# Properties

 Label 1875.2.a.e.1.1 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $1$ Dimension $4$ 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: $$1$$ 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 1.1 Root $$2.70636$$ 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)

 $$f(q)$$ $$=$$ $$q-2.70636 q^{2} -1.00000 q^{3} +5.32440 q^{4} +2.70636 q^{6} -0.470294 q^{7} -8.99702 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.70636 q^{2} -1.00000 q^{3} +5.32440 q^{4} +2.70636 q^{6} -0.470294 q^{7} -8.99702 q^{8} +1.00000 q^{9} -3.18148 q^{11} -5.32440 q^{12} -0.563444 q^{13} +1.27279 q^{14} +13.7004 q^{16} +1.70636 q^{17} -2.70636 q^{18} +3.74010 q^{19} +0.470294 q^{21} +8.61023 q^{22} +2.26981 q^{23} +8.99702 q^{24} +1.52488 q^{26} -1.00000 q^{27} -2.50403 q^{28} -8.32440 q^{29} +5.43656 q^{31} -19.0842 q^{32} +3.18148 q^{33} -4.61803 q^{34} +5.32440 q^{36} -1.02085 q^{37} -10.1221 q^{38} +0.563444 q^{39} +1.47214 q^{41} -1.27279 q^{42} -6.72721 q^{43} -16.9395 q^{44} -6.14292 q^{46} +4.43358 q^{47} -13.7004 q^{48} -6.77882 q^{49} -1.70636 q^{51} -3.00000 q^{52} +7.05161 q^{53} +2.70636 q^{54} +4.23125 q^{56} -3.74010 q^{57} +22.5288 q^{58} -12.8720 q^{59} +0.126888 q^{61} -14.7133 q^{62} -0.470294 q^{63} +24.2480 q^{64} -8.61023 q^{66} +2.79469 q^{67} +9.08535 q^{68} -2.26981 q^{69} +16.0456 q^{71} -8.99702 q^{72} +9.78873 q^{73} +2.76279 q^{74} +19.9138 q^{76} +1.49623 q^{77} -1.52488 q^{78} -4.75499 q^{79} +1.00000 q^{81} -3.98413 q^{82} -11.6905 q^{83} +2.50403 q^{84} +18.2063 q^{86} +8.32440 q^{87} +28.6238 q^{88} -6.20049 q^{89} +0.264985 q^{91} +12.0853 q^{92} -5.43656 q^{93} -11.9989 q^{94} +19.0842 q^{96} -8.45443 q^{97} +18.3460 q^{98} -3.18148 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} - 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.70636 −1.91369 −0.956844 0.290604i $$-0.906144\pi$$
−0.956844 + 0.290604i $$0.906144\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 5.32440 2.66220
$$5$$ 0 0
$$6$$ 2.70636 1.10487
$$7$$ −0.470294 −0.177754 −0.0888772 0.996043i $$-0.528328\pi$$
−0.0888772 + 0.996043i $$0.528328\pi$$
$$8$$ −8.99702 −3.18093
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.18148 −0.959252 −0.479626 0.877473i $$-0.659228\pi$$
−0.479626 + 0.877473i $$0.659228\pi$$
$$12$$ −5.32440 −1.53702
$$13$$ −0.563444 −0.156271 −0.0781356 0.996943i $$-0.524897\pi$$
−0.0781356 + 0.996943i $$0.524897\pi$$
$$14$$ 1.27279 0.340166
$$15$$ 0 0
$$16$$ 13.7004 3.42510
$$17$$ 1.70636 0.413854 0.206927 0.978356i $$-0.433654\pi$$
0.206927 + 0.978356i $$0.433654\pi$$
$$18$$ −2.70636 −0.637896
$$19$$ 3.74010 0.858038 0.429019 0.903295i $$-0.358859\pi$$
0.429019 + 0.903295i $$0.358859\pi$$
$$20$$ 0 0
$$21$$ 0.470294 0.102627
$$22$$ 8.61023 1.83571
$$23$$ 2.26981 0.473287 0.236644 0.971597i $$-0.423953\pi$$
0.236644 + 0.971597i $$0.423953\pi$$
$$24$$ 8.99702 1.83651
$$25$$ 0 0
$$26$$ 1.52488 0.299054
$$27$$ −1.00000 −0.192450
$$28$$ −2.50403 −0.473218
$$29$$ −8.32440 −1.54580 −0.772901 0.634527i $$-0.781195\pi$$
−0.772901 + 0.634527i $$0.781195\pi$$
$$30$$ 0 0
$$31$$ 5.43656 0.976434 0.488217 0.872722i $$-0.337647\pi$$
0.488217 + 0.872722i $$0.337647\pi$$
$$32$$ −19.0842 −3.37364
$$33$$ 3.18148 0.553824
$$34$$ −4.61803 −0.791986
$$35$$ 0 0
$$36$$ 5.32440 0.887399
$$37$$ −1.02085 −0.167827 −0.0839135 0.996473i $$-0.526742\pi$$
−0.0839135 + 0.996473i $$0.526742\pi$$
$$38$$ −10.1221 −1.64202
$$39$$ 0.563444 0.0902233
$$40$$ 0 0
$$41$$ 1.47214 0.229909 0.114955 0.993371i $$-0.463328\pi$$
0.114955 + 0.993371i $$0.463328\pi$$
$$42$$ −1.27279 −0.196395
$$43$$ −6.72721 −1.02589 −0.512945 0.858421i $$-0.671446\pi$$
−0.512945 + 0.858421i $$0.671446\pi$$
$$44$$ −16.9395 −2.55372
$$45$$ 0 0
$$46$$ −6.14292 −0.905724
$$47$$ 4.43358 0.646703 0.323352 0.946279i $$-0.395190\pi$$
0.323352 + 0.946279i $$0.395190\pi$$
$$48$$ −13.7004 −1.97748
$$49$$ −6.77882 −0.968403
$$50$$ 0 0
$$51$$ −1.70636 −0.238938
$$52$$ −3.00000 −0.416025
$$53$$ 7.05161 0.968613 0.484307 0.874898i $$-0.339072\pi$$
0.484307 + 0.874898i $$0.339072\pi$$
$$54$$ 2.70636 0.368289
$$55$$ 0 0
$$56$$ 4.23125 0.565424
$$57$$ −3.74010 −0.495388
$$58$$ 22.5288 2.95818
$$59$$ −12.8720 −1.67579 −0.837894 0.545833i $$-0.816213\pi$$
−0.837894 + 0.545833i $$0.816213\pi$$
$$60$$ 0 0
$$61$$ 0.126888 0.0162464 0.00812319 0.999967i $$-0.497414\pi$$
0.00812319 + 0.999967i $$0.497414\pi$$
$$62$$ −14.7133 −1.86859
$$63$$ −0.470294 −0.0592515
$$64$$ 24.2480 3.03100
$$65$$ 0 0
$$66$$ −8.61023 −1.05985
$$67$$ 2.79469 0.341426 0.170713 0.985321i $$-0.445393\pi$$
0.170713 + 0.985321i $$0.445393\pi$$
$$68$$ 9.08535 1.10176
$$69$$ −2.26981 −0.273253
$$70$$ 0 0
$$71$$ 16.0456 1.90427 0.952134 0.305681i $$-0.0988839\pi$$
0.952134 + 0.305681i $$0.0988839\pi$$
$$72$$ −8.99702 −1.06031
$$73$$ 9.78873 1.14568 0.572842 0.819666i $$-0.305841\pi$$
