LMFDB - Embedded newform 1008.4.bt.d.593.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(17,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.bt (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.12
Character	$\chi$	$=$	1008.593
Dual form			1008.4.bt.d.17.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.35769 + 2.35159i) q^{5} +(8.74105 - 16.3277i) q^{7} +O(q^{10})$	$q+(-1.35769 + 2.35159i) q^{5} +(8.74105 - 16.3277i) q^{7} +(-8.39023 + 4.84410i) q^{11} -67.7228i q^{13} +(-50.1037 - 86.7822i) q^{17} +(59.6837 + 34.4584i) q^{19} +(-126.981 - 73.3122i) q^{23} +(58.8133 + 101.868i) q^{25} +284.951i q^{29} +(-197.116 + 113.805i) q^{31} +(26.5285 + 42.7234i) q^{35} +(150.784 - 261.166i) q^{37} +232.403 q^{41} -173.051 q^{43} +(-191.489 + 331.669i) q^{47} +(-190.188 - 285.443i) q^{49} +(22.2602 - 12.8520i) q^{53} -26.3072i q^{55} +(-371.973 - 644.275i) q^{59} +(-343.708 - 198.440i) q^{61} +(159.257 + 91.9468i) q^{65} +(293.861 + 508.981i) q^{67} +501.792i q^{71} +(616.668 - 356.033i) q^{73} +(5.75368 + 179.336i) q^{77} +(-466.039 + 807.203i) q^{79} -837.407 q^{83} +272.102 q^{85} +(-442.322 + 766.124i) q^{89} +(-1105.76 - 591.968i) q^{91} +(-162.064 + 93.5678i) q^{95} -1175.54i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 24 q^{7}+O(q^{10}) $ 48 * q + 24 * q^7	$ 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) $ 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.35769	+	2.35159i	−0.121436	+	0.210333i	−0.920334	−	0.391133i	$-0.872083\pi$
$5$	−1.35769	+	2.35159i	−0.121436	+	0.210333i	0.798898	+	0.601466i	$0.205417\pi$
$6$		0			0
$7$	8.74105	−	16.3277i	0.471972	−	0.881613i
$7$	8.74105	−	16.3277i	0.471972	−	0.881613i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−8.39023	+	4.84410i	−0.229977	+	0.132777i	−0.610562	−	0.791969i	$-0.709056\pi$
$11$	−8.39023	+	4.84410i	−0.229977	+	0.132777i	0.380584	+	0.924746i	$0.375723\pi$
$12$		0			0
$13$		−	67.7228i		−	1.44484i	−0.691454	−	0.722420i	$-0.743030\pi$
$13$		−	67.7228i		−	1.44484i	0.691454	−	0.722420i	$-0.256970\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−50.1037	−	86.7822i	−0.714820	−	1.23810i	−0.963029	−	0.269397i	$-0.913175\pi$
$17$	−50.1037	−	86.7822i	−0.714820	−	1.23810i	0.248209	−	0.968706i	$-0.420158\pi$
$18$		0			0
$19$	59.6837	+	34.4584i	0.720651	+	0.416068i	0.814992	−	0.579472i	$-0.196741\pi$
$19$	59.6837	+	34.4584i	0.720651	+	0.416068i	−0.0943412	+	0.995540i	$0.530074\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−126.981	−	73.3122i	−1.15119	−	0.664637i	−0.202010	−	0.979384i	$-0.564747\pi$
$23$	−126.981	−	73.3122i	−1.15119	−	0.664637i	−0.949176	+	0.314746i	$0.898081\pi$
$24$		0			0
$25$	58.8133	+	101.868i	0.470507	+	0.814941i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			284.951i			1.82463i	0.409493	+	0.912313i	$0.365706\pi$
$29$			284.951i			1.82463i	−0.409493	+	0.912313i	$0.634294\pi$
$30$		0			0
$31$	−197.116	+	113.805i	−1.14203	+	0.659353i	−0.946933	−	0.321430i	$-0.895837\pi$
$31$	−197.116	+	113.805i	−1.14203	+	0.659353i	−0.195100	+	0.980783i	$0.562503\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	26.5285	+	42.7234i	0.128118	+	0.206331i
$36$		0			0
$37$	150.784	−	261.166i	0.669966	−	1.16042i	−0.307947	−	0.951404i	$-0.599642\pi$
$37$	150.784	−	261.166i	0.669966	−	1.16042i	0.977913	−	0.209012i	$-0.0670248\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	232.403			0.885251			0.442626	−	0.896707i	$-0.354047\pi$
$41$	232.403			0.885251			0.442626	+	0.896707i	$0.354047\pi$
$42$		0			0
$43$	−173.051			−0.613722			−0.306861	−	0.951754i	$-0.599279\pi$
$43$	−173.051			−0.613722			−0.306861	+	0.951754i	$0.599279\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−191.489	+	331.669i	−0.594288	+	1.02934i	0.399358	+	0.916795i	$0.369233\pi$
$47$	−191.489	+	331.669i	−0.594288	+	1.02934i	−0.993647	+	0.112543i	$0.964100\pi$
$48$		0			0
$49$	−190.188	−	285.443i	−0.554484	−	0.832194i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	22.2602	−	12.8520i	0.0576921	−	0.0333085i	−0.470877	−	0.882199i	$-0.656062\pi$
$53$	22.2602	−	12.8520i	0.0576921	−	0.0333085i	0.528569	+	0.848891i	$0.322729\pi$
$54$		0			0
$55$		−	26.3072i		−	0.0644957i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−371.973	−	644.275i	−0.820792	−	1.42165i	−0.905094	−	0.425212i	$-0.860199\pi$
$59$	−371.973	−	644.275i	−0.820792	−	1.42165i	0.0843021	−	0.996440i	$-0.473134\pi$
$60$		0			0
$61$	−343.708	−	198.440i	−0.721432	−	0.416519i	0.0938477	−	0.995587i	$-0.470083\pi$
