LMFDB - Embedded newform 1984.2.a.bb.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1984,2,Mod(1,1984)] 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("1984.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1984 = 2^{6} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1984.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2,0,-2,0,2,0,10,0,10,0,-10,0,8,0,4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.8423197610$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.66862976.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 20x^{2} - 16x - 15 $ x^6 - 2x^5 - 8x^4 + 12x^3 + 20x^2 - 16*x - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 992)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.35608$ of defining polynomial
Character	$\chi$	$=$	1984.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.59078 q^{3} +2.33595 q^{5} -4.18539 q^{7} +3.71216 q^{9} +3.60845 q^{11} +0.590783 q^{13} +6.05193 q^{15} -4.05193 q^{17} +8.35254 q^{19} -10.8434 q^{21} +5.80918 q^{23} +0.456644 q^{25} +1.84504 q^{27} +5.62504 q^{29} +1.00000 q^{31} +9.34871 q^{33} -9.77685 q^{35} -7.30619 q^{37} +1.53059 q^{39} +1.83120 q^{41} +4.23607 q^{43} +8.67140 q^{45} -2.20248 q^{47} +10.5175 q^{49} -10.4977 q^{51} -8.94498 q^{53} +8.42915 q^{55} +21.6396 q^{57} +8.59529 q^{59} +4.17005 q^{61} -15.5368 q^{63} +1.38004 q^{65} -10.0962 q^{67} +15.0503 q^{69} -2.38787 q^{71} +6.64421 q^{73} +1.18306 q^{75} -15.1028 q^{77} +8.04428 q^{79} -6.35637 q^{81} +12.6150 q^{83} -9.46509 q^{85} +14.5733 q^{87} -17.9054 q^{89} -2.47266 q^{91} +2.59078 q^{93} +19.5111 q^{95} -0.777674 q^{97} +13.3951 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 10 q^{9} + 10 q^{11} - 10 q^{13} + 8 q^{15} + 4 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} + 20 q^{25} + 8 q^{27} - 10 q^{29} + 6 q^{31} + 8 q^{33} + 14 q^{35} - 26 q^{37}+ \cdots + 46 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 10 * q^9 + 10 * q^11 - 10 * q^13 + 8 * q^15 + 4 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 + 20 * q^25 + 8 * q^27 - 10 * q^29 + 6 * q^31 + 8 * q^33 + 14 * q^35 - 26 * q^37 + 24 * q^39 + 6 * q^41 + 10 * q^43 - 22 * q^45 + 4 * q^47 + 20 * q^49 + 32 * q^51 - 18 * q^53 + 16 * q^55 + 24 * q^57 + 6 * q^59 - 10 * q^61 - 2 * q^63 + 12 * q^65 - 4 * q^67 + 4 * q^69 + 30 * q^71 + 4 * q^73 - 14 * q^75 + 20 * q^79 + 10 * q^81 + 30 * q^83 + 16 * q^87 - 20 * q^89 - 8 * q^91 + 2 * q^93 + 26 * q^95 + 30 * q^97 + 46 * 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
$3$	2.59078	1.49579	0.747895	−	0.663817i	$-0.231065\pi$
$3$	2.59078	1.49579	0.747895	+	0.663817i	$0.231065\pi$
$4$	0	0
$5$	2.33595	1.04467	0.522333	−	0.852741i	$-0.325062\pi$
$5$	2.33595	1.04467	0.522333	+	0.852741i	$0.325062\pi$
$6$	0	0
$7$	−4.18539	−1.58193	−0.790965	−	0.611862i	$-0.790421\pi$
$7$	−4.18539	−1.58193	−0.790965	+	0.611862i	$0.790421\pi$
$8$	0	0
$9$	3.71216	1.23739
$10$	0	0
$11$	3.60845	1.08799	0.543994	−	0.839089i	$-0.316911\pi$
$11$	3.60845	1.08799	0.543994	+	0.839089i	$0.316911\pi$
$12$	0	0
$13$	0.590783	0.163854	0.0819268	−	0.996638i	$-0.473893\pi$
$13$	0.590783	0.163854	0.0819268	+	0.996638i	$0.473893\pi$
$14$	0	0
$15$	6.05193	1.56260
$16$	0	0
$17$	−4.05193	−0.982737	−0.491369	−	0.870952i	$-0.663503\pi$
$17$	−4.05193	−0.982737	−0.491369	+	0.870952i	$0.663503\pi$
$18$	0	0
$19$	8.35254	1.91620	0.958102	−	0.286427i	$-0.0924677\pi$
$19$	8.35254	1.91620	0.958102	+	0.286427i	$0.0924677\pi$
$20$	0	0
$21$	−10.8434	−2.36623
$22$	0	0
$23$	5.80918	1.21130	0.605649	−	0.795732i	$-0.292914\pi$
$23$	5.80918	1.21130	0.605649	+	0.795732i	$0.292914\pi$
$24$	0	0
$25$	0.456644	0.0913287
$26$	0	0
$27$	1.84504	0.355078
$28$	0	0
$29$	5.62504	1.04454	0.522272	−	0.852779i	$-0.325084\pi$
$29$	5.62504	1.04454	0.522272	+	0.852779i	$0.325084\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	9.34871	1.62740
$34$	0	0
$35$	−9.77685	−1.65259
$36$	0	0
$37$	−7.30619	−1.20113	−0.600565	−	0.799576i	$-0.705058\pi$
$37$	−7.30619	−1.20113	−0.600565	+	0.799576i	$0.705058\pi$
$38$	0	0
$39$	1.53059	0.245091
$40$	0	0
$41$	1.83120	0.285985	0.142993	−	0.989724i	$-0.454327\pi$
$41$	1.83120	0.285985	0.142993	+	0.989724i	$0.454327\pi$
$42$	0	0
$43$	4.23607	0.645994	0.322997	−	0.946400i	$-0.395310\pi$
$43$	4.23607	0.645994	0.322997	+	0.946400i	$0.395310\pi$
$44$	0	0
$45$	8.67140	1.29266
$46$	0	0
$47$	−2.20248	−0.321265	−0.160633	−	0.987014i	$-0.551353\pi$
$47$	−2.20248	−0.321265	−0.160633	+	0.987014i	$0.551353\pi$
$48$	0	0
$49$	10.5175	1.50250
$50$	0	0
$51$	−10.4977	−1.46997
$52$	0	0
$53$	−8.94498	−1.22869	−0.614343	−	0.789039i	$-0.710579\pi$
$53$	−8.94498	−1.22869	−0.614343	+	0.789039i	$0.710579\pi$
$54$	0	0
$55$	8.42915	1.13659
$56$	0	0
$57$	21.6396	2.86624
