LMFDB - Embedded newform 312.4.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [312,4,Mod(1,312)] 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("312.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [3,0,9,0,-4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.4085959218$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.13916.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 6 $ x^3 - x^2 - 16*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.29884$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +12.9600 q^{5} +1.76468 q^{7} +9.00000 q^{9} +25.1954 q^{11} +13.0000 q^{13} +38.8801 q^{15} -62.6216 q^{17} +139.798 q^{19} +5.29404 q^{21} -56.6216 q^{23} +42.9625 q^{25} +27.0000 q^{27} +75.4119 q^{29} -71.2900 q^{31} +75.5861 q^{33} +22.8703 q^{35} +55.7602 q^{37} +39.0000 q^{39} -40.0947 q^{41} +14.7854 q^{43} +116.640 q^{45} +531.136 q^{47} -339.886 q^{49} -187.865 q^{51} +368.053 q^{53} +326.533 q^{55} +419.395 q^{57} +165.872 q^{59} -145.878 q^{61} +15.8821 q^{63} +168.480 q^{65} +901.373 q^{67} -169.865 q^{69} -345.660 q^{71} -292.315 q^{73} +128.888 q^{75} +44.4617 q^{77} -722.695 q^{79} +81.0000 q^{81} +565.364 q^{83} -811.578 q^{85} +226.236 q^{87} -275.116 q^{89} +22.9408 q^{91} -213.870 q^{93} +1811.79 q^{95} -1821.49 q^{97} +226.758 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 4 q^{5} + 6 q^{7} + 27 q^{9} + 32 q^{11} + 39 q^{13} - 12 q^{15} + 158 q^{17} + 70 q^{19} + 18 q^{21} + 176 q^{23} + 209 q^{25} + 81 q^{27} + 222 q^{29} + 54 q^{31} + 96 q^{33} + 496 q^{35}+ \cdots + 288 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 - 4 * q^5 + 6 * q^7 + 27 * q^9 + 32 * q^11 + 39 * q^13 - 12 * q^15 + 158 * q^17 + 70 * q^19 + 18 * q^21 + 176 * q^23 + 209 * q^25 + 81 * q^27 + 222 * q^29 + 54 * q^31 + 96 * q^33 + 496 * q^35 - 90 * q^37 + 117 * q^39 + 104 * q^41 + 140 * q^43 - 36 * q^45 + 328 * q^47 - 65 * q^49 + 474 * q^51 - 358 * q^53 + 32 * q^55 + 210 * q^57 + 1060 * q^59 - 186 * q^61 + 54 * q^63 - 52 * q^65 + 354 * q^67 + 528 * q^69 + 1692 * q^71 - 974 * q^73 + 627 * q^75 - 384 * q^77 + 776 * q^79 + 243 * q^81 + 2340 * q^83 - 2824 * q^85 + 666 * q^87 - 684 * q^89 + 78 * q^91 + 162 * q^93 - 742 * q^97 + 288 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	12.9600	1.15918	0.579590	−	0.814908i	$-0.303213\pi$
$5$	12.9600	1.15918	0.579590	+	0.814908i	$0.303213\pi$
$6$	0	0
$7$	1.76468	0.0952837	0.0476418	−	0.998864i	$-0.484829\pi$
$7$	1.76468	0.0952837	0.0476418	+	0.998864i	$0.484829\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	25.1954	0.690608	0.345304	−	0.938491i	$-0.387776\pi$
$11$	25.1954	0.690608	0.345304	+	0.938491i	$0.387776\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	38.8801	0.669253
$16$	0	0
$17$	−62.6216	−0.893409	−0.446705	−	0.894681i	$-0.647403\pi$
$17$	−62.6216	−0.893409	−0.446705	+	0.894681i	$0.647403\pi$
$18$	0	0
$19$	139.798	1.68799	0.843997	−	0.536347i	$-0.180196\pi$
$19$	139.798	1.68799	0.843997	+	0.536347i	$0.180196\pi$
$20$	0	0
$21$	5.29404	0.0550121
$22$	0	0
$23$	−56.6216	−0.513322	−0.256661	−	0.966501i	$-0.582622\pi$
$23$	−56.6216	−0.513322	−0.256661	+	0.966501i	$0.582622\pi$
$24$	0	0
$25$	42.9625	0.343700
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	75.4119	0.482884	0.241442	−	0.970415i	$-0.422380\pi$
$29$	75.4119	0.482884	0.241442	+	0.970415i	$0.422380\pi$
$30$	0	0
$31$	−71.2900	−0.413034	−0.206517	−	0.978443i	$-0.566213\pi$
$31$	−71.2900	−0.413034	−0.206517	+	0.978443i	$0.566213\pi$
$32$	0	0
$33$	75.5861	0.398723
$34$	0	0
$35$	22.8703	0.110451
$36$	0	0
$37$	55.7602	0.247755	0.123877	−	0.992298i	$-0.460467\pi$
$37$	55.7602	0.247755	0.123877	+	0.992298i	$0.460467\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−40.0947	−0.152725	−0.0763626	−	0.997080i	$-0.524331\pi$
$41$	−40.0947	−0.152725	−0.0763626	+	0.997080i	$0.524331\pi$
$42$	0	0
$43$	14.7854	0.0524362	0.0262181	−	0.999656i	$-0.491654\pi$
$43$	14.7854	0.0524362	0.0262181	+	0.999656i	$0.491654\pi$
$44$	0	0
$45$	116.640	0.386394
$46$	0	0
$47$	531.136	1.64839	0.824193	−	0.566309i	$-0.191629\pi$
$47$	531.136	1.64839	0.824193	+	0.566309i	$0.191629\pi$
$48$	0	0
$49$	−339.886	−0.990921
$50$	0	0
$51$	−187.865	−0.515810
$52$	0	0
$53$	368.053	0.953886	0.476943	−	0.878934i	$-0.341745\pi$
$53$	368.053	0.953886	0.476943	+	0.878934i	$0.341745\pi$
$54$	0	0
$55$	326.533	0.800539
$56$	0	0
$57$	419.395	0.974564
$58$	0	0
$59$	165.872	0.366012	0.183006	−	0.983112i	$-0.441417\pi$
$59$	165.872	0.366012	0.183006	+	0.983112i	$0.441417\pi$
$60$	0	0
$61$	−145.878	−0.306192	−0.153096	−	0.988211i	$-0.548924\pi$
$61$	−145.878	−0.306192	−0.153096	+	0.988211i	$0.548924\pi$
$62$	0	0
$63$	15.8821	0.0317612
$64$	0	0
$65$	168.480	0.321499
