# Properties

 Label 245.2.a.f.1.2 Level $245$ Weight $2$ Character 245.1 Self dual yes Analytic conductor $1.956$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,2,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("245.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.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: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 245.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.41421 q^{2} +2.41421 q^{3} +1.00000 q^{5} +3.41421 q^{6} -2.82843 q^{8} +2.82843 q^{9} +O(q^{10})$$ $$q+1.41421 q^{2} +2.41421 q^{3} +1.00000 q^{5} +3.41421 q^{6} -2.82843 q^{8} +2.82843 q^{9} +1.41421 q^{10} -5.82843 q^{11} +1.58579 q^{13} +2.41421 q^{15} -4.00000 q^{16} -5.24264 q^{17} +4.00000 q^{18} +6.00000 q^{19} -8.24264 q^{22} +4.58579 q^{23} -6.82843 q^{24} +1.00000 q^{25} +2.24264 q^{26} -0.414214 q^{27} +2.65685 q^{29} +3.41421 q^{30} +1.75736 q^{31} -14.0711 q^{33} -7.41421 q^{34} -6.24264 q^{37} +8.48528 q^{38} +3.82843 q^{39} -2.82843 q^{40} +2.24264 q^{41} +2.00000 q^{43} +2.82843 q^{45} +6.48528 q^{46} +1.24264 q^{47} -9.65685 q^{48} +1.41421 q^{50} -12.6569 q^{51} -4.24264 q^{53} -0.585786 q^{54} -5.82843 q^{55} +14.4853 q^{57} +3.75736 q^{58} +6.24264 q^{59} +2.82843 q^{61} +2.48528 q^{62} +8.00000 q^{64} +1.58579 q^{65} -19.8995 q^{66} +0.242641 q^{67} +11.0711 q^{69} -8.82843 q^{71} -8.00000 q^{72} +8.48528 q^{73} -8.82843 q^{74} +2.41421 q^{75} +5.41421 q^{78} -15.4853 q^{79} -4.00000 q^{80} -9.48528 q^{81} +3.17157 q^{82} -5.24264 q^{85} +2.82843 q^{86} +6.41421 q^{87} +16.4853 q^{88} -8.00000 q^{89} +4.00000 q^{90} +4.24264 q^{93} +1.75736 q^{94} +6.00000 q^{95} +4.75736 q^{97} -16.4853 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 4 q^{6}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^6 $$2 q + 2 q^{3} + 2 q^{5} + 4 q^{6} - 6 q^{11} + 6 q^{13} + 2 q^{15} - 8 q^{16} - 2 q^{17} + 8 q^{18} + 12 q^{19} - 8 q^{22} + 12 q^{23} - 8 q^{24} + 2 q^{25} - 4 q^{26} + 2 q^{27} - 6 q^{29} + 4 q^{30} + 12 q^{31} - 14 q^{33} - 12 q^{34} - 4 q^{37} + 2 q^{39} - 4 q^{41} + 4 q^{43} - 4 q^{46} - 6 q^{47} - 8 q^{48} - 14 q^{51} - 4 q^{54} - 6 q^{55} + 12 q^{57} + 16 q^{58} + 4 q^{59} - 12 q^{62} + 16 q^{64} + 6 q^{65} - 20 q^{66} - 8 q^{67} + 8 q^{69} - 12 q^{71} - 16 q^{72} - 12 q^{74} + 2 q^{75} + 8 q^{78} - 14 q^{79} - 8 q^{80} - 2 q^{81} + 12 q^{82} - 2 q^{85} + 10 q^{87} + 16 q^{88} - 16 q^{89} + 8 q^{90} + 12 q^{94} + 12 q^{95} + 18 q^{97} - 16 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^6 - 6 * q^11 + 6 * q^13 + 2 * q^15 - 8 * q^16 - 2 * q^17 + 8 * q^18 + 12 * q^19 - 8 * q^22 + 12 * q^23 - 8 * q^24 + 2 * q^25 - 4 * q^26 + 2 * q^27 - 6 * q^29 + 4 * q^30 + 12 * q^31 - 14 * q^33 - 12 * q^34 - 4 * q^37 + 2 * q^39 - 4 * q^41 + 4 * q^43 - 4 * q^46 - 6 * q^47 - 8 * q^48 - 14 * q^51 - 4 * q^54 - 6 * q^55 + 12 * q^57 + 16 * q^58 + 4 * q^59 - 12 * q^62 + 16 * q^64 + 6 * q^65 - 20 * q^66 - 8 * q^67 + 8 * q^69 - 12 * q^71 - 16 * q^72 - 12 * q^74 + 2 * q^75 + 8 * q^78 - 14 * q^79 - 8 * q^80 - 2 * q^81 + 12 * q^82 - 2 * q^85 + 10 * q^87 + 16 * q^88 - 16 * q^89 + 8 * q^90 + 12 * q^94 + 12 * q^95 + 18 * q^97 - 16 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.41421 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$3$$ 2.41421 1.39385 0.696923 0.717146i $$-0.254552\pi$$
0.696923 + 0.717146i $$0.254552\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 3.41421 1.39385
$$7$$ 0 0
$$8$$ −2.82843 −1.00000
$$9$$ 2.82843 0.942809
$$10$$ 1.41421 0.447214
$$11$$ −5.82843 −1.75734 −0.878668 0.477432i $$-0.841568\pi$$
−0.878668 + 0.477432i $$0.841568\pi$$
$$12$$ 0 0
$$13$$ 1.58579 0.439818 0.219909 0.975520i $$-0.429424\pi$$
0.219909 + 0.975520i $$0.429424\pi$$
$$14$$ 0 0
$$15$$ 2.41421 0.623347
$$16$$ −4.00000 −1.00000
$$17$$ −5.24264 −1.27153 −0.635764 0.771884i $$-0.719315\pi$$
−0.635764 + 0.771884i $$0.719315\pi$$
$$18$$ 4.00000 0.942809
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −8.24264 −1.75734
$$23$$ 4.58579 0.956203 0.478101 0.878305i $$-0.341325\pi$$
0.478101 + 0.878305i $$0.341325\pi$$
$$24$$ −6.82843 −1.39385
$$25$$ 1.00000 0.200000
$$26$$ 2.24264 0.439818
$$27$$ −0.414214 −0.0797154
$$28$$ 0 0
$$29$$ 2.65685 0.493365 0.246683 0.969096i $$-0.420659\pi$$
0.246683 + 0.969096i $$0.420659\pi$$
$$30$$ 3.41421 0.623347
$$31$$ 1.75736 0.315631 0.157816 0.987469i $$-0.449555\pi$$
0.157816 + 0.987469i $$0.449555\pi$$
$$32$$ 0 0
$$33$$ −14.0711 −2.44946
$$34$$ −7.41421 −1.27153
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.24264 −1.02628 −0.513142 0.858304i $$-0.671519\pi$$
−0.513142 + 0.858304i $$0.671519\pi$$
$$38$$ 8.48528 1.37649
$$39$$ 3.82843 0.613039
$$40$$ −2.82843 −0.447214
$$41$$ 2.24264 0.350242 0.175121 0.984547i $$-0.443968\pi$$
0.175121 + 0.984547i $$0.443968\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 0 0
$$45$$ 2.82843 0.421637
$$46$$ 6.48528 0.956203
