Properties

 Label 4840.2.a.w.1.3 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.4752.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 3x^{2} + 4x + 1$$ x^4 - 2*x^3 - 3*x^2 + 4*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$-1.49551$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.0947876 q^{3} -1.00000 q^{5} +0.826838 q^{7} -2.99102 q^{9} +O(q^{10})$$ $$q+0.0947876 q^{3} -1.00000 q^{5} +0.826838 q^{7} -2.99102 q^{9} +2.99102 q^{13} -0.0947876 q^{15} -0.662661 q^{19} +0.0783740 q^{21} +0.473086 q^{23} +1.00000 q^{25} -0.567874 q^{27} -4.00000 q^{29} -4.64469 q^{31} -0.826838 q^{35} +5.98203 q^{37} +0.283511 q^{39} +0.394712 q^{41} +6.96562 q^{43} +2.99102 q^{45} -3.88724 q^{47} -6.31634 q^{49} -0.801440 q^{53} -0.0628121 q^{57} -1.71649 q^{59} -2.58429 q^{61} -2.47309 q^{63} -2.99102 q^{65} +5.69767 q^{67} +0.0448427 q^{69} -5.59086 q^{71} -8.00000 q^{73} +0.0947876 q^{75} -11.6357 q^{79} +8.91922 q^{81} -5.90606 q^{83} -0.379150 q^{87} +4.32532 q^{89} +2.47309 q^{91} -0.440259 q^{93} +0.662661 q^{95} +6.50591 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9 $$4 q - 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9} - 4 q^{13} + 2 q^{15} + 12 q^{21} + 4 q^{23} + 4 q^{25} - 2 q^{27} - 16 q^{29} + 16 q^{31} + 6 q^{35} - 8 q^{37} + 8 q^{39} - 8 q^{41} + 10 q^{43} - 4 q^{45} + 14 q^{47} - 4 q^{49} + 8 q^{53} - 12 q^{57} + 4 q^{61} - 12 q^{63} + 4 q^{65} - 2 q^{67} - 20 q^{69} + 8 q^{71} - 32 q^{73} - 2 q^{75} + 4 q^{79} - 8 q^{81} - 12 q^{83} + 8 q^{87} + 12 q^{89} + 12 q^{91} - 32 q^{93}+O(q^{100})$$ 4 * q - 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9 - 4 * q^13 + 2 * q^15 + 12 * q^21 + 4 * q^23 + 4 * q^25 - 2 * q^27 - 16 * q^29 + 16 * q^31 + 6 * q^35 - 8 * q^37 + 8 * q^39 - 8 * q^41 + 10 * q^43 - 4 * q^45 + 14 * q^47 - 4 * q^49 + 8 * q^53 - 12 * q^57 + 4 * q^61 - 12 * q^63 + 4 * q^65 - 2 * q^67 - 20 * q^69 + 8 * q^71 - 32 * q^73 - 2 * q^75 + 4 * q^79 - 8 * q^81 - 12 * q^83 + 8 * q^87 + 12 * q^89 + 12 * q^91 - 32 * q^93

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0.0947876 0.0547256 0.0273628 0.999626i $$-0.491289\pi$$
0.0273628 + 0.999626i $$0.491289\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0.826838 0.312516 0.156258 0.987716i $$-0.450057\pi$$
0.156258 + 0.987716i $$0.450057\pi$$
$$8$$ 0 0
$$9$$ −2.99102 −0.997005
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 2.99102 0.829558 0.414779 0.909922i $$-0.363859\pi$$
0.414779 + 0.909922i $$0.363859\pi$$
$$14$$ 0 0
$$15$$ −0.0947876 −0.0244740
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −0.662661 −0.152025 −0.0760125 0.997107i $$-0.524219\pi$$
−0.0760125 + 0.997107i $$0.524219\pi$$
$$20$$ 0 0
$$21$$ 0.0783740 0.0171026
$$22$$ 0 0
$$23$$ 0.473086 0.0986453 0.0493227 0.998783i $$-0.484294\pi$$
0.0493227 + 0.998783i $$0.484294\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −0.567874 −0.109287
$$28$$ 0 0
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ −4.64469 −0.834211 −0.417106 0.908858i $$-0.636956\pi$$
−0.417106 + 0.908858i $$0.636956\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −0.826838 −0.139761
$$36$$ 0 0
$$37$$ 5.98203 0.983440 0.491720 0.870753i $$-0.336368\pi$$
0.491720 + 0.870753i $$0.336368\pi$$
$$38$$ 0 0
$$39$$ 0.283511 0.0453981
$$40$$ 0 0
$$41$$ 0.394712 0.0616437 0.0308219 0.999525i $$-0.490188\pi$$
0.0308219 + 0.999525i $$0.490188\pi$$
$$42$$ 0 0
$$43$$ 6.96562 1.06225 0.531123 0.847295i $$-0.321770\pi$$
0.531123 + 0.847295i $$0.321770\pi$$
$$44$$ 0 0
$$45$$ 2.99102 0.445874
$$46$$ 0 0
$$47$$ −3.88724 −0.567013 −0.283506 0.958970i $$-0.591498\pi$$
−0.283506 + 0.958970i $$0.591498\pi$$
$$48$$ 0 0
$$49$$ −6.31634 −0.902334
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −0.801440 −0.110086 −0.0550431 0.998484i $$-0.517530\pi$$
−0.0550431 + 0.998484i $$0.517530\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.0628121 −0.00831966
$$58$$ 0 0
$$59$$ −1.71649 −0.223468 −0.111734 0.993738i $$-0.535640\pi$$
−0.111734 + 0.993738i $$0.535640\pi$$
$$60$$ 0 0
$$61$$ −2.58429 −0.330884 −0.165442 0.986220i $$-0.552905\pi$$
−0.165442 + 0.986220i $$0.552905\pi$$
$$62$$ 0 0
$$63$$ −2.47309 −0.311580
