LMFDB - Embedded newform 1664.2.b.j.833.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(833,1664)] 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("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.11574317056.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 45x^{4} + 4 $ x^8 + 45*x^4 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.3
Root			$0.386289 + 0.386289i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.j.833.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81616i q^{3} -2.70156i q^{5} +2.58874 q^{7} -0.298438 q^{9} +0.772577i q^{11} -1.00000i q^{13} -4.90647 q^{15} -0.701562 q^{17} -4.40490i q^{19} -4.70156i q^{21} +3.63232 q^{23} -2.29844 q^{25} -4.90647i q^{27} -2.00000i q^{29} +5.95005 q^{31} +1.40312 q^{33} -6.99364i q^{35} -6.70156i q^{37} -1.81616 q^{39} -11.4031 q^{41} +10.0839i q^{43} +0.806248i q^{45} +9.31137 q^{47} -0.298438 q^{49} +1.27415i q^{51} +1.40312i q^{53} +2.08717 q^{55} -8.00000 q^{57} +2.85974i q^{59} -0.772577 q^{63} -2.70156 q^{65} +12.6727i q^{67} -6.59688i q^{69} -1.04358 q^{71} -7.40312 q^{73} +4.17433i q^{75} +2.00000i q^{77} -2.08717 q^{79} -9.80625 q^{81} -9.58237i q^{83} +1.89531i q^{85} -3.63232 q^{87} -11.4031 q^{89} -2.58874i q^{91} -10.8062i q^{93} -11.9001 q^{95} +10.0000 q^{97} -0.230566i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 28 q^{9} + 20 q^{17} - 44 q^{25} - 40 q^{33} - 40 q^{41} - 28 q^{49} - 64 q^{57} + 4 q^{65} - 8 q^{73} + 24 q^{81} - 40 q^{89} + 80 q^{97}+O(q^{100}) $ 8 * q - 28 * q^9 + 20 * q^17 - 44 * q^25 - 40 * q^33 - 40 * q^41 - 28 * q^49 - 64 * q^57 + 4 * q^65 - 8 * q^73 + 24 * q^81 - 40 * q^89 + 80 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

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

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

$2$	0	0
$2$	0	0
$3$	− 1.81616i	− 1.04856i	−0.851546	−	0.524280i	$-0.824334\pi$
$3$	− 1.81616i	− 1.04856i	0.851546	−	0.524280i	$-0.175666\pi$
$4$	0	0
$5$	− 2.70156i	− 1.20818i	−0.796918	−	0.604088i	$-0.793538\pi$
$5$	− 2.70156i	− 1.20818i	0.796918	−	0.604088i	$-0.206462\pi$
$6$	0	0
$7$	2.58874	0.978451	0.489225	−	0.872157i	$-0.337280\pi$
$7$	2.58874	0.978451	0.489225	+	0.872157i	$0.337280\pi$
$8$	0	0
$9$	−0.298438	−0.0994793
$10$	0	0
$11$	0.772577i	0.232941i	0.993194	+	0.116470i	$0.0371580\pi$
$11$	0.772577i	0.232941i	−0.993194	+	0.116470i	$0.962842\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−4.90647	−1.26685
$16$	0	0
$17$	−0.701562	−0.170154	−0.0850769	−	0.996374i	$-0.527114\pi$
$17$	−0.701562	−0.170154	−0.0850769	+	0.996374i	$0.527114\pi$
$18$	0	0
$19$	− 4.40490i	− 1.01055i	−0.862958	−	0.505276i	$-0.831391\pi$
$19$	− 4.40490i	− 1.01055i	0.862958	−	0.505276i	$-0.168609\pi$
$20$	0	0
$21$	− 4.70156i	− 1.02596i
$22$	0	0
$23$	3.63232	0.757391	0.378696	−	0.925521i	$-0.376373\pi$
$23$	3.63232	0.757391	0.378696	+	0.925521i	$0.376373\pi$
$24$	0	0
$25$	−2.29844	−0.459688
$26$	0	0
$27$	− 4.90647i	− 0.944251i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	5.95005	1.06866	0.534330	−	0.845276i	$-0.320564\pi$
$31$	5.95005	1.06866	0.534330	+	0.845276i	$0.320564\pi$
$32$	0	0
$33$	1.40312	0.244253
$34$	0	0
$35$	− 6.99364i	− 1.18214i
$36$	0	0
$37$	− 6.70156i	− 1.10173i	−0.834594	−	0.550865i	$-0.814298\pi$
$37$	− 6.70156i	− 1.10173i	0.834594	−	0.550865i	$-0.185702\pi$
$38$	0	0
$39$	−1.81616	−0.290818
$40$	0	0
$41$	−11.4031	−1.78087	−0.890434	−	0.455112i	$-0.849599\pi$
$41$	−11.4031	−1.78087	−0.890434	+	0.455112i	$0.849599\pi$
$42$	0	0
$43$	10.0839i	1.53779i	0.639377	+	0.768894i	$0.279192\pi$
$43$	10.0839i	1.53779i	−0.639377	+	0.768894i	$0.720808\pi$
$44$	0	0
$45$	0.806248i	0.120188i
$46$	0	0
$47$	9.31137	1.35820	0.679101	−	0.734045i	$-0.262370\pi$
$47$	9.31137	1.35820	0.679101	+	0.734045i	$0.262370\pi$
$48$	0	0
$49$	−0.298438	−0.0426340
$50$	0	0
$51$	1.27415i	0.178417i
$52$	0	0
$53$	1.40312i	0.192734i	0.995346	+	0.0963670i	$0.0307222\pi$
$53$	1.40312i	0.192734i	−0.995346	+	0.0963670i	$0.969278\pi$
$54$	0	0
$55$	2.08717	0.281433
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	2.85974i	0.372307i	0.982521	+	0.186153i	$0.0596021\pi$
$59$	2.85974i	0.372307i	−0.982521	+	0.186153i	$0.940398\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−0.772577	−0.0973356
$64$	0	0
$65$	−2.70156	−0.335088
$66$	0	0
$67$	12.6727i	1.54821i	0.633054	+	0.774107i	$0.281801\pi$
$67$	12.6727i	1.54821i	−0.633054	+	0.774107i	$0.718199\pi$
$68$	0	0
$69$	− 6.59688i	− 0.794171i
$70$	0	0
$71$	−1.04358	−0.123850	−0.0619252	−	0.998081i	$-0.519724\pi$
$71$	−1.04358	−0.123850	−0.0619252	+	0.998081i	$0.519724\pi$
$72$	0	0
$73$	−7.40312	−0.866470	−0.433235	−	0.901281i	$-0.642628\pi$
