LMFDB - Embedded newform 2736.3.m.f.1711.1

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} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + \cdots + 1024 $ x^12 - 2x^11 + 50x^10 - 136x^9 + 2215x^8 - 5020x^7 + 18282x^6 - 12094x^5 + 48457x^4 - 30372x^3 + 89392x^2 + 9344*x + 1024
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			1711.1
Root			$-0.820808 + 1.42168i$ of defining polynomial
Character	$\chi$	$=$	2736.1711
Dual form			2736.3.m.f.1711.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.90000 q^{5} -2.78940i q^{7} +O(q^{10})$	$q-8.90000 q^{5} -2.78940i q^{7} +0.270477i q^{11} +7.37703 q^{13} -9.27079 q^{17} -4.35890i q^{19} +23.8911i q^{23} +54.2099 q^{25} +19.2337 q^{29} -6.42293i q^{31} +24.8256i q^{35} +31.6410 q^{37} -17.2511 q^{41} -2.05975i q^{43} -59.4123i q^{47} +41.2193 q^{49} +44.1111 q^{53} -2.40725i q^{55} +54.8364i q^{59} -52.2724 q^{61} -65.6555 q^{65} +42.9738i q^{67} +16.7037i q^{71} -20.3233 q^{73} +0.754469 q^{77} -25.5136i q^{79} -23.6263i q^{83} +82.5100 q^{85} -28.2680 q^{89} -20.5775i q^{91} +38.7942i q^{95} -91.4340 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{5}+O(q^{10}) $ 12 * q + 10 * q^5	$ 12 q + 10 q^{5} + 36 q^{13} + 10 q^{17} + 58 q^{25} - 12 q^{29} + 32 q^{37} - 136 q^{41} - 22 q^{49} + 236 q^{53} - 210 q^{61} + 52 q^{65} - 158 q^{73} + 70 q^{77} + 242 q^{85} - 444 q^{89} + 320 q^{97}+O(q^{100}) $ 12 * q + 10 * q^5 + 36 * q^13 + 10 * q^17 + 58 * q^25 - 12 * q^29 + 32 * q^37 - 136 * q^41 - 22 * q^49 + 236 * q^53 - 210 * q^61 + 52 * q^65 - 158 * q^73 + 70 * q^77 + 242 * q^85 - 444 * q^89 + 320 * 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
$5$	−8.90000	−1.78000	−0.890000	−	0.455961i	$-0.849296\pi$
$5$	−8.90000	−1.78000	−0.890000	+	0.455961i	$0.849296\pi$
$6$	0	0
$7$	− 2.78940i	− 0.398486i	−0.979950	−	0.199243i	$-0.936152\pi$
$7$	− 2.78940i	− 0.398486i	0.979950	−	0.199243i	$-0.0638482\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.270477i	0.0245888i	0.999924	+	0.0122944i	$0.00391354\pi$
$11$	0.270477i	0.0245888i	−0.999924	+	0.0122944i	$0.996086\pi$
$12$	0	0
$13$	7.37703	0.567464	0.283732	−	0.958904i	$-0.408427\pi$
$13$	7.37703	0.567464	0.283732	+	0.958904i	$0.408427\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−9.27079	−0.545341	−0.272670	−	0.962108i	$-0.587907\pi$
$17$	−9.27079	−0.545341	−0.272670	+	0.962108i	$0.587907\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	23.8911i	1.03874i	0.854549	+	0.519372i	$0.173834\pi$
$23$	23.8911i	1.03874i	−0.854549	+	0.519372i	$0.826166\pi$
$24$	0	0
$25$	54.2099	2.16840
$26$	0	0
$27$	0	0
$28$	0	0
$29$	19.2337	0.663232	0.331616	−	0.943415i	$-0.392406\pi$
$29$	19.2337	0.663232	0.331616	+	0.943415i	$0.392406\pi$
$30$	0	0
$31$	− 6.42293i	− 0.207191i	−0.994619	−	0.103596i	$-0.966965\pi$
$31$	− 6.42293i	− 0.207191i	0.994619	−	0.103596i	$-0.0330348\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	24.8256i	0.709304i
$36$	0	0
$37$	31.6410	0.855161	0.427580	−	0.903977i	$-0.359366\pi$
$37$	31.6410	0.855161	0.427580	+	0.903977i	$0.359366\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−17.2511	−0.420758	−0.210379	−	0.977620i	$-0.567470\pi$
$41$	−17.2511	−0.420758	−0.210379	+	0.977620i	$0.567470\pi$
$42$	0	0
$43$	− 2.05975i	− 0.0479012i	−0.999713	−	0.0239506i	$-0.992376\pi$
$43$	− 2.05975i	− 0.0479012i	0.999713	−	0.0239506i	$-0.00762443\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 59.4123i	− 1.26409i	−0.774931	−	0.632045i	$-0.782216\pi$
$47$	− 59.4123i	− 1.26409i	0.774931	−	0.632045i	$-0.217784\pi$
$48$	0	0
$49$	41.2193	0.841209
$50$	0	0
$51$	0	0
$52$	0	0
$53$	44.1111	0.832285	0.416142	−	0.909299i	$-0.363382\pi$
$53$	44.1111	0.832285	0.416142	+	0.909299i	$0.363382\pi$
$54$	0	0
$55$	− 2.40725i	− 0.0437681i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	54.8364i	0.929431i	0.885460	+	0.464716i	$0.153843\pi$
$59$	54.8364i	0.929431i	−0.885460	+	0.464716i	$0.846157\pi$
$60$	0	0
$61$	−52.2724	−0.856925	−0.428462	−	0.903560i	$-0.640945\pi$
$61$	−52.2724	−0.856925	−0.428462	+	0.903560i	$0.640945\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−65.6555	−1.01009
$66$	0	0
$67$	42.9738i	0.641399i	0.947181	+	0.320700i	$0.103918\pi$
$67$	42.9738i	0.641399i	−0.947181	+	0.320700i	$0.896082\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.7037i	0.235263i	0.993057	+	0.117631i	$0.0375301\pi$
$71$	16.7037i	0.235263i	−0.993057	+	0.117631i	$0.962470\pi$
$72$	0	0
$73$	−20.3233	−0.278401	−0.139201	−	0.990264i	$-0.544453\pi$
$73$	−20.3233	−0.278401	−0.139201	+	0.990264i	$0.544453\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.754469	0.00979830
