LMFDB - Embedded newform 3249.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3249,2,Mod(1,3249)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3249, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3249.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

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:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53209 q^{2} +0.347296 q^{4} +2.53209 q^{5} +0.532089 q^{7} +2.53209 q^{8} -3.87939 q^{10} +5.10607 q^{11} -4.06418 q^{13} -0.815207 q^{14} -4.57398 q^{16} -1.94356 q^{17} +0.879385 q^{20} -7.82295 q^{22} -3.04189 q^{23} +1.41147 q^{25} +6.22668 q^{26} +0.184793 q^{28} +1.61081 q^{29} -9.87939 q^{31} +1.94356 q^{32} +2.97771 q^{34} +1.34730 q^{35} -6.10607 q^{37} +6.41147 q^{40} -8.47565 q^{41} -0.177052 q^{43} +1.77332 q^{44} +4.66044 q^{46} -7.55943 q^{47} -6.71688 q^{49} -2.16250 q^{50} -1.41147 q^{52} -9.90167 q^{53} +12.9290 q^{55} +1.34730 q^{56} -2.46791 q^{58} -3.81521 q^{59} -13.9290 q^{61} +15.1361 q^{62} +6.17024 q^{64} -10.2909 q^{65} +10.2121 q^{67} -0.674992 q^{68} -2.06418 q^{70} +3.82295 q^{71} -1.85710 q^{73} +9.35504 q^{74} +2.71688 q^{77} -10.5175 q^{79} -11.5817 q^{80} +12.9855 q^{82} +11.4757 q^{83} -4.92127 q^{85} +0.271259 q^{86} +12.9290 q^{88} -5.92396 q^{89} -2.16250 q^{91} -1.05644 q^{92} +11.5817 q^{94} +6.80066 q^{97} +10.2909 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 3 q^{7} + 3 q^{8} - 6 q^{10} + 3 q^{11} - 3 q^{13} - 6 q^{14} - 6 q^{16} + 9 q^{17} - 3 q^{20} - 3 q^{22} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} + 9 q^{29} - 24 q^{31} - 9 q^{32} + 15 q^{34}+ \cdots + 15 q^{98}+O(q^{100}) $ 3 * q + 3 * q^5 - 3 * q^7 + 3 * q^8 - 6 * q^10 + 3 * q^11 - 3 * q^13 - 6 * q^14 - 6 * q^16 + 9 * q^17 - 3 * q^20 - 3 * q^22 - 6 * q^23 - 6 * q^25 + 12 * q^26 - 3 * q^28 + 9 * q^29 - 24 * q^31 - 9 * q^32 + 15 * q^34 + 3 * q^35 - 6 * q^37 + 9 * q^40 - 6 * q^41 - 21 * q^43 + 12 * q^44 - 9 * q^46 + 3 * q^47 - 12 * q^49 - 9 * q^50 + 6 * q^52 - 18 * q^53 + 6 * q^55 + 3 * q^56 - 12 * q^58 - 15 * q^59 - 9 * q^61 + 3 * q^62 - 3 * q^64 - 15 * q^65 + 6 * q^67 + 3 * q^68 + 3 * q^70 - 9 * q^71 - 6 * q^73 + 3 * q^74 - 9 * q^79 - 3 * q^80 + 21 * q^82 + 15 * q^83 - 6 * q^85 - 18 * q^86 + 6 * q^88 - 9 * q^91 - 18 * q^92 + 3 * q^94 + 6 * q^97 + 15 * q^98

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 $

$2$	−1.53209	−1.08335	−0.541675	−	0.840588i	$-0.682210\pi$
$2$	−1.53209	−1.08335	−0.541675	+	0.840588i	$0.682210\pi$
$3$	0	0
$3$	0	0
$4$	0.347296	0.173648
$5$	2.53209	1.13238	0.566192	−	0.824273i	$-0.308416\pi$
$5$	2.53209	1.13238	0.566192	+	0.824273i	$0.308416\pi$
$6$	0	0
$7$	0.532089	0.201111	0.100555	−	0.994931i	$-0.467938\pi$
$7$	0.532089	0.201111	0.100555	+	0.994931i	$0.467938\pi$
$8$	2.53209	0.895229
$9$	0	0
$10$	−3.87939	−1.22677
$11$	5.10607	1.53954	0.769769	−	0.638323i	$-0.220372\pi$
$11$	5.10607	1.53954	0.769769	+	0.638323i	$0.220372\pi$
$12$	0	0
$13$	−4.06418	−1.12720	−0.563600	−	0.826048i	$-0.690584\pi$
$13$	−4.06418	−1.12720	−0.563600	+	0.826048i	$0.690584\pi$
$14$	−0.815207	−0.217873
$15$	0	0
$16$	−4.57398	−1.14349
$17$	−1.94356	−0.471383	−0.235692	−	0.971828i	$-0.575736\pi$
$17$	−1.94356	−0.471383	−0.235692	+	0.971828i	$0.575736\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0.879385	0.196637
$21$	0	0
$22$	−7.82295	−1.66786
$23$	−3.04189	−0.634278	−0.317139	−	0.948379i	$-0.602722\pi$
$23$	−3.04189	−0.634278	−0.317139	+	0.948379i	$0.602722\pi$
$24$	0	0
$25$	1.41147	0.282295
$26$	6.22668	1.22115
$27$	0	0
$28$	0.184793	0.0349225
$29$	1.61081	0.299121	0.149560	−	0.988753i	$-0.452214\pi$
$29$	1.61081	0.299121	0.149560	+	0.988753i	$0.452214\pi$
$30$	0	0
$31$	−9.87939	−1.77439	−0.887195	−	0.461395i	$-0.847349\pi$
$31$	−9.87939	−1.77439	−0.887195	+	0.461395i	$0.847349\pi$
$32$	1.94356	0.343577
$33$	0	0
$34$	2.97771	0.510673
$35$	1.34730	0.227735
$36$	0	0
$37$	−6.10607	−1.00383	−0.501916	−	0.864917i	$-0.667371\pi$
$37$	−6.10607	−1.00383	−0.501916	+	0.864917i	$0.667371\pi$
$38$	0	0
$39$	0	0
$40$	6.41147	1.01374
$41$	−8.47565	−1.32367	−0.661837	−	0.749648i	$-0.730223\pi$
$41$	−8.47565	−1.32367	−0.661837	+	0.749648i	$0.730223\pi$
$42$	0	0
$43$	−0.177052	−0.0270001	−0.0135001	−	0.999909i	$-0.504297\pi$
$43$	−0.177052	−0.0270001	−0.0135001	+	0.999909i	$0.504297\pi$
$44$	1.77332	0.267338
$45$	0	0
$46$	4.66044	0.687145
$47$	−7.55943	−1.10266	−0.551328	−	0.834289i	$-0.685879\pi$
$47$	−7.55943	−1.10266	−0.551328	+	0.834289i	$0.685879\pi$
$48$	0	0
$49$	−6.71688	−0.959554
$50$	−2.16250	−0.305824
$51$	0	0
$52$	−1.41147	−0.195736
$53$	−9.90167	−1.36010	−0.680050	−	0.733166i	$-0.738042\pi$
$53$	−9.90167	−1.36010	−0.680050	+	0.733166i	$0.738042\pi$
$54$	0	0
$55$	12.9290	1.74335
$56$	1.34730	0.180040
$57$	0	0
$58$	−2.46791	−0.324053
$59$	−3.81521	−0.496698	−0.248349	−	0.968671i	$-0.579888\pi$
$59$	−3.81521	−0.496698	−0.248349	+	0.968671i	$0.579888\pi$
$60$	0	0
$61$	−13.9290	−1.78343	−0.891714	−	0.452600i	$-0.850497\pi$
$61$	−13.9290	−1.78343	−0.891714	+	0.452600i	$0.850497\pi$
