# Properties

 Label 400.4.a.i.1.1 Level $400$ Weight $4$ Character 400.1 Self dual yes Analytic conductor $23.601$ Analytic rank $0$ 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] = [400,4,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, 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("400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 400.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: $$23.6007640023$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 100) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 400.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -26.0000 q^{7} -26.0000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -26.0000 q^{7} -26.0000 q^{9} -45.0000 q^{11} +44.0000 q^{13} +117.000 q^{17} +91.0000 q^{19} +26.0000 q^{21} +18.0000 q^{23} +53.0000 q^{27} +144.000 q^{29} -26.0000 q^{31} +45.0000 q^{33} -214.000 q^{37} -44.0000 q^{39} -459.000 q^{41} +460.000 q^{43} +468.000 q^{47} +333.000 q^{49} -117.000 q^{51} +558.000 q^{53} -91.0000 q^{57} +72.0000 q^{59} -118.000 q^{61} +676.000 q^{63} -251.000 q^{67} -18.0000 q^{69} -108.000 q^{71} +299.000 q^{73} +1170.00 q^{77} +898.000 q^{79} +649.000 q^{81} -927.000 q^{83} -144.000 q^{87} +351.000 q^{89} -1144.00 q^{91} +26.0000 q^{93} +386.000 q^{97} +1170.00 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.192450 −0.0962250 0.995360i $$-0.530677\pi$$
−0.0962250 + 0.995360i $$0.530677\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −26.0000 −1.40387 −0.701934 0.712242i $$-0.747680\pi$$
−0.701934 + 0.712242i $$0.747680\pi$$
$$8$$ 0 0
$$9$$ −26.0000 −0.962963
$$10$$ 0 0
$$11$$ −45.0000 −1.23346 −0.616728 0.787177i $$-0.711542\pi$$
−0.616728 + 0.787177i $$0.711542\pi$$
$$12$$ 0 0
$$13$$ 44.0000 0.938723 0.469362 0.883006i $$-0.344484\pi$$
0.469362 + 0.883006i $$0.344484\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 117.000 1.66922 0.834608 0.550845i $$-0.185694\pi$$
0.834608 + 0.550845i $$0.185694\pi$$
$$18$$ 0 0
$$19$$ 91.0000 1.09878 0.549390 0.835566i $$-0.314860\pi$$
0.549390 + 0.835566i $$0.314860\pi$$
$$20$$ 0 0
$$21$$ 26.0000 0.270175
$$22$$ 0 0
$$23$$ 18.0000 0.163185 0.0815926 0.996666i $$-0.473999\pi$$
0.0815926 + 0.996666i $$0.473999\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 53.0000 0.377772
$$28$$ 0 0
$$29$$ 144.000 0.922073 0.461037 0.887381i $$-0.347478\pi$$
0.461037 + 0.887381i $$0.347478\pi$$
$$30$$ 0 0
$$31$$ −26.0000 −0.150637 −0.0753184 0.997160i $$-0.523997\pi$$
−0.0753184 + 0.997160i $$0.523997\pi$$
$$32$$ 0 0
$$33$$ 45.0000 0.237379
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −214.000 −0.950848 −0.475424 0.879757i $$-0.657705\pi$$
−0.475424 + 0.879757i $$0.657705\pi$$
$$38$$ 0 0
$$39$$ −44.0000 −0.180657
$$40$$ 0 0
$$41$$ −459.000 −1.74838 −0.874192 0.485580i $$-0.838608\pi$$
−0.874192 + 0.485580i $$0.838608\pi$$
$$42$$ 0 0
$$43$$ 460.000 1.63138 0.815690 0.578489i $$-0.196358\pi$$
0.815690 + 0.578489i $$0.196358\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 468.000 1.45244 0.726221 0.687461i $$-0.241275\pi$$
0.726221 + 0.687461i $$0.241275\pi$$
$$48$$ 0 0
$$49$$ 333.000 0.970845
$$50$$ 0 0
$$51$$ −117.000 −0.321241
$$52$$ 0 0
$$53$$ 558.000 1.44617 0.723087 0.690757i $$-0.242723\pi$$
0.723087 + 0.690757i $$0.242723\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −91.0000 −0.211460
$$58$$ 0 0
$$59$$ 72.0000 0.158875 0.0794373 0.996840i $$-0.474688\pi$$
0.0794373 + 0.996840i $$0.474688\pi$$
$$60$$ 0 0
$$61$$ −118.000 −0.247678 −0.123839 0.992302i $$-0.539521\pi$$
−0.123839 + 0.992302i $$0.539521\pi$$
$$62$$ 0 0
$$63$$ 676.000 1.35187
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −251.000 −0.457680 −0.228840 0.973464i $$-0.573493\pi$$
−0.228840 + 0.973464i $$0.573493\pi$$
$$68$$ 0 0
$$69$$ −18.0000 −0.0314050
$$70$$ 0 0
$$71$$ −108.000 −0.180525 −0.0902623 0.995918i $$-0.528771\pi$$
−0.0902623 + 0.995918i $$0.528771\pi$$
$$72$$ 0 0
$$73$$ 299.000 0.479388 0.239694 0.970849i $$-0.422953\pi$$
0.239694 + 0.970849i $$0.422953\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1170.00 1.73161
$$78$$ 0 0
$$79$$ 898.000 1.27890 0.639449 0.768834i $$-0.279163\pi$$
