LMFDB - Embedded newform 672.3.l.a.433.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,3,Mod(433,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.433"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$18.3106737650$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.18
Character	$\chi$	$=$	672.433
Dual form			672.3.l.a.433.31

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +9.09019 q^{5} +(-3.83254 + 5.85761i) q^{7} +3.00000 q^{9} -14.8648i q^{11} +4.25079 q^{13} +15.7447 q^{15} +4.31083i q^{17} +26.9453 q^{19} +(-6.63815 + 10.1457i) q^{21} -9.33910 q^{23} +57.6315 q^{25} +5.19615 q^{27} -34.2521i q^{29} -4.01283i q^{31} -25.7467i q^{33} +(-34.8385 + 53.2468i) q^{35} +64.5914i q^{37} +7.36258 q^{39} -36.7189i q^{41} +74.7186i q^{43} +27.2706 q^{45} +52.5115i q^{47} +(-19.6233 - 44.8991i) q^{49} +7.46658i q^{51} -19.3795i q^{53} -135.124i q^{55} +46.6707 q^{57} -3.71348 q^{59} -79.8128 q^{61} +(-11.4976 + 17.5728i) q^{63} +38.6404 q^{65} -66.3899i q^{67} -16.1758 q^{69} -0.994759 q^{71} -8.78191i q^{73} +99.8206 q^{75} +(87.0725 + 56.9701i) q^{77} +31.1944 q^{79} +9.00000 q^{81} +44.7600 q^{83} +39.1863i q^{85} -59.3264i q^{87} -11.6296i q^{89} +(-16.2913 + 24.8995i) q^{91} -6.95043i q^{93} +244.938 q^{95} +91.4731i q^{97} -44.5945i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 96 q^{9} - 64 q^{23} + 160 q^{25} - 16 q^{49} + 96 q^{57} + 640 q^{71} + 64 q^{79} + 288 q^{81} + 768 q^{95}+O(q^{100}) $ 32 * q + 96 * q^9 - 64 * q^23 + 160 * q^25 - 16 * q^49 + 96 * q^57 + 640 * q^71 + 64 * q^79 + 288 * q^81 + 768 * q^95

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
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$	9.09019			1.81804			0.909019	−	0.416756i	$-0.136833\pi$
$5$	9.09019			1.81804			0.909019	+	0.416756i	$0.136833\pi$
$6$		0			0
$7$	−3.83254	+	5.85761i	−0.547506	+	0.836802i
$7$	−3.83254	+	5.85761i	−0.547506	+	0.836802i
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$		−	14.8648i		−	1.35135i	−0.737200	−	0.675675i	$-0.763852\pi$
$11$		−	14.8648i		−	1.35135i	0.737200	−	0.675675i	$-0.236148\pi$
$12$		0			0
$13$	4.25079			0.326984			0.163492	−	0.986545i	$-0.447724\pi$
$13$	4.25079			0.326984			0.163492	+	0.986545i	$0.447724\pi$
$14$		0			0
$15$	15.7447			1.04964
$16$		0			0
$17$			4.31083i			0.253578i	0.991930	+	0.126789i	$0.0404672\pi$
$17$			4.31083i			0.253578i	−0.991930	+	0.126789i	$0.959533\pi$
$18$		0			0
$19$	26.9453			1.41818			0.709088	−	0.705120i	$-0.249107\pi$
$19$	26.9453			1.41818			0.709088	+	0.705120i	$0.249107\pi$
$20$		0			0
$21$	−6.63815	+	10.1457i	−0.316103	+	0.483128i
$22$		0			0
$23$	−9.33910			−0.406048			−0.203024	−	0.979174i	$-0.565077\pi$
$23$	−9.33910			−0.406048			−0.203024	+	0.979174i	$0.565077\pi$
$24$		0			0
$25$	57.6315			2.30526
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$		−	34.2521i		−	1.18111i	−0.806999	−	0.590553i	$-0.798910\pi$
$29$		−	34.2521i		−	1.18111i	0.806999	−	0.590553i	$-0.201090\pi$
$30$		0			0
$31$		−	4.01283i		−	0.129446i	−0.997903	−	0.0647231i	$-0.979384\pi$
$31$		−	4.01283i		−	0.129446i	0.997903	−	0.0647231i	$-0.0206164\pi$
$32$		0			0
$33$		−	25.7467i		−	0.780202i
$34$		0			0
$35$	−34.8385	+	53.2468i	−0.995386	+	1.52134i
$36$		0			0
$37$			64.5914i			1.74571i	0.487976	+	0.872857i	$0.337735\pi$
$37$			64.5914i			1.74571i	−0.487976	+	0.872857i	$0.662265\pi$
$38$		0			0
$39$	7.36258			0.188784
$40$		0			0
$41$		−	36.7189i		−	0.895583i	−0.894138	−	0.447792i	$-0.852211\pi$
$41$		−	36.7189i		−	0.895583i	0.894138	−	0.447792i	$-0.147789\pi$
$42$		0			0
$43$			74.7186i			1.73764i	0.495127	+	0.868821i	$0.335122\pi$
$43$			74.7186i			1.73764i	−0.495127	+	0.868821i	$0.664878\pi$
$44$		0			0
$45$	27.2706			0.606012
$46$		0			0
$47$			52.5115i			1.11727i	0.829415	+	0.558633i	$0.188674\pi$
$47$			52.5115i			1.11727i	−0.829415	+	0.558633i	$0.811326\pi$
$48$		0			0
$49$	−19.6233	−	44.8991i	−0.400475	−	0.916308i
$50$		0			0
$51$			7.46658i			0.146404i
$52$		0			0
$53$		−	19.3795i		−	0.365650i	−0.983145	−	0.182825i	$-0.941476\pi$
$53$		−	19.3795i		−	0.365650i	0.983145	−	0.182825i	$-0.0585242\pi$
$54$		0			0
$55$		−	135.124i		−	2.45680i
$56$		0			0
$57$	46.6707			0.818784
$58$		0			0
$59$	−3.71348			−0.0629404			−0.0314702	−	0.999505i	$-0.510019\pi$
$59$	−3.71348			−0.0629404			−0.0314702	+	0.999505i	$0.510019\pi$
$60$		0			0
$61$	−79.8128			−1.30841			−0.654203	−	0.756319i	$-0.726996\pi$
$61$	−79.8128			−1.30841			−0.654203	+	0.756319i	$0.726996\pi$
$62$		0			0
$63$	−11.4976	+	17.5728i	−0.182502	+	0.278934i
$64$		0			0
$65$	38.6404			0.594468
$66$		0			0
$67$		−	66.3899i		−	0.990894i	−0.868638	−	0.495447i	$-0.835004\pi$
$67$		−	66.3899i		−	0.990894i	0.868638	−	0.495447i	$-0.164996\pi$
$68$		0			0
$69$	−16.1758			−0.234432
$70$		0			0
$71$	−0.994759			−0.0140107			−0.00700535	−	0.999975i	$-0.502230\pi$
$71$	−0.994759			−0.0140107			−0.00700535	+	0.999975i	$0.502230\pi$
$72$		0			0
$73$		−	8.78191i		−	0.120300i	−0.998189	−	0.0601501i	$-0.980842\pi$
$73$		−	8.78191i		−	0.120300i	0.998189	−	0.0601501i	$-0.0191579\pi$
$74$		0			0
$75$	99.8206			1.33094
$76$		0			0
$77$	87.0725	+	56.9701i	1.13081	+	0.739871i
$78$		0			0
$79$	31.1944			0.394866			0.197433	−	0.980316i	$-0.436740\pi$
$79$	31.1944			0.394866			0.197433	+	0.980316i	$0.436740\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	44.7600			0.539277			0.269639	−	0.962962i	$-0.413096\pi$