$$47$$ 1.24264 0.181258 0.0906289 0.995885i $$-0.471112\pi$$
0.0906289 + 0.995885i $$0.471112\pi$$
$$48$$ −9.65685 −1.39385
$$49$$ 0 0
$$50$$ 1.41421 0.200000
$$51$$ −12.6569 −1.77231
$$52$$ 0 0
$$53$$ −4.24264 −0.582772 −0.291386 0.956606i $$-0.594116\pi$$
−0.291386 + 0.956606i $$0.594116\pi$$
$$54$$ −0.585786 −0.0797154
$$55$$ −5.82843 −0.785905
$$56$$ 0 0
$$57$$ 14.4853 1.91862
$$58$$ 3.75736 0.493365
$$59$$ 6.24264 0.812723 0.406361 0.913712i $$-0.366797\pi$$
0.406361 + 0.913712i $$0.366797\pi$$
$$60$$ 0 0
$$61$$ 2.82843 0.362143 0.181071 0.983470i $$-0.442043\pi$$
0.181071 + 0.983470i $$0.442043\pi$$
$$62$$ 2.48528 0.315631
$$63$$ 0 0
$$64$$ 8.00000 1.00000
$$65$$ 1.58579 0.196693
$$66$$ −19.8995 −2.44946
$$67$$ 0.242641 0.0296433 0.0148216 0.999890i $$-0.495282\pi$$
0.0148216 + 0.999890i $$0.495282\pi$$
$$68$$ 0 0
$$69$$ 11.0711 1.33280
$$70$$ 0 0
$$71$$ −8.82843 −1.04774 −0.523871 0.851798i $$-0.675513\pi$$
−0.523871 + 0.851798i $$0.675513\pi$$
$$72$$ −8.00000 −0.942809
$$73$$ 8.48528 0.993127 0.496564 0.868000i $$-0.334595\pi$$
0.496564 + 0.868000i $$0.334595\pi$$
$$74$$ −8.82843 −1.02628
$$75$$ 2.41421 0.278769
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 5.41421 0.613039
$$79$$ −15.4853 −1.74223 −0.871115 0.491079i $$-0.836603\pi$$
−0.871115 + 0.491079i $$0.836603\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ −9.48528 −1.05392
$$82$$ 3.17157 0.350242
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −5.24264 −0.568644
$$86$$ 2.82843 0.304997
$$87$$ 6.41421 0.687676
$$88$$ 16.4853 1.75734
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ 4.00000 0.421637
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.24264 0.439941
$$94$$ 1.75736 0.181258
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ 4.75736 0.483037 0.241518 0.970396i $$-0.422355\pi$$
0.241518 + 0.970396i $$0.422355\pi$$
$$98$$ 0 0
$$99$$ −16.4853 −1.65683
$$100$$ 0 0
$$101$$ −14.4853 −1.44134 −0.720670 0.693279i $$-0.756166\pi$$
−0.720670 + 0.693279i $$0.756166\pi$$
$$102$$ −17.8995 −1.77231
$$103$$ 10.7574 1.05995 0.529977 0.848012i $$-0.322201\pi$$
0.529977 + 0.848012i $$0.322201\pi$$
$$104$$ −4.48528 −0.439818
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 14.4853 1.40035 0.700173 0.713974i $$-0.253106\pi$$
0.700173 + 0.713974i $$0.253106\pi$$
$$108$$ 0 0
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ −8.24264 −0.785905
$$111$$ −15.0711 −1.43048
$$112$$ 0 0
$$113$$ 1.07107 0.100758 0.0503788 0.998730i $$-0.483957\pi$$
0.0503788 + 0.998730i $$0.483957\pi$$
$$114$$ 20.4853 1.91862
$$115$$ 4.58579 0.427627
$$116$$ 0 0
$$117$$ 4.48528 0.414664
$$118$$ 8.82843 0.812723
$$119$$ 0 0
$$120$$ −6.82843 −0.623347
$$121$$ 22.9706 2.08823
$$122$$ 4.00000 0.362143
$$123$$ 5.41421 0.488183
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −0.242641 −0.0215309 −0.0107654 0.999942i $$-0.503427\pi$$
−0.0107654 + 0.999942i $$0.503427\pi$$
$$128$$ 11.3137 1.00000
$$129$$ 4.82843 0.425119
$$130$$ 2.24264 0.196693
$$131$$ 3.75736 0.328282 0.164141 0.986437i $$-0.447515\pi$$
0.164141 + 0.986437i $$0.447515\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0.343146 0.0296433
$$135$$ −0.414214 −0.0356498
$$136$$ 14.8284 1.27153
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 15.6569 1.33280
$$139$$ 16.2426 1.37768 0.688841 0.724912i $$-0.258120\pi$$
0.688841 + 0.724912i $$0.258120\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ −12.4853 −1.04774
$$143$$ −9.24264 −0.772908
$$144$$ −11.3137 −0.942809
$$145$$ 2.65685 0.220640
$$146$$ 12.0000 0.993127
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −14.8284 −1.21479 −0.607396 0.794399i $$-0.707786\pi$$
−0.607396 + 0.794399i $$0.707786\pi$$
$$150$$ 3.41421 0.278769
$$151$$ 9.48528 0.771901 0.385951 0.922519i $$-0.373874\pi$$
0.385951 + 0.922519i $$0.373874\pi$$
$$152$$ −16.9706 −1.37649
$$153$$ −14.8284 −1.19881
$$154$$ 0 0
$$155$$ 1.75736 0.141154
$$156$$ 0 0
$$157$$ −20.8284 −1.66229 −0.831145 0.556056i $$-0.812314\pi$$
−0.831145 + 0.556056i $$0.812314\pi$$
$$158$$ −21.8995 −1.74223
$$159$$ −10.2426 −0.812294
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −13.4142 −1.05392
$$163$$ −1.75736 −0.137647 −0.0688235 0.997629i $$-0.521925\pi$$
−0.0688235 + 0.997629i $$0.521925\pi$$
$$164$$ 0 0
$$165$$ −14.0711 −1.09543
$$166$$ 0 0
$$167$$ 9.24264 0.715217 0.357609 0.933872i $$-0.383592\pi$$
0.357609 + 0.933872i $$0.383592\pi$$
$$168$$ 0 0
$$169$$ −10.4853 −0.806560
$$170$$ −7.41421 −0.568644
$$171$$ 16.9706 1.29777
$$172$$ 0 0
$$173$$ 1.24264 0.0944762 0.0472381 0.998884i $$-0.484958\pi$$
0.0472381 + 0.998884i $$0.484958\pi$$
$$174$$ 9.07107 0.687676
$$175$$ 0 0
$$176$$ 23.3137 1.75734
$$177$$ 15.0711 1.13281
$$178$$ −11.3137 −0.847998
$$179$$ 2.48528 0.185759 0.0928793 0.995677i $$-0.470393\pi$$
0.0928793 + 0.995677i $$0.470393\pi$$
$$180$$ 0 0
$$181$$ 6.72792 0.500083 0.250041 0.968235i $$-0.419556\pi$$
