LMFDB - Embedded newform 9328.2.a.bm.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9328,2,Mod(1,9328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9328, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9328.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9328 = 2^{4} \cdot 11 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9328.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:	$74.4844550055$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5x^{10} - 7x^{9} + 53x^{8} + 13x^{7} - 189x^{6} - 16x^{5} + 260x^{4} + 32x^{3} - 118x^{2} - 8x + 16 $ x^11 - 5x^10 - 7x^9 + 53x^8 + 13x^7 - 189x^6 - 16x^5 + 260x^4 + 32x^3 - 118x^2 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.15323$ of defining polynomial
Character	$\chi$	$=$	9328.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.15323 q^{3} +1.86010 q^{5} +4.61706 q^{7} +6.94284 q^{9} +O(q^{10})$	$q+3.15323 q^{3} +1.86010 q^{5} +4.61706 q^{7} +6.94284 q^{9} -1.00000 q^{11} +0.224174 q^{13} +5.86531 q^{15} +0.675958 q^{17} +2.35168 q^{19} +14.5586 q^{21} -4.72064 q^{23} -1.54004 q^{25} +12.4327 q^{27} -2.15059 q^{29} +7.11790 q^{31} -3.15323 q^{33} +8.58818 q^{35} +2.53577 q^{37} +0.706870 q^{39} -2.11836 q^{41} -7.05078 q^{43} +12.9144 q^{45} -12.1695 q^{47} +14.3172 q^{49} +2.13145 q^{51} +1.00000 q^{53} -1.86010 q^{55} +7.41538 q^{57} +12.9397 q^{59} -5.17197 q^{61} +32.0555 q^{63} +0.416985 q^{65} -15.9295 q^{67} -14.8852 q^{69} +3.14997 q^{71} +9.88914 q^{73} -4.85608 q^{75} -4.61706 q^{77} +2.66870 q^{79} +18.3746 q^{81} -5.65481 q^{83} +1.25735 q^{85} -6.78130 q^{87} +4.78927 q^{89} +1.03502 q^{91} +22.4444 q^{93} +4.37435 q^{95} -14.7800 q^{97} -6.94284 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{3} + 3 q^{5} + 5 q^{7} + 7 q^{9}+O(q^{10}) $ 11 * q + 6 * q^3 + 3 * q^5 + 5 * q^7 + 7 * q^9	$ 11 q + 6 q^{3} + 3 q^{5} + 5 q^{7} + 7 q^{9} - 11 q^{11} - 13 q^{13} + 8 q^{15} - 7 q^{17} + 21 q^{19} + 6 q^{21} - 11 q^{23} + 4 q^{25} + 6 q^{27} + 5 q^{31} - 6 q^{33} + 25 q^{35} - 4 q^{37} + 19 q^{39} - 11 q^{41} + 16 q^{45} + 17 q^{47} - 2 q^{49} + 18 q^{51} + 11 q^{53} - 3 q^{55} - 5 q^{57} + 19 q^{59} - 2 q^{61} + 36 q^{63} - 13 q^{65} - 25 q^{67} + 3 q^{69} + 30 q^{71} + 5 q^{73} + 5 q^{75} - 5 q^{77} + 23 q^{79} - 9 q^{81} + 19 q^{83} + 2 q^{85} + 7 q^{87} + 6 q^{89} + 20 q^{91} + 43 q^{93} + 50 q^{95} - 35 q^{97} - 7 q^{99}+O(q^{100}) $ 11 * q + 6 * q^3 + 3 * q^5 + 5 * q^7 + 7 * q^9 - 11 * q^11 - 13 * q^13 + 8 * q^15 - 7 * q^17 + 21 * q^19 + 6 * q^21 - 11 * q^23 + 4 * q^25 + 6 * q^27 + 5 * q^31 - 6 * q^33 + 25 * q^35 - 4 * q^37 + 19 * q^39 - 11 * q^41 + 16 * q^45 + 17 * q^47 - 2 * q^49 + 18 * q^51 + 11 * q^53 - 3 * q^55 - 5 * q^57 + 19 * q^59 - 2 * q^61 + 36 * q^63 - 13 * q^65 - 25 * q^67 + 3 * q^69 + 30 * q^71 + 5 * q^73 + 5 * q^75 - 5 * q^77 + 23 * q^79 - 9 * q^81 + 19 * q^83 + 2 * q^85 + 7 * q^87 + 6 * q^89 + 20 * q^91 + 43 * q^93 + 50 * q^95 - 35 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.15323	1.82052	0.910258	−	0.414041i	$-0.135883\pi$
$3$	3.15323	1.82052	0.910258	+	0.414041i	$0.135883\pi$
$4$	0	0
$5$	1.86010	0.831861	0.415931	−	0.909396i	$-0.363456\pi$
$5$	1.86010	0.831861	0.415931	+	0.909396i	$0.363456\pi$
$6$	0	0
$7$	4.61706	1.74508	0.872542	−	0.488539i	$-0.162470\pi$
$7$	4.61706	1.74508	0.872542	+	0.488539i	$0.162470\pi$
$8$	0	0
$9$	6.94284	2.31428
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.224174	0.0621746	0.0310873	−	0.999517i	$-0.490103\pi$
$13$	0.224174	0.0621746	0.0310873	+	0.999517i	$0.490103\pi$
$14$	0	0
$15$	5.86531	1.51442
$16$	0	0
$17$	0.675958	0.163944	0.0819719	−	0.996635i	$-0.473878\pi$
$17$	0.675958	0.163944	0.0819719	+	0.996635i	$0.473878\pi$
$18$	0	0
$19$	2.35168	0.539512	0.269756	−	0.962929i	$-0.413057\pi$
$19$	2.35168	0.539512	0.269756	+	0.962929i	$0.413057\pi$
$20$	0	0
$21$	14.5586	3.17695
$22$	0	0
$23$	−4.72064	−0.984321	−0.492160	−	0.870505i	$-0.663793\pi$
$23$	−4.72064	−0.984321	−0.492160	+	0.870505i	$0.663793\pi$
$24$	0	0
$25$	−1.54004	−0.308007
$26$	0	0
$27$	12.4327	2.39267
$28$	0	0
$29$	−2.15059	−0.399355	−0.199677	−	0.979862i	$-0.563989\pi$
$29$	−2.15059	−0.399355	−0.199677	+	0.979862i	$0.563989\pi$
$30$	0	0
$31$	7.11790	1.27841	0.639206	−	0.769035i	$-0.279263\pi$
$31$	7.11790	1.27841	0.639206	+	0.769035i	$0.279263\pi$
$32$	0	0
$33$	−3.15323	−0.548906
$34$	0	0
$35$	8.58818	1.45167
$36$	0	0
$37$	2.53577	0.416878	0.208439	−	0.978035i	$-0.433162\pi$
$37$	2.53577	0.416878	0.208439	+	0.978035i	$0.433162\pi$
$38$	0	0
$39$	0.706870	0.113190
$40$	0	0
$41$	−2.11836	−0.330833	−0.165416	−	0.986224i	$-0.552897\pi$
$41$	−2.11836	−0.330833	−0.165416	+	0.986224i	$0.552897\pi$
$42$	0	0
$43$	−7.05078	−1.07523	−0.537617	−	0.843189i	$-0.680675\pi$
$43$	−7.05078	−1.07523	−0.537617	+	0.843189i	$0.680675\pi$
$44$	0	0
$45$	12.9144	1.92516
