LMFDB - Embedded newform 9522.2.a.ba.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,1,0,-3,-2,0,-1,-3,0,3,3,0,2,4,0,8,1,0,3,0,0,11,-3,0, -3,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3174)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.70156 q^{5} +1.70156 q^{7} -1.00000 q^{8} +2.70156 q^{10} +1.70156 q^{11} +4.70156 q^{13} -1.70156 q^{14} +1.00000 q^{16} +2.00000 q^{17} +4.00000 q^{19} -2.70156 q^{20} -1.70156 q^{22} +2.29844 q^{25} -4.70156 q^{26} +1.70156 q^{28} -6.40312 q^{29} -5.70156 q^{31} -1.00000 q^{32} -2.00000 q^{34} -4.59688 q^{35} -9.40312 q^{37} -4.00000 q^{38} +2.70156 q^{40} +10.1047 q^{41} -11.4031 q^{43} +1.70156 q^{44} -4.00000 q^{47} -4.10469 q^{49} -2.29844 q^{50} +4.70156 q^{52} +6.40312 q^{53} -4.59688 q^{55} -1.70156 q^{56} +6.40312 q^{58} -13.7016 q^{59} +7.29844 q^{61} +5.70156 q^{62} +1.00000 q^{64} -12.7016 q^{65} -12.0000 q^{67} +2.00000 q^{68} +4.59688 q^{70} -15.4031 q^{71} +6.40312 q^{73} +9.40312 q^{74} +4.00000 q^{76} +2.89531 q^{77} +10.2984 q^{79} -2.70156 q^{80} -10.1047 q^{82} +6.29844 q^{83} -5.40312 q^{85} +11.4031 q^{86} -1.70156 q^{88} -10.7016 q^{89} +8.00000 q^{91} +4.00000 q^{94} -10.8062 q^{95} -17.8062 q^{97} +4.10469 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 3 q^{7} - 2 q^{8} - q^{10} - 3 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} + 4 q^{17} + 8 q^{19} + q^{20} + 3 q^{22} + 11 q^{25} - 3 q^{26} - 3 q^{28} - 5 q^{31} - 2 q^{32}+ \cdots - 11 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^5 - 3 * q^7 - 2 * q^8 - q^10 - 3 * q^11 + 3 * q^13 + 3 * q^14 + 2 * q^16 + 4 * q^17 + 8 * q^19 + q^20 + 3 * q^22 + 11 * q^25 - 3 * q^26 - 3 * q^28 - 5 * q^31 - 2 * q^32 - 4 * q^34 - 22 * q^35 - 6 * q^37 - 8 * q^38 - q^40 + q^41 - 10 * q^43 - 3 * q^44 - 8 * q^47 + 11 * q^49 - 11 * q^50 + 3 * q^52 - 22 * q^55 + 3 * q^56 - 21 * q^59 + 21 * q^61 + 5 * q^62 + 2 * q^64 - 19 * q^65 - 24 * q^67 + 4 * q^68 + 22 * q^70 - 18 * q^71 + 6 * q^74 + 8 * q^76 + 25 * q^77 + 27 * q^79 + q^80 - q^82 + 19 * q^83 + 2 * q^85 + 10 * q^86 + 3 * q^88 - 15 * q^89 + 16 * q^91 + 8 * q^94 + 4 * q^95 - 10 * q^97 - 11 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.70156	−1.20818	−0.604088	−	0.796918i	$-0.706462\pi$
$5$	−2.70156	−1.20818	−0.604088	+	0.796918i	$0.706462\pi$
$6$	0	0
$7$	1.70156	0.643130	0.321565	−	0.946888i	$-0.395791\pi$
$7$	1.70156	0.643130	0.321565	+	0.946888i	$0.395791\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.70156	0.854309
$11$	1.70156	0.513040	0.256520	−	0.966539i	$-0.417424\pi$
$11$	1.70156	0.513040	0.256520	+	0.966539i	$0.417424\pi$
$12$	0	0
$13$	4.70156	1.30398	0.651989	−	0.758228i	$-0.273935\pi$
$13$	4.70156	1.30398	0.651989	+	0.758228i	$0.273935\pi$
$14$	−1.70156	−0.454762
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.70156	−0.604088
$21$	0	0
$22$	−1.70156	−0.362774
$23$	0	0
$23$	0	0
$24$	0	0
$25$	2.29844	0.459688
$26$	−4.70156	−0.922052
$27$	0	0
$28$	1.70156	0.321565
$29$	−6.40312	−1.18903	−0.594515	−	0.804084i	$-0.702656\pi$
$29$	−6.40312	−1.18903	−0.594515	+	0.804084i	$0.702656\pi$
$30$	0	0
$31$	−5.70156	−1.02403	−0.512015	−	0.858976i	$-0.671101\pi$
$31$	−5.70156	−1.02403	−0.512015	+	0.858976i	$0.671101\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−4.59688	−0.777014
$36$	0	0
$37$	−9.40312	−1.54586	−0.772932	−	0.634489i	$-0.781211\pi$
$37$	−9.40312	−1.54586	−0.772932	+	0.634489i	$0.781211\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	2.70156	0.427154
$41$	10.1047	1.57809	0.789043	−	0.614337i	$-0.210577\pi$
$41$	10.1047	1.57809	0.789043	+	0.614337i	$0.210577\pi$
$42$	0	0
$43$	−11.4031	−1.73896	−0.869480	−	0.493968i	$-0.835546\pi$
$43$	−11.4031	−1.73896	−0.869480	+	0.493968i	$0.835546\pi$
$44$	1.70156	0.256520
$45$	0	0
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−4.10469	−0.586384
$50$	−2.29844	−0.325048
$51$	0	0
$52$	4.70156	0.651989
$53$	6.40312	0.879537	0.439768	−	0.898111i	$-0.355061\pi$
$53$	6.40312	0.879537	0.439768	+	0.898111i	$0.355061\pi$
$54$	0	0
$55$	−4.59688	−0.619843
$56$	−1.70156	−0.227381
$57$	0	0
$58$	6.40312	0.840771
$59$	−13.7016	−1.78379	−0.891896	−	0.452241i	$-0.850625\pi$
$59$	−13.7016	−1.78379	−0.891896	+	0.452241i	$0.850625\pi$
$60$	0	0
$61$	7.29844	0.934469	0.467235	−	0.884133i	$-0.345250\pi$
$61$	7.29844	0.934469	0.467235	+	0.884133i	$0.345250\pi$
$62$	5.70156	0.724099
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.7016	−1.57543
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	4.59688	0.549432
$71$	−15.4031	−1.82801	−0.914007	−	0.405698i	$-0.867029\pi$
$71$	−15.4031	−1.82801	−0.914007	+	0.405698i	$0.867029\pi$
$72$	0	0
$73$	6.40312	0.749429	0.374715	−	0.927140i	$-0.377741\pi$
