# Properties

 Label 5175.2.a.o.1.1 Level $5175$ Weight $2$ Character 5175.1 Self dual yes Analytic conductor $41.323$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5175,2,Mod(1,5175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, 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("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5175 = 3^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5175.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: $$41.3225830460$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 345) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{4} +3.00000 q^{7} +O(q^{10})$$ $$q-2.00000 q^{4} +3.00000 q^{7} +4.00000 q^{11} +4.00000 q^{16} -3.00000 q^{17} -8.00000 q^{19} +1.00000 q^{23} -6.00000 q^{28} -9.00000 q^{29} -5.00000 q^{31} +9.00000 q^{37} -7.00000 q^{41} -4.00000 q^{43} -8.00000 q^{44} -2.00000 q^{47} +2.00000 q^{49} +13.0000 q^{53} +3.00000 q^{59} -14.0000 q^{61} -8.00000 q^{64} -13.0000 q^{67} +6.00000 q^{68} +13.0000 q^{71} +4.00000 q^{73} +16.0000 q^{76} +12.0000 q^{77} -1.00000 q^{83} +8.00000 q^{89} -2.00000 q^{92} -10.0000 q^{97} +O(q^{100})$$

## 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 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 0 0
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −6.00000 −1.13389
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.00000 1.47959 0.739795 0.672832i $$-0.234922\pi$$
0.739795 + 0.672832i $$0.234922\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.00000 −1.09322 −0.546608 0.837389i $$-0.684081\pi$$
−0.546608 + 0.837389i $$0.684081\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −8.00000 −1.20605
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 13.0000 1.78569 0.892844 0.450367i $$-0.148707\pi$$
0.892844 + 0.450367i $$0.148707\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −13.0000 −1.58820 −0.794101 0.607785i $$-0.792058\pi$$
−0.794101 + 0.607785i $$0.792058\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13.0000 1.54282 0.771408 0.636341i $$-0.219553\pi$$
0.771408 + 0.636341i $$0.219553\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 16.0000 1.83533
$$77$$ 12.0000 1.36753
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −1.00000 −0.109764 −0.0548821 0.998493i $$-0.517478\pi$$
−0.0548821 + 0.998493i $$0.517478\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −2.00000 −0.208514
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.0000 1.06341 0.531705 0.846930i $$-0.321551\pi$$
0.531705 + 0.846930i $$0.321551\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 12.0000 1.13389
$$113$$ 1.00000 0.0940721 0.0470360 0.998893i $$-0.485022\pi$$
0.0470360 + 0.998893i $$0.485022\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 18.0000 1.67126
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −9.00000 −0.825029
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 10.0000 0.898027
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 18.0000 1.59724 0.798621 0.601834i $$-0.205563\pi$$
0.798621 + 0.601834i $$0.205563\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 0 0
$$133$$ −24.0000 −2.08106
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −3.00000 −0.254457 −0.127228 0.991873i $$-0.540608\pi$$
−0.127228 + 0.991873i $$0.540608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −18.0000 −1.47959
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3.00000 0.239426 0.119713 0.992809i $$-0.461803\pi$$
0.119713 + 0.992809i $$0.461803\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ 14.0000 1.09322
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −22.0000 −1.70241 −0.851206 0.524832i $$-0.824128\pi$$
−0.851206 + 0.524832i $$0.824128\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.00000 0.609994
$$173$$ −24.0000 −1.82469 −0.912343 0.409426i $$-0.865729\pi$$
−0.912343 + 0.409426i $$0.865729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 16.0000 1.20605
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −12.0000 −0.877527
$$188$$ 4.00000 0.291730
