LMFDB - Embedded newform 128.3.c.a.127.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,3,Mod(127,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, 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("128.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	128.c (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:	$3.48774738381$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	128.127
Dual form			128.3.c.a.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.82843i q^{3} -7.65685 q^{5} -1.65685i q^{7} -14.3137 q^{9} +O(q^{10})$	$q+4.82843i q^{3} -7.65685 q^{5} -1.65685i q^{7} -14.3137 q^{9} -1.51472i q^{11} +0.343146 q^{13} -36.9706i q^{15} -13.3137 q^{17} +20.8284i q^{19} +8.00000 q^{21} +33.6569i q^{23} +33.6274 q^{25} -25.6569i q^{27} -39.6569 q^{29} +45.2548i q^{31} +7.31371 q^{33} +12.6863i q^{35} +29.5980 q^{37} +1.65685i q^{39} +24.6274 q^{41} -50.0833i q^{43} +109.598 q^{45} -35.3137i q^{47} +46.2548 q^{49} -64.2843i q^{51} +16.3431 q^{53} +11.5980i q^{55} -100.569 q^{57} +53.1127i q^{59} -34.4020 q^{61} +23.7157i q^{63} -2.62742 q^{65} -62.4853i q^{67} -162.510 q^{69} -40.2843i q^{71} +55.9411 q^{73} +162.368i q^{75} -2.50967 q^{77} +137.941i q^{79} -4.94113 q^{81} +114.652i q^{83} +101.941 q^{85} -191.480i q^{87} -2.56854 q^{89} -0.568542i q^{91} -218.510 q^{93} -159.480i q^{95} -138.569 q^{97} +21.6812i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q - 8 * q^5 - 12 * q^9	$ 4 q - 8 q^{5} - 12 q^{9} + 24 q^{13} - 8 q^{17} + 32 q^{21} + 44 q^{25} - 136 q^{29} - 16 q^{33} - 40 q^{37} + 8 q^{41} + 280 q^{45} + 4 q^{49} + 88 q^{53} - 176 q^{57} - 296 q^{61} + 80 q^{65} - 288 q^{69} + 88 q^{73} + 352 q^{77} + 116 q^{81} + 272 q^{85} + 216 q^{89} - 512 q^{93} - 328 q^{97}+O(q^{100}) $ 4 * q - 8 * q^5 - 12 * q^9 + 24 * q^13 - 8 * q^17 + 32 * q^21 + 44 * q^25 - 136 * q^29 - 16 * q^33 - 40 * q^37 + 8 * q^41 + 280 * q^45 + 4 * q^49 + 88 * q^53 - 176 * q^57 - 296 * q^61 + 80 * q^65 - 288 * q^69 + 88 * q^73 + 352 * q^77 + 116 * q^81 + 272 * q^85 + 216 * q^89 - 512 * q^93 - 328 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$1$	$-1$

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 (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
$3$	4.82843i	1.60948i	0.593630	+	0.804738i	$0.297694\pi$
$3$	4.82843i	1.60948i	−0.593630	+	0.804738i	$0.702306\pi$
$4$	0	0
$5$	−7.65685	−1.53137	−0.765685	−	0.643215i	$-0.777600\pi$
$5$	−7.65685	−1.53137	−0.765685	+	0.643215i	$0.777600\pi$
$6$	0	0
$7$	− 1.65685i	− 0.236693i	−0.992972	−	0.118347i	$-0.962241\pi$
$7$	− 1.65685i	− 0.236693i	0.992972	−	0.118347i	$-0.0377594\pi$
$8$	0	0
$9$	−14.3137	−1.59041
$10$	0	0
$11$	− 1.51472i	− 0.137702i	−0.997627	−	0.0688508i	$-0.978067\pi$
$11$	− 1.51472i	− 0.137702i	0.997627	−	0.0688508i	$-0.0219333\pi$
$12$	0	0
$13$	0.343146	0.0263958	0.0131979	−	0.999913i	$-0.495799\pi$
$13$	0.343146	0.0263958	0.0131979	+	0.999913i	$0.495799\pi$
$14$	0	0
$15$	− 36.9706i	− 2.46470i
$16$	0	0
$17$	−13.3137	−0.783159	−0.391580	−	0.920144i	$-0.628071\pi$
$17$	−13.3137	−0.783159	−0.391580	+	0.920144i	$0.628071\pi$
$18$	0	0
$19$	20.8284i	1.09623i	0.836402	+	0.548117i	$0.184655\pi$
$19$	20.8284i	1.09623i	−0.836402	+	0.548117i	$0.815345\pi$
$20$	0	0
$21$	8.00000	0.380952
$22$	0	0
$23$	33.6569i	1.46334i	0.681658	+	0.731671i	$0.261259\pi$
$23$	33.6569i	1.46334i	−0.681658	+	0.731671i	$0.738741\pi$
$24$	0	0
$25$	33.6274	1.34510
$26$	0	0
$27$	− 25.6569i	− 0.950254i
$28$	0	0
$29$	−39.6569	−1.36748	−0.683739	−	0.729727i	$-0.739647\pi$
$29$	−39.6569	−1.36748	−0.683739	+	0.729727i	$0.739647\pi$
$30$	0	0
$31$	45.2548i	1.45983i	0.683536	+	0.729917i	$0.260441\pi$
$31$	45.2548i	1.45983i	−0.683536	+	0.729917i	$0.739559\pi$
$32$	0	0
$33$	7.31371	0.221628
$34$	0	0
$35$	12.6863i	0.362465i
$36$	0	0
$37$	29.5980	0.799945	0.399973	−	0.916527i	$-0.369020\pi$
$37$	29.5980	0.799945	0.399973	+	0.916527i	$0.369020\pi$
$38$	0	0
$39$	1.65685i	0.0424834i
$40$	0	0
$41$	24.6274	0.600669	0.300334	−	0.953834i	$-0.402902\pi$
$41$	24.6274	0.600669	0.300334	+	0.953834i	$0.402902\pi$
$42$	0	0
$43$	− 50.0833i	− 1.16473i	−0.812929	−	0.582364i	$-0.802128\pi$
$43$	− 50.0833i	− 1.16473i	0.812929	−	0.582364i	$-0.197872\pi$
$44$	0	0
$45$	109.598	2.43551
$46$	0	0
$47$	− 35.3137i	− 0.751355i	−0.926750	−	0.375678i	$-0.877410\pi$
$47$	− 35.3137i	− 0.751355i	0.926750	−	0.375678i	$-0.122590\pi$
$48$	0	0
$49$	46.2548	0.943976
$50$	0	0
$51$	− 64.2843i	− 1.26048i
$52$	0	0
$53$	16.3431	0.308361	0.154181	−	0.988043i	$-0.450726\pi$
$53$	16.3431	0.308361	0.154181	+	0.988043i	$0.450726\pi$
$54$	0	0
$55$	11.5980i	0.210872i
$56$	0	0
$57$	−100.569	−1.76436
$58$	0	0
$59$	53.1127i	0.900215i	0.892974	+	0.450108i	$0.148614\pi$
$59$	53.1127i	0.900215i	−0.892974	+	0.450108i	$0.851386\pi$
$60$	0	0
$61$	−34.4020	−0.563968	−0.281984	−	0.959419i	$-0.590992\pi$
$61$	−34.4020	−0.563968	−0.281984	+	0.959419i	$0.590992\pi$
$62$	0	0
$63$	23.7157i	0.376440i
$64$	0	0
