# Properties

 Label 2475.2.c.e.199.2 Level $2475$ Weight $2$ Character 2475.199 Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.199 Dual form 2475.2.c.e.199.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +1.00000 q^{11} +2.00000i q^{13} -3.00000 q^{14} -1.00000 q^{16} +3.00000i q^{17} +1.00000 q^{19} +1.00000i q^{22} -1.00000i q^{23} -2.00000 q^{26} +3.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -3.00000 q^{34} -1.00000i q^{37} +1.00000i q^{38} -5.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} +1.00000 q^{46} +3.00000i q^{47} -2.00000 q^{49} +2.00000i q^{52} -10.0000i q^{53} -9.00000 q^{56} -6.00000i q^{58} -11.0000 q^{59} +14.0000 q^{61} +4.00000i q^{62} -7.00000 q^{64} -2.00000i q^{67} +3.00000i q^{68} -5.00000 q^{71} +2.00000i q^{73} +1.00000 q^{74} +1.00000 q^{76} +3.00000i q^{77} -5.00000 q^{79} -5.00000i q^{82} -8.00000i q^{83} -4.00000 q^{86} +3.00000i q^{88} +10.0000 q^{89} -6.00000 q^{91} -1.00000i q^{92} -3.00000 q^{94} +17.0000i q^{97} -2.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} + 2 q^{11} - 6 q^{14} - 2 q^{16} + 2 q^{19} - 4 q^{26} - 12 q^{29} + 8 q^{31} - 6 q^{34} - 10 q^{41} + 2 q^{44} + 2 q^{46} - 4 q^{49} - 18 q^{56} - 22 q^{59} + 28 q^{61} - 14 q^{64} - 10 q^{71} + 2 q^{74} + 2 q^{76} - 10 q^{79} - 8 q^{86} + 20 q^{89} - 12 q^{91} - 6 q^{94}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^11 - 6 * q^14 - 2 * q^16 + 2 * q^19 - 4 * q^26 - 12 * q^29 + 8 * q^31 - 6 * q^34 - 10 * q^41 + 2 * q^44 + 2 * q^46 - 4 * q^49 - 18 * q^56 - 22 * q^59 + 28 * q^61 - 14 * q^64 - 10 * q^71 + 2 * q^74 + 2 * q^76 - 10 * q^79 - 8 * q^86 + 20 * q^89 - 12 * q^91 - 6 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.00000i 0.213201i
$$23$$ − 1.00000i − 0.208514i −0.994550 0.104257i $$-0.966753\pi$$
0.994550 0.104257i $$-0.0332465\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 3.00000i 0.566947i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 1.00000i − 0.164399i −0.996616 0.0821995i $$-0.973806\pi$$
0.996616 0.0821995i $$-0.0261945\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ 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
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 3.00000i 0.437595i 0.975770 + 0.218797i $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −9.00000 −1.20268
$$57$$ 0 0
$$58$$ − 6.00000i − 0.787839i
$$59$$ −11.0000 −1.43208 −0.716039 0.698060i $$-0.754047\pi$$
−0.716039 + 0.698060i $$0.754047\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 2.00000i − 0.244339i −0.992509 0.122169i $$-0.961015\pi$$
0.992509 0.122169i $$-0.0389851\pi$$
$$68$$ 3.00000i 0.363803i
$$69$$ 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
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 3.00000i 0.341882i
$$78$$ 0 0
$$79$$ −5.00000 −0.562544 −0.281272 0.959628i $$-0.590756\pi$$
−0.281272 + 0.959628i $$0.590756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 5.00000i − 0.552158i
$$83$$ − 8.00000i − 0.878114i −0.898459 0.439057i $$-0.855313\pi$$
0.898459 0.439057i $$-0.144687\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 3.00000i 0.319801i
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ − 1.00000i − 0.104257i
$$93$$ 0 0
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 17.0000i 1.72609i 0.505128 + 0.863044i $$0.331445\pi$$
−0.505128 + 0.863044i $$0.668555\pi$$
$$98$$ − 2.00000i − 0.202031i
$$99$$ 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
$$103$$ 2.00000i 0.197066i 0.995134 + 0.0985329i $$0.0314150\pi$$
−0.995134 + 0.0985329i $$0.968585\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 18.0000i 1.74013i 0.492941 + 0.870063i $$0.335922\pi$$
−0.492941 + 0.870063i $$0.664078\pi$$
$$108$$ 0 0