$78$	0	0
$79$	− 25.5136i	− 0.322957i	−0.986876	−	0.161478i	$-0.948374\pi$
$79$	− 25.5136i	− 0.322957i	0.986876	−	0.161478i	$-0.0516262\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 23.6263i	− 0.284655i	−0.989820	−	0.142327i	$-0.954541\pi$
$83$	− 23.6263i	− 0.284655i	0.989820	−	0.142327i	$-0.0454586\pi$
$84$	0	0
$85$	82.5100	0.970706
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−28.2680	−0.317617	−0.158809	−	0.987309i	$-0.550765\pi$
$89$	−28.2680	−0.317617	−0.158809	+	0.987309i	$0.550765\pi$
$90$	0	0
$91$	− 20.5775i	− 0.226126i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	38.7942i	0.408360i
$96$	0	0
$97$	−91.4340	−0.942619	−0.471309	−	0.881968i	$-0.656218\pi$
$97$	−91.4340	−0.942619	−0.471309	+	0.881968i	$0.656218\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−147.437	−1.45977	−0.729887	−	0.683567i	$-0.760428\pi$
$101$	−147.437	−1.45977	−0.729887	+	0.683567i	$0.760428\pi$
$102$	0	0
$103$	− 103.703i	− 1.00683i	−0.864045	−	0.503415i	$-0.832077\pi$
$103$	− 103.703i	− 1.00683i	0.864045	−	0.503415i	$-0.167923\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	111.378i	1.04092i	0.853887	+	0.520459i	$0.174239\pi$
$107$	111.378i	1.04092i	−0.853887	+	0.520459i	$0.825761\pi$
$108$	0	0
$109$	60.1516	0.551849	0.275925	−	0.961179i	$-0.411016\pi$
$109$	60.1516	0.551849	0.275925	+	0.961179i	$0.411016\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	74.8741	0.662602	0.331301	−	0.943525i	$-0.392512\pi$
$113$	74.8741	0.662602	0.331301	+	0.943525i	$0.392512\pi$
$114$	0	0
$115$	− 212.631i	− 1.84896i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	25.8599i	0.217310i
$120$	0	0
$121$	120.927	0.999395
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−259.968	−2.07975
$126$	0	0
$127$	− 168.857i	− 1.32958i	−0.747030	−	0.664790i	$-0.768521\pi$
$127$	− 168.857i	− 1.32958i	0.747030	−	0.664790i	$-0.231479\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	118.259i	0.902744i	0.892336	+	0.451372i	$0.149065\pi$
$131$	118.259i	0.902744i	−0.892336	+	0.451372i	$0.850935\pi$
$132$	0	0
$133$	−12.1587	−0.0914189
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−85.7018	−0.625560	−0.312780	−	0.949826i	$-0.601260\pi$
$137$	−85.7018	−0.625560	−0.312780	+	0.949826i	$0.601260\pi$
$138$	0	0
$139$	85.7838i	0.617149i	0.951200	+	0.308575i	$0.0998520\pi$
$139$	85.7838i	0.617149i	−0.951200	+	0.308575i	$0.900148\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.99532i	0.0139533i
$144$	0	0
$145$	−171.180	−1.18055
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.3400	−0.109665	−0.0548323	−	0.998496i	$-0.517462\pi$
$149$	−16.3400	−0.109665	−0.0548323	+	0.998496i	$0.517462\pi$
$150$	0	0
$151$	− 49.9011i	− 0.330471i	−0.986254	−	0.165235i	$-0.947162\pi$
$151$	− 49.9011i	− 0.330471i	0.986254	−	0.165235i	$-0.0528383\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	57.1641i	0.368801i
$156$	0	0
$157$	−294.055	−1.87296	−0.936481	−	0.350718i	$-0.885938\pi$
$157$	−294.055	−1.87296	−0.936481	+	0.350718i	$0.885938\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	66.6418	0.413924
$162$	0	0
$163$	33.7956i	0.207335i	0.994612	+	0.103668i	$0.0330578\pi$
$163$	33.7956i	0.207335i	−0.994612	+	0.103668i	$0.966942\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 274.855i	− 1.64584i	−0.568157	−	0.822920i	$-0.692343\pi$
$167$	− 274.855i	− 1.64584i	0.568157	−	0.822920i	$-0.307657\pi$
$168$	0	0
$169$	−114.579	−0.677985
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−272.490	−1.57509	−0.787543	−	0.616260i	$-0.788647\pi$
$173$	−272.490	−1.57509	−0.787543	+	0.616260i	$0.788647\pi$
$174$	0	0
$175$	− 151.213i	− 0.864075i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 319.797i	− 1.78657i	−0.449486	−	0.893287i	$-0.648393\pi$
$179$	− 319.797i	− 1.78657i	0.449486	−	0.893287i	$-0.351607\pi$
$180$	0	0
$181$	−245.785	−1.35793	−0.678963	−	0.734172i	$-0.737570\pi$
$181$	−245.785	−1.35793	−0.678963	+	0.734172i	$0.737570\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−281.604	−1.52219
$186$	0	0
$187$	− 2.50754i	− 0.0134093i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 336.257i	− 1.76051i	−0.474504	−	0.880253i	$-0.657373\pi$
$191$	− 336.257i	− 1.76051i	0.474504	−	0.880253i	$-0.342627\pi$
$192$	0	0
$193$	−329.129	−1.70533	−0.852665	−	0.522458i	$-0.825015\pi$
$193$	−329.129	−1.70533	−0.852665	+	0.522458i	$0.825015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−276.972	−1.40595	−0.702973	−	0.711216i	$-0.748145\pi$
$197$	−276.972	−1.40595	−0.702973	+	0.711216i	$0.748145\pi$
$198$	0	0
$199$	− 235.145i	− 1.18163i	−0.806806	−	0.590816i	$-0.798806\pi$
