LMFDB - Embedded newform 8464.2.a.ch.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.44882$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.44882 q^{3} +2.99678 q^{5} -1.69224 q^{7} -0.900926 q^{9} +O(q^{10})$	$q-1.44882 q^{3} +2.99678 q^{5} -1.69224 q^{7} -0.900926 q^{9} +5.69500 q^{11} -6.47165 q^{13} -4.34179 q^{15} -5.94707 q^{17} +0.280108 q^{19} +2.45174 q^{21} +3.98068 q^{25} +5.65173 q^{27} +1.20529 q^{29} +2.99338 q^{31} -8.25101 q^{33} -5.07126 q^{35} -0.265368 q^{37} +9.37624 q^{39} +5.81542 q^{41} -5.66327 q^{43} -2.69988 q^{45} +1.51532 q^{47} -4.13634 q^{49} +8.61622 q^{51} -6.60567 q^{53} +17.0666 q^{55} -0.405826 q^{57} -3.21061 q^{59} -8.76330 q^{61} +1.52458 q^{63} -19.3941 q^{65} -3.11516 q^{67} +2.19420 q^{71} +12.6635 q^{73} -5.76729 q^{75} -9.63728 q^{77} +0.653751 q^{79} -5.48555 q^{81} +13.8264 q^{83} -17.8221 q^{85} -1.74624 q^{87} +9.80353 q^{89} +10.9516 q^{91} -4.33686 q^{93} +0.839423 q^{95} -3.82589 q^{97} -5.13077 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * 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
$3$	−1.44882	−0.836476	−0.418238	−	0.908338i	$-0.637352\pi$
$3$	−1.44882	−0.836476	−0.418238	+	0.908338i	$0.637352\pi$
$4$	0	0
$5$	2.99678	1.34020	0.670100	−	0.742271i	$-0.266251\pi$
$5$	2.99678	1.34020	0.670100	+	0.742271i	$0.266251\pi$
$6$	0	0
$7$	−1.69224	−0.639605	−0.319803	−	0.947484i	$-0.603617\pi$
$7$	−1.69224	−0.639605	−0.319803	+	0.947484i	$0.603617\pi$
$8$	0	0
$9$	−0.900926	−0.300309
$10$	0	0
$11$	5.69500	1.71711	0.858553	−	0.512725i	$-0.171364\pi$
$11$	5.69500	1.71711	0.858553	+	0.512725i	$0.171364\pi$
$12$	0	0
$13$	−6.47165	−1.79491	−0.897456	−	0.441104i	$-0.854587\pi$
$13$	−6.47165	−1.79491	−0.897456	+	0.441104i	$0.854587\pi$
$14$	0	0
$15$	−4.34179	−1.12104
$16$	0	0
$17$	−5.94707	−1.44238	−0.721188	−	0.692739i	$-0.756404\pi$
$17$	−5.94707	−1.44238	−0.721188	+	0.692739i	$0.756404\pi$
$18$	0	0
$19$	0.280108	0.0642613	0.0321306	−	0.999484i	$-0.489771\pi$
$19$	0.280108	0.0642613	0.0321306	+	0.999484i	$0.489771\pi$
$20$	0	0
$21$	2.45174	0.535014
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	3.98068	0.796136
$26$	0	0
$27$	5.65173	1.08768
$28$	0	0
$29$	1.20529	0.223816	0.111908	−	0.993719i	$-0.464304\pi$
$29$	1.20529	0.223816	0.111908	+	0.993719i	$0.464304\pi$
$30$	0	0
$31$	2.99338	0.537626	0.268813	−	0.963192i	$-0.413369\pi$
$31$	2.99338	0.537626	0.268813	+	0.963192i	$0.413369\pi$
$32$	0	0
$33$	−8.25101	−1.43632
$34$	0	0
$35$	−5.07126	−0.857199
$36$	0	0
$37$	−0.265368	−0.0436262	−0.0218131	−	0.999762i	$-0.506944\pi$
$37$	−0.265368	−0.0436262	−0.0218131	+	0.999762i	$0.506944\pi$
$38$	0	0
$39$	9.37624	1.50140
$40$	0	0
$41$	5.81542	0.908216	0.454108	−	0.890947i	$-0.349958\pi$
$41$	5.81542	0.908216	0.454108	+	0.890947i	$0.349958\pi$
$42$	0	0
$43$	−5.66327	−0.863640	−0.431820	−	0.901960i	$-0.642129\pi$
$43$	−5.66327	−0.863640	−0.431820	+	0.901960i	$0.642129\pi$
$44$	0	0
$45$	−2.69988	−0.402474
$46$	0	0
$47$	1.51532	0.221032	0.110516	−	0.993874i	$-0.464750\pi$
$47$	1.51532	0.221032	0.110516	+	0.993874i	$0.464750\pi$
$48$	0	0
$49$	−4.13634	−0.590905
$50$	0	0
$51$	8.61622	1.20651
$52$	0	0
$53$	−6.60567	−0.907358	−0.453679	−	0.891165i	$-0.649889\pi$
$53$	−6.60567	−0.907358	−0.453679	+	0.891165i	$0.649889\pi$
$54$	0	0
$55$	17.0666	2.30127
$56$	0	0
$57$	−0.405826	−0.0537530
$58$	0	0
$59$	−3.21061	−0.417985	−0.208993	−	0.977917i	$-0.567018\pi$
$59$	−3.21061	−0.417985	−0.208993	+	0.977917i	$0.567018\pi$
$60$	0	0
$61$	−8.76330	−1.12203	−0.561013	−	0.827807i	$-0.689588\pi$
$61$	−8.76330	−1.12203	−0.561013	+	0.827807i	$0.689588\pi$
$62$	0	0
$63$	1.52458	0.192079
$64$	0	0
$65$	−19.3941	−2.40554
$66$	0	0
$67$	−3.11516	−0.380577	−0.190289	−	0.981728i	$-0.560942\pi$
$67$	−3.11516	−0.380577	−0.190289	+	0.981728i	$0.560942\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.19420	0.260403	0.130202	−	0.991488i	$-0.458438\pi$
$71$	2.19420	0.260403	0.130202	+	0.991488i	$0.458438\pi$
$72$	0	0
$73$	12.6635	1.48216	0.741078	−	0.671419i	$-0.234315\pi$
$73$	12.6635	1.48216	0.741078	+	0.671419i	$0.234315\pi$
$74$	0	0
$75$	−5.76729	−0.665949
$76$	0	0
$77$	−9.63728	−1.09827
$78$	0	0
$79$	0.653751	0.0735528	0.0367764	−	0.999324i	$-0.488291\pi$
$79$	0.653751	0.0735528	0.0367764	+	0.999324i	$0.488291\pi$
$80$	0	0
$81$	−5.48555	−0.609506
$82$	0	0
$83$	13.8264	1.51765	0.758823	−	0.651297i	$-0.225775\pi$
$83$	13.8264	1.51765	0.758823	+	0.651297i	$0.225775\pi$
$84$	0	0
$85$	−17.8221	−1.93307
