LMFDB - Embedded newform 2736.3.m.e.1711.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(1711,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2736.1711");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.m (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} + \cdots + 2052928 $ x^12 - 2x^11 - 2x^10 - 28x^9 - 400x^8 - 520x^7 + 17067x^6 - 3250x^5 - 195494x^4 + 302996x^3 + 602332x^2 - 2263536*x + 2052928
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1711.5
Root			$2.21305 - 0.177784i$ of defining polynomial
Character	$\chi$	$=$	2736.1711
Dual form			2736.3.m.e.1711.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67107 q^{5} -11.9371i q^{7} +O(q^{10})$	$q-2.67107 q^{5} -11.9371i q^{7} +13.4703i q^{11} +18.1943 q^{13} -30.1432 q^{17} +4.35890i q^{19} -20.8182i q^{23} -17.8654 q^{25} +30.7704 q^{29} -18.8390i q^{31} +31.8848i q^{35} +10.5370 q^{37} +24.5166 q^{41} -41.2024i q^{43} -10.4830i q^{47} -93.4939 q^{49} -80.7151 q^{53} -35.9802i q^{55} +0.308836i q^{59} -9.64705 q^{61} -48.5983 q^{65} +102.244i q^{67} -18.1641i q^{71} -30.9821 q^{73} +160.797 q^{77} -121.159i q^{79} +98.3874i q^{83} +80.5145 q^{85} -45.2305 q^{89} -217.187i q^{91} -11.6429i q^{95} +35.4274 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{5}+O(q^{10}) $ 12 * q + 10 * q^5	$ 12 q + 10 q^{5} + 36 q^{13} - 14 q^{17} + 10 q^{25} + 108 q^{29} - 16 q^{37} - 16 q^{41} - 118 q^{49} - 220 q^{53} + 366 q^{61} - 140 q^{65} - 158 q^{73} + 286 q^{77} + 98 q^{85} + 396 q^{89} + 224 q^{97}+O(q^{100}) $ 12 * q + 10 * q^5 + 36 * q^13 - 14 * q^17 + 10 * q^25 + 108 * q^29 - 16 * q^37 - 16 * q^41 - 118 * q^49 - 220 * q^53 + 366 * q^61 - 140 * q^65 - 158 * q^73 + 286 * q^77 + 98 * q^85 + 396 * q^89 + 224 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$1$	$-1$	$1$

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	0
$3$	0	0
$4$	0	0
$5$	−2.67107	−0.534214	−0.267107	−	0.963667i	$-0.586068\pi$
$5$	−2.67107	−0.534214	−0.267107	+	0.963667i	$0.586068\pi$
$6$	0	0
$7$	− 11.9371i	− 1.70530i	−0.522485	−	0.852649i	$-0.674995\pi$
$7$	− 11.9371i	− 1.70530i	0.522485	−	0.852649i	$-0.325005\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	13.4703i	1.22458i	0.790635	+	0.612288i	$0.209751\pi$
$11$	13.4703i	1.22458i	−0.790635	+	0.612288i	$0.790249\pi$
$12$	0	0
$13$	18.1943	1.39956	0.699782	−	0.714357i	$-0.253281\pi$
$13$	18.1943	1.39956	0.699782	+	0.714357i	$0.253281\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−30.1432	−1.77313	−0.886564	−	0.462606i	$-0.846915\pi$
$17$	−30.1432	−1.77313	−0.886564	+	0.462606i	$0.846915\pi$
$18$	0	0
$19$	4.35890i	0.229416i
$19$	4.35890i	0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 20.8182i	− 0.905140i	−0.891729	−	0.452570i	$-0.850507\pi$
$23$	− 20.8182i	− 0.905140i	0.891729	−	0.452570i	$-0.149493\pi$
$24$	0	0
$25$	−17.8654	−0.714616
$26$	0	0
$27$	0	0
$28$	0	0
$29$	30.7704	1.06105	0.530523	−	0.847670i	$-0.321995\pi$
$29$	30.7704	1.06105	0.530523	+	0.847670i	$0.321995\pi$
$30$	0	0
$31$	− 18.8390i	− 0.607708i	−0.952719	−	0.303854i	$-0.901726\pi$
$31$	− 18.8390i	− 0.607708i	0.952719	−	0.303854i	$-0.0982736\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	31.8848i	0.910993i
$36$	0	0
$37$	10.5370	0.284784	0.142392	−	0.989810i	$-0.454521\pi$
$37$	10.5370	0.284784	0.142392	+	0.989810i	$0.454521\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	24.5166	0.597965	0.298982	−	0.954259i	$-0.403353\pi$
$41$	24.5166	0.597965	0.298982	+	0.954259i	$0.403353\pi$
$42$	0	0
$43$	− 41.2024i	− 0.958196i	−0.877762	−	0.479098i	$-0.840964\pi$
$43$	− 41.2024i	− 0.958196i	0.877762	−	0.479098i	$-0.159036\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.4830i	− 0.223042i	−0.993762	−	0.111521i	$-0.964428\pi$
$47$	− 10.4830i	− 0.223042i	0.993762	−	0.111521i	$-0.0355722\pi$
$48$	0	0
$49$	−93.4939	−1.90804
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−80.7151	−1.52293	−0.761464	−	0.648208i	$-0.775519\pi$
$53$	−80.7151	−1.52293	−0.761464	+	0.648208i	$0.775519\pi$
$54$	0	0
$55$	− 35.9802i	− 0.654185i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.308836i	0.00523450i	0.999997	+	0.00261725i	$0.000833098\pi$
$59$	0.308836i	0.00523450i	−0.999997	+	0.00261725i	$0.999167\pi$
$60$	0	0
$61$	−9.64705	−0.158148	−0.0790742	−	0.996869i	$-0.525196\pi$
$61$	−9.64705	−0.158148	−0.0790742	+	0.996869i	$0.525196\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−48.5983	−0.747666
$66$	0	0
$67$	102.244i	1.52603i	0.646383	+	0.763013i	$0.276281\pi$
$67$	102.244i	1.52603i	−0.646383	+	0.763013i	$0.723719\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 18.1641i	− 0.255832i	−0.991785	−	0.127916i	$-0.959171\pi$
