# Properties

 Label 1425.2.c.k.799.3 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 799.3 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.k.799.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} +0.732051i q^{7} +1.73205i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} +0.732051i q^{7} +1.73205i q^{8} -1.00000 q^{9} +1.26795 q^{11} +1.00000i q^{12} +2.73205i q^{13} -1.26795 q^{14} -5.00000 q^{16} -1.73205i q^{18} -1.00000 q^{19} +0.732051 q^{21} +2.19615i q^{22} +3.46410i q^{23} +1.73205 q^{24} -4.73205 q^{26} +1.00000i q^{27} -0.732051i q^{28} +2.19615 q^{29} -4.92820 q^{31} -5.19615i q^{32} -1.26795i q^{33} +1.00000 q^{36} +4.19615i q^{37} -1.73205i q^{38} +2.73205 q^{39} +4.73205 q^{41} +1.26795i q^{42} +6.19615i q^{43} -1.26795 q^{44} -6.00000 q^{46} +3.46410i q^{47} +5.00000i q^{48} +6.46410 q^{49} -2.73205i q^{52} +2.53590i q^{53} -1.73205 q^{54} -1.26795 q^{56} +1.00000i q^{57} +3.80385i q^{58} -9.46410 q^{59} -13.4641 q^{61} -8.53590i q^{62} -0.732051i q^{63} -1.00000 q^{64} +2.19615 q^{66} +8.00000i q^{67} +3.46410 q^{69} +16.3923 q^{71} -1.73205i q^{72} +3.07180i q^{73} -7.26795 q^{74} +1.00000 q^{76} +0.928203i q^{77} +4.73205i q^{78} -2.92820 q^{79} +1.00000 q^{81} +8.19615i q^{82} -0.928203i q^{83} -0.732051 q^{84} -10.7321 q^{86} -2.19615i q^{87} +2.19615i q^{88} -7.26795 q^{89} -2.00000 q^{91} -3.46410i q^{92} +4.92820i q^{93} -6.00000 q^{94} -5.19615 q^{96} +4.19615i q^{97} +11.1962i q^{98} -1.26795 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{9}+O(q^{10})$$ 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 $$0.209785\pi$$
−0.790569 + 0.612372i $$0.790215\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.73205 0.707107
$$7$$ 0.732051i 0.276689i 0.990384 + 0.138345i $$0.0441781\pi$$
−0.990384 + 0.138345i $$0.955822\pi$$
$$8$$ 1.73205i 0.612372i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.26795 0.382301 0.191151 0.981561i $$-0.438778\pi$$
0.191151 + 0.981561i $$0.438778\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 2.73205i 0.757735i 0.925451 + 0.378867i $$0.123686\pi$$
−0.925451 + 0.378867i $$0.876314\pi$$
$$14$$ −1.26795 −0.338874
$$15$$ 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
$$21$$ 0.732051 0.159747
$$22$$ 2.19615i 0.468221i
$$23$$ 3.46410i 0.722315i 0.932505 + 0.361158i $$0.117618\pi$$
−0.932505 + 0.361158i $$0.882382\pi$$
$$24$$ 1.73205 0.353553
$$25$$ 0 0
$$26$$ −4.73205 −0.928032
$$27$$ 1.00000i 0.192450i
$$28$$ − 0.732051i − 0.138345i
$$29$$ 2.19615 0.407815 0.203908 0.978990i $$-0.434636\pi$$
0.203908 + 0.978990i $$0.434636\pi$$
$$30$$ 0 0
$$31$$ −4.92820 −0.885131 −0.442566 0.896736i $$-0.645932\pi$$
−0.442566 + 0.896736i $$0.645932\pi$$
$$32$$ − 5.19615i − 0.918559i
$$33$$ − 1.26795i − 0.220722i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 4.19615i 0.689843i 0.938631 + 0.344922i $$0.112095\pi$$
−0.938631 + 0.344922i $$0.887905\pi$$
$$38$$ − 1.73205i − 0.280976i
$$39$$ 2.73205 0.437478
$$40$$ 0 0
$$41$$ 4.73205 0.739022 0.369511 0.929226i $$-0.379525\pi$$
0.369511 + 0.929226i $$0.379525\pi$$
$$42$$ 1.26795i 0.195649i
$$43$$ 6.19615i 0.944904i 0.881356 + 0.472452i $$0.156631\pi$$
−0.881356 + 0.472452i $$0.843369\pi$$
$$44$$ −1.26795 −0.191151
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ 3.46410i 0.505291i 0.967559 + 0.252646i $$0.0813007\pi$$
−0.967559 + 0.252646i $$0.918699\pi$$
$$48$$ 5.00000i 0.721688i
$$49$$ 6.46410 0.923443
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.73205i − 0.378867i
$$53$$ 2.53590i 0.348332i 0.984716 + 0.174166i $$0.0557230\pi$$
−0.984716 + 0.174166i $$0.944277\pi$$
$$54$$ −1.73205 −0.235702
$$55$$ 0 0
$$56$$ −1.26795 −0.169437
$$57$$ 1.00000i 0.132453i
$$58$$ 3.80385i 0.499470i
$$59$$ −9.46410 −1.23212 −0.616061 0.787699i $$-0.711272\pi$$
−0.616061 + 0.787699i $$0.711272\pi$$
$$60$$ 0 0
$$61$$ −13.4641 −1.72390 −0.861951 0.506992i $$-0.830757\pi$$
−0.861951 + 0.506992i $$0.830757\pi$$
$$62$$ − 8.53590i − 1.08406i
$$63$$ − 0.732051i − 0.0922297i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.19615 0.270328
$$67$$ 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
$$71$$ 16.3923 1.94541 0.972704 0.232048i $$-0.0745426\pi$$
0.972704 + 0.232048i $$0.0745426\pi$$
$$72$$ − 1.73205i − 0.204124i
$$73$$ 3.07180i 0.359527i 0.983710 + 0.179763i $$0.0575332\pi$$
−0.983710 + 0.179763i $$0.942467\pi$$
$$74$$ −7.26795 −0.844882
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0.928203i 0.105779i
$$78$$ 4.73205i 0.535799i
$$79$$ −2.92820 −0.329449 −0.164724 0.986340i $$-0.552673\pi$$
−0.164724 + 0.986340i $$0.552673\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 8.19615i 0.905114i
$$83$$ − 0.928203i − 0.101884i −0.998702 0.0509418i $$-0.983778\pi$$