0.250041 + 0.968235i $$0.419556\pi$$
$$182$$ 0 0
$$183$$ 6.82843 0.504772
$$184$$ −12.9706 −0.956203
$$185$$ −6.24264 −0.458968
$$186$$ 6.00000 0.439941
$$187$$ 30.5563 2.23450
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 8.48528 0.615587
$$191$$ −19.9706 −1.44502 −0.722510 0.691361i $$-0.757011\pi$$
−0.722510 + 0.691361i $$0.757011\pi$$
$$192$$ 19.3137 1.39385
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ 6.72792 0.483037
$$195$$ 3.82843 0.274159
$$196$$ 0 0
$$197$$ 10.5858 0.754206 0.377103 0.926171i $$-0.376920\pi$$
0.377103 + 0.926171i $$0.376920\pi$$
$$198$$ −23.3137 −1.65683
$$199$$ −4.58579 −0.325078 −0.162539 0.986702i $$-0.551968\pi$$
−0.162539 + 0.986702i $$0.551968\pi$$
$$200$$ −2.82843 −0.200000
$$201$$ 0.585786 0.0413182
$$202$$ −20.4853 −1.44134
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2.24264 0.156633
$$206$$ 15.2132 1.05995
$$207$$ 12.9706 0.901516
$$208$$ −6.34315 −0.439818
$$209$$ −34.9706 −2.41896
$$210$$ 0 0
$$211$$ 9.00000 0.619586 0.309793 0.950804i $$-0.399740\pi$$
0.309793 + 0.950804i $$0.399740\pi$$
$$212$$ 0 0
$$213$$ −21.3137 −1.46039
$$214$$ 20.4853 1.40035
$$215$$ 2.00000 0.136399
$$216$$ 1.17157 0.0797154
$$217$$ 0 0
$$218$$ 7.07107 0.478913
$$219$$ 20.4853 1.38427
$$220$$ 0 0
$$221$$ −8.31371 −0.559241
$$222$$ −21.3137 −1.43048
$$223$$ 18.2132 1.21965 0.609823 0.792538i $$-0.291240\pi$$
0.609823 + 0.792538i $$0.291240\pi$$
$$224$$ 0 0
$$225$$ 2.82843 0.188562
$$226$$ 1.51472 0.100758
$$227$$ −9.72792 −0.645665 −0.322832 0.946456i $$-0.604635\pi$$
−0.322832 + 0.946456i $$0.604635\pi$$
$$228$$ 0 0
$$229$$ −30.0416 −1.98521 −0.992603 0.121402i $$-0.961261\pi$$
−0.992603 + 0.121402i $$0.961261\pi$$
$$230$$ 6.48528 0.427627
$$231$$ 0 0
$$232$$ −7.51472 −0.493365
$$233$$ −14.8284 −0.971443 −0.485721 0.874114i $$-0.661443\pi$$
−0.485721 + 0.874114i $$0.661443\pi$$
$$234$$ 6.34315 0.414664
$$235$$ 1.24264 0.0810609
$$236$$ 0 0
$$237$$ −37.3848 −2.42840
$$238$$ 0 0
$$239$$ −0.514719 −0.0332944 −0.0166472 0.999861i $$-0.505299\pi$$
−0.0166472 + 0.999861i $$0.505299\pi$$
$$240$$ −9.65685 −0.623347
$$241$$ 24.7279 1.59287 0.796433 0.604727i $$-0.206718\pi$$
0.796433 + 0.604727i $$0.206718\pi$$
$$242$$ 32.4853 2.08823
$$243$$ −21.6569 −1.38929
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 7.65685 0.488183
$$247$$ 9.51472 0.605407
$$248$$ −4.97056 −0.315631
$$249$$ 0 0
$$250$$ 1.41421 0.0894427
$$251$$ −25.2132 −1.59144 −0.795722 0.605663i $$-0.792908\pi$$
−0.795722 + 0.605663i $$0.792908\pi$$
$$252$$ 0 0
$$253$$ −26.7279 −1.68037
$$254$$ −0.343146 −0.0215309
$$255$$ −12.6569 −0.792603
$$256$$ 0 0
$$257$$ −26.4853 −1.65211 −0.826053 0.563592i $$-0.809419\pi$$
−0.826053 + 0.563592i $$0.809419\pi$$
$$258$$ 6.82843 0.425119
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 7.51472 0.465149
$$262$$ 5.31371 0.328282
$$263$$ 28.6274 1.76524 0.882621 0.470085i $$-0.155777\pi$$
0.882621 + 0.470085i $$0.155777\pi$$
$$264$$ 39.7990 2.44946
$$265$$ −4.24264 −0.260623
$$266$$ 0 0
$$267$$ −19.3137 −1.18198
$$268$$ 0 0
$$269$$ −7.75736 −0.472975 −0.236487 0.971635i $$-0.575996\pi$$
−0.236487 + 0.971635i $$0.575996\pi$$
$$270$$ −0.585786 −0.0356498
$$271$$ −23.3137 −1.41621 −0.708103 0.706109i $$-0.750449\pi$$
−0.708103 + 0.706109i $$0.750449\pi$$
$$272$$ 20.9706 1.27153
$$273$$ 0 0
$$274$$ 16.9706 1.02523
$$275$$ −5.82843 −0.351467
$$276$$ 0 0
$$277$$ −27.2132 −1.63508 −0.817541 0.575870i $$-0.804664\pi$$
−0.817541 + 0.575870i $$0.804664\pi$$
$$278$$ 22.9706 1.37768
$$279$$ 4.97056 0.297580
$$280$$ 0 0
$$281$$ −20.3137 −1.21181 −0.605907 0.795535i $$-0.707190\pi$$
−0.605907 + 0.795535i $$0.707190\pi$$
$$282$$ 4.24264 0.252646
$$283$$ −6.55635 −0.389735 −0.194867 0.980830i $$-0.562428\pi$$
−0.194867 + 0.980830i $$0.562428\pi$$
$$284$$ 0 0
$$285$$ 14.4853 0.858034
$$286$$ −13.0711 −0.772908
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 10.4853 0.616781
$$290$$ 3.75736 0.220640
$$291$$ 11.4853 0.673279
$$292$$ 0 0
$$293$$ 0.272078 0.0158950 0.00794748 0.999968i $$-0.497470\pi$$
0.00794748 + 0.999968i $$0.497470\pi$$
$$294$$ 0 0
$$295$$ 6.24264 0.363461
$$296$$ 17.6569 1.02628
$$297$$ 2.41421 0.140087
$$298$$ −20.9706 −1.21479
$$299$$ 7.27208 0.420555
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 13.4142 0.771901
$$303$$ −34.9706 −2.00901
$$304$$ −24.0000 −1.37649
$$305$$ 2.82843 0.161955
$$306$$ −20.9706 −1.19881
$$307$$ 11.1005 0.633539 0.316770 0.948503i $$-0.397402\pi$$
0.316770 + 0.948503i $$0.397402\pi$$
$$308$$ 0 0
$$309$$ 25.9706 1.47741
$$310$$ 2.48528 0.141154
$$311$$ −10.0000 −0.567048 −0.283524 0.958965i $$-0.591504\pi$$
−0.283524 + 0.958965i $$0.591504\pi$$
$$312$$ −10.8284 −0.613039
$$313$$ 24.2132 1.36861 0.684306 0.729195i $$-0.260105\pi$$
0.684306 + 0.729195i $$0.260105\pi$$
$$314$$ −29.4558 −1.66229
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −11.6569 −0.654714 −0.327357 0.944901i $$-0.606158\pi$$
