# Properties

 Label 1875.2.a.k.1.2 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.44400625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1$$ x^6 - 11*x^4 - x^3 + 29*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 1.2 Root $$-2.16056$$ of defining polynomial Character $$\chi$$ $$=$$ 1875.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.16056 q^{2} +1.00000 q^{3} +2.66802 q^{4} -2.16056 q^{6} +3.16056 q^{7} -1.44329 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.16056 q^{2} +1.00000 q^{3} +2.66802 q^{4} -2.16056 q^{6} +3.16056 q^{7} -1.44329 q^{8} +1.00000 q^{9} +1.53835 q^{11} +2.66802 q^{12} +5.24144 q^{13} -6.82858 q^{14} -2.21771 q^{16} -1.29068 q^{17} -2.16056 q^{18} +5.44587 q^{19} +3.16056 q^{21} -3.32370 q^{22} +6.44244 q^{23} -1.44329 q^{24} -11.3244 q^{26} +1.00000 q^{27} +8.43243 q^{28} -2.36361 q^{29} -4.46542 q^{31} +7.67809 q^{32} +1.53835 q^{33} +2.78860 q^{34} +2.66802 q^{36} +5.95751 q^{37} -11.7661 q^{38} +5.24144 q^{39} -8.53219 q^{41} -6.82858 q^{42} +8.48426 q^{43} +4.10435 q^{44} -13.9193 q^{46} -0.753070 q^{47} -2.21771 q^{48} +2.98914 q^{49} -1.29068 q^{51} +13.9843 q^{52} -9.74991 q^{53} -2.16056 q^{54} -4.56162 q^{56} +5.44587 q^{57} +5.10672 q^{58} +4.11270 q^{59} -10.6939 q^{61} +9.64781 q^{62} +3.16056 q^{63} -12.1535 q^{64} -3.32370 q^{66} +1.89864 q^{67} -3.44357 q^{68} +6.44244 q^{69} -0.0708774 q^{71} -1.44329 q^{72} -4.01123 q^{73} -12.8716 q^{74} +14.5297 q^{76} +4.86205 q^{77} -11.3244 q^{78} +1.61849 q^{79} +1.00000 q^{81} +18.4343 q^{82} -13.1488 q^{83} +8.43243 q^{84} -18.3308 q^{86} -2.36361 q^{87} -2.22029 q^{88} +7.27597 q^{89} +16.5659 q^{91} +17.1885 q^{92} -4.46542 q^{93} +1.62705 q^{94} +7.67809 q^{96} -10.1939 q^{97} -6.45821 q^{98} +1.53835 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{3} + 10 q^{4} + 6 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10})$$ 6 * q + 6 * q^3 + 10 * q^4 + 6 * q^7 + 3 * q^8 + 6 * q^9 $$6 q + 6 q^{3} + 10 q^{4} + 6 q^{7} + 3 q^{8} + 6 q^{9} + 3 q^{11} + 10 q^{12} + 6 q^{13} - 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} + 6 q^{21} + 16 q^{22} + 13 q^{23} + 3 q^{24} - 28 q^{26} + 6 q^{27} + 7 q^{28} - 3 q^{29} - 11 q^{31} + 16 q^{32} + 3 q^{33} + 15 q^{34} + 10 q^{36} + 21 q^{37} - 9 q^{38} + 6 q^{39} - q^{41} - 22 q^{42} + 2 q^{43} + 9 q^{44} + 19 q^{46} + 14 q^{47} + 18 q^{48} - 14 q^{49} + 13 q^{51} + 13 q^{52} + 23 q^{53} - 35 q^{56} + 11 q^{57} + 22 q^{58} + 9 q^{59} + 11 q^{61} - 23 q^{62} + 6 q^{63} - 23 q^{64} + 16 q^{66} + 8 q^{67} + 50 q^{68} + 13 q^{69} - 8 q^{71} + 3 q^{72} + 13 q^{73} - 22 q^{74} - 26 q^{76} - 13 q^{77} - 28 q^{78} - 5 q^{79} + 6 q^{81} - 13 q^{82} - 20 q^{83} + 7 q^{84} - 37 q^{86} - 3 q^{87} + 28 q^{88} - 4 q^{89} + 34 q^{91} + 61 q^{92} - 11 q^{93} + 41 q^{94} + 16 q^{96} - 7 q^{97} - 41 q^{98} + 3 q^{99}+O(q^{100})$$ 6 * q + 6 * q^3 + 10 * q^4 + 6 * q^7 + 3 * q^8 + 6 * q^9 + 3 * q^11 + 10 * q^12 + 6 * q^13 - 22 * q^14 + 18 * q^16 + 13 * q^17 + 11 * q^19 + 6 * q^21 + 16 * q^22 + 13 * q^23 + 3 * q^24 - 28 * q^26 + 6 * q^27 + 7 * q^28 - 3 * q^29 - 11 * q^31 + 16 * q^32 + 3 * q^33 + 15 * q^34 + 10 * q^36 + 21 * q^37 - 9 * q^38 + 6 * q^39 - q^41 - 22 * q^42 + 2 * q^43 + 9 * q^44 + 19 * q^46 + 14 * q^47 + 18 * q^48 - 14 * q^49 + 13 * q^51 + 13 * q^52 + 23 * q^53 - 35 * q^56 + 11 * q^57 + 22 * q^58 + 9 * q^59 + 11 * q^61 - 23 * q^62 + 6 * q^63 - 23 * q^64 + 16 * q^66 + 8 * q^67 + 50 * q^68 + 13 * q^69 - 8 * q^71 + 3 * q^72 + 13 * q^73 - 22 * q^74 - 26 * q^76 - 13 * q^77 - 28 * q^78 - 5 * q^79 + 6 * q^81 - 13 * q^82 - 20 * q^83 + 7 * q^84 - 37 * q^86 - 3 * q^87 + 28 * q^88 - 4 * q^89 + 34 * q^91 + 61 * q^92 - 11 * q^93 + 41 * q^94 + 16 * q^96 - 7 * q^97 - 41 * q^98 + 3 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.16056 −1.52775 −0.763873 0.645366i $$-0.776705\pi$$
−0.763873 + 0.645366i $$0.776705\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 2.66802 1.33401
$$5$$ 0 0
$$6$$ −2.16056 −0.882045
$$7$$ 3.16056 1.19458 0.597290 0.802026i $$-0.296244\pi$$
0.597290 + 0.802026i $$0.296244\pi$$
$$8$$ −1.44329 −0.510282
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.53835 0.463830 0.231915 0.972736i $$-0.425501\pi$$
0.231915 + 0.972736i $$0.425501\pi$$
$$12$$ 2.66802 0.770191
$$13$$ 5.24144 1.45371 0.726857 0.686789i $$-0.240981\pi$$
0.726857 + 0.686789i $$0.240981\pi$$
$$14$$ −6.82858 −1.82501
$$15$$ 0 0
$$16$$ −2.21771 −0.554428
$$17$$ −1.29068 −0.313037 −0.156518 0.987675i $$-0.550027\pi$$
−0.156518 + 0.987675i $$0.550027\pi$$
$$18$$ −2.16056 −0.509249
$$19$$ 5.44587 1.24937 0.624685 0.780877i $$-0.285228\pi$$
0.624685 + 0.780877i $$0.285228\pi$$
$$20$$ 0 0
$$21$$ 3.16056 0.689691
$$22$$ −3.32370 −0.708615
$$23$$ 6.44244 1.34334 0.671671 0.740850i $$-0.265577\pi$$
0.671671 + 0.740850i $$0.265577\pi$$
$$24$$ −1.44329 −0.294611
$$25$$ 0 0
$$26$$ −11.3244 −2.22090
$$27$$ 1.00000 0.192450
$$28$$ 8.43243 1.59358
$$29$$ −2.36361 −0.438912 −0.219456 0.975622i $$-0.570428\pi$$
−0.219456 + 0.975622i $$0.570428\pi$$
$$30$$ 0 0