0.998702 0.0509418i $$-0.0162223\pi$$
$$84$$ −0.732051 −0.0798733
$$85$$ 0 0
$$86$$ −10.7321 −1.15727
$$87$$ − 2.19615i − 0.235452i
$$88$$ 2.19615i 0.234111i
$$89$$ −7.26795 −0.770401 −0.385201 0.922833i $$-0.625868\pi$$
−0.385201 + 0.922833i $$0.625868\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ − 3.46410i − 0.361158i
$$93$$ 4.92820i 0.511031i
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ −5.19615 −0.530330
$$97$$ 4.19615i 0.426055i 0.977046 + 0.213027i $$0.0683323\pi$$
−0.977046 + 0.213027i $$0.931668\pi$$
$$98$$ 11.1962i 1.13098i
$$99$$ −1.26795 −0.127434
$$100$$ 0 0
$$101$$ 10.3923 1.03407 0.517036 0.855963i $$-0.327035\pi$$
0.517036 + 0.855963i $$0.327035\pi$$
$$102$$ 0 0
$$103$$ 17.8564i 1.75944i 0.475488 + 0.879722i $$0.342271\pi$$
−0.475488 + 0.879722i $$0.657729\pi$$
$$104$$ −4.73205 −0.464016
$$105$$ 0 0
$$106$$ −4.39230 −0.426618
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −6.39230 −0.612272 −0.306136 0.951988i $$-0.599036\pi$$
−0.306136 + 0.951988i $$0.599036\pi$$
$$110$$ 0 0
$$111$$ 4.19615 0.398281
$$112$$ − 3.66025i − 0.345861i
$$113$$ − 5.07180i − 0.477115i −0.971128 0.238557i $$-0.923326\pi$$
0.971128 0.238557i $$-0.0766745\pi$$
$$114$$ −1.73205 −0.162221
$$115$$ 0 0
$$116$$ −2.19615 −0.203908
$$117$$ − 2.73205i − 0.252578i
$$118$$ − 16.3923i − 1.50903i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −9.39230 −0.853846
$$122$$ − 23.3205i − 2.11134i
$$123$$ − 4.73205i − 0.426675i
$$124$$ 4.92820 0.442566
$$125$$ 0 0
$$126$$ 1.26795 0.112958
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ − 12.1244i − 1.07165i
$$129$$ 6.19615 0.545541
$$130$$ 0 0
$$131$$ 15.1244 1.32142 0.660711 0.750641i $$-0.270255\pi$$
0.660711 + 0.750641i $$0.270255\pi$$
$$132$$ 1.26795i 0.110361i
$$133$$ − 0.732051i − 0.0634769i
$$134$$ −13.8564 −1.19701
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 7.85641i − 0.671218i −0.942001 0.335609i $$-0.891058\pi$$
0.942001 0.335609i $$-0.108942\pi$$
$$138$$ 6.00000i 0.510754i
$$139$$ −12.3923 −1.05110 −0.525551 0.850762i $$-0.676141\pi$$
−0.525551 + 0.850762i $$0.676141\pi$$
$$140$$ 0 0
$$141$$ 3.46410 0.291730
$$142$$ 28.3923i 2.38263i
$$143$$ 3.46410i 0.289683i
$$144$$ 5.00000 0.416667
$$145$$ 0 0
$$146$$ −5.32051 −0.440328
$$147$$ − 6.46410i − 0.533150i
$$148$$ − 4.19615i − 0.344922i
$$149$$ −7.85641 −0.643622 −0.321811 0.946804i $$-0.604292\pi$$
−0.321811 + 0.946804i $$0.604292\pi$$
$$150$$ 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
$$154$$ −1.60770 −0.129552
$$155$$ 0 0
$$156$$ −2.73205 −0.218739
$$157$$ − 14.3923i − 1.14863i −0.818634 0.574315i $$-0.805268\pi$$
0.818634 0.574315i $$-0.194732\pi$$
$$158$$ − 5.07180i − 0.403490i
$$159$$ 2.53590 0.201110
$$160$$ 0 0
$$161$$ −2.53590 −0.199857
$$162$$ 1.73205i 0.136083i
$$163$$ − 12.7321i − 0.997251i −0.866817 0.498626i $$-0.833838\pi$$
0.866817 0.498626i $$-0.166162\pi$$
$$164$$ −4.73205 −0.369511
$$165$$ 0 0
$$166$$ 1.60770 0.124781
$$167$$ − 3.46410i − 0.268060i −0.990977 0.134030i $$-0.957208\pi$$
0.990977 0.134030i $$-0.0427919\pi$$
$$168$$ 1.26795i 0.0978244i
$$169$$ 5.53590 0.425838
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 6.19615i − 0.472452i
$$173$$ 6.92820i 0.526742i 0.964695 + 0.263371i $$0.0848343\pi$$
−0.964695 + 0.263371i $$0.915166\pi$$
$$174$$ 3.80385 0.288369
$$175$$ 0 0
$$176$$ −6.33975 −0.477876
$$177$$ 9.46410i 0.711365i
$$178$$ − 12.5885i − 0.943545i
$$179$$ 11.3205 0.846135 0.423067 0.906098i $$-0.360953\pi$$
0.423067 + 0.906098i $$0.360953\pi$$
$$180$$ 0 0
$$181$$ 18.3923 1.36709 0.683545 0.729909i $$-0.260437\pi$$
0.683545 + 0.729909i $$0.260437\pi$$
$$182$$ − 3.46410i − 0.256776i
$$183$$ 13.4641i 0.995295i
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ −8.53590 −0.625882
$$187$$ 0 0
$$188$$ − 3.46410i − 0.252646i
$$189$$ −0.732051 −0.0532489
$$190$$ 0 0
$$191$$ 17.6603 1.27785 0.638926 0.769269i $$-0.279379\pi$$
0.638926 + 0.769269i $$0.279379\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 7.12436i 0.512822i 0.966568 + 0.256411i $$0.0825401\pi$$
−0.966568 + 0.256411i $$0.917460\pi$$
$$194$$ −7.26795 −0.521808
$$195$$ 0 0
$$196$$ −6.46410 −0.461722
$$197$$ − 24.0000i − 1.70993i −0.518686 0.854965i $$-0.673579\pi$$
0.518686 0.854965i $$-0.326421\pi$$
$$198$$ − 2.19615i − 0.156074i
$$199$$ −19.3205 −1.36959 −0.684797 0.728734i $$-0.740109\pi$$
−0.684797 + 0.728734i $$0.740109\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 18.0000i 1.26648i
$$203$$ 1.60770i 0.112838i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −30.9282 −2.15487
$$207$$ − 3.46410i − 0.240772i
$$208$$ − 13.6603i − 0.947168i
$$209$$ −1.26795 −0.0877059
$$210$$ 0 0
