# Properties

 Label 2205.4.a.u.1.1 Level $2205$ Weight $4$ Character 2205.1 Self dual yes Analytic conductor $130.099$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$130.099211563$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2205.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.41421 q^{2} +21.3137 q^{4} -5.00000 q^{5} -72.0833 q^{8} +O(q^{10})$$ $$q-5.41421 q^{2} +21.3137 q^{4} -5.00000 q^{5} -72.0833 q^{8} +27.0711 q^{10} +52.2548 q^{11} -30.6569 q^{13} +219.765 q^{16} +37.2254 q^{17} -80.2254 q^{19} -106.569 q^{20} -282.919 q^{22} -25.8335 q^{23} +25.0000 q^{25} +165.983 q^{26} -20.9411 q^{29} +314.558 q^{31} -613.186 q^{32} -201.546 q^{34} +197.147 q^{37} +434.357 q^{38} +360.416 q^{40} +11.3625 q^{41} -33.8335 q^{43} +1113.74 q^{44} +139.868 q^{46} -361.676 q^{47} -135.355 q^{50} -653.411 q^{52} -153.019 q^{53} -261.274 q^{55} +113.380 q^{58} -616.000 q^{59} -15.2649 q^{61} -1703.09 q^{62} +1561.80 q^{64} +153.284 q^{65} -166.510 q^{67} +793.411 q^{68} +952.000 q^{71} +148.489 q^{73} -1067.40 q^{74} -1709.90 q^{76} +857.725 q^{79} -1098.82 q^{80} -61.5189 q^{82} +660.528 q^{83} -186.127 q^{85} +183.182 q^{86} -3766.70 q^{88} -45.7746 q^{89} -550.607 q^{92} +1958.19 q^{94} +401.127 q^{95} -1682.13 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8}+O(q^{10})$$ 2 * q - 8 * q^2 + 20 * q^4 - 10 * q^5 - 48 * q^8 $$2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8} + 40 q^{10} + 14 q^{11} - 50 q^{13} + 168 q^{16} - 50 q^{17} - 36 q^{19} - 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} + 216 q^{26} + 26 q^{29} + 120 q^{31} - 672 q^{32} + 24 q^{34} + 564 q^{37} + 320 q^{38} + 240 q^{40} - 328 q^{41} - 260 q^{43} + 1164 q^{44} + 704 q^{46} - 350 q^{47} - 200 q^{50} - 628 q^{52} + 56 q^{53} - 70 q^{55} - 8 q^{58} - 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} + 250 q^{65} - 152 q^{67} + 908 q^{68} + 1904 q^{71} - 676 q^{73} - 2016 q^{74} - 1768 q^{76} + 1014 q^{79} - 840 q^{80} + 816 q^{82} - 376 q^{83} + 250 q^{85} + 768 q^{86} - 4688 q^{88} - 216 q^{89} - 264 q^{92} + 1928 q^{94} + 180 q^{95} - 2742 q^{97}+O(q^{100})$$ 2 * q - 8 * q^2 + 20 * q^4 - 10 * q^5 - 48 * q^8 + 40 * q^10 + 14 * q^11 - 50 * q^13 + 168 * q^16 - 50 * q^17 - 36 * q^19 - 100 * q^20 - 184 * q^22 - 244 * q^23 + 50 * q^25 + 216 * q^26 + 26 * q^29 + 120 * q^31 - 672 * q^32 + 24 * q^34 + 564 * q^37 + 320 * q^38 + 240 * q^40 - 328 * q^41 - 260 * q^43 + 1164 * q^44 + 704 * q^46 - 350 * q^47 - 200 * q^50 - 628 * q^52 + 56 * q^53 - 70 * q^55 - 8 * q^58 - 1232 * q^59 - 336 * q^61 - 1200 * q^62 + 2128 * q^64 + 250 * q^65 - 152 * q^67 + 908 * q^68 + 1904 * q^71 - 676 * q^73 - 2016 * q^74 - 1768 * q^76 + 1014 * q^79 - 840 * q^80 + 816 * q^82 - 376 * q^83 + 250 * q^85 + 768 * q^86 - 4688 * q^88 - 216 * q^89 - 264 * q^92 + 1928 * q^94 + 180 * q^95 - 2742 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.41421 −1.91421 −0.957107 0.289735i $$-0.906433\pi$$
−0.957107 + 0.289735i $$0.906433\pi$$
$$3$$ 0 0
$$4$$ 21.3137 2.66421
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −72.0833 −3.18566
$$9$$ 0 0
$$10$$ 27.0711 0.856062
$$11$$ 52.2548 1.43231 0.716156 0.697941i $$-0.245900\pi$$
0.716156 + 0.697941i $$0.245900\pi$$
$$12$$ 0 0
$$13$$ −30.6569 −0.654052 −0.327026 0.945015i $$-0.606047\pi$$
−0.327026 + 0.945015i $$0.606047\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 219.765 3.43382
$$17$$ 37.2254 0.531087 0.265544 0.964099i $$-0.414449\pi$$
0.265544 + 0.964099i $$0.414449\pi$$
$$18$$ 0 0
$$19$$ −80.2254 −0.968683 −0.484341 0.874879i $$-0.660941\pi$$
−0.484341 + 0.874879i $$0.660941\pi$$
$$20$$ −106.569 −1.19147
$$21$$ 0 0
$$22$$ −282.919 −2.74175
$$23$$ −25.8335 −0.234202 −0.117101 0.993120i $$-0.537360\pi$$
−0.117101 + 0.993120i $$0.537360\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 165.983 1.25200
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −20.9411 −0.134092 −0.0670460 0.997750i $$-0.521357\pi$$
−0.0670460 + 0.997750i $$0.521357\pi$$
$$30$$ 0 0
$$31$$ 314.558 1.82246 0.911232 0.411894i $$-0.135133\pi$$
0.911232 + 0.411894i $$0.135133\pi$$
$$32$$ −613.186 −3.38741
$$33$$ 0 0
$$34$$ −201.546 −1.01661
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 197.147 0.875968 0.437984 0.898983i $$-0.355693\pi$$
0.437984 + 0.898983i $$0.355693\pi$$
$$38$$ 434.357 1.85427
$$39$$ 0 0
$$40$$ 360.416 1.42467
$$41$$ 11.3625 0.0432810 0.0216405 0.999766i $$-0.493111\pi$$
0.0216405 + 0.999766i $$0.493111\pi$$
$$42$$ 0 0
$$43$$ −33.8335 −0.119990 −0.0599948 0.998199i $$-0.519108\pi$$
−0.0599948 + 0.998199i $$0.519108\pi$$
$$44$$ 1113.74 3.81598
$$45$$ 0 0
$$46$$ 139.868 0.448313
$$47$$ −361.676 −1.12247 −0.561233 0.827658i $$-0.689673\pi$$
−0.561233 + 0.827658i $$0.689673\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −135.355 −0.382843
