LMFDB - Embedded newform 920.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.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:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66908 q^{3} -1.00000 q^{5} +3.66908 q^{7} +4.12398 q^{9} +O(q^{10})$	$q+2.66908 q^{3} -1.00000 q^{5} +3.66908 q^{7} +4.12398 q^{9} -1.21417 q^{11} +2.21417 q^{13} -2.66908 q^{15} +1.21417 q^{17} -2.57889 q^{19} +9.79306 q^{21} +1.00000 q^{23} +1.00000 q^{25} +3.00000 q^{27} +1.45490 q^{29} -6.46214 q^{31} -3.24073 q^{33} -3.66908 q^{35} +4.00000 q^{37} +5.90981 q^{39} -10.9170 q^{41} +6.90981 q^{43} -4.12398 q^{45} +5.45490 q^{47} +6.46214 q^{49} +3.24073 q^{51} +3.81962 q^{53} +1.21417 q^{55} -6.88325 q^{57} -4.24797 q^{59} +6.78583 q^{61} +15.1312 q^{63} -2.21417 q^{65} +12.8567 q^{67} +2.66908 q^{69} -6.91705 q^{71} -15.2214 q^{73} +2.66908 q^{75} -4.45490 q^{77} -4.36471 q^{81} -15.5861 q^{83} -1.21417 q^{85} +3.88325 q^{87} -10.9098 q^{89} +8.12398 q^{91} -17.2480 q^{93} +2.57889 q^{95} +1.69563 q^{97} -5.00724 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{17} + 3 q^{19} + 12 q^{21} + 3 q^{23} + 3 q^{25} + 9 q^{27} + 3 q^{29} + 6 q^{31} - 15 q^{33} - 3 q^{35} + 12 q^{37} + 15 q^{39} - 6 q^{41} + 18 q^{43} - 3 q^{45} + 15 q^{47} - 6 q^{49} + 15 q^{51} + 6 q^{53} - 3 q^{55} - 6 q^{57} + 6 q^{59} + 27 q^{61} + 12 q^{63} + 12 q^{67} + 6 q^{71} - 15 q^{73} - 12 q^{77} - 9 q^{81} - 12 q^{83} + 3 q^{85} - 3 q^{87} - 30 q^{89} + 15 q^{91} - 33 q^{93} - 3 q^{95} + 9 q^{97} + 9 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^17 + 3 * q^19 + 12 * q^21 + 3 * q^23 + 3 * q^25 + 9 * q^27 + 3 * q^29 + 6 * q^31 - 15 * q^33 - 3 * q^35 + 12 * q^37 + 15 * q^39 - 6 * q^41 + 18 * q^43 - 3 * q^45 + 15 * q^47 - 6 * q^49 + 15 * q^51 + 6 * q^53 - 3 * q^55 - 6 * q^57 + 6 * q^59 + 27 * q^61 + 12 * q^63 + 12 * q^67 + 6 * q^71 - 15 * q^73 - 12 * q^77 - 9 * q^81 - 12 * q^83 + 3 * q^85 - 3 * q^87 - 30 * q^89 + 15 * q^91 - 33 * q^93 - 3 * q^95 + 9 * q^97 + 9 * 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
$3$	2.66908	1.54099	0.770497	−	0.637444i	$-0.220008\pi$
$3$	2.66908	1.54099	0.770497	+	0.637444i	$0.220008\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.66908	1.38678	0.693391	−	0.720562i	$-0.256116\pi$
$7$	3.66908	1.38678	0.693391	+	0.720562i	$0.256116\pi$
$8$	0	0
$9$	4.12398	1.37466
$10$	0	0
$11$	−1.21417	−0.366088	−0.183044	−	0.983105i	$-0.558595\pi$
$11$	−1.21417	−0.366088	−0.183044	+	0.983105i	$0.558595\pi$
$12$	0	0
$13$	2.21417	0.614102	0.307051	−	0.951693i	$-0.400658\pi$
$13$	2.21417	0.614102	0.307051	+	0.951693i	$0.400658\pi$
$14$	0	0
$15$	−2.66908	−0.689153
$16$	0	0
$17$	1.21417	0.294481	0.147240	−	0.989101i	$-0.452961\pi$
$17$	1.21417	0.294481	0.147240	+	0.989101i	$0.452961\pi$
$18$	0	0
$19$	−2.57889	−0.591637	−0.295819	−	0.955244i	$-0.595592\pi$
$19$	−2.57889	−0.591637	−0.295819	+	0.955244i	$0.595592\pi$
$20$	0	0
$21$	9.79306	2.13702
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.00000	0.577350
$28$	0	0
$29$	1.45490	0.270169	0.135084	−	0.990834i	$-0.456869\pi$
$29$	1.45490	0.270169	0.135084	+	0.990834i	$0.456869\pi$
$30$	0	0
$31$	−6.46214	−1.16063	−0.580317	−	0.814390i	$-0.697072\pi$
$31$	−6.46214	−1.16063	−0.580317	+	0.814390i	$0.697072\pi$
$32$	0	0
$33$	−3.24073	−0.564139
$34$	0	0
$35$	−3.66908	−0.620188
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	5.90981	0.946327
$40$	0	0
$41$	−10.9170	−1.70496	−0.852478	−	0.522763i	$-0.824901\pi$
$41$	−10.9170	−1.70496	−0.852478	+	0.522763i	$0.824901\pi$
$42$	0	0
$43$	6.90981	1.05374	0.526868	−	0.849947i	$-0.323366\pi$
$43$	6.90981	1.05374	0.526868	+	0.849947i	$0.323366\pi$
$44$	0	0
$45$	−4.12398	−0.614767
$46$	0	0
$47$	5.45490	0.795680	0.397840	−	0.917455i	$-0.369760\pi$
$47$	5.45490	0.795680	0.397840	+	0.917455i	$0.369760\pi$
$48$	0	0
$49$	6.46214	0.923163
$50$	0	0
$51$	3.24073	0.453793
$52$	0	0
$53$	3.81962	0.524665	0.262332	−	0.964978i	$-0.415508\pi$
$53$	3.81962	0.524665	0.262332	+	0.964978i	$0.415508\pi$
$54$	0	0
$55$	1.21417	0.163719
$56$	0	0
$57$	−6.88325	−0.911709
$58$	0	0
$59$	−4.24797	−0.553038	−0.276519	−	0.961008i	$-0.589181\pi$
$59$	−4.24797	−0.553038	−0.276519	+	0.961008i	$0.589181\pi$
$60$	0	0
$61$	6.78583	0.868836	0.434418	−	0.900711i	$-0.356954\pi$
$61$	6.78583	0.868836	0.434418	+	0.900711i	$0.356954\pi$
$62$	0	0
$63$	15.1312	1.90635
$64$	0	0
$65$	−2.21417	−0.274635
$66$	0	0
$67$	12.8567	1.57070	0.785348	−	0.619055i	$-0.212484\pi$
$67$	12.8567	1.57070	0.785348	+	0.619055i	$0.212484\pi$
$68$	0	0
$69$	2.66908	0.321319
$70$	0	0
$71$	−6.91705	−0.820902	−0.410451	−	0.911883i	$-0.634629\pi$
$71$	−6.91705	−0.820902	−0.410451	+	0.911883i	$0.634629\pi$
