LMFDB - Embedded newform 3724.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3724,2,Mod(1,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, 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("3724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3724 = 2^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3724.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:	$29.7362897127$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 4x^{6} + 24x^{5} + x^{4} - 40x^{3} + 6x^{2} + 16x - 1 $ x^8 - 4x^7 - 4x^6 + 24x^5 + x^4 - 40x^3 + 6x^2 + 16x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.923026$ of defining polynomial
Character	$\chi$	$=$	3724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.923026 q^{3} +0.765692 q^{5} -2.14802 q^{9} +0.119073 q^{11} -5.50031 q^{13} -0.706754 q^{15} +5.35523 q^{17} +1.00000 q^{19} +2.47375 q^{23} -4.41372 q^{25} +4.75176 q^{27} +10.0997 q^{29} -0.967066 q^{31} -0.109907 q^{33} +5.66670 q^{37} +5.07693 q^{39} -0.467311 q^{41} +1.66934 q^{43} -1.64472 q^{45} -10.1965 q^{47} -4.94302 q^{51} -9.58561 q^{53} +0.0911731 q^{55} -0.923026 q^{57} -14.0439 q^{59} -15.1285 q^{61} -4.21154 q^{65} -10.0927 q^{67} -2.28334 q^{69} +14.6510 q^{71} -0.289408 q^{73} +4.07398 q^{75} -16.2973 q^{79} +2.05807 q^{81} +8.67026 q^{83} +4.10046 q^{85} -9.32230 q^{87} +1.64821 q^{89} +0.892628 q^{93} +0.765692 q^{95} -5.16198 q^{97} -0.255771 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 8 q^{11} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{25} - 16 q^{27} - 4 q^{29} - 24 q^{31} + 12 q^{33} + 12 q^{37} + 4 q^{39} - 20 q^{41} + 12 q^{43} - 24 q^{45} - 24 q^{47} + 12 q^{51} + 4 q^{53}+ \cdots - 20 q^{97}+O(q^{100}) $ 8 * q - 4 * q^3 - 8 * q^11 - 8 * q^17 + 8 * q^19 + 4 * q^23 + 4 * q^25 - 16 * q^27 - 4 * q^29 - 24 * q^31 + 12 * q^33 + 12 * q^37 + 4 * q^39 - 20 * q^41 + 12 * q^43 - 24 * q^45 - 24 * q^47 + 12 * q^51 + 4 * q^53 - 16 * q^55 - 4 * q^57 - 52 * q^59 + 16 * q^61 - 4 * q^65 + 8 * q^67 - 8 * q^69 + 16 * q^71 - 24 * q^73 - 8 * q^75 - 20 * q^79 + 12 * q^81 - 48 * q^83 - 12 * q^85 - 40 * q^87 - 12 * q^89 + 8 * q^93 - 20 * q^97

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
$3$	−0.923026	−0.532909	−0.266455	−	0.963847i	$-0.585852\pi$
$3$	−0.923026	−0.532909	−0.266455	+	0.963847i	$0.585852\pi$
$4$	0	0
$5$	0.765692	0.342428	0.171214	−	0.985234i	$-0.445231\pi$
$5$	0.765692	0.342428	0.171214	+	0.985234i	$0.445231\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.14802	−0.716007
$10$	0	0
$11$	0.119073	0.0359018	0.0179509	−	0.999839i	$-0.494286\pi$
$11$	0.119073	0.0359018	0.0179509	+	0.999839i	$0.494286\pi$
$12$	0	0
$13$	−5.50031	−1.52551	−0.762755	−	0.646687i	$-0.776154\pi$
$13$	−5.50031	−1.52551	−0.762755	+	0.646687i	$0.776154\pi$
$14$	0	0
$15$	−0.706754	−0.182483
$16$	0	0
$17$	5.35523	1.29883	0.649417	−	0.760433i	$-0.275013\pi$
$17$	5.35523	1.29883	0.649417	+	0.760433i	$0.275013\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.47375	0.515813	0.257906	−	0.966170i	$-0.416967\pi$
$23$	2.47375	0.515813	0.257906	+	0.966170i	$0.416967\pi$
$24$	0	0
$25$	−4.41372	−0.882743
$26$	0	0
$27$	4.75176	0.914477
$28$	0	0
$29$	10.0997	1.87547	0.937734	−	0.347353i	$-0.112920\pi$
$29$	10.0997	1.87547	0.937734	+	0.347353i	$0.112920\pi$
$30$	0	0
$31$	−0.967066	−0.173690	−0.0868451	−	0.996222i	$-0.527679\pi$
$31$	−0.967066	−0.173690	−0.0868451	+	0.996222i	$0.527679\pi$
$32$	0	0
$33$	−0.109907	−0.0191324
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.66670	0.931599	0.465800	−	0.884890i	$-0.345767\pi$
$37$	5.66670	0.931599	0.465800	+	0.884890i	$0.345767\pi$
$38$	0	0
$39$	5.07693	0.812959
$40$	0	0
$41$	−0.467311	−0.0729817	−0.0364909	−	0.999334i	$-0.511618\pi$
$41$	−0.467311	−0.0729817	−0.0364909	+	0.999334i	$0.511618\pi$
$42$	0	0
$43$	1.66934	0.254573	0.127286	−	0.991866i	$-0.459373\pi$
$43$	1.66934	0.254573	0.127286	+	0.991866i	$0.459373\pi$
$44$	0	0
$45$	−1.64472	−0.245181
$46$	0	0
$47$	−10.1965	−1.48731	−0.743653	−	0.668566i	$-0.766908\pi$
$47$	−10.1965	−1.48731	−0.743653	+	0.668566i	$0.766908\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.94302	−0.692161
$52$	0	0
$53$	−9.58561	−1.31669	−0.658343	−	0.752718i	$-0.728742\pi$
$53$	−9.58561	−1.31669	−0.658343	+	0.752718i	$0.728742\pi$
$54$	0	0
$55$	0.0911731	0.0122938
$56$	0	0
$57$	−0.923026	−0.122258
$58$	0	0
$59$	−14.0439	−1.82837	−0.914183	−	0.405301i	$-0.867167\pi$
$59$	−14.0439	−1.82837	−0.914183	+	0.405301i	$0.867167\pi$
$60$	0	0
$61$	−15.1285	−1.93701	−0.968504	−	0.248999i	$-0.919898\pi$
$61$	−15.1285	−1.93701	−0.968504	+	0.248999i	$0.919898\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.21154	−0.522378
