LMFDB - Embedded newform 342.4.a.h.1.1

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			1.1
Root			$-3.77200$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -8.31601 q^{5} +8.08801 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} -8.31601 q^{5} +8.08801 q^{7} -8.00000 q^{8} +16.6320 q^{10} +12.7720 q^{11} -47.0360 q^{13} -16.1760 q^{14} +16.0000 q^{16} +31.4560 q^{17} +19.0000 q^{19} -33.2640 q^{20} -25.5440 q^{22} +19.0360 q^{23} -55.8441 q^{25} +94.0720 q^{26} +32.3520 q^{28} -91.2120 q^{29} +293.968 q^{31} -32.0000 q^{32} -62.9120 q^{34} -67.2599 q^{35} +215.616 q^{37} -38.0000 q^{38} +66.5280 q^{40} +67.7200 q^{41} +308.596 q^{43} +51.0880 q^{44} -38.0720 q^{46} -108.812 q^{47} -277.584 q^{49} +111.688 q^{50} -188.144 q^{52} +682.124 q^{53} -106.212 q^{55} -64.7041 q^{56} +182.424 q^{58} +250.300 q^{59} -317.692 q^{61} -587.936 q^{62} +64.0000 q^{64} +391.152 q^{65} +940.444 q^{67} +125.824 q^{68} +134.520 q^{70} +395.552 q^{71} +975.048 q^{73} -431.232 q^{74} +76.0000 q^{76} +103.300 q^{77} +922.776 q^{79} -133.056 q^{80} -135.440 q^{82} +1163.77 q^{83} -261.588 q^{85} -617.192 q^{86} -102.176 q^{88} -685.136 q^{89} -380.428 q^{91} +76.1441 q^{92} +217.624 q^{94} -158.004 q^{95} +211.256 q^{97} +555.168 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−8.31601	−0.743806	−0.371903	−	0.928272i	$-0.621295\pi$
$5$	−8.31601	−0.743806	−0.371903	+	0.928272i	$0.621295\pi$
$6$	0	0
$7$	8.08801	0.436711	0.218356	−	0.975869i	$-0.429931\pi$
$7$	8.08801	0.436711	0.218356	+	0.975869i	$0.429931\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	16.6320	0.525950
$11$	12.7720	0.350082	0.175041	−	0.984561i	$-0.443994\pi$
$11$	12.7720	0.350082	0.175041	+	0.984561i	$0.443994\pi$
$12$	0	0
$13$	−47.0360	−1.00350	−0.501748	−	0.865014i	$-0.667309\pi$
$13$	−47.0360	−1.00350	−0.501748	+	0.865014i	$0.667309\pi$
$14$	−16.1760	−0.308802
$15$	0	0
$16$	16.0000	0.250000
$17$	31.4560	0.448776	0.224388	−	0.974500i	$-0.427962\pi$
$17$	31.4560	0.448776	0.224388	+	0.974500i	$0.427962\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−33.2640	−0.371903
$21$	0	0
$22$	−25.5440	−0.247545
$23$	19.0360	0.172578	0.0862888	−	0.996270i	$-0.472499\pi$
$23$	19.0360	0.172578	0.0862888	+	0.996270i	$0.472499\pi$
$24$	0	0
$25$	−55.8441	−0.446752
$26$	94.0720	0.709579
$27$	0	0
$28$	32.3520	0.218356
$29$	−91.2120	−0.584057	−0.292028	−	0.956410i	$-0.594330\pi$
$29$	−91.2120	−0.584057	−0.292028	+	0.956410i	$0.594330\pi$
$30$	0	0
$31$	293.968	1.70317	0.851584	−	0.524218i	$-0.175642\pi$
$31$	293.968	1.70317	0.851584	+	0.524218i	$0.175642\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−62.9120	−0.317333
$35$	−67.2599	−0.324829
$36$	0	0
$37$	215.616	0.958029	0.479014	−	0.877807i	$-0.340994\pi$
$37$	215.616	0.958029	0.479014	+	0.877807i	$0.340994\pi$
$38$	−38.0000	−0.162221
$39$	0	0
$40$	66.5280	0.262975
$41$	67.7200	0.257953	0.128977	−	0.991648i	$-0.458831\pi$
$41$	67.7200	0.257953	0.128977	+	0.991648i	$0.458831\pi$
$42$	0	0
$43$	308.596	1.09443	0.547214	−	0.836992i	$-0.315688\pi$
$43$	308.596	1.09443	0.547214	+	0.836992i	$0.315688\pi$
$44$	51.0880	0.175041
$45$	0	0
$46$	−38.0720	−0.122031
$47$	−108.812	−0.337700	−0.168850	−	0.985642i	$-0.554005\pi$
$47$	−108.812	−0.337700	−0.168850	+	0.985642i	$0.554005\pi$
$48$	0	0
$49$	−277.584	−0.809283
$50$	111.688	0.315902
$51$	0	0
$52$	−188.144	−0.501748
$53$	682.124	1.76787	0.883933	−	0.467613i	$-0.154886\pi$
$53$	682.124	1.76787	0.883933	+	0.467613i	$0.154886\pi$
$54$	0	0
$55$	−106.212	−0.260393
$56$	−64.7041	−0.154401
$57$	0	0
$58$	182.424	0.412991
$59$	250.300	0.552310	0.276155	−	0.961113i	$-0.410940\pi$
$59$	250.300	0.552310	0.276155	+	0.961113i	$0.410940\pi$
$60$	0	0
$61$	−317.692	−0.666825	−0.333412	−	0.942781i	$-0.608200\pi$
$61$	−317.692	−0.666825	−0.333412	+	0.942781i	$0.608200\pi$
$62$	−587.936	−1.20432
$63$	0	0
$64$	64.0000	0.125000
$65$	391.152	0.746406
$66$	0	0
$67$	940.444	1.71483	0.857414	−	0.514626i	$-0.172069\pi$
$67$	940.444	1.71483	0.857414	+	0.514626i	$0.172069\pi$
$68$	125.824	0.224388
$69$	0	0
$70$	134.520	0.229689
$71$	395.552	0.661175	0.330587	−	0.943775i	$-0.392753\pi$
$71$	395.552	0.661175	0.330587	+	0.943775i	$0.392753\pi$
$72$	0	0
$73$	975.048	1.56330	0.781649	−	0.623718i	$-0.214379\pi$
$73$	975.048	1.56330	0.781649	+	0.623718i	$0.214379\pi$
$74$	−431.232	−0.677429
$75$	0	0
$76$	76.0000	0.114708
$77$	103.300	0.152885
$78$	0	0
$79$	922.776	1.31418	0.657091	−	0.753811i	$-0.271787\pi$
$79$	922.776	1.31418	0.657091	+	0.753811i	$0.271787\pi$
$80$	−133.056	−0.185952
$81$	0	0
$82$	−135.440	−0.182401
$83$	1163.77	1.53904	0.769519	−	0.638624i	$-0.220496\pi$
$83$	1163.77	1.53904	0.769519	+	0.638624i	$0.220496\pi$
$84$	0	0
$85$	−261.588	−0.333803
$86$	−617.192	−0.773878
$87$	0	0
$88$	−102.176	−0.123773
$89$	−685.136	−0.816003	−0.408002	−	0.912981i	$-0.633774\pi$
