LMFDB - Embedded newform 672.4.h.a.575.1

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			575.1
Character	$\chi$	$=$	672.575
Dual form			672.4.h.a.575.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-5.18300 - 0.369441i) q^{3} -10.4214i q^{5} -7.00000i q^{7} +(26.7270 + 3.82963i) q^{9} -53.7484 q^{11} -40.4940 q^{13} +(-3.85011 + 54.0143i) q^{15} -23.2039i q^{17} +138.453i q^{19} +(-2.58609 + 36.2810i) q^{21} -23.4309 q^{23} +16.3938 q^{25} +(-137.111 - 29.7230i) q^{27} -19.7264i q^{29} -251.588i q^{31} +(278.578 + 19.8569i) q^{33} -72.9500 q^{35} -123.492 q^{37} +(209.880 + 14.9601i) q^{39} -45.1280i q^{41} +229.969i q^{43} +(39.9102 - 278.534i) q^{45} -559.181 q^{47} -49.0000 q^{49} +(-8.57248 + 120.266i) q^{51} -289.888i q^{53} +560.136i q^{55} +(51.1504 - 717.604i) q^{57} +371.528 q^{59} +518.829 q^{61} +(26.8074 - 187.089i) q^{63} +422.005i q^{65} +668.064i q^{67} +(121.442 + 8.65635i) q^{69} +1052.74 q^{71} +1166.35 q^{73} +(-84.9689 - 6.05653i) q^{75} +376.239i q^{77} +1091.01i q^{79} +(699.668 + 204.709i) q^{81} +253.374 q^{83} -241.818 q^{85} +(-7.28774 + 102.242i) q^{87} -190.115i q^{89} +283.458i q^{91} +(-92.9470 + 1303.98i) q^{93} +1442.88 q^{95} +117.271 q^{97} +(-1436.54 - 205.837i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.18300	−	0.369441i	−0.997469	−	0.0710990i
$3$	−5.18300	−	0.369441i	−0.997469	−	0.0710990i
$4$		0			0
$5$		−	10.4214i		−	0.932121i	−0.884753	−	0.466061i	$-0.845673\pi$
$5$		−	10.4214i		−	0.932121i	0.884753	−	0.466061i	$-0.154327\pi$
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		0			0
$9$	26.7270	+	3.82963i	0.989890	+	0.141838i
$10$		0			0
$11$	−53.7484			−1.47325			−0.736626	−	0.676301i	$-0.763582\pi$
$11$	−53.7484			−1.47325			−0.736626	+	0.676301i	$0.763582\pi$
$12$		0			0
$13$	−40.4940			−0.863923			−0.431962	−	0.901892i	$-0.642178\pi$
$13$	−40.4940			−0.863923			−0.431962	+	0.901892i	$0.642178\pi$
$14$		0			0
$15$	−3.85011	+	54.0143i	−0.0662729	+	0.929762i
$16$		0			0
$17$		−	23.2039i		−	0.331045i	−0.986206	−	0.165523i	$-0.947069\pi$
$17$		−	23.2039i		−	0.331045i	0.986206	−	0.165523i	$-0.0529311\pi$
$18$		0			0
$19$			138.453i			1.67176i	0.548914	+	0.835879i	$0.315041\pi$
$19$			138.453i			1.67176i	−0.548914	+	0.835879i	$0.684959\pi$
$20$		0			0
$21$	−2.58609	+	36.2810i	−0.0268729	+	0.377008i
$22$		0			0
$23$	−23.4309			−0.212421			−0.106211	−	0.994344i	$-0.533872\pi$
$23$	−23.4309			−0.212421			−0.106211	+	0.994344i	$0.533872\pi$
$24$		0			0
$25$	16.3938			0.131150
$26$		0			0
$27$	−137.111	−	29.7230i	−0.977300	−	0.211859i
$28$		0			0
$29$		−	19.7264i		−	0.126314i	−0.998004	−	0.0631568i	$-0.979883\pi$
$29$		−	19.7264i		−	0.126314i	0.998004	−	0.0631568i	$-0.0201168\pi$
$30$		0			0
$31$		−	251.588i		−	1.45763i	−0.684710	−	0.728816i	$-0.740071\pi$
$31$		−	251.588i		−	1.45763i	0.684710	−	0.728816i	$-0.259929\pi$
$32$		0			0
$33$	278.578	+	19.8569i	1.46952	+	0.104747i
$34$		0			0
$35$	−72.9500			−0.352309
$36$		0			0
$37$	−123.492			−0.548700			−0.274350	−	0.961630i	$-0.588463\pi$
$37$	−123.492			−0.548700			−0.274350	+	0.961630i	$0.588463\pi$
$38$		0			0
$39$	209.880	+	14.9601i	0.861737	+	0.0614241i
$40$		0			0
$41$		−	45.1280i		−	0.171898i	−0.996300	−	0.0859489i	$-0.972608\pi$
$41$		−	45.1280i		−	0.171898i	0.996300	−	0.0859489i	$-0.0273922\pi$
$42$		0			0
$43$			229.969i			0.815579i	0.913076	+	0.407790i	$0.133700\pi$
$43$			229.969i			0.815579i	−0.913076	+	0.407790i	$0.866300\pi$
$44$		0			0
$45$	39.9102	−	278.534i	0.132210	−	0.922697i
$46$		0			0
$47$	−559.181			−1.73542			−0.867712	−	0.497068i	$-0.834410\pi$
$47$	−559.181			−1.73542			−0.867712	+	0.497068i	$0.834410\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	−8.57248	+	120.266i	−0.0235370	+	0.330208i
$52$		0			0
$53$		−	289.888i		−	0.751304i	−0.926761	−	0.375652i	$-0.877419\pi$
$53$		−	289.888i		−	0.751304i	0.926761	−	0.375652i	$-0.122581\pi$
$54$		0			0
$55$			560.136i			1.37325i
$56$		0			0
$57$	51.1504	−	717.604i	0.118860	−	1.66753i
$58$		0			0
$59$	371.528			0.819810			0.409905	−	0.912128i	$-0.365562\pi$
$59$	371.528			0.819810			0.409905	+	0.912128i	$0.365562\pi$
$60$		0			0
$61$	518.829			1.08900			0.544502	−	0.838760i	$-0.316719\pi$
$61$	518.829			1.08900			0.544502	+	0.838760i	$0.316719\pi$
$62$		0			0
$63$	26.8074	−	187.089i	0.0536098	−	0.374143i
$64$		0			0
$65$			422.005i			0.805281i
$66$		0			0
$67$			668.064i			1.21816i	0.793107	+	0.609082i	$0.208462\pi$
$67$			668.064i			1.21816i	−0.793107	+	0.609082i	$0.791538\pi$
$68$		0			0
$69$	121.442	+	8.65635i	0.211883	+	0.0151029i
$70$		0			0
$71$	1052.74			1.75967			0.879837	−	0.475275i	$-0.157651\pi$
$71$	1052.74			1.75967			0.879837	+	0.475275i	$0.157651\pi$
$72$		0			0
$73$	1166.35			1.87001			0.935006	−	0.354631i	$-0.115394\pi$
$73$	1166.35			1.87001			0.935006	+	0.354631i	$0.115394\pi$
$74$		0			0
$75$	−84.9689	−	6.05653i	−0.130818	−	0.00932464i
$76$		0			0
$77$			376.239i			0.556837i
$78$		0			0
$79$			1091.01i			1.55378i	0.629639	+	0.776888i	$0.283203\pi$
$79$			1091.01i			1.55378i	−0.629639	+	0.776888i	$0.716797\pi$
$80$		0			0
$81$	699.668	+	204.709i	0.959764	+	0.280808i
$82$		0			0
$83$	253.374			0.335077			0.167539	−	0.985866i	$-0.446418\pi$
$83$	253.374			0.335077			0.167539	+	0.985866i	$0.446418\pi$
$84$		0			0
$85$	−241.818			−0.308574
$86$		0			0
$87$	−7.28774	+	102.242i	−0.00898078	+	0.125994i
$88$		0			0
$89$		−	190.115i		−	0.226429i	−0.993571	−	0.113214i	$-0.963885\pi$
$89$		−	190.115i		−	0.226429i	0.993571	−	0.113214i	$-0.0361147\pi$
$90$		0			0
