LMFDB - Embedded newform 2312.4.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152 $ x^8 - 95x^6 + 756x^4 - 1780*x^2 + 1152
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-9.30031$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62097 q^{3} +11.5318 q^{5} +23.0485 q^{7} -20.1305 q^{9} +O(q^{10})$	$q-2.62097 q^{3} +11.5318 q^{5} +23.0485 q^{7} -20.1305 q^{9} +1.19452 q^{11} +58.3731 q^{13} -30.2245 q^{15} -138.971 q^{19} -60.4096 q^{21} +180.911 q^{23} +7.98158 q^{25} +123.528 q^{27} -115.027 q^{29} +210.726 q^{31} -3.13080 q^{33} +265.790 q^{35} +210.661 q^{37} -152.994 q^{39} -297.500 q^{41} -174.746 q^{43} -232.140 q^{45} -199.717 q^{47} +188.234 q^{49} +706.217 q^{53} +13.7749 q^{55} +364.238 q^{57} +182.746 q^{59} +489.499 q^{61} -463.978 q^{63} +673.144 q^{65} -167.043 q^{67} -474.163 q^{69} +818.157 q^{71} +1090.15 q^{73} -20.9195 q^{75} +27.5319 q^{77} -110.196 q^{79} +219.760 q^{81} -118.672 q^{83} +301.483 q^{87} +400.332 q^{89} +1345.41 q^{91} -552.306 q^{93} -1602.58 q^{95} -924.543 q^{97} -24.0463 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 132 q^{9}+O(q^{10}) $ 8 * q + 132 * q^9	$ 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100}) $ 8 * q + 132 * q^9 + 44 * q^13 - 24 * q^15 - 48 * q^19 + 308 * q^21 + 520 * q^25 + 812 * q^33 - 1064 * q^35 - 8 * q^43 + 312 * q^47 + 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 180 * q^69 - 1660 * q^77 + 3156 * q^81 - 2472 * q^83 + 6664 * q^87 + 68 * q^89 + 4036 * q^93

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
$3$	−2.62097	−0.504407	−0.252203	−	0.967674i	$-0.581155\pi$
$3$	−2.62097	−0.504407	−0.252203	+	0.967674i	$0.581155\pi$
$4$	0	0
$5$	11.5318	1.03143	0.515716	−	0.856759i	$-0.327526\pi$
$5$	11.5318	1.03143	0.515716	+	0.856759i	$0.327526\pi$
$6$	0	0
$7$	23.0485	1.24450	0.622251	−	0.782817i	$-0.286218\pi$
$7$	23.0485	1.24450	0.622251	+	0.782817i	$0.286218\pi$
$8$	0	0
$9$	−20.1305	−0.745574
$10$	0	0
$11$	1.19452	0.0327419	0.0163710	−	0.999866i	$-0.494789\pi$
$11$	1.19452	0.0327419	0.0163710	+	0.999866i	$0.494789\pi$
$12$	0	0
$13$	58.3731	1.24537	0.622684	−	0.782474i	$-0.286042\pi$
$13$	58.3731	1.24537	0.622684	+	0.782474i	$0.286042\pi$
$14$	0	0
$15$	−30.2245	−0.520262
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−138.971	−1.67800	−0.839001	−	0.544130i	$-0.816860\pi$
$19$	−138.971	−1.67800	−0.839001	+	0.544130i	$0.816860\pi$
$20$	0	0
$21$	−60.4096	−0.627736
$22$	0	0
$23$	180.911	1.64011	0.820055	−	0.572284i	$-0.193943\pi$
$23$	180.911	1.64011	0.820055	+	0.572284i	$0.193943\pi$
$24$	0	0
$25$	7.98158	0.0638527
$26$	0	0
$27$	123.528	0.880479
$28$	0	0
$29$	−115.027	−0.736551	−0.368275	−	0.929717i	$-0.620052\pi$
$29$	−115.027	−0.736551	−0.368275	+	0.929717i	$0.620052\pi$
$30$	0	0
$31$	210.726	1.22088	0.610442	−	0.792061i	$-0.290992\pi$
$31$	210.726	1.22088	0.610442	+	0.792061i	$0.290992\pi$
$32$	0	0
$33$	−3.13080	−0.0165152
$34$	0	0
$35$	265.790	1.28362
$36$	0	0
$37$	210.661	0.936010	0.468005	−	0.883726i	$-0.344973\pi$
$37$	210.661	0.936010	0.468005	+	0.883726i	$0.344973\pi$
$38$	0	0
$39$	−152.994	−0.628172
$40$	0	0
$41$	−297.500	−1.13321	−0.566605	−	0.823989i	$-0.691743\pi$
$41$	−297.500	−1.13321	−0.566605	+	0.823989i	$0.691743\pi$
$42$	0	0
$43$	−174.746	−0.619734	−0.309867	−	0.950780i	$-0.600285\pi$
$43$	−174.746	−0.619734	−0.309867	+	0.950780i	$0.600285\pi$
$44$	0	0
$45$	−232.140	−0.769009
$46$	0	0
$47$	−199.717	−0.619823	−0.309911	−	0.950765i	$-0.600299\pi$
$47$	−199.717	−0.619823	−0.309911	+	0.950765i	$0.600299\pi$
$48$	0	0
$49$	188.234	0.548787
$50$	0	0
$51$	0	0
$52$	0	0
$53$	706.217	1.83031	0.915154	−	0.403105i	$-0.132069\pi$
$53$	706.217	1.83031	0.915154	+	0.403105i	$0.132069\pi$
$54$	0	0
$55$	13.7749	0.0337711
$56$	0	0
$57$	364.238	0.846396
$58$	0	0
$59$	182.746	0.403247	0.201623	−	0.979463i	$-0.435378\pi$
$59$	182.746	0.403247	0.201623	+	0.979463i	$0.435378\pi$
$60$	0	0
$61$	489.499	1.02744	0.513720	−	0.857958i	$-0.328267\pi$
$61$	489.499	1.02744	0.513720	+	0.857958i	$0.328267\pi$
$62$	0	0
$63$	−463.978	−0.927869
$64$	0	0
$65$	673.144	1.28451
$66$	0	0
$67$	−167.043	−0.304590	−0.152295	−	0.988335i	$-0.548666\pi$
$67$	−167.043	−0.304590	−0.152295	+	0.988335i	$0.548666\pi$
$68$	0	0
$69$	−474.163	−0.827283
$70$	0	0
$71$	818.157	1.36757	0.683784	−	0.729684i	$-0.260333\pi$
$71$	818.157	1.36757	0.683784	+	0.729684i	$0.260333\pi$
$72$	0	0
$73$	1090.15	1.74784	0.873919	−	0.486071i	$-0.161570\pi$
$73$	1090.15	1.74784	0.873919	+	0.486071i	$0.161570\pi$
$74$	0	0
$75$	−20.9195	−0.0322077
$76$	0	0
$77$	27.5319	0.0407474
$78$	0	0
