# Properties

 Label 4275.2.a.a.1.1 Level $4275$ Weight $2$ Character 4275.1 Self dual yes Analytic conductor $34.136$ Analytic rank $0$ 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] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.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: $$34.1360468641$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4275.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{7} +O(q^{10})$$ $$q-2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{7} +3.00000 q^{11} +6.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} +3.00000 q^{17} -1.00000 q^{19} -6.00000 q^{22} +4.00000 q^{23} -12.0000 q^{26} -6.00000 q^{28} +10.0000 q^{29} +2.00000 q^{31} +8.00000 q^{32} -6.00000 q^{34} -8.00000 q^{37} +2.00000 q^{38} +8.00000 q^{41} +1.00000 q^{43} +6.00000 q^{44} -8.00000 q^{46} +3.00000 q^{47} +2.00000 q^{49} +12.0000 q^{52} -6.00000 q^{53} -20.0000 q^{58} +7.00000 q^{61} -4.00000 q^{62} -8.00000 q^{64} -8.00000 q^{67} +6.00000 q^{68} -12.0000 q^{71} +11.0000 q^{73} +16.0000 q^{74} -2.00000 q^{76} -9.00000 q^{77} -16.0000 q^{82} +4.00000 q^{83} -2.00000 q^{86} -10.0000 q^{89} -18.0000 q^{91} +8.00000 q^{92} -6.00000 q^{94} +2.00000 q^{97} -4.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 6.00000 1.60357
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −12.0000 −2.35339
$$27$$ 0 0
$$28$$ −6.00000 −1.13389
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 12.0000 1.66410
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −20.0000 −2.62613
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 16.0000 1.85996
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ −9.00000 −1.02565
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −16.0000 −1.76690
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −18.0000 −1.88691
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −4.00000 −0.404061
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 12.0000 1.16554
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 0 0
$$109$$ 20.0000 1.91565 0.957826 0.287348i $$-0.0927736\pi$$
0.957826 + 0.287348i $$0.0927736\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 12.0000 1.13389
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 20.0000 1.85695
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −9.00000 −0.825029
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −14.0000 −1.26750
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 13.0000 1.13582 0.567908 0.823092i $$-0.307753\pi$$
0.567908 + 0.823092i $$0.307753\pi$$
$$132$$ 0 0
$$133$$ 3.00000 0.260133
$$134$$ 16.0000 1.38219
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.00000 0.256307 0.128154 0.991754i $$-0.459095\pi$$
0.128154 + 0.991754i $$0.459095\pi$$
$$138$$ 0 0
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 24.0000 2.01404
$$143$$ 18.0000 1.50524
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −22.0000 −1.82073
$$147$$ 0 0
$$148$$ −16.0000 −1.31519
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ 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
$$154$$ 18.0000 1.45048
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 0 0
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 16.0000 1.24939
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −12.0000 −0.904534
$$177$$ 0 0
$$178$$ 20.0000 1.49906
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 36.0000 2.66850
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000 0.658145
$$188$$ 6.00000 0.437595
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −4.00000 −0.287183
$$195$$ 0 0
$$196$$ 4.00000 0.285714
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 4.00000 0.281439
$$203$$ −30.0000 −2.10559
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 28.0000 1.95085
$$207$$ 0 0
$$208$$ −24.0000 −1.66410
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ −12.0000 −0.824163
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −6.00000 −0.407307
$$218$$ −40.0000 −2.70914
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 18.0000 1.21081
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −24.0000 −1.60357
