# Properties

 Label 1425.2.c.k.799.1 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$11.3786822880$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 285) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$-0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.k.799.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -2.73205i q^{7} -1.73205i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -2.73205i q^{7} -1.73205i q^{8} -1.00000 q^{9} +4.73205 q^{11} +1.00000i q^{12} -0.732051i q^{13} -4.73205 q^{14} -5.00000 q^{16} +1.73205i q^{18} -1.00000 q^{19} -2.73205 q^{21} -8.19615i q^{22} -3.46410i q^{23} -1.73205 q^{24} -1.26795 q^{26} +1.00000i q^{27} +2.73205i q^{28} -8.19615 q^{29} +8.92820 q^{31} +5.19615i q^{32} -4.73205i q^{33} +1.00000 q^{36} -6.19615i q^{37} +1.73205i q^{38} -0.732051 q^{39} +1.26795 q^{41} +4.73205i q^{42} -4.19615i q^{43} -4.73205 q^{44} -6.00000 q^{46} -3.46410i q^{47} +5.00000i q^{48} -0.464102 q^{49} +0.732051i q^{52} +9.46410i q^{53} +1.73205 q^{54} -4.73205 q^{56} +1.00000i q^{57} +14.1962i q^{58} -2.53590 q^{59} -6.53590 q^{61} -15.4641i q^{62} +2.73205i q^{63} -1.00000 q^{64} -8.19615 q^{66} +8.00000i q^{67} -3.46410 q^{69} -4.39230 q^{71} +1.73205i q^{72} +16.9282i q^{73} -10.7321 q^{74} +1.00000 q^{76} -12.9282i q^{77} +1.26795i q^{78} +10.9282 q^{79} +1.00000 q^{81} -2.19615i q^{82} +12.9282i q^{83} +2.73205 q^{84} -7.26795 q^{86} +8.19615i q^{87} -8.19615i q^{88} -10.7321 q^{89} -2.00000 q^{91} +3.46410i q^{92} -8.92820i q^{93} -6.00000 q^{94} +5.19615 q^{96} -6.19615i q^{97} +0.803848i q^{98} -4.73205 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^9 $$4 q - 4 q^{4} - 4 q^{9} + 12 q^{11} - 12 q^{14} - 20 q^{16} - 4 q^{19} - 4 q^{21} - 12 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{36} + 4 q^{39} + 12 q^{41} - 12 q^{44} - 24 q^{46} + 12 q^{49} - 12 q^{56} - 24 q^{59} - 40 q^{61} - 4 q^{64} - 12 q^{66} + 24 q^{71} - 36 q^{74} + 4 q^{76} + 16 q^{79} + 4 q^{81} + 4 q^{84} - 36 q^{86} - 36 q^{89} - 8 q^{91} - 24 q^{94} - 12 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^9 + 12 * q^11 - 12 * q^14 - 20 * q^16 - 4 * q^19 - 4 * q^21 - 12 * q^26 - 12 * q^29 + 8 * q^31 + 4 * q^36 + 4 * q^39 + 12 * q^41 - 12 * q^44 - 24 * q^46 + 12 * q^49 - 12 * q^56 - 24 * q^59 - 40 * q^61 - 4 * q^64 - 12 * q^66 + 24 * q^71 - 36 * q^74 + 4 * q^76 + 16 * q^79 + 4 * q^81 + 4 * q^84 - 36 * q^86 - 36 * q^89 - 8 * q^91 - 24 * q^94 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.73205i − 1.22474i −0.790569 0.612372i $$-0.790215\pi$$
0.790569 0.612372i $$-0.209785\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.73205 −0.707107
$$7$$ − 2.73205i − 1.03262i −0.856402 0.516309i $$-0.827306\pi$$
0.856402 0.516309i $$-0.172694\pi$$
$$8$$ − 1.73205i − 0.612372i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.73205 1.42677 0.713384 0.700774i $$-0.247162\pi$$
0.713384 + 0.700774i $$0.247162\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 0.732051i − 0.203034i −0.994834 0.101517i $$-0.967630\pi$$
0.994834 0.101517i $$-0.0323697\pi$$
$$14$$ −4.73205 −1.26469
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 1.73205i 0.408248i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −2.73205 −0.596182
$$22$$ − 8.19615i − 1.74743i
$$23$$ − 3.46410i − 0.722315i −0.932505 0.361158i $$-0.882382\pi$$
0.932505 0.361158i $$-0.117618\pi$$
$$24$$ −1.73205 −0.353553
$$25$$ 0 0
$$26$$ −1.26795 −0.248665
$$27$$ 1.00000i 0.192450i
$$28$$ 2.73205i 0.516309i
$$29$$ −8.19615 −1.52199 −0.760994 0.648759i $$-0.775288\pi$$
−0.760994 + 0.648759i $$0.775288\pi$$
$$30$$ 0 0
$$31$$ 8.92820 1.60355 0.801776 0.597624i $$-0.203889\pi$$
0.801776 + 0.597624i $$0.203889\pi$$
$$32$$ 5.19615i 0.918559i
$$33$$ − 4.73205i − 0.823744i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 6.19615i − 1.01864i −0.860577 0.509321i $$-0.829897\pi$$
0.860577 0.509321i $$-0.170103\pi$$
$$38$$ 1.73205i 0.280976i
$$39$$ −0.732051 −0.117222
$$40$$ 0 0
$$41$$ 1.26795 0.198020 0.0990102 0.995086i $$-0.468432\pi$$
0.0990102 + 0.995086i $$0.468432\pi$$
$$42$$ 4.73205i 0.730171i
$$43$$ − 4.19615i − 0.639907i −0.947433 0.319954i $$-0.896333\pi$$
0.947433 0.319954i $$-0.103667\pi$$
$$44$$ −4.73205 −0.713384
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ − 3.46410i − 0.505291i −0.967559 0.252646i $$-0.918699\pi$$
0.967559 0.252646i $$-0.0813007\pi$$
$$48$$ 5.00000i 0.721688i
$$49$$ −0.464102 −0.0663002
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0.732051i 0.101517i
$$53$$ 9.46410i 1.29999i 0.759937 + 0.649997i $$0.225230\pi$$
−0.759937 + 0.649997i $$0.774770\pi$$
$$54$$ 1.73205 0.235702
$$55$$ 0 0
$$56$$ −4.73205 −0.632347
$$57$$ 1.00000i 0.132453i
$$58$$ 14.1962i 1.86405i
$$59$$ −2.53590 −0.330146 −0.165073 0.986281i $$-0.552786\pi$$
−0.165073 + 0.986281i $$0.552786\pi$$
$$60$$ 0 0
$$61$$ −6.53590 −0.836836 −0.418418 0.908255i $$-0.637415\pi$$
