# Properties

 Label 1875.2.a.j.1.1 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $1$ 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: $$1$$ 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 1.1 Root $$2.68704$$ 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)

 $$f(q)$$ $$=$$ $$q-2.68704 q^{2} -1.00000 q^{3} +5.22020 q^{4} +2.68704 q^{6} +1.68704 q^{7} -8.65280 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.68704 q^{2} -1.00000 q^{3} +5.22020 q^{4} +2.68704 q^{6} +1.68704 q^{7} -8.65280 q^{8} +1.00000 q^{9} +1.07882 q^{11} -5.22020 q^{12} +2.67723 q^{13} -4.53315 q^{14} +12.8101 q^{16} -3.93167 q^{17} -2.68704 q^{18} -1.17755 q^{19} -1.68704 q^{21} -2.89883 q^{22} -4.06295 q^{23} +8.65280 q^{24} -7.19384 q^{26} -1.00000 q^{27} +8.80669 q^{28} +5.95595 q^{29} -7.10310 q^{31} -17.1155 q^{32} -1.07882 q^{33} +10.5646 q^{34} +5.22020 q^{36} -4.58187 q^{37} +3.16412 q^{38} -2.67723 q^{39} -11.2614 q^{41} +4.53315 q^{42} +2.58587 q^{43} +5.63164 q^{44} +10.9173 q^{46} -1.91782 q^{47} -12.8101 q^{48} -4.15389 q^{49} +3.93167 q^{51} +13.9757 q^{52} -2.54861 q^{53} +2.68704 q^{54} -14.5976 q^{56} +1.17755 q^{57} -16.0039 q^{58} +1.33599 q^{59} -7.28106 q^{61} +19.0863 q^{62} +1.68704 q^{63} +20.3701 q^{64} +2.89883 q^{66} +12.4451 q^{67} -20.5241 q^{68} +4.06295 q^{69} -5.98480 q^{71} -8.65280 q^{72} -3.30065 q^{73} +12.3117 q^{74} -6.14702 q^{76} +1.82001 q^{77} +7.19384 q^{78} -4.00404 q^{79} +1.00000 q^{81} +30.2600 q^{82} +8.87482 q^{83} -8.80669 q^{84} -6.94834 q^{86} -5.95595 q^{87} -9.33480 q^{88} +15.4436 q^{89} +4.51660 q^{91} -21.2094 q^{92} +7.10310 q^{93} +5.15327 q^{94} +17.1155 q^{96} +10.7682 q^{97} +11.1617 q^{98} +1.07882 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{3} + 10 q^{4} - 6 q^{7} - 3 q^{8} + 6 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.68704 −1.90003 −0.950013 0.312211i $$-0.898931\pi$$
−0.950013 + 0.312211i $$0.898931\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 5.22020 2.61010
$$5$$ 0 0
$$6$$ 2.68704 1.09698
$$7$$ 1.68704 0.637642 0.318821 0.947815i $$-0.396713\pi$$
0.318821 + 0.947815i $$0.396713\pi$$
$$8$$ −8.65280 −3.05923
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.07882 0.325276 0.162638 0.986686i $$-0.448000\pi$$
0.162638 + 0.986686i $$0.448000\pi$$
$$12$$ −5.22020 −1.50694
$$13$$ 2.67723 0.742531 0.371265 0.928527i $$-0.378924\pi$$
0.371265 + 0.928527i $$0.378924\pi$$
$$14$$ −4.53315 −1.21154
$$15$$ 0 0
$$16$$ 12.8101 3.20251
$$17$$ −3.93167 −0.953570 −0.476785 0.879020i $$-0.658198\pi$$
−0.476785 + 0.879020i $$0.658198\pi$$
$$18$$ −2.68704 −0.633342
$$19$$ −1.17755 −0.270148 −0.135074 0.990836i $$-0.543127\pi$$
−0.135074 + 0.990836i $$0.543127\pi$$
$$20$$ 0 0
$$21$$ −1.68704 −0.368143
$$22$$ −2.89883 −0.618032
$$23$$ −4.06295 −0.847183 −0.423591 0.905853i $$-0.639231\pi$$
−0.423591 + 0.905853i $$0.639231\pi$$
$$24$$ 8.65280 1.76625
$$25$$ 0 0
$$26$$ −7.19384 −1.41083
$$27$$ −1.00000 −0.192450
$$28$$ 8.80669 1.66431
$$29$$ 5.95595 1.10599 0.552996 0.833184i $$-0.313484\pi$$
0.552996 + 0.833184i $$0.313484\pi$$
$$30$$ 0 0
$$31$$ −7.10310 −1.27575 −0.637877 0.770138i $$-0.720187\pi$$
−0.637877 + 0.770138i $$0.720187\pi$$
$$32$$ −17.1155 −3.02563
$$33$$ −1.07882 −0.187798
$$34$$ 10.5646 1.81181
$$35$$ 0 0
$$36$$ 5.22020 0.870033
$$37$$ −4.58187 −0.753254 −0.376627 0.926365i $$-0.622916\pi$$
−0.376627 + 0.926365i $$0.622916\pi$$
$$38$$ 3.16412 0.513287
$$39$$ −2.67723 −0.428700
$$40$$ 0 0
$$41$$ −11.2614 −1.75874 −0.879371 0.476137i $$-0.842037\pi$$
−0.879371 + 0.476137i $$0.842037\pi$$
$$42$$ 4.53315 0.699481
$$43$$ 2.58587 0.394342 0.197171 0.980369i $$-0.436825\pi$$
0.197171 + 0.980369i $$0.436825\pi$$
$$44$$ 5.63164 0.849002
$$45$$ 0 0
$$46$$ 10.9173 1.60967
$$47$$ −1.91782 −0.279743 −0.139871 0.990170i $$-0.544669\pi$$
−0.139871 + 0.990170i $$0.544669\pi$$
$$48$$ −12.8101 −1.84897
$$49$$ −4.15389 −0.593413
$$50$$ 0 0
$$51$$ 3.93167 0.550544
$$52$$ 13.9757 1.93808
$$53$$ −2.54861 −0.350078 −0.175039 0.984561i $$-0.556005\pi$$
−0.175039 + 0.984561i $$0.556005\pi$$
$$54$$ 2.68704 0.365660
$$55$$ 0 0
$$56$$ −14.5976 −1.95069
$$57$$ 1.17755 0.155970
$$58$$ −16.0039 −2.10141
$$59$$ 1.33599 0.173931 0.0869654 0.996211i $$-0.472283\pi$$
0.0869654 + 0.996211i $$0.472283\pi$$
$$60$$ 0 0
$$61$$ −7.28106 −0.932245 −0.466122 0.884720i $$-0.654349\pi$$
−0.466122 + 0.884720i $$0.654349\pi$$
$$62$$ 19.0863 2.42397
$$63$$ 1.68704 0.212547
$$64$$ 20.3701 2.54626
$$65$$ 0 0
$$66$$ 2.89883 0.356821
$$67$$ 12.4451 1.52041 0.760203 0.649686i $$-0.225100\pi$$
0.760203 + 0.649686i $$0.225100\pi$$
$$68$$ −20.5241 −2.48891
$$69$$ 4.06295 0.489121
$$70$$ 0 0
$$71$$ −5.98480 −0.710266 −0.355133 0.934816i $$-0.615564\pi$$
−0.355133 + 0.934816i $$0.615564\pi$$
$$72$$ −8.65280 −1.01974