$$31$$ −4.46542 −0.802014 −0.401007 0.916075i $$-0.631340\pi$$
−0.401007 + 0.916075i $$0.631340\pi$$
$$32$$ 7.67809 1.35731
$$33$$ 1.53835 0.267793
$$34$$ 2.78860 0.478241
$$35$$ 0 0
$$36$$ 2.66802 0.444670
$$37$$ 5.95751 0.979408 0.489704 0.871889i $$-0.337105\pi$$
0.489704 + 0.871889i $$0.337105\pi$$
$$38$$ −11.7661 −1.90872
$$39$$ 5.24144 0.839302
$$40$$ 0 0
$$41$$ −8.53219 −1.33250 −0.666252 0.745726i $$-0.732103\pi$$
−0.666252 + 0.745726i $$0.732103\pi$$
$$42$$ −6.82858 −1.05367
$$43$$ 8.48426 1.29384 0.646919 0.762559i $$-0.276057\pi$$
0.646919 + 0.762559i $$0.276057\pi$$
$$44$$ 4.10435 0.618754
$$45$$ 0 0
$$46$$ −13.9193 −2.05228
$$47$$ −0.753070 −0.109847 −0.0549233 0.998491i $$-0.517491\pi$$
−0.0549233 + 0.998491i $$0.517491\pi$$
$$48$$ −2.21771 −0.320099
$$49$$ 2.98914 0.427020
$$50$$ 0 0
$$51$$ −1.29068 −0.180732
$$52$$ 13.9843 1.93927
$$53$$ −9.74991 −1.33925 −0.669626 0.742698i $$-0.733546\pi$$
−0.669626 + 0.742698i $$0.733546\pi$$
$$54$$ −2.16056 −0.294015
$$55$$ 0 0
$$56$$ −4.56162 −0.609572
$$57$$ 5.44587 0.721324
$$58$$ 5.10672 0.670546
$$59$$ 4.11270 0.535428 0.267714 0.963498i $$-0.413732\pi$$
0.267714 + 0.963498i $$0.413732\pi$$
$$60$$ 0 0
$$61$$ −10.6939 −1.36922 −0.684610 0.728910i $$-0.740027\pi$$
−0.684610 + 0.728910i $$0.740027\pi$$
$$62$$ 9.64781 1.22527
$$63$$ 3.16056 0.398193
$$64$$ −12.1535 −1.51919
$$65$$ 0 0
$$66$$ −3.32370 −0.409119
$$67$$ 1.89864 0.231956 0.115978 0.993252i $$-0.463000\pi$$
0.115978 + 0.993252i $$0.463000\pi$$
$$68$$ −3.44357 −0.417594
$$69$$ 6.44244 0.775578
$$70$$ 0 0
$$71$$ −0.0708774 −0.00841160 −0.00420580 0.999991i $$-0.501339\pi$$
−0.00420580 + 0.999991i $$0.501339\pi$$
$$72$$ −1.44329 −0.170094
$$73$$ −4.01123 −0.469478 −0.234739 0.972058i $$-0.575424\pi$$
−0.234739 + 0.972058i $$0.575424\pi$$
$$74$$ −12.8716 −1.49629
$$75$$ 0 0
$$76$$ 14.5297 1.66667
$$77$$ 4.86205 0.554082
$$78$$ −11.3244 −1.28224
$$79$$ 1.61849 0.182094 0.0910472 0.995847i $$-0.470979\pi$$
0.0910472 + 0.995847i $$0.470979\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 18.4343 2.03573
$$83$$ −13.1488 −1.44327 −0.721636 0.692272i $$-0.756610\pi$$
−0.721636 + 0.692272i $$0.756610\pi$$
$$84$$ 8.43243 0.920054
$$85$$ 0 0
$$86$$ −18.3308 −1.97666
$$87$$ −2.36361 −0.253406
$$88$$ −2.22029 −0.236684
$$89$$ 7.27597 0.771251 0.385626 0.922655i $$-0.373986\pi$$
0.385626 + 0.922655i $$0.373986\pi$$
$$90$$ 0 0
$$91$$ 16.5659 1.73658
$$92$$ 17.1885 1.79203
$$93$$ −4.46542 −0.463043
$$94$$ 1.62705 0.167818
$$95$$ 0 0
$$96$$ 7.67809 0.783642
$$97$$ −10.1939 −1.03504 −0.517518 0.855673i $$-0.673144\pi$$
−0.517518 + 0.855673i $$0.673144\pi$$
$$98$$ −6.45821 −0.652378
$$99$$ 1.53835 0.154610
$$100$$ 0 0
$$101$$ 1.08759 0.108219 0.0541096 0.998535i $$-0.482768\pi$$
0.0541096 + 0.998535i $$0.482768\pi$$
$$102$$ 2.78860 0.276112
$$103$$ 4.93756 0.486512 0.243256 0.969962i $$-0.421784\pi$$
0.243256 + 0.969962i $$0.421784\pi$$
$$104$$ −7.56494 −0.741803
$$105$$ 0 0
$$106$$ 21.0653 2.04604
$$107$$ −15.3059 −1.47967 −0.739837 0.672786i $$-0.765098\pi$$
−0.739837 + 0.672786i $$0.765098\pi$$
$$108$$ 2.66802 0.256730
$$109$$ 6.65900 0.637816 0.318908 0.947786i $$-0.396684\pi$$
0.318908 + 0.947786i $$0.396684\pi$$
$$110$$ 0 0
$$111$$ 5.95751 0.565462
$$112$$ −7.00922 −0.662309
$$113$$ 8.97143 0.843961 0.421981 0.906605i $$-0.361335\pi$$
0.421981 + 0.906605i $$0.361335\pi$$
$$114$$ −11.7661 −1.10200
$$115$$ 0 0
$$116$$ −6.30616 −0.585512
$$117$$ 5.24144 0.484571
$$118$$ −8.88573 −0.817998
$$119$$ −4.07928 −0.373947
$$120$$ 0 0
$$121$$ −8.63348 −0.784861
$$122$$ 23.1049 2.09182
$$123$$ −8.53219 −0.769322
$$124$$ −11.9138 −1.06989
$$125$$ 0 0
$$126$$ −6.82858 −0.608338
$$127$$ −10.0185 −0.888995 −0.444497 0.895780i $$-0.646618\pi$$
−0.444497 + 0.895780i $$0.646618\pi$$
$$128$$ 10.9023 0.963635
$$129$$ 8.48426 0.746997
$$130$$ 0 0
$$131$$ −6.37865 −0.557305 −0.278653 0.960392i $$-0.589888\pi$$
−0.278653 + 0.960392i $$0.589888\pi$$
$$132$$ 4.10435 0.357238
$$133$$ 17.2120 1.49247
$$134$$ −4.10214 −0.354371
$$135$$ 0 0
$$136$$ 1.86284 0.159737
$$137$$ 9.17732 0.784071 0.392036 0.919950i $$-0.371771\pi$$
0.392036 + 0.919950i $$0.371771\pi$$
$$138$$ −13.9193 −1.18489
$$139$$ −9.26539 −0.785880 −0.392940 0.919564i $$-0.628542\pi$$
−0.392940 + 0.919564i $$0.628542\pi$$
$$140$$ 0 0
$$141$$ −0.753070 −0.0634200
$$142$$ 0.153135 0.0128508
$$143$$ 8.06317 0.674276
$$144$$ −2.21771 −0.184809
$$145$$ 0 0
$$146$$ 8.66649 0.717244
$$147$$ 2.98914 0.246540
$$148$$ 15.8947 1.30654
$$149$$ −10.0817 −0.825928 −0.412964 0.910747i $$-0.635506\pi$$
−0.412964 + 0.910747i $$0.635506\pi$$
$$150$$ 0 0
$$151$$ 10.7375 0.873804 0.436902 0.899509i $$-0.356076\pi$$
0.436902 + 0.899509i $$0.356076\pi$$
$$152$$ −7.86000 −0.637530
$$153$$ −1.29068 −0.104346
$$154$$ −10.5048 −0.846497
$$155$$ 0 0
$$156$$ 13.9843 1.11964
$$157$$ −17.4417 −1.39200 −0.696001 0.718041i $$-0.745039\pi$$
−0.696001 + 0.718041i $$0.745039\pi$$
$$158$$ −3.49684 −0.278194
$$159$$ −9.74991 −0.773218
$$160$$ 0 0
$$161$$ 20.3617 1.60473
$$162$$ −2.16056 −0.169750
$$163$$ 16.2574 1.27338 0.636689 0.771120i $$-0.280303\pi$$
