# Properties

 Label 1875.2.a.p.1.6 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 1.6 Root $$1.53655$$ 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)

 $$f(q)$$ $$=$$ $$q+1.53655 q^{2} -1.00000 q^{3} +0.360976 q^{4} -1.53655 q^{6} +1.49550 q^{7} -2.51844 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.53655 q^{2} -1.00000 q^{3} +0.360976 q^{4} -1.53655 q^{6} +1.49550 q^{7} -2.51844 q^{8} +1.00000 q^{9} -2.35626 q^{11} -0.360976 q^{12} +1.34951 q^{13} +2.29790 q^{14} -4.59165 q^{16} -2.19405 q^{17} +1.53655 q^{18} +5.71069 q^{19} -1.49550 q^{21} -3.62050 q^{22} +8.79501 q^{23} +2.51844 q^{24} +2.07358 q^{26} -1.00000 q^{27} +0.539839 q^{28} +7.90017 q^{29} -3.69717 q^{31} -2.01841 q^{32} +2.35626 q^{33} -3.37126 q^{34} +0.360976 q^{36} +9.75097 q^{37} +8.77474 q^{38} -1.34951 q^{39} +1.85550 q^{41} -2.29790 q^{42} -8.01874 q^{43} -0.850553 q^{44} +13.5139 q^{46} +6.66298 q^{47} +4.59165 q^{48} -4.76349 q^{49} +2.19405 q^{51} +0.487140 q^{52} +4.17153 q^{53} -1.53655 q^{54} -3.76631 q^{56} -5.71069 q^{57} +12.1390 q^{58} +11.0647 q^{59} -12.2372 q^{61} -5.68088 q^{62} +1.49550 q^{63} +6.08192 q^{64} +3.62050 q^{66} +4.31358 q^{67} -0.792000 q^{68} -8.79501 q^{69} +5.77750 q^{71} -2.51844 q^{72} +6.92684 q^{73} +14.9828 q^{74} +2.06142 q^{76} -3.52377 q^{77} -2.07358 q^{78} -10.6687 q^{79} +1.00000 q^{81} +2.85106 q^{82} +0.224003 q^{83} -0.539839 q^{84} -12.3212 q^{86} -7.90017 q^{87} +5.93408 q^{88} +0.429167 q^{89} +2.01818 q^{91} +3.17479 q^{92} +3.69717 q^{93} +10.2380 q^{94} +2.01841 q^{96} +12.4945 q^{97} -7.31933 q^{98} -2.35626 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.53655 1.08650 0.543251 0.839570i $$-0.317193\pi$$
0.543251 + 0.839570i $$0.317193\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 0.360976 0.180488
$$5$$ 0 0
$$6$$ −1.53655 −0.627293
$$7$$ 1.49550 0.565244 0.282622 0.959231i $$-0.408796\pi$$
0.282622 + 0.959231i $$0.408796\pi$$
$$8$$ −2.51844 −0.890402
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.35626 −0.710438 −0.355219 0.934783i $$-0.615594\pi$$
−0.355219 + 0.934783i $$0.615594\pi$$
$$12$$ −0.360976 −0.104205
$$13$$ 1.34951 0.374286 0.187143 0.982333i $$-0.440077\pi$$
0.187143 + 0.982333i $$0.440077\pi$$
$$14$$ 2.29790 0.614140
$$15$$ 0 0
$$16$$ −4.59165 −1.14791
$$17$$ −2.19405 −0.532135 −0.266068 0.963954i $$-0.585724\pi$$
−0.266068 + 0.963954i $$0.585724\pi$$
$$18$$ 1.53655 0.362168
$$19$$ 5.71069 1.31012 0.655061 0.755576i $$-0.272643\pi$$
0.655061 + 0.755576i $$0.272643\pi$$
$$20$$ 0 0
$$21$$ −1.49550 −0.326344
$$22$$ −3.62050 −0.771892
$$23$$ 8.79501 1.83389 0.916943 0.399018i $$-0.130649\pi$$
0.916943 + 0.399018i $$0.130649\pi$$
$$24$$ 2.51844 0.514074
$$25$$ 0 0
$$26$$ 2.07358 0.406662
$$27$$ −1.00000 −0.192450
$$28$$ 0.539839 0.102020
$$29$$ 7.90017 1.46702 0.733512 0.679676i $$-0.237880\pi$$
0.733512 + 0.679676i $$0.237880\pi$$
$$30$$ 0 0
$$31$$ −3.69717 −0.664031 −0.332016 0.943274i $$-0.607729\pi$$
−0.332016 + 0.943274i $$0.607729\pi$$
$$32$$ −2.01841 −0.356808
$$33$$ 2.35626 0.410171
$$34$$ −3.37126 −0.578167
$$35$$ 0 0
$$36$$ 0.360976 0.0601627
$$37$$ 9.75097 1.60305 0.801525 0.597962i $$-0.204023\pi$$
0.801525 + 0.597962i $$0.204023\pi$$
$$38$$ 8.77474 1.42345
$$39$$ −1.34951 −0.216094
$$40$$ 0 0
$$41$$ 1.85550 0.289780 0.144890 0.989448i $$-0.453717\pi$$
0.144890 + 0.989448i $$0.453717\pi$$
$$42$$ −2.29790 −0.354574
$$43$$ −8.01874 −1.22285 −0.611423 0.791304i $$-0.709403\pi$$
−0.611423 + 0.791304i $$0.709403\pi$$
$$44$$ −0.850553 −0.128226
$$45$$ 0 0
$$46$$ 13.5139 1.99252
$$47$$ 6.66298 0.971895 0.485948 0.873988i $$-0.338475\pi$$
0.485948 + 0.873988i $$0.338475\pi$$
$$48$$ 4.59165 0.662747
$$49$$ −4.76349 −0.680499
$$50$$ 0 0
$$51$$ 2.19405 0.307228
$$52$$ 0.487140 0.0675542
$$53$$ 4.17153 0.573003 0.286502 0.958080i $$-0.407508\pi$$
0.286502 + 0.958080i $$0.407508\pi$$
$$54$$ −1.53655 −0.209098
$$55$$ 0 0
$$56$$ −3.76631 −0.503295
$$57$$ −5.71069 −0.756399
$$58$$ 12.1390 1.59393
$$59$$ 11.0647 1.44050 0.720248 0.693716i $$-0.244028\pi$$
0.720248 + 0.693716i $$0.244028\pi$$
$$60$$ 0 0
$$61$$ −12.2372 −1.56682 −0.783408 0.621508i $$-0.786520\pi$$
−0.783408 + 0.621508i $$0.786520\pi$$
$$62$$ −5.68088 −0.721472
$$63$$ 1.49550 0.188415
$$64$$ 6.08192 0.760239
$$65$$ 0 0
$$66$$ 3.62050 0.445652
$$67$$ 4.31358 0.526988 0.263494 0.964661i $$-0.415125\pi$$
0.263494 + 0.964661i $$0.415125\pi$$
$$68$$ −0.792000 −0.0960442
$$69$$ −8.79501 −1.05879
$$70$$ 0 0
$$71$$ 5.77750 0.685663 0.342832 0.939397i $$-0.388614\pi$$
0.342832 + 0.939397i $$0.388614\pi$$
$$72$$ −2.51844 −0.296801
$$73$$ 6.92684 0.810725 0.405362 0.914156i $$-0.367145\pi$$
0.405362 + 0.914156i $$0.367145\pi$$
$$74$$ 14.9828 1.74172
$$75$$ 0 0
$$76$$ 2.06142 0.236461
$$77$$ −3.52377 −0.401571
$$78$$ −2.07358 −0.234787
