LMFDB - Embedded newform 3249.2.a.z.1.2

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:	$0$
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 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	$q+1.34730 q^{2} -0.184793 q^{4} -0.879385 q^{5} +0.347296 q^{7} -2.94356 q^{8} +O(q^{10})$	$q+1.34730 q^{2} -0.184793 q^{4} -0.879385 q^{5} +0.347296 q^{7} -2.94356 q^{8} -1.18479 q^{10} +2.22668 q^{11} +2.57398 q^{13} +0.467911 q^{14} -3.59627 q^{16} -0.467911 q^{17} +0.162504 q^{20} +3.00000 q^{22} +2.69459 q^{23} -4.22668 q^{25} +3.46791 q^{26} -0.0641778 q^{28} +6.87939 q^{29} -7.10607 q^{31} +1.04189 q^{32} -0.630415 q^{34} -0.305407 q^{35} +4.94356 q^{37} +2.58853 q^{40} -2.47565 q^{41} +3.90167 q^{43} -0.411474 q^{44} +3.63041 q^{46} +7.29086 q^{47} -6.87939 q^{49} -5.69459 q^{50} -0.475652 q^{52} +2.83750 q^{53} -1.95811 q^{55} -1.02229 q^{56} +9.26857 q^{58} +6.30541 q^{59} +9.12836 q^{61} -9.57398 q^{62} +8.59627 q^{64} -2.26352 q^{65} +7.67499 q^{67} +0.0864665 q^{68} -0.411474 q^{70} +9.30541 q^{71} +1.38919 q^{73} +6.66044 q^{74} +0.773318 q^{77} -11.8452 q^{79} +3.16250 q^{80} -3.33544 q^{82} +14.8307 q^{83} +0.411474 q^{85} +5.25671 q^{86} -6.55438 q^{88} +10.2909 q^{89} +0.893933 q^{91} -0.497941 q^{92} +9.82295 q^{94} +9.45336 q^{97} -9.26857 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} + 6 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 + 6 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} + 6 q^{8} + 6 q^{14} + 3 q^{16} - 6 q^{17} + 3 q^{20} + 9 q^{22} + 6 q^{23} - 6 q^{25} + 15 q^{26} + 9 q^{28} + 15 q^{29} - 9 q^{31} - 9 q^{34} - 3 q^{35} + 18 q^{40} + 12 q^{41} + 9 q^{44} + 18 q^{46} + 6 q^{47} - 15 q^{49} - 15 q^{50} + 18 q^{52} + 6 q^{53} - 9 q^{55} + 3 q^{56} + 18 q^{58} + 21 q^{59} + 9 q^{61} - 21 q^{62} + 12 q^{64} - 12 q^{65} + 18 q^{67} - 15 q^{68} + 9 q^{70} + 30 q^{71} - 3 q^{74} + 9 q^{77} - 9 q^{79} + 12 q^{80} + 18 q^{82} - 9 q^{85} - 21 q^{86} - 9 q^{88} + 15 q^{89} + 15 q^{91} + 24 q^{92} + 9 q^{94} + 15 q^{97} - 18 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 + 6 * q^8 + 6 * q^14 + 3 * q^16 - 6 * q^17 + 3 * q^20 + 9 * q^22 + 6 * q^23 - 6 * q^25 + 15 * q^26 + 9 * q^28 + 15 * q^29 - 9 * q^31 - 9 * q^34 - 3 * q^35 + 18 * q^40 + 12 * q^41 + 9 * q^44 + 18 * q^46 + 6 * q^47 - 15 * q^49 - 15 * q^50 + 18 * q^52 + 6 * q^53 - 9 * q^55 + 3 * q^56 + 18 * q^58 + 21 * q^59 + 9 * q^61 - 21 * q^62 + 12 * q^64 - 12 * q^65 + 18 * q^67 - 15 * q^68 + 9 * q^70 + 30 * q^71 - 3 * q^74 + 9 * q^77 - 9 * q^79 + 12 * q^80 + 18 * q^82 - 9 * q^85 - 21 * q^86 - 9 * q^88 + 15 * q^89 + 15 * q^91 + 24 * q^92 + 9 * q^94 + 15 * q^97 - 18 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.34730	0.952682	0.476341	−	0.879261i	$-0.341963\pi$
$2$	1.34730	0.952682	0.476341	+	0.879261i	$0.341963\pi$
$3$	0	0
$3$	0	0
$4$	−0.184793	−0.0923963
$5$	−0.879385	−0.393273	−0.196637	−	0.980476i	$-0.563002\pi$
$5$	−0.879385	−0.393273	−0.196637	+	0.980476i	$0.563002\pi$
$6$	0	0
$7$	0.347296	0.131266	0.0656328	−	0.997844i	$-0.479093\pi$
$7$	0.347296	0.131266	0.0656328	+	0.997844i	$0.479093\pi$
$8$	−2.94356	−1.04071
$9$	0	0
$10$	−1.18479	−0.374664
$11$	2.22668	0.671370	0.335685	−	0.941974i	$-0.391032\pi$
$11$	2.22668	0.671370	0.335685	+	0.941974i	$0.391032\pi$
$12$	0	0
$13$	2.57398	0.713893	0.356947	−	0.934125i	$-0.383818\pi$
$13$	2.57398	0.713893	0.356947	+	0.934125i	$0.383818\pi$
$14$	0.467911	0.125055
$15$	0	0
$16$	−3.59627	−0.899067
$17$	−0.467911	−0.113485	−0.0567426	−	0.998389i	$-0.518071\pi$
$17$	−0.467911	−0.113485	−0.0567426	+	0.998389i	$0.518071\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0.162504	0.0363370
$21$	0	0
$22$	3.00000	0.639602
$23$	2.69459	0.561861	0.280931	−	0.959728i	$-0.409357\pi$
$23$	2.69459	0.561861	0.280931	+	0.959728i	$0.409357\pi$
$24$	0	0
$25$	−4.22668	−0.845336
$26$	3.46791	0.680113
$27$	0	0
$28$	−0.0641778	−0.0121285
$29$	6.87939	1.27747	0.638735	−	0.769427i	$-0.279458\pi$
$29$	6.87939	1.27747	0.638735	+	0.769427i	$0.279458\pi$
$30$	0	0
$31$	−7.10607	−1.27629	−0.638144	−	0.769917i	$-0.720297\pi$
$31$	−7.10607	−1.27629	−0.638144	+	0.769917i	$0.720297\pi$
$32$	1.04189	0.184182
$33$	0	0
$34$	−0.630415	−0.108115
$35$	−0.305407	−0.0516233
$36$	0	0
$37$	4.94356	0.812717	0.406358	−	0.913714i	$-0.366798\pi$
$37$	4.94356	0.812717	0.406358	+	0.913714i	$0.366798\pi$
$38$	0	0
$39$	0	0
$40$	2.58853	0.409282
$41$	−2.47565	−0.386632	−0.193316	−	0.981137i	$-0.561924\pi$
$41$	−2.47565	−0.386632	−0.193316	+	0.981137i	$0.561924\pi$
$42$	0	0
$43$	3.90167	0.595000	0.297500	−	0.954722i	$-0.403847\pi$
$43$	3.90167	0.595000	0.297500	+	0.954722i	$0.403847\pi$
$44$	−0.411474	−0.0620321
$45$	0	0
$46$	3.63041	0.535275
$47$	7.29086	1.06348	0.531741	−	0.846907i	$-0.321538\pi$
$47$	7.29086	1.06348	0.531741	+	0.846907i	$0.321538\pi$
$48$	0	0
$49$	−6.87939	−0.982769
$50$	−5.69459	−0.805337
$51$	0	0
$52$	−0.475652	−0.0659611
$53$	2.83750	0.389760	0.194880	−	0.980827i	$-0.437568\pi$
$53$	2.83750	0.389760	0.194880	+	0.980827i	$0.437568\pi$
$54$	0	0
$55$	−1.95811	−0.264032
$56$	−1.02229	−0.136609
$57$	0	0
$58$	9.26857	1.21702
$59$	6.30541	0.820894	0.410447	−	0.911884i	$-0.365373\pi$
$59$	6.30541	0.820894	0.410447	+	0.911884i	$0.365373\pi$
$60$	0	0
$61$	9.12836	1.16877	0.584383	−	0.811478i	$-0.301337\pi$
$61$	9.12836	1.16877	0.584383	+	0.811478i	$0.301337\pi$
$62$	−9.57398	−1.21590
$63$	0	0
$64$	8.59627	1.07453
$65$	−2.26352	−0.280755