$$109$$ 12.0000 1.14939 0.574696 0.818367i $$-0.305120\pi$$
0.574696 + 0.818367i $$0.305120\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 3.00000i − 0.283473i
$$113$$ − 18.0000i − 1.69330i −0.532152 0.846649i $$-0.678617\pi$$
0.532152 0.846649i $$-0.321383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ − 11.0000i − 1.01263i
$$119$$ −9.00000 −0.825029
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 14.0000i 1.26750i
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 5.00000i 0.443678i 0.975083 + 0.221839i $$0.0712060\pi$$
−0.975083 + 0.221839i $$0.928794\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 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
$$133$$ 3.00000i 0.260133i
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ −9.00000 −0.771744
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 5.00000i − 0.419591i
$$143$$ 2.00000i 0.167248i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ − 1.00000i − 0.0821995i
$$149$$ −7.00000 −0.573462 −0.286731 0.958011i $$-0.592569\pi$$
−0.286731 + 0.958011i $$0.592569\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 3.00000i 0.243332i
$$153$$ 0 0
$$154$$ −3.00000 −0.241747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 10.0000i − 0.798087i −0.916932 0.399043i $$-0.869342\pi$$
0.916932 0.399043i $$-0.130658\pi$$
$$158$$ − 5.00000i − 0.397779i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ 10.0000i 0.783260i 0.920123 + 0.391630i $$0.128089\pi$$
−0.920123 + 0.391630i $$0.871911\pi$$
$$164$$ −5.00000 −0.390434
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ 10.0000i 0.773823i 0.922117 + 0.386912i $$0.126458\pi$$
−0.922117 + 0.386912i $$0.873542\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000i 0.304997i
$$173$$ 9.00000i 0.684257i 0.939653 + 0.342129i $$0.111148\pi$$
−0.939653 + 0.342129i $$0.888852\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 10.0000i 0.749532i
$$179$$ 1.00000 0.0747435 0.0373718 0.999301i $$-0.488101\pi$$
0.0373718 + 0.999301i $$0.488101\pi$$
$$180$$ 0 0
$$181$$ −25.0000 −1.85824 −0.929118 0.369784i $$-0.879432\pi$$
−0.929118 + 0.369784i $$0.879432\pi$$
$$182$$ − 6.00000i − 0.444750i
$$183$$ 0 0
$$184$$ 3.00000 0.221163
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.00000i 0.219382i
$$188$$ 3.00000i 0.218797i
$$189$$ 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
$$193$$ 4.00000i 0.287926i 0.989583 + 0.143963i $$0.0459847\pi$$
−0.989583 + 0.143963i $$0.954015\pi$$
$$194$$ −17.0000 −1.22053
$$195$$ 0 0
$$196$$ −2.00000 −0.142857
$$197$$ 23.0000i 1.63868i 0.573306 + 0.819341i $$0.305660\pi$$
−0.573306 + 0.819341i $$0.694340\pi$$
$$198$$ 0 0
$$199$$ 18.0000 1.27599 0.637993 0.770042i $$-0.279765\pi$$
0.637993 + 0.770042i $$0.279765\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 11.0000i − 0.773957i
$$203$$ − 18.0000i − 1.26335i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2.00000 −0.139347
$$207$$ 0 0
$$208$$ − 2.00000i − 0.138675i
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ − 10.0000i − 0.686803i
$$213$$ 0 0
$$214$$ −18.0000 −1.23045
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.0000i 0.814613i
$$218$$ 12.0000i 0.812743i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ − 22.0000i − 1.47323i −0.676313 0.736614i $$-0.736423\pi$$
0.676313 0.736614i $$-0.263577\pi$$
$$224$$ −15.0000 −1.00223
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ − 8.00000i − 0.530979i −0.964114 0.265489i $$-0.914466\pi$$
0.964114 0.265489i $$-0.0855335\pi$$
$$228$$ 0 0
$$229$$ −11.0000 −0.726900 −0.363450 0.931614i $$-0.618401\pi$$
−0.363450 + 0.931614i $$0.618401\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 18.0000i − 1.18176i