$$211$$ 14.9282 1.02770 0.513850 0.857880i $$-0.328219\pi$$
0.513850 + 0.857880i $$0.328219\pi$$
$$212$$ − 2.53590i − 0.174166i
$$213$$ − 16.3923i − 1.12318i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ −1.73205 −0.117851
$$217$$ − 3.60770i − 0.244906i
$$218$$ − 11.0718i − 0.749877i
$$219$$ 3.07180 0.207573
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 7.26795i 0.487793i
$$223$$ − 9.85641i − 0.660034i −0.943975 0.330017i $$-0.892946\pi$$
0.943975 0.330017i $$-0.107054\pi$$
$$224$$ 3.80385 0.254155
$$225$$ 0 0
$$226$$ 8.78461 0.584344
$$227$$ − 10.3923i − 0.689761i −0.938647 0.344881i $$-0.887919\pi$$
0.938647 0.344881i $$-0.112081\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ 25.4641 1.68272 0.841358 0.540479i $$-0.181757\pi$$
0.841358 + 0.540479i $$0.181757\pi$$
$$230$$ 0 0
$$231$$ 0.928203 0.0610713
$$232$$ 3.80385i 0.249735i
$$233$$ − 19.8564i − 1.30084i −0.759576 0.650418i $$-0.774594\pi$$
0.759576 0.650418i $$-0.225406\pi$$
$$234$$ 4.73205 0.309344
$$235$$ 0 0
$$236$$ 9.46410 0.616061
$$237$$ 2.92820i 0.190207i
$$238$$ 0 0
$$239$$ 20.1962 1.30638 0.653190 0.757194i $$-0.273430\pi$$
0.653190 + 0.757194i $$0.273430\pi$$
$$240$$ 0 0
$$241$$ −16.9282 −1.09044 −0.545221 0.838293i $$-0.683554\pi$$
−0.545221 + 0.838293i $$0.683554\pi$$
$$242$$ − 16.2679i − 1.04574i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 13.4641 0.861951
$$245$$ 0 0
$$246$$ 8.19615 0.522568
$$247$$ − 2.73205i − 0.173836i
$$248$$ − 8.53590i − 0.542030i
$$249$$ −0.928203 −0.0588225
$$250$$ 0 0
$$251$$ −10.0526 −0.634512 −0.317256 0.948340i $$-0.602761\pi$$
−0.317256 + 0.948340i $$0.602761\pi$$
$$252$$ 0.732051i 0.0461149i
$$253$$ 4.39230i 0.276142i
$$254$$ 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 $$-0.269245\pi$$
$$258$$ 10.7321i 0.668148i
$$259$$ −3.07180 −0.190872
$$260$$ 0 0
$$261$$ −2.19615 −0.135938
$$262$$ 26.1962i 1.61840i
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ 2.19615 0.135164
$$265$$ 0 0
$$266$$ 1.26795 0.0777430
$$267$$ 7.26795i 0.444791i
$$268$$ − 8.00000i − 0.488678i
$$269$$ 30.5885 1.86501 0.932506 0.361156i $$-0.117618\pi$$
0.932506 + 0.361156i $$0.117618\pi$$
$$270$$ 0 0
$$271$$ −20.3923 −1.23874 −0.619372 0.785098i $$-0.712613\pi$$
−0.619372 + 0.785098i $$0.712613\pi$$
$$272$$ 0 0
$$273$$ 2.00000i 0.121046i
$$274$$ 13.6077 0.822071
$$275$$ 0 0
$$276$$ −3.46410 −0.208514
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ − 21.4641i − 1.28733i
$$279$$ 4.92820 0.295044
$$280$$ 0 0
$$281$$ 4.73205 0.282290 0.141145 0.989989i $$-0.454922\pi$$
0.141145 + 0.989989i $$0.454922\pi$$
$$282$$ 6.00000i 0.357295i
$$283$$ 26.9808i 1.60384i 0.597432 + 0.801920i $$0.296188\pi$$
−0.597432 + 0.801920i $$0.703812\pi$$
$$284$$ −16.3923 −0.972704
$$285$$ 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
$$291$$ 4.19615 0.245983
$$292$$ − 3.07180i − 0.179763i
$$293$$ 27.7128i 1.61900i 0.587120 + 0.809500i $$0.300262\pi$$
−0.587120 + 0.809500i $$0.699738\pi$$
$$294$$ 11.1962 0.652973
$$295$$ 0 0
$$296$$ −7.26795 −0.422441
$$297$$ 1.26795i 0.0735739i
$$298$$ − 13.6077i − 0.788273i
$$299$$ −9.46410 −0.547323
$$300$$ 0 0
$$301$$ −4.53590 −0.261445
$$302$$ 24.2487i 1.39536i
$$303$$ − 10.3923i − 0.597022i
$$304$$ 5.00000 0.286770
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 11.6077i − 0.662486i −0.943545 0.331243i $$-0.892532\pi$$
0.943545 0.331243i $$-0.107468\pi$$
$$308$$ − 0.928203i − 0.0528893i
$$309$$ 17.8564 1.01582
$$310$$ 0 0
$$311$$ −26.4449 −1.49955 −0.749775 0.661692i $$-0.769838\pi$$
−0.749775 + 0.661692i $$0.769838\pi$$
$$312$$ 4.73205i 0.267900i
$$313$$ 14.3923i 0.813501i 0.913539 + 0.406751i $$0.133338\pi$$
−0.913539 + 0.406751i $$0.866662\pi$$
$$314$$ 24.9282 1.40678
$$315$$ 0 0
$$316$$ 2.92820 0.164724
$$317$$ 23.3205i 1.30981i 0.755711 + 0.654905i $$0.227291\pi$$
−0.755711 + 0.654905i $$0.772709\pi$$
$$318$$ 4.39230i 0.246308i
$$319$$ 2.78461 0.155908
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 4.39230i − 0.244774i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 22.0526 1.22138
$$327$$ 6.39230i 0.353495i
$$328$$ 8.19615i 0.452557i
$$329$$ −2.53590 −0.139809
$$330$$ 0 0
$$331$$ 29.7128 1.63316 0.816582 0.577230i $$-0.195866\pi$$
0.816582 + 0.577230i $$0.195866\pi$$
$$332$$ 0.928203i 0.0509418i
$$333$$ − 4.19615i − 0.229948i
$$334$$ 6.00000 0.328305
$$335$$ 0 0
$$336$$ −3.66025 −0.199683
$$337$$ − 19.1244i − 1.04177i −0.853627 0.520885i $$-0.825602\pi$$
0.853627 0.520885i $$-0.174398\pi$$
$$338$$ 9.58846i 0.521543i
$$339$$ −5.07180 −0.275462
$$340$$ 0 0
$$341$$ −6.24871 −0.338387
$$342$$ 1.73205i 0.0936586i
$$343$$ 9.85641i 0.532196i
$$344$$ −10.7321 −0.578633
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ 12.9282i 0.694022i 0.937861 + 0.347011i $$0.112803\pi$$