$91$			283.458i			0.326532i
$92$		0			0
$93$	−92.9470	+	1303.98i	−0.103636	+	1.45394i
$94$		0			0
$95$	1442.88			1.55828
$96$		0			0
$97$	117.271			0.122754			0.0613768	−	0.998115i	$-0.480451\pi$
$97$	117.271			0.122754			0.0613768	+	0.998115i	$0.480451\pi$
$98$		0			0
$99$	−1436.54	−	205.837i	−1.45836	−	0.208963i
$100$		0			0
$101$			1020.77i			1.00565i	0.864388	+	0.502825i	$0.167706\pi$
$101$			1020.77i			1.00565i	−0.864388	+	0.502825i	$0.832294\pi$
$102$		0			0
$103$		−	182.050i		−	0.174154i	−0.996202	−	0.0870771i	$-0.972247\pi$
$103$		−	182.050i		−	0.174154i	0.996202	−	0.0870771i	$-0.0277527\pi$
$104$		0			0
$105$	378.100	+	26.9507i	0.351417	+	0.0250488i
$106$		0			0
$107$	168.192			0.151960			0.0759800	−	0.997109i	$-0.475791\pi$
$107$	168.192			0.151960			0.0759800	+	0.997109i	$0.475791\pi$
$108$		0			0
$109$	153.395			0.134794			0.0673972	−	0.997726i	$-0.478531\pi$
$109$	153.395			0.134794			0.0673972	+	0.997726i	$0.478531\pi$
$110$		0			0
$111$	640.057	+	45.6229i	0.547311	+	0.0390120i
$112$		0			0
$113$			896.301i			0.746167i	0.927798	+	0.373084i	$0.121700\pi$
$113$			896.301i			0.746167i	−0.927798	+	0.373084i	$0.878300\pi$
$114$		0			0
$115$			244.184i			0.198002i
$116$		0			0
$117$	−1082.28	−	155.077i	−0.855189	−	0.122537i
$118$		0			0
$119$	−162.427			−0.125123
$120$		0			0
$121$	1557.89			1.17047
$122$		0			0
$123$	−16.6721	+	233.899i	−0.0122218	+	0.171463i
$124$		0			0
$125$		−	1473.53i		−	1.05437i
$126$		0			0
$127$			1423.18i			0.994386i	0.867640	+	0.497193i	$0.165636\pi$
$127$			1423.18i			0.994386i	−0.867640	+	0.497193i	$0.834364\pi$
$128$		0			0
$129$	84.9600	−	1191.93i	0.0579869	−	0.813515i
$130$		0			0
$131$	−2597.12			−1.73215			−0.866074	−	0.499916i	$-0.833364\pi$
$131$	−2597.12			−1.73215			−0.866074	+	0.499916i	$0.833364\pi$
$132$		0			0
$133$	969.174			0.631865
$134$		0			0
$135$	−309.757	+	1428.90i	−0.197479	+	0.910962i
$136$		0			0
$137$			2309.65i			1.44034i	0.693796	+	0.720172i	$0.255937\pi$
$137$			2309.65i			1.44034i	−0.693796	+	0.720172i	$0.744063\pi$
$138$		0			0
$139$		−	138.217i		−	0.0843411i	−0.999110	−	0.0421706i	$-0.986573\pi$
$139$		−	138.217i		−	0.0843411i	0.999110	−	0.0421706i	$-0.0134273\pi$
$140$		0			0
$141$	2898.24	+	206.584i	1.73103	+	0.123387i
$142$		0			0
$143$	2176.49			1.27278
$144$		0			0
$145$	−205.577			−0.117740
$146$		0			0
$147$	253.967	+	18.1026i	0.142496	+	0.0101570i
$148$		0			0
$149$			140.228i			0.0771003i	0.999257	+	0.0385502i	$0.0122739\pi$
$149$			140.228i			0.0771003i	−0.999257	+	0.0385502i	$0.987726\pi$
$150$		0			0
$151$		−	871.982i		−	0.469940i	−0.972003	−	0.234970i	$-0.924501\pi$
$151$		−	871.982i		−	0.469940i	0.972003	−	0.234970i	$-0.0754991\pi$
$152$		0			0
$153$	88.8623	−	620.171i	0.0469549	−	0.327698i
$154$		0			0
$155$	−2621.91			−1.35869
$156$		0			0
$157$	1195.25			0.607589			0.303795	−	0.952738i	$-0.401746\pi$
$157$	1195.25			0.607589			0.303795	+	0.952738i	$0.401746\pi$
$158$		0			0
$159$	−107.096	+	1502.49i	−0.0534170	+	0.749403i
$160$		0			0
$161$			164.016i			0.0802876i
$162$		0			0
$163$		−	944.806i		−	0.454006i	−0.973894	−	0.227003i	$-0.927107\pi$
$163$		−	944.806i		−	0.454006i	0.973894	−	0.227003i	$-0.0728927\pi$
$164$		0			0
$165$	206.937	−	2903.18i	0.0976366	−	1.36977i
$166$		0			0
$167$	3198.72			1.48218			0.741091	−	0.671405i	$-0.234309\pi$
$167$	3198.72			1.48218			0.741091	+	0.671405i	$0.234309\pi$
$168$		0			0
$169$	−557.240			−0.253637
$170$		0			0
$171$	−530.225	+	3700.45i	−0.237119	+	1.65486i
$172$		0			0
$173$		−	1742.98i		−	0.765992i	−0.923750	−	0.382996i	$-0.874892\pi$
$173$		−	1742.98i		−	0.765992i	0.923750	−	0.382996i	$-0.125108\pi$
$174$		0			0
$175$		−	114.756i		−	0.0495701i
$176$		0			0
$177$	−1925.63	−	137.258i	−0.817736	−	0.0582877i
$178$		0			0
$179$	1492.58			0.623242			0.311621	−	0.950206i	$-0.399128\pi$
$179$	1492.58			0.623242			0.311621	+	0.950206i	$0.399128\pi$
$180$		0			0
$181$	3032.55			1.24535			0.622673	−	0.782482i	$-0.286047\pi$
$181$	3032.55			1.24535			0.622673	+	0.782482i	$0.286047\pi$
$182$		0			0
$183$	−2689.09	−	191.677i	−1.08625	−	0.0774271i
$184$		0			0
$185$			1286.96i			0.511455i
$186$		0			0
$187$			1247.17i			0.487713i
$188$		0			0
$189$	−208.061	+	959.780i	−0.0800753	+	0.369385i
$190$		0			0
$191$	2864.32			1.08510			0.542552	−	0.840022i	$-0.317458\pi$
$191$	2864.32			1.08510			0.542552	+	0.840022i	$0.317458\pi$
$192$		0			0
$193$	−5136.39			−1.91568			−0.957839	−	0.287306i	$-0.907240\pi$
$193$	−5136.39			−1.91568			−0.957839	+	0.287306i	$0.907240\pi$
$194$		0			0
$195$	155.906	−	2187.25i	0.0572547	−	0.803243i
$196$		0			0
$197$			2218.23i			0.802244i	0.916025	+	0.401122i	$0.131380\pi$
$197$			2218.23i			0.802244i	−0.916025	+	0.401122i	$0.868620\pi$
$198$		0			0
$199$		−	3740.27i		−	1.33237i	−0.745788	−	0.666183i	$-0.767927\pi$
$199$		−	3740.27i		−	1.33237i	0.745788	−	0.666183i	$-0.232073\pi$
$200$		0			0
$201$	246.810	−	3462.58i	0.0866102	−	1.21508i
$202$		0			0
$203$	−138.085			−0.0477421
$204$		0			0
$205$	−470.298			−0.160230
$206$		0			0
$207$	−626.239	−	89.7317i	−0.210273	−	0.0301294i
$208$		0			0
$209$		−	7441.65i		−	2.46292i
$210$		0			0
$211$		−	3469.64i		−	1.13204i	−0.824392	−	0.566019i	$-0.808483\pi$
$211$		−	3469.64i		−	1.13204i	0.824392	−	0.566019i	$-0.191517\pi$
$212$		0			0
$213$	−5456.34	−	388.924i	−1.75522	−	0.125111i
$214$		0			0
$215$	2396.60			0.760219
$216$		0			0
$217$	−1761.12			−0.550933
$218$		0			0
$219$	−6045.19	−	430.898i	−1.86528	−	0.132956i
$220$		0			0
$221$			939.617i			0.285998i