$89$	−685.136	−0.816003	−0.408002	+	0.912981i	$0.633774\pi$
$90$	0	0
$91$	−380.428	−0.438238
$92$	76.1441	0.0862888
$93$	0	0
$94$	217.624	0.238790
$95$	−158.004	−0.170641
$96$	0	0
$97$	211.256	0.221132	0.110566	−	0.993869i	$-0.464734\pi$
$97$	211.256	0.221132	0.110566	+	0.993869i	$0.464734\pi$
$98$	555.168	0.572250
$99$	0	0
$100$	−223.376	−0.223376
$101$	1703.55	1.67831	0.839157	−	0.543889i	$-0.183049\pi$
$101$	1703.55	1.67831	0.839157	+	0.543889i	$0.183049\pi$
$102$	0	0
$103$	−1393.52	−1.33308	−0.666542	−	0.745468i	$-0.732226\pi$
$103$	−1393.52	−1.33308	−0.666542	+	0.745468i	$0.732226\pi$
$104$	376.288	0.354789
$105$	0	0
$106$	−1364.25	−1.25007
$107$	−907.996	−0.820367	−0.410184	−	0.912003i	$-0.634535\pi$
$107$	−907.996	−0.820367	−0.410184	+	0.912003i	$0.634535\pi$
$108$	0	0
$109$	862.077	0.757541	0.378770	−	0.925491i	$-0.376347\pi$
$109$	862.077	0.757541	0.378770	+	0.925491i	$0.376347\pi$
$110$	212.424	0.184126
$111$	0	0
$112$	129.408	0.109178
$113$	−1502.72	−1.25101	−0.625505	−	0.780220i	$-0.715107\pi$
$113$	−1502.72	−1.25101	−0.625505	+	0.780220i	$0.715107\pi$
$114$	0	0
$115$	−158.304	−0.128364
$116$	−364.848	−0.292028
$117$	0	0
$118$	−500.600	−0.390542
$119$	254.416	0.195986
$120$	0	0
$121$	−1167.88	−0.877443
$122$	635.384	0.471516
$123$	0	0
$124$	1175.87	0.851584
$125$	1503.90	1.07610
$126$	0	0
$127$	389.280	0.271992	0.135996	−	0.990709i	$-0.456577\pi$
$127$	389.280	0.271992	0.135996	+	0.990709i	$0.456577\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−782.304	−0.527789
$131$	268.308	0.178948	0.0894739	−	0.995989i	$-0.471481\pi$
$131$	268.308	0.178948	0.0894739	+	0.995989i	$0.471481\pi$
$132$	0	0
$133$	153.672	0.100188
$134$	−1880.89	−1.21257
$135$	0	0
$136$	−251.648	−0.158666
$137$	−2657.33	−1.65716	−0.828580	−	0.559871i	$-0.810851\pi$
$137$	−2657.33	−1.65716	−0.828580	+	0.559871i	$0.810851\pi$
$138$	0	0
$139$	−2859.92	−1.74514	−0.872572	−	0.488486i	$-0.837549\pi$
$139$	−2859.92	−1.74514	−0.872572	+	0.488486i	$0.837549\pi$
$140$	−269.040	−0.162414
$141$	0	0
$142$	−791.104	−0.467521
$143$	−600.744	−0.351306
$144$	0	0
$145$	758.520	0.434425
$146$	−1950.10	−1.10542
$147$	0	0
$148$	862.464	0.479014
$149$	−311.812	−0.171440	−0.0857202	−	0.996319i	$-0.527319\pi$
$149$	−311.812	−0.171440	−0.0857202	+	0.996319i	$0.527319\pi$
$150$	0	0
$151$	−1462.32	−0.788093	−0.394046	−	0.919091i	$-0.628925\pi$
$151$	−1462.32	−0.788093	−0.394046	+	0.919091i	$0.628925\pi$
$152$	−152.000	−0.0811107
$153$	0	0
$154$	−206.600	−0.108106
$155$	−2444.64	−1.26683
$156$	0	0
$157$	4.38395	0.00222852	0.00111426	−	0.999999i	$-0.499645\pi$
$157$	4.38395	0.00222852	0.00111426	+	0.999999i	$0.499645\pi$
$158$	−1845.55	−0.929267
$159$	0	0
$160$	266.112	0.131488
$161$	153.964	0.0753666
$162$	0	0
$163$	−1777.89	−0.854325	−0.427162	−	0.904175i	$-0.640487\pi$
$163$	−1777.89	−0.854325	−0.427162	+	0.904175i	$0.640487\pi$
$164$	270.880	0.128977
$165$	0	0
$166$	−2327.54	−1.08826
$167$	−893.064	−0.413817	−0.206908	−	0.978360i	$-0.566340\pi$
$167$	−893.064	−0.413817	−0.206908	+	0.978360i	$0.566340\pi$
$168$	0	0
$169$	15.3876	0.00700391
$170$	523.176	0.236034
$171$	0	0
$172$	1234.38	0.547214
$173$	2452.56	1.07783	0.538915	−	0.842360i	$-0.318834\pi$
$173$	2452.56	1.07783	0.538915	+	0.842360i	$0.318834\pi$
$174$	0	0
$175$	−451.667	−0.195102
$176$	204.352	0.0875205
$177$	0	0
$178$	1370.27	0.577002
$179$	2064.81	0.862185	0.431092	−	0.902308i	$-0.358128\pi$
$179$	2064.81	0.862185	0.431092	+	0.902308i	$0.358128\pi$
$180$	0	0
$181$	−2518.54	−1.03426	−0.517132	−	0.855906i	$-0.673000\pi$
$181$	−2518.54	−1.03426	−0.517132	+	0.855906i	$0.673000\pi$
$182$	760.855	0.309881
$183$	0	0
$184$	−152.288	−0.0610154
$185$	−1793.06	−0.712588
$186$	0	0
$187$	401.756	0.157109
$188$	−435.249	−0.168850
$189$	0	0
$190$	316.008	0.120661
$191$	4206.38	1.59352	0.796761	−	0.604294i	$-0.206545\pi$
$191$	4206.38	1.59352	0.796761	+	0.604294i	$0.206545\pi$
$192$	0	0
$193$	3245.82	1.21056	0.605282	−	0.796011i	$-0.293060\pi$
$193$	3245.82	1.21056	0.605282	+	0.796011i	$0.293060\pi$
$194$	−422.512	−0.156364
$195$	0	0
$196$	−1110.34	−0.404642
$197$	1734.71	0.627377	0.313688	−	0.949526i	$-0.398435\pi$
$197$	1734.71	0.627377	0.313688	+	0.949526i	$0.398435\pi$
$198$	0	0
$199$	380.792	0.135646	0.0678232	−	0.997697i	$-0.478395\pi$
$199$	380.792	0.135646	0.0678232	+	0.997697i	$0.478395\pi$
$200$	446.752	0.157951
$201$	0	0
$202$	−3407.10	−1.18675
$203$	−737.724	−0.255064
$204$	0	0
$205$	−563.160	−0.191867
$206$	2787.04	0.942633
$207$	0	0
$208$	−752.576	−0.250874
$209$	242.668	0.0803143
$210$	0	0
$211$	1010.44	0.329675	0.164837	−	0.986321i	$-0.447290\pi$
$211$	1010.44	0.329675	0.164837	+	0.986321i	$0.447290\pi$
$212$	2728.50	0.883933
$213$	0	0
$214$	1815.99	0.580087
$215$	−2566.29	−0.814043
$216$	0	0
$217$	2377.62	0.743793
$218$	−1724.15	−0.535662
$219$	0	0
