LMFDB - Embedded newform 672.4.i.c.209.37

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,4,Mod(209,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.209"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.37
Character	$\chi$	$=$	672.209
Dual form			672.4.i.c.209.39

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.04870 - 5.08923i) q^{3} -3.71228i q^{5} +(-15.6071 - 9.97081i) q^{7} +(-24.8005 + 10.6741i) q^{9} -48.1150 q^{11} -69.5724 q^{13} +(-18.8927 + 3.89307i) q^{15} +48.9235 q^{17} +62.2883 q^{19} +(-34.3765 + 89.8847i) q^{21} +63.4952i q^{23} +111.219 q^{25} +(80.3314 + 115.021i) q^{27} -20.2759 q^{29} +77.3838i q^{31} +(50.4582 + 244.868i) q^{33} +(-37.0145 + 57.9382i) q^{35} -151.811i q^{37} +(72.9606 + 354.070i) q^{39} +284.960 q^{41} -441.143i q^{43} +(39.6255 + 92.0663i) q^{45} -615.429 q^{47} +(144.166 + 311.232i) q^{49} +(-51.3061 - 248.983i) q^{51} -129.453 q^{53} +178.616i q^{55} +(-65.3217 - 316.999i) q^{57} -406.934i q^{59} +576.405 q^{61} +(493.494 + 80.6876i) q^{63} +258.272i q^{65} +665.046i q^{67} +(323.142 - 66.5875i) q^{69} +129.482i q^{71} -290.352i q^{73} +(-116.635 - 566.018i) q^{75} +(750.938 + 479.745i) q^{77} +130.921 q^{79} +(501.125 - 529.447i) q^{81} +763.300i q^{83} -181.618i q^{85} +(21.2633 + 103.189i) q^{87} -269.586 q^{89} +(1085.83 + 693.693i) q^{91} +(393.824 - 81.1524i) q^{93} -231.232i q^{95} +1379.26i q^{97} +(1193.27 - 513.586i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) $ 80 * q - 64 * q^7 + 104 * q^9 + 8 * q^15 - 976 * q^25 - 568 * q^39 - 4048 * q^49 - 1448 * q^57 + 2152 * q^63 - 4992 * q^79 + 1568 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	−1.04870	−	5.08923i	−0.201822	−	0.979422i
$3$	−1.04870	−	5.08923i	−0.201822	−	0.979422i
$4$		0			0
$5$		−	3.71228i		−	0.332037i	−0.986123	−	0.166018i	$-0.946909\pi$
$5$		−	3.71228i		−	0.332037i	0.986123	−	0.166018i	$-0.0530911\pi$
$6$		0			0
$7$	−15.6071	−	9.97081i	−0.842707	−	0.538373i
$7$	−15.6071	−	9.97081i	−0.842707	−	0.538373i
$8$		0			0
$9$	−24.8005	+	10.6741i	−0.918535	+	0.395339i
$10$		0			0
$11$	−48.1150			−1.31884			−0.659419	−	0.751776i	$-0.729198\pi$
$11$	−48.1150			−1.31884			−0.659419	+	0.751776i	$0.729198\pi$
$12$		0			0
$13$	−69.5724			−1.48430			−0.742150	−	0.670234i	$-0.766194\pi$
$13$	−69.5724			−1.48430			−0.742150	+	0.670234i	$0.766194\pi$
$14$		0			0
$15$	−18.8927	+	3.89307i	−0.325204	+	0.0670125i
$16$		0			0
$17$	48.9235			0.697982			0.348991	−	0.937126i	$-0.386524\pi$
$17$	48.9235			0.697982			0.348991	+	0.937126i	$0.386524\pi$
$18$		0			0
$19$	62.2883			0.752100			0.376050	−	0.926599i	$-0.377282\pi$
$19$	62.2883			0.752100			0.376050	+	0.926599i	$0.377282\pi$
$20$		0			0
$21$	−34.3765	+	89.8847i	−0.357217	+	0.934021i
$22$		0			0
$23$			63.4952i			0.575638i	0.957685	+	0.287819i	$0.0929301\pi$
$23$			63.4952i			0.575638i	−0.957685	+	0.287819i	$0.907070\pi$
$24$		0			0
$25$	111.219			0.889752
$26$		0			0
$27$	80.3314	+	115.021i	0.572585	+	0.819846i
$28$		0			0
$29$	−20.2759			−0.129832			−0.0649161	−	0.997891i	$-0.520678\pi$
$29$	−20.2759			−0.129832			−0.0649161	+	0.997891i	$0.520678\pi$
$30$		0			0
$31$			77.3838i			0.448340i	0.974550	+	0.224170i	$0.0719671\pi$
$31$			77.3838i			0.448340i	−0.974550	+	0.224170i	$0.928033\pi$
$32$		0			0
$33$	50.4582	+	244.868i	0.266171	+	1.29170i
$34$		0			0
$35$	−37.0145	+	57.9382i	−0.178760	+	0.279810i
$36$		0			0
$37$		−	151.811i		−	0.674527i	−0.941410	−	0.337263i	$-0.890499\pi$
$37$		−	151.811i		−	0.674527i	0.941410	−	0.337263i	$-0.109501\pi$
$38$		0			0
$39$	72.9606	+	354.070i	0.299565	+	1.45376i
$40$		0			0
$41$	284.960			1.08544			0.542722	−	0.839912i	$-0.317394\pi$
$41$	284.960			1.08544			0.542722	+	0.839912i	$0.317394\pi$
$42$		0			0
$43$		−	441.143i		−	1.56450i	−0.622963	−	0.782252i	$-0.714071\pi$
$43$		−	441.143i		−	1.56450i	0.622963	−	0.782252i	$-0.285929\pi$
$44$		0			0
$45$	39.6255	+	92.0663i	0.131267	+	0.304988i
$46$		0			0
$47$	−615.429			−1.90999			−0.954996	−	0.296620i	$-0.904141\pi$
$47$	−615.429			−1.90999			−0.954996	+	0.296620i	$0.904141\pi$
$48$		0			0
$49$	144.166	+	311.232i	0.420309	+	0.907381i
$50$		0			0
$51$	−51.3061	−	248.983i	−0.140868	−	0.683619i
$52$		0			0
$53$	−129.453			−0.335504			−0.167752	−	0.985829i	$-0.553651\pi$
$53$	−129.453			−0.335504			−0.167752	+	0.985829i	$0.553651\pi$
$54$		0			0
$55$			178.616i			0.437903i
$56$		0			0
$57$	−65.3217	−	316.999i	−0.151791	−	0.736624i
$58$		0			0
$59$		−	406.934i		−	0.897938i	−0.893547	−	0.448969i	$-0.851791\pi$
$59$		−	406.934i		−	0.897938i	0.893547	−	0.448969i	$-0.148209\pi$
$60$		0			0
$61$	576.405			1.20985			0.604927	−	0.796281i	$-0.293202\pi$
$61$	576.405			1.20985			0.604927	+	0.796281i	$0.293202\pi$
$62$		0			0
$63$	493.494	+	80.6876i	0.986896	+	0.161360i
$64$		0			0
$65$			258.272i			0.492842i
$66$		0			0
$67$			665.046i			1.21266i	0.795212	+	0.606331i	$0.207359\pi$
$67$			665.046i			1.21266i	−0.795212	+	0.606331i	$0.792641\pi$
$68$		0			0
$69$	323.142	−	66.5875i	0.563792	−	0.116177i
$70$		0			0
$71$			129.482i			0.216432i	0.994127	+	0.108216i	$0.0345139\pi$
$71$			129.482i			0.216432i	−0.994127	+	0.108216i	$0.965486\pi$
$72$		0			0
$73$		−	290.352i		−	0.465523i	−0.972534	−	0.232761i	$-0.925224\pi$
$73$		−	290.352i		−	0.465523i	0.972534	−	0.232761i	$-0.0747762\pi$
$74$		0			0
$75$	−116.635	−	566.018i	−0.179572	−	0.871442i
$76$		0			0
$77$	750.938	+	479.745i	1.11139	+	0.710027i
$78$		0			0
$79$	130.921			0.186452			0.0932261	−	0.995645i	$-0.470282\pi$
$79$	130.921			0.186452			0.0932261	+	0.995645i	$0.470282\pi$
$80$		0			0
$81$	501.125	−	529.447i	0.687415	−	0.726265i
$82$		0			0
$83$			763.300i			1.00943i	0.863285	+	0.504717i	$0.168403\pi$
$83$			763.300i			1.00943i	−0.863285	+	0.504717i	$0.831597\pi$
$84$		0			0
