# Properties

 Label 234.4.a.c.1.1 Level $234$ Weight $4$ Character 234.1 Self dual yes Analytic conductor $13.806$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,4,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("234.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 234.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: $$13.8064469413$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 234.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +4.00000 q^{4} -4.00000 q^{5} +4.00000 q^{7} -8.00000 q^{8} +O(q^{10})$$ $$q-2.00000 q^{2} +4.00000 q^{4} -4.00000 q^{5} +4.00000 q^{7} -8.00000 q^{8} +8.00000 q^{10} -2.00000 q^{11} -13.0000 q^{13} -8.00000 q^{14} +16.0000 q^{16} +6.00000 q^{17} -36.0000 q^{19} -16.0000 q^{20} +4.00000 q^{22} +20.0000 q^{23} -109.000 q^{25} +26.0000 q^{26} +16.0000 q^{28} +14.0000 q^{29} -152.000 q^{31} -32.0000 q^{32} -12.0000 q^{34} -16.0000 q^{35} -258.000 q^{37} +72.0000 q^{38} +32.0000 q^{40} -84.0000 q^{41} -188.000 q^{43} -8.00000 q^{44} -40.0000 q^{46} -254.000 q^{47} -327.000 q^{49} +218.000 q^{50} -52.0000 q^{52} -366.000 q^{53} +8.00000 q^{55} -32.0000 q^{56} -28.0000 q^{58} -550.000 q^{59} -14.0000 q^{61} +304.000 q^{62} +64.0000 q^{64} +52.0000 q^{65} +448.000 q^{67} +24.0000 q^{68} +32.0000 q^{70} -926.000 q^{71} +254.000 q^{73} +516.000 q^{74} -144.000 q^{76} -8.00000 q^{77} +1328.00 q^{79} -64.0000 q^{80} +168.000 q^{82} -186.000 q^{83} -24.0000 q^{85} +376.000 q^{86} +16.0000 q^{88} +336.000 q^{89} -52.0000 q^{91} +80.0000 q^{92} +508.000 q^{94} +144.000 q^{95} +614.000 q^{97} +654.000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ 0 0
$$4$$ 4.00000 0.500000
$$5$$ −4.00000 −0.357771 −0.178885 0.983870i $$-0.557249\pi$$
−0.178885 + 0.983870i $$0.557249\pi$$
$$6$$ 0 0
$$7$$ 4.00000 0.215980 0.107990 0.994152i $$-0.465559\pi$$
0.107990 + 0.994152i $$0.465559\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ 0 0
$$10$$ 8.00000 0.252982
$$11$$ −2.00000 −0.0548202 −0.0274101 0.999624i $$-0.508726\pi$$
−0.0274101 + 0.999624i $$0.508726\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ −8.00000 −0.152721
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 6.00000 0.0856008 0.0428004 0.999084i $$-0.486372\pi$$
0.0428004 + 0.999084i $$0.486372\pi$$
$$18$$ 0 0
$$19$$ −36.0000 −0.434682 −0.217341 0.976096i $$-0.569738\pi$$
−0.217341 + 0.976096i $$0.569738\pi$$
$$20$$ −16.0000 −0.178885
$$21$$ 0 0
$$22$$ 4.00000 0.0387638
$$23$$ 20.0000 0.181317 0.0906584 0.995882i $$-0.471103\pi$$
0.0906584 + 0.995882i $$0.471103\pi$$
$$24$$ 0 0
$$25$$ −109.000 −0.872000
$$26$$ 26.0000 0.196116
$$27$$ 0 0
$$28$$ 16.0000 0.107990
$$29$$ 14.0000 0.0896460 0.0448230 0.998995i $$-0.485728\pi$$
0.0448230 + 0.998995i $$0.485728\pi$$
$$30$$ 0 0
$$31$$ −152.000 −0.880645 −0.440323 0.897840i $$-0.645136\pi$$
−0.440323 + 0.897840i $$0.645136\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 0 0
$$34$$ −12.0000 −0.0605289
$$35$$ −16.0000 −0.0772712
$$36$$ 0 0
$$37$$ −258.000 −1.14635 −0.573175 0.819433i $$-0.694288\pi$$
−0.573175 + 0.819433i $$0.694288\pi$$
$$38$$ 72.0000 0.307367
$$39$$ 0 0
$$40$$ 32.0000 0.126491
$$41$$ −84.0000 −0.319966 −0.159983 0.987120i $$-0.551144\pi$$
−0.159983 + 0.987120i $$0.551144\pi$$
$$42$$ 0 0
$$43$$ −188.000 −0.666738 −0.333369 0.942796i $$-0.608185\pi$$
−0.333369 + 0.942796i $$0.608185\pi$$
$$44$$ −8.00000 −0.0274101
$$45$$ 0 0
$$46$$ −40.0000 −0.128210
$$47$$ −254.000 −0.788292 −0.394146 0.919048i $$-0.628960\pi$$
−0.394146 + 0.919048i $$0.628960\pi$$
$$48$$ 0 0
$$49$$ −327.000 −0.953353
$$50$$ 218.000 0.616597
$$51$$ 0 0
$$52$$ −52.0000 −0.138675
$$53$$ −366.000 −0.948565 −0.474283 0.880373i $$-0.657293\pi$$
−0.474283 + 0.880373i $$0.657293\pi$$
$$54$$ 0 0
$$55$$ 8.00000 0.0196131
$$56$$ −32.0000 −0.0763604
$$57$$ 0 0
$$58$$ −28.0000 −0.0633893
$$59$$ −550.000 −1.21363 −0.606813 0.794845i $$-0.707552\pi$$
−0.606813 + 0.794845i $$0.707552\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −0.0293855 −0.0146928 0.999892i $$-0.504677\pi$$
−0.0146928 + 0.999892i $$0.504677\pi$$
$$62$$ 304.000 0.622710
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ 52.0000 0.0992278
$$66$$ 0 0
$$67$$ 448.000 0.816894 0.408447 0.912782i $$-0.366070\pi$$
0.408447 + 0.912782i $$0.366070\pi$$
$$68$$ 24.0000 0.0428004
$$69$$ 0 0
$$70$$ 32.0000 0.0546390
$$71$$ −926.000 −1.54783 −0.773915 0.633289i $$-0.781704\pi$$
−0.773915 + 0.633289i $$0.781704\pi$$
$$72$$ 0 0
$$73$$ 254.000 0.407239 0.203620 0.979050i $$-0.434729\pi$$
0.203620 + 0.979050i $$0.434729\pi$$
