# Properties

 Label 75.22.a.b.1.1 Level $75$ Weight $22$ Character 75.1 Self dual yes Analytic conductor $209.608$ Analytic rank $1$ 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] = [75,22,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$209.608008215$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 75.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-544.000 q^{2} -59049.0 q^{3} -1.80122e6 q^{4} +3.21227e7 q^{6} -1.27770e9 q^{7} +2.12071e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$$ $$q-544.000 q^{2} -59049.0 q^{3} -1.80122e6 q^{4} +3.21227e7 q^{6} -1.27770e9 q^{7} +2.12071e9 q^{8} +3.48678e9 q^{9} -7.75859e10 q^{11} +1.06360e11 q^{12} +4.34111e11 q^{13} +6.95068e11 q^{14} +2.62376e12 q^{16} -1.28489e13 q^{17} -1.89681e12 q^{18} -2.86053e13 q^{19} +7.54468e13 q^{21} +4.22067e13 q^{22} -2.24022e14 q^{23} -1.25226e14 q^{24} -2.36156e14 q^{26} -2.05891e14 q^{27} +2.30141e15 q^{28} -5.16760e13 q^{29} +8.92111e15 q^{31} -5.87478e15 q^{32} +4.58137e15 q^{33} +6.98981e15 q^{34} -6.28045e15 q^{36} -4.39772e16 q^{37} +1.55613e16 q^{38} -2.56338e16 q^{39} +5.81681e16 q^{41} -4.10431e16 q^{42} +1.61438e17 q^{43} +1.39749e17 q^{44} +1.21868e17 q^{46} +1.60065e17 q^{47} -1.54930e17 q^{48} +1.07397e18 q^{49} +7.58716e17 q^{51} -7.81927e17 q^{52} -2.29953e18 q^{53} +1.12005e17 q^{54} -2.70963e18 q^{56} +1.68911e18 q^{57} +2.81118e16 q^{58} +5.15426e18 q^{59} +1.25169e18 q^{61} -4.85308e18 q^{62} -4.45506e18 q^{63} -2.30654e18 q^{64} -2.49227e18 q^{66} +5.40779e18 q^{67} +2.31437e19 q^{68} +1.32283e19 q^{69} -1.10432e19 q^{71} +7.39447e18 q^{72} +3.77012e19 q^{73} +2.39236e19 q^{74} +5.15242e19 q^{76} +9.91314e19 q^{77} +1.39448e19 q^{78} +6.31554e19 q^{79} +1.21577e19 q^{81} -3.16434e19 q^{82} +1.45158e20 q^{83} -1.35896e20 q^{84} -8.78222e19 q^{86} +3.05142e18 q^{87} -1.64537e20 q^{88} +1.37255e20 q^{89} -5.54663e20 q^{91} +4.03512e20 q^{92} -5.26783e20 q^{93} -8.70752e19 q^{94} +3.46900e20 q^{96} +3.24306e20 q^{97} -5.84238e20 q^{98} -2.70525e20 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −544.000 −0.375650 −0.187825 0.982202i $$-0.560144\pi$$
−0.187825 + 0.982202i $$0.560144\pi$$
$$3$$ −59049.0 −0.577350
$$4$$ −1.80122e6 −0.858887
$$5$$ 0 0
$$6$$ 3.21227e7 0.216882
$$7$$ −1.27770e9 −1.70962 −0.854809 0.518943i $$-0.826326\pi$$
−0.854809 + 0.518943i $$0.826326\pi$$
$$8$$ 2.12071e9 0.698292
$$9$$ 3.48678e9 0.333333
$$10$$ 0 0
$$11$$ −7.75859e10 −0.901903 −0.450951 0.892549i $$-0.648915\pi$$
−0.450951 + 0.892549i $$0.648915\pi$$
$$12$$ 1.06360e11 0.495878
$$13$$ 4.34111e11 0.873364 0.436682 0.899616i $$-0.356153\pi$$
0.436682 + 0.899616i $$0.356153\pi$$
$$14$$ 6.95068e11 0.642219
$$15$$ 0 0
$$16$$ 2.62376e12 0.596573
$$17$$ −1.28489e13 −1.54580 −0.772899 0.634529i $$-0.781194\pi$$
−0.772899 + 0.634529i $$0.781194\pi$$
$$18$$ −1.89681e12 −0.125217
$$19$$ −2.86053e13 −1.07037 −0.535184 0.844736i $$-0.679758\pi$$
−0.535184 + 0.844736i $$0.679758\pi$$
$$20$$ 0 0
$$21$$ 7.54468e13 0.987048
$$22$$ 4.22067e13 0.338800
$$23$$ −2.24022e14 −1.12758 −0.563792 0.825917i $$-0.690658\pi$$
−0.563792 + 0.825917i $$0.690658\pi$$
$$24$$ −1.25226e14 −0.403159
$$25$$ 0 0
$$26$$ −2.36156e14 −0.328080
$$27$$ −2.05891e14 −0.192450
$$28$$ 2.30141e15 1.46837
$$29$$ −5.16760e13 −0.0228092 −0.0114046 0.999935i $$-0.503630\pi$$
−0.0114046 + 0.999935i $$0.503630\pi$$
$$30$$ 0 0
$$31$$ 8.92111e15 1.95488 0.977442 0.211203i $$-0.0677382\pi$$
0.977442 + 0.211203i $$0.0677382\pi$$
$$32$$ −5.87478e15 −0.922395
$$33$$ 4.58137e15 0.520714
$$34$$ 6.98981e15 0.580680
$$35$$ 0 0
$$36$$ −6.28045e15 −0.286296
$$37$$ −4.39772e16 −1.50352 −0.751760 0.659437i $$-0.770795\pi$$
−0.751760 + 0.659437i $$0.770795\pi$$
$$38$$ 1.55613e16 0.402084
$$39$$ −2.56338e16 −0.504237
$$40$$ 0 0
$$41$$ 5.81681e16 0.676791 0.338395 0.941004i $$-0.390116\pi$$
0.338395 + 0.941004i $$0.390116\pi$$
$$42$$ −4.10431e16 −0.370785
$$43$$ 1.61438e17 1.13916 0.569582 0.821934i $$-0.307105\pi$$
0.569582 + 0.821934i $$0.307105\pi$$
$$44$$ 1.39749e17 0.774632
$$45$$ 0 0
$$46$$ 1.21868e17 0.423577
$$47$$ 1.60065e17 0.443883 0.221941 0.975060i $$-0.428761\pi$$
0.221941 + 0.975060i $$0.428761\pi$$
$$48$$ −1.54930e17 −0.344432
$$49$$ 1.07397e18 1.92279
$$50$$ 0 0
$$51$$ 7.58716e17 0.892467
$$52$$ −7.81927e17 −0.750121
$$53$$ −2.29953e18 −1.80610 −0.903050 0.429535i $$-0.858678\pi$$
−0.903050 + 0.429535i $$0.858678\pi$$
$$54$$ 1.12005e17 0.0722940
$$55$$ 0 0
$$56$$ −2.70963e18 −1.19381
$$57$$ 1.68911e18 0.617977
$$58$$ 2.81118e16 0.00856829
$$59$$ 5.15426e18 1.31286 0.656432 0.754385i $$-0.272065\pi$$
0.656432 + 0.754385i $$0.272065\pi$$
$$60$$ 0 0
$$61$$ 1.25169e18 0.224663 0.112332 0.993671i $$-0.464168\pi$$
0.112332 + 0.993671i $$0.464168\pi$$
$$62$$ −4.85308e18 −0.734353
$$63$$ −4.45506e18 −0.569872
$$64$$ −2.30654e18 −0.250075
$$65$$ 0 0
$$66$$ −2.49227e18 −0.195606
$$67$$ 5.40779e18 0.362438 0.181219 0.983443i $$-0.441996\pi$$
0.181219 + 0.983443i $$0.441996\pi$$
$$68$$ 2.31437e19 1.32767
$$69$$ 1.32283e19 0.651011
$$70$$ 0 0
