LMFDB - Embedded newform 1568.3.c.h.97.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,3,Mod(97,1568)] 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("1568.97"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-32,0,0,0, 112,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(53)] == traces)

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

Self dual:	no
Analytic conductor:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 36x^{14} + 522x^{12} + 3644x^{10} + 12219x^{8} + 15156x^{6} + 15478x^{4} - 10992x^{2} + 11025 $ x^16 + 36x^14 + 522x^12 + 3644x^10 + 12219x^8 + 15156x^6 + 15478x^4 - 10992*x^2 + 11025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.9
Root			$-0.707107 - 0.358323i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.h.97.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.506745i q^{3} -5.30261i q^{5} +8.74321 q^{9} +17.0900 q^{11} -21.4744i q^{13} +2.68707 q^{15} +24.0168i q^{17} +12.2203i q^{19} +40.2929 q^{23} -3.11765 q^{25} +8.99128i q^{27} -26.0770 q^{29} -25.2635i q^{31} +8.66025i q^{33} +12.9625 q^{37} +10.8820 q^{39} +33.8721i q^{41} -29.9958 q^{43} -46.3618i q^{45} +55.7476i q^{47} -12.1704 q^{51} -8.72723 q^{53} -90.6214i q^{55} -6.19259 q^{57} -49.9862i q^{59} +3.93621i q^{61} -113.870 q^{65} +105.885 q^{67} +20.4182i q^{69} +35.0232 q^{71} -46.6166i q^{73} -1.57985i q^{75} -86.9995 q^{79} +74.1326 q^{81} -64.0079i q^{83} +127.352 q^{85} -13.2144i q^{87} +42.9863i q^{89} +12.8022 q^{93} +64.7996 q^{95} -28.7493i q^{97} +149.421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9} - 32 q^{25} + 112 q^{29} - 16 q^{37} + 48 q^{53} - 528 q^{57} + 16 q^{65} - 64 q^{81} + 720 q^{85} + 464 q^{93}+O(q^{100}) $ 16 * q - 16 * q^9 - 32 * q^25 + 112 * q^29 - 16 * q^37 + 48 * q^53 - 528 * q^57 + 16 * q^65 - 64 * q^81 + 720 * q^85 + 464 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.506745i	0.168915i	0.996427	+	0.0844575i	$0.0269157\pi$
$3$	0.506745i	0.168915i	−0.996427	+	0.0844575i	$0.973084\pi$
$4$	0	0
$5$	− 5.30261i	− 1.06052i	−0.847835	−	0.530261i	$-0.822094\pi$
$5$	− 5.30261i	− 1.06052i	0.847835	−	0.530261i	$-0.177906\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.74321	0.971468
$10$	0	0
$11$	17.0900	1.55363	0.776817	−	0.629727i	$-0.216833\pi$
$11$	17.0900	1.55363	0.776817	+	0.629727i	$0.216833\pi$
$12$	0	0
$13$	− 21.4744i	− 1.65187i	−0.563762	−	0.825937i	$-0.690647\pi$
$13$	− 21.4744i	− 1.65187i	0.563762	−	0.825937i	$-0.309353\pi$
$14$	0	0
$15$	2.68707	0.179138
$16$	0	0
$17$	24.0168i	1.41276i	0.707835	+	0.706378i	$0.249672\pi$
$17$	24.0168i	1.41276i	−0.707835	+	0.706378i	$0.750328\pi$
$18$	0	0
$19$	12.2203i	0.643175i	0.946880	+	0.321588i	$0.104216\pi$
$19$	12.2203i	0.643175i	−0.946880	+	0.321588i	$0.895784\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	40.2929	1.75186	0.875932	−	0.482435i	$-0.160248\pi$
$23$	40.2929	1.75186	0.875932	+	0.482435i	$0.160248\pi$
$24$	0	0
$25$	−3.11765	−0.124706
$26$	0	0
$27$	8.99128i	0.333010i
$28$	0	0
$29$	−26.0770	−0.899209	−0.449604	−	0.893228i	$-0.648435\pi$
$29$	−26.0770	−0.899209	−0.449604	+	0.893228i	$0.648435\pi$
$30$	0	0
$31$	− 25.2635i	− 0.814953i	−0.913216	−	0.407477i	$-0.866409\pi$
$31$	− 25.2635i	− 0.814953i	0.913216	−	0.407477i	$-0.133591\pi$
$32$	0	0
$33$	8.66025i	0.262432i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	12.9625	0.350339	0.175169	−	0.984538i	$-0.443953\pi$
$37$	12.9625	0.350339	0.175169	+	0.984538i	$0.443953\pi$
$38$	0	0
$39$	10.8820	0.279026
$40$	0	0
$41$	33.8721i	0.826150i	0.910697	+	0.413075i	$0.135545\pi$
$41$	33.8721i	0.826150i	−0.910697	+	0.413075i	$0.864455\pi$
$42$	0	0
$43$	−29.9958	−0.697576	−0.348788	−	0.937202i	$-0.613407\pi$
$43$	−29.9958	−0.697576	−0.348788	+	0.937202i	$0.613407\pi$
$44$	0	0
$45$	− 46.3618i	− 1.03026i
$46$	0	0
$47$	55.7476i	1.18612i	0.805159	+	0.593060i	$0.202080\pi$
$47$	55.7476i	1.18612i	−0.805159	+	0.593060i	$0.797920\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−12.1704	−0.238636
$52$	0	0
$53$	−8.72723	−0.164665	−0.0823324	−	0.996605i	$-0.526237\pi$
$53$	−8.72723	−0.164665	−0.0823324	+	0.996605i	$0.526237\pi$
$54$	0	0
$55$	− 90.6214i	− 1.64766i
$56$	0	0
$57$	−6.19259	−0.108642
$58$	0	0
$59$	− 49.9862i	− 0.847224i	−0.905844	−	0.423612i	$-0.860762\pi$
$59$	− 49.9862i	− 0.847224i	0.905844	−	0.423612i	$-0.139238\pi$
$60$	0	0
$61$	3.93621i	0.0645281i	0.999479	+	0.0322640i	$0.0102717\pi$
$61$	3.93621i	0.0645281i	−0.999479	+	0.0322640i	$0.989728\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−113.870	−1.75185
$66$	0	0
$67$	105.885	1.58038	0.790189	−	0.612864i	$-0.209982\pi$
$67$	105.885	1.58038	0.790189	+	0.612864i	$0.209982\pi$
$68$	0	0
$69$	20.4182i	0.295916i
$70$	0	0
$71$	35.0232	0.493285	0.246642	−	0.969107i	$-0.420673\pi$