$85$		−	181.618i		−	0.231756i
$86$		0			0
$87$	21.2633	+	103.189i	0.0262031	+	0.127161i
$88$		0			0
$89$	−269.586			−0.321080			−0.160540	−	0.987029i	$-0.551324\pi$
$89$	−269.586			−0.321080			−0.160540	+	0.987029i	$0.551324\pi$
$90$		0			0
$91$	1085.83	+	693.693i	1.25083	+	0.799107i
$92$		0			0
$93$	393.824	−	81.1524i	0.439114	−	0.0904851i
$94$		0			0
$95$		−	231.232i		−	0.249725i
$96$		0			0
$97$			1379.26i			1.44374i	0.692030	+	0.721869i	$0.256717\pi$
$97$			1379.26i			1.44374i	−0.692030	+	0.721869i	$0.743283\pi$
$98$		0			0
$99$	1193.27	−	513.586i	1.21140	−	0.521388i
$100$		0			0
$101$		−	117.695i		−	0.115952i	−0.998318	−	0.0579759i	$-0.981535\pi$
$101$		−	117.695i		−	0.115952i	0.998318	−	0.0579759i	$-0.0184647\pi$
$102$		0			0
$103$			1590.34i			1.52137i	0.649122	+	0.760684i	$0.275137\pi$
$103$			1590.34i			1.52137i	−0.649122	+	0.760684i	$0.724863\pi$
$104$		0			0
$105$	333.677	+	127.615i	0.310129	+	0.118609i
$106$		0			0
$107$	−443.102			−0.400339			−0.200169	−	0.979761i	$-0.564149\pi$
$107$	−443.102			−0.400339			−0.200169	+	0.979761i	$0.564149\pi$
$108$		0			0
$109$		−	1655.46i		−	1.45472i	−0.686256	−	0.727360i	$-0.740747\pi$
$109$		−	1655.46i		−	1.45472i	0.686256	−	0.727360i	$-0.259253\pi$
$110$		0			0
$111$	−772.598	+	159.204i	−0.660647	+	0.136135i
$112$		0			0
$113$			73.7875i			0.0614278i	0.999528	+	0.0307139i	$0.00977807\pi$
$113$			73.7875i			0.0614278i	−0.999528	+	0.0307139i	$0.990222\pi$
$114$		0			0
$115$	235.712			0.191133
$116$		0			0
$117$	1725.43	−	742.626i	1.36338	−	0.586801i
$118$		0			0
$119$	−763.556	−	487.807i	−0.588194	−	0.375775i
$120$		0			0
$121$	984.052			0.739332
$122$		0			0
$123$	−298.837	−	1450.22i	−0.219067	−	1.06311i
$124$		0			0
$125$		−	876.912i		−	0.627467i
$126$		0			0
$127$	1960.00			1.36947			0.684733	−	0.728794i	$-0.259919\pi$
$127$	1960.00			1.36947			0.684733	+	0.728794i	$0.259919\pi$
$128$		0			0
$129$	−2245.08	+	462.626i	−1.53231	+	0.315752i
$130$		0			0
$131$			1179.46i			0.786642i	0.919401	+	0.393321i	$0.128674\pi$
$131$			1179.46i			0.786642i	−0.919401	+	0.393321i	$0.871326\pi$
$132$		0			0
$133$	−972.142	−	621.064i	−0.633800	−	0.404911i
$134$		0			0
$135$	426.991	−	298.213i	0.272219	−	0.190119i
$136$		0			0
$137$			2694.83i			1.68054i	0.542165	+	0.840272i	$0.317605\pi$
$137$			2694.83i			1.68054i	−0.542165	+	0.840272i	$0.682395\pi$
$138$		0			0
$139$	2581.98			1.57554			0.787771	−	0.615968i	$-0.211235\pi$
$139$	2581.98			1.57554			0.787771	+	0.615968i	$0.211235\pi$
$140$		0			0
$141$	645.401	+	3132.06i	0.385479	+	1.87069i
$142$		0			0
$143$	3347.47			1.95755
$144$		0			0
$145$			75.2698i			0.0431091i
$146$		0			0
$147$	1432.74	−	1060.08i	0.803881	−	0.594790i
$148$		0			0
$149$	−3252.26			−1.78816			−0.894079	−	0.447910i	$-0.852168\pi$
$149$	−3252.26			−1.78816			−0.894079	+	0.447910i	$0.852168\pi$
$150$		0			0
$151$	−74.9590			−0.0403978			−0.0201989	−	0.999796i	$-0.506430\pi$
$151$	−74.9590			−0.0403978			−0.0201989	+	0.999796i	$0.506430\pi$
$152$		0			0
$153$	−1213.33	+	522.217i	−0.641121	+	0.275939i
$154$		0			0
$155$	287.271			0.148865
$156$		0			0
$157$	706.105			0.358938			0.179469	−	0.983764i	$-0.442562\pi$
$157$	706.105			0.358938			0.179469	+	0.983764i	$0.442562\pi$
$158$		0			0
$159$	135.757	+	658.815i	0.0677123	+	0.328600i
$160$		0			0
$161$	633.099	−	990.979i	0.309908	−	0.485094i
$162$		0			0
$163$		−	1520.38i		−	0.730586i	−0.930893	−	0.365293i	$-0.880969\pi$
$163$		−	1520.38i		−	0.730586i	0.930893	−	0.365293i	$-0.119031\pi$
$164$		0			0
$165$	909.020	−	187.315i	0.428891	−	0.0883786i
$166$		0			0
$167$	1959.42			0.907933			0.453966	−	0.891019i	$-0.350009\pi$
$167$	1959.42			0.907933			0.453966	+	0.891019i	$0.350009\pi$
$168$		0			0
$169$	2643.32			1.20315
$170$		0			0
$171$	−1544.78	+	664.874i	−0.690831	+	0.297334i
$172$		0			0
$173$			3842.54i			1.68869i	0.535800	+	0.844345i	$0.320010\pi$
$173$			3842.54i			1.68869i	−0.535800	+	0.844345i	$0.679990\pi$
$174$		0			0
$175$	−1735.81	−	1108.94i	−0.749800	−	0.479018i
$176$		0			0
$177$	−2070.98	+	426.752i	−0.879461	+	0.181224i
$178$		0			0
$179$	−2750.46			−1.14849			−0.574243	−	0.818685i	$-0.694704\pi$
$179$	−2750.46			−1.14849			−0.574243	+	0.818685i	$0.694704\pi$
$180$		0			0
$181$	−1299.66			−0.533718			−0.266859	−	0.963736i	$-0.585986\pi$
$181$	−1299.66			−0.533718			−0.266859	+	0.963736i	$0.585986\pi$
$182$		0			0
$183$	−604.476	−	2933.45i	−0.244176	−	1.18496i
$184$		0			0
$185$	−563.564			−0.223968
$186$		0			0
$187$	−2353.95			−0.920525
$188$		0			0
$189$	−106.890	−	2596.12i	−0.0411380	−	0.999153i
$190$		0			0
$191$			3914.42i			1.48292i	0.670998	+	0.741459i	$0.265866\pi$
$191$			3914.42i			1.48292i	−0.670998	+	0.741459i	$0.734134\pi$
$192$		0			0
$193$	−3648.18			−1.36063			−0.680316	−	0.732919i	$-0.738157\pi$
$193$	−3648.18			−1.36063			−0.680316	+	0.732919i	$0.738157\pi$
$194$		0			0
$195$	1314.41	−	270.850i	0.482701	−	0.0994666i
$196$		0			0
$197$	−79.0843			−0.0286016			−0.0143008	−	0.999898i	$-0.504552\pi$
$197$	−79.0843			−0.0286016			−0.0143008	+	0.999898i	$0.504552\pi$
$198$		0			0
$199$			710.675i			0.253158i	0.991957	+	0.126579i	$0.0403997\pi$
$199$			710.675i			0.253158i	−0.991957	+	0.126579i	$0.959600\pi$
$200$		0			0
$201$	3384.57	−	697.434i	1.18771	−	0.244742i
$202$		0			0
$203$	316.449	+	202.167i	0.109411	+	0.0698982i
$204$		0			0
$205$		−	1057.85i		−	0.360408i
$206$		0			0
$207$	−677.757	−	1574.71i	−0.227572	−	0.528744i
$208$		0			0
$209$	−2997.00			−0.991898
$210$		0			0
$211$			2273.94i			0.741915i	0.928650	+	0.370958i	$0.120971\pi$
$211$			2273.94i			0.741915i	−0.928650	+	0.370958i	$0.879029\pi$
$212$		0			0
$213$	658.964	−	135.788i	0.211979	−	0.0436809i
$214$		0			0
$215$	−1637.65			−0.519473
$216$		0			0
$217$	771.579	−	1207.74i	0.241374	−	0.377819i
$218$		0			0