$73$	6.40312	0.749429	0.374715	+	0.927140i	$0.377741\pi$
$74$	9.40312	1.09309
$75$	0	0
$76$	4.00000	0.458831
$77$	2.89531	0.329952
$78$	0	0
$79$	10.2984	1.15866	0.579332	−	0.815091i	$-0.303313\pi$
$79$	10.2984	1.15866	0.579332	+	0.815091i	$0.303313\pi$
$80$	−2.70156	−0.302044
$81$	0	0
$82$	−10.1047	−1.11588
$83$	6.29844	0.691343	0.345672	−	0.938356i	$-0.387651\pi$
$83$	6.29844	0.691343	0.345672	+	0.938356i	$0.387651\pi$
$84$	0	0
$85$	−5.40312	−0.586051
$86$	11.4031	1.22963
$87$	0	0
$88$	−1.70156	−0.181387
$89$	−10.7016	−1.13436	−0.567182	−	0.823593i	$-0.691966\pi$
$89$	−10.7016	−1.13436	−0.567182	+	0.823593i	$0.691966\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	0	0
$94$	4.00000	0.412568
$95$	−10.8062	−1.10870
$96$	0	0
$97$	−17.8062	−1.80795	−0.903975	−	0.427585i	$-0.859365\pi$
$97$	−17.8062	−1.80795	−0.903975	+	0.427585i	$0.859365\pi$
$98$	4.10469	0.414636
$99$	0	0
$100$	2.29844	0.229844
$101$	12.4031	1.23416	0.617078	−	0.786902i	$-0.288316\pi$
$101$	12.4031	1.23416	0.617078	+	0.786902i	$0.288316\pi$
$102$	0	0
$103$	5.10469	0.502980	0.251490	−	0.967860i	$-0.419079\pi$
$103$	5.10469	0.502980	0.251490	+	0.967860i	$0.419079\pi$
$104$	−4.70156	−0.461026
$105$	0	0
$106$	−6.40312	−0.621926
$107$	6.80625	0.657985	0.328992	−	0.944333i	$-0.393291\pi$
$107$	6.80625	0.657985	0.328992	+	0.944333i	$0.393291\pi$
$108$	0	0
$109$	3.29844	0.315933	0.157967	−	0.987444i	$-0.449506\pi$
$109$	3.29844	0.315933	0.157967	+	0.987444i	$0.449506\pi$
$110$	4.59688	0.438295
$111$	0	0
$112$	1.70156	0.160783
$113$	12.7016	1.19486	0.597431	−	0.801920i	$-0.296188\pi$
$113$	12.7016	1.19486	0.597431	+	0.801920i	$0.296188\pi$
$114$	0	0
$115$	0	0
$116$	−6.40312	−0.594515
$117$	0	0
$118$	13.7016	1.26133
$119$	3.40312	0.311964
$120$	0	0
$121$	−8.10469	−0.736790
$122$	−7.29844	−0.660770
$123$	0	0
$124$	−5.70156	−0.512015
$125$	7.29844	0.652792
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	12.7016	1.11400
$131$	−5.70156	−0.498148	−0.249074	−	0.968484i	$-0.580126\pi$
$131$	−5.70156	−0.498148	−0.249074	+	0.968484i	$0.580126\pi$
$132$	0	0
$133$	6.80625	0.590177
$134$	12.0000	1.03664
$135$	0	0
$136$	−2.00000	−0.171499
$137$	4.70156	0.401682	0.200841	−	0.979624i	$-0.435633\pi$
$137$	4.70156	0.401682	0.200841	+	0.979624i	$0.435633\pi$
$138$	0	0
$139$	10.8062	0.916574	0.458287	−	0.888804i	$-0.348463\pi$
$139$	10.8062	0.916574	0.458287	+	0.888804i	$0.348463\pi$
$140$	−4.59688	−0.388507
$141$	0	0
$142$	15.4031	1.29260
$143$	8.00000	0.668994
$144$	0	0
$145$	17.2984	1.43656
$146$	−6.40312	−0.529926
$147$	0	0
$148$	−9.40312	−0.772932
$149$	0.104686	0.00857624	0.00428812	−	0.999991i	$-0.498635\pi$
$149$	0.104686	0.00857624	0.00428812	+	0.999991i	$0.498635\pi$
$150$	0	0
$151$	−5.70156	−0.463987	−0.231993	−	0.972717i	$-0.574525\pi$
$151$	−5.70156	−0.463987	−0.231993	+	0.972717i	$0.574525\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	−2.89531	−0.233311
$155$	15.4031	1.23721
$156$	0	0
$157$	7.29844	0.582479	0.291239	−	0.956650i	$-0.405932\pi$
$157$	7.29844	0.582479	0.291239	+	0.956650i	$0.405932\pi$
$158$	−10.2984	−0.819300
$159$	0	0
$160$	2.70156	0.213577
$161$	0	0
$162$	0	0
$163$	11.4031	0.893162	0.446581	−	0.894743i	$-0.352642\pi$
$163$	11.4031	0.893162	0.446581	+	0.894743i	$0.352642\pi$
$164$	10.1047	0.789043
$165$	0	0
$166$	−6.29844	−0.488854
$167$	10.8062	0.836213	0.418106	−	0.908398i	$-0.362694\pi$
$167$	10.8062	0.836213	0.418106	+	0.908398i	$0.362694\pi$
$168$	0	0
$169$	9.10469	0.700360
$170$	5.40312	0.414401
$171$	0	0
$172$	−11.4031	−0.869480
$173$	−16.1047	−1.22442	−0.612208	−	0.790697i	$-0.709719\pi$
$173$	−16.1047	−1.22442	−0.612208	+	0.790697i	$0.709719\pi$
$174$	0	0
$175$	3.91093	0.295639
$176$	1.70156	0.128260
$177$	0	0
$178$	10.7016	0.802116
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−2.59688	−0.193024	−0.0965121	−	0.995332i	$-0.530769\pi$
$181$	−2.59688	−0.193024	−0.0965121	+	0.995332i	$0.530769\pi$
$182$	−8.00000	−0.592999
$183$	0	0
$184$	0	0
$185$	25.4031	1.86767
$186$	0	0
$187$	3.40312	0.248861
$188$	−4.00000	−0.291730
$189$	0	0
$190$	10.8062	0.783968
$191$	−7.40312	−0.535671	−0.267836	−	0.963465i	$-0.586308\pi$
$191$	−7.40312	−0.535671	−0.267836	+	0.963465i	$0.586308\pi$
$192$	0	0
$193$	−24.4031	−1.75658	−0.878288	−	0.478133i	$-0.841314\pi$
$193$	−24.4031	−1.75658	−0.878288	+	0.478133i	$0.841314\pi$
$194$	17.8062	1.27841
$195$	0	0
$196$	−4.10469	−0.293192
$197$	−18.4031	−1.31117	−0.655584	−	0.755122i	$-0.727578\pi$
$197$	−18.4031	−1.31117	−0.655584	+	0.755122i	$0.727578\pi$
$198$	0	0
$199$	13.1047	0.928967	0.464483	−	0.885582i	$-0.346240\pi$
$199$	13.1047	0.928967	0.464483	+	0.885582i	$0.346240\pi$
$200$	−2.29844	−0.162524
$201$	0	0
$202$	−12.4031	−0.872681
$203$	−10.8953	−0.764701