$71$	− 18.1641i	− 0.255832i	0.991785	−	0.127916i	$-0.0408288\pi$
$72$	0	0
$73$	−30.9821	−0.424412	−0.212206	−	0.977225i	$-0.568065\pi$
$73$	−30.9821	−0.424412	−0.212206	+	0.977225i	$0.568065\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	160.797	2.08827
$78$	0	0
$79$	− 121.159i	− 1.53365i	−0.641854	−	0.766827i	$-0.721834\pi$
$79$	− 121.159i	− 1.53365i	0.641854	−	0.766827i	$-0.278166\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	98.3874i	1.18539i	0.805427	+	0.592695i	$0.201936\pi$
$83$	98.3874i	1.18539i	−0.805427	+	0.592695i	$0.798064\pi$
$84$	0	0
$85$	80.5145	0.947229
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−45.2305	−0.508208	−0.254104	−	0.967177i	$-0.581781\pi$
$89$	−45.2305	−0.508208	−0.254104	+	0.967177i	$0.581781\pi$
$90$	0	0
$91$	− 217.187i	− 2.38667i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 11.6429i	− 0.122557i
$96$	0	0
$97$	35.4274	0.365231	0.182616	−	0.983184i	$-0.441544\pi$
$97$	35.4274	0.365231	0.182616	+	0.983184i	$0.441544\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	112.581	1.11466	0.557330	−	0.830291i	$-0.311826\pi$
$101$	112.581	1.11466	0.557330	+	0.830291i	$0.311826\pi$
$102$	0	0
$103$	− 25.9897i	− 0.252327i	−0.992009	−	0.126164i	$-0.959734\pi$
$103$	− 25.9897i	− 0.252327i	0.992009	−	0.126164i	$-0.0402665\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	62.8964i	0.587817i	0.955834	+	0.293909i	$0.0949561\pi$
$107$	62.8964i	0.587817i	−0.955834	+	0.293909i	$0.905044\pi$
$108$	0	0
$109$	−186.691	−1.71276	−0.856381	−	0.516345i	$-0.827292\pi$
$109$	−186.691	−1.71276	−0.856381	+	0.516345i	$0.827292\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−128.670	−1.13867	−0.569337	−	0.822104i	$-0.692800\pi$
$113$	−128.670	−1.13867	−0.569337	+	0.822104i	$0.692800\pi$
$114$	0	0
$115$	55.6069i	0.483538i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	359.822i	3.02371i
$120$	0	0
$121$	−60.4501	−0.499588
$122$	0	0
$123$	0	0
$124$	0	0
$125$	114.496	0.915971
$126$	0	0
$127$	133.134i	1.04830i	0.851625	+	0.524151i	$0.175617\pi$
$127$	133.134i	1.04830i	−0.851625	+	0.524151i	$0.824383\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	53.4427i	0.407959i	0.978975	+	0.203980i	$0.0653876\pi$
$131$	53.4427i	0.407959i	−0.978975	+	0.203980i	$0.934612\pi$
$132$	0	0
$133$	52.0325	0.391222
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−109.828	−0.801661	−0.400831	−	0.916152i	$-0.631278\pi$
$137$	−109.828	−0.801661	−0.400831	+	0.916152i	$0.631278\pi$
$138$	0	0
$139$	121.902i	0.876989i	0.898734	+	0.438495i	$0.144488\pi$
$139$	121.902i	0.876989i	−0.898734	+	0.438495i	$0.855512\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	245.084i	1.71387i
$144$	0	0
$145$	−82.1897	−0.566826
$146$	0	0
$147$	0	0
$148$	0	0
$149$	33.3146	0.223588	0.111794	−	0.993731i	$-0.464340\pi$
$149$	33.3146	0.223588	0.111794	+	0.993731i	$0.464340\pi$
$150$	0	0
$151$	139.889i	0.926417i	0.886249	+	0.463208i	$0.153302\pi$
$151$	139.889i	0.926417i	−0.886249	+	0.463208i	$0.846698\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	50.3201i	0.324646i
$156$	0	0
$157$	−208.515	−1.32812	−0.664062	−	0.747678i	$-0.731169\pi$
$157$	−208.515	−1.32812	−0.664062	+	0.747678i	$0.731169\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−248.509	−1.54353
$162$	0	0
$163$	193.263i	1.18566i	0.805326	+	0.592832i	$0.201990\pi$
$163$	193.263i	1.18566i	−0.805326	+	0.592832i	$0.798010\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 90.1917i	− 0.540070i	−0.962851	−	0.270035i	$-0.912965\pi$
$167$	− 90.1917i	− 0.540070i	0.962851	−	0.270035i	$-0.0870353\pi$
$168$	0	0
$169$	162.034	0.958778
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−236.078	−1.36461	−0.682305	−	0.731067i	$-0.739022\pi$
$173$	−236.078	−1.36461	−0.682305	+	0.731067i	$0.739022\pi$
$174$	0	0
$175$	213.261i	1.21863i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 115.773i	− 0.646776i	−0.946267	−	0.323388i	$-0.895178\pi$
$179$	− 115.773i	− 0.646776i	0.946267	−	0.323388i	$-0.104822\pi$
$180$	0	0
$181$	−238.750	−1.31906	−0.659530	−	0.751678i	$-0.729245\pi$
$181$	−238.750	−1.31906	−0.659530	+	0.751678i	$0.729245\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−28.1451	−0.152136
$186$	0	0
$187$	− 406.039i	− 2.17133i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	340.622i	1.78336i	0.452663	+	0.891681i	$0.350474\pi$
$191$	340.622i	1.78336i	−0.452663	+	0.891681i	$0.649526\pi$
$192$	0	0
$193$	59.0984	0.306209	0.153105	−	0.988210i	$-0.451073\pi$
$193$	59.0984	0.306209	0.153105	+	0.988210i	$0.451073\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	83.6109	0.424421	0.212210	−	0.977224i	$-0.431934\pi$
