LMFDB - Embedded newform 1232.4.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1232,4,Mod(1,1232)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1232.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.6903531271$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{3} +6.48528 q^{5} -7.00000 q^{7} -23.0000 q^{9} +11.0000 q^{11} -45.6569 q^{13} +12.9706 q^{15} -63.6812 q^{17} +39.5147 q^{19} -14.0000 q^{21} +78.9117 q^{23} -82.9411 q^{25} -100.000 q^{27} +256.558 q^{29} +170.818 q^{31} +22.0000 q^{33} -45.3970 q^{35} -223.941 q^{37} -91.3137 q^{39} +307.730 q^{41} +316.000 q^{43} -149.161 q^{45} +576.357 q^{47} +49.0000 q^{49} -127.362 q^{51} +173.951 q^{53} +71.3381 q^{55} +79.0294 q^{57} +82.6375 q^{59} -63.1472 q^{61} +161.000 q^{63} -296.098 q^{65} +349.726 q^{67} +157.823 q^{69} +119.050 q^{71} +573.318 q^{73} -165.882 q^{75} -77.0000 q^{77} -568.362 q^{79} +421.000 q^{81} -510.447 q^{83} -412.991 q^{85} +513.117 q^{87} +1090.10 q^{89} +319.598 q^{91} +341.637 q^{93} +256.264 q^{95} +396.530 q^{97} -253.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 4 q^{5} - 14 q^{7} - 46 q^{9} + 22 q^{11} - 80 q^{13} - 8 q^{15} + 48 q^{17} + 96 q^{19} - 28 q^{21} + 56 q^{23} - 98 q^{25} - 200 q^{27} + 4 q^{29} - 60 q^{31} + 44 q^{33} + 28 q^{35} - 380 q^{37}+ \cdots - 506 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 4 * q^5 - 14 * q^7 - 46 * q^9 + 22 * q^11 - 80 * q^13 - 8 * q^15 + 48 * q^17 + 96 * q^19 - 28 * q^21 + 56 * q^23 - 98 * q^25 - 200 * q^27 + 4 * q^29 - 60 * q^31 + 44 * q^33 + 28 * q^35 - 380 * q^37 - 160 * q^39 + 112 * q^41 + 632 * q^43 + 92 * q^45 + 604 * q^47 + 98 * q^49 + 96 * q^51 + 676 * q^53 - 44 * q^55 + 192 * q^57 + 516 * q^59 - 296 * q^61 + 322 * q^63 + 64 * q^65 + 1152 * q^67 + 112 * q^69 + 1064 * q^71 + 168 * q^73 - 196 * q^75 - 154 * q^77 + 368 * q^79 + 842 * q^81 - 1728 * q^83 - 1584 * q^85 + 8 * q^87 + 1524 * q^89 + 560 * q^91 - 120 * q^93 - 336 * q^95 + 1404 * q^97 - 506 * q^99

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
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	0	0
$5$	6.48528	0.580061	0.290031	−	0.957017i	$-0.406335\pi$
$5$	6.48528	0.580061	0.290031	+	0.957017i	$0.406335\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−23.0000	−0.851852
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−45.6569	−0.974072	−0.487036	−	0.873382i	$-0.661922\pi$
$13$	−45.6569	−0.974072	−0.487036	+	0.873382i	$0.661922\pi$
$14$	0	0
$15$	12.9706	0.223266
$16$	0	0
$17$	−63.6812	−0.908528	−0.454264	−	0.890867i	$-0.650098\pi$
$17$	−63.6812	−0.908528	−0.454264	+	0.890867i	$0.650098\pi$
$18$	0	0
$19$	39.5147	0.477121	0.238560	−	0.971128i	$-0.423324\pi$
$19$	39.5147	0.477121	0.238560	+	0.971128i	$0.423324\pi$
$20$	0	0
$21$	−14.0000	−0.145479
$22$	0	0
$23$	78.9117	0.715401	0.357701	−	0.933836i	$-0.383561\pi$
$23$	78.9117	0.715401	0.357701	+	0.933836i	$0.383561\pi$
$24$	0	0
$25$	−82.9411	−0.663529
$26$	0	0
$27$	−100.000	−0.712778
$28$	0	0
$29$	256.558	1.64282	0.821409	−	0.570340i	$-0.193189\pi$
$29$	256.558	1.64282	0.821409	+	0.570340i	$0.193189\pi$
$30$	0	0
$31$	170.818	0.989673	0.494837	−	0.868986i	$-0.335228\pi$
$31$	170.818	0.989673	0.494837	+	0.868986i	$0.335228\pi$
$32$	0	0
$33$	22.0000	0.116052
$34$	0	0
$35$	−45.3970	−0.219243
$36$	0	0
$37$	−223.941	−0.995019	−0.497509	−	0.867459i	$-0.665752\pi$
$37$	−223.941	−0.995019	−0.497509	+	0.867459i	$0.665752\pi$
$38$	0	0
$39$	−91.3137	−0.374920
$40$	0	0
$41$	307.730	1.17218	0.586090	−	0.810246i	$-0.300667\pi$
$41$	307.730	1.17218	0.586090	+	0.810246i	$0.300667\pi$
$42$	0	0
$43$	316.000	1.12069	0.560344	−	0.828260i	$-0.310669\pi$
$43$	316.000	1.12069	0.560344	+	0.828260i	$0.310669\pi$
$44$	0	0
$45$	−149.161	−0.494126
$46$	0	0
$47$	576.357	1.78873	0.894366	−	0.447337i	$-0.147627\pi$
$47$	576.357	1.78873	0.894366	+	0.447337i	$0.147627\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−127.362	−0.349692
$52$	0	0
$53$	173.951	0.450831	0.225415	−	0.974263i	$-0.427626\pi$
$53$	173.951	0.450831	0.225415	+	0.974263i	$0.427626\pi$
$54$	0	0
$55$	71.3381	0.174895
$56$	0	0
$57$	79.0294	0.183644
$58$	0	0
$59$	82.6375	0.182347	0.0911736	−	0.995835i	$-0.470938\pi$
$59$	82.6375	0.182347	0.0911736	+	0.995835i	$0.470938\pi$
$60$	0	0
$61$	−63.1472	−0.132544	−0.0662719	−	0.997802i	$-0.521110\pi$
$61$	−63.1472	−0.132544	−0.0662719	+	0.997802i	$0.521110\pi$
$62$	0	0
$63$	161.000	0.321970
$64$	0	0
$65$	−296.098	−0.565021
$66$	0	0
$67$	349.726	0.637699	0.318849	−	0.947805i	$-0.396704\pi$
$67$	349.726	0.637699	0.318849	+	0.947805i	$0.396704\pi$
$68$	0	0
$69$	157.823	0.275358
$70$	0	0
$71$	119.050	0.198994	0.0994971	−	0.995038i	$-0.468277\pi$
