LMFDB - Embedded newform 2300.2.a.m.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2300,2,Mod(1,2300)] 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("2300.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.53121.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} - x + 2 $ x^4 - 7*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.66337$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66337 q^{3} +0.750930 q^{7} +4.09352 q^{9} -4.57580 q^{11} +3.84445 q^{13} +7.32673 q^{17} +6.57580 q^{19} +2.00000 q^{21} -1.00000 q^{23} +2.91244 q^{27} -7.00595 q^{29} +4.43015 q^{31} -12.1870 q^{33} -0.860303 q^{37} +10.2392 q^{39} -9.60133 q^{41} -6.57580 q^{43} +10.9184 q^{47} -6.43610 q^{49} +19.5138 q^{51} +1.82487 q^{53} +17.5138 q^{57} +7.57580 q^{59} +9.82487 q^{61} +3.07394 q^{63} +10.1870 q^{67} -2.66337 q^{69} -4.42025 q^{71} +4.50781 q^{73} -3.43610 q^{77} +12.0777 q^{79} -4.52367 q^{81} -11.9025 q^{83} -18.6594 q^{87} +3.86402 q^{89} +2.88691 q^{91} +11.7991 q^{93} +0.537289 q^{97} -18.7311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{7} + 2 q^{9} + q^{11} - q^{13} + 8 q^{17} + 7 q^{19} + 8 q^{21} - 4 q^{23} + 3 q^{27} - 5 q^{29} + 14 q^{31} - 20 q^{33} + 4 q^{37} + 11 q^{39} + 3 q^{41} - 7 q^{43} + 12 q^{47} + q^{49} + 28 q^{51}+ \cdots - 9 q^{99}+O(q^{100}) $ 4 * q + q^7 + 2 * q^9 + q^11 - q^13 + 8 * q^17 + 7 * q^19 + 8 * q^21 - 4 * q^23 + 3 * q^27 - 5 * q^29 + 14 * q^31 - 20 * q^33 + 4 * q^37 + 11 * q^39 + 3 * q^41 - 7 * q^43 + 12 * q^47 + q^49 + 28 * q^51 - 10 * q^53 + 20 * q^57 + 11 * q^59 + 22 * q^61 - 3 * q^63 + 12 * q^67 + 18 * q^71 - 9 * q^73 + 13 * q^77 + 25 * q^79 - 7 * q^83 - 9 * q^87 + 25 * q^91 - 11 * q^93 + 8 * q^97 - 9 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.66337	1.53769	0.768847	−	0.639432i	$-0.220831\pi$
$3$	2.66337	1.53769	0.768847	+	0.639432i	$0.220831\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.750930	0.283825	0.141912	−	0.989879i	$-0.454675\pi$
$7$	0.750930	0.283825	0.141912	+	0.989879i	$0.454675\pi$
$8$	0	0
$9$	4.09352	1.36451
$10$	0	0
$11$	−4.57580	−1.37966	−0.689828	−	0.723973i	$-0.742314\pi$
$11$	−4.57580	−1.37966	−0.689828	+	0.723973i	$0.742314\pi$
$12$	0	0
$13$	3.84445	1.06626	0.533129	−	0.846034i	$-0.321016\pi$
$13$	3.84445	1.06626	0.533129	+	0.846034i	$0.321016\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.32673	1.77699	0.888497	−	0.458883i	$-0.151750\pi$
$17$	7.32673	1.77699	0.888497	+	0.458883i	$0.151750\pi$
$18$	0	0
$19$	6.57580	1.50859	0.754296	−	0.656534i	$-0.227978\pi$
$19$	6.57580	1.50859	0.754296	+	0.656534i	$0.227978\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.91244	0.560499
$28$	0	0
$29$	−7.00595	−1.30097	−0.650486	−	0.759518i	$-0.725435\pi$
$29$	−7.00595	−1.30097	−0.650486	+	0.759518i	$0.725435\pi$
$30$	0	0
$31$	4.43015	0.795679	0.397839	−	0.917455i	$-0.369760\pi$
$31$	4.43015	0.795679	0.397839	+	0.917455i	$0.369760\pi$
$32$	0	0
$33$	−12.1870	−2.12149
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.860303	−0.141433	−0.0707165	−	0.997496i	$-0.522529\pi$
$37$	−0.860303	−0.141433	−0.0707165	+	0.997496i	$0.522529\pi$
$38$	0	0
$39$	10.2392	1.63958
$40$	0	0
$41$	−9.60133	−1.49948	−0.749738	−	0.661735i	$-0.769820\pi$
$41$	−9.60133	−1.49948	−0.749738	+	0.661735i	$0.769820\pi$
$42$	0	0
$43$	−6.57580	−1.00280	−0.501400	−	0.865215i	$-0.667182\pi$
$43$	−6.57580	−1.00280	−0.501400	+	0.865215i	$0.667182\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.9184	1.59261	0.796305	−	0.604895i	$-0.206785\pi$
$47$	10.9184	1.59261	0.796305	+	0.604895i	$0.206785\pi$
$48$	0	0
$49$	−6.43610	−0.919444
$50$	0	0
$51$	19.5138	2.73247
$52$	0	0
$53$	1.82487	0.250666	0.125333	−	0.992115i	$-0.460000\pi$
$53$	1.82487	0.250666	0.125333	+	0.992115i	$0.460000\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	17.5138	2.31975
$58$	0	0
$59$	7.57580	0.986285	0.493143	−	0.869948i	$-0.335848\pi$
$59$	7.57580	0.986285	0.493143	+	0.869948i	$0.335848\pi$
$60$	0	0
$61$	9.82487	1.25795	0.628973	−	0.777427i	$-0.283476\pi$
$61$	9.82487	1.25795	0.628973	+	0.777427i	$0.283476\pi$
$62$	0	0
$63$	3.07394	0.387280
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.1870	1.24454	0.622272	−	0.782801i	$-0.286210\pi$
$67$	10.1870	1.24454	0.622272	+	0.782801i	$0.286210\pi$
$68$	0	0
$69$	−2.66337	−0.320632
$70$	0	0
$71$	−4.42025	−0.524587	−0.262294	−	0.964988i	$-0.584479\pi$
$71$	−4.42025	−0.524587	−0.262294	+	0.964988i	$0.584479\pi$
$72$	0	0
$73$	4.50781	0.527600	0.263800	−	0.964577i	$-0.415024\pi$