0.636689 + 0.771120i $$0.280303\pi$$
$$164$$ −22.7641 −1.77757
$$165$$ 0 0
$$166$$ 28.4089 2.20496
$$167$$ 11.3906 0.881430 0.440715 0.897647i $$-0.354725\pi$$
0.440715 + 0.897647i $$0.354725\pi$$
$$168$$ −4.56162 −0.351936
$$169$$ 14.4727 1.11328
$$170$$ 0 0
$$171$$ 5.44587 0.416456
$$172$$ 22.6362 1.72599
$$173$$ −9.45845 −0.719113 −0.359556 0.933123i $$-0.617072\pi$$
−0.359556 + 0.933123i $$0.617072\pi$$
$$174$$ 5.10672 0.387140
$$175$$ 0 0
$$176$$ −3.41162 −0.257161
$$177$$ 4.11270 0.309129
$$178$$ −15.7202 −1.17828
$$179$$ 4.07747 0.304764 0.152382 0.988322i $$-0.451306\pi$$
0.152382 + 0.988322i $$0.451306\pi$$
$$180$$ 0 0
$$181$$ −11.1202 −0.826558 −0.413279 0.910604i $$-0.635617\pi$$
−0.413279 + 0.910604i $$0.635617\pi$$
$$182$$ −35.7916 −2.65305
$$183$$ −10.6939 −0.790519
$$184$$ −9.29833 −0.685482
$$185$$ 0 0
$$186$$ 9.64781 0.707412
$$187$$ −1.98552 −0.145196
$$188$$ −2.00921 −0.146536
$$189$$ 3.16056 0.229897
$$190$$ 0 0
$$191$$ 3.61135 0.261308 0.130654 0.991428i $$-0.458292\pi$$
0.130654 + 0.991428i $$0.458292\pi$$
$$192$$ −12.1535 −0.877107
$$193$$ 17.0562 1.22774 0.613868 0.789409i $$-0.289613\pi$$
0.613868 + 0.789409i $$0.289613\pi$$
$$194$$ 22.0246 1.58127
$$195$$ 0 0
$$196$$ 7.97508 0.569648
$$197$$ 4.40523 0.313860 0.156930 0.987610i $$-0.449840\pi$$
0.156930 + 0.987610i $$0.449840\pi$$
$$198$$ −3.32370 −0.236205
$$199$$ 16.5956 1.17643 0.588214 0.808705i $$-0.299831\pi$$
0.588214 + 0.808705i $$0.299831\pi$$
$$200$$ 0 0
$$201$$ 1.89864 0.133920
$$202$$ −2.34980 −0.165332
$$203$$ −7.47034 −0.524315
$$204$$ −3.44357 −0.241098
$$205$$ 0 0
$$206$$ −10.6679 −0.743267
$$207$$ 6.44244 0.447780
$$208$$ −11.6240 −0.805980
$$209$$ 8.37767 0.579495
$$210$$ 0 0
$$211$$ 18.5454 1.27671 0.638357 0.769740i $$-0.279614\pi$$
0.638357 + 0.769740i $$0.279614\pi$$
$$212$$ −26.0129 −1.78658
$$213$$ −0.0708774 −0.00485644
$$214$$ 33.0693 2.26057
$$215$$ 0 0
$$216$$ −1.44329 −0.0982037
$$217$$ −14.1132 −0.958069
$$218$$ −14.3872 −0.974422
$$219$$ −4.01123 −0.271053
$$220$$ 0 0
$$221$$ −6.76504 −0.455066
$$222$$ −12.8716 −0.863882
$$223$$ 0.771557 0.0516673 0.0258336 0.999666i $$-0.491776\pi$$
0.0258336 + 0.999666i $$0.491776\pi$$
$$224$$ 24.2671 1.62141
$$225$$ 0 0
$$226$$ −19.3833 −1.28936
$$227$$ −11.4314 −0.758727 −0.379363 0.925248i $$-0.623857\pi$$
−0.379363 + 0.925248i $$0.623857\pi$$
$$228$$ 14.5297 0.962253
$$229$$ 10.2248 0.675671 0.337835 0.941205i $$-0.390305\pi$$
0.337835 + 0.941205i $$0.390305\pi$$
$$230$$ 0 0
$$231$$ 4.86205 0.319899
$$232$$ 3.41139 0.223969
$$233$$ 22.5229 1.47553 0.737763 0.675059i $$-0.235882\pi$$
0.737763 + 0.675059i $$0.235882\pi$$
$$234$$ −11.3244 −0.740302
$$235$$ 0 0
$$236$$ 10.9728 0.714266
$$237$$ 1.61849 0.105132
$$238$$ 8.81353 0.571297
$$239$$ −0.589074 −0.0381040 −0.0190520 0.999818i $$-0.506065\pi$$
−0.0190520 + 0.999818i $$0.506065\pi$$
$$240$$ 0 0
$$241$$ 1.50937 0.0972270 0.0486135 0.998818i $$-0.484520\pi$$
0.0486135 + 0.998818i $$0.484520\pi$$
$$242$$ 18.6531 1.19907
$$243$$ 1.00000 0.0641500
$$244$$ −28.5317 −1.82655
$$245$$ 0 0
$$246$$ 18.4343 1.17533
$$247$$ 28.5442 1.81622
$$248$$ 6.44492 0.409253
$$249$$ −13.1488 −0.833274
$$250$$ 0 0
$$251$$ 11.8953 0.750823 0.375412 0.926858i $$-0.377501\pi$$
0.375412 + 0.926858i $$0.377501\pi$$
$$252$$ 8.43243 0.531193
$$253$$ 9.91073 0.623082
$$254$$ 21.6455 1.35816
$$255$$ 0 0
$$256$$ 0.752056 0.0470035
$$257$$ 18.0492 1.12588 0.562939 0.826498i $$-0.309671\pi$$
0.562939 + 0.826498i $$0.309671\pi$$
$$258$$ −18.3308 −1.14122
$$259$$ 18.8291 1.16998
$$260$$ 0 0
$$261$$ −2.36361 −0.146304
$$262$$ 13.7815 0.851421
$$263$$ 19.1066 1.17817 0.589083 0.808072i $$-0.299489\pi$$
0.589083 + 0.808072i $$0.299489\pi$$
$$264$$ −2.22029 −0.136650
$$265$$ 0 0
$$266$$ −37.1876 −2.28012
$$267$$ 7.27597 0.445282
$$268$$ 5.06562 0.309432
$$269$$ −22.2250 −1.35508 −0.677542 0.735484i $$-0.736955\pi$$
−0.677542 + 0.735484i $$0.736955\pi$$
$$270$$ 0 0
$$271$$ 20.2107 1.22772 0.613858 0.789417i $$-0.289617\pi$$
0.613858 + 0.789417i $$0.289617\pi$$
$$272$$ 2.86237 0.173556
$$273$$ 16.5659 1.00261
$$274$$ −19.8281 −1.19786
$$275$$ 0 0
$$276$$ 17.1885 1.03463
$$277$$ 16.0413 0.963826 0.481913 0.876219i $$-0.339942\pi$$
0.481913 + 0.876219i $$0.339942\pi$$
$$278$$ 20.0184 1.20063
$$279$$ −4.46542 −0.267338
$$280$$ 0 0
$$281$$ 8.37308 0.499496 0.249748 0.968311i $$-0.419652\pi$$
0.249748 + 0.968311i $$0.419652\pi$$
$$282$$ 1.62705 0.0968896
$$283$$ 1.12995 0.0671685 0.0335843 0.999436i $$-0.489308\pi$$
0.0335843 + 0.999436i $$0.489308\pi$$
$$284$$ −0.189102 −0.0112211
$$285$$ 0 0
$$286$$ −17.4210 −1.03012
$$287$$ −26.9665 −1.59178
$$288$$ 7.67809 0.452436
$$289$$ −15.3341 −0.902008
$$290$$ 0 0
$$291$$ −10.1939 −0.597578
$$292$$ −10.7020 −0.626289
$$293$$ 2.37857 0.138958 0.0694789 0.997583i $$-0.477866\pi$$
0.0694789 + 0.997583i $$0.477866\pi$$
$$294$$ −6.45821 −0.376651
$$295$$ 0 0
$$296$$ −8.59844 −0.499774
$$297$$ 1.53835 0.0892642
$$298$$ 21.7822 1.26181
$$299$$ 33.7676 1.95283
$$300$$ 0 0
$$301$$ 26.8150 1.54559
$$302$$ −23.1990 −1.33495