$83$	44.7600			0.539277			0.269639	+	0.962962i	$0.413096\pi$
$84$		0			0
$85$			39.1863i			0.461015i
$86$		0			0
$87$		−	59.3264i		−	0.681912i
$88$		0			0
$89$		−	11.6296i		−	0.130669i	−0.997863	−	0.0653347i	$-0.979188\pi$
$89$		−	11.6296i		−	0.130669i	0.997863	−	0.0653347i	$-0.0208115\pi$
$90$		0			0
$91$	−16.2913	+	24.8995i	−0.179025	+	0.273621i
$92$		0			0
$93$		−	6.95043i		−	0.0747358i
$94$		0			0
$95$	244.938			2.57830
$96$		0			0
$97$			91.4731i			0.943021i	0.881860	+	0.471511i	$0.156291\pi$
$97$			91.4731i			0.943021i	−0.881860	+	0.471511i	$0.843709\pi$
$98$		0			0
$99$		−	44.5945i		−	0.450450i
$100$		0			0
$101$	35.1617			0.348136			0.174068	−	0.984734i	$-0.444309\pi$
$101$	35.1617			0.348136			0.174068	+	0.984734i	$0.444309\pi$
$102$		0			0
$103$		−	45.5116i		−	0.441860i	−0.975290	−	0.220930i	$-0.929091\pi$
$103$		−	45.5116i		−	0.441860i	0.975290	−	0.220930i	$-0.0709093\pi$
$104$		0			0
$105$	−60.3420	+	92.2262i	−0.574686	+	0.878344i
$106$		0			0
$107$			64.2360i			0.600336i	0.953886	+	0.300168i	$0.0970428\pi$
$107$			64.2360i			0.600336i	−0.953886	+	0.300168i	$0.902957\pi$
$108$		0			0
$109$		−	39.3165i		−	0.360702i	−0.983602	−	0.180351i	$-0.942277\pi$
$109$		−	39.3165i		−	0.360702i	0.983602	−	0.180351i	$-0.0577234\pi$
$110$		0			0
$111$			111.876i			1.00789i
$112$		0			0
$113$	−72.5108			−0.641688			−0.320844	−	0.947132i	$-0.603967\pi$
$113$	−72.5108			−0.641688			−0.320844	+	0.947132i	$0.603967\pi$
$114$		0			0
$115$	−84.8941			−0.738210
$116$		0			0
$117$	12.7524			0.108995
$118$		0			0
$119$	−25.2512	−	16.5214i	−0.212195	−	0.138836i
$120$		0			0
$121$	−99.9635			−0.826145
$122$		0			0
$123$		−	63.5990i		−	0.517065i
$124$		0			0
$125$	296.626			2.37301
$126$		0			0
$127$	78.5274			0.618326			0.309163	−	0.951009i	$-0.399951\pi$
$127$	78.5274			0.618326			0.309163	+	0.951009i	$0.399951\pi$
$128$		0			0
$129$			129.416i			1.00323i
$130$		0			0
$131$	−13.2698			−0.101296			−0.0506479	−	0.998717i	$-0.516129\pi$
$131$	−13.2698			−0.101296			−0.0506479	+	0.998717i	$0.516129\pi$
$132$		0			0
$133$	−103.269	+	157.835i	−0.776460	+	1.18673i
$134$		0			0
$135$	47.2340			0.349881
$136$		0			0
$137$	101.465			0.740623			0.370311	−	0.928908i	$-0.379251\pi$
$137$	101.465			0.740623			0.370311	+	0.928908i	$0.379251\pi$
$138$		0			0
$139$	−163.338			−1.17509			−0.587546	−	0.809191i	$-0.699906\pi$
$139$	−163.338			−1.17509			−0.587546	+	0.809191i	$0.699906\pi$
$140$		0			0
$141$			90.9526i			0.645054i
$142$		0			0
$143$		−	63.1873i		−	0.441869i
$144$		0			0
$145$		−	311.358i		−	2.14730i
$146$		0			0
$147$	−33.9885	−	77.7675i	−0.231214	−	0.529030i
$148$		0			0
$149$		−	95.5068i		−	0.640985i	−0.947251	−	0.320492i	$-0.896152\pi$
$149$		−	95.5068i		−	0.640985i	0.947251	−	0.320492i	$-0.103848\pi$
$150$		0			0
$151$	−194.643			−1.28902			−0.644512	−	0.764594i	$-0.722940\pi$
$151$	−194.643			−1.28902			−0.644512	+	0.764594i	$0.722940\pi$
$152$		0			0
$153$			12.9325i			0.0845261i
$154$		0			0
$155$		−	36.4774i		−	0.235338i
$156$		0			0
$157$	−247.603			−1.57709			−0.788545	−	0.614977i	$-0.789165\pi$
$157$	−247.603			−1.57709			−0.788545	+	0.614977i	$0.789165\pi$
$158$		0			0
$159$		−	33.5662i		−	0.211108i
$160$		0			0
$161$	35.7925	−	54.7048i	0.222313	−	0.339781i
$162$		0			0
$163$			42.2893i			0.259444i	0.991550	+	0.129722i	$0.0414085\pi$
$163$			42.2893i			0.259444i	−0.991550	+	0.129722i	$0.958592\pi$
$164$		0			0
$165$		−	234.042i		−	1.41844i
$166$		0			0
$167$		−	218.408i		−	1.30783i	−0.756568	−	0.653915i	$-0.773126\pi$
$167$		−	218.408i		−	1.30783i	0.756568	−	0.653915i	$-0.226874\pi$
$168$		0			0
$169$	−150.931			−0.893082
$170$		0			0
$171$	80.8360			0.472725
$172$		0			0
$173$	−270.029			−1.56086			−0.780431	−	0.625242i	$-0.785000\pi$
$173$	−270.029			−1.56086			−0.780431	+	0.625242i	$0.785000\pi$
$174$		0			0
$175$	−220.875	+	337.583i	−1.26214	+	1.92905i
$176$		0			0
$177$	−6.43194			−0.0363387
$178$		0			0
$179$		−	150.910i		−	0.843071i	−0.906812	−	0.421535i	$-0.861491\pi$
$179$		−	150.910i		−	0.843071i	0.906812	−	0.421535i	$-0.138509\pi$
$180$		0			0
$181$	62.7742			0.346819			0.173409	−	0.984850i	$-0.444522\pi$
$181$	62.7742			0.346819			0.173409	+	0.984850i	$0.444522\pi$
$182$		0			0
$183$	−138.240			−0.755409
$184$		0			0
$185$			587.148i			3.17377i
$186$		0			0
$187$	64.0798			0.342673
$188$		0			0
$189$	−19.9145	+	30.4371i	−0.105368	+	0.161043i
$190$		0			0
$191$	−152.844			−0.800231			−0.400115	−	0.916465i	$-0.631030\pi$
$191$	−152.844			−0.800231			−0.400115	+	0.916465i	$0.631030\pi$
$192$		0			0
$193$	−196.995			−1.02070			−0.510350	−	0.859967i	$-0.670484\pi$
$193$	−196.995			−1.02070			−0.510350	+	0.859967i	$0.670484\pi$
$194$		0			0
$195$	66.9272			0.343216
$196$		0			0
$197$		−	58.4171i		−	0.296533i	−0.988947	−	0.148267i	$-0.952631\pi$
$197$		−	58.4171i		−	0.296533i	0.988947	−	0.148267i	$-0.0473694\pi$
$198$		0			0
$199$		−	31.9014i		−	0.160309i	−0.996782	−	0.0801543i	$-0.974459\pi$
$199$		−	31.9014i		−	0.160309i	0.996782	−	0.0801543i	$-0.0255413\pi$
$200$		0			0
$201$		−	114.991i		−	0.572093i
$202$		0			0
$203$	200.636	+	131.273i	0.988352	+	0.646663i
$204$		0			0
$205$		−	333.782i		−	1.62820i
$206$		0			0
$207$	−28.0173			−0.135349
$208$		0			0
$209$		−	400.538i		−	1.91645i
$210$		0			0
$211$		−	77.3310i		−	0.366498i	−0.983067	−	0.183249i	$-0.941339\pi$
$211$		−	77.3310i		−	0.366498i	0.983067	−	0.183249i	$-0.0586614\pi$
$212$		0			0
