# Properties

 Label 1875.2.a.a.1.2 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $1$ 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: $$1$$ 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 $$-0.618034$$ 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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.23607 q^{11} +1.00000 q^{12} -5.61803 q^{13} -4.47214 q^{14} -1.00000 q^{16} -3.85410 q^{17} -1.00000 q^{18} +1.23607 q^{19} -4.47214 q^{21} +1.23607 q^{22} -4.47214 q^{23} -3.00000 q^{24} +5.61803 q^{26} -1.00000 q^{27} -4.47214 q^{28} +6.61803 q^{29} +2.76393 q^{31} -5.00000 q^{32} +1.23607 q^{33} +3.85410 q^{34} -1.00000 q^{36} +3.09017 q^{37} -1.23607 q^{38} +5.61803 q^{39} -3.61803 q^{41} +4.47214 q^{42} -7.70820 q^{43} +1.23607 q^{44} +4.47214 q^{46} +0.763932 q^{47} +1.00000 q^{48} +13.0000 q^{49} +3.85410 q^{51} +5.61803 q^{52} -3.61803 q^{53} +1.00000 q^{54} +13.4164 q^{56} -1.23607 q^{57} -6.61803 q^{58} -4.00000 q^{59} +1.61803 q^{61} -2.76393 q^{62} +4.47214 q^{63} +7.00000 q^{64} -1.23607 q^{66} -0.763932 q^{67} +3.85410 q^{68} +4.47214 q^{69} -5.23607 q^{71} +3.00000 q^{72} +8.09017 q^{73} -3.09017 q^{74} -1.23607 q^{76} -5.52786 q^{77} -5.61803 q^{78} +1.00000 q^{81} +3.61803 q^{82} -12.4721 q^{83} +4.47214 q^{84} +7.70820 q^{86} -6.61803 q^{87} -3.70820 q^{88} +5.38197 q^{89} -25.1246 q^{91} +4.47214 q^{92} -2.76393 q^{93} -0.763932 q^{94} +5.00000 q^{96} -2.14590 q^{97} -13.0000 q^{98} -1.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 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 $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ −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
$$11$$ −1.23607 −0.372689 −0.186344 0.982485i $$-0.559664\pi$$
−0.186344 + 0.982485i $$0.559664\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −5.61803 −1.55816 −0.779081 0.626923i $$-0.784314\pi$$
−0.779081 + 0.626923i $$0.784314\pi$$
$$14$$ −4.47214 −1.19523
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −3.85410 −0.934757 −0.467379 0.884057i $$-0.654801\pi$$
−0.467379 + 0.884057i $$0.654801\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 1.23607 0.283573 0.141787 0.989897i $$-0.454715\pi$$
0.141787 + 0.989897i $$0.454715\pi$$
$$20$$ 0 0
$$21$$ −4.47214 −0.975900
$$22$$ 1.23607 0.263531
$$23$$ −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
$$26$$ 5.61803 1.10179
$$27$$ −1.00000 −0.192450
$$28$$ −4.47214 −0.845154
$$29$$ 6.61803 1.22894 0.614469 0.788941i $$-0.289370\pi$$
0.614469 + 0.788941i $$0.289370\pi$$
$$30$$ 0 0
$$31$$ 2.76393 0.496417 0.248208 0.968707i $$-0.420158\pi$$
0.248208 + 0.968707i $$0.420158\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 1.23607 0.215172
$$34$$ 3.85410 0.660973
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 3.09017 0.508021 0.254010 0.967201i $$-0.418250\pi$$
0.254010 + 0.967201i $$0.418250\pi$$
$$38$$ −1.23607 −0.200517
$$39$$ 5.61803 0.899605
$$40$$ 0 0
$$41$$ −3.61803 −0.565042 −0.282521 0.959261i $$-0.591171\pi$$
−0.282521 + 0.959261i $$0.591171\pi$$
$$42$$ 4.47214 0.690066
$$43$$ −7.70820 −1.17549 −0.587745 0.809046i $$-0.699984\pi$$
−0.587745 + 0.809046i $$0.699984\pi$$
$$44$$ 1.23607 0.186344
$$45$$ 0 0
$$46$$ 4.47214 0.659380
$$47$$ 0.763932 0.111431 0.0557155 0.998447i $$-0.482256\pi$$
0.0557155 + 0.998447i $$0.482256\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 13.0000 1.85714
$$50$$ 0 0
$$51$$ 3.85410 0.539682
$$52$$ 5.61803 0.779081
$$53$$ −3.61803 −0.496975 −0.248488 0.968635i $$-0.579934\pi$$
−0.248488 + 0.968635i $$0.579934\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 13.4164 1.79284
$$57$$ −1.23607 −0.163721
$$58$$ −6.61803 −0.868990
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 1.61803 0.207168 0.103584 0.994621i $$-0.466969\pi$$
0.103584 + 0.994621i $$0.466969\pi$$
$$62$$ −2.76393 −0.351020
$$63$$ 4.47214 0.563436
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ −1.23607 −0.152149
$$67$$ −0.763932 −0.0933292 −0.0466646 0.998911i $$-0.514859\pi$$
−0.0466646 + 0.998911i $$0.514859\pi$$
$$68$$ 3.85410 0.467379
$$69$$ 4.47214 0.538382
$$70$$ 0 0
$$71$$ −5.23607 −0.621407 −0.310703 0.950507i $$-0.600565\pi$$
−0.310703 + 0.950507i $$0.600565\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 8.09017 0.946883 0.473441 0.880825i $$-0.343012\pi$$
0.473441 + 0.880825i $$0.343012\pi$$
$$74$$ −3.09017 −0.359225
$$75$$ 0 0
$$76$$ −1.23607 −0.141787
$$77$$ −5.52786 −0.629959
$$78$$ −5.61803 −0.636117
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 3.61803 0.399545
$$83$$ −12.4721 −1.36899 −0.684497 0.729015i $$-0.739978\pi$$
−0.684497 + 0.729015i $$0.739978\pi$$
$$84$$ 4.47214 0.487950
$$85$$ 0 0
$$86$$ 7.70820 0.831197
$$87$$ −6.61803 −0.709528
$$88$$ −3.70820 −0.395296
