# Properties

 Label 23.7.b.a.22.1 Level $23$ Weight $7$ Character 23.22 Self dual yes Analytic conductor $5.291$ Analytic rank $0$ Dimension $1$ CM discriminant -23 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,7,Mod(22,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 23.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$5.29124392326$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 22.1 Character $$\chi$$ $$=$$ 23.22

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-7.00000 q^{2} -38.0000 q^{3} -15.0000 q^{4} +266.000 q^{6} +553.000 q^{8} +715.000 q^{9} +O(q^{10})$$ $$q-7.00000 q^{2} -38.0000 q^{3} -15.0000 q^{4} +266.000 q^{6} +553.000 q^{8} +715.000 q^{9} +570.000 q^{12} +1082.00 q^{13} -2911.00 q^{16} -5005.00 q^{18} -12167.0 q^{23} -21014.0 q^{24} +15625.0 q^{25} -7574.00 q^{26} +532.000 q^{27} +30746.0 q^{29} +58754.0 q^{31} -15015.0 q^{32} -10725.0 q^{36} -41116.0 q^{39} +43634.0 q^{41} +85169.0 q^{46} -205342. q^{47} +110618. q^{48} +117649. q^{49} -109375. q^{50} -16230.0 q^{52} -3724.00 q^{54} -215222. q^{58} -253942. q^{59} -411278. q^{62} +291409. q^{64} +462346. q^{69} +667154. q^{71} +395395. q^{72} +725042. q^{73} -593750. q^{75} +287812. q^{78} -541451. q^{81} -305438. q^{82} -1.16835e6 q^{87} +182505. q^{92} -2.23265e6 q^{93} +1.43739e6 q^{94} +570570. q^{96} -823543. q^{98} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −7.00000 −0.875000 −0.437500 0.899218i $$-0.644136\pi$$
−0.437500 + 0.899218i $$0.644136\pi$$
$$3$$ −38.0000 −1.40741 −0.703704 0.710494i $$-0.748472\pi$$
−0.703704 + 0.710494i $$0.748472\pi$$
$$4$$ −15.0000 −0.234375
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 266.000 1.23148
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 553.000 1.08008
$$9$$ 715.000 0.980796
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 570.000 0.329861
$$13$$ 1082.00 0.492490 0.246245 0.969208i $$-0.420803\pi$$
0.246245 + 0.969208i $$0.420803\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −2911.00 −0.710693
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ −5005.00 −0.858196
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −12167.0 −1.00000
$$24$$ −21014.0 −1.52011
$$25$$ 15625.0 1.00000
$$26$$ −7574.00 −0.430929
$$27$$ 532.000 0.0270284
$$28$$ 0 0
$$29$$ 30746.0 1.26065 0.630325 0.776331i $$-0.282922\pi$$
0.630325 + 0.776331i $$0.282922\pi$$
$$30$$ 0 0
$$31$$ 58754.0 1.97221 0.986103 0.166134i $$-0.0531284\pi$$
0.986103 + 0.166134i $$0.0531284\pi$$
$$32$$ −15015.0 −0.458221
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −10725.0 −0.229874
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ −41116.0 −0.693134
$$40$$ 0 0
$$41$$ 43634.0 0.633102 0.316551 0.948576i $$-0.397475\pi$$
0.316551 + 0.948576i $$0.397475\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 85169.0 0.875000
$$47$$ −205342. −1.97781 −0.988904 0.148555i $$-0.952538\pi$$
−0.988904 + 0.148555i $$0.952538\pi$$
$$48$$ 110618. 1.00024
$$49$$ 117649. 1.00000
$$50$$ −109375. −0.875000
$$51$$ 0 0
$$52$$ −16230.0 −0.115427
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ −3724.00 −0.0236499
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −215222. −1.10307
$$59$$ −253942. −1.23646 −0.618228 0.785999i $$-0.712149\pi$$
−0.618228 + 0.785999i $$0.712149\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ −411278. −1.72568
$$63$$ 0 0
$$64$$ 291409. 1.11164
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 462346. 1.40741
$$70$$ 0 0
$$71$$ 667154. 1.86402 0.932011 0.362430i $$-0.118053\pi$$
0.932011 + 0.362430i $$0.118053\pi$$
$$72$$ 395395. 1.05934
$$73$$ 725042. 1.86378 0.931890 0.362741i $$-0.118159\pi$$
0.931890 + 0.362741i $$0.118159\pi$$
$$74$$ 0 0
$$75$$ −593750. −1.40741
