LMFDB - Embedded newform 672.4.h.a.575.22

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,4,Mod(575,672)] 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([1, 0, 1, 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("672.575"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [36,0,0,0,0,0,0,0,-132] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.22
Character	$\chi$	$=$	672.575
Dual form			672.4.h.a.575.21

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.49716 + 4.97579i) q^{3} -20.4606i q^{5} -7.00000i q^{7} +(-22.5170 + 14.8991i) q^{9} -20.3078 q^{11} -21.7749 q^{13} +(101.808 - 30.6328i) q^{15} +29.8439i q^{17} -87.0330i q^{19} +(34.8305 - 10.4801i) q^{21} +77.3830 q^{23} -293.635 q^{25} +(-107.847 - 89.7335i) q^{27} +254.794i q^{29} +155.953i q^{31} +(-30.4041 - 101.047i) q^{33} -143.224 q^{35} -324.488 q^{37} +(-32.6005 - 108.347i) q^{39} +241.980i q^{41} +245.969i q^{43} +(304.845 + 460.711i) q^{45} +295.540 q^{47} -49.0000 q^{49} +(-148.497 + 44.6812i) q^{51} +116.148i q^{53} +415.510i q^{55} +(433.058 - 130.303i) q^{57} -244.728 q^{59} +722.221 q^{61} +(104.294 + 157.619i) q^{63} +445.526i q^{65} -64.5032i q^{67} +(115.855 + 385.042i) q^{69} -960.293 q^{71} -673.227 q^{73} +(-439.620 - 1461.07i) q^{75} +142.155i q^{77} +946.494i q^{79} +(285.031 - 670.968i) q^{81} -360.920 q^{83} +610.624 q^{85} +(-1267.80 + 381.468i) q^{87} +1348.41i q^{89} +152.424i q^{91} +(-775.987 + 233.486i) q^{93} -1780.74 q^{95} -936.093 q^{97} +(457.271 - 302.569i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 132 q^{9} - 120 q^{13} + 28 q^{21} - 756 q^{25} + 40 q^{33} + 672 q^{37} + 304 q^{45} - 1764 q^{49} + 1624 q^{57} + 2472 q^{61} - 1224 q^{69} - 2376 q^{73} + 468 q^{81} + 5160 q^{85} - 648 q^{93}+ \cdots - 4488 q^{97}+O(q^{100}) $ 36 * q - 132 * q^9 - 120 * q^13 + 28 * q^21 - 756 * q^25 + 40 * q^33 + 672 * q^37 + 304 * q^45 - 1764 * q^49 + 1624 * q^57 + 2472 * q^61 - 1224 * q^69 - 2376 * q^73 + 468 * q^81 + 5160 * q^85 - 648 * q^93 - 4488 * q^97

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
$3$	1.49716	+	4.97579i	0.288129	+	0.957592i
$3$	1.49716	+	4.97579i	0.288129	+	0.957592i
$4$		0			0
$5$		−	20.4606i		−	1.83005i	−0.403398	−	0.915025i	$-0.632171\pi$
$5$		−	20.4606i		−	1.83005i	0.403398	−	0.915025i	$-0.367829\pi$
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		0			0
$9$	−22.5170	+	14.8991i	−0.833963	+	0.551820i
$10$		0			0
$11$	−20.3078			−0.556640			−0.278320	−	0.960488i	$-0.589778\pi$
$11$	−20.3078			−0.556640			−0.278320	+	0.960488i	$0.589778\pi$
$12$		0			0
$13$	−21.7749			−0.464558			−0.232279	−	0.972649i	$-0.574618\pi$
$13$	−21.7749			−0.464558			−0.232279	+	0.972649i	$0.574618\pi$
$14$		0			0
$15$	101.808	−	30.6328i	1.75244	−	0.527291i
$16$		0			0
$17$			29.8439i			0.425777i	0.977076	+	0.212889i	$0.0682871\pi$
$17$			29.8439i			0.425777i	−0.977076	+	0.212889i	$0.931713\pi$
$18$		0			0
$19$		−	87.0330i		−	1.05088i	−0.850830	−	0.525440i	$-0.823901\pi$
$19$		−	87.0330i		−	1.05088i	0.850830	−	0.525440i	$-0.176099\pi$
$20$		0			0
$21$	34.8305	−	10.4801i	0.361936	−	0.108903i
$22$		0			0
$23$	77.3830			0.701543			0.350771	−	0.936461i	$-0.385919\pi$
$23$	77.3830			0.701543			0.350771	+	0.936461i	$0.385919\pi$
$24$		0			0
$25$	−293.635			−2.34908
$26$		0			0
$27$	−107.847	−	89.7335i	−0.768707	−	0.639601i
$28$		0			0
$29$			254.794i			1.63152i	0.578393	+	0.815758i	$0.303680\pi$
$29$			254.794i			1.63152i	−0.578393	+	0.815758i	$0.696320\pi$
$30$		0			0
$31$			155.953i			0.903545i	0.892133	+	0.451773i	$0.149208\pi$
$31$			155.953i			0.903545i	−0.892133	+	0.451773i	$0.850792\pi$
$32$		0			0
$33$	−30.4041	−	101.047i	−0.160384	−	0.533034i
$34$		0			0
$35$	−143.224			−0.691694
$36$		0			0
$37$	−324.488			−1.44177			−0.720884	−	0.693055i	$-0.756264\pi$
$37$	−324.488			−1.44177			−0.720884	+	0.693055i	$0.756264\pi$
$38$		0			0
$39$	−32.6005	−	108.347i	−0.133853	−	0.444857i
$40$		0			0
$41$			241.980i			0.921731i	0.887470	+	0.460866i	$0.152461\pi$
$41$			241.980i			0.921731i	−0.887470	+	0.460866i	$0.847539\pi$
$42$		0			0
$43$			245.969i			0.872323i	0.899868	+	0.436161i	$0.143662\pi$
$43$			245.969i			0.872323i	−0.899868	+	0.436161i	$0.856338\pi$
$44$		0			0
$45$	304.845	+	460.711i	1.00986	+	1.52619i
$46$		0			0
$47$	295.540			0.917213			0.458606	−	0.888640i	$-0.348349\pi$
$47$	295.540			0.917213			0.458606	+	0.888640i	$0.348349\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	−148.497	+	44.6812i	−0.407721	+	0.122679i
$52$		0			0
$53$			116.148i			0.301021i	0.988608	+	0.150510i	$0.0480917\pi$
$53$			116.148i			0.301021i	−0.988608	+	0.150510i	$0.951908\pi$
$54$		0			0
$55$			415.510i			1.01868i
$56$		0			0
$57$	433.058	−	130.303i	1.00631	−	0.302789i
$58$		0			0
$59$	−244.728			−0.540016			−0.270008	−	0.962858i	$-0.587026\pi$
$59$	−244.728			−0.540016			−0.270008	+	0.962858i	$0.587026\pi$
$60$		0			0
$61$	722.221			1.51592			0.757959	−	0.652302i	$-0.226197\pi$
$61$	722.221			1.51592			0.757959	+	0.652302i	$0.226197\pi$
$62$		0			0
$63$	104.294	+	157.619i	0.208568	+	0.315208i
$64$		0			0
$65$			445.526i			0.850165i
$66$		0			0
$67$		−	64.5032i		−	0.117617i	−0.998269	−	0.0588083i	$-0.981270\pi$
$67$		−	64.5032i		−	0.117617i	0.998269	−	0.0588083i	$-0.0187301\pi$
$68$		0			0
$69$	115.855	+	385.042i	0.202135	+	0.671791i
$70$		0			0
$71$	−960.293			−1.60515			−0.802577	−	0.596549i	$-0.796538\pi$
$71$	−960.293			−1.60515			−0.802577	+	0.596549i	$0.796538\pi$
$72$		0			0
$73$	−673.227			−1.07939			−0.539693	−	0.841862i	$-0.681460\pi$
$73$	−673.227			−1.07939			−0.539693	+	0.841862i	$0.681460\pi$
$74$		0			0