−0.418418 + 0.908255i $$0.637415\pi$$
$$62$$ − 15.4641i − 1.96394i
$$63$$ 2.73205i 0.344206i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −8.19615 −1.00888
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ −3.46410 −0.417029
$$70$$ 0 0
$$71$$ −4.39230 −0.521271 −0.260635 0.965437i $$-0.583932\pi$$
−0.260635 + 0.965437i $$0.583932\pi$$
$$72$$ 1.73205i 0.204124i
$$73$$ 16.9282i 1.98130i 0.136441 + 0.990648i $$0.456434\pi$$
−0.136441 + 0.990648i $$0.543566\pi$$
$$74$$ −10.7321 −1.24758
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ − 12.9282i − 1.47331i
$$78$$ 1.26795i 0.143567i
$$79$$ 10.9282 1.22952 0.614759 0.788715i $$-0.289253\pi$$
0.614759 + 0.788715i $$0.289253\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 2.19615i − 0.242524i
$$83$$ 12.9282i 1.41905i 0.704678 + 0.709527i $$0.251092\pi$$
−0.704678 + 0.709527i $$0.748908\pi$$
$$84$$ 2.73205 0.298091
$$85$$ 0 0
$$86$$ −7.26795 −0.783723
$$87$$ 8.19615i 0.878720i
$$88$$ − 8.19615i − 0.873713i
$$89$$ −10.7321 −1.13760 −0.568798 0.822478i $$-0.692591\pi$$
−0.568798 + 0.822478i $$0.692591\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 3.46410i 0.361158i
$$93$$ − 8.92820i − 0.925812i
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 5.19615 0.530330
$$97$$ − 6.19615i − 0.629124i −0.949237 0.314562i $$-0.898142\pi$$
0.949237 0.314562i $$-0.101858\pi$$
$$98$$ 0.803848i 0.0812009i
$$99$$ −4.73205 −0.475589
$$100$$ 0 0
$$101$$ −10.3923 −1.03407 −0.517036 0.855963i $$-0.672965\pi$$
−0.517036 + 0.855963i $$0.672965\pi$$
$$102$$ 0 0
$$103$$ − 9.85641i − 0.971181i −0.874187 0.485590i $$-0.838605\pi$$
0.874187 0.485590i $$-0.161395\pi$$
$$104$$ −1.26795 −0.124333
$$105$$ 0 0
$$106$$ 16.3923 1.59216
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 14.3923 1.37853 0.689266 0.724508i $$-0.257933\pi$$
0.689266 + 0.724508i $$0.257933\pi$$
$$110$$ 0 0
$$111$$ −6.19615 −0.588113
$$112$$ 13.6603i 1.29077i
$$113$$ − 18.9282i − 1.78062i −0.455359 0.890308i $$-0.650489\pi$$
0.455359 0.890308i $$-0.349511\pi$$
$$114$$ 1.73205 0.162221
$$115$$ 0 0
$$116$$ 8.19615 0.760994
$$117$$ 0.732051i 0.0676781i
$$118$$ 4.39230i 0.404344i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.3923 1.03566
$$122$$ 11.3205i 1.02491i
$$123$$ − 1.26795i − 0.114327i
$$124$$ −8.92820 −0.801776
$$125$$ 0 0
$$126$$ 4.73205 0.421565
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ 12.1244i 1.07165i
$$129$$ −4.19615 −0.369451
$$130$$ 0 0
$$131$$ −9.12436 −0.797199 −0.398599 0.917125i $$-0.630504\pi$$
−0.398599 + 0.917125i $$0.630504\pi$$
$$132$$ 4.73205i 0.411872i
$$133$$ 2.73205i 0.236899i
$$134$$ 13.8564 1.19701
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 19.8564i 1.69645i 0.529638 + 0.848224i $$0.322328\pi$$
−0.529638 + 0.848224i $$0.677672\pi$$
$$138$$ 6.00000i 0.510754i
$$139$$ 8.39230 0.711826 0.355913 0.934519i $$-0.384170\pi$$
0.355913 + 0.934519i $$0.384170\pi$$
$$140$$ 0 0
$$141$$ −3.46410 −0.291730
$$142$$ 7.60770i 0.638424i
$$143$$ − 3.46410i − 0.289683i
$$144$$ 5.00000 0.416667
$$145$$ 0 0
$$146$$ 29.3205 2.42658
$$147$$ 0.464102i 0.0382785i
$$148$$ 6.19615i 0.509321i
$$149$$ 19.8564 1.62670 0.813350 0.581775i $$-0.197641\pi$$
0.813350 + 0.581775i $$0.197641\pi$$
$$150$$ 0 0
$$151$$ 14.0000 1.13930 0.569652 0.821886i $$-0.307078\pi$$
0.569652 + 0.821886i $$0.307078\pi$$
$$152$$ 1.73205i 0.140488i
$$153$$ 0 0
$$154$$ −22.3923 −1.80442
$$155$$ 0 0
$$156$$ 0.732051 0.0586110
$$157$$ 6.39230i 0.510161i 0.966920 + 0.255081i $$0.0821021\pi$$
−0.966920 + 0.255081i $$0.917898\pi$$
$$158$$ − 18.9282i − 1.50585i
$$159$$ 9.46410 0.750552
$$160$$ 0 0
$$161$$ −9.46410 −0.745876
$$162$$ − 1.73205i − 0.136083i
$$163$$ − 9.26795i − 0.725922i −0.931804 0.362961i $$-0.881766\pi$$
0.931804 0.362961i $$-0.118234\pi$$
$$164$$ −1.26795 −0.0990102
$$165$$ 0 0
$$166$$ 22.3923 1.73798
$$167$$ 3.46410i 0.268060i 0.990977 + 0.134030i $$0.0427919\pi$$
−0.990977 + 0.134030i $$0.957208\pi$$
$$168$$ 4.73205i 0.365086i
$$169$$ 12.4641 0.958777
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 4.19615i 0.319954i
$$173$$ − 6.92820i − 0.526742i −0.964695 0.263371i $$-0.915166\pi$$
0.964695 0.263371i $$-0.0848343\pi$$
$$174$$ 14.1962 1.07621
$$175$$ 0 0
$$176$$ −23.6603 −1.78346
$$177$$ 2.53590i 0.190610i
$$178$$ 18.5885i 1.39326i
$$179$$ −23.3205 −1.74306 −0.871528 0.490345i $$-0.836871\pi$$
−0.871528 + 0.490345i $$0.836871\pi$$
$$180$$ 0 0
$$181$$ −2.39230 −0.177819 −0.0889093 0.996040i $$-0.528338\pi$$
−0.0889093 + 0.996040i $$0.528338\pi$$
$$182$$ 3.46410i 0.256776i
$$183$$ 6.53590i 0.483148i
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ −15.4641 −1.13388
$$187$$ 0 0
$$188$$ 3.46410i 0.252646i
$$189$$ 2.73205 0.198727
$$190$$ 0 0
$$191$$ 0.339746 0.0245832 0.0122916 0.999924i $$-0.496087\pi$$