−0.937861 + 0.347011i $$0.887197\pi$$
$$348$$ 2.19615i 0.117726i
$$349$$ 22.0000 1.17763 0.588817 0.808267i $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ 0 0
$$351$$ −2.73205 −0.145826
$$352$$ − 6.58846i − 0.351166i
$$353$$ − 26.7846i − 1.42560i −0.701367 0.712800i $$-0.747427\pi$$
0.701367 0.712800i $$-0.252573\pi$$
$$354$$ −16.3923 −0.871241
$$355$$ 0 0
$$356$$ 7.26795 0.385201
$$357$$ 0 0
$$358$$ 19.6077i 1.03630i
$$359$$ −17.6603 −0.932073 −0.466036 0.884766i $$-0.654318\pi$$
−0.466036 + 0.884766i $$0.654318\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 31.8564i 1.67434i
$$363$$ 9.39230i 0.492968i
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ −23.3205 −1.21898
$$367$$ 5.80385i 0.302958i 0.988460 + 0.151479i $$0.0484037\pi$$
−0.988460 + 0.151479i $$0.951596\pi$$
$$368$$ − 17.3205i − 0.902894i
$$369$$ −4.73205 −0.246341
$$370$$ 0 0
$$371$$ −1.85641 −0.0963798
$$372$$ − 4.92820i − 0.255515i
$$373$$ − 4.19615i − 0.217269i −0.994082 0.108634i $$-0.965352\pi$$
0.994082 0.108634i $$-0.0346477\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ 6.00000i 0.309016i
$$378$$ − 1.26795i − 0.0652163i
$$379$$ −7.07180 −0.363254 −0.181627 0.983368i $$-0.558136\pi$$
−0.181627 + 0.983368i $$0.558136\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 30.5885i 1.56504i
$$383$$ 30.9282i 1.58036i 0.612877 + 0.790179i $$0.290012\pi$$
−0.612877 + 0.790179i $$0.709988\pi$$
$$384$$ −12.1244 −0.618718
$$385$$ 0 0
$$386$$ −12.3397 −0.628077
$$387$$ − 6.19615i − 0.314968i
$$388$$ − 4.19615i − 0.213027i
$$389$$ 19.8564 1.00676 0.503380 0.864065i $$-0.332090\pi$$
0.503380 + 0.864065i $$0.332090\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 11.1962i 0.565491i
$$393$$ − 15.1244i − 0.762923i
$$394$$ 41.5692 2.09423
$$395$$ 0 0
$$396$$ 1.26795 0.0637168
$$397$$ − 4.92820i − 0.247339i −0.992323 0.123670i $$-0.960534\pi$$
0.992323 0.123670i $$-0.0394663\pi$$
$$398$$ − 33.4641i − 1.67740i
$$399$$ −0.732051 −0.0366484
$$400$$ 0 0
$$401$$ 4.05256 0.202375 0.101188 0.994867i $$-0.467736\pi$$
0.101188 + 0.994867i $$0.467736\pi$$
$$402$$ 13.8564i 0.691095i
$$403$$ − 13.4641i − 0.670695i
$$404$$ −10.3923 −0.517036
$$405$$ 0 0
$$406$$ −2.78461 −0.138198
$$407$$ 5.32051i 0.263728i
$$408$$ 0 0
$$409$$ 5.60770 0.277283 0.138641 0.990343i $$-0.455726\pi$$
0.138641 + 0.990343i $$0.455726\pi$$
$$410$$ 0 0
$$411$$ −7.85641 −0.387528
$$412$$ − 17.8564i − 0.879722i
$$413$$ − 6.92820i − 0.340915i
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ 14.1962 0.696024
$$417$$ 12.3923i 0.606854i
$$418$$ − 2.19615i − 0.107417i
$$419$$ −10.0526 −0.491100 −0.245550 0.969384i $$-0.578968\pi$$
−0.245550 + 0.969384i $$0.578968\pi$$
$$420$$ 0 0
$$421$$ 22.7846 1.11045 0.555227 0.831699i $$-0.312631\pi$$
0.555227 + 0.831699i $$0.312631\pi$$
$$422$$ 25.8564i 1.25867i
$$423$$ − 3.46410i − 0.168430i
$$424$$ −4.39230 −0.213309
$$425$$ 0 0
$$426$$ 28.3923 1.37561
$$427$$ − 9.85641i − 0.476985i
$$428$$ 0 0
$$429$$ 3.46410 0.167248
$$430$$ 0 0
$$431$$ 23.3205 1.12331 0.561655 0.827372i $$-0.310165\pi$$
0.561655 + 0.827372i $$0.310165\pi$$
$$432$$ − 5.00000i − 0.240563i
$$433$$ − 20.5885i − 0.989418i −0.869059 0.494709i $$-0.835275\pi$$
0.869059 0.494709i $$-0.164725\pi$$
$$434$$ 6.24871 0.299948
$$435$$ 0 0
$$436$$ 6.39230 0.306136
$$437$$ − 3.46410i − 0.165710i
$$438$$ 5.32051i 0.254224i
$$439$$ −13.0718 −0.623883 −0.311941 0.950101i $$-0.600979\pi$$
−0.311941 + 0.950101i $$0.600979\pi$$
$$440$$ 0 0
$$441$$ −6.46410 −0.307814
$$442$$ 0 0
$$443$$ − 29.3205i − 1.39306i −0.717528 0.696530i $$-0.754726\pi$$
0.717528 0.696530i $$-0.245274\pi$$
$$444$$ −4.19615 −0.199141
$$445$$ 0 0
$$446$$ 17.0718 0.808373
$$447$$ 7.85641i 0.371595i
$$448$$ − 0.732051i − 0.0345861i
$$449$$ −11.6603 −0.550281 −0.275141 0.961404i $$-0.588724\pi$$
−0.275141 + 0.961404i $$0.588724\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ 5.07180i 0.238557i
$$453$$ − 14.0000i − 0.657777i
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ −1.73205 −0.0811107
$$457$$ 11.4641i 0.536268i 0.963382 + 0.268134i $$0.0864070\pi$$
−0.963382 + 0.268134i $$0.913593\pi$$
$$458$$ 44.1051i 2.06090i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 1.60770i 0.0747967i
$$463$$ − 9.51666i − 0.442277i −0.975242 0.221138i $$-0.929023\pi$$
0.975242 0.221138i $$-0.0709772\pi$$
$$464$$ −10.9808 −0.509769
$$465$$ 0 0
$$466$$ 34.3923 1.59319
$$467$$ 27.4641i 1.27089i 0.772147 + 0.635444i $$0.219183\pi$$
−0.772147 + 0.635444i $$0.780817\pi$$
$$468$$ 2.73205i 0.126289i
$$469$$ −5.85641 −0.270424
$$470$$ 0 0
$$471$$ −14.3923 −0.663162
$$472$$ − 16.3923i − 0.754517i
$$473$$ 7.85641i 0.361238i
$$474$$ −5.07180 −0.232955
