LMFDB - Embedded newform 1344.4.a.bm.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,4,Mod(1,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1344.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,-10,0,-14,0,18,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$79.2985670477$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{137}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 34 $ x^2 - x - 34
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.35235$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +6.70470 q^{5} -7.00000 q^{7} +9.00000 q^{9} -0.704700 q^{11} -35.4094 q^{13} +20.1141 q^{15} +26.7047 q^{17} -30.8188 q^{19} -21.0000 q^{21} -5.88590 q^{23} -80.0470 q^{25} +27.0000 q^{27} -252.094 q^{29} +37.6376 q^{31} -2.11410 q^{33} -46.9329 q^{35} +139.866 q^{37} -106.228 q^{39} -46.7047 q^{41} +254.094 q^{43} +60.3423 q^{45} -607.960 q^{47} +49.0000 q^{49} +80.1141 q^{51} -298.685 q^{53} -4.72480 q^{55} -92.4564 q^{57} +178.591 q^{59} -390.188 q^{61} -63.0000 q^{63} -237.409 q^{65} +349.141 q^{67} -17.6577 q^{69} -348.477 q^{71} -646.416 q^{73} -240.141 q^{75} +4.93290 q^{77} -1047.50 q^{79} +81.0000 q^{81} +278.819 q^{83} +179.047 q^{85} -756.282 q^{87} +439.309 q^{89} +247.866 q^{91} +112.913 q^{93} -206.631 q^{95} -154.309 q^{97} -6.34230 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 10 q^{5} - 14 q^{7} + 18 q^{9} + 22 q^{11} - 24 q^{13} - 30 q^{15} + 30 q^{17} + 32 q^{19} - 42 q^{21} - 82 q^{23} + 74 q^{25} + 54 q^{27} - 36 q^{29} - 112 q^{31} + 66 q^{33} + 70 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 10 * q^5 - 14 * q^7 + 18 * q^9 + 22 * q^11 - 24 * q^13 - 30 * q^15 + 30 * q^17 + 32 * q^19 - 42 * q^21 - 82 * q^23 + 74 * q^25 + 54 * q^27 - 36 * q^29 - 112 * q^31 + 66 * q^33 + 70 * q^35 - 48 * q^37 - 72 * q^39 - 70 * q^41 + 40 * q^43 - 90 * q^45 - 420 * q^47 + 98 * q^49 + 90 * q^51 - 176 * q^53 - 384 * q^55 + 96 * q^57 + 404 * q^59 + 156 * q^61 - 126 * q^63 - 428 * q^65 - 4 * q^67 - 246 * q^69 - 814 * q^71 - 216 * q^73 + 222 * q^75 - 154 * q^77 - 1580 * q^79 + 162 * q^81 + 464 * q^83 + 124 * q^85 - 108 * q^87 - 1158 * q^89 + 168 * q^91 - 336 * q^93 - 1256 * q^95 - 1760 * q^97 + 198 * 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
$5$	6.70470	0.599687	0.299843	−	0.953988i	$-0.403066\pi$
$5$	6.70470	0.599687	0.299843	+	0.953988i	$0.403066\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−0.704700	−0.0193159	−0.00965796	−	0.999953i	$-0.503074\pi$
$11$	−0.704700	−0.0193159	−0.00965796	+	0.999953i	$0.503074\pi$
$12$	0	0
$13$	−35.4094	−0.755446	−0.377723	−	0.925919i	$-0.623293\pi$
$13$	−35.4094	−0.755446	−0.377723	+	0.925919i	$0.623293\pi$
$14$	0	0
$15$	20.1141	0.346229
$16$	0	0
$17$	26.7047	0.380991	0.190495	−	0.981688i	$-0.438991\pi$
$17$	26.7047	0.380991	0.190495	+	0.981688i	$0.438991\pi$
$18$	0	0
$19$	−30.8188	−0.372122	−0.186061	−	0.982538i	$-0.559572\pi$
$19$	−30.8188	−0.372122	−0.186061	+	0.982538i	$0.559572\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−5.88590	−0.0533607	−0.0266803	−	0.999644i	$-0.508494\pi$
$23$	−5.88590	−0.0533607	−0.0266803	+	0.999644i	$0.508494\pi$
$24$	0	0
$25$	−80.0470	−0.640376
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−252.094	−1.61423	−0.807115	−	0.590394i	$-0.798972\pi$
$29$	−252.094	−1.61423	−0.807115	+	0.590394i	$0.798972\pi$
$30$	0	0
$31$	37.6376	0.218062	0.109031	−	0.994038i	$-0.465225\pi$
$31$	37.6376	0.218062	0.109031	+	0.994038i	$0.465225\pi$
$32$	0	0
$33$	−2.11410	−0.0111520
$34$	0	0
$35$	−46.9329	−0.226660
$36$	0	0
$37$	139.866	0.621454	0.310727	−	0.950499i	$-0.399428\pi$
$37$	139.866	0.621454	0.310727	+	0.950499i	$0.399428\pi$
$38$	0	0
$39$	−106.228	−0.436157
$40$	0	0
$41$	−46.7047	−0.177904	−0.0889518	−	0.996036i	$-0.528352\pi$
$41$	−46.7047	−0.177904	−0.0889518	+	0.996036i	$0.528352\pi$
$42$	0	0
$43$	254.094	0.901139	0.450569	−	0.892741i	$-0.351221\pi$
$43$	254.094	0.901139	0.450569	+	0.892741i	$0.351221\pi$
$44$	0	0
$45$	60.3423	0.199896
$46$	0	0
$47$	−607.960	−1.88681	−0.943405	−	0.331643i	$-0.892397\pi$
$47$	−607.960	−1.88681	−0.943405	+	0.331643i	$0.892397\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	80.1141	0.219965
$52$	0	0
$53$	−298.685	−0.774103	−0.387052	−	0.922058i	$-0.626507\pi$
$53$	−298.685	−0.774103	−0.387052	+	0.922058i	$0.626507\pi$
$54$	0	0
$55$	−4.72480	−0.0115835
$56$	0	0
$57$	−92.4564	−0.214845
$58$	0	0
$59$	178.591	0.394077	0.197038	−	0.980396i	$-0.436868\pi$
$59$	178.591	0.394077	0.197038	+	0.980396i	$0.436868\pi$
$60$	0	0
$61$	−390.188	−0.818991	−0.409496	−	0.912312i	$-0.634295\pi$
$61$	−390.188	−0.818991	−0.409496	+	0.912312i	$0.634295\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−237.409	−0.453031