$213$	−1.72297			−0.00808908
$214$		0			0
$215$			679.206i			3.15910i
$216$		0			0
$217$	23.5056	+	15.3793i	0.108321	+	0.0708725i
$218$		0			0
$219$		−	15.2107i		−	0.0694553i
$220$		0			0
$221$			18.3244i			0.0829160i
$222$		0			0
$223$		−	145.310i		−	0.651614i	−0.945436	−	0.325807i	$-0.894364\pi$
$223$		−	145.310i		−	0.651614i	0.945436	−	0.325807i	$-0.105636\pi$
$224$		0			0
$225$	172.894			0.768420
$226$		0			0
$227$	−357.567			−1.57519			−0.787593	−	0.616196i	$-0.788673\pi$
$227$	−357.567			−1.57519			−0.787593	+	0.616196i	$0.788673\pi$
$228$		0			0
$229$	127.298			0.555886			0.277943	−	0.960598i	$-0.410347\pi$
$229$	127.298			0.555886			0.277943	+	0.960598i	$0.410347\pi$
$230$		0			0
$231$	150.814	+	98.6751i	0.652874	+	0.427165i
$232$		0			0
$233$	80.4881			0.345442			0.172721	−	0.984971i	$-0.444744\pi$
$233$	80.4881			0.345442			0.172721	+	0.984971i	$0.444744\pi$
$234$		0			0
$235$			477.340i			2.03123i
$236$		0			0
$237$	54.0303			0.227976
$238$		0			0
$239$	−343.515			−1.43730			−0.718650	−	0.695372i	$-0.755240\pi$
$239$	−343.515			−1.43730			−0.718650	+	0.695372i	$0.755240\pi$
$240$		0			0
$241$			201.772i			0.837229i	0.908164	+	0.418614i	$0.137484\pi$
$241$			201.772i			0.837229i	−0.908164	+	0.418614i	$0.862516\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$	−178.379	−	408.141i	−0.728079	−	1.66588i
$246$		0			0
$247$	114.539			0.463720
$248$		0			0
$249$	77.5266			0.311352
$250$		0			0
$251$	388.278			1.54692			0.773461	−	0.633844i	$-0.218524\pi$
$251$	388.278			1.54692			0.773461	+	0.633844i	$0.218524\pi$
$252$		0			0
$253$			138.824i			0.548712i
$254$		0			0
$255$			67.8726i			0.266167i
$256$		0			0
$257$			290.061i			1.12864i	0.825555	+	0.564322i	$0.190862\pi$
$257$			290.061i			1.12864i	−0.825555	+	0.564322i	$0.809138\pi$
$258$		0			0
$259$	−378.352	−	247.549i	−1.46082	−	0.955788i
$260$		0			0
$261$		−	102.756i		−	0.393702i
$262$		0			0
$263$	−289.474			−1.10066			−0.550331	−	0.834947i	$-0.685498\pi$
$263$	−289.474			−1.10066			−0.550331	+	0.834947i	$0.685498\pi$
$264$		0			0
$265$		−	176.163i		−	0.664765i
$266$		0			0
$267$		−	20.1430i		−	0.0754421i
$268$		0			0
$269$	−130.290			−0.484348			−0.242174	−	0.970233i	$-0.577861\pi$
$269$	−130.290			−0.484348			−0.242174	+	0.970233i	$0.577861\pi$
$270$		0			0
$271$		−	317.150i		−	1.17030i	−0.810927	−	0.585148i	$-0.801036\pi$
$271$		−	317.150i		−	1.17030i	0.810927	−	0.585148i	$-0.198964\pi$
$272$		0			0
$273$	−28.2174	+	43.1271i	−0.103360	+	0.157975i
$274$		0			0
$275$		−	856.683i		−	3.11521i
$276$		0			0
$277$			526.555i			1.90092i	0.310843	+	0.950461i	$0.399389\pi$
$277$			526.555i			1.90092i	−0.310843	+	0.950461i	$0.600611\pi$
$278$		0			0
$279$		−	12.0385i		−	0.0431487i
$280$		0			0
$281$	165.241			0.588046			0.294023	−	0.955798i	$-0.405006\pi$
$281$	165.241			0.588046			0.294023	+	0.955798i	$0.405006\pi$
$282$		0			0
$283$	−170.636			−0.602953			−0.301477	−	0.953474i	$-0.597480\pi$
$283$	−170.636			−0.602953			−0.301477	+	0.953474i	$0.597480\pi$
$284$		0			0
$285$	424.245			1.48858
$286$		0			0
$287$	215.085	+	140.727i	0.749426	+	0.490337i
$288$		0			0
$289$	270.417			0.935698
$290$		0			0
$291$			158.436i			0.544454i
$292$		0			0
$293$	16.1118			0.0549892			0.0274946	−	0.999622i	$-0.491247\pi$
$293$	16.1118			0.0549892			0.0274946	+	0.999622i	$0.491247\pi$
$294$		0			0
$295$	−33.7562			−0.114428
$296$		0			0
$297$		−	77.2400i		−	0.260067i
$298$		0			0
$299$	−39.6985			−0.132771
$300$		0			0
$301$	−437.673	−	286.362i	−1.45406	−	0.951369i
$302$		0			0
$303$	60.9019			0.200996
$304$		0			0
$305$	−725.513			−2.37873
$306$		0			0
$307$	5.96095			0.0194168			0.00970839	−	0.999953i	$-0.496910\pi$
$307$	5.96095			0.0194168			0.00970839	+	0.999953i	$0.496910\pi$
$308$		0			0
$309$		−	78.8284i		−	0.255108i
$310$		0			0
$311$			261.056i			0.839408i	0.907661	+	0.419704i	$0.137866\pi$
$311$			261.056i			0.839408i	−0.907661	+	0.419704i	$0.862134\pi$
$312$		0			0
$313$		−	285.715i		−	0.912826i	−0.889768	−	0.456413i	$-0.849134\pi$
$313$		−	285.715i		−	0.912826i	0.889768	−	0.456413i	$-0.150866\pi$
$314$		0			0
$315$	−104.515	+	159.740i	−0.331795	+	0.507112i
$316$		0			0
$317$			593.885i			1.87345i	0.350060	+	0.936727i	$0.386161\pi$
$317$			593.885i			1.87345i	−0.350060	+	0.936727i	$0.613839\pi$
$318$		0			0
$319$	−509.152			−1.59609
$320$		0			0
$321$			111.260i			0.346604i
$322$		0			0
$323$			116.157i			0.359619i
$324$		0			0
$325$	244.979			0.753782
$326$		0			0
$327$		−	68.0982i		−	0.208252i
$328$		0			0
$329$	−307.592	−	201.253i	−0.934931	−	0.611710i
$330$		0			0
$331$		−	25.5772i		−	0.0772725i	−0.999253	−	0.0386362i	$-0.987699\pi$
$331$		−	25.5772i		−	0.0772725i	0.999253	−	0.0386362i	$-0.0123014\pi$
$332$		0			0
$333$			193.774i			0.581905i
$334$		0			0
$335$		−	603.496i		−	1.80148i
$336$		0			0
$337$	−565.903			−1.67924			−0.839619	−	0.543176i	$-0.817222\pi$
$337$	−565.903			−1.67924			−0.839619	+	0.543176i	$0.817222\pi$
$338$		0			0
$339$	−125.592			−0.370479
$340$		0			0
$341$	−59.6501			−0.174927
$342$		0			0
$343$	338.208	+	57.1319i	0.986030	+	0.166565i
$344$		0			0
$345$	−147.041			−0.426206
$346$		0			0
$347$			85.1612i			0.245421i	0.992442	+	0.122711i	$0.0391587\pi$
$347$			85.1612i			0.245421i	−0.992442	+	0.122711i	$0.960841\pi$
$348$		0			0
$349$	−275.035			−0.788066			−0.394033	−	0.919096i	$-0.628920\pi$
$349$	−275.035			−0.788066			−0.394033	+	0.919096i	$0.628920\pi$
