LMFDB - Embedded newform 384.5.g.a.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(127,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("384.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.g (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:	no
Analytic conductor:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 104 x^{14} - 588 x^{13} + 3846 x^{12} - 15796 x^{11} + 66992 x^{10} - 203224 x^{9} + \cdots + 2196513 $ x^16 - 8x^15 + 104x^14 - 588x^13 + 3846x^12 - 15796x^11 + 66992x^10 - 203224x^9 + 604877x^8 - 1351612x^7 + 2878620x^6 - 4608408x^5 + 6889612x^4 - 7344036x^3 + 7202376x^2 - 4122756*x + 2196513
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{86}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$0.500000 - 1.33456i$ of defining polynomial
Character	$\chi$	$=$	384.127
Dual form			384.5.g.a.127.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} -31.6602 q^{5} +17.9843i q^{7} -27.0000 q^{9} +O(q^{10})$	$q-5.19615i q^{3} -31.6602 q^{5} +17.9843i q^{7} -27.0000 q^{9} +149.516i q^{11} -41.4325 q^{13} +164.511i q^{15} +473.865 q^{17} +162.436i q^{19} +93.4490 q^{21} -714.864i q^{23} +377.369 q^{25} +140.296i q^{27} -907.514 q^{29} -178.280i q^{31} +776.906 q^{33} -569.386i q^{35} +1462.16 q^{37} +215.290i q^{39} -1950.58 q^{41} -2329.40i q^{43} +854.826 q^{45} -2393.71i q^{47} +2077.57 q^{49} -2462.27i q^{51} -3530.12 q^{53} -4733.70i q^{55} +844.042 q^{57} +2351.29i q^{59} -2073.07 q^{61} -485.575i q^{63} +1311.76 q^{65} -4577.69i q^{67} -3714.54 q^{69} -1520.80i q^{71} +5781.95 q^{73} -1960.87i q^{75} -2688.93 q^{77} +6050.35i q^{79} +729.000 q^{81} -6107.73i q^{83} -15002.7 q^{85} +4715.58i q^{87} +2631.72 q^{89} -745.133i q^{91} -926.370 q^{93} -5142.76i q^{95} +11406.0 q^{97} -4036.92i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 96 q^{5} - 432 q^{9}+O(q^{10}) $ 16 * q - 96 * q^5 - 432 * q^9	$ 16 q - 96 q^{5} - 432 q^{9} - 480 q^{13} - 480 q^{17} + 2672 q^{25} + 3360 q^{29} - 1120 q^{37} + 1440 q^{41} + 2592 q^{45} - 2480 q^{49} - 3552 q^{53} - 7488 q^{57} - 18272 q^{61} - 1344 q^{65} - 8480 q^{73} + 17280 q^{77} + 11664 q^{81} + 29888 q^{85} + 18720 q^{89} - 28800 q^{93} + 13088 q^{97}+O(q^{100}) $ 16 * q - 96 * q^5 - 432 * q^9 - 480 * q^13 - 480 * q^17 + 2672 * q^25 + 3360 * q^29 - 1120 * q^37 + 1440 * q^41 + 2592 * q^45 - 2480 * q^49 - 3552 * q^53 - 7488 * q^57 - 18272 * q^61 - 1344 * q^65 - 8480 * q^73 + 17280 * q^77 + 11664 * q^81 + 29888 * q^85 + 18720 * q^89 - 28800 * q^93 + 13088 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 5.19615i	− 0.577350i
$3$	− 5.19615i	− 0.577350i
$4$	0	0
$5$	−31.6602	−1.26641	−0.633204	−	0.773985i	$-0.718261\pi$
$5$	−31.6602	−1.26641	−0.633204	+	0.773985i	$0.718261\pi$
$6$	0	0
$7$	17.9843i	0.367026i	0.983017	+	0.183513i	$0.0587469\pi$
$7$	17.9843i	0.367026i	−0.983017	+	0.183513i	$0.941253\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	149.516i	1.23567i	0.786309	+	0.617833i	$0.211989\pi$
$11$	149.516i	1.23567i	−0.786309	+	0.617833i	$0.788011\pi$
$12$	0	0
$13$	−41.4325	−0.245163	−0.122581	−	0.992458i	$-0.539117\pi$
$13$	−41.4325	−0.245163	−0.122581	+	0.992458i	$0.539117\pi$
$14$	0	0
$15$	164.511i	0.731161i
$16$	0	0
$17$	473.865	1.63967	0.819835	−	0.572599i	$-0.194065\pi$
$17$	473.865	1.63967	0.819835	+	0.572599i	$0.194065\pi$
$18$	0	0
$19$	162.436i	0.449961i	0.974363	+	0.224981i	$0.0722319\pi$
$19$	162.436i	0.449961i	−0.974363	+	0.224981i	$0.927768\pi$
$20$	0	0
$21$	93.4490	0.211903
$22$	0	0
$23$	− 714.864i	− 1.35135i	−0.737200	−	0.675674i	$-0.763853\pi$
$23$	− 714.864i	− 1.35135i	0.737200	−	0.675674i	$-0.236147\pi$
$24$	0	0
$25$	377.369	0.603790
$26$	0	0
$27$	140.296i	0.192450i
$28$	0	0
$29$	−907.514	−1.07909	−0.539544	−	0.841957i	$-0.681404\pi$
$29$	−907.514	−1.07909	−0.539544	+	0.841957i	$0.681404\pi$
$30$	0	0
$31$	− 178.280i	− 0.185515i	−0.995689	−	0.0927575i	$-0.970432\pi$
$31$	− 178.280i	− 0.185515i	0.995689	−	0.0927575i	$-0.0295681\pi$
$32$	0	0
$33$	776.906	0.713413
$34$	0	0
$35$	− 569.386i	− 0.464805i
$36$	0	0
$37$	1462.16	1.06805	0.534024	−	0.845470i	$-0.320679\pi$
$37$	1462.16	1.06805	0.534024	+	0.845470i	$0.320679\pi$
$38$	0	0
$39$	215.290i	0.141545i
$40$	0	0
$41$	−1950.58	−1.16037	−0.580184	−	0.814485i	$-0.697020\pi$
$41$	−1950.58	−1.16037	−0.580184	+	0.814485i	$0.697020\pi$
$42$	0	0
$43$	− 2329.40i	− 1.25981i	−0.776670	−	0.629907i	$-0.783093\pi$
$43$	− 2329.40i	− 1.25981i	0.776670	−	0.629907i	$-0.216907\pi$
$44$	0	0
$45$	854.826	0.422136
$46$	0	0
$47$	− 2393.71i	− 1.08362i	−0.840501	−	0.541809i	$-0.817739\pi$
$47$	− 2393.71i	− 1.08362i	0.840501	−	0.541809i	$-0.182261\pi$
$48$	0	0
$49$	2077.57	0.865292
$50$	0	0
$51$	− 2462.27i	− 0.946664i
$52$	0	0
$53$	−3530.12	−1.25672	−0.628359	−	0.777923i	$-0.716273\pi$
$53$	−3530.12	−1.25672	−0.628359	+	0.777923i	$0.716273\pi$
$54$	0	0
$55$	− 4733.70i	− 1.56486i
$56$	0	0
$57$	844.042	0.259785
$58$	0	0
$59$	2351.29i	0.675463i	0.941243	+	0.337731i	$0.109660\pi$
