# Properties

 Label 2028.4.b.d.337.1 Level $2028$ Weight $4$ Character 2028.337 Analytic conductor $119.656$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,4,Mod(337,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2028.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2028.b (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: $$119.655873492$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 156) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2028.337 Dual form 2028.4.b.d.337.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -2.00000i q^{5} +32.0000i q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -2.00000i q^{5} +32.0000i q^{7} +9.00000 q^{9} +68.0000i q^{11} -6.00000i q^{15} +14.0000 q^{17} +4.00000i q^{19} +96.0000i q^{21} -72.0000 q^{23} +121.000 q^{25} +27.0000 q^{27} +102.000 q^{29} -136.000i q^{31} +204.000i q^{33} +64.0000 q^{35} +386.000i q^{37} +250.000i q^{41} +140.000 q^{43} -18.0000i q^{45} +296.000i q^{47} -681.000 q^{49} +42.0000 q^{51} +526.000 q^{53} +136.000 q^{55} +12.0000i q^{57} -332.000i q^{59} -410.000 q^{61} +288.000i q^{63} +596.000i q^{67} -216.000 q^{69} -880.000i q^{71} -506.000i q^{73} +363.000 q^{75} -2176.00 q^{77} -640.000 q^{79} +81.0000 q^{81} +1380.00i q^{83} -28.0000i q^{85} +306.000 q^{87} -1450.00i q^{89} -408.000i q^{93} +8.00000 q^{95} -446.000i q^{97} +612.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 18 * q^9 $$2 q + 6 q^{3} + 18 q^{9} + 28 q^{17} - 144 q^{23} + 242 q^{25} + 54 q^{27} + 204 q^{29} + 128 q^{35} + 280 q^{43} - 1362 q^{49} + 84 q^{51} + 1052 q^{53} + 272 q^{55} - 820 q^{61} - 432 q^{69} + 726 q^{75} - 4352 q^{77} - 1280 q^{79} + 162 q^{81} + 612 q^{87} + 16 q^{95}+O(q^{100})$$ 2 * q + 6 * q^3 + 18 * q^9 + 28 * q^17 - 144 * q^23 + 242 * q^25 + 54 * q^27 + 204 * q^29 + 128 * q^35 + 280 * q^43 - 1362 * q^49 + 84 * q^51 + 1052 * q^53 + 272 * q^55 - 820 * q^61 - 432 * q^69 + 726 * q^75 - 4352 * q^77 - 1280 * q^79 + 162 * q^81 + 612 * q^87 + 16 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1015$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ − 2.00000i − 0.178885i −0.995992 0.0894427i $$-0.971491\pi$$
0.995992 0.0894427i $$-0.0285086\pi$$
$$6$$ 0 0
$$7$$ 32.0000i 1.72784i 0.503631 + 0.863919i $$0.331997\pi$$
−0.503631 + 0.863919i $$0.668003\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 68.0000i 1.86389i 0.362602 + 0.931944i $$0.381889\pi$$
−0.362602 + 0.931944i $$0.618111\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ − 6.00000i − 0.103280i
$$16$$ 0 0
$$17$$ 14.0000 0.199735 0.0998676 0.995001i $$-0.468158\pi$$
0.0998676 + 0.995001i $$0.468158\pi$$
$$18$$ 0 0
$$19$$ 4.00000i 0.0482980i 0.999708 + 0.0241490i $$0.00768762\pi$$
−0.999708 + 0.0241490i $$0.992312\pi$$
$$20$$ 0 0
$$21$$ 96.0000i 0.997567i
$$22$$ 0 0
$$23$$ −72.0000 −0.652741 −0.326370 0.945242i $$-0.605826\pi$$
−0.326370 + 0.945242i $$0.605826\pi$$
$$24$$ 0 0
$$25$$ 121.000 0.968000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 102.000 0.653135 0.326568 0.945174i $$-0.394108\pi$$
0.326568 + 0.945174i $$0.394108\pi$$
$$30$$ 0 0
$$31$$ − 136.000i − 0.787946i −0.919122 0.393973i $$-0.871100\pi$$
0.919122 0.393973i $$-0.128900\pi$$
$$32$$ 0 0
$$33$$ 204.000i 1.07612i
$$34$$ 0 0
$$35$$ 64.0000 0.309085
$$36$$ 0 0
$$37$$ 386.000i 1.71508i 0.514416 + 0.857541i $$0.328009\pi$$
−0.514416 + 0.857541i $$0.671991\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 250.000i 0.952279i 0.879370 + 0.476140i $$0.157964\pi$$
−0.879370 + 0.476140i $$0.842036\pi$$
$$42$$ 0 0
$$43$$ 140.000 0.496507 0.248253 0.968695i $$-0.420143\pi$$
0.248253 + 0.968695i $$0.420143\pi$$
$$44$$ 0 0
$$45$$ − 18.0000i − 0.0596285i
$$46$$ 0 0
$$47$$ 296.000i 0.918639i 0.888271 + 0.459320i $$0.151907\pi$$
−0.888271 + 0.459320i $$0.848093\pi$$
$$48$$ 0 0
$$49$$ −681.000 −1.98542
$$50$$ 0 0
$$51$$ 42.0000 0.115317
$$52$$ 0 0
$$53$$ 526.000 1.36324 0.681619 0.731707i $$-0.261276\pi$$
0.681619 + 0.731707i $$0.261276\pi$$
$$54$$ 0 0
$$55$$ 136.000 0.333422
$$56$$ 0 0
$$57$$ 12.0000i 0.0278849i
$$58$$ 0 0
$$59$$ − 332.000i − 0.732588i −0.930499 0.366294i $$-0.880626\pi$$
0.930499 0.366294i $$-0.119374\pi$$
$$60$$ 0 0
$$61$$ −410.000 −0.860576 −0.430288 0.902692i $$-0.641588\pi$$
−0.430288 + 0.902692i $$0.641588\pi$$
$$62$$ 0 0