$72$	0	0
$73$	−15.2214	−1.78153	−0.890766	−	0.454463i	$-0.849831\pi$
$73$	−15.2214	−1.78153	−0.890766	+	0.454463i	$0.849831\pi$
$74$	0	0
$75$	2.66908	0.308199
$76$	0	0
$77$	−4.45490	−0.507683
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−4.36471	−0.484968
$82$	0	0
$83$	−15.5861	−1.71080	−0.855400	−	0.517968i	$-0.826688\pi$
$83$	−15.5861	−1.71080	−0.855400	+	0.517968i	$0.826688\pi$
$84$	0	0
$85$	−1.21417	−0.131696
$86$	0	0
$87$	3.88325	0.416329
$88$	0	0
$89$	−10.9098	−1.15644	−0.578219	−	0.815882i	$-0.696252\pi$
$89$	−10.9098	−1.15644	−0.578219	+	0.815882i	$0.696252\pi$
$90$	0	0
$91$	8.12398	0.851625
$92$	0	0
$93$	−17.2480	−1.78853
$94$	0	0
$95$	2.57889	0.264588
$96$	0	0
$97$	1.69563	0.172165	0.0860827	−	0.996288i	$-0.472565\pi$
$97$	1.69563	0.172165	0.0860827	+	0.996288i	$0.472565\pi$
$98$	0	0
$99$	−5.00724	−0.503246
$100$	0	0
$101$	−2.24797	−0.223681	−0.111841	−	0.993726i	$-0.535675\pi$
$101$	−2.24797	−0.223681	−0.111841	+	0.993726i	$0.535675\pi$
$102$	0	0
$103$	−2.78583	−0.274495	−0.137248	−	0.990537i	$-0.543826\pi$
$103$	−2.78583	−0.274495	−0.137248	+	0.990537i	$0.543826\pi$
$104$	0	0
$105$	−9.79306	−0.955705
$106$	0	0
$107$	13.1578	1.27201	0.636005	−	0.771685i	$-0.280586\pi$
$107$	13.1578	1.27201	0.636005	+	0.771685i	$0.280586\pi$
$108$	0	0
$109$	3.42111	0.327683	0.163842	−	0.986487i	$-0.447611\pi$
$109$	3.42111	0.327683	0.163842	+	0.986487i	$0.447611\pi$
$110$	0	0
$111$	10.6763	1.01335
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	9.13122	0.844182
$118$	0	0
$119$	4.45490	0.408380
$120$	0	0
$121$	−9.52578	−0.865980
$122$	0	0
$123$	−29.1385	−2.62733
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.9508	1.77035	0.885175	−	0.465258i	$-0.154038\pi$
$127$	19.9508	1.77035	0.885175	+	0.465258i	$0.154038\pi$
$128$	0	0
$129$	18.4428	1.62380
$130$	0	0
$131$	−9.70287	−0.847744	−0.423872	−	0.905722i	$-0.639329\pi$
$131$	−9.70287	−0.847744	−0.423872	+	0.905722i	$0.639329\pi$
$132$	0	0
$133$	−9.46214	−0.820472
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	2.82685	0.241514	0.120757	−	0.992682i	$-0.461468\pi$
$137$	2.82685	0.241514	0.120757	+	0.992682i	$0.461468\pi$
$138$	0	0
$139$	2.36471	0.200572	0.100286	−	0.994959i	$-0.468024\pi$
$139$	2.36471	0.200572	0.100286	+	0.994959i	$0.468024\pi$
$140$	0	0
$141$	14.5596	1.22614
$142$	0	0
$143$	−2.68840	−0.224815
$144$	0	0
$145$	−1.45490	−0.120823
$146$	0	0
$147$	17.2480	1.42259
$148$	0	0
$149$	−11.9170	−0.976282	−0.488141	−	0.872765i	$-0.662325\pi$
$149$	−11.9170	−0.976282	−0.488141	+	0.872765i	$0.662325\pi$
$150$	0	0
$151$	9.57889	0.779519	0.389759	−	0.920917i	$-0.372558\pi$
$151$	9.57889	0.779519	0.389759	+	0.920917i	$0.372558\pi$
$152$	0	0
$153$	5.00724	0.404811
$154$	0	0
$155$	6.46214	0.519052
$156$	0	0
$157$	−18.7439	−1.49593	−0.747963	−	0.663740i	$-0.768968\pi$
$157$	−18.7439	−1.49593	−0.747963	+	0.663740i	$0.768968\pi$
$158$	0	0
$159$	10.1949	0.808505
$160$	0	0
$161$	3.66908	0.289164
$162$	0	0
$163$	−19.3188	−1.51317	−0.756584	−	0.653896i	$-0.773133\pi$
$163$	−19.3188	−1.51317	−0.756584	+	0.653896i	$0.773133\pi$
$164$	0	0
$165$	3.24073	0.252290
$166$	0	0
$167$	−1.81962	−0.140806	−0.0704031	−	0.997519i	$-0.522429\pi$
$167$	−1.81962	−0.140806	−0.0704031	+	0.997519i	$0.522429\pi$
$168$	0	0
$169$	−8.09743	−0.622879
$170$	0	0
$171$	−10.6353	−0.813301
$172$	0	0
$173$	15.9581	1.21327	0.606635	−	0.794981i	$-0.292519\pi$
$173$	15.9581	1.21327	0.606635	+	0.794981i	$0.292519\pi$
$174$	0	0
$175$	3.66908	0.277356
$176$	0	0
$177$	−11.3382	−0.852228
$178$	0	0
$179$	17.9508	1.34171	0.670854	−	0.741589i	$-0.265928\pi$
$179$	17.9508	1.34171	0.670854	+	0.741589i	$0.265928\pi$
$180$	0	0
$181$	6.33092	0.470574	0.235287	−	0.971926i	$-0.424397\pi$
$181$	6.33092	0.470574	0.235287	+	0.971926i	$0.424397\pi$
$182$	0	0
$183$	18.1119	1.33887
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−1.47422	−0.107806
$188$	0	0
$189$	11.0072	0.800659
$190$	0	0
$191$	−15.1578	−1.09678	−0.548389	−	0.836223i	$-0.684759\pi$
$191$	−15.1578	−1.09678	−0.548389	+	0.836223i	$0.684759\pi$
$192$	0	0
$193$	18.9879	1.36678	0.683390	−	0.730053i	$-0.260505\pi$
$193$	18.9879	1.36678	0.683390	+	0.730053i	$0.260505\pi$
$194$	0	0
$195$	−5.90981	−0.423210
$196$	0	0
$197$	−19.1650	−1.36545	−0.682725	−	0.730675i	$-0.739205\pi$
$197$	−19.1650	−1.36545	−0.682725	+	0.730675i	$0.739205\pi$
$198$	0	0
$199$	−5.76651	−0.408777	−0.204388	−	0.978890i	$-0.565521\pi$
$199$	−5.76651	−0.408777	−0.204388	+	0.978890i	$0.565521\pi$
$200$	0	0
$201$	34.3155	2.42043
$202$	0	0
$203$	5.33816	0.374665
$204$	0	0
$205$	10.9170	0.762480
$206$	0	0
$207$	4.12398	0.286637
$208$	0	0