$199$	− 235.145i	− 1.18163i	0.806806	−	0.590816i	$-0.201194\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 53.6505i	− 0.264288i
$204$	0	0
$205$	153.534	0.748948
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.17898	0.00564107
$210$	0	0
$211$	147.521i	0.699154i	0.936908	+	0.349577i	$0.113675\pi$
$211$	147.521i	0.699154i	−0.936908	+	0.349577i	$0.886325\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.3318i	0.0852640i
$216$	0	0
$217$	−17.9161	−0.0825628
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−68.3909	−0.309461
$222$	0	0
$223$	420.735i	1.88670i	0.331794	+	0.943352i	$0.392346\pi$
$223$	420.735i	1.88670i	−0.331794	+	0.943352i	$0.607654\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 257.749i	− 1.13546i	−0.823215	−	0.567730i	$-0.807822\pi$
$227$	− 257.749i	− 1.13546i	0.823215	−	0.567730i	$-0.192178\pi$
$228$	0	0
$229$	155.183	0.677654	0.338827	−	0.940849i	$-0.389970\pi$
$229$	155.183	0.677654	0.338827	+	0.940849i	$0.389970\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	334.053	1.43370	0.716852	−	0.697225i	$-0.245582\pi$
$233$	334.053	1.43370	0.716852	+	0.697225i	$0.245582\pi$
$234$	0	0
$235$	528.769i	2.25008i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 447.085i	− 1.87065i	−0.353793	−	0.935324i	$-0.615108\pi$
$239$	− 447.085i	− 1.87065i	0.353793	−	0.935324i	$-0.384892\pi$
$240$	0	0
$241$	−321.663	−1.33470	−0.667351	−	0.744744i	$-0.732572\pi$
$241$	−321.663	−1.33470	−0.667351	+	0.744744i	$0.732572\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−366.851	−1.49735
$246$	0	0
$247$	− 32.1557i	− 0.130185i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	231.428i	0.922024i	0.887394	+	0.461012i	$0.152513\pi$
$251$	231.428i	0.922024i	−0.887394	+	0.461012i	$0.847487\pi$
$252$	0	0
$253$	−6.46200	−0.0255415
$254$	0	0
$255$	0	0
$256$	0	0
$257$	383.148	1.49085	0.745424	−	0.666590i	$-0.232247\pi$
$257$	383.148	1.49085	0.745424	+	0.666590i	$0.232247\pi$
$258$	0	0
$259$	− 88.2592i	− 0.340769i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 211.078i	− 0.802579i	−0.915951	−	0.401289i	$-0.868562\pi$
$263$	− 211.078i	− 0.802579i	0.915951	−	0.401289i	$-0.131438\pi$
$264$	0	0
$265$	−392.589	−1.48147
$266$	0	0
$267$	0	0
$268$	0	0
$269$	250.597	0.931586	0.465793	−	0.884894i	$-0.345769\pi$
$269$	250.597	0.931586	0.465793	+	0.884894i	$0.345769\pi$
$270$	0	0
$271$	− 180.019i	− 0.664275i	−0.943231	−	0.332138i	$-0.892230\pi$
$271$	− 180.019i	− 0.664275i	0.943231	−	0.332138i	$-0.107770\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14.6626i	0.0533184i
$276$	0	0
$277$	464.791	1.67795	0.838974	−	0.544172i	$-0.183156\pi$
$277$	464.791	1.67795	0.838974	+	0.544172i	$0.183156\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−216.576	−0.770734	−0.385367	−	0.922763i	$-0.625925\pi$
$281$	−216.576	−0.770734	−0.385367	+	0.922763i	$0.625925\pi$
$282$	0	0
$283$	70.7228i	0.249904i	0.992163	+	0.124952i	$0.0398777\pi$
$283$	70.7228i	0.249904i	−0.992163	+	0.124952i	$0.960122\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	48.1201i	0.167666i
$288$	0	0
$289$	−203.052	−0.702604
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−378.264	−1.29100	−0.645502	−	0.763758i	$-0.723352\pi$
$293$	−378.264	−1.29100	−0.645502	+	0.763758i	$0.723352\pi$
$294$	0	0
$295$	− 488.044i	− 1.65439i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	176.245i	0.589449i
$300$	0	0
$301$	−5.74546	−0.0190879
$302$	0	0
$303$	0	0
$304$	0	0
$305$	465.224	1.52533
$306$	0	0
$307$	122.269i	0.398270i	0.979972	+	0.199135i	$0.0638132\pi$
$307$	122.269i	0.398270i	−0.979972	+	0.199135i	$0.936187\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 280.599i	− 0.902249i	−0.892461	−	0.451125i	$-0.851023\pi$
$311$	− 280.599i	− 0.902249i	0.892461	−	0.451125i	$-0.148977\pi$
$312$	0	0
$313$	405.043	1.29407	0.647033	−	0.762462i	$-0.276009\pi$
$313$	405.043	1.29407	0.647033	+	0.762462i	$0.276009\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	150.448	0.474598	0.237299	−	0.971437i	$-0.423738\pi$
$317$	150.448	0.474598	0.237299	+	0.971437i	$0.423738\pi$
$318$	0	0
$319$	5.20228i	0.0163081i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	40.4104i	0.125110i
$324$	0	0
$325$	399.908	1.23049
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−165.725	−0.503722
$330$	0	0
$331$	411.544i	1.24334i	0.783281	+	0.621668i	$0.213544\pi$
$331$	411.544i	1.24334i	−0.783281	+	0.621668i	$0.786456\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 382.466i	− 1.14169i
$336$	0	0
$337$	−293.791	−0.871783	−0.435892	−	0.899999i	$-0.643567\pi$
