# Properties

 Label 1875.2.a.d.1.2 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1875.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} -3.00000 q^{8} +1.00000 q^{9} +3.23607 q^{11} -1.00000 q^{12} +3.38197 q^{13} +4.47214 q^{14} -1.00000 q^{16} -2.85410 q^{17} +1.00000 q^{18} -3.23607 q^{19} +4.47214 q^{21} +3.23607 q^{22} -4.47214 q^{23} -3.00000 q^{24} +3.38197 q^{26} +1.00000 q^{27} -4.47214 q^{28} +4.38197 q^{29} +7.23607 q^{31} +5.00000 q^{32} +3.23607 q^{33} -2.85410 q^{34} -1.00000 q^{36} +8.09017 q^{37} -3.23607 q^{38} +3.38197 q^{39} -1.38197 q^{41} +4.47214 q^{42} -5.70820 q^{43} -3.23607 q^{44} -4.47214 q^{46} -5.23607 q^{47} -1.00000 q^{48} +13.0000 q^{49} -2.85410 q^{51} -3.38197 q^{52} +1.38197 q^{53} +1.00000 q^{54} -13.4164 q^{56} -3.23607 q^{57} +4.38197 q^{58} -4.00000 q^{59} -0.618034 q^{61} +7.23607 q^{62} +4.47214 q^{63} +7.00000 q^{64} +3.23607 q^{66} +5.23607 q^{67} +2.85410 q^{68} -4.47214 q^{69} -0.763932 q^{71} -3.00000 q^{72} +3.09017 q^{73} +8.09017 q^{74} +3.23607 q^{76} +14.4721 q^{77} +3.38197 q^{78} +1.00000 q^{81} -1.38197 q^{82} +3.52786 q^{83} -4.47214 q^{84} -5.70820 q^{86} +4.38197 q^{87} -9.70820 q^{88} +7.61803 q^{89} +15.1246 q^{91} +4.47214 q^{92} +7.23607 q^{93} -5.23607 q^{94} +5.00000 q^{96} +8.85410 q^{97} +13.0000 q^{98} +3.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 9 q^{13} - 2 q^{16} + q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{22} - 6 q^{24} + 9 q^{26} + 2 q^{27} + 11 q^{29} + 10 q^{31} + 10 q^{32} + 2 q^{33} + q^{34} - 2 q^{36} + 5 q^{37} - 2 q^{38} + 9 q^{39} - 5 q^{41} + 2 q^{43} - 2 q^{44} - 6 q^{47} - 2 q^{48} + 26 q^{49} + q^{51} - 9 q^{52} + 5 q^{53} + 2 q^{54} - 2 q^{57} + 11 q^{58} - 8 q^{59} + q^{61} + 10 q^{62} + 14 q^{64} + 2 q^{66} + 6 q^{67} - q^{68} - 6 q^{71} - 6 q^{72} - 5 q^{73} + 5 q^{74} + 2 q^{76} + 20 q^{77} + 9 q^{78} + 2 q^{81} - 5 q^{82} + 16 q^{83} + 2 q^{86} + 11 q^{87} - 6 q^{88} + 13 q^{89} - 10 q^{91} + 10 q^{93} - 6 q^{94} + 10 q^{96} + 11 q^{97} + 26 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 + 9 * q^13 - 2 * q^16 + q^17 + 2 * q^18 - 2 * q^19 + 2 * q^22 - 6 * q^24 + 9 * q^26 + 2 * q^27 + 11 * q^29 + 10 * q^31 + 10 * q^32 + 2 * q^33 + q^34 - 2 * q^36 + 5 * q^37 - 2 * q^38 + 9 * q^39 - 5 * q^41 + 2 * q^43 - 2 * q^44 - 6 * q^47 - 2 * q^48 + 26 * q^49 + q^51 - 9 * q^52 + 5 * q^53 + 2 * q^54 - 2 * q^57 + 11 * q^58 - 8 * q^59 + q^61 + 10 * q^62 + 14 * q^64 + 2 * q^66 + 6 * q^67 - q^68 - 6 * q^71 - 6 * q^72 - 5 * q^73 + 5 * q^74 + 2 * q^76 + 20 * q^77 + 9 * q^78 + 2 * q^81 - 5 * q^82 + 16 * q^83 + 2 * q^86 + 11 * q^87 - 6 * q^88 + 13 * q^89 - 10 * q^91 + 10 * q^93 - 6 * q^94 + 10 * q^96 + 11 * q^97 + 26 * q^98 + 2 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 4.47214 1.69031 0.845154 0.534522i $$-0.179509\pi$$
0.845154 + 0.534522i $$0.179509\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.23607 0.975711 0.487856 0.872924i $$-0.337779\pi$$
0.487856 + 0.872924i $$0.337779\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 3.38197 0.937989 0.468994 0.883201i $$-0.344616\pi$$
0.468994 + 0.883201i $$0.344616\pi$$
$$14$$ 4.47214 1.19523
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −2.85410 −0.692221 −0.346111 0.938194i $$-0.612498\pi$$
−0.346111 + 0.938194i $$0.612498\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −3.23607 −0.742405 −0.371202 0.928552i $$-0.621054\pi$$
−0.371202 + 0.928552i $$0.621054\pi$$
$$20$$ 0 0
$$21$$ 4.47214 0.975900
$$22$$ 3.23607 0.689932
$$23$$ −4.47214 −0.932505 −0.466252 0.884652i $$-0.654396\pi$$
−0.466252 + 0.884652i $$0.654396\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ 3.38197 0.663258
$$27$$ 1.00000 0.192450
$$28$$ −4.47214 −0.845154
$$29$$ 4.38197 0.813711 0.406855 0.913493i $$-0.366625\pi$$
0.406855 + 0.913493i $$0.366625\pi$$
$$30$$ 0 0
$$31$$ 7.23607 1.29964 0.649818 0.760090i $$-0.274845\pi$$
0.649818 + 0.760090i $$0.274845\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 3.23607 0.563327
$$34$$ −2.85410 −0.489474
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 8.09017 1.33002 0.665008 0.746836i $$-0.268428\pi$$
0.665008 + 0.746836i $$0.268428\pi$$
$$38$$ −3.23607 −0.524960
$$39$$ 3.38197 0.541548
$$40$$ 0 0
$$41$$ −1.38197 −0.215827 −0.107913 0.994160i $$-0.534417\pi$$
−0.107913 + 0.994160i $$0.534417\pi$$
$$42$$ 4.47214 0.690066
$$43$$ −5.70820 −0.870493 −0.435246 0.900311i $$-0.643339\pi$$
−0.435246 + 0.900311i $$0.643339\pi$$
$$44$$ −3.23607 −0.487856
$$45$$ 0 0
$$46$$ −4.47214 −0.659380
$$47$$ −5.23607 −0.763759 −0.381880 0.924212i $$-0.624723\pi$$
−0.381880 + 0.924212i $$0.624723\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 13.0000 1.85714
$$50$$ 0 0
$$51$$ −2.85410 −0.399654
$$52$$ −3.38197 −0.468994
$$53$$ 1.38197 0.189828 0.0949138 0.995485i $$-0.469742\pi$$
0.0949138 + 0.995485i $$0.469742\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −13.4164 −1.79284