$71$	119.050	0.198994	0.0994971	+	0.995038i	$0.468277\pi$
$72$	0	0
$73$	573.318	0.919203	0.459601	−	0.888125i	$-0.347992\pi$
$73$	573.318	0.919203	0.459601	+	0.888125i	$0.347992\pi$
$74$	0	0
$75$	−165.882	−0.255392
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−568.362	−0.809439	−0.404719	−	0.914441i	$-0.632631\pi$
$79$	−568.362	−0.809439	−0.404719	+	0.914441i	$0.632631\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	−510.447	−0.675046	−0.337523	−	0.941317i	$-0.609589\pi$
$83$	−510.447	−0.675046	−0.337523	+	0.941317i	$0.609589\pi$
$84$	0	0
$85$	−412.991	−0.527002
$86$	0	0
$87$	513.117	0.632321
$88$	0	0
$89$	1090.10	1.29832	0.649158	−	0.760654i	$-0.275121\pi$
$89$	1090.10	1.29832	0.649158	+	0.760654i	$0.275121\pi$
$90$	0	0
$91$	319.598	0.368165
$92$	0	0
$93$	341.637	0.380925
$94$	0	0
$95$	256.264	0.276759
$96$	0	0
$97$	396.530	0.415067	0.207534	−	0.978228i	$-0.433456\pi$
$97$	396.530	0.415067	0.207534	+	0.978228i	$0.433456\pi$
$98$	0	0
$99$	−253.000	−0.256843
$100$	0	0
$101$	825.844	0.813609	0.406804	−	0.913515i	$-0.366643\pi$
$101$	825.844	0.813609	0.406804	+	0.913515i	$0.366643\pi$
$102$	0	0
$103$	−1246.19	−1.19214	−0.596071	−	0.802932i	$-0.703272\pi$
$103$	−1246.19	−1.19214	−0.596071	+	0.802932i	$0.703272\pi$
$104$	0	0
$105$	−90.7939	−0.0843865
$106$	0	0
$107$	1368.24	1.23620	0.618099	−	0.786100i	$-0.287903\pi$
$107$	1368.24	1.23620	0.618099	+	0.786100i	$0.287903\pi$
$108$	0	0
$109$	−1312.80	−1.15361	−0.576806	−	0.816881i	$-0.695701\pi$
$109$	−1312.80	−1.15361	−0.576806	+	0.816881i	$0.695701\pi$
$110$	0	0
$111$	−447.882	−0.382983
$112$	0	0
$113$	−312.607	−0.260244	−0.130122	−	0.991498i	$-0.541537\pi$
$113$	−312.607	−0.260244	−0.130122	+	0.991498i	$0.541537\pi$
$114$	0	0
$115$	511.765	0.414976
$116$	0	0
$117$	1050.11	0.829765
$118$	0	0
$119$	445.769	0.343391
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	615.460	0.451172
$124$	0	0
$125$	−1348.56	−0.964949
$126$	0	0
$127$	1592.65	1.11280	0.556398	−	0.830916i	$-0.312183\pi$
$127$	1592.65	1.11280	0.556398	+	0.830916i	$0.312183\pi$
$128$	0	0
$129$	632.000	0.431353
$130$	0	0
$131$	−1291.43	−0.861321	−0.430661	−	0.902514i	$-0.641719\pi$
$131$	−1291.43	−0.861321	−0.430661	+	0.902514i	$0.641719\pi$
$132$	0	0
$133$	−276.603	−0.180335
$134$	0	0
$135$	−648.528	−0.413455
$136$	0	0
$137$	−2107.76	−1.31444	−0.657220	−	0.753699i	$-0.728268\pi$
$137$	−2107.76	−1.31444	−0.657220	+	0.753699i	$0.728268\pi$
$138$	0	0
$139$	−2318.30	−1.41464	−0.707321	−	0.706892i	$-0.750097\pi$
$139$	−2318.30	−1.41464	−0.707321	+	0.706892i	$0.750097\pi$
$140$	0	0
$141$	1152.71	0.688483
$142$	0	0
$143$	−502.225	−0.293694
$144$	0	0
$145$	1663.85	0.952935
$146$	0	0
$147$	98.0000	0.0549857
$148$	0	0
$149$	602.061	0.331025	0.165513	−	0.986208i	$-0.447072\pi$
$149$	602.061	0.331025	0.165513	+	0.986208i	$0.447072\pi$
$150$	0	0
$151$	1928.20	1.03917	0.519584	−	0.854420i	$-0.326087\pi$
$151$	1928.20	1.03917	0.519584	+	0.854420i	$0.326087\pi$
$152$	0	0
$153$	1464.67	0.773931
$154$	0	0
$155$	1107.80	0.574071
$156$	0	0
$157$	2476.45	1.25887	0.629434	−	0.777054i	$-0.283287\pi$
$157$	2476.45	1.25887	0.629434	+	0.777054i	$0.283287\pi$
$158$	0	0
$159$	347.902	0.173525
$160$	0	0
$161$	−552.382	−0.270396
$162$	0	0
$163$	1063.06	0.510829	0.255415	−	0.966832i	$-0.417788\pi$
$163$	1063.06	0.510829	0.255415	+	0.966832i	$0.417788\pi$
$164$	0	0
$165$	142.676	0.0673171
$166$	0	0
$167$	3720.65	1.72403	0.862015	−	0.506883i	$-0.169203\pi$
$167$	3720.65	1.72403	0.862015	+	0.506883i	$0.169203\pi$
$168$	0	0
$169$	−112.452	−0.0511842
$170$	0	0
$171$	−908.839	−0.406436
$172$	0	0
$173$	−909.421	−0.399665	−0.199832	−	0.979830i	$-0.564040\pi$
$173$	−909.421	−0.399665	−0.199832	+	0.979830i	$0.564040\pi$
$174$	0	0
$175$	580.588	0.250790
$176$	0	0
$177$	165.275	0.0701855
$178$	0	0
$179$	−1349.80	−0.563626	−0.281813	−	0.959469i	$-0.590936\pi$
$179$	−1349.80	−0.563626	−0.281813	+	0.959469i	$0.590936\pi$
$180$	0	0
$181$	−4559.47	−1.87239	−0.936196	−	0.351479i	$-0.885679\pi$
$181$	−4559.47	−1.87239	−0.936196	+	0.351479i	$0.885679\pi$
$182$	0	0
$183$	−126.294	−0.0510161
$184$	0	0
$185$	−1452.32	−0.577172
$186$	0	0
$187$	−700.494	−0.273931
$188$	0	0
$189$	700.000	0.269405
$190$	0	0
$191$	2238.60	0.848061	0.424031	−	0.905648i	$-0.360615\pi$
$191$	2238.60	0.848061	0.424031	+	0.905648i	$0.360615\pi$
$192$	0	0
$193$	2966.36	1.10634	0.553169	−	0.833069i	$-0.313418\pi$
$193$	2966.36	1.10634	0.553169	+	0.833069i	$0.313418\pi$
$194$	0	0
$195$	−592.195	−0.217477
$196$	0	0
$197$	3982.18	1.44020	0.720099	−	0.693872i	$-0.244097\pi$
$197$	3982.18	1.44020	0.720099	+	0.693872i	$0.244097\pi$
$198$	0	0
$199$	−1895.03	−0.675052	−0.337526	−	0.941316i	$-0.609590\pi$
