LMFDB - Embedded newform 9200.2.a.cd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
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:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66908 q^{3} +3.66908 q^{7} +4.12398 q^{9} +O(q^{10})$	$q+2.66908 q^{3} +3.66908 q^{7} +4.12398 q^{9} +1.21417 q^{11} -2.21417 q^{13} -1.21417 q^{17} +2.57889 q^{19} +9.79306 q^{21} +1.00000 q^{23} +3.00000 q^{27} +1.45490 q^{29} +6.46214 q^{31} +3.24073 q^{33} -4.00000 q^{37} -5.90981 q^{39} -10.9170 q^{41} +6.90981 q^{43} +5.45490 q^{47} +6.46214 q^{49} -3.24073 q^{51} -3.81962 q^{53} +6.88325 q^{57} +4.24797 q^{59} +6.78583 q^{61} +15.1312 q^{63} +12.8567 q^{67} +2.66908 q^{69} +6.91705 q^{71} +15.2214 q^{73} +4.45490 q^{77} -4.36471 q^{81} -15.5861 q^{83} +3.88325 q^{87} -10.9098 q^{89} -8.12398 q^{91} +17.2480 q^{93} -1.69563 q^{97} +5.00724 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{17} - 3 q^{19} + 12 q^{21} + 3 q^{23} + 9 q^{27} + 3 q^{29} - 6 q^{31} + 15 q^{33} - 12 q^{37} - 15 q^{39} - 6 q^{41} + 18 q^{43} + 15 q^{47} - 6 q^{49} - 15 q^{51} - 6 q^{53} + 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^{87} - 30 q^{89} - 15 q^{91} + 33 q^{93} - 9 q^{97} - 9 q^{99}+O(q^{100}) $ 3 * q + 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^17 - 3 * q^19 + 12 * q^21 + 3 * q^23 + 9 * q^27 + 3 * q^29 - 6 * q^31 + 15 * q^33 - 12 * q^37 - 15 * q^39 - 6 * q^41 + 18 * q^43 + 15 * q^47 - 6 * q^49 - 15 * q^51 - 6 * q^53 + 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^87 - 30 * q^89 - 15 * q^91 + 33 * q^93 - 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
$5$	0	0
$5$	0	0
$6$	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	$-0.441405\pi$
$11$	1.21417	0.366088	0.183044	+	0.983105i	$0.441405\pi$
$12$	0	0
$13$	−2.21417	−0.614102	−0.307051	−	0.951693i	$-0.599342\pi$
$13$	−2.21417	−0.614102	−0.307051	+	0.951693i	$0.599342\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.21417	−0.294481	−0.147240	−	0.989101i	$-0.547039\pi$
$17$	−1.21417	−0.294481	−0.147240	+	0.989101i	$0.547039\pi$
$18$	0	0
$19$	2.57889	0.591637	0.295819	−	0.955244i	$-0.404408\pi$
$19$	2.57889	0.591637	0.295819	+	0.955244i	$0.404408\pi$
$20$	0	0
$21$	9.79306	2.13702
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	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	$-0.302928\pi$
$31$	6.46214	1.16063	0.580317	+	0.814390i	$0.302928\pi$
$32$	0	0
$33$	3.24073	0.564139
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	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
$45$	0	0
$46$	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	$-0.584492\pi$
$53$	−3.81962	−0.524665	−0.262332	+	0.964978i	$0.584492\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.88325	0.911709
$58$	0	0
$59$	4.24797	0.553038	0.276519	−	0.961008i	$-0.410819\pi$
$59$	4.24797	0.553038	0.276519	+	0.961008i	$0.410819\pi$
$60$	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
$65$	0	0
$66$	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	$-0.365371\pi$
$71$	6.91705	0.820902	0.410451	+	0.911883i	$0.365371\pi$
$72$	0	0
$73$	15.2214	1.78153	0.890766	−	0.454463i	$-0.150169\pi$
$73$	15.2214	1.78153	0.890766	+	0.454463i	$0.150169\pi$
$74$	0	0
$75$	0	0
$76$	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
$85$	0	0
$86$	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
$95$	0	0
$96$	0	0
$97$	−1.69563	−0.172165	−0.0860827	−	0.996288i	$-0.527435\pi$
$97$	−1.69563	−0.172165	−0.0860827	+	0.996288i	$0.527435\pi$
$98$	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
$105$	0	0
$106$	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	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	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	$-0.360671\pi$
$131$	9.70287	0.847744	0.423872	+	0.905722i	$0.360671\pi$
$132$	0	0
$133$	9.46214	0.820472