$66$	0	0
$67$	349.141	0.636632	0.318316	−	0.947985i	$-0.396883\pi$
$67$	349.141	0.636632	0.318316	+	0.947985i	$0.396883\pi$
$68$	0	0
$69$	−17.6577	−0.0308078
$70$	0	0
$71$	−348.477	−0.582487	−0.291243	−	0.956649i	$-0.594069\pi$
$71$	−348.477	−0.582487	−0.291243	+	0.956649i	$0.594069\pi$
$72$	0	0
$73$	−646.416	−1.03640	−0.518201	−	0.855259i	$-0.673398\pi$
$73$	−646.416	−1.03640	−0.518201	+	0.855259i	$0.673398\pi$
$74$	0	0
$75$	−240.141	−0.369721
$76$	0	0
$77$	4.93290	0.00730073
$78$	0	0
$79$	−1047.50	−1.49181	−0.745907	−	0.666050i	$-0.767984\pi$
$79$	−1047.50	−1.49181	−0.745907	+	0.666050i	$0.767984\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	278.819	0.368727	0.184363	−	0.982858i	$-0.440978\pi$
$83$	278.819	0.368727	0.184363	+	0.982858i	$0.440978\pi$
$84$	0	0
$85$	179.047	0.228475
$86$	0	0
$87$	−756.282	−0.931976
$88$	0	0
$89$	439.309	0.523221	0.261610	−	0.965174i	$-0.415746\pi$
$89$	439.309	0.523221	0.261610	+	0.965174i	$0.415746\pi$
$90$	0	0
$91$	247.866	0.285532
$92$	0	0
$93$	112.913	0.125898
$94$	0	0
$95$	−206.631	−0.223157
$96$	0	0
$97$	−154.309	−0.161522	−0.0807612	−	0.996733i	$-0.525735\pi$
$97$	−154.309	−0.161522	−0.0807612	+	0.996733i	$0.525735\pi$
$98$	0	0
$99$	−6.34230	−0.00643864
$100$	0	0
$101$	−166.060	−0.163600	−0.0818001	−	0.996649i	$-0.526067\pi$
$101$	−166.060	−0.163600	−0.0818001	+	0.996649i	$0.526067\pi$
$102$	0	0
$103$	−42.4428	−0.0406021	−0.0203010	−	0.999794i	$-0.506462\pi$
$103$	−42.4428	−0.0406021	−0.0203010	+	0.999794i	$0.506462\pi$
$104$	0	0
$105$	−140.799	−0.130862
$106$	0	0
$107$	293.470	0.265148	0.132574	−	0.991173i	$-0.457676\pi$
$107$	293.470	0.265148	0.132574	+	0.991173i	$0.457676\pi$
$108$	0	0
$109$	1176.01	1.03341	0.516705	−	0.856164i	$-0.327158\pi$
$109$	1176.01	1.03341	0.516705	+	0.856164i	$0.327158\pi$
$110$	0	0
$111$	419.597	0.358797
$112$	0	0
$113$	−182.537	−0.151961	−0.0759806	−	0.997109i	$-0.524209\pi$
$113$	−182.537	−0.151961	−0.0759806	+	0.997109i	$0.524209\pi$
$114$	0	0
$115$	−39.4632	−0.0319997
$116$	0	0
$117$	−318.685	−0.251815
$118$	0	0
$119$	−186.933	−0.144001
$120$	0	0
$121$	−1330.50	−0.999627
$122$	0	0
$123$	−140.114	−0.102713
$124$	0	0
$125$	−1374.78	−0.983711
$126$	0	0
$127$	−1374.59	−0.960435	−0.480217	−	0.877150i	$-0.659442\pi$
$127$	−1374.59	−0.960435	−0.480217	+	0.877150i	$0.659442\pi$
$128$	0	0
$129$	762.282	0.520273
$130$	0	0
$131$	1316.91	0.878315	0.439157	−	0.898410i	$-0.355277\pi$
$131$	1316.91	0.878315	0.439157	+	0.898410i	$0.355277\pi$
$132$	0	0
$133$	215.732	0.140649
$134$	0	0
$135$	181.027	0.115410
$136$	0	0
$137$	−2266.98	−1.41373	−0.706866	−	0.707348i	$-0.749891\pi$
$137$	−2266.98	−1.41373	−0.706866	+	0.707348i	$0.749891\pi$
$138$	0	0
$139$	1162.93	0.709627	0.354813	−	0.934937i	$-0.384544\pi$
$139$	1162.93	0.709627	0.354813	+	0.934937i	$0.384544\pi$
$140$	0	0
$141$	−1823.88	−1.08935
$142$	0	0
$143$	24.9530	0.0145921
$144$	0	0
$145$	−1690.21	−0.968032
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	54.0674	0.0297273	0.0148637	−	0.999890i	$-0.495269\pi$
$149$	54.0674	0.0297273	0.0148637	+	0.999890i	$0.495269\pi$
$150$	0	0
$151$	862.014	0.464567	0.232284	−	0.972648i	$-0.425380\pi$
$151$	862.014	0.464567	0.232284	+	0.972648i	$0.425380\pi$
$152$	0	0
$153$	240.342	0.126997
$154$	0	0
$155$	252.349	0.130769
$156$	0	0
$157$	−1459.75	−0.742040	−0.371020	−	0.928625i	$-0.620992\pi$
$157$	−1459.75	−0.742040	−0.371020	+	0.928625i	$0.620992\pi$
$158$	0	0
$159$	−896.054	−0.446929
$160$	0	0
$161$	41.2013	0.0201684
$162$	0	0
$163$	1067.77	0.513094	0.256547	−	0.966532i	$-0.417415\pi$
$163$	1067.77	0.513094	0.256547	+	0.966532i	$0.417415\pi$
$164$	0	0
$165$	−14.1744	−0.00668773
$166$	0	0
$167$	2082.86	0.965129	0.482564	−	0.875861i	$-0.339705\pi$
$167$	2082.86	0.965129	0.482564	+	0.875861i	$0.339705\pi$
$168$	0	0
$169$	−943.174	−0.429301
$170$	0	0
$171$	−277.369	−0.124041
$172$	0	0
$173$	−2210.17	−0.971306	−0.485653	−	0.874152i	$-0.661418\pi$
$173$	−2210.17	−0.971306	−0.485653	+	0.874152i	$0.661418\pi$
$174$	0	0
$175$	560.329	0.242039
$176$	0	0
$177$	535.772	0.227520
$178$	0	0
$179$	934.718	0.390302	0.195151	−	0.980773i	$-0.437480\pi$
$179$	934.718	0.390302	0.195151	+	0.980773i	$0.437480\pi$
$180$	0	0
$181$	−5.85221	−0.00240327	−0.00120163	−	0.999999i	$-0.500382\pi$
$181$	−5.85221	−0.00240327	−0.00120163	+	0.999999i	$0.500382\pi$
$182$	0	0
$183$	−1170.56	−0.472845
$184$	0	0
$185$	937.758	0.372678
$186$	0	0
$187$	−18.8188	−0.00735918
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−2377.34	−0.900618	−0.450309	−	0.892873i	$-0.648686\pi$
$191$	−2377.34	−0.900618	−0.450309	+	0.892873i	$0.648686\pi$
$192$	0	0
$193$	−2233.85	−0.833141	−0.416570	−	0.909103i	$-0.636768\pi$
$193$	−2233.85	−0.833141	−0.416570	+	0.909103i	$0.636768\pi$