$73$	−7.40312	−0.866470	−0.433235	+	0.901281i	$0.642628\pi$
$74$	0	0
$75$	4.17433i	0.482010i
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	−2.08717	−0.234824	−0.117412	−	0.993083i	$-0.537460\pi$
$79$	−2.08717	−0.234824	−0.117412	+	0.993083i	$0.537460\pi$
$80$	0	0
$81$	−9.80625	−1.08958
$82$	0	0
$83$	− 9.58237i	− 1.05180i	−0.850546	−	0.525901i	$-0.823728\pi$
$83$	− 9.58237i	− 1.05180i	0.850546	−	0.525901i	$-0.176272\pi$
$84$	0	0
$85$	1.89531i	0.205576i
$86$	0	0
$87$	−3.63232	−0.389426
$88$	0	0
$89$	−11.4031	−1.20873	−0.604364	−	0.796708i	$-0.706573\pi$
$89$	−11.4031	−1.20873	−0.604364	+	0.796708i	$0.706573\pi$
$90$	0	0
$91$	− 2.58874i	− 0.271373i
$92$	0	0
$93$	− 10.8062i	− 1.12056i
$94$	0	0
$95$	−11.9001	−1.22093
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	− 0.230566i	− 0.0231728i
$100$	0	0
$101$	6.80625i	0.677247i	0.940922	+	0.338624i	$0.109961\pi$
$101$	6.80625i	0.677247i	−0.940922	+	0.338624i	$0.890039\pi$
$102$	0	0
$103$	−13.9873	−1.37821	−0.689103	−	0.724663i	$-0.741995\pi$
$103$	−13.9873	−1.37821	−0.689103	+	0.724663i	$0.741995\pi$
$104$	0	0
$105$	−12.7016	−1.23955
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	− 1.29844i	− 0.124368i	−0.998065	−	0.0621839i	$-0.980193\pi$
$109$	− 1.29844i	− 0.124368i	0.998065	−	0.0621839i	$-0.0198065\pi$
$110$	0	0
$111$	−12.1711	−1.15523
$112$	0	0
$113$	−1.40312	−0.131995	−0.0659974	−	0.997820i	$-0.521023\pi$
$113$	−1.40312	−0.131995	−0.0659974	+	0.997820i	$0.521023\pi$
$114$	0	0
$115$	− 9.81294i	− 0.915061i
$116$	0	0
$117$	0.298438i	0.0275906i
$118$	0	0
$119$	−1.81616	−0.166487
$120$	0	0
$121$	10.4031	0.945739
$122$	0	0
$123$	20.7099i	1.86735i
$124$	0	0
$125$	− 7.29844i	− 0.652792i
$126$	0	0
$127$	10.8970	0.966949	0.483474	−	0.875358i	$-0.339375\pi$
$127$	10.8970	0.966949	0.483474	+	0.875358i	$0.339375\pi$
$128$	0	0
$129$	18.3141	1.61246
$130$	0	0
$131$	4.90647i	0.428680i	0.976759	+	0.214340i	$0.0687601\pi$
$131$	4.90647i	0.428680i	−0.976759	+	0.214340i	$0.931240\pi$
$132$	0	0
$133$	− 11.4031i	− 0.988776i
$134$	0	0
$135$	−13.2551	−1.14082
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	− 12.1711i	− 1.03234i	−0.856486	−	0.516170i	$-0.827357\pi$
$139$	− 12.1711i	− 1.03234i	0.856486	−	0.516170i	$-0.172643\pi$
$140$	0	0
$141$	− 16.9109i	− 1.42416i
$142$	0	0
$143$	0.772577	0.0646062
$144$	0	0
$145$	−5.40312	−0.448705
$146$	0	0
$147$	0.542011i	0.0447043i
$148$	0	0
$149$	12.8062i	1.04913i	0.851371	+	0.524564i	$0.175772\pi$
$149$	12.8062i	1.04913i	−0.851371	+	0.524564i	$0.824228\pi$
$150$	0	0
$151$	23.8406	1.94012	0.970062	−	0.242856i	$-0.0780844\pi$
$151$	23.8406	1.94012	0.970062	+	0.242856i	$0.0780844\pi$
$152$	0	0
$153$	0.209373	0.0169268
$154$	0	0
$155$	− 16.0744i	− 1.29113i
$156$	0	0
$157$	4.59688i	0.366871i	0.983032	+	0.183435i	$0.0587218\pi$
$157$	4.59688i	0.366871i	−0.983032	+	0.183435i	$0.941278\pi$
$158$	0	0
$159$	2.54830	0.202093
$160$	0	0
$161$	9.40312	0.741070
$162$	0	0
$163$	16.8470i	1.31956i	0.751459	+	0.659780i	$0.229351\pi$
$163$	16.8470i	1.31956i	−0.751459	+	0.659780i	$0.770649\pi$
$164$	0	0
$165$	− 3.79063i	− 0.295100i
$166$	0	0
$167$	−14.7598	−1.14215	−0.571076	−	0.820897i	$-0.693474\pi$
$167$	−14.7598	−1.14215	−0.571076	+	0.820897i	$0.693474\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	1.31459i	0.100529i
$172$	0	0
$173$	− 21.4031i	− 1.62725i	−0.581390	−	0.813625i	$-0.697491\pi$
$173$	− 21.4031i	− 1.62725i	0.581390	−	0.813625i	$-0.302509\pi$
$174$	0	0
$175$	−5.95005	−0.449782
$176$	0	0
$177$	5.19375	0.390386
$178$	0	0
$179$	24.0712i	1.79917i	0.436749	+	0.899584i	$0.356130\pi$
$179$	24.0712i	1.79917i	−0.436749	+	0.899584i	$0.643870\pi$
$180$	0	0
$181$	14.0000i	1.04061i	0.853980	+	0.520306i	$0.174182\pi$
$181$	14.0000i	1.04061i	−0.853980	+	0.520306i	$0.825818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−18.1047	−1.33108
$186$	0	0
$187$	− 0.542011i	− 0.0396358i
$188$	0	0
$189$	− 12.7016i	− 0.923903i
$190$	0	0
$191$	5.17748	0.374629	0.187314	−	0.982300i	$-0.440022\pi$
$191$	5.17748	0.374629	0.187314	+	0.982300i	$0.440022\pi$
$192$	0	0
$193$	20.8062	1.49767	0.748833	−	0.662758i	$-0.230614\pi$
$193$	20.8062	1.49767	0.748833	+	0.662758i	$0.230614\pi$
$194$	0	0
$195$	4.90647i	0.351360i
$196$	0	0
$197$	− 0.104686i	− 0.00745859i	−0.999993	−	0.00372930i	$-0.998813\pi$
$197$	− 0.104686i	− 0.00745859i	0.999993	−	0.00372930i	$-0.00118707\pi$
$198$	0	0
$199$	−6.72263	−0.476555	−0.238277	−	0.971197i	$-0.576583\pi$
$199$	−6.72263	−0.476555	−0.238277	+	0.971197i	$0.576583\pi$
$200$	0	0
$201$	23.0156	1.62340
$202$	0	0
$203$	− 5.17748i	− 0.363388i
$204$	0	0
$205$	30.8062i	2.15160i
$206$	0	0