$$57$$ −3.23607 −0.428628
$$58$$ 4.38197 0.575380
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −0.618034 −0.0791311 −0.0395656 0.999217i $$-0.512597\pi$$
−0.0395656 + 0.999217i $$0.512597\pi$$
$$62$$ 7.23607 0.918982
$$63$$ 4.47214 0.563436
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 3.23607 0.398332
$$67$$ 5.23607 0.639688 0.319844 0.947470i $$-0.396370\pi$$
0.319844 + 0.947470i $$0.396370\pi$$
$$68$$ 2.85410 0.346111
$$69$$ −4.47214 −0.538382
$$70$$ 0 0
$$71$$ −0.763932 −0.0906621 −0.0453310 0.998972i $$-0.514434\pi$$
−0.0453310 + 0.998972i $$0.514434\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 3.09017 0.361677 0.180839 0.983513i $$-0.442119\pi$$
0.180839 + 0.983513i $$0.442119\pi$$
$$74$$ 8.09017 0.940463
$$75$$ 0 0
$$76$$ 3.23607 0.371202
$$77$$ 14.4721 1.64925
$$78$$ 3.38197 0.382932
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −1.38197 −0.152613
$$83$$ 3.52786 0.387233 0.193617 0.981077i $$-0.437978\pi$$
0.193617 + 0.981077i $$0.437978\pi$$
$$84$$ −4.47214 −0.487950
$$85$$ 0 0
$$86$$ −5.70820 −0.615531
$$87$$ 4.38197 0.469796
$$88$$ −9.70820 −1.03490
$$89$$ 7.61803 0.807510 0.403755 0.914867i $$-0.367705\pi$$
0.403755 + 0.914867i $$0.367705\pi$$
$$90$$ 0 0
$$91$$ 15.1246 1.58549
$$92$$ 4.47214 0.466252
$$93$$ 7.23607 0.750345
$$94$$ −5.23607 −0.540059
$$95$$ 0 0
$$96$$ 5.00000 0.510310
$$97$$ 8.85410 0.898998 0.449499 0.893281i $$-0.351603\pi$$
0.449499 + 0.893281i $$0.351603\pi$$
$$98$$ 13.0000 1.31320
$$99$$ 3.23607 0.325237
$$100$$ 0 0
$$101$$ −16.5623 −1.64801 −0.824006 0.566582i $$-0.808266\pi$$
−0.824006 + 0.566582i $$0.808266\pi$$
$$102$$ −2.85410 −0.282598
$$103$$ −1.23607 −0.121793 −0.0608967 0.998144i $$-0.519396\pi$$
−0.0608967 + 0.998144i $$0.519396\pi$$
$$104$$ −10.1459 −0.994887
$$105$$ 0 0
$$106$$ 1.38197 0.134228
$$107$$ 16.4721 1.59242 0.796211 0.605019i $$-0.206835\pi$$
0.796211 + 0.605019i $$0.206835\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −19.0902 −1.82851 −0.914253 0.405143i $$-0.867222\pi$$
−0.914253 + 0.405143i $$0.867222\pi$$
$$110$$ 0 0
$$111$$ 8.09017 0.767885
$$112$$ −4.47214 −0.422577
$$113$$ 2.67376 0.251526 0.125763 0.992060i $$-0.459862\pi$$
0.125763 + 0.992060i $$0.459862\pi$$
$$114$$ −3.23607 −0.303086
$$115$$ 0 0
$$116$$ −4.38197 −0.406855
$$117$$ 3.38197 0.312663
$$118$$ −4.00000 −0.368230
$$119$$ −12.7639 −1.17007
$$120$$ 0 0
$$121$$ −0.527864 −0.0479876
$$122$$ −0.618034 −0.0559542
$$123$$ −1.38197 −0.124608
$$124$$ −7.23607 −0.649818
$$125$$ 0 0
$$126$$ 4.47214 0.398410
$$127$$ 9.70820 0.861464 0.430732 0.902480i $$-0.358255\pi$$
0.430732 + 0.902480i $$0.358255\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ −5.70820 −0.502579
$$130$$ 0 0
$$131$$ 14.1803 1.23894 0.619471 0.785020i $$-0.287347\pi$$
0.619471 + 0.785020i $$0.287347\pi$$
$$132$$ −3.23607 −0.281664
$$133$$ −14.4721 −1.25489
$$134$$ 5.23607 0.452327
$$135$$ 0 0
$$136$$ 8.56231 0.734212
$$137$$ −5.38197 −0.459812 −0.229906 0.973213i $$-0.573842\pi$$
−0.229906 + 0.973213i $$0.573842\pi$$
$$138$$ −4.47214 −0.380693
$$139$$ −5.05573 −0.428821 −0.214411 0.976744i $$-0.568783\pi$$
−0.214411 + 0.976744i $$0.568783\pi$$
$$140$$ 0 0
$$141$$ −5.23607 −0.440956
$$142$$ −0.763932 −0.0641078
$$143$$ 10.9443 0.915206
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 3.09017 0.255744
$$147$$ 13.0000 1.07222
$$148$$ −8.09017 −0.665008
$$149$$ −12.1459 −0.995031 −0.497515 0.867455i $$-0.665754\pi$$
−0.497515 + 0.867455i $$0.665754\pi$$
$$150$$ 0 0
$$151$$ 16.4721 1.34048 0.670242 0.742143i $$-0.266190\pi$$
0.670242 + 0.742143i $$0.266190\pi$$
$$152$$ 9.70820 0.787439
$$153$$ −2.85410 −0.230740
$$154$$ 14.4721 1.16620
$$155$$ 0 0
$$156$$ −3.38197 −0.270774
$$157$$ −13.7984 −1.10123 −0.550615 0.834759i $$-0.685607\pi$$
−0.550615 + 0.834759i $$0.685607\pi$$
$$158$$ 0 0
$$159$$ 1.38197 0.109597
$$160$$ 0 0
$$161$$ −20.0000 −1.57622
$$162$$ 1.00000 0.0785674
$$163$$ 2.94427 0.230613 0.115307 0.993330i $$-0.463215\pi$$
0.115307 + 0.993330i $$0.463215\pi$$
$$164$$ 1.38197 0.107913
$$165$$ 0 0
$$166$$ 3.52786 0.273815
$$167$$ −23.4164 −1.81202 −0.906008 0.423261i $$-0.860886\pi$$
−0.906008 + 0.423261i $$0.860886\pi$$
$$168$$ −13.4164 −1.03510
$$169$$ −1.56231 −0.120177
$$170$$ 0 0
$$171$$ −3.23607 −0.247468
$$172$$ 5.70820 0.435246
$$173$$ 9.90983 0.753430 0.376715 0.926329i $$-0.377054\pi$$
0.376715 + 0.926329i $$0.377054\pi$$
$$174$$ 4.38197 0.332196
$$175$$ 0 0
$$176$$ −3.23607 −0.243928
$$177$$ −4.00000 −0.300658
$$178$$ 7.61803 0.570996
$$179$$ 6.18034 0.461940 0.230970 0.972961i $$-0.425810\pi$$
0.230970 + 0.972961i $$0.425810\pi$$
$$180$$ 0 0
$$181$$ −9.79837 −0.728307 −0.364154 0.931339i $$-0.618642\pi$$
−0.364154 + 0.931339i $$0.618642\pi$$
$$182$$ 15.1246 1.12111
$$183$$ −0.618034 −0.0456864
$$184$$ 13.4164 0.989071
$$185$$ 0 0
$$186$$ 7.23607 0.530574
$$187$$ −9.23607 −0.675408
$$188$$ 5.23607 0.381880
$$189$$ 4.47214 0.325300
