# Properties

 Label 1875.2.a.e.1.4 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Learn more

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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.5125.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 6x^{2} + 7x + 11$$ x^4 - 2*x^3 - 6*x^2 + 7*x + 11 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 1.4 Root $$-1.70636$$ 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)

 $$f(q)$$ $$=$$ $$q+1.70636 q^{2} -1.00000 q^{3} +0.911672 q^{4} -1.70636 q^{6} +3.94243 q^{7} -1.85708 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.70636 q^{2} -1.00000 q^{3} +0.911672 q^{4} -1.70636 q^{6} +3.94243 q^{7} -1.85708 q^{8} +1.00000 q^{9} -5.90869 q^{11} -0.911672 q^{12} -3.29066 q^{13} +6.72721 q^{14} -4.99220 q^{16} -2.70636 q^{17} +1.70636 q^{18} -2.35813 q^{19} -3.94243 q^{21} -10.0824 q^{22} +0.584296 q^{23} +1.85708 q^{24} -5.61505 q^{26} -1.00000 q^{27} +3.59420 q^{28} -3.91167 q^{29} +2.70934 q^{31} -4.80433 q^{32} +5.90869 q^{33} -4.61803 q^{34} +0.911672 q^{36} +0.0208515 q^{37} -4.02383 q^{38} +3.29066 q^{39} +1.47214 q^{41} -6.72721 q^{42} -1.27279 q^{43} -5.38679 q^{44} +0.997020 q^{46} -5.43358 q^{47} +4.99220 q^{48} +8.54276 q^{49} +2.70636 q^{51} -3.00000 q^{52} -2.81554 q^{53} -1.70636 q^{54} -7.32142 q^{56} +2.35813 q^{57} -6.67473 q^{58} -4.69033 q^{59} +5.58132 q^{61} +4.62312 q^{62} +3.94243 q^{63} +1.78646 q^{64} +10.0824 q^{66} -6.03076 q^{67} -2.46731 q^{68} -0.584296 q^{69} -8.10138 q^{71} -1.85708 q^{72} -13.3166 q^{73} +0.0355801 q^{74} -2.14984 q^{76} -23.2946 q^{77} +5.61505 q^{78} +16.6648 q^{79} +1.00000 q^{81} +2.51200 q^{82} -0.781641 q^{83} -3.59420 q^{84} -2.17183 q^{86} +3.91167 q^{87} +10.9729 q^{88} -3.47327 q^{89} -12.9732 q^{91} +0.532686 q^{92} -2.70934 q^{93} -9.27165 q^{94} +4.80433 q^{96} +2.45443 q^{97} +14.5770 q^{98} -5.90869 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} - 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^6 - 2 * q^7 - 15 * q^8 + 4 * q^9 $$4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} - 2 q^{7} - 15 q^{8} + 4 q^{9} - 7 q^{11} - 8 q^{12} - q^{13} + 16 q^{14} + 4 q^{16} - 2 q^{17} - 2 q^{18} + 5 q^{19} + 2 q^{21} + 6 q^{22} - q^{23} + 15 q^{24} + 3 q^{26} - 4 q^{27} - 9 q^{28} - 20 q^{29} + 23 q^{31} - 12 q^{32} + 7 q^{33} - 14 q^{34} + 8 q^{36} - 2 q^{37} - 35 q^{38} + q^{39} - 12 q^{41} - 16 q^{42} - 16 q^{43} - 29 q^{44} - 17 q^{46} - 2 q^{47} - 4 q^{48} + 8 q^{49} + 2 q^{51} - 12 q^{52} + 4 q^{53} + 2 q^{54} + 5 q^{56} - 5 q^{57} + 25 q^{58} - 15 q^{59} - 2 q^{61} - 9 q^{62} - 2 q^{63} + 23 q^{64} - 6 q^{66} - 2 q^{67} + 11 q^{68} + q^{69} - 2 q^{71} - 15 q^{72} - 16 q^{73} - 19 q^{74} + 40 q^{76} - 19 q^{77} - 3 q^{78} + 35 q^{79} + 4 q^{81} + 6 q^{82} - 16 q^{83} + 9 q^{84} + 3 q^{86} + 20 q^{87} + 30 q^{88} - 35 q^{89} - 12 q^{91} + 23 q^{92} - 23 q^{93} - 9 q^{94} + 12 q^{96} - 12 q^{97} + q^{98} - 7 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^6 - 2 * q^7 - 15 * q^8 + 4 * q^9 - 7 * q^11 - 8 * q^12 - q^13 + 16 * q^14 + 4 * q^16 - 2 * q^17 - 2 * q^18 + 5 * q^19 + 2 * q^21 + 6 * q^22 - q^23 + 15 * q^24 + 3 * q^26 - 4 * q^27 - 9 * q^28 - 20 * q^29 + 23 * q^31 - 12 * q^32 + 7 * q^33 - 14 * q^34 + 8 * q^36 - 2 * q^37 - 35 * q^38 + q^39 - 12 * q^41 - 16 * q^42 - 16 * q^43 - 29 * q^44 - 17 * q^46 - 2 * q^47 - 4 * q^48 + 8 * q^49 + 2 * q^51 - 12 * q^52 + 4 * q^53 + 2 * q^54 + 5 * q^56 - 5 * q^57 + 25 * q^58 - 15 * q^59 - 2 * q^61 - 9 * q^62 - 2 * q^63 + 23 * q^64 - 6 * q^66 - 2 * q^67 + 11 * q^68 + q^69 - 2 * q^71 - 15 * q^72 - 16 * q^73 - 19 * q^74 + 40 * q^76 - 19 * q^77 - 3 * q^78 + 35 * q^79 + 4 * q^81 + 6 * q^82 - 16 * q^83 + 9 * q^84 + 3 * q^86 + 20 * q^87 + 30 * q^88 - 35 * q^89 - 12 * q^91 + 23 * q^92 - 23 * q^93 - 9 * q^94 + 12 * q^96 - 12 * q^97 + q^98 - 7 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.70636 1.20658 0.603290 0.797522i $$-0.293856\pi$$
0.603290 + 0.797522i $$0.293856\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 0.911672 0.455836
$$5$$ 0 0
$$6$$ −1.70636 −0.696619
$$7$$ 3.94243 1.49010 0.745049 0.667009i $$-0.232426\pi$$
0.745049 + 0.667009i $$0.232426\pi$$
$$8$$ −1.85708 −0.656578
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.90869 −1.78154 −0.890769 0.454457i $$-0.849833\pi$$
−0.890769 + 0.454457i $$0.849833\pi$$
$$12$$ −0.911672 −0.263177
$$13$$ −3.29066 −0.912664 −0.456332 0.889810i $$-0.650837\pi$$
−0.456332 + 0.889810i $$0.650837\pi$$
$$14$$ 6.72721 1.79792
$$15$$ 0 0
$$16$$ −4.99220 −1.24805
$$17$$ −2.70636 −0.656389 −0.328195 0.944610i $$-0.606440\pi$$
−0.328195 + 0.944610i $$0.606440\pi$$
$$18$$ 1.70636 0.402193
$$19$$ −2.35813 −0.540993 −0.270497 0.962721i $$-0.587188\pi$$
−0.270497 + 0.962721i $$0.587188\pi$$
$$20$$ 0 0
$$21$$ −3.94243 −0.860309
$$22$$ −10.0824 −2.14957
$$23$$ 0.584296 0.121834 0.0609170 0.998143i $$-0.480597\pi$$
0.0609170 + 0.998143i $$0.480597\pi$$
$$24$$ 1.85708 0.379075
$$25$$ 0 0
$$26$$ −5.61505 −1.10120
$$27$$ −1.00000 −0.192450
$$28$$ 3.59420 0.679240
$$29$$ −3.91167 −0.726379 −0.363190 0.931715i $$-0.618312\pi$$