$79$	−110.196	−0.156937	−0.0784685	−	0.996917i	$-0.525003\pi$
$79$	−110.196	−0.156937	−0.0784685	+	0.996917i	$0.525003\pi$
$80$	0	0
$81$	219.760	0.301454
$82$	0	0
$83$	−118.672	−0.156939	−0.0784696	−	0.996917i	$-0.525003\pi$
$83$	−118.672	−0.156939	−0.0784696	+	0.996917i	$0.525003\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	301.483	0.371521
$88$	0	0
$89$	400.332	0.476799	0.238399	−	0.971167i	$-0.423377\pi$
$89$	400.332	0.476799	0.238399	+	0.971167i	$0.423377\pi$
$90$	0	0
$91$	1345.41	1.54986
$92$	0	0
$93$	−552.306	−0.615823
$94$	0	0
$95$	−1602.58	−1.73075
$96$	0	0
$97$	−924.543	−0.967764	−0.483882	−	0.875133i	$-0.660774\pi$
$97$	−924.543	−0.967764	−0.483882	+	0.875133i	$0.660774\pi$
$98$	0	0
$99$	−24.0463	−0.0244115
$100$	0	0
$101$	223.816	0.220500	0.110250	−	0.993904i	$-0.464835\pi$
$101$	223.816	0.220500	0.110250	+	0.993904i	$0.464835\pi$
$102$	0	0
$103$	−1280.06	−1.22454	−0.612271	−	0.790648i	$-0.709744\pi$
$103$	−1280.06	−1.22454	−0.612271	+	0.790648i	$0.709744\pi$
$104$	0	0
$105$	−696.629	−0.647467
$106$	0	0
$107$	−1099.03	−0.992969	−0.496485	−	0.868046i	$-0.665376\pi$
$107$	−1099.03	−0.992969	−0.496485	+	0.868046i	$0.665376\pi$
$108$	0	0
$109$	−733.360	−0.644433	−0.322216	−	0.946666i	$-0.604428\pi$
$109$	−733.360	−0.644433	−0.322216	+	0.946666i	$0.604428\pi$
$110$	0	0
$111$	−552.136	−0.472130
$112$	0	0
$113$	1637.39	1.36312	0.681561	−	0.731761i	$-0.261301\pi$
$113$	1637.39	1.36312	0.681561	+	0.731761i	$0.261301\pi$
$114$	0	0
$115$	2086.22	1.69166
$116$	0	0
$117$	−1175.08	−0.928513
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1329.57	−0.998928
$122$	0	0
$123$	779.739	0.571599
$124$	0	0
$125$	−1349.43	−0.965573
$126$	0	0
$127$	1502.71	1.04995	0.524977	−	0.851117i	$-0.324074\pi$
$127$	1502.71	1.04995	0.524977	+	0.851117i	$0.324074\pi$
$128$	0	0
$129$	458.006	0.312598
$130$	0	0
$131$	−702.381	−0.468453	−0.234226	−	0.972182i	$-0.575256\pi$
$131$	−702.381	−0.468453	−0.234226	+	0.972182i	$0.575256\pi$
$132$	0	0
$133$	−3203.07	−2.08828
$134$	0	0
$135$	1424.49	0.908155
$136$	0	0
$137$	18.9889	0.0118418	0.00592092	−	0.999982i	$-0.498115\pi$
$137$	18.9889	0.0118418	0.00592092	+	0.999982i	$0.498115\pi$
$138$	0	0
$139$	−276.969	−0.169009	−0.0845043	−	0.996423i	$-0.526931\pi$
$139$	−276.969	−0.169009	−0.0845043	+	0.996423i	$0.526931\pi$
$140$	0	0
$141$	523.452	0.312643
$142$	0	0
$143$	69.7277	0.0407757
$144$	0	0
$145$	−1326.46	−0.759702
$146$	0	0
$147$	−493.357	−0.276812
$148$	0	0
$149$	128.491	0.0706468	0.0353234	−	0.999376i	$-0.488754\pi$
$149$	128.491	0.0706468	0.0353234	+	0.999376i	$0.488754\pi$
$150$	0	0
$151$	2097.44	1.13038	0.565190	−	0.824961i	$-0.308803\pi$
$151$	2097.44	1.13038	0.565190	+	0.824961i	$0.308803\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2430.04	1.25926
$156$	0	0
$157$	2911.59	1.48007	0.740033	−	0.672571i	$-0.234810\pi$
$157$	2911.59	1.48007	0.740033	+	0.672571i	$0.234810\pi$
$158$	0	0
$159$	−1850.98	−0.923220
$160$	0	0
$161$	4169.73	2.04112
$162$	0	0
$163$	−1574.26	−0.756478	−0.378239	−	0.925708i	$-0.623470\pi$
$163$	−1574.26	−0.756478	−0.378239	+	0.925708i	$0.623470\pi$
$164$	0	0
$165$	−36.1037	−0.0170344
$166$	0	0
$167$	−1290.10	−0.597788	−0.298894	−	0.954286i	$-0.596618\pi$
$167$	−1290.10	−0.597788	−0.298894	+	0.954286i	$0.596618\pi$
$168$	0	0
$169$	1210.41	0.550939
$170$	0	0
$171$	2797.55	1.25107
$172$	0	0
$173$	2370.19	1.04163	0.520816	−	0.853669i	$-0.325628\pi$
$173$	2370.19	1.04163	0.520816	+	0.853669i	$0.325628\pi$
$174$	0	0
$175$	183.964	0.0794648
$176$	0	0
$177$	−478.974	−0.203400
$178$	0	0
$179$	−1719.87	−0.718153	−0.359077	−	0.933308i	$-0.616908\pi$
$179$	−1719.87	−0.718153	−0.359077	+	0.933308i	$0.616908\pi$
$180$	0	0
$181$	−61.7101	−0.0253418	−0.0126709	−	0.999920i	$-0.504033\pi$
$181$	−61.7101	−0.0253418	−0.0126709	+	0.999920i	$0.504033\pi$
$182$	0	0
$183$	−1282.96	−0.518248
$184$	0	0
$185$	2429.29	0.965431
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2847.13	1.09576
$190$	0	0
$191$	−2565.34	−0.971839	−0.485919	−	0.874004i	$-0.661515\pi$
$191$	−2565.34	−0.971839	−0.485919	+	0.874004i	$0.661515\pi$
$192$	0	0
$193$	−1991.65	−0.742810	−0.371405	−	0.928471i	$-0.621124\pi$
$193$	−1991.65	−0.742810	−0.371405	+	0.928471i	$0.621124\pi$
$194$	0	0
$195$	−1764.29	−0.647917
$196$	0	0
$197$	−3831.76	−1.38579	−0.692897	−	0.721036i	$-0.743666\pi$
$197$	−3831.76	−1.38579	−0.692897	+	0.721036i	$0.743666\pi$
$198$	0	0
$199$	2171.38	0.773492	0.386746	−	0.922186i	$-0.373599\pi$
$199$	2171.38	0.773492	0.386746	+	0.922186i	$0.373599\pi$
$200$	0	0
$201$	437.815	0.153637
$202$	0	0
$203$	−2651.20	−0.916639
