LMFDB - Embedded newform 6468.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6468,2,Mod(1,6468)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6468, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6468.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6468 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6468.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:	$51.6472400274$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.126956032.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 18x^{4} - 18x^{3} + 44x^{2} + 56x + 7 $ x^6 - 18x^4 - 18x^3 + 44x^2 + 56x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.03564$ of defining polynomial
Character	$\chi$	$=$	6468.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.07889 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.07889 q^{5} +1.00000 q^{9} -1.00000 q^{11} +0.810917 q^{13} -1.07889 q^{15} -4.76726 q^{17} +6.61480 q^{19} -0.429382 q^{23} -3.83599 q^{25} +1.00000 q^{27} -1.18851 q^{29} -0.206240 q^{31} -1.00000 q^{33} +5.75092 q^{37} +0.810917 q^{39} -6.88022 q^{41} -12.9254 q^{43} -1.07889 q^{45} -11.8478 q^{47} -4.76726 q^{51} +11.7785 q^{53} +1.07889 q^{55} +6.61480 q^{57} +6.60499 q^{59} -13.6819 q^{61} -0.874893 q^{65} +5.37194 q^{67} -0.429382 q^{69} -10.1311 q^{71} +12.2967 q^{73} -3.83599 q^{75} +11.7301 q^{79} +1.00000 q^{81} -9.19245 q^{83} +5.14337 q^{85} -1.18851 q^{87} +7.56155 q^{89} -0.206240 q^{93} -7.13666 q^{95} +5.34779 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 4 q^{5} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 4 * q^5 + 6 * q^9	$ 6 q + 6 q^{3} - 4 q^{5} + 6 q^{9} - 6 q^{11} - 8 q^{13} - 4 q^{15} - 16 q^{17} - 4 q^{19} + 22 q^{25} + 6 q^{27} - 4 q^{29} - 6 q^{33} + 4 q^{37} - 8 q^{39} - 12 q^{41} - 8 q^{43} - 4 q^{45} + 12 q^{47} - 16 q^{51} + 4 q^{53} + 4 q^{55} - 4 q^{57} - 24 q^{59} - 16 q^{61} - 12 q^{65} + 8 q^{67} - 36 q^{71} - 36 q^{73} + 22 q^{75} + 6 q^{81} - 32 q^{83} - 20 q^{85} - 4 q^{87} - 12 q^{89} - 44 q^{95} - 16 q^{97} - 6 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - 4 * q^5 + 6 * q^9 - 6 * q^11 - 8 * q^13 - 4 * q^15 - 16 * q^17 - 4 * q^19 + 22 * q^25 + 6 * q^27 - 4 * q^29 - 6 * q^33 + 4 * q^37 - 8 * q^39 - 12 * q^41 - 8 * q^43 - 4 * q^45 + 12 * q^47 - 16 * q^51 + 4 * q^53 + 4 * q^55 - 4 * q^57 - 24 * q^59 - 16 * q^61 - 12 * q^65 + 8 * q^67 - 36 * q^71 - 36 * q^73 + 22 * q^75 + 6 * q^81 - 32 * q^83 - 20 * q^85 - 4 * q^87 - 12 * q^89 - 44 * q^95 - 16 * q^97 - 6 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.07889	−0.482496	−0.241248	−	0.970464i	$-0.577557\pi$
$5$	−1.07889	−0.482496	−0.241248	+	0.970464i	$0.577557\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.810917	0.224908	0.112454	−	0.993657i	$-0.464129\pi$
$13$	0.810917	0.224908	0.112454	+	0.993657i	$0.464129\pi$
$14$	0	0
$15$	−1.07889	−0.278569
$16$	0	0
$17$	−4.76726	−1.15623	−0.578116	−	0.815955i	$-0.696212\pi$
$17$	−4.76726	−1.15623	−0.578116	+	0.815955i	$0.696212\pi$
$18$	0	0
$19$	6.61480	1.51754	0.758770	−	0.651359i	$-0.225801\pi$
$19$	6.61480	1.51754	0.758770	+	0.651359i	$0.225801\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.429382	−0.0895324	−0.0447662	−	0.998997i	$-0.514254\pi$
$23$	−0.429382	−0.0895324	−0.0447662	+	0.998997i	$0.514254\pi$
$24$	0	0
$25$	−3.83599	−0.767198
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.18851	−0.220701	−0.110350	−	0.993893i	$-0.535197\pi$
$29$	−1.18851	−0.220701	−0.110350	+	0.993893i	$0.535197\pi$
$30$	0	0
$31$	−0.206240	−0.0370418	−0.0185209	−	0.999828i	$-0.505896\pi$
$31$	−0.206240	−0.0370418	−0.0185209	+	0.999828i	$0.505896\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.75092	0.945445	0.472722	−	0.881211i	$-0.343271\pi$
$37$	5.75092	0.945445	0.472722	+	0.881211i	$0.343271\pi$
$38$	0	0
$39$	0.810917	0.129851
$40$	0	0
$41$	−6.88022	−1.07451	−0.537255	−	0.843420i	$-0.680539\pi$
$41$	−6.88022	−1.07451	−0.537255	+	0.843420i	$0.680539\pi$
$42$	0	0
$43$	−12.9254	−1.97110	−0.985549	−	0.169388i	$-0.945821\pi$
$43$	−12.9254	−1.97110	−0.985549	+	0.169388i	$0.945821\pi$
$44$	0	0
$45$	−1.07889	−0.160832
$46$	0	0
$47$	−11.8478	−1.72819	−0.864093	−	0.503332i	$-0.832107\pi$
$47$	−11.8478	−1.72819	−0.864093	+	0.503332i	$0.832107\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.76726	−0.667550
$52$	0	0
$53$	11.7785	1.61791	0.808954	−	0.587872i	$-0.200034\pi$
$53$	11.7785	1.61791	0.808954	+	0.587872i	$0.200034\pi$
$54$	0	0
$55$	1.07889	0.145478
$56$	0	0
$57$	6.61480	0.876152
$58$	0	0
$59$	6.60499	0.859896	0.429948	−	0.902854i	$-0.358532\pi$
$59$	6.60499	0.859896	0.429948	+	0.902854i	$0.358532\pi$
$60$	0	0
$61$	−13.6819	−1.75179	−0.875896	−	0.482501i	$-0.839729\pi$