$86$	0	0
$87$	−1.74624	−0.187217
$88$	0	0
$89$	9.80353	1.03917	0.519586	−	0.854418i	$-0.326086\pi$
$89$	9.80353	1.03917	0.519586	+	0.854418i	$0.326086\pi$
$90$	0	0
$91$	10.9516	1.14804
$92$	0	0
$93$	−4.33686	−0.449711
$94$	0	0
$95$	0.839423	0.0861230
$96$	0	0
$97$	−3.82589	−0.388460	−0.194230	−	0.980956i	$-0.562221\pi$
$97$	−3.82589	−0.388460	−0.194230	+	0.980956i	$0.562221\pi$
$98$	0	0
$99$	−5.13077	−0.515662
$100$	0	0
$101$	19.4995	1.94028	0.970138	−	0.242554i	$-0.0779852\pi$
$101$	19.4995	1.94028	0.970138	+	0.242554i	$0.0779852\pi$
$102$	0	0
$103$	15.9157	1.56822	0.784112	−	0.620619i	$-0.213119\pi$
$103$	15.9157	1.56822	0.784112	+	0.620619i	$0.213119\pi$
$104$	0	0
$105$	7.34733	0.717026
$106$	0	0
$107$	20.1139	1.94448	0.972241	−	0.233983i	$-0.0751761\pi$
$107$	20.1139	1.94448	0.972241	+	0.233983i	$0.0751761\pi$
$108$	0	0
$109$	0.786965	0.0753776	0.0376888	−	0.999290i	$-0.488000\pi$
$109$	0.786965	0.0753776	0.0376888	+	0.999290i	$0.488000\pi$
$110$	0	0
$111$	0.384470	0.0364923
$112$	0	0
$113$	5.02962	0.473147	0.236573	−	0.971614i	$-0.423976\pi$
$113$	5.02962	0.473147	0.236573	+	0.971614i	$0.423976\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.83048	0.539028
$118$	0	0
$119$	10.0638	0.922551
$120$	0	0
$121$	21.4330	1.94845
$122$	0	0
$123$	−8.42548	−0.759700
$124$	0	0
$125$	−3.05467	−0.273218
$126$	0	0
$127$	8.02716	0.712295	0.356148	−	0.934430i	$-0.384090\pi$
$127$	8.02716	0.712295	0.356148	+	0.934430i	$0.384090\pi$
$128$	0	0
$129$	8.20505	0.722414
$130$	0	0
$131$	−10.2717	−0.897441	−0.448720	−	0.893672i	$-0.648120\pi$
$131$	−10.2717	−0.897441	−0.448720	+	0.893672i	$0.648120\pi$
$132$	0	0
$133$	−0.474010	−0.0411018
$134$	0	0
$135$	16.9370	1.45770
$136$	0	0
$137$	3.77645	0.322644	0.161322	−	0.986902i	$-0.448424\pi$
$137$	3.77645	0.322644	0.161322	+	0.986902i	$0.448424\pi$
$138$	0	0
$139$	15.4948	1.31425	0.657127	−	0.753780i	$-0.271772\pi$
$139$	15.4948	1.31425	0.657127	+	0.753780i	$0.271772\pi$
$140$	0	0
$141$	−2.19542	−0.184888
$142$	0	0
$143$	−36.8560	−3.08205
$144$	0	0
$145$	3.61198	0.299958
$146$	0	0
$147$	5.99280	0.494278
$148$	0	0
$149$	−3.46788	−0.284100	−0.142050	−	0.989859i	$-0.545369\pi$
$149$	−3.46788	−0.284100	−0.142050	+	0.989859i	$0.545369\pi$
$150$	0	0
$151$	−7.22099	−0.587636	−0.293818	−	0.955861i	$-0.594926\pi$
$151$	−7.22099	−0.587636	−0.293818	+	0.955861i	$0.594926\pi$
$152$	0	0
$153$	5.35787	0.433158
$154$	0	0
$155$	8.97049	0.720527
$156$	0	0
$157$	−10.5678	−0.843402	−0.421701	−	0.906735i	$-0.638567\pi$
$157$	−10.5678	−0.843402	−0.421701	+	0.906735i	$0.638567\pi$
$158$	0	0
$159$	9.57041	0.758983
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−14.2608	−1.11699	−0.558496	−	0.829507i	$-0.688622\pi$
$163$	−14.2608	−1.11699	−0.558496	+	0.829507i	$0.688622\pi$
$164$	0	0
$165$	−24.7265	−1.92495
$166$	0	0
$167$	−22.2273	−1.72000	−0.860000	−	0.510294i	$-0.829537\pi$
$167$	−22.2273	−1.72000	−0.860000	+	0.510294i	$0.829537\pi$
$168$	0	0
$169$	28.8822	2.22171
$170$	0	0
$171$	−0.252357	−0.0192982
$172$	0	0
$173$	−10.1153	−0.769055	−0.384527	−	0.923114i	$-0.625636\pi$
$173$	−10.1153	−0.769055	−0.384527	+	0.923114i	$0.625636\pi$
$174$	0	0
$175$	−6.73625	−0.509213
$176$	0	0
$177$	4.65158	0.349635
$178$	0	0
$179$	4.02091	0.300537	0.150269	−	0.988645i	$-0.451986\pi$
$179$	4.02091	0.300537	0.150269	+	0.988645i	$0.451986\pi$
$180$	0	0
$181$	5.92063	0.440077	0.220038	−	0.975491i	$-0.429382\pi$
$181$	5.92063	0.440077	0.220038	+	0.975491i	$0.429382\pi$
$182$	0	0
$183$	12.6964	0.938547
$184$	0	0
$185$	−0.795249	−0.0584678
$186$	0	0
$187$	−33.8685	−2.47671
$188$	0	0
$189$	−9.56406	−0.695683
$190$	0	0
$191$	6.09973	0.441361	0.220681	−	0.975346i	$-0.429172\pi$
$191$	6.09973	0.441361	0.220681	+	0.975346i	$0.429172\pi$
$192$	0	0
$193$	−1.88904	−0.135976	−0.0679882	−	0.997686i	$-0.521658\pi$
$193$	−1.88904	−0.135976	−0.0679882	+	0.997686i	$0.521658\pi$
$194$	0	0
$195$	28.0985	2.01218
$196$	0	0
$197$	20.4561	1.45744	0.728719	−	0.684812i	$-0.240116\pi$
$197$	20.4561	1.45744	0.728719	+	0.684812i	$0.240116\pi$
$198$	0	0
$199$	2.27503	0.161273	0.0806364	−	0.996744i	$-0.474305\pi$
$199$	2.27503	0.161273	0.0806364	+	0.996744i	$0.474305\pi$
$200$	0	0
$201$	4.51330	0.318344
$202$	0	0
$203$	−2.03963	−0.143154
$204$	0	0
$205$	17.4275	1.21719
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.59522	0.110343
$210$	0	0
$211$	1.74646	0.120231	0.0601157	−	0.998191i	$-0.480853\pi$
$211$	1.74646	0.120231	0.0601157	+	0.998191i	$0.480853\pi$