$$51$$ 0 0
$$52$$ −653.411 −1.74254
$$53$$ −153.019 −0.396582 −0.198291 0.980143i $$-0.563539\pi$$
−0.198291 + 0.980143i $$0.563539\pi$$
$$54$$ 0 0
$$55$$ −261.274 −0.640549
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 113.380 0.256681
$$59$$ −616.000 −1.35926 −0.679630 0.733555i $$-0.737860\pi$$
−0.679630 + 0.733555i $$0.737860\pi$$
$$60$$ 0 0
$$61$$ −15.2649 −0.0320406 −0.0160203 0.999872i $$-0.505100\pi$$
−0.0160203 + 0.999872i $$0.505100\pi$$
$$62$$ −1703.09 −3.48858
$$63$$ 0 0
$$64$$ 1561.80 3.05040
$$65$$ 153.284 0.292501
$$66$$ 0 0
$$67$$ −166.510 −0.303618 −0.151809 0.988410i $$-0.548510\pi$$
−0.151809 + 0.988410i $$0.548510\pi$$
$$68$$ 793.411 1.41493
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 952.000 1.59129 0.795645 0.605763i $$-0.207132\pi$$
0.795645 + 0.605763i $$0.207132\pi$$
$$72$$ 0 0
$$73$$ 148.489 0.238074 0.119037 0.992890i $$-0.462019\pi$$
0.119037 + 0.992890i $$0.462019\pi$$
$$74$$ −1067.40 −1.67679
$$75$$ 0 0
$$76$$ −1709.90 −2.58078
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 857.725 1.22154 0.610770 0.791808i $$-0.290860\pi$$
0.610770 + 0.791808i $$0.290860\pi$$
$$80$$ −1098.82 −1.53565
$$81$$ 0 0
$$82$$ −61.5189 −0.0828491
$$83$$ 660.528 0.873523 0.436761 0.899577i $$-0.356125\pi$$
0.436761 + 0.899577i $$0.356125\pi$$
$$84$$ 0 0
$$85$$ −186.127 −0.237509
$$86$$ 183.182 0.229686
$$87$$ 0 0
$$88$$ −3766.70 −4.56286
$$89$$ −45.7746 −0.0545180 −0.0272590 0.999628i $$-0.508678\pi$$
−0.0272590 + 0.999628i $$0.508678\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −550.607 −0.623965
$$93$$ 0 0
$$94$$ 1958.19 2.14864
$$95$$ 401.127 0.433208
$$96$$ 0 0
$$97$$ −1682.13 −1.76076 −0.880382 0.474265i $$-0.842714\pi$$
−0.880382 + 0.474265i $$0.842714\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 532.843 0.532843
$$101$$ −434.167 −0.427734 −0.213867 0.976863i $$-0.568606\pi$$
−0.213867 + 0.976863i $$0.568606\pi$$
$$102$$ 0 0
$$103$$ −345.577 −0.330589 −0.165295 0.986244i $$-0.552858\pi$$
−0.165295 + 0.986244i $$0.552858\pi$$
$$104$$ 2209.85 2.08359
$$105$$ 0 0
$$106$$ 828.479 0.759142
$$107$$ −217.119 −0.196165 −0.0980825 0.995178i $$-0.531271\pi$$
−0.0980825 + 0.995178i $$0.531271\pi$$
$$108$$ 0 0
$$109$$ 1734.41 1.52409 0.762047 0.647521i $$-0.224194\pi$$
0.762047 + 0.647521i $$0.224194\pi$$
$$110$$ 1414.59 1.22615
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1854.20 1.54362 0.771809 0.635855i $$-0.219352\pi$$
0.771809 + 0.635855i $$0.219352\pi$$
$$114$$ 0 0
$$115$$ 129.167 0.104738
$$116$$ −446.333 −0.357250
$$117$$ 0 0
$$118$$ 3335.16 2.60191
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1399.57 1.05152
$$122$$ 82.6476 0.0613325
$$123$$ 0 0
$$124$$ 6704.41 4.85543
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 1394.51 0.974352 0.487176 0.873304i $$-0.338027\pi$$
0.487176 + 0.873304i $$0.338027\pi$$
$$128$$ −3550.45 −2.45171
$$129$$ 0 0
$$130$$ −829.914 −0.559910
$$131$$ 1762.42 1.17544 0.587722 0.809063i $$-0.300025\pi$$
0.587722 + 0.809063i $$0.300025\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 901.519 0.581189
$$135$$ 0 0
$$136$$ −2683.33 −1.69186
$$137$$ 922.949 0.575568 0.287784 0.957695i $$-0.407081\pi$$
0.287784 + 0.957695i $$0.407081\pi$$
$$138$$ 0 0
$$139$$ 196.039 0.119624 0.0598122 0.998210i $$-0.480950\pi$$
0.0598122 + 0.998210i $$0.480950\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −5154.33 −3.04607
$$143$$ −1601.97 −0.936807
$$144$$ 0 0
$$145$$ 104.706 0.0599678
$$146$$ −803.954 −0.455724
$$147$$ 0 0
$$148$$ 4201.94 2.33376
$$149$$ −780.372 −0.429064 −0.214532 0.976717i $$-0.568823\pi$$
−0.214532 + 0.976717i $$0.568823\pi$$
$$150$$ 0 0
$$151$$ −2319.43 −1.25002 −0.625008 0.780618i $$-0.714904\pi$$
−0.625008 + 0.780618i $$0.714904\pi$$
$$152$$ 5782.91 3.08589
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1572.79 −0.815030
$$156$$ 0 0
$$157$$ −1022.90 −0.519977 −0.259989 0.965612i $$-0.583719\pi$$
−0.259989 + 0.965612i $$0.583719\pi$$
$$158$$ −4643.91 −2.33829
$$159$$ 0 0
$$160$$ 3065.93 1.51489
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1350.63 −0.649013 −0.324507 0.945883i $$-0.605198\pi$$
−0.324507 + 0.945883i $$0.605198\pi$$
$$164$$ 242.177 0.115310
$$165$$ 0 0
$$166$$ −3576.24 −1.67211
$$167$$ −1230.58 −0.570209 −0.285105 0.958496i $$-0.592028\pi$$
−0.285105 + 0.958496i $$0.592028\pi$$
$$168$$ 0 0
$$169$$ −1257.16 −0.572215
$$170$$ 1007.73 0.454644
$$171$$ 0 0
$$172$$ −721.117 −0.319678
$$173$$ −2487.65 −1.09325 −0.546626 0.837377i $$-0.684088\pi$$
−0.546626 + 0.837377i $$0.684088\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 11483.8 4.91830
$$177$$ 0 0
$$178$$ 247.833 0.104359
$$179$$ −1621.18 −0.676941 −0.338471 0.940977i $$-0.609910\pi$$