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −11.0000 −0.720634 −0.360317 0.932830i $$-0.617331\pi$$
−0.360317 + 0.932830i $$0.617331\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 18.0000 1.16677
$$239$$ 15.0000 0.970269 0.485135 0.874439i $$-0.338771\pi$$
0.485135 + 0.874439i $$0.338771\pi$$
$$240$$ 0 0
$$241$$ 12.0000 0.772988 0.386494 0.922292i $$-0.373686\pi$$
0.386494 + 0.922292i $$0.373686\pi$$
$$242$$ 4.00000 0.257130
$$243$$ 0 0
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.00000 −0.381771
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −27.0000 −1.70422 −0.852112 0.523359i $$-0.824679\pi$$
−0.852112 + 0.523359i $$0.824679\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 8.00000 0.499026 0.249513 0.968371i $$-0.419729\pi$$
0.249513 + 0.968371i $$0.419729\pi$$
$$258$$ 0 0
$$259$$ 24.0000 1.49129
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −26.0000 −1.60629
$$263$$ −21.0000 −1.29492 −0.647458 0.762101i $$-0.724168\pi$$
−0.647458 + 0.762101i $$0.724168\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.00000 −0.367884
$$267$$ 0 0
$$268$$ −16.0000 −0.977356
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ −12.0000 −0.727607
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −13.0000 −0.781094 −0.390547 0.920583i $$-0.627714\pi$$
−0.390547 + 0.920583i $$0.627714\pi$$
$$278$$ 10.0000 0.599760
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ 0 0
$$283$$ −19.0000 −1.12943 −0.564716 0.825285i $$-0.691014\pi$$
−0.564716 + 0.825285i $$0.691014\pi$$
$$284$$ −24.0000 −1.42414
$$285$$ 0 0
$$286$$ −36.0000 −2.12872
$$287$$ −24.0000 −1.41668
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 22.0000 1.28745
$$293$$ 4.00000 0.233682 0.116841 0.993151i $$-0.462723\pi$$
0.116841 + 0.993151i $$0.462723\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 30.0000 1.73785
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ −18.0000 −1.02565
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −7.00000 −0.396934 −0.198467 0.980108i $$-0.563596\pi$$
−0.198467 + 0.980108i $$0.563596\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −12.0000 −0.673987 −0.336994 0.941507i $$-0.609410\pi$$
−0.336994 + 0.941507i $$0.609410\pi$$
$$318$$ 0 0
$$319$$ 30.0000 1.67968
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 24.0000 1.33747
$$323$$ −3.00000 −0.166924
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −32.0000 −1.77232
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −9.00000 −0.496186
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 0 0
$$334$$ −36.0000 −1.96983
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ −46.0000 −2.50207
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −28.0000 −1.50529
$$347$$ 3.00000 0.161048 0.0805242 0.996753i $$-0.474341\pi$$
0.0805242 + 0.996753i $$0.474341\pi$$
$$348$$ 0 0
$$349$$ 25.0000 1.33822 0.669110 0.743164i $$-0.266676\pi$$
0.669110 + 0.743164i $$0.266676\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 24.0000 1.27920
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −20.0000 −1.06000
$$357$$ 0 0
$$358$$ −20.0000 −1.05703
$$359$$ −25.0000 −1.31945 −0.659725 0.751507i $$-0.729327\pi$$
−0.659725 + 0.751507i $$0.729327\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −4.00000 −0.210235
$$363$$ 0 0
$$364$$ −36.0000 −1.88691
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ −16.0000 −0.834058
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 18.0000 0.934513
$$372$$ 0 0
$$373$$ 16.0000 0.828449 0.414224 0.910175i $$-0.364053\pi$$
0.414224 + 0.910175i $$0.364053\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 60.0000 3.09016
$$378$$ 0 0
$$379$$ −30.0000 −1.54100 −0.770498 0.637442i $$-0.779993\pi$$
−0.770498 + 0.637442i $$0.779993\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −6.00000 −0.306987
$$383$$ 14.0000 0.715367 0.357683 0.933843i $$-0.383567\pi$$
0.357683 + 0.933843i $$0.383567\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 8.00000 0.407189
$$387$$ 0 0
$$388$$ 4.00000 0.203069
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 4.00000 0.201517