$66$	0	0
$67$	901.373	1.64358	0.821792	−	0.569787i	$-0.192974\pi$
$67$	901.373	1.64358	0.821792	+	0.569787i	$0.192974\pi$
$68$	0	0
$69$	−169.865	−0.296367
$70$	0	0
$71$	−345.660	−0.577778	−0.288889	−	0.957363i	$-0.593286\pi$
$71$	−345.660	−0.577778	−0.288889	+	0.957363i	$0.593286\pi$
$72$	0	0
$73$	−292.315	−0.468669	−0.234335	−	0.972156i	$-0.575291\pi$
$73$	−292.315	−0.468669	−0.234335	+	0.972156i	$0.575291\pi$
$74$	0	0
$75$	128.888	0.198435
$76$	0	0
$77$	44.4617	0.0658037
$78$	0	0
$79$	−722.695	−1.02924	−0.514618	−	0.857420i	$-0.672066\pi$
$79$	−722.695	−1.02924	−0.514618	+	0.857420i	$0.672066\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	565.364	0.747672	0.373836	−	0.927495i	$-0.378042\pi$
$83$	565.364	0.747672	0.373836	+	0.927495i	$0.378042\pi$
$84$	0	0
$85$	−811.578	−1.03562
$86$	0	0
$87$	226.236	0.278793
$88$	0	0
$89$	−275.116	−0.327665	−0.163833	−	0.986488i	$-0.552386\pi$
$89$	−275.116	−0.327665	−0.163833	+	0.986488i	$0.552386\pi$
$90$	0	0
$91$	22.9408	0.0264269
$92$	0	0
$93$	−213.870	−0.238466
$94$	0	0
$95$	1811.79	1.95669
$96$	0	0
$97$	−1821.49	−1.90664	−0.953319	−	0.301964i	$-0.902358\pi$
$97$	−1821.49	−1.90664	−0.953319	+	0.301964i	$0.902358\pi$
$98$	0	0
$99$	226.758	0.230203
$100$	0	0
$101$	−278.714	−0.274584	−0.137292	−	0.990531i	$-0.543840\pi$
$101$	−278.714	−0.274584	−0.137292	+	0.990531i	$0.543840\pi$
$102$	0	0
$103$	−237.909	−0.227591	−0.113796	−	0.993504i	$-0.536301\pi$
$103$	−237.909	−0.227591	−0.113796	+	0.993504i	$0.536301\pi$
$104$	0	0
$105$	68.6109	0.0637689
$106$	0	0
$107$	922.569	0.833534	0.416767	−	0.909013i	$-0.363163\pi$
$107$	922.569	0.833534	0.416767	+	0.909013i	$0.363163\pi$
$108$	0	0
$109$	−361.546	−0.317705	−0.158853	−	0.987302i	$-0.550779\pi$
$109$	−361.546	−0.317705	−0.158853	+	0.987302i	$0.550779\pi$
$110$	0	0
$111$	167.281	0.143041
$112$	0	0
$113$	−1069.90	−0.890687	−0.445344	−	0.895360i	$-0.646918\pi$
$113$	−1069.90	−0.890687	−0.445344	+	0.895360i	$0.646918\pi$
$114$	0	0
$115$	−733.817	−0.595033
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−110.507	−0.0851273
$120$	0	0
$121$	−696.194	−0.523061
$122$	0	0
$123$	−120.284	−0.0881760
$124$	0	0
$125$	−1063.21	−0.760770
$126$	0	0
$127$	−1753.06	−1.22488	−0.612438	−	0.790518i	$-0.709811\pi$
$127$	−1753.06	−1.22488	−0.612438	+	0.790518i	$0.709811\pi$
$128$	0	0
$129$	44.3563	0.0302741
$130$	0	0
$131$	1731.18	1.15461	0.577304	−	0.816529i	$-0.304105\pi$
$131$	1731.18	1.15461	0.577304	+	0.816529i	$0.304105\pi$
$132$	0	0
$133$	246.699	0.160838
$134$	0	0
$135$	349.921	0.223084
$136$	0	0
$137$	45.9710	0.0286684	0.0143342	−	0.999897i	$-0.495437\pi$
$137$	45.9710	0.0286684	0.0143342	+	0.999897i	$0.495437\pi$
$138$	0	0
$139$	247.025	0.150736	0.0753682	−	0.997156i	$-0.475987\pi$
$139$	247.025	0.150736	0.0753682	+	0.997156i	$0.475987\pi$
$140$	0	0
$141$	1593.41	0.951696
$142$	0	0
$143$	327.540	0.191540
$144$	0	0
$145$	977.341	0.559750
$146$	0	0
$147$	−1019.66	−0.572109
$148$	0	0
$149$	−3389.84	−1.86380	−0.931901	−	0.362712i	$-0.881851\pi$
$149$	−3389.84	−1.86380	−0.931901	+	0.362712i	$0.881851\pi$
$150$	0	0
$151$	−3123.16	−1.68317	−0.841587	−	0.540122i	$-0.818378\pi$
$151$	−3123.16	−1.68317	−0.841587	+	0.540122i	$0.818378\pi$
$152$	0	0
$153$	−563.594	−0.297803
$154$	0	0
$155$	−923.921	−0.478782
$156$	0	0
$157$	−1878.59	−0.954955	−0.477477	−	0.878644i	$-0.658449\pi$
$157$	−1878.59	−0.954955	−0.477477	+	0.878644i	$0.658449\pi$
$158$	0	0
$159$	1104.16	0.550727
$160$	0	0
$161$	−99.9189	−0.0489112
$162$	0	0
$163$	−2230.38	−1.07176	−0.535880	−	0.844294i	$-0.680020\pi$
$163$	−2230.38	−1.07176	−0.535880	+	0.844294i	$0.680020\pi$
$164$	0	0
$165$	979.598	0.462192
$166$	0	0
$167$	−952.254	−0.441243	−0.220622	−	0.975359i	$-0.570809\pi$
$167$	−952.254	−0.441243	−0.220622	+	0.975359i	$0.570809\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	1258.18	0.562665
$172$	0	0
$173$	−3124.52	−1.37314	−0.686569	−	0.727065i	$-0.740884\pi$
$173$	−3124.52	−1.37314	−0.686569	+	0.727065i	$0.740884\pi$
$174$	0	0
$175$	75.8150	0.0327490
$176$	0	0
$177$	497.617	0.211317
$178$	0	0
$179$	−1940.12	−0.810120	−0.405060	−	0.914290i	$-0.632749\pi$
$179$	−1940.12	−0.810120	−0.405060	+	0.914290i	$0.632749\pi$
$180$	0	0
$181$	−61.8517	−0.0254000	−0.0127000	−	0.999919i	$-0.504043\pi$
$181$	−61.8517	−0.0254000	−0.0127000	+	0.999919i	$0.504043\pi$
$182$	0	0
$183$	−437.633	−0.176780
$184$	0	0
$185$	722.654	0.287192
$186$	0	0
$187$	−1577.77	−0.616996
$188$	0	0
$189$	47.6463	0.0183374
$190$	0	0
$191$	1333.55	0.505197	0.252598	−	0.967571i	$-0.418715\pi$
$191$	1333.55	0.505197	0.252598	+	0.967571i	$0.418715\pi$
$192$	0	0
$193$	4030.89	1.50337	0.751683	−	0.659525i	$-0.229242\pi$
$193$	4030.89	1.50337	0.751683	+	0.659525i	$0.229242\pi$