$75$	−439.620	−	1461.07i	−0.676839	−	2.24946i
$76$		0			0
$77$			142.155i			0.210390i
$78$		0			0
$79$			946.494i			1.34796i	0.738749	+	0.673980i	$0.235417\pi$
$79$			946.494i			1.34796i	−0.738749	+	0.673980i	$0.764583\pi$
$80$		0			0
$81$	285.031	−	670.968i	0.390989	−	0.920395i
$82$		0			0
$83$	−360.920			−0.477303			−0.238652	−	0.971105i	$-0.576705\pi$
$83$	−360.920			−0.477303			−0.238652	+	0.971105i	$0.576705\pi$
$84$		0			0
$85$	610.624			0.779193
$86$		0			0
$87$	−1267.80	+	381.468i	−1.56233	+	0.470087i
$88$		0			0
$89$			1348.41i			1.60597i	0.596000	+	0.802984i	$0.296756\pi$
$89$			1348.41i			1.60597i	−0.596000	+	0.802984i	$0.703244\pi$
$90$		0			0
$91$			152.424i			0.175587i
$92$		0			0
$93$	−775.987	+	233.486i	−0.865227	+	0.260338i
$94$		0			0
$95$	−1780.74			−1.92316
$96$		0			0
$97$	−936.093			−0.979854			−0.489927	−	0.871763i	$-0.662977\pi$
$97$	−936.093			−0.979854			−0.489927	+	0.871763i	$0.662977\pi$
$98$		0			0
$99$	457.271	−	302.569i	0.464217	−	0.307165i
$100$		0			0
$101$		−	1178.89i		−	1.16143i	−0.814108	−	0.580713i	$-0.802774\pi$
$101$		−	1178.89i		−	1.16143i	0.814108	−	0.580713i	$-0.197226\pi$
$102$		0			0
$103$		−	1431.11i		−	1.36904i	−0.728994	−	0.684520i	$-0.760012\pi$
$103$		−	1431.11i		−	1.36904i	0.728994	−	0.684520i	$-0.239988\pi$
$104$		0			0
$105$	−214.430	−	712.653i	−0.199297	−	0.662360i
$106$		0			0
$107$	1026.30			0.927253			0.463627	−	0.886031i	$-0.346548\pi$
$107$	1026.30			0.927253			0.463627	+	0.886031i	$0.346548\pi$
$108$		0			0
$109$	−1368.47			−1.20253			−0.601264	−	0.799050i	$-0.705336\pi$
$109$	−1368.47			−1.20253			−0.601264	+	0.799050i	$0.705336\pi$
$110$		0			0
$111$	−485.811	−	1614.58i	−0.415416	−	1.38063i
$112$		0			0
$113$		−	135.630i		−	0.112911i	−0.998405	−	0.0564555i	$-0.982020\pi$
$113$		−	135.630i		−	0.112911i	0.998405	−	0.0564555i	$-0.0179799\pi$
$114$		0			0
$115$		−	1583.30i		−	1.28386i
$116$		0			0
$117$	490.305	−	324.427i	0.387425	−	0.256353i
$118$		0			0
$119$	208.907			0.160929
$120$		0			0
$121$	−918.592			−0.690152
$122$		0			0
$123$	−1204.04	+	362.284i	−0.882642	+	0.265578i
$124$		0			0
$125$			3450.37i			2.46889i
$126$		0			0
$127$		−	2437.65i		−	1.70320i	−0.524192	−	0.851600i	$-0.675632\pi$
$127$		−	2437.65i		−	1.70320i	0.524192	−	0.851600i	$-0.324368\pi$
$128$		0			0
$129$	−1223.89	+	368.255i	−0.835329	+	0.251342i
$130$		0			0
$131$	−566.045			−0.377523			−0.188762	−	0.982023i	$-0.560447\pi$
$131$	−566.045			−0.377523			−0.188762	+	0.982023i	$0.560447\pi$
$132$		0			0
$133$	−609.231			−0.397196
$134$		0			0
$135$	−1836.00	+	2206.60i	−1.17050	+	1.40677i
$136$		0			0
$137$			1563.89i			0.975268i	0.873048	+	0.487634i	$0.162140\pi$
$137$			1563.89i			0.975268i	−0.873048	+	0.487634i	$0.837860\pi$
$138$		0			0
$139$		−	3204.60i		−	1.95547i	−0.209844	−	0.977735i	$-0.567296\pi$
$139$		−	3204.60i		−	1.95547i	0.209844	−	0.977735i	$-0.432704\pi$
$140$		0			0
$141$	442.472	+	1470.55i	0.264276	+	0.878315i
$142$		0			0
$143$	442.200			0.258592
$144$		0			0
$145$	5213.22			2.98576
$146$		0			0
$147$	−73.3610	−	243.814i	−0.0411613	−	0.136799i
$148$		0			0
$149$		−	3019.60i		−	1.66024i	−0.557587	−	0.830118i	$-0.688273\pi$
$149$		−	3019.60i		−	1.66024i	0.557587	−	0.830118i	$-0.311727\pi$
$150$		0			0
$151$			3583.82i			1.93144i	0.259594	+	0.965718i	$0.416411\pi$
$151$			3583.82i			1.93144i	−0.259594	+	0.965718i	$0.583589\pi$
$152$		0			0
$153$	−444.649	−	671.995i	−0.234952	−	0.355082i
$154$		0			0
$155$	3190.88			1.65353
$156$		0			0
$157$	−471.349			−0.239603			−0.119802	−	0.992798i	$-0.538226\pi$
$157$	−471.349			−0.239603			−0.119802	+	0.992798i	$0.538226\pi$
$158$		0			0
$159$	−577.926	+	173.892i	−0.288255	+	0.0867328i
$160$		0			0
$161$		−	541.681i		−	0.265158i
$162$		0			0
$163$			1275.37i			0.612851i	0.951895	+	0.306426i	$0.0991331\pi$
$163$			1275.37i			0.612851i	−0.951895	+	0.306426i	$0.900867\pi$
$164$		0			0
$165$	−2067.49	+	622.086i	−0.975478	+	0.293511i
$166$		0			0
$167$	−762.580			−0.353355			−0.176677	−	0.984269i	$-0.556535\pi$
$167$	−762.580			−0.353355			−0.176677	+	0.984269i	$0.556535\pi$
$168$		0			0
$169$	−1722.86			−0.784185
$170$		0			0
$171$	1296.72	+	1959.72i	0.579897	+	0.876396i
$172$		0			0
$173$		−	570.667i		−	0.250792i	−0.992107	−	0.125396i	$-0.959980\pi$
$173$		−	570.667i		−	0.250792i	0.992107	−	0.125396i	$-0.0400201\pi$
$174$		0			0
$175$			2055.45i			0.887869i
$176$		0			0
$177$	−366.398	−	1217.72i	−0.155594	−	0.517114i
$178$		0			0
$179$	−1158.87			−0.483899			−0.241949	−	0.970289i	$-0.577787\pi$
$179$	−1158.87			−0.483899			−0.241949	+	0.970289i	$0.577787\pi$
$180$		0			0
$181$	892.923			0.366688			0.183344	−	0.983049i	$-0.441308\pi$
$181$	892.923			0.366688			0.183344	+	0.983049i	$0.441308\pi$
$182$		0			0
$183$	1081.28	+	3593.62i	0.436780	+	1.45163i
$184$		0			0
$185$			6639.20i			2.63851i
$186$		0			0
$187$		−	606.065i		−	0.237004i
$188$		0			0
$189$	−628.134	+	754.927i	−0.241746	+	0.290544i
$190$		0			0
$191$	−165.530			−0.0627085			−0.0313542	−	0.999508i	$-0.509982\pi$
$191$	−165.530			−0.0627085			−0.0313542	+	0.999508i	$0.509982\pi$
$192$		0			0
$193$	259.826			0.0969050			0.0484525	−	0.998825i	$-0.484571\pi$
$193$	259.826			0.0969050			0.0484525	+	0.998825i	$0.484571\pi$
$194$		0			0
$195$	−2216.85	+	667.025i	−0.814111	+	0.244957i
$196$		0			0
$197$			1877.65i			0.679070i	0.940593	+	0.339535i	$0.110270\pi$
$197$			1877.65i			0.679070i	−0.940593	+	0.339535i	$0.889730\pi$
$198$		0			0
$199$			1919.19i			0.683659i	0.939762	+	0.341829i	$0.111046\pi$
$199$			1919.19i			0.683659i	−0.939762	+	0.341829i	$0.888954\pi$
$200$		0			0
$201$	320.954	−	96.5717i	0.112629	−	0.0338888i
$202$		0			0