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 7.00000 0.351320 0.175660 0.984451i $$-0.443794\pi$$
0.175660 + 0.984451i $$0.443794\pi$$
$$398$$ 10.0000 0.501255
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 28.0000 1.39825 0.699127 0.714998i $$-0.253572\pi$$
0.699127 + 0.714998i $$0.253572\pi$$
$$402$$ 0 0
$$403$$ 12.0000 0.597763
$$404$$ −4.00000 −0.199007
$$405$$ 0 0
$$406$$ 60.0000 2.97775
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −28.0000 −1.37946
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 48.0000 2.35339
$$417$$ 0 0
$$418$$ 6.00000 0.293470
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 56.0000 2.72604
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −21.0000 −1.01626
$$428$$ −4.00000 −0.193347
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 12.0000 0.576018
$$435$$ 0 0
$$436$$ 40.0000 1.91565
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −36.0000 −1.71235
$$443$$ 39.0000 1.85295 0.926473 0.376361i $$-0.122825\pi$$
0.926473 + 0.376361i $$0.122825\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 0 0
$$448$$ 24.0000 1.13389
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ −12.0000 −0.564433
$$453$$ 0 0
$$454$$ −36.0000 −1.68956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3.00000 −0.140334 −0.0701670 0.997535i $$-0.522353\pi$$
−0.0701670 + 0.997535i $$0.522353\pi$$
$$458$$ 30.0000 1.40181
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 33.0000 1.53696 0.768482 0.639872i $$-0.221013\pi$$
0.768482 + 0.639872i $$0.221013\pi$$
$$462$$ 0 0
$$463$$ 31.0000 1.44069 0.720346 0.693615i $$-0.243983\pi$$
0.720346 + 0.693615i $$0.243983\pi$$
$$464$$ −40.0000 −1.85695
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ −17.0000 −0.786666 −0.393333 0.919396i $$-0.628678\pi$$
−0.393333 + 0.919396i $$0.628678\pi$$
$$468$$ 0 0
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 3.00000 0.137940
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −18.0000 −0.825029
$$477$$ 0 0
$$478$$ −30.0000 −1.37217
$$479$$ 40.0000 1.82765 0.913823 0.406112i $$-0.133116\pi$$
0.913823 + 0.406112i $$0.133116\pi$$
$$480$$ 0 0
$$481$$ −48.0000 −2.18861
$$482$$ −24.0000 −1.09317
$$483$$ 0 0
$$484$$ −4.00000 −0.181818
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 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
$$493$$ 30.0000 1.35113
$$494$$ 12.0000 0.539906
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 36.0000 1.61482
$$498$$ 0 0
$$499$$ −35.0000 −1.56682 −0.783408 0.621508i $$-0.786520\pi$$
−0.783408 + 0.621508i $$0.786520\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 54.0000 2.41014
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −24.0000 −1.06693
$$507$$ 0 0
$$508$$ 4.00000 0.177471
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ −33.0000 −1.45983
$$512$$ −32.0000 −1.41421
$$513$$ 0 0
$$514$$ −16.0000 −0.705730
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 9.00000 0.395820
$$518$$ −48.0000 −2.10900
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 28.0000 1.22670 0.613351 0.789810i $$-0.289821\pi$$
0.613351 + 0.789810i $$0.289821\pi$$
$$522$$ 0 0
$$523$$ −14.0000 −0.612177 −0.306089 0.952003i $$-0.599020\pi$$
−0.306089 + 0.952003i $$0.599020\pi$$
$$524$$ 26.0000 1.13582
$$525$$ 0 0
$$526$$ 42.0000 1.83129
$$527$$ 6.00000 0.261364
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 6.00000 0.260133
$$533$$ 48.0000 2.07911
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −60.0000 −2.58678
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −13.0000 −0.558914 −0.279457 0.960158i $$-0.590154\pi$$
−0.279457 + 0.960158i $$0.590154\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ 0 0
$$544$$ 24.0000 1.02899
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2.00000 0.0855138 0.0427569 0.999086i $$-0.486386\pi$$
0.0427569 + 0.999086i $$0.486386\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −10.0000 −0.426014
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −10.0000 −0.424094
$$557$$ 3.00000 0.127114 0.0635570 0.997978i $$-0.479756\pi$$