$$303$$ 1.08759 0.0624804
$$304$$ −12.0774 −0.692686
$$305$$ 0 0
$$306$$ 2.78860 0.159414
$$307$$ −2.89366 −0.165150 −0.0825748 0.996585i $$-0.526314\pi$$
−0.0825748 + 0.996585i $$0.526314\pi$$
$$308$$ 12.9720 0.739151
$$309$$ 4.93756 0.280888
$$310$$ 0 0
$$311$$ −7.14545 −0.405181 −0.202591 0.979264i $$-0.564936\pi$$
−0.202591 + 0.979264i $$0.564936\pi$$
$$312$$ −7.56494 −0.428280
$$313$$ −23.0318 −1.30184 −0.650918 0.759148i $$-0.725616\pi$$
−0.650918 + 0.759148i $$0.725616\pi$$
$$314$$ 37.6839 2.12663
$$315$$ 0 0
$$316$$ 4.31816 0.242916
$$317$$ 20.7750 1.16684 0.583419 0.812171i $$-0.301715\pi$$
0.583419 + 0.812171i $$0.301715\pi$$
$$318$$ 21.0653 1.18128
$$319$$ −3.63606 −0.203581
$$320$$ 0 0
$$321$$ −15.3059 −0.854291
$$322$$ −43.9927 −2.45162
$$323$$ −7.02890 −0.391098
$$324$$ 2.66802 0.148223
$$325$$ 0 0
$$326$$ −35.1251 −1.94540
$$327$$ 6.65900 0.368243
$$328$$ 12.3145 0.679953
$$329$$ −2.38012 −0.131220
$$330$$ 0 0
$$331$$ −19.2504 −1.05810 −0.529048 0.848592i $$-0.677451\pi$$
−0.529048 + 0.848592i $$0.677451\pi$$
$$332$$ −35.0814 −1.92534
$$333$$ 5.95751 0.326469
$$334$$ −24.6100 −1.34660
$$335$$ 0 0
$$336$$ −7.00922 −0.382384
$$337$$ −21.7064 −1.18243 −0.591213 0.806516i $$-0.701351\pi$$
−0.591213 + 0.806516i $$0.701351\pi$$
$$338$$ −31.2690 −1.70081
$$339$$ 8.97143 0.487261
$$340$$ 0 0
$$341$$ −6.86939 −0.371998
$$342$$ −11.7661 −0.636240
$$343$$ −12.6766 −0.684470
$$344$$ −12.2453 −0.660221
$$345$$ 0 0
$$346$$ 20.4356 1.09862
$$347$$ 1.71494 0.0920626 0.0460313 0.998940i $$-0.485343\pi$$
0.0460313 + 0.998940i $$0.485343\pi$$
$$348$$ −6.30616 −0.338046
$$349$$ −15.5553 −0.832654 −0.416327 0.909215i $$-0.636683\pi$$
−0.416327 + 0.909215i $$0.636683\pi$$
$$350$$ 0 0
$$351$$ 5.24144 0.279767
$$352$$ 11.8116 0.629560
$$353$$ −15.5536 −0.827833 −0.413916 0.910315i $$-0.635839\pi$$
−0.413916 + 0.910315i $$0.635839\pi$$
$$354$$ −8.88573 −0.472271
$$355$$ 0 0
$$356$$ 19.4124 1.02886
$$357$$ −4.07928 −0.215899
$$358$$ −8.80961 −0.465602
$$359$$ −24.5371 −1.29502 −0.647508 0.762059i $$-0.724189\pi$$
−0.647508 + 0.762059i $$0.724189\pi$$
$$360$$ 0 0
$$361$$ 10.6575 0.560924
$$362$$ 24.0259 1.26277
$$363$$ −8.63348 −0.453140
$$364$$ 44.1981 2.31661
$$365$$ 0 0
$$366$$ 23.1049 1.20771
$$367$$ −28.1068 −1.46716 −0.733581 0.679602i $$-0.762153\pi$$
−0.733581 + 0.679602i $$0.762153\pi$$
$$368$$ −14.2875 −0.744786
$$369$$ −8.53219 −0.444168
$$370$$ 0 0
$$371$$ −30.8152 −1.59984
$$372$$ −11.9138 −0.617703
$$373$$ 9.70825 0.502674 0.251337 0.967900i $$-0.419130\pi$$
0.251337 + 0.967900i $$0.419130\pi$$
$$374$$ 4.28984 0.221823
$$375$$ 0 0
$$376$$ 1.08690 0.0560527
$$377$$ −12.3887 −0.638052
$$378$$ −6.82858 −0.351224
$$379$$ −22.2665 −1.14375 −0.571875 0.820340i $$-0.693784\pi$$
−0.571875 + 0.820340i $$0.693784\pi$$
$$380$$ 0 0
$$381$$ −10.0185 −0.513261
$$382$$ −7.80253 −0.399212
$$383$$ 13.6299 0.696458 0.348229 0.937410i $$-0.386783\pi$$
0.348229 + 0.937410i $$0.386783\pi$$
$$384$$ 10.9023 0.556355
$$385$$ 0 0
$$386$$ −36.8510 −1.87567
$$387$$ 8.48426 0.431279
$$388$$ −27.1976 −1.38075
$$389$$ 19.2304 0.975021 0.487510 0.873117i $$-0.337905\pi$$
0.487510 + 0.873117i $$0.337905\pi$$
$$390$$ 0 0
$$391$$ −8.31515 −0.420515
$$392$$ −4.31421 −0.217900
$$393$$ −6.37865 −0.321760
$$394$$ −9.51777 −0.479498
$$395$$ 0 0
$$396$$ 4.10435 0.206251
$$397$$ −22.2281 −1.11560 −0.557799 0.829976i $$-0.688354\pi$$
−0.557799 + 0.829976i $$0.688354\pi$$
$$398$$ −35.8557 −1.79729
$$399$$ 17.2120 0.861678
$$400$$ 0 0
$$401$$ 4.71728 0.235570 0.117785 0.993039i $$-0.462421\pi$$
0.117785 + 0.993039i $$0.462421\pi$$
$$402$$ −4.10214 −0.204596
$$403$$ −23.4052 −1.16590
$$404$$ 2.90171 0.144365
$$405$$ 0 0
$$406$$ 16.1401 0.801020
$$407$$ 9.16474 0.454279
$$408$$ 1.86284 0.0922241
$$409$$ −12.4807 −0.617130 −0.308565 0.951203i $$-0.599849\pi$$
−0.308565 + 0.951203i $$0.599849\pi$$
$$410$$ 0 0
$$411$$ 9.17732 0.452684
$$412$$ 13.1735 0.649012
$$413$$ 12.9984 0.639611
$$414$$ −13.9193 −0.684095
$$415$$ 0 0
$$416$$ 40.2442 1.97314
$$417$$ −9.26539 −0.453728
$$418$$ −18.1005 −0.885322
$$419$$ 34.9484 1.70734 0.853669 0.520815i $$-0.174372\pi$$
0.853669 + 0.520815i $$0.174372\pi$$
$$420$$ 0 0
$$421$$ −39.5601 −1.92804 −0.964020 0.265829i $$-0.914354\pi$$
−0.964020 + 0.265829i $$0.914354\pi$$
$$422$$ −40.0683 −1.95050
$$423$$ −0.753070 −0.0366155
$$424$$ 14.0720 0.683396
$$425$$ 0 0
$$426$$ 0.153135 0.00741940
$$427$$ −33.7989 −1.63564
$$428$$ −40.8364 −1.97390
$$429$$ 8.06317 0.389294
$$430$$ 0 0
$$431$$ −14.0584 −0.677170 −0.338585 0.940936i $$-0.609948\pi$$
−0.338585 + 0.940936i $$0.609948\pi$$
$$432$$ −2.21771 −0.106700
$$433$$ 0.549678 0.0264158 0.0132079 0.999913i $$-0.495796\pi$$
0.0132079 + 0.999913i $$0.495796\pi$$
$$434$$ 30.4925 1.46369
$$435$$ 0 0
$$436$$ 17.7663 0.850853
$$437$$ 35.0847 1.67833
$$438$$ 8.66649 0.414101
$$439$$ −2.97377 −0.141930 −0.0709651 0.997479i $$-0.522608\pi$$
−0.0709651 + 0.997479i $$0.522608\pi$$
$$440$$ 0 0
$$441$$ 2.98914 0.142340
$$442$$ 14.6163 0.695225
$$443$$ 36.6893 1.74316 0.871580 0.490253i $$-0.163095\pi$$