$204$	0	0
$205$	−3430.69	−1.16883
$206$	0	0
$207$	−3641.83	−1.22282
$208$	0	0
$209$	−166.003	−0.0549410
$210$	0	0
$211$	2069.97	0.675369	0.337685	−	0.941259i	$-0.390356\pi$
$211$	2069.97	0.675369	0.337685	+	0.941259i	$0.390356\pi$
$212$	0	0
$213$	−2144.37	−0.689811
$214$	0	0
$215$	−2015.13	−0.639214
$216$	0	0
$217$	4856.91	1.51939
$218$	0	0
$219$	−2857.25	−0.881622
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4297.20	1.29041	0.645206	−	0.764008i	$-0.276771\pi$
$223$	4297.20	1.29041	0.645206	+	0.764008i	$0.276771\pi$
$224$	0	0
$225$	−160.673	−0.0476069
$226$	0	0
$227$	336.979	0.0985291	0.0492646	−	0.998786i	$-0.484312\pi$
$227$	336.979	0.0985291	0.0492646	+	0.998786i	$0.484312\pi$
$228$	0	0
$229$	3076.57	0.887798	0.443899	−	0.896077i	$-0.353595\pi$
$229$	3076.57	0.887798	0.443899	+	0.896077i	$0.353595\pi$
$230$	0	0
$231$	−72.1604	−0.0205533
$232$	0	0
$233$	6537.86	1.83824	0.919119	−	0.393980i	$-0.128902\pi$
$233$	6537.86	1.83824	0.919119	+	0.393980i	$0.128902\pi$
$234$	0	0
$235$	−2303.09	−0.639305
$236$	0	0
$237$	288.821	0.0791601
$238$	0	0
$239$	−3432.31	−0.928943	−0.464472	−	0.885588i	$-0.653756\pi$
$239$	−3432.31	−0.928943	−0.464472	+	0.885588i	$0.653756\pi$
$240$	0	0
$241$	1312.48	0.350807	0.175404	−	0.984497i	$-0.443877\pi$
$241$	1312.48	0.350807	0.175404	+	0.984497i	$0.443877\pi$
$242$	0	0
$243$	−3911.24	−1.03253
$244$	0	0
$245$	2170.67	0.566037
$246$	0	0
$247$	−8112.14	−2.08973
$248$	0	0
$249$	311.037	0.0791612
$250$	0	0
$251$	6498.55	1.63420	0.817101	−	0.576494i	$-0.195580\pi$
$251$	6498.55	1.63420	0.817101	+	0.576494i	$0.195580\pi$
$252$	0	0
$253$	216.102	0.0537004
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4874.66	1.18316	0.591582	−	0.806245i	$-0.298504\pi$
$257$	4874.66	1.18316	0.591582	+	0.806245i	$0.298504\pi$
$258$	0	0
$259$	4855.41	1.16487
$260$	0	0
$261$	2315.55	0.549153
$262$	0	0
$263$	−1482.50	−0.347584	−0.173792	−	0.984782i	$-0.555602\pi$
$263$	−1482.50	−0.347584	−0.173792	+	0.984782i	$0.555602\pi$
$264$	0	0
$265$	8143.92	1.88784
$266$	0	0
$267$	−1049.26	−0.240501
$268$	0	0
$269$	5058.75	1.14661	0.573304	−	0.819343i	$-0.305662\pi$
$269$	5058.75	1.14661	0.573304	+	0.819343i	$0.305662\pi$
$270$	0	0
$271$	1133.70	0.254122	0.127061	−	0.991895i	$-0.459446\pi$
$271$	1133.70	0.254122	0.127061	+	0.991895i	$0.459446\pi$
$272$	0	0
$273$	−3526.29	−0.781761
$274$	0	0
$275$	9.53415	0.00209066
$276$	0	0
$277$	−3698.34	−0.802208	−0.401104	−	0.916033i	$-0.631373\pi$
$277$	−3698.34	−0.802208	−0.401104	+	0.916033i	$0.631373\pi$
$278$	0	0
$279$	−4242.01	−0.910260
$280$	0	0
$281$	−2190.08	−0.464945	−0.232472	−	0.972603i	$-0.574682\pi$
$281$	−2190.08	−0.464945	−0.232472	+	0.972603i	$0.574682\pi$
$282$	0	0
$283$	−8159.66	−1.71393	−0.856963	−	0.515378i	$-0.827652\pi$
$283$	−8159.66	−1.71393	−0.856963	+	0.515378i	$0.827652\pi$
$284$	0	0
$285$	4200.31	0.873000
$286$	0	0
$287$	−6856.92	−1.41028
$288$	0	0
$289$	0	0
$290$	0	0
$291$	2423.20	0.488147
$292$	0	0
$293$	4777.49	0.952574	0.476287	−	0.879290i	$-0.341982\pi$
$293$	4777.49	0.952574	0.476287	+	0.879290i	$0.341982\pi$
$294$	0	0
$295$	2107.39	0.415922
$296$	0	0
$297$	147.556	0.0288286
$298$	0	0
$299$	10560.3	2.04254
$300$	0	0
$301$	−4027.65	−0.771261
$302$	0	0
$303$	−586.615	−0.111222
$304$	0	0
$305$	5644.78	1.05974
$306$	0	0
$307$	531.103	0.0987351	0.0493675	−	0.998781i	$-0.484279\pi$
$307$	531.103	0.0987351	0.0493675	+	0.998781i	$0.484279\pi$
$308$	0	0
$309$	3355.00	0.617667
$310$	0	0
$311$	4767.26	0.869217	0.434609	−	0.900619i	$-0.356887\pi$
$311$	4767.26	0.869217	0.434609	+	0.900619i	$0.356887\pi$
$312$	0	0
$313$	8613.47	1.55547	0.777735	−	0.628593i	$-0.216369\pi$
$313$	8613.47	1.55547	0.777735	+	0.628593i	$0.216369\pi$
$314$	0	0
$315$	−5350.48	−0.957034
$316$	0	0
$317$	2453.20	0.434654	0.217327	−	0.976099i	$-0.430266\pi$
$317$	2453.20	0.434654	0.217327	+	0.976099i	$0.430266\pi$
$318$	0	0
$319$	−137.402	−0.0241161
$320$	0	0
$321$	2880.54	0.500860
$322$	0	0
$323$	0	0
$324$	0	0
$325$	465.909	0.0795200
$326$	0	0
$327$	1922.12	0.325056
$328$	0	0
$329$	−4603.17	−0.771371
$330$	0	0
$331$	11003.5	1.82721	0.913605	−	0.406604i	$-0.133287\pi$
$331$	11003.5	1.82721	0.913605	+	0.406604i	$0.133287\pi$
$332$	0	0
$333$	−4240.70	−0.697865
$334$	0	0
$335$	−1926.30	−0.314164
$336$	0	0
$337$	3664.37	0.592318	0.296159	−	0.955139i	$-0.404294\pi$
$337$	3664.37	0.592318	0.296159	+	0.955139i	$0.404294\pi$
$338$	0	0
$339$	−4291.56	−0.687568
$340$	0	0
$341$	251.716	0.0399741
$342$	0	0
$343$	−3567.12	−0.561535
$344$	0	0
$345$	−5467.94	−0.853286
$346$	0	0
