LMFDB - Embedded newform 9576.2.a.cj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.9248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.662153$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69614 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.69614 q^{5} -1.00000 q^{7} +1.32431 q^{11} -6.14355 q^{17} -1.00000 q^{19} -2.91779 q^{23} -2.12311 q^{25} +1.42696 q^{29} -3.32431 q^{31} -1.69614 q^{35} +9.83969 q^{37} +8.04090 q^{41} +7.16400 q^{43} +7.66906 q^{47} +1.00000 q^{49} -8.81925 q^{53} +2.24621 q^{55} -9.43318 q^{59} +4.64861 q^{61} -11.5705 q^{67} +3.22165 q^{71} -8.04090 q^{73} -1.32431 q^{77} -10.3101 q^{79} +4.77835 q^{83} -10.4203 q^{85} +11.6385 q^{89} -1.69614 q^{95} -6.71659 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} + 4 q^{17} - 4 q^{19} - 4 q^{23} + 8 q^{25} - 4 q^{29} - 8 q^{31} + 4 q^{37} + 8 q^{41} - 12 q^{43} + 8 q^{47} + 4 q^{49} - 12 q^{53} - 24 q^{55} + 8 q^{61} - 8 q^{67} + 12 q^{71} - 8 q^{73} - 20 q^{79} + 20 q^{83} - 4 q^{85} - 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 4 * q^17 - 4 * q^19 - 4 * q^23 + 8 * q^25 - 4 * q^29 - 8 * q^31 + 4 * q^37 + 8 * q^41 - 12 * q^43 + 8 * q^47 + 4 * q^49 - 12 * q^53 - 24 * q^55 + 8 * q^61 - 8 * q^67 + 12 * q^71 - 8 * q^73 - 20 * q^79 + 20 * q^83 - 4 * q^85 - 8 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.69614	0.758537	0.379269	−	0.925287i	$-0.376176\pi$
$5$	1.69614	0.758537	0.379269	+	0.925287i	$0.376176\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.32431	0.399294	0.199647	−	0.979868i	$-0.436021\pi$
$11$	1.32431	0.399294	0.199647	+	0.979868i	$0.436021\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.14355	−1.49003	−0.745015	−	0.667047i	$-0.767558\pi$
$17$	−6.14355	−1.49003	−0.745015	+	0.667047i	$0.767558\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.91779	−0.608401	−0.304201	−	0.952608i	$-0.598389\pi$
$23$	−2.91779	−0.608401	−0.304201	+	0.952608i	$0.598389\pi$
$24$	0	0
$25$	−2.12311	−0.424621
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.42696	0.264981	0.132490	−	0.991184i	$-0.457703\pi$
$29$	1.42696	0.264981	0.132490	+	0.991184i	$0.457703\pi$
$30$	0	0
$31$	−3.32431	−0.597063	−0.298532	−	0.954400i	$-0.596497\pi$
$31$	−3.32431	−0.597063	−0.298532	+	0.954400i	$0.596497\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.69614	−0.286700
$36$	0	0
$37$	9.83969	1.61764	0.808818	−	0.588059i	$-0.200108\pi$
$37$	9.83969	1.61764	0.808818	+	0.588059i	$0.200108\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.04090	1.25578	0.627888	−	0.778303i	$-0.283919\pi$
$41$	8.04090	1.25578	0.627888	+	0.778303i	$0.283919\pi$
$42$	0	0
$43$	7.16400	1.09250	0.546250	−	0.837622i	$-0.316055\pi$
$43$	7.16400	1.09250	0.546250	+	0.837622i	$0.316055\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.66906	1.11865	0.559324	−	0.828949i	$-0.311061\pi$
$47$	7.66906	1.11865	0.559324	+	0.828949i	$0.311061\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.81925	−1.21142	−0.605708	−	0.795687i	$-0.707110\pi$
$53$	−8.81925	−1.21142	−0.605708	+	0.795687i	$0.707110\pi$
$54$	0	0
$55$	2.24621	0.302879
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.43318	−1.22810	−0.614048	−	0.789269i	$-0.710460\pi$
$59$	−9.43318	−1.22810	−0.614048	+	0.789269i	$0.710460\pi$
$60$	0	0
$61$	4.64861	0.595194	0.297597	−	0.954692i	$-0.403815\pi$
$61$	4.64861	0.595194	0.297597	+	0.954692i	$0.403815\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.5705	−1.41356	−0.706782	−	0.707432i	$-0.749854\pi$
$67$	−11.5705	−1.41356	−0.706782	+	0.707432i	$0.749854\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.22165	0.382339	0.191170	−	0.981557i	$-0.438772\pi$
$71$	3.22165	0.382339	0.191170	+	0.981557i	$0.438772\pi$
$72$	0	0
$73$	−8.04090	−0.941116	−0.470558	−	0.882369i	$-0.655947\pi$
$73$	−8.04090	−0.941116	−0.470558	+	0.882369i	$0.655947\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.32431	−0.150919
$78$	0	0
$79$	−10.3101	−1.15997	−0.579987	−	0.814626i	$-0.696942\pi$
$79$	−10.3101	−1.15997	−0.579987	+	0.814626i	$0.696942\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.77835	0.524492	0.262246	−	0.965001i	$-0.415537\pi$
$83$	4.77835	0.524492	0.262246	+	0.965001i	$0.415537\pi$
$84$	0	0
$85$	−10.4203	−1.13024
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.6385	1.23368	0.616839	−	0.787089i	$-0.288413\pi$
