LMFDB - Embedded newform 342.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,4,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("342.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	342.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:	$20.1786532220$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.77200$ of defining polynomial
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +17.3160 q^{5} -26.0880 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} +17.3160 q^{5} -26.0880 q^{7} -8.00000 q^{8} -34.6320 q^{10} +4.22800 q^{11} +64.0360 q^{13} +52.1760 q^{14} +16.0000 q^{16} +48.5440 q^{17} +19.0000 q^{19} +69.2640 q^{20} -8.45600 q^{22} -92.0360 q^{23} +174.844 q^{25} -128.072 q^{26} -104.352 q^{28} +88.2120 q^{29} -81.9681 q^{31} -32.0000 q^{32} -97.0880 q^{34} -451.740 q^{35} -23.6161 q^{37} -38.0000 q^{38} -138.528 q^{40} -17.7200 q^{41} +368.404 q^{43} +16.9120 q^{44} +184.072 q^{46} +497.812 q^{47} +337.584 q^{49} -349.688 q^{50} +256.144 q^{52} +536.876 q^{53} +73.2120 q^{55} +208.704 q^{56} -176.424 q^{58} +36.7000 q^{59} +630.692 q^{61} +163.936 q^{62} +64.0000 q^{64} +1108.85 q^{65} +282.556 q^{67} +194.176 q^{68} +903.480 q^{70} -595.552 q^{71} -597.048 q^{73} +47.2321 q^{74} +76.0000 q^{76} -110.300 q^{77} +427.224 q^{79} +277.056 q^{80} +35.4400 q^{82} -493.768 q^{83} +840.588 q^{85} -736.808 q^{86} -33.8240 q^{88} +921.136 q^{89} -1670.57 q^{91} -368.144 q^{92} -995.624 q^{94} +329.004 q^{95} +1082.74 q^{97} -675.168 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} - 18 q^{7} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 + 9 * q^5 - 18 * q^7 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} - 18 q^{7} - 16 q^{8} - 18 q^{10} + 17 q^{11} + 17 q^{13} + 36 q^{14} + 32 q^{16} + 80 q^{17} + 38 q^{19} + 36 q^{20} - 34 q^{22} - 73 q^{23} + 119 q^{25} - 34 q^{26} - 72 q^{28} - 3 q^{29} + 212 q^{31} - 64 q^{32} - 160 q^{34} - 519 q^{35} + 192 q^{37} - 76 q^{38} - 72 q^{40} + 50 q^{41} + 677 q^{43} + 68 q^{44} + 146 q^{46} + 389 q^{47} + 60 q^{49} - 238 q^{50} + 68 q^{52} + 1219 q^{53} - 33 q^{55} + 144 q^{56} + 6 q^{58} + 287 q^{59} + 313 q^{61} - 424 q^{62} + 128 q^{64} + 1500 q^{65} + 1223 q^{67} + 320 q^{68} + 1038 q^{70} - 200 q^{71} + 378 q^{73} - 384 q^{74} + 152 q^{76} - 7 q^{77} + 1350 q^{79} + 144 q^{80} - 100 q^{82} + 670 q^{83} + 579 q^{85} - 1354 q^{86} - 136 q^{88} + 236 q^{89} - 2051 q^{91} - 292 q^{92} - 778 q^{94} + 171 q^{95} + 1294 q^{97} - 120 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 + 9 * q^5 - 18 * q^7 - 16 * q^8 - 18 * q^10 + 17 * q^11 + 17 * q^13 + 36 * q^14 + 32 * q^16 + 80 * q^17 + 38 * q^19 + 36 * q^20 - 34 * q^22 - 73 * q^23 + 119 * q^25 - 34 * q^26 - 72 * q^28 - 3 * q^29 + 212 * q^31 - 64 * q^32 - 160 * q^34 - 519 * q^35 + 192 * q^37 - 76 * q^38 - 72 * q^40 + 50 * q^41 + 677 * q^43 + 68 * q^44 + 146 * q^46 + 389 * q^47 + 60 * q^49 - 238 * q^50 + 68 * q^52 + 1219 * q^53 - 33 * q^55 + 144 * q^56 + 6 * q^58 + 287 * q^59 + 313 * q^61 - 424 * q^62 + 128 * q^64 + 1500 * q^65 + 1223 * q^67 + 320 * q^68 + 1038 * q^70 - 200 * q^71 + 378 * q^73 - 384 * q^74 + 152 * q^76 - 7 * q^77 + 1350 * q^79 + 144 * q^80 - 100 * q^82 + 670 * q^83 + 579 * q^85 - 1354 * q^86 - 136 * q^88 + 236 * q^89 - 2051 * q^91 - 292 * q^92 - 778 * q^94 + 171 * q^95 + 1294 * q^97 - 120 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	17.3160	1.54879	0.774395	−	0.632702i	$-0.218054\pi$
$5$	17.3160	1.54879	0.774395	+	0.632702i	$0.218054\pi$
$6$	0	0
$7$	−26.0880	−1.40862	−0.704310	−	0.709893i	$-0.748743\pi$
$7$	−26.0880	−1.40862	−0.704310	+	0.709893i	$0.748743\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−34.6320	−1.09516
$11$	4.22800	0.115890	0.0579450	−	0.998320i	$-0.481545\pi$
$11$	4.22800	0.115890	0.0579450	+	0.998320i	$0.481545\pi$
$12$	0	0
$13$	64.0360	1.36618	0.683092	−	0.730332i	$-0.260635\pi$
$13$	64.0360	1.36618	0.683092	+	0.730332i	$0.260635\pi$
$14$	52.1760	0.996045
$15$	0	0
$16$	16.0000	0.250000
$17$	48.5440	0.692568	0.346284	−	0.938130i	$-0.387443\pi$
$17$	48.5440	0.692568	0.346284	+	0.938130i	$0.387443\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	69.2640	0.774395
$21$	0	0
$22$	−8.45600	−0.0819466
$23$	−92.0360	−0.834384	−0.417192	−	0.908818i	$-0.636986\pi$
$23$	−92.0360	−0.834384	−0.417192	+	0.908818i	$0.636986\pi$
$24$	0	0
$25$	174.844	1.39875
$26$	−128.072	−0.966038
$27$	0	0
$28$	−104.352	−0.704310
$29$	88.2120	0.564847	0.282424	−	0.959290i	$-0.408862\pi$
$29$	88.2120	0.564847	0.282424	+	0.959290i	$0.408862\pi$
$30$	0	0
$31$	−81.9681	−0.474900	−0.237450	−	0.971400i	$-0.576312\pi$
$31$	−81.9681	−0.474900	−0.237450	+	0.971400i	$0.576312\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−97.0880	−0.489719
$35$	−451.740	−2.18166
$36$	0	0
$37$	−23.6161	−0.104931	−0.0524656	−	0.998623i	$-0.516708\pi$
$37$	−23.6161	−0.104931	−0.0524656	+	0.998623i	$0.516708\pi$
$38$	−38.0000	−0.162221
$39$	0	0
$40$	−138.528	−0.547580
$41$	−17.7200	−0.0674976	−0.0337488	−	0.999430i	$-0.510745\pi$
$41$	−17.7200	−0.0674976	−0.0337488	+	0.999430i	$0.510745\pi$
$42$	0	0
$43$	368.404	1.30654	0.653268	−	0.757126i	$-0.273397\pi$
$43$	368.404	1.30654	0.653268	+	0.757126i	$0.273397\pi$
$44$	16.9120	0.0579450
$45$	0	0
$46$	184.072	0.589999
$47$	497.812	1.54497	0.772483	−	0.635036i	$-0.219015\pi$