$58$	0	0
$59$	8.59529	1.11901	0.559505	−	0.828827i	$-0.310991\pi$
$59$	8.59529	1.11901	0.559505	+	0.828827i	$0.310991\pi$
$60$	0	0
$61$	4.17005	0.533921	0.266960	−	0.963708i	$-0.413981\pi$
$61$	4.17005	0.533921	0.266960	+	0.963708i	$0.413981\pi$
$62$	0	0
$63$	−15.5368	−1.95746
$64$	0	0
$65$	1.38004	0.171173
$66$	0	0
$67$	−10.0962	−1.23345	−0.616724	−	0.787180i	$-0.711540\pi$
$67$	−10.0962	−1.23345	−0.616724	+	0.787180i	$0.711540\pi$
$68$	0	0
$69$	15.0503	1.81185
$70$	0	0
$71$	−2.38787	−0.283389	−0.141694	−	0.989910i	$-0.545255\pi$
$71$	−2.38787	−0.283389	−0.141694	+	0.989910i	$0.545255\pi$
$72$	0	0
$73$	6.64421	0.777646	0.388823	−	0.921312i	$-0.372882\pi$
$73$	6.64421	0.777646	0.388823	+	0.921312i	$0.372882\pi$
$74$	0	0
$75$	1.18306	0.136608
$76$	0	0
$77$	−15.1028	−1.72112
$78$	0	0
$79$	8.04428	0.905052	0.452526	−	0.891751i	$-0.350523\pi$
$79$	8.04428	0.905052	0.452526	+	0.891751i	$0.350523\pi$
$80$	0	0
$81$	−6.35637	−0.706263
$82$	0	0
$83$	12.6150	1.38468	0.692340	−	0.721572i	$-0.256580\pi$
$83$	12.6150	1.38468	0.692340	+	0.721572i	$0.256580\pi$
$84$	0	0
$85$	−9.46509	−1.02663
$86$	0	0
$87$	14.5733	1.56242
$88$	0	0
$89$	−17.9054	−1.89797	−0.948984	−	0.315325i	$-0.897886\pi$
$89$	−17.9054	−1.89797	−0.948984	+	0.315325i	$0.897886\pi$
$90$	0	0
$91$	−2.47266	−0.259205
$92$	0	0
$93$	2.59078	0.268652
$94$	0	0
$95$	19.5111	2.00179
$96$	0	0
$97$	−0.777674	−0.0789608	−0.0394804	−	0.999220i	$-0.512570\pi$
$97$	−0.777674	−0.0789608	−0.0394804	+	0.999220i	$0.512570\pi$
$98$	0	0
$99$	13.3951	1.34626
$100$	0	0
$101$	9.45799	0.941106	0.470553	−	0.882372i	$-0.344055\pi$
$101$	9.45799	0.941106	0.470553	+	0.882372i	$0.344055\pi$
$102$	0	0
$103$	16.0791	1.58432	0.792161	−	0.610312i	$-0.208956\pi$
$103$	16.0791	1.58432	0.792161	+	0.610312i	$0.208956\pi$
$104$	0	0
$105$	−25.3297	−2.47193
$106$	0	0
$107$	−15.2683	−1.47605	−0.738023	−	0.674776i	$-0.764240\pi$
$107$	−15.2683	−1.47605	−0.738023	+	0.674776i	$0.764240\pi$
$108$	0	0
$109$	15.8883	1.52182	0.760911	−	0.648856i	$-0.224752\pi$
$109$	15.8883	1.52182	0.760911	+	0.648856i	$0.224752\pi$
$110$	0	0
$111$	−18.9287	−1.79664
$112$	0	0
$113$	−14.8739	−1.39922	−0.699608	−	0.714527i	$-0.746642\pi$
$113$	−14.8739	−1.39922	−0.699608	+	0.714527i	$0.746642\pi$
$114$	0	0
$115$	13.5699	1.26540
$116$	0	0
$117$	2.19308	0.202750
$118$	0	0
$119$	16.9589	1.55462
$120$	0	0
$121$	2.02092	0.183720
$122$	0	0
$123$	4.74424	0.427774
$124$	0	0
$125$	−10.6130	−0.949259
$126$	0	0
$127$	−14.3389	−1.27238	−0.636188	−	0.771534i	$-0.719490\pi$
$127$	−14.3389	−1.27238	−0.636188	+	0.771534i	$0.719490\pi$
$128$	0	0
$129$	10.9747	0.966271
$130$	0	0
$131$	−7.07801	−0.618409	−0.309204	−	0.950996i	$-0.600063\pi$
$131$	−7.07801	−0.618409	−0.309204	+	0.950996i	$0.600063\pi$
$132$	0	0
$133$	−34.9587	−3.03130
$134$	0	0
$135$	4.30992	0.370939
$136$	0	0
$137$	−12.0454	−1.02911	−0.514555	−	0.857457i	$-0.672043\pi$
$137$	−12.0454	−1.02911	−0.514555	+	0.857457i	$0.672043\pi$
$138$	0	0
$139$	−15.5220	−1.31656	−0.658280	−	0.752773i	$-0.728716\pi$
$139$	−15.5220	−1.31656	−0.658280	+	0.752773i	$0.728716\pi$
$140$	0	0
$141$	−5.70615	−0.480545
$142$	0	0
$143$	2.13181	0.178271
$144$	0	0
$145$	13.1398	1.09120
$146$	0	0
$147$	27.2486	2.24743
$148$	0	0
$149$	−14.7021	−1.20444	−0.602221	−	0.798330i	$-0.705717\pi$
$149$	−14.7021	−1.20444	−0.602221	+	0.798330i	$0.705717\pi$
$150$	0	0
$151$	−9.29301	−0.756255	−0.378127	−	0.925754i	$-0.623432\pi$
$151$	−9.29301	−0.756255	−0.378127	+	0.925754i	$0.623432\pi$
$152$	0	0
$153$	−15.0414	−1.21602
$154$	0	0
$155$	2.33595	0.187628
$156$	0	0
$157$	7.22290	0.576450	0.288225	−	0.957563i	$-0.406935\pi$
$157$	7.22290	0.576450	0.288225	+	0.957563i	$0.406935\pi$
$158$	0	0
$159$	−23.1745	−1.83786
$160$	0	0
$161$	−24.3137	−1.91619
$162$	0	0
$163$	−4.52792	−0.354654	−0.177327	−	0.984152i	$-0.556745\pi$
$163$	−4.52792	−0.354654	−0.177327	+	0.984152i	$0.556745\pi$
$164$	0	0
$165$	21.8381	1.70009
$166$	0	0
$167$	−19.4123	−1.50217	−0.751085	−	0.660206i	$-0.770469\pi$
$167$	−19.4123	−1.50217	−0.751085	+	0.660206i	$0.770469\pi$
$168$	0	0
$169$	−12.6510	−0.973152
$170$	0	0
$171$	31.0059	2.37108
$172$	0	0
$173$	−1.72658	−0.131269	−0.0656345	−	0.997844i	$-0.520907\pi$
$173$	−1.72658	−0.131269	−0.0656345	+	0.997844i	$0.520907\pi$
$174$	0	0
$175$	−1.91123	−0.144476
$176$	0	0
$177$	22.2685	1.67380
$178$	0	0
$179$	3.61013	0.269834	0.134917	−	0.990857i	$-0.456923\pi$
$179$	3.61013	0.269834	0.134917	+	0.990857i	$0.456923\pi$
$180$	0	0
$181$	−13.2992	−0.988519	−0.494260	−	0.869314i	$-0.664561\pi$
$181$	−13.2992	−0.988519	−0.494260	+	0.869314i	$0.664561\pi$
$182$	0	0
$183$	10.8037	0.798633
$184$	0	0
$185$	−17.0669	−1.25478
$186$	0	0
$187$	−14.6212	−1.06921
$188$	0	0
$189$	−7.72222	−0.561709