$$233$$ − 9.00000i − 0.589610i −0.955557 0.294805i $$-0.904745\pi$$
0.955557 0.294805i $$-0.0952546\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −11.0000 −0.716039
$$237$$ 0 0
$$238$$ − 9.00000i − 0.583383i
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ 4.00000 0.257663 0.128831 0.991667i $$-0.458877\pi$$
0.128831 + 0.991667i $$0.458877\pi$$
$$242$$ 1.00000i 0.0642824i
$$243$$ 0 0
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000i 0.127257i
$$248$$ 12.0000i 0.762001i
$$249$$ 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
$$253$$ − 1.00000i − 0.0628695i
$$254$$ −5.00000 −0.313728
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 0 0
$$259$$ 3.00000 0.186411
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 12.0000i 0.741362i
$$263$$ 18.0000i 1.10993i 0.831875 + 0.554964i $$0.187268\pi$$
−0.831875 + 0.554964i $$0.812732\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −3.00000 −0.183942
$$267$$ 0 0
$$268$$ − 2.00000i − 0.122169i
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ −3.00000 −0.182237 −0.0911185 0.995840i $$-0.529044\pi$$
−0.0911185 + 0.995840i $$0.529044\pi$$
$$272$$ − 3.00000i − 0.181902i
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 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
$$283$$ − 23.0000i − 1.36721i −0.729853 0.683604i $$-0.760412\pi$$
0.729853 0.683604i $$-0.239588\pi$$
$$284$$ −5.00000 −0.296695
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ − 15.0000i − 0.885422i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.00000i 0.117041i
$$293$$ − 13.0000i − 0.759468i −0.925096 0.379734i $$-0.876015\pi$$
0.925096 0.379734i $$-0.123985\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 3.00000 0.174371
$$297$$ 0 0
$$298$$ − 7.00000i − 0.405499i
$$299$$ 2.00000 0.115663
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ − 16.0000i − 0.920697i
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 32.0000i − 1.82634i −0.407583 0.913168i $$-0.633628\pi$$
0.407583 0.913168i $$-0.366372\pi$$
$$308$$ 3.00000i 0.170941i
$$309$$ 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
$$313$$ − 1.00000i − 0.0565233i −0.999601 0.0282617i $$-0.991003\pi$$
0.999601 0.0282617i $$-0.00899717\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 3.00000i 0.167183i
$$323$$ 3.00000i 0.166924i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −10.0000 −0.553849
$$327$$ 0 0
$$328$$ − 15.0000i − 0.828236i
$$329$$ −9.00000 −0.496186
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ − 8.00000i − 0.439057i
$$333$$ 0 0
$$334$$ −10.0000 −0.547176
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 6.00000i − 0.326841i −0.986557 0.163420i $$-0.947747\pi$$
0.986557 0.163420i $$-0.0522527\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ 15.0000i 0.809924i
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ −9.00000 −0.483843
$$347$$ − 10.0000i − 0.536828i −0.963304 0.268414i $$-0.913500\pi$$
0.963304 0.268414i $$-0.0864995\pi$$
$$348$$ 0 0
$$349$$ −8.00000 −0.428230 −0.214115 0.976808i $$-0.568687\pi$$
−0.214115 + 0.976808i $$0.568687\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5.00000i 0.266501i
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ 1.00000i 0.0528516i
$$359$$ 28.0000 1.47778 0.738892 0.673824i $$-0.235349\pi$$
0.738892 + 0.673824i $$0.235349\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ − 25.0000i − 1.31397i
$$363$$ 0 0
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 4.00000i − 0.208798i −0.994535 0.104399i $$-0.966708\pi$$
0.994535 0.104399i $$-0.0332919\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 30.0000 1.55752
$$372$$ 0 0
$$373$$ 6.00000i 0.310668i 0.987862 + 0.155334i $$0.0496454\pi$$