$66$	0	0
$67$	7.67499	0.937650	0.468825	−	0.883291i	$-0.344678\pi$
$67$	7.67499	0.937650	0.468825	+	0.883291i	$0.344678\pi$
$68$	0.0864665	0.0104856
$69$	0	0
$70$	−0.411474	−0.0491806
$71$	9.30541	1.10435	0.552174	−	0.833729i	$-0.313798\pi$
$71$	9.30541	1.10435	0.552174	+	0.833729i	$0.313798\pi$
$72$	0	0
$73$	1.38919	0.162592	0.0812959	−	0.996690i	$-0.474094\pi$
$73$	1.38919	0.162592	0.0812959	+	0.996690i	$0.474094\pi$
$74$	6.66044	0.774261
$75$	0	0
$76$	0	0
$77$	0.773318	0.0881278
$78$	0	0
$79$	−11.8452	−1.33269	−0.666347	−	0.745642i	$-0.732143\pi$
$79$	−11.8452	−1.33269	−0.666347	+	0.745642i	$0.732143\pi$
$80$	3.16250	0.353579
$81$	0	0
$82$	−3.33544	−0.368337
$83$	14.8307	1.62788	0.813940	−	0.580949i	$-0.197319\pi$
$83$	14.8307	1.62788	0.813940	+	0.580949i	$0.197319\pi$
$84$	0	0
$85$	0.411474	0.0446306
$86$	5.25671	0.566846
$87$	0	0
$88$	−6.55438	−0.698699
$89$	10.2909	1.09083	0.545414	−	0.838166i	$-0.316372\pi$
$89$	10.2909	1.09083	0.545414	+	0.838166i	$0.316372\pi$
$90$	0	0
$91$	0.893933	0.0937097
$92$	−0.497941	−0.0519139
$93$	0	0
$94$	9.82295	1.01316
$95$	0	0
$96$	0	0
$97$	9.45336	0.959844	0.479922	−	0.877311i	$-0.340665\pi$
$97$	9.45336	0.959844	0.479922	+	0.877311i	$0.340665\pi$
$98$	−9.26857	−0.936267
$99$	0	0
$100$	0.781059	0.0781059
$101$	9.24897	0.920307	0.460153	−	0.887839i	$-0.347794\pi$
$101$	9.24897	0.920307	0.460153	+	0.887839i	$0.347794\pi$
$102$	0	0
$103$	−5.50980	−0.542897	−0.271448	−	0.962453i	$-0.587503\pi$
$103$	−5.50980	−0.542897	−0.271448	+	0.962453i	$0.587503\pi$
$104$	−7.57667	−0.742953
$105$	0	0
$106$	3.82295	0.371318
$107$	−10.2344	−0.989399	−0.494699	−	0.869064i	$-0.664722\pi$
$107$	−10.2344	−0.989399	−0.494699	+	0.869064i	$0.664722\pi$
$108$	0	0
$109$	−1.82295	−0.174607	−0.0873034	−	0.996182i	$-0.527825\pi$
$109$	−1.82295	−0.174607	−0.0873034	+	0.996182i	$0.527825\pi$
$110$	−2.63816	−0.251538
$111$	0	0
$112$	−1.24897	−0.118017
$113$	17.6878	1.66393	0.831963	−	0.554830i	$-0.187217\pi$
$113$	17.6878	1.66393	0.831963	+	0.554830i	$0.187217\pi$
$114$	0	0
$115$	−2.36959	−0.220965
$116$	−1.27126	−0.118033
$117$	0	0
$118$	8.49525	0.782051
$119$	−0.162504	−0.0148967
$120$	0	0
$121$	−6.04189	−0.549263
$122$	12.2986	1.11346
$123$	0	0
$124$	1.31315	0.117924
$125$	8.11381	0.725721
$126$	0	0
$127$	11.6040	1.02969	0.514845	−	0.857284i	$-0.327850\pi$
$127$	11.6040	1.02969	0.514845	+	0.857284i	$0.327850\pi$
$128$	9.49794	0.839507
$129$	0	0
$130$	−3.04963	−0.267470
$131$	−1.84524	−0.161219	−0.0806096	−	0.996746i	$-0.525687\pi$
$131$	−1.84524	−0.161219	−0.0806096	+	0.996746i	$0.525687\pi$
$132$	0	0
$133$	0	0
$134$	10.3405	0.893282
$135$	0	0
$136$	1.37733	0.118105
$137$	−0.255777	−0.0218525	−0.0109263	−	0.999940i	$-0.503478\pi$
$137$	−0.255777	−0.0218525	−0.0109263	+	0.999940i	$0.503478\pi$
$138$	0	0
$139$	4.26352	0.361627	0.180813	−	0.983517i	$-0.442127\pi$
$139$	4.26352	0.361627	0.180813	+	0.983517i	$0.442127\pi$
$140$	0.0564370	0.00476980
$141$	0	0
$142$	12.5371	1.05209
$143$	5.73143	0.479286
$144$	0	0
$145$	−6.04963	−0.502394
$146$	1.87164	0.154898
$147$	0	0
$148$	−0.913534	−0.0750920
$149$	−16.5594	−1.35660	−0.678301	−	0.734784i	$-0.737283\pi$
$149$	−16.5594	−1.35660	−0.678301	+	0.734784i	$0.737283\pi$
$150$	0	0
$151$	−4.36184	−0.354962	−0.177481	−	0.984124i	$-0.556795\pi$
$151$	−4.36184	−0.354962	−0.177481	+	0.984124i	$0.556795\pi$
$152$	0	0
$153$	0	0
$154$	1.04189	0.0839578
$155$	6.24897	0.501929
$156$	0	0
$157$	−9.61856	−0.767644	−0.383822	−	0.923407i	$-0.625392\pi$
$157$	−9.61856	−0.767644	−0.383822	+	0.923407i	$0.625392\pi$
$158$	−15.9590	−1.26963
$159$	0	0
$160$	−0.916222	−0.0724337
$161$	0.935822	0.0737531
$162$	0	0
$163$	8.35504	0.654417	0.327209	−	0.944952i	$-0.393892\pi$
$163$	8.35504	0.654417	0.327209	+	0.944952i	$0.393892\pi$
$164$	0.457482	0.0357233
$165$	0	0
$166$	19.9813	1.55085
$167$	−4.03508	−0.312244	−0.156122	−	0.987738i	$-0.549899\pi$
$167$	−4.03508	−0.312244	−0.156122	+	0.987738i	$0.549899\pi$
$168$	0	0
$169$	−6.37464	−0.490357
$170$	0.554378	0.0425188
$171$	0	0
$172$	−0.721000	−0.0549758
$173$	20.1438	1.53151	0.765754	−	0.643134i	$-0.222366\pi$
$173$	20.1438	1.53151	0.765754	+	0.643134i	$0.222366\pi$
$174$	0	0
$175$	−1.46791	−0.110964
$176$	−8.00774	−0.603606
$177$	0	0
$178$	13.8648	1.03921
$179$	11.5125	0.860484	0.430242	−	0.902714i	$-0.358428\pi$
$179$	11.5125	0.860484	0.430242	+	0.902714i	$0.358428\pi$
$180$	0	0
$181$	8.53714	0.634561	0.317280	−	0.948332i	$-0.397230\pi$
$181$	8.53714	0.634561	0.317280	+	0.948332i	$0.397230\pi$
$182$	1.20439	0.0892755
$183$	0	0
$184$	−7.93170	−0.584733
$185$	−4.34730	−0.319620
$186$	0	0
$187$	−1.04189	−0.0761905
$188$	−1.34730	−0.0982617
$189$	0	0
$190$	0	0
$191$	−18.3354	−1.32671	−0.663353	−	0.748307i	$-0.730867\pi$
$191$	−18.3354	−1.32671	−0.663353	+	0.748307i	$0.730867\pi$
$192$	0	0
$193$	0.297667	0.0214265	0.0107133	−	0.999943i	$-0.496590\pi$
$193$	0.297667	0.0214265	0.0107133	+	0.999943i	$0.496590\pi$
$194$	12.7365	0.914426
$195$	0	0
$196$	1.27126	0.0908042
$197$	−13.1411	−0.936268	−0.468134	−	0.883657i	$-0.655074\pi$
$197$	−13.1411	−0.936268	−0.468134	+	0.883657i	$0.655074\pi$
$198$	0	0
$199$	−0.256711	−0.0181978	−0.00909888	−	0.999959i	$-0.502896\pi$
$199$	−0.256711	−0.0181978	−0.00909888	+	0.999959i	$0.502896\pi$
$200$	12.4415	0.879747
$201$	0	0
$202$	12.4611	0.876760
$203$	2.38919	0.167688
$204$	0	0
$205$	2.17705	0.152052
$206$	−7.42333	−0.517208
$207$	0	0
$208$	−9.25671	−0.641837