$212$	0	0
$213$	−3.17899	−0.217821
$214$	0	0
$215$	−16.9716	−1.15745
$216$	0	0
$217$	−5.06550	−0.343869
$218$	0	0
$219$	−18.3472	−1.23979
$220$	0	0
$221$	38.4873	2.58894
$222$	0	0
$223$	10.0252	0.671336	0.335668	−	0.941980i	$-0.391038\pi$
$223$	10.0252	0.671336	0.335668	+	0.941980i	$0.391038\pi$
$224$	0	0
$225$	−3.58630	−0.239087
$226$	0	0
$227$	−11.0041	−0.730367	−0.365183	−	0.930936i	$-0.618994\pi$
$227$	−11.0041	−0.730367	−0.365183	+	0.930936i	$0.618994\pi$
$228$	0	0
$229$	10.8863	0.719384	0.359692	−	0.933071i	$-0.382882\pi$
$229$	10.8863	0.719384	0.359692	+	0.933071i	$0.382882\pi$
$230$	0	0
$231$	13.9627	0.918676
$232$	0	0
$233$	10.2437	0.671088	0.335544	−	0.942025i	$-0.391080\pi$
$233$	10.2437	0.671088	0.335544	+	0.942025i	$0.391080\pi$
$234$	0	0
$235$	4.54108	0.296227
$236$	0	0
$237$	−0.947167	−0.0615251
$238$	0	0
$239$	−15.4895	−1.00193	−0.500966	−	0.865467i	$-0.667022\pi$
$239$	−15.4895	−1.00193	−0.500966	+	0.865467i	$0.667022\pi$
$240$	0	0
$241$	6.81957	0.439287	0.219644	−	0.975580i	$-0.429511\pi$
$241$	6.81957	0.439287	0.219644	+	0.975580i	$0.429511\pi$
$242$	0	0
$243$	−9.00763	−0.577839
$244$	0	0
$245$	−12.3957	−0.791931
$246$	0	0
$247$	−1.81276	−0.115343
$248$	0	0
$249$	−20.0319	−1.26947
$250$	0	0
$251$	−8.19606	−0.517331	−0.258665	−	0.965967i	$-0.583283\pi$
$251$	−8.19606	−0.517331	−0.258665	+	0.965967i	$0.583283\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	25.8209	1.61697
$256$	0	0
$257$	−2.54923	−0.159017	−0.0795084	−	0.996834i	$-0.525335\pi$
$257$	−2.54923	−0.159017	−0.0795084	+	0.996834i	$0.525335\pi$
$258$	0	0
$259$	0.449065	0.0279035
$260$	0	0
$261$	−1.08587	−0.0672139
$262$	0	0
$263$	−3.10814	−0.191656	−0.0958280	−	0.995398i	$-0.530550\pi$
$263$	−3.10814	−0.191656	−0.0958280	+	0.995398i	$0.530550\pi$
$264$	0	0
$265$	−19.7957	−1.21604
$266$	0	0
$267$	−14.2035	−0.869242
$268$	0	0
$269$	−26.1929	−1.59701	−0.798504	−	0.601990i	$-0.794375\pi$
$269$	−26.1929	−1.59701	−0.798504	+	0.601990i	$0.794375\pi$
$270$	0	0
$271$	14.7813	0.897902	0.448951	−	0.893556i	$-0.351798\pi$
$271$	14.7813	0.897902	0.448951	+	0.893556i	$0.351798\pi$
$272$	0	0
$273$	−15.8668	−0.960303
$274$	0	0
$275$	22.6700	1.36705
$276$	0	0
$277$	4.59471	0.276069	0.138035	−	0.990427i	$-0.455921\pi$
$277$	4.59471	0.276069	0.138035	+	0.990427i	$0.455921\pi$
$278$	0	0
$279$	−2.69681	−0.161454
$280$	0	0
$281$	−8.72942	−0.520754	−0.260377	−	0.965507i	$-0.583847\pi$
$281$	−8.72942	−0.520754	−0.260377	+	0.965507i	$0.583847\pi$
$282$	0	0
$283$	25.1539	1.49524	0.747622	−	0.664124i	$-0.231195\pi$
$283$	25.1539	1.49524	0.747622	+	0.664124i	$0.231195\pi$
$284$	0	0
$285$	−1.21617	−0.0720398
$286$	0	0
$287$	−9.84106	−0.580899
$288$	0	0
$289$	18.3676	1.08045
$290$	0	0
$291$	5.54301	0.324937
$292$	0	0
$293$	−20.1270	−1.17583	−0.587917	−	0.808921i	$-0.700052\pi$
$293$	−20.1270	−1.17583	−0.587917	+	0.808921i	$0.700052\pi$
$294$	0	0
$295$	−9.62148	−0.560184
$296$	0	0
$297$	32.1866	1.86766
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.58359	0.552389
$302$	0	0
$303$	−28.2513	−1.62299
$304$	0	0
$305$	−26.2617	−1.50374
$306$	0	0
$307$	25.0569	1.43007	0.715037	−	0.699086i	$-0.246410\pi$
$307$	25.0569	1.43007	0.715037	+	0.699086i	$0.246410\pi$
$308$	0	0
$309$	−23.0590	−1.31178
$310$	0	0
$311$	13.2143	0.749315	0.374658	−	0.927163i	$-0.377760\pi$
$311$	13.2143	0.749315	0.374658	+	0.927163i	$0.377760\pi$
$312$	0	0
$313$	−28.7043	−1.62246	−0.811231	−	0.584726i	$-0.801202\pi$
$313$	−28.7043	−1.62246	−0.811231	+	0.584726i	$0.801202\pi$
$314$	0	0
$315$	4.56883	0.257424
$316$	0	0
$317$	−4.58449	−0.257490	−0.128745	−	0.991678i	$-0.541095\pi$
$317$	−4.58449	−0.257490	−0.128745	+	0.991678i	$0.541095\pi$
$318$	0	0
$319$	6.86410	0.384316
$320$	0	0
$321$	−29.1413	−1.62651
$322$	0	0
$323$	−1.66582	−0.0926889
$324$	0	0
$325$	−25.7616	−1.42900
$326$	0	0
$327$	−1.14017	−0.0630515
$328$	0	0
$329$	−2.56428	−0.141373
$330$	0	0
$331$	21.7963	1.19803	0.599017	−	0.800736i	$-0.295558\pi$
$331$	21.7963	1.19803	0.599017	+	0.800736i	$0.295558\pi$
$332$	0	0
$333$	0.239077	0.0131013
$334$	0	0
$335$	−9.33545	−0.510050
$336$	0	0
$337$	16.1795	0.881353	0.440677	−	0.897666i	$-0.354739\pi$
$337$	16.1795	0.881353	0.440677	+	0.897666i	$0.354739\pi$
$338$	0	0
$339$	−7.28701	−0.395776
$340$	0	0
$341$	17.0473	0.923161
$342$	0	0
$343$	18.8453	1.01755
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.6402	1.21539	0.607694	−	0.794171i	$-0.292095\pi$
$347$	22.6402	1.21539	0.607694	+	0.794171i	$0.292095\pi$