$61$	−343.708	−	198.440i	−0.721432	−	0.416519i	−0.815279	+	0.579068i	$0.803417\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	159.257	+	91.9468i	0.303898	+	0.175455i
$66$		0			0
$67$	293.861	+	508.981i	0.535833	+	0.928089i	0.999123	+	0.0418825i	$0.0133355\pi$
$67$	293.861	+	508.981i	0.535833	+	0.928089i	−0.463290	+	0.886207i	$0.653331\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			501.792i			0.838758i	0.907811	+	0.419379i	$0.137752\pi$
$71$			501.792i			0.838758i	−0.907811	+	0.419379i	$0.862248\pi$
$72$		0			0
$73$	616.668	−	356.033i	0.988706	−	0.570830i	0.0838188	−	0.996481i	$-0.473288\pi$
$73$	616.668	−	356.033i	0.988706	−	0.570830i	0.904887	+	0.425651i	$0.139955\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.75368	+	179.336i	0.00851549	+	0.265418i
$78$		0			0
$79$	−466.039	+	807.203i	−0.663715	+	1.14959i	0.315917	+	0.948787i	$0.397688\pi$
$79$	−466.039	+	807.203i	−0.663715	+	1.14959i	−0.979632	+	0.200801i	$0.935646\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−837.407			−1.10744			−0.553719	−	0.832703i	$-0.686792\pi$
$83$	−837.407			−1.10744			−0.553719	+	0.832703i	$0.686792\pi$
$84$		0			0
$85$	272.102			0.347219
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−442.322	+	766.124i	−0.526809	+	0.912460i	0.472703	+	0.881222i	$0.343278\pi$
$89$	−442.322	+	766.124i	−0.526809	+	0.912460i	−0.999512	+	0.0312382i	$0.990055\pi$
$90$		0			0
$91$	−1105.76	−	591.968i	−1.27379	−	0.681925i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−162.064	+	93.5678i	−0.175026	+	0.101051i
$96$		0			0
$97$		−	1175.54i		−	1.23049i	−0.788335	−	0.615247i	$-0.789056\pi$
$97$		−	1175.54i		−	1.23049i	0.788335	−	0.615247i	$-0.210944\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	117.031	+	202.703i	0.115297	+	0.199700i	0.917898	−	0.396816i	$-0.129885\pi$
$101$	117.031	+	202.703i	0.115297	+	0.199700i	−0.802602	+	0.596516i	$0.796551\pi$
$102$		0			0
$103$	−622.050	−	359.141i	−0.595072	−	0.343565i	0.172028	−	0.985092i	$-0.444968\pi$
$103$	−622.050	−	359.141i	−0.595072	−	0.343565i	−0.767101	+	0.641527i	$0.778301\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−156.488	−	90.3486i	−0.141386	−	0.0816292i	0.427638	−	0.903950i	$-0.359346\pi$
$107$	−156.488	−	90.3486i	−0.141386	−	0.0816292i	−0.569024	+	0.822321i	$0.692679\pi$
$108$		0			0
$109$	−993.277	−	1720.41i	−0.872832	−	1.51179i	−0.859054	−	0.511885i	$-0.828947\pi$
$109$	−993.277	−	1720.41i	−0.872832	−	1.51179i	−0.0137784	−	0.999905i	$-0.504386\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			841.537i			0.700576i	0.936642	+	0.350288i	$0.113916\pi$
$113$			841.537i			0.700576i	−0.936642	+	0.350288i	$0.886084\pi$
$114$		0			0
$115$	344.801	−	199.071i	0.279590	−	0.161422i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1854.91	+	59.5117i	−1.42890	+	0.0458439i
$120$		0			0
$121$	−618.569	+	1071.39i	−0.464740	+	0.804954i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−658.825			−0.471417
$126$		0			0
$127$	−4.74209			−0.00331333			−0.00165666	−	0.999999i	$-0.500527\pi$
$127$	−4.74209			−0.00331333			−0.00165666	+	0.999999i	$0.500527\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−602.154	+	1042.96i	−0.401606	+	0.695602i	−0.993920	−	0.110105i	$-0.964881\pi$
$131$	−602.154	+	1042.96i	−0.401606	+	0.695602i	0.592314	+	0.805707i	$0.298215\pi$
$132$		0			0
$133$	1084.32	−	673.295i	0.706939	−	0.438963i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−88.4208	+	51.0498i	−0.0551409	+	0.0318356i	−0.527317	−	0.849669i	$-0.676802\pi$
$137$	−88.4208	+	51.0498i	−0.0551409	+	0.0318356i	0.472176	+	0.881504i	$0.343469\pi$
$138$		0			0
$139$			778.185i			0.474854i	0.971405	+	0.237427i	$0.0763041\pi$
$139$			778.185i			0.474854i	−0.971405	+	0.237427i	$0.923696\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	328.056	+	568.210i	0.191842	+	0.332281i
$144$		0			0
$145$	−670.090	−	386.877i	−0.383779	−	0.221575i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2325.47	−	1342.61i	−1.27859	−	0.738194i	−0.302000	−	0.953308i	$-0.597654\pi$
$149$	−2325.47	−	1342.61i	−1.27859	−	0.738194i	−0.976589	+	0.215114i	$0.930988\pi$
$150$		0			0
$151$	−1570.46	−	2720.12i	−0.846375	−	1.46596i	−0.884422	−	0.466688i	$-0.845447\pi$
$151$	−1570.46	−	2720.12i	−0.846375	−	1.46596i	0.0380473	−	0.999276i	$-0.487886\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	618.049i		−	0.320276i
$156$		0			0
$157$	424.103	−	244.856i	0.215587	−	0.124469i	−0.388318	−	0.921525i	$-0.626944\pi$
$157$	424.103	−	244.856i	0.215587	−	0.124469i	0.603905	+	0.797056i	$0.293611\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2306.96	+	1432.48i	−1.12928	+	0.701210i