$197$	83.6109	0.424421	0.212210	+	0.977224i	$0.431934\pi$
$198$	0	0
$199$	313.520i	1.57548i	0.616011	+	0.787738i	$0.288748\pi$
$199$	313.520i	1.57548i	−0.616011	+	0.787738i	$0.711252\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 367.308i	− 1.80940i
$204$	0	0
$205$	−65.4854	−0.319441
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−58.7159	−0.280937
$210$	0	0
$211$	304.615i	1.44368i	0.692062	+	0.721838i	$0.256702\pi$
$211$	304.615i	1.44368i	−0.692062	+	0.721838i	$0.743298\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	110.054i	0.511881i
$216$	0	0
$217$	−224.882	−1.03632
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−548.435	−2.48161
$222$	0	0
$223$	− 385.048i	− 1.72667i	−0.504629	−	0.863336i	$-0.668371\pi$
$223$	− 385.048i	− 1.72667i	0.504629	−	0.863336i	$-0.331629\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 342.415i	− 1.50843i	−0.656625	−	0.754217i	$-0.728017\pi$
$227$	− 342.415i	− 1.50843i	0.656625	−	0.754217i	$-0.271983\pi$
$228$	0	0
$229$	378.829	1.65428	0.827138	−	0.562000i	$-0.189968\pi$
$229$	378.829	1.65428	0.827138	+	0.562000i	$0.189968\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−397.814	−1.70736	−0.853678	−	0.520801i	$-0.825633\pi$
$233$	−397.814	−1.70736	−0.853678	+	0.520801i	$0.825633\pi$
$234$	0	0
$235$	28.0007i	0.119152i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.50203i	0.0104687i	0.999986	+	0.00523437i	$0.00166616\pi$
$239$	2.50203i	0.0104687i	−0.999986	+	0.00523437i	$0.998334\pi$
$240$	0	0
$241$	−45.1397	−0.187302	−0.0936508	−	0.995605i	$-0.529854\pi$
$241$	−45.1397	−0.187302	−0.0936508	+	0.995605i	$0.529854\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	249.729	1.01930
$246$	0	0
$247$	79.3072i	0.321082i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	295.359i	1.17673i	0.808596	+	0.588364i	$0.200228\pi$
$251$	295.359i	1.17673i	−0.808596	+	0.588364i	$0.799772\pi$
$252$	0	0
$253$	280.428	1.10841
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−490.602	−1.90896	−0.954479	−	0.298278i	$-0.903588\pi$
$257$	−490.602	−1.90896	−0.954479	+	0.298278i	$0.903588\pi$
$258$	0	0
$259$	− 125.781i	− 0.485641i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 252.540i	− 0.960230i	−0.877206	−	0.480115i	$-0.840595\pi$
$263$	− 252.540i	− 0.960230i	0.877206	−	0.480115i	$-0.159405\pi$
$264$	0	0
$265$	215.596	0.813568
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−69.3385	−0.257764	−0.128882	−	0.991660i	$-0.541139\pi$
$269$	−69.3385	−0.257764	−0.128882	+	0.991660i	$0.541139\pi$
$270$	0	0
$271$	− 80.0499i	− 0.295387i	−0.989033	−	0.147694i	$-0.952815\pi$
$271$	− 80.0499i	− 0.295387i	0.989033	−	0.147694i	$-0.0471849\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 240.653i	− 0.875102i
$276$	0	0
$277$	−329.659	−1.19010	−0.595052	−	0.803687i	$-0.702869\pi$
$277$	−329.659	−1.19010	−0.595052	+	0.803687i	$0.702869\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−131.428	−0.467716	−0.233858	−	0.972271i	$-0.575135\pi$
$281$	−131.428	−0.467716	−0.233858	+	0.972271i	$0.575135\pi$
$282$	0	0
$283$	100.493i	0.355098i	0.984112	+	0.177549i	$0.0568169\pi$
$283$	100.493i	0.355098i	−0.984112	+	0.177549i	$0.943183\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 292.656i	− 1.01971i
$288$	0	0
$289$	619.611	2.14398
$290$	0	0
$291$	0	0
$292$	0	0
$293$	187.556	0.640122	0.320061	−	0.947397i	$-0.396297\pi$
$293$	187.556	0.640122	0.320061	+	0.947397i	$0.396297\pi$
$294$	0	0
$295$	− 0.824921i	− 0.00279634i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 378.773i	− 1.26680i
$300$	0	0
$301$	−491.837	−1.63401
$302$	0	0
$303$	0	0
$304$	0	0
$305$	25.7679	0.0844850
$306$	0	0
$307$	− 519.275i	− 1.69145i	−0.533621	−	0.845724i	$-0.679169\pi$
$307$	− 519.275i	− 1.69145i	0.533621	−	0.845724i	$-0.320831\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	410.921i	1.32129i	0.750699	+	0.660645i	$0.229717\pi$
$311$	410.921i	1.32129i	−0.750699	+	0.660645i	$0.770283\pi$
$312$	0	0
$313$	−408.644	−1.30557	−0.652786	−	0.757543i	$-0.726400\pi$
$313$	−408.644	−1.30557	−0.652786	+	0.757543i	$0.726400\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−158.824	−0.501022	−0.250511	−	0.968114i	$-0.580599\pi$
$317$	−158.824	−0.501022	−0.250511	+	0.968114i	$0.580599\pi$
$318$	0	0
$319$	414.487i	1.29933i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 131.391i	− 0.406784i
$324$	0	0
$325$	−325.049	−1.00015
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−125.136	−0.380353
$330$	0	0
$331$	225.123i	0.680130i	0.940402	+	0.340065i	$0.110449\pi$
$331$	225.123i	0.680130i	−0.940402	+	0.340065i	$0.889551\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 273.100i	− 0.815224i