$$74$$ 516.000 0.810592
$$75$$ 0 0
$$76$$ −144.000 −0.217341
$$77$$ −8.00000 −0.0118401
$$78$$ 0 0
$$79$$ 1328.00 1.89129 0.945644 0.325205i $$-0.105433\pi$$
0.945644 + 0.325205i $$0.105433\pi$$
$$80$$ −64.0000 −0.0894427
$$81$$ 0 0
$$82$$ 168.000 0.226250
$$83$$ −186.000 −0.245978 −0.122989 0.992408i $$-0.539248\pi$$
−0.122989 + 0.992408i $$0.539248\pi$$
$$84$$ 0 0
$$85$$ −24.0000 −0.0306255
$$86$$ 376.000 0.471455
$$87$$ 0 0
$$88$$ 16.0000 0.0193819
$$89$$ 336.000 0.400179 0.200089 0.979778i $$-0.435877\pi$$
0.200089 + 0.979778i $$0.435877\pi$$
$$90$$ 0 0
$$91$$ −52.0000 −0.0599020
$$92$$ 80.0000 0.0906584
$$93$$ 0 0
$$94$$ 508.000 0.557406
$$95$$ 144.000 0.155517
$$96$$ 0 0
$$97$$ 614.000 0.642704 0.321352 0.946960i $$-0.395863\pi$$
0.321352 + 0.946960i $$0.395863\pi$$
$$98$$ 654.000 0.674122
$$99$$ 0 0
$$100$$ −436.000 −0.436000
$$101$$ 1606.00 1.58221 0.791104 0.611682i $$-0.209507\pi$$
0.791104 + 0.611682i $$0.209507\pi$$
$$102$$ 0 0
$$103$$ 208.000 0.198979 0.0994896 0.995039i $$-0.468279\pi$$
0.0994896 + 0.995039i $$0.468279\pi$$
$$104$$ 104.000 0.0980581
$$105$$ 0 0
$$106$$ 732.000 0.670737
$$107$$ 248.000 0.224066 0.112033 0.993704i $$-0.464264\pi$$
0.112033 + 0.993704i $$0.464264\pi$$
$$108$$ 0 0
$$109$$ −542.000 −0.476277 −0.238138 0.971231i $$-0.576537\pi$$
−0.238138 + 0.971231i $$0.576537\pi$$
$$110$$ −16.0000 −0.0138685
$$111$$ 0 0
$$112$$ 64.0000 0.0539949
$$113$$ 2042.00 1.69996 0.849979 0.526817i $$-0.176615\pi$$
0.849979 + 0.526817i $$0.176615\pi$$
$$114$$ 0 0
$$115$$ −80.0000 −0.0648699
$$116$$ 56.0000 0.0448230
$$117$$ 0 0
$$118$$ 1100.00 0.858163
$$119$$ 24.0000 0.0184880
$$120$$ 0 0
$$121$$ −1327.00 −0.996995
$$122$$ 28.0000 0.0207787
$$123$$ 0 0
$$124$$ −608.000 −0.440323
$$125$$ 936.000 0.669747
$$126$$ 0 0
$$127$$ −488.000 −0.340968 −0.170484 0.985360i $$-0.554533\pi$$
−0.170484 + 0.985360i $$0.554533\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 0 0
$$130$$ −104.000 −0.0701646
$$131$$ −1744.00 −1.16316 −0.581580 0.813489i $$-0.697565\pi$$
−0.581580 + 0.813489i $$0.697565\pi$$
$$132$$ 0 0
$$133$$ −144.000 −0.0938826
$$134$$ −896.000 −0.577631
$$135$$ 0 0
$$136$$ −48.0000 −0.0302645
$$137$$ 828.000 0.516356 0.258178 0.966097i $$-0.416878\pi$$
0.258178 + 0.966097i $$0.416878\pi$$
$$138$$ 0 0
$$139$$ −404.000 −0.246524 −0.123262 0.992374i $$-0.539336\pi$$
−0.123262 + 0.992374i $$0.539336\pi$$
$$140$$ −64.0000 −0.0386356
$$141$$ 0 0
$$142$$ 1852.00 1.09448
$$143$$ 26.0000 0.0152044
$$144$$ 0 0
$$145$$ −56.0000 −0.0320727
$$146$$ −508.000 −0.287962
$$147$$ 0 0
$$148$$ −1032.00 −0.573175
$$149$$ −2928.00 −1.60987 −0.804937 0.593361i $$-0.797801\pi$$
−0.804937 + 0.593361i $$0.797801\pi$$
$$150$$ 0 0
$$151$$ 1944.00 1.04769 0.523843 0.851815i $$-0.324498\pi$$
0.523843 + 0.851815i $$0.324498\pi$$
$$152$$ 288.000 0.153683
$$153$$ 0 0
$$154$$ 16.0000 0.00837219
$$155$$ 608.000 0.315069
$$156$$ 0 0
$$157$$ 3590.00 1.82492 0.912462 0.409161i $$-0.134178\pi$$
0.912462 + 0.409161i $$0.134178\pi$$
$$158$$ −2656.00 −1.33734
$$159$$ 0 0
$$160$$ 128.000 0.0632456
$$161$$ 80.0000 0.0391608
$$162$$ 0 0
$$163$$ −2284.00 −1.09753 −0.548763 0.835978i $$-0.684901\pi$$
−0.548763 + 0.835978i $$0.684901\pi$$
$$164$$ −336.000 −0.159983
$$165$$ 0 0
$$166$$ 372.000 0.173933
$$167$$ −3174.00 −1.47073 −0.735364 0.677673i $$-0.762989\pi$$
−0.735364 + 0.677673i $$0.762989\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 48.0000 0.0216555
$$171$$ 0 0
$$172$$ −752.000 −0.333369
$$173$$ 1358.00 0.596802 0.298401 0.954441i $$-0.403547\pi$$
0.298401 + 0.954441i $$0.403547\pi$$
$$174$$ 0 0
$$175$$ −436.000 −0.188334
$$176$$ −32.0000 −0.0137051
$$177$$ 0 0
$$178$$ −672.000 −0.282969
$$179$$ −708.000 −0.295634 −0.147817 0.989015i $$-0.547225\pi$$
−0.147817 + 0.989015i $$0.547225\pi$$
$$180$$ 0 0
$$181$$ −546.000 −0.224220 −0.112110 0.993696i $$-0.535761\pi$$
−0.112110 + 0.993696i $$0.535761\pi$$
$$182$$ 104.000 0.0423571
$$183$$ 0 0
$$184$$ −160.000 −0.0641052
$$185$$ 1032.00 0.410131
$$186$$ 0 0
$$187$$ −12.0000 −0.00469266
$$188$$ −1016.00 −0.394146
$$189$$ 0 0
$$190$$ −288.000 −0.109967
$$191$$ 3472.00 1.31531 0.657657 0.753317i $$-0.271547\pi$$
0.657657 + 0.753317i $$0.271547\pi$$
$$192$$ 0 0
$$193$$ −310.000 −0.115618 −0.0578090 0.998328i $$-0.518411\pi$$
−0.0578090 + 0.998328i $$0.518411\pi$$
$$194$$ −1228.00 −0.454460
$$195$$ 0 0
$$196$$ −1308.00 −0.476676
$$197$$ −1020.00 −0.368893 −0.184447 0.982843i $$-0.559049\pi$$
−0.184447 + 0.982843i $$0.559049\pi$$
$$198$$ 0 0
$$199$$ −3256.00 −1.15986 −0.579929 0.814667i $$-0.696920\pi$$
