LMFDB - Embedded newform 9152.2.a.cv.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9152,2,Mod(1,9152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9152.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9152 = 2^{6} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9152.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-1,0,1,0,-4,0,9,0,8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$73.0790879299$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 16x^{6} + 16x^{5} + 68x^{4} - 70x^{3} - 89x^{2} + 91x + 8 $ x^8 - x^7 - 16x^6 + 16x^5 + 68x^4 - 70x^3 - 89x^2 + 91x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4576)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.0818190$ of defining polynomial
Character	$\chi$	$=$	9152.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0818190 q^{3} -2.12622 q^{5} +1.15548 q^{7} -2.99331 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.173966 q^{15} -1.78259 q^{17} +7.57824 q^{19} +0.0945404 q^{21} +3.00509 q^{23} -0.479170 q^{25} -0.490366 q^{27} -7.42086 q^{29} -4.34363 q^{31} +0.0818190 q^{33} -2.45682 q^{35} +5.00270 q^{37} -0.0818190 q^{39} +1.36657 q^{41} +5.81648 q^{43} +6.36444 q^{45} +5.50365 q^{47} -5.66486 q^{49} -0.145850 q^{51} -0.407900 q^{53} -2.12622 q^{55} +0.620044 q^{57} +11.0187 q^{59} -1.51658 q^{61} -3.45871 q^{63} +2.12622 q^{65} +5.69950 q^{67} +0.245874 q^{69} -2.63381 q^{71} -12.1178 q^{73} -0.0392052 q^{75} +1.15548 q^{77} -6.92082 q^{79} +8.93980 q^{81} +5.80856 q^{83} +3.79019 q^{85} -0.607167 q^{87} -1.86021 q^{89} -1.15548 q^{91} -0.355391 q^{93} -16.1130 q^{95} +0.00722180 q^{97} -2.99331 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} + q^{5} - 4 q^{7} + 9 q^{9} + 8 q^{11} - 8 q^{13} - 13 q^{15} - 8 q^{17} - 2 q^{19} + 12 q^{21} - 17 q^{23} + 23 q^{25} + 5 q^{27} - 23 q^{31} - q^{33} + 2 q^{35} + q^{37} + q^{39} - 14 q^{41}+ \cdots + 9 q^{99}+O(q^{100}) $ 8 * q - q^3 + q^5 - 4 * q^7 + 9 * q^9 + 8 * q^11 - 8 * q^13 - 13 * q^15 - 8 * q^17 - 2 * q^19 + 12 * q^21 - 17 * q^23 + 23 * q^25 + 5 * q^27 - 23 * q^31 - q^33 + 2 * q^35 + q^37 + q^39 - 14 * q^41 - 10 * q^43 - 4 * q^45 - 28 * q^47 - 4 * q^49 - 6 * q^51 + 14 * q^53 + q^55 - 36 * q^57 + 31 * q^59 - 16 * q^63 - q^65 - 21 * q^67 - 13 * q^69 - 61 * q^71 + 22 * q^75 - 4 * q^77 + 16 * q^81 + 22 * q^83 + 22 * q^85 - 10 * q^87 + 9 * q^89 + 4 * q^91 - 3 * q^93 - 22 * q^95 - 11 * q^97 + 9 * q^99

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
$3$	0.0818190	0.0472382	0.0236191	−	0.999721i	$-0.492481\pi$
$3$	0.0818190	0.0472382	0.0236191	+	0.999721i	$0.492481\pi$
$4$	0	0
$5$	−2.12622	−0.950876	−0.475438	−	0.879749i	$-0.657710\pi$
$5$	−2.12622	−0.950876	−0.475438	+	0.879749i	$0.657710\pi$
$6$	0	0
$7$	1.15548	0.436732	0.218366	−	0.975867i	$-0.429927\pi$
$7$	1.15548	0.436732	0.218366	+	0.975867i	$0.429927\pi$
$8$	0	0
$9$	−2.99331	−0.997769
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.173966	−0.0449177
$16$	0	0
$17$	−1.78259	−0.432342	−0.216171	−	0.976355i	$-0.569357\pi$
$17$	−1.78259	−0.432342	−0.216171	+	0.976355i	$0.569357\pi$
$18$	0	0
$19$	7.57824	1.73857	0.869284	−	0.494313i	$-0.164580\pi$
$19$	7.57824	1.73857	0.869284	+	0.494313i	$0.164580\pi$
$20$	0	0
$21$	0.0945404	0.0206304
$22$	0	0
$23$	3.00509	0.626605	0.313303	−	0.949653i	$-0.398565\pi$
$23$	3.00509	0.626605	0.313303	+	0.949653i	$0.398565\pi$
$24$	0	0
$25$	−0.479170	−0.0958341
$26$	0	0
$27$	−0.490366	−0.0943710
$28$	0	0
$29$	−7.42086	−1.37802	−0.689010	−	0.724752i	$-0.741954\pi$
$29$	−7.42086	−1.37802	−0.689010	+	0.724752i	$0.741954\pi$
$30$	0	0
$31$	−4.34363	−0.780139	−0.390070	−	0.920785i	$-0.627549\pi$
$31$	−4.34363	−0.780139	−0.390070	+	0.920785i	$0.627549\pi$
$32$	0	0
$33$	0.0818190	0.0142429
$34$	0	0
$35$	−2.45682	−0.415278
$36$	0	0
$37$	5.00270	0.822439	0.411219	−	0.911536i	$-0.365103\pi$
$37$	5.00270	0.822439	0.411219	+	0.911536i	$0.365103\pi$
$38$	0	0
$39$	−0.0818190	−0.0131015
$40$	0	0
$41$	1.36657	0.213422	0.106711	−	0.994290i	$-0.465968\pi$
$41$	1.36657	0.213422	0.106711	+	0.994290i	$0.465968\pi$
$42$	0	0
$43$	5.81648	0.887005	0.443502	−	0.896273i	$-0.353736\pi$
$43$	5.81648	0.887005	0.443502	+	0.896273i	$0.353736\pi$
$44$	0	0
$45$	6.36444	0.948755
$46$	0	0
$47$	5.50365	0.802789	0.401395	−	0.915905i	$-0.368526\pi$
$47$	5.50365	0.802789	0.401395	+	0.915905i	$0.368526\pi$
$48$	0	0
$49$	−5.66486	−0.809266
$50$	0	0
$51$	−0.145850	−0.0204231
$52$	0	0
$53$	−0.407900	−0.0560294	−0.0280147	−	0.999608i	$-0.508919\pi$
$53$	−0.407900	−0.0560294	−0.0280147	+	0.999608i	$0.508919\pi$
$54$	0	0
$55$	−2.12622	−0.286700
$56$	0	0
$57$	0.620044	0.0821269
$58$	0	0
$59$	11.0187	1.43451	0.717257	−	0.696808i	$-0.245397\pi$
$59$	11.0187	1.43451	0.717257	+	0.696808i	$0.245397\pi$
$60$	0	0