0.639449 + 0.768834i $$0.279163\pi$$
$$80$$ 0 0
$$81$$ 649.000 0.890261
$$82$$ 0 0
$$83$$ −927.000 −1.22592 −0.612961 0.790113i $$-0.710022\pi$$
−0.612961 + 0.790113i $$0.710022\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −144.000 −0.177453
$$88$$ 0 0
$$89$$ 351.000 0.418044 0.209022 0.977911i $$-0.432972\pi$$
0.209022 + 0.977911i $$0.432972\pi$$
$$90$$ 0 0
$$91$$ −1144.00 −1.31784
$$92$$ 0 0
$$93$$ 26.0000 0.0289900
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 386.000 0.404045 0.202022 0.979381i $$-0.435249\pi$$
0.202022 + 0.979381i $$0.435249\pi$$
$$98$$ 0 0
$$99$$ 1170.00 1.18777
$$100$$ 0 0
$$101$$ −954.000 −0.939867 −0.469933 0.882702i $$-0.655722\pi$$
−0.469933 + 0.882702i $$0.655722\pi$$
$$102$$ 0 0
$$103$$ 772.000 0.738519 0.369259 0.929326i $$-0.379611\pi$$
0.369259 + 0.929326i $$0.379611\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1197.00 −1.08148 −0.540740 0.841190i $$-0.681856\pi$$
−0.540740 + 0.841190i $$0.681856\pi$$
$$108$$ 0 0
$$109$$ −802.000 −0.704749 −0.352375 0.935859i $$-0.614626\pi$$
−0.352375 + 0.935859i $$0.614626\pi$$
$$110$$ 0 0
$$111$$ 214.000 0.182991
$$112$$ 0 0
$$113$$ 1143.00 0.951543 0.475772 0.879569i $$-0.342169\pi$$
0.475772 + 0.879569i $$0.342169\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1144.00 −0.903956
$$118$$ 0 0
$$119$$ −3042.00 −2.34336
$$120$$ 0 0
$$121$$ 694.000 0.521412
$$122$$ 0 0
$$123$$ 459.000 0.336477
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2374.00 1.65873 0.829364 0.558709i $$-0.188703\pi$$
0.829364 + 0.558709i $$0.188703\pi$$
$$128$$ 0 0
$$129$$ −460.000 −0.313959
$$130$$ 0 0
$$131$$ 1260.00 0.840357 0.420178 0.907442i $$-0.361968\pi$$
0.420178 + 0.907442i $$0.361968\pi$$
$$132$$ 0 0
$$133$$ −2366.00 −1.54254
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −891.000 −0.555644 −0.277822 0.960633i $$-0.589613\pi$$
−0.277822 + 0.960633i $$0.589613\pi$$
$$138$$ 0 0
$$139$$ −389.000 −0.237371 −0.118685 0.992932i $$-0.537868\pi$$
−0.118685 + 0.992932i $$0.537868\pi$$
$$140$$ 0 0
$$141$$ −468.000 −0.279523
$$142$$ 0 0
$$143$$ −1980.00 −1.15787
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −333.000 −0.186839
$$148$$ 0 0
$$149$$ 1296.00 0.712567 0.356283 0.934378i $$-0.384044\pi$$
0.356283 + 0.934378i $$0.384044\pi$$
$$150$$ 0 0
$$151$$ 2710.00 1.46051 0.730254 0.683176i $$-0.239402\pi$$
0.730254 + 0.683176i $$0.239402\pi$$
$$152$$ 0 0
$$153$$ −3042.00 −1.60739
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −1846.00 −0.938388 −0.469194 0.883095i $$-0.655455\pi$$
−0.469194 + 0.883095i $$0.655455\pi$$
$$158$$ 0 0
$$159$$ −558.000 −0.278316
$$160$$ 0 0
$$161$$ −468.000 −0.229090
$$162$$ 0 0
$$163$$ −1475.00 −0.708779 −0.354389 0.935098i $$-0.615311\pi$$
−0.354389 + 0.935098i $$0.615311\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1476.00 0.683930 0.341965 0.939713i $$-0.388908\pi$$
0.341965 + 0.939713i $$0.388908\pi$$
$$168$$ 0 0
$$169$$ −261.000 −0.118798
$$170$$ 0 0
$$171$$ −2366.00 −1.05809
$$172$$ 0 0
$$173$$ 1368.00 0.601197 0.300599 0.953751i $$-0.402814\pi$$
0.300599 + 0.953751i $$0.402814\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −72.0000 −0.0305754
$$178$$ 0 0
$$179$$ 1503.00 0.627595 0.313797 0.949490i $$-0.398399\pi$$
0.313797 + 0.949490i $$0.398399\pi$$
$$180$$ 0 0
$$181$$ 3770.00 1.54819 0.774094 0.633071i $$-0.218206\pi$$
0.774094 + 0.633071i $$0.218206\pi$$
$$182$$ 0 0
$$183$$ 118.000 0.0476656
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −5265.00 −2.05890
$$188$$ 0 0
$$189$$ −1378.00 −0.530343
$$190$$ 0 0
$$191$$ 4122.00 1.56156 0.780779 0.624808i $$-0.214823\pi$$
0.780779 + 0.624808i $$0.214823\pi$$
$$192$$ 0 0
$$193$$ −1963.00 −0.732123 −0.366062 0.930591i $$-0.619294\pi$$
−0.366062 + 0.930591i $$0.619294\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2934.00 1.06111 0.530555 0.847650i $$-0.321983\pi$$
0.530555 + 0.847650i $$0.321983\pi$$
$$198$$ 0 0
$$199$$ −1412.00 −0.502985 −0.251493 0.967859i $$-0.580921\pi$$
−0.251493 + 0.967859i $$0.580921\pi$$
$$200$$ 0 0
$$201$$ 251.000 0.0880805
$$202$$ 0 0
$$203$$ −3744.00 −1.29447