$190$	0	0
$191$	11.1660	0.807947	0.403973	−	0.914771i	$-0.367629\pi$
$191$	11.1660	0.807947	0.403973	+	0.914771i	$0.367629\pi$
$192$	0	0
$193$	−2.27759	−0.163944	−0.0819721	−	0.996635i	$-0.526122\pi$
$193$	−2.27759	−0.163944	−0.0819721	+	0.996635i	$0.526122\pi$
$194$	0	0
$195$	3.57538	0.256038
$196$	0	0
$197$	15.7441	1.12172	0.560862	−	0.827909i	$-0.310470\pi$
$197$	15.7441	1.12172	0.560862	+	0.827909i	$0.310470\pi$
$198$	0	0
$199$	11.7663	0.834088	0.417044	−	0.908886i	$-0.363066\pi$
$199$	11.7663	0.834088	0.417044	+	0.908886i	$0.363066\pi$
$200$	0	0
$201$	−26.1571	−1.84498
$202$	0	0
$203$	−23.5430	−1.65240
$204$	0	0
$205$	4.27759	0.298760
$206$	0	0
$207$	21.5646	1.49884
$208$	0	0
$209$	30.1397	2.08481
$210$	0	0
$211$	24.8247	1.70900	0.854501	−	0.519451i	$-0.173863\pi$
$211$	24.8247	1.70900	0.854501	+	0.519451i	$0.173863\pi$
$212$	0	0
$213$	−6.18647	−0.423890
$214$	0	0
$215$	9.89522	0.674849
$216$	0	0
$217$	−4.18539	−0.284123
$218$	0	0
$219$	17.2137	1.16319
$220$	0	0
$221$	−2.39381	−0.161025
$222$	0	0
$223$	6.22424	0.416806	0.208403	−	0.978043i	$-0.433173\pi$
$223$	6.22424	0.416806	0.208403	+	0.978043i	$0.433173\pi$
$224$	0	0
$225$	1.69513	0.113009
$226$	0	0
$227$	14.8353	0.984651	0.492326	−	0.870411i	$-0.336147\pi$
$227$	14.8353	0.984651	0.492326	+	0.870411i	$0.336147\pi$
$228$	0	0
$229$	3.05694	0.202009	0.101004	−	0.994886i	$-0.467794\pi$
$229$	3.05694	0.202009	0.101004	+	0.994886i	$0.467794\pi$
$230$	0	0
$231$	−39.1280	−2.57444
$232$	0	0
$233$	−3.78937	−0.248250	−0.124125	−	0.992267i	$-0.539612\pi$
$233$	−3.78937	−0.248250	−0.124125	+	0.992267i	$0.539612\pi$
$234$	0	0
$235$	−5.14488	−0.335615
$236$	0	0
$237$	20.8410	1.35377
$238$	0	0
$239$	−15.0581	−0.974028	−0.487014	−	0.873394i	$-0.661914\pi$
$239$	−15.0581	−0.974028	−0.487014	+	0.873394i	$0.661914\pi$
$240$	0	0
$241$	4.12688	0.265836	0.132918	−	0.991127i	$-0.457565\pi$
$241$	4.12688	0.265836	0.132918	+	0.991127i	$0.457565\pi$
$242$	0	0
$243$	−22.0031	−1.41150
$244$	0	0
$245$	24.5683	1.56961
$246$	0	0
$247$	4.93454	0.313977
$248$	0	0
$249$	32.6828	2.07119
$250$	0	0
$251$	2.98083	0.188149	0.0940743	−	0.995565i	$-0.470011\pi$
$251$	2.98083	0.188149	0.0940743	+	0.995565i	$0.470011\pi$
$252$	0	0
$253$	20.9621	1.31788
$254$	0	0
$255$	−24.5220	−1.53563
$256$	0	0
$257$	−15.6279	−0.974840	−0.487420	−	0.873168i	$-0.662062\pi$
$257$	−15.6279	−0.974840	−0.487420	+	0.873168i	$0.662062\pi$
$258$	0	0
$259$	30.5793	1.90010
$260$	0	0
$261$	20.8810	1.29250
$262$	0	0
$263$	−6.38622	−0.393791	−0.196896	−	0.980424i	$-0.563086\pi$
$263$	−6.38622	−0.393791	−0.196896	+	0.980424i	$0.563086\pi$
$264$	0	0
$265$	−20.8950	−1.28357
$266$	0	0
$267$	−46.3890	−2.83896
$268$	0	0
$269$	17.3598	1.05845	0.529223	−	0.848483i	$-0.322484\pi$
$269$	17.3598	1.05845	0.529223	+	0.848483i	$0.322484\pi$
$270$	0	0
$271$	−2.53141	−0.153772	−0.0768862	−	0.997040i	$-0.524498\pi$
$271$	−2.53141	−0.153772	−0.0768862	+	0.997040i	$0.524498\pi$
$272$	0	0
$273$	−6.40612	−0.387716
$274$	0	0
$275$	1.64778	0.0993646
$276$	0	0
$277$	−32.2028	−1.93488	−0.967441	−	0.253098i	$-0.918551\pi$
$277$	−32.2028	−1.93488	−0.967441	+	0.253098i	$0.918551\pi$
$278$	0	0
$279$	3.71216	0.222241
$280$	0	0
$281$	5.84746	0.348830	0.174415	−	0.984672i	$-0.444197\pi$
$281$	5.84746	0.348830	0.174415	+	0.984672i	$0.444197\pi$
$282$	0	0
$283$	−22.8288	−1.35703	−0.678515	−	0.734587i	$-0.737376\pi$
$283$	−22.8288	−1.35703	−0.678515	+	0.734587i	$0.737376\pi$
$284$	0	0
$285$	50.5490	2.99426
$286$	0	0
$287$	−7.66429	−0.452409
$288$	0	0
$289$	−0.581873	−0.0342278
$290$	0	0
$291$	−2.01478	−0.118109
$292$	0	0
$293$	0.781259	0.0456416	0.0228208	−	0.999740i	$-0.492735\pi$
$293$	0.781259	0.0456416	0.0228208	+	0.999740i	$0.492735\pi$
$294$	0	0
$295$	20.0781	1.16899
$296$	0	0
$297$	6.65774	0.386321
$298$	0	0
$299$	3.43197	0.198476
$300$	0	0
$301$	−17.7296	−1.02192
$302$	0	0
$303$	24.5036	1.40770
$304$	0	0
$305$	9.74102	0.557769
$306$	0	0
$307$	−15.2748	−0.871781	−0.435890	−	0.900000i	$-0.643566\pi$
$307$	−15.2748	−0.871781	−0.435890	+	0.900000i	$0.643566\pi$
$308$	0	0
$309$	41.6575	2.36981
$310$	0	0
$311$	19.0256	1.07884	0.539421	−	0.842036i	$-0.318643\pi$
$311$	19.0256	1.07884	0.539421	+	0.842036i	$0.318643\pi$
$312$	0	0
$313$	16.2912	0.920832	0.460416	−	0.887703i	$-0.347700\pi$
$313$	16.2912	0.920832	0.460416	+	0.887703i	$0.347700\pi$
$314$	0	0
$315$	−36.2932	−2.04489
$316$	0	0
$317$	−25.7418	−1.44580	−0.722900	−	0.690952i	$-0.757192\pi$
$317$	−25.7418	−1.44580	−0.722900	+	0.690952i	$0.757192\pi$
$318$	0	0
$319$	20.2977	1.13645
$320$	0	0
$321$	−39.5569	−2.20785
$322$	0	0
$323$	−33.8439	−1.88312
$324$	0	0
$325$	0.269777	0.0149645
$326$	0	0
$327$	41.1631	2.27633
$328$	0	0
$329$	9.21825	0.508219
$330$	0	0