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 287812. 0.606492
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ −541451. −1.01884
$$82$$ −305438. −0.553964
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −1.16835e6 −1.77425
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 182505. 0.234375
$$93$$ −2.23265e6 −2.77570
$$94$$ 1.43739e6 1.73058
$$95$$ 0 0
$$96$$ 570570. 0.644904
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ −823543. −0.875000
$$99$$ 0 0
$$100$$ −234375. −0.234375
$$101$$ 505802. 0.490926 0.245463 0.969406i $$-0.421060\pi$$
0.245463 + 0.969406i $$0.421060\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 598346. 0.531927
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ −7980.00 −0.00633478
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −461190. −0.295465
$$117$$ 773630. 0.483032
$$118$$ 1.77759e6 1.08190
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.77156e6 1.00000
$$122$$ 0 0
$$123$$ −1.65809e6 −0.891032
$$124$$ −881310. −0.462236
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.70490e6 1.32050 0.660252 0.751044i $$-0.270449\pi$$
0.660252 + 0.751044i $$0.270449\pi$$
$$128$$ −1.07890e6 −0.514461
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 3.32143e6 1.47745 0.738723 0.674009i $$-0.235429\pi$$
0.738723 + 0.674009i $$0.235429\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ −3.23642e6 −1.23148
$$139$$ −5.20149e6 −1.93680 −0.968398 0.249411i $$-0.919763\pi$$
−0.968398 + 0.249411i $$0.919763\pi$$
$$140$$ 0 0
$$141$$ 7.80300e6 2.78358
$$142$$ −4.67008e6 −1.63102
$$143$$ 0 0
$$144$$ −2.08136e6 −0.697045
$$145$$ 0 0
$$146$$ −5.07529e6 −1.63081
$$147$$ −4.47066e6 −1.40741
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 4.15625e6 1.23148
$$151$$ 6.18955e6 1.79775 0.898873 0.438208i $$-0.144387\pi$$
0.898873 + 0.438208i $$0.144387\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 616740. 0.162453
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 3.79016e6 0.891481
$$163$$ −5.98500e6 −1.38198 −0.690989 0.722865i $$-0.742825\pi$$
−0.690989 + 0.722865i $$0.742825\pi$$
$$164$$ −654510. −0.148383
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −6.07493e6 −1.30434 −0.652171 0.758072i $$-0.726142\pi$$
−0.652171 + 0.758072i $$0.726142\pi$$
$$168$$ 0 0
$$169$$ −3.65608e6 −0.757454
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 1.96467e6 0.379446 0.189723 0.981838i $$-0.439241\pi$$
0.189723 + 0.981838i $$0.439241\pi$$
$$174$$ 8.17844e6 1.55247
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 9.64980e6 1.74020
$$178$$ 0 0
$$179$$ −3.91917e6 −0.683338 −0.341669 0.939820i $$-0.610992\pi$$
−0.341669 + 0.939820i $$0.610992\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −6.72835e6 −1.08008
$$185$$ 0 0
$$186$$ 1.56286e7 2.42874
$$187$$ 0 0
$$188$$ 3.08013e6 0.463549
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ −1.10735e7 −1.56453
$$193$$ 3.99168e6 0.555244 0.277622 0.960690i $$-0.410454\pi$$
0.277622 + 0.960690i $$0.410454\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −1.76474e6 −0.234375
$$197$$ 1.49813e7 1.95952 0.979760 0.200177i $$-0.0641517\pi$$
0.979760 + 0.200177i $$0.0641517\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 8.64062e6 1.08008
$$201$$ 0 0
$$202$$ −3.54061e6 −0.429561
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −8.69940e6 −0.980796
$$208$$ −3.14970e6 −0.350009
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −1.26968e7 −1.35160 −0.675800 0.737085i $$-0.736202\pi$$
−0.675800 + 0.737085i $$0.736202\pi$$
$$212$$ 0 0
$$213$$ −2.53519e7 −2.62344
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 294196. 0.0291928
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −2.75516e7 −2.62310