$66$	0	0
$67$	−10.0927	−1.23302	−0.616512	−	0.787346i	$-0.711455\pi$
$67$	−10.0927	−1.23302	−0.616512	+	0.787346i	$0.711455\pi$
$68$	0	0
$69$	−2.28334	−0.274881
$70$	0	0
$71$	14.6510	1.73876	0.869379	−	0.494146i	$-0.164519\pi$
$71$	14.6510	1.73876	0.869379	+	0.494146i	$0.164519\pi$
$72$	0	0
$73$	−0.289408	−0.0338727	−0.0169363	−	0.999857i	$-0.505391\pi$
$73$	−0.289408	−0.0338727	−0.0169363	+	0.999857i	$0.505391\pi$
$74$	0	0
$75$	4.07398	0.470422
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.2973	−1.83359	−0.916795	−	0.399359i	$-0.869233\pi$
$79$	−16.2973	−1.83359	−0.916795	+	0.399359i	$0.869233\pi$
$80$	0	0
$81$	2.05807	0.228674
$82$	0	0
$83$	8.67026	0.951685	0.475843	−	0.879530i	$-0.342143\pi$
$83$	8.67026	0.951685	0.475843	+	0.879530i	$0.342143\pi$
$84$	0	0
$85$	4.10046	0.444757
$86$	0	0
$87$	−9.32230	−0.999455
$88$	0	0
$89$	1.64821	0.174710	0.0873550	−	0.996177i	$-0.472159\pi$
$89$	1.64821	0.174710	0.0873550	+	0.996177i	$0.472159\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.892628	0.0925612
$94$	0	0
$95$	0.765692	0.0785584
$96$	0	0
$97$	−5.16198	−0.524119	−0.262060	−	0.965052i	$-0.584402\pi$
$97$	−5.16198	−0.524119	−0.262060	+	0.965052i	$0.584402\pi$
$98$	0	0
$99$	−0.255771	−0.0257060
$100$	0	0
$101$	−4.12868	−0.410819	−0.205410	−	0.978676i	$-0.565853\pi$
$101$	−4.12868	−0.410819	−0.205410	+	0.978676i	$0.565853\pi$
$102$	0	0
$103$	−0.956458	−0.0942426	−0.0471213	−	0.998889i	$-0.515005\pi$
$103$	−0.956458	−0.0942426	−0.0471213	+	0.998889i	$0.515005\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.9095	1.15133	0.575666	−	0.817685i	$-0.304743\pi$
$107$	11.9095	1.15133	0.575666	+	0.817685i	$0.304743\pi$
$108$	0	0
$109$	11.9985	1.14924	0.574622	−	0.818419i	$-0.305149\pi$
$109$	11.9985	1.14924	0.574622	+	0.818419i	$0.305149\pi$
$110$	0	0
$111$	−5.23051	−0.496458
$112$	0	0
$113$	−2.83195	−0.266408	−0.133204	−	0.991089i	$-0.542526\pi$
$113$	−2.83195	−0.266408	−0.133204	+	0.991089i	$0.542526\pi$
$114$	0	0
$115$	1.89413	0.176629
$116$	0	0
$117$	11.8148	1.09228
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9858	−0.998711
$122$	0	0
$123$	0.431340	0.0388927
$124$	0	0
$125$	−7.20801	−0.644704
$126$	0	0
$127$	1.41907	0.125922	0.0629609	−	0.998016i	$-0.479946\pi$
$127$	1.41907	0.125922	0.0629609	+	0.998016i	$0.479946\pi$
$128$	0	0
$129$	−1.54085	−0.135664
$130$	0	0
$131$	−6.75319	−0.590029	−0.295014	−	0.955493i	$-0.595324\pi$
$131$	−6.75319	−0.590029	−0.295014	+	0.955493i	$0.595324\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.63839	0.313142
$136$	0	0
$137$	−2.69392	−0.230157	−0.115079	−	0.993356i	$-0.536712\pi$
$137$	−2.69392	−0.230157	−0.115079	+	0.993356i	$0.536712\pi$
$138$	0	0
$139$	−21.8339	−1.85192	−0.925962	−	0.377616i	$-0.876744\pi$
$139$	−21.8339	−1.85192	−0.925962	+	0.377616i	$0.876744\pi$
$140$	0	0
$141$	9.41159	0.792599
$142$	0	0
$143$	−0.654937	−0.0547686
$144$	0	0
$145$	7.73327	0.642213
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.4268	1.01804	0.509021	−	0.860754i	$-0.330007\pi$
$149$	12.4268	1.01804	0.509021	+	0.860754i	$0.330007\pi$
$150$	0	0
$151$	−15.9677	−1.29943	−0.649716	−	0.760177i	$-0.725112\pi$
$151$	−15.9677	−1.29943	−0.649716	+	0.760177i	$0.725112\pi$
$152$	0	0
$153$	−11.5032	−0.929975
$154$	0	0
$155$	−0.740475	−0.0594764
$156$	0	0
$157$	−17.0680	−1.36217	−0.681086	−	0.732204i	$-0.738492\pi$
$157$	−17.0680	−1.36217	−0.681086	+	0.732204i	$0.738492\pi$
$158$	0	0
$159$	8.84777	0.701674
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−18.8648	−1.47761	−0.738803	−	0.673921i	$-0.764609\pi$
$163$	−18.8648	−1.47761	−0.738803	+	0.673921i	$0.764609\pi$
$164$	0	0
$165$	−0.0841552	−0.00655147
$166$	0	0
$167$	−12.7010	−0.982829	−0.491415	−	0.870926i	$-0.663520\pi$
$167$	−12.7010	−0.982829	−0.491415	+	0.870926i	$0.663520\pi$
$168$	0	0
$169$	17.2534	1.32718
$170$	0	0
$171$	−2.14802	−0.164263
$172$	0	0
$173$	−18.2103	−1.38450	−0.692251	−	0.721657i	$-0.743381\pi$
$173$	−18.2103	−1.38450	−0.692251	+	0.721657i	$0.743381\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.9629	0.974354
$178$	0	0
$179$	19.6574	1.46926	0.734632	−	0.678466i	$-0.237355\pi$
$179$	19.6574	1.46926	0.734632	+	0.678466i	$0.237355\pi$
$180$	0	0
$181$	19.4592	1.44639	0.723194	−	0.690645i	$-0.242673\pi$
$181$	19.4592	1.44639	0.723194	+	0.690645i	$0.242673\pi$
$182$	0	0
$183$	13.9640	1.03225
$184$	0	0
$185$	4.33895	0.319006
$186$	0	0
$187$	0.637662	0.0466305
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.2086	−0.811027	−0.405513	−	0.914089i	$-0.632907\pi$
$191$	−11.2086	−0.811027	−0.405513	+	0.914089i	$0.632907\pi$
$192$	0	0