$$190$$ 0 0
$$191$$ −24.6525 −1.78379 −0.891895 0.452242i $$-0.850624\pi$$
−0.891895 + 0.452242i $$0.850624\pi$$
$$192$$ 7.00000 0.505181
$$193$$ 7.67376 0.552369 0.276185 0.961105i $$-0.410930\pi$$
0.276185 + 0.961105i $$0.410930\pi$$
$$194$$ 8.85410 0.635687
$$195$$ 0 0
$$196$$ −13.0000 −0.928571
$$197$$ 5.38197 0.383449 0.191725 0.981449i $$-0.438592\pi$$
0.191725 + 0.981449i $$0.438592\pi$$
$$198$$ 3.23607 0.229977
$$199$$ −18.6525 −1.32224 −0.661119 0.750281i $$-0.729918\pi$$
−0.661119 + 0.750281i $$0.729918\pi$$
$$200$$ 0 0
$$201$$ 5.23607 0.369324
$$202$$ −16.5623 −1.16532
$$203$$ 19.5967 1.37542
$$204$$ 2.85410 0.199827
$$205$$ 0 0
$$206$$ −1.23607 −0.0861209
$$207$$ −4.47214 −0.310835
$$208$$ −3.38197 −0.234497
$$209$$ −10.4721 −0.724373
$$210$$ 0 0
$$211$$ −17.8885 −1.23150 −0.615749 0.787942i $$-0.711146\pi$$
−0.615749 + 0.787942i $$0.711146\pi$$
$$212$$ −1.38197 −0.0949138
$$213$$ −0.763932 −0.0523438
$$214$$ 16.4721 1.12601
$$215$$ 0 0
$$216$$ −3.00000 −0.204124
$$217$$ 32.3607 2.19679
$$218$$ −19.0902 −1.29295
$$219$$ 3.09017 0.208814
$$220$$ 0 0
$$221$$ −9.65248 −0.649296
$$222$$ 8.09017 0.542977
$$223$$ −14.1803 −0.949586 −0.474793 0.880098i $$-0.657477\pi$$
−0.474793 + 0.880098i $$0.657477\pi$$
$$224$$ 22.3607 1.49404
$$225$$ 0 0
$$226$$ 2.67376 0.177856
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 3.23607 0.214314
$$229$$ −24.9787 −1.65064 −0.825320 0.564665i $$-0.809005\pi$$
−0.825320 + 0.564665i $$0.809005\pi$$
$$230$$ 0 0
$$231$$ 14.4721 0.952197
$$232$$ −13.1459 −0.863070
$$233$$ 14.6180 0.957659 0.478830 0.877908i $$-0.341061\pi$$
0.478830 + 0.877908i $$0.341061\pi$$
$$234$$ 3.38197 0.221086
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ −12.7639 −0.827363
$$239$$ −7.05573 −0.456397 −0.228199 0.973615i $$-0.573284\pi$$
−0.228199 + 0.973615i $$0.573284\pi$$
$$240$$ 0 0
$$241$$ −1.03444 −0.0666343 −0.0333171 0.999445i $$-0.510607\pi$$
−0.0333171 + 0.999445i $$0.510607\pi$$
$$242$$ −0.527864 −0.0339324
$$243$$ 1.00000 0.0641500
$$244$$ 0.618034 0.0395656
$$245$$ 0 0
$$246$$ −1.38197 −0.0881109
$$247$$ −10.9443 −0.696367
$$248$$ −21.7082 −1.37847
$$249$$ 3.52786 0.223569
$$250$$ 0 0
$$251$$ −26.9443 −1.70071 −0.850354 0.526212i $$-0.823612\pi$$
−0.850354 + 0.526212i $$0.823612\pi$$
$$252$$ −4.47214 −0.281718
$$253$$ −14.4721 −0.909855
$$254$$ 9.70820 0.609147
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −12.7984 −0.798341 −0.399170 0.916877i $$-0.630702\pi$$
−0.399170 + 0.916877i $$0.630702\pi$$
$$258$$ −5.70820 −0.355377
$$259$$ 36.1803 2.24814
$$260$$ 0 0
$$261$$ 4.38197 0.271237
$$262$$ 14.1803 0.876064
$$263$$ 11.8885 0.733079 0.366540 0.930402i $$-0.380542\pi$$
0.366540 + 0.930402i $$0.380542\pi$$
$$264$$ −9.70820 −0.597499
$$265$$ 0 0
$$266$$ −14.4721 −0.887344
$$267$$ 7.61803 0.466216
$$268$$ −5.23607 −0.319844
$$269$$ −0.145898 −0.00889556 −0.00444778 0.999990i $$-0.501416\pi$$
−0.00444778 + 0.999990i $$0.501416\pi$$
$$270$$ 0 0
$$271$$ 22.1803 1.34736 0.673680 0.739023i $$-0.264713\pi$$
0.673680 + 0.739023i $$0.264713\pi$$
$$272$$ 2.85410 0.173055
$$273$$ 15.1246 0.915383
$$274$$ −5.38197 −0.325136
$$275$$ 0 0
$$276$$ 4.47214 0.269191
$$277$$ −5.79837 −0.348391 −0.174195 0.984711i $$-0.555732\pi$$
−0.174195 + 0.984711i $$0.555732\pi$$
$$278$$ −5.05573 −0.303222
$$279$$ 7.23607 0.433212
$$280$$ 0 0
$$281$$ −17.3820 −1.03692 −0.518461 0.855102i $$-0.673495\pi$$
−0.518461 + 0.855102i $$0.673495\pi$$
$$282$$ −5.23607 −0.311803
$$283$$ 0.291796 0.0173455 0.00867274 0.999962i $$-0.497239\pi$$
0.00867274 + 0.999962i $$0.497239\pi$$
$$284$$ 0.763932 0.0453310
$$285$$ 0 0
$$286$$ 10.9443 0.647148
$$287$$ −6.18034 −0.364814
$$288$$ 5.00000 0.294628
$$289$$ −8.85410 −0.520830
$$290$$ 0 0
$$291$$ 8.85410 0.519037
$$292$$ −3.09017 −0.180839
$$293$$ 3.79837 0.221903 0.110952 0.993826i $$-0.464610\pi$$
0.110952 + 0.993826i $$0.464610\pi$$
$$294$$ 13.0000 0.758175
$$295$$ 0 0
$$296$$ −24.2705 −1.41069
$$297$$ 3.23607 0.187776
$$298$$ −12.1459 −0.703593
$$299$$ −15.1246 −0.874679
$$300$$ 0 0
$$301$$ −25.5279 −1.47140
$$302$$ 16.4721 0.947865
$$303$$ −16.5623 −0.951480
$$304$$ 3.23607 0.185601
$$305$$ 0 0
$$306$$ −2.85410 −0.163158
$$307$$ 1.34752 0.0769073 0.0384536 0.999260i $$-0.487757\pi$$
0.0384536 + 0.999260i $$0.487757\pi$$
$$308$$ −14.4721 −0.824626
$$309$$ −1.23607 −0.0703175
$$310$$ 0 0
$$311$$ 4.29180 0.243365 0.121683 0.992569i $$-0.461171\pi$$
0.121683 + 0.992569i $$0.461171\pi$$
$$312$$ −10.1459 −0.574398
$$313$$ 8.47214 0.478873 0.239437 0.970912i $$-0.423037\pi$$
0.239437 + 0.970912i $$0.423037\pi$$
$$314$$ −13.7984 −0.778687
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −24.8328 −1.39475 −0.697375 0.716706i $$-0.745649\pi$$
−0.697375 + 0.716706i $$0.745649\pi$$
$$318$$ 1.38197 0.0774968
$$319$$ 14.1803 0.793947
$$320$$ 0 0
$$321$$ 16.4721 0.919385
$$322$$ −20.0000 −1.11456
$$323$$ 9.23607 0.513909
