# Properties

 Label 1875.2.a.m.1.7 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $1$ Dimension $8$ 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: $$8$$ 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 1.7 Root $$2.35083$$ 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)

 $$f(q)$$ $$=$$ $$q+1.35083 q^{2} +1.00000 q^{3} -0.175259 q^{4} +1.35083 q^{6} -1.59580 q^{7} -2.93840 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.35083 q^{2} +1.00000 q^{3} -0.175259 q^{4} +1.35083 q^{6} -1.59580 q^{7} -2.93840 q^{8} +1.00000 q^{9} +3.33277 q^{11} -0.175259 q^{12} -7.05132 q^{13} -2.15565 q^{14} -3.61877 q^{16} -4.09625 q^{17} +1.35083 q^{18} +0.567535 q^{19} -1.59580 q^{21} +4.50200 q^{22} -6.30400 q^{23} -2.93840 q^{24} -9.52513 q^{26} +1.00000 q^{27} +0.279678 q^{28} -2.78357 q^{29} -0.995824 q^{31} +0.988473 q^{32} +3.33277 q^{33} -5.53333 q^{34} -0.175259 q^{36} +3.55334 q^{37} +0.766643 q^{38} -7.05132 q^{39} +1.16293 q^{41} -2.15565 q^{42} -0.117022 q^{43} -0.584098 q^{44} -8.51563 q^{46} -7.64173 q^{47} -3.61877 q^{48} -4.45343 q^{49} -4.09625 q^{51} +1.23581 q^{52} -0.523635 q^{53} +1.35083 q^{54} +4.68910 q^{56} +0.567535 q^{57} -3.76013 q^{58} +0.983998 q^{59} +10.6137 q^{61} -1.34519 q^{62} -1.59580 q^{63} +8.57279 q^{64} +4.50200 q^{66} +15.2159 q^{67} +0.717905 q^{68} -6.30400 q^{69} -10.6639 q^{71} -2.93840 q^{72} -5.55832 q^{73} +4.79996 q^{74} -0.0994657 q^{76} -5.31842 q^{77} -9.52513 q^{78} -14.5969 q^{79} +1.00000 q^{81} +1.57091 q^{82} -5.02398 q^{83} +0.279678 q^{84} -0.158076 q^{86} -2.78357 q^{87} -9.79302 q^{88} +2.82350 q^{89} +11.2525 q^{91} +1.10483 q^{92} -0.995824 q^{93} -10.3227 q^{94} +0.988473 q^{96} +1.70592 q^{97} -6.01583 q^{98} +3.33277 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.35083 0.955181 0.477590 0.878583i $$-0.341510\pi$$
0.477590 + 0.878583i $$0.341510\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −0.175259 −0.0876296
$$5$$ 0 0
$$6$$ 1.35083 0.551474
$$7$$ −1.59580 −0.603155 −0.301577 0.953442i $$-0.597513\pi$$
−0.301577 + 0.953442i $$0.597513\pi$$
$$8$$ −2.93840 −1.03888
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.33277 1.00487 0.502434 0.864616i $$-0.332438\pi$$
0.502434 + 0.864616i $$0.332438\pi$$
$$12$$ −0.175259 −0.0505930
$$13$$ −7.05132 −1.95568 −0.977842 0.209345i $$-0.932867\pi$$
−0.977842 + 0.209345i $$0.932867\pi$$
$$14$$ −2.15565 −0.576122
$$15$$ 0 0
$$16$$ −3.61877 −0.904691
$$17$$ −4.09625 −0.993486 −0.496743 0.867898i $$-0.665471\pi$$
−0.496743 + 0.867898i $$0.665471\pi$$
$$18$$ 1.35083 0.318394
$$19$$ 0.567535 0.130201 0.0651007 0.997879i $$-0.479263\pi$$
0.0651007 + 0.997879i $$0.479263\pi$$
$$20$$ 0 0
$$21$$ −1.59580 −0.348231
$$22$$ 4.50200 0.959830
$$23$$ −6.30400 −1.31448 −0.657238 0.753683i $$-0.728275\pi$$
−0.657238 + 0.753683i $$0.728275\pi$$
$$24$$ −2.93840 −0.599799
$$25$$ 0 0
$$26$$ −9.52513 −1.86803
$$27$$ 1.00000 0.192450
$$28$$ 0.279678 0.0528542
$$29$$ −2.78357 −0.516897 −0.258448 0.966025i $$-0.583211\pi$$
−0.258448 + 0.966025i $$0.583211\pi$$
$$30$$ 0 0
$$31$$ −0.995824 −0.178855 −0.0894276 0.995993i $$-0.528504\pi$$
−0.0894276 + 0.995993i $$0.528504\pi$$
$$32$$ 0.988473 0.174739
$$33$$ 3.33277 0.580161
$$34$$ −5.53333 −0.948959
$$35$$ 0 0
$$36$$ −0.175259 −0.0292099
$$37$$ 3.55334 0.584165 0.292083 0.956393i $$-0.405652\pi$$
0.292083 + 0.956393i $$0.405652\pi$$
$$38$$ 0.766643 0.124366
$$39$$ −7.05132 −1.12911
$$40$$ 0 0
$$41$$ 1.16293 0.181618 0.0908092 0.995868i $$-0.471055\pi$$
0.0908092 + 0.995868i $$0.471055\pi$$
$$42$$ −2.15565 −0.332624
$$43$$ −0.117022 −0.0178456 −0.00892281 0.999960i $$-0.502840\pi$$
−0.00892281 + 0.999960i $$0.502840\pi$$
$$44$$ −0.584098 −0.0880561
$$45$$ 0 0
$$46$$ −8.51563 −1.25556
$$47$$ −7.64173 −1.11466 −0.557331 0.830291i $$-0.688174\pi$$
−0.557331 + 0.830291i $$0.688174\pi$$
$$48$$ −3.61877 −0.522324
$$49$$ −4.45343 −0.636205
$$50$$ 0 0
$$51$$ −4.09625 −0.573589
$$52$$ 1.23581 0.171376
$$53$$ −0.523635 −0.0719268 −0.0359634 0.999353i $$-0.511450\pi$$
−0.0359634 + 0.999353i $$0.511450\pi$$
$$54$$ 1.35083 0.183825
$$55$$ 0 0
$$56$$ 4.68910 0.626607
$$57$$ 0.567535 0.0751718
$$58$$ −3.76013 −0.493730
$$59$$ 0.983998 0.128106 0.0640528 0.997947i $$-0.479597\pi$$
0.0640528 + 0.997947i $$0.479597\pi$$
$$60$$ 0 0
$$61$$ 10.6137 1.35895 0.679473 0.733701i $$-0.262208\pi$$
0.679473 + 0.733701i $$0.262208\pi$$
$$62$$ −1.34519 −0.170839
$$63$$ −1.59580 −0.201052
$$64$$ 8.57279 1.07160
$$65$$ 0 0
$$66$$ 4.50200 0.554158
$$67$$ 15.2159 1.85892 0.929461 0.368920i $$-0.120272\pi$$
0.929461 + 0.368920i $$0.120272\pi$$
$$68$$ 0.717905 0.0870588
$$69$$ −6.30400 −0.758913
$$70$$ 0 0
$$71$$ −10.6639 −1.26558 −0.632788 0.774325i $$-0.718090\pi$$
−0.632788 + 0.774325i $$0.718090\pi$$
$$72$$ −2.93840 −0.346294
$$73$$ −5.55832 −0.650552 −0.325276 0.945619i $$-0.605457\pi$$
−0.325276 + 0.945619i $$0.605457\pi$$