$193$	−14.5231	−1.04539	−0.522696	−	0.852519i	$-0.675074\pi$
$193$	−14.5231	−1.04539	−0.522696	+	0.852519i	$0.675074\pi$
$194$	0	0
$195$	3.88737	0.278380
$196$	0	0
$197$	25.3899	1.80895	0.904477	−	0.426523i	$-0.140262\pi$
$197$	25.3899	1.80895	0.904477	+	0.426523i	$0.140262\pi$
$198$	0	0
$199$	−14.8633	−1.05363	−0.526814	−	0.849980i	$-0.676614\pi$
$199$	−14.8633	−1.05363	−0.526814	+	0.849980i	$0.676614\pi$
$200$	0	0
$201$	9.31586	0.657090
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.357817	−0.0249910
$206$	0	0
$207$	−5.31367	−0.369326
$208$	0	0
$209$	0.119073	0.00823644
$210$	0	0
$211$	−8.04839	−0.554074	−0.277037	−	0.960859i	$-0.589352\pi$
$211$	−8.04839	−0.554074	−0.277037	+	0.960859i	$0.589352\pi$
$212$	0	0
$213$	−13.5233	−0.926600
$214$	0	0
$215$	1.27820	0.0871728
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.267131	0.0180511
$220$	0	0
$221$	−29.4554	−1.98138
$222$	0	0
$223$	13.4292	0.899285	0.449642	−	0.893209i	$-0.351551\pi$
$223$	13.4292	0.899285	0.449642	+	0.893209i	$0.351551\pi$
$224$	0	0
$225$	9.48076	0.632051
$226$	0	0
$227$	−1.38109	−0.0916659	−0.0458330	−	0.998949i	$-0.514594\pi$
$227$	−1.38109	−0.0916659	−0.0458330	+	0.998949i	$0.514594\pi$
$228$	0	0
$229$	13.2347	0.874572	0.437286	−	0.899323i	$-0.355940\pi$
$229$	13.2347	0.874572	0.437286	+	0.899323i	$0.355940\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.27126	−0.0832831	−0.0416416	−	0.999133i	$-0.513259\pi$
$233$	−1.27126	−0.0832831	−0.0416416	+	0.999133i	$0.513259\pi$
$234$	0	0
$235$	−7.80735	−0.509295
$236$	0	0
$237$	15.0428	0.977137
$238$	0	0
$239$	−11.6876	−0.756007	−0.378003	−	0.925804i	$-0.623389\pi$
$239$	−11.6876	−0.756007	−0.378003	+	0.925804i	$0.623389\pi$
$240$	0	0
$241$	7.32444	0.471809	0.235904	−	0.971776i	$-0.424195\pi$
$241$	7.32444	0.471809	0.235904	+	0.971776i	$0.424195\pi$
$242$	0	0
$243$	−16.1549	−1.03634
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.50031	−0.349976
$248$	0	0
$249$	−8.00288	−0.507162
$250$	0	0
$251$	−30.1929	−1.90576	−0.952880	−	0.303346i	$-0.901896\pi$
$251$	−30.1929	−1.90576	−0.952880	+	0.303346i	$0.901896\pi$
$252$	0	0
$253$	0.294556	0.0185186
$254$	0	0
$255$	−3.78483	−0.237015
$256$	0	0
$257$	19.2097	1.19827	0.599135	−	0.800648i	$-0.295511\pi$
$257$	19.2097	1.19827	0.599135	+	0.800648i	$0.295511\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−21.6944	−1.34285
$262$	0	0
$263$	3.74697	0.231048	0.115524	−	0.993305i	$-0.463145\pi$
$263$	3.74697	0.231048	0.115524	+	0.993305i	$0.463145\pi$
$264$	0	0
$265$	−7.33963	−0.450870
$266$	0	0
$267$	−1.52134	−0.0931046
$268$	0	0
$269$	−7.64294	−0.465998	−0.232999	−	0.972477i	$-0.574854\pi$
$269$	−7.64294	−0.465998	−0.232999	+	0.972477i	$0.574854\pi$
$270$	0	0
$271$	−12.8107	−0.778195	−0.389098	−	0.921196i	$-0.627213\pi$
$271$	−12.8107	−0.778195	−0.389098	+	0.921196i	$0.627213\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.525553	−0.0316921
$276$	0	0
$277$	16.7793	1.00817	0.504085	−	0.863654i	$-0.331830\pi$
$277$	16.7793	1.00817	0.504085	+	0.863654i	$0.331830\pi$
$278$	0	0
$279$	2.07728	0.124364
$280$	0	0
$281$	−25.8940	−1.54471	−0.772354	−	0.635193i	$-0.780921\pi$
$281$	−25.8940	−1.54471	−0.772354	+	0.635193i	$0.780921\pi$
$282$	0	0
$283$	−25.0440	−1.48871	−0.744356	−	0.667783i	$-0.767243\pi$
$283$	−25.0440	−1.48871	−0.744356	+	0.667783i	$0.767243\pi$
$284$	0	0
$285$	−0.706754	−0.0418645
$286$	0	0
$287$	0	0
$288$	0	0
$289$	11.6785	0.686970
$290$	0	0
$291$	4.76464	0.279308
$292$	0	0
$293$	−2.93743	−0.171607	−0.0858033	−	0.996312i	$-0.527346\pi$
$293$	−2.93743	−0.171607	−0.0858033	+	0.996312i	$0.527346\pi$
$294$	0	0
$295$	−10.7533	−0.626084
$296$	0	0
$297$	0.565805	0.0328314
$298$	0	0
$299$	−13.6064	−0.786878
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.81088	0.218930
$304$	0	0
$305$	−11.5838	−0.663286
$306$	0	0
$307$	7.86843	0.449075	0.224537	−	0.974465i	$-0.427913\pi$
$307$	7.86843	0.449075	0.224537	+	0.974465i	$0.427913\pi$
$308$	0	0
$309$	0.882836	0.0502228
$310$	0	0
$311$	10.0594	0.570418	0.285209	−	0.958465i	$-0.407937\pi$
$311$	10.0594	0.570418	0.285209	+	0.958465i	$0.407937\pi$
$312$	0	0
$313$	22.7851	1.28789	0.643946	−	0.765071i	$-0.277296\pi$
$313$	22.7851	1.28789	0.643946	+	0.765071i	$0.277296\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.25396	0.238926	0.119463	−	0.992839i	$-0.461883\pi$
$317$	4.25396	0.238926	0.119463	+	0.992839i	$0.461883\pi$
$318$	0	0
$319$	1.20260	0.0673327
$320$	0	0
$321$	−10.9928	−0.613556
$322$	0	0
$323$	5.35523	0.297973
$324$	0	0
$325$	24.2768	1.34663
$326$	0	0
$327$	−11.0749	−0.612443
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−18.8705	−1.03721	−0.518607	−	0.855013i	$-0.673549\pi$