$89$	11.6385	1.23368	0.616839	+	0.787089i	$0.288413\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.69614	−0.174020
$96$	0	0
$97$	−6.71659	−0.681966	−0.340983	−	0.940069i	$-0.610760\pi$
$97$	−6.71659	−0.681966	−0.340983	+	0.940069i	$0.610760\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.7779	−1.37096	−0.685478	−	0.728094i	$-0.740407\pi$
$101$	−13.7779	−1.37096	−0.685478	+	0.728094i	$0.740407\pi$
$102$	0	0
$103$	−13.1911	−1.29976	−0.649878	−	0.760039i	$-0.725180\pi$
$103$	−13.1911	−1.29976	−0.649878	+	0.760039i	$0.725180\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.46786	−0.528598	−0.264299	−	0.964441i	$-0.585141\pi$
$107$	−5.46786	−0.528598	−0.264299	+	0.964441i	$0.585141\pi$
$108$	0	0
$109$	−14.5292	−1.39165	−0.695823	−	0.718214i	$-0.744960\pi$
$109$	−14.5292	−1.39165	−0.695823	+	0.718214i	$0.744960\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.77835	0.261365	0.130683	−	0.991424i	$-0.458283\pi$
$113$	2.77835	0.261365	0.130683	+	0.991424i	$0.458283\pi$
$114$	0	0
$115$	−4.94898	−0.461495
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.14355	0.563179
$120$	0	0
$121$	−9.24621	−0.840565
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0818	−1.08063
$126$	0	0
$127$	1.16400	0.103288	0.0516442	−	0.998666i	$-0.483554\pi$
$127$	1.16400	0.103288	0.0516442	+	0.998666i	$0.483554\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.573035	0.0500663	0.0250332	−	0.999687i	$-0.492031\pi$
$131$	0.573035	0.0500663	0.0250332	+	0.999687i	$0.492031\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.83558	0.327696	0.163848	−	0.986486i	$-0.447609\pi$
$137$	3.83558	0.327696	0.163848	+	0.986486i	$0.447609\pi$
$138$	0	0
$139$	−20.8255	−1.76639	−0.883196	−	0.469004i	$-0.844613\pi$
$139$	−20.8255	−1.76639	−0.883196	+	0.469004i	$0.844613\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.42033	0.200998
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.63438	0.133894	0.0669468	−	0.997757i	$-0.478674\pi$
$149$	1.63438	0.133894	0.0669468	+	0.997757i	$0.478674\pi$
$150$	0	0
$151$	−5.08221	−0.413584	−0.206792	−	0.978385i	$-0.566302\pi$
$151$	−5.08221	−0.413584	−0.206792	+	0.978385i	$0.566302\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.63849	−0.452895
$156$	0	0
$157$	4.24621	0.338885	0.169442	−	0.985540i	$-0.445803\pi$
$157$	4.24621	0.338885	0.169442	+	0.985540i	$0.445803\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.91779	0.229954
$162$	0	0
$163$	−21.9486	−1.71914	−0.859572	−	0.511014i	$-0.829270\pi$
$163$	−21.9486	−1.71914	−0.859572	+	0.511014i	$0.829270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.20532	0.634946	0.317473	−	0.948267i	$-0.397166\pi$
$167$	8.20532	0.634946	0.317473	+	0.948267i	$0.397166\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.39228	−0.105853	−0.0529266	−	0.998598i	$-0.516855\pi$
$173$	−1.39228	−0.105853	−0.0529266	+	0.998598i	$0.516855\pi$
$174$	0	0
$175$	2.12311	0.160492
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.1064	−1.42808	−0.714038	−	0.700107i	$-0.753136\pi$
$179$	−19.1064	−1.42808	−0.714038	+	0.700107i	$0.753136\pi$
$180$	0	0
$181$	19.9895	1.48580	0.742902	−	0.669400i	$-0.233449\pi$
$181$	19.9895	1.48580	0.742902	+	0.669400i	$0.233449\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.6895	1.22704
$186$	0	0
$187$	−8.13595	−0.594960
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.70235	0.412608	0.206304	−	0.978488i	$-0.433856\pi$
$191$	5.70235	0.412608	0.206304	+	0.978488i	$0.433856\pi$
$192$	0	0
$193$	22.9766	1.65389	0.826947	−	0.562281i	$-0.190076\pi$
$193$	22.9766	1.65389	0.826947	+	0.562281i	$0.190076\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.24210	0.444731	0.222366	−	0.974963i	$-0.428622\pi$
$197$	6.24210	0.444731	0.222366	+	0.974963i	$0.428622\pi$
$198$	0	0
$199$	17.7203	1.25616	0.628079	−	0.778150i	$-0.283842\pi$
$199$	17.7203	1.25616	0.628079	+	0.778150i	$0.283842\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.42696	−0.100153
$204$	0	0
$205$	13.6385	0.952554
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.32431	−0.0916042
$210$	0	0
$211$	−14.8268	−1.02072	−0.510361	−	0.859960i	$-0.670488\pi$
$211$	−14.8268	−1.02072	−0.510361	+	0.859960i	$0.670488\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.1512	0.828702
