LMFDB - Embedded newform 3330.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.a (trivial)

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:	yes
Analytic conductor:	$26.5901838731$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.91729$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.91729 q^{7} -1.00000 q^{8} -1.00000 q^{10} -6.51056 q^{11} -0.813457 q^{13} -2.91729 q^{14} +1.00000 q^{16} +2.51056 q^{17} +0.406728 q^{19} +1.00000 q^{20} +6.51056 q^{22} -5.02112 q^{23} +1.00000 q^{25} +0.813457 q^{26} +2.91729 q^{28} +5.32401 q^{29} -8.75186 q^{31} -1.00000 q^{32} -2.51056 q^{34} +2.91729 q^{35} -1.00000 q^{37} -0.406728 q^{38} -1.00000 q^{40} -6.34513 q^{41} -7.32401 q^{43} -6.51056 q^{44} +5.02112 q^{46} +5.42784 q^{47} +1.51056 q^{49} -1.00000 q^{50} -0.813457 q^{52} +2.34513 q^{53} -6.51056 q^{55} -2.91729 q^{56} -5.32401 q^{58} +1.42784 q^{59} -1.32401 q^{61} +8.75186 q^{62} +1.00000 q^{64} -0.813457 q^{65} -5.42784 q^{67} +2.51056 q^{68} -2.91729 q^{70} -14.6480 q^{71} -11.0211 q^{73} +1.00000 q^{74} +0.406728 q^{76} -18.9932 q^{77} -1.75870 q^{79} +1.00000 q^{80} +6.34513 q^{82} +7.05476 q^{83} +2.51056 q^{85} +7.32401 q^{86} +6.51056 q^{88} -6.00000 q^{89} -2.37309 q^{91} -5.02112 q^{92} -5.42784 q^{94} +0.406728 q^{95} +2.34513 q^{97} -1.51056 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 11 q^{11} + q^{14} + 3 q^{16} - q^{17} + 3 q^{20} + 11 q^{22} + 2 q^{23} + 3 q^{25} - q^{28} + 5 q^{29} + 3 q^{31} - 3 q^{32} + q^{34}+ \cdots + 4 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8 - 3 * q^10 - 11 * q^11 + q^14 + 3 * q^16 - q^17 + 3 * q^20 + 11 * q^22 + 2 * q^23 + 3 * q^25 - q^28 + 5 * q^29 + 3 * q^31 - 3 * q^32 + q^34 - q^35 - 3 * q^37 - 3 * q^40 + 9 * q^41 - 11 * q^43 - 11 * q^44 - 2 * q^46 - 2 * q^47 - 4 * q^49 - 3 * q^50 - 21 * q^53 - 11 * q^55 + q^56 - 5 * q^58 - 14 * q^59 + 7 * q^61 - 3 * q^62 + 3 * q^64 + 2 * q^67 - q^68 + q^70 - 22 * q^71 - 16 * q^73 + 3 * q^74 - 7 * q^77 - 26 * q^79 + 3 * q^80 - 9 * q^82 - 2 * q^83 - q^85 + 11 * q^86 + 11 * q^88 - 18 * q^89 - 12 * q^91 + 2 * q^92 + 2 * q^94 - 21 * q^97 + 4 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.91729	1.10263	0.551315	−	0.834297i	$-0.314126\pi$
$7$	2.91729	1.10263	0.551315	+	0.834297i	$0.314126\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−6.51056	−1.96301	−0.981503	−	0.191444i	$-0.938683\pi$
$11$	−6.51056	−1.96301	−0.981503	+	0.191444i	$0.938683\pi$
$12$	0	0
$13$	−0.813457	−0.225612	−0.112806	−	0.993617i	$-0.535984\pi$
$13$	−0.813457	−0.225612	−0.112806	+	0.993617i	$0.535984\pi$
$14$	−2.91729	−0.779677
$15$	0	0
$16$	1.00000	0.250000
$17$	2.51056	0.608900	0.304450	−	0.952528i	$-0.401527\pi$
$17$	2.51056	0.608900	0.304450	+	0.952528i	$0.401527\pi$
$18$	0	0
$19$	0.406728	0.0933099	0.0466550	−	0.998911i	$-0.485144\pi$
$19$	0.406728	0.0933099	0.0466550	+	0.998911i	$0.485144\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	6.51056	1.38806
$23$	−5.02112	−1.04697	−0.523487	−	0.852033i	$-0.675369\pi$
$23$	−5.02112	−1.04697	−0.523487	+	0.852033i	$0.675369\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.813457	0.159532
$27$	0	0
$28$	2.91729	0.551315
$29$	5.32401	0.988645	0.494322	−	0.869279i	$-0.335416\pi$
$29$	5.32401	0.988645	0.494322	+	0.869279i	$0.335416\pi$
$30$	0	0
$31$	−8.75186	−1.57188	−0.785940	−	0.618303i	$-0.787821\pi$
$31$	−8.75186	−1.57188	−0.785940	+	0.618303i	$0.787821\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.51056	−0.430557
$35$	2.91729	0.493111
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−0.406728	−0.0659801
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−6.34513	−0.990943	−0.495471	−	0.868624i	$-0.665005\pi$
$41$	−6.34513	−0.990943	−0.495471	+	0.868624i	$0.665005\pi$
$42$	0	0
$43$	−7.32401	−1.11690	−0.558451	−	0.829538i	$-0.688604\pi$
$43$	−7.32401	−1.11690	−0.558451	+	0.829538i	$0.688604\pi$
$44$	−6.51056	−0.981503
$45$	0	0
$46$	5.02112	0.740323
$47$	5.42784	0.791732	0.395866	−	0.918308i	$-0.370444\pi$
$47$	5.42784	0.791732	0.395866	+	0.918308i	$0.370444\pi$
$48$	0	0
$49$	1.51056	0.215794
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−0.813457	−0.112806
$53$	2.34513	0.322128	0.161064	−	0.986944i	$-0.448507\pi$
$53$	2.34513	0.322128	0.161064	+	0.986944i	$0.448507\pi$
$54$	0	0
$55$	−6.51056	−0.877883
$56$	−2.91729	−0.389839
$57$	0	0
$58$	−5.32401	−0.699077
$59$	1.42784	0.185889	0.0929447	−	0.995671i	$-0.470372\pi$
$59$	1.42784	0.185889	0.0929447	+	0.995671i	$0.470372\pi$
$60$	0	0
$61$	−1.32401	−0.169523	−0.0847613	−	0.996401i	$-0.527013\pi$
$61$	−1.32401	−0.169523	−0.0847613	+	0.996401i	$0.527013\pi$
$62$	8.75186	1.11149
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.813457	−0.100897
$66$	0	0
$67$	−5.42784	−0.663117	−0.331558	−	0.943435i	$-0.607574\pi$
$67$	−5.42784	−0.663117	−0.331558	+	0.943435i	$0.607574\pi$
$68$	2.51056	0.304450
$69$	0	0
$70$	−2.91729	−0.348682
$71$	−14.6480	−1.73840	−0.869201	−	0.494460i	$-0.835366\pi$
$71$	−14.6480	−1.73840	−0.869201	+	0.494460i	$0.835366\pi$