$$74$$ 4.79996 0.557984
$$75$$ 0 0
$$76$$ −0.0994657 −0.0114095
$$77$$ −5.31842 −0.606090
$$78$$ −9.52513 −1.07851
$$79$$ −14.5969 −1.64227 −0.821137 0.570731i $$-0.806660\pi$$
−0.821137 + 0.570731i $$0.806660\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 1.57091 0.173478
$$83$$ −5.02398 −0.551453 −0.275727 0.961236i $$-0.588918\pi$$
−0.275727 + 0.961236i $$0.588918\pi$$
$$84$$ 0.279678 0.0305154
$$85$$ 0 0
$$86$$ −0.158076 −0.0170458
$$87$$ −2.78357 −0.298430
$$88$$ −9.79302 −1.04394
$$89$$ 2.82350 0.299291 0.149645 0.988740i $$-0.452187\pi$$
0.149645 + 0.988740i $$0.452187\pi$$
$$90$$ 0 0
$$91$$ 11.2525 1.17958
$$92$$ 1.10483 0.115187
$$93$$ −0.995824 −0.103262
$$94$$ −10.3227 −1.06470
$$95$$ 0 0
$$96$$ 0.988473 0.100886
$$97$$ 1.70592 0.173210 0.0866049 0.996243i $$-0.472398\pi$$
0.0866049 + 0.996243i $$0.472398\pi$$
$$98$$ −6.01583 −0.607690
$$99$$ 3.33277 0.334956
$$100$$ 0 0
$$101$$ 13.1747 1.31093 0.655464 0.755226i $$-0.272473\pi$$
0.655464 + 0.755226i $$0.272473\pi$$
$$102$$ −5.53333 −0.547882
$$103$$ −10.8720 −1.07125 −0.535624 0.844456i $$-0.679924\pi$$
−0.535624 + 0.844456i $$0.679924\pi$$
$$104$$ 20.7196 2.03173
$$105$$ 0 0
$$106$$ −0.707341 −0.0687031
$$107$$ −9.37236 −0.906060 −0.453030 0.891495i $$-0.649657\pi$$
−0.453030 + 0.891495i $$0.649657\pi$$
$$108$$ −0.175259 −0.0168643
$$109$$ −15.5899 −1.49324 −0.746622 0.665249i $$-0.768326\pi$$
−0.746622 + 0.665249i $$0.768326\pi$$
$$110$$ 0 0
$$111$$ 3.55334 0.337268
$$112$$ 5.77482 0.545669
$$113$$ 10.0481 0.945243 0.472622 0.881265i $$-0.343308\pi$$
0.472622 + 0.881265i $$0.343308\pi$$
$$114$$ 0.766643 0.0718027
$$115$$ 0 0
$$116$$ 0.487847 0.0452954
$$117$$ −7.05132 −0.651895
$$118$$ 1.32921 0.122364
$$119$$ 6.53678 0.599226
$$120$$ 0 0
$$121$$ 0.107347 0.00975880
$$122$$ 14.3373 1.29804
$$123$$ 1.16293 0.104857
$$124$$ 0.174527 0.0156730
$$125$$ 0 0
$$126$$ −2.15565 −0.192041
$$127$$ −0.976784 −0.0866756 −0.0433378 0.999060i $$-0.513799\pi$$
−0.0433378 + 0.999060i $$0.513799\pi$$
$$128$$ 9.60343 0.848832
$$129$$ −0.117022 −0.0103032
$$130$$ 0 0
$$131$$ 10.0616 0.879086 0.439543 0.898221i $$-0.355140\pi$$
0.439543 + 0.898221i $$0.355140\pi$$
$$132$$ −0.584098 −0.0508392
$$133$$ −0.905670 −0.0785316
$$134$$ 20.5541 1.77561
$$135$$ 0 0
$$136$$ 12.0364 1.03212
$$137$$ −4.87244 −0.416281 −0.208140 0.978099i $$-0.566741\pi$$
−0.208140 + 0.978099i $$0.566741\pi$$
$$138$$ −8.51563 −0.724899
$$139$$ 0.185784 0.0157580 0.00787898 0.999969i $$-0.497492\pi$$
0.00787898 + 0.999969i $$0.497492\pi$$
$$140$$ 0 0
$$141$$ −7.64173 −0.643550
$$142$$ −14.4052 −1.20885
$$143$$ −23.5004 −1.96520
$$144$$ −3.61877 −0.301564
$$145$$ 0 0
$$146$$ −7.50834 −0.621395
$$147$$ −4.45343 −0.367313
$$148$$ −0.622755 −0.0511902
$$149$$ 3.88889 0.318590 0.159295 0.987231i $$-0.449078\pi$$
0.159295 + 0.987231i $$0.449078\pi$$
$$150$$ 0 0
$$151$$ −22.1146 −1.79966 −0.899829 0.436242i $$-0.856309\pi$$
−0.899829 + 0.436242i $$0.856309\pi$$
$$152$$ −1.66765 −0.135264
$$153$$ −4.09625 −0.331162
$$154$$ −7.18428 −0.578926
$$155$$ 0 0
$$156$$ 1.23581 0.0989438
$$157$$ 13.6058 1.08586 0.542931 0.839777i $$-0.317314\pi$$
0.542931 + 0.839777i $$0.317314\pi$$
$$158$$ −19.7179 −1.56867
$$159$$ −0.523635 −0.0415269
$$160$$ 0 0
$$161$$ 10.0599 0.792832
$$162$$ 1.35083 0.106131
$$163$$ 8.62895 0.675871 0.337936 0.941169i $$-0.390271\pi$$
0.337936 + 0.941169i $$0.390271\pi$$
$$164$$ −0.203813 −0.0159151
$$165$$ 0 0
$$166$$ −6.78654 −0.526737
$$167$$ 6.59891 0.510639 0.255319 0.966857i $$-0.417819\pi$$
0.255319 + 0.966857i $$0.417819\pi$$
$$168$$ 4.68910 0.361772
$$169$$ 36.7211 2.82470
$$170$$ 0 0
$$171$$ 0.567535 0.0434005
$$172$$ 0.0205091 0.00156381
$$173$$ 13.6595 1.03851 0.519257 0.854618i $$-0.326209\pi$$
0.519257 + 0.854618i $$0.326209\pi$$
$$174$$ −3.76013 −0.285055
$$175$$ 0 0
$$176$$ −12.0605 −0.909095
$$177$$ 0.983998 0.0739618
$$178$$ 3.81407 0.285877
$$179$$ 9.82880 0.734639 0.367320 0.930095i $$-0.380276\pi$$
0.367320 + 0.930095i $$0.380276\pi$$
$$180$$ 0 0
$$181$$ 17.8687 1.32817 0.664085 0.747657i $$-0.268821\pi$$
0.664085 + 0.747657i $$0.268821\pi$$
$$182$$ 15.2002 1.12671
$$183$$ 10.6137 0.784588
$$184$$ 18.5237 1.36559
$$185$$ 0 0
$$186$$ −1.34519 −0.0986340
$$187$$ −13.6518 −0.998322
$$188$$ 1.33928 0.0976773
$$189$$ −1.59580 −0.116077
$$190$$ 0 0
$$191$$ 0.325813 0.0235750 0.0117875 0.999931i $$-0.496248\pi$$
0.0117875 + 0.999931i $$0.496248\pi$$
$$192$$ 8.57279 0.618688
$$193$$ −2.90187 −0.208881 −0.104441 0.994531i $$-0.533305\pi$$
−0.104441 + 0.994531i $$0.533305\pi$$
$$194$$ 2.30441 0.165447
$$195$$ 0 0
$$196$$ 0.780505 0.0557503
$$197$$ −18.1220 −1.29114 −0.645568 0.763702i $$-0.723380\pi$$
−0.645568 + 0.763702i $$0.723380\pi$$
$$198$$ 4.50200 0.319943
$$199$$ 1.53256 0.108640 0.0543201 0.998524i $$-0.482701\pi$$
0.0543201 + 0.998524i $$0.482701\pi$$
$$200$$ 0 0
