# Properties

 Label 2475.2.a.b.1.1 Level $2475$ Weight $2$ Character 2475.1 Self dual yes Analytic conductor $19.763$ 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] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$19.7629745003$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{4} -3.00000 q^{7} +3.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{4} -3.00000 q^{7} +3.00000 q^{8} +1.00000 q^{11} +2.00000 q^{13} +3.00000 q^{14} -1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} -1.00000 q^{22} -1.00000 q^{23} -2.00000 q^{26} +3.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +3.00000 q^{34} +1.00000 q^{37} +1.00000 q^{38} -5.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{46} -3.00000 q^{47} +2.00000 q^{49} -2.00000 q^{52} -10.0000 q^{53} -9.00000 q^{56} -6.00000 q^{58} +11.0000 q^{59} +14.0000 q^{61} -4.00000 q^{62} +7.00000 q^{64} +2.00000 q^{67} +3.00000 q^{68} -5.00000 q^{71} +2.00000 q^{73} -1.00000 q^{74} +1.00000 q^{76} -3.00000 q^{77} +5.00000 q^{79} +5.00000 q^{82} -8.00000 q^{83} -4.00000 q^{86} +3.00000 q^{88} -10.0000 q^{89} -6.00000 q^{91} +1.00000 q^{92} +3.00000 q^{94} -17.0000 q^{97} -2.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 3.00000 0.566947
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.00000 0.164399 0.0821995 0.996616i $$-0.473806\pi$$
0.0821995 + 0.996616i $$0.473806\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −9.00000 −1.20268
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 11.0000 1.43208 0.716039 0.698060i $$-0.245953\pi$$
0.716039 + 0.698060i $$0.245953\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 5.00000 0.552158
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 3.00000 0.319801
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ 1.00000 0.104257
$$93$$ 0 0
$$94$$ 3.00000 0.309426
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −17.0000 −1.72609 −0.863044 0.505128i $$-0.831445\pi$$
−0.863044 + 0.505128i $$0.831445\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −11.0000 −1.09454 −0.547270 0.836956i $$-0.684333\pi$$
−0.547270 + 0.836956i $$0.684333\pi$$
$$102$$ 0 0
$$103$$ 2.00000 0.197066 0.0985329 0.995134i $$-0.468585\pi$$
0.0985329 + 0.995134i $$0.468585\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 0 0
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 3.00000 0.283473
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ −11.0000 −1.01263
$$119$$ 9.00000 0.825029
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −14.0000 −1.26750
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −5.00000 −0.443678 −0.221839 0.975083i $$-0.571206\pi$$
−0.221839 + 0.975083i $$0.571206\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 3.00000 0.260133
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ −9.00000 −0.771744
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 5.00000 0.419591
$$143$$ 2.00000 0.167248
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ −1.00000 −0.0821995
$$149$$ 7.00000 0.573462 0.286731 0.958011i $$-0.407431\pi$$
0.286731 + 0.958011i $$0.407431\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −3.00000 −0.243332
$$153$$ 0 0
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ −5.00000 −0.397779
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ 10.0000 0.783260 0.391630 0.920123i $$-0.371911\pi$$
0.391630 + 0.920123i $$0.371911\pi$$
$$164$$ 5.00000 0.390434
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ −10.0000 −0.773823 −0.386912 0.922117i $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ 9.00000 0.684257 0.342129 0.939653i $$-0.388852\pi$$
