# Properties

 Label 3800.2.d.l.3649.1 Level $3800$ Weight $2$ Character 3800.3649 Analytic conductor $30.343$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.d (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: $$30.3431527681$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.3356224.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 8x^{4} + 16x^{2} + 1$$ x^6 + 8*x^4 + 16*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 760) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3649.1 Root $$0.254102i$$ of defining polynomial Character $$\chi$$ $$=$$ 3800.3649 Dual form 3800.2.d.l.3649.6

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.68133i q^{3} +3.18953i q^{7} -4.18953 q^{9} +O(q^{10})$$ $$q-2.68133i q^{3} +3.18953i q^{7} -4.18953 q^{9} -0.681331i q^{13} +1.18953i q^{17} -1.00000 q^{19} +8.55220 q^{21} -2.17313i q^{23} +3.18953i q^{27} -2.81047 q^{29} +6.37907 q^{31} -7.87086i q^{37} -1.82687 q^{39} +0.983593 q^{41} +1.36266i q^{43} -11.7417i q^{47} -3.17313 q^{49} +3.18953 q^{51} -1.69774i q^{53} +2.68133i q^{57} -11.5358 q^{59} +7.36266 q^{61} -13.3627i q^{63} -7.02759i q^{67} -5.82687 q^{69} -12.7581 q^{71} -5.53579i q^{73} -5.36266 q^{79} -4.01641 q^{81} +2.37907i q^{83} +7.53579i q^{87} -3.01641 q^{89} +2.17313 q^{91} -17.1044i q^{93} -4.88727i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{9}+O(q^{10})$$ 6 * q - 8 * q^9 $$6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100})$$ 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 2.68133i − 1.54807i −0.633145 0.774033i $$-0.718236\pi$$
0.633145 0.774033i $$-0.281764\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.18953i 1.20553i 0.797919 + 0.602765i $$0.205934\pi$$
−0.797919 + 0.602765i $$0.794066\pi$$
$$8$$ 0 0
$$9$$ −4.18953 −1.39651
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ − 0.681331i − 0.188967i −0.995526 0.0944836i $$-0.969880\pi$$
0.995526 0.0944836i $$-0.0301200\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.18953i 0.288504i 0.989541 + 0.144252i $$0.0460777\pi$$
−0.989541 + 0.144252i $$0.953922\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 8.55220 1.86624
$$22$$ 0 0
$$23$$ − 2.17313i − 0.453128i −0.973996 0.226564i $$-0.927251\pi$$
0.973996 0.226564i $$-0.0727493\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 3.18953i 0.613826i
$$28$$ 0 0
$$29$$ −2.81047 −0.521890 −0.260945 0.965354i $$-0.584034\pi$$
−0.260945 + 0.965354i $$0.584034\pi$$
$$30$$ 0 0
$$31$$ 6.37907 1.14571 0.572857 0.819655i $$-0.305835\pi$$
0.572857 + 0.819655i $$0.305835\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 7.87086i − 1.29396i −0.762506 0.646981i $$-0.776031\pi$$
0.762506 0.646981i $$-0.223969\pi$$
$$38$$ 0 0
$$39$$ −1.82687 −0.292534
$$40$$ 0 0
$$41$$ 0.983593 0.153611 0.0768057 0.997046i $$-0.475528\pi$$
0.0768057 + 0.997046i $$0.475528\pi$$
$$42$$ 0 0
$$43$$ 1.36266i 0.207804i 0.994588 + 0.103902i $$0.0331328\pi$$
−0.994588 + 0.103902i $$0.966867\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 11.7417i − 1.71271i −0.516390 0.856354i $$-0.672724\pi$$
0.516390 0.856354i $$-0.327276\pi$$
$$48$$ 0 0
$$49$$ −3.17313 −0.453304
$$50$$ 0 0
$$51$$ 3.18953 0.446624
$$52$$ 0 0
$$53$$ − 1.69774i − 0.233202i −0.993179 0.116601i $$-0.962800\pi$$
0.993179 0.116601i $$-0.0371999\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.68133i 0.355151i
$$58$$ 0 0
$$59$$ −11.5358 −1.50183 −0.750916 0.660398i $$-0.770388\pi$$
−0.750916 + 0.660398i $$0.770388\pi$$
$$60$$ 0 0
$$61$$ 7.36266 0.942692 0.471346 0.881948i $$-0.343768\pi$$
0.471346 + 0.881948i $$0.343768\pi$$
$$62$$ 0 0
$$63$$ − 13.3627i − 1.68354i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 7.02759i − 0.858556i −0.903172 0.429278i $$-0.858768\pi$$
0.903172 0.429278i $$-0.141232\pi$$
$$68$$ 0 0
$$69$$ −5.82687 −0.701473
$$70$$ 0 0
$$71$$ −12.7581 −1.51411 −0.757056 0.653350i $$-0.773363\pi$$
−0.757056 + 0.653350i $$0.773363\pi$$
$$72$$ 0 0
$$73$$ − 5.53579i − 0.647915i −0.946072 0.323958i $$-0.894987\pi$$