$331$	−18.8705	−1.03721	−0.518607	+	0.855013i	$0.673549\pi$
$332$	0	0
$333$	−12.1722	−0.667032
$334$	0	0
$335$	−7.72793	−0.422222
$336$	0	0
$337$	2.29350	0.124935	0.0624673	−	0.998047i	$-0.480103\pi$
$337$	2.29350	0.124935	0.0624673	+	0.998047i	$0.480103\pi$
$338$	0	0
$339$	2.61397	0.141971
$340$	0	0
$341$	−0.115151	−0.00623579
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.74833	−0.0941271
$346$	0	0
$347$	−10.3828	−0.557380	−0.278690	−	0.960381i	$-0.589900\pi$
$347$	−10.3828	−0.557380	−0.278690	+	0.960381i	$0.589900\pi$
$348$	0	0
$349$	3.38565	0.181230	0.0906149	−	0.995886i	$-0.471117\pi$
$349$	3.38565	0.181230	0.0906149	+	0.995886i	$0.471117\pi$
$350$	0	0
$351$	−26.1361	−1.39504
$352$	0	0
$353$	−17.1858	−0.914710	−0.457355	−	0.889284i	$-0.651203\pi$
$353$	−17.1858	−0.914710	−0.457355	+	0.889284i	$0.651203\pi$
$354$	0	0
$355$	11.2182	0.595399
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.8989	−1.05022	−0.525111	−	0.851034i	$-0.675976\pi$
$359$	−19.8989	−1.05022	−0.525111	+	0.851034i	$0.675976\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	10.1402	0.532223
$364$	0	0
$365$	−0.221598	−0.0115990
$366$	0	0
$367$	−12.5795	−0.656647	−0.328324	−	0.944565i	$-0.606484\pi$
$367$	−12.5795	−0.656647	−0.328324	+	0.944565i	$0.606484\pi$
$368$	0	0
$369$	1.00379	0.0522555
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.99804	−0.155232	−0.0776162	−	0.996983i	$-0.524731\pi$
$373$	−2.99804	−0.155232	−0.0776162	+	0.996983i	$0.524731\pi$
$374$	0	0
$375$	6.65318	0.343569
$376$	0	0
$377$	−55.5515	−2.86105
$378$	0	0
$379$	31.1950	1.60238	0.801191	−	0.598409i	$-0.204200\pi$
$379$	31.1950	1.60238	0.801191	+	0.598409i	$0.204200\pi$
$380$	0	0
$381$	−1.30984	−0.0671050
$382$	0	0
$383$	−1.13421	−0.0579555	−0.0289777	−	0.999580i	$-0.509225\pi$
$383$	−1.13421	−0.0579555	−0.0289777	+	0.999580i	$0.509225\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.58579	−0.182276
$388$	0	0
$389$	4.12774	0.209285	0.104642	−	0.994510i	$-0.466630\pi$
$389$	4.12774	0.209285	0.104642	+	0.994510i	$0.466630\pi$
$390$	0	0
$391$	13.2475	0.669955
$392$	0	0
$393$	6.23337	0.314432
$394$	0	0
$395$	−12.4787	−0.627872
$396$	0	0
$397$	29.1663	1.46382	0.731908	−	0.681404i	$-0.238630\pi$
$397$	29.1663	1.46382	0.731908	+	0.681404i	$0.238630\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.74370	0.0870764	0.0435382	−	0.999052i	$-0.486137\pi$
$401$	1.74370	0.0870764	0.0435382	+	0.999052i	$0.486137\pi$
$402$	0	0
$403$	5.31916	0.264966
$404$	0	0
$405$	1.57585	0.0783045
$406$	0	0
$407$	0.674750	0.0334461
$408$	0	0
$409$	10.4484	0.516640	0.258320	−	0.966059i	$-0.416831\pi$
$409$	10.4484	0.516640	0.258320	+	0.966059i	$0.416831\pi$
$410$	0	0
$411$	2.48656	0.122653
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6.63876	0.325884
$416$	0	0
$417$	20.1532	0.986908
$418$	0	0
$419$	9.62283	0.470106	0.235053	−	0.971983i	$-0.424474\pi$
$419$	9.62283	0.470106	0.235053	+	0.971983i	$0.424474\pi$
$420$	0	0
$421$	27.1271	1.32209	0.661046	−	0.750346i	$-0.270113\pi$
$421$	27.1271	1.32209	0.661046	+	0.750346i	$0.270113\pi$
$422$	0	0
$423$	21.9022	1.06492
$424$	0	0
$425$	−23.6365	−1.14654
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.604524	0.0291867
$430$	0	0
$431$	2.20460	0.106192	0.0530960	−	0.998589i	$-0.483091\pi$
$431$	2.20460	0.106192	0.0530960	+	0.998589i	$0.483091\pi$
$432$	0	0
$433$	19.6906	0.946270	0.473135	−	0.880990i	$-0.343122\pi$
$433$	19.6906	0.946270	0.473135	+	0.880990i	$0.343122\pi$
$434$	0	0
$435$	−7.13801	−0.342241
$436$	0	0
$437$	2.47375	0.118336
$438$	0	0
$439$	−26.2182	−1.25133	−0.625663	−	0.780093i	$-0.715172\pi$
$439$	−26.2182	−1.25133	−0.625663	+	0.780093i	$0.715172\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.03641	−0.239287	−0.119644	−	0.992817i	$-0.538175\pi$
$443$	−5.03641	−0.239287	−0.119644	+	0.992817i	$0.538175\pi$
$444$	0	0
$445$	1.26202	0.0598256
$446$	0	0
$447$	−11.4703	−0.542524
$448$	0	0
$449$	−18.9966	−0.896503	−0.448252	−	0.893907i	$-0.647953\pi$
$449$	−18.9966	−0.896503	−0.448252	+	0.893907i	$0.647953\pi$
$450$	0	0
$451$	−0.0556440	−0.00262018
$452$	0	0
$453$	14.7386	0.692479
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.2517	0.994111	0.497056	−	0.867719i	$-0.334414\pi$
$457$	21.2517	0.994111	0.497056	+	0.867719i	$0.334414\pi$
$458$	0	0
$459$	25.4468	1.18775
$460$	0	0
$461$	9.98621	0.465104	0.232552	−	0.972584i	$-0.425292\pi$
$461$	9.98621	0.465104	0.232552	+	0.972584i	$0.425292\pi$
$462$	0	0
$463$	−22.7254	−1.05614	−0.528069	−	0.849201i	$-0.677084\pi$
$463$	−22.7254	−1.05614	−0.528069	+	0.849201i	$0.677084\pi$
$464$	0	0
$465$	0.683478	0.0316955