$194$	0	0
$195$	505.441	0.185617
$196$	0	0
$197$	2339.69	0.846173	0.423087	−	0.906089i	$-0.360947\pi$
$197$	2339.69	0.846173	0.423087	+	0.906089i	$0.360947\pi$
$198$	0	0
$199$	−3332.68	−1.18717	−0.593587	−	0.804770i	$-0.702289\pi$
$199$	−3332.68	−1.18717	−0.593587	+	0.804770i	$0.702289\pi$
$200$	0	0
$201$	2704.12	0.948924
$202$	0	0
$203$	133.078	0.0460110
$204$	0	0
$205$	−519.628	−0.177036
$206$	0	0
$207$	−509.594	−0.171107
$208$	0	0
$209$	3522.26	1.16574
$210$	0	0
$211$	−1037.82	−0.338608	−0.169304	−	0.985564i	$-0.554152\pi$
$211$	−1037.82	−0.338608	−0.169304	+	0.985564i	$0.554152\pi$
$212$	0	0
$213$	−1036.98	−0.333580
$214$	0	0
$215$	191.620	0.0607830
$216$	0	0
$217$	−125.804	−0.0393554
$218$	0	0
$219$	−876.944	−0.270586
$220$	0	0
$221$	−814.080	−0.247787
$222$	0	0
$223$	−2758.72	−0.828418	−0.414209	−	0.910182i	$-0.635942\pi$
$223$	−2758.72	−0.828418	−0.414209	+	0.910182i	$0.635942\pi$
$224$	0	0
$225$	386.663	0.114567
$226$	0	0
$227$	841.159	0.245946	0.122973	−	0.992410i	$-0.460757\pi$
$227$	841.159	0.245946	0.122973	+	0.992410i	$0.460757\pi$
$228$	0	0
$229$	2030.90	0.586050	0.293025	−	0.956105i	$-0.405338\pi$
$229$	2030.90	0.586050	0.293025	+	0.956105i	$0.405338\pi$
$230$	0	0
$231$	133.385	0.0379918
$232$	0	0
$233$	4876.76	1.37119	0.685595	−	0.727983i	$-0.259542\pi$
$233$	4876.76	1.37119	0.685595	+	0.727983i	$0.259542\pi$
$234$	0	0
$235$	6883.54	1.91078
$236$	0	0
$237$	−2168.09	−0.594229
$238$	0	0
$239$	4827.46	1.30654	0.653268	−	0.757127i	$-0.273397\pi$
$239$	4827.46	1.30654	0.653268	+	0.757127i	$0.273397\pi$
$240$	0	0
$241$	4967.73	1.32780	0.663899	−	0.747822i	$-0.268900\pi$
$241$	4967.73	1.32780	0.663899	+	0.747822i	$0.268900\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−4404.93	−1.14866
$246$	0	0
$247$	1817.38	0.468166
$248$	0	0
$249$	1696.09	0.431669
$250$	0	0
$251$	4056.06	1.01999	0.509993	−	0.860179i	$-0.329648\pi$
$251$	4056.06	1.01999	0.509993	+	0.860179i	$0.329648\pi$
$252$	0	0
$253$	−1426.60	−0.354504
$254$	0	0
$255$	−2434.73	−0.597917
$256$	0	0
$257$	5326.27	1.29278	0.646388	−	0.763008i	$-0.276279\pi$
$257$	5326.27	1.29278	0.646388	+	0.763008i	$0.276279\pi$
$258$	0	0
$259$	98.3988	0.0236070
$260$	0	0
$261$	678.707	0.160961
$262$	0	0
$263$	5571.89	1.30638	0.653189	−	0.757195i	$-0.273431\pi$
$263$	5571.89	1.30638	0.653189	+	0.757195i	$0.273431\pi$
$264$	0	0
$265$	4769.98	1.10573
$266$	0	0
$267$	−825.348	−0.189178
$268$	0	0
$269$	197.034	0.0446593	0.0223297	−	0.999751i	$-0.492892\pi$
$269$	197.034	0.0446593	0.0223297	+	0.999751i	$0.492892\pi$
$270$	0	0
$271$	−5389.86	−1.20816	−0.604079	−	0.796925i	$-0.706459\pi$
$271$	−5389.86	−1.20816	−0.604079	+	0.796925i	$0.706459\pi$
$272$	0	0
$273$	68.8225	0.0152576
$274$	0	0
$275$	1082.46	0.237362
$276$	0	0
$277$	−5465.12	−1.18544	−0.592721	−	0.805408i	$-0.701946\pi$
$277$	−5465.12	−1.18544	−0.592721	+	0.805408i	$0.701946\pi$
$278$	0	0
$279$	−641.610	−0.137678
$280$	0	0
$281$	−4794.27	−1.01780	−0.508900	−	0.860825i	$-0.669948\pi$
$281$	−4794.27	−1.01780	−0.508900	+	0.860825i	$0.669948\pi$
$282$	0	0
$283$	2284.81	0.479921	0.239961	−	0.970783i	$-0.422865\pi$
$283$	2284.81	0.479921	0.239961	+	0.970783i	$0.422865\pi$
$284$	0	0
$285$	5435.37	1.12970
$286$	0	0
$287$	−70.7542	−0.0145522
$288$	0	0
$289$	−991.540	−0.201820
$290$	0	0
$291$	−5464.46	−1.10080
$292$	0	0
$293$	−6690.87	−1.33408	−0.667039	−	0.745023i	$-0.732439\pi$
$293$	−6690.87	−1.33408	−0.667039	+	0.745023i	$0.732439\pi$
$294$	0	0
$295$	2149.71	0.424275
$296$	0	0
$297$	680.275	0.132908
$298$	0	0
$299$	−736.080	−0.142370
$300$	0	0
$301$	26.0915	0.00499631
$302$	0	0
$303$	−836.141	−0.158531
$304$	0	0
$305$	−1890.58	−0.354932
$306$	0	0
$307$	−4001.88	−0.743973	−0.371986	−	0.928238i	$-0.621323\pi$
$307$	−4001.88	−0.743973	−0.371986	+	0.928238i	$0.621323\pi$
$308$	0	0
$309$	−713.728	−0.131400
$310$	0	0
$311$	−244.283	−0.0445402	−0.0222701	−	0.999752i	$-0.507089\pi$
$311$	−244.283	−0.0445402	−0.0222701	+	0.999752i	$0.507089\pi$
$312$	0	0
$313$	1849.39	0.333973	0.166986	−	0.985959i	$-0.446596\pi$
$313$	1849.39	0.333973	0.166986	+	0.985959i	$0.446596\pi$
$314$	0	0
$315$	205.833	0.0368170
$316$	0	0
$317$	10115.8	1.79230	0.896151	−	0.443749i	$-0.146352\pi$
$317$	10115.8	1.79230	0.896151	+	0.443749i	$0.146352\pi$
$318$	0	0
$319$	1900.03	0.333484
$320$	0	0
$321$	2767.71	0.481241
$322$	0	0
$323$	−8754.38	−1.50807
$324$	0	0
$325$	558.513	0.0953252
$326$	0	0
$327$	−1084.64	−0.183427
$328$	0	0
$329$	937.284	0.157064
$330$	0	0
$331$	6287.14	1.04403	0.522013	−	0.852938i	$-0.325181\pi$
$331$	6287.14	1.04403	0.522013	+	0.852938i	$0.325181\pi$
$332$	0	0
$333$	501.842	0.0825849
$334$	0	0
$335$	11681.8	1.90521
$336$	0	0
$337$	8013.83	1.29537	0.647687	−	0.761907i	$-0.275736\pi$
