LMFDB - Embedded newform 3328.2.a.bh.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,3,0,-1,0,6] 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:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10278 q^{3} +1.00000 q^{5} +2.10278 q^{7} +1.42166 q^{9} +5.62721 q^{11} +1.00000 q^{13} -2.10278 q^{15} +1.00000 q^{17} +4.00000 q^{19} -4.42166 q^{21} -1.62721 q^{23} -4.00000 q^{25} +3.31889 q^{27} +7.83276 q^{29} +9.62721 q^{31} -11.8328 q^{33} +2.10278 q^{35} +0.421663 q^{37} -2.10278 q^{39} -5.83276 q^{41} +0.475562 q^{43} +1.42166 q^{45} -4.68111 q^{47} -2.57834 q^{49} -2.10278 q^{51} +4.57834 q^{53} +5.62721 q^{55} -8.41110 q^{57} -8.67609 q^{59} -12.6761 q^{61} +2.98944 q^{63} +1.00000 q^{65} +12.2056 q^{67} +3.42166 q^{69} +9.15165 q^{71} +2.57834 q^{73} +8.41110 q^{75} +11.8328 q^{77} -14.3033 q^{79} -11.2439 q^{81} +8.20555 q^{83} +1.00000 q^{85} -16.4705 q^{87} -11.2544 q^{89} +2.10278 q^{91} -20.2439 q^{93} +4.00000 q^{95} +10.6761 q^{97} +8.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{5} - q^{7} + 6 q^{9} + 4 q^{11} + 3 q^{13} + q^{15} + 3 q^{17} + 12 q^{19} - 15 q^{21} + 8 q^{23} - 12 q^{25} + 19 q^{27} - 4 q^{29} + 16 q^{31} - 8 q^{33} - q^{35} + 3 q^{37} + q^{39}+ \cdots + 24 q^{99}+O(q^{100}) $ 3 * q + q^3 + 3 * q^5 - q^7 + 6 * q^9 + 4 * q^11 + 3 * q^13 + q^15 + 3 * q^17 + 12 * q^19 - 15 * q^21 + 8 * q^23 - 12 * q^25 + 19 * q^27 - 4 * q^29 + 16 * q^31 - 8 * q^33 - q^35 + 3 * q^37 + q^39 + 10 * q^41 + 7 * q^43 + 6 * q^45 - 5 * q^47 - 6 * q^49 + q^51 + 12 * q^53 + 4 * q^55 + 4 * q^57 - 2 * q^59 - 14 * q^61 - 22 * q^63 + 3 * q^65 + 22 * q^67 + 12 * q^69 + 9 * q^71 + 6 * q^73 - 4 * q^75 + 8 * q^77 - 6 * q^79 + 23 * q^81 + 10 * q^83 + 3 * q^85 - 40 * q^87 - 8 * q^89 - q^91 - 4 * q^93 + 12 * q^95 + 8 * q^97 + 24 * 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
$3$	−2.10278	−1.21404	−0.607019	−	0.794687i	$-0.707635\pi$
$3$	−2.10278	−1.21404	−0.607019	+	0.794687i	$0.707635\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	2.10278	0.794774	0.397387	−	0.917651i	$-0.369917\pi$
$7$	2.10278	0.794774	0.397387	+	0.917651i	$0.369917\pi$
$8$	0	0
$9$	1.42166	0.473888
$10$	0	0
$11$	5.62721	1.69667	0.848334	−	0.529461i	$-0.177606\pi$
$11$	5.62721	1.69667	0.848334	+	0.529461i	$0.177606\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.10278	−0.542934
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−4.42166	−0.964886
$22$	0	0
$23$	−1.62721	−0.339297	−0.169649	−	0.985505i	$-0.554263\pi$
$23$	−1.62721	−0.339297	−0.169649	+	0.985505i	$0.554263\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	3.31889	0.638720
$28$	0	0
$29$	7.83276	1.45451	0.727254	−	0.686369i	$-0.240796\pi$
$29$	7.83276	1.45451	0.727254	+	0.686369i	$0.240796\pi$
$30$	0	0
$31$	9.62721	1.72910	0.864549	−	0.502548i	$-0.167604\pi$
$31$	9.62721	1.72910	0.864549	+	0.502548i	$0.167604\pi$
$32$	0	0
$33$	−11.8328	−2.05982
$34$	0	0
$35$	2.10278	0.355434
$36$	0	0
$37$	0.421663	0.0693210	0.0346605	−	0.999399i	$-0.488965\pi$
$37$	0.421663	0.0693210	0.0346605	+	0.999399i	$0.488965\pi$
$38$	0	0
$39$	−2.10278	−0.336713
$40$	0	0
$41$	−5.83276	−0.910925	−0.455462	−	0.890255i	$-0.650526\pi$
$41$	−5.83276	−0.910925	−0.455462	+	0.890255i	$0.650526\pi$
$42$	0	0
$43$	0.475562	0.0725225	0.0362613	−	0.999342i	$-0.488455\pi$
$43$	0.475562	0.0725225	0.0362613	+	0.999342i	$0.488455\pi$
$44$	0	0
$45$	1.42166	0.211929
$46$	0	0
$47$	−4.68111	−0.682810	−0.341405	−	0.939916i	$-0.610903\pi$
$47$	−4.68111	−0.682810	−0.341405	+	0.939916i	$0.610903\pi$
$48$	0	0
$49$	−2.57834	−0.368334
$50$	0	0
$51$	−2.10278	−0.294447
$52$	0	0
$53$	4.57834	0.628883	0.314441	−	0.949277i	$-0.398183\pi$
$53$	4.57834	0.628883	0.314441	+	0.949277i	$0.398183\pi$
$54$	0	0
$55$	5.62721	0.758773
$56$	0	0
$57$	−8.41110	−1.11408
$58$	0	0
$59$	−8.67609	−1.12953	−0.564765	−	0.825252i	$-0.691033\pi$
$59$	−8.67609	−1.12953	−0.564765	+	0.825252i	$0.691033\pi$
$60$	0	0
$61$	−12.6761	−1.62301	−0.811503	−	0.584348i	$-0.801350\pi$
$61$	−12.6761	−1.62301	−0.811503	+	0.584348i	$0.801350\pi$
$62$	0	0
$63$	2.98944	0.376634
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	12.2056	1.49115	0.745573	−	0.666424i	$-0.232176\pi$
$67$	12.2056	1.49115	0.745573	+	0.666424i	$0.232176\pi$
$68$	0	0
$69$	3.42166	0.411920
$70$	0	0
$71$	9.15165	1.08610	0.543051	−	0.839700i	$-0.317269\pi$
$71$	9.15165	1.08610	0.543051	+	0.839700i	$0.317269\pi$
$72$	0	0
$73$	2.57834	0.301772	0.150886	−	0.988551i	$-0.451787\pi$
$73$	2.57834	0.301772	0.150886	+	0.988551i	$0.451787\pi$
$74$	0	0
$75$	8.41110	0.971230
$76$	0	0
$77$	11.8328	1.34847
$78$	0	0
$79$	−14.3033	−1.60925	−0.804624	−	0.593785i	$-0.797633\pi$
