# Properties

 Label 55.5.d.a.54.1 Level $55$ Weight $5$ Character 55.54 Self dual yes Analytic conductor $5.685$ Analytic rank $0$ Dimension $1$ CM discriminant -55 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,5,Mod(54,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("55.54");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 55.d (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$5.68534796961$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 54.1 Character $$\chi$$ $$=$$ 55.54

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} -7.00000 q^{4} +25.0000 q^{5} -78.0000 q^{7} +69.0000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{2} -7.00000 q^{4} +25.0000 q^{5} -78.0000 q^{7} +69.0000 q^{8} +81.0000 q^{9} -75.0000 q^{10} +121.000 q^{11} +162.000 q^{13} +234.000 q^{14} -95.0000 q^{16} +402.000 q^{17} -243.000 q^{18} -175.000 q^{20} -363.000 q^{22} +625.000 q^{25} -486.000 q^{26} +546.000 q^{28} -1598.00 q^{31} -819.000 q^{32} -1206.00 q^{34} -1950.00 q^{35} -567.000 q^{36} +1725.00 q^{40} +3522.00 q^{43} -847.000 q^{44} +2025.00 q^{45} +3683.00 q^{49} -1875.00 q^{50} -1134.00 q^{52} +3025.00 q^{55} -5382.00 q^{56} +3442.00 q^{59} +4794.00 q^{62} -6318.00 q^{63} +3977.00 q^{64} +4050.00 q^{65} -2814.00 q^{68} +5850.00 q^{70} -3998.00 q^{71} +5589.00 q^{72} -10638.0 q^{73} -9438.00 q^{77} -2375.00 q^{80} +6561.00 q^{81} +13602.0 q^{83} +10050.0 q^{85} -10566.0 q^{86} +8349.00 q^{88} -15838.0 q^{89} -6075.00 q^{90} -12636.0 q^{91} -11049.0 q^{98} +9801.00 q^{99} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.00000 −0.750000 −0.375000 0.927025i $$-0.622357\pi$$
−0.375000 + 0.927025i $$0.622357\pi$$
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ −7.00000 −0.437500
$$5$$ 25.0000 1.00000
$$6$$ 0 0
$$7$$ −78.0000 −1.59184 −0.795918 0.605404i $$-0.793012\pi$$
−0.795918 + 0.605404i $$0.793012\pi$$
$$8$$ 69.0000 1.07812
$$9$$ 81.0000 1.00000
$$10$$ −75.0000 −0.750000
$$11$$ 121.000 1.00000
$$12$$ 0 0
$$13$$ 162.000 0.958580 0.479290 0.877657i $$-0.340894\pi$$
0.479290 + 0.877657i $$0.340894\pi$$
$$14$$ 234.000 1.19388
$$15$$ 0 0
$$16$$ −95.0000 −0.371094
$$17$$ 402.000 1.39100 0.695502 0.718524i $$-0.255182\pi$$
0.695502 + 0.718524i $$0.255182\pi$$
$$18$$ −243.000 −0.750000
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ −175.000 −0.437500
$$21$$ 0 0
$$22$$ −363.000 −0.750000
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 625.000 1.00000
$$26$$ −486.000 −0.718935
$$27$$ 0 0
$$28$$ 546.000 0.696429
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ −1598.00 −1.66285 −0.831426 0.555636i $$-0.812475\pi$$
−0.831426 + 0.555636i $$0.812475\pi$$
$$32$$ −819.000 −0.799805
$$33$$ 0 0
$$34$$ −1206.00 −1.04325
$$35$$ −1950.00 −1.59184
$$36$$ −567.000 −0.437500
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 1725.00 1.07812
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 3522.00 1.90481 0.952407 0.304830i $$-0.0985997\pi$$
0.952407 + 0.304830i $$0.0985997\pi$$
$$44$$ −847.000 −0.437500
$$45$$ 2025.00 1.00000
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 3683.00 1.53394
$$50$$ −1875.00 −0.750000
$$51$$ 0 0
$$52$$ −1134.00 −0.419379
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 3025.00 1.00000
$$56$$ −5382.00 −1.71620
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3442.00 0.988796 0.494398 0.869236i $$-0.335388\pi$$