$162$		0			0
$163$	1608.04	−	2785.20i	0.772707	−	1.33837i	−0.163367	−	0.986565i	$-0.552236\pi$
$163$	1608.04	−	2785.20i	0.772707	−	1.33837i	0.936074	−	0.351802i	$-0.114431\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1225.29			−0.567760			−0.283880	−	0.958860i	$-0.591622\pi$
$167$	−1225.29			−0.567760			−0.283880	+	0.958860i	$0.591622\pi$
$168$		0			0
$169$	−2389.38			−1.08757
$170$		0			0
$171$		0			0
$172$		0			0
$173$	270.349	−	468.258i	0.118811	−	0.205786i	−0.800486	−	0.599351i	$-0.795425\pi$
$173$	270.349	−	468.258i	0.118811	−	0.205786i	0.919297	+	0.393565i	$0.128759\pi$
$174$		0			0
$175$	2177.36	−	69.8567i	0.940529	−	0.0301753i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3064.67	+	1769.39i	−1.27969	+	0.738829i	−0.976791	−	0.214194i	$-0.931288\pi$
$179$	−3064.67	+	1769.39i	−1.27969	+	0.738829i	−0.302899	+	0.953023i	$0.597954\pi$
$180$		0			0
$181$			1575.92i			0.647166i	0.946200	+	0.323583i	$0.104888\pi$
$181$			1575.92i			0.647166i	−0.946200	+	0.323583i	$0.895112\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	409.437	+	709.166i	0.162716	+	0.281832i
$186$		0			0
$187$	840.763	+	485.415i	0.328784	+	0.189824i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2091.04	+	1207.26i	0.792158	+	0.457353i	0.840722	−	0.541467i	$-0.182131\pi$
$191$	2091.04	+	1207.26i	0.792158	+	0.457353i	−0.0485635	+	0.998820i	$0.515464\pi$
$192$		0			0
$193$	208.561	+	361.238i	0.0777851	+	0.134728i	0.902294	−	0.431121i	$-0.141882\pi$
$193$	208.561	+	361.238i	0.0777851	+	0.134728i	−0.824509	+	0.565849i	$0.808549\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			665.192i			0.240574i	0.992739	+	0.120287i	$0.0383814\pi$
$197$			665.192i			0.240574i	−0.992739	+	0.120287i	$0.961619\pi$
$198$		0			0
$199$	−1901.90	+	1098.06i	−0.677498	+	0.391154i	−0.798912	−	0.601448i	$-0.794591\pi$
$199$	−1901.90	+	1098.06i	−0.677498	+	0.391154i	0.121414	+	0.992602i	$0.461257\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4652.61	+	2490.77i	1.60862	+	0.861173i
$204$		0			0
$205$	−315.532	+	546.518i	−0.107501	+	0.186198i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−667.680			−0.220978
$210$		0			0
$211$	−3218.47			−1.05009			−0.525045	−	0.851075i	$-0.675951\pi$
$211$	−3218.47			−1.05009			−0.525045	+	0.851075i	$0.675951\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	234.950	−	406.946i	0.0745278	−	0.129086i
$216$		0			0
$217$	135.174	+	4213.22i	0.0422867	+	1.31803i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5877.13	+	3393.16i	−1.78886	+	1.03280i
$222$		0			0
$223$		−	3741.43i		−	1.12352i	−0.827300	−	0.561760i	$-0.810125\pi$
$223$		−	3741.43i		−	1.12352i	0.827300	−	0.561760i	$-0.189875\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	166.598	+	288.557i	0.0487116	+	0.0843709i	0.889353	−	0.457221i	$-0.151155\pi$
$227$	166.598	+	288.557i	0.0487116	+	0.0843709i	−0.840642	+	0.541592i	$0.817822\pi$
$228$		0			0
$229$	−3169.55	−	1829.94i	−0.914629	−	0.528061i	−0.0327112	−	0.999465i	$-0.510414\pi$
$229$	−3169.55	−	1829.94i	−0.914629	−	0.528061i	−0.881918	+	0.471404i	$0.843747\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1056.00	+	609.682i	0.296914	+	0.171423i	0.641056	−	0.767495i	$-0.278497\pi$
$233$	1056.00	+	609.682i	0.296914	+	0.171423i	−0.344142	+	0.938918i	$0.611830\pi$
$234$		0			0
$235$	−519.967	−	900.610i	−0.144336	−	0.249997i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1038.71i			0.281123i	0.990072	+	0.140561i	$0.0448907\pi$
$239$			1038.71i			0.281123i	−0.990072	+	0.140561i	$0.955109\pi$
$240$		0			0
$241$	−2063.92	+	1191.60i	−0.551654	+	0.318497i	−0.749789	−	0.661677i	$-0.769845\pi$
$241$	−2063.92	+	1191.60i	−0.551654	+	0.318497i	0.198135	+	0.980175i	$0.436512\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	929.462	−	59.7019i	0.242372	−	0.0155682i
$246$		0			0
$247$	2333.62	−	4041.95i	0.601152	−	1.04123i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2601.02			0.654083			0.327041	−	0.945010i	$-0.393948\pi$
$251$	2601.02			0.654083			0.327041	+	0.945010i	$0.393948\pi$
$252$		0			0
$253$	1420.53			0.352995
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1380.41	−	2390.94i	0.335049	−	0.580321i	−0.648446	−	0.761261i	$-0.724581\pi$
$257$	1380.41	−	2390.94i	0.335049	−	0.580321i	0.983494	+	0.180940i	$0.0579139\pi$
$258$		0			0
$259$	−2946.23	−	4744.82i	−0.706833	−	1.13834i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−463.082	+	267.361i	−0.108574	+	0.0626850i	−0.553304	−	0.832980i	$-0.686633\pi$