$337$	8013.83	1.29537	0.647687	+	0.761907i	$0.275736\pi$
$338$	0	0
$339$	−3209.70	−0.514238
$340$	0	0
$341$	−1796.18	−0.285245
$342$	0	0
$343$	−1205.07	−0.189702
$344$	0	0
$345$	−2201.45	−0.343543
$346$	0	0
$347$	−8648.51	−1.33797	−0.668986	−	0.743275i	$-0.733272\pi$
$347$	−8648.51	−1.33797	−0.668986	+	0.743275i	$0.733272\pi$
$348$	0	0
$349$	−4093.84	−0.627903	−0.313952	−	0.949439i	$-0.601653\pi$
$349$	−4093.84	−0.627903	−0.313952	+	0.949439i	$0.601653\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−2804.97	−0.422928	−0.211464	−	0.977386i	$-0.567823\pi$
$353$	−2804.97	−0.422928	−0.211464	+	0.977386i	$0.567823\pi$
$354$	0	0
$355$	−4479.76	−0.669750
$356$	0	0
$357$	−331.521	−0.0491483
$358$	0	0
$359$	−8549.11	−1.25684	−0.628419	−	0.777875i	$-0.716298\pi$
$359$	−8549.11	−1.25684	−0.628419	+	0.777875i	$0.716298\pi$
$360$	0	0
$361$	12684.5	1.84933
$362$	0	0
$363$	−2088.58	−0.301989
$364$	0	0
$365$	−3788.41	−0.543272
$366$	0	0
$367$	2955.54	0.420375	0.210188	−	0.977661i	$-0.432592\pi$
$367$	2955.54	0.420375	0.210188	+	0.977661i	$0.432592\pi$
$368$	0	0
$369$	−360.852	−0.0509084
$370$	0	0
$371$	649.495	0.0908898
$372$	0	0
$373$	10712.0	1.48698	0.743490	−	0.668747i	$-0.233169\pi$
$373$	10712.0	1.48698	0.743490	+	0.668747i	$0.233169\pi$
$374$	0	0
$375$	−3189.63	−0.439231
$376$	0	0
$377$	980.355	0.133928
$378$	0	0
$379$	3698.52	0.501267	0.250633	−	0.968082i	$-0.419361\pi$
$379$	3698.52	0.501267	0.250633	+	0.968082i	$0.419361\pi$
$380$	0	0
$381$	−5259.19	−0.707183
$382$	0	0
$383$	−477.469	−0.0637011	−0.0318505	−	0.999493i	$-0.510140\pi$
$383$	−477.469	−0.0637011	−0.0318505	+	0.999493i	$0.510140\pi$
$384$	0	0
$385$	576.225	0.0762783
$386$	0	0
$387$	133.069	0.0174787
$388$	0	0
$389$	−1723.82	−0.224682	−0.112341	−	0.993670i	$-0.535835\pi$
$389$	−1723.82	−0.224682	−0.112341	+	0.993670i	$0.535835\pi$
$390$	0	0
$391$	3545.73	0.458607
$392$	0	0
$393$	5193.53	0.666613
$394$	0	0
$395$	−9366.16	−1.19307
$396$	0	0
$397$	5468.16	0.691283	0.345641	−	0.938367i	$-0.387661\pi$
$397$	5468.16	0.691283	0.345641	+	0.938367i	$0.387661\pi$
$398$	0	0
$399$	740.096	0.0928601
$400$	0	0
$401$	−6727.20	−0.837757	−0.418878	−	0.908042i	$-0.637577\pi$
$401$	−6727.20	−0.837757	−0.418878	+	0.908042i	$0.637577\pi$
$402$	0	0
$403$	−926.770	−0.114555
$404$	0	0
$405$	1049.76	0.128798
$406$	0	0
$407$	1404.90	0.171101
$408$	0	0
$409$	3274.63	0.395892	0.197946	−	0.980213i	$-0.436573\pi$
$409$	3274.63	0.395892	0.197946	+	0.980213i	$0.436573\pi$
$410$	0	0
$411$	137.913	0.0165517
$412$	0	0
$413$	292.711	0.0348750
$414$	0	0
$415$	7327.14	0.866687
$416$	0	0
$417$	741.074	0.0870277
$418$	0	0
$419$	−351.961	−0.0410368	−0.0205184	−	0.999789i	$-0.506532\pi$
$419$	−351.961	−0.0410368	−0.0205184	+	0.999789i	$0.506532\pi$
$420$	0	0
$421$	−10956.4	−1.26836	−0.634182	−	0.773184i	$-0.718663\pi$
$421$	−10956.4	−1.26836	−0.634182	+	0.773184i	$0.718663\pi$
$422$	0	0
$423$	4780.22	0.549462
$424$	0	0
$425$	−2690.38	−0.307065
$426$	0	0
$427$	−257.427	−0.0291751
$428$	0	0
$429$	982.619	0.110586
$430$	0	0
$431$	4087.61	0.456829	0.228414	−	0.973564i	$-0.426646\pi$
$431$	4087.61	0.456829	0.228414	+	0.973564i	$0.426646\pi$
$432$	0	0
$433$	−15377.9	−1.70673	−0.853366	−	0.521313i	$-0.825443\pi$
$433$	−15377.9	−1.70673	−0.853366	+	0.521313i	$0.825443\pi$
$434$	0	0
$435$	2932.02	0.323172
$436$	0	0
$437$	−7915.59	−0.866485
$438$	0	0
$439$	−6609.85	−0.718613	−0.359306	−	0.933220i	$-0.616987\pi$
$439$	−6609.85	−0.718613	−0.359306	+	0.933220i	$0.616987\pi$
$440$	0	0
$441$	−3058.97	−0.330307
$442$	0	0
$443$	2487.10	0.266740	0.133370	−	0.991066i	$-0.457420\pi$
$443$	2487.10	0.266740	0.133370	+	0.991066i	$0.457420\pi$
$444$	0	0
$445$	−3565.51	−0.379823
$446$	0	0
$447$	−10169.5	−1.07607
$448$	0	0
$449$	−13228.9	−1.39045	−0.695226	−	0.718792i	$-0.744696\pi$
$449$	−13228.9	−1.39045	−0.695226	+	0.718792i	$0.744696\pi$
$450$	0	0
$451$	−1010.20	−0.105473
$452$	0	0
$453$	−9369.48	−0.971781
$454$	0	0
$455$	297.314	0.0306336
$456$	0	0
$457$	11721.5	1.19980	0.599898	−	0.800076i	$-0.295208\pi$
$457$	11721.5	1.19980	0.599898	+	0.800076i	$0.295208\pi$
$458$	0	0
$459$	−1690.78	−0.171937
$460$	0	0
$461$	−1728.80	−0.174660	−0.0873301	−	0.996179i	$-0.527833\pi$
$461$	−1728.80	−0.174660	−0.0873301	+	0.996179i	$0.527833\pi$
$462$	0	0
$463$	−11326.0	−1.13685	−0.568425	−	0.822735i	$-0.692447\pi$
$463$	−11326.0	−1.13685	−0.568425	+	0.822735i	$0.692447\pi$
$464$	0	0
$465$	−2771.76	−0.276425
$466$	0	0
$467$	−5221.54	−0.517396	−0.258698	−	0.965958i	$-0.583294\pi$
$467$	−5221.54	−0.517396	−0.258698	+	0.965958i	$0.583294\pi$
$468$	0	0
$469$	1590.63	0.156607
$470$	0	0
$471$	−5635.77	−0.551343
$472$	0	0
$473$	372.524	0.0362129
$474$	0	0
$475$	6006.08	0.580164