$220$	−424.848	−0.130197
$221$	−1479.57	−0.450345
$222$	0	0
$223$	3398.70	1.02060	0.510301	−	0.859996i	$-0.329534\pi$
$223$	3398.70	1.02060	0.510301	+	0.859996i	$0.329534\pi$
$224$	−258.816	−0.0772004
$225$	0	0
$226$	3005.44	0.884597
$227$	−5760.80	−1.68439	−0.842197	−	0.539169i	$-0.818738\pi$
$227$	−5760.80	−1.68439	−0.842197	+	0.539169i	$0.818738\pi$
$228$	0	0
$229$	−2179.00	−0.628786	−0.314393	−	0.949293i	$-0.601801\pi$
$229$	−2179.00	−0.628786	−0.314393	+	0.949293i	$0.601801\pi$
$230$	316.607	0.0907673
$231$	0	0
$232$	729.696	0.206495
$233$	2808.49	0.789659	0.394830	−	0.918754i	$-0.370804\pi$
$233$	2808.49	0.789659	0.394830	+	0.918754i	$0.370804\pi$
$234$	0	0
$235$	904.882	0.251183
$236$	1001.20	0.276155
$237$	0	0
$238$	−508.833	−0.138583
$239$	−6285.67	−1.70120	−0.850599	−	0.525815i	$-0.823761\pi$
$239$	−6285.67	−1.70120	−0.850599	+	0.525815i	$0.823761\pi$
$240$	0	0
$241$	1129.22	0.301825	0.150912	−	0.988547i	$-0.451779\pi$
$241$	1129.22	0.301825	0.150912	+	0.988547i	$0.451779\pi$
$242$	2335.75	0.620446
$243$	0	0
$244$	−1270.77	−0.333412
$245$	2308.39	0.601950
$246$	0	0
$247$	−893.684	−0.230218
$248$	−2351.74	−0.602161
$249$	0	0
$250$	−3007.80	−0.760920
$251$	2873.73	0.722661	0.361331	−	0.932438i	$-0.382323\pi$
$251$	2873.73	0.722661	0.361331	+	0.932438i	$0.382323\pi$
$252$	0	0
$253$	243.128	0.0604163
$254$	−778.559	−0.192327
$255$	0	0
$256$	256.000	0.0625000
$257$	3712.18	0.901008	0.450504	−	0.892774i	$-0.351244\pi$
$257$	3712.18	0.901008	0.450504	+	0.892774i	$0.351244\pi$
$258$	0	0
$259$	1743.90	0.418382
$260$	1564.61	0.373203
$261$	0	0
$262$	−536.616	−0.126535
$263$	−1263.04	−0.296130	−0.148065	−	0.988978i	$-0.547304\pi$
$263$	−1263.04	−0.296130	−0.148065	+	0.988978i	$0.547304\pi$
$264$	0	0
$265$	−5672.55	−1.31495
$266$	−307.344	−0.0708439
$267$	0	0
$268$	3761.78	0.857414
$269$	5484.39	1.24308	0.621541	−	0.783381i	$-0.286507\pi$
$269$	5484.39	1.24308	0.621541	+	0.783381i	$0.286507\pi$
$270$	0	0
$271$	−3217.66	−0.721251	−0.360625	−	0.932711i	$-0.617437\pi$
$271$	−3217.66	−0.721251	−0.360625	+	0.932711i	$0.617437\pi$
$272$	503.296	0.112194
$273$	0	0
$274$	5314.66	1.17179
$275$	−713.240	−0.156400
$276$	0	0
$277$	7668.13	1.66330	0.831649	−	0.555302i	$-0.187397\pi$
$277$	7668.13	1.66330	0.831649	+	0.555302i	$0.187397\pi$
$278$	5719.83	1.23400
$279$	0	0
$280$	538.079	0.114844
$281$	−1126.81	−0.239216	−0.119608	−	0.992821i	$-0.538164\pi$
$281$	−1126.81	−0.239216	−0.119608	+	0.992821i	$0.538164\pi$
$282$	0	0
$283$	−1502.63	−0.315625	−0.157813	−	0.987469i	$-0.550444\pi$
$283$	−1502.63	−0.315625	−0.157813	+	0.987469i	$0.550444\pi$
$284$	1582.21	0.330587
$285$	0	0
$286$	1201.49	0.248411
$287$	547.720	0.112651
$288$	0	0
$289$	−3923.52	−0.798600
$290$	−1517.04	−0.307185
$291$	0	0
$292$	3900.19	0.781649
$293$	452.324	0.0901878	0.0450939	−	0.998983i	$-0.485641\pi$
$293$	452.324	0.0901878	0.0450939	+	0.998983i	$0.485641\pi$
$294$	0	0
$295$	−2081.50	−0.410812
$296$	−1724.93	−0.338714
$297$	0	0
$298$	623.623	0.121227
$299$	−895.379	−0.173181
$300$	0	0
$301$	2495.93	0.477950
$302$	2924.64	0.557266
$303$	0	0
$304$	304.000	0.0573539
$305$	2641.93	0.495988
$306$	0	0
$307$	−2333.46	−0.433803	−0.216901	−	0.976194i	$-0.569595\pi$
$307$	−2333.46	−0.433803	−0.216901	+	0.976194i	$0.569595\pi$
$308$	413.200	0.0764424
$309$	0	0
$310$	4889.28	0.895782
$311$	−10476.1	−1.91011	−0.955055	−	0.296429i	$-0.904204\pi$
$311$	−10476.1	−1.91011	−0.955055	+	0.296429i	$0.904204\pi$
$312$	0	0
$313$	4160.33	0.751297	0.375648	−	0.926762i	$-0.377420\pi$
$313$	4160.33	0.751297	0.375648	+	0.926762i	$0.377420\pi$
$314$	−8.76790	−0.00157580
$315$	0	0
$316$	3691.10	0.657091
$317$	7508.56	1.33036	0.665178	−	0.746685i	$-0.268356\pi$
$317$	7508.56	1.33036	0.665178	+	0.746685i	$0.268356\pi$
$318$	0	0
$319$	−1164.96	−0.204468
$320$	−532.224	−0.0929758
$321$	0	0
$322$	−307.927	−0.0532922
$323$	597.664	0.102956
$324$	0	0
$325$	2626.68	0.448314
$326$	3555.78	0.604099
$327$	0	0
$328$	−541.760	−0.0912003
$329$	−880.073	−0.147477
$330$	0	0
$331$	10386.8	1.72480	0.862400	−	0.506227i	$-0.168960\pi$
$331$	10386.8	1.72480	0.862400	+	0.506227i	$0.168960\pi$
$332$	4655.07	0.769519
$333$	0	0
$334$	1786.13	0.292613
$335$	−7820.74	−1.27550
$336$	0	0
$337$	5618.29	0.908153	0.454077	−	0.890963i	$-0.349969\pi$
$337$	5618.29	0.908153	0.454077	+	0.890963i	$0.349969\pi$
$338$	−30.7752	−0.00495251
$339$	0	0
$340$	−1046.35	−0.166901
$341$	3754.56	0.596249
$342$	0	0
$343$	−5019.29	−0.790135
$344$	−2468.77	−0.386939
$345$	0	0
$346$	−4905.12	−0.762142
$347$	−1814.32	−0.280686	−0.140343	−	0.990103i	$-0.544820\pi$
$347$	−1814.32	−0.280686	−0.140343	+	0.990103i	$0.544820\pi$
$348$	0	0
$349$	−816.757	−0.125272	−0.0626361	−	0.998036i	$-0.519951\pi$
$349$	−816.757	−0.125272	−0.0626361	+	0.998036i	$0.519951\pi$
$350$	903.334	0.137958
$351$	0	0
$352$	−408.704	−0.0618864
