# Properties

 Label 1875.2.a.l.1.6 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: 6.6.46840000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9$$ x^6 - x^5 - 11*x^4 + 8*x^3 + 31*x^2 - 15*x - 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$2.78712$$ 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)

 $$f(q)$$ $$=$$ $$q+2.78712 q^{2} -1.00000 q^{3} +5.76803 q^{4} -2.78712 q^{6} -3.15000 q^{7} +10.5020 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.78712 q^{2} -1.00000 q^{3} +5.76803 q^{4} -2.78712 q^{6} -3.15000 q^{7} +10.5020 q^{8} +1.00000 q^{9} +2.94681 q^{11} -5.76803 q^{12} +0.188171 q^{13} -8.77943 q^{14} +17.7341 q^{16} +2.62743 q^{17} +2.78712 q^{18} +1.94100 q^{19} +3.15000 q^{21} +8.21310 q^{22} -1.30447 q^{23} -10.5020 q^{24} +0.524454 q^{26} -1.00000 q^{27} -18.1693 q^{28} +1.23197 q^{29} +2.65174 q^{31} +28.4233 q^{32} -2.94681 q^{33} +7.32297 q^{34} +5.76803 q^{36} +10.1161 q^{37} +5.40980 q^{38} -0.188171 q^{39} +3.85940 q^{41} +8.77943 q^{42} -9.63734 q^{43} +16.9973 q^{44} -3.63570 q^{46} -11.9846 q^{47} -17.7341 q^{48} +2.92250 q^{49} -2.62743 q^{51} +1.08537 q^{52} +0.369521 q^{53} -2.78712 q^{54} -33.0812 q^{56} -1.94100 q^{57} +3.43364 q^{58} -6.90864 q^{59} +5.68643 q^{61} +7.39071 q^{62} -3.15000 q^{63} +43.7507 q^{64} -8.21310 q^{66} +3.24188 q^{67} +15.1551 q^{68} +1.30447 q^{69} +6.69982 q^{71} +10.5020 q^{72} -9.46938 q^{73} +28.1948 q^{74} +11.1957 q^{76} -9.28244 q^{77} -0.524454 q^{78} +0.420137 q^{79} +1.00000 q^{81} +10.7566 q^{82} -12.1635 q^{83} +18.1693 q^{84} -26.8604 q^{86} -1.23197 q^{87} +30.9473 q^{88} -15.1828 q^{89} -0.592737 q^{91} -7.52421 q^{92} -2.65174 q^{93} -33.4025 q^{94} -28.4233 q^{96} +13.9035 q^{97} +8.14536 q^{98} +2.94681 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{2} - 6 q^{3} + 11 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10})$$ 6 * q + q^2 - 6 * q^3 + 11 * q^4 - q^6 - 2 * q^7 + 6 * q^8 + 6 * q^9 $$6 q + q^{2} - 6 q^{3} + 11 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9} - 11 q^{12} - 4 q^{14} + 17 q^{16} + 2 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 9 q^{22} - q^{23} - 6 q^{24} + 37 q^{26} - 6 q^{27} - 44 q^{28} + 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 11 q^{36} - 22 q^{37} + 27 q^{38} + 33 q^{41} + 4 q^{42} - 3 q^{43} - 11 q^{44} - 12 q^{46} - 6 q^{47} - 17 q^{48} + 4 q^{49} - 2 q^{51} - 33 q^{52} + 14 q^{53} - q^{54} - 30 q^{56} + 2 q^{57} - q^{58} - 8 q^{59} + 34 q^{61} + 31 q^{62} - 2 q^{63} + 12 q^{64} - 9 q^{66} + 2 q^{67} + 27 q^{68} + q^{69} - 3 q^{71} + 6 q^{72} - 36 q^{73} + 36 q^{74} + 27 q^{76} + 16 q^{77} - 37 q^{78} + 25 q^{79} + 6 q^{81} + 36 q^{82} - 12 q^{83} + 44 q^{84} - 30 q^{86} - 31 q^{87} + 56 q^{88} + 18 q^{89} + 28 q^{91} + 3 q^{92} + 2 q^{93} - 50 q^{94} - 33 q^{96} + 7 q^{97} + 15 q^{98}+O(q^{100})$$ 6 * q + q^2 - 6 * q^3 + 11 * q^4 - q^6 - 2 * q^7 + 6 * q^8 + 6 * q^9 - 11 * q^12 - 4 * q^14 + 17 * q^16 + 2 * q^17 + q^18 - 2 * q^19 + 2 * q^21 + 9 * q^22 - q^23 - 6 * q^24 + 37 * q^26 - 6 * q^27 - 44 * q^28 + 31 * q^29 - 2 * q^31 + 33 * q^32 + 37 * q^34 + 11 * q^36 - 22 * q^37 + 27 * q^38 + 33 * q^41 + 4 * q^42 - 3 * q^43 - 11 * q^44 - 12 * q^46 - 6 * q^47 - 17 * q^48 + 4 * q^49 - 2 * q^51 - 33 * q^52 + 14 * q^53 - q^54 - 30 * q^56 + 2 * q^57 - q^58 - 8 * q^59 + 34 * q^61 + 31 * q^62 - 2 * q^63 + 12 * q^64 - 9 * q^66 + 2 * q^67 + 27 * q^68 + q^69 - 3 * q^71 + 6 * q^72 - 36 * q^73 + 36 * q^74 + 27 * q^76 + 16 * q^77 - 37 * q^78 + 25 * q^79 + 6 * q^81 + 36 * q^82 - 12 * q^83 + 44 * q^84 - 30 * q^86 - 31 * q^87 + 56 * q^88 + 18 * q^89 + 28 * q^91 + 3 * q^92 + 2 * q^93 - 50 * q^94 - 33 * q^96 + 7 * q^97 + 15 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.78712 1.97079 0.985395 0.170281i $$-0.0544677\pi$$
0.985395 + 0.170281i $$0.0544677\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 5.76803 2.88402
$$5$$ 0 0
$$6$$ −2.78712 −1.13784
$$7$$ −3.15000 −1.19059 −0.595294 0.803508i $$-0.702964\pi$$
−0.595294 + 0.803508i $$0.702964\pi$$
$$8$$ 10.5020 3.71300
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.94681 0.888496 0.444248 0.895904i $$-0.353471\pi$$
0.444248 + 0.895904i $$0.353471\pi$$
$$12$$ −5.76803 −1.66509
$$13$$ 0.188171 0.0521891 0.0260946 0.999659i $$-0.491693\pi$$
0.0260946 + 0.999659i $$0.491693\pi$$
$$14$$ −8.77943 −2.34640
$$15$$ 0 0
$$16$$ 17.7341 4.43354
$$17$$ 2.62743 0.637246 0.318623 0.947882i $$-0.396780\pi$$
0.318623 + 0.947882i $$0.396780\pi$$
$$18$$ 2.78712 0.656930
$$19$$ 1.94100 0.445296 0.222648 0.974899i $$-0.428530\pi$$
0.222648 + 0.974899i $$0.428530\pi$$
$$20$$ 0 0
$$21$$ 3.15000 0.687386
$$22$$ 8.21310 1.75104
$$23$$ −1.30447 −0.272000 −0.136000 0.990709i $$-0.543425\pi$$
−0.136000 + 0.990709i $$0.543425\pi$$
$$24$$ −10.5020 −2.14370
$$25$$ 0 0
$$26$$ 0.524454 0.102854
$$27$$ −1.00000 −0.192450
$$28$$ −18.1693 −3.43368
$$29$$ 1.23197 0.228770 0.114385 0.993436i $$-0.463510\pi$$
0.114385 + 0.993436i $$0.463510\pi$$