−0.338471 + 0.940977i $$0.609910\pi$$
$$180$$ 0 0
$$181$$ −2593.69 −1.06512 −0.532561 0.846392i $$-0.678770\pi$$
−0.532561 + 0.846392i $$0.678770\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1862.16 0.746089
$$185$$ −985.736 −0.391745
$$186$$ 0 0
$$187$$ 1945.21 0.760682
$$188$$ −7708.66 −2.99049
$$189$$ 0 0
$$190$$ −2171.79 −0.829253
$$191$$ 1823.08 0.690645 0.345323 0.938484i $$-0.387769\pi$$
0.345323 + 0.938484i $$0.387769\pi$$
$$192$$ 0 0
$$193$$ −1541.03 −0.574744 −0.287372 0.957819i $$-0.592782\pi$$
−0.287372 + 0.957819i $$0.592782\pi$$
$$194$$ 9107.39 3.37048
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −701.243 −0.253612 −0.126806 0.991928i $$-0.540473\pi$$
−0.126806 + 0.991928i $$0.540473\pi$$
$$198$$ 0 0
$$199$$ −3294.96 −1.17374 −0.586868 0.809682i $$-0.699639\pi$$
−0.586868 + 0.809682i $$0.699639\pi$$
$$200$$ −1802.08 −0.637132
$$201$$ 0 0
$$202$$ 2350.67 0.818775
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −56.8124 −0.0193559
$$206$$ 1871.03 0.632819
$$207$$ 0 0
$$208$$ −6737.29 −2.24590
$$209$$ −4192.16 −1.38746
$$210$$ 0 0
$$211$$ 4082.35 1.33195 0.665974 0.745975i $$-0.268016\pi$$
0.665974 + 0.745975i $$0.268016\pi$$
$$212$$ −3261.41 −1.05658
$$213$$ 0 0
$$214$$ 1175.53 0.375502
$$215$$ 169.167 0.0536610
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −9390.46 −2.91744
$$219$$ 0 0
$$220$$ −5568.72 −1.70656
$$221$$ −1141.21 −0.347359
$$222$$ 0 0
$$223$$ −747.161 −0.224366 −0.112183 0.993688i $$-0.535784\pi$$
−0.112183 + 0.993688i $$0.535784\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −10039.1 −2.95481
$$227$$ 1665.67 0.487025 0.243513 0.969898i $$-0.421700\pi$$
0.243513 + 0.969898i $$0.421700\pi$$
$$228$$ 0 0
$$229$$ 6628.35 1.91272 0.956362 0.292183i $$-0.0943816\pi$$
0.956362 + 0.292183i $$0.0943816\pi$$
$$230$$ −699.340 −0.200492
$$231$$ 0 0
$$232$$ 1509.50 0.427172
$$233$$ 432.431 0.121586 0.0607929 0.998150i $$-0.480637\pi$$
0.0607929 + 0.998150i $$0.480637\pi$$
$$234$$ 0 0
$$235$$ 1808.38 0.501982
$$236$$ −13129.2 −3.62136
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5580.44 −1.51033 −0.755165 0.655535i $$-0.772443\pi$$
−0.755165 + 0.655535i $$0.772443\pi$$
$$240$$ 0 0
$$241$$ 6296.87 1.68306 0.841529 0.540212i $$-0.181656\pi$$
0.841529 + 0.540212i $$0.181656\pi$$
$$242$$ −7577.56 −2.01283
$$243$$ 0 0
$$244$$ −325.352 −0.0853629
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2459.46 0.633569
$$248$$ −22674.4 −5.80575
$$249$$ 0 0
$$250$$ 676.777 0.171212
$$251$$ 311.921 0.0784393 0.0392197 0.999231i $$-0.487513\pi$$
0.0392197 + 0.999231i $$0.487513\pi$$
$$252$$ 0 0
$$253$$ −1349.92 −0.335451
$$254$$ −7550.17 −1.86512
$$255$$ 0 0
$$256$$ 6728.46 1.64269
$$257$$ −7861.39 −1.90809 −0.954046 0.299659i $$-0.903127\pi$$
−0.954046 + 0.299659i $$0.903127\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 3267.06 0.779285
$$261$$ 0 0
$$262$$ −9542.11 −2.25005
$$263$$ −5227.09 −1.22554 −0.612769 0.790262i $$-0.709944\pi$$
−0.612769 + 0.790262i $$0.709944\pi$$
$$264$$ 0 0
$$265$$ 765.097 0.177357
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −3548.94 −0.808903
$$269$$ 1281.71 0.290510 0.145255 0.989394i $$-0.453600\pi$$
0.145255 + 0.989394i $$0.453600\pi$$
$$270$$ 0 0
$$271$$ −4704.14 −1.05445 −0.527226 0.849725i $$-0.676768\pi$$
−0.527226 + 0.849725i $$0.676768\pi$$
$$272$$ 8180.82 1.82366
$$273$$ 0 0
$$274$$ −4997.04 −1.10176
$$275$$ 1306.37 0.286462
$$276$$ 0 0
$$277$$ 8958.56 1.94321 0.971603 0.236619i $$-0.0760393\pi$$
0.971603 + 0.236619i $$0.0760393\pi$$
$$278$$ −1061.40 −0.228987
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 370.904 0.0787412 0.0393706 0.999225i $$-0.487465\pi$$
0.0393706 + 0.999225i $$0.487465\pi$$
$$282$$ 0 0
$$283$$ 5822.26 1.22296 0.611479 0.791261i $$-0.290575\pi$$
0.611479 + 0.791261i $$0.290575\pi$$
$$284$$ 20290.7 4.23954
$$285$$ 0 0
$$286$$ 8673.40 1.79325
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −3527.27 −0.717946
$$290$$ −566.899 −0.114791
$$291$$ 0 0
$$292$$ 3164.86 0.634279
$$293$$ 7443.79 1.48420 0.742100 0.670289i $$-0.233830\pi$$
0.742100 + 0.670289i $$0.233830\pi$$
$$294$$ 0 0
$$295$$ 3080.00 0.607880
$$296$$ −14211.0 −2.79053
$$297$$ 0 0
$$298$$ 4225.10 0.821320
$$299$$ 791.973 0.153181
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 12557.9 2.39280
$$303$$ 0 0
$$304$$ −17630.7 −3.32628
$$305$$ 76.3247 0.0143290
$$306$$ 0 0
$$307$$ 761.674 0.141600 0.0707998 0.997491i $$-0.477445\pi$$
0.0707998 + 0.997491i $$0.477445\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 8515.43 1.56014
$$311$$ 7718.69 1.40735 0.703677 0.710520i $$-0.251540\pi$$
0.703677 + 0.710520i $$0.251540\pi$$
$$312$$ 0 0
$$313$$ −8556.00 −1.54509 −0.772546 0.634959i $$-0.781017\pi$$