−0.579929 + 0.814667i $$0.696920\pi$$
$$200$$ 872.000 0.308299
$$201$$ 0 0
$$202$$ −3212.00 −1.11879
$$203$$ 56.0000 0.0193617
$$204$$ 0 0
$$205$$ 336.000 0.114474
$$206$$ −416.000 −0.140699
$$207$$ 0 0
$$208$$ −208.000 −0.0693375
$$209$$ 72.0000 0.0238294
$$210$$ 0 0
$$211$$ −4564.00 −1.48909 −0.744547 0.667570i $$-0.767334\pi$$
−0.744547 + 0.667570i $$0.767334\pi$$
$$212$$ −1464.00 −0.474283
$$213$$ 0 0
$$214$$ −496.000 −0.158439
$$215$$ 752.000 0.238539
$$216$$ 0 0
$$217$$ −608.000 −0.190202
$$218$$ 1084.00 0.336779
$$219$$ 0 0
$$220$$ 32.0000 0.00980654
$$221$$ −78.0000 −0.0237414
$$222$$ 0 0
$$223$$ −72.0000 −0.0216210 −0.0108105 0.999942i $$-0.503441\pi$$
−0.0108105 + 0.999942i $$0.503441\pi$$
$$224$$ −128.000 −0.0381802
$$225$$ 0 0
$$226$$ −4084.00 −1.20205
$$227$$ −2694.00 −0.787696 −0.393848 0.919176i $$-0.628856\pi$$
−0.393848 + 0.919176i $$0.628856\pi$$
$$228$$ 0 0
$$229$$ 5922.00 1.70889 0.854447 0.519538i $$-0.173896\pi$$
0.854447 + 0.519538i $$0.173896\pi$$
$$230$$ 160.000 0.0458699
$$231$$ 0 0
$$232$$ −112.000 −0.0316947
$$233$$ 5122.00 1.44014 0.720072 0.693900i $$-0.244109\pi$$
0.720072 + 0.693900i $$0.244109\pi$$
$$234$$ 0 0
$$235$$ 1016.00 0.282028
$$236$$ −2200.00 −0.606813
$$237$$ 0 0
$$238$$ −48.0000 −0.0130730
$$239$$ −5022.00 −1.35919 −0.679595 0.733588i $$-0.737844\pi$$
−0.679595 + 0.733588i $$0.737844\pi$$
$$240$$ 0 0
$$241$$ −1218.00 −0.325553 −0.162777 0.986663i $$-0.552045\pi$$
−0.162777 + 0.986663i $$0.552045\pi$$
$$242$$ 2654.00 0.704982
$$243$$ 0 0
$$244$$ −56.0000 −0.0146928
$$245$$ 1308.00 0.341082
$$246$$ 0 0
$$247$$ 468.000 0.120559
$$248$$ 1216.00 0.311355
$$249$$ 0 0
$$250$$ −1872.00 −0.473583
$$251$$ 2112.00 0.531109 0.265554 0.964096i $$-0.414445\pi$$
0.265554 + 0.964096i $$0.414445\pi$$
$$252$$ 0 0
$$253$$ −40.0000 −0.00993984
$$254$$ 976.000 0.241101
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ −2814.00 −0.683006 −0.341503 0.939881i $$-0.610936\pi$$
−0.341503 + 0.939881i $$0.610936\pi$$
$$258$$ 0 0
$$259$$ −1032.00 −0.247588
$$260$$ 208.000 0.0496139
$$261$$ 0 0
$$262$$ 3488.00 0.822478
$$263$$ 4044.00 0.948151 0.474076 0.880484i $$-0.342782\pi$$
0.474076 + 0.880484i $$0.342782\pi$$
$$264$$ 0 0
$$265$$ 1464.00 0.339369
$$266$$ 288.000 0.0663850
$$267$$ 0 0
$$268$$ 1792.00 0.408447
$$269$$ 1470.00 0.333188 0.166594 0.986026i $$-0.446723\pi$$
0.166594 + 0.986026i $$0.446723\pi$$
$$270$$ 0 0
$$271$$ −1844.00 −0.413340 −0.206670 0.978411i $$-0.566263\pi$$
−0.206670 + 0.978411i $$0.566263\pi$$
$$272$$ 96.0000 0.0214002
$$273$$ 0 0
$$274$$ −1656.00 −0.365119
$$275$$ 218.000 0.0478033
$$276$$ 0 0
$$277$$ 5766.00 1.25071 0.625353 0.780342i $$-0.284955\pi$$
0.625353 + 0.780342i $$0.284955\pi$$
$$278$$ 808.000 0.174319
$$279$$ 0 0
$$280$$ 128.000 0.0273195
$$281$$ 7468.00 1.58542 0.792711 0.609598i $$-0.208669\pi$$
0.792711 + 0.609598i $$0.208669\pi$$
$$282$$ 0 0
$$283$$ 1228.00 0.257940 0.128970 0.991648i $$-0.458833\pi$$
0.128970 + 0.991648i $$0.458833\pi$$
$$284$$ −3704.00 −0.773915
$$285$$ 0 0
$$286$$ −52.0000 −0.0107511
$$287$$ −336.000 −0.0691061
$$288$$ 0 0
$$289$$ −4877.00 −0.992673
$$290$$ 112.000 0.0226788
$$291$$ 0 0
$$292$$ 1016.00 0.203620
$$293$$ −6608.00 −1.31755 −0.658777 0.752338i $$-0.728926\pi$$
−0.658777 + 0.752338i $$0.728926\pi$$
$$294$$ 0 0
$$295$$ 2200.00 0.434200
$$296$$ 2064.00 0.405296
$$297$$ 0 0
$$298$$ 5856.00 1.13835
$$299$$ −260.000 −0.0502883
$$300$$ 0 0
$$301$$ −752.000 −0.144002
$$302$$ −3888.00 −0.740825
$$303$$ 0 0
$$304$$ −576.000 −0.108671
$$305$$ 56.0000 0.0105133
$$306$$ 0 0
$$307$$ 7664.00 1.42478 0.712390 0.701784i $$-0.247613\pi$$
0.712390 + 0.701784i $$0.247613\pi$$
$$308$$ −32.0000 −0.00592003
$$309$$ 0 0
$$310$$ −1216.00 −0.222788
$$311$$ 2340.00 0.426653 0.213327 0.976981i $$-0.431570\pi$$
0.213327 + 0.976981i $$0.431570\pi$$
$$312$$ 0 0
$$313$$ 6710.00 1.21173 0.605865 0.795567i $$-0.292827\pi$$
0.605865 + 0.795567i $$0.292827\pi$$
$$314$$ −7180.00 −1.29042
$$315$$ 0 0
$$316$$ 5312.00 0.945644
$$317$$ −4164.00 −0.737771 −0.368886 0.929475i $$-0.620261\pi$$
−0.368886 + 0.929475i $$0.620261\pi$$
$$318$$ 0 0
$$319$$ −28.0000 −0.00491442
$$320$$ −256.000 −0.0447214
$$321$$ 0 0
$$322$$ −160.000 −0.0276908
$$323$$ −216.000 −0.0372092
$$324$$ 0 0
$$325$$ 1417.00 0.241849
$$326$$ 4568.00 0.776068
$$327$$ 0 0
$$328$$ 672.000 0.113125
$$329$$ −1016.00 −0.170255
$$330$$ 0 0
$$331$$ −10072.0 −1.67253 −0.836265 0.548326i $$-0.815265\pi$$
−0.836265 + 0.548326i $$0.815265\pi$$
$$332$$ −744.000 −0.122989
$$333$$ 0 0