$331$	5.63438	0.309693	0.154847	−	0.987939i	$-0.450512\pi$
$331$	5.63438	0.309693	0.154847	+	0.987939i	$0.450512\pi$
$332$	0	0
$333$	−27.1217	−1.48626
$334$	0	0
$335$	−23.5842	−1.28854
$336$	0	0
$337$	25.5336	1.39090	0.695452	−	0.718572i	$-0.255204\pi$
$337$	25.5336	1.39090	0.695452	+	0.718572i	$0.255204\pi$
$338$	0	0
$339$	−38.5350	−2.09293
$340$	0	0
$341$	3.60845	0.195409
$342$	0	0
$343$	−14.7222	−0.794922
$344$	0	0
$345$	35.1568	1.89278
$346$	0	0
$347$	15.2285	0.817508	0.408754	−	0.912644i	$-0.365963\pi$
$347$	15.2285	0.817508	0.408754	+	0.912644i	$0.365963\pi$
$348$	0	0
$349$	−11.2642	−0.602961	−0.301480	−	0.953472i	$-0.597481\pi$
$349$	−11.2642	−0.602961	−0.301480	+	0.953472i	$0.597481\pi$
$350$	0	0
$351$	1.09002	0.0581809
$352$	0	0
$353$	−2.56154	−0.136337	−0.0681684	−	0.997674i	$-0.521716\pi$
$353$	−2.56154	−0.136337	−0.0681684	+	0.997674i	$0.521716\pi$
$354$	0	0
$355$	−5.57795	−0.296047
$356$	0	0
$357$	43.9369	2.32539
$358$	0	0
$359$	14.1224	0.745350	0.372675	−	0.927962i	$-0.378441\pi$
$359$	14.1224	0.745350	0.372675	+	0.927962i	$0.378441\pi$
$360$	0	0
$361$	50.7649	2.67184
$362$	0	0
$363$	5.23576	0.274806
$364$	0	0
$365$	15.5205	0.812381
$366$	0	0
$367$	−10.4225	−0.544049	−0.272024	−	0.962290i	$-0.587693\pi$
$367$	−10.4225	−0.544049	−0.272024	+	0.962290i	$0.587693\pi$
$368$	0	0
$369$	6.79770	0.353874
$370$	0	0
$371$	37.4382	1.94370
$372$	0	0
$373$	5.27612	0.273187	0.136593	−	0.990627i	$-0.456385\pi$
$373$	5.27612	0.273187	0.136593	+	0.990627i	$0.456385\pi$
$374$	0	0
$375$	−27.4961	−1.41989
$376$	0	0
$377$	3.32318	0.171152
$378$	0	0
$379$	−32.6977	−1.67957	−0.839785	−	0.542919i	$-0.817319\pi$
$379$	−32.6977	−1.67957	−0.839785	+	0.542919i	$0.817319\pi$
$380$	0	0
$381$	−37.1491	−1.90321
$382$	0	0
$383$	6.24390	0.319049	0.159524	−	0.987194i	$-0.449004\pi$
$383$	6.24390	0.319049	0.159524	+	0.987194i	$0.449004\pi$
$384$	0	0
$385$	−35.2793	−1.79800
$386$	0	0
$387$	15.7249	0.799344
$388$	0	0
$389$	−28.1180	−1.42564	−0.712820	−	0.701347i	$-0.752583\pi$
$389$	−28.1180	−1.42564	−0.712820	+	0.701347i	$0.752583\pi$
$390$	0	0
$391$	−23.5384	−1.19039
$392$	0	0
$393$	−18.3376	−0.925009
$394$	0	0
$395$	18.7910	0.945477
$396$	0	0
$397$	6.39681	0.321047	0.160523	−	0.987032i	$-0.448682\pi$
$397$	6.39681	0.321047	0.160523	+	0.987032i	$0.448682\pi$
$398$	0	0
$399$	−90.5703	−4.53419
$400$	0	0
$401$	24.9467	1.24578	0.622890	−	0.782310i	$-0.285959\pi$
$401$	24.9467	1.24578	0.622890	+	0.782310i	$0.285959\pi$
$402$	0	0
$403$	0.590783	0.0294290
$404$	0	0
$405$	−14.8481	−0.737809
$406$	0	0
$407$	−26.3640	−1.30682
$408$	0	0
$409$	−40.1661	−1.98609	−0.993043	−	0.117748i	$-0.962432\pi$
$409$	−40.1661	−1.98609	−0.993043	+	0.117748i	$0.962432\pi$
$410$	0	0
$411$	−31.2071	−1.53933
$412$	0	0
$413$	−35.9746	−1.77020
$414$	0	0
$415$	29.4680	1.44653
$416$	0	0
$417$	−40.2142	−1.96930
$418$	0	0
$419$	−2.87636	−0.140520	−0.0702598	−	0.997529i	$-0.522383\pi$
$419$	−2.87636	−0.140520	−0.0702598	+	0.997529i	$0.522383\pi$
$420$	0	0
$421$	−20.3338	−0.991009	−0.495504	−	0.868605i	$-0.665017\pi$
$421$	−20.3338	−0.991009	−0.495504	+	0.868605i	$0.665017\pi$
$422$	0	0
$423$	−8.17596	−0.397529
$424$	0	0
$425$	−1.85029	−0.0897521
$426$	0	0
$427$	−17.4533	−0.844625
$428$	0	0
$429$	5.52306	0.266656
$430$	0	0
$431$	1.49740	0.0721274	0.0360637	−	0.999349i	$-0.488518\pi$
$431$	1.49740	0.0721274	0.0360637	+	0.999349i	$0.488518\pi$
$432$	0	0
$433$	−0.112797	−0.00542068	−0.00271034	−	0.999996i	$-0.500863\pi$
$433$	−0.112797	−0.00542068	−0.00271034	+	0.999996i	$0.500863\pi$
$434$	0	0
$435$	34.0424	1.63221
$436$	0	0
$437$	48.5214	2.32109
$438$	0	0
$439$	−18.3181	−0.874275	−0.437137	−	0.899395i	$-0.644008\pi$
$439$	−18.3181	−0.874275	−0.437137	+	0.899395i	$0.644008\pi$
$440$	0	0
$441$	39.0426	1.85917
$442$	0	0
$443$	12.2107	0.580146	0.290073	−	0.957005i	$-0.406320\pi$
$443$	12.2107	0.580146	0.290073	+	0.957005i	$0.406320\pi$
$444$	0	0
$445$	−41.8260	−1.98274
$446$	0	0
$447$	−38.0899	−1.80159
$448$	0	0
$449$	−1.80460	−0.0851641	−0.0425821	−	0.999093i	$-0.513558\pi$
$449$	−1.80460	−0.0851641	−0.0425821	+	0.999093i	$0.513558\pi$
$450$	0	0
$451$	6.60780	0.311149
$452$	0	0
$453$	−24.0762	−1.13120
$454$	0	0
$455$	−5.77600	−0.270783
$456$	0	0
$457$	28.8808	1.35099	0.675494	−	0.737365i	$-0.263930\pi$
$457$	28.8808	1.35099	0.675494	+	0.737365i	$0.263930\pi$
$458$	0	0
$459$	−7.47598	−0.348949
$460$	0	0
$461$	13.5615	0.631622	0.315811	−	0.948822i	$-0.397723\pi$
$461$	13.5615	0.631622	0.315811	+	0.948822i	$0.397723\pi$
$462$	0	0
$463$	7.12205	0.330990	0.165495	−	0.986211i	$-0.447078\pi$
$463$	7.12205	0.330990	0.165495	+	0.986211i	$0.447078\pi$
$464$	0	0
$465$	6.05193	0.280651
$466$	0	0
$467$	24.1983	1.11977	0.559883	−	0.828572i	$-0.310846\pi$
