LMFDB - Embedded newform 128.3.c.a.127.2

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			127.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	128.127
Dual form			128.3.c.a.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.828427i q^{3} +3.65685 q^{5} +9.65685i q^{7} +8.31371 q^{9} +O(q^{10})$	$q-0.828427i q^{3} +3.65685 q^{5} +9.65685i q^{7} +8.31371 q^{9} -18.4853i q^{11} +11.6569 q^{13} -3.02944i q^{15} +9.31371 q^{17} +15.1716i q^{19} +8.00000 q^{21} +22.3431i q^{23} -11.6274 q^{25} -14.3431i q^{27} -28.3431 q^{29} -45.2548i q^{31} -15.3137 q^{33} +35.3137i q^{35} -49.5980 q^{37} -9.65685i q^{39} -20.6274 q^{41} +46.0833i q^{43} +30.4020 q^{45} -12.6863i q^{47} -44.2548 q^{49} -7.71573i q^{51} +27.6569 q^{53} -67.5980i q^{55} +12.5685 q^{57} -9.11270i q^{59} -113.598 q^{61} +80.2843i q^{63} +42.6274 q^{65} -45.5147i q^{67} +18.5097 q^{69} +16.2843i q^{71} -11.9411 q^{73} +9.63247i q^{75} +178.510 q^{77} +70.0589i q^{79} +62.9411 q^{81} -94.6518i q^{83} +34.0589 q^{85} +23.4802i q^{87} +110.569 q^{89} +112.569i q^{91} -37.4903 q^{93} +55.4802i q^{95} -25.4315 q^{97} -153.681i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 0.828427i	− 0.276142i	−0.990422	−	0.138071i	$-0.955910\pi$
$3$	− 0.828427i	− 0.276142i	0.990422	−	0.138071i	$-0.0440903\pi$
$4$	0	0
$5$	3.65685	0.731371	0.365685	−	0.930739i	$-0.380835\pi$
$5$	3.65685	0.731371	0.365685	+	0.930739i	$0.380835\pi$
$6$	0	0
$7$	9.65685i	1.37955i	0.724024	+	0.689775i	$0.242291\pi$
$7$	9.65685i	1.37955i	−0.724024	+	0.689775i	$0.757709\pi$
$8$	0	0
$9$	8.31371	0.923745
$10$	0	0
$11$	− 18.4853i	− 1.68048i	−0.542215	−	0.840240i	$-0.682414\pi$
$11$	− 18.4853i	− 1.68048i	0.542215	−	0.840240i	$-0.317586\pi$
$12$	0	0
$13$	11.6569	0.896681	0.448341	−	0.893863i	$-0.352015\pi$
$13$	11.6569	0.896681	0.448341	+	0.893863i	$0.352015\pi$
$14$	0	0
$15$	− 3.02944i	− 0.201962i
$16$	0	0
$17$	9.31371	0.547865	0.273933	−	0.961749i	$-0.411675\pi$
$17$	9.31371	0.547865	0.273933	+	0.961749i	$0.411675\pi$
$18$	0	0
$19$	15.1716i	0.798504i	0.916841	+	0.399252i	$0.130730\pi$
$19$	15.1716i	0.798504i	−0.916841	+	0.399252i	$0.869270\pi$
$20$	0	0
$21$	8.00000	0.380952
$22$	0	0
$23$	22.3431i	0.971441i	0.874114	+	0.485721i	$0.161443\pi$
$23$	22.3431i	0.971441i	−0.874114	+	0.485721i	$0.838557\pi$
$24$	0	0
$25$	−11.6274	−0.465097
$26$	0	0
$27$	− 14.3431i	− 0.531228i
$28$	0	0
$29$	−28.3431	−0.977350	−0.488675	−	0.872466i	$-0.662520\pi$
$29$	−28.3431	−0.977350	−0.488675	+	0.872466i	$0.662520\pi$
$30$	0	0
$31$	− 45.2548i	− 1.45983i	−0.683536	−	0.729917i	$-0.739559\pi$
$31$	− 45.2548i	− 1.45983i	0.683536	−	0.729917i	$-0.260441\pi$
$32$	0	0
$33$	−15.3137	−0.464052
$34$	0	0
$35$	35.3137i	1.00896i
$36$	0	0
$37$	−49.5980	−1.34049	−0.670243	−	0.742142i	$-0.733810\pi$
$37$	−49.5980	−1.34049	−0.670243	+	0.742142i	$0.733810\pi$
$38$	0	0
$39$	− 9.65685i	− 0.247612i
$40$	0	0
$41$	−20.6274	−0.503108	−0.251554	−	0.967843i	$-0.580942\pi$
$41$	−20.6274	−0.503108	−0.251554	+	0.967843i	$0.580942\pi$
$42$	0	0
$43$	46.0833i	1.07170i	0.844312	+	0.535852i	$0.180009\pi$
$43$	46.0833i	1.07170i	−0.844312	+	0.535852i	$0.819991\pi$
$44$	0	0
$45$	30.4020	0.675600
$46$	0	0
$47$	− 12.6863i	− 0.269921i	−0.990851	−	0.134961i	$-0.956909\pi$
$47$	− 12.6863i	− 0.269921i	0.990851	−	0.134961i	$-0.0430908\pi$
$48$	0	0
$49$	−44.2548	−0.903160
$50$	0	0
$51$	− 7.71573i	− 0.151289i
$52$	0	0
$53$	27.6569	0.521827	0.260914	−	0.965362i	$-0.415976\pi$
$53$	27.6569	0.521827	0.260914	+	0.965362i	$0.415976\pi$
$54$	0	0
$55$	− 67.5980i	− 1.22905i
$56$	0	0
$57$	12.5685	0.220501
$58$	0	0
$59$	− 9.11270i	− 0.154453i	−0.997014	−	0.0772263i	$-0.975394\pi$
$59$	− 9.11270i	− 0.154453i	0.997014	−	0.0772263i	$-0.0246064\pi$
$60$	0	0
$61$	−113.598	−1.86226	−0.931131	−	0.364685i	$-0.881177\pi$
$61$	−113.598	−1.86226	−0.931131	+	0.364685i	$0.881177\pi$
$62$	0	0
$63$	80.2843i	1.27435i
$64$	0	0
$65$	42.6274	0.655806
$66$	0	0
$67$	− 45.5147i	− 0.679324i	−0.940548	−	0.339662i	$-0.889687\pi$
$67$	− 45.5147i	− 0.679324i	0.940548	−	0.339662i	$-0.110313\pi$
$68$	0	0
$69$	18.5097	0.268256
$70$	0	0
$71$	16.2843i	0.229356i	0.993403	+	0.114678i	$0.0365836\pi$
$71$	16.2843i	0.229356i	−0.993403	+	0.114678i	$0.963416\pi$
$72$	0	0
$73$	−11.9411	−0.163577	−0.0817885	−	0.996650i	$-0.526063\pi$
$73$	−11.9411	−0.163577	−0.0817885	+	0.996650i	$0.526063\pi$
$74$	0	0
$75$	9.63247i	0.128433i
$76$	0	0
$77$	178.510	2.31831
$78$	0	0
$79$	70.0589i	0.886821i	0.896319	+	0.443411i	$0.146232\pi$
$79$	70.0589i	0.886821i	−0.896319	+	0.443411i	$0.853768\pi$
$80$	0	0
$81$	62.9411	0.777051
$82$	0	0