$347$	−3173.37	−0.490938	−0.245469	−	0.969404i	$-0.578942\pi$
$347$	−3173.37	−0.490938	−0.245469	+	0.969404i	$0.578942\pi$
$348$	0	0
$349$	2185.47	0.335202	0.167601	−	0.985855i	$-0.446398\pi$
$349$	2185.47	0.335202	0.167601	+	0.985855i	$0.446398\pi$
$350$	0	0
$351$	7210.70	1.09652
$352$	0	0
$353$	1216.36	0.183400	0.0917002	−	0.995787i	$-0.470770\pi$
$353$	1216.36	0.183400	0.0917002	+	0.995787i	$0.470770\pi$
$354$	0	0
$355$	9434.80	1.41055
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7448.93	1.09510	0.547548	−	0.836774i	$-0.315561\pi$
$359$	7448.93	1.09510	0.547548	+	0.836774i	$0.315561\pi$
$360$	0	0
$361$	12453.8	1.81569
$362$	0	0
$363$	3484.78	0.503866
$364$	0	0
$365$	12571.3	1.80278
$366$	0	0
$367$	−2374.59	−0.337746	−0.168873	−	0.985638i	$-0.554013\pi$
$367$	−2374.59	−0.337746	−0.168873	+	0.985638i	$0.554013\pi$
$368$	0	0
$369$	5988.81	0.844892
$370$	0	0
$371$	16277.2	2.27782
$372$	0	0
$373$	12231.7	1.69795	0.848974	−	0.528435i	$-0.177221\pi$
$373$	12231.7	1.69795	0.848974	+	0.528435i	$0.177221\pi$
$374$	0	0
$375$	3536.82	0.487041
$376$	0	0
$377$	−6714.48	−0.917276
$378$	0	0
$379$	14129.9	1.91505	0.957523	−	0.288358i	$-0.0931094\pi$
$379$	14129.9	1.91505	0.957523	+	0.288358i	$0.0931094\pi$
$380$	0	0
$381$	−3938.57	−0.529604
$382$	0	0
$383$	−13306.4	−1.77526	−0.887631	−	0.460556i	$-0.847650\pi$
$383$	−13306.4	−1.77526	−0.887631	+	0.460556i	$0.847650\pi$
$384$	0	0
$385$	317.491	0.0420282
$386$	0	0
$387$	3517.73	0.462058
$388$	0	0
$389$	8007.84	1.04374	0.521868	−	0.853026i	$-0.325235\pi$
$389$	8007.84	1.04374	0.521868	+	0.853026i	$0.325235\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1840.92	0.236291
$394$	0	0
$395$	−1270.76	−0.161870
$396$	0	0
$397$	13185.2	1.66687	0.833434	−	0.552619i	$-0.186372\pi$
$397$	13185.2	1.66687	0.833434	+	0.552619i	$0.186372\pi$
$398$	0	0
$399$	8395.15	1.05334
$400$	0	0
$401$	−3205.81	−0.399228	−0.199614	−	0.979875i	$-0.563969\pi$
$401$	−3205.81	−0.399228	−0.199614	+	0.979875i	$0.563969\pi$
$402$	0	0
$403$	12300.7	1.52045
$404$	0	0
$405$	2534.22	0.310929
$406$	0	0
$407$	251.638	0.0306468
$408$	0	0
$409$	−15693.3	−1.89727	−0.948636	−	0.316369i	$-0.897536\pi$
$409$	−15693.3	−1.89727	−0.948636	+	0.316369i	$0.897536\pi$
$410$	0	0
$411$	−49.7695	−0.00597311
$412$	0	0
$413$	4212.03	0.501842
$414$	0	0
$415$	−1368.50	−0.161872
$416$	0	0
$417$	725.929	0.0852491
$418$	0	0
$419$	6910.65	0.805745	0.402873	−	0.915256i	$-0.368012\pi$
$419$	6910.65	0.805745	0.402873	+	0.915256i	$0.368012\pi$
$420$	0	0
$421$	9225.05	1.06794	0.533968	−	0.845505i	$-0.320700\pi$
$421$	9225.05	1.06794	0.533968	+	0.845505i	$0.320700\pi$
$422$	0	0
$423$	4020.40	0.462124
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11282.2	1.27865
$428$	0	0
$429$	−182.755	−0.0205675
$430$	0	0
$431$	−8822.74	−0.986024	−0.493012	−	0.870022i	$-0.664104\pi$
$431$	−8822.74	−0.986024	−0.493012	+	0.870022i	$0.664104\pi$
$432$	0	0
$433$	3190.95	0.354151	0.177075	−	0.984197i	$-0.443336\pi$
$433$	3190.95	0.354151	0.177075	+	0.984197i	$0.443336\pi$
$434$	0	0
$435$	3476.63	0.383199
$436$	0	0
$437$	−25141.3	−2.75211
$438$	0	0
$439$	−15069.5	−1.63834	−0.819168	−	0.573553i	$-0.805565\pi$
$439$	−15069.5	−1.63834	−0.819168	+	0.573553i	$0.805565\pi$
$440$	0	0
$441$	−3789.24	−0.409161
$442$	0	0
$443$	−3026.24	−0.324562	−0.162281	−	0.986745i	$-0.551885\pi$
$443$	−3026.24	−0.324562	−0.162281	+	0.986745i	$0.551885\pi$
$444$	0	0
$445$	4616.53	0.491786
$446$	0	0
$447$	−336.771	−0.0356347
$448$	0	0
$449$	4641.02	0.487802	0.243901	−	0.969800i	$-0.421573\pi$
$449$	4641.02	0.487802	0.243901	+	0.969800i	$0.421573\pi$
$450$	0	0
$451$	−355.369	−0.0371035
$452$	0	0
$453$	−5497.34	−0.570171
$454$	0	0
$455$	15515.0	1.59858
$456$	0	0
$457$	−17393.2	−1.78035	−0.890174	−	0.455620i	$-0.849418\pi$
$457$	−17393.2	−1.78035	−0.890174	+	0.455620i	$0.849418\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2451.93	−0.247718	−0.123859	−	0.992300i	$-0.539527\pi$
$461$	−2451.93	−0.247718	−0.123859	+	0.992300i	$0.539527\pi$
$462$	0	0
$463$	−14586.3	−1.46411	−0.732054	−	0.681246i	$-0.761438\pi$
$463$	−14586.3	−1.46411	−0.732054	+	0.681246i	$0.761438\pi$
$464$	0	0
$465$	−6369.07	−0.635179
$466$	0	0
$467$	−15195.6	−1.50572	−0.752859	−	0.658182i	$-0.771326\pi$
$467$	−15195.6	−1.50572	−0.752859	+	0.658182i	$0.771326\pi$
$468$	0	0
$469$	−3850.09	−0.379063
$470$	0	0
$471$	−7631.21	−0.746555
$472$	0	0
$473$	−208.738	−0.0202913
$474$	0	0
$475$	−1109.21	−0.107145
$476$	0	0
$477$	−14216.5	−1.36463
$478$	0	0
$479$	−4422.10	−0.421819	−0.210909	−	0.977506i	$-0.567642\pi$
$479$	−4422.10	−0.421819	−0.210909	+	0.977506i	$0.567642\pi$