$467$	24.1983	1.11977	0.559883	+	0.828572i	$0.310846\pi$
$468$	0	0
$469$	42.2566	1.95123
$470$	0	0
$471$	18.7130	0.862248
$472$	0	0
$473$	15.2856	0.702835
$474$	0	0
$475$	3.81413	0.175004
$476$	0	0
$477$	−33.2051	−1.52036
$478$	0	0
$479$	11.0054	0.502849	0.251425	−	0.967877i	$-0.419101\pi$
$479$	11.0054	0.502849	0.251425	+	0.967877i	$0.419101\pi$
$480$	0	0
$481$	−4.31637	−0.196810
$482$	0	0
$483$	−62.9915	−2.86621
$484$	0	0
$485$	−1.81660	−0.0824878
$486$	0	0
$487$	−31.9464	−1.44763	−0.723815	−	0.689994i	$-0.757613\pi$
$487$	−31.9464	−1.44763	−0.723815	+	0.689994i	$0.757613\pi$
$488$	0	0
$489$	−11.7309	−0.530488
$490$	0	0
$491$	13.1678	0.594253	0.297126	−	0.954838i	$-0.403972\pi$
$491$	13.1678	0.594253	0.297126	+	0.954838i	$0.403972\pi$
$492$	0	0
$493$	−22.7923	−1.02651
$494$	0	0
$495$	31.2903	1.40639
$496$	0	0
$497$	9.99419	0.448301
$498$	0	0
$499$	−20.0524	−0.897669	−0.448835	−	0.893615i	$-0.648161\pi$
$499$	−20.0524	−0.897669	−0.448835	+	0.893615i	$0.648161\pi$
$500$	0	0
$501$	−50.2931	−2.24693
$502$	0	0
$503$	−4.41825	−0.197000	−0.0985000	−	0.995137i	$-0.531404\pi$
$503$	−4.41825	−0.197000	−0.0985000	+	0.995137i	$0.531404\pi$
$504$	0	0
$505$	22.0934	0.983142
$506$	0	0
$507$	−32.7759	−1.45563
$508$	0	0
$509$	21.5134	0.953567	0.476783	−	0.879021i	$-0.341803\pi$
$509$	21.5134	0.953567	0.476783	+	0.879021i	$0.341803\pi$
$510$	0	0
$511$	−27.8086	−1.23018
$512$	0	0
$513$	15.4108	0.680403
$514$	0	0
$515$	37.5599	1.65509
$516$	0	0
$517$	−7.94755	−0.349533
$518$	0	0
$519$	−4.47318	−0.196351
$520$	0	0
$521$	10.0055	0.438349	0.219175	−	0.975686i	$-0.429664\pi$
$521$	10.0055	0.438349	0.219175	+	0.975686i	$0.429664\pi$
$522$	0	0
$523$	−8.29044	−0.362516	−0.181258	−	0.983436i	$-0.558017\pi$
$523$	−8.29044	−0.362516	−0.181258	+	0.983436i	$0.558017\pi$
$524$	0	0
$525$	−4.95159	−0.216105
$526$	0	0
$527$	−4.05193	−0.176505
$528$	0	0
$529$	10.7466	0.467244
$530$	0	0
$531$	31.9070	1.38465
$532$	0	0
$533$	1.08184	0.0468598
$534$	0	0
$535$	−35.6660	−1.54198
$536$	0	0
$537$	9.35306	0.403615
$538$	0	0
$539$	37.9519	1.63471
$540$	0	0
$541$	−32.1617	−1.38274	−0.691370	−	0.722501i	$-0.742993\pi$
$541$	−32.1617	−1.38274	−0.691370	+	0.722501i	$0.742993\pi$
$542$	0	0
$543$	−34.4553	−1.47862
$544$	0	0
$545$	37.1142	1.58980
$546$	0	0
$547$	−34.3034	−1.46671	−0.733353	−	0.679848i	$-0.762046\pi$
$547$	−34.3034	−1.46671	−0.733353	+	0.679848i	$0.762046\pi$
$548$	0	0
$549$	15.4799	0.660666
$550$	0	0
$551$	46.9834	2.00156
$552$	0	0
$553$	−33.6684	−1.43173
$554$	0	0
$555$	−44.2165	−1.87689
$556$	0	0
$557$	14.5221	0.615321	0.307660	−	0.951496i	$-0.400454\pi$
$557$	14.5221	0.615321	0.307660	+	0.951496i	$0.400454\pi$
$558$	0	0
$559$	2.50260	0.105849
$560$	0	0
$561$	−37.8803	−1.59931
$562$	0	0
$563$	−29.7051	−1.25192	−0.625961	−	0.779854i	$-0.715293\pi$
$563$	−29.7051	−1.25192	−0.625961	+	0.779854i	$0.715293\pi$
$564$	0	0
$565$	−34.7446	−1.46172
$566$	0	0
$567$	26.6039	1.11726
$568$	0	0
$569$	−30.2231	−1.26702	−0.633509	−	0.773735i	$-0.718386\pi$
$569$	−30.2231	−1.26702	−0.633509	+	0.773735i	$0.718386\pi$
$570$	0	0
$571$	29.6827	1.24218	0.621092	−	0.783738i	$-0.286690\pi$
$571$	29.6827	1.24218	0.621092	+	0.783738i	$0.286690\pi$
$572$	0	0
$573$	28.9288	1.20852
$574$	0	0
$575$	2.65273	0.110626
$576$	0	0
$577$	−5.06863	−0.211010	−0.105505	−	0.994419i	$-0.533646\pi$
$577$	−5.06863	−0.211010	−0.105505	+	0.994419i	$0.533646\pi$
$578$	0	0
$579$	−5.90073	−0.245226
$580$	0	0
$581$	−52.7989	−2.19047
$582$	0	0
$583$	−32.2775	−1.33680
$584$	0	0
$585$	5.12291	0.211806
$586$	0	0
$587$	27.7667	1.14605	0.573027	−	0.819537i	$-0.305769\pi$
$587$	27.7667	1.14605	0.573027	+	0.819537i	$0.305769\pi$
$588$	0	0
$589$	8.35254	0.344160
$590$	0	0
$591$	40.7897	1.67786
$592$	0	0
$593$	22.8397	0.937914	0.468957	−	0.883221i	$-0.344630\pi$
$593$	22.8397	0.937914	0.468957	+	0.883221i	$0.344630\pi$
$594$	0	0
$595$	39.6151	1.62406
$596$	0	0
$597$	30.4838	1.24762
$598$	0	0
$599$	34.7834	1.42121	0.710606	−	0.703590i	$-0.248421\pi$
$599$	34.7834	1.42121	0.710606	+	0.703590i	$0.248421\pi$
$600$	0	0
$601$	−14.9159	−0.608433	−0.304217	−	0.952603i	$-0.598395\pi$
$601$	−14.9159	−0.608433	−0.304217	+	0.952603i	$0.598395\pi$
$602$	0	0
$603$	−37.4787	−1.52625
$604$	0	0
$605$	4.72075	0.191926
$606$	0	0
$607$	−26.7231	−1.08466	−0.542328	−	0.840167i	$-0.682457\pi$
$607$	−26.7231	−1.08466	−0.542328	+	0.840167i	$0.682457\pi$
$608$	0	0
$609$	−60.9948	−2.47164
$610$	0	0
$611$	−1.30119	−0.0526405
$612$	0	0
$613$	16.9767	0.685683	0.342842	−	0.939393i	$-0.388611\pi$
$613$	16.9767	0.685683	0.342842	+	0.939393i	$0.388611\pi$
$614$	0	0
$615$	11.0823	0.446881
$616$	0	0
$617$	21.6792	0.872773	0.436387	−	0.899759i	$-0.356258\pi$
$617$	21.6792	0.872773	0.436387	+	0.899759i	$0.356258\pi$