$83$	− 94.6518i	− 1.14038i	−0.821512	−	0.570192i	$-0.806869\pi$
$83$	− 94.6518i	− 1.14038i	0.821512	−	0.570192i	$-0.193131\pi$
$84$	0	0
$85$	34.0589	0.400693
$86$	0	0
$87$	23.4802i	0.269888i
$88$	0	0
$89$	110.569	1.24234	0.621172	−	0.783675i	$-0.286657\pi$
$89$	110.569	1.24234	0.621172	+	0.783675i	$0.286657\pi$
$90$	0	0
$91$	112.569i	1.23702i
$92$	0	0
$93$	−37.4903	−0.403122
$94$	0	0
$95$	55.4802i	0.584002i
$96$	0	0
$97$	−25.4315	−0.262180	−0.131090	−	0.991370i	$-0.541848\pi$
$97$	−25.4315	−0.262180	−0.131090	+	0.991370i	$0.541848\pi$
$98$	0	0
$99$	− 153.681i	− 1.55234i
$100$	0	0
$101$	−33.5980	−0.332653	−0.166327	−	0.986071i	$-0.553191\pi$
$101$	−33.5980	−0.332653	−0.166327	+	0.986071i	$0.553191\pi$
$102$	0	0
$103$	− 99.5980i	− 0.966971i	−0.875352	−	0.483485i	$-0.839371\pi$
$103$	− 99.5980i	− 0.966971i	0.875352	−	0.483485i	$-0.160629\pi$
$104$	0	0
$105$	29.2548	0.278617
$106$	0	0
$107$	− 114.485i	− 1.06996i	−0.844866	−	0.534978i	$-0.820320\pi$
$107$	− 114.485i	− 1.06996i	0.844866	−	0.534978i	$-0.179680\pi$
$108$	0	0
$109$	70.4020	0.645890	0.322945	−	0.946418i	$-0.395327\pi$
$109$	70.4020	0.645890	0.322945	+	0.946418i	$0.395327\pi$
$110$	0	0
$111$	41.0883i	0.370165i
$112$	0	0
$113$	12.5097	0.110705	0.0553525	−	0.998467i	$-0.482372\pi$
$113$	12.5097	0.110705	0.0553525	+	0.998467i	$0.482372\pi$
$114$	0	0
$115$	81.7056i	0.710484i
$116$	0	0
$117$	96.9117	0.828305
$118$	0	0
$119$	89.9411i	0.755808i
$120$	0	0
$121$	−220.706	−1.82401
$122$	0	0
$123$	17.0883i	0.138929i
$124$	0	0
$125$	−133.941	−1.07153
$126$	0	0
$127$	− 57.3726i	− 0.451753i	−0.974156	−	0.225876i	$-0.927475\pi$
$127$	− 57.3726i	− 0.451753i	0.974156	−	0.225876i	$-0.0725245\pi$
$128$	0	0
$129$	38.1766	0.295943
$130$	0	0
$131$	198.368i	1.51426i	0.653267	+	0.757128i	$0.273398\pi$
$131$	198.368i	1.51426i	−0.653267	+	0.757128i	$0.726602\pi$
$132$	0	0
$133$	−146.510	−1.10158
$134$	0	0
$135$	− 52.4508i	− 0.388524i
$136$	0	0
$137$	53.8823	0.393301	0.196651	−	0.980474i	$-0.436994\pi$
$137$	53.8823	0.393301	0.196651	+	0.980474i	$0.436994\pi$
$138$	0	0
$139$	103.456i	0.744287i	0.928175	+	0.372143i	$0.121377\pi$
$139$	103.456i	0.744287i	−0.928175	+	0.372143i	$0.878623\pi$
$140$	0	0
$141$	−10.5097	−0.0745367
$142$	0	0
$143$	− 215.480i	− 1.50685i
$144$	0	0
$145$	−103.647	−0.714805
$146$	0	0
$147$	36.6619i	0.249401i
$148$	0	0
$149$	−15.0883	−0.101264	−0.0506319	−	0.998717i	$-0.516124\pi$
$149$	−15.0883	−0.101264	−0.0506319	+	0.998717i	$0.516124\pi$
$150$	0	0
$151$	− 158.676i	− 1.05084i	−0.850844	−	0.525418i	$-0.823909\pi$
$151$	− 158.676i	− 1.05084i	0.850844	−	0.525418i	$-0.176091\pi$
$152$	0	0
$153$	77.4315	0.506088
$154$	0	0
$155$	− 165.490i	− 1.06768i
$156$	0	0
$157$	−124.108	−0.790495	−0.395247	−	0.918575i	$-0.629341\pi$
$157$	−124.108	−0.790495	−0.395247	+	0.918575i	$0.629341\pi$
$158$	0	0
$159$	− 22.9117i	− 0.144099i
$160$	0	0
$161$	−215.765	−1.34015
$162$	0	0
$163$	121.681i	0.746511i	0.927729	+	0.373255i	$0.121758\pi$
$163$	121.681i	0.746511i	−0.927729	+	0.373255i	$0.878242\pi$
$164$	0	0
$165$	−56.0000	−0.339394
$166$	0	0
$167$	330.794i	1.98080i	0.138224	+	0.990401i	$0.455861\pi$
$167$	330.794i	1.98080i	−0.138224	+	0.990401i	$0.544139\pi$
$168$	0	0
$169$	−33.1177	−0.195963
$170$	0	0
$171$	126.132i	0.737614i
$172$	0	0
$173$	27.6569	0.159866	0.0799331	−	0.996800i	$-0.474529\pi$
$173$	27.6569	0.159866	0.0799331	+	0.996800i	$0.474529\pi$
$174$	0	0
$175$	− 112.284i	− 0.641624i
$176$	0	0
$177$	−7.54921	−0.0426509
$178$	0	0
$179$	91.8579i	0.513172i	0.966521	+	0.256586i	$0.0825978\pi$
$179$	91.8579i	0.513172i	−0.966521	+	0.256586i	$0.917402\pi$
$180$	0	0
$181$	−84.1076	−0.464683	−0.232342	−	0.972634i	$-0.574639\pi$
$181$	−84.1076	−0.464683	−0.232342	+	0.972634i	$0.574639\pi$
$182$	0	0
$183$	94.1076i	0.514249i
$184$	0	0
$185$	−181.373	−0.980392
$186$	0	0
$187$	− 172.167i	− 0.920677i
$188$	0	0
$189$	138.510	0.732855
$190$	0	0
$191$	− 289.137i	− 1.51381i	−0.653527	−	0.756903i	$-0.726711\pi$
$191$	− 289.137i	− 1.51381i	0.653527	−	0.756903i	$-0.273289\pi$
$192$	0	0
$193$	225.078	1.16621	0.583104	−	0.812397i	$-0.301838\pi$
$193$	225.078	1.16621	0.583104	+	0.812397i	$0.301838\pi$
$194$	0	0
$195$	− 35.3137i	− 0.181096i
$196$	0	0
$197$	238.167	1.20897	0.604484	−	0.796618i	$-0.293380\pi$
$197$	238.167	1.20897	0.604484	+	0.796618i	$0.293380\pi$
$198$	0	0
$199$	− 27.8335i	− 0.139867i	−0.997552	−	0.0699334i	$-0.977721\pi$
$199$	− 27.8335i	− 0.139867i	0.997552	−	0.0699334i	$-0.0222787\pi$
$200$	0	0
$201$	−37.7056	−0.187590
$202$	0	0
$203$	− 273.706i	− 1.34830i
$204$	0	0
$205$	−75.4315	−0.367958
$206$	0	0