$61$	−1.51658	−0.194179	−0.0970893	−	0.995276i	$-0.530953\pi$
$61$	−1.51658	−0.194179	−0.0970893	+	0.995276i	$0.530953\pi$
$62$	0	0
$63$	−3.45871	−0.435757
$64$	0	0
$65$	2.12622	0.263726
$66$	0	0
$67$	5.69950	0.696305	0.348152	−	0.937438i	$-0.386809\pi$
$67$	5.69950	0.696305	0.348152	+	0.937438i	$0.386809\pi$
$68$	0	0
$69$	0.245874	0.0295997
$70$	0	0
$71$	−2.63381	−0.312575	−0.156288	−	0.987712i	$-0.549953\pi$
$71$	−2.63381	−0.312575	−0.156288	+	0.987712i	$0.549953\pi$
$72$	0	0
$73$	−12.1178	−1.41829	−0.709143	−	0.705065i	$-0.750918\pi$
$73$	−12.1178	−1.41829	−0.709143	+	0.705065i	$0.750918\pi$
$74$	0	0
$75$	−0.0392052	−0.00452703
$76$	0	0
$77$	1.15548	0.131680
$78$	0	0
$79$	−6.92082	−0.778653	−0.389326	−	0.921100i	$-0.627292\pi$
$79$	−6.92082	−0.778653	−0.389326	+	0.921100i	$0.627292\pi$
$80$	0	0
$81$	8.93980	0.993311
$82$	0	0
$83$	5.80856	0.637573	0.318786	−	0.947827i	$-0.396725\pi$
$83$	5.80856	0.637573	0.318786	+	0.947827i	$0.396725\pi$
$84$	0	0
$85$	3.79019	0.411104
$86$	0	0
$87$	−0.607167	−0.0650952
$88$	0	0
$89$	−1.86021	−0.197182	−0.0985912	−	0.995128i	$-0.531434\pi$
$89$	−1.86021	−0.197182	−0.0985912	+	0.995128i	$0.531434\pi$
$90$	0	0
$91$	−1.15548	−0.121128
$92$	0	0
$93$	−0.355391	−0.0368524
$94$	0	0
$95$	−16.1130	−1.65316
$96$	0	0
$97$	0.00722180	0.000733263 0	0.000366631	−	1.00000i	$-0.499883\pi$
$97$	0.00722180	0.000733263 0	0.000366631		1.00000i	$0.499883\pi$
$98$	0	0
$99$	−2.99331	−0.300839
$100$	0	0
$101$	−18.0936	−1.80038	−0.900191	−	0.435495i	$-0.856573\pi$
$101$	−18.0936	−1.80038	−0.900191	+	0.435495i	$0.856573\pi$
$102$	0	0
$103$	−12.2254	−1.20461	−0.602304	−	0.798267i	$-0.705751\pi$
$103$	−12.2254	−1.20461	−0.602304	+	0.798267i	$0.705751\pi$
$104$	0	0
$105$	−0.201014	−0.0196170
$106$	0	0
$107$	−0.810358	−0.0783403	−0.0391701	−	0.999233i	$-0.512471\pi$
$107$	−0.810358	−0.0783403	−0.0391701	+	0.999233i	$0.512471\pi$
$108$	0	0
$109$	−3.34991	−0.320863	−0.160431	−	0.987047i	$-0.551289\pi$
$109$	−3.34991	−0.320863	−0.160431	+	0.987047i	$0.551289\pi$
$110$	0	0
$111$	0.409316	0.0388505
$112$	0	0
$113$	9.79318	0.921265	0.460632	−	0.887591i	$-0.347623\pi$
$113$	9.79318	0.921265	0.460632	+	0.887591i	$0.347623\pi$
$114$	0	0
$115$	−6.38950	−0.595824
$116$	0	0
$117$	2.99331	0.276731
$118$	0	0
$119$	−2.05976	−0.188818
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.111811	0.0100817
$124$	0	0
$125$	11.6499	1.04200
$126$	0	0
$127$	1.49762	0.132892	0.0664461	−	0.997790i	$-0.478834\pi$
$127$	1.49762	0.132892	0.0664461	+	0.997790i	$0.478834\pi$
$128$	0	0
$129$	0.475898	0.0419005
$130$	0	0
$131$	20.6370	1.80307	0.901533	−	0.432711i	$-0.142443\pi$
$131$	20.6370	1.80307	0.901533	+	0.432711i	$0.142443\pi$
$132$	0	0
$133$	8.75653	0.759288
$134$	0	0
$135$	1.04263	0.0897352
$136$	0	0
$137$	6.34808	0.542353	0.271177	−	0.962530i	$-0.412587\pi$
$137$	6.34808	0.542353	0.271177	+	0.962530i	$0.412587\pi$
$138$	0	0
$139$	−13.4926	−1.14443	−0.572213	−	0.820105i	$-0.693915\pi$
$139$	−13.4926	−1.14443	−0.572213	+	0.820105i	$0.693915\pi$
$140$	0	0
$141$	0.450303	0.0379223
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	15.7784	1.31033
$146$	0	0
$147$	−0.463493	−0.0382283
$148$	0	0
$149$	9.35318	0.766242	0.383121	−	0.923698i	$-0.374849\pi$
$149$	9.35318	0.766242	0.383121	+	0.923698i	$0.374849\pi$
$150$	0	0
$151$	−13.1271	−1.06827	−0.534135	−	0.845399i	$-0.679362\pi$
$151$	−13.1271	−1.06827	−0.534135	+	0.845399i	$0.679362\pi$
$152$	0	0
$153$	5.33585	0.431378
$154$	0	0
$155$	9.23553	0.741816
$156$	0	0
$157$	−14.5722	−1.16299	−0.581493	−	0.813552i	$-0.697531\pi$
$157$	−14.5722	−1.16299	−0.581493	+	0.813552i	$0.697531\pi$
$158$	0	0
$159$	−0.0333740	−0.00264673
$160$	0	0
$161$	3.47233	0.273658
$162$	0	0
$163$	−0.712050	−0.0557720	−0.0278860	−	0.999611i	$-0.508878\pi$
$163$	−0.712050	−0.0557720	−0.0278860	+	0.999611i	$0.508878\pi$
$164$	0	0
$165$	−0.173966	−0.0135432
$166$	0	0
$167$	17.5231	1.35598	0.677989	−	0.735072i	$-0.262852\pi$
$167$	17.5231	1.35598	0.677989	+	0.735072i	$0.262852\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−22.6840	−1.73469
$172$	0	0
$173$	15.1199	1.14954	0.574771	−	0.818315i	$-0.305091\pi$
$173$	15.1199	1.14954	0.574771	+	0.818315i	$0.305091\pi$
$174$	0	0
$175$	−0.553673	−0.0418538
$176$	0	0
$177$	0.901540	0.0677639
$178$	0	0
$179$	−10.8705	−0.812502	−0.406251	−	0.913762i	$-0.633164\pi$
$179$	−10.8705	−0.812502	−0.406251	+	0.913762i	$0.633164\pi$
$180$	0	0
$181$	−12.1507	−0.903158	−0.451579	−	0.892231i	$-0.649139\pi$
$181$	−12.1507	−0.903158	−0.451579	+	0.892231i	$0.649139\pi$
$182$	0	0
$183$	−0.124085	−0.00917265
$184$	0	0
$185$	−10.6369	−0.782038
$186$	0	0
$187$	−1.78259	−0.130356
$188$	0	0
$189$	−0.566610	−0.0412148
$190$	0	0
$191$	−22.0247	−1.59365	−0.796827	−	0.604208i	$-0.793490\pi$