$61$	−13.6819	−1.75179	−0.875896	+	0.482501i	$0.839729\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.874893	−0.108517
$66$	0	0
$67$	5.37194	0.656287	0.328144	−	0.944628i	$-0.393577\pi$
$67$	5.37194	0.656287	0.328144	+	0.944628i	$0.393577\pi$
$68$	0	0
$69$	−0.429382	−0.0516915
$70$	0	0
$71$	−10.1311	−1.20234	−0.601168	−	0.799122i	$-0.705298\pi$
$71$	−10.1311	−1.20234	−0.601168	+	0.799122i	$0.705298\pi$
$72$	0	0
$73$	12.2967	1.43922	0.719612	−	0.694377i	$-0.244320\pi$
$73$	12.2967	1.43922	0.719612	+	0.694377i	$0.244320\pi$
$74$	0	0
$75$	−3.83599	−0.442942
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.7301	1.31974	0.659869	−	0.751380i	$-0.270612\pi$
$79$	11.7301	1.31974	0.659869	+	0.751380i	$0.270612\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.19245	−1.00900	−0.504501	−	0.863411i	$-0.668324\pi$
$83$	−9.19245	−1.00900	−0.504501	+	0.863411i	$0.668324\pi$
$84$	0	0
$85$	5.14337	0.557876
$86$	0	0
$87$	−1.18851	−0.127422
$88$	0	0
$89$	7.56155	0.801523	0.400762	−	0.916182i	$-0.368746\pi$
$89$	7.56155	0.801523	0.400762	+	0.916182i	$0.368746\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.206240	−0.0213861
$94$	0	0
$95$	−7.13666	−0.732206
$96$	0	0
$97$	5.34779	0.542985	0.271493	−	0.962441i	$-0.412483\pi$
$97$	5.34779	0.542985	0.271493	+	0.962441i	$0.412483\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−17.6689	−1.75812	−0.879062	−	0.476707i	$-0.841830\pi$
$101$	−17.6689	−1.75812	−0.879062	+	0.476707i	$0.841830\pi$
$102$	0	0
$103$	−11.7707	−1.15980	−0.579900	−	0.814687i	$-0.696908\pi$
$103$	−11.7707	−1.15980	−0.579900	+	0.814687i	$0.696908\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.933007	−0.0901972	−0.0450986	−	0.998983i	$-0.514360\pi$
$107$	−0.933007	−0.0901972	−0.0450986	+	0.998983i	$0.514360\pi$
$108$	0	0
$109$	12.5720	1.20418	0.602088	−	0.798430i	$-0.294336\pi$
$109$	12.5720	1.20418	0.602088	+	0.798430i	$0.294336\pi$
$110$	0	0
$111$	5.75092	0.545853
$112$	0	0
$113$	−20.3249	−1.91201	−0.956004	−	0.293352i	$-0.905229\pi$
$113$	−20.3249	−1.91201	−0.956004	+	0.293352i	$0.905229\pi$
$114$	0	0
$115$	0.463257	0.0431990
$116$	0	0
$117$	0.810917	0.0749693
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.88022	−0.620368
$124$	0	0
$125$	9.53309	0.852665
$126$	0	0
$127$	−9.54862	−0.847303	−0.423651	−	0.905825i	$-0.639252\pi$
$127$	−9.54862	−0.847303	−0.423651	+	0.905825i	$0.639252\pi$
$128$	0	0
$129$	−12.9254	−1.13801
$130$	0	0
$131$	−3.83859	−0.335379	−0.167690	−	0.985840i	$-0.553631\pi$
$131$	−3.83859	−0.335379	−0.167690	+	0.985840i	$0.553631\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.07889	−0.0928563
$136$	0	0
$137$	−5.51641	−0.471299	−0.235649	−	0.971838i	$-0.575722\pi$
$137$	−5.51641	−0.471299	−0.235649	+	0.971838i	$0.575722\pi$
$138$	0	0
$139$	−12.7769	−1.08373	−0.541863	−	0.840467i	$-0.682281\pi$
$139$	−12.7769	−1.08373	−0.541863	+	0.840467i	$0.682281\pi$
$140$	0	0
$141$	−11.8478	−0.997769
$142$	0	0
$143$	−0.810917	−0.0678123
$144$	0	0
$145$	1.28228	0.106487
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.362846	−0.0297255	−0.0148628	−	0.999890i	$-0.504731\pi$
$149$	−0.362846	−0.0297255	−0.0148628	+	0.999890i	$0.504731\pi$
$150$	0	0
$151$	10.8843	0.885754	0.442877	−	0.896582i	$-0.353958\pi$
$151$	10.8843	0.885754	0.442877	+	0.896582i	$0.353958\pi$
$152$	0	0
$153$	−4.76726	−0.385410
$154$	0	0
$155$	0.222511	0.0178725
$156$	0	0
$157$	−7.03886	−0.561762	−0.280881	−	0.959743i	$-0.590627\pi$
$157$	−7.03886	−0.561762	−0.280881	+	0.959743i	$0.590627\pi$
$158$	0	0
$159$	11.7785	0.934100
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.51452	−0.745234	−0.372617	−	0.927985i	$-0.621540\pi$
$163$	−9.51452	−0.745234	−0.372617	+	0.927985i	$0.621540\pi$
$164$	0	0
$165$	1.07889	0.0839917
$166$	0	0
$167$	−4.74495	−0.367176	−0.183588	−	0.983003i	$-0.558771\pi$
$167$	−4.74495	−0.367176	−0.183588	+	0.983003i	$0.558771\pi$
$168$	0	0
$169$	−12.3424	−0.949416
$170$	0	0
$171$	6.61480	0.505847
$172$	0	0
$173$	2.52557	0.192015	0.0960076	−	0.995381i	$-0.469393\pi$
$173$	2.52557	0.192015	0.0960076	+	0.995381i	$0.469393\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.60499	0.496461
$178$	0	0
$179$	22.8290	1.70632	0.853162	−	0.521647i	$-0.174682\pi$
$179$	22.8290	1.70632	0.853162	+	0.521647i	$0.174682\pi$
$180$	0	0
$181$	−15.9129	−1.18280	−0.591399	−	0.806379i	$-0.701424\pi$
$181$	−15.9129	−1.18280	−0.591399	+	0.806379i	$0.701424\pi$
$182$	0	0
$183$	−13.6819	−1.01140
$184$	0	0
$185$	−6.20462	−0.456173
$186$	0	0
$187$	4.76726	0.348617
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.9631	−1.15505	−0.577523	−	0.816374i	$-0.695981\pi$
$191$	−15.9631	−1.15505	−0.577523	+	0.816374i	$0.695981\pi$
$192$	0	0