$65$	−2.62742	−0.0404218
$66$	0	0
$67$	− 62.4853i	− 0.932616i	−0.884622	−	0.466308i	$-0.845584\pi$
$67$	− 62.4853i	− 0.932616i	0.884622	−	0.466308i	$-0.154416\pi$
$68$	0	0
$69$	−162.510	−2.35521
$70$	0	0
$71$	− 40.2843i	− 0.567384i	−0.958915	−	0.283692i	$-0.908441\pi$
$71$	− 40.2843i	− 0.567384i	0.958915	−	0.283692i	$-0.0915593\pi$
$72$	0	0
$73$	55.9411	0.766317	0.383158	−	0.923683i	$-0.374836\pi$
$73$	55.9411	0.766317	0.383158	+	0.923683i	$0.374836\pi$
$74$	0	0
$75$	162.368i	2.16490i
$76$	0	0
$77$	−2.50967	−0.0325931
$78$	0	0
$79$	137.941i	1.74609i	0.487639	+	0.873045i	$0.337858\pi$
$79$	137.941i	1.74609i	−0.487639	+	0.873045i	$0.662142\pi$
$80$	0	0
$81$	−4.94113	−0.0610015
$82$	0	0
$83$	114.652i	1.38135i	0.723167	+	0.690674i	$0.242686\pi$
$83$	114.652i	1.38135i	−0.723167	+	0.690674i	$0.757314\pi$
$84$	0	0
$85$	101.941	1.19931
$86$	0	0
$87$	− 191.480i	− 2.20092i
$88$	0	0
$89$	−2.56854	−0.0288600	−0.0144300	−	0.999896i	$-0.504593\pi$
$89$	−2.56854	−0.0288600	−0.0144300	+	0.999896i	$0.504593\pi$
$90$	0	0
$91$	− 0.568542i	− 0.00624772i
$92$	0	0
$93$	−218.510	−2.34957
$94$	0	0
$95$	− 159.480i	− 1.67874i
$96$	0	0
$97$	−138.569	−1.42854	−0.714271	−	0.699869i	$-0.753242\pi$
$97$	−138.569	−1.42854	−0.714271	+	0.699869i	$0.753242\pi$
$98$	0	0
$99$	21.6812i	0.219002i
$100$	0	0
$101$	45.5980	0.451465	0.225733	−	0.974189i	$-0.427522\pi$
$101$	45.5980	0.451465	0.225733	+	0.974189i	$0.427522\pi$
$102$	0	0
$103$	− 20.4020i	− 0.198078i	−0.995084	−	0.0990389i	$-0.968423\pi$
$103$	− 20.4020i	− 0.198078i	0.995084	−	0.0990389i	$-0.0315768\pi$
$104$	0	0
$105$	−61.2548	−0.583379
$106$	0	0
$107$	− 97.5147i	− 0.911353i	−0.890146	−	0.455676i	$-0.849397\pi$
$107$	− 97.5147i	− 0.911353i	0.890146	−	0.455676i	$-0.150603\pi$
$108$	0	0
$109$	149.598	1.37246	0.686229	−	0.727385i	$-0.259265\pi$
$109$	149.598	1.37246	0.686229	+	0.727385i	$0.259265\pi$
$110$	0	0
$111$	142.912i	1.28749i
$112$	0	0
$113$	−168.510	−1.49124	−0.745618	−	0.666374i	$-0.767846\pi$
$113$	−168.510	−1.49124	−0.745618	+	0.666374i	$0.767846\pi$
$114$	0	0
$115$	− 257.706i	− 2.24092i
$116$	0	0
$117$	−4.91169	−0.0419802
$118$	0	0
$119$	22.0589i	0.185369i
$120$	0	0
$121$	118.706	0.981038
$122$	0	0
$123$	118.912i	0.966762i
$124$	0	0
$125$	−66.0589	−0.528471
$126$	0	0
$127$	− 102.627i	− 0.808090i	−0.914739	−	0.404045i	$-0.867604\pi$
$127$	− 102.627i	− 0.808090i	0.914739	−	0.404045i	$-0.132396\pi$
$128$	0	0
$129$	241.823	1.87460
$130$	0	0
$131$	45.6325i	0.348339i	0.984716	+	0.174170i	$0.0557241\pi$
$131$	45.6325i	0.348339i	−0.984716	+	0.174170i	$0.944276\pi$
$132$	0	0
$133$	34.5097	0.259471
$134$	0	0
$135$	196.451i	1.45519i
$136$	0	0
$137$	−81.8823	−0.597681	−0.298840	−	0.954303i	$-0.596600\pi$
$137$	−81.8823	−0.597681	−0.298840	+	0.954303i	$0.596600\pi$
$138$	0	0
$139$	52.5442i	0.378016i	0.981976	+	0.189008i	$0.0605271\pi$
$139$	52.5442i	0.378016i	−0.981976	+	0.189008i	$0.939473\pi$
$140$	0	0
$141$	170.510	1.20929
$142$	0	0
$143$	− 0.519769i	− 0.00363475i
$144$	0	0
$145$	303.647	2.09412
$146$	0	0
$147$	223.338i	1.51931i
$148$	0	0
$149$	−116.912	−0.784642	−0.392321	−	0.919828i	$-0.628328\pi$
$149$	−116.912	−0.784642	−0.392321	+	0.919828i	$0.628328\pi$
$150$	0	0
$151$	214.676i	1.42170i	0.703345	+	0.710848i	$0.251689\pi$
$151$	214.676i	1.42170i	−0.703345	+	0.710848i	$0.748311\pi$
$152$	0	0
$153$	190.569	1.24555
$154$	0	0
$155$	− 346.510i	− 2.23555i
$156$	0	0
$157$	136.108	0.866928	0.433464	−	0.901171i	$-0.357291\pi$
$157$	136.108	0.866928	0.433464	+	0.901171i	$0.357291\pi$
$158$	0	0
$159$	78.9117i	0.496300i
$160$	0	0
$161$	55.7645	0.346363
$162$	0	0
$163$	− 53.6812i	− 0.329333i	−0.986349	−	0.164666i	$-0.947345\pi$
$163$	− 53.6812i	− 0.329333i	0.986349	−	0.164666i	$-0.0526547\pi$
$164$	0	0
$165$	−56.0000	−0.339394
$166$	0	0
$167$	93.2061i	0.558120i	0.960274	+	0.279060i	$0.0900229\pi$
$167$	93.2061i	0.558120i	−0.960274	+	0.279060i	$0.909977\pi$
$168$	0	0
$169$	−168.882	−0.999303
$170$	0	0
$171$	− 298.132i	− 1.74346i
$172$	0	0
$173$	16.3431	0.0944691	0.0472345	−	0.998884i	$-0.484959\pi$
$173$	16.3431	0.0944691	0.0472345	+	0.998884i	$0.484959\pi$
$174$	0	0
$175$	− 55.7157i	− 0.318376i
$176$	0	0
$177$	−256.451	−1.44887
$178$	0	0
$179$	120.142i	0.671185i	0.942007	+	0.335593i	$0.108937\pi$
$179$	120.142i	0.671185i	−0.942007	+	0.335593i	$0.891063\pi$
$180$	0	0
$181$	176.108	0.972970	0.486485	−	0.873689i	$-0.338279\pi$
$181$	176.108	0.972970	0.486485	+	0.873689i	$0.338279\pi$
$182$	0	0
$183$	− 166.108i	− 0.907692i
$184$	0	0
$185$	−226.627	−1.22501
$186$	0	0
$187$	20.1665i	0.107842i
$188$	0	0
$189$	−42.5097	−0.224919
$190$	0	0
$191$	− 62.8629i	− 0.329125i	−0.986367	−	0.164563i	$-0.947379\pi$
$191$	− 62.8629i	− 0.329125i	0.986367	−	0.164563i	$-0.0526213\pi$
$192$	0	0
$193$	−69.0782	−0.357918	−0.178959	−	0.983857i	$-0.557273\pi$
$193$	−69.0782	−0.357918	−0.178959	+	0.983857i	$0.557273\pi$
$194$	0	0
$195$	− 12.6863i	− 0.0650579i
$196$	0	0
