# Properties

 Label 1875.2.a.p.1.8 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.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: $$14.9719503790$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.8.5444000000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1$$ x^8 - 4*x^7 - 2*x^6 + 20*x^5 - 4*x^4 - 30*x^3 + 7*x^2 + 12*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.8 Root $$2.53767$$ of defining polynomial Character $$\chi$$ $$=$$ 1875.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.53767 q^{2} -1.00000 q^{3} +4.43979 q^{4} -2.53767 q^{6} +1.04054 q^{7} +6.19138 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.53767 q^{2} -1.00000 q^{3} +4.43979 q^{4} -2.53767 q^{6} +1.04054 q^{7} +6.19138 q^{8} +1.00000 q^{9} +2.97101 q^{11} -4.43979 q^{12} +5.66922 q^{13} +2.64054 q^{14} +6.83213 q^{16} -5.08361 q^{17} +2.53767 q^{18} -5.37156 q^{19} -1.04054 q^{21} +7.53945 q^{22} +3.86039 q^{23} -6.19138 q^{24} +14.3866 q^{26} -1.00000 q^{27} +4.61976 q^{28} +0.679696 q^{29} +0.850111 q^{31} +4.95495 q^{32} -2.97101 q^{33} -12.9006 q^{34} +4.43979 q^{36} -1.61763 q^{37} -13.6313 q^{38} -5.66922 q^{39} +1.16529 q^{41} -2.64054 q^{42} +5.68601 q^{43} +13.1906 q^{44} +9.79640 q^{46} +3.28640 q^{47} -6.83213 q^{48} -5.91729 q^{49} +5.08361 q^{51} +25.1701 q^{52} +12.6861 q^{53} -2.53767 q^{54} +6.44235 q^{56} +5.37156 q^{57} +1.72485 q^{58} +3.21187 q^{59} -5.42093 q^{61} +2.15730 q^{62} +1.04054 q^{63} -1.09021 q^{64} -7.53945 q^{66} -0.929140 q^{67} -22.5702 q^{68} -3.86039 q^{69} -1.41358 q^{71} +6.19138 q^{72} +11.3234 q^{73} -4.10501 q^{74} -23.8486 q^{76} +3.09144 q^{77} -14.3866 q^{78} -1.44707 q^{79} +1.00000 q^{81} +2.95713 q^{82} -11.4756 q^{83} -4.61976 q^{84} +14.4292 q^{86} -0.679696 q^{87} +18.3946 q^{88} +9.07225 q^{89} +5.89903 q^{91} +17.1393 q^{92} -0.850111 q^{93} +8.33982 q^{94} -4.95495 q^{96} -6.02928 q^{97} -15.0161 q^{98} +2.97101 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10})$$ 8 * q + 4 * q^2 - 8 * q^3 + 4 * q^4 - 4 * q^6 + 8 * q^7 + 12 * q^8 + 8 * q^9 $$8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9} + 2 q^{11} - 4 q^{12} + 16 q^{13} + 6 q^{14} + 16 q^{17} + 4 q^{18} - 14 q^{19} - 8 q^{21} + 12 q^{22} + 14 q^{23} - 12 q^{24} + 6 q^{26} - 8 q^{27} + 16 q^{28} + 2 q^{29} - 22 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{34} + 4 q^{36} + 28 q^{37} - 16 q^{38} - 16 q^{39} + 8 q^{41} - 6 q^{42} + 20 q^{43} + 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{51} + 16 q^{52} + 44 q^{53} - 4 q^{54} + 30 q^{56} + 14 q^{57} + 8 q^{58} + 14 q^{59} - 20 q^{61} + 16 q^{62} + 8 q^{63} + 6 q^{64} - 12 q^{66} + 16 q^{67} - 2 q^{68} - 14 q^{69} + 16 q^{71} + 12 q^{72} + 24 q^{73} + 26 q^{74} - 16 q^{76} + 46 q^{77} - 6 q^{78} - 30 q^{79} + 8 q^{81} + 16 q^{82} + 12 q^{83} - 16 q^{84} + 32 q^{86} - 2 q^{87} + 32 q^{88} + 16 q^{89} - 12 q^{91} - 2 q^{92} + 22 q^{93} + 14 q^{94} + 2 q^{96} + 16 q^{97} + 4 q^{98} + 2 q^{99}+O(q^{100})$$ 8 * q + 4 * q^2 - 8 * q^3 + 4 * q^4 - 4 * q^6 + 8 * q^7 + 12 * q^8 + 8 * q^9 + 2 * q^11 - 4 * q^12 + 16 * q^13 + 6 * q^14 + 16 * q^17 + 4 * q^18 - 14 * q^19 - 8 * q^21 + 12 * q^22 + 14 * q^23 - 12 * q^24 + 6 * q^26 - 8 * q^27 + 16 * q^28 + 2 * q^29 - 22 * q^31 - 2 * q^32 - 2 * q^33 - 12 * q^34 + 4 * q^36 + 28 * q^37 - 16 * q^38 - 16 * q^39 + 8 * q^41 - 6 * q^42 + 20 * q^43 + 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^51 + 16 * q^52 + 44 * q^53 - 4 * q^54 + 30 * q^56 + 14 * q^57 + 8 * q^58 + 14 * q^59 - 20 * q^61 + 16 * q^62 + 8 * q^63 + 6 * q^64 - 12 * q^66 + 16 * q^67 - 2 * q^68 - 14 * q^69 + 16 * q^71 + 12 * q^72 + 24 * q^73 + 26 * q^74 - 16 * q^76 + 46 * q^77 - 6 * q^78 - 30 * q^79 + 8 * q^81 + 16 * q^82 + 12 * q^83 - 16 * q^84 + 32 * q^86 - 2 * q^87 + 32 * q^88 + 16 * q^89 - 12 * q^91 - 2 * q^92 + 22 * q^93 + 14 * q^94 + 2 * q^96 + 16 * q^97 + 4 * q^98 + 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.53767 1.79441 0.897203 0.441618i $$-0.145595\pi$$
0.897203 + 0.441618i $$0.145595\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 4.43979 2.21989
$$5$$ 0 0
$$6$$ −2.53767 −1.03600
$$7$$ 1.04054 0.393285 0.196643 0.980475i $$-0.436996\pi$$
0.196643 + 0.980475i $$0.436996\pi$$
$$8$$ 6.19138 2.18898
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.97101 0.895792 0.447896 0.894086i $$-0.352173\pi$$
0.447896 + 0.894086i $$0.352173\pi$$
$$12$$ −4.43979 −1.28166
$$13$$ 5.66922 1.57236 0.786180 0.617998i $$-0.212056\pi$$
0.786180 + 0.617998i $$0.212056\pi$$
$$14$$ 2.64054 0.705714
$$15$$ 0 0
$$16$$ 6.83213 1.70803
$$17$$ −5.08361 −1.23296 −0.616479 0.787372i $$-0.711441\pi$$
−0.616479 + 0.787372i $$0.711441\pi$$
$$18$$ 2.53767 0.598135
$$19$$ −5.37156 −1.23232 −0.616160 0.787621i $$-0.711313\pi$$
−0.616160 + 0.787621i $$0.711313\pi$$
$$20$$ 0 0
$$21$$ −1.04054 −0.227063
$$22$$ 7.53945 1.60742
$$23$$ 3.86039 0.804946 0.402473 0.915432i $$-0.368151\pi$$
0.402473 + 0.915432i $$0.368151\pi$$
$$24$$ −6.19138 −1.26381
$$25$$ 0 0
$$26$$ 14.3866 2.82145
$$27$$ −1.00000 −0.192450
$$28$$ 4.61976 0.873052