$$201$$ 15.2159 1.07325
$$202$$ 17.7967 1.25217
$$203$$ 4.44202 0.311769
$$204$$ 0.717905 0.0502634
$$205$$ 0 0
$$206$$ −14.6862 −1.02324
$$207$$ −6.30400 −0.438158
$$208$$ 25.5171 1.76929
$$209$$ 1.89146 0.130835
$$210$$ 0 0
$$211$$ −11.3698 −0.782727 −0.391363 0.920236i $$-0.627996\pi$$
−0.391363 + 0.920236i $$0.627996\pi$$
$$212$$ 0.0917718 0.00630291
$$213$$ −10.6639 −0.730681
$$214$$ −12.6605 −0.865451
$$215$$ 0 0
$$216$$ −2.93840 −0.199933
$$217$$ 1.58913 0.107877
$$218$$ −21.0593 −1.42632
$$219$$ −5.55832 −0.375596
$$220$$ 0 0
$$221$$ 28.8839 1.94294
$$222$$ 4.79996 0.322152
$$223$$ 16.9507 1.13510 0.567550 0.823339i $$-0.307891\pi$$
0.567550 + 0.823339i $$0.307891\pi$$
$$224$$ −1.57740 −0.105395
$$225$$ 0 0
$$226$$ 13.5732 0.902878
$$227$$ 14.1117 0.936627 0.468314 0.883562i $$-0.344862\pi$$
0.468314 + 0.883562i $$0.344862\pi$$
$$228$$ −0.0994657 −0.00658728
$$229$$ 0.0619945 0.00409671 0.00204835 0.999998i $$-0.499348\pi$$
0.00204835 + 0.999998i $$0.499348\pi$$
$$230$$ 0 0
$$231$$ −5.31842 −0.349926
$$232$$ 8.17927 0.536995
$$233$$ −26.0191 −1.70457 −0.852283 0.523081i $$-0.824782\pi$$
−0.852283 + 0.523081i $$0.824782\pi$$
$$234$$ −9.52513 −0.622677
$$235$$ 0 0
$$236$$ −0.172455 −0.0112258
$$237$$ −14.5969 −0.948168
$$238$$ 8.83008 0.572369
$$239$$ 19.4970 1.26116 0.630579 0.776125i $$-0.282817\pi$$
0.630579 + 0.776125i $$0.282817\pi$$
$$240$$ 0 0
$$241$$ −4.09860 −0.264014 −0.132007 0.991249i $$-0.542142\pi$$
−0.132007 + 0.991249i $$0.542142\pi$$
$$242$$ 0.145007 0.00932141
$$243$$ 1.00000 0.0641500
$$244$$ −1.86015 −0.119084
$$245$$ 0 0
$$246$$ 1.57091 0.100158
$$247$$ −4.00187 −0.254633
$$248$$ 2.92613 0.185810
$$249$$ −5.02398 −0.318382
$$250$$ 0 0
$$251$$ −1.02933 −0.0649704 −0.0324852 0.999472i $$-0.510342\pi$$
−0.0324852 + 0.999472i $$0.510342\pi$$
$$252$$ 0.279678 0.0176181
$$253$$ −21.0098 −1.32087
$$254$$ −1.31947 −0.0827909
$$255$$ 0 0
$$256$$ −4.17298 −0.260811
$$257$$ −18.5597 −1.15772 −0.578862 0.815426i $$-0.696503\pi$$
−0.578862 + 0.815426i $$0.696503\pi$$
$$258$$ −0.158076 −0.00984140
$$259$$ −5.67041 −0.352342
$$260$$ 0 0
$$261$$ −2.78357 −0.172299
$$262$$ 13.5915 0.839686
$$263$$ 12.3938 0.764231 0.382116 0.924114i $$-0.375196\pi$$
0.382116 + 0.924114i $$0.375196\pi$$
$$264$$ −9.79302 −0.602719
$$265$$ 0 0
$$266$$ −1.22341 −0.0750119
$$267$$ 2.82350 0.172796
$$268$$ −2.66673 −0.162897
$$269$$ −5.30032 −0.323166 −0.161583 0.986859i $$-0.551660\pi$$
−0.161583 + 0.986859i $$0.551660\pi$$
$$270$$ 0 0
$$271$$ 0.797428 0.0484403 0.0242201 0.999707i $$-0.492290\pi$$
0.0242201 + 0.999707i $$0.492290\pi$$
$$272$$ 14.8234 0.898798
$$273$$ 11.2525 0.681031
$$274$$ −6.58184 −0.397624
$$275$$ 0 0
$$276$$ 1.10483 0.0665032
$$277$$ −4.74425 −0.285054 −0.142527 0.989791i $$-0.545523\pi$$
−0.142527 + 0.989791i $$0.545523\pi$$
$$278$$ 0.250962 0.0150517
$$279$$ −0.995824 −0.0596184
$$280$$ 0 0
$$281$$ −18.7398 −1.11792 −0.558961 0.829194i $$-0.688800\pi$$
−0.558961 + 0.829194i $$0.688800\pi$$
$$282$$ −10.3227 −0.614707
$$283$$ 11.7612 0.699130 0.349565 0.936912i $$-0.386329\pi$$
0.349565 + 0.936912i $$0.386329\pi$$
$$284$$ 1.86895 0.110902
$$285$$ 0 0
$$286$$ −31.7451 −1.87712
$$287$$ −1.85579 −0.109544
$$288$$ 0.988473 0.0582463
$$289$$ −0.220750 −0.0129853
$$290$$ 0 0
$$291$$ 1.70592 0.100003
$$292$$ 0.974146 0.0570076
$$293$$ 22.2819 1.30172 0.650860 0.759198i $$-0.274408\pi$$
0.650860 + 0.759198i $$0.274408\pi$$
$$294$$ −6.01583 −0.350850
$$295$$ 0 0
$$296$$ −10.4411 −0.606879
$$297$$ 3.33277 0.193387
$$298$$ 5.25322 0.304311
$$299$$ 44.4515 2.57070
$$300$$ 0 0
$$301$$ 0.186743 0.0107637
$$302$$ −29.8730 −1.71900
$$303$$ 13.1747 0.756865
$$304$$ −2.05378 −0.117792
$$305$$ 0 0
$$306$$ −5.53333 −0.316320
$$307$$ −15.3063 −0.873574 −0.436787 0.899565i $$-0.643884\pi$$
−0.436787 + 0.899565i $$0.643884\pi$$
$$308$$ 0.932102 0.0531115
$$309$$ −10.8720 −0.618486
$$310$$ 0 0
$$311$$ −12.8545 −0.728913 −0.364456 0.931220i $$-0.618745\pi$$
−0.364456 + 0.931220i $$0.618745\pi$$
$$312$$ 20.7196 1.17302
$$313$$ −10.5072 −0.593901 −0.296951 0.954893i $$-0.595970\pi$$
−0.296951 + 0.954893i $$0.595970\pi$$
$$314$$ 18.3791 1.03720
$$315$$ 0 0
$$316$$ 2.55823 0.143912
$$317$$ −19.4806 −1.09414 −0.547071 0.837086i $$-0.684257\pi$$
−0.547071 + 0.837086i $$0.684257\pi$$
$$318$$ −0.707341 −0.0396657
$$319$$ −9.27701 −0.519413
$$320$$ 0 0
$$321$$ −9.37236 −0.523114
$$322$$ 13.5892 0.757298
$$323$$ −2.32476 −0.129353
$$324$$ −0.175259 −0.00973662
$$325$$ 0 0
$$326$$ 11.6562 0.645579
$$327$$ −15.5899 −0.862125
$$328$$ −3.41714 −0.188680
$$329$$ 12.1947 0.672313
$$330$$ 0 0
$$331$$ −14.4925 −0.796579 −0.398289 0.917260i $$-0.630396\pi$$
−0.398289 + 0.917260i $$0.630396\pi$$
$$332$$ 0.880498 0.0483236
$$333$$ 3.55334 0.194722
$$334$$ 8.91400 0.487753
$$335$$ 0 0
$$336$$ 5.77482 0.315042
$$337$$ 9.33225 0.508360 0.254180 0.967157i $$-0.418194\pi$$