0.572842 + 0.819666i $$0.305841\pi$$
$$74$$ 2.76279 0.321168
$$75$$ 0 0
$$76$$ 19.9138 2.28427
$$77$$ 1.49623 0.170511
$$78$$ −1.52488 −0.172659
$$79$$ −4.75499 −0.534978 −0.267489 0.963561i $$-0.586194\pi$$
−0.267489 + 0.963561i $$0.586194\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −3.98413 −0.439974
$$83$$ −11.6905 −1.28320 −0.641599 0.767040i $$-0.721729\pi$$
−0.641599 + 0.767040i $$0.721729\pi$$
$$84$$ 2.50403 0.273212
$$85$$ 0 0
$$86$$ 18.2063 1.96323
$$87$$ 8.32440 0.892469
$$88$$ 28.6238 3.05131
$$89$$ −6.20049 −0.657250 −0.328625 0.944460i $$-0.606585\pi$$
−0.328625 + 0.944460i $$0.606585\pi$$
$$90$$ 0 0
$$91$$ 0.264985 0.0277779
$$92$$ 12.0853 1.25998
$$93$$ −5.43656 −0.563745
$$94$$ −11.9989 −1.23759
$$95$$ 0 0
$$96$$ 19.0842 1.94777
$$97$$ −8.45443 −0.858417 −0.429209 0.903205i $$-0.641207\pi$$
−0.429209 + 0.903205i $$0.641207\pi$$
$$98$$ 18.3460 1.85322
$$99$$ −3.18148 −0.319751
$$100$$ 0 0
$$101$$ −5.83325 −0.580430 −0.290215 0.956961i $$-0.593727\pi$$
−0.290215 + 0.956961i $$0.593727\pi$$
$$102$$ 4.61803 0.457254
$$103$$ 14.0794 1.38728 0.693642 0.720320i $$-0.256005\pi$$
0.693642 + 0.720320i $$0.256005\pi$$
$$104$$ 5.06932 0.497088
$$105$$ 0 0
$$106$$ −19.0842 −1.85362
$$107$$ 8.61207 0.832561 0.416280 0.909236i $$-0.363334\pi$$
0.416280 + 0.909236i $$0.363334\pi$$
$$108$$ −5.32440 −0.512340
$$109$$ 16.6875 1.59837 0.799187 0.601082i $$-0.205264\pi$$
0.799187 + 0.601082i $$0.205264\pi$$
$$110$$ 0 0
$$111$$ 1.02085 0.0968949
$$112$$ −6.44322 −0.608827
$$113$$ −11.7758 −1.10778 −0.553889 0.832590i $$-0.686857\pi$$
−0.553889 + 0.832590i $$0.686857\pi$$
$$114$$ 10.1221 0.948018
$$115$$ 0 0
$$116$$ −44.3224 −4.11523
$$117$$ −0.563444 −0.0520904
$$118$$ 34.8362 3.20693
$$119$$ −0.802492 −0.0735643
$$120$$ 0 0
$$121$$ −0.878197 −0.0798361
$$122$$ −0.343406 −0.0310905
$$123$$ −1.47214 −0.132738
$$124$$ 28.9464 2.59946
$$125$$ 0 0
$$126$$ 1.27279 0.113389
$$127$$ −1.77882 −0.157845 −0.0789225 0.996881i $$-0.525148\pi$$
−0.0789225 + 0.996881i $$0.525148\pi$$
$$128$$ −27.4554 −2.42674
$$129$$ 6.72721 0.592298
$$130$$ 0 0
$$131$$ −5.91860 −0.517110 −0.258555 0.965996i $$-0.583246\pi$$
−0.258555 + 0.965996i $$0.583246\pi$$
$$132$$ 16.9395 1.47439
$$133$$ −1.75895 −0.152520
$$134$$ −7.56344 −0.653382
$$135$$ 0 0
$$136$$ −15.3522 −1.31644
$$137$$ 5.11611 0.437098 0.218549 0.975826i $$-0.429868\pi$$
0.218549 + 0.975826i $$0.429868\pi$$
$$138$$ 6.14292 0.522920
$$139$$ −8.23817 −0.698753 −0.349376 0.936982i $$-0.613607\pi$$
−0.349376 + 0.936982i $$0.613607\pi$$
$$140$$ 0 0
$$141$$ −4.43358 −0.373374
$$142$$ −43.4253 −3.64417
$$143$$ 1.79259 0.149904
$$144$$ 13.7004 1.14170
$$145$$ 0 0
$$146$$ −26.4918 −2.19248
$$147$$ 6.77882 0.559108
$$148$$ −5.43542 −0.446789
$$149$$ −10.0585 −0.824027 −0.412014 0.911178i $$-0.635174\pi$$
−0.412014 + 0.911178i $$0.635174\pi$$
$$150$$ 0 0
$$151$$ −11.9810 −0.974999 −0.487500 0.873123i $$-0.662091\pi$$
−0.487500 + 0.873123i $$0.662091\pi$$
$$152$$ −33.6498 −2.72936
$$153$$ 1.70636 0.137951
$$154$$ −4.04934 −0.326305
$$155$$ 0 0
$$156$$ 3.00000 0.240192
$$157$$ −19.5960 −1.56393 −0.781967 0.623319i $$-0.785784\pi$$
−0.781967 + 0.623319i $$0.785784\pi$$
$$158$$ 12.8687 1.02378
$$159$$ −7.05161 −0.559229
$$160$$ 0 0
$$161$$ −1.06748 −0.0841290
$$162$$ −2.70636 −0.212632
$$163$$ −11.8927 −0.931505 −0.465753 0.884915i $$-0.654216\pi$$
−0.465753 + 0.884915i $$0.654216\pi$$
$$164$$ 7.83824 0.612063
$$165$$ 0 0
$$166$$ 31.6387 2.45564
$$167$$ 5.99378 0.463812 0.231906 0.972738i $$-0.425504\pi$$
0.231906 + 0.972738i $$0.425504\pi$$
$$168$$ −4.23125 −0.326448
$$169$$ −12.6825 −0.975579
$$170$$ 0 0
$$171$$ 3.74010 0.286013
$$172$$ −35.8184 −2.73112
$$173$$ −17.3244 −1.31715 −0.658575 0.752515i $$-0.728840\pi$$
−0.658575 + 0.752515i $$0.728840\pi$$
$$174$$ −22.5288 −1.70791
$$175$$ 0 0
$$176$$ −43.5875 −3.28553
$$177$$ 12.8720 0.967517
$$178$$ 16.7808 1.25777
$$179$$ 10.5765 0.790524 0.395262 0.918568i $$-0.370654\pi$$
0.395262 + 0.918568i $$0.370654\pi$$
$$180$$ 0 0
$$181$$ 11.9799 0.890455 0.445228 0.895417i $$-0.353123\pi$$
0.445228 + 0.895417i $$0.353123\pi$$
$$182$$ −0.717144 −0.0531583
$$183$$ −0.126888 −0.00937986
$$184$$ −20.4215 −1.50549
$$185$$ 0 0
$$186$$ 14.7133 1.07883
$$187$$ −5.42875 −0.396990
$$188$$ 23.6061 1.72165
$$189$$ 0.470294 0.0342089
$$190$$ 0 0
$$191$$ 18.2657 1.32166 0.660829 0.750536i $$-0.270205\pi$$
0.660829 + 0.750536i $$0.270205\pi$$
$$192$$ −24.2480 −1.74995
$$193$$ −25.2063 −1.81439 −0.907194 0.420713i $$-0.861780\pi$$
−0.907194 + 0.420713i $$0.861780\pi$$
$$194$$ 22.8807 1.64274
$$195$$ 0 0
$$196$$ −36.0931 −2.57808
$$197$$ −12.0923 −0.861539 −0.430769 0.902462i $$-0.641758\pi$$
−0.430769 + 0.902462i $$0.641758\pi$$
$$198$$ 8.61023 0.611903
$$199$$ −0.544434 −0.0385939 −0.0192970 0.999814i $$-0.506143\pi$$