$46$	0	0
$47$	−12.1695	−1.77510	−0.887549	−	0.460713i	$-0.847594\pi$
$47$	−12.1695	−1.77510	−0.887549	+	0.460713i	$0.847594\pi$
$48$	0	0
$49$	14.3172	2.04532
$50$	0	0
$51$	2.13145	0.298463
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−1.86010	−0.250816
$56$	0	0
$57$	7.41538	0.982191
$58$	0	0
$59$	12.9397	1.68461	0.842303	−	0.539005i	$-0.181200\pi$
$59$	12.9397	1.68461	0.842303	+	0.539005i	$0.181200\pi$
$60$	0	0
$61$	−5.17197	−0.662203	−0.331101	−	0.943595i	$-0.607420\pi$
$61$	−5.17197	−0.662203	−0.331101	+	0.943595i	$0.607420\pi$
$62$	0	0
$63$	32.0555	4.03862
$64$	0	0
$65$	0.416985	0.0517206
$66$	0	0
$67$	−15.9295	−1.94609	−0.973046	−	0.230610i	$-0.925928\pi$
$67$	−15.9295	−1.94609	−0.973046	+	0.230610i	$0.925928\pi$
$68$	0	0
$69$	−14.8852	−1.79197
$70$	0	0
$71$	3.14997	0.373832	0.186916	−	0.982376i	$-0.440151\pi$
$71$	3.14997	0.373832	0.186916	+	0.982376i	$0.440151\pi$
$72$	0	0
$73$	9.88914	1.15744	0.578718	−	0.815528i	$-0.303553\pi$
$73$	9.88914	1.15744	0.578718	+	0.815528i	$0.303553\pi$
$74$	0	0
$75$	−4.85608	−0.560732
$76$	0	0
$77$	−4.61706	−0.526163
$78$	0	0
$79$	2.66870	0.300252	0.150126	−	0.988667i	$-0.452032\pi$
$79$	2.66870	0.300252	0.150126	+	0.988667i	$0.452032\pi$
$80$	0	0
$81$	18.3746	2.04162
$82$	0	0
$83$	−5.65481	−0.620696	−0.310348	−	0.950623i	$-0.600445\pi$
$83$	−5.65481	−0.620696	−0.310348	+	0.950623i	$0.600445\pi$
$84$	0	0
$85$	1.25735	0.136379
$86$	0	0
$87$	−6.78130	−0.727032
$88$	0	0
$89$	4.78927	0.507661	0.253831	−	0.967249i	$-0.418309\pi$
$89$	4.78927	0.507661	0.253831	+	0.967249i	$0.418309\pi$
$90$	0	0
$91$	1.03502	0.108500
$92$	0	0
$93$	22.4444	2.32737
$94$	0	0
$95$	4.37435	0.448799
$96$	0	0
$97$	−14.7800	−1.50068	−0.750339	−	0.661053i	$-0.770110\pi$
$97$	−14.7800	−1.50068	−0.750339	+	0.661053i	$0.770110\pi$
$98$	0	0
$99$	−6.94284	−0.697782
$100$	0	0
$101$	14.6134	1.45408	0.727042	−	0.686593i	$-0.240894\pi$
$101$	14.6134	1.45408	0.727042	+	0.686593i	$0.240894\pi$
$102$	0	0
$103$	−10.3058	−1.01546	−0.507730	−	0.861516i	$-0.669515\pi$
$103$	−10.3058	−1.01546	−0.507730	+	0.861516i	$0.669515\pi$
$104$	0	0
$105$	27.0805	2.64279
$106$	0	0
$107$	8.31517	0.803858	0.401929	−	0.915671i	$-0.368340\pi$
$107$	8.31517	0.803858	0.401929	+	0.915671i	$0.368340\pi$
$108$	0	0
$109$	−14.2637	−1.36622	−0.683108	−	0.730318i	$-0.739372\pi$
$109$	−14.2637	−1.36622	−0.683108	+	0.730318i	$0.739372\pi$
$110$	0	0
$111$	7.99587	0.758934
$112$	0	0
$113$	−3.98946	−0.375297	−0.187649	−	0.982236i	$-0.560087\pi$
$113$	−3.98946	−0.375297	−0.187649	+	0.982236i	$0.560087\pi$
$114$	0	0
$115$	−8.78084	−0.818818
$116$	0	0
$117$	1.55640	0.143889
$118$	0	0
$119$	3.12094	0.286096
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.67968	−0.602286
$124$	0	0
$125$	−12.1651	−1.08808
$126$	0	0
$127$	8.64093	0.766759	0.383379	−	0.923591i	$-0.374760\pi$
$127$	8.64093	0.766759	0.383379	+	0.923591i	$0.374760\pi$
$128$	0	0
$129$	−22.2327	−1.95748
$130$	0	0
$131$	15.1005	1.31934	0.659669	−	0.751556i	$-0.270696\pi$
$131$	15.1005	1.31934	0.659669	+	0.751556i	$0.270696\pi$
$132$	0	0
$133$	10.8578	0.941494
$134$	0	0
$135$	23.1260	1.99037
$136$	0	0
$137$	2.69116	0.229921	0.114960	−	0.993370i	$-0.463326\pi$
$137$	2.69116	0.229921	0.114960	+	0.993370i	$0.463326\pi$
$138$	0	0
$139$	−16.6628	−1.41332	−0.706660	−	0.707554i	$-0.749799\pi$
$139$	−16.6628	−1.41332	−0.706660	+	0.707554i	$0.749799\pi$
$140$	0	0
$141$	−38.3731	−3.23160
$142$	0	0
$143$	−0.224174	−0.0187463
$144$	0	0
$145$	−4.00031	−0.332208
$146$	0	0
$147$	45.1455	3.72354
$148$	0	0
$149$	15.8857	1.30141	0.650704	−	0.759331i	$-0.274474\pi$
$149$	15.8857	1.30141	0.650704	+	0.759331i	$0.274474\pi$
$150$	0	0
$151$	−13.7819	−1.12155	−0.560776	−	0.827968i	$-0.689497\pi$
$151$	−13.7819	−1.12155	−0.560776	+	0.827968i	$0.689497\pi$
$152$	0	0
$153$	4.69307	0.379412
$154$	0	0
$155$	13.2400	1.06346
$156$	0	0
$157$	−2.04562	−0.163258	−0.0816290	−	0.996663i	$-0.526012\pi$
$157$	−2.04562	−0.163258	−0.0816290	+	0.996663i	$0.526012\pi$
$158$	0	0
$159$	3.15323	0.250067
$160$	0	0
$161$	−21.7954	−1.71772
$162$	0	0
$163$	−2.91104	−0.228010	−0.114005	−	0.993480i	$-0.536368\pi$
$163$	−2.91104	−0.228010	−0.114005	+	0.993480i	$0.536368\pi$
$164$	0	0
$165$	−5.86531	−0.456614
$166$	0	0
$167$	20.1280	1.55755	0.778777	−	0.627301i	$-0.215840\pi$
$167$	20.1280	1.55755	0.778777	+	0.627301i	$0.215840\pi$
$168$	0	0
$169$	−12.9497	−0.996134
$170$	0	0
$171$	16.3273	1.24858
$172$	0	0
$173$	−6.58507	−0.500654	−0.250327	−	0.968161i	$-0.580538\pi$
$173$	−6.58507	−0.500654	−0.250327	+	0.968161i	$0.580538\pi$
$174$	0	0
$175$	−7.11043	−0.537498
$176$	0	0
$177$	40.8018	3.06685
$178$	0	0
$179$	7.21286	0.539114	0.269557	−	0.962984i	$-0.413123\pi$
$179$	7.21286	0.539114	0.269557	+	0.962984i	$0.413123\pi$
$180$	0	0
$181$	−3.14630	−0.233862	−0.116931	−	0.993140i	$-0.537306\pi$