$204$	0	0
$205$	−27.2984	−1.90661
$206$	−5.10469	−0.355660
$207$	0	0
$208$	4.70156	0.325995
$209$	6.80625	0.470798
$210$	0	0
$211$	−14.2094	−0.978214	−0.489107	−	0.872224i	$-0.662677\pi$
$211$	−14.2094	−0.978214	−0.489107	+	0.872224i	$0.662677\pi$
$212$	6.40312	0.439768
$213$	0	0
$214$	−6.80625	−0.465266
$215$	30.8062	2.10097
$216$	0	0
$217$	−9.70156	−0.658585
$218$	−3.29844	−0.223398
$219$	0	0
$220$	−4.59688	−0.309921
$221$	9.40312	0.632523
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−1.70156	−0.113690
$225$	0	0
$226$	−12.7016	−0.844895
$227$	−14.2984	−0.949021	−0.474510	−	0.880250i	$-0.657375\pi$
$227$	−14.2984	−0.949021	−0.474510	+	0.880250i	$0.657375\pi$
$228$	0	0
$229$	20.2094	1.33547	0.667736	−	0.744398i	$-0.267263\pi$
$229$	20.2094	1.33547	0.667736	+	0.744398i	$0.267263\pi$
$230$	0	0
$231$	0	0
$232$	6.40312	0.420386
$233$	9.50781	0.622877	0.311439	−	0.950266i	$-0.399189\pi$
$233$	9.50781	0.622877	0.311439	+	0.950266i	$0.399189\pi$
$234$	0	0
$235$	10.8062	0.704922
$236$	−13.7016	−0.891896
$237$	0	0
$238$	−3.40312	−0.220592
$239$	−10.8062	−0.698998	−0.349499	−	0.936937i	$-0.613648\pi$
$239$	−10.8062	−0.698998	−0.349499	+	0.936937i	$0.613648\pi$
$240$	0	0
$241$	6.70156	0.431686	0.215843	−	0.976428i	$-0.430750\pi$
$241$	6.70156	0.431686	0.215843	+	0.976428i	$0.430750\pi$
$242$	8.10469	0.520989
$243$	0	0
$244$	7.29844	0.467235
$245$	11.0891	0.708454
$246$	0	0
$247$	18.8062	1.19661
$248$	5.70156	0.362050
$249$	0	0
$250$	−7.29844	−0.461594
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.29844	−0.0809943	−0.0404972	−	0.999180i	$-0.512894\pi$
$257$	−1.29844	−0.0809943	−0.0404972	+	0.999180i	$0.512894\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	−12.7016	−0.787717
$261$	0	0
$262$	5.70156	0.352244
$263$	−14.8062	−0.912992	−0.456496	−	0.889725i	$-0.650896\pi$
$263$	−14.8062	−0.912992	−0.456496	+	0.889725i	$0.650896\pi$
$264$	0	0
$265$	−17.2984	−1.06263
$266$	−6.80625	−0.417318
$267$	0	0
$268$	−12.0000	−0.733017
$269$	11.7016	0.713457	0.356728	−	0.934208i	$-0.383892\pi$
$269$	11.7016	0.713457	0.356728	+	0.934208i	$0.383892\pi$
$270$	0	0
$271$	−28.5078	−1.73173	−0.865863	−	0.500281i	$-0.833230\pi$
$271$	−28.5078	−1.73173	−0.865863	+	0.500281i	$0.833230\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−4.70156	−0.284032
$275$	3.91093	0.235838
$276$	0	0
$277$	−16.8062	−1.00979	−0.504895	−	0.863181i	$-0.668469\pi$
$277$	−16.8062	−1.00979	−0.504895	+	0.863181i	$0.668469\pi$
$278$	−10.8062	−0.648116
$279$	0	0
$280$	4.59688	0.274716
$281$	−9.40312	−0.560943	−0.280472	−	0.959862i	$-0.590491\pi$
$281$	−9.40312	−0.560943	−0.280472	+	0.959862i	$0.590491\pi$
$282$	0	0
$283$	25.6125	1.52250	0.761252	−	0.648456i	$-0.224585\pi$
$283$	25.6125	1.52250	0.761252	+	0.648456i	$0.224585\pi$
$284$	−15.4031	−0.914007
$285$	0	0
$286$	−8.00000	−0.473050
$287$	17.1938	1.01492
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−17.2984	−1.01580
$291$	0	0
$292$	6.40312	0.374715
$293$	−15.2094	−0.888541	−0.444271	−	0.895893i	$-0.646537\pi$
$293$	−15.2094	−0.888541	−0.444271	+	0.895893i	$0.646537\pi$
$294$	0	0
$295$	37.0156	2.15513
$296$	9.40312	0.546545
$297$	0	0
$298$	−0.104686	−0.00606432
$299$	0	0
$300$	0	0
$301$	−19.4031	−1.11838
$302$	5.70156	0.328088
$303$	0	0
$304$	4.00000	0.229416
$305$	−19.7172	−1.12900
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	2.89531	0.164976
$309$	0	0
$310$	−15.4031	−0.874839
$311$	−30.8062	−1.74686	−0.873431	−	0.486948i	$-0.838110\pi$
$311$	−30.8062	−1.74686	−0.873431	+	0.486948i	$0.838110\pi$
$312$	0	0
$313$	−21.2984	−1.20386	−0.601929	−	0.798549i	$-0.705601\pi$
$313$	−21.2984	−1.20386	−0.601929	+	0.798549i	$0.705601\pi$
$314$	−7.29844	−0.411875
$315$	0	0
$316$	10.2984	0.579332
$317$	−18.4031	−1.03362	−0.516811	−	0.856099i	$-0.672881\pi$
$317$	−18.4031	−1.03362	−0.516811	+	0.856099i	$0.672881\pi$
$318$	0	0
$319$	−10.8953	−0.610020
$320$	−2.70156	−0.151022
$321$	0	0
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	10.8062	0.599423
$326$	−11.4031	−0.631561
$327$	0	0
$328$	−10.1047	−0.557938
$329$	−6.80625	−0.375241
$330$	0	0
$331$	3.40312	0.187053	0.0935263	−	0.995617i	$-0.470186\pi$
$331$	3.40312	0.187053	0.0935263	+	0.995617i	$0.470186\pi$
$332$	6.29844	0.345672
$333$	0	0
$334$	−10.8062	−0.591292
$335$	32.4187	1.77123
$336$	0	0
$337$	−5.29844	−0.288624	−0.144312	−	0.989532i	$-0.546097\pi$
$337$	−5.29844	−0.288624	−0.144312	+	0.989532i	$0.546097\pi$
$338$	−9.10469	−0.495230
$339$	0	0
$340$	−5.40312	−0.293026
$341$	−9.70156	−0.525369
$342$	0	0
$343$	−18.8953	−1.02025
$344$	11.4031	0.614815
$345$	0	0
$346$	16.1047	0.865793
$347$	−9.70156	−0.520807	−0.260404	−	0.965500i	$-0.583856\pi$
$347$	−9.70156	−0.520807	−0.260404	+	0.965500i	$0.583856\pi$