0.494398 + 0.869236i $$0.335388\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 4794.00 1.24714
$$63$$ −6318.00 −1.59184
$$64$$ 3977.00 0.970947
$$65$$ 4050.00 0.958580
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ −2814.00 −0.608564
$$69$$ 0 0
$$70$$ 5850.00 1.19388
$$71$$ −3998.00 −0.793097 −0.396548 0.918014i $$-0.629792\pi$$
−0.396548 + 0.918014i $$0.629792\pi$$
$$72$$ 5589.00 1.07812
$$73$$ −10638.0 −1.99625 −0.998123 0.0612334i $$-0.980497\pi$$
−0.998123 + 0.0612334i $$0.980497\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −9438.00 −1.59184
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ −2375.00 −0.371094
$$81$$ 6561.00 1.00000
$$82$$ 0 0
$$83$$ 13602.0 1.97445 0.987226 0.159326i $$-0.0509321\pi$$
0.987226 + 0.159326i $$0.0509321\pi$$
$$84$$ 0 0
$$85$$ 10050.0 1.39100
$$86$$ −10566.0 −1.42861
$$87$$ 0 0
$$88$$ 8349.00 1.07812
$$89$$ −15838.0 −1.99950 −0.999748 0.0224705i $$-0.992847\pi$$
−0.999748 + 0.0224705i $$0.992847\pi$$
$$90$$ −6075.00 −0.750000
$$91$$ −12636.0 −1.52590
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ −11049.0 −1.15046
$$99$$ 9801.00 1.00000
$$100$$ −4375.00 −0.437500
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 11178.0 1.03347
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1602.00 0.139925 0.0699624 0.997550i $$-0.477712\pi$$
0.0699624 + 0.997550i $$0.477712\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ −9075.00 −0.750000
$$111$$ 0 0
$$112$$ 7410.00 0.590721
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 13122.0 0.958580
$$118$$ −10326.0 −0.741597
$$119$$ −31356.0 −2.21425
$$120$$ 0 0
$$121$$ 14641.0 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 11186.0 0.727497
$$125$$ 15625.0 1.00000
$$126$$ 18954.0 1.19388
$$127$$ −31278.0 −1.93924 −0.969620 0.244616i $$-0.921338\pi$$
−0.969620 + 0.244616i $$0.921338\pi$$
$$128$$ 1173.00 0.0715942
$$129$$ 0 0
$$130$$ −12150.0 −0.718935
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 27738.0 1.49968
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 13650.0 0.696429
$$141$$ 0 0
$$142$$ 11994.0 0.594822
$$143$$ 19602.0 0.958580
$$144$$ −7695.00 −0.371094
$$145$$ 0 0
$$146$$ 31914.0 1.49719
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 32562.0 1.39100
$$154$$ 28314.0 1.19388
$$155$$ −39950.0 −1.66285
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ −20475.0 −0.799805
$$161$$ 0 0
$$162$$ −19683.0 −0.750000
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −40806.0 −1.48084
$$167$$ −7758.00 −0.278174 −0.139087 0.990280i $$-0.544417\pi$$
−0.139087 + 0.990280i $$0.544417\pi$$
$$168$$ 0 0
$$169$$ −2317.00 −0.0811246
$$170$$ −30150.0 −1.04325
$$171$$ 0 0
$$172$$ −24654.0 −0.833356
$$173$$ −3678.00 −0.122891 −0.0614454 0.998110i $$-0.519571\pi$$
−0.0614454 + 0.998110i $$0.519571\pi$$
$$174$$ 0 0
$$175$$ −48750.0 −1.59184
$$176$$ −11495.0 −0.371094
$$177$$ 0 0
$$178$$ 47514.0 1.49962
$$179$$ −62638.0 −1.95493 −0.977466 0.211091i $$-0.932298\pi$$
−0.977466 + 0.211091i $$0.932298\pi$$
$$180$$ −14175.0 −0.437500
$$181$$ 51442.0 1.57022 0.785110 0.619356i $$-0.212606\pi$$
0.785110 + 0.619356i $$0.212606\pi$$
$$182$$ 37908.0 1.14443
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 48642.0 1.39100
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 41282.0 1.13160 0.565801 0.824542i $$-0.308567\pi$$
0.565801 + 0.824542i $$0.308567\pi$$
$$192$$ 0 0
$$193$$ −73518.0 −1.97369 −0.986845 0.161668i $$-0.948313\pi$$
