# Properties

 Label 6525.2.a.ba.1.1 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ 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] = [6525,2,Mod(1,6525)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 87) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.618034 q^{2} -1.61803 q^{4} -0.236068 q^{7} +2.23607 q^{8} +O(q^{10})$$ $$q-0.618034 q^{2} -1.61803 q^{4} -0.236068 q^{7} +2.23607 q^{8} +0.236068 q^{11} +5.47214 q^{13} +0.145898 q^{14} +1.85410 q^{16} +3.00000 q^{17} -7.23607 q^{19} -0.145898 q^{22} -7.70820 q^{23} -3.38197 q^{26} +0.381966 q^{28} +1.00000 q^{29} +3.70820 q^{31} -5.61803 q^{32} -1.85410 q^{34} -5.23607 q^{37} +4.47214 q^{38} -2.00000 q^{41} -4.00000 q^{43} -0.381966 q^{44} +4.76393 q^{46} +4.70820 q^{47} -6.94427 q^{49} -8.85410 q^{52} +11.2361 q^{53} -0.527864 q^{56} -0.618034 q^{58} -4.47214 q^{59} -5.23607 q^{61} -2.29180 q^{62} -0.236068 q^{64} +13.1803 q^{67} -4.85410 q^{68} +5.23607 q^{71} -6.76393 q^{73} +3.23607 q^{74} +11.7082 q^{76} -0.0557281 q^{77} -12.7639 q^{79} +1.23607 q^{82} +2.94427 q^{83} +2.47214 q^{86} +0.527864 q^{88} -5.00000 q^{89} -1.29180 q^{91} +12.4721 q^{92} -2.90983 q^{94} -18.6525 q^{97} +4.29180 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} + 4 q^{7}+O(q^{10})$$ 2 * q + q^2 - q^4 + 4 * q^7 $$2 q + q^{2} - q^{4} + 4 q^{7} - 4 q^{11} + 2 q^{13} + 7 q^{14} - 3 q^{16} + 6 q^{17} - 10 q^{19} - 7 q^{22} - 2 q^{23} - 9 q^{26} + 3 q^{28} + 2 q^{29} - 6 q^{31} - 9 q^{32} + 3 q^{34} - 6 q^{37} - 4 q^{41} - 8 q^{43} - 3 q^{44} + 14 q^{46} - 4 q^{47} + 4 q^{49} - 11 q^{52} + 18 q^{53} - 10 q^{56} + q^{58} - 6 q^{61} - 18 q^{62} + 4 q^{64} + 4 q^{67} - 3 q^{68} + 6 q^{71} - 18 q^{73} + 2 q^{74} + 10 q^{76} - 18 q^{77} - 30 q^{79} - 2 q^{82} - 12 q^{83} - 4 q^{86} + 10 q^{88} - 10 q^{89} - 16 q^{91} + 16 q^{92} - 17 q^{94} - 6 q^{97} + 22 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 + 4 * q^7 - 4 * q^11 + 2 * q^13 + 7 * q^14 - 3 * q^16 + 6 * q^17 - 10 * q^19 - 7 * q^22 - 2 * q^23 - 9 * q^26 + 3 * q^28 + 2 * q^29 - 6 * q^31 - 9 * q^32 + 3 * q^34 - 6 * q^37 - 4 * q^41 - 8 * q^43 - 3 * q^44 + 14 * q^46 - 4 * q^47 + 4 * q^49 - 11 * q^52 + 18 * q^53 - 10 * q^56 + q^58 - 6 * q^61 - 18 * q^62 + 4 * q^64 + 4 * q^67 - 3 * q^68 + 6 * q^71 - 18 * q^73 + 2 * q^74 + 10 * q^76 - 18 * q^77 - 30 * q^79 - 2 * q^82 - 12 * q^83 - 4 * q^86 + 10 * q^88 - 10 * q^89 - 16 * q^91 + 16 * q^92 - 17 * q^94 - 6 * q^97 + 22 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.618034 −0.437016 −0.218508 0.975835i $$-0.570119\pi$$
−0.218508 + 0.975835i $$0.570119\pi$$
$$3$$ 0 0
$$4$$ −1.61803 −0.809017
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.236068 −0.0892253 −0.0446127 0.999004i $$-0.514205\pi$$
−0.0446127 + 0.999004i $$0.514205\pi$$
$$8$$ 2.23607 0.790569
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0.236068 0.0711772 0.0355886 0.999367i $$-0.488669\pi$$
0.0355886 + 0.999367i $$0.488669\pi$$
$$12$$ 0 0
$$13$$ 5.47214 1.51770 0.758849 0.651267i $$-0.225762\pi$$
0.758849 + 0.651267i $$0.225762\pi$$
$$14$$ 0.145898 0.0389929
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ −7.23607 −1.66007 −0.830034 0.557713i $$-0.811679\pi$$
−0.830034 + 0.557713i $$0.811679\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −0.145898 −0.0311056
$$23$$ −7.70820 −1.60727 −0.803636 0.595121i $$-0.797104\pi$$
−0.803636 + 0.595121i $$0.797104\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −3.38197 −0.663258
$$27$$ 0 0
$$28$$ 0.381966 0.0721848
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 3.70820 0.666013 0.333007 0.942925i $$-0.391937\pi$$
0.333007 + 0.942925i $$0.391937\pi$$
$$32$$ −5.61803 −0.993137
$$33$$ 0 0
$$34$$ −1.85410 −0.317976
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.23607 −0.860804 −0.430402 0.902637i $$-0.641628\pi$$
−0.430402 + 0.902637i $$0.641628\pi$$
$$38$$ 4.47214 0.725476
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −0.381966 −0.0575835
$$45$$ 0 0
$$46$$ 4.76393 0.702403
$$47$$ 4.70820 0.686762 0.343381 0.939196i $$-0.388428\pi$$
0.343381 + 0.939196i $$0.388428\pi$$
$$48$$ 0 0
$$49$$ −6.94427 −0.992039
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −8.85410 −1.22784
$$53$$ 11.2361 1.54339 0.771696 0.635991i $$-0.219409\pi$$
0.771696 + 0.635991i $$0.219409\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −0.527864 −0.0705388
$$57$$ 0 0
$$58$$ −0.618034 −0.0811518
$$59$$ −4.47214 −0.582223 −0.291111 0.956689i $$-0.594025\pi$$
−0.291111 + 0.956689i $$0.594025\pi$$
$$60$$ 0 0
$$61$$ −5.23607 −0.670410 −0.335205 0.942145i $$-0.608806\pi$$
−0.335205 + 0.942145i $$0.608806\pi$$
$$62$$ −2.29180 −0.291058