$191$	−22.0247	−1.59365	−0.796827	+	0.604208i	$0.793490\pi$
$192$	0	0
$193$	−21.3417	−1.53621	−0.768105	−	0.640324i	$-0.778800\pi$
$193$	−21.3417	−1.53621	−0.768105	+	0.640324i	$0.778800\pi$
$194$	0	0
$195$	0.173966	0.0124579
$196$	0	0
$197$	−4.30142	−0.306463	−0.153232	−	0.988190i	$-0.548968\pi$
$197$	−4.30142	−0.306463	−0.153232	+	0.988190i	$0.548968\pi$
$198$	0	0
$199$	−5.94319	−0.421302	−0.210651	−	0.977561i	$-0.567558\pi$
$199$	−5.94319	−0.421302	−0.210651	+	0.977561i	$0.567558\pi$
$200$	0	0
$201$	0.466327	0.0328922
$202$	0	0
$203$	−8.57468	−0.601825
$204$	0	0
$205$	−2.90563	−0.202938
$206$	0	0
$207$	−8.99516	−0.625207
$208$	0	0
$209$	7.57824	0.524198
$210$	0	0
$211$	16.5964	1.14255	0.571273	−	0.820760i	$-0.306450\pi$
$211$	16.5964	1.14255	0.571273	+	0.820760i	$0.306450\pi$
$212$	0	0
$213$	−0.215495	−0.0147655
$214$	0	0
$215$	−12.3671	−0.843432
$216$	0	0
$217$	−5.01899	−0.340711
$218$	0	0
$219$	−0.991470	−0.0669973
$220$	0	0
$221$	1.78259	0.119910
$222$	0	0
$223$	−5.42392	−0.363213	−0.181606	−	0.983371i	$-0.558130\pi$
$223$	−5.42392	−0.363213	−0.181606	+	0.983371i	$0.558130\pi$
$224$	0	0
$225$	1.43430	0.0956202
$226$	0	0
$227$	26.5716	1.76362	0.881808	−	0.471609i	$-0.156326\pi$
$227$	26.5716	1.76362	0.881808	+	0.471609i	$0.156326\pi$
$228$	0	0
$229$	−25.5474	−1.68822	−0.844111	−	0.536168i	$-0.819871\pi$
$229$	−25.5474	−1.68822	−0.844111	+	0.536168i	$0.819871\pi$
$230$	0	0
$231$	0.0945404	0.00622030
$232$	0	0
$233$	−9.31557	−0.610283	−0.305142	−	0.952307i	$-0.598704\pi$
$233$	−9.31557	−0.610283	−0.305142	+	0.952307i	$0.598704\pi$
$234$	0	0
$235$	−11.7020	−0.763353
$236$	0	0
$237$	−0.566254	−0.0367822
$238$	0	0
$239$	1.05081	0.0679711	0.0339856	−	0.999422i	$-0.489180\pi$
$239$	1.05081	0.0679711	0.0339856	+	0.999422i	$0.489180\pi$
$240$	0	0
$241$	15.2468	0.982130	0.491065	−	0.871123i	$-0.336608\pi$
$241$	15.2468	0.982130	0.491065	+	0.871123i	$0.336608\pi$
$242$	0	0
$243$	2.20254	0.141293
$244$	0	0
$245$	12.0448	0.769512
$246$	0	0
$247$	−7.57824	−0.482192
$248$	0	0
$249$	0.475251	0.0301178
$250$	0	0
$251$	−24.0214	−1.51622	−0.758108	−	0.652129i	$-0.773876\pi$
$251$	−24.0214	−1.51622	−0.758108	+	0.652129i	$0.773876\pi$
$252$	0	0
$253$	3.00509	0.188929
$254$	0	0
$255$	0.310110	0.0194198
$256$	0	0
$257$	−24.6383	−1.53689	−0.768446	−	0.639914i	$-0.778970\pi$
$257$	−24.6383	−1.53689	−0.768446	+	0.639914i	$0.778970\pi$
$258$	0	0
$259$	5.78053	0.359185
$260$	0	0
$261$	22.2129	1.37494
$262$	0	0
$263$	−5.36547	−0.330849	−0.165424	−	0.986222i	$-0.552899\pi$
$263$	−5.36547	−0.330849	−0.165424	+	0.986222i	$0.552899\pi$
$264$	0	0
$265$	0.867287	0.0532770
$266$	0	0
$267$	−0.152201	−0.00931454
$268$	0	0
$269$	−19.1979	−1.17052	−0.585260	−	0.810846i	$-0.699007\pi$
$269$	−19.1979	−1.17052	−0.585260	+	0.810846i	$0.699007\pi$
$270$	0	0
$271$	−4.38291	−0.266243	−0.133121	−	0.991100i	$-0.542500\pi$
$271$	−4.38291	−0.266243	−0.133121	+	0.991100i	$0.542500\pi$
$272$	0	0
$273$	−0.0945404	−0.00572185
$274$	0	0
$275$	−0.479170	−0.0288951
$276$	0	0
$277$	2.79860	0.168151	0.0840757	−	0.996459i	$-0.473206\pi$
$277$	2.79860	0.168151	0.0840757	+	0.996459i	$0.473206\pi$
$278$	0	0
$279$	13.0018	0.778398
$280$	0	0
$281$	−16.1165	−0.961433	−0.480716	−	0.876876i	$-0.659623\pi$
$281$	−16.1165	−0.961433	−0.480716	+	0.876876i	$0.659623\pi$
$282$	0	0
$283$	−28.7229	−1.70740	−0.853700	−	0.520765i	$-0.825647\pi$
$283$	−28.7229	−1.70740	−0.853700	+	0.520765i	$0.825647\pi$
$284$	0	0
$285$	−1.31835	−0.0780925
$286$	0	0
$287$	1.57904	0.0932081
$288$	0	0
$289$	−13.8224	−0.813080
$290$	0	0
$291$	0.000590880 0	3.46380e−5 0
$292$	0	0
$293$	13.1606	0.768849	0.384425	−	0.923156i	$-0.374400\pi$
$293$	13.1606	0.768849	0.384425	+	0.923156i	$0.374400\pi$
$294$	0	0
$295$	−23.4283	−1.36405
$296$	0	0
$297$	−0.490366	−0.0284539
$298$	0	0
$299$	−3.00509	−0.173789
$300$	0	0
$301$	6.72084	0.387383
$302$	0	0
$303$	−1.48040	−0.0850468
$304$	0	0
$305$	3.22460	0.184640
$306$	0	0
$307$	−14.0743	−0.803263	−0.401631	−	0.915801i	$-0.631557\pi$
$307$	−14.0743	−0.803263	−0.401631	+	0.915801i	$0.631557\pi$
$308$	0	0
$309$	−1.00027	−0.0569035
$310$	0	0
$311$	−19.4539	−1.10313	−0.551566	−	0.834131i	$-0.685969\pi$
$311$	−19.4539	−1.10313	−0.551566	+	0.834131i	$0.685969\pi$
$312$	0	0
$313$	16.9771	0.959603	0.479802	−	0.877377i	$-0.340709\pi$
$313$	16.9771	0.959603	0.479802	+	0.877377i	$0.340709\pi$
$314$	0	0
$315$	7.35400	0.414351
$316$	0	0
$317$	−8.68416	−0.487751	−0.243875	−	0.969807i	$-0.578419\pi$
$317$	−8.68416	−0.487751	−0.243875	+	0.969807i	$0.578419\pi$
$318$	0	0
$319$	−7.42086	−0.415489
$320$	0	0
$321$	−0.0663027	−0.00370065
$322$	0	0
$323$	−13.5089	−0.751657
$324$	0	0
$325$	0.479170	0.0265796
$326$	0	0
$327$	−0.274086	−0.0151570
$328$	0	0
$329$	6.35937	0.350603
$330$	0	0
