LMFDB - Embedded newform 6498.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6498, 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("6498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.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:	$51.8867912334$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	6498.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.467911 q^{5} -4.41147 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +0.467911 q^{5} -4.41147 q^{7} +1.00000 q^{8} +0.467911 q^{10} +5.53209 q^{11} -5.98545 q^{13} -4.41147 q^{14} +1.00000 q^{16} +4.34730 q^{17} +0.467911 q^{20} +5.53209 q^{22} +3.16250 q^{23} -4.78106 q^{25} -5.98545 q^{26} -4.41147 q^{28} -1.50980 q^{29} +2.18479 q^{31} +1.00000 q^{32} +4.34730 q^{34} -2.06418 q^{35} +2.75877 q^{37} +0.467911 q^{40} -1.95811 q^{41} +3.59627 q^{43} +5.53209 q^{44} +3.16250 q^{46} +9.24897 q^{47} +12.4611 q^{49} -4.78106 q^{50} -5.98545 q^{52} -2.67499 q^{53} +2.58853 q^{55} -4.41147 q^{56} -1.50980 q^{58} +1.32770 q^{59} -6.63816 q^{61} +2.18479 q^{62} +1.00000 q^{64} -2.80066 q^{65} +2.49020 q^{67} +4.34730 q^{68} -2.06418 q^{70} -7.59627 q^{71} +3.16519 q^{73} +2.75877 q^{74} -24.4047 q^{77} -1.26352 q^{79} +0.467911 q^{80} -1.95811 q^{82} +17.9368 q^{83} +2.03415 q^{85} +3.59627 q^{86} +5.53209 q^{88} +12.4807 q^{89} +26.4047 q^{91} +3.16250 q^{92} +9.24897 q^{94} +8.31046 q^{97} +12.4611 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 3 q^{8} + 6 q^{10} + 12 q^{11} - 3 q^{14} + 3 q^{16} + 12 q^{17} + 6 q^{20} + 12 q^{22} + 12 q^{23} + 3 q^{25} - 3 q^{28} - 6 q^{29} + 3 q^{31} + 3 q^{32} + 12 q^{34} + 3 q^{35} - 3 q^{37} + 6 q^{40} - 9 q^{41} - 3 q^{43} + 12 q^{44} + 12 q^{46} + 15 q^{47} + 3 q^{50} - 3 q^{53} + 18 q^{55} - 3 q^{56} - 6 q^{58} - 3 q^{61} + 3 q^{62} + 3 q^{64} + 6 q^{65} + 6 q^{67} + 12 q^{68} + 3 q^{70} - 9 q^{71} + 3 q^{73} - 3 q^{74} - 21 q^{77} - 9 q^{79} + 6 q^{80} - 9 q^{82} - 3 q^{83} + 27 q^{85} - 3 q^{86} + 12 q^{88} + 3 q^{89} + 27 q^{91} + 12 q^{92} + 15 q^{94} + 12 q^{97}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 3 * q^8 + 6 * q^10 + 12 * q^11 - 3 * q^14 + 3 * q^16 + 12 * q^17 + 6 * q^20 + 12 * q^22 + 12 * q^23 + 3 * q^25 - 3 * q^28 - 6 * q^29 + 3 * q^31 + 3 * q^32 + 12 * q^34 + 3 * q^35 - 3 * q^37 + 6 * q^40 - 9 * q^41 - 3 * q^43 + 12 * q^44 + 12 * q^46 + 15 * q^47 + 3 * q^50 - 3 * q^53 + 18 * q^55 - 3 * q^56 - 6 * q^58 - 3 * q^61 + 3 * q^62 + 3 * q^64 + 6 * q^65 + 6 * q^67 + 12 * q^68 + 3 * q^70 - 9 * q^71 + 3 * q^73 - 3 * q^74 - 21 * q^77 - 9 * q^79 + 6 * q^80 - 9 * q^82 - 3 * q^83 + 27 * q^85 - 3 * q^86 + 12 * q^88 + 3 * q^89 + 27 * q^91 + 12 * q^92 + 15 * q^94 + 12 * q^97

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.467911	0.209256	0.104628	−	0.994511i	$-0.466635\pi$
$5$	0.467911	0.209256	0.104628	+	0.994511i	$0.466635\pi$
$6$	0	0
$7$	−4.41147	−1.66738	−0.833690	−	0.552232i	$-0.813776\pi$
$7$	−4.41147	−1.66738	−0.833690	+	0.552232i	$0.813776\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.467911	0.147966
$11$	5.53209	1.66799	0.833994	−	0.551774i	$-0.186049\pi$
$11$	5.53209	1.66799	0.833994	+	0.551774i	$0.186049\pi$
$12$	0	0
$13$	−5.98545	−1.66007	−0.830033	−	0.557714i	$-0.811678\pi$
$13$	−5.98545	−1.66007	−0.830033	+	0.557714i	$0.811678\pi$
$14$	−4.41147	−1.17902
$15$	0	0
$16$	1.00000	0.250000
$17$	4.34730	1.05437	0.527187	−	0.849749i	$-0.323247\pi$
$17$	4.34730	1.05437	0.527187	+	0.849749i	$0.323247\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0.467911	0.104628
$21$	0	0
$22$	5.53209	1.17945
$23$	3.16250	0.659428	0.329714	−	0.944081i	$-0.393048\pi$
$23$	3.16250	0.659428	0.329714	+	0.944081i	$0.393048\pi$
$24$	0	0
$25$	−4.78106	−0.956212
$26$	−5.98545	−1.17384
$27$	0	0
$28$	−4.41147	−0.833690
$29$	−1.50980	−0.280363	−0.140181	−	0.990126i	$-0.544769\pi$
$29$	−1.50980	−0.280363	−0.140181	+	0.990126i	$0.544769\pi$
$30$	0	0
$31$	2.18479	0.392400	0.196200	−	0.980564i	$-0.437140\pi$
$31$	2.18479	0.392400	0.196200	+	0.980564i	$0.437140\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.34730	0.745555
$35$	−2.06418	−0.348910
$36$	0	0
$37$	2.75877	0.453539	0.226770	−	0.973948i	$-0.427184\pi$
$37$	2.75877	0.453539	0.226770	+	0.973948i	$0.427184\pi$
$38$	0	0
$39$	0	0
$40$	0.467911	0.0739832
$41$	−1.95811	−0.305806	−0.152903	−	0.988241i	$-0.548862\pi$
$41$	−1.95811	−0.305806	−0.152903	+	0.988241i	$0.548862\pi$
$42$	0	0
$43$	3.59627	0.548426	0.274213	−	0.961669i	$-0.411583\pi$
$43$	3.59627	0.548426	0.274213	+	0.961669i	$0.411583\pi$
$44$	5.53209	0.833994
$45$	0	0
$46$	3.16250	0.466286
$47$	9.24897	1.34910	0.674550	−	0.738229i	$-0.264337\pi$
$47$	9.24897	1.34910	0.674550	+	0.738229i	$0.264337\pi$
$48$	0	0
$49$	12.4611	1.78016
$50$	−4.78106	−0.676144
$51$	0	0
$52$	−5.98545	−0.830033
$53$	−2.67499	−0.367438	−0.183719	−	0.982979i	$-0.558814\pi$
$53$	−2.67499	−0.367438	−0.183719	+	0.982979i	$0.558814\pi$
$54$	0	0
$55$	2.58853	0.349037
$56$	−4.41147	−0.589508
$57$	0	0
$58$	−1.50980	−0.198246
$59$	1.32770	0.172851	0.0864256	−	0.996258i	$-0.472455\pi$
$59$	1.32770	0.172851	0.0864256	+	0.996258i	$0.472455\pi$
