LMFDB - Embedded newform 2736.3.o.n.721.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (of order $2$, degree $1$, not 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:	$8$
Coefficient field:	8.0.184143974400.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 $ x^8 - 4x^7 - 22x^6 + 80x^5 + 215x^4 - 568x^3 - 1022x^2 + 1320*x + 2628
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.4
Root			$-2.87998 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.n.721.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.94975 q^{5} -5.84873 q^{7} +O(q^{10})$	$q-2.94975 q^{5} -5.84873 q^{7} +18.6246 q^{11} +1.55708i q^{13} -12.7477 q^{17} +(17.7256 - 6.84117i) q^{19} -15.8205 q^{23} -16.2990 q^{25} +46.0273i q^{29} -37.5811i q^{31} +17.2523 q^{35} -17.6705i q^{37} +33.3798i q^{41} +50.6939 q^{43} -15.1733 q^{47} -14.7924 q^{49} -75.3754i q^{53} -54.9380 q^{55} +53.0940i q^{59} +11.3011 q^{61} -4.59299i q^{65} +12.1999i q^{67} +20.8673i q^{71} -34.2076 q^{73} -108.930 q^{77} -128.601i q^{79} +66.9672 q^{83} +37.6026 q^{85} +25.1600i q^{89} -9.10693i q^{91} +(-52.2862 + 20.1797i) q^{95} -117.436i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5} + 12 q^{7}+O(q^{10}) $ 8 * q - 4 * q^5 + 12 * q^7	$ 8 q - 4 q^{5} + 12 q^{7} + 4 q^{11} - 4 q^{17} + 36 q^{19} - 56 q^{23} + 140 q^{25} + 236 q^{35} - 100 q^{43} - 188 q^{47} - 36 q^{49} - 28 q^{55} - 180 q^{61} - 356 q^{73} - 68 q^{77} + 136 q^{83} + 148 q^{85} - 140 q^{95}+O(q^{100}) $ 8 * q - 4 * q^5 + 12 * q^7 + 4 * q^11 - 4 * q^17 + 36 * q^19 - 56 * q^23 + 140 * q^25 + 236 * q^35 - 100 * q^43 - 188 * q^47 - 36 * q^49 - 28 * q^55 - 180 * q^61 - 356 * q^73 - 68 * q^77 + 136 * q^83 + 148 * q^85 - 140 * q^95

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
$5$	−2.94975			−0.589950			−0.294975	−	0.955505i	$-0.595311\pi$
$5$	−2.94975			−0.589950			−0.294975	+	0.955505i	$0.595311\pi$
$6$		0			0
$7$	−5.84873			−0.835533			−0.417766	−	0.908554i	$-0.637187\pi$
$7$	−5.84873			−0.835533			−0.417766	+	0.908554i	$0.637187\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18.6246			1.69315			0.846574	−	0.532271i	$-0.178661\pi$
$11$	18.6246			1.69315			0.846574	+	0.532271i	$0.178661\pi$
$12$		0			0
$13$			1.55708i			0.119775i	0.998205	+	0.0598876i	$0.0190742\pi$
$13$			1.55708i			0.119775i	−0.998205	+	0.0598876i	$0.980926\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.7477			−0.749865			−0.374933	−	0.927052i	$-0.622334\pi$
$17$	−12.7477			−0.749865			−0.374933	+	0.927052i	$0.622334\pi$
$18$		0			0
$19$	17.7256	−	6.84117i	0.932929	−	0.360061i
$19$	17.7256	−	6.84117i	0.932929	−	0.360061i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−15.8205			−0.687850			−0.343925	−	0.938997i	$-0.611757\pi$
$23$	−15.8205			−0.687850			−0.343925	+	0.938997i	$0.611757\pi$
$24$		0			0
$25$	−16.2990			−0.651959
$26$		0			0
$27$		0			0
$28$		0			0
$29$			46.0273i			1.58715i	0.608474	+	0.793573i	$0.291782\pi$
$29$			46.0273i			1.58715i	−0.608474	+	0.793573i	$0.708218\pi$
$30$		0			0
$31$		−	37.5811i		−	1.21229i	−0.795353	−	0.606147i	$-0.792714\pi$
$31$		−	37.5811i		−	1.21229i	0.795353	−	0.606147i	$-0.207286\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	17.2523			0.492923
$36$		0			0
$37$		−	17.6705i		−	0.477580i	−0.971071	−	0.238790i	$-0.923249\pi$
$37$		−	17.6705i		−	0.477580i	0.971071	−	0.238790i	$-0.0767507\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			33.3798i			0.814141i	0.913397	+	0.407071i	$0.133450\pi$
$41$			33.3798i			0.814141i	−0.913397	+	0.407071i	$0.866550\pi$
$42$		0			0
$43$	50.6939			1.17893			0.589464	−	0.807795i	$-0.299339\pi$
$43$	50.6939			1.17893			0.589464	+	0.807795i	$0.299339\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−15.1733			−0.322837			−0.161418	−	0.986886i	$-0.551607\pi$
$47$	−15.1733			−0.322837			−0.161418	+	0.986886i	$0.551607\pi$
$48$		0			0
$49$	−14.7924			−0.301885
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	75.3754i		−	1.42218i	−0.703102	−	0.711089i	$-0.748202\pi$
$53$		−	75.3754i		−	1.42218i	0.703102	−	0.711089i	$-0.251798\pi$
$54$		0			0
$55$	−54.9380			−0.998872
$56$		0			0
$57$		0			0
$58$		0			0
$59$			53.0940i			0.899899i	0.893054	+	0.449949i	$0.148558\pi$
$59$			53.0940i			0.899899i	−0.893054	+	0.449949i	$0.851442\pi$
$60$		0			0
$61$	11.3011			0.185263			0.0926316	−	0.995700i	$-0.470472\pi$
$61$	11.3011			0.185263			0.0926316	+	0.995700i	$0.470472\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	4.59299i		−	0.0706614i
$66$		0			0
$67$			12.1999i			0.182088i	0.995847	+	0.0910440i	$0.0290204\pi$
$67$			12.1999i			0.182088i	−0.995847	+	0.0910440i	$0.970980\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			20.8673i			0.293905i	0.989144	+	0.146953i	$0.0469465\pi$
$71$			20.8673i			0.293905i	−0.989144	+	0.146953i	$0.953054\pi$
$72$		0			0
$73$	−34.2076			−0.468598			−0.234299	−	0.972165i	$-0.575279\pi$
$73$	−34.2076			−0.468598			−0.234299	+	0.972165i	$0.575279\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−108.930			−1.41468