$197$	45.8335	0.232657	0.116329	−	0.993211i	$-0.462887\pi$
$197$	45.8335	0.232657	0.116329	+	0.993211i	$0.462887\pi$
$198$	0	0
$199$	− 220.167i	− 1.10636i	−0.833060	−	0.553182i	$-0.813413\pi$
$199$	− 220.167i	− 1.10636i	0.833060	−	0.553182i	$-0.186587\pi$
$200$	0	0
$201$	301.706	1.50102
$202$	0	0
$203$	65.7056i	0.323673i
$204$	0	0
$205$	−188.569	−0.919847
$206$	0	0
$207$	− 481.754i	− 2.32732i
$208$	0	0
$209$	31.5492	0.150953
$210$	0	0
$211$	30.7696i	0.145827i	0.997338	+	0.0729136i	$0.0232297\pi$
$211$	30.7696i	0.145827i	−0.997338	+	0.0729136i	$0.976770\pi$
$212$	0	0
$213$	194.510	0.913191
$214$	0	0
$215$	383.480i	1.78363i
$216$	0	0
$217$	74.9807	0.345533
$218$	0	0
$219$	270.108i	1.23337i
$220$	0	0
$221$	−4.56854	−0.0206721
$222$	0	0
$223$	210.745i	0.945046i	0.881318	+	0.472523i	$0.156657\pi$
$223$	210.745i	0.945046i	−0.881318	+	0.472523i	$0.843343\pi$
$224$	0	0
$225$	−481.333	−2.13926
$226$	0	0
$227$	403.220i	1.77630i	0.459553	+	0.888151i	$0.348010\pi$
$227$	403.220i	1.77630i	−0.459553	+	0.888151i	$0.651990\pi$
$228$	0	0
$229$	−359.186	−1.56850	−0.784249	−	0.620447i	$-0.786951\pi$
$229$	−359.186	−1.56850	−0.784249	+	0.620447i	$0.786951\pi$
$230$	0	0
$231$	− 12.1177i	− 0.0524578i
$232$	0	0
$233$	210.451	0.903222	0.451611	−	0.892215i	$-0.350849\pi$
$233$	210.451	0.903222	0.451611	+	0.892215i	$0.350849\pi$
$234$	0	0
$235$	270.392i	1.15060i
$236$	0	0
$237$	−666.039	−2.81029
$238$	0	0
$239$	− 332.215i	− 1.39002i	−0.718999	−	0.695011i	$-0.755399\pi$
$239$	− 332.215i	− 1.39002i	0.718999	−	0.695011i	$-0.244601\pi$
$240$	0	0
$241$	71.7056	0.297534	0.148767	−	0.988872i	$-0.452470\pi$
$241$	71.7056	0.297534	0.148767	+	0.988872i	$0.452470\pi$
$242$	0	0
$243$	− 254.770i	− 1.04843i
$244$	0	0
$245$	−354.167	−1.44558
$246$	0	0
$247$	7.14719i	0.0289360i
$248$	0	0
$249$	−553.588	−2.22324
$250$	0	0
$251$	− 50.6518i	− 0.201800i	−0.994897	−	0.100900i	$-0.967828\pi$
$251$	− 50.6518i	− 0.201800i	0.994897	−	0.100900i	$-0.0321722\pi$
$252$	0	0
$253$	50.9807	0.201505
$254$	0	0
$255$	492.215i	1.93026i
$256$	0	0
$257$	338.000	1.31518	0.657588	−	0.753378i	$-0.271577\pi$
$257$	338.000	1.31518	0.657588	+	0.753378i	$0.271577\pi$
$258$	0	0
$259$	− 49.0395i	− 0.189342i
$260$	0	0
$261$	567.637	2.17485
$262$	0	0
$263$	− 258.794i	− 0.984007i	−0.870593	−	0.492004i	$-0.836265\pi$
$263$	− 258.794i	− 0.984007i	0.870593	−	0.492004i	$-0.163735\pi$
$264$	0	0
$265$	−125.137	−0.472215
$266$	0	0
$267$	− 12.4020i	− 0.0464495i
$268$	0	0
$269$	399.637	1.48564	0.742819	−	0.669492i	$-0.233488\pi$
$269$	399.637	1.48564	0.742819	+	0.669492i	$0.233488\pi$
$270$	0	0
$271$	253.823i	0.936618i	0.883565	+	0.468309i	$0.155137\pi$
$271$	253.823i	0.936618i	−0.883565	+	0.468309i	$0.844863\pi$
$272$	0	0
$273$	2.74517	0.0100556
$274$	0	0
$275$	− 50.9361i	− 0.185222i
$276$	0	0
$277$	−505.696	−1.82562	−0.912808	−	0.408389i	$-0.866091\pi$
$277$	−505.696	−1.82562	−0.912808	+	0.408389i	$0.866091\pi$
$278$	0	0
$279$	− 647.765i	− 2.32174i
$280$	0	0
$281$	245.196	0.872583	0.436292	−	0.899805i	$-0.356292\pi$
$281$	245.196	0.872583	0.436292	+	0.899805i	$0.356292\pi$
$282$	0	0
$283$	22.1522i	0.0782765i	0.999234	+	0.0391382i	$0.0124613\pi$
$283$	22.1522i	0.0782765i	−0.999234	+	0.0391382i	$0.987539\pi$
$284$	0	0
$285$	770.039	2.70189
$286$	0	0
$287$	− 40.8040i	− 0.142174i
$288$	0	0
$289$	−111.745	−0.386661
$290$	0	0
$291$	− 669.068i	− 2.29920i
$292$	0	0
$293$	232.343	0.792980	0.396490	−	0.918039i	$-0.370228\pi$
$293$	232.343	0.792980	0.396490	+	0.918039i	$0.370228\pi$
$294$	0	0
$295$	− 406.676i	− 1.37856i
$296$	0	0
$297$	−38.8629	−0.130852
$298$	0	0
$299$	11.5492i	0.0386261i
$300$	0	0
$301$	−82.9807	−0.275683
$302$	0	0
$303$	220.167i	0.726622i
$304$	0	0
$305$	263.411	0.863643
$306$	0	0
$307$	408.240i	1.32977i	0.746945	+	0.664885i	$0.231520\pi$
$307$	408.240i	1.32977i	−0.746945	+	0.664885i	$0.768480\pi$
$308$	0	0
$309$	98.5097	0.318802
$310$	0	0
$311$	544.715i	1.75149i	0.482770	+	0.875747i	$0.339631\pi$
$311$	544.715i	1.75149i	−0.482770	+	0.875747i	$0.660369\pi$
$312$	0	0
$313$	−327.373	−1.04592	−0.522959	−	0.852358i	$-0.675172\pi$
$313$	−327.373	−1.04592	−0.522959	+	0.852358i	$0.675172\pi$
$314$	0	0
$315$	− 181.588i	− 0.576469i
$316$	0	0
$317$	120.343	0.379631	0.189816	−	0.981820i	$-0.439211\pi$
$317$	120.343	0.379631	0.189816	+	0.981820i	$0.439211\pi$
$318$	0	0
$319$	60.0690i	0.188304i
$320$	0	0
$321$	470.843	1.46680
$322$	0	0
$323$	− 277.304i	− 0.858525i
$324$	0	0
$325$	11.5391	0.0355049
$326$	0	0
$327$	722.323i	2.20894i
$328$	0	0
$329$	−58.5097	−0.177841
$330$	0	0
$331$	− 61.1615i	− 0.184778i	−0.995723	−	0.0923889i	$-0.970550\pi$
$331$	− 61.1615i	− 0.184778i	0.995723	−	0.0923889i	$-0.0294503\pi$
$332$	0	0
$333$	−423.657	−1.27224
$334$	0	0
$335$	478.441i	1.42818i
$336$	0	0
$337$	−58.7838	−0.174433	−0.0872164	−	0.996189i	$-0.527797\pi$
$337$	−58.7838	−0.174433	−0.0872164	+	0.996189i	$0.527797\pi$
$338$	0	0
$339$	− 813.637i	− 2.40011i
$340$	0	0
$341$	68.5483	0.201022
