LMFDB - Embedded newform 1984.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1984,2,Mod(1,1984)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 1984 = 2^{6} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1984.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:	$15.8423197610$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+3.23607 q^{3} -1.00000 q^{5} +0.236068 q^{7} +7.47214 q^{9} +O(q^{10})$	$q+3.23607 q^{3} -1.00000 q^{5} +0.236068 q^{7} +7.47214 q^{9} -2.00000 q^{11} +3.23607 q^{13} -3.23607 q^{15} +0.763932 q^{17} +2.23607 q^{19} +0.763932 q^{21} +5.70820 q^{23} -4.00000 q^{25} +14.4721 q^{27} -2.76393 q^{29} +1.00000 q^{31} -6.47214 q^{33} -0.236068 q^{35} +2.00000 q^{37} +10.4721 q^{39} +7.00000 q^{41} -1.23607 q^{43} -7.47214 q^{45} +2.47214 q^{47} -6.94427 q^{49} +2.47214 q^{51} +10.4721 q^{53} +2.00000 q^{55} +7.23607 q^{57} -2.23607 q^{59} -8.18034 q^{61} +1.76393 q^{63} -3.23607 q^{65} -8.00000 q^{67} +18.4721 q^{69} -9.18034 q^{71} +8.47214 q^{73} -12.9443 q^{75} -0.472136 q^{77} -11.7082 q^{79} +24.4164 q^{81} +14.9443 q^{83} -0.763932 q^{85} -8.94427 q^{87} +11.7082 q^{89} +0.763932 q^{91} +3.23607 q^{93} -2.23607 q^{95} -15.9443 q^{97} -14.9443 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^7 + 6 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} + 6 q^{17} + 6 q^{21} - 2 q^{23} - 8 q^{25} + 20 q^{27} - 10 q^{29} + 2 q^{31} - 4 q^{33} + 4 q^{35} + 4 q^{37} + 12 q^{39} + 14 q^{41} + 2 q^{43} - 6 q^{45} - 4 q^{47} + 4 q^{49} - 4 q^{51} + 12 q^{53} + 4 q^{55} + 10 q^{57} + 6 q^{61} + 8 q^{63} - 2 q^{65} - 16 q^{67} + 28 q^{69} + 4 q^{71} + 8 q^{73} - 8 q^{75} + 8 q^{77} - 10 q^{79} + 22 q^{81} + 12 q^{83} - 6 q^{85} + 10 q^{89} + 6 q^{91} + 2 q^{93} - 14 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^13 - 2 * q^15 + 6 * q^17 + 6 * q^21 - 2 * q^23 - 8 * q^25 + 20 * q^27 - 10 * q^29 + 2 * q^31 - 4 * q^33 + 4 * q^35 + 4 * q^37 + 12 * q^39 + 14 * q^41 + 2 * q^43 - 6 * q^45 - 4 * q^47 + 4 * q^49 - 4 * q^51 + 12 * q^53 + 4 * q^55 + 10 * q^57 + 6 * q^61 + 8 * q^63 - 2 * q^65 - 16 * q^67 + 28 * q^69 + 4 * q^71 + 8 * q^73 - 8 * q^75 + 8 * q^77 - 10 * q^79 + 22 * q^81 + 12 * q^83 - 6 * q^85 + 10 * q^89 + 6 * q^91 + 2 * q^93 - 14 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

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
$3$	3.23607	1.86834	0.934172	−	0.356822i	$-0.116140\pi$
$3$	3.23607	1.86834	0.934172	+	0.356822i	$0.116140\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	3.23607	0.897524	0.448762	−	0.893651i	$-0.351865\pi$
$13$	3.23607	0.897524	0.448762	+	0.893651i	$0.351865\pi$
$14$	0	0
$15$	−3.23607	−0.835549
$16$	0	0
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	0	0
$19$	2.23607	0.512989	0.256495	−	0.966546i	$-0.417432\pi$
$19$	2.23607	0.512989	0.256495	+	0.966546i	$0.417432\pi$
$20$	0	0
$21$	0.763932	0.166704
$22$	0	0
$23$	5.70820	1.19024	0.595121	−	0.803636i	$-0.297104\pi$
$23$	5.70820	1.19024	0.595121	+	0.803636i	$0.297104\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	14.4721	2.78516
$28$	0	0
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−6.47214	−1.12665
$34$	0	0
$35$	−0.236068	−0.0399028
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	10.4721	1.67688
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−1.23607	−0.188499	−0.0942493	−	0.995549i	$-0.530045\pi$
$43$	−1.23607	−0.188499	−0.0942493	+	0.995549i	$0.530045\pi$
$44$	0	0
$45$	−7.47214	−1.11388
$46$	0	0
$47$	2.47214	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$47$	2.47214	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	2.47214	0.346168
$52$	0	0
$53$	10.4721	1.43846	0.719229	−	0.694773i	$-0.244495\pi$
$53$	10.4721	1.43846	0.719229	+	0.694773i	$0.244495\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	7.23607	0.958441
$58$	0	0
$59$	−2.23607	−0.291111	−0.145556	−	0.989350i	$-0.546497\pi$
$59$	−2.23607	−0.291111	−0.145556	+	0.989350i	$0.546497\pi$
$60$	0	0
$61$	−8.18034	−1.04739	−0.523693	−	0.851907i	$-0.675446\pi$
$61$	−8.18034	−1.04739	−0.523693	+	0.851907i	$0.675446\pi$
$62$	0	0
$63$	1.76393	0.222235
$64$	0	0
$65$	−3.23607	−0.401385
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	18.4721	2.22378
$70$	0	0
$71$	−9.18034	−1.08951	−0.544753	−	0.838597i	$-0.683377\pi$
$71$	−9.18034	−1.08951	−0.544753	+	0.838597i	$0.683377\pi$
$72$	0	0
$73$	8.47214	0.991589	0.495794	−	0.868440i	$-0.334877\pi$
$73$	8.47214	0.991589	0.495794	+	0.868440i	$0.334877\pi$
$74$	0	0
$75$	−12.9443	−1.49468
$76$	0	0
$77$	−0.472136	−0.0538049
$78$	0	0
$79$	−11.7082	−1.31728	−0.658638	−	0.752460i	$-0.728867\pi$
$79$	−11.7082	−1.31728	−0.658638	+	0.752460i	$0.728867\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	14.9443	1.64035	0.820173	−	0.572115i	$-0.193877\pi$