−0.987862 + 0.155334i $$0.950355\pi$$
$$374$$ −3.00000 −0.155126
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ − 12.0000i − 0.618031i
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 19.0000i 0.972125i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −4.00000 −0.203595
$$387$$ 0 0
$$388$$ 17.0000i 0.863044i
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 3.00000 0.151717
$$392$$ − 6.00000i − 0.303046i
$$393$$ 0 0
$$394$$ −23.0000 −1.15872
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6.00000i 0.301131i 0.988600 + 0.150566i $$0.0481095\pi$$
−0.988600 + 0.150566i $$0.951890\pi$$
$$398$$ 18.0000i 0.902258i
$$399$$ 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
$$403$$ 8.00000i 0.398508i
$$404$$ −11.0000 −0.547270
$$405$$ 0 0
$$406$$ 18.0000 0.893325
$$407$$ − 1.00000i − 0.0495682i
$$408$$ 0 0
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 2.00000i 0.0985329i
$$413$$ − 33.0000i − 1.62382i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −10.0000 −0.490290
$$417$$ 0 0
$$418$$ 1.00000i 0.0489116i
$$419$$ 23.0000 1.12362 0.561812 0.827265i $$-0.310105\pi$$
0.561812 + 0.827265i $$0.310105\pi$$
$$420$$ 0 0
$$421$$ 9.00000 0.438633 0.219317 0.975654i $$-0.429617\pi$$
0.219317 + 0.975654i $$0.429617\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ 0 0
$$424$$ 30.0000 1.45693
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 42.0000i 2.03252i
$$428$$ 18.0000i 0.870063i
$$429$$ 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
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ −12.0000 −0.576018
$$435$$ 0 0
$$436$$ 12.0000 0.574696
$$437$$ − 1.00000i − 0.0478365i
$$438$$ 0 0
$$439$$ 35.0000 1.67046 0.835229 0.549902i $$-0.185335\pi$$
0.835229 + 0.549902i $$0.185335\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 6.00000i − 0.285391i
$$443$$ 31.0000i 1.47285i 0.676517 + 0.736427i $$0.263489\pi$$
−0.676517 + 0.736427i $$0.736511\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 22.0000 1.04173
$$447$$ 0 0
$$448$$ − 21.0000i − 0.992157i
$$449$$ 8.00000 0.377543 0.188772 0.982021i $$-0.439549\pi$$
0.188772 + 0.982021i $$0.439549\pi$$
$$450$$ 0 0
$$451$$ −5.00000 −0.235441
$$452$$ − 18.0000i − 0.846649i
$$453$$ 0 0
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 28.0000i − 1.30978i −0.755722 0.654892i $$-0.772714\pi$$
0.755722 0.654892i $$-0.227286\pi$$
$$458$$ − 11.0000i − 0.513996i
$$459$$ 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
$$463$$ 24.0000i 1.11537i 0.830051 + 0.557687i $$0.188311\pi$$
−0.830051 + 0.557687i $$0.811689\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 9.00000 0.416917
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ 0 0
$$469$$ 6.00000 0.277054
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 33.0000i − 1.51895i
$$473$$ 4.00000i 0.183920i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −9.00000 −0.412514
$$477$$ 0 0
$$478$$ 20.0000i 0.914779i
$$479$$ 18.0000 0.822441 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 4.00000i 0.182195i
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 26.0000i 1.17817i 0.808070 + 0.589086i $$0.200512\pi$$
−0.808070 + 0.589086i $$0.799488\pi$$
$$488$$ 42.0000i 1.90125i
$$489$$ 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
$$493$$ − 18.0000i − 0.810679i
$$494$$ −2.00000 −0.0899843
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ − 15.0000i − 0.672842i
$$498$$ 0 0
$$499$$ −24.0000 −1.07439 −0.537194 0.843459i $$-0.680516\pi$$
−0.537194 + 0.843459i $$0.680516\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 12.0000i 0.535586i
$$503$$ − 20.0000i − 0.891756i −0.895094 0.445878i $$-0.852892\pi$$
0.895094 0.445878i $$-0.147108\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1.00000 0.0444554