$331$	−27.4303	−1.50771	−0.753853	−	0.657043i	$-0.771807\pi$
$331$	−27.4303	−1.50771	−0.753853	+	0.657043i	$0.771807\pi$
$332$	0	0
$333$	−14.9746	−0.820604
$334$	0	0
$335$	−12.1184	−0.662100
$336$	0	0
$337$	−19.8824	−1.08306	−0.541531	−	0.840681i	$-0.682155\pi$
$337$	−19.8824	−1.08306	−0.541531	+	0.840681i	$0.682155\pi$
$338$	0	0
$339$	0.801268	0.0435189
$340$	0	0
$341$	−4.34363	−0.235221
$342$	0	0
$343$	−14.6340	−0.790163
$344$	0	0
$345$	−0.522783	−0.0281457
$346$	0	0
$347$	−5.10143	−0.273859	−0.136929	−	0.990581i	$-0.543723\pi$
$347$	−5.10143	−0.273859	−0.136929	+	0.990581i	$0.543723\pi$
$348$	0	0
$349$	−2.26115	−0.121037	−0.0605184	−	0.998167i	$-0.519275\pi$
$349$	−2.26115	−0.121037	−0.0605184	+	0.998167i	$0.519275\pi$
$350$	0	0
$351$	0.490366	0.0261738
$352$	0	0
$353$	−0.835846	−0.0444876	−0.0222438	−	0.999753i	$-0.507081\pi$
$353$	−0.835846	−0.0444876	−0.0222438	+	0.999753i	$0.507081\pi$
$354$	0	0
$355$	5.60006	0.297220
$356$	0	0
$357$	−0.168527	−0.00891941
$358$	0	0
$359$	3.39733	0.179304	0.0896521	−	0.995973i	$-0.471424\pi$
$359$	3.39733	0.179304	0.0896521	+	0.995973i	$0.471424\pi$
$360$	0	0
$361$	38.4298	2.02262
$362$	0	0
$363$	0.0818190	0.00429438
$364$	0	0
$365$	25.7653	1.34861
$366$	0	0
$367$	−2.96848	−0.154954	−0.0774768	−	0.996994i	$-0.524686\pi$
$367$	−2.96848	−0.154954	−0.0774768	+	0.996994i	$0.524686\pi$
$368$	0	0
$369$	−4.09055	−0.212946
$370$	0	0
$371$	−0.471322	−0.0244698
$372$	0	0
$373$	−4.86138	−0.251713	−0.125856	−	0.992048i	$-0.540168\pi$
$373$	−4.86138	−0.251713	−0.125856	+	0.992048i	$0.540168\pi$
$374$	0	0
$375$	0.953187	0.0492223
$376$	0	0
$377$	7.42086	0.382194
$378$	0	0
$379$	1.41059	0.0724571	0.0362286	−	0.999344i	$-0.488466\pi$
$379$	1.41059	0.0724571	0.0362286	+	0.999344i	$0.488466\pi$
$380$	0	0
$381$	0.122534	0.00627759
$382$	0	0
$383$	−27.0646	−1.38294	−0.691469	−	0.722406i	$-0.743036\pi$
$383$	−27.0646	−1.38294	−0.691469	+	0.722406i	$0.743036\pi$
$384$	0	0
$385$	−2.45682	−0.125211
$386$	0	0
$387$	−17.4105	−0.885025
$388$	0	0
$389$	5.69774	0.288887	0.144443	−	0.989513i	$-0.453861\pi$
$389$	5.69774	0.288887	0.144443	+	0.989513i	$0.453861\pi$
$390$	0	0
$391$	−5.35686	−0.270908
$392$	0	0
$393$	1.68850	0.0851736
$394$	0	0
$395$	14.7152	0.740402
$396$	0	0
$397$	−6.67743	−0.335131	−0.167565	−	0.985861i	$-0.553590\pi$
$397$	−6.67743	−0.335131	−0.167565	+	0.985861i	$0.553590\pi$
$398$	0	0
$399$	0.716451	0.0358674
$400$	0	0
$401$	8.71848	0.435380	0.217690	−	0.976018i	$-0.430148\pi$
$401$	8.71848	0.435380	0.217690	+	0.976018i	$0.430148\pi$
$402$	0	0
$403$	4.34363	0.216372
$404$	0	0
$405$	−19.0080	−0.944516
$406$	0	0
$407$	5.00270	0.247975
$408$	0	0
$409$	12.7688	0.631378	0.315689	−	0.948863i	$-0.397764\pi$
$409$	12.7688	0.631378	0.315689	+	0.948863i	$0.397764\pi$
$410$	0	0
$411$	0.519394	0.0256198
$412$	0	0
$413$	12.7319	0.626498
$414$	0	0
$415$	−12.3503	−0.606253
$416$	0	0
$417$	−1.10395	−0.0540607
$418$	0	0
$419$	22.1037	1.07984	0.539918	−	0.841718i	$-0.318455\pi$
$419$	22.1037	1.07984	0.539918	+	0.841718i	$0.318455\pi$
$420$	0	0
$421$	18.6607	0.909467	0.454733	−	0.890628i	$-0.349735\pi$
$421$	18.6607	0.909467	0.454733	+	0.890628i	$0.349735\pi$
$422$	0	0
$423$	−16.4741	−0.800998
$424$	0	0
$425$	0.854166	0.0414331
$426$	0	0
$427$	−1.75239	−0.0848039
$428$	0	0
$429$	−0.0818190	−0.00395026
$430$	0	0
$431$	8.31911	0.400717	0.200359	−	0.979723i	$-0.435789\pi$
$431$	8.31911	0.400717	0.200359	+	0.979723i	$0.435789\pi$
$432$	0	0
$433$	2.31947	0.111466	0.0557332	−	0.998446i	$-0.482250\pi$
$433$	2.31947	0.111466	0.0557332	+	0.998446i	$0.482250\pi$
$434$	0	0
$435$	1.29097	0.0618975
$436$	0	0
$437$	22.7733	1.08940
$438$	0	0
$439$	−8.88062	−0.423849	−0.211924	−	0.977286i	$-0.567973\pi$
$439$	−8.88062	−0.423849	−0.211924	+	0.977286i	$0.567973\pi$
$440$	0	0
$441$	16.9567	0.807460
$442$	0	0
$443$	20.9064	0.993293	0.496646	−	0.867953i	$-0.334565\pi$
$443$	20.9064	0.993293	0.496646	+	0.867953i	$0.334565\pi$
$444$	0	0
$445$	3.95523	0.187496
$446$	0	0
$447$	0.765268	0.0361959
$448$	0	0
$449$	0.806925	0.0380811	0.0190406	−	0.999819i	$-0.493939\pi$
$449$	0.806925	0.0380811	0.0190406	+	0.999819i	$0.493939\pi$
$450$	0	0
$451$	1.36657	0.0643491
$452$	0	0
$453$	−1.07405	−0.0504631
$454$	0	0
$455$	2.45682	0.115177
$456$	0	0
$457$	−21.0713	−0.985673	−0.492837	−	0.870122i	$-0.664040\pi$
$457$	−21.0713	−0.985673	−0.492837	+	0.870122i	$0.664040\pi$
$458$	0	0
$459$	0.874124	0.0408006
$460$	0	0
$461$	6.31724	0.294223	0.147112	−	0.989120i	$-0.453002\pi$
$461$	6.31724	0.294223	0.147112	+	0.989120i	$0.453002\pi$
$462$	0	0
$463$	−19.4379	−0.903357	−0.451678	−	0.892181i	$-0.649175\pi$
$463$	−19.4379	−0.903357	−0.451678	+	0.892181i	$0.649175\pi$
$464$	0	0
$465$	0.755642	0.0350421
$466$	0	0
$467$	32.8919	1.52205	0.761027	−	0.648720i	$-0.224696\pi$
