# Properties

 Label 256.3.c.a.255.1 Level $256$ Weight $3$ Character 256.255 Self dual yes Analytic conductor $6.975$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,3,Mod(255,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 256.c (of order $$2$$, degree $$1$$, not 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: $$6.97549476762$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 255.1 Character $$\chi$$ $$=$$ 256.255

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-8.00000 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-8.00000 q^{5} +9.00000 q^{9} +24.0000 q^{13} +30.0000 q^{17} +39.0000 q^{25} -40.0000 q^{29} +24.0000 q^{37} -18.0000 q^{41} -72.0000 q^{45} +49.0000 q^{49} +56.0000 q^{53} +120.000 q^{61} -192.000 q^{65} -110.000 q^{73} +81.0000 q^{81} -240.000 q^{85} -78.0000 q^{89} -130.000 q^{97} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 0 0
$$5$$ −8.00000 −1.60000 −0.800000 0.600000i $$-0.795167\pi$$
−0.800000 + 0.600000i $$0.795167\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 9.00000 1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 24.0000 1.84615 0.923077 0.384615i $$-0.125666\pi$$
0.923077 + 0.384615i $$0.125666\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 30.0000 1.76471 0.882353 0.470588i $$-0.155958\pi$$
0.882353 + 0.470588i $$0.155958\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 39.0000 1.56000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −40.0000 −1.37931 −0.689655 0.724138i $$-0.742238\pi$$
−0.689655 + 0.724138i $$0.742238\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 24.0000 0.648649 0.324324 0.945946i $$-0.394863\pi$$
0.324324 + 0.945946i $$0.394863\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −18.0000 −0.439024 −0.219512 0.975610i $$-0.570447\pi$$
−0.219512 + 0.975610i $$0.570447\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ −72.0000 −1.60000
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 49.0000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 56.0000 1.05660 0.528302 0.849057i $$-0.322829\pi$$
0.528302 + 0.849057i $$0.322829\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 120.000 1.96721 0.983607 0.180328i $$-0.0577159\pi$$
0.983607 + 0.180328i $$0.0577159\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −192.000 −2.95385
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ −110.000 −1.50685 −0.753425 0.657534i $$-0.771599\pi$$
−0.753425 + 0.657534i $$0.771599\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 81.0000 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ −240.000 −2.82353
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −78.0000 −0.876404 −0.438202 0.898876i $$-0.644385\pi$$
−0.438202 + 0.898876i $$0.644385\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −130.000 −1.34021 −0.670103 0.742268i $$-0.733750\pi$$
−0.670103 + 0.742268i $$0.733750\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −40.0000 −0.396040 −0.198020 0.980198i $$-0.563451\pi$$
−0.198020 + 0.980198i $$0.563451\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 120.000 1.10092 0.550459 0.834862i $$-0.314453\pi$$
0.550459 + 0.834862i $$0.314453\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −30.0000 −0.265487 −0.132743 0.991150i $$-0.542379\pi$$
−0.132743 + 0.991150i $$0.542379\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 216.000 1.84615
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 121.000 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −112.000 −0.896000
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −210.000 −1.53285 −0.766423 0.642336i $$-0.777965\pi$$
−0.766423 + 0.642336i $$0.777965\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 320.000 2.20690
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 280.000 1.87919 0.939597 0.342282i $$-0.111200\pi$$
0.939597 + 0.342282i $$0.111200\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 270.000 1.76471
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −264.000 −1.68153 −0.840764 0.541401i $$-0.817894\pi$$
−0.840764 + 0.541401i $$0.817894\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 407.000 2.40828
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −104.000 −0.601156 −0.300578 0.953757i $$-0.597180\pi$$
−0.300578 + 0.953757i $$0.597180\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ −360.000 −1.98895 −0.994475 0.104972i $$-0.966525\pi$$
−0.994475 + 0.104972i $$0.966525\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −192.000 −1.03784
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ 190.000 0.984456 0.492228 0.870466i $$-0.336183\pi$$