$337$	−293.791	−0.871783	−0.435892	+	0.899999i	$0.643567\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.73726	0.00509460
$342$	0	0
$343$	− 251.657i	− 0.733695i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	332.604i	0.958512i	0.877675	+	0.479256i	$0.159093\pi$
$347$	332.604i	0.958512i	−0.877675	+	0.479256i	$0.840907\pi$
$348$	0	0
$349$	−4.98236	−0.0142761	−0.00713806	−	0.999975i	$-0.502272\pi$
$349$	−4.98236	−0.0142761	−0.00713806	+	0.999975i	$0.502272\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−357.751	−1.01346	−0.506730	−	0.862105i	$-0.669146\pi$
$353$	−357.751	−1.01346	−0.506730	+	0.862105i	$0.669146\pi$
$354$	0	0
$355$	− 148.662i	− 0.418768i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 379.442i	− 1.05694i	−0.848952	−	0.528471i	$-0.822766\pi$
$359$	− 379.442i	− 1.05694i	0.848952	−	0.528471i	$-0.177234\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	180.877	0.495554
$366$	0	0
$367$	317.133i	0.864124i	0.901844	+	0.432062i	$0.142214\pi$
$367$	317.133i	0.864124i	−0.901844	+	0.432062i	$0.857786\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 123.043i	− 0.331653i
$372$	0	0
$373$	−619.617	−1.66117	−0.830586	−	0.556891i	$-0.811994\pi$
$373$	−619.617	−1.66117	−0.830586	+	0.556891i	$0.811994\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	141.888	0.376360
$378$	0	0
$379$	503.578i	1.32870i	0.747421	+	0.664351i	$0.231292\pi$
$379$	503.578i	1.32870i	−0.747421	+	0.664351i	$0.768708\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	357.174i	0.932568i	0.884635	+	0.466284i	$0.154407\pi$
$383$	357.174i	0.932568i	−0.884635	+	0.466284i	$0.845593\pi$
$384$	0	0
$385$	−6.71477	−0.0174410
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−340.019	−0.874084	−0.437042	−	0.899441i	$-0.643974\pi$
$389$	−340.019	−0.874084	−0.437042	+	0.899441i	$0.643974\pi$
$390$	0	0
$391$	− 221.489i	− 0.566469i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	227.071i	0.574863i
$396$	0	0
$397$	294.916	0.742862	0.371431	−	0.928460i	$-0.378867\pi$
$397$	294.916	0.742862	0.371431	+	0.928460i	$0.378867\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−55.5356	−0.138493	−0.0692463	−	0.997600i	$-0.522059\pi$
$401$	−55.5356	−0.138493	−0.0692463	+	0.997600i	$0.522059\pi$
$402$	0	0
$403$	− 47.3822i	− 0.117574i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.55816i	0.0210274i
$408$	0	0
$409$	−275.415	−0.673386	−0.336693	−	0.941615i	$-0.609308\pi$
$409$	−275.415	−0.673386	−0.336693	+	0.941615i	$0.609308\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	152.961	0.370365
$414$	0	0
$415$	210.274i	0.506685i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 33.9969i	− 0.0811381i	−0.999177	−	0.0405691i	$-0.987083\pi$
$419$	− 33.9969i	− 0.0811381i	0.999177	−	0.0405691i	$-0.0129171\pi$
$420$	0	0
$421$	148.897	0.353675	0.176838	−	0.984240i	$-0.443413\pi$
$421$	148.897	0.353675	0.176838	+	0.984240i	$0.443413\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−502.569	−1.18252
$426$	0	0
$427$	145.809i	0.341472i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	307.522i	0.713508i	0.934198	+	0.356754i	$0.116116\pi$
$431$	307.522i	0.713508i	−0.934198	+	0.356754i	$0.883884\pi$
$432$	0	0
$433$	151.782	0.350535	0.175268	−	0.984521i	$-0.443921\pi$
$433$	151.782	0.350535	0.175268	+	0.984521i	$0.443921\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	104.139	0.238304
$438$	0	0
$439$	316.456i	0.720857i	0.932787	+	0.360428i	$0.117370\pi$
$439$	316.456i	0.720857i	−0.932787	+	0.360428i	$0.882630\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 671.256i	− 1.51525i	−0.652689	−	0.757626i	$-0.726359\pi$
$443$	− 671.256i	− 1.51525i	0.652689	−	0.757626i	$-0.273641\pi$
$444$	0	0
$445$	251.585	0.565359
$446$	0	0
$447$	0	0
$448$	0	0
$449$	756.843	1.68562	0.842809	−	0.538212i	$-0.180900\pi$
$449$	756.843	1.68562	0.842809	+	0.538212i	$0.180900\pi$
$450$	0	0
$451$	− 4.66602i	− 0.0103459i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	183.139i	0.402504i
$456$	0	0
$457$	−454.168	−0.993803	−0.496902	−	0.867807i	$-0.665529\pi$
$457$	−454.168	−0.993803	−0.496902	+	0.867807i	$0.665529\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−599.386	−1.30019	−0.650093	−	0.759855i	$-0.725270\pi$
$461$	−599.386	−1.30019	−0.650093	+	0.759855i	$0.725270\pi$
$462$	0	0
$463$	277.842i	0.600091i	0.953925	+	0.300046i	$0.0970019\pi$
$463$	277.842i	0.600091i	−0.953925	+	0.300046i	$0.902998\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 377.100i	− 0.807494i	−0.914871	−	0.403747i	$-0.867708\pi$
$467$	− 377.100i	− 0.807494i	0.914871	−	0.403747i	$-0.132292\pi$
$468$	0	0
$469$	119.871	0.255588