$216$	0	0
$217$	3.32431	0.225669
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−18.9219	−1.26710	−0.633552	−	0.773700i	$-0.718404\pi$
$223$	−18.9219	−1.26710	−0.633552	+	0.773700i	$0.718404\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.3381	0.752538	0.376269	−	0.926511i	$-0.377207\pi$
$227$	11.3381	0.752538	0.376269	+	0.926511i	$0.377207\pi$
$228$	0	0
$229$	−15.9867	−1.05643	−0.528217	−	0.849110i	$-0.677139\pi$
$229$	−15.9867	−1.05643	−0.528217	+	0.849110i	$0.677139\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.87648	−0.647029	−0.323515	−	0.946223i	$-0.604865\pi$
$233$	−9.87648	−0.647029	−0.323515	+	0.946223i	$0.604865\pi$
$234$	0	0
$235$	13.0078	0.848536
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.6155	0.751346	0.375673	−	0.926752i	$-0.377412\pi$
$239$	11.6155	0.751346	0.375673	+	0.926752i	$0.377412\pi$
$240$	0	0
$241$	−14.3142	−0.922058	−0.461029	−	0.887385i	$-0.652520\pi$
$241$	−14.3142	−0.922058	−0.461029	+	0.887385i	$0.652520\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.69614	0.108362
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.0779	1.58290	0.791451	−	0.611233i	$-0.209326\pi$
$251$	25.0779	1.58290	0.791451	+	0.611233i	$0.209326\pi$
$252$	0	0
$253$	−3.86405	−0.242931
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.7304	1.79215	0.896077	−	0.443899i	$-0.146405\pi$
$257$	28.7304	1.79215	0.896077	+	0.443899i	$0.146405\pi$
$258$	0	0
$259$	−9.83969	−0.611409
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.09464	0.314149	0.157074	−	0.987587i	$-0.449794\pi$
$263$	5.09464	0.314149	0.157074	+	0.987587i	$0.449794\pi$
$264$	0	0
$265$	−14.9587	−0.918905
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.1410	0.801223	0.400612	−	0.916248i	$-0.368798\pi$
$269$	13.1410	0.801223	0.400612	+	0.916248i	$0.368798\pi$
$270$	0	0
$271$	−13.2972	−0.807749	−0.403875	−	0.914814i	$-0.632337\pi$
$271$	−13.2972	−0.807749	−0.403875	+	0.914814i	$0.632337\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.81164	−0.169548
$276$	0	0
$277$	23.2049	1.39425	0.697124	−	0.716951i	$-0.254463\pi$
$277$	23.2049	1.39425	0.697124	+	0.716951i	$0.254463\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.4036	−1.57511	−0.787553	−	0.616247i	$-0.788652\pi$
$281$	−26.4036	−1.57511	−0.787553	+	0.616247i	$0.788652\pi$
$282$	0	0
$283$	−28.7795	−1.71077	−0.855383	−	0.517997i	$-0.826678\pi$
$283$	−28.7795	−1.71077	−0.855383	+	0.517997i	$0.826678\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.04090	−0.474639
$288$	0	0
$289$	20.7433	1.22019
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.3689	1.54049	0.770244	−	0.637750i	$-0.220135\pi$
$293$	26.3689	1.54049	0.770244	+	0.637750i	$0.220135\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−7.16400	−0.412926
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.88470	0.451477
$306$	0	0
$307$	−9.73082	−0.555367	−0.277684	−	0.960673i	$-0.589567\pi$
$307$	−9.73082	−0.555367	−0.277684	+	0.960673i	$0.589567\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.4844	−1.55850	−0.779249	−	0.626715i	$-0.784399\pi$
$311$	−27.4844	−1.55850	−0.779249	+	0.626715i	$0.784399\pi$
$312$	0	0
$313$	22.9766	1.29872	0.649358	−	0.760483i	$-0.275038\pi$
$313$	22.9766	1.29872	0.649358	+	0.760483i	$0.275038\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.1983	−1.69610	−0.848052	−	0.529913i	$-0.822224\pi$
$317$	−30.1983	−1.69610	−0.848052	+	0.529913i	$0.822224\pi$
$318$	0	0
$319$	1.88974	0.105805
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.14355	0.341836
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.66906	−0.422809
$330$	0	0
$331$	−24.3266	−1.33711	−0.668556	−	0.743662i	$-0.733087\pi$
$331$	−24.3266	−1.33711	−0.668556	+	0.743662i	$0.733087\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−19.6252	−1.07224
$336$	0	0
$337$	14.3689	0.782724	0.391362	−	0.920237i	$-0.372004\pi$
$337$	14.3689	0.782724	0.391362	+	0.920237i	$0.372004\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.40240	−0.238403
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.0138	−0.967032	−0.483516	−	0.875335i	$-0.660641\pi$
$347$	−18.0138	−0.967032	−0.483516	+	0.875335i	$0.660641\pi$
$348$	0	0
$349$	−10.4924	−0.561646	−0.280823	−	0.959760i	$-0.590607\pi$