−0.0192970 + 0.999814i $$0.506143\pi$$
$$200$$ 0 0
$$201$$ −2.79469 −0.197122
$$202$$ 15.7869 1.11076
$$203$$ 3.91492 0.274773
$$204$$ −9.08535 −0.636102
$$205$$ 0 0
$$206$$ −38.1039 −2.65483
$$207$$ 2.26981 0.157762
$$208$$ −7.71941 −0.535245
$$209$$ −11.8990 −0.823074
$$210$$ 0 0
$$211$$ −3.12207 −0.214932 −0.107466 0.994209i $$-0.534274\pi$$
−0.107466 + 0.994209i $$0.534274\pi$$
$$212$$ 37.5456 2.57864
$$213$$ −16.0456 −1.09943
$$214$$ −23.3074 −1.59326
$$215$$ 0 0
$$216$$ 8.99702 0.612170
$$217$$ −2.55678 −0.173566
$$218$$ −45.1625 −3.05879
$$219$$ −9.78873 −0.661461
$$220$$ 0 0
$$221$$ −0.961440 −0.0646734
$$222$$ −2.76279 −0.185427
$$223$$ −0.339125 −0.0227095 −0.0113547 0.999936i $$-0.503614\pi$$
−0.0113547 + 0.999936i $$0.503614\pi$$
$$224$$ 8.97519 0.599680
$$225$$ 0 0
$$226$$ 31.8697 2.11994
$$227$$ 1.78847 0.118705 0.0593524 0.998237i $$-0.481096\pi$$
0.0593524 + 0.998237i $$0.481096\pi$$
$$228$$ −19.9138 −1.31882
$$229$$ −27.9833 −1.84919 −0.924593 0.380957i $$-0.875595\pi$$
−0.924593 + 0.380957i $$0.875595\pi$$
$$230$$ 0 0
$$231$$ −1.49623 −0.0984448
$$232$$ 74.8948 4.91708
$$233$$ 6.77400 0.443780 0.221890 0.975072i $$-0.428777\pi$$
0.221890 + 0.975072i $$0.428777\pi$$
$$234$$ 1.52488 0.0996848
$$235$$ 0 0
$$236$$ −68.5355 −4.46128
$$237$$ 4.75499 0.308870
$$238$$ 2.17183 0.140779
$$239$$ −6.48334 −0.419373 −0.209686 0.977769i $$-0.567244\pi$$
−0.209686 + 0.977769i $$0.567244\pi$$
$$240$$ 0 0
$$241$$ −7.44857 −0.479804 −0.239902 0.970797i $$-0.577115\pi$$
−0.239902 + 0.970797i $$0.577115\pi$$
$$242$$ 2.37672 0.152781
$$243$$ −1.00000 −0.0641500
$$244$$ 0.675604 0.0432511
$$245$$ 0 0
$$246$$ 3.98413 0.254019
$$247$$ −2.10734 −0.134087
$$248$$ −48.9128 −3.10597
$$249$$ 11.6905 0.740855
$$250$$ 0 0
$$251$$ −4.60217 −0.290486 −0.145243 0.989396i $$-0.546396\pi$$
−0.145243 + 0.989396i $$0.546396\pi$$
$$252$$ −2.50403 −0.157739
$$253$$ −7.22134 −0.454002
$$254$$ 4.81414 0.302066
$$255$$ 0 0
$$256$$ 25.8083 1.61302
$$257$$ 5.79485 0.361473 0.180736 0.983532i $$-0.442152\pi$$
0.180736 + 0.983532i $$0.442152\pi$$
$$258$$ −18.2063 −1.13347
$$259$$ 0.480101 0.0298320
$$260$$ 0 0
$$261$$ −8.32440 −0.515267
$$262$$ 16.0179 0.989587
$$263$$ −14.2989 −0.881707 −0.440854 0.897579i $$-0.645324\pi$$
−0.440854 + 0.897579i $$0.645324\pi$$
$$264$$ −28.6238 −1.76167
$$265$$ 0 0
$$266$$ 4.76035 0.291876
$$267$$ 6.20049 0.379464
$$268$$ 14.8800 0.908943
$$269$$ −19.2205 −1.17189 −0.585946 0.810350i $$-0.699277\pi$$
−0.585946 + 0.810350i $$0.699277\pi$$
$$270$$ 0 0
$$271$$ −23.2433 −1.41193 −0.705964 0.708248i $$-0.749486\pi$$
−0.705964 + 0.708248i $$0.749486\pi$$
$$272$$ 23.3778 1.41749
$$273$$ −0.264985 −0.0160376
$$274$$ −13.8460 −0.836470
$$275$$ 0 0
$$276$$ −12.0853 −0.727452
$$277$$ −21.8910 −1.31530 −0.657651 0.753323i $$-0.728450\pi$$
−0.657651 + 0.753323i $$0.728450\pi$$
$$278$$ 22.2955 1.33719
$$279$$ 5.43656 0.325478
$$280$$ 0 0
$$281$$ 27.1138 1.61748 0.808738 0.588169i $$-0.200151\pi$$
0.808738 + 0.588169i $$0.200151\pi$$
$$282$$ 11.9989 0.714522
$$283$$ −16.3074 −0.969374 −0.484687 0.874688i $$-0.661066\pi$$
−0.484687 + 0.874688i $$0.661066\pi$$
$$284$$ 85.4334 5.06954
$$285$$ 0 0
$$286$$ −4.85139 −0.286868
$$287$$ −0.692337 −0.0408674
$$288$$ −19.0842 −1.12455
$$289$$ −14.0883 −0.828725
$$290$$ 0 0
$$291$$ 8.45443 0.495607
$$292$$ 52.1191 3.05004
$$293$$ −5.46407 −0.319214 −0.159607 0.987181i $$-0.551023\pi$$
−0.159607 + 0.987181i $$0.551023\pi$$
$$294$$ −18.3460 −1.06996
$$295$$ 0 0
$$296$$ 9.18462 0.533845
$$297$$ 3.18148 0.184608
$$298$$ 27.2220 1.57693
$$299$$ −1.27891 −0.0739612
$$300$$ 0 0
$$301$$ 3.16377 0.182357
$$302$$ 32.4249 1.86584
$$303$$ 5.83325 0.335111
$$304$$ 51.2409 2.93887
$$305$$ 0 0
$$306$$ −4.61803 −0.263995
$$307$$ 21.4194 1.22247 0.611235 0.791450i $$-0.290673\pi$$
0.611235 + 0.791450i $$0.290673\pi$$
$$308$$ 7.96652 0.453935
$$309$$ −14.0794 −0.800948
$$310$$ 0 0
$$311$$ −2.85200 −0.161722 −0.0808610 0.996725i $$-0.525767\pi$$
−0.0808610 + 0.996725i $$0.525767\pi$$
$$312$$ −5.06932 −0.286994
$$313$$ −14.4375 −0.816057 −0.408029 0.912969i $$-0.633784\pi$$
−0.408029 + 0.912969i $$0.633784\pi$$
$$314$$ 53.0340 2.99288
$$315$$ 0 0
$$316$$ −25.3175 −1.42422
$$317$$ −22.6102 −1.26992 −0.634959 0.772546i $$-0.718983\pi$$
−0.634959 + 0.772546i $$0.718983\pi$$
$$318$$ 19.0842 1.07019
$$319$$ 26.4839 1.48281
$$320$$ 0 0
$$321$$ −8.61207 −0.480679
$$322$$ 2.88898 0.160996
$$323$$ 6.38197 0.355102
$$324$$ 5.32440 0.295800
$$325$$ 0 0
$$326$$ 32.1859 1.78261
$$327$$ −16.6875 −0.922822
$$328$$ −13.2448 −0.731324
$$329$$ −2.08508 −0.114954
$$330$$ 0 0
$$331$$ 3.86645 0.212519 0.106260 0.994338i $$-0.466113\pi$$
0.106260 + 0.994338i $$0.466113\pi$$
$$332$$ −62.2448 −3.41613
$$333$$ −1.02085 −0.0559423
$$334$$ −16.2213 −0.887592
$$335$$ 0 0
$$336$$ 6.44322 0.351506