$181$	−3.14630	−0.233862	−0.116931	+	0.993140i	$0.537306\pi$
$182$	0	0
$183$	−16.3084	−1.20555
$184$	0	0
$185$	4.71678	0.346785
$186$	0	0
$187$	−0.675958	−0.0494309
$188$	0	0
$189$	57.4024	4.17541
$190$	0	0
$191$	3.80081	0.275017	0.137508	−	0.990501i	$-0.456091\pi$
$191$	3.80081	0.275017	0.137508	+	0.990501i	$0.456091\pi$
$192$	0	0
$193$	−8.67632	−0.624535	−0.312268	−	0.949994i	$-0.601089\pi$
$193$	−8.67632	−0.624535	−0.312268	+	0.949994i	$0.601089\pi$
$194$	0	0
$195$	1.31485	0.0941582
$196$	0	0
$197$	4.55566	0.324577	0.162289	−	0.986743i	$-0.448112\pi$
$197$	4.55566	0.324577	0.162289	+	0.986743i	$0.448112\pi$
$198$	0	0
$199$	6.31575	0.447712	0.223856	−	0.974622i	$-0.428136\pi$
$199$	6.31575	0.447712	0.223856	+	0.974622i	$0.428136\pi$
$200$	0	0
$201$	−50.2292	−3.54289
$202$	0	0
$203$	−9.92940	−0.696907
$204$	0	0
$205$	−3.94036	−0.275207
$206$	0	0
$207$	−32.7746	−2.27799
$208$	0	0
$209$	−2.35168	−0.162669
$210$	0	0
$211$	−24.3594	−1.67697	−0.838486	−	0.544923i	$-0.816559\pi$
$211$	−24.3594	−1.67697	−0.838486	+	0.544923i	$0.816559\pi$
$212$	0	0
$213$	9.93256	0.680568
$214$	0	0
$215$	−13.1151	−0.894446
$216$	0	0
$217$	32.8638	2.23094
$218$	0	0
$219$	31.1827	2.10713
$220$	0	0
$221$	0.151532	0.0101931
$222$	0	0
$223$	−13.7282	−0.919312	−0.459656	−	0.888097i	$-0.652027\pi$
$223$	−13.7282	−0.919312	−0.459656	+	0.888097i	$0.652027\pi$
$224$	0	0
$225$	−10.6922	−0.712815
$226$	0	0
$227$	15.9311	1.05738	0.528692	−	0.848814i	$-0.322683\pi$
$227$	15.9311	1.05738	0.528692	+	0.848814i	$0.322683\pi$
$228$	0	0
$229$	21.6324	1.42951	0.714753	−	0.699377i	$-0.246539\pi$
$229$	21.6324	1.42951	0.714753	+	0.699377i	$0.246539\pi$
$230$	0	0
$231$	−14.5586	−0.957888
$232$	0	0
$233$	−3.91297	−0.256347	−0.128173	−	0.991752i	$-0.540911\pi$
$233$	−3.91297	−0.256347	−0.128173	+	0.991752i	$0.540911\pi$
$234$	0	0
$235$	−22.6364	−1.47664
$236$	0	0
$237$	8.41501	0.546614
$238$	0	0
$239$	1.31882	0.0853073	0.0426537	−	0.999090i	$-0.486419\pi$
$239$	1.31882	0.0853073	0.0426537	+	0.999090i	$0.486419\pi$
$240$	0	0
$241$	−17.9964	−1.15925	−0.579626	−	0.814883i	$-0.696801\pi$
$241$	−17.9964	−1.15925	−0.579626	+	0.814883i	$0.696801\pi$
$242$	0	0
$243$	20.6411	1.32413
$244$	0	0
$245$	26.6314	1.70142
$246$	0	0
$247$	0.527184	0.0335439
$248$	0	0
$249$	−17.8309	−1.12999
$250$	0	0
$251$	30.9864	1.95584	0.977922	−	0.208968i	$-0.0670105\pi$
$251$	30.9864	1.95584	0.977922	+	0.208968i	$0.0670105\pi$
$252$	0	0
$253$	4.72064	0.296784
$254$	0	0
$255$	3.96470	0.248279
$256$	0	0
$257$	−10.4593	−0.652435	−0.326217	−	0.945295i	$-0.605774\pi$
$257$	−10.4593	−0.652435	−0.326217	+	0.945295i	$0.605774\pi$
$258$	0	0
$259$	11.7078	0.727488
$260$	0	0
$261$	−14.9312	−0.924219
$262$	0	0
$263$	8.88100	0.547626	0.273813	−	0.961783i	$-0.411715\pi$
$263$	8.88100	0.547626	0.273813	+	0.961783i	$0.411715\pi$
$264$	0	0
$265$	1.86010	0.114265
$266$	0	0
$267$	15.1017	0.924206
$268$	0	0
$269$	−21.7305	−1.32493	−0.662467	−	0.749091i	$-0.730491\pi$
$269$	−21.7305	−1.32493	−0.662467	+	0.749091i	$0.730491\pi$
$270$	0	0
$271$	20.5171	1.24632	0.623162	−	0.782093i	$-0.285848\pi$
$271$	20.5171	1.24632	0.623162	+	0.782093i	$0.285848\pi$
$272$	0	0
$273$	3.26366	0.197526
$274$	0	0
$275$	1.54004	0.0928676
$276$	0	0
$277$	−5.38062	−0.323290	−0.161645	−	0.986849i	$-0.551680\pi$
$277$	−5.38062	−0.323290	−0.161645	+	0.986849i	$0.551680\pi$
$278$	0	0
$279$	49.4185	2.95861
$280$	0	0
$281$	−14.4398	−0.861404	−0.430702	−	0.902494i	$-0.641734\pi$
$281$	−14.4398	−0.861404	−0.430702	+	0.902494i	$0.641734\pi$
$282$	0	0
$283$	21.3021	1.26628	0.633140	−	0.774037i	$-0.281766\pi$
$283$	21.3021	1.26628	0.633140	+	0.774037i	$0.281766\pi$
$284$	0	0
$285$	13.7933	0.817046
$286$	0	0
$287$	−9.78060	−0.577331
$288$	0	0
$289$	−16.5431	−0.973122
$290$	0	0
$291$	−46.6046	−2.73201
$292$	0	0
$293$	−28.9496	−1.69125	−0.845626	−	0.533776i	$-0.820772\pi$
$293$	−28.9496	−1.69125	−0.845626	+	0.533776i	$0.820772\pi$
$294$	0	0
$295$	24.0691	1.40136
$296$	0	0
$297$	−12.4327	−0.721418
$298$	0	0
$299$	−1.05824	−0.0611997
$300$	0	0
$301$	−32.5539	−1.87637
$302$	0	0
$303$	46.0793	2.64719
$304$	0	0
$305$	−9.62037	−0.550861
$306$	0	0
$307$	20.1028	1.14732	0.573662	−	0.819092i	$-0.305522\pi$
$307$	20.1028	1.14732	0.573662	+	0.819092i	$0.305522\pi$
$308$	0	0
$309$	−32.4965	−1.84866
$310$	0	0
$311$	−15.6640	−0.888226	−0.444113	−	0.895971i	$-0.646481\pi$
$311$	−15.6640	−0.888226	−0.444113	+	0.895971i	$0.646481\pi$
$312$	0	0
$313$	−7.99003	−0.451623	−0.225812	−	0.974171i	$-0.572503\pi$
$313$	−7.99003	−0.451623	−0.225812	+	0.974171i	$0.572503\pi$
$314$	0	0
$315$	59.6264	3.35957
$316$	0	0
$317$	9.91715	0.557002	0.278501	−	0.960436i	$-0.410162\pi$
$317$	9.91715	0.557002	0.278501	+	0.960436i	$0.410162\pi$
$318$	0	0
$319$	2.15059	0.120410
$320$	0	0
$321$	26.2196	1.46344
$322$	0	0
$323$	1.58963	0.0884497