$348$	0	0
$349$	20.0876	1.07527	0.537633	−	0.843179i	$-0.319319\pi$
$349$	20.0876	1.07527	0.537633	+	0.843179i	$0.319319\pi$
$350$	0	0
$351$	−36.5760	−1.95228
$352$	0	0
$353$	−11.9058	−0.633681	−0.316841	−	0.948479i	$-0.602622\pi$
$353$	−11.9058	−0.633681	−0.316841	+	0.948479i	$0.602622\pi$
$354$	0	0
$355$	6.57553	0.348993
$356$	0	0
$357$	−14.5807	−0.771692
$358$	0	0
$359$	−21.0744	−1.11227	−0.556133	−	0.831093i	$-0.687715\pi$
$359$	−21.0744	−1.11227	−0.556133	+	0.831093i	$0.687715\pi$
$360$	0	0
$361$	−18.9215	−0.995870
$362$	0	0
$363$	−31.0525	−1.62983
$364$	0	0
$365$	37.9498	1.98639
$366$	0	0
$367$	20.0368	1.04591	0.522957	−	0.852359i	$-0.324829\pi$
$367$	20.0368	1.04591	0.522957	+	0.852359i	$0.324829\pi$
$368$	0	0
$369$	−5.23926	−0.272745
$370$	0	0
$371$	11.1783	0.580351
$372$	0	0
$373$	36.8053	1.90571	0.952854	−	0.303430i	$-0.0981318\pi$
$373$	36.8053	1.90571	0.952854	+	0.303430i	$0.0981318\pi$
$374$	0	0
$375$	4.42566	0.228540
$376$	0	0
$377$	−7.80019	−0.401730
$378$	0	0
$379$	−28.7886	−1.47877	−0.739386	−	0.673282i	$-0.764884\pi$
$379$	−28.7886	−1.47877	−0.739386	+	0.673282i	$0.764884\pi$
$380$	0	0
$381$	−11.6299	−0.595818
$382$	0	0
$383$	5.86326	0.299598	0.149799	−	0.988716i	$-0.452137\pi$
$383$	5.86326	0.299598	0.149799	+	0.988716i	$0.452137\pi$
$384$	0	0
$385$	−28.8808	−1.47190
$386$	0	0
$387$	5.10219	0.259359
$388$	0	0
$389$	−2.66338	−0.135039	−0.0675194	−	0.997718i	$-0.521508\pi$
$389$	−2.66338	−0.135039	−0.0675194	+	0.997718i	$0.521508\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	14.8818	0.750687
$394$	0	0
$395$	1.95915	0.0985754
$396$	0	0
$397$	1.89393	0.0950536	0.0475268	−	0.998870i	$-0.484866\pi$
$397$	1.89393	0.0950536	0.0475268	+	0.998870i	$0.484866\pi$
$398$	0	0
$399$	0.686754	0.0343807
$400$	0	0
$401$	21.2684	1.06209	0.531046	−	0.847343i	$-0.321799\pi$
$401$	21.2684	1.06209	0.531046	+	0.847343i	$0.321799\pi$
$402$	0	0
$403$	−19.3721	−0.964992
$404$	0	0
$405$	−16.4390	−0.816860
$406$	0	0
$407$	−1.51127	−0.0749108
$408$	0	0
$409$	4.90159	0.242368	0.121184	−	0.992630i	$-0.461331\pi$
$409$	4.90159	0.242368	0.121184	+	0.992630i	$0.461331\pi$
$410$	0	0
$411$	−5.47138	−0.269883
$412$	0	0
$413$	5.43310	0.267346
$414$	0	0
$415$	41.4347	2.03395
$416$	0	0
$417$	−22.4492	−1.09934
$418$	0	0
$419$	6.78193	0.331319	0.165659	−	0.986183i	$-0.447025\pi$
$419$	6.78193	0.331319	0.165659	+	0.986183i	$0.447025\pi$
$420$	0	0
$421$	22.9562	1.11882	0.559408	−	0.828892i	$-0.311028\pi$
$421$	22.9562	1.11882	0.559408	+	0.828892i	$0.311028\pi$
$422$	0	0
$423$	−1.36519	−0.0663778
$424$	0	0
$425$	−23.6734	−1.14833
$426$	0	0
$427$	14.8296	0.717653
$428$	0	0
$429$	53.3977	2.57806
$430$	0	0
$431$	23.6912	1.14117	0.570583	−	0.821240i	$-0.306717\pi$
$431$	23.6912	1.14117	0.570583	+	0.821240i	$0.306717\pi$
$432$	0	0
$433$	23.1997	1.11491	0.557453	−	0.830209i	$-0.311779\pi$
$433$	23.1997	1.11491	0.557453	+	0.830209i	$0.311779\pi$
$434$	0	0
$435$	−5.23310	−0.250908
$436$	0	0
$437$	0	0
$438$	0	0
$439$	26.3942	1.25973	0.629863	−	0.776706i	$-0.283111\pi$
$439$	26.3942	1.25973	0.629863	+	0.776706i	$0.283111\pi$
$440$	0	0
$441$	3.72653	0.177454
$442$	0	0
$443$	0.119780	0.00569093	0.00284546	−	0.999996i	$-0.499094\pi$
$443$	0.119780	0.00569093	0.00284546	+	0.999996i	$0.499094\pi$
$444$	0	0
$445$	29.3790	1.39270
$446$	0	0
$447$	5.02433	0.237643
$448$	0	0
$449$	−38.2146	−1.80346	−0.901730	−	0.432300i	$-0.857702\pi$
$449$	−38.2146	−1.80346	−0.901730	+	0.432300i	$0.857702\pi$
$450$	0	0
$451$	33.1188	1.55950
$452$	0	0
$453$	10.4619	0.491543
$454$	0	0
$455$	32.8194	1.53860
$456$	0	0
$457$	−19.2031	−0.898284	−0.449142	−	0.893460i	$-0.648270\pi$
$457$	−19.2031	−0.898284	−0.449142	+	0.893460i	$0.648270\pi$
$458$	0	0
$459$	−33.6112	−1.56884
$460$	0	0
$461$	14.9421	0.695925	0.347962	−	0.937508i	$-0.386874\pi$
$461$	14.9421	0.695925	0.347962	+	0.937508i	$0.386874\pi$
$462$	0	0
$463$	−30.0954	−1.39865	−0.699327	−	0.714802i	$-0.746517\pi$
$463$	−30.0954	−1.39865	−0.699327	+	0.714802i	$0.746517\pi$
$464$	0	0
$465$	−12.9966	−0.602703
$466$	0	0
$467$	10.3071	0.476957	0.238479	−	0.971148i	$-0.423351\pi$
$467$	10.3071	0.476957	0.238479	+	0.971148i	$0.423351\pi$
$468$	0	0
$469$	5.27159	0.243419
$470$	0	0
$471$	15.3108	0.705485
$472$	0	0
$473$	−32.2523	−1.48296
$474$	0	0
$475$	1.11502	0.0511608
$476$	0	0
$477$	5.95121	0.272487
$478$	0	0
$479$	−1.40898	−0.0643779	−0.0321889	−	0.999482i	$-0.510248\pi$
$479$	−1.40898	−0.0643779	−0.0321889	+	0.999482i	$0.510248\pi$
$480$	0	0