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.82685	−0.241514	−0.120757	−	0.992682i	$-0.538532\pi$
$137$	−2.82685	−0.241514	−0.120757	+	0.992682i	$0.538532\pi$
$138$	0	0
$139$	−2.36471	−0.200572	−0.100286	−	0.994959i	$-0.531976\pi$
$139$	−2.36471	−0.200572	−0.100286	+	0.994959i	$0.531976\pi$
$140$	0	0
$141$	14.5596	1.22614
$142$	0	0
$143$	−2.68840	−0.224815
$144$	0	0
$145$	0	0
$146$	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	$-0.627442\pi$
$151$	−9.57889	−0.779519	−0.389759	+	0.920917i	$0.627442\pi$
$152$	0	0
$153$	−5.00724	−0.404811
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.7439	1.49593	0.747963	−	0.663740i	$-0.231032\pi$
$157$	18.7439	1.49593	0.747963	+	0.663740i	$0.231032\pi$
$158$	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
$165$	0	0
$166$	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	$-0.707481\pi$
$173$	−15.9581	−1.21327	−0.606635	+	0.794981i	$0.707481\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.3382	0.852228
$178$	0	0
$179$	−17.9508	−1.34171	−0.670854	−	0.741589i	$-0.734072\pi$
$179$	−17.9508	−1.34171	−0.670854	+	0.741589i	$0.734072\pi$
$180$	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
$185$	0	0
$186$	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	$-0.315241\pi$
$191$	15.1578	1.09678	0.548389	+	0.836223i	$0.315241\pi$
$192$	0	0
$193$	−18.9879	−1.36678	−0.683390	−	0.730053i	$-0.739495\pi$
$193$	−18.9879	−1.36678	−0.683390	+	0.730053i	$0.739495\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.1650	1.36545	0.682725	−	0.730675i	$-0.260795\pi$
$197$	19.1650	1.36545	0.682725	+	0.730675i	$0.260795\pi$
$198$	0	0
$199$	5.76651	0.408777	0.204388	−	0.978890i	$-0.434479\pi$
$199$	5.76651	0.408777	0.204388	+	0.978890i	$0.434479\pi$
$200$	0	0
$201$	34.3155	2.42043
$202$	0	0
$203$	5.33816	0.374665
$204$	0	0
$205$	0	0
$206$	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	$-0.295513\pi$
$211$	17.4057	1.19826	0.599130	+	0.800652i	$0.295513\pi$
$212$	0	0
$213$	18.4621	1.26501
$214$	0	0
$215$	0	0
$216$	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
$225$	0	0
$226$	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	$-0.280065\pi$
$233$	19.4549	1.27453	0.637267	+	0.770643i	$0.280065\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.7029	−1.53321	−0.766606	−	0.642118i	$-0.778056\pi$
$239$	−23.7029	−1.53321	−0.766606	+	0.642118i	$0.778056\pi$
$240$	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
$245$	0	0
$246$	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	$-0.740836\pi$
$251$	−21.7511	−1.37292	−0.686460	+	0.727168i	$0.740836\pi$
$252$	0	0
$253$	1.21417	0.0763345
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.3116	1.01749	0.508745	−	0.860917i	$-0.330110\pi$
$257$	16.3116	1.01749	0.508745	+	0.860917i	$0.330110\pi$
$258$	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
$265$	0	0
$266$	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	$-0.762536\pi$
$271$	−24.1795	−1.46880	−0.734400	+	0.678717i	$0.762536\pi$
$272$	0	0
$273$	−21.6836	−1.31235
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.2359	1.27594	0.637970	−	0.770061i	$-0.279774\pi$
$277$	21.2359	1.27594	0.637970	+	0.770061i	$0.279774\pi$
$278$	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
$285$	0	0
$286$	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	$-0.251644\pi$
$293$	24.0821	1.40689	0.703444	+	0.710750i	$0.251644\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	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	$-0.473133\pi$
$311$	2.97345	0.168609	0.0843043	+	0.996440i	$0.473133\pi$
$312$	0	0
$313$	−15.4887	−0.875473	−0.437736	−	0.899103i	$-0.644220\pi$
$313$	−15.4887	−0.875473	−0.437736	+	0.899103i	$0.644220\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.82685	−0.495765	−0.247883	−	0.968790i	$-0.579735\pi$
$317$	−8.82685	−0.495765	−0.247883	+	0.968790i	$0.579735\pi$