−0.986845 + 0.161668i $$0.948313\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −25781.0 −0.671101
$$197$$ −70398.0 −1.81396 −0.906980 0.421173i $$-0.861619\pi$$
−0.906980 + 0.421173i $$0.861619\pi$$
$$198$$ −29403.0 −0.750000
$$199$$ −47518.0 −1.19992 −0.599960 0.800030i $$-0.704817\pi$$
−0.599960 + 0.800030i $$0.704817\pi$$
$$200$$ 43125.0 1.07812
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −15390.0 −0.355723
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ −4806.00 −0.104944
$$215$$ 88050.0 1.90481
$$216$$ 0 0
$$217$$ 124644. 2.64699
$$218$$ 0 0
$$219$$ 0 0
$$220$$ −21175.0 −0.437500
$$221$$ 65124.0 1.33339
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 63882.0 1.27316
$$225$$ 50625.0 1.00000
$$226$$ 0 0
$$227$$ −66078.0 −1.28235 −0.641173 0.767396i $$-0.721552\pi$$
−0.641173 + 0.767396i $$0.721552\pi$$
$$228$$ 0 0
$$229$$ 73202.0 1.39589 0.697946 0.716150i $$-0.254097\pi$$
0.697946 + 0.716150i $$0.254097\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −60558.0 −1.11547 −0.557737 0.830018i $$-0.688330\pi$$
−0.557737 + 0.830018i $$0.688330\pi$$
$$234$$ −39366.0 −0.718935
$$235$$ 0 0
$$236$$ −24094.0 −0.432598
$$237$$ 0 0
$$238$$ 94068.0 1.66069
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ −43923.0 −0.750000
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 92075.0 1.53394
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −110262. −1.79276
$$249$$ 0 0
$$250$$ −46875.0 −0.750000
$$251$$ −46478.0 −0.737734 −0.368867 0.929482i $$-0.620254\pi$$
−0.368867 + 0.929482i $$0.620254\pi$$
$$252$$ 44226.0 0.696429
$$253$$ 0 0
$$254$$ 93834.0 1.45443
$$255$$ 0 0
$$256$$ −67151.0 −1.02464
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −28350.0 −0.419379
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 117042. 1.69212 0.846058 0.533091i $$-0.178969\pi$$
0.846058 + 0.533091i $$0.178969\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −27758.0 −0.383604 −0.191802 0.981434i $$-0.561433\pi$$
−0.191802 + 0.981434i $$0.561433\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ −38190.0 −0.516193
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 75625.0 1.00000
$$276$$ 0 0
$$277$$ −15678.0 −0.204330 −0.102165 0.994767i $$-0.532577\pi$$
−0.102165 + 0.994767i $$0.532577\pi$$
$$278$$ 0 0
$$279$$ −129438. −1.66285
$$280$$ −134550. −1.71620
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ −135678. −1.69409 −0.847045 0.531521i $$-0.821621\pi$$
−0.847045 + 0.531521i $$0.821621\pi$$
$$284$$ 27986.0 0.346980
$$285$$ 0 0
$$286$$ −58806.0 −0.718935
$$287$$ 0 0
$$288$$ −66339.0 −0.799805
$$289$$ 78083.0 0.934891
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 74466.0 0.873358
$$293$$ 171522. 1.99795 0.998975 0.0452665i $$-0.0144137\pi$$
0.998975 + 0.0452665i $$0.0144137\pi$$
$$294$$ 0 0
$$295$$ 86050.0 0.988796
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −274716. −3.03215
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −97686.0 −1.04325
$$307$$ 40482.0 0.429522 0.214761 0.976667i $$-0.431103\pi$$
0.214761 + 0.976667i $$0.431103\pi$$
$$308$$ 66066.0 0.696429
$$309$$ 0 0
$$310$$ 119850. 1.24714
$$311$$ −31838.0 −0.329174 −0.164587 0.986363i $$-0.552629\pi$$
−0.164587 + 0.986363i $$0.552629\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ −157950. −1.59184
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 99425.0 0.970947
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −45927.0 −0.437500