0.0122916 + 0.999924i $$0.496087\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 17.1244i − 1.23264i −0.787497 0.616319i $$-0.788623\pi$$
0.787497 0.616319i $$-0.211377\pi$$
$$194$$ −10.7321 −0.770516
$$195$$ 0 0
$$196$$ 0.464102 0.0331501
$$197$$ − 24.0000i − 1.70993i −0.518686 0.854965i $$-0.673579\pi$$
0.518686 0.854965i $$-0.326421\pi$$
$$198$$ 8.19615i 0.582475i
$$199$$ 15.3205 1.08604 0.543021 0.839719i $$-0.317280\pi$$
0.543021 + 0.839719i $$0.317280\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 18.0000i 1.26648i
$$203$$ 22.3923i 1.57163i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −17.0718 −1.18945
$$207$$ 3.46410i 0.240772i
$$208$$ 3.66025i 0.253793i
$$209$$ −4.73205 −0.327323
$$210$$ 0 0
$$211$$ 1.07180 0.0737855 0.0368928 0.999319i $$-0.488254\pi$$
0.0368928 + 0.999319i $$0.488254\pi$$
$$212$$ − 9.46410i − 0.649997i
$$213$$ 4.39230i 0.300956i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 1.73205 0.117851
$$217$$ − 24.3923i − 1.65586i
$$218$$ − 24.9282i − 1.68835i
$$219$$ 16.9282 1.14390
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 10.7321i 0.720288i
$$223$$ 17.8564i 1.19575i 0.801588 + 0.597877i $$0.203989\pi$$
−0.801588 + 0.597877i $$0.796011\pi$$
$$224$$ 14.1962 0.948520
$$225$$ 0 0
$$226$$ −32.7846 −2.18080
$$227$$ 10.3923i 0.689761i 0.938647 + 0.344881i $$0.112081\pi$$
−0.938647 + 0.344881i $$0.887919\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ 18.5359 1.22489 0.612443 0.790515i $$-0.290187\pi$$
0.612443 + 0.790515i $$0.290187\pi$$
$$230$$ 0 0
$$231$$ −12.9282 −0.850613
$$232$$ 14.1962i 0.932023i
$$233$$ 7.85641i 0.514690i 0.966320 + 0.257345i $$0.0828477\pi$$
−0.966320 + 0.257345i $$0.917152\pi$$
$$234$$ 1.26795 0.0828884
$$235$$ 0 0
$$236$$ 2.53590 0.165073
$$237$$ − 10.9282i − 0.709863i
$$238$$ 0 0
$$239$$ 9.80385 0.634158 0.317079 0.948399i $$-0.397298\pi$$
0.317079 + 0.948399i $$0.397298\pi$$
$$240$$ 0 0
$$241$$ −3.07180 −0.197872 −0.0989359 0.995094i $$-0.531544\pi$$
−0.0989359 + 0.995094i $$0.531544\pi$$
$$242$$ − 19.7321i − 1.26842i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 6.53590 0.418418
$$245$$ 0 0
$$246$$ −2.19615 −0.140022
$$247$$ 0.732051i 0.0465793i
$$248$$ − 15.4641i − 0.981971i
$$249$$ 12.9282 0.819292
$$250$$ 0 0
$$251$$ 28.0526 1.77066 0.885331 0.464961i $$-0.153932\pi$$
0.885331 + 0.464961i $$0.153932\pi$$
$$252$$ − 2.73205i − 0.172103i
$$253$$ − 16.3923i − 1.03058i
$$254$$ −6.92820 −0.434714
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ − 24.0000i − 1.49708i −0.663090 0.748539i $$-0.730755\pi$$
0.663090 0.748539i $$-0.269245\pi$$
$$258$$ 7.26795i 0.452483i
$$259$$ −16.9282 −1.05187
$$260$$ 0 0
$$261$$ 8.19615 0.507329
$$262$$ 15.8038i 0.976365i
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ −8.19615 −0.504438
$$265$$ 0 0
$$266$$ 4.73205 0.290141
$$267$$ 10.7321i 0.656791i
$$268$$ − 8.00000i − 0.488678i
$$269$$ −0.588457 −0.0358789 −0.0179394 0.999839i $$-0.505711\pi$$
−0.0179394 + 0.999839i $$0.505711\pi$$
$$270$$ 0 0
$$271$$ 0.392305 0.0238308 0.0119154 0.999929i $$-0.496207\pi$$
0.0119154 + 0.999929i $$0.496207\pi$$
$$272$$ 0 0
$$273$$ 2.00000i 0.121046i
$$274$$ 34.3923 2.07772
$$275$$ 0 0
$$276$$ 3.46410 0.208514
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ − 14.5359i − 0.871805i
$$279$$ −8.92820 −0.534518
$$280$$ 0 0
$$281$$ 1.26795 0.0756395 0.0378198 0.999285i $$-0.487959\pi$$
0.0378198 + 0.999285i $$0.487959\pi$$
$$282$$ 6.00000i 0.357295i
$$283$$ − 24.9808i − 1.48495i −0.669873 0.742476i $$-0.733651\pi$$
0.669873 0.742476i $$-0.266349\pi$$
$$284$$ 4.39230 0.260635
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ − 3.46410i − 0.204479i
$$288$$ − 5.19615i − 0.306186i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −6.19615 −0.363225
$$292$$ − 16.9282i − 0.990648i
$$293$$ − 27.7128i − 1.61900i −0.587120 0.809500i $$-0.699738\pi$$
0.587120 0.809500i $$-0.300262\pi$$
$$294$$ 0.803848 0.0468813
$$295$$ 0 0
$$296$$ −10.7321 −0.623788
$$297$$ 4.73205i 0.274581i
$$298$$ − 34.3923i − 1.99229i
$$299$$ −2.53590 −0.146655
$$300$$ 0 0
$$301$$ −11.4641 −0.660780
$$302$$ − 24.2487i − 1.39536i
$$303$$ 10.3923i 0.597022i
$$304$$ 5.00000 0.286770
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 32.3923i − 1.84873i −0.381514 0.924363i $$-0.624597\pi$$
0.381514 0.924363i $$-0.375403\pi$$
$$308$$ 12.9282i 0.736653i
$$309$$ −9.85641 −0.560711
$$310$$ 0 0
$$311$$ 32.4449 1.83978 0.919890 0.392177i $$-0.128278\pi$$
0.919890 + 0.392177i $$0.128278\pi$$
$$312$$ 1.26795i 0.0717835i
$$313$$ − 6.39230i − 0.361314i −0.983546 0.180657i $$-0.942178\pi$$
0.983546 0.180657i $$-0.0578225\pi$$
$$314$$ 11.0718 0.624818
$$315$$ 0 0
$$316$$ −10.9282 −0.614759
$$317$$ − 11.3205i − 0.635823i −0.948120 0.317912i $$-0.897018\pi$$
0.948120 0.317912i $$-0.102982\pi$$
$$318$$ − 16.3923i − 0.919235i
$$319$$ −38.7846 −2.17152
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 16.3923i 0.913507i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −16.0526 −0.889069