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 2.94427 0.163068
$$327$$ −19.0902 −1.05569
$$328$$ 4.14590 0.228919
$$329$$ −23.4164 −1.29099
$$330$$ 0 0
$$331$$ 10.9443 0.601552 0.300776 0.953695i $$-0.402754\pi$$
0.300776 + 0.953695i $$0.402754\pi$$
$$332$$ −3.52786 −0.193617
$$333$$ 8.09017 0.443339
$$334$$ −23.4164 −1.28129
$$335$$ 0 0
$$336$$ −4.47214 −0.243975
$$337$$ −14.9443 −0.814066 −0.407033 0.913413i $$-0.633437\pi$$
−0.407033 + 0.913413i $$0.633437\pi$$
$$338$$ −1.56231 −0.0849782
$$339$$ 2.67376 0.145219
$$340$$ 0 0
$$341$$ 23.4164 1.26807
$$342$$ −3.23607 −0.174987
$$343$$ 26.8328 1.44884
$$344$$ 17.1246 0.923297
$$345$$ 0 0
$$346$$ 9.90983 0.532756
$$347$$ −33.7082 −1.80955 −0.904776 0.425889i $$-0.859962\pi$$
−0.904776 + 0.425889i $$0.859962\pi$$
$$348$$ −4.38197 −0.234898
$$349$$ 25.0344 1.34006 0.670031 0.742333i $$-0.266281\pi$$
0.670031 + 0.742333i $$0.266281\pi$$
$$350$$ 0 0
$$351$$ 3.38197 0.180516
$$352$$ 16.1803 0.862415
$$353$$ −30.3607 −1.61594 −0.807968 0.589226i $$-0.799433\pi$$
−0.807968 + 0.589226i $$0.799433\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ −7.61803 −0.403755
$$357$$ −12.7639 −0.675539
$$358$$ 6.18034 0.326641
$$359$$ −10.5836 −0.558581 −0.279290 0.960207i $$-0.590099\pi$$
−0.279290 + 0.960207i $$0.590099\pi$$
$$360$$ 0 0
$$361$$ −8.52786 −0.448835
$$362$$ −9.79837 −0.514991
$$363$$ −0.527864 −0.0277057
$$364$$ −15.1246 −0.792745
$$365$$ 0 0
$$366$$ −0.618034 −0.0323052
$$367$$ −6.00000 −0.313197 −0.156599 0.987662i $$-0.550053\pi$$
−0.156599 + 0.987662i $$0.550053\pi$$
$$368$$ 4.47214 0.233126
$$369$$ −1.38197 −0.0719423
$$370$$ 0 0
$$371$$ 6.18034 0.320867
$$372$$ −7.23607 −0.375173
$$373$$ 29.4164 1.52312 0.761562 0.648092i $$-0.224433\pi$$
0.761562 + 0.648092i $$0.224433\pi$$
$$374$$ −9.23607 −0.477586
$$375$$ 0 0
$$376$$ 15.7082 0.810089
$$377$$ 14.8197 0.763251
$$378$$ 4.47214 0.230022
$$379$$ 23.4164 1.20282 0.601410 0.798941i $$-0.294606\pi$$
0.601410 + 0.798941i $$0.294606\pi$$
$$380$$ 0 0
$$381$$ 9.70820 0.497366
$$382$$ −24.6525 −1.26133
$$383$$ 20.8328 1.06451 0.532254 0.846585i $$-0.321345\pi$$
0.532254 + 0.846585i $$0.321345\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ 7.67376 0.390584
$$387$$ −5.70820 −0.290164
$$388$$ −8.85410 −0.449499
$$389$$ −37.0902 −1.88055 −0.940273 0.340421i $$-0.889430\pi$$
−0.940273 + 0.340421i $$0.889430\pi$$
$$390$$ 0 0
$$391$$ 12.7639 0.645500
$$392$$ −39.0000 −1.96980
$$393$$ 14.1803 0.715304
$$394$$ 5.38197 0.271140
$$395$$ 0 0
$$396$$ −3.23607 −0.162619
$$397$$ 10.9443 0.549277 0.274639 0.961548i $$-0.411442\pi$$
0.274639 + 0.961548i $$0.411442\pi$$
$$398$$ −18.6525 −0.934964
$$399$$ −14.4721 −0.724513
$$400$$ 0 0
$$401$$ 33.4508 1.67046 0.835228 0.549904i $$-0.185336\pi$$
0.835228 + 0.549904i $$0.185336\pi$$
$$402$$ 5.23607 0.261151
$$403$$ 24.4721 1.21904
$$404$$ 16.5623 0.824006
$$405$$ 0 0
$$406$$ 19.5967 0.972570
$$407$$ 26.1803 1.29771
$$408$$ 8.56231 0.423897
$$409$$ 25.7984 1.27565 0.637824 0.770182i $$-0.279835\pi$$
0.637824 + 0.770182i $$0.279835\pi$$
$$410$$ 0 0
$$411$$ −5.38197 −0.265473
$$412$$ 1.23607 0.0608967
$$413$$ −17.8885 −0.880238
$$414$$ −4.47214 −0.219793
$$415$$ 0 0
$$416$$ 16.9098 0.829073
$$417$$ −5.05573 −0.247580
$$418$$ −10.4721 −0.512209
$$419$$ 32.9443 1.60943 0.804717 0.593659i $$-0.202317\pi$$
0.804717 + 0.593659i $$0.202317\pi$$
$$420$$ 0 0
$$421$$ 23.1459 1.12806 0.564031 0.825754i $$-0.309250\pi$$
0.564031 + 0.825754i $$0.309250\pi$$
$$422$$ −17.8885 −0.870801
$$423$$ −5.23607 −0.254586
$$424$$ −4.14590 −0.201343
$$425$$ 0 0
$$426$$ −0.763932 −0.0370126
$$427$$ −2.76393 −0.133756
$$428$$ −16.4721 −0.796211
$$429$$ 10.9443 0.528394
$$430$$ 0 0
$$431$$ −10.6525 −0.513112 −0.256556 0.966529i $$-0.582588\pi$$
−0.256556 + 0.966529i $$0.582588\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 24.9787 1.20040 0.600200 0.799850i $$-0.295088\pi$$
0.600200 + 0.799850i $$0.295088\pi$$
$$434$$ 32.3607 1.55336
$$435$$ 0 0
$$436$$ 19.0902 0.914253
$$437$$ 14.4721 0.692296
$$438$$ 3.09017 0.147654
$$439$$ −6.18034 −0.294972 −0.147486 0.989064i $$-0.547118\pi$$
−0.147486 + 0.989064i $$0.547118\pi$$
$$440$$ 0 0
$$441$$ 13.0000 0.619048
$$442$$ −9.65248 −0.459121
$$443$$ −30.7639 −1.46164 −0.730819 0.682571i $$-0.760862\pi$$
−0.730819 + 0.682571i $$0.760862\pi$$
$$444$$ −8.09017 −0.383942
$$445$$ 0 0
$$446$$ −14.1803 −0.671459
$$447$$ −12.1459 −0.574481
$$448$$ 31.3050 1.47902
$$449$$ −7.79837 −0.368028 −0.184014 0.982924i $$-0.558909\pi$$
−0.184014 + 0.982924i $$0.558909\pi$$
$$450$$ 0 0
$$451$$ −4.47214 −0.210585
$$452$$ −2.67376 −0.125763
$$453$$ 16.4721 0.773928
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 9.70820 0.454628
$$457$$ 22.3607 1.04599 0.522994 0.852336i $$-0.324815\pi$$
0.522994 + 0.852336i $$0.324815\pi$$
$$458$$ −24.9787 −1.16718
$$459$$ −2.85410 −0.133218
$$460$$ 0 0
$$461$$ −8.20163 −0.381988 −0.190994 0.981591i $$-0.561171\pi$$