$353$	−11090.4	−1.67219	−0.836095	−	0.548585i	$-0.815167\pi$
$353$	−11090.4	−1.67219	−0.836095	+	0.548585i	$0.815167\pi$
$354$	0	0
$355$	−3289.41	−0.491786
$356$	−2740.55	−0.408002
$357$	0	0
$358$	−4129.62	−0.609657
$359$	3211.68	0.472161	0.236081	−	0.971733i	$-0.424137\pi$
$359$	3211.68	0.472161	0.236081	+	0.971733i	$0.424137\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	5037.09	0.731335
$363$	0	0
$364$	−1521.71	−0.219119
$365$	−8108.51	−1.16279
$366$	0	0
$367$	8077.81	1.14893	0.574466	−	0.818528i	$-0.305210\pi$
$367$	8077.81	1.14893	0.574466	+	0.818528i	$0.305210\pi$
$368$	304.576	0.0431444
$369$	0	0
$370$	3586.13	0.503876
$371$	5517.02	0.772048
$372$	0	0
$373$	−5088.15	−0.706312	−0.353156	−	0.935564i	$-0.614892\pi$
$373$	−5088.15	−0.706312	−0.353156	+	0.935564i	$0.614892\pi$
$374$	−803.512	−0.111093
$375$	0	0
$376$	870.497	0.119395
$377$	4290.25	0.586099
$378$	0	0
$379$	2547.00	0.345199	0.172600	−	0.984992i	$-0.444783\pi$
$379$	2547.00	0.345199	0.172600	+	0.984992i	$0.444783\pi$
$380$	−632.016	−0.0853204
$381$	0	0
$382$	−8412.76	−1.12679
$383$	7056.11	0.941384	0.470692	−	0.882297i	$-0.344004\pi$
$383$	7056.11	0.941384	0.470692	+	0.882297i	$0.344004\pi$
$384$	0	0
$385$	−859.044	−0.113717
$386$	−6491.63	−0.855999
$387$	0	0
$388$	845.023	0.110566
$389$	−4728.25	−0.616277	−0.308138	−	0.951342i	$-0.599706\pi$
$389$	−4728.25	−0.616277	−0.308138	+	0.951342i	$0.599706\pi$
$390$	0	0
$391$	598.797	0.0774488
$392$	2220.67	0.286125
$393$	0	0
$394$	−3469.43	−0.443622
$395$	−7673.81	−0.977497
$396$	0	0
$397$	740.837	0.0936563	0.0468281	−	0.998903i	$-0.485089\pi$
$397$	740.837	0.0936563	0.0468281	+	0.998903i	$0.485089\pi$
$398$	−761.584	−0.0959165
$399$	0	0
$400$	−893.505	−0.111688
$401$	−1879.58	−0.234070	−0.117035	−	0.993128i	$-0.537339\pi$
$401$	−1879.58	−0.234070	−0.117035	+	0.993128i	$0.537339\pi$
$402$	0	0
$403$	−13827.1	−1.70912
$404$	6814.21	0.839157
$405$	0	0
$406$	1475.45	0.180358
$407$	2753.85	0.335389
$408$	0	0
$409$	−1715.45	−0.207393	−0.103697	−	0.994609i	$-0.533067\pi$
$409$	−1715.45	−0.207393	−0.103697	+	0.994609i	$0.533067\pi$
$410$	1126.32	0.135671
$411$	0	0
$412$	−5574.08	−0.666542
$413$	2024.43	0.241200
$414$	0	0
$415$	−9677.90	−1.14475
$416$	1505.15	0.177395
$417$	0	0
$418$	−485.336	−0.0567908
$419$	−2497.15	−0.291155	−0.145578	−	0.989347i	$-0.546504\pi$
$419$	−2497.15	−0.291155	−0.145578	+	0.989347i	$0.546504\pi$
$420$	0	0
$421$	6582.52	0.762024	0.381012	−	0.924570i	$-0.375576\pi$
$421$	6582.52	0.762024	0.381012	+	0.924570i	$0.375576\pi$
$422$	−2020.87	−0.233115
$423$	0	0
$424$	−5456.99	−0.625035
$425$	−1756.63	−0.200492
$426$	0	0
$427$	−2569.50	−0.291210
$428$	−3631.99	−0.410184
$429$	0	0
$430$	5132.57	0.575615
$431$	−8875.72	−0.991946	−0.495973	−	0.868338i	$-0.665188\pi$
$431$	−8875.72	−0.991946	−0.495973	+	0.868338i	$0.665188\pi$
$432$	0	0
$433$	−3636.90	−0.403645	−0.201822	−	0.979422i	$-0.564686\pi$
$433$	−3636.90	−0.403645	−0.201822	+	0.979422i	$0.564686\pi$
$434$	−4755.23	−0.525941
$435$	0	0
$436$	3448.31	0.378770
$437$	361.684	0.0395920
$438$	0	0
$439$	−10979.4	−1.19366	−0.596829	−	0.802368i	$-0.703573\pi$
$439$	−10979.4	−1.19366	−0.596829	+	0.802368i	$0.703573\pi$
$440$	849.696	0.0920629
$441$	0	0
$442$	2959.13	0.318442
$443$	−1300.16	−0.139442	−0.0697208	−	0.997567i	$-0.522211\pi$
$443$	−1300.16	−0.139442	−0.0697208	+	0.997567i	$0.522211\pi$
$444$	0	0
$445$	5697.60	0.606948
$446$	−6797.41	−0.721674
$447$	0	0
$448$	517.632	0.0545889
$449$	15875.2	1.66859	0.834296	−	0.551317i	$-0.185875\pi$
$449$	15875.2	1.66859	0.834296	+	0.551317i	$0.185875\pi$
$450$	0	0
$451$	864.920	0.0903049
$452$	−6010.88	−0.625505
$453$	0	0
$454$	11521.6	1.19105
$455$	3163.64	0.325964
$456$	0	0
$457$	3115.66	0.318916	0.159458	−	0.987205i	$-0.449025\pi$
$457$	3115.66	0.318916	0.159458	+	0.987205i	$0.449025\pi$
$458$	4357.99	0.444619
$459$	0	0
$460$	−633.215	−0.0641822
$461$	−13479.7	−1.36185	−0.680924	−	0.732354i	$-0.738422\pi$
$461$	−13479.7	−1.36185	−0.680924	+	0.732354i	$0.738422\pi$
$462$	0	0
$463$	7946.19	0.797604	0.398802	−	0.917037i	$-0.369426\pi$
$463$	7946.19	0.797604	0.398802	+	0.917037i	$0.369426\pi$
$464$	−1459.39	−0.146014
$465$	0	0
$466$	−5616.99	−0.558373
$467$	9148.37	0.906501	0.453250	−	0.891383i	$-0.350264\pi$
$467$	9148.37	0.906501	0.453250	+	0.891383i	$0.350264\pi$
$468$	0	0
$469$	7606.32	0.748885
$470$	−1809.76	−0.177613
$471$	0	0
$472$	−2002.40	−0.195271
$473$	3941.39	0.383140
$474$	0	0
$475$	−1061.04	−0.102492
$476$	1017.67	0.0979929
$477$	0	0
$478$	12571.3	1.20293
$479$	−7664.64	−0.731120	−0.365560	−	0.930788i	$-0.619122\pi$
$479$	−7664.64	−0.731120	−0.365560	+	0.930788i	$0.619122\pi$
$480$	0	0
$481$	−10141.7	−0.961378
$482$	−2258.45	−0.213422
$483$	0	0
$484$	−4671.50	−0.438721
$485$	−1756.80	−0.164479
$486$	0	0
$487$	−5347.21	−0.497547	−0.248774	−	0.968562i	$-0.580027\pi$