$83$	14.9443	1.64035	0.820173	+	0.572115i	$0.193877\pi$
$84$	0	0
$85$	−0.763932	−0.0828601
$86$	0	0
$87$	−8.94427	−0.958927
$88$	0	0
$89$	11.7082	1.24107	0.620534	−	0.784180i	$-0.286916\pi$
$89$	11.7082	1.24107	0.620534	+	0.784180i	$0.286916\pi$
$90$	0	0
$91$	0.763932	0.0800818
$92$	0	0
$93$	3.23607	0.335565
$94$	0	0
$95$	−2.23607	−0.229416
$96$	0	0
$97$	−15.9443	−1.61890	−0.809448	−	0.587192i	$-0.800233\pi$
$97$	−15.9443	−1.61890	−0.809448	+	0.587192i	$0.800233\pi$
$98$	0	0
$99$	−14.9443	−1.50196
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	6.23607	0.614458	0.307229	−	0.951636i	$-0.400598\pi$
$103$	6.23607	0.614458	0.307229	+	0.951636i	$0.400598\pi$
$104$	0	0
$105$	−0.763932	−0.0745521
$106$	0	0
$107$	−5.76393	−0.557220	−0.278610	−	0.960404i	$-0.589874\pi$
$107$	−5.76393	−0.557220	−0.278610	+	0.960404i	$0.589874\pi$
$108$	0	0
$109$	13.9443	1.33562	0.667810	−	0.744332i	$-0.267232\pi$
$109$	13.9443	1.33562	0.667810	+	0.744332i	$0.267232\pi$
$110$	0	0
$111$	6.47214	0.614308
$112$	0	0
$113$	3.47214	0.326631	0.163316	−	0.986574i	$-0.447781\pi$
$113$	3.47214	0.326631	0.163316	+	0.986574i	$0.447781\pi$
$114$	0	0
$115$	−5.70820	−0.532293
$116$	0	0
$117$	24.1803	2.23547
$118$	0	0
$119$	0.180340	0.0165317
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	22.6525	2.04250
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	12.4721	1.10672	0.553362	−	0.832941i	$-0.313345\pi$
$127$	12.4721	1.10672	0.553362	+	0.832941i	$0.313345\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0.527864	0.0457716
$134$	0	0
$135$	−14.4721	−1.24556
$136$	0	0
$137$	6.29180	0.537544	0.268772	−	0.963204i	$-0.413382\pi$
$137$	6.29180	0.537544	0.268772	+	0.963204i	$0.413382\pi$
$138$	0	0
$139$	−13.4164	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−6.47214	−0.541227
$144$	0	0
$145$	2.76393	0.229532
$146$	0	0
$147$	−22.4721	−1.85347
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−14.1803	−1.15398	−0.576990	−	0.816751i	$-0.695773\pi$
$151$	−14.1803	−1.15398	−0.576990	+	0.816751i	$0.695773\pi$
$152$	0	0
$153$	5.70820	0.461481
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−20.8885	−1.66709	−0.833544	−	0.552454i	$-0.813692\pi$
$157$	−20.8885	−1.66709	−0.833544	+	0.552454i	$0.813692\pi$
$158$	0	0
$159$	33.8885	2.68754
$160$	0	0
$161$	1.34752	0.106200
$162$	0	0
$163$	−10.7082	−0.838731	−0.419366	−	0.907817i	$-0.637747\pi$
$163$	−10.7082	−0.838731	−0.419366	+	0.907817i	$0.637747\pi$
$164$	0	0
$165$	6.47214	0.503855
$166$	0	0
$167$	−6.47214	−0.500829	−0.250414	−	0.968139i	$-0.580567\pi$
$167$	−6.47214	−0.500829	−0.250414	+	0.968139i	$0.580567\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	16.7082	1.27771
$172$	0	0
$173$	−2.94427	−0.223849	−0.111924	−	0.993717i	$-0.535701\pi$
$173$	−2.94427	−0.223849	−0.111924	+	0.993717i	$0.535701\pi$
$174$	0	0
$175$	−0.944272	−0.0713802
$176$	0	0
$177$	−7.23607	−0.543896
$178$	0	0
$179$	−1.70820	−0.127677	−0.0638386	−	0.997960i	$-0.520334\pi$
$179$	−1.70820	−0.127677	−0.0638386	+	0.997960i	$0.520334\pi$
$180$	0	0
$181$	4.18034	0.310722	0.155361	−	0.987858i	$-0.450346\pi$
$181$	4.18034	0.310722	0.155361	+	0.987858i	$0.450346\pi$
$182$	0	0
$183$	−26.4721	−1.95688
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−1.52786	−0.111728
$188$	0	0
$189$	3.41641	0.248507
$190$	0	0
$191$	−19.1803	−1.38784	−0.693920	−	0.720052i	$-0.744118\pi$
$191$	−19.1803	−1.38784	−0.693920	+	0.720052i	$0.744118\pi$
$192$	0	0
$193$	3.47214	0.249930	0.124965	−	0.992161i	$-0.460118\pi$
$193$	3.47214	0.249930	0.124965	+	0.992161i	$0.460118\pi$
$194$	0	0
$195$	−10.4721	−0.749925
$196$	0	0
$197$	−11.4164	−0.813385	−0.406693	−	0.913565i	$-0.633318\pi$
$197$	−11.4164	−0.813385	−0.406693	+	0.913565i	$0.633318\pi$
$198$	0	0
$199$	−18.9443	−1.34292	−0.671462	−	0.741039i	$-0.734333\pi$
$199$	−18.9443	−1.34292	−0.671462	+	0.741039i	$0.734333\pi$
$200$	0	0
$201$	−25.8885	−1.82604
$202$	0	0
$203$	−0.652476	−0.0457948
$204$	0	0
$205$	−7.00000	−0.488901
$206$	0	0
$207$	42.6525	2.96455
$208$	0	0
$209$	−4.47214	−0.309344
$210$	0	0
$211$	−23.1803	−1.59580	−0.797900	−	0.602790i	$-0.794056\pi$
$211$	−23.1803	−1.59580	−0.797900	+	0.602790i	$0.794056\pi$
$212$	0	0
$213$	−29.7082	−2.03557
$214$	0	0
$215$	1.23607	0.0842991
$216$	0	0
$217$	0.236068	0.0160253
$218$	0	0
$219$	27.4164	1.85263
$220$	0	0
$221$	2.47214	0.166294
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−29.8885	−1.99257
$226$	0	0
$227$	6.47214	0.429571	0.214785	−	0.976661i	$-0.431095\pi$
$227$	6.47214	0.429571	0.214785	+	0.976661i	$0.431095\pi$
$228$	0	0
$229$	13.4164	0.886581	0.443291	−	0.896378i	$-0.353811\pi$
$229$	13.4164	0.886581	0.443291	+	0.896378i	$0.353811\pi$
$230$	0	0
$231$	−1.52786	−0.100526
$232$	0	0