$$64$$ 0 0
$$65$$ −2.99102 −0.370990
$$66$$ 0 0
$$67$$ 5.69767 0.696081 0.348040 0.937480i $$-0.386847\pi$$
0.348040 + 0.937480i $$0.386847\pi$$
$$68$$ 0 0
$$69$$ 0.0448427 0.00539843
$$70$$ 0 0
$$71$$ −5.59086 −0.663514 −0.331757 0.943365i $$-0.607641\pi$$
−0.331757 + 0.943365i $$0.607641\pi$$
$$72$$ 0 0
$$73$$ −8.00000 −0.936329 −0.468165 0.883641i $$-0.655085\pi$$
−0.468165 + 0.883641i $$0.655085\pi$$
$$74$$ 0 0
$$75$$ 0.0947876 0.0109451
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −11.6357 −1.30912 −0.654560 0.756010i $$-0.727146\pi$$
−0.654560 + 0.756010i $$0.727146\pi$$
$$80$$ 0 0
$$81$$ 8.91922 0.991024
$$82$$ 0 0
$$83$$ −5.90606 −0.648275 −0.324137 0.946010i $$-0.605074\pi$$
−0.324137 + 0.946010i $$0.605074\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −0.379150 −0.0406492
$$88$$ 0 0
$$89$$ 4.32532 0.458483 0.229242 0.973370i $$-0.426375\pi$$
0.229242 + 0.973370i $$0.426375\pi$$
$$90$$ 0 0
$$91$$ 2.47309 0.259250
$$92$$ 0 0
$$93$$ −0.440259 −0.0456527
$$94$$ 0 0
$$95$$ 0.662661 0.0679876
$$96$$ 0 0
$$97$$ 6.50591 0.660575 0.330288 0.943880i $$-0.392854\pi$$
0.330288 + 0.943880i $$0.392854\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.54248 0.352489 0.176245 0.984346i $$-0.443605\pi$$
0.176245 + 0.984346i $$0.443605\pi$$
$$102$$ 0 0
$$103$$ 16.4371 1.61960 0.809800 0.586706i $$-0.199576\pi$$
0.809800 + 0.586706i $$0.199576\pi$$
$$104$$ 0 0
$$105$$ −0.0783740 −0.00764852
$$106$$ 0 0
$$107$$ −17.4387 −1.68586 −0.842932 0.538021i $$-0.819172\pi$$
−0.842932 + 0.538021i $$0.819172\pi$$
$$108$$ 0 0
$$109$$ −3.24100 −0.310431 −0.155216 0.987881i $$-0.549607\pi$$
−0.155216 + 0.987881i $$0.549607\pi$$
$$110$$ 0 0
$$111$$ 0.567022 0.0538194
$$112$$ 0 0
$$113$$ −12.3612 −1.16284 −0.581421 0.813603i $$-0.697503\pi$$
−0.581421 + 0.813603i $$0.697503\pi$$
$$114$$ 0 0
$$115$$ −0.473086 −0.0441155
$$116$$ 0 0
$$117$$ −8.94617 −0.827074
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 0.0374138 0.00337349
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 2.73290 0.242506 0.121253 0.992622i $$-0.461309\pi$$
0.121253 + 0.992622i $$0.461309\pi$$
$$128$$ 0 0
$$129$$ 0.660254 0.0581321
$$130$$ 0 0
$$131$$ 9.57290 0.836388 0.418194 0.908358i $$-0.362663\pi$$
0.418194 + 0.908358i $$0.362663\pi$$
$$132$$ 0 0
$$133$$ −0.547914 −0.0475102
$$134$$ 0 0
$$135$$ 0.567874 0.0488748
$$136$$ 0 0
$$137$$ −6.12676 −0.523445 −0.261722 0.965143i $$-0.584290\pi$$
−0.261722 + 0.965143i $$0.584290\pi$$
$$138$$ 0 0
$$139$$ −20.9551 −1.77739 −0.888693 0.458502i $$-0.848386\pi$$
−0.888693 + 0.458502i $$0.848386\pi$$
$$140$$ 0 0
$$141$$ −0.368462 −0.0310301
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ 0 0
$$147$$ −0.598710 −0.0493808
$$148$$ 0 0
$$149$$ −11.9096 −0.975673 −0.487837 0.872935i $$-0.662214\pi$$
−0.487837 + 0.872935i $$0.662214\pi$$
$$150$$ 0 0
$$151$$ −16.2984 −1.32634 −0.663171 0.748468i $$-0.730790\pi$$
−0.663171 + 0.748468i $$0.730790\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.64469 0.373071
$$156$$ 0 0
$$157$$ −11.5849 −0.924577 −0.462288 0.886730i $$-0.652971\pi$$
−0.462288 + 0.886730i $$0.652971\pi$$
$$158$$ 0 0
$$159$$ −0.0759666 −0.00602454
$$160$$ 0 0
$$161$$ 0.391166 0.0308282
$$162$$ 0 0
$$163$$ −1.63967 −0.128429 −0.0642145 0.997936i $$-0.520454\pi$$
−0.0642145 + 0.997936i $$0.520454\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −11.7550 −0.909632 −0.454816 0.890585i $$-0.650295\pi$$
−0.454816 + 0.890585i $$0.650295\pi$$
$$168$$ 0 0
$$169$$ −4.05383 −0.311833
$$170$$ 0 0
$$171$$ 1.98203 0.151570
$$172$$ 0 0
$$173$$ −14.4240 −1.09664 −0.548318 0.836270i $$-0.684732\pi$$
−0.548318 + 0.836270i $$0.684732\pi$$
$$174$$ 0 0
$$175$$ 0.826838 0.0625031
$$176$$ 0 0
$$177$$ −0.162702 −0.0122294
$$178$$ 0 0
$$179$$ 6.86425 0.513058 0.256529 0.966536i $$-0.417421\pi$$
0.256529 + 0.966536i $$0.417421\pi$$
$$180$$ 0 0
$$181$$ −23.9742 −1.78199 −0.890994 0.454016i $$-0.849991\pi$$
−0.890994 + 0.454016i $$0.849991\pi$$
$$182$$ 0 0
$$183$$ −0.244958 −0.0181078
$$184$$ 0 0
$$185$$ −5.98203 −0.439808
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −0.469540 −0.0341540
$$190$$ 0 0
