LMFDB - Embedded newform 312.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,4,Mod(1,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.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:	$18.4085959218$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.36248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 54x - 90 $ x^3 - x^2 - 54*x - 90
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.54849$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +7.92181 q^{5} +28.1158 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +7.92181 q^{5} +28.1158 q^{7} +9.00000 q^{9} -34.3503 q^{11} +13.0000 q^{13} -23.7654 q^{15} +34.0000 q^{17} +72.9728 q^{19} -84.3473 q^{21} -120.776 q^{23} -62.2450 q^{25} -27.0000 q^{27} +197.919 q^{29} -23.9594 q^{31} +103.051 q^{33} +222.728 q^{35} +396.959 q^{37} -39.0000 q^{39} +438.896 q^{41} +278.198 q^{43} +71.2963 q^{45} -353.848 q^{47} +447.496 q^{49} -102.000 q^{51} +324.728 q^{53} -272.117 q^{55} -218.919 q^{57} +571.658 q^{59} -554.336 q^{61} +253.042 q^{63} +102.983 q^{65} +490.314 q^{67} +362.327 q^{69} -678.338 q^{71} +332.918 q^{73} +186.735 q^{75} -965.786 q^{77} +341.918 q^{79} +81.0000 q^{81} -787.045 q^{83} +269.341 q^{85} -593.756 q^{87} -327.600 q^{89} +365.505 q^{91} +71.8781 q^{93} +578.077 q^{95} +1087.18 q^{97} -309.153 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 16 q^{5} - 22 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 + 16 * q^5 - 22 * q^7 + 27 * q^9	$ 3 q - 9 q^{3} + 16 q^{5} - 22 q^{7} + 27 q^{9} - 20 q^{11} + 39 q^{13} - 48 q^{15} + 102 q^{17} - 38 q^{19} + 66 q^{21} + 32 q^{23} + 161 q^{25} - 81 q^{27} + 350 q^{29} + 50 q^{31} + 60 q^{33} + 232 q^{35} + 542 q^{37} - 117 q^{39} + 500 q^{41} + 420 q^{43} + 144 q^{45} + 324 q^{47} + 1119 q^{49} - 306 q^{51} + 538 q^{53} + 896 q^{55} + 114 q^{57} + 112 q^{59} + 1206 q^{61} - 198 q^{63} + 208 q^{65} + 950 q^{67} - 96 q^{69} - 1008 q^{71} + 1834 q^{73} - 483 q^{75} - 792 q^{77} - 272 q^{79} + 243 q^{81} - 1640 q^{83} + 544 q^{85} - 1050 q^{87} - 1576 q^{89} - 286 q^{91} - 150 q^{93} - 2760 q^{95} + 402 q^{97} - 180 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 + 16 * q^5 - 22 * q^7 + 27 * q^9 - 20 * q^11 + 39 * q^13 - 48 * q^15 + 102 * q^17 - 38 * q^19 + 66 * q^21 + 32 * q^23 + 161 * q^25 - 81 * q^27 + 350 * q^29 + 50 * q^31 + 60 * q^33 + 232 * q^35 + 542 * q^37 - 117 * q^39 + 500 * q^41 + 420 * q^43 + 144 * q^45 + 324 * q^47 + 1119 * q^49 - 306 * q^51 + 538 * q^53 + 896 * q^55 + 114 * q^57 + 112 * q^59 + 1206 * q^61 - 198 * q^63 + 208 * q^65 + 950 * q^67 - 96 * q^69 - 1008 * q^71 + 1834 * q^73 - 483 * q^75 - 792 * q^77 - 272 * q^79 + 243 * q^81 - 1640 * q^83 + 544 * q^85 - 1050 * q^87 - 1576 * q^89 - 286 * q^91 - 150 * q^93 - 2760 * q^95 + 402 * q^97 - 180 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	7.92181	0.708548	0.354274	−	0.935142i	$-0.384728\pi$
$5$	7.92181	0.708548	0.354274	+	0.935142i	$0.384728\pi$
$6$	0	0
$7$	28.1158	1.51811	0.759054	−	0.651027i	$-0.225662\pi$
$7$	28.1158	1.51811	0.759054	+	0.651027i	$0.225662\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−34.3503	−0.941547	−0.470774	−	0.882254i	$-0.656025\pi$
$11$	−34.3503	−0.941547	−0.470774	+	0.882254i	$0.656025\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−23.7654	−0.409080
$16$	0	0
$17$	34.0000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	34.0000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	72.9728	0.881111	0.440556	−	0.897725i	$-0.354781\pi$
$19$	72.9728	0.881111	0.440556	+	0.897725i	$0.354781\pi$
$20$	0	0
$21$	−84.3473	−0.876480
$22$	0	0
$23$	−120.776	−1.09493	−0.547467	−	0.836827i	$-0.684408\pi$
$23$	−120.776	−1.09493	−0.547467	+	0.836827i	$0.684408\pi$
$24$	0	0
$25$	−62.2450	−0.497960
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	197.919	1.26733	0.633665	−	0.773607i	$-0.281550\pi$
$29$	197.919	1.26733	0.633665	+	0.773607i	$0.281550\pi$
$30$	0	0
$31$	−23.9594	−0.138814	−0.0694069	−	0.997588i	$-0.522111\pi$
$31$	−23.9594	−0.138814	−0.0694069	+	0.997588i	$0.522111\pi$
$32$	0	0
$33$	103.051	0.543602
$34$	0	0
$35$	222.728	1.07565
$36$	0	0
$37$	396.959	1.76378	0.881888	−	0.471460i	$-0.156273\pi$
$37$	396.959	1.76378	0.881888	+	0.471460i	$0.156273\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	438.896	1.67181	0.835903	−	0.548877i	$-0.184944\pi$
$41$	438.896	1.67181	0.835903	+	0.548877i	$0.184944\pi$
$42$	0	0
$43$	278.198	0.986625	0.493312	−	0.869852i	$-0.335786\pi$
$43$	278.198	0.986625	0.493312	+	0.869852i	$0.335786\pi$
$44$	0	0
$45$	71.2963	0.236183
$46$	0	0
$47$	−353.848	−1.09817	−0.549085	−	0.835767i	$-0.685024\pi$
$47$	−353.848	−1.09817	−0.549085	+	0.835767i	$0.685024\pi$
$48$	0	0
$49$	447.496	1.30465
$50$	0	0
$51$	−102.000	−0.280056
$52$	0	0
$53$	324.728	0.841599	0.420800	−	0.907154i	$-0.361750\pi$