$194$	0	0
$195$	−712.228	−0.261558
$196$	0	0
$197$	−2358.95	−0.853139	−0.426570	−	0.904455i	$-0.640278\pi$
$197$	−2358.95	−0.853139	−0.426570	+	0.904455i	$0.640278\pi$
$198$	0	0
$199$	−367.759	−0.131004	−0.0655018	−	0.997852i	$-0.520865\pi$
$199$	−367.759	−0.131004	−0.0655018	+	0.997852i	$0.520865\pi$
$200$	0	0
$201$	1047.42	0.367560
$202$	0	0
$203$	1764.66	0.610122
$204$	0	0
$205$	−313.141	−0.106686
$206$	0	0
$207$	−52.9731	−0.0177869
$208$	0	0
$209$	21.7180	0.00718787
$210$	0	0
$211$	1873.13	0.611144	0.305572	−	0.952169i	$-0.401152\pi$
$211$	1873.13	0.611144	0.305572	+	0.952169i	$0.401152\pi$
$212$	0	0
$213$	−1045.43	−0.336299
$214$	0	0
$215$	1703.62	0.540401
$216$	0	0
$217$	−263.463	−0.0824196
$218$	0	0
$219$	−1939.25	−0.598367
$220$	0	0
$221$	−945.597	−0.287818
$222$	0	0
$223$	3352.83	1.00683	0.503413	−	0.864046i	$-0.332077\pi$
$223$	3352.83	1.00683	0.503413	+	0.864046i	$0.332077\pi$
$224$	0	0
$225$	−720.423	−0.213459
$226$	0	0
$227$	−1006.08	−0.294167	−0.147084	−	0.989124i	$-0.546989\pi$
$227$	−1006.08	−0.294167	−0.147084	+	0.989124i	$0.546989\pi$
$228$	0	0
$229$	−1550.82	−0.447515	−0.223758	−	0.974645i	$-0.571832\pi$
$229$	−1550.82	−0.447515	−0.223758	+	0.974645i	$0.571832\pi$
$230$	0	0
$231$	14.7987	0.00421508
$232$	0	0
$233$	592.215	0.166512	0.0832559	−	0.996528i	$-0.473468\pi$
$233$	592.215	0.166512	0.0832559	+	0.996528i	$0.473468\pi$
$234$	0	0
$235$	−4076.19	−1.13149
$236$	0	0
$237$	−3142.51	−0.861299
$238$	0	0
$239$	5096.56	1.37937	0.689684	−	0.724111i	$-0.257749\pi$
$239$	5096.56	1.37937	0.689684	+	0.724111i	$0.257749\pi$
$240$	0	0
$241$	−4499.44	−1.20263	−0.601316	−	0.799011i	$-0.705357\pi$
$241$	−4499.44	−1.20263	−0.601316	+	0.799011i	$0.705357\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	328.530	0.0856695
$246$	0	0
$247$	1091.28	0.281118
$248$	0	0
$249$	836.456	0.212885
$250$	0	0
$251$	−1602.75	−0.403047	−0.201523	−	0.979484i	$-0.564589\pi$
$251$	−1602.75	−0.403047	−0.201523	+	0.979484i	$0.564589\pi$
$252$	0	0
$253$	4.14779	0.00103071
$254$	0	0
$255$	537.141	0.131910
$256$	0	0
$257$	−4953.82	−1.20238	−0.601188	−	0.799107i	$-0.705306\pi$
$257$	−4953.82	−1.20238	−0.601188	+	0.799107i	$0.705306\pi$
$258$	0	0
$259$	−979.061	−0.234888
$260$	0	0
$261$	−2268.85	−0.538077
$262$	0	0
$263$	3675.40	0.861730	0.430865	−	0.902416i	$-0.358208\pi$
$263$	3675.40	0.861730	0.430865	+	0.902416i	$0.358208\pi$
$264$	0	0
$265$	−2002.59	−0.464219
$266$	0	0
$267$	1317.93	0.302082
$268$	0	0
$269$	−1989.64	−0.450969	−0.225485	−	0.974247i	$-0.572397\pi$
$269$	−1989.64	−0.450969	−0.225485	+	0.974247i	$0.572397\pi$
$270$	0	0
$271$	−1638.85	−0.367354	−0.183677	−	0.982987i	$-0.558800\pi$
$271$	−1638.85	−0.367354	−0.183677	+	0.982987i	$0.558800\pi$
$272$	0	0
$273$	743.597	0.164852
$274$	0	0
$275$	56.4091	0.0123694
$276$	0	0
$277$	7810.87	1.69426	0.847130	−	0.531386i	$-0.178329\pi$
$277$	7810.87	1.69426	0.847130	+	0.531386i	$0.178329\pi$
$278$	0	0
$279$	338.738	0.0726872
$280$	0	0
$281$	3018.34	0.640779	0.320389	−	0.947286i	$-0.396186\pi$
$281$	3018.34	0.640779	0.320389	+	0.947286i	$0.396186\pi$
$282$	0	0
$283$	−7348.13	−1.54347	−0.771734	−	0.635946i	$-0.780610\pi$
$283$	−7348.13	−1.54347	−0.771734	+	0.635946i	$0.780610\pi$
$284$	0	0
$285$	−619.892	−0.128839
$286$	0	0
$287$	326.933	0.0672413
$288$	0	0
$289$	−4199.86	−0.854846
$290$	0	0
$291$	−462.926	−0.0932550
$292$	0	0
$293$	−4211.91	−0.839804	−0.419902	−	0.907569i	$-0.637936\pi$
$293$	−4211.91	−0.839804	−0.419902	+	0.907569i	$0.637936\pi$
$294$	0	0
$295$	1197.40	0.236322
$296$	0	0
$297$	−19.0269	−0.00371735
$298$	0	0
$299$	208.416	0.0403111
$300$	0	0
$301$	−1778.66	−0.340598
$302$	0	0
$303$	−498.181	−0.0944546
$304$	0	0
$305$	−2616.09	−0.491138
$306$	0	0
$307$	−7553.16	−1.40417	−0.702087	−	0.712091i	$-0.747748\pi$
$307$	−7553.16	−1.40417	−0.702087	+	0.712091i	$0.747748\pi$
$308$	0	0
$309$	−127.328	−0.0234416
$310$	0	0
$311$	2656.58	0.484375	0.242188	−	0.970229i	$-0.422135\pi$
$311$	2656.58	0.484375	0.242188	+	0.970229i	$0.422135\pi$
$312$	0	0
$313$	−10819.2	−1.95379	−0.976893	−	0.213731i	$-0.931439\pi$
$313$	−10819.2	−1.95379	−0.976893	+	0.213731i	$0.931439\pi$
$314$	0	0
$315$	−422.396	−0.0755534
$316$	0	0
$317$	−1704.91	−0.302074	−0.151037	−	0.988528i	$-0.548261\pi$
$317$	−1704.91	−0.302074	−0.151037	+	0.988528i	$0.548261\pi$
$318$	0	0
$319$	177.651	0.0311803
$320$	0	0
$321$	880.409	0.153083
$322$	0	0
$323$	−823.007	−0.141775
$324$	0	0
$325$	2834.42	0.483770
$326$	0	0
$327$	3528.04	0.596639
$328$	0	0
$329$	4255.72	0.713147
$330$	0	0
$331$	75.1144	0.0124733	0.00623665	−	0.999981i	$-0.498015\pi$
$331$	75.1144	0.0124733	0.00623665	+	0.999981i	$0.498015\pi$
$332$	0	0
$333$	1258.79	0.207151
$334$	0	0