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 1.24647e6 0.112400 0.0561999 0.998420i $$-0.482102\pi$$
0.0561999 + 0.998420i $$0.482102\pi$$
$$224$$ 0 0
$$225$$ 1.11719e7 0.980796
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.70025e7 1.36160
$$233$$ 7.80984e6 0.617411 0.308706 0.951158i $$-0.400104\pi$$
0.308706 + 0.951158i $$0.400104\pi$$
$$234$$ −5.41541e6 −0.422653
$$235$$ 0 0
$$236$$ 3.80913e6 0.289794
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 2.69836e7 1.97654 0.988271 0.152712i $$-0.0488005\pi$$
0.988271 + 0.152712i $$0.0488005\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ −1.24009e7 −0.875000
$$243$$ 2.01873e7 1.40689
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 1.16066e7 0.779653
$$247$$ 0 0
$$248$$ 3.24910e7 2.13014
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −1.89343e7 −1.15544
$$255$$ 0 0
$$256$$ −1.10979e7 −0.661484
$$257$$ −2.31777e7 −1.36543 −0.682716 0.730684i $$-0.739201\pi$$
−0.682716 + 0.730684i $$0.739201\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.19834e7 1.23644
$$262$$ −2.32500e7 −1.29277
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3.79262e7 −1.94842 −0.974210 0.225643i $$-0.927552\pi$$
−0.974210 + 0.225643i $$0.927552\pi$$
$$270$$ 0 0
$$271$$ 3.96187e7 1.99064 0.995320 0.0966371i $$-0.0308086\pi$$
0.995320 + 0.0966371i $$0.0308086\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ −6.93519e6 −0.329861
$$277$$ −3.91296e7 −1.84105 −0.920527 0.390680i $$-0.872240\pi$$
−0.920527 + 0.390680i $$0.872240\pi$$
$$278$$ 3.64105e7 1.69470
$$279$$ 4.20091e7 1.93433
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ −5.46210e7 −2.43563
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ −1.00073e7 −0.436880
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −1.07357e7 −0.449422
$$289$$ 2.41376e7 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −1.08756e7 −0.436823
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 3.12946e7 1.23148
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.31647e7 −0.492490
$$300$$ 8.90625e6 0.329861
$$301$$ 0 0
$$302$$ −4.33269e7 −1.57303
$$303$$ −1.92205e7 −0.690934
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5.07075e7 1.75250 0.876248 0.481860i $$-0.160039\pi$$
0.876248 + 0.481860i $$0.160039\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2.58677e7 −0.859958 −0.429979 0.902839i $$-0.641479\pi$$
−0.429979 + 0.902839i $$0.641479\pi$$
$$312$$ −2.27371e7 −0.748639
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.13691e7 −1.92651 −0.963257 0.268582i $$-0.913445\pi$$
−0.963257 + 0.268582i $$0.913445\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 8.12176e6 0.238790
$$325$$ 1.69062e7 0.492490
$$326$$ 4.18950e7 1.20923
$$327$$ 0 0
$$328$$ 2.41296e7 0.683799
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 7.25286e7 1.99998 0.999989 0.00477828i $$-0.00152098\pi$$
0.999989 + 0.00477828i $$0.00152098\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 4.25245e7 1.14130
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 2.55926e7 0.662772
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −1.37527e7 −0.332016
$$347$$ 708554. 0.0169584 0.00847919 0.999964i $$-0.497301\pi$$
0.00847919 + 0.999964i $$0.497301\pi$$
$$348$$ 1.75252e7 0.415840
$$349$$ 9.50019e6 0.223489 0.111744 0.993737i $$-0.464356\pi$$
0.111744 + 0.993737i $$0.464356\pi$$
$$350$$ 0 0
$$351$$ 575624. 0.0133112
$$352$$ 0 0
$$353$$ −7.62365e7 −1.73316 −0.866580 0.499038i $$-0.833687\pi$$
−0.866580 + 0.499038i $$0.833687\pi$$
$$354$$ −6.75486e7 −1.52267
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 2.74342e7 0.597921
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 4.70459e7 1.00000
$$362$$ 0 0
$$363$$ −6.73193e7 −1.40741
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 3.54181e7 0.710693