$$79$$ −10.6687 −1.20033 −0.600163 0.799878i $$-0.704898\pi$$
−0.600163 + 0.799878i $$0.704898\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.85106 0.314846
$$83$$ 0.224003 0.0245875 0.0122938 0.999924i $$-0.496087\pi$$
0.0122938 + 0.999924i $$0.496087\pi$$
$$84$$ −0.539839 −0.0589012
$$85$$ 0 0
$$86$$ −12.3212 −1.32863
$$87$$ −7.90017 −0.846987
$$88$$ 5.93408 0.632575
$$89$$ 0.429167 0.0454916 0.0227458 0.999741i $$-0.492759\pi$$
0.0227458 + 0.999741i $$0.492759\pi$$
$$90$$ 0 0
$$91$$ 2.01818 0.211563
$$92$$ 3.17479 0.330995
$$93$$ 3.69717 0.383379
$$94$$ 10.2380 1.05597
$$95$$ 0 0
$$96$$ 2.01841 0.206003
$$97$$ 12.4945 1.26863 0.634315 0.773075i $$-0.281282\pi$$
0.634315 + 0.773075i $$0.281282\pi$$
$$98$$ −7.31933 −0.739364
$$99$$ −2.35626 −0.236813
$$100$$ 0 0
$$101$$ 8.19767 0.815698 0.407849 0.913049i $$-0.366279\pi$$
0.407849 + 0.913049i $$0.366279\pi$$
$$102$$ 3.37126 0.333805
$$103$$ 2.50005 0.246337 0.123169 0.992386i $$-0.460694\pi$$
0.123169 + 0.992386i $$0.460694\pi$$
$$104$$ −3.39865 −0.333265
$$105$$ 0 0
$$106$$ 6.40975 0.622570
$$107$$ −1.81004 −0.174983 −0.0874914 0.996165i $$-0.527885\pi$$
−0.0874914 + 0.996165i $$0.527885\pi$$
$$108$$ −0.360976 −0.0347350
$$109$$ −2.94778 −0.282346 −0.141173 0.989985i $$-0.545087\pi$$
−0.141173 + 0.989985i $$0.545087\pi$$
$$110$$ 0 0
$$111$$ −9.75097 −0.925521
$$112$$ −6.86679 −0.648851
$$113$$ 13.8365 1.30163 0.650813 0.759238i $$-0.274428\pi$$
0.650813 + 0.759238i $$0.274428\pi$$
$$114$$ −8.77474 −0.821829
$$115$$ 0 0
$$116$$ 2.85178 0.264781
$$117$$ 1.34951 0.124762
$$118$$ 17.0014 1.56510
$$119$$ −3.28119 −0.300787
$$120$$ 0 0
$$121$$ −5.44806 −0.495278
$$122$$ −18.8031 −1.70235
$$123$$ −1.85550 −0.167304
$$124$$ −1.33459 −0.119850
$$125$$ 0 0
$$126$$ 2.29790 0.204713
$$127$$ 5.73995 0.509338 0.254669 0.967028i $$-0.418033\pi$$
0.254669 + 0.967028i $$0.418033\pi$$
$$128$$ 13.3820 1.18281
$$129$$ 8.01874 0.706011
$$130$$ 0 0
$$131$$ −4.35756 −0.380722 −0.190361 0.981714i $$-0.560966\pi$$
−0.190361 + 0.981714i $$0.560966\pi$$
$$132$$ 0.850553 0.0740311
$$133$$ 8.54031 0.740539
$$134$$ 6.62802 0.572574
$$135$$ 0 0
$$136$$ 5.52558 0.473814
$$137$$ 1.97461 0.168702 0.0843510 0.996436i $$-0.473118\pi$$
0.0843510 + 0.996436i $$0.473118\pi$$
$$138$$ −13.5139 −1.15038
$$139$$ 1.67910 0.142419 0.0712095 0.997461i $$-0.477314\pi$$
0.0712095 + 0.997461i $$0.477314\pi$$
$$140$$ 0 0
$$141$$ −6.66298 −0.561124
$$142$$ 8.87740 0.744975
$$143$$ −3.17978 −0.265907
$$144$$ −4.59165 −0.382637
$$145$$ 0 0
$$146$$ 10.6434 0.880855
$$147$$ 4.76349 0.392886
$$148$$ 3.51987 0.289331
$$149$$ −7.38524 −0.605023 −0.302511 0.953146i $$-0.597825\pi$$
−0.302511 + 0.953146i $$0.597825\pi$$
$$150$$ 0 0
$$151$$ −4.26137 −0.346785 −0.173393 0.984853i $$-0.555473\pi$$
−0.173393 + 0.984853i $$0.555473\pi$$
$$152$$ −14.3820 −1.16653
$$153$$ −2.19405 −0.177378
$$154$$ −5.41444 −0.436308
$$155$$ 0 0
$$156$$ −0.487140 −0.0390024
$$157$$ 16.0573 1.28152 0.640758 0.767743i $$-0.278620\pi$$
0.640758 + 0.767743i $$0.278620\pi$$
$$158$$ −16.3930 −1.30416
$$159$$ −4.17153 −0.330824
$$160$$ 0 0
$$161$$ 13.1529 1.03659
$$162$$ 1.53655 0.120723
$$163$$ −22.2938 −1.74618 −0.873092 0.487556i $$-0.837889\pi$$
−0.873092 + 0.487556i $$0.837889\pi$$
$$164$$ 0.669790 0.0523018
$$165$$ 0 0
$$166$$ 0.344191 0.0267144
$$167$$ −6.46601 −0.500355 −0.250177 0.968200i $$-0.580489\pi$$
−0.250177 + 0.968200i $$0.580489\pi$$
$$168$$ 3.76631 0.290577
$$169$$ −11.1788 −0.859910
$$170$$ 0 0
$$171$$ 5.71069 0.436707
$$172$$ −2.89458 −0.220709
$$173$$ −11.8180 −0.898509 −0.449255 0.893404i $$-0.648310\pi$$
−0.449255 + 0.893404i $$0.648310\pi$$
$$174$$ −12.1390 −0.920254
$$175$$ 0 0
$$176$$ 10.8191 0.815520
$$177$$ −11.0647 −0.831671
$$178$$ 0.659435 0.0494268
$$179$$ 15.5746 1.16410 0.582049 0.813154i $$-0.302251\pi$$
0.582049 + 0.813154i $$0.302251\pi$$
$$180$$ 0 0
$$181$$ −14.5797 −1.08370 −0.541851 0.840475i $$-0.682276\pi$$
−0.541851 + 0.840475i $$0.682276\pi$$
$$182$$ 3.10103 0.229864
$$183$$ 12.2372 0.904602
$$184$$ −22.1497 −1.63290
$$185$$ 0 0
$$186$$ 5.68088 0.416542
$$187$$ 5.16974 0.378049
$$188$$ 2.40518 0.175416
$$189$$ −1.49550 −0.108781
$$190$$ 0 0
$$191$$ −20.9884 −1.51867 −0.759333 0.650702i $$-0.774475\pi$$
−0.759333 + 0.650702i $$0.774475\pi$$
$$192$$ −6.08192 −0.438924
$$193$$ 22.7094 1.63466 0.817328 0.576173i $$-0.195454\pi$$
0.817328 + 0.576173i $$0.195454\pi$$
$$194$$ 19.1985 1.37837
$$195$$ 0 0
$$196$$ −1.71951 −0.122822
$$197$$ 1.35341 0.0964268 0.0482134 0.998837i $$-0.484647\pi$$
0.0482134 + 0.998837i $$0.484647\pi$$
$$198$$ −3.62050 −0.257297
$$199$$ −8.96061 −0.635201 −0.317600 0.948225i $$-0.602877\pi$$
−0.317600 + 0.948225i $$0.602877\pi$$
$$200$$ 0 0
$$201$$ −4.31358 −0.304257
$$202$$ 12.5961 0.886259
$$203$$ 11.8147 0.829228
$$204$$ 0.792000 0.0554511
$$205$$ 0 0