$209$	0	0
$210$	0	0
$211$	2.44831	0.168549	0.0842743	−	0.996443i	$-0.473143\pi$
$211$	2.44831	0.168549	0.0842743	+	0.996443i	$0.473143\pi$
$212$	−0.524348	−0.0360124
$213$	0	0
$214$	−13.7888	−0.942583
$215$	−3.43107	−0.233997
$216$	0	0
$217$	−2.46791	−0.167533
$218$	−2.45605	−0.166345
$219$	0	0
$220$	0.361844	0.0243955
$221$	−1.20439	−0.0810162
$222$	0	0
$223$	−8.50980	−0.569858	−0.284929	−	0.958549i	$-0.591970\pi$
$223$	−8.50980	−0.569858	−0.284929	+	0.958549i	$0.591970\pi$
$224$	0.361844	0.0241767
$225$	0	0
$226$	23.8307	1.58519
$227$	14.1506	0.939211	0.469606	−	0.882876i	$-0.344396\pi$
$227$	14.1506	0.939211	0.469606	+	0.882876i	$0.344396\pi$
$228$	0	0
$229$	−20.5330	−1.35686	−0.678430	−	0.734665i	$-0.737339\pi$
$229$	−20.5330	−1.35686	−0.678430	+	0.734665i	$0.737339\pi$
$230$	−3.19253	−0.210509
$231$	0	0
$232$	−20.2499	−1.32947
$233$	−17.6509	−1.15635	−0.578176	−	0.815912i	$-0.696235\pi$
$233$	−17.6509	−1.15635	−0.578176	+	0.815912i	$0.696235\pi$
$234$	0	0
$235$	−6.41147	−0.418238
$236$	−1.16519	−0.0758475
$237$	0	0
$238$	−0.218941	−0.0141918
$239$	−2.35235	−0.152161	−0.0760804	−	0.997102i	$-0.524241\pi$
$239$	−2.35235	−0.152161	−0.0760804	+	0.997102i	$0.524241\pi$
$240$	0	0
$241$	13.8007	0.888979	0.444489	−	0.895784i	$-0.353385\pi$
$241$	13.8007	0.888979	0.444489	+	0.895784i	$0.353385\pi$
$242$	−8.14022	−0.523273
$243$	0	0
$244$	−1.68685	−0.107990
$245$	6.04963	0.386497
$246$	0	0
$247$	0	0
$248$	20.9172	1.32824
$249$	0	0
$250$	10.9317	0.691382
$251$	4.16519	0.262905	0.131452	−	0.991322i	$-0.458036\pi$
$251$	4.16519	0.262905	0.131452	+	0.991322i	$0.458036\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	15.6340	0.980967
$255$	0	0
$256$	−4.39599	−0.274750
$257$	−0.667252	−0.0416220	−0.0208110	−	0.999783i	$-0.506625\pi$
$257$	−0.667252	−0.0416220	−0.0208110	+	0.999783i	$0.506625\pi$
$258$	0	0
$259$	1.71688	0.106682
$260$	0.418281	0.0259407
$261$	0	0
$262$	−2.48608	−0.153591
$263$	−11.3996	−0.702930	−0.351465	−	0.936201i	$-0.614316\pi$
$263$	−11.3996	−0.702930	−0.351465	+	0.936201i	$0.614316\pi$
$264$	0	0
$265$	−2.49525	−0.153282
$266$	0	0
$267$	0	0
$268$	−1.41828	−0.0866353
$269$	−19.3901	−1.18224	−0.591118	−	0.806585i	$-0.701313\pi$
$269$	−19.3901	−1.18224	−0.591118	+	0.806585i	$0.701313\pi$
$270$	0	0
$271$	−13.3942	−0.813642	−0.406821	−	0.913508i	$-0.633363\pi$
$271$	−13.3942	−0.813642	−0.406821	+	0.913508i	$0.633363\pi$
$272$	1.68273	0.102031
$273$	0	0
$274$	−0.344608	−0.0208185
$275$	−9.41147	−0.567533
$276$	0	0
$277$	17.7469	1.06631	0.533154	−	0.846018i	$-0.321007\pi$
$277$	17.7469	1.06631	0.533154	+	0.846018i	$0.321007\pi$
$278$	5.74422	0.344516
$279$	0	0
$280$	0.898986	0.0537247
$281$	−18.2790	−1.09043	−0.545217	−	0.838295i	$-0.683553\pi$
$281$	−18.2790	−1.09043	−0.545217	+	0.838295i	$0.683553\pi$
$282$	0	0
$283$	7.68779	0.456991	0.228496	−	0.973545i	$-0.426619\pi$
$283$	7.68779	0.456991	0.228496	+	0.973545i	$0.426619\pi$
$284$	−1.71957	−0.102038
$285$	0	0
$286$	7.72193	0.456608
$287$	−0.859785	−0.0507515
$288$	0	0
$289$	−16.7811	−0.987121
$290$	−8.15064	−0.478622
$291$	0	0
$292$	−0.256711	−0.0150229
$293$	10.5030	0.613591	0.306796	−	0.951775i	$-0.400743\pi$
$293$	10.5030	0.613591	0.306796	+	0.951775i	$0.400743\pi$
$294$	0	0
$295$	−5.54488	−0.322836
$296$	−14.5517	−0.845800
$297$	0	0
$298$	−22.3105	−1.29241
$299$	6.93582	0.401109
$300$	0	0
$301$	1.35504	0.0781030
$302$	−5.87670	−0.338166
$303$	0	0
$304$	0	0
$305$	−8.02734	−0.459644
$306$	0	0
$307$	−11.6955	−0.667499	−0.333749	−	0.942662i	$-0.608314\pi$
$307$	−11.6955	−0.667499	−0.333749	+	0.942662i	$0.608314\pi$
$308$	−0.142903	−0.00814268
$309$	0	0
$310$	8.41921	0.478179
$311$	−15.9659	−0.905340	−0.452670	−	0.891678i	$-0.649529\pi$
$311$	−15.9659	−0.905340	−0.452670	+	0.891678i	$0.649529\pi$
$312$	0	0
$313$	−26.6287	−1.50514	−0.752570	−	0.658512i	$-0.771186\pi$
$313$	−26.6287	−1.50514	−0.752570	+	0.658512i	$0.771186\pi$
$314$	−12.9590	−0.731321
$315$	0	0
$316$	2.18891	0.123136
$317$	−29.5321	−1.65869	−0.829344	−	0.558739i	$-0.811285\pi$
$317$	−29.5321	−1.65869	−0.829344	+	0.558739i	$0.811285\pi$
$318$	0	0
$319$	15.3182	0.857655
$320$	−7.55943	−0.422585
$321$	0	0
$322$	1.26083	0.0702633
$323$	0	0
$324$	0	0
$325$	−10.8794	−0.603480
$326$	11.2567	0.623452
$327$	0	0
$328$	7.28724	0.402370
$329$	2.53209	0.139599
$330$	0	0
$331$	27.6655	1.52063	0.760317	−	0.649553i	$-0.225044\pi$
$331$	27.6655	1.52063	0.760317	+	0.649553i	$0.225044\pi$
$332$	−2.74060	−0.150410
$333$	0	0
$334$	−5.43645	−0.297469
$335$	−6.74928	−0.368752
$336$	0	0
$337$	17.8598	0.972884	0.486442	−	0.873713i	$-0.338294\pi$
$337$	17.8598	0.972884	0.486442	+	0.873713i	$0.338294\pi$
$338$	−8.58853	−0.467154
$339$	0	0
$340$	−0.0760373	−0.00412370
$341$	−15.8229	−0.856861
$342$	0	0
$343$	−4.82026	−0.260270
$344$	−11.4848	−0.619220
$345$	0	0
$346$	27.1397	1.45904
$347$	5.80066	0.311396	0.155698	−	0.987805i	$-0.450237\pi$
$347$	5.80066	0.311396	0.155698	+	0.987805i	$0.450237\pi$
$348$	0	0
$349$	5.37227	0.287571	0.143786	−	0.989609i	$-0.454072\pi$
$349$	5.37227	0.287571	0.143786	+	0.989609i	$0.454072\pi$
$350$	−1.97771	−0.105713
$351$	0	0
$352$	2.31996	0.123654
$353$	25.2344	1.34309	0.671546	−	0.740963i	$-0.265630\pi$
$353$	25.2344	1.34309	0.671546	+	0.740963i	$0.265630\pi$
$354$	0	0
$355$	−8.18304	−0.434311
$356$	−1.90167	−0.100789
$357$	0	0
$358$	15.5107	0.819768
$359$	−6.68685	−0.352919	−0.176459	−	0.984308i	$-0.556464\pi$