$348$	0	0
$349$	4.80625	0.257273	0.128636	−	0.991692i	$-0.458940\pi$
$349$	4.80625	0.257273	0.128636	+	0.991692i	$0.458940\pi$
$350$	−3.91093	−0.209048
$351$	0	0
$352$	−1.70156	−0.0906936
$353$	24.9109	1.32588	0.662938	−	0.748675i	$-0.269309\pi$
$353$	24.9109	1.32588	0.662938	+	0.748675i	$0.269309\pi$
$354$	0	0
$355$	41.6125	2.20856
$356$	−10.7016	−0.567182
$357$	0	0
$358$	0	0
$359$	10.8062	0.570332	0.285166	−	0.958478i	$-0.407951\pi$
$359$	10.8062	0.570332	0.285166	+	0.958478i	$0.407951\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	2.59688	0.136489
$363$	0	0
$364$	8.00000	0.419314
$365$	−17.2984	−0.905442
$366$	0	0
$367$	−13.1047	−0.684059	−0.342030	−	0.939689i	$-0.611114\pi$
$367$	−13.1047	−0.684059	−0.342030	+	0.939689i	$0.611114\pi$
$368$	0	0
$369$	0	0
$370$	−25.4031	−1.32065
$371$	10.8953	0.565657
$372$	0	0
$373$	32.8062	1.69864	0.849322	−	0.527876i	$-0.177011\pi$
$373$	32.8062	1.69864	0.849322	+	0.527876i	$0.177011\pi$
$374$	−3.40312	−0.175971
$375$	0	0
$376$	4.00000	0.206284
$377$	−30.1047	−1.55047
$378$	0	0
$379$	−26.2094	−1.34629	−0.673143	−	0.739513i	$-0.735056\pi$
$379$	−26.2094	−1.34629	−0.673143	+	0.739513i	$0.735056\pi$
$380$	−10.8062	−0.554349
$381$	0	0
$382$	7.40312	0.378777
$383$	0.596876	0.0304989	0.0152495	−	0.999884i	$-0.495146\pi$
$383$	0.596876	0.0304989	0.0152495	+	0.999884i	$0.495146\pi$
$384$	0	0
$385$	−7.82187	−0.398639
$386$	24.4031	1.24209
$387$	0	0
$388$	−17.8062	−0.903975
$389$	−39.1047	−1.98269	−0.991343	−	0.131296i	$-0.958086\pi$
$389$	−39.1047	−1.98269	−0.991343	+	0.131296i	$0.958086\pi$
$390$	0	0
$391$	0	0
$392$	4.10469	0.207318
$393$	0	0
$394$	18.4031	0.927136
$395$	−27.8219	−1.39987
$396$	0	0
$397$	−34.7016	−1.74162	−0.870811	−	0.491618i	$-0.836405\pi$
$397$	−34.7016	−1.74162	−0.870811	+	0.491618i	$0.836405\pi$
$398$	−13.1047	−0.656879
$399$	0	0
$400$	2.29844	0.114922
$401$	23.5078	1.17392	0.586962	−	0.809614i	$-0.300324\pi$
$401$	23.5078	1.17392	0.586962	+	0.809614i	$0.300324\pi$
$402$	0	0
$403$	−26.8062	−1.33531
$404$	12.4031	0.617078
$405$	0	0
$406$	10.8953	0.540725
$407$	−16.0000	−0.793091
$408$	0	0
$409$	26.5078	1.31073	0.655363	−	0.755314i	$-0.272515\pi$
$409$	26.5078	1.31073	0.655363	+	0.755314i	$0.272515\pi$
$410$	27.2984	1.34817
$411$	0	0
$412$	5.10469	0.251490
$413$	−23.3141	−1.14721
$414$	0	0
$415$	−17.0156	−0.835264
$416$	−4.70156	−0.230513
$417$	0	0
$418$	−6.80625	−0.332904
$419$	4.50781	0.220221	0.110110	−	0.993919i	$-0.464880\pi$
$419$	4.50781	0.220221	0.110110	+	0.993919i	$0.464880\pi$
$420$	0	0
$421$	−19.6125	−0.955855	−0.477927	−	0.878399i	$-0.658612\pi$
$421$	−19.6125	−0.955855	−0.477927	+	0.878399i	$0.658612\pi$
$422$	14.2094	0.691701
$423$	0	0
$424$	−6.40312	−0.310963
$425$	4.59688	0.222981
$426$	0	0
$427$	12.4187	0.600985
$428$	6.80625	0.328992
$429$	0	0
$430$	−30.8062	−1.48561
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	−0.104686	−0.00503091	−0.00251545	−	0.999997i	$-0.500801\pi$
$433$	−0.104686	−0.00503091	−0.00251545	+	0.999997i	$0.500801\pi$
$434$	9.70156	0.465690
$435$	0	0
$436$	3.29844	0.157967
$437$	0	0
$438$	0	0
$439$	−6.29844	−0.300608	−0.150304	−	0.988640i	$-0.548025\pi$
$439$	−6.29844	−0.300608	−0.150304	+	0.988640i	$0.548025\pi$
$440$	4.59688	0.219147
$441$	0	0
$442$	−9.40312	−0.447261
$443$	23.9109	1.13604	0.568021	−	0.823014i	$-0.307709\pi$
$443$	23.9109	1.13604	0.568021	+	0.823014i	$0.307709\pi$
$444$	0	0
$445$	28.9109	1.37051
$446$	0	0
$447$	0	0
$448$	1.70156	0.0803913
$449$	35.7172	1.68560	0.842799	−	0.538228i	$-0.180906\pi$
$449$	35.7172	1.68560	0.842799	+	0.538228i	$0.180906\pi$
$450$	0	0
$451$	17.1938	0.809622
$452$	12.7016	0.597431
$453$	0	0
$454$	14.2984	0.671059
$455$	−21.6125	−1.01321
$456$	0	0
$457$	−12.6125	−0.589988	−0.294994	−	0.955499i	$-0.595318\pi$
$457$	−12.6125	−0.589988	−0.294994	+	0.955499i	$0.595318\pi$
$458$	−20.2094	−0.944322
$459$	0	0
$460$	0	0
$461$	−7.00000	−0.326023	−0.163011	−	0.986624i	$-0.552121\pi$
$461$	−7.00000	−0.326023	−0.163011	+	0.986624i	$0.552121\pi$
$462$	0	0
$463$	−29.1047	−1.35261	−0.676305	−	0.736622i	$-0.736420\pi$
$463$	−29.1047	−1.35261	−0.676305	+	0.736622i	$0.736420\pi$
$464$	−6.40312	−0.297258
$465$	0	0
$466$	−9.50781	−0.440441
$467$	39.9109	1.84686	0.923429	−	0.383770i	$-0.125374\pi$
$467$	39.9109	1.84686	0.923429	+	0.383770i	$0.125374\pi$
$468$	0	0
$469$	−20.4187	−0.942850
$470$	−10.8062	−0.498455
$471$	0	0
$472$	13.7016	0.630666
$473$	−19.4031	−0.892157
$474$	0	0
$475$	9.19375	0.421838
$476$	3.40312	0.155982
$477$	0	0
$478$	10.8062	0.494266
$479$	−11.4031	−0.521022	−0.260511	−	0.965471i	$-0.583891\pi$
$479$	−11.4031	−0.521022	−0.260511	+	0.965471i	$0.583891\pi$
$480$	0	0
$481$	−44.2094	−2.01577
$482$	−6.70156	−0.305248