$$73$$ −3.30065 −0.386312 −0.193156 0.981168i $$-0.561872\pi$$
−0.193156 + 0.981168i $$0.561872\pi$$
$$74$$ 12.3117 1.43120
$$75$$ 0 0
$$76$$ −6.14702 −0.705112
$$77$$ 1.82001 0.207410
$$78$$ 7.19384 0.814542
$$79$$ −4.00404 −0.450490 −0.225245 0.974302i $$-0.572318\pi$$
−0.225245 + 0.974302i $$0.572318\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 30.2600 3.34166
$$83$$ 8.87482 0.974138 0.487069 0.873364i $$-0.338066\pi$$
0.487069 + 0.873364i $$0.338066\pi$$
$$84$$ −8.80669 −0.960889
$$85$$ 0 0
$$86$$ −6.94834 −0.749259
$$87$$ −5.95595 −0.638545
$$88$$ −9.33480 −0.995093
$$89$$ 15.4436 1.63702 0.818509 0.574493i $$-0.194801\pi$$
0.818509 + 0.574493i $$0.194801\pi$$
$$90$$ 0 0
$$91$$ 4.51660 0.473469
$$92$$ −21.2094 −2.21123
$$93$$ 7.10310 0.736557
$$94$$ 5.15327 0.531519
$$95$$ 0 0
$$96$$ 17.1155 1.74685
$$97$$ 10.7682 1.09335 0.546673 0.837346i $$-0.315894\pi$$
0.546673 + 0.837346i $$0.315894\pi$$
$$98$$ 11.1617 1.12750
$$99$$ 1.07882 0.108425
$$100$$ 0 0
$$101$$ −13.5654 −1.34981 −0.674905 0.737905i $$-0.735815\pi$$
−0.674905 + 0.737905i $$0.735815\pi$$
$$102$$ −10.5646 −1.04605
$$103$$ 1.36904 0.134895 0.0674476 0.997723i $$-0.478514\pi$$
0.0674476 + 0.997723i $$0.478514\pi$$
$$104$$ −23.1656 −2.27157
$$105$$ 0 0
$$106$$ 6.84822 0.665158
$$107$$ 11.0115 1.06452 0.532262 0.846579i $$-0.321342\pi$$
0.532262 + 0.846579i $$0.321342\pi$$
$$108$$ −5.22020 −0.502314
$$109$$ 3.44890 0.330344 0.165172 0.986265i $$-0.447182\pi$$
0.165172 + 0.986265i $$0.447182\pi$$
$$110$$ 0 0
$$111$$ 4.58187 0.434891
$$112$$ 21.6111 2.04206
$$113$$ 6.05194 0.569319 0.284660 0.958629i $$-0.408119\pi$$
0.284660 + 0.958629i $$0.408119\pi$$
$$114$$ −3.16412 −0.296347
$$115$$ 0 0
$$116$$ 31.0912 2.88675
$$117$$ 2.67723 0.247510
$$118$$ −3.58986 −0.330473
$$119$$ −6.63289 −0.608036
$$120$$ 0 0
$$121$$ −9.83615 −0.894196
$$122$$ 19.5645 1.77129
$$123$$ 11.2614 1.01541
$$124$$ −37.0796 −3.32984
$$125$$ 0 0
$$126$$ −4.53315 −0.403845
$$127$$ −20.2362 −1.79567 −0.897835 0.440332i $$-0.854861\pi$$
−0.897835 + 0.440332i $$0.854861\pi$$
$$128$$ −20.5042 −1.81233
$$129$$ −2.58587 −0.227673
$$130$$ 0 0
$$131$$ −3.05011 −0.266490 −0.133245 0.991083i $$-0.542540\pi$$
−0.133245 + 0.991083i $$0.542540\pi$$
$$132$$ −5.63164 −0.490171
$$133$$ −1.98657 −0.172257
$$134$$ −33.4404 −2.88881
$$135$$ 0 0
$$136$$ 34.0200 2.91719
$$137$$ −20.5100 −1.75229 −0.876145 0.482048i $$-0.839893\pi$$
−0.876145 + 0.482048i $$0.839893\pi$$
$$138$$ −10.9173 −0.929343
$$139$$ 18.3184 1.55374 0.776871 0.629659i $$-0.216806\pi$$
0.776871 + 0.629659i $$0.216806\pi$$
$$140$$ 0 0
$$141$$ 1.91782 0.161510
$$142$$ 16.0814 1.34952
$$143$$ 2.88825 0.241527
$$144$$ 12.8101 1.06750
$$145$$ 0 0
$$146$$ 8.86899 0.734003
$$147$$ 4.15389 0.342607
$$148$$ −23.9182 −1.96607
$$149$$ 0.705938 0.0578327 0.0289163 0.999582i $$-0.490794\pi$$
0.0289163 + 0.999582i $$0.490794\pi$$
$$150$$ 0 0
$$151$$ 7.04538 0.573345 0.286673 0.958029i $$-0.407451\pi$$
0.286673 + 0.958029i $$0.407451\pi$$
$$152$$ 10.1891 0.826443
$$153$$ −3.93167 −0.317857
$$154$$ −4.89045 −0.394083
$$155$$ 0 0
$$156$$ −13.9757 −1.11895
$$157$$ −5.12880 −0.409323 −0.204662 0.978833i $$-0.565609\pi$$
−0.204662 + 0.978833i $$0.565609\pi$$
$$158$$ 10.7590 0.855942
$$159$$ 2.54861 0.202118
$$160$$ 0 0
$$161$$ −6.85436 −0.540199
$$162$$ −2.68704 −0.211114
$$163$$ −24.2613 −1.90029 −0.950146 0.311806i $$-0.899066\pi$$
−0.950146 + 0.311806i $$0.899066\pi$$
$$164$$ −58.7869 −4.59049
$$165$$ 0 0
$$166$$ −23.8470 −1.85089
$$167$$ 4.54890 0.352004 0.176002 0.984390i $$-0.443683\pi$$
0.176002 + 0.984390i $$0.443683\pi$$
$$168$$ 14.5976 1.12623
$$169$$ −5.83243 −0.448648
$$170$$ 0 0
$$171$$ −1.17755 −0.0900492
$$172$$ 13.4988 1.02927
$$173$$ −2.67729 −0.203550 −0.101775 0.994807i $$-0.532452\pi$$
−0.101775 + 0.994807i $$0.532452\pi$$
$$174$$ 16.0039 1.21325
$$175$$ 0 0
$$176$$ 13.8197 1.04170
$$177$$ −1.33599 −0.100419
$$178$$ −41.4976 −3.11038
$$179$$ −15.6266 −1.16799 −0.583995 0.811757i $$-0.698511\pi$$
−0.583995 + 0.811757i $$0.698511\pi$$
$$180$$ 0 0
$$181$$ 0.202843 0.0150772 0.00753859 0.999972i $$-0.497600\pi$$
0.00753859 + 0.999972i $$0.497600\pi$$
$$182$$ −12.1363 −0.899603
$$183$$ 7.28106 0.538232
$$184$$ 35.1559 2.59172
$$185$$ 0 0
$$186$$ −19.0863 −1.39948
$$187$$ −4.24155 −0.310173
$$188$$ −10.0114 −0.730156
$$189$$ −1.68704 −0.122714
$$190$$ 0 0
$$191$$ −7.91975 −0.573053 −0.286526 0.958072i $$-0.592501\pi$$
−0.286526 + 0.958072i $$0.592501\pi$$
$$192$$ −20.3701 −1.47008
$$193$$ −16.4269 −1.18243 −0.591216 0.806513i $$-0.701352\pi$$
−0.591216 + 0.806513i $$0.701352\pi$$
$$194$$ −28.9346 −2.07738
$$195$$ 0 0
$$196$$ −21.6841 −1.54887
$$197$$ −17.1369 −1.22095 −0.610475 0.792035i $$-0.709022\pi$$
−0.610475 + 0.792035i $$0.709022\pi$$
$$198$$ −2.89883 −0.206011
$$199$$ −2.01848 −0.143086 −0.0715430 0.997438i $$-0.522792\pi$$