$$89$$ 5.38197 0.570487 0.285244 0.958455i $$-0.407925\pi$$
0.285244 + 0.958455i $$0.407925\pi$$
$$90$$ 0 0
$$91$$ −25.1246 −2.63377
$$92$$ 4.47214 0.466252
$$93$$ −2.76393 −0.286606
$$94$$ −0.763932 −0.0787936
$$95$$ 0 0
$$96$$ 5.00000 0.510310
$$97$$ −2.14590 −0.217883 −0.108941 0.994048i $$-0.534746\pi$$
−0.108941 + 0.994048i $$0.534746\pi$$
$$98$$ −13.0000 −1.31320
$$99$$ −1.23607 −0.124230
$$100$$ 0 0
$$101$$ 3.56231 0.354463 0.177231 0.984169i $$-0.443286\pi$$
0.177231 + 0.984169i $$0.443286\pi$$
$$102$$ −3.85410 −0.381613
$$103$$ −3.23607 −0.318859 −0.159430 0.987209i $$-0.550966\pi$$
−0.159430 + 0.987209i $$0.550966\pi$$
$$104$$ −16.8541 −1.65268
$$105$$ 0 0
$$106$$ 3.61803 0.351415
$$107$$ −7.52786 −0.727746 −0.363873 0.931449i $$-0.618546\pi$$
−0.363873 + 0.931449i $$0.618546\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −7.90983 −0.757624 −0.378812 0.925474i $$-0.623667\pi$$
−0.378812 + 0.925474i $$0.623667\pi$$
$$110$$ 0 0
$$111$$ −3.09017 −0.293306
$$112$$ −4.47214 −0.422577
$$113$$ −18.3262 −1.72399 −0.861994 0.506919i $$-0.830784\pi$$
−0.861994 + 0.506919i $$0.830784\pi$$
$$114$$ 1.23607 0.115768
$$115$$ 0 0
$$116$$ −6.61803 −0.614469
$$117$$ −5.61803 −0.519387
$$118$$ 4.00000 0.368230
$$119$$ −17.2361 −1.58003
$$120$$ 0 0
$$121$$ −9.47214 −0.861103
$$122$$ −1.61803 −0.146490
$$123$$ 3.61803 0.326227
$$124$$ −2.76393 −0.248208
$$125$$ 0 0
$$126$$ −4.47214 −0.398410
$$127$$ 3.70820 0.329050 0.164525 0.986373i $$-0.447391\pi$$
0.164525 + 0.986373i $$0.447391\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 7.70820 0.678670
$$130$$ 0 0
$$131$$ −8.18034 −0.714720 −0.357360 0.933967i $$-0.616323\pi$$
−0.357360 + 0.933967i $$0.616323\pi$$
$$132$$ −1.23607 −0.107586
$$133$$ 5.52786 0.479327
$$134$$ 0.763932 0.0659937
$$135$$ 0 0
$$136$$ −11.5623 −0.991460
$$137$$ 7.61803 0.650853 0.325426 0.945567i $$-0.394492\pi$$
0.325426 + 0.945567i $$0.394492\pi$$
$$138$$ −4.47214 −0.380693
$$139$$ −22.9443 −1.94611 −0.973054 0.230578i $$-0.925938\pi$$
−0.973054 + 0.230578i $$0.925938\pi$$
$$140$$ 0 0
$$141$$ −0.763932 −0.0643347
$$142$$ 5.23607 0.439401
$$143$$ 6.94427 0.580709
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −8.09017 −0.669547
$$147$$ −13.0000 −1.07222
$$148$$ −3.09017 −0.254010
$$149$$ −18.8541 −1.54459 −0.772294 0.635265i $$-0.780891\pi$$
−0.772294 + 0.635265i $$0.780891\pi$$
$$150$$ 0 0
$$151$$ 7.52786 0.612609 0.306304 0.951934i $$-0.400907\pi$$
0.306304 + 0.951934i $$0.400907\pi$$
$$152$$ 3.70820 0.300775
$$153$$ −3.85410 −0.311586
$$154$$ 5.52786 0.445448
$$155$$ 0 0
$$156$$ −5.61803 −0.449803
$$157$$ −10.7984 −0.861804 −0.430902 0.902399i $$-0.641805\pi$$
−0.430902 + 0.902399i $$0.641805\pi$$
$$158$$ 0 0
$$159$$ 3.61803 0.286929
$$160$$ 0 0
$$161$$ −20.0000 −1.57622
$$162$$ −1.00000 −0.0785674
$$163$$ 14.9443 1.17053 0.585263 0.810844i $$-0.300991\pi$$
0.585263 + 0.810844i $$0.300991\pi$$
$$164$$ 3.61803 0.282521
$$165$$ 0 0
$$166$$ 12.4721 0.968025
$$167$$ −3.41641 −0.264370 −0.132185 0.991225i $$-0.542199\pi$$
−0.132185 + 0.991225i $$0.542199\pi$$
$$168$$ −13.4164 −1.03510
$$169$$ 18.5623 1.42787
$$170$$ 0 0
$$171$$ 1.23607 0.0945245
$$172$$ 7.70820 0.587745
$$173$$ −21.0902 −1.60346 −0.801728 0.597689i $$-0.796086\pi$$
−0.801728 + 0.597689i $$0.796086\pi$$
$$174$$ 6.61803 0.501712
$$175$$ 0 0
$$176$$ 1.23607 0.0931721
$$177$$ 4.00000 0.300658
$$178$$ −5.38197 −0.403395
$$179$$ −16.1803 −1.20938 −0.604688 0.796463i $$-0.706702\pi$$
−0.604688 + 0.796463i $$0.706702\pi$$
$$180$$ 0 0
$$181$$ 14.7984 1.09995 0.549977 0.835180i $$-0.314636\pi$$
0.549977 + 0.835180i $$0.314636\pi$$
$$182$$ 25.1246 1.86236
$$183$$ −1.61803 −0.119609
$$184$$ −13.4164 −0.989071
$$185$$ 0 0
$$186$$ 2.76393 0.202661
$$187$$ 4.76393 0.348373
$$188$$ −0.763932 −0.0557155
$$189$$ −4.47214 −0.325300
$$190$$ 0 0
$$191$$ 6.65248 0.481356 0.240678 0.970605i $$-0.422630\pi$$
0.240678 + 0.970605i $$0.422630\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ −23.3262 −1.67906 −0.839530 0.543314i $$-0.817169\pi$$
−0.839530 + 0.543314i $$0.817169\pi$$
$$194$$ 2.14590 0.154067
$$195$$ 0 0
$$196$$ −13.0000 −0.928571
$$197$$ −7.61803 −0.542762 −0.271381 0.962472i $$-0.587480\pi$$
−0.271381 + 0.962472i $$0.587480\pi$$
$$198$$ 1.23607 0.0878435
$$199$$ 12.6525 0.896910 0.448455 0.893805i $$-0.351974\pi$$
0.448455 + 0.893805i $$0.351974\pi$$
$$200$$ 0 0
$$201$$ 0.763932 0.0538836
$$202$$ −3.56231 −0.250643
$$203$$ 29.5967 2.07728
$$204$$ −3.85410 −0.269841
$$205$$ 0 0
$$206$$ 3.23607 0.225468
$$207$$ −4.47214 −0.310835
$$208$$ 5.61803 0.389541
$$209$$ −1.52786 −0.105685
$$210$$ 0 0
$$211$$ 17.8885 1.23150 0.615749 0.787942i $$-0.288854\pi$$
0.615749 + 0.787942i $$0.288854\pi$$
$$212$$ 3.61803 0.248488
$$213$$ 5.23607 0.358769