$$30$$ 0 0
$$31$$ 2.65174 0.476266 0.238133 0.971233i $$-0.423465\pi$$
0.238133 + 0.971233i $$0.423465\pi$$
$$32$$ 28.4233 5.02457
$$33$$ −2.94681 −0.512973
$$34$$ 7.32297 1.25588
$$35$$ 0 0
$$36$$ 5.76803 0.961339
$$37$$ 10.1161 1.66308 0.831540 0.555466i $$-0.187460\pi$$
0.831540 + 0.555466i $$0.187460\pi$$
$$38$$ 5.40980 0.877585
$$39$$ −0.188171 −0.0301314
$$40$$ 0 0
$$41$$ 3.85940 0.602737 0.301368 0.953508i $$-0.402557\pi$$
0.301368 + 0.953508i $$0.402557\pi$$
$$42$$ 8.77943 1.35469
$$43$$ −9.63734 −1.46968 −0.734840 0.678240i $$-0.762743\pi$$
−0.734840 + 0.678240i $$0.762743\pi$$
$$44$$ 16.9973 2.56244
$$45$$ 0 0
$$46$$ −3.63570 −0.536055
$$47$$ −11.9846 −1.74814 −0.874068 0.485804i $$-0.838527\pi$$
−0.874068 + 0.485804i $$0.838527\pi$$
$$48$$ −17.7341 −2.55970
$$49$$ 2.92250 0.417500
$$50$$ 0 0
$$51$$ −2.62743 −0.367914
$$52$$ 1.08537 0.150514
$$53$$ 0.369521 0.0507576 0.0253788 0.999678i $$-0.491921\pi$$
0.0253788 + 0.999678i $$0.491921\pi$$
$$54$$ −2.78712 −0.379279
$$55$$ 0 0
$$56$$ −33.0812 −4.42066
$$57$$ −1.94100 −0.257092
$$58$$ 3.43364 0.450859
$$59$$ −6.90864 −0.899428 −0.449714 0.893173i $$-0.648474\pi$$
−0.449714 + 0.893173i $$0.648474\pi$$
$$60$$ 0 0
$$61$$ 5.68643 0.728073 0.364037 0.931385i $$-0.381398\pi$$
0.364037 + 0.931385i $$0.381398\pi$$
$$62$$ 7.39071 0.938621
$$63$$ −3.15000 −0.396863
$$64$$ 43.7507 5.46884
$$65$$ 0 0
$$66$$ −8.21310 −1.01096
$$67$$ 3.24188 0.396058 0.198029 0.980196i $$-0.436546\pi$$
0.198029 + 0.980196i $$0.436546\pi$$
$$68$$ 15.1551 1.83783
$$69$$ 1.30447 0.157039
$$70$$ 0 0
$$71$$ 6.69982 0.795122 0.397561 0.917576i $$-0.369857\pi$$
0.397561 + 0.917576i $$0.369857\pi$$
$$72$$ 10.5020 1.23767
$$73$$ −9.46938 −1.10831 −0.554153 0.832415i $$-0.686958\pi$$
−0.554153 + 0.832415i $$0.686958\pi$$
$$74$$ 28.1948 3.27758
$$75$$ 0 0
$$76$$ 11.1957 1.28424
$$77$$ −9.28244 −1.05783
$$78$$ −0.524454 −0.0593827
$$79$$ 0.420137 0.0472691 0.0236345 0.999721i $$-0.492476\pi$$
0.0236345 + 0.999721i $$0.492476\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.7566 1.18787
$$83$$ −12.1635 −1.33512 −0.667559 0.744557i $$-0.732661\pi$$
−0.667559 + 0.744557i $$0.732661\pi$$
$$84$$ 18.1693 1.98243
$$85$$ 0 0
$$86$$ −26.8604 −2.89643
$$87$$ −1.23197 −0.132081
$$88$$ 30.9473 3.29899
$$89$$ −15.1828 −1.60937 −0.804687 0.593699i $$-0.797667\pi$$
−0.804687 + 0.593699i $$0.797667\pi$$
$$90$$ 0 0
$$91$$ −0.592737 −0.0621358
$$92$$ −7.52421 −0.784453
$$93$$ −2.65174 −0.274972
$$94$$ −33.4025 −3.44521
$$95$$ 0 0
$$96$$ −28.4233 −2.90094
$$97$$ 13.9035 1.41169 0.705845 0.708367i $$-0.250568\pi$$
0.705845 + 0.708367i $$0.250568\pi$$
$$98$$ 8.14536 0.822805
$$99$$ 2.94681 0.296165
$$100$$ 0 0
$$101$$ −9.67188 −0.962388 −0.481194 0.876614i $$-0.659797\pi$$
−0.481194 + 0.876614i $$0.659797\pi$$
$$102$$ −7.32297 −0.725082
$$103$$ 8.69434 0.856679 0.428339 0.903618i $$-0.359099\pi$$
0.428339 + 0.903618i $$0.359099\pi$$
$$104$$ 1.97616 0.193778
$$105$$ 0 0
$$106$$ 1.02990 0.100033
$$107$$ −2.56585 −0.248050 −0.124025 0.992279i $$-0.539580\pi$$
−0.124025 + 0.992279i $$0.539580\pi$$
$$108$$ −5.76803 −0.555029
$$109$$ −15.2352 −1.45926 −0.729632 0.683840i $$-0.760309\pi$$
−0.729632 + 0.683840i $$0.760309\pi$$
$$110$$ 0 0
$$111$$ −10.1161 −0.960179
$$112$$ −55.8626 −5.27852
$$113$$ 2.20391 0.207326 0.103663 0.994612i $$-0.466944\pi$$
0.103663 + 0.994612i $$0.466944\pi$$
$$114$$ −5.40980 −0.506674
$$115$$ 0 0
$$116$$ 7.10602 0.659778
$$117$$ 0.188171 0.0173964
$$118$$ −19.2552 −1.77258
$$119$$ −8.27641 −0.758697
$$120$$ 0 0
$$121$$ −2.31633 −0.210575
$$122$$ 15.8488 1.43488
$$123$$ −3.85940 −0.347990
$$124$$ 15.2953 1.37356
$$125$$ 0 0
$$126$$ −8.77943 −0.782133
$$127$$ −14.9689 −1.32827 −0.664136 0.747611i $$-0.731201\pi$$
−0.664136 + 0.747611i $$0.731201\pi$$
$$128$$ 65.0920 5.75337
$$129$$ 9.63734 0.848521
$$130$$ 0 0
$$131$$ 11.2957 0.986911 0.493456 0.869771i $$-0.335734\pi$$
0.493456 + 0.869771i $$0.335734\pi$$
$$132$$ −16.9973 −1.47942
$$133$$ −6.11415 −0.530164
$$134$$ 9.03550 0.780548
$$135$$ 0 0
$$136$$ 27.5932 2.36610
$$137$$ 4.17877 0.357017 0.178508 0.983938i $$-0.442873\pi$$
0.178508 + 0.983938i $$0.442873\pi$$
$$138$$ 3.63570 0.309492
$$139$$ −15.3552 −1.30241 −0.651207 0.758900i $$-0.725737\pi$$
−0.651207 + 0.758900i $$0.725737\pi$$
$$140$$ 0 0
$$141$$ 11.9846 1.00929
$$142$$ 18.6732 1.56702
$$143$$ 0.554502 0.0463698
$$144$$ 17.7341 1.47785
$$145$$ 0 0
$$146$$ −26.3923 −2.18424
$$147$$ −2.92250 −0.241044
$$148$$ 58.3501 4.79635
$$149$$ 3.43364 0.281294 0.140647 0.990060i $$-0.455082\pi$$
0.140647 + 0.990060i $$0.455082\pi$$
$$150$$ 0 0
$$151$$ 10.7716 0.876581 0.438290 0.898833i $$-0.355584\pi$$
0.438290 + 0.898833i $$0.355584\pi$$
$$152$$ 20.3843 1.65338
$$153$$ 2.62743 0.212415
$$154$$ −25.8713 −2.08477
$$155$$ 0 0
$$156$$ −1.08537 −0.0868995
$$157$$ −13.4045 −1.06980 −0.534900 0.844916i $$-0.679651\pi$$
−0.534900 + 0.844916i $$0.679651\pi$$
$$158$$ 1.17097 0.0931574
$$159$$ −0.369521 −0.0293049