0.492228 + 0.870466i $$0.336183\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 56.0000 0.284264 0.142132 0.989848i $$-0.454604\pi$$
0.142132 + 0.989848i $$0.454604\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 144.000 0.702439
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 720.000 3.25792
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 351.000 1.56000
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 120.000 0.524017 0.262009 0.965066i $$-0.415615\pi$$
0.262009 + 0.965066i $$0.415615\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 210.000 0.901288 0.450644 0.892704i $$-0.351194\pi$$
0.450644 + 0.892704i $$0.351194\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −418.000 −1.73444 −0.867220 0.497925i $$-0.834095\pi$$
−0.867220 + 0.497925i $$0.834095\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −392.000 −1.60000
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −510.000 −1.98444 −0.992218 0.124514i $$-0.960263\pi$$
−0.992218 + 0.124514i $$0.960263\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −360.000 −1.37931
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ −448.000 −1.69057
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −520.000 −1.93309 −0.966543 0.256506i $$-0.917429\pi$$
−0.966543 + 0.256506i $$0.917429\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 504.000 1.81949 0.909747 0.415162i $$-0.136275\pi$$
0.909747 + 0.415162i $$0.136275\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −462.000 −1.64413 −0.822064 0.569395i $$-0.807178\pi$$
−0.822064 + 0.569395i $$0.807178\pi$$
$$282$$ 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
$$289$$ 611.000 2.11419
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −136.000 −0.464164 −0.232082 0.972696i $$-0.574554\pi$$
−0.232082 + 0.972696i $$0.574554\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −960.000 −3.14754
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ −50.0000 −0.159744 −0.0798722 0.996805i $$-0.525451\pi$$
−0.0798722 + 0.996805i $$0.525451\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −616.000 −1.94322 −0.971609 0.236593i $$-0.923969\pi$$
−0.971609 + 0.236593i $$0.923969\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 936.000 2.88000
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 216.000 0.648649
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −350.000 −1.03858 −0.519288 0.854599i $$-0.673803\pi$$
−0.519288 + 0.854599i $$0.673803\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 0 0
$$349$$ −360.000 −1.03152 −0.515759 0.856734i $$-0.672490\pi$$
−0.515759 + 0.856734i $$0.672490\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 450.000 1.27479 0.637394 0.770538i $$-0.280012\pi$$
0.637394 + 0.770538i $$0.280012\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 880.000 2.41096
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ −162.000 −0.439024
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 504.000 1.35121 0.675603 0.737265i $$-0.263883\pi$$
0.675603 + 0.737265i $$0.263883\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −960.000 −2.54642
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −680.000 −1.74807 −0.874036 0.485861i $$-0.838506\pi$$
−0.874036 + 0.485861i $$0.838506\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −456.000 −1.14861 −0.574307 0.818640i $$-0.694729\pi$$
−0.574307 + 0.818640i $$0.694729\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 798.000 1.99002 0.995012 0.0997506i $$-0.0318045\pi$$
0.995012 + 0.0997506i $$0.0318045\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −648.000 −1.60000
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 782.000 1.91198 0.955990 0.293399i $$-0.0947863\pi$$
0.955990 + 0.293399i $$0.0947863\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ −840.000 −1.99525 −0.997625 0.0688836i $$-0.978056\pi$$
−0.997625 + 0.0688836i $$0.978056\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1170.00 2.75294
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ −290.000 −0.669746 −0.334873 0.942263i $$-0.608693\pi$$
−0.334873 + 0.942263i $$0.608693\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ 441.000 1.00000
$$442$$ 0 0
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 0 0
$$445$$ 624.000 1.40225
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 702.000 1.56347 0.781737 0.623608i $$-0.214334\pi$$
0.781737 + 0.623608i $$0.214334\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −850.000 −1.85996 −0.929978 0.367615i $$-0.880174\pi$$
−0.929978 + 0.367615i $$0.880174\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 760.000 1.64859 0.824295 0.566161i $$-0.191572\pi$$