$47$	497.812	1.54497	0.772483	+	0.635036i	$0.219015\pi$
$48$	0	0
$49$	337.584	0.984210
$50$	−349.688	−0.989067
$51$	0	0
$52$	256.144	0.683092
$53$	536.876	1.39143	0.695713	−	0.718320i	$-0.255089\pi$
$53$	536.876	1.39143	0.695713	+	0.718320i	$0.255089\pi$
$54$	0	0
$55$	73.2120	0.179489
$56$	208.704	0.498022
$57$	0	0
$58$	−176.424	−0.399407
$59$	36.7000	0.0809818	0.0404909	−	0.999180i	$-0.487108\pi$
$59$	36.7000	0.0809818	0.0404909	+	0.999180i	$0.487108\pi$
$60$	0	0
$61$	630.692	1.32380	0.661901	−	0.749592i	$-0.269750\pi$
$61$	630.692	1.32380	0.661901	+	0.749592i	$0.269750\pi$
$62$	163.936	0.335805
$63$	0	0
$64$	64.0000	0.125000
$65$	1108.85	2.11593
$66$	0	0
$67$	282.556	0.515219	0.257610	−	0.966249i	$-0.417065\pi$
$67$	282.556	0.515219	0.257610	+	0.966249i	$0.417065\pi$
$68$	194.176	0.346284
$69$	0	0
$70$	903.480	1.54266
$71$	−595.552	−0.995480	−0.497740	−	0.867326i	$-0.665837\pi$
$71$	−595.552	−0.995480	−0.497740	+	0.867326i	$0.665837\pi$
$72$	0	0
$73$	−597.048	−0.957250	−0.478625	−	0.878020i	$-0.658865\pi$
$73$	−597.048	−0.957250	−0.478625	+	0.878020i	$0.658865\pi$
$74$	47.2321	0.0741976
$75$	0	0
$76$	76.0000	0.114708
$77$	−110.300	−0.163245
$78$	0	0
$79$	427.224	0.608436	0.304218	−	0.952602i	$-0.401605\pi$
$79$	427.224	0.608436	0.304218	+	0.952602i	$0.401605\pi$
$80$	277.056	0.387198
$81$	0	0
$82$	35.4400	0.0477280
$83$	−493.768	−0.652989	−0.326495	−	0.945199i	$-0.605868\pi$
$83$	−493.768	−0.652989	−0.326495	+	0.945199i	$0.605868\pi$
$84$	0	0
$85$	840.588	1.07264
$86$	−736.808	−0.923861
$87$	0	0
$88$	−33.8240	−0.0409733
$89$	921.136	1.09708	0.548541	−	0.836124i	$-0.315184\pi$
$89$	921.136	1.09708	0.548541	+	0.836124i	$0.315184\pi$
$90$	0	0
$91$	−1670.57	−1.92443
$92$	−368.144	−0.417192
$93$	0	0
$94$	−995.624	−1.09246
$95$	329.004	0.355317
$96$	0	0
$97$	1082.74	1.13336	0.566680	−	0.823938i	$-0.308227\pi$
$97$	1082.74	1.13336	0.566680	+	0.823938i	$0.308227\pi$
$98$	−675.168	−0.695942
$99$	0	0
$100$	699.376	0.699376
$101$	712.448	0.701893	0.350947	−	0.936395i	$-0.385860\pi$
$101$	712.448	0.701893	0.350947	+	0.936395i	$0.385860\pi$
$102$	0	0
$103$	−26.4797	−0.0253313	−0.0126656	−	0.999920i	$-0.504032\pi$
$103$	−26.4797	−0.0253313	−0.0126656	+	0.999920i	$0.504032\pi$
$104$	−512.288	−0.483019
$105$	0	0
$106$	−1073.75	−0.983887
$107$	740.996	0.669484	0.334742	−	0.942310i	$-0.391351\pi$
$107$	740.996	0.669484	0.334742	+	0.942310i	$0.391351\pi$
$108$	0	0
$109$	−1983.08	−1.74261	−0.871304	−	0.490744i	$-0.836725\pi$
$109$	−1983.08	−1.74261	−0.871304	+	0.490744i	$0.836725\pi$
$110$	−146.424	−0.126918
$111$	0	0
$112$	−417.408	−0.352155
$113$	718.720	0.598332	0.299166	−	0.954201i	$-0.403292\pi$
$113$	718.720	0.598332	0.299166	+	0.954201i	$0.403292\pi$
$114$	0	0
$115$	−1593.70	−1.29229
$116$	352.848	0.282424
$117$	0	0
$118$	−73.3999	−0.0572628
$119$	−1266.42	−0.975565
$120$	0	0
$121$	−1313.12	−0.986570
$122$	−1261.38	−0.936069
$123$	0	0
$124$	−327.872	−0.237450
$125$	863.100	0.617584
$126$	0	0
$127$	2610.72	1.82413	0.912063	−	0.410050i	$-0.134489\pi$
$127$	2610.72	1.82413	0.912063	+	0.410050i	$0.134489\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−2217.70	−1.49619
$131$	1216.69	0.811472	0.405736	−	0.913990i	$-0.367015\pi$
$131$	1216.69	0.811472	0.405736	+	0.913990i	$0.367015\pi$
$132$	0	0
$133$	−495.672	−0.323160
$134$	−565.112	−0.364315
$135$	0	0
$136$	−388.352	−0.244860
$137$	−1170.67	−0.730053	−0.365026	−	0.930997i	$-0.618940\pi$
$137$	−1170.67	−0.730053	−0.365026	+	0.930997i	$0.618940\pi$
$138$	0	0
$139$	−271.083	−0.165417	−0.0827086	−	0.996574i	$-0.526357\pi$
$139$	−271.083	−0.165417	−0.0827086	+	0.996574i	$0.526357\pi$
$140$	−1806.96	−1.09083
$141$	0	0
$142$	1191.10	0.703910
$143$	270.744	0.158327
$144$	0	0
$145$	1527.48	0.874830
$146$	1194.10	0.676878
$147$	0	0
$148$	−94.4642	−0.0524656
$149$	−1841.19	−1.01232	−0.506161	−	0.862439i	$-0.668936\pi$
$149$	−1841.19	−1.01232	−0.506161	+	0.862439i	$0.668936\pi$
$150$	0	0
$151$	3322.32	1.79051	0.895254	−	0.445557i	$-0.146994\pi$
$151$	3322.32	1.79051	0.895254	+	0.445557i	$0.146994\pi$
$152$	−152.000	−0.0811107
$153$	0	0
$154$	220.600	0.115432
$155$	−1419.36	−0.735521
$156$	0	0
$157$	243.616	0.123839	0.0619194	−	0.998081i	$-0.480278\pi$
$157$	243.616	0.123839	0.0619194	+	0.998081i	$0.480278\pi$
$158$	−854.448	−0.430229
$159$	0	0
$160$	−554.112	−0.273790
$161$	2401.04	1.17533
$162$	0	0
$163$	−2598.11	−1.24847	−0.624233	−	0.781238i	$-0.714588\pi$
$163$	−2598.11	−1.24847	−0.624233	+	0.781238i	$0.714588\pi$
$164$	−70.8801	−0.0337488
$165$	0	0
$166$	987.537	0.461733
$167$	491.064	0.227543	0.113772	−	0.993507i	$-0.463707\pi$
$167$	491.064	0.227543	0.113772	+	0.993507i	$0.463707\pi$
$168$	0	0
$169$	1903.61	0.866460
$170$	−1681.18	−0.758473
$171$	0	0
$172$	1473.62	0.653268
$173$	−1648.56	−0.724496	−0.362248	−	0.932082i	$-0.617991\pi$
$173$	−1648.56	−0.724496	−0.362248	+	0.932082i	$0.617991\pi$
$174$	0	0
$175$	−4561.33	−1.97031
$176$	67.6480	0.0289725
$177$	0	0
$178$	−1842.27	−0.775754
$179$	−2326.81	−0.971586	−0.485793	−	0.874074i	$-0.661469\pi$
$179$	−2326.81	−0.971586	−0.485793	+	0.874074i	$0.661469\pi$
$180$	0	0
$181$	−4637.46	−1.90442	−0.952208	−	0.305449i	$-0.901193\pi$
$181$	−4637.46	−1.90442	−0.952208	+	0.305449i	$0.901193\pi$
$182$	3341.14	1.36078
$183$	0	0
$184$	736.288	0.294999
$185$	−408.936	−0.162516