$$214$$ 7.52786 0.514594
$$215$$ 0 0
$$216$$ −3.00000 −0.204124
$$217$$ 12.3607 0.839098
$$218$$ 7.90983 0.535721
$$219$$ −8.09017 −0.546683
$$220$$ 0 0
$$221$$ 21.6525 1.45650
$$222$$ 3.09017 0.207399
$$223$$ −8.18034 −0.547796 −0.273898 0.961759i $$-0.588313\pi$$
−0.273898 + 0.961759i $$0.588313\pi$$
$$224$$ −22.3607 −1.49404
$$225$$ 0 0
$$226$$ 18.3262 1.21904
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 1.23607 0.0818606
$$229$$ 21.9787 1.45239 0.726197 0.687487i $$-0.241286\pi$$
0.726197 + 0.687487i $$0.241286\pi$$
$$230$$ 0 0
$$231$$ 5.52786 0.363707
$$232$$ 19.8541 1.30349
$$233$$ −12.3820 −0.811170 −0.405585 0.914057i $$-0.632932\pi$$
−0.405585 + 0.914057i $$0.632932\pi$$
$$234$$ 5.61803 0.367262
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 17.2361 1.11725
$$239$$ −24.9443 −1.61351 −0.806755 0.590886i $$-0.798778\pi$$
−0.806755 + 0.590886i $$0.798778\pi$$
$$240$$ 0 0
$$241$$ 28.0344 1.80586 0.902929 0.429791i $$-0.141413\pi$$
0.902929 + 0.429791i $$0.141413\pi$$
$$242$$ 9.47214 0.608892
$$243$$ −1.00000 −0.0641500
$$244$$ −1.61803 −0.103584
$$245$$ 0 0
$$246$$ −3.61803 −0.230677
$$247$$ −6.94427 −0.441853
$$248$$ 8.29180 0.526530
$$249$$ 12.4721 0.790390
$$250$$ 0 0
$$251$$ −9.05573 −0.571592 −0.285796 0.958290i $$-0.592258\pi$$
−0.285796 + 0.958290i $$0.592258\pi$$
$$252$$ −4.47214 −0.281718
$$253$$ 5.52786 0.347534
$$254$$ −3.70820 −0.232673
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −11.7984 −0.735962 −0.367981 0.929833i $$-0.619951\pi$$
−0.367981 + 0.929833i $$0.619951\pi$$
$$258$$ −7.70820 −0.479892
$$259$$ 13.8197 0.858712
$$260$$ 0 0
$$261$$ 6.61803 0.409646
$$262$$ 8.18034 0.505383
$$263$$ 23.8885 1.47303 0.736515 0.676421i $$-0.236470\pi$$
0.736515 + 0.676421i $$0.236470\pi$$
$$264$$ 3.70820 0.228224
$$265$$ 0 0
$$266$$ −5.52786 −0.338935
$$267$$ −5.38197 −0.329371
$$268$$ 0.763932 0.0466646
$$269$$ −6.85410 −0.417902 −0.208951 0.977926i $$-0.567005\pi$$
−0.208951 + 0.977926i $$0.567005\pi$$
$$270$$ 0 0
$$271$$ −0.180340 −0.0109549 −0.00547743 0.999985i $$-0.501744\pi$$
−0.00547743 + 0.999985i $$0.501744\pi$$
$$272$$ 3.85410 0.233689
$$273$$ 25.1246 1.52061
$$274$$ −7.61803 −0.460222
$$275$$ 0 0
$$276$$ −4.47214 −0.269191
$$277$$ −18.7984 −1.12948 −0.564742 0.825267i $$-0.691024\pi$$
−0.564742 + 0.825267i $$0.691024\pi$$
$$278$$ 22.9443 1.37611
$$279$$ 2.76393 0.165472
$$280$$ 0 0
$$281$$ −19.6180 −1.17031 −0.585157 0.810920i $$-0.698967\pi$$
−0.585157 + 0.810920i $$0.698967\pi$$
$$282$$ 0.763932 0.0454915
$$283$$ −13.7082 −0.814868 −0.407434 0.913235i $$-0.633576\pi$$
−0.407434 + 0.913235i $$0.633576\pi$$
$$284$$ 5.23607 0.310703
$$285$$ 0 0
$$286$$ −6.94427 −0.410623
$$287$$ −16.1803 −0.955095
$$288$$ −5.00000 −0.294628
$$289$$ −2.14590 −0.126229
$$290$$ 0 0
$$291$$ 2.14590 0.125795
$$292$$ −8.09017 −0.473441
$$293$$ 20.7984 1.21505 0.607527 0.794299i $$-0.292162\pi$$
0.607527 + 0.794299i $$0.292162\pi$$
$$294$$ 13.0000 0.758175
$$295$$ 0 0
$$296$$ 9.27051 0.538837
$$297$$ 1.23607 0.0717239
$$298$$ 18.8541 1.09219
$$299$$ 25.1246 1.45299
$$300$$ 0 0
$$301$$ −34.4721 −1.98694
$$302$$ −7.52786 −0.433180
$$303$$ −3.56231 −0.204649
$$304$$ −1.23607 −0.0708934
$$305$$ 0 0
$$306$$ 3.85410 0.220324
$$307$$ −32.6525 −1.86358 −0.931788 0.363004i $$-0.881751\pi$$
−0.931788 + 0.363004i $$0.881751\pi$$
$$308$$ 5.52786 0.314979
$$309$$ 3.23607 0.184093
$$310$$ 0 0
$$311$$ 17.7082 1.00414 0.502070 0.864827i $$-0.332572\pi$$
0.502070 + 0.864827i $$0.332572\pi$$
$$312$$ 16.8541 0.954176
$$313$$ 0.472136 0.0266867 0.0133434 0.999911i $$-0.495753\pi$$
0.0133434 + 0.999911i $$0.495753\pi$$
$$314$$ 10.7984 0.609387
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −28.8328 −1.61941 −0.809706 0.586836i $$-0.800373\pi$$
−0.809706 + 0.586836i $$0.800373\pi$$
$$318$$ −3.61803 −0.202889
$$319$$ −8.18034 −0.458011
$$320$$ 0 0
$$321$$ 7.52786 0.420164
$$322$$ 20.0000 1.11456
$$323$$ −4.76393 −0.265072
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −14.9443 −0.827687
$$327$$ 7.90983 0.437415
$$328$$ −10.8541 −0.599318
$$329$$ 3.41641 0.188353
$$330$$ 0 0
$$331$$ −6.94427 −0.381692 −0.190846 0.981620i $$-0.561123\pi$$
−0.190846 + 0.981620i $$0.561123\pi$$
$$332$$ 12.4721 0.684497
$$333$$ 3.09017 0.169340
$$334$$ 3.41641 0.186938
$$335$$ 0 0
$$336$$ 4.47214 0.243975
$$337$$ −2.94427 −0.160385 −0.0801924 0.996779i $$-0.525553\pi$$
−0.0801924 + 0.996779i $$0.525553\pi$$
$$338$$ −18.5623 −1.00966
$$339$$ 18.3262 0.995345
$$340$$ 0 0
$$341$$ −3.41641 −0.185009
$$342$$ −1.23607 −0.0668389
$$343$$ 26.8328 1.44884
$$344$$ −23.1246 −1.24680
$$345$$ 0 0
$$346$$ 21.0902 1.13381
$$347$$ 20.2918 1.08932 0.544660 0.838657i $$-0.316659\pi$$
0.544660 + 0.838657i $$0.316659\pi$$
$$348$$ 6.61803 0.354764