$60$	0	0
$61$	−6.63816	−0.849929	−0.424964	−	0.905210i	$-0.639713\pi$
$61$	−6.63816	−0.849929	−0.424964	+	0.905210i	$0.639713\pi$
$62$	2.18479	0.277469
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.80066	−0.347379
$66$	0	0
$67$	2.49020	0.304226	0.152113	−	0.988363i	$-0.451392\pi$
$67$	2.49020	0.304226	0.152113	+	0.988363i	$0.451392\pi$
$68$	4.34730	0.527187
$69$	0	0
$70$	−2.06418	−0.246716
$71$	−7.59627	−0.901511	−0.450755	−	0.892647i	$-0.648845\pi$
$71$	−7.59627	−0.901511	−0.450755	+	0.892647i	$0.648845\pi$
$72$	0	0
$73$	3.16519	0.370458	0.185229	−	0.982695i	$-0.440697\pi$
$73$	3.16519	0.370458	0.185229	+	0.982695i	$0.440697\pi$
$74$	2.75877	0.320701
$75$	0	0
$76$	0	0
$77$	−24.4047	−2.78117
$78$	0	0
$79$	−1.26352	−0.142157	−0.0710785	−	0.997471i	$-0.522644\pi$
$79$	−1.26352	−0.142157	−0.0710785	+	0.997471i	$0.522644\pi$
$80$	0.467911	0.0523141
$81$	0	0
$82$	−1.95811	−0.216237
$83$	17.9368	1.96881	0.984407	−	0.175904i	$-0.0562848\pi$
$83$	17.9368	1.96881	0.984407	+	0.175904i	$0.0562848\pi$
$84$	0	0
$85$	2.03415	0.220634
$86$	3.59627	0.387795
$87$	0	0
$88$	5.53209	0.589723
$89$	12.4807	1.32295	0.661476	−	0.749966i	$-0.269930\pi$
$89$	12.4807	1.32295	0.661476	+	0.749966i	$0.269930\pi$
$90$	0	0
$91$	26.4047	2.76796
$92$	3.16250	0.329714
$93$	0	0
$94$	9.24897	0.953958
$95$	0	0
$96$	0	0
$97$	8.31046	0.843799	0.421900	−	0.906643i	$-0.361363\pi$
$97$	8.31046	0.843799	0.421900	+	0.906643i	$0.361363\pi$
$98$	12.4611	1.25876
$99$	0	0
$100$	−4.78106	−0.478106
$101$	−1.24123	−0.123507	−0.0617535	−	0.998091i	$-0.519669\pi$
$101$	−1.24123	−0.123507	−0.0617535	+	0.998091i	$0.519669\pi$
$102$	0	0
$103$	−3.38413	−0.333449	−0.166724	−	0.986004i	$-0.553319\pi$
$103$	−3.38413	−0.333449	−0.166724	+	0.986004i	$0.553319\pi$
$104$	−5.98545	−0.586922
$105$	0	0
$106$	−2.67499	−0.259818
$107$	15.8161	1.52900	0.764502	−	0.644621i	$-0.222985\pi$
$107$	15.8161	1.52900	0.764502	+	0.644621i	$0.222985\pi$
$108$	0	0
$109$	5.81521	0.556996	0.278498	−	0.960437i	$-0.410163\pi$
$109$	5.81521	0.556996	0.278498	+	0.960437i	$0.410163\pi$
$110$	2.58853	0.246806
$111$	0	0
$112$	−4.41147	−0.416845
$113$	12.1361	1.14167	0.570834	−	0.821066i	$-0.306620\pi$
$113$	12.1361	1.14167	0.570834	+	0.821066i	$0.306620\pi$
$114$	0	0
$115$	1.47977	0.137989
$116$	−1.50980	−0.140181
$117$	0	0
$118$	1.32770	0.122224
$119$	−19.1780	−1.75804
$120$	0	0
$121$	19.6040	1.78218
$122$	−6.63816	−0.600990
$123$	0	0
$124$	2.18479	0.196200
$125$	−4.57667	−0.409349
$126$	0	0
$127$	−0.453363	−0.0402295	−0.0201147	−	0.999798i	$-0.506403\pi$
$127$	−0.453363	−0.0402295	−0.0201147	+	0.999798i	$0.506403\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.80066	−0.245634
$131$	10.8084	0.944334	0.472167	−	0.881509i	$-0.343472\pi$
$131$	10.8084	0.944334	0.472167	+	0.881509i	$0.343472\pi$
$132$	0	0
$133$	0	0
$134$	2.49020	0.215120
$135$	0	0
$136$	4.34730	0.372778
$137$	−10.9709	−0.937308	−0.468654	−	0.883382i	$-0.655261\pi$
$137$	−10.9709	−0.937308	−0.468654	+	0.883382i	$0.655261\pi$
$138$	0	0
$139$	−13.4611	−1.14176	−0.570878	−	0.821035i	$-0.693397\pi$
$139$	−13.4611	−1.14176	−0.570878	+	0.821035i	$0.693397\pi$
$140$	−2.06418	−0.174455
$141$	0	0
$142$	−7.59627	−0.637465
$143$	−33.1121	−2.76897
$144$	0	0
$145$	−0.706452	−0.0586677
$146$	3.16519	0.261953
$147$	0	0
$148$	2.75877	0.226770
$149$	−8.18984	−0.670938	−0.335469	−	0.942051i	$-0.608895\pi$
$149$	−8.18984	−0.670938	−0.335469	+	0.942051i	$0.608895\pi$
$150$	0	0
$151$	−21.1411	−1.72044	−0.860221	−	0.509921i	$-0.829675\pi$
$151$	−21.1411	−1.72044	−0.860221	+	0.509921i	$0.829675\pi$
$152$	0	0
$153$	0	0
$154$	−24.4047	−1.96658
$155$	1.02229	0.0821122
$156$	0	0
$157$	19.6382	1.56730	0.783648	−	0.621205i	$-0.213357\pi$
$157$	19.6382	1.56730	0.783648	+	0.621205i	$0.213357\pi$
$158$	−1.26352	−0.100520
$159$	0	0
$160$	0.467911	0.0369916
$161$	−13.9513	−1.09952
$162$	0	0
$163$	4.55169	0.356516	0.178258	−	0.983984i	$-0.442954\pi$
$163$	4.55169	0.356516	0.178258	+	0.983984i	$0.442954\pi$
$164$	−1.95811	−0.152903
$165$	0	0
$166$	17.9368	1.39216
$167$	19.1411	1.48119	0.740593	−	0.671954i	$-0.234545\pi$
$167$	19.1411	1.48119	0.740593	+	0.671954i	$0.234545\pi$
$168$	0	0
$169$	22.8256	1.75582
$170$	2.03415	0.156012
$171$	0	0
$172$	3.59627	0.274213
$173$	5.55169	0.422087	0.211044	−	0.977477i	$-0.432314\pi$
$173$	5.55169	0.422087	0.211044	+	0.977477i	$0.432314\pi$
$174$	0	0
$175$	21.0915	1.59437
$176$	5.53209	0.416997
$177$	0	0
$178$	12.4807	0.935468
$179$	−8.34998	−0.624107	−0.312054	−	0.950065i	$-0.601017\pi$
$179$	−8.34998	−0.624107	−0.312054	+	0.950065i	$0.601017\pi$
$180$	0	0
$181$	12.5885	0.935698	0.467849	−	0.883808i	$-0.345029\pi$
$181$	12.5885	0.935698	0.467849	+	0.883808i	$0.345029\pi$
$182$	26.4047	1.95724
$183$	0	0
$184$	3.16250	0.233143
$185$	1.29086	0.0949059
$186$	0	0
$187$	24.0496	1.75868
$188$	9.24897	0.674550
$189$	0	0
$190$	0	0
$191$	6.71688	0.486016	0.243008	−	0.970024i	$-0.421866\pi$
$191$	6.71688	0.486016	0.243008	+	0.970024i	$0.421866\pi$
$192$	0	0
$193$	12.5321	0.902079	0.451040	−	0.892504i	$-0.351053\pi$
$193$	12.5321	0.902079	0.451040	+	0.892504i	$0.351053\pi$