−0.0715430 + 0.997438i $$0.522792\pi$$
$$200$$ 0 0
$$201$$ −12.4451 −0.877806
$$202$$ 36.4509 2.56467
$$203$$ 10.0479 0.705227
$$204$$ 20.5241 1.43697
$$205$$ 0 0
$$206$$ −3.67866 −0.256304
$$207$$ −4.06295 −0.282394
$$208$$ 34.2955 2.37796
$$209$$ −1.27036 −0.0878725
$$210$$ 0 0
$$211$$ −23.8213 −1.63993 −0.819963 0.572416i $$-0.806006\pi$$
−0.819963 + 0.572416i $$0.806006\pi$$
$$212$$ −13.3042 −0.913738
$$213$$ 5.98480 0.410072
$$214$$ −29.5884 −2.02262
$$215$$ 0 0
$$216$$ 8.65280 0.588749
$$217$$ −11.9832 −0.813475
$$218$$ −9.26733 −0.627663
$$219$$ 3.30065 0.223037
$$220$$ 0 0
$$221$$ −10.5260 −0.708055
$$222$$ −12.3117 −0.826305
$$223$$ 13.1112 0.877989 0.438995 0.898490i $$-0.355335\pi$$
0.438995 + 0.898490i $$0.355335\pi$$
$$224$$ −28.8746 −1.92927
$$225$$ 0 0
$$226$$ −16.2618 −1.08172
$$227$$ −26.2807 −1.74431 −0.872155 0.489229i $$-0.837278\pi$$
−0.872155 + 0.489229i $$0.837278\pi$$
$$228$$ 6.14702 0.407096
$$229$$ 1.99605 0.131903 0.0659513 0.997823i $$-0.478992\pi$$
0.0659513 + 0.997823i $$0.478992\pi$$
$$230$$ 0 0
$$231$$ −1.82001 −0.119748
$$232$$ −51.5357 −3.38348
$$233$$ −0.525548 −0.0344298 −0.0172149 0.999852i $$-0.505480\pi$$
−0.0172149 + 0.999852i $$0.505480\pi$$
$$234$$ −7.19384 −0.470276
$$235$$ 0 0
$$236$$ 6.97412 0.453976
$$237$$ 4.00404 0.260090
$$238$$ 17.8229 1.15528
$$239$$ −7.00971 −0.453421 −0.226710 0.973962i $$-0.572797\pi$$
−0.226710 + 0.973962i $$0.572797\pi$$
$$240$$ 0 0
$$241$$ 2.31554 0.149157 0.0745786 0.997215i $$-0.476239\pi$$
0.0745786 + 0.997215i $$0.476239\pi$$
$$242$$ 26.4302 1.69899
$$243$$ −1.00000 −0.0641500
$$244$$ −38.0086 −2.43325
$$245$$ 0 0
$$246$$ −30.2600 −1.92931
$$247$$ −3.15257 −0.200593
$$248$$ 61.4617 3.90282
$$249$$ −8.87482 −0.562419
$$250$$ 0 0
$$251$$ −6.17885 −0.390005 −0.195003 0.980803i $$-0.562472\pi$$
−0.195003 + 0.980803i $$0.562472\pi$$
$$252$$ 8.80669 0.554769
$$253$$ −4.38318 −0.275568
$$254$$ 54.3755 3.41182
$$255$$ 0 0
$$256$$ 14.3555 0.897217
$$257$$ 13.2239 0.824882 0.412441 0.910984i $$-0.364676\pi$$
0.412441 + 0.910984i $$0.364676\pi$$
$$258$$ 6.94834 0.432585
$$259$$ −7.72980 −0.480306
$$260$$ 0 0
$$261$$ 5.95595 0.368664
$$262$$ 8.19578 0.506337
$$263$$ −6.97643 −0.430185 −0.215092 0.976594i $$-0.569005\pi$$
−0.215092 + 0.976594i $$0.569005\pi$$
$$264$$ 9.33480 0.574517
$$265$$ 0 0
$$266$$ 5.33800 0.327294
$$267$$ −15.4436 −0.945133
$$268$$ 64.9656 3.96841
$$269$$ 23.7199 1.44623 0.723113 0.690730i $$-0.242711\pi$$
0.723113 + 0.690730i $$0.242711\pi$$
$$270$$ 0 0
$$271$$ 28.9910 1.76108 0.880539 0.473975i $$-0.157181\pi$$
0.880539 + 0.473975i $$0.157181\pi$$
$$272$$ −50.3649 −3.05382
$$273$$ −4.51660 −0.273357
$$274$$ 55.1113 3.32940
$$275$$ 0 0
$$276$$ 21.2094 1.27665
$$277$$ −8.58667 −0.515923 −0.257962 0.966155i $$-0.583051\pi$$
−0.257962 + 0.966155i $$0.583051\pi$$
$$278$$ −49.2222 −2.95215
$$279$$ −7.10310 −0.425251
$$280$$ 0 0
$$281$$ 17.1661 1.02405 0.512023 0.858972i $$-0.328896\pi$$
0.512023 + 0.858972i $$0.328896\pi$$
$$282$$ −5.15327 −0.306872
$$283$$ −9.62005 −0.571853 −0.285926 0.958252i $$-0.592301\pi$$
−0.285926 + 0.958252i $$0.592301\pi$$
$$284$$ −31.2419 −1.85386
$$285$$ 0 0
$$286$$ −7.76084 −0.458908
$$287$$ −18.9985 −1.12145
$$288$$ −17.1155 −1.00854
$$289$$ −1.54198 −0.0907048
$$290$$ 0 0
$$291$$ −10.7682 −0.631243
$$292$$ −17.2301 −1.00831
$$293$$ 16.8202 0.982649 0.491324 0.870977i $$-0.336513\pi$$
0.491324 + 0.870977i $$0.336513\pi$$
$$294$$ −11.1617 −0.650962
$$295$$ 0 0
$$296$$ 39.6460 2.30438
$$297$$ −1.07882 −0.0625994
$$298$$ −1.89688 −0.109884
$$299$$ −10.8775 −0.629059
$$300$$ 0 0
$$301$$ 4.36247 0.251449
$$302$$ −18.9312 −1.08937
$$303$$ 13.5654 0.779313
$$304$$ −15.0844 −0.865151
$$305$$ 0 0
$$306$$ 10.5646 0.603936
$$307$$ 11.3212 0.646134 0.323067 0.946376i $$-0.395286\pi$$
0.323067 + 0.946376i $$0.395286\pi$$
$$308$$ 9.50081 0.541359
$$309$$ −1.36904 −0.0778818
$$310$$ 0 0
$$311$$ 18.5635 1.05264 0.526318 0.850288i $$-0.323572\pi$$
0.526318 + 0.850288i $$0.323572\pi$$
$$312$$ 23.1656 1.31149
$$313$$ −9.69951 −0.548249 −0.274124 0.961694i $$-0.588388\pi$$
−0.274124 + 0.961694i $$0.588388\pi$$
$$314$$ 13.7813 0.777724
$$315$$ 0 0
$$316$$ −20.9019 −1.17582
$$317$$ 7.47647 0.419920 0.209960 0.977710i $$-0.432667\pi$$
0.209960 + 0.977710i $$0.432667\pi$$
$$318$$ −6.84822 −0.384029
$$319$$ 6.42538 0.359752
$$320$$ 0 0
$$321$$ −11.0115 −0.614604
$$322$$ 18.4180 1.02639
$$323$$ 4.62972 0.257605
$$324$$ 5.22020 0.290011
$$325$$ 0 0
$$326$$ 65.1911 3.61060
$$327$$ −3.44890 −0.190724
$$328$$ 97.4430 5.38039
$$329$$ −3.23544 −0.178376
$$330$$ 0 0
$$331$$ −25.2570 −1.38825 −0.694125 0.719854i $$-0.744209\pi$$
−0.694125 + 0.719854i $$0.744209\pi$$
$$332$$ 46.3283 2.54259
$$333$$ −4.58187 −0.251085
$$334$$ −12.2231 −0.668817
$$335$$ 0 0
$$336$$ −21.6111 −1.17898