$233$	17.9443	1.17557	0.587784	−	0.809018i	$-0.300000\pi$
$233$	17.9443	1.17557	0.587784	+	0.809018i	$0.300000\pi$
$234$	0	0
$235$	−2.47214	−0.161264
$236$	0	0
$237$	−37.8885	−2.46113
$238$	0	0
$239$	−11.7082	−0.757341	−0.378670	−	0.925532i	$-0.623619\pi$
$239$	−11.7082	−0.757341	−0.378670	+	0.925532i	$0.623619\pi$
$240$	0	0
$241$	14.3607	0.925053	0.462526	−	0.886606i	$-0.346943\pi$
$241$	14.3607	0.925053	0.462526	+	0.886606i	$0.346943\pi$
$242$	0	0
$243$	35.5967	2.28353
$244$	0	0
$245$	6.94427	0.443653
$246$	0	0
$247$	7.23607	0.460420
$248$	0	0
$249$	48.3607	3.06473
$250$	0	0
$251$	1.81966	0.114856	0.0574280	−	0.998350i	$-0.481710\pi$
$251$	1.81966	0.114856	0.0574280	+	0.998350i	$0.481710\pi$
$252$	0	0
$253$	−11.4164	−0.717743
$254$	0	0
$255$	−2.47214	−0.154811
$256$	0	0
$257$	1.94427	0.121280	0.0606402	−	0.998160i	$-0.480686\pi$
$257$	1.94427	0.121280	0.0606402	+	0.998160i	$0.480686\pi$
$258$	0	0
$259$	0.472136	0.0293371
$260$	0	0
$261$	−20.6525	−1.27836
$262$	0	0
$263$	−23.2361	−1.43280	−0.716399	−	0.697691i	$-0.754211\pi$
$263$	−23.2361	−1.43280	−0.716399	+	0.697691i	$0.754211\pi$
$264$	0	0
$265$	−10.4721	−0.643298
$266$	0	0
$267$	37.8885	2.31874
$268$	0	0
$269$	11.0557	0.674080	0.337040	−	0.941490i	$-0.390574\pi$
$269$	11.0557	0.674080	0.337040	+	0.941490i	$0.390574\pi$
$270$	0	0
$271$	−14.1803	−0.861394	−0.430697	−	0.902497i	$-0.641732\pi$
$271$	−14.1803	−0.861394	−0.430697	+	0.902497i	$0.641732\pi$
$272$	0	0
$273$	2.47214	0.149620
$274$	0	0
$275$	8.00000	0.482418
$276$	0	0
$277$	12.6525	0.760214	0.380107	−	0.924943i	$-0.375887\pi$
$277$	12.6525	0.760214	0.380107	+	0.924943i	$0.375887\pi$
$278$	0	0
$279$	7.47214	0.447345
$280$	0	0
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	0	0
$283$	13.8885	0.825588	0.412794	−	0.910824i	$-0.364553\pi$
$283$	13.8885	0.825588	0.412794	+	0.910824i	$0.364553\pi$
$284$	0	0
$285$	−7.23607	−0.428628
$286$	0	0
$287$	1.65248	0.0975426
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	−51.5967	−3.02465
$292$	0	0
$293$	0.472136	0.0275825	0.0137912	−	0.999905i	$-0.495610\pi$
$293$	0.472136	0.0275825	0.0137912	+	0.999905i	$0.495610\pi$
$294$	0	0
$295$	2.23607	0.130189
$296$	0	0
$297$	−28.9443	−1.67952
$298$	0	0
$299$	18.4721	1.06827
$300$	0	0
$301$	−0.291796	−0.0168188
$302$	0	0
$303$	9.70820	0.557722
$304$	0	0
$305$	8.18034	0.468405
$306$	0	0
$307$	28.7082	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$307$	28.7082	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$308$	0	0
$309$	20.1803	1.14802
$310$	0	0
$311$	−29.1803	−1.65467	−0.827333	−	0.561712i	$-0.810143\pi$
$311$	−29.1803	−1.65467	−0.827333	+	0.561712i	$0.810143\pi$
$312$	0	0
$313$	16.7639	0.947553	0.473777	−	0.880645i	$-0.342890\pi$
$313$	16.7639	0.947553	0.473777	+	0.880645i	$0.342890\pi$
$314$	0	0
$315$	−1.76393	−0.0993863
$316$	0	0
$317$	−4.05573	−0.227792	−0.113896	−	0.993493i	$-0.536333\pi$
$317$	−4.05573	−0.227792	−0.113896	+	0.993493i	$0.536333\pi$
$318$	0	0
$319$	5.52786	0.309501
$320$	0	0
$321$	−18.6525	−1.04108
$322$	0	0
$323$	1.70820	0.0950470
$324$	0	0
$325$	−12.9443	−0.718019
$326$	0	0
$327$	45.1246	2.49540
$328$	0	0
$329$	0.583592	0.0321745
$330$	0	0
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	0	0
$333$	14.9443	0.818941
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−14.7639	−0.804243	−0.402121	−	0.915586i	$-0.631727\pi$
$337$	−14.7639	−0.804243	−0.402121	+	0.915586i	$0.631727\pi$
$338$	0	0
$339$	11.2361	0.610259
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	−18.4721	−0.994506
$346$	0	0
$347$	−24.1803	−1.29807	−0.649034	−	0.760759i	$-0.724827\pi$
$347$	−24.1803	−1.29807	−0.649034	+	0.760759i	$0.724827\pi$
$348$	0	0
$349$	7.88854	0.422264	0.211132	−	0.977458i	$-0.432285\pi$
$349$	7.88854	0.422264	0.211132	+	0.977458i	$0.432285\pi$
$350$	0	0
$351$	46.8328	2.49975
$352$	0	0
$353$	7.41641	0.394736	0.197368	−	0.980330i	$-0.436761\pi$
$353$	7.41641	0.394736	0.197368	+	0.980330i	$0.436761\pi$
$354$	0	0
$355$	9.18034	0.487242
$356$	0	0
$357$	0.583592	0.0308870
$358$	0	0
$359$	22.2361	1.17357	0.586787	−	0.809741i	$-0.300392\pi$
$359$	22.2361	1.17357	0.586787	+	0.809741i	$0.300392\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	0	0
$363$	−22.6525	−1.18895
$364$	0	0
$365$	−8.47214	−0.443452
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	52.3050	2.72289
$370$	0	0
$371$	2.47214	0.128347
$372$	0	0
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	0	0
$375$	29.1246	1.50399
$376$	0	0
$377$	−8.94427	−0.460653
$378$	0	0
$379$	2.11146	0.108458	0.0542291	−	0.998529i	$-0.482730\pi$
$379$	2.11146	0.108458	0.0542291	+	0.998529i	$0.482730\pi$
$380$	0	0
$381$	40.3607	2.06774
$382$	0	0
$383$	−23.8885	−1.22065	−0.610324	−	0.792152i	$-0.708961\pi$