−0.772546 + 0.634959i $$0.781017\pi$$
$$314$$ 5538.21 0.995348
$$315$$ 0 0
$$316$$ 18281.3 3.25444
$$317$$ 7780.95 1.37862 0.689309 0.724468i $$-0.257914\pi$$
0.689309 + 0.724468i $$0.257914\pi$$
$$318$$ 0 0
$$319$$ −1094.28 −0.192062
$$320$$ −7809.02 −1.36418
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2986.42 −0.514455
$$324$$ 0 0
$$325$$ −766.421 −0.130810
$$326$$ 7312.58 1.24235
$$327$$ 0 0
$$328$$ −819.045 −0.137879
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4932.12 −0.819015 −0.409507 0.912307i $$-0.634299\pi$$
−0.409507 + 0.912307i $$0.634299\pi$$
$$332$$ 14078.3 2.32725
$$333$$ 0 0
$$334$$ 6662.61 1.09150
$$335$$ 832.548 0.135782
$$336$$ 0 0
$$337$$ −7121.13 −1.15108 −0.575538 0.817775i $$-0.695207\pi$$
−0.575538 + 0.817775i $$0.695207\pi$$
$$338$$ 6806.52 1.09534
$$339$$ 0 0
$$340$$ −3967.06 −0.632776
$$341$$ 16437.2 2.61034
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2438.83 0.382246
$$345$$ 0 0
$$346$$ 13468.7 2.09272
$$347$$ −9540.58 −1.47598 −0.737991 0.674811i $$-0.764225\pi$$
−0.737991 + 0.674811i $$0.764225\pi$$
$$348$$ 0 0
$$349$$ −1281.65 −0.196576 −0.0982880 0.995158i $$-0.531337\pi$$
−0.0982880 + 0.995158i $$0.531337\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −32041.9 −4.85182
$$353$$ 5798.07 0.874221 0.437110 0.899408i $$-0.356002\pi$$
0.437110 + 0.899408i $$0.356002\pi$$
$$354$$ 0 0
$$355$$ −4760.00 −0.711647
$$356$$ −975.627 −0.145247
$$357$$ 0 0
$$358$$ 8777.40 1.29581
$$359$$ −2267.29 −0.333323 −0.166662 0.986014i $$-0.553299\pi$$
−0.166662 + 0.986014i $$0.553299\pi$$
$$360$$ 0 0
$$361$$ −422.886 −0.0616541
$$362$$ 14042.8 2.03887
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −742.447 −0.106470
$$366$$ 0 0
$$367$$ 7372.85 1.04866 0.524332 0.851514i $$-0.324315\pi$$
0.524332 + 0.851514i $$0.324315\pi$$
$$368$$ −5677.28 −0.804209
$$369$$ 0 0
$$370$$ 5336.98 0.749883
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6447.14 0.894961 0.447480 0.894294i $$-0.352321\pi$$
0.447480 + 0.894294i $$0.352321\pi$$
$$374$$ −10531.8 −1.45611
$$375$$ 0 0
$$376$$ 26070.8 3.57579
$$377$$ 641.989 0.0877032
$$378$$ 0 0
$$379$$ −4247.57 −0.575680 −0.287840 0.957678i $$-0.592937\pi$$
−0.287840 + 0.957678i $$0.592937\pi$$
$$380$$ 8549.50 1.15416
$$381$$ 0 0
$$382$$ −9870.53 −1.32204
$$383$$ −6681.86 −0.891454 −0.445727 0.895169i $$-0.647055\pi$$
−0.445727 + 0.895169i $$0.647055\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 8343.45 1.10018
$$387$$ 0 0
$$388$$ −35852.4 −4.69105
$$389$$ 6371.78 0.830494 0.415247 0.909709i $$-0.363695\pi$$
0.415247 + 0.909709i $$0.363695\pi$$
$$390$$ 0 0
$$391$$ −961.661 −0.124382
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 3796.68 0.485467
$$395$$ −4288.62 −0.546289
$$396$$ 0 0
$$397$$ −4247.93 −0.537021 −0.268510 0.963277i $$-0.586531\pi$$
−0.268510 + 0.963277i $$0.586531\pi$$
$$398$$ 17839.6 2.24678
$$399$$ 0 0
$$400$$ 5494.11 0.686764
$$401$$ 8833.62 1.10008 0.550038 0.835140i $$-0.314613\pi$$
0.550038 + 0.835140i $$0.314613\pi$$
$$402$$ 0 0
$$403$$ −9643.37 −1.19199
$$404$$ −9253.70 −1.13958
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 10301.9 1.25466
$$408$$ 0 0
$$409$$ 319.205 0.0385908 0.0192954 0.999814i $$-0.493858\pi$$
0.0192954 + 0.999814i $$0.493858\pi$$
$$410$$ 307.595 0.0370512
$$411$$ 0 0
$$412$$ −7365.53 −0.880761
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −3302.64 −0.390651
$$416$$ 18798.3 2.21554
$$417$$ 0 0
$$418$$ 22697.3 2.65589
$$419$$ −12789.2 −1.49115 −0.745577 0.666420i $$-0.767826\pi$$
−0.745577 + 0.666420i $$0.767826\pi$$
$$420$$ 0 0
$$421$$ −6747.40 −0.781112 −0.390556 0.920579i $$-0.627717\pi$$
−0.390556 + 0.920579i $$0.627717\pi$$
$$422$$ −22102.7 −2.54963
$$423$$ 0 0
$$424$$ 11030.1 1.26337
$$425$$ 930.635 0.106217
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −4627.60 −0.522625
$$429$$ 0 0
$$430$$ −915.908 −0.102719
$$431$$ 5184.75 0.579444 0.289722 0.957111i $$-0.406437\pi$$
0.289722 + 0.957111i $$0.406437\pi$$
$$432$$ 0 0
$$433$$ 4242.03 0.470806 0.235403 0.971898i $$-0.424359\pi$$
0.235403 + 0.971898i $$0.424359\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 36966.7 4.06051
$$437$$ 2072.50 0.226868
$$438$$ 0 0
$$439$$ 5434.12 0.590789 0.295394 0.955375i $$-0.404549\pi$$
0.295394 + 0.955375i $$0.404549\pi$$
$$440$$ 18833.5 2.04057
$$441$$ 0 0
$$442$$ 6178.77 0.664919
$$443$$ 11493.8 1.23270 0.616350 0.787472i $$-0.288611\pi$$
0.616350 + 0.787472i $$0.288611\pi$$
$$444$$ 0 0
$$445$$ 228.873 0.0243812
$$446$$ 4045.29 0.429484
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 16849.3 1.77098 0.885489 0.464661i $$-0.153824\pi$$
0.885489 + 0.464661i $$0.153824\pi$$
$$450$$ 0 0
$$451$$ 593.745 0.0619919
$$452$$ 39520.0 4.11253
$$453$$ 0 0
$$454$$ −9018.32 −0.932270