$$63$$ 0 0
$$64$$ −0.236068 −0.0295085
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 13.1803 1.61023 0.805117 0.593115i $$-0.202102\pi$$
0.805117 + 0.593115i $$0.202102\pi$$
$$68$$ −4.85410 −0.588646
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.23607 0.621407 0.310703 0.950507i $$-0.399435\pi$$
0.310703 + 0.950507i $$0.399435\pi$$
$$72$$ 0 0
$$73$$ −6.76393 −0.791658 −0.395829 0.918324i $$-0.629543\pi$$
−0.395829 + 0.918324i $$0.629543\pi$$
$$74$$ 3.23607 0.376185
$$75$$ 0 0
$$76$$ 11.7082 1.34302
$$77$$ −0.0557281 −0.00635081
$$78$$ 0 0
$$79$$ −12.7639 −1.43605 −0.718027 0.696015i $$-0.754955\pi$$
−0.718027 + 0.696015i $$0.754955\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1.23607 0.136501
$$83$$ 2.94427 0.323176 0.161588 0.986858i $$-0.448338\pi$$
0.161588 + 0.986858i $$0.448338\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.47214 0.266577
$$87$$ 0 0
$$88$$ 0.527864 0.0562705
$$89$$ −5.00000 −0.529999 −0.264999 0.964249i $$-0.585372\pi$$
−0.264999 + 0.964249i $$0.585372\pi$$
$$90$$ 0 0
$$91$$ −1.29180 −0.135417
$$92$$ 12.4721 1.30031
$$93$$ 0 0
$$94$$ −2.90983 −0.300126
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −18.6525 −1.89387 −0.946936 0.321422i $$-0.895839\pi$$
−0.946936 + 0.321422i $$0.895839\pi$$
$$98$$ 4.29180 0.433537
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 7.47214 0.743505 0.371753 0.928332i $$-0.378757\pi$$
0.371753 + 0.928332i $$0.378757\pi$$
$$102$$ 0 0
$$103$$ 4.94427 0.487174 0.243587 0.969879i $$-0.421676\pi$$
0.243587 + 0.969879i $$0.421676\pi$$
$$104$$ 12.2361 1.19985
$$105$$ 0 0
$$106$$ −6.94427 −0.674487
$$107$$ 9.70820 0.938527 0.469264 0.883058i $$-0.344519\pi$$
0.469264 + 0.883058i $$0.344519\pi$$
$$108$$ 0 0
$$109$$ 9.47214 0.907266 0.453633 0.891189i $$-0.350128\pi$$
0.453633 + 0.891189i $$0.350128\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −0.437694 −0.0413582
$$113$$ 19.0000 1.78737 0.893685 0.448695i $$-0.148111\pi$$
0.893685 + 0.448695i $$0.148111\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.61803 −0.150231
$$117$$ 0 0
$$118$$ 2.76393 0.254441
$$119$$ −0.708204 −0.0649209
$$120$$ 0 0
$$121$$ −10.9443 −0.994934
$$122$$ 3.23607 0.292980
$$123$$ 0 0
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −12.4721 −1.10672 −0.553362 0.832941i $$-0.686655\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$128$$ 11.3820 1.00603
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −8.70820 −0.760839 −0.380420 0.924814i $$-0.624220\pi$$
−0.380420 + 0.924814i $$0.624220\pi$$
$$132$$ 0 0
$$133$$ 1.70820 0.148120
$$134$$ −8.14590 −0.703698
$$135$$ 0 0
$$136$$ 6.70820 0.575224
$$137$$ 3.52786 0.301406 0.150703 0.988579i $$-0.451846\pi$$
0.150703 + 0.988579i $$0.451846\pi$$
$$138$$ 0 0
$$139$$ 1.18034 0.100115 0.0500576 0.998746i $$-0.484060\pi$$
0.0500576 + 0.998746i $$0.484060\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −3.23607 −0.271565
$$143$$ 1.29180 0.108025
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 4.18034 0.345967
$$147$$ 0 0
$$148$$ 8.47214 0.696405
$$149$$ 23.4164 1.91835 0.959173 0.282819i $$-0.0912694\pi$$
0.959173 + 0.282819i $$0.0912694\pi$$
$$150$$ 0 0
$$151$$ 3.05573 0.248672 0.124336 0.992240i $$-0.460320\pi$$
0.124336 + 0.992240i $$0.460320\pi$$
$$152$$ −16.1803 −1.31240
$$153$$ 0 0
$$154$$ 0.0344419 0.00277540
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 7.88854 0.627579
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.81966 0.143409
$$162$$ 0 0
$$163$$ 6.00000 0.469956 0.234978 0.972001i $$-0.424498\pi$$
0.234978 + 0.972001i $$0.424498\pi$$
$$164$$ 3.23607 0.252694
$$165$$ 0 0
$$166$$ −1.81966 −0.141233
$$167$$ −6.47214 −0.500829 −0.250414 0.968139i $$-0.580567\pi$$
−0.250414 + 0.968139i $$0.580567\pi$$
$$168$$ 0 0
$$169$$ 16.9443 1.30341
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 6.47214 0.493496
$$173$$ −7.05573 −0.536437 −0.268219 0.963358i $$-0.586435\pi$$
−0.268219 + 0.963358i $$0.586435\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0.437694 0.0329924
$$177$$ 0 0
$$178$$ 3.09017 0.231618
$$179$$ −17.2361 −1.28828 −0.644142 0.764906i $$-0.722785\pi$$
−0.644142 + 0.764906i $$0.722785\pi$$
$$180$$ 0 0
$$181$$ −3.00000 −0.222988 −0.111494 0.993765i $$-0.535564\pi$$
−0.111494 + 0.993765i $$0.535564\pi$$
$$182$$ 0.798374 0.0591794
$$183$$ 0 0
$$184$$ −17.2361 −1.27066
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.708204 0.0517890
$$188$$ −7.61803 −0.555602
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −8.47214 −0.609838 −0.304919 0.952378i $$-0.598629\pi$$
−0.304919 + 0.952378i $$0.598629\pi$$
$$194$$ 11.5279 0.827652