0.946072 0.323958i $$-0.105013\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −5.36266 −0.603347 −0.301673 0.953411i $$-0.597545\pi$$
−0.301673 + 0.953411i $$0.597545\pi$$
$$80$$ 0 0
$$81$$ −4.01641 −0.446267
$$82$$ 0 0
$$83$$ 2.37907i 0.261137i 0.991439 + 0.130568i $$0.0416802\pi$$
−0.991439 + 0.130568i $$0.958320\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 7.53579i 0.807921i
$$88$$ 0 0
$$89$$ −3.01641 −0.319738 −0.159869 0.987138i $$-0.551107\pi$$
−0.159869 + 0.987138i $$0.551107\pi$$
$$90$$ 0 0
$$91$$ 2.17313 0.227806
$$92$$ 0 0
$$93$$ − 17.1044i − 1.77364i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 4.88727i − 0.496227i −0.968731 0.248114i $$-0.920189\pi$$
0.968731 0.248114i $$-0.0798106\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.1208 1.20606 0.603032 0.797717i $$-0.293959\pi$$
0.603032 + 0.797717i $$0.293959\pi$$
$$102$$ 0 0
$$103$$ − 17.8709i − 1.76087i −0.474168 0.880434i $$-0.657251\pi$$
0.474168 0.880434i $$-0.342749\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 14.4231i − 1.39433i −0.716911 0.697165i $$-0.754445\pi$$
0.716911 0.697165i $$-0.245555\pi$$
$$108$$ 0 0
$$109$$ −10.2059 −0.977552 −0.488776 0.872409i $$-0.662556\pi$$
−0.488776 + 0.872409i $$0.662556\pi$$
$$110$$ 0 0
$$111$$ −21.1044 −2.00314
$$112$$ 0 0
$$113$$ − 1.49180i − 0.140336i −0.997535 0.0701682i $$-0.977646\pi$$
0.997535 0.0701682i $$-0.0223536\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.85446i 0.263895i
$$118$$ 0 0
$$119$$ −3.79406 −0.347801
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ − 2.63734i − 0.237801i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.24993i − 0.732063i −0.930602 0.366032i $$-0.880716\pi$$
0.930602 0.366032i $$-0.119284\pi$$
$$128$$ 0 0
$$129$$ 3.65375 0.321694
$$130$$ 0 0
$$131$$ 8.75814 0.765202 0.382601 0.923914i $$-0.375028\pi$$
0.382601 + 0.923914i $$0.375028\pi$$
$$132$$ 0 0
$$133$$ − 3.18953i − 0.276568i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 11.9149i 1.01795i 0.860780 + 0.508977i $$0.169976\pi$$
−0.860780 + 0.508977i $$0.830024\pi$$
$$138$$ 0 0
$$139$$ 23.4835 1.99184 0.995920 0.0902352i $$-0.0287619\pi$$
0.995920 + 0.0902352i $$0.0287619\pi$$
$$140$$ 0 0
$$141$$ −31.4835 −2.65139
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 8.50820i 0.701745i
$$148$$ 0 0
$$149$$ 18.8461 1.54393 0.771967 0.635662i $$-0.219273\pi$$
0.771967 + 0.635662i $$0.219273\pi$$
$$150$$ 0 0
$$151$$ −16.0880 −1.30922 −0.654611 0.755966i $$-0.727167\pi$$
−0.654611 + 0.755966i $$0.727167\pi$$
$$152$$ 0 0
$$153$$ − 4.98359i − 0.402900i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.7417i 1.09671i 0.836246 + 0.548355i $$0.184746\pi$$
−0.836246 + 0.548355i $$0.815254\pi$$
$$158$$ 0 0
$$159$$ −4.55220 −0.361013
$$160$$ 0 0
$$161$$ 6.93126 0.546260
$$162$$ 0 0
$$163$$ − 14.6373i − 1.14648i −0.819386 0.573242i $$-0.805685\pi$$
0.819386 0.573242i $$-0.194315\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 17.9588i − 1.38970i −0.719156 0.694849i $$-0.755471\pi$$
0.719156 0.694849i $$-0.244529\pi$$
$$168$$ 0 0
$$169$$ 12.5358 0.964291
$$170$$ 0 0
$$171$$ 4.18953 0.320382
$$172$$ 0 0
$$173$$ 6.85446i 0.521135i 0.965456 + 0.260567i $$0.0839096\pi$$
−0.965456 + 0.260567i $$0.916090\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 30.9313i 2.32494i
$$178$$ 0 0
$$179$$ −5.01641 −0.374944 −0.187472 0.982270i $$-0.560029\pi$$
−0.187472 + 0.982270i $$0.560029\pi$$
$$180$$ 0 0
$$181$$ 16.7253 1.24318 0.621592 0.783341i $$-0.286486\pi$$
0.621592 + 0.783341i $$0.286486\pi$$
$$182$$ 0 0
$$183$$ − 19.7417i − 1.45935i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −10.1731 −0.739986
$$190$$ 0 0
$$191$$ −11.2775 −0.816013 −0.408006 0.912979i $$-0.633776\pi$$
−0.408006 + 0.912979i $$0.633776\pi$$
$$192$$ 0 0
$$193$$ 13.5798i 0.977494i 0.872426 + 0.488747i $$0.162546\pi$$
−0.872426 + 0.488747i $$0.837454\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 0 0
$$199$$ −25.6566 −1.81875 −0.909374 0.415980i $$-0.863438\pi$$
−0.909374 + 0.415980i $$0.863438\pi$$
$$200$$ 0 0
$$201$$ −18.8433 −1.32910
$$202$$ 0 0