$72$	0	0
$73$	−11.0211	−1.28992	−0.644962	−	0.764215i	$-0.723127\pi$
$73$	−11.0211	−1.28992	−0.644962	+	0.764215i	$0.723127\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	0.406728	0.0466550
$77$	−18.9932	−2.16447
$78$	0	0
$79$	−1.75870	−0.197869	−0.0989346	−	0.995094i	$-0.531543\pi$
$79$	−1.75870	−0.197869	−0.0989346	+	0.995094i	$0.531543\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	6.34513	0.700702
$83$	7.05476	0.774360	0.387180	−	0.922004i	$-0.373449\pi$
$83$	7.05476	0.774360	0.387180	+	0.922004i	$0.373449\pi$
$84$	0	0
$85$	2.51056	0.272308
$86$	7.32401	0.789769
$87$	0	0
$88$	6.51056	0.694028
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−2.37309	−0.248767
$92$	−5.02112	−0.523487
$93$	0	0
$94$	−5.42784	−0.559839
$95$	0.406728	0.0417295
$96$	0	0
$97$	2.34513	0.238112	0.119056	−	0.992888i	$-0.462013\pi$
$97$	2.34513	0.238112	0.119056	+	0.992888i	$0.462013\pi$
$98$	−1.51056	−0.152589
$99$	0	0
$100$	1.00000	0.100000
$101$	8.20766	0.816693	0.408346	−	0.912827i	$-0.366106\pi$
$101$	8.20766	0.816693	0.408346	+	0.912827i	$0.366106\pi$
$102$	0	0
$103$	−8.81346	−0.868416	−0.434208	−	0.900813i	$-0.642972\pi$
$103$	−8.81346	−0.868416	−0.434208	+	0.900813i	$0.642972\pi$
$104$	0.813457	0.0797660
$105$	0	0
$106$	−2.34513	−0.227779
$107$	−15.2624	−1.47547	−0.737737	−	0.675089i	$-0.764105\pi$
$107$	−15.2624	−1.47547	−0.737737	+	0.675089i	$0.764105\pi$
$108$	0	0
$109$	4.30290	0.412143	0.206072	−	0.978537i	$-0.433932\pi$
$109$	4.30290	0.412143	0.206072	+	0.978537i	$0.433932\pi$
$110$	6.51056	0.620757
$111$	0	0
$112$	2.91729	0.275658
$113$	9.15859	0.861567	0.430784	−	0.902455i	$-0.358237\pi$
$113$	9.15859	0.861567	0.430784	+	0.902455i	$0.358237\pi$
$114$	0	0
$115$	−5.02112	−0.468221
$116$	5.32401	0.494322
$117$	0	0
$118$	−1.42784	−0.131444
$119$	7.32401	0.671391
$120$	0	0
$121$	31.3874	2.85340
$122$	1.32401	0.119871
$123$	0	0
$124$	−8.75186	−0.785940
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.61439	−0.764403	−0.382202	−	0.924079i	$-0.624834\pi$
$127$	−8.61439	−0.764403	−0.382202	+	0.924079i	$0.624834\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.813457	0.0713449
$131$	−13.4278	−1.17320	−0.586598	−	0.809878i	$-0.699533\pi$
$131$	−13.4278	−1.17320	−0.586598	+	0.809878i	$0.699533\pi$
$132$	0	0
$133$	1.18654	0.102886
$134$	5.42784	0.468894
$135$	0	0
$136$	−2.51056	−0.215279
$137$	14.6903	1.25507	0.627537	−	0.778587i	$-0.284063\pi$
$137$	14.6903	1.25507	0.627537	+	0.778587i	$0.284063\pi$
$138$	0	0
$139$	17.3662	1.47299	0.736493	−	0.676445i	$-0.236481\pi$
$139$	17.3662	1.47299	0.736493	+	0.676445i	$0.236481\pi$
$140$	2.91729	0.246556
$141$	0	0
$142$	14.6480	1.22924
$143$	5.29606	0.442879
$144$	0	0
$145$	5.32401	0.442135
$146$	11.0211	0.912114
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	0.207658	0.0170120	0.00850601	−	0.999964i	$-0.497292\pi$
$149$	0.207658	0.0170120	0.00850601	+	0.999964i	$0.497292\pi$
$150$	0	0
$151$	0.813457	0.0661982	0.0330991	−	0.999452i	$-0.489462\pi$
$151$	0.813457	0.0661982	0.0330991	+	0.999452i	$0.489462\pi$
$152$	−0.406728	−0.0329900
$153$	0	0
$154$	18.9932	1.53051
$155$	−8.75186	−0.702966
$156$	0	0
$157$	5.32401	0.424903	0.212451	−	0.977172i	$-0.431855\pi$
$157$	5.32401	0.424903	0.212451	+	0.977172i	$0.431855\pi$
$158$	1.75870	0.139915
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−14.6480	−1.15443
$162$	0	0
$163$	15.2008	1.19062	0.595310	−	0.803496i	$-0.297029\pi$
$163$	15.2008	1.19062	0.595310	+	0.803496i	$0.297029\pi$
$164$	−6.34513	−0.495471
$165$	0	0
$166$	−7.05476	−0.547555
$167$	12.4826	0.965933	0.482966	−	0.875639i	$-0.339559\pi$
$167$	12.4826	0.965933	0.482966	+	0.875639i	$0.339559\pi$
$168$	0	0
$169$	−12.3383	−0.949099
$170$	−2.51056	−0.192551
$171$	0	0
$172$	−7.32401	−0.558451
$173$	−23.0354	−1.75135	−0.875674	−	0.482903i	$-0.839583\pi$
$173$	−23.0354	−1.75135	−0.875674	+	0.482903i	$0.839583\pi$
$174$	0	0
$175$	2.91729	0.220526
$176$	−6.51056	−0.490752
$177$	0	0
$178$	6.00000	0.449719
$179$	−24.2835	−1.81504	−0.907518	−	0.420013i	$-0.862026\pi$
$179$	−24.2835	−1.81504	−0.907518	+	0.420013i	$0.862026\pi$
$180$	0	0
$181$	2.16543	0.160955	0.0804775	−	0.996756i	$-0.474355\pi$
$181$	2.16543	0.160955	0.0804775	+	0.996756i	$0.474355\pi$
$182$	2.37309	0.175905
$183$	0	0
$184$	5.02112	0.370162
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−16.3451	−1.19527
$188$	5.42784	0.395866
$189$	0	0
$190$	−0.406728	−0.0295072
$191$	19.3326	1.39886	0.699429	−	0.714702i	$-0.253438\pi$
$191$	19.3326	1.39886	0.699429	+	0.714702i	$0.253438\pi$
$192$	0	0
$193$	−20.0422	−1.44267	−0.721336	−	0.692586i	$-0.756471\pi$
$193$	−20.0422	−1.44267	−0.721336	+	0.692586i	$0.756471\pi$
$194$	−2.34513	−0.168370
$195$	0	0
$196$	1.51056	0.107897
$197$	−15.6269	−1.11337	−0.556686	−	0.830723i	$-0.687927\pi$
$197$	−15.6269	−1.11337	−0.556686	+	0.830723i	$0.687927\pi$
$198$	0	0
$199$	20.2835	1.43786	0.718931	−	0.695082i	$-0.244632\pi$