$209$	3.13122	0.216591
$210$	0	0
$211$	−17.4057	−1.19826	−0.599130	−	0.800652i	$-0.704487\pi$
$211$	−17.4057	−1.19826	−0.599130	+	0.800652i	$0.704487\pi$
$212$	0	0
$213$	−18.4621	−1.26501
$214$	0	0
$215$	−6.90981	−0.471245
$216$	0	0
$217$	−23.7101	−1.60955
$218$	0	0
$219$	−40.6272	−2.74533
$220$	0	0
$221$	2.68840	0.180841
$222$	0	0
$223$	−14.6763	−0.982799	−0.491399	−	0.870934i	$-0.663514\pi$
$223$	−14.6763	−0.982799	−0.491399	+	0.870934i	$0.663514\pi$
$224$	0	0
$225$	4.12398	0.274932
$226$	0	0
$227$	16.4283	1.09039	0.545194	−	0.838310i	$-0.316456\pi$
$227$	16.4283	1.09039	0.545194	+	0.838310i	$0.316456\pi$
$228$	0	0
$229$	19.3526	1.27886	0.639429	−	0.768850i	$-0.279171\pi$
$229$	19.3526	1.27886	0.639429	+	0.768850i	$0.279171\pi$
$230$	0	0
$231$	−11.8905	−0.782337
$232$	0	0
$233$	−19.4549	−1.27453	−0.637267	−	0.770643i	$-0.719935\pi$
$233$	−19.4549	−1.27453	−0.637267	+	0.770643i	$0.719935\pi$
$234$	0	0
$235$	−5.45490	−0.355839
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.7029	1.53321	0.766606	−	0.642118i	$-0.221944\pi$
$239$	23.7029	1.53321	0.766606	+	0.642118i	$0.221944\pi$
$240$	0	0
$241$	0.233492	0.0150405	0.00752027	−	0.999972i	$-0.497606\pi$
$241$	0.233492	0.0150405	0.00752027	+	0.999972i	$0.497606\pi$
$242$	0	0
$243$	−20.6498	−1.32468
$244$	0	0
$245$	−6.46214	−0.412851
$246$	0	0
$247$	−5.71011	−0.363325
$248$	0	0
$249$	−41.6006	−2.63633
$250$	0	0
$251$	21.7511	1.37292	0.686460	−	0.727168i	$-0.259164\pi$
$251$	21.7511	1.37292	0.686460	+	0.727168i	$0.259164\pi$
$252$	0	0
$253$	−1.21417	−0.0763345
$254$	0	0
$255$	−3.24073	−0.202942
$256$	0	0
$257$	−16.3116	−1.01749	−0.508745	−	0.860917i	$-0.669890\pi$
$257$	−16.3116	−1.01749	−0.508745	+	0.860917i	$0.669890\pi$
$258$	0	0
$259$	14.6763	0.911942
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−2.53786	−0.156491	−0.0782455	−	0.996934i	$-0.524932\pi$
$263$	−2.53786	−0.156491	−0.0782455	+	0.996934i	$0.524932\pi$
$264$	0	0
$265$	−3.81962	−0.234637
$266$	0	0
$267$	−29.1191	−1.78206
$268$	0	0
$269$	−2.54510	−0.155177	−0.0775886	−	0.996985i	$-0.524722\pi$
$269$	−2.54510	−0.155177	−0.0775886	+	0.996985i	$0.524722\pi$
$270$	0	0
$271$	24.1795	1.46880	0.734400	−	0.678717i	$-0.237464\pi$
$271$	24.1795	1.46880	0.734400	+	0.678717i	$0.237464\pi$
$272$	0	0
$273$	21.6836	1.31235
$274$	0	0
$275$	−1.21417	−0.0732175
$276$	0	0
$277$	−21.2359	−1.27594	−0.637970	−	0.770061i	$-0.720226\pi$
$277$	−21.2359	−1.27594	−0.637970	+	0.770061i	$0.720226\pi$
$278$	0	0
$279$	−26.6498	−1.59548
$280$	0	0
$281$	−13.8872	−0.828441	−0.414220	−	0.910177i	$-0.635946\pi$
$281$	−13.8872	−0.828441	−0.414220	+	0.910177i	$0.635946\pi$
$282$	0	0
$283$	−29.5330	−1.75556	−0.877778	−	0.479068i	$-0.840975\pi$
$283$	−29.5330	−1.75556	−0.877778	+	0.479068i	$0.840975\pi$
$284$	0	0
$285$	6.88325	0.407729
$286$	0	0
$287$	−40.0555	−2.36440
$288$	0	0
$289$	−15.5258	−0.913281
$290$	0	0
$291$	4.52578	0.265306
$292$	0	0
$293$	−24.0821	−1.40689	−0.703444	−	0.710750i	$-0.748356\pi$
$293$	−24.0821	−1.40689	−0.703444	+	0.710750i	$0.748356\pi$
$294$	0	0
$295$	4.24797	0.247326
$296$	0	0
$297$	−3.64252	−0.211361
$298$	0	0
$299$	2.21417	0.128049
$300$	0	0
$301$	25.3526	1.46130
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−6.78583	−0.388555
$306$	0	0
$307$	19.9436	1.13824	0.569121	−	0.822254i	$-0.307284\pi$
$307$	19.9436	1.13824	0.569121	+	0.822254i	$0.307284\pi$
$308$	0	0
$309$	−7.43559	−0.422996
$310$	0	0
$311$	−2.97345	−0.168609	−0.0843043	−	0.996440i	$-0.526867\pi$
$311$	−2.97345	−0.168609	−0.0843043	+	0.996440i	$0.526867\pi$
$312$	0	0
$313$	15.4887	0.875473	0.437736	−	0.899103i	$-0.355780\pi$
$313$	15.4887	0.875473	0.437736	+	0.899103i	$0.355780\pi$
$314$	0	0
$315$	−15.1312	−0.852548
$316$	0	0
$317$	8.82685	0.495765	0.247883	−	0.968790i	$-0.420265\pi$
$317$	8.82685	0.495765	0.247883	+	0.968790i	$0.420265\pi$
$318$	0	0
$319$	−1.76651	−0.0989055
$320$	0	0
$321$	35.1191	1.96016
$322$	0	0
$323$	−3.13122	−0.174226
$324$	0	0
$325$	2.21417	0.122820
$326$	0	0
$327$	9.13122	0.504958
$328$	0	0
$329$	20.0145	1.10343
$330$	0	0
$331$	17.4018	0.956489	0.478245	−	0.878227i	$-0.341273\pi$
$331$	17.4018	0.956489	0.478245	+	0.878227i	$0.341273\pi$
$332$	0	0
$333$	16.4959	0.903972
$334$	0	0
$335$	−12.8567	−0.702437
$336$	0	0
$337$	26.9541	1.46828	0.734142	−	0.678995i	$-0.237584\pi$
$337$	26.9541	1.46828	0.734142	+	0.678995i	$0.237584\pi$
$338$	0	0
$339$	−26.6908	−1.44964
$340$	0	0
$341$	7.84617	0.424894
$342$	0	0
$343$	−1.97345	−0.106556
$344$	0	0
$345$	−2.66908	−0.143698
$346$	0	0
$347$	−10.1240	−0.543484	−0.271742	−	0.962370i	$-0.587600\pi$
$347$	−10.1240	−0.543484	−0.271742	+	0.962370i	$0.587600\pi$
$348$	0	0