$466$	0	0
$467$	−25.8641	−1.19685	−0.598424	−	0.801179i	$-0.704206\pi$
$467$	−25.8641	−1.19685	−0.598424	+	0.801179i	$0.704206\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	15.7542	0.725914
$472$	0	0
$473$	0.198773	0.00913961
$474$	0	0
$475$	−4.41372	−0.202515
$476$	0	0
$477$	20.5901	0.942757
$478$	0	0
$479$	10.8037	0.493632	0.246816	−	0.969062i	$-0.420616\pi$
$479$	10.8037	0.493632	0.246816	+	0.969062i	$0.420616\pi$
$480$	0	0
$481$	−31.1686	−1.42116
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.95249	−0.179473
$486$	0	0
$487$	25.4905	1.15508	0.577542	−	0.816361i	$-0.304012\pi$
$487$	25.4905	1.15508	0.577542	+	0.816361i	$0.304012\pi$
$488$	0	0
$489$	17.4127	0.787431
$490$	0	0
$491$	−25.9732	−1.17215	−0.586077	−	0.810255i	$-0.699328\pi$
$491$	−25.9732	−1.17215	−0.586077	+	0.810255i	$0.699328\pi$
$492$	0	0
$493$	54.0863	2.43592
$494$	0	0
$495$	−0.195842	−0.00880244
$496$	0	0
$497$	0	0
$498$	0	0
$499$	30.3627	1.35922	0.679610	−	0.733574i	$-0.262149\pi$
$499$	30.3627	1.35922	0.679610	+	0.733574i	$0.262149\pi$
$500$	0	0
$501$	11.7233	0.523759
$502$	0	0
$503$	−16.6196	−0.741031	−0.370515	−	0.928826i	$-0.620819\pi$
$503$	−16.6196	−0.741031	−0.370515	+	0.928826i	$0.620819\pi$
$504$	0	0
$505$	−3.16130	−0.140676
$506$	0	0
$507$	−15.9253	−0.707268
$508$	0	0
$509$	25.9516	1.15028	0.575142	−	0.818054i	$-0.304947\pi$
$509$	25.9516	1.15028	0.575142	+	0.818054i	$0.304947\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.75176	0.209795
$514$	0	0
$515$	−0.732353	−0.0322713
$516$	0	0
$517$	−1.21412	−0.0533969
$518$	0	0
$519$	16.8086	0.737814
$520$	0	0
$521$	16.0064	0.701255	0.350628	−	0.936515i	$-0.385968\pi$
$521$	16.0064	0.701255	0.350628	+	0.936515i	$0.385968\pi$
$522$	0	0
$523$	12.9789	0.567526	0.283763	−	0.958894i	$-0.408417\pi$
$523$	12.9789	0.567526	0.283763	+	0.958894i	$0.408417\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.17886	−0.225595
$528$	0	0
$529$	−16.8806	−0.733937
$530$	0	0
$531$	30.1667	1.30912
$532$	0	0
$533$	2.57035	0.111334
$534$	0	0
$535$	9.11900	0.394249
$536$	0	0
$537$	−18.1443	−0.782985
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−45.0955	−1.93880	−0.969402	−	0.245477i	$-0.921055\pi$
$541$	−45.0955	−1.93880	−0.969402	+	0.245477i	$0.921055\pi$
$542$	0	0
$543$	−17.9613	−0.770794
$544$	0	0
$545$	9.18714	0.393534
$546$	0	0
$547$	−22.6028	−0.966427	−0.483214	−	0.875502i	$-0.660531\pi$
$547$	−22.6028	−0.966427	−0.483214	+	0.875502i	$0.660531\pi$
$548$	0	0
$549$	32.4964	1.38691
$550$	0	0
$551$	10.0997	0.430262
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.00496	−0.170001
$556$	0	0
$557$	−5.17022	−0.219069	−0.109535	−	0.993983i	$-0.534936\pi$
$557$	−5.17022	−0.219069	−0.109535	+	0.993983i	$0.534936\pi$
$558$	0	0
$559$	−9.18190	−0.388353
$560$	0	0
$561$	−0.588579	−0.0248498
$562$	0	0
$563$	3.05312	0.128674	0.0643368	−	0.997928i	$-0.479507\pi$
$563$	3.05312	0.128674	0.0643368	+	0.997928i	$0.479507\pi$
$564$	0	0
$565$	−2.16840	−0.0912255
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.5344	1.28007	0.640035	−	0.768346i	$-0.278920\pi$
$569$	30.5344	1.28007	0.640035	+	0.768346i	$0.278920\pi$
$570$	0	0
$571$	25.1860	1.05400	0.527002	−	0.849864i	$-0.323316\pi$
$571$	25.1860	1.05400	0.527002	+	0.849864i	$0.323316\pi$
$572$	0	0
$573$	10.3458	0.432204
$574$	0	0
$575$	−10.9184	−0.455330
$576$	0	0
$577$	−27.0127	−1.12455	−0.562277	−	0.826949i	$-0.690075\pi$
$577$	−27.0127	−1.12455	−0.562277	+	0.826949i	$0.690075\pi$
$578$	0	0
$579$	13.4052	0.557100
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.14139	−0.0472714
$584$	0	0
$585$	9.04649	0.374026
$586$	0	0
$587$	0.197187	0.00813879	0.00406940	−	0.999992i	$-0.498705\pi$
$587$	0.197187	0.00813879	0.00406940	+	0.999992i	$0.498705\pi$
$588$	0	0
$589$	−0.967066	−0.0398473
$590$	0	0
$591$	−23.4355	−0.964008
$592$	0	0
$593$	31.3399	1.28698	0.643488	−	0.765456i	$-0.277486\pi$
$593$	31.3399	1.28698	0.643488	+	0.765456i	$0.277486\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13.7192	0.561489
$598$	0	0
$599$	−25.6476	−1.04793	−0.523967	−	0.851739i	$-0.675548\pi$
$599$	−25.6476	−1.04793	−0.523967	+	0.851739i	$0.675548\pi$
$600$	0	0
$601$	9.63125	0.392867	0.196433	−	0.980517i	$-0.437064\pi$
$601$	9.63125	0.392867	0.196433	+	0.980517i	$0.437064\pi$
$602$	0	0
$603$	21.6794	0.882854
$604$	0	0
$605$	−8.41176	−0.341987
$606$	0	0
$607$	−40.6672	−1.65063	−0.825315	−	0.564673i	$-0.809002\pi$
$607$	−40.6672	−1.65063	−0.825315	+	0.564673i	$0.809002\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	56.0836	2.26890
$612$	0	0
$613$	−26.2934	−1.06198	−0.530990	−	0.847378i	$-0.678180\pi$