$$160$$ 0 0
$$161$$ 4.10907 0.323840
$$162$$ 2.78712 0.218977
$$163$$ −15.4178 −1.20762 −0.603808 0.797129i $$-0.706351\pi$$
−0.603808 + 0.797129i $$0.706351\pi$$
$$164$$ 22.2611 1.73830
$$165$$ 0 0
$$166$$ −33.9011 −2.63124
$$167$$ 17.4045 1.34680 0.673402 0.739276i $$-0.264832\pi$$
0.673402 + 0.739276i $$0.264832\pi$$
$$168$$ 33.0812 2.55227
$$169$$ −12.9646 −0.997276
$$170$$ 0 0
$$171$$ 1.94100 0.148432
$$172$$ −55.5885 −4.23858
$$173$$ −19.2737 −1.46535 −0.732676 0.680578i $$-0.761729\pi$$
−0.732676 + 0.680578i $$0.761729\pi$$
$$174$$ −3.43364 −0.260303
$$175$$ 0 0
$$176$$ 52.2591 3.93918
$$177$$ 6.90864 0.519285
$$178$$ −42.3163 −3.17174
$$179$$ 23.7940 1.77845 0.889223 0.457475i $$-0.151246\pi$$
0.889223 + 0.457475i $$0.151246\pi$$
$$180$$ 0 0
$$181$$ −10.6891 −0.794516 −0.397258 0.917707i $$-0.630038\pi$$
−0.397258 + 0.917707i $$0.630038\pi$$
$$182$$ −1.65203 −0.122457
$$183$$ −5.68643 −0.420353
$$184$$ −13.6995 −1.00994
$$185$$ 0 0
$$186$$ −7.39071 −0.541913
$$187$$ 7.74253 0.566190
$$188$$ −69.1277 −5.04165
$$189$$ 3.15000 0.229129
$$190$$ 0 0
$$191$$ −14.3581 −1.03892 −0.519458 0.854496i $$-0.673866\pi$$
−0.519458 + 0.854496i $$0.673866\pi$$
$$192$$ −43.7507 −3.15744
$$193$$ 1.95508 0.140730 0.0703648 0.997521i $$-0.477584\pi$$
0.0703648 + 0.997521i $$0.477584\pi$$
$$194$$ 38.7508 2.78214
$$195$$ 0 0
$$196$$ 16.8571 1.20408
$$197$$ −1.17958 −0.0840417 −0.0420209 0.999117i $$-0.513380\pi$$
−0.0420209 + 0.999117i $$0.513380\pi$$
$$198$$ 8.21310 0.583680
$$199$$ −6.55820 −0.464899 −0.232449 0.972608i $$-0.574674\pi$$
−0.232449 + 0.972608i $$0.574674\pi$$
$$200$$ 0 0
$$201$$ −3.24188 −0.228664
$$202$$ −26.9567 −1.89666
$$203$$ −3.88069 −0.272371
$$204$$ −15.1551 −1.06107
$$205$$ 0 0
$$206$$ 24.2322 1.68833
$$207$$ −1.30447 −0.0906667
$$208$$ 3.33705 0.231382
$$209$$ 5.71975 0.395643
$$210$$ 0 0
$$211$$ 12.3573 0.850715 0.425357 0.905026i $$-0.360148\pi$$
0.425357 + 0.905026i $$0.360148\pi$$
$$212$$ 2.13141 0.146386
$$213$$ −6.69982 −0.459064
$$214$$ −7.15134 −0.488856
$$215$$ 0 0
$$216$$ −10.5020 −0.714568
$$217$$ −8.35298 −0.567037
$$218$$ −42.4622 −2.87591
$$219$$ 9.46938 0.639881
$$220$$ 0 0
$$221$$ 0.494405 0.0332573
$$222$$ −28.1948 −1.89231
$$223$$ −7.30693 −0.489308 −0.244654 0.969610i $$-0.578674\pi$$
−0.244654 + 0.969610i $$0.578674\pi$$
$$224$$ −89.5333 −5.98219
$$225$$ 0 0
$$226$$ 6.14256 0.408597
$$227$$ −21.0202 −1.39516 −0.697579 0.716508i $$-0.745739\pi$$
−0.697579 + 0.716508i $$0.745739\pi$$
$$228$$ −11.1957 −0.741457
$$229$$ 19.5544 1.29219 0.646094 0.763258i $$-0.276401\pi$$
0.646094 + 0.763258i $$0.276401\pi$$
$$230$$ 0 0
$$231$$ 9.28244 0.610740
$$232$$ 12.9381 0.849425
$$233$$ 12.1472 0.795792 0.397896 0.917430i $$-0.369740\pi$$
0.397896 + 0.917430i $$0.369740\pi$$
$$234$$ 0.524454 0.0342846
$$235$$ 0 0
$$236$$ −39.8493 −2.59397
$$237$$ −0.420137 −0.0272908
$$238$$ −23.0673 −1.49523
$$239$$ 13.4517 0.870119 0.435059 0.900402i $$-0.356727\pi$$
0.435059 + 0.900402i $$0.356727\pi$$
$$240$$ 0 0
$$241$$ 19.9370 1.28426 0.642129 0.766597i $$-0.278051\pi$$
0.642129 + 0.766597i $$0.278051\pi$$
$$242$$ −6.45588 −0.415000
$$243$$ −1.00000 −0.0641500
$$244$$ 32.7995 2.09978
$$245$$ 0 0
$$246$$ −10.7566 −0.685816
$$247$$ 0.365239 0.0232396
$$248$$ 27.8485 1.76838
$$249$$ 12.1635 0.770830
$$250$$ 0 0
$$251$$ −2.76013 −0.174218 −0.0871088 0.996199i $$-0.527763\pi$$
−0.0871088 + 0.996199i $$0.527763\pi$$
$$252$$ −18.1693 −1.14456
$$253$$ −3.84401 −0.241671
$$254$$ −41.7200 −2.61775
$$255$$ 0 0
$$256$$ 93.9177 5.86986
$$257$$ −17.2848 −1.07820 −0.539098 0.842243i $$-0.681235\pi$$
−0.539098 + 0.842243i $$0.681235\pi$$
$$258$$ 26.8604 1.67226
$$259$$ −31.8658 −1.98004
$$260$$ 0 0
$$261$$ 1.23197 0.0762568
$$262$$ 31.4825 1.94500
$$263$$ −3.96578 −0.244541 −0.122270 0.992497i $$-0.539017\pi$$
−0.122270 + 0.992497i $$0.539017\pi$$
$$264$$ −30.9473 −1.90467
$$265$$ 0 0
$$266$$ −17.0409 −1.04484
$$267$$ 15.1828 0.929173
$$268$$ 18.6993 1.14224
$$269$$ 1.19564 0.0728993 0.0364496 0.999335i $$-0.488395\pi$$
0.0364496 + 0.999335i $$0.488395\pi$$
$$270$$ 0 0
$$271$$ −2.18295 −0.132605 −0.0663023 0.997800i $$-0.521120\pi$$
−0.0663023 + 0.997800i $$0.521120\pi$$
$$272$$ 46.5953 2.82525
$$273$$ 0.592737 0.0358741
$$274$$ 11.6467 0.703605
$$275$$ 0 0
$$276$$ 7.52421 0.452904
$$277$$ 1.95174 0.117268 0.0586342 0.998280i $$-0.481325\pi$$
0.0586342 + 0.998280i $$0.481325\pi$$
$$278$$ −42.7969 −2.56679
$$279$$ 2.65174 0.158755
$$280$$ 0 0
$$281$$ 2.30131 0.137284 0.0686422 0.997641i $$-0.478133\pi$$
0.0686422 + 0.997641i $$0.478133\pi$$
$$282$$ 33.4025 1.98909
$$283$$ −10.6459 −0.632835 −0.316417 0.948620i $$-0.602480\pi$$
−0.316417 + 0.948620i $$0.602480\pi$$
$$284$$ 38.6448 2.29315
$$285$$ 0 0
$$286$$ 1.54546 0.0913852
$$287$$ −12.1571 −0.717611
$$288$$ 28.4233 1.67486
$$289$$ −10.0966 −0.593918
$$290$$ 0 0
$$291$$ −13.9035 −0.815039
$$292$$ −54.6197 −3.19638
$$293$$ 14.9591 0.873921 0.436960 0.899481i $$-0.356055\pi$$