$71$	35.0232	0.493285	0.246642	+	0.969107i	$0.420673\pi$
$72$	0	0
$73$	− 46.6166i	− 0.638584i	−0.947656	−	0.319292i	$-0.896555\pi$
$73$	− 46.6166i	− 0.638584i	0.947656	−	0.319292i	$-0.103445\pi$
$74$	0	0
$75$	− 1.57985i	− 0.0210647i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−86.9995	−1.10126	−0.550630	−	0.834750i	$-0.685612\pi$
$79$	−86.9995	−1.10126	−0.550630	+	0.834750i	$0.685612\pi$
$80$	0	0
$81$	74.1326	0.915217
$82$	0	0
$83$	− 64.0079i	− 0.771180i	−0.922670	−	0.385590i	$-0.873998\pi$
$83$	− 64.0079i	− 0.771180i	0.922670	−	0.385590i	$-0.126002\pi$
$84$	0	0
$85$	127.352	1.49826
$86$	0	0
$87$	− 13.2144i	− 0.151890i
$88$	0	0
$89$	42.9863i	0.482992i	0.970402	+	0.241496i	$0.0776381\pi$
$89$	42.9863i	0.482992i	−0.970402	+	0.241496i	$0.922362\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12.8022	0.137658
$94$	0	0
$95$	64.7996	0.682101
$96$	0	0
$97$	− 28.7493i	− 0.296384i	−0.988959	−	0.148192i	$-0.952655\pi$
$97$	− 28.7493i	− 0.296384i	0.988959	−	0.148192i	$-0.0473454\pi$
$98$	0	0
$99$	149.421	1.50930
$100$	0	0
$101$	61.4690i	0.608604i	0.952576	+	0.304302i	$0.0984232\pi$
$101$	61.4690i	0.608604i	−0.952576	+	0.304302i	$0.901577\pi$
$102$	0	0
$103$	− 59.7284i	− 0.579887i	−0.957044	−	0.289944i	$-0.906363\pi$
$103$	− 59.7284i	− 0.579887i	0.957044	−	0.289944i	$-0.0936366\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	131.676	1.23062	0.615308	−	0.788287i	$-0.289032\pi$
$107$	131.676	1.23062	0.615308	+	0.788287i	$0.289032\pi$
$108$	0	0
$109$	−37.8091	−0.346872	−0.173436	−	0.984845i	$-0.555487\pi$
$109$	−37.8091	−0.346872	−0.173436	+	0.984845i	$0.555487\pi$
$110$	0	0
$111$	6.56870i	0.0591774i
$112$	0	0
$113$	−46.4443	−0.411012	−0.205506	−	0.978656i	$-0.565884\pi$
$113$	−46.4443	−0.411012	−0.205506	+	0.978656i	$0.565884\pi$
$114$	0	0
$115$	− 213.657i	− 1.85789i
$116$	0	0
$117$	− 187.755i	− 1.60474i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	171.067	1.41378
$122$	0	0
$123$	−17.1645	−0.139549
$124$	0	0
$125$	− 116.034i	− 0.928268i
$126$	0	0
$127$	−12.7816	−0.100642	−0.0503211	−	0.998733i	$-0.516024\pi$
$127$	−12.7816	−0.100642	−0.0503211	+	0.998733i	$0.516024\pi$
$128$	0	0
$129$	− 15.2002i	− 0.117831i
$130$	0	0
$131$	− 1.07311i	− 0.00819166i	−0.999992	−	0.00409583i	$-0.998696\pi$
$131$	− 1.07311i	− 0.00819166i	0.999992	−	0.00409583i	$-0.00130375\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	47.6772	0.353165
$136$	0	0
$137$	−66.8659	−0.488072	−0.244036	−	0.969766i	$-0.578472\pi$
$137$	−66.8659	−0.488072	−0.244036	+	0.969766i	$0.578472\pi$
$138$	0	0
$139$	− 221.071i	− 1.59044i	−0.606322	−	0.795219i	$-0.707356\pi$
$139$	− 221.071i	− 1.59044i	0.606322	−	0.795219i	$-0.292644\pi$
$140$	0	0
$141$	−28.2498	−0.200353
$142$	0	0
$143$	− 366.996i	− 2.56641i
$144$	0	0
$145$	138.276i	0.953630i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−39.5212	−0.265243	−0.132621	−	0.991167i	$-0.542339\pi$
$149$	−39.5212	−0.265243	−0.132621	+	0.991167i	$0.542339\pi$
$150$	0	0
$151$	222.725	1.47500	0.737499	−	0.675348i	$-0.236006\pi$
$151$	222.725	1.47500	0.737499	+	0.675348i	$0.236006\pi$
$152$	0	0
$153$	209.984i	1.37245i
$154$	0	0
$155$	−133.963	−0.864275
$156$	0	0
$157$	202.166i	1.28768i	0.765160	+	0.643840i	$0.222660\pi$
$157$	202.166i	1.28768i	−0.765160	+	0.643840i	$0.777340\pi$
$158$	0	0
$159$	− 4.42248i	− 0.0278144i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.63017	−0.0161360	−0.00806802	−	0.999967i	$-0.502568\pi$
$163$	−2.63017	−0.0161360	−0.00806802	+	0.999967i	$0.502568\pi$
$164$	0	0
$165$	45.9219	0.278315
$166$	0	0
$167$	− 133.004i	− 0.796434i	−0.917291	−	0.398217i	$-0.869629\pi$
$167$	− 133.004i	− 0.796434i	0.917291	−	0.398217i	$-0.130371\pi$
$168$	0	0
$169$	−292.148	−1.72869
$170$	0	0
$171$	106.845i	0.624824i
$172$	0	0
$173$	97.0857i	0.561189i	0.959826	+	0.280594i	$0.0905316\pi$
$173$	97.0857i	0.561189i	−0.959826	+	0.280594i	$0.909468\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	25.3303	0.143109
$178$	0	0
$179$	84.0371	0.469481	0.234740	−	0.972058i	$-0.424576\pi$
$179$	84.0371	0.469481	0.234740	+	0.972058i	$0.424576\pi$
$180$	0	0
$181$	− 214.347i	− 1.18424i	−0.805851	−	0.592119i	$-0.798292\pi$
$181$	− 214.347i	− 1.18424i	0.805851	−	0.592119i	$-0.201708\pi$
$182$	0	0
$183$	−1.99466	−0.0108998
$184$	0	0
$185$	− 68.7352i	− 0.371542i
$186$	0	0
$187$	410.447i	2.19490i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	54.7374	0.286583	0.143292	−	0.989681i	$-0.454231\pi$
$191$	54.7374	0.286583	0.143292	+	0.989681i	$0.454231\pi$
$192$	0	0
$193$	−348.300	−1.80466	−0.902332	−	0.431042i	$-0.858146\pi$
$193$	−348.300	−1.80466	−0.902332	+	0.431042i	$0.858146\pi$
$194$	0	0
$195$	− 57.7031i	− 0.295913i
$196$	0	0
$197$	161.606	0.820337	0.410169	−	0.912010i	$-0.365470\pi$
$197$	161.606	0.820337	0.410169	+	0.912010i	$0.365470\pi$