$$334$$ 6348.00 1.03996
$$335$$ −1792.00 −0.292261
$$336$$ 0 0
$$337$$ 2990.00 0.483311 0.241655 0.970362i $$-0.422310\pi$$
0.241655 + 0.970362i $$0.422310\pi$$
$$338$$ −338.000 −0.0543928
$$339$$ 0 0
$$340$$ −96.0000 −0.0153127
$$341$$ 304.000 0.0482772
$$342$$ 0 0
$$343$$ −2680.00 −0.421885
$$344$$ 1504.00 0.235727
$$345$$ 0 0
$$346$$ −2716.00 −0.422003
$$347$$ −6564.00 −1.01549 −0.507743 0.861508i $$-0.669520\pi$$
−0.507743 + 0.861508i $$0.669520\pi$$
$$348$$ 0 0
$$349$$ −674.000 −0.103376 −0.0516882 0.998663i $$-0.516460\pi$$
−0.0516882 + 0.998663i $$0.516460\pi$$
$$350$$ 872.000 0.133172
$$351$$ 0 0
$$352$$ 64.0000 0.00969094
$$353$$ 10732.0 1.61815 0.809075 0.587706i $$-0.199969\pi$$
0.809075 + 0.587706i $$0.199969\pi$$
$$354$$ 0 0
$$355$$ 3704.00 0.553769
$$356$$ 1344.00 0.200089
$$357$$ 0 0
$$358$$ 1416.00 0.209044
$$359$$ 4842.00 0.711841 0.355921 0.934516i $$-0.384167\pi$$
0.355921 + 0.934516i $$0.384167\pi$$
$$360$$ 0 0
$$361$$ −5563.00 −0.811051
$$362$$ 1092.00 0.158548
$$363$$ 0 0
$$364$$ −208.000 −0.0299510
$$365$$ −1016.00 −0.145698
$$366$$ 0 0
$$367$$ −6280.00 −0.893224 −0.446612 0.894728i $$-0.647370\pi$$
−0.446612 + 0.894728i $$0.647370\pi$$
$$368$$ 320.000 0.0453292
$$369$$ 0 0
$$370$$ −2064.00 −0.290006
$$371$$ −1464.00 −0.204871
$$372$$ 0 0
$$373$$ 6434.00 0.893136 0.446568 0.894750i $$-0.352646\pi$$
0.446568 + 0.894750i $$0.352646\pi$$
$$374$$ 24.0000 0.00331821
$$375$$ 0 0
$$376$$ 2032.00 0.278703
$$377$$ −182.000 −0.0248633
$$378$$ 0 0
$$379$$ −9068.00 −1.22900 −0.614501 0.788916i $$-0.710643\pi$$
−0.614501 + 0.788916i $$0.710643\pi$$
$$380$$ 576.000 0.0777584
$$381$$ 0 0
$$382$$ −6944.00 −0.930068
$$383$$ −3162.00 −0.421855 −0.210928 0.977502i $$-0.567648\pi$$
−0.210928 + 0.977502i $$0.567648\pi$$
$$384$$ 0 0
$$385$$ 32.0000 0.00423603
$$386$$ 620.000 0.0817543
$$387$$ 0 0
$$388$$ 2456.00 0.321352
$$389$$ 3666.00 0.477824 0.238912 0.971041i $$-0.423209\pi$$
0.238912 + 0.971041i $$0.423209\pi$$
$$390$$ 0 0
$$391$$ 120.000 0.0155209
$$392$$ 2616.00 0.337061
$$393$$ 0 0
$$394$$ 2040.00 0.260847
$$395$$ −5312.00 −0.676647
$$396$$ 0 0
$$397$$ 11054.0 1.39744 0.698721 0.715394i $$-0.253753\pi$$
0.698721 + 0.715394i $$0.253753\pi$$
$$398$$ 6512.00 0.820143
$$399$$ 0 0
$$400$$ −1744.00 −0.218000
$$401$$ 5328.00 0.663510 0.331755 0.943366i $$-0.392359\pi$$
0.331755 + 0.943366i $$0.392359\pi$$
$$402$$ 0 0
$$403$$ 1976.00 0.244247
$$404$$ 6424.00 0.791104
$$405$$ 0 0
$$406$$ −112.000 −0.0136908
$$407$$ 516.000 0.0628432
$$408$$ 0 0
$$409$$ −12074.0 −1.45971 −0.729854 0.683603i $$-0.760412\pi$$
−0.729854 + 0.683603i $$0.760412\pi$$
$$410$$ −672.000 −0.0809456
$$411$$ 0 0
$$412$$ 832.000 0.0994896
$$413$$ −2200.00 −0.262118
$$414$$ 0 0
$$415$$ 744.000 0.0880037
$$416$$ 416.000 0.0490290
$$417$$ 0 0
$$418$$ −144.000 −0.0168499
$$419$$ −13584.0 −1.58382 −0.791911 0.610636i $$-0.790914\pi$$
−0.791911 + 0.610636i $$0.790914\pi$$
$$420$$ 0 0
$$421$$ −7406.00 −0.857355 −0.428677 0.903458i $$-0.641020\pi$$
−0.428677 + 0.903458i $$0.641020\pi$$
$$422$$ 9128.00 1.05295
$$423$$ 0 0
$$424$$ 2928.00 0.335369
$$425$$ −654.000 −0.0746439
$$426$$ 0 0
$$427$$ −56.0000 −0.00634667
$$428$$ 992.000 0.112033
$$429$$ 0 0
$$430$$ −1504.00 −0.168673
$$431$$ 10134.0 1.13257 0.566285 0.824210i $$-0.308380\pi$$
0.566285 + 0.824210i $$0.308380\pi$$
$$432$$ 0 0
$$433$$ 9406.00 1.04393 0.521967 0.852966i $$-0.325198\pi$$
0.521967 + 0.852966i $$0.325198\pi$$
$$434$$ 1216.00 0.134493
$$435$$ 0 0
$$436$$ −2168.00 −0.238138
$$437$$ −720.000 −0.0788153
$$438$$ 0 0
$$439$$ 4088.00 0.444441 0.222220 0.974996i $$-0.428670\pi$$
0.222220 + 0.974996i $$0.428670\pi$$
$$440$$ −64.0000 −0.00693427
$$441$$ 0 0
$$442$$ 156.000 0.0167877
$$443$$ 5328.00 0.571424 0.285712 0.958315i $$-0.407770\pi$$
0.285712 + 0.958315i $$0.407770\pi$$
$$444$$ 0 0
$$445$$ −1344.00 −0.143172
$$446$$ 144.000 0.0152883
$$447$$ 0 0
$$448$$ 256.000 0.0269975
$$449$$ −13160.0 −1.38320 −0.691602 0.722279i $$-0.743095\pi$$
−0.691602 + 0.722279i $$0.743095\pi$$
$$450$$ 0 0
$$451$$ 168.000 0.0175406
$$452$$ 8168.00 0.849979
$$453$$ 0 0
$$454$$ 5388.00 0.556985
$$455$$ 208.000 0.0214312
$$456$$ 0 0
$$457$$ −9146.00 −0.936175 −0.468087 0.883682i $$-0.655057\pi$$
−0.468087 + 0.883682i $$0.655057\pi$$
$$458$$ −11844.0 −1.20837
$$459$$ 0 0
$$460$$ −320.000 −0.0324349
$$461$$ −5580.00 −0.563745 −0.281873 0.959452i $$-0.590956\pi$$
−0.281873 + 0.959452i $$0.590956\pi$$
$$462$$ 0 0
$$463$$ 14788.0 1.48436 0.742178 0.670203i $$-0.233793\pi$$
0.742178 + 0.670203i $$0.233793\pi$$