$207$	185.754i	0.897364i
$208$	0	0
$209$	280.451	1.34187
$210$	0	0
$211$	− 42.7696i	− 0.202699i	−0.994851	−	0.101350i	$-0.967684\pi$
$211$	− 42.7696i	− 0.202699i	0.994851	−	0.101350i	$-0.0323161\pi$
$212$	0	0
$213$	13.4903	0.0633349
$214$	0	0
$215$	168.520i	0.783813i
$216$	0	0
$217$	437.019	2.01391
$218$	0	0
$219$	9.89235i	0.0451706i
$220$	0	0
$221$	108.569	0.491260
$222$	0	0
$223$	301.255i	1.35092i	0.737397	+	0.675459i	$0.236055\pi$
$223$	301.255i	1.35092i	−0.737397	+	0.675459i	$0.763945\pi$
$224$	0	0
$225$	−96.6670	−0.429631
$226$	0	0
$227$	80.7797i	0.355858i	0.984043	+	0.177929i	$0.0569397\pi$
$227$	80.7797i	0.355858i	−0.984043	+	0.177929i	$0.943060\pi$
$228$	0	0
$229$	195.186	0.852340	0.426170	−	0.904643i	$-0.359863\pi$
$229$	195.186	0.852340	0.426170	+	0.904643i	$0.359863\pi$
$230$	0	0
$231$	− 147.882i	− 0.640183i
$232$	0	0
$233$	−38.4508	−0.165025	−0.0825124	−	0.996590i	$-0.526294\pi$
$233$	−38.4508	−0.165025	−0.0825124	+	0.996590i	$0.526294\pi$
$234$	0	0
$235$	− 46.3919i	− 0.197412i
$236$	0	0
$237$	58.0387	0.244889
$238$	0	0
$239$	188.215i	0.787512i	0.919215	+	0.393756i	$0.128824\pi$
$239$	188.215i	0.787512i	−0.919215	+	0.393756i	$0.871176\pi$
$240$	0	0
$241$	−267.706	−1.11081	−0.555406	−	0.831579i	$-0.687437\pi$
$241$	−267.706	−1.11081	−0.555406	+	0.831579i	$0.687437\pi$
$242$	0	0
$243$	− 181.230i	− 0.745804i
$244$	0	0
$245$	−161.833	−0.660545
$246$	0	0
$247$	176.853i	0.716003i
$248$	0	0
$249$	−78.4121	−0.314908
$250$	0	0
$251$	158.652i	0.632079i	0.948746	+	0.316039i	$0.102353\pi$
$251$	158.652i	0.632079i	−0.948746	+	0.316039i	$0.897647\pi$
$252$	0	0
$253$	413.019	1.63249
$254$	0	0
$255$	− 28.2153i	− 0.110648i
$256$	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
$259$	− 478.960i	− 1.84927i
$260$	0	0
$261$	−235.637	−0.902822
$262$	0	0
$263$	− 21.2061i	− 0.0806314i	−0.999187	−	0.0403157i	$-0.987164\pi$
$263$	− 21.2061i	− 0.0806314i	0.999187	−	0.0403157i	$-0.0128364\pi$
$264$	0	0
$265$	101.137	0.381649
$266$	0	0
$267$	− 91.5980i	− 0.343064i
$268$	0	0
$269$	−403.637	−1.50051	−0.750254	−	0.661150i	$-0.770069\pi$
$269$	−403.637	−1.50051	−0.750254	+	0.661150i	$0.770069\pi$
$270$	0	0
$271$	50.1766i	0.185154i	0.995706	+	0.0925768i	$0.0295104\pi$
$271$	50.1766i	0.185154i	−0.995706	+	0.0925768i	$0.970490\pi$
$272$	0	0
$273$	93.2548	0.341593
$274$	0	0
$275$	214.936i	0.781586i
$276$	0	0
$277$	229.696	0.829226	0.414613	−	0.909998i	$-0.363917\pi$
$277$	229.696	0.829226	0.414613	+	0.909998i	$0.363917\pi$
$278$	0	0
$279$	− 376.235i	− 1.34851i
$280$	0	0
$281$	86.8040	0.308911	0.154456	−	0.988000i	$-0.450638\pi$
$281$	86.8040	0.308911	0.154456	+	0.988000i	$0.450638\pi$
$282$	0	0
$283$	389.848i	1.37755i	0.724973	+	0.688777i	$0.241852\pi$
$283$	389.848i	1.37755i	−0.724973	+	0.688777i	$0.758148\pi$
$284$	0	0
$285$	45.9613	0.161268
$286$	0	0
$287$	− 199.196i	− 0.694063i
$288$	0	0
$289$	−202.255	−0.699844
$290$	0	0
$291$	21.0681i	0.0723990i
$292$	0	0
$293$	243.657	0.831593	0.415797	−	0.909458i	$-0.363503\pi$
$293$	243.657	0.831593	0.415797	+	0.909458i	$0.363503\pi$
$294$	0	0
$295$	− 33.3238i	− 0.112962i
$296$	0	0
$297$	−265.137	−0.892717
$298$	0	0
$299$	260.451i	0.871073i
$300$	0	0
$301$	−445.019	−1.47847
$302$	0	0
$303$	27.8335i	0.0918597i
$304$	0	0
$305$	−415.411	−1.36200
$306$	0	0
$307$	− 276.240i	− 0.899804i	−0.893078	−	0.449902i	$-0.851459\pi$
$307$	− 276.240i	− 0.899804i	0.893078	−	0.449902i	$-0.148541\pi$
$308$	0	0
$309$	−82.5097	−0.267022
$310$	0	0
$311$	− 552.715i	− 1.77722i	−0.458665	−	0.888609i	$-0.651672\pi$
$311$	− 552.715i	− 1.77722i	0.458665	−	0.888609i	$-0.348328\pi$
$312$	0	0
$313$	−372.627	−1.19050	−0.595251	−	0.803539i	$-0.702948\pi$
$313$	−372.627	−1.19050	−0.595251	+	0.803539i	$0.702948\pi$
$314$	0	0
$315$	293.588i	0.932025i
$316$	0	0
$317$	131.657	0.415321	0.207661	−	0.978201i	$-0.433415\pi$
$317$	131.657	0.415321	0.207661	+	0.978201i	$0.433415\pi$
$318$	0	0
$319$	523.931i	1.64242i
$320$	0	0
$321$	−94.8427	−0.295460
$322$	0	0
$323$	141.304i	0.437472i
$324$	0	0
$325$	−135.539	−0.417043
$326$	0	0
$327$	− 58.3229i	− 0.178358i
$328$	0	0
$329$	122.510	0.372370
$330$	0	0
$331$	329.161i	0.994446i	0.867623	+	0.497223i	$0.165647\pi$
$331$	329.161i	0.994446i	−0.867623	+	0.497223i	$0.834353\pi$
$332$	0	0
$333$	−412.343	−1.23827
$334$	0	0
$335$	− 166.441i	− 0.496838i
$336$	0	0
$337$	574.784	1.70559	0.852795	−	0.522246i	$-0.174906\pi$
$337$	574.784	1.70559	0.852795	+	0.522246i	$0.174906\pi$
$338$	0	0
$339$	− 10.3633i	− 0.0305703i
$340$	0	0
$341$	−836.548	−2.45322
$342$	0	0
$343$	45.8234i	0.133596i
$344$	0	0
$345$	67.6872	0.196195