$263$	−463.082	+	267.361i	−0.108574	+	0.0626850i	0.444730	+	0.895665i	$0.353300\pi$
$264$		0			0
$265$			69.7961i			0.0161794i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1271.06	−	2201.55i	−0.288097	−	0.498999i	0.685259	−	0.728300i	$-0.259689\pi$
$269$	−1271.06	−	2201.55i	−0.288097	−	0.498999i	−0.973355	+	0.229301i	$0.926356\pi$
$270$		0			0
$271$	7707.74	+	4450.06i	1.72772	+	0.997499i	0.899201	+	0.437537i	$0.144149\pi$
$271$	7707.74	+	4450.06i	1.72772	+	0.997499i	0.828518	+	0.559962i	$0.189184\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−986.915	−	569.796i	−0.216412	−	0.124945i
$276$		0			0
$277$	−1185.18	−	2052.80i	−0.257078	−	0.445273i	0.708380	−	0.705832i	$-0.249427\pi$
$277$	−1185.18	−	2052.80i	−0.257078	−	0.445273i	−0.965458	+	0.260559i	$0.916093\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	6135.53i		−	1.30255i	−0.758844	−	0.651273i	$-0.774235\pi$
$281$		−	6135.53i		−	1.30255i	0.758844	−	0.651273i	$-0.225765\pi$
$282$		0			0
$283$	1108.84	−	640.186i	0.232910	−	0.134470i	−0.379004	−	0.925395i	$-0.623733\pi$
$283$	1108.84	−	640.186i	0.232910	−	0.134470i	0.611914	+	0.790925i	$0.290400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2031.45	−	3794.61i	0.417814	−	0.780449i
$288$		0			0
$289$	−2564.26	+	4441.43i	−0.521934	+	0.904016i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3293.61			−0.656706			−0.328353	−	0.944555i	$-0.606493\pi$
$293$	−3293.61			−0.656706			−0.328353	+	0.944555i	$0.606493\pi$
$294$		0			0
$295$	2020.10			0.398694
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4964.91	+	8599.48i	−0.960295	+	1.66328i
$300$		0			0
$301$	−1512.65	+	2825.53i	−0.289660	+	0.541065i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	933.301	−	538.842i	0.175215	−	0.101161i
$306$		0			0
$307$			3320.45i			0.617290i	0.951177	+	0.308645i	$0.0998756\pi$
$307$			3320.45i			0.617290i	−0.951177	+	0.308645i	$0.900124\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3578.36	−	6197.90i	−0.652444	−	1.13007i	−0.982528	−	0.186115i	$-0.940410\pi$
$311$	−3578.36	−	6197.90i	−0.652444	−	1.13007i	0.330084	−	0.943952i	$-0.392923\pi$
$312$		0			0
$313$	1971.44	+	1138.21i	0.356014	+	0.205545i	0.667331	−	0.744761i	$-0.267437\pi$
$313$	1971.44	+	1138.21i	0.356014	+	0.205545i	−0.311317	+	0.950306i	$0.600770\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7987.14	−	4611.37i	−1.41515	−	0.817037i	−0.419282	−	0.907856i	$-0.637718\pi$
$317$	−7987.14	−	4611.37i	−1.41515	−	0.817037i	−0.995867	+	0.0908197i	$0.971051\pi$
$318$		0			0
$319$	−1380.33	−	2390.81i	−0.242269	−	0.419623i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	6905.97i		−	1.18965i
$324$		0			0
$325$	6898.77	−	3983.01i	1.17746	−	0.679807i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3741.58	+	6025.71i	0.626990	+	1.00975i
$330$		0			0
$331$	304.578	−	527.545i	0.0505774	−	0.0876027i	−0.839628	−	0.543161i	$-0.817227\pi$
$331$	304.578	−	527.545i	0.0505774	−	0.0876027i	0.890206	+	0.455559i	$0.150561\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1595.89			−0.260277
$336$		0			0
$337$	5810.77			0.939267			0.469633	−	0.882862i	$-0.344386\pi$
$337$	5810.77			0.939267			0.469633	+	0.882862i	$0.344386\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1102.56	−	1909.70i	0.175094	−	0.303273i
$342$		0			0
$343$	−6323.07	+	610.270i	−0.995375	+	0.0960685i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4341.59	−	2506.62i	0.671668	−	0.387788i	−0.125040	−	0.992152i	$-0.539906\pi$
$347$	4341.59	−	2506.62i	0.671668	−	0.387788i	0.796708	+	0.604364i	$0.206573\pi$
$348$		0			0
$349$		−	7354.17i		−	1.12796i	−0.825787	−	0.563982i	$-0.809269\pi$
$349$		−	7354.17i		−	1.12796i	0.825787	−	0.563982i	$-0.190731\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4948.60	+	8571.22i	0.746140	+	1.29235i	0.949660	+	0.313281i	$0.101428\pi$
$353$	4948.60	+	8571.22i	0.746140	+	1.29235i	−0.203521	+	0.979071i	$0.565238\pi$
$354$		0			0
$355$	−1180.01	−	681.280i	−0.176418	−	0.101855i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9868.73	+	5697.71i	1.45084	+	0.837643i	0.998529	−	0.0542171i	$-0.0172663\pi$
$359$	9868.73	+	5697.71i	1.45084	+	0.837643i	0.452311	+	0.891860i	$0.350600\pi$
$360$		0			0
$361$	−1054.74	−	1826.86i	−0.153775	−	0.266346i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1933.54i			0.277277i
$366$		0			0
$367$	−5837.69	+	3370.39i	−0.830313	+	0.479382i	−0.853960	−	0.520339i	$-0.825806\pi$
$367$	−5837.69	+	3370.39i	−0.830313	+	0.479382i	0.0236465	+	0.999720i	$0.492472\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−15.2652	−	475.798i	−0.00213619	−	0.0665828i