$$203$$ − 8.96408i − 0.629155i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 9.10439i 0.632799i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 16.2939 1.12172 0.560860 0.827911i $$-0.310471\pi$$
0.560860 + 0.827911i $$0.310471\pi$$
$$212$$ 0 0
$$213$$ 34.2088i 2.34395i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 20.3463i 1.38119i
$$218$$ 0 0
$$219$$ −14.8433 −1.00302
$$220$$ 0 0
$$221$$ 0.810466 0.0545178
$$222$$ 0 0
$$223$$ − 24.2499i − 1.62390i −0.583730 0.811948i $$-0.698407\pi$$
0.583730 0.811948i $$-0.301593\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 5.31867i − 0.353012i −0.984300 0.176506i $$-0.943520\pi$$
0.984300 0.176506i $$-0.0564796\pi$$
$$228$$ 0 0
$$229$$ −11.7089 −0.773747 −0.386873 0.922133i $$-0.626445\pi$$
−0.386873 + 0.922133i $$0.626445\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 27.5163i 1.80265i 0.433142 + 0.901325i $$0.357405\pi$$
−0.433142 + 0.901325i $$0.642595\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 14.3791i 0.934021i
$$238$$ 0 0
$$239$$ −14.9313 −0.965823 −0.482912 0.875669i $$-0.660421\pi$$
−0.482912 + 0.875669i $$0.660421\pi$$
$$240$$ 0 0
$$241$$ −9.32985 −0.600988 −0.300494 0.953784i $$-0.597152\pi$$
−0.300494 + 0.953784i $$0.597152\pi$$
$$242$$ 0 0
$$243$$ 20.3379i 1.30468i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.681331i 0.0433520i
$$248$$ 0 0
$$249$$ 6.37907 0.404257
$$250$$ 0 0
$$251$$ 1.27468 0.0804569 0.0402285 0.999191i $$-0.487191\pi$$
0.0402285 + 0.999191i $$0.487191\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 31.2887i − 1.95174i −0.218363 0.975868i $$-0.570072\pi$$
0.218363 0.975868i $$-0.429928\pi$$
$$258$$ 0 0
$$259$$ 25.1044 1.55991
$$260$$ 0 0
$$261$$ 11.7745 0.728826
$$262$$ 0 0
$$263$$ 25.5163i 1.57340i 0.617335 + 0.786700i $$0.288212\pi$$
−0.617335 + 0.786700i $$0.711788\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 8.08798i 0.494977i
$$268$$ 0 0
$$269$$ −26.7581 −1.63147 −0.815736 0.578424i $$-0.803668\pi$$
−0.815736 + 0.578424i $$0.803668\pi$$
$$270$$ 0 0
$$271$$ 14.1731 0.860956 0.430478 0.902601i $$-0.358345\pi$$
0.430478 + 0.902601i $$0.358345\pi$$
$$272$$ 0 0
$$273$$ − 5.82687i − 0.352658i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 6.08798i − 0.365791i −0.983132 0.182896i $$-0.941453\pi$$
0.983132 0.182896i $$-0.0585471\pi$$
$$278$$ 0 0
$$279$$ −26.7253 −1.60000
$$280$$ 0 0
$$281$$ −15.7089 −0.937115 −0.468558 0.883433i $$-0.655226\pi$$
−0.468558 + 0.883433i $$0.655226\pi$$
$$282$$ 0 0
$$283$$ 0.346255i 0.0205827i 0.999947 + 0.0102913i $$0.00327590\pi$$
−0.999947 + 0.0102913i $$0.996724\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3.13720i 0.185183i
$$288$$ 0 0
$$289$$ 15.5850 0.916765
$$290$$ 0 0
$$291$$ −13.1044 −0.768193
$$292$$ 0 0
$$293$$ 6.71414i 0.392244i 0.980579 + 0.196122i $$0.0628349\pi$$
−0.980579 + 0.196122i $$0.937165\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.48062 −0.0856264
$$300$$ 0 0
$$301$$ −4.34625 −0.250514
$$302$$ 0 0
$$303$$ − 32.4999i − 1.86707i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 17.6126i 1.00520i 0.864518 + 0.502602i $$0.167624\pi$$
−0.864518 + 0.502602i $$0.832376\pi$$
$$308$$ 0 0
$$309$$ −47.9177 −2.72594
$$310$$ 0 0
$$311$$ −6.58501 −0.373402 −0.186701 0.982417i $$-0.559779\pi$$
−0.186701 + 0.982417i $$0.559779\pi$$
$$312$$ 0 0
$$313$$ − 18.6178i − 1.05234i −0.850379 0.526171i $$-0.823627\pi$$
0.850379 0.526171i $$-0.176373\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 17.7857i 0.998946i 0.866330 + 0.499473i $$0.166473\pi$$
−0.866330 + 0.499473i $$0.833527\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −38.6730 −2.15852
$$322$$ 0 0
$$323$$ − 1.18953i − 0.0661874i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 27.3655i 1.51332i
$$328$$ 0 0
$$329$$ 37.4506 2.06472
$$330$$ 0 0
$$331$$ −3.53579 −0.194345 −0.0971723 0.995268i $$-0.530980\pi$$
−0.0971723 + 0.995268i $$0.530980\pi$$
$$332$$ 0 0
$$333$$ 32.9753i 1.80703i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7.87086i 0.428753i 0.976751 + 0.214377i $$0.0687720\pi$$