$203$	1783.56			0.616655
$204$		0			0
$205$	4951.06			1.68681
$206$		0			0
$207$	−1742.43	+	1152.94i	−0.585061	+	0.387125i
$208$		0			0
$209$			1767.45i			0.584962i
$210$		0			0
$211$		−	3895.07i		−	1.27084i	−0.772166	−	0.635421i	$-0.780827\pi$
$211$		−	3895.07i		−	1.27084i	0.772166	−	0.635421i	$-0.219173\pi$
$212$		0			0
$213$	−1437.72	−	4778.22i	−0.462491	−	1.53708i
$214$		0			0
$215$	5032.66			1.59639
$216$		0			0
$217$	1091.67			0.341508
$218$		0			0
$219$	−1007.93	−	3349.84i	−0.311003	−	1.03361i
$220$		0			0
$221$		−	649.847i		−	0.197798i
$222$		0			0
$223$			2913.46i			0.874887i	0.899246	+	0.437444i	$0.144116\pi$
$223$			2913.46i			0.874887i	−0.899246	+	0.437444i	$0.855884\pi$
$224$		0			0
$225$	6611.78	−	4374.91i	1.95905	−	1.29627i
$226$		0			0
$227$	−1254.02			−0.366663			−0.183332	−	0.983051i	$-0.558688\pi$
$227$	−1254.02			−0.366663			−0.183332	+	0.983051i	$0.558688\pi$
$228$		0			0
$229$	1946.83			0.561792			0.280896	−	0.959738i	$-0.409368\pi$
$229$	1946.83			0.561792			0.280896	+	0.959738i	$0.409368\pi$
$230$		0			0
$231$	−707.332	+	212.829i	−0.201468	+	0.0606195i
$232$		0			0
$233$			2118.26i			0.595587i	0.954630	+	0.297794i	$0.0962507\pi$
$233$			2118.26i			0.595587i	−0.954630	+	0.297794i	$0.903749\pi$
$234$		0			0
$235$		−	6046.93i		−	1.67854i
$236$		0			0
$237$	−4709.56	+	1417.06i	−1.29080	+	0.388387i
$238$		0			0
$239$	6507.44			1.76122			0.880609	−	0.473844i	$-0.157134\pi$
$239$	6507.44			1.76122			0.880609	+	0.473844i	$0.157134\pi$
$240$		0			0
$241$	−4705.98			−1.25784			−0.628919	−	0.777471i	$-0.716502\pi$
$241$	−4705.98			−1.25784			−0.628919	+	0.777471i	$0.716502\pi$
$242$		0			0
$243$	3765.34	+	413.706i	0.994018	+	0.109215i
$244$		0			0
$245$			1002.57i			0.261436i
$246$		0			0
$247$			1895.13i			0.488195i
$248$		0			0
$249$	−540.357	−	1795.86i	−0.137525	−	0.457061i
$250$		0			0
$251$	3963.90			0.996809			0.498404	−	0.866945i	$-0.333919\pi$
$251$	3963.90			0.996809			0.498404	+	0.866945i	$0.333919\pi$
$252$		0			0
$253$	−1571.48			−0.390506
$254$		0			0
$255$	914.203	+	3038.34i	0.224508	+	0.746149i
$256$		0			0
$257$		−	38.6112i		−	0.00937160i	−0.999989	−	0.00468580i	$-0.998508\pi$
$257$		−	38.6112i		−	0.00937160i	0.999989	−	0.00468580i	$-0.00149154\pi$
$258$		0			0
$259$			2271.41i			0.544937i
$260$		0			0
$261$	−3796.21	−	5737.19i	−0.900304	−	1.36062i
$262$		0			0
$263$	42.4838			0.00996069			0.00498034	−	0.999988i	$-0.498415\pi$
$263$	42.4838			0.00996069			0.00498034	+	0.999988i	$0.498415\pi$
$264$		0			0
$265$	2376.45			0.550883
$266$		0			0
$267$	−6709.41	+	2018.79i	−1.53786	+	0.462726i
$268$		0			0
$269$			899.050i			0.203777i	0.994796	+	0.101889i	$0.0324885\pi$
$269$			899.050i			0.203777i	−0.994796	+	0.101889i	$0.967511\pi$
$270$		0			0
$271$			276.446i			0.0619665i	0.999520	+	0.0309832i	$0.00986385\pi$
$271$			276.446i			0.0619665i	−0.999520	+	0.0309832i	$0.990136\pi$
$272$		0			0
$273$	−758.430	+	228.204i	−0.168140	+	0.0505916i
$274$		0			0
$275$	5963.09			1.30759
$276$		0			0
$277$	−4653.56			−1.00941			−0.504703	−	0.863293i	$-0.668398\pi$
$277$	−4653.56			−1.00941			−0.504703	+	0.863293i	$0.668398\pi$
$278$		0			0
$279$	−2323.56	−	3511.58i	−0.498594	−	0.753523i
$280$		0			0
$281$		−	8261.72i		−	1.75392i	−0.480559	−	0.876962i	$-0.659566\pi$
$281$		−	8261.72i		−	1.75392i	0.480559	−	0.876962i	$-0.340434\pi$
$282$		0			0
$283$		−	4598.84i		−	0.965981i	−0.875626	−	0.482990i	$-0.839551\pi$
$283$		−	4598.84i		−	0.965981i	0.875626	−	0.482990i	$-0.160449\pi$
$284$		0			0
$285$	−2666.07	−	8860.61i	−0.554120	−	1.84161i
$286$		0			0
$287$	1693.86			0.348382
$288$		0			0
$289$	4022.34			0.818714
$290$		0			0
$291$	−1401.48	−	4657.80i	−0.282324	−	0.938300i
$292$		0			0
$293$			6278.55i			1.25187i	0.779877	+	0.625933i	$0.215282\pi$
$293$			6278.55i			1.25187i	−0.779877	+	0.625933i	$0.784718\pi$
$294$		0			0
$295$			5007.28i			0.988255i
$296$		0			0
$297$	2190.13	+	1822.29i	0.427893	+	0.356027i
$298$		0			0
$299$	−1685.00			−0.325907
$300$		0			0
$301$	1721.78			0.329707
$302$		0			0
$303$	5865.92	−	1764.99i	1.11217	−	0.334641i
$304$		0			0
$305$		−	14777.1i		−	2.77420i
$306$		0			0
$307$			2025.63i			0.376576i	0.982114	+	0.188288i	$0.0602938\pi$
$307$			2025.63i			0.376576i	−0.982114	+	0.188288i	$0.939706\pi$
$308$		0			0
$309$	7120.89	−	2142.60i	1.31098	−	0.394461i
$310$		0			0
$311$	5442.54			0.992342			0.496171	−	0.868225i	$-0.334739\pi$
$311$	5442.54			0.992342			0.496171	+	0.868225i	$0.334739\pi$
$312$		0			0
$313$	−7840.20			−1.41583			−0.707914	−	0.706298i	$-0.750364\pi$
$313$	−7840.20			−1.41583			−0.707914	+	0.706298i	$0.750364\pi$
$314$		0			0
$315$	3224.98	−	2133.92i	0.576847	−	0.381690i
$316$		0			0
$317$		−	9145.28i		−	1.62035i	−0.586190	−	0.810174i	$-0.699373\pi$
$317$		−	9145.28i		−	1.62035i	0.586190	−	0.810174i	$-0.300627\pi$
$318$		0			0
$319$		−	5174.30i		−	0.908167i
$320$		0			0
$321$	1536.54	+	5106.65i	0.267169	+	0.887930i
$322$		0			0
$323$	2597.40			0.447441
$324$		0			0
$325$	6393.86			1.09129
$326$		0			0
$327$	−2048.82	−	6809.22i	−0.346483	−	1.15153i
$328$		0			0
$329$		−	2068.78i		−	0.346674i
$330$		0			0
$331$		−	45.8768i		−	0.00761818i	−0.999993	−	0.00380909i	$-0.998788\pi$
$331$		−	45.8768i		−	0.00761818i	0.999993	−	0.00380909i	$-0.00121247\pi$
$332$		0			0
$333$	7306.49	−	4834.59i	1.20238	−	0.795597i
$334$		0			0
$335$	−1319.77			−0.215244
$336$		0			0
$337$	−8222.68			−1.32913			−0.664567	−	0.747229i	$-0.731384\pi$
$337$	−8222.68			−1.32913			−0.664567	+	0.747229i	$0.731384\pi$
$338$		0			0
$339$	674.864	−	203.060i	0.108123	−	0.0325330i
$340$		0			0
$341$		−	3167.06i		−	0.502949i
$342$		0			0