$$195$$ 0 0
$$196$$ 11.2361 0.802576
$$197$$ −19.2361 −1.37051 −0.685257 0.728302i $$-0.740310\pi$$
−0.685257 + 0.728302i $$0.740310\pi$$
$$198$$ 0 0
$$199$$ −15.6525 −1.10957 −0.554787 0.831992i $$-0.687200\pi$$
−0.554787 + 0.831992i $$0.687200\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −4.61803 −0.324924
$$203$$ −0.236068 −0.0165687
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −3.05573 −0.212903
$$207$$ 0 0
$$208$$ 10.1459 0.703491
$$209$$ −1.70820 −0.118159
$$210$$ 0 0
$$211$$ 10.9443 0.753435 0.376717 0.926328i $$-0.377053\pi$$
0.376717 + 0.926328i $$0.377053\pi$$
$$212$$ −18.1803 −1.24863
$$213$$ 0 0
$$214$$ −6.00000 −0.410152
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.875388 −0.0594252
$$218$$ −5.85410 −0.396490
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 16.4164 1.10429
$$222$$ 0 0
$$223$$ −15.1803 −1.01655 −0.508275 0.861195i $$-0.669717\pi$$
−0.508275 + 0.861195i $$0.669717\pi$$
$$224$$ 1.32624 0.0886130
$$225$$ 0 0
$$226$$ −11.7426 −0.781109
$$227$$ −10.9443 −0.726397 −0.363198 0.931712i $$-0.618315\pi$$
−0.363198 + 0.931712i $$0.618315\pi$$
$$228$$ 0 0
$$229$$ −16.1803 −1.06923 −0.534613 0.845097i $$-0.679543\pi$$
−0.534613 + 0.845097i $$0.679543\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.23607 0.146805
$$233$$ 17.4164 1.14099 0.570493 0.821302i $$-0.306752\pi$$
0.570493 + 0.821302i $$0.306752\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 7.23607 0.471028
$$237$$ 0 0
$$238$$ 0.437694 0.0283715
$$239$$ −19.5967 −1.26761 −0.633804 0.773494i $$-0.718507\pi$$
−0.633804 + 0.773494i $$0.718507\pi$$
$$240$$ 0 0
$$241$$ 10.4164 0.670980 0.335490 0.942044i $$-0.391098\pi$$
0.335490 + 0.942044i $$0.391098\pi$$
$$242$$ 6.76393 0.434802
$$243$$ 0 0
$$244$$ 8.47214 0.542373
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −39.5967 −2.51948
$$248$$ 8.29180 0.526530
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 9.18034 0.579458 0.289729 0.957109i $$-0.406435\pi$$
0.289729 + 0.957109i $$0.406435\pi$$
$$252$$ 0 0
$$253$$ −1.81966 −0.114401
$$254$$ 7.70820 0.483656
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ −15.4164 −0.961649 −0.480825 0.876817i $$-0.659663\pi$$
−0.480825 + 0.876817i $$0.659663\pi$$
$$258$$ 0 0
$$259$$ 1.23607 0.0768055
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 5.38197 0.332499
$$263$$ −22.8328 −1.40793 −0.703966 0.710234i $$-0.748589\pi$$
−0.703966 + 0.710234i $$0.748589\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −1.05573 −0.0647308
$$267$$ 0 0
$$268$$ −21.3262 −1.30271
$$269$$ 5.00000 0.304855 0.152428 0.988315i $$-0.451291\pi$$
0.152428 + 0.988315i $$0.451291\pi$$
$$270$$ 0 0
$$271$$ −26.9443 −1.63675 −0.818374 0.574686i $$-0.805124\pi$$
−0.818374 + 0.574686i $$0.805124\pi$$
$$272$$ 5.56231 0.337264
$$273$$ 0 0
$$274$$ −2.18034 −0.131719
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3.00000 −0.180253 −0.0901263 0.995930i $$-0.528727\pi$$
−0.0901263 + 0.995930i $$0.528727\pi$$
$$278$$ −0.729490 −0.0437519
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 21.4164 1.27760 0.638798 0.769375i $$-0.279432\pi$$
0.638798 + 0.769375i $$0.279432\pi$$
$$282$$ 0 0
$$283$$ −7.41641 −0.440860 −0.220430 0.975403i $$-0.570746\pi$$
−0.220430 + 0.975403i $$0.570746\pi$$
$$284$$ −8.47214 −0.502729
$$285$$ 0 0
$$286$$ −0.798374 −0.0472088
$$287$$ 0.472136 0.0278693
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 10.9443 0.640465
$$293$$ −19.9443 −1.16516 −0.582578 0.812775i $$-0.697956\pi$$
−0.582578 + 0.812775i $$0.697956\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −11.7082 −0.680526
$$297$$ 0 0
$$298$$ −14.4721 −0.838348
$$299$$ −42.1803 −2.43935
$$300$$ 0 0
$$301$$ 0.944272 0.0544269
$$302$$ −1.88854 −0.108673
$$303$$ 0 0
$$304$$ −13.4164 −0.769484
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 12.6525 0.722115 0.361057 0.932544i $$-0.382416\pi$$
0.361057 + 0.932544i $$0.382416\pi$$
$$308$$ 0.0901699 0.00513791
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −28.7082 −1.62789 −0.813946 0.580940i $$-0.802685\pi$$
−0.813946 + 0.580940i $$0.802685\pi$$
$$312$$ 0 0
$$313$$ 11.0000 0.621757 0.310878 0.950450i $$-0.399377\pi$$
0.310878 + 0.950450i $$0.399377\pi$$
$$314$$ −1.23607 −0.0697554
$$315$$ 0 0
$$316$$ 20.6525 1.16179
$$317$$ −1.47214 −0.0826834 −0.0413417 0.999145i $$-0.513163\pi$$
−0.0413417 + 0.999145i $$0.513163\pi$$
$$318$$ 0 0
$$319$$ 0.236068 0.0132173
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −1.12461 −0.0626722
$$323$$ −21.7082 −1.20788
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −3.70820 −0.205378
$$327$$ 0 0
$$328$$ −4.47214 −0.246932