0.824295 + 0.566161i $$0.191572\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 504.000 1.05660
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 576.000 1.19751
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1040.00 2.14433
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ −1200.00 −2.43408
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 320.000 0.633663
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 440.000 0.864440 0.432220 0.901768i $$-0.357730\pi$$
0.432220 + 0.901768i $$0.357730\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 558.000 1.07102 0.535509 0.844530i $$-0.320120\pi$$
0.535509 + 0.844530i $$0.320120\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 529.000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −432.000 −0.810507
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −840.000 −1.55268 −0.776340 0.630314i $$-0.782926\pi$$
−0.776340 + 0.630314i $$0.782926\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −960.000 −1.76147
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 1080.00 1.96721
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1064.00 −1.91023 −0.955117 0.296230i $$-0.904271\pi$$
−0.955117 + 0.296230i $$0.904271\pi$$
$$558$$ 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
$$565$$ 240.000 0.424779
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 462.000 0.811951 0.405975 0.913884i $$-0.366932\pi$$
0.405975 + 0.913884i $$0.366932\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −1150.00 −1.99307 −0.996534 0.0831889i $$-0.973490\pi$$
−0.996534 + 0.0831889i $$0.973490\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −1728.00 −2.95385
$$586$$ 0 0
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 930.000 1.56830 0.784148 0.620573i $$-0.213100\pi$$
0.784148 + 0.620573i $$0.213100\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ −1102.00 −1.83361 −0.916805 0.399334i $$-0.869241\pi$$
−0.916805 + 0.399334i $$0.869241\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −968.000 −1.60000
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −1224.00 −1.99674 −0.998369 0.0570962i $$-0.981816\pi$$
−0.998369 + 0.0570962i $$0.981816\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 210.000 0.340357 0.170178 0.985413i $$-0.445566\pi$$
0.170178 + 0.985413i $$0.445566\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −79.0000 −0.126400
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 720.000 1.14467
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1176.00 1.84615
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1218.00 −1.90016 −0.950078 0.312012i $$-0.898997\pi$$
−0.950078 + 0.312012i $$0.898997\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1144.00 1.75191 0.875957 0.482389i $$-0.160231\pi$$
0.875957 + 0.482389i $$0.160231\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −990.000 −1.50685
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 600.000 0.907716 0.453858 0.891074i $$-0.350047\pi$$
0.453858 + 0.891074i $$0.350047\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −770.000 −1.14413 −0.572065 0.820208i $$-0.693858\pi$$
−0.572065 + 0.820208i $$0.693858\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −104.000 −0.153619 −0.0768095 0.997046i $$-0.524473\pi$$
−0.0768095 + 0.997046i $$0.524473\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 1680.00 2.45255
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 1344.00 1.95065
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −540.000 −0.774749
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −520.000 −0.741797 −0.370899 0.928673i $$-0.620950\pi$$
−0.370899 + 0.928673i $$0.620950\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1320.00 −1.86178 −0.930889 0.365303i $$-0.880965\pi$$
−0.930889 + 0.365303i $$0.880965\pi$$
$$710$$ 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 $$\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −1560.00 −2.15172
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 729.000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 216.000 0.294679 0.147340 0.989086i $$-0.452929\pi$$
0.147340 + 0.989086i $$0.452929\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ −2240.00 −3.00671
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −936.000 −1.23646 −0.618230 0.785997i $$-0.712150\pi$$
−0.618230 + 0.785997i $$0.712150\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 78.0000 0.102497 0.0512484 0.998686i $$-0.483680\pi$$
0.0512484 + 0.998686i $$0.483680\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −2160.00 −2.82353
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −962.000 −1.25098 −0.625488 0.780234i $$-0.715100\pi$$