$199$	−1895.03	−0.675052	−0.337526	+	0.941316i	$0.609590\pi$
$200$	0	0
$201$	699.452	0.245450
$202$	0	0
$203$	−1795.91	−0.620927
$204$	0	0
$205$	1995.72	0.679936
$206$	0	0
$207$	−1814.97	−0.609416
$208$	0	0
$209$	434.662	0.143857
$210$	0	0
$211$	1506.79	0.491621	0.245810	−	0.969318i	$-0.420946\pi$
$211$	1506.79	0.491621	0.245810	+	0.969318i	$0.420946\pi$
$212$	0	0
$213$	238.099	0.0765929
$214$	0	0
$215$	2049.35	0.650067
$216$	0	0
$217$	−1195.73	−0.374061
$218$	0	0
$219$	1146.64	0.353801
$220$	0	0
$221$	2907.49	0.884971
$222$	0	0
$223$	−4289.51	−1.28810	−0.644051	−	0.764983i	$-0.722748\pi$
$223$	−4289.51	−1.28810	−0.644051	+	0.764983i	$0.722748\pi$
$224$	0	0
$225$	1907.65	0.565228
$226$	0	0
$227$	2848.72	0.832934	0.416467	−	0.909151i	$-0.363268\pi$
$227$	2848.72	0.832934	0.416467	+	0.909151i	$0.363268\pi$
$228$	0	0
$229$	3273.70	0.944681	0.472341	−	0.881416i	$-0.343409\pi$
$229$	3273.70	0.944681	0.472341	+	0.881416i	$0.343409\pi$
$230$	0	0
$231$	−154.000	−0.0438634
$232$	0	0
$233$	4773.81	1.34224	0.671122	−	0.741347i	$-0.265813\pi$
$233$	4773.81	1.34224	0.671122	+	0.741347i	$0.265813\pi$
$234$	0	0
$235$	3737.84	1.03757
$236$	0	0
$237$	−1136.72	−0.311553
$238$	0	0
$239$	−1976.50	−0.534934	−0.267467	−	0.963567i	$-0.586187\pi$
$239$	−1976.50	−0.534934	−0.267467	+	0.963567i	$0.586187\pi$
$240$	0	0
$241$	−2236.73	−0.597845	−0.298923	−	0.954277i	$-0.596627\pi$
$241$	−2236.73	−0.597845	−0.298923	+	0.954277i	$0.596627\pi$
$242$	0	0
$243$	3542.00	0.935059
$244$	0	0
$245$	317.779	0.0828659
$246$	0	0
$247$	−1804.12	−0.464750
$248$	0	0
$249$	−1020.89	−0.259825
$250$	0	0
$251$	6364.56	1.60051	0.800254	−	0.599662i	$-0.204698\pi$
$251$	6364.56	1.60051	0.800254	+	0.599662i	$0.204698\pi$
$252$	0	0
$253$	868.029	0.215702
$254$	0	0
$255$	−825.982	−0.202843
$256$	0	0
$257$	7947.46	1.92898	0.964492	−	0.264110i	$-0.0850783\pi$
$257$	7947.46	1.92898	0.964492	+	0.264110i	$0.0850783\pi$
$258$	0	0
$259$	1567.59	0.376082
$260$	0	0
$261$	−5900.84	−1.39944
$262$	0	0
$263$	7330.82	1.71877	0.859387	−	0.511325i	$-0.170845\pi$
$263$	7330.82	1.71877	0.859387	+	0.511325i	$0.170845\pi$
$264$	0	0
$265$	1128.12	0.261510
$266$	0	0
$267$	2180.20	0.499722
$268$	0	0
$269$	−3504.02	−0.794216	−0.397108	−	0.917772i	$-0.629986\pi$
$269$	−3504.02	−0.794216	−0.397108	+	0.917772i	$0.629986\pi$
$270$	0	0
$271$	−36.4996	−0.00818152	−0.00409076	−	0.999992i	$-0.501302\pi$
$271$	−36.4996	−0.00818152	−0.00409076	+	0.999992i	$0.501302\pi$
$272$	0	0
$273$	639.196	0.141707
$274$	0	0
$275$	−912.352	−0.200062
$276$	0	0
$277$	−4965.14	−1.07699	−0.538495	−	0.842629i	$-0.681007\pi$
$277$	−4965.14	−1.07699	−0.538495	+	0.842629i	$0.681007\pi$
$278$	0	0
$279$	−3928.82	−0.843055
$280$	0	0
$281$	−4560.42	−0.968156	−0.484078	−	0.875025i	$-0.660845\pi$
$281$	−4560.42	−0.968156	−0.484078	+	0.875025i	$0.660845\pi$
$282$	0	0
$283$	917.744	0.192771	0.0963855	−	0.995344i	$-0.469272\pi$
$283$	917.744	0.192771	0.0963855	+	0.995344i	$0.469272\pi$
$284$	0	0
$285$	512.528	0.106525
$286$	0	0
$287$	−2154.11	−0.443042
$288$	0	0
$289$	−857.700	−0.174578
$290$	0	0
$291$	793.060	0.159759
$292$	0	0
$293$	−3983.97	−0.794355	−0.397177	−	0.917742i	$-0.630010\pi$
$293$	−3983.97	−0.794355	−0.397177	+	0.917742i	$0.630010\pi$
$294$	0	0
$295$	535.928	0.105773
$296$	0	0
$297$	−1100.00	−0.214911
$298$	0	0
$299$	−3602.86	−0.696852
$300$	0	0
$301$	−2212.00	−0.423580
$302$	0	0
$303$	1651.69	0.313158
$304$	0	0
$305$	−409.527	−0.0768835
$306$	0	0
$307$	7992.52	1.48585	0.742927	−	0.669372i	$-0.233437\pi$
$307$	7992.52	1.48585	0.742927	+	0.669372i	$0.233437\pi$
$308$	0	0
$309$	−2492.38	−0.458856
$310$	0	0
$311$	−7217.65	−1.31600	−0.657999	−	0.753019i	$-0.728597\pi$
$311$	−7217.65	−1.31600	−0.657999	+	0.753019i	$0.728597\pi$
$312$	0	0
$313$	−8415.62	−1.51974	−0.759871	−	0.650074i	$-0.774738\pi$
$313$	−8415.62	−1.51974	−0.759871	+	0.650074i	$0.774738\pi$
$314$	0	0
$315$	1044.13	0.186762
$316$	0	0
$317$	632.365	0.112041	0.0560207	−	0.998430i	$-0.482159\pi$
$317$	632.365	0.112041	0.0560207	+	0.998430i	$0.482159\pi$
$318$	0	0
$319$	2822.14	0.495328
$320$	0	0
$321$	2736.49	0.475813
$322$	0	0
$323$	−2516.35	−0.433478
$324$	0	0
$325$	3786.83	0.646325
$326$	0	0
$327$	−2625.60	−0.444025
$328$	0	0
$329$	−4034.50	−0.676077
$330$	0	0
$331$	−425.759	−0.0707004	−0.0353502	−	0.999375i	$-0.511255\pi$
$331$	−425.759	−0.0707004	−0.0353502	+	0.999375i	$0.511255\pi$
$332$	0	0
$333$	5150.65	0.847609
$334$	0	0
$335$	2268.07	0.369904
$336$	0	0
$337$	1369.56	0.221379	0.110689	−	0.993855i	$-0.464694\pi$
$337$	1369.56	0.221379	0.110689	+	0.993855i	$0.464694\pi$
$338$	0	0
$339$	−625.214	−0.100168
$340$	0	0
$341$	1879.00	0.298398