−0.976751 + 0.214377i $$0.931228\pi$$
$$338$$ 0 0
$$339$$ −4.00000 −0.217250
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 12.2059i 0.659059i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 15.6537i − 0.840337i −0.907446 0.420169i $$-0.861971\pi$$
0.907446 0.420169i $$-0.138029\pi$$
$$348$$ 0 0
$$349$$ 21.4178 1.14647 0.573235 0.819391i $$-0.305688\pi$$
0.573235 + 0.819391i $$0.305688\pi$$
$$350$$ 0 0
$$351$$ 2.17313 0.115993
$$352$$ 0 0
$$353$$ 3.15672i 0.168015i 0.996465 + 0.0840076i $$0.0267720\pi$$
−0.996465 + 0.0840076i $$0.973228\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 10.1731i 0.538419i
$$358$$ 0 0
$$359$$ −17.8269 −0.940866 −0.470433 0.882436i $$-0.655902\pi$$
−0.470433 + 0.882436i $$0.655902\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 29.4946i 1.54807i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 19.2252i − 1.00355i −0.864999 0.501773i $$-0.832681\pi$$
0.864999 0.501773i $$-0.167319\pi$$
$$368$$ 0 0
$$369$$ −4.12080 −0.214520
$$370$$ 0 0
$$371$$ 5.41499 0.281132
$$372$$ 0 0
$$373$$ 9.95601i 0.515503i 0.966211 + 0.257751i $$0.0829815\pi$$
−0.966211 + 0.257751i $$0.917018\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.91486i 0.0986201i
$$378$$ 0 0
$$379$$ 19.6238 1.00801 0.504003 0.863702i $$-0.331860\pi$$
0.504003 + 0.863702i $$0.331860\pi$$
$$380$$ 0 0
$$381$$ −22.1208 −1.13328
$$382$$ 0 0
$$383$$ 11.9037i 0.608250i 0.952632 + 0.304125i $$0.0983640\pi$$
−0.952632 + 0.304125i $$0.901636\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 5.70892i − 0.290201i
$$388$$ 0 0
$$389$$ −10.3463 −0.524576 −0.262288 0.964990i $$-0.584477\pi$$
−0.262288 + 0.964990i $$0.584477\pi$$
$$390$$ 0 0
$$391$$ 2.58501 0.130730
$$392$$ 0 0
$$393$$ − 23.4835i − 1.18458i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 26.1536i − 1.31261i −0.754495 0.656306i $$-0.772118\pi$$
0.754495 0.656306i $$-0.227882\pi$$
$$398$$ 0 0
$$399$$ −8.55220 −0.428145
$$400$$ 0 0
$$401$$ −32.7253 −1.63422 −0.817112 0.576479i $$-0.804426\pi$$
−0.817112 + 0.576479i $$0.804426\pi$$
$$402$$ 0 0
$$403$$ − 4.34625i − 0.216502i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −4.03281 −0.199410 −0.0997049 0.995017i $$-0.531790\pi$$
−0.0997049 + 0.995017i $$0.531790\pi$$
$$410$$ 0 0
$$411$$ 31.9477 1.57586
$$412$$ 0 0
$$413$$ − 36.7938i − 1.81050i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 62.9669i − 3.08350i
$$418$$ 0 0
$$419$$ 15.8297 0.773332 0.386666 0.922220i $$-0.373627\pi$$
0.386666 + 0.922220i $$0.373627\pi$$
$$420$$ 0 0
$$421$$ 6.05233 0.294973 0.147486 0.989064i $$-0.452882\pi$$
0.147486 + 0.989064i $$0.452882\pi$$
$$422$$ 0 0
$$423$$ 49.1924i 2.39182i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 23.4835i 1.13644i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 25.7089 1.23835 0.619177 0.785251i $$-0.287466\pi$$
0.619177 + 0.785251i $$0.287466\pi$$
$$432$$ 0 0
$$433$$ 2.57383i 0.123690i 0.998086 + 0.0618452i $$0.0196985\pi$$
−0.998086 + 0.0618452i $$0.980301\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.17313i 0.103955i
$$438$$ 0 0
$$439$$ −26.8133 −1.27973 −0.639865 0.768488i $$-0.721010\pi$$
−0.639865 + 0.768488i $$0.721010\pi$$
$$440$$ 0 0
$$441$$ 13.2939 0.633044
$$442$$ 0 0
$$443$$ 10.7909i 0.512693i 0.966585 + 0.256347i $$0.0825189\pi$$
−0.966585 + 0.256347i $$0.917481\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 50.5327i − 2.39011i
$$448$$ 0 0
$$449$$ 29.4835 1.39141 0.695705 0.718327i $$-0.255092\pi$$
0.695705 + 0.718327i $$0.255092\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 43.1372i 2.02676i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 11.8269i 0.553238i 0.960980 + 0.276619i $$0.0892140\pi$$
−0.960980 + 0.276619i $$0.910786\pi$$
$$458$$ 0 0
$$459$$ −3.79406 −0.177092
$$460$$ 0 0
$$461$$ −7.45065 −0.347011 −0.173506 0.984833i $$-0.555509\pi$$
−0.173506 + 0.984833i $$0.555509\pi$$
$$462$$ 0 0
$$463$$ − 37.4506i − 1.74048i −0.492629 0.870240i $$-0.663964\pi$$