$198$	0	0
$199$	137.624i	0.691579i	0.938312	+	0.345789i	$0.112389\pi$
$199$	137.624i	0.691579i	−0.938312	+	0.345789i	$0.887611\pi$
$200$	0	0
$201$	53.6568i	0.266949i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	179.611	0.876150
$206$	0	0
$207$	352.289	1.70188
$208$	0	0
$209$	208.845i	0.999258i
$210$	0	0
$211$	101.563	0.481341	0.240670	−	0.970607i	$-0.422633\pi$
$211$	101.563	0.481341	0.240670	+	0.970607i	$0.422633\pi$
$212$	0	0
$213$	17.7478i	0.0833232i
$214$	0	0
$215$	159.056i	0.739795i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	23.6227	0.107866
$220$	0	0
$221$	515.747	2.33370
$222$	0	0
$223$	180.573i	0.809744i	0.914373	+	0.404872i	$0.132684\pi$
$223$	180.573i	0.809744i	−0.914373	+	0.404872i	$0.867316\pi$
$224$	0	0
$225$	−27.2582	−0.121148
$226$	0	0
$227$	− 50.7082i	− 0.223384i	−0.993743	−	0.111692i	$-0.964373\pi$
$227$	− 50.7082i	− 0.223384i	0.993743	−	0.111692i	$-0.0356270\pi$
$228$	0	0
$229$	− 199.533i	− 0.871323i	−0.900111	−	0.435661i	$-0.856514\pi$
$229$	− 199.533i	− 0.871323i	0.900111	−	0.435661i	$-0.143486\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−177.158	−0.760335	−0.380167	−	0.924918i	$-0.624134\pi$
$233$	−177.158	−0.760335	−0.380167	+	0.924918i	$0.624134\pi$
$234$	0	0
$235$	295.608	1.25790
$236$	0	0
$237$	− 44.0866i	− 0.186019i
$238$	0	0
$239$	135.694	0.567756	0.283878	−	0.958860i	$-0.408379\pi$
$239$	135.694	0.567756	0.283878	+	0.958860i	$0.408379\pi$
$240$	0	0
$241$	− 316.241i	− 1.31220i	−0.754673	−	0.656101i	$-0.772205\pi$
$241$	− 316.241i	− 1.31220i	0.754673	−	0.656101i	$-0.227795\pi$
$242$	0	0
$243$	118.488i	0.487604i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	262.424	1.06244
$248$	0	0
$249$	32.4357	0.130264
$250$	0	0
$251$	− 40.1231i	− 0.159853i	−0.996801	−	0.0799265i	$-0.974531\pi$
$251$	− 40.1231i	− 0.159853i	0.996801	−	0.0799265i	$-0.0254686\pi$
$252$	0	0
$253$	688.603	2.72175
$254$	0	0
$255$	64.5349i	0.253078i
$256$	0	0
$257$	− 173.673i	− 0.675771i	−0.941187	−	0.337886i	$-0.890288\pi$
$257$	− 173.673i	− 0.675771i	0.941187	−	0.337886i	$-0.109712\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−227.997	−0.873552
$262$	0	0
$263$	−364.581	−1.38624	−0.693120	−	0.720822i	$-0.743764\pi$
$263$	−364.581	−1.38624	−0.693120	+	0.720822i	$0.743764\pi$
$264$	0	0
$265$	46.2771i	0.174631i
$266$	0	0
$267$	−21.7831	−0.0815846
$268$	0	0
$269$	− 17.4215i	− 0.0647638i	−0.999476	−	0.0323819i	$-0.989691\pi$
$269$	− 17.4215i	− 0.0647638i	0.999476	−	0.0323819i	$-0.0103093\pi$
$270$	0	0
$271$	130.264i	0.480680i	0.970689	+	0.240340i	$0.0772590\pi$
$271$	130.264i	0.480680i	−0.970689	+	0.240340i	$0.922741\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−53.2805	−0.193747
$276$	0	0
$277$	−249.190	−0.899601	−0.449801	−	0.893129i	$-0.648505\pi$
$277$	−249.190	−0.899601	−0.449801	+	0.893129i	$0.648505\pi$
$278$	0	0
$279$	− 220.885i	− 0.791701i
$280$	0	0
$281$	−197.454	−0.702684	−0.351342	−	0.936247i	$-0.614275\pi$
$281$	−197.454	−0.702684	−0.351342	+	0.936247i	$0.614275\pi$
$282$	0	0
$283$	− 214.330i	− 0.757350i	−0.925530	−	0.378675i	$-0.876380\pi$
$283$	− 214.330i	− 0.757350i	0.925530	−	0.378675i	$-0.123620\pi$
$284$	0	0
$285$	32.8369i	0.115217i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−287.809	−0.995879
$290$	0	0
$291$	14.5685	0.0500637
$292$	0	0
$293$	− 71.8385i	− 0.245182i	−0.992457	−	0.122591i	$-0.960880\pi$
$293$	− 71.8385i	− 0.245182i	0.992457	−	0.122591i	$-0.0391204\pi$
$294$	0	0
$295$	−265.057	−0.898499
$296$	0	0
$297$	153.661i	0.517376i
$298$	0	0
$299$	− 865.263i	− 2.89386i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−31.1491	−0.102802
$304$	0	0
$305$	20.8722	0.0684334
$306$	0	0
$307$	507.046i	1.65162i	0.563951	+	0.825808i	$0.309281\pi$
$307$	507.046i	1.65162i	−0.563951	+	0.825808i	$0.690719\pi$
$308$	0	0
$309$	30.2671	0.0979516
$310$	0	0
$311$	− 310.718i	− 0.999092i	−0.866287	−	0.499546i	$-0.833500\pi$
$311$	− 310.718i	− 0.999092i	0.866287	−	0.499546i	$-0.166500\pi$
$312$	0	0
$313$	548.907i	1.75370i	0.480767	+	0.876848i	$0.340358\pi$
$313$	548.907i	1.75370i	−0.480767	+	0.876848i	$0.659642\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	141.930	0.447730	0.223865	−	0.974620i	$-0.428133\pi$
$317$	141.930	0.447730	0.223865	+	0.974620i	$0.428133\pi$
$318$	0	0
$319$	−445.656	−1.39704
$320$	0	0
$321$	66.7261i	0.207869i
$322$	0	0
$323$	−293.494	−0.908649
$324$	0	0
$325$	66.9495i	0.205998i
$326$	0	0
$327$	− 19.1596i	− 0.0585919i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−36.8650	−0.111375	−0.0556873	−	0.998448i	$-0.517735\pi$
$331$	−36.8650	−0.111375	−0.0556873	+	0.998448i	$0.517735\pi$
$332$	0	0
$333$	113.334	0.340343
$334$	0	0
$335$	− 561.468i	− 1.67602i
$336$	0	0
$337$	−541.604	−1.60713	−0.803567	−	0.595214i	$-0.797067\pi$