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.557116	0.00117783
$474$	0	0
$475$	− 236.296i	− 0.497465i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 65.7826i	− 0.137333i	−0.997640	−	0.0686666i	$-0.978126\pi$
$479$	− 65.7826i	− 0.137333i	0.997640	−	0.0686666i	$-0.0218745\pi$
$480$	0	0
$481$	233.416	0.485273
$482$	0	0
$483$	0	0
$484$	0	0
$485$	813.762	1.67786
$486$	0	0
$487$	77.8127i	0.159780i	0.996804	+	0.0798898i	$0.0254568\pi$
$487$	77.8127i	0.159780i	−0.996804	+	0.0798898i	$0.974543\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 450.685i	− 0.917892i	−0.888464	−	0.458946i	$-0.848227\pi$
$491$	− 450.685i	− 0.917892i	0.888464	−	0.458946i	$-0.151773\pi$
$492$	0	0
$493$	−178.312	−0.361687
$494$	0	0
$495$	0	0
$496$	0	0
$497$	46.5932	0.0937488
$498$	0	0
$499$	851.013i	1.70544i	0.522371	+	0.852718i	$0.325048\pi$
$499$	851.013i	1.70544i	−0.522371	+	0.852718i	$0.674952\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	245.544i	0.488159i	0.969755	+	0.244080i	$0.0784858\pi$
$503$	245.544i	0.488159i	−0.969755	+	0.244080i	$0.921514\pi$
$504$	0	0
$505$	1312.19	2.59840
$506$	0	0
$507$	0	0
$508$	0	0
$509$	509.002	1.00000	0.500002	−	0.866024i	$-0.333333\pi$
$509$	509.002	1.00000	0.500002	+	0.866024i	$0.333333\pi$
$510$	0	0
$511$	56.6898i	0.110939i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	922.961i	1.79216i
$516$	0	0
$517$	16.0697	0.0310825
$518$	0	0
$519$	0	0
$520$	0	0
$521$	47.7953	0.0917376	0.0458688	−	0.998947i	$-0.485394\pi$
$521$	47.7953	0.0917376	0.0458688	+	0.998947i	$0.485394\pi$
$522$	0	0
$523$	− 375.574i	− 0.718114i	−0.933316	−	0.359057i	$-0.883098\pi$
$523$	− 375.574i	− 0.718114i	0.933316	−	0.359057i	$-0.116902\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	59.5457i	0.112990i
$528$	0	0
$529$	−41.7843	−0.0789873
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−127.262	−0.238765
$534$	0	0
$535$	− 991.266i	− 1.85283i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11.1489i	0.0206844i
$540$	0	0
$541$	220.703	0.407953	0.203977	−	0.978976i	$-0.434613\pi$
$541$	220.703	0.407953	0.203977	+	0.978976i	$0.434613\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−535.349	−0.982291
$546$	0	0
$547$	− 746.194i	− 1.36416i	−0.731279	−	0.682079i	$-0.761076\pi$
$547$	− 746.194i	− 1.36416i	0.731279	−	0.682079i	$-0.238924\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 83.8378i	− 0.152156i
$552$	0	0
$553$	−71.1676	−0.128694
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−173.653	−0.311764	−0.155882	−	0.987776i	$-0.549822\pi$
$557$	−173.653	−0.311764	−0.155882	+	0.987776i	$0.549822\pi$
$558$	0	0
$559$	− 15.1948i	− 0.0271822i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	623.274i	1.10706i	0.832830	+	0.553530i	$0.186719\pi$
$563$	623.274i	1.10706i	−0.832830	+	0.553530i	$0.813281\pi$
$564$	0	0
$565$	−666.379	−1.17943
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−789.796	−1.38804	−0.694021	−	0.719955i	$-0.744162\pi$
$569$	−789.796	−1.38804	−0.694021	+	0.719955i	$0.744162\pi$
$570$	0	0
$571$	− 1017.55i	− 1.78205i	−0.453950	−	0.891027i	$-0.649985\pi$
$571$	− 1017.55i	− 1.78205i	0.453950	−	0.891027i	$-0.350015\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1295.13i	2.25241i
$576$	0	0
$577$	−250.762	−0.434596	−0.217298	−	0.976105i	$-0.569724\pi$
$577$	−250.762	−0.434596	−0.217298	+	0.976105i	$0.569724\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−65.9033	−0.113431
$582$	0	0
$583$	11.9311i	0.0204649i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 378.322i	− 0.644501i	−0.946654	−	0.322250i	$-0.895561\pi$
$587$	− 378.322i	− 0.644501i	0.946654	−	0.322250i	$-0.104439\pi$
$588$	0	0
$589$	−27.9969	−0.0475330
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−414.449	−0.698903	−0.349451	−	0.936955i	$-0.613632\pi$
$593$	−414.449	−0.698903	−0.349451	+	0.936955i	$0.613632\pi$
$594$	0	0
$595$	− 230.153i	− 0.386812i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 538.060i	− 0.898263i	−0.893466	−	0.449132i	$-0.851733\pi$
$599$	− 538.060i	− 0.898263i	0.893466	−	0.449132i	$-0.148267\pi$
$600$	0	0
$601$	−21.9493	−0.0365213	−0.0182606	−	0.999833i	$-0.505813\pi$
$601$	−21.9493	−0.0365213	−0.0182606	+	0.999833i	$0.505813\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1076.25	−1.77892
$606$	0	0
$607$	1096.78i	1.80689i	0.428703	+	0.903446i	$0.358971\pi$
$607$	1096.78i	1.80689i	−0.428703	+	0.903446i	$0.641029\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 438.286i	− 0.717326i
$612$	0	0
$613$	−681.420	−1.11162	−0.555808	−	0.831311i	$-0.687591\pi$