0.254180 + 0.967157i $$0.418194\pi$$
$$338$$ 49.6039 2.69810
$$339$$ 10.0481 0.545737
$$340$$ 0 0
$$341$$ −3.31885 −0.179726
$$342$$ 0.766643 0.0414553
$$343$$ 18.2774 0.986884
$$344$$ 0.343857 0.0185395
$$345$$ 0 0
$$346$$ 18.4517 0.991968
$$347$$ −1.05341 −0.0565499 −0.0282750 0.999600i $$-0.509001\pi$$
−0.0282750 + 0.999600i $$0.509001\pi$$
$$348$$ 0.487847 0.0261513
$$349$$ −13.0715 −0.699700 −0.349850 0.936806i $$-0.613767\pi$$
−0.349850 + 0.936806i $$0.613767\pi$$
$$350$$ 0 0
$$351$$ −7.05132 −0.376371
$$352$$ 3.29435 0.175590
$$353$$ 33.9473 1.80683 0.903415 0.428767i $$-0.141052\pi$$
0.903415 + 0.428767i $$0.141052\pi$$
$$354$$ 1.32921 0.0706469
$$355$$ 0 0
$$356$$ −0.494845 −0.0262267
$$357$$ 6.53678 0.345963
$$358$$ 13.2770 0.701713
$$359$$ −6.09450 −0.321656 −0.160828 0.986982i $$-0.551416\pi$$
−0.160828 + 0.986982i $$0.551416\pi$$
$$360$$ 0 0
$$361$$ −18.6779 −0.983048
$$362$$ 24.1376 1.26864
$$363$$ 0.107347 0.00563424
$$364$$ −1.97210 −0.103366
$$365$$ 0 0
$$366$$ 14.3373 0.749423
$$367$$ −21.8636 −1.14127 −0.570636 0.821203i $$-0.693303\pi$$
−0.570636 + 0.821203i $$0.693303\pi$$
$$368$$ 22.8127 1.18919
$$369$$ 1.16293 0.0605395
$$370$$ 0 0
$$371$$ 0.835615 0.0433830
$$372$$ 0.174527 0.00904882
$$373$$ −24.4559 −1.26628 −0.633140 0.774037i $$-0.718234\pi$$
−0.633140 + 0.774037i $$0.718234\pi$$
$$374$$ −18.4413 −0.953578
$$375$$ 0 0
$$376$$ 22.4545 1.15800
$$377$$ 19.6279 1.01089
$$378$$ −2.15565 −0.110875
$$379$$ −6.27821 −0.322490 −0.161245 0.986914i $$-0.551551\pi$$
−0.161245 + 0.986914i $$0.551551\pi$$
$$380$$ 0 0
$$381$$ −0.976784 −0.0500422
$$382$$ 0.440118 0.0225184
$$383$$ 24.6876 1.26148 0.630738 0.775996i $$-0.282752\pi$$
0.630738 + 0.775996i $$0.282752\pi$$
$$384$$ 9.60343 0.490073
$$385$$ 0 0
$$386$$ −3.91993 −0.199519
$$387$$ −0.117022 −0.00594854
$$388$$ −0.298978 −0.0151783
$$389$$ −12.9236 −0.655251 −0.327626 0.944808i $$-0.606248\pi$$
−0.327626 + 0.944808i $$0.606248\pi$$
$$390$$ 0 0
$$391$$ 25.8228 1.30591
$$392$$ 13.0860 0.660942
$$393$$ 10.0616 0.507541
$$394$$ −24.4797 −1.23327
$$395$$ 0 0
$$396$$ −0.584098 −0.0293520
$$397$$ −29.0214 −1.45654 −0.728272 0.685288i $$-0.759676\pi$$
−0.728272 + 0.685288i $$0.759676\pi$$
$$398$$ 2.07022 0.103771
$$399$$ −0.905670 −0.0453402
$$400$$ 0 0
$$401$$ 23.3926 1.16817 0.584084 0.811693i $$-0.301454\pi$$
0.584084 + 0.811693i $$0.301454\pi$$
$$402$$ 20.5541 1.02515
$$403$$ 7.02187 0.349784
$$404$$ −2.30898 −0.114876
$$405$$ 0 0
$$406$$ 6.00041 0.297795
$$407$$ 11.8425 0.587009
$$408$$ 12.0364 0.595892
$$409$$ −15.9918 −0.790742 −0.395371 0.918521i $$-0.629384\pi$$
−0.395371 + 0.918521i $$0.629384\pi$$
$$410$$ 0 0
$$411$$ −4.87244 −0.240340
$$412$$ 1.90542 0.0938731
$$413$$ −1.57026 −0.0772675
$$414$$ −8.51563 −0.418521
$$415$$ 0 0
$$416$$ −6.97004 −0.341734
$$417$$ 0.185784 0.00909787
$$418$$ 2.55504 0.124971
$$419$$ 32.3769 1.58172 0.790858 0.611999i $$-0.209634\pi$$
0.790858 + 0.611999i $$0.209634\pi$$
$$420$$ 0 0
$$421$$ 18.0520 0.879801 0.439900 0.898047i $$-0.355014\pi$$
0.439900 + 0.898047i $$0.355014\pi$$
$$422$$ −15.3586 −0.747646
$$423$$ −7.64173 −0.371554
$$424$$ 1.53865 0.0747235
$$425$$ 0 0
$$426$$ −14.4052 −0.697932
$$427$$ −16.9373 −0.819654
$$428$$ 1.64259 0.0793977
$$429$$ −23.5004 −1.13461
$$430$$ 0 0
$$431$$ 33.1353 1.59607 0.798035 0.602612i $$-0.205873\pi$$
0.798035 + 0.602612i $$0.205873\pi$$
$$432$$ −3.61877 −0.174108
$$433$$ −22.6653 −1.08922 −0.544612 0.838688i $$-0.683323\pi$$
−0.544612 + 0.838688i $$0.683323\pi$$
$$434$$ 2.14665 0.103042
$$435$$ 0 0
$$436$$ 2.73228 0.130852
$$437$$ −3.57774 −0.171147
$$438$$ −7.50834 −0.358762
$$439$$ −8.50436 −0.405891 −0.202945 0.979190i $$-0.565051\pi$$
−0.202945 + 0.979190i $$0.565051\pi$$
$$440$$ 0 0
$$441$$ −4.45343 −0.212068
$$442$$ 39.0173 1.85586
$$443$$ 6.35768 0.302063 0.151031 0.988529i $$-0.451741\pi$$
0.151031 + 0.988529i $$0.451741\pi$$
$$444$$ −0.622755 −0.0295547
$$445$$ 0 0
$$446$$ 22.8975 1.08423
$$447$$ 3.88889 0.183938
$$448$$ −13.6804 −0.646340
$$449$$ −6.25726 −0.295298 −0.147649 0.989040i $$-0.547171\pi$$
−0.147649 + 0.989040i $$0.547171\pi$$
$$450$$ 0 0
$$451$$ 3.87576 0.182502
$$452$$ −1.76102 −0.0828313
$$453$$ −22.1146 −1.03903
$$454$$ 19.0625 0.894648
$$455$$ 0 0
$$456$$ −1.66765 −0.0780947
$$457$$ −11.0441 −0.516620 −0.258310 0.966062i $$-0.583166\pi$$
−0.258310 + 0.966062i $$0.583166\pi$$
$$458$$ 0.0837440 0.00391310
$$459$$ −4.09625 −0.191196
$$460$$ 0 0
$$461$$ −23.6622 −1.10206 −0.551029 0.834486i $$-0.685765\pi$$
−0.551029 + 0.834486i $$0.685765\pi$$
$$462$$ −7.18428 −0.334243
$$463$$ −6.22442 −0.289273 −0.144637 0.989485i $$-0.546201\pi$$
−0.144637 + 0.989485i $$0.546201\pi$$
$$464$$ 10.0731 0.467632
$$465$$ 0 0
$$466$$ −35.1473 −1.62817
$$467$$ 4.94679 0.228910 0.114455 0.993428i $$-0.463488\pi$$
0.114455 + 0.993428i $$0.463488\pi$$
$$468$$ 1.23581 0.0571252