$222$		0			0
$223$		−	142.329i		−	0.0427400i	−0.999772	−	0.0213700i	$-0.993197\pi$
$223$		−	142.329i		−	0.0427400i	0.999772	−	0.0213700i	$-0.00680280\pi$
$224$		0			0
$225$	438.157	+	62.7820i	0.129824	+	0.0186021i
$226$		0			0
$227$	−3729.72			−1.09053			−0.545264	−	0.838264i	$-0.683571\pi$
$227$	−3729.72			−1.09053			−0.545264	+	0.838264i	$0.683571\pi$
$228$		0			0
$229$	1178.54			0.340086			0.170043	−	0.985437i	$-0.445609\pi$
$229$	1178.54			0.340086			0.170043	+	0.985437i	$0.445609\pi$
$230$		0			0
$231$	138.998	−	1950.05i	0.0395905	−	0.555427i
$232$		0			0
$233$			3975.79i			1.11787i	0.829213	+	0.558933i	$0.188789\pi$
$233$			3975.79i			1.11787i	−0.829213	+	0.558933i	$0.811211\pi$
$234$		0			0
$235$			5827.47i			1.61763i
$236$		0			0
$237$	403.064	−	5654.71i	0.110472	−	1.54984i
$238$		0			0
$239$	−3272.80			−0.885774			−0.442887	−	0.896577i	$-0.646046\pi$
$239$	−3272.80			−0.885774			−0.442887	+	0.896577i	$0.646046\pi$
$240$		0			0
$241$	−511.709			−0.136772			−0.0683860	−	0.997659i	$-0.521785\pi$
$241$	−511.709			−0.136772			−0.0683860	+	0.997659i	$0.521785\pi$
$242$		0			0
$243$	−3550.75	−	1319.49i	−0.937370	−	0.348336i
$244$		0			0
$245$			510.650i			0.133160i
$246$		0			0
$247$		−	5606.53i		−	1.44427i
$248$		0			0
$249$	−1313.24	−	93.6068i	−0.334229	−	0.0238237i
$250$		0			0
$251$	3238.52			0.814397			0.407198	−	0.913340i	$-0.366506\pi$
$251$	3238.52			0.814397			0.407198	+	0.913340i	$0.366506\pi$
$252$		0			0
$253$	1259.38			0.312950
$254$		0			0
$255$	1253.34	+	89.3375i	0.307793	+	0.0219393i
$256$		0			0
$257$			7057.09i			1.71288i	0.516251	+	0.856438i	$0.327327\pi$
$257$			7057.09i			1.71288i	−0.516251	+	0.856438i	$0.672673\pi$
$258$		0			0
$259$			864.441i			0.207389i
$260$		0			0
$261$	75.5447	−	527.228i	0.0179161	−	0.125037i
$262$		0			0
$263$	4755.41			1.11495			0.557473	−	0.830195i	$-0.311771\pi$
$263$	4755.41			1.11495			0.557473	+	0.830195i	$0.311771\pi$
$264$		0			0
$265$	−3021.04			−0.700307
$266$		0			0
$267$	−70.2363	+	985.366i	−0.0160988	+	0.225856i
$268$		0			0
$269$			152.943i			0.0346658i	0.999850	+	0.0173329i	$0.00551752\pi$
$269$			152.943i			0.0346658i	−0.999850	+	0.0173329i	$0.994482\pi$
$270$		0			0
$271$			7405.36i			1.65994i	0.557808	+	0.829970i	$0.311642\pi$
$271$			7405.36i			1.65994i	−0.557808	+	0.829970i	$0.688358\pi$
$272$		0			0
$273$	104.721	−	1469.16i	0.0232161	−	0.325706i
$274$		0			0
$275$	−881.139			−0.193217
$276$		0			0
$277$	−3128.91			−0.678693			−0.339346	−	0.940661i	$-0.610206\pi$
$277$	−3128.91			−0.678693			−0.339346	+	0.940661i	$0.610206\pi$
$278$		0			0
$279$	963.489	−	6724.20i	0.206748	−	1.44289i
$280$		0			0
$281$			6334.25i			1.34473i	0.740219	+	0.672366i	$0.234722\pi$
$281$			6334.25i			1.34473i	−0.740219	+	0.672366i	$0.765278\pi$
$282$		0			0
$283$		−	7926.87i		−	1.66503i	−0.554002	−	0.832515i	$-0.686900\pi$
$283$		−	7926.87i		−	1.66503i	0.554002	−	0.832515i	$-0.313100\pi$
$284$		0			0
$285$	−7478.46	−	533.060i	−1.55434	−	0.110792i
$286$		0			0
$287$	−315.896			−0.0649713
$288$		0			0
$289$	4374.58			0.890409
$290$		0			0
$291$	−607.817	−	43.3248i	−0.122443	−	0.00872765i
$292$		0			0
$293$			9212.99i			1.83696i	0.395470	+	0.918479i	$0.370582\pi$
$293$			9212.99i			1.83696i	−0.395470	+	0.918479i	$0.629418\pi$
$294$		0			0
$295$		−	3871.85i		−	0.764163i
$296$		0			0
$297$	7369.52	+	1597.57i	1.43981	+	0.312122i
$298$		0			0
$299$	948.810			0.183515
$300$		0			0
$301$	1609.78			0.308260
$302$		0			0
$303$	377.116	−	5290.67i	0.0715008	−	1.00311i
$304$		0			0
$305$		−	5406.94i		−	1.01508i
$306$		0			0
$307$		−	7367.55i		−	1.36967i	−0.728699	−	0.684834i	$-0.759875\pi$
$307$		−	7367.55i		−	1.36967i	0.728699	−	0.684834i	$-0.240125\pi$
$308$		0			0
$309$	−67.2567	+	943.564i	−0.0123822	+	0.173714i
$310$		0			0
$311$	4776.99			0.870991			0.435495	−	0.900191i	$-0.356573\pi$
$311$	4776.99			0.870991			0.435495	+	0.900191i	$0.356573\pi$
$312$		0			0
$313$	−4555.75			−0.822704			−0.411352	−	0.911476i	$-0.634943\pi$
$313$	−4555.75			−0.822704			−0.411352	+	0.911476i	$0.634943\pi$
$314$		0			0
$315$	−1949.74	−	279.372i	−0.348747	−	0.0499708i
$316$		0			0
$317$			3955.67i			0.700860i	0.936589	+	0.350430i	$0.113965\pi$
$317$			3955.67i			0.700860i	−0.936589	+	0.350430i	$0.886035\pi$
$318$		0			0
$319$			1060.26i			0.186092i
$320$		0			0
$321$	−871.738	−	62.1370i	−0.151575	−	0.0108042i
$322$		0			0
$323$	3212.66			0.553427
$324$		0			0
$325$	−663.848			−0.113304
$326$		0			0
$327$	−795.047	−	56.6705i	−0.134453	−	0.00958375i
$328$		0			0
$329$			3914.27i			0.655928i
$330$		0			0
$331$			10762.9i			1.78726i	0.448802	+	0.893631i	$0.351851\pi$
$331$			10762.9i			1.78726i	−0.448802	+	0.893631i	$0.648149\pi$
$332$		0			0
$333$	−3300.56	−	472.927i	−0.543152	−	0.0778266i
$334$		0			0
$335$	6962.18			1.13548
$336$		0			0
$337$	−12223.9			−1.97590			−0.987950	−	0.154773i	$-0.950535\pi$
$337$	−12223.9			−1.97590			−0.987950	+	0.154773i	$0.950535\pi$
$338$		0			0
$339$	331.131	−	4645.53i	0.0530518	−	0.744279i
$340$		0			0
$341$			13522.5i			2.14746i
$342$		0			0
$343$			343.000i			0.0539949i
$344$		0			0
$345$	90.2115	−	1265.60i	0.0140778	−	0.197501i
$346$		0			0
$347$	688.805			0.106562			0.0532810	−	0.998580i	$-0.483032\pi$
$347$	688.805			0.106562			0.0532810	+	0.998580i	$0.483032\pi$
$348$		0			0
$349$	−2607.05			−0.399863			−0.199931	−	0.979810i	$-0.564072\pi$
$349$	−2607.05			−0.399863			−0.199931	+	0.979810i	$0.564072\pi$
$350$		0			0
$351$	5552.18	+	1203.60i	0.844312	+	0.183030i
$352$		0			0