$$63$$ 288.000i 0.575946i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 596.000i 1.08676i 0.839487 + 0.543381i $$0.182856\pi$$
−0.839487 + 0.543381i $$0.817144\pi$$
$$68$$ 0 0
$$69$$ −216.000 −0.376860
$$70$$ 0 0
$$71$$ − 880.000i − 1.47094i −0.677557 0.735470i $$-0.736961\pi$$
0.677557 0.735470i $$-0.263039\pi$$
$$72$$ 0 0
$$73$$ − 506.000i − 0.811272i −0.914035 0.405636i $$-0.867050\pi$$
0.914035 0.405636i $$-0.132950\pi$$
$$74$$ 0 0
$$75$$ 363.000 0.558875
$$76$$ 0 0
$$77$$ −2176.00 −3.22050
$$78$$ 0 0
$$79$$ −640.000 −0.911464 −0.455732 0.890117i $$-0.650622\pi$$
−0.455732 + 0.890117i $$0.650622\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1380.00i 1.82500i 0.409081 + 0.912498i $$0.365849\pi$$
−0.409081 + 0.912498i $$0.634151\pi$$
$$84$$ 0 0
$$85$$ − 28.0000i − 0.0357297i
$$86$$ 0 0
$$87$$ 306.000 0.377088
$$88$$ 0 0
$$89$$ − 1450.00i − 1.72696i −0.504381 0.863481i $$-0.668279\pi$$
0.504381 0.863481i $$-0.331721\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 408.000i − 0.454921i
$$94$$ 0 0
$$95$$ 8.00000 0.00863982
$$96$$ 0 0
$$97$$ − 446.000i − 0.466850i −0.972375 0.233425i $$-0.925007\pi$$
0.972375 0.233425i $$-0.0749933\pi$$
$$98$$ 0 0
$$99$$ 612.000i 0.621296i
$$100$$ 0 0
$$101$$ 610.000 0.600963 0.300482 0.953788i $$-0.402853\pi$$
0.300482 + 0.953788i $$0.402853\pi$$
$$102$$ 0 0
$$103$$ 1352.00 1.29336 0.646682 0.762760i $$-0.276156\pi$$
0.646682 + 0.762760i $$0.276156\pi$$
$$104$$ 0 0
$$105$$ 192.000 0.178450
$$106$$ 0 0
$$107$$ −732.000 −0.661356 −0.330678 0.943744i $$-0.607277\pi$$
−0.330678 + 0.943744i $$0.607277\pi$$
$$108$$ 0 0
$$109$$ − 1514.00i − 1.33041i −0.746660 0.665206i $$-0.768344\pi$$
0.746660 0.665206i $$-0.231656\pi$$
$$110$$ 0 0
$$111$$ 1158.00i 0.990203i
$$112$$ 0 0
$$113$$ −1518.00 −1.26373 −0.631865 0.775079i $$-0.717710\pi$$
−0.631865 + 0.775079i $$0.717710\pi$$
$$114$$ 0 0
$$115$$ 144.000i 0.116766i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 448.000i 0.345110i
$$120$$ 0 0
$$121$$ −3293.00 −2.47408
$$122$$ 0 0
$$123$$ 750.000i 0.549799i
$$124$$ 0 0
$$125$$ − 492.000i − 0.352047i
$$126$$ 0 0
$$127$$ 96.0000 0.0670758 0.0335379 0.999437i $$-0.489323\pi$$
0.0335379 + 0.999437i $$0.489323\pi$$
$$128$$ 0 0
$$129$$ 420.000 0.286658
$$130$$ 0 0
$$131$$ −2548.00 −1.69939 −0.849694 0.527276i $$-0.823213\pi$$
−0.849694 + 0.527276i $$0.823213\pi$$
$$132$$ 0 0
$$133$$ −128.000 −0.0834512
$$134$$ 0 0
$$135$$ − 54.0000i − 0.0344265i
$$136$$ 0 0
$$137$$ 230.000i 0.143432i 0.997425 + 0.0717162i $$0.0228476\pi$$
−0.997425 + 0.0717162i $$0.977152\pi$$
$$138$$ 0 0
$$139$$ 516.000 0.314867 0.157434 0.987530i $$-0.449678\pi$$
0.157434 + 0.987530i $$0.449678\pi$$
$$140$$ 0 0
$$141$$ 888.000i 0.530377i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ − 204.000i − 0.116836i
$$146$$ 0 0
$$147$$ −2043.00 −1.14628
$$148$$ 0 0
$$149$$ − 1842.00i − 1.01277i −0.862308 0.506384i $$-0.830982\pi$$
0.862308 0.506384i $$-0.169018\pi$$
$$150$$ 0 0
$$151$$ 528.000i 0.284556i 0.989827 + 0.142278i $$0.0454428\pi$$
−0.989827 + 0.142278i $$0.954557\pi$$
$$152$$ 0 0
$$153$$ 126.000 0.0665784
$$154$$ 0 0
$$155$$ −272.000 −0.140952
$$156$$ 0 0
$$157$$ −1306.00 −0.663886 −0.331943 0.943299i $$-0.607704\pi$$
−0.331943 + 0.943299i $$0.607704\pi$$
$$158$$ 0 0
$$159$$ 1578.00 0.787066
$$160$$ 0 0
$$161$$ − 2304.00i − 1.12783i
$$162$$ 0 0
$$163$$ 3772.00i 1.81255i 0.422687 + 0.906276i $$0.361087\pi$$
−0.422687 + 0.906276i $$0.638913\pi$$
$$164$$ 0 0
$$165$$ 408.000 0.192502
$$166$$ 0 0
$$167$$ − 384.000i − 0.177933i −0.996035 0.0889665i $$-0.971644\pi$$
0.996035 0.0889665i $$-0.0283564\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 36.0000i 0.0160993i
$$172$$ 0 0
$$173$$ −1462.00 −0.642508 −0.321254 0.946993i $$-0.604104\pi$$
−0.321254 + 0.946993i $$0.604104\pi$$
$$174$$ 0 0
$$175$$ 3872.00i 1.67255i
$$176$$ 0 0
$$177$$ − 996.000i − 0.422960i
$$178$$ 0 0
$$179$$ 1332.00 0.556192 0.278096 0.960553i $$-0.410297\pi$$
0.278096 + 0.960553i $$0.410297\pi$$
$$180$$ 0 0
$$181$$ −2030.00 −0.833639 −0.416820 0.908989i $$-0.636855\pi$$
−0.416820 + 0.908989i $$0.636855\pi$$
$$182$$ 0 0
$$183$$ −1230.00 −0.496854
$$184$$ 0 0
$$185$$ 772.000 0.306803
$$186$$ 0 0
$$187$$ 952.000i 0.372284i
$$188$$ 0 0
$$189$$ 864.000i 0.332522i
$$190$$ 0 0
$$191$$ −16.0000 −0.00606136 −0.00303068 0.999995i $$-0.500965\pi$$