$$206$$ 3.84145 0.267646
$$207$$ 8.79501 0.611295
$$208$$ −6.19646 −0.429647
$$209$$ −13.4558 −0.930760
$$210$$ 0 0
$$211$$ 5.83983 0.402031 0.201015 0.979588i $$-0.435576\pi$$
0.201015 + 0.979588i $$0.435576\pi$$
$$212$$ 1.50582 0.103420
$$213$$ −5.77750 −0.395868
$$214$$ −2.78121 −0.190119
$$215$$ 0 0
$$216$$ 2.51844 0.171358
$$217$$ −5.52910 −0.375340
$$218$$ −4.52939 −0.306769
$$219$$ −6.92684 −0.468072
$$220$$ 0 0
$$221$$ −2.96088 −0.199171
$$222$$ −14.9828 −1.00558
$$223$$ −7.82097 −0.523731 −0.261865 0.965104i $$-0.584338\pi$$
−0.261865 + 0.965104i $$0.584338\pi$$
$$224$$ −3.01853 −0.201684
$$225$$ 0 0
$$226$$ 21.2604 1.41422
$$227$$ 16.3090 1.08246 0.541232 0.840873i $$-0.317958\pi$$
0.541232 + 0.840873i $$0.317958\pi$$
$$228$$ −2.06142 −0.136521
$$229$$ −21.7088 −1.43456 −0.717278 0.696787i $$-0.754612\pi$$
−0.717278 + 0.696787i $$0.754612\pi$$
$$230$$ 0 0
$$231$$ 3.52377 0.231847
$$232$$ −19.8961 −1.30624
$$233$$ −13.5341 −0.886646 −0.443323 0.896362i $$-0.646201\pi$$
−0.443323 + 0.896362i $$0.646201\pi$$
$$234$$ 2.07358 0.135554
$$235$$ 0 0
$$236$$ 3.99408 0.259993
$$237$$ 10.6687 0.693009
$$238$$ −5.04171 −0.326805
$$239$$ −10.5338 −0.681377 −0.340689 0.940176i $$-0.610660\pi$$
−0.340689 + 0.940176i $$0.610660\pi$$
$$240$$ 0 0
$$241$$ 19.4838 1.25506 0.627530 0.778592i $$-0.284066\pi$$
0.627530 + 0.778592i $$0.284066\pi$$
$$242$$ −8.37120 −0.538121
$$243$$ −1.00000 −0.0641500
$$244$$ −4.41735 −0.282792
$$245$$ 0 0
$$246$$ −2.85106 −0.181777
$$247$$ 7.70661 0.490360
$$248$$ 9.31109 0.591255
$$249$$ −0.224003 −0.0141956
$$250$$ 0 0
$$251$$ −20.9446 −1.32201 −0.661007 0.750380i $$-0.729871\pi$$
−0.661007 + 0.750380i $$0.729871\pi$$
$$252$$ 0.539839 0.0340066
$$253$$ −20.7233 −1.30286
$$254$$ 8.81971 0.553398
$$255$$ 0 0
$$256$$ 8.39819 0.524887
$$257$$ −1.67121 −0.104247 −0.0521237 0.998641i $$-0.516599\pi$$
−0.0521237 + 0.998641i $$0.516599\pi$$
$$258$$ 12.3212 0.767083
$$259$$ 14.5825 0.906115
$$260$$ 0 0
$$261$$ 7.90017 0.489008
$$262$$ −6.69559 −0.413655
$$263$$ 7.55667 0.465964 0.232982 0.972481i $$-0.425152\pi$$
0.232982 + 0.972481i $$0.425152\pi$$
$$264$$ −5.93408 −0.365217
$$265$$ 0 0
$$266$$ 13.1226 0.804597
$$267$$ −0.429167 −0.0262646
$$268$$ 1.55710 0.0951151
$$269$$ −11.2841 −0.688005 −0.344002 0.938969i $$-0.611783\pi$$
−0.344002 + 0.938969i $$0.611783\pi$$
$$270$$ 0 0
$$271$$ −10.9752 −0.666696 −0.333348 0.942804i $$-0.608178\pi$$
−0.333348 + 0.942804i $$0.608178\pi$$
$$272$$ 10.0743 0.610845
$$273$$ −2.01818 −0.122146
$$274$$ 3.03407 0.183295
$$275$$ 0 0
$$276$$ −3.17479 −0.191100
$$277$$ −5.75112 −0.345551 −0.172776 0.984961i $$-0.555274\pi$$
−0.172776 + 0.984961i $$0.555274\pi$$
$$278$$ 2.58001 0.154739
$$279$$ −3.69717 −0.221344
$$280$$ 0 0
$$281$$ 8.49962 0.507045 0.253522 0.967330i $$-0.418411\pi$$
0.253522 + 0.967330i $$0.418411\pi$$
$$282$$ −10.2380 −0.609663
$$283$$ −17.2248 −1.02391 −0.511955 0.859013i $$-0.671078\pi$$
−0.511955 + 0.859013i $$0.671078\pi$$
$$284$$ 2.08554 0.123754
$$285$$ 0 0
$$286$$ −4.88588 −0.288908
$$287$$ 2.77489 0.163796
$$288$$ −2.01841 −0.118936
$$289$$ −12.1861 −0.716832
$$290$$ 0 0
$$291$$ −12.4945 −0.732443
$$292$$ 2.50042 0.146326
$$293$$ 9.38764 0.548432 0.274216 0.961668i $$-0.411582\pi$$
0.274216 + 0.961668i $$0.411582\pi$$
$$294$$ 7.31933 0.426872
$$295$$ 0 0
$$296$$ −24.5572 −1.42736
$$297$$ 2.35626 0.136724
$$298$$ −11.3478 −0.657359
$$299$$ 11.8689 0.686397
$$300$$ 0 0
$$301$$ −11.9920 −0.691207
$$302$$ −6.54779 −0.376783
$$303$$ −8.19767 −0.470944
$$304$$ −26.2215 −1.50390
$$305$$ 0 0
$$306$$ −3.37126 −0.192722
$$307$$ 10.6465 0.607627 0.303814 0.952731i $$-0.401740\pi$$
0.303814 + 0.952731i $$0.401740\pi$$
$$308$$ −1.27200 −0.0724788
$$309$$ −2.50005 −0.142223
$$310$$ 0 0
$$311$$ −27.3572 −1.55128 −0.775641 0.631174i $$-0.782573\pi$$
−0.775641 + 0.631174i $$0.782573\pi$$
$$312$$ 3.39865 0.192410
$$313$$ −1.99509 −0.112769 −0.0563846 0.998409i $$-0.517957\pi$$
−0.0563846 + 0.998409i $$0.517957\pi$$
$$314$$ 24.6729 1.39237
$$315$$ 0 0
$$316$$ −3.85116 −0.216645
$$317$$ 8.64876 0.485763 0.242881 0.970056i $$-0.421907\pi$$
0.242881 + 0.970056i $$0.421907\pi$$
$$318$$ −6.40975 −0.359441
$$319$$ −18.6148 −1.04223
$$320$$ 0 0
$$321$$ 1.81004 0.101026
$$322$$ 20.2101 1.12626
$$323$$ −12.5295 −0.697162
$$324$$ 0.360976 0.0200542
$$325$$ 0 0
$$326$$ −34.2554 −1.89723
$$327$$ 2.94778 0.163012
$$328$$ −4.67295 −0.258020
$$329$$ 9.96446 0.549358
$$330$$ 0 0
$$331$$ −3.94331 −0.216744 −0.108372 0.994110i $$-0.534564\pi$$
−0.108372 + 0.994110i $$0.534564\pi$$
$$332$$ 0.0808598 0.00443776
$$333$$ 9.75097 0.534350
$$334$$ −9.93532 −0.543637
$$335$$ 0 0
$$336$$ 6.86679 0.374614
$$337$$ 3.95471 0.215427 0.107713 0.994182i $$-0.465647\pi$$
0.107713 + 0.994182i $$0.465647\pi$$
$$338$$ −17.1768 −0.934295
$$339$$ −13.8365 −0.751494
$$340$$ 0 0