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2.00000 0.144715 0.0723575 0.997379i $$-0.476948\pi$$
0.0723575 + 0.997379i $$0.476948\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −4.00000 −0.285714
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ −18.0000 −1.27599 −0.637993 0.770042i $$-0.720235\pi$$
−0.637993 + 0.770042i $$0.720235\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −27.0000 −1.89503
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −32.0000 −2.21349
$$210$$ 0 0
$$211$$ 7.00000 0.481900 0.240950 0.970538i $$-0.422541\pi$$
0.240950 + 0.970538i $$0.422541\pi$$
$$212$$ −26.0000 −1.78569
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −15.0000 −1.01827
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −13.0000 −0.840900 −0.420450 0.907316i $$-0.638128\pi$$
−0.420450 + 0.907316i $$0.638128\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 28.0000 1.79252
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 27.0000 1.67770
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −23.0000 −1.41824 −0.709120 0.705087i $$-0.750908\pi$$
−0.709120 + 0.705087i $$0.750908\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 26.0000 1.58820
$$269$$ 17.0000 1.03651 0.518254 0.855227i $$-0.326582\pi$$
0.518254 + 0.855227i $$0.326582\pi$$
$$270$$ 0 0
$$271$$ −23.0000 −1.39715 −0.698575 0.715537i $$-0.746182\pi$$
−0.698575 + 0.715537i $$0.746182\pi$$
$$272$$ −12.0000 −0.727607
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −24.0000 −1.43172 −0.715860 0.698244i $$-0.753965\pi$$
−0.715860 + 0.698244i $$0.753965\pi$$
$$282$$ 0 0
$$283$$ 5.00000 0.297219 0.148610 0.988896i $$-0.452520\pi$$
0.148610 + 0.988896i $$0.452520\pi$$
$$284$$ −26.0000 −1.54282
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −21.0000 −1.23959
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −8.00000 −0.468165
$$293$$ 19.0000 1.10999 0.554996 0.831853i $$-0.312720\pi$$
0.554996 + 0.831853i $$0.312720\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −32.0000 −1.83533
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ −24.0000 −1.36753
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −25.0000 −1.41308 −0.706542 0.707671i $$-0.749746\pi$$
−0.706542 + 0.707671i $$0.749746\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 0 0
$$319$$ −36.0000 −2.01561
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ 7.00000 0.384755 0.192377 0.981321i $$-0.438380\pi$$
0.192377 + 0.981321i $$0.438380\pi$$
$$332$$ 2.00000 0.109764
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −10.0000 −0.544735 −0.272367 0.962193i $$-0.587807\pi$$
−0.272367 + 0.962193i $$0.587807\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −20.0000 −1.08306
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ 0 0
$$349$$ −1.00000 −0.0535288 −0.0267644 0.999642i $$-0.508520\pi$$
−0.0267644 + 0.999642i $$0.508520\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −16.0000 −0.847998
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 2.00000 0.105556 0.0527780 0.998606i $$-0.483192\pi$$
0.0527780 + 0.998606i $$0.483192\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −31.0000 −1.61819 −0.809093 0.587680i $$-0.800041\pi$$
−0.809093 + 0.587680i $$0.800041\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 39.0000 2.02478
$$372$$ 0 0
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −23.0000 −1.17525 −0.587623 0.809135i $$-0.699936\pi$$
−0.587623 + 0.809135i $$0.699936\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 20.0000 1.01535
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −3.00000 −0.151717
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −8.00000 −0.401508 −0.200754 0.979642i $$-0.564339\pi$$
−0.200754 + 0.979642i $$0.564339\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 36.0000 1.78445
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 16.0000 0.788263
$$413$$ 9.00000 0.442861
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 4.00000 0.194948 0.0974740 0.995238i $$-0.468924\pi$$
0.0974740 + 0.995238i $$0.468924\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −42.0000 −2.03252