−0.363190 + 0.931715i $$0.618312\pi$$
$$30$$ 0 0
$$31$$ 2.70934 0.486612 0.243306 0.969950i $$-0.421768\pi$$
0.243306 + 0.969950i $$0.421768\pi$$
$$32$$ −4.80433 −0.849294
$$33$$ 5.90869 1.02857
$$34$$ −4.61803 −0.791986
$$35$$ 0 0
$$36$$ 0.911672 0.151945
$$37$$ 0.0208515 0.00342796 0.00171398 0.999999i $$-0.499454\pi$$
0.00171398 + 0.999999i $$0.499454\pi$$
$$38$$ −4.02383 −0.652752
$$39$$ 3.29066 0.526927
$$40$$ 0 0
$$41$$ 1.47214 0.229909 0.114955 0.993371i $$-0.463328\pi$$
0.114955 + 0.993371i $$0.463328\pi$$
$$42$$ −6.72721 −1.03803
$$43$$ −1.27279 −0.194098 −0.0970491 0.995280i $$-0.530940\pi$$
−0.0970491 + 0.995280i $$0.530940\pi$$
$$44$$ −5.38679 −0.812089
$$45$$ 0 0
$$46$$ 0.997020 0.147003
$$47$$ −5.43358 −0.792568 −0.396284 0.918128i $$-0.629701\pi$$
−0.396284 + 0.918128i $$0.629701\pi$$
$$48$$ 4.99220 0.720562
$$49$$ 8.54276 1.22039
$$50$$ 0 0
$$51$$ 2.70636 0.378967
$$52$$ −3.00000 −0.416025
$$53$$ −2.81554 −0.386744 −0.193372 0.981125i $$-0.561942\pi$$
−0.193372 + 0.981125i $$0.561942\pi$$
$$54$$ −1.70636 −0.232206
$$55$$ 0 0
$$56$$ −7.32142 −0.978365
$$57$$ 2.35813 0.312343
$$58$$ −6.67473 −0.876435
$$59$$ −4.69033 −0.610629 −0.305315 0.952252i $$-0.598762\pi$$
−0.305315 + 0.952252i $$0.598762\pi$$
$$60$$ 0 0
$$61$$ 5.58132 0.714614 0.357307 0.933987i $$-0.383695\pi$$
0.357307 + 0.933987i $$0.383695\pi$$
$$62$$ 4.62312 0.587137
$$63$$ 3.94243 0.496700
$$64$$ 1.78646 0.223308
$$65$$ 0 0
$$66$$ 10.0824 1.24105
$$67$$ −6.03076 −0.736774 −0.368387 0.929672i $$-0.620090\pi$$
−0.368387 + 0.929672i $$0.620090\pi$$
$$68$$ −2.46731 −0.299206
$$69$$ −0.584296 −0.0703409
$$70$$ 0 0
$$71$$ −8.10138 −0.961457 −0.480728 0.876870i $$-0.659628\pi$$
−0.480728 + 0.876870i $$0.659628\pi$$
$$72$$ −1.85708 −0.218859
$$73$$ −13.3166 −1.55859 −0.779295 0.626658i $$-0.784422\pi$$
−0.779295 + 0.626658i $$0.784422\pi$$
$$74$$ 0.0355801 0.00413611
$$75$$ 0 0
$$76$$ −2.14984 −0.246604
$$77$$ −23.2946 −2.65467
$$78$$ 5.61505 0.635780
$$79$$ 16.6648 1.87494 0.937469 0.348067i $$-0.113162\pi$$
0.937469 + 0.348067i $$0.113162\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.51200 0.277404
$$83$$ −0.781641 −0.0857962 −0.0428981 0.999079i $$-0.513659\pi$$
−0.0428981 + 0.999079i $$0.513659\pi$$
$$84$$ −3.59420 −0.392160
$$85$$ 0 0
$$86$$ −2.17183 −0.234195
$$87$$ 3.91167 0.419375
$$88$$ 10.9729 1.16972
$$89$$ −3.47327 −0.368166 −0.184083 0.982911i $$-0.558932\pi$$
−0.184083 + 0.982911i $$0.558932\pi$$
$$90$$ 0 0
$$91$$ −12.9732 −1.35996
$$92$$ 0.532686 0.0555363
$$93$$ −2.70934 −0.280946
$$94$$ −9.27165 −0.956297
$$95$$ 0 0
$$96$$ 4.80433 0.490340
$$97$$ 2.45443 0.249209 0.124605 0.992206i $$-0.460234\pi$$
0.124605 + 0.992206i $$0.460234\pi$$
$$98$$ 14.5770 1.47250
$$99$$ −5.90869 −0.593846
$$100$$ 0 0
$$101$$ −6.87495 −0.684083 −0.342042 0.939685i $$-0.611118\pi$$
−0.342042 + 0.939685i $$0.611118\pi$$
$$102$$ 4.61803 0.457254
$$103$$ −11.7532 −1.15807 −0.579036 0.815302i $$-0.696571\pi$$
−0.579036 + 0.815302i $$0.696571\pi$$
$$104$$ 6.11102 0.599235
$$105$$ 0 0
$$106$$ −4.80433 −0.466638
$$107$$ −5.66780 −0.547927 −0.273964 0.961740i $$-0.588335\pi$$
−0.273964 + 0.961740i $$0.588335\pi$$
$$108$$ −0.911672 −0.0877257
$$109$$ −1.36128 −0.130387 −0.0651934 0.997873i $$-0.520766\pi$$
−0.0651934 + 0.997873i $$0.520766\pi$$
$$110$$ 0 0
$$111$$ −0.0208515 −0.00197913
$$112$$ −19.6814 −1.85972
$$113$$ 10.6857 1.00522 0.502612 0.864512i $$-0.332373\pi$$
0.502612 + 0.864512i $$0.332373\pi$$
$$114$$ 4.02383 0.376866
$$115$$ 0 0
$$116$$ −3.56616 −0.331110
$$117$$ −3.29066 −0.304221
$$118$$ −8.00341 −0.736773
$$119$$ −10.6696 −0.978085
$$120$$ 0 0
$$121$$ 23.9126 2.17388
$$122$$ 9.52375 0.862239
$$123$$ −1.47214 −0.132738
$$124$$ 2.47003 0.221815
$$125$$ 0 0
$$126$$ 6.72721 0.599308
$$127$$ 13.5428 1.20173 0.600863 0.799352i $$-0.294824\pi$$
0.600863 + 0.799352i $$0.294824\pi$$
$$128$$ 12.6570 1.11873
$$129$$ 1.27279 0.112063
$$130$$ 0 0
$$131$$ 4.59236 0.401236 0.200618 0.979670i $$-0.435705\pi$$
0.200618 + 0.979670i $$0.435705\pi$$
$$132$$ 5.38679 0.468860
$$133$$ −9.29678 −0.806133
$$134$$ −10.2907 −0.888977
$$135$$ 0 0
$$136$$ 5.02594 0.430970
$$137$$ −15.2620 −1.30392 −0.651961 0.758253i $$-0.726053\pi$$
−0.651961 + 0.758253i $$0.726053\pi$$
$$138$$ −0.997020 −0.0848720
$$139$$ 18.2382 1.54694 0.773471 0.633832i $$-0.218519\pi$$
0.773471 + 0.633832i $$0.218519\pi$$
$$140$$ 0 0
$$141$$ 5.43358 0.457590
$$142$$ −13.8239 −1.16007
$$143$$ 19.4435 1.62595
$$144$$ −4.99220 −0.416017
$$145$$ 0 0
$$146$$ −22.7229 −1.88056
$$147$$ −8.54276 −0.704595
$$148$$ 0.0190097 0.00156259
$$149$$ 14.7323 1.20692 0.603458 0.797394i $$-0.293789\pi$$
0.603458 + 0.797394i $$0.293789\pi$$
$$150$$ 0 0
$$151$$ −17.4354 −1.41887 −0.709437 0.704769i $$-0.751051\pi$$
−0.709437 + 0.704769i $$0.751051\pi$$
$$152$$ 4.37925 0.355204
$$153$$ −2.70636 −0.218796
$$154$$ −39.7490 −3.20307
$$155$$ 0 0
$$156$$ 3.00000 0.240192
$$157$$ −17.9105 −1.42942 −0.714708 0.699423i $$-0.753440\pi$$
−0.714708 + 0.699423i $$0.753440\pi$$
$$158$$ 28.4362 2.26226
$$159$$ 2.81554 0.223287
$$160$$ 0 0
$$161$$ 2.30354 0.181545