0.436960 + 0.899481i $$0.356055\pi$$
$$294$$ −8.14536 −0.475047
$$295$$ 0 0
$$296$$ 106.239 6.17502
$$297$$ −2.94681 −0.170991
$$298$$ 9.56995 0.554373
$$299$$ −0.245462 −0.0141954
$$300$$ 0 0
$$301$$ 30.3576 1.74978
$$302$$ 30.0217 1.72756
$$303$$ 9.67188 0.555635
$$304$$ 34.4220 1.97424
$$305$$ 0 0
$$306$$ 7.32297 0.418626
$$307$$ −3.07333 −0.175404 −0.0877021 0.996147i $$-0.527952\pi$$
−0.0877021 + 0.996147i $$0.527952\pi$$
$$308$$ −53.5414 −3.05081
$$309$$ −8.69434 −0.494604
$$310$$ 0 0
$$311$$ 24.9095 1.41249 0.706244 0.707968i $$-0.250388\pi$$
0.706244 + 0.707968i $$0.250388\pi$$
$$312$$ −1.97616 −0.111878
$$313$$ 9.28562 0.524854 0.262427 0.964952i $$-0.415477\pi$$
0.262427 + 0.964952i $$0.415477\pi$$
$$314$$ −37.3601 −2.10835
$$315$$ 0 0
$$316$$ 2.42336 0.136325
$$317$$ 20.7685 1.16647 0.583237 0.812302i $$-0.301786\pi$$
0.583237 + 0.812302i $$0.301786\pi$$
$$318$$ −1.02990 −0.0577538
$$319$$ 3.63037 0.203261
$$320$$ 0 0
$$321$$ 2.56585 0.143212
$$322$$ 11.4525 0.638221
$$323$$ 5.09984 0.283763
$$324$$ 5.76803 0.320446
$$325$$ 0 0
$$326$$ −42.9713 −2.37996
$$327$$ 15.2352 0.842507
$$328$$ 40.5312 2.23796
$$329$$ 37.7515 2.08131
$$330$$ 0 0
$$331$$ −33.9172 −1.86426 −0.932128 0.362129i $$-0.882050\pi$$
−0.932128 + 0.362129i $$0.882050\pi$$
$$332$$ −70.1595 −3.85050
$$333$$ 10.1161 0.554360
$$334$$ 48.5085 2.65427
$$335$$ 0 0
$$336$$ 55.8626 3.04755
$$337$$ 17.7792 0.968494 0.484247 0.874931i $$-0.339094\pi$$
0.484247 + 0.874931i $$0.339094\pi$$
$$338$$ −36.1339 −1.96542
$$339$$ −2.20391 −0.119700
$$340$$ 0 0
$$341$$ 7.81416 0.423161
$$342$$ 5.40980 0.292528
$$343$$ 12.8441 0.693518
$$344$$ −101.211 −5.45693
$$345$$ 0 0
$$346$$ −53.7181 −2.88790
$$347$$ −12.2251 −0.656277 −0.328138 0.944630i $$-0.606421\pi$$
−0.328138 + 0.944630i $$0.606421\pi$$
$$348$$ −7.10602 −0.380923
$$349$$ 12.8425 0.687441 0.343720 0.939072i $$-0.388313\pi$$
0.343720 + 0.939072i $$0.388313\pi$$
$$350$$ 0 0
$$351$$ −0.188171 −0.0100438
$$352$$ 83.7579 4.46431
$$353$$ 17.4163 0.926976 0.463488 0.886103i $$-0.346598\pi$$
0.463488 + 0.886103i $$0.346598\pi$$
$$354$$ 19.2552 1.02340
$$355$$ 0 0
$$356$$ −87.5749 −4.64146
$$357$$ 8.27641 0.438034
$$358$$ 66.3167 3.50494
$$359$$ −3.82947 −0.202111 −0.101056 0.994881i $$-0.532222\pi$$
−0.101056 + 0.994881i $$0.532222\pi$$
$$360$$ 0 0
$$361$$ −15.2325 −0.801712
$$362$$ −29.7918 −1.56582
$$363$$ 2.31633 0.121576
$$364$$ −3.41893 −0.179201
$$365$$ 0 0
$$366$$ −15.8488 −0.828428
$$367$$ −29.6282 −1.54658 −0.773289 0.634054i $$-0.781390\pi$$
−0.773289 + 0.634054i $$0.781390\pi$$
$$368$$ −23.1336 −1.20592
$$369$$ 3.85940 0.200912
$$370$$ 0 0
$$371$$ −1.16399 −0.0604314
$$372$$ −15.2953 −0.793025
$$373$$ 11.4595 0.593350 0.296675 0.954978i $$-0.404122\pi$$
0.296675 + 0.954978i $$0.404122\pi$$
$$374$$ 21.5794 1.11584
$$375$$ 0 0
$$376$$ −125.862 −6.49083
$$377$$ 0.231820 0.0119393
$$378$$ 8.77943 0.451565
$$379$$ −2.19307 −0.112650 −0.0563251 0.998412i $$-0.517938\pi$$
−0.0563251 + 0.998412i $$0.517938\pi$$
$$380$$ 0 0
$$381$$ 14.9689 0.766879
$$382$$ −40.0177 −2.04749
$$383$$ −10.1997 −0.521181 −0.260591 0.965449i $$-0.583917\pi$$
−0.260591 + 0.965449i $$0.583917\pi$$
$$384$$ −65.0920 −3.32171
$$385$$ 0 0
$$386$$ 5.44904 0.277349
$$387$$ −9.63734 −0.489894
$$388$$ 80.1960 4.07134
$$389$$ 28.5482 1.44745 0.723726 0.690088i $$-0.242428\pi$$
0.723726 + 0.690088i $$0.242428\pi$$
$$390$$ 0 0
$$391$$ −3.42740 −0.173331
$$392$$ 30.6920 1.55018
$$393$$ −11.2957 −0.569794
$$394$$ −3.28764 −0.165629
$$395$$ 0 0
$$396$$ 16.9973 0.854146
$$397$$ 37.1827 1.86614 0.933072 0.359688i $$-0.117117\pi$$
0.933072 + 0.359688i $$0.117117\pi$$
$$398$$ −18.2785 −0.916218
$$399$$ 6.11415 0.306090
$$400$$ 0 0
$$401$$ 7.38648 0.368863 0.184432 0.982845i $$-0.440956\pi$$
0.184432 + 0.982845i $$0.440956\pi$$
$$402$$ −9.03550 −0.450650
$$403$$ 0.498979 0.0248559
$$404$$ −55.7877 −2.77554
$$405$$ 0 0
$$406$$ −10.8160 −0.536787
$$407$$ 29.8102 1.47764
$$408$$ −27.5932 −1.36607
$$409$$ −5.19942 −0.257095 −0.128547 0.991703i $$-0.541031\pi$$
−0.128547 + 0.991703i $$0.541031\pi$$
$$410$$ 0 0
$$411$$ −4.17877 −0.206124
$$412$$ 50.1492 2.47068
$$413$$ 21.7622 1.07085
$$414$$ −3.63570 −0.178685
$$415$$ 0 0
$$416$$ 5.34842 0.262228
$$417$$ 15.3552 0.751949
$$418$$ 15.9416 0.779730
$$419$$ −14.9205 −0.728914 −0.364457 0.931220i $$-0.618745\pi$$
−0.364457 + 0.931220i $$0.618745\pi$$
$$420$$ 0 0
$$421$$ −0.0666925 −0.00325039 −0.00162520 0.999999i $$-0.500517\pi$$
−0.00162520 + 0.999999i $$0.500517\pi$$
$$422$$ 34.4414 1.67658
$$423$$ −11.9846 −0.582712
$$424$$ 3.88069 0.188463
$$425$$ 0 0
$$426$$ −18.6732 −0.904719
$$427$$ −17.9123 −0.866835
$$428$$ −14.7999 −0.715382
$$429$$ −0.554502 −0.0267716
$$430$$ 0 0
$$431$$ 27.6996 1.33424 0.667121 0.744950i $$-0.267527\pi$$
0.667121 + 0.744950i $$0.267527\pi$$
$$432$$ −17.7341 −0.853235
$$433$$ −30.5626 −1.46874 −0.734372 0.678748i $$-0.762523\pi$$
−0.734372 + 0.678748i $$0.762523\pi$$
$$434$$ −23.2807 −1.11751