$$428$$ −22.0000 −1.06341
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −20.0000 −0.963366 −0.481683 0.876346i $$-0.659974\pi$$
−0.481683 + 0.876346i $$0.659974\pi$$
$$432$$ 0 0
$$433$$ 37.0000 1.77811 0.889053 0.457804i $$-0.151364\pi$$
0.889053 + 0.457804i $$0.151364\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 32.0000 1.53252
$$437$$ −8.00000 −0.382692
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 6.00000 0.285069 0.142534 0.989790i $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −24.0000 −1.13389
$$449$$ 11.0000 0.519122 0.259561 0.965727i $$-0.416422\pi$$
0.259561 + 0.965727i $$0.416422\pi$$
$$450$$ 0 0
$$451$$ −28.0000 −1.31847
$$452$$ −2.00000 −0.0940721
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 35.0000 1.63723 0.818615 0.574342i $$-0.194742\pi$$
0.818615 + 0.574342i $$0.194742\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −26.0000 −1.21094 −0.605470 0.795868i $$-0.707015\pi$$
−0.605470 + 0.795868i $$0.707015\pi$$
$$462$$ 0 0
$$463$$ 20.0000 0.929479 0.464739 0.885448i $$-0.346148\pi$$
0.464739 + 0.885448i $$0.346148\pi$$
$$464$$ −36.0000 −1.67126
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −13.0000 −0.601568 −0.300784 0.953692i $$-0.597248\pi$$
−0.300784 + 0.953692i $$0.597248\pi$$
$$468$$ 0 0
$$469$$ −39.0000 −1.80085
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 18.0000 0.825029
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3.00000 −0.135388 −0.0676941 0.997706i $$-0.521564\pi$$
−0.0676941 + 0.997706i $$0.521564\pi$$
$$492$$ 0 0
$$493$$ 27.0000 1.21602
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −20.0000 −0.898027
$$497$$ 39.0000 1.74939
$$498$$ 0 0
$$499$$ 21.0000 0.940089 0.470045 0.882643i $$-0.344238\pi$$
0.470045 + 0.882643i $$0.344238\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 21.0000 0.936344 0.468172 0.883637i $$-0.344913\pi$$
0.468172 + 0.883637i $$0.344913\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −36.0000 −1.59724
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −8.00000 −0.351840
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 12.0000 0.525730 0.262865 0.964833i $$-0.415333\pi$$
0.262865 + 0.964833i $$0.415333\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 15.0000 0.653410
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 48.0000 2.08106
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 8.00000 0.344584
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ −4.00000 −0.170872
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 72.0000 3.06730
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 6.00000 0.254457
$$557$$ 21.0000 0.889799 0.444899 0.895581i $$-0.353239\pi$$
0.444899 + 0.895581i $$0.353239\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −11.0000 −0.463595 −0.231797 0.972764i $$-0.574461\pi$$
−0.231797 + 0.972764i $$0.574461\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −22.0000 −0.920671 −0.460336 0.887745i $$-0.652271\pi$$
−0.460336 + 0.887745i $$0.652271\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 20.0000 0.832611 0.416305 0.909225i $$-0.363325\pi$$
0.416305 + 0.909225i $$0.363325\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3.00000 −0.124461
$$582$$ 0 0
$$583$$ 52.0000 2.15362
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.0000 0.577842 0.288921 0.957353i $$-0.406704\pi$$
0.288921 + 0.957353i $$0.406704\pi$$
$$588$$ 0 0
$$589$$ 40.0000 1.64817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 36.0000 1.47959
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −28.0000 −1.14692
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 4.00000 0.163436 0.0817178 0.996656i $$-0.473959\pi$$
0.0817178 + 0.996656i $$0.473959\pi$$
$$600$$ 0 0
$$601$$ −41.0000 −1.67242 −0.836212 0.548406i $$-0.815235\pi$$
−0.836212 + 0.548406i $$0.815235\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 10.0000 0.405887 0.202944 0.979190i $$-0.434949\pi$$
0.202944 + 0.979190i $$0.434949\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −14.0000 −0.565455 −0.282727 0.959200i $$-0.591239\pi$$