$193$	−4.05634	−0.291982	−0.145991	−	0.989286i	$-0.546637\pi$
$193$	−4.05634	−0.291982	−0.145991	+	0.989286i	$0.546637\pi$
$194$	0	0
$195$	−0.874893	−0.0626524
$196$	0	0
$197$	22.6601	1.61447	0.807234	−	0.590231i	$-0.200963\pi$
$197$	22.6601	1.61447	0.807234	+	0.590231i	$0.200963\pi$
$198$	0	0
$199$	−19.3187	−1.36946	−0.684731	−	0.728795i	$-0.740081\pi$
$199$	−19.3187	−1.36946	−0.684731	+	0.728795i	$0.740081\pi$
$200$	0	0
$201$	5.37194	0.378908
$202$	0	0
$203$	0	0
$204$	0	0
$205$	7.42302	0.518446
$206$	0	0
$207$	−0.429382	−0.0298441
$208$	0	0
$209$	−6.61480	−0.457555
$210$	0	0
$211$	−0.569826	−0.0392284	−0.0196142	−	0.999808i	$-0.506244\pi$
$211$	−0.569826	−0.0392284	−0.0196142	+	0.999808i	$0.506244\pi$
$212$	0	0
$213$	−10.1311	−0.694169
$214$	0	0
$215$	13.9451	0.951046
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.2967	0.830936
$220$	0	0
$221$	−3.86586	−0.260046
$222$	0	0
$223$	10.9670	0.734406	0.367203	−	0.930141i	$-0.380315\pi$
$223$	10.9670	0.734406	0.367203	+	0.930141i	$0.380315\pi$
$224$	0	0
$225$	−3.83599	−0.255733
$226$	0	0
$227$	−9.68357	−0.642721	−0.321361	−	0.946957i	$-0.604140\pi$
$227$	−9.68357	−0.642721	−0.321361	+	0.946957i	$0.604140\pi$
$228$	0	0
$229$	−9.66502	−0.638683	−0.319341	−	0.947640i	$-0.603462\pi$
$229$	−9.66502	−0.638683	−0.319341	+	0.947640i	$0.603462\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.479790	−0.0314321	−0.0157160	−	0.999876i	$-0.505003\pi$
$233$	−0.479790	−0.0314321	−0.0157160	+	0.999876i	$0.505003\pi$
$234$	0	0
$235$	12.7826	0.833842
$236$	0	0
$237$	11.7301	0.761952
$238$	0	0
$239$	−3.76029	−0.243233	−0.121616	−	0.992577i	$-0.538808\pi$
$239$	−3.76029	−0.243233	−0.121616	+	0.992577i	$0.538808\pi$
$240$	0	0
$241$	−19.1062	−1.23074	−0.615368	−	0.788240i	$-0.710993\pi$
$241$	−19.1062	−1.23074	−0.615368	+	0.788240i	$0.710993\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.36406	0.341307
$248$	0	0
$249$	−9.19245	−0.582548
$250$	0	0
$251$	6.91665	0.436575	0.218288	−	0.975884i	$-0.429953\pi$
$251$	6.91665	0.436575	0.218288	+	0.975884i	$0.429953\pi$
$252$	0	0
$253$	0.429382	0.0269950
$254$	0	0
$255$	5.14337	0.322090
$256$	0	0
$257$	−21.2676	−1.32664	−0.663319	−	0.748336i	$-0.730853\pi$
$257$	−21.2676	−1.32664	−0.663319	+	0.748336i	$0.730853\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.18851	−0.0735670
$262$	0	0
$263$	−21.8326	−1.34626	−0.673129	−	0.739525i	$-0.735050\pi$
$263$	−21.8326	−1.34626	−0.673129	+	0.739525i	$0.735050\pi$
$264$	0	0
$265$	−12.7078	−0.780633
$266$	0	0
$267$	7.56155	0.462760
$268$	0	0
$269$	−0.00482536	−0.000294207 0	−0.000147104	−	1.00000i	$-0.500047\pi$
$269$	−0.00482536	−0.000294207 0	−0.000147104		1.00000i	$0.500047\pi$
$270$	0	0
$271$	6.02746	0.366142	0.183071	−	0.983100i	$-0.441396\pi$
$271$	6.02746	0.366142	0.183071	+	0.983100i	$0.441396\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.83599	0.231319
$276$	0	0
$277$	2.69584	0.161978	0.0809888	−	0.996715i	$-0.474192\pi$
$277$	2.69584	0.161978	0.0809888	+	0.996715i	$0.474192\pi$
$278$	0	0
$279$	−0.206240	−0.0123473
$280$	0	0
$281$	−5.57537	−0.332599	−0.166299	−	0.986075i	$-0.553182\pi$
$281$	−5.57537	−0.332599	−0.166299	+	0.986075i	$0.553182\pi$
$282$	0	0
$283$	19.2064	1.14170	0.570851	−	0.821054i	$-0.306614\pi$
$283$	19.2064	1.14170	0.570851	+	0.821054i	$0.306614\pi$
$284$	0	0
$285$	−7.13666	−0.422739
$286$	0	0
$287$	0	0
$288$	0	0
$289$	5.72680	0.336871
$290$	0	0
$291$	5.34779	0.313493
$292$	0	0
$293$	−8.59490	−0.502119	−0.251060	−	0.967972i	$-0.580779\pi$
$293$	−8.59490	−0.502119	−0.251060	+	0.967972i	$0.580779\pi$
$294$	0	0
$295$	−7.12607	−0.414896
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−0.348194	−0.0201366
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−17.6689	−1.01505
$304$	0	0
$305$	14.7613	0.845231
$306$	0	0
$307$	−29.4032	−1.67813	−0.839065	−	0.544031i	$-0.816897\pi$
$307$	−29.4032	−1.67813	−0.839065	+	0.544031i	$0.816897\pi$
$308$	0	0
$309$	−11.7707	−0.669611
$310$	0	0
$311$	−22.2173	−1.25983	−0.629915	−	0.776664i	$-0.716910\pi$
$311$	−22.2173	−1.25983	−0.629915	+	0.776664i	$0.716910\pi$
$312$	0	0
$313$	−10.9278	−0.617676	−0.308838	−	0.951115i	$-0.599940\pi$
$313$	−10.9278	−0.617676	−0.308838	+	0.951115i	$0.599940\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.2541	0.969087	0.484544	−	0.874767i	$-0.338986\pi$
$317$	17.2541	0.969087	0.484544	+	0.874767i	$0.338986\pi$
$318$	0	0
$319$	1.18851	0.0665438
$320$	0	0
$321$	−0.933007	−0.0520754
$322$	0	0
$323$	−31.5345	−1.75463
$324$	0	0
$325$	−3.11067	−0.172549
$326$	0	0
$327$	12.5720	0.695231
$328$	0	0
$329$	0	0
$330$	0	0
$331$	32.1404	1.76660	0.883299	−	0.468810i	$-0.155317\pi$
$331$	32.1404	1.76660	0.883299	+	0.468810i	$0.155317\pi$