$$507$$ 0 0
$$508$$ 5.00000i 0.221839i
$$509$$ 26.0000 1.15243 0.576215 0.817298i $$-0.304529\pi$$
0.576215 + 0.817298i $$0.304529\pi$$
$$510$$ 0 0
$$511$$ −6.00000 −0.265424
$$512$$ − 11.0000i − 0.486136i
$$513$$ 0 0
$$514$$ −12.0000 −0.529297
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.00000i 0.131940i
$$518$$ 3.00000i 0.131812i
$$519$$ 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
$$523$$ 7.00000i 0.306089i 0.988219 + 0.153044i $$0.0489077\pi$$
−0.988219 + 0.153044i $$0.951092\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −18.0000 −0.784837
$$527$$ 12.0000i 0.522728i
$$528$$ 0 0
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 3.00000i 0.130066i
$$533$$ − 10.0000i − 0.433148i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 6.00000 0.259161
$$537$$ 0 0
$$538$$ 24.0000i 1.03471i
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ 32.0000 1.37579 0.687894 0.725811i $$-0.258536\pi$$
0.687894 + 0.725811i $$0.258536\pi$$
$$542$$ − 3.00000i − 0.128861i
$$543$$ 0 0
$$544$$ −15.0000 −0.643120
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 33.0000i 1.41098i 0.708721 + 0.705489i $$0.249273\pi$$
−0.708721 + 0.705489i $$0.750727\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ − 15.0000i − 0.637865i
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 22.0000i 0.932170i 0.884740 + 0.466085i $$0.154336\pi$$
−0.884740 + 0.466085i $$0.845664\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 21.0000i − 0.885832i
$$563$$ − 18.0000i − 0.758610i −0.925272 0.379305i $$-0.876163\pi$$
0.925272 0.379305i $$-0.123837\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 23.0000 0.966762
$$567$$ 0 0
$$568$$ − 15.0000i − 0.629386i
$$569$$ −37.0000 −1.55112 −0.775560 0.631273i $$-0.782533\pi$$
−0.775560 + 0.631273i $$0.782533\pi$$
$$570$$ 0 0
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ 2.00000i 0.0836242i
$$573$$ 0 0
$$574$$ 15.0000 0.626088
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 43.0000i 1.79011i 0.445952 + 0.895057i $$0.352865\pi$$
−0.445952 + 0.895057i $$0.647135\pi$$
$$578$$ 8.00000i 0.332756i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ − 10.0000i − 0.414158i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 13.0000 0.537025
$$587$$ 9.00000i 0.371470i 0.982600 + 0.185735i $$0.0594666\pi$$
−0.982600 + 0.185735i $$0.940533\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.00000i 0.0410997i
$$593$$ 14.0000i 0.574911i 0.957794 + 0.287456i $$0.0928094\pi$$
−0.957794 + 0.287456i $$0.907191\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −7.00000 −0.286731
$$597$$ 0 0
$$598$$ 2.00000i 0.0817861i
$$599$$ 27.0000 1.10319 0.551595 0.834112i $$-0.314019\pi$$
0.551595 + 0.834112i $$0.314019\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ − 12.0000i − 0.489083i
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 40.0000i − 1.62355i −0.583970 0.811775i $$-0.698502\pi$$
0.583970 0.811775i $$-0.301498\pi$$
$$608$$ 5.00000i 0.202777i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ 0 0
$$613$$ 16.0000i 0.646234i 0.946359 + 0.323117i $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ 32.0000 1.29141
$$615$$ 0 0
$$616$$ −9.00000 −0.362620
$$617$$ − 28.0000i − 1.12724i −0.826035 0.563619i $$-0.809409\pi$$
0.826035 0.563619i $$-0.190591\pi$$
$$618$$ 0 0
$$619$$ −34.0000 −1.36658 −0.683288 0.730149i $$-0.739451\pi$$
−0.683288 + 0.730149i $$0.739451\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 12.0000i − 0.481156i
$$623$$ 30.0000i 1.20192i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1.00000 0.0399680
$$627$$ 0 0
$$628$$ − 10.0000i − 0.399043i