$62$	15.1361	1.92229
$63$	0	0
$64$	6.17024	0.771281
$65$	−10.2909	−1.27642
$66$	0	0
$67$	10.2121	1.24761	0.623805	−	0.781580i	$-0.285586\pi$
$67$	10.2121	1.24761	0.623805	+	0.781580i	$0.285586\pi$
$68$	−0.674992	−0.0818548
$69$	0	0
$70$	−2.06418	−0.246716
$71$	3.82295	0.453700	0.226850	−	0.973930i	$-0.427157\pi$
$71$	3.82295	0.453700	0.226850	+	0.973930i	$0.427157\pi$
$72$	0	0
$73$	−1.85710	−0.217357	−0.108678	−	0.994077i	$-0.534662\pi$
$73$	−1.85710	−0.217357	−0.108678	+	0.994077i	$0.534662\pi$
$74$	9.35504	1.08750
$75$	0	0
$76$	0	0
$77$	2.71688	0.309617
$78$	0	0
$79$	−10.5175	−1.18332	−0.591658	−	0.806189i	$-0.701526\pi$
$79$	−10.5175	−1.18332	−0.591658	+	0.806189i	$0.701526\pi$
$80$	−11.5817	−1.29488
$81$	0	0
$82$	12.9855	1.43400
$83$	11.4757	1.25962	0.629808	−	0.776751i	$-0.283133\pi$
$83$	11.4757	1.25962	0.629808	+	0.776751i	$0.283133\pi$
$84$	0	0
$85$	−4.92127	−0.533787
$86$	0.271259	0.0292506
$87$	0	0
$88$	12.9290	1.37824
$89$	−5.92396	−0.627939	−0.313969	−	0.949433i	$-0.601659\pi$
$89$	−5.92396	−0.627939	−0.313969	+	0.949433i	$0.601659\pi$
$90$	0	0
$91$	−2.16250	−0.226692
$92$	−1.05644	−0.110141
$93$	0	0
$94$	11.5817	1.19456
$95$	0	0
$96$	0	0
$97$	6.80066	0.690502	0.345251	−	0.938510i	$-0.387794\pi$
$97$	6.80066	0.690502	0.345251	+	0.938510i	$0.387794\pi$
$98$	10.2909	1.03953
$99$	0	0
$100$	0.490200	0.0490200
$101$	8.75877	0.871530	0.435765	−	0.900060i	$-0.356478\pi$
$101$	8.75877	0.871530	0.435765	+	0.900060i	$0.356478\pi$
$102$	0	0
$103$	3.84524	0.378882	0.189441	−	0.981892i	$-0.439332\pi$
$103$	3.84524	0.378882	0.189441	+	0.981892i	$0.439332\pi$
$104$	−10.2909	−1.00910
$105$	0	0
$106$	15.1702	1.47346
$107$	−5.67499	−0.548622	−0.274311	−	0.961641i	$-0.588450\pi$
$107$	−5.67499	−0.548622	−0.274311	+	0.961641i	$0.588450\pi$
$108$	0	0
$109$	2.08647	0.199847	0.0999236	−	0.994995i	$-0.468140\pi$
$109$	2.08647	0.199847	0.0999236	+	0.994995i	$0.468140\pi$
$110$	−19.8084	−1.88866
$111$	0	0
$112$	−2.43376	−0.229969
$113$	7.27631	0.684498	0.342249	−	0.939609i	$-0.388811\pi$
$113$	7.27631	0.684498	0.342249	+	0.939609i	$0.388811\pi$
$114$	0	0
$115$	−7.70233	−0.718246
$116$	0.559430	0.0519418
$117$	0	0
$118$	5.84524	0.538098
$119$	−1.03415	−0.0948002
$120$	0	0
$121$	15.0719	1.37017
$122$	21.3405	1.93208
$123$	0	0
$124$	−3.43107	−0.308120
$125$	−9.08647	−0.812718
$126$	0	0
$127$	−1.82976	−0.162365	−0.0811823	−	0.996699i	$-0.525870\pi$
$127$	−1.82976	−0.162365	−0.0811823	+	0.996699i	$0.525870\pi$
$128$	−13.3405	−1.17914
$129$	0	0
$130$	15.7665	1.38281
$131$	8.78611	0.767646	0.383823	−	0.923407i	$-0.374607\pi$
$131$	8.78611	0.767646	0.383823	+	0.923407i	$0.374607\pi$
$132$	0	0
$133$	0	0
$134$	−15.6459	−1.35160
$135$	0	0
$136$	−4.92127	−0.421996
$137$	12.6304	1.07909	0.539545	−	0.841957i	$-0.318596\pi$
$137$	12.6304	1.07909	0.539545	+	0.841957i	$0.318596\pi$
$138$	0	0
$139$	−4.04963	−0.343485	−0.171743	−	0.985142i	$-0.554940\pi$
$139$	−4.04963	−0.343485	−0.171743	+	0.985142i	$0.554940\pi$
$140$	0.467911	0.0395457
$141$	0	0
$142$	−5.85710	−0.491517
$143$	−20.7520	−1.73537
$144$	0	0
$145$	4.07873	0.338720
$146$	2.84524	0.235473
$147$	0	0
$148$	−2.12061	−0.174313
$149$	−6.06923	−0.497211	−0.248605	−	0.968605i	$-0.579972\pi$
$149$	−6.06923	−0.497211	−0.248605	+	0.968605i	$0.579972\pi$
$150$	0	0
$151$	0.162504	0.0132244	0.00661219	−	0.999978i	$-0.497895\pi$
$151$	0.162504	0.0132244	0.00661219	+	0.999978i	$0.497895\pi$
$152$	0	0
$153$	0	0
$154$	−4.16250	−0.335424
$155$	−25.0155	−2.00929
$156$	0	0
$157$	6.66044	0.531561	0.265781	−	0.964034i	$-0.414370\pi$
$157$	6.66044	0.531561	0.265781	+	0.964034i	$0.414370\pi$
$158$	16.1138	1.28195
$159$	0	0
$160$	4.92127	0.389061
$161$	−1.61856	−0.127560
$162$	0	0
$163$	−0.448311	−0.0351144	−0.0175572	−	0.999846i	$-0.505589\pi$
$163$	−0.448311	−0.0351144	−0.0175572	+	0.999846i	$0.505589\pi$
$164$	−2.94356	−0.229854
$165$	0	0
$166$	−17.5817	−1.36461
$167$	3.38919	0.262263	0.131132	−	0.991365i	$-0.458139\pi$
$167$	3.38919	0.262263	0.131132	+	0.991365i	$0.458139\pi$
$168$	0	0
$169$	3.51754	0.270580
$170$	7.53983	0.578279
$171$	0	0
$172$	−0.0614894	−0.00468852
$173$	−1.61587	−0.122852	−0.0614260	−	0.998112i	$-0.519565\pi$
$173$	−1.61587	−0.122852	−0.0614260	+	0.998112i	$0.519565\pi$
$174$	0	0
$175$	0.751030	0.0567725
$176$	−23.3550	−1.76045
$177$	0	0
$178$	9.07604	0.680278
$179$	2.11381	0.157993	0.0789967	−	0.996875i	$-0.474828\pi$
$179$	2.11381	0.157993	0.0789967	+	0.996875i	$0.474828\pi$
$180$	0	0
$181$	−8.75196	−0.650528	−0.325264	−	0.945623i	$-0.605453\pi$
$181$	−8.75196	−0.650528	−0.325264	+	0.945623i	$0.605453\pi$
$182$	3.31315	0.245587
$183$	0	0
$184$	−7.70233	−0.567824
$185$	−15.4611	−1.13672
$186$	0	0
$187$	−9.92396	−0.725712
$188$	−2.62536	−0.191474
$189$	0	0
$190$	0	0
$191$	19.0719	1.38000	0.689998	−	0.723811i	$-0.257611\pi$
$191$	19.0719	1.38000	0.689998	+	0.723811i	$0.257611\pi$
$192$	0	0
$193$	23.8357	1.71573	0.857867	−	0.513872i	$-0.171789\pi$
$193$	23.8357	1.71573	0.857867	+	0.513872i	$0.171789\pi$
$194$	−10.4192	−0.748056
$195$	0	0