$353$		−	6979.37i		−	1.05234i	−0.850381	−	0.526168i	$-0.823628\pi$
$353$		−	6979.37i		−	1.05234i	0.850381	−	0.526168i	$-0.176372\pi$
$354$		0			0
$355$		−	10971.0i		−	1.64023i
$356$		0			0
$357$	841.861	+	60.0073i	0.124807	+	0.00889615i
$358$		0			0
$359$	−9153.20			−1.34565			−0.672824	−	0.739803i	$-0.734919\pi$
$359$	−9153.20			−1.34565			−0.672824	+	0.739803i	$0.734919\pi$
$360$		0			0
$361$	−12310.3			−1.79477
$362$		0			0
$363$	−8074.57	−	575.551i	−1.16751	−	0.0832192i
$364$		0			0
$365$		−	12155.0i		−	1.74308i
$366$		0			0
$367$		−	4536.80i		−	0.645284i	−0.946521	−	0.322642i	$-0.895429\pi$
$367$		−	4536.80i		−	0.645284i	0.946521	−	0.322642i	$-0.104571\pi$
$368$		0			0
$369$	172.824	−	1206.14i	0.0243817	−	0.170160i
$370$		0			0
$371$	−2029.21			−0.283966
$372$		0			0
$373$	−5368.30			−0.745202			−0.372601	−	0.927992i	$-0.621534\pi$
$373$	−5368.30			−0.745202			−0.372601	+	0.927992i	$0.621534\pi$
$374$		0			0
$375$	−544.381	+	7637.29i	−0.0749646	+	1.05170i
$376$		0			0
$377$			798.799i			0.109125i
$378$		0			0
$379$		−	9684.95i		−	1.31262i	−0.754492	−	0.656309i	$-0.772117\pi$
$379$		−	9684.95i		−	1.31262i	0.754492	−	0.656309i	$-0.227883\pi$
$380$		0			0
$381$	525.783	−	7376.36i	0.0706999	−	0.991870i
$382$		0			0
$383$	4960.01			0.661735			0.330868	−	0.943677i	$-0.392659\pi$
$383$	4960.01			0.661735			0.330868	+	0.943677i	$0.392659\pi$
$384$		0			0
$385$	3920.95			0.519039
$386$		0			0
$387$	−880.695	+	6146.38i	−0.115680	+	0.807334i
$388$		0			0
$389$			145.505i			0.0189650i	0.999955	+	0.00948251i	$0.00301842\pi$
$389$			145.505i			0.0189650i	−0.999955	+	0.00948251i	$0.996982\pi$
$390$		0			0
$391$			543.688i			0.0703210i
$392$		0			0
$393$	13460.9	+	959.483i	1.72776	+	0.123154i
$394$		0			0
$395$	11369.9			1.44831
$396$		0			0
$397$	−3493.53			−0.441651			−0.220825	−	0.975313i	$-0.570875\pi$
$397$	−3493.53			−0.441651			−0.220825	+	0.975313i	$0.570875\pi$
$398$		0			0
$399$	−5023.23	−	358.053i	−0.630266	−	0.0449250i
$400$		0			0
$401$		−	599.088i		−	0.0746060i	−0.999304	−	0.0373030i	$-0.988123\pi$
$401$		−	599.088i		−	0.0746060i	0.999304	−	0.0373030i	$-0.0118767\pi$
$402$		0			0
$403$			10187.8i			1.25928i
$404$		0			0
$405$	2133.36	−	7291.54i	0.261747	−	0.894616i
$406$		0			0
$407$	6637.48			0.808373
$408$		0			0
$409$	5305.67			0.641439			0.320719	−	0.947174i	$-0.396075\pi$
$409$	5305.67			0.641439			0.320719	+	0.947174i	$0.396075\pi$
$410$		0			0
$411$	853.282	−	11970.9i	0.102407	−	1.43670i
$412$		0			0
$413$		−	2600.70i		−	0.309859i
$414$		0			0
$415$		−	2640.52i		−	0.312333i
$416$		0			0
$417$	−51.0631	+	716.379i	−0.00599657	+	0.0841277i
$418$		0			0
$419$	5439.40			0.634206			0.317103	−	0.948391i	$-0.397290\pi$
$419$	5439.40			0.634206			0.317103	+	0.948391i	$0.397290\pi$
$420$		0			0
$421$	46.1434			0.00534179			0.00267089	−	0.999996i	$-0.499150\pi$
$421$	46.1434			0.00534179			0.00267089	+	0.999996i	$0.499150\pi$
$422$		0			0
$423$	−14945.2	−	2141.46i	−1.71788	−	0.246149i
$424$		0			0
$425$		−	380.399i		−	0.0434166i
$426$		0			0
$427$		−	3631.80i		−	0.411605i
$428$		0			0
$429$	−11280.7	−	804.084i	−1.26956	−	0.0904931i
$430$		0			0
$431$	−4710.85			−0.526481			−0.263241	−	0.964730i	$-0.584791\pi$
$431$	−4710.85			−0.526481			−0.263241	+	0.964730i	$0.584791\pi$
$432$		0			0
$433$	15397.5			1.70890			0.854451	−	0.519531i	$-0.173893\pi$
$433$	15397.5			1.70890			0.854451	+	0.519531i	$0.173893\pi$
$434$		0			0
$435$	1065.51	+	75.9487i	0.117442	+	0.00837117i
$436$		0			0
$437$		−	3244.09i		−	0.355116i
$438$		0			0
$439$			14074.5i			1.53016i	0.643936	+	0.765080i	$0.277300\pi$
$439$			14074.5i			1.53016i	−0.643936	+	0.765080i	$0.722700\pi$
$440$		0			0
$441$	−1309.62	−	187.652i	−0.141413	−	0.0202626i
$442$		0			0
$443$	8176.03			0.876874			0.438437	−	0.898762i	$-0.355532\pi$
$443$	8176.03			0.876874			0.438437	+	0.898762i	$0.355532\pi$
$444$		0			0
$445$	−1981.27			−0.211059
$446$		0			0
$447$	51.8061	−	726.803i	0.00548175	−	0.0769052i
$448$		0			0
$449$			10401.4i			1.09326i	0.837374	+	0.546631i	$0.184090\pi$
$449$			10401.4i			1.09326i	−0.837374	+	0.546631i	$0.815910\pi$
$450$		0			0
$451$			2425.56i			0.253249i
$452$		0			0
$453$	−322.146	+	4519.48i	−0.0334122	+	0.468750i
$454$		0			0
$455$	2954.03			0.304368
$456$		0			0
$457$	−13676.5			−1.39991			−0.699957	−	0.714185i	$-0.746798\pi$
$457$	−13676.5			−1.39991			−0.699957	+	0.714185i	$0.746798\pi$
$458$		0			0
$459$	−689.690	+	3181.52i	−0.0701351	+	0.323531i
$460$		0			0
$461$			8754.98i			0.884512i	0.896889	+	0.442256i	$0.145822\pi$
$461$			8754.98i			0.884512i	−0.896889	+	0.442256i	$0.854178\pi$
$462$		0			0
$463$			1335.87i			0.134089i	0.997750	+	0.0670445i	$0.0213569\pi$
$463$			1335.87i			0.134089i	−0.997750	+	0.0670445i	$0.978643\pi$
$464$		0			0
$465$	13589.4	+	968.641i	1.35525	+	0.0966014i
$466$		0			0
$467$	−519.731			−0.0514996			−0.0257498	−	0.999668i	$-0.508197\pi$
$467$	−519.731			−0.0514996			−0.0257498	+	0.999668i	$0.508197\pi$
$468$		0			0
$469$	4676.45			0.460423
$470$		0			0
$471$	−6195.00	−	441.576i	−0.606052	−	0.0431990i
$472$		0			0
$473$		−	12360.5i		−	1.20155i
$474$		0			0
$475$			2269.77i			0.219251i
$476$		0			0
$477$	1110.16	−	7747.83i	0.106564	−	0.743708i
$478$		0			0
$479$	−10277.1			−0.980319			−0.490160	−	0.871633i	$-0.663062\pi$
$479$	−10277.1			−0.980319			−0.490160	+	0.871633i	$0.663062\pi$
$480$		0			0
$481$	5000.66			0.474035
$482$		0			0
$483$	60.5944	−	850.097i	0.00570837	−	0.0800844i
$484$		0			0
$485$		−	1222.13i		−	0.114421i
$486$		0			0