$383$	−23.8885	−1.22065	−0.610324	+	0.792152i	$0.708961\pi$
$384$	0	0
$385$	0.472136	0.0240623
$386$	0	0
$387$	−9.23607	−0.469496
$388$	0	0
$389$	17.8885	0.906985	0.453493	−	0.891260i	$-0.350178\pi$
$389$	17.8885	0.906985	0.453493	+	0.891260i	$0.350178\pi$
$390$	0	0
$391$	4.36068	0.220529
$392$	0	0
$393$	−38.8328	−1.95886
$394$	0	0
$395$	11.7082	0.589104
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	0	0
$399$	1.70820	0.0855172
$400$	0	0
$401$	38.1803	1.90664	0.953318	−	0.301969i	$-0.0976441\pi$
$401$	38.1803	1.90664	0.953318	+	0.301969i	$0.0976441\pi$
$402$	0	0
$403$	3.23607	0.161200
$404$	0	0
$405$	−24.4164	−1.21326
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	−3.81966	−0.188870	−0.0944350	−	0.995531i	$-0.530104\pi$
$409$	−3.81966	−0.188870	−0.0944350	+	0.995531i	$0.530104\pi$
$410$	0	0
$411$	20.3607	1.00432
$412$	0	0
$413$	−0.527864	−0.0259745
$414$	0	0
$415$	−14.9443	−0.733585
$416$	0	0
$417$	−43.4164	−2.12611
$418$	0	0
$419$	10.1246	0.494620	0.247310	−	0.968936i	$-0.420453\pi$
$419$	10.1246	0.494620	0.247310	+	0.968936i	$0.420453\pi$
$420$	0	0
$421$	−29.3607	−1.43095	−0.715476	−	0.698637i	$-0.753790\pi$
$421$	−29.3607	−1.43095	−0.715476	+	0.698637i	$0.753790\pi$
$422$	0	0
$423$	18.4721	0.898146
$424$	0	0
$425$	−3.05573	−0.148225
$426$	0	0
$427$	−1.93112	−0.0934533
$428$	0	0
$429$	−20.9443	−1.01120
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	10.1803	0.489236	0.244618	−	0.969620i	$-0.421337\pi$
$433$	10.1803	0.489236	0.244618	+	0.969620i	$0.421337\pi$
$434$	0	0
$435$	8.94427	0.428845
$436$	0	0
$437$	12.7639	0.610582
$438$	0	0
$439$	−1.18034	−0.0563345	−0.0281673	−	0.999603i	$-0.508967\pi$
$439$	−1.18034	−0.0563345	−0.0281673	+	0.999603i	$0.508967\pi$
$440$	0	0
$441$	−51.8885	−2.47088
$442$	0	0
$443$	−30.7082	−1.45899	−0.729495	−	0.683986i	$-0.760245\pi$
$443$	−30.7082	−1.45899	−0.729495	+	0.683986i	$0.760245\pi$
$444$	0	0
$445$	−11.7082	−0.555022
$446$	0	0
$447$	−32.3607	−1.53061
$448$	0	0
$449$	−31.3050	−1.47737	−0.738686	−	0.674050i	$-0.764553\pi$
$449$	−31.3050	−1.47737	−0.738686	+	0.674050i	$0.764553\pi$
$450$	0	0
$451$	−14.0000	−0.659234
$452$	0	0
$453$	−45.8885	−2.15603
$454$	0	0
$455$	−0.763932	−0.0358137
$456$	0	0
$457$	−3.05573	−0.142941	−0.0714705	−	0.997443i	$-0.522769\pi$
$457$	−3.05573	−0.142941	−0.0714705	+	0.997443i	$0.522769\pi$
$458$	0	0
$459$	11.0557	0.516037
$460$	0	0
$461$	−34.3607	−1.60034	−0.800168	−	0.599776i	$-0.795256\pi$
$461$	−34.3607	−1.60034	−0.800168	+	0.599776i	$0.795256\pi$
$462$	0	0
$463$	−2.58359	−0.120070	−0.0600349	−	0.998196i	$-0.519121\pi$
$463$	−2.58359	−0.120070	−0.0600349	+	0.998196i	$0.519121\pi$
$464$	0	0
$465$	−3.23607	−0.150069
$466$	0	0
$467$	−4.70820	−0.217870	−0.108935	−	0.994049i	$-0.534744\pi$
$467$	−4.70820	−0.217870	−0.108935	+	0.994049i	$0.534744\pi$
$468$	0	0
$469$	−1.88854	−0.0872049
$470$	0	0
$471$	−67.5967	−3.11469
$472$	0	0
$473$	2.47214	0.113669
$474$	0	0
$475$	−8.94427	−0.410391
$476$	0	0
$477$	78.2492	3.58279
$478$	0	0
$479$	23.2918	1.06423	0.532115	−	0.846672i	$-0.321398\pi$
$479$	23.2918	1.06423	0.532115	+	0.846672i	$0.321398\pi$
$480$	0	0
$481$	6.47214	0.295104
$482$	0	0
$483$	4.36068	0.198418
$484$	0	0
$485$	15.9443	0.723992
$486$	0	0
$487$	−19.2361	−0.871669	−0.435835	−	0.900027i	$-0.643547\pi$
$487$	−19.2361	−0.871669	−0.435835	+	0.900027i	$0.643547\pi$
$488$	0	0
$489$	−34.6525	−1.56704
$490$	0	0
$491$	−4.36068	−0.196795	−0.0983974	−	0.995147i	$-0.531372\pi$
$491$	−4.36068	−0.196795	−0.0983974	+	0.995147i	$0.531372\pi$
$492$	0	0
$493$	−2.11146	−0.0950952
$494$	0	0
$495$	14.9443	0.671695
$496$	0	0
$497$	−2.16718	−0.0972115
$498$	0	0
$499$	6.58359	0.294722	0.147361	−	0.989083i	$-0.452922\pi$
$499$	6.58359	0.294722	0.147361	+	0.989083i	$0.452922\pi$
$500$	0	0
$501$	−20.9443	−0.935721
$502$	0	0
$503$	29.6525	1.32214	0.661069	−	0.750325i	$-0.270103\pi$
$503$	29.6525	1.32214	0.661069	+	0.750325i	$0.270103\pi$
$504$	0	0
$505$	−3.00000	−0.133498
$506$	0	0
$507$	−8.18034	−0.363302
$508$	0	0
$509$	−29.5967	−1.31185	−0.655926	−	0.754825i	$-0.727722\pi$
$509$	−29.5967	−1.31185	−0.655926	+	0.754825i	$0.727722\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	32.3607	1.42876
$514$	0	0
$515$	−6.23607	−0.274794
$516$	0	0
$517$	−4.94427	−0.217449
$518$	0	0
$519$	−9.52786	−0.418227
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	17.7082	0.774326	0.387163	−	0.922011i	$-0.373455\pi$
$523$	17.7082	0.774326	0.387163	+	0.922011i	$0.373455\pi$
$524$	0	0
$525$	−3.05573	−0.133363
$526$	0	0
$527$	0.763932	0.0332774
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	−16.7082	−0.725074
$532$	0	0
$533$	22.6525	0.981188
$534$	0	0
$535$	5.76393	0.249197
$536$	0	0
$537$	−5.52786	−0.238545
$538$	0	0
$539$	13.8885	0.598222