$219$	−1477.67	+	304.493i	−0.455944	+	0.0939530i
$220$		0			0
$221$	−3403.72			−1.03602
$222$		0			0
$223$			3196.79i			0.959968i	0.877277	+	0.479984i	$0.159358\pi$
$223$			3196.79i			0.959968i	−0.877277	+	0.479984i	$0.840642\pi$
$224$		0			0
$225$	−2758.28	+	1187.17i	−0.817268	+	0.351753i
$226$		0			0
$227$			3103.70i			0.907487i	0.891132	+	0.453744i	$0.149912\pi$
$227$			3103.70i			0.907487i	−0.891132	+	0.453744i	$0.850088\pi$
$228$		0			0
$229$	693.650			0.200165			0.100082	−	0.994979i	$-0.468089\pi$
$229$	693.650			0.200165			0.100082	+	0.994979i	$0.468089\pi$
$230$		0			0
$231$	1654.02	−	4324.80i	0.471112	−	1.23182i
$232$		0			0
$233$			3188.80i			0.896589i	0.893886	+	0.448294i	$0.147968\pi$
$233$			3188.80i			0.896589i	−0.893886	+	0.448294i	$0.852032\pi$
$234$		0			0
$235$			2284.65i			0.634187i
$236$		0			0
$237$	−137.297	−	666.285i	−0.0376302	−	0.182615i
$238$		0			0
$239$			1879.03i			0.508554i	0.967131	+	0.254277i	$0.0818375\pi$
$239$			1879.03i			0.508554i	−0.967131	+	0.254277i	$0.918162\pi$
$240$		0			0
$241$		−	3281.84i		−	0.877187i	−0.898685	−	0.438594i	$-0.855477\pi$
$241$		−	3281.84i		−	0.877187i	0.898685	−	0.438594i	$-0.144523\pi$
$242$		0			0
$243$	−3220.01	−	1995.11i	−0.850056	−	0.526692i
$244$		0			0
$245$	1155.38	−	535.185i	0.301284	−	0.139558i
$246$		0			0
$247$	−4333.54			−1.11634
$248$		0			0
$249$	3884.60	−	800.472i	0.988662	−	0.203726i
$250$		0			0
$251$		−	1680.42i		−	0.422578i	−0.977424	−	0.211289i	$-0.932234\pi$
$251$		−	1680.42i		−	0.422578i	0.977424	−	0.211289i	$-0.0677661\pi$
$252$		0			0
$253$		−	3055.07i		−	0.759173i
$254$		0			0
$255$	−924.295	+	190.463i	−0.226987	+	0.0467735i
$256$		0			0
$257$	6706.25			1.62772			0.813861	−	0.581059i	$-0.197362\pi$
$257$	6706.25			1.62772			0.813861	+	0.581059i	$0.197362\pi$
$258$		0			0
$259$	−1513.67	+	2369.33i	−0.363147	+	0.568428i
$260$		0			0
$261$	502.851	−	216.428i	0.119256	−	0.0513277i
$262$		0			0
$263$			4758.38i			1.11564i	0.829961	+	0.557822i	$0.188363\pi$
$263$			4758.38i			1.11564i	−0.829961	+	0.557822i	$0.811637\pi$
$264$		0			0
$265$			480.566i			0.111400i
$266$		0			0
$267$	282.715	+	1371.99i	0.0648011	+	0.314473i
$268$		0			0
$269$		−	2644.09i		−	0.599306i	−0.954048	−	0.299653i	$-0.903129\pi$
$269$		−	2644.09i		−	0.599306i	0.954048	−	0.299653i	$-0.0968708\pi$
$270$		0			0
$271$		−	7621.80i		−	1.70846i	−0.519899	−	0.854228i	$-0.674030\pi$
$271$		−	7621.80i		−	1.70846i	0.519899	−	0.854228i	$-0.325970\pi$
$272$		0			0
$273$	2391.65	−	6253.49i	0.530218	−	1.38637i
$274$		0			0
$275$	−5351.30			−1.17344
$276$		0			0
$277$			3715.63i			0.805960i	0.915209	+	0.402980i	$0.132026\pi$
$277$			3715.63i			0.805960i	−0.915209	+	0.402980i	$0.867974\pi$
$278$		0			0
$279$	−826.006	−	1919.15i	−0.177246	−	0.411816i
$280$		0			0
$281$		−	7823.45i		−	1.66088i	−0.557106	−	0.830441i	$-0.688088\pi$
$281$		−	7823.45i		−	1.66088i	0.557106	−	0.830441i	$-0.311912\pi$
$282$		0			0
$283$	−4118.81			−0.865151			−0.432575	−	0.901598i	$-0.642395\pi$
$283$	−4118.81			−0.865151			−0.432575	+	0.901598i	$0.642395\pi$
$284$		0			0
$285$	−1176.79	+	242.493i	−0.244586	+	0.0504001i
$286$		0			0
$287$	−4447.41	−	2841.28i	−0.914711	−	0.584374i
$288$		0			0
$289$	−2519.49			−0.512821
$290$		0			0
$291$	7019.37	−	1446.43i	1.41403	−	0.291379i
$292$		0			0
$293$			6032.10i			1.20273i	0.798976	+	0.601363i	$0.205376\pi$
$293$			6032.10i			1.20273i	−0.798976	+	0.601363i	$0.794624\pi$
$294$		0			0
$295$	−1510.66			−0.298148
$296$		0			0
$297$	−3865.14	−	5534.24i	−0.755146	−	1.08124i
$298$		0			0
$299$		−	4417.51i		−	0.854420i
$300$		0			0
$301$	−4398.55	+	6884.98i	−0.842286	+	1.31842i
$302$		0			0
$303$	−598.979	+	123.427i	−0.113566	+	0.0234017i
$304$		0			0
$305$		−	2139.78i		−	0.401716i
$306$		0			0
$307$	2031.61			0.377688			0.188844	−	0.982007i	$-0.439526\pi$
$307$	2031.61			0.377688			0.188844	+	0.982007i	$0.439526\pi$
$308$		0			0
$309$	8093.61	−	1667.79i	1.49006	−	0.307046i
$310$		0			0
$311$	6093.07			1.11095			0.555477	−	0.831532i	$-0.312536\pi$
$311$	6093.07			1.11095			0.555477	+	0.831532i	$0.312536\pi$
$312$		0			0
$313$		−	2761.91i		−	0.498762i	−0.968405	−	0.249381i	$-0.919773\pi$
$313$		−	2761.91i		−	0.498762i	0.968405	−	0.249381i	$-0.0802271\pi$
$314$		0			0
$315$	299.535	−	1831.99i	0.0535775	−	0.327686i
$316$		0			0
$317$	−3870.63			−0.685792			−0.342896	−	0.939373i	$-0.611408\pi$
$317$	−3870.63			−0.685792			−0.342896	+	0.939373i	$0.611408\pi$
$318$		0			0
$319$	975.573			0.171228
$320$		0			0
$321$	464.681	+	2255.05i	0.0807974	+	0.392101i
$322$		0			0
$323$	3047.36			0.524953
$324$		0			0
$325$	−7737.77			−1.32066
$326$		0			0
$327$	−8425.02	+	1736.08i	−1.42478	+	0.293595i
$328$		0			0
$329$	9605.09	+	6136.33i	1.60956	+	1.02829i
$330$		0			0
$331$		−	4264.16i		−	0.708095i	−0.935227	−	0.354048i	$-0.884805\pi$
$331$		−	4264.16i		−	0.708095i	0.935227	−	0.354048i	$-0.115195\pi$
$332$		0			0
$333$	1620.45	+	3764.97i	0.266667	+	0.619577i
$334$		0			0
$335$	2468.84			0.402648
$336$		0			0
$337$	−3256.10			−0.526323			−0.263162	−	0.964752i	$-0.584765\pi$
$337$	−3256.10			−0.526323			−0.263162	+	0.964752i	$0.584765\pi$
$338$		0			0
$339$	375.521	−	77.3809i	0.0601637	−	0.0123975i
$340$		0			0
$341$		−	3723.32i		−	0.591288i
$342$		0			0
$343$	853.212	−	6294.89i	0.134312	−	0.990939i
$344$		0			0
$345$	−247.192	−	1199.59i	−0.0385749	−	0.187200i
$346$		0			0
$347$	−6076.58			−0.940081			−0.470041	−	0.882645i	$-0.655761\pi$
$347$	−6076.58			−0.940081			−0.470041	+	0.882645i	$0.655761\pi$
$348$		0			0
$349$	6858.46			1.05193			0.525967	−	0.850505i	$-0.323704\pi$
$349$	6858.46			1.05193			0.525967	+	0.850505i	$0.323704\pi$
$350$		0			0
$351$	−5588.85	−	8002.30i	−0.849887	−	1.21690i
$352$		0			0
$353$	−6340.22			−0.955965			−0.477983	−	0.878369i	$-0.658632\pi$