$207$	−1.08402	−0.0753447
$208$	0	0
$209$	3.40312	0.235399
$210$	0	0
$211$	− 8.53879i	− 0.587835i	−0.955831	−	0.293917i	$-0.905041\pi$
$211$	− 8.53879i	− 0.587835i	0.955831	−	0.293917i	$-0.0949590\pi$
$212$	0	0
$213$	1.89531i	0.129865i
$214$	0	0
$215$	27.2424	1.85792
$216$	0	0
$217$	15.4031	1.04563
$218$	0	0
$219$	13.4453i	0.908546i
$220$	0	0
$221$	0.701562i	0.0471922i
$222$	0	0
$223$	20.2083	1.35325	0.676625	−	0.736328i	$-0.263442\pi$
$223$	20.2083	1.35325	0.676625	+	0.736328i	$0.263442\pi$
$224$	0	0
$225$	0.685941	0.0457294
$226$	0	0
$227$	− 0.230566i	− 0.0153032i	−0.999971	−	0.00765161i	$-0.997564\pi$
$227$	− 0.230566i	− 0.0153032i	0.999971	−	0.00765161i	$-0.00243561\pi$
$228$	0	0
$229$	− 22.9109i	− 1.51400i	−0.653417	−	0.756999i	$-0.726665\pi$
$229$	− 22.9109i	− 1.51400i	0.653417	−	0.756999i	$-0.273335\pi$
$230$	0	0
$231$	3.63232	0.238989
$232$	0	0
$233$	−21.5078	−1.40902	−0.704512	−	0.709692i	$-0.748834\pi$
$233$	−21.5078	−1.40902	−0.704512	+	0.709692i	$0.748834\pi$
$234$	0	0
$235$	− 25.1552i	− 1.64095i
$236$	0	0
$237$	3.79063i	0.246228i
$238$	0	0
$239$	−27.9341	−1.80691	−0.903453	−	0.428686i	$-0.858977\pi$
$239$	−27.9341	−1.80691	−0.903453	+	0.428686i	$0.858977\pi$
$240$	0	0
$241$	15.4031	0.992202	0.496101	−	0.868265i	$-0.334764\pi$
$241$	15.4031	0.992202	0.496101	+	0.868265i	$0.334764\pi$
$242$	0	0
$243$	3.09031i	0.198243i
$244$	0	0
$245$	0.806248i	0.0515093i
$246$	0	0
$247$	−4.40490	−0.280277
$248$	0	0
$249$	−17.4031	−1.10288
$250$	0	0
$251$	− 26.4294i	− 1.66821i	−0.551607	−	0.834104i	$-0.685985\pi$
$251$	− 26.4294i	− 1.66821i	0.551607	−	0.834104i	$-0.314015\pi$
$252$	0	0
$253$	2.80625i	0.176427i
$254$	0	0
$255$	3.44219	0.215559
$256$	0	0
$257$	10.7016	0.667545	0.333773	−	0.942654i	$-0.391678\pi$
$257$	10.7016	0.667545	0.333773	+	0.942654i	$0.391678\pi$
$258$	0	0
$259$	− 17.3486i	− 1.07799i
$260$	0	0
$261$	0.596876i	0.0369457i
$262$	0	0
$263$	19.1647	1.18175	0.590874	−	0.806764i	$-0.298783\pi$
$263$	19.1647	1.18175	0.590874	+	0.806764i	$0.298783\pi$
$264$	0	0
$265$	3.79063	0.232856
$266$	0	0
$267$	20.7099i	1.26743i
$268$	0	0
$269$	27.4031i	1.67080i	0.549644	+	0.835399i	$0.314763\pi$
$269$	27.4031i	1.67080i	−0.549644	+	0.835399i	$0.685237\pi$
$270$	0	0
$271$	23.8406	1.44822	0.724108	−	0.689686i	$-0.242252\pi$
$271$	23.8406	1.44822	0.724108	+	0.689686i	$0.242252\pi$
$272$	0	0
$273$	−4.70156	−0.284551
$274$	0	0
$275$	− 1.77572i	− 0.107080i
$276$	0	0
$277$	12.0000i	0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$	12.0000i	0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$	0	0
$279$	−1.77572	−0.106310
$280$	0	0
$281$	11.4031	0.680253	0.340127	−	0.940380i	$-0.389530\pi$
$281$	11.4031	0.680253	0.340127	+	0.940380i	$0.389530\pi$
$282$	0	0
$283$	− 1.54515i	− 0.0918499i	−0.998945	−	0.0459250i	$-0.985376\pi$
$283$	− 1.54515i	− 0.0918499i	0.998945	−	0.0459250i	$-0.0146235\pi$
$284$	0	0
$285$	21.6125i	1.28021i
$286$	0	0
$287$	−29.5197	−1.74249
$288$	0	0
$289$	−16.5078	−0.971048
$290$	0	0
$291$	− 18.1616i	− 1.06465i
$292$	0	0
$293$	8.10469i	0.473481i	0.971573	+	0.236740i	$0.0760791\pi$
$293$	8.10469i	0.473481i	−0.971573	+	0.236740i	$0.923921\pi$
$294$	0	0
$295$	7.72577	0.449812
$296$	0	0
$297$	3.79063	0.219955
$298$	0	0
$299$	− 3.63232i	− 0.210063i
$300$	0	0
$301$	26.1047i	1.50465i
$302$	0	0
$303$	12.3612	0.710135
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.5696i	1.34519i	0.740011	+	0.672595i	$0.234820\pi$
$307$	23.5696i	1.34519i	−0.740011	+	0.672595i	$0.765180\pi$
$308$	0	0
$309$	25.4031i	1.44513i
$310$	0	0
$311$	23.8002	1.34959	0.674793	−	0.738007i	$-0.264233\pi$
$311$	23.8002	1.34959	0.674793	+	0.738007i	$0.264233\pi$
$312$	0	0
$313$	6.10469	0.345057	0.172529	−	0.985005i	$-0.444806\pi$
$313$	6.10469	0.345057	0.172529	+	0.985005i	$0.444806\pi$
$314$	0	0
$315$	2.08717i	0.117598i
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	1.54515	0.0865121
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.09031i	0.171949i
$324$	0	0
$325$	2.29844i	0.127494i
$326$	0	0
$327$	−2.35817	−0.130407
$328$	0	0
$329$	24.1047	1.32893
$330$	0	0
$331$	− 18.3922i	− 1.01093i	−0.862849	−	0.505463i	$-0.831322\pi$
$331$	− 18.3922i	− 1.01093i	0.862849	−	0.505463i	$-0.168678\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	34.2360	1.87051
$336$	0	0
$337$	27.5078	1.49845	0.749223	−	0.662318i	$-0.230427\pi$
$337$	27.5078	1.49845	0.749223	+	0.662318i	$0.230427\pi$
$338$	0	0
$339$	2.54830i	0.138405i
$340$	0	0
$341$	4.59688i	0.248935i
$342$	0	0
$343$	−18.8937	−1.02017
$344$	0	0
$345$	−17.8219	−0.959497
$346$	0	0
$347$	− 16.3454i	− 0.877469i	−0.898617	−	0.438735i	$-0.855427\pi$
$347$	− 16.3454i	− 0.877469i	0.898617	−	0.438735i	$-0.144573\pi$