$$29$$ 0.679696 0.126216 0.0631082 0.998007i $$-0.479899\pi$$
0.0631082 + 0.998007i $$0.479899\pi$$
$$30$$ 0 0
$$31$$ 0.850111 0.152684 0.0763422 0.997082i $$-0.475676\pi$$
0.0763422 + 0.997082i $$0.475676\pi$$
$$32$$ 4.95495 0.875921
$$33$$ −2.97101 −0.517186
$$34$$ −12.9006 −2.21243
$$35$$ 0 0
$$36$$ 4.43979 0.739964
$$37$$ −1.61763 −0.265936 −0.132968 0.991120i $$-0.542451\pi$$
−0.132968 + 0.991120i $$0.542451\pi$$
$$38$$ −13.6313 −2.21128
$$39$$ −5.66922 −0.907802
$$40$$ 0 0
$$41$$ 1.16529 0.181988 0.0909939 0.995851i $$-0.470996\pi$$
0.0909939 + 0.995851i $$0.470996\pi$$
$$42$$ −2.64054 −0.407444
$$43$$ 5.68601 0.867109 0.433554 0.901127i $$-0.357259\pi$$
0.433554 + 0.901127i $$0.357259\pi$$
$$44$$ 13.1906 1.98856
$$45$$ 0 0
$$46$$ 9.79640 1.44440
$$47$$ 3.28640 0.479371 0.239686 0.970851i $$-0.422956\pi$$
0.239686 + 0.970851i $$0.422956\pi$$
$$48$$ −6.83213 −0.986133
$$49$$ −5.91729 −0.845327
$$50$$ 0 0
$$51$$ 5.08361 0.711848
$$52$$ 25.1701 3.49047
$$53$$ 12.6861 1.74257 0.871287 0.490773i $$-0.163286\pi$$
0.871287 + 0.490773i $$0.163286\pi$$
$$54$$ −2.53767 −0.345334
$$55$$ 0 0
$$56$$ 6.44235 0.860896
$$57$$ 5.37156 0.711481
$$58$$ 1.72485 0.226484
$$59$$ 3.21187 0.418150 0.209075 0.977900i $$-0.432955\pi$$
0.209075 + 0.977900i $$0.432955\pi$$
$$60$$ 0 0
$$61$$ −5.42093 −0.694079 −0.347039 0.937851i $$-0.612813\pi$$
−0.347039 + 0.937851i $$0.612813\pi$$
$$62$$ 2.15730 0.273978
$$63$$ 1.04054 0.131095
$$64$$ −1.09021 −0.136276
$$65$$ 0 0
$$66$$ −7.53945 −0.928042
$$67$$ −0.929140 −0.113513 −0.0567563 0.998388i $$-0.518076\pi$$
−0.0567563 + 0.998388i $$0.518076\pi$$
$$68$$ −22.5702 −2.73703
$$69$$ −3.86039 −0.464736
$$70$$ 0 0
$$71$$ −1.41358 −0.167761 −0.0838807 0.996476i $$-0.526731\pi$$
−0.0838807 + 0.996476i $$0.526731\pi$$
$$72$$ 6.19138 0.729661
$$73$$ 11.3234 1.32530 0.662650 0.748929i $$-0.269432\pi$$
0.662650 + 0.748929i $$0.269432\pi$$
$$74$$ −4.10501 −0.477198
$$75$$ 0 0
$$76$$ −23.8486 −2.73562
$$77$$ 3.09144 0.352302
$$78$$ −14.3866 −1.62897
$$79$$ −1.44707 −0.162809 −0.0814043 0.996681i $$-0.525941\pi$$
−0.0814043 + 0.996681i $$0.525941\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.95713 0.326560
$$83$$ −11.4756 −1.25961 −0.629806 0.776752i $$-0.716866\pi$$
−0.629806 + 0.776752i $$0.716866\pi$$
$$84$$ −4.61976 −0.504057
$$85$$ 0 0
$$86$$ 14.4292 1.55594
$$87$$ −0.679696 −0.0728711
$$88$$ 18.3946 1.96087
$$89$$ 9.07225 0.961657 0.480828 0.876815i $$-0.340336\pi$$
0.480828 + 0.876815i $$0.340336\pi$$
$$90$$ 0 0
$$91$$ 5.89903 0.618386
$$92$$ 17.1393 1.78689
$$93$$ −0.850111 −0.0881524
$$94$$ 8.33982 0.860187
$$95$$ 0 0
$$96$$ −4.95495 −0.505713
$$97$$ −6.02928 −0.612181 −0.306091 0.952002i $$-0.599021\pi$$
−0.306091 + 0.952002i $$0.599021\pi$$
$$98$$ −15.0161 −1.51686
$$99$$ 2.97101 0.298597
$$100$$ 0 0
$$101$$ −15.3408 −1.52647 −0.763236 0.646120i $$-0.776390\pi$$
−0.763236 + 0.646120i $$0.776390\pi$$
$$102$$ 12.9006 1.27734
$$103$$ −12.0590 −1.18820 −0.594102 0.804390i $$-0.702493\pi$$
−0.594102 + 0.804390i $$0.702493\pi$$
$$104$$ 35.1003 3.44187
$$105$$ 0 0
$$106$$ 32.1933 3.12689
$$107$$ 6.49787 0.628173 0.314086 0.949394i $$-0.398302\pi$$
0.314086 + 0.949394i $$0.398302\pi$$
$$108$$ −4.43979 −0.427219
$$109$$ −2.31057 −0.221313 −0.110656 0.993859i $$-0.535295\pi$$
−0.110656 + 0.993859i $$0.535295\pi$$
$$110$$ 0 0
$$111$$ 1.61763 0.153538
$$112$$ 7.10907 0.671744
$$113$$ −3.87281 −0.364323 −0.182162 0.983269i $$-0.558309\pi$$
−0.182162 + 0.983269i $$0.558309\pi$$
$$114$$ 13.6313 1.27669
$$115$$ 0 0
$$116$$ 3.01771 0.280187
$$117$$ 5.66922 0.524120
$$118$$ 8.15067 0.750330
$$119$$ −5.28968 −0.484904
$$120$$ 0 0
$$121$$ −2.17312 −0.197556
$$122$$ −13.7565 −1.24546
$$123$$ −1.16529 −0.105071
$$124$$ 3.77431 0.338943
$$125$$ 0 0
$$126$$ 2.64054 0.235238
$$127$$ −11.6938 −1.03765 −0.518827 0.854879i $$-0.673631\pi$$
−0.518827 + 0.854879i $$0.673631\pi$$
$$128$$ −12.6765 −1.12045
$$129$$ −5.68601 −0.500625
$$130$$ 0 0
$$131$$ −7.96210 −0.695652 −0.347826 0.937559i $$-0.613080\pi$$
−0.347826 + 0.937559i $$0.613080\pi$$
$$132$$ −13.1906 −1.14810
$$133$$ −5.58930 −0.484654
$$134$$ −2.35785 −0.203688
$$135$$ 0 0
$$136$$ −31.4746 −2.69892
$$137$$ 11.0513 0.944176 0.472088 0.881551i $$-0.343500\pi$$
0.472088 + 0.881551i $$0.343500\pi$$
$$138$$ −9.79640 −0.833925
$$139$$ −12.2698 −1.04071 −0.520357 0.853949i $$-0.674201\pi$$
−0.520357 + 0.853949i $$0.674201\pi$$
$$140$$ 0 0
$$141$$ −3.28640 −0.276765
$$142$$ −3.58721 −0.301032
$$143$$ 16.8433 1.40851
$$144$$ 6.83213 0.569344
$$145$$ 0 0
$$146$$ 28.7350 2.37813
$$147$$ 5.91729 0.488050
$$148$$ −7.18192 −0.590350
$$149$$ −4.62832 −0.379167 −0.189584 0.981865i $$-0.560714\pi$$
−0.189584 + 0.981865i $$0.560714\pi$$
$$150$$ 0 0
$$151$$ −4.67249 −0.380242 −0.190121 0.981761i $$-0.560888\pi$$
−0.190121 + 0.981761i $$0.560888\pi$$
$$152$$ −33.2574 −2.69753
$$153$$ −5.08361 −0.410986
$$154$$ 7.84506 0.632173
$$155$$ 0 0
$$156$$ −25.1701 −2.01522
$$157$$ −14.9726 −1.19494 −0.597472 0.801890i $$-0.703828\pi$$
−0.597472 + 0.801890i $$0.703828\pi$$
$$158$$ −3.67220 −0.292145
$$159$$ −12.6861 −1.00608