$$327$$ − 14.3923i − 0.795896i
$$328$$ − 2.19615i − 0.121262i
$$329$$ −9.46410 −0.521773
$$330$$ 0 0
$$331$$ −25.7128 −1.41330 −0.706652 0.707561i $$-0.749795\pi$$
−0.706652 + 0.707561i $$0.749795\pi$$
$$332$$ − 12.9282i − 0.709527i
$$333$$ 6.19615i 0.339547i
$$334$$ 6.00000 0.328305
$$335$$ 0 0
$$336$$ 13.6603 0.745228
$$337$$ 5.12436i 0.279141i 0.990212 + 0.139571i $$0.0445723\pi$$
−0.990212 + 0.139571i $$0.955428\pi$$
$$338$$ − 21.5885i − 1.17426i
$$339$$ −18.9282 −1.02804
$$340$$ 0 0
$$341$$ 42.2487 2.28790
$$342$$ − 1.73205i − 0.0936586i
$$343$$ − 17.8564i − 0.964155i
$$344$$ −7.26795 −0.391862
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ − 0.928203i − 0.0498286i −0.999690 0.0249143i $$-0.992069\pi$$
0.999690 0.0249143i $$-0.00793128\pi$$
$$348$$ − 8.19615i − 0.439360i
$$349$$ 22.0000 1.17763 0.588817 0.808267i $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ 0 0
$$351$$ 0.732051 0.0390740
$$352$$ 24.5885i 1.31057i
$$353$$ 14.7846i 0.786905i 0.919345 + 0.393453i $$0.128719\pi$$
−0.919345 + 0.393453i $$0.871281\pi$$
$$354$$ 4.39230 0.233448
$$355$$ 0 0
$$356$$ 10.7321 0.568798
$$357$$ 0 0
$$358$$ 40.3923i 2.13480i
$$359$$ −0.339746 −0.0179311 −0.00896555 0.999960i $$-0.502854\pi$$
−0.00896555 + 0.999960i $$0.502854\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 4.14359i 0.217782i
$$363$$ − 11.3923i − 0.597941i
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 11.3205 0.591732
$$367$$ 16.1962i 0.845432i 0.906262 + 0.422716i $$0.138923\pi$$
−0.906262 + 0.422716i $$0.861077\pi$$
$$368$$ 17.3205i 0.902894i
$$369$$ −1.26795 −0.0660068
$$370$$ 0 0
$$371$$ 25.8564 1.34240
$$372$$ 8.92820i 0.462906i
$$373$$ 6.19615i 0.320825i 0.987050 + 0.160412i $$0.0512824\pi$$
−0.987050 + 0.160412i $$0.948718\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ 6.00000i 0.309016i
$$378$$ − 4.73205i − 0.243390i
$$379$$ −20.9282 −1.07501 −0.537505 0.843261i $$-0.680633\pi$$
−0.537505 + 0.843261i $$0.680633\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ − 0.588457i − 0.0301081i
$$383$$ 17.0718i 0.872328i 0.899867 + 0.436164i $$0.143663\pi$$
−0.899867 + 0.436164i $$0.856337\pi$$
$$384$$ 12.1244 0.618718
$$385$$ 0 0
$$386$$ −29.6603 −1.50967
$$387$$ 4.19615i 0.213302i
$$388$$ 6.19615i 0.314562i
$$389$$ −7.85641 −0.398336 −0.199168 0.979965i $$-0.563824\pi$$
−0.199168 + 0.979965i $$0.563824\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0.803848i 0.0406004i
$$393$$ 9.12436i 0.460263i
$$394$$ −41.5692 −2.09423
$$395$$ 0 0
$$396$$ 4.73205 0.237795
$$397$$ 8.92820i 0.448094i 0.974578 + 0.224047i $$0.0719269\pi$$
−0.974578 + 0.224047i $$0.928073\pi$$
$$398$$ − 26.5359i − 1.33012i
$$399$$ 2.73205 0.136774
$$400$$ 0 0
$$401$$ −34.0526 −1.70050 −0.850252 0.526376i $$-0.823550\pi$$
−0.850252 + 0.526376i $$0.823550\pi$$
$$402$$ − 13.8564i − 0.691095i
$$403$$ − 6.53590i − 0.325576i
$$404$$ 10.3923 0.517036
$$405$$ 0 0
$$406$$ 38.7846 1.92485
$$407$$ − 29.3205i − 1.45336i
$$408$$ 0 0
$$409$$ 26.3923 1.30502 0.652508 0.757782i $$-0.273717\pi$$
0.652508 + 0.757782i $$0.273717\pi$$
$$410$$ 0 0
$$411$$ 19.8564 0.979444
$$412$$ 9.85641i 0.485590i
$$413$$ 6.92820i 0.340915i
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ 3.80385 0.186499
$$417$$ − 8.39230i − 0.410973i
$$418$$ 8.19615i 0.400887i
$$419$$ 28.0526 1.37046 0.685229 0.728328i $$-0.259702\pi$$
0.685229 + 0.728328i $$0.259702\pi$$
$$420$$ 0 0
$$421$$ −18.7846 −0.915506 −0.457753 0.889079i $$-0.651346\pi$$
−0.457753 + 0.889079i $$0.651346\pi$$
$$422$$ − 1.85641i − 0.0903685i
$$423$$ 3.46410i 0.168430i
$$424$$ 16.3923 0.796081
$$425$$ 0 0
$$426$$ 7.60770 0.368594
$$427$$ 17.8564i 0.864132i
$$428$$ 0 0
$$429$$ −3.46410 −0.167248
$$430$$ 0 0
$$431$$ −11.3205 −0.545290 −0.272645 0.962115i $$-0.587898\pi$$
−0.272645 + 0.962115i $$0.587898\pi$$
$$432$$ − 5.00000i − 0.240563i
$$433$$ 10.5885i 0.508849i 0.967093 + 0.254424i $$0.0818860\pi$$
−0.967093 + 0.254424i $$0.918114\pi$$
$$434$$ −42.2487 −2.02800
$$435$$ 0 0
$$436$$ −14.3923 −0.689266
$$437$$ 3.46410i 0.165710i
$$438$$ − 29.3205i − 1.40099i
$$439$$ −26.9282 −1.28521 −0.642607 0.766196i $$-0.722147\pi$$
−0.642607 + 0.766196i $$0.722147\pi$$
$$440$$ 0 0
$$441$$ 0.464102 0.0221001
$$442$$ 0 0
$$443$$ 5.32051i 0.252785i 0.991980 + 0.126392i $$0.0403399\pi$$
−0.991980 + 0.126392i $$0.959660\pi$$
$$444$$ 6.19615 0.294056
$$445$$ 0 0
$$446$$ 30.9282 1.46449
$$447$$ − 19.8564i − 0.939176i
$$448$$ 2.73205i 0.129077i
$$449$$ 5.66025 0.267124 0.133562 0.991040i $$-0.457358\pi$$
0.133562 + 0.991040i $$0.457358\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ 18.9282i 0.890308i
$$453$$ − 14.0000i − 0.657777i
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ 1.73205 0.0811107
$$457$$ 4.53590i 0.212180i 0.994357 + 0.106090i $$0.0338332\pi$$
−0.994357 + 0.106090i $$0.966167\pi$$