0.0635570 + 0.997978i $$0.479756\pi$$
$$558$$ 0 0
$$559$$ 6.00000 0.253773
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 4.00000 0.168730
$$563$$ 44.0000 1.85438 0.927189 0.374593i $$-0.122217\pi$$
0.927189 + 0.374593i $$0.122217\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 38.0000 1.59726
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 36.0000 1.50524
$$573$$ 0 0
$$574$$ 48.0000 2.00348
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −3.00000 −0.124892 −0.0624458 0.998048i $$-0.519890\pi$$
−0.0624458 + 0.998048i $$0.519890\pi$$
$$578$$ 16.0000 0.665512
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ −18.0000 −0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −8.00000 −0.330477
$$587$$ −37.0000 −1.52715 −0.763577 0.645717i $$-0.776559\pi$$
−0.763577 + 0.645717i $$0.776559\pi$$
$$588$$ 0 0
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 32.0000 1.31519
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −30.0000 −1.22885
$$597$$ 0 0
$$598$$ −48.0000 −1.96287
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 6.00000 0.244542
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −18.0000 −0.730597 −0.365299 0.930890i $$-0.619033\pi$$
−0.365299 + 0.930890i $$0.619033\pi$$
$$608$$ −8.00000 −0.324443
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 18.0000 0.728202
$$612$$ 0 0
$$613$$ −9.00000 −0.363507 −0.181753 0.983344i $$-0.558177\pi$$
−0.181753 + 0.983344i $$0.558177\pi$$
$$614$$ −24.0000 −0.968561
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 23.0000 0.925945 0.462973 0.886373i $$-0.346783\pi$$
0.462973 + 0.886373i $$0.346783\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 14.0000 0.561349
$$623$$ 30.0000 1.20192
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 28.0000 1.11911
$$627$$ 0 0
$$628$$ 4.00000 0.159617
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 24.0000 0.953162
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 12.0000 0.475457
$$638$$ −60.0000 −2.37542
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ 1.00000 0.0394362 0.0197181 0.999806i $$-0.493723\pi$$
0.0197181 + 0.999806i $$0.493723\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ −27.0000 −1.06148 −0.530740 0.847535i $$-0.678086\pi$$
−0.530740 + 0.847535i $$0.678086\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 32.0000 1.25322
$$653$$ −1.00000 −0.0391330 −0.0195665 0.999809i $$-0.506229\pi$$
−0.0195665 + 0.999809i $$0.506229\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −32.0000 −1.24939
$$657$$ 0 0
$$658$$ 18.0000 0.701713
$$659$$ −10.0000 −0.389545 −0.194772 0.980848i $$-0.562397\pi$$
−0.194772 + 0.980848i $$0.562397\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ −24.0000 −0.932786
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 40.0000 1.54881
$$668$$ 36.0000 1.39288
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 21.0000 0.810696
$$672$$ 0 0
$$673$$ 16.0000 0.616755 0.308377 0.951264i $$-0.400214\pi$$
0.308377 + 0.951264i $$0.400214\pi$$
$$674$$ −44.0000 −1.69482
$$675$$ 0 0
$$676$$ 46.0000 1.76923
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 0 0
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −12.0000 −0.459504
$$683$$ −6.00000 −0.229584 −0.114792 0.993390i $$-0.536620\pi$$
−0.114792 + 0.993390i $$0.536620\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −30.0000 −1.14541
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ 17.0000 0.646710 0.323355 0.946278i $$-0.395189\pi$$
0.323355 + 0.946278i $$0.395189\pi$$
$$692$$ 28.0000 1.06440
$$693$$ 0 0
$$694$$ −6.00000 −0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 24.0000 0.909065
$$698$$ −50.0000 −1.89253
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ −24.0000 −0.904534
$$705$$ 0 0
$$706$$ −28.0000 −1.05379
$$707$$ 6.00000 0.225653
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 8.00000 0.299602
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ 0 0
$$718$$ 50.0000 1.86598
$$719$$ 35.0000 1.30528 0.652640 0.757668i $$-0.273661\pi$$
0.652640 + 0.757668i $$0.273661\pi$$
$$720$$ 0 0
$$721$$ 42.0000 1.56416
$$722$$ −2.00000 −0.0744323