0.871580 + 0.490253i $$0.163095\pi$$
$$444$$ 15.8947 0.754331
$$445$$ 0 0
$$446$$ −1.66700 −0.0789345
$$447$$ −10.0817 −0.476850
$$448$$ −38.4120 −1.81480
$$449$$ −17.8432 −0.842071 −0.421036 0.907044i $$-0.638333\pi$$
−0.421036 + 0.907044i $$0.638333\pi$$
$$450$$ 0 0
$$451$$ −13.1255 −0.618056
$$452$$ 23.9359 1.12585
$$453$$ 10.7375 0.504491
$$454$$ 24.6982 1.15914
$$455$$ 0 0
$$456$$ −7.86000 −0.368078
$$457$$ 24.1784 1.13102 0.565509 0.824742i $$-0.308680\pi$$
0.565509 + 0.824742i $$0.308680\pi$$
$$458$$ −22.0912 −1.03225
$$459$$ −1.29068 −0.0602439
$$460$$ 0 0
$$461$$ −37.3874 −1.74130 −0.870651 0.491901i $$-0.836302\pi$$
−0.870651 + 0.491901i $$0.836302\pi$$
$$462$$ −10.5048 −0.488725
$$463$$ −20.4314 −0.949530 −0.474765 0.880113i $$-0.657467\pi$$
−0.474765 + 0.880113i $$0.657467\pi$$
$$464$$ 5.24181 0.243345
$$465$$ 0 0
$$466$$ −48.6622 −2.25423
$$467$$ −31.2124 −1.44434 −0.722169 0.691716i $$-0.756855\pi$$
−0.722169 + 0.691716i $$0.756855\pi$$
$$468$$ 13.9843 0.646422
$$469$$ 6.00078 0.277090
$$470$$ 0 0
$$471$$ −17.4417 −0.803673
$$472$$ −5.93584 −0.273219
$$473$$ 13.0518 0.600121
$$474$$ −3.49684 −0.160615
$$475$$ 0 0
$$476$$ −10.8836 −0.498849
$$477$$ −9.74991 −0.446418
$$478$$ 1.27273 0.0582133
$$479$$ 11.0970 0.507033 0.253516 0.967331i $$-0.418413\pi$$
0.253516 + 0.967331i $$0.418413\pi$$
$$480$$ 0 0
$$481$$ 31.2259 1.42378
$$482$$ −3.26108 −0.148538
$$483$$ 20.3617 0.926490
$$484$$ −23.0343 −1.04701
$$485$$ 0 0
$$486$$ −2.16056 −0.0980050
$$487$$ −7.13702 −0.323409 −0.161705 0.986839i $$-0.551699\pi$$
−0.161705 + 0.986839i $$0.551699\pi$$
$$488$$ 15.4345 0.698688
$$489$$ 16.2574 0.735185
$$490$$ 0 0
$$491$$ 7.23348 0.326442 0.163221 0.986590i $$-0.447812\pi$$
0.163221 + 0.986590i $$0.447812\pi$$
$$492$$ −22.7641 −1.02628
$$493$$ 3.05067 0.137395
$$494$$ −61.6715 −2.77473
$$495$$ 0 0
$$496$$ 9.90303 0.444659
$$497$$ −0.224012 −0.0100483
$$498$$ 28.4089 1.27303
$$499$$ 16.0169 0.717014 0.358507 0.933527i $$-0.383286\pi$$
0.358507 + 0.933527i $$0.383286\pi$$
$$500$$ 0 0
$$501$$ 11.3906 0.508894
$$502$$ −25.7005 −1.14707
$$503$$ 33.7963 1.50690 0.753452 0.657503i $$-0.228387\pi$$
0.753452 + 0.657503i $$0.228387\pi$$
$$504$$ −4.56162 −0.203191
$$505$$ 0 0
$$506$$ −21.4127 −0.951912
$$507$$ 14.4727 0.642753
$$508$$ −26.7294 −1.18593
$$509$$ −36.4270 −1.61460 −0.807300 0.590142i $$-0.799072\pi$$
−0.807300 + 0.590142i $$0.799072\pi$$
$$510$$ 0 0
$$511$$ −12.6777 −0.560829
$$512$$ −23.4294 −1.03544
$$513$$ 5.44587 0.240441
$$514$$ −38.9964 −1.72006
$$515$$ 0 0
$$516$$ 22.6362 0.996502
$$517$$ −1.15849 −0.0509502
$$518$$ −40.6813 −1.78743
$$519$$ −9.45845 −0.415180
$$520$$ 0 0
$$521$$ 25.4993 1.11714 0.558572 0.829456i $$-0.311349\pi$$
0.558572 + 0.829456i $$0.311349\pi$$
$$522$$ 5.10672 0.223515
$$523$$ 7.03174 0.307477 0.153738 0.988112i $$-0.450869\pi$$
0.153738 + 0.988112i $$0.450869\pi$$
$$524$$ −17.0184 −0.743450
$$525$$ 0 0
$$526$$ −41.2811 −1.79994
$$527$$ 5.76345 0.251060
$$528$$ −3.41162 −0.148472
$$529$$ 18.5050 0.804565
$$530$$ 0 0
$$531$$ 4.11270 0.178476
$$532$$ 45.9220 1.99097
$$533$$ −44.7210 −1.93708
$$534$$ −15.7202 −0.680278
$$535$$ 0 0
$$536$$ −2.74030 −0.118363
$$537$$ 4.07747 0.175956
$$538$$ 48.0185 2.07022
$$539$$ 4.59834 0.198065
$$540$$ 0 0
$$541$$ −27.1476 −1.16717 −0.583584 0.812053i $$-0.698350\pi$$
−0.583584 + 0.812053i $$0.698350\pi$$
$$542$$ −43.6665 −1.87564
$$543$$ −11.1202 −0.477214
$$544$$ −9.90999 −0.424887
$$545$$ 0 0
$$546$$ −35.7916 −1.53174
$$547$$ −44.0769 −1.88459 −0.942296 0.334782i $$-0.891337\pi$$
−0.942296 + 0.334782i $$0.891337\pi$$
$$548$$ 24.4853 1.04596
$$549$$ −10.6939 −0.456407
$$550$$ 0 0
$$551$$ −12.8719 −0.548363
$$552$$ −9.29833 −0.395763
$$553$$ 5.11533 0.217526
$$554$$ −34.6581 −1.47248
$$555$$ 0 0
$$556$$ −24.7202 −1.04837
$$557$$ 9.85667 0.417641 0.208820 0.977954i $$-0.433038\pi$$
0.208820 + 0.977954i $$0.433038\pi$$
$$558$$ 9.64781 0.408425
$$559$$ 44.4697 1.88087
$$560$$ 0 0
$$561$$ −1.98552 −0.0838289
$$562$$ −18.0905 −0.763103
$$563$$ 12.3917 0.522249 0.261125 0.965305i $$-0.415907\pi$$
0.261125 + 0.965305i $$0.415907\pi$$
$$564$$ −2.00921 −0.0846028
$$565$$ 0 0
$$566$$ −2.44132 −0.102616
$$567$$ 3.16056 0.132731
$$568$$ 0.102297 0.00429228
$$569$$ −11.7158 −0.491150 −0.245575 0.969378i $$-0.578977\pi$$
−0.245575 + 0.969378i $$0.578977\pi$$
$$570$$ 0 0
$$571$$ 28.9797 1.21276 0.606382 0.795173i $$-0.292620\pi$$
0.606382 + 0.795173i $$0.292620\pi$$
$$572$$ 21.5127 0.899491
$$573$$ 3.61135 0.150866
$$574$$ 58.2628 2.43184
$$575$$ 0 0
$$576$$ −12.1535 −0.506398
$$577$$ 32.8024 1.36558 0.682790 0.730614i $$-0.260766\pi$$
0.682790 + 0.730614i $$0.260766\pi$$
$$578$$ 33.1303 1.37804
$$579$$ 17.0562 0.708833
$$580$$ 0 0
$$581$$ −41.5577 −1.72410
$$582$$ 22.0246 0.912947
$$583$$ −14.9988 −0.621186
$$584$$ 5.78938 0.239566
$$585$$ 0 0
$$586$$ −5.13905 −0.212292
$$587$$ −18.6395 −0.769334 −0.384667 0.923055i $$-0.625684\pi$$
−0.384667 + 0.923055i $$0.625684\pi$$
$$588$$ 7.97508 0.328887
$$589$$ −24.3181 −1.00201
$$590$$ 0 0
$$591$$ 4.40523 0.181207