$$325$$ 101250. 0.958580
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 162802. 1.48595 0.742974 0.669320i $$-0.233414\pi$$
0.742974 + 0.669320i $$0.233414\pi$$
$$332$$ −95214.0 −0.863823
$$333$$ 0 0
$$334$$ 23274.0 0.208631
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −68718.0 −0.605077 −0.302539 0.953137i $$-0.597834\pi$$
−0.302539 + 0.953137i $$0.597834\pi$$
$$338$$ 6951.00 0.0608435
$$339$$ 0 0
$$340$$ −70350.0 −0.608564
$$341$$ −193358. −1.66285
$$342$$ 0 0
$$343$$ −99996.0 −0.849952
$$344$$ 243018. 2.05363
$$345$$ 0 0
$$346$$ 11034.0 0.0921681
$$347$$ 71682.0 0.595321 0.297660 0.954672i $$-0.403794\pi$$
0.297660 + 0.954672i $$0.403794\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 146250. 1.19388
$$351$$ 0 0
$$352$$ −99099.0 −0.799805
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ −99950.0 −0.793097
$$356$$ 110866. 0.874779
$$357$$ 0 0
$$358$$ 187914. 1.46620
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 139725. 1.07812
$$361$$ 130321. 1.00000
$$362$$ −154326. −1.17767
$$363$$ 0 0
$$364$$ 88452.0 0.667582
$$365$$ −265950. −1.99625
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 214722. 1.54333 0.771665 0.636029i $$-0.219424\pi$$
0.771665 + 0.636029i $$0.219424\pi$$
$$374$$ −145926. −1.04325
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 2162.00 0.0150514 0.00752571 0.999972i $$-0.497604\pi$$
0.00752571 + 0.999972i $$0.497604\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −123846. −0.848702
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ −235950. −1.59184
$$386$$ 220554. 1.48027
$$387$$ 285282. 1.90481
$$388$$ 0 0
$$389$$ −292238. −1.93125 −0.965623 0.259948i $$-0.916295\pi$$
−0.965623 + 0.259948i $$0.916295\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 254127. 1.65378
$$393$$ 0 0
$$394$$ 211194. 1.36047
$$395$$ 0 0
$$396$$ −68607.0 −0.437500
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 142554. 0.899939
$$399$$ 0 0
$$400$$ −59375.0 −0.371094
$$401$$ 289922. 1.80299 0.901493 0.432793i $$-0.142472\pi$$
0.901493 + 0.432793i $$0.142472\pi$$
$$402$$ 0 0
$$403$$ −258876. −1.59398
$$404$$ 0 0
$$405$$ 164025. 1.00000
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −268476. −1.57400
$$414$$ 0 0
$$415$$ 340050. 1.97445
$$416$$ −132678. −0.766677
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −155758. −0.887202 −0.443601 0.896224i $$-0.646299\pi$$
−0.443601 + 0.896224i $$0.646299\pi$$
$$420$$ 0 0
$$421$$ −335438. −1.89255 −0.946277 0.323358i $$-0.895188\pi$$
−0.946277 + 0.323358i $$0.895188\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 251250. 1.39100
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −11214.0 −0.0612171
$$429$$ 0 0
$$430$$ −264150. −1.42861
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ −373932. −1.98524
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 208725. 1.07812
$$441$$ 298323. 1.53394
$$442$$ −195372. −1.00004
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 0 0
$$445$$ −395950. −1.99950
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −310206. −1.54559
$$449$$ 118082. 0.585721 0.292861 0.956155i $$-0.405393\pi$$
0.292861 + 0.956155i $$0.405393\pi$$
$$450$$ −151875. −0.750000
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 198234. 0.961759
$$455$$ −315900. −1.52590
$$456$$ 0 0
$$457$$ −237198. −1.13574 −0.567870 0.823119i $$-0.692232\pi$$
−0.567870 + 0.823119i $$0.692232\pi$$