$$469$$ −24.2815 −1.12122
$$470$$ 0 0
$$471$$ 13.6058 0.626923
$$472$$ −2.89138 −0.133087
$$473$$ −0.390006 −0.0179325
$$474$$ −19.7179 −0.905671
$$475$$ 0 0
$$476$$ −1.14563 −0.0525099
$$477$$ −0.523635 −0.0239756
$$478$$ 26.3372 1.20463
$$479$$ −30.0898 −1.37484 −0.687419 0.726261i $$-0.741256\pi$$
−0.687419 + 0.726261i $$0.741256\pi$$
$$480$$ 0 0
$$481$$ −25.0557 −1.14244
$$482$$ −5.53651 −0.252181
$$483$$ 10.0599 0.457742
$$484$$ −0.0188135 −0.000855159 0
$$485$$ 0 0
$$486$$ 1.35083 0.0612749
$$487$$ −34.2499 −1.55201 −0.776006 0.630726i $$-0.782757\pi$$
−0.776006 + 0.630726i $$0.782757\pi$$
$$488$$ −31.1874 −1.41179
$$489$$ 8.62895 0.390215
$$490$$ 0 0
$$491$$ −10.4193 −0.470218 −0.235109 0.971969i $$-0.575545\pi$$
−0.235109 + 0.971969i $$0.575545\pi$$
$$492$$ −0.203813 −0.00918861
$$493$$ 11.4022 0.513530
$$494$$ −5.40584 −0.243220
$$495$$ 0 0
$$496$$ 3.60365 0.161809
$$497$$ 17.0175 0.763338
$$498$$ −6.78654 −0.304112
$$499$$ 8.83514 0.395515 0.197757 0.980251i $$-0.436634\pi$$
0.197757 + 0.980251i $$0.436634\pi$$
$$500$$ 0 0
$$501$$ 6.59891 0.294818
$$502$$ −1.39044 −0.0620585
$$503$$ 21.3734 0.952994 0.476497 0.879176i $$-0.341906\pi$$
0.476497 + 0.879176i $$0.341906\pi$$
$$504$$ 4.68910 0.208869
$$505$$ 0 0
$$506$$ −28.3806 −1.26167
$$507$$ 36.7211 1.63084
$$508$$ 0.171190 0.00759535
$$509$$ −16.7800 −0.743759 −0.371879 0.928281i $$-0.621286\pi$$
−0.371879 + 0.928281i $$0.621286\pi$$
$$510$$ 0 0
$$511$$ 8.86994 0.392383
$$512$$ −24.8438 −1.09795
$$513$$ 0.567535 0.0250573
$$514$$ −25.0710 −1.10584
$$515$$ 0 0
$$516$$ 0.0205091 0.000902863 0
$$517$$ −25.4681 −1.12009
$$518$$ −7.65976 −0.336550
$$519$$ 13.6595 0.599586
$$520$$ 0 0
$$521$$ −4.60508 −0.201752 −0.100876 0.994899i $$-0.532165\pi$$
−0.100876 + 0.994899i $$0.532165\pi$$
$$522$$ −3.76013 −0.164577
$$523$$ 12.9634 0.566852 0.283426 0.958994i $$-0.408529\pi$$
0.283426 + 0.958994i $$0.408529\pi$$
$$524$$ −1.76339 −0.0770340
$$525$$ 0 0
$$526$$ 16.7418 0.729979
$$527$$ 4.07914 0.177690
$$528$$ −12.0605 −0.524866
$$529$$ 16.7405 0.727846
$$530$$ 0 0
$$531$$ 0.983998 0.0427019
$$532$$ 0.158727 0.00688169
$$533$$ −8.20015 −0.355188
$$534$$ 3.81407 0.165051
$$535$$ 0 0
$$536$$ −44.7106 −1.93120
$$537$$ 9.82880 0.424144
$$538$$ −7.15983 −0.308682
$$539$$ −14.8423 −0.639301
$$540$$ 0 0
$$541$$ −27.3641 −1.17647 −0.588237 0.808688i $$-0.700178\pi$$
−0.588237 + 0.808688i $$0.700178\pi$$
$$542$$ 1.07719 0.0462692
$$543$$ 17.8687 0.766819
$$544$$ −4.04903 −0.173601
$$545$$ 0 0
$$546$$ 15.2002 0.650507
$$547$$ 27.6453 1.18203 0.591015 0.806661i $$-0.298728\pi$$
0.591015 + 0.806661i $$0.298728\pi$$
$$548$$ 0.853941 0.0364785
$$549$$ 10.6137 0.452982
$$550$$ 0 0
$$551$$ −1.57978 −0.0673007
$$552$$ 18.5237 0.788422
$$553$$ 23.2936 0.990545
$$554$$ −6.40868 −0.272279
$$555$$ 0 0
$$556$$ −0.0325603 −0.00138086
$$557$$ −6.17333 −0.261572 −0.130786 0.991411i $$-0.541750\pi$$
−0.130786 + 0.991411i $$0.541750\pi$$
$$558$$ −1.34519 −0.0569464
$$559$$ 0.825156 0.0349004
$$560$$ 0 0
$$561$$ −13.6518 −0.576381
$$562$$ −25.3143 −1.06782
$$563$$ 5.69934 0.240198 0.120099 0.992762i $$-0.461679\pi$$
0.120099 + 0.992762i $$0.461679\pi$$
$$564$$ 1.33928 0.0563940
$$565$$ 0 0
$$566$$ 15.8874 0.667796
$$567$$ −1.59580 −0.0670172
$$568$$ 31.3350 1.31479
$$569$$ 20.1708 0.845605 0.422802 0.906222i $$-0.361046\pi$$
0.422802 + 0.906222i $$0.361046\pi$$
$$570$$ 0 0
$$571$$ −15.7554 −0.659341 −0.329671 0.944096i $$-0.606938\pi$$
−0.329671 + 0.944096i $$0.606938\pi$$
$$572$$ 4.11866 0.172210
$$573$$ 0.325813 0.0136111
$$574$$ −2.50686 −0.104634
$$575$$ 0 0
$$576$$ 8.57279 0.357200
$$577$$ −16.1062 −0.670509 −0.335255 0.942128i $$-0.608822\pi$$
−0.335255 + 0.942128i $$0.608822\pi$$
$$578$$ −0.298195 −0.0124033
$$579$$ −2.90187 −0.120598
$$580$$ 0 0
$$581$$ 8.01725 0.332611
$$582$$ 2.30441 0.0955207
$$583$$ −1.74515 −0.0722769
$$584$$ 16.3326 0.675847
$$585$$ 0 0
$$586$$ 30.0990 1.24338
$$587$$ 2.00072 0.0825786 0.0412893 0.999147i $$-0.486853\pi$$
0.0412893 + 0.999147i $$0.486853\pi$$
$$588$$ 0.780505 0.0321875
$$589$$ −0.565165 −0.0232872
$$590$$ 0 0
$$591$$ −18.1220 −0.745438
$$592$$ −12.8587 −0.528489
$$593$$ −26.8231 −1.10149 −0.550747 0.834672i $$-0.685657\pi$$
−0.550747 + 0.834672i $$0.685657\pi$$
$$594$$ 4.50200 0.184719
$$595$$ 0 0
$$596$$ −0.681563 −0.0279179
$$597$$ 1.53256 0.0627234
$$598$$ 60.0464 2.45548
$$599$$ −44.8025 −1.83058 −0.915290 0.402796i $$-0.868038\pi$$
−0.915290 + 0.402796i $$0.868038\pi$$
$$600$$ 0 0
$$601$$ −14.2298 −0.580446 −0.290223 0.956959i $$-0.593729\pi$$
−0.290223 + 0.956959i $$0.593729\pi$$
$$602$$ 0.252258 0.0102813
$$603$$ 15.2159 0.619641
$$604$$ 3.87578 0.157703
$$605$$ 0 0
$$606$$ 17.7967 0.722943
$$607$$ −20.3346 −0.825356 −0.412678 0.910877i $$-0.635406\pi$$
−0.412678 + 0.910877i $$0.635406\pi$$
$$608$$ 0.560993 0.0227513
$$609$$ 4.44202 0.180000
$$610$$ 0 0