$$464$$ 224.000 0.0224115
$$465$$ 0 0
$$466$$ −10244.0 −1.01834
$$467$$ −12376.0 −1.22632 −0.613162 0.789957i $$-0.710103\pi$$
−0.613162 + 0.789957i $$0.710103\pi$$
$$468$$ 0 0
$$469$$ 1792.00 0.176433
$$470$$ −2032.00 −0.199424
$$471$$ 0 0
$$472$$ 4400.00 0.429081
$$473$$ 376.000 0.0365507
$$474$$ 0 0
$$475$$ 3924.00 0.379043
$$476$$ 96.0000 0.00924402
$$477$$ 0 0
$$478$$ 10044.0 0.961092
$$479$$ −834.000 −0.0795541 −0.0397771 0.999209i $$-0.512665\pi$$
−0.0397771 + 0.999209i $$0.512665\pi$$
$$480$$ 0 0
$$481$$ 3354.00 0.317940
$$482$$ 2436.00 0.230201
$$483$$ 0 0
$$484$$ −5308.00 −0.498497
$$485$$ −2456.00 −0.229941
$$486$$ 0 0
$$487$$ −13192.0 −1.22749 −0.613744 0.789505i $$-0.710337\pi$$
−0.613744 + 0.789505i $$0.710337\pi$$
$$488$$ 112.000 0.0103893
$$489$$ 0 0
$$490$$ −2616.00 −0.241181
$$491$$ −16568.0 −1.52282 −0.761409 0.648272i $$-0.775492\pi$$
−0.761409 + 0.648272i $$0.775492\pi$$
$$492$$ 0 0
$$493$$ 84.0000 0.00767377
$$494$$ −936.000 −0.0852482
$$495$$ 0 0
$$496$$ −2432.00 −0.220161
$$497$$ −3704.00 −0.334300
$$498$$ 0 0
$$499$$ −10136.0 −0.909318 −0.454659 0.890666i $$-0.650239\pi$$
−0.454659 + 0.890666i $$0.650239\pi$$
$$500$$ 3744.00 0.334874
$$501$$ 0 0
$$502$$ −4224.00 −0.375550
$$503$$ −10412.0 −0.922959 −0.461479 0.887151i $$-0.652681\pi$$
−0.461479 + 0.887151i $$0.652681\pi$$
$$504$$ 0 0
$$505$$ −6424.00 −0.566068
$$506$$ 80.0000 0.00702853
$$507$$ 0 0
$$508$$ −1952.00 −0.170484
$$509$$ 4180.00 0.363999 0.181999 0.983299i $$-0.441743\pi$$
0.181999 + 0.983299i $$0.441743\pi$$
$$510$$ 0 0
$$511$$ 1016.00 0.0879554
$$512$$ −512.000 −0.0441942
$$513$$ 0 0
$$514$$ 5628.00 0.482958
$$515$$ −832.000 −0.0711889
$$516$$ 0 0
$$517$$ 508.000 0.0432143
$$518$$ 2064.00 0.175071
$$519$$ 0 0
$$520$$ −416.000 −0.0350823
$$521$$ 14610.0 1.22855 0.614276 0.789091i $$-0.289448\pi$$
0.614276 + 0.789091i $$0.289448\pi$$
$$522$$ 0 0
$$523$$ −2172.00 −0.181596 −0.0907982 0.995869i $$-0.528942\pi$$
−0.0907982 + 0.995869i $$0.528942\pi$$
$$524$$ −6976.00 −0.581580
$$525$$ 0 0
$$526$$ −8088.00 −0.670444
$$527$$ −912.000 −0.0753840
$$528$$ 0 0
$$529$$ −11767.0 −0.967124
$$530$$ −2928.00 −0.239970
$$531$$ 0 0
$$532$$ −576.000 −0.0469413
$$533$$ 1092.00 0.0887425
$$534$$ 0 0
$$535$$ −992.000 −0.0801643
$$536$$ −3584.00 −0.288816
$$537$$ 0 0
$$538$$ −2940.00 −0.235599
$$539$$ 654.000 0.0522630
$$540$$ 0 0
$$541$$ −11758.0 −0.934410 −0.467205 0.884149i $$-0.654739\pi$$
−0.467205 + 0.884149i $$0.654739\pi$$
$$542$$ 3688.00 0.292275
$$543$$ 0 0
$$544$$ −192.000 −0.0151322
$$545$$ 2168.00 0.170398
$$546$$ 0 0
$$547$$ 340.000 0.0265765 0.0132883 0.999912i $$-0.495770\pi$$
0.0132883 + 0.999912i $$0.495770\pi$$
$$548$$ 3312.00 0.258178
$$549$$ 0 0
$$550$$ −436.000 −0.0338020
$$551$$ −504.000 −0.0389676
$$552$$ 0 0
$$553$$ 5312.00 0.408480
$$554$$ −11532.0 −0.884382
$$555$$ 0 0
$$556$$ −1616.00 −0.123262
$$557$$ 3768.00 0.286634 0.143317 0.989677i $$-0.454223\pi$$
0.143317 + 0.989677i $$0.454223\pi$$
$$558$$ 0 0
$$559$$ 2444.00 0.184920
$$560$$ −256.000 −0.0193178
$$561$$ 0 0
$$562$$ −14936.0 −1.12106
$$563$$ 10172.0 0.761454 0.380727 0.924687i $$-0.375674\pi$$
0.380727 + 0.924687i $$0.375674\pi$$
$$564$$ 0 0
$$565$$ −8168.00 −0.608195
$$566$$ −2456.00 −0.182391
$$567$$ 0 0
$$568$$ 7408.00 0.547241
$$569$$ 5506.00 0.405665 0.202833 0.979213i $$-0.434985\pi$$
0.202833 + 0.979213i $$0.434985\pi$$
$$570$$ 0 0
$$571$$ 2340.00 0.171499 0.0857495 0.996317i $$-0.472672\pi$$
0.0857495 + 0.996317i $$0.472672\pi$$
$$572$$ 104.000 0.00760220
$$573$$ 0 0
$$574$$ 672.000 0.0488654
$$575$$ −2180.00 −0.158108
$$576$$ 0 0
$$577$$ −20094.0 −1.44978 −0.724891 0.688864i $$-0.758110\pi$$
−0.724891 + 0.688864i $$0.758110\pi$$
$$578$$ 9754.00 0.701925
$$579$$ 0 0
$$580$$ −224.000 −0.0160364
$$581$$ −744.000 −0.0531262
$$582$$ 0 0
$$583$$ 732.000 0.0520006
$$584$$ −2032.00 −0.143981
$$585$$ 0 0
$$586$$ 13216.0 0.931652
$$587$$ 7118.00 0.500496 0.250248 0.968182i $$-0.419488\pi$$
0.250248 + 0.968182i $$0.419488\pi$$
$$588$$ 0 0
$$589$$ 5472.00 0.382801
$$590$$ −4400.00 −0.307026
$$591$$ 0 0
$$592$$ −4128.00 −0.286587
$$593$$ 10328.0 0.715211 0.357606 0.933873i $$-0.383593\pi$$
0.357606 + 0.933873i $$0.383593\pi$$
$$594$$ 0 0
$$595$$ −96.0000 −0.00661448
$$596$$ −11712.0 −0.804937
$$597$$ 0 0
$$598$$ 520.000 0.0355592
$$599$$ 19732.0 1.34596 0.672978 0.739662i $$-0.265015\pi$$
0.672978 + 0.739662i $$0.265015\pi$$
$$600$$ 0 0
$$601$$ −12026.0 −0.816224 −0.408112 0.912932i $$-0.633813\pi$$
−0.408112 + 0.912932i $$0.633813\pi$$
$$602$$ 1504.00 0.101825
$$603$$ 0 0