$$337$$ −13.0240 −0.709463 −0.354731 0.934968i $$-0.615428\pi$$
−0.354731 + 0.934968i $$0.615428\pi$$
$$338$$ 15.6720 0.852443
$$339$$ −6.05194 −0.328697
$$340$$ 0 0
$$341$$ −7.66295 −0.414972
$$342$$ 3.16412 0.171096
$$343$$ −18.8171 −1.01603
$$344$$ −22.3750 −1.20638
$$345$$ 0 0
$$346$$ 7.19398 0.386751
$$347$$ −6.00149 −0.322177 −0.161089 0.986940i $$-0.551500\pi$$
−0.161089 + 0.986940i $$0.551500\pi$$
$$348$$ −31.0912 −1.66666
$$349$$ −22.7183 −1.21608 −0.608042 0.793905i $$-0.708045\pi$$
−0.608042 + 0.793905i $$0.708045\pi$$
$$350$$ 0 0
$$351$$ −2.67723 −0.142900
$$352$$ −18.4646 −0.984164
$$353$$ −4.85493 −0.258402 −0.129201 0.991618i $$-0.541241\pi$$
−0.129201 + 0.991618i $$0.541241\pi$$
$$354$$ 3.58986 0.190799
$$355$$ 0 0
$$356$$ 80.6186 4.27278
$$357$$ 6.63289 0.351050
$$358$$ 41.9895 2.21921
$$359$$ 28.8910 1.52481 0.762403 0.647102i $$-0.224019\pi$$
0.762403 + 0.647102i $$0.224019\pi$$
$$360$$ 0 0
$$361$$ −17.6134 −0.927020
$$362$$ −0.545047 −0.0286470
$$363$$ 9.83615 0.516264
$$364$$ 23.5776 1.23580
$$365$$ 0 0
$$366$$ −19.5645 −1.02265
$$367$$ −14.7089 −0.767800 −0.383900 0.923375i $$-0.625419\pi$$
−0.383900 + 0.923375i $$0.625419\pi$$
$$368$$ −52.0465 −2.71311
$$369$$ −11.2614 −0.586247
$$370$$ 0 0
$$371$$ −4.29961 −0.223225
$$372$$ 37.0796 1.92249
$$373$$ −33.0391 −1.71070 −0.855350 0.518051i $$-0.826658\pi$$
−0.855350 + 0.518051i $$0.826658\pi$$
$$374$$ 11.3972 0.589337
$$375$$ 0 0
$$376$$ 16.5945 0.855797
$$377$$ 15.9455 0.821233
$$378$$ 4.53315 0.233160
$$379$$ −15.1556 −0.778493 −0.389247 0.921134i $$-0.627265\pi$$
−0.389247 + 0.921134i $$0.627265\pi$$
$$380$$ 0 0
$$381$$ 20.2362 1.03673
$$382$$ 21.2807 1.08882
$$383$$ −8.00811 −0.409195 −0.204598 0.978846i $$-0.565589\pi$$
−0.204598 + 0.978846i $$0.565589\pi$$
$$384$$ 20.5042 1.04635
$$385$$ 0 0
$$386$$ 44.1397 2.24665
$$387$$ 2.58587 0.131447
$$388$$ 56.2121 2.85374
$$389$$ 1.07463 0.0544858 0.0272429 0.999629i $$-0.491327\pi$$
0.0272429 + 0.999629i $$0.491327\pi$$
$$390$$ 0 0
$$391$$ 15.9742 0.807848
$$392$$ 35.9428 1.81538
$$393$$ 3.05011 0.153858
$$394$$ 46.0475 2.31984
$$395$$ 0 0
$$396$$ 5.63164 0.283001
$$397$$ −24.2139 −1.21526 −0.607630 0.794220i $$-0.707880\pi$$
−0.607630 + 0.794220i $$0.707880\pi$$
$$398$$ 5.42373 0.271867
$$399$$ 1.98657 0.0994529
$$400$$ 0 0
$$401$$ −1.99317 −0.0995340 −0.0497670 0.998761i $$-0.515848\pi$$
−0.0497670 + 0.998761i $$0.515848\pi$$
$$402$$ 33.4404 1.66785
$$403$$ −19.0166 −0.947287
$$404$$ −70.8142 −3.52314
$$405$$ 0 0
$$406$$ −26.9992 −1.33995
$$407$$ −4.94300 −0.245015
$$408$$ −34.0200 −1.68424
$$409$$ 6.66478 0.329552 0.164776 0.986331i $$-0.447310\pi$$
0.164776 + 0.986331i $$0.447310\pi$$
$$410$$ 0 0
$$411$$ 20.5100 1.01168
$$412$$ 7.14664 0.352090
$$413$$ 2.25387 0.110906
$$414$$ 10.9173 0.536556
$$415$$ 0 0
$$416$$ −45.8223 −2.24662
$$417$$ −18.3184 −0.897054
$$418$$ 3.41351 0.166960
$$419$$ 21.3540 1.04321 0.521606 0.853186i $$-0.325333\pi$$
0.521606 + 0.853186i $$0.325333\pi$$
$$420$$ 0 0
$$421$$ 22.3367 1.08863 0.544313 0.838882i $$-0.316790\pi$$
0.544313 + 0.838882i $$0.316790\pi$$
$$422$$ 64.0088 3.11590
$$423$$ −1.91782 −0.0932476
$$424$$ 22.0526 1.07097
$$425$$ 0 0
$$426$$ −16.0814 −0.779147
$$427$$ −12.2835 −0.594438
$$428$$ 57.4823 2.77851
$$429$$ −2.88825 −0.139446
$$430$$ 0 0
$$431$$ −30.4129 −1.46494 −0.732469 0.680800i $$-0.761632\pi$$
−0.732469 + 0.680800i $$0.761632\pi$$
$$432$$ −12.8101 −0.616324
$$433$$ 0.576639 0.0277115 0.0138558 0.999904i $$-0.495589\pi$$
0.0138558 + 0.999904i $$0.495589\pi$$
$$434$$ 32.1994 1.54562
$$435$$ 0 0
$$436$$ 18.0039 0.862231
$$437$$ 4.78431 0.228864
$$438$$ −8.86899 −0.423777
$$439$$ 12.8045 0.611125 0.305563 0.952172i $$-0.401155\pi$$
0.305563 + 0.952172i $$0.401155\pi$$
$$440$$ 0 0
$$441$$ −4.15389 −0.197804
$$442$$ 28.2838 1.34532
$$443$$ 14.3147 0.680110 0.340055 0.940405i $$-0.389554\pi$$
0.340055 + 0.940405i $$0.389554\pi$$
$$444$$ 23.9182 1.13511
$$445$$ 0 0
$$446$$ −35.2303 −1.66820
$$447$$ −0.705938 −0.0333897
$$448$$ 34.3652 1.62360
$$449$$ −11.9663 −0.564724 −0.282362 0.959308i $$-0.591118\pi$$
−0.282362 + 0.959308i $$0.591118\pi$$
$$450$$ 0 0
$$451$$ −12.1490 −0.572076
$$452$$ 31.5923 1.48598
$$453$$ −7.04538 −0.331021
$$454$$ 70.6173 3.31424
$$455$$ 0 0
$$456$$ −10.1891 −0.477147
$$457$$ −8.22154 −0.384587 −0.192294 0.981337i $$-0.561593\pi$$
−0.192294 + 0.981337i $$0.561593\pi$$
$$458$$ −5.36346 −0.250618
$$459$$ 3.93167 0.183515
$$460$$ 0 0
$$461$$ 1.27216 0.0592505 0.0296253 0.999561i $$-0.490569\pi$$
0.0296253 + 0.999561i $$0.490569\pi$$
$$462$$ 4.89045 0.227524
$$463$$ 32.6764 1.51860 0.759300 0.650740i $$-0.225541\pi$$
0.759300 + 0.650740i $$0.225541\pi$$
$$464$$ 76.2960 3.54195
$$465$$ 0 0
$$466$$ 1.41217 0.0654175
$$467$$ 14.4347 0.667958 0.333979 0.942581i $$-0.391609\pi$$