$332$	0	0
$333$	5.75092	0.315148
$334$	0	0
$335$	−5.79575	−0.316656
$336$	0	0
$337$	−23.8992	−1.30187	−0.650935	−	0.759133i	$-0.725623\pi$
$337$	−23.8992	−1.30187	−0.650935	+	0.759133i	$0.725623\pi$
$338$	0	0
$339$	−20.3249	−1.10390
$340$	0	0
$341$	0.206240	0.0111685
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0.463257	0.0249409
$346$	0	0
$347$	7.03555	0.377688	0.188844	−	0.982007i	$-0.439526\pi$
$347$	7.03555	0.377688	0.188844	+	0.982007i	$0.439526\pi$
$348$	0	0
$349$	−8.81755	−0.471993	−0.235996	−	0.971754i	$-0.575835\pi$
$349$	−8.81755	−0.471993	−0.235996	+	0.971754i	$0.575835\pi$
$350$	0	0
$351$	0.810917	0.0432836
$352$	0	0
$353$	−7.19765	−0.383092	−0.191546	−	0.981484i	$-0.561350\pi$
$353$	−7.19765	−0.383092	−0.191546	+	0.981484i	$0.561350\pi$
$354$	0	0
$355$	10.9303	0.580122
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.55771	0.240547	0.120273	−	0.992741i	$-0.461623\pi$
$359$	4.55771	0.240547	0.120273	+	0.992741i	$0.461623\pi$
$360$	0	0
$361$	24.7556	1.30293
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−13.2669	−0.694419
$366$	0	0
$367$	17.2072	0.898209	0.449105	−	0.893479i	$-0.351743\pi$
$367$	17.2072	0.898209	0.449105	+	0.893479i	$0.351743\pi$
$368$	0	0
$369$	−6.88022	−0.358170
$370$	0	0
$371$	0	0
$372$	0	0
$373$	20.8576	1.07997	0.539984	−	0.841675i	$-0.318430\pi$
$373$	20.8576	1.07997	0.539984	+	0.841675i	$0.318430\pi$
$374$	0	0
$375$	9.53309	0.492286
$376$	0	0
$377$	−0.963784	−0.0496374
$378$	0	0
$379$	7.97059	0.409422	0.204711	−	0.978822i	$-0.434375\pi$
$379$	7.97059	0.409422	0.204711	+	0.978822i	$0.434375\pi$
$380$	0	0
$381$	−9.54862	−0.489191
$382$	0	0
$383$	19.4294	0.992796	0.496398	−	0.868095i	$-0.334656\pi$
$383$	19.4294	0.992796	0.496398	+	0.868095i	$0.334656\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.9254	−0.657033
$388$	0	0
$389$	−30.3548	−1.53905	−0.769526	−	0.638616i	$-0.779507\pi$
$389$	−30.3548	−1.53905	−0.769526	+	0.638616i	$0.779507\pi$
$390$	0	0
$391$	2.04698	0.103520
$392$	0	0
$393$	−3.83859	−0.193631
$394$	0	0
$395$	−12.6555	−0.636768
$396$	0	0
$397$	−35.0452	−1.75887	−0.879435	−	0.476019i	$-0.842079\pi$
$397$	−35.0452	−1.75887	−0.879435	+	0.476019i	$0.842079\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−29.7546	−1.48587	−0.742937	−	0.669362i	$-0.766568\pi$
$401$	−29.7546	−1.48587	−0.742937	+	0.669362i	$0.766568\pi$
$402$	0	0
$403$	−0.167243	−0.00833099
$404$	0	0
$405$	−1.07889	−0.0536106
$406$	0	0
$407$	−5.75092	−0.285062
$408$	0	0
$409$	13.2224	0.653805	0.326902	−	0.945058i	$-0.393995\pi$
$409$	13.2224	0.653805	0.326902	+	0.945058i	$0.393995\pi$
$410$	0	0
$411$	−5.51641	−0.272104
$412$	0	0
$413$	0	0
$414$	0	0
$415$	9.91767	0.486839
$416$	0	0
$417$	−12.7769	−0.625690
$418$	0	0
$419$	15.3604	0.750406	0.375203	−	0.926943i	$-0.377573\pi$
$419$	15.3604	0.750406	0.375203	+	0.926943i	$0.377573\pi$
$420$	0	0
$421$	11.7644	0.573363	0.286681	−	0.958026i	$-0.407448\pi$
$421$	11.7644	0.573363	0.286681	+	0.958026i	$0.407448\pi$
$422$	0	0
$423$	−11.8478	−0.576062
$424$	0	0
$425$	18.2872	0.887058
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.810917	−0.0391515
$430$	0	0
$431$	22.0274	1.06102	0.530511	−	0.847678i	$-0.322000\pi$
$431$	22.0274	1.06102	0.530511	+	0.847678i	$0.322000\pi$
$432$	0	0
$433$	−1.59285	−0.0765477	−0.0382739	−	0.999267i	$-0.512186\pi$
$433$	−1.59285	−0.0765477	−0.0382739	+	0.999267i	$0.512186\pi$
$434$	0	0
$435$	1.28228	0.0614804
$436$	0	0
$437$	−2.84028	−0.135869
$438$	0	0
$439$	2.81404	0.134307	0.0671534	−	0.997743i	$-0.478608\pi$
$439$	2.81404	0.134307	0.0671534	+	0.997743i	$0.478608\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.81941	0.276488	0.138244	−	0.990398i	$-0.455854\pi$
$443$	5.81941	0.276488	0.138244	+	0.990398i	$0.455854\pi$
$444$	0	0
$445$	−8.15811	−0.386731
$446$	0	0
$447$	−0.362846	−0.0171620
$448$	0	0
$449$	29.1514	1.37574	0.687871	−	0.725833i	$-0.258546\pi$
$449$	29.1514	1.37574	0.687871	+	0.725833i	$0.258546\pi$
$450$	0	0
$451$	6.88022	0.323977
$452$	0	0
$453$	10.8843	0.511390
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.8420	1.16206	0.581031	−	0.813882i	$-0.302650\pi$
$457$	24.8420	1.16206	0.581031	+	0.813882i	$0.302650\pi$
$458$	0	0
$459$	−4.76726	−0.222517
$460$	0	0
$461$	26.2571	1.22291	0.611457	−	0.791278i	$-0.290584\pi$
$461$	26.2571	1.22291	0.611457	+	0.791278i	$0.290584\pi$
$462$	0	0
$463$	−0.633989	−0.0294640	−0.0147320	−	0.999891i	$-0.504690\pi$
$463$	−0.633989	−0.0294640	−0.0147320	+	0.999891i	$0.504690\pi$
$464$	0	0
$465$	0.222511	0.0103187
$466$	0	0
$467$	−3.22434	−0.149205	−0.0746023	−	0.997213i	$-0.523769\pi$
$467$	−3.22434	−0.149205	−0.0746023	+	0.997213i	$0.523769\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−7.03886	−0.324334