$487$	−5347.21	−0.497547	−0.248774	+	0.968562i	$0.580027\pi$
$488$	2541.54	0.235758
$489$	0	0
$490$	−4616.78	−0.425643
$491$	−13647.2	−1.25436	−0.627178	−	0.778876i	$-0.715790\pi$
$491$	−13647.2	−1.25436	−0.627178	+	0.778876i	$0.715790\pi$
$492$	0	0
$493$	−2869.17	−0.262111
$494$	1787.37	0.162789
$495$	0	0
$496$	4703.49	0.425792
$497$	3199.23	0.288743
$498$	0	0
$499$	−19351.6	−1.73607	−0.868034	−	0.496504i	$-0.834617\pi$
$499$	−19351.6	−1.73607	−0.868034	+	0.496504i	$0.834617\pi$
$500$	6015.60	0.538052
$501$	0	0
$502$	−5747.45	−0.510999
$503$	19259.1	1.70720	0.853600	−	0.520929i	$-0.174414\pi$
$503$	19259.1	1.70720	0.853600	+	0.520929i	$0.174414\pi$
$504$	0	0
$505$	−14166.7	−1.24834
$506$	−486.256	−0.0427208
$507$	0	0
$508$	1557.12	0.135996
$509$	−3595.77	−0.313123	−0.156561	−	0.987668i	$-0.550041\pi$
$509$	−3595.77	−0.313123	−0.156561	+	0.987668i	$0.550041\pi$
$510$	0	0
$511$	7886.20	0.682710
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−7424.35	−0.637109
$515$	11588.5	0.991556
$516$	0	0
$517$	−1389.75	−0.118223
$518$	−3487.81	−0.295841
$519$	0	0
$520$	−3129.21	−0.263895
$521$	15211.0	1.27909	0.639544	−	0.768754i	$-0.279123\pi$
$521$	15211.0	1.27909	0.639544	+	0.768754i	$0.279123\pi$
$522$	0	0
$523$	18307.1	1.53062	0.765310	−	0.643662i	$-0.222586\pi$
$523$	18307.1	1.53062	0.765310	+	0.643662i	$0.222586\pi$
$524$	1073.23	0.0894739
$525$	0	0
$526$	2526.07	0.209395
$527$	9247.06	0.764342
$528$	0	0
$529$	−11804.6	−0.970217
$530$	11345.1	0.929810
$531$	0	0
$532$	614.689	0.0500942
$533$	−3185.28	−0.258855
$534$	0	0
$535$	7550.90	0.610194
$536$	−7523.55	−0.606284
$537$	0	0
$538$	−10968.8	−0.878992
$539$	−3545.31	−0.283316
$540$	0	0
$541$	9102.17	0.723351	0.361676	−	0.932304i	$-0.382205\pi$
$541$	9102.17	0.723351	0.361676	+	0.932304i	$0.382205\pi$
$542$	6435.32	0.510001
$543$	0	0
$544$	−1006.59	−0.0793332
$545$	−7169.03	−0.563464
$546$	0	0
$547$	−9218.75	−0.720595	−0.360297	−	0.932837i	$-0.617325\pi$
$547$	−9218.75	−0.720595	−0.360297	+	0.932837i	$0.617325\pi$
$548$	−10629.3	−0.828580
$549$	0	0
$550$	1426.48	0.110592
$551$	−1733.03	−0.133992
$552$	0	0
$553$	7463.42	0.573918
$554$	−15336.3	−1.17613
$555$	0	0
$556$	−11439.7	−0.872572
$557$	13435.1	1.02202	0.511010	−	0.859575i	$-0.329272\pi$
$557$	13435.1	1.02202	0.511010	+	0.859575i	$0.329272\pi$
$558$	0	0
$559$	−14515.1	−1.09825
$560$	−1076.16	−0.0812071
$561$	0	0
$562$	2253.62	0.169151
$563$	−11941.5	−0.893916	−0.446958	−	0.894555i	$-0.647493\pi$
$563$	−11941.5	−0.893916	−0.446958	+	0.894555i	$0.647493\pi$
$564$	0	0
$565$	12496.6	0.930508
$566$	3005.26	0.223181
$567$	0	0
$568$	−3164.42	−0.233761
$569$	6378.91	0.469979	0.234989	−	0.971998i	$-0.424494\pi$
$569$	6378.91	0.469979	0.234989	+	0.971998i	$0.424494\pi$
$570$	0	0
$571$	24903.9	1.82521	0.912605	−	0.408843i	$-0.134068\pi$
$571$	24903.9	1.82521	0.912605	+	0.408843i	$0.134068\pi$
$572$	−2402.98	−0.175653
$573$	0	0
$574$	−1095.44	−0.0796564
$575$	−1063.05	−0.0770995
$576$	0	0
$577$	−11414.7	−0.823568	−0.411784	−	0.911281i	$-0.635094\pi$
$577$	−11414.7	−0.823568	−0.411784	+	0.911281i	$0.635094\pi$
$578$	7847.04	0.564695
$579$	0	0
$580$	3034.08	0.217213
$581$	9412.57	0.672115
$582$	0	0
$583$	8712.09	0.618899
$584$	−7800.39	−0.552709
$585$	0	0
$586$	−904.648	−0.0637724
$587$	−20732.1	−1.45776	−0.728881	−	0.684641i	$-0.759959\pi$
$587$	−20732.1	−1.45776	−0.728881	+	0.684641i	$0.759959\pi$
$588$	0	0
$589$	5585.39	0.390734
$590$	4162.99	0.290488
$591$	0	0
$592$	3449.86	0.239507
$593$	−18010.5	−1.24722	−0.623611	−	0.781735i	$-0.714335\pi$
$593$	−18010.5	−1.24722	−0.623611	+	0.781735i	$0.714335\pi$
$594$	0	0
$595$	−2115.73	−0.145775
$596$	−1247.25	−0.0857202
$597$	0	0
$598$	1790.76	0.122457
$599$	−27944.7	−1.90616	−0.953080	−	0.302719i	$-0.902106\pi$
$599$	−27944.7	−1.90616	−0.953080	+	0.302719i	$0.902106\pi$
$600$	0	0
$601$	−11598.1	−0.787179	−0.393590	−	0.919286i	$-0.628767\pi$
$601$	−11598.1	−0.787179	−0.393590	+	0.919286i	$0.628767\pi$
$602$	−4991.85	−0.337961
$603$	0	0
$604$	−5849.28	−0.394046
$605$	9712.06	0.652647
$606$	0	0
$607$	20170.5	1.34876	0.674379	−	0.738385i	$-0.264411\pi$
$607$	20170.5	1.34876	0.674379	+	0.738385i	$0.264411\pi$
$608$	−608.000	−0.0405554
$609$	0	0
$610$	−5283.86	−0.350717
$611$	5118.09	0.338880
$612$	0	0
$613$	14618.3	0.963174	0.481587	−	0.876398i	$-0.340061\pi$
$613$	14618.3	0.963174	0.481587	+	0.876398i	$0.340061\pi$
$614$	4666.91	0.306745
$615$	0	0
$616$	−826.400	−0.0540530
$617$	17538.1	1.14434	0.572171	−	0.820134i	$-0.306101\pi$
$617$	17538.1	1.14434	0.572171	+	0.820134i	$0.306101\pi$
$618$	0	0
$619$	−8815.75	−0.572431	−0.286216	−	0.958165i	$-0.592397\pi$
$619$	−8815.75	−0.572431	−0.286216	+	0.958165i	$0.592397\pi$
$620$	−9778.56	−0.633414
$621$	0	0
$622$	20952.2	1.35065
$623$	−5541.39	−0.356358
$624$	0	0
$625$	−5525.94	−0.353660
$626$	−8320.66	−0.531247
$627$	0	0