$$329$$ −1.11146 −0.0612766
$$330$$ 0 0
$$331$$ −20.7639 −1.14129 −0.570644 0.821197i $$-0.693307\pi$$
−0.570644 + 0.821197i $$0.693307\pi$$
$$332$$ −4.76393 −0.261455
$$333$$ 0 0
$$334$$ 4.00000 0.218870
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8.18034 0.445612 0.222806 0.974863i $$-0.428478\pi$$
0.222806 + 0.974863i $$0.428478\pi$$
$$338$$ −10.4721 −0.569609
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0.875388 0.0474049
$$342$$ 0 0
$$343$$ 3.29180 0.177740
$$344$$ −8.94427 −0.482243
$$345$$ 0 0
$$346$$ 4.36068 0.234432
$$347$$ 25.2361 1.35474 0.677372 0.735641i $$-0.263119\pi$$
0.677372 + 0.735641i $$0.263119\pi$$
$$348$$ 0 0
$$349$$ −13.4164 −0.718164 −0.359082 0.933306i $$-0.616910\pi$$
−0.359082 + 0.933306i $$0.616910\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.32624 −0.0706887
$$353$$ −3.23607 −0.172239 −0.0861193 0.996285i $$-0.527447\pi$$
−0.0861193 + 0.996285i $$0.527447\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 8.09017 0.428778
$$357$$ 0 0
$$358$$ 10.6525 0.563001
$$359$$ 14.4721 0.763810 0.381905 0.924202i $$-0.375268\pi$$
0.381905 + 0.924202i $$0.375268\pi$$
$$360$$ 0 0
$$361$$ 33.3607 1.75583
$$362$$ 1.85410 0.0974494
$$363$$ 0 0
$$364$$ 2.09017 0.109555
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −13.5279 −0.706149 −0.353074 0.935595i $$-0.614864\pi$$
−0.353074 + 0.935595i $$0.614864\pi$$
$$368$$ −14.2918 −0.745011
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.65248 −0.137710
$$372$$ 0 0
$$373$$ 11.5279 0.596890 0.298445 0.954427i $$-0.403532\pi$$
0.298445 + 0.954427i $$0.403532\pi$$
$$374$$ −0.437694 −0.0226326
$$375$$ 0 0
$$376$$ 10.5279 0.542933
$$377$$ 5.47214 0.281829
$$378$$ 0 0
$$379$$ 1.70820 0.0877445 0.0438723 0.999037i $$-0.486031\pi$$
0.0438723 + 0.999037i $$0.486031\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 7.41641 0.379456
$$383$$ 11.2361 0.574136 0.287068 0.957910i $$-0.407319\pi$$
0.287068 + 0.957910i $$0.407319\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 5.23607 0.266509
$$387$$ 0 0
$$388$$ 30.1803 1.53217
$$389$$ −17.3607 −0.880221 −0.440111 0.897944i $$-0.645061\pi$$
−0.440111 + 0.897944i $$0.645061\pi$$
$$390$$ 0 0
$$391$$ −23.1246 −1.16946
$$392$$ −15.5279 −0.784276
$$393$$ 0 0
$$394$$ 11.8885 0.598936
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −6.94427 −0.348523 −0.174262 0.984699i $$-0.555754\pi$$
−0.174262 + 0.984699i $$0.555754\pi$$
$$398$$ 9.67376 0.484902
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −28.1803 −1.40726 −0.703630 0.710567i $$-0.748439\pi$$
−0.703630 + 0.710567i $$0.748439\pi$$
$$402$$ 0 0
$$403$$ 20.2918 1.01081
$$404$$ −12.0902 −0.601508
$$405$$ 0 0
$$406$$ 0.145898 0.00724080
$$407$$ −1.23607 −0.0612696
$$408$$ 0 0
$$409$$ −4.47214 −0.221133 −0.110566 0.993869i $$-0.535266\pi$$
−0.110566 + 0.993869i $$0.535266\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 1.05573 0.0519490
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −30.7426 −1.50728
$$417$$ 0 0
$$418$$ 1.05573 0.0516373
$$419$$ −27.2361 −1.33057 −0.665284 0.746590i $$-0.731690\pi$$
−0.665284 + 0.746590i $$0.731690\pi$$
$$420$$ 0 0
$$421$$ −35.8885 −1.74910 −0.874550 0.484935i $$-0.838843\pi$$
−0.874550 + 0.484935i $$0.838843\pi$$
$$422$$ −6.76393 −0.329263
$$423$$ 0 0
$$424$$ 25.1246 1.22016
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1.23607 0.0598175
$$428$$ −15.7082 −0.759285
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 6.65248 0.319698 0.159849 0.987142i $$-0.448899\pi$$
0.159849 + 0.987142i $$0.448899\pi$$
$$434$$ 0.541020 0.0259698
$$435$$ 0 0
$$436$$ −15.3262 −0.733994
$$437$$ 55.7771 2.66818
$$438$$ 0 0
$$439$$ 13.2918 0.634383 0.317191 0.948362i $$-0.397260\pi$$
0.317191 + 0.948362i $$0.397260\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −10.1459 −0.482591
$$443$$ −3.76393 −0.178830 −0.0894149 0.995994i $$-0.528500\pi$$
−0.0894149 + 0.995994i $$0.528500\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 9.38197 0.444249
$$447$$ 0 0
$$448$$ 0.0557281 0.00263290
$$449$$ 19.4721 0.918947 0.459473 0.888192i $$-0.348038\pi$$
0.459473 + 0.888192i $$0.348038\pi$$
$$450$$ 0 0
$$451$$ −0.472136 −0.0222320
$$452$$ −30.7426 −1.44601
$$453$$ 0 0
$$454$$ 6.76393 0.317447
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.47214 0.0688636 0.0344318 0.999407i $$-0.489038\pi$$
0.0344318 + 0.999407i $$0.489038\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 15.8885 0.740003 0.370002 0.929031i $$-0.379357\pi$$
0.370002 + 0.929031i $$0.379357\pi$$
$$462$$ 0 0
$$463$$ −15.1803 −0.705490 −0.352745 0.935719i $$-0.614752\pi$$
−0.352745 + 0.935719i $$0.614752\pi$$
