# Properties

 Label 435.3.b.b.434.1 Level $435$ Weight $3$ Character 435.434 Self dual yes Analytic conductor $11.853$ Analytic rank $0$ Dimension $1$ CM discriminant -435 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,3,Mod(434,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("435.434");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$435 = 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 435.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: $$11.8528914997$$ 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 434.1 Character $$\chi$$ $$=$$ 435.434

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +9.00000 q^{9} +7.00000 q^{11} -12.0000 q^{12} -15.0000 q^{15} +16.0000 q^{16} +20.0000 q^{20} -41.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +29.0000 q^{29} -21.0000 q^{33} +36.0000 q^{36} +71.0000 q^{37} -53.0000 q^{41} +59.0000 q^{43} +28.0000 q^{44} +45.0000 q^{45} -48.0000 q^{48} +49.0000 q^{49} +19.0000 q^{53} +35.0000 q^{55} -60.0000 q^{60} +64.0000 q^{64} +123.000 q^{69} -1.00000 q^{73} -75.0000 q^{75} +80.0000 q^{80} +81.0000 q^{81} +79.0000 q^{83} -87.0000 q^{87} -62.0000 q^{89} -164.000 q^{92} -49.0000 q^{97} +63.0000 q^{99} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/435\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$146$$ $$262$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$3$$ −3.00000 −1.00000
$$4$$ 4.00000 1.00000
$$5$$ 5.00000 1.00000
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 9.00000 1.00000
$$10$$ 0 0
$$11$$ 7.00000 0.636364 0.318182 0.948030i $$-0.396928\pi$$
0.318182 + 0.948030i $$0.396928\pi$$
$$12$$ −12.0000 −1.00000
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −1.00000
$$16$$ 16.0000 1.00000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 20.0000 1.00000
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −41.0000 −1.78261 −0.891304 0.453406i $$-0.850209\pi$$
−0.891304 + 0.453406i $$0.850209\pi$$
$$24$$ 0 0
$$25$$ 25.0000 1.00000
$$26$$ 0 0
$$27$$ −27.0000 −1.00000
$$28$$ 0 0
$$29$$ 29.0000 1.00000
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ −21.0000 −0.636364
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 36.0000 1.00000
$$37$$ 71.0000 1.91892 0.959459 0.281847i $$-0.0909469\pi$$
0.959459 + 0.281847i $$0.0909469\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −53.0000 −1.29268 −0.646341 0.763048i $$-0.723702\pi$$
−0.646341 + 0.763048i $$0.723702\pi$$
$$42$$ 0 0
$$43$$ 59.0000 1.37209 0.686047 0.727558i $$-0.259345\pi$$
0.686047 + 0.727558i $$0.259345\pi$$
$$44$$ 28.0000 0.636364
$$45$$ 45.0000 1.00000
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ −48.0000 −1.00000
$$49$$ 49.0000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 19.0000 0.358491 0.179245 0.983804i $$-0.442634\pi$$
0.179245 + 0.983804i $$0.442634\pi$$
$$54$$ 0 0
$$55$$ 35.0000 0.636364
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ −60.0000 −1.00000
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 64.0000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 123.000 1.78261
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ −1.00000 −0.0136986 −0.00684932 0.999977i $$-0.502180\pi$$
−0.00684932 + 0.999977i $$0.502180\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −1.00000
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 80.0000 1.00000
$$81$$ 81.0000 1.00000
$$82$$ 0 0
$$83$$ 79.0000 0.951807 0.475904 0.879497i $$-0.342121\pi$$
0.475904 + 0.879497i $$0.342121\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −87.0000 −1.00000
$$88$$ 0 0
$$89$$ −62.0000 −0.696629 −0.348315 0.937378i $$-0.613246\pi$$
−0.348315 + 0.937378i $$0.613246\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −164.000 −1.78261
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −49.0000 −0.505155 −0.252577 0.967577i $$-0.581278\pi$$
−0.252577 + 0.967577i $$0.581278\pi$$
$$98$$ 0 0
$$99$$ 63.0000 0.636364
$$100$$ 100.000 1.00000
$$101$$ −173.000 −1.71287 −0.856436 0.516254i $$-0.827326\pi$$
−0.856436 + 0.516254i $$0.827326\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −134.000 −1.25234 −0.626168 0.779688i $$-0.715378\pi$$