$472$	0	0
$473$	12.9254	0.594309
$474$	0	0
$475$	−25.3743	−1.16425
$476$	0	0
$477$	11.7785	0.539303
$478$	0	0
$479$	−24.6890	−1.12807	−0.564034	−	0.825752i	$-0.690751\pi$
$479$	−24.6890	−1.12807	−0.564034	+	0.825752i	$0.690751\pi$
$480$	0	0
$481$	4.66352	0.212638
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.76969	−0.261988
$486$	0	0
$487$	11.2781	0.511058	0.255529	−	0.966801i	$-0.417750\pi$
$487$	11.2781	0.511058	0.255529	+	0.966801i	$0.417750\pi$
$488$	0	0
$489$	−9.51452	−0.430261
$490$	0	0
$491$	−10.2248	−0.461437	−0.230719	−	0.973021i	$-0.574108\pi$
$491$	−10.2248	−0.461437	−0.230719	+	0.973021i	$0.574108\pi$
$492$	0	0
$493$	5.66594	0.255181
$494$	0	0
$495$	1.07889	0.0484926
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−6.39594	−0.286321	−0.143161	−	0.989699i	$-0.545727\pi$
$499$	−6.39594	−0.286321	−0.143161	+	0.989699i	$0.545727\pi$
$500$	0	0
$501$	−4.74495	−0.211989
$502$	0	0
$503$	−9.94117	−0.443255	−0.221627	−	0.975131i	$-0.571137\pi$
$503$	−9.94117	−0.443255	−0.221627	+	0.975131i	$0.571137\pi$
$504$	0	0
$505$	19.0629	0.848287
$506$	0	0
$507$	−12.3424	−0.548146
$508$	0	0
$509$	25.7188	1.13997	0.569983	−	0.821656i	$-0.306950\pi$
$509$	25.7188	1.13997	0.569983	+	0.821656i	$0.306950\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.61480	0.292051
$514$	0	0
$515$	12.6993	0.559599
$516$	0	0
$517$	11.8478	0.521068
$518$	0	0
$519$	2.52557	0.110860
$520$	0	0
$521$	−32.4602	−1.42211	−0.711054	−	0.703137i	$-0.751782\pi$
$521$	−32.4602	−1.42211	−0.711054	+	0.703137i	$0.751782\pi$
$522$	0	0
$523$	−22.6486	−0.990354	−0.495177	−	0.868792i	$-0.664897\pi$
$523$	−22.6486	−0.990354	−0.495177	+	0.868792i	$0.664897\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.983199	0.0428288
$528$	0	0
$529$	−22.8156	−0.991984
$530$	0	0
$531$	6.60499	0.286632
$532$	0	0
$533$	−5.57929	−0.241666
$534$	0	0
$535$	1.00661	0.0435198
$536$	0	0
$537$	22.8290	0.985146
$538$	0	0
$539$	0	0
$540$	0	0
$541$	26.8910	1.15613	0.578067	−	0.815990i	$-0.303807\pi$
$541$	26.8910	1.15613	0.578067	+	0.815990i	$0.303807\pi$
$542$	0	0
$543$	−15.9129	−0.682889
$544$	0	0
$545$	−13.5638	−0.581009
$546$	0	0
$547$	−41.7290	−1.78420	−0.892101	−	0.451836i	$-0.850769\pi$
$547$	−41.7290	−1.78420	−0.892101	+	0.451836i	$0.850769\pi$
$548$	0	0
$549$	−13.6819	−0.583930
$550$	0	0
$551$	−7.86176	−0.334922
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.20462	−0.263372
$556$	0	0
$557$	2.27321	0.0963192	0.0481596	−	0.998840i	$-0.484664\pi$
$557$	2.27321	0.0963192	0.0481596	+	0.998840i	$0.484664\pi$
$558$	0	0
$559$	−10.4814	−0.443316
$560$	0	0
$561$	4.76726	0.201274
$562$	0	0
$563$	−37.7561	−1.59123	−0.795615	−	0.605803i	$-0.792852\pi$
$563$	−37.7561	−1.59123	−0.795615	+	0.605803i	$0.792852\pi$
$564$	0	0
$565$	21.9284	0.922536
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.22233	−0.0931651	−0.0465825	−	0.998914i	$-0.514833\pi$
$569$	−2.22233	−0.0931651	−0.0465825	+	0.998914i	$0.514833\pi$
$570$	0	0
$571$	38.1979	1.59853	0.799265	−	0.600978i	$-0.205222\pi$
$571$	38.1979	1.59853	0.799265	+	0.600978i	$0.205222\pi$
$572$	0	0
$573$	−15.9631	−0.666866
$574$	0	0
$575$	1.64711	0.0686891
$576$	0	0
$577$	−28.1586	−1.17226	−0.586130	−	0.810217i	$-0.699349\pi$
$577$	−28.1586	−1.17226	−0.586130	+	0.810217i	$0.699349\pi$
$578$	0	0
$579$	−4.05634	−0.168576
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−11.7785	−0.487818
$584$	0	0
$585$	−0.874893	−0.0361724
$586$	0	0
$587$	−38.3664	−1.58355	−0.791776	−	0.610811i	$-0.790843\pi$
$587$	−38.3664	−1.58355	−0.791776	+	0.610811i	$0.790843\pi$
$588$	0	0
$589$	−1.36423	−0.0562123
$590$	0	0
$591$	22.6601	0.932114
$592$	0	0
$593$	−21.1270	−0.867583	−0.433791	−	0.901013i	$-0.642825\pi$
$593$	−21.1270	−0.867583	−0.433791	+	0.901013i	$0.642825\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.3187	−0.790660
$598$	0	0
$599$	25.5769	1.04504	0.522522	−	0.852626i	$-0.324991\pi$
$599$	25.5769	1.04504	0.522522	+	0.852626i	$0.324991\pi$
$600$	0	0
$601$	−3.71491	−0.151534	−0.0757672	−	0.997126i	$-0.524141\pi$
$601$	−3.71491	−0.151534	−0.0757672	+	0.997126i	$0.524141\pi$
$602$	0	0
$603$	5.37194	0.218762
$604$	0	0
$605$	−1.07889	−0.0438632
$606$	0	0
$607$	−2.33388	−0.0947293	−0.0473646	−	0.998878i	$-0.515082\pi$
$607$	−2.33388	−0.0947293	−0.0473646	+	0.998878i	$0.515082\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.60763	−0.388683
$612$	0	0
$613$	7.42194	0.299770	0.149885	−	0.988703i	$-0.452110\pi$
$613$	7.42194	0.299770	0.149885	+	0.988703i	$0.452110\pi$
$614$	0	0
$615$	7.42302	0.299325
$616$	0	0
$617$	20.6685	0.832082	0.416041	−	0.909346i	$-0.363417\pi$
$617$	20.6685	0.832082	0.416041	+	0.909346i	$0.363417\pi$