$$458$$ − 32.1051i − 1.50017i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 22.3923i 1.04178i
$$463$$ 35.5167i 1.65060i 0.564695 + 0.825300i $$0.308994\pi$$
−0.564695 + 0.825300i $$0.691006\pi$$
$$464$$ 40.9808 1.90248
$$465$$ 0 0
$$466$$ 13.6077 0.630364
$$467$$ 20.5359i 0.950288i 0.879908 + 0.475144i $$0.157604\pi$$
−0.879908 + 0.475144i $$0.842396\pi$$
$$468$$ − 0.732051i − 0.0338391i
$$469$$ 21.8564 1.00924
$$470$$ 0 0
$$471$$ 6.39230 0.294542
$$472$$ 4.39230i 0.202172i
$$473$$ − 19.8564i − 0.912999i
$$474$$ −18.9282 −0.869401
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 9.46410i − 0.433331i
$$478$$ − 16.9808i − 0.776682i
$$479$$ −25.5167 −1.16589 −0.582943 0.812513i $$-0.698099\pi$$
−0.582943 + 0.812513i $$0.698099\pi$$
$$480$$ 0 0
$$481$$ −4.53590 −0.206819
$$482$$ 5.32051i 0.242343i
$$483$$ 9.46410i 0.430632i
$$484$$ −11.3923 −0.517832
$$485$$ 0 0
$$486$$ −1.73205 −0.0785674
$$487$$ − 32.3923i − 1.46784i −0.679238 0.733918i $$-0.737690\pi$$
0.679238 0.733918i $$-0.262310\pi$$
$$488$$ 11.3205i 0.512455i
$$489$$ −9.26795 −0.419111
$$490$$ 0 0
$$491$$ 16.0526 0.724442 0.362221 0.932092i $$-0.382019\pi$$
0.362221 + 0.932092i $$0.382019\pi$$
$$492$$ 1.26795i 0.0571636i
$$493$$ 0 0
$$494$$ 1.26795 0.0570477
$$495$$ 0 0
$$496$$ −44.6410 −2.00444
$$497$$ 12.0000i 0.538274i
$$498$$ − 22.3923i − 1.00342i
$$499$$ −10.5359 −0.471652 −0.235826 0.971795i $$-0.575779\pi$$
−0.235826 + 0.971795i $$0.575779\pi$$
$$500$$ 0 0
$$501$$ 3.46410 0.154765
$$502$$ − 48.5885i − 2.16861i
$$503$$ 23.0718i 1.02872i 0.857574 + 0.514360i $$0.171971\pi$$
−0.857574 + 0.514360i $$0.828029\pi$$
$$504$$ 4.73205 0.210782
$$505$$ 0 0
$$506$$ −28.3923 −1.26219
$$507$$ − 12.4641i − 0.553550i
$$508$$ 4.00000i 0.177471i
$$509$$ 10.0526 0.445572 0.222786 0.974867i $$-0.428485\pi$$
0.222786 + 0.974867i $$0.428485\pi$$
$$510$$ 0 0
$$511$$ 46.2487 2.04592
$$512$$ − 8.66025i − 0.382733i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ −41.5692 −1.83354
$$515$$ 0 0
$$516$$ 4.19615 0.184725
$$517$$ − 16.3923i − 0.720933i
$$518$$ 29.3205i 1.28827i
$$519$$ −6.92820 −0.304114
$$520$$ 0 0
$$521$$ 37.2679 1.63274 0.816369 0.577530i $$-0.195983\pi$$
0.816369 + 0.577530i $$0.195983\pi$$
$$522$$ − 14.1962i − 0.621349i
$$523$$ − 8.67949i − 0.379528i −0.981830 0.189764i $$-0.939228\pi$$
0.981830 0.189764i $$-0.0607722\pi$$
$$524$$ 9.12436 0.398599
$$525$$ 0 0
$$526$$ −10.3923 −0.453126
$$527$$ 0 0
$$528$$ 23.6603i 1.02968i
$$529$$ 11.0000 0.478261
$$530$$ 0 0
$$531$$ 2.53590 0.110049
$$532$$ − 2.73205i − 0.118449i
$$533$$ − 0.928203i − 0.0402049i
$$534$$ 18.5885 0.804401
$$535$$ 0 0
$$536$$ 13.8564 0.598506
$$537$$ 23.3205i 1.00635i
$$538$$ 1.01924i 0.0439425i
$$539$$ −2.19615 −0.0945950
$$540$$ 0 0
$$541$$ 41.7128 1.79337 0.896687 0.442665i $$-0.145967\pi$$
0.896687 + 0.442665i $$0.145967\pi$$
$$542$$ − 0.679492i − 0.0291867i
$$543$$ 2.39230i 0.102664i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 3.46410 0.148250
$$547$$ 43.3205i 1.85225i 0.377215 + 0.926126i $$0.376882\pi$$
−0.377215 + 0.926126i $$0.623118\pi$$
$$548$$ − 19.8564i − 0.848224i
$$549$$ 6.53590 0.278945
$$550$$ 0 0
$$551$$ 8.19615 0.349168
$$552$$ 6.00000i 0.255377i
$$553$$ − 29.8564i − 1.26962i
$$554$$ 3.46410 0.147176
$$555$$ 0 0
$$556$$ −8.39230 −0.355913
$$557$$ 0.928203i 0.0393292i 0.999807 + 0.0196646i $$0.00625985\pi$$
−0.999807 + 0.0196646i $$0.993740\pi$$
$$558$$ 15.4641i 0.654648i
$$559$$ −3.07180 −0.129923
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 2.19615i − 0.0926391i
$$563$$ 27.4641i 1.15747i 0.815514 + 0.578737i $$0.196454\pi$$
−0.815514 + 0.578737i $$0.803546\pi$$
$$564$$ 3.46410 0.145865
$$565$$ 0 0
$$566$$ −43.2679 −1.81869
$$567$$ − 2.73205i − 0.114735i
$$568$$ 7.60770i 0.319212i
$$569$$ −22.0526 −0.924491 −0.462246 0.886752i $$-0.652956\pi$$
−0.462246 + 0.886752i $$0.652956\pi$$
$$570$$ 0 0
$$571$$ −34.2487 −1.43326 −0.716632 0.697452i $$-0.754317\pi$$
−0.716632 + 0.697452i $$0.754317\pi$$
$$572$$ 3.46410i 0.144841i
$$573$$ − 0.339746i − 0.0141931i
$$574$$ −6.00000 −0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 15.1769i 0.631823i 0.948789 + 0.315912i $$0.102310\pi$$
−0.948789 + 0.315912i $$0.897690\pi$$
$$578$$ − 29.4449i − 1.22474i
$$579$$ −17.1244 −0.711664
$$580$$ 0 0
$$581$$ 35.3205 1.46534
$$582$$ 10.7321i 0.444858i
$$583$$ 44.7846i 1.85479i
$$584$$ 29.3205 1.21329
$$585$$ 0 0
$$586$$ −48.0000 −1.98286
$$587$$ − 3.46410i − 0.142979i −0.997441 0.0714894i $$-0.977225\pi$$
0.997441 0.0714894i $$-0.0227752\pi$$
$$588$$ − 0.464102i − 0.0191392i
$$589$$ −8.92820 −0.367880
$$590$$ 0 0
$$591$$ −24.0000 −0.987228
$$592$$ 30.9808i 1.27330i
$$593$$ 38.7846i 1.59269i 0.604841 + 0.796347i $$0.293237\pi$$
−0.604841 + 0.796347i $$0.706763\pi$$
$$594$$ 8.19615 0.336292
$$595$$ 0 0
$$596$$ −19.8564 −0.813350
$$597$$ − 15.3205i − 0.627027i