$$71$$ −1.10432e19 −0.402609 −0.201305 0.979529i $$-0.564518\pi$$
−0.201305 + 0.979529i $$0.564518\pi$$
$$72$$ 7.39447e18 0.232764
$$73$$ 3.77012e19 1.02675 0.513375 0.858164i $$-0.328395\pi$$
0.513375 + 0.858164i $$0.328395\pi$$
$$74$$ 2.39236e19 0.564798
$$75$$ 0 0
$$76$$ 5.15242e19 0.919325
$$77$$ 9.91314e19 1.54191
$$78$$ 1.39448e19 0.189417
$$79$$ 6.31554e19 0.750457 0.375229 0.926932i $$-0.377564\pi$$
0.375229 + 0.926932i $$0.377564\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ −3.16434e19 −0.254237
$$83$$ 1.45158e20 1.02688 0.513442 0.858124i $$-0.328370\pi$$
0.513442 + 0.858124i $$0.328370\pi$$
$$84$$ −1.35896e20 −0.847762
$$85$$ 0 0
$$86$$ −8.78222e19 −0.427928
$$87$$ 3.05142e18 0.0131689
$$88$$ −1.64537e20 −0.629791
$$89$$ 1.37255e20 0.466588 0.233294 0.972406i $$-0.425050\pi$$
0.233294 + 0.972406i $$0.425050\pi$$
$$90$$ 0 0
$$91$$ −5.54663e20 −1.49312
$$92$$ 4.03512e20 0.968467
$$93$$ −5.26783e20 −1.12865
$$94$$ −8.70752e19 −0.166745
$$95$$ 0 0
$$96$$ 3.46900e20 0.532545
$$97$$ 3.24306e20 0.446532 0.223266 0.974758i $$-0.428328\pi$$
0.223266 + 0.974758i $$0.428328\pi$$
$$98$$ −5.84238e20 −0.722298
$$99$$ −2.70525e20 −0.300634
$$100$$ 0 0
$$101$$ 1.14557e21 1.03192 0.515962 0.856611i $$-0.327435\pi$$
0.515962 + 0.856611i $$0.327435\pi$$
$$102$$ −4.12741e20 −0.335256
$$103$$ −8.95787e20 −0.656771 −0.328385 0.944544i $$-0.606504\pi$$
−0.328385 + 0.944544i $$0.606504\pi$$
$$104$$ 9.20624e20 0.609863
$$105$$ 0 0
$$106$$ 1.25094e21 0.678463
$$107$$ −1.25783e21 −0.618146 −0.309073 0.951038i $$-0.600019\pi$$
−0.309073 + 0.951038i $$0.600019\pi$$
$$108$$ 3.70854e20 0.165293
$$109$$ −1.66725e21 −0.674562 −0.337281 0.941404i $$-0.609507\pi$$
−0.337281 + 0.941404i $$0.609507\pi$$
$$110$$ 0 0
$$111$$ 2.59681e21 0.868057
$$112$$ −3.35237e21 −1.01991
$$113$$ 6.51198e21 1.80463 0.902317 0.431073i $$-0.141865\pi$$
0.902317 + 0.431073i $$0.141865\pi$$
$$114$$ −9.18877e20 −0.232143
$$115$$ 0 0
$$116$$ 9.30797e19 0.0195905
$$117$$ 1.51365e21 0.291121
$$118$$ −2.80392e21 −0.493178
$$119$$ 1.64170e22 2.64272
$$120$$ 0 0
$$121$$ −1.38067e21 −0.186571
$$122$$ −6.80917e20 −0.0843949
$$123$$ −3.43477e21 −0.390745
$$124$$ −1.60688e22 −1.67902
$$125$$ 0 0
$$126$$ 2.42355e21 0.214073
$$127$$ 7.99589e21 0.650022 0.325011 0.945710i $$-0.394632\pi$$
0.325011 + 0.945710i $$0.394632\pi$$
$$128$$ 1.35751e22 1.01634
$$129$$ −9.53274e21 −0.657697
$$130$$ 0 0
$$131$$ −5.53406e21 −0.324859 −0.162430 0.986720i $$-0.551933\pi$$
−0.162430 + 0.986720i $$0.551933\pi$$
$$132$$ −8.25204e21 −0.447234
$$133$$ 3.65489e22 1.82992
$$134$$ −2.94184e21 −0.136150
$$135$$ 0 0
$$136$$ −2.72489e22 −1.07942
$$137$$ −5.02831e21 −0.184440 −0.0922201 0.995739i $$-0.529396\pi$$
−0.0922201 + 0.995739i $$0.529396\pi$$
$$138$$ −7.19619e21 −0.244552
$$139$$ 3.06514e21 0.0965594 0.0482797 0.998834i $$-0.484626\pi$$
0.0482797 + 0.998834i $$0.484626\pi$$
$$140$$ 0 0
$$141$$ −9.45166e21 −0.256276
$$142$$ 6.00752e21 0.151240
$$143$$ −3.36809e22 −0.787690
$$144$$ 9.14847e21 0.198858
$$145$$ 0 0
$$146$$ −2.05094e22 −0.385699
$$147$$ −6.34167e22 −1.11012
$$148$$ 7.92124e22 1.29135
$$149$$ 3.18365e22 0.483581 0.241791 0.970328i $$-0.422265\pi$$
0.241791 + 0.970328i $$0.422265\pi$$
$$150$$ 0 0
$$151$$ 6.56248e22 0.866583 0.433292 0.901254i $$-0.357352\pi$$
0.433292 + 0.901254i $$0.357352\pi$$
$$152$$ −6.06635e22 −0.747429
$$153$$ −4.48014e22 −0.515266
$$154$$ −5.39275e22 −0.579219
$$155$$ 0 0
$$156$$ 4.61720e22 0.433083
$$157$$ 6.59284e22 0.578264 0.289132 0.957289i $$-0.406633\pi$$
0.289132 + 0.957289i $$0.406633\pi$$
$$158$$ −3.43565e22 −0.281910
$$159$$ 1.35785e23 1.04275
$$160$$ 0 0
$$161$$ 2.86233e23 1.92774
$$162$$ −6.61377e21 −0.0417389
$$163$$ 8.79055e21 0.0520052 0.0260026 0.999662i $$-0.491722\pi$$
0.0260026 + 0.999662i $$0.491722\pi$$
$$164$$ −1.04773e23 −0.581286
$$165$$ 0 0
$$166$$ −7.89661e22 −0.385750
$$167$$ 1.51028e23 0.692683 0.346341 0.938109i $$-0.387424\pi$$
0.346341 + 0.938109i $$0.387424\pi$$
$$168$$ 1.60001e23 0.689247
$$169$$ −5.86123e22 −0.237235
$$170$$ 0 0
$$171$$ −9.97404e22 −0.356789
$$172$$ −2.90784e23 −0.978414
$$173$$ −6.14868e23 −1.94669 −0.973347 0.229338i $$-0.926344\pi$$
−0.973347 + 0.229338i $$0.926344\pi$$
$$174$$ −1.65997e21 −0.00494690
$$175$$ 0 0
$$176$$ −2.03567e23 −0.538051
$$177$$ −3.04354e23 −0.757983
$$178$$ −7.46667e22 −0.175274
$$179$$ 4.83508e23 1.07016 0.535078 0.844803i $$-0.320282\pi$$
0.535078 + 0.844803i $$0.320282\pi$$
$$180$$ 0 0
$$181$$ −9.28497e23 −1.82876 −0.914378 0.404862i $$-0.867320\pi$$
−0.914378 + 0.404862i $$0.867320\pi$$
$$182$$ 3.01737e23 0.560891
$$183$$ −7.39108e22 −0.129709
$$184$$ −4.75087e23 −0.787382
$$185$$ 0 0
$$186$$ 2.86570e23 0.423979
$$187$$ 9.96895e23 1.39416
$$188$$ −2.88311e23 −0.381245
$$189$$ 2.63067e23 0.329016
$$190$$ 0 0
$$191$$ −5.96634e23 −0.668125 −0.334062 0.942551i $$-0.608420\pi$$
−0.334062 + 0.942551i $$0.608420\pi$$
$$192$$ 1.36199e23 0.144381
$$193$$ 8.56373e23 0.859629 0.429815 0.902917i $$-0.358579\pi$$
0.429815 + 0.902917i $$0.358579\pi$$
$$194$$ −1.76423e23 −0.167740
$$195$$ 0 0
$$196$$ −1.93445e24 −1.65146