$353$	−6340.22			−0.955965			−0.477983	+	0.878369i	$0.658632\pi$
$354$		0			0
$355$	480.674			0.0718635
$356$		0			0
$357$	−1681.82	+	4397.47i	−0.249331	+	0.651930i
$358$		0			0
$359$		−	79.4813i		−	0.0116849i	−0.999983	−	0.00584243i	$-0.998140\pi$
$359$		−	79.4813i		−	0.0116849i	0.999983	−	0.00584243i	$-0.00185971\pi$
$360$		0			0
$361$	−2979.17			−0.434345
$362$		0			0
$363$	−1031.97	−	5008.06i	−0.149214	−	0.724119i
$364$		0			0
$365$	−1077.87			−0.154571
$366$		0			0
$367$		−	11025.8i		−	1.56824i	−0.620610	−	0.784120i	$-0.713115\pi$
$367$		−	11025.8i		−	1.56824i	0.620610	−	0.784120i	$-0.286885\pi$
$368$		0			0
$369$	−7067.13	+	3041.70i	−0.997019	+	0.429118i
$370$		0			0
$371$	2020.39	+	1290.75i	0.282732	+	0.180626i
$372$		0			0
$373$			5947.81i			0.825646i	0.910811	+	0.412823i	$0.135457\pi$
$373$			5947.81i			0.825646i	−0.910811	+	0.412823i	$0.864543\pi$
$374$		0			0
$375$	−4462.80	+	919.617i	−0.614555	+	0.126637i
$376$		0			0
$377$	1410.64			0.192710
$378$		0			0
$379$			4855.67i			0.658098i	0.944313	+	0.329049i	$0.106728\pi$
$379$			4855.67i			0.658098i	−0.944313	+	0.329049i	$0.893272\pi$
$380$		0			0
$381$	−2055.46	−	9974.91i	−0.276389	−	1.34129i
$382$		0			0
$383$	−4314.03			−0.575553			−0.287777	−	0.957698i	$-0.592916\pi$
$383$	−4314.03			−0.575553			−0.287777	+	0.957698i	$0.592916\pi$
$384$		0			0
$385$	1780.95	−	2787.69i	0.235755	−	0.369023i
$386$		0			0
$387$	4708.82	+	10940.5i	0.618509	+	1.43705i
$388$		0			0
$389$	−1492.00			−0.194466			−0.0972330	−	0.995262i	$-0.530999\pi$
$389$	−1492.00			−0.194466			−0.0972330	+	0.995262i	$0.530999\pi$
$390$		0			0
$391$			3106.41i			0.401785i
$392$		0			0
$393$	6002.55	−	1236.90i	0.770454	−	0.158762i
$394$		0			0
$395$		−	486.015i		−	0.0619090i
$396$		0			0
$397$	−2577.45			−0.325840			−0.162920	−	0.986639i	$-0.552091\pi$
$397$	−2577.45			−0.325840			−0.162920	+	0.986639i	$0.552091\pi$
$398$		0			0
$399$	−2141.25	+	5598.76i	−0.268663	+	0.702478i
$400$		0			0
$401$			3185.37i			0.396682i	0.980133	+	0.198341i	$0.0635554\pi$
$401$			3185.37i			0.396682i	−0.980133	+	0.198341i	$0.936445\pi$
$402$		0			0
$403$		−	5383.78i		−	0.665471i
$404$		0			0
$405$	−1965.46	−	1860.32i	−0.241147	−	0.228247i
$406$		0			0
$407$			7304.36i			0.889592i
$408$		0			0
$409$		−	9157.04i		−	1.10706i	−0.832830	−	0.553529i	$-0.813281\pi$
$409$		−	9157.04i		−	1.10706i	0.832830	−	0.553529i	$-0.186719\pi$
$410$		0			0
$411$	13714.6	−	2826.06i	1.64596	−	0.339172i
$412$		0			0
$413$	−4057.47	+	6351.09i	−0.483426	+	0.756699i
$414$		0			0
$415$	2833.58			0.335169
$416$		0			0
$417$	−2707.72	−	13140.3i	−0.317980	−	1.54312i
$418$		0			0
$419$			16278.2i			1.89796i	0.315339	+	0.948979i	$0.397882\pi$
$419$			16278.2i			1.89796i	−0.315339	+	0.948979i	$0.602118\pi$
$420$		0			0
$421$			14.4956i			0.00167808i	1.00000		0.000839038i	$0.000267074\pi$
$421$			14.4956i			0.00167808i	−1.00000		0.000839038i	$0.999733\pi$
$422$		0			0
$423$	15262.9	−	6569.18i	1.75439	−	0.755093i
$424$		0			0
$425$	5441.22			0.621031
$426$		0			0
$427$	−8996.03	−	5747.22i	−1.01955	−	0.651353i
$428$		0			0
$429$	−3510.50	−	17036.1i	−0.395078	−	1.91727i
$430$		0			0
$431$		−	6800.15i		−	0.759981i	−0.924990	−	0.379991i	$-0.875927\pi$
$431$		−	6800.15i		−	0.759981i	0.924990	−	0.379991i	$-0.124073\pi$
$432$		0			0
$433$			11155.5i			1.23811i	0.785349	+	0.619053i	$0.212483\pi$
$433$			11155.5i			1.23811i	−0.785349	+	0.619053i	$0.787517\pi$
$434$		0			0
$435$	383.065	−	78.9355i	0.0422220	−	0.00870038i
$436$		0			0
$437$			3955.01i			0.432938i
$438$		0			0
$439$			12426.7i			1.35101i	0.737357	+	0.675504i	$0.236074\pi$
$439$			12426.7i			1.35101i	−0.737357	+	0.675504i	$0.763926\pi$
$440$		0			0
$441$	−6897.51	−	6179.84i	−0.744792	−	0.667297i
$442$		0			0
$443$	−10870.0			−1.16580			−0.582900	−	0.812544i	$-0.698082\pi$
$443$	−10870.0			−1.16580			−0.582900	+	0.812544i	$0.698082\pi$
$444$		0			0
$445$			1000.78i			0.106610i
$446$		0			0
$447$	3410.64	+	16551.5i	0.360890	+	1.75136i
$448$		0			0
$449$			11786.4i			1.23883i	0.785063	+	0.619416i	$0.212631\pi$
$449$			11786.4i			1.23883i	−0.785063	+	0.619416i	$0.787369\pi$
$450$		0			0
$451$	−13710.8			−1.43153
$452$		0			0
$453$	78.6095	+	381.483i	0.00815319	+	0.0395665i
$454$		0			0
$455$	2575.18	−	4030.90i	0.265333	−	0.415321i
$456$		0			0
$457$	−1151.19			−0.117834			−0.0589171	−	0.998263i	$-0.518765\pi$
$457$	−1151.19			−0.117834			−0.0589171	+	0.998263i	$0.518765\pi$
$458$		0			0
$459$	3930.09	+	5627.24i	0.399654	+	0.572238i
$460$		0			0
$461$		−	1149.15i		−	0.116098i	−0.998314	−	0.0580491i	$-0.981512\pi$
$461$		−	1149.15i		−	0.116098i	0.998314	−	0.0580491i	$-0.0184880\pi$
$462$		0			0
$463$	−511.258			−0.0513179			−0.0256590	−	0.999671i	$-0.508168\pi$
$463$	−511.258			−0.0513179			−0.0256590	+	0.999671i	$0.508168\pi$
$464$		0			0
$465$	−301.261	−	1461.99i	−0.0300444	−	0.145802i
$466$		0			0
$467$			7738.29i			0.766779i	0.923587	+	0.383389i	$0.125243\pi$
$467$			7738.29i			0.766779i	−0.923587	+	0.383389i	$0.874757\pi$
$468$		0			0
$469$	6631.05	−	10379.5i	0.652864	−	1.02192i
$470$		0			0
$471$	−740.492	−	3593.53i	−0.0724418	−	0.351552i
$472$		0			0
$473$			21225.6i			2.06333i
$474$		0			0
$475$	6927.64			0.669183
$476$		0			0
$477$	3210.49	−	1381.80i	0.308173	−	0.132638i
$478$		0			0
$479$	15169.4			1.44699			0.723493	−	0.690332i	$-0.242535\pi$
$479$	15169.4			1.44699			0.723493	+	0.690332i	$0.242535\pi$
$480$		0			0
$481$			10561.8i			1.00120i
$482$		0			0
$483$	−5707.25	−	2182.74i	−0.537658	−	0.205628i
$484$		0			0
$485$	5120.20			0.479374
$486$		0			0
$487$	7078.96			0.658683			0.329341	−	0.944211i	$-0.393173\pi$
$487$	7078.96			0.658683			0.329341	+	0.944211i	$0.393173\pi$
$488$		0			0
$489$	−7737.57	+	1594.42i	−0.715552	+	0.147449i
$490$		0			0