$337$	−541.604	−1.60713	−0.803567	+	0.595214i	$0.797067\pi$
$338$	0	0
$339$	− 23.5354i	− 0.0694260i
$340$	0	0
$341$	− 431.753i	− 1.26614i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	108.270	0.313825
$346$	0	0
$347$	−244.402	−0.704328	−0.352164	−	0.935938i	$-0.614554\pi$
$347$	−244.402	−0.704328	−0.352164	+	0.935938i	$0.614554\pi$
$348$	0	0
$349$	− 190.205i	− 0.545001i	−0.962156	−	0.272501i	$-0.912149\pi$
$349$	− 190.205i	− 0.545001i	0.962156	−	0.272501i	$-0.0878507\pi$
$350$	0	0
$351$	193.082	0.550091
$352$	0	0
$353$	− 394.767i	− 1.11832i	−0.829060	−	0.559160i	$-0.811124\pi$
$353$	− 394.767i	− 1.11832i	0.829060	−	0.559160i	$-0.188876\pi$
$354$	0	0
$355$	− 185.714i	− 0.523139i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−296.325	−0.825418	−0.412709	−	0.910863i	$-0.635417\pi$
$359$	−296.325	−0.825418	−0.412709	+	0.910863i	$0.635417\pi$
$360$	0	0
$361$	211.664	0.586326
$362$	0	0
$363$	86.6873i	0.238808i
$364$	0	0
$365$	−247.190	−0.677232
$366$	0	0
$367$	602.598i	1.64196i	0.570960	+	0.820978i	$0.306571\pi$
$367$	602.598i	1.64196i	−0.570960	+	0.820978i	$0.693429\pi$
$368$	0	0
$369$	296.151i	0.802578i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	107.000	0.286862	0.143431	−	0.989660i	$-0.454186\pi$
$373$	107.000	0.286862	0.143431	+	0.989660i	$0.454186\pi$
$374$	0	0
$375$	58.7994	0.156798
$376$	0	0
$377$	559.988i	1.48538i
$378$	0	0
$379$	539.901	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$379$	539.901	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$380$	0	0
$381$	− 6.47699i	− 0.0170000i
$382$	0	0
$383$	− 133.620i	− 0.348878i	−0.984668	−	0.174439i	$-0.944189\pi$
$383$	− 133.620i	− 0.348878i	0.984668	−	0.174439i	$-0.0558112\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−262.259	−0.677673
$388$	0	0
$389$	−571.255	−1.46852	−0.734260	−	0.678868i	$-0.762471\pi$
$389$	−571.255	−1.46852	−0.734260	+	0.678868i	$0.762471\pi$
$390$	0	0
$391$	967.707i	2.47495i
$392$	0	0
$393$	0.543792	0.00138369
$394$	0	0
$395$	461.324i	1.16791i
$396$	0	0
$397$	− 199.373i	− 0.502198i	−0.967961	−	0.251099i	$-0.919208\pi$
$397$	− 199.373i	− 0.502198i	0.967961	−	0.251099i	$-0.0807919\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	634.422	1.58210	0.791050	−	0.611752i	$-0.209535\pi$
$401$	634.422	1.58210	0.791050	+	0.611752i	$0.209535\pi$
$402$	0	0
$403$	−542.519	−1.34620
$404$	0	0
$405$	− 393.096i	− 0.970608i
$406$	0	0
$407$	221.529	0.544298
$408$	0	0
$409$	689.821i	1.68661i	0.537439	+	0.843303i	$0.319392\pi$
$409$	689.821i	1.68661i	−0.537439	+	0.843303i	$0.680608\pi$
$410$	0	0
$411$	− 33.8840i	− 0.0824428i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−339.409	−0.817853
$416$	0	0
$417$	112.027	0.268649
$418$	0	0
$419$	43.8224i	0.104588i	0.998632	+	0.0522940i	$0.0166533\pi$
$419$	43.8224i	0.104588i	−0.998632	+	0.0522940i	$0.983347\pi$
$420$	0	0
$421$	−357.611	−0.849433	−0.424717	−	0.905326i	$-0.639626\pi$
$421$	−357.611	−0.849433	−0.424717	+	0.905326i	$0.639626\pi$
$422$	0	0
$423$	487.413i	1.15228i
$424$	0	0
$425$	− 74.8761i	− 0.176179i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	185.973	0.433505
$430$	0	0
$431$	−286.518	−0.664774	−0.332387	−	0.943143i	$-0.607854\pi$
$431$	−286.518	−0.664774	−0.332387	+	0.943143i	$0.607854\pi$
$432$	0	0
$433$	− 407.880i	− 0.941986i	−0.882137	−	0.470993i	$-0.843896\pi$
$433$	− 407.880i	− 0.941986i	0.882137	−	0.470993i	$-0.156104\pi$
$434$	0	0
$435$	−70.0708	−0.161082
$436$	0	0
$437$	492.392i	1.12675i
$438$	0	0
$439$	54.0687i	0.123163i	0.998102	+	0.0615817i	$0.0196145\pi$
$439$	54.0687i	0.123163i	−0.998102	+	0.0615817i	$0.980386\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−316.995	−0.715564	−0.357782	−	0.933805i	$-0.616467\pi$
$443$	−316.995	−0.715564	−0.357782	+	0.933805i	$0.616467\pi$
$444$	0	0
$445$	227.940	0.512224
$446$	0	0
$447$	− 20.0272i	− 0.0448035i
$448$	0	0
$449$	544.261	1.21216	0.606081	−	0.795403i	$-0.292741\pi$
$449$	544.261	1.21216	0.606081	+	0.795403i	$0.292741\pi$
$450$	0	0
$451$	578.874i	1.28353i
$452$	0	0
$453$	112.865i	0.249149i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−687.416	−1.50419	−0.752096	−	0.659053i	$-0.770957\pi$
$457$	−687.416	−1.50419	−0.752096	+	0.659053i	$0.770957\pi$
$458$	0	0
$459$	−215.942	−0.470462
$460$	0	0
$461$	− 676.260i	− 1.46694i	−0.679721	−	0.733470i	$-0.737899\pi$
$461$	− 676.260i	− 1.46694i	0.679721	−	0.733470i	$-0.262101\pi$
$462$	0	0
$463$	−559.738	−1.20894	−0.604469	−	0.796629i	$-0.706615\pi$
$463$	−559.738	−1.20894	−0.604469	+	0.796629i	$0.706615\pi$
$464$	0	0
$465$	− 67.8849i	− 0.145989i
$466$	0	0
$467$	369.373i	0.790948i	0.918477	+	0.395474i	$0.129420\pi$
$467$	369.373i	0.790948i	−0.918477	+	0.395474i	$0.870580\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−102.447	−0.217509
$472$	0	0
$473$	−512.627	−1.08378
$474$	0	0