$$464$$ 1.85410 0.0860745
$$465$$ 0 0
$$466$$ −10.7639 −0.498630
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ −3.11146 −0.143674
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −10.0000 −0.460287
$$473$$ −0.944272 −0.0434177
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 1.14590 0.0525222
$$477$$ 0 0
$$478$$ 12.1115 0.553965
$$479$$ −14.4721 −0.661249 −0.330624 0.943762i $$-0.607259\pi$$
−0.330624 + 0.943762i $$0.607259\pi$$
$$480$$ 0 0
$$481$$ −28.6525 −1.30644
$$482$$ −6.43769 −0.293229
$$483$$ 0 0
$$484$$ 17.7082 0.804918
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −36.9443 −1.67410 −0.837052 0.547123i $$-0.815723\pi$$
−0.837052 + 0.547123i $$0.815723\pi$$
$$488$$ −11.7082 −0.530005
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 8.00000 0.361035 0.180517 0.983572i $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 0 0
$$493$$ 3.00000 0.135113
$$494$$ 24.4721 1.10105
$$495$$ 0 0
$$496$$ 6.87539 0.308714
$$497$$ −1.23607 −0.0554452
$$498$$ 0 0
$$499$$ 3.29180 0.147361 0.0736805 0.997282i $$-0.476525\pi$$
0.0736805 + 0.997282i $$0.476525\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −5.67376 −0.253232
$$503$$ 27.5410 1.22799 0.613997 0.789309i $$-0.289561\pi$$
0.613997 + 0.789309i $$0.289561\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1.12461 0.0499951
$$507$$ 0 0
$$508$$ 20.1803 0.895358
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 1.59675 0.0706360
$$512$$ −18.7082 −0.826794
$$513$$ 0 0
$$514$$ 9.52786 0.420256
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.11146 0.0488818
$$518$$ −0.763932 −0.0335652
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 31.4164 1.37638 0.688189 0.725532i $$-0.258406\pi$$
0.688189 + 0.725532i $$0.258406\pi$$
$$522$$ 0 0
$$523$$ −14.1246 −0.617626 −0.308813 0.951123i $$-0.599932\pi$$
−0.308813 + 0.951123i $$0.599932\pi$$
$$524$$ 14.0902 0.615532
$$525$$ 0 0
$$526$$ 14.1115 0.615289
$$527$$ 11.1246 0.484596
$$528$$ 0 0
$$529$$ 36.4164 1.58332
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −2.76393 −0.119832
$$533$$ −10.9443 −0.474049
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 29.4721 1.27300
$$537$$ 0 0
$$538$$ −3.09017 −0.133227
$$539$$ −1.63932 −0.0706105
$$540$$ 0 0
$$541$$ 8.18034 0.351700 0.175850 0.984417i $$-0.443733\pi$$
0.175850 + 0.984417i $$0.443733\pi$$
$$542$$ 16.6525 0.715285
$$543$$ 0 0
$$544$$ −16.8541 −0.722614
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −3.65248 −0.156169 −0.0780843 0.996947i $$-0.524880\pi$$
−0.0780843 + 0.996947i $$0.524880\pi$$
$$548$$ −5.70820 −0.243842
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −7.23607 −0.308267
$$552$$ 0 0
$$553$$ 3.01316 0.128132
$$554$$ 1.85410 0.0787732
$$555$$ 0 0
$$556$$ −1.90983 −0.0809948
$$557$$ −25.4164 −1.07693 −0.538464 0.842649i $$-0.680995\pi$$
−0.538464 + 0.842649i $$0.680995\pi$$
$$558$$ 0 0
$$559$$ −21.8885 −0.925787
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −13.2361 −0.558330
$$563$$ −45.0689 −1.89943 −0.949713 0.313120i $$-0.898626\pi$$
−0.949713 + 0.313120i $$0.898626\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.58359 0.192663
$$567$$ 0 0
$$568$$ 11.7082 0.491265
$$569$$ −35.2492 −1.47772 −0.738862 0.673857i $$-0.764637\pi$$
−0.738862 + 0.673857i $$0.764637\pi$$
$$570$$ 0 0
$$571$$ 15.4164 0.645157 0.322578 0.946543i $$-0.395450\pi$$
0.322578 + 0.946543i $$0.395450\pi$$
$$572$$ −2.09017 −0.0873944
$$573$$ 0 0
$$574$$ −0.291796 −0.0121793
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 40.9443 1.70453 0.852266 0.523108i $$-0.175228\pi$$
0.852266 + 0.523108i $$0.175228\pi$$
$$578$$ 4.94427 0.205655
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −0.695048 −0.0288355
$$582$$ 0 0
$$583$$ 2.65248 0.109854
$$584$$ −15.1246 −0.625861
$$585$$ 0 0
$$586$$ 12.3262 0.509192
$$587$$ −5.41641 −0.223559 −0.111780 0.993733i $$-0.535655\pi$$
−0.111780 + 0.993733i $$0.535655\pi$$
$$588$$ 0 0
$$589$$ −26.8328 −1.10563
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −9.70820 −0.399005
$$593$$ −15.5967 −0.640482 −0.320241 0.947336i $$-0.603764\pi$$
−0.320241 + 0.947336i $$0.603764\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −37.8885 −1.55198
$$597$$ 0 0
$$598$$ 26.0689 1.06604
$$599$$ −33.2918 −1.36027 −0.680133 0.733089i $$-0.738078\pi$$
−0.680133 + 0.733089i $$0.738078\pi$$
$$600$$ 0 0
$$601$$ 8.18034 0.333683 0.166842 0.985984i $$-0.446643\pi$$
0.166842 + 0.985984i $$0.446643\pi$$
$$602$$ −0.583592 −0.0237854
$$603$$ 0 0
$$604$$ −4.94427 −0.201180
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −1.41641 −0.0574902 −0.0287451 0.999587i $$-0.509151\pi$$