$78$		0			0
$79$		−	128.601i		−	1.62786i	−0.580965	−	0.813929i	$-0.697325\pi$
$79$		−	128.601i		−	1.62786i	0.580965	−	0.813929i	$-0.302675\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	66.9672			0.806834			0.403417	−	0.915016i	$-0.367822\pi$
$83$	66.9672			0.806834			0.403417	+	0.915016i	$0.367822\pi$
$84$		0			0
$85$	37.6026			0.442383
$86$		0			0
$87$		0			0
$88$		0			0
$89$			25.1600i			0.282697i	0.989960	+	0.141348i	$0.0451437\pi$
$89$			25.1600i			0.282697i	−0.989960	+	0.141348i	$0.954856\pi$
$90$		0			0
$91$		−	9.10693i		−	0.100076i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−52.2862	+	20.1797i	−0.550381	+	0.212418i
$96$		0			0
$97$		−	117.436i		−	1.21068i	−0.795968	−	0.605339i	$-0.793038\pi$
$97$		−	117.436i		−	1.21068i	0.795968	−	0.605339i	$-0.206962\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	45.8852			0.454309			0.227154	−	0.973859i	$-0.427058\pi$
$101$	45.8852			0.454309			0.227154	+	0.973859i	$0.427058\pi$
$102$		0			0
$103$			169.904i			1.64956i	0.565455	+	0.824779i	$0.308701\pi$
$103$			169.904i			1.64956i	−0.565455	+	0.824779i	$0.691299\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			183.761i			1.71739i	0.512486	+	0.858696i	$0.328725\pi$
$107$			183.761i			1.71739i	−0.512486	+	0.858696i	$0.671275\pi$
$108$		0			0
$109$		−	59.0488i		−	0.541732i	−0.962617	−	0.270866i	$-0.912690\pi$
$109$		−	59.0488i		−	0.541732i	0.962617	−	0.270866i	$-0.0873101\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			103.714i			0.917825i	0.888481	+	0.458913i	$0.151761\pi$
$113$			103.714i			0.917825i	−0.888481	+	0.458913i	$0.848239\pi$
$114$		0			0
$115$	46.6667			0.405797
$116$		0			0
$117$		0			0
$118$		0			0
$119$	74.5579			0.626537
$120$		0			0
$121$	225.877			1.86675
$122$		0			0
$123$		0			0
$124$		0			0
$125$	121.822			0.974573
$126$		0			0
$127$		−	27.7707i		−	0.218667i	−0.994005	−	0.109333i	$-0.965128\pi$
$127$		−	27.7707i		−	0.218667i	0.994005	−	0.109333i	$-0.0348716\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	102.402			0.781696			0.390848	−	0.920455i	$-0.372182\pi$
$131$	102.402			0.781696			0.390848	+	0.920455i	$0.372182\pi$
$132$		0			0
$133$	−103.672	+	40.0121i	−0.779492	+	0.300843i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	243.826			1.77975			0.889875	−	0.456204i	$-0.150791\pi$
$137$	243.826			1.77975			0.889875	+	0.456204i	$0.150791\pi$
$138$		0			0
$139$	183.616			1.32098			0.660490	−	0.750835i	$-0.270349\pi$
$139$	183.616			1.32098			0.660490	+	0.750835i	$0.270349\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			29.0000i			0.202797i
$144$		0			0
$145$		−	135.769i		−	0.936337i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−154.434			−1.03647			−0.518234	−	0.855239i	$-0.673411\pi$
$149$	−154.434			−1.03647			−0.518234	+	0.855239i	$0.673411\pi$
$150$		0			0
$151$			193.877i			1.28395i	0.766724	+	0.641977i	$0.221886\pi$
$151$			193.877i			1.28395i	−0.766724	+	0.641977i	$0.778114\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			110.855i			0.715193i
$156$		0			0
$157$	129.392			0.824153			0.412076	−	0.911149i	$-0.364804\pi$
$157$	129.392			0.824153			0.412076	+	0.911149i	$0.364804\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	92.5301			0.574721
$162$		0			0
$163$	211.815			1.29948			0.649738	−	0.760158i	$-0.274879\pi$
$163$	211.815			1.29948			0.649738	+	0.760158i	$0.274879\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	315.483i		−	1.88912i	−0.328339	−	0.944560i	$-0.606489\pi$
$167$		−	315.483i		−	1.88912i	0.328339	−	0.944560i	$-0.393511\pi$
$168$		0			0
$169$	166.576			0.985654
$170$		0			0
$171$		0			0
$172$		0			0
$173$			236.367i			1.36629i	0.730285	+	0.683143i	$0.239387\pi$
$173$			236.367i			1.36629i	−0.730285	+	0.683143i	$0.760613\pi$
$174$		0			0
$175$	95.3283			0.544733
$176$		0			0
$177$		0			0
$178$		0			0
$179$			184.348i			1.02988i	0.857227	+	0.514939i	$0.172185\pi$
$179$			184.348i			1.02988i	−0.857227	+	0.514939i	$0.827815\pi$
$180$		0			0
$181$			80.2606i			0.443429i	0.975112	+	0.221714i	$0.0711653\pi$
$181$			80.2606i			0.443429i	−0.975112	+	0.221714i	$0.928835\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			52.1234i			0.281748i
$186$		0			0
$187$	−237.421			−1.26963
$188$		0			0
$189$		0			0
$190$		0			0
$191$	362.279			1.89675			0.948375	−	0.317150i	$-0.102726\pi$
$191$	362.279			1.89675			0.948375	+	0.317150i	$0.102726\pi$
$192$		0			0
$193$			271.802i			1.40830i	0.710052	+	0.704149i	$0.248672\pi$
$193$			271.802i			1.40830i	−0.710052	+	0.704149i	$0.751328\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	118.230			0.600150			0.300075	−	0.953916i	$-0.402988\pi$
$197$	118.230			0.600150			0.300075	+	0.953916i	$0.402988\pi$
$198$		0			0
$199$	−185.073			−0.930013			−0.465007	−	0.885307i	$-0.653948\pi$
$199$	−185.073			−0.930013			−0.465007	+	0.885307i	$0.653948\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	269.201i		−	1.32611i
$204$		0			0