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	1023.53	0.159724
$346$	0	0
$347$	5531.76	0.855794	0.427897	−	0.903827i	$-0.359255\pi$
$347$	5531.76	0.855794	0.427897	+	0.903827i	$0.359255\pi$
$348$	0	0
$349$	10171.7	1.56011	0.780055	−	0.625711i	$-0.215191\pi$
$349$	10171.7	1.56011	0.780055	+	0.625711i	$0.215191\pi$
$350$	0	0
$351$	4565.69	0.694297
$352$	0	0
$353$	9626.38	1.45145	0.725723	−	0.687987i	$-0.241505\pi$
$353$	9626.38	1.45145	0.725723	+	0.687987i	$0.241505\pi$
$354$	0	0
$355$	772.070	0.115429
$356$	0	0
$357$	891.537	0.132171
$358$	0	0
$359$	−8491.00	−1.24829	−0.624147	−	0.781307i	$-0.714553\pi$
$359$	−8491.00	−1.24829	−0.624147	+	0.781307i	$0.714553\pi$
$360$	0	0
$361$	−5297.59	−0.772356
$362$	0	0
$363$	242.000	0.0349909
$364$	0	0
$365$	3718.13	0.533194
$366$	0	0
$367$	−2948.02	−0.419307	−0.209653	−	0.977776i	$-0.567234\pi$
$367$	−2948.02	−0.419307	−0.209653	+	0.977776i	$0.567234\pi$
$368$	0	0
$369$	−7077.79	−0.998523
$370$	0	0
$371$	−1217.66	−0.170398
$372$	0	0
$373$	−8295.57	−1.15155	−0.575775	−	0.817608i	$-0.695300\pi$
$373$	−8295.57	−1.15155	−0.575775	+	0.817608i	$0.695300\pi$
$374$	0	0
$375$	−2697.11	−0.371409
$376$	0	0
$377$	−11713.7	−1.60022
$378$	0	0
$379$	12137.3	1.64499	0.822495	−	0.568773i	$-0.192582\pi$
$379$	12137.3	1.64499	0.822495	+	0.568773i	$0.192582\pi$
$380$	0	0
$381$	3185.31	0.428316
$382$	0	0
$383$	9053.44	1.20786	0.603929	−	0.797038i	$-0.293601\pi$
$383$	9053.44	1.20786	0.603929	+	0.797038i	$0.293601\pi$
$384$	0	0
$385$	−499.367	−0.0661041
$386$	0	0
$387$	−7268.00	−0.954659
$388$	0	0
$389$	−1130.94	−0.147406	−0.0737030	−	0.997280i	$-0.523482\pi$
$389$	−1130.94	−0.147406	−0.0737030	+	0.997280i	$0.523482\pi$
$390$	0	0
$391$	−5025.19	−0.649962
$392$	0	0
$393$	−2582.87	−0.331523
$394$	0	0
$395$	−3685.98	−0.469524
$396$	0	0
$397$	−13113.4	−1.65779	−0.828896	−	0.559403i	$-0.811030\pi$
$397$	−13113.4	−1.65779	−0.828896	+	0.559403i	$0.811030\pi$
$398$	0	0
$399$	−553.206	−0.0694109
$400$	0	0
$401$	12805.1	1.59465	0.797327	−	0.603548i	$-0.206247\pi$
$401$	12805.1	1.59465	0.797327	+	0.603548i	$0.206247\pi$
$402$	0	0
$403$	−7799.03	−0.964013
$404$	0	0
$405$	2730.30	0.334987
$406$	0	0
$407$	−2463.35	−0.300009
$408$	0	0
$409$	9184.62	1.11039	0.555196	−	0.831720i	$-0.312643\pi$
$409$	9184.62	1.11039	0.555196	+	0.831720i	$0.312643\pi$
$410$	0	0
$411$	−4215.53	−0.505928
$412$	0	0
$413$	−578.463	−0.0689208
$414$	0	0
$415$	−3310.39	−0.391568
$416$	0	0
$417$	−4636.59	−0.544496
$418$	0	0
$419$	6249.25	0.728629	0.364315	−	0.931276i	$-0.381303\pi$
$419$	6249.25	0.728629	0.364315	+	0.931276i	$0.381303\pi$
$420$	0	0
$421$	4703.21	0.544467	0.272233	−	0.962231i	$-0.412238\pi$
$421$	4703.21	0.544467	0.272233	+	0.962231i	$0.412238\pi$
$422$	0	0
$423$	−13256.2	−1.52373
$424$	0	0
$425$	5281.79	0.602834
$426$	0	0
$427$	442.030	0.0500968
$428$	0	0
$429$	−1004.45	−0.113043
$430$	0	0
$431$	−7554.41	−0.844276	−0.422138	−	0.906532i	$-0.638720\pi$
$431$	−7554.41	−0.844276	−0.422138	+	0.906532i	$0.638720\pi$
$432$	0	0
$433$	2631.98	0.292113	0.146057	−	0.989276i	$-0.453342\pi$
$433$	2631.98	0.292113	0.146057	+	0.989276i	$0.453342\pi$
$434$	0	0
$435$	3327.71	0.366785
$436$	0	0
$437$	3118.17	0.341333
$438$	0	0
$439$	−12551.2	−1.36455	−0.682274	−	0.731097i	$-0.739009\pi$
$439$	−12551.2	−1.36455	−0.682274	+	0.731097i	$0.739009\pi$
$440$	0	0
$441$	−1127.00	−0.121693
$442$	0	0
$443$	−11226.2	−1.20400	−0.602000	−	0.798496i	$-0.705629\pi$
$443$	−11226.2	−1.20400	−0.602000	+	0.798496i	$0.705629\pi$
$444$	0	0
$445$	7069.59	0.753103
$446$	0	0
$447$	1204.12	0.127412
$448$	0	0
$449$	−11177.4	−1.17482	−0.587410	−	0.809289i	$-0.699853\pi$
$449$	−11177.4	−1.17482	−0.587410	+	0.809289i	$0.699853\pi$
$450$	0	0
$451$	3385.03	0.353425
$452$	0	0
$453$	3856.39	0.399976
$454$	0	0
$455$	2072.68	0.213558
$456$	0	0
$457$	7040.35	0.720643	0.360321	−	0.932828i	$-0.382667\pi$
$457$	7040.35	0.720643	0.360321	+	0.932828i	$0.382667\pi$
$458$	0	0
$459$	6368.12	0.647579
$460$	0	0
$461$	−1483.35	−0.149862	−0.0749312	−	0.997189i	$-0.523874\pi$
$461$	−1483.35	−0.149862	−0.0749312	+	0.997189i	$0.523874\pi$
$462$	0	0
$463$	3106.19	0.311786	0.155893	−	0.987774i	$-0.450174\pi$
$463$	3106.19	0.311786	0.155893	+	0.987774i	$0.450174\pi$
$464$	0	0
$465$	2215.61	0.220960
$466$	0	0
$467$	9605.04	0.951752	0.475876	−	0.879512i	$-0.342131\pi$
$467$	9605.04	0.951752	0.475876	+	0.879512i	$0.342131\pi$
$468$	0	0
$469$	−2448.08	−0.241027
$470$	0	0
$471$	4952.90	0.484539
$472$	0	0
$473$	3476.00	0.337900
$474$	0	0
$475$	−3277.40	−0.316584
$476$	0	0
$477$	−4000.88	−0.384041
$478$	0	0
$479$	−9118.53	−0.869805	−0.434902	−	0.900478i	$-0.643217\pi$
$479$	−9118.53	−0.869805	−0.434902	+	0.900478i	$0.643217\pi$