$335$	2340.89	0.381780
$336$	0	0
$337$	−1474.97	−0.238417	−0.119208	−	0.992869i	$-0.538036\pi$
$337$	−1474.97	−0.238417	−0.119208	+	0.992869i	$0.538036\pi$
$338$	0	0
$339$	−547.610	−0.0877349
$340$	0	0
$341$	−26.5232	−0.00421206
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−118.390	−0.0184750
$346$	0	0
$347$	73.8327	0.0114223	0.00571116	−	0.999984i	$-0.498182\pi$
$347$	73.8327	0.0114223	0.00571116	+	0.999984i	$0.498182\pi$
$348$	0	0
$349$	−3803.45	−0.583364	−0.291682	−	0.956515i	$-0.594215\pi$
$349$	−3803.45	−0.583364	−0.291682	+	0.956515i	$0.594215\pi$
$350$	0	0
$351$	−956.054	−0.145386
$352$	0	0
$353$	7569.15	1.14126	0.570631	−	0.821207i	$-0.306699\pi$
$353$	7569.15	1.14126	0.570631	+	0.821207i	$0.306699\pi$
$354$	0	0
$355$	−2336.43	−0.349309
$356$	0	0
$357$	−560.799	−0.0831390
$358$	0	0
$359$	5094.05	0.748896	0.374448	−	0.927248i	$-0.377832\pi$
$359$	5094.05	0.748896	0.374448	+	0.927248i	$0.377832\pi$
$360$	0	0
$361$	−5909.20	−0.861525
$362$	0	0
$363$	−3991.51	−0.577135
$364$	0	0
$365$	−4334.03	−0.621516
$366$	0	0
$367$	−8647.95	−1.23003	−0.615013	−	0.788517i	$-0.710849\pi$
$367$	−8647.95	−1.23003	−0.615013	+	0.788517i	$0.710849\pi$
$368$	0	0
$369$	−420.342	−0.0593012
$370$	0	0
$371$	2090.79	0.292584
$372$	0	0
$373$	13760.1	1.91011	0.955053	−	0.296436i	$-0.0957982\pi$
$373$	13760.1	1.91011	0.955053	+	0.296436i	$0.0957982\pi$
$374$	0	0
$375$	−4124.34	−0.567946
$376$	0	0
$377$	8926.50	1.21946
$378$	0	0
$379$	−2462.66	−0.333768	−0.166884	−	0.985977i	$-0.553371\pi$
$379$	−2462.66	−0.333768	−0.166884	+	0.985977i	$0.553371\pi$
$380$	0	0
$381$	−4123.77	−0.554507
$382$	0	0
$383$	2553.40	0.340659	0.170330	−	0.985387i	$-0.445517\pi$
$383$	2553.40	0.340659	0.170330	+	0.985387i	$0.445517\pi$
$384$	0	0
$385$	33.0736	0.00437815
$386$	0	0
$387$	2286.85	0.300380
$388$	0	0
$389$	5247.13	0.683907	0.341954	−	0.939717i	$-0.388911\pi$
$389$	5247.13	0.683907	0.341954	+	0.939717i	$0.388911\pi$
$390$	0	0
$391$	−157.181	−0.0203299
$392$	0	0
$393$	3950.74	0.507095
$394$	0	0
$395$	−7023.20	−0.894621
$396$	0	0
$397$	11500.6	1.45390	0.726952	−	0.686688i	$-0.240936\pi$
$397$	11500.6	1.45390	0.726952	+	0.686688i	$0.240936\pi$
$398$	0	0
$399$	647.195	0.0812037
$400$	0	0
$401$	−288.322	−0.0359055	−0.0179528	−	0.999839i	$-0.505715\pi$
$401$	−288.322	−0.0359055	−0.0179528	+	0.999839i	$0.505715\pi$
$402$	0	0
$403$	−1332.72	−0.164734
$404$	0	0
$405$	543.081	0.0666318
$406$	0	0
$407$	−98.5634	−0.0120039
$408$	0	0
$409$	13567.8	1.64030	0.820151	−	0.572147i	$-0.193889\pi$
$409$	13567.8	1.64030	0.820151	+	0.572147i	$0.193889\pi$
$410$	0	0
$411$	−6800.94	−0.816218
$412$	0	0
$413$	−1250.13	−0.148947
$414$	0	0
$415$	1869.40	0.221121
$416$	0	0
$417$	3488.78	0.409703
$418$	0	0
$419$	12094.2	1.41012	0.705059	−	0.709148i	$-0.250920\pi$
$419$	12094.2	1.41012	0.705059	+	0.709148i	$0.250920\pi$
$420$	0	0
$421$	−2030.79	−0.235094	−0.117547	−	0.993067i	$-0.537503\pi$
$421$	−2030.79	−0.235094	−0.117547	+	0.993067i	$0.537503\pi$
$422$	0	0
$423$	−5471.64	−0.628937
$424$	0	0
$425$	−2137.63	−0.243977
$426$	0	0
$427$	2731.32	0.309550
$428$	0	0
$429$	74.8590	0.00842477
$430$	0	0
$431$	11236.7	1.25580	0.627902	−	0.778292i	$-0.283914\pi$
$431$	11236.7	1.25580	0.627902	+	0.778292i	$0.283914\pi$
$432$	0	0
$433$	−2847.80	−0.316066	−0.158033	−	0.987434i	$-0.550515\pi$
$433$	−2847.80	−0.316066	−0.158033	+	0.987434i	$0.550515\pi$
$434$	0	0
$435$	−5070.64	−0.558894
$436$	0	0
$437$	181.396	0.0198567
$438$	0	0
$439$	−596.831	−0.0648866	−0.0324433	−	0.999474i	$-0.510329\pi$
$439$	−596.831	−0.0648866	−0.0324433	+	0.999474i	$0.510329\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	10883.5	1.16725	0.583624	−	0.812024i	$-0.301634\pi$
$443$	10883.5	1.16725	0.583624	+	0.812024i	$0.301634\pi$
$444$	0	0
$445$	2945.43	0.313768
$446$	0	0
$447$	162.202	0.0171631
$448$	0	0
$449$	−6593.68	−0.693040	−0.346520	−	0.938043i	$-0.612637\pi$
$449$	−6593.68	−0.693040	−0.346520	+	0.938043i	$0.612637\pi$
$450$	0	0
$451$	32.9128	0.00343637
$452$	0	0
$453$	2586.04	0.268218
$454$	0	0
$455$	1661.87	0.171230
$456$	0	0
$457$	14848.9	1.51992	0.759960	−	0.649970i	$-0.225218\pi$
$457$	14848.9	1.51992	0.759960	+	0.649970i	$0.225218\pi$
$458$	0	0
$459$	721.027	0.0733217
$460$	0	0
$461$	−1723.59	−0.174134	−0.0870668	−	0.996202i	$-0.527749\pi$
$461$	−1723.59	−0.174134	−0.0870668	+	0.996202i	$0.527749\pi$
$462$	0	0
$463$	−17899.2	−1.79664	−0.898321	−	0.439339i	$-0.855213\pi$
$463$	−17899.2	−1.79664	−0.898321	+	0.439339i	$0.855213\pi$
$464$	0	0
$465$	757.046	0.0754993
$466$	0	0
$467$	17921.1	1.77579	0.887893	−	0.460051i	$-0.152169\pi$
$467$	17921.1	1.77579	0.887893	+	0.460051i	$0.152169\pi$
$468$	0	0
$469$	−2443.99	−0.240624
$470$	0	0
$471$	−4379.24	−0.428417
$472$	0	0