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ 34.0000 1.35352 0.676759 0.736204i $$-0.263384\pi$$
0.676759 + 0.736204i $$0.263384\pi$$
$$632$$ − 15.0000i − 0.596668i
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 4.00000i − 0.158486i
$$638$$ − 6.00000i − 0.237542i
$$639$$ 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
$$643$$ 14.0000i 0.552106i 0.961142 + 0.276053i $$0.0890266\pi$$
−0.961142 + 0.276053i $$0.910973\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 0 0
$$646$$ −3.00000 −0.118033
$$647$$ − 13.0000i − 0.511083i −0.966798 0.255541i $$-0.917746\pi$$
0.966798 0.255541i $$-0.0822537\pi$$
$$648$$ 0 0
$$649$$ −11.0000 −0.431788
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 10.0000i 0.391630i
$$653$$ 4.00000i 0.156532i 0.996933 + 0.0782660i $$0.0249384\pi$$
−0.996933 + 0.0782660i $$0.975062\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 5.00000 0.195217
$$657$$ 0 0
$$658$$ − 9.00000i − 0.350857i
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 35.0000 1.36134 0.680671 0.732589i $$-0.261688\pi$$
0.680671 + 0.732589i $$0.261688\pi$$
$$662$$ 4.00000i 0.155464i
$$663$$ 0 0
$$664$$ 24.0000 0.931381
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.00000i 0.232321i
$$668$$ 10.0000i 0.386912i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 14.0000 0.540464
$$672$$ 0 0
$$673$$ 4.00000i 0.154189i 0.997024 + 0.0770943i $$0.0245643\pi$$
−0.997024 + 0.0770943i $$0.975436\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ − 14.0000i − 0.538064i −0.963131 0.269032i $$-0.913296\pi$$
0.963131 0.269032i $$-0.0867037\pi$$
$$678$$ 0 0
$$679$$ −51.0000 −1.95720
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 4.00000i 0.153168i
$$683$$ − 47.0000i − 1.79841i −0.437533 0.899203i $$-0.644148\pi$$
0.437533 0.899203i $$-0.355852\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 0 0
$$688$$ − 4.00000i − 0.152499i
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ −40.0000 −1.52167 −0.760836 0.648944i $$-0.775211\pi$$
−0.760836 + 0.648944i $$0.775211\pi$$
$$692$$ 9.00000i 0.342129i
$$693$$ 0 0
$$694$$ 10.0000 0.379595
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 15.0000i − 0.568166i
$$698$$ − 8.00000i − 0.302804i
$$699$$ 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
$$703$$ − 1.00000i − 0.0377157i
$$704$$ −7.00000 −0.263822
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ − 33.0000i − 1.24109i
$$708$$ 0 0
$$709$$ −35.0000 −1.31445 −0.657226 0.753693i $$-0.728270\pi$$
−0.657226 + 0.753693i $$0.728270\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 30.0000i 1.12430i
$$713$$ − 4.00000i − 0.149801i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 1.00000 0.0373718
$$717$$ 0 0
$$718$$ 28.0000i 1.04495i
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ −6.00000 −0.223452
$$722$$ − 18.0000i − 0.669891i
$$723$$ 0 0
$$724$$ −25.0000 −0.929118
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 28.0000i − 1.03846i −0.854634 0.519231i $$-0.826218\pi$$
0.854634 0.519231i $$-0.173782\pi$$
$$728$$ − 18.0000i − 0.667124i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ − 6.00000i − 0.221615i −0.993842 0.110808i $$-0.964656\pi$$
0.993842 0.110808i $$-0.0353437\pi$$
$$734$$ 4.00000 0.147643
$$735$$ 0 0
$$736$$ 5.00000 0.184302
$$737$$ − 2.00000i − 0.0736709i
$$738$$ 0 0
$$739$$ −1.00000 −0.0367856 −0.0183928 0.999831i $$-0.505855\pi$$
−0.0183928 + 0.999831i $$0.505855\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 30.0000i 1.10133i
$$743$$ 14.0000i 0.513610i 0.966463 + 0.256805i $$0.0826698\pi$$
−0.966463 + 0.256805i $$0.917330\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −6.00000 −0.219676
$$747$$ 0 0
$$748$$ 3.00000i 0.109691i
$$749$$ −54.0000 −1.97312