$79$	−14.3033	−1.60925	−0.804624	+	0.593785i	$0.797633\pi$
$80$	0	0
$81$	−11.2439	−1.24932
$82$	0	0
$83$	8.20555	0.900676	0.450338	−	0.892858i	$-0.351303\pi$
$83$	8.20555	0.900676	0.450338	+	0.892858i	$0.351303\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−16.4705	−1.76583
$88$	0	0
$89$	−11.2544	−1.19297	−0.596483	−	0.802625i	$-0.703436\pi$
$89$	−11.2544	−1.19297	−0.596483	+	0.802625i	$0.703436\pi$
$90$	0	0
$91$	2.10278	0.220431
$92$	0	0
$93$	−20.2439	−2.09919
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	10.6761	1.08399	0.541996	−	0.840381i	$-0.317669\pi$
$97$	10.6761	1.08399	0.541996	+	0.840381i	$0.317669\pi$
$98$	0	0
$99$	8.00000	0.804030
$100$	0	0
$101$	−9.15667	−0.911123	−0.455562	−	0.890204i	$-0.650562\pi$
$101$	−9.15667	−0.911123	−0.455562	+	0.890204i	$0.650562\pi$
$102$	0	0
$103$	−5.36222	−0.528356	−0.264178	−	0.964474i	$-0.585101\pi$
$103$	−5.36222	−0.528356	−0.264178	+	0.964474i	$0.585101\pi$
$104$	0	0
$105$	−4.42166	−0.431510
$106$	0	0
$107$	6.37279	0.616081	0.308040	−	0.951373i	$-0.400327\pi$
$107$	6.37279	0.616081	0.308040	+	0.951373i	$0.400327\pi$
$108$	0	0
$109$	−14.0872	−1.34931	−0.674654	−	0.738134i	$-0.735707\pi$
$109$	−14.0872	−1.34931	−0.674654	+	0.738134i	$0.735707\pi$
$110$	0	0
$111$	−0.886662	−0.0841583
$112$	0	0
$113$	−4.84333	−0.455622	−0.227811	−	0.973705i	$-0.573157\pi$
$113$	−4.84333	−0.455622	−0.227811	+	0.973705i	$0.573157\pi$
$114$	0	0
$115$	−1.62721	−0.151738
$116$	0	0
$117$	1.42166	0.131433
$118$	0	0
$119$	2.10278	0.192761
$120$	0	0
$121$	20.6655	1.87868
$122$	0	0
$123$	12.2650	1.10590
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	10.0383	0.890756	0.445378	−	0.895343i	$-0.353069\pi$
$127$	10.0383	0.890756	0.445378	+	0.895343i	$0.353069\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	8.47556	0.740513	0.370257	−	0.928929i	$-0.379270\pi$
$131$	8.47556	0.740513	0.370257	+	0.928929i	$0.379270\pi$
$132$	0	0
$133$	8.41110	0.729335
$134$	0	0
$135$	3.31889	0.285644
$136$	0	0
$137$	19.0872	1.63073	0.815364	−	0.578948i	$-0.196537\pi$
$137$	19.0872	1.63073	0.815364	+	0.578948i	$0.196537\pi$
$138$	0	0
$139$	−3.99498	−0.338850	−0.169425	−	0.985543i	$-0.554191\pi$
$139$	−3.99498	−0.338850	−0.169425	+	0.985543i	$0.554191\pi$
$140$	0	0
$141$	9.84333	0.828958
$142$	0	0
$143$	5.62721	0.470571
$144$	0	0
$145$	7.83276	0.650476
$146$	0	0
$147$	5.42166	0.447171
$148$	0	0
$149$	3.15667	0.258605	0.129302	−	0.991605i	$-0.458726\pi$
$149$	3.15667	0.258605	0.129302	+	0.991605i	$0.458726\pi$
$150$	0	0
$151$	2.36776	0.192686	0.0963429	−	0.995348i	$-0.469285\pi$
$151$	2.36776	0.192686	0.0963429	+	0.995348i	$0.469285\pi$
$152$	0	0
$153$	1.42166	0.114935
$154$	0	0
$155$	9.62721	0.773276
$156$	0	0
$157$	15.9305	1.27139	0.635697	−	0.771939i	$-0.280713\pi$
$157$	15.9305	1.27139	0.635697	+	0.771939i	$0.280713\pi$
$158$	0	0
$159$	−9.62721	−0.763488
$160$	0	0
$161$	−3.42166	−0.269665
$162$	0	0
$163$	−3.52946	−0.276449	−0.138224	−	0.990401i	$-0.544140\pi$
$163$	−3.52946	−0.276449	−0.138224	+	0.990401i	$0.544140\pi$
$164$	0	0
$165$	−11.8328	−0.921179
$166$	0	0
$167$	−15.7250	−1.21683	−0.608417	−	0.793617i	$-0.708195\pi$
$167$	−15.7250	−1.21683	−0.608417	+	0.793617i	$0.708195\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.68665	0.434869
$172$	0	0
$173$	14.4111	1.09566	0.547828	−	0.836591i	$-0.315455\pi$
$173$	14.4111	1.09566	0.547828	+	0.836591i	$0.315455\pi$
$174$	0	0
$175$	−8.41110	−0.635819
$176$	0	0
$177$	18.2439	1.37129
$178$	0	0
$179$	1.69167	0.126442	0.0632209	−	0.998000i	$-0.479863\pi$
$179$	1.69167	0.126442	0.0632209	+	0.998000i	$0.479863\pi$
$180$	0	0
$181$	17.2544	1.28251	0.641256	−	0.767327i	$-0.278414\pi$
$181$	17.2544	1.28251	0.641256	+	0.767327i	$0.278414\pi$
$182$	0	0
$183$	26.6550	1.97039
$184$	0	0
$185$	0.421663	0.0310013
$186$	0	0
$187$	5.62721	0.411503
$188$	0	0
$189$	6.97887	0.507638
$190$	0	0
$191$	6.03831	0.436917	0.218459	−	0.975846i	$-0.429897\pi$
$191$	6.03831	0.436917	0.218459	+	0.975846i	$0.429897\pi$
$192$	0	0
$193$	13.2544	0.954074	0.477037	−	0.878883i	$-0.341711\pi$
$193$	13.2544	0.954074	0.477037	+	0.878883i	$0.341711\pi$
$194$	0	0
$195$	−2.10278	−0.150583
$196$	0	0
$197$	−22.9305	−1.63373	−0.816866	−	0.576828i	$-0.804290\pi$
$197$	−22.9305	−1.63373	−0.816866	+	0.576828i	$0.804290\pi$
$198$	0	0
$199$	0.951124	0.0674234	0.0337117	−	0.999432i	$-0.489267\pi$
$199$	0.951124	0.0674234	0.0337117	+	0.999432i	$0.489267\pi$
$200$	0	0
$201$	−25.6655	−1.81031
$202$	0	0
$203$	16.4705	1.15601
$204$	0	0
$205$	−5.83276	−0.407378
$206$	0	0
$207$	−2.31335	−0.160789
$208$	0	0
$209$	22.5089	1.55697
$210$	0	0
$211$	11.7300	0.807526	0.403763	−	0.914864i	$-0.367702\pi$