$349$	7.38732	0.395434	0.197717	−	0.980259i	$-0.436647\pi$
$349$	7.38732	0.395434	0.197717	+	0.980259i	$0.436647\pi$
$350$	0	0
$351$	6.64252	0.354552
$352$	0	0
$353$	13.8833	0.738931	0.369466	−	0.929244i	$-0.379541\pi$
$353$	13.8833	0.738931	0.369466	+	0.929244i	$0.379541\pi$
$354$	0	0
$355$	6.91705	0.367119
$356$	0	0
$357$	11.8905	0.629312
$358$	0	0
$359$	15.0902	0.796430	0.398215	−	0.917292i	$-0.369630\pi$
$359$	15.0902	0.796430	0.398215	+	0.917292i	$0.369630\pi$
$360$	0	0
$361$	−12.3493	−0.649965
$362$	0	0
$363$	−25.4251	−1.33447
$364$	0	0
$365$	15.2214	0.796725
$366$	0	0
$367$	−24.0289	−1.25430	−0.627150	−	0.778898i	$-0.715779\pi$
$367$	−24.0289	−1.25430	−0.627150	+	0.778898i	$0.715779\pi$
$368$	0	0
$369$	−45.0217	−2.34374
$370$	0	0
$371$	14.0145	0.727595
$372$	0	0
$373$	30.6908	1.58911	0.794554	−	0.607193i	$-0.207704\pi$
$373$	30.6908	1.58911	0.794554	+	0.607193i	$0.207704\pi$
$374$	0	0
$375$	−2.66908	−0.137831
$376$	0	0
$377$	3.22141	0.165911
$378$	0	0
$379$	2.07087	0.106374	0.0531869	−	0.998585i	$-0.483062\pi$
$379$	2.07087	0.106374	0.0531869	+	0.998585i	$0.483062\pi$
$380$	0	0
$381$	53.2504	2.72810
$382$	0	0
$383$	−1.58612	−0.0810472	−0.0405236	−	0.999179i	$-0.512903\pi$
$383$	−1.58612	−0.0810472	−0.0405236	+	0.999179i	$0.512903\pi$
$384$	0	0
$385$	4.45490	0.227043
$386$	0	0
$387$	28.4959	1.44853
$388$	0	0
$389$	21.2142	1.07560	0.537801	−	0.843072i	$-0.319255\pi$
$389$	21.2142	1.07560	0.537801	+	0.843072i	$0.319255\pi$
$390$	0	0
$391$	1.21417	0.0614035
$392$	0	0
$393$	−25.8977	−1.30637
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.93965	−0.0973485	−0.0486742	−	0.998815i	$-0.515500\pi$
$397$	−1.93965	−0.0973485	−0.0486742	+	0.998815i	$0.515500\pi$
$398$	0	0
$399$	−25.2552	−1.26434
$400$	0	0
$401$	−5.69893	−0.284591	−0.142295	−	0.989824i	$-0.545448\pi$
$401$	−5.69893	−0.284591	−0.142295	+	0.989824i	$0.545448\pi$
$402$	0	0
$403$	−14.3083	−0.712748
$404$	0	0
$405$	4.36471	0.216884
$406$	0	0
$407$	−4.85670	−0.240738
$408$	0	0
$409$	3.53786	0.174936	0.0874679	−	0.996167i	$-0.472122\pi$
$409$	3.53786	0.174936	0.0874679	+	0.996167i	$0.472122\pi$
$410$	0	0
$411$	7.54510	0.372172
$412$	0	0
$413$	−15.5861	−0.766943
$414$	0	0
$415$	15.5861	0.765093
$416$	0	0
$417$	6.31160	0.309081
$418$	0	0
$419$	28.2624	1.38071	0.690355	−	0.723471i	$-0.257454\pi$
$419$	28.2624	1.38071	0.690355	+	0.723471i	$0.257454\pi$
$420$	0	0
$421$	34.0974	1.66181	0.830904	−	0.556417i	$-0.187824\pi$
$421$	34.0974	1.66181	0.830904	+	0.556417i	$0.187824\pi$
$422$	0	0
$423$	22.4959	1.09379
$424$	0	0
$425$	1.21417	0.0588961
$426$	0	0
$427$	24.8977	1.20489
$428$	0	0
$429$	−7.17554	−0.346438
$430$	0	0
$431$	−13.2850	−0.639918	−0.319959	−	0.947431i	$-0.603669\pi$
$431$	−13.2850	−0.639918	−0.319959	+	0.947431i	$0.603669\pi$
$432$	0	0
$433$	11.0603	0.531526	0.265763	−	0.964038i	$-0.414376\pi$
$433$	11.0603	0.531526	0.265763	+	0.964038i	$0.414376\pi$
$434$	0	0
$435$	−3.88325	−0.186188
$436$	0	0
$437$	−2.57889	−0.123365
$438$	0	0
$439$	−19.1916	−0.915963	−0.457982	−	0.888962i	$-0.651427\pi$
$439$	−19.1916	−0.915963	−0.457982	+	0.888962i	$0.651427\pi$
$440$	0	0
$441$	26.6498	1.26904
$442$	0	0
$443$	24.2818	1.15366	0.576831	−	0.816864i	$-0.304289\pi$
$443$	24.2818	1.15366	0.576831	+	0.816864i	$0.304289\pi$
$444$	0	0
$445$	10.9098	0.517175
$446$	0	0
$447$	−31.8075	−1.50444
$448$	0	0
$449$	14.8413	0.700406	0.350203	−	0.936674i	$-0.386113\pi$
$449$	14.8413	0.700406	0.350203	+	0.936674i	$0.386113\pi$
$450$	0	0
$451$	13.2552	0.624163
$452$	0	0
$453$	25.5668	1.20123
$454$	0	0
$455$	−8.12398	−0.380858
$456$	0	0
$457$	15.5861	0.729088	0.364544	−	0.931186i	$-0.381225\pi$
$457$	15.5861	0.729088	0.364544	+	0.931186i	$0.381225\pi$
$458$	0	0
$459$	3.64252	0.170019
$460$	0	0
$461$	−14.1312	−0.658157	−0.329078	−	0.944303i	$-0.606738\pi$
$461$	−14.1312	−0.658157	−0.329078	+	0.944303i	$0.606738\pi$
$462$	0	0
$463$	26.7584	1.24357	0.621784	−	0.783189i	$-0.286408\pi$
$463$	26.7584	1.24357	0.621784	+	0.783189i	$0.286408\pi$
$464$	0	0
$465$	17.2480	0.799855
$466$	0	0
$467$	26.6908	1.23510	0.617551	−	0.786531i	$-0.288125\pi$
$467$	26.6908	1.23510	0.617551	+	0.786531i	$0.288125\pi$
$468$	0	0
$469$	47.1722	2.17821
$470$	0	0
$471$	−50.0289	−2.30521
$472$	0	0
$473$	−8.38972	−0.385760
$474$	0	0
$475$	−2.57889	−0.118327
$476$	0	0
$477$	15.7520	0.721236
$478$	0	0
$479$	4.90981	0.224335	0.112167	−	0.993689i	$-0.464221\pi$
$479$	4.90981	0.224335	0.112167	+	0.993689i	$0.464221\pi$
$480$	0	0
$481$	8.85670	0.403831
$482$	0	0
$483$	9.79306	0.445600
$484$	0	0
$485$	−1.69563	−0.0769947
$486$	0	0
$487$	16.7786	0.760310	0.380155	−	0.924923i	$-0.375871\pi$