$349$	−10.4924	−0.561646	−0.280823	+	0.959760i	$0.590607\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.6562	0.673622	0.336811	−	0.941572i	$-0.390652\pi$
$353$	12.6562	0.673622	0.336811	+	0.941572i	$0.390652\pi$
$354$	0	0
$355$	5.46437	0.290019
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.3050	−1.91611	−0.958053	−	0.286590i	$-0.907478\pi$
$359$	−36.3050	−1.91611	−0.958053	+	0.286590i	$0.907478\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−13.6385	−0.713871
$366$	0	0
$367$	−23.0634	−1.20390	−0.601951	−	0.798533i	$-0.705610\pi$
$367$	−23.0634	−1.20390	−0.601951	+	0.798533i	$0.705610\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.81925	0.457872
$372$	0	0
$373$	−15.1369	−0.783760	−0.391880	−	0.920016i	$-0.628175\pi$
$373$	−15.1369	−0.783760	−0.391880	+	0.920016i	$0.628175\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−4.42948	−0.227527	−0.113764	−	0.993508i	$-0.536291\pi$
$379$	−4.42948	−0.227527	−0.113764	+	0.993508i	$0.536291\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.98674	−0.305908	−0.152954	−	0.988233i	$-0.548879\pi$
$383$	−5.98674	−0.305908	−0.152954	+	0.988233i	$0.548879\pi$
$384$	0	0
$385$	−2.24621	−0.114478
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−26.9858	−1.36823	−0.684116	−	0.729373i	$-0.739812\pi$
$389$	−26.9858	−1.36823	−0.684116	+	0.729373i	$0.739812\pi$
$390$	0	0
$391$	17.9256	0.906537
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−17.4873	−0.879883
$396$	0	0
$397$	−14.1360	−0.709463	−0.354732	−	0.934968i	$-0.615428\pi$
$397$	−14.1360	−0.709463	−0.354732	+	0.934968i	$0.615428\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.33505	−0.216482	−0.108241	−	0.994125i	$-0.534522\pi$
$401$	−4.33505	−0.216482	−0.108241	+	0.994125i	$0.534522\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13.0308	0.645912
$408$	0	0
$409$	19.5191	0.965157	0.482578	−	0.875853i	$-0.339700\pi$
$409$	19.5191	0.965157	0.482578	+	0.875853i	$0.339700\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.43318	0.464176
$414$	0	0
$415$	8.10476	0.397847
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.9552	0.535196	0.267598	−	0.963531i	$-0.413770\pi$
$419$	10.9552	0.535196	0.267598	+	0.963531i	$0.413770\pi$
$420$	0	0
$421$	−1.49843	−0.0730290	−0.0365145	−	0.999333i	$-0.511626\pi$
$421$	−1.49843	−0.0730290	−0.0365145	+	0.999333i	$0.511626\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.0434	0.632698
$426$	0	0
$427$	−4.64861	−0.224962
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.64471	0.368233	0.184116	−	0.982904i	$-0.441058\pi$
$431$	7.64471	0.368233	0.184116	+	0.982904i	$0.441058\pi$
$432$	0	0
$433$	−26.3418	−1.26591	−0.632954	−	0.774190i	$-0.718158\pi$
$433$	−26.3418	−1.26591	−0.632954	+	0.774190i	$0.718158\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.91779	0.139577
$438$	0	0
$439$	5.29584	0.252757	0.126378	−	0.991982i	$-0.459665\pi$
$439$	5.29584	0.252757	0.126378	+	0.991982i	$0.459665\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.9118	0.565946	0.282973	−	0.959128i	$-0.408679\pi$
$443$	11.9118	0.565946	0.282973	+	0.959128i	$0.408679\pi$
$444$	0	0
$445$	19.7405	0.935791
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.8091	0.934850	0.467425	−	0.884033i	$-0.345182\pi$
$449$	19.8091	0.934850	0.467425	+	0.884033i	$0.345182\pi$
$450$	0	0
$451$	10.6486	0.501424
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.9077	0.837685	0.418843	−	0.908059i	$-0.362436\pi$
$457$	17.9077	0.837685	0.418843	+	0.908059i	$0.362436\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.49083	0.0694347	0.0347173	−	0.999397i	$-0.488947\pi$
$461$	1.49083	0.0694347	0.0347173	+	0.999397i	$0.488947\pi$
$462$	0	0
$463$	−18.7028	−0.869192	−0.434596	−	0.900626i	$-0.643109\pi$
$463$	−18.7028	−0.869192	−0.434596	+	0.900626i	$0.643109\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.1574	−1.02532	−0.512660	−	0.858591i	$-0.671340\pi$
$467$	−22.1574	−1.02532	−0.512660	+	0.858591i	$0.671340\pi$
$468$	0	0
$469$	11.5705	0.534277
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.48734	0.436228
$474$	0	0
$475$	2.12311	0.0974148
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.9387	−0.773947	−0.386973	−	0.922091i	$-0.626480\pi$
$479$	−16.9387	−0.773947	−0.386973	+	0.922091i	$0.626480\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.3923	−0.517297