$$349$$ −4.03444 −0.215959 −0.107979 0.994153i $$-0.534438\pi$$
−0.107979 + 0.994153i $$0.534438\pi$$
$$350$$ 0 0
$$351$$ 5.61803 0.299868
$$352$$ 6.18034 0.329413
$$353$$ −14.3607 −0.764342 −0.382171 0.924092i $$-0.624823\pi$$
−0.382171 + 0.924092i $$0.624823\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ −5.38197 −0.285244
$$357$$ 17.2361 0.912229
$$358$$ 16.1803 0.855158
$$359$$ −37.4164 −1.97476 −0.987381 0.158361i $$-0.949379\pi$$
−0.987381 + 0.158361i $$0.949379\pi$$
$$360$$ 0 0
$$361$$ −17.4721 −0.919586
$$362$$ −14.7984 −0.777785
$$363$$ 9.47214 0.497158
$$364$$ 25.1246 1.31689
$$365$$ 0 0
$$366$$ 1.61803 0.0845760
$$367$$ 6.00000 0.313197 0.156599 0.987662i $$-0.449947\pi$$
0.156599 + 0.987662i $$0.449947\pi$$
$$368$$ 4.47214 0.233126
$$369$$ −3.61803 −0.188347
$$370$$ 0 0
$$371$$ −16.1803 −0.840041
$$372$$ 2.76393 0.143303
$$373$$ −2.58359 −0.133773 −0.0668867 0.997761i $$-0.521307\pi$$
−0.0668867 + 0.997761i $$0.521307\pi$$
$$374$$ −4.76393 −0.246337
$$375$$ 0 0
$$376$$ 2.29180 0.118190
$$377$$ −37.1803 −1.91488
$$378$$ 4.47214 0.230022
$$379$$ −3.41641 −0.175489 −0.0877445 0.996143i $$-0.527966\pi$$
−0.0877445 + 0.996143i $$0.527966\pi$$
$$380$$ 0 0
$$381$$ −3.70820 −0.189977
$$382$$ −6.65248 −0.340370
$$383$$ 32.8328 1.67768 0.838839 0.544379i $$-0.183235\pi$$
0.838839 + 0.544379i $$0.183235\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ 23.3262 1.18727
$$387$$ −7.70820 −0.391830
$$388$$ 2.14590 0.108941
$$389$$ −25.9098 −1.31368 −0.656840 0.754030i $$-0.728107\pi$$
−0.656840 + 0.754030i $$0.728107\pi$$
$$390$$ 0 0
$$391$$ 17.2361 0.871665
$$392$$ 39.0000 1.96980
$$393$$ 8.18034 0.412644
$$394$$ 7.61803 0.383791
$$395$$ 0 0
$$396$$ 1.23607 0.0621148
$$397$$ 6.94427 0.348523 0.174262 0.984699i $$-0.444246\pi$$
0.174262 + 0.984699i $$0.444246\pi$$
$$398$$ −12.6525 −0.634211
$$399$$ −5.52786 −0.276739
$$400$$ 0 0
$$401$$ −22.4508 −1.12114 −0.560571 0.828106i $$-0.689418\pi$$
−0.560571 + 0.828106i $$0.689418\pi$$
$$402$$ −0.763932 −0.0381015
$$403$$ −15.5279 −0.773498
$$404$$ −3.56231 −0.177231
$$405$$ 0 0
$$406$$ −29.5967 −1.46886
$$407$$ −3.81966 −0.189334
$$408$$ 11.5623 0.572419
$$409$$ 1.20163 0.0594166 0.0297083 0.999559i $$-0.490542\pi$$
0.0297083 + 0.999559i $$0.490542\pi$$
$$410$$ 0 0
$$411$$ −7.61803 −0.375770
$$412$$ 3.23607 0.159430
$$413$$ −17.8885 −0.880238
$$414$$ 4.47214 0.219793
$$415$$ 0 0
$$416$$ 28.0902 1.37723
$$417$$ 22.9443 1.12359
$$418$$ 1.52786 0.0747303
$$419$$ 15.0557 0.735520 0.367760 0.929921i $$-0.380125\pi$$
0.367760 + 0.929921i $$0.380125\pi$$
$$420$$ 0 0
$$421$$ 29.8541 1.45500 0.727500 0.686108i $$-0.240682\pi$$
0.727500 + 0.686108i $$0.240682\pi$$
$$422$$ −17.8885 −0.870801
$$423$$ 0.763932 0.0371436
$$424$$ −10.8541 −0.527122
$$425$$ 0 0
$$426$$ −5.23607 −0.253688
$$427$$ 7.23607 0.350178
$$428$$ 7.52786 0.363873
$$429$$ −6.94427 −0.335273
$$430$$ 0 0
$$431$$ 20.6525 0.994795 0.497397 0.867523i $$-0.334289\pi$$
0.497397 + 0.867523i $$0.334289\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 21.9787 1.05623 0.528115 0.849173i $$-0.322899\pi$$
0.528115 + 0.849173i $$0.322899\pi$$
$$434$$ −12.3607 −0.593332
$$435$$ 0 0
$$436$$ 7.90983 0.378812
$$437$$ −5.52786 −0.264434
$$438$$ 8.09017 0.386563
$$439$$ 16.1803 0.772245 0.386123 0.922447i $$-0.373814\pi$$
0.386123 + 0.922447i $$0.373814\pi$$
$$440$$ 0 0
$$441$$ 13.0000 0.619048
$$442$$ −21.6525 −1.02990
$$443$$ 35.2361 1.67412 0.837058 0.547114i $$-0.184274\pi$$
0.837058 + 0.547114i $$0.184274\pi$$
$$444$$ 3.09017 0.146653
$$445$$ 0 0
$$446$$ 8.18034 0.387350
$$447$$ 18.8541 0.891768
$$448$$ 31.3050 1.47902
$$449$$ 16.7984 0.792764 0.396382 0.918086i $$-0.370266\pi$$
0.396382 + 0.918086i $$0.370266\pi$$
$$450$$ 0 0
$$451$$ 4.47214 0.210585
$$452$$ 18.3262 0.861994
$$453$$ −7.52786 −0.353690
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ −3.70820 −0.173653
$$457$$ 22.3607 1.04599 0.522994 0.852336i $$-0.324815\pi$$
0.522994 + 0.852336i $$0.324815\pi$$
$$458$$ −21.9787 −1.02700
$$459$$ 3.85410 0.179894
$$460$$ 0 0
$$461$$ −32.7984 −1.52757 −0.763786 0.645469i $$-0.776662\pi$$
−0.763786 + 0.645469i $$0.776662\pi$$
$$462$$ −5.52786 −0.257180
$$463$$ −18.7639 −0.872034 −0.436017 0.899938i $$-0.643611\pi$$
−0.436017 + 0.899938i $$0.643611\pi$$
$$464$$ −6.61803 −0.307235
$$465$$ 0 0
$$466$$ 12.3820 0.573583
$$467$$ −28.1803 −1.30403 −0.652015 0.758206i $$-0.726076\pi$$
−0.652015 + 0.758206i $$0.726076\pi$$
$$468$$ 5.61803 0.259694
$$469$$ −3.41641 −0.157755
$$470$$ 0 0
$$471$$ 10.7984 0.497563
$$472$$ −12.0000 −0.552345
$$473$$ 9.52786 0.438092
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 17.2361 0.790014
$$477$$ −3.61803 −0.165658
$$478$$ 24.9443 1.14092
$$479$$ 18.9443 0.865586 0.432793 0.901493i $$-0.357528\pi$$