$480$	0	0
$481$	12296.9	1.16568
$482$	0	0
$483$	−10928.8	−1.02956
$484$	0	0
$485$	−10661.6	−0.998183
$486$	0	0
$487$	17772.7	1.65371	0.826856	−	0.562414i	$-0.190127\pi$
$487$	17772.7	1.65371	0.826856	+	0.562414i	$0.190127\pi$
$488$	0	0
$489$	4126.11	0.381573
$490$	0	0
$491$	−4845.34	−0.445350	−0.222675	−	0.974893i	$-0.571479\pi$
$491$	−4845.34	−0.445350	−0.222675	+	0.974893i	$0.571479\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−277.296	−0.0251788
$496$	0	0
$497$	18857.3	1.70194
$498$	0	0
$499$	1060.23	0.0951153	0.0475576	−	0.998868i	$-0.484856\pi$
$499$	1060.23	0.0951153	0.0475576	+	0.998868i	$0.484856\pi$
$500$	0	0
$501$	3381.31	0.301528
$502$	0	0
$503$	−2803.19	−0.248485	−0.124243	−	0.992252i	$-0.539650\pi$
$503$	−2803.19	−0.248485	−0.124243	+	0.992252i	$0.539650\pi$
$504$	0	0
$505$	2580.99	0.227431
$506$	0	0
$507$	−3172.46	−0.277898
$508$	0	0
$509$	1658.97	0.144465	0.0722323	−	0.997388i	$-0.476988\pi$
$509$	1658.97	0.144465	0.0722323	+	0.997388i	$0.476988\pi$
$510$	0	0
$511$	25126.3	2.17519
$512$	0	0
$513$	−17166.7	−1.47745
$514$	0	0
$515$	−14761.3	−1.26303
$516$	0	0
$517$	−238.565	−0.0202942
$518$	0	0
$519$	−6212.21	−0.525406
$520$	0	0
$521$	12553.6	1.05563	0.527815	−	0.849359i	$-0.323011\pi$
$521$	12553.6	1.05563	0.527815	+	0.849359i	$0.323011\pi$
$522$	0	0
$523$	−12031.6	−1.00593	−0.502967	−	0.864306i	$-0.667758\pi$
$523$	−12031.6	−1.00593	−0.502967	+	0.864306i	$0.667758\pi$
$524$	0	0
$525$	−482.164	−0.0400826
$526$	0	0
$527$	0	0
$528$	0	0
$529$	20561.8	1.68996
$530$	0	0
$531$	−3678.78	−0.300650
$532$	0	0
$533$	−17366.0	−1.41126
$534$	0	0
$535$	−12673.8	−1.02418
$536$	0	0
$537$	4507.75	0.362241
$538$	0	0
$539$	224.849	0.0179683
$540$	0	0
$541$	−10632.7	−0.844984	−0.422492	−	0.906367i	$-0.638845\pi$
$541$	−10632.7	−0.844984	−0.422492	+	0.906367i	$0.638845\pi$
$542$	0	0
$543$	161.741	0.0127826
$544$	0	0
$545$	−8456.94	−0.664689
$546$	0	0
$547$	17370.0	1.35774	0.678872	−	0.734257i	$-0.262469\pi$
$547$	17370.0	1.35774	0.678872	+	0.734257i	$0.262469\pi$
$548$	0	0
$549$	−9853.85	−0.766033
$550$	0	0
$551$	15985.4	1.23593
$552$	0	0
$553$	−2539.86	−0.195309
$554$	0	0
$555$	−6367.10	−0.486970
$556$	0	0
$557$	1084.77	0.0825190	0.0412595	−	0.999148i	$-0.486863\pi$
$557$	1084.77	0.0825190	0.0412595	+	0.999148i	$0.486863\pi$
$558$	0	0
$559$	−10200.5	−0.771797
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5494.27	0.411289	0.205645	−	0.978627i	$-0.434071\pi$
$563$	5494.27	0.411289	0.205645	+	0.978627i	$0.434071\pi$
$564$	0	0
$565$	18882.0	1.40597
$566$	0	0
$567$	5065.14	0.375160
$568$	0	0
$569$	11612.4	0.855569	0.427784	−	0.903881i	$-0.359294\pi$
$569$	11612.4	0.855569	0.427784	+	0.903881i	$0.359294\pi$
$570$	0	0
$571$	−3413.97	−0.250210	−0.125105	−	0.992143i	$-0.539927\pi$
$571$	−3413.97	−0.250210	−0.125105	+	0.992143i	$0.539927\pi$
$572$	0	0
$573$	6723.68	0.490202
$574$	0	0
$575$	1443.96	0.104725
$576$	0	0
$577$	1355.70	0.0978141	0.0489070	−	0.998803i	$-0.484426\pi$
$577$	1355.70	0.0978141	0.0489070	+	0.998803i	$0.484426\pi$
$578$	0	0
$579$	5220.07	0.374678
$580$	0	0
$581$	−2735.22	−0.195311
$582$	0	0
$583$	843.589	0.0599278
$584$	0	0
$585$	−13550.7	−0.957698
$586$	0	0
$587$	−25232.2	−1.77418	−0.887092	−	0.461593i	$-0.847278\pi$
$587$	−25232.2	−1.77418	−0.887092	+	0.461593i	$0.847278\pi$
$588$	0	0
$589$	−29284.7	−2.04865
$590$	0	0
$591$	10042.9	0.699004
$592$	0	0
$593$	14327.4	0.992171	0.496086	−	0.868274i	$-0.334770\pi$
$593$	14327.4	0.992171	0.496086	+	0.868274i	$0.334770\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5691.13	−0.390155
$598$	0	0
$599$	−2523.52	−0.172134	−0.0860671	−	0.996289i	$-0.527430\pi$
$599$	−2523.52	−0.172134	−0.0860671	+	0.996289i	$0.527430\pi$
$600$	0	0
$601$	17286.6	1.17327	0.586634	−	0.809853i	$-0.300453\pi$
$601$	17286.6	1.17327	0.586634	+	0.809853i	$0.300453\pi$
$602$	0	0
$603$	3362.66	0.227095
$604$	0	0
$605$	−15332.3	−1.03033
$606$	0	0
$607$	6413.65	0.428867	0.214433	−	0.976739i	$-0.431210\pi$
$607$	6413.65	0.428867	0.214433	+	0.976739i	$0.431210\pi$
$608$	0	0
$609$	6948.73	0.462359
$610$	0	0
$611$	−11658.1	−0.771907
$612$	0	0
$613$	485.788	0.0320078	0.0160039	−	0.999872i	$-0.494906\pi$
$613$	485.788	0.0320078	0.0160039	+	0.999872i	$0.494906\pi$
$614$	0	0
$615$	8991.76	0.589566
$616$	0	0
$617$	6830.32	0.445670	0.222835	−	0.974856i	$-0.428469\pi$
$617$	6830.32	0.445670	0.222835	+	0.974856i	$0.428469\pi$
$618$	0	0
$619$	10667.8	0.692689	0.346344	−	0.938107i	$-0.387423\pi$
$619$	10667.8	0.692689	0.346344	+	0.938107i	$0.387423\pi$
$620$	0	0
$621$	22347.5	1.44408
$622$	0	0