$613$	−681.420	−1.11162	−0.555808	+	0.831311i	$0.687591\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	146.398	0.237273	0.118637	−	0.992938i	$-0.462148\pi$
$617$	146.398	0.237273	0.118637	+	0.992938i	$0.462148\pi$
$618$	0	0
$619$	− 1171.04i	− 1.89183i	−0.324414	−	0.945915i	$-0.605167\pi$
$619$	− 1171.04i	− 1.89183i	0.324414	−	0.945915i	$-0.394833\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	78.8506i	0.126566i
$624$	0	0
$625$	958.470	1.53355
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−293.337	−0.466354
$630$	0	0
$631$	0.379351i	0 0.000601190i	1.00000		0.000300595i	$9.56824e-5\pi$
$631$	0.379351i	0 0.000601190i	−1.00000		0.000300595i	$0.999904\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1502.82i	2.36665i
$636$	0	0
$637$	304.076	0.477356
$638$	0	0
$639$	0	0
$640$	0	0
$641$	122.649	0.191340	0.0956700	−	0.995413i	$-0.469501\pi$
$641$	122.649	0.191340	0.0956700	+	0.995413i	$0.469501\pi$
$642$	0	0
$643$	− 1158.37i	− 1.80150i	−0.434335	−	0.900751i	$-0.643017\pi$
$643$	− 1158.37i	− 1.80150i	0.434335	−	0.900751i	$-0.356983\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	179.547i	0.277507i	0.990327	+	0.138754i	$0.0443097\pi$
$647$	179.547i	0.277507i	−0.990327	+	0.138754i	$0.955690\pi$
$648$	0	0
$649$	−14.8320	−0.0228536
$650$	0	0
$651$	0	0
$652$	0	0
$653$	266.762	0.408517	0.204259	−	0.978917i	$-0.434522\pi$
$653$	266.762	0.408517	0.204259	+	0.978917i	$0.434522\pi$
$654$	0	0
$655$	− 1052.51i	− 1.60688i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.1621i	0.0609441i	0.999536	+	0.0304720i	$0.00970105\pi$
$659$	40.1621i	0.0609441i	−0.999536	+	0.0304720i	$0.990299\pi$
$660$	0	0
$661$	−90.7035	−0.137222	−0.0686108	−	0.997644i	$-0.521857\pi$
$661$	−90.7035	−0.137222	−0.0686108	+	0.997644i	$0.521857\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	108.212	0.162726
$666$	0	0
$667$	459.515i	0.688927i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 14.1385i	− 0.0210708i
$672$	0	0
$673$	−760.379	−1.12983	−0.564917	−	0.825147i	$-0.691092\pi$
$673$	−760.379	−1.12983	−0.564917	+	0.825147i	$0.691092\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	191.850	0.283382	0.141691	−	0.989911i	$-0.454746\pi$
$677$	191.850	0.283382	0.141691	+	0.989911i	$0.454746\pi$
$678$	0	0
$679$	255.046i	0.375620i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1243.23i	1.82024i	0.414342	+	0.910121i	$0.364012\pi$
$683$	1243.23i	1.82024i	−0.414342	+	0.910121i	$0.635988\pi$
$684$	0	0
$685$	762.746	1.11350
$686$	0	0
$687$	0	0
$688$	0	0
$689$	325.409	0.472291
$690$	0	0
$691$	245.587i	0.355409i	0.984084	+	0.177704i	$0.0568671\pi$
$691$	245.587i	0.355409i	−0.984084	+	0.177704i	$0.943133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 763.475i	− 1.09853i
$696$	0	0
$697$	159.931	0.229456
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−511.775	−0.730064	−0.365032	−	0.930995i	$-0.618942\pi$
$701$	−511.775	−0.730064	−0.365032	+	0.930995i	$0.618942\pi$
$702$	0	0
$703$	− 137.920i	− 0.196187i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	411.261i	0.581699i
$708$	0	0
$709$	−813.026	−1.14672	−0.573361	−	0.819303i	$-0.694361\pi$
$709$	−813.026	−1.14672	−0.573361	+	0.819303i	$0.694361\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	153.451	0.215219
$714$	0	0
$715$	− 17.7583i	− 0.0248368i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	156.564i	0.217753i	0.994055	+	0.108877i	$0.0347253\pi$
$719$	156.564i	0.217753i	−0.994055	+	0.108877i	$0.965275\pi$
$720$	0	0
$721$	−289.270	−0.401207
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1042.66	1.43815
$726$	0	0
$727$	− 759.958i	− 1.04533i	−0.852537	−	0.522667i	$-0.824937\pi$
$727$	− 759.958i	− 1.04533i	0.852537	−	0.522667i	$-0.175063\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19.0955i	0.0261224i
$732$	0	0
$733$	655.205	0.893868	0.446934	−	0.894567i	$-0.352516\pi$
$733$	655.205	0.893868	0.446934	+	0.894567i	$0.352516\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.6234	−0.0157713
$738$	0	0
$739$	322.374i	0.436230i	0.975923	+	0.218115i	$0.0699908\pi$
$739$	322.374i	0.436230i	−0.975923	+	0.218115i	$0.930009\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1325.78i	− 1.78436i	−0.451680	−	0.892180i	$-0.649175\pi$
$743$	− 1325.78i	− 1.78436i	0.451680	−	0.892180i	$-0.350825\pi$
$744$	0	0
$745$	145.426	0.195203
$746$	0	0
$747$	0	0
$748$	0	0
$749$	310.678	0.414791
$750$	0	0
$751$	1165.72i	1.55222i	0.630599	+	0.776109i	$0.282809\pi$
$751$	1165.72i	1.55222i	−0.630599	+	0.776109i	$0.717191\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	444.119i	0.588237i
$756$	0	0
$757$	959.947	1.26809	0.634047	−	0.773295i	$-0.281393\pi$