$186$	0	0
$187$	205.244	0.0802616
$188$	1991.25	0.772483
$189$	0	0
$190$	−658.008	−0.251247
$191$	−5260.38	−1.99281	−0.996407	−	0.0846903i	$-0.973010\pi$
$191$	−5260.38	−1.99281	−0.996407	+	0.0846903i	$0.973010\pi$
$192$	0	0
$193$	16.1833	0.00603575	0.00301787	−	0.999995i	$-0.499039\pi$
$193$	16.1833	0.00603575	0.00301787	+	0.999995i	$0.499039\pi$
$194$	−2165.49	−0.801407
$195$	0	0
$196$	1350.34	0.492105
$197$	−3784.71	−1.36878	−0.684390	−	0.729116i	$-0.739931\pi$
$197$	−3784.71	−1.36878	−0.684390	+	0.729116i	$0.739931\pi$
$198$	0	0
$199$	73.2079	0.0260783	0.0130391	−	0.999915i	$-0.495849\pi$
$199$	73.2079	0.0260783	0.0130391	+	0.999915i	$0.495849\pi$
$200$	−1398.75	−0.494534
$201$	0	0
$202$	−1424.90	−0.496313
$203$	−2301.28	−0.795655
$204$	0	0
$205$	−306.840	−0.104540
$206$	52.9594	0.0179119
$207$	0	0
$208$	1024.58	0.341546
$209$	80.3320	0.0265870
$210$	0	0
$211$	−2945.44	−0.961006	−0.480503	−	0.876993i	$-0.659546\pi$
$211$	−2945.44	−0.961006	−0.480503	+	0.876993i	$0.659546\pi$
$212$	2147.50	0.695713
$213$	0	0
$214$	−1481.99	−0.473397
$215$	6379.29	2.02355
$216$	0	0
$217$	2138.38	0.668954
$218$	3966.15	1.23221
$219$	0	0
$220$	292.848	0.0897446
$221$	3108.57	0.946175
$222$	0	0
$223$	3125.30	0.938499	0.469250	−	0.883066i	$-0.344524\pi$
$223$	3125.30	0.938499	0.469250	+	0.883066i	$0.344524\pi$
$224$	834.816	0.249011
$225$	0	0
$226$	−1437.44	−0.423085
$227$	3577.80	1.04611	0.523055	−	0.852299i	$-0.324792\pi$
$227$	3577.80	1.04611	0.523055	+	0.852299i	$0.324792\pi$
$228$	0	0
$229$	−4802.00	−1.38570	−0.692850	−	0.721082i	$-0.743645\pi$
$229$	−4802.00	−1.38570	−0.692850	+	0.721082i	$0.743645\pi$
$230$	3187.39	0.913785
$231$	0	0
$232$	−705.696	−0.199704
$233$	−5829.49	−1.63907	−0.819534	−	0.573031i	$-0.805768\pi$
$233$	−5829.49	−1.63907	−0.819534	+	0.573031i	$0.805768\pi$
$234$	0	0
$235$	8620.12	2.39283
$236$	146.800	0.0404909
$237$	0	0
$238$	2532.83	0.689828
$239$	−1364.33	−0.369251	−0.184625	−	0.982809i	$-0.559107\pi$
$239$	−1364.33	−0.369251	−0.184625	+	0.982809i	$0.559107\pi$
$240$	0	0
$241$	−2647.22	−0.707563	−0.353782	−	0.935328i	$-0.615104\pi$
$241$	−2647.22	−0.707563	−0.353782	+	0.935328i	$0.615104\pi$
$242$	2626.25	0.697610
$243$	0	0
$244$	2522.77	0.661901
$245$	5845.61	1.52434
$246$	0	0
$247$	1216.68	0.313424
$248$	655.745	0.167903
$249$	0	0
$250$	−1726.20	−0.436698
$251$	−1970.73	−0.495582	−0.247791	−	0.968814i	$-0.579705\pi$
$251$	−1970.73	−0.495582	−0.247791	+	0.968814i	$0.579705\pi$
$252$	0	0
$253$	−389.128	−0.0966967
$254$	−5221.44	−1.28985
$255$	0	0
$256$	256.000	0.0625000
$257$	7915.82	1.92131	0.960653	−	0.277752i	$-0.0895892\pi$
$257$	7915.82	1.92131	0.960653	+	0.277752i	$0.0895892\pi$
$258$	0	0
$259$	616.096	0.147808
$260$	4435.39	1.05797
$261$	0	0
$262$	−2433.38	−0.573798
$263$	−3287.96	−0.770892	−0.385446	−	0.922730i	$-0.625952\pi$
$263$	−3287.96	−0.770892	−0.385446	+	0.922730i	$0.625952\pi$
$264$	0	0
$265$	9296.55	2.15503
$266$	991.344	0.228508
$267$	0	0
$268$	1130.22	0.257610
$269$	4749.61	1.07654	0.538269	−	0.842773i	$-0.319078\pi$
$269$	4749.61	1.07654	0.538269	+	0.842773i	$0.319078\pi$
$270$	0	0
$271$	242.661	0.0543933	0.0271967	−	0.999630i	$-0.491342\pi$
$271$	242.661	0.0543933	0.0271967	+	0.999630i	$0.491342\pi$
$272$	776.704	0.173142
$273$	0	0
$274$	2341.34	0.516225
$275$	739.240	0.162101
$276$	0	0
$277$	−4131.13	−0.896086	−0.448043	−	0.894012i	$-0.647879\pi$
$277$	−4131.13	−0.896086	−0.448043	+	0.894012i	$0.647879\pi$
$278$	542.167	0.116968
$279$	0	0
$280$	3613.92	0.771332
$281$	−1007.19	−0.213822	−0.106911	−	0.994269i	$-0.534096\pi$
$281$	−1007.19	−0.213822	−0.106911	+	0.994269i	$0.534096\pi$
$282$	0	0
$283$	2333.63	0.490176	0.245088	−	0.969501i	$-0.421183\pi$
$283$	2333.63	0.490176	0.245088	+	0.969501i	$0.421183\pi$
$284$	−2382.21	−0.497740
$285$	0	0
$286$	−541.488	−0.111954
$287$	462.280	0.0950785
$288$	0	0
$289$	−2556.48	−0.520350
$290$	−3054.96	−0.618598
$291$	0	0
$292$	−2388.19	−0.478625
$293$	1588.68	0.316763	0.158381	−	0.987378i	$-0.449372\pi$
$293$	1588.68	0.316763	0.158381	+	0.987378i	$0.449372\pi$
$294$	0	0
$295$	635.497	0.125424
$296$	188.928	0.0370988
$297$	0	0
$298$	3682.38	0.715820
$299$	−5893.62	−1.13992
$300$	0	0
$301$	−9610.93	−1.84041
$302$	−6644.64	−1.26608
$303$	0	0
$304$	304.000	0.0573539
$305$	10921.1	2.05029
$306$	0	0
$307$	4057.46	0.754304	0.377152	−	0.926151i	$-0.376903\pi$
$307$	4057.46	0.754304	0.377152	+	0.926151i	$0.376903\pi$
$308$	−441.200	−0.0816224
$309$	0	0
$310$	2838.72	0.520092
$311$	−2871.92	−0.523638	−0.261819	−	0.965117i	$-0.584322\pi$
$311$	−2871.92	−0.523638	−0.261819	+	0.965117i	$0.584322\pi$
$312$	0	0
$313$	4322.67	0.780612	0.390306	−	0.920685i	$-0.372369\pi$
$313$	4322.67	0.780612	0.390306	+	0.920685i	$0.372369\pi$
$314$	−487.232	−0.0875672
$315$	0	0
$316$	1708.90	0.304218
$317$	−2513.56	−0.445349	−0.222674	−	0.974893i	$-0.571479\pi$
$317$	−2513.56	−0.445349	−0.222674	+	0.974893i	$0.571479\pi$
$318$	0	0
$319$	372.960	0.0654601
$320$	1108.22	0.193599
$321$	0	0
$322$	−4802.07	−0.831084
$323$	922.336	0.158886
$324$	0	0
$325$	11196.3	1.91095
$326$	5196.22	0.882798
$327$	0	0
$328$	141.760	0.0238640
$329$	−12986.9	−2.17627
$330$	0	0
$331$	−4573.78	−0.759509	−0.379754	−	0.925087i	$-0.623992\pi$
$331$	−4573.78	−0.759509	−0.379754	+	0.925087i	$0.623992\pi$
$332$	−1975.07	−0.326495