$359$	−6.68685	−0.352919	−0.176459	+	0.984308i	$0.556464\pi$
$360$	0	0
$361$	0	0
$362$	11.5021	0.604535
$363$	0	0
$364$	−0.165192	−0.00865842
$365$	−1.22163	−0.0639430
$366$	0	0
$367$	8.11886	0.423801	0.211901	−	0.977291i	$-0.432035\pi$
$367$	8.11886	0.423801	0.211901	+	0.977291i	$0.432035\pi$
$368$	−9.69047	−0.505151
$369$	0	0
$370$	−5.85710	−0.304496
$371$	0.985452	0.0511621
$372$	0	0
$373$	34.8976	1.80693	0.903463	−	0.428665i	$-0.141016\pi$
$373$	34.8976	1.80693	0.903463	+	0.428665i	$0.141016\pi$
$374$	−1.40373	−0.0725853
$375$	0	0
$376$	−21.4611	−1.10677
$377$	17.7074	0.911977
$378$	0	0
$379$	1.70140	0.0873950	0.0436975	−	0.999045i	$-0.486086\pi$
$379$	1.70140	0.0873950	0.0436975	+	0.999045i	$0.486086\pi$
$380$	0	0
$381$	0	0
$382$	−24.7033	−1.26393
$383$	−2.93676	−0.150061	−0.0750306	−	0.997181i	$-0.523905\pi$
$383$	−2.93676	−0.150061	−0.0750306	+	0.997181i	$0.523905\pi$
$384$	0	0
$385$	−0.680045	−0.0346583
$386$	0.401045	0.0204127
$387$	0	0
$388$	−1.74691	−0.0886860
$389$	24.5672	1.24561	0.622803	−	0.782379i	$-0.285994\pi$
$389$	24.5672	1.24561	0.622803	+	0.782379i	$0.285994\pi$
$390$	0	0
$391$	−1.26083	−0.0637629
$392$	20.2499	1.02277
$393$	0	0
$394$	−17.7050	−0.891966
$395$	10.4165	0.524112
$396$	0	0
$397$	−31.8357	−1.59779	−0.798895	−	0.601470i	$-0.794582\pi$
$397$	−31.8357	−1.59779	−0.798895	+	0.601470i	$0.794582\pi$
$398$	−0.345866	−0.0173367
$399$	0	0
$400$	15.2003	0.760014
$401$	0.0864665	0.00431793	0.00215896	−	0.999998i	$-0.499313\pi$
$401$	0.0864665	0.00431793	0.00215896	+	0.999998i	$0.499313\pi$
$402$	0	0
$403$	−18.2909	−0.911133
$404$	−1.70914	−0.0850329
$405$	0	0
$406$	3.21894	0.159753
$407$	11.0077	0.545633
$408$	0	0
$409$	20.0060	0.989232	0.494616	−	0.869112i	$-0.335309\pi$
$409$	20.0060	0.989232	0.494616	+	0.869112i	$0.335309\pi$
$410$	2.93313	0.144857
$411$	0	0
$412$	1.01817	0.0501616
$413$	2.18984	0.107755
$414$	0	0
$415$	−13.0419	−0.640201
$416$	2.68180	0.131486
$417$	0	0
$418$	0	0
$419$	−25.4097	−1.24135	−0.620673	−	0.784070i	$-0.713141\pi$
$419$	−25.4097	−1.24135	−0.620673	+	0.784070i	$0.713141\pi$
$420$	0	0
$421$	4.36959	0.212961	0.106480	−	0.994315i	$-0.466042\pi$
$421$	4.36959	0.212961	0.106480	+	0.994315i	$0.466042\pi$
$422$	3.29860	0.160573
$423$	0	0
$424$	−8.35235	−0.405626
$425$	1.97771	0.0959331
$426$	0	0
$427$	3.17024	0.153419
$428$	1.89124	0.0914168
$429$	0	0
$430$	−4.62267	−0.222925
$431$	38.3063	1.84515	0.922576	−	0.385816i	$-0.126080\pi$
$431$	38.3063	1.84515	0.922576	+	0.385816i	$0.126080\pi$
$432$	0	0
$433$	−18.1310	−0.871322	−0.435661	−	0.900111i	$-0.643485\pi$
$433$	−18.1310	−0.871322	−0.435661	+	0.900111i	$0.643485\pi$
$434$	−3.32501	−0.159605
$435$	0	0
$436$	0.336867	0.0161330
$437$	0	0
$438$	0	0
$439$	−6.09059	−0.290688	−0.145344	−	0.989381i	$-0.546429\pi$
$439$	−6.09059	−0.290688	−0.145344	+	0.989381i	$0.546429\pi$
$440$	5.76382	0.274779
$441$	0	0
$442$	−1.62267	−0.0771827
$443$	29.8931	1.42026	0.710132	−	0.704068i	$-0.248635\pi$
$443$	29.8931	1.42026	0.710132	+	0.704068i	$0.248635\pi$
$444$	0	0
$445$	−9.04963	−0.428994
$446$	−11.4652	−0.542894
$447$	0	0
$448$	2.98545	0.141049
$449$	11.2499	0.530916	0.265458	−	0.964122i	$-0.414477\pi$
$449$	11.2499	0.530916	0.265458	+	0.964122i	$0.414477\pi$
$450$	0	0
$451$	−5.51249	−0.259573
$452$	−3.26857	−0.153741
$453$	0	0
$454$	19.0651	0.894770
$455$	−0.786112	−0.0368535
$456$	0	0
$457$	−23.3901	−1.09414	−0.547072	−	0.837086i	$-0.684258\pi$
$457$	−23.3901	−1.09414	−0.547072	+	0.837086i	$0.684258\pi$
$458$	−27.6641	−1.29266
$459$	0	0
$460$	0.437882	0.0204163
$461$	36.6236	1.70573	0.852866	−	0.522130i	$-0.174863\pi$
$461$	36.6236	1.70573	0.852866	+	0.522130i	$0.174863\pi$
$462$	0	0
$463$	−42.9864	−1.99775	−0.998873	−	0.0474549i	$-0.984889\pi$
$463$	−42.9864	−1.99775	−0.998873	+	0.0474549i	$0.984889\pi$
$464$	−24.7401	−1.14853
$465$	0	0
$466$	−23.7811	−1.10164
$467$	−25.5963	−1.18445	−0.592227	−	0.805771i	$-0.701751\pi$
$467$	−25.5963	−1.18445	−0.592227	+	0.805771i	$0.701751\pi$
$468$	0	0
$469$	2.66550	0.123081
$470$	−8.63816	−0.398448
$471$	0	0
$472$	−18.5604	−0.854310
$473$	8.68779	0.399465
$474$	0	0
$475$	0	0
$476$	0.0300295	0.00137640
$477$	0	0
$478$	−3.16931	−0.144961
$479$	−38.1762	−1.74432	−0.872158	−	0.489224i	$-0.837280\pi$
$479$	−38.1762	−1.74432	−0.872158	+	0.489224i	$0.837280\pi$
$480$	0	0
$481$	12.7246	0.580193
$482$	18.5936	0.846914
$483$	0	0
$484$	1.11650	0.0507498
$485$	−8.31315	−0.377481
$486$	0	0
$487$	7.76382	0.351812	0.175906	−	0.984407i	$-0.443714\pi$
$487$	7.76382	0.351812	0.175906	+	0.984407i	$0.443714\pi$
$488$	−26.8699	−1.21634
$489$	0	0
$490$	8.15064	0.368209
$491$	36.7229	1.65728	0.828640	−	0.559782i	$-0.189115\pi$
$491$	36.7229	1.65728	0.828640	+	0.559782i	$0.189115\pi$
$492$	0	0
$493$	−3.21894	−0.144974
$494$	0	0
$495$	0	0
$496$	25.5553	1.14747
$497$	3.23173	0.144963
$498$	0	0
$499$	4.92633	0.220533	0.110266	−	0.993902i	$-0.464830\pi$
$499$	4.92633	0.220533	0.110266	+	0.993902i	$0.464830\pi$
$500$	−1.49937	−0.0670539
$501$	0	0
$502$	5.61175	0.250465
$503$	−32.9495	−1.46915	−0.734574	−	0.678529i	$-0.762618\pi$
$503$	−32.9495	−1.46915	−0.734574	+	0.678529i	$0.762618\pi$
$504$	0	0
$505$	−8.13341	−0.361932
$506$	8.08378	0.359368
$507$	0	0
$508$	−2.14433	−0.0951394
$509$	36.9350	1.63712	0.818558	−	0.574424i	$-0.194774\pi$
$509$	36.9350	1.63712	0.818558	+	0.574424i	$0.194774\pi$
$510$	0	0
$511$	0.482459	0.0213427
$512$	−24.9186	−1.10126
$513$	0	0
$514$	−0.898986	−0.0396526