$$160$$ 0 0
$$161$$ 4.01687 0.316574
$$162$$ 2.53767 0.199378
$$163$$ −11.9112 −0.932958 −0.466479 0.884532i $$-0.654478\pi$$
−0.466479 + 0.884532i $$0.654478\pi$$
$$164$$ 5.17364 0.403994
$$165$$ 0 0
$$166$$ −29.1214 −2.26026
$$167$$ 10.6081 0.820881 0.410440 0.911887i $$-0.365375\pi$$
0.410440 + 0.911887i $$0.365375\pi$$
$$168$$ −6.44235 −0.497038
$$169$$ 19.1401 1.47231
$$170$$ 0 0
$$171$$ −5.37156 −0.410774
$$172$$ 25.2447 1.92489
$$173$$ 1.25338 0.0952928 0.0476464 0.998864i $$-0.484828\pi$$
0.0476464 + 0.998864i $$0.484828\pi$$
$$174$$ −1.72485 −0.130760
$$175$$ 0 0
$$176$$ 20.2983 1.53004
$$177$$ −3.21187 −0.241419
$$178$$ 23.0224 1.72560
$$179$$ −20.8113 −1.55551 −0.777755 0.628567i $$-0.783642\pi$$
−0.777755 + 0.628567i $$0.783642\pi$$
$$180$$ 0 0
$$181$$ −6.92706 −0.514884 −0.257442 0.966294i $$-0.582880\pi$$
−0.257442 + 0.966294i $$0.582880\pi$$
$$182$$ 14.9698 1.10964
$$183$$ 5.42093 0.400726
$$184$$ 23.9011 1.76201
$$185$$ 0 0
$$186$$ −2.15730 −0.158181
$$187$$ −15.1034 −1.10447
$$188$$ 14.5909 1.06415
$$189$$ −1.04054 −0.0756878
$$190$$ 0 0
$$191$$ 8.16415 0.590737 0.295369 0.955383i $$-0.404558\pi$$
0.295369 + 0.955383i $$0.404558\pi$$
$$192$$ 1.09021 0.0786788
$$193$$ 13.9629 1.00507 0.502537 0.864556i $$-0.332400\pi$$
0.502537 + 0.864556i $$0.332400\pi$$
$$194$$ −15.3004 −1.09850
$$195$$ 0 0
$$196$$ −26.2715 −1.87653
$$197$$ −5.76250 −0.410561 −0.205281 0.978703i $$-0.565811\pi$$
−0.205281 + 0.978703i $$0.565811\pi$$
$$198$$ 7.53945 0.535805
$$199$$ −26.5748 −1.88384 −0.941919 0.335841i $$-0.890980\pi$$
−0.941919 + 0.335841i $$0.890980\pi$$
$$200$$ 0 0
$$201$$ 0.929140 0.0655365
$$202$$ −38.9301 −2.73911
$$203$$ 0.707248 0.0496391
$$204$$ 22.5702 1.58023
$$205$$ 0 0
$$206$$ −30.6017 −2.13212
$$207$$ 3.86039 0.268315
$$208$$ 38.7329 2.68564
$$209$$ −15.9589 −1.10390
$$210$$ 0 0
$$211$$ −26.4594 −1.82154 −0.910771 0.412912i $$-0.864512\pi$$
−0.910771 + 0.412912i $$0.864512\pi$$
$$212$$ 56.3237 3.86833
$$213$$ 1.41358 0.0968571
$$214$$ 16.4895 1.12720
$$215$$ 0 0
$$216$$ −6.19138 −0.421270
$$217$$ 0.884570 0.0600486
$$218$$ −5.86348 −0.397125
$$219$$ −11.3234 −0.765163
$$220$$ 0 0
$$221$$ −28.8201 −1.93865
$$222$$ 4.10501 0.275510
$$223$$ 27.4456 1.83789 0.918946 0.394383i $$-0.129042\pi$$
0.918946 + 0.394383i $$0.129042\pi$$
$$224$$ 5.15581 0.344487
$$225$$ 0 0
$$226$$ −9.82792 −0.653744
$$227$$ −0.130161 −0.00863907 −0.00431954 0.999991i $$-0.501375\pi$$
−0.00431954 + 0.999991i $$0.501375\pi$$
$$228$$ 23.8486 1.57941
$$229$$ 2.82530 0.186701 0.0933504 0.995633i $$-0.470242\pi$$
0.0933504 + 0.995633i $$0.470242\pi$$
$$230$$ 0 0
$$231$$ −3.09144 −0.203402
$$232$$ 4.20826 0.276286
$$233$$ −7.98709 −0.523252 −0.261626 0.965169i $$-0.584259\pi$$
−0.261626 + 0.965169i $$0.584259\pi$$
$$234$$ 14.3866 0.940484
$$235$$ 0 0
$$236$$ 14.2600 0.928247
$$237$$ 1.44707 0.0939976
$$238$$ −13.4235 −0.870115
$$239$$ −19.0619 −1.23301 −0.616506 0.787350i $$-0.711452\pi$$
−0.616506 + 0.787350i $$0.711452\pi$$
$$240$$ 0 0
$$241$$ 21.1199 1.36045 0.680226 0.733002i $$-0.261882\pi$$
0.680226 + 0.733002i $$0.261882\pi$$
$$242$$ −5.51466 −0.354496
$$243$$ −1.00000 −0.0641500
$$244$$ −24.0678 −1.54078
$$245$$ 0 0
$$246$$ −2.95713 −0.188540
$$247$$ −30.4526 −1.93765
$$248$$ 5.26336 0.334224
$$249$$ 11.4756 0.727238
$$250$$ 0 0
$$251$$ 30.2224 1.90762 0.953811 0.300408i $$-0.0971228\pi$$
0.953811 + 0.300408i $$0.0971228\pi$$
$$252$$ 4.61976 0.291017
$$253$$ 11.4692 0.721065
$$254$$ −29.6750 −1.86197
$$255$$ 0 0
$$256$$ −29.9884 −1.87427
$$257$$ −5.10215 −0.318263 −0.159132 0.987257i $$-0.550869\pi$$
−0.159132 + 0.987257i $$0.550869\pi$$
$$258$$ −14.4292 −0.898325
$$259$$ −1.68320 −0.104589
$$260$$ 0 0
$$261$$ 0.679696 0.0420721
$$262$$ −20.2052 −1.24828
$$263$$ −6.41540 −0.395591 −0.197795 0.980243i $$-0.563378\pi$$
−0.197795 + 0.980243i $$0.563378\pi$$
$$264$$ −18.3946 −1.13211
$$265$$ 0 0
$$266$$ −14.1838 −0.869666
$$267$$ −9.07225 −0.555213
$$268$$ −4.12518 −0.251986
$$269$$ −17.4592 −1.06450 −0.532252 0.846586i $$-0.678654\pi$$
−0.532252 + 0.846586i $$0.678654\pi$$
$$270$$ 0 0
$$271$$ −0.0951857 −0.00578212 −0.00289106 0.999996i $$-0.500920\pi$$
−0.00289106 + 0.999996i $$0.500920\pi$$
$$272$$ −34.7319 −2.10593
$$273$$ −5.89903 −0.357025
$$274$$ 28.0446 1.69424
$$275$$ 0 0
$$276$$ −17.1393 −1.03166
$$277$$ 18.4007 1.10559 0.552796 0.833316i $$-0.313561\pi$$
0.552796 + 0.833316i $$0.313561\pi$$
$$278$$ −31.1369 −1.86747
$$279$$ 0.850111 0.0508948
$$280$$ 0 0
$$281$$ 17.9361 1.06998 0.534988 0.844860i $$-0.320316\pi$$
0.534988 + 0.844860i $$0.320316\pi$$
$$282$$ −8.33982 −0.496629
$$283$$ 22.2399 1.32203 0.661014 0.750374i $$-0.270126\pi$$
0.661014 + 0.750374i $$0.270126\pi$$
$$284$$ −6.27601 −0.372413
$$285$$ 0 0
$$286$$ 42.7428 2.52743
$$287$$ 1.21253 0.0715732
$$288$$ 4.95495 0.291974
$$289$$ 8.84312 0.520184
$$290$$ 0 0
$$291$$ 6.02928 0.353443
$$292$$ 50.2734 2.94203
$$293$$ 18.7316 1.09431 0.547155 0.837031i $$-0.315711\pi$$
0.547155 + 0.837031i $$0.315711\pi$$
$$294$$ 15.0161 0.875759
$$295$$ 0 0
$$296$$ −10.0154 −0.582130