$59$	2351.29i	0.675463i	−0.941243	+	0.337731i	$0.890340\pi$
$60$	0	0
$61$	−2073.07	−0.557126	−0.278563	−	0.960418i	$-0.589858\pi$
$61$	−2073.07	−0.557126	−0.278563	+	0.960418i	$0.589858\pi$
$62$	0	0
$63$	− 485.575i	− 0.122342i
$64$	0	0
$65$	1311.76	0.310476
$66$	0	0
$67$	− 4577.69i	− 1.01976i	−0.860246	−	0.509879i	$-0.829690\pi$
$67$	− 4577.69i	− 1.01976i	0.860246	−	0.509879i	$-0.170310\pi$
$68$	0	0
$69$	−3714.54	−0.780202
$70$	0	0
$71$	− 1520.80i	− 0.301686i	−0.988558	−	0.150843i	$-0.951801\pi$
$71$	− 1520.80i	− 0.301686i	0.988558	−	0.150843i	$-0.0481988\pi$
$72$	0	0
$73$	5781.95	1.08500	0.542498	−	0.840057i	$-0.317479\pi$
$73$	5781.95	1.08500	0.542498	+	0.840057i	$0.317479\pi$
$74$	0	0
$75$	− 1960.87i	− 0.348598i
$76$	0	0
$77$	−2688.93	−0.453522
$78$	0	0
$79$	6050.35i	0.969451i	0.874666	+	0.484726i	$0.161081\pi$
$79$	6050.35i	0.969451i	−0.874666	+	0.484726i	$0.838919\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	− 6107.73i	− 0.886592i	−0.896375	−	0.443296i	$-0.853809\pi$
$83$	− 6107.73i	− 0.886592i	0.896375	−	0.443296i	$-0.146191\pi$
$84$	0	0
$85$	−15002.7	−2.07649
$86$	0	0
$87$	4715.58i	0.623012i
$88$	0	0
$89$	2631.72	0.332246	0.166123	−	0.986105i	$-0.446875\pi$
$89$	2631.72	0.332246	0.166123	+	0.986105i	$0.446875\pi$
$90$	0	0
$91$	− 745.133i	− 0.0899811i
$92$	0	0
$93$	−926.370	−0.107107
$94$	0	0
$95$	− 5142.76i	− 0.569835i
$96$	0	0
$97$	11406.0	1.21225	0.606124	−	0.795370i	$-0.292723\pi$
$97$	11406.0	1.21225	0.606124	+	0.795370i	$0.292723\pi$
$98$	0	0
$99$	− 4036.92i	− 0.411889i
$100$	0	0
$101$	−15069.8	−1.47729	−0.738643	−	0.674097i	$-0.764533\pi$
$101$	−15069.8	−1.47729	−0.738643	+	0.674097i	$0.764533\pi$
$102$	0	0
$103$	− 16620.8i	− 1.56667i	−0.621600	−	0.783335i	$-0.713517\pi$
$103$	− 16620.8i	− 1.56667i	0.621600	−	0.783335i	$-0.286483\pi$
$104$	0	0
$105$	−2958.62	−0.268355
$106$	0	0
$107$	− 19998.9i	− 1.74678i	−0.487021	−	0.873390i	$-0.661916\pi$
$107$	− 19998.9i	− 1.74678i	0.487021	−	0.873390i	$-0.338084\pi$
$108$	0	0
$109$	21468.7	1.80698	0.903490	−	0.428608i	$-0.140996\pi$
$109$	21468.7	1.80698	0.903490	+	0.428608i	$0.140996\pi$
$110$	0	0
$111$	− 7597.59i	− 0.616637i
$112$	0	0
$113$	15625.4	1.22370	0.611849	−	0.790974i	$-0.290426\pi$
$113$	15625.4	1.22370	0.611849	+	0.790974i	$0.290426\pi$
$114$	0	0
$115$	22632.7i	1.71136i
$116$	0	0
$117$	1118.68	0.0817209
$118$	0	0
$119$	8522.11i	0.601802i
$120$	0	0
$121$	−7713.94	−0.526872
$122$	0	0
$123$	10135.5i	0.669939i
$124$	0	0
$125$	7840.05	0.501763
$126$	0	0
$127$	− 23551.0i	− 1.46016i	−0.683361	−	0.730081i	$-0.739482\pi$
$127$	− 23551.0i	− 1.46016i	0.683361	−	0.730081i	$-0.260518\pi$
$128$	0	0
$129$	−12103.9	−0.727354
$130$	0	0
$131$	− 18659.9i	− 1.08734i	−0.839298	−	0.543672i	$-0.817034\pi$
$131$	− 18659.9i	− 1.08734i	0.839298	−	0.543672i	$-0.182966\pi$
$132$	0	0
$133$	−2921.29	−0.165147
$134$	0	0
$135$	− 4441.80i	− 0.243720i
$136$	0	0
$137$	−7944.70	−0.423288	−0.211644	−	0.977347i	$-0.567882\pi$
$137$	−7944.70	−0.423288	−0.211644	+	0.977347i	$0.567882\pi$
$138$	0	0
$139$	− 15815.5i	− 0.818566i	−0.912407	−	0.409283i	$-0.865779\pi$
$139$	− 15815.5i	− 0.818566i	0.912407	−	0.409283i	$-0.134221\pi$
$140$	0	0
$141$	−12438.1	−0.625628
$142$	0	0
$143$	− 6194.81i	− 0.302939i
$144$	0	0
$145$	28732.1	1.36657
$146$	0	0
$147$	− 10795.3i	− 0.499577i
$148$	0	0
$149$	12844.9	0.578571	0.289286	−	0.957243i	$-0.406582\pi$
$149$	12844.9	0.578571	0.289286	+	0.957243i	$0.406582\pi$
$150$	0	0
$151$	− 15024.8i	− 0.658952i	−0.944164	−	0.329476i	$-0.893128\pi$
$151$	− 15024.8i	− 0.658952i	0.944164	−	0.329476i	$-0.106872\pi$
$152$	0	0
$153$	−12794.4	−0.546557
$154$	0	0
$155$	5644.38i	0.234938i
$156$	0	0
$157$	5914.31	0.239941	0.119971	−	0.992777i	$-0.461720\pi$
$157$	5914.31	0.239941	0.119971	+	0.992777i	$0.461720\pi$
$158$	0	0
$159$	18343.1i	0.725567i
$160$	0	0
$161$	12856.3	0.495980
$162$	0	0
$163$	25753.8i	0.969317i	0.874703	+	0.484658i	$0.161056\pi$
$163$	25753.8i	0.969317i	−0.874703	+	0.484658i	$0.838944\pi$
$164$	0	0
$165$	−24597.0	−0.903472
$166$	0	0
$167$	32893.2i	1.17943i	0.807610	+	0.589717i	$0.200761\pi$
$167$	32893.2i	1.17943i	−0.807610	+	0.589717i	$0.799239\pi$
$168$	0	0
$169$	−26844.3	−0.939895
$170$	0	0
$171$	− 4385.77i	− 0.149987i
$172$	0	0
$173$	32636.4	1.09046	0.545231	−	0.838286i	$-0.316442\pi$
$173$	32636.4	1.09046	0.545231	+	0.838286i	$0.316442\pi$
$174$	0	0
$175$	6786.70i	0.221607i
$176$	0	0
$177$	12217.6	0.389978
$178$	0	0
$179$	41844.8i	1.30598i	0.757368	+	0.652988i	$0.226485\pi$
$179$	41844.8i	1.30598i	−0.757368	+	0.652988i	$0.773515\pi$
$180$	0	0
$181$	28443.4	0.868208	0.434104	−	0.900863i	$-0.357065\pi$
$181$	28443.4	0.868208	0.434104	+	0.900863i	$0.357065\pi$
$182$	0	0
$183$	10772.0i	0.321657i
$184$	0	0
$185$	−46292.2	−1.35258
$186$	0	0
$187$	70850.2i	2.02609i
$188$	0	0
$189$	−2523.12	−0.0706342
$190$	0	0
$191$	40793.1i	1.11820i	0.829100	+	0.559100i	$0.188853\pi$