$333$	0	0
$334$	−982.129	−0.160897
$335$	4892.74	0.797967
$336$	0	0
$337$	9001.71	1.45506	0.727529	−	0.686077i	$-0.240669\pi$
$337$	9001.71	1.45506	0.727529	+	0.686077i	$0.240669\pi$
$338$	−3807.22	−0.612680
$339$	0	0
$340$	3362.35	0.536321
$341$	−346.561	−0.0550361
$342$	0	0
$343$	141.289	0.0222417
$344$	−2947.23	−0.461931
$345$	0	0
$346$	3297.12	0.512296
$347$	−9358.68	−1.44784	−0.723920	−	0.689884i	$-0.757661\pi$
$347$	−9358.68	−1.44784	−0.723920	+	0.689884i	$0.757661\pi$
$348$	0	0
$349$	5787.76	0.887712	0.443856	−	0.896098i	$-0.353610\pi$
$349$	5787.76	0.887712	0.443856	+	0.896098i	$0.353610\pi$
$350$	9122.67	1.39322
$351$	0	0
$352$	−135.296	−0.0204866
$353$	−5784.59	−0.872188	−0.436094	−	0.899901i	$-0.643639\pi$
$353$	−5784.59	−0.872188	−0.436094	+	0.899901i	$0.643639\pi$
$354$	0	0
$355$	−10312.6	−1.54179
$356$	3684.55	0.548541
$357$	0	0
$358$	4653.62	0.687015
$359$	10132.3	1.48959	0.744796	−	0.667292i	$-0.232547\pi$
$359$	10132.3	1.48959	0.744796	+	0.667292i	$0.232547\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	9274.91	1.34663
$363$	0	0
$364$	−6682.29	−0.962217
$365$	−10338.5	−1.48258
$366$	0	0
$367$	−6993.81	−0.994752	−0.497376	−	0.867535i	$-0.665703\pi$
$367$	−6993.81	−0.994752	−0.497376	+	0.867535i	$0.665703\pi$
$368$	−1472.58	−0.208596
$369$	0	0
$370$	817.871	0.114917
$371$	−14006.0	−1.95999
$372$	0	0
$373$	6523.15	0.905512	0.452756	−	0.891634i	$-0.350441\pi$
$373$	6523.15	0.905512	0.452756	+	0.891634i	$0.350441\pi$
$374$	−410.488	−0.0567535
$375$	0	0
$376$	−3982.50	−0.546228
$377$	5648.75	0.771685
$378$	0	0
$379$	−9782.00	−1.32577	−0.662886	−	0.748720i	$-0.730669\pi$
$379$	−9782.00	−1.32577	−0.662886	+	0.748720i	$0.730669\pi$
$380$	1316.02	0.177658
$381$	0	0
$382$	10520.8	1.40913
$383$	−9878.11	−1.31788	−0.658940	−	0.752196i	$-0.728995\pi$
$383$	−9878.11	−1.31788	−0.658940	+	0.752196i	$0.728995\pi$
$384$	0	0
$385$	−1909.96	−0.252832
$386$	−32.3666	−0.00426792
$387$	0	0
$388$	4330.98	0.566680
$389$	7891.25	1.02854	0.514270	−	0.857628i	$-0.328063\pi$
$389$	7891.25	1.02854	0.514270	+	0.857628i	$0.328063\pi$
$390$	0	0
$391$	−4467.80	−0.577868
$392$	−2700.67	−0.347971
$393$	0	0
$394$	7569.43	0.967874
$395$	7397.81	0.942340
$396$	0	0
$397$	−2787.84	−0.352437	−0.176219	−	0.984351i	$-0.556387\pi$
$397$	−2787.84	−0.352437	−0.176219	+	0.984351i	$0.556387\pi$
$398$	−146.416	−0.0184401
$399$	0	0
$400$	2797.50	0.349688
$401$	−1264.42	−0.157461	−0.0787306	−	0.996896i	$-0.525087\pi$
$401$	−1264.42	−0.157461	−0.0787306	+	0.996896i	$0.525087\pi$
$402$	0	0
$403$	−5248.91	−0.648801
$404$	2849.79	0.350947
$405$	0	0
$406$	4602.55	0.562613
$407$	−99.8486	−0.0121605
$408$	0	0
$409$	−8140.55	−0.984166	−0.492083	−	0.870548i	$-0.663764\pi$
$409$	−8140.55	−0.984166	−0.492083	+	0.870548i	$0.663764\pi$
$410$	613.680	0.0739207
$411$	0	0
$412$	−105.919	−0.0126656
$413$	−957.429	−0.114073
$414$	0	0
$415$	−8550.10	−1.01134
$416$	−2049.15	−0.241510
$417$	0	0
$418$	−160.664	−0.0187998
$419$	9601.15	1.11944	0.559722	−	0.828680i	$-0.310908\pi$
$419$	9601.15	1.11944	0.559722	+	0.828680i	$0.310908\pi$
$420$	0	0
$421$	5702.48	0.660147	0.330074	−	0.943955i	$-0.392926\pi$
$421$	5702.48	0.660147	0.330074	+	0.943955i	$0.392926\pi$
$422$	5890.87	0.679534
$423$	0	0
$424$	−4295.01	−0.491943
$425$	8487.63	0.968731
$426$	0	0
$427$	−16453.5	−1.86473
$428$	2963.99	0.334742
$429$	0	0
$430$	−12758.6	−1.43087
$431$	4025.72	0.449912	0.224956	−	0.974369i	$-0.427776\pi$
$431$	4025.72	0.449912	0.224956	+	0.974369i	$0.427776\pi$
$432$	0	0
$433$	−1347.10	−0.149510	−0.0747548	−	0.997202i	$-0.523817\pi$
$433$	−1347.10	−0.149510	−0.0747548	+	0.997202i	$0.523817\pi$
$434$	−4276.77	−0.473022
$435$	0	0
$436$	−7932.31	−0.871304
$437$	−1748.68	−0.191421
$438$	0	0
$439$	4109.36	0.446763	0.223381	−	0.974731i	$-0.428290\pi$
$439$	4109.36	0.446763	0.223381	+	0.974731i	$0.428290\pi$
$440$	−585.696	−0.0634590
$441$	0	0
$442$	−6217.13	−0.669047
$443$	−6964.84	−0.746974	−0.373487	−	0.927635i	$-0.621838\pi$
$443$	−6964.84	−0.746974	−0.373487	+	0.927635i	$0.621838\pi$
$444$	0	0
$445$	15950.4	1.69915
$446$	−6250.59	−0.663619
$447$	0	0
$448$	−1669.63	−0.176078
$449$	−3041.21	−0.319652	−0.159826	−	0.987145i	$-0.551093\pi$
$449$	−3041.21	−0.319652	−0.159826	+	0.987145i	$0.551093\pi$
$450$	0	0
$451$	−74.9202	−0.00782229
$452$	2874.88	0.299166
$453$	0	0
$454$	−7155.60	−0.739711
$455$	−28927.6	−2.98055
$456$	0	0
$457$	11984.3	1.22670	0.613352	−	0.789810i	$-0.289821\pi$
$457$	11984.3	1.22670	0.613352	+	0.789810i	$0.289821\pi$
$458$	9604.01	0.979838
$459$	0	0
$460$	−6374.79	−0.646143
$461$	12126.7	1.22515	0.612577	−	0.790411i	$-0.290133\pi$
$461$	12126.7	1.22515	0.612577	+	0.790411i	$0.290133\pi$
$462$	0	0
$463$	−6399.19	−0.642323	−0.321162	−	0.947024i	$-0.604073\pi$
$463$	−6399.19	−0.642323	−0.321162	+	0.947024i	$0.604073\pi$
$464$	1411.39	0.141212
$465$	0	0
$466$	11659.0	1.15900
$467$	−993.366	−0.0984315	−0.0492157	−	0.998788i	$-0.515672\pi$
$467$	−993.366	−0.0984315	−0.0492157	+	0.998788i	$0.515672\pi$
$468$	0	0
$469$	−7371.32	−0.725748
$470$	−17240.2	−1.69198
$471$	0	0
$472$	−293.600	−0.0286314
$473$	1557.61	0.151414
$474$	0	0
$475$	3322.04	0.320896
$476$	−5065.67	−0.487782
$477$	0	0
$478$	2728.65	0.261100
$479$	−6639.36	−0.633320	−0.316660	−	0.948539i	$-0.602561\pi$
$479$	−6639.36	−0.633320	−0.316660	+	0.948539i	$0.602561\pi$