$$611$$ 53.8843 2.17993
$$612$$ 0.717905 0.0290196
$$613$$ −20.7921 −0.839783 −0.419892 0.907574i $$-0.637932\pi$$
−0.419892 + 0.907574i $$0.637932\pi$$
$$614$$ −20.6761 −0.834421
$$615$$ 0 0
$$616$$ 15.6277 0.629657
$$617$$ 4.94634 0.199132 0.0995660 0.995031i $$-0.468255\pi$$
0.0995660 + 0.995031i $$0.468255\pi$$
$$618$$ −14.6862 −0.590766
$$619$$ −45.9527 −1.84700 −0.923498 0.383603i $$-0.874683\pi$$
−0.923498 + 0.383603i $$0.874683\pi$$
$$620$$ 0 0
$$621$$ −6.30400 −0.252971
$$622$$ −17.3643 −0.696243
$$623$$ −4.50574 −0.180519
$$624$$ 25.5171 1.02150
$$625$$ 0 0
$$626$$ −14.1934 −0.567283
$$627$$ 1.89146 0.0755377
$$628$$ −2.38454 −0.0951537
$$629$$ −14.5554 −0.580360
$$630$$ 0 0
$$631$$ −21.6636 −0.862414 −0.431207 0.902253i $$-0.641912\pi$$
−0.431207 + 0.902253i $$0.641912\pi$$
$$632$$ 42.8915 1.70613
$$633$$ −11.3698 −0.451908
$$634$$ −26.3150 −1.04510
$$635$$ 0 0
$$636$$ 0.0917718 0.00363899
$$637$$ 31.4026 1.24421
$$638$$ −12.5317 −0.496133
$$639$$ −10.6639 −0.421859
$$640$$ 0 0
$$641$$ 36.3870 1.43720 0.718600 0.695424i $$-0.244783\pi$$
0.718600 + 0.695424i $$0.244783\pi$$
$$642$$ −12.6605 −0.499668
$$643$$ −1.01349 −0.0399682 −0.0199841 0.999800i $$-0.506362\pi$$
−0.0199841 + 0.999800i $$0.506362\pi$$
$$644$$ −1.76309 −0.0694755
$$645$$ 0 0
$$646$$ −3.14036 −0.123556
$$647$$ 13.7007 0.538629 0.269315 0.963052i $$-0.413203\pi$$
0.269315 + 0.963052i $$0.413203\pi$$
$$648$$ −2.93840 −0.115431
$$649$$ 3.27944 0.128729
$$650$$ 0 0
$$651$$ 1.58913 0.0622830
$$652$$ −1.51230 −0.0592263
$$653$$ −23.8372 −0.932822 −0.466411 0.884568i $$-0.654453\pi$$
−0.466411 + 0.884568i $$0.654453\pi$$
$$654$$ −21.0593 −0.823485
$$655$$ 0 0
$$656$$ −4.20835 −0.164309
$$657$$ −5.55832 −0.216851
$$658$$ 16.4729 0.642181
$$659$$ 14.7758 0.575583 0.287791 0.957693i $$-0.407079\pi$$
0.287791 + 0.957693i $$0.407079\pi$$
$$660$$ 0 0
$$661$$ 7.55471 0.293844 0.146922 0.989148i $$-0.453063\pi$$
0.146922 + 0.989148i $$0.453063\pi$$
$$662$$ −19.5769 −0.760877
$$663$$ 28.8839 1.12176
$$664$$ 14.7625 0.572895
$$665$$ 0 0
$$666$$ 4.79996 0.185995
$$667$$ 17.5477 0.679448
$$668$$ −1.15652 −0.0447471
$$669$$ 16.9507 0.655351
$$670$$ 0 0
$$671$$ 35.3730 1.36556
$$672$$ −1.57740 −0.0608496
$$673$$ −12.1947 −0.470069 −0.235035 0.971987i $$-0.575520\pi$$
−0.235035 + 0.971987i $$0.575520\pi$$
$$674$$ 12.6063 0.485576
$$675$$ 0 0
$$676$$ −6.43571 −0.247527
$$677$$ 17.9507 0.689900 0.344950 0.938621i $$-0.387896\pi$$
0.344950 + 0.938621i $$0.387896\pi$$
$$678$$ 13.5732 0.521277
$$679$$ −2.72230 −0.104472
$$680$$ 0 0
$$681$$ 14.1117 0.540762
$$682$$ −4.48320 −0.171671
$$683$$ 10.0273 0.383685 0.191842 0.981426i $$-0.438554\pi$$
0.191842 + 0.981426i $$0.438554\pi$$
$$684$$ −0.0994657 −0.00380317
$$685$$ 0 0
$$686$$ 24.6896 0.942653
$$687$$ 0.0619945 0.00236524
$$688$$ 0.423474 0.0161448
$$689$$ 3.69231 0.140666
$$690$$ 0 0
$$691$$ 19.1296 0.727722 0.363861 0.931453i $$-0.381458\pi$$
0.363861 + 0.931453i $$0.381458\pi$$
$$692$$ −2.39396 −0.0910045
$$693$$ −5.31842 −0.202030
$$694$$ −1.42297 −0.0540154
$$695$$ 0 0
$$696$$ 8.17927 0.310034
$$697$$ −4.76363 −0.180435
$$698$$ −17.6573 −0.668340
$$699$$ −26.0191 −0.984131
$$700$$ 0 0
$$701$$ −3.81920 −0.144249 −0.0721246 0.997396i $$-0.522978\pi$$
−0.0721246 + 0.997396i $$0.522978\pi$$
$$702$$ −9.52513 −0.359503
$$703$$ 2.01664 0.0760592
$$704$$ 28.5711 1.07681
$$705$$ 0 0
$$706$$ 45.8570 1.72585
$$707$$ −21.0241 −0.790692
$$708$$ −0.172455 −0.00648124
$$709$$ 38.0535 1.42913 0.714565 0.699569i $$-0.246625\pi$$
0.714565 + 0.699569i $$0.246625\pi$$
$$710$$ 0 0
$$711$$ −14.5969 −0.547425
$$712$$ −8.29660 −0.310928
$$713$$ 6.27768 0.235101
$$714$$ 8.83008 0.330457
$$715$$ 0 0
$$716$$ −1.72259 −0.0643761
$$717$$ 19.4970 0.728130
$$718$$ −8.23264 −0.307239
$$719$$ 19.4340 0.724766 0.362383 0.932029i $$-0.381963\pi$$
0.362383 + 0.932029i $$0.381963\pi$$
$$720$$ 0 0
$$721$$ 17.3495 0.646129
$$722$$ −25.2307 −0.938988
$$723$$ −4.09860 −0.152429
$$724$$ −3.13165 −0.116387
$$725$$ 0 0
$$726$$ 0.145007 0.00538172
$$727$$ −38.0739 −1.41208 −0.706042 0.708170i $$-0.749521\pi$$
−0.706042 + 0.708170i $$0.749521\pi$$
$$728$$ −33.0643 −1.22544
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0.479350 0.0177294
$$732$$ −1.86015 −0.0687531
$$733$$ 16.9591 0.626400 0.313200 0.949687i $$-0.398599\pi$$
0.313200 + 0.949687i $$0.398599\pi$$
$$734$$ −29.5341 −1.09012
$$735$$ 0 0
$$736$$ −6.23134 −0.229690
$$737$$ 50.7112 1.86797
$$738$$ 1.57091 0.0578261
$$739$$ 13.2040 0.485716 0.242858 0.970062i $$-0.421915\pi$$
0.242858 + 0.970062i $$0.421915\pi$$
$$740$$ 0 0
$$741$$ −4.00187 −0.147012
$$742$$ 1.12877 0.0414386
$$743$$ 42.2364 1.54950 0.774752 0.632265i $$-0.217875\pi$$
0.774752 + 0.632265i $$0.217875\pi$$
$$744$$ 2.92613 0.107277
$$745$$ 0 0
$$746$$ −33.0358 −1.20953
$$747$$ −5.02398 −0.183818
$$748$$ 2.39261 0.0874825
$$749$$ 14.9564 0.546494
$$750$$ 0 0