$515$	4.84524	0.213507
$516$	0	0
$517$	16.2344	0.713989
$518$	2.31315	0.101634
$519$	0	0
$520$	6.66281	0.292183
$521$	−9.29179	−0.407081	−0.203540	−	0.979067i	$-0.565245\pi$
$521$	−9.29179	−0.407081	−0.203540	+	0.979067i	$0.565245\pi$
$522$	0	0
$523$	−28.4151	−1.24251	−0.621253	−	0.783610i	$-0.713376\pi$
$523$	−28.4151	−1.24251	−0.621253	+	0.783610i	$0.713376\pi$
$524$	0.340986	0.0148960
$525$	0	0
$526$	−15.3587	−0.669669
$527$	3.32501	0.144840
$528$	0	0
$529$	−15.7392	−0.684312
$530$	−3.36184	−0.146029
$531$	0	0
$532$	0	0
$533$	−6.37227	−0.276014
$534$	0	0
$535$	9.00000	0.389104
$536$	−22.5918	−0.975818
$537$	0	0
$538$	−26.1242	−1.12630
$539$	−15.3182	−0.659802
$540$	0	0
$541$	14.9855	0.644275	0.322137	−	0.946693i	$-0.395599\pi$
$541$	14.9855	0.644275	0.322137	+	0.946693i	$0.395599\pi$
$542$	−18.0460	−0.775142
$543$	0	0
$544$	−0.487511	−0.0209019
$545$	1.60307	0.0686681
$546$	0	0
$547$	−3.88713	−0.166202	−0.0831008	−	0.996541i	$-0.526482\pi$
$547$	−3.88713	−0.166202	−0.0831008	+	0.996541i	$0.526482\pi$
$548$	0.0472658	0.00201909
$549$	0	0
$550$	−12.6800	−0.540679
$551$	0	0
$552$	0	0
$553$	−4.11381	−0.174937
$554$	23.9103	1.01585
$555$	0	0
$556$	−0.787866	−0.0334130
$557$	−13.2044	−0.559488	−0.279744	−	0.960075i	$-0.590250\pi$
$557$	−13.2044	−0.559488	−0.279744	+	0.960075i	$0.590250\pi$
$558$	0	0
$559$	10.0428	0.424766
$560$	1.09833	0.0464127
$561$	0	0
$562$	−24.6272	−1.03884
$563$	−10.7128	−0.451489	−0.225745	−	0.974187i	$-0.572481\pi$
$563$	−10.7128	−0.451489	−0.225745	+	0.974187i	$0.572481\pi$
$564$	0	0
$565$	−15.5544	−0.654378
$566$	10.3577	0.435368
$567$	0	0
$568$	−27.3911	−1.14930
$569$	13.4706	0.564717	0.282358	−	0.959309i	$-0.408883\pi$
$569$	13.4706	0.564717	0.282358	+	0.959309i	$0.408883\pi$
$570$	0	0
$571$	12.6655	0.530035	0.265017	−	0.964244i	$-0.414622\pi$
$571$	12.6655	0.530035	0.265017	+	0.964244i	$0.414622\pi$
$572$	−1.05913	−0.0442843
$573$	0	0
$574$	−1.15839	−0.0483501
$575$	−11.3892	−0.474962
$576$	0	0
$577$	−10.5544	−0.439384	−0.219692	−	0.975569i	$-0.570505\pi$
$577$	−10.5544	−0.439384	−0.219692	+	0.975569i	$0.570505\pi$
$578$	−22.6091	−0.940413
$579$	0	0
$580$	1.11793	0.0464194
$581$	5.15064	0.213685
$582$	0	0
$583$	6.31820	0.261673
$584$	−4.08915	−0.169210
$585$	0	0
$586$	14.1506	0.584558
$587$	19.1548	0.790602	0.395301	−	0.918552i	$-0.370640\pi$
$587$	19.1548	0.790602	0.395301	+	0.918552i	$0.370640\pi$
$588$	0	0
$589$	0	0
$590$	−7.47060	−0.307560
$591$	0	0
$592$	−17.7784	−0.730687
$593$	8.69459	0.357044	0.178522	−	0.983936i	$-0.442868\pi$
$593$	8.69459	0.357044	0.178522	+	0.983936i	$0.442868\pi$
$594$	0	0
$595$	0.142903	0.00585847
$596$	3.06006	0.125345
$597$	0	0
$598$	9.34461	0.382129
$599$	−19.8316	−0.810298	−0.405149	−	0.914251i	$-0.632780\pi$
$599$	−19.8316	−0.810298	−0.405149	+	0.914251i	$0.632780\pi$
$600$	0	0
$601$	−33.7615	−1.37716	−0.688579	−	0.725161i	$-0.741765\pi$
$601$	−33.7615	−1.37716	−0.688579	+	0.725161i	$0.741765\pi$
$602$	1.82564	0.0744074
$603$	0	0
$604$	0.806036	0.0327971
$605$	5.31315	0.216010
$606$	0	0
$607$	35.2850	1.43217	0.716087	−	0.698011i	$-0.245932\pi$
$607$	35.2850	1.43217	0.716087	+	0.698011i	$0.245932\pi$
$608$	0	0
$609$	0	0
$610$	−10.8152	−0.437895
$611$	18.7665	0.759212
$612$	0	0
$613$	18.4534	0.745324	0.372662	−	0.927967i	$-0.378445\pi$
$613$	18.4534	0.745324	0.372662	+	0.927967i	$0.378445\pi$
$614$	−15.7573	−0.635914
$615$	0	0
$616$	−2.27631	−0.0917152
$617$	35.6854	1.43664	0.718320	−	0.695712i	$-0.244911\pi$
$617$	35.6854	1.43664	0.718320	+	0.695712i	$0.244911\pi$
$618$	0	0
$619$	3.65951	0.147088	0.0735441	−	0.997292i	$-0.476569\pi$
$619$	3.65951	0.147088	0.0735441	+	0.997292i	$0.476569\pi$
$620$	−1.15476	−0.0463764
$621$	0	0
$622$	−21.5107	−0.862502
$623$	3.57398	0.143188
$624$	0	0
$625$	13.9982	0.559930
$626$	−35.8767	−1.43392
$627$	0	0
$628$	1.77744	0.0709275
$629$	−2.31315	−0.0922313
$630$	0	0
$631$	−0.793852	−0.0316028	−0.0158014	−	0.999875i	$-0.505030\pi$
$631$	−0.793852	−0.0316028	−0.0158014	+	0.999875i	$0.505030\pi$
$632$	34.8672	1.38694
$633$	0	0
$634$	−39.7885	−1.58020
$635$	−10.2044	−0.404949
$636$	0	0
$637$	−17.7074	−0.701592
$638$	20.6382	0.817072
$639$	0	0
$640$	−8.35235	−0.330156
$641$	−29.3824	−1.16053	−0.580267	−	0.814426i	$-0.697052\pi$
$641$	−29.3824	−1.16053	−0.580267	+	0.814426i	$0.697052\pi$
$642$	0	0
$643$	22.2139	0.876030	0.438015	−	0.898968i	$-0.355682\pi$
$643$	22.2139	0.876030	0.438015	+	0.898968i	$0.355682\pi$
$644$	−0.172933	−0.00681451
$645$	0	0
$646$	0	0
$647$	−11.2591	−0.442640	−0.221320	−	0.975201i	$-0.571037\pi$
$647$	−11.2591	−0.442640	−0.221320	+	0.975201i	$0.571037\pi$
$648$	0	0
$649$	14.0401	0.551123
$650$	−14.6578	−0.574924
$651$	0	0
$652$	−1.54395	−0.0604657
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	0	0
$655$	1.62267	0.0634031
$656$	8.90310	0.347608
$657$	0	0
$658$	3.41147	0.132993
$659$	−28.0259	−1.09173	−0.545867	−	0.837872i	$-0.683800\pi$
$659$	−28.0259	−1.09173	−0.545867	+	0.837872i	$0.683800\pi$
$660$	0	0
$661$	11.3678	0.442157	0.221079	−	0.975256i	$-0.429042\pi$
$661$	11.3678	0.442157	0.221079	+	0.975256i	$0.429042\pi$
$662$	37.2736	1.44868
$663$	0	0
$664$	−43.6551	−1.69415
$665$	0	0
$666$	0	0
$667$	18.5371	0.717761
$668$	0.745653	0.0288502
$669$	0	0
$670$	−9.09327	−0.351304
$671$	20.3259	0.784674
$672$	0	0
$673$	−16.5672	−0.638618	−0.319309	−	0.947651i	$-0.603451\pi$
$673$	−16.5672	−0.638618	−0.319309	+	0.947651i	$0.603451\pi$
$674$	24.0624	0.926850