$211$	11.7300	0.807526	0.403763	+	0.914864i	$0.367702\pi$
$212$	0	0
$213$	−19.2439	−1.31857
$214$	0	0
$215$	0.475562	0.0324331
$216$	0	0
$217$	20.2439	1.37424
$218$	0	0
$219$	−5.42166	−0.366362
$220$	0	0
$221$	1.00000	0.0672673
$222$	0	0
$223$	4.68111	0.313470	0.156735	−	0.987641i	$-0.449903\pi$
$223$	4.68111	0.313470	0.156735	+	0.987641i	$0.449903\pi$
$224$	0	0
$225$	−5.68665	−0.379110
$226$	0	0
$227$	11.7350	0.778880	0.389440	−	0.921052i	$-0.372669\pi$
$227$	11.7350	0.778880	0.389440	+	0.921052i	$0.372669\pi$
$228$	0	0
$229$	5.84333	0.386138	0.193069	−	0.981185i	$-0.438156\pi$
$229$	5.84333	0.386138	0.193069	+	0.981185i	$0.438156\pi$
$230$	0	0
$231$	−24.8816	−1.63709
$232$	0	0
$233$	25.2439	1.65378	0.826890	−	0.562363i	$-0.190108\pi$
$233$	25.2439	1.65378	0.826890	+	0.562363i	$0.190108\pi$
$234$	0	0
$235$	−4.68111	−0.305362
$236$	0	0
$237$	30.0766	1.95369
$238$	0	0
$239$	−11.4650	−0.741609	−0.370805	−	0.928711i	$-0.620918\pi$
$239$	−11.4650	−0.741609	−0.370805	+	0.928711i	$0.620918\pi$
$240$	0	0
$241$	−2.09775	−0.135128	−0.0675640	−	0.997715i	$-0.521523\pi$
$241$	−2.09775	−0.135128	−0.0675640	+	0.997715i	$0.521523\pi$
$242$	0	0
$243$	13.6867	0.877999
$244$	0	0
$245$	−2.57834	−0.164724
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−17.2544	−1.09345
$250$	0	0
$251$	14.7839	0.933151	0.466575	−	0.884481i	$-0.345488\pi$
$251$	14.7839	0.933151	0.466575	+	0.884481i	$0.345488\pi$
$252$	0	0
$253$	−9.15667	−0.575675
$254$	0	0
$255$	−2.10278	−0.131681
$256$	0	0
$257$	3.31335	0.206681	0.103340	−	0.994646i	$-0.467047\pi$
$257$	3.31335	0.206681	0.103340	+	0.994646i	$0.467047\pi$
$258$	0	0
$259$	0.886662	0.0550945
$260$	0	0
$261$	11.1355	0.689273
$262$	0	0
$263$	−17.4217	−1.07427	−0.537133	−	0.843498i	$-0.680493\pi$
$263$	−17.4217	−1.07427	−0.537133	+	0.843498i	$0.680493\pi$
$264$	0	0
$265$	4.57834	0.281245
$266$	0	0
$267$	23.6655	1.44831
$268$	0	0
$269$	3.32391	0.202662	0.101331	−	0.994853i	$-0.467690\pi$
$269$	3.32391	0.202662	0.101331	+	0.994853i	$0.467690\pi$
$270$	0	0
$271$	−14.0333	−0.852462	−0.426231	−	0.904614i	$-0.640159\pi$
$271$	−14.0333	−0.852462	−0.426231	+	0.904614i	$0.640159\pi$
$272$	0	0
$273$	−4.42166	−0.267611
$274$	0	0
$275$	−22.5089	−1.35733
$276$	0	0
$277$	−1.01056	−0.0607188	−0.0303594	−	0.999539i	$-0.509665\pi$
$277$	−1.01056	−0.0607188	−0.0303594	+	0.999539i	$0.509665\pi$
$278$	0	0
$279$	13.6867	0.819398
$280$	0	0
$281$	−3.58890	−0.214096	−0.107048	−	0.994254i	$-0.534140\pi$
$281$	−3.58890	−0.214096	−0.107048	+	0.994254i	$0.534140\pi$
$282$	0	0
$283$	−13.2927	−0.790171	−0.395086	−	0.918644i	$-0.629285\pi$
$283$	−13.2927	−0.790171	−0.395086	+	0.918644i	$0.629285\pi$
$284$	0	0
$285$	−8.41110	−0.498231
$286$	0	0
$287$	−12.2650	−0.723979
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−22.4494	−1.31601
$292$	0	0
$293$	9.24386	0.540032	0.270016	−	0.962856i	$-0.412971\pi$
$293$	9.24386	0.540032	0.270016	+	0.962856i	$0.412971\pi$
$294$	0	0
$295$	−8.67609	−0.505141
$296$	0	0
$297$	18.6761	1.08370
$298$	0	0
$299$	−1.62721	−0.0941042
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	19.2544	1.10614
$304$	0	0
$305$	−12.6761	−0.725831
$306$	0	0
$307$	−8.26499	−0.471708	−0.235854	−	0.971789i	$-0.575789\pi$
$307$	−8.26499	−0.471708	−0.235854	+	0.971789i	$0.575789\pi$
$308$	0	0
$309$	11.2756	0.641444
$310$	0	0
$311$	23.6655	1.34195	0.670974	−	0.741480i	$-0.265876\pi$
$311$	23.6655	1.34195	0.670974	+	0.741480i	$0.265876\pi$
$312$	0	0
$313$	0.735011	0.0415453	0.0207726	−	0.999784i	$-0.493387\pi$
$313$	0.735011	0.0415453	0.0207726	+	0.999784i	$0.493387\pi$
$314$	0	0
$315$	2.98944	0.168436
$316$	0	0
$317$	−11.1567	−0.626621	−0.313311	−	0.949651i	$-0.601438\pi$
$317$	−11.1567	−0.626621	−0.313311	+	0.949651i	$0.601438\pi$
$318$	0	0
$319$	44.0766	2.46782
$320$	0	0
$321$	−13.4005	−0.747945
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	29.6222	1.63811
$328$	0	0
$329$	−9.84333	−0.542680
$330$	0	0
$331$	−7.79445	−0.428422	−0.214211	−	0.976787i	$-0.568718\pi$
$331$	−7.79445	−0.428422	−0.214211	+	0.976787i	$0.568718\pi$
$332$	0	0
$333$	0.599463	0.0328503
$334$	0	0
$335$	12.2056	0.666860
$336$	0	0
$337$	19.2439	1.04828	0.524140	−	0.851632i	$-0.324387\pi$
$337$	19.2439	1.04828	0.524140	+	0.851632i	$0.324387\pi$
$338$	0	0
$339$	10.1844	0.553142
$340$	0	0
$341$	54.1744	2.93371
$342$	0	0
$343$	−20.1411	−1.08752
$344$	0	0
$345$	3.42166	0.184216
$346$	0	0
$347$	32.6116	1.75068	0.875342	−	0.483504i	$-0.160636\pi$
$347$	32.6116	1.75068	0.875342	+	0.483504i	$0.160636\pi$
$348$	0	0
$349$	−31.8222	−1.70340	−0.851702	−	0.524027i	$-0.824429\pi$
$349$	−31.8222	−1.70340	−0.851702	+	0.524027i	$0.824429\pi$
$350$	0	0
$351$	3.31889	0.177149