$$297$$ −2.97101 −0.172395
$$298$$ −11.7452 −0.680380
$$299$$ 21.8854 1.26566
$$300$$ 0 0
$$301$$ 5.91650 0.341021
$$302$$ −11.8573 −0.682309
$$303$$ 15.3408 0.881309
$$304$$ −36.6992 −2.10484
$$305$$ 0 0
$$306$$ −12.9006 −0.737475
$$307$$ 7.03850 0.401708 0.200854 0.979621i $$-0.435628\pi$$
0.200854 + 0.979621i $$0.435628\pi$$
$$308$$ 13.7253 0.782073
$$309$$ 12.0590 0.686010
$$310$$ 0 0
$$311$$ −29.2790 −1.66026 −0.830130 0.557570i $$-0.811734\pi$$
−0.830130 + 0.557570i $$0.811734\pi$$
$$312$$ −35.1003 −1.98716
$$313$$ 15.6354 0.883764 0.441882 0.897073i $$-0.354311\pi$$
0.441882 + 0.897073i $$0.354311\pi$$
$$314$$ −37.9956 −2.14422
$$315$$ 0 0
$$316$$ −6.42470 −0.361418
$$317$$ 6.24144 0.350554 0.175277 0.984519i $$-0.443918\pi$$
0.175277 + 0.984519i $$0.443918\pi$$
$$318$$ −32.1933 −1.80531
$$319$$ 2.01938 0.113064
$$320$$ 0 0
$$321$$ −6.49787 −0.362676
$$322$$ 10.1935 0.568062
$$323$$ 27.3069 1.51940
$$324$$ 4.43979 0.246655
$$325$$ 0 0
$$326$$ −30.2268 −1.67411
$$327$$ 2.31057 0.127775
$$328$$ 7.21476 0.398369
$$329$$ 3.41962 0.188530
$$330$$ 0 0
$$331$$ 21.4575 1.17941 0.589705 0.807619i $$-0.299244\pi$$
0.589705 + 0.807619i $$0.299244\pi$$
$$332$$ −50.9493 −2.79621
$$333$$ −1.61763 −0.0886455
$$334$$ 26.9199 1.47299
$$335$$ 0 0
$$336$$ −7.10907 −0.387832
$$337$$ 34.7511 1.89301 0.946507 0.322683i $$-0.104585\pi$$
0.946507 + 0.322683i $$0.104585\pi$$
$$338$$ 48.5712 2.64193
$$339$$ 3.87281 0.210342
$$340$$ 0 0
$$341$$ 2.52569 0.136774
$$342$$ −13.6313 −0.737094
$$343$$ −13.4409 −0.725740
$$344$$ 35.2043 1.89809
$$345$$ 0 0
$$346$$ 3.18067 0.170994
$$347$$ 28.7241 1.54199 0.770995 0.636841i $$-0.219759\pi$$
0.770995 + 0.636841i $$0.219759\pi$$
$$348$$ −3.01771 −0.161766
$$349$$ −12.2834 −0.657515 −0.328758 0.944414i $$-0.606630\pi$$
−0.328758 + 0.944414i $$0.606630\pi$$
$$350$$ 0 0
$$351$$ −5.66922 −0.302601
$$352$$ 14.7212 0.784643
$$353$$ −26.9779 −1.43589 −0.717945 0.696100i $$-0.754917\pi$$
−0.717945 + 0.696100i $$0.754917\pi$$
$$354$$ −8.15067 −0.433203
$$355$$ 0 0
$$356$$ 40.2789 2.13478
$$357$$ 5.28968 0.279960
$$358$$ −52.8123 −2.79122
$$359$$ 16.4243 0.866839 0.433419 0.901192i $$-0.357307\pi$$
0.433419 + 0.901192i $$0.357307\pi$$
$$360$$ 0 0
$$361$$ 9.85366 0.518614
$$362$$ −17.5786 −0.923912
$$363$$ 2.17312 0.114059
$$364$$ 26.1904 1.37275
$$365$$ 0 0
$$366$$ 13.7565 0.719066
$$367$$ −29.8953 −1.56052 −0.780262 0.625453i $$-0.784914\pi$$
−0.780262 + 0.625453i $$0.784914\pi$$
$$368$$ 26.3747 1.37487
$$369$$ 1.16529 0.0606626
$$370$$ 0 0
$$371$$ 13.2004 0.685329
$$372$$ −3.77431 −0.195689
$$373$$ −24.8307 −1.28568 −0.642842 0.765999i $$-0.722245\pi$$
−0.642842 + 0.765999i $$0.722245\pi$$
$$374$$ −38.3276 −1.98187
$$375$$ 0 0
$$376$$ 20.3474 1.04934
$$377$$ 3.85335 0.198458
$$378$$ −2.64054 −0.135815
$$379$$ 19.3966 0.996338 0.498169 0.867080i $$-0.334006\pi$$
0.498169 + 0.867080i $$0.334006\pi$$
$$380$$ 0 0
$$381$$ 11.6938 0.599090
$$382$$ 20.7179 1.06002
$$383$$ −11.2508 −0.574891 −0.287446 0.957797i $$-0.592806\pi$$
−0.287446 + 0.957797i $$0.592806\pi$$
$$384$$ 12.6765 0.646895
$$385$$ 0 0
$$386$$ 35.4334 1.80351
$$387$$ 5.68601 0.289036
$$388$$ −26.7687 −1.35898
$$389$$ 14.9017 0.755549 0.377774 0.925898i $$-0.376690\pi$$
0.377774 + 0.925898i $$0.376690\pi$$
$$390$$ 0 0
$$391$$ −19.6247 −0.992464
$$392$$ −36.6362 −1.85041
$$393$$ 7.96210 0.401635
$$394$$ −14.6233 −0.736713
$$395$$ 0 0
$$396$$ 13.1906 0.662854
$$397$$ 24.0966 1.20937 0.604687 0.796463i $$-0.293298\pi$$
0.604687 + 0.796463i $$0.293298\pi$$
$$398$$ −67.4382 −3.38037
$$399$$ 5.58930 0.279815
$$400$$ 0 0
$$401$$ 13.4580 0.672059 0.336030 0.941851i $$-0.390916\pi$$
0.336030 + 0.941851i $$0.390916\pi$$
$$402$$ 2.35785 0.117599
$$403$$ 4.81947 0.240075
$$404$$ −68.1101 −3.38860
$$405$$ 0 0
$$406$$ 1.79476 0.0890727
$$407$$ −4.80598 −0.238224
$$408$$ 31.4746 1.55822
$$409$$ −35.4737 −1.75406 −0.877030 0.480435i $$-0.840479\pi$$
−0.877030 + 0.480435i $$0.840479\pi$$
$$410$$ 0 0
$$411$$ −11.0513 −0.545120
$$412$$ −53.5392 −2.63769
$$413$$ 3.34206 0.164452
$$414$$ 9.79640 0.481467
$$415$$ 0 0
$$416$$ 28.0907 1.37726
$$417$$ 12.2698 0.600857
$$418$$ −40.4986 −1.98085
$$419$$ 15.8120 0.772466 0.386233 0.922401i $$-0.373776\pi$$
0.386233 + 0.922401i $$0.373776\pi$$
$$420$$ 0 0
$$421$$ 11.9813 0.583935 0.291967 0.956428i $$-0.405690\pi$$
0.291967 + 0.956428i $$0.405690\pi$$
$$422$$ −67.1454 −3.26859
$$423$$ 3.28640 0.159790
$$424$$ 78.5447 3.81447
$$425$$ 0 0
$$426$$ 3.58721 0.173801
$$427$$ −5.64067 −0.272971
$$428$$ 28.8492 1.39448
$$429$$ −16.8433 −0.813202
$$430$$ 0 0
$$431$$ 14.6428 0.705317 0.352659 0.935752i $$-0.385278\pi$$
0.352659 + 0.935752i $$0.385278\pi$$
$$432$$ −6.83213 −0.328711
$$433$$ −4.27293 −0.205344 −0.102672 0.994715i $$-0.532739\pi$$
−0.102672 + 0.994715i $$0.532739\pi$$
$$434$$ 2.24475 0.107751
$$435$$ 0 0
$$436$$ −10.2584 −0.491291
$$437$$ −20.7363 −0.991952
$$438$$ −28.7350 −1.37301
$$439$$ −20.3073 −0.969214 −0.484607 0.874732i $$-0.661037\pi$$
−0.484607 + 0.874732i $$0.661037\pi$$
$$440$$ 0 0
$$441$$ −5.91729 −0.281776
$$442$$ −73.1361 −3.47873