$53$	324.728	0.841599	0.420800	+	0.907154i	$0.361750\pi$
$54$	0	0
$55$	−272.117	−0.667131
$56$	0	0
$57$	−218.919	−0.508710
$58$	0	0
$59$	571.658	1.26142	0.630708	−	0.776020i	$-0.282765\pi$
$59$	571.658	1.26142	0.630708	+	0.776020i	$0.282765\pi$
$60$	0	0
$61$	−554.336	−1.16353	−0.581766	−	0.813356i	$-0.697638\pi$
$61$	−554.336	−1.16353	−0.581766	+	0.813356i	$0.697638\pi$
$62$	0	0
$63$	253.042	0.506036
$64$	0	0
$65$	102.983	0.196516
$66$	0	0
$67$	490.314	0.894051	0.447026	−	0.894521i	$-0.352483\pi$
$67$	490.314	0.894051	0.447026	+	0.894521i	$0.352483\pi$
$68$	0	0
$69$	362.327	0.632161
$70$	0	0
$71$	−678.338	−1.13386	−0.566929	−	0.823767i	$-0.691869\pi$
$71$	−678.338	−1.13386	−0.566929	+	0.823767i	$0.691869\pi$
$72$	0	0
$73$	332.918	0.533768	0.266884	−	0.963729i	$-0.414006\pi$
$73$	332.918	0.533768	0.266884	+	0.963729i	$0.414006\pi$
$74$	0	0
$75$	186.735	0.287497
$76$	0	0
$77$	−965.786	−1.42937
$78$	0	0
$79$	341.918	0.486947	0.243473	−	0.969908i	$-0.421713\pi$
$79$	341.918	0.486947	0.243473	+	0.969908i	$0.421713\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−787.045	−1.04084	−0.520418	−	0.853912i	$-0.674224\pi$
$83$	−787.045	−1.04084	−0.520418	+	0.853912i	$0.674224\pi$
$84$	0	0
$85$	269.341	0.343696
$86$	0	0
$87$	−593.756	−0.731694
$88$	0	0
$89$	−327.600	−0.390174	−0.195087	−	0.980786i	$-0.562499\pi$
$89$	−327.600	−0.390174	−0.195087	+	0.980786i	$0.562499\pi$
$90$	0	0
$91$	365.505	0.421048
$92$	0	0
$93$	71.8781	0.0801442
$94$	0	0
$95$	578.077	0.624310
$96$	0	0
$97$	1087.18	1.13800	0.569001	−	0.822337i	$-0.307330\pi$
$97$	1087.18	1.13800	0.569001	+	0.822337i	$0.307330\pi$
$98$	0	0
$99$	−309.153	−0.313849
$100$	0	0
$101$	−230.015	−0.226608	−0.113304	−	0.993560i	$-0.536143\pi$
$101$	−230.015	−0.226608	−0.113304	+	0.993560i	$0.536143\pi$
$102$	0	0
$103$	−684.238	−0.654563	−0.327281	−	0.944927i	$-0.606132\pi$
$103$	−684.238	−0.654563	−0.327281	+	0.944927i	$0.606132\pi$
$104$	0	0
$105$	−668.183	−0.621028
$106$	0	0
$107$	−1448.78	−1.30896	−0.654481	−	0.756078i	$-0.727113\pi$
$107$	−1448.78	−1.30896	−0.654481	+	0.756078i	$0.727113\pi$
$108$	0	0
$109$	2174.36	1.91070	0.955348	−	0.295483i	$-0.0954806\pi$
$109$	2174.36	1.91070	0.955348	+	0.295483i	$0.0954806\pi$
$110$	0	0
$111$	−1190.88	−1.01832
$112$	0	0
$113$	−1481.63	−1.23345	−0.616724	−	0.787179i	$-0.711541\pi$
$113$	−1481.63	−1.23345	−0.616724	+	0.787179i	$0.711541\pi$
$114$	0	0
$115$	−956.763	−0.775814
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	955.936	0.736391
$120$	0	0
$121$	−151.054	−0.113489
$122$	0	0
$123$	−1316.69	−0.965218
$124$	0	0
$125$	−1483.32	−1.06138
$126$	0	0
$127$	−902.481	−0.630569	−0.315284	−	0.948997i	$-0.602100\pi$
$127$	−902.481	−0.630569	−0.315284	+	0.948997i	$0.602100\pi$
$128$	0	0
$129$	−834.595	−0.569628
$130$	0	0
$131$	304.072	0.202801	0.101400	−	0.994846i	$-0.467668\pi$
$131$	304.072	0.202801	0.101400	+	0.994846i	$0.467668\pi$
$132$	0	0
$133$	2051.69	1.33762
$134$	0	0
$135$	−213.889	−0.136360
$136$	0	0
$137$	1127.71	0.703263	0.351631	−	0.936139i	$-0.385627\pi$
$137$	1127.71	0.703263	0.351631	+	0.936139i	$0.385627\pi$
$138$	0	0
$139$	−1777.54	−1.08467	−0.542334	−	0.840163i	$-0.682459\pi$
$139$	−1777.54	−1.08467	−0.542334	+	0.840163i	$0.682459\pi$
$140$	0	0
$141$	1061.54	0.634029
$142$	0	0
$143$	−446.554	−0.261138
$144$	0	0
$145$	1567.87	0.897964
$146$	0	0
$147$	−1342.49	−0.753242
$148$	0	0
$149$	632.083	0.347532	0.173766	−	0.984787i	$-0.444406\pi$
$149$	632.083	0.347532	0.173766	+	0.984787i	$0.444406\pi$
$150$	0	0
$151$	1799.99	0.970076	0.485038	−	0.874493i	$-0.338806\pi$
$151$	1799.99	0.970076	0.485038	+	0.874493i	$0.338806\pi$
$152$	0	0
$153$	306.000	0.161690
$154$	0	0
$155$	−189.802	−0.0983563
$156$	0	0
$157$	−2235.15	−1.13620	−0.568102	−	0.822958i	$-0.692322\pi$
$157$	−2235.15	−1.13620	−0.568102	+	0.822958i	$0.692322\pi$
$158$	0	0
$159$	−974.183	−0.485898
$160$	0	0
$161$	−3395.70	−1.66223
$162$	0	0
$163$	3311.03	1.59104	0.795521	−	0.605925i	$-0.207197\pi$
$163$	3311.03	1.59104	0.795521	+	0.605925i	$0.207197\pi$
$164$	0	0
$165$	816.350	0.385168
$166$	0	0
$167$	−761.161	−0.352697	−0.176349	−	0.984328i	$-0.556429\pi$
$167$	−761.161	−0.352697	−0.176349	+	0.984328i	$0.556429\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	656.756	0.293704
$172$	0	0
$173$	−2777.78	−1.22076	−0.610379	−	0.792110i	$-0.708983\pi$
$173$	−2777.78	−1.22076	−0.610379	+	0.792110i	$0.708983\pi$
$174$	0	0
$175$	−1750.07	−0.755957
$176$	0	0
$177$	−1714.97	−0.728279
$178$	0	0
$179$	−465.740	−0.194475	−0.0972376	−	0.995261i	$-0.531001\pi$
$179$	−465.740	−0.194475	−0.0972376	+	0.995261i	$0.531001\pi$
$180$	0	0
$181$	1079.31	0.443229	0.221614	−	0.975134i	$-0.428867\pi$
$181$	1079.31	0.443229	0.221614	+	0.975134i	$0.428867\pi$
$182$	0	0
$183$	1663.01	0.671766
$184$	0	0
$185$	3144.63	1.24972
$186$	0	0
$187$	−1167.91	−0.456717
$188$	0	0