$$162$$ 1.70636 0.134064
$$163$$ −21.7598 −1.70436 −0.852180 0.523249i $$-0.824720\pi$$
−0.852180 + 0.523249i $$0.824720\pi$$
$$164$$ 1.34210 0.104801
$$165$$ 0 0
$$166$$ −1.33376 −0.103520
$$167$$ 22.6030 1.74907 0.874535 0.484962i $$-0.161167\pi$$
0.874535 + 0.484962i $$0.161167\pi$$
$$168$$ 7.32142 0.564860
$$169$$ −2.17157 −0.167044
$$170$$ 0 0
$$171$$ −2.35813 −0.180331
$$172$$ −1.16036 −0.0884769
$$173$$ −12.9117 −0.981656 −0.490828 0.871256i $$-0.663306\pi$$
−0.490828 + 0.871256i $$0.663306\pi$$
$$174$$ 6.67473 0.506010
$$175$$ 0 0
$$176$$ 29.4974 2.22345
$$177$$ 4.69033 0.352547
$$178$$ −5.92666 −0.444222
$$179$$ −6.43060 −0.480645 −0.240323 0.970693i $$-0.577253\pi$$
−0.240323 + 0.970693i $$0.577253\pi$$
$$180$$ 0 0
$$181$$ 14.7071 1.09317 0.546584 0.837404i $$-0.315928\pi$$
0.546584 + 0.837404i $$0.315928\pi$$
$$182$$ −22.1370 −1.64090
$$183$$ −5.58132 −0.412583
$$184$$ −1.08508 −0.0799935
$$185$$ 0 0
$$186$$ −4.62312 −0.338984
$$187$$ 15.9911 1.16938
$$188$$ −4.95364 −0.361281
$$189$$ −3.94243 −0.286770
$$190$$ 0 0
$$191$$ 6.71303 0.485737 0.242869 0.970059i $$-0.421912\pi$$
0.242869 + 0.970059i $$0.421912\pi$$
$$192$$ −1.78646 −0.128927
$$193$$ −4.82817 −0.347539 −0.173769 0.984786i $$-0.555595\pi$$
−0.173769 + 0.984786i $$0.555595\pi$$
$$194$$ 4.18814 0.300691
$$195$$ 0 0
$$196$$ 7.78819 0.556299
$$197$$ 14.3841 1.02482 0.512411 0.858740i $$-0.328752\pi$$
0.512411 + 0.858740i $$0.328752\pi$$
$$198$$ −10.0824 −0.716523
$$199$$ −8.72608 −0.618575 −0.309288 0.950969i $$-0.600091\pi$$
−0.309288 + 0.950969i $$0.600091\pi$$
$$200$$ 0 0
$$201$$ 6.03076 0.425377
$$202$$ −11.7312 −0.825402
$$203$$ −15.4215 −1.08238
$$204$$ 2.46731 0.172747
$$205$$ 0 0
$$206$$ −20.0551 −1.39731
$$207$$ 0.584296 0.0406114
$$208$$ 16.4276 1.13905
$$209$$ 13.9335 0.963800
$$210$$ 0 0
$$211$$ 2.97617 0.204888 0.102444 0.994739i $$-0.467334\pi$$
0.102444 + 0.994739i $$0.467334\pi$$
$$212$$ −2.56685 −0.176292
$$213$$ 8.10138 0.555097
$$214$$ −9.67132 −0.661118
$$215$$ 0 0
$$216$$ 1.85708 0.126358
$$217$$ 10.6814 0.725100
$$218$$ −2.32283 −0.157322
$$219$$ 13.3166 0.899852
$$220$$ 0 0
$$221$$ 8.90571 0.599063
$$222$$ −0.0355801 −0.00238798
$$223$$ 0.304683 0.0204031 0.0102015 0.999948i $$-0.496753\pi$$
0.0102015 + 0.999948i $$0.496753\pi$$
$$224$$ −18.9408 −1.26553
$$225$$ 0 0
$$226$$ 18.2336 1.21288
$$227$$ 9.57221 0.635330 0.317665 0.948203i $$-0.397101\pi$$
0.317665 + 0.948203i $$0.397101\pi$$
$$228$$ 2.14984 0.142377
$$229$$ 12.1292 0.801517 0.400759 0.916184i $$-0.368747\pi$$
0.400759 + 0.916184i $$0.368747\pi$$
$$230$$ 0 0
$$231$$ 23.2946 1.53267
$$232$$ 7.26430 0.476924
$$233$$ −20.1002 −1.31681 −0.658405 0.752664i $$-0.728769\pi$$
−0.658405 + 0.752664i $$0.728769\pi$$
$$234$$ −5.61505 −0.367068
$$235$$ 0 0
$$236$$ −4.27604 −0.278347
$$237$$ −16.6648 −1.08250
$$238$$ −18.2063 −1.18014
$$239$$ 17.6637 1.14257 0.571284 0.820752i $$-0.306445\pi$$
0.571284 + 0.820752i $$0.306445\pi$$
$$240$$ 0 0
$$241$$ 29.5387 1.90276 0.951379 0.308024i $$-0.0996676\pi$$
0.951379 + 0.308024i $$0.0996676\pi$$
$$242$$ 40.8036 2.62296
$$243$$ −1.00000 −0.0641500
$$244$$ 5.08833 0.325747
$$245$$ 0 0
$$246$$ −2.51200 −0.160159
$$247$$ 7.75981 0.493745
$$248$$ −5.03147 −0.319499
$$249$$ 0.781641 0.0495345
$$250$$ 0 0
$$251$$ 1.89396 0.119546 0.0597729 0.998212i $$-0.480962\pi$$
0.0597729 + 0.998212i $$0.480962\pi$$
$$252$$ 3.59420 0.226413
$$253$$ −3.45242 −0.217052
$$254$$ 23.1088 1.44998
$$255$$ 0 0
$$256$$ 18.0245 1.12653
$$257$$ −22.1211 −1.37988 −0.689938 0.723869i $$-0.742362\pi$$
−0.689938 + 0.723869i $$0.742362\pi$$
$$258$$ 2.17183 0.135213
$$259$$ 0.0822054 0.00510799
$$260$$ 0 0
$$261$$ −3.91167 −0.242126
$$262$$ 7.83623 0.484124
$$263$$ 19.7153 1.21570 0.607849 0.794053i $$-0.292033\pi$$
0.607849 + 0.794053i $$0.292033\pi$$
$$264$$ −10.9729 −0.675337
$$265$$ 0 0
$$266$$ −15.8637 −0.972664
$$267$$ 3.47327 0.212561
$$268$$ −5.49807 −0.335848
$$269$$ 18.1647 1.10752 0.553762 0.832675i $$-0.313192\pi$$
0.553762 + 0.832675i $$0.313192\pi$$
$$270$$ 0 0
$$271$$ 22.5695 1.37100 0.685500 0.728073i $$-0.259584\pi$$
0.685500 + 0.728073i $$0.259584\pi$$
$$272$$ 13.5107 0.819206
$$273$$ 12.9732 0.785173
$$274$$ −26.0425 −1.57329
$$275$$ 0 0
$$276$$ −0.532686 −0.0320639
$$277$$ −8.25491 −0.495990 −0.247995 0.968761i $$-0.579772\pi$$
−0.247995 + 0.968761i $$0.579772\pi$$
$$278$$ 31.1209 1.86651
$$279$$ 2.70934 0.162204
$$280$$ 0 0
$$281$$ 1.28129 0.0764355 0.0382177 0.999269i $$-0.487832\pi$$
0.0382177 + 0.999269i $$0.487832\pi$$
$$282$$ 9.27165 0.552119
$$283$$ −2.67132 −0.158794 −0.0793968 0.996843i $$-0.525299\pi$$
−0.0793968 + 0.996843i $$0.525299\pi$$
$$284$$ −7.38580 −0.438266
$$285$$ 0 0
$$286$$ 33.1776 1.96183
$$287$$ 5.80379 0.342587
$$288$$ −4.80433 −0.283098
$$289$$ −9.67560 −0.569153
$$290$$ 0 0
$$291$$ −2.45443 −0.143881
$$292$$ −12.1404 −0.710461
$$293$$ −17.6605 −1.03174 −0.515870 0.856667i $$-0.672531\pi$$
−0.515870 + 0.856667i $$0.672531\pi$$
$$294$$ −14.5770 −0.850150
$$295$$ 0 0
$$296$$ −0.0387229 −0.00225072
$$297$$ 5.90869 0.342857
$$298$$ 25.1386 1.45624