0.333979 + 0.942581i $$0.391609\pi$$
$$468$$ 13.9757 0.646026
$$469$$ 20.9953 0.969474
$$470$$ 0 0
$$471$$ 5.12880 0.236323
$$472$$ −11.5600 −0.532094
$$473$$ 2.78968 0.128270
$$474$$ −10.7590 −0.494178
$$475$$ 0 0
$$476$$ −34.6250 −1.58703
$$477$$ −2.54861 −0.116693
$$478$$ 18.8354 0.861511
$$479$$ −16.8618 −0.770436 −0.385218 0.922826i $$-0.625874\pi$$
−0.385218 + 0.922826i $$0.625874\pi$$
$$480$$ 0 0
$$481$$ −12.2667 −0.559314
$$482$$ −6.22196 −0.283403
$$483$$ 6.85436 0.311884
$$484$$ −51.3466 −2.33394
$$485$$ 0 0
$$486$$ 2.68704 0.121887
$$487$$ 39.8235 1.80457 0.902287 0.431135i $$-0.141887\pi$$
0.902287 + 0.431135i $$0.141887\pi$$
$$488$$ 63.0016 2.85195
$$489$$ 24.2613 1.09713
$$490$$ 0 0
$$491$$ 14.0255 0.632961 0.316480 0.948599i $$-0.397499\pi$$
0.316480 + 0.948599i $$0.397499\pi$$
$$492$$ 58.7869 2.65032
$$493$$ −23.4168 −1.05464
$$494$$ 8.47108 0.381132
$$495$$ 0 0
$$496$$ −90.9911 −4.08562
$$497$$ −10.0966 −0.452895
$$498$$ 23.8470 1.06861
$$499$$ 15.2315 0.681857 0.340929 0.940089i $$-0.389259\pi$$
0.340929 + 0.940089i $$0.389259\pi$$
$$500$$ 0 0
$$501$$ −4.54890 −0.203230
$$502$$ 16.6028 0.741020
$$503$$ 23.7330 1.05820 0.529101 0.848559i $$-0.322529\pi$$
0.529101 + 0.848559i $$0.322529\pi$$
$$504$$ −14.5976 −0.650231
$$505$$ 0 0
$$506$$ 11.7778 0.523586
$$507$$ 5.83243 0.259027
$$508$$ −105.637 −4.68688
$$509$$ −12.5097 −0.554482 −0.277241 0.960800i $$-0.589420\pi$$
−0.277241 + 0.960800i $$0.589420\pi$$
$$510$$ 0 0
$$511$$ −5.56834 −0.246329
$$512$$ 2.43464 0.107597
$$513$$ 1.17755 0.0519899
$$514$$ −35.5331 −1.56730
$$515$$ 0 0
$$516$$ −13.4988 −0.594249
$$517$$ −2.06898 −0.0909936
$$518$$ 20.7703 0.912595
$$519$$ 2.67729 0.117520
$$520$$ 0 0
$$521$$ −27.9638 −1.22512 −0.612559 0.790425i $$-0.709860\pi$$
−0.612559 + 0.790425i $$0.709860\pi$$
$$522$$ −16.0039 −0.700471
$$523$$ 15.7013 0.686569 0.343285 0.939231i $$-0.388460\pi$$
0.343285 + 0.939231i $$0.388460\pi$$
$$524$$ −15.9222 −0.695564
$$525$$ 0 0
$$526$$ 18.7460 0.817362
$$527$$ 27.9270 1.21652
$$528$$ −13.8197 −0.601426
$$529$$ −6.49247 −0.282281
$$530$$ 0 0
$$531$$ 1.33599 0.0579769
$$532$$ −10.3703 −0.449609
$$533$$ −30.1495 −1.30592
$$534$$ 41.4976 1.79578
$$535$$ 0 0
$$536$$ −107.685 −4.65127
$$537$$ 15.6266 0.674340
$$538$$ −63.7363 −2.74787
$$539$$ −4.48129 −0.193023
$$540$$ 0 0
$$541$$ −23.9783 −1.03091 −0.515455 0.856917i $$-0.672377\pi$$
−0.515455 + 0.856917i $$0.672377\pi$$
$$542$$ −77.9000 −3.34609
$$543$$ −0.202843 −0.00870482
$$544$$ 67.2927 2.88515
$$545$$ 0 0
$$546$$ 12.1363 0.519386
$$547$$ −5.62681 −0.240585 −0.120293 0.992738i $$-0.538383\pi$$
−0.120293 + 0.992738i $$0.538383\pi$$
$$548$$ −107.066 −4.57365
$$549$$ −7.28106 −0.310748
$$550$$ 0 0
$$551$$ −7.01341 −0.298781
$$552$$ −35.1559 −1.49633
$$553$$ −6.75498 −0.287251
$$554$$ 23.0728 0.980268
$$555$$ 0 0
$$556$$ 95.6254 4.05542
$$557$$ −42.4247 −1.79759 −0.898796 0.438366i $$-0.855557\pi$$
−0.898796 + 0.438366i $$0.855557\pi$$
$$558$$ 19.0863 0.807989
$$559$$ 6.92298 0.292811
$$560$$ 0 0
$$561$$ 4.24155 0.179079
$$562$$ −46.1261 −1.94571
$$563$$ 5.78597 0.243850 0.121925 0.992539i $$-0.461093\pi$$
0.121925 + 0.992539i $$0.461093\pi$$
$$564$$ 10.0114 0.421556
$$565$$ 0 0
$$566$$ 25.8495 1.08653
$$567$$ 1.68704 0.0708491
$$568$$ 51.7853 2.17286
$$569$$ −22.0056 −0.922522 −0.461261 0.887265i $$-0.652603\pi$$
−0.461261 + 0.887265i $$0.652603\pi$$
$$570$$ 0 0
$$571$$ 33.6143 1.40672 0.703358 0.710835i $$-0.251683\pi$$
0.703358 + 0.710835i $$0.251683\pi$$
$$572$$ 15.0772 0.630410
$$573$$ 7.91975 0.330852
$$574$$ 51.0499 2.13078
$$575$$ 0 0
$$576$$ 20.3701 0.848754
$$577$$ 40.2350 1.67501 0.837503 0.546433i $$-0.184015\pi$$
0.837503 + 0.546433i $$0.184015\pi$$
$$578$$ 4.14337 0.172341
$$579$$ 16.4269 0.682677
$$580$$ 0 0
$$581$$ 14.9722 0.621151
$$582$$ 28.9346 1.19938
$$583$$ −2.74948 −0.113872
$$584$$ 28.5599 1.18182
$$585$$ 0 0
$$586$$ −45.1967 −1.86706
$$587$$ −2.82299 −0.116517 −0.0582587 0.998302i $$-0.518555\pi$$
−0.0582587 + 0.998302i $$0.518555\pi$$
$$588$$ 21.6841 0.894238
$$589$$ 8.36423 0.344642
$$590$$ 0 0
$$591$$ 17.1369 0.704916
$$592$$ −58.6939 −2.41231
$$593$$ −24.6805 −1.01351 −0.506754 0.862091i $$-0.669155\pi$$
−0.506754 + 0.862091i $$0.669155\pi$$
$$594$$ 2.89883 0.118940
$$595$$ 0 0
$$596$$ 3.68513 0.150949
$$597$$ 2.01848 0.0826108
$$598$$ 29.2282 1.19523
$$599$$ −7.71547 −0.315245 −0.157623 0.987499i $$-0.550383\pi$$
−0.157623 + 0.987499i $$0.550383\pi$$
$$600$$ 0 0
$$601$$ 22.6506 0.923938 0.461969 0.886896i $$-0.347143\pi$$
0.461969 + 0.886896i $$0.347143\pi$$
$$602$$ −11.7222 −0.477759
$$603$$ 12.4451 0.506802
$$604$$ 36.7783 1.49649
$$605$$ 0 0
$$606$$ −36.4509 −1.48072
$$607$$ −24.6423 −1.00020 −0.500099 0.865968i $$-0.666703\pi$$
−0.500099 + 0.865968i $$0.666703\pi$$
$$608$$ 20.1543 0.817367