$$337$$ −17.4853 −0.952484 −0.476242 0.879314i $$-0.658001\pi$$
−0.476242 + 0.879314i $$0.658001\pi$$
$$338$$ 34.3235 1.86695
$$339$$ 11.7758 0.639576
$$340$$ 0 0
$$341$$ −17.2963 −0.936646
$$342$$ −10.1221 −0.547339
$$343$$ 6.48010 0.349893
$$344$$ 60.5249 3.26328
$$345$$ 0 0
$$346$$ 46.8861 2.52061
$$347$$ 23.0941 1.23976 0.619879 0.784698i $$-0.287182\pi$$
0.619879 + 0.784698i $$0.287182\pi$$
$$348$$ 44.3224 2.37593
$$349$$ −9.32650 −0.499236 −0.249618 0.968344i $$-0.580305\pi$$
−0.249618 + 0.968344i $$0.580305\pi$$
$$350$$ 0 0
$$351$$ 0.563444 0.0300744
$$352$$ 60.7160 3.23617
$$353$$ −31.9471 −1.70037 −0.850186 0.526483i $$-0.823510\pi$$
−0.850186 + 0.526483i $$0.823510\pi$$
$$354$$ −34.8362 −1.85152
$$355$$ 0 0
$$356$$ −33.0139 −1.74973
$$357$$ 0.802492 0.0424724
$$358$$ −28.6238 −1.51282
$$359$$ −9.58954 −0.506117 −0.253058 0.967451i $$-0.581436\pi$$
−0.253058 + 0.967451i $$0.581436\pi$$
$$360$$ 0 0
$$361$$ −5.01165 −0.263771
$$362$$ −32.4218 −1.70405
$$363$$ 0.878197 0.0460934
$$364$$ 1.41088 0.0739503
$$365$$ 0 0
$$366$$ 0.343406 0.0179501
$$367$$ −1.06423 −0.0555525 −0.0277763 0.999614i $$-0.508843\pi$$
−0.0277763 + 0.999614i $$0.508843\pi$$
$$368$$ 31.0973 1.62106
$$369$$ 1.47214 0.0766363
$$370$$ 0 0
$$371$$ −3.31633 −0.172175
$$372$$ −28.9464 −1.50080
$$373$$ 4.12022 0.213337 0.106669 0.994295i $$-0.465982\pi$$
0.106669 + 0.994295i $$0.465982\pi$$
$$374$$ 14.6922 0.759714
$$375$$ 0 0
$$376$$ −39.8890 −2.05712
$$377$$ 4.69033 0.241564
$$378$$ −1.27279 −0.0654651
$$379$$ 10.3563 0.531967 0.265984 0.963978i $$-0.414303\pi$$
0.265984 + 0.963978i $$0.414303\pi$$
$$380$$ 0 0
$$381$$ 1.77882 0.0911319
$$382$$ −49.4336 −2.52924
$$383$$ −1.19935 −0.0612839 −0.0306420 0.999530i $$-0.509755\pi$$
−0.0306420 + 0.999530i $$0.509755\pi$$
$$384$$ 27.4554 1.40108
$$385$$ 0 0
$$386$$ 68.2173 3.47217
$$387$$ −6.72721 −0.341963
$$388$$ −45.0147 −2.28528
$$389$$ 12.6043 0.639062 0.319531 0.947576i $$-0.396475\pi$$
0.319531 + 0.947576i $$0.396475\pi$$
$$390$$ 0 0
$$391$$ 3.87311 0.195872
$$392$$ 60.9892 3.08042
$$393$$ 5.91860 0.298554
$$394$$ 32.7261 1.64872
$$395$$ 0 0
$$396$$ −16.9395 −0.851239
$$397$$ 1.31335 0.0659152 0.0329576 0.999457i $$-0.489507\pi$$
0.0329576 + 0.999457i $$0.489507\pi$$
$$398$$ 1.47344 0.0738567
$$399$$ 1.75895 0.0880575
$$400$$ 0 0
$$401$$ 16.1042 0.804205 0.402102 0.915595i $$-0.368280\pi$$
0.402102 + 0.915595i $$0.368280\pi$$
$$402$$ 7.56344 0.377230
$$403$$ −3.06320 −0.152589
$$404$$ −31.0585 −1.54522
$$405$$ 0 0
$$406$$ −10.5952 −0.525830
$$407$$ 3.24782 0.160988
$$408$$ 15.3522 0.760046
$$409$$ 21.1600 1.04630 0.523148 0.852242i $$-0.324758\pi$$
0.523148 + 0.852242i $$0.324758\pi$$
$$410$$ 0 0
$$411$$ −5.11611 −0.252359
$$412$$ 74.9642 3.69322
$$413$$ 6.05361 0.297879
$$414$$ −6.14292 −0.301908
$$415$$ 0 0
$$416$$ 10.7529 0.527204
$$417$$ 8.23817 0.403425
$$418$$ 32.2031 1.57511
$$419$$ −6.67094 −0.325897 −0.162948 0.986635i $$-0.552100\pi$$
−0.162948 + 0.986635i $$0.552100\pi$$
$$420$$ 0 0
$$421$$ 27.2980 1.33042 0.665212 0.746655i $$-0.268341\pi$$
0.665212 + 0.746655i $$0.268341\pi$$
$$422$$ 8.44944 0.411312
$$423$$ 4.43358 0.215568
$$424$$ −63.4435 −3.08109
$$425$$ 0 0
$$426$$ 43.4253 2.10396
$$427$$ −0.0596749 −0.00288787
$$428$$ 45.8541 2.21644
$$429$$ −1.79259 −0.0865468
$$430$$ 0 0
$$431$$ 12.2974 0.592346 0.296173 0.955134i $$-0.404290\pi$$
0.296173 + 0.955134i $$0.404290\pi$$
$$432$$ −13.7004 −0.659161
$$433$$ 33.8452 1.62649 0.813247 0.581918i $$-0.197698\pi$$
0.813247 + 0.581918i $$0.197698\pi$$
$$434$$ 6.91957 0.332150
$$435$$ 0 0
$$436$$ 88.8509 4.25519
$$437$$ 8.48930 0.406098
$$438$$ 26.4918 1.26583
$$439$$ −20.9654 −1.00062 −0.500312 0.865845i $$-0.666781\pi$$
−0.500312 + 0.865845i $$0.666781\pi$$
$$440$$ 0 0
$$441$$ −6.77882 −0.322801
$$442$$ 2.60200 0.123765
$$443$$ −6.04847 −0.287371 −0.143686 0.989623i $$-0.545895\pi$$
−0.143686 + 0.989623i $$0.545895\pi$$
$$444$$ 5.43542 0.257954
$$445$$ 0 0
$$446$$ 0.917795 0.0434588
$$447$$ 10.0585 0.475752
$$448$$ −11.4037 −0.538773
$$449$$ 15.5896 0.735721 0.367860 0.929881i $$-0.380090\pi$$
0.367860 + 0.929881i $$0.380090\pi$$
$$450$$ 0 0
$$451$$ −4.68357 −0.220541
$$452$$ −62.6993 −2.94912
$$453$$ 11.9810 0.562916
$$454$$ −4.84024 −0.227164
$$455$$ 0 0
$$456$$ 33.6498 1.57579
$$457$$ 2.50193 0.117035 0.0585176 0.998286i $$-0.481363\pi$$
0.0585176 + 0.998286i $$0.481363\pi$$
$$458$$ 75.7328 3.53876
$$459$$ −1.70636 −0.0796462
$$460$$ 0 0
$$461$$ −0.153963 −0.00717078 −0.00358539 0.999994i $$-0.501141\pi$$
−0.00358539 + 0.999994i $$0.501141\pi$$
$$462$$ 4.04934 0.188392
$$463$$ 11.6327 0.540616 0.270308 0.962774i $$-0.412875\pi$$
0.270308 + 0.962774i $$0.412875\pi$$
$$464$$ −114.048 −5.29453
$$465$$ 0 0
$$466$$ −18.3329 −0.849255
$$467$$ −0.470294 −0.0217626 −0.0108813 0.999941i $$-0.503464\pi$$