$480$	0	0
$481$	10224.4	0.969220
$482$	0	0
$483$	−1104.76	−0.104076
$484$	0	0
$485$	2571.61	0.240764
$486$	0	0
$487$	−14045.1	−1.30687	−0.653434	−	0.756983i	$-0.726672\pi$
$487$	−14045.1	−1.30687	−0.653434	+	0.756983i	$0.726672\pi$
$488$	0	0
$489$	2126.12	0.196618
$490$	0	0
$491$	1034.26	0.0950620	0.0475310	−	0.998870i	$-0.484865\pi$
$491$	1034.26	0.0950620	0.0475310	+	0.998870i	$0.484865\pi$
$492$	0	0
$493$	−16338.0	−1.49255
$494$	0	0
$495$	−1640.78	−0.148985
$496$	0	0
$497$	−833.347	−0.0752128
$498$	0	0
$499$	14332.3	1.28577	0.642886	−	0.765962i	$-0.277737\pi$
$499$	14332.3	1.28577	0.642886	+	0.765962i	$0.277737\pi$
$500$	0	0
$501$	7441.31	0.663579
$502$	0	0
$503$	5007.72	0.443903	0.221951	−	0.975058i	$-0.428757\pi$
$503$	5007.72	0.443903	0.221951	+	0.975058i	$0.428757\pi$
$504$	0	0
$505$	5355.83	0.471943
$506$	0	0
$507$	−224.903	−0.0197008
$508$	0	0
$509$	−9234.40	−0.804140	−0.402070	−	0.915609i	$-0.631709\pi$
$509$	−9234.40	−0.804140	−0.402070	+	0.915609i	$0.631709\pi$
$510$	0	0
$511$	−4013.23	−0.347426
$512$	0	0
$513$	−3951.47	−0.340081
$514$	0	0
$515$	−8081.89	−0.691516
$516$	0	0
$517$	6339.93	0.539323
$518$	0	0
$519$	−1818.84	−0.153831
$520$	0	0
$521$	−5141.26	−0.432328	−0.216164	−	0.976357i	$-0.569355\pi$
$521$	−5141.26	−0.432328	−0.216164	+	0.976357i	$0.569355\pi$
$522$	0	0
$523$	19766.3	1.65262	0.826308	−	0.563218i	$-0.190437\pi$
$523$	19766.3	1.65262	0.826308	+	0.563218i	$0.190437\pi$
$524$	0	0
$525$	1161.18	0.0965293
$526$	0	0
$527$	−10877.9	−0.899146
$528$	0	0
$529$	−5939.95	−0.488201
$530$	0	0
$531$	−1900.66	−0.155333
$532$	0	0
$533$	−14050.0	−1.14179
$534$	0	0
$535$	8873.45	0.717070
$536$	0	0
$537$	−2699.61	−0.216940
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−16805.9	−1.33556	−0.667782	−	0.744356i	$-0.732756\pi$
$541$	−16805.9	−1.33556	−0.667782	+	0.744356i	$0.732756\pi$
$542$	0	0
$543$	−9118.95	−0.720684
$544$	0	0
$545$	−8513.89	−0.669165
$546$	0	0
$547$	−16284.0	−1.27286	−0.636428	−	0.771336i	$-0.719589\pi$
$547$	−16284.0	−1.27286	−0.636428	+	0.771336i	$0.719589\pi$
$548$	0	0
$549$	1452.39	0.112908
$550$	0	0
$551$	10137.8	0.783823
$552$	0	0
$553$	3978.53	0.305939
$554$	0	0
$555$	−2904.64	−0.222154
$556$	0	0
$557$	−13972.6	−1.06291	−0.531454	−	0.847087i	$-0.678354\pi$
$557$	−13972.6	−1.06291	−0.531454	+	0.847087i	$0.678354\pi$
$558$	0	0
$559$	−14427.6	−1.09163
$560$	0	0
$561$	−1400.99	−0.105436
$562$	0	0
$563$	−6396.33	−0.478816	−0.239408	−	0.970919i	$-0.576953\pi$
$563$	−6396.33	−0.478816	−0.239408	+	0.970919i	$0.576953\pi$
$564$	0	0
$565$	−2027.35	−0.150958
$566$	0	0
$567$	−2947.00	−0.218276
$568$	0	0
$569$	−15320.3	−1.12876	−0.564378	−	0.825517i	$-0.690884\pi$
$569$	−15320.3	−1.12876	−0.564378	+	0.825517i	$0.690884\pi$
$570$	0	0
$571$	14347.1	1.05150	0.525750	−	0.850639i	$-0.323785\pi$
$571$	14347.1	1.05150	0.525750	+	0.850639i	$0.323785\pi$
$572$	0	0
$573$	4477.21	0.326419
$574$	0	0
$575$	−6545.02	−0.474689
$576$	0	0
$577$	−19882.3	−1.43451	−0.717256	−	0.696810i	$-0.754602\pi$
$577$	−19882.3	−1.43451	−0.717256	+	0.696810i	$0.754602\pi$
$578$	0	0
$579$	5932.72	0.425829
$580$	0	0
$581$	3573.13	0.255143
$582$	0	0
$583$	1913.46	0.135931
$584$	0	0
$585$	6810.24	0.481314
$586$	0	0
$587$	−17868.5	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$587$	−17868.5	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$588$	0	0
$589$	6749.84	0.472194
$590$	0	0
$591$	7964.37	0.554332
$592$	0	0
$593$	24594.0	1.70312	0.851562	−	0.524253i	$-0.175656\pi$
$593$	24594.0	1.70312	0.851562	+	0.524253i	$0.175656\pi$
$594$	0	0
$595$	2890.94	0.199188
$596$	0	0
$597$	−3790.07	−0.259828
$598$	0	0
$599$	−13645.6	−0.930789	−0.465394	−	0.885103i	$-0.654087\pi$
$599$	−13645.6	−0.930789	−0.465394	+	0.885103i	$0.654087\pi$
$600$	0	0
$601$	−23143.9	−1.57081	−0.785407	−	0.618980i	$-0.787546\pi$
$601$	−23143.9	−1.57081	−0.785407	+	0.618980i	$0.787546\pi$
$602$	0	0
$603$	−8043.69	−0.543225
$604$	0	0
$605$	784.719	0.0527328
$606$	0	0
$607$	11968.7	0.800321	0.400160	−	0.916445i	$-0.368954\pi$
$607$	11968.7	0.800321	0.400160	+	0.916445i	$0.368954\pi$
$608$	0	0
$609$	−3591.82	−0.238995
$610$	0	0
$611$	−26314.7	−1.74235
$612$	0	0
$613$	7079.21	0.466438	0.233219	−	0.972424i	$-0.425074\pi$
$613$	7079.21	0.466438	0.233219	+	0.972424i	$0.425074\pi$
$614$	0	0
$615$	3991.43	0.261707
$616$	0	0
$617$	−24592.0	−1.60460	−0.802300	−	0.596921i	$-0.796390\pi$
$617$	−24592.0	−1.60460	−0.802300	+	0.596921i	$0.796390\pi$
$618$	0	0
$619$	5460.23	0.354548	0.177274	−	0.984162i	$-0.443272\pi$
$619$	5460.23	0.354548	0.177274	+	0.984162i	$0.443272\pi$
$620$	0	0
$621$	−7891.17	−0.509922
$622$	0	0
$623$	−7630.68	−0.490717