−0.626168 + 0.779688i $$0.715378\pi$$
$$108$$ −108.000 −1.00000
$$109$$ −217.000 −1.99083 −0.995413 0.0956727i $$-0.969500\pi$$
−0.995413 + 0.0956727i $$0.969500\pi$$
$$110$$ 0 0
$$111$$ −213.000 −1.91892
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ −205.000 −1.78261
$$116$$ 116.000 1.00000
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −72.0000 −0.595041
$$122$$ 0 0
$$123$$ 159.000 1.29268
$$124$$ 0 0
$$125$$ 125.000 1.00000
$$126$$ 0 0
$$127$$ −109.000 −0.858268 −0.429134 0.903241i $$-0.641181\pi$$
−0.429134 + 0.903241i $$0.641181\pi$$
$$128$$ 0 0
$$129$$ −177.000 −1.37209
$$130$$ 0 0
$$131$$ 202.000 1.54198 0.770992 0.636844i $$-0.219761\pi$$
0.770992 + 0.636844i $$0.219761\pi$$
$$132$$ −84.0000 −0.636364
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −135.000 −1.00000
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ −157.000 −1.12950 −0.564748 0.825263i $$-0.691027\pi$$
−0.564748 + 0.825263i $$0.691027\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 144.000 1.00000
$$145$$ 145.000 1.00000
$$146$$ 0 0
$$147$$ −147.000 −1.00000
$$148$$ 284.000 1.91892
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ −133.000 −0.880795 −0.440397 0.897803i $$-0.645162\pi$$
−0.440397 + 0.897803i $$0.645162\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 266.000 1.69427 0.847134 0.531380i $$-0.178326\pi$$
0.847134 + 0.531380i $$0.178326\pi$$
$$158$$ 0 0
$$159$$ −57.0000 −0.358491
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −181.000 −1.11043 −0.555215 0.831707i $$-0.687364\pi$$
−0.555215 + 0.831707i $$0.687364\pi$$
$$164$$ −212.000 −1.29268
$$165$$ −105.000 −0.636364
$$166$$ 0 0
$$167$$ −14.0000 −0.0838323 −0.0419162 0.999121i $$-0.513346\pi$$
−0.0419162 + 0.999121i $$0.513346\pi$$
$$168$$ 0 0
$$169$$ 169.000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 236.000 1.37209
$$173$$ 259.000 1.49711 0.748555 0.663073i $$-0.230748\pi$$
0.748555 + 0.663073i $$0.230748\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 112.000 0.636364
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 180.000 1.00000
$$181$$ −73.0000 −0.403315 −0.201657 0.979456i $$-0.564633\pi$$
−0.201657 + 0.979456i $$0.564633\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 355.000 1.91892
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −353.000 −1.84817 −0.924084 0.382190i $$-0.875170\pi$$
−0.924084 + 0.382190i $$0.875170\pi$$
$$192$$ −192.000 −1.00000
$$193$$ 194.000 1.00518 0.502591 0.864525i $$-0.332380\pi$$
0.502591 + 0.864525i $$0.332380\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 196.000 1.00000
$$197$$ −389.000 −1.97462 −0.987310 0.158807i $$-0.949235\pi$$
−0.987310 + 0.158807i $$0.949235\pi$$
$$198$$ 0 0
$$199$$ −37.0000 −0.185930 −0.0929648 0.995669i $$-0.529634\pi$$
−0.0929648 + 0.995669i $$0.529634\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −265.000 −1.29268
$$206$$ 0 0
$$207$$ −369.000 −1.78261
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 76.0000 0.358491
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 295.000 1.37209
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 3.00000 0.0136986
$$220$$ 140.000 0.636364
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 225.000 1.00000
$$226$$ 0 0
$$227$$ −329.000 −1.44934 −0.724670 0.689096i $$-0.758008\pi$$
−0.724670 + 0.689096i $$0.758008\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 379.000 1.62661 0.813305 0.581838i $$-0.197666\pi$$
0.813305 + 0.581838i $$0.197666\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ −240.000 −1.00000
$$241$$ 47.0000 0.195021 0.0975104 0.995235i $$-0.468912\pi$$
0.0975104 + 0.995235i $$0.468912\pi$$
$$242$$ 0 0
$$243$$ −243.000 −1.00000
$$244$$ 0 0
$$245$$ 245.000 1.00000
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −237.000 −0.951807
$$250$$ 0 0
$$251$$ −38.0000 −0.151394 −0.0756972 0.997131i $$-0.524118\pi$$
−0.0756972 + 0.997131i $$0.524118\pi$$
$$252$$ 0 0
$$253$$ −287.000 −1.13439
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 256.000 1.00000
$$257$$ −269.000 −1.04669 −0.523346 0.852120i $$-0.675317\pi$$