$350$		0			0
$351$	22.0877			0.0629280
$352$		0			0
$353$			145.204i			0.411342i	0.978621	+	0.205671i	$0.0659376\pi$
$353$			145.204i			0.411342i	−0.978621	+	0.205671i	$0.934062\pi$
$354$		0			0
$355$	−9.04255			−0.0254720
$356$		0			0
$357$	−43.7363	−	28.6160i	−0.122511	−	0.0801568i
$358$		0			0
$359$	40.1294			0.111781			0.0558906	−	0.998437i	$-0.482200\pi$
$359$	40.1294			0.111781			0.0558906	+	0.998437i	$0.482200\pi$
$360$		0			0
$361$	365.052			1.01122
$362$		0			0
$363$	−173.142			−0.476975
$364$		0			0
$365$		−	79.8292i		−	0.218710i
$366$		0			0
$367$		−	373.965i		−	1.01898i	−0.860477	−	0.509489i	$-0.829834\pi$
$367$		−	373.965i		−	1.01898i	0.860477	−	0.509489i	$-0.170166\pi$
$368$		0			0
$369$		−	110.157i		−	0.298528i
$370$		0			0
$371$	113.517	+	74.2725i	0.305977	+	0.200195i
$372$		0			0
$373$		−	222.942i		−	0.597699i	−0.954300	−	0.298849i	$-0.903397\pi$
$373$		−	222.942i		−	0.597699i	0.954300	−	0.298849i	$-0.0966028\pi$
$374$		0			0
$375$	513.771			1.37006
$376$		0			0
$377$		−	145.598i		−	0.386203i
$378$		0			0
$379$			445.711i			1.17602i	0.808854	+	0.588010i	$0.200088\pi$
$379$			445.711i			1.17602i	−0.808854	+	0.588010i	$0.799912\pi$
$380$		0			0
$381$	136.013			0.356991
$382$		0			0
$383$		−	463.682i		−	1.21066i	−0.795975	−	0.605329i	$-0.793042\pi$
$383$		−	463.682i		−	1.21066i	0.795975	−	0.605329i	$-0.206958\pi$
$384$		0			0
$385$	791.505	+	517.869i	2.05586	+	1.34511i
$386$		0			0
$387$			224.156i			0.579214i
$388$		0			0
$389$			170.836i			0.439167i	0.975594	+	0.219584i	$0.0704699\pi$
$389$			170.836i			0.439167i	−0.975594	+	0.219584i	$0.929530\pi$
$390$		0			0
$391$		−	40.2593i		−	0.102965i
$392$		0			0
$393$	−22.9839			−0.0584832
$394$		0			0
$395$	283.563			0.717881
$396$		0			0
$397$	537.521			1.35396			0.676979	−	0.736003i	$-0.263289\pi$
$397$	537.521			1.35396			0.676979	+	0.736003i	$0.263289\pi$
$398$		0			0
$399$	−178.867	+	273.379i	−0.448289	+	0.685160i
$400$		0			0
$401$	490.979			1.22439			0.612193	−	0.790709i	$-0.290288\pi$
$401$	490.979			1.22439			0.612193	+	0.790709i	$0.290288\pi$
$402$		0			0
$403$		−	17.0577i		−	0.0423268i
$404$		0			0
$405$	81.8117			0.202004
$406$		0			0
$407$	960.141			2.35907
$408$		0			0
$409$			426.048i			1.04168i	0.853654	+	0.520841i	$0.174382\pi$
$409$			426.048i			1.04168i	−0.853654	+	0.520841i	$0.825618\pi$
$410$		0			0
$411$	175.743			0.427599
$412$		0			0
$413$	14.2321	−	21.7521i	0.0344602	−	0.0526686i
$414$		0			0
$415$	406.877			0.980426
$416$		0			0
$417$	−282.909			−0.678440
$418$		0			0
$419$	725.389			1.73124			0.865619	−	0.500704i	$-0.166925\pi$
$419$	725.389			1.73124			0.865619	+	0.500704i	$0.166925\pi$
$420$		0			0
$421$			30.0999i			0.0714963i	0.999361	+	0.0357481i	$0.0113814\pi$
$421$			30.0999i			0.0714963i	−0.999361	+	0.0357481i	$0.988619\pi$
$422$		0			0
$423$			157.535i			0.372422i
$424$		0			0
$425$			248.440i			0.584564i
$426$		0			0
$427$	305.886	−	467.513i	0.716360	−	1.09488i
$428$		0			0
$429$		−	109.444i		−	0.255113i
$430$		0			0
$431$	265.508			0.616027			0.308013	−	0.951382i	$-0.400336\pi$
$431$	265.508			0.616027			0.308013	+	0.951382i	$0.400336\pi$
$432$		0			0
$433$		−	716.150i		−	1.65393i	−0.562256	−	0.826963i	$-0.690067\pi$
$433$		−	716.150i		−	1.65393i	0.562256	−	0.826963i	$-0.309933\pi$
$434$		0			0
$435$		−	539.288i		−	1.23974i
$436$		0			0
$437$	−251.645			−0.575847
$438$		0			0
$439$			294.401i			0.670617i	0.942108	+	0.335308i	$0.108840\pi$
$439$			294.401i			0.670617i	−0.942108	+	0.335308i	$0.891160\pi$
$440$		0			0
$441$	−58.8698	−	134.697i	−0.133492	−	0.305436i
$442$		0			0
$443$			488.734i			1.10324i	0.834096	+	0.551619i	$0.185990\pi$
$443$			488.734i			1.10324i	−0.834096	+	0.551619i	$0.814010\pi$
$444$		0			0
$445$		−	105.715i		−	0.237562i
$446$		0			0
$447$		−	165.423i		−	0.370073i
$448$		0			0
$449$	430.945			0.959788			0.479894	−	0.877326i	$-0.340675\pi$
$449$	430.945			0.959788			0.479894	+	0.877326i	$0.340675\pi$
$450$		0			0
$451$	−545.821			−1.21025
$452$		0			0
$453$	−337.131			−0.744219
$454$		0			0
$455$	−148.091	+	226.341i	−0.325475	+	0.497452i
$456$		0			0
$457$	−699.398			−1.53041			−0.765206	−	0.643785i	$-0.777363\pi$
$457$	−699.398			−1.53041			−0.765206	+	0.643785i	$0.777363\pi$
$458$		0			0
$459$			22.3997i			0.0488012i
$460$		0			0
$461$	387.880			0.841389			0.420695	−	0.907202i	$-0.361786\pi$
$461$	387.880			0.841389			0.420695	+	0.907202i	$0.361786\pi$
$462$		0			0
$463$	293.086			0.633015			0.316508	−	0.948590i	$-0.397490\pi$
$463$	293.086			0.633015			0.316508	+	0.948590i	$0.397490\pi$
$464$		0			0
$465$		−	63.1807i		−	0.135872i
$466$		0			0
$467$	−588.136			−1.25939			−0.629696	−	0.776842i	$-0.716820\pi$
$467$	−588.136			−1.25939			−0.629696	+	0.776842i	$0.716820\pi$
$468$		0			0
$469$	388.886	+	254.442i	0.829182	+	0.542520i
$470$		0			0
$471$	−428.861			−0.910534
$472$		0			0
$473$	1110.68			2.34816
$474$		0			0
$475$	1552.90			3.26926
$476$		0			0
$477$		−	58.1384i		−	0.121883i
$478$		0			0
$479$			389.912i			0.814012i	0.913425	+	0.407006i	$0.133427\pi$
$479$			389.912i			0.814012i	−0.913425	+	0.407006i	$0.866573\pi$
$480$		0			0
$481$			274.564i			0.570820i
$482$		0			0
$483$	61.9944	−	94.7515i	0.128353	−	0.196173i
$484$		0			0
$485$			831.507i			1.71445i
$486$		0			0
$487$	307.232			0.630867			0.315433	−	0.948948i	$-0.397850\pi$
$487$	307.232			0.630867			0.315433	+	0.948948i	$0.397850\pi$