$$723$$ 0 0
$$724$$ 4.00000 0.148659
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 7.00000 0.259616 0.129808 0.991539i $$-0.458564\pi$$
0.129808 + 0.991539i $$0.458564\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 3.00000 0.110959
$$732$$ 0 0
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ 32.0000 1.17954
$$737$$ −24.0000 −0.884051
$$738$$ 0 0
$$739$$ −45.0000 −1.65535 −0.827676 0.561206i $$-0.810337\pi$$
−0.827676 + 0.561206i $$0.810337\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −36.0000 −1.32160
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −32.0000 −1.17160
$$747$$ 0 0
$$748$$ 18.0000 0.658145
$$749$$ 6.00000 0.219235
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 0 0
$$754$$ −120.000 −4.37014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −23.0000 −0.835949 −0.417975 0.908459i $$-0.637260\pi$$
−0.417975 + 0.908459i $$0.637260\pi$$
$$758$$ 60.0000 2.17930
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 13.0000 0.471250 0.235625 0.971844i $$-0.424286\pi$$
0.235625 + 0.971844i $$0.424286\pi$$
$$762$$ 0 0
$$763$$ −60.0000 −2.17215
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ −28.0000 −1.01168
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −45.0000 −1.62274 −0.811371 0.584532i $$-0.801278\pi$$
−0.811371 + 0.584532i $$0.801278\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −8.00000 −0.287926
$$773$$ −36.0000 −1.29483 −0.647415 0.762138i $$-0.724150\pi$$
−0.647415 + 0.762138i $$0.724150\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −30.0000 −1.07555
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ −36.0000 −1.28818
$$782$$ −24.0000 −0.858238
$$783$$ 0 0
$$784$$ −8.00000 −0.285714
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −8.00000 −0.285169 −0.142585 0.989783i $$-0.545541\pi$$
−0.142585 + 0.989783i $$0.545541\pi$$
$$788$$ −4.00000 −0.142494
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ 42.0000 1.49146
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 0 0
$$799$$ 9.00000 0.318397
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −56.0000 −1.97743
$$803$$ 33.0000 1.16454
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −24.0000 −0.845364
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −5.00000 −0.175791 −0.0878953 0.996130i $$-0.528014\pi$$
−0.0878953 + 0.996130i $$0.528014\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ −60.0000 −2.10559
$$813$$ 0 0
$$814$$ 48.0000 1.68240
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.00000 −0.0349856
$$818$$ −20.0000 −0.699284
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ 0 0
$$823$$ −19.0000 −0.662298 −0.331149 0.943578i $$-0.607436\pi$$
−0.331149 + 0.943578i $$0.607436\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −52.0000 −1.80822 −0.904109 0.427303i $$-0.859464\pi$$
−0.904109 + 0.427303i $$0.859464\pi$$
$$828$$ 0 0
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −48.0000 −1.66410
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −6.00000 −0.207514
$$837$$ 0 0
$$838$$ 40.0000 1.38178
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −4.00000 −0.137849
$$843$$ 0 0
$$844$$ −56.0000 −1.92760
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 6.00000 0.206162
$$848$$ 24.0000 0.824163
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −32.0000 −1.09695
$$852$$ 0 0
$$853$$ −34.0000 −1.16414 −0.582069 0.813139i $$-0.697757\pi$$
−0.582069 + 0.813139i $$0.697757\pi$$
$$854$$ 42.0000 1.43721
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 48.0000 1.63965 0.819824 0.572615i $$-0.194071\pi$$
0.819824 + 0.572615i $$0.194071\pi$$
$$858$$ 0 0
$$859$$ 35.0000 1.19418 0.597092 0.802173i $$-0.296323\pi$$
0.597092 + 0.802173i $$0.296323\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −36.0000 −1.22616
$$863$$ 44.0000 1.49778 0.748889 0.662696i $$-0.230588\pi$$
0.748889 + 0.662696i $$0.230588\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −52.0000 −1.76703
$$867$$ 0 0
$$868$$ −12.0000 −0.407307
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 8.00000 0.270604