$491$	−8864.05			−0.814723			−0.407361	−	0.913267i	$-0.633551\pi$
$491$	−8864.05			−0.814723			−0.407361	+	0.913267i	$0.633551\pi$
$492$		0			0
$493$	−991.967			−0.0906206
$494$		0			0
$495$	−1906.58	−	4429.77i	−0.173120	−	0.402229i
$496$		0			0
$497$	1291.04	−	2020.85i	0.116521	−	0.182389i
$498$		0			0
$499$		−	9294.42i		−	0.833818i	−0.908948	−	0.416909i	$-0.863113\pi$
$499$		−	9294.42i		−	0.833818i	0.908948	−	0.416909i	$-0.136887\pi$
$500$		0			0
$501$	−2054.85	−	9971.95i	−0.183241	−	0.889249i
$502$		0			0
$503$	−3930.02			−0.348372			−0.174186	−	0.984713i	$-0.555729\pi$
$503$	−3930.02			−0.348372			−0.174186	+	0.984713i	$0.555729\pi$
$504$		0			0
$505$	−436.919			−0.0385003
$506$		0			0
$507$	−2772.05	−	13452.4i	−0.242822	−	1.17839i
$508$		0			0
$509$			4139.73i			0.360491i	0.983622	+	0.180246i	$0.0576893\pi$
$509$			4139.73i			0.360491i	−0.983622	+	0.180246i	$0.942311\pi$
$510$		0			0
$511$	−2895.05	+	4531.57i	−0.250625	+	0.392299i
$512$		0			0
$513$	5003.70	+	7164.47i	0.430641	+	0.616606i
$514$		0			0
$515$	5903.80			0.505150
$516$		0			0
$517$	29611.4			2.51897
$518$		0			0
$519$	19555.6	−	4029.68i	1.65394	−	0.340815i
$520$		0			0
$521$	−15730.9			−1.32281			−0.661406	−	0.750028i	$-0.730040\pi$
$521$	−15730.9			−1.32281			−0.661406	+	0.750028i	$0.730040\pi$
$522$		0			0
$523$	12687.5			1.06078			0.530388	−	0.847755i	$-0.322046\pi$
$523$	12687.5			1.06078			0.530388	+	0.847755i	$0.322046\pi$
$524$		0			0
$525$	−3823.32	+	9996.88i	−0.317835	+	0.831047i
$526$		0			0
$527$			3785.89i			0.312933i
$528$		0			0
$529$	8135.36			0.668641
$530$		0			0
$531$	4343.68	+	10092.2i	0.354990	+	0.824788i
$532$		0			0
$533$	−19825.3			−1.61113
$534$		0			0
$535$			1644.92i			0.132927i
$536$		0			0
$537$	2884.41	+	13997.7i	0.231790	+	1.12485i
$538$		0			0
$539$	−6936.54	−	14974.9i	−0.554319	−	1.19669i
$540$		0			0
$541$			16674.9i			1.32516i	0.748991	+	0.662580i	$0.230538\pi$
$541$			16674.9i			1.32516i	−0.748991	+	0.662580i	$0.769462\pi$
$542$		0			0
$543$	1362.95	+	6614.26i	0.107716	+	0.522735i
$544$		0			0
$545$	−6145.54			−0.483020
$546$		0			0
$547$			3533.79i			0.276223i	0.990417	+	0.138111i	$0.0441032\pi$
$547$			3533.79i			0.276223i	−0.990417	+	0.138111i	$0.955897\pi$
$548$		0			0
$549$	−14295.1	+	6152.63i	−1.11129	+	0.478302i
$550$		0			0
$551$	−1262.95			−0.0976469
$552$		0			0
$553$	−2043.30	−	1305.39i	−0.157125	−	0.100381i
$554$		0			0
$555$	591.009	+	2868.10i	0.0452017	+	0.219359i
$556$		0			0
$557$	8507.84			0.647197			0.323599	−	0.946194i	$-0.395107\pi$
$557$	8507.84			0.647197			0.323599	+	0.946194i	$0.395107\pi$
$558$		0			0
$559$			30691.4i			2.32219i
$560$		0			0
$561$	2468.59	+	11979.8i	0.185783	+	0.901582i
$562$		0			0
$563$			12127.1i			0.907808i	0.891051	+	0.453904i	$0.149969\pi$
$563$			12127.1i			0.907808i	−0.891051	+	0.453904i	$0.850031\pi$
$564$		0			0
$565$	273.920			0.0203963
$566$		0			0
$567$	−13100.2	+	3266.54i	−0.970290	+	0.241943i
$568$		0			0
$569$		−	16103.5i		−	1.18646i	−0.805034	−	0.593228i	$-0.797853\pi$
$569$		−	16103.5i		−	1.18646i	0.805034	−	0.593228i	$-0.202147\pi$
$570$		0			0
$571$		−	16824.4i		−	1.23307i	−0.787329	−	0.616533i	$-0.788537\pi$
$571$		−	16824.4i		−	1.23307i	0.787329	−	0.616533i	$-0.211463\pi$
$572$		0			0
$573$	19921.4	−	4105.05i	1.45240	−	0.299286i
$574$		0			0
$575$			7061.87i			0.512175i
$576$		0			0
$577$		−	12917.3i		−	0.931986i	−0.884788	−	0.465993i	$-0.845697\pi$
$577$		−	12917.3i		−	0.931986i	0.884788	−	0.465993i	$-0.154303\pi$
$578$		0			0
$579$	3825.85	+	18566.4i	0.274606	+	1.33263i
$580$		0			0
$581$	7610.71	−	11912.9i	0.543452	−	0.850657i
$582$		0			0
$583$	6228.63			0.442476
$584$		0			0
$585$	−2756.84	−	6405.27i	−0.194840	−	0.452693i
$586$		0			0
$587$		−	15149.6i		−	1.06523i	−0.846357	−	0.532617i	$-0.821209\pi$
$587$		−	15149.6i		−	1.06523i	0.846357	−	0.532617i	$-0.178791\pi$
$588$		0			0
$589$			4820.10i			0.337197i
$590$		0			0
$591$	82.9357	+	402.478i	0.00577245	+	0.0280131i
$592$		0			0
$593$	5091.10			0.352558			0.176279	−	0.984340i	$-0.443594\pi$
$593$	5091.10			0.352558			0.176279	+	0.984340i	$0.443594\pi$
$594$		0			0
$595$	−1810.88	+	2834.54i	−0.124771	+	0.195302i
$596$		0			0
$597$	3616.78	−	745.285i	0.247948	−	0.0510929i
$598$		0			0
$599$		−	18901.6i		−	1.28931i	−0.764472	−	0.644657i	$-0.777000\pi$
$599$		−	18901.6i		−	1.28931i	0.764472	−	0.644657i	$-0.223000\pi$
$600$		0			0
$601$			28238.3i			1.91658i	0.285803	+	0.958288i	$0.407740\pi$
$601$			28238.3i			1.91658i	−0.285803	+	0.958288i	$0.592260\pi$
$602$		0			0
$603$	−7098.80	−	16493.5i	−0.479412	−	1.11387i
$604$		0			0
$605$		−	3653.08i		−	0.245486i
$606$		0			0
$607$		−	11435.7i		−	0.764683i	−0.924021	−	0.382342i	$-0.875118\pi$
$607$		−	11435.7i		−	0.764683i	0.924021	−	0.382342i	$-0.124882\pi$
$608$		0			0
$609$	697.013	−	1822.49i	0.0463783	−	0.121266i
$610$		0			0
$611$	42816.9			2.83500
$612$		0			0
$613$		−	577.455i		−	0.0380476i	−0.999819	−	0.0190238i	$-0.993944\pi$
$613$		−	577.455i		−	0.0380476i	0.999819	−	0.0190238i	$-0.00605583\pi$
$614$		0			0
$615$	−5383.65	+	1109.37i	−0.352991	+	0.0727383i
$616$		0			0
$617$		−	22282.8i		−	1.45393i	−0.686677	−	0.726963i	$-0.740931\pi$
$617$		−	22282.8i		−	1.45393i	0.686677	−	0.726963i	$-0.259069\pi$
$618$		0			0
$619$	16401.1			1.06497			0.532484	−	0.846440i	$-0.321259\pi$
$619$	16401.1			1.06497			0.532484	+	0.846440i	$0.321259\pi$
$620$		0			0
$621$	−7303.30	+	5100.66i	−0.471934	+	0.329601i
$622$		0			0
$623$	4207.48	+	2688.00i	0.270576	+	0.172861i
$624$		0			0
$625$	10647.0			0.681409
$626$		0			0
$627$	3142.95	+	15252.4i	0.200187	+	0.971487i
$628$		0			0
$629$		−	7427.10i		−	0.470808i
$630$		0			0
$631$	−6542.50			−0.412762			−0.206381	−	0.978472i	$-0.566169\pi$
$631$	−6542.50			−0.412762			−0.206381	+	0.978472i	$0.566169\pi$