$618$	0	0
$619$	33.7441	1.35629	0.678145	−	0.734928i	$-0.262784\pi$
$619$	33.7441	1.35629	0.678145	+	0.734928i	$0.262784\pi$
$620$	0	0
$621$	−0.429382	−0.0172305
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.89477	0.355791
$626$	0	0
$627$	−6.61480	−0.264170
$628$	0	0
$629$	−27.4161	−1.09315
$630$	0	0
$631$	7.54524	0.300371	0.150186	−	0.988658i	$-0.452013\pi$
$631$	7.54524	0.300371	0.150186	+	0.988658i	$0.452013\pi$
$632$	0	0
$633$	−0.569826	−0.0226486
$634$	0	0
$635$	10.3019	0.408820
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.1311	−0.400779
$640$	0	0
$641$	−9.14407	−0.361169	−0.180585	−	0.983559i	$-0.557799\pi$
$641$	−9.14407	−0.361169	−0.180585	+	0.983559i	$0.557799\pi$
$642$	0	0
$643$	−5.24782	−0.206954	−0.103477	−	0.994632i	$-0.532997\pi$
$643$	−5.24782	−0.206954	−0.103477	+	0.994632i	$0.532997\pi$
$644$	0	0
$645$	13.9451	0.549087
$646$	0	0
$647$	34.3148	1.34905	0.674527	−	0.738250i	$-0.264347\pi$
$647$	34.3148	1.34905	0.674527	+	0.738250i	$0.264347\pi$
$648$	0	0
$649$	−6.60499	−0.259268
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.9734	1.52515	0.762573	−	0.646902i	$-0.223936\pi$
$653$	38.9734	1.52515	0.762573	+	0.646902i	$0.223936\pi$
$654$	0	0
$655$	4.14143	0.161819
$656$	0	0
$657$	12.2967	0.479741
$658$	0	0
$659$	−4.45780	−0.173651	−0.0868256	−	0.996224i	$-0.527672\pi$
$659$	−4.45780	−0.173651	−0.0868256	+	0.996224i	$0.527672\pi$
$660$	0	0
$661$	43.0808	1.67565	0.837824	−	0.545940i	$-0.183828\pi$
$661$	43.0808	1.67565	0.837824	+	0.545940i	$0.183828\pi$
$662$	0	0
$663$	−3.86586	−0.150137
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.510325	0.0197599
$668$	0	0
$669$	10.9670	0.424010
$670$	0	0
$671$	13.6819	0.528185
$672$	0	0
$673$	37.6811	1.45250	0.726249	−	0.687432i	$-0.241262\pi$
$673$	37.6811	1.45250	0.726249	+	0.687432i	$0.241262\pi$
$674$	0	0
$675$	−3.83599	−0.147647
$676$	0	0
$677$	−23.2420	−0.893261	−0.446630	−	0.894719i	$-0.647376\pi$
$677$	−23.2420	−0.893261	−0.446630	+	0.894719i	$0.647376\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−9.68357	−0.371075
$682$	0	0
$683$	−34.1311	−1.30599	−0.652994	−	0.757363i	$-0.726487\pi$
$683$	−34.1311	−1.30599	−0.652994	+	0.757363i	$0.726487\pi$
$684$	0	0
$685$	5.95162	0.227400
$686$	0	0
$687$	−9.66502	−0.368744
$688$	0	0
$689$	9.55143	0.363881
$690$	0	0
$691$	49.9453	1.90001	0.950004	−	0.312238i	$-0.101079\pi$
$691$	49.9453	1.90001	0.950004	+	0.312238i	$0.101079\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.7850	0.522893
$696$	0	0
$697$	32.7998	1.24238
$698$	0	0
$699$	−0.479790	−0.0181473
$700$	0	0
$701$	−17.4618	−0.659525	−0.329762	−	0.944064i	$-0.606969\pi$
$701$	−17.4618	−0.659525	−0.329762	+	0.944064i	$0.606969\pi$
$702$	0	0
$703$	38.0412	1.43475
$704$	0	0
$705$	12.7826	0.481419
$706$	0	0
$707$	0	0
$708$	0	0
$709$	40.4492	1.51910	0.759552	−	0.650447i	$-0.225418\pi$
$709$	40.4492	1.51910	0.759552	+	0.650447i	$0.225418\pi$
$710$	0	0
$711$	11.7301	0.439913
$712$	0	0
$713$	0.0885557	0.00331644
$714$	0	0
$715$	0.874893	0.0327191
$716$	0	0
$717$	−3.76029	−0.140431
$718$	0	0
$719$	25.1684	0.938622	0.469311	−	0.883033i	$-0.344502\pi$
$719$	25.1684	0.938622	0.469311	+	0.883033i	$0.344502\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−19.1062	−0.710566
$724$	0	0
$725$	4.55911	0.169321
$726$	0	0
$727$	21.3688	0.792526	0.396263	−	0.918137i	$-0.370307\pi$
$727$	21.3688	0.792526	0.396263	+	0.918137i	$0.370307\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	61.6186	2.27905
$732$	0	0
$733$	−2.32117	−0.0857344	−0.0428672	−	0.999081i	$-0.513649\pi$
$733$	−2.32117	−0.0857344	−0.0428672	+	0.999081i	$0.513649\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.37194	−0.197878
$738$	0	0
$739$	−44.2338	−1.62717	−0.813584	−	0.581447i	$-0.802487\pi$
$739$	−44.2338	−1.62717	−0.813584	+	0.581447i	$0.802487\pi$
$740$	0	0
$741$	5.36406	0.197054
$742$	0	0
$743$	−7.59083	−0.278481	−0.139240	−	0.990259i	$-0.544466\pi$
$743$	−7.59083	−0.278481	−0.139240	+	0.990259i	$0.544466\pi$
$744$	0	0
$745$	0.391472	0.0143424
$746$	0	0
$747$	−9.19245	−0.336334
$748$	0	0
$749$	0	0
$750$	0	0
$751$	35.0974	1.28072	0.640362	−	0.768074i	$-0.278785\pi$
$751$	35.0974	1.28072	0.640362	+	0.768074i	$0.278785\pi$
$752$	0	0
$753$	6.91665	0.252057
$754$	0	0
$755$	−11.7430	−0.427372
$756$	0	0
$757$	−6.09387	−0.221486	−0.110743	−	0.993849i	$-0.535323\pi$
$757$	−6.09387	−0.221486	−0.110743	+	0.993849i	$0.535323\pi$
$758$	0	0
$759$	0.429382	0.0155856
$760$	0	0
$761$	−18.6427	−0.675798	−0.337899	−	0.941182i	$-0.609716\pi$
$761$	−18.6427	−0.675798	−0.337899	+	0.941182i	$0.609716\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.14337	0.185959
$766$	0	0
$767$	5.35610	0.193398