0.342129 + 0.939653i $$0.388852\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ −1.00000 −0.0747435 −0.0373718 0.999301i $$-0.511899\pi$$
−0.0373718 + 0.999301i $$0.511899\pi$$
$$180$$ 0 0
$$181$$ −25.0000 −1.85824 −0.929118 0.369784i $$-0.879432\pi$$
−0.929118 + 0.369784i $$0.879432\pi$$
$$182$$ 6.00000 0.444750
$$183$$ 0 0
$$184$$ −3.00000 −0.221163
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3.00000 −0.219382
$$188$$ 3.00000 0.218797
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 19.0000 1.37479 0.687396 0.726283i $$-0.258754\pi$$
0.687396 + 0.726283i $$0.258754\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 17.0000 1.22053
$$195$$ 0 0
$$196$$ −2.00000 −0.142857
$$197$$ −23.0000 −1.63868 −0.819341 0.573306i $$-0.805660\pi$$
−0.819341 + 0.573306i $$0.805660\pi$$
$$198$$ 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
$$202$$ 11.0000 0.773957
$$203$$ −18.0000 −1.26335
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2.00000 −0.139347
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 0 0
$$214$$ 18.0000 1.23045
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −12.0000 −0.814613
$$218$$ 12.0000 0.812743
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ −22.0000 −1.47323 −0.736614 0.676313i $$-0.763577\pi$$
−0.736614 + 0.676313i $$0.763577\pi$$
$$224$$ 15.0000 1.00223
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 0 0
$$229$$ 11.0000 0.726900 0.363450 0.931614i $$-0.381599\pi$$
0.363450 + 0.931614i $$0.381599\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 18.0000 1.18176
$$233$$ −9.00000 −0.589610 −0.294805 0.955557i $$-0.595255\pi$$
−0.294805 + 0.955557i $$0.595255\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −11.0000 −0.716039
$$237$$ 0 0
$$238$$ −9.00000 −0.583383
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ 4.00000 0.257663 0.128831 0.991667i $$-0.458877\pi$$
0.128831 + 0.991667i $$0.458877\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 12.0000 0.762001
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ −1.00000 −0.0628695
$$254$$ 5.00000 0.313728
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ −3.00000 −0.186411
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −12.0000 −0.741362
$$263$$ 18.0000 1.10993 0.554964 0.831875i $$-0.312732\pi$$
0.554964 + 0.831875i $$0.312732\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −3.00000 −0.183942
$$267$$ 0 0
$$268$$ −2.00000 −0.122169
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ −3.00000 −0.182237 −0.0911185 0.995840i $$-0.529044\pi$$
−0.0911185 + 0.995840i $$0.529044\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −21.0000 −1.25275 −0.626377 0.779520i $$-0.715463\pi$$
−0.626377 + 0.779520i $$0.715463\pi$$
$$282$$ 0 0
$$283$$ −23.0000 −1.36721 −0.683604 0.729853i $$-0.739588\pi$$
−0.683604 + 0.729853i $$0.739588\pi$$
$$284$$ 5.00000 0.296695
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 15.0000 0.885422
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −2.00000 −0.117041
$$293$$ −13.0000 −0.759468 −0.379734 0.925096i $$-0.623985\pi$$
−0.379734 + 0.925096i $$0.623985\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 3.00000 0.174371
$$297$$ 0 0
$$298$$ −7.00000 −0.405499
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 32.0000 1.82634 0.913168 0.407583i $$-0.133628\pi$$
0.913168 + 0.407583i $$0.133628\pi$$
$$308$$ 3.00000 0.170941
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −3.00000 −0.167183
$$323$$ 3.00000 0.166924
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −10.0000 −0.553849
$$327$$ 0 0
$$328$$ −15.0000 −0.828236
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 0 0
$$334$$ 10.0000 0.547176
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.00000 0.326841 0.163420 0.986557i $$-0.447747\pi$$