$348$	0	0
$349$	2.70156i	0.144611i	0.997383	+	0.0723057i	$0.0230357\pi$
$349$	2.70156i	0.144611i	−0.997383	+	0.0723057i	$0.976964\pi$
$350$	0	0
$351$	−4.90647	−0.261888
$352$	0	0
$353$	−24.8062	−1.32030	−0.660152	−	0.751132i	$-0.729508\pi$
$353$	−24.8062	−1.32030	−0.660152	+	0.751132i	$0.729508\pi$
$354$	0	0
$355$	2.81930i	0.149633i
$356$	0	0
$357$	3.29844i	0.174572i
$358$	0	0
$359$	−5.95005	−0.314032	−0.157016	−	0.987596i	$-0.550187\pi$
$359$	−5.95005	−0.314032	−0.157016	+	0.987596i	$0.550187\pi$
$360$	0	0
$361$	−0.403124	−0.0212171
$362$	0	0
$363$	− 18.8937i	− 0.991664i
$364$	0	0
$365$	20.0000i	1.04685i
$366$	0	0
$367$	−36.2423	−1.89183	−0.945917	−	0.324409i	$-0.894835\pi$
$367$	−36.2423	−1.89183	−0.945917	+	0.324409i	$0.894835\pi$
$368$	0	0
$369$	3.40312	0.177160
$370$	0	0
$371$	3.63232i	0.188581i
$372$	0	0
$373$	− 3.40312i	− 0.176207i	−0.996111	−	0.0881035i	$-0.971919\pi$
$373$	− 3.40312i	− 0.176207i	0.996111	−	0.0881035i	$-0.0280806\pi$
$374$	0	0
$375$	−13.2551	−0.684492
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	− 3.86289i	− 0.198423i	−0.995066	−	0.0992116i	$-0.968368\pi$
$379$	− 3.86289i	− 0.198423i	0.995066	−	0.0992116i	$-0.0316321\pi$
$380$	0	0
$381$	− 19.7906i	− 1.01390i
$382$	0	0
$383$	16.0340	0.819299	0.409649	−	0.912243i	$-0.365651\pi$
$383$	16.0340	0.819299	0.409649	+	0.912243i	$0.365651\pi$
$384$	0	0
$385$	5.40312	0.275369
$386$	0	0
$387$	− 3.00943i	− 0.152978i
$388$	0	0
$389$	− 19.4031i	− 0.983777i	−0.870658	−	0.491889i	$-0.836307\pi$
$389$	− 19.4031i	− 0.983777i	0.870658	−	0.491889i	$-0.163693\pi$
$390$	0	0
$391$	−2.54830	−0.128873
$392$	0	0
$393$	8.91093	0.449497
$394$	0	0
$395$	5.63861i	0.283709i
$396$	0	0
$397$	12.8062i	0.642727i	0.946956	+	0.321364i	$0.104141\pi$
$397$	12.8062i	0.642727i	−0.946956	+	0.321364i	$0.895859\pi$
$398$	0	0
$399$	−20.7099	−1.03679
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	− 5.95005i	− 0.296393i
$404$	0	0
$405$	26.4922i	1.31641i
$406$	0	0
$407$	5.17748	0.256638
$408$	0	0
$409$	36.8062	1.81995	0.909976	−	0.414661i	$-0.136100\pi$
$409$	36.8062	1.81995	0.909976	+	0.414661i	$0.136100\pi$
$410$	0	0
$411$	25.4262i	1.25418i
$412$	0	0
$413$	7.40312i	0.364284i
$414$	0	0
$415$	−25.8874	−1.27076
$416$	0	0
$417$	−22.1047	−1.08247
$418$	0	0
$419$	− 19.4358i	− 0.949499i	−0.880121	−	0.474749i	$-0.842539\pi$
$419$	− 19.4358i	− 0.949499i	0.880121	−	0.474749i	$-0.157461\pi$
$420$	0	0
$421$	4.10469i	0.200050i	0.994985	+	0.100025i	$0.0318923\pi$
$421$	4.10469i	0.200050i	−0.994985	+	0.100025i	$0.968108\pi$
$422$	0	0
$423$	−2.77886	−0.135113
$424$	0	0
$425$	1.61250	0.0782176
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 1.40312i	− 0.0677435i
$430$	0	0
$431$	15.0309	0.724011	0.362005	−	0.932176i	$-0.382092\pi$
$431$	15.0309	0.724011	0.362005	+	0.932176i	$0.382092\pi$
$432$	0	0
$433$	22.9109	1.10103	0.550515	−	0.834826i	$-0.314432\pi$
$433$	22.9109	1.10103	0.550515	+	0.834826i	$0.314432\pi$
$434$	0	0
$435$	9.81294i	0.470494i
$436$	0	0
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	−17.6196	−0.840937	−0.420469	−	0.907307i	$-0.638134\pi$
$439$	−17.6196	−0.840937	−0.420469	+	0.907307i	$0.638134\pi$
$440$	0	0
$441$	0.0890652	0.00424120
$442$	0	0
$443$	22.5261i	1.07025i	0.844774	+	0.535123i	$0.179735\pi$
$443$	22.5261i	1.07025i	−0.844774	+	0.535123i	$0.820265\pi$
$444$	0	0
$445$	30.8062i	1.46036i
$446$	0	0
$447$	23.2582	1.10008
$448$	0	0
$449$	27.6125	1.30311	0.651557	−	0.758600i	$-0.274116\pi$
$449$	27.6125	1.30311	0.651557	+	0.758600i	$0.274116\pi$
$450$	0	0
$451$	− 8.80980i	− 0.414837i
$452$	0	0
$453$	− 43.2984i	− 2.03434i
$454$	0	0
$455$	−6.99364	−0.327867
$456$	0	0
$457$	11.4031	0.533416	0.266708	−	0.963777i	$-0.414064\pi$
$457$	11.4031	0.533416	0.266708	+	0.963777i	$0.414064\pi$
$458$	0	0
$459$	3.44219i	0.160668i
$460$	0	0
$461$	− 10.9109i	− 0.508173i	−0.967182	−	0.254086i	$-0.918225\pi$
$461$	− 10.9109i	− 0.508173i	0.967182	−	0.254086i	$-0.0817748\pi$
$462$	0	0
$463$	14.7598	0.685948	0.342974	−	0.939345i	$-0.388566\pi$
$463$	14.7598	0.685948	0.342974	+	0.939345i	$0.388566\pi$
$464$	0	0
$465$	−29.1938	−1.35383
$466$	0	0
$467$	2.62918i	0.121664i	0.998148	+	0.0608319i	$0.0193754\pi$
$467$	2.62918i	0.121664i	−0.998148	+	0.0608319i	$0.980625\pi$
$468$	0	0
$469$	32.8062i	1.51485i
$470$	0	0
$471$	8.34866	0.384686
$472$	0	0
$473$	−7.79063	−0.358213
$474$	0	0
$475$	10.1244i	0.464539i
$476$	0	0
$477$	− 0.418745i	− 0.0191730i
$478$	0	0
$479$	18.1212	0.827977	0.413989	−	0.910282i	$-0.364135\pi$
$479$	18.1212	0.827977	0.413989	+	0.910282i	$0.364135\pi$
$480$	0	0
$481$	−6.70156	−0.305565
$482$	0	0
$483$	− 17.0776i	− 0.777057i