$$341$$ 8.71148 0.471753
$$342$$ 8.77474 0.474483
$$343$$ −17.5923 −0.949893
$$344$$ 20.1947 1.08882
$$345$$ 0 0
$$346$$ −18.1590 −0.976233
$$347$$ −9.99596 −0.536611 −0.268306 0.963334i $$-0.586464\pi$$
−0.268306 + 0.963334i $$0.586464\pi$$
$$348$$ −2.85178 −0.152871
$$349$$ 18.4534 0.987789 0.493895 0.869522i $$-0.335573\pi$$
0.493895 + 0.869522i $$0.335573\pi$$
$$350$$ 0 0
$$351$$ −1.34951 −0.0720313
$$352$$ 4.75589 0.253490
$$353$$ 9.32398 0.496265 0.248133 0.968726i $$-0.420183\pi$$
0.248133 + 0.968726i $$0.420183\pi$$
$$354$$ −17.0014 −0.903613
$$355$$ 0 0
$$356$$ 0.154919 0.00821070
$$357$$ 3.28119 0.173659
$$358$$ 23.9311 1.26480
$$359$$ 28.9438 1.52759 0.763797 0.645457i $$-0.223333\pi$$
0.763797 + 0.645457i $$0.223333\pi$$
$$360$$ 0 0
$$361$$ 13.6119 0.716418
$$362$$ −22.4024 −1.17744
$$363$$ 5.44806 0.285949
$$364$$ 0.728516 0.0381846
$$365$$ 0 0
$$366$$ 18.8031 0.982852
$$367$$ −3.99869 −0.208730 −0.104365 0.994539i $$-0.533281\pi$$
−0.104365 + 0.994539i $$0.533281\pi$$
$$368$$ −40.3836 −2.10514
$$369$$ 1.85550 0.0965932
$$370$$ 0 0
$$371$$ 6.23850 0.323887
$$372$$ 1.33459 0.0691953
$$373$$ −3.17260 −0.164271 −0.0821354 0.996621i $$-0.526174\pi$$
−0.0821354 + 0.996621i $$0.526174\pi$$
$$374$$ 7.94355 0.410751
$$375$$ 0 0
$$376$$ −16.7803 −0.865377
$$377$$ 10.6613 0.549086
$$378$$ −2.29790 −0.118191
$$379$$ 28.5206 1.46500 0.732501 0.680766i $$-0.238353\pi$$
0.732501 + 0.680766i $$0.238353\pi$$
$$380$$ 0 0
$$381$$ −5.73995 −0.294067
$$382$$ −32.2497 −1.65004
$$383$$ −11.4611 −0.585636 −0.292818 0.956168i $$-0.594593\pi$$
−0.292818 + 0.956168i $$0.594593\pi$$
$$384$$ −13.3820 −0.682896
$$385$$ 0 0
$$386$$ 34.8940 1.77606
$$387$$ −8.01874 −0.407616
$$388$$ 4.51024 0.228973
$$389$$ 34.2463 1.73636 0.868179 0.496252i $$-0.165291\pi$$
0.868179 + 0.496252i $$0.165291\pi$$
$$390$$ 0 0
$$391$$ −19.2967 −0.975876
$$392$$ 11.9966 0.605917
$$393$$ 4.35756 0.219810
$$394$$ 2.07958 0.104768
$$395$$ 0 0
$$396$$ −0.850553 −0.0427419
$$397$$ −20.0333 −1.00545 −0.502723 0.864448i $$-0.667668\pi$$
−0.502723 + 0.864448i $$0.667668\pi$$
$$398$$ −13.7684 −0.690148
$$399$$ −8.54031 −0.427550
$$400$$ 0 0
$$401$$ 4.98200 0.248789 0.124395 0.992233i $$-0.460301\pi$$
0.124395 + 0.992233i $$0.460301\pi$$
$$402$$ −6.62802 −0.330576
$$403$$ −4.98935 −0.248537
$$404$$ 2.95916 0.147224
$$405$$ 0 0
$$406$$ 18.1538 0.900958
$$407$$ −22.9758 −1.13887
$$408$$ −5.52558 −0.273557
$$409$$ 23.8591 1.17976 0.589878 0.807493i $$-0.299176\pi$$
0.589878 + 0.807493i $$0.299176\pi$$
$$410$$ 0 0
$$411$$ −1.97461 −0.0974001
$$412$$ 0.902460 0.0444610
$$413$$ 16.5472 0.814233
$$414$$ 13.5139 0.664174
$$415$$ 0 0
$$416$$ −2.72386 −0.133548
$$417$$ −1.67910 −0.0822257
$$418$$ −20.6755 −1.01127
$$419$$ 0.482550 0.0235741 0.0117871 0.999931i $$-0.496248\pi$$
0.0117871 + 0.999931i $$0.496248\pi$$
$$420$$ 0 0
$$421$$ −17.7183 −0.863537 −0.431769 0.901984i $$-0.642110\pi$$
−0.431769 + 0.901984i $$0.642110\pi$$
$$422$$ 8.97317 0.436807
$$423$$ 6.66298 0.323965
$$424$$ −10.5057 −0.510203
$$425$$ 0 0
$$426$$ −8.87740 −0.430112
$$427$$ −18.3007 −0.885634
$$428$$ −0.653380 −0.0315823
$$429$$ 3.17978 0.153521
$$430$$ 0 0
$$431$$ 26.8061 1.29121 0.645603 0.763673i $$-0.276606\pi$$
0.645603 + 0.763673i $$0.276606\pi$$
$$432$$ 4.59165 0.220916
$$433$$ −9.23115 −0.443621 −0.221810 0.975090i $$-0.571197\pi$$
−0.221810 + 0.975090i $$0.571197\pi$$
$$434$$ −8.49573 −0.407808
$$435$$ 0 0
$$436$$ −1.06408 −0.0509601
$$437$$ 50.2255 2.40261
$$438$$ −10.6434 −0.508562
$$439$$ 1.00240 0.0478421 0.0239211 0.999714i $$-0.492385\pi$$
0.0239211 + 0.999714i $$0.492385\pi$$
$$440$$ 0 0
$$441$$ −4.76349 −0.226833
$$442$$ −4.54954 −0.216399
$$443$$ −26.2872 −1.24894 −0.624471 0.781048i $$-0.714685\pi$$
−0.624471 + 0.781048i $$0.714685\pi$$
$$444$$ −3.51987 −0.167046
$$445$$ 0 0
$$446$$ −12.0173 −0.569035
$$447$$ 7.38524 0.349310
$$448$$ 9.09548 0.429721
$$449$$ 4.75449 0.224378 0.112189 0.993687i $$-0.464214\pi$$
0.112189 + 0.993687i $$0.464214\pi$$
$$450$$ 0 0
$$451$$ −4.37202 −0.205870
$$452$$ 4.99464 0.234928
$$453$$ 4.26137 0.200216
$$454$$ 25.0595 1.17610
$$455$$ 0 0
$$456$$ 14.3820 0.673499
$$457$$ 15.9703 0.747059 0.373529 0.927618i $$-0.378148\pi$$
0.373529 + 0.927618i $$0.378148\pi$$
$$458$$ −33.3566 −1.55865
$$459$$ 2.19405 0.102409
$$460$$ 0 0
$$461$$ 39.7558 1.85161 0.925806 0.377998i $$-0.123387\pi$$
0.925806 + 0.377998i $$0.123387\pi$$
$$462$$ 5.41444 0.251902
$$463$$ −26.1209 −1.21394 −0.606971 0.794724i $$-0.707616\pi$$
−0.606971 + 0.794724i $$0.707616\pi$$
$$464$$ −36.2748 −1.68402
$$465$$ 0 0
$$466$$ −20.7957 −0.963343
$$467$$ 3.85204 0.178251 0.0891256 0.996020i $$-0.471593\pi$$
0.0891256 + 0.996020i $$0.471593\pi$$
$$468$$ 0.487140 0.0225181
$$469$$ 6.45095 0.297877
$$470$$ 0 0
$$471$$ −16.0573 −0.739883
$$472$$ −27.8657 −1.28262