$$435$$ 0 0
$$436$$ −87.8770 −4.20854
$$437$$ −2.53197 −0.121120
$$438$$ 26.3923 1.26107
$$439$$ 11.0968 0.529623 0.264812 0.964300i $$-0.414690\pi$$
0.264812 + 0.964300i $$0.414690\pi$$
$$440$$ 0 0
$$441$$ 2.92250 0.139167
$$442$$ 1.37797 0.0655432
$$443$$ −9.82831 −0.466957 −0.233479 0.972362i $$-0.575011\pi$$
−0.233479 + 0.972362i $$0.575011\pi$$
$$444$$ −58.3501 −2.76917
$$445$$ 0 0
$$446$$ −20.3653 −0.964324
$$447$$ −3.43364 −0.162405
$$448$$ −137.815 −6.51114
$$449$$ −19.1301 −0.902805 −0.451402 0.892320i $$-0.649076\pi$$
−0.451402 + 0.892320i $$0.649076\pi$$
$$450$$ 0 0
$$451$$ 11.3729 0.535529
$$452$$ 12.7122 0.597933
$$453$$ −10.7716 −0.506094
$$454$$ −58.5858 −2.74957
$$455$$ 0 0
$$456$$ −20.3843 −0.954582
$$457$$ 18.7578 0.877452 0.438726 0.898621i $$-0.355430\pi$$
0.438726 + 0.898621i $$0.355430\pi$$
$$458$$ 54.5003 2.54663
$$459$$ −2.62743 −0.122638
$$460$$ 0 0
$$461$$ −37.6208 −1.75217 −0.876087 0.482153i $$-0.839855\pi$$
−0.876087 + 0.482153i $$0.839855\pi$$
$$462$$ 25.8713 1.20364
$$463$$ −19.9275 −0.926111 −0.463056 0.886329i $$-0.653247\pi$$
−0.463056 + 0.886329i $$0.653247\pi$$
$$464$$ 21.8479 1.01426
$$465$$ 0 0
$$466$$ 33.8558 1.56834
$$467$$ 27.3667 1.26638 0.633190 0.773997i $$-0.281745\pi$$
0.633190 + 0.773997i $$0.281745\pi$$
$$468$$ 1.08537 0.0501714
$$469$$ −10.2119 −0.471542
$$470$$ 0 0
$$471$$ 13.4045 0.617649
$$472$$ −72.5542 −3.33958
$$473$$ −28.3994 −1.30581
$$474$$ −1.17097 −0.0537845
$$475$$ 0 0
$$476$$ −47.7386 −2.18810
$$477$$ 0.369521 0.0169192
$$478$$ 37.4915 1.71482
$$479$$ −38.5742 −1.76250 −0.881250 0.472650i $$-0.843297\pi$$
−0.881250 + 0.472650i $$0.843297\pi$$
$$480$$ 0 0
$$481$$ 1.90356 0.0867946
$$482$$ 55.5669 2.53100
$$483$$ −4.10907 −0.186969
$$484$$ −13.3607 −0.607303
$$485$$ 0 0
$$486$$ −2.78712 −0.126426
$$487$$ −18.4995 −0.838294 −0.419147 0.907918i $$-0.637671\pi$$
−0.419147 + 0.907918i $$0.637671\pi$$
$$488$$ 59.7187 2.70334
$$489$$ 15.4178 0.697218
$$490$$ 0 0
$$491$$ 19.7211 0.890000 0.445000 0.895531i $$-0.353204\pi$$
0.445000 + 0.895531i $$0.353204\pi$$
$$492$$ −22.2611 −1.00361
$$493$$ 3.23691 0.145783
$$494$$ 1.01796 0.0458004
$$495$$ 0 0
$$496$$ 47.0263 2.11154
$$497$$ −21.1044 −0.946663
$$498$$ 33.9011 1.51915
$$499$$ −36.2906 −1.62459 −0.812296 0.583245i $$-0.801783\pi$$
−0.812296 + 0.583245i $$0.801783\pi$$
$$500$$ 0 0
$$501$$ −17.4045 −0.777578
$$502$$ −7.69280 −0.343347
$$503$$ 2.66163 0.118676 0.0593382 0.998238i $$-0.481101\pi$$
0.0593382 + 0.998238i $$0.481101\pi$$
$$504$$ −33.0812 −1.47355
$$505$$ 0 0
$$506$$ −10.7137 −0.476283
$$507$$ 12.9646 0.575778
$$508$$ −86.3410 −3.83076
$$509$$ 22.8322 1.01202 0.506010 0.862528i $$-0.331120\pi$$
0.506010 + 0.862528i $$0.331120\pi$$
$$510$$ 0 0
$$511$$ 29.8285 1.31954
$$512$$ 131.576 5.81488
$$513$$ −1.94100 −0.0856972
$$514$$ −48.1748 −2.12490
$$515$$ 0 0
$$516$$ 55.5885 2.44715
$$517$$ −35.3163 −1.55321
$$518$$ −88.8137 −3.90225
$$519$$ 19.2737 0.846021
$$520$$ 0 0
$$521$$ −12.3029 −0.539001 −0.269501 0.963000i $$-0.586859\pi$$
−0.269501 + 0.963000i $$0.586859\pi$$
$$522$$ 3.43364 0.150286
$$523$$ −33.3902 −1.46005 −0.730025 0.683420i $$-0.760492\pi$$
−0.730025 + 0.683420i $$0.760492\pi$$
$$524$$ 65.1541 2.84627
$$525$$ 0 0
$$526$$ −11.0531 −0.481939
$$527$$ 6.96726 0.303499
$$528$$ −52.2591 −2.27429
$$529$$ −21.2984 −0.926016
$$530$$ 0 0
$$531$$ −6.90864 −0.299809
$$532$$ −35.2666 −1.52900
$$533$$ 0.726225 0.0314563
$$534$$ 42.3163 1.83121
$$535$$ 0 0
$$536$$ 34.0461 1.47057
$$537$$ −23.7940 −1.02679
$$538$$ 3.33238 0.143669
$$539$$ 8.61204 0.370947
$$540$$ 0 0
$$541$$ 5.98673 0.257389 0.128695 0.991684i $$-0.458921\pi$$
0.128695 + 0.991684i $$0.458921\pi$$
$$542$$ −6.08413 −0.261336
$$543$$ 10.6891 0.458714
$$544$$ 74.6802 3.20189
$$545$$ 0 0
$$546$$ 1.65203 0.0707003
$$547$$ 4.47889 0.191503 0.0957516 0.995405i $$-0.469475\pi$$
0.0957516 + 0.995405i $$0.469475\pi$$
$$548$$ 24.1033 1.02964
$$549$$ 5.68643 0.242691
$$550$$ 0 0
$$551$$ 2.39124 0.101870
$$552$$ 13.6995 0.583088
$$553$$ −1.32343 −0.0562780
$$554$$ 5.43972 0.231112
$$555$$ 0 0
$$556$$ −88.5695 −3.75618
$$557$$ −14.4278 −0.611326 −0.305663 0.952140i $$-0.598878\pi$$
−0.305663 + 0.952140i $$0.598878\pi$$
$$558$$ 7.39071 0.312874
$$559$$ −1.81346 −0.0767014
$$560$$ 0 0
$$561$$ −7.74253 −0.326890
$$562$$ 6.41401 0.270559
$$563$$ 27.6973 1.16730 0.583652 0.812004i $$-0.301623\pi$$
0.583652 + 0.812004i $$0.301623\pi$$
$$564$$ 69.1277 2.91080
$$565$$ 0 0
$$566$$ −29.6715 −1.24719
$$567$$ −3.15000 −0.132288
$$568$$ 70.3612 2.95229
$$569$$ 35.6179 1.49318 0.746590 0.665285i $$-0.231690\pi$$
0.746590 + 0.665285i $$0.231690\pi$$
$$570$$ 0 0
$$571$$ 5.85044 0.244833 0.122417 0.992479i $$-0.460936\pi$$
0.122417 + 0.992479i $$0.460936\pi$$
$$572$$ 3.19839 0.133731
$$573$$ 14.3581 0.599818
$$574$$ −33.8833 −1.41426
$$575$$ 0 0
$$576$$ 43.7507 1.82295
$$577$$ 12.0524 0.501746 0.250873 0.968020i $$-0.419282\pi$$
0.250873 + 0.968020i $$0.419282\pi$$