$199$	20.2835	1.43786	0.718931	+	0.695082i	$0.244632\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−8.20766	−0.577489
$203$	15.5317	1.09011
$204$	0	0
$205$	−6.34513	−0.443163
$206$	8.81346	0.614063
$207$	0	0
$208$	−0.813457	−0.0564031
$209$	−2.64803	−0.183168
$210$	0	0
$211$	18.7182	1.28862	0.644308	−	0.764766i	$-0.277146\pi$
$211$	18.7182	1.28862	0.644308	+	0.764766i	$0.277146\pi$
$212$	2.34513	0.161064
$213$	0	0
$214$	15.2624	1.04332
$215$	−7.32401	−0.499494
$216$	0	0
$217$	−25.5317	−1.73320
$218$	−4.30290	−0.291429
$219$	0	0
$220$	−6.51056	−0.438942
$221$	−2.04223	−0.137375
$222$	0	0
$223$	−15.7307	−1.05341	−0.526704	−	0.850049i	$-0.676572\pi$
$223$	−15.7307	−1.05341	−0.526704	+	0.850049i	$0.676572\pi$
$224$	−2.91729	−0.194919
$225$	0	0
$226$	−9.15859	−0.609220
$227$	−12.8836	−0.855117	−0.427559	−	0.903988i	$-0.640626\pi$
$227$	−12.8836	−0.855117	−0.427559	+	0.903988i	$0.640626\pi$
$228$	0	0
$229$	5.25383	0.347183	0.173591	−	0.984818i	$-0.444463\pi$
$229$	5.25383	0.347183	0.173591	+	0.984818i	$0.444463\pi$
$230$	5.02112	0.331083
$231$	0	0
$232$	−5.32401	−0.349539
$233$	−28.3172	−1.85512	−0.927560	−	0.373675i	$-0.878098\pi$
$233$	−28.3172	−1.85512	−0.927560	+	0.373675i	$0.878098\pi$
$234$	0	0
$235$	5.42784	0.354073
$236$	1.42784	0.0929447
$237$	0	0
$238$	−7.32401	−0.474745
$239$	−14.3788	−0.930085	−0.465043	−	0.885288i	$-0.653961\pi$
$239$	−14.3788	−0.930085	−0.465043	+	0.885288i	$0.653961\pi$
$240$	0	0
$241$	18.2749	1.17719	0.588596	−	0.808427i	$-0.299681\pi$
$241$	18.2749	1.17719	0.588596	+	0.808427i	$0.299681\pi$
$242$	−31.3874	−2.01766
$243$	0	0
$244$	−1.32401	−0.0847613
$245$	1.51056	0.0965060
$246$	0	0
$247$	−0.330856	−0.0210519
$248$	8.75186	0.555744
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	1.22019	0.0770174	0.0385087	−	0.999258i	$-0.487739\pi$
$251$	1.22019	0.0770174	0.0385087	+	0.999258i	$0.487739\pi$
$252$	0	0
$253$	32.6903	2.05522
$254$	8.61439	0.540515
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.1306	−1.69236	−0.846181	−	0.532895i	$-0.821104\pi$
$257$	−27.1306	−1.69236	−0.846181	+	0.532895i	$0.821104\pi$
$258$	0	0
$259$	−2.91729	−0.181271
$260$	−0.813457	−0.0504485
$261$	0	0
$262$	13.4278	0.829575
$263$	20.2133	1.24641	0.623204	−	0.782059i	$-0.285831\pi$
$263$	20.2133	1.24641	0.623204	+	0.782059i	$0.285831\pi$
$264$	0	0
$265$	2.34513	0.144060
$266$	−1.18654	−0.0727516
$267$	0	0
$268$	−5.42784	−0.331558
$269$	0.207658	0.0126611	0.00633057	−	0.999980i	$-0.497985\pi$
$269$	0.207658	0.0126611	0.00633057	+	0.999980i	$0.497985\pi$
$270$	0	0
$271$	−30.6480	−1.86174	−0.930868	−	0.365357i	$-0.880947\pi$
$271$	−30.6480	−1.86174	−0.930868	+	0.365357i	$0.880947\pi$
$272$	2.51056	0.152225
$273$	0	0
$274$	−14.6903	−0.887471
$275$	−6.51056	−0.392601
$276$	0	0
$277$	−16.8135	−1.01022	−0.505111	−	0.863054i	$-0.668549\pi$
$277$	−16.8135	−1.01022	−0.505111	+	0.863054i	$0.668549\pi$
$278$	−17.3662	−1.04156
$279$	0	0
$280$	−2.91729	−0.174341
$281$	14.2749	0.851572	0.425786	−	0.904824i	$-0.359998\pi$
$281$	14.2749	0.851572	0.425786	+	0.904824i	$0.359998\pi$
$282$	0	0
$283$	−13.2288	−0.786369	−0.393184	−	0.919460i	$-0.628627\pi$
$283$	−13.2288	−0.786369	−0.393184	+	0.919460i	$0.628627\pi$
$284$	−14.6480	−0.869201
$285$	0	0
$286$	−5.29606	−0.313162
$287$	−18.5106	−1.09264
$288$	0	0
$289$	−10.6971	−0.629241
$290$	−5.32401	−0.312637
$291$	0	0
$292$	−11.0211	−0.644962
$293$	−2.34513	−0.137004	−0.0685020	−	0.997651i	$-0.521822\pi$
$293$	−2.34513	−0.137004	−0.0685020	+	0.997651i	$0.521822\pi$
$294$	0	0
$295$	1.42784	0.0831323
$296$	1.00000	0.0581238
$297$	0	0
$298$	−0.207658	−0.0120293
$299$	4.08446	0.236210
$300$	0	0
$301$	−21.3662	−1.23153
$302$	−0.813457	−0.0468092
$303$	0	0
$304$	0.406728	0.0233275
$305$	−1.32401	−0.0758128
$306$	0	0
$307$	−23.2624	−1.32766	−0.663828	−	0.747885i	$-0.731069\pi$
$307$	−23.2624	−1.32766	−0.663828	+	0.747885i	$0.731069\pi$
$308$	−18.9932	−1.08224
$309$	0	0
$310$	8.75186	0.497072
$311$	31.3999	1.78052	0.890262	−	0.455449i	$-0.150521\pi$
$311$	31.3999	1.78052	0.890262	+	0.455449i	$0.150521\pi$
$312$	0	0
$313$	−8.04223	−0.454574	−0.227287	−	0.973828i	$-0.572986\pi$
$313$	−8.04223	−0.454574	−0.227287	+	0.973828i	$0.572986\pi$
$314$	−5.32401	−0.300452
$315$	0	0
$316$	−1.75870	−0.0989346
$317$	12.3029	0.691000	0.345500	−	0.938419i	$-0.387709\pi$
$317$	12.3029	0.691000	0.345500	+	0.938419i	$0.387709\pi$
$318$	0	0
$319$	−34.6623	−1.94072
$320$	1.00000	0.0559017
$321$	0	0
$322$	14.6480	0.816303
$323$	1.02112	0.0568164
$324$	0	0
$325$	−0.813457	−0.0451225
$326$	−15.2008	−0.841895
$327$	0	0
$328$	6.34513	0.350351
$329$	15.8346	0.872988
$330$	0	0
$331$	−6.77981	−0.372652	−0.186326	−	0.982488i	$-0.559658\pi$
$331$	−6.77981	−0.372652	−0.186326	+	0.982488i	$0.559658\pi$
$332$	7.05476	0.387180
$333$	0	0
$334$	−12.4826	−0.683018
$335$	−5.42784	−0.296555
$336$	0	0
$337$	10.6903	0.582336	0.291168	−	0.956672i	$-0.405956\pi$
$337$	10.6903	0.582336	0.291168	+	0.956672i	$0.405956\pi$
$338$	12.3383	0.671114
$339$	0	0
$340$	2.51056	0.136154
$341$	56.9795	3.08561