$324$	0	0
$325$	−0.345235	−0.0191502
$326$	0	0
$327$	−44.9767	−2.48722
$328$	0	0
$329$	−56.1871	−3.09770
$330$	0	0
$331$	−20.6525	−1.13517	−0.567583	−	0.823316i	$-0.692121\pi$
$331$	−20.6525	−1.13517	−0.567583	+	0.823316i	$0.692121\pi$
$332$	0	0
$333$	17.6055	0.964774
$334$	0	0
$335$	−29.6303	−1.61888
$336$	0	0
$337$	−28.6584	−1.56112	−0.780562	−	0.625078i	$-0.785067\pi$
$337$	−28.6584	−1.56112	−0.780562	+	0.625078i	$0.785067\pi$
$338$	0	0
$339$	−12.5797	−0.683235
$340$	0	0
$341$	−7.11790	−0.385456
$342$	0	0
$343$	33.7841	1.82417
$344$	0	0
$345$	−27.6880	−1.49067
$346$	0	0
$347$	3.07618	0.165138	0.0825690	−	0.996585i	$-0.473688\pi$
$347$	3.07618	0.165138	0.0825690	+	0.996585i	$0.473688\pi$
$348$	0	0
$349$	−6.23829	−0.333928	−0.166964	−	0.985963i	$-0.553396\pi$
$349$	−6.23829	−0.333928	−0.166964	+	0.985963i	$0.553396\pi$
$350$	0	0
$351$	2.78708	0.148763
$352$	0	0
$353$	2.40764	0.128146	0.0640728	−	0.997945i	$-0.479591\pi$
$353$	2.40764	0.128146	0.0640728	+	0.997945i	$0.479591\pi$
$354$	0	0
$355$	5.85925	0.310977
$356$	0	0
$357$	9.84102	0.520842
$358$	0	0
$359$	14.6943	0.775534	0.387767	−	0.921757i	$-0.373246\pi$
$359$	14.6943	0.775534	0.387767	+	0.921757i	$0.373246\pi$
$360$	0	0
$361$	−13.4696	−0.708927
$362$	0	0
$363$	3.15323	0.165502
$364$	0	0
$365$	18.3948	0.962826
$366$	0	0
$367$	−13.7268	−0.716532	−0.358266	−	0.933620i	$-0.616632\pi$
$367$	−13.7268	−0.716532	−0.358266	+	0.933620i	$0.616632\pi$
$368$	0	0
$369$	−14.7075	−0.765640
$370$	0	0
$371$	4.61706	0.239706
$372$	0	0
$373$	24.5464	1.27097	0.635483	−	0.772115i	$-0.280801\pi$
$373$	24.5464	1.27097	0.635483	+	0.772115i	$0.280801\pi$
$374$	0	0
$375$	−38.3594	−1.98087
$376$	0	0
$377$	−0.482106	−0.0248297
$378$	0	0
$379$	−14.6735	−0.753727	−0.376864	−	0.926269i	$-0.622997\pi$
$379$	−14.6735	−0.753727	−0.376864	+	0.926269i	$0.622997\pi$
$380$	0	0
$381$	27.2468	1.39590
$382$	0	0
$383$	1.64437	0.0840234	0.0420117	−	0.999117i	$-0.486623\pi$
$383$	1.64437	0.0840234	0.0420117	+	0.999117i	$0.486623\pi$
$384$	0	0
$385$	−8.58818	−0.437694
$386$	0	0
$387$	−48.9525	−2.48839
$388$	0	0
$389$	−6.97281	−0.353536	−0.176768	−	0.984253i	$-0.556564\pi$
$389$	−6.97281	−0.353536	−0.176768	+	0.984253i	$0.556564\pi$
$390$	0	0
$391$	−3.19095	−0.161373
$392$	0	0
$393$	47.6154	2.40188
$394$	0	0
$395$	4.96404	0.249768
$396$	0	0
$397$	22.4816	1.12832	0.564158	−	0.825667i	$-0.309201\pi$
$397$	22.4816	1.12832	0.564158	+	0.825667i	$0.309201\pi$
$398$	0	0
$399$	34.2372	1.71401
$400$	0	0
$401$	−20.6890	−1.03316	−0.516579	−	0.856240i	$-0.672795\pi$
$401$	−20.6890	−1.03316	−0.516579	+	0.856240i	$0.672795\pi$
$402$	0	0
$403$	1.59565	0.0794848
$404$	0	0
$405$	34.1785	1.69834
$406$	0	0
$407$	−2.53577	−0.125694
$408$	0	0
$409$	−39.1393	−1.93531	−0.967657	−	0.252271i	$-0.918822\pi$
$409$	−39.1393	−1.93531	−0.967657	+	0.252271i	$0.918822\pi$
$410$	0	0
$411$	8.48583	0.418575
$412$	0	0
$413$	59.7433	2.93978
$414$	0	0
$415$	−10.5185	−0.516332
$416$	0	0
$417$	−52.5416	−2.57297
$418$	0	0
$419$	−6.70022	−0.327327	−0.163664	−	0.986516i	$-0.552331\pi$
$419$	−6.70022	−0.327327	−0.163664	+	0.986516i	$0.552331\pi$
$420$	0	0
$421$	−30.2440	−1.47400	−0.737001	−	0.675892i	$-0.763759\pi$
$421$	−30.2440	−1.47400	−0.737001	+	0.675892i	$0.763759\pi$
$422$	0	0
$423$	−84.4907	−4.10808
$424$	0	0
$425$	−1.04100	−0.0504959
$426$	0	0
$427$	−23.8793	−1.15560
$428$	0	0
$429$	−0.706870	−0.0341280
$430$	0	0
$431$	17.1244	0.824853	0.412426	−	0.910991i	$-0.364681\pi$
$431$	17.1244	0.824853	0.412426	+	0.910991i	$0.364681\pi$
$432$	0	0
$433$	−18.3260	−0.880690	−0.440345	−	0.897829i	$-0.645144\pi$
$433$	−18.3260	−0.880690	−0.440345	+	0.897829i	$0.645144\pi$
$434$	0	0
$435$	−12.6139	−0.604789
$436$	0	0
$437$	−11.1014	−0.531053
$438$	0	0
$439$	29.0105	1.38460	0.692298	−	0.721612i	$-0.256598\pi$
$439$	29.0105	1.38460	0.692298	+	0.721612i	$0.256598\pi$
$440$	0	0
$441$	99.4023	4.73344
$442$	0	0
$443$	−4.45078	−0.211463	−0.105732	−	0.994395i	$-0.533718\pi$
$443$	−4.45078	−0.211463	−0.105732	+	0.994395i	$0.533718\pi$
$444$	0	0
$445$	8.90851	0.422304
$446$	0	0
$447$	50.0913	2.36924
$448$	0	0
$449$	35.8115	1.69005	0.845024	−	0.534728i	$-0.179586\pi$
$449$	35.8115	1.69005	0.845024	+	0.534728i	$0.179586\pi$
$450$	0	0
$451$	2.11836	0.0997498
$452$	0	0
$453$	−43.4573	−2.04180
$454$	0	0
$455$	1.92524	0.0902568
$456$	0	0
$457$	13.6583	0.638910	0.319455	−	0.947601i	$-0.396500\pi$
$457$	13.6583	0.638910	0.319455	+	0.947601i	$0.396500\pi$
$458$	0	0
$459$	8.40397	0.392264
$460$	0	0
$461$	3.55729	0.165679	0.0828397	−	0.996563i	$-0.473601\pi$
$461$	3.55729	0.165679	0.0828397	+	0.996563i	$0.473601\pi$
$462$	0	0
$463$	27.7547	1.28987	0.644936	−	0.764237i	$-0.276884\pi$
$463$	27.7547	1.28987	0.644936	+	0.764237i	$0.276884\pi$
$464$	0	0
$465$	41.7487	1.93605
$466$	0	0