0.163420 + 0.986557i $$0.447747\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ −9.00000 −0.483843
$$347$$ 10.0000 0.536828 0.268414 0.963304i $$-0.413500\pi$$
0.268414 + 0.963304i $$0.413500\pi$$
$$348$$ 0 0
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5.00000 −0.266501
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ 1.00000 0.0528516
$$359$$ −28.0000 −1.47778 −0.738892 0.673824i $$-0.764651\pi$$
−0.738892 + 0.673824i $$0.764651\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 25.0000 1.31397
$$363$$ 0 0
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 4.00000 0.208798 0.104399 0.994535i $$-0.466708\pi$$
0.104399 + 0.994535i $$0.466708\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 30.0000 1.55752
$$372$$ 0 0
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 3.00000 0.155126
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −19.0000 −0.972125
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −4.00000 −0.203595
$$387$$ 0 0
$$388$$ 17.0000 0.863044
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 3.00000 0.151717
$$392$$ 6.00000 0.303046
$$393$$ 0 0
$$394$$ 23.0000 1.15872
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −6.00000 −0.301131 −0.150566 0.988600i $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ 18.0000 0.902258
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −38.0000 −1.89763 −0.948815 0.315833i $$-0.897716\pi$$
−0.948815 + 0.315833i $$0.897716\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ 11.0000 0.547270
$$405$$ 0 0
$$406$$ 18.0000 0.893325
$$407$$ 1.00000 0.0495682
$$408$$ 0 0
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −2.00000 −0.0985329
$$413$$ −33.0000 −1.62382
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −10.0000 −0.490290
$$417$$ 0 0
$$418$$ 1.00000 0.0489116
$$419$$ −23.0000 −1.12362 −0.561812 0.827265i $$-0.689895\pi$$
−0.561812 + 0.827265i $$0.689895\pi$$
$$420$$ 0 0
$$421$$ 9.00000 0.438633 0.219317 0.975654i $$-0.429617\pi$$
0.219317 + 0.975654i $$0.429617\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 0 0
$$424$$ −30.0000 −1.45693
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −42.0000 −2.03252
$$428$$ 18.0000 0.870063
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 12.0000 0.576018
$$435$$ 0 0
$$436$$ 12.0000 0.574696
$$437$$ 1.00000 0.0478365
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6.00000 0.285391
$$443$$ 31.0000 1.47285 0.736427 0.676517i $$-0.236511\pi$$
0.736427 + 0.676517i $$0.236511\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 22.0000 1.04173
$$447$$ 0 0
$$448$$ −21.0000 −0.992157
$$449$$ −8.00000 −0.377543 −0.188772 0.982021i $$-0.560451\pi$$
−0.188772 + 0.982021i $$0.560451\pi$$
$$450$$ 0 0
$$451$$ −5.00000 −0.235441
$$452$$ 18.0000 0.846649
$$453$$ 0 0
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 28.0000 1.30978 0.654892 0.755722i $$-0.272714\pi$$
0.654892 + 0.755722i $$0.272714\pi$$
$$458$$ −11.0000 −0.513996
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 34.0000 1.58354 0.791769 0.610821i $$-0.209160\pi$$
0.791769 + 0.610821i $$0.209160\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 9.00000 0.416917
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ −6.00000 −0.277054
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 33.0000 1.51895
$$473$$ 4.00000 0.183920
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −9.00000 −0.412514
$$477$$ 0 0
$$478$$ 20.0000 0.914779
$$479$$ −18.0000 −0.822441 −0.411220 0.911536i $$-0.634897\pi$$
−0.411220 + 0.911536i $$0.634897\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ −4.00000 −0.182195