−0.327357 + 0.944901i $$0.606158\pi$$
$$318$$ −14.4853 −0.812294
$$319$$ −15.4853 −0.867009
$$320$$ 8.00000 0.447214
$$321$$ 34.9706 1.95187
$$322$$ 0 0
$$323$$ −31.4558 −1.75025
$$324$$ 0 0
$$325$$ 1.58579 0.0879636
$$326$$ −2.48528 −0.137647
$$327$$ 12.0711 0.667532
$$328$$ −6.34315 −0.350242
$$329$$ 0 0
$$330$$ −19.8995 −1.09543
$$331$$ 23.4558 1.28925 0.644625 0.764499i $$-0.277014\pi$$
0.644625 + 0.764499i $$0.277014\pi$$
$$332$$ 0 0
$$333$$ −17.6569 −0.967590
$$334$$ 13.0711 0.715217
$$335$$ 0.242641 0.0132569
$$336$$ 0 0
$$337$$ 13.7574 0.749411 0.374706 0.927144i $$-0.377744\pi$$
0.374706 + 0.927144i $$0.377744\pi$$
$$338$$ −14.8284 −0.806560
$$339$$ 2.58579 0.140441
$$340$$ 0 0
$$341$$ −10.2426 −0.554670
$$342$$ 24.0000 1.29777
$$343$$ 0 0
$$344$$ −5.65685 −0.304997
$$345$$ 11.0711 0.596046
$$346$$ 1.75736 0.0944762
$$347$$ −1.07107 −0.0574979 −0.0287490 0.999587i $$-0.509152\pi$$
−0.0287490 + 0.999587i $$0.509152\pi$$
$$348$$ 0 0
$$349$$ 22.9706 1.22959 0.614793 0.788688i $$-0.289240\pi$$
0.614793 + 0.788688i $$0.289240\pi$$
$$350$$ 0 0
$$351$$ −0.656854 −0.0350603
$$352$$ 0 0
$$353$$ 36.2132 1.92743 0.963717 0.266925i $$-0.0860078\pi$$
0.963717 + 0.266925i $$0.0860078\pi$$
$$354$$ 21.3137 1.13281
$$355$$ −8.82843 −0.468564
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 3.51472 0.185759
$$359$$ 11.3137 0.597115 0.298557 0.954392i $$-0.403495\pi$$
0.298557 + 0.954392i $$0.403495\pi$$
$$360$$ −8.00000 −0.421637
$$361$$ 17.0000 0.894737
$$362$$ 9.51472 0.500083
$$363$$ 55.4558 2.91068
$$364$$ 0 0
$$365$$ 8.48528 0.444140
$$366$$ 9.65685 0.504772
$$367$$ 29.8701 1.55920 0.779602 0.626275i $$-0.215421\pi$$
0.779602 + 0.626275i $$0.215421\pi$$
$$368$$ −18.3431 −0.956203
$$369$$ 6.34315 0.330211
$$370$$ −8.82843 −0.458968
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0.485281 0.0251269 0.0125635 0.999921i $$-0.496001\pi$$
0.0125635 + 0.999921i $$0.496001\pi$$
$$374$$ 43.2132 2.23450
$$375$$ 2.41421 0.124669
$$376$$ −3.51472 −0.181258
$$377$$ 4.21320 0.216991
$$378$$ 0 0
$$379$$ 2.00000 0.102733 0.0513665 0.998680i $$-0.483642\pi$$
0.0513665 + 0.998680i $$0.483642\pi$$
$$380$$ 0 0
$$381$$ −0.585786 −0.0300107
$$382$$ −28.2426 −1.44502
$$383$$ −12.4853 −0.637968 −0.318984 0.947760i $$-0.603342\pi$$
−0.318984 + 0.947760i $$0.603342\pi$$
$$384$$ 27.3137 1.39385
$$385$$ 0 0
$$386$$ −22.6274 −1.15171
$$387$$ 5.65685 0.287554
$$388$$ 0 0
$$389$$ −35.1421 −1.78178 −0.890889 0.454222i $$-0.849917\pi$$
−0.890889 + 0.454222i $$0.849917\pi$$
$$390$$ 5.41421 0.274159
$$391$$ −24.0416 −1.21584
$$392$$ 0 0
$$393$$ 9.07107 0.457575
$$394$$ 14.9706 0.754206
$$395$$ −15.4853 −0.779149
$$396$$ 0 0
$$397$$ 4.41421 0.221543 0.110772 0.993846i $$-0.464668\pi$$
0.110772 + 0.993846i $$0.464668\pi$$
$$398$$ −6.48528 −0.325078
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ 6.17157 0.308194 0.154097 0.988056i $$-0.450753\pi$$
0.154097 + 0.988056i $$0.450753\pi$$
$$402$$ 0.828427 0.0413182
$$403$$ 2.78680 0.138820
$$404$$ 0 0
$$405$$ −9.48528 −0.471327
$$406$$ 0 0
$$407$$ 36.3848 1.80353
$$408$$ 35.7990 1.77231
$$409$$ −14.4853 −0.716251 −0.358126 0.933673i $$-0.616584\pi$$
−0.358126 + 0.933673i $$0.616584\pi$$
$$410$$ 3.17157 0.156633
$$411$$ 28.9706 1.42901
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 18.3431 0.901516
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 39.2132 1.92028
$$418$$ −49.4558 −2.41896
$$419$$ −6.72792 −0.328681 −0.164340 0.986404i $$-0.552550\pi$$
−0.164340 + 0.986404i $$0.552550\pi$$
$$420$$ 0 0
$$421$$ −19.0000 −0.926003 −0.463002 0.886357i $$-0.653228\pi$$
−0.463002 + 0.886357i $$0.653228\pi$$
$$422$$ 12.7279 0.619586
$$423$$ 3.51472 0.170891
$$424$$ 12.0000 0.582772
$$425$$ −5.24264 −0.254305
$$426$$ −30.1421 −1.46039
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −22.3137 −1.07732
$$430$$ 2.82843 0.136399
$$431$$ 10.7990 0.520169 0.260085 0.965586i $$-0.416250\pi$$
0.260085 + 0.965586i $$0.416250\pi$$
$$432$$ 1.65685 0.0797154
$$433$$ −22.9706 −1.10389 −0.551947 0.833879i $$-0.686115\pi$$
−0.551947 + 0.833879i $$0.686115\pi$$
$$434$$ 0 0
$$435$$ 6.41421 0.307538
$$436$$ 0 0
$$437$$ 27.5147 1.31621
$$438$$ 28.9706 1.38427
$$439$$ 6.38478 0.304729 0.152364 0.988324i $$-0.451311\pi$$
0.152364 + 0.988324i $$0.451311\pi$$
$$440$$ 16.4853 0.785905
$$441$$ 0 0
$$442$$ −11.7574 −0.559241
$$443$$ 20.8284 0.989588 0.494794 0.869010i $$-0.335243\pi$$
0.494794 + 0.869010i $$0.335243\pi$$
$$444$$ 0 0
$$445$$ −8.00000 −0.379236
$$446$$ 25.7574 1.21965
$$447$$ −35.7990 −1.69323
$$448$$ 0 0
$$449$$ −29.8284 −1.40769 −0.703845 0.710353i $$-0.748535\pi$$
−0.703845 + 0.710353i $$0.748535\pi$$
$$450$$ 4.00000 0.188562
$$451$$ −13.0711 −0.615493
$$452$$ 0 0
$$453$$ 22.8995 1.07591
$$454$$ −13.7574 −0.645665
$$455$$ 0 0
$$456$$ −40.9706 −1.91862
$$457$$ −20.2426 −0.946911 −0.473455 0.880818i $$-0.656993\pi$$