$189$	−759.126	−0.292160
$190$	0	0
$191$	1551.62	0.587809	0.293904	−	0.955835i	$-0.405045\pi$
$191$	1551.62	0.587809	0.293904	+	0.955835i	$0.405045\pi$
$192$	0	0
$193$	−4296.51	−1.60243	−0.801216	−	0.598376i	$-0.795813\pi$
$193$	−4296.51	−1.60243	−0.801216	+	0.598376i	$0.795813\pi$
$194$	0	0
$195$	−308.950	−0.113458
$196$	0	0
$197$	−2549.16	−0.921931	−0.460965	−	0.887418i	$-0.652497\pi$
$197$	−2549.16	−0.921931	−0.460965	+	0.887418i	$0.652497\pi$
$198$	0	0
$199$	−2553.13	−0.909481	−0.454741	−	0.890624i	$-0.650268\pi$
$199$	−2553.13	−0.909481	−0.454741	+	0.890624i	$0.650268\pi$
$200$	0	0
$201$	−1470.94	−0.516181
$202$	0	0
$203$	5564.64	1.92395
$204$	0	0
$205$	3476.85	1.18455
$206$	0	0
$207$	−1086.98	−0.364978
$208$	0	0
$209$	−2506.64	−0.829608
$210$	0	0
$211$	1499.67	0.489296	0.244648	−	0.969612i	$-0.421328\pi$
$211$	1499.67	0.489296	0.244648	+	0.969612i	$0.421328\pi$
$212$	0	0
$213$	2035.01	0.654633
$214$	0	0
$215$	2203.83	0.699071
$216$	0	0
$217$	−673.636	−0.210735
$218$	0	0
$219$	−998.753	−0.308171
$220$	0	0
$221$	442.000	0.134535
$222$	0	0
$223$	730.772	0.219445	0.109722	−	0.993962i	$-0.465004\pi$
$223$	730.772	0.219445	0.109722	+	0.993962i	$0.465004\pi$
$224$	0	0
$225$	−560.205	−0.165987
$226$	0	0
$227$	−2286.29	−0.668487	−0.334243	−	0.942487i	$-0.608481\pi$
$227$	−2286.29	−0.668487	−0.334243	+	0.942487i	$0.608481\pi$
$228$	0	0
$229$	5207.95	1.50284	0.751422	−	0.659822i	$-0.229368\pi$
$229$	5207.95	1.50284	0.751422	+	0.659822i	$0.229368\pi$
$230$	0	0
$231$	2897.36	0.825248
$232$	0	0
$233$	−2204.69	−0.619888	−0.309944	−	0.950755i	$-0.600310\pi$
$233$	−2204.69	−0.619888	−0.309944	+	0.950755i	$0.600310\pi$
$234$	0	0
$235$	−2803.11	−0.778106
$236$	0	0
$237$	−1025.75	−0.281139
$238$	0	0
$239$	−2840.87	−0.768872	−0.384436	−	0.923152i	$-0.625604\pi$
$239$	−2840.87	−0.768872	−0.384436	+	0.923152i	$0.625604\pi$
$240$	0	0
$241$	−5032.05	−1.34499	−0.672495	−	0.740101i	$-0.734777\pi$
$241$	−5032.05	−1.34499	−0.672495	+	0.740101i	$0.734777\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	3544.98	0.924409
$246$	0	0
$247$	948.647	0.244376
$248$	0	0
$249$	2361.13	0.600927
$250$	0	0
$251$	−3635.11	−0.914129	−0.457064	−	0.889434i	$-0.651099\pi$
$251$	−3635.11	−0.914129	−0.457064	+	0.889434i	$0.651099\pi$
$252$	0	0
$253$	4148.69	1.03093
$254$	0	0
$255$	−808.024	−0.198433
$256$	0	0
$257$	2614.57	0.634600	0.317300	−	0.948325i	$-0.397224\pi$
$257$	2614.57	0.634600	0.317300	+	0.948325i	$0.397224\pi$
$258$	0	0
$259$	11160.8	2.67760
$260$	0	0
$261$	1781.27	0.422444
$262$	0	0
$263$	5565.62	1.30491	0.652454	−	0.757828i	$-0.273739\pi$
$263$	5565.62	1.30491	0.652454	+	0.757828i	$0.273739\pi$
$264$	0	0
$265$	2572.43	0.596313
$266$	0	0
$267$	982.800	0.225267
$268$	0	0
$269$	3483.33	0.789526	0.394763	−	0.918783i	$-0.370827\pi$
$269$	3483.33	0.789526	0.394763	+	0.918783i	$0.370827\pi$
$270$	0	0
$271$	−4604.75	−1.03217	−0.516086	−	0.856537i	$-0.672611\pi$
$271$	−4604.75	−1.03217	−0.516086	+	0.856537i	$0.672611\pi$
$272$	0	0
$273$	−1096.51	−0.243092
$274$	0	0
$275$	2138.14	0.468853
$276$	0	0
$277$	886.536	0.192299	0.0961494	−	0.995367i	$-0.469347\pi$
$277$	886.536	0.192299	0.0961494	+	0.995367i	$0.469347\pi$
$278$	0	0
$279$	−215.634	−0.0462713
$280$	0	0
$281$	−6898.11	−1.46444	−0.732219	−	0.681070i	$-0.761515\pi$
$281$	−6898.11	−1.46444	−0.732219	+	0.681070i	$0.761515\pi$
$282$	0	0
$283$	9209.50	1.93445	0.967223	−	0.253929i	$-0.0817231\pi$
$283$	9209.50	1.93445	0.967223	+	0.253929i	$0.0817231\pi$
$284$	0	0
$285$	−1734.23	−0.360445
$286$	0	0
$287$	12339.9	2.53798
$288$	0	0
$289$	−3757.00	−0.764706
$290$	0	0
$291$	−3261.53	−0.657026
$292$	0	0
$293$	−3994.94	−0.796542	−0.398271	−	0.917268i	$-0.630390\pi$
$293$	−3994.94	−0.796542	−0.398271	+	0.917268i	$0.630390\pi$
$294$	0	0
$295$	4528.57	0.893774
$296$	0	0
$297$	927.459	0.181201
$298$	0	0
$299$	−1570.09	−0.303680
$300$	0	0
$301$	7821.76	1.49780
$302$	0	0
$303$	690.045	0.130832
$304$	0	0
$305$	−4391.34	−0.824418
$306$	0	0
$307$	736.716	0.136960	0.0684798	−	0.997653i	$-0.478185\pi$
$307$	736.716	0.136960	0.0684798	+	0.997653i	$0.478185\pi$
$308$	0	0
$309$	2052.71	0.377912
$310$	0	0
$311$	2985.84	0.544410	0.272205	−	0.962239i	$-0.412247\pi$
$311$	2985.84	0.544410	0.272205	+	0.962239i	$0.412247\pi$
$312$	0	0
$313$	46.3671	0.00837324	0.00418662	−	0.999991i	$-0.498667\pi$
$313$	46.3671	0.00837324	0.00418662	+	0.999991i	$0.498667\pi$
$314$	0	0
$315$	2004.55	0.358551
$316$	0	0
$317$	−1917.45	−0.339731	−0.169866	−	0.985467i	$-0.554333\pi$
$317$	−1917.45	−0.339731	−0.169866	+	0.985467i	$0.554333\pi$
$318$	0	0
$319$	−6798.58	−1.19325
$320$	0	0
$321$	4346.34	0.755730
$322$	0	0
$323$	2481.08	0.427402
$324$	0	0
$325$	−809.185	−0.138109
$326$	0	0
$327$	−6523.08	−1.10314
$328$	0	0
$329$	−9948.70	−1.66714
$330$	0	0
$331$	−6408.79	−1.06423	−0.532113	−	0.846673i	$-0.678602\pi$