$352$	0	0
$353$	20.0978	1.06970	0.534848	−	0.844948i	$-0.320369\pi$
$353$	20.0978	1.06970	0.534848	+	0.844948i	$0.320369\pi$
$354$	0	0
$355$	9.15165	0.485719
$356$	0	0
$357$	−4.42166	−0.234019
$358$	0	0
$359$	−8.68614	−0.458437	−0.229218	−	0.973375i	$-0.573617\pi$
$359$	−8.68614	−0.458437	−0.229218	+	0.973375i	$0.573617\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−43.4550	−2.28079
$364$	0	0
$365$	2.57834	0.134956
$366$	0	0
$367$	−9.36222	−0.488704	−0.244352	−	0.969687i	$-0.578575\pi$
$367$	−9.36222	−0.488704	−0.244352	+	0.969687i	$0.578575\pi$
$368$	0	0
$369$	−8.29222	−0.431676
$370$	0	0
$371$	9.62721	0.499820
$372$	0	0
$373$	−21.9789	−1.13802	−0.569011	−	0.822330i	$-0.692674\pi$
$373$	−21.9789	−1.13802	−0.569011	+	0.822330i	$0.692674\pi$
$374$	0	0
$375$	18.9250	0.977282
$376$	0	0
$377$	7.83276	0.403408
$378$	0	0
$379$	−4.05944	−0.208519	−0.104260	−	0.994550i	$-0.533247\pi$
$379$	−4.05944	−0.208519	−0.104260	+	0.994550i	$0.533247\pi$
$380$	0	0
$381$	−21.1083	−1.08141
$382$	0	0
$383$	−7.39551	−0.377893	−0.188947	−	0.981987i	$-0.560507\pi$
$383$	−7.39551	−0.377893	−0.188947	+	0.981987i	$0.560507\pi$
$384$	0	0
$385$	11.8328	0.603053
$386$	0	0
$387$	0.676089	0.0343675
$388$	0	0
$389$	−13.4217	−0.680505	−0.340253	−	0.940334i	$-0.610513\pi$
$389$	−13.4217	−0.680505	−0.340253	+	0.940334i	$0.610513\pi$
$390$	0	0
$391$	−1.62721	−0.0822917
$392$	0	0
$393$	−17.8222	−0.899011
$394$	0	0
$395$	−14.3033	−0.719677
$396$	0	0
$397$	33.1355	1.66303	0.831513	−	0.555506i	$-0.187475\pi$
$397$	33.1355	1.66303	0.831513	+	0.555506i	$0.187475\pi$
$398$	0	0
$399$	−17.6867	−0.885440
$400$	0	0
$401$	−14.2650	−0.712360	−0.356180	−	0.934417i	$-0.615921\pi$
$401$	−14.2650	−0.712360	−0.356180	+	0.934417i	$0.615921\pi$
$402$	0	0
$403$	9.62721	0.479566
$404$	0	0
$405$	−11.2439	−0.558712
$406$	0	0
$407$	2.37279	0.117615
$408$	0	0
$409$	−5.93051	−0.293245	−0.146623	−	0.989193i	$-0.546840\pi$
$409$	−5.93051	−0.293245	−0.146623	+	0.989193i	$0.546840\pi$
$410$	0	0
$411$	−40.1361	−1.97977
$412$	0	0
$413$	−18.2439	−0.897722
$414$	0	0
$415$	8.20555	0.402795
$416$	0	0
$417$	8.40054	0.411376
$418$	0	0
$419$	25.5628	1.24882	0.624411	−	0.781096i	$-0.285339\pi$
$419$	25.5628	1.24882	0.624411	+	0.781096i	$0.285339\pi$
$420$	0	0
$421$	−26.6655	−1.29960	−0.649799	−	0.760106i	$-0.725147\pi$
$421$	−26.6655	−1.29960	−0.649799	+	0.760106i	$0.725147\pi$
$422$	0	0
$423$	−6.65496	−0.323575
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−26.6550	−1.28992
$428$	0	0
$429$	−11.8328	−0.571291
$430$	0	0
$431$	−12.4061	−0.597580	−0.298790	−	0.954319i	$-0.596583\pi$
$431$	−12.4061	−0.597580	−0.298790	+	0.954319i	$0.596583\pi$
$432$	0	0
$433$	0.108315	0.00520526	0.00260263	−	0.999997i	$-0.499172\pi$
$433$	0.108315	0.00520526	0.00260263	+	0.999997i	$0.499172\pi$
$434$	0	0
$435$	−16.4705	−0.789702
$436$	0	0
$437$	−6.50885	−0.311361
$438$	0	0
$439$	6.24386	0.298003	0.149002	−	0.988837i	$-0.452394\pi$
$439$	6.24386	0.298003	0.149002	+	0.988837i	$0.452394\pi$
$440$	0	0
$441$	−3.66553	−0.174549
$442$	0	0
$443$	14.0333	0.666742	0.333371	−	0.942796i	$-0.391814\pi$
$443$	14.0333	0.666742	0.333371	+	0.942796i	$0.391814\pi$
$444$	0	0
$445$	−11.2544	−0.533511
$446$	0	0
$447$	−6.63778	−0.313956
$448$	0	0
$449$	5.51941	0.260477	0.130239	−	0.991483i	$-0.458426\pi$
$449$	5.51941	0.260477	0.130239	+	0.991483i	$0.458426\pi$
$450$	0	0
$451$	−32.8222	−1.54554
$452$	0	0
$453$	−4.97887	−0.233928
$454$	0	0
$455$	2.10278	0.0985796
$456$	0	0
$457$	20.0766	0.939145	0.469572	−	0.882894i	$-0.344408\pi$
$457$	20.0766	0.939145	0.469572	+	0.882894i	$0.344408\pi$
$458$	0	0
$459$	3.31889	0.154912
$460$	0	0
$461$	12.4217	0.578535	0.289267	−	0.957248i	$-0.406588\pi$
$461$	12.4217	0.578535	0.289267	+	0.957248i	$0.406588\pi$
$462$	0	0
$463$	−11.1184	−0.516714	−0.258357	−	0.966050i	$-0.583181\pi$
$463$	−11.1184	−0.516714	−0.258357	+	0.966050i	$0.583181\pi$
$464$	0	0
$465$	−20.2439	−0.938787
$466$	0	0
$467$	−1.09724	−0.0507740	−0.0253870	−	0.999678i	$-0.508082\pi$
$467$	−1.09724	−0.0507740	−0.0253870	+	0.999678i	$0.508082\pi$
$468$	0	0
$469$	25.6655	1.18512
$470$	0	0
$471$	−33.4983	−1.54352
$472$	0	0
$473$	2.67609	0.123047
$474$	0	0
$475$	−16.0000	−0.734130
$476$	0	0
$477$	6.50885	0.298020
$478$	0	0
$479$	14.9844	0.684655	0.342328	−	0.939581i	$-0.388785\pi$
$479$	14.9844	0.684655	0.342328	+	0.939581i	$0.388785\pi$
$480$	0	0
$481$	0.421663	0.0192262
$482$	0	0
$483$	7.19499	0.327383
$484$	0	0
$485$	10.6761	0.484776
$486$	0	0
$487$	23.1950	1.05107	0.525533	−	0.850773i	$-0.323866\pi$
$487$	23.1950	1.05107	0.525533	+	0.850773i	$0.323866\pi$
$488$	0	0
$489$	7.42166	0.335619
$490$	0	0
$491$	22.9844	1.03727	0.518636	−	0.854995i	$-0.326440\pi$