$346$	0	0
$347$	− 480.406i	− 1.38446i	−0.721679	−	0.692228i	$-0.756629\pi$
$347$	− 480.406i	− 1.38446i	0.721679	−	0.692228i	$-0.243371\pi$
$348$	0	0
$349$	547.186	1.56787	0.783934	−	0.620844i	$-0.213210\pi$
$349$	547.186	1.56787	0.783934	+	0.620844i	$0.213210\pi$
$350$	0	0
$351$	− 167.196i	− 0.476342i
$352$	0	0
$353$	−624.274	−1.76848	−0.884241	−	0.467031i	$-0.845324\pi$
$353$	−624.274	−1.76848	−0.884241	+	0.467031i	$0.845324\pi$
$354$	0	0
$355$	59.5492i	0.167744i
$356$	0	0
$357$	74.5097	0.208711
$358$	0	0
$359$	− 280.617i	− 0.781664i	−0.920462	−	0.390832i	$-0.872187\pi$
$359$	− 280.617i	− 0.781664i	0.920462	−	0.390832i	$-0.127813\pi$
$360$	0	0
$361$	130.823	0.362392
$362$	0	0
$363$	182.839i	0.503687i
$364$	0	0
$365$	−43.6670	−0.119635
$366$	0	0
$367$	− 103.196i	− 0.281188i	−0.990067	−	0.140594i	$-0.955099\pi$
$367$	− 103.196i	− 0.281188i	0.990067	−	0.140594i	$-0.0449012\pi$
$368$	0	0
$369$	−171.490	−0.464743
$370$	0	0
$371$	267.078i	0.719887i
$372$	0	0
$373$	177.147	0.474925	0.237463	−	0.971397i	$-0.423684\pi$
$373$	177.147	0.474925	0.237463	+	0.971397i	$0.423684\pi$
$374$	0	0
$375$	110.960i	0.295895i
$376$	0	0
$377$	−330.392	−0.876371
$378$	0	0
$379$	− 356.759i	− 0.941318i	−0.882315	−	0.470659i	$-0.844016\pi$
$379$	− 356.759i	− 0.941318i	0.882315	−	0.470659i	$-0.155984\pi$
$380$	0	0
$381$	−47.5290	−0.124748
$382$	0	0
$383$	106.039i	0.276863i	0.990372	+	0.138432i	$0.0442061\pi$
$383$	106.039i	0.276863i	−0.990372	+	0.138432i	$0.955794\pi$
$384$	0	0
$385$	652.784	1.69554
$386$	0	0
$387$	383.123i	0.989981i
$388$	0	0
$389$	520.912	1.33910	0.669552	−	0.742765i	$-0.266486\pi$
$389$	520.912	1.33910	0.669552	+	0.742765i	$0.266486\pi$
$390$	0	0
$391$	208.098i	0.532219i
$392$	0	0
$393$	164.333	0.418150
$394$	0	0
$395$	256.195i	0.648595i
$396$	0	0
$397$	80.4407	0.202621	0.101311	−	0.994855i	$-0.467696\pi$
$397$	80.4407	0.202621	0.101311	+	0.994855i	$0.467696\pi$
$398$	0	0
$399$	121.373i	0.304192i
$400$	0	0
$401$	3.82338	0.00953460	0.00476730	−	0.999989i	$-0.498483\pi$
$401$	3.82338	0.00953460	0.00476730	+	0.999989i	$0.498483\pi$
$402$	0	0
$403$	− 527.529i	− 1.30900i
$404$	0	0
$405$	230.167	0.568312
$406$	0	0
$407$	916.833i	2.25266i
$408$	0	0
$409$	378.902	0.926410	0.463205	−	0.886251i	$-0.346699\pi$
$409$	378.902	0.926410	0.463205	+	0.886251i	$0.346699\pi$
$410$	0	0
$411$	− 44.6375i	− 0.108607i
$412$	0	0
$413$	88.0000	0.213075
$414$	0	0
$415$	− 346.128i	− 0.834043i
$416$	0	0
$417$	85.7056	0.205529
$418$	0	0
$419$	− 603.907i	− 1.44130i	−0.693297	−	0.720652i	$-0.743842\pi$
$419$	− 603.907i	− 1.44130i	0.693297	−	0.720652i	$-0.256158\pi$
$420$	0	0
$421$	184.676	0.438661	0.219330	−	0.975651i	$-0.429613\pi$
$421$	184.676	0.438661	0.219330	+	0.975651i	$0.429613\pi$
$422$	0	0
$423$	− 105.470i	− 0.249338i
$424$	0	0
$425$	−108.294	−0.254810
$426$	0	0
$427$	− 1097.00i	− 2.56908i
$428$	0	0
$429$	−178.510	−0.416106
$430$	0	0
$431$	− 162.843i	− 0.377825i	−0.981994	−	0.188913i	$-0.939504\pi$
$431$	− 162.843i	− 0.377825i	0.981994	−	0.188913i	$-0.0604963\pi$
$432$	0	0
$433$	205.862	0.475432	0.237716	−	0.971335i	$-0.423601\pi$
$433$	205.862	0.475432	0.237716	+	0.971335i	$0.423601\pi$
$434$	0	0
$435$	85.8638i	0.197388i
$436$	0	0
$437$	−338.981	−0.775699
$438$	0	0
$439$	197.872i	0.450734i	0.974274	+	0.225367i	$0.0723581\pi$
$439$	197.872i	0.450734i	−0.974274	+	0.225367i	$0.927642\pi$
$440$	0	0
$441$	−367.922	−0.834290
$442$	0	0
$443$	− 26.1522i	− 0.0590344i	−0.999564	−	0.0295172i	$-0.990603\pi$
$443$	− 26.1522i	− 0.0590344i	0.999564	−	0.0295172i	$-0.00939698\pi$
$444$	0	0
$445$	404.333	0.908614
$446$	0	0
$447$	12.4996i	0.0279632i
$448$	0	0
$449$	321.078	0.715096	0.357548	−	0.933895i	$-0.383613\pi$
$449$	321.078	0.715096	0.357548	+	0.933895i	$0.383613\pi$
$450$	0	0
$451$	381.304i	0.845463i
$452$	0	0
$453$	−131.452	−0.290180
$454$	0	0
$455$	411.647i	0.904718i
$456$	0	0
$457$	357.882	0.783112	0.391556	−	0.920154i	$-0.371937\pi$
$457$	357.882	0.783112	0.391556	+	0.920154i	$0.371937\pi$
$458$	0	0
$459$	− 133.588i	− 0.291041i
$460$	0	0
$461$	240.912	0.522585	0.261293	−	0.965260i	$-0.415851\pi$
$461$	240.912	0.522585	0.261293	+	0.965260i	$0.415851\pi$
$462$	0	0
$463$	176.098i	0.380340i	0.981751	+	0.190170i	$0.0609040\pi$
$463$	176.098i	0.380340i	−0.981751	+	0.190170i	$0.939096\pi$
$464$	0	0
$465$	−137.097	−0.294832
$466$	0	0
$467$	749.092i	1.60405i	0.597289	+	0.802026i	$0.296245\pi$
$467$	749.092i	1.60405i	−0.597289	+	0.802026i	$0.703755\pi$
$468$	0	0
$469$	439.529	0.937162
$470$	0	0
$471$	102.814i	0.218289i
$472$	0	0
$473$	851.862	1.80098
$474$	0	0
$475$	− 176.406i	− 0.371381i