$194$	8.31046	0.596656
$195$	0	0
$196$	12.4611	0.890079
$197$	4.07192	0.290112	0.145056	−	0.989423i	$-0.453664\pi$
$197$	4.07192	0.290112	0.145056	+	0.989423i	$0.453664\pi$
$198$	0	0
$199$	−7.15570	−0.507254	−0.253627	−	0.967302i	$-0.581624\pi$
$199$	−7.15570	−0.507254	−0.253627	+	0.967302i	$0.581624\pi$
$200$	−4.78106	−0.338072
$201$	0	0
$202$	−1.24123	−0.0873326
$203$	6.66044	0.467472
$204$	0	0
$205$	−0.916222	−0.0639917
$206$	−3.38413	−0.235784
$207$	0	0
$208$	−5.98545	−0.415016
$209$	0	0
$210$	0	0
$211$	2.65270	0.182620	0.0913098	−	0.995823i	$-0.470895\pi$
$211$	2.65270	0.182620	0.0913098	+	0.995823i	$0.470895\pi$
$212$	−2.67499	−0.183719
$213$	0	0
$214$	15.8161	1.08117
$215$	1.68273	0.114761
$216$	0	0
$217$	−9.63816	−0.654281
$218$	5.81521	0.393856
$219$	0	0
$220$	2.58853	0.174518
$221$	−26.0205	−1.75033
$222$	0	0
$223$	−13.9504	−0.934186	−0.467093	−	0.884208i	$-0.654699\pi$
$223$	−13.9504	−0.934186	−0.467093	+	0.884208i	$0.654699\pi$
$224$	−4.41147	−0.294754
$225$	0	0
$226$	12.1361	0.807281
$227$	−23.6117	−1.56717	−0.783583	−	0.621287i	$-0.786610\pi$
$227$	−23.6117	−1.56717	−0.783583	+	0.621287i	$0.786610\pi$
$228$	0	0
$229$	8.61081	0.569019	0.284509	−	0.958673i	$-0.408169\pi$
$229$	8.61081	0.569019	0.284509	+	0.958673i	$0.408169\pi$
$230$	1.47977	0.0975732
$231$	0	0
$232$	−1.50980	−0.0991232
$233$	15.4415	1.01161	0.505803	−	0.862649i	$-0.331196\pi$
$233$	15.4415	1.01161	0.505803	+	0.862649i	$0.331196\pi$
$234$	0	0
$235$	4.32770	0.282308
$236$	1.32770	0.0864256
$237$	0	0
$238$	−19.1780	−1.24312
$239$	−7.75877	−0.501873	−0.250937	−	0.968004i	$-0.580739\pi$
$239$	−7.75877	−0.501873	−0.250937	+	0.968004i	$0.580739\pi$
$240$	0	0
$241$	13.2858	0.855814	0.427907	−	0.903823i	$-0.359251\pi$
$241$	13.2858	0.855814	0.427907	+	0.903823i	$0.359251\pi$
$242$	19.6040	1.26019
$243$	0	0
$244$	−6.63816	−0.424964
$245$	5.83069	0.372509
$246$	0	0
$247$	0	0
$248$	2.18479	0.138734
$249$	0	0
$250$	−4.57667	−0.289454
$251$	17.6509	1.11412	0.557059	−	0.830473i	$-0.311930\pi$
$251$	17.6509	1.11412	0.557059	+	0.830473i	$0.311930\pi$
$252$	0	0
$253$	17.4953	1.09992
$254$	−0.453363	−0.0284465
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.90167	0.492893	0.246446	−	0.969156i	$-0.420737\pi$
$257$	7.90167	0.492893	0.246446	+	0.969156i	$0.420737\pi$
$258$	0	0
$259$	−12.1702	−0.756222
$260$	−2.80066	−0.173690
$261$	0	0
$262$	10.8084	0.667745
$263$	25.3705	1.56441	0.782207	−	0.623019i	$-0.214094\pi$
$263$	25.3705	1.56441	0.782207	+	0.623019i	$0.214094\pi$
$264$	0	0
$265$	−1.25166	−0.0768888
$266$	0	0
$267$	0	0
$268$	2.49020	0.152113
$269$	20.1702	1.22980	0.614901	−	0.788604i	$-0.289196\pi$
$269$	20.1702	1.22980	0.614901	+	0.788604i	$0.289196\pi$
$270$	0	0
$271$	20.5107	1.24594	0.622969	−	0.782246i	$-0.285926\pi$
$271$	20.5107	1.24594	0.622969	+	0.782246i	$0.285926\pi$
$272$	4.34730	0.263594
$273$	0	0
$274$	−10.9709	−0.662777
$275$	−26.4492	−1.59495
$276$	0	0
$277$	9.13516	0.548879	0.274439	−	0.961604i	$-0.411508\pi$
$277$	9.13516	0.548879	0.274439	+	0.961604i	$0.411508\pi$
$278$	−13.4611	−0.807343
$279$	0	0
$280$	−2.06418	−0.123358
$281$	−3.43107	−0.204681	−0.102340	−	0.994749i	$-0.532633\pi$
$281$	−3.43107	−0.204681	−0.102340	+	0.994749i	$0.532633\pi$
$282$	0	0
$283$	11.2422	0.668277	0.334139	−	0.942524i	$-0.391555\pi$
$283$	11.2422	0.668277	0.334139	+	0.942524i	$0.391555\pi$
$284$	−7.59627	−0.450755
$285$	0	0
$286$	−33.1121	−1.95796
$287$	8.63816	0.509894
$288$	0	0
$289$	1.89899	0.111705
$290$	−0.706452	−0.0414843
$291$	0	0
$292$	3.16519	0.185229
$293$	−14.5321	−0.848974	−0.424487	−	0.905434i	$-0.639545\pi$
$293$	−14.5321	−0.848974	−0.424487	+	0.905434i	$0.639545\pi$
$294$	0	0
$295$	0.621244	0.0361702
$296$	2.75877	0.160350
$297$	0	0
$298$	−8.18984	−0.474425
$299$	−18.9290	−1.09469
$300$	0	0
$301$	−15.8648	−0.914434
$302$	−21.1411	−1.21654
$303$	0	0
$304$	0	0
$305$	−3.10607	−0.177853
$306$	0	0
$307$	25.7033	1.46696	0.733481	−	0.679709i	$-0.237894\pi$
$307$	25.7033	1.46696	0.733481	+	0.679709i	$0.237894\pi$
$308$	−24.4047	−1.39058
$309$	0	0
$310$	1.02229	0.0580621
$311$	−5.83750	−0.331014	−0.165507	−	0.986209i	$-0.552926\pi$
$311$	−5.83750	−0.331014	−0.165507	+	0.986209i	$0.552926\pi$
$312$	0	0
$313$	−16.5107	−0.933242	−0.466621	−	0.884457i	$-0.654529\pi$
$313$	−16.5107	−0.933242	−0.466621	+	0.884457i	$0.654529\pi$
$314$	19.6382	1.10825
$315$	0	0
$316$	−1.26352	−0.0710785
$317$	−26.9495	−1.51364	−0.756819	−	0.653625i	$-0.773247\pi$
$317$	−26.9495	−1.51364	−0.756819	+	0.653625i	$0.773247\pi$
$318$	0	0
$319$	−8.35235	−0.467642
$320$	0.467911	0.0261570
$321$	0	0
$322$	−13.9513	−0.777476
$323$	0	0
$324$	0	0
$325$	28.6168	1.58737
$326$	4.55169	0.252095
$327$	0	0
$328$	−1.95811	−0.108119
$329$	−40.8016	−2.24946
$330$	0	0
$331$	−28.4884	−1.56587	−0.782933	−	0.622106i	$-0.786277\pi$
$331$	−28.4884	−1.56587	−0.782933	+	0.622106i	$0.786277\pi$
$332$	17.9368	0.984407
$333$	0	0
$334$	19.1411	1.04736
$335$	1.16519	0.0636612
$336$	0	0
$337$	−16.6878	−0.909042	−0.454521	−	0.890736i	$-0.650189\pi$
$337$	−16.6878	−0.909042	−0.454521	+	0.890736i	$0.650189\pi$