$476$	0	0
$477$	3312.48	0.317962
$478$	0	0
$479$	−8525.01	−0.813189	−0.406595	−	0.913609i	$-0.633284\pi$
$479$	−8525.01	−0.813189	−0.406595	+	0.913609i	$0.633284\pi$
$480$	0	0
$481$	724.883	0.0687148
$482$	0	0
$483$	−299.757	−0.0282389
$484$	0	0
$485$	−23606.5	−2.21014
$486$	0	0
$487$	681.285	0.0633921	0.0316961	−	0.999498i	$-0.489909\pi$
$487$	681.285	0.0633921	0.0316961	+	0.999498i	$0.489909\pi$
$488$	0	0
$489$	−6691.14	−0.618781
$490$	0	0
$491$	20581.2	1.89169	0.945844	−	0.324623i	$-0.105237\pi$
$491$	20581.2	1.89169	0.945844	+	0.324623i	$0.105237\pi$
$492$	0	0
$493$	−4722.41	−0.431413
$494$	0	0
$495$	2938.79	0.266846
$496$	0	0
$497$	−609.978	−0.0550528
$498$	0	0
$499$	−1204.64	−0.108071	−0.0540353	−	0.998539i	$-0.517208\pi$
$499$	−1204.64	−0.108071	−0.0540353	+	0.998539i	$0.517208\pi$
$500$	0	0
$501$	−2856.76	−0.254752
$502$	0	0
$503$	1026.69	0.0910099	0.0455050	−	0.998964i	$-0.485510\pi$
$503$	1026.69	0.0910099	0.0455050	+	0.998964i	$0.485510\pi$
$504$	0	0
$505$	−3612.14	−0.318293
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−1921.88	−0.167359	−0.0836797	−	0.996493i	$-0.526667\pi$
$509$	−1921.88	−0.167359	−0.0836797	+	0.996493i	$0.526667\pi$
$510$	0	0
$511$	−515.842	−0.0446565
$512$	0	0
$513$	3774.55	0.324855
$514$	0	0
$515$	−3083.31	−0.263819
$516$	0	0
$517$	13382.2	1.13839
$518$	0	0
$519$	−9373.56	−0.792782
$520$	0	0
$521$	10057.8	0.845756	0.422878	−	0.906187i	$-0.361020\pi$
$521$	10057.8	0.845756	0.422878	+	0.906187i	$0.361020\pi$
$522$	0	0
$523$	4124.89	0.344874	0.172437	−	0.985021i	$-0.444836\pi$
$523$	4124.89	0.344874	0.172437	+	0.985021i	$0.444836\pi$
$524$	0	0
$525$	227.445	0.0189076
$526$	0	0
$527$	4464.29	0.369009
$528$	0	0
$529$	−8961.00	−0.736500
$530$	0	0
$531$	1492.85	0.122004
$532$	0	0
$533$	−521.231	−0.0423584
$534$	0	0
$535$	11956.5	0.966217
$536$	0	0
$537$	−5820.36	−0.467723
$538$	0	0
$539$	−8563.55	−0.684338
$540$	0	0
$541$	4880.50	0.387854	0.193927	−	0.981016i	$-0.437878\pi$
$541$	4880.50	0.387854	0.193927	+	0.981016i	$0.437878\pi$
$542$	0	0
$543$	−185.555	−0.0146647
$544$	0	0
$545$	−4685.65	−0.368278
$546$	0	0
$547$	9833.32	0.768633	0.384316	−	0.923201i	$-0.374437\pi$
$547$	9833.32	0.768633	0.384316	+	0.923201i	$0.374437\pi$
$548$	0	0
$549$	−1312.90	−0.102064
$550$	0	0
$551$	10542.4	0.815106
$552$	0	0
$553$	−1275.32	−0.0980693
$554$	0	0
$555$	2167.96	0.165811
$556$	0	0
$557$	671.131	0.0510534	0.0255267	−	0.999674i	$-0.491874\pi$
$557$	671.131	0.0510534	0.0255267	+	0.999674i	$0.491874\pi$
$558$	0	0
$559$	192.211	0.0145432
$560$	0	0
$561$	−4733.32	−0.356223
$562$	0	0
$563$	−7438.83	−0.556855	−0.278428	−	0.960457i	$-0.589813\pi$
$563$	−7438.83	−0.556855	−0.278428	+	0.960457i	$0.589813\pi$
$564$	0	0
$565$	−13865.9	−1.03247
$566$	0	0
$567$	142.939	0.0105871
$568$	0	0
$569$	19242.7	1.41774	0.708870	−	0.705339i	$-0.249205\pi$
$569$	19242.7	1.41774	0.708870	+	0.705339i	$0.249205\pi$
$570$	0	0
$571$	12917.3	0.946713	0.473357	−	0.880871i	$-0.343042\pi$
$571$	12917.3	0.946713	0.473357	+	0.880871i	$0.343042\pi$
$572$	0	0
$573$	4000.66	0.291675
$574$	0	0
$575$	−2432.60	−0.176429
$576$	0	0
$577$	−13343.4	−0.962727	−0.481363	−	0.876521i	$-0.659858\pi$
$577$	−13343.4	−0.962727	−0.481363	+	0.876521i	$0.659858\pi$
$578$	0	0
$579$	12092.7	0.867969
$580$	0	0
$581$	997.686	0.0712409
$582$	0	0
$583$	9273.23	0.658761
$584$	0	0
$585$	1516.32	0.107166
$586$	0	0
$587$	20799.6	1.46251	0.731254	−	0.682105i	$-0.238935\pi$
$587$	20799.6	1.46251	0.731254	+	0.682105i	$0.238935\pi$
$588$	0	0
$589$	−9966.22	−0.697200
$590$	0	0
$591$	7019.08	0.488538
$592$	0	0
$593$	24109.9	1.66960	0.834801	−	0.550551i	$-0.185582\pi$
$593$	24109.9	1.66960	0.834801	+	0.550551i	$0.185582\pi$
$594$	0	0
$595$	−1432.17	−0.0986780
$596$	0	0
$597$	−9998.04	−0.685415
$598$	0	0
$599$	25368.9	1.73046	0.865228	−	0.501378i	$-0.167173\pi$
$599$	25368.9	1.73046	0.865228	+	0.501378i	$0.167173\pi$
$600$	0	0
$601$	29275.8	1.98700	0.993498	−	0.113853i	$-0.0363192\pi$
$601$	29275.8	1.98700	0.993498	+	0.113853i	$0.0363192\pi$
$602$	0	0
$603$	8112.35	0.547862
$604$	0	0
$605$	−9022.70	−0.606322
$606$	0	0
$607$	−12370.9	−0.827218	−0.413609	−	0.910455i	$-0.635732\pi$
$607$	−12370.9	−0.827218	−0.413609	+	0.910455i	$0.635732\pi$
$608$	0	0
$609$	399.233	0.0265645
$610$	0	0
$611$	6904.77	0.457180
$612$	0	0
$613$	18862.5	1.24282	0.621411	−	0.783485i	$-0.286560\pi$
$613$	18862.5	1.24282	0.621411	+	0.783485i	$0.286560\pi$
$614$	0	0
$615$	−1558.89	−0.102212
$616$	0	0
$617$	−24546.7	−1.60164	−0.800822	−	0.598903i	$-0.795604\pi$
$617$	−24546.7	−1.60164	−0.800822	+	0.598903i	$0.795604\pi$
$618$	0	0
$619$	−19413.6	−1.26058	−0.630289	−	0.776361i	$-0.717064\pi$
$619$	−19413.6	−1.26058	−0.630289	+	0.776361i	$0.717064\pi$