$476$	0	0
$477$	229.931	0.482036
$478$	0	0
$479$	120.235i	0.251014i	0.992093	+	0.125507i	$0.0400557\pi$
$479$	120.235i	0.251014i	−0.992093	+	0.125507i	$0.959944\pi$
$480$	0	0
$481$	−578.156	−1.20199
$482$	0	0
$483$	178.745i	0.370073i
$484$	0	0
$485$	−92.9991	−0.191751
$486$	0	0
$487$	624.284i	1.28190i	0.767584	+	0.640949i	$0.221459\pi$
$487$	624.284i	1.28190i	−0.767584	+	0.640949i	$0.778541\pi$
$488$	0	0
$489$	100.804	0.206143
$490$	0	0
$491$	− 279.602i	− 0.569455i	−0.958609	−	0.284727i	$-0.908097\pi$
$491$	− 279.602i	− 0.569455i	0.958609	−	0.284727i	$-0.0919030\pi$
$492$	0	0
$493$	−263.980	−0.535456
$494$	0	0
$495$	− 561.990i	− 1.13533i
$496$	0	0
$497$	−157.255	−0.316408
$498$	0	0
$499$	260.759i	0.522564i	0.965263	+	0.261282i	$0.0841453\pi$
$499$	260.759i	0.522564i	−0.965263	+	0.261282i	$0.915855\pi$
$500$	0	0
$501$	274.039	0.546983
$502$	0	0
$503$	− 735.813i	− 1.46285i	−0.681922	−	0.731425i	$-0.738856\pi$
$503$	− 735.813i	− 1.46285i	0.681922	−	0.731425i	$-0.261144\pi$
$504$	0	0
$505$	−122.863	−0.243293
$506$	0	0
$507$	27.4356i	0.0541137i
$508$	0	0
$509$	−508.108	−0.998247	−0.499123	−	0.866531i	$-0.666345\pi$
$509$	−508.108	−0.998247	−0.499123	+	0.866531i	$0.666345\pi$
$510$	0	0
$511$	− 115.314i	− 0.225663i
$512$	0	0
$513$	217.608	0.424187
$514$	0	0
$515$	− 364.215i	− 0.707214i
$516$	0	0
$517$	−234.510	−0.453597
$518$	0	0
$519$	− 22.9117i	− 0.0441458i
$520$	0	0
$521$	−973.724	−1.86895	−0.934476	−	0.356026i	$-0.884131\pi$
$521$	−973.724	−1.86895	−0.934476	+	0.356026i	$0.884131\pi$
$522$	0	0
$523$	− 65.8192i	− 0.125849i	−0.998018	−	0.0629247i	$-0.979957\pi$
$523$	− 65.8192i	− 0.125849i	0.998018	−	0.0629247i	$-0.0200428\pi$
$524$	0	0
$525$	−93.0193	−0.177180
$526$	0	0
$527$	− 421.490i	− 0.799792i
$528$	0	0
$529$	29.7838	0.0563022
$530$	0	0
$531$	− 75.7603i	− 0.142675i
$532$	0	0
$533$	−240.451	−0.451127
$534$	0	0
$535$	− 418.656i	− 0.782535i
$536$	0	0
$537$	76.0975	0.141709
$538$	0	0
$539$	818.063i	1.51774i
$540$	0	0
$541$	−524.108	−0.968776	−0.484388	−	0.874853i	$-0.660958\pi$
$541$	−524.108	−0.968776	−0.484388	+	0.874853i	$0.660958\pi$
$542$	0	0
$543$	69.6771i	0.128319i
$544$	0	0
$545$	257.450	0.472385
$546$	0	0
$547$	552.642i	1.01031i	0.863027	+	0.505157i	$0.168565\pi$
$547$	552.642i	1.01031i	−0.863027	+	0.505157i	$0.831435\pi$
$548$	0	0
$549$	−944.420	−1.72026
$550$	0	0
$551$	− 430.010i	− 0.780418i
$552$	0	0
$553$	−676.548	−1.22341
$554$	0	0
$555$	150.254i	0.270728i
$556$	0	0
$557$	374.638	0.672599	0.336299	−	0.941755i	$-0.390825\pi$
$557$	374.638	0.672599	0.336299	+	0.941755i	$0.390825\pi$
$558$	0	0
$559$	537.186i	0.960976i
$560$	0	0
$561$	−142.627	−0.254238
$562$	0	0
$563$	− 479.220i	− 0.851191i	−0.904914	−	0.425595i	$-0.860065\pi$
$563$	− 479.220i	− 0.851191i	0.904914	−	0.425595i	$-0.139935\pi$
$564$	0	0
$565$	45.7460	0.0809664
$566$	0	0
$567$	607.813i	1.07198i
$568$	0	0
$569$	−696.705	−1.22444	−0.612219	−	0.790689i	$-0.709723\pi$
$569$	−696.705	−1.22444	−0.612219	+	0.790689i	$0.709723\pi$
$570$	0	0
$571$	307.103i	0.537833i	0.963164	+	0.268916i	$0.0866656\pi$
$571$	307.103i	0.537833i	−0.963164	+	0.268916i	$0.913334\pi$
$572$	0	0
$573$	−239.529	−0.418026
$574$	0	0
$575$	− 259.793i	− 0.451814i
$576$	0	0
$577$	−714.313	−1.23798	−0.618989	−	0.785400i	$-0.712457\pi$
$577$	−714.313	−1.23798	−0.618989	+	0.785400i	$0.712457\pi$
$578$	0	0
$579$	− 186.461i	− 0.322040i
$580$	0	0
$581$	914.039	1.57322
$582$	0	0
$583$	− 511.245i	− 0.876921i
$584$	0	0
$585$	354.392	0.605798
$586$	0	0
$587$	533.279i	0.908482i	0.890879	+	0.454241i	$0.150090\pi$
$587$	533.279i	0.908482i	−0.890879	+	0.454241i	$0.849910\pi$
$588$	0	0
$589$	686.587	1.16568
$590$	0	0
$591$	− 197.304i	− 0.333847i
$592$	0	0
$593$	29.4517	0.0496655	0.0248328	−	0.999692i	$-0.492095\pi$
$593$	29.4517	0.0496655	0.0248328	+	0.999692i	$0.492095\pi$
$594$	0	0
$595$	328.902i	0.552776i
$596$	0	0
$597$	−23.0580	−0.0386231
$598$	0	0
$599$	− 699.265i	− 1.16739i	−0.811974	−	0.583694i	$-0.801607\pi$
$599$	− 699.265i	− 1.16739i	0.811974	−	0.583694i	$-0.198393\pi$
$600$	0	0
$601$	473.549	0.787935	0.393968	−	0.919124i	$-0.371102\pi$
$601$	473.549	0.787935	0.393968	+	0.919124i	$0.371102\pi$
$602$	0	0
$603$	− 378.396i	− 0.627523i
$604$	0	0
$605$	−807.088	−1.33403
$606$	0	0
$607$	− 696.431i	− 1.14733i	−0.819089	−	0.573666i	$-0.805521\pi$
$607$	− 696.431i	− 1.14733i	0.819089	−	0.573666i	$-0.194479\pi$
$608$	0	0
$609$	−226.745	−0.372324
$610$	0	0
$611$	− 147.882i	− 0.242033i
$612$	0	0
$613$	−828.343	−1.35129	−0.675647	−	0.737225i	$-0.736136\pi$
$613$	−828.343	−1.35129	−0.675647	+	0.737225i	$0.736136\pi$