$484$	0	0
$485$	− 27.0156i	− 1.22672i
$486$	0	0
$487$	12.1307	0.549693	0.274847	−	0.961488i	$-0.411373\pi$
$487$	12.1307	0.549693	0.274847	+	0.961488i	$0.411373\pi$
$488$	0	0
$489$	30.5969	1.38364
$490$	0	0
$491$	27.7035i	1.25024i	0.780527	+	0.625122i	$0.214951\pi$
$491$	27.7035i	1.25024i	−0.780527	+	0.625122i	$0.785049\pi$
$492$	0	0
$493$	1.40312i	0.0631935i
$494$	0	0
$495$	−0.622889	−0.0279968
$496$	0	0
$497$	−2.70156	−0.121182
$498$	0	0
$499$	− 14.7598i	− 0.660742i	−0.943851	−	0.330371i	$-0.892826\pi$
$499$	− 14.7598i	− 0.660742i	0.943851	−	0.330371i	$-0.107174\pi$
$500$	0	0
$501$	26.8062i	1.19761i
$502$	0	0
$503$	−23.3391	−1.04064	−0.520319	−	0.853972i	$-0.674187\pi$
$503$	−23.3391	−1.04064	−0.520319	+	0.853972i	$0.674187\pi$
$504$	0	0
$505$	18.3875	0.818233
$506$	0	0
$507$	1.81616i	0.0806585i
$508$	0	0
$509$	− 35.6125i	− 1.57850i	−0.614074	−	0.789248i	$-0.710471\pi$
$509$	− 35.6125i	− 1.57850i	0.614074	−	0.789248i	$-0.289529\pi$
$510$	0	0
$511$	−19.1647	−0.847798
$512$	0	0
$513$	−21.6125	−0.954215
$514$	0	0
$515$	37.7875i	1.66512i
$516$	0	0
$517$	7.19375i	0.316381i
$518$	0	0
$519$	−38.8715	−1.70627
$520$	0	0
$521$	6.10469	0.267451	0.133726	−	0.991018i	$-0.457306\pi$
$521$	6.10469	0.267451	0.133726	+	0.991018i	$0.457306\pi$
$522$	0	0
$523$	41.4198i	1.81116i	0.424174	+	0.905581i	$0.360564\pi$
$523$	41.4198i	1.81116i	−0.424174	+	0.905581i	$0.639436\pi$
$524$	0	0
$525$	10.8062i	0.471623i
$526$	0	0
$527$	−4.17433	−0.181837
$528$	0	0
$529$	−9.80625	−0.426359
$530$	0	0
$531$	− 0.853456i	− 0.0370368i
$532$	0	0
$533$	11.4031i	0.493924i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	43.7172	1.88654
$538$	0	0
$539$	− 0.230566i	− 0.00993120i
$540$	0	0
$541$	25.2984i	1.08766i	0.839194	+	0.543832i	$0.183027\pi$
$541$	25.2984i	1.08766i	−0.839194	+	0.543832i	$0.816973\pi$
$542$	0	0
$543$	25.4262	1.09114
$544$	0	0
$545$	−3.50781	−0.150258
$546$	0	0
$547$	12.7131i	0.543574i	0.962357	+	0.271787i	$0.0876146\pi$
$547$	12.7131i	0.543574i	−0.962357	+	0.271787i	$0.912385\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.80980	−0.375310
$552$	0	0
$553$	−5.40312	−0.229764
$554$	0	0
$555$	32.8810i	1.39572i
$556$	0	0
$557$	− 38.7016i	− 1.63984i	−0.572480	−	0.819919i	$-0.694018\pi$
$557$	− 38.7016i	− 1.63984i	0.572480	−	0.819919i	$-0.305982\pi$
$558$	0	0
$559$	10.0839	0.426505
$560$	0	0
$561$	−0.984379	−0.0415605
$562$	0	0
$563$	19.8969i	0.838554i	0.907858	+	0.419277i	$0.137716\pi$
$563$	19.8969i	0.838554i	−0.907858	+	0.419277i	$0.862284\pi$
$564$	0	0
$565$	3.79063i	0.159473i
$566$	0	0
$567$	−25.3858	−1.06610
$568$	0	0
$569$	−28.1047	−1.17821	−0.589105	−	0.808057i	$-0.700520\pi$
$569$	−28.1047	−1.17821	−0.589105	+	0.808057i	$0.700520\pi$
$570$	0	0
$571$	− 0.813017i	− 0.0340237i	−0.999855	−	0.0170118i	$-0.994585\pi$
$571$	− 0.813017i	− 0.0340237i	0.999855	−	0.0170118i	$-0.00541530\pi$
$572$	0	0
$573$	− 9.40312i	− 0.392821i
$574$	0	0
$575$	−8.34866	−0.348163
$576$	0	0
$577$	1.79063	0.0745448	0.0372724	−	0.999305i	$-0.488133\pi$
$577$	1.79063	0.0745448	0.0372724	+	0.999305i	$0.488133\pi$
$578$	0	0
$579$	− 37.7875i	− 1.57039i
$580$	0	0
$581$	− 24.8062i	− 1.02914i
$582$	0	0
$583$	−1.08402	−0.0448956
$584$	0	0
$585$	0.806248	0.0333343
$586$	0	0
$587$	30.2923i	1.25030i	0.780506	+	0.625148i	$0.214961\pi$
$587$	30.2923i	1.25030i	−0.780506	+	0.625148i	$0.785039\pi$
$588$	0	0
$589$	− 26.2094i	− 1.07994i
$590$	0	0
$591$	−0.190127	−0.00782079
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	4.90647i	0.201146i
$596$	0	0
$597$	12.2094i	0.499696i
$598$	0	0
$599$	25.3454	1.03558	0.517792	−	0.855507i	$-0.326754\pi$
$599$	25.3454	1.03558	0.517792	+	0.855507i	$0.326754\pi$
$600$	0	0
$601$	34.3141	1.39970	0.699850	−	0.714290i	$-0.253250\pi$
$601$	34.3141	1.39970	0.699850	+	0.714290i	$0.253250\pi$
$602$	0	0
$603$	− 3.78201i	− 0.154015i
$604$	0	0
$605$	− 28.1047i	− 1.14262i
$606$	0	0
$607$	−33.6940	−1.36760	−0.683799	−	0.729670i	$-0.739674\pi$
$607$	−33.6940	−1.36760	−0.683799	+	0.729670i	$0.739674\pi$
$608$	0	0
$609$	−9.40312	−0.381034
$610$	0	0
$611$	− 9.31137i	− 0.376698i
$612$	0	0
$613$	46.4187i	1.87484i	0.348207	+	0.937418i	$0.386791\pi$
$613$	46.4187i	1.87484i	−0.348207	+	0.937418i	$0.613209\pi$
$614$	0	0
$615$	55.9491	2.25608
$616$	0	0
$617$	−1.79063	−0.0720879	−0.0360440	−	0.999350i	$-0.511476\pi$
$617$	−1.79063	−0.0720879	−0.0360440	+	0.999350i	$0.511476\pi$
$618$	0	0
$619$	− 15.2210i	− 0.611783i	−0.952066	−	0.305891i	$-0.901046\pi$
$619$	− 15.2210i	− 0.611783i	0.952066	−	0.305891i	$-0.0989545\pi$
$620$	0	0
$621$	− 17.8219i	− 0.715167i
$622$	0	0
$623$	−29.5197	−1.18268
$624$	0	0
$625$	−31.2094	−1.24837
$626$	0	0