$$609$$ −10.0479 −0.407163
$$610$$ 0 0
$$611$$ −5.13445 −0.207718
$$612$$ −20.5241 −0.829637
$$613$$ −41.8034 −1.68842 −0.844211 0.536011i $$-0.819931\pi$$
−0.844211 + 0.536011i $$0.819931\pi$$
$$614$$ −30.4205 −1.22767
$$615$$ 0 0
$$616$$ −15.7482 −0.634513
$$617$$ 15.9390 0.641680 0.320840 0.947133i $$-0.396035\pi$$
0.320840 + 0.947133i $$0.396035\pi$$
$$618$$ 3.67866 0.147977
$$619$$ −47.1700 −1.89592 −0.947961 0.318385i $$-0.896859\pi$$
−0.947961 + 0.318385i $$0.896859\pi$$
$$620$$ 0 0
$$621$$ 4.06295 0.163040
$$622$$ −49.8808 −2.00004
$$623$$ 26.0540 1.04383
$$624$$ −34.2955 −1.37292
$$625$$ 0 0
$$626$$ 26.0630 1.04169
$$627$$ 1.27036 0.0507332
$$628$$ −26.7734 −1.06837
$$629$$ 18.0144 0.718280
$$630$$ 0 0
$$631$$ 32.3634 1.28836 0.644182 0.764872i $$-0.277198\pi$$
0.644182 + 0.764872i $$0.277198\pi$$
$$632$$ 34.6462 1.37815
$$633$$ 23.8213 0.946812
$$634$$ −20.0896 −0.797859
$$635$$ 0 0
$$636$$ 13.3042 0.527547
$$637$$ −11.1209 −0.440627
$$638$$ −17.2653 −0.683539
$$639$$ −5.98480 −0.236755
$$640$$ 0 0
$$641$$ 18.0042 0.711122 0.355561 0.934653i $$-0.384290\pi$$
0.355561 + 0.934653i $$0.384290\pi$$
$$642$$ 29.5884 1.16776
$$643$$ 21.8891 0.863224 0.431612 0.902059i $$-0.357945\pi$$
0.431612 + 0.902059i $$0.357945\pi$$
$$644$$ −35.7811 −1.40997
$$645$$ 0 0
$$646$$ −12.4403 −0.489455
$$647$$ 19.0263 0.748003 0.374001 0.927428i $$-0.377985\pi$$
0.374001 + 0.927428i $$0.377985\pi$$
$$648$$ −8.65280 −0.339914
$$649$$ 1.44129 0.0565755
$$650$$ 0 0
$$651$$ 11.9832 0.469660
$$652$$ −126.649 −4.95995
$$653$$ −2.31971 −0.0907774 −0.0453887 0.998969i $$-0.514453\pi$$
−0.0453887 + 0.998969i $$0.514453\pi$$
$$654$$ 9.26733 0.362381
$$655$$ 0 0
$$656$$ −144.260 −5.63239
$$657$$ −3.30065 −0.128771
$$658$$ 8.69378 0.338919
$$659$$ 7.19180 0.280153 0.140076 0.990141i $$-0.455265\pi$$
0.140076 + 0.990141i $$0.455265\pi$$
$$660$$ 0 0
$$661$$ 43.0214 1.67334 0.836669 0.547709i $$-0.184500\pi$$
0.836669 + 0.547709i $$0.184500\pi$$
$$662$$ 67.8666 2.63771
$$663$$ 10.5260 0.408796
$$664$$ −76.7920 −2.98011
$$665$$ 0 0
$$666$$ 12.3117 0.477067
$$667$$ −24.1987 −0.936977
$$668$$ 23.7461 0.918765
$$669$$ −13.1112 −0.506907
$$670$$ 0 0
$$671$$ −7.85494 −0.303237
$$672$$ 28.8746 1.11386
$$673$$ −29.8031 −1.14882 −0.574412 0.818566i $$-0.694769\pi$$
−0.574412 + 0.818566i $$0.694769\pi$$
$$674$$ 34.9961 1.34800
$$675$$ 0 0
$$676$$ −30.4464 −1.17102
$$677$$ −26.7346 −1.02749 −0.513746 0.857942i $$-0.671743\pi$$
−0.513746 + 0.857942i $$0.671743\pi$$
$$678$$ 16.2618 0.624532
$$679$$ 18.1664 0.697163
$$680$$ 0 0
$$681$$ 26.2807 1.00708
$$682$$ 20.5907 0.788457
$$683$$ 28.4950 1.09033 0.545165 0.838329i $$-0.316467\pi$$
0.545165 + 0.838329i $$0.316467\pi$$
$$684$$ −6.14702 −0.235037
$$685$$ 0 0
$$686$$ 50.5623 1.93048
$$687$$ −1.99605 −0.0761540
$$688$$ 33.1251 1.26288
$$689$$ −6.82321 −0.259944
$$690$$ 0 0
$$691$$ 34.6469 1.31803 0.659015 0.752130i $$-0.270973\pi$$
0.659015 + 0.752130i $$0.270973\pi$$
$$692$$ −13.9760 −0.531286
$$693$$ 1.82001 0.0691365
$$694$$ 16.1263 0.612145
$$695$$ 0 0
$$696$$ 51.5357 1.95345
$$697$$ 44.2763 1.67708
$$698$$ 61.0451 2.31059
$$699$$ 0.525548 0.0198781
$$700$$ 0 0
$$701$$ 0.973305 0.0367612 0.0183806 0.999831i $$-0.494149\pi$$
0.0183806 + 0.999831i $$0.494149\pi$$
$$702$$ 7.19384 0.271514
$$703$$ 5.39536 0.203490
$$704$$ 21.9756 0.828237
$$705$$ 0 0
$$706$$ 13.0454 0.490971
$$707$$ −22.8854 −0.860696
$$708$$ −6.97412 −0.262103
$$709$$ −11.3242 −0.425291 −0.212645 0.977129i $$-0.568208\pi$$
−0.212645 + 0.977129i $$0.568208\pi$$
$$710$$ 0 0
$$711$$ −4.00404 −0.150163
$$712$$ −133.630 −5.00801
$$713$$ 28.8595 1.08080
$$714$$ −17.8229 −0.667004
$$715$$ 0 0
$$716$$ −81.5742 −3.04857
$$717$$ 7.00971 0.261783
$$718$$ −77.6312 −2.89717
$$719$$ −30.7320 −1.14611 −0.573056 0.819516i $$-0.694242\pi$$
−0.573056 + 0.819516i $$0.694242\pi$$
$$720$$ 0 0
$$721$$ 2.30962 0.0860149
$$722$$ 47.3279 1.76136
$$723$$ −2.31554 −0.0861159
$$724$$ 1.05888 0.0393529
$$725$$ 0 0
$$726$$ −26.4302 −0.980915
$$727$$ −6.56836 −0.243607 −0.121803 0.992554i $$-0.538868\pi$$
−0.121803 + 0.992554i $$0.538868\pi$$
$$728$$ −39.0813 −1.44845
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −10.1668 −0.376032
$$732$$ 38.0086 1.40484
$$733$$ −31.9276 −1.17927 −0.589636 0.807669i $$-0.700729\pi$$
−0.589636 + 0.807669i $$0.700729\pi$$
$$734$$ 39.5235 1.45884
$$735$$ 0 0
$$736$$ 69.5395 2.56326
$$737$$ 13.4259 0.494551
$$738$$ 30.2600 1.11389
$$739$$ −15.3072 −0.563083 −0.281542 0.959549i $$-0.590846\pi$$
−0.281542 + 0.959549i $$0.590846\pi$$
$$740$$ 0 0
$$741$$ 3.15257 0.115812
$$742$$ 11.5532 0.424132
$$743$$ 16.5455 0.606995 0.303498 0.952832i $$-0.401846\pi$$
0.303498 + 0.952832i $$0.401846\pi$$
$$744$$ −61.4617 −2.25330
$$745$$ 0 0
$$746$$ 88.7774 3.25037
$$747$$ 8.87482 0.324713
$$748$$ −22.1417 −0.809582