$487$	16.7786	0.760310	0.380155	+	0.924923i	$0.375871\pi$
$488$	0	0
$489$	−51.5635	−2.33178
$490$	0	0
$491$	41.4839	1.87214	0.936070	−	0.351814i	$-0.114435\pi$
$491$	41.4839	1.87214	0.936070	+	0.351814i	$0.114435\pi$
$492$	0	0
$493$	1.76651	0.0795595
$494$	0	0
$495$	5.00724	0.225059
$496$	0	0
$497$	−25.3792	−1.13841
$498$	0	0
$499$	−6.47751	−0.289973	−0.144987	−	0.989434i	$-0.546314\pi$
$499$	−6.47751	−0.289973	−0.144987	+	0.989434i	$0.546314\pi$
$500$	0	0
$501$	−4.85670	−0.216981
$502$	0	0
$503$	37.3825	1.66680	0.833401	−	0.552669i	$-0.186390\pi$
$503$	37.3825	1.66680	0.833401	+	0.552669i	$0.186390\pi$
$504$	0	0
$505$	2.24797	0.100033
$506$	0	0
$507$	−21.6127	−0.959853
$508$	0	0
$509$	−15.0410	−0.666682	−0.333341	−	0.942806i	$-0.608176\pi$
$509$	−15.0410	−0.666682	−0.333341	+	0.942806i	$0.608176\pi$
$510$	0	0
$511$	−55.8486	−2.47060
$512$	0	0
$513$	−7.73666	−0.341582
$514$	0	0
$515$	2.78583	0.122758
$516$	0	0
$517$	−6.62321	−0.291288
$518$	0	0
$519$	42.5934	1.86964
$520$	0	0
$521$	15.9469	0.698646	0.349323	−	0.937002i	$-0.386412\pi$
$521$	15.9469	0.698646	0.349323	+	0.937002i	$0.386412\pi$
$522$	0	0
$523$	10.8567	0.474730	0.237365	−	0.971420i	$-0.423716\pi$
$523$	10.8567	0.474730	0.237365	+	0.971420i	$0.423716\pi$
$524$	0	0
$525$	9.79306	0.427404
$526$	0	0
$527$	−7.84617	−0.341785
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−17.5185	−0.760240
$532$	0	0
$533$	−24.1722	−1.04702
$534$	0	0
$535$	−13.1578	−0.568860
$536$	0	0
$537$	47.9122	2.06756
$538$	0	0
$539$	−7.84617	−0.337958
$540$	0	0
$541$	−30.3792	−1.30610	−0.653052	−	0.757313i	$-0.726512\pi$
$541$	−30.3792	−1.30610	−0.653052	+	0.757313i	$0.726512\pi$
$542$	0	0
$543$	16.8977	0.725151
$544$	0	0
$545$	−3.42111	−0.146544
$546$	0	0
$547$	28.4090	1.21468	0.607341	−	0.794441i	$-0.292236\pi$
$547$	28.4090	1.21468	0.607341	+	0.794441i	$0.292236\pi$
$548$	0	0
$549$	27.9846	1.19435
$550$	0	0
$551$	−3.75203	−0.159842
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−10.6763	−0.453184
$556$	0	0
$557$	36.7584	1.55750	0.778751	−	0.627333i	$-0.215853\pi$
$557$	36.7584	1.55750	0.778751	+	0.627333i	$0.215853\pi$
$558$	0	0
$559$	15.2995	0.647101
$560$	0	0
$561$	−3.93481	−0.166128
$562$	0	0
$563$	−3.03708	−0.127998	−0.0639989	−	0.997950i	$-0.520385\pi$
$563$	−3.03708	−0.127998	−0.0639989	+	0.997950i	$0.520385\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	−16.0145	−0.672545
$568$	0	0
$569$	−32.2093	−1.35029	−0.675143	−	0.737687i	$-0.735918\pi$
$569$	−32.2093	−1.35029	−0.675143	+	0.737687i	$0.735918\pi$
$570$	0	0
$571$	27.7777	1.16246	0.581230	−	0.813739i	$-0.302572\pi$
$571$	27.7777	1.16246	0.581230	+	0.813739i	$0.302572\pi$
$572$	0	0
$573$	−40.4573	−1.69013
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	11.2890	0.469967	0.234984	−	0.971999i	$-0.424496\pi$
$577$	11.2890	0.469967	0.234984	+	0.971999i	$0.424496\pi$
$578$	0	0
$579$	50.6803	2.10620
$580$	0	0
$581$	−57.1867	−2.37251
$582$	0	0
$583$	−4.63768	−0.192073
$584$	0	0
$585$	−9.13122	−0.377530
$586$	0	0
$587$	23.6609	0.976592	0.488296	−	0.872678i	$-0.337619\pi$
$587$	23.6609	0.976592	0.488296	+	0.872678i	$0.337619\pi$
$588$	0	0
$589$	16.6651	0.686675
$590$	0	0
$591$	−51.1529	−2.10415
$592$	0	0
$593$	−17.8341	−0.732358	−0.366179	−	0.930544i	$-0.619334\pi$
$593$	−17.8341	−0.732358	−0.366179	+	0.930544i	$0.619334\pi$
$594$	0	0
$595$	−4.45490	−0.182633
$596$	0	0
$597$	−15.3913	−0.629923
$598$	0	0
$599$	−23.8308	−0.973700	−0.486850	−	0.873486i	$-0.661854\pi$
$599$	−23.8308	−0.973700	−0.486850	+	0.873486i	$0.661854\pi$
$600$	0	0
$601$	12.5297	0.511098	0.255549	−	0.966796i	$-0.417744\pi$
$601$	12.5297	0.511098	0.255549	+	0.966796i	$0.417744\pi$
$602$	0	0
$603$	53.0208	2.15917
$604$	0	0
$605$	9.52578	0.387278
$606$	0	0
$607$	−23.1722	−0.940533	−0.470266	−	0.882525i	$-0.655842\pi$
$607$	−23.1722	−0.940533	−0.470266	+	0.882525i	$0.655842\pi$
$608$	0	0
$609$	14.2480	0.577357
$610$	0	0
$611$	12.0781	0.488628
$612$	0	0
$613$	−44.5780	−1.80049	−0.900244	−	0.435386i	$-0.856612\pi$
$613$	−44.5780	−1.80049	−0.900244	+	0.435386i	$0.856612\pi$
$614$	0	0
$615$	29.1385	1.17498
$616$	0	0
$617$	−32.1385	−1.29385	−0.646923	−	0.762556i	$-0.723944\pi$
$617$	−32.1385	−1.29385	−0.646923	+	0.762556i	$0.723944\pi$
$618$	0	0
$619$	31.5708	1.26894	0.634468	−	0.772949i	$-0.281219\pi$
$619$	31.5708	1.26894	0.634468	+	0.772949i	$0.281219\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	−40.0289	−1.60373
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.35748	0.333765
$628$	0	0
$629$	4.85670	0.193649
$630$	0	0
$631$	24.0145	0.956001	0.478001	−	0.878360i	$-0.341362\pi$
$631$	24.0145	0.956001	0.478001	+	0.878360i	$0.341362\pi$