0.432793 + 0.901493i $$0.357528\pi$$
$$480$$ 0 0
$$481$$ −17.3607 −0.791579
$$482$$ −28.0344 −1.27693
$$483$$ 20.0000 0.910032
$$484$$ 9.47214 0.430552
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −9.70820 −0.439921 −0.219960 0.975509i $$-0.570593\pi$$
−0.219960 + 0.975509i $$0.570593\pi$$
$$488$$ 4.85410 0.219735
$$489$$ −14.9443 −0.675803
$$490$$ 0 0
$$491$$ 5.88854 0.265746 0.132873 0.991133i $$-0.457580\pi$$
0.132873 + 0.991133i $$0.457580\pi$$
$$492$$ −3.61803 −0.163114
$$493$$ −25.5066 −1.14876
$$494$$ 6.94427 0.312438
$$495$$ 0 0
$$496$$ −2.76393 −0.124104
$$497$$ −23.4164 −1.05037
$$498$$ −12.4721 −0.558890
$$499$$ −6.00000 −0.268597 −0.134298 0.990941i $$-0.542878\pi$$
−0.134298 + 0.990941i $$0.542878\pi$$
$$500$$ 0 0
$$501$$ 3.41641 0.152634
$$502$$ 9.05573 0.404177
$$503$$ −8.58359 −0.382723 −0.191362 0.981520i $$-0.561290\pi$$
−0.191362 + 0.981520i $$0.561290\pi$$
$$504$$ 13.4164 0.597614
$$505$$ 0 0
$$506$$ −5.52786 −0.245744
$$507$$ −18.5623 −0.824381
$$508$$ −3.70820 −0.164525
$$509$$ 20.1459 0.892951 0.446476 0.894796i $$-0.352679\pi$$
0.446476 + 0.894796i $$0.352679\pi$$
$$510$$ 0 0
$$511$$ 36.1803 1.60052
$$512$$ 11.0000 0.486136
$$513$$ −1.23607 −0.0545737
$$514$$ 11.7984 0.520404
$$515$$ 0 0
$$516$$ −7.70820 −0.339335
$$517$$ −0.944272 −0.0415290
$$518$$ −13.8197 −0.607201
$$519$$ 21.0902 0.925756
$$520$$ 0 0
$$521$$ −27.0344 −1.18440 −0.592200 0.805791i $$-0.701741\pi$$
−0.592200 + 0.805791i $$0.701741\pi$$
$$522$$ −6.61803 −0.289663
$$523$$ 25.7082 1.12414 0.562071 0.827089i $$-0.310005\pi$$
0.562071 + 0.827089i $$0.310005\pi$$
$$524$$ 8.18034 0.357360
$$525$$ 0 0
$$526$$ −23.8885 −1.04159
$$527$$ −10.6525 −0.464029
$$528$$ −1.23607 −0.0537930
$$529$$ −3.00000 −0.130435
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ −5.52786 −0.239663
$$533$$ 20.3262 0.880427
$$534$$ 5.38197 0.232900
$$535$$ 0 0
$$536$$ −2.29180 −0.0989905
$$537$$ 16.1803 0.698233
$$538$$ 6.85410 0.295501
$$539$$ −16.0689 −0.692136
$$540$$ 0 0
$$541$$ −19.6738 −0.845841 −0.422921 0.906167i $$-0.638995\pi$$
−0.422921 + 0.906167i $$0.638995\pi$$
$$542$$ 0.180340 0.00774626
$$543$$ −14.7984 −0.635059
$$544$$ 19.2705 0.826216
$$545$$ 0 0
$$546$$ −25.1246 −1.07523
$$547$$ −42.3607 −1.81121 −0.905606 0.424120i $$-0.860583\pi$$
−0.905606 + 0.424120i $$0.860583\pi$$
$$548$$ −7.61803 −0.325426
$$549$$ 1.61803 0.0690560
$$550$$ 0 0
$$551$$ 8.18034 0.348494
$$552$$ 13.4164 0.571040
$$553$$ 0 0
$$554$$ 18.7984 0.798666
$$555$$ 0 0
$$556$$ 22.9443 0.973054
$$557$$ 11.2705 0.477547 0.238773 0.971075i $$-0.423255\pi$$
0.238773 + 0.971075i $$0.423255\pi$$
$$558$$ −2.76393 −0.117007
$$559$$ 43.3050 1.83160
$$560$$ 0 0
$$561$$ −4.76393 −0.201133
$$562$$ 19.6180 0.827537
$$563$$ 26.4721 1.11567 0.557834 0.829953i $$-0.311633\pi$$
0.557834 + 0.829953i $$0.311633\pi$$
$$564$$ 0.763932 0.0321673
$$565$$ 0 0
$$566$$ 13.7082 0.576199
$$567$$ 4.47214 0.187812
$$568$$ −15.7082 −0.659102
$$569$$ 3.27051 0.137107 0.0685535 0.997647i $$-0.478162\pi$$
0.0685535 + 0.997647i $$0.478162\pi$$
$$570$$ 0 0
$$571$$ −16.7639 −0.701549 −0.350774 0.936460i $$-0.614082\pi$$
−0.350774 + 0.936460i $$0.614082\pi$$
$$572$$ −6.94427 −0.290355
$$573$$ −6.65248 −0.277911
$$574$$ 16.1803 0.675354
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 13.0557 0.543517 0.271759 0.962365i $$-0.412395\pi$$
0.271759 + 0.962365i $$0.412395\pi$$
$$578$$ 2.14590 0.0892576
$$579$$ 23.3262 0.969405
$$580$$ 0 0
$$581$$ −55.7771 −2.31402
$$582$$ −2.14590 −0.0889503
$$583$$ 4.47214 0.185217
$$584$$ 24.2705 1.00432
$$585$$ 0 0
$$586$$ −20.7984 −0.859173
$$587$$ 41.4164 1.70944 0.854719 0.519091i $$-0.173729\pi$$
0.854719 + 0.519091i $$0.173729\pi$$
$$588$$ 13.0000 0.536111
$$589$$ 3.41641 0.140771
$$590$$ 0 0
$$591$$ 7.61803 0.313364
$$592$$ −3.09017 −0.127005
$$593$$ −9.74265 −0.400083 −0.200041 0.979787i $$-0.564108\pi$$
−0.200041 + 0.979787i $$0.564108\pi$$
$$594$$ −1.23607 −0.0507165
$$595$$ 0 0
$$596$$ 18.8541 0.772294
$$597$$ −12.6525 −0.517831
$$598$$ −25.1246 −1.02742
$$599$$ −3.52786 −0.144145 −0.0720723 0.997399i $$-0.522961\pi$$
−0.0720723 + 0.997399i $$0.522961\pi$$
$$600$$ 0 0
$$601$$ −8.67376 −0.353810 −0.176905 0.984228i $$-0.556609\pi$$
−0.176905 + 0.984228i $$0.556609\pi$$
$$602$$ 34.4721 1.40498
$$603$$ −0.763932 −0.0311097
$$604$$ −7.52786 −0.306304
$$605$$ 0 0
$$606$$ 3.56231 0.144709
$$607$$ 34.7639 1.41102 0.705512 0.708698i $$-0.250717\pi$$
0.705512 + 0.708698i $$0.250717\pi$$
$$608$$ −6.18034 −0.250646
$$609$$ −29.5967 −1.19932
$$610$$ 0 0
$$611$$ −4.29180 −0.173627
$$612$$ 3.85410 0.155793
$$613$$ −1.49342 −0.0603188 −0.0301594 0.999545i $$-0.509601\pi$$
−0.0301594 + 0.999545i $$0.509601\pi$$
$$614$$ 32.6525 1.31775
$$615$$ 0 0
$$616$$ −16.5836 −0.668172