$613$	−26.2934	−1.06198	−0.530990	+	0.847378i	$0.678180\pi$
$614$	0	0
$615$	0.330274	0.0133179
$616$	0	0
$617$	−34.1210	−1.37366	−0.686830	−	0.726818i	$-0.740998\pi$
$617$	−34.1210	−1.37366	−0.686830	+	0.726818i	$0.740998\pi$
$618$	0	0
$619$	5.26389	0.211574	0.105787	−	0.994389i	$-0.466264\pi$
$619$	5.26389	0.211574	0.105787	+	0.994389i	$0.466264\pi$
$620$	0	0
$621$	11.7547	0.471699
$622$	0	0
$623$	0	0
$624$	0	0
$625$	16.5495	0.661978
$626$	0	0
$627$	−0.109907	−0.00438928
$628$	0	0
$629$	30.3465	1.20999
$630$	0	0
$631$	2.76371	0.110022	0.0550108	−	0.998486i	$-0.482481\pi$
$631$	2.76371	0.110022	0.0550108	+	0.998486i	$0.482481\pi$
$632$	0	0
$633$	7.42887	0.295271
$634$	0	0
$635$	1.08657	0.0431192
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−31.4707	−1.24496
$640$	0	0
$641$	−23.9787	−0.947102	−0.473551	−	0.880766i	$-0.657028\pi$
$641$	−23.9787	−0.947102	−0.473551	+	0.880766i	$0.657028\pi$
$642$	0	0
$643$	44.5701	1.75767	0.878837	−	0.477122i	$-0.158320\pi$
$643$	44.5701	1.75767	0.878837	+	0.477122i	$0.158320\pi$
$644$	0	0
$645$	−1.17982	−0.0464552
$646$	0	0
$647$	12.9229	0.508050	0.254025	−	0.967198i	$-0.418245\pi$
$647$	12.9229	0.508050	0.254025	+	0.967198i	$0.418245\pi$
$648$	0	0
$649$	−1.67225	−0.0656416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.0803	0.864068	0.432034	−	0.901857i	$-0.357796\pi$
$653$	22.0803	0.864068	0.432034	+	0.901857i	$0.357796\pi$
$654$	0	0
$655$	−5.17087	−0.202042
$656$	0	0
$657$	0.621655	0.0242531
$658$	0	0
$659$	9.29365	0.362029	0.181015	−	0.983480i	$-0.442062\pi$
$659$	9.29365	0.362029	0.181015	+	0.983480i	$0.442062\pi$
$660$	0	0
$661$	−9.74697	−0.379113	−0.189557	−	0.981870i	$-0.560705\pi$
$661$	−9.74697	−0.379113	−0.189557	+	0.981870i	$0.560705\pi$
$662$	0	0
$663$	27.1881	1.05590
$664$	0	0
$665$	0	0
$666$	0	0
$667$	24.9842	0.967390
$668$	0	0
$669$	−12.3955	−0.479237
$670$	0	0
$671$	−1.80139	−0.0695420
$672$	0	0
$673$	−25.3321	−0.976481	−0.488241	−	0.872709i	$-0.662361\pi$
$673$	−25.3321	−0.976481	−0.488241	+	0.872709i	$0.662361\pi$
$674$	0	0
$675$	−20.9729	−0.807248
$676$	0	0
$677$	−2.32926	−0.0895206	−0.0447603	−	0.998998i	$-0.514252\pi$
$677$	−2.32926	−0.0895206	−0.0447603	+	0.998998i	$0.514252\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.27478	0.0488496
$682$	0	0
$683$	29.1505	1.11541	0.557706	−	0.830039i	$-0.311682\pi$
$683$	29.1505	1.11541	0.557706	+	0.830039i	$0.311682\pi$
$684$	0	0
$685$	−2.06272	−0.0788123
$686$	0	0
$687$	−12.2160	−0.466068
$688$	0	0
$689$	52.7238	2.00862
$690$	0	0
$691$	−42.6192	−1.62131	−0.810656	−	0.585523i	$-0.800889\pi$
$691$	−42.6192	−1.62131	−0.810656	+	0.585523i	$0.800889\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.7180	−0.634151
$696$	0	0
$697$	−2.50256	−0.0947912
$698$	0	0
$699$	1.17341	0.0443824
$700$	0	0
$701$	−45.6821	−1.72539	−0.862695	−	0.505725i	$-0.831225\pi$
$701$	−45.6821	−1.72539	−0.862695	+	0.505725i	$0.831225\pi$
$702$	0	0
$703$	5.66670	0.213724
$704$	0	0
$705$	7.20639	0.271408
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−30.8071	−1.15699	−0.578493	−	0.815687i	$-0.696359\pi$
$709$	−30.8071	−1.15699	−0.578493	+	0.815687i	$0.696359\pi$
$710$	0	0
$711$	35.0070	1.31286
$712$	0	0
$713$	−2.39228	−0.0895916
$714$	0	0
$715$	−0.501480	−0.0187543
$716$	0	0
$717$	10.7879	0.402883
$718$	0	0
$719$	3.87006	0.144329	0.0721645	−	0.997393i	$-0.477009\pi$
$719$	3.87006	0.144329	0.0721645	+	0.997393i	$0.477009\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−6.76065	−0.251431
$724$	0	0
$725$	−44.5772	−1.65556
$726$	0	0
$727$	9.25902	0.343398	0.171699	−	0.985149i	$-0.445074\pi$
$727$	9.25902	0.343398	0.171699	+	0.985149i	$0.445074\pi$
$728$	0	0
$729$	8.73722	0.323601
$730$	0	0
$731$	8.93972	0.330648
$732$	0	0
$733$	38.4175	1.41898	0.709491	−	0.704714i	$-0.248925\pi$
$733$	38.4175	1.41898	0.709491	+	0.704714i	$0.248925\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.20177	−0.0442677
$738$	0	0
$739$	19.5447	0.718965	0.359482	−	0.933152i	$-0.382953\pi$
$739$	19.5447	0.718965	0.359482	+	0.933152i	$0.382953\pi$
$740$	0	0
$741$	5.07693	0.186506
$742$	0	0
$743$	24.1543	0.886136	0.443068	−	0.896488i	$-0.353890\pi$
$743$	24.1543	0.886136	0.443068	+	0.896488i	$0.353890\pi$
$744$	0	0
$745$	9.51510	0.348606
$746$	0	0
$747$	−18.6239	−0.681414
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−13.3795	−0.488223	−0.244112	−	0.969747i	$-0.578496\pi$
$751$	−13.3795	−0.488223	−0.244112	+	0.969747i	$0.578496\pi$
$752$	0	0
$753$	27.8689	1.01560
$754$	0	0
$755$	−12.2263	−0.444962
$756$	0	0
$757$	−16.3578	−0.594535	−0.297267	−	0.954794i	$-0.596075\pi$
$757$	−16.3578	−0.594535	−0.297267	+	0.954794i	$0.596075\pi$
$758$	0	0