$338$	22.8256	1.24155
$339$	0	0
$340$	2.03415	0.110317
$341$	12.0865	0.654519
$342$	0	0
$343$	−24.0915	−1.30082
$344$	3.59627	0.193898
$345$	0	0
$346$	5.55169	0.298461
$347$	18.7297	1.00546	0.502731	−	0.864443i	$-0.332329\pi$
$347$	18.7297	1.00546	0.502731	+	0.864443i	$0.332329\pi$
$348$	0	0
$349$	−1.64084	−0.0878324	−0.0439162	−	0.999035i	$-0.513983\pi$
$349$	−1.64084	−0.0878324	−0.0439162	+	0.999035i	$0.513983\pi$
$350$	21.0915	1.12739
$351$	0	0
$352$	5.53209	0.294861
$353$	11.9067	0.633731	0.316866	−	0.948470i	$-0.397370\pi$
$353$	11.9067	0.633731	0.316866	+	0.948470i	$0.397370\pi$
$354$	0	0
$355$	−3.55438	−0.188647
$356$	12.4807	0.661476
$357$	0	0
$358$	−8.34998	−0.441310
$359$	−2.29591	−0.121174	−0.0605868	−	0.998163i	$-0.519297\pi$
$359$	−2.29591	−0.121174	−0.0605868	+	0.998163i	$0.519297\pi$
$360$	0	0
$361$	0	0
$362$	12.5885	0.661638
$363$	0	0
$364$	26.4047	1.38398
$365$	1.48103	0.0775206
$366$	0	0
$367$	−21.4953	−1.12204	−0.561022	−	0.827801i	$-0.689592\pi$
$367$	−21.4953	−1.12204	−0.561022	+	0.827801i	$0.689592\pi$
$368$	3.16250	0.164857
$369$	0	0
$370$	1.29086	0.0671086
$371$	11.8007	0.612660
$372$	0	0
$373$	−27.9813	−1.44882	−0.724409	−	0.689370i	$-0.757887\pi$
$373$	−27.9813	−1.44882	−0.724409	+	0.689370i	$0.757887\pi$
$374$	24.0496	1.24358
$375$	0	0
$376$	9.24897	0.476979
$377$	9.03684	0.465421
$378$	0	0
$379$	−8.88981	−0.456639	−0.228320	−	0.973586i	$-0.573323\pi$
$379$	−8.88981	−0.456639	−0.228320	+	0.973586i	$0.573323\pi$
$380$	0	0
$381$	0	0
$382$	6.71688	0.343666
$383$	−25.0155	−1.27823	−0.639116	−	0.769111i	$-0.720700\pi$
$383$	−25.0155	−1.27823	−0.639116	+	0.769111i	$0.720700\pi$
$384$	0	0
$385$	−11.4192	−0.581977
$386$	12.5321	0.637867
$387$	0	0
$388$	8.31046	0.421900
$389$	23.3327	1.18302	0.591509	−	0.806299i	$-0.298533\pi$
$389$	23.3327	1.18302	0.591509	+	0.806299i	$0.298533\pi$
$390$	0	0
$391$	13.7483	0.695284
$392$	12.4611	0.629381
$393$	0	0
$394$	4.07192	0.205140
$395$	−0.591214	−0.0297472
$396$	0	0
$397$	3.78880	0.190154	0.0950772	−	0.995470i	$-0.469690\pi$
$397$	3.78880	0.190154	0.0950772	+	0.995470i	$0.469690\pi$
$398$	−7.15570	−0.358683
$399$	0	0
$400$	−4.78106	−0.239053
$401$	−21.8598	−1.09163	−0.545813	−	0.837907i	$-0.683779\pi$
$401$	−21.8598	−1.09163	−0.545813	+	0.837907i	$0.683779\pi$
$402$	0	0
$403$	−13.0770	−0.651410
$404$	−1.24123	−0.0617535
$405$	0	0
$406$	6.66044	0.330552
$407$	15.2618	0.756498
$408$	0	0
$409$	7.91622	0.391432	0.195716	−	0.980661i	$-0.437297\pi$
$409$	7.91622	0.391432	0.195716	+	0.980661i	$0.437297\pi$
$410$	−0.916222	−0.0452490
$411$	0	0
$412$	−3.38413	−0.166724
$413$	−5.85710	−0.288209
$414$	0	0
$415$	8.39281	0.411987
$416$	−5.98545	−0.293461
$417$	0	0
$418$	0	0
$419$	−6.22937	−0.304325	−0.152162	−	0.988356i	$-0.548624\pi$
$419$	−6.22937	−0.304325	−0.152162	+	0.988356i	$0.548624\pi$
$420$	0	0
$421$	3.57129	0.174054	0.0870270	−	0.996206i	$-0.472263\pi$
$421$	3.57129	0.174054	0.0870270	+	0.996206i	$0.472263\pi$
$422$	2.65270	0.129132
$423$	0	0
$424$	−2.67499	−0.129909
$425$	−20.7847	−1.00821
$426$	0	0
$427$	29.2841	1.41715
$428$	15.8161	0.764502
$429$	0	0
$430$	1.68273	0.0811486
$431$	−25.8084	−1.24315	−0.621573	−	0.783356i	$-0.713506\pi$
$431$	−25.8084	−1.24315	−0.621573	+	0.783356i	$0.713506\pi$
$432$	0	0
$433$	−26.0993	−1.25425	−0.627125	−	0.778918i	$-0.715769\pi$
$433$	−26.0993	−1.25425	−0.627125	+	0.778918i	$0.715769\pi$
$434$	−9.63816	−0.462646
$435$	0	0
$436$	5.81521	0.278498
$437$	0	0
$438$	0	0
$439$	−22.6313	−1.08014	−0.540068	−	0.841622i	$-0.681601\pi$
$439$	−22.6313	−1.08014	−0.540068	+	0.841622i	$0.681601\pi$
$440$	2.58853	0.123403
$441$	0	0
$442$	−26.0205	−1.23767
$443$	−14.4124	−0.684754	−0.342377	−	0.939563i	$-0.611232\pi$
$443$	−14.4124	−0.684754	−0.342377	+	0.939563i	$0.611232\pi$
$444$	0	0
$445$	5.83986	0.276836
$446$	−13.9504	−0.660569
$447$	0	0
$448$	−4.41147	−0.208423
$449$	36.2722	1.71179	0.855895	−	0.517149i	$-0.173007\pi$
$449$	36.2722	1.71179	0.855895	+	0.517149i	$0.173007\pi$
$450$	0	0
$451$	−10.8324	−0.510080
$452$	12.1361	0.570834
$453$	0	0
$454$	−23.6117	−1.10815
$455$	12.3550	0.579213
$456$	0	0
$457$	−40.4543	−1.89237	−0.946186	−	0.323623i	$-0.895099\pi$
$457$	−40.4543	−1.89237	−0.946186	+	0.323623i	$0.895099\pi$
$458$	8.61081	0.402357
$459$	0	0
$460$	1.47977	0.0689947
$461$	39.2746	1.82920	0.914599	−	0.404362i	$-0.132506\pi$
$461$	39.2746	1.82920	0.914599	+	0.404362i	$0.132506\pi$
$462$	0	0
$463$	20.8334	0.968209	0.484105	−	0.875010i	$-0.339145\pi$
$463$	20.8334	0.968209	0.484105	+	0.875010i	$0.339145\pi$
$464$	−1.50980	−0.0700907
$465$	0	0
$466$	15.4415	0.715314
$467$	11.9240	0.551775	0.275888	−	0.961190i	$-0.411028\pi$
$467$	11.9240	0.551775	0.275888	+	0.961190i	$0.411028\pi$
$468$	0	0
$469$	−10.9855	−0.507261
$470$	4.32770	0.199622
$471$	0	0
$472$	1.32770	0.0611122
$473$	19.8949	0.914767
$474$	0	0
$475$	0	0
$476$	−19.1780	−0.879022
$477$	0	0
$478$	−7.75877	−0.354878
$479$	7.40373	0.338285	0.169143	−	0.985592i	$-0.445900\pi$
$479$	7.40373	0.338285	0.169143	+	0.985592i	$0.445900\pi$
$480$	0	0
$481$	−16.5125	−0.752905
$482$	13.2858	0.605152
$483$	0	0