$675$	0	0
$676$	1.17799	0.0453071
$677$	9.04963	0.347806	0.173903	−	0.984763i	$-0.444362\pi$
$677$	9.04963	0.347806	0.173903	+	0.984763i	$0.444362\pi$
$678$	0	0
$679$	3.28312	0.125995
$680$	−1.21120	−0.0464474
$681$	0	0
$682$	−21.3182	−0.816316
$683$	−8.73143	−0.334099	−0.167049	−	0.985949i	$-0.553424\pi$
$683$	−8.73143	−0.334099	−0.167049	+	0.985949i	$0.553424\pi$
$684$	0	0
$685$	0.224927	0.00859402
$686$	−6.49432	−0.247954
$687$	0	0
$688$	−14.0315	−0.534944
$689$	7.30365	0.278247
$690$	0	0
$691$	34.7202	1.32082	0.660409	−	0.750906i	$-0.270383\pi$
$691$	34.7202	1.32082	0.660409	+	0.750906i	$0.270383\pi$
$692$	−3.72243	−0.141506
$693$	0	0
$694$	7.81521	0.296661
$695$	−3.74928	−0.142218
$696$	0	0
$697$	1.15839	0.0438770
$698$	7.23804	0.273964
$699$	0	0
$700$	0.271259	0.0102526
$701$	−39.4151	−1.48869	−0.744344	−	0.667797i	$-0.767238\pi$
$701$	−39.4151	−1.48869	−0.744344	+	0.667797i	$0.767238\pi$
$702$	0	0
$703$	0	0
$704$	19.1411	0.721409
$705$	0	0
$706$	33.9982	1.27954
$707$	3.21213	0.120805
$708$	0	0
$709$	−41.1215	−1.54435	−0.772176	−	0.635409i	$-0.780832\pi$
$709$	−41.1215	−1.54435	−0.772176	+	0.635409i	$0.780832\pi$
$710$	−11.0250	−0.413760
$711$	0	0
$712$	−30.2918	−1.13523
$713$	−19.1480	−0.717097
$714$	0	0
$715$	−5.04013	−0.188490
$716$	−2.12742	−0.0795055
$717$	0	0
$718$	−9.00917	−0.336219
$719$	42.3955	1.58109	0.790543	−	0.612407i	$-0.209799\pi$
$719$	42.3955	1.58109	0.790543	+	0.612407i	$0.209799\pi$
$720$	0	0
$721$	−1.91353	−0.0712637
$722$	0	0
$723$	0	0
$724$	−1.57760	−0.0586310
$725$	−29.0770	−1.07989
$726$	0	0
$727$	51.6563	1.91583	0.957914	−	0.287057i	$-0.0926769\pi$
$727$	51.6563	1.91583	0.957914	+	0.287057i	$0.0926769\pi$
$728$	−2.63135	−0.0975243
$729$	0	0
$730$	−1.64590	−0.0609174
$731$	−1.82564	−0.0675236
$732$	0	0
$733$	−22.9162	−0.846430	−0.423215	−	0.906029i	$-0.639098\pi$
$733$	−22.9162	−0.846430	−0.423215	+	0.906029i	$0.639098\pi$
$734$	10.9385	0.403748
$735$	0	0
$736$	2.80747	0.103485
$737$	17.0898	0.629510
$738$	0	0
$739$	28.1266	1.03465	0.517327	−	0.855788i	$-0.326927\pi$
$739$	28.1266	1.03465	0.517327	+	0.855788i	$0.326927\pi$
$740$	0.803348	0.0295317
$741$	0	0
$742$	1.32770	0.0487413
$743$	−6.13247	−0.224979	−0.112489	−	0.993653i	$-0.535882\pi$
$743$	−6.13247	−0.224979	−0.112489	+	0.993653i	$0.535882\pi$
$744$	0	0
$745$	14.5621	0.533515
$746$	47.0173	1.72143
$747$	0	0
$748$	0.192533	0.00703972
$749$	−3.55438	−0.129874
$750$	0	0
$751$	5.64084	0.205837	0.102919	−	0.994690i	$-0.467182\pi$
$751$	5.64084	0.205837	0.102919	+	0.994690i	$0.467182\pi$
$752$	−26.2199	−0.956140
$753$	0	0
$754$	23.8571	0.868824
$755$	3.83574	0.139597
$756$	0	0
$757$	15.6919	0.570332	0.285166	−	0.958478i	$-0.407951\pi$
$757$	15.6919	0.570332	0.285166	+	0.958478i	$0.407951\pi$
$758$	2.29229	0.0832597
$759$	0	0
$760$	0	0
$761$	−4.86484	−0.176350	−0.0881751	−	0.996105i	$-0.528104\pi$
$761$	−4.86484	−0.176350	−0.0881751	+	0.996105i	$0.528104\pi$
$762$	0	0
$763$	−0.633103	−0.0229199
$764$	3.38825	0.122583
$765$	0	0
$766$	−3.95668	−0.142961
$767$	16.2300	0.586031
$768$	0	0
$769$	22.5321	0.812528	0.406264	−	0.913756i	$-0.366831\pi$
$769$	22.5321	0.812528	0.406264	+	0.913756i	$0.366831\pi$
$770$	−0.916222	−0.0330183
$771$	0	0
$772$	−0.0550065	−0.00197973
$773$	26.4320	0.950693	0.475347	−	0.879799i	$-0.342323\pi$
$773$	26.4320	0.950693	0.475347	+	0.879799i	$0.342323\pi$
$774$	0	0
$775$	30.0351	1.07889
$776$	−27.8266	−0.998916
$777$	0	0
$778$	33.0993	1.18667
$779$	0	0
$780$	0	0
$781$	20.7202	0.741426
$782$	−1.69871	−0.0607458
$783$	0	0
$784$	24.7401	0.883575
$785$	8.45842	0.301894
$786$	0	0
$787$	−15.5577	−0.554571	−0.277286	−	0.960788i	$-0.589435\pi$
$787$	−15.5577	−0.554571	−0.277286	+	0.960788i	$0.589435\pi$
$788$	2.42839	0.0865077
$789$	0	0
$790$	14.0341	0.499313
$791$	6.14290	0.218417
$792$	0	0
$793$	23.4962	0.834374
$794$	−42.8922	−1.52219
$795$	0	0
$796$	0.0474383	0.00168141
$797$	−33.4935	−1.18640	−0.593200	−	0.805055i	$-0.702136\pi$
$797$	−33.4935	−1.18640	−0.593200	+	0.805055i	$0.702136\pi$
$798$	0	0
$799$	−3.41147	−0.120689
$800$	−4.40373	−0.155695
$801$	0	0
$802$	0.116496	0.00411362
$803$	3.09327	0.109159
$804$	0	0
$805$	−0.822948	−0.0290051
$806$	−24.6432	−0.868020
$807$	0	0
$808$	−27.2249	−0.957770
$809$	−41.1162	−1.44557	−0.722784	−	0.691074i	$-0.757138\pi$
$809$	−41.1162	−1.44557	−0.722784	+	0.691074i	$0.757138\pi$
$810$	0	0
$811$	16.6878	0.585987	0.292994	−	0.956114i	$-0.405349\pi$
$811$	16.6878	0.585987	0.292994	+	0.956114i	$0.405349\pi$
$812$	−0.441504	−0.0154937
$813$	0	0
$814$	14.8307	0.519815
$815$	−7.34730	−0.257365
$816$	0	0
$817$	0	0
$818$	26.9540	0.942424
$819$	0	0
$820$	−0.402303	−0.0140490
$821$	31.3901	1.09552	0.547761	−	0.836635i	$-0.315480\pi$
$821$	31.3901	1.09552	0.547761	+	0.836635i	$0.315480\pi$
$822$	0	0
$823$	46.3259	1.61482	0.807410	−	0.589990i	$-0.200868\pi$
$823$	46.3259	1.61482	0.807410	+	0.589990i	$0.200868\pi$
$824$	16.2184	0.564996
$825$	0	0
$826$	2.95037	0.102657
$827$	−40.7588	−1.41732	−0.708661	−	0.705549i	$-0.750700\pi$
$827$	−40.7588	−1.41732	−0.708661	+	0.705549i	$0.750700\pi$
$828$	0	0
$829$	−35.4834	−1.23239	−0.616195	−	0.787594i	$-0.711327\pi$
$829$	−35.4834	−1.23239	−0.616195	+	0.787594i	$0.711327\pi$
$830$	−17.5713	−0.609908
$831$	0	0
$832$	22.1266	0.767102
$833$	3.21894	0.111530
$834$	0	0
$835$	3.54839	0.122797
$836$	0	0
$837$	0	0
$838$	−34.2344	−1.18261
$839$	38.2026	1.31890	0.659451	−	0.751748i	$-0.270789\pi$