$480$	0	0
$481$	−1512.28	−0.143355
$482$	5294.45	0.500323
$483$	0	0
$484$	−5252.50	−0.493285
$485$	18748.8	1.75534
$486$	0	0
$487$	−11088.8	−1.03179	−0.515894	−	0.856652i	$-0.672540\pi$
$487$	−11088.8	−1.03179	−0.515894	+	0.856652i	$0.672540\pi$
$488$	−5045.54	−0.468034
$489$	0	0
$490$	−11691.2	−1.07787
$491$	13215.2	1.21465	0.607324	−	0.794454i	$-0.292243\pi$
$491$	13215.2	1.21465	0.607324	+	0.794454i	$0.292243\pi$
$492$	0	0
$493$	4282.17	0.391195
$494$	−2433.37	−0.221624
$495$	0	0
$496$	−1311.49	−0.118725
$497$	15536.8	1.40225
$498$	0	0
$499$	410.640	0.0368393	0.0184196	−	0.999830i	$-0.494137\pi$
$499$	410.640	0.0368393	0.0184196	+	0.999830i	$0.494137\pi$
$500$	3452.40	0.308792
$501$	0	0
$502$	3941.45	0.350429
$503$	9407.88	0.833950	0.416975	−	0.908918i	$-0.363090\pi$
$503$	9407.88	0.833950	0.416975	+	0.908918i	$0.363090\pi$
$504$	0	0
$505$	12336.7	1.08709
$506$	778.256	0.0683749
$507$	0	0
$508$	10442.9	0.912063
$509$	−10482.2	−0.912803	−0.456402	−	0.889774i	$-0.650862\pi$
$509$	−10482.2	−0.912803	−0.456402	+	0.889774i	$0.650862\pi$
$510$	0	0
$511$	15575.8	1.34840
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−15831.6	−1.35857
$515$	−458.523	−0.0392329
$516$	0	0
$517$	2104.75	0.179046
$518$	−1232.19	−0.104516
$519$	0	0
$520$	−8870.79	−0.748096
$521$	3181.02	0.267492	0.133746	−	0.991016i	$-0.457299\pi$
$521$	3181.02	0.267492	0.133746	+	0.991016i	$0.457299\pi$
$522$	0	0
$523$	−4360.12	−0.364541	−0.182270	−	0.983248i	$-0.558345\pi$
$523$	−4360.12	−0.364541	−0.182270	+	0.983248i	$0.558345\pi$
$524$	4866.77	0.405736
$525$	0	0
$526$	6575.93	0.545103
$527$	−3979.06	−0.328900
$528$	0	0
$529$	−3696.37	−0.303803
$530$	−18593.1	−1.52383
$531$	0	0
$532$	−1982.69	−0.161580
$533$	−1134.72	−0.0922142
$534$	0	0
$535$	12831.1	1.03689
$536$	−2260.45	−0.182158
$537$	0	0
$538$	−9499.22	−0.761227
$539$	1427.31	0.114060
$540$	0	0
$541$	−23681.2	−1.88195	−0.940973	−	0.338481i	$-0.890087\pi$
$541$	−23681.2	−1.88195	−0.940973	+	0.338481i	$0.890087\pi$
$542$	−485.322	−0.0384619
$543$	0	0
$544$	−1553.41	−0.122430
$545$	−34339.0	−2.69894
$546$	0	0
$547$	−7373.25	−0.576339	−0.288169	−	0.957579i	$-0.593047\pi$
$547$	−7373.25	−0.576339	−0.288169	+	0.957579i	$0.593047\pi$
$548$	−4682.69	−0.365026
$549$	0	0
$550$	−1478.48	−0.114623
$551$	1676.03	0.129585
$552$	0	0
$553$	−11145.4	−0.857055
$554$	8262.27	0.633628
$555$	0	0
$556$	−1084.33	−0.0827086
$557$	−4772.14	−0.363020	−0.181510	−	0.983389i	$-0.558098\pi$
$557$	−4772.14	−0.363020	−0.181510	+	0.983389i	$0.558098\pi$
$558$	0	0
$559$	23591.1	1.78497
$560$	−7227.84	−0.545414
$561$	0	0
$562$	2014.38	0.151195
$563$	−7276.49	−0.544702	−0.272351	−	0.962198i	$-0.587801\pi$
$563$	−7276.49	−0.544702	−0.272351	+	0.962198i	$0.587801\pi$
$564$	0	0
$565$	12445.4	0.926691
$566$	−4667.26	−0.346607
$567$	0	0
$568$	4764.42	0.351955
$569$	10685.1	0.787245	0.393622	−	0.919272i	$-0.371222\pi$
$569$	10685.1	0.787245	0.393622	+	0.919272i	$0.371222\pi$
$570$	0	0
$571$	14856.1	1.08881	0.544404	−	0.838823i	$-0.316756\pi$
$571$	14856.1	1.08881	0.544404	+	0.838823i	$0.316756\pi$
$572$	1082.98	0.0791635
$573$	0	0
$574$	−924.560	−0.0672306
$575$	−16092.0	−1.16710
$576$	0	0
$577$	3212.67	0.231794	0.115897	−	0.993261i	$-0.463026\pi$
$577$	3212.67	0.231794	0.115897	+	0.993261i	$0.463026\pi$
$578$	5112.96	0.367943
$579$	0	0
$580$	6109.92	0.437415
$581$	12881.4	0.919814
$582$	0	0
$583$	2269.91	0.161252
$584$	4776.39	0.338439
$585$	0	0
$586$	−3177.35	−0.223985
$587$	22321.1	1.56949	0.784745	−	0.619818i	$-0.212794\pi$
$587$	22321.1	1.56949	0.784745	+	0.619818i	$0.212794\pi$
$588$	0	0
$589$	−1557.39	−0.108950
$590$	−1270.99	−0.0886881
$591$	0	0
$592$	−377.857	−0.0262328
$593$	8202.50	0.568021	0.284010	−	0.958821i	$-0.408335\pi$
$593$	8202.50	0.568021	0.284010	+	0.958821i	$0.408335\pi$
$594$	0	0
$595$	−21929.3	−1.51095
$596$	−7364.75	−0.506161
$597$	0	0
$598$	11787.2	0.806047
$599$	−10583.3	−0.721906	−0.360953	−	0.932584i	$-0.617548\pi$
$599$	−10583.3	−0.721906	−0.360953	+	0.932584i	$0.617548\pi$
$600$	0	0
$601$	−9051.94	−0.614370	−0.307185	−	0.951650i	$-0.599387\pi$
$601$	−9051.94	−0.614370	−0.307185	+	0.951650i	$0.599387\pi$
$602$	19221.9	1.30137
$603$	0	0
$604$	13289.3	0.895254
$605$	−22738.1	−1.52799
$606$	0	0
$607$	8123.48	0.543199	0.271599	−	0.962410i	$-0.412447\pi$
$607$	8123.48	0.543199	0.271599	+	0.962410i	$0.412447\pi$
$608$	−608.000	−0.0405554
$609$	0	0
$610$	−21842.1	−1.44977
$611$	31877.9	2.11071
$612$	0	0
$613$	22384.7	1.47490	0.737448	−	0.675404i	$-0.236031\pi$
$613$	22384.7	1.47490	0.737448	+	0.675404i	$0.236031\pi$
$614$	−8114.91	−0.533373
$615$	0	0
$616$	882.400	0.0577158
$617$	−11349.1	−0.740517	−0.370259	−	0.928929i	$-0.620731\pi$
$617$	−11349.1	−0.740517	−0.370259	+	0.928929i	$0.620731\pi$
$618$	0	0
$619$	−9106.25	−0.591294	−0.295647	−	0.955297i	$-0.595535\pi$
$619$	−9106.25	−0.591294	−0.295647	+	0.955297i	$0.595535\pi$
$620$	−5677.44	−0.367760
$621$	0	0
$622$	5743.84	0.370268
$623$	−24030.6	−1.54537
$624$	0	0
$625$	−6910.06	−0.442244
$626$	−8645.34	−0.551976
$627$	0	0
$628$	974.464	0.0619194
$629$	−1146.42	−0.0726720
$630$	0	0
$631$	−27784.2	−1.75289	−0.876444	−	0.481505i	$-0.840090\pi$
$631$	−27784.2	−1.75289	−0.876444	+	0.481505i	$0.840090\pi$
$632$	−3417.79	−0.215115
$633$	0	0
$634$	5027.12	0.314909
$635$	45207.3	2.82519