$$191$$ 12.4551 0.901221 0.450610 0.892721i $$-0.351206\pi$$
0.450610 + 0.892721i $$0.351206\pi$$
$$192$$ 0 0
$$193$$ −5.19261 −0.373772 −0.186886 0.982382i $$-0.559839\pi$$
−0.186886 + 0.982382i $$0.559839\pi$$
$$194$$ 0 0
$$195$$ −0.283511 −0.0203027
$$196$$ 0 0
$$197$$ −8.68783 −0.618982 −0.309491 0.950902i $$-0.600159\pi$$
−0.309491 + 0.950902i $$0.600159\pi$$
$$198$$ 0 0
$$199$$ 13.5149 0.958046 0.479023 0.877802i $$-0.340991\pi$$
0.479023 + 0.877802i $$0.340991\pi$$
$$200$$ 0 0
$$201$$ 0.540068 0.0380935
$$202$$ 0 0
$$203$$ −3.30735 −0.232131
$$204$$ 0 0
$$205$$ −0.394712 −0.0275679
$$206$$ 0 0
$$207$$ −1.41501 −0.0983499
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 18.0028 1.23937 0.619683 0.784852i $$-0.287261\pi$$
0.619683 + 0.784852i $$0.287261\pi$$
$$212$$ 0 0
$$213$$ −0.529945 −0.0363112
$$214$$ 0 0
$$215$$ −6.96562 −0.475051
$$216$$ 0 0
$$217$$ −3.84041 −0.260704
$$218$$ 0 0
$$219$$ −0.758301 −0.0512412
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 14.2784 0.956153 0.478076 0.878318i $$-0.341334\pi$$
0.478076 + 0.878318i $$0.341334\pi$$
$$224$$ 0 0
$$225$$ −2.99102 −0.199401
$$226$$ 0 0
$$227$$ −13.3627 −0.886916 −0.443458 0.896295i $$-0.646248\pi$$
−0.443458 + 0.896295i $$0.646248\pi$$
$$228$$ 0 0
$$229$$ 1.46713 0.0969508 0.0484754 0.998824i $$-0.484564\pi$$
0.0484754 + 0.998824i $$0.484564\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 12.0269 0.787907 0.393953 0.919130i $$-0.371107\pi$$
0.393953 + 0.919130i $$0.371107\pi$$
$$234$$ 0 0
$$235$$ 3.88724 0.253576
$$236$$ 0 0
$$237$$ −1.10292 −0.0716424
$$238$$ 0 0
$$239$$ −27.0238 −1.74803 −0.874014 0.485902i $$-0.838491\pi$$
−0.874014 + 0.485902i $$0.838491\pi$$
$$240$$ 0 0
$$241$$ −2.94377 −0.189625 −0.0948123 0.995495i $$-0.530225\pi$$
−0.0948123 + 0.995495i $$0.530225\pi$$
$$242$$ 0 0
$$243$$ 2.54905 0.163522
$$244$$ 0 0
$$245$$ 6.31634 0.403536
$$246$$ 0 0
$$247$$ −1.98203 −0.126114
$$248$$ 0 0
$$249$$ −0.559822 −0.0354773
$$250$$ 0 0
$$251$$ −24.8982 −1.57156 −0.785781 0.618505i $$-0.787739\pi$$
−0.785781 + 0.618505i $$0.787739\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 13.4761 0.840617 0.420309 0.907381i $$-0.361922\pi$$
0.420309 + 0.907381i $$0.361922\pi$$
$$258$$ 0 0
$$259$$ 4.94617 0.307340
$$260$$ 0 0
$$261$$ 11.9641 0.740557
$$262$$ 0 0
$$263$$ 4.83427 0.298094 0.149047 0.988830i $$-0.452379\pi$$
0.149047 + 0.988830i $$0.452379\pi$$
$$264$$ 0 0
$$265$$ 0.801440 0.0492321
$$266$$ 0 0
$$267$$ 0.409987 0.0250908
$$268$$ 0 0
$$269$$ −18.8373 −1.14853 −0.574265 0.818669i $$-0.694712\pi$$
−0.574265 + 0.818669i $$0.694712\pi$$
$$270$$ 0 0
$$271$$ 16.9102 1.02722 0.513612 0.858023i $$-0.328307\pi$$
0.513612 + 0.858023i $$0.328307\pi$$
$$272$$ 0 0
$$273$$ 0.234418 0.0141876
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3.87919 −0.233078 −0.116539 0.993186i $$-0.537180\pi$$
−0.116539 + 0.993186i $$0.537180\pi$$
$$278$$ 0 0
$$279$$ 13.8923 0.831713
$$280$$ 0 0
$$281$$ −17.6029 −1.05010 −0.525050 0.851071i $$-0.675953\pi$$
−0.525050 + 0.851071i $$0.675953\pi$$
$$282$$ 0 0
$$283$$ −0.781996 −0.0464848 −0.0232424 0.999730i $$-0.507399\pi$$
−0.0232424 + 0.999730i $$0.507399\pi$$
$$284$$ 0 0
$$285$$ 0.0628121 0.00372067
$$286$$ 0 0
$$287$$ 0.326363 0.0192646
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 0.616680 0.0361504
$$292$$ 0 0
$$293$$ −8.18362 −0.478092 −0.239046 0.971008i $$-0.576835\pi$$
−0.239046 + 0.971008i $$0.576835\pi$$
$$294$$ 0 0
$$295$$ 1.71649 0.0999378
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1.41501 0.0818320
$$300$$ 0 0
$$301$$ 5.75944 0.331969
$$302$$ 0 0
$$303$$ 0.335783 0.0192902
$$304$$ 0 0
$$305$$ 2.58429 0.147976
$$306$$ 0 0
$$307$$ 27.4490 1.56660 0.783298 0.621647i $$-0.213536\pi$$
0.783298 + 0.621647i $$0.213536\pi$$
$$308$$ 0 0
$$309$$ 1.55804 0.0886337
$$310$$ 0 0
$$311$$ 26.6907 1.51349 0.756745 0.653711i $$-0.226789\pi$$
0.756745 + 0.653711i $$0.226789\pi$$
$$312$$ 0 0
$$313$$ −19.7296 −1.11519 −0.557593 0.830115i $$-0.688275\pi$$
−0.557593 + 0.830115i $$0.688275\pi$$
$$314$$ 0 0
$$315$$ 2.47309 0.139343
$$316$$ 0 0
$$317$$ 9.24170 0.519066 0.259533 0.965734i $$-0.416431\pi$$