$624$	0	0
$625$	1621.87	0.103800
$626$	0	0
$627$	869.324	0.0553707
$628$	0	0
$629$	14260.8	0.904002
$630$	0	0
$631$	23358.4	1.47367	0.736833	−	0.676075i	$-0.236320\pi$
$631$	23358.4	1.47367	0.736833	+	0.676075i	$0.236320\pi$
$632$	0	0
$633$	3013.59	0.189225
$634$	0	0
$635$	10328.8	0.645490
$636$	0	0
$637$	−2237.19	−0.139153
$638$	0	0
$639$	−2738.14	−0.169514
$640$	0	0
$641$	9328.68	0.574822	0.287411	−	0.957807i	$-0.407205\pi$
$641$	9328.68	0.574822	0.287411	+	0.957807i	$0.407205\pi$
$642$	0	0
$643$	267.859	0.0164282	0.00821409	−	0.999966i	$-0.497385\pi$
$643$	267.859	0.0164282	0.00821409	+	0.999966i	$0.497385\pi$
$644$	0	0
$645$	4098.70	0.250211
$646$	0	0
$647$	−11518.0	−0.699877	−0.349939	−	0.936773i	$-0.613798\pi$
$647$	−11518.0	−0.699877	−0.349939	+	0.936773i	$0.613798\pi$
$648$	0	0
$649$	909.013	0.0549798
$650$	0	0
$651$	−2391.46	−0.143976
$652$	0	0
$653$	−12565.0	−0.752994	−0.376497	−	0.926418i	$-0.622871\pi$
$653$	−12565.0	−0.752994	−0.376497	+	0.926418i	$0.622871\pi$
$654$	0	0
$655$	−8375.31	−0.499619
$656$	0	0
$657$	−13186.3	−0.783024
$658$	0	0
$659$	4020.26	0.237643	0.118822	−	0.992916i	$-0.462088\pi$
$659$	4020.26	0.237643	0.118822	+	0.992916i	$0.462088\pi$
$660$	0	0
$661$	−2080.20	−0.122406	−0.0612031	−	0.998125i	$-0.519494\pi$
$661$	−2080.20	−0.122406	−0.0612031	+	0.998125i	$0.519494\pi$
$662$	0	0
$663$	5814.97	0.340626
$664$	0	0
$665$	−1793.85	−0.104605
$666$	0	0
$667$	20245.5	1.17527
$668$	0	0
$669$	−8579.02	−0.495791
$670$	0	0
$671$	−694.619	−0.0399634
$672$	0	0
$673$	−17535.3	−1.00436	−0.502182	−	0.864762i	$-0.667469\pi$
$673$	−17535.3	−1.00436	−0.502182	+	0.864762i	$0.667469\pi$
$674$	0	0
$675$	8294.11	0.472949
$676$	0	0
$677$	12560.8	0.713074	0.356537	−	0.934281i	$-0.383957\pi$
$677$	12560.8	0.713074	0.356537	+	0.934281i	$0.383957\pi$
$678$	0	0
$679$	−2775.71	−0.156881
$680$	0	0
$681$	5697.43	0.320596
$682$	0	0
$683$	4961.51	0.277961	0.138980	−	0.990295i	$-0.455618\pi$
$683$	4961.51	0.277961	0.138980	+	0.990295i	$0.455618\pi$
$684$	0	0
$685$	−13669.4	−0.762456
$686$	0	0
$687$	6547.40	0.363608
$688$	0	0
$689$	−7942.07	−0.439142
$690$	0	0
$691$	15429.4	0.849441	0.424720	−	0.905325i	$-0.360372\pi$
$691$	15429.4	0.849441	0.424720	+	0.905325i	$0.360372\pi$
$692$	0	0
$693$	1771.00	0.0970775
$694$	0	0
$695$	−15034.8	−0.820579
$696$	0	0
$697$	−19596.6	−1.06496
$698$	0	0
$699$	9547.62	0.516630
$700$	0	0
$701$	20446.3	1.10163	0.550817	−	0.834626i	$-0.314316\pi$
$701$	20446.3	1.10163	0.550817	+	0.834626i	$0.314316\pi$
$702$	0	0
$703$	−8848.97	−0.474744
$704$	0	0
$705$	7475.68	0.399362
$706$	0	0
$707$	−5780.91	−0.307515
$708$	0	0
$709$	−28090.1	−1.48794	−0.743968	−	0.668216i	$-0.767058\pi$
$709$	−28090.1	−1.48794	−0.743968	+	0.668216i	$0.767058\pi$
$710$	0	0
$711$	13072.3	0.689522
$712$	0	0
$713$	13479.6	0.708013
$714$	0	0
$715$	−3257.07	−0.170360
$716$	0	0
$717$	−3953.00	−0.205896
$718$	0	0
$719$	−24923.7	−1.29277	−0.646383	−	0.763013i	$-0.723719\pi$
$719$	−24923.7	−1.29277	−0.646383	+	0.763013i	$0.723719\pi$
$720$	0	0
$721$	8723.32	0.450587
$722$	0	0
$723$	−4473.47	−0.230111
$724$	0	0
$725$	−21279.2	−1.09006
$726$	0	0
$727$	24025.9	1.22568	0.612842	−	0.790205i	$-0.290026\pi$
$727$	24025.9	1.22568	0.612842	+	0.790205i	$0.290026\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	0	0
$731$	−20123.3	−1.01818
$732$	0	0
$733$	31004.0	1.56229	0.781145	−	0.624350i	$-0.214636\pi$
$733$	31004.0	1.56229	0.781145	+	0.624350i	$0.214636\pi$
$734$	0	0
$735$	635.558	0.0318951
$736$	0	0
$737$	3846.98	0.192273
$738$	0	0
$739$	5542.14	0.275874	0.137937	−	0.990441i	$-0.455953\pi$
$739$	5542.14	0.275874	0.137937	+	0.990441i	$0.455953\pi$
$740$	0	0
$741$	−3608.24	−0.178882
$742$	0	0
$743$	−35227.3	−1.73939	−0.869694	−	0.493591i	$-0.835684\pi$
$743$	−35227.3	−1.73939	−0.869694	+	0.493591i	$0.835684\pi$
$744$	0	0
$745$	3904.53	0.192015
$746$	0	0
$747$	11740.3	0.575039
$748$	0	0
$749$	−9577.71	−0.467239
$750$	0	0
$751$	−19148.6	−0.930415	−0.465207	−	0.885202i	$-0.654020\pi$
$751$	−19148.6	−0.930415	−0.465207	+	0.885202i	$0.654020\pi$
$752$	0	0
$753$	12729.1	0.616036
$754$	0	0
$755$	12504.9	0.602781
$756$	0	0
$757$	12899.6	0.619345	0.309673	−	0.950843i	$-0.399781\pi$
$757$	12899.6	0.619345	0.309673	+	0.950843i	$0.399781\pi$
$758$	0	0
$759$	1736.06	0.0830236
$760$	0	0
$761$	4568.70	0.217628	0.108814	−	0.994062i	$-0.465295\pi$
$761$	4568.70	0.217628	0.108814	+	0.994062i	$0.465295\pi$
$762$	0	0
$763$	9189.62	0.436024
$764$	0	0
$765$	9498.79	0.448927
$766$	0	0
$767$	−3772.97	−0.177619
$768$	0	0
$769$	6294.43	0.295166	0.147583	−	0.989050i	$-0.452851\pi$
$769$	6294.43	0.295166	0.147583	+	0.989050i	$0.452851\pi$
$770$	0	0
$771$	15894.9	0.742467