$191$	40793.1i	1.11820i	−0.829100	+	0.559100i	$0.811147\pi$
$192$	0	0
$193$	59016.5	1.58438	0.792189	−	0.610276i	$-0.208941\pi$
$193$	59016.5	1.58438	0.792189	+	0.610276i	$0.208941\pi$
$194$	0	0
$195$	− 6816.11i	− 0.179253i
$196$	0	0
$197$	−52172.7	−1.34435	−0.672173	−	0.740395i	$-0.734639\pi$
$197$	−52172.7	−1.34435	−0.672173	+	0.740395i	$0.734639\pi$
$198$	0	0
$199$	7520.13i	0.189897i	0.995482	+	0.0949487i	$0.0302687\pi$
$199$	7520.13i	0.189897i	−0.995482	+	0.0949487i	$0.969731\pi$
$200$	0	0
$201$	−23786.4	−0.588758
$202$	0	0
$203$	− 16321.0i	− 0.396054i
$204$	0	0
$205$	61755.8	1.46950
$206$	0	0
$207$	19301.3i	0.450450i
$208$	0	0
$209$	−24286.7	−0.556002
$210$	0	0
$211$	− 20160.9i	− 0.452839i	−0.974030	−	0.226420i	$-0.927298\pi$
$211$	− 20160.9i	− 0.452839i	0.974030	−	0.226420i	$-0.0727021\pi$
$212$	0	0
$213$	−7902.31	−0.174179
$214$	0	0
$215$	73749.2i	1.59544i
$216$	0	0
$217$	3206.23	0.0680888
$218$	0	0
$219$	− 30043.9i	− 0.626423i
$220$	0	0
$221$	−19633.4	−0.401986
$222$	0	0
$223$	− 28211.4i	− 0.567302i	−0.958928	−	0.283651i	$-0.908454\pi$
$223$	− 28211.4i	− 0.567302i	0.958928	−	0.283651i	$-0.0915458\pi$
$224$	0	0
$225$	−10189.0	−0.201263
$226$	0	0
$227$	17681.6i	0.343139i	0.985172	+	0.171569i	$0.0548837\pi$
$227$	17681.6i	0.343139i	−0.985172	+	0.171569i	$0.945116\pi$
$228$	0	0
$229$	31709.2	0.604664	0.302332	−	0.953203i	$-0.402235\pi$
$229$	31709.2	0.604664	0.302332	+	0.953203i	$0.402235\pi$
$230$	0	0
$231$	13972.1i	0.261841i
$232$	0	0
$233$	−96886.3	−1.78464	−0.892319	−	0.451404i	$-0.850923\pi$
$233$	−96886.3	−1.78464	−0.892319	+	0.451404i	$0.850923\pi$
$234$	0	0
$235$	75785.5i	1.37230i
$236$	0	0
$237$	31438.5	0.559713
$238$	0	0
$239$	− 23638.0i	− 0.413824i	−0.978360	−	0.206912i	$-0.933659\pi$
$239$	− 23638.0i	− 0.413824i	0.978360	−	0.206912i	$-0.0663414\pi$
$240$	0	0
$241$	−100583.	−1.73178	−0.865888	−	0.500238i	$-0.833246\pi$
$241$	−100583.	−1.73178	−0.865888	+	0.500238i	$0.833246\pi$
$242$	0	0
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	−65776.2	−1.09581
$246$	0	0
$247$	− 6730.13i	− 0.110314i
$248$	0	0
$249$	−31736.7	−0.511874
$250$	0	0
$251$	− 58712.9i	− 0.931936i	−0.884802	−	0.465968i	$-0.845706\pi$
$251$	− 58712.9i	− 0.931936i	0.884802	−	0.465968i	$-0.154294\pi$
$252$	0	0
$253$	106883.	1.66982
$254$	0	0
$255$	77956.1i	1.19886i
$256$	0	0
$257$	−101128.	−1.53110	−0.765552	−	0.643374i	$-0.777534\pi$
$257$	−101128.	−1.53110	−0.765552	+	0.643374i	$0.777534\pi$
$258$	0	0
$259$	26295.8i	0.392001i
$260$	0	0
$261$	24502.9	0.359696
$262$	0	0
$263$	− 92966.2i	− 1.34404i	−0.740531	−	0.672022i	$-0.765426\pi$
$263$	− 92966.2i	− 1.34404i	0.740531	−	0.672022i	$-0.234574\pi$
$264$	0	0
$265$	111764.	1.59152
$266$	0	0
$267$	− 13674.8i	− 0.191823i
$268$	0	0
$269$	71178.9	0.983664	0.491832	−	0.870690i	$-0.336327\pi$
$269$	71178.9	0.983664	0.491832	+	0.870690i	$0.336327\pi$
$270$	0	0
$271$	73161.8i	0.996198i	0.867120	+	0.498099i	$0.165968\pi$
$271$	73161.8i	0.996198i	−0.867120	+	0.498099i	$0.834032\pi$
$272$	0	0
$273$	−3871.83	−0.0519506
$274$	0	0
$275$	56422.6i	0.746084i
$276$	0	0
$277$	67041.3	0.873741	0.436870	−	0.899524i	$-0.356087\pi$
$277$	67041.3	0.873741	0.436870	+	0.899524i	$0.356087\pi$
$278$	0	0
$279$	4813.56i	0.0618383i
$280$	0	0
$281$	15324.8	0.194081	0.0970406	−	0.995280i	$-0.469062\pi$
$281$	15324.8	0.194081	0.0970406	+	0.995280i	$0.469062\pi$
$282$	0	0
$283$	95061.9i	1.18695i	0.804851	+	0.593477i	$0.202245\pi$
$283$	95061.9i	1.18695i	−0.804851	+	0.593477i	$0.797755\pi$
$284$	0	0
$285$	−26722.5	−0.328994
$286$	0	0
$287$	− 35079.8i	− 0.425885i
$288$	0	0
$289$	141027.	1.68852
$290$	0	0
$291$	− 59267.6i	− 0.699892i
$292$	0	0
$293$	30864.3	0.359519	0.179759	−	0.983711i	$-0.442468\pi$
$293$	30864.3	0.359519	0.179759	+	0.983711i	$0.442468\pi$
$294$	0	0
$295$	− 74442.2i	− 0.855411i
$296$	0	0
$297$	−20976.5	−0.237804
$298$	0	0
$299$	29618.6i	0.331300i
$300$	0	0
$301$	41892.5	0.462385
$302$	0	0
$303$	78304.9i	0.852911i
$304$	0	0
$305$	65633.8	0.705550
$306$	0	0
$307$	− 68623.0i	− 0.728103i	−0.931379	−	0.364052i	$-0.881393\pi$
$307$	− 68623.0i	− 0.728103i	0.931379	−	0.364052i	$-0.118607\pi$
$308$	0	0
$309$	−86364.2	−0.904517
$310$	0	0
$311$	111446.i	1.15224i	0.817366	+	0.576119i	$0.195433\pi$
$311$	111446.i	1.15224i	−0.817366	+	0.576119i	$0.804567\pi$
$312$	0	0
$313$	−34038.5	−0.347442	−0.173721	−	0.984795i	$-0.555579\pi$
$313$	−34038.5	−0.347442	−0.173721	+	0.984795i	$0.555579\pi$
$314$	0	0
$315$	15373.4i	0.154935i
$316$	0	0
$317$	−93886.0	−0.934292	−0.467146	−	0.884180i	$-0.654718\pi$
$317$	−93886.0	−0.934292	−0.467146	+	0.884180i	$0.654718\pi$
$318$	0	0
$319$	− 135688.i	− 1.33339i
$320$	0	0
$321$	−103917.	−1.00850
$322$	0	0
$323$	76972.7i	0.737788i
$324$	0	0
$325$	−15635.3	−0.148027
$326$	0	0
$327$	− 111555.i	− 1.04326i
$328$	0	0
$329$	43049.2	0.397716
$330$	0	0
$331$	190986.i	1.74319i	0.490223	+	0.871597i	$0.336915\pi$
$331$	190986.i	1.74319i	−0.490223	+	0.871597i	$0.663085\pi$
$332$	0	0