$467$	29.0945	1.34633	0.673167	−	0.739490i	$-0.264933\pi$
$467$	29.0945	1.34633	0.673167	+	0.739490i	$0.264933\pi$
$468$	0	0
$469$	−73.5472	−3.39609
$470$	0	0
$471$	−6.45029	−0.297214
$472$	0	0
$473$	7.05078	0.324195
$474$	0	0
$475$	−3.62167	−0.166173
$476$	0	0
$477$	6.94284	0.317891
$478$	0	0
$479$	41.7913	1.90949	0.954746	−	0.297424i	$-0.0961273\pi$
$479$	41.7913	1.90949	0.954746	+	0.297424i	$0.0961273\pi$
$480$	0	0
$481$	0.568453	0.0259192
$482$	0	0
$483$	−68.7260	−3.12714
$484$	0	0
$485$	−27.4922	−1.24836
$486$	0	0
$487$	8.82792	0.400031	0.200016	−	0.979793i	$-0.435901\pi$
$487$	8.82792	0.400031	0.200016	+	0.979793i	$0.435901\pi$
$488$	0	0
$489$	−9.17916	−0.415096
$490$	0	0
$491$	−11.9923	−0.541207	−0.270603	−	0.962691i	$-0.587223\pi$
$491$	−11.9923	−0.541207	−0.270603	+	0.962691i	$0.587223\pi$
$492$	0	0
$493$	−1.45371	−0.0654717
$494$	0	0
$495$	−12.9144	−0.580458
$496$	0	0
$497$	14.5436	0.652369
$498$	0	0
$499$	−27.9016	−1.24905	−0.624525	−	0.781005i	$-0.714707\pi$
$499$	−27.9016	−1.24905	−0.624525	+	0.781005i	$0.714707\pi$
$500$	0	0
$501$	63.4683	2.83555
$502$	0	0
$503$	−14.2757	−0.636520	−0.318260	−	0.948003i	$-0.603099\pi$
$503$	−14.2757	−0.636520	−0.318260	+	0.948003i	$0.603099\pi$
$504$	0	0
$505$	27.1823	1.20960
$506$	0	0
$507$	−40.8335	−1.81348
$508$	0	0
$509$	−12.4935	−0.553765	−0.276882	−	0.960904i	$-0.589301\pi$
$509$	−12.4935	−0.553765	−0.276882	+	0.960904i	$0.589301\pi$
$510$	0	0
$511$	45.6587	2.01982
$512$	0	0
$513$	29.2377	1.29087
$514$	0	0
$515$	−19.1698	−0.844721
$516$	0	0
$517$	12.1695	0.535212
$518$	0	0
$519$	−20.7642	−0.911448
$520$	0	0
$521$	31.7598	1.39142	0.695712	−	0.718321i	$-0.255089\pi$
$521$	31.7598	1.39142	0.695712	+	0.718321i	$0.255089\pi$
$522$	0	0
$523$	−4.43603	−0.193974	−0.0969871	−	0.995286i	$-0.530921\pi$
$523$	−4.43603	−0.193974	−0.0969871	+	0.995286i	$0.530921\pi$
$524$	0	0
$525$	−22.4208	−0.978524
$526$	0	0
$527$	4.81140	0.209588
$528$	0	0
$529$	−0.715602	−0.0311131
$530$	0	0
$531$	89.8383	3.89865
$532$	0	0
$533$	−0.474881	−0.0205694
$534$	0	0
$535$	15.4670	0.668698
$536$	0	0
$537$	22.7438	0.981467
$538$	0	0
$539$	−14.3172	−0.616687
$540$	0	0
$541$	−3.79015	−0.162951	−0.0814756	−	0.996675i	$-0.525963\pi$
$541$	−3.79015	−0.162951	−0.0814756	+	0.996675i	$0.525963\pi$
$542$	0	0
$543$	−9.92099	−0.425750
$544$	0	0
$545$	−26.5319	−1.13650
$546$	0	0
$547$	5.58166	0.238654	0.119327	−	0.992855i	$-0.461926\pi$
$547$	5.58166	0.238654	0.119327	+	0.992855i	$0.461926\pi$
$548$	0	0
$549$	−35.9082	−1.53252
$550$	0	0
$551$	−5.05750	−0.215457
$552$	0	0
$553$	12.3215	0.523965
$554$	0	0
$555$	14.8731	0.631328
$556$	0	0
$557$	32.5496	1.37917	0.689585	−	0.724205i	$-0.257793\pi$
$557$	32.5496	1.37917	0.689585	+	0.724205i	$0.257793\pi$
$558$	0	0
$559$	−1.58060	−0.0668522
$560$	0	0
$561$	−2.13145	−0.0899898
$562$	0	0
$563$	−37.0509	−1.56151	−0.780755	−	0.624838i	$-0.785165\pi$
$563$	−37.0509	−1.56151	−0.780755	+	0.624838i	$0.785165\pi$
$564$	0	0
$565$	−7.42079	−0.312195
$566$	0	0
$567$	84.8364	3.56279
$568$	0	0
$569$	−26.9966	−1.13175	−0.565877	−	0.824490i	$-0.691462\pi$
$569$	−26.9966	−1.13175	−0.565877	+	0.824490i	$0.691462\pi$
$570$	0	0
$571$	14.8095	0.619760	0.309880	−	0.950776i	$-0.399711\pi$
$571$	14.8095	0.619760	0.309880	+	0.950776i	$0.399711\pi$
$572$	0	0
$573$	11.9848	0.500673
$574$	0	0
$575$	7.26994	0.303178
$576$	0	0
$577$	24.5757	1.02310	0.511550	−	0.859253i	$-0.329071\pi$
$577$	24.5757	1.02310	0.511550	+	0.859253i	$0.329071\pi$
$578$	0	0
$579$	−27.3584	−1.13698
$580$	0	0
$581$	−26.1086	−1.08317
$582$	0	0
$583$	−1.00000	−0.0414158
$584$	0	0
$585$	2.89506	0.119696
$586$	0	0
$587$	−13.6289	−0.562523	−0.281262	−	0.959631i	$-0.590753\pi$
$587$	−13.6289	−0.562523	−0.281262	+	0.959631i	$0.590753\pi$
$588$	0	0
$589$	16.7390	0.689719
$590$	0	0
$591$	14.3650	0.590898
$592$	0	0
$593$	−45.9429	−1.88665	−0.943325	−	0.331870i	$-0.892320\pi$
$593$	−45.9429	−1.88665	−0.943325	+	0.331870i	$0.892320\pi$
$594$	0	0
$595$	5.80525	0.237992
$596$	0	0
$597$	19.9150	0.815067
$598$	0	0
$599$	−25.3982	−1.03774	−0.518871	−	0.854852i	$-0.673648\pi$
$599$	−25.3982	−1.03774	−0.518871	+	0.854852i	$0.673648\pi$
$600$	0	0
$601$	−3.96580	−0.161768	−0.0808842	−	0.996724i	$-0.525774\pi$
$601$	−3.96580	−0.161768	−0.0808842	+	0.996724i	$0.525774\pi$
$602$	0	0
$603$	−110.596	−4.50381
$604$	0	0
$605$	1.86010	0.0756237
$606$	0	0
$607$	25.8236	1.04815	0.524073	−	0.851673i	$-0.324412\pi$
$607$	25.8236	1.04815	0.524073	+	0.851673i	$0.324412\pi$
$608$	0	0
$609$	−31.3097	−1.26873
$610$	0	0
$611$	−2.72807	−0.110366
$612$	0	0
$613$	33.1204	1.33772	0.668861	−	0.743388i	$-0.266782\pi$
$613$	33.1204	1.33772	0.668861	+	0.743388i	$0.266782\pi$
$614$	0	0
$615$	−12.4249	−0.501019
$616$	0	0
$617$	17.9170	0.721310	0.360655	−	0.932699i	$-0.382553\pi$
$617$	17.9170	0.721310	0.360655	+	0.932699i	$0.382553\pi$
$618$	0	0