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −468.000 −0.157141
$$208$$ 0 0
$$209$$ −4095.00 −1.35530
$$210$$ 0 0
$$211$$ −3419.00 −1.11552 −0.557758 0.830004i $$-0.688338\pi$$
−0.557758 + 0.830004i $$0.688338\pi$$
$$212$$ 0 0
$$213$$ 108.000 0.0347420
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 676.000 0.211474
$$218$$ 0 0
$$219$$ −299.000 −0.0922582
$$220$$ 0 0
$$221$$ 5148.00 1.56693
$$222$$ 0 0
$$223$$ 100.000 0.0300291 0.0150146 0.999887i $$-0.495221\pi$$
0.0150146 + 0.999887i $$0.495221\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −4212.00 −1.23154 −0.615771 0.787925i $$-0.711156\pi$$
−0.615771 + 0.787925i $$0.711156\pi$$
$$228$$ 0 0
$$229$$ −3484.00 −1.00537 −0.502684 0.864470i $$-0.667654\pi$$
−0.502684 + 0.864470i $$0.667654\pi$$
$$230$$ 0 0
$$231$$ −1170.00 −0.333248
$$232$$ 0 0
$$233$$ 918.000 0.258112 0.129056 0.991637i $$-0.458805\pi$$
0.129056 + 0.991637i $$0.458805\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −898.000 −0.246124
$$238$$ 0 0
$$239$$ −3744.00 −1.01330 −0.506651 0.862151i $$-0.669117\pi$$
−0.506651 + 0.862151i $$0.669117\pi$$
$$240$$ 0 0
$$241$$ −4231.00 −1.13088 −0.565441 0.824789i $$-0.691294\pi$$
−0.565441 + 0.824789i $$0.691294\pi$$
$$242$$ 0 0
$$243$$ −2080.00 −0.549103
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4004.00 1.03145
$$248$$ 0 0
$$249$$ 927.000 0.235929
$$250$$ 0 0
$$251$$ 2925.00 0.735555 0.367778 0.929914i $$-0.380119\pi$$
0.367778 + 0.929914i $$0.380119\pi$$
$$252$$ 0 0
$$253$$ −810.000 −0.201282
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −18.0000 −0.00436891 −0.00218445 0.999998i $$-0.500695\pi$$
−0.00218445 + 0.999998i $$0.500695\pi$$
$$258$$ 0 0
$$259$$ 5564.00 1.33487
$$260$$ 0 0
$$261$$ −3744.00 −0.887923
$$262$$ 0 0
$$263$$ 6786.00 1.59104 0.795518 0.605929i $$-0.207199\pi$$
0.795518 + 0.605929i $$0.207199\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −351.000 −0.0804526
$$268$$ 0 0
$$269$$ 7632.00 1.72986 0.864928 0.501896i $$-0.167364\pi$$
0.864928 + 0.501896i $$0.167364\pi$$
$$270$$ 0 0
$$271$$ −650.000 −0.145700 −0.0728500 0.997343i $$-0.523209\pi$$
−0.0728500 + 0.997343i $$0.523209\pi$$
$$272$$ 0 0
$$273$$ 1144.00 0.253619
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3232.00 −0.701054 −0.350527 0.936553i $$-0.613998\pi$$
−0.350527 + 0.936553i $$0.613998\pi$$
$$278$$ 0 0
$$279$$ 676.000 0.145058
$$280$$ 0 0
$$281$$ 4446.00 0.943865 0.471933 0.881635i $$-0.343557\pi$$
0.471933 + 0.881635i $$0.343557\pi$$
$$282$$ 0 0
$$283$$ −2483.00 −0.521551 −0.260776 0.965399i $$-0.583978\pi$$
−0.260776 + 0.965399i $$0.583978\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 11934.0 2.45450
$$288$$ 0 0
$$289$$ 8776.00 1.78628
$$290$$ 0 0
$$291$$ −386.000 −0.0777585
$$292$$ 0 0
$$293$$ −4050.00 −0.807521 −0.403760 0.914865i $$-0.632297\pi$$
−0.403760 + 0.914865i $$0.632297\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2385.00 −0.465965
$$298$$ 0 0
$$299$$ 792.000 0.153186
$$300$$ 0 0
$$301$$ −11960.0 −2.29024
$$302$$ 0 0
$$303$$ 954.000 0.180877
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2321.00 −0.431487 −0.215743 0.976450i $$-0.569217\pi$$
−0.215743 + 0.976450i $$0.569217\pi$$
$$308$$ 0 0
$$309$$ −772.000 −0.142128
$$310$$ 0 0
$$311$$ 3258.00 0.594033 0.297016 0.954872i $$-0.404008\pi$$
0.297016 + 0.954872i $$0.404008\pi$$
$$312$$ 0 0
$$313$$ 3626.00 0.654804 0.327402 0.944885i $$-0.393827\pi$$
0.327402 + 0.944885i $$0.393827\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3852.00 0.682492 0.341246 0.939974i $$-0.389151\pi$$
0.341246 + 0.939974i $$0.389151\pi$$
$$318$$ 0 0
$$319$$ −6480.00 −1.13734
$$320$$ 0 0
$$321$$ 1197.00 0.208131
$$322$$ 0 0
$$323$$ 10647.0 1.83410
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 802.000 0.135629
$$328$$ 0 0
$$329$$ −12168.0 −2.03904
$$330$$ 0 0
$$331$$ −7553.00 −1.25423 −0.627115 0.778926i $$-0.715765\pi$$
−0.627115 + 0.778926i $$0.715765\pi$$
$$332$$ 0 0
$$333$$ 5564.00 0.915632
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −109.000 −0.0176190 −0.00880951 0.999961i $$-0.502804\pi$$
−0.00880951 + 0.999961i $$0.502804\pi$$
$$338$$ 0 0
$$339$$ −1143.00 −0.183125
$$340$$ 0 0