$318$	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
$325$	0	0
$326$	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	$-0.658727\pi$
$331$	−17.4018	−0.956489	−0.478245	+	0.878227i	$0.658727\pi$
$332$	0	0
$333$	−16.4959	−0.903972
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.9541	−1.46828	−0.734142	−	0.678995i	$-0.762416\pi$
$337$	−26.9541	−1.46828	−0.734142	+	0.678995i	$0.762416\pi$
$338$	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
$345$	0	0
$346$	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	$-0.620459\pi$
$353$	−13.8833	−0.738931	−0.369466	+	0.929244i	$0.620459\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−11.8905	−0.629312
$358$	0	0
$359$	−15.0902	−0.796430	−0.398215	−	0.917292i	$-0.630370\pi$
$359$	−15.0902	−0.796430	−0.398215	+	0.917292i	$0.630370\pi$
$360$	0	0
$361$	−12.3493	−0.649965
$362$	0	0
$363$	−25.4251	−1.33447
$364$	0	0
$365$	0	0
$366$	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	$-0.792296\pi$
$373$	−30.6908	−1.58911	−0.794554	+	0.607193i	$0.792296\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.22141	−0.165911
$378$	0	0
$379$	−2.07087	−0.106374	−0.0531869	−	0.998585i	$-0.516938\pi$
$379$	−2.07087	−0.106374	−0.0531869	+	0.998585i	$0.516938\pi$
$380$	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
$385$	0	0
$386$	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	$-0.484500\pi$
$397$	1.93965	0.0973485	0.0486742	+	0.998815i	$0.484500\pi$
$398$	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
$405$	0	0
$406$	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
$415$	0	0
$416$	0	0
$417$	−6.31160	−0.309081
$418$	0	0
$419$	−28.2624	−1.38071	−0.690355	−	0.723471i	$-0.742546\pi$
$419$	−28.2624	−1.38071	−0.690355	+	0.723471i	$0.742546\pi$
$420$	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
$425$	0	0
$426$	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	$-0.396331\pi$
$431$	13.2850	0.639918	0.319959	+	0.947431i	$0.396331\pi$
$432$	0	0
$433$	−11.0603	−0.531526	−0.265763	−	0.964038i	$-0.585624\pi$
$433$	−11.0603	−0.531526	−0.265763	+	0.964038i	$0.585624\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.57889	0.123365
$438$	0	0
$439$	19.1916	0.915963	0.457982	−	0.888962i	$-0.348573\pi$
$439$	19.1916	0.915963	0.457982	+	0.888962i	$0.348573\pi$
$440$	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
$445$	0	0
$446$	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
$455$	0	0
$456$	0	0
$457$	−15.5861	−0.729088	−0.364544	−	0.931186i	$-0.618775\pi$
$457$	−15.5861	−0.729088	−0.364544	+	0.931186i	$0.618775\pi$
$458$	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
$465$	0	0
$466$	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
$475$	0	0
$476$	0	0
$477$	−15.7520	−0.721236
$478$	0	0
$479$	−4.90981	−0.224335	−0.112167	−	0.993689i	$-0.535779\pi$
$479$	−4.90981	−0.224335	−0.112167	+	0.993689i	$0.535779\pi$
$480$	0	0
$481$	8.85670	0.403831
$482$	0	0
$483$	9.79306	0.445600
$484$	0	0
$485$	0	0
$486$	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	$-0.885565\pi$
$491$	−41.4839	−1.87214	−0.936070	+	0.351814i	$0.885565\pi$
$492$	0	0
$493$	−1.76651	−0.0795595
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.3792	1.13841
$498$	0	0
$499$	6.47751	0.289973	0.144987	−	0.989434i	$-0.453686\pi$
$499$	6.47751	0.289973	0.144987	+	0.989434i	$0.453686\pi$
$500$	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
$505$	0	0
$506$	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
$515$	0	0
$516$	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
$525$	0	0
$526$	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
$535$	0	0
$536$	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
$545$	0	0
$546$	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
$555$	0	0
$556$	0	0
$557$	−36.7584	−1.55750	−0.778751	−	0.627333i	$-0.784147\pi$
$557$	−36.7584	−1.55750	−0.778751	+	0.627333i	$0.784147\pi$