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 15348.5 1.57106 0.785528 0.618826i $$-0.212391\pi$$
0.785528 + 0.618826i $$0.212391\pi$$
$$458$$ −35887.3 −3.66136
$$459$$ 0 0
$$460$$ 2753.04 0.279046
$$461$$ 14038.4 1.41830 0.709148 0.705059i $$-0.249080\pi$$
0.709148 + 0.705059i $$0.249080\pi$$
$$462$$ 0 0
$$463$$ −8661.23 −0.869377 −0.434689 0.900581i $$-0.643142\pi$$
−0.434689 + 0.900581i $$0.643142\pi$$
$$464$$ −4602.12 −0.460448
$$465$$ 0 0
$$466$$ −2341.27 −0.232741
$$467$$ 7014.71 0.695079 0.347539 0.937665i $$-0.387017\pi$$
0.347539 + 0.937665i $$0.387017\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −9790.96 −0.960901
$$471$$ 0 0
$$472$$ 44403.3 4.33014
$$473$$ −1767.96 −0.171863
$$474$$ 0 0
$$475$$ −2005.63 −0.193737
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 30213.7 2.89109
$$479$$ 18134.7 1.72984 0.864922 0.501907i $$-0.167368\pi$$
0.864922 + 0.501907i $$0.167368\pi$$
$$480$$ 0 0
$$481$$ −6043.91 −0.572929
$$482$$ −34092.6 −3.22173
$$483$$ 0 0
$$484$$ 29830.0 2.80146
$$485$$ 8410.63 0.787438
$$486$$ 0 0
$$487$$ 16537.8 1.53881 0.769405 0.638761i $$-0.220553\pi$$
0.769405 + 0.638761i $$0.220553\pi$$
$$488$$ 1100.35 0.102070
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −220.608 −0.0202768 −0.0101384 0.999949i $$-0.503227\pi$$
−0.0101384 + 0.999949i $$0.503227\pi$$
$$492$$ 0 0
$$493$$ −779.542 −0.0712146
$$494$$ −13316.0 −1.21279
$$495$$ 0 0
$$496$$ 69128.8 6.25801
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 5939.04 0.532801 0.266401 0.963862i $$-0.414166\pi$$
0.266401 + 0.963862i $$0.414166\pi$$
$$500$$ −2664.21 −0.238295
$$501$$ 0 0
$$502$$ −1688.81 −0.150150
$$503$$ −11604.8 −1.02869 −0.514345 0.857584i $$-0.671965\pi$$
−0.514345 + 0.857584i $$0.671965\pi$$
$$504$$ 0 0
$$505$$ 2170.83 0.191289
$$506$$ 7308.78 0.642124
$$507$$ 0 0
$$508$$ 29722.2 2.59588
$$509$$ −1867.67 −0.162639 −0.0813193 0.996688i $$-0.525913\pi$$
−0.0813193 + 0.996688i $$0.525913\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −8025.75 −0.692757
$$513$$ 0 0
$$514$$ 42563.2 3.65250
$$515$$ 1727.88 0.147844
$$516$$ 0 0
$$517$$ −18899.3 −1.60772
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −11049.2 −0.931809
$$521$$ 6117.21 0.514395 0.257197 0.966359i $$-0.417201\pi$$
0.257197 + 0.966359i $$0.417201\pi$$
$$522$$ 0 0
$$523$$ 16685.6 1.39505 0.697524 0.716561i $$-0.254285\pi$$
0.697524 + 0.716561i $$0.254285\pi$$
$$524$$ 37563.7 3.13164
$$525$$ 0 0
$$526$$ 28300.6 2.34594
$$527$$ 11709.6 0.967887
$$528$$ 0 0
$$529$$ −11499.6 −0.945149
$$530$$ −4142.40 −0.339499
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −348.338 −0.0283081
$$534$$ 0 0
$$535$$ 1085.59 0.0877276
$$536$$ 12002.6 0.967223
$$537$$ 0 0
$$538$$ −6939.44 −0.556097
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 9309.03 0.739790 0.369895 0.929074i $$-0.379394\pi$$
0.369895 + 0.929074i $$0.379394\pi$$
$$542$$ 25469.2 2.01845
$$543$$ 0 0
$$544$$ −22826.1 −1.79901
$$545$$ −8672.05 −0.681596
$$546$$ 0 0
$$547$$ 10894.7 0.851598 0.425799 0.904818i $$-0.359993\pi$$
0.425799 + 0.904818i $$0.359993\pi$$
$$548$$ 19671.5 1.53344
$$549$$ 0 0
$$550$$ −7072.97 −0.548350
$$551$$ 1680.01 0.129893
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −48503.6 −3.71971
$$555$$ 0 0
$$556$$ 4178.31 0.318705
$$557$$ 7873.90 0.598973 0.299486 0.954101i $$-0.403185\pi$$
0.299486 + 0.954101i $$0.403185\pi$$
$$558$$ 0 0
$$559$$ 1037.23 0.0784796
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −2008.15 −0.150728
$$563$$ 21770.7 1.62971 0.814854 0.579666i $$-0.196817\pi$$
0.814854 + 0.579666i $$0.196817\pi$$
$$564$$ 0 0
$$565$$ −9271.02 −0.690327
$$566$$ −31522.9 −2.34100
$$567$$ 0 0
$$568$$ −68623.3 −5.06931
$$569$$ 12381.3 0.912213 0.456106 0.889925i $$-0.349244\pi$$
0.456106 + 0.889925i $$0.349244\pi$$
$$570$$ 0 0
$$571$$ −5768.38 −0.422765 −0.211383 0.977403i $$-0.567797\pi$$
−0.211383 + 0.977403i $$0.567797\pi$$
$$572$$ −34143.9 −2.49585
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −645.837 −0.0468405
$$576$$ 0 0
$$577$$ −4733.38 −0.341513 −0.170757 0.985313i $$-0.554621\pi$$
−0.170757 + 0.985313i $$0.554621\pi$$
$$578$$ 19097.4 1.37430
$$579$$ 0 0
$$580$$ 2231.67 0.159767
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −7996.00 −0.568028
$$584$$ −10703.6 −0.758422
$$585$$ 0 0
$$586$$ −40302.3 −2.84108
$$587$$ −8441.67 −0.593569 −0.296785 0.954944i $$-0.595914\pi$$
−0.296785 + 0.954944i $$0.595914\pi$$
$$588$$ 0 0
$$589$$ −25235.6 −1.76539
$$590$$ −16675.8 −1.16361
$$591$$ 0 0
$$592$$ 43326.0 3.00792
$$593$$ 18939.9 1.31158 0.655791 0.754943i $$-0.272335\pi$$
0.655791 + 0.754943i $$0.272335\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −16632.6 −1.14312
$$597$$ 0 0
$$598$$ −4287.91 −0.293220