$336$	0	0
$337$	506.422	1.50273	0.751367	−	0.659884i	$-0.229395\pi$
$337$	506.422	1.50273	0.751367	+	0.659884i	$0.229395\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	253.767	0.744186
$342$	0	0
$343$	531.127i	1.54848i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 517.793i	− 1.49220i	−0.665835	−	0.746099i	$-0.731925\pi$
$347$	− 517.793i	− 1.49220i	0.665835	−	0.746099i	$-0.268075\pi$
$348$	0	0
$349$	−238.053	−0.682099	−0.341050	−	0.940045i	$-0.610782\pi$
$349$	−238.053	−0.682099	−0.341050	+	0.940045i	$0.610782\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−590.354	−1.67239	−0.836195	−	0.548432i	$-0.815225\pi$
$353$	−590.354	−1.67239	−0.836195	+	0.548432i	$0.815225\pi$
$354$	0	0
$355$	48.5175i	0.136669i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 570.927i	− 1.59033i	−0.606396	−	0.795163i	$-0.707385\pi$
$359$	− 570.927i	− 1.59033i	0.606396	−	0.795163i	$-0.292615\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	82.7552	0.226727
$366$	0	0
$367$	360.851i	0.983244i	0.870809	+	0.491622i	$0.163596\pi$
$367$	360.851i	0.983244i	−0.870809	+	0.491622i	$0.836404\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	963.503i	2.59704i
$372$	0	0
$373$	−410.201	−1.09974	−0.549868	−	0.835252i	$-0.685322\pi$
$373$	−410.201	−1.09974	−0.549868	+	0.835252i	$0.685322\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	559.846	1.48500
$378$	0	0
$379$	− 283.727i	− 0.748620i	−0.927304	−	0.374310i	$-0.877880\pi$
$379$	− 283.727i	− 0.748620i	0.927304	−	0.374310i	$-0.122120\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	374.650i	0.978199i	0.872228	+	0.489099i	$0.162674\pi$
$383$	374.650i	0.978199i	−0.872228	+	0.489099i	$0.837326\pi$
$384$	0	0
$385$	−429.499	−1.11558
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−630.581	−1.62103	−0.810515	−	0.585717i	$-0.800813\pi$
$389$	−630.581	−1.62103	−0.810515	+	0.585717i	$0.800813\pi$
$390$	0	0
$391$	627.527i	1.60493i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	323.623i	0.819299i
$396$	0	0
$397$	−391.563	−0.986305	−0.493153	−	0.869943i	$-0.664156\pi$
$397$	−391.563	−0.986305	−0.493153	+	0.869943i	$0.664156\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−543.822	−1.35617	−0.678083	−	0.734986i	$-0.737189\pi$
$401$	−543.822	−1.35617	−0.678083	+	0.734986i	$0.737189\pi$
$402$	0	0
$403$	− 342.762i	− 0.850527i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	141.937i	0.348740i
$408$	0	0
$409$	575.794	1.40781	0.703905	−	0.710294i	$-0.251438\pi$
$409$	575.794	1.40781	0.703905	+	0.710294i	$0.251438\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.68660	0.00892638
$414$	0	0
$415$	− 262.799i	− 0.633251i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	329.743i	0.786975i	0.919330	+	0.393488i	$0.128732\pi$
$419$	329.743i	0.786975i	−0.919330	+	0.393488i	$0.871268\pi$
$420$	0	0
$421$	142.972	0.339600	0.169800	−	0.985479i	$-0.445688\pi$
$421$	142.972	0.339600	0.169800	+	0.985479i	$0.445688\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	538.520	1.26711
$426$	0	0
$427$	115.158i	0.269690i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 62.3758i	− 0.144723i	−0.997378	−	0.0723617i	$-0.976946\pi$
$431$	− 62.3758i	− 0.144723i	0.997378	−	0.0723617i	$-0.0230536\pi$
$432$	0	0
$433$	578.660	1.33640	0.668199	−	0.743983i	$-0.267066\pi$
$433$	578.660	1.33640	0.668199	+	0.743983i	$0.267066\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	90.7445	0.207653
$438$	0	0
$439$	317.049i	0.722208i	0.932526	+	0.361104i	$0.117600\pi$
$439$	317.049i	0.722208i	−0.932526	+	0.361104i	$0.882400\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	503.269i	1.13605i	0.823012	+	0.568024i	$0.192292\pi$
$443$	503.269i	1.13605i	−0.823012	+	0.568024i	$0.807708\pi$
$444$	0	0
$445$	120.814	0.271492
$446$	0	0
$447$	0	0
$448$	0	0
$449$	373.907	0.832756	0.416378	−	0.909192i	$-0.363299\pi$
$449$	373.907	0.832756	0.416378	+	0.909192i	$0.363299\pi$
$450$	0	0
$451$	330.246i	0.732254i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	580.122i	1.27499i
$456$	0	0
$457$	588.994	1.28883	0.644414	−	0.764677i	$-0.277101\pi$
$457$	588.994	1.28883	0.644414	+	0.764677i	$0.277101\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−524.278	−1.13726	−0.568631	−	0.822593i	$-0.692527\pi$
$461$	−524.278	−1.13726	−0.568631	+	0.822593i	$0.692527\pi$
$462$	0	0
$463$	− 296.528i	− 0.640449i	−0.947342	−	0.320225i	$-0.896242\pi$
$463$	− 296.528i	− 0.640449i	0.947342	−	0.320225i	$-0.103758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	441.111i	0.944563i	0.881448	+	0.472281i	$0.156569\pi$