0.259533 + 0.965734i $$0.416431\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −1.65297 −0.0922599
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 2.99102 0.165912
$$326$$ 0 0
$$327$$ −0.307206 −0.0169885
$$328$$ 0 0
$$329$$ −3.21412 −0.177200
$$330$$ 0 0
$$331$$ 19.3833 1.06540 0.532702 0.846303i $$-0.321177\pi$$
0.532702 + 0.846303i $$0.321177\pi$$
$$332$$ 0 0
$$333$$ −17.8923 −0.980494
$$334$$ 0 0
$$335$$ −5.69767 −0.311297
$$336$$ 0 0
$$337$$ −29.9192 −1.62980 −0.814902 0.579599i $$-0.803209\pi$$
−0.814902 + 0.579599i $$0.803209\pi$$
$$338$$ 0 0
$$339$$ −1.17169 −0.0636373
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −11.0105 −0.594509
$$344$$ 0 0
$$345$$ −0.0448427 −0.00241425
$$346$$ 0 0
$$347$$ −18.7892 −1.00866 −0.504328 0.863512i $$-0.668260\pi$$
−0.504328 + 0.863512i $$0.668260\pi$$
$$348$$ 0 0
$$349$$ −2.74033 −0.146687 −0.0733433 0.997307i $$-0.523367\pi$$
−0.0733433 + 0.997307i $$0.523367\pi$$
$$350$$ 0 0
$$351$$ −1.69852 −0.0906603
$$352$$ 0 0
$$353$$ 0.631538 0.0336134 0.0168067 0.999859i $$-0.494650\pi$$
0.0168067 + 0.999859i $$0.494650\pi$$
$$354$$ 0 0
$$355$$ 5.59086 0.296732
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −0.0299849 −0.00158254 −0.000791272 1.00000i $$-0.500252\pi$$
−0.000791272 1.00000i $$0.500252\pi$$
$$360$$ 0 0
$$361$$ −18.5609 −0.976888
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 8.00000 0.418739
$$366$$ 0 0
$$367$$ −5.90521 −0.308250 −0.154125 0.988051i $$-0.549256\pi$$
−0.154125 + 0.988051i $$0.549256\pi$$
$$368$$ 0 0
$$369$$ −1.18059 −0.0614591
$$370$$ 0 0
$$371$$ −0.662661 −0.0344037
$$372$$ 0 0
$$373$$ −5.38332 −0.278738 −0.139369 0.990241i $$-0.544507\pi$$
−0.139369 + 0.990241i $$0.544507\pi$$
$$374$$ 0 0
$$375$$ −0.0947876 −0.00489481
$$376$$ 0 0
$$377$$ −11.9641 −0.616181
$$378$$ 0 0
$$379$$ 17.8940 0.919156 0.459578 0.888138i $$-0.348001\pi$$
0.459578 + 0.888138i $$0.348001\pi$$
$$380$$ 0 0
$$381$$ 0.259045 0.0132713
$$382$$ 0 0
$$383$$ 11.8045 0.603180 0.301590 0.953438i $$-0.402483\pi$$
0.301590 + 0.953438i $$0.402483\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −20.8343 −1.05907
$$388$$ 0 0
$$389$$ 12.5418 0.635893 0.317947 0.948109i $$-0.397007\pi$$
0.317947 + 0.948109i $$0.397007\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0.907392 0.0457719
$$394$$ 0 0
$$395$$ 11.6357 0.585456
$$396$$ 0 0
$$397$$ −18.1077 −0.908797 −0.454399 0.890798i $$-0.650146\pi$$
−0.454399 + 0.890798i $$0.650146\pi$$
$$398$$ 0 0
$$399$$ −0.0519354 −0.00260002
$$400$$ 0 0
$$401$$ −24.2058 −1.20878 −0.604389 0.796689i $$-0.706583\pi$$
−0.604389 + 0.796689i $$0.706583\pi$$
$$402$$ 0 0
$$403$$ −13.8923 −0.692027
$$404$$ 0 0
$$405$$ −8.91922 −0.443200
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −4.82902 −0.238780 −0.119390 0.992847i $$-0.538094\pi$$
−0.119390 + 0.992847i $$0.538094\pi$$
$$410$$ 0 0
$$411$$ −0.580741 −0.0286458
$$412$$ 0 0
$$413$$ −1.41926 −0.0698372
$$414$$ 0 0
$$415$$ 5.90606 0.289917
$$416$$ 0 0
$$417$$ −1.98628 −0.0972686
$$418$$ 0 0
$$419$$ −16.7327 −0.817445 −0.408722 0.912659i $$-0.634026\pi$$
−0.408722 + 0.912659i $$0.634026\pi$$
$$420$$ 0 0
$$421$$ 14.1447 0.689368 0.344684 0.938719i $$-0.387986\pi$$
0.344684 + 0.938719i $$0.387986\pi$$
$$422$$ 0 0
$$423$$ 11.6268 0.565315
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −2.13679 −0.103406
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 11.7973 0.568255 0.284127 0.958787i $$-0.408296\pi$$
0.284127 + 0.958787i $$0.408296\pi$$
$$432$$ 0 0
$$433$$ 0.717816 0.0344961 0.0172480 0.999851i $$-0.494510\pi$$
0.0172480 + 0.999851i $$0.494510\pi$$
$$434$$ 0 0
$$435$$ 0.379150 0.0181789
$$436$$ 0 0
$$437$$ −0.313496 −0.0149966
$$438$$ 0 0
$$439$$ 13.0359 0.622168 0.311084 0.950382i $$-0.399308\pi$$
0.311084 + 0.950382i $$0.399308\pi$$
$$440$$ 0 0
$$441$$ 18.8923 0.899632
$$442$$ 0 0
$$443$$ 12.3783 0.588111 0.294055 0.955788i $$-0.404995\pi$$
0.294055 + 0.955788i $$0.404995\pi$$
$$444$$ 0 0
$$445$$ −4.32532 −0.205040
$$446$$ 0 0
$$447$$ −1.12888 −0.0533943
$$448$$ 0 0
$$449$$ 23.1788 1.09388 0.546938 0.837173i $$-0.315793\pi$$
0.546938 + 0.837173i $$0.315793\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −1.54488 −0.0725849