$343$			343.000i			0.0539949i
$344$		0			0
$345$	7878.18	−	2370.46i	1.22941	−	0.369917i
$346$		0			0
$347$	3395.84			0.525355			0.262678	−	0.964884i	$-0.415394\pi$
$347$	3395.84			0.525355			0.262678	+	0.964884i	$0.415394\pi$
$348$		0			0
$349$	−1106.81			−0.169761			−0.0848803	−	0.996391i	$-0.527051\pi$
$349$	−1106.81			−0.169761			−0.0848803	+	0.996391i	$0.527051\pi$
$350$		0			0
$351$	2348.35	+	1953.93i	0.357109	+	0.297132i
$352$		0			0
$353$			9738.66i			1.46838i	0.678946	+	0.734188i	$0.262437\pi$
$353$			9738.66i			1.46838i	−0.678946	+	0.734188i	$0.737563\pi$
$354$		0			0
$355$			19648.2i			2.93751i
$356$		0			0
$357$	312.768	+	1039.48i	0.0463682	+	0.154104i
$358$		0			0
$359$	−5628.36			−0.827447			−0.413724	−	0.910403i	$-0.635772\pi$
$359$	−5628.36			−0.827447			−0.413724	+	0.910403i	$0.635772\pi$
$360$		0			0
$361$	−715.737			−0.104350
$362$		0			0
$363$	−1375.28	−	4570.72i	−0.198853	−	0.660884i
$364$		0			0
$365$			13774.6i			1.97533i
$366$		0			0
$367$			18.2412i			0.00259450i	0.999999	+	0.00129725i	$0.000412928\pi$
$367$			18.2412i			0.00259450i	−0.999999	+	0.00129725i	$0.999587\pi$
$368$		0			0
$369$	−3605.30	−	5448.67i	−0.508630	−	0.768690i
$370$		0			0
$371$	813.033			0.113775
$372$		0			0
$373$	6437.22			0.893584			0.446792	−	0.894638i	$-0.352566\pi$
$373$	6437.22			0.893584			0.446792	+	0.894638i	$0.352566\pi$
$374$		0			0
$375$	−17168.3	+	5165.77i	−2.36418	+	0.711358i
$376$		0			0
$377$		−	5548.09i		−	0.757935i
$378$		0			0
$379$			1912.12i			0.259154i	0.991569	+	0.129577i	$0.0413619\pi$
$379$			1912.12i			0.259154i	−0.991569	+	0.129577i	$0.958638\pi$
$380$		0			0
$381$	12129.2	−	3649.56i	1.63097	−	0.490742i
$382$		0			0
$383$	−7362.83			−0.982306			−0.491153	−	0.871073i	$-0.663424\pi$
$383$	−7362.83			−0.982306			−0.491153	+	0.871073i	$0.663424\pi$
$384$		0			0
$385$	2908.57			0.385024
$386$		0			0
$387$	−3664.72	−	5538.48i	−0.481365	−	0.727485i
$388$		0			0
$389$			11862.8i			1.54619i	0.634291	+	0.773094i	$0.281292\pi$
$389$			11862.8i			1.54619i	−0.634291	+	0.773094i	$0.718708\pi$
$390$		0			0
$391$			2309.41i			0.298701i
$392$		0			0
$393$	−847.461	−	2816.52i	−0.108776	−	0.361513i
$394$		0			0
$395$	19365.8			2.46683
$396$		0			0
$397$	4925.05			0.622622			0.311311	−	0.950308i	$-0.399232\pi$
$397$	4925.05			0.622622			0.311311	+	0.950308i	$0.399232\pi$
$398$		0			0
$399$	−912.118	−	3031.41i	−0.114444	−	0.380351i
$400$		0			0
$401$			1630.09i			0.202999i	0.994836	+	0.101500i	$0.0323641\pi$
$401$			1630.09i			0.202999i	−0.994836	+	0.101500i	$0.967636\pi$
$402$		0			0
$403$		−	3395.84i		−	0.419750i
$404$		0			0
$405$	−13728.4	−	5831.90i	−1.68437	−	0.715529i
$406$		0			0
$407$	6589.64			0.802546
$408$		0			0
$409$	−9699.95			−1.17269			−0.586347	−	0.810060i	$-0.699434\pi$
$409$	−9699.95			−1.17269			−0.586347	+	0.810060i	$0.699434\pi$
$410$		0			0
$411$	−7781.57	+	2341.39i	−0.933909	+	0.281003i
$412$		0			0
$413$			1713.10i			0.204107i
$414$		0			0
$415$			7384.64i			0.873488i
$416$		0			0
$417$	15945.4	−	4797.80i	1.87254	−	0.563428i
$418$		0			0
$419$	3120.60			0.363846			0.181923	−	0.983313i	$-0.441768\pi$
$419$	3120.60			0.363846			0.181923	+	0.983313i	$0.441768\pi$
$420$		0			0
$421$	−8807.80			−1.01963			−0.509817	−	0.860283i	$-0.670287\pi$
$421$	−8807.80			−1.01963			−0.509817	+	0.860283i	$0.670287\pi$
$422$		0			0
$423$	−6654.68	+	4403.30i	−0.764922	+	0.506136i
$424$		0			0
$425$		−	8763.22i		−	1.00019i
$426$		0			0
$427$		−	5055.55i		−	0.572963i
$428$		0			0
$429$	662.045	+	2200.29i	0.0745078	+	0.247625i
$430$		0			0
$431$	−2743.35			−0.306595			−0.153298	−	0.988180i	$-0.548989\pi$
$431$	−2743.35			−0.306595			−0.153298	+	0.988180i	$0.548989\pi$
$432$		0			0
$433$	−3612.22			−0.400906			−0.200453	−	0.979703i	$-0.564241\pi$
$433$	−3612.22			−0.400906			−0.200453	+	0.979703i	$0.564241\pi$
$434$		0			0
$435$	7805.05	+	25939.9i	0.860283	+	2.85913i
$436$		0			0
$437$		−	6734.88i		−	0.737237i
$438$		0			0
$439$			9777.00i			1.06294i	0.847077	+	0.531470i	$0.178360\pi$
$439$			9777.00i			1.06294i	−0.847077	+	0.531470i	$0.821640\pi$
$440$		0			0
$441$	1103.33	−	730.058i	0.119138	−	0.0788314i
$442$		0			0
$443$	14301.0			1.53378			0.766888	−	0.641781i	$-0.221804\pi$
$443$	14301.0			1.53378			0.766888	+	0.641781i	$0.221804\pi$
$444$		0			0
$445$	27589.2			2.93900
$446$		0			0
$447$	15024.9	−	4520.83i	1.58983	−	0.478363i
$448$		0			0
$449$			12280.8i			1.29080i	0.763846	+	0.645398i	$0.223309\pi$
$449$			12280.8i			1.29080i	−0.763846	+	0.645398i	$0.776691\pi$
$450$		0			0
$451$		−	4914.09i		−	0.513072i
$452$		0			0
$453$	−17832.3	+	5365.56i	−1.84953	+	0.556503i
$454$		0			0
$455$	3118.68			0.321332
$456$		0			0
$457$	−11290.1			−1.15565			−0.577823	−	0.816162i	$-0.696098\pi$
$457$	−11290.1			−1.15565			−0.577823	+	0.816162i	$0.696098\pi$
$458$		0			0
$459$	2678.00	−	3218.57i	0.272327	−	0.327298i
$460$		0			0
$461$			13225.8i			1.33619i	0.744074	+	0.668097i	$0.232891\pi$
$461$			13225.8i			1.33619i	−0.744074	+	0.668097i	$0.767109\pi$
$462$		0			0
$463$		−	13749.5i		−	1.38011i	−0.723755	−	0.690057i	$-0.757585\pi$
$463$		−	13749.5i		−	1.38011i	0.723755	−	0.690057i	$-0.242415\pi$
$464$		0			0
$465$	4777.27	+	15877.1i	0.476431	+	1.58341i
$466$		0			0
$467$	−18155.9			−1.79904			−0.899521	−	0.436877i	$-0.856084\pi$
$467$	−18155.9			−1.79904			−0.899521	+	0.436877i	$0.856084\pi$
$468$		0			0
$469$	−451.522			−0.0444549
$470$		0			0
$471$	−705.686	−	2345.33i	−0.0690367	−	0.229442i
$472$		0			0
$473$		−	4995.09i		−	0.485570i
$474$		0			0
$475$			25555.9i			2.46860i
$476$		0			0
$477$	−1730.50	−	2615.29i	−0.166109	−	0.251040i
$478$		0			0