$205$		−	98.4620i		−	0.480303i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	330.133	−	127.414i	1.57959	−	0.609637i
$210$		0			0
$211$		−	417.799i		−	1.98009i	−0.140756	−	0.990044i	$-0.544953\pi$
$211$		−	417.799i		−	1.98009i	0.140756	−	0.990044i	$-0.455047\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−149.534			−0.695508
$216$		0			0
$217$			219.802i			1.01291i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	19.8492i		−	0.0898153i
$222$		0			0
$223$			172.345i			0.772848i	0.922321	+	0.386424i	$0.126290\pi$
$223$			172.345i			0.772848i	−0.922321	+	0.386424i	$0.873710\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			12.5692i			0.0553710i	0.999617	+	0.0276855i	$0.00881370\pi$
$227$			12.5692i			0.0553710i	−0.999617	+	0.0276855i	$0.991186\pi$
$228$		0			0
$229$	240.867			1.05182			0.525911	−	0.850539i	$-0.323724\pi$
$229$	240.867			1.05182			0.525911	+	0.850539i	$0.323724\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	222.034			0.952936			0.476468	−	0.879192i	$-0.341917\pi$
$233$	222.034			0.952936			0.476468	+	0.879192i	$0.341917\pi$
$234$		0			0
$235$	44.7576			0.190458
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.0203			0.0419259			0.0209630	−	0.999780i	$-0.493327\pi$
$239$	10.0203			0.0419259			0.0209630	+	0.999780i	$0.493327\pi$
$240$		0			0
$241$			420.230i			1.74369i	0.489780	+	0.871846i	$0.337077\pi$
$241$			420.230i			1.74369i	−0.489780	+	0.871846i	$0.662923\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	43.6338			0.178097
$246$		0			0
$247$	10.6522	+	27.6002i	0.0431264	+	0.111742i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	185.394			0.738621			0.369310	−	0.929306i	$-0.379594\pi$
$251$	185.394			0.738621			0.369310	+	0.929306i	$0.379594\pi$
$252$		0			0
$253$	−294.652			−1.16463
$254$		0			0
$255$		0			0
$256$		0			0
$257$			26.5370i			0.103257i	0.998666	+	0.0516285i	$0.0164412\pi$
$257$			26.5370i			0.103257i	−0.998666	+	0.0516285i	$0.983559\pi$
$258$		0			0
$259$			103.350i			0.399034i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	71.2398			0.270874			0.135437	−	0.990786i	$-0.456756\pi$
$263$	71.2398			0.270874			0.135437	+	0.990786i	$0.456756\pi$
$264$		0			0
$265$			222.339i			0.839014i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	164.593i		−	0.611869i	−0.952052	−	0.305935i	$-0.901031\pi$
$269$		−	164.593i		−	0.611869i	0.952052	−	0.305935i	$-0.0989689\pi$
$270$		0			0
$271$	156.397			0.577110			0.288555	−	0.957463i	$-0.406825\pi$
$271$	156.397			0.577110			0.288555	+	0.957463i	$0.406825\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−303.562			−1.10386
$276$		0			0
$277$	−278.050			−1.00379			−0.501896	−	0.864928i	$-0.667364\pi$
$277$	−278.050			−1.00379			−0.501896	+	0.864928i	$0.667364\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			64.5425i			0.229688i	0.993384	+	0.114844i	$0.0366369\pi$
$281$			64.5425i			0.229688i	−0.993384	+	0.114844i	$0.963363\pi$
$282$		0			0
$283$	−209.171			−0.739119			−0.369560	−	0.929207i	$-0.620491\pi$
$283$	−209.171			−0.739119			−0.369560	+	0.929207i	$0.620491\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	195.229i		−	0.680242i
$288$		0			0
$289$	−126.496			−0.437702
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	75.5171i		−	0.257738i	−0.991662	−	0.128869i	$-0.958865\pi$
$293$		−	75.5171i		−	0.257738i	0.991662	−	0.128869i	$-0.0411346\pi$
$294$		0			0
$295$		−	156.614i		−	0.530895i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	24.6338i		−	0.0823874i
$300$		0			0
$301$	−296.495			−0.985033
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−33.3353			−0.109296
$306$		0			0
$307$		−	46.7862i		−	0.152398i	−0.997093	−	0.0761990i	$-0.975722\pi$
$307$		−	46.7862i		−	0.152398i	0.997093	−	0.0761990i	$-0.0242784\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	230.261			0.740388			0.370194	−	0.928954i	$-0.379291\pi$
$311$	230.261			0.740388			0.370194	+	0.928954i	$0.379291\pi$
$312$		0			0
$313$	63.4469			0.202706			0.101353	−	0.994851i	$-0.467683\pi$
$313$	63.4469			0.202706			0.101353	+	0.994851i	$0.467683\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			261.464i			0.824809i	0.911001	+	0.412404i	$0.135311\pi$
$317$			261.464i			0.824809i	−0.911001	+	0.412404i	$0.864689\pi$
$318$		0			0
$319$			857.240i			2.68727i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−225.961	+	87.2092i	−0.699571	+	0.269998i
$324$		0			0
$325$		−	25.3788i		−	0.0780885i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	88.7447			0.269741
$330$		0			0
$331$		−	587.914i		−	1.77618i	−0.459673	−	0.888088i	$-0.652033\pi$
$331$		−	587.914i		−	1.77618i	0.459673	−	0.888088i	$-0.347967\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	35.9866i		−	0.107423i
$336$		0			0
$337$		−	628.812i		−	1.86591i	−0.359990	−	0.932956i	$-0.617220\pi$
$337$		−	628.812i		−	1.86591i	0.359990	−	0.932956i	$-0.382780\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	699.934i		−	2.05259i
$342$		0			0
$343$	373.104			1.08777
$344$		0			0