$618$	0	0
$619$	18.1844	0.730892	0.365446	−	0.930833i	$-0.380917\pi$
$619$	18.1844	0.730892	0.365446	+	0.930833i	$0.380917\pi$
$620$	0	0
$621$	10.7182	0.430106
$622$	0	0
$623$	74.9411	3.00245
$624$	0	0
$625$	−27.0747	−1.08299
$626$	0	0
$627$	78.0855	3.11843
$628$	0	0
$629$	29.6042	1.18039
$630$	0	0
$631$	41.7611	1.66248	0.831242	−	0.555911i	$-0.187630\pi$
$631$	41.7611	1.66248	0.831242	+	0.555911i	$0.187630\pi$
$632$	0	0
$633$	64.3153	2.55631
$634$	0	0
$635$	−33.4950	−1.32921
$636$	0	0
$637$	6.21357	0.246190
$638$	0	0
$639$	−8.86416	−0.350661
$640$	0	0
$641$	−13.9708	−0.551815	−0.275908	−	0.961184i	$-0.588978\pi$
$641$	−13.9708	−0.551815	−0.275908	+	0.961184i	$0.588978\pi$
$642$	0	0
$643$	−20.8811	−0.823470	−0.411735	−	0.911304i	$-0.635077\pi$
$643$	−20.8811	−0.823470	−0.411735	+	0.911304i	$0.635077\pi$
$644$	0	0
$645$	25.6364	1.00943
$646$	0	0
$647$	−17.1044	−0.672443	−0.336222	−	0.941783i	$-0.609149\pi$
$647$	−17.1044	−0.672443	−0.336222	+	0.941783i	$0.609149\pi$
$648$	0	0
$649$	31.0157	1.21747
$650$	0	0
$651$	−10.8434	−0.424988
$652$	0	0
$653$	−38.4890	−1.50619	−0.753096	−	0.657911i	$-0.771440\pi$
$653$	−38.4890	−1.50619	−0.753096	+	0.657911i	$0.771440\pi$
$654$	0	0
$655$	−16.5339	−0.646031
$656$	0	0
$657$	24.6643	0.962248
$658$	0	0
$659$	−34.0971	−1.32823	−0.664117	−	0.747628i	$-0.731192\pi$
$659$	−34.0971	−1.32823	−0.664117	+	0.747628i	$0.731192\pi$
$660$	0	0
$661$	−47.1245	−1.83293	−0.916466	−	0.400112i	$-0.868971\pi$
$661$	−47.1245	−1.83293	−0.916466	+	0.400112i	$0.868971\pi$
$662$	0	0
$663$	−6.20184	−0.240860
$664$	0	0
$665$	−81.6615	−3.16670
$666$	0	0
$667$	32.6769	1.26525
$668$	0	0
$669$	16.1257	0.623454
$670$	0	0
$671$	15.0474	0.580900
$672$	0	0
$673$	47.8262	1.84356	0.921782	−	0.387708i	$-0.126733\pi$
$673$	47.8262	1.84356	0.921782	+	0.387708i	$0.126733\pi$
$674$	0	0
$675$	0.842526	0.0324288
$676$	0	0
$677$	13.6139	0.523224	0.261612	−	0.965173i	$-0.415746\pi$
$677$	13.6139	0.523224	0.261612	+	0.965173i	$0.415746\pi$
$678$	0	0
$679$	3.25487	0.124910
$680$	0	0
$681$	38.4350	1.47283
$682$	0	0
$683$	43.9946	1.68341	0.841703	−	0.539940i	$-0.181553\pi$
$683$	43.9946	1.68341	0.841703	+	0.539940i	$0.181553\pi$
$684$	0	0
$685$	−28.1375	−1.07508
$686$	0	0
$687$	7.91988	0.302162
$688$	0	0
$689$	−5.28454	−0.201325
$690$	0	0
$691$	−22.6781	−0.862717	−0.431358	−	0.902181i	$-0.641966\pi$
$691$	−22.6781	−0.862717	−0.431358	+	0.902181i	$0.641966\pi$
$692$	0	0
$693$	−56.0639	−2.12969
$694$	0	0
$695$	−36.2586	−1.37537
$696$	0	0
$697$	−7.41989	−0.281049
$698$	0	0
$699$	−9.81743	−0.371329
$700$	0	0
$701$	25.6257	0.967870	0.483935	−	0.875104i	$-0.339207\pi$
$701$	25.6257	0.967870	0.483935	+	0.875104i	$0.339207\pi$
$702$	0	0
$703$	−61.0252	−2.30161
$704$	0	0
$705$	−13.3293	−0.502009
$706$	0	0
$707$	−39.5854	−1.48876
$708$	0	0
$709$	−23.1594	−0.869768	−0.434884	−	0.900486i	$-0.643211\pi$
$709$	−23.1594	−0.869768	−0.434884	+	0.900486i	$0.643211\pi$
$710$	0	0
$711$	29.8616	1.11990
$712$	0	0
$713$	5.80918	0.217556
$714$	0	0
$715$	4.97980	0.186234
$716$	0	0
$717$	−39.0123	−1.45694
$718$	0	0
$719$	−9.06976	−0.338245	−0.169122	−	0.985595i	$-0.554093\pi$
$719$	−9.06976	−0.338245	−0.169122	+	0.985595i	$0.554093\pi$
$720$	0	0
$721$	−67.2974	−2.50629
$722$	0	0
$723$	10.6919	0.397634
$724$	0	0
$725$	2.56864	0.0953969
$726$	0	0
$727$	−25.9628	−0.962906	−0.481453	−	0.876472i	$-0.659891\pi$
$727$	−25.9628	−0.962906	−0.481453	+	0.876472i	$0.659891\pi$
$728$	0	0
$729$	−37.9361	−1.40504
$730$	0	0
$731$	−17.1642	−0.634843
$732$	0	0
$733$	42.8101	1.58123	0.790614	−	0.612315i	$-0.209762\pi$
$733$	42.8101	1.58123	0.790614	+	0.612315i	$0.209762\pi$
$734$	0	0
$735$	63.6512	2.34781
$736$	0	0
$737$	−36.4317	−1.34198
$738$	0	0
$739$	44.5199	1.63769	0.818846	−	0.574013i	$-0.194614\pi$
$739$	44.5199	1.63769	0.818846	+	0.574013i	$0.194614\pi$
$740$	0	0
$741$	12.7843	0.469644
$742$	0	0
$743$	17.7159	0.649932	0.324966	−	0.945726i	$-0.394647\pi$
$743$	17.7159	0.649932	0.324966	+	0.945726i	$0.394647\pi$
$744$	0	0
$745$	−34.3433	−1.25824
$746$	0	0
$747$	46.8290	1.71338
$748$	0	0
$749$	63.9040	2.33500
$750$	0	0
$751$	−51.6951	−1.88638	−0.943190	−	0.332255i	$-0.892191\pi$
$751$	−51.6951	−1.88638	−0.943190	+	0.332255i	$0.892191\pi$
$752$	0	0
$753$	7.72269	0.281431
$754$	0	0
$755$	−21.7080	−0.790034
$756$	0	0
$757$	28.6125	1.03994	0.519970	−	0.854184i	$-0.325943\pi$
$757$	28.6125	1.03994	0.519970	+	0.854184i	$0.325943\pi$
$758$	0	0
$759$	54.3084	1.97127
$760$	0	0
$761$	13.9263	0.504829	0.252415	−	0.967619i	$-0.418775\pi$
$761$	13.9263	0.504829	0.252415	+	0.967619i	$0.418775\pi$
$762$	0	0
$763$	−66.4988	−2.40742
$764$	0	0
$765$	−35.1359	−1.27034
$766$	0	0
$767$	5.07795	0.183354
$768$	0	0
$769$	26.9989	0.973606	0.486803	−	0.873512i	$-0.338163\pi$