$$578$$ −28.1404 −1.17049
$$579$$ −1.95508 −0.0812503
$$580$$ 0 0
$$581$$ 38.3150 1.58958
$$582$$ −38.7508 −1.60627
$$583$$ 1.08891 0.0450979
$$584$$ −99.4470 −4.11515
$$585$$ 0 0
$$586$$ 41.6928 1.72232
$$587$$ 40.2350 1.66068 0.830339 0.557259i $$-0.188147\pi$$
0.830339 + 0.557259i $$0.188147\pi$$
$$588$$ −16.8571 −0.695174
$$589$$ 5.14702 0.212079
$$590$$ 0 0
$$591$$ 1.17958 0.0485215
$$592$$ 179.401 7.37332
$$593$$ 33.4086 1.37193 0.685964 0.727636i $$-0.259381\pi$$
0.685964 + 0.727636i $$0.259381\pi$$
$$594$$ −8.21310 −0.336988
$$595$$ 0 0
$$596$$ 19.8053 0.811258
$$597$$ 6.55820 0.268409
$$598$$ −0.684132 −0.0279763
$$599$$ −24.6421 −1.00685 −0.503424 0.864040i $$-0.667927\pi$$
−0.503424 + 0.864040i $$0.667927\pi$$
$$600$$ 0 0
$$601$$ 16.6327 0.678461 0.339230 0.940703i $$-0.389833\pi$$
0.339230 + 0.940703i $$0.389833\pi$$
$$602$$ 84.6103 3.44846
$$603$$ 3.24188 0.132019
$$604$$ 62.1310 2.52807
$$605$$ 0 0
$$606$$ 26.9567 1.09504
$$607$$ 29.4057 1.19354 0.596770 0.802412i $$-0.296450\pi$$
0.596770 + 0.802412i $$0.296450\pi$$
$$608$$ 55.1695 2.23742
$$609$$ 3.88069 0.157254
$$610$$ 0 0
$$611$$ −2.25515 −0.0912337
$$612$$ 15.1551 0.612609
$$613$$ 12.1702 0.491549 0.245775 0.969327i $$-0.420958\pi$$
0.245775 + 0.969327i $$0.420958\pi$$
$$614$$ −8.56574 −0.345685
$$615$$ 0 0
$$616$$ −97.4838 −3.92774
$$617$$ −39.1024 −1.57420 −0.787102 0.616823i $$-0.788419\pi$$
−0.787102 + 0.616823i $$0.788419\pi$$
$$618$$ −24.2322 −0.974761
$$619$$ −4.72717 −0.190001 −0.0950006 0.995477i $$-0.530285\pi$$
−0.0950006 + 0.995477i $$0.530285\pi$$
$$620$$ 0 0
$$621$$ 1.30447 0.0523464
$$622$$ 69.4258 2.78372
$$623$$ 47.8258 1.91610
$$624$$ −3.33705 −0.133589
$$625$$ 0 0
$$626$$ 25.8801 1.03438
$$627$$ −5.71975 −0.228425
$$628$$ −77.3179 −3.08532
$$629$$ 26.5794 1.05979
$$630$$ 0 0
$$631$$ −33.8710 −1.34838 −0.674191 0.738557i $$-0.735508\pi$$
−0.674191 + 0.738557i $$0.735508\pi$$
$$632$$ 4.41226 0.175510
$$633$$ −12.3573 −0.491160
$$634$$ 57.8842 2.29888
$$635$$ 0 0
$$636$$ −2.13141 −0.0845158
$$637$$ 0.549929 0.0217890
$$638$$ 10.1183 0.400586
$$639$$ 6.69982 0.265041
$$640$$ 0 0
$$641$$ 15.9571 0.630269 0.315135 0.949047i $$-0.397950\pi$$
0.315135 + 0.949047i $$0.397950\pi$$
$$642$$ 7.15134 0.282241
$$643$$ −10.2436 −0.403968 −0.201984 0.979389i $$-0.564739\pi$$
−0.201984 + 0.979389i $$0.564739\pi$$
$$644$$ 23.7013 0.933960
$$645$$ 0 0
$$646$$ 14.2139 0.559237
$$647$$ −6.00244 −0.235980 −0.117990 0.993015i $$-0.537645\pi$$
−0.117990 + 0.993015i $$0.537645\pi$$
$$648$$ 10.5020 0.412556
$$649$$ −20.3584 −0.799138
$$650$$ 0 0
$$651$$ 8.35298 0.327379
$$652$$ −88.9305 −3.48279
$$653$$ 33.1069 1.29557 0.647787 0.761822i $$-0.275695\pi$$
0.647787 + 0.761822i $$0.275695\pi$$
$$654$$ 42.4622 1.66040
$$655$$ 0 0
$$656$$ 68.4431 2.67226
$$657$$ −9.46938 −0.369436
$$658$$ 105.218 4.10183
$$659$$ 19.5593 0.761922 0.380961 0.924591i $$-0.375593\pi$$
0.380961 + 0.924591i $$0.375593\pi$$
$$660$$ 0 0
$$661$$ 30.5290 1.18744 0.593721 0.804671i $$-0.297658\pi$$
0.593721 + 0.804671i $$0.297658\pi$$
$$662$$ −94.5312 −3.67406
$$663$$ −0.494405 −0.0192011
$$664$$ −127.741 −4.95730
$$665$$ 0 0
$$666$$ 28.1948 1.09253
$$667$$ −1.60706 −0.0622255
$$668$$ 100.390 3.88421
$$669$$ 7.30693 0.282502
$$670$$ 0 0
$$671$$ 16.7568 0.646890
$$672$$ 89.5333 3.45382
$$673$$ 4.19687 0.161778 0.0808888 0.996723i $$-0.474224\pi$$
0.0808888 + 0.996723i $$0.474224\pi$$
$$674$$ 49.5527 1.90870
$$675$$ 0 0
$$676$$ −74.7802 −2.87616
$$677$$ 9.66481 0.371449 0.185724 0.982602i $$-0.440537\pi$$
0.185724 + 0.982602i $$0.440537\pi$$
$$678$$ −6.14256 −0.235903
$$679$$ −43.7961 −1.68074
$$680$$ 0 0
$$681$$ 21.0202 0.805495
$$682$$ 21.7790 0.833961
$$683$$ 9.03781 0.345822 0.172911 0.984937i $$-0.444683\pi$$
0.172911 + 0.984937i $$0.444683\pi$$
$$684$$ 11.1957 0.428080
$$685$$ 0 0
$$686$$ 35.7981 1.36678
$$687$$ −19.5544 −0.746045
$$688$$ −170.910 −6.51588
$$689$$ 0.0695329 0.00264899
$$690$$ 0 0
$$691$$ −32.1359 −1.22251 −0.611254 0.791435i $$-0.709334\pi$$
−0.611254 + 0.791435i $$0.709334\pi$$
$$692$$ −111.171 −4.22610
$$693$$ −9.28244 −0.352611
$$694$$ −34.0728 −1.29338
$$695$$ 0 0
$$696$$ −12.9381 −0.490416
$$697$$ 10.1403 0.384091
$$698$$ 35.7935 1.35480
$$699$$ −12.1472 −0.459451
$$700$$ 0 0
$$701$$ −8.54116 −0.322595 −0.161298 0.986906i $$-0.551568\pi$$
−0.161298 + 0.986906i $$0.551568\pi$$
$$702$$ −0.524454 −0.0197942
$$703$$ 19.6354 0.740562
$$704$$ 128.925 4.85904
$$705$$ 0 0
$$706$$ 48.5413 1.82688
$$707$$ 30.4664 1.14581
$$708$$ 39.8493 1.49763
$$709$$ 5.04613 0.189511 0.0947556 0.995501i $$-0.469793\pi$$
0.0947556 + 0.995501i $$0.469793\pi$$
$$710$$ 0 0
$$711$$ 0.420137 0.0157564
$$712$$ −159.449 −5.97561
$$713$$ −3.45910 −0.129544
$$714$$ 23.0673 0.863274
$$715$$ 0 0
$$716$$ 137.244 5.12907
$$717$$ −13.4517 −0.502363
$$718$$ −10.6732 −0.398319
$$719$$ −19.5253 −0.728170 −0.364085 0.931366i $$-0.618618\pi$$
−0.364085 + 0.931366i $$0.618618\pi$$
$$720$$ 0 0
$$721$$ −27.3872 −1.01995