$487$		−	4494.88i		−	0.418239i	−0.977890	−	0.209120i	$-0.932940\pi$
$487$		−	4494.88i		−	0.418239i	0.977890	−	0.209120i	$-0.0670598\pi$
$488$		0			0
$489$	−349.050	+	4896.93i	−0.0322793	+	0.452857i
$490$		0			0
$491$	−5758.70			−0.529301			−0.264650	−	0.964344i	$-0.585257\pi$
$491$	−5758.70			−0.529301			−0.264650	+	0.964344i	$0.585257\pi$
$492$		0			0
$493$	−457.729			−0.0418156
$494$		0			0
$495$	−2145.11	+	14970.8i	−0.194779	+	1.35936i
$496$		0			0
$497$		−	7369.16i		−	0.665094i
$498$		0			0
$499$			18455.1i			1.65564i	0.560996	+	0.827818i	$0.310418\pi$
$499$			18455.1i			1.65564i	−0.560996	+	0.827818i	$0.689582\pi$
$500$		0			0
$501$	−16579.0	−	1181.74i	−1.47843	−	0.105382i
$502$		0			0
$503$	−18376.5			−1.62896			−0.814480	−	0.580192i	$-0.802977\pi$
$503$	−18376.5			−1.62896			−0.814480	+	0.580192i	$0.802977\pi$
$504$		0			0
$505$	10637.9			0.937389
$506$		0			0
$507$	2888.17	+	205.867i	0.252995	+	0.0180333i
$508$		0			0
$509$			3769.35i			0.328238i	0.986441	+	0.164119i	$0.0524782\pi$
$509$			3769.35i			0.328238i	−0.986441	+	0.164119i	$0.947522\pi$
$510$		0			0
$511$		−	8164.45i		−	0.706798i
$512$		0			0
$513$	4115.26	−	18983.5i	0.354177	−	1.63381i
$514$		0			0
$515$	−1897.22			−0.162333
$516$		0			0
$517$	30055.1			2.55672
$518$		0			0
$519$	−643.930	+	9033.89i	−0.0544613	+	0.764054i
$520$		0			0
$521$			12313.8i			1.03546i	0.855543	+	0.517731i	$0.173223\pi$
$521$			12313.8i			1.03546i	−0.855543	+	0.517731i	$0.826777\pi$
$522$		0			0
$523$			14859.7i			1.24239i	0.783655	+	0.621196i	$0.213353\pi$
$523$			14859.7i			1.24239i	−0.783655	+	0.621196i	$0.786647\pi$
$524$		0			0
$525$	−42.3957	+	594.782i	−0.00352438	+	0.0494446i
$526$		0			0
$527$	−5837.82			−0.482542
$528$		0			0
$529$	−11618.0			−0.954877
$530$		0			0
$531$	9929.84	+	1422.81i	0.811522	+	0.116280i
$532$		0			0
$533$			1827.41i			0.148507i
$534$		0			0
$535$		−	1752.80i		−	0.141645i
$536$		0			0
$537$	−7736.02	−	551.419i	−0.621665	−	0.0443119i
$538$		0			0
$539$	2633.67			0.210464
$540$		0			0
$541$	7032.83			0.558900			0.279450	−	0.960160i	$-0.409848\pi$
$541$	7032.83			0.558900			0.279450	+	0.960160i	$0.409848\pi$
$542$		0			0
$543$	−15717.7	−	1120.35i	−1.24219	−	0.0885428i
$544$		0			0
$545$		−	1598.60i		−	0.125645i
$546$		0			0
$547$		−	13364.0i		−	1.04461i	−0.852758	−	0.522307i	$-0.825072\pi$
$547$		−	13364.0i		−	1.04461i	0.852758	−	0.522307i	$-0.174928\pi$
$548$		0			0
$549$	13866.7	+	1986.92i	1.07799	+	0.154462i
$550$		0			0
$551$	2731.18			0.211166
$552$		0			0
$553$	7637.07			0.587272
$554$		0			0
$555$	475.456	−	6670.31i	0.0363639	−	0.510160i
$556$		0			0
$557$			8860.41i			0.674018i	0.941501	+	0.337009i	$0.109415\pi$
$557$			8860.41i			0.674018i	−0.941501	+	0.337009i	$0.890585\pi$
$558$		0			0
$559$		−	9312.35i		−	0.704598i
$560$		0			0
$561$	460.757	−	6464.10i	0.0346759	−	0.486479i
$562$		0			0
$563$	−15178.5			−1.13623			−0.568113	−	0.822950i	$-0.692326\pi$
$563$	−15178.5			−1.13623			−0.568113	+	0.822950i	$0.692326\pi$
$564$		0			0
$565$	9340.74			0.695518
$566$		0			0
$567$	1432.96	−	4897.68i	0.106136	−	0.362757i
$568$		0			0
$569$			1604.35i			0.118203i	0.998252	+	0.0591017i	$0.0188236\pi$
$569$			1604.35i			0.118203i	−0.998252	+	0.0591017i	$0.981176\pi$
$570$		0			0
$571$			15669.0i			1.14838i	0.818721	+	0.574191i	$0.194683\pi$
$571$			15669.0i			1.14838i	−0.818721	+	0.574191i	$0.805317\pi$
$572$		0			0
$573$	−14845.8	−	1058.20i	−1.08236	−	0.0771498i
$574$		0			0
$575$	−384.121			−0.0278590
$576$		0			0
$577$	−4500.34			−0.324700			−0.162350	−	0.986733i	$-0.551907\pi$
$577$	−4500.34			−0.324700			−0.162350	+	0.986733i	$0.551907\pi$
$578$		0			0
$579$	26621.9	+	1897.60i	1.91083	+	0.136203i
$580$		0			0
$581$		−	1773.62i		−	0.126647i
$582$		0			0
$583$			15581.0i			1.10686i
$584$		0			0
$585$	−1616.12	+	11278.9i	−0.114220	+	0.797140i
$586$		0			0
$587$	−5704.29			−0.401093			−0.200546	−	0.979684i	$-0.564272\pi$
$587$	−5704.29			−0.401093			−0.200546	+	0.979684i	$0.564272\pi$
$588$		0			0
$589$	34833.2			2.43680
$590$		0			0
$591$	819.505	−	11497.1i	0.0570388	−	0.800214i
$592$		0			0
$593$		−	20645.1i		−	1.42967i	−0.699295	−	0.714833i	$-0.746503\pi$
$593$		−	20645.1i		−	1.42967i	0.699295	−	0.714833i	$-0.253497\pi$
$594$		0			0
$595$			1692.72i			0.116630i
$596$		0			0
$597$	−1381.81	+	19385.8i	−0.0947299	+	1.32899i
$598$		0			0
$599$	6562.46			0.447637			0.223819	−	0.974631i	$-0.428148\pi$
$599$	6562.46			0.447637			0.223819	+	0.974631i	$0.428148\pi$
$600$		0			0
$601$	−4595.21			−0.311885			−0.155942	−	0.987766i	$-0.549841\pi$
$601$	−4595.21			−0.311885			−0.155942	+	0.987766i	$0.549841\pi$
$602$		0			0
$603$	−2558.44	+	17855.4i	−0.172782	+	1.20585i
$604$		0			0
$605$		−	16235.5i		−	1.09102i
$606$		0			0
$607$			22748.8i			1.52116i	0.649244	+	0.760580i	$0.275085\pi$
$607$			22748.8i			1.52116i	−0.649244	+	0.760580i	$0.724915\pi$
$608$		0			0
$609$	715.693	+	51.0142i	0.0476213	+	0.00339441i
$610$		0			0
$611$	22643.4			1.49927
$612$		0			0
$613$	4586.44			0.302194			0.151097	−	0.988519i	$-0.451719\pi$
$613$	4586.44			0.302194			0.151097	+	0.988519i	$0.451719\pi$
$614$		0			0
$615$	2437.56	+	173.748i	0.159824	+	0.0113922i
$616$		0			0
$617$			21776.7i			1.42090i	0.703748	+	0.710450i	$0.251509\pi$
$617$			21776.7i			1.42090i	−0.703748	+	0.710450i	$0.748491\pi$
$618$		0			0
$619$			1712.83i			0.111219i	0.998453	+	0.0556093i	$0.0177101\pi$
$619$			1712.83i			0.111219i	−0.998453	+	0.0556093i	$0.982290\pi$
$620$		0			0
$621$	3212.65	+	696.438i	0.207599	+	0.0450034i
$622$		0			0
$623$	−1330.80			−0.0855820