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ 42.0000 1.90125
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ −18.0000 −0.810679
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 15.0000 0.672842
$$498$$ 0 0
$$499$$ 24.0000 1.07439 0.537194 0.843459i $$-0.319484\pi$$
0.537194 + 0.843459i $$0.319484\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ −20.0000 −0.891756 −0.445878 0.895094i $$-0.647108\pi$$
−0.445878 + 0.895094i $$0.647108\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1.00000 0.0444554
$$507$$ 0 0
$$508$$ 5.00000 0.221839
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ −6.00000 −0.265424
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ 12.0000 0.529297
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3.00000 −0.131940
$$518$$ 3.00000 0.131812
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ 0 0
$$523$$ 7.00000 0.306089 0.153044 0.988219i $$-0.451092\pi$$
0.153044 + 0.988219i $$0.451092\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −18.0000 −0.784837
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −3.00000 −0.130066
$$533$$ −10.0000 −0.433148
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 6.00000 0.259161
$$537$$ 0 0
$$538$$ 24.0000 1.03471
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ 32.0000 1.37579 0.687894 0.725811i $$-0.258536\pi$$
0.687894 + 0.725811i $$0.258536\pi$$
$$542$$ 3.00000 0.128861
$$543$$ 0 0
$$544$$ 15.0000 0.643120
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −33.0000 −1.41098 −0.705489 0.708721i $$-0.749273\pi$$
−0.705489 + 0.708721i $$0.749273\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ −15.0000 −0.637865
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −22.0000 −0.932170 −0.466085 0.884740i $$-0.654336\pi$$
−0.466085 + 0.884740i $$0.654336\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 21.0000 0.885832
$$563$$ −18.0000 −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 23.0000 0.966762
$$567$$ 0 0
$$568$$ −15.0000 −0.629386
$$569$$ 37.0000 1.55112 0.775560 0.631273i $$-0.217467\pi$$
0.775560 + 0.631273i $$0.217467\pi$$
$$570$$ 0 0
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ 0 0
$$574$$ −15.0000 −0.626088
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −43.0000 −1.79011 −0.895057 0.445952i $$-0.852865\pi$$
−0.895057 + 0.445952i $$0.852865\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ −10.0000 −0.414158
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ 13.0000 0.537025
$$587$$ −9.00000 −0.371470 −0.185735 0.982600i $$-0.559467\pi$$
−0.185735 + 0.982600i $$0.559467\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −1.00000 −0.0410997
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −7.00000 −0.286731
$$597$$ 0 0
$$598$$ 2.00000 0.0817861
$$599$$ −27.0000 −1.10319 −0.551595 0.834112i $$-0.685981\pi$$
−0.551595 + 0.834112i $$0.685981\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 12.0000 0.489083
$$603$$ 0 0
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ 0 0
$$613$$ 16.0000 0.646234 0.323117 0.946359i $$-0.395269\pi$$
0.323117 + 0.946359i $$0.395269\pi$$
$$614$$ −32.0000 −1.29141
$$615$$ 0 0
$$616$$ −9.00000 −0.362620
$$617$$ 28.0000 1.12724 0.563619 0.826035i $$-0.309409\pi$$
0.563619 + 0.826035i $$0.309409\pi$$
$$618$$ 0 0
$$619$$ 34.0000 1.36658 0.683288 0.730149i $$-0.260549\pi$$
0.683288 + 0.730149i $$0.260549\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ 30.0000 1.20192
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1.00000 0.0399680
$$627$$ 0 0
$$628$$ −10.0000 −0.399043
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ 34.0000 1.35352 0.676759 0.736204i $$-0.263384\pi$$
0.676759 + 0.736204i $$0.263384\pi$$