−0.282727 + 0.959200i $$0.591239\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −33.0000 −1.32853 −0.664265 0.747497i $$-0.731255\pi$$
−0.664265 + 0.747497i $$0.731255\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 24.0000 0.961540
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −6.00000 −0.239426
$$629$$ −27.0000 −1.07656
$$630$$ 0 0
$$631$$ 38.0000 1.51276 0.756378 0.654135i $$-0.226967\pi$$
0.756378 + 0.654135i $$0.226967\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 20.0000 0.789953 0.394976 0.918691i $$-0.370753\pi$$
0.394976 + 0.918691i $$0.370753\pi$$
$$642$$ 0 0
$$643$$ −13.0000 −0.512670 −0.256335 0.966588i $$-0.582515\pi$$
−0.256335 + 0.966588i $$0.582515\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 16.0000 0.626128 0.313064 0.949732i $$-0.398644\pi$$
0.313064 + 0.949732i $$0.398644\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −28.0000 −1.09322
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9.00000 −0.348481
$$668$$ 44.0000 1.70241
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −56.0000 −2.16186
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 26.0000 1.00000
$$677$$ −45.0000 −1.72949 −0.864745 0.502211i $$-0.832520\pi$$
−0.864745 + 0.502211i $$0.832520\pi$$
$$678$$ 0 0
$$679$$ −30.0000 −1.15129
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 26.0000 0.994862 0.497431 0.867503i $$-0.334277\pi$$
0.497431 + 0.867503i $$0.334277\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −16.0000 −0.609994
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 48.0000 1.82469
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 21.0000 0.795432
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ −72.0000 −2.71553
$$704$$ −32.0000 −1.20605
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −27.0000 −1.01544
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −5.00000 −0.187251
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 7.00000 0.261056 0.130528 0.991445i $$-0.458333\pi$$
0.130528 + 0.991445i $$0.458333\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 37.0000 1.37225 0.686127 0.727482i $$-0.259309\pi$$
0.686127 + 0.727482i $$0.259309\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ −51.0000 −1.88373 −0.941864 0.335994i $$-0.890928\pi$$
−0.941864 + 0.335994i $$0.890928\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −52.0000 −1.91544
$$738$$ 0 0
$$739$$ 27.0000 0.993211 0.496606 0.867976i $$-0.334580\pi$$
0.496606 + 0.867976i $$0.334580\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 24.0000 0.877527
$$749$$ 33.0000 1.20579
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −45.0000 −1.63555 −0.817776 0.575536i $$-0.804793\pi$$
−0.817776 + 0.575536i $$0.804793\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 41.0000 1.48625 0.743124 0.669153i $$-0.233343\pi$$
0.743124 + 0.669153i $$0.233343\pi$$
$$762$$ 0 0
$$763$$ −48.0000 −1.73772
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 6.00000 0.216366 0.108183 0.994131i $$-0.465497\pi$$
0.108183 + 0.994131i $$0.465497\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8.00000 0.287926
$$773$$ −46.0000 −1.65451 −0.827253 0.561830i $$-0.810097\pi$$
−0.827253 + 0.561830i $$0.810097\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 56.0000 2.00641
$$780$$ 0 0
$$781$$ 52.0000 1.86071
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 8.00000 0.285714
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 27.0000 0.962446 0.481223 0.876598i $$-0.340193\pi$$
0.481223 + 0.876598i $$0.340193\pi$$
$$788$$ −4.00000 −0.142494
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3.00000 0.106668
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 36.0000 1.27599
$$797$$ 27.0000 0.956389 0.478195 0.878254i $$-0.341291\pi$$
0.478195 + 0.878254i $$0.341291\pi$$
$$798$$ 0 0
$$799$$ 6.00000 0.212265
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 16.0000 0.564628
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 0 0
$$811$$ −25.0000 −0.877869 −0.438934 0.898519i $$-0.644644\pi$$