$73$	4.50781	0.527600	0.263800	+	0.964577i	$0.415024\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.43610	−0.391580
$78$	0	0
$79$	12.0777	1.35884	0.679422	−	0.733748i	$-0.262231\pi$
$79$	12.0777	1.35884	0.679422	+	0.733748i	$0.262231\pi$
$80$	0	0
$81$	−4.52367	−0.502630
$82$	0	0
$83$	−11.9025	−1.30647	−0.653236	−	0.757154i	$-0.726589\pi$
$83$	−11.9025	−1.30647	−0.653236	+	0.757154i	$0.726589\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−18.6594	−2.00050
$88$	0	0
$89$	3.86402	0.409585	0.204793	−	0.978805i	$-0.434348\pi$
$89$	3.86402	0.409585	0.204793	+	0.978805i	$0.434348\pi$
$90$	0	0
$91$	2.88691	0.302630
$92$	0	0
$93$	11.7991	1.22351
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.537289	0.0545535	0.0272767	−	0.999628i	$-0.491316\pi$
$97$	0.537289	0.0545535	0.0272767	+	0.999628i	$0.491316\pi$
$98$	0	0
$99$	−18.7311	−1.88255
$100$	0	0
$101$	2.78636	0.277253	0.138627	−	0.990345i	$-0.455731\pi$
$101$	2.78636	0.277253	0.138627	+	0.990345i	$0.455731\pi$
$102$	0	0
$103$	−4.25279	−0.419040	−0.209520	−	0.977804i	$-0.567190\pi$
$103$	−4.25279	−0.419040	−0.209520	+	0.977804i	$0.567190\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.68889	−0.549966	−0.274983	−	0.961449i	$-0.588672\pi$
$107$	−5.68889	−0.549966	−0.274983	+	0.961449i	$0.588672\pi$
$108$	0	0
$109$	8.68518	0.831889	0.415944	−	0.909390i	$-0.363451\pi$
$109$	8.68518	0.831889	0.415944	+	0.909390i	$0.363451\pi$
$110$	0	0
$111$	−2.29130	−0.217481
$112$	0	0
$113$	−8.96457	−0.843316	−0.421658	−	0.906755i	$-0.638552\pi$
$113$	−8.96457	−0.843316	−0.421658	+	0.906755i	$0.638552\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	15.7373	1.45491
$118$	0	0
$119$	5.50186	0.504355
$120$	0	0
$121$	9.93796	0.903451
$122$	0	0
$123$	−25.5719	−2.30574
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.430151	0.0381698	0.0190849	−	0.999818i	$-0.493925\pi$
$127$	0.430151	0.0381698	0.0190849	+	0.999818i	$0.493925\pi$
$128$	0	0
$129$	−17.5138	−1.54200
$130$	0	0
$131$	13.4920	1.17880	0.589399	−	0.807842i	$-0.299365\pi$
$131$	13.4920	1.17880	0.589399	+	0.807842i	$0.299365\pi$
$132$	0	0
$133$	4.93796	0.428176
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.65346	−0.739315	−0.369658	−	0.929168i	$-0.620525\pi$
$137$	−8.65346	−0.739315	−0.369658	+	0.929168i	$0.620525\pi$
$138$	0	0
$139$	8.77050	0.743904	0.371952	−	0.928252i	$-0.378689\pi$
$139$	8.77050	0.743904	0.371952	+	0.928252i	$0.378689\pi$
$140$	0	0
$141$	29.0797	2.44895
$142$	0	0
$143$	−17.5914	−1.47107
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−17.1417	−1.41382
$148$	0	0
$149$	−0.143415	−0.0117490	−0.00587452	−	0.999983i	$-0.501870\pi$
$149$	−0.143415	−0.0117490	−0.00587452	+	0.999983i	$0.501870\pi$
$150$	0	0
$151$	−20.7943	−1.69222	−0.846109	−	0.533010i	$-0.821061\pi$
$151$	−20.7943	−1.69222	−0.846109	+	0.533010i	$0.821061\pi$
$152$	0	0
$153$	29.9921	2.42472
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.6654	−1.48966	−0.744829	−	0.667255i	$-0.767469\pi$
$157$	−18.6654	−1.48966	−0.744829	+	0.667255i	$0.767469\pi$
$158$	0	0
$159$	4.86030	0.385447
$160$	0	0
$161$	−0.750930	−0.0591815
$162$	0	0
$163$	−3.21959	−0.252178	−0.126089	−	0.992019i	$-0.540242\pi$
$163$	−3.21959	−0.252178	−0.126089	+	0.992019i	$0.540242\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.3001	−0.797048	−0.398524	−	0.917158i	$-0.630477\pi$
$167$	−10.3001	−0.797048	−0.398524	+	0.917158i	$0.630477\pi$
$168$	0	0
$169$	1.77977	0.136905
$170$	0	0
$171$	26.9182	2.05848
$172$	0	0
$173$	−18.1516	−1.38004	−0.690020	−	0.723790i	$-0.742398\pi$
$173$	−18.1516	−1.38004	−0.690020	+	0.723790i	$0.742398\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.1771	1.51661
$178$	0	0
$179$	−0.935729	−0.0699396	−0.0349698	−	0.999388i	$-0.511134\pi$
$179$	−0.935729	−0.0699396	−0.0349698	+	0.999388i	$0.511134\pi$
$180$	0	0
$181$	15.6572	1.16379	0.581895	−	0.813264i	$-0.302312\pi$
$181$	15.6572	1.16379	0.581895	+	0.813264i	$0.302312\pi$
$182$	0	0
$183$	26.1672	1.93434
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−33.5257	−2.45164
$188$	0	0
$189$	2.18703	0.159083
$190$	0	0
$191$	12.8988	0.933326	0.466663	−	0.884435i	$-0.345456\pi$
$191$	12.8988	0.933326	0.466663	+	0.884435i	$0.345456\pi$
$192$	0	0
$193$	−15.5356	−1.11828	−0.559138	−	0.829075i	$-0.688868\pi$
$193$	−15.5356	−1.11828	−0.559138	+	0.829075i	$0.688868\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.27088	0.660523	0.330261	−	0.943890i	$-0.392863\pi$
$197$	9.27088	0.660523	0.330261	+	0.943890i	$0.392863\pi$
$198$	0	0
$199$	5.89063	0.417575	0.208788	−	0.977961i	$-0.433048\pi$