$$443$$ −19.0543 −0.905299 −0.452649 0.891689i $$-0.649521\pi$$
−0.452649 + 0.891689i $$0.649521\pi$$
$$444$$ 7.18192 0.340839
$$445$$ 0 0
$$446$$ 69.6479 3.29793
$$447$$ 4.62832 0.218912
$$448$$ −1.13440 −0.0535952
$$449$$ 29.4793 1.39122 0.695608 0.718421i $$-0.255135\pi$$
0.695608 + 0.718421i $$0.255135\pi$$
$$450$$ 0 0
$$451$$ 3.46209 0.163023
$$452$$ −17.1944 −0.808759
$$453$$ 4.67249 0.219533
$$454$$ −0.330305 −0.0155020
$$455$$ 0 0
$$456$$ 33.2574 1.55742
$$457$$ −28.3015 −1.32389 −0.661945 0.749553i $$-0.730269\pi$$
−0.661945 + 0.749553i $$0.730269\pi$$
$$458$$ 7.16968 0.335017
$$459$$ 5.08361 0.237283
$$460$$ 0 0
$$461$$ −16.5575 −0.771157 −0.385579 0.922675i $$-0.625998\pi$$
−0.385579 + 0.922675i $$0.625998\pi$$
$$462$$ −7.84506 −0.364985
$$463$$ −9.20933 −0.427994 −0.213997 0.976834i $$-0.568648\pi$$
−0.213997 + 0.976834i $$0.568648\pi$$
$$464$$ 4.64377 0.215582
$$465$$ 0 0
$$466$$ −20.2686 −0.938926
$$467$$ −10.6123 −0.491078 −0.245539 0.969387i $$-0.578965\pi$$
−0.245539 + 0.969387i $$0.578965\pi$$
$$468$$ 25.1701 1.16349
$$469$$ −0.966803 −0.0446428
$$470$$ 0 0
$$471$$ 14.9726 0.689902
$$472$$ 19.8859 0.915323
$$473$$ 16.8932 0.776749
$$474$$ 3.67220 0.168670
$$475$$ 0 0
$$476$$ −23.4850 −1.07644
$$477$$ 12.6861 0.580858
$$478$$ −48.3729 −2.21253
$$479$$ 33.6061 1.53550 0.767751 0.640749i $$-0.221376\pi$$
0.767751 + 0.640749i $$0.221376\pi$$
$$480$$ 0 0
$$481$$ −9.17069 −0.418147
$$482$$ 53.5954 2.44120
$$483$$ −4.01687 −0.182774
$$484$$ −9.64818 −0.438554
$$485$$ 0 0
$$486$$ −2.53767 −0.115111
$$487$$ −29.2486 −1.32538 −0.662690 0.748894i $$-0.730585\pi$$
−0.662690 + 0.748894i $$0.730585\pi$$
$$488$$ −33.5630 −1.51933
$$489$$ 11.9112 0.538644
$$490$$ 0 0
$$491$$ 13.9549 0.629778 0.314889 0.949129i $$-0.398033\pi$$
0.314889 + 0.949129i $$0.398033\pi$$
$$492$$ −5.17364 −0.233246
$$493$$ −3.45531 −0.155619
$$494$$ −77.2787 −3.47693
$$495$$ 0 0
$$496$$ 5.80807 0.260790
$$497$$ −1.47088 −0.0659782
$$498$$ 29.1214 1.30496
$$499$$ −35.7533 −1.60054 −0.800268 0.599642i $$-0.795310\pi$$
−0.800268 + 0.599642i $$0.795310\pi$$
$$500$$ 0 0
$$501$$ −10.6081 −0.473936
$$502$$ 76.6946 3.42305
$$503$$ 11.6443 0.519193 0.259596 0.965717i $$-0.416410\pi$$
0.259596 + 0.965717i $$0.416410\pi$$
$$504$$ 6.44235 0.286965
$$505$$ 0 0
$$506$$ 29.1052 1.29388
$$507$$ −19.1401 −0.850040
$$508$$ −51.9178 −2.30348
$$509$$ −8.53362 −0.378246 −0.189123 0.981953i $$-0.560564\pi$$
−0.189123 + 0.981953i $$0.560564\pi$$
$$510$$ 0 0
$$511$$ 11.7824 0.521222
$$512$$ −50.7478 −2.24276
$$513$$ 5.37156 0.237160
$$514$$ −12.9476 −0.571094
$$515$$ 0 0
$$516$$ −25.2447 −1.11133
$$517$$ 9.76393 0.429417
$$518$$ −4.27141 −0.187675
$$519$$ −1.25338 −0.0550173
$$520$$ 0 0
$$521$$ 1.26530 0.0554336 0.0277168 0.999616i $$-0.491176\pi$$
0.0277168 + 0.999616i $$0.491176\pi$$
$$522$$ 1.72485 0.0754945
$$523$$ −27.2204 −1.19027 −0.595133 0.803627i $$-0.702901\pi$$
−0.595133 + 0.803627i $$0.702901\pi$$
$$524$$ −35.3500 −1.54427
$$525$$ 0 0
$$526$$ −16.2802 −0.709850
$$527$$ −4.32163 −0.188253
$$528$$ −20.2983 −0.883370
$$529$$ −8.09742 −0.352062
$$530$$ 0 0
$$531$$ 3.21187 0.139383
$$532$$ −24.8153 −1.07588
$$533$$ 6.60629 0.286150
$$534$$ −23.0224 −0.996277
$$535$$ 0 0
$$536$$ −5.75266 −0.248477
$$537$$ 20.8113 0.898074
$$538$$ −44.3057 −1.91015
$$539$$ −17.5803 −0.757237
$$540$$ 0 0
$$541$$ 30.3830 1.30627 0.653134 0.757242i $$-0.273454\pi$$
0.653134 + 0.757242i $$0.273454\pi$$
$$542$$ −0.241550 −0.0103755
$$543$$ 6.92706 0.297269
$$544$$ −25.1891 −1.07997
$$545$$ 0 0
$$546$$ −14.9698 −0.640648
$$547$$ −11.3768 −0.486439 −0.243219 0.969971i $$-0.578203\pi$$
−0.243219 + 0.969971i $$0.578203\pi$$
$$548$$ 49.0654 2.09597
$$549$$ −5.42093 −0.231360
$$550$$ 0 0
$$551$$ −3.65103 −0.155539
$$552$$ −23.9011 −1.01730
$$553$$ −1.50573 −0.0640303
$$554$$ 46.6950 1.98388
$$555$$ 0 0
$$556$$ −54.4755 −2.31028
$$557$$ 45.7532 1.93862 0.969312 0.245833i $$-0.0790616\pi$$
0.969312 + 0.245833i $$0.0790616\pi$$
$$558$$ 2.15730 0.0913259
$$559$$ 32.2353 1.36341
$$560$$ 0 0
$$561$$ 15.1034 0.637668
$$562$$ 45.5158 1.91997
$$563$$ −8.94289 −0.376898 −0.188449 0.982083i $$-0.560346\pi$$
−0.188449 + 0.982083i $$0.560346\pi$$
$$564$$ −14.5909 −0.614389
$$565$$ 0 0
$$566$$ 56.4377 2.37225
$$567$$ 1.04054 0.0436984
$$568$$ −8.75203 −0.367227
$$569$$ −2.28908 −0.0959632 −0.0479816 0.998848i $$-0.515279\pi$$
−0.0479816 + 0.998848i $$0.515279\pi$$
$$570$$ 0 0
$$571$$ −30.7595 −1.28725 −0.643623 0.765342i $$-0.722570\pi$$
−0.643623 + 0.765342i $$0.722570\pi$$
$$572$$ 74.7806 3.12674
$$573$$ −8.16415 −0.341062
$$574$$ 3.07700 0.128431
$$575$$ 0 0
$$576$$ −1.09021 −0.0454252
$$577$$ −6.59227 −0.274440 −0.137220 0.990541i $$-0.543817\pi$$
−0.137220 + 0.990541i $$0.543817\pi$$
$$578$$ 22.4410 0.933421
$$579$$ −13.9629 −0.580280
$$580$$ 0 0
$$581$$ −11.9408 −0.495387
$$582$$ 15.3004 0.634220
$$583$$ 37.6906 1.56098
$$584$$ 70.1073 2.90106
$$585$$ 0 0
$$586$$ 47.5346 1.96364
$$587$$ −5.53771 −0.228566 −0.114283 0.993448i $$-0.536457\pi$$
−0.114283 + 0.993448i $$0.536457\pi$$
$$588$$ 26.2715 1.08342