$$592$$ −13.2120 −0.543012
$$593$$ −1.55882 −0.0640129 −0.0320064 0.999488i $$-0.510190\pi$$
−0.0320064 + 0.999488i $$0.510190\pi$$
$$594$$ −3.32370 −0.136373
$$595$$ 0 0
$$596$$ −26.8983 −1.10180
$$597$$ 16.5956 0.679212
$$598$$ −72.9570 −2.98343
$$599$$ −31.5470 −1.28898 −0.644489 0.764614i $$-0.722930\pi$$
−0.644489 + 0.764614i $$0.722930\pi$$
$$600$$ 0 0
$$601$$ −32.3999 −1.32162 −0.660809 0.750554i $$-0.729787\pi$$
−0.660809 + 0.750554i $$0.729787\pi$$
$$602$$ −57.9354 −2.36127
$$603$$ 1.89864 0.0773188
$$604$$ 28.6478 1.16566
$$605$$ 0 0
$$606$$ −2.34980 −0.0954542
$$607$$ −24.8410 −1.00827 −0.504133 0.863626i $$-0.668188\pi$$
−0.504133 + 0.863626i $$0.668188\pi$$
$$608$$ 41.8139 1.69578
$$609$$ −7.47034 −0.302713
$$610$$ 0 0
$$611$$ −3.94717 −0.159685
$$612$$ −3.44357 −0.139198
$$613$$ 3.94558 0.159360 0.0796802 0.996820i $$-0.474610\pi$$
0.0796802 + 0.996820i $$0.474610\pi$$
$$614$$ 6.25192 0.252307
$$615$$ 0 0
$$616$$ −7.01737 −0.282738
$$617$$ 17.4821 0.703801 0.351900 0.936037i $$-0.385536\pi$$
0.351900 + 0.936037i $$0.385536\pi$$
$$618$$ −10.6679 −0.429126
$$619$$ −14.3869 −0.578260 −0.289130 0.957290i $$-0.593366\pi$$
−0.289130 + 0.957290i $$0.593366\pi$$
$$620$$ 0 0
$$621$$ 6.44244 0.258526
$$622$$ 15.4382 0.619014
$$623$$ 22.9961 0.921321
$$624$$ −11.6240 −0.465333
$$625$$ 0 0
$$626$$ 49.7617 1.98888
$$627$$ 8.37767 0.334572
$$628$$ −46.5349 −1.85694
$$629$$ −7.68926 −0.306591
$$630$$ 0 0
$$631$$ −17.6758 −0.703664 −0.351832 0.936063i $$-0.614441\pi$$
−0.351832 + 0.936063i $$0.614441\pi$$
$$632$$ −2.33596 −0.0929194
$$633$$ 18.5454 0.737112
$$634$$ −44.8855 −1.78263
$$635$$ 0 0
$$636$$ −26.0129 −1.03148
$$637$$ 15.6674 0.620764
$$638$$ 7.85594 0.311019
$$639$$ −0.0708774 −0.00280387
$$640$$ 0 0
$$641$$ −2.74127 −0.108274 −0.0541369 0.998534i $$-0.517241\pi$$
−0.0541369 + 0.998534i $$0.517241\pi$$
$$642$$ 33.0693 1.30514
$$643$$ −20.9276 −0.825303 −0.412651 0.910889i $$-0.635397\pi$$
−0.412651 + 0.910889i $$0.635397\pi$$
$$644$$ 54.3254 2.14072
$$645$$ 0 0
$$646$$ 15.1864 0.597499
$$647$$ 1.84667 0.0726002 0.0363001 0.999341i $$-0.488443\pi$$
0.0363001 + 0.999341i $$0.488443\pi$$
$$648$$ −1.44329 −0.0566980
$$649$$ 6.32678 0.248348
$$650$$ 0 0
$$651$$ −14.1132 −0.553141
$$652$$ 43.3751 1.69870
$$653$$ 18.4133 0.720567 0.360284 0.932843i $$-0.382680\pi$$
0.360284 + 0.932843i $$0.382680\pi$$
$$654$$ −14.3872 −0.562583
$$655$$ 0 0
$$656$$ 18.9220 0.738778
$$657$$ −4.01123 −0.156493
$$658$$ 5.14240 0.200472
$$659$$ −13.3800 −0.521210 −0.260605 0.965446i $$-0.583922\pi$$
−0.260605 + 0.965446i $$0.583922\pi$$
$$660$$ 0 0
$$661$$ 39.1834 1.52406 0.762030 0.647542i $$-0.224203\pi$$
0.762030 + 0.647542i $$0.224203\pi$$
$$662$$ 41.5915 1.61650
$$663$$ −6.76504 −0.262732
$$664$$ 18.9776 0.736476
$$665$$ 0 0
$$666$$ −12.8716 −0.498763
$$667$$ −15.2274 −0.589608
$$668$$ 30.3903 1.17584
$$669$$ 0.771557 0.0298301
$$670$$ 0 0
$$671$$ −16.4510 −0.635086
$$672$$ 24.2671 0.936122
$$673$$ −19.3911 −0.747473 −0.373736 0.927535i $$-0.621923\pi$$
−0.373736 + 0.927535i $$0.621923\pi$$
$$674$$ 46.8981 1.80645
$$675$$ 0 0
$$676$$ 38.6133 1.48513
$$677$$ 45.6836 1.75576 0.877882 0.478876i $$-0.158956\pi$$
0.877882 + 0.478876i $$0.158956\pi$$
$$678$$ −19.3833 −0.744412
$$679$$ −32.2185 −1.23643
$$680$$ 0 0
$$681$$ −11.4314 −0.438051
$$682$$ 14.8417 0.568319
$$683$$ −11.5893 −0.443451 −0.221725 0.975109i $$-0.571169\pi$$
−0.221725 + 0.975109i $$0.571169\pi$$
$$684$$ 14.5297 0.555557
$$685$$ 0 0
$$686$$ 27.3885 1.04570
$$687$$ 10.2248 0.390099
$$688$$ −18.8157 −0.717340
$$689$$ −51.1035 −1.94689
$$690$$ 0 0
$$691$$ −21.6424 −0.823317 −0.411658 0.911338i $$-0.635050\pi$$
−0.411658 + 0.911338i $$0.635050\pi$$
$$692$$ −25.2353 −0.959303
$$693$$ 4.86205 0.184694
$$694$$ −3.70522 −0.140648
$$695$$ 0 0
$$696$$ 3.41139 0.129308
$$697$$ 11.0124 0.417123
$$698$$ 33.6081 1.27208
$$699$$ 22.5229 0.851896
$$700$$ 0 0
$$701$$ −8.61904 −0.325537 −0.162768 0.986664i $$-0.552042\pi$$
−0.162768 + 0.986664i $$0.552042\pi$$
$$702$$ −11.3244 −0.427413
$$703$$ 32.4438 1.22364
$$704$$ −18.6964 −0.704648
$$705$$ 0 0
$$706$$ 33.6044 1.26472
$$707$$ 3.43739 0.129276
$$708$$ 10.9728 0.412381
$$709$$ 16.0001 0.600895 0.300447 0.953798i $$-0.402864\pi$$
0.300447 + 0.953798i $$0.402864\pi$$
$$710$$ 0 0
$$711$$ 1.61849 0.0606981
$$712$$ −10.5014 −0.393555
$$713$$ −28.7682 −1.07738
$$714$$ 8.81353 0.329838
$$715$$ 0 0
$$716$$ 10.8788 0.406558
$$717$$ −0.589074 −0.0219994
$$718$$ 53.0138 1.97846
$$719$$ −48.5838 −1.81187 −0.905935 0.423416i $$-0.860831\pi$$
−0.905935 + 0.423416i $$0.860831\pi$$
$$720$$ 0 0
$$721$$ 15.6055 0.581177
$$722$$ −23.0263 −0.856949
$$723$$ 1.50937 0.0561340
$$724$$ −29.6689 −1.10264
$$725$$ 0 0
$$726$$ 18.6531 0.692283
$$727$$ −25.4328 −0.943251 −0.471625 0.881799i $$-0.656333\pi$$
−0.471625 + 0.881799i $$0.656333\pi$$
$$728$$ −23.9094 −0.886143
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −10.9505 −0.405019
$$732$$ −28.5317 −1.05456
$$733$$ 18.6190 0.687708 0.343854 0.939023i $$-0.388267\pi$$
0.343854 + 0.939023i $$0.388267\pi$$