$199$	5.89063	0.417575	0.208788	+	0.977961i	$0.433048\pi$
$200$	0	0
$201$	27.1318	1.91373
$202$	0	0
$203$	−5.26098	−0.369248
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.09352	−0.284519
$208$	0	0
$209$	−30.0896	−2.08134
$210$	0	0
$211$	−14.7274	−1.01388	−0.506938	−	0.861982i	$-0.669223\pi$
$211$	−14.7274	−1.01388	−0.506938	+	0.861982i	$0.669223\pi$
$212$	0	0
$213$	−11.7727	−0.806655
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.32673	0.225833
$218$	0	0
$219$	12.0060	0.811287
$220$	0	0
$221$	28.1672	1.89473
$222$	0	0
$223$	3.42048	0.229052	0.114526	−	0.993420i	$-0.463465\pi$
$223$	3.42048	0.229052	0.114526	+	0.993420i	$0.463465\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.4214	1.55453	0.777267	−	0.629171i	$-0.216606\pi$
$227$	23.4214	1.55453	0.777267	+	0.629171i	$0.216606\pi$
$228$	0	0
$229$	−22.3032	−1.47384	−0.736919	−	0.675981i	$-0.763720\pi$
$229$	−22.3032	−1.47384	−0.736919	+	0.675981i	$0.763720\pi$
$230$	0	0
$231$	−9.15160	−0.602131
$232$	0	0
$233$	−14.3817	−0.942179	−0.471089	−	0.882085i	$-0.656139\pi$
$233$	−14.3817	−0.942179	−0.471089	+	0.882085i	$0.656139\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	32.1672	2.08949
$238$	0	0
$239$	−20.0337	−1.29587	−0.647937	−	0.761694i	$-0.724368\pi$
$239$	−20.0337	−1.29587	−0.647937	+	0.761694i	$0.724368\pi$
$240$	0	0
$241$	−19.2950	−1.24290	−0.621451	−	0.783453i	$-0.713456\pi$
$241$	−19.2950	−1.24290	−0.621451	+	0.783453i	$0.713456\pi$
$242$	0	0
$243$	−20.7855	−1.33339
$244$	0	0
$245$	0	0
$246$	0	0
$247$	25.2803	1.60855
$248$	0	0
$249$	−31.7008	−2.00896
$250$	0	0
$251$	−15.4095	−0.972639	−0.486319	−	0.873781i	$-0.661661\pi$
$251$	−15.4095	−0.972639	−0.486319	+	0.873781i	$0.661661\pi$
$252$	0	0
$253$	4.57580	0.287678
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.1454	−1.38139	−0.690697	−	0.723145i	$-0.742696\pi$
$257$	−22.1454	−1.38139	−0.690697	+	0.723145i	$0.742696\pi$
$258$	0	0
$259$	−0.646027	−0.0401422
$260$	0	0
$261$	−28.6790	−1.77518
$262$	0	0
$263$	17.2344	1.06272	0.531358	−	0.847147i	$-0.321682\pi$
$263$	17.2344	1.06272	0.531358	+	0.847147i	$0.321682\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.2913	0.629817
$268$	0	0
$269$	18.1477	1.10648	0.553241	−	0.833021i	$-0.313391\pi$
$269$	18.1477	1.10648	0.553241	+	0.833021i	$0.313391\pi$
$270$	0	0
$271$	−27.3883	−1.66372	−0.831861	−	0.554985i	$-0.812724\pi$
$271$	−27.3883	−1.66372	−0.831861	+	0.554985i	$0.812724\pi$
$272$	0	0
$273$	7.68889	0.465353
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−25.7567	−1.54757	−0.773784	−	0.633450i	$-0.781638\pi$
$277$	−25.7567	−1.54757	−0.773784	+	0.633450i	$0.781638\pi$
$278$	0	0
$279$	18.1349	1.08571
$280$	0	0
$281$	−20.5100	−1.22353	−0.611763	−	0.791041i	$-0.709540\pi$
$281$	−20.5100	−1.22353	−0.611763	+	0.791041i	$0.709540\pi$
$282$	0	0
$283$	20.1479	1.19767	0.598834	−	0.800873i	$-0.295631\pi$
$283$	20.1479	1.19767	0.598834	+	0.800873i	$0.295631\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.20992	−0.425588
$288$	0	0
$289$	36.6810	2.15771
$290$	0	0
$291$	1.43100	0.0838866
$292$	0	0
$293$	−6.33045	−0.369829	−0.184914	−	0.982755i	$-0.559201\pi$
$293$	−6.33045	−0.369829	−0.184914	+	0.982755i	$0.559201\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−13.3267	−0.773295
$298$	0	0
$299$	−3.84445	−0.222330
$300$	0	0
$301$	−4.93796	−0.284620
$302$	0	0
$303$	7.42110	0.426331
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−26.8722	−1.53368	−0.766839	−	0.641839i	$-0.778172\pi$
$307$	−26.8722	−1.53368	−0.766839	+	0.641839i	$0.778172\pi$
$308$	0	0
$309$	−11.3267	−0.644355
$310$	0	0
$311$	−9.49899	−0.538638	−0.269319	−	0.963051i	$-0.586799\pi$
$311$	−9.49899	−0.538638	−0.269319	+	0.963051i	$0.586799\pi$
$312$	0	0
$313$	−31.6810	−1.79072	−0.895358	−	0.445347i	$-0.853080\pi$
$313$	−31.6810	−1.79072	−0.895358	+	0.445347i	$0.853080\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.4398	1.14801	0.574007	−	0.818850i	$-0.305388\pi$
$317$	20.4398	1.14801	0.574007	+	0.818850i	$0.305388\pi$
$318$	0	0
$319$	32.0579	1.79490
$320$	0	0
$321$	−15.1516	−0.845680
$322$	0	0
$323$	48.1791	2.68076
$324$	0	0
$325$	0	0
$326$	0	0
$327$	23.1318	1.27919
$328$	0	0
$329$	8.19894	0.452022
$330$	0	0
$331$	−2.26949	−0.124743	−0.0623713	−	0.998053i	$-0.519866\pi$
$331$	−2.26949	−0.124743	−0.0623713	+	0.998053i	$0.519866\pi$
$332$	0	0
$333$	−3.52166	−0.192986
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.11617	0.224222	0.112111	−	0.993696i	$-0.464239\pi$
$337$	4.11617	0.224222	0.112111	+	0.993696i	$0.464239\pi$
$338$	0	0
$339$	−23.8759	−1.29676
$340$	0	0