−0.0287451 + 0.999587i $$0.509151\pi$$
$$608$$ 40.6525 1.64868
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 25.7639 1.04230
$$612$$ 0 0
$$613$$ −5.58359 −0.225519 −0.112760 0.993622i $$-0.535969\pi$$
−0.112760 + 0.993622i $$0.535969\pi$$
$$614$$ −7.81966 −0.315576
$$615$$ 0 0
$$616$$ −0.124612 −0.00502075
$$617$$ 12.4721 0.502109 0.251055 0.967973i $$-0.419223\pi$$
0.251055 + 0.967973i $$0.419223\pi$$
$$618$$ 0 0
$$619$$ −1.70820 −0.0686585 −0.0343293 0.999411i $$-0.510929\pi$$
−0.0343293 + 0.999411i $$0.510929\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 17.7426 0.711415
$$623$$ 1.18034 0.0472893
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −6.79837 −0.271718
$$627$$ 0 0
$$628$$ −3.23607 −0.129133
$$629$$ −15.7082 −0.626327
$$630$$ 0 0
$$631$$ −23.6525 −0.941590 −0.470795 0.882243i $$-0.656033\pi$$
−0.470795 + 0.882243i $$0.656033\pi$$
$$632$$ −28.5410 −1.13530
$$633$$ 0 0
$$634$$ 0.909830 0.0361340
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −38.0000 −1.50561
$$638$$ −0.145898 −0.00577616
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −38.3050 −1.51295 −0.756477 0.654020i $$-0.773081\pi$$
−0.756477 + 0.654020i $$0.773081\pi$$
$$642$$ 0 0
$$643$$ −0.708204 −0.0279288 −0.0139644 0.999902i $$-0.504445\pi$$
−0.0139644 + 0.999902i $$0.504445\pi$$
$$644$$ −2.94427 −0.116021
$$645$$ 0 0
$$646$$ 13.4164 0.527862
$$647$$ 14.1803 0.557487 0.278743 0.960366i $$-0.410082\pi$$
0.278743 + 0.960366i $$0.410082\pi$$
$$648$$ 0 0
$$649$$ −1.05573 −0.0414410
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −9.70820 −0.380203
$$653$$ 9.00000 0.352197 0.176099 0.984373i $$-0.443652\pi$$
0.176099 + 0.984373i $$0.443652\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −3.70820 −0.144781
$$657$$ 0 0
$$658$$ 0.686918 0.0267788
$$659$$ −24.5967 −0.958153 −0.479077 0.877773i $$-0.659028\pi$$
−0.479077 + 0.877773i $$0.659028\pi$$
$$660$$ 0 0
$$661$$ −23.0000 −0.894596 −0.447298 0.894385i $$-0.647614\pi$$
−0.447298 + 0.894385i $$0.647614\pi$$
$$662$$ 12.8328 0.498762
$$663$$ 0 0
$$664$$ 6.58359 0.255493
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −7.70820 −0.298463
$$668$$ 10.4721 0.405179
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −1.23607 −0.0477179
$$672$$ 0 0
$$673$$ −30.3050 −1.16817 −0.584085 0.811692i $$-0.698547\pi$$
−0.584085 + 0.811692i $$0.698547\pi$$
$$674$$ −5.05573 −0.194739
$$675$$ 0 0
$$676$$ −27.4164 −1.05448
$$677$$ −0.416408 −0.0160039 −0.00800193 0.999968i $$-0.502547\pi$$
−0.00800193 + 0.999968i $$0.502547\pi$$
$$678$$ 0 0
$$679$$ 4.40325 0.168981
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −0.541020 −0.0207167
$$683$$ 10.8328 0.414506 0.207253 0.978287i $$-0.433548\pi$$
0.207253 + 0.978287i $$0.433548\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −2.03444 −0.0776754
$$687$$ 0 0
$$688$$ −7.41641 −0.282748
$$689$$ 61.4853 2.34240
$$690$$ 0 0
$$691$$ −32.5967 −1.24004 −0.620019 0.784587i $$-0.712875\pi$$
−0.620019 + 0.784587i $$0.712875\pi$$
$$692$$ 11.4164 0.433987
$$693$$ 0 0
$$694$$ −15.5967 −0.592044
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −6.00000 −0.227266
$$698$$ 8.29180 0.313849
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 26.5410 1.00244 0.501220 0.865320i $$-0.332885\pi$$
0.501220 + 0.865320i $$0.332885\pi$$
$$702$$ 0 0
$$703$$ 37.8885 1.42899
$$704$$ −0.0557281 −0.00210033
$$705$$ 0 0
$$706$$ 2.00000 0.0752710
$$707$$ −1.76393 −0.0663395
$$708$$ 0 0
$$709$$ −30.0000 −1.12667 −0.563337 0.826227i $$-0.690483\pi$$
−0.563337 + 0.826227i $$0.690483\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −11.1803 −0.419001
$$713$$ −28.5836 −1.07046
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 27.8885 1.04224
$$717$$ 0 0
$$718$$ −8.94427 −0.333797
$$719$$ 50.2492 1.87398 0.936990 0.349356i $$-0.113600\pi$$
0.936990 + 0.349356i $$0.113600\pi$$
$$720$$ 0 0
$$721$$ −1.16718 −0.0434682
$$722$$ −20.6180 −0.767324
$$723$$ 0 0
$$724$$ 4.85410 0.180401
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 14.7639 0.547564 0.273782 0.961792i $$-0.411725\pi$$
0.273782 + 0.961792i $$0.411725\pi$$
$$728$$ −2.88854 −0.107057
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ −18.4721 −0.682284 −0.341142 0.940012i $$-0.610814\pi$$
−0.341142 + 0.940012i $$0.610814\pi$$
$$734$$ 8.36068 0.308598
$$735$$ 0 0
$$736$$ 43.3050 1.59624
$$737$$ 3.11146 0.114612
$$738$$ 0 0
$$739$$ 6.18034 0.227347 0.113674 0.993518i $$-0.463738\pi$$
0.113674 + 0.993518i $$0.463738\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 1.63932 0.0601813
$$743$$ −10.3475 −0.379614 −0.189807 0.981821i $$-0.560786\pi$$