$614$	0	0
$615$	62.4895i	0.101609i
$616$	0	0
$617$	885.862	1.43576	0.717878	−	0.696168i	$-0.245113\pi$
$617$	885.862	1.43576	0.717878	+	0.696168i	$0.245113\pi$
$618$	0	0
$619$	1217.40i	1.96672i	0.181679	+	0.983358i	$0.441847\pi$
$619$	1217.40i	1.96672i	−0.181679	+	0.983358i	$0.558153\pi$
$620$	0	0
$621$	320.471	0.516056
$622$	0	0
$623$	1067.74i	1.71388i
$624$	0	0
$625$	−199.118	−0.318588
$626$	0	0
$627$	− 232.333i	− 0.370547i
$628$	0	0
$629$	−461.941	−0.734406
$630$	0	0
$631$	261.677i	0.414702i	0.978267	+	0.207351i	$0.0664842\pi$
$631$	261.677i	0.414702i	−0.978267	+	0.207351i	$0.933516\pi$
$632$	0	0
$633$	−35.4315	−0.0559739
$634$	0	0
$635$	− 209.803i	− 0.330399i
$636$	0	0
$637$	−515.872	−0.809846
$638$	0	0
$639$	135.383i	0.211866i
$640$	0	0
$641$	923.588	1.44085	0.720427	−	0.693530i	$-0.243946\pi$
$641$	923.588	1.44085	0.720427	+	0.693530i	$0.243946\pi$
$642$	0	0
$643$	− 416.260i	− 0.647372i	−0.946165	−	0.323686i	$-0.895078\pi$
$643$	− 416.260i	− 0.647372i	0.946165	−	0.323686i	$-0.104922\pi$
$644$	0	0
$645$	139.606	0.216444
$646$	0	0
$647$	− 745.283i	− 1.15191i	−0.817483	−	0.575953i	$-0.804631\pi$
$647$	− 745.283i	− 1.15191i	0.817483	−	0.575953i	$-0.195369\pi$
$648$	0	0
$649$	−168.451	−0.259554
$650$	0	0
$651$	− 362.039i	− 0.556127i
$652$	0	0
$653$	928.441	1.42181	0.710904	−	0.703289i	$-0.248286\pi$
$653$	928.441	1.42181	0.710904	+	0.703289i	$0.248286\pi$
$654$	0	0
$655$	725.401i	1.10748i
$656$	0	0
$657$	−99.2750	−0.151104
$658$	0	0
$659$	− 1085.71i	− 1.64751i	−0.566945	−	0.823756i	$-0.691875\pi$
$659$	− 1085.71i	− 1.64751i	0.566945	−	0.823756i	$-0.308125\pi$
$660$	0	0
$661$	251.421	0.380365	0.190183	−	0.981749i	$-0.439092\pi$
$661$	251.421	0.380365	0.190183	+	0.981749i	$0.439092\pi$
$662$	0	0
$663$	− 89.9411i	− 0.135658i
$664$	0	0
$665$	−535.765	−0.805661
$666$	0	0
$667$	− 633.275i	− 0.949438i
$668$	0	0
$669$	249.568	0.373046
$670$	0	0
$671$	2099.89i	3.12949i
$672$	0	0
$673$	173.001	0.257059	0.128530	−	0.991706i	$-0.458974\pi$
$673$	173.001	0.257059	0.128530	+	0.991706i	$0.458974\pi$
$674$	0	0
$675$	166.774i	0.247072i
$676$	0	0
$677$	856.441	1.26505	0.632526	−	0.774539i	$-0.282018\pi$
$677$	856.441	1.26505	0.632526	+	0.774539i	$0.282018\pi$
$678$	0	0
$679$	− 245.588i	− 0.361691i
$680$	0	0
$681$	66.9201	0.0982673
$682$	0	0
$683$	762.201i	1.11596i	0.829854	+	0.557980i	$0.188424\pi$
$683$	762.201i	1.11596i	−0.829854	+	0.557980i	$0.811576\pi$
$684$	0	0
$685$	197.040	0.287649
$686$	0	0
$687$	− 161.697i	− 0.235367i
$688$	0	0
$689$	322.392	0.467913
$690$	0	0
$691$	− 217.161i	− 0.314271i	−0.987577	−	0.157136i	$-0.949774\pi$
$691$	− 217.161i	− 0.314271i	0.987577	−	0.157136i	$-0.0502260\pi$
$692$	0	0
$693$	1484.08	2.14153
$694$	0	0
$695$	378.323i	0.544350i
$696$	0	0
$697$	−192.118	−0.275635
$698$	0	0
$699$	31.8537i	0.0455704i
$700$	0	0
$701$	−663.088	−0.945918	−0.472959	−	0.881085i	$-0.656814\pi$
$701$	−663.088	−0.945918	−0.472959	+	0.881085i	$0.656814\pi$
$702$	0	0
$703$	− 752.479i	− 1.07038i
$704$	0	0
$705$	−38.4323	−0.0545139
$706$	0	0
$707$	− 324.451i	− 0.458912i
$708$	0	0
$709$	−912.656	−1.28724	−0.643622	−	0.765344i	$-0.722569\pi$
$709$	−912.656	−1.28724	−0.643622	+	0.765344i	$0.722569\pi$
$710$	0	0
$711$	582.449i	0.819197i
$712$	0	0
$713$	1011.14	1.41814
$714$	0	0
$715$	− 787.980i	− 1.10207i
$716$	0	0
$717$	155.923	0.217465
$718$	0	0
$719$	105.275i	0.146419i	0.997317	+	0.0732093i	$0.0233241\pi$
$719$	105.275i	0.146419i	−0.997317	+	0.0732093i	$0.976676\pi$
$720$	0	0
$721$	961.803	1.33398
$722$	0	0
$723$	221.775i	0.306742i
$724$	0	0
$725$	329.558	0.454562
$726$	0	0
$727$	− 518.246i	− 0.712855i	−0.934323	−	0.356428i	$-0.883995\pi$
$727$	− 518.246i	− 0.712855i	0.934323	−	0.356428i	$-0.116005\pi$
$728$	0	0
$729$	416.334	0.571103
$730$	0	0
$731$	429.206i	0.587149i
$732$	0	0
$733$	−469.911	−0.641079	−0.320539	−	0.947235i	$-0.603864\pi$
$733$	−469.911	−0.641079	−0.320539	+	0.947235i	$0.603864\pi$
$734$	0	0
$735$	134.067i	0.182404i
$736$	0	0
$737$	−841.352	−1.14159
$738$	0	0
$739$	− 334.278i	− 0.452339i	−0.974088	−	0.226169i	$-0.927380\pi$
$739$	− 334.278i	− 0.452339i	0.974088	−	0.226169i	$-0.0726203\pi$
$740$	0	0
$741$	146.510	0.197719
$742$	0	0
$743$	937.852i	1.26225i	0.775681	+	0.631125i	$0.217407\pi$
$743$	937.852i	1.26225i	−0.775681	+	0.631125i	$0.782593\pi$
$744$	0	0
$745$	−55.1758	−0.0740614
$746$	0	0
$747$	− 786.908i	− 1.05342i
$748$	0	0
$749$	1105.57	1.47606
$750$	0	0
$751$	− 1193.67i	− 1.58943i	−0.606980	−	0.794717i	$-0.707619\pi$
$751$	− 1193.67i	− 1.58943i	0.606980	−	0.794717i	$-0.292381\pi$
$752$	0	0
$753$	131.431	0.174544