$$299$$ −1.92272 −0.111194
$$300$$ 0 0
$$301$$ −5.01787 −0.289225
$$302$$ −29.7511 −1.71199
$$303$$ 6.87495 0.394956
$$304$$ 11.7723 0.675186
$$305$$ 0 0
$$306$$ −4.61803 −0.263995
$$307$$ 28.5593 1.62997 0.814983 0.579484i $$-0.196746\pi$$
0.814983 + 0.579484i $$0.196746\pi$$
$$308$$ −21.2370 −1.21009
$$309$$ 11.7532 0.668613
$$310$$ 0 0
$$311$$ −29.3283 −1.66306 −0.831529 0.555482i $$-0.812534\pi$$
−0.831529 + 0.555482i $$0.812534\pi$$
$$312$$ −6.11102 −0.345968
$$313$$ 17.4933 0.988777 0.494388 0.869241i $$-0.335392\pi$$
0.494388 + 0.869241i $$0.335392\pi$$
$$314$$ −30.5619 −1.72471
$$315$$ 0 0
$$316$$ 15.1928 0.854665
$$317$$ −3.91763 −0.220036 −0.110018 0.993930i $$-0.535091\pi$$
−0.110018 + 0.993930i $$0.535091\pi$$
$$318$$ 4.80433 0.269414
$$319$$ 23.1129 1.29407
$$320$$ 0 0
$$321$$ 5.66780 0.316346
$$322$$ 3.93068 0.219048
$$323$$ 6.38197 0.355102
$$324$$ 0.911672 0.0506484
$$325$$ 0 0
$$326$$ −37.1301 −2.05645
$$327$$ 1.36128 0.0752788
$$328$$ −2.73388 −0.150953
$$329$$ −21.4215 −1.18101
$$330$$ 0 0
$$331$$ −17.5534 −0.964820 −0.482410 0.875945i $$-0.660238\pi$$
−0.482410 + 0.875945i $$0.660238\pi$$
$$332$$ −0.712600 −0.0391090
$$333$$ 0.0208515 0.00114265
$$334$$ 38.5689 2.11039
$$335$$ 0 0
$$336$$ 19.6814 1.07371
$$337$$ 14.0476 0.765221 0.382611 0.923910i $$-0.375025\pi$$
0.382611 + 0.923910i $$0.375025\pi$$
$$338$$ −3.70549 −0.201552
$$339$$ −10.6857 −0.580366
$$340$$ 0 0
$$341$$ −16.0087 −0.866918
$$342$$ −4.02383 −0.217584
$$343$$ 6.08221 0.328408
$$344$$ 2.36367 0.127440
$$345$$ 0 0
$$346$$ −22.0320 −1.18445
$$347$$ 1.03050 0.0553199 0.0276599 0.999617i $$-0.491194\pi$$
0.0276599 + 0.999617i $$0.491194\pi$$
$$348$$ 3.56616 0.191166
$$349$$ 21.5626 1.15422 0.577109 0.816667i $$-0.304181\pi$$
0.577109 + 0.816667i $$0.304181\pi$$
$$350$$ 0 0
$$351$$ 3.29066 0.175642
$$352$$ 28.3873 1.51305
$$353$$ −7.15625 −0.380888 −0.190444 0.981698i $$-0.560993\pi$$
−0.190444 + 0.981698i $$0.560993\pi$$
$$354$$ 8.00341 0.425376
$$355$$ 0 0
$$356$$ −3.16649 −0.167823
$$357$$ 10.6696 0.564697
$$358$$ −10.9729 −0.579937
$$359$$ 27.1518 1.43302 0.716510 0.697577i $$-0.245738\pi$$
0.716510 + 0.697577i $$0.245738\pi$$
$$360$$ 0 0
$$361$$ −13.4392 −0.707326
$$362$$ 25.0956 1.31899
$$363$$ −23.9126 −1.25509
$$364$$ −11.8273 −0.619918
$$365$$ 0 0
$$366$$ −9.52375 −0.497814
$$367$$ −21.4423 −1.11928 −0.559641 0.828735i $$-0.689061\pi$$
−0.559641 + 0.828735i $$0.689061\pi$$
$$368$$ −2.91692 −0.152055
$$369$$ 1.47214 0.0766363
$$370$$ 0 0
$$371$$ −11.1001 −0.576287
$$372$$ −2.47003 −0.128065
$$373$$ −6.39073 −0.330900 −0.165450 0.986218i $$-0.552908\pi$$
−0.165450 + 0.986218i $$0.552908\pi$$
$$374$$ 27.2865 1.41095
$$375$$ 0 0
$$376$$ 10.0906 0.520383
$$377$$ 12.8720 0.662940
$$378$$ −6.72721 −0.346011
$$379$$ −0.154667 −0.00794469 −0.00397235 0.999992i $$-0.501264\pi$$
−0.00397235 + 0.999992i $$0.501264\pi$$
$$380$$ 0 0
$$381$$ −13.5428 −0.693816
$$382$$ 11.4549 0.586081
$$383$$ 4.25508 0.217424 0.108712 0.994073i $$-0.465327\pi$$
0.108712 + 0.994073i $$0.465327\pi$$
$$384$$ −12.6570 −0.645901
$$385$$ 0 0
$$386$$ −8.23860 −0.419334
$$387$$ −1.27279 −0.0646994
$$388$$ 2.23763 0.113599
$$389$$ −20.3682 −1.03271 −0.516354 0.856375i $$-0.672711\pi$$
−0.516354 + 0.856375i $$0.672711\pi$$
$$390$$ 0 0
$$391$$ −1.58132 −0.0799706
$$392$$ −15.8646 −0.801283
$$393$$ −4.59236 −0.231654
$$394$$ 24.5444 1.23653
$$395$$ 0 0
$$396$$ −5.38679 −0.270696
$$397$$ 1.95716 0.0982270 0.0491135 0.998793i $$-0.484360\pi$$
0.0491135 + 0.998793i $$0.484360\pi$$
$$398$$ −14.8898 −0.746360
$$399$$ 9.29678 0.465421
$$400$$ 0 0
$$401$$ −32.8337 −1.63964 −0.819818 0.572624i $$-0.805925\pi$$
−0.819818 + 0.572624i $$0.805925\pi$$
$$402$$ 10.2907 0.513251
$$403$$ −8.91552 −0.444114
$$404$$ −6.26770 −0.311830
$$405$$ 0 0
$$406$$ −26.3147 −1.30597
$$407$$ −0.123205 −0.00610704
$$408$$ −5.02594 −0.248821
$$409$$ −39.5764 −1.95693 −0.978464 0.206417i $$-0.933820\pi$$
−0.978464 + 0.206417i $$0.933820\pi$$
$$410$$ 0 0
$$411$$ 15.2620 0.752819
$$412$$ −10.7150 −0.527891
$$413$$ −18.4913 −0.909898
$$414$$ 0.997020 0.0490009
$$415$$ 0 0
$$416$$ 15.8094 0.775121
$$417$$ −18.2382 −0.893127
$$418$$ 23.7756 1.16290
$$419$$ 19.5595 0.955544 0.477772 0.878484i $$-0.341445\pi$$
0.477772 + 0.878484i $$0.341445\pi$$
$$420$$ 0 0
$$421$$ −40.3325 −1.96568 −0.982842 0.184451i $$-0.940949\pi$$
−0.982842 + 0.184451i $$0.940949\pi$$
$$422$$ 5.07842 0.247214
$$423$$ −5.43358 −0.264189
$$424$$ 5.22869 0.253928
$$425$$ 0 0
$$426$$ 13.8239 0.669769
$$427$$ 22.0039 1.06485
$$428$$ −5.16718 −0.249765
$$429$$ −19.4435 −0.938740
$$430$$ 0 0
$$431$$ 13.7370 0.661689 0.330844 0.943685i $$-0.392666\pi$$
0.330844 + 0.943685i $$0.392666\pi$$
$$432$$ 4.99220 0.240187
$$433$$ 12.4253 0.597124 0.298562 0.954390i $$-0.403493\pi$$
0.298562 + 0.954390i $$0.403493\pi$$
$$434$$ 18.2263 0.874892
$$435$$ 0 0
$$436$$ −1.24104 −0.0594349
$$437$$ −1.37785 −0.0659114
$$438$$ 22.7229 1.08574
$$439$$ 10.9654 0.523349 0.261675 0.965156i $$-0.415725\pi$$
0.261675 + 0.965156i $$0.415725\pi$$
$$440$$ 0 0
$$441$$ 8.54276 0.406798
$$442$$ 15.1964 0.722818