$473$	−179.060	−0.0174063
$474$	0	0
$475$	2466.95	0.238298
$476$	0	0
$477$	−2688.16	−0.258034
$478$	0	0
$479$	−1507.10	−0.143760	−0.0718801	−	0.997413i	$-0.522900\pi$
$479$	−1507.10	−0.143760	−0.0718801	+	0.997413i	$0.522900\pi$
$480$	0	0
$481$	−4952.56	−0.469475
$482$	0	0
$483$	123.604	0.0116442
$484$	0	0
$485$	−1034.59	−0.0968628
$486$	0	0
$487$	−7618.50	−0.708885	−0.354443	−	0.935078i	$-0.615329\pi$
$487$	−7618.50	−0.708885	−0.354443	+	0.935078i	$0.615329\pi$
$488$	0	0
$489$	3203.32	0.296235
$490$	0	0
$491$	−7346.20	−0.675212	−0.337606	−	0.941287i	$-0.609617\pi$
$491$	−7346.20	−0.675212	−0.337606	+	0.941287i	$0.609617\pi$
$492$	0	0
$493$	−6732.09	−0.615007
$494$	0	0
$495$	−42.5232	−0.00386116
$496$	0	0
$497$	2439.34	0.220159
$498$	0	0
$499$	−18775.3	−1.68436	−0.842182	−	0.539193i	$-0.818729\pi$
$499$	−18775.3	−1.68436	−0.842182	+	0.539193i	$0.818729\pi$
$500$	0	0
$501$	6248.58	0.557217
$502$	0	0
$503$	2556.27	0.226597	0.113299	−	0.993561i	$-0.463858\pi$
$503$	2556.27	0.226597	0.113299	+	0.993561i	$0.463858\pi$
$504$	0	0
$505$	−1113.38	−0.0981088
$506$	0	0
$507$	−2829.52	−0.247857
$508$	0	0
$509$	−1902.18	−0.165644	−0.0828219	−	0.996564i	$-0.526393\pi$
$509$	−1902.18	−0.165644	−0.0828219	+	0.996564i	$0.526393\pi$
$510$	0	0
$511$	4524.91	0.391723
$512$	0	0
$513$	−832.108	−0.0716149
$514$	0	0
$515$	−284.566	−0.0243485
$516$	0	0
$517$	428.429	0.0364454
$518$	0	0
$519$	−6630.50	−0.560784
$520$	0	0
$521$	14253.1	1.19854	0.599272	−	0.800545i	$-0.295457\pi$
$521$	14253.1	1.19854	0.599272	+	0.800545i	$0.295457\pi$
$522$	0	0
$523$	10544.3	0.881591	0.440796	−	0.897608i	$-0.354696\pi$
$523$	10544.3	0.881591	0.440796	+	0.897608i	$0.354696\pi$
$524$	0	0
$525$	1680.99	0.139741
$526$	0	0
$527$	1005.10	0.0830795
$528$	0	0
$529$	−12132.4	−0.997153
$530$	0	0
$531$	1607.32	0.131359
$532$	0	0
$533$	1653.79	0.134397
$534$	0	0
$535$	1967.63	0.159005
$536$	0	0
$537$	2804.15	0.225341
$538$	0	0
$539$	−34.5303	−0.00275942
$540$	0	0
$541$	7563.14	0.601044	0.300522	−	0.953775i	$-0.402839\pi$
$541$	7563.14	0.601044	0.300522	+	0.953775i	$0.402839\pi$
$542$	0	0
$543$	−17.5566	−0.00138753
$544$	0	0
$545$	7884.82	0.619722
$546$	0	0
$547$	−9059.07	−0.708113	−0.354057	−	0.935224i	$-0.615198\pi$
$547$	−9059.07	−0.708113	−0.354057	+	0.935224i	$0.615198\pi$
$548$	0	0
$549$	−3511.69	−0.272997
$550$	0	0
$551$	7769.23	0.600691
$552$	0	0
$553$	7332.52	0.563853
$554$	0	0
$555$	2813.27	0.215166
$556$	0	0
$557$	23719.2	1.80434	0.902169	−	0.431384i	$-0.141974\pi$
$557$	23719.2	1.80434	0.902169	+	0.431384i	$0.141974\pi$
$558$	0	0
$559$	−8997.32	−0.680762
$560$	0	0
$561$	−56.4564	−0.00424883
$562$	0	0
$563$	14719.8	1.10189	0.550947	−	0.834540i	$-0.314267\pi$
$563$	14719.8	1.10189	0.550947	+	0.834540i	$0.314267\pi$
$564$	0	0
$565$	−1223.85	−0.0911291
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−5184.65	−0.381989	−0.190994	−	0.981591i	$-0.561171\pi$
$569$	−5184.65	−0.381989	−0.190994	+	0.981591i	$0.561171\pi$
$570$	0	0
$571$	13173.6	0.965496	0.482748	−	0.875759i	$-0.339639\pi$
$571$	13173.6	0.965496	0.482748	+	0.875759i	$0.339639\pi$
$572$	0	0
$573$	−7132.01	−0.519972
$574$	0	0
$575$	471.149	0.0341709
$576$	0	0
$577$	2354.29	0.169862	0.0849311	−	0.996387i	$-0.472933\pi$
$577$	2354.29	0.169862	0.0849311	+	0.996387i	$0.472933\pi$
$578$	0	0
$579$	−6701.56	−0.481014
$580$	0	0
$581$	−1951.73	−0.139366
$582$	0	0
$583$	210.483	0.0149525
$584$	0	0
$585$	−2136.68	−0.151010
$586$	0	0
$587$	17138.5	1.20508	0.602541	−	0.798088i	$-0.294155\pi$
$587$	17138.5	1.20508	0.602541	+	0.798088i	$0.294155\pi$
$588$	0	0
$589$	−1159.95	−0.0811455
$590$	0	0
$591$	−7076.86	−0.492560
$592$	0	0
$593$	−20553.0	−1.42329	−0.711645	−	0.702540i	$-0.752049\pi$
$593$	−20553.0	−1.42329	−0.711645	+	0.702540i	$0.752049\pi$
$594$	0	0
$595$	−1253.33	−0.0863554
$596$	0	0
$597$	−1103.28	−0.0756350
$598$	0	0
$599$	−19280.3	−1.31514	−0.657571	−	0.753392i	$-0.728416\pi$
$599$	−19280.3	−1.31514	−0.657571	+	0.753392i	$0.728416\pi$
$600$	0	0
$601$	24736.4	1.67890	0.839451	−	0.543436i	$-0.182877\pi$
$601$	24736.4	1.67890	0.839451	+	0.543436i	$0.182877\pi$
$602$	0	0
$603$	3142.27	0.212211
$604$	0	0
$605$	−8920.63	−0.599463
$606$	0	0
$607$	−4926.07	−0.329395	−0.164698	−	0.986344i	$-0.552665\pi$
$607$	−4926.07	−0.329395	−0.164698	+	0.986344i	$0.552665\pi$
$608$	0	0
$609$	5293.97	0.352254
$610$	0	0
$611$	21527.5	1.42538
$612$	0	0
$613$	28344.4	1.86757	0.933784	−	0.357837i	$-0.116486\pi$
$613$	28344.4	1.86757	0.933784	+	0.357837i	$0.116486\pi$
$614$	0	0
$615$	−939.423	−0.0615954
$616$	0	0
$617$	21881.8	1.42776	0.713879	−	0.700269i	$-0.246937\pi$
$617$	21881.8	1.42776	0.713879	+	0.700269i	$0.246937\pi$
$618$	0	0
$619$	8355.39	0.542538	0.271269	−	0.962504i	$-0.412557\pi$