$$197$$ −1.10520e24 −0.894430 −0.447215 0.894427i $$-0.647584\pi$$
−0.447215 + 0.894427i $$0.647584\pi$$
$$198$$ 1.47166e23 0.112933
$$199$$ 1.53187e24 1.11497 0.557487 0.830186i $$-0.311766\pi$$
0.557487 + 0.830186i $$0.311766\pi$$
$$200$$ 0 0
$$201$$ −3.19324e23 −0.209254
$$202$$ −6.23191e23 −0.387643
$$203$$ 6.60264e22 0.0389950
$$204$$ −1.36661e24 −0.766528
$$205$$ 0 0
$$206$$ 4.87308e23 0.246716
$$207$$ −7.81117e23 −0.375861
$$208$$ 1.13900e24 0.521026
$$209$$ 2.21937e24 0.965368
$$210$$ 0 0
$$211$$ 2.15918e24 0.849811 0.424906 0.905238i $$-0.360307\pi$$
0.424906 + 0.905238i $$0.360307\pi$$
$$212$$ 4.14195e24 1.55124
$$213$$ 6.52092e23 0.232446
$$214$$ 6.84258e23 0.232207
$$215$$ 0 0
$$216$$ −4.36636e23 −0.134386
$$217$$ −1.13985e25 −3.34210
$$218$$ 9.06983e23 0.253400
$$219$$ −2.22622e24 −0.592795
$$220$$ 0 0
$$221$$ −5.57785e24 −1.35005
$$222$$ −1.41266e24 −0.326086
$$223$$ −7.86309e24 −1.73138 −0.865689 0.500582i $$-0.833119\pi$$
−0.865689 + 0.500582i $$0.833119\pi$$
$$224$$ 7.50620e24 1.57694
$$225$$ 0 0
$$226$$ −3.54252e24 −0.677912
$$227$$ −5.06979e24 −0.926229 −0.463115 0.886298i $$-0.653268\pi$$
−0.463115 + 0.886298i $$0.653268\pi$$
$$228$$ −3.04246e24 −0.530772
$$229$$ −6.10932e24 −1.01793 −0.508967 0.860786i $$-0.669973\pi$$
−0.508967 + 0.860786i $$0.669973\pi$$
$$230$$ 0 0
$$231$$ −5.85361e24 −0.890221
$$232$$ −1.09590e23 −0.0159275
$$233$$ −2.40865e24 −0.334609 −0.167304 0.985905i $$-0.553506\pi$$
−0.167304 + 0.985905i $$0.553506\pi$$
$$234$$ −8.23426e23 −0.109360
$$235$$ 0 0
$$236$$ −9.28393e24 −1.12760
$$237$$ −3.72926e24 −0.433277
$$238$$ −8.93087e24 −0.992740
$$239$$ −8.12956e23 −0.0864748 −0.0432374 0.999065i $$-0.513767\pi$$
−0.0432374 + 0.999065i $$0.513767\pi$$
$$240$$ 0 0
$$241$$ 1.98089e24 0.193055 0.0965277 0.995330i $$-0.469226\pi$$
0.0965277 + 0.995330i $$0.469226\pi$$
$$242$$ 7.51087e23 0.0700856
$$243$$ −7.17898e23 −0.0641500
$$244$$ −2.25456e24 −0.192960
$$245$$ 0 0
$$246$$ 1.86851e24 0.146784
$$247$$ −1.24179e25 −0.934821
$$248$$ 1.89191e25 1.36508
$$249$$ −8.57145e24 −0.592872
$$250$$ 0 0
$$251$$ −1.71966e25 −1.09363 −0.546813 0.837255i $$-0.684159\pi$$
−0.546813 + 0.837255i $$0.684159\pi$$
$$252$$ 8.02452e24 0.489456
$$253$$ 1.73810e25 1.01697
$$254$$ −4.34976e24 −0.244181
$$255$$ 0 0
$$256$$ −2.54768e24 −0.131712
$$257$$ −1.42771e25 −0.708503 −0.354251 0.935150i $$-0.615264\pi$$
−0.354251 + 0.935150i $$0.615264\pi$$
$$258$$ 5.18581e24 0.247064
$$259$$ 5.61895e25 2.57044
$$260$$ 0 0
$$261$$ −1.80183e23 −0.00760307
$$262$$ 3.01053e24 0.122034
$$263$$ −1.23443e25 −0.480765 −0.240382 0.970678i $$-0.577273\pi$$
−0.240382 + 0.970678i $$0.577273\pi$$
$$264$$ 9.71577e24 0.363610
$$265$$ 0 0
$$266$$ −1.98826e25 −0.687410
$$267$$ −8.10477e24 −0.269384
$$268$$ −9.74059e24 −0.311293
$$269$$ −4.23127e24 −0.130039 −0.0650193 0.997884i $$-0.520711\pi$$
−0.0650193 + 0.997884i $$0.520711\pi$$
$$270$$ 0 0
$$271$$ −5.48006e25 −1.55815 −0.779073 0.626933i $$-0.784310\pi$$
−0.779073 + 0.626933i $$0.784310\pi$$
$$272$$ −3.37124e25 −0.922182
$$273$$ 3.27523e25 0.862053
$$274$$ 2.73540e24 0.0692851
$$275$$ 0 0
$$276$$ −2.38270e25 −0.559144
$$277$$ −1.74992e25 −0.395349 −0.197675 0.980268i $$-0.563339\pi$$
−0.197675 + 0.980268i $$0.563339\pi$$
$$278$$ −1.66743e24 −0.0362726
$$279$$ 3.11060e25 0.651628
$$280$$ 0 0
$$281$$ 2.52311e25 0.490365 0.245183 0.969477i $$-0.421152\pi$$
0.245183 + 0.969477i $$0.421152\pi$$
$$282$$ 5.14171e24 0.0962701
$$283$$ 1.19713e25 0.215966 0.107983 0.994153i $$-0.465561\pi$$
0.107983 + 0.994153i $$0.465561\pi$$
$$284$$ 1.98912e25 0.345796
$$285$$ 0 0
$$286$$ 1.83224e25 0.295896
$$287$$ −7.43213e25 −1.15705
$$288$$ −2.04841e25 −0.307465
$$289$$ 9.60027e25 1.38949
$$290$$ 0 0
$$291$$ −1.91500e25 −0.257805
$$292$$ −6.79080e25 −0.881863
$$293$$ −6.89938e25 −0.864371 −0.432186 0.901785i $$-0.642257\pi$$
−0.432186 + 0.901785i $$0.642257\pi$$
$$294$$ 3.44987e25 0.417019
$$295$$ 0 0
$$296$$ −9.32629e25 −1.04990
$$297$$ 1.59743e25 0.173571
$$298$$ −1.73191e25 −0.181657
$$299$$ −9.72505e25 −0.984791
$$300$$ 0 0
$$301$$ −2.06269e26 −1.94754
$$302$$ −3.56999e25 −0.325532
$$303$$ −6.76449e25 −0.595782
$$304$$ −7.50532e25 −0.638553
$$305$$ 0 0
$$306$$ 2.43720e25 0.193560
$$307$$ 6.78268e25 0.520533 0.260267 0.965537i $$-0.416190\pi$$
0.260267 + 0.965537i $$0.416190\pi$$
$$308$$ −1.78557e26 −1.32432
$$309$$ 5.28953e25 0.379187
$$310$$ 0 0
$$311$$ 1.00215e26 0.671347 0.335673 0.941978i $$-0.391036\pi$$
0.335673 + 0.941978i $$0.391036\pi$$
$$312$$ −5.43619e25 −0.352105
$$313$$ 2.29792e26 1.43920 0.719599 0.694390i $$-0.244326\pi$$
0.719599 + 0.694390i $$0.244326\pi$$
$$314$$ −3.58651e25 −0.217225
$$315$$ 0 0
$$316$$ −1.13756e26 −0.644558
$$317$$ −2.06457e26 −1.13164 −0.565818 0.824530i $$-0.691439\pi$$
−0.565818 + 0.824530i $$0.691439\pi$$
$$318$$ −7.38669e25 −0.391711
$$319$$ 4.00933e24 0.0205717
$$320$$ 0 0
$$321$$ 7.42734e25 0.356887
$$322$$ −1.55711e26 −0.724155
$$323$$ 3.67547e26 1.65457
$$324$$ −2.18986e25 −0.0954319
$$325$$ 0 0
$$326$$ −4.78206e24 −0.0195358
$$327$$ 9.84493e25 0.389459