$479$	3212.69			0.306454			0.153227	−	0.988191i	$-0.451033\pi$
$479$	3212.69			0.306454			0.153227	+	0.988191i	$0.451033\pi$
$480$		0			0
$481$	7065.67			0.669786
$482$		0			0
$483$	2695.29	−	810.985i	0.253913	−	0.0763998i
$484$		0			0
$485$			19153.0i			1.79318i
$486$		0			0
$487$		−	19360.3i		−	1.80144i	−0.434404	−	0.900718i	$-0.643041\pi$
$487$		−	19360.3i		−	1.80144i	0.434404	−	0.900718i	$-0.356959\pi$
$488$		0			0
$489$	−6345.98	+	1909.44i	−0.586861	+	0.176580i
$490$		0			0
$491$	−4235.18			−0.389269			−0.194634	−	0.980876i	$-0.562352\pi$
$491$	−4235.18			−0.389269			−0.194634	+	0.980876i	$0.562352\pi$
$492$		0			0
$493$	−7604.04			−0.694662
$494$		0			0
$495$	−6190.74	−	9356.03i	−0.562127	−	0.849540i
$496$		0			0
$497$			6722.05i			0.606691i
$498$		0			0
$499$		−	19854.7i		−	1.78120i	−0.454788	−	0.890600i	$-0.650285\pi$
$499$		−	19854.7i		−	1.78120i	0.454788	−	0.890600i	$-0.349715\pi$
$500$		0			0
$501$	−1141.71	−	3794.44i	−0.101812	−	0.338370i
$502$		0			0
$503$	21829.5			1.93505			0.967526	−	0.252770i	$-0.0813417\pi$
$503$	21829.5			1.93505			0.967526	+	0.252770i	$0.0813417\pi$
$504$		0			0
$505$	−24120.8			−2.12547
$506$		0			0
$507$	−2579.40	−	8572.57i	−0.225947	−	0.750929i
$508$		0			0
$509$		−	10217.9i		−	0.889781i	−0.895585	−	0.444890i	$-0.853243\pi$
$509$		−	10217.9i		−	0.889781i	0.895585	−	0.444890i	$-0.146757\pi$
$510$		0			0
$511$			4712.59i			0.407970i
$512$		0			0
$513$	−7809.77	+	9386.22i	−0.672144	+	0.807820i
$514$		0			0
$515$	−29281.3			−2.50541
$516$		0			0
$517$	−6001.78			−0.510557
$518$		0			0
$519$	2839.52	−	854.381i	0.240156	−	0.0722604i
$520$		0			0
$521$		−	6347.75i		−	0.533781i	−0.963727	−	0.266890i	$-0.914004\pi$
$521$		−	6347.75i		−	0.533781i	0.963727	−	0.266890i	$-0.0859962\pi$
$522$		0			0
$523$			9870.31i			0.825236i	0.910904	+	0.412618i	$0.135386\pi$
$523$			9870.31i			0.825236i	−0.910904	+	0.412618i	$0.864614\pi$
$524$		0			0
$525$	−10227.5	+	3077.34i	−0.850216	+	0.255821i
$526$		0			0
$527$	−4654.23			−0.384709
$528$		0			0
$529$	−6178.87			−0.507838
$530$		0			0
$531$	5510.55	−	3646.24i	0.450353	−	0.297991i
$532$		0			0
$533$		−	5269.09i		−	0.428198i
$534$		0			0
$535$		−	20998.7i		−	1.69692i
$536$		0			0
$537$	−1735.01	−	5766.28i	−0.139425	−	0.463377i
$538$		0			0
$539$	995.083			0.0795200
$540$		0			0
$541$	6586.32			0.523416			0.261708	−	0.965147i	$-0.415714\pi$
$541$	6586.32			0.523416			0.261708	+	0.965147i	$0.415714\pi$
$542$		0			0
$543$	1336.85	+	4443.00i	0.105653	+	0.351137i
$544$		0			0
$545$			27999.7i			2.20069i
$546$		0			0
$547$		−	1579.47i		−	0.123461i	−0.998093	−	0.0617306i	$-0.980338\pi$
$547$		−	1579.47i		−	0.123461i	0.998093	−	0.0617306i	$-0.0196620\pi$
$548$		0			0
$549$	−16262.3	+	10760.5i	−1.26422	+	0.836514i
$550$		0			0
$551$	22175.4			1.71453
$552$		0			0
$553$	6625.46			0.509481
$554$		0			0
$555$	−33035.3	+	9939.97i	−2.52661	+	0.760231i
$556$		0			0
$557$		−	15675.2i		−	1.19242i	−0.802828	−	0.596211i	$-0.796672\pi$
$557$		−	15675.2i		−	1.19242i	0.802828	−	0.596211i	$-0.203328\pi$
$558$		0			0
$559$		−	5355.93i		−	0.405245i
$560$		0			0
$561$	3015.65	−	907.378i	0.226953	−	0.0682879i
$562$		0			0
$563$	−10592.5			−0.792929			−0.396465	−	0.918050i	$-0.629763\pi$
$563$	−10592.5			−0.792929			−0.396465	+	0.918050i	$0.629763\pi$
$564$		0			0
$565$	−2775.06			−0.206633
$566$		0			0
$567$	−4696.78	−	1995.22i	−0.347877	−	0.147780i
$568$		0			0
$569$		−	6998.96i		−	0.515662i	−0.966190	−	0.257831i	$-0.916992\pi$
$569$		−	6998.96i		−	0.515662i	0.966190	−	0.257831i	$-0.0830077\pi$
$570$		0			0
$571$		−	7891.16i		−	0.578345i	−0.957277	−	0.289172i	$-0.906620\pi$
$571$		−	7891.16i		−	0.578345i	0.957277	−	0.289172i	$-0.0933801\pi$
$572$		0			0
$573$	−247.825	−	823.642i	−0.0180681	−	0.0600491i
$574$		0			0
$575$	−22722.4			−1.64798
$576$		0			0
$577$	6592.33			0.475636			0.237818	−	0.971310i	$-0.423568\pi$
$577$	6592.33			0.475636			0.237818	+	0.971310i	$0.423568\pi$
$578$		0			0
$579$	389.001	+	1292.84i	0.0279211	+	0.0927954i
$580$		0			0
$581$			2526.44i			0.180404i
$582$		0			0
$583$		−	2358.70i		−	0.167560i
$584$		0			0
$585$	−6637.96	−	10031.9i	−0.469138	−	0.709006i
$586$		0			0
$587$	−6607.11			−0.464573			−0.232287	−	0.972647i	$-0.574621\pi$
$587$	−6607.11			−0.464573			−0.232287	+	0.972647i	$0.574621\pi$
$588$		0			0
$589$	13573.0			0.949518
$590$		0			0
$591$	−9342.78	+	2811.15i	−0.650272	+	0.195660i
$592$		0			0
$593$			20893.4i			1.44686i	0.690398	+	0.723429i	$0.257435\pi$
$593$			20893.4i			1.44686i	−0.690398	+	0.723429i	$0.742565\pi$
$594$		0			0
$595$		−	4274.36i		−	0.294507i
$596$		0			0
$597$	−9549.51	+	2873.35i	−0.654666	+	0.196982i
$598$		0			0
$599$	−20486.5			−1.39742			−0.698712	−	0.715403i	$-0.746243\pi$
$599$	−20486.5			−1.39742			−0.698712	+	0.715403i	$0.746243\pi$
$600$		0			0
$601$	14843.4			1.00744			0.503722	−	0.863866i	$-0.331964\pi$
$601$	14843.4			1.00744			0.503722	+	0.863866i	$0.331964\pi$
$602$		0			0
$603$	961.042	+	1452.42i	0.0649032	+	0.0980879i
$604$		0			0
$605$			18794.9i			1.26301i
$606$		0			0
$607$		−	23443.2i		−	1.56760i	−0.621014	−	0.783799i	$-0.713279\pi$
$607$		−	23443.2i		−	1.56760i	0.621014	−	0.783799i	$-0.286721\pi$
$608$		0			0
$609$	2670.27	+	8874.60i	0.177676	+	0.590504i
$610$		0			0
$611$	−6435.35			−0.426099
$612$		0			0
$613$	−2857.55			−0.188280			−0.0941399	−	0.995559i	$-0.530010\pi$
$613$	−2857.55			−0.188280			−0.0941399	+	0.995559i	$0.530010\pi$
$614$		0			0
$615$	7412.54	+	24635.4i	0.486020	+	1.61528i
$616$		0			0
$617$			19473.7i			1.27064i	0.772250	+	0.635319i	$0.219131\pi$
$617$			19473.7i			1.27064i	−0.772250	+	0.635319i	$0.780869\pi$
$618$		0			0