−0.0108813 + 0.999941i $$0.503464\pi$$
$$468$$ −3.00000 −0.138675
$$469$$ −1.31433 −0.0606900
$$470$$ 0 0
$$471$$ 19.5960 0.902938
$$472$$ 115.809 5.33056
$$473$$ 21.4025 0.984087
$$474$$ −12.8687 −0.591080
$$475$$ 0 0
$$476$$ −4.27279 −0.195843
$$477$$ 7.05161 0.322871
$$478$$ 17.5463 0.802548
$$479$$ 12.4114 0.567092 0.283546 0.958959i $$-0.408489\pi$$
0.283546 + 0.958959i $$0.408489\pi$$
$$480$$ 0 0
$$481$$ 0.575193 0.0262265
$$482$$ 20.1585 0.918196
$$483$$ 1.06748 0.0485719
$$484$$ −4.67587 −0.212539
$$485$$ 0 0
$$486$$ 2.70636 0.122763
$$487$$ −34.5646 −1.56627 −0.783135 0.621852i $$-0.786381\pi$$
−0.783135 + 0.621852i $$0.786381\pi$$
$$488$$ −1.14162 −0.0516786
$$489$$ 11.8927 0.537805
$$490$$ 0 0
$$491$$ −24.8658 −1.12218 −0.561088 0.827756i $$-0.689617\pi$$
−0.561088 + 0.827756i $$0.689617\pi$$
$$492$$ −7.83824 −0.353375
$$493$$ −14.2044 −0.639736
$$494$$ 5.70322 0.256600
$$495$$ 0 0
$$496$$ 74.4830 3.34439
$$497$$ −7.54618 −0.338492
$$498$$ −31.6387 −1.41776
$$499$$ 12.6037 0.564220 0.282110 0.959382i $$-0.408966\pi$$
0.282110 + 0.959382i $$0.408966\pi$$
$$500$$ 0 0
$$501$$ −5.99378 −0.267782
$$502$$ 12.4551 0.555900
$$503$$ −29.9927 −1.33731 −0.668655 0.743573i $$-0.733130\pi$$
−0.668655 + 0.743573i $$0.733130\pi$$
$$504$$ 4.23125 0.188475
$$505$$ 0 0
$$506$$ 19.5436 0.868817
$$507$$ 12.6825 0.563251
$$508$$ −9.47116 −0.420215
$$509$$ −0.466989 −0.0206989 −0.0103495 0.999946i $$-0.503294\pi$$
−0.0103495 + 0.999946i $$0.503294\pi$$
$$510$$ 0 0
$$511$$ −4.60358 −0.203651
$$512$$ −14.9358 −0.660074
$$513$$ −3.74010 −0.165129
$$514$$ −15.6830 −0.691746
$$515$$ 0 0
$$516$$ 35.8184 1.57681
$$517$$ −14.1053 −0.620351
$$518$$ −1.29933 −0.0570891
$$519$$ 17.3244 0.760457
$$520$$ 0 0
$$521$$ −17.2819 −0.757133 −0.378566 0.925574i $$-0.623583\pi$$
−0.378566 + 0.925574i $$0.623583\pi$$
$$522$$ 22.5288 0.986060
$$523$$ −40.6479 −1.77741 −0.888705 0.458480i $$-0.848394\pi$$
−0.888705 + 0.458480i $$0.848394\pi$$
$$524$$ −31.5130 −1.37665
$$525$$ 0 0
$$526$$ 38.6980 1.68731
$$527$$ 9.27673 0.404101
$$528$$ 43.5875 1.89690
$$529$$ −17.8480 −0.775999
$$530$$ 0 0
$$531$$ −12.8720 −0.558596
$$532$$ −9.36533 −0.406039
$$533$$ −0.829466 −0.0359282
$$534$$ −16.7808 −0.726175
$$535$$ 0 0
$$536$$ −25.1439 −1.08605
$$537$$ −10.5765 −0.456409
$$538$$ 52.0175 2.24264
$$539$$ 21.5667 0.928943
$$540$$ 0 0
$$541$$ −1.98099 −0.0851694 −0.0425847 0.999093i $$-0.513559\pi$$
−0.0425847 + 0.999093i $$0.513559\pi$$
$$542$$ 62.9047 2.70199
$$543$$ −11.9799 −0.514105
$$544$$ −32.5646 −1.39619
$$545$$ 0 0
$$546$$ 0.717144 0.0306909
$$547$$ −35.0543 −1.49881 −0.749406 0.662111i $$-0.769661\pi$$
−0.749406 + 0.662111i $$0.769661\pi$$
$$548$$ 27.2402 1.16364
$$549$$ 0.126888 0.00541546
$$550$$ 0 0
$$551$$ −31.1341 −1.32636
$$552$$ 20.4215 0.869196
$$553$$ 2.23625 0.0950948
$$554$$ 59.2449 2.51708
$$555$$ 0 0
$$556$$ −43.8633 −1.86022
$$557$$ −19.2383 −0.815154 −0.407577 0.913171i $$-0.633626\pi$$
−0.407577 + 0.913171i $$0.633626\pi$$
$$558$$ −14.7133 −0.622863
$$559$$ 3.79041 0.160317
$$560$$ 0 0
$$561$$ 5.42875 0.229202
$$562$$ −73.3798 −3.09534
$$563$$ 29.1741 1.22954 0.614771 0.788706i $$-0.289248\pi$$
0.614771 + 0.788706i $$0.289248\pi$$
$$564$$ −23.6061 −0.993997
$$565$$ 0 0
$$566$$ 44.1337 1.85508
$$567$$ −0.470294 −0.0197505
$$568$$ −144.363 −6.05734
$$569$$ −32.9112 −1.37971 −0.689855 0.723947i $$-0.742326\pi$$
−0.689855 + 0.723947i $$0.742326\pi$$
$$570$$ 0 0
$$571$$ 26.6349 1.11464 0.557319 0.830299i $$-0.311830\pi$$
0.557319 + 0.830299i $$0.311830\pi$$
$$572$$ 9.54443 0.399073
$$573$$ −18.2657 −0.763060
$$574$$ 1.87371 0.0782073
$$575$$ 0 0
$$576$$ 24.2480 1.01033
$$577$$ −34.5156 −1.43690 −0.718452 0.695577i $$-0.755149\pi$$
−0.718452 + 0.695577i $$0.755149\pi$$
$$578$$ 38.1281 1.58592
$$579$$ 25.2063 1.04754
$$580$$ 0 0
$$581$$ 5.49797 0.228094
$$582$$ −22.8807 −0.948437
$$583$$ −22.4345 −0.929144
$$584$$ −88.0694 −3.64434
$$585$$ 0 0
$$586$$ 14.7878 0.610877
$$587$$ 22.2755 0.919408 0.459704 0.888072i $$-0.347955\pi$$
0.459704 + 0.888072i $$0.347955\pi$$
$$588$$ 36.0931 1.48846
$$589$$ 20.3333 0.837818
$$590$$ 0 0
$$591$$ 12.0923 0.497410
$$592$$ −13.9861 −0.574824
$$593$$ 30.9158 1.26956 0.634779 0.772693i $$-0.281091\pi$$
0.634779 + 0.772693i $$0.281091\pi$$
$$594$$ −8.61023 −0.353282
$$595$$ 0 0
$$596$$ −53.5556 −2.19372
$$597$$ 0.544434 0.0222822
$$598$$ 3.46119 0.141539
$$599$$ −24.0276 −0.981742 −0.490871 0.871232i $$-0.663321\pi$$
−0.490871 + 0.871232i $$0.663321\pi$$
$$600$$ 0 0
$$601$$ −32.0387 −1.30688 −0.653442 0.756977i $$-0.726676\pi$$
−0.653442 + 0.756977i $$0.726676\pi$$
$$602$$ −8.56231 −0.348974
$$603$$ 2.79469 0.113809
$$604$$ −63.7915 −2.59564
$$605$$ 0 0
$$606$$ −15.7869 −0.641299
$$607$$ −45.5915 −1.85050 −0.925251 0.379356i $$-0.876145\pi$$
−0.925251 + 0.379356i $$0.876145\pi$$