$768$	0	0
$769$	28.6603	1.03352	0.516758	−	0.856131i	$-0.327139\pi$
$769$	28.6603	1.03352	0.516758	+	0.856131i	$0.327139\pi$
$770$	0	0
$771$	−21.2676	−0.765935
$772$	0	0
$773$	−25.7753	−0.927072	−0.463536	−	0.886078i	$-0.653420\pi$
$773$	−25.7753	−0.927072	−0.463536	+	0.886078i	$0.653420\pi$
$774$	0	0
$775$	0.791134	0.0284184
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−45.5113	−1.63061
$780$	0	0
$781$	10.1311	0.362518
$782$	0	0
$783$	−1.18851	−0.0424739
$784$	0	0
$785$	7.59418	0.271048
$786$	0	0
$787$	1.34955	0.0481063	0.0240532	−	0.999711i	$-0.492343\pi$
$787$	1.34955	0.0481063	0.0240532	+	0.999711i	$0.492343\pi$
$788$	0	0
$789$	−21.8326	−0.777262
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−11.0949	−0.393992
$794$	0	0
$795$	−12.7078	−0.450699
$796$	0	0
$797$	55.8305	1.97762	0.988809	−	0.149187i	$-0.0476657\pi$
$797$	55.8305	1.97762	0.988809	+	0.149187i	$0.0476657\pi$
$798$	0	0
$799$	56.4818	1.99818
$800$	0	0
$801$	7.56155	0.267174
$802$	0	0
$803$	−12.2967	−0.433942
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.00482536	−0.000169861 0
$808$	0	0
$809$	−39.0159	−1.37173	−0.685864	−	0.727730i	$-0.740575\pi$
$809$	−39.0159	−1.37173	−0.685864	+	0.727730i	$0.740575\pi$
$810$	0	0
$811$	−14.0720	−0.494134	−0.247067	−	0.968998i	$-0.579467\pi$
$811$	−14.0720	−0.494134	−0.247067	+	0.968998i	$0.579467\pi$
$812$	0	0
$813$	6.02746	0.211392
$814$	0	0
$815$	10.2651	0.359572
$816$	0	0
$817$	−85.4987	−2.99122
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.9008	0.694544	0.347272	−	0.937764i	$-0.387108\pi$
$821$	19.9008	0.694544	0.347272	+	0.937764i	$0.387108\pi$
$822$	0	0
$823$	49.1058	1.71172	0.855860	−	0.517207i	$-0.173028\pi$
$823$	49.1058	1.71172	0.855860	+	0.517207i	$0.173028\pi$
$824$	0	0
$825$	3.83599	0.133552
$826$	0	0
$827$	−30.9652	−1.07677	−0.538383	−	0.842700i	$-0.680965\pi$
$827$	−30.9652	−1.07677	−0.538383	+	0.842700i	$0.680965\pi$
$828$	0	0
$829$	27.1749	0.943824	0.471912	−	0.881646i	$-0.343564\pi$
$829$	27.1749	0.943824	0.471912	+	0.881646i	$0.343564\pi$
$830$	0	0
$831$	2.69584	0.0935178
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.11930	0.177161
$836$	0	0
$837$	−0.206240	−0.00712869
$838$	0	0
$839$	28.6058	0.987582	0.493791	−	0.869581i	$-0.335611\pi$
$839$	28.6058	0.987582	0.493791	+	0.869581i	$0.335611\pi$
$840$	0	0
$841$	−27.5874	−0.951291
$842$	0	0
$843$	−5.57537	−0.192026
$844$	0	0
$845$	13.3161	0.458089
$846$	0	0
$847$	0	0
$848$	0	0
$849$	19.2064	0.659162
$850$	0	0
$851$	−2.46934	−0.0846479
$852$	0	0
$853$	−42.9222	−1.46963	−0.734814	−	0.678269i	$-0.762730\pi$
$853$	−42.9222	−1.46963	−0.734814	+	0.678269i	$0.762730\pi$
$854$	0	0
$855$	−7.13666	−0.244069
$856$	0	0
$857$	−54.4384	−1.85958	−0.929789	−	0.368092i	$-0.880011\pi$
$857$	−54.4384	−1.85958	−0.929789	+	0.368092i	$0.880011\pi$
$858$	0	0
$859$	−12.4937	−0.426278	−0.213139	−	0.977022i	$-0.568369\pi$
$859$	−12.4937	−0.426278	−0.213139	+	0.977022i	$0.568369\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.77949	−0.0605746	−0.0302873	−	0.999541i	$-0.509642\pi$
$863$	−1.77949	−0.0605746	−0.0302873	+	0.999541i	$0.509642\pi$
$864$	0	0
$865$	−2.72481	−0.0926465
$866$	0	0
$867$	5.72680	0.194492
$868$	0	0
$869$	−11.7301	−0.397916
$870$	0	0
$871$	4.35620	0.147604
$872$	0	0
$873$	5.34779	0.180995
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−45.6320	−1.54088	−0.770441	−	0.637511i	$-0.779964\pi$
$877$	−45.6320	−1.54088	−0.770441	+	0.637511i	$0.779964\pi$
$878$	0	0
$879$	−8.59490	−0.289899
$880$	0	0
$881$	19.1615	0.645568	0.322784	−	0.946473i	$-0.395381\pi$
$881$	19.1615	0.645568	0.322784	+	0.946473i	$0.395381\pi$
$882$	0	0
$883$	39.1074	1.31607	0.658034	−	0.752988i	$-0.271388\pi$
$883$	39.1074	1.31607	0.658034	+	0.752988i	$0.271388\pi$
$884$	0	0
$885$	−7.12607	−0.239540
$886$	0	0
$887$	48.4208	1.62581	0.812906	−	0.582395i	$-0.197884\pi$
$887$	48.4208	1.62581	0.812906	+	0.582395i	$0.197884\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−78.3712	−2.62259
$894$	0	0
$895$	−24.6301	−0.823293
$896$	0	0
$897$	−0.348194	−0.0116258
$898$	0	0
$899$	0.245118	0.00817515
$900$	0	0
$901$	−56.1514	−1.87068
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.1683	0.570695
$906$	0	0
$907$	55.0502	1.82791	0.913955	−	0.405815i	$-0.133012\pi$
$907$	55.0502	1.82791	0.913955	+	0.405815i	$0.133012\pi$
$908$	0	0
$909$	−17.6689	−0.586041
$910$	0	0
$911$	−51.8765	−1.71874	−0.859372	−	0.511351i	$-0.829145\pi$
$911$	−51.8765	−1.71874	−0.859372	+	0.511351i	$0.829145\pi$
$912$	0	0
$913$	9.19245	0.304226
$914$	0	0
$915$	14.7613	0.487995
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.02706	0.0668666	0.0334333	−	0.999441i	$-0.489356\pi$