$$589$$ −4.56642 −0.188156
$$590$$ 0 0
$$591$$ 5.76250 0.237038
$$592$$ −11.0518 −0.454228
$$593$$ −1.88122 −0.0772524 −0.0386262 0.999254i $$-0.512298\pi$$
−0.0386262 + 0.999254i $$0.512298\pi$$
$$594$$ −7.53945 −0.309347
$$595$$ 0 0
$$596$$ −20.5488 −0.841710
$$597$$ 26.5748 1.08763
$$598$$ 55.5380 2.27112
$$599$$ 17.8272 0.728400 0.364200 0.931321i $$-0.381342\pi$$
0.364200 + 0.931321i $$0.381342\pi$$
$$600$$ 0 0
$$601$$ 33.0994 1.35015 0.675077 0.737747i $$-0.264110\pi$$
0.675077 + 0.737747i $$0.264110\pi$$
$$602$$ 15.0141 0.611931
$$603$$ −0.929140 −0.0378375
$$604$$ −20.7449 −0.844097
$$605$$ 0 0
$$606$$ 38.9301 1.58143
$$607$$ −23.4603 −0.952226 −0.476113 0.879384i $$-0.657955\pi$$
−0.476113 + 0.879384i $$0.657955\pi$$
$$608$$ −26.6158 −1.07941
$$609$$ −0.707248 −0.0286591
$$610$$ 0 0
$$611$$ 18.6314 0.753744
$$612$$ −22.5702 −0.912345
$$613$$ 47.2281 1.90752 0.953762 0.300564i $$-0.0971750\pi$$
0.953762 + 0.300564i $$0.0971750\pi$$
$$614$$ 17.8614 0.720828
$$615$$ 0 0
$$616$$ 19.1403 0.771184
$$617$$ 16.2698 0.654999 0.327499 0.944851i $$-0.393794\pi$$
0.327499 + 0.944851i $$0.393794\pi$$
$$618$$ 30.6017 1.23098
$$619$$ 35.4599 1.42525 0.712627 0.701543i $$-0.247505\pi$$
0.712627 + 0.701543i $$0.247505\pi$$
$$620$$ 0 0
$$621$$ −3.86039 −0.154912
$$622$$ −74.3005 −2.97918
$$623$$ 9.44000 0.378206
$$624$$ −38.7329 −1.55056
$$625$$ 0 0
$$626$$ 39.6775 1.58583
$$627$$ 15.9589 0.637339
$$628$$ −66.4752 −2.65265
$$629$$ 8.22339 0.327888
$$630$$ 0 0
$$631$$ −29.2364 −1.16388 −0.581941 0.813231i $$-0.697706\pi$$
−0.581941 + 0.813231i $$0.697706\pi$$
$$632$$ −8.95939 −0.356385
$$633$$ 26.4594 1.05167
$$634$$ 15.8387 0.629037
$$635$$ 0 0
$$636$$ −56.3237 −2.23338
$$637$$ −33.5464 −1.32916
$$638$$ 5.12453 0.202882
$$639$$ −1.41358 −0.0559205
$$640$$ 0 0
$$641$$ −15.7160 −0.620745 −0.310373 0.950615i $$-0.600454\pi$$
−0.310373 + 0.950615i $$0.600454\pi$$
$$642$$ −16.4895 −0.650788
$$643$$ 8.08055 0.318666 0.159333 0.987225i $$-0.449066\pi$$
0.159333 + 0.987225i $$0.449066\pi$$
$$644$$ 17.8340 0.702760
$$645$$ 0 0
$$646$$ 69.2961 2.72642
$$647$$ −11.0193 −0.433214 −0.216607 0.976259i $$-0.569499\pi$$
−0.216607 + 0.976259i $$0.569499\pi$$
$$648$$ 6.19138 0.243220
$$649$$ 9.54248 0.374575
$$650$$ 0 0
$$651$$ −0.884570 −0.0346691
$$652$$ −52.8832 −2.07107
$$653$$ 31.0861 1.21649 0.608246 0.793748i $$-0.291873\pi$$
0.608246 + 0.793748i $$0.291873\pi$$
$$654$$ 5.86348 0.229280
$$655$$ 0 0
$$656$$ 7.96142 0.310841
$$657$$ 11.3234 0.441767
$$658$$ 8.67788 0.338299
$$659$$ 32.4252 1.26311 0.631553 0.775333i $$-0.282418\pi$$
0.631553 + 0.775333i $$0.282418\pi$$
$$660$$ 0 0
$$661$$ 15.6076 0.607067 0.303534 0.952821i $$-0.401834\pi$$
0.303534 + 0.952821i $$0.401834\pi$$
$$662$$ 54.4521 2.11634
$$663$$ 28.8201 1.11928
$$664$$ −71.0499 −2.75727
$$665$$ 0 0
$$666$$ −4.10501 −0.159066
$$667$$ 2.62389 0.101597
$$668$$ 47.0978 1.82227
$$669$$ −27.4456 −1.06111
$$670$$ 0 0
$$671$$ −16.1056 −0.621750
$$672$$ −5.15581 −0.198890
$$673$$ −31.0271 −1.19601 −0.598003 0.801494i $$-0.704039\pi$$
−0.598003 + 0.801494i $$0.704039\pi$$
$$674$$ 88.1870 3.39684
$$675$$ 0 0
$$676$$ 84.9778 3.26838
$$677$$ 23.0893 0.887394 0.443697 0.896177i $$-0.353667\pi$$
0.443697 + 0.896177i $$0.353667\pi$$
$$678$$ 9.82792 0.377439
$$679$$ −6.27369 −0.240762
$$680$$ 0 0
$$681$$ 0.130161 0.00498777
$$682$$ 6.40936 0.245427
$$683$$ 41.3541 1.58237 0.791185 0.611577i $$-0.209464\pi$$
0.791185 + 0.611577i $$0.209464\pi$$
$$684$$ −23.8486 −0.911873
$$685$$ 0 0
$$686$$ −34.1086 −1.30227
$$687$$ −2.82530 −0.107792
$$688$$ 38.8476 1.48105
$$689$$ 71.9205 2.73995
$$690$$ 0 0
$$691$$ 4.46909 0.170012 0.0850061 0.996380i $$-0.472909\pi$$
0.0850061 + 0.996380i $$0.472909\pi$$
$$692$$ 5.56475 0.211540
$$693$$ 3.09144 0.117434
$$694$$ 72.8924 2.76696
$$695$$ 0 0
$$696$$ −4.20826 −0.159514
$$697$$ −5.92389 −0.224383
$$698$$ −31.1713 −1.17985
$$699$$ 7.98709 0.302099
$$700$$ 0 0
$$701$$ −25.2265 −0.952791 −0.476396 0.879231i $$-0.658057\pi$$
−0.476396 + 0.879231i $$0.658057\pi$$
$$702$$ −14.3866 −0.542988
$$703$$ 8.68919 0.327719
$$704$$ −3.23901 −0.122075
$$705$$ 0 0
$$706$$ −68.4611 −2.57657
$$707$$ −15.9627 −0.600339
$$708$$ −14.2600 −0.535924
$$709$$ 1.55990 0.0585834 0.0292917 0.999571i $$-0.490675\pi$$
0.0292917 + 0.999571i $$0.490675\pi$$
$$710$$ 0 0
$$711$$ −1.44707 −0.0542695
$$712$$ 56.1698 2.10505
$$713$$ 3.28176 0.122903
$$714$$ 13.4235 0.502361
$$715$$ 0 0
$$716$$ −92.3978 −3.45307
$$717$$ 19.0619 0.711880
$$718$$ 41.6794 1.55546
$$719$$ 28.9403 1.07929 0.539645 0.841893i $$-0.318558\pi$$
0.539645 + 0.841893i $$0.318558\pi$$
$$720$$ 0 0
$$721$$ −12.5478 −0.467304
$$722$$ 25.0054 0.930604
$$723$$ −21.1199 −0.785458
$$724$$ −30.7547 −1.14299
$$725$$ 0 0
$$726$$ 5.51466 0.204668
$$727$$ −36.9595 −1.37075 −0.685375 0.728190i $$-0.740362\pi$$
−0.685375 + 0.728190i $$0.740362\pi$$
$$728$$ 36.5231 1.35364
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −28.9055 −1.06911
$$732$$ 24.0678 0.889570
$$733$$ −4.16161 −0.153713 −0.0768564 0.997042i $$-0.524488\pi$$