$342$	0	0
$343$	−16.0143	−0.864689
$344$	7.32401	0.394884
$345$	0	0
$346$	23.0354	1.23839
$347$	−5.62691	−0.302069	−0.151034	−	0.988529i	$-0.548260\pi$
$347$	−5.62691	−0.302069	−0.151034	+	0.988529i	$0.548260\pi$
$348$	0	0
$349$	15.1306	0.809924	0.404962	−	0.914334i	$-0.367285\pi$
$349$	15.1306	0.809924	0.404962	+	0.914334i	$0.367285\pi$
$350$	−2.91729	−0.155935
$351$	0	0
$352$	6.51056	0.347014
$353$	−35.7816	−1.90446	−0.952230	−	0.305381i	$-0.901216\pi$
$353$	−35.7816	−1.90446	−0.952230	+	0.305381i	$0.901216\pi$
$354$	0	0
$355$	−14.6480	−0.777437
$356$	−6.00000	−0.317999
$357$	0	0
$358$	24.2835	1.28342
$359$	−4.48260	−0.236583	−0.118291	−	0.992979i	$-0.537742\pi$
$359$	−4.48260	−0.236583	−0.118291	+	0.992979i	$0.537742\pi$
$360$	0	0
$361$	−18.8346	−0.991293
$362$	−2.16543	−0.113812
$363$	0	0
$364$	−2.37309	−0.124384
$365$	−11.0211	−0.576872
$366$	0	0
$367$	−21.5653	−1.12570	−0.562850	−	0.826559i	$-0.690295\pi$
$367$	−21.5653	−1.12570	−0.562850	+	0.826559i	$0.690295\pi$
$368$	−5.02112	−0.261744
$369$	0	0
$370$	1.00000	0.0519875
$371$	6.84141	0.355188
$372$	0	0
$373$	16.9230	0.876238	0.438119	−	0.898917i	$-0.355645\pi$
$373$	16.9230	0.876238	0.438119	+	0.898917i	$0.355645\pi$
$374$	16.3451	0.845187
$375$	0	0
$376$	−5.42784	−0.279920
$377$	−4.33086	−0.223050
$378$	0	0
$379$	31.2710	1.60628	0.803142	−	0.595788i	$-0.203160\pi$
$379$	31.2710	1.60628	0.803142	+	0.595788i	$0.203160\pi$
$380$	0.406728	0.0208647
$381$	0	0
$382$	−19.3326	−0.989142
$383$	−1.62691	−0.0831314	−0.0415657	−	0.999136i	$-0.513235\pi$
$383$	−1.62691	−0.0831314	−0.0415657	+	0.999136i	$0.513235\pi$
$384$	0	0
$385$	−18.9932	−0.967981
$386$	20.0422	1.02012
$387$	0	0
$388$	2.34513	0.119056
$389$	31.6412	1.60427	0.802136	−	0.597141i	$-0.203697\pi$
$389$	31.6412	1.60427	0.802136	+	0.597141i	$0.203697\pi$
$390$	0	0
$391$	−12.6058	−0.637503
$392$	−1.51056	−0.0762947
$393$	0	0
$394$	15.6269	0.787273
$395$	−1.75870	−0.0884898
$396$	0	0
$397$	−39.2961	−1.97221	−0.986106	−	0.166116i	$-0.946878\pi$
$397$	−39.2961	−1.97221	−0.986106	+	0.166116i	$0.946878\pi$
$398$	−20.2835	−1.01672
$399$	0	0
$400$	1.00000	0.0500000
$401$	9.66914	0.482854	0.241427	−	0.970419i	$-0.422385\pi$
$401$	9.66914	0.482854	0.241427	+	0.970419i	$0.422385\pi$
$402$	0	0
$403$	7.11926	0.354636
$404$	8.20766	0.408346
$405$	0	0
$406$	−15.5317	−0.770824
$407$	6.51056	0.322716
$408$	0	0
$409$	−26.2749	−1.29921	−0.649606	−	0.760271i	$-0.725066\pi$
$409$	−26.2749	−1.29921	−0.649606	+	0.760271i	$0.725066\pi$
$410$	6.34513	0.313364
$411$	0	0
$412$	−8.81346	−0.434208
$413$	4.16543	0.204967
$414$	0	0
$415$	7.05476	0.346304
$416$	0.813457	0.0398830
$417$	0	0
$418$	2.64803	0.129519
$419$	30.1940	1.47507	0.737536	−	0.675308i	$-0.235989\pi$
$419$	30.1940	1.47507	0.737536	+	0.675308i	$0.235989\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−18.7182	−0.911188
$423$	0	0
$424$	−2.34513	−0.113890
$425$	2.51056	0.121780
$426$	0	0
$427$	−3.86253	−0.186921
$428$	−15.2624	−0.737737
$429$	0	0
$430$	7.32401	0.353195
$431$	−25.5653	−1.23144	−0.615719	−	0.787966i	$-0.711134\pi$
$431$	−25.5653	−1.23144	−0.615719	+	0.787966i	$0.711134\pi$
$432$	0	0
$433$	32.0422	1.53985	0.769926	−	0.638134i	$-0.220293\pi$
$433$	32.0422	1.53985	0.769926	+	0.638134i	$0.220293\pi$
$434$	25.5317	1.22556
$435$	0	0
$436$	4.30290	0.206072
$437$	−2.04223	−0.0976931
$438$	0	0
$439$	3.93840	0.187970	0.0939848	−	0.995574i	$-0.470039\pi$
$439$	3.93840	0.187970	0.0939848	+	0.995574i	$0.470039\pi$
$440$	6.51056	0.310379
$441$	0	0
$442$	2.04223	0.0971390
$443$	0.0758724	0.00360481	0.00180240	−	0.999998i	$-0.499426\pi$
$443$	0.0758724	0.00360481	0.00180240	+	0.999998i	$0.499426\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	15.7307	0.744872
$447$	0	0
$448$	2.91729	0.137829
$449$	−19.2961	−0.910637	−0.455319	−	0.890329i	$-0.650475\pi$
$449$	−19.2961	−0.910637	−0.455319	+	0.890329i	$0.650475\pi$
$450$	0	0
$451$	41.3103	1.94523
$452$	9.15859	0.430784
$453$	0	0
$454$	12.8836	0.604659
$455$	−2.37309	−0.111252
$456$	0	0
$457$	3.00684	0.140654	0.0703271	−	0.997524i	$-0.477596\pi$
$457$	3.00684	0.140654	0.0703271	+	0.997524i	$0.477596\pi$
$458$	−5.25383	−0.245495
$459$	0	0
$460$	−5.02112	−0.234111
$461$	0.633755	0.0295169	0.0147585	−	0.999891i	$-0.495302\pi$
$461$	0.633755	0.0295169	0.0147585	+	0.999891i	$0.495302\pi$
$462$	0	0
$463$	18.0422	0.838494	0.419247	−	0.907872i	$-0.362294\pi$
$463$	18.0422	0.838494	0.419247	+	0.907872i	$0.362294\pi$
$464$	5.32401	0.247161
$465$	0	0
$466$	28.3172	1.31177
$467$	23.4758	1.08633	0.543164	−	0.839626i	$-0.317226\pi$
$467$	23.4758	1.08633	0.543164	+	0.839626i	$0.317226\pi$
$468$	0	0
$469$	−15.8346	−0.731173
$470$	−5.42784	−0.250368
$471$	0	0
$472$	−1.42784	−0.0657218
$473$	47.6834	2.19249
$474$	0	0
$475$	0.406728	0.0186620
$476$	7.32401	0.335696
$477$	0	0
$478$	14.3788	0.657670
$479$	8.19907	0.374625	0.187313	−	0.982300i	$-0.440022\pi$
$479$	8.19907	0.374625	0.187313	+	0.982300i	$0.440022\pi$
$480$	0	0
$481$	0.813457	0.0370904
$482$	−18.2749	−0.832401