$759$	−0.271883	−0.00986874
$760$	0	0
$761$	2.49687	0.0905116	0.0452558	−	0.998975i	$-0.485590\pi$
$761$	2.49687	0.0905116	0.0452558	+	0.998975i	$0.485590\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−8.80788	−0.318449
$766$	0	0
$767$	77.2460	2.78919
$768$	0	0
$769$	−7.39362	−0.266621	−0.133310	−	0.991074i	$-0.542561\pi$
$769$	−7.39362	−0.266621	−0.133310	+	0.991074i	$0.542561\pi$
$770$	0	0
$771$	−17.7311	−0.638570
$772$	0	0
$773$	28.4954	1.02491	0.512455	−	0.858714i	$-0.328736\pi$
$773$	28.4954	1.02491	0.512455	+	0.858714i	$0.328736\pi$
$774$	0	0
$775$	4.26836	0.153324
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.467311	−0.0167432
$780$	0	0
$781$	1.74454	0.0624245
$782$	0	0
$783$	47.9914	1.71507
$784$	0	0
$785$	−13.0688	−0.466446
$786$	0	0
$787$	25.0290	0.892186	0.446093	−	0.894987i	$-0.352815\pi$
$787$	25.0290	0.892186	0.446093	+	0.894987i	$0.352815\pi$
$788$	0	0
$789$	−3.45855	−0.123128
$790$	0	0
$791$	0	0
$792$	0	0
$793$	83.2114	2.95493
$794$	0	0
$795$	6.77467	0.240273
$796$	0	0
$797$	34.3723	1.21753	0.608764	−	0.793352i	$-0.291666\pi$
$797$	34.3723	1.21753	0.608764	+	0.793352i	$0.291666\pi$
$798$	0	0
$799$	−54.6043	−1.93176
$800$	0	0
$801$	−3.54039	−0.125094
$802$	0	0
$803$	−0.0344606	−0.00121609
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.05464	0.248335
$808$	0	0
$809$	25.4630	0.895233	0.447616	−	0.894226i	$-0.352273\pi$
$809$	25.4630	0.895233	0.447616	+	0.894226i	$0.352273\pi$
$810$	0	0
$811$	17.1159	0.601021	0.300511	−	0.953778i	$-0.402843\pi$
$811$	17.1159	0.601021	0.300511	+	0.953778i	$0.402843\pi$
$812$	0	0
$813$	11.8246	0.414708
$814$	0	0
$815$	−14.4447	−0.505974
$816$	0	0
$817$	1.66934	0.0584030
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.4155	0.991709	0.495854	−	0.868406i	$-0.334855\pi$
$821$	28.4155	0.991709	0.495854	+	0.868406i	$0.334855\pi$
$822$	0	0
$823$	50.8663	1.77309	0.886544	−	0.462644i	$-0.153099\pi$
$823$	50.8663	1.77309	0.886544	+	0.462644i	$0.153099\pi$
$824$	0	0
$825$	0.485100	0.0168890
$826$	0	0
$827$	−26.0730	−0.906647	−0.453323	−	0.891346i	$-0.649762\pi$
$827$	−26.0730	−0.906647	−0.453323	+	0.891346i	$0.649762\pi$
$828$	0	0
$829$	−9.68048	−0.336217	−0.168108	−	0.985769i	$-0.553766\pi$
$829$	−9.68048	−0.336217	−0.168108	+	0.985769i	$0.553766\pi$
$830$	0	0
$831$	−15.4877	−0.537263
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−9.72503	−0.336548
$836$	0	0
$837$	−4.59527	−0.158836
$838$	0	0
$839$	−26.6744	−0.920902	−0.460451	−	0.887685i	$-0.652312\pi$
$839$	−26.6744	−0.920902	−0.460451	+	0.887685i	$0.652312\pi$
$840$	0	0
$841$	73.0041	2.51738
$842$	0	0
$843$	23.9009	0.823189
$844$	0	0
$845$	13.2108	0.454465
$846$	0	0
$847$	0	0
$848$	0	0
$849$	23.1163	0.793349
$850$	0	0
$851$	14.0180	0.480531
$852$	0	0
$853$	−21.1426	−0.723908	−0.361954	−	0.932196i	$-0.617890\pi$
$853$	−21.1426	−0.723908	−0.361954	+	0.932196i	$0.617890\pi$
$854$	0	0
$855$	−1.64472	−0.0562484
$856$	0	0
$857$	17.3680	0.593280	0.296640	−	0.954989i	$-0.404134\pi$
$857$	17.3680	0.593280	0.296640	+	0.954989i	$0.404134\pi$
$858$	0	0
$859$	27.5373	0.939561	0.469781	−	0.882783i	$-0.344333\pi$
$859$	27.5373	0.939561	0.469781	+	0.882783i	$0.344333\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.8743	1.62966	0.814830	−	0.579700i	$-0.196830\pi$
$863$	47.8743	1.62966	0.814830	+	0.579700i	$0.196830\pi$
$864$	0	0
$865$	−13.9435	−0.474092
$866$	0	0
$867$	−10.7795	−0.366093
$868$	0	0
$869$	−1.94056	−0.0658291
$870$	0	0
$871$	55.5131	1.88099
$872$	0	0
$873$	11.0880	0.375273
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32.3823	−1.09347	−0.546736	−	0.837305i	$-0.684130\pi$
$877$	−32.3823	−1.09347	−0.546736	+	0.837305i	$0.684130\pi$
$878$	0	0
$879$	2.71133	0.0914508
$880$	0	0
$881$	−19.7003	−0.663720	−0.331860	−	0.943329i	$-0.607676\pi$
$881$	−19.7003	−0.663720	−0.331860	+	0.943329i	$0.607676\pi$
$882$	0	0
$883$	6.65506	0.223960	0.111980	−	0.993710i	$-0.464281\pi$
$883$	6.65506	0.223960	0.111980	+	0.993710i	$0.464281\pi$
$884$	0	0
$885$	9.92562	0.333646
$886$	0	0
$887$	−23.9760	−0.805035	−0.402518	−	0.915412i	$-0.631865\pi$
$887$	−23.9760	−0.805035	−0.402518	+	0.915412i	$0.631865\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0.245060	0.00820982
$892$	0	0
$893$	−10.1965	−0.341211
$894$	0	0
$895$	15.0515	0.503117
$896$	0	0
$897$	12.5591	0.419335
$898$	0	0
$899$	−9.76709	−0.325751
$900$	0	0
$901$	−51.3332	−1.71016
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.8997	0.495284
$906$	0	0
$907$	47.5318	1.57827	0.789133	−	0.614222i	$-0.210530\pi$
$907$	47.5318	1.57827	0.789133	+	0.614222i	$0.210530\pi$
$908$	0	0
$909$	8.86850	0.294150
$910$	0	0