$$458$$ −219606. −1.04692
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 181674. 0.836606
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ −91854.0 −0.419379
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 237498. 1.06605
$$473$$ 426162. 1.90481
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 219492. 0.968735
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −102487. −0.437500
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ −276225. −1.15046
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 245025. 1.00000
$$496$$ 151810. 0.617074
$$497$$ 311844. 1.26248
$$498$$ 0 0
$$499$$ 466322. 1.87277 0.936386 0.350972i $$-0.114149\pi$$
0.936386 + 0.350972i $$0.114149\pi$$
$$500$$ −109375. −0.437500
$$501$$ 0 0
$$502$$ 139434. 0.553301
$$503$$ −381198. −1.50666 −0.753329 0.657644i $$-0.771553\pi$$
−0.753329 + 0.657644i $$0.771553\pi$$
$$504$$ −435942. −1.71620
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 218946. 0.848417
$$509$$ −499118. −1.92649 −0.963247 0.268617i $$-0.913433\pi$$
−0.963247 + 0.268617i $$0.913433\pi$$
$$510$$ 0 0
$$511$$ 829764. 3.17770
$$512$$ 182685. 0.696888
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 279450. 1.03347
$$521$$ −249118. −0.917761 −0.458881 0.888498i $$-0.651749\pi$$
−0.458881 + 0.888498i $$0.651749\pi$$
$$522$$ 0 0
$$523$$ 483522. 1.76772 0.883859 0.467754i $$-0.154937\pi$$
0.883859 + 0.467754i $$0.154937\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −351126. −1.26909
$$527$$ −642396. −2.31303
$$528$$ 0 0
$$529$$ 279841. 1.00000
$$530$$ 0 0
$$531$$ 278802. 0.988796
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 40050.0 0.139925
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 83274.0 0.287703
$$539$$ 445643. 1.53394
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ −329238. −1.11253
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 450402. 1.50531 0.752654 0.658416i $$-0.228773\pi$$
0.752654 + 0.658416i $$0.228773\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ −226875. −0.750000
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 47034.0 0.153247
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −266718. −0.859690 −0.429845 0.902903i $$-0.641432\pi$$
−0.429845 + 0.902903i $$0.641432\pi$$
$$558$$ 388314. 1.24714
$$559$$ 570564. 1.82592
$$560$$ 185250. 0.590721
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 21282.0 0.0671422 0.0335711 0.999436i $$-0.489312\pi$$
0.0335711 + 0.999436i $$0.489312\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 407034. 1.27057
$$567$$ −511758. −1.59184
$$568$$ −275862. −0.855057
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ −137214. −0.419379
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 322137. 0.970947
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ −234249. −0.701168
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1.06096e6 −3.14301
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −734022. −2.15220
$$585$$ 328050. 0.958580
$$586$$ −514566. −1.49846
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −258150. −0.741597
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −690798. −1.96445 −0.982227 0.187699i $$-0.939897\pi$$
−0.982227 + 0.187699i $$0.939897\pi$$
$$594$$ 0 0
$$595$$ −783900. −2.21425
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −74398.0 −0.207352 −0.103676 0.994611i $$-0.533060\pi$$