$333$	−39478.2	−0.356016
$334$	0	0
$335$	144931.i	1.29143i
$336$	0	0
$337$	69378.3	0.610891	0.305446	−	0.952210i	$-0.401195\pi$
$337$	69378.3	0.610891	0.305446	+	0.952210i	$0.401195\pi$
$338$	0	0
$339$	− 81192.0i	− 0.706503i
$340$	0	0
$341$	26655.6	0.229235
$342$	0	0
$343$	80543.7i	0.684611i
$344$	0	0
$345$	117603.	0.988054
$346$	0	0
$347$	− 200109.i	− 1.66191i	−0.556337	−	0.830956i	$-0.687794\pi$
$347$	− 200109.i	− 1.66191i	0.556337	−	0.830956i	$-0.312206\pi$
$348$	0	0
$349$	−142378.	−1.16894	−0.584471	−	0.811415i	$-0.698698\pi$
$349$	−142378.	−1.16894	−0.584471	+	0.811415i	$0.698698\pi$
$350$	0	0
$351$	− 5812.82i	− 0.0471816i
$352$	0	0
$353$	−33608.4	−0.269711	−0.134855	−	0.990865i	$-0.543057\pi$
$353$	−33608.4	−0.269711	−0.134855	+	0.990865i	$0.543057\pi$
$354$	0	0
$355$	48148.9i	0.382058i
$356$	0	0
$357$	44282.2	0.347450
$358$	0	0
$359$	88008.1i	0.682863i	0.939907	+	0.341432i	$0.110912\pi$
$359$	88008.1i	0.682863i	−0.939907	+	0.341432i	$0.889088\pi$
$360$	0	0
$361$	103936.	0.797535
$362$	0	0
$363$	40082.8i	0.304190i
$364$	0	0
$365$	−183058.	−1.37405
$366$	0	0
$367$	− 226272.i	− 1.67996i	−0.542617	−	0.839980i	$-0.682566\pi$
$367$	− 226272.i	− 1.67996i	0.542617	−	0.839980i	$-0.317434\pi$
$368$	0	0
$369$	52665.7	0.386790
$370$	0	0
$371$	− 63486.7i	− 0.461248i
$372$	0	0
$373$	74885.5	0.538245	0.269123	−	0.963106i	$-0.413266\pi$
$373$	74885.5	0.538245	0.269123	+	0.963106i	$0.413266\pi$
$374$	0	0
$375$	− 40738.1i	− 0.289693i
$376$	0	0
$377$	37600.6	0.264552
$378$	0	0
$379$	− 224182.i	− 1.56071i	−0.625337	−	0.780355i	$-0.715039\pi$
$379$	− 224182.i	− 1.56071i	0.625337	−	0.780355i	$-0.284961\pi$
$380$	0	0
$381$	−122374.	−0.843025
$382$	0	0
$383$	− 251582.i	− 1.71507i	−0.514428	−	0.857534i	$-0.671996\pi$
$383$	− 251582.i	− 1.71507i	0.514428	−	0.857534i	$-0.328004\pi$
$384$	0	0
$385$	85132.1	0.574344
$386$	0	0
$387$	62893.7i	0.419938i
$388$	0	0
$389$	159429.	1.05358	0.526792	−	0.849994i	$-0.323395\pi$
$389$	159429.	1.05358	0.526792	+	0.849994i	$0.323395\pi$
$390$	0	0
$391$	− 338749.i	− 2.21577i
$392$	0	0
$393$	−96959.8	−0.627778
$394$	0	0
$395$	− 191555.i	− 1.22772i
$396$	0	0
$397$	−33989.3	−0.215656	−0.107828	−	0.994170i	$-0.534390\pi$
$397$	−33989.3	−0.215656	−0.107828	+	0.994170i	$0.534390\pi$
$398$	0	0
$399$	15179.5i	0.0953479i
$400$	0	0
$401$	27026.2	0.168072	0.0840361	−	0.996463i	$-0.473219\pi$
$401$	27026.2	0.168072	0.0840361	+	0.996463i	$0.473219\pi$
$402$	0	0
$403$	7386.58i	0.0454814i
$404$	0	0
$405$	−23080.3	−0.140712
$406$	0	0
$407$	218615.i	1.31975i
$408$	0	0
$409$	−24143.5	−0.144329	−0.0721646	−	0.997393i	$-0.522991\pi$
$409$	−24143.5	−0.144329	−0.0721646	+	0.997393i	$0.522991\pi$
$410$	0	0
$411$	41281.8i	0.244386i
$412$	0	0
$413$	−42286.1	−0.247912
$414$	0	0
$415$	193372.i	1.12279i
$416$	0	0
$417$	−82179.9	−0.472600
$418$	0	0
$419$	− 117690.i	− 0.670366i	−0.942153	−	0.335183i	$-0.891202\pi$
$419$	− 117690.i	− 0.670366i	0.942153	−	0.335183i	$-0.108798\pi$
$420$	0	0
$421$	−120697.	−0.680977	−0.340488	−	0.940249i	$-0.610592\pi$
$421$	−120697.	−0.680977	−0.340488	+	0.940249i	$0.610592\pi$
$422$	0	0
$423$	64630.3i	0.361206i
$424$	0	0
$425$	178822.	0.990017
$426$	0	0
$427$	− 37282.6i	− 0.204480i
$428$	0	0
$429$	−32189.2	−0.174902
$430$	0	0
$431$	− 257974.i	− 1.38874i	−0.719618	−	0.694370i	$-0.755683\pi$
$431$	− 257974.i	− 1.38874i	0.719618	−	0.694370i	$-0.244317\pi$
$432$	0	0
$433$	−348461.	−1.85857	−0.929284	−	0.369366i	$-0.879575\pi$
$433$	−348461.	−1.85857	−0.929284	+	0.369366i	$0.879575\pi$
$434$	0	0
$435$	− 149296.i	− 0.788988i
$436$	0	0
$437$	116120.	0.608054
$438$	0	0
$439$	− 160986.i	− 0.835332i	−0.908601	−	0.417666i	$-0.862848\pi$
$439$	− 160986.i	− 0.835332i	0.908601	−	0.417666i	$-0.137152\pi$
$440$	0	0
$441$	−56094.3	−0.288431
$442$	0	0
$443$	− 335523.i	− 1.70968i	−0.518890	−	0.854841i	$-0.673655\pi$
$443$	− 335523.i	− 1.70968i	0.518890	−	0.854841i	$-0.326345\pi$
$444$	0	0
$445$	−83320.9	−0.420760
$446$	0	0
$447$	− 66743.9i	− 0.334038i
$448$	0	0
$449$	43243.6	0.214501	0.107250	−	0.994232i	$-0.465795\pi$
$449$	43243.6	0.214501	0.107250	+	0.994232i	$0.465795\pi$
$450$	0	0
$451$	− 291642.i	− 1.43383i
$452$	0	0
$453$	−78071.0	−0.380446
$454$	0	0
$455$	23591.1i	0.113953i
$456$	0	0
$457$	175968.	0.842562	0.421281	−	0.906930i	$-0.361581\pi$
$457$	175968.	0.842562	0.421281	+	0.906930i	$0.361581\pi$
$458$	0	0
$459$	66481.4i	0.315555i
$460$	0	0
$461$	−257784.	−1.21298	−0.606491	−	0.795090i	$-0.707424\pi$
$461$	−257784.	−1.21298	−0.606491	+	0.795090i	$0.707424\pi$
$462$	0	0
$463$	345945.i	1.61378i	0.590699	+	0.806892i	$0.298852\pi$
$463$	345945.i	1.61378i	−0.590699	+	0.806892i	$0.701148\pi$
$464$	0	0
$465$	29329.1	0.135641
$466$	0	0
$467$	156168.i	0.716076i	0.933707	+	0.358038i	$0.116554\pi$
$467$	156168.i	0.716076i	−0.933707	+	0.358038i	$0.883446\pi$
$468$	0	0
$469$	82326.5	0.374278
$470$	0	0
$471$	− 30731.6i	− 0.138530i
$472$	0	0
$473$	348281.	1.55671
$474$	0	0
$475$	61298.3i	0.271682i