$839$	38.2026	1.31890	0.659451	+	0.751748i	$0.270789\pi$
$840$	0	0
$841$	18.3259	0.631929
$842$	5.88713	0.202884
$843$	0	0
$844$	−0.452430	−0.0155733
$845$	5.60576	0.192844
$846$	0	0
$847$	−2.09833	−0.0720993
$848$	−10.2044	−0.350420
$849$	0	0
$850$	2.66456	0.0913938
$851$	13.3209	0.456634
$852$	0	0
$853$	−25.6016	−0.876584	−0.438292	−	0.898833i	$-0.644416\pi$
$853$	−25.6016	−0.876584	−0.438292	+	0.898833i	$0.644416\pi$
$854$	4.27126	0.146159
$855$	0	0
$856$	30.1257	1.02967
$857$	21.0865	0.720300	0.360150	−	0.932894i	$-0.382726\pi$
$857$	21.0865	0.720300	0.360150	+	0.932894i	$0.382726\pi$
$858$	0	0
$859$	−19.5672	−0.667623	−0.333812	−	0.942640i	$-0.608335\pi$
$859$	−19.5672	−0.667623	−0.333812	+	0.942640i	$0.608335\pi$
$860$	0.634037	0.0216205
$861$	0	0
$862$	51.6100	1.75784
$863$	−4.94894	−0.168464	−0.0842319	−	0.996446i	$-0.526844\pi$
$863$	−4.94894	−0.168464	−0.0842319	+	0.996446i	$0.526844\pi$
$864$	0	0
$865$	−17.7142	−0.602301
$866$	−24.4279	−0.830093
$867$	0	0
$868$	0.456052	0.0154794
$869$	−26.3756	−0.894730
$870$	0	0
$871$	19.7553	0.669381
$872$	5.36596	0.181714
$873$	0	0
$874$	0	0
$875$	2.81790	0.0952623
$876$	0	0
$877$	1.21987	0.0411922	0.0205961	−	0.999788i	$-0.493444\pi$
$877$	1.21987	0.0411922	0.0205961	+	0.999788i	$0.493444\pi$
$878$	−8.20582	−0.276933
$879$	0	0
$880$	7.04189	0.237382
$881$	−46.5030	−1.56673	−0.783363	−	0.621565i	$-0.786497\pi$
$881$	−46.5030	−1.56673	−0.783363	+	0.621565i	$0.786497\pi$
$882$	0	0
$883$	12.9249	0.434957	0.217479	−	0.976065i	$-0.430217\pi$
$883$	12.9249	0.434957	0.217479	+	0.976065i	$0.430217\pi$
$884$	0.222563	0.00748560
$885$	0	0
$886$	40.2749	1.35306
$887$	23.2243	0.779796	0.389898	−	0.920858i	$-0.372510\pi$
$887$	23.2243	0.779796	0.389898	+	0.920858i	$0.372510\pi$
$888$	0	0
$889$	4.03003	0.135163
$890$	−12.1925	−0.408695
$891$	0	0
$892$	1.57255	0.0526528
$893$	0	0
$894$	0	0
$895$	−10.1239	−0.338405
$896$	3.29860	0.110198
$897$	0	0
$898$	15.1570	0.505794
$899$	−48.8854	−1.63042
$900$	0	0
$901$	−1.32770	−0.0442320
$902$	−7.42696	−0.247291
$903$	0	0
$904$	−52.0651	−1.73166
$905$	−7.50744	−0.249556
$906$	0	0
$907$	−39.9968	−1.32807	−0.664036	−	0.747700i	$-0.731158\pi$
$907$	−39.9968	−1.32807	−0.664036	+	0.747700i	$0.731158\pi$
$908$	−2.61493	−0.0867796
$909$	0	0
$910$	−1.05913	−0.0351097
$911$	18.7997	0.622863	0.311431	−	0.950269i	$-0.399192\pi$
$911$	18.7997	0.622863	0.311431	+	0.950269i	$0.399192\pi$
$912$	0	0
$913$	33.0232	1.09291
$914$	−31.5134	−1.04237
$915$	0	0
$916$	3.79435	0.125369
$917$	−0.640844	−0.0211625
$918$	0	0
$919$	39.8316	1.31392	0.656962	−	0.753924i	$-0.271841\pi$
$919$	39.8316	1.31392	0.656962	+	0.753924i	$0.271841\pi$
$920$	6.97502	0.229960
$921$	0	0
$922$	49.3429	1.62502
$923$	23.9519	0.788387
$924$	0	0
$925$	−20.8949	−0.687019
$926$	−57.9154	−1.90322
$927$	0	0
$928$	7.16756	0.235287
$929$	−26.9540	−0.884332	−0.442166	−	0.896933i	$-0.645790\pi$
$929$	−26.9540	−0.884332	−0.442166	+	0.896933i	$0.645790\pi$
$930$	0	0
$931$	0	0
$932$	3.26176	0.106843
$933$	0	0
$934$	−34.4858	−1.12841
$935$	0.916222	0.0299637
$936$	0	0
$937$	2.62361	0.0857095	0.0428548	−	0.999081i	$-0.486355\pi$
$937$	2.62361	0.0857095	0.0428548	+	0.999081i	$0.486355\pi$
$938$	3.59121	0.117257
$939$	0	0
$940$	1.18479	0.0386437
$941$	18.6696	0.608612	0.304306	−	0.952574i	$-0.401575\pi$
$941$	18.6696	0.608612	0.304306	+	0.952574i	$0.401575\pi$
$942$	0	0
$943$	−6.67087	−0.217234
$944$	−22.6759	−0.738039
$945$	0	0
$946$	11.7050	0.380563
$947$	−8.39961	−0.272951	−0.136475	−	0.990643i	$-0.543577\pi$
$947$	−8.39961	−0.272951	−0.136475	+	0.990643i	$0.543577\pi$
$948$	0	0
$949$	3.57573	0.116073
$950$	0	0
$951$	0	0
$952$	0.478340	0.0155031
$953$	−33.6928	−1.09142	−0.545709	−	0.837975i	$-0.683740\pi$
$953$	−33.6928	−1.09142	−0.545709	+	0.837975i	$0.683740\pi$
$954$	0	0
$955$	16.1239	0.521758
$956$	0.434696	0.0140591
$957$	0	0
$958$	−51.4347	−1.66178
$959$	−0.0888306	−0.00286849
$960$	0	0
$961$	19.4962	0.628909
$962$	17.1438	0.552739
$963$	0	0
$964$	−2.55026	−0.0821383
$965$	−0.261764	−0.00842647
$966$	0	0
$967$	11.7433	0.377639	0.188819	−	0.982012i	$-0.439534\pi$
$967$	11.7433	0.377639	0.188819	+	0.982012i	$0.439534\pi$
$968$	17.7847	0.571621
$969$	0	0
$970$	−11.2003	−0.359619
$971$	−12.8093	−0.411071	−0.205536	−	0.978650i	$-0.565894\pi$
$971$	−12.8093	−0.411071	−0.205536	+	0.978650i	$0.565894\pi$
$972$	0	0
$973$	1.48070	0.0474692
$974$	10.4602	0.335165
$975$	0	0
$976$	−32.8280	−1.05080
$977$	−14.5276	−0.464781	−0.232390	−	0.972623i	$-0.574655\pi$
$977$	−14.5276	−0.464781	−0.232390	+	0.972623i	$0.574655\pi$
$978$	0	0
$979$	22.9145	0.732350
$980$	−1.11793	−0.0357108
$981$	0	0
$982$	49.4766	1.57886
$983$	37.0502	1.18172	0.590860	−	0.806774i	$-0.298789\pi$
$983$	37.0502	1.18172	0.590860	+	0.806774i	$0.298789\pi$
$984$	0	0
$985$	11.5561	0.368209
$986$	−4.33687	−0.138114
$987$	0	0
$988$	0	0
$989$	10.5134	0.334307
$990$	0	0
$991$	3.43140	0.109002	0.0545010	−	0.998514i	$-0.482643\pi$
$991$	3.43140	0.109002	0.0545010	+	0.998514i	$0.482643\pi$
$992$	−7.40373	−0.235069
$993$	0	0
$994$	4.35410	0.138104
$995$	0.225748	0.00715669
$996$	0	0
$997$	12.7760	0.404620	0.202310	−	0.979322i	$-0.435155\pi$
$997$	12.7760	0.404620	0.202310	+	0.979322i	$0.435155\pi$
$998$	6.63722	0.210098
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.z.1.2		3
3.2	odd	2		361.2.a.g.1.2		3
12.11	even	2		5776.2.a.br.1.3		3
15.14	odd	2		9025.2.a.bd.1.2		3
19.6	even	9		171.2.u.c.55.1		6