$627$	− 6.18062i	− 0.246830i
$628$	0	0
$629$	4.70156i	0.187464i
$630$	0	0
$631$	37.2859	1.48433	0.742164	−	0.670218i	$-0.233799\pi$
$631$	37.2859	1.48433	0.742164	+	0.670218i	$0.233799\pi$
$632$	0	0
$633$	−15.5078	−0.616380
$634$	0	0
$635$	− 29.4388i	− 1.16824i
$636$	0	0
$637$	0.298438i	0.0118245i
$638$	0	0
$639$	0.311445	0.0123206
$640$	0	0
$641$	−8.20937	−0.324251	−0.162125	−	0.986770i	$-0.551835\pi$
$641$	−8.20937	−0.324251	−0.162125	+	0.986770i	$0.551835\pi$
$642$	0	0
$643$	− 19.9373i	− 0.786251i	−0.919485	−	0.393126i	$-0.871394\pi$
$643$	− 19.9373i	− 0.786251i	0.919485	−	0.393126i	$-0.128606\pi$
$644$	0	0
$645$	− 49.4766i	− 1.94814i
$646$	0	0
$647$	−14.5293	−0.571205	−0.285603	−	0.958348i	$-0.592194\pi$
$647$	−14.5293	−0.571205	−0.285603	+	0.958348i	$0.592194\pi$
$648$	0	0
$649$	−2.20937	−0.0867255
$650$	0	0
$651$	− 27.9745i	− 1.09641i
$652$	0	0
$653$	23.4031i	0.915835i	0.888995	+	0.457918i	$0.151404\pi$
$653$	23.4031i	0.915835i	−0.888995	+	0.457918i	$0.848596\pi$
$654$	0	0
$655$	13.2551	0.517921
$656$	0	0
$657$	2.20937	0.0861958
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	10.0000i	0.388955i	0.980907	+	0.194477i	$0.0623011\pi$
$661$	10.0000i	0.388955i	−0.980907	+	0.194477i	$0.937699\pi$
$662$	0	0
$663$	1.27415	0.0494839
$664$	0	0
$665$	−30.8062	−1.19462
$666$	0	0
$667$	− 7.26464i	− 0.281288i
$668$	0	0
$669$	− 36.7016i	− 1.41896i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−46.3141	−1.78528	−0.892638	−	0.450774i	$-0.851148\pi$
$673$	−46.3141	−1.78528	−0.892638	+	0.450774i	$0.851148\pi$
$674$	0	0
$675$	11.2772i	0.434060i
$676$	0	0
$677$	15.0156i	0.577097i	0.957465	+	0.288549i	$0.0931727\pi$
$677$	15.0156i	0.577097i	−0.957465	+	0.288549i	$0.906827\pi$
$678$	0	0
$679$	25.8874	0.993466
$680$	0	0
$681$	−0.418745	−0.0160464
$682$	0	0
$683$	− 46.8278i	− 1.79182i	−0.444238	−	0.895909i	$-0.646526\pi$
$683$	− 46.8278i	− 1.79182i	0.444238	−	0.895909i	$-0.353474\pi$
$684$	0	0
$685$	37.8219i	1.44510i
$686$	0	0
$687$	−41.6099	−1.58752
$688$	0	0
$689$	1.40312	0.0534548
$690$	0	0
$691$	− 1.31459i	− 0.0500093i	−0.999687	−	0.0250046i	$-0.992040\pi$
$691$	− 1.31459i	− 0.0500093i	0.999687	−	0.0250046i	$-0.00796006\pi$
$692$	0	0
$693$	− 0.596876i	− 0.0226734i
$694$	0	0
$695$	−32.8810	−1.24725
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	39.0616i	1.47745i
$700$	0	0
$701$	− 8.80625i	− 0.332607i	−0.986075	−	0.166304i	$-0.946817\pi$
$701$	− 8.80625i	− 0.332607i	0.986075	−	0.166304i	$-0.0531832\pi$
$702$	0	0
$703$	−29.5197	−1.11336
$704$	0	0
$705$	−45.6859	−1.72063
$706$	0	0
$707$	17.6196i	0.662653i
$708$	0	0
$709$	14.0000i	0.525781i	0.964826	+	0.262891i	$0.0846758\pi$
$709$	14.0000i	0.525781i	−0.964826	+	0.262891i	$0.915324\pi$
$710$	0	0
$711$	0.622889	0.0233602
$712$	0	0
$713$	21.6125	0.809394
$714$	0	0
$715$	− 2.08717i	− 0.0780556i
$716$	0	0
$717$	50.7328i	1.89465i
$718$	0	0
$719$	−5.63861	−0.210285	−0.105142	−	0.994457i	$-0.533530\pi$
$719$	−5.63861	−0.210285	−0.105142	+	0.994457i	$0.533530\pi$
$720$	0	0
$721$	−36.2094	−1.34851
$722$	0	0
$723$	− 27.9745i	− 1.04038i
$724$	0	0
$725$	4.59688i	0.170724i
$726$	0	0
$727$	15.0713	0.558963	0.279482	−	0.960151i	$-0.409837\pi$
$727$	15.0713	0.558963	0.279482	+	0.960151i	$0.409837\pi$
$728$	0	0
$729$	−23.8062	−0.881713
$730$	0	0
$731$	− 7.07451i	− 0.261660i
$732$	0	0
$733$	− 39.1203i	− 1.44494i	−0.691401	−	0.722471i	$-0.743006\pi$
$733$	− 39.1203i	− 1.44494i	0.691401	−	0.722471i	$-0.256994\pi$
$734$	0	0
$735$	1.46428	0.0540106
$736$	0	0
$737$	−9.79063	−0.360642
$738$	0	0
$739$	4.40490i	0.162037i	0.996713	+	0.0810184i	$0.0258173\pi$
$739$	4.40490i	0.162037i	−0.996713	+	0.0810184i	$0.974183\pi$
$740$	0	0
$741$	8.00000i	0.293887i
$742$	0	0
$743$	−21.2115	−0.778173	−0.389087	−	0.921201i	$-0.627209\pi$
$743$	−21.2115	−0.778173	−0.389087	+	0.921201i	$0.627209\pi$
$744$	0	0
$745$	34.5969	1.26753
$746$	0	0
$747$	2.85974i	0.104633i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	17.6196	0.642948	0.321474	−	0.946918i	$-0.395822\pi$
$751$	17.6196	0.642948	0.321474	+	0.946918i	$0.395822\pi$
$752$	0	0
$753$	−48.0000	−1.74922
$754$	0	0
$755$	− 64.4070i	− 2.34401i
$756$	0	0
$757$	28.2094i	1.02529i	0.858602	+	0.512644i	$0.171334\pi$
$757$	28.2094i	1.02529i	−0.858602	+	0.512644i	$0.828666\pi$
$758$	0	0
$759$	5.09660	0.184995
$760$	0	0
$761$	11.4031	0.413363	0.206681	−	0.978408i	$-0.433734\pi$
$761$	11.4031	0.413363	0.206681	+	0.978408i	$0.433734\pi$
$762$	0	0
$763$	− 3.36131i	− 0.121688i
$764$	0	0
$765$	− 0.565633i	− 0.0204505i
$766$	0	0
$767$	2.85974	0.103259
$768$	0	0
$769$	−27.4031	−0.988182	−0.494091	−	0.869410i	$-0.664499\pi$