$$749$$ 18.5769 0.678786
$$750$$ 0 0
$$751$$ 46.0279 1.67958 0.839791 0.542911i $$-0.182678\pi$$
0.839791 + 0.542911i $$0.182678\pi$$
$$752$$ −24.5674 −0.895880
$$753$$ 6.17885 0.225170
$$754$$ −42.8461 −1.56036
$$755$$ 0 0
$$756$$ −8.80669 −0.320296
$$757$$ −26.5282 −0.964184 −0.482092 0.876121i $$-0.660123\pi$$
−0.482092 + 0.876121i $$0.660123\pi$$
$$758$$ 40.7239 1.47916
$$759$$ 4.38318 0.159099
$$760$$ 0 0
$$761$$ −28.4241 −1.03037 −0.515187 0.857078i $$-0.672277\pi$$
−0.515187 + 0.857078i $$0.672277\pi$$
$$762$$ −54.3755 −1.96982
$$763$$ 5.81843 0.210641
$$764$$ −41.3426 −1.49572
$$765$$ 0 0
$$766$$ 21.5181 0.777481
$$767$$ 3.57675 0.129149
$$768$$ −14.3555 −0.518009
$$769$$ −8.95952 −0.323089 −0.161544 0.986865i $$-0.551647\pi$$
−0.161544 + 0.986865i $$0.551647\pi$$
$$770$$ 0 0
$$771$$ −13.2239 −0.476246
$$772$$ −85.7515 −3.08626
$$773$$ 23.1042 0.831001 0.415500 0.909593i $$-0.363607\pi$$
0.415500 + 0.909593i $$0.363607\pi$$
$$774$$ −6.94834 −0.249753
$$775$$ 0 0
$$776$$ −93.1752 −3.34479
$$777$$ 7.72980 0.277305
$$778$$ −2.88757 −0.103524
$$779$$ 13.2609 0.475120
$$780$$ 0 0
$$781$$ −6.45651 −0.231032
$$782$$ −42.9232 −1.53493
$$783$$ −5.95595 −0.212848
$$784$$ −53.2115 −1.90041
$$785$$ 0 0
$$786$$ −8.19578 −0.292334
$$787$$ −2.03390 −0.0725009 −0.0362504 0.999343i $$-0.511541\pi$$
−0.0362504 + 0.999343i $$0.511541\pi$$
$$788$$ −89.4578 −3.18680
$$789$$ 6.97643 0.248367
$$790$$ 0 0
$$791$$ 10.2099 0.363022
$$792$$ −9.33480 −0.331698
$$793$$ −19.4931 −0.692220
$$794$$ 65.0637 2.30902
$$795$$ 0 0
$$796$$ −10.5368 −0.373469
$$797$$ −12.1923 −0.431874 −0.215937 0.976407i $$-0.569281\pi$$
−0.215937 + 0.976407i $$0.569281\pi$$
$$798$$ −5.33800 −0.188963
$$799$$ 7.54024 0.266754
$$800$$ 0 0
$$801$$ 15.4436 0.545673
$$802$$ 5.35572 0.189117
$$803$$ −3.56080 −0.125658
$$804$$ −64.9656 −2.29116
$$805$$ 0 0
$$806$$ 51.0985 1.79987
$$807$$ −23.7199 −0.834979
$$808$$ 117.379 4.12938
$$809$$ 8.83355 0.310571 0.155286 0.987870i $$-0.450370\pi$$
0.155286 + 0.987870i $$0.450370\pi$$
$$810$$ 0 0
$$811$$ −13.9184 −0.488741 −0.244370 0.969682i $$-0.578581\pi$$
−0.244370 + 0.969682i $$0.578581\pi$$
$$812$$ 52.4522 1.84071
$$813$$ −28.9910 −1.01676
$$814$$ 13.2820 0.465535
$$815$$ 0 0
$$816$$ 50.3649 1.76312
$$817$$ −3.04498 −0.106530
$$818$$ −17.9085 −0.626158
$$819$$ 4.51660 0.157823
$$820$$ 0 0
$$821$$ −29.4930 −1.02931 −0.514656 0.857397i $$-0.672080\pi$$
−0.514656 + 0.857397i $$0.672080\pi$$
$$822$$ −55.1113 −1.92223
$$823$$ −41.9661 −1.46285 −0.731423 0.681924i $$-0.761143\pi$$
−0.731423 + 0.681924i $$0.761143\pi$$
$$824$$ −11.8460 −0.412675
$$825$$ 0 0
$$826$$ −6.05624 −0.210724
$$827$$ 37.4674 1.30287 0.651435 0.758704i $$-0.274167\pi$$
0.651435 + 0.758704i $$0.274167\pi$$
$$828$$ −21.2094 −0.737077
$$829$$ −9.55614 −0.331899 −0.165949 0.986134i $$-0.553069\pi$$
−0.165949 + 0.986134i $$0.553069\pi$$
$$830$$ 0 0
$$831$$ 8.58667 0.297868
$$832$$ 54.5355 1.89068
$$833$$ 16.3317 0.565860
$$834$$ 49.2222 1.70443
$$835$$ 0 0
$$836$$ −6.63152 −0.229356
$$837$$ 7.10310 0.245519
$$838$$ −57.3792 −1.98213
$$839$$ 54.7310 1.88952 0.944761 0.327759i $$-0.106293\pi$$
0.944761 + 0.327759i $$0.106293\pi$$
$$840$$ 0 0
$$841$$ 6.47334 0.223219
$$842$$ −60.0197 −2.06842
$$843$$ −17.1661 −0.591233
$$844$$ −124.352 −4.28037
$$845$$ 0 0
$$846$$ 5.15327 0.177173
$$847$$ −16.5940 −0.570177
$$848$$ −32.6478 −1.12113
$$849$$ 9.62005 0.330159
$$850$$ 0 0
$$851$$ 18.6159 0.638144
$$852$$ 31.2419 1.07033
$$853$$ −15.3067 −0.524092 −0.262046 0.965055i $$-0.584397\pi$$
−0.262046 + 0.965055i $$0.584397\pi$$
$$854$$ 33.0062 1.12945
$$855$$ 0 0
$$856$$ −95.2806 −3.25662
$$857$$ −8.66910 −0.296131 −0.148065 0.988978i $$-0.547305\pi$$
−0.148065 + 0.988978i $$0.547305\pi$$
$$858$$ 7.76084 0.264951
$$859$$ −30.4078 −1.03750 −0.518750 0.854926i $$-0.673603\pi$$
−0.518750 + 0.854926i $$0.673603\pi$$
$$860$$ 0 0
$$861$$ 18.9985 0.647468
$$862$$ 81.7208 2.78342
$$863$$ −18.0425 −0.614175 −0.307088 0.951681i $$-0.599354\pi$$
−0.307088 + 0.951681i $$0.599354\pi$$
$$864$$ 17.1155 0.582283
$$865$$ 0 0
$$866$$ −1.54945 −0.0526526
$$867$$ 1.54198 0.0523684
$$868$$ −62.5548 −2.12325
$$869$$ −4.31963 −0.146533
$$870$$ 0 0
$$871$$ 33.3183 1.12895
$$872$$ −29.8426 −1.01060
$$873$$ 10.7682 0.364449
$$874$$ −12.8556 −0.434848
$$875$$ 0 0
$$876$$ 17.2301 0.582149
$$877$$ 22.3057 0.753211 0.376605 0.926374i $$-0.377091\pi$$
0.376605 + 0.926374i $$0.377091\pi$$
$$878$$ −34.4062 −1.16115
$$879$$ −16.8202 −0.567332
$$880$$ 0 0
$$881$$ 7.75956 0.261426 0.130713 0.991420i $$-0.458273\pi$$
0.130713 + 0.991420i $$0.458273\pi$$
$$882$$ 11.1617 0.375833
$$883$$ −19.9527 −0.671462 −0.335731 0.941958i $$-0.608983\pi$$
−0.335731 + 0.941958i $$0.608983\pi$$
$$884$$ −54.9477 −1.84809
$$885$$ 0 0
$$886$$ −38.4641 −1.29223