19.16	even	9		171.2.u.c.28.1		6
19.18	odd	2		3249.2.a.s.1.2		3
57.2	even	18		361.2.e.b.99.1		6
57.5	odd	18		361.2.e.g.234.1		6
57.8	even	6		361.2.c.h.292.2		6
57.11	odd	6		361.2.c.i.292.2		6
57.14	even	18		361.2.e.a.234.1		6
57.17	odd	18		361.2.e.f.99.1		6
57.23	odd	18		361.2.e.g.54.1		6
57.26	odd	6		361.2.c.i.68.2		6
57.29	even	18		361.2.e.b.62.1		6
57.32	even	18		361.2.e.h.245.1		6
57.35	odd	18		19.2.e.a.9.1	✓	6
57.41	even	18		361.2.e.h.28.1		6
57.44	odd	18		19.2.e.a.17.1	yes	6
57.47	odd	18		361.2.e.f.62.1		6
57.50	even	6		361.2.c.h.68.2		6
57.53	even	18		361.2.e.a.54.1		6
57.56	even	2		361.2.a.h.1.2		3
228.35	even	18		304.2.u.b.161.1		6
228.215	even	18		304.2.u.b.17.1		6
228.227	odd	2		5776.2.a.bi.1.1		3
285.44	odd	18		475.2.l.a.226.1		6
285.92	even	36		475.2.u.a.199.2		12
285.149	odd	18		475.2.l.a.351.1		6
285.158	even	36		475.2.u.a.74.2		12
285.263	even	36		475.2.u.a.199.1		12
285.272	even	36		475.2.u.a.74.1		12
285.284	even	2		9025.2.a.x.1.2		3
399.44	odd	18		931.2.v.b.606.1		6
399.101	even	18		931.2.x.b.226.1		6
399.149	odd	18		931.2.x.a.655.1		6
399.158	odd	18		931.2.x.a.226.1		6
399.206	even	18		931.2.v.a.275.1		6
399.215	even	18		931.2.v.a.606.1		6
399.263	odd	18		931.2.v.b.275.1		6
399.272	even	18		931.2.w.a.834.1		6
399.320	even	18		931.2.x.b.655.1		6
399.377	even	18		931.2.w.a.883.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.e.a.9.1	✓	6	57.35	odd	18
19.2.e.a.17.1	yes	6	57.44	odd	18
171.2.u.c.28.1		6	19.16	even	9
171.2.u.c.55.1		6	19.6	even	9
304.2.u.b.17.1		6	228.215	even	18
304.2.u.b.161.1		6	228.35	even	18
361.2.a.g.1.2		3	3.2	odd	2
361.2.a.h.1.2		3	57.56	even	2
361.2.c.h.68.2		6	57.50	even	6
361.2.c.h.292.2		6	57.8	even	6
361.2.c.i.68.2		6	57.26	odd	6
361.2.c.i.292.2		6	57.11	odd	6
361.2.e.a.54.1		6	57.53	even	18
361.2.e.a.234.1		6	57.14	even	18
361.2.e.b.62.1		6	57.29	even	18
361.2.e.b.99.1		6	57.2	even	18
361.2.e.f.62.1		6	57.47	odd	18
361.2.e.f.99.1		6	57.17	odd	18
361.2.e.g.54.1		6	57.23	odd	18
361.2.e.g.234.1		6	57.5	odd	18
361.2.e.h.28.1		6	57.41	even	18
361.2.e.h.245.1		6	57.32	even	18
475.2.l.a.226.1		6	285.44	odd	18
475.2.l.a.351.1		6	285.149	odd	18
475.2.u.a.74.1		12	285.272	even	36
475.2.u.a.74.2		12	285.158	even	36
475.2.u.a.199.1		12	285.263	even	36
475.2.u.a.199.2		12	285.92	even	36
931.2.v.a.275.1		6	399.206	even	18
931.2.v.a.606.1		6	399.215	even	18
931.2.v.b.275.1		6	399.263	odd	18
931.2.v.b.606.1		6	399.44	odd	18
931.2.w.a.834.1		6	399.272	even	18
931.2.w.a.883.1		6	399.377	even	18
931.2.x.a.226.1		6	399.158	odd	18
931.2.x.a.655.1		6	399.149	odd	18
931.2.x.b.226.1		6	399.101	even	18
931.2.x.b.655.1		6	399.320	even	18
3249.2.a.s.1.2		3	19.18	odd	2
3249.2.a.z.1.2		3	1.1	even	1	trivial
5776.2.a.bi.1.1		3	228.227	odd	2
5776.2.a.br.1.3		3	12.11	even	2
9025.2.a.x.1.2		3	285.284	even	2
9025.2.a.bd.1.2		3	15.14	odd	2

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)

\(f(q)\)	\(=\)	\(q+1.34730 q^{2} -0.184793 q^{4} -0.879385 q^{5} +0.347296 q^{7} -2.94356 q^{8} +O(q^{10})\)	\(q+1.34730 q^{2} -0.184793 q^{4} -0.879385 q^{5} +0.347296 q^{7} -2.94356 q^{8} -1.18479 q^{10} +2.22668 q^{11} +2.57398 q^{13} +0.467911 q^{14} -3.59627 q^{16} -0.467911 q^{17} +0.162504 q^{20} +3.00000 q^{22} +2.69459 q^{23} -4.22668 q^{25} +3.46791 q^{26} -0.0641778 q^{28} +6.87939 q^{29} -7.10607 q^{31} +1.04189 q^{32} -0.630415 q^{34} -0.305407 q^{35} +4.94356 q^{37} +2.58853 q^{40} -2.47565 q^{41} +3.90167 q^{43} -0.411474 q^{44} +3.63041 q^{46} +7.29086 q^{47} -6.87939 q^{49} -5.69459 q^{50} -0.475652 q^{52} +2.83750 q^{53} -1.95811 q^{55} -1.02229 q^{56} +9.26857 q^{58} +6.30541 q^{59} +9.12836 q^{61} -9.57398 q^{62} +8.59627 q^{64} -2.26352 q^{65} +7.67499 q^{67} +0.0864665 q^{68} -0.411474 q^{70} +9.30541 q^{71} +1.38919 q^{73} +6.66044 q^{74} +0.773318 q^{77} -11.8452 q^{79} +3.16250 q^{80} -3.33544 q^{82} +14.8307 q^{83} +0.411474 q^{85} +5.25671 q^{86} -6.55438 q^{88} +10.2909 q^{89} +0.893933 q^{91} -0.497941 q^{92} +9.82295 q^{94} +9.45336 q^{97} -9.26857 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} + 6 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 + 6 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} + 6 q^{8} + 6 q^{14} + 3 q^{16} - 6 q^{17} + 3 q^{20} + 9 q^{22} + 6 q^{23} - 6 q^{25} + 15 q^{26} + 9 q^{28} + 15 q^{29} - 9 q^{31} - 9 q^{34} - 3 q^{35} + 18 q^{40} + 12 q^{41} + 9 q^{44} + 18 q^{46} + 6 q^{47} - 15 q^{49} - 15 q^{50} + 18 q^{52} + 6 q^{53} - 9 q^{55} + 3 q^{56} + 18 q^{58} + 21 q^{59} + 9 q^{61} - 21 q^{62} + 12 q^{64} - 12 q^{65} + 18 q^{67} - 15 q^{68} + 9 q^{70} + 30 q^{71} - 3 q^{74} + 9 q^{77} - 9 q^{79} + 12 q^{80} + 18 q^{82} - 9 q^{85} - 21 q^{86} - 9 q^{88} + 15 q^{89} + 15 q^{91} + 24 q^{92} + 9 q^{94} + 15 q^{97} - 18 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 + 6 * q^8 + 6 * q^14 + 3 * q^16 - 6 * q^17 + 3 * q^20 + 9 * q^22 + 6 * q^23 - 6 * q^25 + 15 * q^26 + 9 * q^28 + 15 * q^29 - 9 * q^31 - 9 * q^34 - 3 * q^35 + 18 * q^40 + 12 * q^41 + 9 * q^44 + 18 * q^46 + 6 * q^47 - 15 * q^49 - 15 * q^50 + 18 * q^52 + 6 * q^53 - 9 * q^55 + 3 * q^56 + 18 * q^58 + 21 * q^59 + 9 * q^61 - 21 * q^62 + 12 * q^64 - 12 * q^65 + 18 * q^67 - 15 * q^68 + 9 * q^70 + 30 * q^71 - 3 * q^74 + 9 * q^77 - 9 * q^79 + 12 * q^80 + 18 * q^82 - 9 * q^85 - 21 * q^86 - 9 * q^88 + 15 * q^89 + 15 * q^91 + 24 * q^92 + 9 * q^94 + 15 * q^97 - 18 * q^98

Introduction

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