$$473$$ 18.8942 0.868756
$$474$$ 16.3930 0.752956
$$475$$ 0 0
$$476$$ −1.18443 −0.0542884
$$477$$ 4.17153 0.191001
$$478$$ −16.1857 −0.740318
$$479$$ −14.2698 −0.652004 −0.326002 0.945369i $$-0.605702\pi$$
−0.326002 + 0.945369i $$0.605702\pi$$
$$480$$ 0 0
$$481$$ 13.1590 0.599998
$$482$$ 29.9377 1.36363
$$483$$ −13.1529 −0.598478
$$484$$ −1.96662 −0.0893919
$$485$$ 0 0
$$486$$ −1.53655 −0.0696992
$$487$$ −24.8222 −1.12480 −0.562401 0.826865i $$-0.690122\pi$$
−0.562401 + 0.826865i $$0.690122\pi$$
$$488$$ 30.8187 1.39510
$$489$$ 22.2938 1.00816
$$490$$ 0 0
$$491$$ 11.4893 0.518507 0.259253 0.965809i $$-0.416524\pi$$
0.259253 + 0.965809i $$0.416524\pi$$
$$492$$ −0.669790 −0.0301965
$$493$$ −17.3334 −0.780656
$$494$$ 11.8416 0.532777
$$495$$ 0 0
$$496$$ 16.9761 0.762250
$$497$$ 8.64023 0.387567
$$498$$ −0.344191 −0.0154236
$$499$$ −4.17487 −0.186893 −0.0934465 0.995624i $$-0.529788\pi$$
−0.0934465 + 0.995624i $$0.529788\pi$$
$$500$$ 0 0
$$501$$ 6.46601 0.288880
$$502$$ −32.1824 −1.43637
$$503$$ −38.7163 −1.72627 −0.863137 0.504970i $$-0.831503\pi$$
−0.863137 + 0.504970i $$0.831503\pi$$
$$504$$ −3.76631 −0.167765
$$505$$ 0 0
$$506$$ −31.8423 −1.41556
$$507$$ 11.1788 0.496469
$$508$$ 2.07199 0.0919296
$$509$$ 30.1797 1.33769 0.668845 0.743402i $$-0.266789\pi$$
0.668845 + 0.743402i $$0.266789\pi$$
$$510$$ 0 0
$$511$$ 10.3591 0.458258
$$512$$ −13.8597 −0.612519
$$513$$ −5.71069 −0.252133
$$514$$ −2.56790 −0.113265
$$515$$ 0 0
$$516$$ 2.89458 0.127427
$$517$$ −15.6997 −0.690471
$$518$$ 22.4067 0.984496
$$519$$ 11.8180 0.518755
$$520$$ 0 0
$$521$$ −25.4856 −1.11654 −0.558272 0.829658i $$-0.688536\pi$$
−0.558272 + 0.829658i $$0.688536\pi$$
$$522$$ 12.1390 0.531309
$$523$$ 3.89180 0.170176 0.0850882 0.996373i $$-0.472883\pi$$
0.0850882 + 0.996373i $$0.472883\pi$$
$$524$$ −1.57298 −0.0687158
$$525$$ 0 0
$$526$$ 11.6112 0.506271
$$527$$ 8.11178 0.353355
$$528$$ −10.8191 −0.470841
$$529$$ 54.3522 2.36314
$$530$$ 0 0
$$531$$ 11.0647 0.480166
$$532$$ 3.08285 0.133659
$$533$$ 2.50400 0.108460
$$534$$ −0.659435 −0.0285366
$$535$$ 0 0
$$536$$ −10.8635 −0.469231
$$537$$ −15.5746 −0.672092
$$538$$ −17.3386 −0.747519
$$539$$ 11.2240 0.483452
$$540$$ 0 0
$$541$$ 40.2148 1.72897 0.864484 0.502661i $$-0.167646\pi$$
0.864484 + 0.502661i $$0.167646\pi$$
$$542$$ −16.8639 −0.724367
$$543$$ 14.5797 0.625675
$$544$$ 4.42849 0.189870
$$545$$ 0 0
$$546$$ −3.10103 −0.132712
$$547$$ 7.37923 0.315513 0.157757 0.987478i $$-0.449574\pi$$
0.157757 + 0.987478i $$0.449574\pi$$
$$548$$ 0.712786 0.0304487
$$549$$ −12.2372 −0.522272
$$550$$ 0 0
$$551$$ 45.1154 1.92198
$$552$$ 22.1497 0.942753
$$553$$ −15.9550 −0.678478
$$554$$ −8.83686 −0.375442
$$555$$ 0 0
$$556$$ 0.606114 0.0257050
$$557$$ 1.52499 0.0646160 0.0323080 0.999478i $$-0.489714\pi$$
0.0323080 + 0.999478i $$0.489714\pi$$
$$558$$ −5.68088 −0.240491
$$559$$ −10.8213 −0.457694
$$560$$ 0 0
$$561$$ −5.16974 −0.218267
$$562$$ 13.0601 0.550905
$$563$$ 13.7955 0.581411 0.290706 0.956813i $$-0.406110\pi$$
0.290706 + 0.956813i $$0.406110\pi$$
$$564$$ −2.40518 −0.101276
$$565$$ 0 0
$$566$$ −26.4667 −1.11248
$$567$$ 1.49550 0.0628049
$$568$$ −14.5503 −0.610516
$$569$$ −8.49904 −0.356298 −0.178149 0.984004i $$-0.557011\pi$$
−0.178149 + 0.984004i $$0.557011\pi$$
$$570$$ 0 0
$$571$$ 39.3230 1.64562 0.822809 0.568319i $$-0.192406\pi$$
0.822809 + 0.568319i $$0.192406\pi$$
$$572$$ −1.14783 −0.0479930
$$573$$ 20.9884 0.876803
$$574$$ 4.26374 0.177965
$$575$$ 0 0
$$576$$ 6.08192 0.253413
$$577$$ 24.5832 1.02341 0.511707 0.859160i $$-0.329013\pi$$
0.511707 + 0.859160i $$0.329013\pi$$
$$578$$ −18.7246 −0.778840
$$579$$ −22.7094 −0.943769
$$580$$ 0 0
$$581$$ 0.334995 0.0138980
$$582$$ −19.1985 −0.795802
$$583$$ −9.82918 −0.407083
$$584$$ −17.4448 −0.721871
$$585$$ 0 0
$$586$$ 14.4245 0.595872
$$587$$ −9.24270 −0.381487 −0.190744 0.981640i $$-0.561090\pi$$
−0.190744 + 0.981640i $$0.561090\pi$$
$$588$$ 1.71951 0.0709113
$$589$$ −21.1134 −0.869962
$$590$$ 0 0
$$591$$ −1.35341 −0.0556720
$$592$$ −44.7730 −1.84016
$$593$$ −6.07888 −0.249630 −0.124815 0.992180i $$-0.539834\pi$$
−0.124815 + 0.992180i $$0.539834\pi$$
$$594$$ 3.62050 0.148551
$$595$$ 0 0
$$596$$ −2.66590 −0.109199
$$597$$ 8.96061 0.366733
$$598$$ 18.2372 0.745773
$$599$$ 6.40129 0.261550 0.130775 0.991412i $$-0.458254\pi$$
0.130775 + 0.991412i $$0.458254\pi$$
$$600$$ 0 0
$$601$$ −38.4675 −1.56912 −0.784560 0.620052i $$-0.787111\pi$$
−0.784560 + 0.620052i $$0.787111\pi$$
$$602$$ −18.4263 −0.750999
$$603$$ 4.31358 0.175663
$$604$$ −1.53825 −0.0625906
$$605$$ 0 0
$$606$$ −12.5961 −0.511682
$$607$$ 5.22464 0.212062 0.106031 0.994363i $$-0.466186\pi$$
0.106031 + 0.994363i $$0.466186\pi$$
$$608$$ −11.5265 −0.467462
$$609$$ −11.8147 −0.478755
$$610$$ 0 0
$$611$$ 8.99173 0.363766
$$612$$ −0.792000 −0.0320147