$769$	−27.4031	−0.988182	−0.494091	+	0.869410i	$0.664499\pi$
$770$	0	0
$771$	− 19.4358i	− 0.699961i
$772$	0	0
$773$	− 42.7016i	− 1.53587i	−0.640529	−	0.767934i	$-0.721285\pi$
$773$	− 42.7016i	− 1.53587i	0.640529	−	0.767934i	$-0.278715\pi$
$774$	0	0
$775$	−13.6758	−0.491250
$776$	0	0
$777$	−31.5078	−1.13034
$778$	0	0
$779$	50.2296i	1.79966i
$780$	0	0
$781$	− 0.806248i	− 0.0288498i
$782$	0	0
$783$	−9.81294	−0.350686
$784$	0	0
$785$	12.4187	0.443244
$786$	0	0
$787$	26.6600i	0.950325i	0.879898	+	0.475162i	$0.157611\pi$
$787$	26.6600i	0.950325i	−0.879898	+	0.475162i	$0.842389\pi$
$788$	0	0
$789$	− 34.8062i	− 1.23914i
$790$	0	0
$791$	−3.63232	−0.129150
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 6.88439i	− 0.244164i
$796$	0	0
$797$	− 46.2094i	− 1.63682i	−0.574635	−	0.818410i	$-0.694856\pi$
$797$	− 46.2094i	− 1.63682i	0.574635	−	0.818410i	$-0.305144\pi$
$798$	0	0
$799$	−6.53250	−0.231103
$800$	0	0
$801$	3.40312	0.120243
$802$	0	0
$803$	− 5.71949i	− 0.201836i
$804$	0	0
$805$	− 25.4031i	− 0.895342i
$806$	0	0
$807$	49.7685	1.75193
$808$	0	0
$809$	−31.5078	−1.10776	−0.553878	−	0.832598i	$-0.686853\pi$
$809$	−31.5078	−1.10776	−0.553878	+	0.832598i	$0.686853\pi$
$810$	0	0
$811$	− 38.0989i	− 1.33783i	−0.743337	−	0.668917i	$-0.766758\pi$
$811$	− 38.0989i	− 1.33783i	0.743337	−	0.668917i	$-0.233242\pi$
$812$	0	0
$813$	− 43.2984i	− 1.51854i
$814$	0	0
$815$	45.5133	1.59426
$816$	0	0
$817$	44.4187	1.55402
$818$	0	0
$819$	0.772577i	0.0269960i
$820$	0	0
$821$	18.4922i	0.645382i	0.946504	+	0.322691i	$0.104587\pi$
$821$	18.4922i	0.645382i	−0.946504	+	0.322691i	$0.895413\pi$
$822$	0	0
$823$	−34.1552	−1.19057	−0.595287	−	0.803513i	$-0.702962\pi$
$823$	−34.1552	−1.19057	−0.595287	+	0.803513i	$0.702962\pi$
$824$	0	0
$825$	−3.22499	−0.112280
$826$	0	0
$827$	2.85974i	0.0994430i	0.998763	+	0.0497215i	$0.0158334\pi$
$827$	2.85974i	0.0994430i	−0.998763	+	0.0497215i	$0.984167\pi$
$828$	0	0
$829$	37.4031i	1.29906i	0.760334	+	0.649532i	$0.225035\pi$
$829$	37.4031i	1.29906i	−0.760334	+	0.649532i	$0.774965\pi$
$830$	0	0
$831$	21.7939	0.756023
$832$	0	0
$833$	0.209373	0.00725433
$834$	0	0
$835$	39.8746i	1.37992i
$836$	0	0
$837$	− 29.1938i	− 1.00908i
$838$	0	0
$839$	53.0893	1.83285	0.916424	−	0.400209i	$-0.131063\pi$
$839$	53.0893	1.83285	0.916424	+	0.400209i	$0.131063\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 20.7099i	− 0.713287i
$844$	0	0
$845$	2.70156i	0.0929366i
$846$	0	0
$847$	26.9310	0.925359
$848$	0	0
$849$	−2.80625	−0.0963102
$850$	0	0
$851$	− 24.3422i	− 0.834441i
$852$	0	0
$853$	− 21.2984i	− 0.729245i	−0.931156	−	0.364622i	$-0.881198\pi$
$853$	− 21.2984i	− 0.729245i	0.931156	−	0.364622i	$-0.118802\pi$
$854$	0	0
$855$	3.55144	0.121457
$856$	0	0
$857$	−50.4187	−1.72227	−0.861136	−	0.508375i	$-0.830246\pi$
$857$	−50.4187	−1.72227	−0.861136	+	0.508375i	$0.830246\pi$
$858$	0	0
$859$	38.3295i	1.30779i	0.756587	+	0.653893i	$0.226865\pi$
$859$	38.3295i	1.30779i	−0.756587	+	0.653893i	$0.773135\pi$
$860$	0	0
$861$	53.6125i	1.82711i
$862$	0	0
$863$	21.2115	0.722047	0.361023	−	0.932557i	$-0.382427\pi$
$863$	21.2115	0.722047	0.361023	+	0.932557i	$0.382427\pi$
$864$	0	0
$865$	−57.8219	−1.96600
$866$	0	0
$867$	29.9808i	1.01820i
$868$	0	0
$869$	− 1.61250i	− 0.0547002i
$870$	0	0
$871$	12.6727	0.429397
$872$	0	0
$873$	−2.98438	−0.101006
$874$	0	0
$875$	− 18.8937i	− 0.638725i
$876$	0	0
$877$	− 48.3141i	− 1.63145i	−0.578440	−	0.815725i	$-0.696338\pi$
$877$	− 48.3141i	− 1.63145i	0.578440	−	0.815725i	$-0.303662\pi$
$878$	0	0
$879$	14.7194	0.496473
$880$	0	0
$881$	−38.9109	−1.31094	−0.655471	−	0.755220i	$-0.727530\pi$
$881$	−38.9109	−1.31094	−0.655471	+	0.755220i	$0.727530\pi$
$882$	0	0
$883$	11.1680i	0.375832i	0.982185	+	0.187916i	$0.0601733\pi$
$883$	11.1680i	0.375832i	−0.982185	+	0.187916i	$0.939827\pi$
$884$	0	0
$885$	− 14.0312i	− 0.471655i
$886$	0	0
$887$	−42.9650	−1.44262	−0.721311	−	0.692611i	$-0.756460\pi$
$887$	−42.9650	−1.44262	−0.721311	+	0.692611i	$0.756460\pi$
$888$	0	0
$889$	28.2094	0.946112
$890$	0	0
$891$	− 7.57609i	− 0.253808i
$892$	0	0
$893$	− 41.0156i	− 1.37254i
$894$	0	0
$895$	65.0299	2.17371
$896$	0	0
$897$	−6.59688	−0.220263
$898$	0	0
$899$	− 11.9001i	− 0.396891i
$900$	0	0
$901$	− 0.984379i	− 0.0327944i
$902$	0	0
$903$	47.4103	1.57772
$904$	0	0
$905$	37.8219	1.25724
$906$	0	0
$907$	− 41.6908i	− 1.38432i	−0.721744	−	0.692160i	$-0.756659\pi$
$907$	− 41.6908i	− 1.38432i	0.721744	−	0.692160i	$-0.243341\pi$
$908$	0	0
$909$	− 2.03124i	− 0.0673721i
$910$	0	0
$911$	19.7068	0.652914	0.326457	−	0.945212i	$-0.394145\pi$
$911$	19.7068	0.652914	0.326457	+	0.945212i	$0.394145\pi$
$912$	0	0
$913$	7.40312	0.245008