$481$	1.71737	0.0783052
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.4653	−0.520614
$486$	0	0
$487$	8.59118	0.389303	0.194652	−	0.980872i	$-0.437642\pi$
$487$	8.59118	0.389303	0.194652	+	0.980872i	$0.437642\pi$
$488$	0	0
$489$	20.6613	0.934337
$490$	0	0
$491$	30.1963	1.36274	0.681370	−	0.731939i	$-0.261385\pi$
$491$	30.1963	1.36274	0.681370	+	0.731939i	$0.261385\pi$
$492$	0	0
$493$	−7.16793	−0.322827
$494$	0	0
$495$	−15.3758	−0.691090
$496$	0	0
$497$	−3.71310	−0.166555
$498$	0	0
$499$	19.3363	0.865610	0.432805	−	0.901488i	$-0.357524\pi$
$499$	19.3363	0.865610	0.432805	+	0.901488i	$0.357524\pi$
$500$	0	0
$501$	32.2033	1.43874
$502$	0	0
$503$	13.9588	0.622393	0.311196	−	0.950346i	$-0.399270\pi$
$503$	13.9588	0.622393	0.311196	+	0.950346i	$0.399270\pi$
$504$	0	0
$505$	58.4358	2.60036
$506$	0	0
$507$	−41.8451	−1.85841
$508$	0	0
$509$	17.0388	0.755232	0.377616	−	0.925962i	$-0.376744\pi$
$509$	17.0388	0.755232	0.377616	+	0.925962i	$0.376744\pi$
$510$	0	0
$511$	−21.4297	−0.947995
$512$	0	0
$513$	1.58310	0.0698955
$514$	0	0
$515$	47.6960	2.10174
$516$	0	0
$517$	8.62974	0.379535
$518$	0	0
$519$	14.6553	0.643296
$520$	0	0
$521$	22.5632	0.988511	0.494256	−	0.869317i	$-0.335441\pi$
$521$	22.5632	0.988511	0.494256	+	0.869317i	$0.335441\pi$
$522$	0	0
$523$	−4.74739	−0.207589	−0.103794	−	0.994599i	$-0.533098\pi$
$523$	−4.74739	−0.207589	−0.103794	+	0.994599i	$0.533098\pi$
$524$	0	0
$525$	9.75961	0.425944
$526$	0	0
$527$	−17.8018	−0.775459
$528$	0	0
$529$	0	0
$530$	0	0
$531$	2.89252	0.125525
$532$	0	0
$533$	−37.6353	−1.63017
$534$	0	0
$535$	60.2768	2.60599
$536$	0	0
$537$	−5.82557	−0.251392
$538$	0	0
$539$	−23.5564	−1.01465
$540$	0	0
$541$	41.4003	1.77994	0.889969	−	0.456021i	$-0.150726\pi$
$541$	41.4003	1.77994	0.889969	+	0.456021i	$0.150726\pi$
$542$	0	0
$543$	−8.57791	−0.368113
$544$	0	0
$545$	2.35836	0.101021
$546$	0	0
$547$	−31.9821	−1.36746	−0.683729	−	0.729736i	$-0.739643\pi$
$547$	−31.9821	−1.36746	−0.683729	+	0.729736i	$0.739643\pi$
$548$	0	0
$549$	7.89508	0.336954
$550$	0	0
$551$	0.337611	0.0143827
$552$	0	0
$553$	−1.10630	−0.0470447
$554$	0	0
$555$	1.15217	0.0489069
$556$	0	0
$557$	−14.4711	−0.613158	−0.306579	−	0.951845i	$-0.599184\pi$
$557$	−14.4711	−0.613158	−0.306579	+	0.951845i	$0.599184\pi$
$558$	0	0
$559$	36.6507	1.55016
$560$	0	0
$561$	49.0693	2.07171
$562$	0	0
$563$	27.5811	1.16241	0.581203	−	0.813759i	$-0.302582\pi$
$563$	27.5811	1.16241	0.581203	+	0.813759i	$0.302582\pi$
$564$	0	0
$565$	15.0727	0.634112
$566$	0	0
$567$	9.28285	0.389843
$568$	0	0
$569$	18.3129	0.767717	0.383858	−	0.923392i	$-0.374595\pi$
$569$	18.3129	0.767717	0.383858	+	0.923392i	$0.374595\pi$
$570$	0	0
$571$	37.1755	1.55575	0.777873	−	0.628422i	$-0.216299\pi$
$571$	37.1755	1.55575	0.777873	+	0.628422i	$0.216299\pi$
$572$	0	0
$573$	−8.83740	−0.369188
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.5138	0.729107	0.364554	−	0.931182i	$-0.381222\pi$
$577$	17.5138	0.729107	0.364554	+	0.931182i	$0.381222\pi$
$578$	0	0
$579$	2.73688	0.113741
$580$	0	0
$581$	−23.3975	−0.970694
$582$	0	0
$583$	−37.6192	−1.55803
$584$	0	0
$585$	17.4726	0.722405
$586$	0	0
$587$	−48.2469	−1.99136	−0.995680	−	0.0928476i	$-0.970403\pi$
$587$	−48.2469	−1.99136	−0.995680	+	0.0928476i	$0.970403\pi$
$588$	0	0
$589$	0.838470	0.0345486
$590$	0	0
$591$	−29.6372	−1.21911
$592$	0	0
$593$	−32.2135	−1.32285	−0.661425	−	0.750011i	$-0.730048\pi$
$593$	−32.2135	−1.32285	−0.661425	+	0.750011i	$0.730048\pi$
$594$	0	0
$595$	30.1591	1.23640
$596$	0	0
$597$	−3.29611	−0.134901
$598$	0	0
$599$	−17.5873	−0.718599	−0.359299	−	0.933222i	$-0.616984\pi$
$599$	−17.5873	−0.718599	−0.359299	+	0.933222i	$0.616984\pi$
$600$	0	0
$601$	1.68245	0.0686286	0.0343143	−	0.999411i	$-0.489075\pi$
$601$	1.68245	0.0686286	0.0343143	+	0.999411i	$0.489075\pi$
$602$	0	0
$603$	2.80653	0.114291
$604$	0	0
$605$	64.2299	2.61132
$606$	0	0
$607$	−15.4725	−0.628011	−0.314006	−	0.949421i	$-0.601671\pi$
$607$	−15.4725	−0.628011	−0.314006	+	0.949421i	$0.601671\pi$
$608$	0	0
$609$	2.95505	0.119745
$610$	0	0
$611$	−9.80661	−0.396733
$612$	0	0
$613$	−21.7704	−0.879297	−0.439648	−	0.898170i	$-0.644897\pi$
$613$	−21.7704	−0.879297	−0.439648	+	0.898170i	$0.644897\pi$
$614$	0	0
$615$	−25.2493	−1.01815
$616$	0	0
$617$	−10.0755	−0.405625	−0.202813	−	0.979218i	$-0.565008\pi$
$617$	−10.0755	−0.405625	−0.202813	+	0.979218i	$0.565008\pi$
$618$	0	0
$619$	25.6075	1.02925	0.514627	−	0.857414i	$-0.327931\pi$