−0.00303068 + 0.999995i $$0.500965\pi$$
$$192$$ 0 0
$$193$$ 2078.00i 0.775014i 0.921867 + 0.387507i $$0.126664\pi$$
−0.921867 + 0.387507i $$0.873336\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3486.00i 1.26075i 0.776292 + 0.630374i $$0.217098\pi$$
−0.776292 + 0.630374i $$0.782902\pi$$
$$198$$ 0 0
$$199$$ −568.000 −0.202334 −0.101167 0.994869i $$-0.532258\pi$$
−0.101167 + 0.994869i $$0.532258\pi$$
$$200$$ 0 0
$$201$$ 1788.00i 0.627442i
$$202$$ 0 0
$$203$$ 3264.00i 1.12851i
$$204$$ 0 0
$$205$$ 500.000 0.170349
$$206$$ 0 0
$$207$$ −648.000 −0.217580
$$208$$ 0 0
$$209$$ −272.000 −0.0900222
$$210$$ 0 0
$$211$$ 3804.00 1.24113 0.620564 0.784156i $$-0.286904\pi$$
0.620564 + 0.784156i $$0.286904\pi$$
$$212$$ 0 0
$$213$$ − 2640.00i − 0.849248i
$$214$$ 0 0
$$215$$ − 280.000i − 0.0888179i
$$216$$ 0 0
$$217$$ 4352.00 1.36144
$$218$$ 0 0
$$219$$ − 1518.00i − 0.468388i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 5912.00i 1.77532i 0.460498 + 0.887661i $$0.347671\pi$$
−0.460498 + 0.887661i $$0.652329\pi$$
$$224$$ 0 0
$$225$$ 1089.00 0.322667
$$226$$ 0 0
$$227$$ − 3308.00i − 0.967223i −0.875283 0.483612i $$-0.839325\pi$$
0.875283 0.483612i $$-0.160675\pi$$
$$228$$ 0 0
$$229$$ 6050.00i 1.74583i 0.487872 + 0.872915i $$0.337773\pi$$
−0.487872 + 0.872915i $$0.662227\pi$$
$$230$$ 0 0
$$231$$ −6528.00 −1.85935
$$232$$ 0 0
$$233$$ −4794.00 −1.34792 −0.673960 0.738768i $$-0.735408\pi$$
−0.673960 + 0.738768i $$0.735408\pi$$
$$234$$ 0 0
$$235$$ 592.000 0.164331
$$236$$ 0 0
$$237$$ −1920.00 −0.526234
$$238$$ 0 0
$$239$$ − 4440.00i − 1.20167i −0.799372 0.600836i $$-0.794834\pi$$
0.799372 0.600836i $$-0.205166\pi$$
$$240$$ 0 0
$$241$$ − 1330.00i − 0.355489i −0.984077 0.177744i $$-0.943120\pi$$
0.984077 0.177744i $$-0.0568800\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 1362.00i 0.355163i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 4140.00i 1.05366i
$$250$$ 0 0
$$251$$ 5116.00 1.28653 0.643265 0.765644i $$-0.277579\pi$$
0.643265 + 0.765644i $$0.277579\pi$$
$$252$$ 0 0
$$253$$ − 4896.00i − 1.21664i
$$254$$ 0 0
$$255$$ − 84.0000i − 0.0206286i
$$256$$ 0 0
$$257$$ −642.000 −0.155824 −0.0779122 0.996960i $$-0.524825\pi$$
−0.0779122 + 0.996960i $$0.524825\pi$$
$$258$$ 0 0
$$259$$ −12352.0 −2.96338
$$260$$ 0 0
$$261$$ 918.000 0.217712
$$262$$ 0 0
$$263$$ 4264.00 0.999732 0.499866 0.866103i $$-0.333383\pi$$
0.499866 + 0.866103i $$0.333383\pi$$
$$264$$ 0 0
$$265$$ − 1052.00i − 0.243864i
$$266$$ 0 0
$$267$$ − 4350.00i − 0.997062i
$$268$$ 0 0
$$269$$ −3850.00 −0.872634 −0.436317 0.899793i $$-0.643717\pi$$
−0.436317 + 0.899793i $$0.643717\pi$$
$$270$$ 0 0
$$271$$ − 2936.00i − 0.658115i −0.944310 0.329058i $$-0.893269\pi$$
0.944310 0.329058i $$-0.106731\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 8228.00i 1.80424i
$$276$$ 0 0
$$277$$ 2066.00 0.448137 0.224068 0.974573i $$-0.428066\pi$$
0.224068 + 0.974573i $$0.428066\pi$$
$$278$$ 0 0
$$279$$ − 1224.00i − 0.262649i
$$280$$ 0 0
$$281$$ 214.000i 0.0454312i 0.999742 + 0.0227156i $$0.00723122\pi$$
−0.999742 + 0.0227156i $$0.992769\pi$$
$$282$$ 0 0
$$283$$ 2620.00 0.550328 0.275164 0.961397i $$-0.411268\pi$$
0.275164 + 0.961397i $$0.411268\pi$$
$$284$$ 0 0
$$285$$ 24.0000 0.00498820
$$286$$ 0 0
$$287$$ −8000.00 −1.64538
$$288$$ 0 0
$$289$$ −4717.00 −0.960106
$$290$$ 0 0
$$291$$ − 1338.00i − 0.269536i
$$292$$ 0 0
$$293$$ 1154.00i 0.230094i 0.993360 + 0.115047i $$0.0367018\pi$$
−0.993360 + 0.115047i $$0.963298\pi$$
$$294$$ 0 0
$$295$$ −664.000 −0.131049
$$296$$ 0 0
$$297$$ 1836.00i 0.358705i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 4480.00i 0.857883i
$$302$$ 0 0
$$303$$ 1830.00 0.346966
$$304$$ 0 0
$$305$$ 820.000i 0.153944i
$$306$$ 0 0
$$307$$ 4076.00i 0.757751i 0.925448 + 0.378876i $$0.123689\pi$$
−0.925448 + 0.378876i $$0.876311\pi$$
$$308$$ 0 0
$$309$$ 4056.00 0.746724
$$310$$ 0 0
$$311$$ 6456.00 1.17713 0.588563 0.808451i $$-0.299694\pi$$
0.588563 + 0.808451i $$0.299694\pi$$
$$312$$ 0 0
$$313$$ −5526.00 −0.997917 −0.498958 0.866626i $$-0.666284\pi$$
−0.498958 + 0.866626i $$0.666284\pi$$
$$314$$ 0 0
$$315$$ 576.000 0.103028
$$316$$ 0 0
$$317$$ − 10458.0i − 1.85293i −0.376377 0.926467i $$-0.622830\pi$$
0.376377 0.926467i $$-0.377170\pi$$
$$318$$ 0 0
$$319$$ 6936.00i 1.21737i
$$320$$ 0 0
$$321$$ −2196.00 −0.381834
$$322$$ 0 0