$$617$$ 19.9787 0.804313 0.402156 0.915571i $$-0.368261\pi$$
0.402156 + 0.915571i $$0.368261\pi$$
$$618$$ −3.23607 −0.130174
$$619$$ −16.3607 −0.657591 −0.328796 0.944401i $$-0.606643\pi$$
−0.328796 + 0.944401i $$0.606643\pi$$
$$620$$ 0 0
$$621$$ 4.47214 0.179461
$$622$$ −17.7082 −0.710034
$$623$$ 24.0689 0.964299
$$624$$ −5.61803 −0.224901
$$625$$ 0 0
$$626$$ −0.472136 −0.0188703
$$627$$ 1.52786 0.0610170
$$628$$ 10.7984 0.430902
$$629$$ −11.9098 −0.474876
$$630$$ 0 0
$$631$$ 23.7082 0.943809 0.471904 0.881650i $$-0.343567\pi$$
0.471904 + 0.881650i $$0.343567\pi$$
$$632$$ 0 0
$$633$$ −17.8885 −0.711006
$$634$$ 28.8328 1.14510
$$635$$ 0 0
$$636$$ −3.61803 −0.143464
$$637$$ −73.0344 −2.89373
$$638$$ 8.18034 0.323863
$$639$$ −5.23607 −0.207136
$$640$$ 0 0
$$641$$ 21.0557 0.831651 0.415826 0.909444i $$-0.363493\pi$$
0.415826 + 0.909444i $$0.363493\pi$$
$$642$$ −7.52786 −0.297101
$$643$$ 21.8885 0.863200 0.431600 0.902065i $$-0.357949\pi$$
0.431600 + 0.902065i $$0.357949\pi$$
$$644$$ 20.0000 0.788110
$$645$$ 0 0
$$646$$ 4.76393 0.187434
$$647$$ 34.3607 1.35086 0.675429 0.737425i $$-0.263959\pi$$
0.675429 + 0.737425i $$0.263959\pi$$
$$648$$ 3.00000 0.117851
$$649$$ 4.94427 0.194080
$$650$$ 0 0
$$651$$ −12.3607 −0.484453
$$652$$ −14.9443 −0.585263
$$653$$ 42.1033 1.64763 0.823815 0.566858i $$-0.191841\pi$$
0.823815 + 0.566858i $$0.191841\pi$$
$$654$$ −7.90983 −0.309299
$$655$$ 0 0
$$656$$ 3.61803 0.141260
$$657$$ 8.09017 0.315628
$$658$$ −3.41641 −0.133185
$$659$$ −33.4164 −1.30172 −0.650859 0.759198i $$-0.725591\pi$$
−0.650859 + 0.759198i $$0.725591\pi$$
$$660$$ 0 0
$$661$$ −1.41641 −0.0550919 −0.0275459 0.999621i $$-0.508769\pi$$
−0.0275459 + 0.999621i $$0.508769\pi$$
$$662$$ 6.94427 0.269897
$$663$$ −21.6525 −0.840912
$$664$$ −37.4164 −1.45204
$$665$$ 0 0
$$666$$ −3.09017 −0.119742
$$667$$ −29.5967 −1.14599
$$668$$ 3.41641 0.132185
$$669$$ 8.18034 0.316270
$$670$$ 0 0
$$671$$ −2.00000 −0.0772091
$$672$$ 22.3607 0.862582
$$673$$ −5.20163 −0.200508 −0.100254 0.994962i $$-0.531966\pi$$
−0.100254 + 0.994962i $$0.531966\pi$$
$$674$$ 2.94427 0.113409
$$675$$ 0 0
$$676$$ −18.5623 −0.713935
$$677$$ −10.5836 −0.406760 −0.203380 0.979100i $$-0.565193\pi$$
−0.203380 + 0.979100i $$0.565193\pi$$
$$678$$ −18.3262 −0.703815
$$679$$ −9.59675 −0.368289
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 3.41641 0.130821
$$683$$ 19.4164 0.742948 0.371474 0.928443i $$-0.378852\pi$$
0.371474 + 0.928443i $$0.378852\pi$$
$$684$$ −1.23607 −0.0472622
$$685$$ 0 0
$$686$$ −26.8328 −1.02448
$$687$$ −21.9787 −0.838540
$$688$$ 7.70820 0.293873
$$689$$ 20.3262 0.774368
$$690$$ 0 0
$$691$$ −16.7639 −0.637730 −0.318865 0.947800i $$-0.603302\pi$$
−0.318865 + 0.947800i $$0.603302\pi$$
$$692$$ 21.0902 0.801728
$$693$$ −5.52786 −0.209986
$$694$$ −20.2918 −0.770266
$$695$$ 0 0
$$696$$ −19.8541 −0.752568
$$697$$ 13.9443 0.528177
$$698$$ 4.03444 0.152706
$$699$$ 12.3820 0.468329
$$700$$ 0 0
$$701$$ 20.5623 0.776628 0.388314 0.921527i $$-0.373058\pi$$
0.388314 + 0.921527i $$0.373058\pi$$
$$702$$ −5.61803 −0.212039
$$703$$ 3.81966 0.144061
$$704$$ −8.65248 −0.326102
$$705$$ 0 0
$$706$$ 14.3607 0.540471
$$707$$ 15.9311 0.599151
$$708$$ −4.00000 −0.150329
$$709$$ −40.2705 −1.51239 −0.756195 0.654346i $$-0.772944\pi$$
−0.756195 + 0.654346i $$0.772944\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 16.1459 0.605093
$$713$$ −12.3607 −0.462911
$$714$$ −17.2361 −0.645044
$$715$$ 0 0
$$716$$ 16.1803 0.604688
$$717$$ 24.9443 0.931561
$$718$$ 37.4164 1.39637
$$719$$ −0.111456 −0.00415661 −0.00207831 0.999998i $$-0.500662\pi$$
−0.00207831 + 0.999998i $$0.500662\pi$$
$$720$$ 0 0
$$721$$ −14.4721 −0.538971
$$722$$ 17.4721 0.650246
$$723$$ −28.0344 −1.04261
$$724$$ −14.7984 −0.549977
$$725$$ 0 0
$$726$$ −9.47214 −0.351544
$$727$$ −46.6525 −1.73024 −0.865122 0.501561i $$-0.832759\pi$$
−0.865122 + 0.501561i $$0.832759\pi$$
$$728$$ −75.3738 −2.79354
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 29.7082 1.09880
$$732$$ 1.61803 0.0598043
$$733$$ −29.4164 −1.08652 −0.543260 0.839565i $$-0.682810\pi$$
−0.543260 + 0.839565i $$0.682810\pi$$
$$734$$ −6.00000 −0.221464
$$735$$ 0 0
$$736$$ 22.3607 0.824226
$$737$$ 0.944272 0.0347827
$$738$$ 3.61803 0.133182
$$739$$ 4.29180 0.157876 0.0789381 0.996880i $$-0.474847\pi$$
0.0789381 + 0.996880i $$0.474847\pi$$
$$740$$ 0 0
$$741$$ 6.94427 0.255104
$$742$$ 16.1803 0.593999
$$743$$ −41.1246 −1.50872 −0.754358 0.656463i $$-0.772052\pi$$
−0.754358 + 0.656463i $$0.772052\pi$$
$$744$$ −8.29180 −0.303992
$$745$$ 0 0
$$746$$ 2.58359 0.0945920
$$747$$ −12.4721 −0.456332
$$748$$ −4.76393 −0.174187
$$749$$ −33.6656 −1.23012
$$750$$ 0 0
$$751$$ 36.6525 1.33747 0.668734 0.743502i $$-0.266837\pi$$
0.668734 + 0.743502i $$0.266837\pi$$