$341$	−20.2715	−1.09776
$342$	0	0
$343$	−10.0896	−0.544785
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.9499	−1.39306	−0.696531	−	0.717527i	$-0.745274\pi$
$347$	−25.9499	−1.39306	−0.696531	+	0.717527i	$0.745274\pi$
$348$	0	0
$349$	18.8482	1.00892	0.504460	−	0.863435i	$-0.331692\pi$
$349$	18.8482	1.00892	0.504460	+	0.863435i	$0.331692\pi$
$350$	0	0
$351$	11.1967	0.597636
$352$	0	0
$353$	−11.8807	−0.632347	−0.316174	−	0.948701i	$-0.602398\pi$
$353$	−11.8807	−0.632347	−0.316174	+	0.948701i	$0.602398\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	14.6535	0.775544
$358$	0	0
$359$	15.6957	0.828387	0.414194	−	0.910189i	$-0.364064\pi$
$359$	15.6957	0.828387	0.414194	+	0.910189i	$0.364064\pi$
$360$	0	0
$361$	24.2412	1.27585
$362$	0	0
$363$	26.4684	1.38923
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.0858	0.578677	0.289338	−	0.957227i	$-0.406565\pi$
$367$	11.0858	0.578677	0.289338	+	0.957227i	$0.406565\pi$
$368$	0	0
$369$	−39.3032	−2.04604
$370$	0	0
$371$	1.37035	0.0711451
$372$	0	0
$373$	30.5100	1.57975	0.789875	−	0.613268i	$-0.210145\pi$
$373$	30.5100	1.57975	0.789875	+	0.613268i	$0.210145\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−26.9340	−1.38717
$378$	0	0
$379$	12.5299	0.643615	0.321808	−	0.946805i	$-0.395710\pi$
$379$	12.5299	0.643615	0.321808	+	0.946805i	$0.395710\pi$
$380$	0	0
$381$	1.14565	0.0586935
$382$	0	0
$383$	−17.4480	−0.891552	−0.445776	−	0.895145i	$-0.647072\pi$
$383$	−17.4480	−0.891552	−0.445776	+	0.895145i	$0.647072\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−26.9182	−1.36833
$388$	0	0
$389$	8.98391	0.455502	0.227751	−	0.973719i	$-0.426863\pi$
$389$	8.98391	0.455502	0.227751	+	0.973719i	$0.426863\pi$
$390$	0	0
$391$	−7.32673	−0.370529
$392$	0	0
$393$	35.9340	1.81263
$394$	0	0
$395$	0	0
$396$	0	0
$397$	18.7334	0.940200	0.470100	−	0.882613i	$-0.344218\pi$
$397$	18.7334	0.940200	0.470100	+	0.882613i	$0.344218\pi$
$398$	0	0
$399$	13.1516	0.658404
$400$	0	0
$401$	−30.6852	−1.53234	−0.766172	−	0.642635i	$-0.777841\pi$
$401$	−30.6852	−1.53234	−0.766172	+	0.642635i	$0.777841\pi$
$402$	0	0
$403$	17.0315	0.848399
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.93658	0.195129
$408$	0	0
$409$	−7.05957	−0.349073	−0.174537	−	0.984651i	$-0.555843\pi$
$409$	−7.05957	−0.349073	−0.174537	+	0.984651i	$0.555843\pi$
$410$	0	0
$411$	−23.0473	−1.13684
$412$	0	0
$413$	5.68889	0.279932
$414$	0	0
$415$	0	0
$416$	0	0
$417$	23.3591	1.14390
$418$	0	0
$419$	10.5486	0.515331	0.257665	−	0.966234i	$-0.417047\pi$
$419$	10.5486	0.515331	0.257665	+	0.966234i	$0.417047\pi$
$420$	0	0
$421$	−28.4902	−1.38853	−0.694265	−	0.719720i	$-0.744270\pi$
$421$	−28.4902	−1.38853	−0.694265	+	0.719720i	$0.744270\pi$
$422$	0	0
$423$	44.6946	2.17313
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.37779	0.357036
$428$	0	0
$429$	−46.8524	−2.26206
$430$	0	0
$431$	39.1437	1.88549	0.942743	−	0.333520i	$-0.108237\pi$
$431$	39.1437	1.88549	0.942743	+	0.333520i	$0.108237\pi$
$432$	0	0
$433$	31.7932	1.52788	0.763941	−	0.645286i	$-0.223262\pi$
$433$	31.7932	1.52788	0.763941	+	0.645286i	$0.223262\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.57580	−0.314563
$438$	0	0
$439$	35.1928	1.67966	0.839829	−	0.542851i	$-0.182655\pi$
$439$	35.1928	1.67966	0.839829	+	0.542851i	$0.182655\pi$
$440$	0	0
$441$	−26.3463	−1.25459
$442$	0	0
$443$	17.7034	0.841112	0.420556	−	0.907267i	$-0.361835\pi$
$443$	17.7034	0.841112	0.420556	+	0.907267i	$0.361835\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.381967	−0.0180664
$448$	0	0
$449$	4.44354	0.209704	0.104852	−	0.994488i	$-0.466563\pi$
$449$	4.44354	0.209704	0.104852	+	0.994488i	$0.466563\pi$
$450$	0	0
$451$	43.9338	2.06876
$452$	0	0
$453$	−55.3829	−2.60211
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.2797	1.55676	0.778379	−	0.627795i	$-0.216042\pi$
$457$	33.2797	1.55676	0.778379	+	0.627795i	$0.216042\pi$
$458$	0	0
$459$	21.3386	0.996002
$460$	0	0
$461$	−39.8440	−1.85572	−0.927860	−	0.372930i	$-0.878353\pi$
$461$	−39.8440	−1.85572	−0.927860	+	0.372930i	$0.878353\pi$
$462$	0	0
$463$	−24.7274	−1.14918	−0.574590	−	0.818441i	$-0.694838\pi$
$463$	−24.7274	−1.14918	−0.574590	+	0.818441i	$0.694838\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.47154	0.299467	0.149733	−	0.988726i	$-0.452158\pi$
$467$	6.47154	0.299467	0.149733	+	0.988726i	$0.452158\pi$
$468$	0	0
$469$	7.64974	0.353232
$470$	0	0
$471$	−49.7127	−2.29064
$472$	0	0
$473$	30.0896	1.38352
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.47015	0.342035
$478$	0	0
$479$	0.658570	0.0300908	0.0150454	−	0.999887i	$-0.495211\pi$