$467$	32.8919	1.52205	0.761027	+	0.648720i	$0.224696\pi$
$468$	0	0
$469$	6.58567	0.304098
$470$	0	0
$471$	−1.19228	−0.0549373
$472$	0	0
$473$	5.81648	0.267442
$474$	0	0
$475$	−3.63127	−0.166614
$476$	0	0
$477$	1.22097	0.0559044
$478$	0	0
$479$	15.0384	0.687120	0.343560	−	0.939131i	$-0.388367\pi$
$479$	15.0384	0.687120	0.343560	+	0.939131i	$0.388367\pi$
$480$	0	0
$481$	−5.00270	−0.228103
$482$	0	0
$483$	0.284103	0.0129271
$484$	0	0
$485$	−0.0153552	−0.000697242 0
$486$	0	0
$487$	−6.69852	−0.303539	−0.151769	−	0.988416i	$-0.548497\pi$
$487$	−6.69852	−0.303539	−0.151769	+	0.988416i	$0.548497\pi$
$488$	0	0
$489$	−0.0582592	−0.00263457
$490$	0	0
$491$	−36.2726	−1.63696	−0.818480	−	0.574536i	$-0.805183\pi$
$491$	−36.2726	−1.63696	−0.818480	+	0.574536i	$0.805183\pi$
$492$	0	0
$493$	13.2284	0.595776
$494$	0	0
$495$	6.36444	0.286060
$496$	0	0
$497$	−3.04332	−0.136511
$498$	0	0
$499$	5.20742	0.233116	0.116558	−	0.993184i	$-0.462814\pi$
$499$	5.20742	0.233116	0.116558	+	0.993184i	$0.462814\pi$
$500$	0	0
$501$	1.43372	0.0640540
$502$	0	0
$503$	−9.59145	−0.427661	−0.213831	−	0.976871i	$-0.568594\pi$
$503$	−9.59145	−0.427661	−0.213831	+	0.976871i	$0.568594\pi$
$504$	0	0
$505$	38.4711	1.71194
$506$	0	0
$507$	0.0818190	0.00363371
$508$	0	0
$509$	−2.30267	−0.102064	−0.0510319	−	0.998697i	$-0.516251\pi$
$509$	−2.30267	−0.102064	−0.0510319	+	0.998697i	$0.516251\pi$
$510$	0	0
$511$	−14.0020	−0.619410
$512$	0	0
$513$	−3.71611	−0.164070
$514$	0	0
$515$	25.9940	1.14543
$516$	0	0
$517$	5.50365	0.242050
$518$	0	0
$519$	1.23709	0.0543023
$520$	0	0
$521$	−21.0663	−0.922932	−0.461466	−	0.887158i	$-0.652676\pi$
$521$	−21.0663	−0.922932	−0.461466	+	0.887158i	$0.652676\pi$
$522$	0	0
$523$	18.1126	0.792011	0.396005	−	0.918248i	$-0.370396\pi$
$523$	18.1126	0.792011	0.396005	+	0.918248i	$0.370396\pi$
$524$	0	0
$525$	−0.0453010	−0.00197710
$526$	0	0
$527$	7.74293	0.337287
$528$	0	0
$529$	−13.9694	−0.607366
$530$	0	0
$531$	−32.9824	−1.43131
$532$	0	0
$533$	−1.36657	−0.0591926
$534$	0	0
$535$	1.72300	0.0744919
$536$	0	0
$537$	−0.889416	−0.0383811
$538$	0	0
$539$	−5.66486	−0.244003
$540$	0	0
$541$	−32.6055	−1.40182	−0.700909	−	0.713250i	$-0.747222\pi$
$541$	−32.6055	−1.40182	−0.700909	+	0.713250i	$0.747222\pi$
$542$	0	0
$543$	−0.994162	−0.0426636
$544$	0	0
$545$	7.12265	0.305101
$546$	0	0
$547$	−7.11533	−0.304230	−0.152115	−	0.988363i	$-0.548608\pi$
$547$	−7.11533	−0.304230	−0.152115	+	0.988363i	$0.548608\pi$
$548$	0	0
$549$	4.53960	0.193745
$550$	0	0
$551$	−56.2371	−2.39578
$552$	0	0
$553$	−7.99688	−0.340062
$554$	0	0
$555$	−0.870297	−0.0369421
$556$	0	0
$557$	−5.86169	−0.248368	−0.124184	−	0.992259i	$-0.539631\pi$
$557$	−5.86169	−0.248368	−0.124184	+	0.992259i	$0.539631\pi$
$558$	0	0
$559$	−5.81648	−0.246011
$560$	0	0
$561$	−0.145850	−0.00615779
$562$	0	0
$563$	15.0844	0.635733	0.317866	−	0.948136i	$-0.397034\pi$
$563$	15.0844	0.635733	0.317866	+	0.948136i	$0.397034\pi$
$564$	0	0
$565$	−20.8225	−0.876009
$566$	0	0
$567$	10.3298	0.433810
$568$	0	0
$569$	−34.8108	−1.45934	−0.729672	−	0.683797i	$-0.760327\pi$
$569$	−34.8108	−1.45934	−0.729672	+	0.683797i	$0.760327\pi$
$570$	0	0
$571$	15.2782	0.639371	0.319686	−	0.947524i	$-0.396423\pi$
$571$	15.2782	0.639371	0.319686	+	0.947524i	$0.396423\pi$
$572$	0	0
$573$	−1.80204	−0.0752814
$574$	0	0
$575$	−1.43995	−0.0600501
$576$	0	0
$577$	−19.2354	−0.800780	−0.400390	−	0.916345i	$-0.631125\pi$
$577$	−19.2354	−0.800780	−0.400390	+	0.916345i	$0.631125\pi$
$578$	0	0
$579$	−1.74616	−0.0725678
$580$	0	0
$581$	6.71170	0.278448
$582$	0	0
$583$	−0.407900	−0.0168935
$584$	0	0
$585$	−6.36444	−0.263137
$586$	0	0
$587$	−40.9161	−1.68879	−0.844394	−	0.535722i	$-0.820040\pi$
$587$	−40.9161	−1.68879	−0.844394	+	0.535722i	$0.820040\pi$
$588$	0	0
$589$	−32.9171	−1.35633
$590$	0	0
$591$	−0.351938	−0.0144768
$592$	0	0
$593$	19.0482	0.782215	0.391108	−	0.920345i	$-0.372092\pi$
$593$	19.0482	0.782215	0.391108	+	0.920345i	$0.372092\pi$
$594$	0	0
$595$	4.37950	0.179542
$596$	0	0
$597$	−0.486266	−0.0199015
$598$	0	0
$599$	21.2427	0.867952	0.433976	−	0.900925i	$-0.357110\pi$
$599$	21.2427	0.867952	0.433976	+	0.900925i	$0.357110\pi$
$600$	0	0
$601$	−28.7245	−1.17170	−0.585848	−	0.810421i	$-0.699238\pi$
$601$	−28.7245	−1.17170	−0.585848	+	0.810421i	$0.699238\pi$
$602$	0	0
$603$	−17.0603	−0.694751
$604$	0	0
$605$	−2.12622	−0.0864433
$606$	0	0
$607$	−12.3354	−0.500679	−0.250339	−	0.968158i	$-0.580542\pi$
$607$	−12.3354	−0.500679	−0.250339	+	0.968158i	$0.580542\pi$
$608$	0	0
$609$	−0.701572	−0.0284291
$610$	0	0
$611$	−5.50365	−0.222654
$612$	0	0
$613$	24.1201	0.974202	0.487101	−	0.873346i	$-0.338054\pi$
$613$	24.1201	0.974202	0.487101	+	0.873346i	$0.338054\pi$
$614$	0	0
$615$	−0.237736	−0.00958642
$616$	0	0