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.7016i	0.419443i
$918$	0	0
$919$	−28.4357	−0.938006	−0.469003	−	0.883196i	$-0.655387\pi$
$919$	−28.4357	−0.938006	−0.469003	+	0.883196i	$0.655387\pi$
$920$	0	0
$921$	42.8062	1.41051
$922$	0	0
$923$	1.04358i	0.0343499i
$924$	0	0
$925$	15.4031i	0.506452i
$926$	0	0
$927$	4.17433	0.137103
$928$	0	0
$929$	20.5969	0.675762	0.337881	−	0.941189i	$-0.390290\pi$
$929$	20.5969	0.675762	0.337881	+	0.941189i	$0.390290\pi$
$930$	0	0
$931$	1.31459i	0.0430839i
$932$	0	0
$933$	− 43.2250i	− 1.41512i
$934$	0	0
$935$	−1.46428	−0.0478870
$936$	0	0
$937$	−3.61250	−0.118015	−0.0590076	−	0.998258i	$-0.518794\pi$
$937$	−3.61250	−0.118015	−0.0590076	+	0.998258i	$0.518794\pi$
$938$	0	0
$939$	− 11.0871i	− 0.361813i
$940$	0	0
$941$	52.5234i	1.71221i	0.516798	+	0.856107i	$0.327124\pi$
$941$	52.5234i	1.71221i	−0.516798	+	0.856107i	$0.672876\pi$
$942$	0	0
$943$	−41.4198	−1.34881
$944$	0	0
$945$	−34.3141	−1.11624
$946$	0	0
$947$	37.0958i	1.20545i	0.797949	+	0.602725i	$0.205919\pi$
$947$	37.0958i	1.20545i	−0.797949	+	0.602725i	$0.794081\pi$
$948$	0	0
$949$	7.40312i	0.240316i
$950$	0	0
$951$	−10.8970	−0.353358
$952$	0	0
$953$	−13.2984	−0.430779	−0.215389	−	0.976528i	$-0.569102\pi$
$953$	−13.2984	−0.430779	−0.215389	+	0.976528i	$0.569102\pi$
$954$	0	0
$955$	− 13.9873i	− 0.452617i
$956$	0	0
$957$	− 2.80625i	− 0.0907131i
$958$	0	0
$959$	−36.2423	−1.17033
$960$	0	0
$961$	4.40312	0.142036
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 56.2094i	− 1.80944i
$966$	0	0
$967$	2.04673	0.0658183	0.0329091	−	0.999458i	$-0.489523\pi$
$967$	2.04673	0.0658183	0.0329091	+	0.999458i	$0.489523\pi$
$968$	0	0
$969$	5.61250	0.180299
$970$	0	0
$971$	1.81616i	0.0582834i	0.999575	+	0.0291417i	$0.00927740\pi$
$971$	1.81616i	0.0582834i	−0.999575	+	0.0291417i	$0.990723\pi$
$972$	0	0
$973$	− 31.5078i	− 1.01009i
$974$	0	0
$975$	4.17433	0.133686
$976$	0	0
$977$	7.19375	0.230149	0.115074	−	0.993357i	$-0.463289\pi$
$977$	7.19375	0.230149	0.115074	+	0.993357i	$0.463289\pi$
$978$	0	0
$979$	− 8.80980i	− 0.281562i
$980$	0	0
$981$	0.387503i	0.0123720i
$982$	0	0
$983$	3.04987	0.0972758	0.0486379	−	0.998816i	$-0.484512\pi$
$983$	3.04987	0.0972758	0.0486379	+	0.998816i	$0.484512\pi$
$984$	0	0
$985$	−0.282817	−0.00901129
$986$	0	0
$987$	− 43.7780i	− 1.39347i
$988$	0	0
$989$	36.6281i	1.16471i
$990$	0	0
$991$	−40.9587	−1.30109	−0.650547	−	0.759466i	$-0.725461\pi$
$991$	−40.9587	−1.30109	−0.650547	+	0.759466i	$0.725461\pi$
$992$	0	0
$993$	−33.4031	−1.06002
$994$	0	0
$995$	18.1616i	0.575761i
$996$	0	0
$997$	− 12.2094i	− 0.386675i	−0.981132	−	0.193337i	$-0.938069\pi$
$997$	− 12.2094i	− 0.386675i	0.981132	−	0.193337i	$-0.0619312\pi$
$998$	0	0
$999$	−32.8810	−1.04031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1664.2.b.j.833.3	✓	8
4.3	odd	2	inner	1664.2.b.j.833.5	yes	8
8.3	odd	2	inner	1664.2.b.j.833.4	yes	8
8.5	even	2	inner	1664.2.b.j.833.6	yes	8
16.3	odd	4		3328.2.a.bl.1.2		4
16.5	even	4		3328.2.a.bk.1.2		4
16.11	odd	4		3328.2.a.bk.1.3		4
16.13	even	4		3328.2.a.bl.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.j.833.3	✓	8	1.1	even	1	trivial
1664.2.b.j.833.4	yes	8	8.3	odd	2	inner
1664.2.b.j.833.5	yes	8	4.3	odd	2	inner
1664.2.b.j.833.6	yes	8	8.5	even	2	inner
3328.2.a.bk.1.2		4	16.5	even	4
3328.2.a.bk.1.3		4	16.11	odd	4
3328.2.a.bl.1.2		4	16.3	odd	4
3328.2.a.bl.1.3		4	16.13	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.81616i q^{3} -2.70156i q^{5} +2.58874 q^{7} -0.298438 q^{9} +0.772577i q^{11} -1.00000i q^{13} -4.90647 q^{15} -0.701562 q^{17} -4.40490i q^{19} -4.70156i q^{21} +3.63232 q^{23} -2.29844 q^{25} -4.90647i q^{27} -2.00000i q^{29} +5.95005 q^{31} +1.40312 q^{33} -6.99364i q^{35} -6.70156i q^{37} -1.81616 q^{39} -11.4031 q^{41} +10.0839i q^{43} +0.806248i q^{45} +9.31137 q^{47} -0.298438 q^{49} +1.27415i q^{51} +1.40312i q^{53} +2.08717 q^{55} -8.00000 q^{57} +2.85974i q^{59} -0.772577 q^{63} -2.70156 q^{65} +12.6727i q^{67} -6.59688i q^{69} -1.04358 q^{71} -7.40312 q^{73} +4.17433i q^{75} +2.00000i q^{77} -2.08717 q^{79} -9.80625 q^{81} -9.58237i q^{83} +1.89531i q^{85} -3.63232 q^{87} -11.4031 q^{89} -2.58874i q^{91} -10.8062i q^{93} -11.9001 q^{95} +10.0000 q^{97} -0.230566i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 28 q^{9} + 20 q^{17} - 44 q^{25} - 40 q^{33} - 40 q^{41} - 28 q^{49} - 64 q^{57} + 4 q^{65} - 8 q^{73} + 24 q^{81} - 40 q^{89} + 80 q^{97}+O(q^{100}) \) 8 * q - 28 * q^9 + 20 * q^17 - 44 * q^25 - 40 * q^33 - 40 * q^41 - 28 * q^49 - 64 * q^57 + 4 * q^65 - 8 * q^73 + 24 * q^81 - 40 * q^89 + 80 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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