$331$	−6408.79	−1.06423	−0.532113	+	0.846673i	$0.678602\pi$
$332$	0	0
$333$	3572.63	0.587925
$334$	0	0
$335$	3884.17	0.633478
$336$	0	0
$337$	−9489.38	−1.53389	−0.766943	−	0.641715i	$-0.778223\pi$
$337$	−9489.38	−1.53389	−0.766943	+	0.641715i	$0.778223\pi$
$338$	0	0
$339$	4444.88	0.712132
$340$	0	0
$341$	823.013	0.130700
$342$	0	0
$343$	2937.99	0.462497
$344$	0	0
$345$	2870.29	0.447916
$346$	0	0
$347$	7257.13	1.12272	0.561359	−	0.827572i	$-0.310279\pi$
$347$	7257.13	1.12272	0.561359	+	0.827572i	$0.310279\pi$
$348$	0	0
$349$	−12407.2	−1.90298	−0.951491	−	0.307678i	$-0.900448\pi$
$349$	−12407.2	−1.90298	−0.951491	+	0.307678i	$0.900448\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	1494.97	0.225409	0.112705	−	0.993629i	$-0.464049\pi$
$353$	1494.97	0.225409	0.112705	+	0.993629i	$0.464049\pi$
$354$	0	0
$355$	−5373.66	−0.803393
$356$	0	0
$357$	−2867.81	−0.425155
$358$	0	0
$359$	−7884.41	−1.15912	−0.579559	−	0.814930i	$-0.696775\pi$
$359$	−7884.41	−1.15912	−0.579559	+	0.814930i	$0.696775\pi$
$360$	0	0
$361$	−1533.96	−0.223643
$362$	0	0
$363$	453.162	0.0655229
$364$	0	0
$365$	2637.31	0.378200
$366$	0	0
$367$	8736.23	1.24258	0.621291	−	0.783580i	$-0.286609\pi$
$367$	8736.23	1.24258	0.621291	+	0.783580i	$0.286609\pi$
$368$	0	0
$369$	3950.07	0.557269
$370$	0	0
$371$	9129.96	1.27764
$372$	0	0
$373$	−7079.36	−0.982722	−0.491361	−	0.870956i	$-0.663500\pi$
$373$	−7079.36	−0.982722	−0.491361	+	0.870956i	$0.663500\pi$
$374$	0	0
$375$	4449.96	0.612786
$376$	0	0
$377$	2572.94	0.351494
$378$	0	0
$379$	−9890.04	−1.34041	−0.670207	−	0.742174i	$-0.733795\pi$
$379$	−9890.04	−1.34041	−0.670207	+	0.742174i	$0.733795\pi$
$380$	0	0
$381$	2707.44	0.364059
$382$	0	0
$383$	14499.5	1.93444	0.967222	−	0.253934i	$-0.0817245\pi$
$383$	14499.5	1.93444	0.967222	+	0.253934i	$0.0817245\pi$
$384$	0	0
$385$	−7650.77	−1.01278
$386$	0	0
$387$	2503.79	0.328875
$388$	0	0
$389$	1488.74	0.194041	0.0970207	−	0.995282i	$-0.469069\pi$
$389$	1488.74	0.194041	0.0970207	+	0.995282i	$0.469069\pi$
$390$	0	0
$391$	−4106.38	−0.531121
$392$	0	0
$393$	−912.216	−0.117087
$394$	0	0
$395$	2708.61	0.345025
$396$	0	0
$397$	5868.21	0.741857	0.370928	−	0.928662i	$-0.379040\pi$
$397$	5868.21	0.741857	0.370928	+	0.928662i	$0.379040\pi$
$398$	0	0
$399$	−6155.06	−0.772277
$400$	0	0
$401$	−4669.34	−0.581485	−0.290742	−	0.956801i	$-0.593902\pi$
$401$	−4669.34	−0.581485	−0.290742	+	0.956801i	$0.593902\pi$
$402$	0	0
$403$	−311.472	−0.0385000
$404$	0	0
$405$	641.666	0.0787275
$406$	0	0
$407$	−13635.7	−1.66068
$408$	0	0
$409$	7875.82	0.952162	0.476081	−	0.879401i	$-0.342057\pi$
$409$	7875.82	0.952162	0.476081	+	0.879401i	$0.342057\pi$
$410$	0	0
$411$	−3383.14	−0.406029
$412$	0	0
$413$	16072.6	1.91497
$414$	0	0
$415$	−6234.82	−0.737482
$416$	0	0
$417$	5332.62	0.626234
$418$	0	0
$419$	−6372.72	−0.743026	−0.371513	−	0.928428i	$-0.621161\pi$
$419$	−6372.72	−0.743026	−0.371513	+	0.928428i	$0.621161\pi$
$420$	0	0
$421$	−12188.4	−1.41099	−0.705494	−	0.708716i	$-0.749275\pi$
$421$	−12188.4	−1.41099	−0.705494	+	0.708716i	$0.749275\pi$
$422$	0	0
$423$	−3184.63	−0.366057
$424$	0	0
$425$	−2116.33	−0.241546
$426$	0	0
$427$	−15585.6	−1.76637
$428$	0	0
$429$	1339.66	0.150768
$430$	0	0
$431$	13498.9	1.50862	0.754312	−	0.656516i	$-0.227970\pi$
$431$	13498.9	1.50862	0.754312	+	0.656516i	$0.227970\pi$
$432$	0	0
$433$	−8403.29	−0.932647	−0.466324	−	0.884614i	$-0.654422\pi$
$433$	−8403.29	−0.932647	−0.466324	+	0.884614i	$0.654422\pi$
$434$	0	0
$435$	−4703.62	−0.518440
$436$	0	0
$437$	−8813.35	−0.964760
$438$	0	0
$439$	−11361.0	−1.23515	−0.617573	−	0.786514i	$-0.711884\pi$
$439$	−11361.0	−1.23515	−0.617573	+	0.786514i	$0.711884\pi$
$440$	0	0
$441$	4027.46	0.434884
$442$	0	0
$443$	−13294.7	−1.42585	−0.712924	−	0.701242i	$-0.752629\pi$
$443$	−13294.7	−1.42585	−0.712924	+	0.701242i	$0.752629\pi$
$444$	0	0
$445$	−2595.18	−0.276457
$446$	0	0
$447$	−1896.25	−0.200648
$448$	0	0
$449$	−883.910	−0.0929049	−0.0464525	−	0.998921i	$-0.514792\pi$
$449$	−883.910	−0.0929049	−0.0464525	+	0.998921i	$0.514792\pi$
$450$	0	0
$451$	−15076.2	−1.57408
$452$	0	0
$453$	−5399.98	−0.560073
$454$	0	0
$455$	2895.46	0.298332
$456$	0	0
$457$	6942.43	0.710619	0.355310	−	0.934749i	$-0.384375\pi$
$457$	6942.43	0.710619	0.355310	+	0.934749i	$0.384375\pi$
$458$	0	0
$459$	−918.000	−0.0933520
$460$	0	0
$461$	2502.43	0.252820	0.126410	−	0.991978i	$-0.459655\pi$
$461$	2502.43	0.252820	0.126410	+	0.991978i	$0.459655\pi$
$462$	0	0
$463$	−12495.9	−1.25429	−0.627143	−	0.778904i	$-0.715776\pi$
$463$	−12495.9	−1.25429	−0.627143	+	0.778904i	$0.715776\pi$
$464$	0	0
$465$	569.405	0.0567860
$466$	0	0
$467$	−2418.96	−0.239691	−0.119846	−	0.992793i	$-0.538240\pi$
$467$	−2418.96	−0.239691	−0.119846	+	0.992793i	$0.538240\pi$
$468$	0	0
$469$	13785.6	1.35727
$470$	0	0
$471$	6705.44	0.655988
$472$	0	0
$473$	−9556.21	−0.928954
$474$	0	0