$$598$$ 4.39230i 0.179615i
$$599$$ 13.8564 0.566157 0.283079 0.959097i $$-0.408644\pi$$
0.283079 + 0.959097i $$0.408644\pi$$
$$600$$ 0 0
$$601$$ −47.1769 −1.92439 −0.962193 0.272368i $$-0.912193\pi$$
−0.962193 + 0.272368i $$0.912193\pi$$
$$602$$ 19.8564i 0.809287i
$$603$$ − 8.00000i − 0.325785i
$$604$$ −14.0000 −0.569652
$$605$$ 0 0
$$606$$ 18.0000 0.731200
$$607$$ − 11.6077i − 0.471142i −0.971857 0.235571i $$-0.924304\pi$$
0.971857 0.235571i $$-0.0756960\pi$$
$$608$$ − 5.19615i − 0.210732i
$$609$$ 22.3923 0.907382
$$610$$ 0 0
$$611$$ −2.53590 −0.102591
$$612$$ 0 0
$$613$$ − 42.3923i − 1.71221i −0.516803 0.856105i $$-0.672878\pi$$
0.516803 0.856105i $$-0.327122\pi$$
$$614$$ −56.1051 −2.26422
$$615$$ 0 0
$$616$$ −22.3923 −0.902212
$$617$$ 27.7128i 1.11568i 0.829950 + 0.557838i $$0.188369\pi$$
−0.829950 + 0.557838i $$0.811631\pi$$
$$618$$ 17.0718i 0.686728i
$$619$$ 15.3205 0.615783 0.307892 0.951421i $$-0.400377\pi$$
0.307892 + 0.951421i $$0.400377\pi$$
$$620$$ 0 0
$$621$$ 3.46410 0.139010
$$622$$ − 56.1962i − 2.25326i
$$623$$ 29.3205i 1.17470i
$$624$$ 3.66025 0.146527
$$625$$ 0 0
$$626$$ −11.0718 −0.442518
$$627$$ 4.73205i 0.188980i
$$628$$ − 6.39230i − 0.255081i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −34.9282 −1.39047 −0.695235 0.718783i $$-0.744700\pi$$
−0.695235 + 0.718783i $$0.744700\pi$$
$$632$$ − 18.9282i − 0.752923i
$$633$$ − 1.07180i − 0.0426001i
$$634$$ −19.6077 −0.778721
$$635$$ 0 0
$$636$$ −9.46410 −0.375276
$$637$$ 0.339746i 0.0134612i
$$638$$ 67.1769i 2.65956i
$$639$$ 4.39230 0.173757
$$640$$ 0 0
$$641$$ 48.5885 1.91913 0.959564 0.281489i $$-0.0908284\pi$$
0.959564 + 0.281489i $$0.0908284\pi$$
$$642$$ 0 0
$$643$$ 12.1962i 0.480969i 0.970653 + 0.240485i $$0.0773064\pi$$
−0.970653 + 0.240485i $$0.922694\pi$$
$$644$$ 9.46410 0.372938
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 4.14359i − 0.162901i −0.996677 0.0814507i $$-0.974045\pi$$
0.996677 0.0814507i $$-0.0259553\pi$$
$$648$$ − 1.73205i − 0.0680414i
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ −24.3923 −0.956010
$$652$$ 9.26795i 0.362961i
$$653$$ − 17.0718i − 0.668071i −0.942560 0.334036i $$-0.891589\pi$$
0.942560 0.334036i $$-0.108411\pi$$
$$654$$ −24.9282 −0.974770
$$655$$ 0 0
$$656$$ −6.33975 −0.247525
$$657$$ − 16.9282i − 0.660432i
$$658$$ 16.3923i 0.639039i
$$659$$ −5.07180 −0.197569 −0.0987846 0.995109i $$-0.531495\pi$$
−0.0987846 + 0.995109i $$0.531495\pi$$
$$660$$ 0 0
$$661$$ 39.1769 1.52381 0.761903 0.647692i $$-0.224265\pi$$
0.761903 + 0.647692i $$0.224265\pi$$
$$662$$ 44.5359i 1.73094i
$$663$$ 0 0
$$664$$ 22.3923 0.868990
$$665$$ 0 0
$$666$$ 10.7321 0.415859
$$667$$ 28.3923i 1.09935i
$$668$$ − 3.46410i − 0.134030i
$$669$$ 17.8564 0.690369
$$670$$ 0 0
$$671$$ −30.9282 −1.19397
$$672$$ − 14.1962i − 0.547628i
$$673$$ − 17.1244i − 0.660095i −0.943964 0.330048i $$-0.892935\pi$$
0.943964 0.330048i $$-0.107065\pi$$
$$674$$ 8.87564 0.341877
$$675$$ 0 0
$$676$$ −12.4641 −0.479389
$$677$$ 0.679492i 0.0261150i 0.999915 + 0.0130575i $$0.00415645\pi$$
−0.999915 + 0.0130575i $$0.995844\pi$$
$$678$$ 32.7846i 1.25909i
$$679$$ −16.9282 −0.649645
$$680$$ 0 0
$$681$$ 10.3923 0.398234
$$682$$ − 73.1769i − 2.80209i
$$683$$ 5.07180i 0.194067i 0.995281 + 0.0970335i $$0.0309354\pi$$
−0.995281 + 0.0970335i $$0.969065\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ −30.9282 −1.18084
$$687$$ − 18.5359i − 0.707189i
$$688$$ 20.9808i 0.799884i
$$689$$ 6.92820 0.263944
$$690$$ 0 0
$$691$$ 12.3923 0.471425 0.235713 0.971823i $$-0.424258\pi$$
0.235713 + 0.971823i $$0.424258\pi$$
$$692$$ 6.92820i 0.263371i
$$693$$ 12.9282i 0.491102i
$$694$$ −1.60770 −0.0610273
$$695$$ 0 0
$$696$$ 14.1962 0.538104
$$697$$ 0 0
$$698$$ − 38.1051i − 1.44230i
$$699$$ 7.85641 0.297157
$$700$$ 0 0
$$701$$ −33.7128 −1.27332 −0.636658 0.771147i $$-0.719684\pi$$
−0.636658 + 0.771147i $$0.719684\pi$$
$$702$$ − 1.26795i − 0.0478557i
$$703$$ 6.19615i 0.233692i
$$704$$ −4.73205 −0.178346
$$705$$ 0 0
$$706$$ 25.6077 0.963758
$$707$$ 28.3923i 1.06780i
$$708$$ − 2.53590i − 0.0953049i
$$709$$ 29.1769 1.09576 0.547881 0.836556i $$-0.315435\pi$$
0.547881 + 0.836556i $$0.315435\pi$$
$$710$$ 0 0
$$711$$ −10.9282 −0.409840
$$712$$ 18.5885i 0.696632i
$$713$$ − 30.9282i − 1.15827i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 23.3205 0.871528
$$717$$ − 9.80385i − 0.366131i
$$718$$ 0.588457i 0.0219610i
$$719$$ 11.6603 0.434854 0.217427 0.976077i $$-0.430234\pi$$
0.217427 + 0.976077i $$0.430234\pi$$
$$720$$ 0 0
$$721$$ −26.9282 −1.00286
$$722$$ − 1.73205i − 0.0644603i
$$723$$ 3.07180i 0.114241i
$$724$$ 2.39230 0.0889093
$$725$$ 0 0
$$726$$ −19.7321 −0.732325
$$727$$ 25.6603i 0.951686i 0.879530 + 0.475843i $$0.157857\pi$$
−0.879530 + 0.475843i $$0.842143\pi$$
$$728$$ 3.46410i 0.128388i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ − 6.53590i − 0.241574i
$$733$$ 18.7846i 0.693825i 0.937897 + 0.346913i $$0.112770\pi$$