$540$	0	0
$541$	25.3607	1.09034	0.545170	−	0.838325i	$-0.316465\pi$
$541$	25.3607	1.09034	0.545170	+	0.838325i	$0.316465\pi$
$542$	0	0
$543$	13.5279	0.580536
$544$	0	0
$545$	−13.9443	−0.597307
$546$	0	0
$547$	12.1246	0.518411	0.259205	−	0.965822i	$-0.416539\pi$
$547$	12.1246	0.518411	0.259205	+	0.965822i	$0.416539\pi$
$548$	0	0
$549$	−61.1246	−2.60873
$550$	0	0
$551$	−6.18034	−0.263291
$552$	0	0
$553$	−2.76393	−0.117534
$554$	0	0
$555$	−6.47214	−0.274727
$556$	0	0
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−4.94427	−0.208747
$562$	0	0
$563$	−27.5410	−1.16072	−0.580358	−	0.814362i	$-0.697087\pi$
$563$	−27.5410	−1.16072	−0.580358	+	0.814362i	$0.697087\pi$
$564$	0	0
$565$	−3.47214	−0.146074
$566$	0	0
$567$	5.76393	0.242062
$568$	0	0
$569$	5.52786	0.231740	0.115870	−	0.993264i	$-0.463034\pi$
$569$	5.52786	0.231740	0.115870	+	0.993264i	$0.463034\pi$
$570$	0	0
$571$	−28.1803	−1.17931	−0.589655	−	0.807655i	$-0.700736\pi$
$571$	−28.1803	−1.17931	−0.589655	+	0.807655i	$0.700736\pi$
$572$	0	0
$573$	−62.0689	−2.59296
$574$	0	0
$575$	−22.8328	−0.952194
$576$	0	0
$577$	−28.8328	−1.20033	−0.600163	−	0.799878i	$-0.704898\pi$
$577$	−28.8328	−1.20033	−0.600163	+	0.799878i	$0.704898\pi$
$578$	0	0
$579$	11.2361	0.466955
$580$	0	0
$581$	3.52786	0.146360
$582$	0	0
$583$	−20.9443	−0.867423
$584$	0	0
$585$	−24.1803	−0.999734
$586$	0	0
$587$	6.47214	0.267134	0.133567	−	0.991040i	$-0.457357\pi$
$587$	6.47214	0.267134	0.133567	+	0.991040i	$0.457357\pi$
$588$	0	0
$589$	2.23607	0.0921356
$590$	0	0
$591$	−36.9443	−1.51968
$592$	0	0
$593$	−6.52786	−0.268067	−0.134034	−	0.990977i	$-0.542793\pi$
$593$	−6.52786	−0.268067	−0.134034	+	0.990977i	$0.542793\pi$
$594$	0	0
$595$	−0.180340	−0.00739321
$596$	0	0
$597$	−61.3050	−2.50904
$598$	0	0
$599$	−14.5967	−0.596407	−0.298203	−	0.954502i	$-0.596387\pi$
$599$	−14.5967	−0.596407	−0.298203	+	0.954502i	$0.596387\pi$
$600$	0	0
$601$	30.5410	1.24579	0.622897	−	0.782304i	$-0.285956\pi$
$601$	30.5410	1.24579	0.622897	+	0.782304i	$0.285956\pi$
$602$	0	0
$603$	−59.7771	−2.43431
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	22.4721	0.912116	0.456058	−	0.889950i	$-0.349261\pi$
$607$	22.4721	0.912116	0.456058	+	0.889950i	$0.349261\pi$
$608$	0	0
$609$	−2.11146	−0.0855605
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	43.8885	1.77264	0.886321	−	0.463072i	$-0.153253\pi$
$613$	43.8885	1.77264	0.886321	+	0.463072i	$0.153253\pi$
$614$	0	0
$615$	−22.6525	−0.913436
$616$	0	0
$617$	32.4721	1.30728	0.653639	−	0.756806i	$-0.273241\pi$
$617$	32.4721	1.30728	0.653639	+	0.756806i	$0.273241\pi$
$618$	0	0
$619$	−6.18034	−0.248409	−0.124204	−	0.992257i	$-0.539638\pi$
$619$	−6.18034	−0.248409	−0.124204	+	0.992257i	$0.539638\pi$
$620$	0	0
$621$	82.6099	3.31502
$622$	0	0
$623$	2.76393	0.110735
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−14.4721	−0.577961
$628$	0	0
$629$	1.52786	0.0609199
$630$	0	0
$631$	34.3607	1.36788	0.683939	−	0.729540i	$-0.260266\pi$
$631$	34.3607	1.36788	0.683939	+	0.729540i	$0.260266\pi$
$632$	0	0
$633$	−75.0132	−2.98151
$634$	0	0
$635$	−12.4721	−0.494942
$636$	0	0
$637$	−22.4721	−0.890378
$638$	0	0
$639$	−68.5967	−2.71365
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	−19.5279	−0.770104	−0.385052	−	0.922895i	$-0.625816\pi$
$643$	−19.5279	−0.770104	−0.385052	+	0.922895i	$0.625816\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	−0.944272	−0.0371232	−0.0185616	−	0.999828i	$-0.505909\pi$
$647$	−0.944272	−0.0371232	−0.0185616	+	0.999828i	$0.505909\pi$
$648$	0	0
$649$	4.47214	0.175547
$650$	0	0
$651$	0.763932	0.0299409
$652$	0	0
$653$	47.3050	1.85119	0.925593	−	0.378521i	$-0.123567\pi$
$653$	47.3050	1.85119	0.925593	+	0.378521i	$0.123567\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	63.3050	2.46976
$658$	0	0
$659$	−25.6525	−0.999279	−0.499639	−	0.866234i	$-0.666534\pi$
$659$	−25.6525	−0.999279	−0.499639	+	0.866234i	$0.666534\pi$
$660$	0	0
$661$	0.639320	0.0248667	0.0124333	−	0.999923i	$-0.496042\pi$
$661$	0.639320	0.0248667	0.0124333	+	0.999923i	$0.496042\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	−0.527864	−0.0204697
$666$	0	0
$667$	−15.7771	−0.610891
$668$	0	0
$669$	12.9443	0.500454
$670$	0	0
$671$	16.3607	0.631597
$672$	0	0
$673$	−29.0132	−1.11837	−0.559187	−	0.829041i	$-0.688887\pi$
$673$	−29.0132	−1.11837	−0.559187	+	0.829041i	$0.688887\pi$
$674$	0	0
$675$	−57.8885	−2.22813
$676$	0	0
$677$	46.7214	1.79565	0.897824	−	0.440355i	$-0.145147\pi$
$677$	46.7214	1.79565	0.897824	+	0.440355i	$0.145147\pi$
$678$	0	0
$679$	−3.76393	−0.144446
$680$	0	0
$681$	20.9443	0.802586
$682$	0	0
$683$	−5.18034	−0.198220	−0.0991101	−	0.995076i	$-0.531600\pi$
$683$	−5.18034	−0.198220	−0.0991101	+	0.995076i	$0.531600\pi$
$684$	0	0
$685$	−6.29180	−0.240397