$$722$$ −42.4549 −1.58001
$$723$$ −19.9370 −0.741467
$$724$$ −61.6552 −2.29140
$$725$$ 0 0
$$726$$ 6.45588 0.239600
$$727$$ 40.4464 1.50007 0.750037 0.661395i $$-0.230035\pi$$
0.750037 + 0.661395i $$0.230035\pi$$
$$728$$ −6.22490 −0.230710
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −25.3215 −0.936548
$$732$$ −32.7995 −1.21231
$$733$$ 29.3862 1.08540 0.542702 0.839926i $$-0.317401\pi$$
0.542702 + 0.839926i $$0.317401\pi$$
$$734$$ −82.5772 −3.04798
$$735$$ 0 0
$$736$$ −37.0772 −1.36668
$$737$$ 9.55318 0.351896
$$738$$ 10.7566 0.395956
$$739$$ −30.7254 −1.13025 −0.565126 0.825005i $$-0.691172\pi$$
−0.565126 + 0.825005i $$0.691172\pi$$
$$740$$ 0 0
$$741$$ −0.365239 −0.0134174
$$742$$ −3.24418 −0.119098
$$743$$ 41.4841 1.52190 0.760952 0.648808i $$-0.224732\pi$$
0.760952 + 0.648808i $$0.224732\pi$$
$$744$$ −27.8485 −1.02097
$$745$$ 0 0
$$746$$ 31.9390 1.16937
$$747$$ −12.1635 −0.445039
$$748$$ 44.6592 1.63290
$$749$$ 8.08244 0.295326
$$750$$ 0 0
$$751$$ 38.6123 1.40898 0.704491 0.709713i $$-0.251175\pi$$
0.704491 + 0.709713i $$0.251175\pi$$
$$752$$ −212.537 −7.75042
$$753$$ 2.76013 0.100585
$$754$$ 0.646109 0.0235299
$$755$$ 0 0
$$756$$ 18.1693 0.660811
$$757$$ −23.0860 −0.839074 −0.419537 0.907738i $$-0.637807\pi$$
−0.419537 + 0.907738i $$0.637807\pi$$
$$758$$ −6.11234 −0.222010
$$759$$ 3.84401 0.139529
$$760$$ 0 0
$$761$$ −31.7092 −1.14946 −0.574728 0.818344i $$-0.694892\pi$$
−0.574728 + 0.818344i $$0.694892\pi$$
$$762$$ 41.7200 1.51136
$$763$$ 47.9908 1.73738
$$764$$ −82.8180 −2.99625
$$765$$ 0 0
$$766$$ −28.4278 −1.02714
$$767$$ −1.30000 −0.0469404
$$768$$ −93.9177 −3.38896
$$769$$ 38.6475 1.39366 0.696832 0.717235i $$-0.254592\pi$$
0.696832 + 0.717235i $$0.254592\pi$$
$$770$$ 0 0
$$771$$ 17.2848 0.622497
$$772$$ 11.2770 0.405867
$$773$$ 38.9958 1.40258 0.701291 0.712875i $$-0.252607\pi$$
0.701291 + 0.712875i $$0.252607\pi$$
$$774$$ −26.8604 −0.965478
$$775$$ 0 0
$$776$$ 146.014 5.24161
$$777$$ 31.8658 1.14318
$$778$$ 79.5672 2.85262
$$779$$ 7.49109 0.268396
$$780$$ 0 0
$$781$$ 19.7431 0.706463
$$782$$ −9.55256 −0.341599
$$783$$ −1.23197 −0.0440269
$$784$$ 51.8281 1.85100
$$785$$ 0 0
$$786$$ −31.4825 −1.12294
$$787$$ −8.11107 −0.289128 −0.144564 0.989495i $$-0.546178\pi$$
−0.144564 + 0.989495i $$0.546178\pi$$
$$788$$ −6.80387 −0.242378
$$789$$ 3.96578 0.141186
$$790$$ 0 0
$$791$$ −6.94231 −0.246840
$$792$$ 30.9473 1.09966
$$793$$ 1.07002 0.0379975
$$794$$ 103.633 3.67778
$$795$$ 0 0
$$796$$ −37.8279 −1.34078
$$797$$ −19.0152 −0.673554 −0.336777 0.941584i $$-0.609337\pi$$
−0.336777 + 0.941584i $$0.609337\pi$$
$$798$$ 17.0409 0.603240
$$799$$ −31.4888 −1.11399
$$800$$ 0 0
$$801$$ −15.1828 −0.536458
$$802$$ 20.5870 0.726952
$$803$$ −27.9044 −0.984726
$$804$$ −18.6993 −0.659472
$$805$$ 0 0
$$806$$ 1.39071 0.0489858
$$807$$ −1.19564 −0.0420884
$$808$$ −101.574 −3.57335
$$809$$ 49.9078 1.75467 0.877333 0.479882i $$-0.159321\pi$$
0.877333 + 0.479882i $$0.159321\pi$$
$$810$$ 0 0
$$811$$ 30.9727 1.08760 0.543799 0.839215i $$-0.316985\pi$$
0.543799 + 0.839215i $$0.316985\pi$$
$$812$$ −22.3840 −0.785523
$$813$$ 2.18295 0.0765593
$$814$$ 83.0847 2.91212
$$815$$ 0 0
$$816$$ −46.5953 −1.63116
$$817$$ −18.7061 −0.654443
$$818$$ −14.4914 −0.506680
$$819$$ −0.592737 −0.0207119
$$820$$ 0 0
$$821$$ 18.9246 0.660473 0.330236 0.943898i $$-0.392872\pi$$
0.330236 + 0.943898i $$0.392872\pi$$
$$822$$ −11.6467 −0.406227
$$823$$ −9.31122 −0.324569 −0.162284 0.986744i $$-0.551886\pi$$
−0.162284 + 0.986744i $$0.551886\pi$$
$$824$$ 91.3076 3.18085
$$825$$ 0 0
$$826$$ 60.6539 2.11042
$$827$$ 23.7383 0.825461 0.412731 0.910853i $$-0.364575\pi$$
0.412731 + 0.910853i $$0.364575\pi$$
$$828$$ −7.52421 −0.261484
$$829$$ 1.91351 0.0664590 0.0332295 0.999448i $$-0.489421\pi$$
0.0332295 + 0.999448i $$0.489421\pi$$
$$830$$ 0 0
$$831$$ −1.95174 −0.0677050
$$832$$ 8.23260 0.285414
$$833$$ 7.67867 0.266050
$$834$$ 42.7969 1.48193
$$835$$ 0 0
$$836$$ 32.9917 1.14104
$$837$$ −2.65174 −0.0916575
$$838$$ −41.5852 −1.43654
$$839$$ 40.5105 1.39858 0.699289 0.714839i $$-0.253500\pi$$
0.699289 + 0.714839i $$0.253500\pi$$
$$840$$ 0 0
$$841$$ −27.4823 −0.947664
$$842$$ −0.185880 −0.00640584
$$843$$ −2.30131 −0.0792612
$$844$$ 71.2776 2.45348
$$845$$ 0 0
$$846$$ −33.4025 −1.14840
$$847$$ 7.29643 0.250708
$$848$$ 6.55314 0.225036
$$849$$ 10.6459 0.365367
$$850$$ 0 0
$$851$$ −13.1961 −0.452358
$$852$$ −38.6448 −1.32395
$$853$$ 18.7735 0.642792 0.321396 0.946945i $$-0.395848\pi$$
0.321396 + 0.946945i $$0.395848\pi$$
$$854$$ −49.9236 −1.70835
$$855$$ 0 0
$$856$$ −26.9465 −0.921012
$$857$$ −10.0276 −0.342536 −0.171268 0.985225i $$-0.554786\pi$$
−0.171268 + 0.985225i $$0.554786\pi$$
$$858$$ −1.54546 −0.0527613
$$859$$ −28.0443 −0.956858 −0.478429 0.878126i $$-0.658794\pi$$
−0.478429 + 0.878126i $$0.658794\pi$$
$$860$$ 0 0
$$861$$ 12.1571 0.414313
$$862$$ 77.2020 2.62951
$$863$$ 9.84666 0.335184 0.167592 0.985856i $$-0.446401\pi$$
0.167592 + 0.985856i $$0.446401\pi$$