$483$	0	0
$484$	−8.10469	−0.368395
$485$	48.1047	2.18432
$486$	0	0
$487$	29.7016	1.34591	0.672953	−	0.739685i	$-0.265026\pi$
$487$	29.7016	1.34591	0.672953	+	0.739685i	$0.265026\pi$
$488$	−7.29844	−0.330385
$489$	0	0
$490$	−11.0891	−0.500953
$491$	18.2984	0.825797	0.412898	−	0.910777i	$-0.364516\pi$
$491$	18.2984	0.825797	0.412898	+	0.910777i	$0.364516\pi$
$492$	0	0
$493$	−12.8062	−0.576764
$494$	−18.8062	−0.846133
$495$	0	0
$496$	−5.70156	−0.256008
$497$	−26.2094	−1.17565
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	7.29844	0.326396
$501$	0	0
$502$	4.00000	0.178529
$503$	−3.40312	−0.151738	−0.0758689	−	0.997118i	$-0.524173\pi$
$503$	−3.40312	−0.151738	−0.0758689	+	0.997118i	$0.524173\pi$
$504$	0	0
$505$	−33.5078	−1.49108
$506$	0	0
$507$	0	0
$508$	−4.00000	−0.177471
$509$	3.20937	0.142253	0.0711265	−	0.997467i	$-0.477341\pi$
$509$	3.20937	0.142253	0.0711265	+	0.997467i	$0.477341\pi$
$510$	0	0
$511$	10.8953	0.481980
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	1.29844	0.0572716
$515$	−13.7906	−0.607688
$516$	0	0
$517$	−6.80625	−0.299338
$518$	16.0000	0.703000
$519$	0	0
$520$	12.7016	0.557000
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	6.20937	0.271517	0.135758	−	0.990742i	$-0.456653\pi$
$523$	6.20937	0.271517	0.135758	+	0.990742i	$0.456653\pi$
$524$	−5.70156	−0.249074
$525$	0	0
$526$	14.8062	0.645583
$527$	−11.4031	−0.496728
$528$	0	0
$529$	0	0
$530$	17.2984	0.751396
$531$	0	0
$532$	6.80625	0.295088
$533$	47.5078	2.05779
$534$	0	0
$535$	−18.3875	−0.794961
$536$	12.0000	0.518321
$537$	0	0
$538$	−11.7016	−0.504490
$539$	−6.98438	−0.300838
$540$	0	0
$541$	−4.49219	−0.193134	−0.0965672	−	0.995326i	$-0.530786\pi$
$541$	−4.49219	−0.193134	−0.0965672	+	0.995326i	$0.530786\pi$
$542$	28.5078	1.22452
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	−8.91093	−0.381703
$546$	0	0
$547$	−21.6125	−0.924084	−0.462042	−	0.886858i	$-0.652883\pi$
$547$	−21.6125	−0.924084	−0.462042	+	0.886858i	$0.652883\pi$
$548$	4.70156	0.200841
$549$	0	0
$550$	−3.91093	−0.166763
$551$	−25.6125	−1.09113
$552$	0	0
$553$	17.5234	0.745172
$554$	16.8062	0.714029
$555$	0	0
$556$	10.8062	0.458287
$557$	−13.0000	−0.550828	−0.275414	−	0.961326i	$-0.588815\pi$
$557$	−13.0000	−0.550828	−0.275414	+	0.961326i	$0.588815\pi$
$558$	0	0
$559$	−53.6125	−2.26757
$560$	−4.59688	−0.194253
$561$	0	0
$562$	9.40312	0.396647
$563$	−11.9109	−0.501986	−0.250993	−	0.967989i	$-0.580757\pi$
$563$	−11.9109	−0.501986	−0.250993	+	0.967989i	$0.580757\pi$
$564$	0	0
$565$	−34.3141	−1.44360
$566$	−25.6125	−1.07657
$567$	0	0
$568$	15.4031	0.646301
$569$	−7.89531	−0.330989	−0.165494	−	0.986211i	$-0.552922\pi$
$569$	−7.89531	−0.330989	−0.165494	+	0.986211i	$0.552922\pi$
$570$	0	0
$571$	−44.4187	−1.85887	−0.929433	−	0.368990i	$-0.879704\pi$
$571$	−44.4187	−1.85887	−0.929433	+	0.368990i	$0.879704\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	−17.1938	−0.717653
$575$	0	0
$576$	0	0
$577$	−4.29844	−0.178946	−0.0894732	−	0.995989i	$-0.528518\pi$
$577$	−4.29844	−0.178946	−0.0894732	+	0.995989i	$0.528518\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	17.2984	0.718279
$581$	10.7172	0.444624
$582$	0	0
$583$	10.8953	0.451238
$584$	−6.40312	−0.264963
$585$	0	0
$586$	15.2094	0.628293
$587$	−26.8062	−1.10641	−0.553206	−	0.833044i	$-0.686596\pi$
$587$	−26.8062	−1.10641	−0.553206	+	0.833044i	$0.686596\pi$
$588$	0	0
$589$	−22.8062	−0.939715
$590$	−37.0156	−1.52391
$591$	0	0
$592$	−9.40312	−0.386466
$593$	−37.4031	−1.53596	−0.767981	−	0.640472i	$-0.778739\pi$
$593$	−37.4031	−1.53596	−0.767981	+	0.640472i	$0.778739\pi$
$594$	0	0
$595$	−9.19375	−0.376907
$596$	0.104686	0.00428812
$597$	0	0
$598$	0	0
$599$	−46.8062	−1.91245	−0.956226	−	0.292630i	$-0.905470\pi$
$599$	−46.8062	−1.91245	−0.956226	+	0.292630i	$0.905470\pi$
$600$	0	0
$601$	9.80625	0.400005	0.200003	−	0.979795i	$-0.435905\pi$
$601$	9.80625	0.400005	0.200003	+	0.979795i	$0.435905\pi$
$602$	19.4031	0.790812
$603$	0	0
$604$	−5.70156	−0.231993
$605$	21.8953	0.890171
$606$	0	0
$607$	19.9109	0.808160	0.404080	−	0.914724i	$-0.367592\pi$
$607$	19.9109	0.808160	0.404080	+	0.914724i	$0.367592\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	19.7172	0.798325
$611$	−18.8062	−0.760819
$612$	0	0
$613$	−10.3141	−0.416581	−0.208290	−	0.978067i	$-0.566790\pi$
$613$	−10.3141	−0.416581	−0.208290	+	0.978067i	$0.566790\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	−2.89531	−0.116656
$617$	13.4031	0.539589	0.269795	−	0.962918i	$-0.413044\pi$
$617$	13.4031	0.539589	0.269795	+	0.962918i	$0.413044\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	15.4031	0.618604
$621$	0	0
$622$	30.8062	1.23522
$623$	−18.2094	−0.729543
$624$	0	0
$625$	−31.2094	−1.24837