$$608$$ −71.3769 −2.89471
$$609$$ −3.91492 −0.158640
$$610$$ 0 0
$$611$$ −2.49807 −0.101061
$$612$$ 9.08535 0.367253
$$613$$ 10.8564 0.438485 0.219242 0.975670i $$-0.429641\pi$$
0.219242 + 0.975670i $$0.429641\pi$$
$$614$$ −57.9686 −2.33942
$$615$$ 0 0
$$616$$ −13.4616 −0.542384
$$617$$ 16.1054 0.648380 0.324190 0.945992i $$-0.394908\pi$$
0.324190 + 0.945992i $$0.394908\pi$$
$$618$$ 38.1039 1.53276
$$619$$ 38.3581 1.54174 0.770872 0.636991i $$-0.219821\pi$$
0.770872 + 0.636991i $$0.219821\pi$$
$$620$$ 0 0
$$621$$ −2.26981 −0.0910842
$$622$$ 7.71854 0.309485
$$623$$ 2.91605 0.116829
$$624$$ 7.71941 0.309024
$$625$$ 0 0
$$626$$ 39.0732 1.56168
$$627$$ 11.8990 0.475202
$$628$$ −104.337 −4.16350
$$629$$ −1.74194 −0.0694558
$$630$$ 0 0
$$631$$ 22.7292 0.904833 0.452417 0.891807i $$-0.350562\pi$$
0.452417 + 0.891807i $$0.350562\pi$$
$$632$$ 42.7808 1.70173
$$633$$ 3.12207 0.124091
$$634$$ 61.1915 2.43022
$$635$$ 0 0
$$636$$ −37.5456 −1.48878
$$637$$ 3.81949 0.151334
$$638$$ −71.6750 −2.83764
$$639$$ 16.0456 0.634756
$$640$$ 0 0
$$641$$ 26.9930 1.06616 0.533080 0.846065i $$-0.321035\pi$$
0.533080 + 0.846065i $$0.321035\pi$$
$$642$$ 23.3074 0.919869
$$643$$ 25.9062 1.02164 0.510821 0.859687i $$-0.329341\pi$$
0.510821 + 0.859687i $$0.329341\pi$$
$$644$$ −5.68367 −0.223968
$$645$$ 0 0
$$646$$ −17.2719 −0.679554
$$647$$ 21.6623 0.851632 0.425816 0.904810i $$-0.359987\pi$$
0.425816 + 0.904810i $$0.359987\pi$$
$$648$$ −8.99702 −0.353436
$$649$$ 40.9519 1.60750
$$650$$ 0 0
$$651$$ 2.55678 0.100208
$$652$$ −63.3212 −2.47985
$$653$$ −11.1775 −0.437411 −0.218705 0.975791i $$-0.570183\pi$$
−0.218705 + 0.975791i $$0.570183\pi$$
$$654$$ 45.1625 1.76599
$$655$$ 0 0
$$656$$ 20.1689 0.787461
$$657$$ 9.78873 0.381895
$$658$$ 5.64299 0.219987
$$659$$ 13.4744 0.524888 0.262444 0.964947i $$-0.415471\pi$$
0.262444 + 0.964947i $$0.415471\pi$$
$$660$$ 0 0
$$661$$ −23.2622 −0.904793 −0.452397 0.891817i $$-0.649431\pi$$
−0.452397 + 0.891817i $$0.649431\pi$$
$$662$$ −10.4640 −0.406695
$$663$$ 0.961440 0.0373392
$$664$$ 105.180 4.08176
$$665$$ 0 0
$$666$$ 2.76279 0.107056
$$667$$ −18.8948 −0.731608
$$668$$ 31.9132 1.23476
$$669$$ 0.339125 0.0131113
$$670$$ 0 0
$$671$$ −0.403693 −0.0155844
$$672$$ −8.97519 −0.346226
$$673$$ −18.8392 −0.726198 −0.363099 0.931751i $$-0.618281\pi$$
−0.363099 + 0.931751i $$0.618281\pi$$
$$674$$ 47.3215 1.82276
$$675$$ 0 0
$$676$$ −67.5268 −2.59719
$$677$$ 42.2440 1.62357 0.811785 0.583957i $$-0.198496\pi$$
0.811785 + 0.583957i $$0.198496\pi$$
$$678$$ −31.8697 −1.22395
$$679$$ 3.97607 0.152587
$$680$$ 0 0
$$681$$ −1.78847 −0.0685342
$$682$$ 46.8100 1.79245
$$683$$ −48.9502 −1.87303 −0.936513 0.350633i $$-0.885966\pi$$
−0.936513 + 0.350633i $$0.885966\pi$$
$$684$$ 19.9138 0.761422
$$685$$ 0 0
$$686$$ −17.5375 −0.669585
$$687$$ 27.9833 1.06763
$$688$$ −92.1655 −3.51378
$$689$$ −3.97319 −0.151366
$$690$$ 0 0
$$691$$ −14.1917 −0.539878 −0.269939 0.962877i $$-0.587004\pi$$
−0.269939 + 0.962877i $$0.587004\pi$$
$$692$$ −92.2419 −3.50651
$$693$$ 1.49623 0.0568371
$$694$$ −62.5010 −2.37251
$$695$$ 0 0
$$696$$ −74.8948 −2.83888
$$697$$ 2.51200 0.0951487
$$698$$ 25.2409 0.955382
$$699$$ −6.77400 −0.256216
$$700$$ 0 0
$$701$$ −9.00786 −0.340222 −0.170111 0.985425i $$-0.554413\pi$$
−0.170111 + 0.985425i $$0.554413\pi$$
$$702$$ −1.52488 −0.0575530
$$703$$ −3.81809 −0.144002
$$704$$ −77.1444 −2.90749
$$705$$ 0 0
$$706$$ 86.4604 3.25398
$$707$$ 2.74334 0.103174
$$708$$ 68.5355 2.57572
$$709$$ 49.6994 1.86650 0.933249 0.359229i $$-0.116960\pi$$
0.933249 + 0.359229i $$0.116960\pi$$
$$710$$ 0 0
$$711$$ −4.75499 −0.178326
$$712$$ 55.7859 2.09067
$$713$$ 12.3399 0.462134
$$714$$ −2.17183 −0.0812789
$$715$$ 0 0
$$716$$ 56.3134 2.10453
$$717$$ 6.48334 0.242125
$$718$$ 25.9528 0.968549
$$719$$ −2.13706 −0.0796988 −0.0398494 0.999206i $$-0.512688\pi$$
−0.0398494 + 0.999206i $$0.512688\pi$$
$$720$$ 0 0
$$721$$ −6.62145 −0.246596
$$722$$ 13.5633 0.504775
$$723$$ 7.44857 0.277015
$$724$$ 63.7855 2.37057
$$725$$ 0 0
$$726$$ −2.37672 −0.0882083
$$727$$ 41.4634 1.53779 0.768895 0.639375i $$-0.220807\pi$$
0.768895 + 0.639375i $$0.220807\pi$$
$$728$$ −2.38407 −0.0883596
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −11.4791 −0.424568
$$732$$ −0.675604 −0.0249710
$$733$$ −32.5210 −1.20119 −0.600596 0.799553i $$-0.705070\pi$$
−0.600596 + 0.799553i $$0.705070\pi$$
$$734$$ 2.88020 0.106310
$$735$$ 0 0
$$736$$ −43.3175 −1.59670
$$737$$ −8.89125 −0.327513
$$738$$ −3.98413 −0.146658
$$739$$ −35.3175 −1.29917 −0.649587 0.760287i $$-0.725058\pi$$
−0.649587 + 0.760287i $$0.725058\pi$$
$$740$$ 0 0
$$741$$ 2.10734 0.0774150
$$742$$ 8.97519 0.329490
$$743$$ 36.0897 1.32400 0.662001 0.749503i $$-0.269708\pi$$
0.662001 + 0.749503i $$0.269708\pi$$
$$744$$ 48.9128 1.79323
$$745$$ 0 0
$$746$$ −11.1508 −0.408261
$$747$$ −11.6905 −0.427733