$619$	−26.5229	−1.06605	−0.533023	−	0.846101i	$-0.678944\pi$
$619$	−26.5229	−1.06605	−0.533023	+	0.846101i	$0.678944\pi$
$620$	0	0
$621$	−58.6902	−2.35516
$622$	0	0
$623$	22.1123	0.885912
$624$	0	0
$625$	−14.9281	−0.597125
$626$	0	0
$627$	−7.41538	−0.296142
$628$	0	0
$629$	1.71407	0.0683446
$630$	0	0
$631$	12.5530	0.499726	0.249863	−	0.968281i	$-0.419614\pi$
$631$	12.5530	0.499726	0.249863	+	0.968281i	$0.419614\pi$
$632$	0	0
$633$	−76.8108	−3.05296
$634$	0	0
$635$	16.0730	0.637837
$636$	0	0
$637$	3.20954	0.127167
$638$	0	0
$639$	21.8697	0.865153
$640$	0	0
$641$	6.61337	0.261212	0.130606	−	0.991434i	$-0.458308\pi$
$641$	6.61337	0.261212	0.130606	+	0.991434i	$0.458308\pi$
$642$	0	0
$643$	−21.2042	−0.836211	−0.418105	−	0.908399i	$-0.637306\pi$
$643$	−21.2042	−0.836211	−0.418105	+	0.908399i	$0.637306\pi$
$644$	0	0
$645$	−41.3550	−1.62835
$646$	0	0
$647$	−26.5488	−1.04374	−0.521870	−	0.853025i	$-0.674765\pi$
$647$	−26.5488	−1.04374	−0.521870	+	0.853025i	$0.674765\pi$
$648$	0	0
$649$	−12.9397	−0.507928
$650$	0	0
$651$	103.627	4.06146
$652$	0	0
$653$	0.493076	0.0192956	0.00964778	−	0.999953i	$-0.496929\pi$
$653$	0.493076	0.0192956	0.00964778	+	0.999953i	$0.496929\pi$
$654$	0	0
$655$	28.0884	1.09751
$656$	0	0
$657$	68.6588	2.67863
$658$	0	0
$659$	44.9118	1.74952	0.874758	−	0.484561i	$-0.161021\pi$
$659$	44.9118	1.74952	0.874758	+	0.484561i	$0.161021\pi$
$660$	0	0
$661$	9.58037	0.372633	0.186317	−	0.982490i	$-0.440345\pi$
$661$	9.58037	0.372633	0.186317	+	0.982490i	$0.440345\pi$
$662$	0	0
$663$	0.477815	0.0185568
$664$	0	0
$665$	20.1966	0.783192
$666$	0	0
$667$	10.1522	0.393093
$668$	0	0
$669$	−43.2883	−1.67362
$670$	0	0
$671$	5.17197	0.199662
$672$	0	0
$673$	33.7890	1.30247	0.651235	−	0.758876i	$-0.274251\pi$
$673$	33.7890	1.30247	0.651235	+	0.758876i	$0.274251\pi$
$674$	0	0
$675$	−19.1468	−0.736960
$676$	0	0
$677$	−14.8021	−0.568892	−0.284446	−	0.958692i	$-0.591810\pi$
$677$	−14.8021	−0.568892	−0.284446	+	0.958692i	$0.591810\pi$
$678$	0	0
$679$	−68.2400	−2.61881
$680$	0	0
$681$	50.2343	1.92498
$682$	0	0
$683$	30.1861	1.15504	0.577520	−	0.816377i	$-0.304021\pi$
$683$	30.1861	1.15504	0.577520	+	0.816377i	$0.304021\pi$
$684$	0	0
$685$	5.00581	0.191262
$686$	0	0
$687$	68.2118	2.60244
$688$	0	0
$689$	0.224174	0.00854033
$690$	0	0
$691$	−11.9737	−0.455499	−0.227750	−	0.973720i	$-0.573137\pi$
$691$	−11.9737	−0.455499	−0.227750	+	0.973720i	$0.573137\pi$
$692$	0	0
$693$	−32.0555	−1.21769
$694$	0	0
$695$	−30.9944	−1.17569
$696$	0	0
$697$	−1.43192	−0.0542380
$698$	0	0
$699$	−12.3385	−0.466684
$700$	0	0
$701$	24.4913	0.925024	0.462512	−	0.886613i	$-0.346948\pi$
$701$	24.4913	0.925024	0.462512	+	0.886613i	$0.346948\pi$
$702$	0	0
$703$	5.96332	0.224911
$704$	0	0
$705$	−71.3777	−2.68824
$706$	0	0
$707$	67.4708	2.53750
$708$	0	0
$709$	37.3104	1.40122	0.700610	−	0.713544i	$-0.252911\pi$
$709$	37.3104	1.40122	0.700610	+	0.713544i	$0.252911\pi$
$710$	0	0
$711$	18.5283	0.694867
$712$	0	0
$713$	−33.6010	−1.25837
$714$	0	0
$715$	−0.416985	−0.0155944
$716$	0	0
$717$	4.15854	0.155303
$718$	0	0
$719$	6.50843	0.242724	0.121362	−	0.992608i	$-0.461274\pi$
$719$	6.50843	0.242724	0.121362	+	0.992608i	$0.461274\pi$
$720$	0	0
$721$	−47.5824	−1.77206
$722$	0	0
$723$	−56.7468	−2.11044
$724$	0	0
$725$	3.31198	0.123004
$726$	0	0
$727$	13.1898	0.489184	0.244592	−	0.969626i	$-0.421346\pi$
$727$	13.1898	0.489184	0.244592	+	0.969626i	$0.421346\pi$
$728$	0	0
$729$	9.96242	0.368978
$730$	0	0
$731$	−4.76603	−0.176278
$732$	0	0
$733$	−7.81707	−0.288730	−0.144365	−	0.989525i	$-0.546114\pi$
$733$	−7.81707	−0.288730	−0.144365	+	0.989525i	$0.546114\pi$
$734$	0	0
$735$	83.9750	3.09747
$736$	0	0
$737$	15.9295	0.586769
$738$	0	0
$739$	−34.2438	−1.25968	−0.629840	−	0.776725i	$-0.716879\pi$
$739$	−34.2438	−1.25968	−0.629840	+	0.776725i	$0.716879\pi$
$740$	0	0
$741$	1.66233	0.0610673
$742$	0	0
$743$	41.6247	1.52706	0.763531	−	0.645771i	$-0.223464\pi$
$743$	41.6247	1.52706	0.763531	+	0.645771i	$0.223464\pi$
$744$	0	0
$745$	29.5490	1.08259
$746$	0	0
$747$	−39.2604	−1.43646
$748$	0	0
$749$	38.3916	1.40280
$750$	0	0
$751$	−5.16294	−0.188398	−0.0941991	−	0.995553i	$-0.530029\pi$
$751$	−5.16294	−0.188398	−0.0941991	+	0.995553i	$0.530029\pi$
$752$	0	0
$753$	97.7072	3.56065
$754$	0	0
$755$	−25.6356	−0.932976
$756$	0	0
$757$	−33.2938	−1.21008	−0.605042	−	0.796194i	$-0.706844\pi$
$757$	−33.2938	−1.21008	−0.605042	+	0.796194i	$0.706844\pi$
$758$	0	0
$759$	14.8852	0.540300
$760$	0	0
$761$	32.4428	1.17605	0.588025	−	0.808843i	$-0.299906\pi$
$761$	32.4428	1.17605	0.588025	+	0.808843i	$0.299906\pi$
$762$	0	0
$763$	−65.8564	−2.38416
$764$	0	0
$765$	8.72957	0.315618
$766$	0	0
$767$	2.90074	0.104740
$768$	0	0
$769$	−33.3574	−1.20290	−0.601449	−	0.798911i	$-0.705410\pi$
$769$	−33.3574	−1.20290	−0.601449	+	0.798911i	$0.705410\pi$
$770$	0	0