$372$		0			0
$373$	−4281.55	+	7415.87i	−0.594344	+	1.02943i	0.399295	+	0.916823i	$0.369255\pi$
$373$	−4281.55	+	7415.87i	−0.594344	+	1.02943i	−0.993639	+	0.112612i	$0.964078\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19297.7			2.63629
$378$		0			0
$379$	5665.90			0.767910			0.383955	−	0.923352i	$-0.374562\pi$
$379$	5665.90			0.767910			0.383955	+	0.923352i	$0.374562\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3810.82	−	6600.53i	0.508417	−	0.880604i	−0.491535	−	0.870858i	$-0.663564\pi$
$383$	3810.82	−	6600.53i	0.508417	−	0.880604i	0.999952	−	0.00974677i	$-0.00310254\pi$
$384$		0			0
$385$	−429.537	−	229.953i	−0.0568603	−	0.0304402i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11743.2	−	6779.95i	1.53060	−	0.883694i	0.531269	−	0.847203i	$-0.321715\pi$
$389$	11743.2	−	6779.95i	1.53060	−	0.883694i	0.999334	−	0.0364904i	$-0.0116178\pi$
$390$		0			0
$391$			14692.9i			1.90038i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1265.48	−	2191.87i	−0.161198	−	0.279202i
$396$		0			0
$397$	65.9737	+	38.0899i	0.00834036	+	0.00481531i	0.504164	−	0.863608i	$-0.331801\pi$
$397$	65.9737	+	38.0899i	0.00834036	+	0.00481531i	−0.495824	+	0.868423i	$0.665134\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3304.95	+	1908.11i	0.411574	+	0.237622i	0.691466	−	0.722409i	$-0.256965\pi$
$401$	3304.95	+	1908.11i	0.411574	+	0.237622i	−0.279892	+	0.960032i	$0.590299\pi$
$402$		0			0
$403$	7707.19	+	13349.2i	0.952661	+	1.65006i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2921.65i			0.355826i
$408$		0			0
$409$	7641.81	−	4412.00i	0.923871	−	0.533397i	0.0390032	−	0.999239i	$-0.487582\pi$
$409$	7641.81	−	4412.00i	0.923871	−	0.533397i	0.884868	+	0.465842i	$0.154248\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−13771.0	+	441.818i	−1.64074	+	0.0526403i
$414$		0			0
$415$	1136.94	−	1969.24i	0.134483	−	0.232931i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9891.78			1.15333			0.576665	−	0.816981i	$-0.304354\pi$
$419$	9891.78			1.15333			0.576665	+	0.816981i	$0.304354\pi$
$420$		0			0
$421$	9584.37			1.10953			0.554767	−	0.832006i	$-0.312807\pi$
$421$	9584.37			1.10953			0.554767	+	0.832006i	$0.312807\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5893.53	−	10207.9i	0.672655	−	1.16507i
$426$		0			0
$427$	−6244.44	+	3877.39i	−0.707704	+	0.439439i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11923.9	+	6884.25i	−1.33260	+	0.769380i	−0.985698	−	0.168520i	$-0.946101\pi$
$431$	−11923.9	+	6884.25i	−1.33260	+	0.769380i	−0.346906	+	0.937900i	$0.612768\pi$
$432$		0			0
$433$			3459.71i			0.383979i	0.981397	+	0.191990i	$0.0614940\pi$
$433$			3459.71i			0.383979i	−0.981397	+	0.191990i	$0.938506\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5052.44	−	8751.09i	−0.553069	−	0.957943i
$438$		0			0
$439$	13148.8	+	7591.49i	1.42952	+	0.825335i	0.997083	−	0.0763287i	$-0.0243198\pi$
$439$	13148.8	+	7591.49i	1.42952	+	0.825335i	0.432439	+	0.901663i	$0.357653\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7369.30	−	4254.67i	−0.790353	−	0.456310i	0.0497341	−	0.998762i	$-0.484163\pi$
$443$	−7369.30	−	4254.67i	−0.790353	−	0.456310i	−0.840087	+	0.542452i	$0.817496\pi$
$444$		0			0
$445$	−1201.07	−	2080.32i	−0.127947	−	0.221611i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			14011.3i			1.47269i	0.676608	+	0.736343i	$0.263449\pi$
$449$			14011.3i			1.47269i	−0.676608	+	0.736343i	$0.736551\pi$
$450$		0			0
$451$	−1949.92	+	1125.79i	−0.203588	+	0.117541i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2893.35	−	1796.58i	0.298115	−	0.185110i
$456$		0			0
$457$	−8286.20	+	14352.1i	−0.848167	+	1.46907i	0.0346758	+	0.999399i	$0.488960\pi$
$457$	−8286.20	+	14352.1i	−0.848167	+	1.46907i	−0.882842	+	0.469669i	$0.844373\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16366.1			1.65346			0.826731	−	0.562598i	$-0.190198\pi$
$461$	16366.1			1.65346			0.826731	+	0.562598i	$0.190198\pi$
$462$		0			0
$463$	−13247.2			−1.32969			−0.664847	−	0.746980i	$-0.731503\pi$
$463$	−13247.2			−1.32969			−0.664847	+	0.746980i	$0.731503\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5570.15	−	9647.78i	0.551939	−	0.955987i	−0.446195	−	0.894936i	$-0.647221\pi$
$467$	5570.15	−	9647.78i	0.551939	−	0.955987i	0.998135	−	0.0610512i	$-0.0194453\pi$
$468$		0			0
$469$	10879.2	−	349.039i	1.07111	−	0.0343649i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1451.94	−	838.277i	0.141142	−	0.0814884i
$474$		0			0