−0.523346 + 0.852120i $$0.675317\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 261.000 1.00000
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 95.0000 0.358491
$$266$$ 0 0
$$267$$ 186.000 0.696629
$$268$$ 0 0
$$269$$ −422.000 −1.56877 −0.784387 0.620272i $$-0.787022\pi$$
−0.784387 + 0.620272i $$0.787022\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 175.000 0.636364
$$276$$ 492.000 1.78261
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 289.000 1.00000
$$290$$ 0 0
$$291$$ 147.000 0.505155
$$292$$ −4.00000 −0.0136986
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −189.000 −0.636364
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −300.000 −1.00000
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 519.000 1.71287
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −469.000 −1.52769 −0.763844 0.645401i $$-0.776690\pi$$
−0.763844 + 0.645401i $$0.776690\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −593.000 −1.90675 −0.953376 0.301784i $$-0.902418\pi$$
−0.953376 + 0.301784i $$0.902418\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 203.000 0.636364
$$320$$ 320.000 1.00000
$$321$$ 402.000 1.25234
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 324.000 1.00000
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 651.000 1.99083
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 316.000 0.951807
$$333$$ 639.000 1.91892
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −94.0000 −0.278932 −0.139466 0.990227i $$-0.544539\pi$$
−0.139466 + 0.990227i $$0.544539\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 615.000 1.78261
$$346$$ 0 0
$$347$$ −89.0000 −0.256484 −0.128242 0.991743i $$-0.540933\pi$$
−0.128242 + 0.991743i $$0.540933\pi$$
$$348$$ −348.000 −1.00000
$$349$$ 263.000 0.753582 0.376791 0.926298i $$-0.377028\pi$$
0.376791 + 0.926298i $$0.377028\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −686.000 −1.94334 −0.971671 0.236336i $$-0.924053\pi$$
−0.971671 + 0.236336i $$0.924053\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −248.000 −0.696629
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 703.000 1.95822 0.979109 0.203338i $$-0.0651790\pi$$
0.979109 + 0.203338i $$0.0651790\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 0 0
$$363$$ 216.000 0.595041
$$364$$ 0 0
$$365$$ −5.00000 −0.0136986
$$366$$ 0 0
$$367$$ −589.000 −1.60490 −0.802452 0.596716i $$-0.796472\pi$$
−0.802452 + 0.596716i $$0.796472\pi$$
$$368$$ −656.000 −1.78261
$$369$$ −477.000 −1.29268
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ −375.000 −1.00000
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 327.000 0.858268
$$382$$ 0 0
$$383$$ 679.000 1.77285 0.886423 0.462876i $$-0.153183\pi$$
0.886423 + 0.462876i $$0.153183\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 531.000 1.37209
$$388$$ −196.000 −0.505155
$$389$$ 643.000 1.65296 0.826478 0.562969i $$-0.190341\pi$$
0.826478 + 0.562969i $$0.190341\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −606.000 −1.54198
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 252.000 0.636364
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 400.000 1.00000
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −692.000 −1.71287
$$405$$ 405.000 1.00000
$$406$$ 0 0
$$407$$ 497.000 1.22113
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 395.000 0.951807
$$416$$ 0 0
$$417$$ 471.000 1.12950
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −536.000 −1.25234
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ −432.000 −1.00000
$$433$$ −721.000 −1.66513 −0.832564 0.553930i $$-0.813128\pi$$
−0.832564 + 0.553930i $$0.813128\pi$$
$$434$$ 0 0
$$435$$ −435.000 −1.00000
$$436$$ −868.000 −1.99083
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −862.000 −1.96355 −0.981777 0.190038i $$-0.939139\pi$$
−0.981777 + 0.190038i $$0.939139\pi$$
$$440$$ 0 0
$$441$$ 441.000 1.00000
$$442$$ 0 0
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ −852.000 −1.91892
$$445$$ −310.000 −0.696629
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 523.000 1.16481 0.582405 0.812899i $$-0.302112\pi$$