$619$			19016.0i			1.23476i	0.786664	+	0.617381i	$0.211806\pi$
$619$			19016.0i			1.23476i	−0.786664	+	0.617381i	$0.788194\pi$
$620$		0			0
$621$	−8345.50	−	6943.85i	−0.539281	−	0.448707i
$622$		0			0
$623$	9438.87			0.606999
$624$		0			0
$625$	33892.2			2.16910
$626$		0			0
$627$	−8794.46	+	2646.16i	−0.560155	+	0.168545i
$628$		0			0
$629$		−	9683.98i		−	0.613872i
$630$		0			0
$631$		−	30755.6i		−	1.94035i	−0.242399	−	0.970177i	$-0.577934\pi$
$631$		−	30755.6i		−	1.94035i	0.242399	−	0.970177i	$-0.422066\pi$
$632$		0			0
$633$	19381.0	−	5831.55i	1.21695	−	0.366166i
$634$		0			0
$635$	−49875.7			−3.11694
$636$		0			0
$637$	1066.97			0.0663655
$638$		0			0
$639$	21622.9	−	14307.5i	1.33864	−	0.885756i
$640$		0			0
$641$		−	25291.0i		−	1.55840i	−0.626775	−	0.779201i	$-0.715625\pi$
$641$		−	25291.0i		−	1.55840i	0.626775	−	0.779201i	$-0.284375\pi$
$642$		0			0
$643$			11833.0i			0.725738i	0.931840	+	0.362869i	$0.118203\pi$
$643$			11833.0i			0.725738i	−0.931840	+	0.362869i	$0.881797\pi$
$644$		0			0
$645$	7534.71	+	25041.5i	0.459968	+	1.52869i
$646$		0			0
$647$	26639.0			1.61868			0.809342	−	0.587338i	$-0.199824\pi$
$647$	26639.0			1.61868			0.809342	+	0.587338i	$0.199824\pi$
$648$		0			0
$649$	4969.90			0.300594
$650$		0			0
$651$	1634.40	+	5431.91i	0.0983984	+	0.327025i
$652$		0			0
$653$		−	15802.1i		−	0.946992i	−0.880796	−	0.473496i	$-0.842992\pi$
$653$		−	15802.1i		−	0.946992i	0.880796	−	0.473496i	$-0.157008\pi$
$654$		0			0
$655$			11581.6i			0.690887i
$656$		0			0
$657$	15159.0	−	10030.5i	0.900169	−	0.595627i
$658$		0			0
$659$	127.301			0.00752496			0.00376248	−	0.999993i	$-0.498802\pi$
$659$	127.301			0.00752496			0.00376248	+	0.999993i	$0.498802\pi$
$660$		0			0
$661$	7619.97			0.448385			0.224192	−	0.974545i	$-0.428026\pi$
$661$	7619.97			0.448385			0.224192	+	0.974545i	$0.428026\pi$
$662$		0			0
$663$	3233.50	−	972.927i	0.189410	−	0.0569915i
$664$		0			0
$665$			12465.2i			0.726888i
$666$		0			0
$667$			19716.7i			1.14458i
$668$		0			0
$669$	−14496.8	+	4361.93i	−0.837785	+	0.252081i
$670$		0			0
$671$	−14666.7			−0.843820
$672$		0			0
$673$	−17220.6			−0.986338			−0.493169	−	0.869933i	$-0.664162\pi$
$673$	−17220.6			−0.986338			−0.493169	+	0.869933i	$0.664162\pi$
$674$		0			0
$675$	31667.6	+	26348.9i	1.80576	+	1.50247i
$676$		0			0
$677$			2038.86i			0.115745i	0.998324	+	0.0578727i	$0.0184317\pi$
$677$			2038.86i			0.115745i	−0.998324	+	0.0578727i	$0.981568\pi$
$678$		0			0
$679$			6552.65i			0.370350i
$680$		0			0
$681$	−1877.48	−	6239.76i	−0.105646	−	0.351113i
$682$		0			0
$683$	7126.18			0.399233			0.199616	−	0.979874i	$-0.436030\pi$
$683$	7126.18			0.399233			0.199616	+	0.979874i	$0.436030\pi$
$684$		0			0
$685$	31998.0			1.78479
$686$		0			0
$687$	2914.73	+	9687.04i	0.161869	+	0.537967i
$688$		0			0
$689$		−	2529.10i		−	0.139842i
$690$		0			0
$691$		−	11456.6i		−	0.630725i	−0.948971	−	0.315362i	$-0.897874\pi$
$691$		−	11456.6i		−	0.630725i	0.948971	−	0.315362i	$-0.102126\pi$
$692$		0			0
$693$	−2117.98	−	3200.90i	−0.116097	−	0.175458i
$694$		0			0
$695$	−65567.9			−3.57861
$696$		0			0
$697$	−7221.64			−0.392452
$698$		0			0
$699$	−10540.0	+	3171.38i	−0.570329	+	0.171606i
$700$		0			0
$701$			12593.4i			0.678525i	0.940692	+	0.339262i	$0.110177\pi$
$701$			12593.4i			0.678525i	−0.940692	+	0.339262i	$0.889823\pi$
$702$		0			0
$703$			28241.1i			1.51513i
$704$		0			0
$705$	30088.2	−	9053.24i	1.60736	−	0.483638i
$706$		0			0
$707$	−8252.24			−0.438978
$708$		0			0
$709$	−29125.4			−1.54278			−0.771389	−	0.636364i	$-0.780437\pi$
$709$	−29125.4			−1.54278			−0.771389	+	0.636364i	$0.780437\pi$
$710$		0			0
$711$	−14101.9	−	21312.2i	−0.743832	−	1.12415i
$712$		0			0
$713$			12068.1i			0.633875i
$714$		0			0
$715$		−	9047.66i		−	0.473236i
$716$		0			0
$717$	9742.69	+	32379.6i	0.507458	+	1.68653i
$718$		0			0
$719$	−30927.1			−1.60415			−0.802077	−	0.597220i	$-0.796272\pi$
$719$	−30927.1			−1.60415			−0.802077	+	0.597220i	$0.796272\pi$
$720$		0			0
$721$	−10017.8			−0.517449
$722$		0			0
$723$	−7045.63	−	23416.0i	−0.362420	−	1.20450i
$724$		0			0
$725$		−	74816.3i		−	3.83256i
$726$		0			0
$727$		−	3678.29i		−	0.187648i	−0.995589	−	0.0938242i	$-0.970091\pi$
$727$		−	3678.29i		−	0.187648i	0.995589	−	0.0938242i	$-0.0299092\pi$
$728$		0			0
$729$	3578.80	+	19354.9i	0.181822	+	0.983331i
$730$		0			0
$731$	−7340.67			−0.371415
$732$		0			0
$733$	3527.22			0.177736			0.0888682	−	0.996043i	$-0.471675\pi$
$733$	3527.22			0.177736			0.0888682	+	0.996043i	$0.471675\pi$
$734$		0			0
$735$	−4988.57	+	1501.01i	−0.250349	+	0.0753272i
$736$		0			0
$737$			1309.92i			0.0654701i
$738$		0			0
$739$		−	7635.17i		−	0.380060i	−0.981778	−	0.190030i	$-0.939142\pi$
$739$		−	7635.17i		−	0.380060i	0.981778	−	0.190030i	$-0.0608585\pi$
$740$		0			0
$741$	−9429.78	+	2837.32i	−0.467492	+	0.140663i
$742$		0			0
$743$	12182.7			0.601534			0.300767	−	0.953698i	$-0.402757\pi$
$743$	12182.7			0.601534			0.300767	+	0.953698i	$0.402757\pi$
$744$		0			0
$745$	−61782.7			−3.03832
$746$		0			0
$747$	8126.85	−	5377.40i	0.398053	−	0.263385i
$748$		0			0
$749$		−	7184.09i		−	0.350469i
$750$		0			0
$751$		−	12153.7i		−	0.590540i	−0.955414	−	0.295270i	$-0.904590\pi$
$751$		−	12153.7i		−	0.590540i	0.955414	−	0.295270i	$-0.0954097\pi$
$752$		0			0
$753$	5934.60	+	19723.5i	0.287210	+	0.954536i
$754$		0			0
$755$	73326.9			3.53462
$756$		0			0
$757$	14004.1			0.672375			0.336187	−	0.941795i	$-0.390862\pi$
$757$	14004.1			0.672375			0.336187	+	0.941795i	$0.390862\pi$
$758$		0			0
$759$	−2352.76	−	7819.36i	−0.112516	−	0.373946i
$760$		0			0
$761$			5193.12i			0.247373i	0.992321	+	0.123686i	$0.0394717\pi$