−0.625488 + 0.780234i $$0.715100\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1496.00 1.93532 0.967658 0.252264i $$-0.0811751\pi$$
0.967658 + 0.252264i $$0.0811751\pi$$
$$774$$ 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
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 2112.00 2.69045
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 2880.00 3.63178
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1144.00 1.43538 0.717691 0.696361i $$-0.245199\pi$$
0.717691 + 0.696361i $$0.245199\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −702.000 −0.876404
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 1518.00 1.87639 0.938195 0.346106i $$-0.112496\pi$$
0.938195 + 0.346106i $$0.112496\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1400.00 1.70524 0.852619 0.522533i $$-0.175013\pi$$
0.852619 + 0.522533i $$0.175013\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ 1080.00 1.30277 0.651387 0.758745i $$-0.274187\pi$$
0.651387 + 0.758745i $$0.274187\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1470.00 1.76471
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 759.000 0.902497
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −3256.00 −3.85325
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 1656.00 1.94138 0.970692 0.240328i $$-0.0772551\pi$$
0.970692 + 0.240328i $$0.0772551\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −1650.00 −1.92532 −0.962660 0.270712i $$-0.912741\pi$$
−0.962660 + 0.270712i $$0.912741\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ 832.000 0.961850
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −1170.00 −1.34021
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 696.000 0.793615 0.396807 0.917902i $$-0.370118\pi$$
0.396807 + 0.917902i $$0.370118\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 738.000 0.837684 0.418842 0.908059i $$-0.362436\pi$$
0.418842 + 0.908059i $$0.362436\pi$$
$$882$$ 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
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 1680.00 1.86459
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 2880.00 3.18232
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ −360.000 −0.396040
$$910$$ 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
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 936.000 1.01189
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −258.000 −0.277718 −0.138859 0.990312i $$-0.544343\pi$$
−0.138859 + 0.990312i $$0.544343\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −430.000 −0.458911 −0.229456 0.973319i $$-0.573695\pi$$
−0.229456 + 0.973319i $$0.573695\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1160.00 −1.23273 −0.616366 0.787460i $$-0.711396\pi$$
−0.616366 + 0.787460i $$0.711396\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$948$$ 0 0
$$949$$ −2640.00 −2.78188
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 1230.00 1.29066 0.645331 0.763903i $$-0.276720\pi$$
0.645331 + 0.763903i $$0.276720\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 961.000 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −1520.00 −1.57513
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −1890.00 −1.93449 −0.967247 0.253838i $$-0.918307\pi$$
−0.967247 + 0.253838i $$0.918307\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 1080.00 1.10092
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ −448.000 −0.454822
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −744.000 −0.746239 −0.373119 0.927783i $$-0.621712\pi$$
−0.373119 + 0.927783i $$0.621712\pi$$
$$998$$ 0 0
$$999$$ 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 256.3.c.a.255.1 1
3.2 odd 2 2304.3.g.f.1279.1 1
4.3 odd 2 CM 256.3.c.a.255.1 1
8.3 odd 2 256.3.c.b.255.1 1
8.5 even 2 256.3.c.b.255.1 1
12.11 even 2 2304.3.g.f.1279.1 1
16.3 odd 4 128.3.d.a.63.2 yes 2
16.5 even 4 128.3.d.a.63.1 2
16.11 odd 4 128.3.d.a.63.1 2
16.13 even 4 128.3.d.a.63.2 yes 2
24.5 odd 2 2304.3.g.a.1279.1 1
24.11 even 2 2304.3.g.a.1279.1 1
48.5 odd 4 1152.3.b.a.703.2 2
48.11 even 4 1152.3.b.a.703.2 2
48.29 odd 4 1152.3.b.a.703.1 2
48.35 even 4 1152.3.b.a.703.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
128.3.d.a.63.1 2 16.5 even 4
128.3.d.a.63.1 2 16.11 odd 4
128.3.d.a.63.2 yes 2 16.3 odd 4
128.3.d.a.63.2 yes 2 16.13 even 4
256.3.c.a.255.1 1 1.1 even 1 trivial
256.3.c.a.255.1 1 4.3 odd 2 CM
256.3.c.b.255.1 1 8.3 odd 2
256.3.c.b.255.1 1 8.5 even 2
1152.3.b.a.703.1 2 48.29 odd 4
1152.3.b.a.703.1 2 48.35 even 4
1152.3.b.a.703.2 2 48.5 odd 4
1152.3.b.a.703.2 2 48.11 even 4
2304.3.g.a.1279.1 1 24.5 odd 2
2304.3.g.a.1279.1 1 24.11 even 2
2304.3.g.f.1279.1 1 3.2 odd 2
2304.3.g.f.1279.1 1 12.11 even 2