$769$	26.9989	0.973606	0.486803	+	0.873512i	$0.338163\pi$
$770$	0	0
$771$	−40.4884	−1.45815
$772$	0	0
$773$	−25.8722	−0.930559	−0.465279	−	0.885164i	$-0.654046\pi$
$773$	−25.8722	−0.930559	−0.465279	+	0.885164i	$0.654046\pi$
$774$	0	0
$775$	0.456644	0.0164031
$776$	0	0
$777$	79.2242	2.84215
$778$	0	0
$779$	15.2952	0.548006
$780$	0	0
$781$	−8.61653	−0.308324
$782$	0	0
$783$	10.3784	0.370895
$784$	0	0
$785$	16.8723	0.602199
$786$	0	0
$787$	53.0820	1.89217	0.946084	−	0.323922i	$-0.105002\pi$
$787$	53.0820	1.89217	0.946084	+	0.323922i	$0.105002\pi$
$788$	0	0
$789$	−16.5453	−0.589029
$790$	0	0
$791$	62.2530	2.21346
$792$	0	0
$793$	2.46360	0.0874849
$794$	0	0
$795$	−54.1344	−1.91995
$796$	0	0
$797$	21.6195	0.765801	0.382900	−	0.923790i	$-0.374925\pi$
$797$	21.6195	0.765801	0.382900	+	0.923790i	$0.374925\pi$
$798$	0	0
$799$	8.92430	0.315719
$800$	0	0
$801$	−66.4676	−2.34852
$802$	0	0
$803$	23.9753	0.846070
$804$	0	0
$805$	−56.7955	−2.00178
$806$	0	0
$807$	44.9755	1.58321
$808$	0	0
$809$	−21.7721	−0.765465	−0.382733	−	0.923859i	$-0.625017\pi$
$809$	−21.7721	−0.765465	−0.382733	+	0.923859i	$0.625017\pi$
$810$	0	0
$811$	11.6372	0.408636	0.204318	−	0.978905i	$-0.434502\pi$
$811$	11.6372	0.408636	0.204318	+	0.978905i	$0.434502\pi$
$812$	0	0
$813$	−6.55834	−0.230011
$814$	0	0
$815$	−10.5770	−0.370495
$816$	0	0
$817$	35.3819	1.23786
$818$	0	0
$819$	−9.17889	−0.320736
$820$	0	0
$821$	−11.0104	−0.384264	−0.192132	−	0.981369i	$-0.561540\pi$
$821$	−11.0104	−0.384264	−0.192132	+	0.981369i	$0.561540\pi$
$822$	0	0
$823$	14.2358	0.496229	0.248115	−	0.968731i	$-0.420189\pi$
$823$	14.2358	0.496229	0.248115	+	0.968731i	$0.420189\pi$
$824$	0	0
$825$	4.26903	0.148629
$826$	0	0
$827$	35.9702	1.25081	0.625403	−	0.780302i	$-0.284935\pi$
$827$	35.9702	1.25081	0.625403	+	0.780302i	$0.284935\pi$
$828$	0	0
$829$	16.0660	0.557994	0.278997	−	0.960292i	$-0.409998\pi$
$829$	16.0660	0.557994	0.278997	+	0.960292i	$0.409998\pi$
$830$	0	0
$831$	−83.4306	−2.89417
$832$	0	0
$833$	−42.6162	−1.47656
$834$	0	0
$835$	−45.3461	−1.56927
$836$	0	0
$837$	1.84504	0.0637740
$838$	0	0
$839$	−53.8030	−1.85749	−0.928743	−	0.370724i	$-0.879110\pi$
$839$	−53.8030	−1.85749	−0.928743	+	0.370724i	$0.879110\pi$
$840$	0	0
$841$	2.64112	0.0910731
$842$	0	0
$843$	15.1495	0.521776
$844$	0	0
$845$	−29.5520	−1.01662
$846$	0	0
$847$	−8.45833	−0.290632
$848$	0	0
$849$	−59.1444	−2.02983
$850$	0	0
$851$	−42.4430	−1.45493
$852$	0	0
$853$	27.9176	0.955882	0.477941	−	0.878392i	$-0.341383\pi$
$853$	27.9176	0.955882	0.477941	+	0.878392i	$0.341383\pi$
$854$	0	0
$855$	72.4282	2.47699
$856$	0	0
$857$	25.1699	0.859786	0.429893	−	0.902880i	$-0.358551\pi$
$857$	25.1699	0.859786	0.429893	+	0.902880i	$0.358551\pi$
$858$	0	0
$859$	−33.4304	−1.14063	−0.570315	−	0.821426i	$-0.693179\pi$
$859$	−33.4304	−1.14063	−0.570315	+	0.821426i	$0.693179\pi$
$860$	0	0
$861$	−19.8565	−0.676708
$862$	0	0
$863$	−23.9078	−0.813832	−0.406916	−	0.913466i	$-0.633396\pi$
$863$	−23.9078	−0.813832	−0.406916	+	0.913466i	$0.633396\pi$
$864$	0	0
$865$	−4.03319	−0.137132
$866$	0	0
$867$	−1.50751	−0.0511976
$868$	0	0
$869$	29.0274	0.984686
$870$	0	0
$871$	−5.96466	−0.202105
$872$	0	0
$873$	−2.88685	−0.0977050
$874$	0	0
$875$	44.4197	1.50166
$876$	0	0
$877$	−4.11548	−0.138970	−0.0694849	−	0.997583i	$-0.522136\pi$
$877$	−4.11548	−0.138970	−0.0694849	+	0.997583i	$0.522136\pi$
$878$	0	0
$879$	2.02407	0.0682702
$880$	0	0
$881$	27.2406	0.917759	0.458879	−	0.888499i	$-0.348251\pi$
$881$	27.2406	0.917759	0.458879	+	0.888499i	$0.348251\pi$
$882$	0	0
$883$	−35.3208	−1.18864	−0.594320	−	0.804229i	$-0.702579\pi$
$883$	−35.3208	−1.18864	−0.594320	+	0.804229i	$0.702579\pi$
$884$	0	0
$885$	52.0181	1.74857
$886$	0	0
$887$	31.8903	1.07077	0.535386	−	0.844608i	$-0.320166\pi$
$887$	31.8903	1.07077	0.535386	+	0.844608i	$0.320166\pi$
$888$	0	0
$889$	60.0141	2.01281
$890$	0	0
$891$	−22.9366	−0.768406
$892$	0	0
$893$	−18.3963	−0.615609
$894$	0	0
$895$	8.43307	0.281886
$896$	0	0
$897$	8.89148	0.296878
$898$	0	0
$899$	5.62504	0.187606
$900$	0	0
$901$	36.2444	1.20748
$902$	0	0
$903$	−45.9336	−1.52857
$904$	0	0
$905$	−31.0661	−1.03267
$906$	0	0
$907$	6.92754	0.230025	0.115013	−	0.993364i	$-0.463309\pi$
$907$	6.92754	0.230025	0.115013	+	0.993364i	$0.463309\pi$
$908$	0	0
$909$	35.1096	1.16451
$910$	0	0
$911$	−39.3906	−1.30507	−0.652534	−	0.757760i	$-0.726294\pi$
$911$	−39.3906	−1.30507	−0.652534	+	0.757760i	$0.726294\pi$
$912$	0	0
$913$	45.5207	1.50652
$914$	0	0
$915$	25.2369	0.834305
$916$	0	0
$917$	29.6243	0.978279
$918$	0	0
$919$	16.0485	0.529391	0.264695	−	0.964332i	$-0.414729\pi$
$919$	16.0485	0.529391	0.264695	+	0.964332i	$0.414729\pi$
$920$	0	0
$921$	−39.5738	−1.30400
$922$	0	0