$$752$$ −0.763932 −0.0278577
$$753$$ 9.05573 0.330009
$$754$$ 37.1803 1.35403
$$755$$ 0 0
$$756$$ 4.47214 0.162650
$$757$$ −29.6180 −1.07649 −0.538243 0.842790i $$-0.680912\pi$$
−0.538243 + 0.842790i $$0.680912\pi$$
$$758$$ 3.41641 0.124090
$$759$$ −5.52786 −0.200649
$$760$$ 0 0
$$761$$ 32.7426 1.18692 0.593460 0.804863i $$-0.297762\pi$$
0.593460 + 0.804863i $$0.297762\pi$$
$$762$$ 3.70820 0.134334
$$763$$ −35.3738 −1.28062
$$764$$ −6.65248 −0.240678
$$765$$ 0 0
$$766$$ −32.8328 −1.18630
$$767$$ 22.4721 0.811422
$$768$$ 17.0000 0.613435
$$769$$ −50.3607 −1.81605 −0.908026 0.418913i $$-0.862411\pi$$
−0.908026 + 0.418913i $$0.862411\pi$$
$$770$$ 0 0
$$771$$ 11.7984 0.424908
$$772$$ 23.3262 0.839530
$$773$$ −28.9230 −1.04029 −0.520144 0.854079i $$-0.674122\pi$$
−0.520144 + 0.854079i $$0.674122\pi$$
$$774$$ 7.70820 0.277066
$$775$$ 0 0
$$776$$ −6.43769 −0.231100
$$777$$ −13.8197 −0.495778
$$778$$ 25.9098 0.928912
$$779$$ −4.47214 −0.160231
$$780$$ 0 0
$$781$$ 6.47214 0.231591
$$782$$ −17.2361 −0.616361
$$783$$ −6.61803 −0.236509
$$784$$ −13.0000 −0.464286
$$785$$ 0 0
$$786$$ −8.18034 −0.291783
$$787$$ 14.1803 0.505475 0.252737 0.967535i $$-0.418669\pi$$
0.252737 + 0.967535i $$0.418669\pi$$
$$788$$ 7.61803 0.271381
$$789$$ −23.8885 −0.850455
$$790$$ 0 0
$$791$$ −81.9574 −2.91407
$$792$$ −3.70820 −0.131765
$$793$$ −9.09017 −0.322801
$$794$$ −6.94427 −0.246443
$$795$$ 0 0
$$796$$ −12.6525 −0.448455
$$797$$ 49.1033 1.73933 0.869665 0.493643i $$-0.164335\pi$$
0.869665 + 0.493643i $$0.164335\pi$$
$$798$$ 5.52786 0.195684
$$799$$ −2.94427 −0.104161
$$800$$ 0 0
$$801$$ 5.38197 0.190162
$$802$$ 22.4508 0.792767
$$803$$ −10.0000 −0.352892
$$804$$ −0.763932 −0.0269418
$$805$$ 0 0
$$806$$ 15.5279 0.546946
$$807$$ 6.85410 0.241276
$$808$$ 10.6869 0.375964
$$809$$ 28.2148 0.991979 0.495989 0.868329i $$-0.334805\pi$$
0.495989 + 0.868329i $$0.334805\pi$$
$$810$$ 0 0
$$811$$ 4.76393 0.167284 0.0836421 0.996496i $$-0.473345\pi$$
0.0836421 + 0.996496i $$0.473345\pi$$
$$812$$ −29.5967 −1.03864
$$813$$ 0.180340 0.00632480
$$814$$ 3.81966 0.133879
$$815$$ 0 0
$$816$$ −3.85410 −0.134921
$$817$$ −9.52786 −0.333338
$$818$$ −1.20163 −0.0420139
$$819$$ −25.1246 −0.877925
$$820$$ 0 0
$$821$$ 22.9443 0.800761 0.400380 0.916349i $$-0.368878\pi$$
0.400380 + 0.916349i $$0.368878\pi$$
$$822$$ 7.61803 0.265709
$$823$$ 19.2361 0.670527 0.335264 0.942124i $$-0.391175\pi$$
0.335264 + 0.942124i $$0.391175\pi$$
$$824$$ −9.70820 −0.338201
$$825$$ 0 0
$$826$$ 17.8885 0.622422
$$827$$ 36.0689 1.25424 0.627119 0.778923i $$-0.284234\pi$$
0.627119 + 0.778923i $$0.284234\pi$$
$$828$$ 4.47214 0.155417
$$829$$ 11.5066 0.399640 0.199820 0.979833i $$-0.435964\pi$$
0.199820 + 0.979833i $$0.435964\pi$$
$$830$$ 0 0
$$831$$ 18.7984 0.652108
$$832$$ −39.3262 −1.36339
$$833$$ −50.1033 −1.73598
$$834$$ −22.9443 −0.794495
$$835$$ 0 0
$$836$$ 1.52786 0.0528423
$$837$$ −2.76393 −0.0955355
$$838$$ −15.0557 −0.520091
$$839$$ 14.0689 0.485712 0.242856 0.970062i $$-0.421916\pi$$
0.242856 + 0.970062i $$0.421916\pi$$
$$840$$ 0 0
$$841$$ 14.7984 0.510289
$$842$$ −29.8541 −1.02884
$$843$$ 19.6180 0.675681
$$844$$ −17.8885 −0.615749
$$845$$ 0 0
$$846$$ −0.763932 −0.0262645
$$847$$ −42.3607 −1.45553
$$848$$ 3.61803 0.124244
$$849$$ 13.7082 0.470464
$$850$$ 0 0
$$851$$ −13.8197 −0.473732
$$852$$ −5.23607 −0.179385
$$853$$ −23.1459 −0.792500 −0.396250 0.918143i $$-0.629689\pi$$
−0.396250 + 0.918143i $$0.629689\pi$$
$$854$$ −7.23607 −0.247613
$$855$$ 0 0
$$856$$ −22.5836 −0.771891
$$857$$ 12.4721 0.426040 0.213020 0.977048i $$-0.431670\pi$$
0.213020 + 0.977048i $$0.431670\pi$$
$$858$$ 6.94427 0.237074
$$859$$ −23.4164 −0.798958 −0.399479 0.916742i $$-0.630809\pi$$
−0.399479 + 0.916742i $$0.630809\pi$$
$$860$$ 0 0
$$861$$ 16.1803 0.551425
$$862$$ −20.6525 −0.703426
$$863$$ −10.8754 −0.370203 −0.185101 0.982719i $$-0.559261\pi$$
−0.185101 + 0.982719i $$0.559261\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ −21.9787 −0.746867
$$867$$ 2.14590 0.0728785
$$868$$ −12.3607 −0.419549
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.29180 0.145422
$$872$$ −23.7295 −0.803582
$$873$$ −2.14590 −0.0726276
$$874$$ 5.52786 0.186983
$$875$$ 0 0
$$876$$ 8.09017 0.273342
$$877$$ 33.2148 1.12158 0.560792 0.827957i $$-0.310497\pi$$
0.560792 + 0.827957i $$0.310497\pi$$
$$878$$ −16.1803 −0.546060
$$879$$ −20.7984 −0.701512
$$880$$ 0 0
$$881$$ 42.9443 1.44683 0.723415 0.690414i $$-0.242572\pi$$
0.723415 + 0.690414i $$0.242572\pi$$
$$882$$ −13.0000 −0.437733
$$883$$ −22.1803 −0.746428 −0.373214 0.927745i $$-0.621744\pi$$
−0.373214 + 0.927745i $$0.621744\pi$$
$$884$$ −21.6525 −0.728252
$$885$$ 0 0
$$886$$ −35.2361 −1.18378
$$887$$ 23.1246 0.776448 0.388224 0.921565i $$-0.373089\pi$$
0.388224 + 0.921565i $$0.373089\pi$$