−0.473455 + 0.880818i $$0.656993\pi$$
$$458$$ −42.4853 −1.98521
$$459$$ 2.17157 0.101360
$$460$$ 0 0
$$461$$ 36.9706 1.72189 0.860945 0.508697i $$-0.169873\pi$$
0.860945 + 0.508697i $$0.169873\pi$$
$$462$$ 0 0
$$463$$ 29.4558 1.36893 0.684465 0.729046i $$-0.260036\pi$$
0.684465 + 0.729046i $$0.260036\pi$$
$$464$$ −10.6274 −0.493365
$$465$$ 4.24264 0.196748
$$466$$ −20.9706 −0.971443
$$467$$ 19.7279 0.912899 0.456450 0.889749i $$-0.349121\pi$$
0.456450 + 0.889749i $$0.349121\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 1.75736 0.0810609
$$471$$ −50.2843 −2.31698
$$472$$ −17.6569 −0.812723
$$473$$ −11.6569 −0.535983
$$474$$ −52.8701 −2.42840
$$475$$ 6.00000 0.275299
$$476$$ 0 0
$$477$$ −12.0000 −0.549442
$$478$$ −0.727922 −0.0332944
$$479$$ 20.2426 0.924910 0.462455 0.886643i $$-0.346969\pi$$
0.462455 + 0.886643i $$0.346969\pi$$
$$480$$ 0 0
$$481$$ −9.89949 −0.451378
$$482$$ 34.9706 1.59287
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.75736 0.216021
$$486$$ −30.6274 −1.38929
$$487$$ −31.6985 −1.43640 −0.718198 0.695839i $$-0.755033\pi$$
−0.718198 + 0.695839i $$0.755033\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ −4.24264 −0.191859
$$490$$ 0 0
$$491$$ 19.2843 0.870287 0.435143 0.900361i $$-0.356698\pi$$
0.435143 + 0.900361i $$0.356698\pi$$
$$492$$ 0 0
$$493$$ −13.9289 −0.627328
$$494$$ 13.4558 0.605407
$$495$$ −16.4853 −0.740958
$$496$$ −7.02944 −0.315631
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 3.00000 0.134298 0.0671492 0.997743i $$-0.478610\pi$$
0.0671492 + 0.997743i $$0.478610\pi$$
$$500$$ 0 0
$$501$$ 22.3137 0.996903
$$502$$ −35.6569 −1.59144
$$503$$ −32.7574 −1.46058 −0.730289 0.683138i $$-0.760615\pi$$
−0.730289 + 0.683138i $$0.760615\pi$$
$$504$$ 0 0
$$505$$ −14.4853 −0.644587
$$506$$ −37.7990 −1.68037
$$507$$ −25.3137 −1.12422
$$508$$ 0 0
$$509$$ 17.2132 0.762962 0.381481 0.924377i $$-0.375414\pi$$
0.381481 + 0.924377i $$0.375414\pi$$
$$510$$ −17.8995 −0.792603
$$511$$ 0 0
$$512$$ −22.6274 −1.00000
$$513$$ −2.48528 −0.109728
$$514$$ −37.4558 −1.65211
$$515$$ 10.7574 0.474026
$$516$$ 0 0
$$517$$ −7.24264 −0.318531
$$518$$ 0 0
$$519$$ 3.00000 0.131685
$$520$$ −4.48528 −0.196693
$$521$$ −18.9706 −0.831115 −0.415558 0.909567i $$-0.636414\pi$$
−0.415558 + 0.909567i $$0.636414\pi$$
$$522$$ 10.6274 0.465149
$$523$$ 15.5147 0.678411 0.339206 0.940712i $$-0.389842\pi$$
0.339206 + 0.940712i $$0.389842\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 40.4853 1.76524
$$527$$ −9.21320 −0.401333
$$528$$ 56.2843 2.44946
$$529$$ −1.97056 −0.0856766
$$530$$ −6.00000 −0.260623
$$531$$ 17.6569 0.766242
$$532$$ 0 0
$$533$$ 3.55635 0.154043
$$534$$ −27.3137 −1.18198
$$535$$ 14.4853 0.626253
$$536$$ −0.686292 −0.0296433
$$537$$ 6.00000 0.258919
$$538$$ −10.9706 −0.472975
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 11.9706 0.514655 0.257327 0.966324i $$-0.417158\pi$$
0.257327 + 0.966324i $$0.417158\pi$$
$$542$$ −32.9706 −1.41621
$$543$$ 16.2426 0.697038
$$544$$ 0 0
$$545$$ 5.00000 0.214176
$$546$$ 0 0
$$547$$ −7.51472 −0.321306 −0.160653 0.987011i $$-0.551360\pi$$
−0.160653 + 0.987011i $$0.551360\pi$$
$$548$$ 0 0
$$549$$ 8.00000 0.341432
$$550$$ −8.24264 −0.351467
$$551$$ 15.9411 0.679115
$$552$$ −31.3137 −1.33280
$$553$$ 0 0
$$554$$ −38.4853 −1.63508
$$555$$ −15.0711 −0.639731
$$556$$ 0 0
$$557$$ 31.7990 1.34737 0.673683 0.739020i $$-0.264711\pi$$
0.673683 + 0.739020i $$0.264711\pi$$
$$558$$ 7.02944 0.297580
$$559$$ 3.17157 0.134143
$$560$$ 0 0
$$561$$ 73.7696 3.11455
$$562$$ −28.7279 −1.21181
$$563$$ 35.9411 1.51474 0.757369 0.652987i $$-0.226484\pi$$
0.757369 + 0.652987i $$0.226484\pi$$
$$564$$ 0 0
$$565$$ 1.07107 0.0450602
$$566$$ −9.27208 −0.389735
$$567$$ 0 0
$$568$$ 24.9706 1.04774
$$569$$ −2.14214 −0.0898030 −0.0449015 0.998991i $$-0.514297\pi$$
−0.0449015 + 0.998991i $$0.514297\pi$$
$$570$$ 20.4853 0.858034
$$571$$ −34.4853 −1.44316 −0.721582 0.692329i $$-0.756585\pi$$
−0.721582 + 0.692329i $$0.756585\pi$$
$$572$$ 0 0
$$573$$ −48.2132 −2.01414
$$574$$ 0 0
$$575$$ 4.58579 0.191241
$$576$$ 22.6274 0.942809
$$577$$ −9.72792 −0.404979 −0.202489 0.979284i $$-0.564903\pi$$
−0.202489 + 0.979284i $$0.564903\pi$$
$$578$$ 14.8284 0.616781
$$579$$ −38.6274 −1.60530
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 16.2426 0.673279
$$583$$ 24.7279 1.02413
$$584$$ −24.0000 −0.993127
$$585$$ 4.48528 0.185444
$$586$$ 0.384776 0.0158950
$$587$$ 13.4558 0.555382 0.277691 0.960670i $$-0.410431\pi$$
0.277691 + 0.960670i $$0.410431\pi$$
$$588$$ 0 0
$$589$$ 10.5442 0.434464
$$590$$ 8.82843 0.363461
$$591$$ 25.5563 1.05125
$$592$$ 24.9706 1.02628
$$593$$ 10.7574 0.441752 0.220876 0.975302i $$-0.429108\pi$$
0.220876 + 0.975302i $$0.429108\pi$$
$$594$$ 3.41421 0.140087
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −11.0711 −0.453109
$$598$$ 10.2843 0.420555
$$599$$ −12.1716 −0.497317 −0.248658 0.968591i $$-0.579990\pi$$
−0.248658 + 0.968591i $$0.579990\pi$$