$623$	9227.05	0.593377
$624$	0	0
$625$	−16559.0	−1.05978
$626$	0	0
$627$	435.090	0.0277126
$628$	0	0
$629$	0	0
$630$	0	0
$631$	15251.2	0.962188	0.481094	−	0.876669i	$-0.340240\pi$
$631$	15251.2	0.962188	0.481094	+	0.876669i	$0.340240\pi$
$632$	0	0
$633$	−5425.35	−0.340661
$634$	0	0
$635$	17328.9	1.08296
$636$	0	0
$637$	10987.8	0.683442
$638$	0	0
$639$	−16469.9	−1.01962
$640$	0	0
$641$	12000.9	0.739480	0.369740	−	0.929135i	$-0.379447\pi$
$641$	12000.9	0.739480	0.369740	+	0.929135i	$0.379447\pi$
$642$	0	0
$643$	−5642.30	−0.346051	−0.173025	−	0.984917i	$-0.555354\pi$
$643$	−5642.30	−0.346051	−0.173025	+	0.984917i	$0.555354\pi$
$644$	0	0
$645$	5281.62	0.322424
$646$	0	0
$647$	7790.69	0.473391	0.236695	−	0.971584i	$-0.423936\pi$
$647$	7790.69	0.473391	0.236695	+	0.971584i	$0.423936\pi$
$648$	0	0
$649$	218.294	0.0132031
$650$	0	0
$651$	−12729.8	−0.766393
$652$	0	0
$653$	6755.02	0.404815	0.202408	−	0.979301i	$-0.435123\pi$
$653$	6755.02	0.404815	0.202408	+	0.979301i	$0.435123\pi$
$654$	0	0
$655$	−8099.69	−0.483177
$656$	0	0
$657$	−21945.2	−1.30314
$658$	0	0
$659$	−14303.3	−0.845492	−0.422746	−	0.906248i	$-0.638934\pi$
$659$	−14303.3	−0.845492	−0.422746	+	0.906248i	$0.638934\pi$
$660$	0	0
$661$	26136.0	1.53793	0.768966	−	0.639289i	$-0.220771\pi$
$661$	26136.0	1.53793	0.768966	+	0.639289i	$0.220771\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−36937.0	−2.15392
$666$	0	0
$667$	−20809.6	−1.20802
$668$	0	0
$669$	−11262.9	−0.650893
$670$	0	0
$671$	584.715	0.0336404
$672$	0	0
$673$	19026.5	1.08977	0.544887	−	0.838509i	$-0.316572\pi$
$673$	19026.5	1.08977	0.544887	+	0.838509i	$0.316572\pi$
$674$	0	0
$675$	985.948	0.0562210
$676$	0	0
$677$	−25683.6	−1.45805	−0.729026	−	0.684486i	$-0.760026\pi$
$677$	−25683.6	−1.45805	−0.729026	+	0.684486i	$0.760026\pi$
$678$	0	0
$679$	−21309.3	−1.20439
$680$	0	0
$681$	−883.214	−0.0496988
$682$	0	0
$683$	28196.4	1.57966	0.789829	−	0.613327i	$-0.210169\pi$
$683$	28196.4	1.57966	0.789829	+	0.613327i	$0.210169\pi$
$684$	0	0
$685$	218.976	0.0122141
$686$	0	0
$687$	−8063.62	−0.447811
$688$	0	0
$689$	41224.0	2.27941
$690$	0	0
$691$	−35037.4	−1.92892	−0.964462	−	0.264221i	$-0.914885\pi$
$691$	−35037.4	−1.92892	−0.964462	+	0.264221i	$0.914885\pi$
$692$	0	0
$693$	−554.230	−0.0303802
$694$	0	0
$695$	−3193.94	−0.174321
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−17135.6	−0.927220
$700$	0	0
$701$	−3539.51	−0.190707	−0.0953535	−	0.995443i	$-0.530398\pi$
$701$	−3539.51	−0.190707	−0.0953535	+	0.995443i	$0.530398\pi$
$702$	0	0
$703$	−29275.6	−1.57063
$704$	0	0
$705$	6036.33	0.322470
$706$	0	0
$707$	5158.62	0.274413
$708$	0	0
$709$	7264.39	0.384795	0.192398	−	0.981317i	$-0.438374\pi$
$709$	7264.39	0.384795	0.192398	+	0.981317i	$0.438374\pi$
$710$	0	0
$711$	2218.30	0.117008
$712$	0	0
$713$	38122.6	2.00239
$714$	0	0
$715$	804.084	0.0420574
$716$	0	0
$717$	8995.99	0.468565
$718$	0	0
$719$	−9704.92	−0.503383	−0.251692	−	0.967807i	$-0.580987\pi$
$719$	−9704.92	−0.503383	−0.251692	+	0.967807i	$0.580987\pi$
$720$	0	0
$721$	−29503.4	−1.52395
$722$	0	0
$723$	−3439.99	−0.176949
$724$	0	0
$725$	−918.097	−0.0470307
$726$	0	0
$727$	10937.3	0.557967	0.278983	−	0.960296i	$-0.410003\pi$
$727$	10937.3	0.557967	0.278983	+	0.960296i	$0.410003\pi$
$728$	0	0
$729$	4317.73	0.219364
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−14196.3	−0.715351	−0.357675	−	0.933846i	$-0.616431\pi$
$733$	−14196.3	−0.715351	−0.357675	+	0.933846i	$0.616431\pi$
$734$	0	0
$735$	−5689.27	−0.285513
$736$	0	0
$737$	−199.536	−0.00997287
$738$	0	0
$739$	−8477.05	−0.421966	−0.210983	−	0.977490i	$-0.567667\pi$
$739$	−8477.05	−0.421966	−0.210983	+	0.977490i	$0.567667\pi$
$740$	0	0
$741$	21261.7	1.05407
$742$	0	0
$743$	−5251.86	−0.259316	−0.129658	−	0.991559i	$-0.541388\pi$
$743$	−5251.86	−0.259316	−0.129658	+	0.991559i	$0.541388\pi$
$744$	0	0
$745$	1481.72	0.0728674
$746$	0	0
$747$	2388.93	0.117010
$748$	0	0
$749$	−25331.1	−1.23575
$750$	0	0
$751$	−16018.0	−0.778302	−0.389151	−	0.921174i	$-0.627232\pi$
$751$	−16018.0	−0.778302	−0.389151	+	0.921174i	$0.627232\pi$
$752$	0	0
$753$	−17032.5	−0.824303
$754$	0	0
$755$	24187.2	1.16591
$756$	0	0
$757$	9344.23	0.448642	0.224321	−	0.974515i	$-0.427984\pi$
$757$	9344.23	0.448642	0.224321	+	0.974515i	$0.427984\pi$
$758$	0	0
$759$	−566.397	−0.0270868
$760$	0	0
$761$	35043.1	1.66927	0.834633	−	0.550807i	$-0.185680\pi$
$761$	35043.1	1.66927	0.834633	+	0.550807i	$0.185680\pi$
$762$	0	0
$763$	−16902.9	−0.801998
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10667.5	0.502190
$768$	0	0
$769$	−6719.33	−0.315091	−0.157546	−	0.987512i	$-0.550358\pi$