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ − 3.00000i − 0.109399i
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 42.0000i 1.52652i 0.646094 + 0.763258i $$0.276401\pi$$
−0.646094 + 0.763258i $$0.723599\pi$$
$$758$$ 16.0000i 0.581146i
$$759$$ 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
$$763$$ 36.0000i 1.30329i
$$764$$ 19.0000 0.687396
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 22.0000i − 0.794374i
$$768$$ 0 0
$$769$$ 28.0000 1.00971 0.504853 0.863205i $$-0.331547\pi$$
0.504853 + 0.863205i $$0.331547\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 4.00000i 0.143963i
$$773$$ − 44.0000i − 1.58257i −0.611448 0.791285i $$-0.709412\pi$$
0.611448 0.791285i $$-0.290588\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −51.0000 −1.83079
$$777$$ 0 0
$$778$$ 6.00000i 0.215110i
$$779$$ −5.00000 −0.179144
$$780$$ 0 0
$$781$$ −5.00000 −0.178914
$$782$$ 3.00000i 0.107280i
$$783$$ 0 0
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 43.0000i − 1.53278i −0.642373 0.766392i $$-0.722050\pi$$
0.642373 0.766392i $$-0.277950\pi$$
$$788$$ 23.0000i 0.819341i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 54.0000 1.92002
$$792$$ 0 0
$$793$$ 28.0000i 0.994309i
$$794$$ −6.00000 −0.212932
$$795$$ 0 0
$$796$$ 18.0000 0.637993
$$797$$ 22.0000i 0.779280i 0.920967 + 0.389640i $$0.127401\pi$$
−0.920967 + 0.389640i $$0.872599\pi$$
$$798$$ 0 0
$$799$$ −9.00000 −0.318397
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 38.0000i − 1.34183i
$$803$$ 2.00000i 0.0705785i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ − 33.0000i − 1.16094i
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ −21.0000 −0.737410 −0.368705 0.929547i $$-0.620199\pi$$
−0.368705 + 0.929547i $$0.620199\pi$$
$$812$$ − 18.0000i − 0.631676i
$$813$$ 0 0
$$814$$ 1.00000 0.0350500
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.00000i 0.139942i
$$818$$ − 6.00000i − 0.209785i
$$819$$ 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
$$823$$ − 42.0000i − 1.46403i −0.681290 0.732014i $$-0.738581\pi$$
0.681290 0.732014i $$-0.261419\pi$$
$$824$$ −6.00000 −0.209020
$$825$$ 0 0
$$826$$ 33.0000 1.14822
$$827$$ − 48.0000i − 1.66912i −0.550914 0.834562i $$-0.685721\pi$$
0.550914 0.834562i $$-0.314279\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 14.0000i − 0.485363i
$$833$$ − 6.00000i − 0.207888i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 1.00000 0.0345857
$$837$$ 0 0
$$838$$ 23.0000i 0.794522i
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 9.00000i 0.310160i
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 3.00000i 0.103081i
$$848$$ 10.0000i 0.343401i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.00000 −0.0342796
$$852$$ 0 0
$$853$$ − 40.0000i − 1.36957i −0.728743 0.684787i $$-0.759895\pi$$
0.728743 0.684787i $$-0.240105\pi$$
$$854$$ −42.0000 −1.43721
$$855$$ 0 0
$$856$$ −54.0000 −1.84568
$$857$$ 39.0000i 1.33221i 0.745856 + 0.666107i $$0.232041\pi$$
−0.745856 + 0.666107i $$0.767959\pi$$
$$858$$ 0 0
$$859$$ 26.0000 0.887109 0.443554 0.896248i $$-0.353717\pi$$
0.443554 + 0.896248i $$0.353717\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 36.0000i 1.22616i
$$863$$ − 4.00000i − 0.136162i −0.997680 0.0680808i $$-0.978312\pi$$
0.997680 0.0680808i $$-0.0216876\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ 0 0
$$868$$ 12.0000i 0.407307i
$$869$$ −5.00000 −0.169613
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 36.0000i 1.21911i
$$873$$ 0 0
$$874$$ 1.00000 0.0338255
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 52.0000i 1.75592i 0.478738 + 0.877958i $$0.341094\pi$$
−0.478738 + 0.877958i $$0.658906\pi$$