$483$	0	0
$484$	31.3874	1.42670
$485$	2.34513	0.106487
$486$	0	0
$487$	0.953831	0.0432222	0.0216111	−	0.999766i	$-0.493120\pi$
$487$	0.953831	0.0432222	0.0216111	+	0.999766i	$0.493120\pi$
$488$	1.32401	0.0599353
$489$	0	0
$490$	−1.51056	−0.0682400
$491$	−19.3383	−0.872725	−0.436362	−	0.899771i	$-0.643733\pi$
$491$	−19.3383	−0.872725	−0.436362	+	0.899771i	$0.643733\pi$
$492$	0	0
$493$	13.3662	0.601985
$494$	0.330856	0.0148859
$495$	0	0
$496$	−8.75186	−0.392970
$497$	−42.7325	−1.91681
$498$	0	0
$499$	−5.63550	−0.252280	−0.126140	−	0.992012i	$-0.540259\pi$
$499$	−5.63550	−0.252280	−0.126140	+	0.992012i	$0.540259\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−1.22019	−0.0544595
$503$	33.6269	1.49935	0.749675	−	0.661806i	$-0.230210\pi$
$503$	33.6269	1.49935	0.749675	+	0.661806i	$0.230210\pi$
$504$	0	0
$505$	8.20766	0.365236
$506$	−32.6903	−1.45326
$507$	0	0
$508$	−8.61439	−0.382202
$509$	18.6903	0.828431	0.414216	−	0.910179i	$-0.364056\pi$
$509$	18.6903	0.828431	0.414216	+	0.910179i	$0.364056\pi$
$510$	0	0
$511$	−32.1517	−1.42231
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	27.1306	1.19668
$515$	−8.81346	−0.388367
$516$	0	0
$517$	−35.3383	−1.55418
$518$	2.91729	0.128178
$519$	0	0
$520$	0.813457	0.0356724
$521$	−25.1586	−1.10222	−0.551109	−	0.834433i	$-0.685795\pi$
$521$	−25.1586	−1.10222	−0.551109	+	0.834433i	$0.685795\pi$
$522$	0	0
$523$	−7.25383	−0.317188	−0.158594	−	0.987344i	$-0.550696\pi$
$523$	−7.25383	−0.317188	−0.158594	+	0.987344i	$0.550696\pi$
$524$	−13.4278	−0.586598
$525$	0	0
$526$	−20.2133	−0.881344
$527$	−21.9720	−0.957117
$528$	0	0
$529$	2.21160	0.0961564
$530$	−2.34513	−0.101866
$531$	0	0
$532$	1.18654	0.0514432
$533$	5.16149	0.223569
$534$	0	0
$535$	−15.2624	−0.659852
$536$	5.42784	0.234447
$537$	0	0
$538$	−0.207658	−0.00895278
$539$	−9.83457	−0.423605
$540$	0	0
$541$	16.0422	0.689709	0.344855	−	0.938656i	$-0.387928\pi$
$541$	16.0422	0.689709	0.344855	+	0.938656i	$0.387928\pi$
$542$	30.6480	1.31645
$543$	0	0
$544$	−2.51056	−0.107639
$545$	4.30290	0.184316
$546$	0	0
$547$	16.0143	0.684721	0.342360	−	0.939569i	$-0.388774\pi$
$547$	16.0143	0.684721	0.342360	+	0.939569i	$0.388774\pi$
$548$	14.6903	0.627537
$549$	0	0
$550$	6.51056	0.277611
$551$	2.16543	0.0922503
$552$	0	0
$553$	−5.13063	−0.218177
$554$	16.8135	0.714335
$555$	0	0
$556$	17.3662	0.736493
$557$	10.8557	0.459970	0.229985	−	0.973194i	$-0.426132\pi$
$557$	10.8557	0.459970	0.229985	+	0.973194i	$0.426132\pi$
$558$	0	0
$559$	5.95777	0.251987
$560$	2.91729	0.123278
$561$	0	0
$562$	−14.2749	−0.602152
$563$	−31.0776	−1.30977	−0.654883	−	0.755731i	$-0.727282\pi$
$563$	−31.0776	−1.30977	−0.654883	+	0.755731i	$0.727282\pi$
$564$	0	0
$565$	9.15859	0.385305
$566$	13.2288	0.556047
$567$	0	0
$568$	14.6480	0.614618
$569$	−30.0845	−1.26121	−0.630603	−	0.776105i	$-0.717192\pi$
$569$	−30.0845	−1.26121	−0.630603	+	0.776105i	$0.717192\pi$
$570$	0	0
$571$	5.55673	0.232542	0.116271	−	0.993218i	$-0.462906\pi$
$571$	5.55673	0.232542	0.116271	+	0.993218i	$0.462906\pi$
$572$	5.29606	0.221439
$573$	0	0
$574$	18.5106	0.772616
$575$	−5.02112	−0.209395
$576$	0	0
$577$	−22.2499	−0.926275	−0.463137	−	0.886286i	$-0.653276\pi$
$577$	−22.2499	−0.926275	−0.463137	+	0.886286i	$0.653276\pi$
$578$	10.6971	0.444941
$579$	0	0
$580$	5.32401	0.221068
$581$	20.5807	0.853833
$582$	0	0
$583$	−15.2681	−0.632340
$584$	11.0211	0.456057
$585$	0	0
$586$	2.34513	0.0968764
$587$	−45.3103	−1.87016	−0.935079	−	0.354440i	$-0.884672\pi$
$587$	−45.3103	−1.87016	−0.935079	+	0.354440i	$0.884672\pi$
$588$	0	0
$589$	−3.55963	−0.146672
$590$	−1.42784	−0.0587834
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	0.232712	0.00955635	0.00477817	−	0.999989i	$-0.498479\pi$
$593$	0.232712	0.00955635	0.00477817	+	0.999989i	$0.498479\pi$
$594$	0	0
$595$	7.32401	0.300255
$596$	0.207658	0.00850601
$597$	0	0
$598$	−4.08446	−0.167026
$599$	−43.4056	−1.77350	−0.886752	−	0.462246i	$-0.847044\pi$
$599$	−43.4056	−1.77350	−0.886752	+	0.462246i	$0.847044\pi$
$600$	0	0
$601$	35.5317	1.44937	0.724684	−	0.689082i	$-0.241986\pi$
$601$	35.5317	1.44937	0.724684	+	0.689082i	$0.241986\pi$
$602$	21.3662	0.870823
$603$	0	0
$604$	0.813457	0.0330991
$605$	31.3874	1.27608
$606$	0	0
$607$	26.7325	1.08504	0.542519	−	0.840043i	$-0.317471\pi$
$607$	26.7325	1.08504	0.542519	+	0.840043i	$0.317471\pi$
$608$	−0.406728	−0.0164950
$609$	0	0
$610$	1.32401	0.0536078
$611$	−4.41532	−0.178625
$612$	0	0
$613$	48.0565	1.94098	0.970492	−	0.241134i	$-0.0775192\pi$
$613$	48.0565	1.94098	0.970492	+	0.241134i	$0.0775192\pi$
$614$	23.2624	0.938795
$615$	0	0
$616$	18.9932	0.765256
$617$	−12.9789	−0.522510	−0.261255	−	0.965270i	$-0.584136\pi$
$617$	−12.9789	−0.522510	−0.261255	+	0.965270i	$0.584136\pi$
$618$	0	0
$619$	25.3662	1.01956	0.509778	−	0.860306i	$-0.329728\pi$
$619$	25.3662	1.01956	0.509778	+	0.860306i	$0.329728\pi$
$620$	−8.75186	−0.351483
$621$	0	0
$622$	−31.3999	−1.25902
$623$	−17.5037	−0.701272
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.04223	0.321432