0.492629 0.870240i $$-0.336036\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 27.4835i 1.27178i 0.771779 + 0.635891i $$0.219367\pi$$
−0.771779 + 0.635891i $$0.780633\pi$$
$$468$$ 0 0
$$469$$ 22.4147 1.03502
$$470$$ 0 0
$$471$$ 36.8461 1.69778
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 7.11273i 0.325669i
$$478$$ 0 0
$$479$$ −10.6597 −0.487054 −0.243527 0.969894i $$-0.578304\pi$$
−0.243527 + 0.969894i $$0.578304\pi$$
$$480$$ 0 0
$$481$$ −5.36266 −0.244516
$$482$$ 0 0
$$483$$ − 18.5850i − 0.845647i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 7.92604i − 0.359163i −0.983743 0.179581i $$-0.942526\pi$$
0.983743 0.179581i $$-0.0574743\pi$$
$$488$$ 0 0
$$489$$ −39.2475 −1.77484
$$490$$ 0 0
$$491$$ −1.68656 −0.0761133 −0.0380567 0.999276i $$-0.512117\pi$$
−0.0380567 + 0.999276i $$0.512117\pi$$
$$492$$ 0 0
$$493$$ − 3.34314i − 0.150568i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 40.6925i − 1.82531i
$$498$$ 0 0
$$499$$ 20.3463 0.910823 0.455412 0.890281i $$-0.349492\pi$$
0.455412 + 0.890281i $$0.349492\pi$$
$$500$$ 0 0
$$501$$ −48.1536 −2.15134
$$502$$ 0 0
$$503$$ 1.06874i 0.0476526i 0.999716 + 0.0238263i $$0.00758487\pi$$
−0.999716 + 0.0238263i $$0.992415\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 33.6126i − 1.49279i
$$508$$ 0 0
$$509$$ 19.0164 0.842887 0.421444 0.906855i $$-0.361524\pi$$
0.421444 + 0.906855i $$0.361524\pi$$
$$510$$ 0 0
$$511$$ 17.6566 0.781081
$$512$$ 0 0
$$513$$ − 3.18953i − 0.140821i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 18.3791 0.806752
$$520$$ 0 0
$$521$$ 9.07158 0.397433 0.198717 0.980057i $$-0.436323\pi$$
0.198717 + 0.980057i $$0.436323\pi$$
$$522$$ 0 0
$$523$$ − 13.4946i − 0.590079i −0.955485 0.295040i $$-0.904667\pi$$
0.955485 0.295040i $$-0.0953329\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 7.58812i 0.330544i
$$528$$ 0 0
$$529$$ 18.2775 0.794675
$$530$$ 0 0
$$531$$ 48.3296 2.09733
$$532$$ 0 0
$$533$$ − 0.670152i − 0.0290275i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 13.4506i 0.580438i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 4.63734 0.199375 0.0996874 0.995019i $$-0.468216\pi$$
0.0996874 + 0.995019i $$0.468216\pi$$
$$542$$ 0 0
$$543$$ − 44.8461i − 1.92453i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 18.1291i 0.775146i 0.921839 + 0.387573i $$0.126686\pi$$
−0.921839 + 0.387573i $$0.873314\pi$$
$$548$$ 0 0
$$549$$ −30.8461 −1.31648
$$550$$ 0 0
$$551$$ 2.81047 0.119730
$$552$$ 0 0
$$553$$ − 17.1044i − 0.727353i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 32.2968i − 1.36846i −0.729267 0.684229i $$-0.760139\pi$$
0.729267 0.684229i $$-0.239861\pi$$
$$558$$ 0 0
$$559$$ 0.928423 0.0392681
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 31.3871i − 1.32281i −0.750029 0.661405i $$-0.769960\pi$$
0.750029 0.661405i $$-0.230040\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 12.8105i − 0.537989i
$$568$$ 0 0
$$569$$ 31.7969 1.33300 0.666498 0.745507i $$-0.267793\pi$$
0.666498 + 0.745507i $$0.267793\pi$$
$$570$$ 0 0
$$571$$ 18.3134 0.766394 0.383197 0.923667i $$-0.374823\pi$$
0.383197 + 0.923667i $$0.374823\pi$$
$$572$$ 0 0
$$573$$ 30.2388i 1.26324i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 26.7282i − 1.11271i −0.830945 0.556354i $$-0.812200\pi$$
0.830945 0.556354i $$-0.187800\pi$$
$$578$$ 0 0
$$579$$ 36.4119 1.51323
$$580$$ 0 0
$$581$$ −7.58812 −0.314808
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 24.3463i − 1.00488i −0.864613 0.502439i $$-0.832436\pi$$
0.864613 0.502439i $$-0.167564\pi$$
$$588$$ 0 0
$$589$$ −6.37907 −0.262845
$$590$$ 0 0
$$591$$ 5.36266 0.220590
$$592$$ 0 0
$$593$$ 12.7253i 0.522566i 0.965262 + 0.261283i $$0.0841456\pi$$
−0.965262 + 0.261283i $$0.915854\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 68.7938i 2.81554i
$$598$$ 0 0
$$599$$ −10.6373 −0.434630 −0.217315 0.976102i $$-0.569730\pi$$
−0.217315 + 0.976102i $$0.569730\pi$$
$$600$$ 0 0
$$601$$ 11.5329 0.470439 0.235219 0.971942i $$-0.424419\pi$$
0.235219 + 0.971942i $$0.424419\pi$$
$$602$$ 0 0