$754$	0	0
$755$	− 580.256i	− 0.768551i
$756$	0	0
$757$	129.383	0.170915	0.0854575	−	0.996342i	$-0.472765\pi$
$757$	129.383	0.170915	0.0854575	+	0.996342i	$0.472765\pi$
$758$	0	0
$759$	− 342.156i	− 0.450799i
$760$	0	0
$761$	1055.45	1.38693	0.693463	−	0.720493i	$-0.256084\pi$
$761$	1055.45	1.38693	0.693463	+	0.720493i	$0.256084\pi$
$762$	0	0
$763$	679.862i	0.891038i
$764$	0	0
$765$	283.156	0.370138
$766$	0	0
$767$	− 106.225i	− 0.138495i
$768$	0	0
$769$	−285.980	−0.371885	−0.185943	−	0.982561i	$-0.559534\pi$
$769$	−285.980	−0.371885	−0.185943	+	0.982561i	$0.559534\pi$
$770$	0	0
$771$	− 280.008i	− 0.363176i
$772$	0	0
$773$	−688.656	−0.890887	−0.445444	−	0.895310i	$-0.646954\pi$
$773$	−688.656	−0.890887	−0.445444	+	0.895310i	$0.646954\pi$
$774$	0	0
$775$	526.197i	0.678964i
$776$	0	0
$777$	−396.784	−0.510661
$778$	0	0
$779$	− 312.950i	− 0.401733i
$780$	0	0
$781$	301.019	0.385428
$782$	0	0
$783$	406.530i	0.519195i
$784$	0	0
$785$	−453.844	−0.578145
$786$	0	0
$787$	1535.27i	1.95079i	0.220472	+	0.975393i	$0.429240\pi$
$787$	1535.27i	1.95079i	−0.220472	+	0.975393i	$0.570760\pi$
$788$	0	0
$789$	−17.5677	−0.0222657
$790$	0	0
$791$	120.804i	0.152723i
$792$	0	0
$793$	−1324.20	−1.66986
$794$	0	0
$795$	− 83.7847i	− 0.105390i
$796$	0	0
$797$	−311.559	−0.390915	−0.195458	−	0.980712i	$-0.562619\pi$
$797$	−311.559	−0.390915	−0.195458	+	0.980712i	$0.562619\pi$
$798$	0	0
$799$	− 118.156i	− 0.147880i
$800$	0	0
$801$	919.235	1.14761
$802$	0	0
$803$	220.735i	0.274888i
$804$	0	0
$805$	−789.019	−0.980148
$806$	0	0
$807$	334.384i	0.414354i
$808$	0	0
$809$	−669.724	−0.827842	−0.413921	−	0.910313i	$-0.635841\pi$
$809$	−669.724	−0.827842	−0.413921	+	0.910313i	$0.635841\pi$
$810$	0	0
$811$	5.94531i	0.00733084i	0.999993	+	0.00366542i	$0.00116674\pi$
$811$	5.94531i	0.00733084i	−0.999993	+	0.00366542i	$0.998833\pi$
$812$	0	0
$813$	41.5677	0.0511288
$814$	0	0
$815$	444.971i	0.545976i
$816$	0	0
$817$	−699.156	−0.855760
$818$	0	0
$819$	935.862i	1.14269i
$820$	0	0
$821$	−804.814	−0.980285	−0.490143	−	0.871642i	$-0.663055\pi$
$821$	−804.814	−0.980285	−0.490143	+	0.871642i	$0.663055\pi$
$822$	0	0
$823$	352.382i	0.428167i	0.976815	+	0.214084i	$0.0686765\pi$
$823$	352.382i	0.428167i	−0.976815	+	0.214084i	$0.931323\pi$
$824$	0	0
$825$	178.059	0.215829
$826$	0	0
$827$	− 51.5248i	− 0.0623033i	−0.999515	−	0.0311516i	$-0.990083\pi$
$827$	− 51.5248i	− 0.0623033i	0.999515	−	0.0311516i	$-0.00991748\pi$
$828$	0	0
$829$	−1243.17	−1.49960	−0.749798	−	0.661666i	$-0.769850\pi$
$829$	−1243.17	−1.49960	−0.749798	+	0.661666i	$0.769850\pi$
$830$	0	0
$831$	− 190.286i	− 0.228984i
$832$	0	0
$833$	−412.177	−0.494810
$834$	0	0
$835$	1209.67i	1.44870i
$836$	0	0
$837$	−649.097	−0.775504
$838$	0	0
$839$	− 1383.20i	− 1.64863i	−0.566128	−	0.824317i	$-0.691559\pi$
$839$	− 1383.20i	− 1.64863i	0.566128	−	0.824317i	$-0.308441\pi$
$840$	0	0
$841$	−37.6661	−0.0447873
$842$	0	0
$843$	− 71.9108i	− 0.0853035i
$844$	0	0
$845$	−121.107	−0.143322
$846$	0	0
$847$	− 2131.32i	− 2.51632i
$848$	0	0
$849$	322.960	0.380401
$850$	0	0
$851$	− 1108.17i	− 1.30220i
$852$	0	0
$853$	886.167	1.03888	0.519441	−	0.854506i	$-0.326140\pi$
$853$	886.167	1.03888	0.519441	+	0.854506i	$0.326140\pi$
$854$	0	0
$855$	461.246i	0.539470i
$856$	0	0
$857$	−409.647	−0.478001	−0.239000	−	0.971019i	$-0.576820\pi$
$857$	−409.647	−0.478001	−0.239000	+	0.971019i	$0.576820\pi$
$858$	0	0
$859$	506.494i	0.589632i	0.955554	+	0.294816i	$0.0952583\pi$
$859$	506.494i	0.589632i	−0.955554	+	0.294816i	$0.904742\pi$
$860$	0	0
$861$	−165.019	−0.191660
$862$	0	0
$863$	− 1226.04i	− 1.42067i	−0.703863	−	0.710335i	$-0.748543\pi$
$863$	− 1226.04i	− 1.42067i	0.703863	−	0.710335i	$-0.251457\pi$
$864$	0	0
$865$	101.137	0.116921
$866$	0	0
$867$	167.553i	0.193257i
$868$	0	0
$869$	1295.06	1.49029
$870$	0	0
$871$	− 530.558i	− 0.609137i
$872$	0	0
$873$	−211.430	−0.242188
$874$	0	0
$875$	− 1293.45i	− 1.47823i
$876$	0	0
$877$	−1052.73	−1.20038	−0.600190	−	0.799857i	$-0.704908\pi$
$877$	−1052.73	−1.20038	−0.600190	+	0.799857i	$0.704908\pi$
$878$	0	0
$879$	− 201.852i	− 0.229638i
$880$	0	0
$881$	−149.294	−0.169459	−0.0847296	−	0.996404i	$-0.527003\pi$
$881$	−149.294	−0.169459	−0.0847296	+	0.996404i	$0.527003\pi$
$882$	0	0
$883$	1621.05i	1.83585i	0.396758	+	0.917923i	$0.370135\pi$
$883$	1621.05i	1.83585i	−0.396758	+	0.917923i	$0.629865\pi$
$884$	0	0
$885$	−27.6063	−0.0311936
$886$	0	0
$887$	266.420i	0.300361i	0.988659	+	0.150181i	$0.0479855\pi$
$887$	266.420i	0.300361i	−0.988659	+	0.150181i	$0.952014\pi$
$888$	0	0
$889$	554.039	0.623216
$890$	0	0
$891$	− 1163.48i	− 1.30582i