$769$	−6719.33	−0.315091	−0.157546	+	0.987512i	$0.550358\pi$
$770$	0	0
$771$	−12776.4	−0.596796
$772$	0	0
$773$	−18716.9	−0.870892	−0.435446	−	0.900215i	$-0.643409\pi$
$773$	−18716.9	−0.870892	−0.435446	+	0.900215i	$0.643409\pi$
$774$	0	0
$775$	1681.92	0.0779568
$776$	0	0
$777$	−12725.9	−0.587567
$778$	0	0
$779$	41343.7	1.90153
$780$	0	0
$781$	977.304	0.0447768
$782$	0	0
$783$	−14209.0	−0.648518
$784$	0	0
$785$	33575.8	1.52659
$786$	0	0
$787$	−15916.1	−0.720898	−0.360449	−	0.932779i	$-0.617377\pi$
$787$	−15916.1	−0.720898	−0.360449	+	0.932779i	$0.617377\pi$
$788$	0	0
$789$	3885.59	0.175324
$790$	0	0
$791$	37739.4	1.69641
$792$	0	0
$793$	28573.5	1.27954
$794$	0	0
$795$	−21345.0	−0.952239
$796$	0	0
$797$	−39802.4	−1.76898	−0.884489	−	0.466562i	$-0.845492\pi$
$797$	−39802.4	−1.76898	−0.884489	+	0.466562i	$0.845492\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−8058.88	−0.355489
$802$	0	0
$803$	1302.20	0.0572276
$804$	0	0
$805$	48084.3	2.10528
$806$	0	0
$807$	−13258.9	−0.578357
$808$	0	0
$809$	−12250.3	−0.532384	−0.266192	−	0.963920i	$-0.585766\pi$
$809$	−12250.3	−0.532384	−0.266192	+	0.963920i	$0.585766\pi$
$810$	0	0
$811$	−15936.1	−0.690002	−0.345001	−	0.938602i	$-0.612121\pi$
$811$	−15936.1	−0.690002	−0.345001	+	0.938602i	$0.612121\pi$
$812$	0	0
$813$	−2971.39	−0.128181
$814$	0	0
$815$	−18154.0	−0.780256
$816$	0	0
$817$	24284.6	1.03992
$818$	0	0
$819$	−27083.8	−1.15554
$820$	0	0
$821$	−10389.6	−0.441656	−0.220828	−	0.975313i	$-0.570876\pi$
$821$	−10389.6	−0.441656	−0.220828	+	0.975313i	$0.570876\pi$
$822$	0	0
$823$	1915.45	0.0811281	0.0405640	−	0.999177i	$-0.487085\pi$
$823$	1915.45	0.0811281	0.0405640	+	0.999177i	$0.487085\pi$
$824$	0	0
$825$	−24.9888	−0.00105454
$826$	0	0
$827$	1406.72	0.0591492	0.0295746	−	0.999563i	$-0.490585\pi$
$827$	1406.72	0.0591492	0.0295746	+	0.999563i	$0.490585\pi$
$828$	0	0
$829$	−22131.8	−0.927226	−0.463613	−	0.886038i	$-0.653447\pi$
$829$	−22131.8	−0.927226	−0.463613	+	0.886038i	$0.653447\pi$
$830$	0	0
$831$	9693.25	0.404639
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−14877.1	−0.616578
$836$	0	0
$837$	26030.5	1.07496
$838$	0	0
$839$	5716.82	0.235240	0.117620	−	0.993059i	$-0.462473\pi$
$839$	5716.82	0.235240	0.117620	+	0.993059i	$0.462473\pi$
$840$	0	0
$841$	−11157.8	−0.457493
$842$	0	0
$843$	5740.16	0.234521
$844$	0	0
$845$	13958.2	0.568257
$846$	0	0
$847$	−30644.7	−1.24317
$848$	0	0
$849$	21386.3	0.864516
$850$	0	0
$851$	38110.8	1.53516
$852$	0	0
$853$	−26801.3	−1.07580	−0.537900	−	0.843008i	$-0.680782\pi$
$853$	−26801.3	−1.07580	−0.537900	+	0.843008i	$0.680782\pi$
$854$	0	0
$855$	32260.6	1.29040
$856$	0	0
$857$	−6701.95	−0.267135	−0.133567	−	0.991040i	$-0.542643\pi$
$857$	−6701.95	−0.267135	−0.133567	+	0.991040i	$0.542643\pi$
$858$	0	0
$859$	−33950.5	−1.34852	−0.674259	−	0.738495i	$-0.735537\pi$
$859$	−33950.5	−1.34852	−0.674259	+	0.738495i	$0.735537\pi$
$860$	0	0
$861$	17971.8	0.711357
$862$	0	0
$863$	−43090.0	−1.69965	−0.849827	−	0.527062i	$-0.823294\pi$
$863$	−43090.0	−1.69965	−0.849827	+	0.527062i	$0.823294\pi$
$864$	0	0
$865$	27332.5	1.07437
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−131.631	−0.00513842
$870$	0	0
$871$	−9750.81	−0.379327
$872$	0	0
$873$	18611.5	0.721540
$874$	0	0
$875$	−31102.3	−1.20166
$876$	0	0
$877$	−38515.6	−1.48299	−0.741493	−	0.670960i	$-0.765882\pi$
$877$	−38515.6	−1.48299	−0.741493	+	0.670960i	$0.765882\pi$
$878$	0	0
$879$	−12521.7	−0.480485
$880$	0	0
$881$	−32178.8	−1.23057	−0.615284	−	0.788305i	$-0.710959\pi$
$881$	−32178.8	−1.23057	−0.615284	+	0.788305i	$0.710959\pi$
$882$	0	0
$883$	−33578.7	−1.27974	−0.639871	−	0.768482i	$-0.721012\pi$
$883$	−33578.7	−1.27974	−0.639871	+	0.768482i	$0.721012\pi$
$884$	0	0
$885$	−5523.41	−0.209794
$886$	0	0
$887$	17336.8	0.656273	0.328137	−	0.944630i	$-0.393579\pi$
$887$	17336.8	0.656273	0.328137	+	0.944630i	$0.393579\pi$
$888$	0	0
$889$	34635.3	1.30667
$890$	0	0
$891$	262.507	0.00987018
$892$	0	0
$893$	27754.7	1.04006
$894$	0	0
$895$	−19833.2	−0.740726
$896$	0	0
$897$	−27678.4	−1.03027
$898$	0	0
$899$	−24239.1	−0.899244
$900$	0	0
$901$	0	0
$902$	0	0
$903$	10556.4	0.389029
$904$	0	0
$905$	−711.626	−0.0261384
$906$	0	0
$907$	−2865.79	−0.104914	−0.0524570	−	0.998623i	$-0.516705\pi$
$907$	−2865.79	−0.104914	−0.0524570	+	0.998623i	$0.516705\pi$
$908$	0	0
$909$	−4505.52	−0.164399
$910$	0	0
$911$	−23972.0	−0.871819	−0.435910	−	0.899990i	$-0.643573\pi$
$911$	−23972.0	−0.871819	−0.435910	+	0.899990i	$0.643573\pi$
$912$	0	0
$913$	−141.756	−0.00513849
$914$	0	0
$915$	−14794.8	−0.534538