$$604$$ 7776.00 0.523843
$$605$$ 5308.00 0.356696
$$606$$ 0 0
$$607$$ 17016.0 1.13782 0.568911 0.822399i $$-0.307365\pi$$
0.568911 + 0.822399i $$0.307365\pi$$
$$608$$ 1152.00 0.0768417
$$609$$ 0 0
$$610$$ −112.000 −0.00743401
$$611$$ 3302.00 0.218633
$$612$$ 0 0
$$613$$ 11654.0 0.767864 0.383932 0.923361i $$-0.374570\pi$$
0.383932 + 0.923361i $$0.374570\pi$$
$$614$$ −15328.0 −1.00747
$$615$$ 0 0
$$616$$ 64.0000 0.00418609
$$617$$ −11612.0 −0.757669 −0.378834 0.925465i $$-0.623675\pi$$
−0.378834 + 0.925465i $$0.623675\pi$$
$$618$$ 0 0
$$619$$ 4024.00 0.261290 0.130645 0.991429i $$-0.458295\pi$$
0.130645 + 0.991429i $$0.458295\pi$$
$$620$$ 2432.00 0.157535
$$621$$ 0 0
$$622$$ −4680.00 −0.301690
$$623$$ 1344.00 0.0864305
$$624$$ 0 0
$$625$$ 9881.00 0.632384
$$626$$ −13420.0 −0.856823
$$627$$ 0 0
$$628$$ 14360.0 0.912462
$$629$$ −1548.00 −0.0981285
$$630$$ 0 0
$$631$$ −1088.00 −0.0686412 −0.0343206 0.999411i $$-0.510927\pi$$
−0.0343206 + 0.999411i $$0.510927\pi$$
$$632$$ −10624.0 −0.668671
$$633$$ 0 0
$$634$$ 8328.00 0.521683
$$635$$ 1952.00 0.121989
$$636$$ 0 0
$$637$$ 4251.00 0.264412
$$638$$ 56.0000 0.00347502
$$639$$ 0 0
$$640$$ 512.000 0.0316228
$$641$$ 7078.00 0.436138 0.218069 0.975933i $$-0.430024\pi$$
0.218069 + 0.975933i $$0.430024\pi$$
$$642$$ 0 0
$$643$$ 8336.00 0.511259 0.255630 0.966775i $$-0.417717\pi$$
0.255630 + 0.966775i $$0.417717\pi$$
$$644$$ 320.000 0.0195804
$$645$$ 0 0
$$646$$ 432.000 0.0263109
$$647$$ −32.0000 −0.00194444 −0.000972218 1.00000i $$-0.500309\pi$$
−0.000972218 1.00000i $$0.500309\pi$$
$$648$$ 0 0
$$649$$ 1100.00 0.0665312
$$650$$ −2834.00 −0.171013
$$651$$ 0 0
$$652$$ −9136.00 −0.548763
$$653$$ 15822.0 0.948182 0.474091 0.880476i $$-0.342777\pi$$
0.474091 + 0.880476i $$0.342777\pi$$
$$654$$ 0 0
$$655$$ 6976.00 0.416145
$$656$$ −1344.00 −0.0799914
$$657$$ 0 0
$$658$$ 2032.00 0.120388
$$659$$ −21540.0 −1.27326 −0.636631 0.771169i $$-0.719672\pi$$
−0.636631 + 0.771169i $$0.719672\pi$$
$$660$$ 0 0
$$661$$ 8270.00 0.486635 0.243317 0.969947i $$-0.421764\pi$$
0.243317 + 0.969947i $$0.421764\pi$$
$$662$$ 20144.0 1.18266
$$663$$ 0 0
$$664$$ 1488.00 0.0869663
$$665$$ 576.000 0.0335885
$$666$$ 0 0
$$667$$ 280.000 0.0162543
$$668$$ −12696.0 −0.735364
$$669$$ 0 0
$$670$$ 3584.00 0.206660
$$671$$ 28.0000 0.00161092
$$672$$ 0 0
$$673$$ 8482.00 0.485820 0.242910 0.970049i $$-0.421898\pi$$
0.242910 + 0.970049i $$0.421898\pi$$
$$674$$ −5980.00 −0.341752
$$675$$ 0 0
$$676$$ 676.000 0.0384615
$$677$$ −2550.00 −0.144763 −0.0723814 0.997377i $$-0.523060\pi$$
−0.0723814 + 0.997377i $$0.523060\pi$$
$$678$$ 0 0
$$679$$ 2456.00 0.138811
$$680$$ 192.000 0.0108277
$$681$$ 0 0
$$682$$ −608.000 −0.0341371
$$683$$ 31534.0 1.76664 0.883320 0.468771i $$-0.155303\pi$$
0.883320 + 0.468771i $$0.155303\pi$$
$$684$$ 0 0
$$685$$ −3312.00 −0.184737
$$686$$ 5360.00 0.298317
$$687$$ 0 0
$$688$$ −3008.00 −0.166684
$$689$$ 4758.00 0.263085
$$690$$ 0 0
$$691$$ 33832.0 1.86256 0.931281 0.364302i $$-0.118693\pi$$
0.931281 + 0.364302i $$0.118693\pi$$
$$692$$ 5432.00 0.298401
$$693$$ 0 0
$$694$$ 13128.0 0.718058
$$695$$ 1616.00 0.0881991
$$696$$ 0 0
$$697$$ −504.000 −0.0273893
$$698$$ 1348.00 0.0730982
$$699$$ 0 0
$$700$$ −1744.00 −0.0941671
$$701$$ −19422.0 −1.04645 −0.523223 0.852196i $$-0.675271\pi$$
−0.523223 + 0.852196i $$0.675271\pi$$
$$702$$ 0 0
$$703$$ 9288.00 0.498298
$$704$$ −128.000 −0.00685253
$$705$$ 0 0
$$706$$ −21464.0 −1.14420
$$707$$ 6424.00 0.341725
$$708$$ 0 0
$$709$$ −1894.00 −0.100325 −0.0501627 0.998741i $$-0.515974\pi$$
−0.0501627 + 0.998741i $$0.515974\pi$$
$$710$$ −7408.00 −0.391574
$$711$$ 0 0
$$712$$ −2688.00 −0.141485
$$713$$ −3040.00 −0.159676
$$714$$ 0 0
$$715$$ −104.000 −0.00543969
$$716$$ −2832.00 −0.147817
$$717$$ 0 0
$$718$$ −9684.00 −0.503348
$$719$$ 20156.0 1.04547 0.522734 0.852496i $$-0.324912\pi$$
0.522734 + 0.852496i $$0.324912\pi$$
$$720$$ 0 0
$$721$$ 832.000 0.0429754
$$722$$ 11126.0 0.573500
$$723$$ 0 0
$$724$$ −2184.00 −0.112110
$$725$$ −1526.00 −0.0781713
$$726$$ 0 0
$$727$$ 11128.0 0.567696 0.283848 0.958869i $$-0.408389\pi$$
0.283848 + 0.958869i $$0.408389\pi$$
$$728$$ 416.000 0.0211786
$$729$$ 0 0
$$730$$ 2032.00 0.103024
$$731$$ −1128.00 −0.0570733
$$732$$ 0 0
$$733$$ 16202.0 0.816418 0.408209 0.912888i $$-0.366153\pi$$
0.408209 + 0.912888i $$0.366153\pi$$
$$734$$ 12560.0 0.631605
$$735$$ 0 0
$$736$$ −640.000 −0.0320526
$$737$$ −896.000 −0.0447823
$$738$$ 0 0
$$739$$ −5328.00 −0.265215 −0.132607 0.991169i $$-0.542335\pi$$
−0.132607 + 0.991169i $$0.542335\pi$$
$$740$$ 4128.00 0.205065