$757$	959.947	1.26809	0.634047	+	0.773295i	$0.281393\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−542.826	−0.713306	−0.356653	−	0.934237i	$-0.616082\pi$
$761$	−542.826	−0.713306	−0.356653	+	0.934237i	$0.616082\pi$
$762$	0	0
$763$	− 167.787i	− 0.219904i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	404.530i	0.527418i
$768$	0	0
$769$	−951.457	−1.23727	−0.618633	−	0.785680i	$-0.712313\pi$
$769$	−951.457	−1.23727	−0.618633	+	0.785680i	$0.712313\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	881.670	1.14058	0.570291	−	0.821443i	$-0.306830\pi$
$773$	881.670	1.14058	0.570291	+	0.821443i	$0.306830\pi$
$774$	0	0
$775$	− 348.187i	− 0.449273i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	75.1956i	0.0965284i
$780$	0	0
$781$	−4.51796	−0.00578484
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2617.09	3.33387
$786$	0	0
$787$	− 652.411i	− 0.828985i	−0.910053	−	0.414493i	$-0.863959\pi$
$787$	− 652.411i	− 0.828985i	0.910053	−	0.414493i	$-0.136041\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 208.854i	− 0.264037i
$792$	0	0
$793$	−385.615	−0.486274
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−584.554	−0.733442	−0.366721	−	0.930331i	$-0.619520\pi$
$797$	−584.554	−0.733442	−0.366721	+	0.930331i	$0.619520\pi$
$798$	0	0
$799$	550.799i	0.689360i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 5.49699i	− 0.00684556i
$804$	0	0
$805$	−593.112	−0.736785
$806$	0	0
$807$	0	0
$808$	0	0
$809$	917.370	1.13396	0.566978	−	0.823733i	$-0.308112\pi$
$809$	917.370	1.13396	0.566978	+	0.823733i	$0.308112\pi$
$810$	0	0
$811$	677.208i	0.835028i	0.908671	+	0.417514i	$0.137099\pi$
$811$	677.208i	0.835028i	−0.908671	+	0.417514i	$0.862901\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 300.781i	− 0.369056i
$816$	0	0
$817$	−8.97824	−0.0109893
$818$	0	0
$819$	0	0
$820$	0	0
$821$	304.673	0.371100	0.185550	−	0.982635i	$-0.440593\pi$
$821$	304.673	0.371100	0.185550	+	0.982635i	$0.440593\pi$
$822$	0	0
$823$	− 1510.65i	− 1.83554i	−0.397112	−	0.917770i	$-0.629987\pi$
$823$	− 1510.65i	− 1.83554i	0.397112	−	0.917770i	$-0.370013\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 278.384i	− 0.336620i	−0.985734	−	0.168310i	$-0.946169\pi$
$827$	− 278.384i	− 0.336620i	0.985734	−	0.168310i	$-0.0538309\pi$
$828$	0	0
$829$	361.034	0.435505	0.217752	−	0.976004i	$-0.430127\pi$
$829$	361.034	0.435505	0.217752	+	0.976004i	$0.430127\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−382.135	−0.458746
$834$	0	0
$835$	2446.21i	2.92959i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 568.835i	− 0.677991i	−0.940788	−	0.338996i	$-0.889913\pi$
$839$	− 568.835i	− 0.677991i	0.940788	−	0.338996i	$-0.110087\pi$
$840$	0	0
$841$	−471.064	−0.560124
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1019.76	1.20681
$846$	0	0
$847$	− 337.313i	− 0.398245i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	755.937i	0.888293i
$852$	0	0
$853$	24.6871	0.0289415	0.0144708	−	0.999895i	$-0.495394\pi$
$853$	24.6871	0.0289415	0.0144708	+	0.999895i	$0.495394\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−689.321	−0.804342	−0.402171	−	0.915565i	$-0.631744\pi$
$857$	−689.321	−0.804342	−0.402171	+	0.915565i	$0.631744\pi$
$858$	0	0
$859$	− 466.648i	− 0.543246i	−0.962404	−	0.271623i	$-0.912440\pi$
$859$	− 466.648i	− 0.543246i	0.962404	−	0.271623i	$-0.0875603\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	739.477i	0.856868i	0.903573	+	0.428434i	$0.140934\pi$
$863$	739.477i	0.856868i	−0.903573	+	0.428434i	$0.859066\pi$
$864$	0	0
$865$	2425.16	2.80365
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.90085	0.00794114
$870$	0	0
$871$	317.019i	0.363971i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	725.156i	0.828749i
$876$	0	0
$877$	692.804	0.789970	0.394985	−	0.918688i	$-0.370750\pi$
$877$	692.804	0.789970	0.394985	+	0.918688i	$0.370750\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	347.623	0.394577	0.197289	−	0.980345i	$-0.436786\pi$
$881$	347.623	0.394577	0.197289	+	0.980345i	$0.436786\pi$
$882$	0	0
$883$	− 783.477i	− 0.887290i	−0.896203	−	0.443645i	$-0.853685\pi$
$883$	− 783.477i	− 0.887290i	0.896203	−	0.443645i	$-0.146315\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 539.971i	− 0.608761i	−0.952551	−	0.304380i	$-0.901551\pi$
$887$	− 539.971i	− 0.608761i	0.952551	−	0.304380i	$-0.0984494\pi$
$888$	0	0
$889$	−471.009	−0.529819
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−258.972	−0.290002
$894$	0	0
$895$	2846.19i	3.18010i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 123.537i	− 0.137416i
$900$	0	0
$901$	−408.945	−0.453879