$626$	21.2984	0.851257
$627$	0	0
$628$	7.29844	0.291239
$629$	−18.8062	−0.749854
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−10.2984	−0.409650
$633$	0	0
$634$	18.4031	0.730881
$635$	10.8062	0.428833
$636$	0	0
$637$	−19.2984	−0.764632
$638$	10.8953	0.431350
$639$	0	0
$640$	2.70156	0.106789
$641$	25.8953	1.02280	0.511402	−	0.859342i	$-0.329126\pi$
$641$	25.8953	1.02280	0.511402	+	0.859342i	$0.329126\pi$
$642$	0	0
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	0	0
$646$	−8.00000	−0.314756
$647$	25.0156	0.983466	0.491733	−	0.870746i	$-0.336364\pi$
$647$	25.0156	0.983466	0.491733	+	0.870746i	$0.336364\pi$
$648$	0	0
$649$	−23.3141	−0.915157
$650$	−10.8062	−0.423856
$651$	0	0
$652$	11.4031	0.446581
$653$	−11.0000	−0.430463	−0.215232	−	0.976563i	$-0.569051\pi$
$653$	−11.0000	−0.430463	−0.215232	+	0.976563i	$0.569051\pi$
$654$	0	0
$655$	15.4031	0.601850
$656$	10.1047	0.394522
$657$	0	0
$658$	6.80625	0.265335
$659$	22.8062	0.888405	0.444203	−	0.895926i	$-0.353487\pi$
$659$	22.8062	0.888405	0.444203	+	0.895926i	$0.353487\pi$
$660$	0	0
$661$	43.1203	1.67719	0.838593	−	0.544759i	$-0.183379\pi$
$661$	43.1203	1.67719	0.838593	+	0.544759i	$0.183379\pi$
$662$	−3.40312	−0.132266
$663$	0	0
$664$	−6.29844	−0.244427
$665$	−18.3875	−0.713037
$666$	0	0
$667$	0	0
$668$	10.8062	0.418106
$669$	0	0
$670$	−32.4187	−1.25245
$671$	12.4187	0.479420
$672$	0	0
$673$	−14.5078	−0.559235	−0.279618	−	0.960111i	$-0.590208\pi$
$673$	−14.5078	−0.559235	−0.279618	+	0.960111i	$0.590208\pi$
$674$	5.29844	0.204088
$675$	0	0
$676$	9.10469	0.350180
$677$	11.0000	0.422764	0.211382	−	0.977403i	$-0.432204\pi$
$677$	11.0000	0.422764	0.211382	+	0.977403i	$0.432204\pi$
$678$	0	0
$679$	−30.2984	−1.16275
$680$	5.40312	0.207200
$681$	0	0
$682$	9.70156	0.371492
$683$	9.61250	0.367812	0.183906	−	0.982944i	$-0.441126\pi$
$683$	9.61250	0.367812	0.183906	+	0.982944i	$0.441126\pi$
$684$	0	0
$685$	−12.7016	−0.485302
$686$	18.8953	0.721426
$687$	0	0
$688$	−11.4031	−0.434740
$689$	30.1047	1.14690
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	−16.1047	−0.612208
$693$	0	0
$694$	9.70156	0.368266
$695$	−29.1938	−1.10738
$696$	0	0
$697$	20.2094	0.765485
$698$	−4.80625	−0.181919
$699$	0	0
$700$	3.91093	0.147819
$701$	−4.89531	−0.184893	−0.0924467	−	0.995718i	$-0.529469\pi$
$701$	−4.89531	−0.184893	−0.0924467	+	0.995718i	$0.529469\pi$
$702$	0	0
$703$	−37.6125	−1.41858
$704$	1.70156	0.0641300
$705$	0	0
$706$	−24.9109	−0.937535
$707$	21.1047	0.793723
$708$	0	0
$709$	−26.9109	−1.01066	−0.505331	−	0.862926i	$-0.668629\pi$
$709$	−26.9109	−1.01066	−0.505331	+	0.862926i	$0.668629\pi$
$710$	−41.6125	−1.56169
$711$	0	0
$712$	10.7016	0.401058
$713$	0	0
$714$	0	0
$715$	−21.6125	−0.808262
$716$	0	0
$717$	0	0
$718$	−10.8062	−0.403286
$719$	−3.40312	−0.126915	−0.0634576	−	0.997985i	$-0.520213\pi$
$719$	−3.40312	−0.126915	−0.0634576	+	0.997985i	$0.520213\pi$
$720$	0	0
$721$	8.68594	0.323481
$722$	3.00000	0.111648
$723$	0	0
$724$	−2.59688	−0.0965121
$725$	−14.7172	−0.546582
$726$	0	0
$727$	−13.6125	−0.504860	−0.252430	−	0.967615i	$-0.581230\pi$
$727$	−13.6125	−0.504860	−0.252430	+	0.967615i	$0.581230\pi$
$728$	−8.00000	−0.296500
$729$	0	0
$730$	17.2984	0.640244
$731$	−22.8062	−0.843520
$732$	0	0
$733$	24.3141	0.898060	0.449030	−	0.893517i	$-0.351770\pi$
$733$	24.3141	0.898060	0.449030	+	0.893517i	$0.351770\pi$
$734$	13.1047	0.483703
$735$	0	0
$736$	0	0
$737$	−20.4187	−0.752134
$738$	0	0
$739$	−37.6125	−1.38360	−0.691799	−	0.722090i	$-0.743182\pi$
$739$	−37.6125	−1.38360	−0.691799	+	0.722090i	$0.743182\pi$
$740$	25.4031	0.933837
$741$	0	0
$742$	−10.8953	−0.399980
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	−0.282817	−0.0103616
$746$	−32.8062	−1.20112
$747$	0	0
$748$	3.40312	0.124431
$749$	11.5813	0.423170
$750$	0	0
$751$	51.3141	1.87248	0.936238	−	0.351366i	$-0.114283\pi$
$751$	51.3141	1.87248	0.936238	+	0.351366i	$0.114283\pi$
$752$	−4.00000	−0.145865
$753$	0	0
$754$	30.1047	1.09635
$755$	15.4031	0.560577
$756$	0	0
$757$	30.7016	1.11587	0.557934	−	0.829886i	$-0.311594\pi$
$757$	30.7016	1.11587	0.557934	+	0.829886i	$0.311594\pi$
$758$	26.2094	0.951967
$759$	0	0
$760$	10.8062	0.391984
$761$	−3.50781	−0.127158	−0.0635790	−	0.997977i	$-0.520251\pi$
$761$	−3.50781	−0.127158	−0.0635790	+	0.997977i	$0.520251\pi$
$762$	0	0
$763$	5.61250	0.203186
$764$	−7.40312	−0.267836
$765$	0	0
$766$	−0.596876	−0.0215660
$767$	−64.4187	−2.32603
$768$	0	0
$769$	−37.9109	−1.36710	−0.683552	−	0.729902i	$-0.739566\pi$
$769$	−37.9109	−1.36710	−0.683552	+	0.729902i	$0.739566\pi$
$770$	7.82187	0.281881
$771$	0	0
$772$	−24.4031	−0.878288
$773$	28.2984	1.01782	0.508912	−	0.860819i	$-0.330048\pi$
$773$	28.2984	1.01782	0.508912	+	0.860819i	$0.330048\pi$