$$741$$ 0 0
$$742$$ 2928.00 0.144866
$$743$$ 20482.0 1.01132 0.505661 0.862732i $$-0.331249\pi$$
0.505661 + 0.862732i $$0.331249\pi$$
$$744$$ 0 0
$$745$$ 11712.0 0.575966
$$746$$ −12868.0 −0.631543
$$747$$ 0 0
$$748$$ −48.0000 −0.00234633
$$749$$ 992.000 0.0483937
$$750$$ 0 0
$$751$$ 8040.00 0.390657 0.195329 0.980738i $$-0.437423\pi$$
0.195329 + 0.980738i $$0.437423\pi$$
$$752$$ −4064.00 −0.197073
$$753$$ 0 0
$$754$$ 364.000 0.0175810
$$755$$ −7776.00 −0.374831
$$756$$ 0 0
$$757$$ −15822.0 −0.759657 −0.379829 0.925057i $$-0.624017\pi$$
−0.379829 + 0.925057i $$0.624017\pi$$
$$758$$ 18136.0 0.869036
$$759$$ 0 0
$$760$$ −1152.00 −0.0549835
$$761$$ 1452.00 0.0691655 0.0345828 0.999402i $$-0.488990\pi$$
0.0345828 + 0.999402i $$0.488990\pi$$
$$762$$ 0 0
$$763$$ −2168.00 −0.102866
$$764$$ 13888.0 0.657657
$$765$$ 0 0
$$766$$ 6324.00 0.298297
$$767$$ 7150.00 0.336599
$$768$$ 0 0
$$769$$ 32298.0 1.51456 0.757279 0.653091i $$-0.226528\pi$$
0.757279 + 0.653091i $$0.226528\pi$$
$$770$$ −64.0000 −0.00299532
$$771$$ 0 0
$$772$$ −1240.00 −0.0578090
$$773$$ −18736.0 −0.871781 −0.435891 0.900000i $$-0.643567\pi$$
−0.435891 + 0.900000i $$0.643567\pi$$
$$774$$ 0 0
$$775$$ 16568.0 0.767923
$$776$$ −4912.00 −0.227230
$$777$$ 0 0
$$778$$ −7332.00 −0.337873
$$779$$ 3024.00 0.139083
$$780$$ 0 0
$$781$$ 1852.00 0.0848525
$$782$$ −240.000 −0.0109749
$$783$$ 0 0
$$784$$ −5232.00 −0.238338
$$785$$ −14360.0 −0.652905
$$786$$ 0 0
$$787$$ −40816.0 −1.84871 −0.924354 0.381536i $$-0.875395\pi$$
−0.924354 + 0.381536i $$0.875395\pi$$
$$788$$ −4080.00 −0.184447
$$789$$ 0 0
$$790$$ 10624.0 0.478462
$$791$$ 8168.00 0.367156
$$792$$ 0 0
$$793$$ 182.000 0.00815008
$$794$$ −22108.0 −0.988141
$$795$$ 0 0
$$796$$ −13024.0 −0.579929
$$797$$ −4518.00 −0.200798 −0.100399 0.994947i $$-0.532012\pi$$
−0.100399 + 0.994947i $$0.532012\pi$$
$$798$$ 0 0
$$799$$ −1524.00 −0.0674784
$$800$$ 3488.00 0.154149
$$801$$ 0 0
$$802$$ −10656.0 −0.469173
$$803$$ −508.000 −0.0223249
$$804$$ 0 0
$$805$$ −320.000 −0.0140106
$$806$$ −3952.00 −0.172709
$$807$$ 0 0
$$808$$ −12848.0 −0.559395
$$809$$ 5058.00 0.219814 0.109907 0.993942i $$-0.464945\pi$$
0.109907 + 0.993942i $$0.464945\pi$$
$$810$$ 0 0
$$811$$ −22564.0 −0.976978 −0.488489 0.872570i $$-0.662452\pi$$
−0.488489 + 0.872570i $$0.662452\pi$$
$$812$$ 224.000 0.00968086
$$813$$ 0 0
$$814$$ −1032.00 −0.0444368
$$815$$ 9136.00 0.392663
$$816$$ 0 0
$$817$$ 6768.00 0.289819
$$818$$ 24148.0 1.03217
$$819$$ 0 0
$$820$$ 1344.00 0.0572372
$$821$$ −32584.0 −1.38513 −0.692564 0.721357i $$-0.743519\pi$$
−0.692564 + 0.721357i $$0.743519\pi$$
$$822$$ 0 0
$$823$$ −9288.00 −0.393389 −0.196695 0.980465i $$-0.563021\pi$$
−0.196695 + 0.980465i $$0.563021\pi$$
$$824$$ −1664.00 −0.0703497
$$825$$ 0 0
$$826$$ 4400.00 0.185346
$$827$$ −20586.0 −0.865593 −0.432796 0.901492i $$-0.642473\pi$$
−0.432796 + 0.901492i $$0.642473\pi$$
$$828$$ 0 0
$$829$$ −46118.0 −1.93214 −0.966070 0.258280i $$-0.916844\pi$$
−0.966070 + 0.258280i $$0.916844\pi$$
$$830$$ −1488.00 −0.0622280
$$831$$ 0 0
$$832$$ −832.000 −0.0346688
$$833$$ −1962.00 −0.0816078
$$834$$ 0 0
$$835$$ 12696.0 0.526183
$$836$$ 288.000 0.0119147
$$837$$ 0 0
$$838$$ 27168.0 1.11993
$$839$$ 39230.0 1.61427 0.807133 0.590369i $$-0.201018\pi$$
0.807133 + 0.590369i $$0.201018\pi$$
$$840$$ 0 0
$$841$$ −24193.0 −0.991964
$$842$$ 14812.0 0.606241
$$843$$ 0 0
$$844$$ −18256.0 −0.744547
$$845$$ −676.000 −0.0275208
$$846$$ 0 0
$$847$$ −5308.00 −0.215331
$$848$$ −5856.00 −0.237141
$$849$$ 0 0
$$850$$ 1308.00 0.0527812
$$851$$ −5160.00 −0.207853
$$852$$ 0 0
$$853$$ −18674.0 −0.749573 −0.374786 0.927111i $$-0.622284\pi$$
−0.374786 + 0.927111i $$0.622284\pi$$
$$854$$ 112.000 0.00448778
$$855$$ 0 0
$$856$$ −1984.00 −0.0792193
$$857$$ −41678.0 −1.66125 −0.830626 0.556830i $$-0.812017\pi$$
−0.830626 + 0.556830i $$0.812017\pi$$
$$858$$ 0 0
$$859$$ −14740.0 −0.585474 −0.292737 0.956193i $$-0.594566\pi$$
−0.292737 + 0.956193i $$0.594566\pi$$
$$860$$ 3008.00 0.119270
$$861$$ 0 0
$$862$$ −20268.0 −0.800848
$$863$$ 24982.0 0.985396 0.492698 0.870200i $$-0.336011\pi$$
0.492698 + 0.870200i $$0.336011\pi$$
$$864$$ 0 0
$$865$$ −5432.00 −0.213519
$$866$$ −18812.0 −0.738173
$$867$$ 0 0
$$868$$ −2432.00 −0.0951008
$$869$$ −2656.00 −0.103681
$$870$$ 0 0
$$871$$ −5824.00 −0.226566
$$872$$ 4336.00 0.168389
$$873$$ 0 0
$$874$$ 1440.00 0.0557308
$$875$$ 3744.00 0.144652
$$876$$ 0 0
$$877$$ 1134.00 0.0436630 0.0218315 0.999762i $$-0.493050\pi$$
0.0218315 + 0.999762i $$0.493050\pi$$
$$878$$ −8176.00 −0.314267
$$879$$ 0 0