$$599$$ −22655.3 −1.54536 −0.772681 0.634794i $$-0.781085\pi$$
−0.772681 + 0.634794i $$0.781085\pi$$
$$600$$ 0 0
$$601$$ 15947.4 1.08237 0.541187 0.840902i $$-0.317975\pi$$
0.541187 + 0.840902i $$0.317975\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −49435.6 −3.33031
$$605$$ −6997.84 −0.470252
$$606$$ 0 0
$$607$$ 25993.2 1.73811 0.869053 0.494719i $$-0.164729\pi$$
0.869053 + 0.494719i $$0.164729\pi$$
$$608$$ 49193.1 3.28132
$$609$$ 0 0
$$610$$ −413.238 −0.0274287
$$611$$ 11087.9 0.734152
$$612$$ 0 0
$$613$$ 665.408 0.0438427 0.0219213 0.999760i $$-0.493022\pi$$
0.0219213 + 0.999760i $$0.493022\pi$$
$$614$$ −4123.87 −0.271052
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18401.3 −1.20066 −0.600330 0.799752i $$-0.704964\pi$$
−0.600330 + 0.799752i $$0.704964\pi$$
$$618$$ 0 0
$$619$$ 11150.6 0.724040 0.362020 0.932170i $$-0.382087\pi$$
0.362020 + 0.932170i $$0.382087\pi$$
$$620$$ −33522.0 −2.17142
$$621$$ 0 0
$$622$$ −41790.6 −2.69397
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 46324.0 2.95764
$$627$$ 0 0
$$628$$ −21801.8 −1.38533
$$629$$ 7338.88 0.465215
$$630$$ 0 0
$$631$$ 5381.79 0.339534 0.169767 0.985484i $$-0.445699\pi$$
0.169767 + 0.985484i $$0.445699\pi$$
$$632$$ −61827.6 −3.89141
$$633$$ 0 0
$$634$$ −42127.7 −2.63897
$$635$$ −6972.55 −0.435744
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 5924.64 0.367647
$$639$$ 0 0
$$640$$ 17752.2 1.09644
$$641$$ 19455.1 1.19880 0.599398 0.800451i $$-0.295407\pi$$
0.599398 + 0.800451i $$0.295407\pi$$
$$642$$ 0 0
$$643$$ 14695.8 0.901317 0.450658 0.892696i $$-0.351189\pi$$
0.450658 + 0.892696i $$0.351189\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 16169.1 0.984777
$$647$$ −12694.8 −0.771383 −0.385691 0.922628i $$-0.626037\pi$$
−0.385691 + 0.922628i $$0.626037\pi$$
$$648$$ 0 0
$$649$$ −32189.0 −1.94688
$$650$$ 4149.57 0.250399
$$651$$ 0 0
$$652$$ −28786.8 −1.72911
$$653$$ 12385.6 0.742247 0.371124 0.928583i $$-0.378973\pi$$
0.371124 + 0.928583i $$0.378973\pi$$
$$654$$ 0 0
$$655$$ −8812.09 −0.525675
$$656$$ 2497.07 0.148619
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 2072.18 0.122489 0.0612447 0.998123i $$-0.480493\pi$$
0.0612447 + 0.998123i $$0.480493\pi$$
$$660$$ 0 0
$$661$$ −1074.36 −0.0632193 −0.0316096 0.999500i $$-0.510063\pi$$
−0.0316096 + 0.999500i $$0.510063\pi$$
$$662$$ 26703.6 1.56777
$$663$$ 0 0
$$664$$ −47613.0 −2.78275
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 540.982 0.0314047
$$668$$ −26228.2 −1.51916
$$669$$ 0 0
$$670$$ −4507.59 −0.259916
$$671$$ −797.667 −0.0458921
$$672$$ 0 0
$$673$$ 26195.2 1.50037 0.750186 0.661226i $$-0.229964\pi$$
0.750186 + 0.661226i $$0.229964\pi$$
$$674$$ 38555.3 2.20341
$$675$$ 0 0
$$676$$ −26794.7 −1.52450
$$677$$ −4228.44 −0.240047 −0.120024 0.992771i $$-0.538297\pi$$
−0.120024 + 0.992771i $$0.538297\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 13416.6 0.756624
$$681$$ 0 0
$$682$$ −88994.5 −4.99674
$$683$$ −27525.5 −1.54207 −0.771036 0.636792i $$-0.780261\pi$$
−0.771036 + 0.636792i $$0.780261\pi$$
$$684$$ 0 0
$$685$$ −4614.74 −0.257402
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −7435.40 −0.412023
$$689$$ 4691.09 0.259385
$$690$$ 0 0
$$691$$ 33324.4 1.83462 0.917309 0.398177i $$-0.130357\pi$$
0.917309 + 0.398177i $$0.130357\pi$$
$$692$$ −53021.1 −2.91266
$$693$$ 0 0
$$694$$ 51654.8 2.82534
$$695$$ −980.193 −0.0534976
$$696$$ 0 0
$$697$$ 422.973 0.0229860
$$698$$ 6939.12 0.376289
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 33262.9 1.79219 0.896094 0.443864i $$-0.146393\pi$$
0.896094 + 0.443864i $$0.146393\pi$$
$$702$$ 0 0
$$703$$ −15816.2 −0.848534
$$704$$ 81611.8 4.36912
$$705$$ 0 0
$$706$$ −31392.0 −1.67345
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 13703.0 0.725851 0.362926 0.931818i $$-0.381778\pi$$
0.362926 + 0.931818i $$0.381778\pi$$
$$710$$ 25771.7 1.36224
$$711$$ 0 0
$$712$$ 3299.58 0.173676
$$713$$ −8126.14 −0.426825
$$714$$ 0 0
$$715$$ 8009.84 0.418953
$$716$$ −34553.3 −1.80352
$$717$$ 0 0
$$718$$ 12275.6 0.638052
$$719$$ −8074.93 −0.418838 −0.209419 0.977826i $$-0.567157\pi$$
−0.209419 + 0.977826i $$0.567157\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 2289.59 0.118019
$$723$$ 0 0
$$724$$ −55281.1 −2.83771
$$725$$ −523.528 −0.0268184
$$726$$ 0 0
$$727$$ 3668.70 0.187159 0.0935794 0.995612i $$-0.470169\pi$$
0.0935794 + 0.995612i $$0.470169\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 4019.77 0.203806
$$731$$ −1259.46 −0.0637250
$$732$$ 0 0
$$733$$ 14980.3 0.754857 0.377428 0.926039i $$-0.376808\pi$$
0.377428 + 0.926039i $$0.376808\pi$$
$$734$$ −39918.2 −2.00737
$$735$$ 0 0
$$736$$ 15840.7 0.793338
$$737$$ −8700.94 −0.434875
$$738$$ 0 0
$$739$$ 6530.59 0.325077 0.162538 0.986702i $$-0.448032\pi$$
0.162538 + 0.986702i $$0.448032\pi$$