$491$	22.9844	1.03727	0.518636	+	0.854995i	$0.326440\pi$
$492$	0	0
$493$	7.83276	0.352770
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	19.2439	0.863205
$498$	0	0
$499$	13.4983	0.604266	0.302133	−	0.953266i	$-0.402301\pi$
$499$	13.4983	0.604266	0.302133	+	0.953266i	$0.402301\pi$
$500$	0	0
$501$	33.0661	1.47728
$502$	0	0
$503$	−17.5678	−0.783308	−0.391654	−	0.920112i	$-0.628097\pi$
$503$	−17.5678	−0.783308	−0.391654	+	0.920112i	$0.628097\pi$
$504$	0	0
$505$	−9.15667	−0.407467
$506$	0	0
$507$	−2.10278	−0.0933875
$508$	0	0
$509$	−8.31335	−0.368483	−0.184241	−	0.982881i	$-0.558983\pi$
$509$	−8.31335	−0.368483	−0.184241	+	0.982881i	$0.558983\pi$
$510$	0	0
$511$	5.42166	0.239840
$512$	0	0
$513$	13.2756	0.586130
$514$	0	0
$515$	−5.36222	−0.236288
$516$	0	0
$517$	−26.3416	−1.15850
$518$	0	0
$519$	−30.3033	−1.33017
$520$	0	0
$521$	−16.0872	−0.704793	−0.352396	−	0.935851i	$-0.614633\pi$
$521$	−16.0872	−0.704793	−0.352396	+	0.935851i	$0.614633\pi$
$522$	0	0
$523$	−9.09724	−0.397794	−0.198897	−	0.980020i	$-0.563736\pi$
$523$	−9.09724	−0.397794	−0.198897	+	0.980020i	$0.563736\pi$
$524$	0	0
$525$	17.6867	0.771909
$526$	0	0
$527$	9.62721	0.419368
$528$	0	0
$529$	−20.3522	−0.884877
$530$	0	0
$531$	−12.3345	−0.535271
$532$	0	0
$533$	−5.83276	−0.252645
$534$	0	0
$535$	6.37279	0.275520
$536$	0	0
$537$	−3.55721	−0.153505
$538$	0	0
$539$	−14.5089	−0.624940
$540$	0	0
$541$	22.0872	0.949602	0.474801	−	0.880093i	$-0.342520\pi$
$541$	22.0872	0.949602	0.474801	+	0.880093i	$0.342520\pi$
$542$	0	0
$543$	−36.2822	−1.55702
$544$	0	0
$545$	−14.0872	−0.603429
$546$	0	0
$547$	−44.2772	−1.89315	−0.946577	−	0.322477i	$-0.895484\pi$
$547$	−44.2772	−1.89315	−0.946577	+	0.322477i	$0.895484\pi$
$548$	0	0
$549$	−18.0211	−0.769123
$550$	0	0
$551$	31.3311	1.33475
$552$	0	0
$553$	−30.0766	−1.27899
$554$	0	0
$555$	−0.886662	−0.0376367
$556$	0	0
$557$	−36.9094	−1.56390	−0.781951	−	0.623341i	$-0.785775\pi$
$557$	−36.9094	−1.56390	−0.781951	+	0.623341i	$0.785775\pi$
$558$	0	0
$559$	0.475562	0.0201141
$560$	0	0
$561$	−11.8328	−0.499580
$562$	0	0
$563$	1.28057	0.0539698	0.0269849	−	0.999636i	$-0.491409\pi$
$563$	1.28057	0.0539698	0.0269849	+	0.999636i	$0.491409\pi$
$564$	0	0
$565$	−4.84333	−0.203760
$566$	0	0
$567$	−23.6433	−0.992926
$568$	0	0
$569$	37.5089	1.57245	0.786226	−	0.617938i	$-0.212032\pi$
$569$	37.5089	1.57245	0.786226	+	0.617938i	$0.212032\pi$
$570$	0	0
$571$	29.7583	1.24534	0.622672	−	0.782483i	$-0.286047\pi$
$571$	29.7583	1.24534	0.622672	+	0.782483i	$0.286047\pi$
$572$	0	0
$573$	−12.6972	−0.530434
$574$	0	0
$575$	6.50885	0.271438
$576$	0	0
$577$	−20.4605	−0.851781	−0.425891	−	0.904775i	$-0.640039\pi$
$577$	−20.4605	−0.851781	−0.425891	+	0.904775i	$0.640039\pi$
$578$	0	0
$579$	−27.8711	−1.15828
$580$	0	0
$581$	17.2544	0.715834
$582$	0	0
$583$	25.7633	1.06701
$584$	0	0
$585$	1.42166	0.0587785
$586$	0	0
$587$	−9.62721	−0.397358	−0.198679	−	0.980065i	$-0.563665\pi$
$587$	−9.62721	−0.397358	−0.198679	+	0.980065i	$0.563665\pi$
$588$	0	0
$589$	38.5089	1.58673
$590$	0	0
$591$	48.2177	1.98341
$592$	0	0
$593$	−28.6550	−1.17672	−0.588359	−	0.808600i	$-0.700226\pi$
$593$	−28.6550	−1.17672	−0.588359	+	0.808600i	$0.700226\pi$
$594$	0	0
$595$	2.10278	0.0862054
$596$	0	0
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	−39.0661	−1.59620	−0.798098	−	0.602528i	$-0.794160\pi$
$599$	−39.0661	−1.59620	−0.798098	+	0.602528i	$0.794160\pi$
$600$	0	0
$601$	−19.8222	−0.808564	−0.404282	−	0.914634i	$-0.632479\pi$
$601$	−19.8222	−0.808564	−0.404282	+	0.914634i	$0.632479\pi$
$602$	0	0
$603$	17.3522	0.706635
$604$	0	0
$605$	20.6655	0.840173
$606$	0	0
$607$	−12.8917	−0.523257	−0.261629	−	0.965169i	$-0.584260\pi$
$607$	−12.8917	−0.523257	−0.261629	+	0.965169i	$0.584260\pi$
$608$	0	0
$609$	−34.6338	−1.40343
$610$	0	0
$611$	−4.68111	−0.189378
$612$	0	0
$613$	−0.313348	−0.0126560	−0.00632801	−	0.999980i	$-0.502014\pi$
$613$	−0.313348	−0.0126560	−0.00632801	+	0.999980i	$0.502014\pi$
$614$	0	0
$615$	12.2650	0.494572
$616$	0	0
$617$	−6.31335	−0.254166	−0.127083	−	0.991892i	$-0.540561\pi$
$617$	−6.31335	−0.254166	−0.127083	+	0.991892i	$0.540561\pi$
$618$	0	0
$619$	33.1638	1.33297	0.666483	−	0.745520i	$-0.267799\pi$
$619$	33.1638	1.33297	0.666483	+	0.745520i	$0.267799\pi$
$620$	0	0
$621$	−5.40054	−0.216716
$622$	0	0
$623$	−23.6655	−0.948139
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−47.3311	−1.89022
$628$	0	0
$629$	0.421663	0.0168128
$630$	0	0
$631$	9.42669	0.375270	0.187635	−	0.982239i	$-0.439918\pi$
$631$	9.42669	0.375270	0.187635	+	0.982239i	$0.439918\pi$
$632$	0	0
$633$	−24.6655	−0.980367
$634$	0	0
$635$	10.0383	0.398358
$636$	0	0