$$613$$ 29.8153 1.20423 0.602114 0.798410i $$-0.294325\pi$$
0.602114 + 0.798410i $$0.294325\pi$$
$$614$$ 16.3588 0.660189
$$615$$ 0 0
$$616$$ 8.87439 0.357559
$$617$$ 23.3025 0.938124 0.469062 0.883165i $$-0.344592\pi$$
0.469062 + 0.883165i $$0.344592\pi$$
$$618$$ −3.84145 −0.154526
$$619$$ −0.236492 −0.00950543 −0.00475272 0.999989i $$-0.501513\pi$$
−0.00475272 + 0.999989i $$0.501513\pi$$
$$620$$ 0 0
$$621$$ −8.79501 −0.352932
$$622$$ −42.0356 −1.68547
$$623$$ 0.641818 0.0257139
$$624$$ 6.19646 0.248057
$$625$$ 0 0
$$626$$ −3.06555 −0.122524
$$627$$ 13.4558 0.537374
$$628$$ 5.79632 0.231298
$$629$$ −21.3941 −0.853039
$$630$$ 0 0
$$631$$ −17.8789 −0.711748 −0.355874 0.934534i $$-0.615817\pi$$
−0.355874 + 0.934534i $$0.615817\pi$$
$$632$$ 26.8685 1.06877
$$633$$ −5.83983 −0.232112
$$634$$ 13.2892 0.527782
$$635$$ 0 0
$$636$$ −1.50582 −0.0597098
$$637$$ −6.42836 −0.254701
$$638$$ −28.6025 −1.13239
$$639$$ 5.77750 0.228554
$$640$$ 0 0
$$641$$ 10.8680 0.429259 0.214630 0.976695i $$-0.431146\pi$$
0.214630 + 0.976695i $$0.431146\pi$$
$$642$$ 2.78121 0.109765
$$643$$ 3.09039 0.121873 0.0609366 0.998142i $$-0.480591\pi$$
0.0609366 + 0.998142i $$0.480591\pi$$
$$644$$ 4.74789 0.187093
$$645$$ 0 0
$$646$$ −19.2522 −0.757468
$$647$$ −5.53705 −0.217684 −0.108842 0.994059i $$-0.534714\pi$$
−0.108842 + 0.994059i $$0.534714\pi$$
$$648$$ −2.51844 −0.0989335
$$649$$ −26.0712 −1.02338
$$650$$ 0 0
$$651$$ 5.52910 0.216703
$$652$$ −8.04753 −0.315166
$$653$$ 38.8754 1.52131 0.760657 0.649154i $$-0.224877\pi$$
0.760657 + 0.649154i $$0.224877\pi$$
$$654$$ 4.52939 0.177113
$$655$$ 0 0
$$656$$ −8.51978 −0.332642
$$657$$ 6.92684 0.270242
$$658$$ 15.3109 0.596879
$$659$$ −20.5389 −0.800082 −0.400041 0.916497i $$-0.631004\pi$$
−0.400041 + 0.916497i $$0.631004\pi$$
$$660$$ 0 0
$$661$$ 38.4254 1.49457 0.747287 0.664502i $$-0.231356\pi$$
0.747287 + 0.664502i $$0.231356\pi$$
$$662$$ −6.05907 −0.235493
$$663$$ 2.96088 0.114991
$$664$$ −0.564137 −0.0218928
$$665$$ 0 0
$$666$$ 14.9828 0.580572
$$667$$ 69.4821 2.69036
$$668$$ −2.33408 −0.0903081
$$669$$ 7.82097 0.302376
$$670$$ 0 0
$$671$$ 28.8340 1.11313
$$672$$ 3.01853 0.116442
$$673$$ −12.9819 −0.500417 −0.250208 0.968192i $$-0.580499\pi$$
−0.250208 + 0.968192i $$0.580499\pi$$
$$674$$ 6.07660 0.234062
$$675$$ 0 0
$$676$$ −4.03529 −0.155204
$$677$$ 7.23957 0.278239 0.139120 0.990276i $$-0.455573\pi$$
0.139120 + 0.990276i $$0.455573\pi$$
$$678$$ −21.2604 −0.816500
$$679$$ 18.6855 0.717086
$$680$$ 0 0
$$681$$ −16.3090 −0.624961
$$682$$ 13.3856 0.512561
$$683$$ 24.8800 0.952005 0.476003 0.879444i $$-0.342085\pi$$
0.476003 + 0.879444i $$0.342085\pi$$
$$684$$ 2.06142 0.0788205
$$685$$ 0 0
$$686$$ −27.0313 −1.03206
$$687$$ 21.7088 0.828241
$$688$$ 36.8192 1.40372
$$689$$ 5.62950 0.214467
$$690$$ 0 0
$$691$$ −33.2706 −1.26567 −0.632836 0.774286i $$-0.718109\pi$$
−0.632836 + 0.774286i $$0.718109\pi$$
$$692$$ −4.26604 −0.162170
$$693$$ −3.52377 −0.133857
$$694$$ −15.3593 −0.583029
$$695$$ 0 0
$$696$$ 19.8961 0.754159
$$697$$ −4.07105 −0.154202
$$698$$ 28.3546 1.07324
$$699$$ 13.5341 0.511905
$$700$$ 0 0
$$701$$ −13.2163 −0.499173 −0.249586 0.968353i $$-0.580295\pi$$
−0.249586 + 0.968353i $$0.580295\pi$$
$$702$$ −2.07358 −0.0782622
$$703$$ 55.6847 2.10019
$$704$$ −14.3305 −0.540103
$$705$$ 0 0
$$706$$ 14.3267 0.539194
$$707$$ 12.2596 0.461069
$$708$$ −3.99408 −0.150107
$$709$$ 6.25068 0.234749 0.117375 0.993088i $$-0.462552\pi$$
0.117375 + 0.993088i $$0.462552\pi$$
$$710$$ 0 0
$$711$$ −10.6687 −0.400109
$$712$$ −1.08083 −0.0405058
$$713$$ −32.5167 −1.21776
$$714$$ 5.04171 0.188681
$$715$$ 0 0
$$716$$ 5.62205 0.210106
$$717$$ 10.5338 0.393393
$$718$$ 44.4735 1.65973
$$719$$ 27.5016 1.02564 0.512818 0.858497i $$-0.328602\pi$$
0.512818 + 0.858497i $$0.328602\pi$$
$$720$$ 0 0
$$721$$ 3.73882 0.139241
$$722$$ 20.9154 0.778390
$$723$$ −19.4838 −0.724610
$$724$$ −5.26293 −0.195595
$$725$$ 0 0
$$726$$ 8.37120 0.310684
$$727$$ −22.0397 −0.817406 −0.408703 0.912668i $$-0.634019\pi$$
−0.408703 + 0.912668i $$0.634019\pi$$
$$728$$ −5.08266 −0.188376
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 17.5935 0.650720
$$732$$ 4.41735 0.163270
$$733$$ 34.7134 1.28217 0.641085 0.767470i $$-0.278485\pi$$
0.641085 + 0.767470i $$0.278485\pi$$
$$734$$ −6.14417 −0.226785
$$735$$ 0 0
$$736$$ −17.7519 −0.654345
$$737$$ −10.1639 −0.374392
$$738$$ 2.85106 0.104949
$$739$$ −7.58593 −0.279053 −0.139526 0.990218i $$-0.544558\pi$$
−0.139526 + 0.990218i $$0.544558\pi$$
$$740$$ 0 0
$$741$$ −7.70661 −0.283109
$$742$$ 9.58575 0.351904
$$743$$ 27.5328 1.01008 0.505040 0.863096i $$-0.331478\pi$$
0.505040 + 0.863096i $$0.331478\pi$$
$$744$$ −9.31109 −0.341361
$$745$$ 0 0
$$746$$ −4.87484 −0.178481
$$747$$ 0.224003 0.00819584
$$748$$ 1.86615 0.0682334
$$749$$ −2.70690 −0.0989081
$$750$$ 0 0
$$751$$ 4.24930 0.155059 0.0775296 0.996990i $$-0.475297\pi$$