$476$	0	0
$477$	95313.3	0.418906
$478$	0	0
$479$	− 311999.i	− 1.35982i	−0.733294	−	0.679912i	$-0.762018\pi$
$479$	− 311999.i	− 1.35982i	0.733294	−	0.679912i	$-0.237982\pi$
$480$	0	0
$481$	−60580.8	−0.261845
$482$	0	0
$483$	− 66803.3i	− 0.286354i
$484$	0	0
$485$	−361118.	−1.53520
$486$	0	0
$487$	− 258932.i	− 1.09176i	−0.837862	−	0.545881i	$-0.816195\pi$
$487$	− 258932.i	− 1.09176i	0.837862	−	0.545881i	$-0.183805\pi$
$488$	0	0
$489$	133821.	0.559635
$490$	0	0
$491$	59302.0i	0.245984i	0.992408	+	0.122992i	$0.0392489\pi$
$491$	59302.0i	0.245984i	−0.992408	+	0.122992i	$0.960751\pi$
$492$	0	0
$493$	−430039.	−1.76935
$494$	0	0
$495$	127810.i	0.521620i
$496$	0	0
$497$	27350.5	0.110727
$498$	0	0
$499$	90707.0i	0.364284i	0.983272	+	0.182142i	$0.0583030\pi$
$499$	90707.0i	0.364284i	−0.983272	+	0.182142i	$0.941697\pi$
$500$	0	0
$501$	170918.	0.680947
$502$	0	0
$503$	185242.i	0.732157i	0.930584	+	0.366078i	$0.119300\pi$
$503$	185242.i	0.732157i	−0.930584	+	0.366078i	$0.880700\pi$
$504$	0	0
$505$	477113.	1.87085
$506$	0	0
$507$	139487.i	0.542649i
$508$	0	0
$509$	157900.	0.609464	0.304732	−	0.952438i	$-0.401433\pi$
$509$	157900.	0.609464	0.304732	+	0.952438i	$0.401433\pi$
$510$	0	0
$511$	103984.i	0.398222i
$512$	0	0
$513$	−22789.1	−0.0865951
$514$	0	0
$515$	526218.i	1.98404i
$516$	0	0
$517$	357898.	1.33899
$518$	0	0
$519$	− 169584.i	− 0.629578i
$520$	0	0
$521$	−57852.3	−0.213130	−0.106565	−	0.994306i	$-0.533985\pi$
$521$	−57852.3	−0.213130	−0.106565	+	0.994306i	$0.533985\pi$
$522$	0	0
$523$	− 141511.i	− 0.517352i	−0.965964	−	0.258676i	$-0.916714\pi$
$523$	− 141511.i	− 0.517352i	0.965964	−	0.258676i	$-0.0832861\pi$
$524$	0	0
$525$	35264.8	0.127945
$526$	0	0
$527$	− 84480.6i	− 0.304184i
$528$	0	0
$529$	−231189.	−0.826144
$530$	0	0
$531$	− 63484.7i	− 0.225154i
$532$	0	0
$533$	80817.4	0.284479
$534$	0	0
$535$	633169.i	2.21214i
$536$	0	0
$537$	217432.	0.754006
$538$	0	0
$539$	310629.i	1.06921i
$540$	0	0
$541$	42028.3	0.143598	0.0717989	−	0.997419i	$-0.477126\pi$
$541$	42028.3	0.143598	0.0717989	+	0.997419i	$0.477126\pi$
$542$	0	0
$543$	− 147796.i	− 0.501260i
$544$	0	0
$545$	−679705.	−2.28838
$546$	0	0
$547$	194495.i	0.650031i	0.945709	+	0.325016i	$0.105370\pi$
$547$	194495.i	0.650031i	−0.945709	+	0.325016i	$0.894630\pi$
$548$	0	0
$549$	55972.8	0.185709
$550$	0	0
$551$	− 147413.i	− 0.485548i
$552$	0	0
$553$	−108811.	−0.355814
$554$	0	0
$555$	240541.i	0.780915i
$556$	0	0
$557$	231877.	0.747390	0.373695	−	0.927552i	$-0.378091\pi$
$557$	231877.	0.747390	0.373695	+	0.927552i	$0.378091\pi$
$558$	0	0
$559$	96512.7i	0.308860i
$560$	0	0
$561$	368149.	1.16976
$562$	0	0
$563$	62902.4i	0.198450i	0.995065	+	0.0992249i	$0.0316363\pi$
$563$	62902.4i	0.198450i	−0.995065	+	0.0992249i	$0.968364\pi$
$564$	0	0
$565$	−494704.	−1.54970
$566$	0	0
$567$	13110.5i	0.0407807i
$568$	0	0
$569$	−256828.	−0.793264	−0.396632	−	0.917978i	$-0.629821\pi$
$569$	−256828.	−0.793264	−0.396632	+	0.917978i	$0.629821\pi$
$570$	0	0
$571$	− 75452.1i	− 0.231419i	−0.993283	−	0.115709i	$-0.963086\pi$
$571$	− 75452.1i	− 0.231419i	0.993283	−	0.115709i	$-0.0369142\pi$
$572$	0	0
$573$	211967.	0.645593
$574$	0	0
$575$	− 269767.i	− 0.815931i
$576$	0	0
$577$	256938.	0.771751	0.385876	−	0.922551i	$-0.373899\pi$
$577$	256938.	0.771751	0.385876	+	0.922551i	$0.373899\pi$
$578$	0	0
$579$	− 306659.i	− 0.914741i
$580$	0	0
$581$	109843.	0.325402
$582$	0	0
$583$	− 527809.i	− 1.55289i
$584$	0	0
$585$	−35417.6	−0.103492
$586$	0	0
$587$	− 240005.i	− 0.696536i	−0.937395	−	0.348268i	$-0.886770\pi$
$587$	− 240005.i	− 0.696536i	0.937395	−	0.348268i	$-0.113230\pi$
$588$	0	0
$589$	28959.1	0.0834746
$590$	0	0
$591$	271097.i	0.776158i
$592$	0	0
$593$	102869.	0.292533	0.146267	−	0.989245i	$-0.453274\pi$
$593$	102869.	0.292533	0.146267	+	0.989245i	$0.453274\pi$
$594$	0	0
$595$	− 269812.i	− 0.762127i
$596$	0	0
$597$	39075.7	0.109637
$598$	0	0
$599$	598610.i	1.66836i	0.551491	+	0.834181i	$0.314059\pi$
$599$	598610.i	1.66836i	−0.551491	+	0.834181i	$0.685941\pi$
$600$	0	0
$601$	−649510.	−1.79819	−0.899097	−	0.437749i	$-0.855776\pi$
$601$	−649510.	−1.79819	−0.899097	+	0.437749i	$0.855776\pi$
$602$	0	0
$603$	123598.i	0.339919i
$604$	0	0
$605$	244225.	0.667235
$606$	0	0
$607$	442679.i	1.20147i	0.799450	+	0.600733i	$0.205125\pi$
$607$	442679.i	1.20147i	−0.799450	+	0.600733i	$0.794875\pi$
$608$	0	0
$609$	−84806.3	−0.228662
$610$	0	0
$611$	99177.5i	0.265663i
$612$	0	0
$613$	601837.	1.60161	0.800807	−	0.598922i	$-0.204404\pi$
$613$	601837.	1.60161	0.800807	+	0.598922i	$0.204404\pi$
$614$	0	0
$615$	− 320892.i	− 0.848417i
$616$	0	0
$617$	−68843.1	−0.180838	−0.0904191	−	0.995904i	$-0.528821\pi$
$617$	−68843.1	−0.180838	−0.0904191	+	0.995904i	$0.528821\pi$
$618$	0	0
$619$	− 253349.i	− 0.661208i	−0.943770	−	0.330604i	$-0.892748\pi$
$619$	− 253349.i	− 0.661208i	0.943770	−	0.330604i	$-0.107252\pi$
$620$	0	0
$621$	100293.	0.260067
$622$	0	0
$623$	47329.6i	0.121943i
$624$	0	0