$467$	441.111i	0.944563i	−0.881448	+	0.472281i	$0.843431\pi$
$468$	0	0
$469$	1220.49	2.60233
$470$	0	0
$471$	0	0
$472$	0	0
$473$	555.011	1.17338
$474$	0	0
$475$	− 77.8735i	− 0.163944i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 318.846i	− 0.665648i	−0.942989	−	0.332824i	$-0.891998\pi$
$479$	− 318.846i	− 0.665648i	0.942989	−	0.332824i	$-0.108002\pi$
$480$	0	0
$481$	191.714	0.398573
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−94.6291	−0.195111
$486$	0	0
$487$	− 492.673i	− 1.01165i	−0.862636	−	0.505824i	$-0.831188\pi$
$487$	− 492.673i	− 1.01165i	0.862636	−	0.505824i	$-0.168812\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 333.780i	− 0.679797i	−0.940462	−	0.339899i	$-0.889607\pi$
$491$	− 333.780i	− 0.679797i	0.940462	−	0.339899i	$-0.110393\pi$
$492$	0	0
$493$	−927.517	−1.88137
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−216.826	−0.436270
$498$	0	0
$499$	727.621i	1.45816i	0.684429	+	0.729079i	$0.260051\pi$
$499$	727.621i	1.45816i	−0.684429	+	0.729079i	$0.739949\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	171.182i	0.340322i	0.985416	+	0.170161i	$0.0544288\pi$
$503$	171.182i	0.340322i	−0.985416	+	0.170161i	$0.945571\pi$
$504$	0	0
$505$	−300.710	−0.595466
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−387.484	−0.761265	−0.380633	−	0.924726i	$-0.624294\pi$
$509$	−387.484	−0.761265	−0.380633	+	0.924726i	$0.624294\pi$
$510$	0	0
$511$	369.835i	0.723748i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	69.4203i	0.134797i
$516$	0	0
$517$	141.209	0.273132
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−36.4359	−0.0699346	−0.0349673	−	0.999388i	$-0.511133\pi$
$521$	−36.4359	−0.0699346	−0.0349673	+	0.999388i	$0.511133\pi$
$522$	0	0
$523$	365.186i	0.698253i	0.937076	+	0.349126i	$0.113522\pi$
$523$	365.186i	0.698253i	−0.937076	+	0.349126i	$0.886478\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	567.866i	1.07755i
$528$	0	0
$529$	95.6020	0.180722
$530$	0	0
$531$	0	0
$532$	0	0
$533$	446.062	0.836890
$534$	0	0
$535$	− 168.001i	− 0.314020i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1259.39i	− 2.33654i
$540$	0	0
$541$	115.152	0.212850	0.106425	−	0.994321i	$-0.466060\pi$
$541$	115.152	0.212850	0.106425	+	0.994321i	$0.466060\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	498.664	0.914980
$546$	0	0
$547$	− 587.574i	− 1.07418i	−0.843526	−	0.537088i	$-0.819524\pi$
$547$	− 587.574i	− 1.07418i	0.843526	−	0.537088i	$-0.180476\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	134.125i	0.243421i
$552$	0	0
$553$	−1446.28	−2.61534
$554$	0	0
$555$	0	0
$556$	0	0
$557$	393.804	0.707009	0.353505	−	0.935433i	$-0.384990\pi$
$557$	393.804	0.707009	0.353505	+	0.935433i	$0.384990\pi$
$558$	0	0
$559$	− 749.650i	− 1.34106i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	377.848i	0.671133i	0.942016	+	0.335567i	$0.108928\pi$
$563$	377.848i	0.671133i	−0.942016	+	0.335567i	$0.891072\pi$
$564$	0	0
$565$	343.687	0.608295
$566$	0	0
$567$	0	0
$568$	0	0
$569$	771.267	1.35548	0.677739	−	0.735302i	$-0.262960\pi$
$569$	771.267	1.35548	0.677739	+	0.735302i	$0.262960\pi$
$570$	0	0
$571$	− 193.651i	− 0.339144i	−0.985518	−	0.169572i	$-0.945762\pi$
$571$	− 193.651i	− 0.339144i	0.985518	−	0.169572i	$-0.0542385\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	371.926i	0.646827i
$576$	0	0
$577$	−426.852	−0.739779	−0.369889	−	0.929076i	$-0.620604\pi$
$577$	−426.852	−0.739779	−0.369889	+	0.929076i	$0.620604\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1174.46	2.02144
$582$	0	0
$583$	− 1087.26i	− 1.86494i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 855.854i	− 1.45801i	−0.684506	−	0.729007i	$-0.739982\pi$
$587$	− 855.854i	− 1.45801i	0.684506	−	0.729007i	$-0.260018\pi$
$588$	0	0
$589$	82.1171	0.139418
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1101.32	1.85721	0.928604	−	0.371073i	$-0.121010\pi$
$593$	1101.32	1.85721	0.928604	+	0.371073i	$0.121010\pi$
$594$	0	0
$595$	− 961.108i	− 1.61531i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1038.20i	1.73322i	0.498986	+	0.866610i	$0.333706\pi$
$599$	1038.20i	1.73322i	−0.498986	+	0.866610i	$0.666294\pi$
$600$	0	0
$601$	458.884	0.763534	0.381767	−	0.924259i	$-0.375316\pi$
$601$	458.884	0.763534	0.381767	+	0.924259i	$0.375316\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	161.466	0.266887
$606$	0	0
$607$	− 787.255i	− 1.29696i	−0.761232	−	0.648480i	$-0.775405\pi$
$607$	− 787.255i	− 1.29696i	0.761232	−	0.648480i	$-0.224595\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 190.731i	− 0.312161i
$612$	0	0
$613$	412.585	0.673059	0.336529	−	0.941673i	$-0.390747\pi$