$771$	−32.9806	−1.18777
$772$	0	0
$773$	26.5636	0.955428	0.477714	−	0.878515i	$-0.341466\pi$
$773$	26.5636	0.955428	0.477714	+	0.878515i	$0.341466\pi$
$774$	0	0
$775$	−10.9618	−0.393760
$776$	0	0
$777$	36.9174	1.32440
$778$	0	0
$779$	−4.98171	−0.178488
$780$	0	0
$781$	−3.14997	−0.112715
$782$	0	0
$783$	−26.7376	−0.955524
$784$	0	0
$785$	−3.80505	−0.135808
$786$	0	0
$787$	4.50150	0.160461	0.0802307	−	0.996776i	$-0.474434\pi$
$787$	4.50150	0.160461	0.0802307	+	0.996776i	$0.474434\pi$
$788$	0	0
$789$	28.0038	0.996962
$790$	0	0
$791$	−18.4196	−0.654925
$792$	0	0
$793$	−1.15942	−0.0411722
$794$	0	0
$795$	5.86531	0.208021
$796$	0	0
$797$	42.7960	1.51591	0.757955	−	0.652306i	$-0.226198\pi$
$797$	42.7960	1.51591	0.757955	+	0.652306i	$0.226198\pi$
$798$	0	0
$799$	−8.22604	−0.291016
$800$	0	0
$801$	33.2511	1.17487
$802$	0	0
$803$	−9.88914	−0.348980
$804$	0	0
$805$	−40.5417	−1.42891
$806$	0	0
$807$	−68.5213	−2.41206
$808$	0	0
$809$	−25.6361	−0.901318	−0.450659	−	0.892696i	$-0.648811\pi$
$809$	−25.6361	−0.901318	−0.450659	+	0.892696i	$0.648811\pi$
$810$	0	0
$811$	29.2508	1.02713	0.513567	−	0.858049i	$-0.328324\pi$
$811$	29.2508	1.02713	0.513567	+	0.858049i	$0.328324\pi$
$812$	0	0
$813$	64.6950	2.26895
$814$	0	0
$815$	−5.41481	−0.189673
$816$	0	0
$817$	−16.5812	−0.580102
$818$	0	0
$819$	7.18600	0.251099
$820$	0	0
$821$	−35.8326	−1.25057	−0.625284	−	0.780397i	$-0.715017\pi$
$821$	−35.8326	−1.25057	−0.625284	+	0.780397i	$0.715017\pi$
$822$	0	0
$823$	36.6338	1.27697	0.638487	−	0.769633i	$-0.279561\pi$
$823$	36.6338	1.27697	0.638487	+	0.769633i	$0.279561\pi$
$824$	0	0
$825$	4.85608	0.169067
$826$	0	0
$827$	−7.95018	−0.276455	−0.138227	−	0.990401i	$-0.544140\pi$
$827$	−7.95018	−0.276455	−0.138227	+	0.990401i	$0.544140\pi$
$828$	0	0
$829$	33.4773	1.16271	0.581357	−	0.813648i	$-0.302522\pi$
$829$	33.4773	1.16271	0.581357	+	0.813648i	$0.302522\pi$
$830$	0	0
$831$	−16.9663	−0.588555
$832$	0	0
$833$	9.67784	0.335317
$834$	0	0
$835$	37.4401	1.29567
$836$	0	0
$837$	88.4946	3.05882
$838$	0	0
$839$	−17.8703	−0.616952	−0.308476	−	0.951232i	$-0.599819\pi$
$839$	−17.8703	−0.616952	−0.308476	+	0.951232i	$0.599819\pi$
$840$	0	0
$841$	−24.3750	−0.840516
$842$	0	0
$843$	−45.5318	−1.56820
$844$	0	0
$845$	−24.0878	−0.828645
$846$	0	0
$847$	4.61706	0.158644
$848$	0	0
$849$	67.1705	2.30529
$850$	0	0
$851$	−11.9705	−0.410342
$852$	0	0
$853$	−27.1773	−0.930532	−0.465266	−	0.885171i	$-0.654041\pi$
$853$	−27.1773	−0.930532	−0.465266	+	0.885171i	$0.654041\pi$
$854$	0	0
$855$	30.3704	1.03865
$856$	0	0
$857$	27.3380	0.933847	0.466923	−	0.884298i	$-0.345362\pi$
$857$	27.3380	0.933847	0.466923	+	0.884298i	$0.345362\pi$
$858$	0	0
$859$	11.4337	0.390113	0.195056	−	0.980792i	$-0.437511\pi$
$859$	11.4337	0.390113	0.195056	+	0.980792i	$0.437511\pi$
$860$	0	0
$861$	−30.8405	−1.05104
$862$	0	0
$863$	17.8233	0.606714	0.303357	−	0.952877i	$-0.401893\pi$
$863$	17.8233	0.606714	0.303357	+	0.952877i	$0.401893\pi$
$864$	0	0
$865$	−12.2489	−0.416474
$866$	0	0
$867$	−52.1641	−1.77159
$868$	0	0
$869$	−2.66870	−0.0905293
$870$	0	0
$871$	−3.57096	−0.120997
$872$	0	0
$873$	−102.615	−3.47299
$874$	0	0
$875$	−56.1670	−1.89879
$876$	0	0
$877$	33.5350	1.13239	0.566197	−	0.824270i	$-0.308414\pi$
$877$	33.5350	1.13239	0.566197	+	0.824270i	$0.308414\pi$
$878$	0	0
$879$	−91.2845	−3.07895
$880$	0	0
$881$	52.9584	1.78421	0.892107	−	0.451824i	$-0.149227\pi$
$881$	52.9584	1.78421	0.892107	+	0.451824i	$0.149227\pi$
$882$	0	0
$883$	37.6154	1.26586	0.632929	−	0.774210i	$-0.281853\pi$
$883$	37.6154	1.26586	0.632929	+	0.774210i	$0.281853\pi$
$884$	0	0
$885$	75.8954	2.55120
$886$	0	0
$887$	−8.38568	−0.281564	−0.140782	−	0.990041i	$-0.544962\pi$
$887$	−8.38568	−0.281564	−0.140782	+	0.990041i	$0.544962\pi$
$888$	0	0
$889$	39.8957	1.33806
$890$	0	0
$891$	−18.3746	−0.615571
$892$	0	0
$893$	−28.6186	−0.957687
$894$	0	0
$895$	13.4166	0.448468
$896$	0	0
$897$	−3.33688	−0.111415
$898$	0	0
$899$	−15.3077	−0.510540
$900$	0	0
$901$	0.675958	0.0225194
$902$	0	0
$903$	−102.650	−3.41597
$904$	0	0
$905$	−5.85242	−0.194541
$906$	0	0
$907$	−13.2511	−0.439996	−0.219998	−	0.975500i	$-0.570605\pi$
$907$	−13.2511	−0.439996	−0.219998	+	0.975500i	$0.570605\pi$
$908$	0	0
$909$	101.458	3.36516
$910$	0	0
$911$	1.82403	0.0604328	0.0302164	−	0.999543i	$-0.490380\pi$
$911$	1.82403	0.0604328	0.0302164	+	0.999543i	$0.490380\pi$
$912$	0	0
$913$	5.65481	0.187147
$914$	0	0
$915$	−30.3352	−1.00285
$916$	0	0
$917$	69.7200	2.30236
$918$	0	0
$919$	26.3722	0.869938	0.434969	−	0.900446i	$-0.356759\pi$
$919$	26.3722	0.869938	0.434969	+	0.900446i	$0.356759\pi$
$920$	0	0
$921$	63.3886	2.08872
$922$	0	0
$923$	0.706139	0.0232429
$924$	0	0
$925$	−3.90518	−0.128401
$926$	0	0
$927$	−71.5515	−2.35006
$928$	0	0
$929$	−34.2843	−1.12483	−0.562416	−	0.826854i	$-0.690128\pi$