$488$		0			0
$489$			73.2473i			0.149790i
$490$		0			0
$491$			751.084i			1.52970i	0.644207	+	0.764851i	$0.277188\pi$
$491$			751.084i			1.52970i	−0.644207	+	0.764851i	$0.722812\pi$
$492$		0			0
$493$	147.655			0.299503
$494$		0			0
$495$		−	405.373i		−	0.818934i
$496$		0			0
$497$	3.81246	−	5.82692i	0.00767094	−	0.0117242i
$498$		0			0
$499$			252.238i			0.505487i	0.967533	+	0.252744i	$0.0813329\pi$
$499$			252.238i			0.505487i	−0.967533	+	0.252744i	$0.918667\pi$
$500$		0			0
$501$		−	378.293i		−	0.755076i
$502$		0			0
$503$		−	840.895i		−	1.67176i	−0.548912	−	0.835880i	$-0.684958\pi$
$503$		−	840.895i		−	1.67176i	0.548912	−	0.835880i	$-0.315042\pi$
$504$		0			0
$505$	319.626			0.632924
$506$		0			0
$507$	−261.420			−0.515621
$508$		0			0
$509$	532.050			1.04528			0.522642	−	0.852552i	$-0.324946\pi$
$509$	532.050			1.04528			0.522642	+	0.852552i	$0.324946\pi$
$510$		0			0
$511$	51.4410	+	33.6570i	0.100667	+	0.0658650i
$512$		0			0
$513$	140.012			0.272928
$514$		0			0
$515$		−	413.709i		−	0.803318i
$516$		0			0
$517$	780.576			1.50982
$518$		0			0
$519$	−467.704			−0.901164
$520$		0			0
$521$		−	673.016i		−	1.29178i	−0.763432	−	0.645889i	$-0.776487\pi$
$521$		−	673.016i		−	1.29178i	0.763432	−	0.645889i	$-0.223513\pi$
$522$		0			0
$523$	−131.940			−0.252276			−0.126138	−	0.992013i	$-0.540258\pi$
$523$	−131.940			−0.252276			−0.126138	+	0.992013i	$0.540258\pi$
$524$		0			0
$525$	−382.567	+	584.711i	−0.728698	+	1.11373i
$526$		0			0
$527$	17.2986			0.0328247
$528$		0			0
$529$	−441.781			−0.835125
$530$		0			0
$531$	−11.1404			−0.0209801
$532$		0			0
$533$		−	156.084i		−	0.292841i
$534$		0			0
$535$			583.917i			1.09143i
$536$		0			0
$537$		−	261.383i		−	0.486747i
$538$		0			0
$539$	−667.418	+	291.697i	−1.23825	+	0.541182i
$540$		0			0
$541$		−	413.321i		−	0.763995i	−0.924163	−	0.381997i	$-0.875236\pi$
$541$		−	413.321i		−	0.763995i	0.924163	−	0.381997i	$-0.124764\pi$
$542$		0			0
$543$	108.728			0.200236
$544$		0			0
$545$		−	357.395i		−	0.655770i
$546$		0			0
$547$		−	869.058i		−	1.58877i	−0.607414	−	0.794385i	$-0.707793\pi$
$547$		−	869.058i		−	1.58877i	0.607414	−	0.794385i	$-0.292207\pi$
$548$		0			0
$549$	−239.438			−0.436136
$550$		0			0
$551$		−	922.935i		−	1.67502i
$552$		0			0
$553$	−119.554	+	182.725i	−0.216191	+	0.330425i
$554$		0			0
$555$			1016.97i			1.83238i
$556$		0			0
$557$		−	238.943i		−	0.428982i	−0.976726	−	0.214491i	$-0.931191\pi$
$557$		−	238.943i		−	0.428982i	0.976726	−	0.214491i	$-0.0688092\pi$
$558$		0			0
$559$			317.613i			0.568180i
$560$		0			0
$561$	110.990			0.197842
$562$		0			0
$563$	−349.331			−0.620481			−0.310240	−	0.950658i	$-0.600410\pi$
$563$	−349.331			−0.620481			−0.310240	+	0.950658i	$0.600410\pi$
$564$		0			0
$565$	−659.136			−1.16661
$566$		0			0
$567$	−34.4929	+	52.7185i	−0.0608340	+	0.0929780i
$568$		0			0
$569$	−905.015			−1.59054			−0.795268	−	0.606258i	$-0.792670\pi$
$569$	−905.015			−1.59054			−0.795268	+	0.606258i	$0.792670\pi$
$570$		0			0
$571$			469.382i			0.822035i	0.911627	+	0.411017i	$0.134826\pi$
$571$			469.382i			0.822035i	−0.911627	+	0.411017i	$0.865174\pi$
$572$		0			0
$573$	−264.734			−0.462013
$574$		0			0
$575$	−538.226			−0.936045
$576$		0			0
$577$			607.599i			1.05303i	0.850165	+	0.526516i	$0.176502\pi$
$577$			607.599i			1.05303i	−0.850165	+	0.526516i	$0.823498\pi$
$578$		0			0
$579$	−341.205			−0.589301
$580$		0			0
$581$	−171.544	+	262.187i	−0.295257	+	0.451268i
$582$		0			0
$583$	−288.073			−0.494121
$584$		0			0
$585$	115.921			0.198156
$586$		0			0
$587$	−374.034			−0.637196			−0.318598	−	0.947890i	$-0.603212\pi$
$587$	−374.034			−0.637196			−0.318598	+	0.947890i	$0.603212\pi$
$588$		0			0
$589$		−	108.127i		−	0.183577i
$590$		0			0
$591$		−	101.181i		−	0.171204i
$592$		0			0
$593$		−	1159.14i		−	1.95471i	−0.211614	−	0.977353i	$-0.567872\pi$
$593$		−	1159.14i		−	1.95471i	0.211614	−	0.977353i	$-0.432128\pi$
$594$		0			0
$595$	−229.538	−	150.183i	−0.385778	−	0.252408i
$596$		0			0
$597$		−	55.2548i		−	0.0925542i
$598$		0			0
$599$	−154.662			−0.258200			−0.129100	−	0.991632i	$-0.541209\pi$
$599$	−154.662			−0.258200			−0.129100	+	0.991632i	$0.541209\pi$
$600$		0			0
$601$		−	924.074i		−	1.53756i	−0.639513	−	0.768780i	$-0.720864\pi$
$601$		−	924.074i		−	1.53756i	0.639513	−	0.768780i	$-0.279136\pi$
$602$		0			0
$603$		−	199.170i		−	0.330298i
$604$		0			0
$605$	−908.687			−1.50196
$606$		0			0
$607$			61.7356i			0.101706i	0.998706	+	0.0508531i	$0.0161940\pi$
$607$			61.7356i			0.101706i	−0.998706	+	0.0508531i	$0.983806\pi$
$608$		0			0
$609$	347.511	+	227.371i	0.570626	+	0.373351i
$610$		0			0
$611$			223.215i			0.365328i
$612$		0			0
$613$			661.600i			1.07928i	0.841895	+	0.539641i	$0.181440\pi$
$613$			661.600i			1.07928i	−0.841895	+	0.539641i	$0.818560\pi$
$614$		0			0
$615$		−	578.127i		−	0.940044i
$616$		0			0
$617$	354.372			0.574347			0.287174	−	0.957879i	$-0.407284\pi$
$617$	354.372			0.574347			0.287174	+	0.957879i	$0.407284\pi$
$618$		0			0
$619$	636.341			1.02801			0.514007	−	0.857786i	$-0.328161\pi$
$619$	636.341			1.02801			0.514007	+	0.857786i	$0.328161\pi$
$620$		0			0
$621$	−48.5274			−0.0781439
$622$		0			0
$623$	68.1216	+	44.5708i	0.109344	+	0.0715423i
$624$		0			0
$625$	1255.60			2.00896
$626$		0			0
$627$		−	693.753i		−	1.10646i
$628$		0			0
$629$	−278.443			−0.442675
$630$		0			0
$631$	199.202			0.315693			0.157847	−	0.987464i	$-0.449545\pi$