$761$			5193.12i			0.247373i	−0.992321	+	0.123686i	$0.960528\pi$
$762$		0			0
$763$			9579.28i			0.454513i
$764$		0			0
$765$	−13749.4	+	9097.77i	−0.649818	+	0.429974i
$766$		0			0
$767$	5328.93			0.250869
$768$		0			0
$769$	6212.79			0.291338			0.145669	−	0.989333i	$-0.453467\pi$
$769$	6212.79			0.291338			0.145669	+	0.989333i	$0.453467\pi$
$770$		0			0
$771$	192.121	−	57.8073i	0.00897417	−	0.00270023i
$772$		0			0
$773$			35834.8i			1.66739i	0.552228	+	0.833693i	$0.313778\pi$
$773$			35834.8i			1.66739i	−0.552228	+	0.833693i	$0.686222\pi$
$774$		0			0
$775$		−	45793.1i		−	2.12250i
$776$		0			0
$777$	−11302.1	+	3400.68i	−0.521827	+	0.157012i
$778$		0			0
$779$	21060.3			0.968629
$780$		0			0
$781$	19501.5			0.893492
$782$		0			0
$783$	22863.5	−	27478.6i	1.04352	−	1.25416i
$784$		0			0
$785$			9644.07i			0.438486i
$786$		0			0
$787$			13358.7i			0.605063i	0.953139	+	0.302532i	$0.0978318\pi$
$787$			13358.7i			0.605063i	−0.953139	+	0.302532i	$0.902168\pi$
$788$		0			0
$789$	63.6051	+	211.390i	0.00286996	+	0.00953827i
$790$		0			0
$791$	−949.407			−0.0426764
$792$		0			0
$793$	−15726.3			−0.704232
$794$		0			0
$795$	3557.93	+	11824.7i	0.158725	+	0.527521i
$796$		0			0
$797$		−	28233.5i		−	1.25481i	−0.778695	−	0.627403i	$-0.784118\pi$
$797$		−	28233.5i		−	1.25481i	0.778695	−	0.627403i	$-0.215882\pi$
$798$		0			0
$799$			8820.08i			0.390528i
$800$		0			0
$801$	−20090.2	−	30362.2i	−0.886206	−	1.33932i
$802$		0			0
$803$	13671.8			0.600830
$804$		0			0
$805$	−11083.1			−0.485253
$806$		0			0
$807$	−4473.49	+	1346.02i	−0.195135	+	0.0587141i
$808$		0			0
$809$		−	15336.0i		−	0.666482i	−0.942842	−	0.333241i	$-0.891858\pi$
$809$		−	15336.0i		−	0.666482i	0.942842	−	0.333241i	$-0.108142\pi$
$810$		0			0
$811$		−	10086.1i		−	0.436709i	−0.975869	−	0.218355i	$-0.929931\pi$
$811$		−	10086.1i		−	0.436709i	0.975869	−	0.218355i	$-0.0700690\pi$
$812$		0			0
$813$	−1375.54	+	413.885i	−0.0593386	+	0.0178543i
$814$		0			0
$815$	26094.8			1.12155
$816$		0			0
$817$	21407.4			0.916707
$818$		0			0
$819$	−2270.99	−	3432.13i	−0.0968922	−	0.146433i
$820$		0			0
$821$		−	24168.3i		−	1.02738i	−0.857975	−	0.513691i	$-0.828278\pi$
$821$		−	24168.3i		−	1.02738i	0.857975	−	0.513691i	$-0.171722\pi$
$822$		0			0
$823$			41459.1i			1.75598i	0.478679	+	0.877990i	$0.341116\pi$
$823$			41459.1i			1.75598i	−0.478679	+	0.877990i	$0.658884\pi$
$824$		0			0
$825$	8927.72	+	29671.1i	0.376755	+	1.25214i
$826$		0			0
$827$	−14214.2			−0.597674			−0.298837	−	0.954304i	$-0.596599\pi$
$827$	−14214.2			−0.597674			−0.298837	+	0.954304i	$0.596599\pi$
$828$		0			0
$829$	10720.2			0.449127			0.224564	−	0.974459i	$-0.427904\pi$
$829$	10720.2			0.449127			0.224564	+	0.974459i	$0.427904\pi$
$830$		0			0
$831$	−6967.14	−	23155.1i	−0.290839	−	0.966598i
$832$		0			0
$833$		−	1462.35i		−	0.0608253i
$834$		0			0
$835$			15602.8i			0.646657i
$836$		0			0
$837$	13994.2	−	16819.0i	0.577908	−	0.694562i
$838$		0			0
$839$	4364.99			0.179614			0.0898071	−	0.995959i	$-0.471375\pi$
$839$	4364.99			0.179614			0.0898071	+	0.995959i	$0.471375\pi$
$840$		0			0
$841$	−40530.8			−1.66185
$842$		0			0
$843$	41108.6	−	12369.1i	1.67954	−	0.505357i
$844$		0			0
$845$			35250.6i			1.43510i
$846$		0			0
$847$			6430.15i			0.260853i
$848$		0			0
$849$	22882.9	−	6885.21i	0.925015	−	0.278327i
$850$		0			0
$851$	−25109.8			−1.01146
$852$		0			0
$853$	−44761.3			−1.79672			−0.898358	−	0.439263i	$-0.855239\pi$
$853$	−44761.3			−1.79672			−0.898358	+	0.439263i	$0.855239\pi$
$854$		0			0
$855$	40097.0	−	26531.6i	1.60385	−	1.06124i
$856$		0			0
$857$		−	27438.9i		−	1.09369i	−0.837233	−	0.546846i	$-0.815828\pi$
$857$		−	27438.9i		−	1.09369i	0.837233	−	0.546846i	$-0.184172\pi$
$858$		0			0
$859$			3336.47i			0.132525i	0.997802	+	0.0662623i	$0.0211074\pi$
$859$			3336.47i			0.132525i	−0.997802	+	0.0662623i	$0.978893\pi$
$860$		0			0
$861$	2535.99	+	8428.31i	0.100379	+	0.333607i
$862$		0			0
$863$	−2569.38			−0.101347			−0.0506737	−	0.998715i	$-0.516137\pi$
$863$	−2569.38			−0.101347			−0.0506737	+	0.998715i	$0.516137\pi$
$864$		0			0
$865$	−11676.2			−0.458961
$866$		0			0
$867$	6022.10	+	20014.3i	0.235895	+	0.783993i
$868$		0			0
$869$		−	19221.2i		−	0.750328i
$870$		0			0
$871$			1404.55i			0.0546398i
$872$		0			0
$873$	21078.0	−	13947.0i	0.817162	−	0.540703i
$874$		0			0
$875$	24152.6			0.933151
$876$		0			0
$877$	−43209.1			−1.66370			−0.831851	−	0.554999i	$-0.812719\pi$
$877$	−43209.1			−1.66370			−0.831851	+	0.554999i	$0.812719\pi$
$878$		0			0
$879$	−31240.8	+	9400.02i	−1.19878	+	0.360699i
$880$		0			0
$881$		−	38789.8i		−	1.48339i	−0.670740	−	0.741693i	$-0.734023\pi$
$881$		−	38789.8i		−	1.48339i	0.670740	−	0.741693i	$-0.265977\pi$
$882$		0			0
$883$		−	2928.14i		−	0.111596i	−0.998442	−	0.0557982i	$-0.982230\pi$
$883$		−	2928.14i		−	0.111596i	0.998442	−	0.0557982i	$-0.0177703\pi$
$884$		0			0
$885$	−24915.2	+	7496.72i	−0.946345	+	0.284745i
$886$		0			0
$887$	19695.8			0.745570			0.372785	−	0.927918i	$-0.378403\pi$
$887$	19695.8			0.745570			0.372785	+	0.927918i	$0.378403\pi$
$888$		0			0
$889$	−17063.6			−0.643749
$890$		0			0
$891$	−5788.36	+	13625.9i	−0.217640	+	0.512329i
$892$		0			0
$893$		−	25721.8i		−	0.963881i
$894$		0			0
$895$			23711.1i			0.885558i
$896$		0			0
$897$	−2522.73	−	8384.23i	−0.0939035	−	0.312086i
$898$		0			0
$899$	−39735.7			−1.47415
$900$		0			0
$901$	−3466.30			−0.128168
$902$		0			0
$903$	2577.79	+	8567.22i	0.0949982	+	0.315725i
$904$		0			0
$905$		−	18269.7i		−	0.671056i
$906$		0			0
$907$			36489.9i			1.33586i	0.744223	+	0.667931i	$0.232820\pi$
$907$			36489.9i			1.33586i	−0.744223	+	0.667931i	$0.767180\pi$