$$443$$ −8.13187 −0.386357 −0.193178 0.981164i $$-0.561880\pi$$
−0.193178 + 0.981164i $$0.561880\pi$$
$$444$$ −0.0190097 −0.000902160 0
$$445$$ 0 0
$$446$$ 0.519899 0.0246180
$$447$$ −14.7323 −0.696814
$$448$$ 7.04300 0.332751
$$449$$ −32.9503 −1.55502 −0.777511 0.628869i $$-0.783518\pi$$
−0.777511 + 0.628869i $$0.783518\pi$$
$$450$$ 0 0
$$451$$ −8.69840 −0.409592
$$452$$ 9.74183 0.458217
$$453$$ 17.4354 0.819187
$$454$$ 16.3337 0.766577
$$455$$ 0 0
$$456$$ −4.37925 −0.205077
$$457$$ 22.8800 1.07028 0.535142 0.844762i $$-0.320258\pi$$
0.535142 + 0.844762i $$0.320258\pi$$
$$458$$ 20.6967 0.967095
$$459$$ 2.70636 0.126322
$$460$$ 0 0
$$461$$ 12.0425 0.560875 0.280438 0.959872i $$-0.409520\pi$$
0.280438 + 0.959872i $$0.409520\pi$$
$$462$$ 39.7490 1.84929
$$463$$ 27.2002 1.26410 0.632049 0.774928i $$-0.282214\pi$$
0.632049 + 0.774928i $$0.282214\pi$$
$$464$$ 19.5278 0.906557
$$465$$ 0 0
$$466$$ −34.2983 −1.58884
$$467$$ 3.94243 0.182434 0.0912170 0.995831i $$-0.470924\pi$$
0.0912170 + 0.995831i $$0.470924\pi$$
$$468$$ −3.00000 −0.138675
$$469$$ −23.7758 −1.09787
$$470$$ 0 0
$$471$$ 17.9105 0.825274
$$472$$ 8.71033 0.400926
$$473$$ 7.52050 0.345793
$$474$$ −28.4362 −1.30612
$$475$$ 0 0
$$476$$ −9.72721 −0.445846
$$477$$ −2.81554 −0.128915
$$478$$ 30.1406 1.37860
$$479$$ 19.9493 0.911505 0.455752 0.890107i $$-0.349370\pi$$
0.455752 + 0.890107i $$0.349370\pi$$
$$480$$ 0 0
$$481$$ −0.0686150 −0.00312857
$$482$$ 50.4038 2.29583
$$483$$ −2.30354 −0.104815
$$484$$ 21.8005 0.990931
$$485$$ 0 0
$$486$$ −1.70636 −0.0774022
$$487$$ 11.0023 0.498560 0.249280 0.968431i $$-0.419806\pi$$
0.249280 + 0.968431i $$0.419806\pi$$
$$488$$ −10.3650 −0.469200
$$489$$ 21.7598 0.984013
$$490$$ 0 0
$$491$$ −33.2933 −1.50251 −0.751253 0.660014i $$-0.770550\pi$$
−0.751253 + 0.660014i $$0.770550\pi$$
$$492$$ −1.34210 −0.0605068
$$493$$ 10.5864 0.476788
$$494$$ 13.2411 0.595743
$$495$$ 0 0
$$496$$ −13.5256 −0.607316
$$497$$ −31.9391 −1.43267
$$498$$ 1.33376 0.0597673
$$499$$ −41.1448 −1.84189 −0.920946 0.389690i $$-0.872582\pi$$
−0.920946 + 0.389690i $$0.872582\pi$$
$$500$$ 0 0
$$501$$ −22.6030 −1.00983
$$502$$ 3.23179 0.144242
$$503$$ −32.0761 −1.43020 −0.715102 0.699020i $$-0.753620\pi$$
−0.715102 + 0.699020i $$0.753620\pi$$
$$504$$ −7.32142 −0.326122
$$505$$ 0 0
$$506$$ −5.89108 −0.261891
$$507$$ 2.17157 0.0964429
$$508$$ 12.3465 0.547790
$$509$$ −27.0953 −1.20098 −0.600489 0.799633i $$-0.705028\pi$$
−0.600489 + 0.799633i $$0.705028\pi$$
$$510$$ 0 0
$$511$$ −52.4997 −2.32245
$$512$$ 5.44234 0.240520
$$513$$ 2.35813 0.104114
$$514$$ −37.7466 −1.66493
$$515$$ 0 0
$$516$$ 1.16036 0.0510822
$$517$$ 32.1053 1.41199
$$518$$ 0.140272 0.00616321
$$519$$ 12.9117 0.566759
$$520$$ 0 0
$$521$$ 25.9556 1.13714 0.568569 0.822636i $$-0.307497\pi$$
0.568569 + 0.822636i $$0.307497\pi$$
$$522$$ −6.67473 −0.292145
$$523$$ 1.79382 0.0784381 0.0392190 0.999231i $$-0.487513\pi$$
0.0392190 + 0.999231i $$0.487513\pi$$
$$524$$ 4.18673 0.182898
$$525$$ 0 0
$$526$$ 33.6414 1.46684
$$527$$ −7.33246 −0.319407
$$528$$ −29.4974 −1.28371
$$529$$ −22.6586 −0.985156
$$530$$ 0 0
$$531$$ −4.69033 −0.203543
$$532$$ −8.47561 −0.367464
$$533$$ −4.84430 −0.209830
$$534$$ 5.92666 0.256472
$$535$$ 0 0
$$536$$ 11.1996 0.483750
$$537$$ 6.43060 0.277501
$$538$$ 30.9956 1.33632
$$539$$ −50.4765 −2.17418
$$540$$ 0 0
$$541$$ −7.43542 −0.319674 −0.159837 0.987143i $$-0.551097\pi$$
−0.159837 + 0.987143i $$0.551097\pi$$
$$542$$ 38.5117 1.65422
$$543$$ −14.7071 −0.631141
$$544$$ 13.0023 0.557468
$$545$$ 0 0
$$546$$ 22.1370 0.947374
$$547$$ −19.4868 −0.833194 −0.416597 0.909091i $$-0.636777\pi$$
−0.416597 + 0.909091i $$0.636777\pi$$
$$548$$ −13.9139 −0.594374
$$549$$ 5.58132 0.238205
$$550$$ 0 0
$$551$$ 9.22425 0.392966
$$552$$ 1.08508 0.0461843
$$553$$ 65.6999 2.79384
$$554$$ −14.0859 −0.598451
$$555$$ 0 0
$$556$$ 16.6272 0.705152
$$557$$ 26.3285 1.11557 0.557787 0.829984i $$-0.311650\pi$$
0.557787 + 0.829984i $$0.311650\pi$$
$$558$$ 4.62312 0.195712
$$559$$ 4.18830 0.177146
$$560$$ 0 0
$$561$$ −15.9911 −0.675143
$$562$$ 2.18635 0.0922255
$$563$$ −36.7708 −1.54971 −0.774853 0.632141i $$-0.782176\pi$$
−0.774853 + 0.632141i $$0.782176\pi$$
$$564$$ 4.95364 0.208586
$$565$$ 0 0
$$566$$ −4.55824 −0.191597
$$567$$ 3.94243 0.165567
$$568$$ 15.0449 0.631271
$$569$$ −36.2823 −1.52103 −0.760516 0.649320i $$-0.775054\pi$$
−0.760516 + 0.649320i $$0.775054\pi$$
$$570$$ 0 0
$$571$$ −3.36442 −0.140797 −0.0703983 0.997519i $$-0.522427\pi$$
−0.0703983 + 0.997519i $$0.522427\pi$$
$$572$$ 17.7261 0.741164
$$573$$ −6.71303 −0.280441
$$574$$ 9.90337 0.413359
$$575$$ 0 0
$$576$$ 1.78646 0.0744359
$$577$$ 6.88442 0.286602 0.143301 0.989679i $$-0.454228\pi$$
0.143301 + 0.989679i $$0.454228\pi$$
$$578$$ −16.5101 −0.686729
$$579$$ 4.82817 0.200652
$$580$$ 0 0
$$581$$ −3.08156 −0.127845
$$582$$ −4.18814 −0.173604
$$583$$ 16.6362 0.689000
$$584$$ 24.7300 1.02334
$$585$$ 0 0
$$586$$ −30.1353 −1.24488
$$587$$ 14.7377 0.608288 0.304144 0.952626i $$-0.401630\pi$$
0.304144 + 0.952626i $$0.401630\pi$$
$$588$$ −7.78819 −0.321180
$$589$$ −6.38899 −0.263254