$$454$$ 0 0
$$455$$ −2.47309 −0.115940
$$456$$ 0 0
$$457$$ 22.8102 1.06702 0.533509 0.845794i $$-0.320873\pi$$
0.533509 + 0.845794i $$0.320873\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −16.4527 −0.766279 −0.383140 0.923690i $$-0.625157\pi$$
−0.383140 + 0.923690i $$0.625157\pi$$
$$462$$ 0 0
$$463$$ 36.1708 1.68100 0.840499 0.541813i $$-0.182262\pi$$
0.840499 + 0.541813i $$0.182262\pi$$
$$464$$ 0 0
$$465$$ 0.440259 0.0204165
$$466$$ 0 0
$$467$$ 23.9332 1.10750 0.553749 0.832684i $$-0.313197\pi$$
0.553749 + 0.832684i $$0.313197\pi$$
$$468$$ 0 0
$$469$$ 4.71105 0.217536
$$470$$ 0 0
$$471$$ −1.09811 −0.0505980
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −0.662661 −0.0304050
$$476$$ 0 0
$$477$$ 2.39712 0.109757
$$478$$ 0 0
$$479$$ 15.5101 0.708674 0.354337 0.935118i $$-0.384707\pi$$
0.354337 + 0.935118i $$0.384707\pi$$
$$480$$ 0 0
$$481$$ 17.8923 0.815821
$$482$$ 0 0
$$483$$ 0.0370777 0.00168709
$$484$$ 0 0
$$485$$ −6.50591 −0.295418
$$486$$ 0 0
$$487$$ −41.3114 −1.87200 −0.936000 0.352000i $$-0.885502\pi$$
−0.936000 + 0.352000i $$0.885502\pi$$
$$488$$ 0 0
$$489$$ −0.155420 −0.00702835
$$490$$ 0 0
$$491$$ 3.79727 0.171368 0.0856842 0.996322i $$-0.472692\pi$$
0.0856842 + 0.996322i $$0.472692\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −4.62274 −0.207358
$$498$$ 0 0
$$499$$ 3.62255 0.162168 0.0810839 0.996707i $$-0.474162\pi$$
0.0810839 + 0.996707i $$0.474162\pi$$
$$500$$ 0 0
$$501$$ −1.11423 −0.0497802
$$502$$ 0 0
$$503$$ −16.6641 −0.743017 −0.371509 0.928430i $$-0.621159\pi$$
−0.371509 + 0.928430i $$0.621159\pi$$
$$504$$ 0 0
$$505$$ −3.54248 −0.157638
$$506$$ 0 0
$$507$$ −0.384253 −0.0170653
$$508$$ 0 0
$$509$$ −34.4969 −1.52905 −0.764525 0.644594i $$-0.777026\pi$$
−0.764525 + 0.644594i $$0.777026\pi$$
$$510$$ 0 0
$$511$$ −6.61471 −0.292617
$$512$$ 0 0
$$513$$ 0.376308 0.0166144
$$514$$ 0 0
$$515$$ −16.4371 −0.724307
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −1.36722 −0.0600141
$$520$$ 0 0
$$521$$ −9.44480 −0.413784 −0.206892 0.978364i $$-0.566335\pi$$
−0.206892 + 0.978364i $$0.566335\pi$$
$$522$$ 0 0
$$523$$ 8.38946 0.366846 0.183423 0.983034i $$-0.441282\pi$$
0.183423 + 0.983034i $$0.441282\pi$$
$$524$$ 0 0
$$525$$ 0.0783740 0.00342052
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −22.7762 −0.990269
$$530$$ 0 0
$$531$$ 5.13404 0.222799
$$532$$ 0 0
$$533$$ 1.18059 0.0511371
$$534$$ 0 0
$$535$$ 17.4387 0.753941
$$536$$ 0 0
$$537$$ 0.650646 0.0280774
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 25.8725 1.11235 0.556174 0.831066i $$-0.312269\pi$$
0.556174 + 0.831066i $$0.312269\pi$$
$$542$$ 0 0
$$543$$ −2.27246 −0.0975204
$$544$$ 0 0
$$545$$ 3.24100 0.138829
$$546$$ 0 0
$$547$$ −26.7804 −1.14505 −0.572523 0.819889i $$-0.694035\pi$$
−0.572523 + 0.819889i $$0.694035\pi$$
$$548$$ 0 0
$$549$$ 7.72964 0.329893
$$550$$ 0 0
$$551$$ 2.65065 0.112921
$$552$$ 0 0
$$553$$ −9.62085 −0.409120
$$554$$ 0 0
$$555$$ −0.567022 −0.0240688
$$556$$ 0 0
$$557$$ −35.5490 −1.50626 −0.753129 0.657873i $$-0.771456\pi$$
−0.753129 + 0.657873i $$0.771456\pi$$
$$558$$ 0 0
$$559$$ 20.8343 0.881196
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −25.1893 −1.06160 −0.530801 0.847497i $$-0.678109\pi$$
−0.530801 + 0.847497i $$0.678109\pi$$
$$564$$ 0 0
$$565$$ 12.3612 0.520039
$$566$$ 0 0
$$567$$ 7.37475 0.309710
$$568$$ 0 0
$$569$$ −7.98444 −0.334725 −0.167363 0.985895i $$-0.553525\pi$$
−0.167363 + 0.985895i $$0.553525\pi$$
$$570$$ 0 0
$$571$$ 19.8892 0.832339 0.416169 0.909287i $$-0.363372\pi$$
0.416169 + 0.909287i $$0.363372\pi$$
$$572$$ 0 0
$$573$$ 1.18059 0.0493199
$$574$$ 0 0
$$575$$ 0.473086 0.0197291
$$576$$ 0 0
$$577$$ 16.3444 0.680424 0.340212 0.940349i $$-0.389501\pi$$
0.340212 + 0.940349i $$0.389501\pi$$
$$578$$ 0 0
$$579$$ −0.492195 −0.0204549
$$580$$ 0 0
$$581$$ −4.88336 −0.202596
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 8.94617 0.369879
$$586$$ 0 0
$$587$$ −37.4781 −1.54689 −0.773444 0.633865i $$-0.781467\pi$$
−0.773444 + 0.633865i $$0.781467\pi$$
$$588$$ 0 0
$$589$$ 3.07786 0.126821
$$590$$ 0 0
$$591$$ −0.823499 −0.0338742
$$592$$ 0 0
$$593$$ −28.0028 −1.14994 −0.574969 0.818175i $$-0.694986\pi$$