$637$	−2.57834	−0.102157
$638$	0	0
$639$	13.0106	0.514690
$640$	0	0
$641$	−3.68665	−0.145614	−0.0728070	−	0.997346i	$-0.523196\pi$
$641$	−3.68665	−0.145614	−0.0728070	+	0.997346i	$0.523196\pi$
$642$	0	0
$643$	10.3033	0.406323	0.203161	−	0.979145i	$-0.434878\pi$
$643$	10.3033	0.406323	0.203161	+	0.979145i	$0.434878\pi$
$644$	0	0
$645$	−1.00000	−0.0393750
$646$	0	0
$647$	29.8328	1.17285	0.586423	−	0.810005i	$-0.300535\pi$
$647$	29.8328	1.17285	0.586423	+	0.810005i	$0.300535\pi$
$648$	0	0
$649$	−48.8222	−1.91644
$650$	0	0
$651$	−42.5683	−1.66838
$652$	0	0
$653$	7.47002	0.292325	0.146162	−	0.989261i	$-0.453308\pi$
$653$	7.47002	0.292325	0.146162	+	0.989261i	$0.453308\pi$
$654$	0	0
$655$	8.47556	0.331168
$656$	0	0
$657$	3.66553	0.143006
$658$	0	0
$659$	9.09724	0.354378	0.177189	−	0.984177i	$-0.443300\pi$
$659$	9.09724	0.354378	0.177189	+	0.984177i	$0.443300\pi$
$660$	0	0
$661$	−42.6832	−1.66019	−0.830093	−	0.557626i	$-0.811713\pi$
$661$	−42.6832	−1.66019	−0.830093	+	0.557626i	$0.811713\pi$
$662$	0	0
$663$	−2.10278	−0.0816650
$664$	0	0
$665$	8.41110	0.326168
$666$	0	0
$667$	−12.7456	−0.493511
$668$	0	0
$669$	−9.84333	−0.380565
$670$	0	0
$671$	−71.3311	−2.75370
$672$	0	0
$673$	−18.6655	−0.719503	−0.359752	−	0.933048i	$-0.617139\pi$
$673$	−18.6655	−0.719503	−0.359752	+	0.933048i	$0.617139\pi$
$674$	0	0
$675$	−13.2756	−0.510976
$676$	0	0
$677$	−11.9022	−0.457441	−0.228720	−	0.973492i	$-0.573454\pi$
$677$	−11.9022	−0.457441	−0.228720	+	0.973492i	$0.573454\pi$
$678$	0	0
$679$	22.4494	0.861529
$680$	0	0
$681$	−24.6761	−0.945590
$682$	0	0
$683$	−4.95112	−0.189449	−0.0947247	−	0.995504i	$-0.530197\pi$
$683$	−4.95112	−0.189449	−0.0947247	+	0.995504i	$0.530197\pi$
$684$	0	0
$685$	19.0872	0.729284
$686$	0	0
$687$	−12.2872	−0.468786
$688$	0	0
$689$	4.57834	0.174421
$690$	0	0
$691$	−34.7244	−1.32098	−0.660490	−	0.750835i	$-0.729651\pi$
$691$	−34.7244	−1.32098	−0.660490	+	0.750835i	$0.729651\pi$
$692$	0	0
$693$	16.8222	0.639023
$694$	0	0
$695$	−3.99498	−0.151538
$696$	0	0
$697$	−5.83276	−0.220932
$698$	0	0
$699$	−53.0822	−2.00775
$700$	0	0
$701$	−30.2650	−1.14309	−0.571546	−	0.820570i	$-0.693656\pi$
$701$	−30.2650	−1.14309	−0.571546	+	0.820570i	$0.693656\pi$
$702$	0	0
$703$	1.68665	0.0636133
$704$	0	0
$705$	9.84333	0.370721
$706$	0	0
$707$	−19.2544	−0.724137
$708$	0	0
$709$	49.3311	1.85267	0.926333	−	0.376705i	$-0.122943\pi$
$709$	49.3311	1.85267	0.926333	+	0.376705i	$0.122943\pi$
$710$	0	0
$711$	−20.3345	−0.762602
$712$	0	0
$713$	−15.6655	−0.586679
$714$	0	0
$715$	5.62721	0.210446
$716$	0	0
$717$	24.1083	0.900342
$718$	0	0
$719$	39.3411	1.46718	0.733588	−	0.679595i	$-0.237844\pi$
$719$	39.3411	1.46718	0.733588	+	0.679595i	$0.237844\pi$
$720$	0	0
$721$	−11.2756	−0.419923
$722$	0	0
$723$	4.41110	0.164051
$724$	0	0
$725$	−31.3311	−1.16361
$726$	0	0
$727$	−38.4494	−1.42601	−0.713005	−	0.701159i	$-0.752666\pi$
$727$	−38.4494	−1.42601	−0.713005	+	0.701159i	$0.752666\pi$
$728$	0	0
$729$	4.95164	0.183394
$730$	0	0
$731$	0.475562	0.0175893
$732$	0	0
$733$	−42.3522	−1.56431	−0.782157	−	0.623082i	$-0.785880\pi$
$733$	−42.3522	−1.56431	−0.782157	+	0.623082i	$0.785880\pi$
$734$	0	0
$735$	5.42166	0.199981
$736$	0	0
$737$	68.6832	2.52998
$738$	0	0
$739$	−14.5783	−0.536273	−0.268136	−	0.963381i	$-0.586408\pi$
$739$	−14.5783	−0.536273	−0.268136	+	0.963381i	$0.586408\pi$
$740$	0	0
$741$	−8.41110	−0.308989
$742$	0	0
$743$	2.78891	0.102315	0.0511576	−	0.998691i	$-0.483709\pi$
$743$	2.78891	0.102315	0.0511576	+	0.998691i	$0.483709\pi$
$744$	0	0
$745$	3.15667	0.115652
$746$	0	0
$747$	11.6655	0.426819
$748$	0	0
$749$	13.4005	0.489645
$750$	0	0
$751$	−34.2439	−1.24958	−0.624788	−	0.780794i	$-0.714815\pi$
$751$	−34.2439	−1.24958	−0.624788	+	0.780794i	$0.714815\pi$
$752$	0	0
$753$	−31.0872	−1.13288
$754$	0	0
$755$	2.36776	0.0861717
$756$	0	0
$757$	−18.4605	−0.670958	−0.335479	−	0.942048i	$-0.608898\pi$
$757$	−18.4605	−0.670958	−0.335479	+	0.942048i	$0.608898\pi$
$758$	0	0
$759$	19.2544	0.698891
$760$	0	0
$761$	−17.6172	−0.638622	−0.319311	−	0.947650i	$-0.603451\pi$
$761$	−17.6172	−0.638622	−0.319311	+	0.947650i	$0.603451\pi$
$762$	0	0
$763$	−29.6222	−1.07240
$764$	0	0
$765$	1.42166	0.0514003
$766$	0	0
$767$	−8.67609	−0.313275
$768$	0	0
$769$	−41.3522	−1.49120	−0.745599	−	0.666395i	$-0.767836\pi$
$769$	−41.3522	−1.49120	−0.745599	+	0.666395i	$0.767836\pi$
$770$	0	0
$771$	−6.96723	−0.250919
$772$	0	0
$773$	22.6867	0.815982	0.407991	−	0.912986i	$-0.366229\pi$
$773$	22.6867	0.815982	0.407991	+	0.912986i	$0.366229\pi$
$774$	0	0
$775$	−38.5089	−1.38328
$776$	0	0
$777$	−1.86445	−0.0668868
$778$	0	0
$779$	−23.3311	−0.835922
$780$	0	0
$781$	51.4983	1.84275
$782$	0	0