$632$	0	0
$633$	−46.4573	−1.84651
$634$	0	0
$635$	−19.9508	−0.791725
$636$	0	0
$637$	14.3083	0.566916
$638$	0	0
$639$	−28.5258	−1.12846
$640$	0	0
$641$	−10.4428	−0.412467	−0.206233	−	0.978503i	$-0.566121\pi$
$641$	−10.4428	−0.412467	−0.206233	+	0.978503i	$0.566121\pi$
$642$	0	0
$643$	4.39940	0.173495	0.0867477	−	0.996230i	$-0.472353\pi$
$643$	4.39940	0.173495	0.0867477	+	0.996230i	$0.472353\pi$
$644$	0	0
$645$	−18.4428	−0.726186
$646$	0	0
$647$	−23.6498	−0.929768	−0.464884	−	0.885372i	$-0.653904\pi$
$647$	−23.6498	−0.929768	−0.464884	+	0.885372i	$0.653904\pi$
$648$	0	0
$649$	5.15777	0.202460
$650$	0	0
$651$	−63.2842	−2.48030
$652$	0	0
$653$	40.9194	1.60130	0.800651	−	0.599131i	$-0.204487\pi$
$653$	40.9194	1.60130	0.800651	+	0.599131i	$0.204487\pi$
$654$	0	0
$655$	9.70287	0.379123
$656$	0	0
$657$	−62.7728	−2.44900
$658$	0	0
$659$	48.8567	1.90319	0.951593	−	0.307360i	$-0.0994455\pi$
$659$	48.8567	1.90319	0.951593	+	0.307360i	$0.0994455\pi$
$660$	0	0
$661$	−12.1529	−0.472694	−0.236347	−	0.971669i	$-0.575950\pi$
$661$	−12.1529	−0.472694	−0.236347	+	0.971669i	$0.575950\pi$
$662$	0	0
$663$	7.17554	0.278675
$664$	0	0
$665$	9.46214	0.366926
$666$	0	0
$667$	1.45490	0.0563341
$668$	0	0
$669$	−39.1722	−1.51449
$670$	0	0
$671$	−8.23918	−0.318070
$672$	0	0
$673$	9.20694	0.354901	0.177451	−	0.984130i	$-0.443215\pi$
$673$	9.20694	0.354901	0.177451	+	0.984130i	$0.443215\pi$
$674$	0	0
$675$	3.00000	0.115470
$676$	0	0
$677$	−39.8196	−1.53039	−0.765196	−	0.643797i	$-0.777358\pi$
$677$	−39.8196	−1.53039	−0.765196	+	0.643797i	$0.777358\pi$
$678$	0	0
$679$	6.22141	0.238756
$680$	0	0
$681$	43.8486	1.68028
$682$	0	0
$683$	−5.60544	−0.214486	−0.107243	−	0.994233i	$-0.534202\pi$
$683$	−5.60544	−0.214486	−0.107243	+	0.994233i	$0.534202\pi$
$684$	0	0
$685$	−2.82685	−0.108009
$686$	0	0
$687$	51.6537	1.97071
$688$	0	0
$689$	8.45730	0.322197
$690$	0	0
$691$	−2.84223	−0.108123	−0.0540617	−	0.998538i	$-0.517217\pi$
$691$	−2.84223	−0.108123	−0.0540617	+	0.998538i	$0.517217\pi$
$692$	0	0
$693$	−18.3719	−0.697893
$694$	0	0
$695$	−2.36471	−0.0896986
$696$	0	0
$697$	−13.2552	−0.502077
$698$	0	0
$699$	−51.9267	−1.96405
$700$	0	0
$701$	−24.4130	−0.922065	−0.461033	−	0.887383i	$-0.652521\pi$
$701$	−24.4130	−0.922065	−0.461033	+	0.887383i	$0.652521\pi$
$702$	0	0
$703$	−10.3155	−0.389058
$704$	0	0
$705$	−14.5596	−0.548345
$706$	0	0
$707$	−8.24797	−0.310197
$708$	0	0
$709$	0.123983	0.00465629	0.00232814	−	0.999997i	$-0.499259\pi$
$709$	0.123983	0.00465629	0.00232814	+	0.999997i	$0.499259\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.46214	−0.242009
$714$	0	0
$715$	2.68840	0.100540
$716$	0	0
$717$	63.2648	2.36267
$718$	0	0
$719$	47.1569	1.75865	0.879327	−	0.476218i	$-0.157993\pi$
$719$	47.1569	1.75865	0.879327	+	0.476218i	$0.157993\pi$
$720$	0	0
$721$	−10.2214	−0.380665
$722$	0	0
$723$	0.623208	0.0231774
$724$	0	0
$725$	1.45490	0.0540338
$726$	0	0
$727$	43.2962	1.60577	0.802884	−	0.596135i	$-0.203298\pi$
$727$	43.2962	1.60577	0.802884	+	0.596135i	$0.203298\pi$
$728$	0	0
$729$	−42.0217	−1.55636
$730$	0	0
$731$	8.38972	0.310305
$732$	0	0
$733$	11.5861	0.427943	0.213972	−	0.976840i	$-0.431360\pi$
$733$	11.5861	0.427943	0.213972	+	0.976840i	$0.431360\pi$
$734$	0	0
$735$	−17.2480	−0.636201
$736$	0	0
$737$	−15.6103	−0.575012
$738$	0	0
$739$	−30.3792	−1.11752	−0.558758	−	0.829331i	$-0.688722\pi$
$739$	−30.3792	−1.11752	−0.558758	+	0.829331i	$0.688722\pi$
$740$	0	0
$741$	−15.2407	−0.559882
$742$	0	0
$743$	35.5708	1.30496	0.652482	−	0.757804i	$-0.273728\pi$
$743$	35.5708	1.30496	0.652482	+	0.757804i	$0.273728\pi$
$744$	0	0
$745$	11.9170	0.436607
$746$	0	0
$747$	−64.2769	−2.35177
$748$	0	0
$749$	48.2769	1.76400
$750$	0	0
$751$	−24.9774	−0.911438	−0.455719	−	0.890124i	$-0.650618\pi$
$751$	−24.9774	−0.911438	−0.455719	+	0.890124i	$0.650618\pi$
$752$	0	0
$753$	58.0555	2.11566
$754$	0	0
$755$	−9.57889	−0.348611
$756$	0	0
$757$	9.35263	0.339927	0.169964	−	0.985450i	$-0.445635\pi$
$757$	9.35263	0.339927	0.169964	+	0.985450i	$0.445635\pi$
$758$	0	0
$759$	−3.24073	−0.117631
$760$	0	0
$761$	−18.1466	−0.657813	−0.328907	−	0.944362i	$-0.606680\pi$
$761$	−18.1466	−0.657813	−0.328907	+	0.944362i	$0.606680\pi$
$762$	0	0
$763$	12.5523	0.454425
$764$	0	0
$765$	−5.00724	−0.181037
$766$	0	0
$767$	−9.40574	−0.339622
$768$	0	0
$769$	44.3445	1.59910	0.799552	−	0.600597i	$-0.205070\pi$
$769$	44.3445	1.59910	0.799552	+	0.600597i	$0.205070\pi$
$770$	0	0
$771$	−43.5370	−1.56795
$772$	0	0
$773$	−10.8036	−0.388578	−0.194289	−	0.980944i	$-0.562240\pi$
$773$	−10.8036	−0.388578	−0.194289	+	0.980944i	$0.562240\pi$
$774$	0	0
$775$	−6.46214	−0.232127
$776$	0	0
$777$	39.1722	1.40530