$558$	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
$565$	0	0
$566$	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	$-0.697428\pi$
$571$	−27.7777	−1.16246	−0.581230	+	0.813739i	$0.697428\pi$
$572$	0	0
$573$	40.4573	1.69013
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.2890	−0.469967	−0.234984	−	0.971999i	$-0.575504\pi$
$577$	−11.2890	−0.469967	−0.234984	+	0.971999i	$0.575504\pi$
$578$	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
$585$	0	0
$586$	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	$-0.380666\pi$
$593$	17.8341	0.732358	0.366179	+	0.930544i	$0.380666\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.3913	0.629923
$598$	0	0
$599$	23.8308	0.973700	0.486850	−	0.873486i	$-0.338146\pi$
$599$	23.8308	0.973700	0.486850	+	0.873486i	$0.338146\pi$
$600$	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
$605$	0	0
$606$	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	$-0.143388\pi$
$613$	44.5780	1.80049	0.900244	+	0.435386i	$0.143388\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.1385	1.29385	0.646923	−	0.762556i	$-0.276056\pi$
$617$	32.1385	1.29385	0.646923	+	0.762556i	$0.276056\pi$
$618$	0	0
$619$	−31.5708	−1.26894	−0.634468	−	0.772949i	$-0.718781\pi$
$619$	−31.5708	−1.26894	−0.634468	+	0.772949i	$0.718781\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	−40.0289	−1.60373
$624$	0	0
$625$	0	0
$626$	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	$-0.658638\pi$
$631$	−24.0145	−0.956001	−0.478001	+	0.878360i	$0.658638\pi$
$632$	0	0
$633$	46.4573	1.84651
$634$	0	0
$635$	0	0
$636$	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
$645$	0	0
$646$	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	$-0.795513\pi$
$653$	−40.9194	−1.60130	−0.800651	+	0.599131i	$0.795513\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	62.7728	2.44900
$658$	0	0
$659$	−48.8567	−1.90319	−0.951593	−	0.307360i	$-0.900555\pi$
$659$	−48.8567	−1.90319	−0.951593	+	0.307360i	$0.900555\pi$
$660$	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
$665$	0	0
$666$	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	$-0.556785\pi$
$673$	−9.20694	−0.354901	−0.177451	+	0.984130i	$0.556785\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.8196	1.53039	0.765196	−	0.643797i	$-0.222642\pi$
$677$	39.8196	1.53039	0.765196	+	0.643797i	$0.222642\pi$
$678$	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
$685$	0	0
$686$	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	$-0.482783\pi$
$691$	2.84223	0.108123	0.0540617	+	0.998538i	$0.482783\pi$
$692$	0	0
$693$	18.3719	0.697893
$694$	0	0
$695$	0	0
$696$	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
$705$	0	0
$706$	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
$715$	0	0
$716$	0	0
$717$	−63.2648	−2.36267
$718$	0	0
$719$	−47.1569	−1.75865	−0.879327	−	0.476218i	$-0.842007\pi$
$719$	−47.1569	−1.75865	−0.879327	+	0.476218i	$0.842007\pi$
$720$	0	0
$721$	−10.2214	−0.380665
$722$	0	0
$723$	0.623208	0.0231774
$724$	0	0
$725$	0	0
$726$	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	$-0.568640\pi$
$733$	−11.5861	−0.427943	−0.213972	+	0.976840i	$0.568640\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.6103	0.575012
$738$	0	0
$739$	30.3792	1.11752	0.558758	−	0.829331i	$-0.311278\pi$
$739$	30.3792	1.11752	0.558758	+	0.829331i	$0.311278\pi$
$740$	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
$745$	0	0
$746$	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	$-0.349382\pi$
$751$	24.9774	0.911438	0.455719	+	0.890124i	$0.349382\pi$
$752$	0	0
$753$	−58.0555	−2.11566
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.35263	−0.339927	−0.169964	−	0.985450i	$-0.554365\pi$
$757$	−9.35263	−0.339927	−0.169964	+	0.985450i	$0.554365\pi$
$758$	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
$765$	0	0
$766$	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	$-0.437760\pi$