$475$	−4542.19	−0.438758
$476$	0	0
$477$	2922.55	0.280533
$478$	0	0
$479$	−20679.9	−1.97263	−0.986314	−	0.164876i	$-0.947278\pi$
$479$	−20679.9	−1.97263	−0.986314	+	0.164876i	$0.947278\pi$
$480$	0	0
$481$	5160.47	0.489183
$482$	0	0
$483$	10187.1	0.959689
$484$	0	0
$485$	8612.41	0.806329
$486$	0	0
$487$	−426.771	−0.0397101	−0.0198551	−	0.999803i	$-0.506320\pi$
$487$	−426.771	−0.0397101	−0.0198551	+	0.999803i	$0.506320\pi$
$488$	0	0
$489$	−9933.09	−0.918589
$490$	0	0
$491$	−2265.88	−0.208264	−0.104132	−	0.994563i	$-0.533207\pi$
$491$	−2265.88	−0.208264	−0.104132	+	0.994563i	$0.533207\pi$
$492$	0	0
$493$	6729.24	0.614746
$494$	0	0
$495$	−2449.05	−0.222377
$496$	0	0
$497$	−19072.0	−1.72132
$498$	0	0
$499$	−2733.56	−0.245233	−0.122616	−	0.992454i	$-0.539128\pi$
$499$	−2733.56	−0.245233	−0.122616	+	0.992454i	$0.539128\pi$
$500$	0	0
$501$	2283.48	0.203630
$502$	0	0
$503$	14428.0	1.27896	0.639478	−	0.768810i	$-0.279151\pi$
$503$	14428.0	1.27896	0.639478	+	0.768810i	$0.279151\pi$
$504$	0	0
$505$	−1822.14	−0.160562
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	7551.28	0.657573	0.328786	−	0.944404i	$-0.393360\pi$
$509$	7551.28	0.657573	0.328786	+	0.944404i	$0.393360\pi$
$510$	0	0
$511$	9360.23	0.810317
$512$	0	0
$513$	−1970.27	−0.169570
$514$	0	0
$515$	−5420.40	−0.463789
$516$	0	0
$517$	12154.8	1.03398
$518$	0	0
$519$	8333.35	0.704805
$520$	0	0
$521$	8272.18	0.695606	0.347803	−	0.937568i	$-0.386928\pi$
$521$	8272.18	0.695606	0.347803	+	0.937568i	$0.386928\pi$
$522$	0	0
$523$	17148.4	1.43374	0.716871	−	0.697206i	$-0.245574\pi$
$523$	17148.4	1.43374	0.716871	+	0.697206i	$0.245574\pi$
$524$	0	0
$525$	5250.20	0.436452
$526$	0	0
$527$	−814.619	−0.0673346
$528$	0	0
$529$	2419.80	0.198882
$530$	0	0
$531$	5144.92	0.420472
$532$	0	0
$533$	5705.65	0.463676
$534$	0	0
$535$	−11477.0	−0.927462
$536$	0	0
$537$	1397.22	0.112280
$538$	0	0
$539$	−15371.6	−1.22839
$540$	0	0
$541$	−18087.8	−1.43744	−0.718719	−	0.695301i	$-0.755271\pi$
$541$	−18087.8	−1.43744	−0.718719	+	0.695301i	$0.755271\pi$
$542$	0	0
$543$	−3237.93	−0.255898
$544$	0	0
$545$	17224.9	1.35382
$546$	0	0
$547$	21150.2	1.65323	0.826615	−	0.562767i	$-0.190263\pi$
$547$	21150.2	1.65323	0.826615	+	0.562767i	$0.190263\pi$
$548$	0	0
$549$	−4989.02	−0.387844
$550$	0	0
$551$	14442.7	1.11666
$552$	0	0
$553$	9613.29	0.739238
$554$	0	0
$555$	−9433.90	−0.721526
$556$	0	0
$557$	22308.5	1.69702	0.848510	−	0.529180i	$-0.177500\pi$
$557$	22308.5	1.69702	0.848510	+	0.529180i	$0.177500\pi$
$558$	0	0
$559$	3616.58	0.273641
$560$	0	0
$561$	3503.74	0.263686
$562$	0	0
$563$	−25664.1	−1.92116	−0.960580	−	0.278005i	$-0.910327\pi$
$563$	−25664.1	−1.92116	−0.960580	+	0.278005i	$0.910327\pi$
$564$	0	0
$565$	−11737.2	−0.873957
$566$	0	0
$567$	2277.38	0.168679
$568$	0	0
$569$	−7436.74	−0.547917	−0.273958	−	0.961742i	$-0.588333\pi$
$569$	−7436.74	−0.547917	−0.273958	+	0.961742i	$0.588333\pi$
$570$	0	0
$571$	1490.38	0.109230	0.0546151	−	0.998507i	$-0.482607\pi$
$571$	1490.38	0.109230	0.0546151	+	0.998507i	$0.482607\pi$
$572$	0	0
$573$	−4654.87	−0.339372
$574$	0	0
$575$	7517.69	0.545234
$576$	0	0
$577$	15069.7	1.08728	0.543640	−	0.839318i	$-0.317046\pi$
$577$	15069.7	1.08728	0.543640	+	0.839318i	$0.317046\pi$
$578$	0	0
$579$	12889.5	0.925164
$580$	0	0
$581$	−22128.4	−1.58010
$582$	0	0
$583$	−11154.5	−0.792406
$584$	0	0
$585$	926.851	0.0655053
$586$	0	0
$587$	19986.6	1.40534	0.702670	−	0.711516i	$-0.251991\pi$
$587$	19986.6	1.40534	0.702670	+	0.711516i	$0.251991\pi$
$588$	0	0
$589$	−1748.38	−0.122311
$590$	0	0
$591$	7647.49	0.532277
$592$	0	0
$593$	150.196	0.0104010	0.00520051	−	0.999986i	$-0.498345\pi$
$593$	150.196	0.0104010	0.00520051	+	0.999986i	$0.498345\pi$
$594$	0	0
$595$	7572.74	0.521768
$596$	0	0
$597$	7659.40	0.525089
$598$	0	0
$599$	−9108.92	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$599$	−9108.92	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$600$	0	0
$601$	−18079.8	−1.22710	−0.613551	−	0.789655i	$-0.710260\pi$
$601$	−18079.8	−1.22710	−0.613551	+	0.789655i	$0.710260\pi$
$602$	0	0
$603$	4412.83	0.298017
$604$	0	0
$605$	−1196.62	−0.0804124
$606$	0	0
$607$	16907.8	1.13058	0.565292	−	0.824891i	$-0.308763\pi$
$607$	16907.8	1.13058	0.565292	+	0.824891i	$0.308763\pi$
$608$	0	0
$609$	−16693.9	−1.11079
$610$	0	0
$611$	−4600.02	−0.304578
$612$	0	0
$613$	21824.3	1.43797	0.718984	−	0.695027i	$-0.244607\pi$
$613$	21824.3	1.43797	0.718984	+	0.695027i	$0.244607\pi$
$614$	0	0
$615$	−10430.5	−0.683903
$616$	0	0
$617$	−4521.73	−0.295038	−0.147519	−	0.989059i	$-0.547129\pi$
$617$	−4521.73	−0.295038	−0.147519	+	0.989059i	$0.547129\pi$
$618$	0	0
$619$	−22546.8	−1.46403	−0.732014	−	0.681289i	$-0.761420\pi$
$619$	−22546.8	−1.46403	−0.732014	+	0.681289i	$0.761420\pi$
$620$	0	0
$621$	3260.95	0.210720
$622$	0	0
$623$	−9210.72	−0.592327
$624$	0	0
$625$	−3969.94	−0.254076