−0.103676 + 0.994611i $$0.533060\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 824148. 2.27411
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 366025. 1.00000
$$606$$ 0 0
$$607$$ −361518. −0.981189 −0.490594 0.871388i $$-0.663220\pi$$
−0.490594 + 0.871388i $$0.663220\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −227934. −0.608564
$$613$$ 138882. 0.369594 0.184797 0.982777i $$-0.440837\pi$$
0.184797 + 0.982777i $$0.440837\pi$$
$$614$$ −121446. −0.322141
$$615$$ 0 0
$$616$$ −651222. −1.71620
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ 76402.0 0.199399 0.0996996 0.995018i $$-0.468212\pi$$
0.0996996 + 0.995018i $$0.468212\pi$$
$$620$$ 279650. 0.727497
$$621$$ 0 0
$$622$$ 95514.0 0.246880
$$623$$ 1.23536e6 3.18287
$$624$$ 0 0
$$625$$ 390625. 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 473850. 1.19388
$$631$$ 289442. 0.726947 0.363474 0.931605i $$-0.381591\pi$$
0.363474 + 0.931605i $$0.381591\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −781950. −1.93924
$$636$$ 0 0
$$637$$ 596646. 1.47041
$$638$$ 0 0
$$639$$ −323838. −0.793097
$$640$$ 29325.0 0.0715942
$$641$$ −730558. −1.77803 −0.889014 0.457880i $$-0.848609\pi$$
−0.889014 + 0.457880i $$0.848609\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 452709. 1.07812
$$649$$ 416482. 0.988796
$$650$$ −303750. −0.718935
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −861678. −1.99625
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ −678478. −1.55286 −0.776431 0.630202i $$-0.782972\pi$$
−0.776431 + 0.630202i $$0.782972\pi$$
$$662$$ −488406. −1.11446
$$663$$ 0 0
$$664$$ 938538. 2.12871
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 54306.0 0.121701
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 18642.0 0.0411588 0.0205794 0.999788i $$-0.493449\pi$$
0.0205794 + 0.999788i $$0.493449\pi$$
$$674$$ 206154. 0.453808
$$675$$ 0 0
$$676$$ 16219.0 0.0354920
$$677$$ 895362. 1.95354 0.976768 0.214300i $$-0.0687472\pi$$
0.976768 + 0.214300i $$0.0687472\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 693450. 1.49968
$$681$$ 0 0
$$682$$ 580074. 1.24714
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 299988. 0.637464
$$687$$ 0 0
$$688$$ −334590. −0.706864
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −597358. −1.25106 −0.625531 0.780200i $$-0.715117\pi$$
−0.625531 + 0.780200i $$0.715117\pi$$
$$692$$ 25746.0 0.0537647
$$693$$ −764478. −1.59184
$$694$$ −215046. −0.446491
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 341250. 0.696429
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 481217. 0.970947
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 949042. 1.88796 0.943980 0.330002i $$-0.107049\pi$$
0.943980 + 0.330002i $$0.107049\pi$$
$$710$$ 299850. 0.594822
$$711$$ 0 0
$$712$$ −1.09282e6 −2.15571
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 490050. 0.958580
$$716$$ 438466. 0.855283
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −518398. −1.00278 −0.501390 0.865221i $$-0.667178\pi$$
−0.501390 + 0.865221i $$0.667178\pi$$
$$720$$ −192375. −0.371094
$$721$$ 0 0
$$722$$ −390963. −0.750000
$$723$$ 0 0
$$724$$ −360094. −0.686972
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ −871884. −1.64511
$$729$$ 531441. 1.00000
$$730$$ 797850. 1.49719
$$731$$ 1.41584e6 2.64960
$$732$$ 0 0
$$733$$ −720798. −1.34155 −0.670773 0.741663i $$-0.734038\pi$$
−0.670773 + 0.741663i $$0.734038\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 216882. 0.392867 0.196434 0.980517i $$-0.437064\pi$$