$$341$$ 1170.00 0.185804
$$342$$ 0 0
$$343$$ 260.000 0.0409291
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2835.00 0.438590 0.219295 0.975659i $$-0.429624\pi$$
0.219295 + 0.975659i $$0.429624\pi$$
$$348$$ 0 0
$$349$$ 2990.00 0.458599 0.229299 0.973356i $$-0.426356\pi$$
0.229299 + 0.973356i $$0.426356\pi$$
$$350$$ 0 0
$$351$$ 2332.00 0.354624
$$352$$ 0 0
$$353$$ 9126.00 1.37600 0.688000 0.725711i $$-0.258489\pi$$
0.688000 + 0.725711i $$0.258489\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 3042.00 0.450980
$$358$$ 0 0
$$359$$ 9594.00 1.41045 0.705226 0.708983i $$-0.250846\pi$$
0.705226 + 0.708983i $$0.250846\pi$$
$$360$$ 0 0
$$361$$ 1422.00 0.207319
$$362$$ 0 0
$$363$$ −694.000 −0.100346
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −9764.00 −1.38876 −0.694382 0.719606i $$-0.744322\pi$$
−0.694382 + 0.719606i $$0.744322\pi$$
$$368$$ 0 0
$$369$$ 11934.0 1.68363
$$370$$ 0 0
$$371$$ −14508.0 −2.03024
$$372$$ 0 0
$$373$$ 6722.00 0.933115 0.466558 0.884491i $$-0.345494\pi$$
0.466558 + 0.884491i $$0.345494\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6336.00 0.865572
$$378$$ 0 0
$$379$$ 13537.0 1.83469 0.917347 0.398089i $$-0.130326\pi$$
0.917347 + 0.398089i $$0.130326\pi$$
$$380$$ 0 0
$$381$$ −2374.00 −0.319222
$$382$$ 0 0
$$383$$ 8658.00 1.15510 0.577550 0.816355i $$-0.304009\pi$$
0.577550 + 0.816355i $$0.304009\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −11960.0 −1.57096
$$388$$ 0 0
$$389$$ −8874.00 −1.15663 −0.578316 0.815813i $$-0.696290\pi$$
−0.578316 + 0.815813i $$0.696290\pi$$
$$390$$ 0 0
$$391$$ 2106.00 0.272391
$$392$$ 0 0
$$393$$ −1260.00 −0.161727
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 5876.00 0.742841 0.371421 0.928465i $$-0.378871\pi$$
0.371421 + 0.928465i $$0.378871\pi$$
$$398$$ 0 0
$$399$$ 2366.00 0.296863
$$400$$ 0 0
$$401$$ −1755.00 −0.218555 −0.109277 0.994011i $$-0.534854\pi$$
−0.109277 + 0.994011i $$0.534854\pi$$
$$402$$ 0 0
$$403$$ −1144.00 −0.141406
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9630.00 1.17283
$$408$$ 0 0
$$409$$ 4589.00 0.554796 0.277398 0.960755i $$-0.410528\pi$$
0.277398 + 0.960755i $$0.410528\pi$$
$$410$$ 0 0
$$411$$ 891.000 0.106934
$$412$$ 0 0
$$413$$ −1872.00 −0.223039
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 389.000 0.0456820
$$418$$ 0 0
$$419$$ −5409.00 −0.630661 −0.315330 0.948982i $$-0.602115\pi$$
−0.315330 + 0.948982i $$0.602115\pi$$
$$420$$ 0 0
$$421$$ 12116.0 1.40261 0.701304 0.712863i $$-0.252602\pi$$
0.701304 + 0.712863i $$0.252602\pi$$
$$422$$ 0 0
$$423$$ −12168.0 −1.39865
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3068.00 0.347707
$$428$$ 0 0
$$429$$ 1980.00 0.222833
$$430$$ 0 0
$$431$$ −9126.00 −1.01992 −0.509958 0.860199i $$-0.670339\pi$$
−0.509958 + 0.860199i $$0.670339\pi$$
$$432$$ 0 0
$$433$$ 629.000 0.0698102 0.0349051 0.999391i $$-0.488887\pi$$
0.0349051 + 0.999391i $$0.488887\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1638.00 0.179305
$$438$$ 0 0
$$439$$ −4472.00 −0.486189 −0.243094 0.970003i $$-0.578162\pi$$
−0.243094 + 0.970003i $$0.578162\pi$$
$$440$$ 0 0
$$441$$ −8658.00 −0.934888
$$442$$ 0 0
$$443$$ 3393.00 0.363897 0.181948 0.983308i $$-0.441760\pi$$
0.181948 + 0.983308i $$0.441760\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −1296.00 −0.137134
$$448$$ 0 0
$$449$$ −5031.00 −0.528792 −0.264396 0.964414i $$-0.585173\pi$$
−0.264396 + 0.964414i $$0.585173\pi$$
$$450$$ 0 0
$$451$$ 20655.0 2.15655
$$452$$ 0 0
$$453$$ −2710.00 −0.281075
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6487.00 −0.664002 −0.332001 0.943279i $$-0.607724\pi$$
−0.332001 + 0.943279i $$0.607724\pi$$
$$458$$ 0 0
$$459$$ 6201.00 0.630584
$$460$$ 0 0
$$461$$ 2700.00 0.272780 0.136390 0.990655i $$-0.456450\pi$$
0.136390 + 0.990655i $$0.456450\pi$$
$$462$$ 0 0
$$463$$ 2932.00 0.294302 0.147151 0.989114i $$-0.452990\pi$$
0.147151 + 0.989114i $$0.452990\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 15660.0 1.55173 0.775866 0.630898i $$-0.217314\pi$$
0.775866 + 0.630898i $$0.217314\pi$$
$$468$$ 0 0
$$469$$ 6526.00 0.642522
$$470$$ 0 0
$$471$$ 1846.00 0.180593
$$472$$ 0 0