$342$	0	0
$343$	− 157.823i	− 0.460126i
$344$	0	0
$345$	1244.31	3.60670
$346$	0	0
$347$	396.406i	1.14238i	0.820818	+	0.571190i	$0.193518\pi$
$347$	396.406i	1.14238i	−0.820818	+	0.571190i	$0.806482\pi$
$348$	0	0
$349$	−7.18586	−0.0205899	−0.0102949	−	0.999947i	$-0.503277\pi$
$349$	−7.18586	−0.0205899	−0.0102949	+	0.999947i	$0.503277\pi$
$350$	0	0
$351$	− 8.80404i	− 0.0250827i
$352$	0	0
$353$	−171.726	−0.486475	−0.243238	−	0.969967i	$-0.578210\pi$
$353$	−171.726	−0.486475	−0.243238	+	0.969967i	$0.578210\pi$
$354$	0	0
$355$	308.451i	0.868875i
$356$	0	0
$357$	−106.510	−0.298346
$358$	0	0
$359$	160.617i	0.447402i	0.974658	+	0.223701i	$0.0718139\pi$
$359$	160.617i	0.447402i	−0.974658	+	0.223701i	$0.928186\pi$
$360$	0	0
$361$	−72.8234	−0.201727
$362$	0	0
$363$	573.161i	1.57896i
$364$	0	0
$365$	−428.333	−1.17352
$366$	0	0
$367$	55.1960i	0.150398i	0.997169	+	0.0751989i	$0.0239592\pi$
$367$	55.1960i	0.150398i	−0.997169	+	0.0751989i	$0.976041\pi$
$368$	0	0
$369$	−352.510	−0.955311
$370$	0	0
$371$	− 27.0782i	− 0.0729871i
$372$	0	0
$373$	346.853	0.929900	0.464950	−	0.885337i	$-0.346072\pi$
$373$	346.853	0.929900	0.464950	+	0.885337i	$0.346072\pi$
$374$	0	0
$375$	− 318.960i	− 0.850561i
$376$	0	0
$377$	−13.6081	−0.0360957
$378$	0	0
$379$	112.759i	0.297518i	0.988873	+	0.148759i	$0.0475279\pi$
$379$	112.759i	0.297518i	−0.988873	+	0.148759i	$0.952472\pi$
$380$	0	0
$381$	495.529	1.30060
$382$	0	0
$383$	− 618.039i	− 1.61368i	−0.590771	−	0.806839i	$-0.701176\pi$
$383$	− 618.039i	− 1.61368i	0.590771	−	0.806839i	$-0.298824\pi$
$384$	0	0
$385$	19.2162	0.0499121
$386$	0	0
$387$	716.877i	1.85240i
$388$	0	0
$389$	419.088	1.07735	0.538674	−	0.842514i	$-0.318925\pi$
$389$	419.088	1.07735	0.538674	+	0.842514i	$0.318925\pi$
$390$	0	0
$391$	− 448.098i	− 1.14603i
$392$	0	0
$393$	−220.333	−0.560644
$394$	0	0
$395$	− 1056.20i	− 2.67391i
$396$	0	0
$397$	−564.441	−1.42176	−0.710882	−	0.703311i	$-0.751704\pi$
$397$	−564.441	−1.42176	−0.710882	+	0.703311i	$0.751704\pi$
$398$	0	0
$399$	166.627i	0.417613i
$400$	0	0
$401$	−199.823	−0.498313	−0.249156	−	0.968463i	$-0.580153\pi$
$401$	−199.823	−0.498313	−0.249156	+	0.968463i	$0.580153\pi$
$402$	0	0
$403$	15.5290i	0.0385335i
$404$	0	0
$405$	37.8335	0.0934160
$406$	0	0
$407$	− 44.8326i	− 0.110154i
$408$	0	0
$409$	−118.902	−0.290713	−0.145356	−	0.989379i	$-0.546433\pi$
$409$	−118.902	−0.290713	−0.145356	+	0.989379i	$0.546433\pi$
$410$	0	0
$411$	− 395.362i	− 0.961953i
$412$	0	0
$413$	88.0000	0.213075
$414$	0	0
$415$	− 877.872i	− 2.11535i
$416$	0	0
$417$	−253.706	−0.608407
$418$	0	0
$419$	− 304.093i	− 0.725760i	−0.931836	−	0.362880i	$-0.881794\pi$
$419$	− 304.093i	− 0.725760i	0.931836	−	0.362880i	$-0.118206\pi$
$420$	0	0
$421$	−188.676	−0.448162	−0.224081	−	0.974571i	$-0.571938\pi$
$421$	−188.676	−0.448162	−0.224081	+	0.974571i	$0.571938\pi$
$422$	0	0
$423$	505.470i	1.19496i
$424$	0	0
$425$	−447.706	−1.05343
$426$	0	0
$427$	56.9991i	0.133487i
$428$	0	0
$429$	2.50967	0.00585004
$430$	0	0
$431$	402.843i	0.934670i	0.884080	+	0.467335i	$0.154786\pi$
$431$	402.843i	0.934670i	−0.884080	+	0.467335i	$0.845214\pi$
$432$	0	0
$433$	−721.862	−1.66712	−0.833559	−	0.552431i	$-0.813700\pi$
$433$	−721.862	−1.66712	−0.833559	+	0.552431i	$0.813700\pi$
$434$	0	0
$435$	1466.14i	3.37043i
$436$	0	0
$437$	−701.019	−1.60416
$438$	0	0
$439$	− 333.872i	− 0.760529i	−0.924878	−	0.380264i	$-0.875833\pi$
$439$	− 333.872i	− 0.760529i	0.924878	−	0.380264i	$-0.124167\pi$
$440$	0	0
$441$	−662.078	−1.50131
$442$	0	0
$443$	− 393.848i	− 0.889047i	−0.895767	−	0.444523i	$-0.853373\pi$
$443$	− 393.848i	− 0.889047i	0.895767	−	0.444523i	$-0.146627\pi$
$444$	0	0
$445$	19.6670	0.0441954
$446$	0	0
$447$	− 564.500i	− 1.26286i
$448$	0	0
$449$	26.9218	0.0599594	0.0299797	−	0.999551i	$-0.490456\pi$
$449$	26.9218	0.0599594	0.0299797	+	0.999551i	$0.490456\pi$
$450$	0	0
$451$	− 37.3036i	− 0.0827131i
$452$	0	0
$453$	−1036.55	−2.28819
$454$	0	0
$455$	4.35325i	0.00956758i
$456$	0	0
$457$	222.118	0.486034	0.243017	−	0.970022i	$-0.421863\pi$
$457$	222.118	0.486034	0.243017	+	0.970022i	$0.421863\pi$
$458$	0	0
$459$	341.588i	0.744200i
$460$	0	0
$461$	139.088	0.301710	0.150855	−	0.988556i	$-0.451797\pi$
$461$	139.088	0.301710	0.150855	+	0.988556i	$0.451797\pi$
$462$	0	0
$463$	− 480.098i	− 1.03693i	−0.855100	−	0.518464i	$-0.826504\pi$
$463$	− 480.098i	− 1.03693i	0.855100	−	0.518464i	$-0.173496\pi$
$464$	0	0
$465$	1673.10	3.59806
$466$	0	0
$467$	− 105.092i	− 0.225037i	−0.993650	−	0.112519i	$-0.964108\pi$
$467$	− 105.092i	− 0.225037i	0.993650	−	0.112519i	$-0.0358918\pi$
$468$	0	0
$469$	−103.529	−0.220744
$470$	0	0
$471$	657.186i	1.39530i
$472$	0	0
$473$	−75.8620	−0.160385
$474$	0	0
$475$	700.406i	1.47454i
$476$	0	0
$477$	−233.931	−0.490421
$478$	0	0
$479$	391.765i	0.817880i	0.912561	+	0.408940i	$0.134101\pi$
$479$	391.765i	0.817880i	−0.912561	+	0.408940i	$0.865899\pi$
$480$	0	0
$481$	10.1564	0.0211152
$482$	0	0
$483$	269.255i	0.557463i