−0.0768564 + 0.997042i $$0.524488\pi$$
$$734$$ −75.8646 −2.80021
$$735$$ 0 0
$$736$$ 19.1280 0.705069
$$737$$ −2.76048 −0.101684
$$738$$ 2.95713 0.108853
$$739$$ 18.9577 0.697370 0.348685 0.937240i $$-0.386628\pi$$
0.348685 + 0.937240i $$0.386628\pi$$
$$740$$ 0 0
$$741$$ 30.4526 1.11870
$$742$$ 33.4982 1.22976
$$743$$ −11.7060 −0.429452 −0.214726 0.976674i $$-0.568886\pi$$
−0.214726 + 0.976674i $$0.568886\pi$$
$$744$$ −5.26336 −0.192964
$$745$$ 0 0
$$746$$ −63.0122 −2.30704
$$747$$ −11.4756 −0.419871
$$748$$ −67.0561 −2.45181
$$749$$ 6.76126 0.247051
$$750$$ 0 0
$$751$$ −4.95672 −0.180873 −0.0904367 0.995902i $$-0.528826\pi$$
−0.0904367 + 0.995902i $$0.528826\pi$$
$$752$$ 22.4531 0.818782
$$753$$ −30.2224 −1.10137
$$754$$ 9.77854 0.356113
$$755$$ 0 0
$$756$$ −4.61976 −0.168019
$$757$$ 18.6020 0.676101 0.338051 0.941128i $$-0.390232\pi$$
0.338051 + 0.941128i $$0.390232\pi$$
$$758$$ 49.2223 1.78784
$$759$$ −11.4692 −0.416307
$$760$$ 0 0
$$761$$ −11.3585 −0.411747 −0.205873 0.978579i $$-0.566003\pi$$
−0.205873 + 0.978579i $$0.566003\pi$$
$$762$$ 29.6750 1.07501
$$763$$ −2.40423 −0.0870391
$$764$$ 36.2471 1.31137
$$765$$ 0 0
$$766$$ −28.5510 −1.03159
$$767$$ 18.2088 0.657481
$$768$$ 29.9884 1.08211
$$769$$ 7.27477 0.262335 0.131168 0.991360i $$-0.458127\pi$$
0.131168 + 0.991360i $$0.458127\pi$$
$$770$$ 0 0
$$771$$ 5.10215 0.183749
$$772$$ 61.9925 2.23116
$$773$$ 9.32288 0.335321 0.167660 0.985845i $$-0.446379\pi$$
0.167660 + 0.985845i $$0.446379\pi$$
$$774$$ 14.4292 0.518648
$$775$$ 0 0
$$776$$ −37.3296 −1.34005
$$777$$ 1.68320 0.0603844
$$778$$ 37.8158 1.35576
$$779$$ −6.25943 −0.224267
$$780$$ 0 0
$$781$$ −4.19977 −0.150279
$$782$$ −49.8011 −1.78088
$$783$$ −0.679696 −0.0242904
$$784$$ −40.4277 −1.44385
$$785$$ 0 0
$$786$$ 20.2052 0.720696
$$787$$ −14.8658 −0.529907 −0.264953 0.964261i $$-0.585357\pi$$
−0.264953 + 0.964261i $$0.585357\pi$$
$$788$$ −25.5843 −0.911402
$$789$$ 6.41540 0.228394
$$790$$ 0 0
$$791$$ −4.02979 −0.143283
$$792$$ 18.3946 0.653625
$$793$$ −30.7324 −1.09134
$$794$$ 61.1493 2.17011
$$795$$ 0 0
$$796$$ −117.986 −4.18192
$$797$$ 9.57546 0.339180 0.169590 0.985515i $$-0.445756\pi$$
0.169590 + 0.985515i $$0.445756\pi$$
$$798$$ 14.1838 0.502102
$$799$$ −16.7068 −0.591044
$$800$$ 0 0
$$801$$ 9.07225 0.320552
$$802$$ 34.1520 1.20595
$$803$$ 33.6418 1.18719
$$804$$ 4.12518 0.145484
$$805$$ 0 0
$$806$$ 12.2302 0.430792
$$807$$ 17.4592 0.614592
$$808$$ −94.9810 −3.34142
$$809$$ 41.1436 1.44653 0.723266 0.690569i $$-0.242640\pi$$
0.723266 + 0.690569i $$0.242640\pi$$
$$810$$ 0 0
$$811$$ 43.4398 1.52538 0.762688 0.646766i $$-0.223879\pi$$
0.762688 + 0.646766i $$0.223879\pi$$
$$812$$ 3.14003 0.110193
$$813$$ 0.0951857 0.00333831
$$814$$ −12.1960 −0.427470
$$815$$ 0 0
$$816$$ 34.7319 1.21586
$$817$$ −30.5428 −1.06856
$$818$$ −90.0206 −3.14750
$$819$$ 5.89903 0.206129
$$820$$ 0 0
$$821$$ 17.4617 0.609416 0.304708 0.952446i $$-0.401441\pi$$
0.304708 + 0.952446i $$0.401441\pi$$
$$822$$ −28.0446 −0.978167
$$823$$ 2.22456 0.0775432 0.0387716 0.999248i $$-0.487656\pi$$
0.0387716 + 0.999248i $$0.487656\pi$$
$$824$$ −74.6616 −2.60096
$$825$$ 0 0
$$826$$ 8.48106 0.295094
$$827$$ −31.4550 −1.09380 −0.546898 0.837199i $$-0.684192\pi$$
−0.546898 + 0.837199i $$0.684192\pi$$
$$828$$ 17.1393 0.595632
$$829$$ 53.2947 1.85100 0.925500 0.378748i $$-0.123645\pi$$
0.925500 + 0.378748i $$0.123645\pi$$
$$830$$ 0 0
$$831$$ −18.4007 −0.638314
$$832$$ −6.18061 −0.214274
$$833$$ 30.0812 1.04225
$$834$$ 31.1369 1.07818
$$835$$ 0 0
$$836$$ −70.8543 −2.45055
$$837$$ −0.850111 −0.0293841
$$838$$ 40.1256 1.38612
$$839$$ −8.37331 −0.289079 −0.144539 0.989499i $$-0.546170\pi$$
−0.144539 + 0.989499i $$0.546170\pi$$
$$840$$ 0 0
$$841$$ −28.5380 −0.984069
$$842$$ 30.4047 1.04782
$$843$$ −17.9361 −0.617750
$$844$$ −117.474 −4.04363
$$845$$ 0 0
$$846$$ 8.33982 0.286729
$$847$$ −2.26121 −0.0776960
$$848$$ 86.6733 2.97637
$$849$$ −22.2399 −0.763273
$$850$$ 0 0
$$851$$ −6.24467 −0.214064
$$852$$ 6.27601 0.215013
$$853$$ 1.22370 0.0418989 0.0209494 0.999781i $$-0.493331\pi$$
0.0209494 + 0.999781i $$0.493331\pi$$
$$854$$ −14.3142 −0.489821
$$855$$ 0 0
$$856$$ 40.2308 1.37506
$$857$$ 14.3684 0.490816 0.245408 0.969420i $$-0.421078\pi$$
0.245408 + 0.969420i $$0.421078\pi$$
$$858$$ −42.7428 −1.45921
$$859$$ −10.3090 −0.351738 −0.175869 0.984414i $$-0.556274\pi$$
−0.175869 + 0.984414i $$0.556274\pi$$
$$860$$ 0 0
$$861$$ −1.21253 −0.0413228
$$862$$ 37.1586 1.26563
$$863$$ −30.2724 −1.03048 −0.515242 0.857045i $$-0.672298\pi$$
−0.515242 + 0.857045i $$0.672298\pi$$
$$864$$ −4.95495 −0.168571
$$865$$ 0 0
$$866$$ −10.8433 −0.368470
$$867$$ −8.84312 −0.300328
$$868$$ 3.92730 0.133301
$$869$$ −4.29927 −0.145843
$$870$$ 0 0
$$871$$ −5.26750 −0.178482
$$872$$ −14.3056 −0.484450
$$873$$ −6.02928 −0.204060
$$874$$ −52.6220 −1.77996
$$875$$ 0 0
$$876$$ −50.2734 −1.69858
$$877$$ −30.6345 −1.03445 −0.517227 0.855848i $$-0.673036\pi$$
−0.517227 + 0.855848i $$0.673036\pi$$
$$878$$ −51.5333 −1.73916
$$879$$ −18.7316 −0.631800
$$880$$ 0 0
$$881$$ 15.2211 0.512813 0.256406 0.966569i $$-0.417461\pi$$