$486$	0	0
$487$	10.4460	0.473354	0.236677	−	0.971588i	$-0.423942\pi$
$487$	10.4460	0.473354	0.236677	+	0.971588i	$0.423942\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.66557	0.255684	0.127842	−	0.991795i	$-0.459195\pi$
$491$	5.66557	0.255684	0.127842	+	0.991795i	$0.459195\pi$
$492$	0	0
$493$	−8.76663	−0.394829
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.22165	−0.144511
$498$	0	0
$499$	29.8438	1.33599	0.667996	−	0.744165i	$-0.267152\pi$
$499$	29.8438	1.33599	0.667996	+	0.744165i	$0.267152\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.02868	−0.402569	−0.201284	−	0.979533i	$-0.564512\pi$
$503$	−9.02868	−0.402569	−0.201284	+	0.979533i	$0.564512\pi$
$504$	0	0
$505$	−23.3693	−1.03992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−23.9256	−1.06048	−0.530242	−	0.847846i	$-0.677899\pi$
$509$	−23.9256	−1.06048	−0.530242	+	0.847846i	$0.677899\pi$
$510$	0	0
$511$	8.04090	0.355708
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−22.3739	−0.985913
$516$	0	0
$517$	10.1562	0.446669
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.5742	−1.16424	−0.582119	−	0.813104i	$-0.697776\pi$
$521$	−26.5742	−1.16424	−0.582119	+	0.813104i	$0.697776\pi$
$522$	0	0
$523$	−9.73082	−0.425499	−0.212750	−	0.977107i	$-0.568242\pi$
$523$	−9.73082	−0.425499	−0.212750	+	0.977107i	$0.568242\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.4231	0.889642
$528$	0	0
$529$	−14.4865	−0.629848
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−9.27426	−0.400961
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.32431	0.0570419
$540$	0	0
$541$	−2.49242	−0.107158	−0.0535788	−	0.998564i	$-0.517063\pi$
$541$	−2.49242	−0.107158	−0.0535788	+	0.998564i	$0.517063\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−24.6436	−1.05561
$546$	0	0
$547$	−5.12722	−0.219224	−0.109612	−	0.993974i	$-0.534961\pi$
$547$	−5.12722	−0.219224	−0.109612	+	0.993974i	$0.534961\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.42696	−0.0607907
$552$	0	0
$553$	10.3101	0.438429
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.96238	0.337377	0.168688	−	0.985669i	$-0.446047\pi$
$557$	7.96238	0.337377	0.168688	+	0.985669i	$0.446047\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18.9899	−0.800328	−0.400164	−	0.916444i	$-0.631047\pi$
$563$	−18.9899	−0.800328	−0.400164	+	0.916444i	$0.631047\pi$
$564$	0	0
$565$	4.71247	0.198255
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.71407	0.407235	0.203618	−	0.979051i	$-0.434730\pi$
$569$	9.71407	0.407235	0.203618	+	0.979051i	$0.434730\pi$
$570$	0	0
$571$	17.7079	0.741051	0.370525	−	0.928822i	$-0.379178\pi$
$571$	17.7079	0.741051	0.370525	+	0.928822i	$0.379178\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.19478	0.258340
$576$	0	0
$577$	−38.1636	−1.58877	−0.794385	−	0.607414i	$-0.792207\pi$
$577$	−38.1636	−1.58877	−0.794385	+	0.607414i	$0.792207\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.77835	−0.198239
$582$	0	0
$583$	−11.6794	−0.483711
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.9186	1.73016	0.865082	−	0.501631i	$-0.167266\pi$
$587$	41.9186	1.73016	0.865082	+	0.501631i	$0.167266\pi$
$588$	0	0
$589$	3.32431	0.136976
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.0200	0.904254	0.452127	−	0.891954i	$-0.350665\pi$
$593$	22.0200	0.904254	0.452127	+	0.891954i	$0.350665\pi$
$594$	0	0
$595$	10.4203	0.427192
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.0880	0.575620	0.287810	−	0.957687i	$-0.407073\pi$
$599$	14.0880	0.575620	0.287810	+	0.957687i	$0.407073\pi$
$600$	0	0
$601$	−5.01696	−0.204646	−0.102323	−	0.994751i	$-0.532628\pi$
$601$	−5.01696	−0.204646	−0.102323	+	0.994751i	$0.532628\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−15.6829	−0.637600
$606$	0	0
$607$	−35.9215	−1.45801	−0.729004	−	0.684509i	$-0.760016\pi$
$607$	−35.9215	−1.45801	−0.729004	+	0.684509i	$0.760016\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	29.5053	1.19171	0.595853	−	0.803093i	$-0.296814\pi$
$613$	29.5053	1.19171	0.595853	+	0.803093i	$0.296814\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.3822	0.498487	0.249244	−	0.968441i	$-0.419818\pi$
$617$	12.3822	0.498487	0.249244	+	0.968441i	$0.419818\pi$
$618$	0	0
$619$	26.5333	1.06646	0.533232	−	0.845969i	$-0.320977\pi$