−0.189807 + 0.981821i $$0.560786\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −7.12461 −0.260851
$$747$$ 0 0
$$748$$ −1.14590 −0.0418982
$$749$$ −2.29180 −0.0837404
$$750$$ 0 0
$$751$$ −43.7771 −1.59745 −0.798724 0.601697i $$-0.794491\pi$$
−0.798724 + 0.601697i $$0.794491\pi$$
$$752$$ 8.72949 0.318332
$$753$$ 0 0
$$754$$ −3.38197 −0.123164
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 20.9443 0.761233 0.380616 0.924733i $$-0.375712\pi$$
0.380616 + 0.924733i $$0.375712\pi$$
$$758$$ −1.05573 −0.0383458
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −28.1803 −1.02154 −0.510768 0.859718i $$-0.670639\pi$$
−0.510768 + 0.859718i $$0.670639\pi$$
$$762$$ 0 0
$$763$$ −2.23607 −0.0809511
$$764$$ 19.4164 0.702461
$$765$$ 0 0
$$766$$ −6.94427 −0.250907
$$767$$ −24.4721 −0.883638
$$768$$ 0 0
$$769$$ 17.2361 0.621549 0.310774 0.950484i $$-0.399412\pi$$
0.310774 + 0.950484i $$0.399412\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 13.7082 0.493369
$$773$$ 28.4721 1.02407 0.512036 0.858964i $$-0.328892\pi$$
0.512036 + 0.858964i $$0.328892\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −41.7082 −1.49724
$$777$$ 0 0
$$778$$ 10.7295 0.384671
$$779$$ 14.4721 0.518518
$$780$$ 0 0
$$781$$ 1.23607 0.0442300
$$782$$ 14.2918 0.511074
$$783$$ 0 0
$$784$$ −12.8754 −0.459835
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 46.4721 1.65655 0.828276 0.560320i $$-0.189322\pi$$
0.828276 + 0.560320i $$0.189322\pi$$
$$788$$ 31.1246 1.10877
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.48529 −0.159479
$$792$$ 0 0
$$793$$ −28.6525 −1.01748
$$794$$ 4.29180 0.152310
$$795$$ 0 0
$$796$$ 25.3262 0.897665
$$797$$ 9.05573 0.320770 0.160385 0.987055i $$-0.448726\pi$$
0.160385 + 0.987055i $$0.448726\pi$$
$$798$$ 0 0
$$799$$ 14.1246 0.499693
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 17.4164 0.614995
$$803$$ −1.59675 −0.0563480
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −12.5410 −0.441739
$$807$$ 0 0
$$808$$ 16.7082 0.587793
$$809$$ −36.3050 −1.27641 −0.638207 0.769865i $$-0.720324\pi$$
−0.638207 + 0.769865i $$0.720324\pi$$
$$810$$ 0 0
$$811$$ −23.6525 −0.830551 −0.415275 0.909696i $$-0.636315\pi$$
−0.415275 + 0.909696i $$0.636315\pi$$
$$812$$ 0.381966 0.0134044
$$813$$ 0 0
$$814$$ 0.763932 0.0267758
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 28.9443 1.01263
$$818$$ 2.76393 0.0966386
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25.4164 −0.887039 −0.443519 0.896265i $$-0.646270\pi$$
−0.443519 + 0.896265i $$0.646270\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 11.0557 0.385145
$$825$$ 0 0
$$826$$ −0.652476 −0.0227025
$$827$$ 1.16718 0.0405870 0.0202935 0.999794i $$-0.493540\pi$$
0.0202935 + 0.999794i $$0.493540\pi$$
$$828$$ 0 0
$$829$$ 30.2492 1.05060 0.525299 0.850917i $$-0.323953\pi$$
0.525299 + 0.850917i $$0.323953\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.29180 −0.0447850
$$833$$ −20.8328 −0.721814
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2.76393 0.0955926
$$837$$ 0 0
$$838$$ 16.8328 0.581480
$$839$$ 35.6525 1.23086 0.615430 0.788191i $$-0.288982\pi$$
0.615430 + 0.788191i $$0.288982\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 22.1803 0.764385
$$843$$ 0 0
$$844$$ −17.7082 −0.609542
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.58359 0.0887733
$$848$$ 20.8328 0.715402
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 40.3607 1.38355
$$852$$ 0 0
$$853$$ −7.41641 −0.253933 −0.126966 0.991907i $$-0.540524\pi$$
−0.126966 + 0.991907i $$0.540524\pi$$
$$854$$ −0.763932 −0.0261412
$$855$$ 0 0
$$856$$ 21.7082 0.741971
$$857$$ −31.5967 −1.07932 −0.539662 0.841882i $$-0.681448\pi$$
−0.539662 + 0.841882i $$0.681448\pi$$
$$858$$ 0 0
$$859$$ −36.8328 −1.25672 −0.628360 0.777923i $$-0.716273\pi$$
−0.628360 + 0.777923i $$0.716273\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −4.94427 −0.168403
$$863$$ 37.0132 1.25994 0.629971 0.776618i $$-0.283067\pi$$
0.629971 + 0.776618i $$0.283067\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −4.11146 −0.139713
$$867$$ 0 0
$$868$$ 1.41641 0.0480760
$$869$$ −3.01316 −0.102214
$$870$$ 0 0
$$871$$ 72.1246 2.44385
$$872$$ 21.1803 0.717257
$$873$$ 0 0
$$874$$ −34.4721 −1.16604
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −21.4164 −0.723181 −0.361590 0.932337i $$-0.617766\pi$$
−0.361590 + 0.932337i $$0.617766\pi$$
$$878$$ −8.21478 −0.277235
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −22.5279 −0.758983 −0.379492 0.925195i $$-0.623901\pi$$
−0.379492 + 0.925195i $$0.623901\pi$$
$$882$$ 0 0
$$883$$ −27.4164 −0.922636 −0.461318 0.887235i $$-0.652623\pi$$
−0.461318 + 0.887235i $$0.652623\pi$$