$617$	18.7965	0.756721	0.378360	−	0.925658i	$-0.376488\pi$
$617$	18.7965	0.756721	0.378360	+	0.925658i	$0.376488\pi$
$618$	0	0
$619$	14.9163	0.599536	0.299768	−	0.954012i	$-0.403091\pi$
$619$	14.9163	0.599536	0.299768	+	0.954012i	$0.403091\pi$
$620$	0	0
$621$	−1.47360	−0.0591334
$622$	0	0
$623$	−2.14945	−0.0861157
$624$	0	0
$625$	−22.3745	−0.894982
$626$	0	0
$627$	0.620044	0.0247622
$628$	0	0
$629$	−8.91778	−0.355575
$630$	0	0
$631$	45.8637	1.82581	0.912903	−	0.408176i	$-0.133835\pi$
$631$	45.8637	1.82581	0.912903	+	0.408176i	$0.133835\pi$
$632$	0	0
$633$	1.35790	0.0539719
$634$	0	0
$635$	−3.18427	−0.126364
$636$	0	0
$637$	5.66486	0.224450
$638$	0	0
$639$	7.88378	0.311878
$640$	0	0
$641$	21.7671	0.859748	0.429874	−	0.902889i	$-0.358558\pi$
$641$	21.7671	0.859748	0.429874	+	0.902889i	$0.358558\pi$
$642$	0	0
$643$	1.52696	0.0602175	0.0301088	−	0.999547i	$-0.490415\pi$
$643$	1.52696	0.0602175	0.0301088	+	0.999547i	$0.490415\pi$
$644$	0	0
$645$	−1.01187	−0.0398422
$646$	0	0
$647$	38.9091	1.52967	0.764836	−	0.644225i	$-0.222820\pi$
$647$	38.9091	1.52967	0.764836	+	0.644225i	$0.222820\pi$
$648$	0	0
$649$	11.0187	0.432522
$650$	0	0
$651$	−0.410649	−0.0160946
$652$	0	0
$653$	−20.5645	−0.804750	−0.402375	−	0.915475i	$-0.631815\pi$
$653$	−20.5645	−0.804750	−0.402375	+	0.915475i	$0.631815\pi$
$654$	0	0
$655$	−43.8789	−1.71449
$656$	0	0
$657$	36.2724	1.41512
$658$	0	0
$659$	−50.0671	−1.95034	−0.975169	−	0.221463i	$-0.928917\pi$
$659$	−50.0671	−1.95034	−0.975169	+	0.221463i	$0.928917\pi$
$660$	0	0
$661$	5.22780	0.203338	0.101669	−	0.994818i	$-0.467582\pi$
$661$	5.22780	0.203338	0.101669	+	0.994818i	$0.467582\pi$
$662$	0	0
$663$	0.145850	0.00566434
$664$	0	0
$665$	−18.6183	−0.721989
$666$	0	0
$667$	−22.3004	−0.863475
$668$	0	0
$669$	−0.443780	−0.0171575
$670$	0	0
$671$	−1.51658	−0.0585471
$672$	0	0
$673$	1.62268	0.0625497	0.0312748	−	0.999511i	$-0.490043\pi$
$673$	1.62268	0.0625497	0.0312748	+	0.999511i	$0.490043\pi$
$674$	0	0
$675$	0.234969	0.00904396
$676$	0	0
$677$	−37.5035	−1.44138	−0.720688	−	0.693259i	$-0.756174\pi$
$677$	−37.5035	−1.44138	−0.720688	+	0.693259i	$0.756174\pi$
$678$	0	0
$679$	0.00834467	0.000320239 0
$680$	0	0
$681$	2.17406	0.0833101
$682$	0	0
$683$	−25.1893	−0.963843	−0.481921	−	0.876214i	$-0.660061\pi$
$683$	−25.1893	−0.963843	−0.481921	+	0.876214i	$0.660061\pi$
$684$	0	0
$685$	−13.4975	−0.515711
$686$	0	0
$687$	−2.09027	−0.0797486
$688$	0	0
$689$	0.407900	0.0155398
$690$	0	0
$691$	−12.5056	−0.475737	−0.237868	−	0.971297i	$-0.576449\pi$
$691$	−12.5056	−0.475737	−0.237868	+	0.971297i	$0.576449\pi$
$692$	0	0
$693$	−3.45871	−0.131386
$694$	0	0
$695$	28.6883	1.08821
$696$	0	0
$697$	−2.43603	−0.0922714
$698$	0	0
$699$	−0.762190	−0.0288287
$700$	0	0
$701$	14.4303	0.545025	0.272512	−	0.962152i	$-0.412145\pi$
$701$	14.4303	0.545025	0.272512	+	0.962152i	$0.412145\pi$
$702$	0	0
$703$	37.9117	1.42987
$704$	0	0
$705$	−0.957445	−0.0360595
$706$	0	0
$707$	−20.9069	−0.786283
$708$	0	0
$709$	2.14339	0.0804967	0.0402483	−	0.999190i	$-0.487185\pi$
$709$	2.14339	0.0804967	0.0402483	+	0.999190i	$0.487185\pi$
$710$	0	0
$711$	20.7161	0.776915
$712$	0	0
$713$	−13.0530	−0.488839
$714$	0	0
$715$	2.12622	0.0795163
$716$	0	0
$717$	0.0859761	0.00321084
$718$	0	0
$719$	−28.8092	−1.07440	−0.537201	−	0.843454i	$-0.680518\pi$
$719$	−28.8092	−1.07440	−0.537201	+	0.843454i	$0.680518\pi$
$720$	0	0
$721$	−14.1263	−0.526090
$722$	0	0
$723$	1.24747	0.0463941
$724$	0	0
$725$	3.55586	0.132061
$726$	0	0
$727$	19.1139	0.708895	0.354447	−	0.935076i	$-0.384669\pi$
$727$	19.1139	0.708895	0.354447	+	0.935076i	$0.384669\pi$
$728$	0	0
$729$	−26.6392	−0.986636
$730$	0	0
$731$	−10.3684	−0.383490
$732$	0	0
$733$	17.8002	0.657464	0.328732	−	0.944423i	$-0.393379\pi$
$733$	17.8002	0.657464	0.328732	+	0.944423i	$0.393379\pi$
$734$	0	0
$735$	0.985490	0.0363504
$736$	0	0
$737$	5.69950	0.209944
$738$	0	0
$739$	−9.07208	−0.333722	−0.166861	−	0.985980i	$-0.553363\pi$
$739$	−9.07208	−0.333722	−0.166861	+	0.985980i	$0.553363\pi$
$740$	0	0
$741$	−0.620044	−0.0227779
$742$	0	0
$743$	−20.7249	−0.760322	−0.380161	−	0.924920i	$-0.624131\pi$
$743$	−20.7249	−0.760322	−0.380161	+	0.924920i	$0.624131\pi$
$744$	0	0
$745$	−19.8870	−0.728602
$746$	0	0
$747$	−17.3868	−0.636150
$748$	0	0
$749$	−0.936355	−0.0342137
$750$	0	0
$751$	8.63566	0.315120	0.157560	−	0.987509i	$-0.449637\pi$
$751$	8.63566	0.315120	0.157560	+	0.987509i	$0.449637\pi$
$752$	0	0
$753$	−1.96541	−0.0716234
$754$	0	0
$755$	27.9112	1.01579
$756$	0	0
$757$	−36.8692	−1.34003	−0.670017	−	0.742345i	$-0.733713\pi$
$757$	−36.8692	−1.34003	−0.670017	+	0.742345i	$0.733713\pi$
$758$	0	0
$759$	0.245874	0.00892465
$760$	0	0
$761$	15.4227	0.559072	0.279536	−	0.960135i	$-0.409819\pi$
$761$	15.4227	0.559072	0.279536	+	0.960135i	$0.409819\pi$
$762$	0	0