$$734$$ 60.7264 2.24145
$$735$$ 0 0
$$736$$ 49.4656 1.82333
$$737$$ 2.92078 0.107588
$$738$$ 18.4343 0.678576
$$739$$ 46.8872 1.72478 0.862388 0.506249i $$-0.168968\pi$$
0.862388 + 0.506249i $$0.168968\pi$$
$$740$$ 0 0
$$741$$ 28.5442 1.04860
$$742$$ 66.5780 2.44416
$$743$$ 42.7061 1.56674 0.783368 0.621558i $$-0.213500\pi$$
0.783368 + 0.621558i $$0.213500\pi$$
$$744$$ 6.44492 0.236282
$$745$$ 0 0
$$746$$ −20.9753 −0.767959
$$747$$ −13.1488 −0.481091
$$748$$ −5.29742 −0.193693
$$749$$ −48.3751 −1.76759
$$750$$ 0 0
$$751$$ −2.64893 −0.0966608 −0.0483304 0.998831i $$-0.515390\pi$$
−0.0483304 + 0.998831i $$0.515390\pi$$
$$752$$ 1.67009 0.0609021
$$753$$ 11.8953 0.433488
$$754$$ 26.7666 0.974781
$$755$$ 0 0
$$756$$ 8.43243 0.306685
$$757$$ −52.9153 −1.92324 −0.961620 0.274383i $$-0.911526\pi$$
−0.961620 + 0.274383i $$0.911526\pi$$
$$758$$ 48.1080 1.74736
$$759$$ 9.91073 0.359737
$$760$$ 0 0
$$761$$ 50.7787 1.84073 0.920363 0.391064i $$-0.127893\pi$$
0.920363 + 0.391064i $$0.127893\pi$$
$$762$$ 21.6455 0.784133
$$763$$ 21.0462 0.761922
$$764$$ 9.63514 0.348587
$$765$$ 0 0
$$766$$ −29.4483 −1.06401
$$767$$ 21.5565 0.778358
$$768$$ 0.752056 0.0271375
$$769$$ 24.5805 0.886396 0.443198 0.896424i $$-0.353844\pi$$
0.443198 + 0.896424i $$0.353844\pi$$
$$770$$ 0 0
$$771$$ 18.0492 0.650026
$$772$$ 45.5064 1.63781
$$773$$ 20.0112 0.719754 0.359877 0.933000i $$-0.382819\pi$$
0.359877 + 0.933000i $$0.382819\pi$$
$$774$$ −18.3308 −0.658885
$$775$$ 0 0
$$776$$ 14.7128 0.528159
$$777$$ 18.8291 0.675489
$$778$$ −41.5485 −1.48958
$$779$$ −46.4653 −1.66479
$$780$$ 0 0
$$781$$ −0.109034 −0.00390155
$$782$$ 17.9654 0.642441
$$783$$ −2.36361 −0.0844686
$$784$$ −6.62905 −0.236752
$$785$$ 0 0
$$786$$ 13.7815 0.491568
$$787$$ −8.99272 −0.320556 −0.160278 0.987072i $$-0.551239\pi$$
−0.160278 + 0.987072i $$0.551239\pi$$
$$788$$ 11.7532 0.418692
$$789$$ 19.1066 0.680215
$$790$$ 0 0
$$791$$ 28.3547 1.00818
$$792$$ −2.22029 −0.0788947
$$793$$ −56.0517 −1.99045
$$794$$ 48.0252 1.70435
$$795$$ 0 0
$$796$$ 44.2773 1.56937
$$797$$ 37.7564 1.33740 0.668700 0.743533i $$-0.266851\pi$$
0.668700 + 0.743533i $$0.266851\pi$$
$$798$$ −37.1876 −1.31643
$$799$$ 0.971976 0.0343860
$$800$$ 0 0
$$801$$ 7.27597 0.257084
$$802$$ −10.1920 −0.359891
$$803$$ −6.17067 −0.217758
$$804$$ 5.06562 0.178651
$$805$$ 0 0
$$806$$ 50.5684 1.78120
$$807$$ −22.2250 −0.782358
$$808$$ −1.56971 −0.0552223
$$809$$ −48.9001 −1.71924 −0.859618 0.510937i $$-0.829298\pi$$
−0.859618 + 0.510937i $$0.829298\pi$$
$$810$$ 0 0
$$811$$ −18.0551 −0.633999 −0.317000 0.948426i $$-0.602675\pi$$
−0.317000 + 0.948426i $$0.602675\pi$$
$$812$$ −19.9310 −0.699441
$$813$$ 20.2107 0.708822
$$814$$ −19.8010 −0.694024
$$815$$ 0 0
$$816$$ 2.86237 0.100203
$$817$$ 46.2042 1.61648
$$818$$ 26.9653 0.942819
$$819$$ 16.5659 0.578859
$$820$$ 0 0
$$821$$ 50.6097 1.76629 0.883146 0.469099i $$-0.155421\pi$$
0.883146 + 0.469099i $$0.155421\pi$$
$$822$$ −19.8281 −0.691586
$$823$$ 26.8070 0.934433 0.467217 0.884143i $$-0.345257\pi$$
0.467217 + 0.884143i $$0.345257\pi$$
$$824$$ −7.12635 −0.248258
$$825$$ 0 0
$$826$$ −28.0839 −0.977163
$$827$$ 21.3649 0.742929 0.371465 0.928447i $$-0.378856\pi$$
0.371465 + 0.928447i $$0.378856\pi$$
$$828$$ 17.1885 0.597343
$$829$$ −48.1123 −1.67101 −0.835504 0.549484i $$-0.814824\pi$$
−0.835504 + 0.549484i $$0.814824\pi$$
$$830$$ 0 0
$$831$$ 16.0413 0.556465
$$832$$ −63.7021 −2.20847
$$833$$ −3.85803 −0.133673
$$834$$ 20.0184 0.693182
$$835$$ 0 0
$$836$$ 22.3518 0.773052
$$837$$ −4.46542 −0.154348
$$838$$ −75.5080 −2.60838
$$839$$ −41.8540 −1.44496 −0.722481 0.691391i $$-0.756998\pi$$
−0.722481 + 0.691391i $$0.756998\pi$$
$$840$$ 0 0
$$841$$ −23.4133 −0.807357
$$842$$ 85.4719 2.94556
$$843$$ 8.37308 0.288384
$$844$$ 49.4794 1.70315
$$845$$ 0 0
$$846$$ 1.62705 0.0559393
$$847$$ −27.2866 −0.937579
$$848$$ 21.6225 0.742520
$$849$$ 1.12995 0.0387798
$$850$$ 0 0
$$851$$ 38.3809 1.31568
$$852$$ −0.189102 −0.00647853
$$853$$ 58.2023 1.99281 0.996404 0.0847260i $$-0.0270015\pi$$
0.996404 + 0.0847260i $$0.0270015\pi$$
$$854$$ 73.0245 2.49885
$$855$$ 0 0
$$856$$ 22.0909 0.755051
$$857$$ 3.22045 0.110008 0.0550042 0.998486i $$-0.482483\pi$$
0.0550042 + 0.998486i $$0.482483\pi$$
$$858$$ −17.4210 −0.594742
$$859$$ 49.4431 1.68698 0.843488 0.537147i $$-0.180498\pi$$
0.843488 + 0.537147i $$0.180498\pi$$
$$860$$ 0 0
$$861$$ −26.9665 −0.919016
$$862$$ 30.3740 1.03454
$$863$$ −21.4924 −0.731611 −0.365806 0.930691i $$-0.619207\pi$$
−0.365806 + 0.930691i $$0.619207\pi$$
$$864$$ 7.67809 0.261214
$$865$$ 0 0
$$866$$ −1.18761 −0.0403567
$$867$$ −15.3341 −0.520775
$$868$$ −37.6544 −1.27807
$$869$$ 2.48981 0.0844609
$$870$$ 0 0
$$871$$ 9.95163 0.337198
$$872$$ −9.61090 −0.325466
$$873$$ −10.1939 −0.345012
$$874$$ −75.8026 −2.56406
$$875$$ 0 0
$$876$$ −10.7020 −0.361588
$$877$$ −15.3793 −0.519321 −0.259661 0.965700i $$-0.583611\pi$$
−0.259661 + 0.965700i $$0.583611\pi$$
$$878$$ 6.42501 0.216833
$$879$$ 2.37857 0.0802273
$$880$$ 0 0
$$881$$ 18.9275 0.637682 0.318841 0.947808i $$-0.396706\pi$$
0.318841 + 0.947808i $$0.396706\pi$$