$$751$$ −1.04801 −0.0382426 −0.0191213 0.999817i $$-0.506087\pi$$
−0.0191213 + 0.999817i $$0.506087\pi$$
$$752$$ 27.6536 1.00842
$$753$$ −1.02933 −0.0375107
$$754$$ 26.5139 0.965579
$$755$$ 0 0
$$756$$ 0.279678 0.0101718
$$757$$ 17.0074 0.618146 0.309073 0.951038i $$-0.399981\pi$$
0.309073 + 0.951038i $$0.399981\pi$$
$$758$$ −8.48079 −0.308036
$$759$$ −21.0098 −0.762607
$$760$$ 0 0
$$761$$ −24.1244 −0.874508 −0.437254 0.899338i $$-0.644049\pi$$
−0.437254 + 0.899338i $$0.644049\pi$$
$$762$$ −1.31947 −0.0477993
$$763$$ 24.8783 0.900657
$$764$$ −0.0571018 −0.00206587
$$765$$ 0 0
$$766$$ 33.3487 1.20494
$$767$$ −6.93848 −0.250534
$$768$$ −4.17298 −0.150579
$$769$$ 49.7819 1.79518 0.897591 0.440830i $$-0.145316\pi$$
0.897591 + 0.440830i $$0.145316\pi$$
$$770$$ 0 0
$$771$$ −18.5597 −0.668412
$$772$$ 0.508580 0.0183042
$$773$$ −31.4743 −1.13205 −0.566025 0.824388i $$-0.691520\pi$$
−0.566025 + 0.824388i $$0.691520\pi$$
$$774$$ −0.158076 −0.00568193
$$775$$ 0 0
$$776$$ −5.01268 −0.179945
$$777$$ −5.67041 −0.203425
$$778$$ −17.4575 −0.625883
$$779$$ 0.660001 0.0236470
$$780$$ 0 0
$$781$$ −35.5404 −1.27174
$$782$$ 34.8822 1.24738
$$783$$ −2.78357 −0.0994768
$$784$$ 16.1159 0.575569
$$785$$ 0 0
$$786$$ 13.5915 0.484793
$$787$$ −19.1415 −0.682321 −0.341161 0.940005i $$-0.610820\pi$$
−0.341161 + 0.940005i $$0.610820\pi$$
$$788$$ 3.17604 0.113142
$$789$$ 12.3938 0.441229
$$790$$ 0 0
$$791$$ −16.0347 −0.570128
$$792$$ −9.79302 −0.347980
$$793$$ −74.8406 −2.65767
$$794$$ −39.2030 −1.39126
$$795$$ 0 0
$$796$$ −0.268595 −0.00952009
$$797$$ −7.21437 −0.255546 −0.127773 0.991803i $$-0.540783\pi$$
−0.127773 + 0.991803i $$0.540783\pi$$
$$798$$ −1.22341 −0.0433081
$$799$$ 31.3024 1.10740
$$800$$ 0 0
$$801$$ 2.82350 0.0997636
$$802$$ 31.5994 1.11581
$$803$$ −18.5246 −0.653718
$$804$$ −2.66673 −0.0940484
$$805$$ 0 0
$$806$$ 9.48535 0.334107
$$807$$ −5.30032 −0.186580
$$808$$ −38.7125 −1.36190
$$809$$ 1.05941 0.0372469 0.0186234 0.999827i $$-0.494072\pi$$
0.0186234 + 0.999827i $$0.494072\pi$$
$$810$$ 0 0
$$811$$ 21.6400 0.759883 0.379941 0.925011i $$-0.375944\pi$$
0.379941 + 0.925011i $$0.375944\pi$$
$$812$$ −0.778504 −0.0273202
$$813$$ 0.797428 0.0279670
$$814$$ 15.9971 0.560700
$$815$$ 0 0
$$816$$ 14.8234 0.518922
$$817$$ −0.0664138 −0.00232353
$$818$$ −21.6022 −0.755302
$$819$$ 11.2525 0.393193
$$820$$ 0 0
$$821$$ −23.1828 −0.809085 −0.404543 0.914519i $$-0.632569\pi$$
−0.404543 + 0.914519i $$0.632569\pi$$
$$822$$ −6.58184 −0.229568
$$823$$ −27.9769 −0.975214 −0.487607 0.873063i $$-0.662130\pi$$
−0.487607 + 0.873063i $$0.662130\pi$$
$$824$$ 31.9463 1.11290
$$825$$ 0 0
$$826$$ −2.12116 −0.0738044
$$827$$ −41.3045 −1.43630 −0.718150 0.695889i $$-0.755011\pi$$
−0.718150 + 0.695889i $$0.755011\pi$$
$$828$$ 1.10483 0.0383956
$$829$$ 47.5687 1.65213 0.826064 0.563577i $$-0.190575\pi$$
0.826064 + 0.563577i $$0.190575\pi$$
$$830$$ 0 0
$$831$$ −4.74425 −0.164576
$$832$$ −60.4495 −2.09571
$$833$$ 18.2424 0.632060
$$834$$ 0.250962 0.00869011
$$835$$ 0 0
$$836$$ −0.331496 −0.0114650
$$837$$ −0.995824 −0.0344207
$$838$$ 43.7357 1.51083
$$839$$ 49.0322 1.69278 0.846389 0.532565i $$-0.178772\pi$$
0.846389 + 0.532565i $$0.178772\pi$$
$$840$$ 0 0
$$841$$ −21.2517 −0.732818
$$842$$ 24.3852 0.840369
$$843$$ −18.7398 −0.645433
$$844$$ 1.99266 0.0685900
$$845$$ 0 0
$$846$$ −10.3227 −0.354901
$$847$$ −0.171304 −0.00588606
$$848$$ 1.89491 0.0650715
$$849$$ 11.7612 0.403643
$$850$$ 0 0
$$851$$ −22.4003 −0.767871
$$852$$ 1.86895 0.0640293
$$853$$ −32.5376 −1.11407 −0.557033 0.830490i $$-0.688061\pi$$
−0.557033 + 0.830490i $$0.688061\pi$$
$$854$$ −22.8794 −0.782918
$$855$$ 0 0
$$856$$ 27.5398 0.941290
$$857$$ −30.4813 −1.04122 −0.520610 0.853794i $$-0.674296\pi$$
−0.520610 + 0.853794i $$0.674296\pi$$
$$858$$ −31.7451 −1.08376
$$859$$ 4.12215 0.140646 0.0703231 0.997524i $$-0.477597\pi$$
0.0703231 + 0.997524i $$0.477597\pi$$
$$860$$ 0 0
$$861$$ −1.85579 −0.0632452
$$862$$ 44.7601 1.52453
$$863$$ −18.4184 −0.626969 −0.313485 0.949593i $$-0.601496\pi$$
−0.313485 + 0.949593i $$0.601496\pi$$
$$864$$ 0.988473 0.0336285
$$865$$ 0 0
$$866$$ −30.6169 −1.04041
$$867$$ −0.220750 −0.00749705
$$868$$ −0.278510 −0.00945325
$$869$$ −48.6479 −1.65027
$$870$$ 0 0
$$871$$ −107.292 −3.63546
$$872$$ 45.8095 1.55131
$$873$$ 1.70592 0.0577366
$$874$$ −4.83292 −0.163476
$$875$$ 0 0
$$876$$ 0.974146 0.0329133
$$877$$ −18.9126 −0.638634 −0.319317 0.947648i $$-0.603453\pi$$
−0.319317 + 0.947648i $$0.603453\pi$$
$$878$$ −11.4879 −0.387699
$$879$$ 22.2819 0.751548
$$880$$ 0 0
$$881$$ 34.6336 1.16684 0.583418 0.812172i $$-0.301715\pi$$
0.583418 + 0.812172i $$0.301715\pi$$
$$882$$ −6.01583 −0.202563
$$883$$ −32.6874 −1.10002 −0.550009 0.835158i $$-0.685376\pi$$
−0.550009 + 0.835158i $$0.685376\pi$$
$$884$$ −5.06218 −0.170259
$$885$$ 0 0
$$886$$ 8.58814 0.288524
$$887$$ 48.7918 1.63827 0.819134 0.573602i $$-0.194455\pi$$