$619$	26.5333	1.06646	0.533232	+	0.845969i	$0.320977\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.6385	−0.466286
$624$	0	0
$625$	−9.87689	−0.395076
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−60.4507	−2.41033
$630$	0	0
$631$	20.0588	0.798529	0.399265	−	0.916836i	$-0.369265\pi$
$631$	20.0588	0.798529	0.399265	+	0.916836i	$0.369265\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.97431	0.0783481
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−43.0237	−1.69934	−0.849668	−	0.527319i	$-0.823197\pi$
$641$	−43.0237	−1.69934	−0.849668	+	0.527319i	$0.823197\pi$
$642$	0	0
$643$	31.9665	1.26064	0.630318	−	0.776337i	$-0.282925\pi$
$643$	31.9665	1.26064	0.630318	+	0.776337i	$0.282925\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.3820	−0.604727	−0.302364	−	0.953193i	$-0.597776\pi$
$647$	−15.3820	−0.604727	−0.302364	+	0.953193i	$0.597776\pi$
$648$	0	0
$649$	−12.4924	−0.490370
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.3013	0.677054	0.338527	−	0.940957i	$-0.390071\pi$
$653$	17.3013	0.677054	0.338527	+	0.940957i	$0.390071\pi$
$654$	0	0
$655$	0.971949	0.0379772
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−46.0421	−1.79354	−0.896772	−	0.442492i	$-0.854094\pi$
$659$	−46.0421	−1.79354	−0.896772	+	0.442492i	$0.854094\pi$
$660$	0	0
$661$	5.69963	0.221690	0.110845	−	0.993838i	$-0.464644\pi$
$661$	5.69963	0.221690	0.110845	+	0.993838i	$0.464644\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.69614	0.0657735
$666$	0	0
$667$	−4.16358	−0.161215
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.15619	0.237657
$672$	0	0
$673$	−18.2095	−0.701925	−0.350963	−	0.936389i	$-0.614146\pi$
$673$	−18.2095	−0.701925	−0.350963	+	0.936389i	$0.614146\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.1126	1.65695	0.828475	−	0.560026i	$-0.189209\pi$
$677$	43.1126	1.65695	0.828475	+	0.560026i	$0.189209\pi$
$678$	0	0
$679$	6.71659	0.257759
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.60381	0.0613681	0.0306841	−	0.999529i	$-0.490231\pi$
$683$	1.60381	0.0613681	0.0306841	+	0.999529i	$0.490231\pi$
$684$	0	0
$685$	6.50569	0.248569
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−18.3739	−0.698977	−0.349489	−	0.936941i	$-0.613645\pi$
$691$	−18.3739	−0.698977	−0.349489	+	0.936941i	$0.613645\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−35.3229	−1.33987
$696$	0	0
$697$	−49.3997	−1.87115
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.5958	1.15559	0.577794	−	0.816183i	$-0.303914\pi$
$701$	30.5958	1.15559	0.577794	+	0.816183i	$0.303914\pi$
$702$	0	0
$703$	−9.83969	−0.371111
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.7779	0.518172
$708$	0	0
$709$	25.2334	0.947659	0.473829	−	0.880617i	$-0.342871\pi$
$709$	25.2334	0.947659	0.473829	+	0.880617i	$0.342871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.69963	0.363254
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.8459	−0.814715	−0.407357	−	0.913269i	$-0.633550\pi$
$719$	−21.8459	−0.814715	−0.407357	+	0.913269i	$0.633550\pi$
$720$	0	0
$721$	13.1911	0.491262
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.02960	−0.112516
$726$	0	0
$727$	39.8889	1.47940	0.739699	−	0.672938i	$-0.234968\pi$
$727$	39.8889	1.47940	0.739699	+	0.672938i	$0.234968\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−44.0124	−1.62786
$732$	0	0
$733$	−34.7713	−1.28431	−0.642154	−	0.766576i	$-0.721959\pi$
$733$	−34.7713	−1.28431	−0.642154	+	0.766576i	$0.721959\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15.3229	−0.564427
$738$	0	0
$739$	−3.02805	−0.111389	−0.0556943	−	0.998448i	$-0.517737\pi$
$739$	−3.02805	−0.111389	−0.0556943	+	0.998448i	$0.517737\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25.0737	−0.919864	−0.459932	−	0.887954i	$-0.652126\pi$
$743$	−25.0737	−0.919864	−0.459932	+	0.887954i	$0.652126\pi$
$744$	0	0
$745$	2.77214	0.101563
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.46786	0.199791
$750$	0	0
$751$	12.7616	0.465677	0.232839	−	0.972515i	$-0.425199\pi$
$751$	12.7616	0.465677	0.232839	+	0.972515i	$0.425199\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.62014	−0.313719
$756$	0	0
$757$	16.3198	0.593152	0.296576	−	0.955009i	$-0.404155\pi$
$757$	16.3198	0.593152	0.296576	+	0.955009i	$0.404155\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	24.6492	0.893534	0.446767	−	0.894650i	$-0.352575\pi$