−0.574969 + 0.818175i $$0.694986\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1.28104 0.0524297
$$598$$ 0 0
$$599$$ 27.8522 1.13801 0.569004 0.822335i $$-0.307329\pi$$
0.569004 + 0.822335i $$0.307329\pi$$
$$600$$ 0 0
$$601$$ 26.9282 1.09842 0.549212 0.835683i $$-0.314928\pi$$
0.549212 + 0.835683i $$0.314928\pi$$
$$602$$ 0 0
$$603$$ −17.0418 −0.693996
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −15.3833 −0.624390 −0.312195 0.950018i $$-0.601064\pi$$
−0.312195 + 0.950018i $$0.601064\pi$$
$$608$$ 0 0
$$609$$ −0.313496 −0.0127035
$$610$$ 0 0
$$611$$ −11.6268 −0.470370
$$612$$ 0 0
$$613$$ 29.0967 1.17520 0.587602 0.809150i $$-0.300072\pi$$
0.587602 + 0.809150i $$0.300072\pi$$
$$614$$ 0 0
$$615$$ −0.0374138 −0.00150867
$$616$$ 0 0
$$617$$ −17.9832 −0.723975 −0.361988 0.932183i $$-0.617902\pi$$
−0.361988 + 0.932183i $$0.617902\pi$$
$$618$$ 0 0
$$619$$ −1.85641 −0.0746153 −0.0373076 0.999304i $$-0.511878\pi$$
−0.0373076 + 0.999304i $$0.511878\pi$$
$$620$$ 0 0
$$621$$ −0.268653 −0.0107807
$$622$$ 0 0
$$623$$ 3.57634 0.143283
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −2.61471 −0.104090 −0.0520449 0.998645i $$-0.516574\pi$$
−0.0520449 + 0.998645i $$0.516574\pi$$
$$632$$ 0 0
$$633$$ 1.70645 0.0678251
$$634$$ 0 0
$$635$$ −2.73290 −0.108452
$$636$$ 0 0
$$637$$ −18.8923 −0.748539
$$638$$ 0 0
$$639$$ 16.7224 0.661526
$$640$$ 0 0
$$641$$ 31.3157 1.23690 0.618448 0.785826i $$-0.287762\pi$$
0.618448 + 0.785826i $$0.287762\pi$$
$$642$$ 0 0
$$643$$ 2.67383 0.105445 0.0527227 0.998609i $$-0.483210\pi$$
0.0527227 + 0.998609i $$0.483210\pi$$
$$644$$ 0 0
$$645$$ −0.660254 −0.0259975
$$646$$ 0 0
$$647$$ 41.4002 1.62761 0.813804 0.581139i $$-0.197393\pi$$
0.813804 + 0.581139i $$0.197393\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −0.364023 −0.0142672
$$652$$ 0 0
$$653$$ −26.2535 −1.02738 −0.513690 0.857976i $$-0.671722\pi$$
−0.513690 + 0.857976i $$0.671722\pi$$
$$654$$ 0 0
$$655$$ −9.57290 −0.374044
$$656$$ 0 0
$$657$$ 23.9281 0.933525
$$658$$ 0 0
$$659$$ −7.50781 −0.292463 −0.146231 0.989250i $$-0.546714\pi$$
−0.146231 + 0.989250i $$0.546714\pi$$
$$660$$ 0 0
$$661$$ 10.4161 0.405141 0.202571 0.979268i $$-0.435070\pi$$
0.202571 + 0.979268i $$0.435070\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0.547914 0.0212472
$$666$$ 0 0
$$667$$ −1.89235 −0.0732719
$$668$$ 0 0
$$669$$ 1.35342 0.0523261
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −21.8881 −0.843724 −0.421862 0.906660i $$-0.638623\pi$$
−0.421862 + 0.906660i $$0.638623\pi$$
$$674$$ 0 0
$$675$$ −0.567874 −0.0218575
$$676$$ 0 0
$$677$$ 18.7666 0.721261 0.360630 0.932709i $$-0.382562\pi$$
0.360630 + 0.932709i $$0.382562\pi$$
$$678$$ 0 0
$$679$$ 5.37934 0.206440
$$680$$ 0 0
$$681$$ −1.26662 −0.0485370
$$682$$ 0 0
$$683$$ −44.8734 −1.71703 −0.858517 0.512785i $$-0.828614\pi$$
−0.858517 + 0.512785i $$0.828614\pi$$
$$684$$ 0 0
$$685$$ 6.12676 0.234092
$$686$$ 0 0
$$687$$ 0.139066 0.00530570
$$688$$ 0 0
$$689$$ −2.39712 −0.0913230
$$690$$ 0 0
$$691$$ −36.2936 −1.38067 −0.690336 0.723489i $$-0.742537\pi$$
−0.690336 + 0.723489i $$0.742537\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 20.9551 0.794871
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 1.14000 0.0431187
$$700$$ 0 0
$$701$$ 23.5490 0.889432 0.444716 0.895672i $$-0.353305\pi$$
0.444716 + 0.895672i $$0.353305\pi$$
$$702$$ 0 0
$$703$$ −3.96406 −0.149507
$$704$$ 0 0
$$705$$ 0.368462 0.0138771
$$706$$ 0 0
$$707$$ 2.92905 0.110158
$$708$$ 0 0
$$709$$ 10.9024 0.409448 0.204724 0.978820i $$-0.434370\pi$$
0.204724 + 0.978820i $$0.434370\pi$$
$$710$$ 0 0
$$711$$ 34.8026 1.30520
$$712$$ 0 0
$$713$$ −2.19734 −0.0822910
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −2.56152 −0.0956619
$$718$$ 0 0
$$719$$ 18.9419 0.706414 0.353207 0.935545i $$-0.385091\pi$$
0.353207 + 0.935545i $$0.385091\pi$$
$$720$$ 0 0
$$721$$ 13.5909 0.506150
$$722$$ 0 0
$$723$$ −0.279032 −0.0103773
$$724$$ 0 0
$$725$$ −4.00000 −0.148556
$$726$$ 0 0
$$727$$ 29.3739 1.08942 0.544708 0.838626i $$-0.316640\pi$$
0.544708 + 0.838626i $$0.316640\pi$$
$$728$$ 0 0
$$729$$ −26.5160 −0.982075