$772$	0	0
$773$	−4218.70	−0.196295	−0.0981475	−	0.995172i	$-0.531292\pi$
$773$	−4218.70	−0.196295	−0.0981475	+	0.995172i	$0.531292\pi$
$774$	0	0
$775$	−14167.9	−0.656677
$776$	0	0
$777$	3135.18	0.144754
$778$	0	0
$779$	12159.9	0.559271
$780$	0	0
$781$	1309.55	0.0599990
$782$	0	0
$783$	−25655.8	−1.17096
$784$	0	0
$785$	16060.5	0.730221
$786$	0	0
$787$	−10453.6	−0.473482	−0.236741	−	0.971573i	$-0.576079\pi$
$787$	−10453.6	−0.473482	−0.236741	+	0.971573i	$0.576079\pi$
$788$	0	0
$789$	14661.6	0.661557
$790$	0	0
$791$	2188.25	0.0983631
$792$	0	0
$793$	2883.10	0.129107
$794$	0	0
$795$	2256.25	0.100655
$796$	0	0
$797$	2131.75	0.0947436	0.0473718	−	0.998877i	$-0.484915\pi$
$797$	2131.75	0.0947436	0.0473718	+	0.998877i	$0.484915\pi$
$798$	0	0
$799$	−36703.2	−1.62511
$800$	0	0
$801$	−25072.2	−1.10597
$802$	0	0
$803$	6306.50	0.277150
$804$	0	0
$805$	−3582.35	−0.156846
$806$	0	0
$807$	−7008.05	−0.305694
$808$	0	0
$809$	−35641.7	−1.54894	−0.774472	−	0.632608i	$-0.781984\pi$
$809$	−35641.7	−1.54894	−0.774472	+	0.632608i	$0.781984\pi$
$810$	0	0
$811$	8034.96	0.347898	0.173949	−	0.984755i	$-0.444347\pi$
$811$	8034.96	0.347898	0.173949	+	0.984755i	$0.444347\pi$
$812$	0	0
$813$	−72.9991	−0.00314907
$814$	0	0
$815$	6894.23	0.296312
$816$	0	0
$817$	12486.7	0.534703
$818$	0	0
$819$	−7350.75	−0.313622
$820$	0	0
$821$	37824.5	1.60790	0.803950	−	0.594697i	$-0.202728\pi$
$821$	37824.5	1.60790	0.803950	+	0.594697i	$0.202728\pi$
$822$	0	0
$823$	−19245.1	−0.815118	−0.407559	−	0.913179i	$-0.633620\pi$
$823$	−19245.1	−0.815118	−0.407559	+	0.913179i	$0.633620\pi$
$824$	0	0
$825$	−1824.70	−0.0770037
$826$	0	0
$827$	31294.1	1.31584	0.657921	−	0.753087i	$-0.271436\pi$
$827$	31294.1	1.31584	0.657921	+	0.753087i	$0.271436\pi$
$828$	0	0
$829$	−13121.7	−0.549740	−0.274870	−	0.961481i	$-0.588635\pi$
$829$	−13121.7	−0.549740	−0.274870	+	0.961481i	$0.588635\pi$
$830$	0	0
$831$	−9930.27	−0.414533
$832$	0	0
$833$	−3120.38	−0.129790
$834$	0	0
$835$	24129.5	1.00004
$836$	0	0
$837$	−17081.8	−0.705418
$838$	0	0
$839$	5784.11	0.238009	0.119005	−	0.992894i	$-0.462030\pi$
$839$	5784.11	0.238009	0.119005	+	0.992894i	$0.462030\pi$
$840$	0	0
$841$	41433.2	1.69885
$842$	0	0
$843$	−9120.84	−0.372643
$844$	0	0
$845$	−729.281	−0.0296900
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	1835.49	0.0741976
$850$	0	0
$851$	−17671.6	−0.711837
$852$	0	0
$853$	39103.1	1.56959	0.784797	−	0.619753i	$-0.212767\pi$
$853$	39103.1	1.56959	0.784797	+	0.619753i	$0.212767\pi$
$854$	0	0
$855$	−5894.07	−0.235758
$856$	0	0
$857$	5793.14	0.230910	0.115455	−	0.993313i	$-0.463167\pi$
$857$	5793.14	0.230910	0.115455	+	0.993313i	$0.463167\pi$
$858$	0	0
$859$	−16365.8	−0.650049	−0.325025	−	0.945706i	$-0.605373\pi$
$859$	−16365.8	−0.650049	−0.325025	+	0.945706i	$0.605373\pi$
$860$	0	0
$861$	−4308.22	−0.170527
$862$	0	0
$863$	20999.3	0.828303	0.414151	−	0.910208i	$-0.364078\pi$
$863$	20999.3	0.828303	0.414151	+	0.910208i	$0.364078\pi$
$864$	0	0
$865$	−5897.85	−0.231830
$866$	0	0
$867$	−1715.40	−0.0671949
$868$	0	0
$869$	−6251.98	−0.244055
$870$	0	0
$871$	−15967.4	−0.621164
$872$	0	0
$873$	−9120.19	−0.353576
$874$	0	0
$875$	9439.90	0.364716
$876$	0	0
$877$	22042.8	0.848727	0.424364	−	0.905492i	$-0.360498\pi$
$877$	22042.8	0.848727	0.424364	+	0.905492i	$0.360498\pi$
$878$	0	0
$879$	−7967.94	−0.305747
$880$	0	0
$881$	−7746.91	−0.296254	−0.148127	−	0.988968i	$-0.547324\pi$
$881$	−7746.91	−0.296254	−0.148127	+	0.988968i	$0.547324\pi$
$882$	0	0
$883$	12901.9	0.491715	0.245857	−	0.969306i	$-0.420930\pi$
$883$	12901.9	0.491715	0.245857	+	0.969306i	$0.420930\pi$
$884$	0	0
$885$	1071.86	0.0407119
$886$	0	0
$887$	22230.9	0.841532	0.420766	−	0.907169i	$-0.361761\pi$
$887$	22230.9	0.841532	0.420766	+	0.907169i	$0.361761\pi$
$888$	0	0
$889$	−11148.6	−0.420598
$890$	0	0
$891$	4631.00	0.174124
$892$	0	0
$893$	22774.6	0.853441
$894$	0	0
$895$	−8753.85	−0.326937
$896$	0	0
$897$	−7205.72	−0.268218
$898$	0	0
$899$	43824.9	1.62585
$900$	0	0
$901$	−11077.4	−0.409592
$902$	0	0
$903$	−4424.00	−0.163036
$904$	0	0
$905$	−29569.5	−1.08610
$906$	0	0
$907$	35025.3	1.28225	0.641123	−	0.767438i	$-0.278469\pi$
$907$	35025.3	1.28225	0.641123	+	0.767438i	$0.278469\pi$
$908$	0	0
$909$	−18994.4	−0.693074
$910$	0	0
$911$	−9605.98	−0.349353	−0.174676	−	0.984626i	$-0.555888\pi$
$911$	−9605.98	−0.349353	−0.174676	+	0.984626i	$0.555888\pi$
$912$	0	0
$913$	−5614.91	−0.203534
$914$	0	0
$915$	−819.055	−0.0295925
$916$	0	0
$917$	9040.04	0.325549
$918$	0	0
$919$	791.576	0.0284132	0.0142066	−	0.999899i	$-0.495478\pi$
$919$	791.576	0.0284132	0.0142066	+	0.999899i	$0.495478\pi$
$920$	0	0
$921$	15985.0	0.571906
$922$	0	0
$923$	−5435.43	−0.193835