$484$	0	0
$485$	1061.00	2.18763
$486$	0	0
$487$	567.716i	1.16574i	0.812565	+	0.582870i	$0.198070\pi$
$487$	567.716i	1.16574i	−0.812565	+	0.582870i	$0.801930\pi$
$488$	0	0
$489$	259.196	0.530053
$490$	0	0
$491$	755.602i	1.53890i	0.638704	+	0.769452i	$0.279471\pi$
$491$	755.602i	1.53890i	−0.638704	+	0.769452i	$0.720529\pi$
$492$	0	0
$493$	527.980	1.07095
$494$	0	0
$495$	− 166.010i	− 0.335374i
$496$	0	0
$497$	−66.7452	−0.134296
$498$	0	0
$499$	− 208.759i	− 0.418356i	−0.977878	−	0.209178i	$-0.932921\pi$
$499$	− 208.759i	− 0.418356i	0.977878	−	0.209178i	$-0.0670787\pi$
$500$	0	0
$501$	−450.039	−0.898281
$502$	0	0
$503$	− 136.187i	− 0.270749i	−0.990795	−	0.135374i	$-0.956776\pi$
$503$	− 136.187i	− 0.270749i	0.990795	−	0.135374i	$-0.0432237\pi$
$504$	0	0
$505$	−349.137	−0.691361
$506$	0	0
$507$	− 815.436i	− 1.60835i
$508$	0	0
$509$	−247.892	−0.487018	−0.243509	−	0.969899i	$-0.578299\pi$
$509$	−247.892	−0.487018	−0.243509	+	0.969899i	$0.578299\pi$
$510$	0	0
$511$	− 92.6863i	− 0.181382i
$512$	0	0
$513$	534.392	1.04170
$514$	0	0
$515$	156.215i	0.303331i
$516$	0	0
$517$	−53.4903	−0.103463
$518$	0	0
$519$	78.9117i	0.152046i
$520$	0	0
$521$	881.724	1.69237	0.846184	−	0.532890i	$-0.178894\pi$
$521$	881.724	1.69237	0.846184	+	0.532890i	$0.178894\pi$
$522$	0	0
$523$	− 818.181i	− 1.56440i	−0.623028	−	0.782200i	$-0.714098\pi$
$523$	− 818.181i	− 1.56440i	0.623028	−	0.782200i	$-0.285902\pi$
$524$	0	0
$525$	269.019	0.512418
$526$	0	0
$527$	− 602.510i	− 1.14328i
$528$	0	0
$529$	−603.784	−1.14137
$530$	0	0
$531$	− 760.240i	− 1.43171i
$532$	0	0
$533$	8.45079	0.0158551
$534$	0	0
$535$	746.656i	1.39562i
$536$	0	0
$537$	−580.098	−1.08026
$538$	0	0
$539$	− 70.0631i	− 0.129987i
$540$	0	0
$541$	−263.892	−0.487786	−0.243893	−	0.969802i	$-0.578425\pi$
$541$	−263.892	−0.487786	−0.243893	+	0.969802i	$0.578425\pi$
$542$	0	0
$543$	850.323i	1.56597i
$544$	0	0
$545$	−1145.45	−2.10174
$546$	0	0
$547$	− 52.6417i	− 0.0962371i	−0.998842	−	0.0481186i	$-0.984677\pi$
$547$	− 52.6417i	− 0.0962371i	0.998842	−	0.0481186i	$-0.0153225\pi$
$548$	0	0
$549$	492.420	0.896941
$550$	0	0
$551$	− 825.990i	− 1.49907i
$552$	0	0
$553$	228.548	0.413288
$554$	0	0
$555$	− 1094.25i	− 1.97163i
$556$	0	0
$557$	725.362	1.30227	0.651133	−	0.758963i	$-0.274294\pi$
$557$	725.362	1.30227	0.651133	+	0.758963i	$0.274294\pi$
$558$	0	0
$559$	− 17.1859i	− 0.0307439i
$560$	0	0
$561$	−97.3726	−0.173570
$562$	0	0
$563$	− 156.780i	− 0.278472i	−0.990259	−	0.139236i	$-0.955535\pi$
$563$	− 156.780i	− 0.278472i	0.990259	−	0.139236i	$-0.0444646\pi$
$564$	0	0
$565$	1290.25	2.28364
$566$	0	0
$567$	8.18672i	0.0144387i
$568$	0	0
$569$	796.705	1.40018	0.700092	−	0.714053i	$-0.253142\pi$
$569$	796.705	1.40018	0.700092	+	0.714053i	$0.253142\pi$
$570$	0	0
$571$	− 151.103i	− 0.264628i	−0.991208	−	0.132314i	$-0.957759\pi$
$571$	− 151.103i	− 0.264628i	0.991208	−	0.132314i	$-0.0422407\pi$
$572$	0	0
$573$	303.529	0.529719
$574$	0	0
$575$	1131.79i	1.96834i
$576$	0	0
$577$	462.313	0.801235	0.400618	−	0.916245i	$-0.368796\pi$
$577$	462.313	0.801235	0.400618	+	0.916245i	$0.368796\pi$
$578$	0	0
$579$	− 333.539i	− 0.576061i
$580$	0	0
$581$	189.961	0.326956
$582$	0	0
$583$	− 24.7553i	− 0.0424619i
$584$	0	0
$585$	37.6081	0.0642873
$586$	0	0
$587$	278.721i	0.474822i	0.971409	+	0.237411i	$0.0762989\pi$
$587$	278.721i	0.474822i	−0.971409	+	0.237411i	$0.923701\pi$
$588$	0	0
$589$	−942.587	−1.60032
$590$	0	0
$591$	221.304i	0.374456i
$592$	0	0
$593$	934.548	1.57597	0.787983	−	0.615696i	$-0.211125\pi$
$593$	934.548	1.57597	0.787983	+	0.615696i	$0.211125\pi$
$594$	0	0
$595$	− 168.902i	− 0.283868i
$596$	0	0
$597$	1063.06	1.78067
$598$	0	0
$599$	− 1004.74i	− 1.67735i	−0.544629	−	0.838677i	$-0.683330\pi$
$599$	− 1004.74i	− 1.67735i	0.544629	−	0.838677i	$-0.316670\pi$
$600$	0	0
$601$	722.451	1.20208	0.601041	−	0.799218i	$-0.294753\pi$
$601$	722.451	1.20208	0.601041	+	0.799218i	$0.294753\pi$
$602$	0	0
$603$	894.396i	1.48324i
$604$	0	0
$605$	−908.912	−1.50233
$606$	0	0
$607$	344.431i	0.567431i	0.958909	+	0.283715i	$0.0915671\pi$
$607$	344.431i	0.567431i	−0.958909	+	0.283715i	$0.908433\pi$
$608$	0	0
$609$	−317.255	−0.520944
$610$	0	0
$611$	− 12.1177i	− 0.0198326i
$612$	0	0
$613$	−839.657	−1.36975	−0.684875	−	0.728660i	$-0.740143\pi$
$613$	−839.657	−1.36975	−0.684875	+	0.728660i	$0.740143\pi$
$614$	0	0
$615$	− 910.489i	− 1.48047i
$616$	0	0
$617$	−41.8620	−0.0678477	−0.0339239	−	0.999424i	$-0.510800\pi$
$617$	−41.8620	−0.0678477	−0.0339239	+	0.999424i	$0.510800\pi$
$618$	0	0
$619$	1098.60i	1.77480i	0.460997	+	0.887401i	$0.347492\pi$
$619$	1098.60i	1.77480i	−0.460997	+	0.887401i	$0.652508\pi$
$620$	0	0
$621$	863.529	1.39055
$622$	0	0
$623$	4.25570i	0.00683098i
$624$	0	0
$625$	−334.882	−0.535812
$626$	0	0
$627$	152.333i	0.242955i
$628$	0	0
$629$	−394.059	−0.626485
$630$	0	0
$631$	1042.32i	1.65186i	0.563774	+	0.825929i	$0.309349\pi$
$631$	1042.32i	1.65186i	−0.563774	+	0.825929i	$0.690651\pi$
$632$	0	0