$$369$$ 3.11983e7 0.620943
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 3.34898e7 0.650554
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −1.13554e8 −2.13619
$$377$$ 3.32672e7 0.620857
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ −1.02786e8 −1.85849
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 4.09983e7 0.724056
$$385$$ 0 0
$$386$$ −2.79418e7 −0.485839
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 6.50599e7 1.08008
$$393$$ −1.26214e8 −2.07937
$$394$$ −1.04869e8 −1.71458
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.02325e8 1.63535 0.817676 0.575679i $$-0.195262\pi$$
0.817676 + 0.575679i $$0.195262\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −4.54844e7 −0.710693
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 6.35718e7 0.971291
$$404$$ −7.58703e6 −0.115061
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −1.11816e8 −1.63431 −0.817153 0.576421i $$-0.804449\pi$$
−0.817153 + 0.576421i $$0.804449\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 6.08958e7 0.858196
$$415$$ 0 0
$$416$$ −1.62462e7 −0.225669
$$417$$ 1.97657e8 2.72586
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 8.88779e7 1.18265
$$423$$ −1.46820e8 −1.93983
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 1.77463e8 2.29551
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ −1.54865e6 −0.0192089
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 1.92861e8 2.29521
$$439$$ −1.16094e8 −1.37220 −0.686099 0.727509i $$-0.740678\pi$$
−0.686099 + 0.727509i $$0.740678\pi$$
$$440$$ 0 0
$$441$$ 8.41190e7 0.980796
$$442$$ 0 0
$$443$$ 1.42821e8 1.64279 0.821393 0.570363i $$-0.193197\pi$$
0.821393 + 0.570363i $$0.193197\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −8.72526e6 −0.0983499
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1.58038e8 1.74591 0.872955 0.487801i $$-0.162201\pi$$
0.872955 + 0.487801i $$0.162201\pi$$
$$450$$ −7.82031e7 −0.858196
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −2.35203e8 −2.53016
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −1.86172e8 −1.90026 −0.950129 0.311856i $$-0.899049\pi$$
−0.950129 + 0.311856i $$0.899049\pi$$
$$462$$ 0 0
$$463$$ −6.20833e7 −0.625506 −0.312753 0.949834i $$-0.601251\pi$$
−0.312753 + 0.949834i $$0.601251\pi$$
$$464$$ −8.95016e7 −0.895936
$$465$$ 0 0
$$466$$ −5.46689e7 −0.540235
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ −1.16044e7 −0.113211
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −1.40430e8 −1.33547
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −1.88885e8 −1.72947
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −2.65734e7 −0.234375
$$485$$ 0 0
$$486$$ −1.41311e8 −1.23103
$$487$$ 1.55985e8 1.35050 0.675252 0.737587i $$-0.264035\pi$$
0.675252 + 0.737587i $$0.264035\pi$$
$$488$$ 0 0
$$489$$ 2.27430e8 1.94501
$$490$$ 0 0
$$491$$ 1.90755e8 1.61151 0.805754 0.592250i $$-0.201760\pi$$
0.805754 + 0.592250i $$0.201760\pi$$
$$492$$ 2.48714e7 0.208836
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −1.71033e8 −1.40163
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 757946. 0.00610010 0.00305005 0.999995i $$-0.499029\pi$$
0.00305005 + 0.999995i $$0.499029\pi$$
$$500$$ 0 0
$$501$$ 2.30847e8 1.83574
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.38931e8 1.06605
$$508$$ −4.05735e7 −0.309493
$$509$$ −2.00351e8 −1.51928 −0.759641 0.650343i $$-0.774625\pi$$
−0.759641 + 0.650343i $$0.774625\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.46735e8 1.09326
$$513$$ 0 0
$$514$$ 1.62244e8 1.19475
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −7.46573e7 −0.534036
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ −1.53884e8 −1.08189