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 2.53590i − 0.116111i
$$478$$ 34.9808i 1.59998i
$$479$$ 19.5167 0.891739 0.445869 0.895098i $$-0.352895\pi$$
0.445869 + 0.895098i $$0.352895\pi$$
$$480$$ 0 0
$$481$$ −11.4641 −0.522718
$$482$$ − 29.3205i − 1.33551i
$$483$$ 2.53590i 0.115387i
$$484$$ 9.39230 0.426923
$$485$$ 0 0
$$486$$ 1.73205 0.0785674
$$487$$ − 11.6077i − 0.525995i −0.964797 0.262997i $$-0.915289\pi$$
0.964797 0.262997i $$-0.0847111\pi$$
$$488$$ − 23.3205i − 1.05567i
$$489$$ −12.7321 −0.575763
$$490$$ 0 0
$$491$$ −22.0526 −0.995218 −0.497609 0.867401i $$-0.665789\pi$$
−0.497609 + 0.867401i $$0.665789\pi$$
$$492$$ 4.73205i 0.213337i
$$493$$ 0 0
$$494$$ 4.73205 0.212905
$$495$$ 0 0
$$496$$ 24.6410 1.10641
$$497$$ 12.0000i 0.538274i
$$498$$ − 1.60770i − 0.0720425i
$$499$$ −17.4641 −0.781801 −0.390900 0.920433i $$-0.627836\pi$$
−0.390900 + 0.920433i $$0.627836\pi$$
$$500$$ 0 0
$$501$$ −3.46410 −0.154765
$$502$$ − 17.4115i − 0.777115i
$$503$$ 36.9282i 1.64655i 0.567645 + 0.823274i $$0.307855\pi$$
−0.567645 + 0.823274i $$0.692145\pi$$
$$504$$ 1.26795 0.0564789
$$505$$ 0 0
$$506$$ −7.60770 −0.338203
$$507$$ − 5.53590i − 0.245858i
$$508$$ 4.00000i 0.177471i
$$509$$ −28.0526 −1.24341 −0.621704 0.783252i $$-0.713559\pi$$
−0.621704 + 0.783252i $$0.713559\pi$$
$$510$$ 0 0
$$511$$ −2.24871 −0.0994771
$$512$$ 8.66025i 0.382733i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ 41.5692 1.83354
$$515$$ 0 0
$$516$$ −6.19615 −0.272770
$$517$$ 4.39230i 0.193173i
$$518$$ − 5.32051i − 0.233770i
$$519$$ 6.92820 0.304114
$$520$$ 0 0
$$521$$ 40.7321 1.78450 0.892252 0.451538i $$-0.149125\pi$$
0.892252 + 0.451538i $$0.149125\pi$$
$$522$$ − 3.80385i − 0.166490i
$$523$$ − 43.3205i − 1.89427i −0.320830 0.947137i $$-0.603962\pi$$
0.320830 0.947137i $$-0.396038\pi$$
$$524$$ −15.1244 −0.660711
$$525$$ 0 0
$$526$$ 10.3923 0.453126
$$527$$ 0 0
$$528$$ 6.33975i 0.275902i
$$529$$ 11.0000 0.478261
$$530$$ 0 0
$$531$$ 9.46410 0.410707
$$532$$ 0.732051i 0.0317384i
$$533$$ 12.9282i 0.559983i
$$534$$ −12.5885 −0.544756
$$535$$ 0 0
$$536$$ −13.8564 −0.598506
$$537$$ − 11.3205i − 0.488516i
$$538$$ 52.9808i 2.28416i
$$539$$ 8.19615 0.353033
$$540$$ 0 0
$$541$$ −13.7128 −0.589560 −0.294780 0.955565i $$-0.595246\pi$$
−0.294780 + 0.955565i $$0.595246\pi$$
$$542$$ − 35.3205i − 1.51715i
$$543$$ − 18.3923i − 0.789289i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −3.46410 −0.148250
$$547$$ 8.67949i 0.371108i 0.982634 + 0.185554i $$0.0594080\pi$$
−0.982634 + 0.185554i $$0.940592\pi$$
$$548$$ 7.85641i 0.335609i
$$549$$ 13.4641 0.574634
$$550$$ 0 0
$$551$$ −2.19615 −0.0935592
$$552$$ 6.00000i 0.255377i
$$553$$ − 2.14359i − 0.0911549i
$$554$$ −3.46410 −0.147176
$$555$$ 0 0
$$556$$ 12.3923 0.525551
$$557$$ − 12.9282i − 0.547786i −0.961760 0.273893i $$-0.911689\pi$$
0.961760 0.273893i $$-0.0883113\pi$$
$$558$$ 8.53590i 0.361353i
$$559$$ −16.9282 −0.715987
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 8.19615i 0.345734i
$$563$$ 20.5359i 0.865485i 0.901518 + 0.432742i $$0.142454\pi$$
−0.901518 + 0.432742i $$0.857546\pi$$
$$564$$ −3.46410 −0.145865
$$565$$ 0 0
$$566$$ −46.7321 −1.96429
$$567$$ 0.732051i 0.0307432i
$$568$$ 28.3923i 1.19131i
$$569$$ 16.0526 0.672958 0.336479 0.941691i $$-0.390764\pi$$
0.336479 + 0.941691i $$0.390764\pi$$
$$570$$ 0 0
$$571$$ 14.2487 0.596290 0.298145 0.954521i $$-0.403632\pi$$
0.298145 + 0.954521i $$0.403632\pi$$
$$572$$ − 3.46410i − 0.144841i
$$573$$ − 17.6603i − 0.737768i
$$574$$ −6.00000 −0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 47.1769i − 1.96400i −0.188879 0.982000i $$-0.560485\pi$$
0.188879 0.982000i $$-0.439515\pi$$
$$578$$ 29.4449i 1.22474i
$$579$$ 7.12436 0.296078
$$580$$ 0 0
$$581$$ 0.679492 0.0281901
$$582$$ 7.26795i 0.301266i
$$583$$ 3.21539i 0.133168i
$$584$$ −5.32051 −0.220164
$$585$$ 0 0
$$586$$ −48.0000 −1.98286
$$587$$ 3.46410i 0.142979i 0.997441 + 0.0714894i $$0.0227752\pi$$
−0.997441 + 0.0714894i $$0.977225\pi$$
$$588$$ 6.46410i 0.266575i
$$589$$ 4.92820 0.203063
$$590$$ 0 0
$$591$$ −24.0000 −0.987228
$$592$$ − 20.9808i − 0.862304i
$$593$$ − 2.78461i − 0.114350i −0.998364 0.0571751i $$-0.981791\pi$$
0.998364 0.0571751i $$-0.0182093\pi$$
$$594$$ −2.19615 −0.0901092
$$595$$ 0 0
$$596$$ 7.85641 0.321811
$$597$$ 19.3205i 0.790736i
$$598$$ − 16.3923i − 0.670331i
$$599$$ −13.8564 −0.566157 −0.283079 0.959097i $$-0.591356\pi$$
−0.283079 + 0.959097i $$0.591356\pi$$
$$600$$ 0 0
$$601$$ 15.1769 0.619079 0.309540 0.950887i $$-0.399825\pi$$
0.309540 + 0.950887i $$0.399825\pi$$
$$602$$ − 7.85641i − 0.320203i
$$603$$ − 8.00000i − 0.325785i
$$604$$ −14.0000 −0.569652
$$605$$ 0 0
$$606$$ 18.0000 0.731200
$$607$$ − 32.3923i − 1.31476i −0.753558 0.657382i $$-0.771664\pi$$