$345$		0			0
$346$		0			0
$347$	113.695			0.327651			0.163825	−	0.986489i	$-0.447617\pi$
$347$	113.695			0.327651			0.163825	+	0.986489i	$0.447617\pi$
$348$		0			0
$349$	359.962			1.03141			0.515705	−	0.856766i	$-0.327530\pi$
$349$	359.962			1.03141			0.515705	+	0.856766i	$0.327530\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−353.032			−1.00009			−0.500046	−	0.865999i	$-0.666684\pi$
$353$	−353.032			−1.00009			−0.500046	+	0.865999i	$0.666684\pi$
$354$		0			0
$355$		−	61.5532i		−	0.173389i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	386.538			1.07671			0.538354	−	0.842719i	$-0.319046\pi$
$359$	386.538			1.07671			0.538354	+	0.842719i	$0.319046\pi$
$360$		0			0
$361$	267.397	−	242.528i	0.740711	−	0.671823i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	100.904			0.276449
$366$		0			0
$367$	−584.274			−1.59203			−0.796014	−	0.605278i	$-0.793062\pi$
$367$	−584.274			−1.59203			−0.796014	+	0.605278i	$0.793062\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			440.850i			1.18828i
$372$		0			0
$373$			199.190i			0.534020i	0.963694	+	0.267010i	$0.0860358\pi$
$373$			199.190i			0.534020i	−0.963694	+	0.267010i	$0.913964\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−71.6680			−0.190101
$378$		0			0
$379$			519.527i			1.37078i	0.728174	+	0.685392i	$0.240369\pi$
$379$			519.527i			1.37078i	−0.728174	+	0.685392i	$0.759631\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	575.644i		−	1.50299i	−0.659741	−	0.751493i	$-0.729334\pi$
$383$		−	575.644i		−	1.50299i	0.659741	−	0.751493i	$-0.270666\pi$
$384$		0			0
$385$	321.317			0.834591
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.5845			−0.0349216			−0.0174608	−	0.999848i	$-0.505558\pi$
$389$	−13.5845			−0.0349216			−0.0174608	+	0.999848i	$0.505558\pi$
$390$		0			0
$391$	201.676			0.515795
$392$		0			0
$393$		0			0
$394$		0			0
$395$			379.340i			0.960355i
$396$		0			0
$397$	−220.020			−0.554206			−0.277103	−	0.960840i	$-0.589374\pi$
$397$	−220.020			−0.554206			−0.277103	+	0.960840i	$0.589374\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			331.524i			0.826744i	0.910562	+	0.413372i	$0.135649\pi$
$401$			331.524i			0.826744i	−0.910562	+	0.413372i	$0.864351\pi$
$402$		0			0
$403$	58.5167			0.145203
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	329.106i		−	0.808613i
$408$		0			0
$409$		−	13.7890i		−	0.0337139i	−0.999858	−	0.0168570i	$-0.994634\pi$
$409$		−	13.7890i		−	0.0337139i	0.999858	−	0.0168570i	$-0.00536599\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	310.533i		−	0.751895i
$414$		0			0
$415$	−197.537			−0.475992
$416$		0			0
$417$		0			0
$418$		0			0
$419$	283.447			0.676485			0.338242	−	0.941059i	$-0.390168\pi$
$419$	283.447			0.676485			0.338242	+	0.941059i	$0.390168\pi$
$420$		0			0
$421$			595.014i			1.41333i	0.707546	+	0.706667i	$0.249802\pi$
$421$			595.014i			1.41333i	−0.707546	+	0.706667i	$0.750198\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	207.775			0.488881
$426$		0			0
$427$	−66.0968			−0.154794
$428$		0			0
$429$		0			0
$430$		0			0
$431$			318.279i			0.738466i	0.929337	+	0.369233i	$0.120380\pi$
$431$			318.279i			0.738466i	−0.929337	+	0.369233i	$0.879620\pi$
$432$		0			0
$433$		−	180.258i		−	0.416301i	−0.978097	−	0.208150i	$-0.933256\pi$
$433$		−	180.258i		−	0.416301i	0.978097	−	0.208150i	$-0.0667444\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−280.429	+	108.231i	−0.641715	+	0.247668i
$438$		0			0
$439$			732.716i			1.66906i	0.550965	+	0.834528i	$0.314260\pi$
$439$			732.716i			1.66906i	−0.550965	+	0.834528i	$0.685740\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−112.171			−0.253207			−0.126603	−	0.991953i	$-0.540408\pi$
$443$	−112.171			−0.253207			−0.126603	+	0.991953i	$0.540408\pi$
$444$		0			0
$445$		−	74.2157i		−	0.166777i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	104.449i		−	0.232626i	−0.993213	−	0.116313i	$-0.962892\pi$
$449$		−	104.449i		−	0.232626i	0.993213	−	0.116313i	$-0.0371076\pi$
$450$		0			0
$451$			621.686i			1.37846i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			26.8632i			0.0590399i
$456$		0			0
$457$	641.023			1.40268			0.701338	−	0.712828i	$-0.252586\pi$
$457$	641.023			1.40268			0.701338	+	0.712828i	$0.252586\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	351.382			0.762217			0.381109	−	0.924530i	$-0.375542\pi$
$461$	351.382			0.762217			0.381109	+	0.924530i	$0.375542\pi$
$462$		0			0
$463$	509.568			1.10058			0.550289	−	0.834974i	$-0.314517\pi$
$463$	509.568			1.10058			0.550289	+	0.834974i	$0.314517\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.6920			0.0528736			0.0264368	−	0.999650i	$-0.491584\pi$
$467$	24.6920			0.0528736			0.0264368	+	0.999650i	$0.491584\pi$
$468$		0			0
$469$		−	71.3539i		−	0.152140i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	944.155			1.99610
$474$		0			0
$475$	−288.910	+	111.504i	−0.608231	+	0.234745i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	872.814			1.82216			0.911080	−	0.412230i	$-0.135250\pi$