$196$	−2.33275	−0.166625
$197$	4.38919	0.312717	0.156358	−	0.987700i	$-0.450025\pi$
$197$	4.38919	0.312717	0.156358	+	0.987700i	$0.450025\pi$
$198$	0	0
$199$	−20.0378	−1.42044	−0.710220	−	0.703980i	$-0.751405\pi$
$199$	−20.0378	−1.42044	−0.710220	+	0.703980i	$0.751405\pi$
$200$	3.57398	0.252718
$201$	0	0
$202$	−13.4192	−0.944173
$203$	0.857097	0.0601564
$204$	0	0
$205$	−21.4611	−1.49891
$206$	−5.89124	−0.410462
$207$	0	0
$208$	18.5895	1.28895
$209$	0	0
$210$	0	0
$211$	−27.1925	−1.87201	−0.936006	−	0.351985i	$-0.885507\pi$
$211$	−27.1925	−1.87201	−0.936006	+	0.351985i	$0.885507\pi$
$212$	−3.43882	−0.236179
$213$	0	0
$214$	8.69459	0.594350
$215$	−0.448311	−0.0305745
$216$	0	0
$217$	−5.25671	−0.356849
$218$	−3.19665	−0.216505
$219$	0	0
$220$	4.49020	0.302729
$221$	7.89899	0.531343
$222$	0	0
$223$	12.0128	0.804436	0.402218	−	0.915544i	$-0.368239\pi$
$223$	12.0128	0.804436	0.402218	+	0.915544i	$0.368239\pi$
$224$	1.03415	0.0690969
$225$	0	0
$226$	−11.1480	−0.741551
$227$	4.34049	0.288088	0.144044	−	0.989571i	$-0.453989\pi$
$227$	4.34049	0.288088	0.144044	+	0.989571i	$0.453989\pi$
$228$	0	0
$229$	3.29591	0.217800	0.108900	−	0.994053i	$-0.465267\pi$
$229$	3.29591	0.217800	0.108900	+	0.994053i	$0.465267\pi$
$230$	11.8007	0.778112
$231$	0	0
$232$	4.07873	0.267781
$233$	15.4311	1.01092	0.505462	−	0.862849i	$-0.331322\pi$
$233$	15.4311	1.01092	0.505462	+	0.862849i	$0.331322\pi$
$234$	0	0
$235$	−19.1411	−1.24863
$236$	−1.32501	−0.0862507
$237$	0	0
$238$	1.58441	0.102702
$239$	−15.8229	−1.02350	−0.511751	−	0.859134i	$-0.671003\pi$
$239$	−15.8229	−1.02350	−0.511751	+	0.859134i	$0.671003\pi$
$240$	0	0
$241$	23.6459	1.52317	0.761583	−	0.648067i	$-0.224422\pi$
$241$	23.6459	1.52317	0.761583	+	0.648067i	$0.224422\pi$
$242$	−23.0915	−1.48438
$243$	0	0
$244$	−4.83750	−0.309689
$245$	−17.0077	−1.08658
$246$	0	0
$247$	0	0
$248$	−25.0155	−1.58848
$249$	0	0
$250$	13.9213	0.880459
$251$	−19.4115	−1.22524	−0.612621	−	0.790377i	$-0.709885\pi$
$251$	−19.4115	−1.22524	−0.612621	+	0.790377i	$0.709885\pi$
$252$	0	0
$253$	−15.5321	−0.976494
$254$	2.80335	0.175898
$255$	0	0
$256$	8.09833	0.506145
$257$	12.2959	0.766998	0.383499	−	0.923541i	$-0.374719\pi$
$257$	12.2959	0.766998	0.383499	+	0.923541i	$0.374719\pi$
$258$	0	0
$259$	−3.24897	−0.201881
$260$	−3.57398	−0.221649
$261$	0	0
$262$	−13.4611	−0.831630
$263$	−6.34049	−0.390971	−0.195486	−	0.980707i	$-0.562628\pi$
$263$	−6.34049	−0.390971	−0.195486	+	0.980707i	$0.562628\pi$
$264$	0	0
$265$	−25.0719	−1.54016
$266$	0	0
$267$	0	0
$268$	3.54664	0.216645
$269$	−2.32501	−0.141758	−0.0708791	−	0.997485i	$-0.522580\pi$
$269$	−2.32501	−0.141758	−0.0708791	+	0.997485i	$0.522580\pi$
$270$	0	0
$271$	14.9436	0.907757	0.453878	−	0.891064i	$-0.350040\pi$
$271$	14.9436	0.907757	0.453878	+	0.891064i	$0.350040\pi$
$272$	8.88981	0.539024
$273$	0	0
$274$	−19.3509	−1.16903
$275$	7.20708	0.434603
$276$	0	0
$277$	−23.9659	−1.43997	−0.719984	−	0.693990i	$-0.755851\pi$
$277$	−23.9659	−1.43997	−0.719984	+	0.693990i	$0.755851\pi$
$278$	6.20439	0.372115
$279$	0	0
$280$	3.41147	0.203875
$281$	25.5604	1.52480	0.762402	−	0.647104i	$-0.224020\pi$
$281$	25.5604	1.52480	0.762402	+	0.647104i	$0.224020\pi$
$282$	0	0
$283$	−7.51073	−0.446467	−0.223233	−	0.974765i	$-0.571661\pi$
$283$	−7.51073	−0.446467	−0.223233	+	0.974765i	$0.571661\pi$
$284$	1.32770	0.0787843
$285$	0	0
$286$	31.7939	1.88001
$287$	−4.50980	−0.266205
$288$	0	0
$289$	−13.2226	−0.777798
$290$	−6.24897	−0.366952
$291$	0	0
$292$	−0.644963	−0.0377436
$293$	14.7665	0.862669	0.431334	−	0.902192i	$-0.358043\pi$
$293$	14.7665	0.862669	0.431334	+	0.902192i	$0.358043\pi$
$294$	0	0
$295$	−9.66044	−0.562453
$296$	−15.4611	−0.898658
$297$	0	0
$298$	9.29860	0.538653
$299$	12.3628	0.714958
$300$	0	0
$301$	−0.0942073	−0.00543002
$302$	−0.248970	−0.0143266
$303$	0	0
$304$	0	0
$305$	−35.2695	−2.01953
$306$	0	0
$307$	−5.51249	−0.314614	−0.157307	−	0.987550i	$-0.550281\pi$
$307$	−5.51249	−0.314614	−0.157307	+	0.987550i	$0.550281\pi$
$308$	0.943563	0.0537645
$309$	0	0
$310$	38.3259	2.17677
$311$	−17.3996	−0.986642	−0.493321	−	0.869847i	$-0.664217\pi$
$311$	−17.3996	−0.986642	−0.493321	+	0.869847i	$0.664217\pi$
$312$	0	0
$313$	15.9436	0.901183	0.450592	−	0.892730i	$-0.351213\pi$
$313$	15.9436	0.901183	0.450592	+	0.892730i	$0.351213\pi$
$314$	−10.2044	−0.575867
$315$	0	0
$316$	−3.65270	−0.205481
$317$	1.97359	0.110848	0.0554240	−	0.998463i	$-0.482349\pi$
$317$	1.97359	0.110848	0.0554240	+	0.998463i	$0.482349\pi$
$318$	0	0
$319$	8.22493	0.460507
$320$	15.6236	0.873386
$321$	0	0
$322$	2.47977	0.138192
$323$	0	0
$324$	0	0
$325$	−5.73648	−0.318203
$326$	0.686852	0.0380412
$327$	0	0
$328$	−21.4611	−1.18499
$329$	−4.02229	−0.221756
$330$	0	0
$331$	0.837496	0.0460330	0.0230165	−	0.999735i	$-0.492673\pi$
$331$	0.837496	0.0460330	0.0230165	+	0.999735i	$0.492673\pi$
$332$	3.98545	0.218730
$333$	0	0
$334$	−5.19253	−0.284123
$335$	25.8580	1.41278
$336$	0	0
$337$	3.32770	0.181271	0.0906356	−	0.995884i	$-0.471110\pi$
$337$	3.32770	0.181271	0.0906356	+	0.995884i	$0.471110\pi$