0.753558 0.657382i $$-0.228336\pi$$
$$608$$ 5.19615i 0.210732i
$$609$$ 1.60770 0.0651471
$$610$$ 0 0
$$611$$ −9.46410 −0.382877
$$612$$ 0 0
$$613$$ − 21.6077i − 0.872727i −0.899771 0.436363i $$-0.856266\pi$$
0.899771 0.436363i $$-0.143734\pi$$
$$614$$ 20.1051 0.811377
$$615$$ 0 0
$$616$$ −1.60770 −0.0647759
$$617$$ − 27.7128i − 1.11568i −0.829950 0.557838i $$-0.811631\pi$$
0.829950 0.557838i $$-0.188369\pi$$
$$618$$ 30.9282i 1.24411i
$$619$$ −19.3205 −0.776557 −0.388278 0.921542i $$-0.626930\pi$$
−0.388278 + 0.921542i $$0.626930\pi$$
$$620$$ 0 0
$$621$$ −3.46410 −0.139010
$$622$$ − 45.8038i − 1.83657i
$$623$$ − 5.32051i − 0.213162i
$$624$$ −13.6603 −0.546848
$$625$$ 0 0
$$626$$ −24.9282 −0.996331
$$627$$ 1.26795i 0.0506370i
$$628$$ 14.3923i 0.574315i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −21.0718 −0.838855 −0.419427 0.907789i $$-0.637769\pi$$
−0.419427 + 0.907789i $$0.637769\pi$$
$$632$$ − 5.07180i − 0.201745i
$$633$$ − 14.9282i − 0.593343i
$$634$$ −40.3923 −1.60418
$$635$$ 0 0
$$636$$ −2.53590 −0.100555
$$637$$ 17.6603i 0.699725i
$$638$$ 4.82309i 0.190948i
$$639$$ −16.3923 −0.648470
$$640$$ 0 0
$$641$$ 17.4115 0.687715 0.343857 0.939022i $$-0.388266\pi$$
0.343857 + 0.939022i $$0.388266\pi$$
$$642$$ 0 0
$$643$$ 1.80385i 0.0711368i 0.999367 + 0.0355684i $$0.0113242\pi$$
−0.999367 + 0.0355684i $$0.988676\pi$$
$$644$$ 2.53590 0.0999284
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 31.8564i − 1.25240i −0.779661 0.626202i $$-0.784608\pi$$
0.779661 0.626202i $$-0.215392\pi$$
$$648$$ 1.73205i 0.0680414i
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ −3.60770 −0.141397
$$652$$ 12.7321i 0.498626i
$$653$$ − 30.9282i − 1.21031i −0.796106 0.605157i $$-0.793110\pi$$
0.796106 0.605157i $$-0.206890\pi$$
$$654$$ −11.0718 −0.432942
$$655$$ 0 0
$$656$$ −23.6603 −0.923778
$$657$$ − 3.07180i − 0.119842i
$$658$$ − 4.39230i − 0.171230i
$$659$$ −18.9282 −0.737338 −0.368669 0.929561i $$-0.620186\pi$$
−0.368669 + 0.929561i $$0.620186\pi$$
$$660$$ 0 0
$$661$$ −23.1769 −0.901477 −0.450739 0.892656i $$-0.648839\pi$$
−0.450739 + 0.892656i $$0.648839\pi$$
$$662$$ 51.4641i 2.00021i
$$663$$ 0 0
$$664$$ 1.60770 0.0623907
$$665$$ 0 0
$$666$$ 7.26795 0.281627
$$667$$ 7.60770i 0.294571i
$$668$$ 3.46410i 0.134030i
$$669$$ −9.85641 −0.381071
$$670$$ 0 0
$$671$$ −17.0718 −0.659049
$$672$$ − 3.80385i − 0.146737i
$$673$$ 7.12436i 0.274624i 0.990528 + 0.137312i $$0.0438463\pi$$
−0.990528 + 0.137312i $$0.956154\pi$$
$$674$$ 33.1244 1.27590
$$675$$ 0 0
$$676$$ −5.53590 −0.212919
$$677$$ 35.3205i 1.35748i 0.734380 + 0.678739i $$0.237473\pi$$
−0.734380 + 0.678739i $$0.762527\pi$$
$$678$$ − 8.78461i − 0.337371i
$$679$$ −3.07180 −0.117885
$$680$$ 0 0
$$681$$ −10.3923 −0.398234
$$682$$ − 10.8231i − 0.414437i
$$683$$ 18.9282i 0.724268i 0.932126 + 0.362134i $$0.117952\pi$$
−0.932126 + 0.362134i $$0.882048\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ −17.0718 −0.651804
$$687$$ − 25.4641i − 0.971516i
$$688$$ − 30.9808i − 1.18113i
$$689$$ −6.92820 −0.263944
$$690$$ 0 0
$$691$$ −8.39230 −0.319258 −0.159629 0.987177i $$-0.551030\pi$$
−0.159629 + 0.987177i $$0.551030\pi$$
$$692$$ − 6.92820i − 0.263371i
$$693$$ − 0.928203i − 0.0352595i
$$694$$ −22.3923 −0.850000
$$695$$ 0 0
$$696$$ 3.80385 0.144184
$$697$$ 0 0
$$698$$ 38.1051i 1.44230i
$$699$$ −19.8564 −0.751038
$$700$$ 0 0
$$701$$ 21.7128 0.820082 0.410041 0.912067i $$-0.365514\pi$$
0.410041 + 0.912067i $$0.365514\pi$$
$$702$$ − 4.73205i − 0.178600i
$$703$$ − 4.19615i − 0.158261i
$$704$$ −1.26795 −0.0477876
$$705$$ 0 0
$$706$$ 46.3923 1.74600
$$707$$ 7.60770i 0.286117i
$$708$$ − 9.46410i − 0.355683i
$$709$$ −33.1769 −1.24599 −0.622993 0.782228i $$-0.714083\pi$$
−0.622993 + 0.782228i $$0.714083\pi$$
$$710$$ 0 0
$$711$$ 2.92820 0.109816
$$712$$ − 12.5885i − 0.471772i
$$713$$ − 17.0718i − 0.639344i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −11.3205 −0.423067
$$717$$ − 20.1962i − 0.754239i
$$718$$ − 30.5885i − 1.14155i
$$719$$ −5.66025 −0.211092 −0.105546 0.994414i $$-0.533659\pi$$
−0.105546 + 0.994414i $$0.533659\pi$$
$$720$$ 0 0
$$721$$ −13.0718 −0.486819
$$722$$ 1.73205i 0.0644603i
$$723$$ 16.9282i 0.629567i
$$724$$ −18.3923 −0.683545
$$725$$ 0 0
$$726$$ −16.2679 −0.603760
$$727$$ 8.33975i 0.309304i 0.987969 + 0.154652i $$0.0494256\pi$$
−0.987969 + 0.154652i $$0.950574\pi$$
$$728$$ − 3.46410i − 0.128388i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ − 13.4641i − 0.497648i
$$733$$ − 22.7846i − 0.841569i −0.907161 0.420784i $$-0.861755\pi$$
0.907161 0.420784i $$-0.138245\pi$$
$$734$$ −10.0526 −0.371047
$$735$$ 0 0
$$736$$ 18.0000 0.663489
$$737$$ 10.1436i 0.373644i
$$738$$ − 8.19615i − 0.301705i