$919$	2.02706	0.0668666	0.0334333	+	0.999441i	$0.489356\pi$
$920$	0	0
$921$	−29.4032	−0.968869
$922$	0	0
$923$	−8.21546	−0.270415
$924$	0	0
$925$	−22.0605	−0.725343
$926$	0	0
$927$	−11.7707	−0.386600
$928$	0	0
$929$	28.8988	0.948137	0.474069	−	0.880488i	$-0.342785\pi$
$929$	28.8988	0.948137	0.474069	+	0.880488i	$0.342785\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−22.2173	−0.727363
$934$	0	0
$935$	−5.14337	−0.168206
$936$	0	0
$937$	10.8511	0.354491	0.177246	−	0.984167i	$-0.443281\pi$
$937$	10.8511	0.354491	0.177246	+	0.984167i	$0.443281\pi$
$938$	0	0
$939$	−10.9278	−0.356616
$940$	0	0
$941$	13.1149	0.427532	0.213766	−	0.976885i	$-0.431427\pi$
$941$	13.1149	0.427532	0.213766	+	0.976885i	$0.431427\pi$
$942$	0	0
$943$	2.95424	0.0962034
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.8073	−1.35856	−0.679278	−	0.733881i	$-0.737707\pi$
$947$	−41.8073	−1.35856	−0.679278	+	0.733881i	$0.737707\pi$
$948$	0	0
$949$	9.97163	0.323693
$950$	0	0
$951$	17.2541	0.559503
$952$	0	0
$953$	22.3916	0.725337	0.362668	−	0.931918i	$-0.381866\pi$
$953$	22.3916	0.725337	0.362668	+	0.931918i	$0.381866\pi$
$954$	0	0
$955$	17.2224	0.557304
$956$	0	0
$957$	1.18851	0.0384191
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.9575	−0.998628
$962$	0	0
$963$	−0.933007	−0.0300657
$964$	0	0
$965$	4.37635	0.140880
$966$	0	0
$967$	29.1727	0.938132	0.469066	−	0.883163i	$-0.344591\pi$
$967$	29.1727	0.938132	0.469066	+	0.883163i	$0.344591\pi$
$968$	0	0
$969$	−31.5345	−1.01303
$970$	0	0
$971$	27.4125	0.879708	0.439854	−	0.898069i	$-0.355030\pi$
$971$	27.4125	0.879708	0.439854	+	0.898069i	$0.355030\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.11067	−0.0996212
$976$	0	0
$977$	−22.9130	−0.733051	−0.366525	−	0.930408i	$-0.619453\pi$
$977$	−22.9130	−0.733051	−0.366525	+	0.930408i	$0.619453\pi$
$978$	0	0
$979$	−7.56155	−0.241668
$980$	0	0
$981$	12.5720	0.401392
$982$	0	0
$983$	32.2828	1.02966	0.514830	−	0.857292i	$-0.327855\pi$
$983$	32.2828	1.02966	0.514830	+	0.857292i	$0.327855\pi$
$984$	0	0
$985$	−24.4479	−0.778974
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.54992	0.176477
$990$	0	0
$991$	−8.87252	−0.281845	−0.140923	−	0.990021i	$-0.545007\pi$
$991$	−8.87252	−0.281845	−0.140923	+	0.990021i	$0.545007\pi$
$992$	0	0
$993$	32.1404	1.01995
$994$	0	0
$995$	20.8428	0.660760
$996$	0	0
$997$	11.0659	0.350459	0.175230	−	0.984528i	$-0.443933\pi$
$997$	11.0659	0.350459	0.175230	+	0.984528i	$0.443933\pi$
$998$	0	0
$999$	5.75092	0.181951

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6468.2.a.be.1.3	yes	6
7.6	odd	2		6468.2.a.bd.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6468.2.a.bd.1.4	✓	6	7.6	odd	2
6468.2.a.be.1.3	yes	6	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:	\( N \)	\(=\)	\( 6468 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6468.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.07889 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.07889 q^{5} +1.00000 q^{9} -1.00000 q^{11} +0.810917 q^{13} -1.07889 q^{15} -4.76726 q^{17} +6.61480 q^{19} -0.429382 q^{23} -3.83599 q^{25} +1.00000 q^{27} -1.18851 q^{29} -0.206240 q^{31} -1.00000 q^{33} +5.75092 q^{37} +0.810917 q^{39} -6.88022 q^{41} -12.9254 q^{43} -1.07889 q^{45} -11.8478 q^{47} -4.76726 q^{51} +11.7785 q^{53} +1.07889 q^{55} +6.61480 q^{57} +6.60499 q^{59} -13.6819 q^{61} -0.874893 q^{65} +5.37194 q^{67} -0.429382 q^{69} -10.1311 q^{71} +12.2967 q^{73} -3.83599 q^{75} +11.7301 q^{79} +1.00000 q^{81} -9.19245 q^{83} +5.14337 q^{85} -1.18851 q^{87} +7.56155 q^{89} -0.206240 q^{93} -7.13666 q^{95} +5.34779 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 4 q^{5} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 4 * q^5 + 6 * q^9	\( 6 q + 6 q^{3} - 4 q^{5} + 6 q^{9} - 6 q^{11} - 8 q^{13} - 4 q^{15} - 16 q^{17} - 4 q^{19} + 22 q^{25} + 6 q^{27} - 4 q^{29} - 6 q^{33} + 4 q^{37} - 8 q^{39} - 12 q^{41} - 8 q^{43} - 4 q^{45} + 12 q^{47} - 16 q^{51} + 4 q^{53} + 4 q^{55} - 4 q^{57} - 24 q^{59} - 16 q^{61} - 12 q^{65} + 8 q^{67} - 36 q^{71} - 36 q^{73} + 22 q^{75} + 6 q^{81} - 32 q^{83} - 20 q^{85} - 4 q^{87} - 12 q^{89} - 44 q^{95} - 16 q^{97} - 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - 4 * q^5 + 6 * q^9 - 6 * q^11 - 8 * q^13 - 4 * q^15 - 16 * q^17 - 4 * q^19 + 22 * q^25 + 6 * q^27 - 4 * q^29 - 6 * q^33 + 4 * q^37 - 8 * q^39 - 12 * q^41 - 8 * q^43 - 4 * q^45 + 12 * q^47 - 16 * q^51 + 4 * q^53 + 4 * q^55 - 4 * q^57 - 24 * q^59 - 16 * q^61 - 12 * q^65 + 8 * q^67 - 36 * q^71 - 36 * q^73 + 22 * q^75 + 6 * q^81 - 32 * q^83 - 20 * q^85 - 4 * q^87 - 12 * q^89 - 44 * q^95 - 16 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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