$774$	0	0
$775$	−13.1047	−0.470734
$776$	17.8062	0.639207
$777$	0	0
$778$	39.1047	1.40197
$779$	40.4187	1.44815
$780$	0	0
$781$	−26.2094	−0.937845
$782$	0	0
$783$	0	0
$784$	−4.10469	−0.146596
$785$	−19.7172	−0.703736
$786$	0	0
$787$	51.2250	1.82597	0.912987	−	0.407989i	$-0.133770\pi$
$787$	51.2250	1.82597	0.912987	+	0.407989i	$0.133770\pi$
$788$	−18.4031	−0.655584
$789$	0	0
$790$	27.8219	0.989858
$791$	21.6125	0.768452
$792$	0	0
$793$	34.3141	1.21853
$794$	34.7016	1.23151
$795$	0	0
$796$	13.1047	0.464483
$797$	20.8062	0.736995	0.368498	−	0.929629i	$-0.379872\pi$
$797$	20.8062	0.736995	0.368498	+	0.929629i	$0.379872\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	−2.29844	−0.0812621
$801$	0	0
$802$	−23.5078	−0.830090
$803$	10.8953	0.384487
$804$	0	0
$805$	0	0
$806$	26.8062	0.944210
$807$	0	0
$808$	−12.4031	−0.436340
$809$	34.5969	1.21636	0.608181	−	0.793799i	$-0.291900\pi$
$809$	34.5969	1.21636	0.608181	+	0.793799i	$0.291900\pi$
$810$	0	0
$811$	45.0156	1.58071	0.790356	−	0.612648i	$-0.209896\pi$
$811$	45.0156	1.58071	0.790356	+	0.612648i	$0.209896\pi$
$812$	−10.8953	−0.382351
$813$	0	0
$814$	16.0000	0.560800
$815$	−30.8062	−1.07910
$816$	0	0
$817$	−45.6125	−1.59578
$818$	−26.5078	−0.926824
$819$	0	0
$820$	−27.2984	−0.953303
$821$	21.5969	0.753736	0.376868	−	0.926267i	$-0.377001\pi$
$821$	21.5969	0.753736	0.376868	+	0.926267i	$0.377001\pi$
$822$	0	0
$823$	−13.6125	−0.474502	−0.237251	−	0.971448i	$-0.576246\pi$
$823$	−13.6125	−0.474502	−0.237251	+	0.971448i	$0.576246\pi$
$824$	−5.10469	−0.177830
$825$	0	0
$826$	23.3141	0.811200
$827$	−30.8062	−1.07124	−0.535619	−	0.844460i	$-0.679922\pi$
$827$	−30.8062	−1.07124	−0.535619	+	0.844460i	$0.679922\pi$
$828$	0	0
$829$	9.89531	0.343678	0.171839	−	0.985125i	$-0.445029\pi$
$829$	9.89531	0.343678	0.171839	+	0.985125i	$0.445029\pi$
$830$	17.0156	0.590621
$831$	0	0
$832$	4.70156	0.162997
$833$	−8.20937	−0.284438
$834$	0	0
$835$	−29.1938	−1.01029
$836$	6.80625	0.235399
$837$	0	0
$838$	−4.50781	−0.155720
$839$	9.61250	0.331860	0.165930	−	0.986138i	$-0.446937\pi$
$839$	9.61250	0.331860	0.165930	+	0.986138i	$0.446937\pi$
$840$	0	0
$841$	12.0000	0.413793
$842$	19.6125	0.675891
$843$	0	0
$844$	−14.2094	−0.489107
$845$	−24.5969	−0.846158
$846$	0	0
$847$	−13.7906	−0.473852
$848$	6.40312	0.219884
$849$	0	0
$850$	−4.59688	−0.157672
$851$	0	0
$852$	0	0
$853$	42.4187	1.45239	0.726195	−	0.687489i	$-0.241287\pi$
$853$	42.4187	1.45239	0.726195	+	0.687489i	$0.241287\pi$
$854$	−12.4187	−0.424961
$855$	0	0
$856$	−6.80625	−0.232633
$857$	−29.8953	−1.02120	−0.510602	−	0.859817i	$-0.670578\pi$
$857$	−29.8953	−1.02120	−0.510602	+	0.859817i	$0.670578\pi$
$858$	0	0
$859$	−18.2094	−0.621296	−0.310648	−	0.950525i	$-0.600546\pi$
$859$	−18.2094	−0.621296	−0.310648	+	0.950525i	$0.600546\pi$
$860$	30.8062	1.05048
$861$	0	0
$862$	−8.00000	−0.272481
$863$	22.2094	0.756016	0.378008	−	0.925802i	$-0.376609\pi$
$863$	22.2094	0.756016	0.378008	+	0.925802i	$0.376609\pi$
$864$	0	0
$865$	43.5078	1.47931
$866$	0.104686	0.00355739
$867$	0	0
$868$	−9.70156	−0.329292
$869$	17.5234	0.594442
$870$	0	0
$871$	−56.4187	−1.91168
$872$	−3.29844	−0.111699
$873$	0	0
$874$	0	0
$875$	12.4187	0.419830
$876$	0	0
$877$	−12.2094	−0.412281	−0.206141	−	0.978522i	$-0.566090\pi$
$877$	−12.2094	−0.412281	−0.206141	+	0.978522i	$0.566090\pi$
$878$	6.29844	0.212562
$879$	0	0
$880$	−4.59688	−0.154961
$881$	5.89531	0.198618	0.0993091	−	0.995057i	$-0.468337\pi$
$881$	5.89531	0.198618	0.0993091	+	0.995057i	$0.468337\pi$
$882$	0	0
$883$	3.40312	0.114524	0.0572621	−	0.998359i	$-0.481763\pi$
$883$	3.40312	0.114524	0.0572621	+	0.998359i	$0.481763\pi$
$884$	9.40312	0.316261
$885$	0	0
$886$	−23.9109	−0.803304
$887$	−22.8062	−0.765759	−0.382879	−	0.923798i	$-0.625068\pi$
$887$	−22.8062	−0.765759	−0.382879	+	0.923798i	$0.625068\pi$
$888$	0	0
$889$	−6.80625	−0.228274
$890$	−28.9109	−0.969097
$891$	0	0
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	0	0
$896$	−1.70156	−0.0568452
$897$	0	0
$898$	−35.7172	−1.19190
$899$	36.5078	1.21760
$900$	0	0
$901$	12.8062	0.426638
$902$	−17.1938	−0.572489
$903$	0	0
$904$	−12.7016	−0.422448
$905$	7.01562	0.233207
$906$	0	0
$907$	19.4031	0.644270	0.322135	−	0.946694i	$-0.395599\pi$
$907$	19.4031	0.644270	0.322135	+	0.946694i	$0.395599\pi$
$908$	−14.2984	−0.474510
$909$	0	0
$910$	21.6125	0.716447
$911$	−58.2094	−1.92856	−0.964281	−	0.264880i	$-0.914668\pi$
$911$	−58.2094	−1.92856	−0.964281	+	0.264880i	$0.914668\pi$
$912$	0	0
$913$	10.7172	0.354687
$914$	12.6125	0.417184
$915$	0	0
$916$	20.2094	0.667736
$917$	−9.70156	−0.320374
$918$	0	0
$919$	−29.1938	−0.963013	−0.481507	−	0.876443i	$-0.659910\pi$
$919$	−29.1938	−0.963013	−0.481507	+	0.876443i	$0.659910\pi$