$619$	8355.39	0.542538	0.271269	+	0.962504i	$0.412557\pi$
$620$	0	0
$621$	−158.919	−0.0102693
$622$	0	0
$623$	−3075.16	−0.197759
$624$	0	0
$625$	788.397	0.0504574
$626$	0	0
$627$	65.1540	0.00414992
$628$	0	0
$629$	3735.07	0.236768
$630$	0	0
$631$	−10780.0	−0.680100	−0.340050	−	0.940407i	$-0.610444\pi$
$631$	−10780.0	−0.680100	−0.340050	+	0.940407i	$0.610444\pi$
$632$	0	0
$633$	5619.38	0.352844
$634$	0	0
$635$	−9216.22	−0.575960
$636$	0	0
$637$	−1735.06	−0.107921
$638$	0	0
$639$	−3136.29	−0.194162
$640$	0	0
$641$	−20161.7	−1.24234	−0.621171	−	0.783675i	$-0.713343\pi$
$641$	−20161.7	−1.24234	−0.621171	+	0.783675i	$0.713343\pi$
$642$	0	0
$643$	18277.8	1.12100	0.560502	−	0.828153i	$-0.310608\pi$
$643$	18277.8	1.12100	0.560502	+	0.828153i	$0.310608\pi$
$644$	0	0
$645$	5110.87	0.312001
$646$	0	0
$647$	2534.86	0.154027	0.0770136	−	0.997030i	$-0.475462\pi$
$647$	2534.86	0.154027	0.0770136	+	0.997030i	$0.475462\pi$
$648$	0	0
$649$	−125.853	−0.00761195
$650$	0	0
$651$	−790.390	−0.0475850
$652$	0	0
$653$	16869.8	1.01098	0.505488	−	0.862833i	$-0.331312\pi$
$653$	16869.8	1.01098	0.505488	+	0.862833i	$0.331312\pi$
$654$	0	0
$655$	8829.51	0.526713
$656$	0	0
$657$	−5817.75	−0.345467
$658$	0	0
$659$	−7724.65	−0.456616	−0.228308	−	0.973589i	$-0.573319\pi$
$659$	−7724.65	−0.456616	−0.228308	+	0.973589i	$0.573319\pi$
$660$	0	0
$661$	−6387.26	−0.375848	−0.187924	−	0.982184i	$-0.560176\pi$
$661$	−6387.26	−0.375848	−0.187924	+	0.982184i	$0.560176\pi$
$662$	0	0
$663$	−2836.79	−0.166172
$664$	0	0
$665$	1446.42	0.0843453
$666$	0	0
$667$	1483.80	0.0861364
$668$	0	0
$669$	10058.5	0.581291
$670$	0	0
$671$	274.965	0.0158196
$672$	0	0
$673$	14558.1	0.833837	0.416919	−	0.908944i	$-0.363110\pi$
$673$	14558.1	0.833837	0.416919	+	0.908944i	$0.363110\pi$
$674$	0	0
$675$	−2161.27	−0.123240
$676$	0	0
$677$	32914.0	1.86852	0.934259	−	0.356595i	$-0.116063\pi$
$677$	32914.0	1.86852	0.934259	+	0.356595i	$0.116063\pi$
$678$	0	0
$679$	1080.16	0.0610497
$680$	0	0
$681$	−3018.24	−0.169837
$682$	0	0
$683$	27334.4	1.53137	0.765683	−	0.643218i	$-0.222401\pi$
$683$	27334.4	1.53137	0.765683	+	0.643218i	$0.222401\pi$
$684$	0	0
$685$	−15199.4	−0.847796
$686$	0	0
$687$	−4652.46	−0.258373
$688$	0	0
$689$	10576.2	0.584794
$690$	0	0
$691$	25255.2	1.39038	0.695190	−	0.718826i	$-0.255320\pi$
$691$	25255.2	1.39038	0.695190	+	0.718826i	$0.255320\pi$
$692$	0	0
$693$	44.3961	0.00243358
$694$	0	0
$695$	7797.07	0.425554
$696$	0	0
$697$	−1247.23	−0.0677796
$698$	0	0
$699$	1776.64	0.0961357
$700$	0	0
$701$	−16418.5	−0.884620	−0.442310	−	0.896862i	$-0.645841\pi$
$701$	−16418.5	−0.884620	−0.442310	+	0.896862i	$0.645841\pi$
$702$	0	0
$703$	−4310.50	−0.231257
$704$	0	0
$705$	−12228.6	−0.653269
$706$	0	0
$707$	1162.42	0.0618351
$708$	0	0
$709$	13186.2	0.698475	0.349238	−	0.937034i	$-0.386441\pi$
$709$	13186.2	0.698475	0.349238	+	0.937034i	$0.386441\pi$
$710$	0	0
$711$	−9427.53	−0.497271
$712$	0	0
$713$	−221.531	−0.0116359
$714$	0	0
$715$	167.302	0.00875071
$716$	0	0
$717$	15289.7	0.796378
$718$	0	0
$719$	17244.2	0.894438	0.447219	−	0.894425i	$-0.352415\pi$
$719$	17244.2	0.894438	0.447219	+	0.894425i	$0.352415\pi$
$720$	0	0
$721$	297.100	0.0153461
$722$	0	0
$723$	−13498.3	−0.694340
$724$	0	0
$725$	20179.4	1.03371
$726$	0	0
$727$	3297.69	0.168232	0.0841160	−	0.996456i	$-0.473193\pi$
$727$	3297.69	0.168232	0.0841160	+	0.996456i	$0.473193\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	6785.50	0.343325
$732$	0	0
$733$	−14191.5	−0.715112	−0.357556	−	0.933892i	$-0.616390\pi$
$733$	−14191.5	−0.715112	−0.357556	+	0.933892i	$0.616390\pi$
$734$	0	0
$735$	985.591	0.0494613
$736$	0	0
$737$	−246.040	−0.0122971
$738$	0	0
$739$	−31437.7	−1.56489	−0.782446	−	0.622718i	$-0.786028\pi$
$739$	−31437.7	−1.56489	−0.782446	+	0.622718i	$0.786028\pi$
$740$	0	0
$741$	3273.83	0.162304
$742$	0	0
$743$	9095.98	0.449124	0.224562	−	0.974460i	$-0.427905\pi$
$743$	9095.98	0.449124	0.224562	+	0.974460i	$0.427905\pi$
$744$	0	0
$745$	362.506	0.0178271
$746$	0	0
$747$	2509.37	0.122909
$748$	0	0
$749$	−2054.29	−0.100216
$750$	0	0
$751$	19791.9	0.961676	0.480838	−	0.876810i	$-0.340333\pi$
$751$	19791.9	0.961676	0.480838	+	0.876810i	$0.340333\pi$
$752$	0	0
$753$	−4808.25	−0.232699
$754$	0	0
$755$	5779.54	0.278595
$756$	0	0
$757$	−11194.5	−0.537479	−0.268739	−	0.963213i	$-0.586607\pi$
$757$	−11194.5	−0.537479	−0.268739	+	0.963213i	$0.586607\pi$
$758$	0	0
$759$	12.4434	0.000595080 0
$760$	0	0
$761$	24445.6	1.16446	0.582230	−	0.813024i	$-0.302180\pi$
$761$	24445.6	1.16446	0.582230	+	0.813024i	$0.302180\pi$
$762$	0	0
$763$	−8232.10	−0.390592
$764$	0	0
$765$	1611.42	0.0761583
$766$	0	0
$767$	−6323.79	−0.297704
$768$	0	0
$769$	33296.7	1.56139	0.780695	−	0.624912i	$-0.214865\pi$