$761$	24.6492	0.893534	0.446767	+	0.894650i	$0.352575\pi$
$762$	0	0
$763$	14.5292	0.525993
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	12.1103	0.436707	0.218354	−	0.975870i	$-0.429931\pi$
$769$	12.1103	0.436707	0.218354	+	0.975870i	$0.429931\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35.4456	1.27489	0.637445	−	0.770496i	$-0.279991\pi$
$773$	35.4456	1.27489	0.637445	+	0.770496i	$0.279991\pi$
$774$	0	0
$775$	7.05785	0.253526
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.04090	−0.288095
$780$	0	0
$781$	4.26645	0.152666
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.20217	0.257057
$786$	0	0
$787$	24.5890	0.876501	0.438251	−	0.898853i	$-0.355598\pi$
$787$	24.5890	0.876501	0.438251	+	0.898853i	$0.355598\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.77835	−0.0987868
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.85393	0.313622	0.156811	−	0.987629i	$-0.449879\pi$
$797$	8.85393	0.313622	0.156811	+	0.987629i	$0.449879\pi$
$798$	0	0
$799$	−47.1153	−1.66682
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.6486	−0.375781
$804$	0	0
$805$	4.94898	0.174429
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.05924	0.177874	0.0889368	−	0.996037i	$-0.471653\pi$
$809$	5.05924	0.177874	0.0889368	+	0.996037i	$0.471653\pi$
$810$	0	0
$811$	24.2095	0.850111	0.425056	−	0.905167i	$-0.360255\pi$
$811$	24.2095	0.850111	0.425056	+	0.905167i	$0.360255\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−37.2279	−1.30404
$816$	0	0
$817$	−7.16400	−0.250637
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32.3372	−1.12857	−0.564287	−	0.825579i	$-0.690849\pi$
$821$	−32.3372	−1.12857	−0.564287	+	0.825579i	$0.690849\pi$
$822$	0	0
$823$	43.2734	1.50842	0.754208	−	0.656635i	$-0.228021\pi$
$823$	43.2734	1.50842	0.754208	+	0.656635i	$0.228021\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.975438	0.0339193	0.0169597	−	0.999856i	$-0.494601\pi$
$827$	0.975438	0.0339193	0.0169597	+	0.999856i	$0.494601\pi$
$828$	0	0
$829$	−20.0436	−0.696144	−0.348072	−	0.937468i	$-0.613163\pi$
$829$	−20.0436	−0.696144	−0.348072	+	0.937468i	$0.613163\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.14355	−0.212862
$834$	0	0
$835$	13.9174	0.481631
$836$	0	0
$837$	0	0
$838$	0	0
$839$	47.4947	1.63970	0.819850	−	0.572578i	$-0.194057\pi$
$839$	47.4947	1.63970	0.819850	+	0.572578i	$0.194057\pi$
$840$	0	0
$841$	−26.9638	−0.929785
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22.0498	−0.758537
$846$	0	0
$847$	9.24621	0.317704
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−28.7102	−0.984172
$852$	0	0
$853$	4.64861	0.159166	0.0795828	−	0.996828i	$-0.474641\pi$
$853$	4.64861	0.159166	0.0795828	+	0.996828i	$0.474641\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.3974	0.696761	0.348380	−	0.937353i	$-0.386732\pi$
$857$	20.3974	0.696761	0.348380	+	0.937353i	$0.386732\pi$
$858$	0	0
$859$	5.07949	0.173310	0.0866549	−	0.996238i	$-0.472382\pi$
$859$	5.07949	0.173310	0.0866549	+	0.996238i	$0.472382\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.5119	−0.425910	−0.212955	−	0.977062i	$-0.568309\pi$
$863$	−12.5119	−0.425910	−0.212955	+	0.977062i	$0.568309\pi$
$864$	0	0
$865$	−2.36151	−0.0802936
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.6537	−0.463170
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.0818	0.408439
$876$	0	0
$877$	52.6601	1.77821	0.889103	−	0.457707i	$-0.151329\pi$
$877$	52.6601	1.77821	0.889103	+	0.457707i	$0.151329\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.5892	−0.794739	−0.397369	−	0.917659i	$-0.630077\pi$
$881$	−23.5892	−0.794739	−0.397369	+	0.917659i	$0.630077\pi$
$882$	0	0
$883$	−41.9256	−1.41091	−0.705454	−	0.708755i	$-0.749257\pi$
$883$	−41.9256	−1.41091	−0.705454	+	0.708755i	$0.749257\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.31280	−0.0776561	−0.0388281	−	0.999246i	$-0.512362\pi$
$887$	−2.31280	−0.0776561	−0.0388281	+	0.999246i	$0.512362\pi$
$888$	0	0
$889$	−1.16400	−0.0390394
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7.66906	−0.256635
$894$	0	0
$895$	−32.4071	−1.08325
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.74367	−0.158210
$900$	0	0
$901$	54.1815	1.80505
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33.9049	1.12704