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ −4.98215e7 −0.346277
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.48036e8 1.00000
$$530$$ 0 0
$$531$$ −1.81569e8 −1.21271
$$532$$ 0 0
$$533$$ 4.72120e7 0.311796
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 1.48929e8 0.961735
$$538$$ 2.65483e8 1.70487
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −3.15808e8 −1.99449 −0.997245 0.0741740i $$-0.976368\pi$$
−0.997245 + 0.0741740i $$0.976368\pi$$
$$542$$ −2.77331e8 −1.74181
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1.27720e8 0.780361 0.390180 0.920738i $$-0.372413\pi$$
0.390180 + 0.920738i $$0.372413\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 2.55677e8 1.52011
$$553$$ 0 0
$$554$$ 2.73907e8 1.61092
$$555$$ 0 0
$$556$$ 7.80224e7 0.453936
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ −2.94064e8 −1.69254
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ −1.17045e8 −0.652402
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 3.68936e8 2.01329
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.90109e8 −1.00000
$$576$$ 2.08357e8 1.09029
$$577$$ 9.00812e7 0.468929 0.234464 0.972125i $$-0.424666\pi$$
0.234464 + 0.972125i $$0.424666\pi$$
$$578$$ −1.68963e8 −0.875000
$$579$$ −1.51684e8 −0.781455
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 4.00948e8 2.01303
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −3.99910e8 −1.97719 −0.988594 0.150604i $$-0.951878\pi$$
−0.988594 + 0.150604i $$0.951878\pi$$
$$588$$ 6.70599e7 0.329861
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −5.69288e8 −2.75784
$$592$$ 0 0
$$593$$ 2.78321e8 1.33470 0.667348 0.744746i $$-0.267429\pi$$
0.667348 + 0.744746i $$0.267429\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 9.21529e7 0.430929
$$599$$ 1.62397e8 0.755611 0.377806 0.925885i $$-0.376679\pi$$
0.377806 + 0.925885i $$0.376679\pi$$
$$600$$ −3.28344e8 −1.52011
$$601$$ −4.34057e8 −1.99951 −0.999755 0.0221248i $$-0.992957\pi$$
−0.999755 + 0.0221248i $$0.992957\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −9.28433e7 −0.421347
$$605$$ 0 0
$$606$$ 1.34543e8 0.604567
$$607$$ −3.69151e8 −1.65059 −0.825294 0.564704i $$-0.808990\pi$$
−0.825294 + 0.564704i $$0.808990\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2.22180e8 −0.974050
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ −3.54953e8 −1.53343
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ −6.47284e6 −0.0270284
$$622$$ 1.81074e8 0.752463
$$623$$ 0 0
$$624$$ 1.19689e8 0.492606
$$625$$ 2.44141e8 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 4.82480e8 1.90225
$$634$$ 4.29584e8 1.68570
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.27296e8 0.492490
$$638$$ 0 0
$$639$$ 4.77015e8 1.82822
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.76738e8 1.02178 0.510888 0.859648i $$-0.329317\pi$$
0.510888 + 0.859648i $$0.329317\pi$$
$$648$$ −2.99422e8 −1.10042
$$649$$ 0 0
$$650$$ −1.18344e8 −0.430929
$$651$$ 0 0
$$652$$ 8.97750e7 0.323901
$$653$$ −7.17455e7 −0.257665 −0.128832 0.991666i $$-0.541123\pi$$
−0.128832 + 0.991666i $$0.541123\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −1.27019e8 −0.449941
$$657$$ 5.18405e8 1.82799
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ −5.07700e8 −1.74998
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −3.74087e8 −1.26065
$$668$$ 9.11239e7 0.305705
$$669$$ −4.73657e7 −0.158192
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 5.18377e8 1.70059 0.850297 0.526304i $$-0.176422\pi$$
0.850297 + 0.526304i $$0.176422\pi$$
$$674$$ 0 0
$$675$$ 8.31250e6 0.0270284
$$676$$ 5.48413e7 0.177528
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 6.13389e8 1.92519 0.962595 0.270945i $$-0.0873361\pi$$