$769$	33296.7	1.56139	0.780695	+	0.624912i	$0.214865\pi$
$770$	0	0
$771$	−14861.5	−0.694192
$772$	0	0
$773$	31129.8	1.44846	0.724230	−	0.689559i	$-0.242195\pi$
$773$	31129.8	1.44846	0.724230	+	0.689559i	$0.242195\pi$
$774$	0	0
$775$	−3012.78	−0.139641
$776$	0	0
$777$	−2937.18	−0.135612
$778$	0	0
$779$	1439.38	0.0662018
$780$	0	0
$781$	245.571	0.0112513
$782$	0	0
$783$	−6806.54	−0.310659
$784$	0	0
$785$	−9787.15	−0.444992
$786$	0	0
$787$	−22918.3	−1.03805	−0.519027	−	0.854758i	$-0.673706\pi$
$787$	−22918.3	−1.03805	−0.519027	+	0.854758i	$0.673706\pi$
$788$	0	0
$789$	11026.2	0.497520
$790$	0	0
$791$	1277.76	0.0574359
$792$	0	0
$793$	13816.3	0.618704
$794$	0	0
$795$	−6007.77	−0.268017
$796$	0	0
$797$	−40462.6	−1.79832	−0.899158	−	0.437624i	$-0.855820\pi$
$797$	−40462.6	−1.79832	−0.899158	+	0.437624i	$0.855820\pi$
$798$	0	0
$799$	−16235.4	−0.718857
$800$	0	0
$801$	3953.78	0.174407
$802$	0	0
$803$	455.529	0.0200190
$804$	0	0
$805$	276.242	0.0120947
$806$	0	0
$807$	−5968.93	−0.260367
$808$	0	0
$809$	−8286.03	−0.360100	−0.180050	−	0.983657i	$-0.557626\pi$
$809$	−8286.03	−0.360100	−0.180050	+	0.983657i	$0.557626\pi$
$810$	0	0
$811$	15527.8	0.672323	0.336162	−	0.941804i	$-0.390871\pi$
$811$	15527.8	0.672323	0.336162	+	0.941804i	$0.390871\pi$
$812$	0	0
$813$	−4916.54	−0.212092
$814$	0	0
$815$	7159.09	0.307696
$816$	0	0
$817$	−7830.87	−0.335334
$818$	0	0
$819$	2230.79	0.0951773
$820$	0	0
$821$	13165.6	0.559663	0.279831	−	0.960049i	$-0.409721\pi$
$821$	13165.6	0.559663	0.279831	+	0.960049i	$0.409721\pi$
$822$	0	0
$823$	−9659.67	−0.409131	−0.204565	−	0.978853i	$-0.565578\pi$
$823$	−9659.67	−0.409131	−0.204565	+	0.978853i	$0.565578\pi$
$824$	0	0
$825$	169.227	0.00714150
$826$	0	0
$827$	−31937.7	−1.34290	−0.671452	−	0.741048i	$-0.734329\pi$
$827$	−31937.7	−1.34290	−0.671452	+	0.741048i	$0.734329\pi$
$828$	0	0
$829$	−29987.5	−1.25635	−0.628173	−	0.778074i	$-0.716197\pi$
$829$	−29987.5	−1.25635	−0.628173	+	0.778074i	$0.716197\pi$
$830$	0	0
$831$	23432.6	0.978181
$832$	0	0
$833$	1308.53	0.0544272
$834$	0	0
$835$	13964.9	0.578775
$836$	0	0
$837$	1016.22	0.0419660
$838$	0	0
$839$	25864.3	1.06428	0.532141	−	0.846656i	$-0.321387\pi$
$839$	25864.3	1.06428	0.532141	+	0.846656i	$0.321387\pi$
$840$	0	0
$841$	39162.4	1.60574
$842$	0	0
$843$	9055.01	0.369954
$844$	0	0
$845$	−6323.70	−0.257446
$846$	0	0
$847$	9313.52	0.377823
$848$	0	0
$849$	−22044.4	−0.891121
$850$	0	0
$851$	−823.236	−0.0331612
$852$	0	0
$853$	9652.50	0.387451	0.193725	−	0.981056i	$-0.437943\pi$
$853$	9652.50	0.387451	0.193725	+	0.981056i	$0.437943\pi$
$854$	0	0
$855$	−1859.68	−0.0743855
$856$	0	0
$857$	7058.37	0.281341	0.140671	−	0.990056i	$-0.455074\pi$
$857$	7058.37	0.281341	0.140671	+	0.990056i	$0.455074\pi$
$858$	0	0
$859$	−5146.18	−0.204407	−0.102203	−	0.994764i	$-0.532589\pi$
$859$	−5146.18	−0.204407	−0.102203	+	0.994764i	$0.532589\pi$
$860$	0	0
$861$	980.799	0.0388218
$862$	0	0
$863$	23233.4	0.916426	0.458213	−	0.888842i	$-0.348490\pi$
$863$	23233.4	0.916426	0.458213	+	0.888842i	$0.348490\pi$
$864$	0	0
$865$	−14818.5	−0.582479
$866$	0	0
$867$	−12599.6	−0.493546
$868$	0	0
$869$	738.176	0.0288158
$870$	0	0
$871$	−12362.9	−0.480941
$872$	0	0
$873$	−1388.78	−0.0538408
$874$	0	0
$875$	9623.45	0.371808
$876$	0	0
$877$	−17998.3	−0.692997	−0.346498	−	0.938051i	$-0.612629\pi$
$877$	−17998.3	−0.692997	−0.346498	+	0.938051i	$0.612629\pi$
$878$	0	0
$879$	−12635.7	−0.484861
$880$	0	0
$881$	10400.3	0.397723	0.198861	−	0.980028i	$-0.436276\pi$
$881$	10400.3	0.397723	0.198861	+	0.980028i	$0.436276\pi$
$882$	0	0
$883$	−35365.0	−1.34782	−0.673912	−	0.738812i	$-0.735387\pi$
$883$	−35365.0	−1.34782	−0.673912	+	0.738812i	$0.735387\pi$
$884$	0	0
$885$	3592.19	0.136441
$886$	0	0
$887$	39216.8	1.48452	0.742261	−	0.670111i	$-0.233754\pi$
$887$	39216.8	1.48452	0.742261	+	0.670111i	$0.233754\pi$
$888$	0	0
$889$	9622.13	0.363010
$890$	0	0
$891$	−57.0807	−0.00214621
$892$	0	0
$893$	18736.6	0.702123
$894$	0	0
$895$	6267.01	0.234059
$896$	0	0
$897$	625.249	0.0232736
$898$	0	0
$899$	−9488.21	−0.352002
$900$	0	0
$901$	−7976.28	−0.294926
$902$	0	0
$903$	−5335.97	−0.196645
$904$	0	0
$905$	−39.2373	−0.00144121
$906$	0	0
$907$	−42804.2	−1.56702	−0.783511	−	0.621378i	$-0.786573\pi$
$907$	−42804.2	−1.56702	−0.783511	+	0.621378i	$0.786573\pi$
$908$	0	0
$909$	−1494.54	−0.0545334
$910$	0	0
$911$	−32448.5	−1.18010	−0.590048	−	0.807368i	$-0.700891\pi$
$911$	−32448.5	−1.18010	−0.590048	+	0.807368i	$0.700891\pi$
$912$	0	0
$913$	−196.484	−0.00712230
$914$	0	0
$915$	−7848.28	−0.283559
$916$	0	0
$917$	−9218.39	−0.331972
$918$	0	0
$919$	21991.6	0.789375	0.394688	−	0.918815i	$-0.370853\pi$
$919$	21991.6	0.789375	0.394688	+	0.918815i	$0.370853\pi$
$920$	0	0