$$473$$ −20700.0 −2.01223
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −14508.0 −1.39261
$$478$$ 0 0
$$479$$ −10134.0 −0.966669 −0.483334 0.875436i $$-0.660574\pi$$
−0.483334 + 0.875436i $$0.660574\pi$$
$$480$$ 0 0
$$481$$ −9416.00 −0.892583
$$482$$ 0 0
$$483$$ 468.000 0.0440885
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −1898.00 −0.176605 −0.0883025 0.996094i $$-0.528144\pi$$
−0.0883025 + 0.996094i $$0.528144\pi$$
$$488$$ 0 0
$$489$$ 1475.00 0.136405
$$490$$ 0 0
$$491$$ 6300.00 0.579053 0.289526 0.957170i $$-0.406502\pi$$
0.289526 + 0.957170i $$0.406502\pi$$
$$492$$ 0 0
$$493$$ 16848.0 1.53914
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2808.00 0.253433
$$498$$ 0 0
$$499$$ −18044.0 −1.61876 −0.809379 0.587286i $$-0.800196\pi$$
−0.809379 + 0.587286i $$0.800196\pi$$
$$500$$ 0 0
$$501$$ −1476.00 −0.131622
$$502$$ 0 0
$$503$$ 6876.00 0.609514 0.304757 0.952430i $$-0.401425\pi$$
0.304757 + 0.952430i $$0.401425\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 261.000 0.0228628
$$508$$ 0 0
$$509$$ −4806.00 −0.418511 −0.209256 0.977861i $$-0.567104\pi$$
−0.209256 + 0.977861i $$0.567104\pi$$
$$510$$ 0 0
$$511$$ −7774.00 −0.672997
$$512$$ 0 0
$$513$$ 4823.00 0.415089
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −21060.0 −1.79152
$$518$$ 0 0
$$519$$ −1368.00 −0.115700
$$520$$ 0 0
$$521$$ 7749.00 0.651612 0.325806 0.945437i $$-0.394364\pi$$
0.325806 + 0.945437i $$0.394364\pi$$
$$522$$ 0 0
$$523$$ −8153.00 −0.681655 −0.340828 0.940126i $$-0.610707\pi$$
−0.340828 + 0.940126i $$0.610707\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −3042.00 −0.251445
$$528$$ 0 0
$$529$$ −11843.0 −0.973371
$$530$$ 0 0
$$531$$ −1872.00 −0.152990
$$532$$ 0 0
$$533$$ −20196.0 −1.64125
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −1503.00 −0.120781
$$538$$ 0 0
$$539$$ −14985.0 −1.19749
$$540$$ 0 0
$$541$$ −10576.0 −0.840476 −0.420238 0.907414i $$-0.638053\pi$$
−0.420238 + 0.907414i $$0.638053\pi$$
$$542$$ 0 0
$$543$$ −3770.00 −0.297949
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −7553.00 −0.590389 −0.295195 0.955437i $$-0.595385\pi$$
−0.295195 + 0.955437i $$0.595385\pi$$
$$548$$ 0 0
$$549$$ 3068.00 0.238505
$$550$$ 0 0
$$551$$ 13104.0 1.01316
$$552$$ 0 0
$$553$$ −23348.0 −1.79540
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 13500.0 1.02695 0.513477 0.858103i $$-0.328357\pi$$
0.513477 + 0.858103i $$0.328357\pi$$
$$558$$ 0 0
$$559$$ 20240.0 1.53141
$$560$$ 0 0
$$561$$ 5265.00 0.396236
$$562$$ 0 0
$$563$$ −23184.0 −1.73550 −0.867752 0.496997i $$-0.834436\pi$$
−0.867752 + 0.496997i $$0.834436\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −16874.0 −1.24981
$$568$$ 0 0
$$569$$ −8055.00 −0.593468 −0.296734 0.954960i $$-0.595897\pi$$
−0.296734 + 0.954960i $$0.595897\pi$$
$$570$$ 0 0
$$571$$ −3068.00 −0.224854 −0.112427 0.993660i $$-0.535862\pi$$
−0.112427 + 0.993660i $$0.535862\pi$$
$$572$$ 0 0
$$573$$ −4122.00 −0.300522
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 12419.0 0.896031 0.448015 0.894026i $$-0.352131\pi$$
0.448015 + 0.894026i $$0.352131\pi$$
$$578$$ 0 0
$$579$$ 1963.00 0.140897
$$580$$ 0 0
$$581$$ 24102.0 1.72103
$$582$$ 0 0
$$583$$ −25110.0 −1.78379
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12393.0 0.871403 0.435702 0.900091i $$-0.356500\pi$$
0.435702 + 0.900091i $$0.356500\pi$$
$$588$$ 0 0
$$589$$ −2366.00 −0.165517
$$590$$ 0 0
$$591$$ −2934.00 −0.204211
$$592$$ 0 0
$$593$$ 23751.0 1.64475 0.822375 0.568946i $$-0.192649\pi$$
0.822375 + 0.568946i $$0.192649\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1412.00 0.0967995
$$598$$ 0 0
$$599$$ 11610.0 0.791939 0.395970 0.918264i $$-0.370409\pi$$
0.395970 + 0.918264i $$0.370409\pi$$
$$600$$ 0 0
$$601$$ 26675.0 1.81048 0.905238 0.424905i $$-0.139693\pi$$
0.905238 + 0.424905i $$0.139693\pi$$
$$602$$ 0 0
$$603$$ 6526.00 0.440728
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −17264.0 −1.15441 −0.577203 0.816601i $$-0.695856\pi$$
−0.577203 + 0.816601i $$0.695856\pi$$
$$608$$ 0 0
$$609$$ 3744.00 0.249121
$$610$$ 0 0
$$611$$ 20592.0 1.36344
$$612$$ 0 0
$$613$$ −26026.0 −1.71481 −0.857406 0.514640i $$-0.827926\pi$$