−0.190994 + 0.981591i $$0.561171\pi$$
$$462$$ 14.4721 0.673305
$$463$$ 23.2361 1.07987 0.539936 0.841706i $$-0.318448\pi$$
0.539936 + 0.841706i $$0.318448\pi$$
$$464$$ −4.38197 −0.203428
$$465$$ 0 0
$$466$$ 14.6180 0.677167
$$467$$ 5.81966 0.269302 0.134651 0.990893i $$-0.457009\pi$$
0.134651 + 0.990893i $$0.457009\pi$$
$$468$$ −3.38197 −0.156331
$$469$$ 23.4164 1.08127
$$470$$ 0 0
$$471$$ −13.7984 −0.635796
$$472$$ 12.0000 0.552345
$$473$$ −18.4721 −0.849350
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 12.7639 0.585034
$$477$$ 1.38197 0.0632759
$$478$$ −7.05573 −0.322721
$$479$$ 1.05573 0.0482374 0.0241187 0.999709i $$-0.492322\pi$$
0.0241187 + 0.999709i $$0.492322\pi$$
$$480$$ 0 0
$$481$$ 27.3607 1.24754
$$482$$ −1.03444 −0.0471175
$$483$$ −20.0000 −0.910032
$$484$$ 0.527864 0.0239938
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −3.70820 −0.168035 −0.0840174 0.996464i $$-0.526775\pi$$
−0.0840174 + 0.996464i $$0.526775\pi$$
$$488$$ 1.85410 0.0839313
$$489$$ 2.94427 0.133145
$$490$$ 0 0
$$491$$ −29.8885 −1.34885 −0.674426 0.738343i $$-0.735609\pi$$
−0.674426 + 0.738343i $$0.735609\pi$$
$$492$$ 1.38197 0.0623038
$$493$$ −12.5066 −0.563268
$$494$$ −10.9443 −0.492406
$$495$$ 0 0
$$496$$ −7.23607 −0.324909
$$497$$ −3.41641 −0.153247
$$498$$ 3.52786 0.158087
$$499$$ −6.00000 −0.268597 −0.134298 0.990941i $$-0.542878\pi$$
−0.134298 + 0.990941i $$0.542878\pi$$
$$500$$ 0 0
$$501$$ −23.4164 −1.04617
$$502$$ −26.9443 −1.20258
$$503$$ 35.4164 1.57914 0.789570 0.613661i $$-0.210304\pi$$
0.789570 + 0.613661i $$0.210304\pi$$
$$504$$ −13.4164 −0.597614
$$505$$ 0 0
$$506$$ −14.4721 −0.643365
$$507$$ −1.56231 −0.0693844
$$508$$ −9.70820 −0.430732
$$509$$ 26.8541 1.19029 0.595144 0.803619i $$-0.297095\pi$$
0.595144 + 0.803619i $$0.297095\pi$$
$$510$$ 0 0
$$511$$ 13.8197 0.611346
$$512$$ −11.0000 −0.486136
$$513$$ −3.23607 −0.142876
$$514$$ −12.7984 −0.564512
$$515$$ 0 0
$$516$$ 5.70820 0.251290
$$517$$ −16.9443 −0.745208
$$518$$ 36.1803 1.58967
$$519$$ 9.90983 0.434993
$$520$$ 0 0
$$521$$ 2.03444 0.0891305 0.0445653 0.999006i $$-0.485810\pi$$
0.0445653 + 0.999006i $$0.485810\pi$$
$$522$$ 4.38197 0.191793
$$523$$ −12.2918 −0.537483 −0.268741 0.963212i $$-0.586608\pi$$
−0.268741 + 0.963212i $$0.586608\pi$$
$$524$$ −14.1803 −0.619471
$$525$$ 0 0
$$526$$ 11.8885 0.518365
$$527$$ −20.6525 −0.899636
$$528$$ −3.23607 −0.140832
$$529$$ −3.00000 −0.130435
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 14.4721 0.627447
$$533$$ −4.67376 −0.202443
$$534$$ 7.61803 0.329665
$$535$$ 0 0
$$536$$ −15.7082 −0.678491
$$537$$ 6.18034 0.266701
$$538$$ −0.145898 −0.00629011
$$539$$ 42.0689 1.81204
$$540$$ 0 0
$$541$$ −35.3262 −1.51879 −0.759397 0.650628i $$-0.774506\pi$$
−0.759397 + 0.650628i $$0.774506\pi$$
$$542$$ 22.1803 0.952727
$$543$$ −9.79837 −0.420488
$$544$$ −14.2705 −0.611843
$$545$$ 0 0
$$546$$ 15.1246 0.647274
$$547$$ −2.36068 −0.100935 −0.0504677 0.998726i $$-0.516071\pi$$
−0.0504677 + 0.998726i $$0.516071\pi$$
$$548$$ 5.38197 0.229906
$$549$$ −0.618034 −0.0263770
$$550$$ 0 0
$$551$$ −14.1803 −0.604103
$$552$$ 13.4164 0.571040
$$553$$ 0 0
$$554$$ −5.79837 −0.246349
$$555$$ 0 0
$$556$$ 5.05573 0.214411
$$557$$ 22.2705 0.943632 0.471816 0.881697i $$-0.343599\pi$$
0.471816 + 0.881697i $$0.343599\pi$$
$$558$$ 7.23607 0.306327
$$559$$ −19.3050 −0.816512
$$560$$ 0 0
$$561$$ −9.23607 −0.389947
$$562$$ −17.3820 −0.733214
$$563$$ −17.5279 −0.738711 −0.369356 0.929288i $$-0.620422\pi$$
−0.369356 + 0.929288i $$0.620422\pi$$
$$564$$ 5.23607 0.220478
$$565$$ 0 0
$$566$$ 0.291796 0.0122651
$$567$$ 4.47214 0.187812
$$568$$ 2.29180 0.0961616
$$569$$ −30.2705 −1.26901 −0.634503 0.772920i $$-0.718795\pi$$
−0.634503 + 0.772920i $$0.718795\pi$$
$$570$$ 0 0
$$571$$ −21.2361 −0.888702 −0.444351 0.895853i $$-0.646566\pi$$
−0.444351 + 0.895853i $$0.646566\pi$$
$$572$$ −10.9443 −0.457603
$$573$$ −24.6525 −1.02987
$$574$$ −6.18034 −0.257962
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ −30.9443 −1.28823 −0.644113 0.764930i $$-0.722774\pi$$
−0.644113 + 0.764930i $$0.722774\pi$$
$$578$$ −8.85410 −0.368282
$$579$$ 7.67376 0.318911
$$580$$ 0 0
$$581$$ 15.7771 0.654544
$$582$$ 8.85410 0.367014
$$583$$ 4.47214 0.185217
$$584$$ −9.27051 −0.383616
$$585$$ 0 0
$$586$$ 3.79837 0.156909
$$587$$ −14.5836 −0.601929 −0.300965 0.953635i $$-0.597309\pi$$
−0.300965 + 0.953635i $$0.597309\pi$$
$$588$$ −13.0000 −0.536111
$$589$$ −23.4164 −0.964856
$$590$$ 0 0
$$591$$ 5.38197 0.221384
$$592$$ −8.09017 −0.332504
$$593$$ −32.7426 −1.34458 −0.672290 0.740288i $$-0.734689\pi$$
−0.672290 + 0.740288i $$0.734689\pi$$
$$594$$ 3.23607 0.132777
$$595$$ 0 0
$$596$$ 12.1459 0.497515
$$597$$ −18.6525 −0.763395
$$598$$ −15.1246 −0.618491
$$599$$ −12.4721 −0.509598 −0.254799 0.966994i $$-0.582009\pi$$
−0.254799 + 0.966994i $$0.582009\pi$$
$$600$$ 0 0
$$601$$ −24.3262 −0.992288 −0.496144 0.868240i $$-0.665251\pi$$
−0.496144 + 0.868240i $$0.665251\pi$$