$475$	− 38.0987i	− 0.0802077i
$476$	0	0
$477$	−76.3040	−0.159967
$478$	0	0
$479$	312.187i	0.651747i	0.945413	+	0.325874i	$0.105658\pi$
$479$	312.187i	0.651747i	−0.945413	+	0.325874i	$0.894342\pi$
$480$	0	0
$481$	− 278.362i	− 0.578715i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−152.446	−0.314322
$486$	0	0
$487$	−280.389	−0.575748	−0.287874	−	0.957668i	$-0.592948\pi$
$487$	−280.389	−0.575748	−0.287874	+	0.957668i	$0.592948\pi$
$488$	0	0
$489$	− 1.33283i	− 0.00272562i
$490$	0	0
$491$	−423.804	−0.863145	−0.431573	−	0.902078i	$-0.642041\pi$
$491$	−423.804	−0.863145	−0.431573	+	0.902078i	$0.642041\pi$
$492$	0	0
$493$	− 626.288i	− 1.27036i
$494$	0	0
$495$	− 792.322i	− 1.60065i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	166.049	0.332763	0.166381	−	0.986061i	$-0.446792\pi$
$499$	166.049	0.332763	0.166381	+	0.986061i	$0.446792\pi$
$500$	0	0
$501$	67.3993	0.134530
$502$	0	0
$503$	632.164i	1.25679i	0.777896	+	0.628393i	$0.216287\pi$
$503$	632.164i	1.25679i	−0.777896	+	0.628393i	$0.783713\pi$
$504$	0	0
$505$	325.946	0.645438
$506$	0	0
$507$	− 148.045i	− 0.292001i
$508$	0	0
$509$	− 310.676i	− 0.610365i	−0.952294	−	0.305183i	$-0.901283\pi$
$509$	− 310.676i	− 0.610365i	0.952294	−	0.305183i	$-0.0987175\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−109.876	−0.214184
$514$	0	0
$515$	−316.716	−0.614983
$516$	0	0
$517$	952.725i	1.84279i
$518$	0	0
$519$	−49.1977	−0.0947932
$520$	0	0
$521$	− 67.0382i	− 0.128672i	−0.997928	−	0.0643361i	$-0.979507\pi$
$521$	− 67.0382i	− 0.128672i	0.997928	−	0.0643361i	$-0.0204930\pi$
$522$	0	0
$523$	534.301i	1.02161i	0.859697	+	0.510804i	$0.170652\pi$
$523$	534.301i	1.02161i	−0.859697	+	0.510804i	$0.829348\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	606.751	1.15133
$528$	0	0
$529$	1094.51	2.06902
$530$	0	0
$531$	− 437.040i	− 0.823051i
$532$	0	0
$533$	727.383	1.36470
$534$	0	0
$535$	− 698.226i	− 1.30509i
$536$	0	0
$537$	42.5854i	0.0793023i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	151.341	0.279744	0.139872	−	0.990170i	$-0.455331\pi$
$541$	151.341	0.279744	0.139872	+	0.990170i	$0.455331\pi$
$542$	0	0
$543$	108.619	0.200035
$544$	0	0
$545$	200.487i	0.367865i
$546$	0	0
$547$	−775.543	−1.41781	−0.708906	−	0.705303i	$-0.750811\pi$
$547$	−775.543	−1.41781	−0.708906	+	0.705303i	$0.750811\pi$
$548$	0	0
$549$	34.4151i	0.0626869i
$550$	0	0
$551$	− 318.670i	− 0.578349i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	34.8312	0.0627590
$556$	0	0
$557$	64.4399	0.115691	0.0578455	−	0.998326i	$-0.481577\pi$
$557$	64.4399	0.115691	0.0578455	+	0.998326i	$0.481577\pi$
$558$	0	0
$559$	644.140i	1.15231i
$560$	0	0
$561$	−207.992	−0.370752
$562$	0	0
$563$	911.853i	1.61963i	0.586683	+	0.809816i	$0.300433\pi$
$563$	911.853i	1.61963i	−0.586683	+	0.809816i	$0.699567\pi$
$564$	0	0
$565$	246.276i	0.435887i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	236.004	0.414770	0.207385	−	0.978259i	$-0.433505\pi$
$569$	236.004	0.414770	0.207385	+	0.978259i	$0.433505\pi$
$570$	0	0
$571$	−829.540	−1.45278	−0.726392	−	0.687280i	$-0.758804\pi$
$571$	−829.540	−1.45278	−0.726392	+	0.687280i	$0.758804\pi$
$572$	0	0
$573$	27.7379i	0.0484082i
$574$	0	0
$575$	−125.619	−0.218468
$576$	0	0
$577$	− 439.047i	− 0.760914i	−0.924799	−	0.380457i	$-0.875767\pi$
$577$	− 439.047i	− 0.760914i	0.924799	−	0.380457i	$-0.124233\pi$
$578$	0	0
$579$	− 176.499i	− 0.304835i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−149.148	−0.255829
$584$	0	0
$585$	−995.591	−1.70186
$586$	0	0
$587$	871.738i	1.48507i	0.669805	+	0.742537i	$0.266378\pi$
$587$	871.738i	1.48507i	−0.669805	+	0.742537i	$0.733622\pi$
$588$	0	0
$589$	308.729	0.524158
$590$	0	0
$591$	81.8933i	0.138567i
$592$	0	0
$593$	765.147i	1.29030i	0.764056	+	0.645149i	$0.223205\pi$
$593$	765.147i	1.29030i	−0.764056	+	0.645149i	$0.776795\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−69.7404	−0.116818
$598$	0	0
$599$	292.799	0.488813	0.244407	−	0.969673i	$-0.421407\pi$
$599$	292.799	0.488813	0.244407	+	0.969673i	$0.421407\pi$
$600$	0	0
$601$	748.440i	1.24532i	0.782491	+	0.622662i	$0.213949\pi$
$601$	748.440i	1.24532i	−0.782491	+	0.622662i	$0.786051\pi$
$602$	0	0
$603$	925.777	1.53529
$604$	0	0
$605$	− 907.101i	− 1.49934i
$606$	0	0
$607$	62.0541i	0.102231i	0.998693	+	0.0511154i	$0.0162776\pi$
$607$	62.0541i	0.102231i	−0.998693	+	0.0511154i	$0.983722\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1197.14	1.95932
$612$	0	0
$613$	44.6560	0.0728482	0.0364241	−	0.999336i	$-0.488403\pi$
$613$	44.6560	0.0728482	0.0364241	+	0.999336i	$0.488403\pi$
$614$	0	0
$615$	91.0168i	0.147995i
$616$	0	0
$617$	−832.160	−1.34872	−0.674360	−	0.738403i	$-0.735580\pi$
$617$	−832.160	−1.34872	−0.674360	+	0.738403i	$0.735580\pi$
$618$	0	0
$619$	249.870i	0.403668i	0.979420	+	0.201834i	$0.0646901\pi$