0.0775296 + 0.996990i $$0.475297\pi$$
$$752$$ −30.5940 −1.11565
$$753$$ 20.9446 0.763265
$$754$$ 16.3816 0.596584
$$755$$ 0 0
$$756$$ −0.539839 −0.0196337
$$757$$ −45.6609 −1.65957 −0.829787 0.558081i $$-0.811538\pi$$
−0.829787 + 0.558081i $$0.811538\pi$$
$$758$$ 43.8232 1.59173
$$759$$ 20.7233 0.752208
$$760$$ 0 0
$$761$$ −40.1268 −1.45460 −0.727298 0.686322i $$-0.759224\pi$$
−0.727298 + 0.686322i $$0.759224\pi$$
$$762$$ −8.81971 −0.319504
$$763$$ −4.40839 −0.159594
$$764$$ −7.57631 −0.274101
$$765$$ 0 0
$$766$$ −17.6105 −0.636295
$$767$$ 14.9318 0.539157
$$768$$ −8.39819 −0.303044
$$769$$ −37.8350 −1.36437 −0.682183 0.731181i $$-0.738969\pi$$
−0.682183 + 0.731181i $$0.738969\pi$$
$$770$$ 0 0
$$771$$ 1.67121 0.0601872
$$772$$ 8.19755 0.295036
$$773$$ −26.2148 −0.942881 −0.471441 0.881898i $$-0.656266\pi$$
−0.471441 + 0.881898i $$0.656266\pi$$
$$774$$ −12.3212 −0.442875
$$775$$ 0 0
$$776$$ −31.4667 −1.12959
$$777$$ −14.5825 −0.523146
$$778$$ 52.6211 1.88656
$$779$$ 10.5962 0.379646
$$780$$ 0 0
$$781$$ −13.6133 −0.487121
$$782$$ −29.6503 −1.06029
$$783$$ −7.90017 −0.282329
$$784$$ 21.8723 0.781153
$$785$$ 0 0
$$786$$ 6.69559 0.238824
$$787$$ −2.41254 −0.0859977 −0.0429988 0.999075i $$-0.513691\pi$$
−0.0429988 + 0.999075i $$0.513691\pi$$
$$788$$ 0.488551 0.0174039
$$789$$ −7.55667 −0.269025
$$790$$ 0 0
$$791$$ 20.6924 0.735737
$$792$$ 5.93408 0.210858
$$793$$ −16.5142 −0.586437
$$794$$ −30.7822 −1.09242
$$795$$ 0 0
$$796$$ −3.23457 −0.114646
$$797$$ 8.11230 0.287353 0.143676 0.989625i $$-0.454108\pi$$
0.143676 + 0.989625i $$0.454108\pi$$
$$798$$ −13.1226 −0.464534
$$799$$ −14.6189 −0.517180
$$800$$ 0 0
$$801$$ 0.429167 0.0151639
$$802$$ 7.65507 0.270310
$$803$$ −16.3214 −0.575970
$$804$$ −1.55710 −0.0549148
$$805$$ 0 0
$$806$$ −7.66638 −0.270037
$$807$$ 11.2841 0.397220
$$808$$ −20.6453 −0.726299
$$809$$ −40.7091 −1.43125 −0.715627 0.698482i $$-0.753859\pi$$
−0.715627 + 0.698482i $$0.753859\pi$$
$$810$$ 0 0
$$811$$ −49.4990 −1.73814 −0.869072 0.494685i $$-0.835283\pi$$
−0.869072 + 0.494685i $$0.835283\pi$$
$$812$$ 4.26482 0.149666
$$813$$ 10.9752 0.384917
$$814$$ −35.3033 −1.23738
$$815$$ 0 0
$$816$$ −10.0743 −0.352671
$$817$$ −45.7925 −1.60208
$$818$$ 36.6606 1.28181
$$819$$ 2.01818 0.0705210
$$820$$ 0 0
$$821$$ 42.2114 1.47319 0.736594 0.676335i $$-0.236433\pi$$
0.736594 + 0.676335i $$0.236433\pi$$
$$822$$ −3.03407 −0.105825
$$823$$ −49.2349 −1.71622 −0.858111 0.513464i $$-0.828362\pi$$
−0.858111 + 0.513464i $$0.828362\pi$$
$$824$$ −6.29622 −0.219339
$$825$$ 0 0
$$826$$ 25.4255 0.884666
$$827$$ 51.0011 1.77348 0.886742 0.462266i $$-0.152963\pi$$
0.886742 + 0.462266i $$0.152963\pi$$
$$828$$ 3.17479 0.110332
$$829$$ −38.2342 −1.32793 −0.663964 0.747764i $$-0.731127\pi$$
−0.663964 + 0.747764i $$0.731127\pi$$
$$830$$ 0 0
$$831$$ 5.75112 0.199504
$$832$$ 8.20758 0.284547
$$833$$ 10.4513 0.362117
$$834$$ −2.58001 −0.0893384
$$835$$ 0 0
$$836$$ −4.85724 −0.167991
$$837$$ 3.69717 0.127793
$$838$$ 0.741461 0.0256133
$$839$$ −16.7086 −0.576845 −0.288422 0.957503i $$-0.593131\pi$$
−0.288422 + 0.957503i $$0.593131\pi$$
$$840$$ 0 0
$$841$$ 33.4127 1.15216
$$842$$ −27.2250 −0.938236
$$843$$ −8.49962 −0.292742
$$844$$ 2.10804 0.0725618
$$845$$ 0 0
$$846$$ 10.2380 0.351989
$$847$$ −8.14755 −0.279953
$$848$$ −19.1542 −0.657757
$$849$$ 17.2248 0.591154
$$850$$ 0 0
$$851$$ 85.7599 2.93981
$$852$$ −2.08554 −0.0714495
$$853$$ −18.2644 −0.625361 −0.312681 0.949858i $$-0.601227\pi$$
−0.312681 + 0.949858i $$0.601227\pi$$
$$854$$ −28.1199 −0.962244
$$855$$ 0 0
$$856$$ 4.55846 0.155805
$$857$$ −53.4773 −1.82675 −0.913375 0.407119i $$-0.866534\pi$$
−0.913375 + 0.407119i $$0.866534\pi$$
$$858$$ 4.88588 0.166801
$$859$$ −18.7575 −0.639998 −0.319999 0.947418i $$-0.603683\pi$$
−0.319999 + 0.947418i $$0.603683\pi$$
$$860$$ 0 0
$$861$$ −2.77489 −0.0945678
$$862$$ 41.1889 1.40290
$$863$$ −51.2363 −1.74410 −0.872051 0.489414i $$-0.837211\pi$$
−0.872051 + 0.489414i $$0.837211\pi$$
$$864$$ 2.01841 0.0686677
$$865$$ 0 0
$$866$$ −14.1841 −0.481995
$$867$$ 12.1861 0.413863
$$868$$ −1.99588 −0.0677444
$$869$$ 25.1383 0.852757
$$870$$ 0 0
$$871$$ 5.82121 0.197244
$$872$$ 7.42378 0.251401
$$873$$ 12.4945 0.422876
$$874$$ 77.1739 2.61045
$$875$$ 0 0
$$876$$ −2.50042 −0.0844815
$$877$$ −6.06306 −0.204735 −0.102368 0.994747i $$-0.532642\pi$$
−0.102368 + 0.994747i $$0.532642\pi$$
$$878$$ 1.54024 0.0519806
$$879$$ −9.38764 −0.316637
$$880$$ 0 0
$$881$$ 22.6698 0.763765 0.381883 0.924211i $$-0.375276\pi$$
0.381883 + 0.924211i $$0.375276\pi$$
$$882$$ −7.31933 −0.246455
$$883$$ −5.53899 −0.186402 −0.0932009 0.995647i $$-0.529710\pi$$
−0.0932009 + 0.995647i $$0.529710\pi$$
$$884$$ −1.06881 −0.0359480
$$885$$ 0 0
$$886$$ −40.3915 −1.35698
$$887$$ −10.9716 −0.368390 −0.184195 0.982890i $$-0.558968\pi$$