$475$			8106.45i			0.783051i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9679.64	−	16765.6i	−0.923328	−	1.59925i	−0.794229	−	0.607619i	$-0.792125\pi$
$479$	−9679.64	−	16765.6i	−0.923328	−	1.59925i	−0.129099	−	0.991632i	$-0.541208\pi$
$480$		0			0
$481$	−17686.9	−	10211.5i	−1.67662	−	0.967995i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2764.39	+	1596.02i	0.258813	+	0.149426i
$486$		0			0
$487$	737.757	+	1277.83i	0.0686467	+	0.118900i	0.898306	−	0.439371i	$-0.144799\pi$
$487$	737.757	+	1277.83i	0.0686467	+	0.118900i	−0.829659	+	0.558270i	$0.811465\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	2838.87i		−	0.260930i	−0.991453	−	0.130465i	$-0.958353\pi$
$491$		−	2838.87i		−	0.260930i	0.991453	−	0.130465i	$-0.0416470\pi$
$492$		0			0
$493$	24728.7	−	14277.1i	2.25908	−	1.30428i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8193.12	+	4386.19i	0.739460	+	0.395870i
$498$		0			0
$499$	−1154.22	+	1999.17i	−0.103547	+	0.179349i	−0.913144	−	0.407638i	$-0.866353\pi$
$499$	−1154.22	+	1999.17i	−0.103547	+	0.179349i	0.809597	+	0.586987i	$0.199686\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4336.61			0.384413			0.192207	−	0.981354i	$-0.438436\pi$
$503$	4336.61			0.384413			0.192207	+	0.981354i	$0.438436\pi$
$504$		0			0
$505$	−635.567			−0.0560047
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6133.44	−	10623.4i	0.534106	−	0.925098i	−0.465100	−	0.885258i	$-0.653982\pi$
$509$	6133.44	−	10623.4i	0.534106	−	0.925098i	0.999206	−	0.0398402i	$-0.0126849\pi$
$510$		0			0
$511$	−422.886	−	13180.9i	−0.0366093	−	1.14107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1689.11	−	975.207i	0.144526	−	0.0834422i
$516$		0			0
$517$		−	3710.37i		−	0.315632i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−605.649	−	1049.01i	−0.0509289	−	0.0882114i	0.839437	−	0.543457i	$-0.182885\pi$
$521$	−605.649	−	1049.01i	−0.0509289	−	0.0882114i	−0.890366	+	0.455245i	$0.849552\pi$
$522$		0			0
$523$	9043.21	+	5221.10i	0.756084	+	0.436526i	0.827888	−	0.560893i	$-0.189542\pi$
$523$	9043.21	+	5221.10i	0.756084	+	0.436526i	−0.0718038	+	0.997419i	$0.522876\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	19752.5	+	11404.1i	1.63270	+	0.942637i
$528$		0			0
$529$	4665.87	+	8081.52i	0.383486	+	0.664216i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	15739.0i		−	1.27905i
$534$		0			0
$535$	424.926	−	245.331i	0.0343386	−	0.0198254i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2978.44	+	1473.64i	0.238015	+	0.117763i
$540$		0			0
$541$	7747.12	−	13418.4i	0.615665	−	1.06636i	−0.374603	−	0.927185i	$-0.622221\pi$
$541$	7747.12	−	13418.4i	0.615665	−	1.06636i	0.990268	−	0.139177i	$-0.0444457\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5394.27			0.423972
$546$		0			0
$547$	2321.63			0.181473			0.0907366	−	0.995875i	$-0.471078\pi$
$547$	2321.63			0.181473			0.0907366	+	0.995875i	$0.471078\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9818.97	+	17006.9i	−0.759169	+	1.31492i
$552$		0			0
$553$	9106.11	+	14665.1i	0.700237	+	1.12771i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17567.9	+	10142.9i	−1.33641	+	0.771574i	−0.986272	−	0.165126i	$-0.947197\pi$
$557$	−17567.9	+	10142.9i	−1.33641	+	0.771574i	−0.350133	+	0.936700i	$0.613864\pi$
$558$		0			0
$559$			11719.5i			0.886730i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12955.8	−	22440.1i	−0.969843	−	1.67982i	−0.695999	−	0.718043i	$-0.745038\pi$
$563$	−12955.8	−	22440.1i	−0.969843	−	1.67982i	−0.273843	−	0.961774i	$-0.588295\pi$
$564$		0			0
$565$	−1978.95	−	1142.55i	−0.147354	−	0.0850751i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8399.50	+	4849.46i	0.618850	+	0.357293i	0.776421	−	0.630215i	$-0.217033\pi$
$569$	8399.50	+	4849.46i	0.618850	+	0.357293i	−0.157571	+	0.987508i	$0.550366\pi$
$570$		0			0
$571$	−5472.11	−	9477.97i	−0.401052	−	0.694642i	0.592801	−	0.805349i	$-0.298022\pi$
$571$	−5472.11	−	9477.97i	−0.401052	−	0.694642i	−0.993853	+	0.110707i	$0.964689\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	17246.9i		−	1.25087i
$576$		0			0
$577$	−3895.27	+	2248.93i	−0.281043	+	0.162261i	−0.633896	−	0.773419i	$-0.718545\pi$
$577$	−3895.27	+	2248.93i	−0.281043	+	0.162261i	0.352852	+	0.935679i	$0.385212\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7319.82	+	13672.9i	−0.522680	+	0.976332i
$582$		0			0
$583$	−124.512	+	215.662i	−0.00884524	+	0.0153204i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11846.6			−0.832985			−0.416493	−	0.909139i	$-0.636741\pi$
$587$	−11846.6			−0.832985			−0.416493	+	0.909139i	$0.636741\pi$
$588$		0			0
$589$	−15686.1			−1.09734