$$748$$ −28.9048 −1.05687
$$749$$ −4.05021 −0.147991
$$750$$ 0 0
$$751$$ −4.62810 −0.168882 −0.0844409 0.996428i $$-0.526910\pi$$
−0.0844409 + 0.996428i $$0.526910\pi$$
$$752$$ 60.7418 2.21502
$$753$$ 4.60217 0.167712
$$754$$ −12.6937 −0.462279
$$755$$ 0 0
$$756$$ 2.50403 0.0910708
$$757$$ −14.6243 −0.531530 −0.265765 0.964038i $$-0.585625\pi$$
−0.265765 + 0.964038i $$0.585625\pi$$
$$758$$ −28.0279 −1.01802
$$759$$ 7.22134 0.262118
$$760$$ 0 0
$$761$$ 12.1893 0.441861 0.220930 0.975290i $$-0.429091\pi$$
0.220930 + 0.975290i $$0.429091\pi$$
$$762$$ −4.81414 −0.174398
$$763$$ −7.84804 −0.284118
$$764$$ 97.2538 3.51852
$$765$$ 0 0
$$766$$ 3.24587 0.117278
$$767$$ 7.25264 0.261878
$$768$$ −25.8083 −0.931276
$$769$$ 13.7961 0.497500 0.248750 0.968568i $$-0.419980\pi$$
0.248750 + 0.968568i $$0.419980\pi$$
$$770$$ 0 0
$$771$$ −5.79485 −0.208697
$$772$$ −134.208 −4.83026
$$773$$ −22.6539 −0.814803 −0.407402 0.913249i $$-0.633565\pi$$
−0.407402 + 0.913249i $$0.633565\pi$$
$$774$$ 18.2063 0.654411
$$775$$ 0 0
$$776$$ 76.0647 2.73056
$$777$$ −0.480101 −0.0172235
$$778$$ −34.1117 −1.22296
$$779$$ 5.50594 0.197271
$$780$$ 0 0
$$781$$ −51.0489 −1.82667
$$782$$ −10.4820 −0.374837
$$783$$ 8.32440 0.297490
$$784$$ −92.8726 −3.31688
$$785$$ 0 0
$$786$$ −16.0179 −0.571339
$$787$$ 47.6857 1.69981 0.849905 0.526936i $$-0.176659\pi$$
0.849905 + 0.526936i $$0.176659\pi$$
$$788$$ −64.3841 −2.29359
$$789$$ 14.2989 0.509054
$$790$$ 0 0
$$791$$ 5.53811 0.196913
$$792$$ 28.6238 1.01710
$$793$$ −0.0714945 −0.00253884
$$794$$ −3.55440 −0.126141
$$795$$ 0 0
$$796$$ −2.89878 −0.102745
$$797$$ −27.2131 −0.963938 −0.481969 0.876188i $$-0.660078\pi$$
−0.481969 + 0.876188i $$0.660078\pi$$
$$798$$ −4.76035 −0.168515
$$799$$ 7.56529 0.267641
$$800$$ 0 0
$$801$$ −6.20049 −0.219083
$$802$$ −43.5838 −1.53900
$$803$$ −31.1426 −1.09900
$$804$$ −14.8800 −0.524778
$$805$$ 0 0
$$806$$ 8.29012 0.292007
$$807$$ 19.2205 0.676592
$$808$$ 52.4819 1.84631
$$809$$ −19.6155 −0.689644 −0.344822 0.938668i $$-0.612061\pi$$
−0.344822 + 0.938668i $$0.612061\pi$$
$$810$$ 0 0
$$811$$ 21.0710 0.739903 0.369951 0.929051i $$-0.379374\pi$$
0.369951 + 0.929051i $$0.379374\pi$$
$$812$$ 20.8446 0.731501
$$813$$ 23.2433 0.815177
$$814$$ −8.78977 −0.308081
$$815$$ 0 0
$$816$$ −23.3778 −0.818388
$$817$$ −25.1605 −0.880253
$$818$$ −57.2667 −2.00228
$$819$$ 0.264985 0.00925931
$$820$$ 0 0
$$821$$ 41.4342 1.44606 0.723031 0.690815i $$-0.242748\pi$$
0.723031 + 0.690815i $$0.242748\pi$$
$$822$$ 13.8460 0.482936
$$823$$ −6.04093 −0.210574 −0.105287 0.994442i $$-0.533576\pi$$
−0.105287 + 0.994442i $$0.533576\pi$$
$$824$$ −126.673 −4.41285
$$825$$ 0 0
$$826$$ −16.3833 −0.570047
$$827$$ 2.99361 0.104098 0.0520491 0.998645i $$-0.483425\pi$$
0.0520491 + 0.998645i $$0.483425\pi$$
$$828$$ 12.0853 0.419995
$$829$$ 26.8530 0.932642 0.466321 0.884616i $$-0.345579\pi$$
0.466321 + 0.884616i $$0.345579\pi$$
$$830$$ 0 0
$$831$$ 21.8910 0.759390
$$832$$ −13.6624 −0.473658
$$833$$ −11.5671 −0.400777
$$834$$ −22.2955 −0.772029
$$835$$ 0 0
$$836$$ −63.3552 −2.19119
$$837$$ −5.43656 −0.187915
$$838$$ 18.0540 0.623665
$$839$$ 12.4595 0.430150 0.215075 0.976598i $$-0.431000\pi$$
0.215075 + 0.976598i $$0.431000\pi$$
$$840$$ 0 0
$$841$$ 40.2956 1.38950
$$842$$ −73.8783 −2.54601
$$843$$ −27.1138 −0.933850
$$844$$ −16.6231 −0.572191
$$845$$ 0 0
$$846$$ −11.9989 −0.412529
$$847$$ 0.413011 0.0141912
$$848$$ 96.6099 3.31760
$$849$$ 16.3074 0.559668
$$850$$ 0 0
$$851$$ −2.31714 −0.0794304
$$852$$ −85.4334 −2.92690
$$853$$ 5.69736 0.195074 0.0975369 0.995232i $$-0.468904\pi$$
0.0975369 + 0.995232i $$0.468904\pi$$
$$854$$ 0.161502 0.00552648
$$855$$ 0 0
$$856$$ −77.4830 −2.64831
$$857$$ 7.41982 0.253456 0.126728 0.991937i $$-0.459552\pi$$
0.126728 + 0.991937i $$0.459552\pi$$
$$858$$ 4.85139 0.165624
$$859$$ −5.68253 −0.193885 −0.0969427 0.995290i $$-0.530906\pi$$
−0.0969427 + 0.995290i $$0.530906\pi$$
$$860$$ 0 0
$$861$$ 0.692337 0.0235948
$$862$$ −33.2813 −1.13356
$$863$$ 34.5747 1.17693 0.588467 0.808521i $$-0.299732\pi$$
0.588467 + 0.808521i $$0.299732\pi$$
$$864$$ 19.0842 0.649258
$$865$$ 0 0
$$866$$ −91.5973 −3.11260
$$867$$ 14.0883 0.478465
$$868$$ −13.6133 −0.462066
$$869$$ 15.1279 0.513179
$$870$$ 0 0
$$871$$ −1.57465 −0.0533550
$$872$$ −150.138 −5.08431
$$873$$ −8.45443 −0.286139
$$874$$ −22.9751 −0.777145
$$875$$ 0 0
$$876$$ −52.1191 −1.76094
$$877$$ 5.30013 0.178973 0.0894863 0.995988i $$-0.471477\pi$$
0.0894863 + 0.995988i $$0.471477\pi$$
$$878$$ 56.7399 1.91488
$$879$$ 5.46407 0.184299
$$880$$ 0 0
$$881$$ 7.61288 0.256484 0.128242 0.991743i $$-0.459067\pi$$
0.128242 + 0.991743i $$0.459067\pi$$
$$882$$ 18.3460 0.617740
$$883$$ 52.8334 1.77799 0.888993 0.457921i $$-0.151406\pi$$
0.888993 + 0.457921i $$0.151406\pi$$
$$884$$ −5.11909 −0.172174
$$885$$ 0 0
$$886$$ 16.3693 0.549939