$484$	19.6040	0.891091
$485$	3.88856	0.176570
$486$	0	0
$487$	22.5868	1.02350	0.511752	−	0.859133i	$-0.328997\pi$
$487$	22.5868	1.02350	0.511752	+	0.859133i	$0.328997\pi$
$488$	−6.63816	−0.300495
$489$	0	0
$490$	5.83069	0.263404
$491$	−16.0351	−0.723653	−0.361827	−	0.932245i	$-0.617847\pi$
$491$	−16.0351	−0.723653	−0.361827	+	0.932245i	$0.617847\pi$
$492$	0	0
$493$	−6.56355	−0.295607
$494$	0	0
$495$	0	0
$496$	2.18479	0.0981001
$497$	33.5107	1.50316
$498$	0	0
$499$	−35.1215	−1.57226	−0.786128	−	0.618063i	$-0.787917\pi$
$499$	−35.1215	−1.57226	−0.786128	+	0.618063i	$0.787917\pi$
$500$	−4.57667	−0.204675
$501$	0	0
$502$	17.6509	0.787800
$503$	−40.1275	−1.78920	−0.894599	−	0.446869i	$-0.852539\pi$
$503$	−40.1275	−1.78920	−0.894599	+	0.446869i	$0.852539\pi$
$504$	0	0
$505$	−0.580785	−0.0258446
$506$	17.4953	0.777759
$507$	0	0
$508$	−0.453363	−0.0201147
$509$	4.29086	0.190189	0.0950945	−	0.995468i	$-0.469685\pi$
$509$	4.29086	0.190189	0.0950945	+	0.995468i	$0.469685\pi$
$510$	0	0
$511$	−13.9632	−0.617694
$512$	1.00000	0.0441942
$513$	0	0
$514$	7.90167	0.348528
$515$	−1.58347	−0.0697762
$516$	0	0
$517$	51.1661	2.25028
$518$	−12.1702	−0.534730
$519$	0	0
$520$	−2.80066	−0.122817
$521$	8.77063	0.384248	0.192124	−	0.981371i	$-0.438462\pi$
$521$	8.77063	0.384248	0.192124	+	0.981371i	$0.438462\pi$
$522$	0	0
$523$	−29.9982	−1.31173	−0.655866	−	0.754877i	$-0.727696\pi$
$523$	−29.9982	−1.31173	−0.655866	+	0.754877i	$0.727696\pi$
$524$	10.8084	0.472167
$525$	0	0
$526$	25.3705	1.10621
$527$	9.49794	0.413737
$528$	0	0
$529$	−12.9986	−0.565155
$530$	−1.25166	−0.0543686
$531$	0	0
$532$	0	0
$533$	11.7202	0.507657
$534$	0	0
$535$	7.40055	0.319954
$536$	2.49020	0.107560
$537$	0	0
$538$	20.1702	0.869601
$539$	68.9359	2.96928
$540$	0	0
$541$	5.00681	0.215259	0.107630	−	0.994191i	$-0.465674\pi$
$541$	5.00681	0.215259	0.107630	+	0.994191i	$0.465674\pi$
$542$	20.5107	0.881011
$543$	0	0
$544$	4.34730	0.186389
$545$	2.72100	0.116555
$546$	0	0
$547$	−13.5794	−0.580611	−0.290306	−	0.956934i	$-0.593757\pi$
$547$	−13.5794	−0.580611	−0.290306	+	0.956934i	$0.593757\pi$
$548$	−10.9709	−0.468654
$549$	0	0
$550$	−26.4492	−1.12780
$551$	0	0
$552$	0	0
$553$	5.57398	0.237030
$554$	9.13516	0.388116
$555$	0	0
$556$	−13.4611	−0.570878
$557$	23.7270	1.00534	0.502672	−	0.864477i	$-0.332350\pi$
$557$	23.7270	1.00534	0.502672	+	0.864477i	$0.332350\pi$
$558$	0	0
$559$	−21.5253	−0.910422
$560$	−2.06418	−0.0872274
$561$	0	0
$562$	−3.43107	−0.144731
$563$	−14.7442	−0.621395	−0.310697	−	0.950509i	$-0.600563\pi$
$563$	−14.7442	−0.621395	−0.310697	+	0.950509i	$0.600563\pi$
$564$	0	0
$565$	5.67861	0.238901
$566$	11.2422	0.472543
$567$	0	0
$568$	−7.59627	−0.318732
$569$	23.9668	1.00474	0.502370	−	0.864653i	$-0.332462\pi$
$569$	23.9668	1.00474	0.502370	+	0.864653i	$0.332462\pi$
$570$	0	0
$571$	−41.0847	−1.71934	−0.859671	−	0.510848i	$-0.829331\pi$
$571$	−41.0847	−1.71934	−0.859671	+	0.510848i	$0.829331\pi$
$572$	−33.1121	−1.38448
$573$	0	0
$574$	8.63816	0.360550
$575$	−15.1201	−0.630553
$576$	0	0
$577$	−7.45512	−0.310361	−0.155180	−	0.987886i	$-0.549596\pi$
$577$	−7.45512	−0.310361	−0.155180	+	0.987886i	$0.549596\pi$
$578$	1.89899	0.0789874
$579$	0	0
$580$	−0.706452	−0.0293338
$581$	−79.1275	−3.28276
$582$	0	0
$583$	−14.7983	−0.612883
$584$	3.16519	0.130977
$585$	0	0
$586$	−14.5321	−0.600315
$587$	−5.91952	−0.244325	−0.122162	−	0.992510i	$-0.538983\pi$
$587$	−5.91952	−0.244325	−0.122162	+	0.992510i	$0.538983\pi$
$588$	0	0
$589$	0	0
$590$	0.621244	0.0255762
$591$	0	0
$592$	2.75877	0.113385
$593$	5.01455	0.205923	0.102961	−	0.994685i	$-0.467168\pi$
$593$	5.01455	0.205923	0.102961	+	0.994685i	$0.467168\pi$
$594$	0	0
$595$	−8.97359	−0.367881
$596$	−8.18984	−0.335469
$597$	0	0
$598$	−18.9290	−0.774065
$599$	4.80604	0.196369	0.0981847	−	0.995168i	$-0.468696\pi$
$599$	4.80604	0.196369	0.0981847	+	0.995168i	$0.468696\pi$
$600$	0	0
$601$	−8.84936	−0.360973	−0.180486	−	0.983577i	$-0.557767\pi$
$601$	−8.84936	−0.360973	−0.180486	+	0.983577i	$0.557767\pi$
$602$	−15.8648	−0.646602
$603$	0	0
$604$	−21.1411	−0.860221
$605$	9.17293	0.372933
$606$	0	0
$607$	−0.715948	−0.0290594	−0.0145297	−	0.999894i	$-0.504625\pi$
$607$	−0.715948	−0.0290594	−0.0145297	+	0.999894i	$0.504625\pi$
$608$	0	0
$609$	0	0
$610$	−3.10607	−0.125761
$611$	−55.3593	−2.23960
$612$	0	0
$613$	24.4097	0.985899	0.492950	−	0.870058i	$-0.335919\pi$
$613$	24.4097	0.985899	0.492950	+	0.870058i	$0.335919\pi$
$614$	25.7033	1.03730
$615$	0	0
$616$	−24.4047	−0.983292
$617$	43.4593	1.74961	0.874804	−	0.484477i	$-0.160990\pi$
$617$	43.4593	1.74961	0.874804	+	0.484477i	$0.160990\pi$
$618$	0	0
$619$	23.7297	0.953776	0.476888	−	0.878964i	$-0.341765\pi$
$619$	23.7297	0.953776	0.476888	+	0.878964i	$0.341765\pi$
$620$	1.02229	0.0410561
$621$	0	0
$622$	−5.83750	−0.234062
$623$	−55.0583	−2.20586
$624$	0	0
$625$	21.7638	0.870553
$626$	−16.5107	−0.659902
$627$	0	0
$628$	19.6382	0.783648
$629$	11.9932	0.478200
$630$	0	0
$631$	−25.8161	−1.02772	−0.513862	−	0.857873i	$-0.671786\pi$
$631$	−25.8161	−1.02772	−0.513862	+	0.857873i	$0.671786\pi$
$632$	−1.26352	−0.0502601
$633$	0	0