$$600$$ −6.82843 −0.278769
$$601$$ 22.9706 0.936989 0.468494 0.883466i $$-0.344797\pi$$
0.468494 + 0.883466i $$0.344797\pi$$
$$602$$ 0 0
$$603$$ 0.686292 0.0279480
$$604$$ 0 0
$$605$$ 22.9706 0.933886
$$606$$ −49.4558 −2.00901
$$607$$ −24.8995 −1.01064 −0.505320 0.862932i $$-0.668625\pi$$
−0.505320 + 0.862932i $$0.668625\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 4.00000 0.161955
$$611$$ 1.97056 0.0797204
$$612$$ 0 0
$$613$$ 43.9411 1.77477 0.887383 0.461034i $$-0.152521\pi$$
0.887383 + 0.461034i $$0.152521\pi$$
$$614$$ 15.6985 0.633539
$$615$$ 5.41421 0.218322
$$616$$ 0 0
$$617$$ 7.41421 0.298485 0.149242 0.988801i $$-0.452316\pi$$
0.149242 + 0.988801i $$0.452316\pi$$
$$618$$ 36.7279 1.47741
$$619$$ 31.0711 1.24885 0.624426 0.781084i $$-0.285333\pi$$
0.624426 + 0.781084i $$0.285333\pi$$
$$620$$ 0 0
$$621$$ −1.89949 −0.0762241
$$622$$ −14.1421 −0.567048
$$623$$ 0 0
$$624$$ −15.3137 −0.613039
$$625$$ 1.00000 0.0400000
$$626$$ 34.2426 1.36861
$$627$$ −84.4264 −3.37167
$$628$$ 0 0
$$629$$ 32.7279 1.30495
$$630$$ 0 0
$$631$$ 8.45584 0.336622 0.168311 0.985734i $$-0.446169\pi$$
0.168311 + 0.985734i $$0.446169\pi$$
$$632$$ 43.7990 1.74223
$$633$$ 21.7279 0.863607
$$634$$ −16.4853 −0.654714
$$635$$ −0.242641 −0.00962890
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −21.8995 −0.867009
$$639$$ −24.9706 −0.987820
$$640$$ 11.3137 0.447214
$$641$$ −0.686292 −0.0271069 −0.0135534 0.999908i $$-0.504314\pi$$
−0.0135534 + 0.999908i $$0.504314\pi$$
$$642$$ 49.4558 1.95187
$$643$$ 27.7279 1.09348 0.546741 0.837302i $$-0.315868\pi$$
0.546741 + 0.837302i $$0.315868\pi$$
$$644$$ 0 0
$$645$$ 4.82843 0.190119
$$646$$ −44.4853 −1.75025
$$647$$ −28.4853 −1.11987 −0.559936 0.828536i $$-0.689174\pi$$
−0.559936 + 0.828536i $$0.689174\pi$$
$$648$$ 26.8284 1.05392
$$649$$ −36.3848 −1.42823
$$650$$ 2.24264 0.0879636
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1.02944 0.0402850 0.0201425 0.999797i $$-0.493588\pi$$
0.0201425 + 0.999797i $$0.493588\pi$$
$$654$$ 17.0711 0.667532
$$655$$ 3.75736 0.146812
$$656$$ −8.97056 −0.350242
$$657$$ 24.0000 0.936329
$$658$$ 0 0
$$659$$ −13.9706 −0.544216 −0.272108 0.962267i $$-0.587721\pi$$
−0.272108 + 0.962267i $$0.587721\pi$$
$$660$$ 0 0
$$661$$ −37.4558 −1.45686 −0.728432 0.685118i $$-0.759750\pi$$
−0.728432 + 0.685118i $$0.759750\pi$$
$$662$$ 33.1716 1.28925
$$663$$ −20.0711 −0.779496
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −24.9706 −0.967590
$$667$$ 12.1838 0.471757
$$668$$ 0 0
$$669$$ 43.9706 1.70000
$$670$$ 0.343146 0.0132569
$$671$$ −16.4853 −0.636407
$$672$$ 0 0
$$673$$ 20.4853 0.789650 0.394825 0.918756i $$-0.370805\pi$$
0.394825 + 0.918756i $$0.370805\pi$$
$$674$$ 19.4558 0.749411
$$675$$ −0.414214 −0.0159431
$$676$$ 0 0
$$677$$ 1.78680 0.0686722 0.0343361 0.999410i $$-0.489068\pi$$
0.0343361 + 0.999410i $$0.489068\pi$$
$$678$$ 3.65685 0.140441
$$679$$ 0 0
$$680$$ 14.8284 0.568644
$$681$$ −23.4853 −0.899958
$$682$$ −14.4853 −0.554670
$$683$$ −7.79899 −0.298420 −0.149210 0.988806i $$-0.547673\pi$$
−0.149210 + 0.988806i $$0.547673\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ 0 0
$$687$$ −72.5269 −2.76707
$$688$$ −8.00000 −0.304997
$$689$$ −6.72792 −0.256313
$$690$$ 15.6569 0.596046
$$691$$ 9.17157 0.348903 0.174452 0.984666i $$-0.444185\pi$$
0.174452 + 0.984666i $$0.444185\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −1.51472 −0.0574979
$$695$$ 16.2426 0.616118
$$696$$ −18.1421 −0.687676
$$697$$ −11.7574 −0.445342
$$698$$ 32.4853 1.22959
$$699$$ −35.7990 −1.35404
$$700$$ 0 0
$$701$$ −4.45584 −0.168295 −0.0841475 0.996453i $$-0.526817\pi$$
−0.0841475 + 0.996453i $$0.526817\pi$$
$$702$$ −0.928932 −0.0350603
$$703$$ −37.4558 −1.41267
$$704$$ −46.6274 −1.75734
$$705$$ 3.00000 0.112987
$$706$$ 51.2132 1.92743
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 17.0000 0.638448 0.319224 0.947679i $$-0.396578\pi$$
0.319224 + 0.947679i $$0.396578\pi$$
$$710$$ −12.4853 −0.468564
$$711$$ −43.7990 −1.64259
$$712$$ 22.6274 0.847998
$$713$$ 8.05887 0.301807
$$714$$ 0 0
$$715$$ −9.24264 −0.345655
$$716$$ 0 0
$$717$$ −1.24264 −0.0464073
$$718$$ 16.0000 0.597115
$$719$$ 17.2132 0.641944 0.320972 0.947089i $$-0.395990\pi$$
0.320972 + 0.947089i $$0.395990\pi$$
$$720$$ −11.3137 −0.421637
$$721$$ 0 0
$$722$$ 24.0416 0.894737
$$723$$ 59.6985 2.22021
$$724$$ 0 0
$$725$$ 2.65685 0.0986731
$$726$$ 78.4264 2.91068
$$727$$ −29.3137 −1.08719 −0.543593 0.839349i $$-0.682936\pi$$
−0.543593 + 0.839349i $$0.682936\pi$$
$$728$$ 0 0
$$729$$ −23.8284 −0.882534
$$730$$ 12.0000 0.444140
$$731$$ −10.4853 −0.387812
$$732$$ 0 0
$$733$$ −44.6985 −1.65098 −0.825488 0.564420i $$-0.809100\pi$$
−0.825488 + 0.564420i $$0.809100\pi$$
$$734$$ 42.2426 1.55920
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.41421 −0.0520932
$$738$$ 8.97056 0.330211
$$739$$ −3.97056 −0.146060 −0.0730298 0.997330i $$-0.523267\pi$$