$636$	0	0
$637$	21617.5	1.34461
$638$	−745.921	−0.0462873
$639$	0	0
$640$	−2216.45	−0.136895
$641$	16958.3	1.04495	0.522476	−	0.852654i	$-0.325008\pi$
$641$	16958.3	1.04495	0.522476	+	0.852654i	$0.325008\pi$
$642$	0	0
$643$	4754.37	0.291592	0.145796	−	0.989315i	$-0.453426\pi$
$643$	4754.37	0.291592	0.145796	+	0.989315i	$0.453426\pi$
$644$	9604.15	0.587665
$645$	0	0
$646$	−1844.67	−0.112349
$647$	11254.0	0.683831	0.341916	−	0.939731i	$-0.388924\pi$
$647$	11254.0	0.683831	0.341916	+	0.939731i	$0.388924\pi$
$648$	0	0
$649$	155.167	0.00938498
$650$	−22392.6	−1.35125
$651$	0	0
$652$	−10392.4	−0.624233
$653$	15515.1	0.929793	0.464896	−	0.885365i	$-0.346092\pi$
$653$	15515.1	0.929793	0.464896	+	0.885365i	$0.346092\pi$
$654$	0	0
$655$	21068.2	1.25680
$656$	−283.520	−0.0168744
$657$	0	0
$658$	25973.9	1.53885
$659$	−17203.2	−1.01691	−0.508453	−	0.861090i	$-0.669783\pi$
$659$	−17203.2	−1.01691	−0.508453	+	0.861090i	$0.669783\pi$
$660$	0	0
$661$	2305.65	0.135672	0.0678361	−	0.997696i	$-0.478390\pi$
$661$	2305.65	0.135672	0.0678361	+	0.997696i	$0.478390\pi$
$662$	9147.55	0.537054
$663$	0	0
$664$	3950.15	0.230867
$665$	−8583.06	−0.500507
$666$	0	0
$667$	−8118.69	−0.471299
$668$	1964.26	0.113772
$669$	0	0
$670$	−9785.48	−0.564248
$671$	2666.57	0.153415
$672$	0	0
$673$	−14242.8	−0.815782	−0.407891	−	0.913031i	$-0.633736\pi$
$673$	−14242.8	−0.815782	−0.407891	+	0.913031i	$0.633736\pi$
$674$	−18003.4	−1.02888
$675$	0	0
$676$	7614.45	0.433230
$677$	13480.0	0.765256	0.382628	−	0.923902i	$-0.375019\pi$
$677$	13480.0	0.765256	0.382628	+	0.923902i	$0.375019\pi$
$678$	0	0
$679$	−28246.6	−1.59647
$680$	−6724.71	−0.379236
$681$	0	0
$682$	693.122	0.0389164
$683$	−27626.1	−1.54771	−0.773854	−	0.633365i	$-0.781673\pi$
$683$	−27626.1	−1.54771	−0.773854	+	0.633365i	$0.781673\pi$
$684$	0	0
$685$	−20271.4	−1.13070
$686$	−282.578	−0.0157272
$687$	0	0
$688$	5894.46	0.326634
$689$	34379.4	1.90094
$690$	0	0
$691$	17419.7	0.959009	0.479505	−	0.877539i	$-0.340816\pi$
$691$	17419.7	0.959009	0.479505	+	0.877539i	$0.340816\pi$
$692$	−6594.24	−0.362248
$693$	0	0
$694$	18717.4	1.02378
$695$	−4694.08	−0.256197
$696$	0	0
$697$	−860.201	−0.0467467
$698$	−11575.5	−0.627707
$699$	0	0
$700$	−18245.3	−0.985155
$701$	−5069.39	−0.273136	−0.136568	−	0.990631i	$-0.543607\pi$
$701$	−5069.39	−0.273136	−0.136568	+	0.990631i	$0.543607\pi$
$702$	0	0
$703$	−448.705	−0.0240729
$704$	270.592	0.0144862
$705$	0	0
$706$	11569.2	0.616730
$707$	−18586.3	−0.988701
$708$	0	0
$709$	−16758.9	−0.887719	−0.443860	−	0.896096i	$-0.646391\pi$
$709$	−16758.9	−0.887719	−0.443860	+	0.896096i	$0.646391\pi$
$710$	20625.2	1.09021
$711$	0	0
$712$	−7369.09	−0.387877
$713$	7544.02	0.396249
$714$	0	0
$715$	4688.21	0.245215
$716$	−9307.24	−0.485793
$717$	0	0
$718$	−20264.6	−1.05330
$719$	3885.84	0.201554	0.100777	−	0.994909i	$-0.467867\pi$
$719$	3885.84	0.201554	0.100777	+	0.994909i	$0.467867\pi$
$720$	0	0
$721$	690.803	0.0356822
$722$	−722.000	−0.0372161
$723$	0	0
$724$	−18549.8	−0.952208
$725$	15423.4	0.790081
$726$	0	0
$727$	6468.37	0.329984	0.164992	−	0.986295i	$-0.447240\pi$
$727$	6468.37	0.329984	0.164992	+	0.986295i	$0.447240\pi$
$728$	13364.6	0.680390
$729$	0	0
$730$	20677.0	1.04834
$731$	17883.8	0.904865
$732$	0	0
$733$	25245.5	1.27212	0.636059	−	0.771640i	$-0.280563\pi$
$733$	25245.5	1.27212	0.636059	+	0.771640i	$0.280563\pi$
$734$	13987.6	0.703396
$735$	0	0
$736$	2945.15	0.147500
$737$	1194.65	0.0597087
$738$	0	0
$739$	3229.28	0.160746	0.0803728	−	0.996765i	$-0.474389\pi$
$739$	3229.28	0.160746	0.0803728	+	0.996765i	$0.474389\pi$
$740$	−1635.74	−0.0812582
$741$	0	0
$742$	28012.0	1.38592
$743$	−18876.2	−0.932033	−0.466016	−	0.884776i	$-0.654311\pi$
$743$	−18876.2	−0.932033	−0.466016	+	0.884776i	$0.654311\pi$
$744$	0	0
$745$	−31882.0	−1.56788
$746$	−13046.3	−0.640294
$747$	0	0
$748$	820.976	0.0401308
$749$	−19331.1	−0.943049
$750$	0	0
$751$	24895.8	1.20967	0.604833	−	0.796352i	$-0.293240\pi$
$751$	24895.8	1.20967	0.604833	+	0.796352i	$0.293240\pi$
$752$	7964.99	0.386241
$753$	0	0
$754$	−11297.5	−0.545664
$755$	57529.3	2.77312
$756$	0	0
$757$	−36203.2	−1.73821	−0.869107	−	0.494624i	$-0.835306\pi$
$757$	−36203.2	−1.73821	−0.869107	+	0.494624i	$0.835306\pi$
$758$	19564.0	0.937462
$759$	0	0
$760$	−2632.03	−0.125624
$761$	−11417.5	−0.543868	−0.271934	−	0.962316i	$-0.587663\pi$
$761$	−11417.5	−0.543868	−0.271934	+	0.962316i	$0.587663\pi$
$762$	0	0
$763$	51734.5	2.45467
$764$	−21041.5	−0.996407
$765$	0	0
$766$	19756.2	0.931881
$767$	2350.12	0.110636
$768$	0	0
$769$	39414.5	1.84828	0.924138	−	0.382058i	$-0.124785\pi$
$769$	39414.5	1.84828	0.924138	+	0.382058i	$0.124785\pi$
$770$	3819.91	0.178779
$771$	0	0
$772$	64.7332	0.00301787
$773$	14268.5	0.663910	0.331955	−	0.943295i	$-0.392292\pi$
$773$	14268.5	0.663910	0.331955	+	0.943295i	$0.392292\pi$
$774$	0	0
$775$	−14331.6	−0.664268
$776$	−8661.95	−0.400704
$777$	0	0
$778$	−15782.5	−0.727288
$779$	−336.680	−0.0154850
$780$	0	0
$781$	−2517.99	−0.115366
$782$	8935.59	0.408614
$783$	0	0
$784$	5401.35	0.246053
$785$	4218.46	0.191800
$786$	0	0
$787$	−2922.28	−0.132361	−0.0661804	−	0.997808i	$-0.521081\pi$
$787$	−2922.28	−0.132361	−0.0661804	+	0.997808i	$0.521081\pi$
$788$	−15138.9	−0.684390
$789$	0	0
$790$	−14795.6	−0.666335
$791$	−18750.0	−0.842823
$792$	0	0
$793$	40387.0	1.80856