$624$		0			0
$625$	−13307.0			−0.851650
$626$		0			0
$627$	−2749.25	+	38570.1i	−0.175111	+	2.45669i
$628$		0			0
$629$			2865.49i			0.181645i
$630$		0			0
$631$		−	14179.2i		−	0.894556i	−0.894395	−	0.447278i	$-0.852393\pi$
$631$		−	14179.2i		−	0.894556i	0.894395	−	0.447278i	$-0.147607\pi$
$632$		0			0
$633$	−1281.83	+	17983.1i	−0.0804867	+	1.12917i
$634$		0			0
$635$	14831.6			0.926889
$636$		0			0
$637$	1984.20			0.123418
$638$		0			0
$639$	28136.5	+	4031.59i	1.74188	+	0.249589i
$640$		0			0
$641$			18660.8i			1.14985i	0.818205	+	0.574927i	$0.194969\pi$
$641$			18660.8i			1.14985i	−0.818205	+	0.574927i	$0.805031\pi$
$642$		0			0
$643$		−	19724.7i		−	1.20974i	−0.796322	−	0.604872i	$-0.793224\pi$
$643$		−	19724.7i		−	1.20974i	0.796322	−	0.604872i	$-0.206776\pi$
$644$		0			0
$645$	−12421.6	−	885.404i	−0.758295	−	0.0540508i
$646$		0			0
$647$	16930.3			1.02875			0.514373	−	0.857567i	$-0.328025\pi$
$647$	16930.3			1.02875			0.514373	+	0.857567i	$0.328025\pi$
$648$		0			0
$649$	−19969.0			−1.20779
$650$		0			0
$651$	9127.87	+	650.629i	0.549538	+	0.0391708i
$652$		0			0
$653$		−	24102.3i		−	1.44440i	−0.691683	−	0.722201i	$-0.743131\pi$
$653$		−	24102.3i		−	1.44440i	0.691683	−	0.722201i	$-0.256869\pi$
$654$		0			0
$655$			27065.7i			1.61457i
$656$		0			0
$657$	31173.1	+	4466.69i	1.85111	+	0.265239i
$658$		0			0
$659$	21973.3			1.29887			0.649437	−	0.760415i	$-0.275005\pi$
$659$	21973.3			1.29887			0.649437	+	0.760415i	$0.275005\pi$
$660$		0			0
$661$	18542.2			1.09109			0.545543	−	0.838083i	$-0.316324\pi$
$661$	18542.2			1.09109			0.545543	+	0.838083i	$0.316324\pi$
$662$		0			0
$663$	347.133	−	4870.04i	0.0203342	−	0.285274i
$664$		0			0
$665$		−	10100.2i		−	0.588975i
$666$		0			0
$667$			462.207i			0.0268317i
$668$		0			0
$669$	−52.5820	+	737.689i	−0.00303877	+	0.0426319i
$670$		0			0
$671$	−27886.2			−1.60438
$672$		0			0
$673$	−2534.71			−0.145179			−0.0725897	−	0.997362i	$-0.523126\pi$
$673$	−2534.71			−0.145179			−0.0725897	+	0.997362i	$0.523126\pi$
$674$		0			0
$675$	−2247.77	−	487.273i	−0.128173	−	0.0277854i
$676$		0			0
$677$		−	2515.84i		−	0.142824i	−0.997447	−	0.0714118i	$-0.977250\pi$
$677$		−	2515.84i		−	0.142824i	0.997447	−	0.0714118i	$-0.0227505\pi$
$678$		0			0
$679$		−	820.899i		−	0.0463965i
$680$		0			0
$681$	19331.1	+	1377.91i	1.08777	+	0.0775355i
$682$		0			0
$683$	16518.6			0.925427			0.462714	−	0.886508i	$-0.346876\pi$
$683$	16518.6			0.925427			0.462714	+	0.886508i	$0.346876\pi$
$684$		0			0
$685$	24069.9			1.34257
$686$		0			0
$687$	−6108.35	−	435.399i	−0.339226	−	0.0241798i
$688$		0			0
$689$			11738.7i			0.649069i
$690$		0			0
$691$		−	6000.93i		−	0.330371i	−0.986263	−	0.165185i	$-0.947178\pi$
$691$		−	6000.93i		−	0.330371i	0.986263	−	0.165185i	$-0.0528222\pi$
$692$		0			0
$693$	−1440.86	+	10055.8i	−0.0789807	+	0.551207i
$694$		0			0
$695$	−1440.42			−0.0786161
$696$		0			0
$697$	−1047.15			−0.0569060
$698$		0			0
$699$	1468.82	−	20606.5i	0.0794792	−	1.11504i
$700$		0			0
$701$		−	16428.0i		−	0.885129i	−0.896737	−	0.442565i	$-0.854069\pi$
$701$		−	16428.0i		−	0.885129i	0.896737	−	0.442565i	$-0.145931\pi$
$702$		0			0
$703$		−	17097.8i		−	0.917293i
$704$		0			0
$705$	2152.91	−	30203.8i	0.115012	−	1.61353i
$706$		0			0
$707$	7145.41			0.380100
$708$		0			0
$709$	28983.4			1.53525			0.767627	−	0.640897i	$-0.221438\pi$
$709$	28983.4			1.53525			0.767627	+	0.640897i	$0.221438\pi$
$710$		0			0
$711$	−4178.17	+	29159.5i	−0.220385	+	1.53807i
$712$		0			0
$713$			5894.94i			0.309631i
$714$		0			0
$715$		−	22682.1i		−	1.18638i
$716$		0			0
$717$	16962.9	+	1209.11i	0.883532	+	0.0629776i
$718$		0			0
$719$	−19080.7			−0.989696			−0.494848	−	0.868980i	$-0.664776\pi$
$719$	−19080.7			−0.989696			−0.494848	+	0.868980i	$0.664776\pi$
$720$		0			0
$721$	−1274.35			−0.0658241
$722$		0			0
$723$	2652.19	+	189.046i	0.136426	+	0.00972436i
$724$		0			0
$725$		−	323.390i		−	0.0165661i
$726$		0			0
$727$			11317.7i			0.577372i	0.957424	+	0.288686i	$0.0932184\pi$
$727$			11317.7i			0.577372i	−0.957424	+	0.288686i	$0.906782\pi$
$728$		0			0
$729$	17916.1	+	8150.74i	0.910231	+	0.414100i
$730$		0			0
$731$	5336.17			0.269994
$732$		0			0
$733$	−17072.2			−0.860267			−0.430133	−	0.902765i	$-0.641533\pi$
$733$	−17072.2			−0.860267			−0.430133	+	0.902765i	$0.641533\pi$
$734$		0			0
$735$	188.655	−	2646.70i	0.00946756	−	0.132823i
$736$		0			0
$737$		−	35907.4i		−	1.79466i
$738$		0			0
$739$			22709.1i			1.13041i	0.824952	+	0.565203i	$0.191202\pi$
$739$			22709.1i			1.13041i	−0.824952	+	0.565203i	$0.808798\pi$
$740$		0			0
$741$	−2071.28	+	29058.6i	−0.102686	+	1.44061i
$742$		0			0
$743$	−37840.2			−1.86840			−0.934201	−	0.356746i	$-0.883886\pi$
$743$	−37840.2			−1.86840			−0.934201	+	0.356746i	$0.883886\pi$
$744$		0			0
$745$	1461.38			0.0718668
$746$		0			0
$747$	6771.93	+	970.329i	0.331690	+	0.0475267i
$748$		0			0
$749$		−	1177.34i		−	0.0574355i
$750$		0			0
$751$		−	13193.6i		−	0.641068i	−0.947237	−	0.320534i	$-0.896138\pi$
$751$		−	13193.6i		−	0.641068i	0.947237	−	0.320534i	$-0.103862\pi$
$752$		0			0
$753$	−16785.3	−	1196.44i	−0.812335	−	0.0579028i
$754$		0			0
$755$	−9087.30			−0.438041
$756$		0			0
$757$	−34477.6			−1.65536			−0.827680	−	0.561200i	$-0.810340\pi$
$757$	−34477.6			−1.65536			−0.827680	+	0.561200i	$0.810340\pi$
$758$		0			0
$759$	−6527.34	−	465.265i	−0.312158	−	0.0222504i
$760$		0			0
$761$			7770.28i			0.370134i	0.982726	+	0.185067i	$0.0592503\pi$
$761$			7770.28i			0.370134i	−0.982726	+	0.185067i	$0.940750\pi$
$762$		0			0