$633$	−148.569	−0.234705
$634$	0	0
$635$	785.803i	1.23749i
$636$	0	0
$637$	15.8721	0.0249170
$638$	0	0
$639$	576.617i	0.902375i
$640$	0	0
$641$	448.412	0.699551	0.349775	−	0.936834i	$-0.386258\pi$
$641$	448.412	0.699551	0.349775	+	0.936834i	$0.386258\pi$
$642$	0	0
$643$	− 523.740i	− 0.814526i	−0.913311	−	0.407263i	$-0.866483\pi$
$643$	− 523.740i	− 0.814526i	0.913311	−	0.407263i	$-0.133517\pi$
$644$	0	0
$645$	−1851.61	−2.87071
$646$	0	0
$647$	465.283i	0.719140i	0.933118	+	0.359570i	$0.117077\pi$
$647$	465.283i	0.719140i	−0.933118	+	0.359570i	$0.882923\pi$
$648$	0	0
$649$	80.4508	0.123961
$650$	0	0
$651$	362.039i	0.556127i
$652$	0	0
$653$	283.559	0.434241	0.217120	−	0.976145i	$-0.430334\pi$
$653$	283.559	0.434241	0.217120	+	0.976145i	$0.430334\pi$
$654$	0	0
$655$	− 349.401i	− 0.533437i
$656$	0	0
$657$	−800.725	−1.21876
$658$	0	0
$659$	209.710i	0.318224i	0.987261	+	0.159112i	$0.0508631\pi$
$659$	209.710i	0.318224i	−0.987261	+	0.159112i	$0.949137\pi$
$660$	0	0
$661$	−31.4214	−0.0475361	−0.0237680	−	0.999718i	$-0.507566\pi$
$661$	−31.4214	−0.0475361	−0.0237680	+	0.999718i	$0.507566\pi$
$662$	0	0
$663$	− 22.0589i	− 0.0332713i
$664$	0	0
$665$	−264.235	−0.397347
$666$	0	0
$667$	− 1334.72i	− 2.00109i
$668$	0	0
$669$	−1017.57	−1.52103
$670$	0	0
$671$	52.1094i	0.0776593i
$672$	0	0
$673$	1327.00	1.97177	0.985883	−	0.167433i	$-0.0535478\pi$
$673$	1327.00	1.97177	0.985883	+	0.167433i	$0.0535478\pi$
$674$	0	0
$675$	− 862.774i	− 1.27818i
$676$	0	0
$677$	211.559	0.312495	0.156248	−	0.987718i	$-0.450060\pi$
$677$	211.559	0.312495	0.156248	+	0.987718i	$0.450060\pi$
$678$	0	0
$679$	229.588i	0.338126i
$680$	0	0
$681$	−1946.92	−2.85891
$682$	0	0
$683$	801.799i	1.17394i	0.809610	+	0.586969i	$0.199679\pi$
$683$	801.799i	1.17394i	−0.809610	+	0.586969i	$0.800321\pi$
$684$	0	0
$685$	626.960	0.915271
$686$	0	0
$687$	− 1734.30i	− 2.52446i
$688$	0	0
$689$	5.60808	0.00813945
$690$	0	0
$691$	173.161i	0.250595i	0.992119	+	0.125298i	$0.0399886\pi$
$691$	173.161i	0.250595i	−0.992119	+	0.125298i	$0.960011\pi$
$692$	0	0
$693$	35.9227	0.0518364
$694$	0	0
$695$	− 402.323i	− 0.578882i
$696$	0	0
$697$	−327.882	−0.470419
$698$	0	0
$699$	1016.15i	1.45371i
$700$	0	0
$701$	−764.912	−1.09117	−0.545586	−	0.838055i	$-0.683693\pi$
$701$	−764.912	−1.09117	−0.545586	+	0.838055i	$0.683693\pi$
$702$	0	0
$703$	616.479i	0.876927i
$704$	0	0
$705$	−1305.57	−1.85187
$706$	0	0
$707$	− 75.5492i	− 0.106859i
$708$	0	0
$709$	252.656	0.356355	0.178178	−	0.983998i	$-0.442980\pi$
$709$	252.656	0.356355	0.178178	+	0.983998i	$0.442980\pi$
$710$	0	0
$711$	− 1974.45i	− 2.77700i
$712$	0	0
$713$	−1523.14	−2.13623
$714$	0	0
$715$	3.97980i	0.00556615i
$716$	0	0
$717$	1604.08	2.23721
$718$	0	0
$719$	806.725i	1.12201i	0.827813	+	0.561005i	$0.189585\pi$
$719$	806.725i	1.12201i	−0.827813	+	0.561005i	$0.810415\pi$
$720$	0	0
$721$	−33.8032	−0.0468837
$722$	0	0
$723$	346.225i	0.478873i
$724$	0	0
$725$	−1333.56	−1.83939
$726$	0	0
$727$	− 1185.75i	− 1.63102i	−0.578740	−	0.815512i	$-0.696455\pi$
$727$	− 1185.75i	− 1.63102i	0.578740	−	0.815512i	$-0.303545\pi$
$728$	0	0
$729$	1185.67	1.62643
$730$	0	0
$731$	666.794i	0.912167i
$732$	0	0
$733$	785.911	1.07218	0.536092	−	0.844160i	$-0.319900\pi$
$733$	785.911	1.07218	0.536092	+	0.844160i	$0.319900\pi$
$734$	0	0
$735$	− 1710.07i	− 2.32662i
$736$	0	0
$737$	−94.6476	−0.128423
$738$	0	0
$739$	1074.28i	1.45369i	0.686800	+	0.726846i	$0.259015\pi$
$739$	1074.28i	1.45369i	−0.686800	+	0.726846i	$0.740985\pi$
$740$	0	0
$741$	−34.5097	−0.0465718
$742$	0	0
$743$	− 385.852i	− 0.519316i	−0.965701	−	0.259658i	$-0.916390\pi$
$743$	− 385.852i	− 0.519316i	0.965701	−	0.259658i	$-0.0836099\pi$
$744$	0	0
$745$	895.176	1.20158
$746$	0	0
$747$	− 1641.09i	− 2.19691i
$748$	0	0
$749$	−161.568	−0.215711
$750$	0	0
$751$	729.665i	0.971592i	0.874072	+	0.485796i	$0.161470\pi$
$751$	729.665i	0.971592i	−0.874072	+	0.485796i	$0.838530\pi$
$752$	0	0
$753$	244.569	0.324792
$754$	0	0
$755$	− 1643.74i	− 2.17714i
$756$	0	0
$757$	570.617	0.753788	0.376894	−	0.926256i	$-0.376992\pi$
$757$	570.617	0.753788	0.376894	+	0.926256i	$0.376992\pi$
$758$	0	0
$759$	246.156i	0.324317i
$760$	0	0
$761$	−347.450	−0.456570	−0.228285	−	0.973594i	$-0.573312\pi$
$761$	−347.450	−0.456570	−0.228285	+	0.973594i	$0.573312\pi$
$762$	0	0
$763$	− 247.862i	− 0.324852i
$764$	0	0
$765$	−1459.16	−1.90739
$766$	0	0
$767$	18.2254i	0.0237619i
$768$	0	0
$769$	505.980	0.657971	0.328986	−	0.944335i	$-0.393293\pi$
$769$	505.980	0.657971	0.328986	+	0.944335i	$0.393293\pi$
$770$	0	0
$771$	1632.01i	2.11674i
$772$	0	0
$773$	476.656	0.616631	0.308316	−	0.951284i	$-0.400235\pi$
$773$	476.656	0.616631	0.308316	+	0.951284i	$0.400235\pi$
$774$	0	0
$775$	1521.80i	1.96362i
$776$	0	0
$777$	236.784	0.304741
$778$	0	0
$779$	512.950i	0.658473i
$780$	0	0
$781$	−61.0193	−0.0781298
$782$	0	0
$783$	1017.47i	1.29945i
$784$	0	0
$785$	−1042.16	−1.32759
$786$	0	0
$787$	884.731i	1.12418i	0.827076	+	0.562091i	$0.190003\pi$