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 25.2654 0.933197 0.466599 0.884469i $$-0.345479\pi$$
0.466599 + 0.884469i $$0.345479\pi$$
$$734$$ 0 0
$$735$$ 0.598710 0.0220838
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 13.5161 0.497199 0.248600 0.968606i $$-0.420030\pi$$
0.248600 + 0.968606i $$0.420030\pi$$
$$740$$ 0 0
$$741$$ −0.187872 −0.00690165
$$742$$ 0 0
$$743$$ 12.1594 0.446084 0.223042 0.974809i $$-0.428401\pi$$
0.223042 + 0.974809i $$0.428401\pi$$
$$744$$ 0 0
$$745$$ 11.9096 0.436334
$$746$$ 0 0
$$747$$ 17.6651 0.646333
$$748$$ 0 0
$$749$$ −14.4190 −0.526858
$$750$$ 0 0
$$751$$ 37.6069 1.37229 0.686147 0.727463i $$-0.259301\pi$$
0.686147 + 0.727463i $$0.259301\pi$$
$$752$$ 0 0
$$753$$ −2.36004 −0.0860047
$$754$$ 0 0
$$755$$ 16.2984 0.593158
$$756$$ 0 0
$$757$$ 32.3886 1.17718 0.588592 0.808430i $$-0.299682\pi$$
0.588592 + 0.808430i $$0.299682\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 52.9640 1.91994 0.959971 0.280098i $$-0.0903670\pi$$
0.959971 + 0.280098i $$0.0903670\pi$$
$$762$$ 0 0
$$763$$ −2.67978 −0.0970145
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −5.13404 −0.185380
$$768$$ 0 0
$$769$$ 10.6387 0.383643 0.191821 0.981430i $$-0.438561\pi$$
0.191821 + 0.981430i $$0.438561\pi$$
$$770$$ 0 0
$$771$$ 1.27737 0.0460033
$$772$$ 0 0
$$773$$ 29.7953 1.07166 0.535831 0.844325i $$-0.319998\pi$$
0.535831 + 0.844325i $$0.319998\pi$$
$$774$$ 0 0
$$775$$ −4.64469 −0.166842
$$776$$ 0 0
$$777$$ 0.468836 0.0168194
$$778$$ 0 0
$$779$$ −0.261561 −0.00937138
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 2.27150 0.0811766
$$784$$ 0 0
$$785$$ 11.5849 0.413483
$$786$$ 0 0
$$787$$ −15.5486 −0.554249 −0.277125 0.960834i $$-0.589381\pi$$
−0.277125 + 0.960834i $$0.589381\pi$$
$$788$$ 0 0
$$789$$ 0.458229 0.0163134
$$790$$ 0 0
$$791$$ −10.2207 −0.363406
$$792$$ 0 0
$$793$$ −7.72964 −0.274488
$$794$$ 0 0
$$795$$ 0.0759666 0.00269426
$$796$$ 0 0
$$797$$ 22.6399 0.801946 0.400973 0.916090i $$-0.368672\pi$$
0.400973 + 0.916090i $$0.368672\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −12.9371 −0.457110
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −0.391166 −0.0137868
$$806$$ 0 0
$$807$$ −1.78554 −0.0628541
$$808$$ 0 0
$$809$$ −29.1638 −1.02534 −0.512672 0.858585i $$-0.671344\pi$$
−0.512672 + 0.858585i $$0.671344\pi$$
$$810$$ 0 0
$$811$$ 50.2445 1.76432 0.882161 0.470948i $$-0.156088\pi$$
0.882161 + 0.470948i $$0.156088\pi$$
$$812$$ 0 0
$$813$$ 1.60288 0.0562155
$$814$$ 0 0
$$815$$ 1.63967 0.0574352
$$816$$ 0 0
$$817$$ −4.61585 −0.161488
$$818$$ 0 0
$$819$$ −7.39704 −0.258473
$$820$$ 0 0
$$821$$ 54.7708 1.91151 0.955756 0.294160i $$-0.0950399\pi$$
0.955756 + 0.294160i $$0.0950399\pi$$
$$822$$ 0 0
$$823$$ 24.4919 0.853734 0.426867 0.904314i $$-0.359617\pi$$
0.426867 + 0.904314i $$0.359617\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −44.3069 −1.54070 −0.770352 0.637619i $$-0.779919\pi$$
−0.770352 + 0.637619i $$0.779919\pi$$
$$828$$ 0 0
$$829$$ −50.9052 −1.76801 −0.884006 0.467476i $$-0.845163\pi$$
−0.884006 + 0.467476i $$0.845163\pi$$
$$830$$ 0 0
$$831$$ −0.367699 −0.0127553
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 11.7550 0.406800
$$836$$ 0 0
$$837$$ 2.63760 0.0911688
$$838$$ 0 0
$$839$$ 43.3354 1.49610 0.748051 0.663641i $$-0.230990\pi$$
0.748051 + 0.663641i $$0.230990\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 0 0
$$843$$ −1.66853 −0.0574674
$$844$$ 0 0
$$845$$ 4.05383 0.139456
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −0.0741235 −0.00254391
$$850$$ 0 0
$$851$$ 2.83002 0.0970117
$$852$$ 0 0
$$853$$ −11.1195 −0.380724 −0.190362 0.981714i $$-0.560966\pi$$
−0.190362 + 0.981714i $$0.560966\pi$$
$$854$$ 0 0
$$855$$ −1.98203 −0.0677840
$$856$$ 0 0
$$857$$ −29.0430 −0.992088 −0.496044 0.868297i $$-0.665215\pi$$
−0.496044 + 0.868297i $$0.665215\pi$$
$$858$$ 0 0
$$859$$ −39.4993 −1.34770 −0.673850 0.738869i $$-0.735360\pi$$
−0.673850 + 0.738869i $$0.735360\pi$$
$$860$$ 0 0
$$861$$ 0.0309352 0.00105427
$$862$$ 0 0
$$863$$ 31.1810 1.06141 0.530707 0.847556i $$-0.321927\pi$$
0.530707 + 0.847556i $$0.321927\pi$$
$$864$$ 0 0
$$865$$ 14.4240 0.490430
$$866$$ 0 0
$$867$$ −1.61139 −0.0547256