$338$	−5.38919	−0.293133
$339$	0	0
$340$	−1.70914	−0.0926912
$341$	−50.4448	−2.73174
$342$	0	0
$343$	−7.29860	−0.394087
$344$	−0.448311	−0.0241713
$345$	0	0
$346$	2.47565	0.133092
$347$	−4.00774	−0.215147	−0.107573	−	0.994197i	$-0.534308\pi$
$347$	−4.00774	−0.215147	−0.107573	+	0.994197i	$0.534308\pi$
$348$	0	0
$349$	−27.9026	−1.49359	−0.746796	−	0.665053i	$-0.768409\pi$
$349$	−27.9026	−1.49359	−0.746796	+	0.665053i	$0.768409\pi$
$350$	−1.15064	−0.0615045
$351$	0	0
$352$	9.92396	0.528949
$353$	−12.9436	−0.688916	−0.344458	−	0.938802i	$-0.611937\pi$
$353$	−12.9436	−0.688916	−0.344458	+	0.938802i	$0.611937\pi$
$354$	0	0
$355$	9.68004	0.513763
$356$	−2.05737	−0.109040
$357$	0	0
$358$	−3.23854	−0.171162
$359$	−11.4466	−0.604126	−0.302063	−	0.953288i	$-0.597675\pi$
$359$	−11.4466	−0.604126	−0.302063	+	0.953288i	$0.597675\pi$
$360$	0	0
$361$	0	0
$362$	13.4088	0.704750
$363$	0	0
$364$	−0.751030	−0.0393647
$365$	−4.70233	−0.246131
$366$	0	0
$367$	−28.6013	−1.49298	−0.746488	−	0.665398i	$-0.768262\pi$
$367$	−28.6013	−1.49298	−0.746488	+	0.665398i	$0.768262\pi$
$368$	13.9135	0.725293
$369$	0	0
$370$	23.6878	1.23147
$371$	−5.26857	−0.273531
$372$	0	0
$373$	12.5662	0.650655	0.325328	−	0.945601i	$-0.394525\pi$
$373$	12.5662	0.650655	0.325328	+	0.945601i	$0.394525\pi$
$374$	15.2044	0.786200
$375$	0	0
$376$	−19.1411	−0.987129
$377$	−6.54664	−0.337169
$378$	0	0
$379$	2.19934	0.112973	0.0564863	−	0.998403i	$-0.482010\pi$
$379$	2.19934	0.112973	0.0564863	+	0.998403i	$0.482010\pi$
$380$	0	0
$381$	0	0
$382$	−29.2199	−1.49502
$383$	−5.92808	−0.302911	−0.151455	−	0.988464i	$-0.548396\pi$
$383$	−5.92808	−0.302911	−0.151455	+	0.988464i	$0.548396\pi$
$384$	0	0
$385$	6.87939	0.350606
$386$	−36.5185	−1.85874
$387$	0	0
$388$	2.36184	0.119904
$389$	33.1189	1.67919	0.839596	−	0.543211i	$-0.182792\pi$
$389$	33.1189	1.67919	0.839596	+	0.543211i	$0.182792\pi$
$390$	0	0
$391$	5.91210	0.298988
$392$	−17.0077	−0.859021
$393$	0	0
$394$	−6.72462	−0.338782
$395$	−26.6313	−1.33997
$396$	0	0
$397$	−26.0496	−1.30739	−0.653697	−	0.756757i	$-0.726783\pi$
$397$	−26.0496	−1.30739	−0.653697	+	0.756757i	$0.726783\pi$
$398$	30.6996	1.53883
$399$	0	0
$400$	−6.45605	−0.322803
$401$	−15.9682	−0.797415	−0.398707	−	0.917078i	$-0.630541\pi$
$401$	−15.9682	−0.797415	−0.398707	+	0.917078i	$0.630541\pi$
$402$	0	0
$403$	40.1516	2.00009
$404$	3.04189	0.151340
$405$	0	0
$406$	−1.31315	−0.0651704
$407$	−31.1780	−1.54544
$408$	0	0
$409$	2.94862	0.145800	0.0728998	−	0.997339i	$-0.476775\pi$
$409$	2.94862	0.145800	0.0728998	+	0.997339i	$0.476775\pi$
$410$	32.8803	1.62384
$411$	0	0
$412$	1.33544	0.0657922
$413$	−2.03003	−0.0998912
$414$	0	0
$415$	29.0574	1.42637
$416$	−7.89899	−0.387280
$417$	0	0
$418$	0	0
$419$	33.4962	1.63640	0.818198	−	0.574937i	$-0.194973\pi$
$419$	33.4962	1.63640	0.818198	+	0.574937i	$0.194973\pi$
$420$	0	0
$421$	−18.0178	−0.878136	−0.439068	−	0.898454i	$-0.644691\pi$
$421$	−18.0178	−0.878136	−0.439068	+	0.898454i	$0.644691\pi$
$422$	41.6614	2.02804
$423$	0	0
$424$	−25.0719	−1.21760
$425$	−2.74329	−0.133069
$426$	0	0
$427$	−7.41147	−0.358666
$428$	−1.97090	−0.0952672
$429$	0	0
$430$	0.686852	0.0331229
$431$	21.3209	1.02699	0.513496	−	0.858092i	$-0.328350\pi$
$431$	21.3209	1.02699	0.513496	+	0.858092i	$0.328350\pi$
$432$	0	0
$433$	22.0523	1.05977	0.529883	−	0.848071i	$-0.322236\pi$
$433$	22.0523	1.05977	0.529883	+	0.848071i	$0.322236\pi$
$434$	8.05375	0.386592
$435$	0	0
$436$	0.724622	0.0347031
$437$	0	0
$438$	0	0
$439$	3.38682	0.161644	0.0808221	−	0.996729i	$-0.474245\pi$
$439$	3.38682	0.161644	0.0808221	+	0.996729i	$0.474245\pi$
$440$	32.7374	1.56070
$441$	0	0
$442$	−12.1019	−0.575631
$443$	30.1644	1.43315	0.716576	−	0.697509i	$-0.245708\pi$
$443$	30.1644	1.43315	0.716576	+	0.697509i	$0.245708\pi$
$444$	0	0
$445$	−15.0000	−0.711068
$446$	−18.4047	−0.871486
$447$	0	0
$448$	3.28312	0.155113
$449$	−30.9118	−1.45882	−0.729409	−	0.684078i	$-0.760205\pi$
$449$	−30.9118	−1.45882	−0.729409	+	0.684078i	$0.760205\pi$
$450$	0	0
$451$	−43.2772	−2.03785
$452$	2.52704	0.118862
$453$	0	0
$454$	−6.65002	−0.312101
$455$	−5.47565	−0.256703
$456$	0	0
$457$	−39.0479	−1.82658	−0.913291	−	0.407307i	$-0.866468\pi$
$457$	−39.0479	−1.82658	−0.913291	+	0.407307i	$0.866468\pi$
$458$	−5.04963	−0.235954
$459$	0	0
$460$	−2.67499	−0.124722
$461$	1.50744	0.0702083	0.0351041	−	0.999384i	$-0.488824\pi$
$461$	1.50744	0.0702083	0.0351041	+	0.999384i	$0.488824\pi$
$462$	0	0
$463$	−3.28312	−0.152579	−0.0762897	−	0.997086i	$-0.524307\pi$
$463$	−3.28312	−0.152579	−0.0762897	+	0.997086i	$0.524307\pi$
$464$	−7.36783	−0.342043
$465$	0	0
$466$	−23.6418	−1.09518
$467$	22.8871	1.05909	0.529545	−	0.848282i	$-0.322363\pi$
$467$	22.8871	1.05909	0.529545	+	0.848282i	$0.322363\pi$
$468$	0	0
$469$	5.43376	0.250908
$470$	29.3259	1.35270
$471$	0	0
$472$	−9.66044	−0.444658
$473$	−0.904038	−0.0415677
$474$	0	0
$475$	0	0
$476$	−0.359156	−0.0164619
$477$	0	0
$478$	24.2422	1.10881
$479$	9.65776	0.441274	0.220637	−	0.975356i	$-0.429186\pi$