$625$	−484073.	−1.23923
$626$	0	0
$627$	126198.i	0.321008i
$628$	0	0
$629$	692865.	1.75125
$630$	0	0
$631$	549097.i	1.37908i	0.724247	+	0.689541i	$0.242188\pi$
$631$	549097.i	1.37908i	−0.724247	+	0.689541i	$0.757812\pi$
$632$	0	0
$633$	−104759.	−0.261447
$634$	0	0
$635$	745628.i	1.84916i
$636$	0	0
$637$	−86078.8	−0.212137
$638$	0	0
$639$	41061.6i	0.100562i
$640$	0	0
$641$	248289.	0.604286	0.302143	−	0.953263i	$-0.402298\pi$
$641$	248289.	0.604286	0.302143	+	0.953263i	$0.402298\pi$
$642$	0	0
$643$	− 518101.i	− 1.25312i	−0.779374	−	0.626560i	$-0.784463\pi$
$643$	− 518101.i	− 1.25312i	0.779374	−	0.626560i	$-0.215537\pi$
$644$	0	0
$645$	383212.	0.921128
$646$	0	0
$647$	− 739762.i	− 1.76719i	−0.468251	−	0.883595i	$-0.655116\pi$
$647$	− 739762.i	− 1.76719i	0.468251	−	0.883595i	$-0.344884\pi$
$648$	0	0
$649$	−351554.	−0.834647
$650$	0	0
$651$	− 16660.1i	− 0.0393111i
$652$	0	0
$653$	−335872.	−0.787677	−0.393838	−	0.919180i	$-0.628853\pi$
$653$	−335872.	−0.787677	−0.393838	+	0.919180i	$0.628853\pi$
$654$	0	0
$655$	590777.i	1.37702i
$656$	0	0
$657$	−156113.	−0.361665
$658$	0	0
$659$	764651.i	1.76073i	0.474299	+	0.880364i	$0.342702\pi$
$659$	764651.i	1.76073i	−0.474299	+	0.880364i	$0.657298\pi$
$660$	0	0
$661$	−401983.	−0.920035	−0.460017	−	0.887910i	$-0.652157\pi$
$661$	−401983.	−0.920035	−0.460017	+	0.887910i	$0.652157\pi$
$662$	0	0
$663$	102018.i	0.232087i
$664$	0	0
$665$	92488.7	0.209144
$666$	0	0
$667$	648749.i	1.45823i
$668$	0	0
$669$	−146591.	−0.327532
$670$	0	0
$671$	− 309956.i	− 0.688423i
$672$	0	0
$673$	−434435.	−0.959167	−0.479583	−	0.877496i	$-0.659212\pi$
$673$	−434435.	−0.959167	−0.479583	+	0.877496i	$0.659212\pi$
$674$	0	0
$675$	52943.4i	0.116199i
$676$	0	0
$677$	147048.	0.320835	0.160417	−	0.987049i	$-0.448716\pi$
$677$	147048.	0.320835	0.160417	+	0.987049i	$0.448716\pi$
$678$	0	0
$679$	205129.i	0.444927i
$680$	0	0
$681$	91876.2	0.198111
$682$	0	0
$683$	− 260690.i	− 0.558833i	−0.960170	−	0.279417i	$-0.909859\pi$
$683$	− 260690.i	− 0.558833i	0.960170	−	0.279417i	$-0.0901410\pi$
$684$	0	0
$685$	251531.	0.536056
$686$	0	0
$687$	− 164766.i	− 0.349103i
$688$	0	0
$689$	146262.	0.308101
$690$	0	0
$691$	− 388693.i	− 0.814050i	−0.913417	−	0.407025i	$-0.866566\pi$
$691$	− 388693.i	− 0.814050i	0.913417	−	0.407025i	$-0.133434\pi$
$692$	0	0
$693$	72601.1	0.151174
$694$	0	0
$695$	500723.i	1.03664i
$696$	0	0
$697$	−924311.	−1.90262
$698$	0	0
$699$	503436.i	1.03036i
$700$	0	0
$701$	174569.	0.355247	0.177623	−	0.984099i	$-0.443159\pi$
$701$	174569.	0.355247	0.177623	+	0.984099i	$0.443159\pi$
$702$	0	0
$703$	237507.i	0.480580i
$704$	0	0
$705$	393793.	0.792300
$706$	0	0
$707$	− 271019.i	− 0.542202i
$708$	0	0
$709$	−512632.	−1.01980	−0.509898	−	0.860235i	$-0.670317\pi$
$709$	−512632.	−1.01980	−0.509898	+	0.860235i	$0.670317\pi$
$710$	0	0
$711$	− 163359.i	− 0.323150i
$712$	0	0
$713$	−127446.	−0.250696
$714$	0	0
$715$	196129.i	0.383645i
$716$	0	0
$717$	−122827.	−0.238921
$718$	0	0
$719$	680262.i	1.31589i	0.753068	+	0.657943i	$0.228573\pi$
$719$	680262.i	1.31589i	−0.753068	+	0.657943i	$0.771427\pi$
$720$	0	0
$721$	298913.	0.575008
$722$	0	0
$723$	522646.i	0.999841i
$724$	0	0
$725$	−342467.	−0.651543
$726$	0	0
$727$	394991.i	0.747340i	0.927562	+	0.373670i	$0.121901\pi$
$727$	394991.i	0.747340i	−0.927562	+	0.373670i	$0.878099\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	− 1.10382e6i	− 2.06568i
$732$	0	0
$733$	−33150.6	−0.0616998	−0.0308499	−	0.999524i	$-0.509821\pi$
$733$	−33150.6	−0.0616998	−0.0308499	+	0.999524i	$0.509821\pi$
$734$	0	0
$735$	341783.i	0.632668i
$736$	0	0
$737$	684437.	1.26008
$738$	0	0
$739$	105852.i	0.193825i	0.995293	+	0.0969127i	$0.0308968\pi$
$739$	105852.i	0.193825i	−0.995293	+	0.0969127i	$0.969103\pi$
$740$	0	0
$741$	−34970.8	−0.0636896
$742$	0	0
$743$	745078.i	1.34966i	0.737973	+	0.674830i	$0.235783\pi$
$743$	745078.i	1.34966i	−0.737973	+	0.674830i	$0.764217\pi$
$744$	0	0
$745$	−406671.	−0.732708
$746$	0	0
$747$	164909.i	0.295531i
$748$	0	0
$749$	359666.	0.641114
$750$	0	0
$751$	296353.i	0.525447i	0.964871	+	0.262724i	$0.0846207\pi$
$751$	296353.i	0.525447i	−0.964871	+	0.262724i	$0.915379\pi$
$752$	0	0
$753$	−305081.	−0.538053
$754$	0	0
$755$	475687.i	0.834503i
$756$	0	0
$757$	−898177.	−1.56737	−0.783683	−	0.621161i	$-0.786661\pi$
$757$	−898177.	−1.56737	−0.783683	+	0.621161i	$0.786661\pi$
$758$	0	0
$759$	− 555382.i	− 0.964069i
$760$	0	0
$761$	−140171.	−0.242042	−0.121021	−	0.992650i	$-0.538617\pi$
$761$	−140171.	−0.242042	−0.121021	+	0.992650i	$0.538617\pi$
$762$	0	0
$763$	386100.i	0.663209i
$764$	0	0
$765$	405072.	0.692164
$766$	0	0
$767$	− 97419.6i	− 0.165598i
$768$	0	0
$769$	122951.	0.207911	0.103956	−	0.994582i	$-0.466850\pi$
$769$	122951.	0.207911	0.103956	+	0.994582i	$0.466850\pi$
$770$	0	0
$771$	525476.i	0.883983i
$772$	0	0
$773$	−9337.35	−0.0156266	−0.00781330	−	0.999969i	$-0.502487\pi$
$773$	−9337.35	−0.0156266	−0.00781330	+	0.999969i	$0.502487\pi$
$774$	0	0