$686$	0	0
$687$	43.4164	1.65644
$688$	0	0
$689$	33.8885	1.29105
$690$	0	0
$691$	−3.18034	−0.120986	−0.0604929	−	0.998169i	$-0.519267\pi$
$691$	−3.18034	−0.120986	−0.0604929	+	0.998169i	$0.519267\pi$
$692$	0	0
$693$	−3.52786	−0.134012
$694$	0	0
$695$	13.4164	0.508913
$696$	0	0
$697$	5.34752	0.202552
$698$	0	0
$699$	58.0689	2.19637
$700$	0	0
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	4.47214	0.168670
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	0.708204	0.0266348
$708$	0	0
$709$	−25.5279	−0.958719	−0.479360	−	0.877619i	$-0.659131\pi$
$709$	−25.5279	−0.958719	−0.479360	+	0.877619i	$0.659131\pi$
$710$	0	0
$711$	−87.4853	−3.28095
$712$	0	0
$713$	5.70820	0.213774
$714$	0	0
$715$	6.47214	0.242044
$716$	0	0
$717$	−37.8885	−1.41497
$718$	0	0
$719$	−13.8197	−0.515386	−0.257693	−	0.966227i	$-0.582962\pi$
$719$	−13.8197	−0.515386	−0.257693	+	0.966227i	$0.582962\pi$
$720$	0	0
$721$	1.47214	0.0548252
$722$	0	0
$723$	46.4721	1.72832
$724$	0	0
$725$	11.0557	0.410599
$726$	0	0
$727$	−44.2361	−1.64062	−0.820312	−	0.571916i	$-0.806200\pi$
$727$	−44.2361	−1.64062	−0.820312	+	0.571916i	$0.806200\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	−0.944272	−0.0349252
$732$	0	0
$733$	−3.47214	−0.128246	−0.0641231	−	0.997942i	$-0.520425\pi$
$733$	−3.47214	−0.128246	−0.0641231	+	0.997942i	$0.520425\pi$
$734$	0	0
$735$	22.4721	0.828897
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	−6.18034	−0.227347	−0.113674	−	0.993518i	$-0.536262\pi$
$739$	−6.18034	−0.227347	−0.113674	+	0.993518i	$0.536262\pi$
$740$	0	0
$741$	23.4164	0.860223
$742$	0	0
$743$	50.1803	1.84094	0.920469	−	0.390815i	$-0.127807\pi$
$743$	50.1803	1.84094	0.920469	+	0.390815i	$0.127807\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	111.666	4.08563
$748$	0	0
$749$	−1.36068	−0.0497182
$750$	0	0
$751$	−21.5410	−0.786043	−0.393021	−	0.919529i	$-0.628570\pi$
$751$	−21.5410	−0.786043	−0.393021	+	0.919529i	$0.628570\pi$
$752$	0	0
$753$	5.88854	0.214590
$754$	0	0
$755$	14.1803	0.516075
$756$	0	0
$757$	−8.65248	−0.314480	−0.157240	−	0.987560i	$-0.550260\pi$
$757$	−8.65248	−0.314480	−0.157240	+	0.987560i	$0.550260\pi$
$758$	0	0
$759$	−36.9443	−1.34099
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	3.29180	0.119171
$764$	0	0
$765$	−5.70820	−0.206381
$766$	0	0
$767$	−7.23607	−0.261279
$768$	0	0
$769$	−47.3607	−1.70787	−0.853935	−	0.520380i	$-0.825790\pi$
$769$	−47.3607	−1.70787	−0.853935	+	0.520380i	$0.825790\pi$
$770$	0	0
$771$	6.29180	0.226594
$772$	0	0
$773$	11.1246	0.400124	0.200062	−	0.979783i	$-0.435886\pi$
$773$	11.1246	0.400124	0.200062	+	0.979783i	$0.435886\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	1.52786	0.0548118
$778$	0	0
$779$	15.6525	0.560808
$780$	0	0
$781$	18.3607	0.656997
$782$	0	0
$783$	−40.0000	−1.42948
$784$	0	0
$785$	20.8885	0.745544
$786$	0	0
$787$	−7.34752	−0.261911	−0.130955	−	0.991388i	$-0.541804\pi$
$787$	−7.34752	−0.261911	−0.130955	+	0.991388i	$0.541804\pi$
$788$	0	0
$789$	−75.1935	−2.67696
$790$	0	0
$791$	0.819660	0.0291438
$792$	0	0
$793$	−26.4721	−0.940053
$794$	0	0
$795$	−33.8885	−1.20190
$796$	0	0
$797$	55.4164	1.96295	0.981475	−	0.191591i	$-0.0613646\pi$
$797$	55.4164	1.96295	0.981475	+	0.191591i	$0.0613646\pi$
$798$	0	0
$799$	1.88854	0.0668119
$800$	0	0
$801$	87.4853	3.09114
$802$	0	0
$803$	−16.9443	−0.597950
$804$	0	0
$805$	−1.34752	−0.0474940
$806$	0	0
$807$	35.7771	1.25941
$808$	0	0
$809$	23.4164	0.823277	0.411639	−	0.911347i	$-0.364957\pi$
$809$	23.4164	0.823277	0.411639	+	0.911347i	$0.364957\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	−45.8885	−1.60938
$814$	0	0
$815$	10.7082	0.375092
$816$	0	0
$817$	−2.76393	−0.0966977
$818$	0	0
$819$	5.70820	0.199461
$820$	0	0
$821$	−30.5410	−1.06589	−0.532944	−	0.846150i	$-0.678915\pi$
$821$	−30.5410	−1.06589	−0.532944	+	0.846150i	$0.678915\pi$
$822$	0	0
$823$	−14.2918	−0.498181	−0.249090	−	0.968480i	$-0.580132\pi$
$823$	−14.2918	−0.498181	−0.249090	+	0.968480i	$0.580132\pi$
$824$	0	0
$825$	25.8885	0.901323
$826$	0	0
$827$	−17.3475	−0.603233	−0.301616	−	0.953429i	$-0.597526\pi$
$827$	−17.3475	−0.603233	−0.301616	+	0.953429i	$0.597526\pi$
$828$	0	0
$829$	−16.8328	−0.584628	−0.292314	−	0.956322i	$-0.594425\pi$
$829$	−16.8328	−0.584628	−0.292314	+	0.956322i	$0.594425\pi$
$830$	0	0
$831$	40.9443	1.42034
$832$	0	0
$833$	−5.30495	−0.183806
$834$	0	0
$835$	6.47214	0.223978
$836$	0	0
$837$	14.4721	0.500230
$838$	0	0
$839$	28.9443	0.999267	0.499634	−	0.866237i	$-0.333468\pi$
$839$	28.9443	0.999267	0.499634	+	0.866237i	$0.333468\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	55.0132	1.89475
$844$	0	0
$845$	2.52786	0.0869612
$846$	0	0
$847$	−1.65248	−0.0567797
$848$	0	0
$849$	44.9443	1.54248
$850$	0	0
$851$	11.4164	0.391349
