# Properties

 Label 368.7.f.a.321.1 Level $368$ Weight $7$ Character 368.321 Self dual yes Analytic conductor $84.660$ Analytic rank $0$ Dimension $1$ CM discriminant -23 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,7,Mod(321,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("368.321");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$368 = 2^{4} \cdot 23$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 368.f (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: $$84.6599027721$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 23) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 321.1 Character $$\chi$$ $$=$$ 368.321

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+38.0000 q^{3} +715.000 q^{9} +O(q^{10})$$ $$q+38.0000 q^{3} +715.000 q^{9} +1082.00 q^{13} +12167.0 q^{23} +15625.0 q^{25} -532.000 q^{27} +30746.0 q^{29} -58754.0 q^{31} +41116.0 q^{39} +43634.0 q^{41} +205342. q^{47} +117649. q^{49} +253942. q^{59} +462346. q^{69} -667154. q^{71} +725042. q^{73} +593750. q^{75} -541451. q^{81} +1.16835e6 q^{87} -2.23265e6 q^{93} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/368\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$97$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 38.0000 1.40741 0.703704 0.710494i $$-0.251528\pi$$
0.703704 + 0.710494i $$0.251528\pi$$
$$4$$ 0 0
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 715.000 0.980796
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 1082.00 0.492490 0.246245 0.969208i $$-0.420803\pi$$
0.246245 + 0.969208i $$0.420803\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 12167.0 1.00000
$$24$$ 0 0
$$25$$ 15625.0 1.00000
$$26$$ 0 0
$$27$$ −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 $$-0.946872\pi$$
−0.986103 + 0.166134i $$0.946872\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 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
$$46$$ 0 0
$$47$$ 205342. 1.97781 0.988904 0.148555i $$-0.0474621\pi$$
0.988904 + 0.148555i $$0.0474621\pi$$
$$48$$ 0 0
$$49$$ 117649. 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 253942. 1.23646 0.618228 0.785999i $$-0.287851\pi$$
0.618228 + 0.785999i $$0.287851\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 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 $$-0.881947\pi$$
−0.932011 + 0.362430i $$0.881947\pi$$
$$72$$ 0 0
$$73$$ 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
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ −541451. −1.01884
$$82$$ 0 0
$$83$$ 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
$$92$$ 0 0
$$93$$ −2.23265e6 −2.77570
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 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 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 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
$$116$$ 0 0
$$117$$ 773630. 0.483032
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.77156e6 1.00000
$$122$$ 0 0
$$123$$ 1.65809e6 0.891032
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2.70490e6 −1.32050 −0.660252 0.751044i $$-0.729551\pi$$
−0.660252 + 0.751044i $$0.729551\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −3.32143e6 −1.47745 −0.738723 0.674009i $$-0.764571\pi$$
−0.738723 + 0.674009i $$0.764571\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 5.20149e6 1.93680 0.968398 0.249411i $$-0.0802372\pi$$
0.968398 + 0.249411i $$0.0802372\pi$$
$$140$$ 0 0
$$141$$ 7.80300e6 2.78358
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.47066e6 1.40741
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ −6.18955e6 −1.79775 −0.898873 0.438208i $$-0.855613\pi$$
−0.898873 + 0.438208i $$0.855613\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 5.98500e6 1.38198 0.690989 0.722865i $$-0.257175\pi$$
0.690989 + 0.722865i $$0.257175\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.07493e6 1.30434 0.652171 0.758072i $$-0.273858\pi$$
0.652171 + 0.758072i $$0.273858\pi$$
$$168$$ 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 $$0.439241\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 9.64980e6 1.74020
$$178$$ 0 0
$$179$$ 3.91917e6 0.683338 0.341669 0.939820i $$-0.389008\pi$$
0.341669 + 0.939820i $$0.389008\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 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
$$193$$ 3.99168e6 0.555244 0.277622 0.960690i $$-0.410454\pi$$
0.277622 + 0.960690i $$0.410454\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 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 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 8.69940e6 0.980796
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1.26968e7 1.35160 0.675800 0.737085i $$-0.263798\pi$$
0.675800 + 0.737085i $$0.263798\pi$$
$$212$$ 0 0
$$213$$ −2.53519e7 −2.62344
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 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 $$-0.517898\pi$$
−0.0561999 + 0.998420i $$0.517898\pi$$
$$224$$ 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
$$232$$ 0 0
$$233$$ 7.80984e6 0.617411 0.308706 0.951158i $$-0.400104\pi$$
0.308706 + 0.951158i $$0.400104\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2.69836e7 −1.97654 −0.988271 0.152712i $$-0.951199\pi$$