$892$	0	0
$893$	192.471	0.215533
$894$	0	0
$895$	335.911i	0.375319i
$896$	0	0
$897$	215.765	0.240540
$898$	0	0
$899$	1282.66i	1.42677i
$900$	0	0
$901$	257.588	0.285891
$902$	0	0
$903$	368.666i	0.408268i
$904$	0	0
$905$	−307.569	−0.339856
$906$	0	0
$907$	1060.43i	1.16917i	0.811333	+	0.584584i	$0.198742\pi$
$907$	1060.43i	1.16917i	−0.811333	+	0.584584i	$0.801258\pi$
$908$	0	0
$909$	−279.324	−0.307287
$910$	0	0
$911$	558.018i	0.612534i	0.951946	+	0.306267i	$0.0990800\pi$
$911$	558.018i	0.612534i	−0.951946	+	0.306267i	$0.900920\pi$
$912$	0	0
$913$	−1749.67	−1.91639
$914$	0	0
$915$	344.138i	0.376107i
$916$	0	0
$917$	−1915.61	−2.08899
$918$	0	0
$919$	1271.68i	1.38376i	0.722013	+	0.691880i	$0.243217\pi$
$919$	1271.68i	1.38376i	−0.722013	+	0.691880i	$0.756783\pi$
$920$	0	0
$921$	−228.844	−0.248474
$922$	0	0
$923$	189.823i	0.205659i
$924$	0	0
$925$	576.696	0.623456
$926$	0	0
$927$	− 828.029i	− 0.893235i
$928$	0	0
$929$	439.040	0.472594	0.236297	−	0.971681i	$-0.424066\pi$
$929$	439.040	0.472594	0.236297	+	0.971681i	$0.424066\pi$
$930$	0	0
$931$	− 671.415i	− 0.721177i
$932$	0	0
$933$	−457.884	−0.490765
$934$	0	0
$935$	− 629.588i	− 0.673356i
$936$	0	0
$937$	−1009.43	−1.07730	−0.538651	−	0.842529i	$-0.681066\pi$
$937$	−1009.43	−1.07730	−0.538651	+	0.842529i	$0.681066\pi$
$938$	0	0
$939$	308.695i	0.328748i
$940$	0	0
$941$	1689.54	1.79547	0.897735	−	0.440536i	$-0.145212\pi$
$941$	1689.54	1.79547	0.897735	+	0.440536i	$0.145212\pi$
$942$	0	0
$943$	− 460.881i	− 0.488740i
$944$	0	0
$945$	506.510	0.535989
$946$	0	0
$947$	61.5635i	0.0650090i	0.999472	+	0.0325045i	$0.0103483\pi$
$947$	61.5635i	0.0650090i	−0.999472	+	0.0325045i	$0.989652\pi$
$948$	0	0
$949$	−139.196	−0.146676
$950$	0	0
$951$	− 109.068i	− 0.114688i
$952$	0	0
$953$	873.960	0.917061	0.458531	−	0.888679i	$-0.348376\pi$
$953$	873.960	0.917061	0.458531	+	0.888679i	$0.348376\pi$
$954$	0	0
$955$	− 1057.33i	− 1.10715i
$956$	0	0
$957$	434.039	0.453541
$958$	0	0
$959$	520.333i	0.542579i
$960$	0	0
$961$	−1087.00	−1.13111
$962$	0	0
$963$	− 951.797i	− 0.988367i
$964$	0	0
$965$	823.078	0.852931
$966$	0	0
$967$	722.363i	0.747015i	0.927627	+	0.373507i	$0.121845\pi$
$967$	722.363i	0.747015i	−0.927627	+	0.373507i	$0.878155\pi$
$968$	0	0
$969$	117.060	0.120805
$970$	0	0
$971$	475.338i	0.489535i	0.969582	+	0.244767i	$0.0787116\pi$
$971$	475.338i	0.489535i	−0.969582	+	0.244767i	$0.921288\pi$
$972$	0	0
$973$	−999.058	−1.02678
$974$	0	0
$975$	112.284i	0.115163i
$976$	0	0
$977$	926.333	0.948140	0.474070	−	0.880487i	$-0.342784\pi$
$977$	926.333	0.948140	0.474070	+	0.880487i	$0.342784\pi$
$978$	0	0
$979$	− 2043.89i	− 2.08773i
$980$	0	0
$981$	585.302	0.596638
$982$	0	0
$983$	− 248.991i	− 0.253297i	−0.991948	−	0.126648i	$-0.959578\pi$
$983$	− 248.991i	− 0.253297i	0.991948	−	0.126648i	$-0.0404220\pi$
$984$	0	0
$985$	870.940	0.884203
$986$	0	0
$987$	− 101.490i	− 0.102827i
$988$	0	0
$989$	−1029.65	−1.04110
$990$	0	0
$991$	680.902i	0.687085i	0.939137	+	0.343543i	$0.111627\pi$
$991$	680.902i	0.687085i	−0.939137	+	0.343543i	$0.888373\pi$
$992$	0	0
$993$	272.686	0.274609
$994$	0	0
$995$	− 101.783i	− 0.102294i
$996$	0	0
$997$	146.950	0.147393	0.0736963	−	0.997281i	$-0.476520\pi$
$997$	146.950	0.147393	0.0736963	+	0.997281i	$0.476520\pi$
$998$	0	0
$999$	711.391i	0.712103i

Twists

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

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

\(f(q)\)	\(=\)	\(q-0.828427i q^{3} +3.65685 q^{5} +9.65685i q^{7} +8.31371 q^{9} +O(q^{10})\)	\(q-0.828427i q^{3} +3.65685 q^{5} +9.65685i q^{7} +8.31371 q^{9} -18.4853i q^{11} +11.6569 q^{13} -3.02944i q^{15} +9.31371 q^{17} +15.1716i q^{19} +8.00000 q^{21} +22.3431i q^{23} -11.6274 q^{25} -14.3431i q^{27} -28.3431 q^{29} -45.2548i q^{31} -15.3137 q^{33} +35.3137i q^{35} -49.5980 q^{37} -9.65685i q^{39} -20.6274 q^{41} +46.0833i q^{43} +30.4020 q^{45} -12.6863i q^{47} -44.2548 q^{49} -7.71573i q^{51} +27.6569 q^{53} -67.5980i q^{55} +12.5685 q^{57} -9.11270i q^{59} -113.598 q^{61} +80.2843i q^{63} +42.6274 q^{65} -45.5147i q^{67} +18.5097 q^{69} +16.2843i q^{71} -11.9411 q^{73} +9.63247i q^{75} +178.510 q^{77} +70.0589i q^{79} +62.9411 q^{81} -94.6518i q^{83} +34.0589 q^{85} +23.4802i q^{87} +110.569 q^{89} +112.569i q^{91} -37.4903 q^{93} +55.4802i q^{95} -25.4315 q^{97} -153.681i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} - 12 q^{9}+O(q^{10}) \) 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

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Database

Properties

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