$$603$$ 29.4423i 1.19898i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 18.2827i 0.742074i 0.928618 + 0.371037i $$0.120998\pi$$
−0.928618 + 0.371037i $$0.879002\pi$$
$$608$$ 0 0
$$609$$ −24.0357 −0.973974
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ − 6.84612i − 0.276512i −0.990397 0.138256i $$-0.955850\pi$$
0.990397 0.138256i $$-0.0441497\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 15.0820i − 0.607180i −0.952803 0.303590i $$-0.901815\pi$$
0.952803 0.303590i $$-0.0981853\pi$$
$$618$$ 0 0
$$619$$ 42.7253 1.71728 0.858638 0.512583i $$-0.171311\pi$$
0.858638 + 0.512583i $$0.171311\pi$$
$$620$$ 0 0
$$621$$ 6.93126 0.278142
$$622$$ 0 0
$$623$$ − 9.62093i − 0.385454i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 9.36266 0.373314
$$630$$ 0 0
$$631$$ 26.7909 1.06653 0.533265 0.845948i $$-0.320965\pi$$
0.533265 + 0.845948i $$0.320965\pi$$
$$632$$ 0 0
$$633$$ − 43.6894i − 1.73650i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.16195i 0.0856595i
$$638$$ 0 0
$$639$$ 53.4506 2.11447
$$640$$ 0 0
$$641$$ 28.1208 1.11070 0.555352 0.831615i $$-0.312583\pi$$
0.555352 + 0.831615i $$0.312583\pi$$
$$642$$ 0 0
$$643$$ 15.1372i 0.596953i 0.954417 + 0.298477i $$0.0964785\pi$$
−0.954417 + 0.298477i $$0.903522\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 1.06874i − 0.0420164i −0.999779 0.0210082i $$-0.993312\pi$$
0.999779 0.0210082i $$-0.00668761\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 54.5550 2.13818
$$652$$ 0 0
$$653$$ − 21.3298i − 0.834701i −0.908745 0.417351i $$-0.862959\pi$$
0.908745 0.417351i $$-0.137041\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 23.1924i 0.904821i
$$658$$ 0 0
$$659$$ −45.4639 −1.77102 −0.885512 0.464617i $$-0.846192\pi$$
−0.885512 + 0.464617i $$0.846192\pi$$
$$660$$ 0 0
$$661$$ 28.6074 1.11270 0.556349 0.830949i $$-0.312202\pi$$
0.556349 + 0.830949i $$0.312202\pi$$
$$662$$ 0 0
$$663$$ − 2.17313i − 0.0843973i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.10750i 0.236483i
$$668$$ 0 0
$$669$$ −65.0221 −2.51390
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 47.1840i 1.81881i 0.415911 + 0.909405i $$0.363463\pi$$
−0.415911 + 0.909405i $$0.636537\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 20.8573i − 0.801611i −0.916163 0.400806i $$-0.868730\pi$$
0.916163 0.400806i $$-0.131270\pi$$
$$678$$ 0 0
$$679$$ 15.5881 0.598217
$$680$$ 0 0
$$681$$ −14.2611 −0.546487
$$682$$ 0 0
$$683$$ − 17.2887i − 0.661534i −0.943713 0.330767i $$-0.892693\pi$$
0.943713 0.330767i $$-0.107307\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 31.3955i 1.19781i
$$688$$ 0 0
$$689$$ −1.15672 −0.0440675
$$690$$ 0 0
$$691$$ −35.8297 −1.36303 −0.681513 0.731806i $$-0.738678\pi$$
−0.681513 + 0.731806i $$0.738678\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.17002i 0.0443176i
$$698$$ 0 0
$$699$$ 73.7802 2.79062
$$700$$ 0 0
$$701$$ −38.0880 −1.43856 −0.719282 0.694719i $$-0.755529\pi$$
−0.719282 + 0.694719i $$0.755529\pi$$
$$702$$ 0 0
$$703$$ 7.87086i 0.296855i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 38.6597i 1.45395i
$$708$$ 0 0
$$709$$ 15.2747 0.573653 0.286826 0.957983i $$-0.407400\pi$$
0.286826 + 0.957983i $$0.407400\pi$$
$$710$$ 0 0
$$711$$ 22.4671 0.842580
$$712$$ 0 0
$$713$$ − 13.8625i − 0.519156i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 40.0357i 1.49516i
$$718$$ 0 0
$$719$$ 27.8597 1.03899 0.519495 0.854473i $$-0.326120\pi$$
0.519495 + 0.854473i $$0.326120\pi$$
$$720$$ 0 0
$$721$$ 56.9997 2.12278
$$722$$ 0 0
$$723$$ 25.0164i 0.930370i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 39.4311i 1.46242i 0.682153 + 0.731210i $$0.261044\pi$$
−0.682153 + 0.731210i $$0.738956\pi$$
$$728$$ 0 0
$$729$$ 42.4835 1.57346
$$730$$ 0 0
$$731$$ −1.62093 −0.0599523
$$732$$ 0 0
$$733$$ 33.1372i 1.22395i 0.790877 + 0.611975i $$0.209625\pi$$
−0.790877 + 0.611975i $$0.790375\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 25.8625 0.951368 0.475684 0.879616i $$-0.342201\pi$$
0.475684 + 0.879616i $$0.342201\pi$$
$$740$$ 0 0
$$741$$ 1.82687 0.0671118
$$742$$ 0 0
$$743$$ − 28.6842i − 1.05232i −0.850386 0.526160i $$-0.823631\pi$$