$775$	− 67277.3i	− 0.112012i
$776$	0	0
$777$	136637.	0.226322
$778$	0	0
$779$	− 316844.i	− 0.522121i
$780$	0	0
$781$	227384.	0.372784
$782$	0	0
$783$	− 127321.i	− 0.207671i
$784$	0	0
$785$	−187248.	−0.303863
$786$	0	0
$787$	− 290223.i	− 0.468579i	−0.972167	−	0.234290i	$-0.924724\pi$
$787$	− 290223.i	− 0.468579i	0.972167	−	0.234290i	$-0.0752764\pi$
$788$	0	0
$789$	−483067.	−0.775984
$790$	0	0
$791$	281012.i	0.449129i
$792$	0	0
$793$	85892.4	0.136587
$794$	0	0
$795$	− 580745.i	− 0.918864i
$796$	0	0
$797$	945225.	1.48805	0.744027	−	0.668150i	$-0.232913\pi$
$797$	945225.	1.48805	0.744027	+	0.668150i	$0.232913\pi$
$798$	0	0
$799$	− 1.13430e6i	− 1.77678i
$800$	0	0
$801$	−71056.5	−0.110749
$802$	0	0
$803$	864492.i	1.34069i
$804$	0	0
$805$	−407033.	−0.628113
$806$	0	0
$807$	− 369856.i	− 0.567919i
$808$	0	0
$809$	342876.	0.523890	0.261945	−	0.965083i	$-0.415636\pi$
$809$	342876.	0.523890	0.261945	+	0.965083i	$0.415636\pi$
$810$	0	0
$811$	− 429529.i	− 0.653056i	−0.945188	−	0.326528i	$-0.894121\pi$
$811$	− 429529.i	− 0.653056i	0.945188	−	0.326528i	$-0.105879\pi$
$812$	0	0
$813$	380160.	0.575155
$814$	0	0
$815$	− 815370.i	− 1.22755i
$816$	0	0
$817$	378378.	0.566868
$818$	0	0
$819$	20118.6i	0.0299937i
$820$	0	0
$821$	−14283.3	−0.0211906	−0.0105953	−	0.999944i	$-0.503373\pi$
$821$	−14283.3	−0.0211906	−0.0105953	+	0.999944i	$0.503373\pi$
$822$	0	0
$823$	112641.i	0.166301i	0.996537	+	0.0831507i	$0.0264983\pi$
$823$	112641.i	0.166301i	−0.996537	+	0.0831507i	$0.973502\pi$
$824$	0	0
$825$	293180.	0.430752
$826$	0	0
$827$	470382.i	0.687765i	0.939013	+	0.343882i	$0.111742\pi$
$827$	470382.i	0.687765i	−0.939013	+	0.343882i	$0.888258\pi$
$828$	0	0
$829$	−17182.5	−0.0250021	−0.0125011	−	0.999922i	$-0.503979\pi$
$829$	−17182.5	−0.0250021	−0.0125011	+	0.999922i	$0.503979\pi$
$830$	0	0
$831$	− 348357.i	− 0.504454i
$832$	0	0
$833$	984486.	1.41879
$834$	0	0
$835$	− 1.04141e6i	− 1.49365i
$836$	0	0
$837$	25012.0	0.0357024
$838$	0	0
$839$	− 1.21028e6i	− 1.71933i	−0.510855	−	0.859667i	$-0.670671\pi$
$839$	− 1.21028e6i	− 1.71933i	0.510855	−	0.859667i	$-0.329329\pi$
$840$	0	0
$841$	116300.	0.164433
$842$	0	0
$843$	− 79630.2i	− 0.112053i
$844$	0	0
$845$	849898.	1.19029
$846$	0	0
$847$	− 138730.i	− 0.193376i
$848$	0	0
$849$	493956.	0.685288
$850$	0	0
$851$	− 1.04524e6i	− 1.44330i
$852$	0	0
$853$	806787.	1.10882	0.554410	−	0.832244i	$-0.312944\pi$
$853$	806787.	1.10882	0.554410	+	0.832244i	$0.312944\pi$
$854$	0	0
$855$	138854.i	0.189945i
$856$	0	0
$857$	132458.	0.180350	0.0901749	−	0.995926i	$-0.471257\pi$
$857$	132458.	0.180350	0.0901749	+	0.995926i	$0.471257\pi$
$858$	0	0
$859$	− 641297.i	− 0.869106i	−0.900646	−	0.434553i	$-0.856906\pi$
$859$	− 641297.i	− 0.869106i	0.900646	−	0.434553i	$-0.143094\pi$
$860$	0	0
$861$	−182280.	−0.245885
$862$	0	0
$863$	− 126234.i	− 0.169495i	−0.996402	−	0.0847473i	$-0.972992\pi$
$863$	− 126234.i	− 0.169495i	0.996402	−	0.0847473i	$-0.0270083\pi$
$864$	0	0
$865$	−1.03328e6	−1.38097
$866$	0	0
$867$	− 732797.i	− 0.974868i
$868$	0	0
$869$	−904622.	−1.19792
$870$	0	0
$871$	189665.i	0.250007i
$872$	0	0
$873$	−307963.	−0.404083
$874$	0	0
$875$	140998.i	0.184160i
$876$	0	0
$877$	−747782.	−0.972245	−0.486122	−	0.873891i	$-0.661589\pi$
$877$	−747782.	−0.972245	−0.486122	+	0.873891i	$0.661589\pi$
$878$	0	0
$879$	− 160376.i	− 0.207568i
$880$	0	0
$881$	58212.7	0.0750007	0.0375004	−	0.999297i	$-0.488060\pi$
$881$	58212.7	0.0750007	0.0375004	+	0.999297i	$0.488060\pi$
$882$	0	0
$883$	453537.i	0.581689i	0.956770	+	0.290845i	$0.0939363\pi$
$883$	453537.i	0.581689i	−0.956770	+	0.290845i	$0.906064\pi$
$884$	0	0
$885$	−386813.	−0.493872
$886$	0	0
$887$	398206.i	0.506129i	0.967449	+	0.253064i	$0.0814384\pi$
$887$	398206.i	0.506129i	−0.967449	+	0.253064i	$0.918562\pi$
$888$	0	0
$889$	423547.	0.535917
$890$	0	0
$891$	108997.i	0.137296i
$892$	0	0
$893$	388825.	0.487586
$894$	0	0
$895$	− 1.32481e6i	− 1.65390i
$896$	0	0
$897$	153903.	0.191276
$898$	0	0
$899$	161792.i	0.200187i
$900$	0	0
$901$	−1.67280e6	−2.06060
$902$	0	0
$903$	− 217680.i	− 0.266958i
$904$	0	0
$905$	−900523.	−1.09951
$906$	0	0
$907$	38258.4i	0.0465063i	0.999730	+	0.0232532i	$0.00740238\pi$
$907$	38258.4i	0.0465063i	−0.999730	+	0.0232532i	$0.992598\pi$
$908$	0	0
$909$	406884.	0.492428
$910$	0	0
$911$	165280.i	0.199152i	0.995030	+	0.0995758i	$0.0317486\pi$
$911$	165280.i	0.199152i	−0.995030	+	0.0995758i	$0.968251\pi$
$912$	0	0
$913$	913202.	1.09553
$914$	0	0
$915$	− 341043.i	− 0.407349i
$916$	0	0
$917$	335585.	0.399084
$918$	0	0
$919$	615312.i	0.728558i	0.931290	+	0.364279i	$0.118685\pi$
$919$	615312.i	0.728558i	−0.931290	+	0.364279i	$0.881315\pi$
$920$	0	0
$921$	−356576.	−0.420371
$922$	0	0
$923$	63010.6i	0.0739622i
$924$	0	0
$925$	551772.	0.644877
$926$	0	0
$927$	448762.i	0.522223i
$928$	0	0
$929$	297330.	0.344514	0.172257	−	0.985052i	$-0.444894\pi$
$929$	297330.	0.344514	0.172257	+	0.985052i	$0.444894\pi$
$930$	0	0
$931$	337471.i	0.389348i
$932$	0	0