−0.857406 + 0.514640i $$0.827926\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 5022.00 0.327679 0.163840 0.986487i $$-0.447612\pi$$
0.163840 + 0.986487i $$0.447612\pi$$
$$618$$ 0 0
$$619$$ −7820.00 −0.507774 −0.253887 0.967234i $$-0.581709\pi$$
−0.253887 + 0.967234i $$0.581709\pi$$
$$620$$ 0 0
$$621$$ 954.000 0.0616469
$$622$$ 0 0
$$623$$ −9126.00 −0.586879
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 4095.00 0.260827
$$628$$ 0 0
$$629$$ −25038.0 −1.58717
$$630$$ 0 0
$$631$$ −15002.0 −0.946466 −0.473233 0.880937i $$-0.656913\pi$$
−0.473233 + 0.880937i $$0.656913\pi$$
$$632$$ 0 0
$$633$$ 3419.00 0.214681
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 14652.0 0.911355
$$638$$ 0 0
$$639$$ 2808.00 0.173838
$$640$$ 0 0
$$641$$ −918.000 −0.0565660 −0.0282830 0.999600i $$-0.509004\pi$$
−0.0282830 + 0.999600i $$0.509004\pi$$
$$642$$ 0 0
$$643$$ 23452.0 1.43835 0.719173 0.694831i $$-0.244521\pi$$
0.719173 + 0.694831i $$0.244521\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −20556.0 −1.24906 −0.624528 0.781002i $$-0.714709\pi$$
−0.624528 + 0.781002i $$0.714709\pi$$
$$648$$ 0 0
$$649$$ −3240.00 −0.195965
$$650$$ 0 0
$$651$$ −676.000 −0.0406982
$$652$$ 0 0
$$653$$ 30654.0 1.83703 0.918517 0.395381i $$-0.129387\pi$$
0.918517 + 0.395381i $$0.129387\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −7774.00 −0.461633
$$658$$ 0 0
$$659$$ −8919.00 −0.527215 −0.263608 0.964630i $$-0.584912\pi$$
−0.263608 + 0.964630i $$0.584912\pi$$
$$660$$ 0 0
$$661$$ −22912.0 −1.34822 −0.674110 0.738631i $$-0.735473\pi$$
−0.674110 + 0.738631i $$0.735473\pi$$
$$662$$ 0 0
$$663$$ −5148.00 −0.301556
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2592.00 0.150469
$$668$$ 0 0
$$669$$ −100.000 −0.00577911
$$670$$ 0 0
$$671$$ 5310.00 0.305500
$$672$$ 0 0
$$673$$ −1222.00 −0.0699920 −0.0349960 0.999387i $$-0.511142\pi$$
−0.0349960 + 0.999387i $$0.511142\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 144.000 0.00817484 0.00408742 0.999992i $$-0.498699\pi$$
0.00408742 + 0.999992i $$0.498699\pi$$
$$678$$ 0 0
$$679$$ −10036.0 −0.567226
$$680$$ 0 0
$$681$$ 4212.00 0.237011
$$682$$ 0 0
$$683$$ 12519.0 0.701356 0.350678 0.936496i $$-0.385951\pi$$
0.350678 + 0.936496i $$0.385951\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 3484.00 0.193483
$$688$$ 0 0
$$689$$ 24552.0 1.35756
$$690$$ 0 0
$$691$$ −11873.0 −0.653647 −0.326824 0.945085i $$-0.605978\pi$$
−0.326824 + 0.945085i $$0.605978\pi$$
$$692$$ 0 0
$$693$$ −30420.0 −1.66748
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −53703.0 −2.91843
$$698$$ 0 0
$$699$$ −918.000 −0.0496737
$$700$$ 0 0
$$701$$ −3060.00 −0.164871 −0.0824355 0.996596i $$-0.526270\pi$$
−0.0824355 + 0.996596i $$0.526270\pi$$
$$702$$ 0 0
$$703$$ −19474.0 −1.04477
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 24804.0 1.31945
$$708$$ 0 0
$$709$$ 4004.00 0.212092 0.106046 0.994361i $$-0.466181\pi$$
0.106046 + 0.994361i $$0.466181\pi$$
$$710$$ 0 0
$$711$$ −23348.0 −1.23153
$$712$$ 0 0
$$713$$ −468.000 −0.0245817
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 3744.00 0.195010
$$718$$ 0 0
$$719$$ −10314.0 −0.534975 −0.267488 0.963561i $$-0.586193\pi$$
−0.267488 + 0.963561i $$0.586193\pi$$
$$720$$ 0 0
$$721$$ −20072.0 −1.03678
$$722$$ 0 0
$$723$$ 4231.00 0.217638
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −9872.00 −0.503621 −0.251810 0.967777i $$-0.581026\pi$$
−0.251810 + 0.967777i $$0.581026\pi$$
$$728$$ 0 0
$$729$$ −15443.0 −0.784586
$$730$$ 0 0
$$731$$ 53820.0 2.72313
$$732$$ 0 0
$$733$$ 7436.00 0.374700 0.187350 0.982293i $$-0.440010\pi$$
0.187350 + 0.982293i $$0.440010\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 11295.0 0.564527
$$738$$ 0 0
$$739$$ 16900.0 0.841240 0.420620 0.907237i $$-0.361813\pi$$
0.420620 + 0.907237i $$0.361813\pi$$
$$740$$ 0 0
$$741$$ −4004.00 −0.198503
$$742$$ 0 0
$$743$$ 23058.0 1.13851 0.569257 0.822160i $$-0.307231\pi$$
0.569257 + 0.822160i $$0.307231\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 24102.0 1.18052
$$748$$ 0 0
$$749$$ 31122.0 1.51826
$$750$$ 0 0
$$751$$ 8224.00 0.399598 0.199799 0.979837i $$-0.435971\pi$$