$$602$$ −25.5279 −1.04044
$$603$$ 5.23607 0.213229
$$604$$ −16.4721 −0.670242
$$605$$ 0 0
$$606$$ −16.5623 −0.672798
$$607$$ −39.2361 −1.59254 −0.796271 0.604940i $$-0.793197\pi$$
−0.796271 + 0.604940i $$0.793197\pi$$
$$608$$ −16.1803 −0.656199
$$609$$ 19.5967 0.794100
$$610$$ 0 0
$$611$$ −17.7082 −0.716397
$$612$$ 2.85410 0.115370
$$613$$ 39.5066 1.59566 0.797828 0.602885i $$-0.205982\pi$$
0.797828 + 0.602885i $$0.205982\pi$$
$$614$$ 1.34752 0.0543816
$$615$$ 0 0
$$616$$ −43.4164 −1.74930
$$617$$ 26.9787 1.08612 0.543061 0.839693i $$-0.317265\pi$$
0.543061 + 0.839693i $$0.317265\pi$$
$$618$$ −1.23607 −0.0497219
$$619$$ 28.3607 1.13991 0.569956 0.821675i $$-0.306960\pi$$
0.569956 + 0.821675i $$0.306960\pi$$
$$620$$ 0 0
$$621$$ −4.47214 −0.179461
$$622$$ 4.29180 0.172085
$$623$$ 34.0689 1.36494
$$624$$ −3.38197 −0.135387
$$625$$ 0 0
$$626$$ 8.47214 0.338615
$$627$$ −10.4721 −0.418217
$$628$$ 13.7984 0.550615
$$629$$ −23.0902 −0.920665
$$630$$ 0 0
$$631$$ 10.2918 0.409710 0.204855 0.978792i $$-0.434328\pi$$
0.204855 + 0.978792i $$0.434328\pi$$
$$632$$ 0 0
$$633$$ −17.8885 −0.711006
$$634$$ −24.8328 −0.986237
$$635$$ 0 0
$$636$$ −1.38197 −0.0547985
$$637$$ 43.9656 1.74198
$$638$$ 14.1803 0.561405
$$639$$ −0.763932 −0.0302207
$$640$$ 0 0
$$641$$ 38.9443 1.53821 0.769103 0.639125i $$-0.220703\pi$$
0.769103 + 0.639125i $$0.220703\pi$$
$$642$$ 16.4721 0.650103
$$643$$ 13.8885 0.547711 0.273855 0.961771i $$-0.411701\pi$$
0.273855 + 0.961771i $$0.411701\pi$$
$$644$$ 20.0000 0.788110
$$645$$ 0 0
$$646$$ 9.23607 0.363388
$$647$$ 10.3607 0.407320 0.203660 0.979042i $$-0.434716\pi$$
0.203660 + 0.979042i $$0.434716\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ −12.9443 −0.508107
$$650$$ 0 0
$$651$$ 32.3607 1.26832
$$652$$ −2.94427 −0.115307
$$653$$ 45.1033 1.76503 0.882515 0.470285i $$-0.155849\pi$$
0.882515 + 0.470285i $$0.155849\pi$$
$$654$$ −19.0902 −0.746485
$$655$$ 0 0
$$656$$ 1.38197 0.0539567
$$657$$ 3.09017 0.120559
$$658$$ −23.4164 −0.912867
$$659$$ −6.58359 −0.256460 −0.128230 0.991744i $$-0.540930\pi$$
−0.128230 + 0.991744i $$0.540930\pi$$
$$660$$ 0 0
$$661$$ 25.4164 0.988584 0.494292 0.869296i $$-0.335427\pi$$
0.494292 + 0.869296i $$0.335427\pi$$
$$662$$ 10.9443 0.425361
$$663$$ −9.65248 −0.374871
$$664$$ −10.5836 −0.410723
$$665$$ 0 0
$$666$$ 8.09017 0.313488
$$667$$ −19.5967 −0.758789
$$668$$ 23.4164 0.906008
$$669$$ −14.1803 −0.548244
$$670$$ 0 0
$$671$$ −2.00000 −0.0772091
$$672$$ 22.3607 0.862582
$$673$$ 29.7984 1.14864 0.574321 0.818630i $$-0.305266\pi$$
0.574321 + 0.818630i $$0.305266\pi$$
$$674$$ −14.9443 −0.575632
$$675$$ 0 0
$$676$$ 1.56231 0.0600887
$$677$$ 37.4164 1.43803 0.719015 0.694995i $$-0.244593\pi$$
0.719015 + 0.694995i $$0.244593\pi$$
$$678$$ 2.67376 0.102685
$$679$$ 39.5967 1.51958
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 23.4164 0.896661
$$683$$ 7.41641 0.283781 0.141890 0.989882i $$-0.454682\pi$$
0.141890 + 0.989882i $$0.454682\pi$$
$$684$$ 3.23607 0.123734
$$685$$ 0 0
$$686$$ 26.8328 1.02448
$$687$$ −24.9787 −0.952997
$$688$$ 5.70820 0.217623
$$689$$ 4.67376 0.178056
$$690$$ 0 0
$$691$$ −21.2361 −0.807858 −0.403929 0.914790i $$-0.632356\pi$$
−0.403929 + 0.914790i $$0.632356\pi$$
$$692$$ −9.90983 −0.376715
$$693$$ 14.4721 0.549751
$$694$$ −33.7082 −1.27955
$$695$$ 0 0
$$696$$ −13.1459 −0.498294
$$697$$ 3.94427 0.149400
$$698$$ 25.0344 0.947568
$$699$$ 14.6180 0.552905
$$700$$ 0 0
$$701$$ 0.437694 0.0165315 0.00826574 0.999966i $$-0.497369\pi$$
0.00826574 + 0.999966i $$0.497369\pi$$
$$702$$ 3.38197 0.127644
$$703$$ −26.1803 −0.987410
$$704$$ 22.6525 0.853747
$$705$$ 0 0
$$706$$ −30.3607 −1.14264
$$707$$ −74.0689 −2.78565
$$708$$ 4.00000 0.150329
$$709$$ −6.72949 −0.252731 −0.126366 0.991984i $$-0.540331\pi$$
−0.126366 + 0.991984i $$0.540331\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −22.8541 −0.856494
$$713$$ −32.3607 −1.21192
$$714$$ −12.7639 −0.477678
$$715$$ 0 0
$$716$$ −6.18034 −0.230970
$$717$$ −7.05573 −0.263501
$$718$$ −10.5836 −0.394976
$$719$$ −35.8885 −1.33842 −0.669208 0.743075i $$-0.733367\pi$$
−0.669208 + 0.743075i $$0.733367\pi$$
$$720$$ 0 0
$$721$$ −5.52786 −0.205868
$$722$$ −8.52786 −0.317374
$$723$$ −1.03444 −0.0384713
$$724$$ 9.79837 0.364154
$$725$$ 0 0
$$726$$ −0.527864 −0.0195909
$$727$$ 15.3475 0.569208 0.284604 0.958645i $$-0.408138\pi$$
0.284604 + 0.958645i $$0.408138\pi$$
$$728$$ −45.3738 −1.68167
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 16.2918 0.602574
$$732$$ 0.618034 0.0228432
$$733$$ 2.58359 0.0954272 0.0477136 0.998861i $$-0.484807\pi$$
0.0477136 + 0.998861i $$0.484807\pi$$
$$734$$ −6.00000 −0.221464
$$735$$ 0 0
$$736$$ −22.3607 −0.824226
$$737$$ 16.9443 0.624150
$$738$$ −1.38197 −0.0508709
$$739$$ 17.7082 0.651407 0.325703 0.945472i $$-0.394399\pi$$
0.325703 + 0.945472i $$0.394399\pi$$
$$740$$ 0 0
$$741$$ −10.9443 −0.402048
$$742$$ 6.18034 0.226887