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13185.0	−	22837.1i	0.913059	−	1.58147i	0.103342	−	0.994646i	$-0.467046\pi$
$593$	13185.0	−	22837.1i	0.913059	−	1.58147i	0.809718	−	0.586819i	$-0.199620\pi$
$594$		0			0
$595$	2378.46	−	4442.80i	0.163878	−	0.306113i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7357.48	−	4247.84i	0.501867	−	0.289753i	−0.227617	−	0.973751i	$-0.573093\pi$
$599$	7357.48	−	4247.84i	0.501867	−	0.289753i	0.729484	+	0.683998i	$0.239760\pi$
$600$		0			0
$601$			18469.7i			1.25357i	0.779194	+	0.626783i	$0.215629\pi$
$601$			18469.7i			1.25357i	−0.779194	+	0.626783i	$0.784371\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1679.66	−	2909.25i	−0.112872	−	0.195500i
$606$		0			0
$607$	7595.64	+	4385.35i	0.507904	+	0.293238i	0.731972	−	0.681335i	$-0.238600\pi$
$607$	7595.64	+	4385.35i	0.507904	+	0.293238i	−0.224068	+	0.974574i	$0.571934\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	22461.6	+	12968.2i	1.48723	+	0.858652i
$612$		0			0
$613$	−7598.87	−	13161.6i	−0.500678	−	0.867199i	−1.00000		0.000782617i	$-0.999751\pi$
$613$	−7598.87	−	13161.6i	−0.500678	−	0.867199i	0.499322	−	0.866416i	$-0.333582\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			10948.4i			0.714370i	0.934034	+	0.357185i	$0.116263\pi$
$617$			10948.4i			0.714370i	−0.934034	+	0.357185i	$0.883737\pi$
$618$		0			0
$619$	−5133.29	+	2963.71i	−0.333319	+	0.192442i	−0.657314	−	0.753617i	$-0.728307\pi$
$619$	−5133.29	+	2963.71i	−0.333319	+	0.192442i	0.323995	+	0.946059i	$0.394974\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8642.69	+	13918.8i	0.555798	+	0.895098i
$624$		0			0
$625$	−6457.18	+	11184.2i	−0.413260	+	0.715787i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−30219.4			−1.91562
$630$		0			0
$631$	−5562.62			−0.350942			−0.175471	−	0.984485i	$-0.556145\pi$
$631$	−5562.62			−0.350942			−0.175471	+	0.984485i	$0.556145\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.43831	−	11.1515i	0.000402357	−	0.000696902i
$636$		0			0
$637$	−19331.0	+	12880.1i	−1.20239	+	0.801142i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6618.59	−	3821.24i	0.407829	−	0.235460i	−0.282027	−	0.959406i	$-0.591007\pi$
$641$	6618.59	−	3821.24i	0.407829	−	0.235460i	0.689857	+	0.723946i	$0.257674\pi$
$642$		0			0
$643$		−	487.471i		−	0.0298974i	−0.999888	−	0.0149487i	$-0.995242\pi$
$643$		−	487.471i		−	0.0298974i	0.999888	−	0.0149487i	$-0.00475849\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1526.57	+	2644.09i	0.0927598	+	0.160665i	0.908671	−	0.417512i	$-0.137098\pi$
$647$	1526.57	+	2644.09i	0.0927598	+	0.160665i	−0.815912	+	0.578177i	$0.803764\pi$
$648$		0			0
$649$	6241.87	+	3603.75i	0.377527	+	0.217965i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18630.0	−	10756.0i	−1.11646	−	0.644588i	−0.175964	−	0.984397i	$-0.556304\pi$
$653$	−18630.0	−	10756.0i	−1.11646	−	0.644588i	−0.940494	+	0.339809i	$0.889638\pi$
$654$		0			0
$655$	−1635.08	−	2832.04i	−0.0975388	−	0.168942i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			12939.8i			0.764891i	0.923978	+	0.382445i	$0.124918\pi$
$659$			12939.8i			0.764891i	−0.923978	+	0.382445i	$0.875082\pi$
$660$		0			0
$661$	2207.26	−	1274.36i	0.129883	−	0.0749878i	−0.433651	−	0.901081i	$-0.642775\pi$
$661$	2207.26	−	1274.36i	0.129883	−	0.0749878i	0.563534	+	0.826093i	$0.309442\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	111.137	+	3464.02i	0.00648077	+	0.201998i
$666$		0			0
$667$	20890.4	−	36183.3i	1.21271	−	2.10048i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3845.06			0.221217
$672$		0			0
$673$	20784.3			1.19045			0.595227	−	0.803558i	$-0.297062\pi$
$673$	20784.3			1.19045			0.595227	+	0.803558i	$0.297062\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3072.44	−	5321.62i	0.174422	−	0.302107i	−0.765539	−	0.643389i	$-0.777528\pi$
$677$	3072.44	−	5321.62i	0.174422	−	0.302107i	0.939961	+	0.341282i	$0.110861\pi$
$678$		0			0
$679$	−19193.9	−	10275.4i	−1.08482	−	0.580759i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7362.49	+	4250.74i	−0.412471	+	0.238141i	−0.691851	−	0.722040i	$-0.743205\pi$
$683$	−7362.49	+	4250.74i	−0.412471	+	0.238141i	0.279380	+	0.960181i	$0.409871\pi$
$684$		0			0
$685$		−	277.240i		−	0.0154639i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−870.371	−	1507.53i	−0.0481255	−	0.0833559i
$690$		0			0
$691$	23895.8	+	13796.2i	1.31554	+	0.759528i	0.983008	−	0.183564i	$-0.0587635\pi$
$691$	23895.8	+	13796.2i	1.31554	+	0.759528i	0.332533	+	0.943092i	$0.392097\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1829.97	−	1056.54i	−0.0998775	−	0.0576643i