$852$	0	0
$853$	−10.5836	−0.362375	−0.181188	−	0.983449i	$-0.557994\pi$
$853$	−10.5836	−0.362375	−0.181188	+	0.983449i	$0.557994\pi$
$854$	0	0
$855$	−16.7082	−0.571409
$856$	0	0
$857$	−55.6656	−1.90150	−0.950751	−	0.309956i	$-0.899686\pi$
$857$	−55.6656	−1.90150	−0.950751	+	0.309956i	$0.899686\pi$
$858$	0	0
$859$	−2.11146	−0.0720420	−0.0360210	−	0.999351i	$-0.511468\pi$
$859$	−2.11146	−0.0720420	−0.0360210	+	0.999351i	$0.511468\pi$
$860$	0	0
$861$	5.34752	0.182243
$862$	0	0
$863$	−9.81966	−0.334265	−0.167133	−	0.985934i	$-0.553451\pi$
$863$	−9.81966	−0.334265	−0.167133	+	0.985934i	$0.553451\pi$
$864$	0	0
$865$	2.94427	0.100108
$866$	0	0
$867$	−53.1246	−1.80421
$868$	0	0
$869$	23.4164	0.794347
$870$	0	0
$871$	−25.8885	−0.877200
$872$	0	0
$873$	−119.138	−4.03220
$874$	0	0
$875$	2.12461	0.0718250
$876$	0	0
$877$	18.0557	0.609699	0.304849	−	0.952401i	$-0.401394\pi$
$877$	18.0557	0.609699	0.304849	+	0.952401i	$0.401394\pi$
$878$	0	0
$879$	1.52786	0.0515336
$880$	0	0
$881$	−20.3607	−0.685969	−0.342984	−	0.939341i	$-0.611438\pi$
$881$	−20.3607	−0.685969	−0.342984	+	0.939341i	$0.611438\pi$
$882$	0	0
$883$	31.7771	1.06938	0.534692	−	0.845047i	$-0.320428\pi$
$883$	31.7771	1.06938	0.534692	+	0.845047i	$0.320428\pi$
$884$	0	0
$885$	7.23607	0.243238
$886$	0	0
$887$	27.0689	0.908884	0.454442	−	0.890776i	$-0.349839\pi$
$887$	27.0689	0.908884	0.454442	+	0.890776i	$0.349839\pi$
$888$	0	0
$889$	2.94427	0.0987477
$890$	0	0
$891$	−48.8328	−1.63596
$892$	0	0
$893$	5.52786	0.184983
$894$	0	0
$895$	1.70820	0.0570990
$896$	0	0
$897$	59.7771	1.99590
$898$	0	0
$899$	−2.76393	−0.0921823
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	−0.944272	−0.0314234
$904$	0	0
$905$	−4.18034	−0.138959
$906$	0	0
$907$	24.2361	0.804745	0.402373	−	0.915476i	$-0.368186\pi$
$907$	24.2361	0.804745	0.402373	+	0.915476i	$0.368186\pi$
$908$	0	0
$909$	22.4164	0.743505
$910$	0	0
$911$	18.1803	0.602342	0.301171	−	0.953570i	$-0.402623\pi$
$911$	18.1803	0.602342	0.301171	+	0.953570i	$0.402623\pi$
$912$	0	0
$913$	−29.8885	−0.989166
$914$	0	0
$915$	26.4721	0.875142
$916$	0	0
$917$	−2.83282	−0.0935478
$918$	0	0
$919$	−14.4721	−0.477392	−0.238696	−	0.971094i	$-0.576720\pi$
$919$	−14.4721	−0.477392	−0.238696	+	0.971094i	$0.576720\pi$
$920$	0	0
$921$	92.9017	3.06122
$922$	0	0
$923$	−29.7082	−0.977857
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	46.5967	1.53044
$928$	0	0
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	−15.5279	−0.508905
$932$	0	0
$933$	−94.4296	−3.09149
$934$	0	0
$935$	1.52786	0.0499665
$936$	0	0
$937$	9.05573	0.295838	0.147919	−	0.988999i	$-0.452743\pi$
$937$	9.05573	0.295838	0.147919	+	0.988999i	$0.452743\pi$
$938$	0	0
$939$	54.2492	1.77036
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	39.9574	1.30119
$944$	0	0
$945$	−3.41641	−0.111136
$946$	0	0
$947$	13.0557	0.424254	0.212127	−	0.977242i	$-0.431961\pi$
$947$	13.0557	0.424254	0.212127	+	0.977242i	$0.431961\pi$
$948$	0	0
$949$	27.4164	0.889974
$950$	0	0
$951$	−13.1246	−0.425595
$952$	0	0
$953$	45.7082	1.48063	0.740317	−	0.672258i	$-0.234675\pi$
$953$	45.7082	1.48063	0.740317	+	0.672258i	$0.234675\pi$
$954$	0	0
$955$	19.1803	0.620661
$956$	0	0
$957$	17.8885	0.578254
$958$	0	0
$959$	1.48529	0.0479626
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−43.0689	−1.38788
$964$	0	0
$965$	−3.47214	−0.111772
$966$	0	0
$967$	60.3607	1.94107	0.970534	−	0.240963i	$-0.0774632\pi$
$967$	60.3607	1.94107	0.970534	+	0.240963i	$0.0774632\pi$
$968$	0	0
$969$	5.52786	0.177581
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	−3.16718	−0.101535
$974$	0	0
$975$	−41.8885	−1.34151
$976$	0	0
$977$	−47.2492	−1.51164	−0.755818	−	0.654781i	$-0.772761\pi$
$977$	−47.2492	−1.51164	−0.755818	+	0.654781i	$0.772761\pi$
$978$	0	0
$979$	−23.4164	−0.748392
$980$	0	0
$981$	104.193	3.32664
$982$	0	0
$983$	39.5279	1.26074	0.630372	−	0.776294i	$-0.282903\pi$
$983$	39.5279	1.26074	0.630372	+	0.776294i	$0.282903\pi$
$984$	0	0
$985$	11.4164	0.363757
$986$	0	0
$987$	1.88854	0.0601130
$988$	0	0
$989$	−7.05573	−0.224359
$990$	0	0
$991$	−16.5410	−0.525443	−0.262721	−	0.964872i	$-0.584620\pi$
$991$	−16.5410	−0.525443	−0.262721	+	0.964872i	$0.584620\pi$
$992$	0	0
$993$	−6.47214	−0.205387
$994$	0	0
$995$	18.9443	0.600574
$996$	0	0
$997$	29.3607	0.929862	0.464931	−	0.885347i	$-0.346079\pi$
$997$	29.3607	0.929862	0.464931	+	0.885347i	$0.346079\pi$
$998$	0	0
$999$	28.9443	0.915756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1984.2.a.r.1.2		2
4.3	odd	2		1984.2.a.n.1.1		2
8.3	odd	2		496.2.a.i.1.2		2
8.5	even	2		31.2.a.a.1.2	✓	2
24.5	odd	2		279.2.a.a.1.1		2
24.11	even	2		4464.2.a.bf.1.1		2
40.13	odd	4		775.2.b.d.249.1		4
40.29	even	2		775.2.a.d.1.1		2
40.37	odd	4		775.2.b.d.249.4		4