$911$	−3.12922	−0.103676	−0.0518379	−	0.998656i	$-0.516508\pi$
$911$	−3.12922	−0.103676	−0.0518379	+	0.998656i	$0.516508\pi$
$912$	0	0
$913$	1.03239	0.0341672
$914$	0	0
$915$	10.6921	0.353471
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.84381	0.0608217	0.0304109	−	0.999537i	$-0.490318\pi$
$919$	1.84381	0.0608217	0.0304109	+	0.999537i	$0.490318\pi$
$920$	0	0
$921$	−7.26276	−0.239316
$922$	0	0
$923$	−80.5852	−2.65249
$924$	0	0
$925$	−25.0112	−0.822363
$926$	0	0
$927$	2.05449	0.0674784
$928$	0	0
$929$	−39.6561	−1.30107	−0.650537	−	0.759474i	$-0.725456\pi$
$929$	−39.6561	−1.30107	−0.650537	+	0.759474i	$0.725456\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−9.28511	−0.303981
$934$	0	0
$935$	0.488253	0.0159676
$936$	0	0
$937$	37.7052	1.23178	0.615888	−	0.787834i	$-0.288797\pi$
$937$	37.7052	1.23178	0.615888	+	0.787834i	$0.288797\pi$
$938$	0	0
$939$	−21.0313	−0.686329
$940$	0	0
$941$	−32.6930	−1.06576	−0.532881	−	0.846190i	$-0.678891\pi$
$941$	−32.6930	−1.06576	−0.532881	+	0.846190i	$0.678891\pi$
$942$	0	0
$943$	−1.15601	−0.0376449
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.6213	−1.06005	−0.530024	−	0.847982i	$-0.677817\pi$
$947$	−32.6213	−1.06005	−0.530024	+	0.847982i	$0.677817\pi$
$948$	0	0
$949$	1.59183	0.0516731
$950$	0	0
$951$	−3.92652	−0.127326
$952$	0	0
$953$	−29.4299	−0.953329	−0.476665	−	0.879085i	$-0.658154\pi$
$953$	−29.4299	−0.953329	−0.476665	+	0.879085i	$0.658154\pi$
$954$	0	0
$955$	−8.58235	−0.277718
$956$	0	0
$957$	−1.11003	−0.0358822
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0648	−0.969832
$962$	0	0
$963$	−25.5818	−0.824363
$964$	0	0
$965$	−11.1202	−0.357972
$966$	0	0
$967$	46.8694	1.50722	0.753610	−	0.657322i	$-0.228311\pi$
$967$	46.8694	1.50722	0.753610	+	0.657322i	$0.228311\pi$
$968$	0	0
$969$	−4.94302	−0.158793
$970$	0	0
$971$	−20.4120	−0.655052	−0.327526	−	0.944842i	$-0.606215\pi$
$971$	−20.4120	−0.655052	−0.327526	+	0.944842i	$0.606215\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−22.4081	−0.717634
$976$	0	0
$977$	−28.7020	−0.918258	−0.459129	−	0.888370i	$-0.651838\pi$
$977$	−28.7020	−0.918258	−0.459129	+	0.888370i	$0.651838\pi$
$978$	0	0
$979$	0.196257	0.00627240
$980$	0	0
$981$	−25.7730	−0.822868
$982$	0	0
$983$	8.79170	0.280412	0.140206	−	0.990122i	$-0.455224\pi$
$983$	8.79170	0.280412	0.140206	+	0.990122i	$0.455224\pi$
$984$	0	0
$985$	19.4408	0.619436
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.12954	0.131312
$990$	0	0
$991$	−9.59881	−0.304916	−0.152458	−	0.988310i	$-0.548719\pi$
$991$	−9.59881	−0.304916	−0.152458	+	0.988310i	$0.548719\pi$
$992$	0	0
$993$	17.4179	0.552741
$994$	0	0
$995$	−11.3807	−0.360792
$996$	0	0
$997$	13.9770	0.442656	0.221328	−	0.975199i	$-0.428961\pi$
$997$	13.9770	0.442656	0.221328	+	0.975199i	$0.428961\pi$
$998$	0	0
$999$	26.9268	0.851926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3724.2.a.o.1.4	✓	8
7.6	odd	2		3724.2.a.p.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3724.2.a.o.1.4	✓	8	1.1	even	1	trivial
3724.2.a.p.1.5	yes	8	7.6	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:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3724.a (trivial)

\(f(q)\)	\(=\)	\(q-0.923026 q^{3} +0.765692 q^{5} -2.14802 q^{9} +0.119073 q^{11} -5.50031 q^{13} -0.706754 q^{15} +5.35523 q^{17} +1.00000 q^{19} +2.47375 q^{23} -4.41372 q^{25} +4.75176 q^{27} +10.0997 q^{29} -0.967066 q^{31} -0.109907 q^{33} +5.66670 q^{37} +5.07693 q^{39} -0.467311 q^{41} +1.66934 q^{43} -1.64472 q^{45} -10.1965 q^{47} -4.94302 q^{51} -9.58561 q^{53} +0.0911731 q^{55} -0.923026 q^{57} -14.0439 q^{59} -15.1285 q^{61} -4.21154 q^{65} -10.0927 q^{67} -2.28334 q^{69} +14.6510 q^{71} -0.289408 q^{73} +4.07398 q^{75} -16.2973 q^{79} +2.05807 q^{81} +8.67026 q^{83} +4.10046 q^{85} -9.32230 q^{87} +1.64821 q^{89} +0.892628 q^{93} +0.765692 q^{95} -5.16198 q^{97} -0.255771 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 8 q^{11} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{25} - 16 q^{27} - 4 q^{29} - 24 q^{31} + 12 q^{33} + 12 q^{37} + 4 q^{39} - 20 q^{41} + 12 q^{43} - 24 q^{45} - 24 q^{47} + 12 q^{51} + 4 q^{53}+ \cdots - 20 q^{97}+O(q^{100}) \) 8 * q - 4 * q^3 - 8 * q^11 - 8 * q^17 + 8 * q^19 + 4 * q^23 + 4 * q^25 - 16 * q^27 - 4 * q^29 - 24 * q^31 + 12 * q^33 + 12 * q^37 + 4 * q^39 - 20 * q^41 + 12 * q^43 - 24 * q^45 - 24 * q^47 + 12 * q^51 + 4 * q^53 - 16 * q^55 - 4 * q^57 - 52 * q^59 + 16 * q^61 - 4 * q^65 + 8 * q^67 - 8 * q^69 + 16 * q^71 - 24 * q^73 - 8 * q^75 - 20 * q^79 + 12 * q^81 - 48 * q^83 - 12 * q^85 - 40 * q^87 - 12 * q^89 + 8 * q^93 - 20 * q^97

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