$626$	0	0
$627$	7519.93	0.478974
$628$	0	0
$629$	13496.6	0.855557
$630$	0	0
$631$	−17477.4	−1.10264	−0.551319	−	0.834295i	$-0.685875\pi$
$631$	−17477.4	−1.10264	−0.551319	+	0.834295i	$0.685875\pi$
$632$	0	0
$633$	−4499.01	−0.282495
$634$	0	0
$635$	−7149.28	−0.446788
$636$	0	0
$637$	5817.45	0.361846
$638$	0	0
$639$	−6105.04	−0.377953
$640$	0	0
$641$	6669.57	0.410971	0.205485	−	0.978660i	$-0.434123\pi$
$641$	6669.57	0.410971	0.205485	+	0.978660i	$0.434123\pi$
$642$	0	0
$643$	9255.94	0.567681	0.283840	−	0.958872i	$-0.408391\pi$
$643$	9255.94	0.567681	0.283840	+	0.958872i	$0.408391\pi$
$644$	0	0
$645$	−6611.50	−0.403609
$646$	0	0
$647$	3783.74	0.229914	0.114957	−	0.993370i	$-0.463327\pi$
$647$	3783.74	0.229914	0.114957	+	0.993370i	$0.463327\pi$
$648$	0	0
$649$	−19636.7	−1.18768
$650$	0	0
$651$	2020.91	0.121668
$652$	0	0
$653$	25312.0	1.51690	0.758450	−	0.651731i	$-0.225957\pi$
$653$	25312.0	1.51690	0.758450	+	0.651731i	$0.225957\pi$
$654$	0	0
$655$	2408.80	0.143694
$656$	0	0
$657$	2996.26	0.177923
$658$	0	0
$659$	−3897.14	−0.230366	−0.115183	−	0.993344i	$-0.536745\pi$
$659$	−3897.14	−0.230366	−0.115183	+	0.993344i	$0.536745\pi$
$660$	0	0
$661$	−9657.09	−0.568256	−0.284128	−	0.958786i	$-0.591704\pi$
$661$	−9657.09	−0.568256	−0.284128	+	0.958786i	$0.591704\pi$
$662$	0	0
$663$	−1326.00	−0.0776736
$664$	0	0
$665$	16253.1	0.947770
$666$	0	0
$667$	−23903.8	−1.38764
$668$	0	0
$669$	−2192.32	−0.126696
$670$	0	0
$671$	19041.6	1.09552
$672$	0	0
$673$	14381.4	0.823719	0.411859	−	0.911247i	$-0.364879\pi$
$673$	14381.4	0.823719	0.411859	+	0.911247i	$0.364879\pi$
$674$	0	0
$675$	1680.61	0.0958324
$676$	0	0
$677$	16665.1	0.946075	0.473037	−	0.881042i	$-0.343158\pi$
$677$	16665.1	0.946075	0.473037	+	0.881042i	$0.343158\pi$
$678$	0	0
$679$	30566.8	1.72761
$680$	0	0
$681$	6858.87	0.385951
$682$	0	0
$683$	−3034.96	−0.170029	−0.0850144	−	0.996380i	$-0.527094\pi$
$683$	−3034.96	−0.170029	−0.0850144	+	0.996380i	$0.527094\pi$
$684$	0	0
$685$	8933.52	0.498295
$686$	0	0
$687$	−15623.9	−0.867667
$688$	0	0
$689$	4221.46	0.233418
$690$	0	0
$691$	20539.9	1.13079	0.565394	−	0.824821i	$-0.308724\pi$
$691$	20539.9	1.13079	0.565394	+	0.824821i	$0.308724\pi$
$692$	0	0
$693$	−8692.08	−0.476457
$694$	0	0
$695$	−14081.3	−0.768539
$696$	0	0
$697$	14922.5	0.810945
$698$	0	0
$699$	6614.07	0.357893
$700$	0	0
$701$	34803.0	1.87516	0.937582	−	0.347765i	$-0.113059\pi$
$701$	34803.0	1.87516	0.937582	+	0.347765i	$0.113059\pi$
$702$	0	0
$703$	28967.2	1.55408
$704$	0	0
$705$	8409.34	0.449240
$706$	0	0
$707$	−6467.05	−0.344015
$708$	0	0
$709$	−5040.88	−0.267016	−0.133508	−	0.991048i	$-0.542624\pi$
$709$	−5040.88	−0.267016	−0.133508	+	0.991048i	$0.542624\pi$
$710$	0	0
$711$	3077.26	0.162316
$712$	0	0
$713$	2893.71	0.151992
$714$	0	0
$715$	−3537.52	−0.185029
$716$	0	0
$717$	8522.60	0.443909
$718$	0	0
$719$	−2200.35	−0.114129	−0.0570647	−	0.998370i	$-0.518174\pi$
$719$	−2200.35	−0.114129	−0.0570647	+	0.998370i	$0.518174\pi$
$720$	0	0
$721$	−19237.9	−0.993697
$722$	0	0
$723$	15096.1	0.776531
$724$	0	0
$725$	−12319.5	−0.631080
$726$	0	0
$727$	−32277.7	−1.64665	−0.823325	−	0.567570i	$-0.807884\pi$
$727$	−32277.7	−1.64665	−0.823325	+	0.567570i	$0.807884\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	9458.75	0.478583
$732$	0	0
$733$	−13519.0	−0.681223	−0.340611	−	0.940204i	$-0.610634\pi$
$733$	−13519.0	−0.681223	−0.340611	+	0.940204i	$0.610634\pi$
$734$	0	0
$735$	−10634.9	−0.533708
$736$	0	0
$737$	−16842.5	−0.841791
$738$	0	0
$739$	31235.3	1.55482	0.777409	−	0.628996i	$-0.216534\pi$
$739$	31235.3	1.55482	0.777409	+	0.628996i	$0.216534\pi$
$740$	0	0
$741$	−2845.94	−0.141091
$742$	0	0
$743$	−18064.8	−0.891968	−0.445984	−	0.895041i	$-0.647146\pi$
$743$	−18064.8	−0.891968	−0.445984	+	0.895041i	$0.647146\pi$
$744$	0	0
$745$	5007.24	0.246243
$746$	0	0
$747$	−7083.40	−0.346945
$748$	0	0
$749$	−40733.6	−1.98715
$750$	0	0
$751$	22079.7	1.07284	0.536419	−	0.843952i	$-0.319777\pi$
$751$	22079.7	1.07284	0.536419	+	0.843952i	$0.319777\pi$
$752$	0	0
$753$	10905.3	0.527772
$754$	0	0
$755$	14259.2	0.687345
$756$	0	0
$757$	32266.6	1.54921	0.774604	−	0.632447i	$-0.217949\pi$
$757$	32266.6	1.54921	0.774604	+	0.632447i	$0.217949\pi$
$758$	0	0
$759$	−12446.1	−0.595209
$760$	0	0
$761$	22684.1	1.08055	0.540276	−	0.841488i	$-0.318320\pi$
$761$	22684.1	1.08055	0.540276	+	0.841488i	$0.318320\pi$
$762$	0	0
$763$	61133.8	2.90064
$764$	0	0
$765$	2424.07	0.114565
$766$	0	0
$767$	7431.56	0.349854
$768$	0	0
$769$	16945.7	0.794641	0.397321	−	0.917680i	$-0.369940\pi$
$769$	16945.7	0.794641	0.397321	+	0.917680i	$0.369940\pi$
$770$	0	0
$771$	−7843.71	−0.366387
$772$	0	0
$773$	3828.25	0.178128	0.0890639	−	0.996026i	$-0.471612\pi$
$773$	3828.25	0.178128	0.0890639	+	0.996026i	$0.471612\pi$
$774$	0	0
$775$	1491.35	0.0691238
$776$	0	0
$777$	−33482.4	−1.54591
$778$	0	0
$779$	32027.5	1.47305