$763$	−3.87076	−0.140131
$764$	0	0
$765$	−11.3452	−0.410187
$766$	0	0
$767$	−11.0187	−0.397863
$768$	0	0
$769$	36.5443	1.31782	0.658911	−	0.752221i	$-0.271017\pi$
$769$	36.5443	1.31782	0.658911	+	0.752221i	$0.271017\pi$
$770$	0	0
$771$	−2.01588	−0.0726001
$772$	0	0
$773$	−50.1473	−1.80367	−0.901836	−	0.432078i	$-0.857780\pi$
$773$	−50.1473	−1.80367	−0.901836	+	0.432078i	$0.857780\pi$
$774$	0	0
$775$	2.08134	0.0747639
$776$	0	0
$777$	0.472957	0.0169673
$778$	0	0
$779$	10.3562	0.371049
$780$	0	0
$781$	−2.63381	−0.0942450
$782$	0	0
$783$	3.63894	0.130045
$784$	0	0
$785$	30.9837	1.10586
$786$	0	0
$787$	−35.6537	−1.27092	−0.635459	−	0.772135i	$-0.719189\pi$
$787$	−35.6537	−1.27092	−0.635459	+	0.772135i	$0.719189\pi$
$788$	0	0
$789$	−0.438997	−0.0156287
$790$	0	0
$791$	11.3158	0.402345
$792$	0	0
$793$	1.51658	0.0538555
$794$	0	0
$795$	0.0709606	0.00251671
$796$	0	0
$797$	−16.1666	−0.572652	−0.286326	−	0.958132i	$-0.592434\pi$
$797$	−16.1666	−0.572652	−0.286326	+	0.958132i	$0.592434\pi$
$798$	0	0
$799$	−9.81077	−0.347080
$800$	0	0
$801$	5.56819	0.196742
$802$	0	0
$803$	−12.1178	−0.427629
$804$	0	0
$805$	−7.38296	−0.260215
$806$	0	0
$807$	−1.57076	−0.0552933
$808$	0	0
$809$	−14.3356	−0.504012	−0.252006	−	0.967726i	$-0.581090\pi$
$809$	−14.3356	−0.504012	−0.252006	+	0.967726i	$0.581090\pi$
$810$	0	0
$811$	−53.2835	−1.87104	−0.935518	−	0.353278i	$-0.885067\pi$
$811$	−53.2835	−1.87104	−0.935518	+	0.353278i	$0.885067\pi$
$812$	0	0
$813$	−0.358605	−0.0125768
$814$	0	0
$815$	1.51398	0.0530323
$816$	0	0
$817$	44.0787	1.54212
$818$	0	0
$819$	3.45871	0.120857
$820$	0	0
$821$	2.87576	0.100365	0.0501823	−	0.998740i	$-0.484020\pi$
$821$	2.87576	0.100365	0.0501823	+	0.998740i	$0.484020\pi$
$822$	0	0
$823$	38.8566	1.35446	0.677228	−	0.735773i	$-0.263181\pi$
$823$	38.8566	1.35446	0.677228	+	0.735773i	$0.263181\pi$
$824$	0	0
$825$	−0.0392052	−0.00136495
$826$	0	0
$827$	6.52970	0.227060	0.113530	−	0.993535i	$-0.463784\pi$
$827$	6.52970	0.227060	0.113530	+	0.993535i	$0.463784\pi$
$828$	0	0
$829$	51.7728	1.79814	0.899071	−	0.437802i	$-0.144243\pi$
$829$	51.7728	1.79814	0.899071	+	0.437802i	$0.144243\pi$
$830$	0	0
$831$	0.228978	0.00794317
$832$	0	0
$833$	10.0981	0.349880
$834$	0	0
$835$	−37.2580	−1.28937
$836$	0	0
$837$	2.12997	0.0736225
$838$	0	0
$839$	−18.5912	−0.641841	−0.320921	−	0.947106i	$-0.603992\pi$
$839$	−18.5912	−0.641841	−0.320921	+	0.947106i	$0.603992\pi$
$840$	0	0
$841$	26.0692	0.898938
$842$	0	0
$843$	−1.31864	−0.0454164
$844$	0	0
$845$	−2.12622	−0.0731443
$846$	0	0
$847$	1.15548	0.0397029
$848$	0	0
$849$	−2.35008	−0.0806545
$850$	0	0
$851$	15.0336	0.515345
$852$	0	0
$853$	31.5418	1.07997	0.539985	−	0.841674i	$-0.318430\pi$
$853$	31.5418	1.07997	0.539985	+	0.841674i	$0.318430\pi$
$854$	0	0
$855$	48.2313	1.64947
$856$	0	0
$857$	−23.8577	−0.814965	−0.407483	−	0.913213i	$-0.633593\pi$
$857$	−23.8577	−0.814965	−0.407483	+	0.913213i	$0.633593\pi$
$858$	0	0
$859$	−3.72951	−0.127249	−0.0636247	−	0.997974i	$-0.520266\pi$
$859$	−3.72951	−0.127249	−0.0636247	+	0.997974i	$0.520266\pi$
$860$	0	0
$861$	0.129196	0.00440298
$862$	0	0
$863$	14.2870	0.486337	0.243168	−	0.969984i	$-0.421813\pi$
$863$	14.2870	0.486337	0.243168	+	0.969984i	$0.421813\pi$
$864$	0	0
$865$	−32.1482	−1.09307
$866$	0	0
$867$	−1.13093	−0.0384084
$868$	0	0
$869$	−6.92082	−0.234773
$870$	0	0
$871$	−5.69950	−0.193120
$872$	0	0
$873$	−0.0216171	−0.000731627 0
$874$	0	0
$875$	13.4613	0.455075
$876$	0	0
$877$	55.5050	1.87427	0.937135	−	0.348967i	$-0.113467\pi$
$877$	55.5050	1.87427	0.937135	+	0.348967i	$0.113467\pi$
$878$	0	0
$879$	1.07679	0.0363191
$880$	0	0
$881$	−51.5573	−1.73701	−0.868505	−	0.495681i	$-0.834919\pi$
$881$	−51.5573	−1.73701	−0.868505	+	0.495681i	$0.834919\pi$
$882$	0	0
$883$	37.1539	1.25033	0.625164	−	0.780494i	$-0.285032\pi$
$883$	37.1539	1.25033	0.625164	+	0.780494i	$0.285032\pi$
$884$	0	0
$885$	−1.91688	−0.0644351
$886$	0	0
$887$	51.9115	1.74302	0.871508	−	0.490381i	$-0.163142\pi$
$887$	51.9115	1.74302	0.871508	+	0.490381i	$0.163142\pi$
$888$	0	0
$889$	1.73047	0.0580382
$890$	0	0
$891$	8.93980	0.299494
$892$	0	0
$893$	41.7080	1.39570
$894$	0	0
$895$	23.1132	0.772589
$896$	0	0
$897$	−0.245874	−0.00820949
$898$	0	0
$899$	32.2335	1.07505
$900$	0	0
$901$	0.727120	0.0242239
$902$	0	0
$903$	0.549892	0.0182993
$904$	0	0
$905$	25.8352	0.858792
$906$	0	0
$907$	−50.4270	−1.67440	−0.837201	−	0.546896i	$-0.815809\pi$
$907$	−50.4270	−1.67440	−0.837201	+	0.546896i	$0.815809\pi$
$908$	0	0
$909$	54.1597	1.79636
$910$	0	0
$911$	58.3276	1.93248	0.966239	−	0.257646i	$-0.0829468\pi$
$911$	58.3276	1.93248	0.966239	+	0.257646i	$0.0829468\pi$
$912$	0	0
$913$	5.80856	0.192235
$914$	0	0
$915$	0.263833	0.00872206
$916$	0	0
$917$	23.8457	0.787455
$918$	0	0