$479$	872.814			1.82216			0.911080	+	0.412230i	$0.135250\pi$
$480$		0			0
$481$	27.5143			0.0572022
$482$		0			0
$483$		0			0
$484$		0			0
$485$			346.406i			0.714239i
$486$		0			0
$487$		−	64.1843i		−	0.131795i	−0.997826	−	0.0658977i	$-0.979009\pi$
$487$		−	64.1843i		−	0.131795i	0.997826	−	0.0658977i	$-0.0209911\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	223.170			0.454521			0.227261	−	0.973834i	$-0.427023\pi$
$491$	223.170			0.454521			0.227261	+	0.973834i	$0.427023\pi$
$492$		0			0
$493$		−	586.742i		−	1.19015i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	122.047i		−	0.245567i
$498$		0			0
$499$	540.219			1.08260			0.541302	−	0.840828i	$-0.317932\pi$
$499$	540.219			1.08260			0.541302	+	0.840828i	$0.317932\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−575.174			−1.14349			−0.571744	−	0.820432i	$-0.693733\pi$
$503$	−575.174			−1.14349			−0.571744	+	0.820432i	$0.693733\pi$
$504$		0			0
$505$	−135.350			−0.268019
$506$		0			0
$507$		0			0
$508$		0			0
$509$			467.013i			0.917511i	0.888563	+	0.458756i	$0.151705\pi$
$509$			467.013i			0.917511i	−0.888563	+	0.458756i	$0.848295\pi$
$510$		0			0
$511$	200.071			0.391529
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	501.176i		−	0.973157i
$516$		0			0
$517$	−282.598			−0.546611
$518$		0			0
$519$		0			0
$520$		0			0
$521$			90.3378i			0.173393i	0.996235	+	0.0866965i	$0.0276311\pi$
$521$			90.3378i			0.173393i	−0.996235	+	0.0866965i	$0.972369\pi$
$522$		0			0
$523$		−	872.076i		−	1.66745i	−0.552180	−	0.833725i	$-0.686204\pi$
$523$		−	872.076i		−	1.66745i	0.552180	−	0.833725i	$-0.313796\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			479.073i			0.909057i
$528$		0			0
$529$	−278.710			−0.526863
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−51.9749			−0.0975140
$534$		0			0
$535$		−	542.049i		−	1.01318i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−275.502			−0.511136
$540$		0			0
$541$	−149.284			−0.275941			−0.137971	−	0.990436i	$-0.544058\pi$
$541$	−149.284			−0.275941			−0.137971	+	0.990436i	$0.544058\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			174.179i			0.319595i
$546$		0			0
$547$		−	750.900i		−	1.37276i	−0.727243	−	0.686380i	$-0.759199\pi$
$547$		−	750.900i		−	1.37276i	0.727243	−	0.686380i	$-0.240801\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	314.880	+	815.863i	0.571470	+	1.48069i
$552$		0			0
$553$			752.151i			1.36013i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1024.62			−1.83954			−0.919770	−	0.392457i	$-0.871625\pi$
$557$	−1024.62			−1.83954			−0.919770	+	0.392457i	$0.871625\pi$
$558$		0			0
$559$			78.9344i			0.141206i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			879.509i			1.56218i	0.624417	+	0.781091i	$0.285337\pi$
$563$			879.509i			1.56218i	−0.624417	+	0.781091i	$0.714663\pi$
$564$		0			0
$565$		−	305.931i		−	0.541471i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	583.249i		−	1.02504i	−0.858675	−	0.512521i	$-0.828712\pi$
$569$		−	583.249i		−	1.02504i	0.858675	−	0.512521i	$-0.171288\pi$
$570$		0			0
$571$	927.321			1.62403			0.812015	−	0.583637i	$-0.198371\pi$
$571$	927.321			1.62403			0.812015	+	0.583637i	$0.198371\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	257.859			0.448450
$576$		0			0
$577$	−983.592			−1.70467			−0.852333	−	0.523000i	$-0.824813\pi$
$577$	−983.592			−1.70467			−0.852333	+	0.523000i	$0.824813\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−391.673			−0.674136
$582$		0			0
$583$		−	1403.84i		−	2.40796i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−566.698			−0.965415			−0.482707	−	0.875782i	$-0.660347\pi$
$587$	−566.698			−0.965415			−0.482707	+	0.875782i	$0.660347\pi$
$588$		0			0
$589$	−257.099	−	666.149i	−0.436500	−	1.13098i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−553.999			−0.934230			−0.467115	−	0.884196i	$-0.654707\pi$
$593$	−553.999			−0.934230			−0.467115	+	0.884196i	$0.654707\pi$
$594$		0			0
$595$	−219.927			−0.369626
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	324.382i		−	0.541540i	−0.962644	−	0.270770i	$-0.912722\pi$
$599$		−	324.382i		−	0.541540i	0.962644	−	0.270770i	$-0.0872783\pi$
$600$		0			0
$601$		−	166.488i		−	0.277018i	−0.990361	−	0.138509i	$-0.955769\pi$
$601$		−	166.488i		−	0.277018i	0.990361	−	0.138509i	$-0.0442309\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−666.279			−1.10129
$606$		0			0
$607$			130.347i			0.214739i	0.994219	+	0.107370i	$0.0342428\pi$
$607$			130.347i			0.214739i	−0.994219	+	0.107370i	$0.965757\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	23.6261i		−	0.0386679i
$612$		0			0
$613$	776.760			1.26715			0.633573	−	0.773683i	$-0.281588\pi$
$613$	776.760			1.26715			0.633573	+	0.773683i	$0.281588\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−121.953			−0.197654			−0.0988270	−	0.995105i	$-0.531509\pi$
$617$	−121.953			−0.197654			−0.0988270	+	0.995105i	$0.531509\pi$
$618$		0			0
$619$	−899.812			−1.45365			−0.726827	−	0.686821i	$-0.759006\pi$