$$323$$ 56.0000i 0.00964682i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 4542.00i − 0.768114i
$$328$$ 0 0
$$329$$ −9472.00 −1.58726
$$330$$ 0 0
$$331$$ 2348.00i 0.389903i 0.980813 + 0.194951i $$0.0624549\pi$$
−0.980813 + 0.194951i $$0.937545\pi$$
$$332$$ 0 0
$$333$$ 3474.00i 0.571694i
$$334$$ 0 0
$$335$$ 1192.00 0.194406
$$336$$ 0 0
$$337$$ −5298.00 −0.856381 −0.428191 0.903688i $$-0.640849\pi$$
−0.428191 + 0.903688i $$0.640849\pi$$
$$338$$ 0 0
$$339$$ −4554.00 −0.729615
$$340$$ 0 0
$$341$$ 9248.00 1.46864
$$342$$ 0 0
$$343$$ − 10816.0i − 1.70265i
$$344$$ 0 0
$$345$$ 432.000i 0.0674148i
$$346$$ 0 0
$$347$$ 9876.00 1.52787 0.763936 0.645292i $$-0.223264\pi$$
0.763936 + 0.645292i $$0.223264\pi$$
$$348$$ 0 0
$$349$$ 5370.00i 0.823638i 0.911266 + 0.411819i $$0.135106\pi$$
−0.911266 + 0.411819i $$0.864894\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 7330.00i 1.10520i 0.833446 + 0.552601i $$0.186365\pi$$
−0.833446 + 0.552601i $$0.813635\pi$$
$$354$$ 0 0
$$355$$ −1760.00 −0.263130
$$356$$ 0 0
$$357$$ 1344.00i 0.199249i
$$358$$ 0 0
$$359$$ − 7488.00i − 1.10084i −0.834888 0.550420i $$-0.814468\pi$$
0.834888 0.550420i $$-0.185532\pi$$
$$360$$ 0 0
$$361$$ 6843.00 0.997667
$$362$$ 0 0
$$363$$ −9879.00 −1.42841
$$364$$ 0 0
$$365$$ −1012.00 −0.145125
$$366$$ 0 0
$$367$$ 1504.00 0.213919 0.106959 0.994263i $$-0.465889\pi$$
0.106959 + 0.994263i $$0.465889\pi$$
$$368$$ 0 0
$$369$$ 2250.00i 0.317426i
$$370$$ 0 0
$$371$$ 16832.0i 2.35546i
$$372$$ 0 0
$$373$$ 6702.00 0.930339 0.465169 0.885222i $$-0.345993\pi$$
0.465169 + 0.885222i $$0.345993\pi$$
$$374$$ 0 0
$$375$$ − 1476.00i − 0.203254i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 5700.00i − 0.772531i −0.922388 0.386266i $$-0.873765\pi$$
0.922388 0.386266i $$-0.126235\pi$$
$$380$$ 0 0
$$381$$ 288.000 0.0387262
$$382$$ 0 0
$$383$$ 2328.00i 0.310588i 0.987868 + 0.155294i $$0.0496325\pi$$
−0.987868 + 0.155294i $$0.950367\pi$$
$$384$$ 0 0
$$385$$ 4352.00i 0.576100i
$$386$$ 0 0
$$387$$ 1260.00 0.165502
$$388$$ 0 0
$$389$$ 11554.0 1.50594 0.752971 0.658054i $$-0.228620\pi$$
0.752971 + 0.658054i $$0.228620\pi$$
$$390$$ 0 0
$$391$$ −1008.00 −0.130375
$$392$$ 0 0
$$393$$ −7644.00 −0.981142
$$394$$ 0 0
$$395$$ 1280.00i 0.163048i
$$396$$ 0 0
$$397$$ − 6486.00i − 0.819957i −0.912095 0.409979i $$-0.865536\pi$$
0.912095 0.409979i $$-0.134464\pi$$
$$398$$ 0 0
$$399$$ −384.000 −0.0481806
$$400$$ 0 0
$$401$$ − 7698.00i − 0.958653i −0.877637 0.479326i $$-0.840881\pi$$
0.877637 0.479326i $$-0.159119\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 162.000i − 0.0198762i
$$406$$ 0 0
$$407$$ −26248.0 −3.19672
$$408$$ 0 0
$$409$$ 3338.00i 0.403554i 0.979432 + 0.201777i $$0.0646716\pi$$
−0.979432 + 0.201777i $$0.935328\pi$$
$$410$$ 0 0
$$411$$ 690.000i 0.0828107i
$$412$$ 0 0
$$413$$ 10624.0 1.26579
$$414$$ 0 0
$$415$$ 2760.00 0.326465
$$416$$ 0 0
$$417$$ 1548.00 0.181789
$$418$$ 0 0
$$419$$ −52.0000 −0.00606293 −0.00303146 0.999995i $$-0.500965\pi$$
−0.00303146 + 0.999995i $$0.500965\pi$$
$$420$$ 0 0
$$421$$ − 5858.00i − 0.678151i −0.940759 0.339075i $$-0.889886\pi$$
0.940759 0.339075i $$-0.110114\pi$$
$$422$$ 0 0
$$423$$ 2664.00i 0.306213i
$$424$$ 0 0
$$425$$ 1694.00 0.193344
$$426$$ 0 0
$$427$$ − 13120.0i − 1.48694i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8840.00i 0.987953i 0.869475 + 0.493977i $$0.164457\pi$$
−0.869475 + 0.493977i $$0.835543\pi$$
$$432$$ 0 0
$$433$$ −11346.0 −1.25925 −0.629624 0.776900i $$-0.716791\pi$$
−0.629624 + 0.776900i $$0.716791\pi$$
$$434$$ 0 0
$$435$$ − 612.000i − 0.0674555i
$$436$$ 0 0
$$437$$ − 288.000i − 0.0315261i
$$438$$ 0 0
$$439$$ −16456.0 −1.78907 −0.894535 0.446997i $$-0.852493\pi$$
−0.894535 + 0.446997i $$0.852493\pi$$
$$440$$ 0 0
$$441$$ −6129.00 −0.661808
$$442$$ 0 0
$$443$$ −3788.00 −0.406260 −0.203130 0.979152i $$-0.565111\pi$$
−0.203130 + 0.979152i $$0.565111\pi$$
$$444$$ 0 0
$$445$$ −2900.00 −0.308929
$$446$$ 0 0
$$447$$ − 5526.00i − 0.584722i
$$448$$ 0 0
$$449$$ − 546.000i − 0.0573883i −0.999588 0.0286941i $$-0.990865\pi$$
0.999588 0.0286941i $$-0.00913488\pi$$
$$450$$ 0 0
$$451$$ −17000.0 −1.77494
$$452$$ 0 0
$$453$$ 1584.00i 0.164289i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3546.00i 0.362965i 0.983394 + 0.181482i $$0.0580895\pi$$
−0.983394 + 0.181482i $$0.941910\pi$$
$$458$$ 0 0