$479$	0.658570	0.0300908	0.0150454	+	0.999887i	$0.495211\pi$
$480$	0	0
$481$	−3.30739	−0.150804
$482$	0	0
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.25703	0.283533	0.141767	−	0.989900i	$-0.454722\pi$
$487$	6.25703	0.283533	0.141767	+	0.989900i	$0.454722\pi$
$488$	0	0
$489$	−8.57496	−0.387773
$490$	0	0
$491$	20.7309	0.935572	0.467786	−	0.883842i	$-0.345052\pi$
$491$	20.7309	0.935572	0.467786	+	0.883842i	$0.345052\pi$
$492$	0	0
$493$	−51.3307	−2.31182
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.31930	−0.148891
$498$	0	0
$499$	−22.4684	−1.00583	−0.502913	−	0.864337i	$-0.667738\pi$
$499$	−22.4684	−1.00583	−0.502913	+	0.864337i	$0.667738\pi$
$500$	0	0
$501$	−27.4330	−1.22562
$502$	0	0
$503$	−22.9499	−1.02328	−0.511642	−	0.859199i	$-0.670963\pi$
$503$	−22.9499	−1.02328	−0.511642	+	0.859199i	$0.670963\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.74018	0.210519
$508$	0	0
$509$	−25.9369	−1.14963	−0.574816	−	0.818283i	$-0.694926\pi$
$509$	−25.9369	−1.14963	−0.574816	+	0.818283i	$0.694926\pi$
$510$	0	0
$511$	3.38505	0.149746
$512$	0	0
$513$	19.1516	0.845564
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−49.9604	−2.19726
$518$	0	0
$519$	−48.3444	−2.12208
$520$	0	0
$521$	−0.454522	−0.0199130	−0.00995648	−	0.999950i	$-0.503169\pi$
$521$	−0.454522	−0.0199130	−0.00995648	+	0.999950i	$0.503169\pi$
$522$	0	0
$523$	6.23345	0.272570	0.136285	−	0.990670i	$-0.456484\pi$
$523$	6.23345	0.272570	0.136285	+	0.990670i	$0.456484\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.4585	1.41392
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	31.0117	1.34579
$532$	0	0
$533$	−36.9118	−1.59883
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.49219	−0.107546
$538$	0	0
$539$	29.4503	1.26852
$540$	0	0
$541$	35.3355	1.51919	0.759597	−	0.650394i	$-0.225396\pi$
$541$	35.3355	1.51919	0.759597	+	0.650394i	$0.225396\pi$
$542$	0	0
$543$	41.7008	1.78955
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−38.5047	−1.64634	−0.823171	−	0.567793i	$-0.807797\pi$
$547$	−38.5047	−1.64634	−0.823171	+	0.567793i	$0.807797\pi$
$548$	0	0
$549$	40.2183	1.71647
$550$	0	0
$551$	−46.0698	−1.96264
$552$	0	0
$553$	9.06947	0.385673
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.6770	0.664255	0.332128	−	0.943234i	$-0.392234\pi$
$557$	15.6770	0.664255	0.332128	+	0.943234i	$0.392234\pi$
$558$	0	0
$559$	−25.2803	−1.06924
$560$	0	0
$561$	−89.2911	−3.76987
$562$	0	0
$563$	−10.4596	−0.440821	−0.220410	−	0.975407i	$-0.570740\pi$
$563$	−10.4596	−0.440821	−0.220410	+	0.975407i	$0.570740\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.39696	−0.142659
$568$	0	0
$569$	−11.3148	−0.474342	−0.237171	−	0.971468i	$-0.576220\pi$
$569$	−11.3148	−0.474342	−0.237171	+	0.971468i	$0.576220\pi$
$570$	0	0
$571$	−35.8363	−1.49970	−0.749852	−	0.661606i	$-0.769875\pi$
$571$	−35.8363	−1.49970	−0.749852	+	0.661606i	$0.769875\pi$
$572$	0	0
$573$	34.3543	1.43517
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−23.5305	−0.979586	−0.489793	−	0.871839i	$-0.662928\pi$
$577$	−23.5305	−0.979586	−0.489793	+	0.871839i	$0.662928\pi$
$578$	0	0
$579$	−41.3769	−1.71957
$580$	0	0
$581$	−8.93796	−0.370809
$582$	0	0
$583$	−8.35026	−0.345832
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.2369	−0.876542	−0.438271	−	0.898843i	$-0.644409\pi$
$587$	−21.2369	−0.876542	−0.438271	+	0.898843i	$0.644409\pi$
$588$	0	0
$589$	29.1318	1.20035
$590$	0	0
$591$	24.6917	1.01568
$592$	0	0
$593$	−12.9454	−0.531604	−0.265802	−	0.964028i	$-0.585637\pi$
$593$	−12.9454	−0.531604	−0.265802	+	0.964028i	$0.585637\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.6889	0.642104
$598$	0	0
$599$	−24.9536	−1.01958	−0.509788	−	0.860300i	$-0.670276\pi$
$599$	−24.9536	−1.01958	−0.509788	+	0.860300i	$0.670276\pi$
$600$	0	0
$601$	−33.9365	−1.38430	−0.692149	−	0.721755i	$-0.743336\pi$
$601$	−33.9365	−1.38430	−0.692149	+	0.721755i	$0.743336\pi$
$602$	0	0
$603$	41.7008	1.69819
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.66090	−0.351535	−0.175768	−	0.984432i	$-0.556241\pi$
$607$	−8.66090	−0.351535	−0.175768	+	0.984432i	$0.556241\pi$
$608$	0	0
$609$	−14.0119	−0.567791
$610$	0	0
$611$	41.9752	1.69813
$612$	0	0
$613$	−13.4503	−0.543254	−0.271627	−	0.962403i	$-0.587562\pi$
$613$	−13.4503	−0.543254	−0.271627	+	0.962403i	$0.587562\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.880108	0.0354318	0.0177159	−	0.999843i	$-0.494361\pi$
$617$	0.880108	0.0354318	0.0177159	+	0.999843i	$0.494361\pi$
$618$	0	0
$619$	16.1360	0.648560	0.324280	−	0.945961i	$-0.394878\pi$