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −7.00000 −0.235836 −0.117918 0.993023i $$-0.537622\pi$$
−0.117918 + 0.993023i $$0.537622\pi$$
$$882$$ 0 0
$$883$$ 21.0000 0.706706 0.353353 0.935490i $$-0.385041\pi$$
0.353353 + 0.935490i $$0.385041\pi$$
$$884$$ 36.0000 1.21081
$$885$$ 0 0
$$886$$ −78.0000 −2.62046
$$887$$ −52.0000 −1.74599 −0.872995 0.487730i $$-0.837825\pi$$
−0.872995 + 0.487730i $$0.837825\pi$$
$$888$$ 0 0
$$889$$ −6.00000 −0.201234
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −8.00000 −0.267860
$$893$$ −3.00000 −0.100391
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −40.0000 −1.33482
$$899$$ 20.0000 0.667037
$$900$$ 0 0
$$901$$ −18.0000 −0.599667
$$902$$ −48.0000 −1.59823
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 2.00000 0.0664089 0.0332045 0.999449i $$-0.489429\pi$$
0.0332045 + 0.999449i $$0.489429\pi$$
$$908$$ 36.0000 1.19470
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −2.00000 −0.0662630 −0.0331315 0.999451i $$-0.510548\pi$$
−0.0331315 + 0.999451i $$0.510548\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ −30.0000 −0.991228
$$917$$ −39.0000 −1.28789
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −66.0000 −2.17359
$$923$$ −72.0000 −2.36991
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −62.0000 −2.03745
$$927$$ 0 0
$$928$$ 80.0000 2.62613
$$929$$ 50.0000 1.64045 0.820223 0.572043i $$-0.193849\pi$$
0.820223 + 0.572043i $$0.193849\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ −22.0000 −0.720634
$$933$$ 0 0
$$934$$ 34.0000 1.11251
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −53.0000 −1.73143 −0.865717 0.500533i $$-0.833137\pi$$
−0.865717 + 0.500533i $$0.833137\pi$$
$$938$$ −48.0000 −1.56726
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −2.00000 −0.0651981 −0.0325991 0.999469i $$-0.510378\pi$$
−0.0325991 + 0.999469i $$0.510378\pi$$
$$942$$ 0 0
$$943$$ 32.0000 1.04206
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 0 0
$$949$$ 66.0000 2.14245
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −16.0000 −0.518291 −0.259145 0.965838i $$-0.583441\pi$$
−0.259145 + 0.965838i $$0.583441\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 30.0000 0.970269
$$957$$ 0 0
$$958$$ −80.0000 −2.58468
$$959$$ −9.00000 −0.290625
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 96.0000 3.09516
$$963$$ 0 0
$$964$$ 24.0000 0.772988
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 15.0000 0.480878
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −28.0000 −0.896258
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −16.0000 −0.510581
$$983$$ 4.00000 0.127580 0.0637901 0.997963i $$-0.479681\pi$$
0.0637901 + 0.997963i $$0.479681\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −60.0000 −1.91079
$$987$$ 0 0
$$988$$ −12.0000 −0.381771
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 16.0000 0.508001
$$993$$ 0 0
$$994$$ −72.0000 −2.28370
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 7.00000 0.221692 0.110846 0.993838i $$-0.464644\pi$$
0.110846 + 0.993838i $$0.464644\pi$$
$$998$$ 70.0000 2.21581
$$999$$ 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 4275.2.a.a.1.1 1
3.2 odd 2 1425.2.a.i.1.1 1
5.4 even 2 171.2.a.c.1.1 1
15.2 even 4 1425.2.c.a.799.2 2
15.8 even 4 1425.2.c.a.799.1 2
15.14 odd 2 57.2.a.b.1.1 1
20.19 odd 2 2736.2.a.h.1.1 1
35.34 odd 2 8379.2.a.q.1.1 1
60.59 even 2 912.2.a.d.1.1 1
95.94 odd 2 3249.2.a.a.1.1 1
105.104 even 2 2793.2.a.a.1.1 1
120.29 odd 2 3648.2.a.h.1.1 1
120.59 even 2 3648.2.a.y.1.1 1
165.164 even 2 6897.2.a.g.1.1 1
195.194 odd 2 9633.2.a.p.1.1 1
285.284 even 2 1083.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.b.1.1 1 15.14 odd 2
171.2.a.c.1.1 1 5.4 even 2
912.2.a.d.1.1 1 60.59 even 2
1083.2.a.d.1.1 1 285.284 even 2
1425.2.a.i.1.1 1 3.2 odd 2
1425.2.c.a.799.1 2 15.8 even 4
1425.2.c.a.799.2 2 15.2 even 4
2736.2.a.h.1.1 1 20.19 odd 2
2793.2.a.a.1.1 1 105.104 even 2
3249.2.a.a.1.1 1 95.94 odd 2
3648.2.a.h.1.1 1 120.29 odd 2
3648.2.a.y.1.1 1 120.59 even 2
4275.2.a.a.1.1 1 1.1 even 1 trivial
6897.2.a.g.1.1 1 165.164 even 2
8379.2.a.q.1.1 1 35.34 odd 2
9633.2.a.p.1.1 1 195.194 odd 2