$921$	−22659.5	−0.810700
$922$	0	0
$923$	12339.3	0.440037
$924$	0	0
$925$	−11195.8	−0.397964
$926$	0	0
$927$	−381.985	−0.0135340
$928$	0	0
$929$	−1281.57	−0.0452606	−0.0226303	−	0.999744i	$-0.507204\pi$
$929$	−1281.57	−0.0452606	−0.0226303	+	0.999744i	$0.507204\pi$
$930$	0	0
$931$	−1510.12	−0.0531603
$932$	0	0
$933$	7969.73	0.279654
$934$	0	0
$935$	−126.174	−0.00441320
$936$	0	0
$937$	23582.2	0.822194	0.411097	−	0.911592i	$-0.365146\pi$
$937$	23582.2	0.822194	0.411097	+	0.911592i	$0.365146\pi$
$938$	0	0
$939$	−32457.5	−1.12802
$940$	0	0
$941$	−27439.5	−0.950586	−0.475293	−	0.879828i	$-0.657658\pi$
$941$	−27439.5	−0.950586	−0.475293	+	0.879828i	$0.657658\pi$
$942$	0	0
$943$	274.899	0.00949305
$944$	0	0
$945$	−1267.19	−0.0436208
$946$	0	0
$947$	−17543.6	−0.601996	−0.300998	−	0.953625i	$-0.597320\pi$
$947$	−17543.6	−0.601996	−0.300998	+	0.953625i	$0.597320\pi$
$948$	0	0
$949$	22889.2	0.782945
$950$	0	0
$951$	−5114.74	−0.174403
$952$	0	0
$953$	−13884.1	−0.471930	−0.235965	−	0.971762i	$-0.575825\pi$
$953$	−13884.1	−0.471930	−0.235965	+	0.971762i	$0.575825\pi$
$954$	0	0
$955$	−15939.3	−0.540088
$956$	0	0
$957$	532.952	0.0180020
$958$	0	0
$959$	15868.9	0.534340
$960$	0	0
$961$	−28374.4	−0.952449
$962$	0	0
$963$	2641.23	0.0883825
$964$	0	0
$965$	−14977.3	−0.499623
$966$	0	0
$967$	−39807.7	−1.32382	−0.661908	−	0.749585i	$-0.730253\pi$
$967$	−39807.7	−1.32382	−0.661908	+	0.749585i	$0.730253\pi$
$968$	0	0
$969$	−2469.02	−0.0818538
$970$	0	0
$971$	20963.7	0.692851	0.346425	−	0.938077i	$-0.387395\pi$
$971$	20963.7	0.692851	0.346425	+	0.938077i	$0.387395\pi$
$972$	0	0
$973$	−8140.48	−0.268214
$974$	0	0
$975$	8503.25	0.279305
$976$	0	0
$977$	−38199.7	−1.25089	−0.625443	−	0.780270i	$-0.715082\pi$
$977$	−38199.7	−1.25089	−0.625443	+	0.780270i	$0.715082\pi$
$978$	0	0
$979$	−309.581	−0.0101065
$980$	0	0
$981$	10584.1	0.344470
$982$	0	0
$983$	−18877.5	−0.612510	−0.306255	−	0.951950i	$-0.599076\pi$
$983$	−18877.5	−0.612510	−0.306255	+	0.951950i	$0.599076\pi$
$984$	0	0
$985$	−15816.1	−0.511616
$986$	0	0
$987$	12767.2	0.411736
$988$	0	0
$989$	−1495.57	−0.0480854
$990$	0	0
$991$	1894.95	0.0607418	0.0303709	−	0.999539i	$-0.490331\pi$
$991$	1894.95	0.0607418	0.0303709	+	0.999539i	$0.490331\pi$
$992$	0	0
$993$	225.343	0.00720146
$994$	0	0
$995$	−2465.71	−0.0785611
$996$	0	0
$997$	−60576.0	−1.92424	−0.962118	−	0.272635i	$-0.912105\pi$
$997$	−60576.0	−1.92424	−0.962118	+	0.272635i	$0.912105\pi$
$998$	0	0
$999$	3776.38	0.119599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bm.1.2		2
4.3	odd	2		1344.4.a.be.1.2		2
8.3	odd	2		672.4.a.m.1.1	yes	2
8.5	even	2		672.4.a.h.1.1	✓	2
24.5	odd	2		2016.4.a.i.1.2		2
24.11	even	2		2016.4.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.h.1.1	✓	2	8.5	even	2
672.4.a.m.1.1	yes	2	8.3	odd	2
1344.4.a.be.1.2		2	4.3	odd	2
1344.4.a.bm.1.2		2	1.1	even	1	trivial
2016.4.a.i.1.2		2	24.5	odd	2
2016.4.a.j.1.2		2	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +6.70470 q^{5} -7.00000 q^{7} +9.00000 q^{9} -0.704700 q^{11} -35.4094 q^{13} +20.1141 q^{15} +26.7047 q^{17} -30.8188 q^{19} -21.0000 q^{21} -5.88590 q^{23} -80.0470 q^{25} +27.0000 q^{27} -252.094 q^{29} +37.6376 q^{31} -2.11410 q^{33} -46.9329 q^{35} +139.866 q^{37} -106.228 q^{39} -46.7047 q^{41} +254.094 q^{43} +60.3423 q^{45} -607.960 q^{47} +49.0000 q^{49} +80.1141 q^{51} -298.685 q^{53} -4.72480 q^{55} -92.4564 q^{57} +178.591 q^{59} -390.188 q^{61} -63.0000 q^{63} -237.409 q^{65} +349.141 q^{67} -17.6577 q^{69} -348.477 q^{71} -646.416 q^{73} -240.141 q^{75} +4.93290 q^{77} -1047.50 q^{79} +81.0000 q^{81} +278.819 q^{83} +179.047 q^{85} -756.282 q^{87} +439.309 q^{89} +247.866 q^{91} +112.913 q^{93} -206.631 q^{95} -154.309 q^{97} -6.34230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 10 q^{5} - 14 q^{7} + 18 q^{9} + 22 q^{11} - 24 q^{13} - 30 q^{15} + 30 q^{17} + 32 q^{19} - 42 q^{21} - 82 q^{23} + 74 q^{25} + 54 q^{27} - 36 q^{29} - 112 q^{31} + 66 q^{33} + 70 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 10 * q^5 - 14 * q^7 + 18 * q^9 + 22 * q^11 - 24 * q^13 - 30 * q^15 + 30 * q^17 + 32 * q^19 - 42 * q^21 - 82 * q^23 + 74 * q^25 + 54 * q^27 - 36 * q^29 - 112 * q^31 + 66 * q^33 + 70 * q^35 - 48 * q^37 - 72 * q^39 - 70 * q^41 + 40 * q^43 - 90 * q^45 - 420 * q^47 + 98 * q^49 + 90 * q^51 - 176 * q^53 - 384 * q^55 + 96 * q^57 + 404 * q^59 + 156 * q^61 - 126 * q^63 - 428 * q^65 - 4 * q^67 - 246 * q^69 - 814 * q^71 - 216 * q^73 + 222 * q^75 - 154 * q^77 - 1580 * q^79 + 162 * q^81 + 464 * q^83 + 124 * q^85 - 108 * q^87 - 1158 * q^89 + 168 * q^91 - 336 * q^93 - 1256 * q^95 - 1760 * q^97 + 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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