$933$	579088.	0.665245
$934$	0	0
$935$	− 2.24313e6i	− 2.56585i
$936$	0	0
$937$	−81207.3	−0.0924945	−0.0462473	−	0.998930i	$-0.514726\pi$
$937$	−81207.3	−0.0924945	−0.0462473	+	0.998930i	$0.514726\pi$
$938$	0	0
$939$	176869.i	0.200596i
$940$	0	0
$941$	−1.46600e6	−1.65560	−0.827798	−	0.561026i	$-0.810407\pi$
$941$	−1.46600e6	−1.65560	−0.827798	+	0.561026i	$0.810407\pi$
$942$	0	0
$943$	1.39440e6i	1.56806i
$944$	0	0
$945$	79882.6	0.0894517
$946$	0	0
$947$	− 756376.i	− 0.843408i	−0.906733	−	0.421704i	$-0.861432\pi$
$947$	− 756376.i	− 0.843408i	0.906733	−	0.421704i	$-0.138568\pi$
$948$	0	0
$949$	−239560.	−0.266001
$950$	0	0
$951$	487846.i	0.539414i
$952$	0	0
$953$	−149406.	−0.164506	−0.0822529	−	0.996611i	$-0.526212\pi$
$953$	−149406.	−0.164506	−0.0822529	+	0.996611i	$0.526212\pi$
$954$	0	0
$955$	− 1.29152e6i	− 1.41610i
$956$	0	0
$957$	−705053.	−0.769836
$958$	0	0
$959$	− 142880.i	− 0.155358i
$960$	0	0
$961$	891737.	0.965584
$962$	0	0
$963$	539970.i	0.582260i
$964$	0	0
$965$	−1.86847e6	−2.00647
$966$	0	0
$967$	− 939551.i	− 1.00477i	−0.864643	−	0.502386i	$-0.832456\pi$
$967$	− 939551.i	− 1.00477i	0.864643	−	0.502386i	$-0.167544\pi$
$968$	0	0
$969$	399962.	0.425962
$970$	0	0
$971$	− 176146.i	− 0.186825i	−0.995627	−	0.0934125i	$-0.970222\pi$
$971$	− 176146.i	− 0.186825i	0.995627	−	0.0934125i	$-0.0297775\pi$
$972$	0	0
$973$	284431.	0.300435
$974$	0	0
$975$	81243.6i	0.0854633i
$976$	0	0
$977$	940921.	0.985744	0.492872	−	0.870102i	$-0.335947\pi$
$977$	940921.	0.985744	0.492872	+	0.870102i	$0.335947\pi$
$978$	0	0
$979$	393484.i	0.410546i
$980$	0	0
$981$	−579656.	−0.602327
$982$	0	0
$983$	− 57050.8i	− 0.0590411i	−0.999564	−	0.0295206i	$-0.990602\pi$
$983$	− 57050.8i	− 0.0590411i	0.999564	−	0.0295206i	$-0.00939805\pi$
$984$	0	0
$985$	1.65180e6	1.70249
$986$	0	0
$987$	− 223690.i	− 0.229622i
$988$	0	0
$989$	−1.66520e6	−1.70245
$990$	0	0
$991$	− 626565.i	− 0.637997i	−0.947755	−	0.318998i	$-0.896654\pi$
$991$	− 626565.i	− 0.637997i	0.947755	−	0.318998i	$-0.103346\pi$
$992$	0	0
$993$	992393.	1.00643
$994$	0	0
$995$	− 238089.i	− 0.240488i
$996$	0	0
$997$	902090.	0.907527	0.453763	−	0.891122i	$-0.350081\pi$
$997$	902090.	0.907527	0.453763	+	0.891122i	$0.350081\pi$
$998$	0	0
$999$	205135.i	0.205546i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.g.a.127.3	✓	16
3.2	odd	2		1152.5.g.f.127.12		16
4.3	odd	2	inner	384.5.g.a.127.11	yes	16
8.3	odd	2		384.5.g.b.127.6	yes	16
8.5	even	2		384.5.g.b.127.14	yes	16
12.11	even	2		1152.5.g.f.127.11		16
16.3	odd	4		768.5.b.h.127.7		16
16.5	even	4		768.5.b.h.127.2		16
16.11	odd	4		768.5.b.i.127.10		16
16.13	even	4		768.5.b.i.127.15		16
24.5	odd	2		1152.5.g.c.127.6		16
24.11	even	2		1152.5.g.c.127.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.g.a.127.3	✓	16	1.1	even	1	trivial
384.5.g.a.127.11	yes	16	4.3	odd	2	inner
384.5.g.b.127.6	yes	16	8.3	odd	2
384.5.g.b.127.14	yes	16	8.5	even	2
768.5.b.h.127.2		16	16.5	even	4
768.5.b.h.127.7		16	16.3	odd	4
768.5.b.i.127.10		16	16.11	odd	4
768.5.b.i.127.15		16	16.13	even	4
1152.5.g.c.127.5		16	24.11	even	2
1152.5.g.c.127.6		16	24.5	odd	2
1152.5.g.f.127.11		16	12.11	even	2
1152.5.g.f.127.12		16	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} -31.6602 q^{5} +17.9843i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q-5.19615i q^{3} -31.6602 q^{5} +17.9843i q^{7} -27.0000 q^{9} +149.516i q^{11} -41.4325 q^{13} +164.511i q^{15} +473.865 q^{17} +162.436i q^{19} +93.4490 q^{21} -714.864i q^{23} +377.369 q^{25} +140.296i q^{27} -907.514 q^{29} -178.280i q^{31} +776.906 q^{33} -569.386i q^{35} +1462.16 q^{37} +215.290i q^{39} -1950.58 q^{41} -2329.40i q^{43} +854.826 q^{45} -2393.71i q^{47} +2077.57 q^{49} -2462.27i q^{51} -3530.12 q^{53} -4733.70i q^{55} +844.042 q^{57} +2351.29i q^{59} -2073.07 q^{61} -485.575i q^{63} +1311.76 q^{65} -4577.69i q^{67} -3714.54 q^{69} -1520.80i q^{71} +5781.95 q^{73} -1960.87i q^{75} -2688.93 q^{77} +6050.35i q^{79} +729.000 q^{81} -6107.73i q^{83} -15002.7 q^{85} +4715.58i q^{87} +2631.72 q^{89} -745.133i q^{91} -926.370 q^{93} -5142.76i q^{95} +11406.0 q^{97} -4036.92i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 96 q^{5} - 432 q^{9}+O(q^{10}) \) 16 * q - 96 * q^5 - 432 * q^9	\( 16 q - 96 q^{5} - 432 q^{9} - 480 q^{13} - 480 q^{17} + 2672 q^{25} + 3360 q^{29} - 1120 q^{37} + 1440 q^{41} + 2592 q^{45} - 2480 q^{49} - 3552 q^{53} - 7488 q^{57} - 18272 q^{61} - 1344 q^{65} - 8480 q^{73} + 17280 q^{77} + 11664 q^{81} + 29888 q^{85} + 18720 q^{89} - 28800 q^{93} + 13088 q^{97}+O(q^{100}) \) 16 * q - 96 * q^5 - 432 * q^9 - 480 * q^13 - 480 * q^17 + 2672 * q^25 + 3360 * q^29 - 1120 * q^37 + 1440 * q^41 + 2592 * q^45 - 2480 * q^49 - 3552 * q^53 - 7488 * q^57 - 18272 * q^61 - 1344 * q^65 - 8480 * q^73 + 17280 * q^77 + 11664 * q^81 + 29888 * q^85 + 18720 * q^89 - 28800 * q^93 + 13088 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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