0.962595 + 0.270945i $$0.0873361\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −6.23542e8 −1.88987 −0.944934 0.327261i $$-0.893875\pi$$
−0.944934 + 0.327261i $$0.893875\pi$$
$$692$$ −2.94700e7 −0.0889327
$$693$$ 0 0
$$694$$ −4.95988e6 −0.0148386
$$695$$ 0 0
$$696$$ −6.46096e8 −1.91633
$$697$$ 0 0
$$698$$ −6.65013e7 −0.195553
$$699$$ −2.96774e8 −0.868949
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ −4.02937e6 −0.0116473
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 5.33655e8 1.51652
$$707$$ 0 0
$$708$$ −1.44747e8 −0.407859
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −7.14860e8 −1.97221
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 5.87876e7 0.160157
$$717$$ −1.02538e9 −2.78180
$$718$$ 0 0
$$719$$ 6.96357e8 1.87346 0.936732 0.350047i $$-0.113834\pi$$
0.936732 + 0.350047i $$0.113834\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −3.29321e8 −0.875000
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 4.80406e8 1.26065
$$726$$ 4.71235e8 1.23148
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ −3.72400e8 −0.961229
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 1.82688e8 0.458221
$$737$$ 0 0
$$738$$ −2.18388e8 −0.543325
$$739$$ 1.26013e8 0.312236 0.156118 0.987738i $$-0.450102\pi$$
0.156118 + 0.987738i $$0.450102\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ −1.23466e9 −2.99797
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 5.97751e8 1.40562
$$753$$ 0 0
$$754$$ −2.32870e8 −0.543250
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.49422e8 1.47358 0.736789 0.676123i $$-0.236341\pi$$
0.736789 + 0.676123i $$0.236341\pi$$
$$762$$ 7.19503e8 1.62618
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −2.74765e8 −0.608942
$$768$$ 4.21718e8 0.930977
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 8.80751e8 1.92172
$$772$$ −5.98752e7 −0.130135
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 9.18031e8 1.97221
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 1.63569e7 0.0340734
$$784$$ −3.42476e8 −0.710693
$$785$$ 0 0
$$786$$ 8.83501e8 1.81945
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ −2.24719e8 −0.459262
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −7.16276e8 −1.43093
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −2.34609e8 −0.458221
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.45003e8 −0.849880
$$807$$ 1.44120e9 2.74222
$$808$$ 2.79709e8 0.530239
$$809$$ −2.83551e8 −0.535531 −0.267766 0.963484i $$-0.586285\pi$$
−0.267766 + 0.963484i $$0.586285\pi$$
$$810$$ 0 0
$$811$$ −1.05551e9 −1.97878 −0.989392 0.145268i $$-0.953596\pi$$
−0.989392 + 0.145268i $$0.953596\pi$$
$$812$$ 0 0
$$813$$ −1.50551e9 −2.80164
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 7.82711e8 1.43002
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −5.08386e8 −0.918679 −0.459340 0.888261i $$-0.651914\pi$$
−0.459340 + 0.888261i $$0.651914\pi$$
$$822$$ 0 0
$$823$$ −8.05686e8 −1.44533 −0.722664 0.691199i $$-0.757083\pi$$
−0.722664 + 0.691199i $$0.757083\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 1.30491e8 0.229874
$$829$$ −1.11478e9 −1.95671 −0.978357 0.206925i $$-0.933655\pi$$
−0.978357 + 0.206925i $$0.933655\pi$$
$$830$$ 0 0
$$831$$ 1.48693e9 2.59111
$$832$$ 3.15305e8 0.547470
$$833$$ 0 0
$$834$$ −1.38360e9 −2.38513
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 3.12571e7 0.0533056
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 3.50493e8 0.589239
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 1.90453e8 0.316781
$$845$$ 0 0
$$846$$ 1.02774e9 1.69735
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 3.80278e8 0.614868
$$853$$ −6.36633e8 −1.02575 −0.512876 0.858463i $$-0.671420\pi$$
−0.512876 + 0.858463i $$0.671420\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −5.12572e8 −0.814353 −0.407177 0.913349i $$-0.633487\pi$$