$619$	16.1360	0.648560	0.324280	+	0.945961i	$0.394878\pi$
$620$	0	0
$621$	−2.91244	−0.116872
$622$	0	0
$623$	2.90161	0.116250
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−80.1395	−3.20046
$628$	0	0
$629$	−6.30321	−0.251325
$630$	0	0
$631$	40.3611	1.60675	0.803374	−	0.595474i	$-0.203036\pi$
$631$	40.3611	1.60675	0.803374	+	0.595474i	$0.203036\pi$
$632$	0	0
$633$	−39.2245	−1.55903
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−24.7433	−0.980364
$638$	0	0
$639$	−18.0944	−0.715802
$640$	0	0
$641$	−9.34654	−0.369166	−0.184583	−	0.982817i	$-0.559093\pi$
$641$	−9.34654	−0.369166	−0.184583	+	0.982817i	$0.559093\pi$
$642$	0	0
$643$	−1.51887	−0.0598985	−0.0299492	−	0.999551i	$-0.509535\pi$
$643$	−1.51887	−0.0598985	−0.0299492	+	0.999551i	$0.509535\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.8969	1.48988	0.744940	−	0.667132i	$-0.232478\pi$
$647$	37.8969	1.48988	0.744940	+	0.667132i	$0.232478\pi$
$648$	0	0
$649$	−34.6654	−1.36073
$650$	0	0
$651$	8.86030	0.347263
$652$	0	0
$653$	29.6005	1.15836	0.579178	−	0.815201i	$-0.303374\pi$
$653$	29.6005	1.15836	0.579178	+	0.815201i	$0.303374\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	18.4528	0.719913
$658$	0	0
$659$	19.0326	0.741406	0.370703	−	0.928751i	$-0.379117\pi$
$659$	19.0326	0.741406	0.370703	+	0.928751i	$0.379117\pi$
$660$	0	0
$661$	−19.5257	−0.759461	−0.379730	−	0.925097i	$-0.623983\pi$
$661$	−19.5257	−0.759461	−0.379730	+	0.925097i	$0.623983\pi$
$662$	0	0
$663$	75.0196	2.91352
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.00595	0.271272
$668$	0	0
$669$	9.10999	0.352213
$670$	0	0
$671$	−44.9567	−1.73553
$672$	0	0
$673$	20.6685	0.796711	0.398356	−	0.917231i	$-0.369581\pi$
$673$	20.6685	0.796711	0.398356	+	0.917231i	$0.369581\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.9765	0.883058	0.441529	−	0.897247i	$-0.354436\pi$
$677$	22.9765	0.883058	0.441529	+	0.897247i	$0.354436\pi$
$678$	0	0
$679$	0.403467	0.0154836
$680$	0	0
$681$	62.3798	2.39040
$682$	0	0
$683$	−19.8805	−0.760706	−0.380353	−	0.924841i	$-0.624197\pi$
$683$	−19.8805	−0.760706	−0.380353	+	0.924841i	$0.624197\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−59.4016	−2.26631
$688$	0	0
$689$	7.01562	0.267274
$690$	0	0
$691$	29.3038	1.11477	0.557385	−	0.830254i	$-0.311805\pi$
$691$	29.3038	1.11477	0.557385	+	0.830254i	$0.311805\pi$
$692$	0	0
$693$	−14.0658	−0.534314
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−70.3464	−2.66456
$698$	0	0
$699$	−38.3038	−1.44878
$700$	0	0
$701$	−11.9757	−0.452317	−0.226158	−	0.974091i	$-0.572617\pi$
$701$	−11.9757	−0.452317	−0.226158	+	0.974091i	$0.572617\pi$
$702$	0	0
$703$	−5.65718	−0.213365
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.09236	0.0786913
$708$	0	0
$709$	−45.8244	−1.72097	−0.860486	−	0.509474i	$-0.829840\pi$
$709$	−45.8244	−1.72097	−0.860486	+	0.509474i	$0.829840\pi$
$710$	0	0
$711$	49.4401	1.85415
$712$	0	0
$713$	−4.43015	−0.165910
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−53.3571	−1.99266
$718$	0	0
$719$	14.6003	0.544498	0.272249	−	0.962227i	$-0.412233\pi$
$719$	14.6003	0.544498	0.272249	+	0.962227i	$0.412233\pi$
$720$	0	0
$721$	−3.19354	−0.118934
$722$	0	0
$723$	−51.3897	−1.91120
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.40376	0.126238	0.0631192	−	0.998006i	$-0.479895\pi$
$727$	3.40376	0.126238	0.0631192	+	0.998006i	$0.479895\pi$
$728$	0	0
$729$	−41.7884	−1.54772
$730$	0	0
$731$	−48.1791	−1.78197
$732$	0	0
$733$	−5.41909	−0.200159	−0.100079	−	0.994979i	$-0.531910\pi$
$733$	−5.41909	−0.200159	−0.100079	+	0.994979i	$0.531910\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−46.6139	−1.71704
$738$	0	0
$739$	−0.235218	−0.00865265	−0.00432633	−	0.999991i	$-0.501377\pi$
$739$	−0.235218	−0.00865265	−0.00432633	+	0.999991i	$0.501377\pi$
$740$	0	0
$741$	67.3307	2.47346
$742$	0	0
$743$	34.6660	1.27177	0.635886	−	0.771783i	$-0.280635\pi$
$743$	34.6660	1.27177	0.635886	+	0.771783i	$0.280635\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−48.7232	−1.78269
$748$	0	0
$749$	−4.27196	−0.156094
$750$	0	0
$751$	29.6697	1.08266	0.541332	−	0.840809i	$-0.317920\pi$
$751$	29.6697	1.08266	0.541332	+	0.840809i	$0.317920\pi$
$752$	0	0
$753$	−41.0411	−1.49562
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.0119	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$757$	18.0119	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$758$	0	0
$759$	12.1870	0.442361
$760$	0	0
$761$	22.8700	0.829036	0.414518	−	0.910041i	$-0.363950\pi$
$761$	22.8700	0.829036	0.414518	+	0.910041i	$0.363950\pi$