$929$	−34.2843	−1.12483	−0.562416	+	0.826854i	$0.690128\pi$
$930$	0	0
$931$	33.6695	1.10347
$932$	0	0
$933$	−49.3923	−1.61703
$934$	0	0
$935$	−1.25735	−0.0411197
$936$	0	0
$937$	−31.3936	−1.02558	−0.512792	−	0.858513i	$-0.671389\pi$
$937$	−31.3936	−1.02558	−0.512792	+	0.858513i	$0.671389\pi$
$938$	0	0
$939$	−25.1944	−0.822188
$940$	0	0
$941$	−13.0046	−0.423938	−0.211969	−	0.977276i	$-0.567988\pi$
$941$	−13.0046	−0.423938	−0.211969	+	0.977276i	$0.567988\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	0	0
$945$	106.774	3.47336
$946$	0	0
$947$	−6.07712	−0.197480	−0.0987399	−	0.995113i	$-0.531481\pi$
$947$	−6.07712	−0.197480	−0.0987399	+	0.995113i	$0.531481\pi$
$948$	0	0
$949$	2.21688	0.0719631
$950$	0	0
$951$	31.2710	1.01403
$952$	0	0
$953$	56.0711	1.81632	0.908161	−	0.418622i	$-0.137487\pi$
$953$	56.0711	1.81632	0.908161	+	0.418622i	$0.137487\pi$
$954$	0	0
$955$	7.06988	0.228776
$956$	0	0
$957$	6.78130	0.219208
$958$	0	0
$959$	12.4252	0.401231
$960$	0	0
$961$	19.6645	0.634338
$962$	0	0
$963$	57.7309	1.86035
$964$	0	0
$965$	−16.1388	−0.519527
$966$	0	0
$967$	−4.33845	−0.139515	−0.0697576	−	0.997564i	$-0.522223\pi$
$967$	−4.33845	−0.139515	−0.0697576	+	0.997564i	$0.522223\pi$
$968$	0	0
$969$	5.01248	0.161024
$970$	0	0
$971$	35.8151	1.14936	0.574680	−	0.818378i	$-0.305127\pi$
$971$	35.8151	1.14936	0.574680	+	0.818378i	$0.305127\pi$
$972$	0	0
$973$	−76.9331	−2.46636
$974$	0	0
$975$	−1.08861	−0.0348633
$976$	0	0
$977$	−54.6925	−1.74977	−0.874884	−	0.484332i	$-0.839063\pi$
$977$	−54.6925	−1.74977	−0.874884	+	0.484332i	$0.839063\pi$
$978$	0	0
$979$	−4.78927	−0.153066
$980$	0	0
$981$	−99.0307	−3.16181
$982$	0	0
$983$	−13.3236	−0.424958	−0.212479	−	0.977166i	$-0.568154\pi$
$983$	−13.3236	−0.424958	−0.212479	+	0.977166i	$0.568154\pi$
$984$	0	0
$985$	8.47397	0.270003
$986$	0	0
$987$	−177.171	−5.63941
$988$	0	0
$989$	33.2842	1.05838
$990$	0	0
$991$	−24.1614	−0.767511	−0.383755	−	0.923435i	$-0.625369\pi$
$991$	−24.1614	−0.767511	−0.383755	+	0.923435i	$0.625369\pi$
$992$	0	0
$993$	−65.1221	−2.06659
$994$	0	0
$995$	11.7479	0.372434
$996$	0	0
$997$	50.9957	1.61505	0.807526	−	0.589832i	$-0.200806\pi$
$997$	50.9957	1.61505	0.807526	+	0.589832i	$0.200806\pi$
$998$	0	0
$999$	31.5265	0.997453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9328.2.a.bm.1.11		11
4.3	odd	2		4664.2.a.k.1.1	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4664.2.a.k.1.1	✓	11	4.3	odd	2
9328.2.a.bm.1.11		11	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:	\( N \)	\(=\)	\( 9328 = 2^{4} \cdot 11 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9328.a (trivial)

\(f(q)\)	\(=\)	\(q+3.15323 q^{3} +1.86010 q^{5} +4.61706 q^{7} +6.94284 q^{9} +O(q^{10})\)	\(q+3.15323 q^{3} +1.86010 q^{5} +4.61706 q^{7} +6.94284 q^{9} -1.00000 q^{11} +0.224174 q^{13} +5.86531 q^{15} +0.675958 q^{17} +2.35168 q^{19} +14.5586 q^{21} -4.72064 q^{23} -1.54004 q^{25} +12.4327 q^{27} -2.15059 q^{29} +7.11790 q^{31} -3.15323 q^{33} +8.58818 q^{35} +2.53577 q^{37} +0.706870 q^{39} -2.11836 q^{41} -7.05078 q^{43} +12.9144 q^{45} -12.1695 q^{47} +14.3172 q^{49} +2.13145 q^{51} +1.00000 q^{53} -1.86010 q^{55} +7.41538 q^{57} +12.9397 q^{59} -5.17197 q^{61} +32.0555 q^{63} +0.416985 q^{65} -15.9295 q^{67} -14.8852 q^{69} +3.14997 q^{71} +9.88914 q^{73} -4.85608 q^{75} -4.61706 q^{77} +2.66870 q^{79} +18.3746 q^{81} -5.65481 q^{83} +1.25735 q^{85} -6.78130 q^{87} +4.78927 q^{89} +1.03502 q^{91} +22.4444 q^{93} +4.37435 q^{95} -14.7800 q^{97} -6.94284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{3} + 3 q^{5} + 5 q^{7} + 7 q^{9}+O(q^{10}) \) 11 * q + 6 * q^3 + 3 * q^5 + 5 * q^7 + 7 * q^9	\( 11 q + 6 q^{3} + 3 q^{5} + 5 q^{7} + 7 q^{9} - 11 q^{11} - 13 q^{13} + 8 q^{15} - 7 q^{17} + 21 q^{19} + 6 q^{21} - 11 q^{23} + 4 q^{25} + 6 q^{27} + 5 q^{31} - 6 q^{33} + 25 q^{35} - 4 q^{37} + 19 q^{39} - 11 q^{41} + 16 q^{45} + 17 q^{47} - 2 q^{49} + 18 q^{51} + 11 q^{53} - 3 q^{55} - 5 q^{57} + 19 q^{59} - 2 q^{61} + 36 q^{63} - 13 q^{65} - 25 q^{67} + 3 q^{69} + 30 q^{71} + 5 q^{73} + 5 q^{75} - 5 q^{77} + 23 q^{79} - 9 q^{81} + 19 q^{83} + 2 q^{85} + 7 q^{87} + 6 q^{89} + 20 q^{91} + 43 q^{93} + 50 q^{95} - 35 q^{97} - 7 q^{99}+O(q^{100}) \) 11 * q + 6 * q^3 + 3 * q^5 + 5 * q^7 + 7 * q^9 - 11 * q^11 - 13 * q^13 + 8 * q^15 - 7 * q^17 + 21 * q^19 + 6 * q^21 - 11 * q^23 + 4 * q^25 + 6 * q^27 + 5 * q^31 - 6 * q^33 + 25 * q^35 - 4 * q^37 + 19 * q^39 - 11 * q^41 + 16 * q^45 + 17 * q^47 - 2 * q^49 + 18 * q^51 + 11 * q^53 - 3 * q^55 - 5 * q^57 + 19 * q^59 - 2 * q^61 + 36 * q^63 - 13 * q^65 - 25 * q^67 + 3 * q^69 + 30 * q^71 + 5 * q^73 + 5 * q^75 - 5 * q^77 + 23 * q^79 - 9 * q^81 + 19 * q^83 + 2 * q^85 + 7 * q^87 + 6 * q^89 + 20 * q^91 + 43 * q^93 + 50 * q^95 - 35 * q^97 - 7 * q^99

Introduction

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