$631$	199.202			0.315693			0.157847	+	0.987464i	$0.449545\pi$
$632$		0			0
$633$		−	133.941i		−	0.211598i
$634$		0			0
$635$	713.829			1.12414
$636$		0			0
$637$	−83.4144	−	190.856i	−0.130949	−	0.299618i
$638$		0			0
$639$	−2.98428			−0.00467023
$640$		0			0
$641$	19.2310			0.0300016			0.0150008	−	0.999887i	$-0.495225\pi$
$641$	19.2310			0.0300016			0.0150008	+	0.999887i	$0.495225\pi$
$642$		0			0
$643$	−783.597			−1.21866			−0.609329	−	0.792918i	$-0.708561\pi$
$643$	−783.597			−1.21866			−0.609329	+	0.792918i	$0.708561\pi$
$644$		0			0
$645$			1176.42i			1.82391i
$646$		0			0
$647$			321.191i			0.496431i	0.968705	+	0.248216i	$0.0798442\pi$
$647$			321.191i			0.496431i	−0.968705	+	0.248216i	$0.920156\pi$
$648$		0			0
$649$			55.2003i			0.0850544i
$650$		0			0
$651$	40.7129	+	26.6378i	0.0625390	+	0.0409183i
$652$		0			0
$653$			567.084i			0.868428i	0.900810	+	0.434214i	$0.142974\pi$
$653$			567.084i			0.868428i	−0.900810	+	0.434214i	$0.857026\pi$
$654$		0			0
$655$	−120.625			−0.184160
$656$		0			0
$657$		−	26.3457i		−	0.0401001i
$658$		0			0
$659$		−	76.3245i		−	0.115819i	−0.998322	−	0.0579094i	$-0.981557\pi$
$659$		−	76.3245i		−	0.115819i	0.998322	−	0.0579094i	$-0.0184434\pi$
$660$		0			0
$661$	−422.507			−0.639193			−0.319597	−	0.947554i	$-0.603547\pi$
$661$	−422.507			−0.639193			−0.319597	+	0.947554i	$0.603547\pi$
$662$		0			0
$663$			31.7388i			0.0478716i
$664$		0			0
$665$	−938.735	+	1434.75i	−1.41163	+	2.15752i
$666$		0			0
$667$			319.884i			0.479586i
$668$		0			0
$669$		−	251.684i		−	0.376209i
$670$		0			0
$671$			1186.41i			1.76811i
$672$		0			0
$673$	523.984			0.778579			0.389289	−	0.921115i	$-0.372721\pi$
$673$	523.984			0.778579			0.389289	+	0.921115i	$0.372721\pi$
$674$		0			0
$675$	299.462			0.443647
$676$		0			0
$677$	523.084			0.772650			0.386325	−	0.922363i	$-0.373744\pi$
$677$	523.084			0.772650			0.386325	+	0.922363i	$0.373744\pi$
$678$		0			0
$679$	−535.814	−	350.574i	−0.789122	−	0.516310i
$680$		0			0
$681$	−619.324			−0.909434
$682$		0			0
$683$		−	243.110i		−	0.355945i	−0.984036	−	0.177972i	$-0.943046\pi$
$683$		−	243.110i		−	0.355945i	0.984036	−	0.177972i	$-0.0569537\pi$
$684$		0			0
$685$	922.339			1.34648
$686$		0			0
$687$	220.486			0.320941
$688$		0			0
$689$		−	82.3779i		−	0.119562i
$690$		0			0
$691$	1129.66			1.63482			0.817409	−	0.576057i	$-0.195410\pi$
$691$	1129.66			1.63482			0.817409	+	0.576057i	$0.195410\pi$
$692$		0			0
$693$	261.218	+	170.910i	0.376937	+	0.246624i
$694$		0			0
$695$	−1484.77			−2.13636
$696$		0			0
$697$	158.289			0.227100
$698$		0			0
$699$	139.409			0.199441
$700$		0			0
$701$		−	1291.80i		−	1.84280i	−0.388616	−	0.921400i	$-0.627047\pi$
$701$		−	1291.80i		−	1.84280i	0.388616	−	0.921400i	$-0.372953\pi$
$702$		0			0
$703$			1740.44i			2.47573i
$704$		0			0
$705$			826.776i			1.17273i
$706$		0			0
$707$	−134.759	+	205.964i	−0.190606	+	0.291321i
$708$		0			0
$709$		−	748.254i		−	1.05537i	−0.849442	−	0.527683i	$-0.823061\pi$
$709$		−	748.254i		−	1.05537i	0.849442	−	0.527683i	$-0.176939\pi$
$710$		0			0
$711$	93.5832			0.131622
$712$		0			0
$713$			37.4762i			0.0525613i
$714$		0			0
$715$		−	574.384i		−	0.803334i
$716$		0			0
$717$	−594.985			−0.829826
$718$		0			0
$719$			612.811i			0.852310i	0.904650	+	0.426155i	$0.140132\pi$
$719$			612.811i			0.852310i	−0.904650	+	0.426155i	$0.859868\pi$
$720$		0			0
$721$	266.589	+	174.425i	0.369750	+	0.241921i
$722$		0			0
$723$			349.480i			0.483374i
$724$		0			0
$725$		−	1974.00i		−	2.72276i
$726$		0			0
$727$		−	138.017i		−	0.189845i	−0.995485	−	0.0949223i	$-0.969740\pi$
$727$		−	138.017i		−	0.189845i	0.995485	−	0.0949223i	$-0.0302603\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$	−322.099			−0.440628
$732$		0			0
$733$	644.555			0.879338			0.439669	−	0.898160i	$-0.355096\pi$
$733$	644.555			0.879338			0.439669	+	0.898160i	$0.355096\pi$
$734$		0			0
$735$	−308.962	−	706.921i	−0.420356	−	0.961797i
$736$		0			0
$737$	−986.875			−1.33904
$738$		0			0
$739$		−	712.278i		−	0.963841i	−0.876215	−	0.481920i	$-0.839939\pi$
$739$		−	712.278i		−	0.963841i	0.876215	−	0.481920i	$-0.160061\pi$
$740$		0			0
$741$	198.387			0.267729
$742$		0			0
$743$	−10.1719			−0.0136903			−0.00684515	−	0.999977i	$-0.502179\pi$
$743$	−10.1719			−0.0136903			−0.00684515	+	0.999977i	$0.502179\pi$
$744$		0			0
$745$		−	868.174i		−	1.16533i
$746$		0			0
$747$	134.280			0.179759
$748$		0			0
$749$	−376.270	−	246.187i	−0.502363	−	0.328688i
$750$		0			0
$751$	1079.30			1.43715			0.718573	−	0.695451i	$-0.244796\pi$
$751$	1079.30			1.43715			0.718573	+	0.695451i	$0.244796\pi$
$752$		0			0
$753$	672.516			0.893116
$754$		0			0
$755$	−1769.34			−2.34349
$756$		0			0
$757$		−	1200.34i		−	1.58566i	−0.609445	−	0.792828i	$-0.708608\pi$
$757$		−	1200.34i		−	1.58566i	0.609445	−	0.792828i	$-0.291392\pi$
$758$		0			0
$759$			240.451i			0.316799i
$760$		0			0
$761$			965.128i			1.26824i	0.773236	+	0.634118i	$0.218637\pi$
$761$			965.128i			1.26824i	−0.773236	+	0.634118i	$0.781363\pi$
$762$		0			0
$763$	230.301	+	150.682i	0.301836	+	0.197486i
$764$		0			0
$765$			117.559i			0.153672i
$766$		0			0
$767$	−15.7852			−0.0205805
$768$		0			0
$769$			999.555i			1.29981i	0.760015	+	0.649906i	$0.225192\pi$
$769$			999.555i			1.29981i	−0.760015	+	0.649906i	$0.774808\pi$
$770$		0			0
$771$			502.401i			0.651623i
$772$		0			0
$773$	1048.92			1.35694			0.678471	−	0.734628i	$-0.262643\pi$