−0.937897 + 0.346913i $$0.887230\pi$$
$$734$$ 28.0526 1.03544
$$735$$ 0 0
$$736$$ 18.0000 0.663489
$$737$$ 37.8564i 1.39446i
$$738$$ 2.19615i 0.0808415i
$$739$$ −6.14359 −0.225996 −0.112998 0.993595i $$-0.536045\pi$$
−0.112998 + 0.993595i $$0.536045\pi$$
$$740$$ 0 0
$$741$$ 0.732051 0.0268926
$$742$$ − 44.7846i − 1.64409i
$$743$$ 3.21539i 0.117961i 0.998259 + 0.0589806i $$0.0187850\pi$$
−0.998259 + 0.0589806i $$0.981215\pi$$
$$744$$ −15.4641 −0.566941
$$745$$ 0 0
$$746$$ 10.7321 0.392928
$$747$$ − 12.9282i − 0.473018i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ 17.3205i 0.631614i
$$753$$ − 28.0526i − 1.02229i
$$754$$ 10.3923 0.378465
$$755$$ 0 0
$$756$$ −2.73205 −0.0993637
$$757$$ 32.2487i 1.17210i 0.810275 + 0.586050i $$0.199318\pi$$
−0.810275 + 0.586050i $$0.800682\pi$$
$$758$$ 36.2487i 1.31661i
$$759$$ −16.3923 −0.595003
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 6.92820i 0.250982i
$$763$$ − 39.3205i − 1.42350i
$$764$$ −0.339746 −0.0122916
$$765$$ 0 0
$$766$$ 29.5692 1.06838
$$767$$ 1.85641i 0.0670310i
$$768$$ − 19.0000i − 0.685603i
$$769$$ 20.6410 0.744334 0.372167 0.928166i $$-0.378615\pi$$
0.372167 + 0.928166i $$0.378615\pi$$
$$770$$ 0 0
$$771$$ −24.0000 −0.864339
$$772$$ 17.1244i 0.616319i
$$773$$ 25.1769i 0.905551i 0.891625 + 0.452775i $$0.149566\pi$$
−0.891625 + 0.452775i $$0.850434\pi$$
$$774$$ 7.26795 0.261241
$$775$$ 0 0
$$776$$ −10.7321 −0.385258
$$777$$ 16.9282i 0.607296i
$$778$$ 13.6077i 0.487860i
$$779$$ −1.26795 −0.0454290
$$780$$ 0 0
$$781$$ −20.7846 −0.743732
$$782$$ 0 0
$$783$$ − 8.19615i − 0.292907i
$$784$$ 2.32051 0.0828753
$$785$$ 0 0
$$786$$ 15.8038 0.563705
$$787$$ 8.67949i 0.309390i 0.987962 + 0.154695i $$0.0494396\pi$$
−0.987962 + 0.154695i $$0.950560\pi$$
$$788$$ 24.0000i 0.854965i
$$789$$ −6.00000 −0.213606
$$790$$ 0 0
$$791$$ −51.7128 −1.83870
$$792$$ 8.19615i 0.291238i
$$793$$ 4.78461i 0.169906i
$$794$$ 15.4641 0.548800
$$795$$ 0 0
$$796$$ −15.3205 −0.543021
$$797$$ 44.7846i 1.58635i 0.608992 + 0.793176i $$0.291574\pi$$
−0.608992 + 0.793176i $$0.708426\pi$$
$$798$$ − 4.73205i − 0.167513i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 10.7321 0.379198
$$802$$ 58.9808i 2.08268i
$$803$$ 80.1051i 2.82685i
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ −11.3205 −0.398748
$$807$$ 0.588457i 0.0207147i
$$808$$ 18.0000i 0.633238i
$$809$$ −14.7846 −0.519799 −0.259900 0.965636i $$-0.583689\pi$$
−0.259900 + 0.965636i $$0.583689\pi$$
$$810$$ 0 0
$$811$$ 37.5692 1.31923 0.659617 0.751602i $$-0.270719\pi$$
0.659617 + 0.751602i $$0.270719\pi$$
$$812$$ − 22.3923i − 0.785816i
$$813$$ − 0.392305i − 0.0137587i
$$814$$ −50.7846 −1.78000
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.19615i 0.146805i
$$818$$ − 45.7128i − 1.59831i
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −32.5359 −1.13551 −0.567755 0.823197i $$-0.692188\pi$$
−0.567755 + 0.823197i $$0.692188\pi$$
$$822$$ − 34.3923i − 1.19957i
$$823$$ − 12.9808i − 0.452481i −0.974071 0.226240i $$-0.927356\pi$$
0.974071 0.226240i $$-0.0726435\pi$$
$$824$$ −17.0718 −0.594724
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ − 5.32051i − 0.185012i −0.995712 0.0925061i $$-0.970512\pi$$
0.995712 0.0925061i $$-0.0294878\pi$$
$$828$$ − 3.46410i − 0.120386i
$$829$$ −34.1051 −1.18452 −0.592260 0.805747i $$-0.701764\pi$$
−0.592260 + 0.805747i $$0.701764\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ 0.732051i 0.0253793i
$$833$$ 0 0
$$834$$ −14.5359 −0.503337
$$835$$ 0 0
$$836$$ 4.73205 0.163661
$$837$$ 8.92820i 0.308604i
$$838$$ − 48.5885i − 1.67846i
$$839$$ 19.6077 0.676933 0.338466 0.940978i $$-0.390092\pi$$
0.338466 + 0.940978i $$0.390092\pi$$
$$840$$ 0 0
$$841$$ 38.1769 1.31645
$$842$$ 32.5359i 1.12126i
$$843$$ − 1.26795i − 0.0436705i
$$844$$ −1.07180 −0.0368928
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ − 31.1244i − 1.06945i
$$848$$ − 47.3205i − 1.62499i
$$849$$ −24.9808 −0.857338
$$850$$ 0 0
$$851$$ −21.4641 −0.735780
$$852$$ − 4.39230i − 0.150478i
$$853$$ − 27.1769i − 0.930520i −0.885174 0.465260i $$-0.845961\pi$$
0.885174 0.465260i $$-0.154039\pi$$
$$854$$ 30.9282 1.05834
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 42.2487i − 1.44319i −0.692316 0.721594i $$-0.743410\pi$$
0.692316 0.721594i $$-0.256590\pi$$
$$858$$ 6.00000i 0.204837i
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 0 0
$$861$$ −3.46410 −0.118056
$$862$$ 19.6077i 0.667841i
$$863$$ 12.0000i 0.408485i 0.978920 + 0.204242i $$0.0654731\pi$$
−0.978920 + 0.204242i $$0.934527\pi$$
$$864$$ −5.19615 −0.176777
$$865$$ 0 0
$$866$$ 18.3397 0.623210
$$867$$ − 17.0000i − 0.577350i
$$868$$ 24.3923i 0.827929i
$$869$$ 51.7128 1.75424
$$870$$ 0 0
$$871$$ 5.85641 0.198437
$$872$$ − 24.9282i − 0.844175i
$$873$$ 6.19615i 0.209708i
$$874$$ 6.00000 0.202953
$$875$$ 0 0
$$876$$ −16.9282 −0.571951
$$877$$ 53.1244i 1.79388i 0.442150 + 0.896941i $$0.354216\pi$$