$613$	412.585	0.673059	0.336529	+	0.941673i	$0.390747\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−82.9712	−0.134475	−0.0672376	−	0.997737i	$-0.521419\pi$
$617$	−82.9712	−0.134475	−0.0672376	+	0.997737i	$0.521419\pi$
$618$	0	0
$619$	409.774i	0.661993i	0.943632	+	0.330997i	$0.107385\pi$
$619$	409.774i	0.661993i	−0.943632	+	0.330997i	$0.892615\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	539.920i	0.866645i
$624$	0	0
$625$	140.807	0.225292
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−317.619	−0.504959
$630$	0	0
$631$	− 80.5496i	− 0.127654i	−0.997961	−	0.0638269i	$-0.979669\pi$
$631$	− 80.5496i	− 0.127654i	0.997961	−	0.0638269i	$-0.0203306\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 355.611i	− 0.560017i
$636$	0	0
$637$	−1701.06	−2.67042
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−512.618	−0.799716	−0.399858	−	0.916577i	$-0.630941\pi$
$641$	−512.618	−0.799716	−0.399858	+	0.916577i	$0.630941\pi$
$642$	0	0
$643$	959.566i	1.49233i	0.665763	+	0.746163i	$0.268106\pi$
$643$	959.566i	1.49233i	−0.665763	+	0.746163i	$0.731894\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 634.956i	− 0.981385i	−0.871333	−	0.490693i	$-0.836744\pi$
$647$	− 634.956i	− 0.981385i	0.871333	−	0.490693i	$-0.163256\pi$
$648$	0	0
$649$	−4.16012	−0.00641005
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−545.514	−0.835396	−0.417698	−	0.908586i	$-0.637163\pi$
$653$	−545.514	−0.835396	−0.417698	+	0.908586i	$0.637163\pi$
$654$	0	0
$655$	− 142.749i	− 0.217937i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 69.3817i	− 0.105283i	−0.998613	−	0.0526417i	$-0.983236\pi$
$659$	− 69.3817i	− 0.105283i	0.998613	−	0.0526417i	$-0.0167641\pi$
$660$	0	0
$661$	−359.160	−0.543358	−0.271679	−	0.962388i	$-0.587579\pi$
$661$	−359.160	−0.543358	−0.271679	+	0.962388i	$0.587579\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−138.982	−0.208996
$666$	0	0
$667$	− 640.584i	− 0.960396i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 129.949i	− 0.193665i
$672$	0	0
$673$	851.299	1.26493	0.632466	−	0.774588i	$-0.282043\pi$
$673$	851.299	1.26493	0.632466	+	0.774588i	$0.282043\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1072.72	−1.58452	−0.792261	−	0.610182i	$-0.791096\pi$
$677$	−1072.72	−1.58452	−0.792261	+	0.610182i	$0.791096\pi$
$678$	0	0
$679$	− 422.900i	− 0.622828i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	337.261i	0.493794i	0.969042	+	0.246897i	$0.0794110\pi$
$683$	337.261i	0.493794i	−0.969042	+	0.246897i	$0.920589\pi$
$684$	0	0
$685$	293.357	0.428258
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1468.56	−2.13143
$690$	0	0
$691$	− 1236.01i	− 1.78872i	−0.447347	−	0.894360i	$-0.647631\pi$
$691$	− 1236.01i	− 1.78872i	0.447347	−	0.894360i	$-0.352369\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 325.607i	− 0.468500i
$696$	0	0
$697$	−739.007	−1.06027
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1068.67	1.52449	0.762244	−	0.647290i	$-0.224098\pi$
$701$	1068.67	1.52449	0.762244	+	0.647290i	$0.224098\pi$
$702$	0	0
$703$	45.9298i	0.0653339i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1343.88i	− 1.90083i
$708$	0	0
$709$	−965.886	−1.36232	−0.681161	−	0.732134i	$-0.738525\pi$
$709$	−965.886	−1.36232	−0.681161	+	0.732134i	$0.738525\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−392.194	−0.550061
$714$	0	0
$715$	− 654.636i	− 0.915574i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 645.354i	− 0.897572i	−0.893639	−	0.448786i	$-0.851857\pi$
$719$	− 645.354i	− 0.897572i	0.893639	−	0.448786i	$-0.148143\pi$
$720$	0	0
$721$	−310.241	−0.430293
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−549.725	−0.758241
$726$	0	0
$727$	452.586i	0.622539i	0.950322	+	0.311270i	$0.100754\pi$
$727$	452.586i	0.622539i	−0.950322	+	0.311270i	$0.899246\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1241.97i	1.69900i
$732$	0	0
$733$	−447.521	−0.610533	−0.305267	−	0.952267i	$-0.598746\pi$
$733$	−447.521	−0.610533	−0.305267	+	0.952267i	$0.598746\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1377.26	−1.86874
$738$	0	0
$739$	− 862.597i	− 1.16725i	−0.812024	−	0.583625i	$-0.801634\pi$
$739$	− 862.597i	− 1.16725i	0.812024	−	0.583625i	$-0.198366\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	460.088i	0.619230i	0.950862	+	0.309615i	$0.100200\pi$
$743$	460.088i	0.619230i	−0.950862	+	0.309615i	$0.899800\pi$
$744$	0	0
$745$	−88.9856	−0.119444
$746$	0	0
$747$	0	0
$748$	0	0
$749$	750.800	1.00240
$750$	0	0
$751$	484.817i	0.645562i	0.946474	+	0.322781i	$0.104618\pi$
$751$	484.817i	0.645562i	−0.946474	+	0.322781i	$0.895382\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 373.653i	− 0.494904i