0.582405 + 0.812899i $$0.302112\pi$$
$$450$$ 0 0
$$451$$ −371.000 −0.822616
$$452$$ 0 0
$$453$$ 399.000 0.880795
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ −820.000 −1.78261
$$461$$ −893.000 −1.93709 −0.968547 0.248832i $$-0.919953\pi$$
−0.968547 + 0.248832i $$0.919953\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 464.000 1.00000
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −798.000 −1.69427
$$472$$ 0 0
$$473$$ 413.000 0.873150
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 171.000 0.358491
$$478$$ 0 0
$$479$$ 898.000 1.87474 0.937370 0.348337i $$-0.113253\pi$$
0.937370 + 0.348337i $$0.113253\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −288.000 −0.595041
$$485$$ −245.000 −0.505155
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 543.000 1.11043
$$490$$ 0 0
$$491$$ −518.000 −1.05499 −0.527495 0.849558i $$-0.676869\pi$$
−0.527495 + 0.849558i $$0.676869\pi$$
$$492$$ 636.000 1.29268
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 315.000 0.636364
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −742.000 −1.48697 −0.743487 0.668750i $$-0.766829\pi$$
−0.743487 + 0.668750i $$0.766829\pi$$
$$500$$ 500.000 1.00000
$$501$$ 42.0000 0.0838323
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ −865.000 −1.71287
$$506$$ 0 0
$$507$$ −507.000 −1.00000
$$508$$ −436.000 −0.858268
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ −708.000 −1.37209
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −777.000 −1.49711
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 808.000 1.54198
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ −336.000 −0.636364
$$529$$ 1152.00 2.17769
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −670.000 −1.25234
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 343.000 0.636364
$$540$$ −540.000 −1.00000
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 219.000 0.403315
$$544$$ 0 0
$$545$$ −1085.00 −1.99083
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −1065.00 −1.91892
$$556$$ −628.000 −1.12950
$$557$$ 331.000 0.594255 0.297127 0.954838i $$-0.403971\pi$$
0.297127 + 0.954838i $$0.403971\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −1022.00 −1.79613 −0.898067 0.439859i $$-0.855028\pi$$
−0.898067 + 0.439859i $$0.855028\pi$$
$$570$$ 0 0
$$571$$ 707.000 1.23818 0.619089 0.785321i $$-0.287502\pi$$
0.619089 + 0.785321i $$0.287502\pi$$
$$572$$ 0 0
$$573$$ 1059.00 1.84817
$$574$$ 0 0
$$575$$ −1025.00 −1.78261
$$576$$ 576.000 1.00000
$$577$$ −574.000 −0.994801 −0.497400 0.867521i $$-0.665712\pi$$
−0.497400 + 0.867521i $$0.665712\pi$$
$$578$$ 0 0
$$579$$ −582.000 −1.00518
$$580$$ 580.000 1.00000
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 133.000 0.228130
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 826.000 1.40716 0.703578 0.710619i $$-0.251585\pi$$
0.703578 + 0.710619i $$0.251585\pi$$
$$588$$ −588.000 −1.00000
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 1167.00 1.97462
$$592$$ 1136.00 1.91892
$$593$$ −206.000 −0.347386 −0.173693 0.984800i $$-0.555570\pi$$
−0.173693 + 0.984800i $$0.555570\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 111.000 0.185930
$$598$$ 0 0
$$599$$ 658.000 1.09850 0.549249 0.835659i $$-0.314914\pi$$
0.549249 + 0.835659i $$0.314914\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −532.000 −0.880795
$$605$$ −360.000 −0.595041
$$606$$ 0 0
$$607$$ 1106.00 1.82208 0.911038 0.412323i $$-0.135282\pi$$
0.911038 + 0.412323i $$0.135282\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 795.000 1.29268
$$616$$ 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
$$621$$ 1107.00 1.78261
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 1064.00 1.69427
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −478.000 −0.757528 −0.378764 0.925493i $$-0.623651\pi$$
−0.378764 + 0.925493i $$0.623651\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −545.000 −0.858268
$$636$$ −228.000 −0.358491
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1253.00 −1.95476 −0.977379 0.211495i $$-0.932167\pi$$