$619$	249.870i	0.403668i	−0.979420	+	0.201834i	$0.935310\pi$
$620$	0	0
$621$	362.284i	0.583389i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−693.221	−1.10915
$626$	0	0
$627$	−105.831	−0.168790
$628$	0	0
$629$	311.319i	0.494943i
$630$	0	0
$631$	−26.0372	−0.0412634	−0.0206317	−	0.999787i	$-0.506568\pi$
$631$	−26.0372	−0.0412634	−0.0206317	+	0.999787i	$0.506568\pi$
$632$	0	0
$633$	51.4665i	0.0813056i
$634$	0	0
$635$	67.7756i	0.106733i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	306.215	0.479210
$640$	0	0
$641$	470.391	0.733839	0.366920	−	0.930253i	$-0.380412\pi$
$641$	470.391	0.733839	0.366920	+	0.930253i	$0.380412\pi$
$642$	0	0
$643$	668.123i	1.03907i	0.854449	+	0.519536i	$0.173895\pi$
$643$	668.123i	1.03907i	−0.854449	+	0.519536i	$0.826105\pi$
$644$	0	0
$645$	−80.6007	−0.124962
$646$	0	0
$647$	1068.22i	1.65104i	0.564374	+	0.825519i	$0.309117\pi$
$647$	1068.22i	1.65104i	−0.564374	+	0.825519i	$0.690883\pi$
$648$	0	0
$649$	− 854.263i	− 1.31628i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−779.835	−1.19423	−0.597117	−	0.802154i	$-0.703687\pi$
$653$	−779.835	−1.19423	−0.597117	+	0.802154i	$0.703687\pi$
$654$	0	0
$655$	−5.69027	−0.00868743
$656$	0	0
$657$	− 407.579i	− 0.620363i
$658$	0	0
$659$	1017.20	1.54355	0.771774	−	0.635897i	$-0.219370\pi$
$659$	1017.20	1.54355	0.771774	+	0.635897i	$0.219370\pi$
$660$	0	0
$661$	945.931i	1.43106i	0.698582	+	0.715530i	$0.253815\pi$
$661$	945.931i	1.43106i	−0.698582	+	0.715530i	$0.746185\pi$
$662$	0	0
$663$	261.352i	0.394196i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1050.72	−1.57529
$668$	0	0
$669$	−91.5044	−0.136778
$670$	0	0
$671$	67.2697i	0.100253i
$672$	0	0
$673$	−76.8911	−0.114251	−0.0571256	−	0.998367i	$-0.518194\pi$
$673$	−76.8911	−0.114251	−0.0571256	+	0.998367i	$0.518194\pi$
$674$	0	0
$675$	− 28.0316i	− 0.0415284i
$676$	0	0
$677$	641.045i	0.946891i	0.880823	+	0.473445i	$0.156990\pi$
$677$	641.045i	0.946891i	−0.880823	+	0.473445i	$0.843010\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	25.6961	0.0377329
$682$	0	0
$683$	1022.92	1.49769	0.748845	−	0.662745i	$-0.230609\pi$
$683$	1022.92	1.49769	0.748845	+	0.662745i	$0.230609\pi$
$684$	0	0
$685$	354.564i	0.517611i
$686$	0	0
$687$	101.112	0.147180
$688$	0	0
$689$	187.412i	0.272006i
$690$	0	0
$691$	− 690.043i	− 0.998615i	−0.866425	−	0.499307i	$-0.833588\pi$
$691$	− 690.043i	− 0.998615i	0.866425	−	0.499307i	$-0.166412\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1172.25	−1.68669
$696$	0	0
$697$	−813.502	−1.16715
$698$	0	0
$699$	− 89.7739i	− 0.128432i
$700$	0	0
$701$	−361.922	−0.516294	−0.258147	−	0.966106i	$-0.583112\pi$
$701$	−361.922	−0.516294	−0.258147	+	0.966106i	$0.583112\pi$
$702$	0	0
$703$	158.406i	0.225329i
$704$	0	0
$705$	149.798i	0.212479i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	521.904	0.736113	0.368056	−	0.929804i	$-0.380023\pi$
$709$	521.904	0.736113	0.368056	+	0.929804i	$0.380023\pi$
$710$	0	0
$711$	−760.655	−1.06984
$712$	0	0
$713$	− 1017.94i	− 1.42769i
$714$	0	0
$715$	−1946.04	−2.72173
$716$	0	0
$717$	68.7621i	0.0959025i
$718$	0	0
$719$	661.810i	0.920459i	0.887800	+	0.460230i	$0.152233\pi$
$719$	661.810i	0.920459i	−0.887800	+	0.460230i	$0.847767\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	160.253	0.221651
$724$	0	0
$725$	81.2990	0.112137
$726$	0	0
$727$	− 1169.62i	− 1.60882i	−0.594071	−	0.804412i	$-0.702480\pi$
$727$	− 1169.62i	− 1.60882i	0.594071	−	0.804412i	$-0.297520\pi$
$728$	0	0
$729$	607.150	0.832854
$730$	0	0
$731$	− 720.404i	− 0.985505i
$732$	0	0
$733$	− 452.412i	− 0.617206i	−0.951191	−	0.308603i	$-0.900139\pi$
$733$	− 452.412i	− 0.617206i	0.951191	−	0.308603i	$-0.0998614\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1809.58	2.45533
$738$	0	0
$739$	5.50756	0.00745272	0.00372636	−	0.999993i	$-0.498814\pi$
$739$	5.50756	0.00745272	0.00372636	+	0.999993i	$0.498814\pi$
$740$	0	0
$741$	132.982i	0.179463i
$742$	0	0
$743$	866.020	1.16557	0.582786	−	0.812626i	$-0.301963\pi$
$743$	866.020	1.16557	0.582786	+	0.812626i	$0.301963\pi$
$744$	0	0
$745$	209.565i	0.281296i
$746$	0	0
$747$	− 559.635i	− 0.749176i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−460.842	−0.613638	−0.306819	−	0.951768i	$-0.599265\pi$
$751$	−460.842	−0.613638	−0.306819	+	0.951768i	$0.599265\pi$
$752$	0	0
$753$	20.3322	0.0270016
$754$	0	0
$755$	− 1181.02i	− 1.56427i
$756$	0	0
$757$	−547.987	−0.723893	−0.361946	−	0.932199i	$-0.617888\pi$
$757$	−547.987	−0.723893	−0.361946	+	0.932199i	$0.617888\pi$
$758$	0	0
$759$	348.946i	0.459745i
$760$	0	0
$761$	− 1303.88i	− 1.71338i	−0.515830	−	0.856691i	$-0.672516\pi$
$761$	− 1303.88i	− 1.71338i	0.515830	−	0.856691i	$-0.327484\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1113.46	1.45551
$766$	0	0
$767$	−1073.42	−1.39951
$768$	0	0