$627$	0	0
$628$	5.32401	0.212451
$629$	−2.51056	−0.100102
$630$	0	0
$631$	−31.6075	−1.25828	−0.629138	−	0.777293i	$-0.716592\pi$
$631$	−31.6075	−1.25828	−0.629138	+	0.777293i	$0.716592\pi$
$632$	1.75870	0.0699573
$633$	0	0
$634$	−12.3029	−0.488611
$635$	−8.61439	−0.341852
$636$	0	0
$637$	−1.22877	−0.0486858
$638$	34.6623	1.37229
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	23.8625	0.942513	0.471257	−	0.881996i	$-0.343801\pi$
$641$	23.8625	0.942513	0.471257	+	0.881996i	$0.343801\pi$
$642$	0	0
$643$	32.8277	1.29460	0.647300	−	0.762236i	$-0.275898\pi$
$643$	32.8277	1.29460	0.647300	+	0.762236i	$0.275898\pi$
$644$	−14.6480	−0.577213
$645$	0	0
$646$	−1.02112	−0.0401752
$647$	13.9578	0.548737	0.274368	−	0.961625i	$-0.411531\pi$
$647$	13.9578	0.548737	0.274368	+	0.961625i	$0.411531\pi$
$648$	0	0
$649$	−9.29606	−0.364902
$650$	0.813457	0.0319064
$651$	0	0
$652$	15.2008	0.595310
$653$	−28.5921	−1.11890	−0.559448	−	0.828865i	$-0.688987\pi$
$653$	−28.5921	−1.11890	−0.559448	+	0.828865i	$0.688987\pi$
$654$	0	0
$655$	−13.4278	−0.524669
$656$	−6.34513	−0.247736
$657$	0	0
$658$	−15.8346	−0.617296
$659$	5.62691	0.219193	0.109597	−	0.993976i	$-0.465044\pi$
$659$	5.62691	0.219193	0.109597	+	0.993976i	$0.465044\pi$
$660$	0	0
$661$	13.3240	0.518244	0.259122	−	0.965845i	$-0.416567\pi$
$661$	13.3240	0.518244	0.259122	+	0.965845i	$0.416567\pi$
$662$	6.77981	0.263505
$663$	0	0
$664$	−7.05476	−0.273778
$665$	1.18654	0.0460122
$666$	0	0
$667$	−26.7325	−1.03509
$668$	12.4826	0.482966
$669$	0	0
$670$	5.42784	0.209696
$671$	8.62007	0.332774
$672$	0	0
$673$	29.6691	1.14366	0.571831	−	0.820372i	$-0.306233\pi$
$673$	29.6691	1.14366	0.571831	+	0.820372i	$0.306233\pi$
$674$	−10.6903	−0.411773
$675$	0	0
$676$	−12.3383	−0.474550
$677$	1.25383	0.0481885	0.0240943	−	0.999710i	$-0.492330\pi$
$677$	1.25383	0.0481885	0.0240943	+	0.999710i	$0.492330\pi$
$678$	0	0
$679$	6.84141	0.262549
$680$	−2.51056	−0.0962755
$681$	0	0
$682$	−56.9795	−2.18186
$683$	4.76045	0.182153	0.0910767	−	0.995844i	$-0.470969\pi$
$683$	4.76045	0.182153	0.0910767	+	0.995844i	$0.470969\pi$
$684$	0	0
$685$	14.6903	0.561286
$686$	16.0143	0.611428
$687$	0	0
$688$	−7.32401	−0.279225
$689$	−1.90766	−0.0726761
$690$	0	0
$691$	−20.2219	−0.769279	−0.384639	−	0.923067i	$-0.625674\pi$
$691$	−20.2219	−0.769279	−0.384639	+	0.923067i	$0.625674\pi$
$692$	−23.0354	−0.875674
$693$	0	0
$694$	5.62691	0.213595
$695$	17.3662	0.658739
$696$	0	0
$697$	−15.9298	−0.603385
$698$	−15.1306	−0.572703
$699$	0	0
$700$	2.91729	0.110263
$701$	22.0845	0.834119	0.417059	−	0.908879i	$-0.363061\pi$
$701$	22.0845	0.834119	0.417059	+	0.908879i	$0.363061\pi$
$702$	0	0
$703$	−0.406728	−0.0153401
$704$	−6.51056	−0.245376
$705$	0	0
$706$	35.7816	1.34666
$707$	23.9441	0.900510
$708$	0	0
$709$	5.59896	0.210273	0.105137	−	0.994458i	$-0.466472\pi$
$709$	5.59896	0.210273	0.105137	+	0.994458i	$0.466472\pi$
$710$	14.6480	0.549731
$711$	0	0
$712$	6.00000	0.224860
$713$	43.9441	1.64572
$714$	0	0
$715$	5.29606	0.198061
$716$	−24.2835	−0.907518
$717$	0	0
$718$	4.48260	0.167289
$719$	−40.2921	−1.50264	−0.751321	−	0.659937i	$-0.770583\pi$
$719$	−40.2921	−1.50264	−0.751321	+	0.659937i	$0.770583\pi$
$720$	0	0
$721$	−25.7114	−0.957542
$722$	18.8346	0.700950
$723$	0	0
$724$	2.16543	0.0804775
$725$	5.32401	0.197729
$726$	0	0
$727$	11.2538	0.417381	0.208691	−	0.977982i	$-0.433080\pi$
$727$	11.2538	0.417381	0.208691	+	0.977982i	$0.433080\pi$
$728$	2.37309	0.0879524
$729$	0	0
$730$	11.0211	0.407910
$731$	−18.3874	−0.680081
$732$	0	0
$733$	−42.2892	−1.56199	−0.780994	−	0.624539i	$-0.785287\pi$
$733$	−42.2892	−1.56199	−0.780994	+	0.624539i	$0.785287\pi$
$734$	21.5653	0.795990
$735$	0	0
$736$	5.02112	0.185081
$737$	35.3383	1.30170
$738$	0	0
$739$	−29.3103	−1.07820	−0.539099	−	0.842242i	$-0.681235\pi$
$739$	−29.3103	−1.07820	−0.539099	+	0.842242i	$0.681235\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−6.84141	−0.251156
$743$	4.39245	0.161144	0.0805718	−	0.996749i	$-0.474325\pi$
$743$	4.39245	0.161144	0.0805718	+	0.996749i	$0.474325\pi$
$744$	0	0
$745$	0.207658	0.00760801
$746$	−16.9230	−0.619594
$747$	0	0
$748$	−16.3451	−0.597637
$749$	−44.5248	−1.62690
$750$	0	0
$751$	2.58074	0.0941727	0.0470864	−	0.998891i	$-0.485006\pi$
$751$	2.58074	0.0941727	0.0470864	+	0.998891i	$0.485006\pi$
$752$	5.42784	0.197933
$753$	0	0
$754$	4.33086	0.157720
$755$	0.813457	0.0296047
$756$	0	0
$757$	27.3805	0.995162	0.497581	−	0.867418i	$-0.334222\pi$
$757$	27.3805	0.995162	0.497581	+	0.867418i	$0.334222\pi$
$758$	−31.2710	−1.13581
$759$	0	0
$760$	−0.406728	−0.0147536
$761$	42.9509	1.55697	0.778485	−	0.627663i	$-0.215989\pi$
$761$	42.9509	1.55697	0.778485	+	0.627663i	$0.215989\pi$
$762$	0	0
$763$	12.5528	0.454441
$764$	19.3326	0.699429
$765$	0	0
$766$	1.62691	0.0587828
$767$	−1.16149	−0.0419389
$768$	0	0
$769$	−13.0633	−0.471076	−0.235538	−	0.971865i	$-0.575685\pi$
$769$	−13.0633	−0.471076	−0.235538	+	0.971865i	$0.575685\pi$
$770$	18.9932	0.684466
$771$	0	0