$$328$$ 1.23358e26 0.472597
$$329$$ −2.04515e26 −0.758870
$$330$$ 0 0
$$331$$ −1.09215e26 −0.380268 −0.190134 0.981758i $$-0.560892\pi$$
−0.190134 + 0.981758i $$0.560892\pi$$
$$332$$ −2.61461e26 −0.881978
$$333$$ −1.53339e26 −0.501173
$$334$$ −8.21592e25 −0.260207
$$335$$ 0 0
$$336$$ 1.97954e26 0.588846
$$337$$ 4.76493e26 1.37386 0.686930 0.726724i $$-0.258958\pi$$
0.686930 + 0.726724i $$0.258958\pi$$
$$338$$ 3.18851e25 0.0891173
$$339$$ −3.84526e26 −1.04191
$$340$$ 0 0
$$341$$ −6.92152e26 −1.76312
$$342$$ 5.42588e25 0.134028
$$343$$ −6.58553e26 −1.57762
$$344$$ 3.42363e26 0.795469
$$345$$ 0 0
$$346$$ 3.34488e26 0.731276
$$347$$ 5.67729e26 1.20415 0.602077 0.798438i $$-0.294340\pi$$
0.602077 + 0.798438i $$0.294340\pi$$
$$348$$ −5.49626e24 −0.0113106
$$349$$ −4.85816e26 −0.970075 −0.485037 0.874493i $$-0.661194\pi$$
−0.485037 + 0.874493i $$0.661194\pi$$
$$350$$ 0 0
$$351$$ −8.93796e25 −0.168079
$$352$$ 4.55800e26 0.831910
$$353$$ −1.71788e26 −0.304340 −0.152170 0.988354i $$-0.548626\pi$$
−0.152170 + 0.988354i $$0.548626\pi$$
$$354$$ 1.65568e26 0.284737
$$355$$ 0 0
$$356$$ −2.47226e26 −0.400746
$$357$$ −9.69410e26 −1.52578
$$358$$ −2.63028e26 −0.402004
$$359$$ 1.33028e27 1.97448 0.987239 0.159247i $$-0.0509068\pi$$
0.987239 + 0.159247i $$0.0509068\pi$$
$$360$$ 0 0
$$361$$ 1.04051e26 0.145687
$$362$$ 5.05103e26 0.686973
$$363$$ 8.15275e25 0.107717
$$364$$ 9.99067e26 1.28242
$$365$$ 0 0
$$366$$ 4.02075e25 0.0487254
$$367$$ −3.40629e26 −0.401132 −0.200566 0.979680i $$-0.564278\pi$$
−0.200566 + 0.979680i $$0.564278\pi$$
$$368$$ −5.87780e26 −0.672686
$$369$$ 2.02820e26 0.225597
$$370$$ 0 0
$$371$$ 2.93810e27 3.08774
$$372$$ 9.48849e26 0.969385
$$373$$ 1.62912e27 1.61812 0.809058 0.587729i $$-0.199978\pi$$
0.809058 + 0.587729i $$0.199978\pi$$
$$374$$ −5.42311e26 −0.523717
$$375$$ 0 0
$$376$$ 3.39451e26 0.309960
$$377$$ −2.24331e25 −0.0199207
$$378$$ −1.43108e26 −0.123595
$$379$$ 3.25550e26 0.273467 0.136734 0.990608i $$-0.456340\pi$$
0.136734 + 0.990608i $$0.456340\pi$$
$$380$$ 0 0
$$381$$ −4.72149e26 −0.375290
$$382$$ 3.24569e26 0.250981
$$383$$ 1.61833e27 1.21753 0.608765 0.793350i $$-0.291665\pi$$
0.608765 + 0.793350i $$0.291665\pi$$
$$384$$ −8.01594e26 −0.586782
$$385$$ 0 0
$$386$$ −4.65867e26 −0.322920
$$387$$ 5.62899e26 0.379722
$$388$$ −5.84146e26 −0.383520
$$389$$ −2.02736e27 −1.29557 −0.647784 0.761824i $$-0.724304\pi$$
−0.647784 + 0.761824i $$0.724304\pi$$
$$390$$ 0 0
$$391$$ 2.87844e27 1.74302
$$392$$ 2.27758e27 1.34267
$$393$$ 3.26781e26 0.187558
$$394$$ 6.01230e26 0.335993
$$395$$ 0 0
$$396$$ 4.87275e26 0.258211
$$397$$ −1.57647e27 −0.813551 −0.406775 0.913528i $$-0.633347\pi$$
−0.406775 + 0.913528i $$0.633347\pi$$
$$398$$ −8.33337e26 −0.418840
$$399$$ −2.15818e27 −1.05650
$$400$$ 0 0
$$401$$ 2.38019e27 1.10559 0.552797 0.833316i $$-0.313561\pi$$
0.552797 + 0.833316i $$0.313561\pi$$
$$402$$ 1.73712e26 0.0786063
$$403$$ 3.87275e27 1.70733
$$404$$ −2.06342e27 −0.886306
$$405$$ 0 0
$$406$$ −3.59184e25 −0.0146485
$$407$$ 3.41201e27 1.35603
$$408$$ 1.60902e27 0.623202
$$409$$ 1.28701e27 0.485832 0.242916 0.970047i $$-0.421896\pi$$
0.242916 + 0.970047i $$0.421896\pi$$
$$410$$ 0 0
$$411$$ 2.96916e26 0.106487
$$412$$ 1.61351e27 0.564092
$$413$$ −6.58558e27 −2.24450
$$414$$ 4.24928e26 0.141192
$$415$$ 0 0
$$416$$ −2.55031e27 −0.805587
$$417$$ −1.80993e26 −0.0557486
$$418$$ −1.20733e27 −0.362641
$$419$$ −6.36510e27 −1.86448 −0.932241 0.361839i $$-0.882149\pi$$
−0.932241 + 0.361839i $$0.882149\pi$$
$$420$$ 0 0
$$421$$ 1.16839e27 0.325556 0.162778 0.986663i $$-0.447955\pi$$
0.162778 + 0.986663i $$0.447955\pi$$
$$422$$ −1.17459e27 −0.319232
$$423$$ 5.58111e26 0.147961
$$424$$ −4.87664e27 −1.26119
$$425$$ 0 0
$$426$$ −3.54738e26 −0.0873186
$$427$$ −1.59928e27 −0.384088
$$428$$ 2.26562e27 0.530917
$$429$$ 1.98882e27 0.454773
$$430$$ 0 0
$$431$$ 4.87470e27 1.06154 0.530770 0.847516i $$-0.321903\pi$$
0.530770 + 0.847516i $$0.321903\pi$$
$$432$$ −5.40208e26 −0.114811
$$433$$ −3.42952e27 −0.711395 −0.355698 0.934601i $$-0.615757\pi$$
−0.355698 + 0.934601i $$0.615757\pi$$
$$434$$ 6.20078e27 1.25546
$$435$$ 0 0
$$436$$ 3.00307e27 0.579372
$$437$$ 6.40821e27 1.20693
$$438$$ 1.21106e27 0.222684
$$439$$ 6.10587e27 1.09615 0.548075 0.836429i $$-0.315361\pi$$
0.548075 + 0.836429i $$0.315361\pi$$
$$440$$ 0 0
$$441$$ 3.74469e27 0.640931
$$442$$ 3.03435e27 0.507145
$$443$$ 6.52956e27 1.06572 0.532862 0.846202i $$-0.321116\pi$$
0.532862 + 0.846202i $$0.321116\pi$$
$$444$$ −4.67741e27 −0.745563
$$445$$ 0 0
$$446$$ 4.27752e27 0.650393
$$447$$ −1.87991e27 −0.279196
$$448$$ 2.94706e27 0.427533
$$449$$ 3.56364e27 0.505018 0.252509 0.967595i $$-0.418744\pi$$
0.252509 + 0.967595i $$0.418744\pi$$
$$450$$ 0 0
$$451$$ −4.51302e27 −0.610399
$$452$$ −1.17295e28 −1.54998
$$453$$ −3.87508e27 −0.500322
$$454$$ 2.75797e27 0.347939
$$455$$ 0 0
$$456$$ 3.58212e27 0.431528
$$457$$ 5.64234e25 0.00664262 0.00332131 0.999994i $$-0.498943\pi$$
0.00332131 + 0.999994i $$0.498943\pi$$
$$458$$ 3.32347e27 0.382388
$$459$$ 2.64548e27 0.297489
$$460$$ 0 0