$762$	0	0
$763$	6.52195	0.236111
$764$	0	0
$765$	0	0
$766$	0	0
$767$	29.1248	1.05163
$768$	0	0
$769$	17.5336	0.632277	0.316138	−	0.948713i	$-0.397614\pi$
$769$	17.5336	0.632277	0.316138	+	0.948713i	$0.397614\pi$
$770$	0	0
$771$	−58.9814	−2.12416
$772$	0	0
$773$	−38.9760	−1.40187	−0.700935	−	0.713225i	$-0.747234\pi$
$773$	−38.9760	−1.40187	−0.700935	+	0.713225i	$0.747234\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.72061	−0.0617264
$778$	0	0
$779$	−63.1364	−2.26210
$780$	0	0
$781$	20.2262	0.723750
$782$	0	0
$783$	−20.4044	−0.729194
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15.6883	0.559226	0.279613	−	0.960113i	$-0.409794\pi$
$787$	15.6883	0.559226	0.279613	+	0.960113i	$0.409794\pi$
$788$	0	0
$789$	45.9014	1.63413
$790$	0	0
$791$	−6.73176	−0.239354
$792$	0	0
$793$	37.7712	1.34129
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.2381	1.00024	0.500122	−	0.865955i	$-0.333288\pi$
$797$	28.2381	1.00024	0.500122	+	0.865955i	$0.333288\pi$
$798$	0	0
$799$	79.9961	2.83006
$800$	0	0
$801$	15.8174	0.558882
$802$	0	0
$803$	−20.6269	−0.727906
$804$	0	0
$805$	0	0
$806$	0	0
$807$	48.3338	1.70143
$808$	0	0
$809$	9.99628	0.351451	0.175725	−	0.984439i	$-0.443773\pi$
$809$	9.99628	0.351451	0.175725	+	0.984439i	$0.443773\pi$
$810$	0	0
$811$	24.1624	0.848455	0.424227	−	0.905556i	$-0.360546\pi$
$811$	24.1624	0.848455	0.424227	+	0.905556i	$0.360546\pi$
$812$	0	0
$813$	−72.9451	−2.55830
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−43.2412	−1.51282
$818$	0	0
$819$	11.8176	0.412941
$820$	0	0
$821$	43.7379	1.52646	0.763232	−	0.646124i	$-0.223611\pi$
$821$	43.7379	1.52646	0.763232	+	0.646124i	$0.223611\pi$
$822$	0	0
$823$	44.9031	1.56523	0.782613	−	0.622509i	$-0.213887\pi$
$823$	44.9031	1.56523	0.782613	+	0.622509i	$0.213887\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.9573	−0.520117	−0.260058	−	0.965593i	$-0.583742\pi$
$827$	−14.9573	−0.520117	−0.260058	+	0.965593i	$0.583742\pi$
$828$	0	0
$829$	32.5965	1.13212	0.566062	−	0.824363i	$-0.308466\pi$
$829$	32.5965	1.13212	0.566062	+	0.824363i	$0.308466\pi$
$830$	0	0
$831$	−68.5994	−2.37969
$832$	0	0
$833$	−47.1556	−1.63385
$834$	0	0
$835$	0	0
$836$	0	0
$837$	12.9025	0.445977
$838$	0	0
$839$	24.8552	0.858097	0.429048	−	0.903282i	$-0.358849\pi$
$839$	24.8552	0.858097	0.429048	+	0.903282i	$0.358849\pi$
$840$	0	0
$841$	20.0834	0.692530
$842$	0	0
$843$	−54.6258	−1.88141
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.46271	0.256422
$848$	0	0
$849$	53.6612	1.84165
$850$	0	0
$851$	0.860303	0.0294908
$852$	0	0
$853$	18.5639	0.635616	0.317808	−	0.948155i	$-0.397053\pi$
$853$	18.5639	0.635616	0.317808	+	0.948155i	$0.397053\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.8042	0.710659	0.355329	−	0.934741i	$-0.384369\pi$
$857$	20.8042	0.710659	0.355329	+	0.934741i	$0.384369\pi$
$858$	0	0
$859$	30.6563	1.04598	0.522990	−	0.852339i	$-0.324816\pi$
$859$	30.6563	1.04598	0.522990	+	0.852339i	$0.324816\pi$
$860$	0	0
$861$	−19.2027	−0.654425
$862$	0	0
$863$	4.51949	0.153845	0.0769226	−	0.997037i	$-0.475491\pi$
$863$	4.51949	0.153845	0.0769226	+	0.997037i	$0.475491\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	97.6949	3.31789
$868$	0	0
$869$	−55.2650	−1.87474
$870$	0	0
$871$	39.1635	1.32700
$872$	0	0
$873$	2.19940	0.0744385
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.1391	−0.409907	−0.204953	−	0.978772i	$-0.565704\pi$
$877$	−12.1391	−0.409907	−0.204953	+	0.978772i	$0.565704\pi$
$878$	0	0
$879$	−16.8603	−0.568684
$880$	0	0
$881$	31.9410	1.07612	0.538061	−	0.842906i	$-0.319157\pi$
$881$	31.9410	1.07612	0.538061	+	0.842906i	$0.319157\pi$
$882$	0	0
$883$	40.2605	1.35487	0.677437	−	0.735581i	$-0.263091\pi$
$883$	40.2605	1.35487	0.677437	+	0.735581i	$0.263091\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.82734	−0.0613560	−0.0306780	−	0.999529i	$-0.509767\pi$
$887$	−1.82734	−0.0613560	−0.0306780	+	0.999529i	$0.509767\pi$
$888$	0	0
$889$	0.323013	0.0108335
$890$	0	0
$891$	20.6994	0.693456
$892$	0	0
$893$	71.7972	2.40260
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−10.2392	−0.341876
$898$	0	0
$899$	−31.0374	−1.03516
$900$	0	0
$901$	13.3704	0.445431
$902$	0	0
$903$	−13.1516	−0.437658
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14.6409	−0.486144	−0.243072	−	0.970008i	$-0.578155\pi$
$907$	−14.6409	−0.486144	−0.243072	+	0.970008i	$0.578155\pi$
$908$	0	0
$909$	11.4060	0.378314
$910$	0	0
$911$	−20.3400	−0.673895	−0.336948	−	0.941523i	$-0.609395\pi$
$911$	−20.3400	−0.673895	−0.336948	+	0.941523i	$0.609395\pi$
$912$	0	0