$479$	9.65776	0.441274	0.220637	+	0.975356i	$0.429186\pi$
$480$	0	0
$481$	24.8161	1.13152
$482$	−36.2276	−1.65012
$483$	0	0
$484$	5.23442	0.237928
$485$	17.2199	0.781914
$486$	0	0
$487$	22.5107	1.02006	0.510029	−	0.860157i	$-0.329635\pi$
$487$	22.5107	1.02006	0.510029	+	0.860157i	$0.329635\pi$
$488$	−35.2695	−1.59658
$489$	0	0
$490$	26.0574	1.17715
$491$	−7.27126	−0.328147	−0.164074	−	0.986448i	$-0.552463\pi$
$491$	−7.27126	−0.328147	−0.164074	+	0.986448i	$0.552463\pi$
$492$	0	0
$493$	−3.13072	−0.141001
$494$	0	0
$495$	0	0
$496$	45.1881	2.02901
$497$	2.03415	0.0912440
$498$	0	0
$499$	−12.8111	−0.573503	−0.286752	−	0.958005i	$-0.592575\pi$
$499$	−12.8111	−0.573503	−0.286752	+	0.958005i	$0.592575\pi$
$500$	−3.15570	−0.141127
$501$	0	0
$502$	29.7401	1.32737
$503$	−8.37052	−0.373223	−0.186611	−	0.982434i	$-0.559751\pi$
$503$	−8.37052	−0.373223	−0.186611	+	0.982434i	$0.559751\pi$
$504$	0	0
$505$	22.1780	0.986907
$506$	23.7965	1.05789
$507$	0	0
$508$	−0.635467	−0.0281943
$509$	13.6827	0.606476	0.303238	−	0.952915i	$-0.401932\pi$
$509$	13.6827	0.606476	0.303238	+	0.952915i	$0.401932\pi$
$510$	0	0
$511$	−0.988140	−0.0437128
$512$	14.2736	0.630811
$513$	0	0
$514$	−18.8384	−0.830928
$515$	9.73648	0.429041
$516$	0	0
$517$	−38.5990	−1.69758
$518$	4.97771	0.218708
$519$	0	0
$520$	−26.0574	−1.14269
$521$	−16.9641	−0.743211	−0.371605	−	0.928391i	$-0.621193\pi$
$521$	−16.9641	−0.743211	−0.371605	+	0.928391i	$0.621193\pi$
$522$	0	0
$523$	3.90673	0.170829	0.0854146	−	0.996345i	$-0.472779\pi$
$523$	3.90673	0.170829	0.0854146	+	0.996345i	$0.472779\pi$
$524$	3.05138	0.133300
$525$	0	0
$526$	9.71419	0.423559
$527$	19.2012	0.836418
$528$	0	0
$529$	−13.7469	−0.597692
$530$	38.4124	1.66853
$531$	0	0
$532$	0	0
$533$	34.4466	1.49205
$534$	0	0
$535$	−14.3696	−0.621251
$536$	25.8580	1.11690
$537$	0	0
$538$	3.56212	0.153574
$539$	−34.2968	−1.47727
$540$	0	0
$541$	13.1976	0.567409	0.283704	−	0.958912i	$-0.408437\pi$
$541$	13.1976	0.567409	0.283704	+	0.958912i	$0.408437\pi$
$542$	−22.8949	−0.983419
$543$	0	0
$544$	−3.77744	−0.161956
$545$	5.28312	0.226304
$546$	0	0
$547$	31.5749	1.35005	0.675023	−	0.737797i	$-0.264134\pi$
$547$	31.5749	1.35005	0.675023	+	0.737797i	$0.264134\pi$
$548$	4.38650	0.187382
$549$	0	0
$550$	−11.0419	−0.470828
$551$	0	0
$552$	0	0
$553$	−5.59627	−0.237977
$554$	36.7178	1.55999
$555$	0	0
$556$	−1.40642	−0.0596456
$557$	−35.6323	−1.50979	−0.754894	−	0.655847i	$-0.772312\pi$
$557$	−35.6323	−1.50979	−0.754894	+	0.655847i	$0.772312\pi$
$558$	0	0
$559$	0.719570	0.0304346
$560$	−6.16250	−0.260413
$561$	0	0
$562$	−39.1607	−1.65190
$563$	−21.9914	−0.926829	−0.463414	−	0.886142i	$-0.653376\pi$
$563$	−21.9914	−0.926829	−0.463414	+	0.886142i	$0.653376\pi$
$564$	0	0
$565$	18.4243	0.775115
$566$	11.5071	0.483680
$567$	0	0
$568$	9.68004	0.406166
$569$	28.7588	1.20563	0.602815	−	0.797881i	$-0.294046\pi$
$569$	28.7588	1.20563	0.602815	+	0.797881i	$0.294046\pi$
$570$	0	0
$571$	7.30365	0.305648	0.152824	−	0.988253i	$-0.451163\pi$
$571$	7.30365	0.305648	0.152824	+	0.988253i	$0.451163\pi$
$572$	−7.20708	−0.301343
$573$	0	0
$574$	6.90941	0.288393
$575$	−4.29355	−0.179053
$576$	0	0
$577$	31.5090	1.31174	0.655868	−	0.754876i	$-0.272303\pi$
$577$	31.5090	1.31174	0.655868	+	0.754876i	$0.272303\pi$
$578$	20.2581	0.842628
$579$	0	0
$580$	1.41653	0.0588181
$581$	6.10607	0.253322
$582$	0	0
$583$	−50.5586	−2.09392
$584$	−4.70233	−0.194584
$585$	0	0
$586$	−22.6236	−0.934573
$587$	−27.2891	−1.12634	−0.563171	−	0.826340i	$-0.690419\pi$
$587$	−27.2891	−1.12634	−0.563171	+	0.826340i	$0.690419\pi$
$588$	0	0
$589$	0	0
$590$	14.8007	0.609334
$591$	0	0
$592$	27.9290	1.14788
$593$	−34.7716	−1.42790	−0.713948	−	0.700198i	$-0.753095\pi$
$593$	−34.7716	−1.42790	−0.713948	+	0.700198i	$0.753095\pi$
$594$	0	0
$595$	−2.61856	−0.107350
$596$	−2.10782	−0.0863397
$597$	0	0
$598$	−18.9409	−0.774550
$599$	15.8990	0.649615	0.324807	−	0.945780i	$-0.394700\pi$
$599$	15.8990	0.649615	0.324807	+	0.945780i	$0.394700\pi$
$600$	0	0
$601$	24.6928	1.00724	0.503621	−	0.863925i	$-0.332001\pi$
$601$	24.6928	1.00724	0.503621	+	0.863925i	$0.332001\pi$
$602$	0.144334	0.00588261
$603$	0	0
$604$	0.0564370	0.00229639
$605$	38.1634	1.55156
$606$	0	0
$607$	−4.01455	−0.162945	−0.0814727	−	0.996676i	$-0.525962\pi$
$607$	−4.01455	−0.162945	−0.0814727	+	0.996676i	$0.525962\pi$
$608$	0	0
$609$	0	0
$610$	54.0360	2.18785
$611$	30.7229	1.24291
$612$	0	0
$613$	−40.2172	−1.62436	−0.812178	−	0.583409i	$-0.801718\pi$
$613$	−40.2172	−1.62436	−0.812178	+	0.583409i	$0.801718\pi$
$614$	8.44562	0.340838
$615$	0	0
$616$	6.87939	0.277178
$617$	22.1803	0.892947	0.446474	−	0.894797i	$-0.352680\pi$
$617$	22.1803	0.892947	0.446474	+	0.894797i	$0.352680\pi$
$618$	0	0
$619$	−23.2354	−0.933908	−0.466954	−	0.884282i	$-0.654649\pi$
$619$	−23.2354	−0.933908	−0.466954	+	0.884282i	$0.654649\pi$
$620$	−8.68779	−0.348910
$621$	0	0
$622$	26.6578	1.06888
$623$	−3.15207	−0.126285
$624$	0	0
$625$	−30.0651	−1.20260
$626$	−24.4270	−0.976298
$627$	0	0
$628$	2.31315	0.0923047
$629$	11.8675	0.473189