$906$	0	0
$907$	−14.4980	−0.481399	−0.240699	−	0.970600i	$-0.577377\pi$
$907$	−14.4980	−0.481399	−0.240699	+	0.970600i	$0.577377\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.1320	0.832662	0.416331	−	0.909213i	$-0.363316\pi$
$911$	25.1320	0.832662	0.416331	+	0.909213i	$0.363316\pi$
$912$	0	0
$913$	6.32800	0.209426
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.573035	−0.0189233
$918$	0	0
$919$	7.17874	0.236805	0.118402	−	0.992966i	$-0.462223\pi$
$919$	7.17874	0.236805	0.118402	+	0.992966i	$0.462223\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−20.8907	−0.686882
$926$	0	0
$927$	0	0
$928$	0	0
$929$	45.1334	1.48078	0.740390	−	0.672178i	$-0.234641\pi$
$929$	45.1334	1.48078	0.740390	+	0.672178i	$0.234641\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13.7997	−0.451299
$936$	0	0
$937$	52.4916	1.71483	0.857413	−	0.514629i	$-0.172071\pi$
$937$	52.4916	1.71483	0.857413	+	0.514629i	$0.172071\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	35.3055	1.15092	0.575462	−	0.817828i	$-0.304822\pi$
$941$	35.3055	1.15092	0.575462	+	0.817828i	$0.304822\pi$
$942$	0	0
$943$	−23.4616	−0.764016
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.7800	0.772747	0.386374	−	0.922342i	$-0.373728\pi$
$947$	23.7800	0.772747	0.386374	+	0.922342i	$0.373728\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.6957	0.540828	0.270414	−	0.962744i	$-0.412840\pi$
$953$	16.6957	0.540828	0.270414	+	0.962744i	$0.412840\pi$
$954$	0	0
$955$	9.67200	0.312978
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.83558	−0.123857
$960$	0	0
$961$	−19.9490	−0.643516
$962$	0	0
$963$	0	0
$964$	0	0
$965$	38.9716	1.25454
$966$	0	0
$967$	−47.7300	−1.53489	−0.767446	−	0.641113i	$-0.778473\pi$
$967$	−47.7300	−1.53489	−0.767446	+	0.641113i	$0.778473\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.6486	−0.598462	−0.299231	−	0.954181i	$-0.596730\pi$
$971$	−18.6486	−0.598462	−0.299231	+	0.954181i	$0.596730\pi$
$972$	0	0
$973$	20.8255	0.667634
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.3728	0.811748	0.405874	−	0.913929i	$-0.366967\pi$
$977$	25.3728	0.811748	0.405874	+	0.913929i	$0.366967\pi$
$978$	0	0
$979$	15.4129	0.492600
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−44.3841	−1.41563	−0.707817	−	0.706396i	$-0.750320\pi$
$983$	−44.3841	−1.41563	−0.707817	+	0.706396i	$0.750320\pi$
$984$	0	0
$985$	10.5875	0.337345
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.9031	−0.664678
$990$	0	0
$991$	−34.7483	−1.10382	−0.551909	−	0.833905i	$-0.686100\pi$
$991$	−34.7483	−1.10382	−0.551909	+	0.833905i	$0.686100\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	30.0561	0.952843
$996$	0	0
$997$	−1.32292	−0.0418972	−0.0209486	−	0.999781i	$-0.506669\pi$
$997$	−1.32292	−0.0418972	−0.0209486	+	0.999781i	$0.506669\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.cj.1.3		4
3.2	odd	2		3192.2.a.x.1.2	✓	4
12.11	even	2		6384.2.a.cb.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.x.1.2	✓	4	3.2	odd	2
6384.2.a.cb.1.2		4	12.11	even	2
9576.2.a.cj.1.3		4	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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+1.69614 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.69614 q^{5} -1.00000 q^{7} +1.32431 q^{11} -6.14355 q^{17} -1.00000 q^{19} -2.91779 q^{23} -2.12311 q^{25} +1.42696 q^{29} -3.32431 q^{31} -1.69614 q^{35} +9.83969 q^{37} +8.04090 q^{41} +7.16400 q^{43} +7.66906 q^{47} +1.00000 q^{49} -8.81925 q^{53} +2.24621 q^{55} -9.43318 q^{59} +4.64861 q^{61} -11.5705 q^{67} +3.22165 q^{71} -8.04090 q^{73} -1.32431 q^{77} -10.3101 q^{79} +4.77835 q^{83} -10.4203 q^{85} +11.6385 q^{89} -1.69614 q^{95} -6.71659 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} + 4 q^{17} - 4 q^{19} - 4 q^{23} + 8 q^{25} - 4 q^{29} - 8 q^{31} + 4 q^{37} + 8 q^{41} - 12 q^{43} + 8 q^{47} + 4 q^{49} - 12 q^{53} - 24 q^{55} + 8 q^{61} - 8 q^{67} + 12 q^{71} - 8 q^{73} - 20 q^{79} + 20 q^{83} - 4 q^{85} - 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 4 * q^17 - 4 * q^19 - 4 * q^23 + 8 * q^25 - 4 * q^29 - 8 * q^31 + 4 * q^37 + 8 * q^41 - 12 * q^43 + 8 * q^47 + 4 * q^49 - 12 * q^53 - 24 * q^55 + 8 * q^61 - 8 * q^67 + 12 * q^71 - 8 * q^73 - 20 * q^79 + 20 * q^83 - 4 * q^85 - 8 * q^97

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