0.819134 + 0.573602i $$0.194455\pi$$
$$888$$ −10.4411 −0.350382
$$889$$ 1.55875 0.0522788
$$890$$ 0 0
$$891$$ 3.33277 0.111652
$$892$$ −2.97076 −0.0994684
$$893$$ −4.33695 −0.145131
$$894$$ 5.25322 0.175694
$$895$$ 0 0
$$896$$ −15.3251 −0.511977
$$897$$ 44.4515 1.48419
$$898$$ −8.45249 −0.282063
$$899$$ 2.77195 0.0924497
$$900$$ 0 0
$$901$$ 2.14494 0.0714582
$$902$$ 5.23549 0.174323
$$903$$ 0.186743 0.00621441
$$904$$ −29.5253 −0.981997
$$905$$ 0 0
$$906$$ −29.8730 −0.992465
$$907$$ 45.0367 1.49542 0.747710 0.664025i $$-0.231153\pi$$
0.747710 + 0.664025i $$0.231153\pi$$
$$908$$ −2.47321 −0.0820762
$$909$$ 13.1747 0.436976
$$910$$ 0 0
$$911$$ 12.9449 0.428882 0.214441 0.976737i $$-0.431207\pi$$
0.214441 + 0.976737i $$0.431207\pi$$
$$912$$ −2.05378 −0.0680073
$$913$$ −16.7438 −0.554137
$$914$$ −14.9187 −0.493466
$$915$$ 0 0
$$916$$ −0.0108651 −0.000358993 0
$$917$$ −16.0563 −0.530225
$$918$$ −5.53333 −0.182627
$$919$$ 14.5719 0.480682 0.240341 0.970689i $$-0.422741\pi$$
0.240341 + 0.970689i $$0.422741\pi$$
$$920$$ 0 0
$$921$$ −15.3063 −0.504358
$$922$$ −31.9636 −1.05266
$$923$$ 75.1948 2.47507
$$924$$ 0.932102 0.0306639
$$925$$ 0 0
$$926$$ −8.40813 −0.276308
$$927$$ −10.8720 −0.357083
$$928$$ −2.75149 −0.0903220
$$929$$ −31.3591 −1.02886 −0.514430 0.857532i $$-0.671996\pi$$
−0.514430 + 0.857532i $$0.671996\pi$$
$$930$$ 0 0
$$931$$ −2.52748 −0.0828347
$$932$$ 4.56008 0.149370
$$933$$ −12.8545 −0.420838
$$934$$ 6.68227 0.218651
$$935$$ 0 0
$$936$$ 20.7196 0.677242
$$937$$ −48.7726 −1.59333 −0.796666 0.604420i $$-0.793405\pi$$
−0.796666 + 0.604420i $$0.793405\pi$$
$$938$$ −32.8002 −1.07097
$$939$$ −10.5072 −0.342889
$$940$$ 0 0
$$941$$ −32.2359 −1.05086 −0.525431 0.850836i $$-0.676096\pi$$
−0.525431 + 0.850836i $$0.676096\pi$$
$$942$$ 18.3791 0.598825
$$943$$ −7.33108 −0.238733
$$944$$ −3.56086 −0.115896
$$945$$ 0 0
$$946$$ −0.526832 −0.0171288
$$947$$ −51.9235 −1.68729 −0.843644 0.536903i $$-0.819594\pi$$
−0.843644 + 0.536903i $$0.819594\pi$$
$$948$$ 2.55823 0.0830875
$$949$$ 39.1935 1.27227
$$950$$ 0 0
$$951$$ −19.4806 −0.631703
$$952$$ −19.2077 −0.622525
$$953$$ −39.7788 −1.28856 −0.644281 0.764789i $$-0.722843\pi$$
−0.644281 + 0.764789i $$0.722843\pi$$
$$954$$ −0.707341 −0.0229010
$$955$$ 0 0
$$956$$ −3.41703 −0.110515
$$957$$ −9.27701 −0.299883
$$958$$ −40.6462 −1.31322
$$959$$ 7.77543 0.251082
$$960$$ 0 0
$$961$$ −30.0083 −0.968011
$$962$$ −33.8460 −1.09124
$$963$$ −9.37236 −0.302020
$$964$$ 0.718317 0.0231354
$$965$$ 0 0
$$966$$ 13.5892 0.437226
$$967$$ −19.8817 −0.639353 −0.319677 0.947527i $$-0.603574\pi$$
−0.319677 + 0.947527i $$0.603574\pi$$
$$968$$ −0.315428 −0.0101382
$$969$$ −2.32476 −0.0746822
$$970$$ 0 0
$$971$$ −41.3841 −1.32808 −0.664039 0.747698i $$-0.731159\pi$$
−0.664039 + 0.747698i $$0.731159\pi$$
$$972$$ −0.175259 −0.00562144
$$973$$ −0.296473 −0.00950449
$$974$$ −46.2658 −1.48245
$$975$$ 0 0
$$976$$ −38.4085 −1.22943
$$977$$ −3.61991 −0.115811 −0.0579056 0.998322i $$-0.518442\pi$$
−0.0579056 + 0.998322i $$0.518442\pi$$
$$978$$ 11.6562 0.372725
$$979$$ 9.41009 0.300748
$$980$$ 0 0
$$981$$ −15.5899 −0.497748
$$982$$ −14.0747 −0.449143
$$983$$ 20.1099 0.641405 0.320702 0.947180i $$-0.396081\pi$$
0.320702 + 0.947180i $$0.396081\pi$$
$$984$$ −3.41714 −0.108935
$$985$$ 0 0
$$986$$ 15.4024 0.490514
$$987$$ 12.1947 0.388160
$$988$$ 0.701364 0.0223134
$$989$$ 0.737705 0.0234576
$$990$$ 0 0
$$991$$ 7.43990 0.236336 0.118168 0.992994i $$-0.462298\pi$$
0.118168 + 0.992994i $$0.462298\pi$$
$$992$$ −0.984345 −0.0312530
$$993$$ −14.4925 −0.459905
$$994$$ 22.9877 0.729126
$$995$$ 0 0
$$996$$ 0.880498 0.0278996
$$997$$ 46.1472 1.46150 0.730748 0.682648i $$-0.239172\pi$$
0.730748 + 0.682648i $$0.239172\pi$$
$$998$$ 11.9348 0.377788
$$999$$ 3.55334 0.112423
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.m.1.7 8
3.2 odd 2 5625.2.a.bd.1.2 8
5.2 odd 4 1875.2.b.h.1249.12 16
5.3 odd 4 1875.2.b.h.1249.5 16
5.4 even 2 1875.2.a.p.1.2 8
15.14 odd 2 5625.2.a.t.1.7 8
25.2 odd 20 75.2.i.a.4.3 16
25.9 even 10 375.2.g.d.151.1 16
25.11 even 5 375.2.g.e.226.4 16
25.12 odd 20 375.2.i.c.349.2 16
25.13 odd 20 75.2.i.a.19.3 yes 16
25.14 even 10 375.2.g.d.226.1 16
25.16 even 5 375.2.g.e.151.4 16
25.23 odd 20 375.2.i.c.274.2 16
75.2 even 20 225.2.m.b.154.2 16
75.38 even 20 225.2.m.b.19.2 16

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.4.3 16 25.2 odd 20
75.2.i.a.19.3 yes 16 25.13 odd 20
225.2.m.b.19.2 16 75.38 even 20
225.2.m.b.154.2 16 75.2 even 20
375.2.g.d.151.1 16 25.9 even 10
375.2.g.d.226.1 16 25.14 even 10
375.2.g.e.151.4 16 25.16 even 5
375.2.g.e.226.4 16 25.11 even 5
375.2.i.c.274.2 16 25.23 odd 20
375.2.i.c.349.2 16 25.12 odd 20
1875.2.a.m.1.7 8 1.1 even 1 trivial
1875.2.a.p.1.2 8 5.4 even 2
1875.2.b.h.1249.5 16 5.3 odd 4
1875.2.b.h.1249.12 16 5.2 odd 4
5625.2.a.t.1.7 8 15.14 odd 2
5625.2.a.bd.1.2 8 3.2 odd 2