$920$	0	0
$921$	0	0
$922$	7.00000	0.230533
$923$	−72.4187	−2.38369
$924$	0	0
$925$	−21.6125	−0.710615
$926$	29.1047	0.956439
$927$	0	0
$928$	6.40312	0.210193
$929$	−12.2094	−0.400577	−0.200288	−	0.979737i	$-0.564188\pi$
$929$	−12.2094	−0.400577	−0.200288	+	0.979737i	$0.564188\pi$
$930$	0	0
$931$	−16.4187	−0.538103
$932$	9.50781	0.311439
$933$	0	0
$934$	−39.9109	−1.30593
$935$	−9.19375	−0.300668
$936$	0	0
$937$	35.1047	1.14682	0.573410	−	0.819269i	$-0.305620\pi$
$937$	35.1047	1.14682	0.573410	+	0.819269i	$0.305620\pi$
$938$	20.4187	0.666696
$939$	0	0
$940$	10.8062	0.352461
$941$	−60.7172	−1.97932	−0.989662	−	0.143421i	$-0.954190\pi$
$941$	−60.7172	−1.97932	−0.989662	+	0.143421i	$0.954190\pi$
$942$	0	0
$943$	0	0
$944$	−13.7016	−0.445948
$945$	0	0
$946$	19.4031	0.630850
$947$	−21.7016	−0.705206	−0.352603	−	0.935773i	$-0.614703\pi$
$947$	−21.7016	−0.705206	−0.352603	+	0.935773i	$0.614703\pi$
$948$	0	0
$949$	30.1047	0.977239
$950$	−9.19375	−0.298285
$951$	0	0
$952$	−3.40312	−0.110296
$953$	−15.7172	−0.509130	−0.254565	−	0.967056i	$-0.581932\pi$
$953$	−15.7172	−0.509130	−0.254565	+	0.967056i	$0.581932\pi$
$954$	0	0
$955$	20.0000	0.647185
$956$	−10.8062	−0.349499
$957$	0	0
$958$	11.4031	0.368418
$959$	8.00000	0.258333
$960$	0	0
$961$	1.50781	0.0486391
$962$	44.2094	1.42537
$963$	0	0
$964$	6.70156	0.215843
$965$	65.9266	2.12225
$966$	0	0
$967$	−31.3141	−1.00699	−0.503496	−	0.863997i	$-0.667953\pi$
$967$	−31.3141	−1.00699	−0.503496	+	0.863997i	$0.667953\pi$
$968$	8.10469	0.260494
$969$	0	0
$970$	−48.1047	−1.54455
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	0	0
$973$	18.3875	0.589476
$974$	−29.7016	−0.951699
$975$	0	0
$976$	7.29844	0.233617
$977$	36.1047	1.15509	0.577546	−	0.816358i	$-0.304011\pi$
$977$	36.1047	1.15509	0.577546	+	0.816358i	$0.304011\pi$
$978$	0	0
$979$	−18.2094	−0.581974
$980$	11.0891	0.354227
$981$	0	0
$982$	−18.2984	−0.583927
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	49.7172	1.58412
$986$	12.8062	0.407834
$987$	0	0
$988$	18.8062	0.598306
$989$	0	0
$990$	0	0
$991$	−3.49219	−0.110933	−0.0554665	−	0.998461i	$-0.517665\pi$
$991$	−3.49219	−0.110933	−0.0554665	+	0.998461i	$0.517665\pi$
$992$	5.70156	0.181025
$993$	0	0
$994$	26.2094	0.831311
$995$	−35.4031	−1.12235
$996$	0	0
$997$	−53.9266	−1.70787	−0.853936	−	0.520379i	$-0.825791\pi$
$997$	−53.9266	−1.70787	−0.853936	+	0.520379i	$0.825791\pi$
$998$	−8.00000	−0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ba.1.1		2
3.2	odd	2		3174.2.a.o.1.2	✓	2
23.22	odd	2		9522.2.a.r.1.2		2
69.68	even	2		3174.2.a.r.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3174.2.a.o.1.2	✓	2	3.2	odd	2
3174.2.a.r.1.1	yes	2	69.68	even	2
9522.2.a.r.1.2		2	23.22	odd	2
9522.2.a.ba.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.70156 q^{5} +1.70156 q^{7} -1.00000 q^{8} +2.70156 q^{10} +1.70156 q^{11} +4.70156 q^{13} -1.70156 q^{14} +1.00000 q^{16} +2.00000 q^{17} +4.00000 q^{19} -2.70156 q^{20} -1.70156 q^{22} +2.29844 q^{25} -4.70156 q^{26} +1.70156 q^{28} -6.40312 q^{29} -5.70156 q^{31} -1.00000 q^{32} -2.00000 q^{34} -4.59688 q^{35} -9.40312 q^{37} -4.00000 q^{38} +2.70156 q^{40} +10.1047 q^{41} -11.4031 q^{43} +1.70156 q^{44} -4.00000 q^{47} -4.10469 q^{49} -2.29844 q^{50} +4.70156 q^{52} +6.40312 q^{53} -4.59688 q^{55} -1.70156 q^{56} +6.40312 q^{58} -13.7016 q^{59} +7.29844 q^{61} +5.70156 q^{62} +1.00000 q^{64} -12.7016 q^{65} -12.0000 q^{67} +2.00000 q^{68} +4.59688 q^{70} -15.4031 q^{71} +6.40312 q^{73} +9.40312 q^{74} +4.00000 q^{76} +2.89531 q^{77} +10.2984 q^{79} -2.70156 q^{80} -10.1047 q^{82} +6.29844 q^{83} -5.40312 q^{85} +11.4031 q^{86} -1.70156 q^{88} -10.7016 q^{89} +8.00000 q^{91} +4.00000 q^{94} -10.8062 q^{95} -17.8062 q^{97} +4.10469 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 3 q^{7} - 2 q^{8} - q^{10} - 3 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} + 4 q^{17} + 8 q^{19} + q^{20} + 3 q^{22} + 11 q^{25} - 3 q^{26} - 3 q^{28} - 5 q^{31} - 2 q^{32}+ \cdots - 11 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^5 - 3 * q^7 - 2 * q^8 - q^10 - 3 * q^11 + 3 * q^13 + 3 * q^14 + 2 * q^16 + 4 * q^17 + 8 * q^19 + q^20 + 3 * q^22 + 11 * q^25 - 3 * q^26 - 3 * q^28 - 5 * q^31 - 2 * q^32 - 4 * q^34 - 22 * q^35 - 6 * q^37 - 8 * q^38 - q^40 + q^41 - 10 * q^43 - 3 * q^44 - 8 * q^47 + 11 * q^49 - 11 * q^50 + 3 * q^52 - 22 * q^55 + 3 * q^56 - 21 * q^59 + 21 * q^61 + 5 * q^62 + 2 * q^64 - 19 * q^65 - 24 * q^67 + 4 * q^68 + 22 * q^70 - 18 * q^71 + 6 * q^74 + 8 * q^76 + 25 * q^77 + 27 * q^79 + q^80 - q^82 + 19 * q^83 + 2 * q^85 + 10 * q^86 + 3 * q^88 - 15 * q^89 + 16 * q^91 + 8 * q^94 + 4 * q^95 - 10 * q^97 - 11 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more