0.199799 + 0.979837i $$0.435971\pi$$
$$752$$ 0 0
$$753$$ −2925.00 −0.141558
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −7696.00 −0.369506 −0.184753 0.982785i $$-0.559148\pi$$
−0.184753 + 0.982785i $$0.559148\pi$$
$$758$$ 0 0
$$759$$ 810.000 0.0387367
$$760$$ 0 0
$$761$$ −6363.00 −0.303099 −0.151550 0.988450i $$-0.548426\pi$$
−0.151550 + 0.988450i $$0.548426\pi$$
$$762$$ 0 0
$$763$$ 20852.0 0.989375
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3168.00 0.149139
$$768$$ 0 0
$$769$$ 8333.00 0.390762 0.195381 0.980727i $$-0.437406\pi$$
0.195381 + 0.980727i $$0.437406\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.000840797 0
$$772$$ 0 0
$$773$$ 32760.0 1.52431 0.762157 0.647392i $$-0.224140\pi$$
0.762157 + 0.647392i $$0.224140\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −5564.00 −0.256895
$$778$$ 0 0
$$779$$ −41769.0 −1.92109
$$780$$ 0 0
$$781$$ 4860.00 0.222669
$$782$$ 0 0
$$783$$ 7632.00 0.348334
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 43732.0 1.98078 0.990392 0.138286i $$-0.0441595\pi$$
0.990392 + 0.138286i $$0.0441595\pi$$
$$788$$ 0 0
$$789$$ −6786.00 −0.306195
$$790$$ 0 0
$$791$$ −29718.0 −1.33584
$$792$$ 0 0
$$793$$ −5192.00 −0.232501
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 16866.0 0.749591 0.374796 0.927107i $$-0.377713\pi$$
0.374796 + 0.927107i $$0.377713\pi$$
$$798$$ 0 0
$$799$$ 54756.0 2.42444
$$800$$ 0 0
$$801$$ −9126.00 −0.402561
$$802$$ 0 0
$$803$$ −13455.0 −0.591303
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −7632.00 −0.332911
$$808$$ 0 0
$$809$$ 16146.0 0.701685 0.350842 0.936434i $$-0.385895\pi$$
0.350842 + 0.936434i $$0.385895\pi$$
$$810$$ 0 0
$$811$$ −32444.0 −1.40476 −0.702382 0.711801i $$-0.747880\pi$$
−0.702382 + 0.711801i $$0.747880\pi$$
$$812$$ 0 0
$$813$$ 650.000 0.0280400
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 41860.0 1.79253
$$818$$ 0 0
$$819$$ 29744.0 1.26903
$$820$$ 0 0
$$821$$ −2574.00 −0.109419 −0.0547096 0.998502i $$-0.517423\pi$$
−0.0547096 + 0.998502i $$0.517423\pi$$
$$822$$ 0 0
$$823$$ 27604.0 1.16916 0.584578 0.811338i $$-0.301260\pi$$
0.584578 + 0.811338i $$0.301260\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −11655.0 −0.490065 −0.245033 0.969515i $$-0.578799\pi$$
−0.245033 + 0.969515i $$0.578799\pi$$
$$828$$ 0 0
$$829$$ 33428.0 1.40049 0.700243 0.713905i $$-0.253075\pi$$
0.700243 + 0.713905i $$0.253075\pi$$
$$830$$ 0 0
$$831$$ 3232.00 0.134918
$$832$$ 0 0
$$833$$ 38961.0 1.62055
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1378.00 −0.0569064
$$838$$ 0 0
$$839$$ 17712.0 0.728827 0.364414 0.931237i $$-0.381269\pi$$
0.364414 + 0.931237i $$0.381269\pi$$
$$840$$ 0 0
$$841$$ −3653.00 −0.149781
$$842$$ 0 0
$$843$$ −4446.00 −0.181647
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −18044.0 −0.731994
$$848$$ 0 0
$$849$$ 2483.00 0.100373
$$850$$ 0 0
$$851$$ −3852.00 −0.155164
$$852$$ 0 0
$$853$$ −10270.0 −0.412237 −0.206118 0.978527i $$-0.566083\pi$$
−0.206118 + 0.978527i $$0.566083\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 38709.0 1.54291 0.771455 0.636284i $$-0.219529\pi$$
0.771455 + 0.636284i $$0.219529\pi$$
$$858$$ 0 0
$$859$$ −15509.0 −0.616019 −0.308009 0.951383i $$-0.599663\pi$$
−0.308009 + 0.951383i $$0.599663\pi$$
$$860$$ 0 0
$$861$$ −11934.0 −0.472369
$$862$$ 0 0
$$863$$ −15912.0 −0.627637 −0.313819 0.949483i $$-0.601608\pi$$
−0.313819 + 0.949483i $$0.601608\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −8776.00 −0.343770
$$868$$ 0 0
$$869$$ −40410.0 −1.57746
$$870$$ 0 0
$$871$$ −11044.0 −0.429635
$$872$$ 0 0
$$873$$ −10036.0 −0.389080
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −10972.0 −0.422461 −0.211230 0.977436i $$-0.567747\pi$$
−0.211230 + 0.977436i $$0.567747\pi$$
$$878$$ 0 0
$$879$$ 4050.00 0.155407
$$880$$ 0 0
$$881$$ −18738.0 −0.716571 −0.358286 0.933612i $$-0.616639\pi$$
−0.358286 + 0.933612i $$0.616639\pi$$
$$882$$ 0 0
$$883$$ 21367.0 0.814334 0.407167 0.913354i $$-0.366517\pi$$
0.407167 + 0.913354i $$0.366517\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −20124.0 −0.761779 −0.380889 0.924621i $$-0.624382\pi$$
−0.380889 + 0.924621i $$0.624382\pi$$