$630$	0	0
$631$	28.0523	1.11675	0.558373	−	0.829590i	$-0.311426\pi$
$631$	28.0523	1.11675	0.558373	+	0.829590i	$0.311426\pi$
$632$	−26.6313	−1.05934
$633$	0	0
$634$	−3.02372	−0.120087
$635$	−4.63310	−0.183859
$636$	0	0
$637$	27.2986	1.08161
$638$	−12.6013	−0.498891
$639$	0	0
$640$	−33.7793	−1.33524
$641$	43.4789	1.71732	0.858658	−	0.512550i	$-0.171299\pi$
$641$	43.4789	1.71732	0.858658	+	0.512550i	$0.171299\pi$
$642$	0	0
$643$	39.3509	1.55185	0.775924	−	0.630826i	$-0.217284\pi$
$643$	39.3509	1.55185	0.775924	+	0.630826i	$0.217284\pi$
$644$	−0.562118	−0.0221506
$645$	0	0
$646$	0	0
$647$	−18.8862	−0.742493	−0.371246	−	0.928534i	$-0.621069\pi$
$647$	−18.8862	−0.742493	−0.371246	+	0.928534i	$0.621069\pi$
$648$	0	0
$649$	−19.4807	−0.764685
$650$	8.78880	0.344725
$651$	0	0
$652$	−0.155697	−0.00609755
$653$	46.7428	1.82919	0.914593	−	0.404375i	$-0.132511\pi$
$653$	46.7428	1.82919	0.914593	+	0.404375i	$0.132511\pi$
$654$	0	0
$655$	22.2472	0.869271
$656$	38.7674	1.51361
$657$	0	0
$658$	6.16250	0.240239
$659$	2.08915	0.0813819	0.0406910	−	0.999172i	$-0.487044\pi$
$659$	2.08915	0.0813819	0.0406910	+	0.999172i	$0.487044\pi$
$660$	0	0
$661$	12.4483	0.484183	0.242092	−	0.970253i	$-0.422167\pi$
$661$	12.4483	0.484183	0.242092	+	0.970253i	$0.422167\pi$
$662$	−1.28312	−0.0498698
$663$	0	0
$664$	29.0574	1.12764
$665$	0	0
$666$	0	0
$667$	−4.89992	−0.189726
$668$	1.17705	0.0455415
$669$	0	0
$670$	−39.6168	−1.53053
$671$	−71.1225	−2.74565
$672$	0	0
$673$	10.8212	0.417126	0.208563	−	0.978009i	$-0.433121\pi$
$673$	10.8212	0.417126	0.208563	+	0.978009i	$0.433121\pi$
$674$	−5.09833	−0.196380
$675$	0	0
$676$	1.22163	0.0469857
$677$	4.36184	0.167639	0.0838196	−	0.996481i	$-0.473288\pi$
$677$	4.36184	0.167639	0.0838196	+	0.996481i	$0.473288\pi$
$678$	0	0
$679$	3.61856	0.138867
$680$	−12.4611	−0.477862
$681$	0	0
$682$	77.2859	2.95943
$683$	−44.6441	−1.70826	−0.854130	−	0.520059i	$-0.825910\pi$
$683$	−44.6441	−1.70826	−0.854130	+	0.520059i	$0.825910\pi$
$684$	0	0
$685$	31.9813	1.22194
$686$	11.1821	0.426935
$687$	0	0
$688$	0.809831	0.0308745
$689$	40.2422	1.53310
$690$	0	0
$691$	24.3209	0.925210	0.462605	−	0.886564i	$-0.346915\pi$
$691$	24.3209	0.925210	0.462605	+	0.886564i	$0.346915\pi$
$692$	−0.561185	−0.0213330
$693$	0	0
$694$	6.14022	0.233079
$695$	−10.2540	−0.388957
$696$	0	0
$697$	16.4730	0.623958
$698$	42.7493	1.61808
$699$	0	0
$700$	0.260830	0.00985844
$701$	23.5594	0.889827	0.444914	−	0.895573i	$-0.353234\pi$
$701$	23.5594	0.889827	0.444914	+	0.895573i	$0.353234\pi$
$702$	0	0
$703$	0	0
$704$	31.5057	1.18742
$705$	0	0
$706$	19.8307	0.746338
$707$	4.66044	0.175274
$708$	0	0
$709$	−6.15333	−0.231093	−0.115547	−	0.993302i	$-0.536862\pi$
$709$	−6.15333	−0.231093	−0.115547	+	0.993302i	$0.536862\pi$
$710$	−14.8307	−0.556586
$711$	0	0
$712$	−15.0000	−0.562149
$713$	30.0520	1.12546
$714$	0	0
$715$	−52.5458	−1.96510
$716$	0.734118	0.0274353
$717$	0	0
$718$	17.5371	0.654480
$719$	−32.5749	−1.21484	−0.607420	−	0.794381i	$-0.707795\pi$
$719$	−32.5749	−1.21484	−0.607420	+	0.794381i	$0.707795\pi$
$720$	0	0
$721$	2.04601	0.0761973
$722$	0	0
$723$	0	0
$724$	−3.03952	−0.112963
$725$	2.27362	0.0844402
$726$	0	0
$727$	32.2300	1.19534	0.597672	−	0.801741i	$-0.296093\pi$
$727$	32.2300	1.19534	0.597672	+	0.801741i	$0.296093\pi$
$728$	−5.47565	−0.202941
$729$	0	0
$730$	7.20439	0.266647
$731$	0.344111	0.0127274
$732$	0	0
$733$	9.78106	0.361272	0.180636	−	0.983550i	$-0.442184\pi$
$733$	9.78106	0.361272	0.180636	+	0.983550i	$0.442184\pi$
$734$	43.8198	1.61742
$735$	0	0
$736$	−5.91210	−0.217923
$737$	52.1438	1.92074
$738$	0	0
$739$	27.7743	1.02169	0.510846	−	0.859672i	$-0.329332\pi$
$739$	27.7743	1.02169	0.510846	+	0.859672i	$0.329332\pi$
$740$	−5.36959	−0.197390
$741$	0	0
$742$	8.07192	0.296329
$743$	−32.3022	−1.18505	−0.592527	−	0.805551i	$-0.701870\pi$
$743$	−32.3022	−1.18505	−0.592527	+	0.805551i	$0.701870\pi$
$744$	0	0
$745$	−15.3678	−0.563034
$746$	−19.2526	−0.704887
$747$	0	0
$748$	−3.44656	−0.126019
$749$	−3.01960	−0.110334
$750$	0	0
$751$	−3.48515	−0.127175	−0.0635874	−	0.997976i	$-0.520254\pi$
$751$	−3.48515	−0.127175	−0.0635874	+	0.997976i	$0.520254\pi$
$752$	34.5767	1.26088
$753$	0	0
$754$	10.0300	0.365272
$755$	0.411474	0.0149751
$756$	0	0
$757$	−36.9341	−1.34239	−0.671196	−	0.741280i	$-0.734219\pi$
$757$	−36.9341	−1.34239	−0.671196	+	0.741280i	$0.734219\pi$
$758$	−3.36959	−0.122389
$759$	0	0
$760$	0	0
$761$	−31.3429	−1.13618	−0.568089	−	0.822967i	$-0.692317\pi$
$761$	−31.3429	−1.13618	−0.568089	+	0.822967i	$0.692317\pi$
$762$	0	0
$763$	1.11019	0.0401914
$764$	6.62361	0.239634
$765$	0	0
$766$	9.08235	0.328159
$767$	15.5057	0.559878
$768$	0	0
$769$	−31.7425	−1.14466	−0.572331	−	0.820022i	$-0.693961\pi$
$769$	−31.7425	−1.14466	−0.572331	+	0.820022i	$0.693961\pi$
$770$	−10.5398	−0.379829
$771$	0	0
$772$	8.27807	0.297934
$773$	10.9572	0.394102	0.197051	−	0.980393i	$-0.436863\pi$
$773$	10.9572	0.394102	0.197051	+	0.980393i	$0.436863\pi$
$774$	0	0
$775$	−13.9445	−0.500901
$776$	17.2199	0.618157
$777$	0	0