$772$	−20.0422	−0.721336
$773$	15.3662	0.552685	0.276343	−	0.961059i	$-0.410878\pi$
$773$	15.3662	0.552685	0.276343	+	0.961059i	$0.410878\pi$
$774$	0	0
$775$	−8.75186	−0.314376
$776$	−2.34513	−0.0841852
$777$	0	0
$778$	−31.6412	−1.13439
$779$	−2.58074	−0.0924648
$780$	0	0
$781$	95.3668	3.41249
$782$	12.6058	0.450782
$783$	0	0
$784$	1.51056	0.0539485
$785$	5.32401	0.190022
$786$	0	0
$787$	10.9316	0.389668	0.194834	−	0.980836i	$-0.437583\pi$
$787$	10.9316	0.389668	0.194834	+	0.980836i	$0.437583\pi$
$788$	−15.6269	−0.556686
$789$	0	0
$790$	1.75870	0.0625717
$791$	26.7182	0.949990
$792$	0	0
$793$	1.07703	0.0382464
$794$	39.2961	1.39456
$795$	0	0
$796$	20.2835	0.718931
$797$	24.2921	0.860471	0.430235	−	0.902717i	$-0.358431\pi$
$797$	24.2921	0.860471	0.430235	+	0.902717i	$0.358431\pi$
$798$	0	0
$799$	13.6269	0.482086
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−9.66914	−0.341429
$803$	71.7536	2.53213
$804$	0	0
$805$	−14.6480	−0.516275
$806$	−7.11926	−0.250765
$807$	0	0
$808$	−8.20766	−0.288744
$809$	40.0422	1.40781	0.703905	−	0.710294i	$-0.251438\pi$
$809$	40.0422	1.40781	0.703905	+	0.710294i	$0.251438\pi$
$810$	0	0
$811$	−40.6343	−1.42686	−0.713432	−	0.700724i	$-0.752860\pi$
$811$	−40.6343	−1.42686	−0.713432	+	0.700724i	$0.752860\pi$
$812$	15.5317	0.545055
$813$	0	0
$814$	−6.51056	−0.228195
$815$	15.2008	0.532461
$816$	0	0
$817$	−2.97888	−0.104218
$818$	26.2749	0.918682
$819$	0	0
$820$	−6.34513	−0.221582
$821$	41.4787	1.44762	0.723808	−	0.690002i	$-0.242390\pi$
$821$	41.4787	1.44762	0.723808	+	0.690002i	$0.242390\pi$
$822$	0	0
$823$	−5.27610	−0.183913	−0.0919566	−	0.995763i	$-0.529312\pi$
$823$	−5.27610	−0.183913	−0.0919566	+	0.995763i	$0.529312\pi$
$824$	8.81346	0.307031
$825$	0	0
$826$	−4.16543	−0.144934
$827$	−1.76438	−0.0613537	−0.0306768	−	0.999529i	$-0.509766\pi$
$827$	−1.76438	−0.0613537	−0.0306768	+	0.999529i	$0.509766\pi$
$828$	0	0
$829$	16.3029	0.566223	0.283112	−	0.959087i	$-0.408633\pi$
$829$	16.3029	0.566223	0.283112	+	0.959087i	$0.408633\pi$
$830$	−7.05476	−0.244874
$831$	0	0
$832$	−0.813457	−0.0282015
$833$	3.79234	0.131397
$834$	0	0
$835$	12.4826	0.431978
$836$	−2.64803	−0.0915840
$837$	0	0
$838$	−30.1940	−1.04303
$839$	−14.3731	−0.496214	−0.248107	−	0.968733i	$-0.579808\pi$
$839$	−14.3731	−0.496214	−0.248107	+	0.968733i	$0.579808\pi$
$840$	0	0
$841$	−0.654870	−0.0225817
$842$	−22.0000	−0.758170
$843$	0	0
$844$	18.7182	0.644308
$845$	−12.3383	−0.424450
$846$	0	0
$847$	91.5659	3.14624
$848$	2.34513	0.0805321
$849$	0	0
$850$	−2.51056	−0.0861114
$851$	5.02112	0.172122
$852$	0	0
$853$	43.2961	1.48243	0.741214	−	0.671268i	$-0.234250\pi$
$853$	43.2961	1.48243	0.741214	+	0.671268i	$0.234250\pi$
$854$	3.86253	0.132173
$855$	0	0
$856$	15.2624	0.521659
$857$	49.0776	1.67646	0.838230	−	0.545317i	$-0.183591\pi$
$857$	49.0776	1.67646	0.838230	+	0.545317i	$0.183591\pi$
$858$	0	0
$859$	15.5374	0.530128	0.265064	−	0.964231i	$-0.414607\pi$
$859$	15.5374	0.530128	0.265064	+	0.964231i	$0.414607\pi$
$860$	−7.32401	−0.249747
$861$	0	0
$862$	25.5653	0.870758
$863$	36.0901	1.22852	0.614261	−	0.789103i	$-0.289454\pi$
$863$	36.0901	1.22852	0.614261	+	0.789103i	$0.289454\pi$
$864$	0	0
$865$	−23.0354	−0.783227
$866$	−32.0422	−1.08884
$867$	0	0
$868$	−25.5317	−0.866601
$869$	11.4501	0.388419
$870$	0	0
$871$	4.41532	0.149607
$872$	−4.30290	−0.145715
$873$	0	0
$874$	2.04223	0.0690795
$875$	2.91729	0.0986223
$876$	0	0
$877$	29.3240	0.990202	0.495101	−	0.868836i	$-0.335131\pi$
$877$	29.3240	0.990202	0.495101	+	0.868836i	$0.335131\pi$
$878$	−3.93840	−0.132915
$879$	0	0
$880$	−6.51056	−0.219471
$881$	−23.8066	−0.802065	−0.401033	−	0.916064i	$-0.631349\pi$
$881$	−23.8066	−0.802065	−0.401033	+	0.916064i	$0.631349\pi$
$882$	0	0
$883$	−3.94699	−0.132827	−0.0664134	−	0.997792i	$-0.521156\pi$
$883$	−3.94699	−0.132827	−0.0664134	+	0.997792i	$0.521156\pi$
$884$	−2.04223	−0.0686876
$885$	0	0
$886$	−0.0758724	−0.00254898
$887$	−1.96346	−0.0659264	−0.0329632	−	0.999457i	$-0.510494\pi$
$887$	−1.96346	−0.0659264	−0.0329632	+	0.999457i	$0.510494\pi$
$888$	0	0
$889$	−25.1306	−0.842854
$890$	6.00000	0.201120
$891$	0	0
$892$	−15.7307	−0.526704
$893$	2.20766	0.0738765
$894$	0	0
$895$	−24.2835	−0.811709
$896$	−2.91729	−0.0974597
$897$	0	0
$898$	19.2961	0.643918
$899$	−46.5950	−1.55403
$900$	0	0
$901$	5.88758	0.196144
$902$	−41.3103	−1.37548
$903$	0	0
$904$	−9.15859	−0.304610
$905$	2.16543	0.0719813
$906$	0	0
$907$	5.89049	0.195590	0.0977952	−	0.995207i	$-0.468821\pi$
$907$	5.89049	0.195590	0.0977952	+	0.995207i	$0.468821\pi$
$908$	−12.8836	−0.427559
$909$	0	0
$910$	2.37309	0.0786670
$911$	27.9527	0.926113	0.463057	−	0.886329i	$-0.346753\pi$
$911$	27.9527	0.926113	0.463057	+	0.886329i	$0.346753\pi$
$912$	0	0
$913$	−45.9304	−1.52007
$914$	−3.00684	−0.0994575
$915$	0	0
$916$	5.25383	0.173591
$917$	−39.1729	−1.29360
$918$	0	0
$919$	17.7587	0.585805	0.292903	−	0.956142i	$-0.405379\pi$
$919$	17.7587	0.585805	0.292903	+	0.956142i	$0.405379\pi$
$920$	5.02112	0.165541
$921$	0	0