0.196434 + 0.980517i $$0.437064\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −644166. −1.15750
$$747$$ 1.10176e6 1.97445
$$748$$ −340494. −0.608564
$$749$$ −124956. −0.222738
$$750$$ 0 0
$$751$$ −899518. −1.59489 −0.797444 0.603393i $$-0.793815\pi$$
−0.797444 + 0.603393i $$0.793815\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ −6486.00 −0.0112886
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −288974. −0.495076
$$765$$ 814050. 1.39100
$$766$$ 0 0
$$767$$ 557604. 0.947840
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 707850. 1.19388
$$771$$ 0 0
$$772$$ 514626. 0.863490
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ −855846. −1.42861
$$775$$ −998750. −1.66285
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 876714. 1.44843
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −483758. −0.793097
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −349885. −0.569237
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −852318. −1.37611 −0.688053 0.725660i $$-0.741535\pi$$
−0.688053 + 0.725660i $$0.741535\pi$$
$$788$$ 492786. 0.793608
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 676269. 1.07812
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 332626. 0.524965
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −511875. −0.799805
$$801$$ −1.28288e6 −1.99950
$$802$$ −869766. −1.35224
$$803$$ −1.28720e6 −1.99625
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 776628. 1.19548
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ −492075. −0.750000
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −1.02352e6 −1.52590
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 805428. 1.18050
$$827$$ −26238.0 −0.0383636 −0.0191818 0.999816i $$-0.506106\pi$$
−0.0191818 + 0.999816i $$0.506106\pi$$
$$828$$ 0 0
$$829$$ −33518.0 −0.0487718 −0.0243859 0.999703i $$-0.507763\pi$$
−0.0243859 + 0.999703i $$0.507763\pi$$
$$830$$ −1.02015e6 −1.48084
$$831$$ 0 0
$$832$$ 644274. 0.930731
$$833$$ 1.48057e6 2.13372
$$834$$ 0 0
$$835$$ −193950. −0.278174
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 467274. 0.665401
$$839$$ 717922. 1.01989 0.509945 0.860207i $$-0.329666\pi$$
0.509945 + 0.860207i $$0.329666\pi$$
$$840$$ 0 0
$$841$$ 707281. 1.00000
$$842$$ 1.00631e6 1.41941
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −57925.0 −0.0811246
$$846$$ 0 0
$$847$$ −1.14200e6 −1.59184
$$848$$ 0 0
$$849$$ 0 0
$$850$$ −753750. −1.04325
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 356802. 0.490376 0.245188 0.969476i $$-0.421150\pi$$
0.245188 + 0.969476i $$0.421150\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 110538. 0.150857
$$857$$ −326478. −0.444521 −0.222260 0.974987i $$-0.571344\pi$$
−0.222260 + 0.974987i $$0.571344\pi$$
$$858$$ 0 0
$$859$$ −1.28392e6 −1.74001 −0.870003 0.493046i $$-0.835884\pi$$
−0.870003 + 0.493046i $$0.835884\pi$$
$$860$$ −616350. −0.833356
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ −91950.0 −0.122891
$$866$$ 0 0
$$867$$ 0 0
$$868$$ −872508. −1.15806
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1.21875e6 −1.59184
$$876$$ 0 0
$$877$$ −1.48208e6 −1.92696 −0.963478 0.267787i $$-0.913708\pi$$
−0.963478 + 0.267787i $$0.913708\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ −287375. −0.371094
$$881$$ 1.53824e6 1.98186 0.990930 0.134381i $$-0.0429046\pi$$