$783$	25.9961	0.929023
$784$	0	0
$785$	15.9305	0.568584
$786$	0	0
$787$	−15.5194	−0.553207	−0.276604	−	0.960984i	$-0.589209\pi$
$787$	−15.5194	−0.553207	−0.276604	+	0.960984i	$0.589209\pi$
$788$	0	0
$789$	36.6338	1.30420
$790$	0	0
$791$	−10.1844	−0.362116
$792$	0	0
$793$	−12.6761	−0.450141
$794$	0	0
$795$	−9.62721	−0.341442
$796$	0	0
$797$	−1.56777	−0.0555334	−0.0277667	−	0.999614i	$-0.508840\pi$
$797$	−1.56777	−0.0555334	−0.0277667	+	0.999614i	$0.508840\pi$
$798$	0	0
$799$	−4.68111	−0.165606
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	14.5089	0.512006
$804$	0	0
$805$	−3.42166	−0.120598
$806$	0	0
$807$	−6.98944	−0.246040
$808$	0	0
$809$	44.4005	1.56104	0.780520	−	0.625131i	$-0.214954\pi$
$809$	44.4005	1.56104	0.780520	+	0.625131i	$0.214954\pi$
$810$	0	0
$811$	−32.9411	−1.15672	−0.578359	−	0.815782i	$-0.696307\pi$
$811$	−32.9411	−1.15672	−0.578359	+	0.815782i	$0.696307\pi$
$812$	0	0
$813$	29.5089	1.03492
$814$	0	0
$815$	−3.52946	−0.123632
$816$	0	0
$817$	1.90225	0.0665512
$818$	0	0
$819$	2.98944	0.104459
$820$	0	0
$821$	30.3522	1.05930	0.529649	−	0.848217i	$-0.322324\pi$
$821$	30.3522	1.05930	0.529649	+	0.848217i	$0.322324\pi$
$822$	0	0
$823$	34.5783	1.20533	0.602663	−	0.797996i	$-0.294107\pi$
$823$	34.5783	1.20533	0.602663	+	0.797996i	$0.294107\pi$
$824$	0	0
$825$	47.3311	1.64786
$826$	0	0
$827$	−14.1744	−0.492891	−0.246446	−	0.969157i	$-0.579263\pi$
$827$	−14.1744	−0.492891	−0.246446	+	0.969157i	$0.579263\pi$
$828$	0	0
$829$	−33.1355	−1.15085	−0.575423	−	0.817856i	$-0.695162\pi$
$829$	−33.1355	−1.15085	−0.575423	+	0.817856i	$0.695162\pi$
$830$	0	0
$831$	2.12499	0.0737149
$832$	0	0
$833$	−2.57834	−0.0893341
$834$	0	0
$835$	−15.7250	−0.544185
$836$	0	0
$837$	31.9516	1.10441
$838$	0	0
$839$	34.4494	1.18933	0.594663	−	0.803975i	$-0.297286\pi$
$839$	34.4494	1.18933	0.594663	+	0.803975i	$0.297286\pi$
$840$	0	0
$841$	32.3522	1.11559
$842$	0	0
$843$	7.54665	0.259920
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	43.4550	1.49313
$848$	0	0
$849$	27.9516	0.959298
$850$	0	0
$851$	−0.686135	−0.0235204
$852$	0	0
$853$	17.1744	0.588040	0.294020	−	0.955799i	$-0.405007\pi$
$853$	17.1744	0.588040	0.294020	+	0.955799i	$0.405007\pi$
$854$	0	0
$855$	5.68665	0.194479
$856$	0	0
$857$	34.4877	1.17808	0.589039	−	0.808104i	$-0.299506\pi$
$857$	34.4877	1.17808	0.589039	+	0.808104i	$0.299506\pi$
$858$	0	0
$859$	−5.29274	−0.180586	−0.0902930	−	0.995915i	$-0.528780\pi$
$859$	−5.29274	−0.180586	−0.0902930	+	0.995915i	$0.528780\pi$
$860$	0	0
$861$	25.7905	0.878938
$862$	0	0
$863$	11.5939	0.394662	0.197331	−	0.980337i	$-0.436773\pi$
$863$	11.5939	0.394662	0.197331	+	0.980337i	$0.436773\pi$
$864$	0	0
$865$	14.4111	0.489992
$866$	0	0
$867$	33.6444	1.14262
$868$	0	0
$869$	−80.4877	−2.73036
$870$	0	0
$871$	12.2056	0.413569
$872$	0	0
$873$	15.1778	0.513691
$874$	0	0
$875$	−18.9250	−0.639781
$876$	0	0
$877$	−10.3522	−0.349568	−0.174784	−	0.984607i	$-0.555923\pi$
$877$	−10.3522	−0.349568	−0.174784	+	0.984607i	$0.555923\pi$
$878$	0	0
$879$	−19.4378	−0.655620
$880$	0	0
$881$	39.7044	1.33767	0.668837	−	0.743409i	$-0.266792\pi$
$881$	39.7044	1.33767	0.668837	+	0.743409i	$0.266792\pi$
$882$	0	0
$883$	−42.2388	−1.42145	−0.710725	−	0.703470i	$-0.751633\pi$
$883$	−42.2388	−1.42145	−0.710725	+	0.703470i	$0.751633\pi$
$884$	0	0
$885$	18.2439	0.613261
$886$	0	0
$887$	−36.1461	−1.21367	−0.606834	−	0.794829i	$-0.707561\pi$
$887$	−36.1461	−1.21367	−0.606834	+	0.794829i	$0.707561\pi$
$888$	0	0
$889$	21.1083	0.707950
$890$	0	0
$891$	−63.2716	−2.11968
$892$	0	0
$893$	−18.7244	−0.626590
$894$	0	0
$895$	1.69167	0.0565465
$896$	0	0
$897$	3.42166	0.114246
$898$	0	0
$899$	75.4077	2.51499
$900$	0	0
$901$	4.57834	0.152527
$902$	0	0
$903$	−2.10278	−0.0699760
$904$	0	0
$905$	17.2544	0.573557
$906$	0	0
$907$	43.5910	1.44742	0.723708	−	0.690106i	$-0.242436\pi$
$907$	43.5910	1.44742	0.723708	+	0.690106i	$0.242436\pi$
$908$	0	0
$909$	−13.0177	−0.431770
$910$	0	0
$911$	53.8993	1.78576	0.892882	−	0.450290i	$-0.148679\pi$
$911$	53.8993	1.78576	0.892882	+	0.450290i	$0.148679\pi$
$912$	0	0
$913$	46.1744	1.52815
$914$	0	0
$915$	26.6550	0.881186
$916$	0	0
$917$	17.8222	0.588541
$918$	0	0
$919$	−6.64782	−0.219291	−0.109646	−	0.993971i	$-0.534972\pi$
$919$	−6.64782	−0.219291	−0.109646	+	0.993971i	$0.534972\pi$
$920$	0	0
$921$	17.3794	0.572671
$922$	0	0
$923$	9.15165	0.301230
$924$	0	0
$925$	−1.68665	−0.0554568
$926$	0	0
$927$	−7.62328	−0.250381
$928$	0	0
$929$	−17.9789	−0.589868	−0.294934	−	0.955518i	$-0.595298\pi$
$929$	−17.9789	−0.589868	−0.294934	+	0.955518i	$0.595298\pi$
$930$	0	0
$931$	−10.3133	−0.338006
$932$	0	0
$933$	−49.7633	−1.62918
$934$	0	0
$935$	5.62721	0.184030