$769$	− 771.004i	− 1.00261i	−0.865272	−	0.501303i	$-0.832854\pi$
$769$	− 771.004i	− 1.00261i	0.865272	−	0.501303i	$-0.167146\pi$
$770$	0	0
$771$	88.0080	0.114148
$772$	0	0
$773$	1065.32i	1.37816i	0.724685	+	0.689080i	$0.241985\pi$
$773$	1065.32i	1.37816i	−0.724685	+	0.689080i	$0.758015\pi$
$774$	0	0
$775$	78.7628i	0.101629i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−413.929	−0.531359
$780$	0	0
$781$	598.546	0.766384
$782$	0	0
$783$	− 234.466i	− 0.299446i
$784$	0	0
$785$	1072.01	1.36561
$786$	0	0
$787$	− 944.631i	− 1.20029i	−0.799890	−	0.600147i	$-0.795109\pi$
$787$	− 944.631i	− 1.20029i	0.799890	−	0.600147i	$-0.204891\pi$
$788$	0	0
$789$	− 184.750i	− 0.234157i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	84.5277	0.106592
$794$	0	0
$795$	−23.4507	−0.0294977
$796$	0	0
$797$	− 1245.89i	− 1.56322i	−0.623765	−	0.781612i	$-0.714398\pi$
$797$	− 1245.89i	− 1.56322i	0.623765	−	0.781612i	$-0.285602\pi$
$798$	0	0
$799$	−1338.88	−1.67570
$800$	0	0
$801$	375.838i	0.469211i
$802$	0	0
$803$	− 796.676i	− 0.992125i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.82824	0.0109396
$808$	0	0
$809$	−177.307	−0.219168	−0.109584	−	0.993978i	$-0.534952\pi$
$809$	−177.307	−0.219168	−0.109584	+	0.993978i	$0.534952\pi$
$810$	0	0
$811$	1212.71i	1.49532i	0.664079	+	0.747662i	$0.268824\pi$
$811$	1212.71i	1.49532i	−0.664079	+	0.747662i	$0.731176\pi$
$812$	0	0
$813$	−66.0108	−0.0811941
$814$	0	0
$815$	13.9468i	0.0171126i
$816$	0	0
$817$	− 366.558i	− 0.448664i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1064.51	−1.29660	−0.648298	−	0.761387i	$-0.724519\pi$
$821$	−1064.51	−1.29660	−0.648298	+	0.761387i	$0.724519\pi$
$822$	0	0
$823$	−285.647	−0.347081	−0.173540	−	0.984827i	$-0.555521\pi$
$823$	−285.647	−0.347081	−0.173540	+	0.984827i	$0.555521\pi$
$824$	0	0
$825$	− 26.9996i	− 0.0327268i
$826$	0	0
$827$	1182.30	1.42962	0.714811	−	0.699317i	$-0.246513\pi$
$827$	1182.30	1.42962	0.714811	+	0.699317i	$0.246513\pi$
$828$	0	0
$829$	− 720.994i	− 0.869715i	−0.900499	−	0.434857i	$-0.856799\pi$
$829$	− 720.994i	− 0.869715i	0.900499	−	0.434857i	$-0.143201\pi$
$830$	0	0
$831$	− 126.276i	− 0.151956i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−705.270	−0.844635
$836$	0	0
$837$	227.152	0.271388
$838$	0	0
$839$	− 548.052i	− 0.653221i	−0.945159	−	0.326610i	$-0.894094\pi$
$839$	− 548.052i	− 0.653221i	0.945159	−	0.326610i	$-0.105906\pi$
$840$	0	0
$841$	−160.988	−0.191424
$842$	0	0
$843$	− 100.059i	− 0.118694i
$844$	0	0
$845$	1549.15i	1.83331i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	108.611	0.127928
$850$	0	0
$851$	522.297	0.613745
$852$	0	0
$853$	1110.62i	1.30201i	0.759072	+	0.651007i	$0.225653\pi$
$853$	1110.62i	1.30201i	−0.759072	+	0.651007i	$0.774347\pi$
$854$	0	0
$855$	566.556	0.662639
$856$	0	0
$857$	145.165i	0.169388i	0.996407	+	0.0846939i	$0.0269912\pi$
$857$	145.165i	0.169388i	−0.996407	+	0.0846939i	$0.973009\pi$
$858$	0	0
$859$	1335.32i	1.55450i	0.629189	+	0.777252i	$0.283387\pi$
$859$	1335.32i	1.55450i	−0.629189	+	0.777252i	$0.716613\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	562.851	0.652203	0.326101	−	0.945335i	$-0.394265\pi$
$863$	562.851	0.652203	0.326101	+	0.945335i	$0.394265\pi$
$864$	0	0
$865$	514.807	0.595153
$866$	0	0
$867$	− 145.846i	− 0.168219i
$868$	0	0
$869$	−1486.82	−1.71095
$870$	0	0
$871$	− 2273.82i	− 2.61059i
$872$	0	0
$873$	− 251.361i	− 0.287928i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−906.537	−1.03368	−0.516840	−	0.856082i	$-0.672892\pi$
$877$	−906.537	−1.03368	−0.516840	+	0.856082i	$0.672892\pi$
$878$	0	0
$879$	36.4038	0.0414150
$880$	0	0
$881$	− 422.614i	− 0.479698i	−0.970810	−	0.239849i	$-0.922902\pi$
$881$	− 422.614i	− 0.479698i	0.970810	−	0.239849i	$-0.0770979\pi$
$882$	0	0
$883$	−310.036	−0.351116	−0.175558	−	0.984469i	$-0.556173\pi$
$883$	−310.036	−0.351116	−0.175558	+	0.984469i	$0.556173\pi$
$884$	0	0
$885$	− 134.316i	− 0.151770i
$886$	0	0
$887$	− 1135.83i	− 1.28053i	−0.768154	−	0.640265i	$-0.778825\pi$
$887$	− 1135.83i	− 1.28053i	0.768154	−	0.640265i	$-0.221175\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1266.92	1.42191
$892$	0	0
$893$	−681.254	−0.762882
$894$	0	0
$895$	− 445.616i	− 0.497895i
$896$	0	0
$897$	438.468	0.488816
$898$	0	0
$899$	658.799i	0.732813i
$900$	0	0
$901$	− 209.601i	− 0.232631i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1136.60	−1.25591
$906$	0	0
$907$	−80.9267	−0.0892246	−0.0446123	−	0.999004i	$-0.514205\pi$
$907$	−80.9267	−0.0892246	−0.0446123	+	0.999004i	$0.514205\pi$
$908$	0	0
$909$	537.436i	0.591239i
$910$	0	0
$911$	1285.92	1.41155	0.705774	−	0.708437i	$-0.250599\pi$
$911$	1285.92	1.41155	0.705774	+	0.708437i	$0.250599\pi$
$912$	0	0
$913$	− 1093.89i	− 1.19813i
$914$	0	0