$619$	−899.812			−1.45365			−0.726827	+	0.686821i	$0.759006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	147.154i		−	0.236202i
$624$		0			0
$625$	48.1309			0.0770095
$626$		0			0
$627$		0			0
$628$		0			0
$629$			225.258i			0.358121i
$630$		0			0
$631$	−471.639			−0.747447			−0.373724	−	0.927540i	$-0.621919\pi$
$631$	−471.639			−0.747447			−0.373724	+	0.927540i	$0.621919\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			81.9165i			0.129002i
$636$		0			0
$637$		−	23.0329i		−	0.0361583i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			989.672i			1.54395i	0.635653	+	0.771975i	$0.280731\pi$
$641$			989.672i			1.54395i	−0.635653	+	0.771975i	$0.719269\pi$
$642$		0			0
$643$	−98.4136			−0.153054			−0.0765269	−	0.997068i	$-0.524383\pi$
$643$	−98.4136			−0.153054			−0.0765269	+	0.997068i	$0.524383\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−713.056			−1.10210			−0.551048	−	0.834473i	$-0.685772\pi$
$647$	−713.056			−1.10210			−0.551048	+	0.834473i	$0.685772\pi$
$648$		0			0
$649$			988.856i			1.52366i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−891.424			−1.36512			−0.682561	−	0.730829i	$-0.739134\pi$
$653$	−891.424			−1.36512			−0.682561	+	0.730829i	$0.739134\pi$
$654$		0			0
$655$	−302.061			−0.461162
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	706.071i		−	1.07143i	−0.844399	−	0.535714i	$-0.820042\pi$
$659$		−	706.071i		−	1.07143i	0.844399	−	0.535714i	$-0.179958\pi$
$660$		0			0
$661$			31.7149i			0.0479802i	0.999712	+	0.0239901i	$0.00763702\pi$
$661$			31.7149i			0.0479802i	−0.999712	+	0.0239901i	$0.992363\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	305.808	−	118.026i	0.459862	−	0.177482i
$666$		0			0
$667$		−	728.176i		−	1.09172i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	210.478			0.313678
$672$		0			0
$673$			261.341i			0.388322i	0.980970	+	0.194161i	$0.0621985\pi$
$673$			261.341i			0.388322i	−0.980970	+	0.194161i	$0.937802\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	316.111i		−	0.466929i	−0.972365	−	0.233465i	$-0.924994\pi$
$677$		−	316.111i		−	0.466929i	0.972365	−	0.233465i	$-0.0750063\pi$
$678$		0			0
$679$			686.850i			1.01156i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	482.119i		−	0.705884i	−0.935645	−	0.352942i	$-0.885181\pi$
$683$		−	482.119i		−	0.705884i	0.935645	−	0.352942i	$-0.114819\pi$
$684$		0			0
$685$	−719.225			−1.04996
$686$		0			0
$687$		0			0
$688$		0			0
$689$	117.365			0.170342
$690$		0			0
$691$	185.162			0.267962			0.133981	−	0.990984i	$-0.457224\pi$
$691$	185.162			0.267962			0.133981	+	0.990984i	$0.457224\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−541.622			−0.779312
$696$		0			0
$697$		−	425.516i		−	0.610496i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	463.049			0.660555			0.330278	−	0.943884i	$-0.392858\pi$
$701$	463.049			0.660555			0.330278	+	0.943884i	$0.392858\pi$
$702$		0			0
$703$	−120.887	−	313.220i	−0.171958	−	0.445548i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−268.370			−0.379590
$708$		0			0
$709$	11.5719			0.0163215			0.00816073	−	0.999967i	$-0.497402\pi$
$709$	11.5719			0.0163215			0.00816073	+	0.999967i	$0.497402\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			594.554i			0.833876i
$714$		0			0
$715$		−	85.5427i		−	0.119640i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1245.89			−1.73280			−0.866402	−	0.499347i	$-0.833573\pi$
$719$	−1245.89			−1.73280			−0.866402	+	0.499347i	$0.833573\pi$
$720$		0			0
$721$		−	993.725i		−	1.37826i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	750.197i		−	1.03475i
$726$		0			0
$727$	−246.205			−0.338658			−0.169329	−	0.985560i	$-0.554160\pi$
$727$	−246.205			−0.338658			−0.169329	+	0.985560i	$0.554160\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−646.231			−0.884037
$732$		0			0
$733$	1390.30			1.89672			0.948362	−	0.317189i	$-0.102739\pi$
$733$	1390.30			1.89672			0.948362	+	0.317189i	$0.102739\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			227.218i			0.308302i
$738$		0			0
$739$	925.346			1.25216			0.626080	−	0.779759i	$-0.284658\pi$
$739$	925.346			1.25216			0.626080	+	0.779759i	$0.284658\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			379.097i			0.510225i	0.966911	+	0.255113i	$0.0821125\pi$
$743$			379.097i			0.510225i	−0.966911	+	0.255113i	$0.917887\pi$
$744$		0			0
$745$	455.541			0.611465
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	1074.77i		−	1.43494i
$750$		0			0
$751$		−	657.073i		−	0.874931i	−0.899235	−	0.437465i	$-0.855876\pi$
$751$		−	657.073i		−	0.874931i	0.899235	−	0.437465i	$-0.144124\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	571.889i		−	0.757469i
$756$		0			0
$757$	−965.336			−1.27521			−0.637606	−	0.770362i	$-0.720075\pi$
$757$	−965.336			−1.27521			−0.637606	+	0.770362i	$0.720075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1199.83			1.57664			0.788322	−	0.615263i	$-0.210950\pi$
$761$	1199.83			1.57664			0.788322	+	0.615263i	$0.210950\pi$