$628$	17.5358	0.00111426
$629$	6782.42	0.429941
$630$	0	0
$631$	−22170.8	−1.39874	−0.699370	−	0.714759i	$-0.746536\pi$
$631$	−22170.8	−1.39874	−0.699370	+	0.714759i	$0.746536\pi$
$632$	−7382.21	−0.464634
$633$	0	0
$634$	−15017.1	−0.940703
$635$	−3237.25	−0.202309
$636$	0	0
$637$	13056.5	0.812112
$638$	2329.92	0.144581
$639$	0	0
$640$	1064.45	0.0657438
$641$	22067.7	1.35978	0.679891	−	0.733313i	$-0.262027\pi$
$641$	22067.7	1.35978	0.679891	+	0.733313i	$0.262027\pi$
$642$	0	0
$643$	−11795.4	−0.723428	−0.361714	−	0.932289i	$-0.617808\pi$
$643$	−11795.4	−0.723428	−0.361714	+	0.932289i	$0.617808\pi$
$644$	615.854	0.0376833
$645$	0	0
$646$	−1195.33	−0.0728012
$647$	9716.04	0.590382	0.295191	−	0.955438i	$-0.404617\pi$
$647$	9716.04	0.590382	0.295191	+	0.955438i	$0.404617\pi$
$648$	0	0
$649$	3196.83	0.193354
$650$	−5253.36	−0.317006
$651$	0	0
$652$	−7111.55	−0.427162
$653$	10311.9	0.617969	0.308985	−	0.951067i	$-0.400011\pi$
$653$	10311.9	0.617969	0.308985	+	0.951067i	$0.400011\pi$
$654$	0	0
$655$	−2231.25	−0.133102
$656$	1083.52	0.0644884
$657$	0	0
$658$	1760.15	0.104282
$659$	−4019.80	−0.237616	−0.118808	−	0.992917i	$-0.537907\pi$
$659$	−4019.80	−0.237616	−0.118808	+	0.992917i	$0.537907\pi$
$660$	0	0
$661$	−22702.6	−1.33590	−0.667951	−	0.744206i	$-0.732828\pi$
$661$	−22702.6	−1.33590	−0.667951	+	0.744206i	$0.732828\pi$
$662$	−20773.6	−1.21962
$663$	0	0
$664$	−9310.15	−0.544132
$665$	−1277.94	−0.0745208
$666$	0	0
$667$	−1736.31	−0.100795
$668$	−3572.26	−0.206908
$669$	0	0
$670$	15641.5	0.901915
$671$	−4057.57	−0.233443
$672$	0	0
$673$	11132.8	0.637652	0.318826	−	0.947813i	$-0.396711\pi$
$673$	11132.8	0.637652	0.318826	+	0.947813i	$0.396711\pi$
$674$	−11236.6	−0.642161
$675$	0	0
$676$	61.5503	0.00350195
$677$	13967.0	0.792903	0.396452	−	0.918056i	$-0.370241\pi$
$677$	13967.0	0.792903	0.396452	+	0.918056i	$0.370241\pi$
$678$	0	0
$679$	1708.64	0.0965707
$680$	2092.71	0.118017
$681$	0	0
$682$	−7509.12	−0.421612
$683$	−1173.88	−0.0657648	−0.0328824	−	0.999459i	$-0.510469\pi$
$683$	−1173.88	−0.0657648	−0.0328824	+	0.999459i	$0.510469\pi$
$684$	0	0
$685$	22098.4	1.23261
$686$	10038.6	0.558709
$687$	0	0
$688$	4937.54	0.273607
$689$	−32084.4	−1.77405
$690$	0	0
$691$	8713.33	0.479697	0.239849	−	0.970810i	$-0.422902\pi$
$691$	8713.33	0.479697	0.239849	+	0.970810i	$0.422902\pi$
$692$	9810.24	0.538915
$693$	0	0
$694$	3628.64	0.198475
$695$	23783.1	1.29805
$696$	0	0
$697$	2130.20	0.115763
$698$	1633.51	0.0885809
$699$	0	0
$700$	−1806.67	−0.0975509
$701$	31003.4	1.67045	0.835223	−	0.549912i	$-0.185339\pi$
$701$	31003.4	1.67045	0.835223	+	0.549912i	$0.185339\pi$
$702$	0	0
$703$	4096.70	0.219787
$704$	817.408	0.0437603
$705$	0	0
$706$	22180.8	1.18242
$707$	13778.3	0.732939
$708$	0	0
$709$	−12145.1	−0.643328	−0.321664	−	0.946854i	$-0.604242\pi$
$709$	−12145.1	−0.643328	−0.321664	+	0.946854i	$0.604242\pi$
$710$	6578.83	0.347745
$711$	0	0
$712$	5481.09	0.288501
$713$	5595.98	0.293929
$714$	0	0
$715$	4995.79	0.261304
$716$	8259.24	0.431092
$717$	0	0
$718$	−6423.36	−0.333868
$719$	−24787.8	−1.28572	−0.642858	−	0.765985i	$-0.722252\pi$
$719$	−24787.8	−1.28572	−0.642858	+	0.765985i	$0.722252\pi$
$720$	0	0
$721$	−11270.8	−0.582173
$722$	−722.000	−0.0372161
$723$	0	0
$724$	−10074.2	−0.517132
$725$	5093.65	0.260929
$726$	0	0
$727$	19335.6	0.986409	0.493204	−	0.869914i	$-0.335826\pi$
$727$	19335.6	0.986409	0.493204	+	0.869914i	$0.335826\pi$
$728$	3043.42	0.154941
$729$	0	0
$730$	16217.0	0.822217
$731$	9707.19	0.491154
$732$	0	0
$733$	20204.5	1.01810	0.509052	−	0.860735i	$-0.329996\pi$
$733$	20204.5	1.01810	0.509052	+	0.860735i	$0.329996\pi$
$734$	−16155.6	−0.812418
$735$	0	0
$736$	−609.153	−0.0305077
$737$	12011.4	0.600331
$738$	0	0
$739$	15643.7	0.778706	0.389353	−	0.921089i	$-0.372699\pi$
$739$	15643.7	0.778706	0.389353	+	0.921089i	$0.372699\pi$
$740$	−7172.26	−0.356294
$741$	0	0
$742$	−11034.0	−0.545920
$743$	4500.20	0.222202	0.111101	−	0.993809i	$-0.464562\pi$
$743$	4500.20	0.222202	0.111101	+	0.993809i	$0.464562\pi$
$744$	0	0
$745$	2593.03	0.127518
$746$	10176.3	0.499438
$747$	0	0
$748$	1607.02	0.0785543
$749$	−7343.88	−0.358264
$750$	0	0
$751$	35080.2	1.70452	0.852261	−	0.523117i	$-0.175231\pi$
$751$	35080.2	1.70452	0.852261	+	0.523117i	$0.175231\pi$
$752$	−1740.99	−0.0844249
$753$	0	0
$754$	−8580.50	−0.414434
$755$	12160.7	0.586188
$756$	0	0
$757$	−10391.8	−0.498938	−0.249469	−	0.968383i	$-0.580256\pi$
$757$	−10391.8	−0.498938	−0.249469	+	0.968383i	$0.580256\pi$
$758$	−5094.00	−0.244093
$759$	0	0
$760$	1264.03	0.0603306
$761$	−11810.5	−0.562590	−0.281295	−	0.959621i	$-0.590764\pi$
$761$	−11810.5	−0.562590	−0.281295	+	0.959621i	$0.590764\pi$
$762$	0	0
$763$	6972.48	0.330827
$764$	16825.5	0.796761
$765$	0	0
$766$	−14112.2	−0.665659
$767$	−11773.1	−0.554241
$768$	0	0
$769$	35125.5	1.64715	0.823574	−	0.567209i	$-0.191977\pi$