$778$	−50.7410	−1.81915
$779$	0	0
$780$	0	0
$781$	19.5202	0.698489
$782$	−9.05787	−0.323909
$783$	0	0
$784$	30.7229	1.09725
$785$	16.8648	0.601932
$786$	0	0
$787$	−46.3542	−1.65235	−0.826175	−	0.563414i	$-0.809488\pi$
$787$	−46.3542	−1.65235	−0.826175	+	0.563414i	$0.809488\pi$
$788$	1.52435	0.0543027
$789$	0	0
$790$	40.8016	1.45166
$791$	3.87164	0.137660
$792$	0	0
$793$	56.6100	2.01028
$794$	39.9103	1.41637
$795$	0	0
$796$	−6.95904	−0.246657
$797$	−3.47296	−0.123019	−0.0615093	−	0.998107i	$-0.519591\pi$
$797$	−3.47296	−0.123019	−0.0615093	+	0.998107i	$0.519591\pi$
$798$	0	0
$799$	14.6922	0.519774
$800$	2.74329	0.0969899
$801$	0	0
$802$	24.4647	0.863880
$803$	−9.48246	−0.334629
$804$	0	0
$805$	−4.09833	−0.144447
$806$	−61.5158	−2.16680
$807$	0	0
$808$	22.1780	0.780219
$809$	51.9495	1.82645	0.913224	−	0.407457i	$-0.133584\pi$
$809$	51.9495	1.82645	0.913224	+	0.407457i	$0.133584\pi$
$810$	0	0
$811$	−16.7769	−0.589118	−0.294559	−	0.955633i	$-0.595173\pi$
$811$	−16.7769	−0.589118	−0.294559	+	0.955633i	$0.595173\pi$
$812$	0.297667	0.0104460
$813$	0	0
$814$	47.7674	1.67425
$815$	−1.13516	−0.0397630
$816$	0	0
$817$	0	0
$818$	−4.51754	−0.157952
$819$	0	0
$820$	−7.45336	−0.260283
$821$	37.2472	1.29994	0.649968	−	0.759961i	$-0.274782\pi$
$821$	37.2472	1.29994	0.649968	+	0.759961i	$0.274782\pi$
$822$	0	0
$823$	−4.39961	−0.153361	−0.0766805	−	0.997056i	$-0.524432\pi$
$823$	−4.39961	−0.153361	−0.0766805	+	0.997056i	$0.524432\pi$
$824$	9.73648	0.339186
$825$	0	0
$826$	3.11019	0.108217
$827$	−13.8625	−0.482045	−0.241023	−	0.970519i	$-0.577483\pi$
$827$	−13.8625	−0.482045	−0.241023	+	0.970519i	$0.577483\pi$
$828$	0	0
$829$	−16.3770	−0.568797	−0.284398	−	0.958706i	$-0.591794\pi$
$829$	−16.3770	−0.568797	−0.284398	+	0.958706i	$0.591794\pi$
$830$	−44.5185	−1.54526
$831$	0	0
$832$	−25.0770	−0.869388
$833$	13.0547	0.452318
$834$	0	0
$835$	8.58172	0.296983
$836$	0	0
$837$	0	0
$838$	−51.3191	−1.77279
$839$	−38.2891	−1.32189	−0.660943	−	0.750436i	$-0.729844\pi$
$839$	−38.2891	−1.32189	−0.660943	+	0.750436i	$0.729844\pi$
$840$	0	0
$841$	−26.4053	−0.910527
$842$	27.6049	0.951329
$843$	0	0
$844$	−9.44387	−0.325071
$845$	8.90673	0.306401
$846$	0	0
$847$	8.01960	0.275557
$848$	45.2900	1.55527
$849$	0	0
$850$	4.20296	0.144160
$851$	18.5740	0.636708
$852$	0	0
$853$	22.0901	0.756350	0.378175	−	0.925734i	$-0.376552\pi$
$853$	22.0901	0.756350	0.378175	+	0.925734i	$0.376552\pi$
$854$	11.3550	0.388561
$855$	0	0
$856$	−14.3696	−0.491142
$857$	45.0077	1.53744	0.768718	−	0.639588i	$-0.220895\pi$
$857$	45.0077	1.53744	0.768718	+	0.639588i	$0.220895\pi$
$858$	0	0
$859$	−15.9727	−0.544980	−0.272490	−	0.962159i	$-0.587847\pi$
$859$	−15.9727	−0.544980	−0.272490	+	0.962159i	$0.587847\pi$
$860$	−0.155697	−0.00530921
$861$	0	0
$862$	−32.6655	−1.11259
$863$	41.2576	1.40443	0.702213	−	0.711967i	$-0.252195\pi$
$863$	41.2576	1.40443	0.702213	+	0.711967i	$0.252195\pi$
$864$	0	0
$865$	−4.09152	−0.139116
$866$	−33.7861	−1.14810
$867$	0	0
$868$	−1.82564	−0.0619661
$869$	−53.7033	−1.82176
$870$	0	0
$871$	−41.5039	−1.40631
$872$	5.28312	0.178909
$873$	0	0
$874$	0	0
$875$	−4.83481	−0.163446
$876$	0	0
$877$	−46.1652	−1.55889	−0.779444	−	0.626472i	$-0.784498\pi$
$877$	−46.1652	−1.55889	−0.779444	+	0.626472i	$0.784498\pi$
$878$	−5.18891	−0.175117
$879$	0	0
$880$	−59.1370	−1.99351
$881$	−18.9581	−0.638715	−0.319357	−	0.947634i	$-0.603467\pi$
$881$	−18.9581	−0.638715	−0.319357	+	0.947634i	$0.603467\pi$
$882$	0	0
$883$	35.0496	1.17951	0.589757	−	0.807581i	$-0.299224\pi$
$883$	35.0496	1.17951	0.589757	+	0.807581i	$0.299224\pi$
$884$	2.74329	0.0922668
$885$	0	0
$886$	−46.2145	−1.55261
$887$	−17.8375	−0.598925	−0.299462	−	0.954108i	$-0.596807\pi$
$887$	−17.8375	−0.598925	−0.299462	+	0.954108i	$0.596807\pi$
$888$	0	0
$889$	−0.973593	−0.0326532
$890$	22.9813	0.770336
$891$	0	0
$892$	4.17200	0.139689
$893$	0	0
$894$	0	0
$895$	5.35235	0.178909
$896$	−7.09833	−0.237138
$897$	0	0
$898$	47.3596	1.58041
$899$	−15.9139	−0.530757
$900$	0	0
$901$	19.2445	0.641128
$902$	66.3046	2.20770
$903$	0	0
$904$	18.4243	0.612782
$905$	−22.1607	−0.736648
$906$	0	0
$907$	14.2513	0.473208	0.236604	−	0.971606i	$-0.423966\pi$
$907$	14.2513	0.473208	0.236604	+	0.971606i	$0.423966\pi$
$908$	1.50744	0.0500260
$909$	0	0
$910$	8.38919	0.278099
$911$	−16.4466	−0.544899	−0.272449	−	0.962170i	$-0.587834\pi$
$911$	−16.4466	−0.544899	−0.272449	+	0.962170i	$0.587834\pi$
$912$	0	0
$913$	58.5954	1.93923
$914$	59.8248	1.97883
$915$	0	0
$916$	1.14466	0.0378206
$917$	4.67499	0.154382
$918$	0	0
$919$	−33.8999	−1.11826	−0.559128	−	0.829082i	$-0.688864\pi$
$919$	−33.8999	−1.11826	−0.559128	+	0.829082i	$0.688864\pi$
$920$	−19.5030	−0.642995
$921$	0	0
$922$	−2.30953	−0.0760602
$923$	−15.5371	−0.511411
$924$	0	0
$925$	−8.61856	−0.283376
$926$	5.03003	0.165297
$927$	0	0
$928$	3.13072	0.102771
$929$	−21.5345	−0.706522	−0.353261	−	0.935525i	$-0.614927\pi$
$929$	−21.5345	−0.706522	−0.353261	+	0.935525i	$0.614927\pi$
$930$	0	0
$931$	0	0
$932$	5.35916	0.175545
$933$	0	0
$934$	−35.0651	−1.14737