$$459$$ 378.000 0.0384391
$$460$$ 0 0
$$461$$ 12918.0i 1.30510i 0.757746 + 0.652550i $$0.226301\pi$$
−0.757746 + 0.652550i $$0.773699\pi$$
$$462$$ 0 0
$$463$$ − 18328.0i − 1.83969i −0.392287 0.919843i $$-0.628316\pi$$
0.392287 0.919843i $$-0.371684\pi$$
$$464$$ 0 0
$$465$$ −816.000 −0.0813787
$$466$$ 0 0
$$467$$ −11980.0 −1.18708 −0.593542 0.804803i $$-0.702271\pi$$
−0.593542 + 0.804803i $$0.702271\pi$$
$$468$$ 0 0
$$469$$ −19072.0 −1.87775
$$470$$ 0 0
$$471$$ −3918.00 −0.383295
$$472$$ 0 0
$$473$$ 9520.00i 0.925434i
$$474$$ 0 0
$$475$$ 484.000i 0.0467525i
$$476$$ 0 0
$$477$$ 4734.00 0.454413
$$478$$ 0 0
$$479$$ 12344.0i 1.17748i 0.808323 + 0.588739i $$0.200375\pi$$
−0.808323 + 0.588739i $$0.799625\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ − 6912.00i − 0.651153i
$$484$$ 0 0
$$485$$ −892.000 −0.0835126
$$486$$ 0 0
$$487$$ − 80.0000i − 0.00744383i −0.999993 0.00372192i $$-0.998815\pi$$
0.999993 0.00372192i $$-0.00118473\pi$$
$$488$$ 0 0
$$489$$ 11316.0i 1.04648i
$$490$$ 0 0
$$491$$ 15660.0 1.43936 0.719680 0.694306i $$-0.244288\pi$$
0.719680 + 0.694306i $$0.244288\pi$$
$$492$$ 0 0
$$493$$ 1428.00 0.130454
$$494$$ 0 0
$$495$$ 1224.00 0.111141
$$496$$ 0 0
$$497$$ 28160.0 2.54155
$$498$$ 0 0
$$499$$ − 60.0000i − 0.00538270i −0.999996 0.00269135i $$-0.999143\pi$$
0.999996 0.00269135i $$-0.000856685\pi$$
$$500$$ 0 0
$$501$$ − 1152.00i − 0.102730i
$$502$$ 0 0
$$503$$ 12248.0 1.08571 0.542854 0.839827i $$-0.317344\pi$$
0.542854 + 0.839827i $$0.317344\pi$$
$$504$$ 0 0
$$505$$ − 1220.00i − 0.107504i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 90.0000i − 0.00783729i −0.999992 0.00391864i $$-0.998753\pi$$
0.999992 0.00391864i $$-0.00124735\pi$$
$$510$$ 0 0
$$511$$ 16192.0 1.40175
$$512$$ 0 0
$$513$$ 108.000i 0.00929496i
$$514$$ 0 0
$$515$$ − 2704.00i − 0.231364i
$$516$$ 0 0
$$517$$ −20128.0 −1.71224
$$518$$ 0 0
$$519$$ −4386.00 −0.370952
$$520$$ 0 0
$$521$$ 9818.00 0.825594 0.412797 0.910823i $$-0.364552\pi$$
0.412797 + 0.910823i $$0.364552\pi$$
$$522$$ 0 0
$$523$$ −20252.0 −1.69323 −0.846614 0.532208i $$-0.821363\pi$$
−0.846614 + 0.532208i $$0.821363\pi$$
$$524$$ 0 0
$$525$$ 11616.0i 0.965645i
$$526$$ 0 0
$$527$$ − 1904.00i − 0.157381i
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ − 2988.00i − 0.244196i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 1464.00i 0.118307i
$$536$$ 0 0
$$537$$ 3996.00 0.321118
$$538$$ 0 0
$$539$$ − 46308.0i − 3.70061i
$$540$$ 0 0
$$541$$ 12634.0i 1.00403i 0.864860 + 0.502013i $$0.167407\pi$$
−0.864860 + 0.502013i $$0.832593\pi$$
$$542$$ 0 0
$$543$$ −6090.00 −0.481302
$$544$$ 0 0
$$545$$ −3028.00 −0.237991
$$546$$ 0 0
$$547$$ 11756.0 0.918922 0.459461 0.888198i $$-0.348043\pi$$
0.459461 + 0.888198i $$0.348043\pi$$
$$548$$ 0 0
$$549$$ −3690.00 −0.286859
$$550$$ 0 0
$$551$$ 408.000i 0.0315452i
$$552$$ 0 0
$$553$$ − 20480.0i − 1.57486i
$$554$$ 0 0
$$555$$ 2316.00 0.177133
$$556$$ 0 0
$$557$$ − 17622.0i − 1.34052i −0.742128 0.670259i $$-0.766183\pi$$
0.742128 0.670259i $$-0.233817\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 2856.00i 0.214938i
$$562$$ 0 0
$$563$$ 23092.0 1.72862 0.864309 0.502961i $$-0.167756\pi$$
0.864309 + 0.502961i $$0.167756\pi$$
$$564$$ 0 0
$$565$$ 3036.00i 0.226063i
$$566$$ 0 0
$$567$$ 2592.00i 0.191982i
$$568$$ 0 0
$$569$$ 1302.00 0.0959274 0.0479637 0.998849i $$-0.484727\pi$$
0.0479637 + 0.998849i $$0.484727\pi$$
$$570$$ 0 0
$$571$$ −24868.0 −1.82258 −0.911290 0.411765i $$-0.864913\pi$$
−0.911290 + 0.411765i $$0.864913\pi$$
$$572$$ 0 0
$$573$$ −48.0000 −0.00349953
$$574$$ 0 0
$$575$$ −8712.00 −0.631853
$$576$$ 0 0
$$577$$ 2562.00i 0.184848i 0.995720 + 0.0924241i $$0.0294616\pi$$
−0.995720 + 0.0924241i $$0.970538\pi$$
$$578$$ 0 0
$$579$$ 6234.00i 0.447455i
$$580$$ 0 0
$$581$$ −44160.0 −3.15330
$$582$$ 0 0
$$583$$ 35768.0i 2.54092i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 13484.0i 0.948116i 0.880494 + 0.474058i $$0.157211\pi$$
−0.880494 + 0.474058i $$0.842789\pi$$
$$588$$ 0 0
$$589$$ 544.000 0.0380562
$$590$$ 0 0
$$591$$ 10458.0i 0.727893i
$$592$$ 0 0
$$593$$ 16974.0i 1.17544i 0.809063 + 0.587722i $$0.199975\pi$$
−0.809063 + 0.587722i $$0.800025\pi$$
$$594$$ 0 0
$$595$$ 896.000 0.0617352
$$596$$ 0 0
$$597$$ −1704.00 −0.116818
$$598$$ 0 0
$$599$$ 3864.00 0.263571 0.131785 0.991278i $$-0.457929\pi$$