$769$	35125.5	1.64715	0.823574	+	0.567209i	$0.191977\pi$
$770$	1718.09	0.0804098
$771$	0	0
$772$	12983.3	0.605282
$773$	−20001.5	−0.930665	−0.465332	−	0.885136i	$-0.654065\pi$
$773$	−20001.5	−0.930665	−0.465332	+	0.885136i	$0.654065\pi$
$774$	0	0
$775$	−16416.4	−0.760895
$776$	−1690.05	−0.0781819
$777$	0	0
$778$	9456.49	0.435773
$779$	1286.68	0.0591786
$780$	0	0
$781$	5051.99	0.231465
$782$	−1197.59	−0.0547646
$783$	0	0
$784$	−4441.35	−0.202321
$785$	−36.4569	−0.00165758
$786$	0	0
$787$	13593.3	0.615690	0.307845	−	0.951437i	$-0.400392\pi$
$787$	13593.3	0.615690	0.307845	+	0.951437i	$0.400392\pi$
$788$	6938.85	0.313688
$789$	0	0
$790$	15347.6	0.691195
$791$	−12154.0	−0.546330
$792$	0	0
$793$	14943.0	0.669156
$794$	−1481.67	−0.0662250
$795$	0	0
$796$	1523.17	0.0678232
$797$	6946.75	0.308741	0.154370	−	0.988013i	$-0.450665\pi$
$797$	6946.75	0.308741	0.154370	+	0.988013i	$0.450665\pi$
$798$	0	0
$799$	−3422.79	−0.151552
$800$	1787.01	0.0789754
$801$	0	0
$802$	3759.17	0.165512
$803$	12453.3	0.547283
$804$	0	0
$805$	−1280.36	−0.0560581
$806$	27654.2	1.20853
$807$	0	0
$808$	−13628.4	−0.593374
$809$	−24987.2	−1.08591	−0.542955	−	0.839762i	$-0.682695\pi$
$809$	−24987.2	−1.08591	−0.542955	+	0.839762i	$0.682695\pi$
$810$	0	0
$811$	−23172.5	−1.00332	−0.501662	−	0.865064i	$-0.667278\pi$
$811$	−23172.5	−1.00332	−0.501662	+	0.865064i	$0.667278\pi$
$812$	−2950.89	−0.127532
$813$	0	0
$814$	−5507.70	−0.237156
$815$	14784.9	0.635452
$816$	0	0
$817$	5863.32	0.251079
$818$	3430.91	0.146649
$819$	0	0
$820$	−2252.64	−0.0959337
$821$	−30703.8	−1.30520	−0.652600	−	0.757703i	$-0.726322\pi$
$821$	−30703.8	−1.30520	−0.652600	+	0.757703i	$0.726322\pi$
$822$	0	0
$823$	−15940.1	−0.675135	−0.337568	−	0.941301i	$-0.609604\pi$
$823$	−15940.1	−0.675135	−0.337568	+	0.941301i	$0.609604\pi$
$824$	11148.2	0.471316
$825$	0	0
$826$	−4048.86	−0.170554
$827$	6662.20	0.280130	0.140065	−	0.990142i	$-0.455269\pi$
$827$	6662.20	0.280130	0.140065	+	0.990142i	$0.455269\pi$
$828$	0	0
$829$	−20606.0	−0.863299	−0.431649	−	0.902041i	$-0.642068\pi$
$829$	−20606.0	−0.863299	−0.431649	+	0.902041i	$0.642068\pi$
$830$	19355.8	0.809458
$831$	0	0
$832$	−3010.31	−0.125437
$833$	−8731.69	−0.363187
$834$	0	0
$835$	7426.73	0.307799
$836$	970.672	0.0401572
$837$	0	0
$838$	4994.31	0.205878
$839$	45717.9	1.88124	0.940618	−	0.339468i	$-0.110247\pi$
$839$	45717.9	1.88124	0.940618	+	0.339468i	$0.110247\pi$
$840$	0	0
$841$	−16069.4	−0.658878
$842$	−13165.0	−0.538833
$843$	0	0
$844$	4041.75	0.164837
$845$	−127.963	−0.00520955
$846$	0	0
$847$	−9445.79	−0.383189
$848$	10914.0	0.441967
$849$	0	0
$850$	3513.26	0.141769
$851$	4104.47	0.165334
$852$	0	0
$853$	17230.4	0.691626	0.345813	−	0.938303i	$-0.387603\pi$
$853$	17230.4	0.691626	0.345813	+	0.938303i	$0.387603\pi$
$854$	5138.99	0.205917
$855$	0	0
$856$	7263.97	0.290044
$857$	−44484.4	−1.77311	−0.886557	−	0.462619i	$-0.846910\pi$
$857$	−44484.4	−1.77311	−0.886557	+	0.462619i	$0.846910\pi$
$858$	0	0
$859$	−23213.4	−0.922039	−0.461019	−	0.887390i	$-0.652516\pi$
$859$	−23213.4	−0.922039	−0.461019	+	0.887390i	$0.652516\pi$
$860$	−10265.1	−0.407022
$861$	0	0
$862$	17751.4	0.701411
$863$	9640.68	0.380270	0.190135	−	0.981758i	$-0.439107\pi$
$863$	9640.68	0.380270	0.190135	+	0.981758i	$0.439107\pi$
$864$	0	0
$865$	−20395.5	−0.801697
$866$	7273.79	0.285420
$867$	0	0
$868$	9510.46	0.371897
$869$	11785.7	0.460072
$870$	0	0
$871$	−44234.8	−1.72082
$872$	−6896.61	−0.267831
$873$	0	0
$874$	−723.369	−0.0279958
$875$	12163.6	0.469947
$876$	0	0
$877$	−9499.62	−0.365769	−0.182885	−	0.983134i	$-0.558543\pi$
$877$	−9499.62	−0.365769	−0.182885	+	0.983134i	$0.558543\pi$
$878$	21958.7	0.844044
$879$	0	0
$880$	−1699.39	−0.0650983
$881$	8252.54	0.315590	0.157795	−	0.987472i	$-0.449561\pi$
$881$	8252.54	0.315590	0.157795	+	0.987472i	$0.449561\pi$
$882$	0	0
$883$	−34768.9	−1.32510	−0.662552	−	0.749016i	$-0.730526\pi$
$883$	−34768.9	−1.32510	−0.662552	+	0.749016i	$0.730526\pi$
$884$	−5918.26	−0.225173
$885$	0	0
$886$	2600.33	0.0986000
$887$	3288.58	0.124487	0.0622433	−	0.998061i	$-0.480175\pi$
$887$	3288.58	0.124487	0.0622433	+	0.998061i	$0.480175\pi$
$888$	0	0
$889$	3148.50	0.118782
$890$	−11395.2	−0.429177
$891$	0	0
$892$	13594.8	0.510301
$893$	−2067.43	−0.0774736
$894$	0	0
$895$	−17171.0	−0.641298
$896$	−1035.26	−0.0386002
$897$	0	0
$898$	−31750.4	−1.17987
$899$	−26813.4	−0.994747
$900$	0	0
$901$	21456.9	0.793377
$902$	−1729.84	−0.0638552
$903$	0	0
$904$	12021.8	0.442299
$905$	20944.2	0.769292
$906$	0	0
$907$	37686.1	1.37966	0.689828	−	0.723973i	$-0.257686\pi$
$907$	37686.1	1.37966	0.689828	+	0.723973i	$0.257686\pi$
$908$	−23043.2	−0.842197
$909$	0	0
$910$	−6327.28	−0.230491
$911$	−15090.9	−0.548828	−0.274414	−	0.961612i	$-0.588484\pi$
$911$	−15090.9	−0.548828	−0.274414	+	0.961612i	$0.588484\pi$
$912$	0	0
$913$	14863.7	0.538790
$914$	−6231.32	−0.225508