$634$	−26.9495	−1.07030
$635$	−0.212134	−0.00841827
$636$	0	0
$637$	−74.5853	−2.95518
$638$	−8.35235	−0.330673
$639$	0	0
$640$	0.467911	0.0184958
$641$	−10.8060	−0.426813	−0.213406	−	0.976964i	$-0.568456\pi$
$641$	−10.8060	−0.426813	−0.213406	+	0.976964i	$0.568456\pi$
$642$	0	0
$643$	14.4047	0.568065	0.284032	−	0.958815i	$-0.408328\pi$
$643$	14.4047	0.568065	0.284032	+	0.958815i	$0.408328\pi$
$644$	−13.9513	−0.549758
$645$	0	0
$646$	0	0
$647$	24.9463	0.980738	0.490369	−	0.871515i	$-0.336862\pi$
$647$	24.9463	0.980738	0.490369	+	0.871515i	$0.336862\pi$
$648$	0	0
$649$	7.34493	0.288314
$650$	28.6168	1.12244
$651$	0	0
$652$	4.55169	0.178258
$653$	−35.6141	−1.39369	−0.696844	−	0.717223i	$-0.745413\pi$
$653$	−35.6141	−1.39369	−0.696844	+	0.717223i	$0.745413\pi$
$654$	0	0
$655$	5.05737	0.197608
$656$	−1.95811	−0.0764514
$657$	0	0
$658$	−40.8016	−1.59061
$659$	−48.9695	−1.90758	−0.953790	−	0.300474i	$-0.902855\pi$
$659$	−48.9695	−1.90758	−0.953790	+	0.300474i	$0.902855\pi$
$660$	0	0
$661$	34.0324	1.32371	0.661853	−	0.749633i	$-0.269770\pi$
$661$	34.0324	1.32371	0.661853	+	0.749633i	$0.269770\pi$
$662$	−28.4884	−1.10723
$663$	0	0
$664$	17.9368	0.696081
$665$	0	0
$666$	0	0
$667$	−4.77475	−0.184879
$668$	19.1411	0.740593
$669$	0	0
$670$	1.16519	0.0450153
$671$	−36.7229	−1.41767
$672$	0	0
$673$	28.2986	1.09083	0.545415	−	0.838166i	$-0.316372\pi$
$673$	28.2986	1.09083	0.545415	+	0.838166i	$0.316372\pi$
$674$	−16.6878	−0.642789
$675$	0	0
$676$	22.8256	0.877909
$677$	25.0051	0.961022	0.480511	−	0.876989i	$-0.340451\pi$
$677$	25.0051	0.961022	0.480511	+	0.876989i	$0.340451\pi$
$678$	0	0
$679$	−36.6614	−1.40693
$680$	2.03415	0.0780060
$681$	0	0
$682$	12.0865	0.462815
$683$	−40.1284	−1.53547	−0.767734	−	0.640768i	$-0.778616\pi$
$683$	−40.1284	−1.53547	−0.767734	+	0.640768i	$0.778616\pi$
$684$	0	0
$685$	−5.13341	−0.196137
$686$	−24.0915	−0.919818
$687$	0	0
$688$	3.59627	0.137106
$689$	16.0110	0.609972
$690$	0	0
$691$	2.88175	0.109627	0.0548135	−	0.998497i	$-0.482544\pi$
$691$	2.88175	0.109627	0.0548135	+	0.998497i	$0.482544\pi$
$692$	5.55169	0.211044
$693$	0	0
$694$	18.7297	0.710969
$695$	−6.29860	−0.238920
$696$	0	0
$697$	−8.51249	−0.322433
$698$	−1.64084	−0.0621069
$699$	0	0
$700$	21.0915	0.797184
$701$	−15.4679	−0.584215	−0.292107	−	0.956386i	$-0.594356\pi$
$701$	−15.4679	−0.584215	−0.292107	+	0.956386i	$0.594356\pi$
$702$	0	0
$703$	0	0
$704$	5.53209	0.208498
$705$	0	0
$706$	11.9067	0.448116
$707$	5.47565	0.205933
$708$	0	0
$709$	46.6486	1.75192	0.875962	−	0.482380i	$-0.160227\pi$
$709$	46.6486	1.75192	0.875962	+	0.482380i	$0.160227\pi$
$710$	−3.55438	−0.133393
$711$	0	0
$712$	12.4807	0.467734
$713$	6.90941	0.258760
$714$	0	0
$715$	−15.4935	−0.579424
$716$	−8.34998	−0.312054
$717$	0	0
$718$	−2.29591	−0.0856827
$719$	11.4953	0.428701	0.214350	−	0.976757i	$-0.431237\pi$
$719$	11.4953	0.428701	0.214350	+	0.976757i	$0.431237\pi$
$720$	0	0
$721$	14.9290	0.555986
$722$	0	0
$723$	0	0
$724$	12.5885	0.467849
$725$	7.21844	0.268086
$726$	0	0
$727$	17.4311	0.646483	0.323241	−	0.946317i	$-0.395227\pi$
$727$	17.4311	0.646483	0.323241	+	0.946317i	$0.395227\pi$
$728$	26.4047	0.978622
$729$	0	0
$730$	1.48103	0.0548153
$731$	15.6340	0.578246
$732$	0	0
$733$	−43.8786	−1.62069	−0.810346	−	0.585952i	$-0.800721\pi$
$733$	−43.8786	−1.62069	−0.810346	+	0.585952i	$0.800721\pi$
$734$	−21.4953	−0.793404
$735$	0	0
$736$	3.16250	0.116571
$737$	13.7760	0.507446
$738$	0	0
$739$	−17.7588	−0.653267	−0.326633	−	0.945151i	$-0.605914\pi$
$739$	−17.7588	−0.653267	−0.326633	+	0.945151i	$0.605914\pi$
$740$	1.29086	0.0474529
$741$	0	0
$742$	11.8007	0.433216
$743$	10.4270	0.382528	0.191264	−	0.981539i	$-0.438741\pi$
$743$	10.4270	0.382528	0.191264	+	0.981539i	$0.438741\pi$
$744$	0	0
$745$	−3.83212	−0.140398
$746$	−27.9813	−1.02447
$747$	0	0
$748$	24.0496	0.879342
$749$	−69.7725	−2.54943
$750$	0	0
$751$	34.0601	1.24287	0.621435	−	0.783466i	$-0.286550\pi$
$751$	34.0601	1.24287	0.621435	+	0.783466i	$0.286550\pi$
$752$	9.24897	0.337275
$753$	0	0
$754$	9.03684	0.329102
$755$	−9.89218	−0.360013
$756$	0	0
$757$	43.2763	1.57290	0.786452	−	0.617651i	$-0.211916\pi$
$757$	43.2763	1.57290	0.786452	+	0.617651i	$0.211916\pi$
$758$	−8.88981	−0.322893
$759$	0	0
$760$	0	0
$761$	−14.1679	−0.513585	−0.256793	−	0.966467i	$-0.582666\pi$
$761$	−14.1679	−0.513585	−0.256793	+	0.966467i	$0.582666\pi$
$762$	0	0
$763$	−25.6536	−0.928724
$764$	6.71688	0.243008
$765$	0	0
$766$	−25.0155	−0.903846
$767$	−7.94686	−0.286945
$768$	0	0
$769$	−18.2395	−0.657732	−0.328866	−	0.944377i	$-0.606666\pi$
$769$	−18.2395	−0.657732	−0.328866	+	0.944377i	$0.606666\pi$
$770$	−11.4192	−0.411520
$771$	0	0
$772$	12.5321	0.451040
$773$	−28.2472	−1.01598	−0.507991	−	0.861362i	$-0.669612\pi$
$773$	−28.2472	−1.01598	−0.507991	+	0.861362i	$0.669612\pi$
$774$	0	0
$775$	−10.4456	−0.375218
$776$	8.31046	0.298328
$777$	0	0
$778$	23.3327	0.836520
$779$	0	0
$780$	0	0
$781$	−42.0232	−1.50371
$782$	13.7483	0.491640
$783$	0	0
$784$	12.4611	0.445039
$785$	9.18891	0.327966
$786$	0	0
$787$	−18.1411	−0.646662	−0.323331	−	0.946286i	$-0.604803\pi$