0.131785 + 0.991278i $$0.457929\pi$$
$$600$$ 0 0
$$601$$ 17546.0 1.19088 0.595438 0.803401i $$-0.296979\pi$$
0.595438 + 0.803401i $$0.296979\pi$$
$$602$$ 0 0
$$603$$ 5364.00i 0.362254i
$$604$$ 0 0
$$605$$ 6586.00i 0.442577i
$$606$$ 0 0
$$607$$ 9296.00 0.621603 0.310801 0.950475i $$-0.399403\pi$$
0.310801 + 0.950475i $$0.399403\pi$$
$$608$$ 0 0
$$609$$ 9792.00i 0.651547i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 6914.00i − 0.455553i −0.973713 0.227776i $$-0.926854\pi$$
0.973713 0.227776i $$-0.0731455\pi$$
$$614$$ 0 0
$$615$$ 1500.00 0.0983510
$$616$$ 0 0
$$617$$ − 25446.0i − 1.66032i −0.557525 0.830160i $$-0.688249\pi$$
0.557525 0.830160i $$-0.311751\pi$$
$$618$$ 0 0
$$619$$ 11236.0i 0.729585i 0.931089 + 0.364792i $$0.118860\pi$$
−0.931089 + 0.364792i $$0.881140\pi$$
$$620$$ 0 0
$$621$$ −1944.00 −0.125620
$$622$$ 0 0
$$623$$ 46400.0 2.98391
$$624$$ 0 0
$$625$$ 14141.0 0.905024
$$626$$ 0 0
$$627$$ −816.000 −0.0519743
$$628$$ 0 0
$$629$$ 5404.00i 0.342562i
$$630$$ 0 0
$$631$$ 29424.0i 1.85634i 0.372156 + 0.928170i $$0.378619\pi$$
−0.372156 + 0.928170i $$0.621381\pi$$
$$632$$ 0 0
$$633$$ 11412.0 0.716566
$$634$$ 0 0
$$635$$ − 192.000i − 0.0119989i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 7920.00i − 0.490314i
$$640$$ 0 0
$$641$$ 5054.00 0.311421 0.155711 0.987803i $$-0.450233\pi$$
0.155711 + 0.987803i $$0.450233\pi$$
$$642$$ 0 0
$$643$$ − 1132.00i − 0.0694273i −0.999397 0.0347136i $$-0.988948\pi$$
0.999397 0.0347136i $$-0.0110519\pi$$
$$644$$ 0 0
$$645$$ − 840.000i − 0.0512790i
$$646$$ 0 0
$$647$$ −1464.00 −0.0889579 −0.0444790 0.999010i $$-0.514163\pi$$
−0.0444790 + 0.999010i $$0.514163\pi$$
$$648$$ 0 0
$$649$$ 22576.0 1.36546
$$650$$ 0 0
$$651$$ 13056.0 0.786029
$$652$$ 0 0
$$653$$ 5494.00 0.329245 0.164622 0.986357i $$-0.447359\pi$$
0.164622 + 0.986357i $$0.447359\pi$$
$$654$$ 0 0
$$655$$ 5096.00i 0.303996i
$$656$$ 0 0
$$657$$ − 4554.00i − 0.270424i
$$658$$ 0 0
$$659$$ 11580.0 0.684511 0.342256 0.939607i $$-0.388809\pi$$
0.342256 + 0.939607i $$0.388809\pi$$
$$660$$ 0 0
$$661$$ 1298.00i 0.0763787i 0.999271 + 0.0381894i $$0.0121590\pi$$
−0.999271 + 0.0381894i $$0.987841\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 256.000i 0.0149282i
$$666$$ 0 0
$$667$$ −7344.00 −0.426328
$$668$$ 0 0
$$669$$ 17736.0i 1.02498i
$$670$$ 0 0
$$671$$ − 27880.0i − 1.60402i
$$672$$ 0 0
$$673$$ −16162.0 −0.925705 −0.462852 0.886435i $$-0.653174\pi$$
−0.462852 + 0.886435i $$0.653174\pi$$
$$674$$ 0 0
$$675$$ 3267.00 0.186292
$$676$$ 0 0
$$677$$ −9890.00 −0.561453 −0.280726 0.959788i $$-0.590575\pi$$
−0.280726 + 0.959788i $$0.590575\pi$$
$$678$$ 0 0
$$679$$ 14272.0 0.806641
$$680$$ 0 0
$$681$$ − 9924.00i − 0.558427i
$$682$$ 0 0
$$683$$ − 27100.0i − 1.51823i −0.650955 0.759116i $$-0.725631\pi$$
0.650955 0.759116i $$-0.274369\pi$$
$$684$$ 0 0
$$685$$ 460.000 0.0256580
$$686$$ 0 0
$$687$$ 18150.0i 1.00796i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 11132.0i − 0.612853i −0.951894 0.306426i $$-0.900867\pi$$
0.951894 0.306426i $$-0.0991333\pi$$
$$692$$ 0 0
$$693$$ −19584.0 −1.07350
$$694$$ 0 0
$$695$$ − 1032.00i − 0.0563252i
$$696$$ 0 0
$$697$$ 3500.00i 0.190204i
$$698$$ 0 0
$$699$$ −14382.0 −0.778222
$$700$$ 0 0
$$701$$ −1862.00 −0.100323 −0.0501617 0.998741i $$-0.515974\pi$$
−0.0501617 + 0.998741i $$0.515974\pi$$
$$702$$ 0 0
$$703$$ −1544.00 −0.0828351
$$704$$ 0 0
$$705$$ 1776.00 0.0948766
$$706$$ 0 0
$$707$$ 19520.0i 1.03837i
$$708$$ 0 0
$$709$$ 7938.00i 0.420477i 0.977650 + 0.210238i $$0.0674240\pi$$
−0.977650 + 0.210238i $$0.932576\pi$$
$$710$$ 0 0
$$711$$ −5760.00 −0.303821
$$712$$ 0 0
$$713$$ 9792.00i 0.514324i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 13320.0i − 0.693786i
$$718$$ 0 0
$$719$$ 24240.0 1.25730 0.628651 0.777688i $$-0.283608\pi$$
0.628651 + 0.777688i $$0.283608\pi$$
$$720$$ 0 0
$$721$$ 43264.0i 2.23472i
$$722$$ 0 0
$$723$$ − 3990.00i − 0.205242i
$$724$$ 0 0
$$725$$ 12342.0 0.632235
$$726$$ 0 0
$$727$$ 13720.0 0.699927 0.349963 0.936763i $$-0.386194\pi$$
0.349963 + 0.936763i $$0.386194\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 1960.00 0.0991699
$$732$$ 0 0
$$733$$ 21958.0i 1.10646i 0.833028 + 0.553231i $$0.186605\pi$$
−0.833028 + 0.553231i $$0.813395\pi$$
$$734$$ 0 0
$$735$$ 4086.00i 0.205054i
$$736$$ 0 0
$$737$$ −40528.0 −2.02560
$$738$$ 0 0