−0.438934 + 0.898519i $$0.644644\pi$$
$$812$$ 54.0000 1.89503
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 32.0000 1.11954
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 37.0000 1.28662 0.643308 0.765607i $$-0.277561\pi$$
0.643308 + 0.765607i $$0.277561\pi$$
$$828$$ 0 0
$$829$$ −27.0000 −0.937749 −0.468874 0.883265i $$-0.655340\pi$$
−0.468874 + 0.883265i $$0.655340\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 64.0000 2.21349
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 6.00000 0.207143 0.103572 0.994622i $$-0.466973\pi$$
0.103572 + 0.994622i $$0.466973\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 0 0
$$844$$ −14.0000 −0.481900
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 15.0000 0.515406
$$848$$ 52.0000 1.78569
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 9.00000 0.308516
$$852$$ 0 0
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 43.0000 1.46714 0.733571 0.679613i $$-0.237852\pi$$
0.733571 + 0.679613i $$0.237852\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 54.0000 1.83818 0.919091 0.394046i $$-0.128925\pi$$
0.919091 + 0.394046i $$0.128925\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 30.0000 1.01827
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ 34.0000 1.14419 0.572096 0.820187i $$-0.306131\pi$$
0.572096 + 0.820187i $$0.306131\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −46.0000 −1.54453 −0.772264 0.635301i $$-0.780876\pi$$
−0.772264 + 0.635301i $$0.780876\pi$$
$$888$$ 0 0
$$889$$ 54.0000 1.81110
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 20.0000 0.669650
$$893$$ 16.0000 0.535420
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 45.0000 1.50083
$$900$$ 0 0
$$901$$ −39.0000 −1.29928
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −53.0000 −1.75984 −0.879918 0.475125i $$-0.842403\pi$$
−0.879918 + 0.475125i $$0.842403\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ −4.00000 −0.132381
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −8.00000 −0.264327
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −57.0000 −1.87011 −0.935055 0.354504i $$-0.884650\pi$$
−0.935055 + 0.354504i $$0.884650\pi$$
$$930$$ 0 0
$$931$$ −16.0000 −0.524379
$$932$$ 16.0000 0.524097
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 0 0
$$943$$ −7.00000 −0.227951
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8.00000 −0.259965 −0.129983 0.991516i $$-0.541492\pi$$
−0.129983 + 0.991516i $$0.541492\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 26.0000 0.840900
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −40.0000 −1.28831
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −10.0000 −0.321578 −0.160789 0.986989i $$-0.551404\pi$$
−0.160789 + 0.986989i $$0.551404\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ −9.00000 −0.288527
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −56.0000 −1.79252
$$977$$ −25.0000 −0.799821 −0.399910 0.916554i $$-0.630959\pi$$
−0.399910 + 0.916554i $$0.630959\pi$$
$$978$$ 0 0
$$979$$ 32.0000 1.02272
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 15.0000 0.478426 0.239213 0.970967i $$-0.423111\pi$$
0.239213 + 0.970967i $$0.423111\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ −33.0000 −1.04828 −0.524140 0.851632i $$-0.675613\pi$$
−0.524140 + 0.851632i $$0.675613\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −8.00000 −0.253363 −0.126681 0.991943i $$-0.540433\pi$$
−0.126681 + 0.991943i $$0.540433\pi$$
$$998$$ 0 0
$$999$$ 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 5175.2.a.o.1.1 1
3.2 odd 2 1725.2.a.k.1.1 1
5.4 even 2 1035.2.a.d.1.1 1
15.2 even 4 1725.2.b.j.1174.2 2
15.8 even 4 1725.2.b.j.1174.1 2
15.14 odd 2 345.2.a.d.1.1 1
60.59 even 2 5520.2.a.h.1.1 1
345.344 even 2 7935.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.d.1.1 1 15.14 odd 2
1035.2.a.d.1.1 1 5.4 even 2
1725.2.a.k.1.1 1 3.2 odd 2
1725.2.b.j.1174.1 2 15.8 even 4
1725.2.b.j.1174.2 2 15.2 even 4
5175.2.a.o.1.1 1 1.1 even 1 trivial
5520.2.a.h.1.1 1 60.59 even 2
7935.2.a.i.1.1 1 345.344 even 2