0.990930 + 0.134381i $$0.0429046\pi$$
$$882$$ −894969. −1.15046
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ −455868. −0.583357
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 918642. 1.16761 0.583807 0.811893i $$-0.301563\pi$$
0.583807 + 0.811893i $$0.301563\pi$$
$$888$$ 0 0
$$889$$ 2.43968e6 3.08695
$$890$$ 1.18785e6 1.49962
$$891$$ 793881. 1.00000
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −1.56595e6 −1.95493
$$896$$ −91494.0 −0.113966
$$897$$ 0 0
$$898$$ −354246. −0.439291
$$899$$ 0 0
$$900$$ −354375. −0.437500
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 1.28605e6 1.57022
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 462546. 0.561026
$$909$$ 0 0
$$910$$ 947700. 1.14443
$$911$$ 1.37472e6 1.65645 0.828225 0.560396i $$-0.189351\pi$$
0.828225 + 0.560396i $$0.189351\pi$$
$$912$$ 0 0
$$913$$ 1.64584e6 1.97445
$$914$$ 711594. 0.851804
$$915$$ 0 0
$$916$$ −512414. −0.610703
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −647676. −0.760246
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1.03616e6 1.20059 0.600297 0.799777i $$-0.295049\pi$$
0.600297 + 0.799777i $$0.295049\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 423906. 0.488020
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 1.21605e6 1.39100
$$936$$ 905418. 1.03347
$$937$$ 1.46008e6 1.66302 0.831511 0.555508i $$-0.187476\pi$$
0.831511 + 0.555508i $$0.187476\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −326990. −0.366936
$$945$$ 0 0
$$946$$ −1.27849e6 −1.42861
$$947$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$948$$ 0 0
$$949$$ −1.72336e6 −1.91356
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −2.16356e6 −2.38724
$$953$$ −1.20392e6 −1.32560 −0.662798 0.748798i $$-0.730631\pi$$
−0.662798 + 0.748798i $$0.730631\pi$$
$$954$$ 0 0
$$955$$ 1.03205e6 1.13160
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.63008e6 1.76507
$$962$$ 0 0
$$963$$ 129762. 0.139925
$$964$$ 0 0
$$965$$ −1.83795e6 −1.97369
$$966$$ 0 0
$$967$$ 1.57432e6 1.68361 0.841803 0.539784i $$-0.181494\pi$$
0.841803 + 0.539784i $$0.181494\pi$$
$$968$$ 1.01023e6 1.07812
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 745202. 0.790379 0.395190 0.918600i $$-0.370679\pi$$
0.395190 + 0.918600i $$0.370679\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ −1.91640e6 −1.99950
$$980$$ −644525. −0.671101
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ −1.75995e6 −1.81396
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ −735075. −0.750000
$$991$$ 1.79168e6 1.82437 0.912186 0.409775i $$-0.134393\pi$$
0.912186 + 0.409775i $$0.134393\pi$$
$$992$$ 1.30876e6 1.32996
$$993$$ 0 0
$$994$$ −935532. −0.946860
$$995$$ −1.18795e6 −1.19992
$$996$$ 0 0
$$997$$ −103038. −0.103659 −0.0518295 0.998656i $$-0.516505\pi$$
−0.0518295 + 0.998656i $$0.516505\pi$$
$$998$$ −1.39897e6 −1.40458
$$999$$ 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 55.5.d.a.54.1 1
5.2 odd 4 275.5.c.c.76.1 2
5.3 odd 4 275.5.c.c.76.2 2
5.4 even 2 55.5.d.b.54.1 yes 1
11.10 odd 2 55.5.d.b.54.1 yes 1
55.32 even 4 275.5.c.c.76.2 2
55.43 even 4 275.5.c.c.76.1 2
55.54 odd 2 CM 55.5.d.a.54.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
55.5.d.a.54.1 1 1.1 even 1 trivial
55.5.d.a.54.1 1 55.54 odd 2 CM
55.5.d.b.54.1 yes 1 5.4 even 2
55.5.d.b.54.1 yes 1 11.10 odd 2
275.5.c.c.76.1 2 5.2 odd 4
275.5.c.c.76.1 2 55.43 even 4
275.5.c.c.76.2 2 5.3 odd 4
275.5.c.c.76.2 2 55.32 even 4