$762$		0			0
$763$			345.361i			0.452635i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−82.6715			−0.107786
$768$		0			0
$769$	771.953			1.00384			0.501920	−	0.864914i	$-0.332627\pi$
$769$	771.953			1.00384			0.501920	+	0.864914i	$0.332627\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1054.17i			1.36374i	0.731475	+	0.681868i	$0.238832\pi$
$773$			1054.17i			1.36374i	−0.731475	+	0.681868i	$0.761168\pi$
$774$		0			0
$775$			612.533i			0.790366i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	228.357	+	591.678i	0.293141	+	0.759536i
$780$		0			0
$781$			388.645i			0.497625i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−381.674			−0.486209
$786$		0			0
$787$			1088.65i			1.38329i	0.722239	+	0.691644i	$0.243113\pi$
$787$			1088.65i			1.38329i	−0.722239	+	0.691644i	$0.756887\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	606.597i		−	0.766873i
$792$		0			0
$793$			17.5966i			0.0221899i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1096.14i			1.37533i	0.726026	+	0.687667i	$0.241365\pi$
$797$			1096.14i			1.37533i	−0.726026	+	0.687667i	$0.758635\pi$
$798$		0			0
$799$	193.425			0.242084
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−637.104			−0.793405
$804$		0			0
$805$	−272.941			−0.339057
$806$		0			0
$807$		0			0
$808$		0			0
$809$	216.962			0.268186			0.134093	−	0.990969i	$-0.457188\pi$
$809$	216.962			0.268186			0.134093	+	0.990969i	$0.457188\pi$
$810$		0			0
$811$		−	133.364i		−	0.164444i	−0.996614	−	0.0822219i	$-0.973798\pi$
$811$		−	133.364i		−	0.164444i	0.996614	−	0.0822219i	$-0.0262016\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−624.800			−0.766626
$816$		0			0
$817$	898.582	−	346.805i	1.09986	−	0.424486i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−703.990			−0.857478			−0.428739	−	0.903428i	$-0.641042\pi$
$821$	−703.990			−0.857478			−0.428739	+	0.903428i	$0.641042\pi$
$822$		0			0
$823$	381.732			0.463830			0.231915	−	0.972736i	$-0.425501\pi$
$823$	381.732			0.463830			0.231915	+	0.972736i	$0.425501\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	385.027i		−	0.465571i	−0.972528	−	0.232785i	$-0.925216\pi$
$827$		−	385.027i		−	0.465571i	0.972528	−	0.232785i	$-0.0747840\pi$
$828$		0			0
$829$		−	1648.22i		−	1.98820i	−0.108462	−	0.994101i	$-0.534593\pi$
$829$		−	1648.22i		−	1.98820i	0.108462	−	0.994101i	$-0.465407\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	188.569			0.226373
$834$		0			0
$835$			930.596i			1.11449i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	234.921i		−	0.280001i	−0.990151	−	0.140001i	$-0.955290\pi$
$839$		−	234.921i		−	0.280001i	0.990151	−	0.140001i	$-0.0447104\pi$
$840$		0			0
$841$	−1277.51			−1.51904
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−491.356			−0.581487
$846$		0			0
$847$	−1321.09			−1.55973
$848$		0			0
$849$		0			0
$850$		0			0
$851$			279.556i			0.328503i
$852$		0			0
$853$	567.207			0.664955			0.332478	−	0.943111i	$-0.392115\pi$
$853$	567.207			0.664955			0.332478	+	0.943111i	$0.392115\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	111.109i		−	0.129649i	−0.997897	−	0.0648243i	$-0.979351\pi$
$857$		−	111.109i		−	0.129649i	0.997897	−	0.0648243i	$-0.0206487\pi$
$858$		0			0
$859$	−635.979			−0.740371			−0.370185	−	0.928958i	$-0.620706\pi$
$859$	−635.979			−0.740371			−0.370185	+	0.928958i	$0.620706\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	824.629i		−	0.955537i	−0.878486	−	0.477769i	$-0.841446\pi$
$863$		−	824.629i		−	0.955537i	0.878486	−	0.477769i	$-0.158554\pi$
$864$		0			0
$865$		−	697.225i		−	0.806040i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	2395.14i		−	2.75620i
$870$		0			0
$871$	−18.9962			−0.0218096
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−712.502			−0.814288
$876$		0			0
$877$			952.668i			1.08628i	0.839642	+	0.543140i	$0.182765\pi$
$877$			952.668i			1.08628i	−0.839642	+	0.543140i	$0.817235\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−642.008			−0.728727			−0.364363	−	0.931257i	$-0.618713\pi$
$881$	−642.008			−0.728727			−0.364363	+	0.931257i	$0.618713\pi$
$882$		0			0
$883$	87.6409			0.0992535			0.0496268	−	0.998768i	$-0.484197\pi$
$883$	87.6409			0.0992535			0.0496268	+	0.998768i	$0.484197\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			522.256i			0.588790i	0.955684	+	0.294395i	$0.0951181\pi$
$887$			522.256i			0.588790i	−0.955684	+	0.294395i	$0.904882\pi$
$888$		0			0
$889$			162.423i			0.182703i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−268.957	+	103.803i	−0.301184	+	0.116241i
$894$		0			0
$895$		−	543.781i		−	0.607577i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1729.76			1.92409
$900$		0			0
$901$			960.864i			1.06644i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	236.749i		−	0.261601i
$906$		0			0
$907$			734.866i			0.810216i	0.914269	+	0.405108i	$0.132766\pi$
$907$			734.866i			0.810216i	−0.914269	+	0.405108i	$0.867234\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			446.769i			0.490416i	0.969470	+	0.245208i	$0.0788563\pi$