0.850386 0.526160i $$-0.176369\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 9.96719i − 0.364680i
$$748$$ 0 0
$$749$$ 46.0028 1.68091
$$750$$ 0 0
$$751$$ −34.2968 −1.25151 −0.625753 0.780021i $$-0.715208\pi$$
−0.625753 + 0.780021i $$0.715208\pi$$
$$752$$ 0 0
$$753$$ − 3.41783i − 0.124553i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 20.6597i − 0.750889i −0.926845 0.375445i $$-0.877490\pi$$
0.926845 0.375445i $$-0.122510\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 38.1236 1.38198 0.690990 0.722864i $$-0.257175\pi$$
0.690990 + 0.722864i $$0.257175\pi$$
$$762$$ 0 0
$$763$$ − 32.5522i − 1.17847i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 7.85969i 0.283797i
$$768$$ 0 0
$$769$$ −20.3267 −0.733001 −0.366500 0.930418i $$-0.619444\pi$$
−0.366500 + 0.930418i $$0.619444\pi$$
$$770$$ 0 0
$$771$$ −83.8953 −3.02142
$$772$$ 0 0
$$773$$ 26.4559i 0.951552i 0.879567 + 0.475776i $$0.157833\pi$$
−0.879567 + 0.475776i $$0.842167\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 67.3132i − 2.41485i
$$778$$ 0 0
$$779$$ −0.983593 −0.0352409
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 8.96408i − 0.320350i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 27.3738i 0.975772i 0.872907 + 0.487886i $$0.162232\pi$$
−0.872907 + 0.487886i $$0.837768\pi$$
$$788$$ 0 0
$$789$$ 68.4176 2.43573
$$790$$ 0 0
$$791$$ 4.75814 0.169180
$$792$$ 0 0
$$793$$ − 5.01641i − 0.178138i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 36.4454i − 1.29096i −0.763776 0.645481i $$-0.776657\pi$$
0.763776 0.645481i $$-0.223343\pi$$
$$798$$ 0 0
$$799$$ 13.9672 0.494124
$$800$$ 0 0
$$801$$ 12.6373 0.446518
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 71.7474i 2.52563i
$$808$$ 0 0
$$809$$ −52.2444 −1.83682 −0.918408 0.395634i $$-0.870525\pi$$
−0.918408 + 0.395634i $$0.870525\pi$$
$$810$$ 0 0
$$811$$ 32.8984 1.15522 0.577610 0.816313i $$-0.303985\pi$$
0.577610 + 0.816313i $$0.303985\pi$$
$$812$$ 0 0
$$813$$ − 38.0028i − 1.33282i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 1.36266i − 0.0476735i
$$818$$ 0 0
$$819$$ −9.10439 −0.318133
$$820$$ 0 0
$$821$$ −4.44470 −0.155121 −0.0775605 0.996988i $$-0.524713\pi$$
−0.0775605 + 0.996988i $$0.524713\pi$$
$$822$$ 0 0
$$823$$ − 27.7774i − 0.968259i −0.874996 0.484129i $$-0.839136\pi$$
0.874996 0.484129i $$-0.160864\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 46.5110i − 1.61735i −0.588257 0.808674i $$-0.700186\pi$$
0.588257 0.808674i $$-0.299814\pi$$
$$828$$ 0 0
$$829$$ −28.2388 −0.980772 −0.490386 0.871505i $$-0.663144\pi$$
−0.490386 + 0.871505i $$0.663144\pi$$
$$830$$ 0 0
$$831$$ −16.3239 −0.566270
$$832$$ 0 0
$$833$$ − 3.77454i − 0.130780i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 20.3463i 0.703269i
$$838$$ 0 0
$$839$$ −9.01641 −0.311281 −0.155640 0.987814i $$-0.549744\pi$$
−0.155640 + 0.987814i $$0.549744\pi$$
$$840$$ 0 0
$$841$$ −21.1013 −0.727630
$$842$$ 0 0
$$843$$ 42.1208i 1.45072i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 35.0849i − 1.20553i
$$848$$ 0 0
$$849$$ 0.928423 0.0318634
$$850$$ 0 0
$$851$$ −17.1044 −0.586331
$$852$$ 0 0
$$853$$ 5.30749i 0.181725i 0.995863 + 0.0908625i $$0.0289624\pi$$
−0.995863 + 0.0908625i $$0.971038\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 38.9258i 1.32968i 0.746986 + 0.664839i $$0.231500\pi$$
−0.746986 + 0.664839i $$0.768500\pi$$
$$858$$ 0 0
$$859$$ −20.1760 −0.688395 −0.344198 0.938897i $$-0.611849\pi$$
−0.344198 + 0.938897i $$0.611849\pi$$
$$860$$ 0 0
$$861$$ 8.41188 0.286676
$$862$$ 0 0
$$863$$ − 30.3051i − 1.03160i −0.856710 0.515799i $$-0.827495\pi$$
0.856710 0.515799i $$-0.172505\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 41.7886i − 1.41921i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −4.78811 −0.162239
$$872$$ 0 0
$$873$$ 20.4754i 0.692987i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 9.43947i − 0.318748i −0.987218 0.159374i $$-0.949052\pi$$
0.987218 0.159374i $$-0.0509476\pi$$
$$878$$ 0 0
$$879$$ 18.0028 0.607221
$$880$$ 0 0