0.256406 + 0.966569i $$0.417461\pi$$
$$882$$ −15.0161 −0.505620
$$883$$ −40.1985 −1.35279 −0.676393 0.736541i $$-0.736458\pi$$
−0.676393 + 0.736541i $$0.736458\pi$$
$$884$$ −127.955 −4.30360
$$885$$ 0 0
$$886$$ −48.3537 −1.62447
$$887$$ 31.8432 1.06919 0.534594 0.845109i $$-0.320464\pi$$
0.534594 + 0.845109i $$0.320464\pi$$
$$888$$ 10.0154 0.336093
$$889$$ −12.1678 −0.408094
$$890$$ 0 0
$$891$$ 2.97101 0.0995325
$$892$$ 121.853 4.07993
$$893$$ −17.6531 −0.590739
$$894$$ 11.7452 0.392817
$$895$$ 0 0
$$896$$ −13.1903 −0.440658
$$897$$ −21.8854 −0.730732
$$898$$ 74.8089 2.49641
$$899$$ 0.577817 0.0192713
$$900$$ 0 0
$$901$$ −64.4914 −2.14852
$$902$$ 8.78565 0.292530
$$903$$ −5.91650 −0.196889
$$904$$ −23.9780 −0.797498
$$905$$ 0 0
$$906$$ 11.8573 0.393931
$$907$$ 28.8507 0.957970 0.478985 0.877823i $$-0.341005\pi$$
0.478985 + 0.877823i $$0.341005\pi$$
$$908$$ −0.577886 −0.0191778
$$909$$ −15.3408 −0.508824
$$910$$ 0 0
$$911$$ 48.3760 1.60277 0.801385 0.598149i $$-0.204097\pi$$
0.801385 + 0.598149i $$0.204097\pi$$
$$912$$ 36.6992 1.21523
$$913$$ −34.0941 −1.12835
$$914$$ −71.8200 −2.37560
$$915$$ 0 0
$$916$$ 12.5437 0.414456
$$917$$ −8.28485 −0.273590
$$918$$ 12.9006 0.425782
$$919$$ −8.38022 −0.276438 −0.138219 0.990402i $$-0.544138\pi$$
−0.138219 + 0.990402i $$0.544138\pi$$
$$920$$ 0 0
$$921$$ −7.03850 −0.231926
$$922$$ −42.0174 −1.38377
$$923$$ −8.01392 −0.263781
$$924$$ −13.7253 −0.451530
$$925$$ 0 0
$$926$$ −23.3703 −0.767995
$$927$$ −12.0590 −0.396068
$$928$$ 3.36786 0.110556
$$929$$ 14.3126 0.469580 0.234790 0.972046i $$-0.424560\pi$$
0.234790 + 0.972046i $$0.424560\pi$$
$$930$$ 0 0
$$931$$ 31.7851 1.04171
$$932$$ −35.4610 −1.16156
$$933$$ 29.2790 0.958552
$$934$$ −26.9305 −0.881194
$$935$$ 0 0
$$936$$ 35.1003 1.14729
$$937$$ 21.9244 0.716240 0.358120 0.933675i $$-0.383418\pi$$
0.358120 + 0.933675i $$0.383418\pi$$
$$938$$ −2.45343 −0.0801074
$$939$$ −15.6354 −0.510241
$$940$$ 0 0
$$941$$ 22.3934 0.730005 0.365003 0.931007i $$-0.381068\pi$$
0.365003 + 0.931007i $$0.381068\pi$$
$$942$$ 37.9956 1.23796
$$943$$ 4.49847 0.146490
$$944$$ 21.9439 0.714213
$$945$$ 0 0
$$946$$ 42.8694 1.39380
$$947$$ −43.5638 −1.41563 −0.707816 0.706397i $$-0.750319\pi$$
−0.707816 + 0.706397i $$0.750319\pi$$
$$948$$ 6.42470 0.208665
$$949$$ 64.1947 2.08385
$$950$$ 0 0
$$951$$ −6.24144 −0.202393
$$952$$ −32.7504 −1.06145
$$953$$ −2.37169 −0.0768266 −0.0384133 0.999262i $$-0.512230\pi$$
−0.0384133 + 0.999262i $$0.512230\pi$$
$$954$$ 32.1933 1.04230
$$955$$ 0 0
$$956$$ −84.6308 −2.73716
$$957$$ −2.01938 −0.0652774
$$958$$ 85.2813 2.75531
$$959$$ 11.4993 0.371331
$$960$$ 0 0
$$961$$ −30.2773 −0.976687
$$962$$ −23.2722 −0.750326
$$963$$ 6.49787 0.209391
$$964$$ 93.7678 3.02006
$$965$$ 0 0
$$966$$ −10.1935 −0.327971
$$967$$ 19.7857 0.636266 0.318133 0.948046i $$-0.396944\pi$$
0.318133 + 0.948046i $$0.396944\pi$$
$$968$$ −13.4546 −0.432447
$$969$$ −27.3069 −0.877225
$$970$$ 0 0
$$971$$ 49.3838 1.58480 0.792401 0.610000i $$-0.208831\pi$$
0.792401 + 0.610000i $$0.208831\pi$$
$$972$$ −4.43979 −0.142406
$$973$$ −12.7672 −0.409298
$$974$$ −74.2234 −2.37827
$$975$$ 0 0
$$976$$ −37.0365 −1.18551
$$977$$ 47.7652 1.52814 0.764072 0.645130i $$-0.223197\pi$$
0.764072 + 0.645130i $$0.223197\pi$$
$$978$$ 30.2268 0.966545
$$979$$ 26.9537 0.861445
$$980$$ 0 0
$$981$$ −2.31057 −0.0737709
$$982$$ 35.4131 1.13008
$$983$$ −51.9282 −1.65625 −0.828127 0.560541i $$-0.810593\pi$$
−0.828127 + 0.560541i $$0.810593\pi$$
$$984$$ −7.21476 −0.229998
$$985$$ 0 0
$$986$$ −8.76846 −0.279245
$$987$$ −3.41962 −0.108848
$$988$$ −135.203 −4.30138
$$989$$ 21.9502 0.697976
$$990$$ 0 0
$$991$$ −16.3790 −0.520297 −0.260148 0.965569i $$-0.583771\pi$$
−0.260148 + 0.965569i $$0.583771\pi$$
$$992$$ 4.21226 0.133739
$$993$$ −21.4575 −0.680933
$$994$$ −3.73262 −0.118392
$$995$$ 0 0
$$996$$ 50.9493 1.61439
$$997$$ 22.2515 0.704713 0.352357 0.935866i $$-0.385380\pi$$
0.352357 + 0.935866i $$0.385380\pi$$
$$998$$ −90.7302 −2.87201
$$999$$ 1.61763 0.0511795
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 1875.2.a.p.1.8 8
3.2 odd 2 5625.2.a.t.1.1 8
5.2 odd 4 1875.2.b.h.1249.16 16
5.3 odd 4 1875.2.b.h.1249.1 16
5.4 even 2 1875.2.a.m.1.1 8
15.14 odd 2 5625.2.a.bd.1.8 8
25.3 odd 20 375.2.i.c.49.1 16
25.4 even 10 375.2.g.e.76.4 16
25.6 even 5 375.2.g.d.301.1 16
25.8 odd 20 75.2.i.a.64.4 yes 16
25.17 odd 20 375.2.i.c.199.1 16
25.19 even 10 375.2.g.e.301.4 16
25.21 even 5 375.2.g.d.76.1 16
25.22 odd 20 75.2.i.a.34.4 16
75.8 even 20 225.2.m.b.64.1 16
75.47 even 20 225.2.m.b.109.1 16

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.34.4 16 25.22 odd 20
75.2.i.a.64.4 yes 16 25.8 odd 20
225.2.m.b.64.1 16 75.8 even 20
225.2.m.b.109.1 16 75.47 even 20
375.2.g.d.76.1 16 25.21 even 5
375.2.g.d.301.1 16 25.6 even 5
375.2.g.e.76.4 16 25.4 even 10
375.2.g.e.301.4 16 25.19 even 10
375.2.i.c.49.1 16 25.3 odd 20
375.2.i.c.199.1 16 25.17 odd 20
1875.2.a.m.1.1 8 5.4 even 2
1875.2.a.p.1.8 8 1.1 even 1 trivial
1875.2.b.h.1249.1 16 5.3 odd 4
1875.2.b.h.1249.16 16 5.2 odd 4
5625.2.a.t.1.1 8 3.2 odd 2
5625.2.a.bd.1.8 8 15.14 odd 2