$923$	−1.41072	−0.0464343
$924$	0	0
$925$	−3.33632	−0.109698
$926$	0	0
$927$	59.6882	1.96042
$928$	0	0
$929$	−15.6914	−0.514817	−0.257409	−	0.966303i	$-0.582869\pi$
$929$	−15.6914	−0.514817	−0.257409	+	0.966303i	$0.582869\pi$
$930$	0	0
$931$	87.8479	2.87910
$932$	0	0
$933$	49.2912	1.61372
$934$	0	0
$935$	−34.1543	−1.11696
$936$	0	0
$937$	−20.0222	−0.654096	−0.327048	−	0.945008i	$-0.606054\pi$
$937$	−20.0222	−0.654096	−0.327048	+	0.945008i	$0.606054\pi$
$938$	0	0
$939$	42.2069	1.37737
$940$	0	0
$941$	0.658193	0.0214565	0.0107282	−	0.999942i	$-0.496585\pi$
$941$	0.658193	0.0214565	0.0107282	+	0.999942i	$0.496585\pi$
$942$	0	0
$943$	10.6378	0.346414
$944$	0	0
$945$	−18.0387	−0.586799
$946$	0	0
$947$	29.8945	0.971439	0.485720	−	0.874115i	$-0.338558\pi$
$947$	29.8945	0.971439	0.485720	+	0.874115i	$0.338558\pi$
$948$	0	0
$949$	3.92529	0.127420
$950$	0	0
$951$	−66.6913	−2.16261
$952$	0	0
$953$	−39.4089	−1.27658	−0.638289	−	0.769797i	$-0.720358\pi$
$953$	−39.4089	−1.27658	−0.638289	+	0.769797i	$0.720358\pi$
$954$	0	0
$955$	26.0833	0.844035
$956$	0	0
$957$	52.5869	1.69989
$958$	0	0
$959$	50.4149	1.62798
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−56.6784	−1.82644
$964$	0	0
$965$	−5.32032	−0.171267
$966$	0	0
$967$	12.5949	0.405025	0.202513	−	0.979280i	$-0.435089\pi$
$967$	12.5949	0.405025	0.202513	+	0.979280i	$0.435089\pi$
$968$	0	0
$969$	−87.6822	−2.81676
$970$	0	0
$971$	−0.183339	−0.00588364	−0.00294182	−	0.999996i	$-0.500936\pi$
$971$	−0.183339	−0.00588364	−0.00294182	+	0.999996i	$0.500936\pi$
$972$	0	0
$973$	64.9657	2.08271
$974$	0	0
$975$	0.698934	0.0223838
$976$	0	0
$977$	47.2332	1.51113	0.755563	−	0.655076i	$-0.227364\pi$
$977$	47.2332	1.51113	0.755563	+	0.655076i	$0.227364\pi$
$978$	0	0
$979$	−64.6107	−2.06497
$980$	0	0
$981$	58.9798	1.88308
$982$	0	0
$983$	−2.25921	−0.0720576	−0.0360288	−	0.999351i	$-0.511471\pi$
$983$	−2.25921	−0.0720576	−0.0360288	+	0.999351i	$0.511471\pi$
$984$	0	0
$985$	36.7775	1.17183
$986$	0	0
$987$	23.8825	0.760188
$988$	0	0
$989$	24.6081	0.782492
$990$	0	0
$991$	−46.2272	−1.46845	−0.734227	−	0.678904i	$-0.762455\pi$
$991$	−46.2272	−1.46845	−0.734227	+	0.678904i	$0.762455\pi$
$992$	0	0
$993$	14.5974	0.463236
$994$	0	0
$995$	27.4853	0.871344
$996$	0	0
$997$	38.6142	1.22292	0.611462	−	0.791274i	$-0.290582\pi$
$997$	38.6142	1.22292	0.611462	+	0.791274i	$0.290582\pi$
$998$	0	0
$999$	−13.4802	−0.426495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1984.2.a.bb.1.5		6
4.3	odd	2		1984.2.a.ba.1.2		6
8.3	odd	2		992.2.a.h.1.5	yes	6
8.5	even	2		992.2.a.g.1.2	✓	6
24.5	odd	2		8928.2.a.br.1.5		6
24.11	even	2		8928.2.a.bq.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
992.2.a.g.1.2	✓	6	8.5	even	2
992.2.a.h.1.5	yes	6	8.3	odd	2
1984.2.a.ba.1.2		6	4.3	odd	2
1984.2.a.bb.1.5		6	1.1	even	1	trivial
8928.2.a.bq.1.5		6	24.11	even	2
8928.2.a.br.1.5		6	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1984.a (trivial)

\(f(q)\)	\(=\)	\(q+2.59078 q^{3} +2.33595 q^{5} -4.18539 q^{7} +3.71216 q^{9} +3.60845 q^{11} +0.590783 q^{13} +6.05193 q^{15} -4.05193 q^{17} +8.35254 q^{19} -10.8434 q^{21} +5.80918 q^{23} +0.456644 q^{25} +1.84504 q^{27} +5.62504 q^{29} +1.00000 q^{31} +9.34871 q^{33} -9.77685 q^{35} -7.30619 q^{37} +1.53059 q^{39} +1.83120 q^{41} +4.23607 q^{43} +8.67140 q^{45} -2.20248 q^{47} +10.5175 q^{49} -10.4977 q^{51} -8.94498 q^{53} +8.42915 q^{55} +21.6396 q^{57} +8.59529 q^{59} +4.17005 q^{61} -15.5368 q^{63} +1.38004 q^{65} -10.0962 q^{67} +15.0503 q^{69} -2.38787 q^{71} +6.64421 q^{73} +1.18306 q^{75} -15.1028 q^{77} +8.04428 q^{79} -6.35637 q^{81} +12.6150 q^{83} -9.46509 q^{85} +14.5733 q^{87} -17.9054 q^{89} -2.47266 q^{91} +2.59078 q^{93} +19.5111 q^{95} -0.777674 q^{97} +13.3951 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 10 q^{9} + 10 q^{11} - 10 q^{13} + 8 q^{15} + 4 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} + 20 q^{25} + 8 q^{27} - 10 q^{29} + 6 q^{31} + 8 q^{33} + 14 q^{35} - 26 q^{37}+ \cdots + 46 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 10 * q^9 + 10 * q^11 - 10 * q^13 + 8 * q^15 + 4 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 + 20 * q^25 + 8 * q^27 - 10 * q^29 + 6 * q^31 + 8 * q^33 + 14 * q^35 - 26 * q^37 + 24 * q^39 + 6 * q^41 + 10 * q^43 - 22 * q^45 + 4 * q^47 + 20 * q^49 + 32 * q^51 - 18 * q^53 + 16 * q^55 + 24 * q^57 + 6 * q^59 - 10 * q^61 - 2 * q^63 + 12 * q^65 - 4 * q^67 + 4 * q^69 + 30 * q^71 + 4 * q^73 - 14 * q^75 + 20 * q^79 + 10 * q^81 + 30 * q^83 + 16 * q^87 - 20 * q^89 - 8 * q^91 + 2 * q^93 + 26 * q^95 + 30 * q^97 + 46 * q^99

Introduction

L-functions

Modular forms

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Database

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