$$880$$ 128.000 0.00490327
$$881$$ −34950.0 −1.33654 −0.668272 0.743917i $$-0.732966\pi$$
−0.668272 + 0.743917i $$0.732966\pi$$
$$882$$ 0 0
$$883$$ −3068.00 −0.116927 −0.0584634 0.998290i $$-0.518620\pi$$
−0.0584634 + 0.998290i $$0.518620\pi$$
$$884$$ −312.000 −0.0118707
$$885$$ 0 0
$$886$$ −10656.0 −0.404058
$$887$$ 14080.0 0.532988 0.266494 0.963837i $$-0.414135\pi$$
0.266494 + 0.963837i $$0.414135\pi$$
$$888$$ 0 0
$$889$$ −1952.00 −0.0736423
$$890$$ 2688.00 0.101238
$$891$$ 0 0
$$892$$ −288.000 −0.0108105
$$893$$ 9144.00 0.342657
$$894$$ 0 0
$$895$$ 2832.00 0.105769
$$896$$ −512.000 −0.0190901
$$897$$ 0 0
$$898$$ 26320.0 0.978073
$$899$$ −2128.00 −0.0789464
$$900$$ 0 0
$$901$$ −2196.00 −0.0811980
$$902$$ −336.000 −0.0124031
$$903$$ 0 0
$$904$$ −16336.0 −0.601026
$$905$$ 2184.00 0.0802195
$$906$$ 0 0
$$907$$ −24876.0 −0.910688 −0.455344 0.890316i $$-0.650484\pi$$
−0.455344 + 0.890316i $$0.650484\pi$$
$$908$$ −10776.0 −0.393848
$$909$$ 0 0
$$910$$ −416.000 −0.0151541
$$911$$ −51456.0 −1.87136 −0.935682 0.352843i $$-0.885215\pi$$
−0.935682 + 0.352843i $$0.885215\pi$$
$$912$$ 0 0
$$913$$ 372.000 0.0134846
$$914$$ 18292.0 0.661975
$$915$$ 0 0
$$916$$ 23688.0 0.854447
$$917$$ −6976.00 −0.251219
$$918$$ 0 0
$$919$$ −31032.0 −1.11388 −0.556938 0.830554i $$-0.688024\pi$$
−0.556938 + 0.830554i $$0.688024\pi$$
$$920$$ 640.000 0.0229350
$$921$$ 0 0
$$922$$ 11160.0 0.398628
$$923$$ 12038.0 0.429291
$$924$$ 0 0
$$925$$ 28122.0 0.999617
$$926$$ −29576.0 −1.04960
$$927$$ 0 0
$$928$$ −448.000 −0.0158473
$$929$$ −50820.0 −1.79478 −0.897390 0.441239i $$-0.854539\pi$$
−0.897390 + 0.441239i $$0.854539\pi$$
$$930$$ 0 0
$$931$$ 11772.0 0.414406
$$932$$ 20488.0 0.720072
$$933$$ 0 0
$$934$$ 24752.0 0.867142
$$935$$ 48.0000 0.00167890
$$936$$ 0 0
$$937$$ 5982.00 0.208563 0.104281 0.994548i $$-0.466746\pi$$
0.104281 + 0.994548i $$0.466746\pi$$
$$938$$ −3584.00 −0.124757
$$939$$ 0 0
$$940$$ 4064.00 0.141014
$$941$$ 20224.0 0.700620 0.350310 0.936634i $$-0.386076\pi$$
0.350310 + 0.936634i $$0.386076\pi$$
$$942$$ 0 0
$$943$$ −1680.00 −0.0580152
$$944$$ −8800.00 −0.303406
$$945$$ 0 0
$$946$$ −752.000 −0.0258453
$$947$$ −8478.00 −0.290917 −0.145458 0.989364i $$-0.546466\pi$$
−0.145458 + 0.989364i $$0.546466\pi$$
$$948$$ 0 0
$$949$$ −3302.00 −0.112948
$$950$$ −7848.00 −0.268024
$$951$$ 0 0
$$952$$ −192.000 −0.00653651
$$953$$ −40918.0 −1.39083 −0.695417 0.718607i $$-0.744780\pi$$
−0.695417 + 0.718607i $$0.744780\pi$$
$$954$$ 0 0
$$955$$ −13888.0 −0.470581
$$956$$ −20088.0 −0.679595
$$957$$ 0 0
$$958$$ 1668.00 0.0562533
$$959$$ 3312.00 0.111522
$$960$$ 0 0
$$961$$ −6687.00 −0.224464
$$962$$ −6708.00 −0.224818
$$963$$ 0 0
$$964$$ −4872.00 −0.162777
$$965$$ 1240.00 0.0413648
$$966$$ 0 0
$$967$$ −4624.00 −0.153772 −0.0768862 0.997040i $$-0.524498\pi$$
−0.0768862 + 0.997040i $$0.524498\pi$$
$$968$$ 10616.0 0.352491
$$969$$ 0 0
$$970$$ 4912.00 0.162593
$$971$$ −15300.0 −0.505665 −0.252832 0.967510i $$-0.581362\pi$$
−0.252832 + 0.967510i $$0.581362\pi$$
$$972$$ 0 0
$$973$$ −1616.00 −0.0532442
$$974$$ 26384.0 0.867965
$$975$$ 0 0
$$976$$ −224.000 −0.00734638
$$977$$ −19584.0 −0.641298 −0.320649 0.947198i $$-0.603901\pi$$
−0.320649 + 0.947198i $$0.603901\pi$$
$$978$$ 0 0
$$979$$ −672.000 −0.0219379
$$980$$ 5232.00 0.170541
$$981$$ 0 0
$$982$$ 33136.0 1.07679
$$983$$ 17582.0 0.570477 0.285238 0.958457i $$-0.407927\pi$$
0.285238 + 0.958457i $$0.407927\pi$$
$$984$$ 0 0
$$985$$ 4080.00 0.131979
$$986$$ −168.000 −0.00542618
$$987$$ 0 0
$$988$$ 1872.00 0.0602796
$$989$$ −3760.00 −0.120891
$$990$$ 0 0
$$991$$ 47904.0 1.53554 0.767770 0.640725i $$-0.221366\pi$$
0.767770 + 0.640725i $$0.221366\pi$$
$$992$$ 4864.00 0.155678
$$993$$ 0 0
$$994$$ 7408.00 0.236386
$$995$$ 13024.0 0.414963
$$996$$ 0 0
$$997$$ −44578.0 −1.41605 −0.708024 0.706189i $$-0.750413\pi$$
−0.708024 + 0.706189i $$0.750413\pi$$
$$998$$ 20272.0 0.642985
$$999$$ 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 234.4.a.c.1.1 1
3.2 odd 2 78.4.a.f.1.1 1
4.3 odd 2 1872.4.a.f.1.1 1
12.11 even 2 624.4.a.c.1.1 1
15.14 odd 2 1950.4.a.a.1.1 1
24.5 odd 2 2496.4.a.c.1.1 1
24.11 even 2 2496.4.a.l.1.1 1
39.5 even 4 1014.4.b.g.337.1 2
39.8 even 4 1014.4.b.g.337.2 2
39.38 odd 2 1014.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.f.1.1 1 3.2 odd 2
234.4.a.c.1.1 1 1.1 even 1 trivial
624.4.a.c.1.1 1 12.11 even 2
1014.4.a.e.1.1 1 39.38 odd 2
1014.4.b.g.337.1 2 39.5 even 4
1014.4.b.g.337.2 2 39.8 even 4
1872.4.a.f.1.1 1 4.3 odd 2
1950.4.a.a.1.1 1 15.14 odd 2
2496.4.a.c.1.1 1 24.5 odd 2
2496.4.a.l.1.1 1 24.11 even 2