$787$	884.731i	1.12418i	−0.827076	+	0.562091i	$0.809997\pi$
$788$	0	0
$789$	1249.57	1.58374
$790$	0	0
$791$	279.196i	0.352966i
$792$	0	0
$793$	−11.8049	−0.0148864
$794$	0	0
$795$	− 604.215i	− 0.760019i
$796$	0	0
$797$	−956.441	−1.20005	−0.600026	−	0.799981i	$-0.704843\pi$
$797$	−956.441	−1.20005	−0.600026	+	0.799981i	$0.704843\pi$
$798$	0	0
$799$	470.156i	0.588431i
$800$	0	0
$801$	36.7654	0.0458993
$802$	0	0
$803$	− 84.7351i	− 0.105523i
$804$	0	0
$805$	−426.981	−0.530411
$806$	0	0
$807$	1929.62i	2.39110i
$808$	0	0
$809$	1185.72	1.46567	0.732833	−	0.680408i	$-0.238198\pi$
$809$	1185.72	1.46567	0.732833	+	0.680408i	$0.238198\pi$
$810$	0	0
$811$	− 1017.95i	− 1.25517i	−0.778547	−	0.627587i	$-0.784043\pi$
$811$	− 1017.95i	− 1.25517i	0.778547	−	0.627587i	$-0.215957\pi$
$812$	0	0
$813$	−1225.57	−1.50746
$814$	0	0
$815$	411.029i	0.504331i
$816$	0	0
$817$	1043.16	1.27681
$818$	0	0
$819$	8.13795i	0.00993645i
$820$	0	0
$821$	−1359.19	−1.65552	−0.827762	−	0.561079i	$-0.810386\pi$
$821$	−1359.19	−1.65552	−0.827762	+	0.561079i	$0.810386\pi$
$822$	0	0
$823$	− 360.382i	− 0.437888i	−0.975737	−	0.218944i	$-0.929739\pi$
$823$	− 360.382i	− 0.437888i	0.975737	−	0.218944i	$-0.0702612\pi$
$824$	0	0
$825$	245.941	0.298110
$826$	0	0
$827$	− 464.475i	− 0.561639i	−0.959761	−	0.280819i	$-0.909394\pi$
$827$	− 464.475i	− 0.561639i	0.959761	−	0.280819i	$-0.0906062\pi$
$828$	0	0
$829$	103.166	0.124446	0.0622230	−	0.998062i	$-0.480181\pi$
$829$	103.166	0.124446	0.0622230	+	0.998062i	$0.480181\pi$
$830$	0	0
$831$	− 2441.71i	− 2.93828i
$832$	0	0
$833$	−615.823	−0.739284
$834$	0	0
$835$	− 713.665i	− 0.854689i
$836$	0	0
$837$	1161.10	1.38721
$838$	0	0
$839$	687.204i	0.819075i	0.912293	+	0.409538i	$0.134310\pi$
$839$	687.204i	0.819075i	−0.912293	+	0.409538i	$0.865690\pi$
$840$	0	0
$841$	731.666	0.869995
$842$	0	0
$843$	1183.91i	1.40440i
$844$	0	0
$845$	1293.11	1.53030
$846$	0	0
$847$	− 196.678i	− 0.232205i
$848$	0	0
$849$	−106.960	−0.125984
$850$	0	0
$851$	996.175i	1.17059i
$852$	0	0
$853$	693.833	0.813404	0.406702	−	0.913561i	$-0.366679\pi$
$853$	693.833	0.813404	0.406702	+	0.913561i	$0.366679\pi$
$854$	0	0
$855$	2282.75i	2.66989i
$856$	0	0
$857$	−2.35325	−0.00274591	−0.00137296	−	0.999999i	$-0.500437\pi$
$857$	−2.35325	−0.00274591	−0.00137296	+	0.999999i	$0.500437\pi$
$858$	0	0
$859$	− 1422.49i	− 1.65599i	−0.560737	−	0.827994i	$-0.689482\pi$
$859$	− 1422.49i	− 1.65599i	0.560737	−	0.827994i	$-0.310518\pi$
$860$	0	0
$861$	197.019	0.228826
$862$	0	0
$863$	− 501.961i	− 0.581647i	−0.956777	−	0.290823i	$-0.906071\pi$
$863$	− 501.961i	− 0.581647i	0.956777	−	0.290823i	$-0.0939292\pi$
$864$	0	0
$865$	−125.137	−0.144667
$866$	0	0
$867$	− 539.553i	− 0.622322i
$868$	0	0
$869$	208.942	0.240440
$870$	0	0
$871$	− 21.4416i	− 0.0246172i
$872$	0	0
$873$	1983.43	2.27197
$874$	0	0
$875$	109.450i	0.125086i
$876$	0	0
$877$	1560.73	1.77963	0.889814	−	0.456324i	$-0.150834\pi$
$877$	1560.73	1.77963	0.889814	+	0.456324i	$0.150834\pi$
$878$	0	0
$879$	1121.85i	1.27628i
$880$	0	0
$881$	665.294	0.755157	0.377579	−	0.925978i	$-0.376757\pi$
$881$	665.294	0.755157	0.377579	+	0.925978i	$0.376757\pi$
$882$	0	0
$883$	− 817.052i	− 0.925314i	−0.886537	−	0.462657i	$-0.846896\pi$
$883$	− 817.052i	− 0.925314i	0.886537	−	0.462657i	$-0.153104\pi$
$884$	0	0
$885$	1963.61	2.21876
$886$	0	0
$887$	− 1170.42i	− 1.31953i	−0.751473	−	0.659764i	$-0.770656\pi$
$887$	− 1170.42i	− 1.31953i	0.751473	−	0.659764i	$-0.229344\pi$
$888$	0	0
$889$	−170.039	−0.191270
$890$	0	0
$891$	7.48441i	0.00840002i
$892$	0	0
$893$	735.529	0.823661
$894$	0	0
$895$	− 919.911i	− 1.02783i
$896$	0	0
$897$	−55.7645	−0.0621678
$898$	0	0
$899$	− 1794.66i	− 1.99629i
$900$	0	0
$901$	−217.588	−0.241496
$902$	0	0
$903$	− 400.666i	− 0.443706i
$904$	0	0
$905$	−1348.43	−1.48998
$906$	0	0
$907$	− 936.435i	− 1.03245i	−0.856452	−	0.516226i	$-0.827336\pi$
$907$	− 936.435i	− 1.03245i	0.856452	−	0.516226i	$-0.172664\pi$
$908$	0	0
$909$	−652.676	−0.718016
$910$	0	0
$911$	− 958.018i	− 1.05161i	−0.850605	−	0.525806i	$-0.823764\pi$
$911$	− 958.018i	− 1.05161i	0.850605	−	0.525806i	$-0.176236\pi$
$912$	0	0
$913$	173.665	0.190214
$914$	0	0
$915$	1271.86i	1.39001i
$916$	0	0
$917$	75.6063	0.0824497
$918$	0	0
$919$	− 255.675i	− 0.278210i	−0.990278	−	0.139105i	$-0.955577\pi$
$919$	− 255.675i	− 0.278210i	0.990278	−	0.139105i	$-0.0444226\pi$
$920$	0	0
$921$	−1971.16	−2.14023
$922$	0	0
$923$	− 13.8234i	− 0.0149766i
$924$	0	0
$925$	995.304	1.07600
$926$	0	0
$927$	292.029i	0.315025i
$928$	0	0
$929$	868.960	0.935372	0.467686	−	0.883895i	$-0.345088\pi$
$929$	868.960	0.935372	0.467686	+	0.883895i	$0.345088\pi$
$930$	0	0
$931$	963.415i	1.03482i
$932$	0	0
$933$	−2630.12	−2.81899
$934$	0	0
$935$	− 154.412i	− 0.165147i
$936$	0	0
$937$	−1122.57	−1.19805	−0.599023	−	0.800732i	$-0.704444\pi$
$937$	−1122.57	−1.19805	−0.599023	+	0.800732i	$0.704444\pi$
$938$	0	0
$939$	− 1580.69i	− 1.68338i
$940$	0	0