$$739$$ − 13348.0i − 0.664430i −0.943204 0.332215i $$-0.892204\pi$$
0.943204 0.332215i $$-0.107796\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 20304.0i 1.00253i 0.865293 + 0.501266i $$0.167132\pi$$
−0.865293 + 0.501266i $$0.832868\pi$$
$$744$$ 0 0
$$745$$ −3684.00 −0.181170
$$746$$ 0 0
$$747$$ 12420.0i 0.608332i
$$748$$ 0 0
$$749$$ − 23424.0i − 1.14272i
$$750$$ 0 0
$$751$$ 3952.00 0.192025 0.0960123 0.995380i $$-0.469391\pi$$
0.0960123 + 0.995380i $$0.469391\pi$$
$$752$$ 0 0
$$753$$ 15348.0 0.742779
$$754$$ 0 0
$$755$$ 1056.00 0.0509030
$$756$$ 0 0
$$757$$ −22386.0 −1.07481 −0.537406 0.843324i $$-0.680596\pi$$
−0.537406 + 0.843324i $$0.680596\pi$$
$$758$$ 0 0
$$759$$ − 14688.0i − 0.702425i
$$760$$ 0 0
$$761$$ − 12458.0i − 0.593433i −0.954966 0.296716i $$-0.904108\pi$$
0.954966 0.296716i $$-0.0958916\pi$$
$$762$$ 0 0
$$763$$ 48448.0 2.29874
$$764$$ 0 0
$$765$$ − 252.000i − 0.0119099i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ − 28126.0i − 1.31892i −0.751740 0.659460i $$-0.770785\pi$$
0.751740 0.659460i $$-0.229215\pi$$
$$770$$ 0 0
$$771$$ −1926.00 −0.0899652
$$772$$ 0 0
$$773$$ − 35778.0i − 1.66474i −0.554219 0.832371i $$-0.686983\pi$$
0.554219 0.832371i $$-0.313017\pi$$
$$774$$ 0 0
$$775$$ − 16456.0i − 0.762732i
$$776$$ 0 0
$$777$$ −37056.0 −1.71091
$$778$$ 0 0
$$779$$ −1000.00 −0.0459932
$$780$$ 0 0
$$781$$ 59840.0 2.74167
$$782$$ 0 0
$$783$$ 2754.00 0.125696
$$784$$ 0 0
$$785$$ 2612.00i 0.118760i
$$786$$ 0 0
$$787$$ − 7252.00i − 0.328470i −0.986421 0.164235i $$-0.947484\pi$$
0.986421 0.164235i $$-0.0525155\pi$$
$$788$$ 0 0
$$789$$ 12792.0 0.577196
$$790$$ 0 0
$$791$$ − 48576.0i − 2.18352i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ − 3156.00i − 0.140795i
$$796$$ 0 0
$$797$$ −4454.00 −0.197953 −0.0989766 0.995090i $$-0.531557\pi$$
−0.0989766 + 0.995090i $$0.531557\pi$$
$$798$$ 0 0
$$799$$ 4144.00i 0.183485i
$$800$$ 0 0
$$801$$ − 13050.0i − 0.575654i
$$802$$ 0 0
$$803$$ 34408.0 1.51212
$$804$$ 0 0
$$805$$ −4608.00 −0.201752
$$806$$ 0 0
$$807$$ −11550.0 −0.503816
$$808$$ 0 0
$$809$$ −34118.0 −1.48273 −0.741363 0.671104i $$-0.765820\pi$$
−0.741363 + 0.671104i $$0.765820\pi$$
$$810$$ 0 0
$$811$$ 16428.0i 0.711301i 0.934619 + 0.355650i $$0.115741\pi$$
−0.934619 + 0.355650i $$0.884259\pi$$
$$812$$ 0 0
$$813$$ − 8808.00i − 0.379963i
$$814$$ 0 0
$$815$$ 7544.00 0.324239
$$816$$ 0 0
$$817$$ 560.000i 0.0239803i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 18738.0i − 0.796542i −0.917268 0.398271i $$-0.869610\pi$$
0.917268 0.398271i $$-0.130390\pi$$
$$822$$ 0 0
$$823$$ −13928.0 −0.589914 −0.294957 0.955510i $$-0.595305\pi$$
−0.294957 + 0.955510i $$0.595305\pi$$
$$824$$ 0 0
$$825$$ 24684.0i 1.04168i
$$826$$ 0 0
$$827$$ − 41804.0i − 1.75776i −0.477043 0.878880i $$-0.658291\pi$$
0.477043 0.878880i $$-0.341709\pi$$
$$828$$ 0 0
$$829$$ 43226.0 1.81098 0.905489 0.424369i $$-0.139504\pi$$
0.905489 + 0.424369i $$0.139504\pi$$
$$830$$ 0 0
$$831$$ 6198.00 0.258732
$$832$$ 0 0
$$833$$ −9534.00 −0.396559
$$834$$ 0 0
$$835$$ −768.000 −0.0318296
$$836$$ 0 0
$$837$$ − 3672.00i − 0.151640i
$$838$$ 0 0
$$839$$ − 10240.0i − 0.421364i −0.977555 0.210682i $$-0.932432\pi$$
0.977555 0.210682i $$-0.0675684\pi$$
$$840$$ 0 0
$$841$$ −13985.0 −0.573414
$$842$$ 0 0
$$843$$ 642.000i 0.0262297i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 105376.i − 4.27481i
$$848$$ 0 0
$$849$$ 7860.00 0.317732
$$850$$ 0 0
$$851$$ − 27792.0i − 1.11950i
$$852$$ 0 0
$$853$$ 25682.0i 1.03087i 0.856928 + 0.515437i $$0.172370\pi$$
−0.856928 + 0.515437i $$0.827630\pi$$
$$854$$ 0 0
$$855$$ 72.0000 0.00287994
$$856$$ 0 0
$$857$$ 21558.0 0.859285 0.429643 0.902999i $$-0.358640\pi$$
0.429643 + 0.902999i $$0.358640\pi$$
$$858$$ 0 0
$$859$$ −14060.0 −0.558465 −0.279232 0.960224i $$-0.590080\pi$$
−0.279232 + 0.960224i $$0.590080\pi$$
$$860$$ 0 0
$$861$$ −24000.0 −0.949963
$$862$$ 0 0
$$863$$ 42008.0i 1.65697i 0.560008 + 0.828487i $$0.310798\pi$$
−0.560008 + 0.828487i $$0.689202\pi$$
$$864$$ 0 0
$$865$$ 2924.00i 0.114935i
$$866$$ 0 0
$$867$$ −14151.0 −0.554317
$$868$$ 0 0
$$869$$ − 43520.0i − 1.69887i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 4014.00i − 0.155617i
$$874$$ 0 0
$$875$$ 15744.0 0.608279
$$876$$ 0 0
$$877$$ 23734.0i 0.913843i 0.889507 + 0.456921i $$0.151048\pi$$
−0.889507 + 0.456921i $$0.848952\pi$$