$$461$$ 3.68168e27 0.395537 0.197768 0.980249i $$-0.436631\pi$$
0.197768 + 0.980249i $$0.436631\pi$$
$$462$$ 3.18436e27 0.334412
$$463$$ 6.25388e27 0.642020 0.321010 0.947076i $$-0.395978\pi$$
0.321010 + 0.947076i $$0.395978\pi$$
$$464$$ −1.35585e26 −0.0136074
$$465$$ 0 0
$$466$$ 1.31031e27 0.125696
$$467$$ 1.81487e26 0.0170223 0.00851114 0.999964i $$-0.497291\pi$$
0.00851114 + 0.999964i $$0.497291\pi$$
$$468$$ −2.72641e27 −0.250040
$$469$$ −6.90952e27 −0.619631
$$470$$ 0 0
$$471$$ −3.89301e27 −0.333861
$$472$$ 1.09307e28 0.916762
$$473$$ −1.25253e28 −1.02742
$$474$$ 2.02872e27 0.162761
$$475$$ 0 0
$$476$$ −2.95706e28 −2.26980
$$477$$ −8.01796e27 −0.602034
$$478$$ 4.42248e26 0.0324843
$$479$$ 4.83395e27 0.347360 0.173680 0.984802i $$-0.444434\pi$$
0.173680 + 0.984802i $$0.444434\pi$$
$$480$$ 0 0
$$481$$ −1.90910e28 −1.31312
$$482$$ −1.07761e27 −0.0725214
$$483$$ −1.69018e28 −1.11298
$$484$$ 2.48689e27 0.160244
$$485$$ 0 0
$$486$$ 3.90537e26 0.0240980
$$487$$ 6.30377e27 0.380668 0.190334 0.981719i $$-0.439043\pi$$
0.190334 + 0.981719i $$0.439043\pi$$
$$488$$ 2.65447e27 0.156881
$$489$$ −5.19073e26 −0.0300252
$$490$$ 0 0
$$491$$ 3.67372e27 0.203587 0.101794 0.994806i $$-0.467542\pi$$
0.101794 + 0.994806i $$0.467542\pi$$
$$492$$ 6.18676e27 0.335606
$$493$$ 6.63981e26 0.0352584
$$494$$ 6.75531e27 0.351166
$$495$$ 0 0
$$496$$ 2.34068e28 1.16623
$$497$$ 1.41099e28 0.688307
$$498$$ 4.66287e27 0.222713
$$499$$ −2.00088e28 −0.935761 −0.467881 0.883792i $$-0.654982\pi$$
−0.467881 + 0.883792i $$0.654982\pi$$
$$500$$ 0 0
$$501$$ −8.91805e27 −0.399921
$$502$$ 9.35496e27 0.410821
$$503$$ −1.40332e28 −0.603521 −0.301761 0.953384i $$-0.597574\pi$$
−0.301761 + 0.953384i $$0.597574\pi$$
$$504$$ −9.44790e27 −0.397937
$$505$$ 0 0
$$506$$ −9.45525e27 −0.382026
$$507$$ 3.46100e27 0.136967
$$508$$ −1.44023e28 −0.558295
$$509$$ −6.58158e27 −0.249916 −0.124958 0.992162i $$-0.539880\pi$$
−0.124958 + 0.992162i $$0.539880\pi$$
$$510$$ 0 0
$$511$$ −4.81708e28 −1.75535
$$512$$ −2.70830e28 −0.966858
$$513$$ 5.88957e27 0.205992
$$514$$ 7.76673e27 0.266149
$$515$$ 0 0
$$516$$ 1.71705e28 0.564887
$$517$$ −1.24188e28 −0.400339
$$518$$ −3.05671e28 −0.965588
$$519$$ 3.63073e28 1.12392
$$520$$ 0 0
$$521$$ −4.65750e28 −1.38470 −0.692351 0.721561i $$-0.743425\pi$$
−0.692351 + 0.721561i $$0.743425\pi$$
$$522$$ 9.80196e25 0.00285610
$$523$$ 3.89578e28 1.11257 0.556284 0.830992i $$-0.312227\pi$$
0.556284 + 0.830992i $$0.312227\pi$$
$$524$$ 9.96804e27 0.279017
$$525$$ 0 0
$$526$$ 6.71532e27 0.180600
$$527$$ −1.14627e29 −3.02186
$$528$$ 1.20204e28 0.310644
$$529$$ 1.07144e28 0.271445
$$530$$ 0 0
$$531$$ 1.79718e28 0.437621
$$532$$ −6.58324e28 −1.57169
$$533$$ 2.52514e28 0.591085
$$534$$ 4.40900e27 0.101194
$$535$$ 0 0
$$536$$ 1.14684e28 0.253088
$$537$$ −2.85507e28 −0.617854
$$538$$ 2.30181e27 0.0488491
$$539$$ −8.33247e28 −1.73417
$$540$$ 0 0
$$541$$ 7.65629e27 0.153267 0.0766333 0.997059i $$-0.475583\pi$$
0.0766333 + 0.997059i $$0.475583\pi$$
$$542$$ 2.98116e28 0.585318
$$543$$ 5.48268e28 1.05583
$$544$$ 7.54846e28 1.42584
$$545$$ 0 0
$$546$$ −1.78172e28 −0.323830
$$547$$ −6.77366e28 −1.20769 −0.603847 0.797100i $$-0.706366\pi$$
−0.603847 + 0.797100i $$0.706366\pi$$
$$548$$ 9.05707e27 0.158413
$$549$$ 4.36436e27 0.0748878
$$550$$ 0 0
$$551$$ 1.47821e27 0.0244142
$$552$$ 2.80534e28 0.454595
$$553$$ −8.06935e28 −1.28299
$$554$$ 9.51957e27 0.148513
$$555$$ 0 0
$$556$$ −5.52097e27 −0.0829336
$$557$$ 5.64828e28 0.832600 0.416300 0.909227i $$-0.363327\pi$$
0.416300 + 0.909227i $$0.363327\pi$$
$$558$$ −1.69217e28 −0.244784
$$559$$ 7.00819e28 0.994906
$$560$$ 0 0
$$561$$ −5.88657e28 −0.804919
$$562$$ −1.37257e28 −0.184206
$$563$$ −1.32120e28 −0.174033 −0.0870166 0.996207i $$-0.527733\pi$$
−0.0870166 + 0.996207i $$0.527733\pi$$
$$564$$ 1.70245e28 0.220112
$$565$$ 0 0
$$566$$ −6.51240e27 −0.0811275
$$567$$ −1.55338e28 −0.189957
$$568$$ −2.34195e28 −0.281139
$$569$$ 3.77888e27 0.0445332 0.0222666 0.999752i $$-0.492912\pi$$
0.0222666 + 0.999752i $$0.492912\pi$$
$$570$$ 0 0
$$571$$ −9.70592e28 −1.10245 −0.551223 0.834358i $$-0.685839\pi$$
−0.551223 + 0.834358i $$0.685839\pi$$
$$572$$ 6.06666e28 0.676536
$$573$$ 3.52306e28 0.385742
$$574$$ 4.04308e28 0.434647
$$575$$ 0 0
$$576$$ −8.04239e27 −0.0833584
$$577$$ 1.40057e27 0.0142547 0.00712737 0.999975i $$-0.497731\pi$$
0.00712737 + 0.999975i $$0.497731\pi$$
$$578$$ −5.22255e28 −0.521964
$$579$$ −5.05680e28 −0.496307
$$580$$ 0 0
$$581$$ −1.85468e29 −1.75558
$$582$$ 1.04176e28 0.0968447
$$583$$ 1.78411e29 1.62893
$$584$$ 7.99534e28 0.716972
$$585$$ 0 0
$$586$$ 3.75326e28 0.324701
$$587$$ 9.16040e28 0.778421 0.389211 0.921149i $$-0.372748\pi$$
0.389211 + 0.921149i $$0.372748\pi$$
$$588$$ 1.14227e29 0.953471
$$589$$ −2.55191e29 −2.09245
$$590$$ 0 0
$$591$$ 6.52610e28 0.516399
$$592$$ −1.15385e29 −0.896959
$$593$$ −1.47495e29 −1.12643 −0.563214 0.826311i $$-0.690435\pi$$
−0.563214 + 0.826311i $$0.690435\pi$$
$$594$$ −8.68999e27 −0.0652021
$$595$$ 0 0
$$596$$ −5.73444e28 −0.415341
$$597$$ −9.04554e28 −0.643730
$$598$$ 5.29043e28 0.369937