$763$		−	1073.77i		−	0.0509475i
$764$		0			0
$765$	−6463.07	−	926.073i	−0.305455	−	0.0437676i
$766$		0			0
$767$	−15044.6			−0.708253
$768$		0			0
$769$	7038.08			0.330039			0.165019	−	0.986290i	$-0.447231\pi$
$769$	7038.08			0.330039			0.165019	+	0.986290i	$0.447231\pi$
$770$		0			0
$771$	2607.18	−	36576.9i	0.121784	−	1.70854i
$772$		0			0
$773$			7104.76i			0.330583i	0.986245	+	0.165291i	$0.0528565\pi$
$773$			7104.76i			0.330583i	−0.986245	+	0.165291i	$0.947144\pi$
$774$		0			0
$775$		−	4124.48i		−	0.191168i
$776$		0			0
$777$	319.360	−	4480.40i	0.0147452	−	0.206864i
$778$		0			0
$779$	6248.12			0.287371
$780$		0			0
$781$	−56583.0			−2.59244
$782$		0			0
$783$	−586.328	+	2704.71i	−0.0267607	+	0.123446i
$784$		0			0
$785$		−	12456.2i		−	0.566347i
$786$		0			0
$787$			22444.0i			1.01657i	0.861189	+	0.508285i	$0.169720\pi$
$787$			22444.0i			1.01657i	−0.861189	+	0.508285i	$0.830280\pi$
$788$		0			0
$789$	−24647.3	−	1756.84i	−1.11212	−	0.0792716i
$790$		0			0
$791$	6274.11			0.282025
$792$		0			0
$793$	−21009.4			−0.940815
$794$		0			0
$795$	15658.1	+	1116.10i	0.698534	+	0.0497911i
$796$		0			0
$797$			705.699i			0.0313640i	0.999877	+	0.0156820i	$0.00499195\pi$
$797$			705.699i			0.0313640i	−0.999877	+	0.0156820i	$0.995008\pi$
$798$		0			0
$799$			12975.2i			0.574504i
$800$		0			0
$801$	728.070	−	5081.21i	0.0321162	−	0.224139i
$802$		0			0
$803$	−62689.5			−2.75500
$804$		0			0
$805$	1709.29			0.0748378
$806$		0			0
$807$	56.5035	−	792.705i	0.00246471	−	0.0345781i
$808$		0			0
$809$			35229.4i			1.53102i	0.643422	+	0.765512i	$0.277514\pi$
$809$			35229.4i			1.53102i	−0.643422	+	0.765512i	$0.722486\pi$
$810$		0			0
$811$		−	22999.5i		−	0.995834i	−0.867225	−	0.497917i	$-0.834098\pi$
$811$		−	22999.5i		−	0.995834i	0.867225	−	0.497917i	$-0.165902\pi$
$812$		0			0
$813$	2735.85	−	38382.0i	0.118020	−	1.65574i
$814$		0			0
$815$	−9846.23			−0.423188
$816$		0			0
$817$	−31840.0			−1.36345
$818$		0			0
$819$	−1085.54	+	7575.98i	−0.0463147	+	0.323231i
$820$		0			0
$821$			29515.7i			1.25470i	0.778739	+	0.627348i	$0.215860\pi$
$821$			29515.7i			1.25470i	−0.778739	+	0.627348i	$0.784140\pi$
$822$		0			0
$823$			21664.3i			0.917581i	0.888544	+	0.458791i	$0.151717\pi$
$823$			21664.3i			0.917581i	−0.888544	+	0.458791i	$0.848283\pi$
$824$		0			0
$825$	4566.95	+	325.529i	0.192728	+	0.0137375i
$826$		0			0
$827$	11507.0			0.483842			0.241921	−	0.970296i	$-0.422222\pi$
$827$	11507.0			0.483842			0.241921	+	0.970296i	$0.422222\pi$
$828$		0			0
$829$	22226.1			0.931175			0.465587	−	0.885002i	$-0.345843\pi$
$829$	22226.1			0.931175			0.465587	+	0.885002i	$0.345843\pi$
$830$		0			0
$831$	16217.1	+	1155.95i	0.676975	+	0.0482544i
$832$		0			0
$833$			1136.99i			0.0472922i
$834$		0			0
$835$		−	33335.2i		−	1.38157i
$836$		0			0
$837$	−7477.97	+	34495.6i	−0.308813	+	1.42454i
$838$		0			0
$839$	−20904.8			−0.860206			−0.430103	−	0.902780i	$-0.641523\pi$
$839$	−20904.8			−0.860206			−0.430103	+	0.902780i	$0.641523\pi$
$840$		0			0
$841$	23999.9			0.984045
$842$		0			0
$843$	2340.13	−	32830.4i	0.0956091	−	1.34133i
$844$		0			0
$845$			5807.23i			0.236420i
$846$		0			0
$847$		−	10905.3i		−	0.442396i
$848$		0			0
$849$	−2928.51	+	41085.0i	−0.118382	+	1.66082i
$850$		0			0
$851$	2893.52			0.116555
$852$		0			0
$853$	8866.60			0.355905			0.177952	−	0.984039i	$-0.443053\pi$
$853$	8866.60			0.355905			0.177952	+	0.984039i	$0.443053\pi$
$854$		0			0
$855$	38564.0	+	5525.71i	1.54253	+	0.221024i
$856$		0			0
$857$		−	5857.25i		−	0.233465i	−0.993163	−	0.116733i	$-0.962758\pi$
$857$		−	5857.25i		−	0.233465i	0.993163	−	0.116733i	$-0.0372421\pi$
$858$		0			0
$859$			32576.7i			1.29395i	0.762511	+	0.646975i	$0.223966\pi$
$859$			32576.7i			1.29395i	−0.762511	+	0.646975i	$0.776034\pi$
$860$		0			0
$861$	1637.29	+	116.705i	0.0648068	+	0.00461939i
$862$		0			0
$863$	−2024.38			−0.0798500			−0.0399250	−	0.999203i	$-0.512712\pi$
$863$	−2024.38			−0.0798500			−0.0399250	+	0.999203i	$0.512712\pi$
$864$		0			0
$865$	−18164.4			−0.713998
$866$		0			0
$867$	−22673.5	−	1616.15i	−0.888156	−	0.0633072i
$868$		0			0
$869$		−	58640.1i		−	2.28910i
$870$		0			0
$871$		−	27052.5i		−	1.05240i
$872$		0			0
$873$	3134.31	+	449.105i	0.121512	+	0.0174111i
$874$		0			0
$875$	−10314.7			−0.398514
$876$		0			0
$877$	10978.9			0.422726			0.211363	−	0.977408i	$-0.432210\pi$
$877$	10978.9			0.422726			0.211363	+	0.977408i	$0.432210\pi$
$878$		0			0
$879$	3403.66	−	47750.9i	0.130606	−	1.83231i
$880$		0			0
$881$		−	27916.7i		−	1.06758i	−0.845617	−	0.533790i	$-0.820767\pi$
$881$		−	27916.7i		−	1.06758i	0.845617	−	0.533790i	$-0.179233\pi$
$882$		0			0
$883$			23688.7i			0.902819i	0.892317	+	0.451410i	$0.149079\pi$
$883$			23688.7i			0.902819i	−0.892317	+	0.451410i	$0.850921\pi$
$884$		0			0
$885$	−1430.42	+	20067.8i	−0.0543312	+	0.762229i
$886$		0			0
$887$	−2921.23			−0.110581			−0.0552905	−	0.998470i	$-0.517608\pi$
$887$	−2921.23			−0.110581			−0.0552905	+	0.998470i	$0.517608\pi$
$888$		0			0
$889$	9962.28			0.375843
$890$		0			0
$891$	−37606.1	−	11002.8i	−1.41397	−	0.413701i
$892$		0			0
$893$		−	77420.5i		−	2.90121i
$894$		0			0
$895$		−	15554.8i		−	0.580937i
$896$		0			0
$897$	−4917.69	−	350.530i	−0.183051	−	0.0130478i
$898$		0			0
$899$	−4962.92			−0.184119
$900$		0			0
$901$	−6726.52			−0.248716
$902$		0			0
$903$	−8343.50	−	594.720i	−0.307480	−	0.0219170i
$904$		0			0
$905$		−	31603.5i		−	1.16081i
$906$		0			0
$907$		−	8272.50i		−	0.302849i	−0.988469	−	0.151424i	$-0.951614\pi$
$907$		−	8272.50i		−	0.302849i	0.988469	−	0.151424i	$-0.0483860\pi$