$$878$$ 35.0000i 1.18119i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 2.00000i 0.0673054i 0.999434 + 0.0336527i $$0.0107140\pi$$
−0.999434 + 0.0336527i $$0.989286\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 0 0
$$886$$ −31.0000 −1.04147
$$887$$ 14.0000i 0.470074i 0.971986 + 0.235037i $$0.0755211\pi$$
−0.971986 + 0.235037i $$0.924479\pi$$
$$888$$ 0 0
$$889$$ −15.0000 −0.503084
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 22.0000i − 0.736614i
$$893$$ 3.00000i 0.100391i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −9.00000 −0.300669
$$897$$ 0 0
$$898$$ 8.00000i 0.266963i
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 30.0000 0.999445
$$902$$ − 5.00000i − 0.166482i
$$903$$ 0 0
$$904$$ 54.0000 1.79601
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 2.00000i − 0.0664089i −0.999449 0.0332045i $$-0.989429\pi$$
0.999449 0.0332045i $$-0.0105712\pi$$
$$908$$ − 8.00000i − 0.265489i
$$909$$ 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
$$913$$ − 8.00000i − 0.264761i
$$914$$ 28.0000 0.926158
$$915$$ 0 0
$$916$$ −11.0000 −0.363450
$$917$$ 36.0000i 1.18882i
$$918$$ 0 0
$$919$$ 39.0000 1.28649 0.643246 0.765660i $$-0.277587\pi$$
0.643246 + 0.765660i $$0.277587\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 34.0000i 1.11973i
$$923$$ − 10.0000i − 0.329154i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −24.0000 −0.788689
$$927$$ 0 0
$$928$$ − 30.0000i − 0.984798i
$$929$$ −32.0000 −1.04989 −0.524943 0.851137i $$-0.675913\pi$$
−0.524943 + 0.851137i $$0.675913\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ − 9.00000i − 0.294805i
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 18.0000i − 0.588034i −0.955800 0.294017i $$-0.905008\pi$$
0.955800 0.294017i $$-0.0949923\pi$$
$$938$$ 6.00000i 0.195907i
$$939$$ 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
$$943$$ 5.00000i 0.162822i
$$944$$ 11.0000 0.358020
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ − 27.0000i − 0.877382i −0.898638 0.438691i $$-0.855442\pi$$
0.898638 0.438691i $$-0.144558\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 27.0000i − 0.875075i
$$953$$ 21.0000i 0.680257i 0.940379 + 0.340128i $$0.110471\pi$$
−0.940379 + 0.340128i $$0.889529\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 20.0000 0.646846
$$957$$ 0 0
$$958$$ 18.0000i 0.581554i
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 2.00000i 0.0644826i
$$963$$ 0 0
$$964$$ 4.00000 0.128831
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 20.0000i − 0.643157i −0.946883 0.321578i $$-0.895787\pi$$
0.946883 0.321578i $$-0.104213\pi$$
$$968$$ 3.00000i 0.0964237i
$$969$$ 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
$$977$$ 46.0000i 1.47167i 0.677161 + 0.735835i $$0.263210\pi$$
−0.677161 + 0.735835i $$0.736790\pi$$
$$978$$ 0 0
$$979$$ 10.0000 0.319601
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 28.0000i 0.893516i
$$983$$ 49.0000i 1.56286i 0.623995 + 0.781429i $$0.285509\pi$$
−0.623995 + 0.781429i $$0.714491\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 18.0000 0.573237
$$987$$ 0 0
$$988$$ 2.00000i 0.0636285i
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ 20.0000i 0.635001i
$$993$$ 0 0
$$994$$ 15.0000 0.475771
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 50.0000i − 1.58352i −0.610835 0.791758i $$-0.709166\pi$$
0.610835 0.791758i $$-0.290834\pi$$
$$998$$ − 24.0000i − 0.759707i
$$999$$ 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.c.e.199.2 2
3.2 odd 2 2475.2.c.c.199.1 2
5.2 odd 4 2475.2.a.b.1.1 1
5.3 odd 4 2475.2.a.k.1.1 yes 1
5.4 even 2 inner 2475.2.c.e.199.1 2
15.2 even 4 2475.2.a.h.1.1 yes 1
15.8 even 4 2475.2.a.d.1.1 yes 1
15.14 odd 2 2475.2.c.c.199.2 2

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