$773$	10.8036	0.388578	0.194289	+	0.980944i	$0.437760\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	0	0
$786$	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
$795$	0	0
$796$	0	0
$797$	−35.0594	−1.24187	−0.620935	−	0.783862i	$-0.713247\pi$
$797$	−35.0594	−1.24187	−0.620935	+	0.783862i	$0.713247\pi$
$798$	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
$805$	0	0
$806$	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	$-0.387161\pi$
$811$	19.7705	0.694235	0.347117	+	0.937822i	$0.387161\pi$
$812$	0	0
$813$	−64.5370	−2.26341
$814$	0	0
$815$	0	0
$816$	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
$825$	0	0
$826$	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
$835$	0	0
$836$	0	0
$837$	19.3864	0.670093
$838$	0	0
$839$	−48.5635	−1.67660	−0.838299	−	0.545210i	$-0.816450\pi$
$839$	−48.5635	−1.67660	−0.838299	+	0.545210i	$0.816450\pi$
$840$	0	0
$841$	−26.8833	−0.927009
$842$	0	0
$843$	−37.0660	−1.27662
$844$	0	0
$845$	0	0
$846$	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	$-0.304552\pi$
$853$	33.6546	1.15231	0.576156	+	0.817340i	$0.304552\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.4081	−0.833766	−0.416883	−	0.908960i	$-0.636878\pi$
$857$	−24.4081	−0.833766	−0.416883	+	0.908960i	$0.636878\pi$
$858$	0	0
$859$	11.1394	0.380070	0.190035	−	0.981777i	$-0.439140\pi$
$859$	11.1394	0.380070	0.190035	+	0.981777i	$0.439140\pi$
$860$	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
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	33.2818	1.12385	0.561923	−	0.827190i	$-0.310062\pi$
$877$	33.2818	1.12385	0.561923	+	0.827190i	$0.310062\pi$
$878$	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
$885$	0	0
$886$	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
$895$	0	0
$896$	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
$905$	0	0
$906$	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	$-0.581775\pi$
$911$	−15.3382	−0.508176	−0.254088	+	0.967181i	$0.581775\pi$
$912$	0	0
$913$	−18.9243	−0.626302
$914$	0	0
$915$	0	0
$916$	0	0
$917$	35.6006	1.17564
$918$	0	0
$919$	38.8036	1.28001	0.640006	−	0.768370i	$-0.278932\pi$
$919$	38.8036	1.28001	0.640006	+	0.768370i	$0.278932\pi$
$920$	0	0
$921$	53.2310	1.75402
$922$	0	0
$923$	−15.3155	−0.504117
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	−43.6836	−1.42708	−0.713540	−	0.700615i	$-0.752909\pi$
$937$	−43.6836	−1.42708	−0.713540	+	0.700615i	$0.752909\pi$
$938$	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
$945$	0	0
$946$	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	$-0.582934\pi$
$953$	−15.9050	−0.515212	−0.257606	+	0.966250i	$0.582934\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	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	$-0.182078\pi$
$971$	52.4009	1.68162	0.840812	+	0.541327i	$0.182078\pi$
$972$	0	0
$973$	−8.67632	−0.278150
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.5258	−0.336750	−0.168375	−	0.985723i	$-0.553852\pi$
$977$	−10.5258	−0.336750	−0.168375	+	0.985723i	$0.553852\pi$
$978$	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
$985$	0	0
$986$	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	$-0.677547\pi$
$991$	−33.3252	−1.05861	−0.529305	+	0.848432i	$0.677547\pi$
$992$	0	0
$993$	−46.4468	−1.47394
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−54.0289	−1.71111	−0.855557	−	0.517709i	$-0.826785\pi$
$997$	−54.0289	−1.71111	−0.855557	+	0.517709i	$0.826785\pi$
$998$	0	0
$999$	−12.0000	−0.379663

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.g.1.3	✓	3	20.19	odd	2
1840.2.a.t.1.1		3	5.4	even	2
4600.2.a.y.1.1		3	4.3	odd	2
4600.2.e.r.4049.1		6	20.3	even	4
4600.2.e.r.4049.6		6	20.7	even	4
7360.2.a.ca.1.3		3	40.29	even	2
7360.2.a.cb.1.1		3	40.19	odd	2
8280.2.a.bo.1.3		3	60.59	even	2
9200.2.a.cd.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more