$913$	54.4636	1.80248
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.1315	0.334572
$918$	0	0
$919$	−1.93488	−0.0638259	−0.0319129	−	0.999491i	$-0.510160\pi$
$919$	−1.93488	−0.0638259	−0.0319129	+	0.999491i	$0.510160\pi$
$920$	0	0
$921$	−71.5705	−2.35833
$922$	0	0
$923$	−16.9934	−0.559345
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.4089	−0.571782
$928$	0	0
$929$	−43.4526	−1.42563	−0.712816	−	0.701351i	$-0.752581\pi$
$929$	−43.4526	−1.42563	−0.712816	+	0.701351i	$0.752581\pi$
$930$	0	0
$931$	−42.3226	−1.38707
$932$	0	0
$933$	−25.2993	−0.828261
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.2797	0.499166	0.249583	−	0.968353i	$-0.419707\pi$
$937$	15.2797	0.499166	0.249583	+	0.968353i	$0.419707\pi$
$938$	0	0
$939$	−84.3781	−2.75357
$940$	0	0
$941$	38.2455	1.24677	0.623384	−	0.781916i	$-0.285757\pi$
$941$	38.2455	1.24677	0.623384	+	0.781916i	$0.285757\pi$
$942$	0	0
$943$	9.60133	0.312662
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.1746	−1.50047	−0.750236	−	0.661170i	$-0.770060\pi$
$947$	−46.1746	−1.50047	−0.750236	+	0.661170i	$0.770060\pi$
$948$	0	0
$949$	17.3300	0.562557
$950$	0	0
$951$	54.4387	1.76530
$952$	0	0
$953$	−5.19865	−0.168401	−0.0842004	−	0.996449i	$-0.526834\pi$
$953$	−5.19865	−0.168401	−0.0842004	+	0.996449i	$0.526834\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	85.3818	2.76000
$958$	0	0
$959$	−6.49814	−0.209836
$960$	0	0
$961$	−11.3738	−0.366895
$962$	0	0
$963$	−23.2876	−0.750432
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−35.3163	−1.13569	−0.567847	−	0.823134i	$-0.692224\pi$
$967$	−35.3163	−1.13569	−0.567847	+	0.823134i	$0.692224\pi$
$968$	0	0
$969$	128.319	4.12219
$970$	0	0
$971$	−10.3277	−0.331430	−0.165715	−	0.986174i	$-0.552993\pi$
$971$	−10.3277	−0.331430	−0.165715	+	0.986174i	$0.552993\pi$
$972$	0	0
$973$	6.58603	0.211138
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.5921	−0.754777	−0.377388	−	0.926055i	$-0.623178\pi$
$977$	−23.5921	−0.754777	−0.377388	+	0.926055i	$0.623178\pi$
$978$	0	0
$979$	−17.6810	−0.565087
$980$	0	0
$981$	35.5529	1.13512
$982$	0	0
$983$	−7.32610	−0.233666	−0.116833	−	0.993152i	$-0.537274\pi$
$983$	−7.32610	−0.233666	−0.116833	+	0.993152i	$0.537274\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	21.8368	0.695072
$988$	0	0
$989$	6.57580	0.209098
$990$	0	0
$991$	61.8864	1.96589	0.982944	−	0.183906i	$-0.0588741\pi$
$991$	61.8864	1.96589	0.982944	+	0.183906i	$0.0588741\pi$
$992$	0	0
$993$	−6.04449	−0.191816
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.8960	0.946817	0.473408	−	0.880843i	$-0.343024\pi$
$997$	29.8960	0.946817	0.473408	+	0.880843i	$0.343024\pi$
$998$	0	0
$999$	−2.50558	−0.0792730

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.a.m.1.4	yes	4
4.3	odd	2		9200.2.a.cm.1.1		4
5.2	odd	4		2300.2.c.j.1749.1		8
5.3	odd	4		2300.2.c.j.1749.8		8
5.4	even	2		2300.2.a.l.1.1	✓	4
20.19	odd	2		9200.2.a.co.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.a.l.1.1	✓	4	5.4	even	2
2300.2.a.m.1.4	yes	4	1.1	even	1	trivial
2300.2.c.j.1749.1		8	5.2	odd	4
2300.2.c.j.1749.8		8	5.3	odd	4
9200.2.a.cm.1.1		4	4.3	odd	2
9200.2.a.co.1.4		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q+2.66337 q^{3} +0.750930 q^{7} +4.09352 q^{9} -4.57580 q^{11} +3.84445 q^{13} +7.32673 q^{17} +6.57580 q^{19} +2.00000 q^{21} -1.00000 q^{23} +2.91244 q^{27} -7.00595 q^{29} +4.43015 q^{31} -12.1870 q^{33} -0.860303 q^{37} +10.2392 q^{39} -9.60133 q^{41} -6.57580 q^{43} +10.9184 q^{47} -6.43610 q^{49} +19.5138 q^{51} +1.82487 q^{53} +17.5138 q^{57} +7.57580 q^{59} +9.82487 q^{61} +3.07394 q^{63} +10.1870 q^{67} -2.66337 q^{69} -4.42025 q^{71} +4.50781 q^{73} -3.43610 q^{77} +12.0777 q^{79} -4.52367 q^{81} -11.9025 q^{83} -18.6594 q^{87} +3.86402 q^{89} +2.88691 q^{91} +11.7991 q^{93} +0.537289 q^{97} -18.7311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{7} + 2 q^{9} + q^{11} - q^{13} + 8 q^{17} + 7 q^{19} + 8 q^{21} - 4 q^{23} + 3 q^{27} - 5 q^{29} + 14 q^{31} - 20 q^{33} + 4 q^{37} + 11 q^{39} + 3 q^{41} - 7 q^{43} + 12 q^{47} + q^{49} + 28 q^{51}+ \cdots - 9 q^{99}+O(q^{100}) \) 4 * q + q^7 + 2 * q^9 + q^11 - q^13 + 8 * q^17 + 7 * q^19 + 8 * q^21 - 4 * q^23 + 3 * q^27 - 5 * q^29 + 14 * q^31 - 20 * q^33 + 4 * q^37 + 11 * q^39 + 3 * q^41 - 7 * q^43 + 12 * q^47 + q^49 + 28 * q^51 - 10 * q^53 + 20 * q^57 + 11 * q^59 + 22 * q^61 - 3 * q^63 + 12 * q^67 + 18 * q^71 - 9 * q^73 + 13 * q^77 + 25 * q^79 - 7 * q^83 - 9 * q^87 + 25 * q^91 - 11 * q^93 + 8 * q^97 - 9 * q^99

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