$941$	−765.537	−0.813536	−0.406768	−	0.913531i	$-0.633344\pi$
$941$	−765.537	−0.813536	−0.406768	+	0.913531i	$0.633344\pi$
$942$	0	0
$943$	828.881i	0.878983i
$944$	0	0
$945$	325.490	0.344434
$946$	0	0
$947$	− 249.563i	− 0.263531i	−0.991281	−	0.131765i	$-0.957935\pi$
$947$	− 249.563i	− 0.263531i	0.991281	−	0.131765i	$-0.0420645\pi$
$948$	0	0
$949$	19.1960	0.0202276
$950$	0	0
$951$	581.068i	0.611007i
$952$	0	0
$953$	−709.960	−0.744973	−0.372487	−	0.928038i	$-0.621495\pi$
$953$	−709.960	−0.744973	−0.372487	+	0.928038i	$0.621495\pi$
$954$	0	0
$955$	481.332i	0.504013i
$956$	0	0
$957$	−290.039	−0.303071
$958$	0	0
$959$	135.667i	0.141467i
$960$	0	0
$961$	−1087.00	−1.13111
$962$	0	0
$963$	1395.80i	1.44943i
$964$	0	0
$965$	528.922	0.548105
$966$	0	0
$967$	1525.64i	1.57770i	0.614585	+	0.788850i	$0.289323\pi$
$967$	1525.64i	1.57770i	−0.614585	+	0.788850i	$0.710677\pi$
$968$	0	0
$969$	1338.94	1.38178
$970$	0	0
$971$	288.662i	0.297283i	0.988891	+	0.148642i	$0.0474901\pi$
$971$	288.662i	0.297283i	−0.988891	+	0.148642i	$0.952510\pi$
$972$	0	0
$973$	87.0580	0.0894738
$974$	0	0
$975$	55.7157i	0.0571443i
$976$	0	0
$977$	541.667	0.554419	0.277209	−	0.960810i	$-0.410590\pi$
$977$	541.667	0.554419	0.277209	+	0.960810i	$0.410590\pi$
$978$	0	0
$979$	3.89062i	0.00397407i
$980$	0	0
$981$	−2141.30	−2.18277
$982$	0	0
$983$	− 1007.01i	− 1.02442i	−0.858859	−	0.512212i	$-0.828826\pi$
$983$	− 1007.01i	− 1.02442i	0.858859	−	0.512212i	$-0.171174\pi$
$984$	0	0
$985$	−350.940	−0.356285
$986$	0	0
$987$	− 282.510i	− 0.286231i
$988$	0	0
$989$	1685.65	1.70439
$990$	0	0
$991$	183.098i	0.184761i	0.995724	+	0.0923806i	$0.0294477\pi$
$991$	183.098i	0.184761i	−0.995724	+	0.0923806i	$0.970552\pi$
$992$	0	0
$993$	295.314	0.297395
$994$	0	0
$995$	1685.78i	1.69425i
$996$	0	0
$997$	−678.950	−0.680993	−0.340497	−	0.940246i	$-0.610595\pi$
$997$	−678.950	−0.680993	−0.340497	+	0.940246i	$0.610595\pi$
$998$	0	0
$999$	− 759.391i	− 0.760151i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.c.a.127.4	yes	4
3.2	odd	2		1152.3.g.b.127.3		4
4.3	odd	2	inner	128.3.c.a.127.1	✓	4
8.3	odd	2		128.3.c.b.127.4	yes	4
8.5	even	2		128.3.c.b.127.1	yes	4
12.11	even	2		1152.3.g.b.127.4		4
16.3	odd	4		256.3.d.e.127.4		4
16.5	even	4		256.3.d.e.127.3		4
16.11	odd	4		256.3.d.d.127.1		4
16.13	even	4		256.3.d.d.127.2		4
24.5	odd	2		1152.3.g.a.127.1		4
24.11	even	2		1152.3.g.a.127.2		4
48.5	odd	4		2304.3.b.j.127.4		4
48.11	even	4		2304.3.b.p.127.4		4
48.29	odd	4		2304.3.b.p.127.1		4
48.35	even	4		2304.3.b.j.127.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.c.a.127.1	✓	4	4.3	odd	2	inner
128.3.c.a.127.4	yes	4	1.1	even	1	trivial
128.3.c.b.127.1	yes	4	8.5	even	2
128.3.c.b.127.4	yes	4	8.3	odd	2
256.3.d.d.127.1		4	16.11	odd	4
256.3.d.d.127.2		4	16.13	even	4
256.3.d.e.127.3		4	16.5	even	4
256.3.d.e.127.4		4	16.3	odd	4
1152.3.g.a.127.1		4	24.5	odd	2
1152.3.g.a.127.2		4	24.11	even	2
1152.3.g.b.127.3		4	3.2	odd	2
1152.3.g.b.127.4		4	12.11	even	2
2304.3.b.j.127.1		4	48.35	even	4
2304.3.b.j.127.4		4	48.5	odd	4
2304.3.b.p.127.1		4	48.29	odd	4
2304.3.b.p.127.4		4	48.11	even	4

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:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	128.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.82843i q^{3} -7.65685 q^{5} -1.65685i q^{7} -14.3137 q^{9} +O(q^{10})\)	\(q+4.82843i q^{3} -7.65685 q^{5} -1.65685i q^{7} -14.3137 q^{9} -1.51472i q^{11} +0.343146 q^{13} -36.9706i q^{15} -13.3137 q^{17} +20.8284i q^{19} +8.00000 q^{21} +33.6569i q^{23} +33.6274 q^{25} -25.6569i q^{27} -39.6569 q^{29} +45.2548i q^{31} +7.31371 q^{33} +12.6863i q^{35} +29.5980 q^{37} +1.65685i q^{39} +24.6274 q^{41} -50.0833i q^{43} +109.598 q^{45} -35.3137i q^{47} +46.2548 q^{49} -64.2843i q^{51} +16.3431 q^{53} +11.5980i q^{55} -100.569 q^{57} +53.1127i q^{59} -34.4020 q^{61} +23.7157i q^{63} -2.62742 q^{65} -62.4853i q^{67} -162.510 q^{69} -40.2843i q^{71} +55.9411 q^{73} +162.368i q^{75} -2.50967 q^{77} +137.941i q^{79} -4.94113 q^{81} +114.652i q^{83} +101.941 q^{85} -191.480i q^{87} -2.56854 q^{89} -0.568542i q^{91} -218.510 q^{93} -159.480i q^{95} -138.569 q^{97} +21.6812i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q - 8 * q^5 - 12 * q^9	\( 4 q - 8 q^{5} - 12 q^{9} + 24 q^{13} - 8 q^{17} + 32 q^{21} + 44 q^{25} - 136 q^{29} - 16 q^{33} - 40 q^{37} + 8 q^{41} + 280 q^{45} + 4 q^{49} + 88 q^{53} - 176 q^{57} - 296 q^{61} + 80 q^{65} - 288 q^{69} + 88 q^{73} + 352 q^{77} + 116 q^{81} + 272 q^{85} + 216 q^{89} - 512 q^{93} - 328 q^{97}+O(q^{100}) \) 4 * q - 8 * q^5 - 12 * q^9 + 24 * q^13 - 8 * q^17 + 32 * q^21 + 44 * q^25 - 136 * q^29 - 16 * q^33 - 40 * q^37 + 8 * q^41 + 280 * q^45 + 4 * q^49 + 88 * q^53 - 176 * q^57 - 296 * q^61 + 80 * q^65 - 288 * q^69 + 88 * q^73 + 352 * q^77 + 116 * q^81 + 272 * q^85 + 216 * q^89 - 512 * q^93 - 328 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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