$632$		0			0
$633$	11572.6	−	2384.68i	0.726648	−	0.149735i
$634$		0			0
$635$		−	7276.09i		−	0.454713i
$636$		0			0
$637$	−10030.0	−	21653.1i	−0.623865	−	1.34683i
$638$		0			0
$639$	−1382.11	−	3211.21i	−0.0855641	−	0.198801i
$640$		0			0
$641$		−	634.500i		−	0.0390971i	−0.999809	−	0.0195486i	$-0.993777\pi$
$641$		−	634.500i		−	0.0390971i	0.999809	−	0.0195486i	$-0.00622290\pi$
$642$		0			0
$643$	4543.63			0.278668			0.139334	−	0.990245i	$-0.455504\pi$
$643$	4543.63			0.278668			0.139334	+	0.990245i	$0.455504\pi$
$644$		0			0
$645$	1717.40	+	8334.36i	0.104841	+	0.508783i
$646$		0			0
$647$	−18802.6			−1.14252			−0.571258	−	0.820771i	$-0.693544\pi$
$647$	−18802.6			−1.14252			−0.571258	+	0.820771i	$0.693544\pi$
$648$		0			0
$649$			19579.6i			1.18423i
$650$		0			0
$651$	−6955.62	−	2660.18i	−0.418759	−	0.160155i
$652$		0			0
$653$	20232.4			1.21249			0.606243	−	0.795279i	$-0.292676\pi$
$653$	20232.4			1.21249			0.606243	+	0.795279i	$0.292676\pi$
$654$		0			0
$655$	4378.50			0.261194
$656$		0			0
$657$	3099.26	+	7200.87i	0.184039	+	0.427599i
$658$		0			0
$659$	13420.3			0.793295			0.396648	−	0.917971i	$-0.370173\pi$
$659$	13420.3			0.793295			0.396648	+	0.917971i	$0.370173\pi$
$660$		0			0
$661$	−8321.89			−0.489689			−0.244844	−	0.969562i	$-0.578737\pi$
$661$	−8321.89			−0.489689			−0.244844	+	0.969562i	$0.578737\pi$
$662$		0			0
$663$	3569.49	+	17322.3i	0.209091	+	1.01470i
$664$		0			0
$665$	−2305.57	+	3608.87i	−0.134445	+	0.210445i
$666$		0			0
$667$		−	1287.42i		−	0.0747364i
$668$		0			0
$669$	16269.2	−	3352.47i	0.940214	−	0.193743i
$670$		0			0
$671$	−27733.7			−1.59560
$672$		0			0
$673$	2322.47			0.133023			0.0665117	−	0.997786i	$-0.478813\pi$
$673$	2322.47			0.133023			0.0665117	+	0.997786i	$0.478813\pi$
$674$		0			0
$675$	8934.37	+	12792.5i	0.509458	+	0.729459i
$676$		0			0
$677$			1313.39i			0.0745606i	0.999305	+	0.0372803i	$0.0118694\pi$
$677$			1313.39i			0.0745606i	−0.999305	+	0.0372803i	$0.988131\pi$
$678$		0			0
$679$	13752.3	−	21526.3i	0.777270	−	1.21665i
$680$		0			0
$681$	15795.4	−	3254.85i	0.888813	−	0.183151i
$682$		0			0
$683$	−20367.6			−1.14106			−0.570532	−	0.821276i	$-0.693263\pi$
$683$	−20367.6			−1.14106			−0.570532	+	0.821276i	$0.693263\pi$
$684$		0			0
$685$	10004.0			0.558002
$686$		0			0
$687$	−727.431	−	3530.14i	−0.0403977	−	0.196046i
$688$		0			0
$689$	9006.35			0.497989
$690$		0			0
$691$	19327.1			1.06402			0.532010	−	0.846738i	$-0.321437\pi$
$691$	19327.1			1.06402			0.532010	+	0.846738i	$0.321437\pi$
$692$		0			0
$693$	−23744.5	−	3882.28i	−1.30155	−	0.212808i
$694$		0			0
$695$		−	9585.03i		−	0.523138i
$696$		0			0
$697$	13941.2			0.757621
$698$		0			0
$699$	16228.5	−	3344.09i	0.878139	−	0.180952i
$700$		0			0
$701$	7068.43			0.380843			0.190422	−	0.981702i	$-0.439015\pi$
$701$	7068.43			0.380843			0.190422	+	0.981702i	$0.439015\pi$
$702$		0			0
$703$		−	9456.02i		−	0.507312i
$704$		0			0
$705$	11627.1	−	2395.91i	0.621137	−	0.127993i
$706$		0			0
$707$	−1173.52	+	1836.89i	−0.0624253	+	0.0977134i
$708$		0			0
$709$			5102.93i			0.270303i	0.990825	+	0.135151i	$0.0431520\pi$
$709$			5102.93i			0.270303i	−0.990825	+	0.135151i	$0.956848\pi$
$710$		0			0
$711$	−3246.89	+	1397.47i	−0.171263	+	0.0737118i
$712$		0			0
$713$	−4913.50			−0.258082
$714$		0			0
$715$		−	12426.8i		−	0.649979i
$716$		0			0
$717$	9562.82	−	1970.54i	0.498089	−	0.102638i
$718$		0			0
$719$	2163.33			0.112209			0.0561046	−	0.998425i	$-0.482132\pi$
$719$	2163.33			0.112209			0.0561046	+	0.998425i	$0.482132\pi$
$720$		0			0
$721$	15857.0	−	24820.7i	0.819064	−	1.28207i
$722$		0			0
$723$	−16702.0	+	3441.67i	−0.859137	+	0.177036i
$724$		0			0
$725$	−2255.06			−0.115518
$726$		0			0
$727$			24985.1i			1.27462i	0.770609	+	0.637309i	$0.219952\pi$
$727$			24985.1i			1.27462i	−0.770609	+	0.637309i	$0.780048\pi$
$728$		0			0
$729$	−6776.74	+	18479.6i	−0.344294	+	0.938862i
$730$		0			0
$731$		−	21582.3i		−	1.09200i
$732$		0			0
$733$	−2600.09			−0.131018			−0.0655092	−	0.997852i	$-0.520867\pi$
$733$	−2600.09			−0.131018			−0.0655092	+	0.997852i	$0.520867\pi$
$734$		0			0
$735$	−3935.33	−	5318.74i	−0.197492	−	0.266918i
$736$		0			0
$737$		−	31998.7i		−	1.59930i
$738$		0			0
$739$			17370.8i			0.864675i	0.901712	+	0.432338i	$0.142311\pi$
$739$			17370.8i			0.864675i	−0.901712	+	0.432338i	$0.857689\pi$
$740$		0			0
$741$	4544.59	+	22054.4i	0.225303	+	1.09337i
$742$		0			0
$743$			28215.9i			1.39319i	0.717463	+	0.696597i	$0.245303\pi$
$743$			28215.9i			1.39319i	−0.717463	+	0.696597i	$0.754697\pi$
$744$		0			0
$745$			12073.3i			0.593734i
$746$		0			0
$747$	−8147.57	−	18930.2i	−0.399068	−	0.927201i
$748$		0			0
$749$	6915.55	+	4418.08i	0.337368	+	0.215532i
$750$		0			0
$751$	33427.0			1.62419			0.812097	−	0.583522i	$-0.198326\pi$
$751$	33427.0			1.62419			0.812097	+	0.583522i	$0.198326\pi$
$752$		0			0
$753$	−8552.02	+	1762.25i	−0.413882	+	0.0852856i
$754$		0			0
$755$			278.269i			0.0134136i
$756$		0			0
$757$		−	24841.4i		−	1.19270i	−0.802723	−	0.596352i	$-0.796616\pi$
$757$		−	24841.4i		−	1.19270i	0.802723	−	0.596352i	$-0.203384\pi$
$758$		0			0
$759$	−15548.0	+	3203.85i	−0.743551	+	0.153218i
$760$		0			0
$761$	−20004.6			−0.952912			−0.476456	−	0.879198i	$-0.658079\pi$
$761$	−20004.6			−0.952912			−0.476456	+	0.879198i	$0.658079\pi$
$762$		0			0
$763$	−16506.3	+	25837.0i	−0.783182	+	1.22590i
$764$		0			0
$765$	1938.62	+	4504.21i	0.0916220	+	0.212876i
$766$		0			0
$767$			28311.4i			1.33281i
$768$		0			0
$769$			9604.90i			0.450405i	0.974312	+	0.225203i	$0.0723044\pi$
$769$			9604.90i			0.450405i	−0.974312	+	0.225203i	$0.927696\pi$
$770$		0			0
$771$	−7032.85	−	34129.6i	−0.328511	−	1.59423i
$772$		0			0
$773$			4885.24i			0.227309i	0.993520	+	0.113655i	$0.0362557\pi$