$$740$$ −21009.7 −1.04369
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −25952.0 −1.28141 −0.640704 0.767788i $$-0.721357\pi$$
−0.640704 + 0.767788i $$0.721357\pi$$
$$744$$ 0 0
$$745$$ 3901.86 0.191883
$$746$$ −34906.2 −1.71315
$$747$$ 0 0
$$748$$ 41459.6 2.02662
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −14093.9 −0.684813 −0.342407 0.939552i $$-0.611242\pi$$
−0.342407 + 0.939552i $$0.611242\pi$$
$$752$$ −79483.6 −3.85435
$$753$$ 0 0
$$754$$ −3475.87 −0.167883
$$755$$ 11597.1 0.559024
$$756$$ 0 0
$$757$$ −2554.41 −0.122644 −0.0613220 0.998118i $$-0.519532\pi$$
−0.0613220 + 0.998118i $$0.519532\pi$$
$$758$$ 22997.2 1.10197
$$759$$ 0 0
$$760$$ −28914.5 −1.38005
$$761$$ 2219.08 0.105705 0.0528527 0.998602i $$-0.483169\pi$$
0.0528527 + 0.998602i $$0.483169\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 38856.5 1.84003
$$765$$ 0 0
$$766$$ 36177.0 1.70643
$$767$$ 18884.6 0.889028
$$768$$ 0 0
$$769$$ 22466.2 1.05352 0.526758 0.850015i $$-0.323408\pi$$
0.526758 + 0.850015i $$0.323408\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −32845.0 −1.53124
$$773$$ 9674.79 0.450165 0.225083 0.974340i $$-0.427735\pi$$
0.225083 + 0.974340i $$0.427735\pi$$
$$774$$ 0 0
$$775$$ 7863.96 0.364493
$$776$$ 121253. 5.60920
$$777$$ 0 0
$$778$$ −34498.2 −1.58974
$$779$$ −911.560 −0.0419256
$$780$$ 0 0
$$781$$ 49746.6 2.27922
$$782$$ 5206.64 0.238093
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 5114.51 0.232541
$$786$$ 0 0
$$787$$ 20942.8 0.948577 0.474288 0.880370i $$-0.342705\pi$$
0.474288 + 0.880370i $$0.342705\pi$$
$$788$$ −14946.1 −0.675676
$$789$$ 0 0
$$790$$ 23219.5 1.04571
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 467.975 0.0209562
$$794$$ 22999.2 1.02797
$$795$$ 0 0
$$796$$ −70227.8 −3.12708
$$797$$ 23526.6 1.04561 0.522807 0.852451i $$-0.324885\pi$$
0.522807 + 0.852451i $$0.324885\pi$$
$$798$$ 0 0
$$799$$ −13463.5 −0.596127
$$800$$ −15329.6 −0.677481
$$801$$ 0 0
$$802$$ −47827.1 −2.10578
$$803$$ 7759.29 0.340996
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 52211.3 2.28172
$$807$$ 0 0
$$808$$ 31296.1 1.36262
$$809$$ 18202.2 0.791047 0.395523 0.918456i $$-0.370563\pi$$
0.395523 + 0.918456i $$0.370563\pi$$
$$810$$ 0 0
$$811$$ 2510.24 0.108689 0.0543443 0.998522i $$-0.482693\pi$$
0.0543443 + 0.998522i $$0.482693\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −55776.7 −2.40168
$$815$$ 6753.13 0.290248
$$816$$ 0 0
$$817$$ 2714.30 0.116232
$$818$$ −1728.24 −0.0738711
$$819$$ 0 0
$$820$$ −1210.88 −0.0515681
$$821$$ −17899.6 −0.760903 −0.380451 0.924801i $$-0.624231\pi$$
−0.380451 + 0.924801i $$0.624231\pi$$
$$822$$ 0 0
$$823$$ 14039.5 0.594637 0.297318 0.954778i $$-0.403908\pi$$
0.297318 + 0.954778i $$0.403908\pi$$
$$824$$ 24910.3 1.05315
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −15127.4 −0.636073 −0.318036 0.948079i $$-0.603023\pi$$
−0.318036 + 0.948079i $$0.603023\pi$$
$$828$$ 0 0
$$829$$ −21986.5 −0.921136 −0.460568 0.887624i $$-0.652354\pi$$
−0.460568 + 0.887624i $$0.652354\pi$$
$$830$$ 17881.2 0.747790
$$831$$ 0 0
$$832$$ −47880.0 −1.99512
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 6152.89 0.255005
$$836$$ −89350.6 −3.69648
$$837$$ 0 0
$$838$$ 69243.5 2.85439
$$839$$ −2276.89 −0.0936914 −0.0468457 0.998902i $$-0.514917\pi$$
−0.0468457 + 0.998902i $$0.514917\pi$$
$$840$$ 0 0
$$841$$ −23950.5 −0.982019
$$842$$ 36531.8 1.49521
$$843$$ 0 0
$$844$$ 87010.1 3.54859
$$845$$ 6285.79 0.255903
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −33628.2 −1.36179
$$849$$ 0 0
$$850$$ −5038.66 −0.203323
$$851$$ −5093.00 −0.205154
$$852$$ 0 0
$$853$$ −13342.6 −0.535570 −0.267785 0.963479i $$-0.586292\pi$$
−0.267785 + 0.963479i $$0.586292\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 15650.6 0.624915
$$857$$ 18690.9 0.745003 0.372502 0.928032i $$-0.378500\pi$$
0.372502 + 0.928032i $$0.378500\pi$$
$$858$$ 0 0
$$859$$ −18318.9 −0.727628 −0.363814 0.931472i $$-0.618526\pi$$
−0.363814 + 0.931472i $$0.618526\pi$$
$$860$$ 3605.58 0.142964
$$861$$ 0 0
$$862$$ −28071.3 −1.10918
$$863$$ −38133.1 −1.50413 −0.752067 0.659087i $$-0.770943\pi$$
−0.752067 + 0.659087i $$0.770943\pi$$
$$864$$ 0 0
$$865$$ 12438.3 0.488917
$$866$$ −22967.3 −0.901223
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 44820.3 1.74962
$$870$$ 0 0
$$871$$ 5104.66 0.198582
$$872$$ −125022. −4.85525
$$873$$ 0 0
$$874$$ −11221.0 −0.434273
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −19707.5 −0.758807 −0.379404 0.925231i $$-0.623871\pi$$
−0.379404 + 0.925231i $$0.623871\pi$$
$$878$$ −29421.5 −1.13090
$$879$$ 0 0
$$880$$ −57418.8 −2.19953
$$881$$ −14091.5 −0.538883 −0.269441 0.963017i $$-0.586839\pi$$
−0.269441 + 0.963017i $$0.586839\pi$$
$$882$$ 0 0