$619$	25.6075	1.02925	0.514627	+	0.857414i	$0.327931\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.5899	−0.664660
$624$	0	0
$625$	−29.0576	−1.16230
$626$	0	0
$627$	−2.31118	−0.0922996
$628$	0	0
$629$	1.57816	0.0629254
$630$	0	0
$631$	18.9069	0.752673	0.376336	−	0.926483i	$-0.377184\pi$
$631$	18.9069	0.752673	0.376336	+	0.926483i	$0.377184\pi$
$632$	0	0
$633$	−2.53031	−0.100571
$634$	0	0
$635$	24.0556	0.954618
$636$	0	0
$637$	26.7689	1.06062
$638$	0	0
$639$	−1.97681	−0.0782014
$640$	0	0
$641$	25.4563	1.00546	0.502731	−	0.864443i	$-0.332328\pi$
$641$	25.4563	1.00546	0.502731	+	0.864443i	$0.332328\pi$
$642$	0	0
$643$	28.9449	1.14147	0.570737	−	0.821133i	$-0.306657\pi$
$643$	28.9449	1.14147	0.570737	+	0.821133i	$0.306657\pi$
$644$	0	0
$645$	24.5887	0.968180
$646$	0	0
$647$	36.4977	1.43487	0.717435	−	0.696625i	$-0.245316\pi$
$647$	36.4977	1.43487	0.717435	+	0.696625i	$0.245316\pi$
$648$	0	0
$649$	−18.2844	−0.717725
$650$	0	0
$651$	7.33899	0.287638
$652$	0	0
$653$	−3.10583	−0.121540	−0.0607702	−	0.998152i	$-0.519356\pi$
$653$	−3.10583	−0.121540	−0.0607702	+	0.998152i	$0.519356\pi$
$654$	0	0
$655$	−30.7819	−1.20275
$656$	0	0
$657$	−11.4089	−0.445104
$658$	0	0
$659$	28.5766	1.11319	0.556593	−	0.830785i	$-0.312108\pi$
$659$	28.5766	1.11319	0.556593	+	0.830785i	$0.312108\pi$
$660$	0	0
$661$	0.829469	0.0322626	0.0161313	−	0.999870i	$-0.494865\pi$
$661$	0.829469	0.0322626	0.0161313	+	0.999870i	$0.494865\pi$
$662$	0	0
$663$	−55.7612	−2.16558
$664$	0	0
$665$	−1.42050	−0.0550847
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−14.5247	−0.561556
$670$	0	0
$671$	−49.9069	−1.92664
$672$	0	0
$673$	−0.665190	−0.0256412	−0.0128206	−	0.999918i	$-0.504081\pi$
$673$	−0.665190	−0.0256412	−0.0128206	+	0.999918i	$0.504081\pi$
$674$	0	0
$675$	22.4978	0.865939
$676$	0	0
$677$	1.54745	0.0594733	0.0297367	−	0.999558i	$-0.490533\pi$
$677$	1.54745	0.0594733	0.0297367	+	0.999558i	$0.490533\pi$
$678$	0	0
$679$	6.47430	0.248461
$680$	0	0
$681$	15.9429	0.610934
$682$	0	0
$683$	−39.7618	−1.52144	−0.760722	−	0.649077i	$-0.775155\pi$
$683$	−39.7618	−1.52144	−0.760722	+	0.649077i	$0.775155\pi$
$684$	0	0
$685$	11.3172	0.432407
$686$	0	0
$687$	−15.7722	−0.601747
$688$	0	0
$689$	42.7495	1.62863
$690$	0	0
$691$	−37.9101	−1.44217	−0.721083	−	0.692848i	$-0.756356\pi$
$691$	−37.9101	−1.44217	−0.721083	+	0.692848i	$0.756356\pi$
$692$	0	0
$693$	8.68247	0.329820
$694$	0	0
$695$	46.4345	1.76136
$696$	0	0
$697$	−34.5847	−1.30999
$698$	0	0
$699$	−14.8413	−0.561349
$700$	0	0
$701$	24.0633	0.908857	0.454429	−	0.890783i	$-0.349843\pi$
$701$	24.0633	0.908857	0.454429	+	0.890783i	$0.349843\pi$
$702$	0	0
$703$	−0.0743318	−0.00280348
$704$	0	0
$705$	−6.57919	−0.247787
$706$	0	0
$707$	−32.9978	−1.24101
$708$	0	0
$709$	−47.2109	−1.77304	−0.886522	−	0.462686i	$-0.846886\pi$
$709$	−47.2109	−1.77304	−0.886522	+	0.462686i	$0.846886\pi$
$710$	0	0
$711$	−0.588981	−0.0220885
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−110.449	−4.13057
$716$	0	0
$717$	22.4414	0.838091
$718$	0	0
$719$	7.41950	0.276701	0.138350	−	0.990383i	$-0.455820\pi$
$719$	7.41950	0.276701	0.138350	+	0.990383i	$0.455820\pi$
$720$	0	0
$721$	−26.9332	−1.00304
$722$	0	0
$723$	−9.88032	−0.367453
$724$	0	0
$725$	4.79786	0.178188
$726$	0	0
$727$	−0.142621	−0.00528953	−0.00264477	−	0.999997i	$-0.500842\pi$
$727$	−0.142621	−0.00528953	−0.00264477	+	0.999997i	$0.500842\pi$
$728$	0	0
$729$	29.5071	1.09285
$730$	0	0
$731$	33.6799	1.24569
$732$	0	0
$733$	−11.7207	−0.432915	−0.216457	−	0.976292i	$-0.569450\pi$
$733$	−11.7207	−0.432915	−0.216457	+	0.976292i	$0.569450\pi$
$734$	0	0
$735$	17.9591	0.662431
$736$	0	0
$737$	−17.7408	−0.653492
$738$	0	0
$739$	38.3686	1.41141	0.705705	−	0.708505i	$-0.250630\pi$
$739$	38.3686	1.41141	0.705705	+	0.708505i	$0.250630\pi$
$740$	0	0
$741$	2.62636	0.0964819
$742$	0	0
$743$	−37.9407	−1.39191	−0.695955	−	0.718086i	$-0.745019\pi$
$743$	−37.9407	−1.39191	−0.695955	+	0.718086i	$0.745019\pi$
$744$	0	0
$745$	−10.3925	−0.380751
$746$	0	0
$747$	−12.4566	−0.455762
$748$	0	0
$749$	−34.0374	−1.24370
$750$	0	0
$751$	39.4716	1.44034	0.720170	−	0.693797i	$-0.244064\pi$
$751$	39.4716	1.44034	0.720170	+	0.693797i	$0.244064\pi$
$752$	0	0
$753$	11.8746	0.432735
$754$	0	0
$755$	−21.6397	−0.787549
$756$	0	0
$757$	−40.1136	−1.45795	−0.728976	−	0.684539i	$-0.760004\pi$
$757$	−40.1136	−1.45795	−0.728976	+	0.684539i	$0.760004\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.793648	−0.0287697	−0.0143849	−	0.999897i	$-0.504579\pi$