$620$	0	0
$621$	−1528.78	−0.0987889
$622$	0	0
$623$	−485.491	−0.0312212
$624$	0	0
$625$	−19149.5	−1.22557
$626$	0	0
$627$	10566.8	0.673042
$628$	0	0
$629$	−3491.79	−0.221346
$630$	0	0
$631$	18875.9	1.19087	0.595433	−	0.803405i	$-0.296980\pi$
$631$	18875.9	1.19087	0.595433	+	0.803405i	$0.296980\pi$
$632$	0	0
$633$	−3113.45	−0.195495
$634$	0	0
$635$	−22719.8	−1.41985
$636$	0	0
$637$	−4418.52	−0.274832
$638$	0	0
$639$	−3110.94	−0.192593
$640$	0	0
$641$	22270.4	1.37228	0.686138	−	0.727472i	$-0.259305\pi$
$641$	22270.4	1.37228	0.686138	+	0.727472i	$0.259305\pi$
$642$	0	0
$643$	−28426.7	−1.74345	−0.871726	−	0.489994i	$-0.836999\pi$
$643$	−28426.7	−1.74345	−0.871726	+	0.489994i	$0.836999\pi$
$644$	0	0
$645$	574.859	0.0350931
$646$	0	0
$647$	1087.22	0.0660635	0.0330317	−	0.999454i	$-0.489484\pi$
$647$	1087.22	0.0660635	0.0330317	+	0.999454i	$0.489484\pi$
$648$	0	0
$649$	4179.21	0.252771
$650$	0	0
$651$	−377.412	−0.0227219
$652$	0	0
$653$	26132.1	1.56605	0.783023	−	0.621992i	$-0.213677\pi$
$653$	26132.1	1.56605	0.783023	+	0.621992i	$0.213677\pi$
$654$	0	0
$655$	22436.1	1.33840
$656$	0	0
$657$	−2630.83	−0.156223
$658$	0	0
$659$	−11713.5	−0.692403	−0.346202	−	0.938160i	$-0.612529\pi$
$659$	−11713.5	−0.692403	−0.346202	+	0.938160i	$0.612529\pi$
$660$	0	0
$661$	−23074.5	−1.35778	−0.678892	−	0.734238i	$-0.737540\pi$
$661$	−23074.5	−1.35778	−0.678892	+	0.734238i	$0.737540\pi$
$662$	0	0
$663$	−2442.24	−0.143060
$664$	0	0
$665$	3197.23	0.186441
$666$	0	0
$667$	−4269.94	−0.247875
$668$	0	0
$669$	−8276.15	−0.478288
$670$	0	0
$671$	−3675.44	−0.211459
$672$	0	0
$673$	32482.6	1.86049	0.930247	−	0.366934i	$-0.119592\pi$
$673$	32482.6	1.86049	0.930247	+	0.366934i	$0.119592\pi$
$674$	0	0
$675$	1159.99	0.0661451
$676$	0	0
$677$	3909.65	0.221950	0.110975	−	0.993823i	$-0.464603\pi$
$677$	3909.65	0.221950	0.110975	+	0.993823i	$0.464603\pi$
$678$	0	0
$679$	−3214.34	−0.181672
$680$	0	0
$681$	2523.48	0.141997
$682$	0	0
$683$	25361.0	1.42081	0.710405	−	0.703793i	$-0.248512\pi$
$683$	25361.0	1.42081	0.710405	+	0.703793i	$0.248512\pi$
$684$	0	0
$685$	595.785	0.0332318
$686$	0	0
$687$	6092.69	0.338356
$688$	0	0
$689$	4784.69	0.264561
$690$	0	0
$691$	24519.5	1.34988	0.674938	−	0.737874i	$-0.264170\pi$
$691$	24519.5	1.34988	0.674938	+	0.737874i	$0.264170\pi$
$692$	0	0
$693$	400.155	0.0219346
$694$	0	0
$695$	3201.45	0.174731
$696$	0	0
$697$	2510.79	0.136446
$698$	0	0
$699$	14630.3	0.791657
$700$	0	0
$701$	−24859.3	−1.33941	−0.669703	−	0.742629i	$-0.733579\pi$
$701$	−24859.3	−1.33941	−0.669703	+	0.742629i	$0.733579\pi$
$702$	0	0
$703$	7795.18	0.418209
$704$	0	0
$705$	20650.6	1.10319
$706$	0	0
$707$	−491.840	−0.0261634
$708$	0	0
$709$	9799.70	0.519091	0.259546	−	0.965731i	$-0.416427\pi$
$709$	9799.70	0.519091	0.259546	+	0.965731i	$0.416427\pi$
$710$	0	0
$711$	−6504.26	−0.343078
$712$	0	0
$713$	4036.55	0.212020
$714$	0	0
$715$	4244.93	0.222030
$716$	0	0
$717$	14482.4	0.754329
$718$	0	0
$719$	−33260.6	−1.72519	−0.862595	−	0.505896i	$-0.831162\pi$
$719$	−33260.6	−1.72519	−0.862595	+	0.505896i	$0.831162\pi$
$720$	0	0
$721$	−419.833	−0.0216857
$722$	0	0
$723$	14903.2	0.766605
$724$	0	0
$725$	3239.89	0.165967
$726$	0	0
$727$	7477.82	0.381482	0.190741	−	0.981640i	$-0.438911\pi$
$727$	7477.82	0.381482	0.190741	+	0.981640i	$0.438911\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−925.887	−0.0468470
$732$	0	0
$733$	−759.810	−0.0382868	−0.0191434	−	0.999817i	$-0.506094\pi$
$733$	−759.810	−0.0382868	−0.0191434	+	0.999817i	$0.506094\pi$
$734$	0	0
$735$	−13214.8	−0.663177
$736$	0	0
$737$	22710.4	1.13507
$738$	0	0
$739$	7873.29	0.391913	0.195956	−	0.980613i	$-0.437219\pi$
$739$	7873.29	0.391913	0.195956	+	0.980613i	$0.437219\pi$
$740$	0	0
$741$	5452.13	0.270295
$742$	0	0
$743$	26778.1	1.32220	0.661099	−	0.750299i	$-0.270090\pi$
$743$	26778.1	1.32220	0.661099	+	0.750299i	$0.270090\pi$
$744$	0	0
$745$	−43932.5	−2.16048
$746$	0	0
$747$	5088.28	0.249224
$748$	0	0
$749$	1628.04	0.0794222
$750$	0	0
$751$	−17975.7	−0.873424	−0.436712	−	0.899601i	$-0.643857\pi$
$751$	−17975.7	−0.873424	−0.436712	+	0.899601i	$0.643857\pi$
$752$	0	0
$753$	12168.2	0.588889
$754$	0	0
$755$	−40476.3	−1.95110
$756$	0	0
$757$	−10835.1	−0.520223	−0.260112	−	0.965579i	$-0.583759\pi$
$757$	−10835.1	−0.520223	−0.260112	+	0.965579i	$0.583759\pi$
$758$	0	0
$759$	−4279.80	−0.204673
$760$	0	0
$761$	−15334.7	−0.730463	−0.365231	−	0.930917i	$-0.619010\pi$
$761$	−15334.7	−0.730463	−0.365231	+	0.930917i	$0.619010\pi$
$762$	0	0
$763$	−638.013	−0.0302721
$764$	0	0
$765$	−7304.20	−0.345208
$766$	0	0
$767$	2156.34	0.101514
$768$	0	0
$769$	−23503.2	−1.10214	−0.551070	−	0.834459i	$-0.685780\pi$
$769$	−23503.2	−1.10214	−0.551070	+	0.834459i	$0.685780\pi$