$$887$$ −7.70778 −0.258802 −0.129401 0.991592i $$-0.541305\pi$$
−0.129401 + 0.991592i $$0.541305\pi$$
$$888$$ −9.18462 −0.308216
$$889$$ 0.836570 0.0280577
$$890$$ 0 0
$$891$$ −3.18148 −0.106584
$$892$$ −1.80563 −0.0604571
$$893$$ 16.5820 0.554896
$$894$$ −27.2220 −0.910441
$$895$$ 0 0
$$896$$ 12.9121 0.431363
$$897$$ 1.27891 0.0427015
$$898$$ −42.1912 −1.40794
$$899$$ −45.2560 −1.50937
$$900$$ 0 0
$$901$$ 12.0326 0.400864
$$902$$ 12.6754 0.422046
$$903$$ −3.16377 −0.105284
$$904$$ 105.947 3.52376
$$905$$ 0 0
$$906$$ −32.4249 −1.07725
$$907$$ 11.3735 0.377650 0.188825 0.982011i $$-0.439532\pi$$
0.188825 + 0.982011i $$0.439532\pi$$
$$908$$ 9.52251 0.316015
$$909$$ −5.83325 −0.193477
$$910$$ 0 0
$$911$$ −18.5896 −0.615902 −0.307951 0.951402i $$-0.599643\pi$$
−0.307951 + 0.951402i $$0.599643\pi$$
$$912$$ −51.2409 −1.69676
$$913$$ 37.1931 1.23091
$$914$$ −6.77112 −0.223969
$$915$$ 0 0
$$916$$ −148.994 −4.92290
$$917$$ 2.78348 0.0919187
$$918$$ 4.61803 0.152418
$$919$$ 7.64799 0.252284 0.126142 0.992012i $$-0.459741\pi$$
0.126142 + 0.992012i $$0.459741\pi$$
$$920$$ 0 0
$$921$$ −21.4194 −0.705793
$$922$$ 0.416680 0.0137226
$$923$$ −9.04083 −0.297582
$$924$$ −7.96652 −0.262079
$$925$$ 0 0
$$926$$ −31.4822 −1.03457
$$927$$ 14.0794 0.462428
$$928$$ 158.865 5.21498
$$929$$ −30.9487 −1.01539 −0.507696 0.861536i $$-0.669503\pi$$
−0.507696 + 0.861536i $$0.669503\pi$$
$$930$$ 0 0
$$931$$ −25.3535 −0.830927
$$932$$ 36.0675 1.18143
$$933$$ 2.85200 0.0933702
$$934$$ 1.27279 0.0416468
$$935$$ 0 0
$$936$$ 5.06932 0.165696
$$937$$ −31.3694 −1.02480 −0.512398 0.858748i $$-0.671243\pi$$
−0.512398 + 0.858748i $$0.671243\pi$$
$$938$$ 3.55704 0.116142
$$939$$ 14.4375 0.471151
$$940$$ 0 0
$$941$$ −3.61810 −0.117947 −0.0589733 0.998260i $$-0.518783\pi$$
−0.0589733 + 0.998260i $$0.518783\pi$$
$$942$$ −53.0340 −1.72794
$$943$$ 3.34146 0.108813
$$944$$ −176.351 −5.73974
$$945$$ 0 0
$$946$$ −57.9229 −1.88323
$$947$$ 11.3216 0.367902 0.183951 0.982935i $$-0.441111\pi$$
0.183951 + 0.982935i $$0.441111\pi$$
$$948$$ 25.3175 0.822273
$$949$$ −5.51540 −0.179038
$$950$$ 0 0
$$951$$ 22.6102 0.733187
$$952$$ 7.22004 0.234003
$$953$$ 17.2182 0.557752 0.278876 0.960327i $$-0.410038\pi$$
0.278876 + 0.960327i $$0.410038\pi$$
$$954$$ −19.0842 −0.617874
$$955$$ 0 0
$$956$$ −34.5199 −1.11645
$$957$$ −26.4839 −0.856102
$$958$$ −33.5898 −1.08524
$$959$$ −2.40608 −0.0776962
$$960$$ 0 0
$$961$$ −1.44386 −0.0465762
$$962$$ −1.55668 −0.0501894
$$963$$ 8.61207 0.277520
$$964$$ −39.6591 −1.27733
$$965$$ 0 0
$$966$$ −2.88898 −0.0929514
$$967$$ 29.9064 0.961723 0.480862 0.876796i $$-0.340324\pi$$
0.480862 + 0.876796i $$0.340324\pi$$
$$968$$ 7.90115 0.253953
$$969$$ −6.38197 −0.205018
$$970$$ 0 0
$$971$$ −25.1927 −0.808472 −0.404236 0.914655i $$-0.632463\pi$$
−0.404236 + 0.914655i $$0.632463\pi$$
$$972$$ −5.32440 −0.170780
$$973$$ 3.87436 0.124206
$$974$$ 93.5443 2.99735
$$975$$ 0 0
$$976$$ 1.73842 0.0556455
$$977$$ −57.0948 −1.82663 −0.913313 0.407259i $$-0.866485\pi$$
−0.913313 + 0.407259i $$0.866485\pi$$
$$978$$ −32.1859 −1.02919
$$979$$ 19.7267 0.630469
$$980$$ 0 0
$$981$$ 16.6875 0.532791
$$982$$ 67.2957 2.14749
$$983$$ 26.4168 0.842566 0.421283 0.906929i $$-0.361580\pi$$
0.421283 + 0.906929i $$0.361580\pi$$
$$984$$ 13.2448 0.422230
$$985$$ 0 0
$$986$$ 38.4423 1.22425
$$987$$ 2.08508 0.0663690
$$988$$ −11.2203 −0.356965
$$989$$ −15.2695 −0.485541
$$990$$ 0 0
$$991$$ −22.5276 −0.715612 −0.357806 0.933796i $$-0.616475\pi$$
−0.357806 + 0.933796i $$0.616475\pi$$
$$992$$ −103.752 −3.29414
$$993$$ −3.86645 −0.122698
$$994$$ 20.4227 0.647768
$$995$$ 0 0
$$996$$ 62.2448 1.97230
$$997$$ 24.1043 0.763392 0.381696 0.924288i $$-0.375340\pi$$
0.381696 + 0.924288i $$0.375340\pi$$
$$998$$ −34.1103 −1.07974
$$999$$ 1.02085 0.0322983
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.1.1 4
3.2 odd 2 5625.2.a.n.1.4 4
5.2 odd 4 1875.2.b.c.1249.1 8
5.3 odd 4 1875.2.b.c.1249.8 8
5.4 even 2 1875.2.a.h.1.4 4
15.14 odd 2 5625.2.a.i.1.1 4
25.3 odd 20 375.2.i.b.49.4 16
25.4 even 10 75.2.g.b.16.1 8
25.6 even 5 375.2.g.b.301.2 8
25.8 odd 20 375.2.i.b.199.1 16
25.17 odd 20 375.2.i.b.199.4 16
25.19 even 10 75.2.g.b.61.1 yes 8
25.21 even 5 375.2.g.b.76.2 8
25.22 odd 20 375.2.i.b.49.1 16
75.29 odd 10 225.2.h.c.91.2 8
75.44 odd 10 225.2.h.c.136.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.16.1 8 25.4 even 10
75.2.g.b.61.1 yes 8 25.19 even 10
225.2.h.c.91.2 8 75.29 odd 10
225.2.h.c.136.2 8 75.44 odd 10
375.2.g.b.76.2 8 25.21 even 5
375.2.g.b.301.2 8 25.6 even 5
375.2.i.b.49.1 16 25.22 odd 20
375.2.i.b.49.4 16 25.3 odd 20
375.2.i.b.199.1 16 25.8 odd 20
375.2.i.b.199.4 16 25.17 odd 20
1875.2.a.e.1.1 4 1.1 even 1 trivial
1875.2.a.h.1.4 4 5.4 even 2
1875.2.b.c.1249.1 8 5.2 odd 4
1875.2.b.c.1249.8 8 5.3 odd 4
5625.2.a.i.1.1 4 15.14 odd 2
5625.2.a.n.1.4 4 3.2 odd 2