−0.0730298 + 0.997330i $$0.523267\pi$$
$$740$$ 0 0
$$741$$ 22.9706 0.843845
$$742$$ 0 0
$$743$$ 36.7279 1.34742 0.673708 0.738997i $$-0.264700\pi$$
0.673708 + 0.738997i $$0.264700\pi$$
$$744$$ −12.0000 −0.439941
$$745$$ −14.8284 −0.543272
$$746$$ 0.686292 0.0251269
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 3.41421 0.124669
$$751$$ 12.5147 0.456669 0.228334 0.973583i $$-0.426672\pi$$
0.228334 + 0.973583i $$0.426672\pi$$
$$752$$ −4.97056 −0.181258
$$753$$ −60.8701 −2.21823
$$754$$ 5.95837 0.216991
$$755$$ 9.48528 0.345205
$$756$$ 0 0
$$757$$ −16.4853 −0.599168 −0.299584 0.954070i $$-0.596848\pi$$
−0.299584 + 0.954070i $$0.596848\pi$$
$$758$$ 2.82843 0.102733
$$759$$ −64.5269 −2.34218
$$760$$ −16.9706 −0.615587
$$761$$ 12.7279 0.461387 0.230693 0.973026i $$-0.425901\pi$$
0.230693 + 0.973026i $$0.425901\pi$$
$$762$$ −0.828427 −0.0300107
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −14.8284 −0.536123
$$766$$ −17.6569 −0.637968
$$767$$ 9.89949 0.357450
$$768$$ 0 0
$$769$$ −3.17157 −0.114370 −0.0571849 0.998364i $$-0.518212\pi$$
−0.0571849 + 0.998364i $$0.518212\pi$$
$$770$$ 0 0
$$771$$ −63.9411 −2.30278
$$772$$ 0 0
$$773$$ 36.2132 1.30250 0.651249 0.758864i $$-0.274245\pi$$
0.651249 + 0.758864i $$0.274245\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 1.75736 0.0631262
$$776$$ −13.4558 −0.483037
$$777$$ 0 0
$$778$$ −49.6985 −1.78178
$$779$$ 13.4558 0.482106
$$780$$ 0 0
$$781$$ 51.4558 1.84123
$$782$$ −34.0000 −1.21584
$$783$$ −1.10051 −0.0393288
$$784$$ 0 0
$$785$$ −20.8284 −0.743398
$$786$$ 12.8284 0.457575
$$787$$ 45.7279 1.63002 0.815012 0.579444i $$-0.196730\pi$$
0.815012 + 0.579444i $$0.196730\pi$$
$$788$$ 0 0
$$789$$ 69.1127 2.46048
$$790$$ −21.8995 −0.779149
$$791$$ 0 0
$$792$$ 46.6274 1.65683
$$793$$ 4.48528 0.159277
$$794$$ 6.24264 0.221543
$$795$$ −10.2426 −0.363269
$$796$$ 0 0
$$797$$ 55.1838 1.95471 0.977355 0.211608i $$-0.0678700\pi$$
0.977355 + 0.211608i $$0.0678700\pi$$
$$798$$ 0 0
$$799$$ −6.51472 −0.230474
$$800$$ 0 0
$$801$$ −22.6274 −0.799500
$$802$$ 8.72792 0.308194
$$803$$ −49.4558 −1.74526
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 3.94113 0.138820
$$807$$ −18.7279 −0.659254
$$808$$ 40.9706 1.44134
$$809$$ 30.5980 1.07577 0.537884 0.843019i $$-0.319224\pi$$
0.537884 + 0.843019i $$0.319224\pi$$
$$810$$ −13.4142 −0.471327
$$811$$ −23.3553 −0.820117 −0.410058 0.912059i $$-0.634492\pi$$
−0.410058 + 0.912059i $$0.634492\pi$$
$$812$$ 0 0
$$813$$ −56.2843 −1.97398
$$814$$ 51.4558 1.80353
$$815$$ −1.75736 −0.0615576
$$816$$ 50.6274 1.77231
$$817$$ 12.0000 0.419827
$$818$$ −20.4853 −0.716251
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.51472 −0.227365 −0.113683 0.993517i $$-0.536265\pi$$
−0.113683 + 0.993517i $$0.536265\pi$$
$$822$$ 40.9706 1.42901
$$823$$ −8.72792 −0.304236 −0.152118 0.988362i $$-0.548609\pi$$
−0.152118 + 0.988362i $$0.548609\pi$$
$$824$$ −30.4264 −1.05995
$$825$$ −14.0711 −0.489892
$$826$$ 0 0
$$827$$ −6.04163 −0.210088 −0.105044 0.994468i $$-0.533498\pi$$
−0.105044 + 0.994468i $$0.533498\pi$$
$$828$$ 0 0
$$829$$ 54.0416 1.87694 0.938472 0.345356i $$-0.112242\pi$$
0.938472 + 0.345356i $$0.112242\pi$$
$$830$$ 0 0
$$831$$ −65.6985 −2.27906
$$832$$ 12.6863 0.439818
$$833$$ 0 0
$$834$$ 55.4558 1.92028
$$835$$ 9.24264 0.319855
$$836$$ 0 0
$$837$$ −0.727922 −0.0251607
$$838$$ −9.51472 −0.328681
$$839$$ −48.7279 −1.68227 −0.841137 0.540822i $$-0.818113\pi$$
−0.841137 + 0.540822i $$0.818113\pi$$
$$840$$ 0 0
$$841$$ −21.9411 −0.756591
$$842$$ −26.8701 −0.926003
$$843$$ −49.0416 −1.68908
$$844$$ 0 0
$$845$$ −10.4853 −0.360705
$$846$$ 4.97056 0.170891
$$847$$ 0 0
$$848$$ 16.9706 0.582772
$$849$$ −15.8284 −0.543230
$$850$$ −7.41421 −0.254305
$$851$$ −28.6274 −0.981335
$$852$$ 0 0
$$853$$ −22.9706 −0.786497 −0.393249 0.919432i $$-0.628649\pi$$
−0.393249 + 0.919432i $$0.628649\pi$$
$$854$$ 0 0
$$855$$ 16.9706 0.580381
$$856$$ −40.9706 −1.40035
$$857$$ −5.51472 −0.188379 −0.0941896 0.995554i $$-0.530026\pi$$
−0.0941896 + 0.995554i $$0.530026\pi$$
$$858$$ −31.5563 −1.07732
$$859$$ 48.7696 1.66400 0.831998 0.554779i $$-0.187197\pi$$
0.831998 + 0.554779i $$0.187197\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 15.2721 0.520169
$$863$$ −18.3848 −0.625825 −0.312913 0.949782i $$-0.601305\pi$$
−0.312913 + 0.949782i $$0.601305\pi$$
$$864$$ 0 0
$$865$$ 1.24264 0.0422511
$$866$$ −32.4853 −1.10389
$$867$$ 25.3137 0.859699
$$868$$ 0 0
$$869$$ 90.2548 3.06169
$$870$$ 9.07107 0.307538
$$871$$ 0.384776 0.0130376
$$872$$ −14.1421 −0.478913
$$873$$ 13.4558 0.455411
$$874$$ 38.9117 1.31621
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 46.9706 1.58608 0.793042 0.609167i $$-0.208496\pi$$
0.793042 + 0.609167i $$0.208496\pi$$
$$878$$ 9.02944 0.304729
$$879$$ 0.656854 0.0221551
$$880$$ 23.3137 0.785905
$$881$$ −19.0294 −0.641118 −0.320559 0.947229i $$-0.603871\pi$$
−0.320559 + 0.947229i $$0.603871\pi$$
$$882$$ 0 0