$$739$$ −33.8564 −1.24543 −0.622714 0.782450i $$-0.713970\pi$$
−0.622714 + 0.782450i $$0.713970\pi$$
$$740$$ 0 0
$$741$$ −2.73205 −0.100364
$$742$$ − 3.21539i − 0.118041i
$$743$$ 44.7846i 1.64299i 0.570217 + 0.821494i $$0.306859\pi$$
−0.570217 + 0.821494i $$0.693141\pi$$
$$744$$ −8.53590 −0.312941
$$745$$ 0 0
$$746$$ 7.26795 0.266099
$$747$$ 0.928203i 0.0339612i
$$748$$ 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
$$753$$ 10.0526i 0.366336i
$$754$$ −10.3923 −0.378465
$$755$$ 0 0
$$756$$ 0.732051 0.0266244
$$757$$ − 16.2487i − 0.590569i −0.955409 0.295285i $$-0.904585\pi$$
0.955409 0.295285i $$-0.0954145\pi$$
$$758$$ − 12.2487i − 0.444893i
$$759$$ 4.39230 0.159431
$$760$$ 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
$$763$$ − 4.67949i − 0.169409i
$$764$$ −17.6603 −0.638926
$$765$$ 0 0
$$766$$ −53.5692 −1.93553
$$767$$ − 25.8564i − 0.933621i
$$768$$ − 19.0000i − 0.685603i
$$769$$ −48.6410 −1.75404 −0.877020 0.480454i $$-0.840472\pi$$
−0.877020 + 0.480454i $$0.840472\pi$$
$$770$$ 0 0
$$771$$ −24.0000 −0.864339
$$772$$ − 7.12436i − 0.256411i
$$773$$ − 37.1769i − 1.33716i −0.743640 0.668580i $$-0.766902\pi$$
0.743640 0.668580i $$-0.233098\pi$$
$$774$$ 10.7321 0.385756
$$775$$ 0 0
$$776$$ −7.26795 −0.260904
$$777$$ 3.07180i 0.110200i
$$778$$ 34.3923i 1.23302i
$$779$$ −4.73205 −0.169543
$$780$$ 0 0
$$781$$ 20.7846 0.743732
$$782$$ 0 0
$$783$$ 2.19615i 0.0784841i
$$784$$ −32.3205 −1.15430
$$785$$ 0 0
$$786$$ 26.1962 0.934386
$$787$$ 43.3205i 1.54421i 0.635495 + 0.772105i $$0.280796\pi$$
−0.635495 + 0.772105i $$0.719204\pi$$
$$788$$ 24.0000i 0.854965i
$$789$$ −6.00000 −0.213606
$$790$$ 0 0
$$791$$ 3.71281 0.132012
$$792$$ − 2.19615i − 0.0780369i
$$793$$ − 36.7846i − 1.30626i
$$794$$ 8.53590 0.302928
$$795$$ 0 0
$$796$$ 19.3205 0.684797
$$797$$ 3.21539i 0.113895i 0.998377 + 0.0569475i $$0.0181368\pi$$
−0.998377 + 0.0569475i $$0.981863\pi$$
$$798$$ − 1.26795i − 0.0448849i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 7.26795 0.256800
$$802$$ 7.01924i 0.247858i
$$803$$ 3.89488i 0.137447i
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 23.3205 0.821430
$$807$$ − 30.5885i − 1.07676i
$$808$$ 18.0000i 0.633238i
$$809$$ 26.7846 0.941697 0.470848 0.882214i $$-0.343948\pi$$
0.470848 + 0.882214i $$0.343948\pi$$
$$810$$ 0 0
$$811$$ −45.5692 −1.60015 −0.800076 0.599899i $$-0.795207\pi$$
−0.800076 + 0.599899i $$0.795207\pi$$
$$812$$ − 1.60770i − 0.0564190i
$$813$$ 20.3923i 0.715189i
$$814$$ −9.21539 −0.322999
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 6.19615i − 0.216776i
$$818$$ 9.71281i 0.339601i
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −39.4641 −1.37731 −0.688653 0.725091i $$-0.741798\pi$$
−0.688653 + 0.725091i $$0.741798\pi$$
$$822$$ − 13.6077i − 0.474623i
$$823$$ 38.9808i 1.35878i 0.733776 + 0.679392i $$0.237756\pi$$
−0.733776 + 0.679392i $$0.762244\pi$$
$$824$$ −30.9282 −1.07744
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 29.3205i 1.01957i 0.860301 + 0.509787i $$0.170276\pi$$
−0.860301 + 0.509787i $$0.829724\pi$$
$$828$$ 3.46410i 0.120386i
$$829$$ 42.1051 1.46237 0.731186 0.682179i $$-0.238967\pi$$
0.731186 + 0.682179i $$0.238967\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ − 2.73205i − 0.0947168i
$$833$$ 0 0
$$834$$ −21.4641 −0.743241
$$835$$ 0 0
$$836$$ 1.26795 0.0438529
$$837$$ − 4.92820i − 0.170344i
$$838$$ − 17.4115i − 0.601472i
$$839$$ 40.3923 1.39450 0.697249 0.716829i $$-0.254407\pi$$
0.697249 + 0.716829i $$0.254407\pi$$
$$840$$ 0 0
$$841$$ −24.1769 −0.833687
$$842$$ 39.4641i 1.36002i
$$843$$ − 4.73205i − 0.162980i
$$844$$ −14.9282 −0.513850
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ − 6.87564i − 0.236250i
$$848$$ − 12.6795i − 0.435416i
$$849$$ 26.9808 0.925977
$$850$$ 0 0
$$851$$ −14.5359 −0.498284
$$852$$ 16.3923i 0.561591i
$$853$$ 35.1769i 1.20443i 0.798332 + 0.602217i $$0.205716\pi$$
−0.798332 + 0.602217i $$0.794284\pi$$
$$854$$ 17.0718 0.584185
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 6.24871i 0.213452i 0.994288 + 0.106726i $$0.0340368\pi$$
−0.994288 + 0.106726i $$0.965963\pi$$
$$858$$ 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
$$862$$ 40.3923i 1.37577i
$$863$$ 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
$$866$$ 35.6603 1.21178
$$867$$ − 17.0000i − 0.577350i
$$868$$ 3.60770i 0.122453i
$$869$$ −3.71281 −0.125949
$$870$$ 0 0
$$871$$ −21.8564 −0.740576
$$872$$ − 11.0718i − 0.374938i
$$873$$ − 4.19615i − 0.142018i
$$874$$ 6.00000 0.202953
$$875$$ 0 0
$$876$$ −3.07180 −0.103786
$$877$$ 28.8756i 0.975061i 0.873106 + 0.487531i $$0.162102\pi$$
−0.873106 + 0.487531i $$0.837898\pi$$
$$878$$ − 22.6410i − 0.764097i
$$879$$ 27.7128 0.934730
$$880$$ 0 0
$$881$$ 15.4641 0.520999 0.260499 0.965474i $$-0.416113\pi$$