$770$	0	0
$771$	15978.8	0.746385
$772$	0	0
$773$	23536.4	1.09514	0.547570	−	0.836760i	$-0.315553\pi$
$773$	23536.4	1.09514	0.547570	+	0.836760i	$0.315553\pi$
$774$	0	0
$775$	−3062.80	−0.141960
$776$	0	0
$777$	295.197	0.0136295
$778$	0	0
$779$	−5605.16	−0.257799
$780$	0	0
$781$	−8709.02	−0.399018
$782$	0	0
$783$	2036.12	0.0929311
$784$	0	0
$785$	−24346.6	−1.10697
$786$	0	0
$787$	23939.3	1.08430	0.542150	−	0.840282i	$-0.317610\pi$
$787$	23939.3	1.08430	0.542150	+	0.840282i	$0.317610\pi$
$788$	0	0
$789$	16715.7	0.754237
$790$	0	0
$791$	−1888.03	−0.0848679
$792$	0	0
$793$	−1896.41	−0.0849224
$794$	0	0
$795$	14309.9	0.638392
$796$	0	0
$797$	13136.7	0.583847	0.291923	−	0.956442i	$-0.405705\pi$
$797$	13136.7	0.583847	0.291923	+	0.956442i	$0.405705\pi$
$798$	0	0
$799$	−33260.6	−1.47268
$800$	0	0
$801$	−2476.04	−0.109222
$802$	0	0
$803$	−7364.98	−0.323667
$804$	0	0
$805$	−1294.95	−0.0566970
$806$	0	0
$807$	591.101	0.0257841
$808$	0	0
$809$	18751.0	0.814896	0.407448	−	0.913228i	$-0.366419\pi$
$809$	18751.0	0.814896	0.407448	+	0.913228i	$0.366419\pi$
$810$	0	0
$811$	−5391.15	−0.233426	−0.116713	−	0.993166i	$-0.537236\pi$
$811$	−5391.15	−0.233426	−0.116713	+	0.993166i	$0.537236\pi$
$812$	0	0
$813$	−16169.6	−0.697530
$814$	0	0
$815$	−28905.8	−1.24236
$816$	0	0
$817$	2066.98	0.0885120
$818$	0	0
$819$	206.467	0.00880898
$820$	0	0
$821$	40576.7	1.72489	0.862446	−	0.506149i	$-0.168932\pi$
$821$	40576.7	1.72489	0.862446	+	0.506149i	$0.168932\pi$
$822$	0	0
$823$	−29261.2	−1.23934	−0.619672	−	0.784861i	$-0.712735\pi$
$823$	−29261.2	−1.23934	−0.619672	+	0.784861i	$0.712735\pi$
$824$	0	0
$825$	3247.37	0.137041
$826$	0	0
$827$	−30518.5	−1.28323	−0.641617	−	0.767026i	$-0.721736\pi$
$827$	−30518.5	−1.28323	−0.641617	+	0.767026i	$0.721736\pi$
$828$	0	0
$829$	−28135.3	−1.17875	−0.589373	−	0.807861i	$-0.700625\pi$
$829$	−28135.3	−1.17875	−0.589373	+	0.807861i	$0.700625\pi$
$830$	0	0
$831$	−16395.4	−0.684415
$832$	0	0
$833$	21284.2	0.885298
$834$	0	0
$835$	−12341.2	−0.511481
$836$	0	0
$837$	−1924.83	−0.0794885
$838$	0	0
$839$	20520.4	0.844388	0.422194	−	0.906506i	$-0.361260\pi$
$839$	20520.4	0.844388	0.422194	+	0.906506i	$0.361260\pi$
$840$	0	0
$841$	−18702.0	−0.766823
$842$	0	0
$843$	−14382.8	−0.587627
$844$	0	0
$845$	2190.25	0.0891678
$846$	0	0
$847$	−1228.56	−0.0498392
$848$	0	0
$849$	6854.42	0.277083
$850$	0	0
$851$	−3157.23	−0.127178
$852$	0	0
$853$	34721.4	1.39371	0.696857	−	0.717210i	$-0.254581\pi$
$853$	34721.4	1.39371	0.696857	+	0.717210i	$0.254581\pi$
$854$	0	0
$855$	16306.1	0.652230
$856$	0	0
$857$	40484.5	1.61368	0.806841	−	0.590769i	$-0.201175\pi$
$857$	40484.5	1.61368	0.806841	+	0.590769i	$0.201175\pi$
$858$	0	0
$859$	−7347.38	−0.291839	−0.145919	−	0.989296i	$-0.546614\pi$
$859$	−7347.38	−0.291839	−0.145919	+	0.989296i	$0.546614\pi$
$860$	0	0
$861$	−212.263	−0.00840173
$862$	0	0
$863$	−43530.1	−1.71701	−0.858505	−	0.512804i	$-0.828607\pi$
$863$	−43530.1	−1.71701	−0.858505	+	0.512804i	$0.828607\pi$
$864$	0	0
$865$	−40493.9	−1.59172
$866$	0	0
$867$	−2974.62	−0.116521
$868$	0	0
$869$	−18208.6	−0.710798
$870$	0	0
$871$	11717.8	0.455848
$872$	0	0
$873$	−16393.4	−0.635546
$874$	0	0
$875$	−1876.22	−0.0724890
$876$	0	0
$877$	−24155.9	−0.930088	−0.465044	−	0.885287i	$-0.653962\pi$
$877$	−24155.9	−0.930088	−0.465044	+	0.885287i	$0.653962\pi$
$878$	0	0
$879$	−20072.6	−0.770230
$880$	0	0
$881$	−21971.9	−0.840239	−0.420120	−	0.907469i	$-0.638012\pi$
$881$	−21971.9	−0.840239	−0.420120	+	0.907469i	$0.638012\pi$
$882$	0	0
$883$	5032.44	0.191795	0.0958976	−	0.995391i	$-0.469428\pi$
$883$	5032.44	0.191795	0.0958976	+	0.995391i	$0.469428\pi$
$884$	0	0
$885$	6449.13	0.244955
$886$	0	0
$887$	−5082.60	−0.192398	−0.0961990	−	0.995362i	$-0.530669\pi$
$887$	−5082.60	−0.192398	−0.0961990	+	0.995362i	$0.530669\pi$
$888$	0	0
$889$	−3093.60	−0.116711
$890$	0	0
$891$	2040.82	0.0767342
$892$	0	0
$893$	74251.8	2.78247
$894$	0	0
$895$	−25144.0	−0.939075
$896$	0	0
$897$	−2208.24	−0.0821973
$898$	0	0
$899$	−5376.12	−0.199448
$900$	0	0
$901$	−23048.1	−0.852211
$902$	0	0
$903$	78.2746	0.00288462
$904$	0	0
$905$	−801.601	−0.0294432
$906$	0	0
$907$	2515.37	0.0920853	0.0460427	−	0.998939i	$-0.485339\pi$
$907$	2515.37	0.0920853	0.0460427	+	0.998939i	$0.485339\pi$
$908$	0	0
$909$	−2508.42	−0.0915282
$910$	0	0
$911$	22288.3	0.810588	0.405294	−	0.914186i	$-0.367169\pi$
$911$	22288.3	0.810588	0.405294	+	0.914186i	$0.367169\pi$
$912$	0	0
$913$	14244.6	0.516348
$914$	0	0
$915$	−5671.74	−0.204920
$916$	0	0
$917$	3054.97	0.110015
$918$	0	0
$919$	−49002.3	−1.75891	−0.879454	−	0.475985i	$-0.842092\pi$
$919$	−49002.3	−1.75891	−0.879454	+	0.475985i	$0.842092\pi$
$920$	0	0
$921$	−12005.7	−0.429533
$922$	0	0
$923$	−4493.58	−0.160247