$$888$$ −9.27051 −0.311098
$$889$$ 16.5836 0.556196
$$890$$ 0 0
$$891$$ −1.23607 −0.0414098
$$892$$ 8.18034 0.273898
$$893$$ 0.944272 0.0315989
$$894$$ −18.8541 −0.630575
$$895$$ 0 0
$$896$$ 13.4164 0.448211
$$897$$ −25.1246 −0.838886
$$898$$ −16.7984 −0.560569
$$899$$ 18.2918 0.610066
$$900$$ 0 0
$$901$$ 13.9443 0.464551
$$902$$ −4.47214 −0.148906
$$903$$ 34.4721 1.14716
$$904$$ −54.9787 −1.82856
$$905$$ 0 0
$$906$$ 7.52786 0.250097
$$907$$ −7.12461 −0.236569 −0.118284 0.992980i $$-0.537739\pi$$
−0.118284 + 0.992980i $$0.537739\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ 3.56231 0.118154
$$910$$ 0 0
$$911$$ 18.1803 0.602342 0.301171 0.953570i $$-0.402623\pi$$
0.301171 + 0.953570i $$0.402623\pi$$
$$912$$ 1.23607 0.0409303
$$913$$ 15.4164 0.510209
$$914$$ −22.3607 −0.739626
$$915$$ 0 0
$$916$$ −21.9787 −0.726197
$$917$$ −36.5836 −1.20810
$$918$$ −3.85410 −0.127204
$$919$$ 49.1935 1.62274 0.811372 0.584530i $$-0.198721\pi$$
0.811372 + 0.584530i $$0.198721\pi$$
$$920$$ 0 0
$$921$$ 32.6525 1.07594
$$922$$ 32.7984 1.08016
$$923$$ 29.4164 0.968253
$$924$$ −5.52786 −0.181853
$$925$$ 0 0
$$926$$ 18.7639 0.616621
$$927$$ −3.23607 −0.106286
$$928$$ −33.0902 −1.08624
$$929$$ −9.09017 −0.298239 −0.149119 0.988819i $$-0.547644\pi$$
−0.149119 + 0.988819i $$0.547644\pi$$
$$930$$ 0 0
$$931$$ 16.0689 0.526636
$$932$$ 12.3820 0.405585
$$933$$ −17.7082 −0.579741
$$934$$ 28.1803 0.922089
$$935$$ 0 0
$$936$$ −16.8541 −0.550894
$$937$$ 48.6869 1.59053 0.795266 0.606260i $$-0.207331\pi$$
0.795266 + 0.606260i $$0.207331\pi$$
$$938$$ 3.41641 0.111550
$$939$$ −0.472136 −0.0154076
$$940$$ 0 0
$$941$$ −10.3262 −0.336626 −0.168313 0.985734i $$-0.553832\pi$$
−0.168313 + 0.985734i $$0.553832\pi$$
$$942$$ −10.7984 −0.351830
$$943$$ 16.1803 0.526904
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −9.52786 −0.309778
$$947$$ −2.58359 −0.0839555 −0.0419777 0.999119i $$-0.513366\pi$$
−0.0419777 + 0.999119i $$0.513366\pi$$
$$948$$ 0 0
$$949$$ −45.4508 −1.47540
$$950$$ 0 0
$$951$$ 28.8328 0.934968
$$952$$ −51.7082 −1.67587
$$953$$ 1.09017 0.0353141 0.0176570 0.999844i $$-0.494379\pi$$
0.0176570 + 0.999844i $$0.494379\pi$$
$$954$$ 3.61803 0.117138
$$955$$ 0 0
$$956$$ 24.9443 0.806755
$$957$$ 8.18034 0.264433
$$958$$ −18.9443 −0.612062
$$959$$ 34.0689 1.10014
$$960$$ 0 0
$$961$$ −23.3607 −0.753570
$$962$$ 17.3607 0.559731
$$963$$ −7.52786 −0.242582
$$964$$ −28.0344 −0.902929
$$965$$ 0 0
$$966$$ −20.0000 −0.643489
$$967$$ 18.8328 0.605623 0.302811 0.953051i $$-0.402075\pi$$
0.302811 + 0.953051i $$0.402075\pi$$
$$968$$ −28.4164 −0.913338
$$969$$ 4.76393 0.153040
$$970$$ 0 0
$$971$$ −33.4164 −1.07238 −0.536192 0.844096i $$-0.680138\pi$$
−0.536192 + 0.844096i $$0.680138\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −102.610 −3.28952
$$974$$ 9.70820 0.311071
$$975$$ 0 0
$$976$$ −1.61803 −0.0517920
$$977$$ 3.25735 0.104212 0.0521060 0.998642i $$-0.483407\pi$$
0.0521060 + 0.998642i $$0.483407\pi$$
$$978$$ 14.9443 0.477865
$$979$$ −6.65248 −0.212614
$$980$$ 0 0
$$981$$ −7.90983 −0.252541
$$982$$ −5.88854 −0.187911
$$983$$ −8.29180 −0.264467 −0.132234 0.991219i $$-0.542215\pi$$
−0.132234 + 0.991219i $$0.542215\pi$$
$$984$$ 10.8541 0.346016
$$985$$ 0 0
$$986$$ 25.5066 0.812295
$$987$$ −3.41641 −0.108745
$$988$$ 6.94427 0.220927
$$989$$ 34.4721 1.09615
$$990$$ 0 0
$$991$$ 37.1246 1.17930 0.589651 0.807658i $$-0.299265\pi$$
0.589651 + 0.807658i $$0.299265\pi$$
$$992$$ −13.8197 −0.438775
$$993$$ 6.94427 0.220370
$$994$$ 23.4164 0.742723
$$995$$ 0 0
$$996$$ −12.4721 −0.395195
$$997$$ 17.4164 0.551583 0.275792 0.961217i $$-0.411060\pi$$
0.275792 + 0.961217i $$0.411060\pi$$
$$998$$ 6.00000 0.189927
$$999$$ −3.09017 −0.0977687
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.a.1.2 2
3.2 odd 2 5625.2.a.h.1.2 2
5.2 odd 4 1875.2.b.b.1249.2 4
5.3 odd 4 1875.2.b.b.1249.3 4
5.4 even 2 1875.2.a.d.1.1 2
15.14 odd 2 5625.2.a.a.1.1 2
25.2 odd 20 375.2.i.a.274.1 8
25.9 even 10 75.2.g.a.31.1 4
25.11 even 5 375.2.g.a.226.1 4
25.12 odd 20 375.2.i.a.349.2 8
25.13 odd 20 375.2.i.a.349.1 8
25.14 even 10 75.2.g.a.46.1 yes 4
25.16 even 5 375.2.g.a.151.1 4
25.23 odd 20 375.2.i.a.274.2 8
75.14 odd 10 225.2.h.a.46.1 4
75.59 odd 10 225.2.h.a.181.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.a.31.1 4 25.9 even 10
75.2.g.a.46.1 yes 4 25.14 even 10
225.2.h.a.46.1 4 75.14 odd 10
225.2.h.a.181.1 4 75.59 odd 10
375.2.g.a.151.1 4 25.16 even 5
375.2.g.a.226.1 4 25.11 even 5
375.2.i.a.274.1 8 25.2 odd 20
375.2.i.a.274.2 8 25.23 odd 20
375.2.i.a.349.1 8 25.13 odd 20
375.2.i.a.349.2 8 25.12 odd 20
1875.2.a.a.1.2 2 1.1 even 1 trivial
1875.2.a.d.1.1 2 5.4 even 2
1875.2.b.b.1249.2 4 5.2 odd 4
1875.2.b.b.1249.3 4 5.3 odd 4
5625.2.a.a.1.1 2 15.14 odd 2
5625.2.a.h.1.2 2 3.2 odd 2