$$599$$ 1.19182e29 0.818899 0.409450 0.912333i $$-0.365721\pi$$
0.409450 + 0.912333i $$0.365721\pi$$
$$600$$ 0 0
$$601$$ 2.64125e29 1.75238 0.876190 0.481966i $$-0.160077\pi$$
0.876190 + 0.481966i $$0.160077\pi$$
$$602$$ 1.12210e29 0.731593
$$603$$ 1.88558e28 0.120813
$$604$$ −1.18205e29 −0.744297
$$605$$ 0 0
$$606$$ 3.67988e28 0.223806
$$607$$ 8.82263e28 0.527371 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$608$$ 1.68050e29 0.987302
$$609$$ −3.89879e27 −0.0225138
$$610$$ 0 0
$$611$$ 6.94859e28 0.387671
$$612$$ 8.06970e28 0.442555
$$613$$ −3.27173e29 −1.76377 −0.881885 0.471465i $$-0.843725\pi$$
−0.881885 + 0.471465i $$0.843725\pi$$
$$614$$ −3.68978e28 −0.195538
$$615$$ 0 0
$$616$$ 2.10229e29 1.07670
$$617$$ −7.46428e28 −0.375832 −0.187916 0.982185i $$-0.560173\pi$$
−0.187916 + 0.982185i $$0.560173\pi$$
$$618$$ −2.87751e28 −0.142442
$$619$$ −8.32478e28 −0.405155 −0.202577 0.979266i $$-0.564932\pi$$
−0.202577 + 0.979266i $$0.564932\pi$$
$$620$$ 0 0
$$621$$ 4.61242e28 0.217004
$$622$$ −5.45167e28 −0.252192
$$623$$ −1.75371e29 −0.797686
$$624$$ −6.72569e28 −0.300814
$$625$$ 0 0
$$626$$ −1.25007e29 −0.540635
$$627$$ −1.31051e29 −0.557355
$$628$$ −1.18751e29 −0.496663
$$629$$ 5.65059e29 2.32414
$$630$$ 0 0
$$631$$ 3.63173e29 1.44479 0.722396 0.691479i $$-0.243041\pi$$
0.722396 + 0.691479i $$0.243041\pi$$
$$632$$ 1.33934e29 0.524038
$$633$$ −1.27497e29 −0.490639
$$634$$ 1.12312e29 0.425099
$$635$$ 0 0
$$636$$ −2.44578e29 −0.895606
$$637$$ 4.66221e29 1.67930
$$638$$ −2.18108e27 −0.00772776
$$639$$ −3.85054e28 −0.134203
$$640$$ 0 0
$$641$$ −4.09788e28 −0.138213 −0.0691067 0.997609i $$-0.522015\pi$$
−0.0691067 + 0.997609i $$0.522015\pi$$
$$642$$ −4.04047e28 −0.134065
$$643$$ 3.44481e29 1.12448 0.562238 0.826976i $$-0.309941\pi$$
0.562238 + 0.826976i $$0.309941\pi$$
$$644$$ −5.15567e29 −1.65571
$$645$$ 0 0
$$646$$ −1.99945e29 −0.621541
$$647$$ 2.99052e29 0.914645 0.457322 0.889301i $$-0.348809\pi$$
0.457322 + 0.889301i $$0.348809\pi$$
$$648$$ 2.57829e28 0.0775880
$$649$$ −3.99898e29 −1.18408
$$650$$ 0 0
$$651$$ 6.73069e29 1.92956
$$652$$ −1.58337e28 −0.0446665
$$653$$ −6.43185e29 −1.78545 −0.892724 0.450603i $$-0.851209\pi$$
−0.892724 + 0.450603i $$0.851209\pi$$
$$654$$ −5.35564e28 −0.146300
$$655$$ 0 0
$$656$$ 1.52619e29 0.403755
$$657$$ 1.31456e29 0.342250
$$658$$ 1.11256e29 0.285070
$$659$$ 1.00447e29 0.253303 0.126652 0.991947i $$-0.459577\pi$$
0.126652 + 0.991947i $$0.459577\pi$$
$$660$$ 0 0
$$661$$ −4.70181e29 −1.14855 −0.574274 0.818663i $$-0.694716\pi$$
−0.574274 + 0.818663i $$0.694716\pi$$
$$662$$ 5.94131e28 0.142848
$$663$$ 3.29367e29 0.779449
$$664$$ 3.07839e29 0.717065
$$665$$ 0 0
$$666$$ 8.34163e28 0.188266
$$667$$ 1.15766e28 0.0257193
$$668$$ −2.72034e29 −0.594936
$$669$$ 4.64307e29 0.999612
$$670$$ 0 0
$$671$$ −9.71132e28 −0.202624
$$672$$ −4.43233e29 −0.910448
$$673$$ 3.92345e29 0.793432 0.396716 0.917941i $$-0.370150\pi$$
0.396716 + 0.917941i $$0.370150\pi$$
$$674$$ −2.59212e29 −0.516091
$$675$$ 0 0
$$676$$ 1.05573e29 0.203758
$$677$$ 1.52718e29 0.290207 0.145104 0.989416i $$-0.453648\pi$$
0.145104 + 0.989416i $$0.453648\pi$$
$$678$$ 2.09182e29 0.391392
$$679$$ −4.14366e29 −0.763399
$$680$$ 0 0
$$681$$ 2.99366e29 0.534759
$$682$$ 3.76531e29 0.662315
$$683$$ 4.36102e29 0.755389 0.377695 0.925930i $$-0.376717\pi$$
0.377695 + 0.925930i $$0.376717\pi$$
$$684$$ 1.79654e29 0.306442
$$685$$ 0 0
$$686$$ 3.58253e29 0.592634
$$687$$ 3.60749e29 0.587705
$$688$$ 4.23574e29 0.679595
$$689$$ −9.98250e29 −1.57738
$$690$$ 0 0
$$691$$ 5.96732e29 0.914660 0.457330 0.889297i $$-0.348806\pi$$
0.457330 + 0.889297i $$0.348806\pi$$
$$692$$ 1.10751e30 1.67199
$$693$$ 3.45650e29 0.513970
$$694$$ −3.08845e29 −0.452341
$$695$$ 0 0
$$696$$ 6.47118e27 0.00919573
$$697$$ −7.47397e29 −1.04618
$$698$$ 2.64284e29 0.364409
$$699$$ 1.42229e29 0.193186
$$700$$ 0 0
$$701$$ 5.37209e29 0.708116 0.354058 0.935223i $$-0.384802\pi$$
0.354058 + 0.935223i $$0.384802\pi$$
$$702$$ 4.86225e28 0.0631390
$$703$$ 1.25798e30 1.60932
$$704$$ 1.78955e29 0.225543
$$705$$ 0 0
$$706$$ 9.34527e28 0.114325
$$707$$ −1.46370e30 −1.76420
$$708$$ 5.48207e29 0.651021
$$709$$ −1.27317e29 −0.148971 −0.0744854 0.997222i $$-0.523731\pi$$
−0.0744854 + 0.997222i $$0.523731\pi$$
$$710$$ 0 0
$$711$$ 2.20209e29 0.250152
$$712$$ 2.91078e29 0.325814
$$713$$ −1.99853e30 −2.20430
$$714$$ 5.27359e29 0.573159
$$715$$ 0 0
$$716$$ −8.70902e29 −0.919142
$$717$$ 4.80043e28 0.0499262
$$718$$ −7.23674e29 −0.741713
$$719$$ −1.86676e30 −1.88553 −0.942767 0.333451i $$-0.891787\pi$$
−0.942767 + 0.333451i $$0.891787\pi$$
$$720$$ 0 0
$$721$$ 1.14455e30 1.12283
$$722$$ −5.66038e28 −0.0547275
$$723$$ −1.16970e29 −0.111461
$$724$$ 1.67242e30 1.57069
$$725$$ 0 0
$$726$$ −4.43509e28 −0.0404640
$$727$$ 1.86570e30 1.67776 0.838882 0.544314i $$-0.183210\pi$$
0.838882 + 0.544314i $$0.183210\pi$$
$$728$$ −1.17628e30 −1.04263
$$729$$ 4.23912e28 0.0370370
$$730$$ 0 0
$$731$$ −2.07430e30 −1.76092
$$732$$ 1.33129e29 0.111406
$$733$$ 4.88128e29 0.402663 0.201331 0.979523i $$-0.435473\pi$$