$756$	0	0
$757$	975.475	1.28861	0.644303	−	0.764770i	$-0.277148\pi$
$757$	975.475	1.28861	0.644303	+	0.764770i	$0.277148\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1144.81	1.50435	0.752173	−	0.658965i	$-0.229006\pi$
$761$	1144.81	1.50435	0.752173	+	0.658965i	$0.229006\pi$
$762$	0	0
$763$	2228.55i	2.92077i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.61906i	0.00732602i
$768$	0	0
$769$	−1139.96	−1.48239	−0.741197	−	0.671288i	$-0.765742\pi$
$769$	−1139.96	−1.48239	−0.741197	+	0.671288i	$0.765742\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	589.169	0.762184	0.381092	−	0.924537i	$-0.375548\pi$
$773$	589.169	0.762184	0.381092	+	0.924537i	$0.375548\pi$
$774$	0	0
$775$	336.566i	0.434278i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	106.865i	0.137183i
$780$	0	0
$781$	244.676	0.313286
$782$	0	0
$783$	0	0
$784$	0	0
$785$	556.959	0.709501
$786$	0	0
$787$	− 442.634i	− 0.562433i	−0.959644	−	0.281216i	$-0.909262\pi$
$787$	− 442.634i	− 0.562433i	0.959644	−	0.281216i	$-0.0907378\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1535.95i	1.94178i
$792$	0	0
$793$	−175.522	−0.221339
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1473.52	1.84883	0.924416	−	0.381385i	$-0.124553\pi$
$797$	1473.52	1.84883	0.924416	+	0.381385i	$0.124553\pi$
$798$	0	0
$799$	315.990i	0.395482i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 417.339i	− 0.519725i
$804$	0	0
$805$	663.784	0.824576
$806$	0	0
$807$	0	0
$808$	0	0
$809$	917.665	1.13432	0.567160	−	0.823607i	$-0.308042\pi$
$809$	917.665	1.13432	0.567160	+	0.823607i	$0.308042\pi$
$810$	0	0
$811$	− 214.327i	− 0.264276i	−0.991231	−	0.132138i	$-0.957816\pi$
$811$	− 214.327i	− 0.264276i	0.991231	−	0.132138i	$-0.0421841\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 516.219i	− 0.633397i
$816$	0	0
$817$	179.597	0.219825
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−751.544	−0.915400	−0.457700	−	0.889107i	$-0.651327\pi$
$821$	−751.544	−0.915400	−0.457700	+	0.889107i	$0.651327\pi$
$822$	0	0
$823$	− 1139.37i	− 1.38441i	−0.721703	−	0.692203i	$-0.756641\pi$
$823$	− 1139.37i	− 1.38441i	0.721703	−	0.692203i	$-0.243359\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 738.372i	− 0.892832i	−0.894826	−	0.446416i	$-0.852700\pi$
$827$	− 738.372i	− 0.892832i	0.894826	−	0.446416i	$-0.147300\pi$
$828$	0	0
$829$	1136.73	1.37120	0.685602	−	0.727977i	$-0.259539\pi$
$829$	1136.73	1.37120	0.685602	+	0.727977i	$0.259539\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2818.20	3.38320
$834$	0	0
$835$	240.908i	0.288513i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1038.94i	− 1.23831i	−0.785268	−	0.619155i	$-0.787475\pi$
$839$	− 1038.94i	− 1.23831i	0.785268	−	0.619155i	$-0.212525\pi$
$840$	0	0
$841$	105.815	0.125821
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−432.803	−0.512192
$846$	0	0
$847$	721.598i	0.851946i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 219.362i	− 0.257769i
$852$	0	0
$853$	−215.622	−0.252781	−0.126390	−	0.991981i	$-0.540339\pi$
$853$	−215.622	−0.252781	−0.126390	+	0.991981i	$0.540339\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	311.997	0.364057	0.182029	−	0.983293i	$-0.441734\pi$
$857$	311.997	0.364057	0.182029	+	0.983293i	$0.441734\pi$
$858$	0	0
$859$	− 240.560i	− 0.280047i	−0.990148	−	0.140023i	$-0.955282\pi$
$859$	− 240.560i	− 0.280047i	0.990148	−	0.140023i	$-0.0447178\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 394.897i	− 0.457587i	−0.973475	−	0.228793i	$-0.926522\pi$
$863$	− 394.897i	− 0.457587i	0.973475	−	0.228793i	$-0.0734780\pi$
$864$	0	0
$865$	630.579	0.728993
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1632.05	1.87808
$870$	0	0
$871$	1860.26i	2.13577i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1366.75i	− 1.56200i
$876$	0	0
$877$	1127.17	1.28525	0.642627	−	0.766179i	$-0.277844\pi$
$877$	1127.17	1.28525	0.642627	+	0.766179i	$0.277844\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1239.88	−1.40736	−0.703679	−	0.710518i	$-0.748461\pi$
$881$	−1239.88	−1.40736	−0.703679	+	0.710518i	$0.748461\pi$
$882$	0	0
$883$	107.542i	0.121792i	0.998144	+	0.0608961i	$0.0193958\pi$
$883$	107.542i	0.121792i	−0.998144	+	0.0608961i	$0.980604\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 64.1710i	− 0.0723461i	−0.999346	−	0.0361731i	$-0.988483\pi$
$887$	− 64.1710i	− 0.0723461i	0.999346	−	0.0361731i	$-0.0115168\pi$
$888$	0	0
$889$	1589.24	1.78767
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45.6942	0.0511693
$894$	0	0
$895$	309.237i	0.345516i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 579.682i	− 0.644807i
$900$	0	0
$901$	2433.01	2.70035