$$864$$ −28.4233 −0.966979
$$865$$ 0 0
$$866$$ −85.1815 −2.89459
$$867$$ 10.0966 0.342899
$$868$$ −48.1802 −1.63534
$$869$$ 1.23806 0.0419984
$$870$$ 0 0
$$871$$ 0.610026 0.0206699
$$872$$ −159.999 −5.41826
$$873$$ 13.9035 0.470563
$$874$$ −7.05690 −0.238703
$$875$$ 0 0
$$876$$ 54.6197 1.84543
$$877$$ 36.9671 1.24829 0.624146 0.781308i $$-0.285447\pi$$
0.624146 + 0.781308i $$0.285447\pi$$
$$878$$ 30.9282 1.04378
$$879$$ −14.9591 −0.504558
$$880$$ 0 0
$$881$$ −20.7137 −0.697863 −0.348932 0.937148i $$-0.613455\pi$$
−0.348932 + 0.937148i $$0.613455\pi$$
$$882$$ 8.14536 0.274268
$$883$$ 12.7254 0.428243 0.214121 0.976807i $$-0.431311\pi$$
0.214121 + 0.976807i $$0.431311\pi$$
$$884$$ 2.85175 0.0959146
$$885$$ 0 0
$$886$$ −27.3927 −0.920275
$$887$$ −21.6903 −0.728287 −0.364144 0.931343i $$-0.618638\pi$$
−0.364144 + 0.931343i $$0.618638\pi$$
$$888$$ −106.239 −3.56515
$$889$$ 47.1520 1.58143
$$890$$ 0 0
$$891$$ 2.94681 0.0987218
$$892$$ −42.1466 −1.41117
$$893$$ −23.2621 −0.778437
$$894$$ −9.56995 −0.320067
$$895$$ 0 0
$$896$$ −205.040 −6.84990
$$897$$ 0.245462 0.00819574
$$898$$ −53.3178 −1.77924
$$899$$ 3.26685 0.108956
$$900$$ 0 0
$$901$$ 0.970891 0.0323451
$$902$$ 31.6976 1.05542
$$903$$ −30.3576 −1.01024
$$904$$ 23.1454 0.769803
$$905$$ 0 0
$$906$$ −30.0217 −0.997406
$$907$$ −3.28462 −0.109064 −0.0545320 0.998512i $$-0.517367\pi$$
−0.0545320 + 0.998512i $$0.517367\pi$$
$$908$$ −121.245 −4.02366
$$909$$ −9.67188 −0.320796
$$910$$ 0 0
$$911$$ 28.8264 0.955060 0.477530 0.878615i $$-0.341532\pi$$
0.477530 + 0.878615i $$0.341532\pi$$
$$912$$ −34.4220 −1.13983
$$913$$ −35.8435 −1.18625
$$914$$ 52.2802 1.72927
$$915$$ 0 0
$$916$$ 112.790 3.72669
$$917$$ −35.5815 −1.17500
$$918$$ −7.32297 −0.241694
$$919$$ −17.3985 −0.573925 −0.286962 0.957942i $$-0.592645\pi$$
−0.286962 + 0.957942i $$0.592645\pi$$
$$920$$ 0 0
$$921$$ 3.07333 0.101270
$$922$$ −104.854 −3.45317
$$923$$ 1.26071 0.0414967
$$924$$ 53.5414 1.76138
$$925$$ 0 0
$$926$$ −55.5404 −1.82517
$$927$$ 8.69434 0.285560
$$928$$ 35.0165 1.14947
$$929$$ −24.3715 −0.799603 −0.399801 0.916602i $$-0.630921\pi$$
−0.399801 + 0.916602i $$0.630921\pi$$
$$930$$ 0 0
$$931$$ 5.67257 0.185911
$$932$$ 70.0657 2.29508
$$933$$ −24.9095 −0.815501
$$934$$ 76.2742 2.49577
$$935$$ 0 0
$$936$$ 1.97616 0.0645928
$$937$$ 14.5784 0.476257 0.238129 0.971234i $$-0.423466\pi$$
0.238129 + 0.971234i $$0.423466\pi$$
$$938$$ −28.4618 −0.929311
$$939$$ −9.28562 −0.303025
$$940$$ 0 0
$$941$$ −4.19686 −0.136814 −0.0684068 0.997658i $$-0.521792\pi$$
−0.0684068 + 0.997658i $$0.521792\pi$$
$$942$$ 37.3601 1.21726
$$943$$ −5.03445 −0.163944
$$944$$ −122.519 −3.98765
$$945$$ 0 0
$$946$$ −79.1525 −2.57347
$$947$$ −5.53251 −0.179782 −0.0898912 0.995952i $$-0.528652\pi$$
−0.0898912 + 0.995952i $$0.528652\pi$$
$$948$$ −2.42336 −0.0787071
$$949$$ −1.78186 −0.0578416
$$950$$ 0 0
$$951$$ −20.7685 −0.673464
$$952$$ −86.9185 −2.81705
$$953$$ 50.4779 1.63514 0.817570 0.575830i $$-0.195321\pi$$
0.817570 + 0.575830i $$0.195321\pi$$
$$954$$ 1.02990 0.0333442
$$955$$ 0 0
$$956$$ 77.5899 2.50944
$$957$$ −3.63037 −0.117353
$$958$$ −107.511 −3.47352
$$959$$ −13.1631 −0.425060
$$960$$ 0 0
$$961$$ −23.9683 −0.773170
$$962$$ 5.30544 0.171054
$$963$$ −2.56585 −0.0826835
$$964$$ 114.998 3.70382
$$965$$ 0 0
$$966$$ −11.4525 −0.368477
$$967$$ −17.0340 −0.547777 −0.273889 0.961761i $$-0.588310\pi$$
−0.273889 + 0.961761i $$0.588310\pi$$
$$968$$ −24.3260 −0.781867
$$969$$ −5.09984 −0.163831
$$970$$ 0 0
$$971$$ 23.4172 0.751494 0.375747 0.926722i $$-0.377386\pi$$
0.375747 + 0.926722i $$0.377386\pi$$
$$972$$ −5.76803 −0.185010
$$973$$ 48.3690 1.55064
$$974$$ −51.5604 −1.65210
$$975$$ 0 0
$$976$$ 100.844 3.22794
$$977$$ −20.7626 −0.664253 −0.332127 0.943235i $$-0.607766\pi$$
−0.332127 + 0.943235i $$0.607766\pi$$
$$978$$ 42.9713 1.37407
$$979$$ −44.7408 −1.42992
$$980$$ 0 0
$$981$$ −15.2352 −0.486422
$$982$$ 54.9650 1.75400
$$983$$ 42.1977 1.34590 0.672948 0.739690i $$-0.265028\pi$$
0.672948 + 0.739690i $$0.265028\pi$$
$$984$$ −40.5312 −1.29209
$$985$$ 0 0
$$986$$ 9.02164 0.287308
$$987$$ −37.7515 −1.20164
$$988$$ 2.10671 0.0670234
$$989$$ 12.5716 0.399753
$$990$$ 0 0
$$991$$ −3.83926 −0.121958 −0.0609791 0.998139i $$-0.519422\pi$$
−0.0609791 + 0.998139i $$0.519422\pi$$
$$992$$ 75.3711 2.39303
$$993$$ 33.9172 1.07633
$$994$$ −58.8206 −1.86567
$$995$$ 0 0
$$996$$ 70.1595 2.22309
$$997$$ 42.2091 1.33678 0.668389 0.743812i $$-0.266984\pi$$
0.668389 + 0.743812i $$0.266984\pi$$
$$998$$ −101.146 −3.20173
$$999$$ −10.1161 −0.320060
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.l.1.6 yes 6
3.2 odd 2 5625.2.a.o.1.1 6
5.2 odd 4 1875.2.b.e.1249.12 12
5.3 odd 4 1875.2.b.e.1249.1 12
5.4 even 2 1875.2.a.i.1.1 6
15.14 odd 2 5625.2.a.r.1.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.1 6 5.4 even 2
1875.2.a.l.1.6 yes 6 1.1 even 1 trivial
1875.2.b.e.1249.1 12 5.3 odd 4
1875.2.b.e.1249.12 12 5.2 odd 4
5625.2.a.o.1.1 6 3.2 odd 2
5625.2.a.r.1.6 6 15.14 odd 2