$773$			4885.24i			0.227309i	−0.993520	+	0.113655i	$0.963744\pi$
$774$		0			0
$775$			8606.55i			0.398911i
$776$		0			0
$777$	13645.4	+	5218.71i	0.630023	+	0.240953i
$778$		0			0
$779$	17749.6			0.816363
$780$		0			0
$781$		−	6230.03i		−	0.285439i
$782$		0			0
$783$	−1628.79	−	2332.16i	−0.0743400	−	0.106442i
$784$		0			0
$785$		−	2621.26i		−	0.119181i
$786$		0			0
$787$	−34277.8			−1.55257			−0.776284	−	0.630384i	$-0.782897\pi$
$787$	−34277.8			−1.55257			−0.776284	+	0.630384i	$0.782897\pi$
$788$		0			0
$789$	24216.5	−	4990.11i	1.09269	−	0.225162i
$790$		0			0
$791$	735.721	−	1151.61i	0.0330711	−	0.0517656i
$792$		0			0
$793$	−40101.9			−1.79579
$794$		0			0
$795$	2445.71	−	503.970i	0.109107	−	0.0224830i
$796$		0			0
$797$			20736.8i			0.921624i	0.887498	+	0.460812i	$0.152442\pi$
$797$			20736.8i			0.921624i	−0.887498	+	0.460812i	$0.847558\pi$
$798$		0			0
$799$	−30109.0			−1.33314
$800$		0			0
$801$	6685.87	−	2877.61i	0.294923	−	0.126935i
$802$		0			0
$803$			13970.3i			0.613949i
$804$		0			0
$805$	−3678.80	−	2350.24i	−0.161069	−	0.102901i
$806$		0			0
$807$	−13456.4	+	2772.86i	−0.586973	+	0.120953i
$808$		0			0
$809$		−	15223.1i		−	0.661576i	−0.943705	−	0.330788i	$-0.892686\pi$
$809$		−	15223.1i		−	0.661576i	0.943705	−	0.330788i	$-0.107314\pi$
$810$		0			0
$811$	34460.7			1.49208			0.746042	−	0.665899i	$-0.231952\pi$
$811$	34460.7			1.49208			0.746042	+	0.665899i	$0.231952\pi$
$812$		0			0
$813$	−38789.1	+	7992.98i	−1.67330	+	0.344805i
$814$		0			0
$815$	−5644.09			−0.242581
$816$		0			0
$817$		−	27478.0i		−	1.17666i
$818$		0			0
$819$	−34333.6	−	5613.63i	−1.46485	−	0.239507i
$820$		0			0
$821$	8486.07			0.360738			0.180369	−	0.983599i	$-0.442271\pi$
$821$	8486.07			0.360738			0.180369	+	0.983599i	$0.442271\pi$
$822$		0			0
$823$	−21594.4			−0.914622			−0.457311	−	0.889307i	$-0.651187\pi$
$823$	−21594.4			−0.914622			−0.457311	+	0.889307i	$0.651187\pi$
$824$		0			0
$825$	5611.91	+	27234.0i	0.236826	+	1.14929i
$826$		0			0
$827$	−11423.5			−0.480330			−0.240165	−	0.970732i	$-0.577202\pi$
$827$	−11423.5			−0.480330			−0.240165	+	0.970732i	$0.577202\pi$
$828$		0			0
$829$	−6869.52			−0.287802			−0.143901	−	0.989592i	$-0.545965\pi$
$829$	−6869.52			−0.287802			−0.143901	+	0.989592i	$0.545965\pi$
$830$		0			0
$831$	18909.7	−	3896.59i	0.789375	−	0.162661i
$832$		0			0
$833$	7053.11	+	15226.5i	0.293368	+	0.633336i
$834$		0			0
$835$		−	7273.94i		−	0.301467i
$836$		0			0
$837$	−8900.78	+	6216.35i	−0.367570	+	0.256713i
$838$		0			0
$839$	29613.1			1.21854			0.609272	−	0.792961i	$-0.291462\pi$
$839$	29613.1			1.21854			0.609272	+	0.792961i	$0.291462\pi$
$840$		0			0
$841$	−23977.9			−0.983144
$842$		0			0
$843$	−39815.3	+	8204.45i	−1.62670	+	0.335203i
$844$		0			0
$845$		−	9812.74i		−	0.399489i
$846$		0			0
$847$	−15358.2	−	9811.79i	−0.623040	−	0.398037i
$848$		0			0
$849$	4319.39	+	20961.5i	0.174607	+	0.847348i
$850$		0			0
$851$	9639.25			0.388283
$852$		0			0
$853$	−15110.8			−0.606545			−0.303273	−	0.952904i	$-0.598079\pi$
$853$	−15110.8			−0.606545			−0.303273	+	0.952904i	$0.598079\pi$
$854$		0			0
$855$	2468.20	+	5734.65i	0.0987260	+	0.229381i
$856$		0			0
$857$	36848.3			1.46875			0.734373	−	0.678747i	$-0.237477\pi$
$857$	36848.3			1.46875			0.734373	+	0.678747i	$0.237477\pi$
$858$		0			0
$859$	−35305.1			−1.40232			−0.701162	−	0.713002i	$-0.747335\pi$
$859$	−35305.1			−1.40232			−0.701162	+	0.713002i	$0.747335\pi$
$860$		0			0
$861$	−9795.91	+	25613.5i	−0.387740	+	1.01383i
$862$		0			0
$863$		−	48135.9i		−	1.89868i	−0.314244	−	0.949342i	$-0.601751\pi$
$863$		−	48135.9i		−	1.89868i	0.314244	−	0.949342i	$-0.398249\pi$
$864$		0			0
$865$	14264.6			0.560707
$866$		0			0
$867$	2642.19	+	12822.3i	0.103499	+	0.502268i
$868$		0			0
$869$	−6299.25			−0.245900
$870$		0			0
$871$		−	46268.9i		−	1.79995i
$872$		0			0
$873$	−14722.4	−	34206.3i	−0.570766	−	1.32612i
$874$		0			0
$875$	−8743.52	+	13686.1i	−0.337811	+	0.528771i
$876$		0			0
$877$			22331.1i			0.859825i	0.902871	+	0.429912i	$0.141456\pi$
$877$			22331.1i			0.859825i	−0.902871	+	0.429912i	$0.858544\pi$
$878$		0			0
$879$	30698.7	−	6325.86i	1.17798	−	0.242737i
$880$		0			0
$881$	9542.16			0.364908			0.182454	−	0.983214i	$-0.441596\pi$
$881$	9542.16			0.364908			0.182454	+	0.983214i	$0.441596\pi$
$882$		0			0
$883$			31638.4i			1.20579i	0.797819	+	0.602897i	$0.205987\pi$
$883$			31638.4i			1.20579i	−0.797819	+	0.602897i	$0.794013\pi$
$884$		0			0
$885$	1584.23	+	7688.07i	0.0601731	+	0.292013i
$886$		0			0
$887$	368.960			0.0139667			0.00698335	−	0.999976i	$-0.497777\pi$
$887$	368.960			0.0139667			0.00698335	+	0.999976i	$0.497777\pi$
$888$		0			0
$889$	−30590.1	−	19542.8i	−1.15406	−	0.737284i
$890$		0			0
$891$	−24111.6	+	25474.3i	−0.906588	+	0.957826i
$892$		0			0
$893$	−38334.0			−1.43651
$894$		0			0
$895$			10210.5i			0.381340i
$896$		0			0
$897$	−22481.7	+	4632.65i	−0.836837	+	0.172441i
$898$		0			0
$899$		−	1569.02i		−	0.0582090i
$900$		0			0
$901$	−6333.29			−0.234176
$902$		0			0
$903$	39652.0	+	15164.9i	1.46128	+	0.558868i
$904$		0			0
$905$			4824.71i			0.177214i
$906$		0			0
$907$			13767.1i			0.504003i	0.967727	+	0.252001i	$0.0810888\pi$
$907$			13767.1i			0.504003i	−0.967727	+	0.252001i	$0.918911\pi$
$908$		0			0
$909$	1256.30	+	2918.90i	0.0458403	+	0.106506i
$910$		0			0
$911$		−	12013.8i		−	0.436921i	−0.975846	−	0.218461i	$-0.929896\pi$
$911$		−	12013.8i		−	0.436921i	0.975846	−	0.218461i	$-0.0701035\pi$
$912$		0			0
$913$		−	36726.1i		−	1.33128i
$914$		0			0
$915$	−10889.8	+	2243.99i	−0.393449	+	0.0810753i
$916$		0			0
$917$	11760.2	−	18408.0i	0.423507	−	0.662908i
$918$		0			0
$919$	−42760.2			−1.53485			−0.767425	−	0.641138i	$-0.778463\pi$
$919$	−42760.2			−1.53485			−0.767425	+	0.641138i	$0.778463\pi$
$920$		0			0
$921$	−2130.55	−	10339.3i	−0.0762259	−	0.369916i