$761$	−0.793648	−0.0287697	−0.0143849	+	0.999897i	$0.504579\pi$
$762$	0	0
$763$	−1.33173	−0.0482119
$764$	0	0
$765$	16.0563	0.580518
$766$	0	0
$767$	20.7779	0.750247
$768$	0	0
$769$	−16.2374	−0.585535	−0.292767	−	0.956184i	$-0.594576\pi$
$769$	−16.2374	−0.585535	−0.292767	+	0.956184i	$0.594576\pi$
$770$	0	0
$771$	3.69338	0.133014
$772$	0	0
$773$	3.99267	0.143607	0.0718033	−	0.997419i	$-0.477125\pi$
$773$	3.99267	0.143607	0.0718033	+	0.997419i	$0.477125\pi$
$774$	0	0
$775$	11.9157	0.428024
$776$	0	0
$777$	−0.650614	−0.0233406
$778$	0	0
$779$	1.62895	0.0583631
$780$	0	0
$781$	12.4959	0.447140
$782$	0	0
$783$	6.81196	0.243440
$784$	0	0
$785$	−31.6693	−1.13033
$786$	0	0
$787$	2.56066	0.0912775	0.0456388	−	0.998958i	$-0.485468\pi$
$787$	2.56066	0.0912775	0.0456388	+	0.998958i	$0.485468\pi$
$788$	0	0
$789$	4.50313	0.160316
$790$	0	0
$791$	−8.51130	−0.302627
$792$	0	0
$793$	56.7130	2.01394
$794$	0	0
$795$	28.6804	1.01719
$796$	0	0
$797$	6.09898	0.216037	0.108018	−	0.994149i	$-0.465549\pi$
$797$	6.09898	0.216037	0.108018	+	0.994149i	$0.465549\pi$
$798$	0	0
$799$	−9.01171	−0.318811
$800$	0	0
$801$	−8.83226	−0.312072
$802$	0	0
$803$	72.1188	2.54502
$804$	0	0
$805$	0	0
$806$	0	0
$807$	37.9487	1.33586
$808$	0	0
$809$	0.421524	0.0148200	0.00741000	−	0.999973i	$-0.497641\pi$
$809$	0.421524	0.0148200	0.00741000	+	0.999973i	$0.497641\pi$
$810$	0	0
$811$	19.4489	0.682945	0.341472	−	0.939892i	$-0.389074\pi$
$811$	19.4489	0.682945	0.341472	+	0.939892i	$0.389074\pi$
$812$	0	0
$813$	−21.4155	−0.751073
$814$	0	0
$815$	−42.7365	−1.49699
$816$	0	0
$817$	−1.58633	−0.0554986
$818$	0	0
$819$	−9.86654	−0.344765
$820$	0	0
$821$	−6.82458	−0.238179	−0.119090	−	0.992883i	$-0.537998\pi$
$821$	−6.82458	−0.238179	−0.119090	+	0.992883i	$0.537998\pi$
$822$	0	0
$823$	−2.27738	−0.0793846	−0.0396923	−	0.999212i	$-0.512638\pi$
$823$	−2.27738	−0.0793846	−0.0396923	+	0.999212i	$0.512638\pi$
$824$	0	0
$825$	−32.8447	−1.14350
$826$	0	0
$827$	−2.32388	−0.0808093	−0.0404047	−	0.999183i	$-0.512865\pi$
$827$	−2.32388	−0.0808093	−0.0404047	+	0.999183i	$0.512865\pi$
$828$	0	0
$829$	−6.89147	−0.239351	−0.119675	−	0.992813i	$-0.538185\pi$
$829$	−6.89147	−0.239351	−0.119675	+	0.992813i	$0.538185\pi$
$830$	0	0
$831$	−6.65689	−0.230925
$832$	0	0
$833$	24.5991	0.852308
$834$	0	0
$835$	−66.6103	−2.30514
$836$	0	0
$837$	16.9178	0.584763
$838$	0	0
$839$	4.94182	0.170611	0.0853053	−	0.996355i	$-0.472813\pi$
$839$	4.94182	0.170611	0.0853053	+	0.996355i	$0.472813\pi$
$840$	0	0
$841$	−27.5473	−0.949906
$842$	0	0
$843$	12.6473	0.435598
$844$	0	0
$845$	86.5537	2.97754
$846$	0	0
$847$	−36.2696	−1.24624
$848$	0	0
$849$	−36.4434	−1.25074
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−12.7407	−0.436234	−0.218117	−	0.975923i	$-0.569991\pi$
$853$	−12.7407	−0.436234	−0.218117	+	0.975923i	$0.569991\pi$
$854$	0	0
$855$	−0.756258	−0.0258635
$856$	0	0
$857$	−18.3428	−0.626577	−0.313289	−	0.949658i	$-0.601431\pi$
$857$	−18.3428	−0.626577	−0.313289	+	0.949658i	$0.601431\pi$
$858$	0	0
$859$	4.07244	0.138950	0.0694750	−	0.997584i	$-0.477868\pi$
$859$	4.07244	0.138950	0.0694750	+	0.997584i	$0.477868\pi$
$860$	0	0
$861$	14.2579	0.485908
$862$	0	0
$863$	15.4592	0.526238	0.263119	−	0.964763i	$-0.415249\pi$
$863$	15.4592	0.526238	0.263119	+	0.964763i	$0.415249\pi$
$864$	0	0
$865$	−30.3134	−1.03069
$866$	0	0
$867$	−26.6114	−0.903769
$868$	0	0
$869$	3.72311	0.126298
$870$	0	0
$871$	20.1602	0.683103
$872$	0	0
$873$	3.44684	0.116658
$874$	0	0
$875$	5.16922	0.174752
$876$	0	0
$877$	52.7400	1.78090	0.890452	−	0.455077i	$-0.150388\pi$
$877$	52.7400	1.78090	0.890452	+	0.455077i	$0.150388\pi$
$878$	0	0
$879$	29.1604	0.983557
$880$	0	0
$881$	11.2468	0.378914	0.189457	−	0.981889i	$-0.439327\pi$
$881$	11.2468	0.378914	0.189457	+	0.981889i	$0.439327\pi$
$882$	0	0
$883$	−8.96433	−0.301674	−0.150837	−	0.988559i	$-0.548197\pi$
$883$	−8.96433	−0.301674	−0.150837	+	0.988559i	$0.548197\pi$
$884$	0	0
$885$	13.9398	0.468580
$886$	0	0
$887$	−55.3587	−1.85876	−0.929381	−	0.369122i	$-0.879659\pi$
$887$	−55.3587	−1.85876	−0.929381	+	0.369122i	$0.879659\pi$
$888$	0	0
$889$	−13.5839	−0.455588
$890$	0	0
$891$	−31.2402	−1.04659
$892$	0	0
$893$	0.424454	0.0142038
$894$	0	0
$895$	12.0498	0.402780
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.60788	0.120329
$900$	0	0
$901$	39.2844	1.30875
$902$	0	0
$903$	−13.8849	−0.462060
$904$	0	0
$905$	17.7428	0.589791
$906$	0	0
$907$	8.99183	0.298569	0.149284	−	0.988794i	$-0.452303\pi$