$$743$$ 0.875388 0.0321149 0.0160574 0.999871i $$-0.494889\pi$$
0.0160574 + 0.999871i $$0.494889\pi$$
$$744$$ −21.7082 −0.795861
$$745$$ 0 0
$$746$$ 29.4164 1.07701
$$747$$ 3.52786 0.129078
$$748$$ 9.23607 0.337704
$$749$$ 73.6656 2.69168
$$750$$ 0 0
$$751$$ 5.34752 0.195134 0.0975670 0.995229i $$-0.468894\pi$$
0.0975670 + 0.995229i $$0.468894\pi$$
$$752$$ 5.23607 0.190940
$$753$$ −26.9443 −0.981904
$$754$$ 14.8197 0.539700
$$755$$ 0 0
$$756$$ −4.47214 −0.162650
$$757$$ 27.3820 0.995214 0.497607 0.867402i $$-0.334212\pi$$
0.497607 + 0.867402i $$0.334212\pi$$
$$758$$ 23.4164 0.850522
$$759$$ −14.4721 −0.525305
$$760$$ 0 0
$$761$$ −9.74265 −0.353171 −0.176585 0.984285i $$-0.556505\pi$$
−0.176585 + 0.984285i $$0.556505\pi$$
$$762$$ 9.70820 0.351691
$$763$$ −85.3738 −3.09074
$$764$$ 24.6525 0.891895
$$765$$ 0 0
$$766$$ 20.8328 0.752720
$$767$$ −13.5279 −0.488463
$$768$$ −17.0000 −0.613435
$$769$$ −5.63932 −0.203359 −0.101680 0.994817i $$-0.532422\pi$$
−0.101680 + 0.994817i $$0.532422\pi$$
$$770$$ 0 0
$$771$$ −12.7984 −0.460922
$$772$$ −7.67376 −0.276185
$$773$$ −35.9230 −1.29206 −0.646030 0.763312i $$-0.723572\pi$$
−0.646030 + 0.763312i $$0.723572\pi$$
$$774$$ −5.70820 −0.205177
$$775$$ 0 0
$$776$$ −26.5623 −0.953531
$$777$$ 36.1803 1.29796
$$778$$ −37.0902 −1.32975
$$779$$ 4.47214 0.160231
$$780$$ 0 0
$$781$$ −2.47214 −0.0884600
$$782$$ 12.7639 0.456437
$$783$$ 4.38197 0.156599
$$784$$ −13.0000 −0.464286
$$785$$ 0 0
$$786$$ 14.1803 0.505796
$$787$$ 8.18034 0.291598 0.145799 0.989314i $$-0.453425\pi$$
0.145799 + 0.989314i $$0.453425\pi$$
$$788$$ −5.38197 −0.191725
$$789$$ 11.8885 0.423243
$$790$$ 0 0
$$791$$ 11.9574 0.425157
$$792$$ −9.70820 −0.344966
$$793$$ −2.09017 −0.0742241
$$794$$ 10.9443 0.388398
$$795$$ 0 0
$$796$$ 18.6525 0.661119
$$797$$ 38.1033 1.34969 0.674845 0.737960i $$-0.264211\pi$$
0.674845 + 0.737960i $$0.264211\pi$$
$$798$$ −14.4721 −0.512308
$$799$$ 14.9443 0.528690
$$800$$ 0 0
$$801$$ 7.61803 0.269170
$$802$$ 33.4508 1.18119
$$803$$ 10.0000 0.352892
$$804$$ −5.23607 −0.184662
$$805$$ 0 0
$$806$$ 24.4721 0.861994
$$807$$ −0.145898 −0.00513585
$$808$$ 49.6869 1.74798
$$809$$ −23.2148 −0.816188 −0.408094 0.912940i $$-0.633806\pi$$
−0.408094 + 0.912940i $$0.633806\pi$$
$$810$$ 0 0
$$811$$ 9.23607 0.324322 0.162161 0.986764i $$-0.448154\pi$$
0.162161 + 0.986764i $$0.448154\pi$$
$$812$$ −19.5967 −0.687711
$$813$$ 22.1803 0.777898
$$814$$ 26.1803 0.917620
$$815$$ 0 0
$$816$$ 2.85410 0.0999136
$$817$$ 18.4721 0.646258
$$818$$ 25.7984 0.902019
$$819$$ 15.1246 0.528497
$$820$$ 0 0
$$821$$ 5.05573 0.176446 0.0882231 0.996101i $$-0.471881\pi$$
0.0882231 + 0.996101i $$0.471881\pi$$
$$822$$ −5.38197 −0.187718
$$823$$ −14.7639 −0.514638 −0.257319 0.966326i $$-0.582839\pi$$
−0.257319 + 0.966326i $$0.582839\pi$$
$$824$$ 3.70820 0.129181
$$825$$ 0 0
$$826$$ −17.8885 −0.622422
$$827$$ 22.0689 0.767410 0.383705 0.923456i $$-0.374648\pi$$
0.383705 + 0.923456i $$0.374648\pi$$
$$828$$ 4.47214 0.155417
$$829$$ −26.5066 −0.920611 −0.460306 0.887760i $$-0.652260\pi$$
−0.460306 + 0.887760i $$0.652260\pi$$
$$830$$ 0 0
$$831$$ −5.79837 −0.201143
$$832$$ 23.6738 0.820740
$$833$$ −37.1033 −1.28555
$$834$$ −5.05573 −0.175066
$$835$$ 0 0
$$836$$ 10.4721 0.362186
$$837$$ 7.23607 0.250115
$$838$$ 32.9443 1.13804
$$839$$ −44.0689 −1.52143 −0.760713 0.649088i $$-0.775151\pi$$
−0.760713 + 0.649088i $$0.775151\pi$$
$$840$$ 0 0
$$841$$ −9.79837 −0.337875
$$842$$ 23.1459 0.797660
$$843$$ −17.3820 −0.598667
$$844$$ 17.8885 0.615749
$$845$$ 0 0
$$846$$ −5.23607 −0.180020
$$847$$ −2.36068 −0.0811139
$$848$$ −1.38197 −0.0474569
$$849$$ 0.291796 0.0100144
$$850$$ 0 0
$$851$$ −36.1803 −1.24025
$$852$$ 0.763932 0.0261719
$$853$$ 29.8541 1.02218 0.511092 0.859526i $$-0.329241\pi$$
0.511092 + 0.859526i $$0.329241\pi$$
$$854$$ −2.76393 −0.0945798
$$855$$ 0 0
$$856$$ −49.4164 −1.68902
$$857$$ −3.52786 −0.120510 −0.0602548 0.998183i $$-0.519191\pi$$
−0.0602548 + 0.998183i $$0.519191\pi$$
$$858$$ 10.9443 0.373631
$$859$$ 3.41641 0.116566 0.0582832 0.998300i $$-0.481437\pi$$
0.0582832 + 0.998300i $$0.481437\pi$$
$$860$$ 0 0
$$861$$ −6.18034 −0.210625
$$862$$ −10.6525 −0.362825
$$863$$ 51.1246 1.74030 0.870151 0.492785i $$-0.164021\pi$$
0.870151 + 0.492785i $$0.164021\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ 24.9787 0.848811
$$867$$ −8.85410 −0.300701
$$868$$ −32.3607 −1.09839
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 17.7082 0.600020
$$872$$ 57.2705 1.93942
$$873$$ 8.85410 0.299666
$$874$$ 14.4721 0.489527
$$875$$ 0 0
$$876$$ −3.09017 −0.104407
$$877$$ 18.2148 0.615069 0.307535 0.951537i $$-0.400496\pi$$
0.307535 + 0.951537i $$0.400496\pi$$
$$878$$ −6.18034 −0.208576
$$879$$ 3.79837 0.128116
$$880$$ 0 0
$$881$$ 25.0557 0.844149 0.422074 0.906561i $$-0.361302\pi$$
0.422074 + 0.906561i $$0.361302\pi$$
$$882$$ 13.0000 0.437733
$$883$$ −0.180340 −0.00606892 −0.00303446 0.999995i $$-0.500966\pi$$