$$884$$ −26.5623 −0.893387
$$885$$ 0 0
$$886$$ 2.32624 0.0781515
$$887$$ 16.8197 0.564749 0.282374 0.959304i $$-0.408878\pi$$
0.282374 + 0.959304i $$0.408878\pi$$
$$888$$ 0 0
$$889$$ 2.94427 0.0987477
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 24.5623 0.822407
$$893$$ −34.0689 −1.14007
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −2.68692 −0.0897636
$$897$$ 0 0
$$898$$ −12.0344 −0.401595
$$899$$ 3.70820 0.123676
$$900$$ 0 0
$$901$$ 33.7082 1.12298
$$902$$ 0.291796 0.00971575
$$903$$ 0 0
$$904$$ 42.4853 1.41304
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 17.7771 0.590279 0.295139 0.955454i $$-0.404634\pi$$
0.295139 + 0.955454i $$0.404634\pi$$
$$908$$ 17.7082 0.587667
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −10.8197 −0.358471 −0.179236 0.983806i $$-0.557362\pi$$
−0.179236 + 0.983806i $$0.557362\pi$$
$$912$$ 0 0
$$913$$ 0.695048 0.0230027
$$914$$ −0.909830 −0.0300945
$$915$$ 0 0
$$916$$ 26.1803 0.865023
$$917$$ 2.05573 0.0678861
$$918$$ 0 0
$$919$$ 58.0132 1.91368 0.956839 0.290619i $$-0.0938614\pi$$
0.956839 + 0.290619i $$0.0938614\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −9.81966 −0.323393
$$923$$ 28.6525 0.943108
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 9.38197 0.308311
$$927$$ 0 0
$$928$$ −5.61803 −0.184421
$$929$$ 35.5279 1.16563 0.582816 0.812604i $$-0.301951\pi$$
0.582816 + 0.812604i $$0.301951\pi$$
$$930$$ 0 0
$$931$$ 50.2492 1.64685
$$932$$ −28.1803 −0.923078
$$933$$ 0 0
$$934$$ 7.41641 0.242672
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −35.3607 −1.15518 −0.577592 0.816326i $$-0.696007\pi$$
−0.577592 + 0.816326i $$0.696007\pi$$
$$938$$ 1.92299 0.0627877
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 10.1115 0.329624 0.164812 0.986325i $$-0.447298\pi$$
0.164812 + 0.986325i $$0.447298\pi$$
$$942$$ 0 0
$$943$$ 15.4164 0.502027
$$944$$ −8.29180 −0.269875
$$945$$ 0 0
$$946$$ 0.583592 0.0189742
$$947$$ −2.12461 −0.0690406 −0.0345203 0.999404i $$-0.510990\pi$$
−0.0345203 + 0.999404i $$0.510990\pi$$
$$948$$ 0 0
$$949$$ −37.0132 −1.20150
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −1.58359 −0.0513245
$$953$$ 41.8885 1.35690 0.678452 0.734645i $$-0.262651\pi$$
0.678452 + 0.734645i $$0.262651\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 31.7082 1.02552
$$957$$ 0 0
$$958$$ 8.94427 0.288976
$$959$$ −0.832816 −0.0268930
$$960$$ 0 0
$$961$$ −17.2492 −0.556427
$$962$$ 17.7082 0.570935
$$963$$ 0 0
$$964$$ −16.8541 −0.542834
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −0.763932 −0.0245664 −0.0122832 0.999925i $$-0.503910\pi$$
−0.0122832 + 0.999925i $$0.503910\pi$$
$$968$$ −24.4721 −0.786564
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0.360680 0.0115748 0.00578738 0.999983i $$-0.498158\pi$$
0.00578738 + 0.999983i $$0.498158\pi$$
$$972$$ 0 0
$$973$$ −0.278640 −0.00893280
$$974$$ 22.8328 0.731611
$$975$$ 0 0
$$976$$ −9.70820 −0.310752
$$977$$ 39.7082 1.27038 0.635189 0.772357i $$-0.280922\pi$$
0.635189 + 0.772357i $$0.280922\pi$$
$$978$$ 0 0
$$979$$ −1.18034 −0.0377238
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −4.94427 −0.157778
$$983$$ −4.94427 −0.157698 −0.0788489 0.996887i $$-0.525124\pi$$
−0.0788489 + 0.996887i $$0.525124\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −1.85410 −0.0590466
$$987$$ 0 0
$$988$$ 64.0689 2.03830
$$989$$ 30.8328 0.980427
$$990$$ 0 0
$$991$$ −26.0132 −0.826335 −0.413168 0.910655i $$-0.635578\pi$$
−0.413168 + 0.910655i $$0.635578\pi$$
$$992$$ −20.8328 −0.661443
$$993$$ 0 0
$$994$$ 0.763932 0.0242305
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 60.1378 1.90458 0.952291 0.305191i $$-0.0987204\pi$$
0.952291 + 0.305191i $$0.0987204\pi$$
$$998$$ −2.03444 −0.0643991
$$999$$ 0 0
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 6525.2.a.ba.1.1 2
3.2 odd 2 2175.2.a.l.1.2 2
5.4 even 2 261.2.a.b.1.2 2
15.2 even 4 2175.2.c.k.349.3 4
15.8 even 4 2175.2.c.k.349.2 4
15.14 odd 2 87.2.a.a.1.1 2
20.19 odd 2 4176.2.a.bn.1.1 2
60.59 even 2 1392.2.a.q.1.2 2
105.104 even 2 4263.2.a.j.1.1 2
120.29 odd 2 5568.2.a.bl.1.1 2
120.59 even 2 5568.2.a.bs.1.1 2
145.144 even 2 7569.2.a.k.1.1 2
435.434 odd 2 2523.2.a.c.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.a.a.1.1 2 15.14 odd 2
261.2.a.b.1.2 2 5.4 even 2
1392.2.a.q.1.2 2 60.59 even 2
2175.2.a.l.1.2 2 3.2 odd 2
2175.2.c.k.349.2 4 15.8 even 4
2175.2.c.k.349.3 4 15.2 even 4
2523.2.a.c.1.2 2 435.434 odd 2
4176.2.a.bn.1.1 2 20.19 odd 2
4263.2.a.j.1.1 2 105.104 even 2
5568.2.a.bl.1.1 2 120.29 odd 2
5568.2.a.bs.1.1 2 120.59 even 2
6525.2.a.ba.1.1 2 1.1 even 1 trivial
7569.2.a.k.1.1 2 145.144 even 2