$$590$$ 0 0
$$591$$ −14.3841 −0.591682
$$592$$ −0.104095 −0.00427826
$$593$$ −5.82561 −0.239229 −0.119615 0.992820i $$-0.538166\pi$$
−0.119615 + 0.992820i $$0.538166\pi$$
$$594$$ 10.0824 0.413685
$$595$$ 0 0
$$596$$ 13.4310 0.550156
$$597$$ 8.72608 0.357134
$$598$$ −3.28085 −0.134164
$$599$$ −27.1527 −1.10943 −0.554715 0.832040i $$-0.687173\pi$$
−0.554715 + 0.832040i $$0.687173\pi$$
$$600$$ 0 0
$$601$$ 20.9140 0.853101 0.426551 0.904464i $$-0.359729\pi$$
0.426551 + 0.904464i $$0.359729\pi$$
$$602$$ −8.56231 −0.348974
$$603$$ −6.03076 −0.245591
$$604$$ −15.8954 −0.646774
$$605$$ 0 0
$$606$$ 11.7312 0.476546
$$607$$ −1.46424 −0.0594318 −0.0297159 0.999558i $$-0.509460\pi$$
−0.0297159 + 0.999558i $$0.509460\pi$$
$$608$$ 11.3293 0.459462
$$609$$ 15.4215 0.624910
$$610$$ 0 0
$$611$$ 17.8800 0.723349
$$612$$ −2.46731 −0.0997353
$$613$$ −34.7105 −1.40194 −0.700971 0.713189i $$-0.747250\pi$$
−0.700971 + 0.713189i $$0.747250\pi$$
$$614$$ 48.7326 1.96669
$$615$$ 0 0
$$616$$ 43.2600 1.74299
$$617$$ −41.9038 −1.68698 −0.843492 0.537142i $$-0.819504\pi$$
−0.843492 + 0.537142i $$0.819504\pi$$
$$618$$ 20.0551 0.806736
$$619$$ −18.7614 −0.754083 −0.377042 0.926196i $$-0.623059\pi$$
−0.377042 + 0.926196i $$0.623059\pi$$
$$620$$ 0 0
$$621$$ −0.584296 −0.0234470
$$622$$ −50.0448 −2.00661
$$623$$ −13.6931 −0.548604
$$624$$ −16.4276 −0.657631
$$625$$ 0 0
$$626$$ 29.8498 1.19304
$$627$$ −13.9335 −0.556450
$$628$$ −16.3285 −0.651579
$$629$$ −0.0564316 −0.00225007
$$630$$ 0 0
$$631$$ 9.88888 0.393670 0.196835 0.980437i $$-0.436934\pi$$
0.196835 + 0.980437i $$0.436934\pi$$
$$632$$ −30.9479 −1.23104
$$633$$ −2.97617 −0.118292
$$634$$ −6.68490 −0.265491
$$635$$ 0 0
$$636$$ 2.56685 0.101782
$$637$$ −28.1113 −1.11381
$$638$$ 39.4389 1.56140
$$639$$ −8.10138 −0.320486
$$640$$ 0 0
$$641$$ −1.81266 −0.0715959 −0.0357979 0.999359i $$-0.511397\pi$$
−0.0357979 + 0.999359i $$0.511397\pi$$
$$642$$ 9.67132 0.381697
$$643$$ −45.7391 −1.80377 −0.901886 0.431974i $$-0.857817\pi$$
−0.901886 + 0.431974i $$0.857817\pi$$
$$644$$ 2.10008 0.0827546
$$645$$ 0 0
$$646$$ 10.8899 0.428459
$$647$$ 25.6771 1.00947 0.504736 0.863274i $$-0.331590\pi$$
0.504736 + 0.863274i $$0.331590\pi$$
$$648$$ −1.85708 −0.0729531
$$649$$ 27.7137 1.08786
$$650$$ 0 0
$$651$$ −10.6814 −0.418637
$$652$$ −19.8378 −0.776909
$$653$$ −35.9684 −1.40755 −0.703775 0.710422i $$-0.748504\pi$$
−0.703775 + 0.710422i $$0.748504\pi$$
$$654$$ 2.32283 0.0908299
$$655$$ 0 0
$$656$$ −7.34919 −0.286938
$$657$$ −13.3166 −0.519530
$$658$$ −36.5528 −1.42498
$$659$$ −32.0924 −1.25014 −0.625072 0.780567i $$-0.714930\pi$$
−0.625072 + 0.780567i $$0.714930\pi$$
$$660$$ 0 0
$$661$$ 16.2064 0.630357 0.315179 0.949032i $$-0.397936\pi$$
0.315179 + 0.949032i $$0.397936\pi$$
$$662$$ −29.9524 −1.16413
$$663$$ −8.90571 −0.345869
$$664$$ 1.45157 0.0563319
$$665$$ 0 0
$$666$$ 0.0355801 0.00137870
$$667$$ −2.28557 −0.0884977
$$668$$ 20.6065 0.797289
$$669$$ −0.304683 −0.0117797
$$670$$ 0 0
$$671$$ −32.9783 −1.27311
$$672$$ 18.9408 0.730655
$$673$$ −34.1608 −1.31680 −0.658401 0.752667i $$-0.728767\pi$$
−0.658401 + 0.752667i $$0.728767\pi$$
$$674$$ 23.9703 0.923301
$$675$$ 0 0
$$676$$ −1.97976 −0.0761446
$$677$$ −9.17514 −0.352629 −0.176315 0.984334i $$-0.556418\pi$$
−0.176315 + 0.984334i $$0.556418\pi$$
$$678$$ −18.2336 −0.700258
$$679$$ 9.67641 0.371346
$$680$$ 0 0
$$681$$ −9.57221 −0.366808
$$682$$ −27.3166 −1.04601
$$683$$ −19.5007 −0.746173 −0.373087 0.927797i $$-0.621701\pi$$
−0.373087 + 0.927797i $$0.621701\pi$$
$$684$$ −2.14984 −0.0822014
$$685$$ 0 0
$$686$$ 10.3784 0.396251
$$687$$ −12.1292 −0.462756
$$688$$ 6.35400 0.242244
$$689$$ 9.26498 0.352968
$$690$$ 0 0
$$691$$ 29.0458 1.10495 0.552477 0.833528i $$-0.313683\pi$$
0.552477 + 0.833528i $$0.313683\pi$$
$$692$$ −11.7712 −0.447474
$$693$$ −23.2946 −0.884889
$$694$$ 1.75840 0.0667479
$$695$$ 0 0
$$696$$ −7.26430 −0.275352
$$697$$ −3.98413 −0.150910
$$698$$ 36.7936 1.39266
$$699$$ 20.1002 0.760261
$$700$$ 0 0
$$701$$ −20.4085 −0.770820 −0.385410 0.922745i $$-0.625940\pi$$
−0.385410 + 0.922745i $$0.625940\pi$$
$$702$$ 5.61505 0.211927
$$703$$ −0.0491705 −0.00185450
$$704$$ −10.5557 −0.397831
$$705$$ 0 0
$$706$$ −12.2111 −0.459573
$$707$$ −27.1040 −1.01935
$$708$$ 4.27604 0.160704
$$709$$ 16.4810 0.618956 0.309478 0.950907i $$-0.399846\pi$$
0.309478 + 0.950907i $$0.399846\pi$$
$$710$$ 0 0
$$711$$ 16.6648 0.624980
$$712$$ 6.45016 0.241730
$$713$$ 1.58306 0.0592859
$$714$$ 18.2063 0.681353
$$715$$ 0 0
$$716$$ −5.86259 −0.219095
$$717$$ −17.6637 −0.659662
$$718$$ 46.3309 1.72905
$$719$$ 31.0813 1.15914 0.579569 0.814923i $$-0.303221\pi$$
0.579569 + 0.814923i $$0.303221\pi$$
$$720$$ 0 0
$$721$$ −46.3360 −1.72564
$$722$$ −22.9321 −0.853446
$$723$$ −29.5387 −1.09856
$$724$$ 13.4080 0.498305
$$725$$ 0 0
$$726$$ −40.8036 −1.51436
$$727$$ 0.953049 0.0353466 0.0176733 0.999844i $$-0.494374\pi$$
0.0176733 + 0.999844i $$0.494374\pi$$
$$728$$ 24.0923 0.892919
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 3.44462 0.127404
$$732$$ −5.08833 −0.188070
$$733$$ 15.3751 0.567893 0.283947 0.958840i $$-0.408356\pi$$