$916$	0	0
$917$	−16188.8	−0.582991
$918$	0	0
$919$	49371.5	1.77216	0.886081	−	0.463530i	$-0.153418\pi$
$919$	49371.5	1.77216	0.886081	+	0.463530i	$0.153418\pi$
$920$	0	0
$921$	−1392.01	−0.0498026
$922$	0	0
$923$	47758.3	1.70313
$924$	0	0
$925$	1681.40	0.0597668
$926$	0	0
$927$	25768.2	0.912986
$928$	0	0
$929$	47434.7	1.67522	0.837612	−	0.546266i	$-0.183951\pi$
$929$	47434.7	1.67522	0.837612	+	0.546266i	$0.183951\pi$
$930$	0	0
$931$	−26159.0	−0.920866
$932$	0	0
$933$	−12494.9	−0.438439
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37330.2	−1.30152	−0.650760	−	0.759283i	$-0.725550\pi$
$937$	−37330.2	−1.30152	−0.650760	+	0.759283i	$0.725550\pi$
$938$	0	0
$939$	−22575.7	−0.784589
$940$	0	0
$941$	−32070.6	−1.11102	−0.555511	−	0.831509i	$-0.687477\pi$
$941$	−32070.6	−1.11102	−0.555511	+	0.831509i	$0.687477\pi$
$942$	0	0
$943$	−53820.9	−1.85859
$944$	0	0
$945$	32832.5	1.13020
$946$	0	0
$947$	−23171.6	−0.795118	−0.397559	−	0.917577i	$-0.630143\pi$
$947$	−23171.6	−0.795118	−0.397559	+	0.917577i	$0.630143\pi$
$948$	0	0
$949$	63635.3	2.17670
$950$	0	0
$951$	−6429.77	−0.219242
$952$	0	0
$953$	−39914.2	−1.35671	−0.678357	−	0.734733i	$-0.737307\pi$
$953$	−39914.2	−1.35671	−0.678357	+	0.734733i	$0.737307\pi$
$954$	0	0
$955$	−29582.8	−1.00239
$956$	0	0
$957$	360.127	0.0121643
$958$	0	0
$959$	437.666	0.0147372
$960$	0	0
$961$	14614.3	0.490560
$962$	0	0
$963$	22124.1	0.740332
$964$	0	0
$965$	−22967.3	−0.766158
$966$	0	0
$967$	40591.6	1.34988	0.674941	−	0.737871i	$-0.264169\pi$
$967$	40591.6	1.34988	0.674941	+	0.737871i	$0.264169\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29860.5	−0.986890	−0.493445	−	0.869777i	$-0.664263\pi$
$971$	−29860.5	−0.986890	−0.493445	+	0.869777i	$0.664263\pi$
$972$	0	0
$973$	−6383.73	−0.210332
$974$	0	0
$975$	−1221.14	−0.0401104
$976$	0	0
$977$	35041.0	1.14745	0.573726	−	0.819048i	$-0.305498\pi$
$977$	35041.0	1.14745	0.573726	+	0.819048i	$0.305498\pi$
$978$	0	0
$979$	478.204	0.0156113
$980$	0	0
$981$	14762.9	0.480472
$982$	0	0
$983$	54091.0	1.75507	0.877535	−	0.479513i	$-0.159187\pi$
$983$	54091.0	1.75507	0.877535	+	0.479513i	$0.159187\pi$
$984$	0	0
$985$	−44187.0	−1.42935
$986$	0	0
$987$	12064.8	0.389085
$988$	0	0
$989$	−31613.5	−1.01643
$990$	0	0
$991$	−15918.2	−0.510251	−0.255126	−	0.966908i	$-0.582117\pi$
$991$	−15918.2	−0.510251	−0.255126	+	0.966908i	$0.582117\pi$
$992$	0	0
$993$	−28839.9	−0.921657
$994$	0	0
$995$	25039.8	0.797805
$996$	0	0
$997$	15294.7	0.485845	0.242923	−	0.970046i	$-0.421894\pi$
$997$	15294.7	0.485845	0.242923	+	0.970046i	$0.421894\pi$
$998$	0	0
$999$	26022.4	0.824138

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.k.1.4		8
17.4	even	4		136.4.b.b.33.5	yes	8
17.13	even	4		136.4.b.b.33.4	✓	8
17.16	even	2	inner	2312.4.a.k.1.5		8
51.38	odd	4		1224.4.c.e.577.6		8
51.47	odd	4		1224.4.c.e.577.3		8
68.47	odd	4		272.4.b.f.33.5		8
68.55	odd	4		272.4.b.f.33.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.b.33.4	✓	8	17.13	even	4
136.4.b.b.33.5	yes	8	17.4	even	4
272.4.b.f.33.4		8	68.55	odd	4
272.4.b.f.33.5		8	68.47	odd	4
1224.4.c.e.577.3		8	51.47	odd	4
1224.4.c.e.577.6		8	51.38	odd	4
2312.4.a.k.1.4		8	1.1	even	1	trivial
2312.4.a.k.1.5		8	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62097 q^{3} +11.5318 q^{5} +23.0485 q^{7} -20.1305 q^{9} +O(q^{10})\)	\(q-2.62097 q^{3} +11.5318 q^{5} +23.0485 q^{7} -20.1305 q^{9} +1.19452 q^{11} +58.3731 q^{13} -30.2245 q^{15} -138.971 q^{19} -60.4096 q^{21} +180.911 q^{23} +7.98158 q^{25} +123.528 q^{27} -115.027 q^{29} +210.726 q^{31} -3.13080 q^{33} +265.790 q^{35} +210.661 q^{37} -152.994 q^{39} -297.500 q^{41} -174.746 q^{43} -232.140 q^{45} -199.717 q^{47} +188.234 q^{49} +706.217 q^{53} +13.7749 q^{55} +364.238 q^{57} +182.746 q^{59} +489.499 q^{61} -463.978 q^{63} +673.144 q^{65} -167.043 q^{67} -474.163 q^{69} +818.157 q^{71} +1090.15 q^{73} -20.9195 q^{75} +27.5319 q^{77} -110.196 q^{79} +219.760 q^{81} -118.672 q^{83} +301.483 q^{87} +400.332 q^{89} +1345.41 q^{91} -552.306 q^{93} -1602.58 q^{95} -924.543 q^{97} -24.0463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 132 q^{9}+O(q^{10}) \) 8 * q + 132 * q^9	\( 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100}) \) 8 * q + 132 * q^9 + 44 * q^13 - 24 * q^15 - 48 * q^19 + 308 * q^21 + 520 * q^25 + 812 * q^33 - 1064 * q^35 - 8 * q^43 + 312 * q^47 + 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 180 * q^69 - 1660 * q^77 + 3156 * q^81 - 2472 * q^83 + 6664 * q^87 + 68 * q^89 + 4036 * q^93

Introduction

L-functions

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