−0.977379 + 0.211495i $$0.932167\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ −885.000 −1.37209
$$646$$ 0 0
$$647$$ 511.000 0.789799 0.394900 0.918724i $$-0.370779\pi$$
0.394900 + 0.918724i $$0.370779\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −724.000 −1.11043
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 1010.00 1.54198
$$656$$ −848.000 −1.29268
$$657$$ −9.00000 −0.0136986
$$658$$ 0 0
$$659$$ 103.000 0.156297 0.0781487 0.996942i $$-0.475099\pi$$
0.0781487 + 0.996942i $$0.475099\pi$$
$$660$$ −420.000 −0.636364
$$661$$ 887.000 1.34191 0.670953 0.741500i $$-0.265885\pi$$
0.670953 + 0.741500i $$0.265885\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1189.00 −1.78261
$$668$$ −56.0000 −0.0838323
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ −675.000 −1.00000
$$676$$ 676.000 1.00000
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 987.000 1.44934
$$682$$ 0 0
$$683$$ 1279.00 1.87262 0.936310 0.351174i $$-0.114217\pi$$
0.936310 + 0.351174i $$0.114217\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 944.000 1.37209
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −358.000 −0.518090 −0.259045 0.965865i $$-0.583408\pi$$
−0.259045 + 0.965865i $$0.583408\pi$$
$$692$$ 1036.00 1.49711
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −785.000 −1.12950
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −1137.00 −1.62661
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 448.000 0.636364
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 983.000 1.38646 0.693230 0.720717i $$-0.256187\pi$$
0.693230 + 0.720717i $$0.256187\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 720.000 1.00000
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −141.000 −0.195021
$$724$$ −292.000 −0.403315
$$725$$ 725.000 1.00000
$$726$$ 0 0
$$727$$ 866.000 1.19120 0.595598 0.803282i $$-0.296915\pi$$
0.595598 + 0.803282i $$0.296915\pi$$
$$728$$ 0 0
$$729$$ 729.000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −886.000 −1.20873 −0.604366 0.796707i $$-0.706573\pi$$
−0.604366 + 0.796707i $$0.706573\pi$$
$$734$$ 0 0
$$735$$ −735.000 −1.00000
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 1420.00 1.91892
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 711.000 0.951807
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ 114.000 0.151394
$$754$$ 0 0
$$755$$ −665.000 −0.880795
$$756$$ 0 0
$$757$$ −1369.00 −1.80845 −0.904227 0.427052i $$-0.859552\pi$$
−0.904227 + 0.427052i $$0.859552\pi$$
$$758$$ 0 0
$$759$$ 861.000 1.13439
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −1412.00 −1.84817
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −768.000 −1.00000
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 807.000 1.04669
$$772$$ 776.000 1.00518
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −783.000 −1.00000
$$784$$ 784.000 1.00000
$$785$$ 1330.00 1.69427
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ −1556.00 −1.97462
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −285.000 −0.358491
$$796$$ −148.000 −0.185930
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −558.000 −0.696629
$$802$$ 0 0
$$803$$ −7.00000 −0.00871731
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1266.00 1.56877
$$808$$ 0 0
$$809$$ −197.000 −0.243511 −0.121755 0.992560i $$-0.538852\pi$$
−0.121755 + 0.992560i $$0.538852\pi$$
$$810$$ 0 0
$$811$$ 1187.00 1.46363 0.731813 0.681506i $$-0.238675\pi$$
0.731813 + 0.681506i $$0.238675\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −905.000 −1.11043
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ −1060.00 −1.29268
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ 674.000 0.818955 0.409478 0.912320i $$-0.365711\pi$$
0.409478 + 0.912320i $$0.365711\pi$$
$$824$$ 0 0
$$825$$ −525.000 −0.636364
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ −1476.00 −1.78261
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −70.0000 −0.0838323
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 178.000 0.212157 0.106079 0.994358i $$-0.466170\pi$$
0.106079 + 0.994358i $$0.466170\pi$$
$$840$$ 0 0