0.283947 + 0.958840i $$0.408356\pi$$
$$734$$ −36.5884 −1.35050
$$735$$ 0 0
$$736$$ −2.80715 −0.103473
$$737$$ 35.6339 1.31259
$$738$$ 2.51200 0.0924679
$$739$$ 5.19285 0.191022 0.0955110 0.995428i $$-0.469551\pi$$
0.0955110 + 0.995428i $$0.469551\pi$$
$$740$$ 0 0
$$741$$ −7.75981 −0.285064
$$742$$ −18.9408 −0.695337
$$743$$ 42.3399 1.55330 0.776650 0.629932i $$-0.216917\pi$$
0.776650 + 0.629932i $$0.216917\pi$$
$$744$$ 5.03147 0.184463
$$745$$ 0 0
$$746$$ −10.9049 −0.399257
$$747$$ −0.781641 −0.0285987
$$748$$ 14.5786 0.533046
$$749$$ −22.3449 −0.816465
$$750$$ 0 0
$$751$$ 22.2461 0.811773 0.405887 0.913923i $$-0.366963\pi$$
0.405887 + 0.913923i $$0.366963\pi$$
$$752$$ 27.1255 0.989165
$$753$$ −1.89396 −0.0690199
$$754$$ 21.9642 0.799891
$$755$$ 0 0
$$756$$ −3.59420 −0.130720
$$757$$ 9.27680 0.337171 0.168585 0.985687i $$-0.446080\pi$$
0.168585 + 0.985687i $$0.446080\pi$$
$$758$$ −0.263917 −0.00958591
$$759$$ 3.45242 0.125315
$$760$$ 0 0
$$761$$ −17.4122 −0.631191 −0.315596 0.948894i $$-0.602204\pi$$
−0.315596 + 0.948894i $$0.602204\pi$$
$$762$$ −23.1088 −0.837145
$$763$$ −5.36674 −0.194289
$$764$$ 6.12008 0.221417
$$765$$ 0 0
$$766$$ 7.26070 0.262340
$$767$$ 15.4343 0.557300
$$768$$ −18.0245 −0.650404
$$769$$ −23.1912 −0.836297 −0.418148 0.908379i $$-0.637321\pi$$
−0.418148 + 0.908379i $$0.637321\pi$$
$$770$$ 0 0
$$771$$ 22.1211 0.796672
$$772$$ −4.40170 −0.158421
$$773$$ 5.50798 0.198108 0.0990541 0.995082i $$-0.468418\pi$$
0.0990541 + 0.995082i $$0.468418\pi$$
$$774$$ −2.17183 −0.0780650
$$775$$ 0 0
$$776$$ −4.55807 −0.163625
$$777$$ −0.0822054 −0.00294910
$$778$$ −34.7555 −1.24605
$$779$$ −3.47149 −0.124379
$$780$$ 0 0
$$781$$ 47.8685 1.71287
$$782$$ −2.69830 −0.0964909
$$783$$ 3.91167 0.139792
$$784$$ −42.6471 −1.52311
$$785$$ 0 0
$$786$$ −7.83623 −0.279509
$$787$$ −25.7971 −0.919568 −0.459784 0.888031i $$-0.652073\pi$$
−0.459784 + 0.888031i $$0.652073\pi$$
$$788$$ 13.1136 0.467151
$$789$$ −19.7153 −0.701883
$$790$$ 0 0
$$791$$ 42.1275 1.49788
$$792$$ 10.9729 0.389906
$$793$$ −18.3662 −0.652203
$$794$$ 3.33962 0.118519
$$795$$ 0 0
$$796$$ −7.95532 −0.281969
$$797$$ −3.70988 −0.131411 −0.0657054 0.997839i $$-0.520930\pi$$
−0.0657054 + 0.997839i $$0.520930\pi$$
$$798$$ 15.8637 0.561568
$$799$$ 14.7052 0.520233
$$800$$ 0 0
$$801$$ −3.47327 −0.122722
$$802$$ −56.0261 −1.97835
$$803$$ 78.6837 2.77669
$$804$$ 5.49807 0.193902
$$805$$ 0 0
$$806$$ −15.2131 −0.535859
$$807$$ −18.1647 −0.639429
$$808$$ 12.7674 0.449154
$$809$$ 37.5040 1.31857 0.659286 0.751893i $$-0.270859\pi$$
0.659286 + 0.751893i $$0.270859\pi$$
$$810$$ 0 0
$$811$$ 45.6159 1.60179 0.800896 0.598804i $$-0.204357\pi$$
0.800896 + 0.598804i $$0.204357\pi$$
$$812$$ −14.0593 −0.493386
$$813$$ −22.5695 −0.791547
$$814$$ −0.210232 −0.00736863
$$815$$ 0 0
$$816$$ −13.5107 −0.472969
$$817$$ 3.00140 0.105006
$$818$$ −67.5317 −2.36119
$$819$$ −12.9732 −0.453320
$$820$$ 0 0
$$821$$ 45.0511 1.57229 0.786147 0.618039i $$-0.212073\pi$$
0.786147 + 0.618039i $$0.212073\pi$$
$$822$$ 26.0425 0.908337
$$823$$ 41.4573 1.44511 0.722556 0.691312i $$-0.242967\pi$$
0.722556 + 0.691312i $$0.242967\pi$$
$$824$$ 21.8266 0.760364
$$825$$ 0 0
$$826$$ −31.5529 −1.09786
$$827$$ 38.6933 1.34550 0.672749 0.739871i $$-0.265113\pi$$
0.672749 + 0.739871i $$0.265113\pi$$
$$828$$ 0.532686 0.0185121
$$829$$ 24.1258 0.837922 0.418961 0.908004i $$-0.362394\pi$$
0.418961 + 0.908004i $$0.362394\pi$$
$$830$$ 0 0
$$831$$ 8.25491 0.286360
$$832$$ −5.87864 −0.203805
$$833$$ −23.1198 −0.801053
$$834$$ −31.1209 −1.07763
$$835$$ 0 0
$$836$$ 12.7028 0.439334
$$837$$ −2.70934 −0.0936486
$$838$$ 33.3756 1.15294
$$839$$ −17.7858 −0.614032 −0.307016 0.951704i $$-0.599331\pi$$
−0.307016 + 0.951704i $$0.599331\pi$$
$$840$$ 0 0
$$841$$ −13.6988 −0.472373
$$842$$ −68.8218 −2.37175
$$843$$ −1.28129 −0.0441300
$$844$$ 2.71329 0.0933953
$$845$$ 0 0
$$846$$ −9.27165 −0.318766
$$847$$ 94.2739 3.23929
$$848$$ 14.0557 0.482676
$$849$$ 2.67132 0.0916796
$$850$$ 0 0
$$851$$ 0.0121834 0.000417642 0
$$852$$ 7.38580 0.253033
$$853$$ −12.8433 −0.439745 −0.219872 0.975529i $$-0.570564\pi$$
−0.219872 + 0.975529i $$0.570564\pi$$
$$854$$ 37.5467 1.28482
$$855$$ 0 0
$$856$$ 10.5256 0.359757
$$857$$ 15.6015 0.532936 0.266468 0.963844i $$-0.414143\pi$$
0.266468 + 0.963844i $$0.414143\pi$$
$$858$$ −33.1776 −1.13267
$$859$$ 4.82843 0.164744 0.0823719 0.996602i $$-0.473750\pi$$
0.0823719 + 0.996602i $$0.473750\pi$$
$$860$$ 0 0
$$861$$ −5.80379 −0.197793
$$862$$ 23.4403 0.798381
$$863$$ 13.1548 0.447796 0.223898 0.974613i $$-0.428122\pi$$
0.223898 + 0.974613i $$0.428122\pi$$
$$864$$ 4.80433 0.163447
$$865$$ 0 0
$$866$$ 21.2021 0.720478
$$867$$ 9.67560 0.328601
$$868$$ 9.73792 0.330527
$$869$$ −98.4673 −3.34027
$$870$$ 0 0
$$871$$ 19.8452 0.672428
$$872$$ 2.52800 0.0856090
$$873$$ 2.45443 0.0830698
$$874$$ −2.35111 −0.0795274
$$875$$ 0 0
$$876$$ 12.1404 0.410185
$$877$$ −6.25253 −0.211133 −0.105567 0.994412i $$-0.533666\pi$$
−0.105567 + 0.994412i $$0.533666\pi$$
$$878$$ 18.7109 0.631463
$$879$$ 17.6605 0.595675
$$880$$ 0 0
$$881$$ −16.7801 −0.565335 −0.282667 0.959218i $$-0.591219\pi$$