$915$	10.5769i	0.0115594i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1712.82	1.86379	0.931894	−	0.362732i	$-0.118156\pi$
$919$	1712.82	1.86379	0.931894	+	0.362732i	$0.118156\pi$
$920$	0	0
$921$	−256.943	−0.278983
$922$	0	0
$923$	− 752.101i	− 0.814845i
$924$	0	0
$925$	−40.4126	−0.0436893
$926$	0	0
$927$	− 522.218i	− 0.563342i
$928$	0	0
$929$	1361.04i	1.46506i	0.680733	+	0.732531i	$0.261661\pi$
$929$	1361.04i	1.46506i	−0.680733	+	0.732531i	$0.738339\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	157.455	0.168762
$934$	0	0
$935$	2176.44	2.32774
$936$	0	0
$937$	− 314.858i	− 0.336028i	−0.985785	−	0.168014i	$-0.946265\pi$
$937$	− 314.858i	− 0.336028i	0.985785	−	0.168014i	$-0.0537354\pi$
$938$	0	0
$939$	−278.156	−0.296226
$940$	0	0
$941$	301.753i	0.320673i	0.987062	+	0.160336i	$0.0512579\pi$
$941$	301.753i	0.320673i	−0.987062	+	0.160336i	$0.948742\pi$
$942$	0	0
$943$	1364.81i	1.44730i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1453.18	−1.53450	−0.767252	−	0.641346i	$-0.778376\pi$
$947$	−1453.18	−1.53450	−0.767252	+	0.641346i	$0.778376\pi$
$948$	0	0
$949$	−1001.06	−1.05486
$950$	0	0
$951$	71.9225i	0.0756282i
$952$	0	0
$953$	977.784	1.02601	0.513003	−	0.858387i	$-0.328533\pi$
$953$	977.784	1.02601	0.513003	+	0.858387i	$0.328533\pi$
$954$	0	0
$955$	− 290.251i	− 0.303928i
$956$	0	0
$957$	− 225.834i	− 0.235981i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	322.753	0.335851
$962$	0	0
$963$	1151.27	1.19550
$964$	0	0
$965$	1846.90i	1.91389i
$966$	0	0
$967$	−181.044	−0.187222	−0.0936109	−	0.995609i	$-0.529841\pi$
$967$	−181.044	−0.187222	−0.0936109	+	0.995609i	$0.529841\pi$
$968$	0	0
$969$	− 148.726i	− 0.153484i
$970$	0	0
$971$	− 243.030i	− 0.250289i	−0.992139	−	0.125144i	$-0.960061\pi$
$971$	− 243.030i	− 0.250289i	0.992139	−	0.125144i	$-0.0399394\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−33.9263	−0.0347962
$976$	0	0
$977$	1431.73	1.46543	0.732717	−	0.680533i	$-0.238252\pi$
$977$	1431.73	1.46543	0.732717	+	0.680533i	$0.238252\pi$
$978$	0	0
$979$	734.635i	0.750393i
$980$	0	0
$981$	−330.573	−0.336975
$982$	0	0
$983$	444.483i	0.452170i	0.974108	+	0.226085i	$0.0725927\pi$
$983$	444.483i	0.452170i	−0.974108	+	0.226085i	$0.927407\pi$
$984$	0	0
$985$	− 856.936i	− 0.869985i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1208.62	−1.22206
$990$	0	0
$991$	1804.72	1.82111	0.910553	−	0.413391i	$-0.135656\pi$
$991$	1804.72	1.82111	0.910553	+	0.413391i	$0.135656\pi$
$992$	0	0
$993$	− 18.6812i	− 0.0188128i
$994$	0	0
$995$	729.767	0.733434
$996$	0	0
$997$	− 1904.31i	− 1.91004i	−0.296548	−	0.955018i	$-0.595835\pi$
$997$	− 1904.31i	− 1.91004i	0.296548	−	0.955018i	$-0.404165\pi$
$998$	0	0
$999$	116.550i	0.116666i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.h.97.9		16
4.3	odd	2	inner	1568.3.c.h.97.7		16
7.2	even	3		224.3.s.a.129.4	yes	16
7.3	odd	6		224.3.s.a.33.4	✓	16
7.6	odd	2	inner	1568.3.c.h.97.8		16
28.3	even	6		224.3.s.a.33.5	yes	16
28.23	odd	6		224.3.s.a.129.5	yes	16
28.27	even	2	inner	1568.3.c.h.97.10		16
56.3	even	6		448.3.s.g.257.4		16
56.37	even	6		448.3.s.g.129.5		16
56.45	odd	6		448.3.s.g.257.5		16
56.51	odd	6		448.3.s.g.129.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.s.a.33.4	✓	16	7.3	odd	6
224.3.s.a.33.5	yes	16	28.3	even	6
224.3.s.a.129.4	yes	16	7.2	even	3
224.3.s.a.129.5	yes	16	28.23	odd	6
448.3.s.g.129.4		16	56.51	odd	6
448.3.s.g.129.5		16	56.37	even	6
448.3.s.g.257.4		16	56.3	even	6
448.3.s.g.257.5		16	56.45	odd	6
1568.3.c.h.97.7		16	4.3	odd	2	inner
1568.3.c.h.97.8		16	7.6	odd	2	inner
1568.3.c.h.97.9		16	1.1	even	1	trivial
1568.3.c.h.97.10		16	28.27	even	2	inner

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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.506745i q^{3} -5.30261i q^{5} +8.74321 q^{9} +17.0900 q^{11} -21.4744i q^{13} +2.68707 q^{15} +24.0168i q^{17} +12.2203i q^{19} +40.2929 q^{23} -3.11765 q^{25} +8.99128i q^{27} -26.0770 q^{29} -25.2635i q^{31} +8.66025i q^{33} +12.9625 q^{37} +10.8820 q^{39} +33.8721i q^{41} -29.9958 q^{43} -46.3618i q^{45} +55.7476i q^{47} -12.1704 q^{51} -8.72723 q^{53} -90.6214i q^{55} -6.19259 q^{57} -49.9862i q^{59} +3.93621i q^{61} -113.870 q^{65} +105.885 q^{67} +20.4182i q^{69} +35.0232 q^{71} -46.6166i q^{73} -1.57985i q^{75} -86.9995 q^{79} +74.1326 q^{81} -64.0079i q^{83} +127.352 q^{85} -13.2144i q^{87} +42.9863i q^{89} +12.8022 q^{93} +64.7996 q^{95} -28.7493i q^{97} +149.421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9} - 32 q^{25} + 112 q^{29} - 16 q^{37} + 48 q^{53} - 528 q^{57} + 16 q^{65} - 64 q^{81} + 720 q^{85} + 464 q^{93}+O(q^{100}) \) 16 * q - 16 * q^9 - 32 * q^25 + 112 * q^29 - 16 * q^37 + 48 * q^53 - 528 * q^57 + 16 * q^65 - 64 * q^81 + 720 * q^85 + 464 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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