$907$	8.99183	0.298569	0.149284	+	0.988794i	$0.452303\pi$
$908$	0	0
$909$	−17.5676	−0.582682
$910$	0	0
$911$	28.2775	0.936874	0.468437	−	0.883497i	$-0.344817\pi$
$911$	28.2775	0.936874	0.468437	+	0.883497i	$0.344817\pi$
$912$	0	0
$913$	78.7413	2.60596
$914$	0	0
$915$	38.0484	1.25784
$916$	0	0
$917$	17.3821	0.574008
$918$	0	0
$919$	−25.9114	−0.854740	−0.427370	−	0.904077i	$-0.640560\pi$
$919$	−25.9114	−0.854740	−0.427370	+	0.904077i	$0.640560\pi$
$920$	0	0
$921$	−36.3029	−1.19622
$922$	0	0
$923$	−14.2001	−0.467401
$924$	0	0
$925$	−1.05635	−0.0347324
$926$	0	0
$927$	−14.3389	−0.470951
$928$	0	0
$929$	18.7663	0.615701	0.307851	−	0.951435i	$-0.400390\pi$
$929$	18.7663	0.615701	0.307851	+	0.951435i	$0.400390\pi$
$930$	0	0
$931$	−1.15862	−0.0379723
$932$	0	0
$933$	−19.1451	−0.626784
$934$	0	0
$935$	−101.496	−3.31929
$936$	0	0
$937$	−31.7473	−1.03714	−0.518569	−	0.855035i	$-0.673535\pi$
$937$	−31.7473	−1.03714	−0.518569	+	0.855035i	$0.673535\pi$
$938$	0	0
$939$	41.5873	1.35715
$940$	0	0
$941$	0.227389	0.00741266	0.00370633	−	0.999993i	$-0.498820\pi$
$941$	0.227389	0.00741266	0.00370633	+	0.999993i	$0.498820\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−28.6614	−0.932355
$946$	0	0
$947$	21.5182	0.699249	0.349625	−	0.936890i	$-0.386309\pi$
$947$	21.5182	0.699249	0.349625	+	0.936890i	$0.386309\pi$
$948$	0	0
$949$	−81.9540	−2.66034
$950$	0	0
$951$	6.64209	0.215384
$952$	0	0
$953$	21.5413	0.697792	0.348896	−	0.937161i	$-0.386557\pi$
$953$	21.5413	0.697792	0.348896	+	0.937161i	$0.386557\pi$
$954$	0	0
$955$	18.2795	0.591512
$956$	0	0
$957$	−9.94484	−0.321471
$958$	0	0
$959$	−6.39064	−0.206364
$960$	0	0
$961$	−22.0397	−0.710958
$962$	0	0
$963$	−18.1211	−0.583944
$964$	0	0
$965$	−5.66105	−0.182235
$966$	0	0
$967$	−18.1310	−0.583054	−0.291527	−	0.956563i	$-0.594163\pi$
$967$	−18.1310	−0.583054	−0.291527	+	0.956563i	$0.594163\pi$
$968$	0	0
$969$	2.41348	0.0775320
$970$	0	0
$971$	31.9704	1.02598	0.512990	−	0.858395i	$-0.328538\pi$
$971$	31.9704	1.02598	0.512990	+	0.858395i	$0.328538\pi$
$972$	0	0
$973$	−26.2209	−0.840603
$974$	0	0
$975$	37.3238	1.19532
$976$	0	0
$977$	19.4648	0.622735	0.311368	−	0.950289i	$-0.399213\pi$
$977$	19.4648	0.622735	0.311368	+	0.950289i	$0.399213\pi$
$978$	0	0
$979$	55.8311	1.78437
$980$	0	0
$981$	−0.708997	−0.0226365
$982$	0	0
$983$	17.8604	0.569659	0.284829	−	0.958578i	$-0.408063\pi$
$983$	17.8604	0.569659	0.284829	+	0.958578i	$0.408063\pi$
$984$	0	0
$985$	61.3025	1.95326
$986$	0	0
$987$	3.71517	0.118255
$988$	0	0
$989$	0	0
$990$	0	0
$991$	40.6094	1.29000	0.645000	−	0.764182i	$-0.276857\pi$
$991$	40.6094	1.29000	0.645000	+	0.764182i	$0.276857\pi$
$992$	0	0
$993$	−31.5789	−1.00213
$994$	0	0
$995$	6.81777	0.216138
$996$	0	0
$997$	42.6851	1.35185	0.675925	−	0.736971i	$-0.263744\pi$
$997$	42.6851	1.35185	0.675925	+	0.736971i	$0.263744\pi$
$998$	0	0
$999$	−1.49979	−0.0474512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.ch.1.5		15
4.3	odd	2		4232.2.a.ba.1.11		15
23.17	odd	22		368.2.m.e.289.2		30
23.19	odd	22		368.2.m.e.177.2		30
23.22	odd	2		8464.2.a.cg.1.5		15
92.19	even	22		184.2.i.b.177.2	yes	30
92.63	even	22		184.2.i.b.105.2	✓	30
92.91	even	2		4232.2.a.bb.1.11		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.105.2	✓	30	92.63	even	22
184.2.i.b.177.2	yes	30	92.19	even	22
368.2.m.e.177.2		30	23.19	odd	22
368.2.m.e.289.2		30	23.17	odd	22
4232.2.a.ba.1.11		15	4.3	odd	2
4232.2.a.bb.1.11		15	92.91	even	2
8464.2.a.cg.1.5		15	23.22	odd	2
8464.2.a.ch.1.5		15	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q-1.44882 q^{3} +2.99678 q^{5} -1.69224 q^{7} -0.900926 q^{9} +O(q^{10})\)	\(q-1.44882 q^{3} +2.99678 q^{5} -1.69224 q^{7} -0.900926 q^{9} +5.69500 q^{11} -6.47165 q^{13} -4.34179 q^{15} -5.94707 q^{17} +0.280108 q^{19} +2.45174 q^{21} +3.98068 q^{25} +5.65173 q^{27} +1.20529 q^{29} +2.99338 q^{31} -8.25101 q^{33} -5.07126 q^{35} -0.265368 q^{37} +9.37624 q^{39} +5.81542 q^{41} -5.66327 q^{43} -2.69988 q^{45} +1.51532 q^{47} -4.13634 q^{49} +8.61622 q^{51} -6.60567 q^{53} +17.0666 q^{55} -0.405826 q^{57} -3.21061 q^{59} -8.76330 q^{61} +1.52458 q^{63} -19.3941 q^{65} -3.11516 q^{67} +2.19420 q^{71} +12.6635 q^{73} -5.76729 q^{75} -9.63728 q^{77} +0.653751 q^{79} -5.48555 q^{81} +13.8264 q^{83} -17.8221 q^{85} -1.74624 q^{87} +9.80353 q^{89} +10.9516 q^{91} -4.33686 q^{93} +0.839423 q^{95} -3.82589 q^{97} -5.13077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

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