$911$			446.769i			0.490416i	−0.969470	+	0.245208i	$0.921144\pi$
$912$		0			0
$913$	1247.24			1.36609
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−598.923			−0.653133
$918$		0			0
$919$	−259.922			−0.282831			−0.141416	−	0.989950i	$-0.545165\pi$
$919$	−259.922			−0.282831			−0.141416	+	0.989950i	$0.545165\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−32.4920			−0.0352026
$924$		0			0
$925$			288.010i			0.311362i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1049.29			1.12948			0.564741	−	0.825268i	$-0.308976\pi$
$929$	1049.29			1.12948			0.564741	+	0.825268i	$0.308976\pi$
$930$		0			0
$931$	−262.204	+	101.197i	−0.281637	+	0.108697i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	700.333			0.749020
$936$		0			0
$937$	−715.553			−0.763664			−0.381832	−	0.924232i	$-0.624707\pi$
$937$	−715.553			−0.763664			−0.381832	+	0.924232i	$0.624707\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	703.553i		−	0.747666i	−0.927496	−	0.373833i	$-0.878043\pi$
$941$		−	703.553i		−	0.747666i	0.927496	−	0.373833i	$-0.121957\pi$
$942$		0			0
$943$		−	528.087i		−	0.560007i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−698.867			−0.737980			−0.368990	−	0.929433i	$-0.620296\pi$
$947$	−698.867			−0.737980			−0.368990	+	0.929433i	$0.620296\pi$
$948$		0			0
$949$		−	53.2640i		−	0.0561264i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			826.216i			0.866963i	0.901162	+	0.433482i	$0.142715\pi$
$953$			826.216i			0.866963i	−0.901162	+	0.433482i	$0.857285\pi$
$954$		0			0
$955$	−1068.63			−1.11899
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1426.07			−1.48704
$960$		0			0
$961$	−451.339			−0.469656
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	801.747i		−	0.830826i
$966$		0			0
$967$	−677.220			−0.700331			−0.350165	−	0.936688i	$-0.613875\pi$
$967$	−677.220			−0.700331			−0.350165	+	0.936688i	$0.613875\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			439.573i			0.452701i	0.974046	+	0.226351i	$0.0726795\pi$
$971$			439.573i			0.452701i	−0.974046	+	0.226351i	$0.927320\pi$
$972$		0			0
$973$	−1073.92			−1.10372
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	305.179i		−	0.312363i	−0.987728	−	0.156182i	$-0.950081\pi$
$977$		−	305.179i		−	0.312363i	0.987728	−	0.156182i	$-0.0499185\pi$
$978$		0			0
$979$			468.595i			0.478647i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			141.357i			0.143801i	0.997412	+	0.0719006i	$0.0229064\pi$
$983$			141.357i			0.143801i	−0.997412	+	0.0719006i	$0.977094\pi$
$984$		0			0
$985$	−348.748			−0.354059
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−802.005			−0.810925
$990$		0			0
$991$		−	29.1163i		−	0.0293807i	−0.999892	−	0.0146904i	$-0.995324\pi$
$991$		−	29.1163i		−	0.0293807i	0.999892	−	0.0146904i	$-0.00467625\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	545.918			0.548661
$996$		0			0
$997$	−897.567			−0.900267			−0.450134	−	0.892961i	$-0.648624\pi$
$997$	−897.567			−0.900267			−0.450134	+	0.892961i	$0.648624\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.n.721.4		8
3.2	odd	2		912.3.o.d.721.7		8
4.3	odd	2		342.3.d.b.37.2		8
12.11	even	2		114.3.d.a.37.6	yes	8
19.18	odd	2	inner	2736.3.o.n.721.3		8
57.56	even	2		912.3.o.d.721.3		8
76.75	even	2		342.3.d.b.37.6		8
228.227	odd	2		114.3.d.a.37.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.3.d.a.37.4	✓	8	228.227	odd	2
114.3.d.a.37.6	yes	8	12.11	even	2
342.3.d.b.37.2		8	4.3	odd	2
342.3.d.b.37.6		8	76.75	even	2
912.3.o.d.721.3		8	57.56	even	2
912.3.o.d.721.7		8	3.2	odd	2
2736.3.o.n.721.3		8	19.18	odd	2	inner
2736.3.o.n.721.4		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.94975 q^{5} -5.84873 q^{7} +O(q^{10})\)	\(q-2.94975 q^{5} -5.84873 q^{7} +18.6246 q^{11} +1.55708i q^{13} -12.7477 q^{17} +(17.7256 - 6.84117i) q^{19} -15.8205 q^{23} -16.2990 q^{25} +46.0273i q^{29} -37.5811i q^{31} +17.2523 q^{35} -17.6705i q^{37} +33.3798i q^{41} +50.6939 q^{43} -15.1733 q^{47} -14.7924 q^{49} -75.3754i q^{53} -54.9380 q^{55} +53.0940i q^{59} +11.3011 q^{61} -4.59299i q^{65} +12.1999i q^{67} +20.8673i q^{71} -34.2076 q^{73} -108.930 q^{77} -128.601i q^{79} +66.9672 q^{83} +37.6026 q^{85} +25.1600i q^{89} -9.10693i q^{91} +(-52.2862 + 20.1797i) q^{95} -117.436i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5} + 12 q^{7}+O(q^{10}) \) 8 * q - 4 * q^5 + 12 * q^7	\( 8 q - 4 q^{5} + 12 q^{7} + 4 q^{11} - 4 q^{17} + 36 q^{19} - 56 q^{23} + 140 q^{25} + 236 q^{35} - 100 q^{43} - 188 q^{47} - 36 q^{49} - 28 q^{55} - 180 q^{61} - 356 q^{73} - 68 q^{77} + 136 q^{83} + 148 q^{85} - 140 q^{95}+O(q^{100}) \) 8 * q - 4 * q^5 + 12 * q^7 + 4 * q^11 - 4 * q^17 + 36 * q^19 - 56 * q^23 + 140 * q^25 + 236 * q^35 - 100 * q^43 - 188 * q^47 - 36 * q^49 - 28 * q^55 - 180 * q^61 - 356 * q^73 - 68 * q^77 + 136 * q^83 + 148 * q^85 - 140 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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