−0.988271 + 0.152712i $$0.951199\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ −2.01873e7 −1.40689
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 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
$$257$$ −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
$$262$$ 0 0
$$263$$ 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 $$-0.969191\pi$$
−0.995320 + 0.0966371i $$0.969191\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3.91296e7 −1.84105 −0.920527 0.390680i $$-0.872240\pi$$
−0.920527 + 0.390680i $$0.872240\pi$$
$$278$$ 0 0
$$279$$ −4.20091e7 −1.93433
$$280$$ 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
$$289$$ 2.41376e7 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1.31647e7 0.492490
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1.92205e7 0.690934
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5.07075e7 −1.75250 −0.876248 0.481860i $$-0.839961\pi$$
−0.876248 + 0.481860i $$0.839961\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2.58677e7 0.859958 0.429979 0.902839i $$-0.358521\pi$$
0.429979 + 0.902839i $$0.358521\pi$$
$$312$$ 0 0
$$313$$ 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
$$324$$ 0 0
$$325$$ 1.69062e7 0.492490
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.25286e7 −1.99998 −0.999989 0.00477828i $$-0.998479\pi$$
−0.999989 + 0.00477828i $$0.998479\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 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
$$347$$ −708554. −0.0169584 −0.00847919 0.999964i $$-0.502699\pi$$
−0.00847919 + 0.999964i $$0.502699\pi$$
$$348$$ 0 0
$$349$$ 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 $$0.833687\pi$$
$$354$$ 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
$$361$$ 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 $$\pi$$
$$368$$ 0 0
$$369$$ 3.11983e7 0.620943
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 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 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −1.26214e8 −2.07937
$$394$$ 0 0
$$395$$ 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
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ −6.35718e7 −0.971291
$$404$$ 0 0
$$405$$ 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
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 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 $$\pi$$
$$422$$ 0 0
$$423$$ 1.46820e8 1.93983
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 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
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 1.16094e8 1.37220 0.686099 0.727509i $$-0.259322\pi$$
0.686099 + 0.727509i $$0.259322\pi$$
$$440$$ 0 0
$$441$$ 8.41190e7 0.980796
$$442$$ 0 0
$$443$$ −1.42821e8 −1.64279 −0.821393 0.570363i $$-0.806803\pi$$
−0.821393 + 0.570363i $$0.806803\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1.58038e8 1.74591 0.872955 0.487801i $$-0.162201\pi$$
0.872955 + 0.487801i $$0.162201\pi$$
$$450$$ 0 0
$$451$$ 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 $$-0.398749\pi$$
0.312753 + 0.949834i $$0.398749\pi$$
$$464$$ 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
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −1.55985e8 −1.35050 −0.675252 0.737587i $$-0.735965\pi$$
−0.675252 + 0.737587i $$0.735965\pi$$
$$488$$ 0 0
$$489$$ 2.27430e8 1.94501
$$490$$ 0 0
$$491$$ −1.90755e8 −1.61151 −0.805754 0.592250i $$-0.798240\pi$$
−0.805754 + 0.592250i $$0.798240\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −757946. −0.00610010 −0.00305005 0.999995i $$-0.500971\pi$$
−0.00305005 + 0.999995i $$0.500971\pi$$
$$500$$ 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
$$508$$ 0 0
$$509$$ −2.00351e8 −1.51928 −0.759641 0.650343i $$-0.774625\pi$$
−0.759641 + 0.650343i $$0.774625\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 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 $$\pi$$
$$522$$ 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
$$529$$ 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
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −3.15808e8 −1.99449 −0.997245 0.0741740i $$-0.976368\pi$$
−0.997245 + 0.0741740i $$0.976368\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.27720e8 −0.780361 −0.390180 0.920738i $$-0.627587\pi$$
−0.390180 + 0.920738i $$0.627587\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 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
$$569$$ 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
$$576$$ 0 0
$$577$$ 9.00812e7 0.468929 0.234464 0.972125i $$-0.424666\pi$$
0.234464 + 0.972125i $$0.424666\pi$$
$$578$$ 0 0
$$579$$ 1.51684e8 0.781455
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 3.99910e8 1.97719 0.988594 0.150604i $$-0.0481217\pi$$
0.988594 + 0.150604i $$0.0481217\pi$$
$$588$$ 0 0
$$589$$ 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
$$598$$ 0 0
$$599$$ −1.62397e8 −0.755611 −0.377806 0.925885i $$-0.623321\pi$$
−0.377806 + 0.925885i $$0.623321\pi$$
$$600$$ 0 0
$$601$$ −4.34057e8 −1.99951 −0.999755 0.0221248i $$-0.992957\pi$$
−0.999755 + 0.0221248i $$0.992957\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 3.69151e8 1.65059 0.825294 0.564704i $$-0.191010\pi$$
0.825294 + 0.564704i $$0.191010\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2.22180e8 0.974050