$922$	−0.633755	−0.0208716
$923$	11.9155	0.392205
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	−18.0422	−0.592904
$927$	0	0
$928$	−5.32401	−0.174769
$929$	−34.2892	−1.12499	−0.562496	−	0.826800i	$-0.690159\pi$
$929$	−34.2892	−1.12499	−0.562496	+	0.826800i	$0.690159\pi$
$930$	0	0
$931$	0.614387	0.0201357
$932$	−28.3172	−0.927560
$933$	0	0
$934$	−23.4758	−0.768150
$935$	−16.3451	−0.534543
$936$	0	0
$937$	−14.0845	−0.460119	−0.230060	−	0.973177i	$-0.573892\pi$
$937$	−14.0845	−0.460119	−0.230060	+	0.973177i	$0.573892\pi$
$938$	15.8346	0.517017
$939$	0	0
$940$	5.42784	0.177037
$941$	−14.2499	−0.464533	−0.232267	−	0.972652i	$-0.574614\pi$
$941$	−14.2499	−0.464533	−0.232267	+	0.972652i	$0.574614\pi$
$942$	0	0
$943$	31.8596	1.03749
$944$	1.42784	0.0464723
$945$	0	0
$946$	−47.6834	−1.55032
$947$	29.8203	0.969029	0.484515	−	0.874783i	$-0.338996\pi$
$947$	29.8203	0.969029	0.484515	+	0.874783i	$0.338996\pi$
$948$	0	0
$949$	8.96520	0.291023
$950$	−0.406728	−0.0131960
$951$	0	0
$952$	−7.32401	−0.237373
$953$	−8.64803	−0.280137	−0.140069	−	0.990142i	$-0.544732\pi$
$953$	−8.64803	−0.280137	−0.140069	+	0.990142i	$0.544732\pi$
$954$	0	0
$955$	19.3326	0.625588
$956$	−14.3788	−0.465043
$957$	0	0
$958$	−8.19907	−0.264900
$959$	42.8557	1.38388
$960$	0	0
$961$	45.5950	1.47081
$962$	−0.813457	−0.0262269
$963$	0	0
$964$	18.2749	0.588596
$965$	−20.0422	−0.645182
$966$	0	0
$967$	−45.0325	−1.44815	−0.724074	−	0.689723i	$-0.757732\pi$
$967$	−45.0325	−1.44815	−0.724074	+	0.689723i	$0.757732\pi$
$968$	−31.3874	−1.00883
$969$	0	0
$970$	−2.34513	−0.0752976
$971$	−26.3029	−0.844100	−0.422050	−	0.906573i	$-0.638689\pi$
$971$	−26.3029	−0.844100	−0.422050	+	0.906573i	$0.638689\pi$
$972$	0	0
$973$	50.6623	1.62416
$974$	−0.953831	−0.0305627
$975$	0	0
$976$	−1.32401	−0.0423807
$977$	−25.6549	−0.820772	−0.410386	−	0.911912i	$-0.634606\pi$
$977$	−25.6549	−0.820772	−0.410386	+	0.911912i	$0.634606\pi$
$978$	0	0
$979$	39.0633	1.24847
$980$	1.51056	0.0482530
$981$	0	0
$982$	19.3383	0.617110
$983$	55.3611	1.76575	0.882873	−	0.469611i	$-0.155606\pi$
$983$	55.3611	1.76575	0.882873	+	0.469611i	$0.155606\pi$
$984$	0	0
$985$	−15.6269	−0.497915
$986$	−13.3662	−0.425668
$987$	0	0
$988$	−0.330856	−0.0105259
$989$	36.7747	1.16937
$990$	0	0
$991$	−3.24814	−0.103181	−0.0515903	−	0.998668i	$-0.516429\pi$
$991$	−3.24814	−0.103181	−0.0515903	+	0.998668i	$0.516429\pi$
$992$	8.75186	0.277872
$993$	0	0
$994$	42.7325	1.35539
$995$	20.2835	0.643031
$996$	0	0
$997$	−12.6480	−0.400567	−0.200284	−	0.979738i	$-0.564186\pi$
$997$	−12.6480	−0.400567	−0.200284	+	0.979738i	$0.564186\pi$
$998$	5.63550	0.178389
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bg.1.3		3
3.2	odd	2		370.2.a.g.1.2	✓	3
12.11	even	2		2960.2.a.u.1.2		3
15.2	even	4		1850.2.b.o.149.5		6
15.8	even	4		1850.2.b.o.149.2		6
15.14	odd	2		1850.2.a.z.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.g.1.2	✓	3	3.2	odd	2
1850.2.a.z.1.2		3	15.14	odd	2
1850.2.b.o.149.2		6	15.8	even	4
1850.2.b.o.149.5		6	15.2	even	4
2960.2.a.u.1.2		3	12.11	even	2
3330.2.a.bg.1.3		3	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.91729 q^{7} -1.00000 q^{8} -1.00000 q^{10} -6.51056 q^{11} -0.813457 q^{13} -2.91729 q^{14} +1.00000 q^{16} +2.51056 q^{17} +0.406728 q^{19} +1.00000 q^{20} +6.51056 q^{22} -5.02112 q^{23} +1.00000 q^{25} +0.813457 q^{26} +2.91729 q^{28} +5.32401 q^{29} -8.75186 q^{31} -1.00000 q^{32} -2.51056 q^{34} +2.91729 q^{35} -1.00000 q^{37} -0.406728 q^{38} -1.00000 q^{40} -6.34513 q^{41} -7.32401 q^{43} -6.51056 q^{44} +5.02112 q^{46} +5.42784 q^{47} +1.51056 q^{49} -1.00000 q^{50} -0.813457 q^{52} +2.34513 q^{53} -6.51056 q^{55} -2.91729 q^{56} -5.32401 q^{58} +1.42784 q^{59} -1.32401 q^{61} +8.75186 q^{62} +1.00000 q^{64} -0.813457 q^{65} -5.42784 q^{67} +2.51056 q^{68} -2.91729 q^{70} -14.6480 q^{71} -11.0211 q^{73} +1.00000 q^{74} +0.406728 q^{76} -18.9932 q^{77} -1.75870 q^{79} +1.00000 q^{80} +6.34513 q^{82} +7.05476 q^{83} +2.51056 q^{85} +7.32401 q^{86} +6.51056 q^{88} -6.00000 q^{89} -2.37309 q^{91} -5.02112 q^{92} -5.42784 q^{94} +0.406728 q^{95} +2.34513 q^{97} -1.51056 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 11 q^{11} + q^{14} + 3 q^{16} - q^{17} + 3 q^{20} + 11 q^{22} + 2 q^{23} + 3 q^{25} - q^{28} + 5 q^{29} + 3 q^{31} - 3 q^{32} + q^{34}+ \cdots + 4 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8 - 3 * q^10 - 11 * q^11 + q^14 + 3 * q^16 - q^17 + 3 * q^20 + 11 * q^22 + 2 * q^23 + 3 * q^25 - q^28 + 5 * q^29 + 3 * q^31 - 3 * q^32 + q^34 - q^35 - 3 * q^37 - 3 * q^40 + 9 * q^41 - 11 * q^43 - 11 * q^44 - 2 * q^46 - 2 * q^47 - 4 * q^49 - 3 * q^50 - 21 * q^53 - 11 * q^55 + q^56 - 5 * q^58 - 14 * q^59 + 7 * q^61 - 3 * q^62 + 3 * q^64 + 2 * q^67 - q^68 + q^70 - 22 * q^71 - 16 * q^73 + 3 * q^74 - 7 * q^77 - 26 * q^79 + 3 * q^80 - 9 * q^82 - 2 * q^83 - q^85 + 11 * q^86 + 11 * q^88 - 18 * q^89 - 12 * q^91 + 2 * q^92 + 2 * q^94 - 21 * q^97 + 4 * q^98

Introduction

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