$935$	−25.1284	−0.821785
$936$	0	0
$937$	7.56624	0.247178	0.123589	−	0.992333i	$-0.460560\pi$
$937$	7.56624	0.247178	0.123589	+	0.992333i	$0.460560\pi$
$938$	−8.32501	−0.271821
$939$	0	0
$940$	−6.64765	−0.216822
$941$	19.1275	0.623540	0.311770	−	0.950158i	$-0.399078\pi$
$941$	19.1275	0.623540	0.311770	+	0.950158i	$0.399078\pi$
$942$	0	0
$943$	25.7820	0.839577
$944$	17.4507	0.567971
$945$	0	0
$946$	1.38507	0.0450324
$947$	−60.2072	−1.95647	−0.978235	−	0.207498i	$-0.933468\pi$
$947$	−60.2072	−1.95647	−0.978235	+	0.207498i	$0.933468\pi$
$948$	0	0
$949$	7.54757	0.245005
$950$	0	0
$951$	0	0
$952$	−2.61856	−0.0848679
$953$	−19.7478	−0.639695	−0.319848	−	0.947469i	$-0.603632\pi$
$953$	−19.7478	−0.639695	−0.319848	+	0.947469i	$0.603632\pi$
$954$	0	0
$955$	48.2918	1.56269
$956$	−5.49525	−0.177729
$957$	0	0
$958$	−14.7965	−0.478055
$959$	6.72050	0.217016
$960$	0	0
$961$	66.6023	2.14846
$962$	−38.0205	−1.22583
$963$	0	0
$964$	8.21213	0.264495
$965$	60.3542	1.94287
$966$	0	0
$967$	−5.16849	−0.166207	−0.0831037	−	0.996541i	$-0.526483\pi$
$967$	−5.16849	−0.166207	−0.0831037	+	0.996541i	$0.526483\pi$
$968$	38.1634	1.22662
$969$	0	0
$970$	−26.3824	−0.847087
$971$	−8.75784	−0.281052	−0.140526	−	0.990077i	$-0.544879\pi$
$971$	−8.75784	−0.281052	−0.140526	+	0.990077i	$0.544879\pi$
$972$	0	0
$973$	−2.15476	−0.0690785
$974$	−34.4884	−1.10508
$975$	0	0
$976$	63.7110	2.03934
$977$	40.7279	1.30300	0.651501	−	0.758648i	$-0.274140\pi$
$977$	40.7279	1.30300	0.651501	+	0.758648i	$0.274140\pi$
$978$	0	0
$979$	−30.2481	−0.966735
$980$	−5.90673	−0.188683
$981$	0	0
$982$	11.1402	0.355499
$983$	3.95037	0.125997	0.0629986	−	0.998014i	$-0.479934\pi$
$983$	3.95037	0.125997	0.0629986	+	0.998014i	$0.479934\pi$
$984$	0	0
$985$	11.1138	0.354115
$986$	4.79654	0.152753
$987$	0	0
$988$	0	0
$989$	0.538572	0.0171256
$990$	0	0
$991$	18.5621	0.589645	0.294823	−	0.955552i	$-0.404739\pi$
$991$	18.5621	0.589645	0.294823	+	0.955552i	$0.404739\pi$
$992$	−19.2012	−0.609639
$993$	0	0
$994$	−3.11650	−0.0988492
$995$	−50.7374	−1.60848
$996$	0	0
$997$	2.73917	0.0867504	0.0433752	−	0.999059i	$-0.486189\pi$
$997$	2.73917	0.0867504	0.0433752	+	0.999059i	$0.486189\pi$
$998$	19.6277	0.621305
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.x.1.1		3
3.2	odd	2		1083.2.a.n.1.3		3
19.2	odd	18		171.2.u.a.118.1		6
19.10	odd	18		171.2.u.a.100.1		6
19.18	odd	2		3249.2.a.w.1.3		3
57.2	even	18		57.2.i.a.4.1	✓	6
57.29	even	18		57.2.i.a.43.1	yes	6
57.56	even	2		1083.2.a.m.1.1		3
228.59	odd	18		912.2.bo.b.289.1		6
228.143	odd	18		912.2.bo.b.385.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.i.a.4.1	✓	6	57.2	even	18
57.2.i.a.43.1	yes	6	57.29	even	18
171.2.u.a.100.1		6	19.10	odd	18
171.2.u.a.118.1		6	19.2	odd	18
912.2.bo.b.289.1		6	228.59	odd	18
912.2.bo.b.385.1		6	228.143	odd	18
1083.2.a.m.1.1		3	57.56	even	2
1083.2.a.n.1.3		3	3.2	odd	2
3249.2.a.w.1.3		3	19.18	odd	2
3249.2.a.x.1.1		3	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:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53209 q^{2} +0.347296 q^{4} +2.53209 q^{5} +0.532089 q^{7} +2.53209 q^{8} -3.87939 q^{10} +5.10607 q^{11} -4.06418 q^{13} -0.815207 q^{14} -4.57398 q^{16} -1.94356 q^{17} +0.879385 q^{20} -7.82295 q^{22} -3.04189 q^{23} +1.41147 q^{25} +6.22668 q^{26} +0.184793 q^{28} +1.61081 q^{29} -9.87939 q^{31} +1.94356 q^{32} +2.97771 q^{34} +1.34730 q^{35} -6.10607 q^{37} +6.41147 q^{40} -8.47565 q^{41} -0.177052 q^{43} +1.77332 q^{44} +4.66044 q^{46} -7.55943 q^{47} -6.71688 q^{49} -2.16250 q^{50} -1.41147 q^{52} -9.90167 q^{53} +12.9290 q^{55} +1.34730 q^{56} -2.46791 q^{58} -3.81521 q^{59} -13.9290 q^{61} +15.1361 q^{62} +6.17024 q^{64} -10.2909 q^{65} +10.2121 q^{67} -0.674992 q^{68} -2.06418 q^{70} +3.82295 q^{71} -1.85710 q^{73} +9.35504 q^{74} +2.71688 q^{77} -10.5175 q^{79} -11.5817 q^{80} +12.9855 q^{82} +11.4757 q^{83} -4.92127 q^{85} +0.271259 q^{86} +12.9290 q^{88} -5.92396 q^{89} -2.16250 q^{91} -1.05644 q^{92} +11.5817 q^{94} +6.80066 q^{97} +10.2909 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7} + 3 q^{8} - 6 q^{10} + 3 q^{11} - 3 q^{13} - 6 q^{14} - 6 q^{16} + 9 q^{17} - 3 q^{20} - 3 q^{22} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} + 9 q^{29} - 24 q^{31} - 9 q^{32} + 15 q^{34}+ \cdots + 15 q^{98}+O(q^{100}) \) 3 * q + 3 * q^5 - 3 * q^7 + 3 * q^8 - 6 * q^10 + 3 * q^11 - 3 * q^13 - 6 * q^14 - 6 * q^16 + 9 * q^17 - 3 * q^20 - 3 * q^22 - 6 * q^23 - 6 * q^25 + 12 * q^26 - 3 * q^28 + 9 * q^29 - 24 * q^31 - 9 * q^32 + 15 * q^34 + 3 * q^35 - 6 * q^37 + 9 * q^40 - 6 * q^41 - 21 * q^43 + 12 * q^44 - 9 * q^46 + 3 * q^47 - 12 * q^49 - 9 * q^50 + 6 * q^52 - 18 * q^53 + 6 * q^55 + 3 * q^56 - 12 * q^58 - 15 * q^59 - 9 * q^61 + 3 * q^62 - 3 * q^64 - 15 * q^65 + 6 * q^67 + 3 * q^68 + 3 * q^70 - 9 * q^71 - 6 * q^73 + 3 * q^74 - 9 * q^79 - 3 * q^80 + 21 * q^82 + 15 * q^83 - 6 * q^85 - 18 * q^86 + 6 * q^88 - 9 * q^91 - 18 * q^92 + 3 * q^94 + 6 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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