$902$	0	0
$903$	0	0
$904$	0	0
$905$	637.717	0.704660
$906$	0	0
$907$	− 542.670i	− 0.598313i	−0.954204	−	0.299157i	$-0.903295\pi$
$907$	− 542.670i	− 0.598313i	0.954204	−	0.299157i	$-0.0967053\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 260.740i	− 0.286213i	−0.989707	−	0.143106i	$-0.954291\pi$
$911$	− 260.740i	− 0.286213i	0.989707	−	0.143106i	$-0.0457091\pi$
$912$	0	0
$913$	−1325.31	−1.45160
$914$	0	0
$915$	0	0
$916$	0	0
$917$	637.949	0.695692
$918$	0	0
$919$	1397.29i	1.52044i	0.649664	+	0.760221i	$0.274909\pi$
$919$	1397.29i	1.52044i	−0.649664	+	0.760221i	$0.725091\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 330.483i	− 0.358053i
$924$	0	0
$925$	−188.248	−0.203511
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1578.09	1.69870	0.849351	−	0.527829i	$-0.176994\pi$
$929$	1578.09	1.69870	0.849351	+	0.527829i	$0.176994\pi$
$930$	0	0
$931$	− 407.530i	− 0.437734i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1084.56i	1.15995i
$936$	0	0
$937$	1419.82	1.51529	0.757643	−	0.652669i	$-0.226351\pi$
$937$	1419.82	1.51529	0.757643	+	0.652669i	$0.226351\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1038.25	1.10335	0.551676	−	0.834058i	$-0.313988\pi$
$941$	1038.25	1.10335	0.551676	+	0.834058i	$0.313988\pi$
$942$	0	0
$943$	− 510.391i	− 0.541242i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1116.92i	− 1.17943i	−0.807613	−	0.589713i	$-0.799241\pi$
$947$	− 1116.92i	− 1.17943i	0.807613	−	0.589713i	$-0.200759\pi$
$948$	0	0
$949$	−563.698	−0.593991
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−367.870	−0.386012	−0.193006	−	0.981198i	$-0.561824\pi$
$953$	−367.870	−0.386012	−0.193006	+	0.981198i	$0.561824\pi$
$954$	0	0
$955$	− 909.825i	− 0.952697i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1311.02i	1.36707i
$960$	0	0
$961$	606.094	0.630690
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−157.856	−0.163581
$966$	0	0
$967$	− 586.624i	− 0.606644i	−0.952888	−	0.303322i	$-0.901904\pi$
$967$	− 586.624i	− 0.606644i	0.952888	−	0.303322i	$-0.0980957\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	227.373i	0.234164i	0.993122	+	0.117082i	$0.0373540\pi$
$971$	227.373i	0.234164i	−0.993122	+	0.117082i	$0.962646\pi$
$972$	0	0
$973$	1455.15	1.49553
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1423.89	−1.45741	−0.728703	−	0.684830i	$-0.759877\pi$
$977$	−1423.89	−1.45741	−0.728703	+	0.684830i	$0.759877\pi$
$978$	0	0
$979$	− 609.270i	− 0.622339i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	393.486i	0.400291i	0.979766	+	0.200146i	$0.0641415\pi$
$983$	393.486i	0.400291i	−0.979766	+	0.200146i	$0.935858\pi$
$984$	0	0
$985$	−223.330	−0.226731
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−857.761	−0.867301
$990$	0	0
$991$	− 1080.95i	− 1.09076i	−0.838188	−	0.545382i	$-0.816385\pi$
$991$	− 1080.95i	− 1.09076i	0.838188	−	0.545382i	$-0.183615\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 837.432i	− 0.841640i
$996$	0	0
$997$	820.061	0.822529	0.411265	−	0.911516i	$-0.365087\pi$
$997$	820.061	0.822529	0.411265	+	0.911516i	$0.365087\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.m.e.1711.5		12
3.2	odd	2		912.3.m.b.799.4	✓	12
4.3	odd	2	inner	2736.3.m.e.1711.6		12
12.11	even	2		912.3.m.b.799.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.3.m.b.799.4	✓	12	3.2	odd	2
912.3.m.b.799.10	yes	12	12.11	even	2
2736.3.m.e.1711.5		12	1.1	even	1	trivial
2736.3.m.e.1711.6		12	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.67107 q^{5} -11.9371i q^{7} +O(q^{10})\)	\(q-2.67107 q^{5} -11.9371i q^{7} +13.4703i q^{11} +18.1943 q^{13} -30.1432 q^{17} +4.35890i q^{19} -20.8182i q^{23} -17.8654 q^{25} +30.7704 q^{29} -18.8390i q^{31} +31.8848i q^{35} +10.5370 q^{37} +24.5166 q^{41} -41.2024i q^{43} -10.4830i q^{47} -93.4939 q^{49} -80.7151 q^{53} -35.9802i q^{55} +0.308836i q^{59} -9.64705 q^{61} -48.5983 q^{65} +102.244i q^{67} -18.1641i q^{71} -30.9821 q^{73} +160.797 q^{77} -121.159i q^{79} +98.3874i q^{83} +80.5145 q^{85} -45.2305 q^{89} -217.187i q^{91} -11.6429i q^{95} +35.4274 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{5}+O(q^{10}) \) 12 * q + 10 * q^5	\( 12 q + 10 q^{5} + 36 q^{13} - 14 q^{17} + 10 q^{25} + 108 q^{29} - 16 q^{37} - 16 q^{41} - 118 q^{49} - 220 q^{53} + 366 q^{61} - 140 q^{65} - 158 q^{73} + 286 q^{77} + 98 q^{85} + 396 q^{89} + 224 q^{97}+O(q^{100}) \) 12 * q + 10 * q^5 + 36 * q^13 - 14 * q^17 + 10 * q^25 + 108 * q^29 - 16 * q^37 - 16 * q^41 - 118 * q^49 - 220 * q^53 + 366 * q^61 - 140 * q^65 - 158 * q^73 + 286 * q^77 + 98 * q^85 + 396 * q^89 + 224 * q^97

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