$780$	0	0
$781$	23301.1	1.06758
$782$	0	0
$783$	−5343.81	−0.243898
$784$	0	0
$785$	−17706.4	−0.805055
$786$	0	0
$787$	−8939.55	−0.404905	−0.202453	−	0.979292i	$-0.564891\pi$
$787$	−8939.55	−0.404905	−0.202453	+	0.979292i	$0.564891\pi$
$788$	0	0
$789$	−16696.9	−0.753389
$790$	0	0
$791$	−41657.1	−1.87251
$792$	0	0
$793$	−7206.37	−0.322706
$794$	0	0
$795$	−7717.29	−0.344282
$796$	0	0
$797$	−43192.3	−1.91964	−0.959818	−	0.280624i	$-0.909459\pi$
$797$	−43192.3	−1.91964	−0.959818	+	0.280624i	$0.909459\pi$
$798$	0	0
$799$	−12030.8	−0.532691
$800$	0	0
$801$	−2948.40	−0.130058
$802$	0	0
$803$	−11435.8	−0.502568
$804$	0	0
$805$	−26900.1	−1.17777
$806$	0	0
$807$	−10450.0	−0.455833
$808$	0	0
$809$	4789.30	0.208137	0.104068	−	0.994570i	$-0.466814\pi$
$809$	4789.30	0.208137	0.104068	+	0.994570i	$0.466814\pi$
$810$	0	0
$811$	−1411.14	−0.0610996	−0.0305498	−	0.999533i	$-0.509726\pi$
$811$	−1411.14	−0.0610996	−0.0305498	+	0.999533i	$0.509726\pi$
$812$	0	0
$813$	13814.2	0.595925
$814$	0	0
$815$	26229.3	1.12733
$816$	0	0
$817$	20300.9	0.869326
$818$	0	0
$819$	3289.54	0.140349
$820$	0	0
$821$	−31013.0	−1.31835	−0.659173	−	0.751992i	$-0.729093\pi$
$821$	−31013.0	−1.31835	−0.659173	+	0.751992i	$0.729093\pi$
$822$	0	0
$823$	−674.392	−0.0285636	−0.0142818	−	0.999898i	$-0.504546\pi$
$823$	−674.392	−0.0285636	−0.0142818	+	0.999898i	$0.504546\pi$
$824$	0	0
$825$	−6414.41	−0.270692
$826$	0	0
$827$	4857.93	0.204264	0.102132	−	0.994771i	$-0.467434\pi$
$827$	4857.93	0.204264	0.102132	+	0.994771i	$0.467434\pi$
$828$	0	0
$829$	9122.13	0.382177	0.191089	−	0.981573i	$-0.438798\pi$
$829$	9122.13	0.382177	0.191089	+	0.981573i	$0.438798\pi$
$830$	0	0
$831$	−2659.61	−0.111024
$832$	0	0
$833$	15214.9	0.632850
$834$	0	0
$835$	−6029.77	−0.249903
$836$	0	0
$837$	646.903	0.0267147
$838$	0	0
$839$	−28731.6	−1.18227	−0.591135	−	0.806573i	$-0.701320\pi$
$839$	−28731.6	−1.18227	−0.591135	+	0.806573i	$0.701320\pi$
$840$	0	0
$841$	14782.8	0.606127
$842$	0	0
$843$	20694.3	0.845493
$844$	0	0
$845$	1338.79	0.0545037
$846$	0	0
$847$	−4246.99	−0.172289
$848$	0	0
$849$	−27628.5	−1.11685
$850$	0	0
$851$	−47943.1	−1.93122
$852$	0	0
$853$	−27008.2	−1.08411	−0.542054	−	0.840344i	$-0.682353\pi$
$853$	−27008.2	−1.08411	−0.542054	+	0.840344i	$0.682353\pi$
$854$	0	0
$855$	5202.69	0.208103
$856$	0	0
$857$	−24046.5	−0.958474	−0.479237	−	0.877686i	$-0.659086\pi$
$857$	−24046.5	−0.958474	−0.479237	+	0.877686i	$0.659086\pi$
$858$	0	0
$859$	743.111	0.0295164	0.0147582	−	0.999891i	$-0.495302\pi$
$859$	743.111	0.0295164	0.0147582	+	0.999891i	$0.495302\pi$
$860$	0	0
$861$	−37019.7	−1.46531
$862$	0	0
$863$	13569.1	0.535223	0.267611	−	0.963527i	$-0.413766\pi$
$863$	13569.1	0.535223	0.267611	+	0.963527i	$0.413766\pi$
$864$	0	0
$865$	−22005.1	−0.864965
$866$	0	0
$867$	11271.0	0.441503
$868$	0	0
$869$	−11745.0	−0.458484
$870$	0	0
$871$	6374.09	0.247965
$872$	0	0
$873$	9784.60	0.379334
$874$	0	0
$875$	−41704.6	−1.61128
$876$	0	0
$877$	5475.90	0.210841	0.105421	−	0.994428i	$-0.466381\pi$
$877$	5475.90	0.210841	0.105421	+	0.994428i	$0.466381\pi$
$878$	0	0
$879$	11984.8	0.459884
$880$	0	0
$881$	43934.9	1.68014	0.840070	−	0.542478i	$-0.182514\pi$
$881$	43934.9	1.68014	0.840070	+	0.542478i	$0.182514\pi$
$882$	0	0
$883$	29202.2	1.11295	0.556474	−	0.830865i	$-0.312154\pi$
$883$	29202.2	1.11295	0.556474	+	0.830865i	$0.312154\pi$
$884$	0	0
$885$	−13585.7	−0.516021
$886$	0	0
$887$	15696.9	0.594196	0.297098	−	0.954847i	$-0.403981\pi$
$887$	15696.9	0.594196	0.297098	+	0.954847i	$0.403981\pi$
$888$	0	0
$889$	−25373.9	−0.957272
$890$	0	0
$891$	−2782.38	−0.104616
$892$	0	0
$893$	−25821.3	−0.967610
$894$	0	0
$895$	−3689.50	−0.137795
$896$	0	0
$897$	4710.26	0.175330
$898$	0	0
$899$	−4742.01	−0.175923
$900$	0	0
$901$	11040.7	0.408236
$902$	0	0
$903$	−23465.3	−0.864757
$904$	0	0
$905$	8550.08	0.314049
$906$	0	0
$907$	−3960.77	−0.145000	−0.0725001	−	0.997368i	$-0.523098\pi$
$907$	−3960.77	−0.145000	−0.0725001	+	0.997368i	$0.523098\pi$
$908$	0	0
$909$	−2070.14	−0.0755359
$910$	0	0
$911$	−1585.33	−0.0576558	−0.0288279	−	0.999584i	$-0.509177\pi$
$911$	−1585.33	−0.0576558	−0.0288279	+	0.999584i	$0.509177\pi$
$912$	0	0
$913$	27035.3	0.979996
$914$	0	0
$915$	13174.0	0.475978
$916$	0	0
$917$	8549.21	0.307873
$918$	0	0
$919$	35033.6	1.25751	0.628754	−	0.777604i	$-0.283565\pi$
$919$	35033.6	1.25751	0.628754	+	0.777604i	$0.283565\pi$
$920$	0	0
$921$	−2210.15	−0.0790736
$922$	0	0
$923$	−8818.40	−0.314476
$924$	0	0
$925$	−24708.7	−0.878289
$926$	0	0
$927$	−6158.14	−0.218188
$928$	0	0
$929$	−10334.1	−0.364964	−0.182482	−	0.983209i	$-0.558413\pi$
$929$	−10334.1	−0.364964	−0.182482	+	0.983209i	$0.558413\pi$
$930$	0	0
$931$	32655.1	1.14954
$932$	0	0
$933$	−8957.51	−0.314315
$934$	0	0
$935$	−9251.97	−0.323606
$936$	0	0
$937$	−28850.8	−1.00589	−0.502943	−	0.864320i	$-0.667749\pi$