$773$	1048.92			1.35694			0.678471	+	0.734628i	$0.262643\pi$
$774$		0			0
$775$		−	231.265i		−	0.298407i
$776$		0			0
$777$	−655.324	−	428.768i	−0.843403	−	0.551825i
$778$		0			0
$779$		−	989.404i		−	1.27009i
$780$		0			0
$781$			14.7869i			0.0189333i
$782$		0			0
$783$		−	177.979i		−	0.227304i
$784$		0			0
$785$	−2250.76			−2.86721
$786$		0			0
$787$	1269.73			1.61338			0.806690	−	0.590975i	$-0.201257\pi$
$787$	1269.73			1.61338			0.806690	+	0.590975i	$0.201257\pi$
$788$		0			0
$789$	−501.384			−0.635467
$790$		0			0
$791$	277.900	−	424.740i	0.351328	−	0.536966i
$792$		0			0
$793$	−339.267			−0.427828
$794$		0			0
$795$		−	305.123i		−	0.383802i
$796$		0			0
$797$	792.144			0.993908			0.496954	−	0.867777i	$-0.334452\pi$
$797$	792.144			0.993908			0.496954	+	0.867777i	$0.334452\pi$
$798$		0			0
$799$	−226.368			−0.283315
$800$		0			0
$801$		−	34.8888i		−	0.0435565i
$802$		0			0
$803$	−130.542			−0.162568
$804$		0			0
$805$	325.360	−	497.277i	0.404174	−	0.617735i
$806$		0			0
$807$	−225.668			−0.279639
$808$		0			0
$809$	1282.81			1.58568			0.792838	−	0.609433i	$-0.208603\pi$
$809$	1282.81			1.58568			0.792838	+	0.609433i	$0.208603\pi$
$810$		0			0
$811$	272.683			0.336231			0.168115	−	0.985767i	$-0.446232\pi$
$811$	272.683			0.336231			0.168115	+	0.985767i	$0.446232\pi$
$812$		0			0
$813$		−	549.320i		−	0.675671i
$814$		0			0
$815$			384.418i			0.471679i
$816$		0			0
$817$			2013.32i			2.46428i
$818$		0			0
$819$	−48.8739	+	74.6984i	−0.0596751	+	0.0912069i
$820$		0			0
$821$			1350.30i			1.64470i	0.568985	+	0.822348i	$0.307336\pi$
$821$			1350.30i			1.64470i	−0.568985	+	0.822348i	$0.692664\pi$
$822$		0			0
$823$	−126.866			−0.154150			−0.0770752	−	0.997025i	$-0.524558\pi$
$823$	−126.866			−0.154150			−0.0770752	+	0.997025i	$0.524558\pi$
$824$		0			0
$825$		−	1483.82i		−	1.79857i
$826$		0			0
$827$			138.892i			0.167946i	0.996468	+	0.0839731i	$0.0267610\pi$
$827$			138.892i			0.167946i	−0.996468	+	0.0839731i	$0.973239\pi$
$828$		0			0
$829$	−951.755			−1.14808			−0.574038	−	0.818829i	$-0.694624\pi$
$829$	−951.755			−1.14808			−0.574038	+	0.818829i	$0.694624\pi$
$830$		0			0
$831$			912.021i			1.09750i
$832$		0			0
$833$	193.552	−	84.5926i	0.232356	−	0.101552i
$834$		0			0
$835$		−	1985.36i		−	2.37768i
$836$		0			0
$837$		−	20.8513i		−	0.0249119i
$838$		0			0
$839$			724.563i			0.863603i	0.901969	+	0.431802i	$0.142122\pi$
$839$			724.563i			0.863603i	−0.901969	+	0.431802i	$0.857878\pi$
$840$		0			0
$841$	−332.206			−0.395013
$842$		0			0
$843$	286.206			0.339509
$844$		0			0
$845$	−1371.99			−1.62366
$846$		0			0
$847$	383.114	−	585.548i	0.452319	−	0.691320i
$848$		0			0
$849$	−295.550			−0.348115
$850$		0			0
$851$		−	603.225i		−	0.708843i
$852$		0			0
$853$	−1219.96			−1.43020			−0.715100	−	0.699022i	$-0.753619\pi$
$853$	−1219.96			−1.43020			−0.715100	+	0.699022i	$0.753619\pi$
$854$		0			0
$855$	734.815			0.859432
$856$		0			0
$857$			169.761i			0.198088i	0.995083	+	0.0990440i	$0.0315785\pi$
$857$			169.761i			0.198088i	−0.995083	+	0.0990440i	$0.968422\pi$
$858$		0			0
$859$	−687.207			−0.800009			−0.400004	−	0.916513i	$-0.630991\pi$
$859$	−687.207			−0.800009			−0.400004	+	0.916513i	$0.630991\pi$
$860$		0			0
$861$	372.538	+	243.746i	0.432681	+	0.283096i
$862$		0			0
$863$	246.094			0.285161			0.142581	−	0.989783i	$-0.454460\pi$
$863$	246.094			0.285161			0.142581	+	0.989783i	$0.454460\pi$
$864$		0			0
$865$	−2454.61			−2.83770
$866$		0			0
$867$	468.376			0.540226
$868$		0			0
$869$		−	463.700i		−	0.533602i
$870$		0			0
$871$		−	282.209i		−	0.324006i
$872$		0			0
$873$			274.419i			0.314340i
$874$		0			0
$875$	−1136.83	+	1737.52i	−1.29924	+	1.98574i
$876$		0			0
$877$			487.171i			0.555497i	0.960654	+	0.277749i	$0.0895882\pi$
$877$			487.171i			0.555497i	−0.960654	+	0.277749i	$0.910412\pi$
$878$		0			0
$879$	27.9065			0.0317480
$880$		0			0
$881$			343.877i			0.390325i	0.980771	+	0.195163i	$0.0625235\pi$
$881$			343.877i			0.390325i	−0.980771	+	0.195163i	$0.937477\pi$
$882$		0			0
$883$			755.288i			0.855366i	0.903929	+	0.427683i	$0.140670\pi$
$883$			755.288i			0.855366i	−0.903929	+	0.427683i	$0.859330\pi$
$884$		0			0
$885$	−58.4675			−0.0660650
$886$		0			0
$887$		−	1633.02i		−	1.84106i	−0.390668	−	0.920532i	$-0.627756\pi$
$887$		−	1633.02i		−	1.84106i	0.390668	−	0.920532i	$-0.372244\pi$
$888$		0			0
$889$	−300.959	+	459.983i	−0.338537	+	0.517416i
$890$		0			0
$891$		−	133.784i		−	0.150150i
$892$		0			0
$893$			1414.94i			1.58448i
$894$		0			0
$895$		−	1371.80i		−	1.53273i
$896$		0			0
$897$	−68.7598			−0.0766553
$898$		0			0
$899$	−137.448			−0.152890
$900$		0			0
$901$	83.5416			0.0927209
$902$		0			0
$903$	−758.071	−	495.993i	−0.839503	−	0.549273i
$904$		0			0
$905$	570.629			0.630530
$906$		0			0
$907$			579.901i			0.639361i	0.947525	+	0.319681i	$0.103576\pi$
$907$			579.901i			0.639361i	−0.947525	+	0.319681i	$0.896424\pi$
$908$		0			0
$909$	105.485			0.116045
$910$		0			0
$911$	−169.237			−0.185770			−0.0928852	−	0.995677i	$-0.529609\pi$
$911$	−169.237			−0.185770			−0.0928852	+	0.995677i	$0.529609\pi$
$912$		0			0
$913$		−	665.350i		−	0.728752i
$914$		0			0
$915$	−1256.63			−1.37336
$916$		0			0
$917$	50.8569	−	77.7291i	0.0554600	−	0.0847645i
$918$		0			0
$919$	−910.454			−0.990701			−0.495351	−	0.868693i	$-0.664960\pi$
$919$	−910.454			−0.990701			−0.495351	+	0.868693i	$0.664960\pi$
$920$		0			0
$921$	10.3247			0.0112103