$$632$$ 15.0000 0.596668
$$633$$ 0 0
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4.00000 0.158486
$$638$$ −6.00000 −0.237542
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −16.0000 −0.631962 −0.315981 0.948766i $$-0.602334\pi$$
−0.315981 + 0.948766i $$0.602334\pi$$
$$642$$ 0 0
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ −3.00000 −0.118217
$$645$$ 0 0
$$646$$ −3.00000 −0.118033
$$647$$ 13.0000 0.511083 0.255541 0.966798i $$-0.417746\pi$$
0.255541 + 0.966798i $$0.417746\pi$$
$$648$$ 0 0
$$649$$ 11.0000 0.431788
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −10.0000 −0.391630
$$653$$ 4.00000 0.156532 0.0782660 0.996933i $$-0.475062\pi$$
0.0782660 + 0.996933i $$0.475062\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 5.00000 0.195217
$$657$$ 0 0
$$658$$ −9.00000 −0.350857
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ 35.0000 1.36134 0.680671 0.732589i $$-0.261688\pi$$
0.680671 + 0.732589i $$0.261688\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ −24.0000 −0.931381
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.00000 −0.232321
$$668$$ 10.0000 0.386912
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 14.0000 0.540464
$$672$$ 0 0
$$673$$ 4.00000 0.154189 0.0770943 0.997024i $$-0.475436\pi$$
0.0770943 + 0.997024i $$0.475436\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 14.0000 0.538064 0.269032 0.963131i $$-0.413296\pi$$
0.269032 + 0.963131i $$0.413296\pi$$
$$678$$ 0 0
$$679$$ 51.0000 1.95720
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −4.00000 −0.153168
$$683$$ −47.0000 −1.79841 −0.899203 0.437533i $$-0.855852\pi$$
−0.899203 + 0.437533i $$0.855852\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −40.0000 −1.52167 −0.760836 0.648944i $$-0.775211\pi$$
−0.760836 + 0.648944i $$0.775211\pi$$
$$692$$ −9.00000 −0.342129
$$693$$ 0 0
$$694$$ −10.0000 −0.379595
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 15.0000 0.568166
$$698$$ −8.00000 −0.302804
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 23.0000 0.868698 0.434349 0.900745i $$-0.356978\pi$$
0.434349 + 0.900745i $$0.356978\pi$$
$$702$$ 0 0
$$703$$ −1.00000 −0.0377157
$$704$$ 7.00000 0.263822
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 33.0000 1.24109
$$708$$ 0 0
$$709$$ 35.0000 1.31445 0.657226 0.753693i $$-0.271730\pi$$
0.657226 + 0.753693i $$0.271730\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −30.0000 −1.12430
$$713$$ −4.00000 −0.149801
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 1.00000 0.0373718
$$717$$ 0 0
$$718$$ 28.0000 1.04495
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ −6.00000 −0.223452
$$722$$ 18.0000 0.669891
$$723$$ 0 0
$$724$$ 25.0000 0.929118
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ −18.0000 −0.667124
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ −4.00000 −0.147643
$$735$$ 0 0
$$736$$ 5.00000 0.184302
$$737$$ 2.00000 0.0736709
$$738$$ 0 0
$$739$$ 1.00000 0.0367856 0.0183928 0.999831i $$-0.494145\pi$$
0.0183928 + 0.999831i $$0.494145\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −30.0000 −1.10133
$$743$$ 14.0000 0.513610 0.256805 0.966463i $$-0.417330\pi$$
0.256805 + 0.966463i $$0.417330\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −6.00000 −0.219676
$$747$$ 0 0
$$748$$ 3.00000 0.109691
$$749$$ 54.0000 1.97312
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 3.00000 0.109399
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −42.0000 −1.52652 −0.763258 0.646094i $$-0.776401\pi$$
−0.763258 + 0.646094i $$0.776401\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ 36.0000 1.30329
$$764$$ −19.0000 −0.687396
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 22.0000 0.794374