$$883$$ 3115.87 0.118751 0.0593757 0.998236i $$-0.481089\pi$$
0.0593757 + 0.998236i $$0.481089\pi$$
$$884$$ −24323.5 −0.925438
$$885$$ 0 0
$$886$$ −62229.8 −2.35965
$$887$$ 38734.6 1.46627 0.733134 0.680084i $$-0.238057\pi$$
0.733134 + 0.680084i $$0.238057\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −1239.17 −0.0466708
$$891$$ 0 0
$$892$$ −15924.8 −0.597759
$$893$$ 29015.6 1.08731
$$894$$ 0 0
$$895$$ 8105.89 0.302737
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −91225.8 −3.39003
$$899$$ −6587.21 −0.244378
$$900$$ 0 0
$$901$$ −5696.21 −0.210619
$$902$$ −3214.66 −0.118666
$$903$$ 0 0
$$904$$ −133657. −4.91744
$$905$$ 12968.4 0.476337
$$906$$ 0 0
$$907$$ 19242.9 0.704464 0.352232 0.935913i $$-0.385423\pi$$
0.352232 + 0.935913i $$0.385423\pi$$
$$908$$ 35501.7 1.29754
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −34613.3 −1.25882 −0.629412 0.777072i $$-0.716704\pi$$
−0.629412 + 0.777072i $$0.716704\pi$$
$$912$$ 0 0
$$913$$ 34515.8 1.25116
$$914$$ −83100.1 −3.00734
$$915$$ 0 0
$$916$$ 141275. 5.09591
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 25826.4 0.927022 0.463511 0.886091i $$-0.346589\pi$$
0.463511 + 0.886091i $$0.346589\pi$$
$$920$$ −9310.81 −0.333661
$$921$$ 0 0
$$922$$ −76007.0 −2.71492
$$923$$ −29185.3 −1.04079
$$924$$ 0 0
$$925$$ 4928.68 0.175194
$$926$$ 46893.8 1.66417
$$927$$ 0 0
$$928$$ 12840.8 0.454224
$$929$$ 19451.6 0.686960 0.343480 0.939160i $$-0.388394\pi$$
0.343480 + 0.939160i $$0.388394\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 9216.70 0.323930
$$933$$ 0 0
$$934$$ −37979.1 −1.33053
$$935$$ −9726.03 −0.340188
$$936$$ 0 0
$$937$$ −34469.1 −1.20177 −0.600884 0.799336i $$-0.705185\pi$$
−0.600884 + 0.799336i $$0.705185\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 38543.3 1.33739
$$941$$ 14156.4 0.490419 0.245209 0.969470i $$-0.421143\pi$$
0.245209 + 0.969470i $$0.421143\pi$$
$$942$$ 0 0
$$943$$ −293.532 −0.0101365
$$944$$ −135375. −4.66746
$$945$$ 0 0
$$946$$ 9572.13 0.328982
$$947$$ 38092.4 1.30711 0.653557 0.756877i $$-0.273276\pi$$
0.653557 + 0.756877i $$0.273276\pi$$
$$948$$ 0 0
$$949$$ −4552.22 −0.155713
$$950$$ 10858.9 0.370853
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −5037.40 −0.171225 −0.0856126 0.996329i $$-0.527285\pi$$
−0.0856126 + 0.996329i $$0.527285\pi$$
$$954$$ 0 0
$$955$$ −9115.39 −0.308866
$$956$$ −118940. −4.02384
$$957$$ 0 0
$$958$$ −98185.1 −3.31129
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 69156.0 2.32137
$$962$$ 32723.0 1.09671
$$963$$ 0 0
$$964$$ 134210. 4.48403
$$965$$ 7705.14 0.257033
$$966$$ 0 0
$$967$$ 11495.3 0.382278 0.191139 0.981563i $$-0.438782\pi$$
0.191139 + 0.981563i $$0.438782\pi$$
$$968$$ −100885. −3.34977
$$969$$ 0 0
$$970$$ −45537.0 −1.50732
$$971$$ 22352.7 0.738757 0.369379 0.929279i $$-0.379571\pi$$
0.369379 + 0.929279i $$0.379571\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −89539.4 −2.94561
$$975$$ 0 0
$$976$$ −3354.69 −0.110022
$$977$$ −14345.7 −0.469765 −0.234882 0.972024i $$-0.575470\pi$$
−0.234882 + 0.972024i $$0.575470\pi$$
$$978$$ 0 0
$$979$$ −2391.94 −0.0780867
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 1194.42 0.0388141
$$983$$ 34460.9 1.11814 0.559070 0.829120i $$-0.311158\pi$$
0.559070 + 0.829120i $$0.311158\pi$$
$$984$$ 0 0
$$985$$ 3506.21 0.113419
$$986$$ 4220.61 0.136320
$$987$$ 0 0
$$988$$ 52420.2 1.68796
$$989$$ 874.036 0.0281019
$$990$$ 0 0
$$991$$ −35189.6 −1.12799 −0.563993 0.825780i $$-0.690735\pi$$
−0.563993 + 0.825780i $$0.690735\pi$$
$$992$$ −192883. −6.17342
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 16474.8 0.524911
$$996$$ 0 0
$$997$$ 50730.0 1.61147 0.805734 0.592277i $$-0.201771\pi$$
0.805734 + 0.592277i $$0.201771\pi$$
$$998$$ −32155.2 −1.01990
$$999$$ 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 2205.4.a.u.1.1 2
3.2 odd 2 245.4.a.k.1.2 2
7.6 odd 2 315.4.a.f.1.1 2
15.14 odd 2 1225.4.a.m.1.1 2
21.2 odd 6 245.4.e.i.116.1 4
21.5 even 6 245.4.e.h.116.1 4
21.11 odd 6 245.4.e.i.226.1 4
21.17 even 6 245.4.e.h.226.1 4
21.20 even 2 35.4.a.b.1.2 2
35.34 odd 2 1575.4.a.z.1.2 2
84.83 odd 2 560.4.a.r.1.2 2
105.62 odd 4 175.4.b.c.99.4 4
105.83 odd 4 175.4.b.c.99.1 4
105.104 even 2 175.4.a.c.1.1 2
168.83 odd 2 2240.4.a.bo.1.1 2
168.125 even 2 2240.4.a.bn.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 21.20 even 2
175.4.a.c.1.1 2 105.104 even 2
175.4.b.c.99.1 4 105.83 odd 4
175.4.b.c.99.4 4 105.62 odd 4
245.4.a.k.1.2 2 3.2 odd 2
245.4.e.h.116.1 4 21.5 even 6
245.4.e.h.226.1 4 21.17 even 6
245.4.e.i.116.1 4 21.2 odd 6
245.4.e.i.226.1 4 21.11 odd 6
315.4.a.f.1.1 2 7.6 odd 2
560.4.a.r.1.2 2 84.83 odd 2
1225.4.a.m.1.1 2 15.14 odd 2
1575.4.a.z.1.2 2 35.34 odd 2
2205.4.a.u.1.1 2 1.1 even 1 trivial
2240.4.a.bn.1.2 2 168.125 even 2
2240.4.a.bo.1.1 2 168.83 odd 2