−0.184195 + 0.982890i $$0.558968\pi$$
$$888$$ 24.5572 0.824086
$$889$$ 8.58408 0.287901
$$890$$ 0 0
$$891$$ −2.35626 −0.0789375
$$892$$ −2.82319 −0.0945272
$$893$$ 38.0502 1.27330
$$894$$ 11.3478 0.379526
$$895$$ 0 0
$$896$$ 20.0127 0.668577
$$897$$ −11.8689 −0.396292
$$898$$ 7.30550 0.243788
$$899$$ −29.2083 −0.974151
$$900$$ 0 0
$$901$$ −9.15254 −0.304915
$$902$$ −6.71781 −0.223679
$$903$$ 11.9920 0.399069
$$904$$ −34.8463 −1.15897
$$905$$ 0 0
$$906$$ 6.54779 0.217536
$$907$$ −19.3907 −0.643856 −0.321928 0.946764i $$-0.604331\pi$$
−0.321928 + 0.946764i $$0.604331\pi$$
$$908$$ 5.88715 0.195372
$$909$$ 8.19767 0.271899
$$910$$ 0 0
$$911$$ −13.7190 −0.454530 −0.227265 0.973833i $$-0.572978\pi$$
−0.227265 + 0.973833i $$0.572978\pi$$
$$912$$ 26.2215 0.868280
$$913$$ −0.527808 −0.0174679
$$914$$ 24.5391 0.811681
$$915$$ 0 0
$$916$$ −7.83636 −0.258920
$$917$$ −6.51671 −0.215201
$$918$$ 3.37126 0.111268
$$919$$ −32.4108 −1.06913 −0.534567 0.845126i $$-0.679525\pi$$
−0.534567 + 0.845126i $$0.679525\pi$$
$$920$$ 0 0
$$921$$ −10.6465 −0.350814
$$922$$ 61.0867 2.01178
$$923$$ 7.79678 0.256634
$$924$$ 1.27200 0.0418457
$$925$$ 0 0
$$926$$ −40.1360 −1.31895
$$927$$ 2.50005 0.0821125
$$928$$ −15.9458 −0.523446
$$929$$ −4.33444 −0.142208 −0.0711042 0.997469i $$-0.522652\pi$$
−0.0711042 + 0.997469i $$0.522652\pi$$
$$930$$ 0 0
$$931$$ −27.2028 −0.891536
$$932$$ −4.88548 −0.160029
$$933$$ 27.3572 0.895633
$$934$$ 5.91884 0.193670
$$935$$ 0 0
$$936$$ −3.39865 −0.111088
$$937$$ −54.5925 −1.78346 −0.891730 0.452567i $$-0.850508\pi$$
−0.891730 + 0.452567i $$0.850508\pi$$
$$938$$ 9.91218 0.323644
$$939$$ 1.99509 0.0651073
$$940$$ 0 0
$$941$$ 4.23853 0.138172 0.0690861 0.997611i $$-0.477992\pi$$
0.0690861 + 0.997611i $$0.477992\pi$$
$$942$$ −24.6729 −0.803885
$$943$$ 16.3191 0.531423
$$944$$ −50.8051 −1.65356
$$945$$ 0 0
$$946$$ 29.0318 0.943906
$$947$$ −12.6069 −0.409671 −0.204835 0.978796i $$-0.565666\pi$$
−0.204835 + 0.978796i $$0.565666\pi$$
$$948$$ 3.85116 0.125080
$$949$$ 9.34781 0.303443
$$950$$ 0 0
$$951$$ −8.64876 −0.280455
$$952$$ 8.26348 0.267821
$$953$$ 31.1635 1.00948 0.504742 0.863270i $$-0.331588\pi$$
0.504742 + 0.863270i $$0.331588\pi$$
$$954$$ 6.40975 0.207523
$$955$$ 0 0
$$956$$ −3.80247 −0.122981
$$957$$ 18.6148 0.601732
$$958$$ −21.9262 −0.708405
$$959$$ 2.95301 0.0953578
$$960$$ 0 0
$$961$$ −17.3309 −0.559062
$$962$$ 20.2194 0.651900
$$963$$ −1.81004 −0.0583276
$$964$$ 7.03319 0.226524
$$965$$ 0 0
$$966$$ −20.2101 −0.650248
$$967$$ −1.55308 −0.0499438 −0.0249719 0.999688i $$-0.507950\pi$$
−0.0249719 + 0.999688i $$0.507950\pi$$
$$968$$ 13.7206 0.440997
$$969$$ 12.5295 0.402507
$$970$$ 0 0
$$971$$ −38.2623 −1.22790 −0.613948 0.789347i $$-0.710419\pi$$
−0.613948 + 0.789347i $$0.710419\pi$$
$$972$$ −0.360976 −0.0115783
$$973$$ 2.51108 0.0805016
$$974$$ −38.1405 −1.22210
$$975$$ 0 0
$$976$$ 56.1890 1.79857
$$977$$ 20.3661 0.651570 0.325785 0.945444i $$-0.394371\pi$$
0.325785 + 0.945444i $$0.394371\pi$$
$$978$$ 34.2554 1.09537
$$979$$ −1.01123 −0.0323190
$$980$$ 0 0
$$981$$ −2.94778 −0.0941152
$$982$$ 17.6539 0.563359
$$983$$ 13.2557 0.422791 0.211395 0.977401i $$-0.432199\pi$$
0.211395 + 0.977401i $$0.432199\pi$$
$$984$$ 4.67295 0.148968
$$985$$ 0 0
$$986$$ −26.6335 −0.848185
$$987$$ −9.96446 −0.317172
$$988$$ 2.78190 0.0885041
$$989$$ −70.5249 −2.24256
$$990$$ 0 0
$$991$$ −2.24081 −0.0711816 −0.0355908 0.999366i $$-0.511331\pi$$
−0.0355908 + 0.999366i $$0.511331\pi$$
$$992$$ 7.46241 0.236932
$$993$$ 3.94331 0.125137
$$994$$ 13.2761 0.421093
$$995$$ 0 0
$$996$$ −0.0808598 −0.00256214
$$997$$ 14.0404 0.444663 0.222331 0.974971i $$-0.428633\pi$$
0.222331 + 0.974971i $$0.428633\pi$$
$$998$$ −6.41489 −0.203060
$$999$$ −9.75097 −0.308507
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.1.6 8
3.2 odd 2 5625.2.a.t.1.3 8
5.2 odd 4 1875.2.b.h.1249.13 16
5.3 odd 4 1875.2.b.h.1249.4 16
5.4 even 2 1875.2.a.m.1.3 8
15.14 odd 2 5625.2.a.bd.1.6 8
25.2 odd 20 75.2.i.a.4.4 16
25.9 even 10 375.2.g.e.151.2 16
25.11 even 5 375.2.g.d.226.3 16
25.12 odd 20 375.2.i.c.349.1 16
25.13 odd 20 75.2.i.a.19.4 yes 16
25.14 even 10 375.2.g.e.226.2 16
25.16 even 5 375.2.g.d.151.3 16
25.23 odd 20 375.2.i.c.274.1 16
75.2 even 20 225.2.m.b.154.1 16
75.38 even 20 225.2.m.b.19.1 16

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.4.4 16 25.2 odd 20
75.2.i.a.19.4 yes 16 25.13 odd 20
225.2.m.b.19.1 16 75.38 even 20
225.2.m.b.154.1 16 75.2 even 20
375.2.g.d.151.3 16 25.16 even 5
375.2.g.d.226.3 16 25.11 even 5
375.2.g.e.151.2 16 25.9 even 10
375.2.g.e.226.2 16 25.14 even 10
375.2.i.c.274.1 16 25.23 odd 20
375.2.i.c.349.1 16 25.12 odd 20
1875.2.a.m.1.3 8 5.4 even 2
1875.2.a.p.1.6 8 1.1 even 1 trivial
1875.2.b.h.1249.4 16 5.3 odd 4
1875.2.b.h.1249.13 16 5.2 odd 4
5625.2.a.t.1.3 8 3.2 odd 2
5625.2.a.bd.1.6 8 15.14 odd 2