0.201331 + 0.979523i $$0.435473\pi$$
$$734$$ 1.85302e29 0.150686
$$735$$ 0 0
$$736$$ 1.31608e30 1.04008
$$737$$ −4.19568e29 −0.326884
$$738$$ −1.10334e29 −0.0847456
$$739$$ 1.67069e30 1.26511 0.632556 0.774515i $$-0.282006\pi$$
0.632556 + 0.774515i $$0.282006\pi$$
$$740$$ 0 0
$$741$$ 7.33262e29 0.539719
$$742$$ −1.59833e30 −1.15991
$$743$$ −1.62834e30 −1.16509 −0.582547 0.812797i $$-0.697944\pi$$
−0.582547 + 0.812797i $$0.697944\pi$$
$$744$$ −1.11715e30 −0.788129
$$745$$ 0 0
$$746$$ −8.86240e29 −0.607846
$$747$$ 5.06136e29 0.342295
$$748$$ −1.79562e30 −1.19743
$$749$$ 1.60712e30 1.05679
$$750$$ 0 0
$$751$$ 5.83073e29 0.372824 0.186412 0.982472i $$-0.440314\pi$$
0.186412 + 0.982472i $$0.440314\pi$$
$$752$$ 4.19971e29 0.264809
$$753$$ 1.01544e30 0.631406
$$754$$ 1.22036e28 0.00748324
$$755$$ 0 0
$$756$$ −4.73840e29 −0.282587
$$757$$ −1.39333e30 −0.819494 −0.409747 0.912199i $$-0.634383\pi$$
−0.409747 + 0.912199i $$0.634383\pi$$
$$758$$ −1.77099e29 −0.102728
$$759$$ −1.02633e30 −0.587148
$$760$$ 0 0
$$761$$ −9.17379e29 −0.510517 −0.255258 0.966873i $$-0.582161\pi$$
−0.255258 + 0.966873i $$0.582161\pi$$
$$762$$ 2.56849e29 0.140978
$$763$$ 2.13024e30 1.15324
$$764$$ 1.07467e30 0.573843
$$765$$ 0 0
$$766$$ −8.80371e29 −0.457366
$$767$$ 2.23752e30 1.14661
$$768$$ 1.50438e29 0.0760438
$$769$$ −1.13366e30 −0.565273 −0.282636 0.959227i $$-0.591209\pi$$
−0.282636 + 0.959227i $$0.591209\pi$$
$$770$$ 0 0
$$771$$ 8.43047e29 0.409054
$$772$$ −1.54251e30 −0.738324
$$773$$ 1.81526e29 0.0857143 0.0428572 0.999081i $$-0.486354\pi$$
0.0428572 + 0.999081i $$0.486354\pi$$
$$774$$ −3.06217e29 −0.142643
$$775$$ 0 0
$$776$$ 6.87761e29 0.311809
$$777$$ −3.31794e30 −1.48405
$$778$$ 1.10289e30 0.486681
$$779$$ −1.66391e30 −0.724415
$$780$$ 0 0
$$781$$ 8.56799e29 0.363114
$$782$$ −1.56587e30 −0.654765
$$783$$ 1.06396e28 0.00438963
$$784$$ 2.81783e30 1.14709
$$785$$ 0 0
$$786$$ −1.77769e29 −0.0704561
$$787$$ −7.88071e29 −0.308199 −0.154099 0.988055i $$-0.549248\pi$$
−0.154099 + 0.988055i $$0.549248\pi$$
$$788$$ 1.99071e30 0.768214
$$789$$ 7.28921e29 0.277570
$$790$$ 0 0
$$791$$ −8.32034e30 −3.08523
$$792$$ −5.73706e29 −0.209930
$$793$$ 5.43371e29 0.196213
$$794$$ 8.57598e29 0.305611
$$795$$ 0 0
$$796$$ −2.75923e30 −0.957636
$$797$$ 3.61162e30 1.23706 0.618528 0.785763i $$-0.287729\pi$$
0.618528 + 0.785763i $$0.287729\pi$$
$$798$$ 1.17405e30 0.396876
$$799$$ −2.05666e30 −0.686153
$$800$$ 0 0
$$801$$ 4.78579e29 0.155529
$$802$$ −1.29482e30 −0.415317
$$803$$ −2.92508e30 −0.926029
$$804$$ 5.75172e29 0.179725
$$805$$ 0 0
$$806$$ −2.10678e30 −0.641358
$$807$$ 2.49853e29 0.0750778
$$808$$ 2.42943e30 0.720584
$$809$$ 6.55296e30 1.91857 0.959286 0.282436i $$-0.0911424\pi$$
0.959286 + 0.282436i $$0.0911424\pi$$
$$810$$ 0 0
$$811$$ −2.95834e30 −0.843975 −0.421988 0.906602i $$-0.638667\pi$$
−0.421988 + 0.906602i $$0.638667\pi$$
$$812$$ −1.18928e29 −0.0334923
$$813$$ 3.23592e30 0.899596
$$814$$ −1.85613e30 −0.509393
$$815$$ 0 0
$$816$$ 1.99069e30 0.532422
$$817$$ −4.61797e30 −1.21933
$$818$$ −7.00132e29 −0.182503
$$819$$ −1.93399e30 −0.497706
$$820$$ 0 0
$$821$$ −6.60712e30 −1.65733 −0.828665 0.559744i $$-0.810899\pi$$
−0.828665 + 0.559744i $$0.810899\pi$$
$$822$$ −1.61523e29 −0.0400018
$$823$$ −4.27938e30 −1.04636 −0.523182 0.852221i $$-0.675255\pi$$
−0.523182 + 0.852221i $$0.675255\pi$$
$$824$$ −1.89971e30 −0.458618
$$825$$ 0 0
$$826$$ 3.58256e30 0.843146
$$827$$ −8.08422e29 −0.187858 −0.0939291 0.995579i $$-0.529943\pi$$
−0.0939291 + 0.995579i $$0.529943\pi$$
$$828$$ 1.40696e30 0.322822
$$829$$ 6.58152e30 1.49109 0.745544 0.666456i $$-0.232190\pi$$
0.745544 + 0.666456i $$0.232190\pi$$
$$830$$ 0 0
$$831$$ 1.03331e30 0.228255
$$832$$ −1.00129e30 −0.218407
$$833$$ −1.37993e31 −2.97225
$$834$$ 9.84603e28 0.0209420
$$835$$ 0 0
$$836$$ −3.99756e30 −0.829141
$$837$$ −1.83678e30 −0.376218
$$838$$ 3.46262e30 0.700393
$$839$$ −8.11970e30 −1.62196 −0.810979 0.585075i $$-0.801065\pi$$
−0.810979 + 0.585075i $$0.801065\pi$$
$$840$$ 0 0
$$841$$ −5.13017e30 −0.999480
$$842$$ −6.35604e29 −0.122295
$$843$$ −1.48987e30 −0.283113
$$844$$ −3.88914e30 −0.729892
$$845$$ 0 0
$$846$$ −3.03613e29 −0.0555816
$$847$$ 1.76409e30 0.318966
$$848$$ −6.03340e30 −1.07747
$$849$$ −7.06894e29 −0.124688
$$850$$ 0 0
$$851$$ 9.85186e30 1.69534
$$852$$ −1.17456e30 −0.199645
$$853$$ −1.29820e30 −0.217960 −0.108980 0.994044i $$-0.534759\pi$$
−0.108980 + 0.994044i $$0.534759\pi$$
$$854$$ 8.70007e29 0.144283
$$855$$ 0 0
$$856$$ −2.66749e30 −0.431646
$$857$$ 1.05399e30 0.168476 0.0842380 0.996446i $$-0.473154\pi$$
0.0842380 + 0.996446i $$0.473154\pi$$
$$858$$ −1.08192e30 −0.170836
$$859$$ 6.98517e30 1.08955 0.544777 0.838581i $$-0.316614\pi$$
0.544777 + 0.838581i $$0.316614\pi$$
$$860$$ 0 0
$$861$$ 4.38860e30 0.668025
$$862$$ −2.65184e30 −0.398768
$$863$$ −1.03987e31 −1.54478 −0.772392 0.635146i $$-0.780940\pi$$
−0.772392 + 0.635146i $$0.780940\pi$$
$$864$$ 1.20956e30 0.177515
$$865$$ 0 0
$$866$$ 1.86566e30 0.267236
$$867$$ −5.66887e30 −0.802224
$$868$$ 2.05311e31 2.87049
$$869$$ −4.89997e30 −0.676839