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 17.0418 0.577440
$$872$$ 0 0
$$873$$ −19.4593 −0.658597
$$874$$ 0 0
$$875$$ −0.826838 −0.0279522
$$876$$ 0 0
$$877$$ 5.04011 0.170192 0.0850962 0.996373i $$-0.472880\pi$$
0.0850962 + 0.996373i $$0.472880\pi$$
$$878$$ 0 0
$$879$$ −0.775706 −0.0261639
$$880$$ 0 0
$$881$$ −19.1788 −0.646150 −0.323075 0.946373i $$-0.604717\pi$$
−0.323075 + 0.946373i $$0.604717\pi$$
$$882$$ 0 0
$$883$$ −7.36535 −0.247864 −0.123932 0.992291i $$-0.539550\pi$$
−0.123932 + 0.992291i $$0.539550\pi$$
$$884$$ 0 0
$$885$$ 0.162702 0.00546916
$$886$$ 0 0
$$887$$ 34.1574 1.14689 0.573446 0.819243i $$-0.305606\pi$$
0.573446 + 0.819243i $$0.305606\pi$$
$$888$$ 0 0
$$889$$ 2.25967 0.0757869
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 2.57593 0.0862001
$$894$$ 0 0
$$895$$ −6.86425 −0.229447
$$896$$ 0 0
$$897$$ 0.134125 0.00447831
$$898$$ 0 0
$$899$$ 18.5788 0.619637
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0.545923 0.0181672
$$904$$ 0 0
$$905$$ 23.9742 0.796929
$$906$$ 0 0
$$907$$ 32.5599 1.08114 0.540568 0.841301i $$-0.318210\pi$$
0.540568 + 0.841301i $$0.318210\pi$$
$$908$$ 0 0
$$909$$ −10.5956 −0.351434
$$910$$ 0 0
$$911$$ 9.47896 0.314052 0.157026 0.987594i $$-0.449809\pi$$
0.157026 + 0.987594i $$0.449809\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0.244958 0.00809807
$$916$$ 0 0
$$917$$ 7.91524 0.261384
$$918$$ 0 0
$$919$$ 33.8624 1.11702 0.558508 0.829499i $$-0.311374\pi$$
0.558508 + 0.829499i $$0.311374\pi$$
$$920$$ 0 0
$$921$$ 2.60182 0.0857330
$$922$$ 0 0
$$923$$ −16.7224 −0.550423
$$924$$ 0 0
$$925$$ 5.98203 0.196688
$$926$$ 0 0
$$927$$ −49.1638 −1.61475
$$928$$ 0 0
$$929$$ 24.9163 0.817477 0.408739 0.912652i $$-0.365969\pi$$
0.408739 + 0.912652i $$0.365969\pi$$
$$930$$ 0 0
$$931$$ 4.18559 0.137177
$$932$$ 0 0
$$933$$ 2.52994 0.0828267
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −14.0132 −0.457790 −0.228895 0.973451i $$-0.573511\pi$$
−0.228895 + 0.973451i $$0.573511\pi$$
$$938$$ 0 0
$$939$$ −1.87013 −0.0610292
$$940$$ 0 0
$$941$$ −33.6453 −1.09681 −0.548403 0.836214i $$-0.684764\pi$$
−0.548403 + 0.836214i $$0.684764\pi$$
$$942$$ 0 0
$$943$$ 0.186733 0.00608086
$$944$$ 0 0
$$945$$ 0.469540 0.0152741
$$946$$ 0 0
$$947$$ −55.9621 −1.81852 −0.909262 0.416225i $$-0.863353\pi$$
−0.909262 + 0.416225i $$0.863353\pi$$
$$948$$ 0 0
$$949$$ −23.9281 −0.776740
$$950$$ 0 0
$$951$$ 0.875998 0.0284062
$$952$$ 0 0
$$953$$ 14.9282 0.483572 0.241786 0.970330i $$-0.422267\pi$$
0.241786 + 0.970330i $$0.422267\pi$$
$$954$$ 0 0
$$955$$ −12.4551 −0.403038
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −5.06584 −0.163585
$$960$$ 0 0
$$961$$ −9.42684 −0.304091
$$962$$ 0 0
$$963$$ 52.1594 1.68081
$$964$$ 0 0
$$965$$ 5.19261 0.167156
$$966$$ 0 0
$$967$$ −31.5986 −1.01614 −0.508072 0.861315i $$-0.669642\pi$$
−0.508072 + 0.861315i $$0.669642\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 56.5871 1.81597 0.907983 0.419006i $$-0.137621\pi$$
0.907983 + 0.419006i $$0.137621\pi$$
$$972$$ 0 0
$$973$$ −17.3265 −0.555461
$$974$$ 0 0
$$975$$ 0.283511 0.00907962
$$976$$ 0 0
$$977$$ 41.6948 1.33394 0.666968 0.745086i $$-0.267592\pi$$
0.666968 + 0.745086i $$0.267592\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 9.69387 0.309501
$$982$$ 0 0
$$983$$ −24.2687 −0.774050 −0.387025 0.922069i $$-0.626497\pi$$
−0.387025 + 0.922069i $$0.626497\pi$$
$$984$$ 0 0
$$985$$ 8.68783 0.276817
$$986$$ 0 0
$$987$$ −0.304659 −0.00969740
$$988$$ 0 0
$$989$$ 3.29534 0.104786
$$990$$ 0 0
$$991$$ 12.6670 0.402381 0.201191 0.979552i $$-0.435519\pi$$
0.201191 + 0.979552i $$0.435519\pi$$
$$992$$ 0 0
$$993$$ 1.83730 0.0583049
$$994$$ 0 0
$$995$$ −13.5149 −0.428451
$$996$$ 0 0
$$997$$ −19.3965 −0.614293 −0.307146 0.951662i $$-0.599374\pi$$
−0.307146 + 0.951662i $$0.599374\pi$$
$$998$$ 0 0
$$999$$ −3.39704 −0.107478
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 4840.2.a.w.1.3 4
4.3 odd 2 9680.2.a.cr.1.2 4
11.10 odd 2 4840.2.a.x.1.3 yes 4
44.43 even 2 9680.2.a.cq.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.w.1.3 4 1.1 even 1 trivial
4840.2.a.x.1.3 yes 4 11.10 odd 2
9680.2.a.cq.1.2 4 44.43 even 2
9680.2.a.cr.1.2 4 4.3 odd 2