$$841$$ 841.000 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 845.000 1.00000
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 304.000 0.358491
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2911.00 −3.42068
$$852$$ 0 0
$$853$$ −1561.00 −1.83001 −0.915006 0.403441i $$-0.867814\pi$$
−0.915006 + 0.403441i $$0.867814\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 931.000 1.08635 0.543174 0.839620i $$-0.317222\pi$$
0.543174 + 0.839620i $$0.317222\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 1180.00 1.37209
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −1406.00 −1.62920 −0.814600 0.580023i $$-0.803044\pi$$
−0.814600 + 0.580023i $$0.803044\pi$$
$$864$$ 0 0
$$865$$ 1295.00 1.49711
$$866$$ 0 0
$$867$$ −867.000 −1.00000
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −441.000 −0.505155
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 12.0000 0.0136986
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 560.000 0.636364
$$881$$ 1747.00 1.98297 0.991487 0.130206i $$-0.0415639\pi$$
0.991487 + 0.130206i $$0.0415639\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 567.000 0.636364
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 900.000 1.00000
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −365.000 −0.403315
$$906$$ 0 0
$$907$$ 1811.00 1.99669 0.998346 0.0574880i $$-0.0183091\pi$$
0.998346 + 0.0574880i $$0.0183091\pi$$
$$908$$ −1316.00 −1.44934
$$909$$ −1557.00 −1.71287
$$910$$ 0 0
$$911$$ 1687.00 1.85181 0.925906 0.377755i $$-0.123304\pi$$
0.925906 + 0.377755i $$0.123304\pi$$
$$912$$ 0 0
$$913$$ 553.000 0.605696
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 98.0000 0.106638 0.0533188 0.998578i $$-0.483020\pi$$
0.0533188 + 0.998578i $$0.483020\pi$$
$$920$$ 0 0
$$921$$ 1407.00 1.52769
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 1775.00 1.91892
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 1516.00 1.62661
$$933$$ 1779.00 1.90675
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 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
$$943$$ 2173.00 2.30435
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 514.000 0.539349 0.269675 0.962951i $$-0.413084\pi$$
0.269675 + 0.962951i $$0.413084\pi$$
$$954$$ 0 0
$$955$$ −1765.00 −1.84817
$$956$$ 0 0
$$957$$ −609.000 −0.636364
$$958$$ 0 0
$$959$$ 0 0
$$960$$ −960.000 −1.00000
$$961$$ 961.000 1.00000
$$962$$ 0 0
$$963$$ −1206.00 −1.25234
$$964$$ 188.000 0.195021
$$965$$ 970.000 1.00518
$$966$$ 0 0
$$967$$ 1691.00 1.74871 0.874354 0.485289i $$-0.161286\pi$$
0.874354 + 0.485289i $$0.161286\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1567.00 1.61380 0.806900 0.590688i $$-0.201144\pi$$
0.806900 + 0.590688i $$0.201144\pi$$
$$972$$ −972.000 −1.00000
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 1171.00 1.19857 0.599284 0.800537i $$-0.295452\pi$$
0.599284 + 0.800537i $$0.295452\pi$$
$$978$$ 0 0
$$979$$ −434.000 −0.443309
$$980$$ 980.000 1.00000
$$981$$ −1953.00 −1.99083
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ −1945.00 −1.97462
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −2419.00 −2.44590
$$990$$ 0 0
$$991$$ −1933.00 −1.95055 −0.975277 0.220984i $$-0.929073\pi$$
−0.975277 + 0.220984i $$0.929073\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −185.000 −0.185930
$$996$$ −948.000 −0.951807
$$997$$ 1631.00 1.63591 0.817954 0.575284i $$-0.195108\pi$$
0.817954 + 0.575284i $$0.195108\pi$$
$$998$$ 0 0
$$999$$ −1917.00 −1.91892
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 435.3.b.b.434.1 yes 1
3.2 odd 2 435.3.b.a.434.1 1
5.4 even 2 435.3.b.c.434.1 yes 1
15.14 odd 2 435.3.b.d.434.1 yes 1
29.28 even 2 435.3.b.d.434.1 yes 1
87.86 odd 2 435.3.b.c.434.1 yes 1
145.144 even 2 435.3.b.a.434.1 1
435.434 odd 2 CM 435.3.b.b.434.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
435.3.b.a.434.1 1 3.2 odd 2
435.3.b.a.434.1 1 145.144 even 2
435.3.b.b.434.1 yes 1 1.1 even 1 trivial
435.3.b.b.434.1 yes 1 435.434 odd 2 CM
435.3.b.c.434.1 yes 1 5.4 even 2
435.3.b.c.434.1 yes 1 87.86 odd 2
435.3.b.d.434.1 yes 1 15.14 odd 2
435.3.b.d.434.1 yes 1 29.28 even 2