$778$	0	0
$779$	28.1538	1.00872
$780$	0	0
$781$	8.39850	0.300522
$782$	0	0
$783$	4.36471	0.155982
$784$	0	0
$785$	18.7439	0.668998
$786$	0	0
$787$	−31.7810	−1.13287	−0.566435	−	0.824107i	$-0.691678\pi$
$787$	−31.7810	−1.13287	−0.566435	+	0.824107i	$0.691678\pi$
$788$	0	0
$789$	−6.77375	−0.241152
$790$	0	0
$791$	−36.6908	−1.30457
$792$	0	0
$793$	15.0250	0.533554
$794$	0	0
$795$	−10.1949	−0.361574
$796$	0	0
$797$	35.0594	1.24187	0.620935	−	0.783862i	$-0.286753\pi$
$797$	35.0594	1.24187	0.620935	+	0.783862i	$0.286753\pi$
$798$	0	0
$799$	6.62321	0.234312
$800$	0	0
$801$	−44.9919	−1.58971
$802$	0	0
$803$	18.4815	0.652196
$804$	0	0
$805$	−3.66908	−0.129318
$806$	0	0
$807$	−6.79306	−0.239127
$808$	0	0
$809$	−51.4911	−1.81033	−0.905165	−	0.425060i	$-0.860253\pi$
$809$	−51.4911	−1.81033	−0.905165	+	0.425060i	$0.860253\pi$
$810$	0	0
$811$	−19.7705	−0.694235	−0.347117	−	0.937822i	$-0.612839\pi$
$811$	−19.7705	−0.694235	−0.347117	+	0.937822i	$0.612839\pi$
$812$	0	0
$813$	64.5370	2.26341
$814$	0	0
$815$	19.3188	0.676709
$816$	0	0
$817$	−17.8196	−0.623429
$818$	0	0
$819$	33.5032	1.17070
$820$	0	0
$821$	31.8486	1.11152	0.555761	−	0.831342i	$-0.312427\pi$
$821$	31.8486	1.11152	0.555761	+	0.831342i	$0.312427\pi$
$822$	0	0
$823$	9.34210	0.325645	0.162823	−	0.986655i	$-0.447940\pi$
$823$	9.34210	0.325645	0.162823	+	0.986655i	$0.447940\pi$
$824$	0	0
$825$	−3.24073	−0.112828
$826$	0	0
$827$	−30.9243	−1.07534	−0.537671	−	0.843155i	$-0.680696\pi$
$827$	−30.9243	−1.07534	−0.537671	+	0.843155i	$0.680696\pi$
$828$	0	0
$829$	−32.5701	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$829$	−32.5701	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$830$	0	0
$831$	−56.6803	−1.96622
$832$	0	0
$833$	7.84617	0.271854
$834$	0	0
$835$	1.81962	0.0629704
$836$	0	0
$837$	−19.3864	−0.670093
$838$	0	0
$839$	48.5635	1.67660	0.838299	−	0.545210i	$-0.183550\pi$
$839$	48.5635	1.67660	0.838299	+	0.545210i	$0.183550\pi$
$840$	0	0
$841$	−26.8833	−0.927009
$842$	0	0
$843$	−37.0660	−1.27662
$844$	0	0
$845$	8.09743	0.278560
$846$	0	0
$847$	−34.9508	−1.20092
$848$	0	0
$849$	−78.8260	−2.70530
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	−33.6546	−1.15231	−0.576156	−	0.817340i	$-0.695448\pi$
$853$	−33.6546	−1.15231	−0.576156	+	0.817340i	$0.695448\pi$
$854$	0	0
$855$	10.6353	0.363719
$856$	0	0
$857$	24.4081	0.833766	0.416883	−	0.908960i	$-0.363122\pi$
$857$	24.4081	0.833766	0.416883	+	0.908960i	$0.363122\pi$
$858$	0	0
$859$	−11.1394	−0.380070	−0.190035	−	0.981777i	$-0.560860\pi$
$859$	−11.1394	−0.380070	−0.190035	+	0.981777i	$0.560860\pi$
$860$	0	0
$861$	−106.911	−3.64353
$862$	0	0
$863$	−15.7173	−0.535025	−0.267512	−	0.963554i	$-0.586202\pi$
$863$	−15.7173	−0.535025	−0.267512	+	0.963554i	$0.586202\pi$
$864$	0	0
$865$	−15.9581	−0.542591
$866$	0	0
$867$	−41.4395	−1.40736
$868$	0	0
$869$	0	0
$870$	0	0
$871$	28.4670	0.964567
$872$	0	0
$873$	6.99276	0.236669
$874$	0	0
$875$	−3.66908	−0.124038
$876$	0	0
$877$	−33.2818	−1.12385	−0.561923	−	0.827190i	$-0.689938\pi$
$877$	−33.2818	−1.12385	−0.561923	+	0.827190i	$0.689938\pi$
$878$	0	0
$879$	−64.2769	−2.16801
$880$	0	0
$881$	8.99187	0.302944	0.151472	−	0.988462i	$-0.451599\pi$
$881$	8.99187	0.302944	0.151472	+	0.988462i	$0.451599\pi$
$882$	0	0
$883$	58.5258	1.96955	0.984775	−	0.173836i	$-0.0556162\pi$
$883$	58.5258	1.96955	0.984775	+	0.173836i	$0.0556162\pi$
$884$	0	0
$885$	11.3382	0.381128
$886$	0	0
$887$	32.8220	1.10206	0.551028	−	0.834487i	$-0.314236\pi$
$887$	32.8220	1.10206	0.551028	+	0.834487i	$0.314236\pi$
$888$	0	0
$889$	73.2012	2.45509
$890$	0	0
$891$	5.29952	0.177541
$892$	0	0
$893$	−14.0676	−0.470754
$894$	0	0
$895$	−17.9508	−0.600030
$896$	0	0
$897$	5.90981	0.197323
$898$	0	0
$899$	−9.40180	−0.313567
$900$	0	0
$901$	4.63768	0.154504
$902$	0	0
$903$	67.6682	2.25186
$904$	0	0
$905$	−6.33092	−0.210447
$906$	0	0
$907$	28.3300	0.940683	0.470341	−	0.882484i	$-0.344131\pi$
$907$	28.3300	0.940683	0.470341	+	0.882484i	$0.344131\pi$
$908$	0	0
$909$	−9.27058	−0.307486
$910$	0	0
$911$	15.3382	0.508176	0.254088	−	0.967181i	$-0.418225\pi$
$911$	15.3382	0.508176	0.254088	+	0.967181i	$0.418225\pi$
$912$	0	0
$913$	18.9243	0.626302
$914$	0	0
$915$	−18.1119	−0.598761
$916$	0	0
$917$	−35.6006	−1.17564
$918$	0	0
$919$	−38.8036	−1.28001	−0.640006	−	0.768370i	$-0.721068\pi$
$919$	−38.8036	−1.28001	−0.640006	+	0.768370i	$0.721068\pi$
$920$	0	0
$921$	53.2310	1.75402
$922$	0	0
$923$	−15.3155	−0.504117
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	−11.4887	−0.377338
$928$	0	0
$929$	4.01053	0.131581	0.0657906	−	0.997833i	$-0.479043\pi$
$929$	4.01053	0.131581	0.0657906	+	0.997833i	$0.479043\pi$