−0.282667 + 0.959218i $$0.591219\pi$$
$$882$$ 14.5770 0.490834
$$883$$ −16.2367 −0.546407 −0.273203 0.961956i $$-0.588083\pi$$
−0.273203 + 0.961956i $$0.588083\pi$$
$$884$$ 8.11909 0.273074
$$885$$ 0 0
$$886$$ −13.8759 −0.466171
$$887$$ −57.6873 −1.93695 −0.968476 0.249108i $$-0.919863\pi$$
−0.968476 + 0.249108i $$0.919863\pi$$
$$888$$ 0.0387229 0.00129945
$$889$$ 53.3914 1.79069
$$890$$ 0 0
$$891$$ −5.90869 −0.197949
$$892$$ 0.277771 0.00930046
$$893$$ 12.8131 0.428774
$$894$$ −25.1386 −0.840762
$$895$$ 0 0
$$896$$ 49.8994 1.66702
$$897$$ 1.92272 0.0641977
$$898$$ −56.2252 −1.87626
$$899$$ −10.5981 −0.353465
$$900$$ 0 0
$$901$$ 7.61988 0.253855
$$902$$ −14.8426 −0.494205
$$903$$ 5.01787 0.166984
$$904$$ −19.8442 −0.660007
$$905$$ 0 0
$$906$$ 29.7511 0.988415
$$907$$ −9.00465 −0.298995 −0.149497 0.988762i $$-0.547766\pi$$
−0.149497 + 0.988762i $$0.547766\pi$$
$$908$$ 8.72672 0.289606
$$909$$ −6.87495 −0.228028
$$910$$ 0 0
$$911$$ 29.9503 0.992298 0.496149 0.868237i $$-0.334747\pi$$
0.496149 + 0.868237i $$0.334747\pi$$
$$912$$ −11.7723 −0.389819
$$913$$ 4.61847 0.152849
$$914$$ 39.0416 1.29138
$$915$$ 0 0
$$916$$ 11.0578 0.365360
$$917$$ 18.1051 0.597882
$$918$$ 4.61803 0.152418
$$919$$ 8.93560 0.294758 0.147379 0.989080i $$-0.452916\pi$$
0.147379 + 0.989080i $$0.452916\pi$$
$$920$$ 0 0
$$921$$ −28.5593 −0.941062
$$922$$ 20.5489 0.676741
$$923$$ 26.6589 0.877487
$$924$$ 21.2370 0.698647
$$925$$ 0 0
$$926$$ 46.4133 1.52524
$$927$$ −11.7532 −0.386024
$$928$$ 18.7930 0.616910
$$929$$ −41.4596 −1.36025 −0.680123 0.733098i $$-0.738074\pi$$
−0.680123 + 0.733098i $$0.738074\pi$$
$$930$$ 0 0
$$931$$ −20.1450 −0.660225
$$932$$ −18.3248 −0.600250
$$933$$ 29.3283 0.960167
$$934$$ 6.72721 0.220121
$$935$$ 0 0
$$936$$ 6.11102 0.199745
$$937$$ −20.8585 −0.681417 −0.340709 0.940169i $$-0.610667\pi$$
−0.340709 + 0.940169i $$0.610667\pi$$
$$938$$ −40.5702 −1.32466
$$939$$ −17.4933 −0.570871
$$940$$ 0 0
$$941$$ 3.67382 0.119763 0.0598816 0.998205i $$-0.480928\pi$$
0.0598816 + 0.998205i $$0.480928\pi$$
$$942$$ 30.5619 0.995759
$$943$$ 0.860163 0.0280107
$$944$$ 23.4151 0.762096
$$945$$ 0 0
$$946$$ 12.8327 0.417227
$$947$$ 31.6997 1.03010 0.515051 0.857160i $$-0.327773\pi$$
0.515051 + 0.857160i $$0.327773\pi$$
$$948$$ −15.1928 −0.493441
$$949$$ 43.8204 1.42247
$$950$$ 0 0
$$951$$ 3.91763 0.127038
$$952$$ 19.8144 0.642189
$$953$$ −25.6214 −0.829960 −0.414980 0.909831i $$-0.636211\pi$$
−0.414980 + 0.909831i $$0.636211\pi$$
$$954$$ −4.80433 −0.155546
$$955$$ 0 0
$$956$$ 16.1035 0.520824
$$957$$ −23.1129 −0.747133
$$958$$ 34.0407 1.09980
$$959$$ −60.1694 −1.94297
$$960$$ 0 0
$$961$$ −23.6595 −0.763209
$$962$$ −0.117082 −0.00377488
$$963$$ −5.66780 −0.182642
$$964$$ 26.9296 0.867345
$$965$$ 0 0
$$966$$ −3.93068 −0.126468
$$967$$ 29.0166 0.933112 0.466556 0.884492i $$-0.345495\pi$$
0.466556 + 0.884492i $$0.345495\pi$$
$$968$$ −44.4077 −1.42732
$$969$$ −6.38197 −0.205018
$$970$$ 0 0
$$971$$ −3.77287 −0.121077 −0.0605386 0.998166i $$-0.519282\pi$$
−0.0605386 + 0.998166i $$0.519282\pi$$
$$972$$ −0.911672 −0.0292419
$$973$$ 71.9027 2.30510
$$974$$ 18.7739 0.601553
$$975$$ 0 0
$$976$$ −27.8630 −0.891874
$$977$$ −36.7167 −1.17467 −0.587336 0.809343i $$-0.699823\pi$$
−0.587336 + 0.809343i $$0.699823\pi$$
$$978$$ 37.1301 1.18729
$$979$$ 20.5225 0.655902
$$980$$ 0 0
$$981$$ −1.36128 −0.0434622
$$982$$ −56.8104 −1.81289
$$983$$ −23.5627 −0.751534 −0.375767 0.926714i $$-0.622621\pi$$
−0.375767 + 0.926714i $$0.622621\pi$$
$$984$$ 2.73388 0.0871528
$$985$$ 0 0
$$986$$ 18.0642 0.575282
$$987$$ 21.4215 0.681854
$$988$$ 7.07440 0.225067
$$989$$ −0.743684 −0.0236478
$$990$$ 0 0
$$991$$ 29.1375 0.925583 0.462791 0.886467i $$-0.346848\pi$$
0.462791 + 0.886467i $$0.346848\pi$$
$$992$$ −13.0166 −0.413277
$$993$$ 17.5534 0.557039
$$994$$ −54.4997 −1.72863
$$995$$ 0 0
$$996$$ 0.712600 0.0225796
$$997$$ −43.9240 −1.39109 −0.695544 0.718484i $$-0.744836\pi$$
−0.695544 + 0.718484i $$0.744836\pi$$
$$998$$ −70.2078 −2.22239
$$999$$ −0.0208515 −0.000659711 0
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.e.1.4 4
3.2 odd 2 5625.2.a.n.1.1 4
5.2 odd 4 1875.2.b.c.1249.6 8
5.3 odd 4 1875.2.b.c.1249.3 8
5.4 even 2 1875.2.a.h.1.1 4
15.14 odd 2 5625.2.a.i.1.4 4
25.3 odd 20 375.2.i.b.49.2 16
25.4 even 10 75.2.g.b.16.2 8
25.6 even 5 375.2.g.b.301.1 8
25.8 odd 20 375.2.i.b.199.3 16
25.17 odd 20 375.2.i.b.199.2 16
25.19 even 10 75.2.g.b.61.2 yes 8
25.21 even 5 375.2.g.b.76.1 8
25.22 odd 20 375.2.i.b.49.3 16
75.29 odd 10 225.2.h.c.91.1 8
75.44 odd 10 225.2.h.c.136.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.16.2 8 25.4 even 10
75.2.g.b.61.2 yes 8 25.19 even 10
225.2.h.c.91.1 8 75.29 odd 10
225.2.h.c.136.1 8 75.44 odd 10
375.2.g.b.76.1 8 25.21 even 5
375.2.g.b.301.1 8 25.6 even 5
375.2.i.b.49.2 16 25.3 odd 20
375.2.i.b.49.3 16 25.22 odd 20
375.2.i.b.199.2 16 25.17 odd 20
375.2.i.b.199.3 16 25.8 odd 20
1875.2.a.e.1.4 4 1.1 even 1 trivial
1875.2.a.h.1.1 4 5.4 even 2
1875.2.b.c.1249.3 8 5.3 odd 4
1875.2.b.c.1249.6 8 5.2 odd 4
5625.2.a.i.1.4 4 15.14 odd 2
5625.2.a.n.1.1 4 3.2 odd 2