$$870$$ 0 0
$$871$$ 2.34758e30 0.316541
$$872$$ −3.53575e30 −0.471041
$$873$$ 1.13079e30 0.148844
$$874$$ −3.48607e30 −0.453384
$$875$$ 0 0
$$876$$ 4.00990e30 0.509144
$$877$$ −5.66163e30 −0.710306 −0.355153 0.934808i $$-0.615571\pi$$
−0.355153 + 0.934808i $$0.615571\pi$$
$$878$$ −3.32159e30 −0.411770
$$879$$ 4.07402e30 0.499045
$$880$$ 0 0
$$881$$ 3.59833e30 0.430382 0.215191 0.976572i $$-0.430963\pi$$
0.215191 + 0.976572i $$0.430963\pi$$
$$882$$ −2.03711e30 −0.240766
$$883$$ 1.24249e31 1.45112 0.725562 0.688157i $$-0.241580\pi$$
0.725562 + 0.688157i $$0.241580\pi$$
$$884$$ 1.00469e31 1.15954
$$885$$ 0 0
$$886$$ −3.55208e30 −0.400340
$$887$$ 9.60349e30 1.06962 0.534812 0.844971i $$-0.320383\pi$$
0.534812 + 0.844971i $$0.320383\pi$$
$$888$$ 5.50708e30 0.606157
$$889$$ −1.02163e31 −1.11129
$$890$$ 0 0
$$891$$ −9.43264e29 −0.100211
$$892$$ 1.41631e31 1.48706
$$893$$ −4.57869e30 −0.475118
$$894$$ 1.02267e30 0.104880
$$895$$ 0 0
$$896$$ −1.73448e31 −1.73754
$$897$$ 5.74254e30 0.568570
$$898$$ −1.93862e30 −0.189710
$$899$$ −4.61007e29 −0.0445894
$$900$$ 0 0
$$901$$ 2.95464e31 2.79187
$$902$$ 2.45509e30 0.229297
$$903$$ 1.21800e31 1.12441
$$904$$ 1.38100e31 1.26016
$$905$$ 0 0
$$906$$ 2.10804e30 0.187946
$$907$$ 1.94141e31 1.71097 0.855483 0.517830i $$-0.173260\pi$$
0.855483 + 0.517830i $$0.173260\pi$$
$$908$$ 9.13179e30 0.795526
$$909$$ 3.99436e30 0.343975
$$910$$ 0 0
$$911$$ −4.71847e30 −0.397062 −0.198531 0.980095i $$-0.563617\pi$$
−0.198531 + 0.980095i $$0.563617\pi$$
$$912$$ 4.43182e30 0.368669
$$913$$ −1.12622e31 −0.926150
$$914$$ −3.06943e28 −0.00249530
$$915$$ 0 0
$$916$$ 1.10042e31 0.874291
$$917$$ 7.07086e30 0.555385
$$918$$ −1.43914e30 −0.111752
$$919$$ −1.97911e31 −1.51935 −0.759673 0.650305i $$-0.774641\pi$$
−0.759673 + 0.650305i $$0.774641\pi$$
$$920$$ 0 0
$$921$$ −4.00510e30 −0.300530
$$922$$ −2.00284e30 −0.148584
$$923$$ −4.79399e30 −0.351624
$$924$$ 1.05436e31 0.764599
$$925$$ 0 0
$$926$$ −3.40211e30 −0.241175
$$927$$ −3.12342e30 −0.218924
$$928$$ 3.03585e29 0.0210391
$$929$$ −2.30269e31 −1.57786 −0.788932 0.614480i $$-0.789366\pi$$
−0.788932 + 0.614480i $$0.789366\pi$$
$$930$$ 0 0
$$931$$ −3.07211e31 −2.05809
$$932$$ 4.33851e30 0.287391
$$933$$ −5.91757e30 −0.387602
$$934$$ −9.87288e28 −0.00639443
$$935$$ 0 0
$$936$$ 3.21002e30 0.203288
$$937$$ −2.58396e31 −1.61816 −0.809078 0.587701i $$-0.800033\pi$$
−0.809078 + 0.587701i $$0.800033\pi$$
$$938$$ 3.75878e30 0.232765
$$939$$ −1.35690e31 −0.830921
$$940$$ 0 0
$$941$$ 2.93678e31 1.75865 0.879326 0.476221i $$-0.157994\pi$$
0.879326 + 0.476221i $$0.157994\pi$$
$$942$$ 2.11780e30 0.125415
$$943$$ −1.30309e31 −0.763138
$$944$$ 1.35235e31 0.783220
$$945$$ 0 0
$$946$$ 6.81377e30 0.385949
$$947$$ 8.23900e29 0.0461530 0.0230765 0.999734i $$-0.492654\pi$$
0.0230765 + 0.999734i $$0.492654\pi$$
$$948$$ 6.71721e30 0.372135
$$949$$ 1.63665e31 0.896728
$$950$$ 0 0
$$951$$ 1.21911e31 0.653350
$$952$$ 3.48158e31 1.84539
$$953$$ 3.34266e31 1.75233 0.876167 0.482008i $$-0.160093\pi$$
0.876167 + 0.482008i $$0.160093\pi$$
$$954$$ 4.36177e30 0.226154
$$955$$ 0 0
$$956$$ 1.46431e30 0.0742720
$$957$$ −2.36747e29 −0.0118771
$$958$$ −2.62967e30 −0.130486
$$959$$ 6.42466e30 0.315322
$$960$$ 0 0
$$961$$ 5.87607e31 2.82157
$$962$$ 1.03855e31 0.493274
$$963$$ −4.38577e30 −0.206049
$$964$$ −3.56802e30 −0.165813
$$965$$ 0 0
$$966$$ 9.19456e30 0.418091
$$967$$ −2.28891e31 −1.02956 −0.514779 0.857323i $$-0.672126\pi$$
−0.514779 + 0.857323i $$0.672126\pi$$
$$968$$ −2.92801e30 −0.130281
$$969$$ −2.17033e31 −0.955268
$$970$$ 0 0
$$971$$ −4.60425e31 −1.98316 −0.991578 0.129508i $$-0.958660\pi$$
−0.991578 + 0.129508i $$0.958660\pi$$
$$972$$ 1.29309e30 0.0550976
$$973$$ −3.91632e30 −0.165080
$$974$$ −3.42925e30 −0.142998
$$975$$ 0 0
$$976$$ 3.28412e30 0.134028
$$977$$ −7.20727e30 −0.290990 −0.145495 0.989359i $$-0.546477\pi$$
−0.145495 + 0.989359i $$0.546477\pi$$
$$978$$ 2.82376e29 0.0112790
$$979$$ −1.06491e31 −0.420817
$$980$$ 0 0
$$981$$ −5.81333e30 −0.224854
$$982$$ −1.99850e30 −0.0764776
$$983$$ −1.58951e31 −0.601800 −0.300900 0.953656i $$-0.597287\pi$$
−0.300900 + 0.953656i $$0.597287\pi$$
$$984$$ −7.28415e30 −0.272854
$$985$$ 0 0
$$986$$ −3.61206e29 −0.0132448
$$987$$ 1.20764e31 0.438134
$$988$$ 2.23672e31 0.802905
$$989$$ −3.61657e31 −1.28450
$$990$$ 0 0
$$991$$ 4.62622e31 1.60862 0.804309 0.594211i $$-0.202536\pi$$
0.804309 + 0.594211i $$0.202536\pi$$
$$992$$ −5.24095e31 −1.80317
$$993$$ 6.44905e30 0.219548
$$994$$ −7.67580e30 −0.258563
$$995$$ 0 0
$$996$$ 1.54390e31 0.509210
$$997$$ 1.13237e31 0.369562 0.184781 0.982780i $$-0.440842\pi$$
0.184781 + 0.982780i $$0.440842\pi$$
$$998$$ 1.08848e31 0.351519
$$999$$ 9.05451e30 0.289352
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 75.22.a.b.1.1 1
5.2 odd 4 75.22.b.c.49.1 2
5.3 odd 4 75.22.b.c.49.2 2
5.4 even 2 15.22.a.a.1.1 1
15.14 odd 2 45.22.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.a.a.1.1 1 5.4 even 2
45.22.a.a.1.1 1 15.14 odd 2
75.22.a.b.1.1 1 1.1 even 1 trivial
75.22.b.c.49.1 2 5.2 odd 4
75.22.b.c.49.2 2 5.3 odd 4