−0.00303446 + 0.999995i $$0.500966\pi$$
$$884$$ 9.65248 0.324648
$$885$$ 0 0
$$886$$ −30.7639 −1.03353
$$887$$ 17.1246 0.574988 0.287494 0.957782i $$-0.407178\pi$$
0.287494 + 0.957782i $$0.407178\pi$$
$$888$$ −24.2705 −0.814465
$$889$$ 43.4164 1.45614
$$890$$ 0 0
$$891$$ 3.23607 0.108412
$$892$$ 14.1803 0.474793
$$893$$ 16.9443 0.567018
$$894$$ −12.1459 −0.406220
$$895$$ 0 0
$$896$$ −13.4164 −0.448211
$$897$$ −15.1246 −0.504996
$$898$$ −7.79837 −0.260235
$$899$$ 31.7082 1.05753
$$900$$ 0 0
$$901$$ −3.94427 −0.131403
$$902$$ −4.47214 −0.148906
$$903$$ −25.5279 −0.849514
$$904$$ −8.02129 −0.266784
$$905$$ 0 0
$$906$$ 16.4721 0.547250
$$907$$ −33.1246 −1.09988 −0.549942 0.835203i $$-0.685350\pi$$
−0.549942 + 0.835203i $$0.685350\pi$$
$$908$$ 20.0000 0.663723
$$909$$ −16.5623 −0.549337
$$910$$ 0 0
$$911$$ −4.18034 −0.138501 −0.0692504 0.997599i $$-0.522061\pi$$
−0.0692504 + 0.997599i $$0.522061\pi$$
$$912$$ 3.23607 0.107157
$$913$$ 11.4164 0.377828
$$914$$ 22.3607 0.739626
$$915$$ 0 0
$$916$$ 24.9787 0.825320
$$917$$ 63.4164 2.09419
$$918$$ −2.85410 −0.0941994
$$919$$ −49.1935 −1.62274 −0.811372 0.584530i $$-0.801279\pi$$
−0.811372 + 0.584530i $$0.801279\pi$$
$$920$$ 0 0
$$921$$ 1.34752 0.0444024
$$922$$ −8.20163 −0.270106
$$923$$ −2.58359 −0.0850400
$$924$$ −14.4721 −0.476098
$$925$$ 0 0
$$926$$ 23.2361 0.763585
$$927$$ −1.23607 −0.0405978
$$928$$ 21.9098 0.719225
$$929$$ 2.09017 0.0685763 0.0342881 0.999412i $$-0.489084\pi$$
0.0342881 + 0.999412i $$0.489084\pi$$
$$930$$ 0 0
$$931$$ −42.0689 −1.37875
$$932$$ −14.6180 −0.478830
$$933$$ 4.29180 0.140507
$$934$$ 5.81966 0.190425
$$935$$ 0 0
$$936$$ −10.1459 −0.331629
$$937$$ 11.6869 0.381795 0.190897 0.981610i $$-0.438860\pi$$
0.190897 + 0.981610i $$0.438860\pi$$
$$938$$ 23.4164 0.764573
$$939$$ 8.47214 0.276478
$$940$$ 0 0
$$941$$ 5.32624 0.173630 0.0868152 0.996224i $$-0.472331\pi$$
0.0868152 + 0.996224i $$0.472331\pi$$
$$942$$ −13.7984 −0.449575
$$943$$ 6.18034 0.201260
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −18.4721 −0.600581
$$947$$ 29.4164 0.955905 0.477952 0.878386i $$-0.341379\pi$$
0.477952 + 0.878386i $$0.341379\pi$$
$$948$$ 0 0
$$949$$ 10.4508 0.339249
$$950$$ 0 0
$$951$$ −24.8328 −0.805259
$$952$$ 38.2918 1.24104
$$953$$ 10.0902 0.326853 0.163426 0.986556i $$-0.447745\pi$$
0.163426 + 0.986556i $$0.447745\pi$$
$$954$$ 1.38197 0.0447428
$$955$$ 0 0
$$956$$ 7.05573 0.228199
$$957$$ 14.1803 0.458385
$$958$$ 1.05573 0.0341090
$$959$$ −24.0689 −0.777225
$$960$$ 0 0
$$961$$ 21.3607 0.689054
$$962$$ 27.3607 0.882144
$$963$$ 16.4721 0.530807
$$964$$ 1.03444 0.0333171
$$965$$ 0 0
$$966$$ −20.0000 −0.643489
$$967$$ 34.8328 1.12015 0.560074 0.828443i $$-0.310773\pi$$
0.560074 + 0.828443i $$0.310773\pi$$
$$968$$ 1.58359 0.0508986
$$969$$ 9.23607 0.296705
$$970$$ 0 0
$$971$$ −6.58359 −0.211278 −0.105639 0.994405i $$-0.533689\pi$$
−0.105639 + 0.994405i $$0.533689\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −22.6099 −0.724840
$$974$$ −3.70820 −0.118819
$$975$$ 0 0
$$976$$ 0.618034 0.0197828
$$977$$ −45.7426 −1.46344 −0.731718 0.681607i $$-0.761281\pi$$
−0.731718 + 0.681607i $$0.761281\pi$$
$$978$$ 2.94427 0.0941474
$$979$$ 24.6525 0.787897
$$980$$ 0 0
$$981$$ −19.0902 −0.609502
$$982$$ −29.8885 −0.953782
$$983$$ 21.7082 0.692384 0.346192 0.938164i $$-0.387475\pi$$
0.346192 + 0.938164i $$0.387475\pi$$
$$984$$ 4.14590 0.132166
$$985$$ 0 0
$$986$$ −12.5066 −0.398291
$$987$$ −23.4164 −0.745352
$$988$$ 10.9443 0.348184
$$989$$ 25.5279 0.811739
$$990$$ 0 0
$$991$$ −3.12461 −0.0992566 −0.0496283 0.998768i $$-0.515804\pi$$
−0.0496283 + 0.998768i $$0.515804\pi$$
$$992$$ 36.1803 1.14873
$$993$$ 10.9443 0.347306
$$994$$ −3.41641 −0.108362
$$995$$ 0 0
$$996$$ −3.52786 −0.111785
$$997$$ 9.41641 0.298221 0.149110 0.988821i $$-0.452359\pi$$
0.149110 + 0.988821i $$0.452359\pi$$
$$998$$ −6.00000 −0.189927
$$999$$ 8.09017 0.255962
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.d.1.2 2
3.2 odd 2 5625.2.a.a.1.2 2
5.2 odd 4 1875.2.b.b.1249.4 4
5.3 odd 4 1875.2.b.b.1249.1 4
5.4 even 2 1875.2.a.a.1.1 2
15.14 odd 2 5625.2.a.h.1.1 2
25.3 odd 20 375.2.i.a.49.1 8
25.4 even 10 375.2.g.a.76.1 4
25.6 even 5 75.2.g.a.61.1 yes 4
25.8 odd 20 375.2.i.a.199.2 8
25.17 odd 20 375.2.i.a.199.1 8
25.19 even 10 375.2.g.a.301.1 4
25.21 even 5 75.2.g.a.16.1 4
25.22 odd 20 375.2.i.a.49.2 8
75.56 odd 10 225.2.h.a.136.1 4
75.71 odd 10 225.2.h.a.91.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.a.16.1 4 25.21 even 5
75.2.g.a.61.1 yes 4 25.6 even 5
225.2.h.a.91.1 4 75.71 odd 10
225.2.h.a.136.1 4 75.56 odd 10
375.2.g.a.76.1 4 25.4 even 10
375.2.g.a.301.1 4 25.19 even 10
375.2.i.a.49.1 8 25.3 odd 20
375.2.i.a.49.2 8 25.22 odd 20
375.2.i.a.199.1 8 25.17 odd 20
375.2.i.a.199.2 8 25.8 odd 20
1875.2.a.a.1.1 2 5.4 even 2
1875.2.a.d.1.2 2 1.1 even 1 trivial
1875.2.b.b.1249.1 4 5.3 odd 4
1875.2.b.b.1249.4 4 5.2 odd 4
5625.2.a.a.1.2 2 3.2 odd 2
5625.2.a.h.1.1 2 15.14 odd 2