$$888$$ 0 0
$$889$$ −61724.0 −2.32864
$$890$$ 0 0
$$891$$ −29205.0 −1.09810
$$892$$ 0 0
$$893$$ 42588.0 1.59592
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −792.000 −0.0294806
$$898$$ 0 0
$$899$$ −3744.00 −0.138898
$$900$$ 0 0
$$901$$ 65286.0 2.41398
$$902$$ 0 0
$$903$$ 11960.0 0.440757
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −23132.0 −0.846842 −0.423421 0.905933i $$-0.639171\pi$$
−0.423421 + 0.905933i $$0.639171\pi$$
$$908$$ 0 0
$$909$$ 24804.0 0.905057
$$910$$ 0 0
$$911$$ −31212.0 −1.13513 −0.567563 0.823330i $$-0.692114\pi$$
−0.567563 + 0.823330i $$0.692114\pi$$
$$912$$ 0 0
$$913$$ 41715.0 1.51212
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −32760.0 −1.17975
$$918$$ 0 0
$$919$$ 6994.00 0.251045 0.125523 0.992091i $$-0.459939\pi$$
0.125523 + 0.992091i $$0.459939\pi$$
$$920$$ 0 0
$$921$$ 2321.00 0.0830397
$$922$$ 0 0
$$923$$ −4752.00 −0.169463
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −20072.0 −0.711166
$$928$$ 0 0
$$929$$ 19422.0 0.685915 0.342958 0.939351i $$-0.388571\pi$$
0.342958 + 0.939351i $$0.388571\pi$$
$$930$$ 0 0
$$931$$ 30303.0 1.06675
$$932$$ 0 0
$$933$$ −3258.00 −0.114322
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 11699.0 0.407887 0.203943 0.978983i $$-0.434624\pi$$
0.203943 + 0.978983i $$0.434624\pi$$
$$938$$ 0 0
$$939$$ −3626.00 −0.126017
$$940$$ 0 0
$$941$$ −42948.0 −1.48785 −0.743924 0.668264i $$-0.767038\pi$$
−0.743924 + 0.668264i $$0.767038\pi$$
$$942$$ 0 0
$$943$$ −8262.00 −0.285310
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 3816.00 0.130943 0.0654717 0.997854i $$-0.479145\pi$$
0.0654717 + 0.997854i $$0.479145\pi$$
$$948$$ 0 0
$$949$$ 13156.0 0.450012
$$950$$ 0 0
$$951$$ −3852.00 −0.131346
$$952$$ 0 0
$$953$$ −43407.0 −1.47544 −0.737718 0.675109i $$-0.764097\pi$$
−0.737718 + 0.675109i $$0.764097\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6480.00 0.218881
$$958$$ 0 0
$$959$$ 23166.0 0.780051
$$960$$ 0 0
$$961$$ −29115.0 −0.977309
$$962$$ 0 0
$$963$$ 31122.0 1.04143
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 43216.0 1.43716 0.718580 0.695445i $$-0.244793\pi$$
0.718580 + 0.695445i $$0.244793\pi$$
$$968$$ 0 0
$$969$$ −10647.0 −0.352973
$$970$$ 0 0
$$971$$ 47619.0 1.57381 0.786903 0.617076i $$-0.211683\pi$$
0.786903 + 0.617076i $$0.211683\pi$$
$$972$$ 0 0
$$973$$ 10114.0 0.333237
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −4671.00 −0.152957 −0.0764783 0.997071i $$-0.524368\pi$$
−0.0764783 + 0.997071i $$0.524368\pi$$
$$978$$ 0 0
$$979$$ −15795.0 −0.515639
$$980$$ 0 0
$$981$$ 20852.0 0.678647
$$982$$ 0 0
$$983$$ −9054.00 −0.293772 −0.146886 0.989153i $$-0.546925\pi$$
−0.146886 + 0.989153i $$0.546925\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 12168.0 0.392413
$$988$$ 0 0
$$989$$ 8280.00 0.266217
$$990$$ 0 0
$$991$$ −8126.00 −0.260475 −0.130238 0.991483i $$-0.541574\pi$$
−0.130238 + 0.991483i $$0.541574\pi$$
$$992$$ 0 0
$$993$$ 7553.00 0.241377
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 38468.0 1.22196 0.610980 0.791646i $$-0.290776\pi$$
0.610980 + 0.791646i $$0.290776\pi$$
$$998$$ 0 0
$$999$$ −11342.0 −0.359204
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 400.4.a.i.1.1 1
4.3 odd 2 100.4.a.c.1.1 yes 1
5.2 odd 4 400.4.c.l.49.2 2
5.3 odd 4 400.4.c.l.49.1 2
5.4 even 2 400.4.a.l.1.1 1
8.3 odd 2 1600.4.a.x.1.1 1
8.5 even 2 1600.4.a.bd.1.1 1
12.11 even 2 900.4.a.p.1.1 1
20.3 even 4 100.4.c.b.49.2 2
20.7 even 4 100.4.c.b.49.1 2
20.19 odd 2 100.4.a.b.1.1 1
40.19 odd 2 1600.4.a.bc.1.1 1
40.29 even 2 1600.4.a.y.1.1 1
60.23 odd 4 900.4.d.a.649.1 2
60.47 odd 4 900.4.d.a.649.2 2
60.59 even 2 900.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
100.4.a.b.1.1 1 20.19 odd 2
100.4.a.c.1.1 yes 1 4.3 odd 2
100.4.c.b.49.1 2 20.7 even 4
100.4.c.b.49.2 2 20.3 even 4
400.4.a.i.1.1 1 1.1 even 1 trivial
400.4.a.l.1.1 1 5.4 even 2
400.4.c.l.49.1 2 5.3 odd 4
400.4.c.l.49.2 2 5.2 odd 4
900.4.a.c.1.1 1 60.59 even 2
900.4.a.p.1.1 1 12.11 even 2
900.4.d.a.649.1 2 60.23 odd 4
900.4.d.a.649.2 2 60.47 odd 4
1600.4.a.x.1.1 1 8.3 odd 2
1600.4.a.y.1.1 1 40.29 even 2
1600.4.a.bc.1.1 1 40.19 odd 2
1600.4.a.bd.1.1 1 8.5 even 2