0.260499 + 0.965474i $$0.416113\pi$$
$$882$$ − 11.1962i − 0.376994i
$$883$$ 14.9808i 0.504143i 0.967709 + 0.252071i $$0.0811118\pi$$
−0.967709 + 0.252071i $$0.918888\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 50.7846 1.70614
$$887$$ − 24.0000i − 0.805841i −0.915235 0.402921i $$-0.867995\pi$$
0.915235 0.402921i $$-0.132005\pi$$
$$888$$ 7.26795i 0.243896i
$$889$$ 2.92820 0.0982088
$$890$$ 0 0
$$891$$ 1.26795 0.0424779
$$892$$ 9.85641i 0.330017i
$$893$$ − 3.46410i − 0.115922i
$$894$$ −13.6077 −0.455109
$$895$$ 0 0
$$896$$ 8.87564 0.296514
$$897$$ 9.46410i 0.315997i
$$898$$ − 20.1962i − 0.673954i
$$899$$ −10.8231 −0.360970
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 10.3923i 0.346026i
$$903$$ 4.53590i 0.150945i
$$904$$ 8.78461 0.292172
$$905$$ 0 0
$$906$$ 24.2487 0.805609
$$907$$ − 11.6077i − 0.385427i −0.981255 0.192714i $$-0.938271\pi$$
0.981255 0.192714i $$-0.0617288\pi$$
$$908$$ 10.3923i 0.344881i
$$909$$ −10.3923 −0.344691
$$910$$ 0 0
$$911$$ −41.0718 −1.36077 −0.680385 0.732855i $$-0.738187\pi$$
−0.680385 + 0.732855i $$0.738187\pi$$
$$912$$ − 5.00000i − 0.165567i
$$913$$ − 1.17691i − 0.0389502i
$$914$$ −19.8564 −0.656792
$$915$$ 0 0
$$916$$ −25.4641 −0.841358
$$917$$ 11.0718i 0.365623i
$$918$$ 0 0
$$919$$ 59.4256 1.96027 0.980135 0.198330i $$-0.0635518\pi$$
0.980135 + 0.198330i $$0.0635518\pi$$
$$920$$ 0 0
$$921$$ −11.6077 −0.382487
$$922$$ 10.3923i 0.342252i
$$923$$ 44.7846i 1.47410i
$$924$$ −0.928203 −0.0305356
$$925$$ 0 0
$$926$$ 16.4833 0.541676
$$927$$ − 17.8564i − 0.586481i
$$928$$ − 11.4115i − 0.374602i
$$929$$ 22.3923 0.734668 0.367334 0.930089i $$-0.380271\pi$$
0.367334 + 0.930089i $$0.380271\pi$$
$$930$$ 0 0
$$931$$ −6.46410 −0.211852
$$932$$ 19.8564i 0.650418i
$$933$$ 26.4449i 0.865766i
$$934$$ −47.5692 −1.55651
$$935$$ 0 0
$$936$$ 4.73205 0.154672
$$937$$ 32.2487i 1.05352i 0.850014 + 0.526760i $$0.176593\pi$$
−0.850014 + 0.526760i $$0.823407\pi$$
$$938$$ − 10.1436i − 0.331200i
$$939$$ 14.3923 0.469675
$$940$$ 0 0
$$941$$ −30.5885 −0.997155 −0.498578 0.866845i $$-0.666144\pi$$
−0.498578 + 0.866845i $$0.666144\pi$$
$$942$$ − 24.9282i − 0.812205i
$$943$$ 16.3923i 0.533807i
$$944$$ 47.3205 1.54015
$$945$$ 0 0
$$946$$ −13.6077 −0.442424
$$947$$ 55.8564i 1.81509i 0.419956 + 0.907545i $$0.362046\pi$$
−0.419956 + 0.907545i $$0.637954\pi$$
$$948$$ − 2.92820i − 0.0951036i
$$949$$ −8.39230 −0.272426
$$950$$ 0 0
$$951$$ 23.3205 0.756219
$$952$$ 0 0
$$953$$ − 10.1436i − 0.328583i −0.986412 0.164292i $$-0.947466\pi$$
0.986412 0.164292i $$-0.0525338\pi$$
$$954$$ 4.39230 0.142206
$$955$$ 0 0
$$956$$ −20.1962 −0.653190
$$957$$ − 2.78461i − 0.0900136i
$$958$$ 33.8038i 1.09215i
$$959$$ 5.75129 0.185719
$$960$$ 0 0
$$961$$ −6.71281 −0.216542
$$962$$ − 19.8564i − 0.640196i
$$963$$ 0 0
$$964$$ 16.9282 0.545221
$$965$$ 0 0
$$966$$ −4.39230 −0.141320
$$967$$ 29.1244i 0.936576i 0.883576 + 0.468288i $$0.155129\pi$$
−0.883576 + 0.468288i $$0.844871\pi$$
$$968$$ − 16.2679i − 0.522872i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 27.7128 0.889346 0.444673 0.895693i $$-0.353320\pi$$
0.444673 + 0.895693i $$0.353320\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 9.07180i − 0.290828i
$$974$$ 20.1051 0.644210
$$975$$ 0 0
$$976$$ 67.3205 2.15488
$$977$$ 51.0333i 1.63270i 0.577557 + 0.816350i $$0.304006\pi$$
−0.577557 + 0.816350i $$0.695994\pi$$
$$978$$ − 22.0526i − 0.705163i
$$979$$ −9.21539 −0.294525
$$980$$ 0 0
$$981$$ 6.39230 0.204091
$$982$$ − 38.1962i − 1.21889i
$$983$$ 6.67949i 0.213043i 0.994310 + 0.106521i $$0.0339713\pi$$
−0.994310 + 0.106521i $$0.966029\pi$$
$$984$$ 8.19615 0.261284
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 2.53590i 0.0807185i
$$988$$ 2.73205i 0.0869181i
$$989$$ −21.4641 −0.682519
$$990$$ 0 0
$$991$$ 26.9282 0.855403 0.427701 0.903920i $$-0.359324\pi$$
0.427701 + 0.903920i $$0.359324\pi$$
$$992$$ 25.6077i 0.813045i
$$993$$ − 29.7128i − 0.942908i
$$994$$ −20.7846 −0.659248
$$995$$ 0 0
$$996$$ 0.928203 0.0294112
$$997$$ − 38.3923i − 1.21590i −0.793977 0.607948i $$-0.791993\pi$$
0.793977 0.607948i $$-0.208007\pi$$
$$998$$ − 30.2487i − 0.957506i
$$999$$ −4.19615 −0.132760
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.799.3 4
5.2 odd 4 1425.2.a.o.1.1 2
5.3 odd 4 285.2.a.e.1.2 2
5.4 even 2 inner 1425.2.c.k.799.2 4
15.2 even 4 4275.2.a.t.1.2 2
15.8 even 4 855.2.a.f.1.1 2
20.3 even 4 4560.2.a.bh.1.1 2
95.18 even 4 5415.2.a.r.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.2 2 5.3 odd 4
855.2.a.f.1.1 2 15.8 even 4
1425.2.a.o.1.1 2 5.2 odd 4
1425.2.c.k.799.2 4 5.4 even 2 inner
1425.2.c.k.799.3 4 1.1 even 1 trivial
4275.2.a.t.1.2 2 15.2 even 4
4560.2.a.bh.1.1 2 20.3 even 4
5415.2.a.r.1.1 2 95.18 even 4