$$768$$ 0 0
$$769$$ −28.0000 −1.00971 −0.504853 0.863205i $$-0.668453\pi$$
−0.504853 + 0.863205i $$0.668453\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ −44.0000 −1.58257 −0.791285 0.611448i $$-0.790588\pi$$
−0.791285 + 0.611448i $$0.790588\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −51.0000 −1.83079
$$777$$ 0 0
$$778$$ 6.00000 0.215110
$$779$$ 5.00000 0.179144
$$780$$ 0 0
$$781$$ −5.00000 −0.178914
$$782$$ −3.00000 −0.107280
$$783$$ 0 0
$$784$$ −2.00000 −0.0714286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 43.0000 1.53278 0.766392 0.642373i $$-0.222050\pi$$
0.766392 + 0.642373i $$0.222050\pi$$
$$788$$ 23.0000 0.819341
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 54.0000 1.92002
$$792$$ 0 0
$$793$$ 28.0000 0.994309
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ 18.0000 0.637993
$$797$$ −22.0000 −0.779280 −0.389640 0.920967i $$-0.627401\pi$$
−0.389640 + 0.920967i $$0.627401\pi$$
$$798$$ 0 0
$$799$$ 9.00000 0.318397
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 38.0000 1.34183
$$803$$ 2.00000 0.0705785
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ −33.0000 −1.16094
$$809$$ −15.0000 −0.527372 −0.263686 0.964609i $$-0.584938\pi$$
−0.263686 + 0.964609i $$0.584938\pi$$
$$810$$ 0 0
$$811$$ −21.0000 −0.737410 −0.368705 0.929547i $$-0.620199\pi$$
−0.368705 + 0.929547i $$0.620199\pi$$
$$812$$ 18.0000 0.631676
$$813$$ 0 0
$$814$$ −1.00000 −0.0350500
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ −6.00000 −0.209785
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ −42.0000 −1.46403 −0.732014 0.681290i $$-0.761419\pi$$
−0.732014 + 0.681290i $$0.761419\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ 33.0000 1.14822
$$827$$ 48.0000 1.66912 0.834562 0.550914i $$-0.185721\pi$$
0.834562 + 0.550914i $$0.185721\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 14.0000 0.485363
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 1.00000 0.0345857
$$837$$ 0 0
$$838$$ 23.0000 0.794522
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −9.00000 −0.310160
$$843$$ 0 0
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −3.00000 −0.103081
$$848$$ 10.0000 0.343401
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.00000 −0.0342796
$$852$$ 0 0
$$853$$ −40.0000 −1.36957 −0.684787 0.728743i $$-0.740105\pi$$
−0.684787 + 0.728743i $$0.740105\pi$$
$$854$$ 42.0000 1.43721
$$855$$ 0 0
$$856$$ −54.0000 −1.84568
$$857$$ −39.0000 −1.33221 −0.666107 0.745856i $$-0.732041\pi$$
−0.666107 + 0.745856i $$0.732041\pi$$
$$858$$ 0 0
$$859$$ −26.0000 −0.887109 −0.443554 0.896248i $$-0.646283\pi$$
−0.443554 + 0.896248i $$0.646283\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −36.0000 −1.22616
$$863$$ −4.00000 −0.136162 −0.0680808 0.997680i $$-0.521688\pi$$
−0.0680808 + 0.997680i $$0.521688\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ 0 0
$$868$$ 12.0000 0.407307
$$869$$ 5.00000 0.169613
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ −36.0000 −1.21911
$$873$$ 0 0
$$874$$ −1.00000 −0.0338255
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −52.0000 −1.75592 −0.877958 0.478738i $$-0.841094\pi$$
−0.877958 + 0.478738i $$0.841094\pi$$
$$878$$ 35.0000 1.18119
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 2.00000 0.0673054 0.0336527 0.999434i $$-0.489286\pi$$
0.0336527 + 0.999434i $$0.489286\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ −31.0000 −1.04147
$$887$$ −14.0000 −0.470074 −0.235037 0.971986i $$-0.575521\pi$$
−0.235037 + 0.971986i $$0.575521\pi$$
$$888$$ 0 0
$$889$$ 15.0000 0.503084
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 22.0000 0.736614
$$893$$ 3.00000 0.100391