$915$	0	0
$916$	−8715.98	−0.314393
$917$	2170.08	0.0781485
$918$	0	0
$919$	43617.4	1.56562	0.782810	−	0.622261i	$-0.213786\pi$
$919$	43617.4	1.56562	0.782810	+	0.622261i	$0.213786\pi$
$920$	1266.43	0.0453836
$921$	0	0
$922$	26959.4	0.962972
$923$	−18605.2	−0.663486
$924$	0	0
$925$	−12040.9	−0.428002
$926$	−15892.4	−0.563991
$927$	0	0
$928$	2918.79	0.103248
$929$	−32446.2	−1.14588	−0.572942	−	0.819596i	$-0.694198\pi$
$929$	−32446.2	−1.14588	−0.572942	+	0.819596i	$0.694198\pi$
$930$	0	0
$931$	−5274.10	−0.185662
$932$	11234.0	0.394830
$933$	0	0
$934$	−18296.7	−0.640993
$935$	−3341.01	−0.116858
$936$	0	0
$937$	−28355.4	−0.988614	−0.494307	−	0.869287i	$-0.664578\pi$
$937$	−28355.4	−0.988614	−0.494307	+	0.869287i	$0.664578\pi$
$938$	−15212.6	−0.529542
$939$	0	0
$940$	3619.53	0.125592
$941$	48970.8	1.69650	0.848248	−	0.529599i	$-0.177658\pi$
$941$	48970.8	1.69650	0.848248	+	0.529599i	$0.177658\pi$
$942$	0	0
$943$	1289.12	0.0445170
$944$	4004.80	0.138078
$945$	0	0
$946$	−7882.78	−0.270921
$947$	9198.84	0.315652	0.157826	−	0.987467i	$-0.449552\pi$
$947$	9198.84	0.315652	0.157826	+	0.987467i	$0.449552\pi$
$948$	0	0
$949$	−45862.4	−1.56876
$950$	2122.07	0.0724728
$951$	0	0
$952$	−2035.33	−0.0692914
$953$	−28428.9	−0.966321	−0.483160	−	0.875532i	$-0.660511\pi$
$953$	−28428.9	−0.966321	−0.483160	+	0.875532i	$0.660511\pi$
$954$	0	0
$955$	−34980.3	−1.18527
$956$	−25142.7	−0.850599
$957$	0	0
$958$	15329.3	0.516980
$959$	−21492.5	−0.723700
$960$	0	0
$961$	56626.2	1.90078
$962$	20283.4	0.679797
$963$	0	0
$964$	4516.90	0.150912
$965$	−26992.2	−0.900426
$966$	0	0
$967$	−22315.5	−0.742109	−0.371054	−	0.928611i	$-0.621004\pi$
$967$	−22315.5	−0.742109	−0.371054	+	0.928611i	$0.621004\pi$
$968$	9343.01	0.310223
$969$	0	0
$970$	3513.61	0.116304
$971$	−208.410	−0.00688795	−0.00344398	−	0.999994i	$-0.501096\pi$
$971$	−208.410	−0.00688795	−0.00344398	+	0.999994i	$0.501096\pi$
$972$	0	0
$973$	−23131.0	−0.762124
$974$	10694.4	0.351819
$975$	0	0
$976$	−5083.08	−0.166706
$977$	−35744.3	−1.17048	−0.585241	−	0.810860i	$-0.699000\pi$
$977$	−35744.3	−1.17048	−0.585241	+	0.810860i	$0.699000\pi$
$978$	0	0
$979$	−8750.56	−0.285668
$980$	9233.56	0.300975
$981$	0	0
$982$	27294.3	0.886963
$983$	36175.5	1.17377	0.586887	−	0.809669i	$-0.300353\pi$
$983$	36175.5	1.17377	0.586887	+	0.809669i	$0.300353\pi$
$984$	0	0
$985$	−14425.9	−0.466647
$986$	5738.33	0.185340
$987$	0	0
$988$	−3574.74	−0.115109
$989$	5874.44	0.188874
$990$	0	0
$991$	34654.3	1.11083	0.555413	−	0.831575i	$-0.312560\pi$
$991$	34654.3	1.11083	0.555413	+	0.831575i	$0.312560\pi$
$992$	−9406.98	−0.301081
$993$	0	0
$994$	−6398.46	−0.204172
$995$	−3166.67	−0.100895
$996$	0	0
$997$	4756.72	0.151100	0.0755501	−	0.997142i	$-0.475929\pi$
$997$	4756.72	0.151100	0.0755501	+	0.997142i	$0.475929\pi$
$998$	38703.3	1.22759
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.4.a.h.1.1		2
3.2	odd	2		38.4.a.c.1.1	✓	2
12.11	even	2		304.4.a.c.1.2		2
15.2	even	4		950.4.b.i.799.4		4
15.8	even	4		950.4.b.i.799.1		4
15.14	odd	2		950.4.a.e.1.2		2
21.20	even	2		1862.4.a.e.1.2		2
24.5	odd	2		1216.4.a.g.1.2		2
24.11	even	2		1216.4.a.p.1.1		2
57.56	even	2		722.4.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.c.1.1	✓	2	3.2	odd	2
304.4.a.c.1.2		2	12.11	even	2
342.4.a.h.1.1		2	1.1	even	1	trivial
722.4.a.f.1.2		2	57.56	even	2
950.4.a.e.1.2		2	15.14	odd	2
950.4.b.i.799.1		4	15.8	even	4
950.4.b.i.799.4		4	15.2	even	4
1216.4.a.g.1.2		2	24.5	odd	2
1216.4.a.p.1.1		2	24.11	even	2
1862.4.a.e.1.2		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -8.31601 q^{5} +8.08801 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -8.31601 q^{5} +8.08801 q^{7} -8.00000 q^{8} +16.6320 q^{10} +12.7720 q^{11} -47.0360 q^{13} -16.1760 q^{14} +16.0000 q^{16} +31.4560 q^{17} +19.0000 q^{19} -33.2640 q^{20} -25.5440 q^{22} +19.0360 q^{23} -55.8441 q^{25} +94.0720 q^{26} +32.3520 q^{28} -91.2120 q^{29} +293.968 q^{31} -32.0000 q^{32} -62.9120 q^{34} -67.2599 q^{35} +215.616 q^{37} -38.0000 q^{38} +66.5280 q^{40} +67.7200 q^{41} +308.596 q^{43} +51.0880 q^{44} -38.0720 q^{46} -108.812 q^{47} -277.584 q^{49} +111.688 q^{50} -188.144 q^{52} +682.124 q^{53} -106.212 q^{55} -64.7041 q^{56} +182.424 q^{58} +250.300 q^{59} -317.692 q^{61} -587.936 q^{62} +64.0000 q^{64} +391.152 q^{65} +940.444 q^{67} +125.824 q^{68} +134.520 q^{70} +395.552 q^{71} +975.048 q^{73} -431.232 q^{74} +76.0000 q^{76} +103.300 q^{77} +922.776 q^{79} -133.056 q^{80} -135.440 q^{82} +1163.77 q^{83} -261.588 q^{85} -617.192 q^{86} -102.176 q^{88} -685.136 q^{89} -380.428 q^{91} +76.1441 q^{92} +217.624 q^{94} -158.004 q^{95} +211.256 q^{97} +555.168 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} - 18 q^{7} - 16 q^{8}+O(q^{10}) \) 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

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