$922$		0			0
$923$		−	9008.37i		−	0.321251i
$924$		0			0
$925$		−	16884.2i		−	0.600161i
$926$		0			0
$927$	−16975.5	−	39441.2i	−0.601456	−	1.39743i
$928$		0			0
$929$	−46059.3			−1.62665			−0.813324	−	0.581811i	$-0.802344\pi$
$929$	−46059.3			−1.62665			−0.813324	+	0.581811i	$0.802344\pi$
$930$		0			0
$931$	8979.85	+	19386.1i	0.316115	+	0.682442i
$932$		0			0
$933$	−6389.81	−	31009.0i	−0.224215	−	1.08809i
$934$		0			0
$935$			8738.54i			0.305648i
$936$		0			0
$937$			5647.59i			0.196904i	0.995142	+	0.0984519i	$0.0313891\pi$
$937$			5647.59i			0.196904i	−0.995142	+	0.0984519i	$0.968611\pi$
$938$		0			0
$939$	−14056.0	+	2896.42i	−0.488499	+	0.100661i
$940$		0			0
$941$		−	41421.7i		−	1.43497i	−0.696572	−	0.717486i	$-0.745293\pi$
$941$		−	41421.7i		−	1.43497i	0.696572	−	0.717486i	$-0.254707\pi$
$942$		0			0
$943$			18093.6i			0.624823i
$944$		0			0
$945$	−9637.54	+	396.805i	−0.331756	+	0.0136593i
$946$		0			0
$947$	21290.6			0.730571			0.365285	−	0.930896i	$-0.380971\pi$
$947$	21290.6			0.730571			0.365285	+	0.930896i	$0.380971\pi$
$948$		0			0
$949$			20200.5i			0.690976i
$950$		0			0
$951$	4059.13	+	19698.5i	0.138408	+	0.671680i
$952$		0			0
$953$			15236.1i			0.517888i	0.965892	+	0.258944i	$0.0833745\pi$
$953$			15236.1i			0.517888i	−0.965892	+	0.258944i	$0.916626\pi$
$954$		0			0
$955$	14531.4			0.492383
$956$		0			0
$957$	−1023.08	−	4964.91i	−0.0345576	−	0.167704i
$958$		0			0
$959$	26869.6	−	42058.6i	0.904760	−	1.41621i
$960$		0			0
$961$	23802.7			0.798991
$962$		0			0
$963$	10989.1	−	4729.73i	0.367726	−	0.158269i
$964$		0			0
$965$			13543.1i			0.451780i
$966$		0			0
$967$	29677.9			0.986945			0.493472	−	0.869761i	$-0.335727\pi$
$967$	29677.9			0.986945			0.493472	+	0.869761i	$0.335727\pi$
$968$		0			0
$969$	−3195.77	−	15508.7i	−0.105947	−	0.514150i
$970$		0			0
$971$		−	39536.5i		−	1.30668i	−0.757065	−	0.653340i	$-0.773367\pi$
$971$		−	39536.5i		−	1.30668i	0.757065	−	0.653340i	$-0.226633\pi$
$972$		0			0
$973$	−40297.3	−	25744.4i	−1.32772	−	0.848230i
$974$		0			0
$975$	8114.60	+	39379.2i	0.266539	+	1.29348i
$976$		0			0
$977$			29672.8i			0.971665i	0.874052	+	0.485832i	$0.161483\pi$
$977$			29672.8i			0.971665i	−0.874052	+	0.485832i	$0.838517\pi$
$978$		0			0
$979$	12971.1			0.423452
$980$		0			0
$981$	17670.6	+	41056.2i	0.575107	+	1.33621i
$982$		0			0
$983$	−4423.26			−0.143520			−0.0717599	−	0.997422i	$-0.522862\pi$
$983$	−4423.26			−0.143520			−0.0717599	+	0.997422i	$0.522862\pi$
$984$		0			0
$985$			293.583i			0.00949679i
$986$		0			0
$987$	21156.3	−	55317.7i	0.682282	−	1.78397i
$988$		0			0
$989$	28010.5			0.900587
$990$		0			0
$991$	−38967.8			−1.24910			−0.624548	−	0.780986i	$-0.714717\pi$
$991$	−38967.8			−1.24910			−0.624548	+	0.780986i	$0.714717\pi$
$992$		0			0
$993$	−21701.3	+	4471.83i	−0.693524	+	0.142909i
$994$		0			0
$995$	2638.23			0.0840577
$996$		0			0
$997$	−43802.8			−1.39142			−0.695712	−	0.718321i	$-0.744911\pi$
$997$	−43802.8			−1.39142			−0.695712	+	0.718321i	$0.744911\pi$
$998$		0			0
$999$	17461.4	−	12195.2i	0.553008	−	0.386224i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.i.c.209.37		80
3.2	odd	2	inner	672.4.i.c.209.40		80
4.3	odd	2		168.4.i.c.125.64	yes	80
7.6	odd	2	inner	672.4.i.c.209.43		80
8.3	odd	2		168.4.i.c.125.19	yes	80
8.5	even	2	inner	672.4.i.c.209.44		80
12.11	even	2		168.4.i.c.125.18	yes	80
21.20	even	2	inner	672.4.i.c.209.42		80
24.5	odd	2	inner	672.4.i.c.209.41		80
24.11	even	2		168.4.i.c.125.61	yes	80
28.27	even	2		168.4.i.c.125.63	yes	80
56.13	odd	2	inner	672.4.i.c.209.38		80
56.27	even	2		168.4.i.c.125.20	yes	80
84.83	odd	2		168.4.i.c.125.17	✓	80
168.83	odd	2		168.4.i.c.125.62	yes	80
168.125	even	2	inner	672.4.i.c.209.39		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.i.c.125.17	✓	80	84.83	odd	2
168.4.i.c.125.18	yes	80	12.11	even	2
168.4.i.c.125.19	yes	80	8.3	odd	2
168.4.i.c.125.20	yes	80	56.27	even	2
168.4.i.c.125.61	yes	80	24.11	even	2
168.4.i.c.125.62	yes	80	168.83	odd	2
168.4.i.c.125.63	yes	80	28.27	even	2
168.4.i.c.125.64	yes	80	4.3	odd	2
672.4.i.c.209.37		80	1.1	even	1	trivial
672.4.i.c.209.38		80	56.13	odd	2	inner
672.4.i.c.209.39		80	168.125	even	2	inner
672.4.i.c.209.40		80	3.2	odd	2	inner
672.4.i.c.209.41		80	24.5	odd	2	inner
672.4.i.c.209.42		80	21.20	even	2	inner
672.4.i.c.209.43		80	7.6	odd	2	inner
672.4.i.c.209.44		80	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.04870 - 5.08923i) q^{3} -3.71228i q^{5} +(-15.6071 - 9.97081i) q^{7} +(-24.8005 + 10.6741i) q^{9} -48.1150 q^{11} -69.5724 q^{13} +(-18.8927 + 3.89307i) q^{15} +48.9235 q^{17} +62.2883 q^{19} +(-34.3765 + 89.8847i) q^{21} +63.4952i q^{23} +111.219 q^{25} +(80.3314 + 115.021i) q^{27} -20.2759 q^{29} +77.3838i q^{31} +(50.4582 + 244.868i) q^{33} +(-37.0145 + 57.9382i) q^{35} -151.811i q^{37} +(72.9606 + 354.070i) q^{39} +284.960 q^{41} -441.143i q^{43} +(39.6255 + 92.0663i) q^{45} -615.429 q^{47} +(144.166 + 311.232i) q^{49} +(-51.3061 - 248.983i) q^{51} -129.453 q^{53} +178.616i q^{55} +(-65.3217 - 316.999i) q^{57} -406.934i q^{59} +576.405 q^{61} +(493.494 + 80.6876i) q^{63} +258.272i q^{65} +665.046i q^{67} +(323.142 - 66.5875i) q^{69} +129.482i q^{71} -290.352i q^{73} +(-116.635 - 566.018i) q^{75} +(750.938 + 479.745i) q^{77} +130.921 q^{79} +(501.125 - 529.447i) q^{81} +763.300i q^{83} -181.618i q^{85} +(21.2633 + 103.189i) q^{87} -269.586 q^{89} +(1085.83 + 693.693i) q^{91} +(393.824 - 81.1524i) q^{93} -231.232i q^{95} +1379.26i q^{97} +(1193.27 - 513.586i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) \) 80 * q - 64 * q^7 + 104 * q^9 + 8 * q^15 - 976 * q^25 - 568 * q^39 - 4048 * q^49 - 1448 * q^57 + 2152 * q^63 - 4992 * q^79 + 1568 * q^81

Introduction

L-functions

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