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −9.00000 −0.300669
$$897$$ 0 0
$$898$$ 8.00000 0.266963
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 30.0000 0.999445
$$902$$ 5.00000 0.166482
$$903$$ 0 0
$$904$$ −54.0000 −1.79601
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 2.00000 0.0664089 0.0332045 0.999449i $$-0.489429\pi$$
0.0332045 + 0.999449i $$0.489429\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −53.0000 −1.75597 −0.877984 0.478690i $$-0.841112\pi$$
−0.877984 + 0.478690i $$0.841112\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ −28.0000 −0.926158
$$915$$ 0 0
$$916$$ −11.0000 −0.363450
$$917$$ −36.0000 −1.18882
$$918$$ 0 0
$$919$$ −39.0000 −1.28649 −0.643246 0.765660i $$-0.722413\pi$$
−0.643246 + 0.765660i $$0.722413\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −34.0000 −1.11973
$$923$$ −10.0000 −0.329154
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −24.0000 −0.788689
$$927$$ 0 0
$$928$$ −30.0000 −0.984798
$$929$$ 32.0000 1.04989 0.524943 0.851137i $$-0.324087\pi$$
0.524943 + 0.851137i $$0.324087\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 9.00000 0.294805
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 18.0000 0.588034 0.294017 0.955800i $$-0.405008\pi$$
0.294017 + 0.955800i $$0.405008\pi$$
$$938$$ 6.00000 0.195907
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 5.00000 0.162995 0.0814977 0.996674i $$-0.474030\pi$$
0.0814977 + 0.996674i $$0.474030\pi$$
$$942$$ 0 0
$$943$$ 5.00000 0.162822
$$944$$ −11.0000 −0.358020
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ 27.0000 0.877382 0.438691 0.898638i $$-0.355442\pi$$
0.438691 + 0.898638i $$0.355442\pi$$
$$948$$ 0 0
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 27.0000 0.875075
$$953$$ 21.0000 0.680257 0.340128 0.940379i $$-0.389529\pi$$
0.340128 + 0.940379i $$0.389529\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 20.0000 0.646846
$$957$$ 0 0
$$958$$ 18.0000 0.581554
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −2.00000 −0.0644826
$$963$$ 0 0
$$964$$ −4.00000 −0.128831
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 20.0000 0.643157 0.321578 0.946883i $$-0.395787\pi$$
0.321578 + 0.946883i $$0.395787\pi$$
$$968$$ 3.00000 0.0964237
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 21.0000 0.673922 0.336961 0.941519i $$-0.390601\pi$$
0.336961 + 0.941519i $$0.390601\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ −46.0000 −1.47167 −0.735835 0.677161i $$-0.763210\pi$$
−0.735835 + 0.677161i $$0.763210\pi$$
$$978$$ 0 0
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −28.0000 −0.893516
$$983$$ 49.0000 1.56286 0.781429 0.623995i $$-0.214491\pi$$
0.781429 + 0.623995i $$0.214491\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 18.0000 0.573237
$$987$$ 0 0
$$988$$ 2.00000 0.0636285
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ −20.0000 −0.635001
$$993$$ 0 0
$$994$$ −15.0000 −0.475771
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 50.0000 1.58352 0.791758 0.610835i $$-0.209166\pi$$
0.791758 + 0.610835i $$0.209166\pi$$
$$998$$ −24.0000 −0.759707
$$999$$ 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 2475.2.a.b.1.1 1
3.2 odd 2 2475.2.a.h.1.1 yes 1
5.2 odd 4 2475.2.c.e.199.1 2
5.3 odd 4 2475.2.c.e.199.2 2
5.4 even 2 2475.2.a.k.1.1 yes 1
15.2 even 4 2475.2.c.c.199.2 2
15.8 even 4 2475.2.c.c.199.1 2
15.14 odd 2 2475.2.a.d.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2475.2.a.b.1.1 1 1.1 even 1 trivial
2475.2.a.d.1.1 yes 1 15.14 odd 2
2475.2.a.h.1.1 yes 1 3.2 odd 2
2475.2.a.k.1.1 yes 1 5.4 even 2
2475.2.c.c.199.1 2 15.8 even 4
2475.2.c.c.199.2 2 15.2 even 4
2475.2.c.e.199.1 2 5.2 odd 4
2475.2.c.e.199.2 2 5.3 odd 4