$908$		0			0
$909$	17564.5	+	26545.1i	0.640898	+	0.968587i
$910$		0			0
$911$	−3110.23			−0.113114			−0.0565569	−	0.998399i	$-0.518012\pi$
$911$	−3110.23			−0.113114			−0.0565569	+	0.998399i	$0.518012\pi$
$912$		0			0
$913$	7329.51			0.265686
$914$		0			0
$915$	73527.6	−	22123.7i	2.65655	−	0.799329i
$916$		0			0
$917$			3962.31i			0.142690i
$918$		0			0
$919$			15010.4i			0.538789i	0.963030	+	0.269394i	$0.0868235\pi$
$919$			15010.4i			0.538789i	−0.963030	+	0.269394i	$0.913177\pi$
$920$		0			0
$921$	−10079.1	+	3032.70i	−0.360606	+	0.108502i
$922$		0			0
$923$	20910.3			0.745687
$924$		0			0
$925$	95281.0			3.38683
$926$		0			0
$927$	21322.3	+	32224.3i	0.755464	+	1.14173i
$928$		0			0
$929$		−	21624.5i		−	0.763699i	−0.924225	−	0.381849i	$-0.875287\pi$
$929$		−	21624.5i		−	0.763699i	0.924225	−	0.381849i	$-0.124713\pi$
$930$		0			0
$931$			4264.62i			0.150126i
$932$		0			0
$933$	8148.37	+	27081.0i	0.285923	+	0.950258i
$934$		0			0
$935$	−12400.4			−0.433730
$936$		0			0
$937$	5630.85			0.196320			0.0981600	−	0.995171i	$-0.468704\pi$
$937$	5630.85			0.196320			0.0981600	+	0.995171i	$0.468704\pi$
$938$		0			0
$939$	−11738.1	−	39011.2i	−0.407942	−	1.35579i
$940$		0			0
$941$			16267.6i			0.563559i	0.959479	+	0.281779i	$0.0909247\pi$
$941$			16267.6i			0.563559i	−0.959479	+	0.281779i	$0.909075\pi$
$942$		0			0
$943$			18725.2i			0.646634i
$944$		0			0
$945$	15446.2	+	12852.0i	0.531710	+	0.442408i
$946$		0			0
$947$	6835.81			0.234566			0.117283	−	0.993099i	$-0.462582\pi$
$947$	6835.81			0.234566			0.117283	+	0.993099i	$0.462582\pi$
$948$		0			0
$949$	14659.4			0.501438
$950$		0			0
$951$	45505.0	−	13692.0i	1.55163	−	0.466869i
$952$		0			0
$953$		−	15089.7i		−	0.512909i	−0.966556	−	0.256455i	$-0.917446\pi$
$953$		−	15089.7i		−	0.512909i	0.966556	−	0.256455i	$-0.0825544\pi$
$954$		0			0
$955$			3386.84i			0.114760i
$956$		0			0
$957$	25746.2	−	7746.77i	0.869653	−	0.261669i
$958$		0			0
$959$	10947.2			0.368617
$960$		0			0
$961$	5469.80			0.183606
$962$		0			0
$963$	−23109.2	+	15291.0i	−0.773295	+	0.511677i
$964$		0			0
$965$		−	5316.18i		−	0.177341i
$966$		0			0
$967$			50988.4i			1.69563i	0.530289	+	0.847817i	$0.322083\pi$
$967$			50988.4i			1.69563i	−0.530289	+	0.847817i	$0.677917\pi$
$968$		0			0
$969$	3888.74	+	12924.1i	0.128921	+	0.428466i
$970$		0			0
$971$	−53178.9			−1.75756			−0.878781	−	0.477225i	$-0.841643\pi$
$971$	−53178.9			−1.75756			−0.878781	+	0.477225i	$0.841643\pi$
$972$		0			0
$973$	−22432.2			−0.739098
$974$		0			0
$975$	9572.66	+	31814.5i	0.314431	+	1.04501i
$976$		0			0
$977$			10968.1i			0.359162i	0.983743	+	0.179581i	$0.0574741\pi$
$977$			10968.1i			0.359162i	−0.983743	+	0.179581i	$0.942526\pi$
$978$		0			0
$979$		−	27383.3i		−	0.893946i
$980$		0			0
$981$	30813.8	−	20389.0i	1.00286	−	0.663579i
$982$		0			0
$983$	32839.7			1.06554			0.532769	−	0.846261i	$-0.321152\pi$
$983$	32839.7			1.06554			0.532769	+	0.846261i	$0.321152\pi$
$984$		0			0
$985$	38417.8			1.24273
$986$		0			0
$987$	10293.8	−	3097.31i	0.331972	−	0.0998868i
$988$		0			0
$989$			19033.8i			0.611971i
$990$		0			0
$991$			9292.22i			0.297858i	0.988848	+	0.148929i	$0.0475826\pi$
$991$			9292.22i			0.297858i	−0.988848	+	0.148929i	$0.952417\pi$
$992$		0			0
$993$	228.273	−	68.6851i	0.00729510	−	0.00219502i
$994$		0			0
$995$	39267.8			1.25113
$996$		0			0
$997$	22458.1			0.713395			0.356697	−	0.934220i	$-0.383903\pi$
$997$	22458.1			0.713395			0.356697	+	0.934220i	$0.383903\pi$
$998$		0			0
$999$	34994.9	+	29117.4i	1.10830	+	0.922156i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.h.a.575.22	yes	36
3.2	odd	2	inner	672.4.h.a.575.16	yes	36
4.3	odd	2	inner	672.4.h.a.575.15	✓	36
12.11	even	2	inner	672.4.h.a.575.21	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.h.a.575.15	✓	36	4.3	odd	2	inner
672.4.h.a.575.16	yes	36	3.2	odd	2	inner
672.4.h.a.575.21	yes	36	12.11	even	2	inner
672.4.h.a.575.22	yes	36	1.1	even	1	trivial

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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.49716 + 4.97579i) q^{3} -20.4606i q^{5} -7.00000i q^{7} +(-22.5170 + 14.8991i) q^{9} -20.3078 q^{11} -21.7749 q^{13} +(101.808 - 30.6328i) q^{15} +29.8439i q^{17} -87.0330i q^{19} +(34.8305 - 10.4801i) q^{21} +77.3830 q^{23} -293.635 q^{25} +(-107.847 - 89.7335i) q^{27} +254.794i q^{29} +155.953i q^{31} +(-30.4041 - 101.047i) q^{33} -143.224 q^{35} -324.488 q^{37} +(-32.6005 - 108.347i) q^{39} +241.980i q^{41} +245.969i q^{43} +(304.845 + 460.711i) q^{45} +295.540 q^{47} -49.0000 q^{49} +(-148.497 + 44.6812i) q^{51} +116.148i q^{53} +415.510i q^{55} +(433.058 - 130.303i) q^{57} -244.728 q^{59} +722.221 q^{61} +(104.294 + 157.619i) q^{63} +445.526i q^{65} -64.5032i q^{67} +(115.855 + 385.042i) q^{69} -960.293 q^{71} -673.227 q^{73} +(-439.620 - 1461.07i) q^{75} +142.155i q^{77} +946.494i q^{79} +(285.031 - 670.968i) q^{81} -360.920 q^{83} +610.624 q^{85} +(-1267.80 + 381.468i) q^{87} +1348.41i q^{89} +152.424i q^{91} +(-775.987 + 233.486i) q^{93} -1780.74 q^{95} -936.093 q^{97} +(457.271 - 302.569i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 132 q^{9} - 120 q^{13} + 28 q^{21} - 756 q^{25} + 40 q^{33} + 672 q^{37} + 304 q^{45} - 1764 q^{49} + 1624 q^{57} + 2472 q^{61} - 1224 q^{69} - 2376 q^{73} + 468 q^{81} + 5160 q^{85} - 648 q^{93}+ \cdots - 4488 q^{97}+O(q^{100}) \) 36 * q - 132 * q^9 - 120 * q^13 + 28 * q^21 - 756 * q^25 + 40 * q^33 + 672 * q^37 + 304 * q^45 - 1764 * q^49 + 1624 * q^57 + 2472 * q^61 - 1224 * q^69 - 2376 * q^73 + 468 * q^81 + 5160 * q^85 - 648 * q^93 - 4488 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more