$919$	31.4216	1.03650	0.518251	−	0.855229i	$-0.326583\pi$
$919$	31.4216	1.03650	0.518251	+	0.855229i	$0.326583\pi$
$920$	0	0
$921$	−1.15154	−0.0379447
$922$	0	0
$923$	2.63381	0.0866928
$924$	0	0
$925$	−2.39715	−0.0788176
$926$	0	0
$927$	36.5945	1.20192
$928$	0	0
$929$	−3.61994	−0.118767	−0.0593833	−	0.998235i	$-0.518913\pi$
$929$	−3.61994	−0.118767	−0.0593833	+	0.998235i	$0.518913\pi$
$930$	0	0
$931$	−42.9297	−1.40696
$932$	0	0
$933$	−1.59170	−0.0521100
$934$	0	0
$935$	3.79019	0.123953
$936$	0	0
$937$	1.27442	0.0416335	0.0208168	−	0.999783i	$-0.493373\pi$
$937$	1.27442	0.0416335	0.0208168	+	0.999783i	$0.493373\pi$
$938$	0	0
$939$	1.38905	0.0453300
$940$	0	0
$941$	−48.4927	−1.58082	−0.790408	−	0.612581i	$-0.790131\pi$
$941$	−48.4927	−1.58082	−0.790408	+	0.612581i	$0.790131\pi$
$942$	0	0
$943$	4.10666	0.133731
$944$	0	0
$945$	1.20474	0.0391902
$946$	0	0
$947$	−17.4674	−0.567615	−0.283808	−	0.958881i	$-0.591598\pi$
$947$	−17.4674	−0.567615	−0.283808	+	0.958881i	$0.591598\pi$
$948$	0	0
$949$	12.1178	0.393362
$950$	0	0
$951$	−0.710529	−0.0230405
$952$	0	0
$953$	−26.0324	−0.843274	−0.421637	−	0.906765i	$-0.638544\pi$
$953$	−26.0324	−0.843274	−0.421637	+	0.906765i	$0.638544\pi$
$954$	0	0
$955$	46.8295	1.51537
$956$	0	0
$957$	−0.607167	−0.0196269
$958$	0	0
$959$	7.33510	0.236863
$960$	0	0
$961$	−12.1329	−0.391383
$962$	0	0
$963$	2.42565	0.0781655
$964$	0	0
$965$	45.3773	1.46075
$966$	0	0
$967$	−48.7405	−1.56739	−0.783694	−	0.621147i	$-0.786667\pi$
$967$	−48.7405	−1.56739	−0.783694	+	0.621147i	$0.786667\pi$
$968$	0	0
$969$	−1.10529	−0.0355069
$970$	0	0
$971$	−40.5560	−1.30150	−0.650751	−	0.759291i	$-0.725546\pi$
$971$	−40.5560	−1.30150	−0.650751	+	0.759291i	$0.725546\pi$
$972$	0	0
$973$	−15.5905	−0.499807
$974$	0	0
$975$	0.0392052	0.00125557
$976$	0	0
$977$	−20.0816	−0.642466	−0.321233	−	0.947000i	$-0.604097\pi$
$977$	−20.0816	−0.642466	−0.321233	+	0.947000i	$0.604097\pi$
$978$	0	0
$979$	−1.86021	−0.0594527
$980$	0	0
$981$	10.0273	0.320147
$982$	0	0
$983$	−25.8729	−0.825219	−0.412609	−	0.910908i	$-0.635383\pi$
$983$	−25.8729	−0.825219	−0.412609	+	0.910908i	$0.635383\pi$
$984$	0	0
$985$	9.14578	0.291409
$986$	0	0
$987$	0.520317	0.0165619
$988$	0	0
$989$	17.4791	0.555802
$990$	0	0
$991$	15.5782	0.494858	0.247429	−	0.968906i	$-0.420414\pi$
$991$	15.5782	0.494858	0.247429	+	0.968906i	$0.420414\pi$
$992$	0	0
$993$	−2.24432	−0.0712213
$994$	0	0
$995$	12.6366	0.400606
$996$	0	0
$997$	35.9606	1.13888	0.569442	−	0.822031i	$-0.307159\pi$
$997$	35.9606	1.13888	0.569442	+	0.822031i	$0.307159\pi$
$998$	0	0
$999$	−2.45315	−0.0776144

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9152.2.a.cv.1.5		8
4.3	odd	2		9152.2.a.cw.1.4		8
8.3	odd	2		4576.2.a.t.1.5	✓	8
8.5	even	2		4576.2.a.u.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4576.2.a.t.1.5	✓	8	8.3	odd	2
4576.2.a.u.1.4	yes	8	8.5	even	2
9152.2.a.cv.1.5		8	1.1	even	1	trivial
9152.2.a.cw.1.4		8	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9152 = 2^{6} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9152.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0818190 q^{3} -2.12622 q^{5} +1.15548 q^{7} -2.99331 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.173966 q^{15} -1.78259 q^{17} +7.57824 q^{19} +0.0945404 q^{21} +3.00509 q^{23} -0.479170 q^{25} -0.490366 q^{27} -7.42086 q^{29} -4.34363 q^{31} +0.0818190 q^{33} -2.45682 q^{35} +5.00270 q^{37} -0.0818190 q^{39} +1.36657 q^{41} +5.81648 q^{43} +6.36444 q^{45} +5.50365 q^{47} -5.66486 q^{49} -0.145850 q^{51} -0.407900 q^{53} -2.12622 q^{55} +0.620044 q^{57} +11.0187 q^{59} -1.51658 q^{61} -3.45871 q^{63} +2.12622 q^{65} +5.69950 q^{67} +0.245874 q^{69} -2.63381 q^{71} -12.1178 q^{73} -0.0392052 q^{75} +1.15548 q^{77} -6.92082 q^{79} +8.93980 q^{81} +5.80856 q^{83} +3.79019 q^{85} -0.607167 q^{87} -1.86021 q^{89} -1.15548 q^{91} -0.355391 q^{93} -16.1130 q^{95} +0.00722180 q^{97} -2.99331 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} + q^{5} - 4 q^{7} + 9 q^{9} + 8 q^{11} - 8 q^{13} - 13 q^{15} - 8 q^{17} - 2 q^{19} + 12 q^{21} - 17 q^{23} + 23 q^{25} + 5 q^{27} - 23 q^{31} - q^{33} + 2 q^{35} + q^{37} + q^{39} - 14 q^{41}+ \cdots + 9 q^{99}+O(q^{100}) \) 8 * q - q^3 + q^5 - 4 * q^7 + 9 * q^9 + 8 * q^11 - 8 * q^13 - 13 * q^15 - 8 * q^17 - 2 * q^19 + 12 * q^21 - 17 * q^23 + 23 * q^25 + 5 * q^27 - 23 * q^31 - q^33 + 2 * q^35 + q^37 + q^39 - 14 * q^41 - 10 * q^43 - 4 * q^45 - 28 * q^47 - 4 * q^49 - 6 * q^51 + 14 * q^53 + q^55 - 36 * q^57 + 31 * q^59 - 16 * q^63 - q^65 - 21 * q^67 - 13 * q^69 - 61 * q^71 + 22 * q^75 - 4 * q^77 + 16 * q^81 + 22 * q^83 + 22 * q^85 - 10 * q^87 + 9 * q^89 + 4 * q^91 - 3 * q^93 - 22 * q^95 - 11 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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