56.13	odd	2		1519.2.a.a.1.2		2
88.21	odd	2		3751.2.a.b.1.1		2
104.77	even	2		5239.2.a.f.1.1		2
120.29	odd	2		6975.2.a.y.1.2		2
136.101	even	2		8959.2.a.b.1.2		2
248.5	even	6		961.2.c.e.521.2		4
248.13	odd	30		961.2.g.e.448.1		8
248.21	odd	30		961.2.g.e.844.1		8
248.29	odd	10		961.2.d.g.531.1		4
248.37	odd	6		961.2.c.c.439.2		4
248.45	even	30		961.2.g.a.816.1		8
248.53	odd	30		961.2.g.d.732.1		8
248.61	odd	2		961.2.a.f.1.2		2
248.69	even	30		961.2.g.a.235.1		8
248.77	odd	10		961.2.d.g.628.1		4
248.85	odd	10		961.2.d.a.374.1		4
248.101	even	10		961.2.d.c.374.1		4
248.109	even	10		961.2.d.d.628.1		4
248.117	odd	30		961.2.g.d.235.1		8
248.133	even	30		961.2.g.a.732.1		8
248.141	odd	30		961.2.g.d.816.1		8
248.149	even	6		961.2.c.e.439.2		4
248.157	even	10		961.2.d.d.531.1		4
248.165	even	30		961.2.g.h.844.1		8
248.173	even	30		961.2.g.h.448.1		8
248.181	odd	6		961.2.c.c.521.2		4
248.189	odd	30		961.2.g.e.846.1		8
248.197	odd	30		961.2.g.d.338.1		8
248.205	even	30		961.2.g.h.547.1		8
248.213	odd	10		961.2.d.a.388.1		4
248.221	even	10		961.2.d.c.388.1		4
248.229	odd	30		961.2.g.e.547.1		8
248.237	even	30		961.2.g.a.338.1		8
248.245	even	30		961.2.g.h.846.1		8
744.557	even	2		8649.2.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.a.a.1.2	✓	2	8.5	even	2
279.2.a.a.1.1		2	24.5	odd	2
496.2.a.i.1.2		2	8.3	odd	2
775.2.a.d.1.1		2	40.29	even	2
775.2.b.d.249.1		4	40.13	odd	4
775.2.b.d.249.4		4	40.37	odd	4
961.2.a.f.1.2		2	248.61	odd	2
961.2.c.c.439.2		4	248.37	odd	6
961.2.c.c.521.2		4	248.181	odd	6
961.2.c.e.439.2		4	248.149	even	6
961.2.c.e.521.2		4	248.5	even	6
961.2.d.a.374.1		4	248.85	odd	10
961.2.d.a.388.1		4	248.213	odd	10
961.2.d.c.374.1		4	248.101	even	10
961.2.d.c.388.1		4	248.221	even	10
961.2.d.d.531.1		4	248.157	even	10
961.2.d.d.628.1		4	248.109	even	10
961.2.d.g.531.1		4	248.29	odd	10
961.2.d.g.628.1		4	248.77	odd	10
961.2.g.a.235.1		8	248.69	even	30
961.2.g.a.338.1		8	248.237	even	30
961.2.g.a.732.1		8	248.133	even	30
961.2.g.a.816.1		8	248.45	even	30
961.2.g.d.235.1		8	248.117	odd	30
961.2.g.d.338.1		8	248.197	odd	30
961.2.g.d.732.1		8	248.53	odd	30
961.2.g.d.816.1		8	248.141	odd	30
961.2.g.e.448.1		8	248.13	odd	30
961.2.g.e.547.1		8	248.229	odd	30
961.2.g.e.844.1		8	248.21	odd	30
961.2.g.e.846.1		8	248.189	odd	30
961.2.g.h.448.1		8	248.173	even	30
961.2.g.h.547.1		8	248.205	even	30
961.2.g.h.844.1		8	248.165	even	30
961.2.g.h.846.1		8	248.245	even	30
1519.2.a.a.1.2		2	56.13	odd	2
1984.2.a.n.1.1		2	4.3	odd	2
1984.2.a.r.1.2		2	1.1	even	1	trivial
3751.2.a.b.1.1		2	88.21	odd	2
4464.2.a.bf.1.1		2	24.11	even	2
5239.2.a.f.1.1		2	104.77	even	2
6975.2.a.y.1.2		2	120.29	odd	2
8649.2.a.c.1.1		2	744.557	even	2
8959.2.a.b.1.2		2	136.101	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.23607 q^{3} -1.00000 q^{5} +0.236068 q^{7} +7.47214 q^{9} +O(q^{10})\)	\(q+3.23607 q^{3} -1.00000 q^{5} +0.236068 q^{7} +7.47214 q^{9} -2.00000 q^{11} +3.23607 q^{13} -3.23607 q^{15} +0.763932 q^{17} +2.23607 q^{19} +0.763932 q^{21} +5.70820 q^{23} -4.00000 q^{25} +14.4721 q^{27} -2.76393 q^{29} +1.00000 q^{31} -6.47214 q^{33} -0.236068 q^{35} +2.00000 q^{37} +10.4721 q^{39} +7.00000 q^{41} -1.23607 q^{43} -7.47214 q^{45} +2.47214 q^{47} -6.94427 q^{49} +2.47214 q^{51} +10.4721 q^{53} +2.00000 q^{55} +7.23607 q^{57} -2.23607 q^{59} -8.18034 q^{61} +1.76393 q^{63} -3.23607 q^{65} -8.00000 q^{67} +18.4721 q^{69} -9.18034 q^{71} +8.47214 q^{73} -12.9443 q^{75} -0.472136 q^{77} -11.7082 q^{79} +24.4164 q^{81} +14.9443 q^{83} -0.763932 q^{85} -8.94427 q^{87} +11.7082 q^{89} +0.763932 q^{91} +3.23607 q^{93} -2.23607 q^{95} -15.9443 q^{97} -14.9443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^7 + 6 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} + 6 q^{17} + 6 q^{21} - 2 q^{23} - 8 q^{25} + 20 q^{27} - 10 q^{29} + 2 q^{31} - 4 q^{33} + 4 q^{35} + 4 q^{37} + 12 q^{39} + 14 q^{41} + 2 q^{43} - 6 q^{45} - 4 q^{47} + 4 q^{49} - 4 q^{51} + 12 q^{53} + 4 q^{55} + 10 q^{57} + 6 q^{61} + 8 q^{63} - 2 q^{65} - 16 q^{67} + 28 q^{69} + 4 q^{71} + 8 q^{73} - 8 q^{75} + 8 q^{77} - 10 q^{79} + 22 q^{81} + 12 q^{83} - 6 q^{85} + 10 q^{89} + 6 q^{91} + 2 q^{93} - 14 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^13 - 2 * q^15 + 6 * q^17 + 6 * q^21 - 2 * q^23 - 8 * q^25 + 20 * q^27 - 10 * q^29 + 2 * q^31 - 4 * q^33 + 4 * q^35 + 4 * q^37 + 12 * q^39 + 14 * q^41 + 2 * q^43 - 6 * q^45 - 4 * q^47 + 4 * q^49 - 4 * q^51 + 12 * q^53 + 4 * q^55 + 10 * q^57 + 6 * q^61 + 8 * q^63 - 2 * q^65 - 16 * q^67 + 28 * q^69 + 4 * q^71 + 8 * q^73 - 8 * q^75 + 8 * q^77 - 10 * q^79 + 22 * q^81 + 12 * q^83 - 6 * q^85 + 10 * q^89 + 6 * q^91 + 2 * q^93 - 14 * q^97 - 12 * q^99

Introduction

L-functions

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