−0.407177 + 0.913349i $$0.633487\pi$$
$$858$$ 0 0
$$859$$ 1.26724e9 1.99931 0.999654 0.0262855i $$-0.00836789\pi$$
0.999654 + 0.0262855i $$0.00836789\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 4.00781e8 0.623555 0.311777 0.950155i $$-0.399076\pi$$
0.311777 + 0.950155i $$0.399076\pi$$
$$864$$ −7.98798e6 −0.0123850
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −9.17228e8 −1.40741
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 4.13274e8 0.614788
$$877$$ −1.86924e8 −0.277119 −0.138560 0.990354i $$-0.544247\pi$$
−0.138560 + 0.990354i $$0.544247\pi$$
$$878$$ 8.12659e8 1.20067
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ −5.88833e8 −0.858196
$$883$$ −1.36875e9 −1.98812 −0.994060 0.108838i $$-0.965287\pi$$
−0.994060 + 0.108838i $$0.965287\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −9.99747e8 −1.43744
$$887$$ −1.39031e9 −1.99223 −0.996117 0.0880391i $$-0.971940\pi$$
−0.996117 + 0.0880391i $$0.971940\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −1.86970e7 −0.0263437
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 5.00258e8 0.693134
$$898$$ −1.10626e9 −1.52767
$$899$$ 1.80645e9 2.48626
$$900$$ −1.67578e8 −0.229874
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 1.64642e9 2.21389
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ 3.61648e8 0.481498
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ −1.92689e9 −2.46648
$$922$$ 1.30321e9 1.66273
$$923$$ 7.21861e8 0.918012
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 4.34583e8 0.547318
$$927$$ 0 0
$$928$$ −4.61651e8 −0.577657
$$929$$ 8.42564e8 1.05089 0.525443 0.850829i $$-0.323900\pi$$
0.525443 + 0.850829i $$0.323900\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −1.17148e8 −0.144706
$$933$$ 9.82974e8 1.21031
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 4.27817e8 0.521712
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ −5.30895e8 −0.633102
$$944$$ 7.39225e8 0.878741
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1.32946e9 −1.56539 −0.782697 0.622403i $$-0.786157\pi$$
−0.782697 + 0.622403i $$0.786157\pi$$
$$948$$ 0 0
$$949$$ 7.84495e8 0.917892
$$950$$ 0 0
$$951$$ 2.33203e9 2.71139
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −4.04754e8 −0.463252
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 2.56453e9 2.88960
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −1.73601e9 −1.91987 −0.959936 0.280219i $$-0.909593\pi$$
−0.959936 + 0.280219i $$0.909593\pi$$
$$968$$ 9.79673e8 1.08008
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ −3.02810e8 −0.329739
$$973$$ 0 0
$$974$$ −1.09189e9 −1.18169
$$975$$ −6.42438e8 −0.693134
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ −1.59201e9 −1.70188
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −1.33529e9 −1.41007
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ −9.16925e8 −0.962384
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −1.86316e9 −1.91439 −0.957194 0.289446i $$-0.906529\pi$$
−0.957194 + 0.289446i $$0.906529\pi$$
$$992$$ −8.82191e8 −0.903707
$$993$$ −2.75609e9 −2.81478
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.64078e9 1.65564 0.827819 0.560995i $$-0.189581\pi$$
0.827819 + 0.560995i $$0.189581\pi$$
$$998$$ −5.30562e6 −0.00533758
$$999$$ 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 23.7.b.a.22.1 1
3.2 odd 2 207.7.d.a.91.1 1
4.3 odd 2 368.7.f.a.321.1 1
23.22 odd 2 CM 23.7.b.a.22.1 1
69.68 even 2 207.7.d.a.91.1 1
92.91 even 2 368.7.f.a.321.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
23.7.b.a.22.1 1 1.1 even 1 trivial
23.7.b.a.22.1 1 23.22 odd 2 CM
207.7.d.a.91.1 1 3.2 odd 2
207.7.d.a.91.1 1 69.68 even 2
368.7.f.a.321.1 1 4.3 odd 2
368.7.f.a.321.1 1 92.91 even 2