$$881$$ −24.4671 −0.824316 −0.412158 0.911112i $$-0.635225\pi$$
−0.412158 + 0.911112i $$0.635225\pi$$
$$882$$ 0 0
$$883$$ 6.12080i 0.205981i 0.994682 + 0.102991i $$0.0328412\pi$$
−0.994682 + 0.102991i $$0.967159\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 45.4200i − 1.52505i −0.646957 0.762526i $$-0.723959\pi$$
0.646957 0.762526i $$-0.276041\pi$$
$$888$$ 0 0
$$889$$ 26.3134 0.882524
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 11.7417i 0.392922i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 3.97003i 0.132555i
$$898$$ 0 0
$$899$$ −17.9282 −0.597937
$$900$$ 0 0
$$901$$ 2.01952 0.0672798
$$902$$ 0 0
$$903$$ 11.6537i 0.387812i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 10.2694i 0.340991i 0.985358 + 0.170496i $$0.0545369\pi$$
−0.985358 + 0.170496i $$0.945463\pi$$
$$908$$ 0 0
$$909$$ −50.7805 −1.68428
$$910$$ 0 0
$$911$$ −10.9180 −0.361728 −0.180864 0.983508i $$-0.557889\pi$$
−0.180864 + 0.983508i $$0.557889\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 27.9344i 0.922474i
$$918$$ 0 0
$$919$$ 38.9313 1.28422 0.642112 0.766611i $$-0.278058\pi$$
0.642112 + 0.766611i $$0.278058\pi$$
$$920$$ 0 0
$$921$$ 47.2252 1.55612
$$922$$ 0 0
$$923$$ 8.69251i 0.286117i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 74.8706i 2.45907i
$$928$$ 0 0
$$929$$ −21.9700 −0.720813 −0.360407 0.932795i $$-0.617362\pi$$
−0.360407 + 0.932795i $$0.617362\pi$$
$$930$$ 0 0
$$931$$ 3.17313 0.103995
$$932$$ 0 0
$$933$$ 17.6566i 0.578051i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 10.6402i − 0.347600i −0.984781 0.173800i $$-0.944395\pi$$
0.984781 0.173800i $$-0.0556045\pi$$
$$938$$ 0 0
$$939$$ −49.9205 −1.62910
$$940$$ 0 0
$$941$$ −20.8656 −0.680200 −0.340100 0.940389i $$-0.610461\pi$$
−0.340100 + 0.940389i $$0.610461\pi$$
$$942$$ 0 0
$$943$$ − 2.13747i − 0.0696057i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 12.6925i − 0.412451i −0.978504 0.206226i $$-0.933882\pi$$
0.978504 0.206226i $$-0.0661181\pi$$
$$948$$ 0 0
$$949$$ −3.77170 −0.122435
$$950$$ 0 0
$$951$$ 47.6894 1.54643
$$952$$ 0 0
$$953$$ 16.7337i 0.542056i 0.962571 + 0.271028i $$0.0873637\pi$$
−0.962571 + 0.271028i $$0.912636\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −38.0028 −1.22718
$$960$$ 0 0
$$961$$ 9.69251 0.312662
$$962$$ 0 0
$$963$$ 60.4259i 1.94720i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 4.62688i − 0.148790i −0.997229 0.0743952i $$-0.976297\pi$$
0.997229 0.0743952i $$-0.0237026\pi$$
$$968$$ 0 0
$$969$$ −3.18953 −0.102463
$$970$$ 0 0
$$971$$ −49.4506 −1.58695 −0.793473 0.608605i $$-0.791729\pi$$
−0.793473 + 0.608605i $$0.791729\pi$$
$$972$$ 0 0
$$973$$ 74.9013i 2.40123i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 22.5962i − 0.722916i −0.932388 0.361458i $$-0.882279\pi$$
0.932388 0.361458i $$-0.117721\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 42.7581 1.36516
$$982$$ 0 0
$$983$$ 32.9424i 1.05070i 0.850886 + 0.525350i $$0.176066\pi$$
−0.850886 + 0.525350i $$0.823934\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 100.418i − 3.19633i
$$988$$ 0 0
$$989$$ 2.96124 0.0941618
$$990$$ 0 0
$$991$$ −49.7802 −1.58132 −0.790660 0.612255i $$-0.790263\pi$$
−0.790660 + 0.612255i $$0.790263\pi$$
$$992$$ 0 0
$$993$$ 9.48062i 0.300858i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 36.4014i 1.15284i 0.817152 + 0.576422i $$0.195552\pi$$
−0.817152 + 0.576422i $$0.804448\pi$$
$$998$$ 0 0
$$999$$ 25.1044 0.794268
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 3800.2.d.l.3649.1 6
5.2 odd 4 760.2.a.j.1.1 3
5.3 odd 4 3800.2.a.x.1.3 3
5.4 even 2 inner 3800.2.d.l.3649.6 6
15.2 even 4 6840.2.a.bg.1.1 3
20.3 even 4 7600.2.a.bq.1.1 3
20.7 even 4 1520.2.a.s.1.3 3
40.27 even 4 6080.2.a.bq.1.1 3
40.37 odd 4 6080.2.a.bv.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.1 3 5.2 odd 4
1520.2.a.s.1.3 3 20.7 even 4
3800.2.a.x.1.3 3 5.3 odd 4
3800.2.d.l.3649.1 6 1.1 even 1 trivial
3800.2.d.l.3649.6 6 5.4 even 2 inner
6080.2.a.bq.1.1 3 40.27 even 4
6080.2.a.bv.1.3 3 40.37 odd 4
6840.2.a.bg.1.1 3 15.2 even 4
7600.2.a.bq.1.1 3 20.3 even 4