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 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
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 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
$$634$$ 0 0
$$635$$ 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 $$-0.670683\pi$$
−0.510888 + 0.859648i $$0.670683\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −7.17455e7 −0.257665 −0.128832 0.991666i $$-0.541123\pi$$
−0.128832 + 0.991666i $$0.541123\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 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 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.74087e8 1.26065
$$668$$ 0 0
$$669$$ −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
$$676$$ 0 0
$$677$$ 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 $$-0.912664\pi$$
−0.962595 + 0.270945i $$0.912664\pi$$
$$684$$ 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 $$-0.106125\pi$$
0.944934 + 0.327261i $$0.106125\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 2.96774e8 0.868949
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 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
$$716$$ 0 0
$$717$$ −1.02538e9 −2.78180
$$718$$ 0 0
$$719$$ −6.96357e8 −1.87346 −0.936732 0.350047i $$-0.886166\pi$$
−0.936732 + 0.350047i $$0.886166\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 4.80406e8 1.26065
$$726$$ 0 0
$$727$$ 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
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −1.26013e8 −0.312236 −0.156118 0.987738i $$-0.549898\pi$$
−0.156118 + 0.987738i $$0.549898\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 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
$$757$$ 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 $$0.236341\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 2.74765e8 0.608942
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ −8.80751e8 −1.92172
$$772$$ 0 0
$$773$$ 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
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 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
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −1.44120e9 −2.74222
$$808$$ 0 0
$$809$$ −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 $$-0.0464044\pi$$
0.989392 + 0.145268i $$0.0464044\pi$$
$$812$$ 0 0
$$813$$ −1.50551e9 −2.80164
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 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 $$-0.242917\pi$$
0.722664 + 0.691199i $$0.242917\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ −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
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 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
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −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 $$-0.991632\pi$$
−0.999654 + 0.0262855i $$0.991632\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −4.00781e8 −0.623555 −0.311777 0.950155i $$-0.600924\pi$$
−0.311777 + 0.950155i $$0.600924\pi$$
$$864$$ 0 0
$$865$$ 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
$$876$$ 0 0
$$877$$ −1.86924e8 −0.277119 −0.138560 0.990354i $$-0.544247\pi$$
−0.138560 + 0.990354i $$0.544247\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 1.36875e9 1.98812 0.994060 0.108838i $$-0.0347129\pi$$
0.994060 + 0.108838i $$0.0347129\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1.39031e9 1.99223 0.996117 0.0880391i $$-0.0280600\pi$$
0.996117 + 0.0880391i $$0.0280600\pi$$
$$888$$ 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
$$897$$ 5.00258e8 0.693134
$$898$$ 0 0
$$899$$ −1.80645e9 −2.48626
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 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
$$922$$ 0 0
$$923$$ −7.21861e8 −0.918012
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 8.42564e8 1.05089 0.525443 0.850829i $$-0.323900\pi$$
0.525443 + 0.850829i $$0.323900\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 9.82974e8 1.21031
$$934$$ 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
$$943$$ 5.30895e8 0.633102
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.32946e9 1.56539 0.782697 0.622403i $$-0.213843\pi$$
0.782697 + 0.622403i $$0.213843\pi$$
$$948$$ 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
$$956$$ 0 0
$$957$$ 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 $$-0.0904071\pi$$
0.959936 + 0.280219i $$0.0904071\pi$$
$$968$$ 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
$$975$$ 6.42438e8 0.693134
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 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 $$-0.0934710\pi$$
0.957194 + 0.289446i $$0.0934710\pi$$
$$992$$ 0 0
$$993$$ −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 $$0.189581\pi$$
$$998$$ 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 368.7.f.a.321.1 1
4.3 odd 2 23.7.b.a.22.1 1
12.11 even 2 207.7.d.a.91.1 1
23.22 odd 2 CM 368.7.f.a.321.1 1
92.91 even 2 23.7.b.a.22.1 1
276.275 odd 2 207.7.d.a.91.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
23.7.b.a.22.1 1 4.3 odd 2
23.7.b.a.22.1 1 92.91 even 2
207.7.d.a.91.1 1 12.11 even 2
207.7.d.a.91.1 1 276.275 odd 2
368.7.f.a.321.1 1 1.1 even 1 trivial
368.7.f.a.321.1 1 23.22 odd 2 CM