$924$	0	0
$925$	2395.60	0.0851533
$926$	0	0
$927$	−2141.18	−0.0758637
$928$	0	0
$929$	9791.91	0.345815	0.172907	−	0.984938i	$-0.444684\pi$
$929$	9791.91	0.345815	0.172907	+	0.984938i	$0.444684\pi$
$930$	0	0
$931$	−47515.4	−1.67267
$932$	0	0
$933$	−732.849	−0.0257153
$934$	0	0
$935$	−20448.0	−0.715209
$936$	0	0
$937$	44528.6	1.55249	0.776246	−	0.630430i	$-0.217122\pi$
$937$	44528.6	1.55249	0.776246	+	0.630430i	$0.217122\pi$
$938$	0	0
$939$	5548.16	0.192819
$940$	0	0
$941$	31492.9	1.09101	0.545505	−	0.838108i	$-0.316338\pi$
$941$	31492.9	1.09101	0.545505	+	0.838108i	$0.316338\pi$
$942$	0	0
$943$	2270.22	0.0783973
$944$	0	0
$945$	617.498	0.0212563
$946$	0	0
$947$	−9774.64	−0.335410	−0.167705	−	0.985837i	$-0.553636\pi$
$947$	−9774.64	−0.335410	−0.167705	+	0.985837i	$0.553636\pi$
$948$	0	0
$949$	−3800.09	−0.129985
$950$	0	0
$951$	30347.4	1.03479
$952$	0	0
$953$	−52688.0	−1.79090	−0.895451	−	0.445159i	$-0.853147\pi$
$953$	−52688.0	−1.79090	−0.895451	+	0.445159i	$0.853147\pi$
$954$	0	0
$955$	17282.9	0.585614
$956$	0	0
$957$	5700.09	0.192537
$958$	0	0
$959$	81.1240	0.00273163
$960$	0	0
$961$	−24708.7	−0.829403
$962$	0	0
$963$	8303.12	0.277845
$964$	0	0
$965$	52240.4	1.74267
$966$	0	0
$967$	25246.9	0.839591	0.419796	−	0.907619i	$-0.362102\pi$
$967$	25246.9	0.839591	0.419796	+	0.907619i	$0.362102\pi$
$968$	0	0
$969$	−26263.1	−0.870685
$970$	0	0
$971$	−11021.4	−0.364257	−0.182129	−	0.983275i	$-0.558299\pi$
$971$	−11021.4	−0.364257	−0.182129	+	0.983275i	$0.558299\pi$
$972$	0	0
$973$	435.919	0.0143627
$974$	0	0
$975$	1675.54	0.0550360
$976$	0	0
$977$	−50522.8	−1.65442	−0.827209	−	0.561894i	$-0.810073\pi$
$977$	−50522.8	−1.65442	−0.827209	+	0.561894i	$0.810073\pi$
$978$	0	0
$979$	−6931.64	−0.226288
$980$	0	0
$981$	−3253.92	−0.105902
$982$	0	0
$983$	30291.7	0.982865	0.491433	−	0.870916i	$-0.336473\pi$
$983$	30291.7	0.982865	0.491433	+	0.870916i	$0.336473\pi$
$984$	0	0
$985$	30322.5	0.980868
$986$	0	0
$987$	2811.85	0.0906811
$988$	0	0
$989$	−837.174	−0.0269167
$990$	0	0
$991$	−59053.5	−1.89293	−0.946466	−	0.322805i	$-0.895374\pi$
$991$	−59053.5	−1.89293	−0.946466	+	0.322805i	$0.895374\pi$
$992$	0	0
$993$	18861.4	0.602769
$994$	0	0
$995$	−43191.7	−1.37615
$996$	0	0
$997$	−6231.88	−0.197960	−0.0989798	−	0.995089i	$-0.531558\pi$
$997$	−6231.88	−0.197960	−0.0989798	+	0.995089i	$0.531558\pi$
$998$	0	0
$999$	1505.53	0.0476804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.h.1.3	✓	3
3.2	odd	2		936.4.a.l.1.1		3
4.3	odd	2		624.4.a.s.1.3		3
8.3	odd	2		2496.4.a.bq.1.1		3
8.5	even	2		2496.4.a.bm.1.1		3
12.11	even	2		1872.4.a.bl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.h.1.3	✓	3	1.1	even	1	trivial
624.4.a.s.1.3		3	4.3	odd	2
936.4.a.l.1.1		3	3.2	odd	2
1872.4.a.bl.1.1		3	12.11	even	2
2496.4.a.bm.1.1		3	8.5	even	2
2496.4.a.bq.1.1		3	8.3	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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +12.9600 q^{5} +1.76468 q^{7} +9.00000 q^{9} +25.1954 q^{11} +13.0000 q^{13} +38.8801 q^{15} -62.6216 q^{17} +139.798 q^{19} +5.29404 q^{21} -56.6216 q^{23} +42.9625 q^{25} +27.0000 q^{27} +75.4119 q^{29} -71.2900 q^{31} +75.5861 q^{33} +22.8703 q^{35} +55.7602 q^{37} +39.0000 q^{39} -40.0947 q^{41} +14.7854 q^{43} +116.640 q^{45} +531.136 q^{47} -339.886 q^{49} -187.865 q^{51} +368.053 q^{53} +326.533 q^{55} +419.395 q^{57} +165.872 q^{59} -145.878 q^{61} +15.8821 q^{63} +168.480 q^{65} +901.373 q^{67} -169.865 q^{69} -345.660 q^{71} -292.315 q^{73} +128.888 q^{75} +44.4617 q^{77} -722.695 q^{79} +81.0000 q^{81} +565.364 q^{83} -811.578 q^{85} +226.236 q^{87} -275.116 q^{89} +22.9408 q^{91} -213.870 q^{93} +1811.79 q^{95} -1821.49 q^{97} +226.758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 4 q^{5} + 6 q^{7} + 27 q^{9} + 32 q^{11} + 39 q^{13} - 12 q^{15} + 158 q^{17} + 70 q^{19} + 18 q^{21} + 176 q^{23} + 209 q^{25} + 81 q^{27} + 222 q^{29} + 54 q^{31} + 96 q^{33} + 496 q^{35}+ \cdots + 288 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 - 4 * q^5 + 6 * q^7 + 27 * q^9 + 32 * q^11 + 39 * q^13 - 12 * q^15 + 158 * q^17 + 70 * q^19 + 18 * q^21 + 176 * q^23 + 209 * q^25 + 81 * q^27 + 222 * q^29 + 54 * q^31 + 96 * q^33 + 496 * q^35 - 90 * q^37 + 117 * q^39 + 104 * q^41 + 140 * q^43 - 36 * q^45 + 328 * q^47 - 65 * q^49 + 474 * q^51 - 358 * q^53 + 32 * q^55 + 210 * q^57 + 1060 * q^59 - 186 * q^61 + 54 * q^63 - 52 * q^65 + 354 * q^67 + 528 * q^69 + 1692 * q^71 - 974 * q^73 + 627 * q^75 - 384 * q^77 + 776 * q^79 + 243 * q^81 + 2340 * q^83 - 2824 * q^85 + 666 * q^87 - 684 * q^89 + 78 * q^91 + 162 * q^93 - 742 * q^97 + 288 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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