$908$		0			0
$909$	−3909.18	+	27282.2i	−0.142640	+	0.995484i
$910$		0			0
$911$	12401.0			0.451004			0.225502	−	0.974243i	$-0.427598\pi$
$911$	12401.0			0.451004			0.225502	+	0.974243i	$0.427598\pi$
$912$		0			0
$913$	−13618.5			−0.493653
$914$		0			0
$915$	−1997.55	+	28024.2i	−0.0721714	+	1.01251i
$916$		0			0
$917$			18179.8i			0.654690i
$918$		0			0
$919$			37708.2i			1.35351i	0.736207	+	0.676756i	$0.236615\pi$
$919$			37708.2i			1.35351i	−0.736207	+	0.676756i	$0.763385\pi$
$920$		0			0
$921$	−2721.88	+	38186.0i	−0.0973820	+	1.36620i
$922$		0			0
$923$	−42629.5			−1.52022
$924$		0			0
$925$	−2024.49			−0.0719621
$926$		0			0
$927$	697.183	−	4865.65i	0.0247017	−	0.172394i
$928$		0			0
$929$		−	44156.2i		−	1.55944i	−0.626129	−	0.779719i	$-0.715362\pi$
$929$		−	44156.2i		−	1.55944i	0.626129	−	0.779719i	$-0.284638\pi$
$930$		0			0
$931$		−	6784.22i		−	0.238822i
$932$		0			0
$933$	−24759.1	−	1764.82i	−0.868786	−	0.0619266i
$934$		0			0
$935$	12997.3			0.454608
$936$		0			0
$937$	9919.81			0.345855			0.172927	−	0.984935i	$-0.444677\pi$
$937$	9919.81			0.345855			0.172927	+	0.984935i	$0.444677\pi$
$938$		0			0
$939$	23612.5	+	1683.08i	0.820622	+	0.0584935i
$940$		0			0
$941$		−	28232.7i		−	0.978067i	−0.872265	−	0.489033i	$-0.837350\pi$
$941$		−	28232.7i		−	0.978067i	0.872265	−	0.489033i	$-0.162650\pi$
$942$		0			0
$943$			1057.39i			0.0365147i
$944$		0			0
$945$	10002.3	+	2168.30i	0.344311	+	0.0746399i
$946$		0			0
$947$	32622.4			1.11941			0.559707	−	0.828691i	$-0.310914\pi$
$947$	32622.4			1.11941			0.559707	+	0.828691i	$0.310914\pi$
$948$		0			0
$949$	−47230.1			−1.61555
$950$		0			0
$951$	1461.39	−	20502.2i	0.0498304	−	0.699086i
$952$		0			0
$953$			8187.98i			0.278315i	0.990270	+	0.139158i	$0.0444395\pi$
$953$			8187.98i			0.278315i	−0.990270	+	0.139158i	$0.955560\pi$
$954$		0			0
$955$		−	29850.3i		−	1.01145i
$956$		0			0
$957$	391.705	−	5495.34i	0.0132309	−	0.185621i
$958$		0			0
$959$	16167.6			0.544399
$960$		0			0
$961$	−33505.6			−1.12469
$962$		0			0
$963$	4495.27	+	644.112i	0.150424	+	0.0215537i
$964$		0			0
$965$			53528.6i			1.78564i
$966$		0			0
$967$		−	19861.9i		−	0.660513i	−0.943891	−	0.330256i	$-0.892865\pi$
$967$		−	19861.9i		−	0.660513i	0.943891	−	0.330256i	$-0.107135\pi$
$968$		0			0
$969$	−16651.2	−	1186.89i	−0.552027	−	0.0393481i
$970$		0			0
$971$	−35197.5			−1.16328			−0.581638	−	0.813448i	$-0.697588\pi$
$971$	−35197.5			−1.16328			−0.581638	+	0.813448i	$0.697588\pi$
$972$		0			0
$973$	−967.519			−0.0318779
$974$		0			0
$975$	3440.73	+	245.253i	0.113017	+	0.00805578i
$976$		0			0
$977$		−	20640.9i		−	0.675906i	−0.941163	−	0.337953i	$-0.890266\pi$
$977$		−	20640.9i		−	0.675906i	0.941163	−	0.337953i	$-0.109734\pi$
$978$		0			0
$979$			10218.4i			0.333586i
$980$		0			0
$981$	4099.80	+	587.447i	0.133432	+	0.0191190i
$982$		0			0
$983$	−1251.84			−0.0406181			−0.0203091	−	0.999794i	$-0.506465\pi$
$983$	−1251.84			−0.0406181			−0.0203091	+	0.999794i	$0.506465\pi$
$984$		0			0
$985$	23117.1			0.747789
$986$		0			0
$987$	1446.09	−	20287.7i	0.0466359	−	0.654269i
$988$		0			0
$989$		−	5388.38i		−	0.173246i
$990$		0			0
$991$			730.192i			0.0234060i	0.999932	+	0.0117030i	$0.00372526\pi$
$991$			730.192i			0.0234060i	−0.999932	+	0.0117030i	$0.996275\pi$
$992$		0			0
$993$	3976.27	−	55784.3i	0.127073	−	1.78274i
$994$		0			0
$995$	−38979.0			−1.24193
$996$		0			0
$997$	14920.1			0.473947			0.236974	−	0.971516i	$-0.423845\pi$
$997$	14920.1			0.473947			0.236974	+	0.971516i	$0.423845\pi$
$998$		0			0
$999$	16932.1	+	3670.55i	0.536245	+	0.116247i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.h.a.575.1	✓	36
3.2	odd	2	inner	672.4.h.a.575.35	yes	36
4.3	odd	2	inner	672.4.h.a.575.36	yes	36
12.11	even	2	inner	672.4.h.a.575.2	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.h.a.575.1	✓	36	1.1	even	1	trivial
672.4.h.a.575.2	yes	36	12.11	even	2	inner
672.4.h.a.575.35	yes	36	3.2	odd	2	inner
672.4.h.a.575.36	yes	36	4.3	odd	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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.18300 - 0.369441i) q^{3} -10.4214i q^{5} -7.00000i q^{7} +(26.7270 + 3.82963i) q^{9} -53.7484 q^{11} -40.4940 q^{13} +(-3.85011 + 54.0143i) q^{15} -23.2039i q^{17} +138.453i q^{19} +(-2.58609 + 36.2810i) q^{21} -23.4309 q^{23} +16.3938 q^{25} +(-137.111 - 29.7230i) q^{27} -19.7264i q^{29} -251.588i q^{31} +(278.578 + 19.8569i) q^{33} -72.9500 q^{35} -123.492 q^{37} +(209.880 + 14.9601i) q^{39} -45.1280i q^{41} +229.969i q^{43} +(39.9102 - 278.534i) q^{45} -559.181 q^{47} -49.0000 q^{49} +(-8.57248 + 120.266i) q^{51} -289.888i q^{53} +560.136i q^{55} +(51.1504 - 717.604i) q^{57} +371.528 q^{59} +518.829 q^{61} +(26.8074 - 187.089i) q^{63} +422.005i q^{65} +668.064i q^{67} +(121.442 + 8.65635i) q^{69} +1052.74 q^{71} +1166.35 q^{73} +(-84.9689 - 6.05653i) q^{75} +376.239i q^{77} +1091.01i q^{79} +(699.668 + 204.709i) q^{81} +253.374 q^{83} -241.818 q^{85} +(-7.28774 + 102.242i) q^{87} -190.115i q^{89} +283.458i q^{91} +(-92.9470 + 1303.98i) q^{93} +1442.88 q^{95} +117.271 q^{97} +(-1436.54 - 205.837i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 132 q^{9} - 120 q^{13} + 28 q^{21} - 756 q^{25} + 40 q^{33} + 672 q^{37} + 304 q^{45} - 1764 q^{49} + 1624 q^{57} + 2472 q^{61} - 1224 q^{69} - 2376 q^{73} + 468 q^{81} + 5160 q^{85} - 648 q^{93}+ \cdots - 4488 q^{97}+O(q^{100}) \) 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

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