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2187.48	2.41711
$906$	0	0
$907$	364.840i	0.402250i	0.979566	+	0.201125i	$0.0644597\pi$
$907$	364.840i	0.402250i	−0.979566	+	0.201125i	$0.935540\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 140.980i	− 0.154753i	−0.997002	−	0.0773764i	$-0.975346\pi$
$911$	− 140.980i	− 0.154753i	0.997002	−	0.0773764i	$-0.0246543\pi$
$912$	0	0
$913$	6.39039	0.00699933
$914$	0	0
$915$	0	0
$916$	0	0
$917$	329.873	0.359730
$918$	0	0
$919$	1653.67i	1.79943i	0.436482	+	0.899713i	$0.356224\pi$
$919$	1653.67i	1.79943i	−0.436482	+	0.899713i	$0.643776\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	123.223i	0.133503i
$924$	0	0
$925$	1715.25	1.85433
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−819.652	−0.882295	−0.441148	−	0.897435i	$-0.645428\pi$
$929$	−819.652	−0.882295	−0.441148	+	0.897435i	$0.645428\pi$
$930$	0	0
$931$	− 179.671i	− 0.192987i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	22.3171i	0.0238685i
$936$	0	0
$937$	−609.090	−0.650043	−0.325021	−	0.945707i	$-0.605371\pi$
$937$	−609.090	−0.650043	−0.325021	+	0.945707i	$0.605371\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−707.610	−0.751977	−0.375988	−	0.926624i	$-0.622697\pi$
$941$	−707.610	−0.751977	−0.375988	+	0.926624i	$0.622697\pi$
$942$	0	0
$943$	− 412.147i	− 0.437059i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 479.473i	− 0.506307i	−0.967426	−	0.253154i	$-0.918532\pi$
$947$	− 479.473i	− 0.506307i	0.967426	−	0.253154i	$-0.0814678\pi$
$948$	0	0
$949$	−149.925	−0.157983
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−743.165	−0.779816	−0.389908	−	0.920854i	$-0.627493\pi$
$953$	−743.165	−0.779816	−0.389908	+	0.920854i	$0.627493\pi$
$954$	0	0
$955$	2992.68i	3.13370i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	239.056i	0.249277i
$960$	0	0
$961$	919.746	0.957072
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2929.24	3.03549
$966$	0	0
$967$	− 1097.27i	− 1.13471i	−0.823473	−	0.567355i	$-0.807967\pi$
$967$	− 1097.27i	− 1.13471i	0.823473	−	0.567355i	$-0.192033\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1062.93i	1.09468i	0.836910	+	0.547340i	$0.184360\pi$
$971$	1062.93i	1.09468i	−0.836910	+	0.547340i	$0.815640\pi$
$972$	0	0
$973$	239.285	0.245925
$974$	0	0
$975$	0	0
$976$	0	0
$977$	685.421	0.701556	0.350778	−	0.936459i	$-0.385917\pi$
$977$	685.421	0.701556	0.350778	+	0.936459i	$0.385917\pi$
$978$	0	0
$979$	− 7.64584i	− 0.00780985i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	747.688i	0.760619i	0.924859	+	0.380309i	$0.124182\pi$
$983$	747.688i	0.760619i	−0.924859	+	0.380309i	$0.875818\pi$
$984$	0	0
$985$	2465.05	2.50258
$986$	0	0
$987$	0	0
$988$	0	0
$989$	49.2097	0.0497570
$990$	0	0
$991$	− 514.441i	− 0.519113i	−0.965728	−	0.259556i	$-0.916424\pi$
$991$	− 514.441i	− 0.519113i	0.965728	−	0.259556i	$-0.0835763\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2092.79i	2.10330i
$996$	0	0
$997$	329.890	0.330883	0.165442	−	0.986220i	$-0.447095\pi$
$997$	329.890	0.330883	0.165442	+	0.986220i	$0.447095\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.m.f.1711.1		12
3.2	odd	2		912.3.m.a.799.6	✓	12
4.3	odd	2	inner	2736.3.m.f.1711.2		12
12.11	even	2		912.3.m.a.799.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.3.m.a.799.6	✓	12	3.2	odd	2
912.3.m.a.799.12	yes	12	12.11	even	2
2736.3.m.f.1711.1		12	1.1	even	1	trivial
2736.3.m.f.1711.2		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)

\(f(q)\)	\(=\)	\(q-8.90000 q^{5} -2.78940i q^{7} +O(q^{10})\)	\(q-8.90000 q^{5} -2.78940i q^{7} +0.270477i q^{11} +7.37703 q^{13} -9.27079 q^{17} -4.35890i q^{19} +23.8911i q^{23} +54.2099 q^{25} +19.2337 q^{29} -6.42293i q^{31} +24.8256i q^{35} +31.6410 q^{37} -17.2511 q^{41} -2.05975i q^{43} -59.4123i q^{47} +41.2193 q^{49} +44.1111 q^{53} -2.40725i q^{55} +54.8364i q^{59} -52.2724 q^{61} -65.6555 q^{65} +42.9738i q^{67} +16.7037i q^{71} -20.3233 q^{73} +0.754469 q^{77} -25.5136i q^{79} -23.6263i q^{83} +82.5100 q^{85} -28.2680 q^{89} -20.5775i q^{91} +38.7942i q^{95} -91.4340 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{5}+O(q^{10}) \) 12 * q + 10 * q^5	\( 12 q + 10 q^{5} + 36 q^{13} + 10 q^{17} + 58 q^{25} - 12 q^{29} + 32 q^{37} - 136 q^{41} - 22 q^{49} + 236 q^{53} - 210 q^{61} + 52 q^{65} - 158 q^{73} + 70 q^{77} + 242 q^{85} - 444 q^{89} + 320 q^{97}+O(q^{100}) \) 12 * q + 10 * q^5 + 36 * q^13 + 10 * q^17 + 58 * q^25 - 12 * q^29 + 32 * q^37 - 136 * q^41 - 22 * q^49 + 236 * q^53 - 210 * q^61 + 52 * q^65 - 158 * q^73 + 70 * q^77 + 242 * q^85 - 444 * q^89 + 320 * q^97

Introduction

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