0.959323i $$-0.591101\pi$$
−0.282311 + 0.959323i $$0.591101\pi$$
$$998$$ −1.29874e10 −0.413586
$$999$$ 8.99687e10 2.85504
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 338.8.a.a.1.1 1
13.5 odd 4 338.8.b.a.337.2 2
13.8 odd 4 338.8.b.a.337.1 2
13.12 even 2 26.8.a.b.1.1 1
39.38 odd 2 234.8.a.a.1.1 1
52.51 odd 2 208.8.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.b.1.1 1 13.12 even 2
208.8.a.e.1.1 1 52.51 odd 2
234.8.a.a.1.1 1 39.38 odd 2
338.8.a.a.1.1 1 1.1 even 1 trivial
338.8.b.a.337.1 2 13.8 odd 4
338.8.b.a.337.2 2 13.5 odd 4