$$887$$ 8.60289 0.288857 0.144428 0.989515i $$-0.453866\pi$$
0.144428 + 0.989515i $$0.453866\pi$$
$$888$$ −39.6460 −1.33043
$$889$$ −34.1393 −1.14499
$$890$$ 0 0
$$891$$ 1.07882 0.0361418
$$892$$ 68.4429 2.29164
$$893$$ 2.25832 0.0755719
$$894$$ 1.89688 0.0634413
$$895$$ 0 0
$$896$$ −34.5915 −1.15562
$$897$$ 10.8775 0.363187
$$898$$ 32.1539 1.07299
$$899$$ −42.3057 −1.41097
$$900$$ 0 0
$$901$$ 10.0203 0.333824
$$902$$ 32.6450 1.08696
$$903$$ −4.36247 −0.145174
$$904$$ −52.3663 −1.74168
$$905$$ 0 0
$$906$$ 18.9312 0.628948
$$907$$ 6.26125 0.207902 0.103951 0.994582i $$-0.466852\pi$$
0.103951 + 0.994582i $$0.466852\pi$$
$$908$$ −137.190 −4.55282
$$909$$ −13.5654 −0.449937
$$910$$ 0 0
$$911$$ 0.905575 0.0300030 0.0150015 0.999887i $$-0.495225\pi$$
0.0150015 + 0.999887i $$0.495225\pi$$
$$912$$ 15.0844 0.499495
$$913$$ 9.57431 0.316863
$$914$$ 22.0916 0.730726
$$915$$ 0 0
$$916$$ 10.4198 0.344279
$$917$$ −5.14567 −0.169925
$$918$$ −10.5646 −0.348682
$$919$$ 43.9933 1.45120 0.725602 0.688114i $$-0.241561\pi$$
0.725602 + 0.688114i $$0.241561\pi$$
$$920$$ 0 0
$$921$$ −11.3212 −0.373046
$$922$$ −3.41836 −0.112578
$$923$$ −16.0227 −0.527394
$$924$$ −9.50081 −0.312554
$$925$$ 0 0
$$926$$ −87.8029 −2.88538
$$927$$ 1.36904 0.0449651
$$928$$ −101.939 −3.34632
$$929$$ −36.7447 −1.20555 −0.602777 0.797910i $$-0.705939\pi$$
−0.602777 + 0.797910i $$0.705939\pi$$
$$930$$ 0 0
$$931$$ 4.89140 0.160309
$$932$$ −2.74347 −0.0898652
$$933$$ −18.5635 −0.607740
$$934$$ −38.7866 −1.26914
$$935$$ 0 0
$$936$$ −23.1656 −0.757190
$$937$$ 24.6490 0.805248 0.402624 0.915366i $$-0.368098\pi$$
0.402624 + 0.915366i $$0.368098\pi$$
$$938$$ −56.4153 −1.84203
$$939$$ 9.69951 0.316532
$$940$$ 0 0
$$941$$ −59.1479 −1.92817 −0.964083 0.265602i $$-0.914429\pi$$
−0.964083 + 0.265602i $$0.914429\pi$$
$$942$$ −13.7813 −0.449019
$$943$$ 45.7546 1.48998
$$944$$ 17.1141 0.557016
$$945$$ 0 0
$$946$$ −7.49600 −0.243716
$$947$$ −30.0960 −0.977989 −0.488995 0.872287i $$-0.662636\pi$$
−0.488995 + 0.872287i $$0.662636\pi$$
$$948$$ 20.9019 0.678861
$$949$$ −8.83661 −0.286849
$$950$$ 0 0
$$951$$ −7.47647 −0.242441
$$952$$ 57.3931 1.86012
$$953$$ −20.6393 −0.668572 −0.334286 0.942472i $$-0.608495\pi$$
−0.334286 + 0.942472i $$0.608495\pi$$
$$954$$ 6.84822 0.221719
$$955$$ 0 0
$$956$$ −36.5921 −1.18347
$$957$$ −6.42538 −0.207703
$$958$$ 45.3084 1.46385
$$959$$ −34.6013 −1.11733
$$960$$ 0 0
$$961$$ 19.4540 0.627549
$$962$$ 32.9612 1.06271
$$963$$ 11.0115 0.354842
$$964$$ 12.0876 0.389315
$$965$$ 0 0
$$966$$ −18.4180 −0.592588
$$967$$ −22.6016 −0.726817 −0.363408 0.931630i $$-0.618387\pi$$
−0.363408 + 0.931630i $$0.618387\pi$$
$$968$$ 85.1103 2.73555
$$969$$ −4.62972 −0.148728
$$970$$ 0 0
$$971$$ 24.1381 0.774628 0.387314 0.921948i $$-0.373403\pi$$
0.387314 + 0.921948i $$0.373403\pi$$
$$972$$ −5.22020 −0.167438
$$973$$ 30.9038 0.990732
$$974$$ −107.007 −3.42874
$$975$$ 0 0
$$976$$ −93.2708 −2.98553
$$977$$ 23.0506 0.737454 0.368727 0.929538i $$-0.379794\pi$$
0.368727 + 0.929538i $$0.379794\pi$$
$$978$$ −65.1911 −2.08458
$$979$$ 16.6608 0.532483
$$980$$ 0 0
$$981$$ 3.44890 0.110115
$$982$$ −37.6870 −1.20264
$$983$$ 29.8713 0.952745 0.476373 0.879243i $$-0.341951\pi$$
0.476373 + 0.879243i $$0.341951\pi$$
$$984$$ −97.4430 −3.10637
$$985$$ 0 0
$$986$$ 62.9220 2.00384
$$987$$ 3.23544 0.102985
$$988$$ −16.4570 −0.523567
$$989$$ −10.5063 −0.334079
$$990$$ 0 0
$$991$$ 16.6560 0.529096 0.264548 0.964372i $$-0.414777\pi$$
0.264548 + 0.964372i $$0.414777\pi$$
$$992$$ 121.573 3.85996
$$993$$ 25.2570 0.801507
$$994$$ 27.1300 0.860513
$$995$$ 0 0
$$996$$ −46.3283 −1.46797
$$997$$ 12.4934 0.395669 0.197835 0.980235i $$-0.436609\pi$$
0.197835 + 0.980235i $$0.436609\pi$$
$$998$$ −40.9278 −1.29555
$$999$$ 4.58187 0.144964
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.j.1.1 6
3.2 odd 2 5625.2.a.p.1.6 6
5.2 odd 4 1875.2.b.f.1249.1 12
5.3 odd 4 1875.2.b.f.1249.12 12
5.4 even 2 1875.2.a.k.1.6 6
15.14 odd 2 5625.2.a.q.1.1 6
25.2 odd 20 375.2.i.d.274.1 24
25.9 even 10 375.2.g.c.151.3 12
25.11 even 5 75.2.g.c.46.1 yes 12
25.12 odd 20 375.2.i.d.349.6 24
25.13 odd 20 375.2.i.d.349.1 24
25.14 even 10 375.2.g.c.226.3 12
25.16 even 5 75.2.g.c.31.1 12
25.23 odd 20 375.2.i.d.274.6 24
75.11 odd 10 225.2.h.d.46.3 12
75.41 odd 10 225.2.h.d.181.3 12

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.31.1 12 25.16 even 5
75.2.g.c.46.1 yes 12 25.11 even 5
225.2.h.d.46.3 12 75.11 odd 10
225.2.h.d.181.3 12 75.41 odd 10
375.2.g.c.151.3 12 25.9 even 10
375.2.g.c.226.3 12 25.14 even 10
375.2.i.d.274.1 24 25.2 odd 20
375.2.i.d.274.6 24 25.23 odd 20
375.2.i.d.349.1 24 25.13 odd 20
375.2.i.d.349.6 24 25.12 odd 20
1875.2.a.j.1.1 6 1.1 even 1 trivial
1875.2.a.k.1.6 6 5.4 even 2
1875.2.b.f.1249.1 12 5.2 odd 4
1875.2.b.f.1249.12 12 5.3 odd 4
5625.2.a.p.1.6 6 3.2 odd 2
5625.2.a.q.1.1 6 15.14 odd 2