$924$	0	0
$925$	18573.9	0.660224
$926$	0	0
$927$	28662.4	1.01553
$928$	0	0
$929$	25050.6	0.884695	0.442348	−	0.896844i	$-0.354146\pi$
$929$	25050.6	0.884695	0.442348	+	0.896844i	$0.354146\pi$
$930$	0	0
$931$	1936.22	0.0681601
$932$	0	0
$933$	−14435.3	−0.506528
$934$	0	0
$935$	−4542.90	−0.158897
$936$	0	0
$937$	14769.7	0.514948	0.257474	−	0.966285i	$-0.417110\pi$
$937$	14769.7	0.514948	0.257474	+	0.966285i	$0.417110\pi$
$938$	0	0
$939$	−16831.2	−0.584949
$940$	0	0
$941$	678.921	0.0235199	0.0117599	−	0.999931i	$-0.496257\pi$
$941$	678.921	0.0235199	0.0117599	+	0.999931i	$0.496257\pi$
$942$	0	0
$943$	24283.5	0.838578
$944$	0	0
$945$	4539.70	0.156271
$946$	0	0
$947$	21354.7	0.732772	0.366386	−	0.930463i	$-0.380595\pi$
$947$	21354.7	0.732772	0.366386	+	0.930463i	$0.380595\pi$
$948$	0	0
$949$	−26175.9	−0.895369
$950$	0	0
$951$	1264.73	0.0431248
$952$	0	0
$953$	−3942.06	−0.133994	−0.0669968	−	0.997753i	$-0.521342\pi$
$953$	−3942.06	−0.133994	−0.0669968	+	0.997753i	$0.521342\pi$
$954$	0	0
$955$	14518.0	0.491927
$956$	0	0
$957$	5644.29	0.190652
$958$	0	0
$959$	14754.3	0.496812
$960$	0	0
$961$	−612.100	−0.0205465
$962$	0	0
$963$	−31469.6	−1.05306
$964$	0	0
$965$	19237.7	0.641743
$966$	0	0
$967$	27117.9	0.901813	0.450907	−	0.892571i	$-0.351101\pi$
$967$	27117.9	0.901813	0.450907	+	0.892571i	$0.351101\pi$
$968$	0	0
$969$	−5032.69	−0.166846
$970$	0	0
$971$	28536.2	0.943120	0.471560	−	0.881834i	$-0.343691\pi$
$971$	28536.2	0.943120	0.471560	+	0.881834i	$0.343691\pi$
$972$	0	0
$973$	16228.1	0.534685
$974$	0	0
$975$	7573.66	0.248771
$976$	0	0
$977$	24692.4	0.808577	0.404288	−	0.914632i	$-0.367519\pi$
$977$	24692.4	0.808577	0.404288	+	0.914632i	$0.367519\pi$
$978$	0	0
$979$	11991.1	0.391457
$980$	0	0
$981$	30194.5	0.982706
$982$	0	0
$983$	−26814.7	−0.870047	−0.435024	−	0.900419i	$-0.643260\pi$
$983$	−26814.7	−0.870047	−0.435024	+	0.900419i	$0.643260\pi$
$984$	0	0
$985$	25825.6	0.835402
$986$	0	0
$987$	−8069.00	−0.260222
$988$	0	0
$989$	24936.1	0.801741
$990$	0	0
$991$	−46160.9	−1.47967	−0.739833	−	0.672790i	$-0.765096\pi$
$991$	−46160.9	−1.47967	−0.739833	+	0.672790i	$0.765096\pi$
$992$	0	0
$993$	−851.519	−0.0272126
$994$	0	0
$995$	−12289.8	−0.391572
$996$	0	0
$997$	−43200.4	−1.37229	−0.686143	−	0.727467i	$-0.740698\pi$
$997$	−43200.4	−1.37229	−0.686143	+	0.727467i	$0.740698\pi$
$998$	0	0
$999$	22394.1	0.709228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.m.1.2		2
4.3	odd	2		77.4.a.b.1.1	✓	2
12.11	even	2		693.4.a.i.1.2		2
20.19	odd	2		1925.4.a.l.1.2		2
28.27	even	2		539.4.a.d.1.1		2
44.43	even	2		847.4.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.b.1.1	✓	2	4.3	odd	2
539.4.a.d.1.1		2	28.27	even	2
693.4.a.i.1.2		2	12.11	even	2
847.4.a.c.1.2		2	44.43	even	2
1232.4.a.m.1.2		2	1.1	even	1	trivial
1925.4.a.l.1.2		2	20.19	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{3} +6.48528 q^{5} -7.00000 q^{7} -23.0000 q^{9} +11.0000 q^{11} -45.6569 q^{13} +12.9706 q^{15} -63.6812 q^{17} +39.5147 q^{19} -14.0000 q^{21} +78.9117 q^{23} -82.9411 q^{25} -100.000 q^{27} +256.558 q^{29} +170.818 q^{31} +22.0000 q^{33} -45.3970 q^{35} -223.941 q^{37} -91.3137 q^{39} +307.730 q^{41} +316.000 q^{43} -149.161 q^{45} +576.357 q^{47} +49.0000 q^{49} -127.362 q^{51} +173.951 q^{53} +71.3381 q^{55} +79.0294 q^{57} +82.6375 q^{59} -63.1472 q^{61} +161.000 q^{63} -296.098 q^{65} +349.726 q^{67} +157.823 q^{69} +119.050 q^{71} +573.318 q^{73} -165.882 q^{75} -77.0000 q^{77} -568.362 q^{79} +421.000 q^{81} -510.447 q^{83} -412.991 q^{85} +513.117 q^{87} +1090.10 q^{89} +319.598 q^{91} +341.637 q^{93} +256.264 q^{95} +396.530 q^{97} -253.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 4 q^{5} - 14 q^{7} - 46 q^{9} + 22 q^{11} - 80 q^{13} - 8 q^{15} + 48 q^{17} + 96 q^{19} - 28 q^{21} + 56 q^{23} - 98 q^{25} - 200 q^{27} + 4 q^{29} - 60 q^{31} + 44 q^{33} + 28 q^{35} - 380 q^{37}+ \cdots - 506 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 4 * q^5 - 14 * q^7 - 46 * q^9 + 22 * q^11 - 80 * q^13 - 8 * q^15 + 48 * q^17 + 96 * q^19 - 28 * q^21 + 56 * q^23 - 98 * q^25 - 200 * q^27 + 4 * q^29 - 60 * q^31 + 44 * q^33 + 28 * q^35 - 380 * q^37 - 160 * q^39 + 112 * q^41 + 632 * q^43 + 92 * q^45 + 604 * q^47 + 98 * q^49 + 96 * q^51 + 676 * q^53 - 44 * q^55 + 192 * q^57 + 516 * q^59 - 296 * q^61 + 322 * q^63 + 64 * q^65 + 1152 * q^67 + 112 * q^69 + 1064 * q^71 + 168 * q^73 - 196 * q^75 - 154 * q^77 + 368 * q^79 + 842 * q^81 - 1728 * q^83 - 1584 * q^85 + 8 * q^87 + 1524 * q^89 + 560 * q^91 - 120 * q^93 - 336 * q^95 + 1404 * q^97 - 506 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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