$$878$$ 0 0
$$879$$ 3462.00i 0.132845i
$$880$$ 0 0
$$881$$ 37550.0 1.43597 0.717986 0.696057i $$-0.245064\pi$$
0.717986 + 0.696057i $$0.245064\pi$$
$$882$$ 0 0
$$883$$ −12556.0 −0.478531 −0.239266 0.970954i $$-0.576907\pi$$
−0.239266 + 0.970954i $$0.576907\pi$$
$$884$$ 0 0
$$885$$ −1992.00 −0.0756614
$$886$$ 0 0
$$887$$ 37368.0 1.41454 0.707269 0.706945i $$-0.249927\pi$$
0.707269 + 0.706945i $$0.249927\pi$$
$$888$$ 0 0
$$889$$ 3072.00i 0.115896i
$$890$$ 0 0
$$891$$ 5508.00i 0.207099i
$$892$$ 0 0
$$893$$ −1184.00 −0.0443685
$$894$$ 0 0
$$895$$ − 2664.00i − 0.0994946i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 13872.0i − 0.514635i
$$900$$ 0 0
$$901$$ 7364.00 0.272287
$$902$$ 0 0
$$903$$ 13440.0i 0.495299i
$$904$$ 0 0
$$905$$ 4060.00i 0.149126i
$$906$$ 0 0
$$907$$ −33364.0 −1.22143 −0.610713 0.791852i $$-0.709117\pi$$
−0.610713 + 0.791852i $$0.709117\pi$$
$$908$$ 0 0
$$909$$ 5490.00 0.200321
$$910$$ 0 0
$$911$$ 50432.0 1.83412 0.917062 0.398745i $$-0.130554\pi$$
0.917062 + 0.398745i $$0.130554\pi$$
$$912$$ 0 0
$$913$$ −93840.0 −3.40159
$$914$$ 0 0
$$915$$ 2460.00i 0.0888799i
$$916$$ 0 0
$$917$$ − 81536.0i − 2.93627i
$$918$$ 0 0
$$919$$ 11864.0 0.425851 0.212926 0.977068i $$-0.431701\pi$$
0.212926 + 0.977068i $$0.431701\pi$$
$$920$$ 0 0
$$921$$ 12228.0i 0.437488i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 46706.0i 1.66020i
$$926$$ 0 0
$$927$$ 12168.0 0.431121
$$928$$ 0 0
$$929$$ − 5950.00i − 0.210133i −0.994465 0.105066i $$-0.966495\pi$$
0.994465 0.105066i $$-0.0335055\pi$$
$$930$$ 0 0
$$931$$ − 2724.00i − 0.0958920i
$$932$$ 0 0
$$933$$ 19368.0 0.679614
$$934$$ 0 0
$$935$$ 1904.00 0.0665962
$$936$$ 0 0
$$937$$ −20806.0 −0.725403 −0.362701 0.931905i $$-0.618146\pi$$
−0.362701 + 0.931905i $$0.618146\pi$$
$$938$$ 0 0
$$939$$ −16578.0 −0.576148
$$940$$ 0 0
$$941$$ − 22346.0i − 0.774133i −0.922052 0.387066i $$-0.873488\pi$$
0.922052 0.387066i $$-0.126512\pi$$
$$942$$ 0 0
$$943$$ − 18000.0i − 0.621591i
$$944$$ 0 0
$$945$$ 1728.00 0.0594834
$$946$$ 0 0
$$947$$ − 31636.0i − 1.08557i −0.839873 0.542783i $$-0.817370\pi$$
0.839873 0.542783i $$-0.182630\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ − 31374.0i − 1.06979i
$$952$$ 0 0
$$953$$ 23190.0 0.788245 0.394123 0.919058i $$-0.371049\pi$$
0.394123 + 0.919058i $$0.371049\pi$$
$$954$$ 0 0
$$955$$ 32.0000i 0.00108429i
$$956$$ 0 0
$$957$$ 20808.0i 0.702850i
$$958$$ 0 0
$$959$$ −7360.00 −0.247828
$$960$$ 0 0
$$961$$ 11295.0 0.379141
$$962$$ 0 0
$$963$$ −6588.00 −0.220452
$$964$$ 0 0
$$965$$ 4156.00 0.138639
$$966$$ 0 0
$$967$$ 304.000i 0.0101096i 0.999987 + 0.00505480i $$0.00160900\pi$$
−0.999987 + 0.00505480i $$0.998391\pi$$
$$968$$ 0 0
$$969$$ 168.000i 0.00556960i
$$970$$ 0 0
$$971$$ −53372.0 −1.76394 −0.881972 0.471302i $$-0.843784\pi$$
−0.881972 + 0.471302i $$0.843784\pi$$
$$972$$ 0 0
$$973$$ 16512.0i 0.544039i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 13650.0i 0.446983i 0.974706 + 0.223491i $$0.0717455\pi$$
−0.974706 + 0.223491i $$0.928255\pi$$
$$978$$ 0 0
$$979$$ 98600.0 3.21887
$$980$$ 0 0
$$981$$ − 13626.0i − 0.443471i
$$982$$ 0 0
$$983$$ − 34992.0i − 1.13537i −0.823245 0.567686i $$-0.807839\pi$$
0.823245 0.567686i $$-0.192161\pi$$
$$984$$ 0 0
$$985$$ 6972.00 0.225529
$$986$$ 0 0
$$987$$ −28416.0 −0.916405
$$988$$ 0 0
$$989$$ −10080.0 −0.324090
$$990$$ 0 0
$$991$$ 18096.0 0.580059 0.290029 0.957018i $$-0.406335\pi$$
0.290029 + 0.957018i $$0.406335\pi$$
$$992$$ 0 0
$$993$$ 7044.00i 0.225110i
$$994$$ 0 0
$$995$$ 1136.00i 0.0361946i
$$996$$ 0 0
$$997$$ −18914.0 −0.600815 −0.300407 0.953811i $$-0.597123\pi$$
−0.300407 + 0.953811i $$0.597123\pi$$
$$998$$ 0 0
$$999$$ 10422.0i 0.330068i
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 2028.4.b.d.337.1 2
13.5 odd 4 156.4.a.b.1.1 1
13.8 odd 4 2028.4.a.b.1.1 1
13.12 even 2 inner 2028.4.b.d.337.2 2
39.5 even 4 468.4.a.a.1.1 1
52.31 even 4 624.4.a.b.1.1 1
104.5 odd 4 2496.4.a.d.1.1 1
104.83 even 4 2496.4.a.m.1.1 1
156.83 odd 4 1872.4.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
156.4.a.b.1.1 1 13.5 odd 4
468.4.a.a.1.1 1 39.5 even 4
624.4.a.b.1.1 1 52.31 even 4
1872.4.a.i.1.1 1 156.83 odd 4
2028.4.a.b.1.1 1 13.8 odd 4
2028.4.b.d.337.1 2 1.1 even 1 trivial
2028.4.b.d.337.2 2 13.12 even 2 inner
2496.4.a.d.1.1 1 104.5 odd 4
2496.4.a.m.1.1 1 104.83 even 4