$$883$$ 48.4853 1.63166 0.815830 0.578292i $$-0.196281\pi$$
0.815830 + 0.578292i $$0.196281\pi$$
$$884$$ 0 0
$$885$$ 15.0711 0.506608
$$886$$ 29.4558 0.989588
$$887$$ −30.9706 −1.03989 −0.519945 0.854200i $$-0.674048\pi$$
−0.519945 + 0.854200i $$0.674048\pi$$
$$888$$ 42.6274 1.43048
$$889$$ 0 0
$$890$$ −11.3137 −0.379236
$$891$$ 55.2843 1.85209
$$892$$ 0 0
$$893$$ 7.45584 0.249500
$$894$$ −50.6274 −1.69323
$$895$$ 2.48528 0.0830738
$$896$$ 0 0
$$897$$ 17.5563 0.586189
$$898$$ −42.1838 −1.40769
$$899$$ 4.66905 0.155721
$$900$$ 0 0
$$901$$ 22.2426 0.741010
$$902$$ −18.4853 −0.615493
$$903$$ 0 0
$$904$$ −3.02944 −0.100758
$$905$$ 6.72792 0.223644
$$906$$ 32.3848 1.07591
$$907$$ 32.1838 1.06864 0.534322 0.845281i $$-0.320567\pi$$
0.534322 + 0.845281i $$0.320567\pi$$
$$908$$ 0 0
$$909$$ −40.9706 −1.35891
$$910$$ 0 0
$$911$$ −5.65685 −0.187420 −0.0937100 0.995600i $$-0.529873\pi$$
−0.0937100 + 0.995600i $$0.529873\pi$$
$$912$$ −57.9411 −1.91862
$$913$$ 0 0
$$914$$ −28.6274 −0.946911
$$915$$ 6.82843 0.225741
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 3.07107 0.101360
$$919$$ −38.4558 −1.26854 −0.634271 0.773111i $$-0.718699\pi$$
−0.634271 + 0.773111i $$0.718699\pi$$
$$920$$ −12.9706 −0.427627
$$921$$ 26.7990 0.883057
$$922$$ 52.2843 1.72189
$$923$$ −14.0000 −0.460816
$$924$$ 0 0
$$925$$ −6.24264 −0.205257
$$926$$ 41.6569 1.36893
$$927$$ 30.4264 0.999334
$$928$$ 0 0
$$929$$ 54.7279 1.79556 0.897782 0.440439i $$-0.145177\pi$$
0.897782 + 0.440439i $$0.145177\pi$$
$$930$$ 6.00000 0.196748
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −24.1421 −0.790378
$$934$$ 27.8995 0.912899
$$935$$ 30.5563 0.999299
$$936$$ −12.6863 −0.414664
$$937$$ −17.4437 −0.569859 −0.284930 0.958548i $$-0.591970\pi$$
−0.284930 + 0.958548i $$0.591970\pi$$
$$938$$ 0 0
$$939$$ 58.4558 1.90763
$$940$$ 0 0
$$941$$ −4.97056 −0.162036 −0.0810179 0.996713i $$-0.525817\pi$$
−0.0810179 + 0.996713i $$0.525817\pi$$
$$942$$ −71.1127 −2.31698
$$943$$ 10.2843 0.334902
$$944$$ −24.9706 −0.812723
$$945$$ 0 0
$$946$$ −16.4853 −0.535983
$$947$$ −43.7574 −1.42192 −0.710962 0.703231i $$-0.751740\pi$$
−0.710962 + 0.703231i $$0.751740\pi$$
$$948$$ 0 0
$$949$$ 13.4558 0.436795
$$950$$ 8.48528 0.275299
$$951$$ −28.1421 −0.912571
$$952$$ 0 0
$$953$$ −29.0122 −0.939797 −0.469899 0.882720i $$-0.655710\pi$$
−0.469899 + 0.882720i $$0.655710\pi$$
$$954$$ −16.9706 −0.549442
$$955$$ −19.9706 −0.646232
$$956$$ 0 0
$$957$$ −37.3848 −1.20848
$$958$$ 28.6274 0.924910
$$959$$ 0 0
$$960$$ 19.3137 0.623347
$$961$$ −27.9117 −0.900377
$$962$$ −14.0000 −0.451378
$$963$$ 40.9706 1.32026
$$964$$ 0 0
$$965$$ −16.0000 −0.515058
$$966$$ 0 0
$$967$$ 24.4264 0.785500 0.392750 0.919645i $$-0.371524\pi$$
0.392750 + 0.919645i $$0.371524\pi$$
$$968$$ −64.9706 −2.08823
$$969$$ −75.9411 −2.43958
$$970$$ 6.72792 0.216021
$$971$$ −42.7279 −1.37120 −0.685602 0.727976i $$-0.740461\pi$$
−0.685602 + 0.727976i $$0.740461\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −44.8284 −1.43640
$$975$$ 3.82843 0.122608
$$976$$ −11.3137 −0.362143
$$977$$ −30.7696 −0.984405 −0.492203 0.870481i $$-0.663808\pi$$
−0.492203 + 0.870481i $$0.663808\pi$$
$$978$$ −6.00000 −0.191859
$$979$$ 46.6274 1.49022
$$980$$ 0 0
$$981$$ 14.1421 0.451524
$$982$$ 27.2721 0.870287
$$983$$ −42.2132 −1.34639 −0.673196 0.739464i $$-0.735079\pi$$
−0.673196 + 0.739464i $$0.735079\pi$$
$$984$$ −15.3137 −0.488183
$$985$$ 10.5858 0.337291
$$986$$ −19.6985 −0.627328
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 9.17157 0.291639
$$990$$ −23.3137 −0.740958
$$991$$ −47.9411 −1.52290 −0.761450 0.648224i $$-0.775512\pi$$
−0.761450 + 0.648224i $$0.775512\pi$$
$$992$$ 0 0
$$993$$ 56.6274 1.79702
$$994$$ 0 0
$$995$$ −4.58579 −0.145379
$$996$$ 0 0
$$997$$ 3.72792 0.118064 0.0590322 0.998256i $$-0.481199\pi$$
0.0590322 + 0.998256i $$0.481199\pi$$
$$998$$ 4.24264 0.134298
$$999$$ 2.58579 0.0818107
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 245.2.a.f.1.2 yes 2
3.2 odd 2 2205.2.a.t.1.1 2
4.3 odd 2 3920.2.a.br.1.1 2
5.2 odd 4 1225.2.b.j.99.3 4
5.3 odd 4 1225.2.b.j.99.2 4
5.4 even 2 1225.2.a.p.1.1 2
7.2 even 3 245.2.e.f.116.1 4
7.3 odd 6 245.2.e.g.226.1 4
7.4 even 3 245.2.e.f.226.1 4
7.5 odd 6 245.2.e.g.116.1 4
7.6 odd 2 245.2.a.e.1.2 2
21.20 even 2 2205.2.a.v.1.1 2
28.27 even 2 3920.2.a.bw.1.2 2
35.13 even 4 1225.2.b.i.99.1 4
35.27 even 4 1225.2.b.i.99.4 4
35.34 odd 2 1225.2.a.r.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.e.1.2 2 7.6 odd 2
245.2.a.f.1.2 yes 2 1.1 even 1 trivial
245.2.e.f.116.1 4 7.2 even 3
245.2.e.f.226.1 4 7.4 even 3
245.2.e.g.116.1 4 7.5 odd 6
245.2.e.g.226.1 4 7.3 odd 6
1225.2.a.p.1.1 2 5.4 even 2
1225.2.a.r.1.1 2 35.34 odd 2
1225.2.b.i.99.1 4 35.13 even 4
1225.2.b.i.99.4 4 35.27 even 4
1225.2.b.j.99.2 4 5.3 odd 4
1225.2.b.j.99.3 4 5.2 odd 4
2205.2.a.t.1.1 2 3.2 odd 2
2205.2.a.v.1.1 2 21.20 even 2
3920.2.a.br.1.1 2 4.3 odd 2
3920.2.a.bw.1.2 2 28.27 even 2