$937$	−28850.8	−1.00589	−0.502943	+	0.864320i	$0.667749\pi$
$938$	0	0
$939$	−139.101	−0.00483429
$940$	0	0
$941$	30834.4	1.06820	0.534099	−	0.845422i	$-0.320651\pi$
$941$	30834.4	1.06820	0.534099	+	0.845422i	$0.320651\pi$
$942$	0	0
$943$	−53008.0	−1.83052
$944$	0	0
$945$	−6013.65	−0.207009
$946$	0	0
$947$	−48971.8	−1.68043	−0.840216	−	0.542251i	$-0.817572\pi$
$947$	−48971.8	−1.68043	−0.840216	+	0.542251i	$0.817572\pi$
$948$	0	0
$949$	4327.93	0.148041
$950$	0	0
$951$	5752.35	0.196144
$952$	0	0
$953$	−3863.47	−0.131322	−0.0656611	−	0.997842i	$-0.520916\pi$
$953$	−3863.47	−0.131322	−0.0656611	+	0.997842i	$0.520916\pi$
$954$	0	0
$955$	12291.7	0.416491
$956$	0	0
$957$	20395.7	0.688924
$958$	0	0
$959$	31706.5	1.06763
$960$	0	0
$961$	−29216.9	−0.980731
$962$	0	0
$963$	−13039.0	−0.436321
$964$	0	0
$965$	−34036.1	−1.13540
$966$	0	0
$967$	−41074.1	−1.36593	−0.682965	−	0.730451i	$-0.739310\pi$
$967$	−41074.1	−1.36593	−0.682965	+	0.730451i	$0.739310\pi$
$968$	0	0
$969$	−7443.23	−0.246761
$970$	0	0
$971$	−34985.3	−1.15626	−0.578132	−	0.815944i	$-0.696218\pi$
$971$	−34985.3	−1.15626	−0.578132	+	0.815944i	$0.696218\pi$
$972$	0	0
$973$	−49976.9	−1.64664
$974$	0	0
$975$	2427.55	0.0797374
$976$	0	0
$977$	33861.8	1.10884	0.554420	−	0.832237i	$-0.312940\pi$
$977$	33861.8	1.10884	0.554420	+	0.832237i	$0.312940\pi$
$978$	0	0
$979$	11253.2	0.367368
$980$	0	0
$981$	19569.2	0.636899
$982$	0	0
$983$	39395.2	1.27824	0.639121	−	0.769106i	$-0.279298\pi$
$983$	39395.2	1.27824	0.639121	+	0.769106i	$0.279298\pi$
$984$	0	0
$985$	−20194.0	−0.653232
$986$	0	0
$987$	29846.1	0.962524
$988$	0	0
$989$	−33599.7	−1.08029
$990$	0	0
$991$	25286.2	0.810539	0.405269	−	0.914197i	$-0.367178\pi$
$991$	25286.2	0.810539	0.405269	+	0.914197i	$0.367178\pi$
$992$	0	0
$993$	19226.4	0.614432
$994$	0	0
$995$	−20225.4	−0.644411
$996$	0	0
$997$	−34643.0	−1.10046	−0.550228	−	0.835014i	$-0.685459\pi$
$997$	−34643.0	−1.10046	−0.550228	+	0.835014i	$0.685459\pi$
$998$	0	0
$999$	−10717.9	−0.339439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.g.1.2	✓	3
3.2	odd	2		936.4.a.k.1.2		3
4.3	odd	2		624.4.a.u.1.2		3
8.3	odd	2		2496.4.a.bk.1.2		3
8.5	even	2		2496.4.a.bo.1.2		3
12.11	even	2		1872.4.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.g.1.2	✓	3	1.1	even	1	trivial
624.4.a.u.1.2		3	4.3	odd	2
936.4.a.k.1.2		3	3.2	odd	2
1872.4.a.bj.1.2		3	12.11	even	2
2496.4.a.bk.1.2		3	8.3	odd	2
2496.4.a.bo.1.2		3	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +7.92181 q^{5} +28.1158 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +7.92181 q^{5} +28.1158 q^{7} +9.00000 q^{9} -34.3503 q^{11} +13.0000 q^{13} -23.7654 q^{15} +34.0000 q^{17} +72.9728 q^{19} -84.3473 q^{21} -120.776 q^{23} -62.2450 q^{25} -27.0000 q^{27} +197.919 q^{29} -23.9594 q^{31} +103.051 q^{33} +222.728 q^{35} +396.959 q^{37} -39.0000 q^{39} +438.896 q^{41} +278.198 q^{43} +71.2963 q^{45} -353.848 q^{47} +447.496 q^{49} -102.000 q^{51} +324.728 q^{53} -272.117 q^{55} -218.919 q^{57} +571.658 q^{59} -554.336 q^{61} +253.042 q^{63} +102.983 q^{65} +490.314 q^{67} +362.327 q^{69} -678.338 q^{71} +332.918 q^{73} +186.735 q^{75} -965.786 q^{77} +341.918 q^{79} +81.0000 q^{81} -787.045 q^{83} +269.341 q^{85} -593.756 q^{87} -327.600 q^{89} +365.505 q^{91} +71.8781 q^{93} +578.077 q^{95} +1087.18 q^{97} -309.153 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 16 q^{5} - 22 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 + 16 * q^5 - 22 * q^7 + 27 * q^9	\( 3 q - 9 q^{3} + 16 q^{5} - 22 q^{7} + 27 q^{9} - 20 q^{11} + 39 q^{13} - 48 q^{15} + 102 q^{17} - 38 q^{19} + 66 q^{21} + 32 q^{23} + 161 q^{25} - 81 q^{27} + 350 q^{29} + 50 q^{31} + 60 q^{33} + 232 q^{35} + 542 q^{37} - 117 q^{39} + 500 q^{41} + 420 q^{43} + 144 q^{45} + 324 q^{47} + 1119 q^{49} - 306 q^{51} + 538 q^{53} + 896 q^{55} + 114 q^{57} + 112 q^{59} + 1206 q^{61} - 198 q^{63} + 208 q^{65} + 950 q^{67} - 96 q^{69} - 1008 q^{71} + 1834 q^{73} - 483 q^{75} - 792 q^{77} - 272 q^{79} + 243 q^{81} - 1640 q^{83} + 544 q^{85} - 1050 q^{87} - 1576 q^{89} - 286 q^{91} - 150 q^{93} - 2760 q^{95} + 402 q^{97} - 180 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 + 16 * q^5 - 22 * q^7 + 27 * q^9 - 20 * q^11 + 39 * q^13 - 48 * q^15 + 102 * q^17 - 38 * q^19 + 66 * q^21 + 32 * q^23 + 161 * q^25 - 81 * q^27 + 350 * q^29 + 50 * q^31 + 60 * q^33 + 232 * q^35 + 542 * q^37 - 117 * q^39 + 500 * q^41 + 420 * q^43 + 144 * q^45 + 324 * q^47 + 1119 * q^49 - 306 * q^51 + 538 * q^53 + 896 * q^55 + 114 * q^57 + 112 * q^59 + 1206 * q^61 - 198 * q^63 + 208 * q^65 + 950 * q^67 - 96 * q^69 - 1008 * q^71 + 1834 * q^73 - 483 * q^75 - 792 * q^77 - 272 * q^79 + 243 * q^81 - 1640 * q^83 + 544 * q^85 - 1050 * q^87 - 1576 * q^89 - 286 * q^91 - 150 * q^93 - 2760 * q^95 + 402 * q^97 - 180 * q^99

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