$794$	5575.67	0.249211
$795$	0	0
$796$	292.832	0.0130391
$797$	7724.25	0.343296	0.171648	−	0.985158i	$-0.445091\pi$
$797$	7724.25	0.343296	0.171648	+	0.985158i	$0.445091\pi$
$798$	0	0
$799$	24165.8	1.06999
$800$	−5595.01	−0.247267
$801$	0	0
$802$	2528.83	0.111342
$803$	−2524.32	−0.110936
$804$	0	0
$805$	41576.4	1.82034
$806$	10497.8	0.458772
$807$	0	0
$808$	−5699.58	−0.248157
$809$	−42980.8	−1.86789	−0.933947	−	0.357412i	$-0.883659\pi$
$809$	−42980.8	−1.86789	−0.933947	+	0.357412i	$0.883659\pi$
$810$	0	0
$811$	28749.5	1.24480	0.622398	−	0.782701i	$-0.286158\pi$
$811$	28749.5	1.24480	0.622398	+	0.782701i	$0.286158\pi$
$812$	−9205.11	−0.397827
$813$	0	0
$814$	199.697	0.00859875
$815$	−44988.9	−1.93361
$816$	0	0
$817$	6999.68	0.299740
$818$	16281.1	0.695911
$819$	0	0
$820$	−1227.36	−0.0522698
$821$	30274.8	1.28696	0.643482	−	0.765461i	$-0.277489\pi$
$821$	30274.8	1.28696	0.643482	+	0.765461i	$0.277489\pi$
$822$	0	0
$823$	17296.1	0.732568	0.366284	−	0.930503i	$-0.380630\pi$
$823$	17296.1	0.732568	0.366284	+	0.930503i	$0.380630\pi$
$824$	211.838	0.00895596
$825$	0	0
$826$	1914.86	0.0806615
$827$	2022.80	0.0850541	0.0425271	−	0.999095i	$-0.486459\pi$
$827$	2022.80	0.0850541	0.0425271	+	0.999095i	$0.486459\pi$
$828$	0	0
$829$	−43239.0	−1.81152	−0.905762	−	0.423786i	$-0.860701\pi$
$829$	−43239.0	−1.81152	−0.905762	+	0.423786i	$0.860701\pi$
$830$	17100.2	0.715128
$831$	0	0
$832$	4098.31	0.170773
$833$	16387.7	0.681632
$834$	0	0
$835$	8503.27	0.352417
$836$	321.328	0.0132935
$837$	0	0
$838$	−19202.3	−0.791567
$839$	−27435.9	−1.12895	−0.564477	−	0.825449i	$-0.690922\pi$
$839$	−27435.9	−1.12895	−0.564477	+	0.825449i	$0.690922\pi$
$840$	0	0
$841$	−16607.6	−0.680948
$842$	−11405.0	−0.466795
$843$	0	0
$844$	−11781.7	−0.480503
$845$	32963.0	1.34196
$846$	0	0
$847$	34256.8	1.38970
$848$	8590.02	0.347857
$849$	0	0
$850$	−16975.3	−0.684996
$851$	2173.53	0.0875530
$852$	0	0
$853$	−20978.4	−0.842071	−0.421035	−	0.907044i	$-0.638333\pi$
$853$	−20978.4	−0.842071	−0.421035	+	0.907044i	$0.638333\pi$
$854$	32907.0	1.31857
$855$	0	0
$856$	−5927.97	−0.236698
$857$	30822.4	1.22856	0.614279	−	0.789089i	$-0.289447\pi$
$857$	30822.4	1.22856	0.614279	+	0.789089i	$0.289447\pi$
$858$	0	0
$859$	−39267.6	−1.55971	−0.779856	−	0.625959i	$-0.784708\pi$
$859$	−39267.6	−1.55971	−0.779856	+	0.625959i	$0.784708\pi$
$860$	25517.1	1.01178
$861$	0	0
$862$	−8051.45	−0.318136
$863$	24131.3	0.951842	0.475921	−	0.879488i	$-0.342115\pi$
$863$	24131.3	0.951842	0.475921	+	0.879488i	$0.342115\pi$
$864$	0	0
$865$	−28546.5	−1.12209
$866$	2694.21	0.105719
$867$	0	0
$868$	8553.54	0.334477
$869$	1806.30	0.0705116
$870$	0	0
$871$	18093.8	0.703885
$872$	15864.6	0.616105
$873$	0	0
$874$	3497.37	0.135355
$875$	−22516.6	−0.869941
$876$	0	0
$877$	39380.6	1.51629	0.758147	−	0.652084i	$-0.226105\pi$
$877$	39380.6	1.51629	0.758147	+	0.652084i	$0.226105\pi$
$878$	−8218.71	−0.315909
$879$	0	0
$880$	1171.39	0.0448723
$881$	−30887.5	−1.18119	−0.590595	−	0.806968i	$-0.701107\pi$
$881$	−30887.5	−1.18119	−0.590595	+	0.806968i	$0.701107\pi$
$882$	0	0
$883$	28191.9	1.07444	0.537221	−	0.843441i	$-0.319474\pi$
$883$	28191.9	1.07444	0.537221	+	0.843441i	$0.319474\pi$
$884$	12434.3	0.473088
$885$	0	0
$886$	13929.7	0.528190
$887$	−2760.58	−0.104500	−0.0522498	−	0.998634i	$-0.516639\pi$
$887$	−2760.58	−0.104500	−0.0522498	+	0.998634i	$0.516639\pi$
$888$	0	0
$889$	−68108.5	−2.56950
$890$	−31900.8	−1.20148
$891$	0	0
$892$	12501.2	0.469250
$893$	9458.43	0.354439
$894$	0	0
$895$	−40291.0	−1.50478
$896$	3339.26	0.124506
$897$	0	0
$898$	6082.42	0.226028
$899$	−7230.57	−0.268246
$900$	0	0
$901$	26062.1	0.963657
$902$	149.840	0.00553120
$903$	0	0
$904$	−5749.76	−0.211542
$905$	−80302.2	−2.94954
$906$	0	0
$907$	−18969.1	−0.694443	−0.347222	−	0.937783i	$-0.612875\pi$
$907$	−18969.1	−0.694443	−0.347222	+	0.937783i	$0.612875\pi$
$908$	14311.2	0.523055
$909$	0	0
$910$	57855.3	2.10756
$911$	48732.9	1.77233	0.886164	−	0.463371i	$-0.153360\pi$
$911$	48732.9	1.77233	0.886164	+	0.463371i	$0.153360\pi$
$912$	0	0
$913$	−2087.65	−0.0756749
$914$	−23968.7	−0.867411
$915$	0	0
$916$	−19208.0	−0.692850
$917$	−31741.1	−1.14306
$918$	0	0
$919$	−35850.4	−1.28683	−0.643414	−	0.765518i	$-0.722483\pi$
$919$	−35850.4	−1.28683	−0.643414	+	0.765518i	$0.722483\pi$
$920$	12749.6	0.456892
$921$	0	0
$922$	−24253.4	−0.866315
$923$	−38136.8	−1.36001
$924$	0	0
$925$	−4129.13	−0.146773
$926$	12798.4	0.454191
$927$	0	0
$928$	−2822.79	−0.0998518
$929$	−22936.8	−0.810044	−0.405022	−	0.914307i	$-0.632736\pi$
$929$	−22936.8	−0.810044	−0.405022	+	0.914307i	$0.632736\pi$
$930$	0	0
$931$	6414.10	0.225793
$932$	−23318.0	−0.819534
$933$	0	0
$934$	1986.73	0.0696016
$935$	3554.01	0.124308
$936$	0	0
$937$	47925.4	1.67092	0.835462	−	0.549548i	$-0.185200\pi$
$937$	47925.4	1.67092	0.835462	+	0.549548i	$0.185200\pi$
$938$	14742.6	0.513181
$939$	0	0
$940$	34480.5	1.19641
$941$	25842.2	0.895251	0.447626	−	0.894221i	$-0.352270\pi$
$941$	25842.2	0.895251	0.447626	+	0.894221i	$0.352270\pi$
$942$	0	0
$943$	1630.88	0.0563189
$944$	587.199	0.0202455
$945$	0	0
$946$	−3115.22	−0.107066
$947$	−36562.8	−1.25463	−0.627314	−	0.778766i	$-0.715846\pi$
$947$	−36562.8	−1.25463	−0.627314	+	0.778766i	$0.715846\pi$
$948$	0	0
$949$	−38232.6	−1.30778
$950$	−6644.07	−0.226908
$951$	0	0
$952$	10131.3	0.344914