$$882$$ −6.45821 −0.217459
$$883$$ −6.31708 −0.212587 −0.106293 0.994335i $$-0.533898\pi$$
−0.106293 + 0.994335i $$0.533898\pi$$
$$884$$ −18.0492 −0.607062
$$885$$ 0 0
$$886$$ −79.2694 −2.66311
$$887$$ −0.0721586 −0.00242285 −0.00121142 0.999999i $$-0.500386\pi$$
−0.00121142 + 0.999999i $$0.500386\pi$$
$$888$$ −8.59844 −0.288545
$$889$$ −31.6639 −1.06197
$$890$$ 0 0
$$891$$ 1.53835 0.0515367
$$892$$ 2.05853 0.0689246
$$893$$ −4.10113 −0.137239
$$894$$ 21.7822 0.728505
$$895$$ 0 0
$$896$$ 34.4573 1.15114
$$897$$ 33.7676 1.12747
$$898$$ 38.5512 1.28647
$$899$$ 10.5545 0.352013
$$900$$ 0 0
$$901$$ 12.5840 0.419235
$$902$$ 28.3584 0.944233
$$903$$ 26.8150 0.892348
$$904$$ −12.9484 −0.430658
$$905$$ 0 0
$$906$$ −23.1990 −0.770735
$$907$$ 25.4951 0.846550 0.423275 0.906001i $$-0.360880\pi$$
0.423275 + 0.906001i $$0.360880\pi$$
$$908$$ −30.4991 −1.01215
$$909$$ 1.08759 0.0360731
$$910$$ 0 0
$$911$$ 3.55803 0.117883 0.0589414 0.998261i $$-0.481227\pi$$
0.0589414 + 0.998261i $$0.481227\pi$$
$$912$$ −12.0774 −0.399922
$$913$$ −20.2275 −0.669434
$$914$$ −52.2389 −1.72791
$$915$$ 0 0
$$916$$ 27.2798 0.901351
$$917$$ −20.1601 −0.665745
$$918$$ 2.78860 0.0920375
$$919$$ 7.96968 0.262896 0.131448 0.991323i $$-0.458037\pi$$
0.131448 + 0.991323i $$0.458037\pi$$
$$920$$ 0 0
$$921$$ −2.89366 −0.0953492
$$922$$ 80.7776 2.66027
$$923$$ −0.371499 −0.0122280
$$924$$ 12.9720 0.426749
$$925$$ 0 0
$$926$$ 44.1434 1.45064
$$927$$ 4.93756 0.162171
$$928$$ −18.1480 −0.595738
$$929$$ 48.5424 1.59263 0.796313 0.604885i $$-0.206781\pi$$
0.796313 + 0.604885i $$0.206781\pi$$
$$930$$ 0 0
$$931$$ 16.2785 0.533505
$$932$$ 60.0916 1.96837
$$933$$ −7.14545 −0.233932
$$934$$ 67.4363 2.20658
$$935$$ 0 0
$$936$$ −7.56494 −0.247268
$$937$$ −31.7296 −1.03656 −0.518280 0.855211i $$-0.673427\pi$$
−0.518280 + 0.855211i $$0.673427\pi$$
$$938$$ −12.9650 −0.423324
$$939$$ −23.0318 −0.751616
$$940$$ 0 0
$$941$$ 41.0068 1.33678 0.668392 0.743809i $$-0.266983\pi$$
0.668392 + 0.743809i $$0.266983\pi$$
$$942$$ 37.6839 1.22781
$$943$$ −54.9681 −1.79001
$$944$$ −9.12079 −0.296856
$$945$$ 0 0
$$946$$ −28.1991 −0.916833
$$947$$ 9.81789 0.319038 0.159519 0.987195i $$-0.449006\pi$$
0.159519 + 0.987195i $$0.449006\pi$$
$$948$$ 4.31816 0.140247
$$949$$ −21.0246 −0.682487
$$950$$ 0 0
$$951$$ 20.7750 0.673674
$$952$$ 5.88761 0.190818
$$953$$ −53.1118 −1.72046 −0.860230 0.509906i $$-0.829680\pi$$
−0.860230 + 0.509906i $$0.829680\pi$$
$$954$$ 21.0653 0.682013
$$955$$ 0 0
$$956$$ −1.57166 −0.0508311
$$957$$ −3.63606 −0.117537
$$958$$ −23.9756 −0.774618
$$959$$ 29.0055 0.936635
$$960$$ 0 0
$$961$$ −11.0600 −0.356774
$$962$$ −67.4654 −2.17517
$$963$$ −15.3059 −0.493225
$$964$$ 4.02702 0.129702
$$965$$ 0 0
$$966$$ −43.9927 −1.41544
$$967$$ 2.44732 0.0787004 0.0393502 0.999225i $$-0.487471\pi$$
0.0393502 + 0.999225i $$0.487471\pi$$
$$968$$ 12.4606 0.400500
$$969$$ −7.02890 −0.225801
$$970$$ 0 0
$$971$$ −47.9683 −1.53938 −0.769688 0.638421i $$-0.779588\pi$$
−0.769688 + 0.638421i $$0.779588\pi$$
$$972$$ 2.66802 0.0855767
$$973$$ −29.2838 −0.938796
$$974$$ 15.4200 0.494088
$$975$$ 0 0
$$976$$ 23.7161 0.759134
$$977$$ −23.6950 −0.758070 −0.379035 0.925382i $$-0.623744\pi$$
−0.379035 + 0.925382i $$0.623744\pi$$
$$978$$ −35.1251 −1.12318
$$979$$ 11.1930 0.357730
$$980$$ 0 0
$$981$$ 6.65900 0.212605
$$982$$ −15.6284 −0.498721
$$983$$ −12.9505 −0.413057 −0.206529 0.978441i $$-0.566217\pi$$
−0.206529 + 0.978441i $$0.566217\pi$$
$$984$$ 12.3145 0.392571
$$985$$ 0 0
$$986$$ −6.59116 −0.209905
$$987$$ −2.38012 −0.0757602
$$988$$ 76.1565 2.42286
$$989$$ 54.6593 1.73807
$$990$$ 0 0
$$991$$ 44.4967 1.41348 0.706742 0.707471i $$-0.250164\pi$$
0.706742 + 0.707471i $$0.250164\pi$$
$$992$$ −34.2859 −1.08858
$$993$$ −19.2504 −0.610891
$$994$$ 0.483992 0.0153513
$$995$$ 0 0
$$996$$ −35.0814 −1.11160
$$997$$ 55.5513 1.75933 0.879663 0.475597i $$-0.157768\pi$$
0.879663 + 0.475597i $$0.157768\pi$$
$$998$$ −34.6054 −1.09542
$$999$$ 5.95751 0.188487
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.k.1.2 6
3.2 odd 2 5625.2.a.q.1.5 6
5.2 odd 4 1875.2.b.f.1249.3 12
5.3 odd 4 1875.2.b.f.1249.10 12
5.4 even 2 1875.2.a.j.1.5 6
15.14 odd 2 5625.2.a.p.1.2 6
25.3 odd 20 375.2.i.d.49.6 24
25.4 even 10 75.2.g.c.16.1 12
25.6 even 5 375.2.g.c.301.3 12
25.8 odd 20 375.2.i.d.199.1 24
25.17 odd 20 375.2.i.d.199.6 24
25.19 even 10 75.2.g.c.61.1 yes 12
25.21 even 5 375.2.g.c.76.3 12
25.22 odd 20 375.2.i.d.49.1 24
75.29 odd 10 225.2.h.d.91.3 12
75.44 odd 10 225.2.h.d.136.3 12

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.16.1 12 25.4 even 10
75.2.g.c.61.1 yes 12 25.19 even 10
225.2.h.d.91.3 12 75.29 odd 10
225.2.h.d.136.3 12 75.44 odd 10
375.2.g.c.76.3 12 25.21 even 5
375.2.g.c.301.3 12 25.6 even 5
375.2.i.d.49.1 24 25.22 odd 20
375.2.i.d.49.6 24 25.3 odd 20
375.2.i.d.199.1 24 25.8 odd 20
375.2.i.d.199.6 24 25.17 odd 20
1875.2.a.j.1.5 6 5.4 even 2
1875.2.a.k.1.2 6 1.1 even 1 trivial
1875.2.b.f.1249.3 12 5.2 odd 4
1875.2.b.f.1249.10 12 5.3 odd 4
5625.2.a.p.1.2 6 15.14 odd 2
5625.2.a.q.1.5 6 3.2 odd 2