$936$	0	0
$937$	22.4877	0.734642	0.367321	−	0.930094i	$-0.380275\pi$
$937$	22.4877	0.734642	0.367321	+	0.930094i	$0.380275\pi$
$938$	0	0
$939$	−1.54556	−0.0504376
$940$	0	0
$941$	15.1955	0.495359	0.247680	−	0.968842i	$-0.420332\pi$
$941$	15.1955	0.495359	0.247680	+	0.968842i	$0.420332\pi$
$942$	0	0
$943$	9.49115	0.309074
$944$	0	0
$945$	6.97887	0.227023
$946$	0	0
$947$	32.0594	1.04179	0.520896	−	0.853620i	$-0.325598\pi$
$947$	32.0594	1.04179	0.520896	+	0.853620i	$0.325598\pi$
$948$	0	0
$949$	2.57834	0.0836964
$950$	0	0
$951$	23.4600	0.760742
$952$	0	0
$953$	−31.5089	−1.02067	−0.510336	−	0.859975i	$-0.670479\pi$
$953$	−31.5089	−1.02067	−0.510336	+	0.859975i	$0.670479\pi$
$954$	0	0
$955$	6.03831	0.195395
$956$	0	0
$957$	−92.6832	−2.99602
$958$	0	0
$959$	40.1361	1.29606
$960$	0	0
$961$	61.6832	1.98978
$962$	0	0
$963$	9.05995	0.291953
$964$	0	0
$965$	13.2544	0.426675
$966$	0	0
$967$	−44.3960	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$967$	−44.3960	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$968$	0	0
$969$	−8.41110	−0.270203
$970$	0	0
$971$	−49.2978	−1.58204	−0.791020	−	0.611790i	$-0.790450\pi$
$971$	−49.2978	−1.58204	−0.791020	+	0.611790i	$0.790450\pi$
$972$	0	0
$973$	−8.40054	−0.269309
$974$	0	0
$975$	8.41110	0.269371
$976$	0	0
$977$	−3.35218	−0.107246	−0.0536228	−	0.998561i	$-0.517077\pi$
$977$	−3.35218	−0.107246	−0.0536228	+	0.998561i	$0.517077\pi$
$978$	0	0
$979$	−63.3311	−2.02407
$980$	0	0
$981$	−20.0272	−0.639420
$982$	0	0
$983$	−21.8972	−0.698413	−0.349207	−	0.937046i	$-0.613549\pi$
$983$	−21.8972	−0.698413	−0.349207	+	0.937046i	$0.613549\pi$
$984$	0	0
$985$	−22.9305	−0.730627
$986$	0	0
$987$	20.6983	0.658834
$988$	0	0
$989$	−0.773841	−0.0246067
$990$	0	0
$991$	−59.7422	−1.89777	−0.948886	−	0.315619i	$-0.897788\pi$
$991$	−59.7422	−1.89777	−0.948886	+	0.315619i	$0.897788\pi$
$992$	0	0
$993$	16.3900	0.520120
$994$	0	0
$995$	0.951124	0.0301527
$996$	0	0
$997$	38.3416	1.21429	0.607146	−	0.794591i	$-0.292314\pi$
$997$	38.3416	1.21429	0.607146	+	0.794591i	$0.292314\pi$
$998$	0	0
$999$	1.39945	0.0442767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bh.1.1		3
4.3	odd	2		3328.2.a.bf.1.3		3
8.3	odd	2		3328.2.a.bg.1.1		3
8.5	even	2		3328.2.a.be.1.3		3
16.3	odd	4		416.2.b.c.209.5		6
16.5	even	4		104.2.b.c.53.2	yes	6
16.11	odd	4		416.2.b.c.209.2		6
16.13	even	4		104.2.b.c.53.1	✓	6
48.5	odd	4		936.2.g.c.469.5		6
48.11	even	4		3744.2.g.c.1873.3		6
48.29	odd	4		936.2.g.c.469.6		6
48.35	even	4		3744.2.g.c.1873.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.c.53.1	✓	6	16.13	even	4
104.2.b.c.53.2	yes	6	16.5	even	4
416.2.b.c.209.2		6	16.11	odd	4
416.2.b.c.209.5		6	16.3	odd	4
936.2.g.c.469.5		6	48.5	odd	4
936.2.g.c.469.6		6	48.29	odd	4
3328.2.a.be.1.3		3	8.5	even	2
3328.2.a.bf.1.3		3	4.3	odd	2
3328.2.a.bg.1.1		3	8.3	odd	2
3328.2.a.bh.1.1		3	1.1	even	1	trivial
3744.2.g.c.1873.3		6	48.11	even	4
3744.2.g.c.1873.6		6	48.35	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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-2.10278 q^{3} +1.00000 q^{5} +2.10278 q^{7} +1.42166 q^{9} +5.62721 q^{11} +1.00000 q^{13} -2.10278 q^{15} +1.00000 q^{17} +4.00000 q^{19} -4.42166 q^{21} -1.62721 q^{23} -4.00000 q^{25} +3.31889 q^{27} +7.83276 q^{29} +9.62721 q^{31} -11.8328 q^{33} +2.10278 q^{35} +0.421663 q^{37} -2.10278 q^{39} -5.83276 q^{41} +0.475562 q^{43} +1.42166 q^{45} -4.68111 q^{47} -2.57834 q^{49} -2.10278 q^{51} +4.57834 q^{53} +5.62721 q^{55} -8.41110 q^{57} -8.67609 q^{59} -12.6761 q^{61} +2.98944 q^{63} +1.00000 q^{65} +12.2056 q^{67} +3.42166 q^{69} +9.15165 q^{71} +2.57834 q^{73} +8.41110 q^{75} +11.8328 q^{77} -14.3033 q^{79} -11.2439 q^{81} +8.20555 q^{83} +1.00000 q^{85} -16.4705 q^{87} -11.2544 q^{89} +2.10278 q^{91} -20.2439 q^{93} +4.00000 q^{95} +10.6761 q^{97} +8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{5} - q^{7} + 6 q^{9} + 4 q^{11} + 3 q^{13} + q^{15} + 3 q^{17} + 12 q^{19} - 15 q^{21} + 8 q^{23} - 12 q^{25} + 19 q^{27} - 4 q^{29} + 16 q^{31} - 8 q^{33} - q^{35} + 3 q^{37} + q^{39}+ \cdots + 24 q^{99}+O(q^{100}) \) 3 * q + q^3 + 3 * q^5 - q^7 + 6 * q^9 + 4 * q^11 + 3 * q^13 + q^15 + 3 * q^17 + 12 * q^19 - 15 * q^21 + 8 * q^23 - 12 * q^25 + 19 * q^27 - 4 * q^29 + 16 * q^31 - 8 * q^33 - q^35 + 3 * q^37 + q^39 + 10 * q^41 + 7 * q^43 + 6 * q^45 - 5 * q^47 - 6 * q^49 + q^51 + 12 * q^53 + 4 * q^55 + 4 * q^57 - 2 * q^59 - 14 * q^61 - 22 * q^63 + 3 * q^65 + 22 * q^67 + 12 * q^69 + 9 * q^71 + 6 * q^73 - 4 * q^75 + 8 * q^77 - 6 * q^79 + 23 * q^81 + 10 * q^83 + 3 * q^85 - 40 * q^87 - 8 * q^89 - q^91 - 4 * q^93 + 12 * q^95 + 8 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more