−0.442150 + 0.896941i $$0.645784\pi$$
$$878$$ 46.6410i 1.57406i
$$879$$ −27.7128 −0.934730
$$880$$ 0 0
$$881$$ 8.53590 0.287582 0.143791 0.989608i $$-0.454071\pi$$
0.143791 + 0.989608i $$0.454071\pi$$
$$882$$ − 0.803848i − 0.0270670i
$$883$$ − 36.9808i − 1.24450i −0.782818 0.622251i $$-0.786218\pi$$
0.782818 0.622251i $$-0.213782\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 9.21539 0.309597
$$887$$ − 24.0000i − 0.805841i −0.915235 0.402921i $$-0.867995\pi$$
0.915235 0.402921i $$-0.132005\pi$$
$$888$$ 10.7321i 0.360144i
$$889$$ −10.9282 −0.366520
$$890$$ 0 0
$$891$$ 4.73205 0.158530
$$892$$ − 17.8564i − 0.597877i
$$893$$ 3.46410i 0.115922i
$$894$$ −34.3923 −1.15025
$$895$$ 0 0
$$896$$ 33.1244 1.10661
$$897$$ 2.53590i 0.0846712i
$$898$$ − 9.80385i − 0.327159i
$$899$$ −73.1769 −2.44059
$$900$$ 0 0
$$901$$ 0 0
$$902$$ − 10.3923i − 0.346026i
$$903$$ 11.4641i 0.381501i
$$904$$ −32.7846 −1.09040
$$905$$ 0 0
$$906$$ −24.2487 −0.805609
$$907$$ − 32.3923i − 1.07557i −0.843082 0.537784i $$-0.819261\pi$$
0.843082 0.537784i $$-0.180739\pi$$
$$908$$ − 10.3923i − 0.344881i
$$909$$ 10.3923 0.344691
$$910$$ 0 0
$$911$$ −54.9282 −1.81985 −0.909926 0.414770i $$-0.863862\pi$$
−0.909926 + 0.414770i $$0.863862\pi$$
$$912$$ − 5.00000i − 0.165567i
$$913$$ 61.1769i 2.02466i
$$914$$ 7.85641 0.259867
$$915$$ 0 0
$$916$$ −18.5359 −0.612443
$$917$$ 24.9282i 0.823202i
$$918$$ 0 0
$$919$$ −51.4256 −1.69637 −0.848187 0.529696i $$-0.822306\pi$$
−0.848187 + 0.529696i $$0.822306\pi$$
$$920$$ 0 0
$$921$$ −32.3923 −1.06736
$$922$$ − 10.3923i − 0.342252i
$$923$$ 3.21539i 0.105836i
$$924$$ 12.9282 0.425307
$$925$$ 0 0
$$926$$ 61.5167 2.02156
$$927$$ 9.85641i 0.323727i
$$928$$ − 42.5885i − 1.39803i
$$929$$ 1.60770 0.0527468 0.0263734 0.999652i $$-0.491604\pi$$
0.0263734 + 0.999652i $$0.491604\pi$$
$$930$$ 0 0
$$931$$ 0.464102 0.0152103
$$932$$ − 7.85641i − 0.257345i
$$933$$ − 32.4449i − 1.06220i
$$934$$ 35.5692 1.16386
$$935$$ 0 0
$$936$$ 1.26795 0.0414442
$$937$$ − 16.2487i − 0.530822i −0.964135 0.265411i $$-0.914492\pi$$
0.964135 0.265411i $$-0.0855077\pi$$
$$938$$ − 37.8564i − 1.23606i
$$939$$ −6.39230 −0.208605
$$940$$ 0 0
$$941$$ 0.588457 0.0191832 0.00959158 0.999954i $$-0.496947\pi$$
0.00959158 + 0.999954i $$0.496947\pi$$
$$942$$ − 11.0718i − 0.360739i
$$943$$ − 4.39230i − 0.143033i
$$944$$ 12.6795 0.412682
$$945$$ 0 0
$$946$$ −34.3923 −1.11819
$$947$$ 28.1436i 0.914544i 0.889327 + 0.457272i $$0.151173\pi$$
−0.889327 + 0.457272i $$0.848827\pi$$
$$948$$ 10.9282i 0.354932i
$$949$$ 12.3923 0.402271
$$950$$ 0 0
$$951$$ −11.3205 −0.367093
$$952$$ 0 0
$$953$$ − 37.8564i − 1.22629i −0.789971 0.613145i $$-0.789904\pi$$
0.789971 0.613145i $$-0.210096\pi$$
$$954$$ −16.3923 −0.530720
$$955$$ 0 0
$$956$$ −9.80385 −0.317079
$$957$$ 38.7846i 1.25373i
$$958$$ 44.1962i 1.42791i
$$959$$ 54.2487 1.75178
$$960$$ 0 0
$$961$$ 48.7128 1.57138
$$962$$ 7.85641i 0.253301i
$$963$$ 0 0
$$964$$ 3.07180 0.0989359
$$965$$ 0 0
$$966$$ 16.3923 0.527414
$$967$$ 4.87564i 0.156790i 0.996922 + 0.0783951i $$0.0249796\pi$$
−0.996922 + 0.0783951i $$0.975020\pi$$
$$968$$ − 19.7321i − 0.634212i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −27.7128 −0.889346 −0.444673 0.895693i $$-0.646680\pi$$
−0.444673 + 0.895693i $$0.646680\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 22.9282i − 0.735044i
$$974$$ −56.1051 −1.79772
$$975$$ 0 0
$$976$$ 32.6795 1.04605
$$977$$ − 39.0333i − 1.24879i −0.781110 0.624393i $$-0.785346\pi$$
0.781110 0.624393i $$-0.214654\pi$$
$$978$$ 16.0526i 0.513304i
$$979$$ −50.7846 −1.62308
$$980$$ 0 0
$$981$$ −14.3923 −0.459511
$$982$$ − 27.8038i − 0.887256i
$$983$$ 41.3205i 1.31792i 0.752178 + 0.658960i $$0.229003\pi$$
−0.752178 + 0.658960i $$0.770997\pi$$
$$984$$ −2.19615 −0.0700108
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 9.46410i 0.301246i
$$988$$ − 0.732051i − 0.0232896i
$$989$$ −14.5359 −0.462215
$$990$$ 0 0
$$991$$ 13.0718 0.415239 0.207620 0.978210i $$-0.433428\pi$$
0.207620 + 0.978210i $$0.433428\pi$$
$$992$$ 46.3923i 1.47296i
$$993$$ 25.7128i 0.815971i
$$994$$ 20.7846 0.659248
$$995$$ 0 0
$$996$$ −12.9282 −0.409646
$$997$$ − 17.6077i − 0.557641i −0.960343 0.278821i $$-0.910056\pi$$
0.960343 0.278821i $$-0.0899435\pi$$
$$998$$ 18.2487i 0.577653i
$$999$$ 6.19615 0.196038
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 1425.2.c.k.799.1 4
5.2 odd 4 1425.2.a.o.1.2 2
5.3 odd 4 285.2.a.e.1.1 2
5.4 even 2 inner 1425.2.c.k.799.4 4
15.2 even 4 4275.2.a.t.1.1 2
15.8 even 4 855.2.a.f.1.2 2
20.3 even 4 4560.2.a.bh.1.2 2
95.18 even 4 5415.2.a.r.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.1 2 5.3 odd 4
855.2.a.f.1.2 2 15.8 even 4
1425.2.a.o.1.2 2 5.2 odd 4
1425.2.c.k.799.1 4 1.1 even 1 trivial
1425.2.c.k.799.4 4 5.4 even 2 inner
4275.2.a.t.1.1 2 15.2 even 4
4560.2.a.bh.1.2 2 20.3 even 4
5415.2.a.r.1.2 2 95.18 even 4