$930$	0	0
$931$	−16.6651	−0.546178
$932$	0	0
$933$	−7.93636	−0.259825
$934$	0	0
$935$	1.47422	0.0482122
$936$	0	0
$937$	43.6836	1.42708	0.713540	−	0.700615i	$-0.247091\pi$
$937$	43.6836	1.42708	0.713540	+	0.700615i	$0.247091\pi$
$938$	0	0
$939$	41.3406	1.34910
$940$	0	0
$941$	−24.8534	−0.810198	−0.405099	−	0.914273i	$-0.632763\pi$
$941$	−24.8534	−0.810198	−0.405099	+	0.914273i	$0.632763\pi$
$942$	0	0
$943$	−10.9170	−0.355508
$944$	0	0
$945$	−11.0072	−0.358065
$946$	0	0
$947$	−22.1690	−0.720394	−0.360197	−	0.932876i	$-0.617291\pi$
$947$	−22.1690	−0.720394	−0.360197	+	0.932876i	$0.617291\pi$
$948$	0	0
$949$	−33.7029	−1.09404
$950$	0	0
$951$	23.5596	0.763971
$952$	0	0
$953$	15.9050	0.515212	0.257606	−	0.966250i	$-0.417066\pi$
$953$	15.9050	0.515212	0.257606	+	0.966250i	$0.417066\pi$
$954$	0	0
$955$	15.1578	0.490494
$956$	0	0
$957$	−4.71495	−0.152413
$958$	0	0
$959$	10.3719	0.334928
$960$	0	0
$961$	10.7593	0.347073
$962$	0	0
$963$	54.2624	1.74858
$964$	0	0
$965$	−18.9879	−0.611243
$966$	0	0
$967$	50.6272	1.62806	0.814030	−	0.580823i	$-0.197269\pi$
$967$	50.6272	1.62806	0.814030	+	0.580823i	$0.197269\pi$
$968$	0	0
$969$	−8.35748	−0.268481
$970$	0	0
$971$	−52.4009	−1.68162	−0.840812	−	0.541327i	$-0.817922\pi$
$971$	−52.4009	−1.68162	−0.840812	+	0.541327i	$0.817922\pi$
$972$	0	0
$973$	8.67632	0.278150
$974$	0	0
$975$	5.90981	0.189265
$976$	0	0
$977$	10.5258	0.336750	0.168375	−	0.985723i	$-0.446148\pi$
$977$	10.5258	0.336750	0.168375	+	0.985723i	$0.446148\pi$
$978$	0	0
$979$	13.2464	0.423357
$980$	0	0
$981$	14.1086	0.450453
$982$	0	0
$983$	9.44767	0.301334	0.150667	−	0.988585i	$-0.451858\pi$
$983$	9.44767	0.301334	0.150667	+	0.988585i	$0.451858\pi$
$984$	0	0
$985$	19.1650	0.610648
$986$	0	0
$987$	53.4202	1.70038
$988$	0	0
$989$	6.90981	0.219719
$990$	0	0
$991$	33.3252	1.05861	0.529305	−	0.848432i	$-0.322453\pi$
$991$	33.3252	1.05861	0.529305	+	0.848432i	$0.322453\pi$
$992$	0	0
$993$	46.4468	1.47394
$994$	0	0
$995$	5.76651	0.182811
$996$	0	0
$997$	54.0289	1.71111	0.855557	−	0.517709i	$-0.173215\pi$
$997$	54.0289	1.71111	0.855557	+	0.517709i	$0.173215\pi$
$998$	0	0
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.g.1.3	✓	3
3.2	odd	2		8280.2.a.bo.1.3		3
4.3	odd	2		1840.2.a.t.1.1		3
5.2	odd	4		4600.2.e.r.4049.1		6
5.3	odd	4		4600.2.e.r.4049.6		6
5.4	even	2		4600.2.a.y.1.1		3
8.3	odd	2		7360.2.a.ca.1.3		3
8.5	even	2		7360.2.a.cb.1.1		3
20.19	odd	2		9200.2.a.cd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.g.1.3	✓	3	1.1	even	1	trivial
1840.2.a.t.1.1		3	4.3	odd	2
4600.2.a.y.1.1		3	5.4	even	2
4600.2.e.r.4049.1		6	5.2	odd	4
4600.2.e.r.4049.6		6	5.3	odd	4
7360.2.a.ca.1.3		3	8.3	odd	2
7360.2.a.cb.1.1		3	8.5	even	2
8280.2.a.bo.1.3		3	3.2	odd	2
9200.2.a.cd.1.3		3	20.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q+2.66908 q^{3} -1.00000 q^{5} +3.66908 q^{7} +4.12398 q^{9} +O(q^{10})\)	\(q+2.66908 q^{3} -1.00000 q^{5} +3.66908 q^{7} +4.12398 q^{9} -1.21417 q^{11} +2.21417 q^{13} -2.66908 q^{15} +1.21417 q^{17} -2.57889 q^{19} +9.79306 q^{21} +1.00000 q^{23} +1.00000 q^{25} +3.00000 q^{27} +1.45490 q^{29} -6.46214 q^{31} -3.24073 q^{33} -3.66908 q^{35} +4.00000 q^{37} +5.90981 q^{39} -10.9170 q^{41} +6.90981 q^{43} -4.12398 q^{45} +5.45490 q^{47} +6.46214 q^{49} +3.24073 q^{51} +3.81962 q^{53} +1.21417 q^{55} -6.88325 q^{57} -4.24797 q^{59} +6.78583 q^{61} +15.1312 q^{63} -2.21417 q^{65} +12.8567 q^{67} +2.66908 q^{69} -6.91705 q^{71} -15.2214 q^{73} +2.66908 q^{75} -4.45490 q^{77} -4.36471 q^{81} -15.5861 q^{83} -1.21417 q^{85} +3.88325 q^{87} -10.9098 q^{89} +8.12398 q^{91} -17.2480 q^{93} +2.57889 q^{95} +1.69563 q^{97} -5.00724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{17} + 3 q^{19} + 12 q^{21} + 3 q^{23} + 3 q^{25} + 9 q^{27} + 3 q^{29} + 6 q^{31} - 15 q^{33} - 3 q^{35} + 12 q^{37} + 15 q^{39} - 6 q^{41} + 18 q^{43} - 3 q^{45} + 15 q^{47} - 6 q^{49} + 15 q^{51} + 6 q^{53} - 3 q^{55} - 6 q^{57} + 6 q^{59} + 27 q^{61} + 12 q^{63} + 12 q^{67} + 6 q^{71} - 15 q^{73} - 12 q^{77} - 9 q^{81} - 12 q^{83} + 3 q^{85} - 3 q^{87} - 30 q^{89} + 15 q^{91} - 33 q^{93} - 3 q^{95} + 9 q^{97} + 9 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^17 + 3 * q^19 + 12 * q^21 + 3 * q^23 + 3 * q^25 + 9 * q^27 + 3 * q^29 + 6 * q^31 - 15 * q^33 - 3 * q^35 + 12 * q^37 + 15 * q^39 - 6 * q^41 + 18 * q^43 - 3 * q^45 + 15 * q^47 - 6 * q^49 + 15 * q^51 + 6 * q^53 - 3 * q^55 - 6 * q^57 + 6 * q^59 + 27 * q^61 + 12 * q^63 + 12 * q^67 + 6 * q^71 - 15 * q^73 - 12 * q^77 - 9 * q^81 - 12 * q^83 + 3 * q^85 - 3 * q^87 - 30 * q^89 + 15 * q^91 - 33 * q^93 - 3 * q^95 + 9 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more