$787$	−18.1411	−0.646662	−0.323331	+	0.946286i	$0.604803\pi$
$788$	4.07192	0.145056
$789$	0	0
$790$	−0.591214	−0.0210345
$791$	−53.5381	−1.90359
$792$	0	0
$793$	39.7324	1.41094
$794$	3.78880	0.134459
$795$	0	0
$796$	−7.15570	−0.253627
$797$	−32.8111	−1.16223	−0.581114	−	0.813822i	$-0.697383\pi$
$797$	−32.8111	−1.16223	−0.581114	+	0.813822i	$0.697383\pi$
$798$	0	0
$799$	40.2080	1.42246
$800$	−4.78106	−0.169036
$801$	0	0
$802$	−21.8598	−0.771896
$803$	17.5101	0.617919
$804$	0	0
$805$	−6.52797	−0.230081
$806$	−13.0770	−0.460617
$807$	0	0
$808$	−1.24123	−0.0436663
$809$	9.34224	0.328456	0.164228	−	0.986422i	$-0.447487\pi$
$809$	9.34224	0.328456	0.164228	+	0.986422i	$0.447487\pi$
$810$	0	0
$811$	15.0077	0.526993	0.263497	−	0.964660i	$-0.415124\pi$
$811$	15.0077	0.526993	0.263497	+	0.964660i	$0.415124\pi$
$812$	6.66044	0.233736
$813$	0	0
$814$	15.2618	0.534925
$815$	2.12979	0.0746031
$816$	0	0
$817$	0	0
$818$	7.91622	0.276784
$819$	0	0
$820$	−0.916222	−0.0319959
$821$	−36.5544	−1.27576	−0.637878	−	0.770137i	$-0.720188\pi$
$821$	−36.5544	−1.27576	−0.637878	+	0.770137i	$0.720188\pi$
$822$	0	0
$823$	22.8367	0.796036	0.398018	−	0.917378i	$-0.369698\pi$
$823$	22.8367	0.796036	0.398018	+	0.917378i	$0.369698\pi$
$824$	−3.38413	−0.117892
$825$	0	0
$826$	−5.85710	−0.203794
$827$	2.43201	0.0845692	0.0422846	−	0.999106i	$-0.486536\pi$
$827$	2.43201	0.0845692	0.0422846	+	0.999106i	$0.486536\pi$
$828$	0	0
$829$	23.3405	0.810649	0.405324	−	0.914173i	$-0.367159\pi$
$829$	23.3405	0.810649	0.405324	+	0.914173i	$0.367159\pi$
$830$	8.39281	0.291319
$831$	0	0
$832$	−5.98545	−0.207508
$833$	54.1721	1.87695
$834$	0	0
$835$	8.95636	0.309947
$836$	0	0
$837$	0	0
$838$	−6.22937	−0.215190
$839$	−1.46698	−0.0506457	−0.0253228	−	0.999679i	$-0.508061\pi$
$839$	−1.46698	−0.0506457	−0.0253228	+	0.999679i	$0.508061\pi$
$840$	0	0
$841$	−26.7205	−0.921397
$842$	3.57129	0.123075
$843$	0	0
$844$	2.65270	0.0913098
$845$	10.6804	0.367416
$846$	0	0
$847$	−86.4826	−2.97158
$848$	−2.67499	−0.0918596
$849$	0	0
$850$	−20.7847	−0.712909
$851$	8.72462	0.299076
$852$	0	0
$853$	−9.33637	−0.319671	−0.159836	−	0.987144i	$-0.551096\pi$
$853$	−9.33637	−0.319671	−0.159836	+	0.987144i	$0.551096\pi$
$854$	29.2841	1.00208
$855$	0	0
$856$	15.8161	0.540585
$857$	34.8452	1.19029	0.595145	−	0.803618i	$-0.297094\pi$
$857$	34.8452	1.19029	0.595145	+	0.803618i	$0.297094\pi$
$858$	0	0
$859$	−23.0787	−0.787436	−0.393718	−	0.919231i	$-0.628811\pi$
$859$	−23.0787	−0.787436	−0.393718	+	0.919231i	$0.628811\pi$
$860$	1.68273	0.0573807
$861$	0	0
$862$	−25.8084	−0.879038
$863$	43.1576	1.46910	0.734550	−	0.678554i	$-0.237393\pi$
$863$	43.1576	1.46910	0.734550	+	0.678554i	$0.237393\pi$
$864$	0	0
$865$	2.59770	0.0883244
$866$	−26.0993	−0.886889
$867$	0	0
$868$	−9.63816	−0.327140
$869$	−6.98990	−0.237116
$870$	0	0
$871$	−14.9050	−0.505036
$872$	5.81521	0.196928
$873$	0	0
$874$	0	0
$875$	20.1898	0.682541
$876$	0	0
$877$	−44.8417	−1.51420	−0.757099	−	0.653300i	$-0.773384\pi$
$877$	−44.8417	−1.51420	−0.757099	+	0.653300i	$0.773384\pi$
$878$	−22.6313	−0.763771
$879$	0	0
$880$	2.58853	0.0872592
$881$	4.59863	0.154932	0.0774659	−	0.996995i	$-0.475317\pi$
$881$	4.59863	0.154932	0.0774659	+	0.996995i	$0.475317\pi$
$882$	0	0
$883$	−7.22575	−0.243166	−0.121583	−	0.992581i	$-0.538797\pi$
$883$	−7.22575	−0.243166	−0.121583	+	0.992581i	$0.538797\pi$
$884$	−26.0205	−0.875165
$885$	0	0
$886$	−14.4124	−0.484194
$887$	46.3637	1.55674	0.778371	−	0.627805i	$-0.216047\pi$
$887$	46.3637	1.55674	0.778371	+	0.627805i	$0.216047\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	5.83986	0.195753
$891$	0	0
$892$	−13.9504	−0.467093
$893$	0	0
$894$	0	0
$895$	−3.90705	−0.130598
$896$	−4.41147	−0.147377
$897$	0	0
$898$	36.2722	1.21042
$899$	−3.29860	−0.110014
$900$	0	0
$901$	−11.6290	−0.387418
$902$	−10.8324	−0.360681
$903$	0	0
$904$	12.1361	0.403641
$905$	5.89031	0.195801
$906$	0	0
$907$	54.3756	1.80551	0.902756	−	0.430154i	$-0.141541\pi$
$907$	54.3756	1.80551	0.902756	+	0.430154i	$0.141541\pi$
$908$	−23.6117	−0.783583
$909$	0	0
$910$	12.3550	0.409565
$911$	−16.8993	−0.559899	−0.279950	−	0.960015i	$-0.590318\pi$
$911$	−16.8993	−0.559899	−0.279950	+	0.960015i	$0.590318\pi$
$912$	0	0
$913$	99.2277	3.28396
$914$	−40.4543	−1.33811
$915$	0	0
$916$	8.61081	0.284509
$917$	−47.6810	−1.57456
$918$	0	0
$919$	−7.70201	−0.254066	−0.127033	−	0.991899i	$-0.540545\pi$
$919$	−7.70201	−0.254066	−0.127033	+	0.991899i	$0.540545\pi$
$920$	1.47977	0.0487866
$921$	0	0
$922$	39.2746	1.29344
$923$	45.4671	1.49657
$924$	0	0
$925$	−13.1898	−0.433679
$926$	20.8334	0.684627
$927$	0	0
$928$	−1.50980	−0.0495616
$929$	18.1465	0.595368	0.297684	−	0.954664i	$-0.403786\pi$
$929$	18.1465	0.595368	0.297684	+	0.954664i	$0.403786\pi$
$930$	0	0
$931$	0	0
$932$	15.4415	0.505803
$933$	0	0
$934$	11.9240	0.390164
$935$	11.2531	0.368015
$936$	0	0
$937$	−10.3378	−0.337721	−0.168861	−	0.985640i	$-0.554009\pi$
$937$	−10.3378	−0.337721	−0.168861	+	0.985640i	$0.554009\pi$
$938$	−10.9855	−0.358688
$939$	0	0
$940$	4.32770	0.141154
$941$	45.1421	1.47159	0.735795	−	0.677204i	$-0.236809\pi$