$953$	−29813.1	−1.01337	−0.506684	−	0.862132i	$-0.669129\pi$
$953$	−29813.1	−1.01337	−0.506684	+	0.862132i	$0.669129\pi$
$954$	0	0
$955$	−91088.7	−3.08645
$956$	−5457.31	−0.184625
$957$	0	0
$958$	13278.7	0.447825
$959$	30540.5	1.02837
$960$	0	0
$961$	−23072.2	−0.774470
$962$	3024.56	0.101368
$963$	0	0
$964$	−10588.9	−0.353782
$965$	280.230	0.00934811
$966$	0	0
$967$	30315.5	1.00815	0.504075	−	0.863660i	$-0.331833\pi$
$967$	30315.5	1.00815	0.504075	+	0.863660i	$0.331833\pi$
$968$	10505.0	0.348805
$969$	0	0
$970$	−37497.6	−1.24121
$971$	−26455.6	−0.874357	−0.437178	−	0.899375i	$-0.644022\pi$
$971$	−26455.6	−0.874357	−0.437178	+	0.899375i	$0.644022\pi$
$972$	0	0
$973$	7072.03	0.233010
$974$	22177.6	0.729584
$975$	0	0
$976$	10091.1	0.330950
$977$	−30207.7	−0.989183	−0.494591	−	0.869126i	$-0.664682\pi$
$977$	−30207.7	−0.989183	−0.494591	+	0.869126i	$0.664682\pi$
$978$	0	0
$979$	3894.56	0.127141
$980$	23382.4	0.762168
$981$	0	0
$982$	−26430.3	−0.858886
$983$	5878.48	0.190737	0.0953685	−	0.995442i	$-0.469597\pi$
$983$	5878.48	0.190737	0.0953685	+	0.995442i	$0.469597\pi$
$984$	0	0
$985$	−65536.1	−2.11995
$986$	−8564.33	−0.276616
$987$	0	0
$988$	4866.74	0.156712
$989$	−33906.4	−1.09015
$990$	0	0
$991$	−42532.3	−1.36335	−0.681676	−	0.731654i	$-0.738749\pi$
$991$	−42532.3	−1.36335	−0.681676	+	0.731654i	$0.738749\pi$
$992$	2622.98	0.0839513
$993$	0	0
$994$	−31073.5	−0.991542
$995$	1267.67	0.0403898
$996$	0	0
$997$	6320.28	0.200767	0.100384	−	0.994949i	$-0.467993\pi$
$997$	6320.28	0.200767	0.100384	+	0.994949i	$0.467993\pi$
$998$	−821.281	−0.0260493
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.4.a.h.1.2		2
3.2	odd	2		38.4.a.c.1.2	✓	2
12.11	even	2		304.4.a.c.1.1		2
15.2	even	4		950.4.b.i.799.3		4
15.8	even	4		950.4.b.i.799.2		4
15.14	odd	2		950.4.a.e.1.1		2
21.20	even	2		1862.4.a.e.1.1		2
24.5	odd	2		1216.4.a.g.1.1		2
24.11	even	2		1216.4.a.p.1.2		2
57.56	even	2		722.4.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.c.1.2	✓	2	3.2	odd	2
304.4.a.c.1.1		2	12.11	even	2
342.4.a.h.1.2		2	1.1	even	1	trivial
722.4.a.f.1.1		2	57.56	even	2
950.4.a.e.1.1		2	15.14	odd	2
950.4.b.i.799.2		4	15.8	even	4
950.4.b.i.799.3		4	15.2	even	4
1216.4.a.g.1.1		2	24.5	odd	2
1216.4.a.p.1.2		2	24.11	even	2
1862.4.a.e.1.1		2	21.20	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +17.3160 q^{5} -26.0880 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +17.3160 q^{5} -26.0880 q^{7} -8.00000 q^{8} -34.6320 q^{10} +4.22800 q^{11} +64.0360 q^{13} +52.1760 q^{14} +16.0000 q^{16} +48.5440 q^{17} +19.0000 q^{19} +69.2640 q^{20} -8.45600 q^{22} -92.0360 q^{23} +174.844 q^{25} -128.072 q^{26} -104.352 q^{28} +88.2120 q^{29} -81.9681 q^{31} -32.0000 q^{32} -97.0880 q^{34} -451.740 q^{35} -23.6161 q^{37} -38.0000 q^{38} -138.528 q^{40} -17.7200 q^{41} +368.404 q^{43} +16.9120 q^{44} +184.072 q^{46} +497.812 q^{47} +337.584 q^{49} -349.688 q^{50} +256.144 q^{52} +536.876 q^{53} +73.2120 q^{55} +208.704 q^{56} -176.424 q^{58} +36.7000 q^{59} +630.692 q^{61} +163.936 q^{62} +64.0000 q^{64} +1108.85 q^{65} +282.556 q^{67} +194.176 q^{68} +903.480 q^{70} -595.552 q^{71} -597.048 q^{73} +47.2321 q^{74} +76.0000 q^{76} -110.300 q^{77} +427.224 q^{79} +277.056 q^{80} +35.4400 q^{82} -493.768 q^{83} +840.588 q^{85} -736.808 q^{86} -33.8240 q^{88} +921.136 q^{89} -1670.57 q^{91} -368.144 q^{92} -995.624 q^{94} +329.004 q^{95} +1082.74 q^{97} -675.168 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} - 18 q^{7} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 + 9 * q^5 - 18 * q^7 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} - 18 q^{7} - 16 q^{8} - 18 q^{10} + 17 q^{11} + 17 q^{13} + 36 q^{14} + 32 q^{16} + 80 q^{17} + 38 q^{19} + 36 q^{20} - 34 q^{22} - 73 q^{23} + 119 q^{25} - 34 q^{26} - 72 q^{28} - 3 q^{29} + 212 q^{31} - 64 q^{32} - 160 q^{34} - 519 q^{35} + 192 q^{37} - 76 q^{38} - 72 q^{40} + 50 q^{41} + 677 q^{43} + 68 q^{44} + 146 q^{46} + 389 q^{47} + 60 q^{49} - 238 q^{50} + 68 q^{52} + 1219 q^{53} - 33 q^{55} + 144 q^{56} + 6 q^{58} + 287 q^{59} + 313 q^{61} - 424 q^{62} + 128 q^{64} + 1500 q^{65} + 1223 q^{67} + 320 q^{68} + 1038 q^{70} - 200 q^{71} + 378 q^{73} - 384 q^{74} + 152 q^{76} - 7 q^{77} + 1350 q^{79} + 144 q^{80} - 100 q^{82} + 670 q^{83} + 579 q^{85} - 1354 q^{86} - 136 q^{88} + 236 q^{89} - 2051 q^{91} - 292 q^{92} - 778 q^{94} + 171 q^{95} + 1294 q^{97} - 120 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 + 9 * q^5 - 18 * q^7 - 16 * q^8 - 18 * q^10 + 17 * q^11 + 17 * q^13 + 36 * q^14 + 32 * q^16 + 80 * q^17 + 38 * q^19 + 36 * q^20 - 34 * q^22 - 73 * q^23 + 119 * q^25 - 34 * q^26 - 72 * q^28 - 3 * q^29 + 212 * q^31 - 64 * q^32 - 160 * q^34 - 519 * q^35 + 192 * q^37 - 76 * q^38 - 72 * q^40 + 50 * q^41 + 677 * q^43 + 68 * q^44 + 146 * q^46 + 389 * q^47 + 60 * q^49 - 238 * q^50 + 68 * q^52 + 1219 * q^53 - 33 * q^55 + 144 * q^56 + 6 * q^58 + 287 * q^59 + 313 * q^61 - 424 * q^62 + 128 * q^64 + 1500 * q^65 + 1223 * q^67 + 320 * q^68 + 1038 * q^70 - 200 * q^71 + 378 * q^73 - 384 * q^74 + 152 * q^76 - 7 * q^77 + 1350 * q^79 + 144 * q^80 - 100 * q^82 + 670 * q^83 + 579 * q^85 - 1354 * q^86 - 136 * q^88 + 236 * q^89 - 2051 * q^91 - 292 * q^92 - 778 * q^94 + 171 * q^95 + 1294 * q^97 - 120 * q^98

Introduction

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