$922$		0			0
$923$	−4.22851			−0.00458127
$924$		0			0
$925$			3722.50i			4.02432i
$926$		0			0
$927$		−	136.535i		−	0.147287i
$928$		0			0
$929$		−	1560.39i		−	1.67965i	−0.542859	−	0.839824i	$-0.682658\pi$
$929$		−	1560.39i		−	1.67965i	0.542859	−	0.839824i	$-0.317342\pi$
$930$		0			0
$931$	−528.756	−	1209.82i	−0.567944	−	1.29949i
$932$		0			0
$933$			452.162i			0.484633i
$934$		0			0
$935$	582.498			0.622992
$936$		0			0
$937$			1529.70i			1.63255i	0.577664	+	0.816275i	$0.303964\pi$
$937$			1529.70i			1.63255i	−0.577664	+	0.816275i	$0.696036\pi$
$938$		0			0
$939$		−	494.872i		−	0.527021i
$940$		0			0
$941$	509.889			0.541859			0.270929	−	0.962599i	$-0.412669\pi$
$941$	509.889			0.541859			0.270929	+	0.962599i	$0.412669\pi$
$942$		0			0
$943$			342.921i			0.363649i
$944$		0			0
$945$	−181.026	+	276.678i	−0.191562	+	0.292781i
$946$		0			0
$947$		−	227.835i		−	0.240586i	−0.992738	−	0.120293i	$-0.961617\pi$
$947$		−	227.835i		−	0.240586i	0.992738	−	0.120293i	$-0.0383833\pi$
$948$		0			0
$949$		−	37.3300i		−	0.0393362i
$950$		0			0
$951$			1028.64i			1.08164i
$952$		0			0
$953$	488.011			0.512079			0.256040	−	0.966666i	$-0.417582\pi$
$953$	488.011			0.512079			0.256040	+	0.966666i	$0.417582\pi$
$954$		0			0
$955$	−1389.38			−1.45485
$956$		0			0
$957$	−881.877			−0.921502
$958$		0			0
$959$	−388.870	+	594.345i	−0.405495	+	0.619755i
$960$		0			0
$961$	944.897			0.983244
$962$		0			0
$963$			192.708i			0.200112i
$964$		0			0
$965$	−1790.72			−1.85567
$966$		0			0
$967$	−976.631			−1.00996			−0.504980	−	0.863131i	$-0.668500\pi$
$967$	−976.631			−1.00996			−0.504980	+	0.863131i	$0.668500\pi$
$968$		0			0
$969$			201.190i			0.207626i
$970$		0			0
$971$	1709.72			1.76078			0.880390	−	0.474250i	$-0.157281\pi$
$971$	1709.72			1.76078			0.880390	+	0.474250i	$0.157281\pi$
$972$		0			0
$973$	625.999	−	956.770i	0.643370	−	0.983320i
$974$		0			0
$975$	424.316			0.435196
$976$		0			0
$977$	−1387.42			−1.42009			−0.710043	−	0.704158i	$-0.751325\pi$
$977$	−1387.42			−1.42009			−0.710043	+	0.704158i	$0.751325\pi$
$978$		0			0
$979$	−172.872			−0.176580
$980$		0			0
$981$		−	117.950i		−	0.120234i
$982$		0			0
$983$		−	910.425i		−	0.926170i	−0.886314	−	0.463085i	$-0.846742\pi$
$983$		−	910.425i		−	0.926170i	0.886314	−	0.463085i	$-0.153258\pi$
$984$		0			0
$985$		−	531.022i		−	0.539108i
$986$		0			0
$987$	−532.765	−	348.580i	−0.539783	−	0.353171i
$988$		0			0
$989$		−	697.804i		−	0.705565i
$990$		0			0
$991$	−474.575			−0.478885			−0.239443	−	0.970910i	$-0.576965\pi$
$991$	−474.575			−0.478885			−0.239443	+	0.970910i	$0.576965\pi$
$992$		0			0
$993$		−	44.3010i		−	0.0446133i
$994$		0			0
$995$		−	289.990i		−	0.291447i
$996$		0			0
$997$	545.379			0.547020			0.273510	−	0.961869i	$-0.411815\pi$
$997$	545.379			0.547020			0.273510	+	0.961869i	$0.411815\pi$
$998$		0			0
$999$			335.627i			0.335963i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.3.l.a.433.18		32
3.2	odd	2		2016.3.l.h.433.1		32
4.3	odd	2		168.3.l.a.13.23	yes	32
7.6	odd	2	inner	672.3.l.a.433.2		32
8.3	odd	2		168.3.l.a.13.22	yes	32
8.5	even	2	inner	672.3.l.a.433.15		32
12.11	even	2		504.3.l.h.181.9		32
21.20	even	2		2016.3.l.h.433.31		32
24.5	odd	2		2016.3.l.h.433.32		32
24.11	even	2		504.3.l.h.181.12		32
28.27	even	2		168.3.l.a.13.24	yes	32
56.13	odd	2	inner	672.3.l.a.433.31		32
56.27	even	2		168.3.l.a.13.21	✓	32
84.83	odd	2		504.3.l.h.181.10		32
168.83	odd	2		504.3.l.h.181.11		32
168.125	even	2		2016.3.l.h.433.2		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.l.a.13.21	✓	32	56.27	even	2
168.3.l.a.13.22	yes	32	8.3	odd	2
168.3.l.a.13.23	yes	32	4.3	odd	2
168.3.l.a.13.24	yes	32	28.27	even	2
504.3.l.h.181.9		32	12.11	even	2
504.3.l.h.181.10		32	84.83	odd	2
504.3.l.h.181.11		32	168.83	odd	2
504.3.l.h.181.12		32	24.11	even	2
672.3.l.a.433.2		32	7.6	odd	2	inner
672.3.l.a.433.15		32	8.5	even	2	inner
672.3.l.a.433.18		32	1.1	even	1	trivial
672.3.l.a.433.31		32	56.13	odd	2	inner
2016.3.l.h.433.1		32	3.2	odd	2
2016.3.l.h.433.2		32	168.125	even	2
2016.3.l.h.433.31		32	21.20	even	2
2016.3.l.h.433.32		32	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +9.09019 q^{5} +(-3.83254 + 5.85761i) q^{7} +3.00000 q^{9} -14.8648i q^{11} +4.25079 q^{13} +15.7447 q^{15} +4.31083i q^{17} +26.9453 q^{19} +(-6.63815 + 10.1457i) q^{21} -9.33910 q^{23} +57.6315 q^{25} +5.19615 q^{27} -34.2521i q^{29} -4.01283i q^{31} -25.7467i q^{33} +(-34.8385 + 53.2468i) q^{35} +64.5914i q^{37} +7.36258 q^{39} -36.7189i q^{41} +74.7186i q^{43} +27.2706 q^{45} +52.5115i q^{47} +(-19.6233 - 44.8991i) q^{49} +7.46658i q^{51} -19.3795i q^{53} -135.124i q^{55} +46.6707 q^{57} -3.71348 q^{59} -79.8128 q^{61} +(-11.4976 + 17.5728i) q^{63} +38.6404 q^{65} -66.3899i q^{67} -16.1758 q^{69} -0.994759 q^{71} -8.78191i q^{73} +99.8206 q^{75} +(87.0725 + 56.9701i) q^{77} +31.1944 q^{79} +9.00000 q^{81} +44.7600 q^{83} +39.1863i q^{85} -59.3264i q^{87} -11.6296i q^{89} +(-16.2913 + 24.8995i) q^{91} -6.95043i q^{93} +244.938 q^{95} +91.4731i q^{97} -44.5945i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 96 q^{9} - 64 q^{23} + 160 q^{25} - 16 q^{49} + 96 q^{57} + 640 q^{71} + 64 q^{79} + 288 q^{81} + 768 q^{95}+O(q^{100}) \) 32 * q + 96 * q^9 - 64 * q^23 + 160 * q^25 - 16 * q^49 + 96 * q^57 + 640 * q^71 + 64 * q^79 + 288 * q^81 + 768 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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