$941$	45.1421	1.47159	0.735795	+	0.677204i	$0.236809\pi$
$942$	0	0
$943$	−6.19253	−0.201657
$944$	1.32770	0.0432128
$945$	0	0
$946$	19.8949	0.646838
$947$	20.1726	0.655522	0.327761	−	0.944761i	$-0.393706\pi$
$947$	20.1726	0.655522	0.327761	+	0.944761i	$0.393706\pi$
$948$	0	0
$949$	−18.9451	−0.614984
$950$	0	0
$951$	0	0
$952$	−19.1780	−0.621562
$953$	−44.7202	−1.44863	−0.724314	−	0.689470i	$-0.757844\pi$
$953$	−44.7202	−1.44863	−0.724314	+	0.689470i	$0.757844\pi$
$954$	0	0
$955$	3.14290	0.101702
$956$	−7.75877	−0.250937
$957$	0	0
$958$	7.40373	0.239204
$959$	48.3979	1.56285
$960$	0	0
$961$	−26.2267	−0.846022
$962$	−16.5125	−0.532384
$963$	0	0
$964$	13.2858	0.427907
$965$	5.86390	0.188766
$966$	0	0
$967$	−11.7000	−0.376246	−0.188123	−	0.982146i	$-0.560240\pi$
$967$	−11.7000	−0.376246	−0.188123	+	0.982146i	$0.560240\pi$
$968$	19.6040	0.630097
$969$	0	0
$970$	3.88856	0.124854
$971$	36.5235	1.17210	0.586048	−	0.810276i	$-0.300683\pi$
$971$	36.5235	1.17210	0.586048	+	0.810276i	$0.300683\pi$
$972$	0	0
$973$	59.3833	1.90374
$974$	22.5868	0.723727
$975$	0	0
$976$	−6.63816	−0.212482
$977$	−50.0438	−1.60104	−0.800521	−	0.599305i	$-0.795444\pi$
$977$	−50.0438	−1.60104	−0.800521	+	0.599305i	$0.795444\pi$
$978$	0	0
$979$	69.0444	2.20667
$980$	5.83069	0.186255
$981$	0	0
$982$	−16.0351	−0.511700
$983$	9.86577	0.314669	0.157335	−	0.987545i	$-0.449710\pi$
$983$	9.86577	0.314669	0.157335	+	0.987545i	$0.449710\pi$
$984$	0	0
$985$	1.90530	0.0607078
$986$	−6.56355	−0.209026
$987$	0	0
$988$	0	0
$989$	11.3732	0.361647
$990$	0	0
$991$	−34.2235	−1.08715	−0.543573	−	0.839362i	$-0.682929\pi$
$991$	−34.2235	−1.08715	−0.543573	+	0.839362i	$0.682929\pi$
$992$	2.18479	0.0693672
$993$	0	0
$994$	33.5107	1.06290
$995$	−3.34823	−0.106146
$996$	0	0
$997$	39.7912	1.26020	0.630099	−	0.776514i	$-0.283014\pi$
$997$	39.7912	1.26020	0.630099	+	0.776514i	$0.283014\pi$
$998$	−35.1215	−1.11175
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bt.1.1		3
3.2	odd	2		2166.2.a.n.1.3		3
19.14	odd	18		342.2.u.d.253.1		6
19.15	odd	18		342.2.u.d.73.1		6
19.18	odd	2		6498.2.a.bo.1.1		3
57.14	even	18		114.2.i.b.25.1	✓	6
57.53	even	18		114.2.i.b.73.1	yes	6
57.56	even	2		2166.2.a.t.1.3		3
228.71	odd	18		912.2.bo.c.481.1		6
228.167	odd	18		912.2.bo.c.529.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.b.25.1	✓	6	57.14	even	18
114.2.i.b.73.1	yes	6	57.53	even	18
342.2.u.d.73.1		6	19.15	odd	18
342.2.u.d.253.1		6	19.14	odd	18
912.2.bo.c.481.1		6	228.71	odd	18
912.2.bo.c.529.1		6	228.167	odd	18
2166.2.a.n.1.3		3	3.2	odd	2
2166.2.a.t.1.3		3	57.56	even	2
6498.2.a.bo.1.1		3	19.18	odd	2
6498.2.a.bt.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:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.467911 q^{5} -4.41147 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.467911 q^{5} -4.41147 q^{7} +1.00000 q^{8} +0.467911 q^{10} +5.53209 q^{11} -5.98545 q^{13} -4.41147 q^{14} +1.00000 q^{16} +4.34730 q^{17} +0.467911 q^{20} +5.53209 q^{22} +3.16250 q^{23} -4.78106 q^{25} -5.98545 q^{26} -4.41147 q^{28} -1.50980 q^{29} +2.18479 q^{31} +1.00000 q^{32} +4.34730 q^{34} -2.06418 q^{35} +2.75877 q^{37} +0.467911 q^{40} -1.95811 q^{41} +3.59627 q^{43} +5.53209 q^{44} +3.16250 q^{46} +9.24897 q^{47} +12.4611 q^{49} -4.78106 q^{50} -5.98545 q^{52} -2.67499 q^{53} +2.58853 q^{55} -4.41147 q^{56} -1.50980 q^{58} +1.32770 q^{59} -6.63816 q^{61} +2.18479 q^{62} +1.00000 q^{64} -2.80066 q^{65} +2.49020 q^{67} +4.34730 q^{68} -2.06418 q^{70} -7.59627 q^{71} +3.16519 q^{73} +2.75877 q^{74} -24.4047 q^{77} -1.26352 q^{79} +0.467911 q^{80} -1.95811 q^{82} +17.9368 q^{83} +2.03415 q^{85} +3.59627 q^{86} +5.53209 q^{88} +12.4807 q^{89} +26.4047 q^{91} +3.16250 q^{92} +9.24897 q^{94} +8.31046 q^{97} +12.4611 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 3 q^{8} + 6 q^{10} + 12 q^{11} - 3 q^{14} + 3 q^{16} + 12 q^{17} + 6 q^{20} + 12 q^{22} + 12 q^{23} + 3 q^{25} - 3 q^{28} - 6 q^{29} + 3 q^{31} + 3 q^{32} + 12 q^{34} + 3 q^{35} - 3 q^{37} + 6 q^{40} - 9 q^{41} - 3 q^{43} + 12 q^{44} + 12 q^{46} + 15 q^{47} + 3 q^{50} - 3 q^{53} + 18 q^{55} - 3 q^{56} - 6 q^{58} - 3 q^{61} + 3 q^{62} + 3 q^{64} + 6 q^{65} + 6 q^{67} + 12 q^{68} + 3 q^{70} - 9 q^{71} + 3 q^{73} - 3 q^{74} - 21 q^{77} - 9 q^{79} + 6 q^{80} - 9 q^{82} - 3 q^{83} + 27 q^{85} - 3 q^{86} + 12 q^{88} + 3 q^{89} + 27 q^{91} + 12 q^{92} + 15 q^{94} + 12 q^{97}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 3 * q^8 + 6 * q^10 + 12 * q^11 - 3 * q^14 + 3 * q^16 + 12 * q^17 + 6 * q^20 + 12 * q^22 + 12 * q^23 + 3 * q^25 - 3 * q^28 - 6 * q^29 + 3 * q^31 + 3 * q^32 + 12 * q^34 + 3 * q^35 - 3 * q^37 + 6 * q^40 - 9 * q^41 - 3 * q^43 + 12 * q^44 + 12 * q^46 + 15 * q^47 + 3 * q^50 - 3 * q^53 + 18 * q^55 - 3 * q^56 - 6 * q^58 - 3 * q^61 + 3 * q^62 + 3 * q^64 + 6 * q^65 + 6 * q^67 + 12 * q^68 + 3 * q^70 - 9 * q^71 + 3 * q^73 - 3 * q^74 - 21 * q^77 - 9 * q^79 + 6 * q^80 - 9 * q^82 - 3 * q^83 + 27 * q^85 - 3 * q^86 + 12 * q^88 + 3 * q^89 + 27 * q^91 + 12 * q^92 + 15 * q^94 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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