$696$		0			0
$697$	−11644.3	−	20168.5i	−0.632795	−	1.09603i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	11627.5i		−	0.626481i	−0.949674	−	0.313240i	$-0.898585\pi$
$701$		−	11627.5i		−	0.626481i	0.949674	−	0.313240i	$-0.101415\pi$
$702$		0			0
$703$	17998.7	−	10391.5i	0.965624	−	0.557503i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4332.65	−	139.005i	0.230475	−	0.00739440i
$708$		0			0
$709$	−12977.4	+	22477.6i	−0.687415	+	1.19064i	0.285256	+	0.958451i	$0.407921\pi$
$709$	−12977.4	+	22477.6i	−0.687415	+	1.19064i	−0.972671	+	0.232187i	$0.925412\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	33373.2			1.75292
$714$		0			0
$715$	−1781.60			−0.0931861
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7484.50	−	12963.5i	0.388213	−	0.672404i	−0.603997	−	0.796987i	$-0.706426\pi$
$719$	7484.50	−	12963.5i	0.388213	−	0.672404i	0.992209	+	0.124583i	$0.0397593\pi$
$720$		0			0
$721$	−11301.3	+	7017.39i	−0.583749	+	0.362471i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−29027.3	+	16758.9i	−1.48696	+	0.858499i
$726$		0			0
$727$			4115.65i			0.209960i	0.994474	+	0.104980i	$0.0334779\pi$
$727$			4115.65i			0.209960i	−0.994474	+	0.104980i	$0.966522\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8670.50	+	15017.7i	0.438700	+	0.759851i
$732$		0			0
$733$	−9055.13	−	5227.98i	−0.456288	−	0.263438i	0.254194	−	0.967153i	$-0.418190\pi$
$733$	−9055.13	−	5227.98i	−0.456288	−	0.263438i	−0.710482	+	0.703715i	$0.751523\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4931.12	−	2846.98i	−0.246459	−	0.142293i
$738$		0			0
$739$	−11879.4	−	20575.7i	−0.591326	−	1.02421i	−0.994054	−	0.108887i	$-0.965271\pi$
$739$	−11879.4	−	20575.7i	−0.591326	−	1.02421i	0.402728	−	0.915320i	$-0.368062\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	22288.6i		−	1.10053i	−0.834992	−	0.550263i	$-0.814528\pi$
$743$		−	22288.6i		−	1.10053i	0.834992	−	0.550263i	$-0.185472\pi$
$744$		0			0
$745$	6314.55	−	3645.71i	0.310533	−	0.179286i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2843.06	+	1765.35i	−0.138696	+	0.0861210i
$750$		0			0
$751$	11424.3	−	19787.5i	0.555099	−	0.961459i	−0.442797	−	0.896622i	$-0.646014\pi$
$751$	11424.3	−	19787.5i	0.555099	−	0.961459i	0.997896	−	0.0648371i	$-0.0206528\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8528.84			0.411121
$756$		0			0
$757$	26376.9			1.26643			0.633214	−	0.773977i	$-0.281735\pi$
$757$	26376.9			1.26643			0.633214	+	0.773977i	$0.281735\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−18855.3	+	32658.4i	−0.898168	+	1.55567i	−0.0683340	+	0.997663i	$0.521768\pi$
$761$	−18855.3	+	32658.4i	−0.898168	+	1.55567i	−0.829834	+	0.558010i	$0.811565\pi$
$762$		0			0
$763$	−36772.6	+	1179.79i	−1.74477	+	0.0559779i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−43632.1	+	25191.0i	−2.05406	+	1.18591i
$768$		0			0
$769$			34719.6i			1.62812i	0.580783	+	0.814059i	$0.302747\pi$
$769$			34719.6i			1.62812i	−0.580783	+	0.814059i	$0.697253\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2092.57	−	3624.43i	−0.0973666	−	0.168644i	0.813227	−	0.581946i	$-0.197709\pi$
$773$	−2092.57	−	3624.43i	−0.0973666	−	0.168644i	−0.910594	+	0.413302i	$0.864375\pi$
$774$		0			0
$775$	−23186.1	−	13386.5i	−1.07467	−	0.620460i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13870.7	+	8008.24i	0.637957	+	0.368325i
$780$		0			0
$781$	−2430.73	−	4210.15i	−0.111368	−	0.192895i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1329.76i			0.0604600i
$786$		0			0
$787$	−21264.4	+	12277.0i	−0.963144	+	0.556071i	−0.897139	−	0.441748i	$-0.854359\pi$
$787$	−21264.4	+	12277.0i	−0.963144	+	0.556071i	−0.0660047	+	0.997819i	$0.521025\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	13740.4	+	7355.91i	0.617638	+	0.330653i
$792$		0			0
$793$	−13438.9	+	23276.9i	−0.601803	+	1.04235i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11823.7			0.525493			0.262747	−	0.964865i	$-0.415372\pi$
$797$	11823.7			0.525493			0.262747	+	0.964865i	$0.415372\pi$
$798$		0			0
$799$	38377.3			1.69924
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3449.32	+	5974.41i	−0.151587	+	0.262556i
$804$		0			0
$805$	−236.451	−	7369.91i	−0.0103525	−	0.322677i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24768.5	+	14300.1i	−1.07641	+	0.621465i	−0.929926	−	0.367748i	$-0.880129\pi$
$809$	−24768.5	+	14300.1i	−1.07641	+	0.621465i	−0.146484	+	0.989213i	$0.546796\pi$
$810$		0			0
$811$		−	1116.56i		−	0.0483449i	−0.999708	−	0.0241724i	$-0.992305\pi$
$811$		−	1116.56i		−	0.0483449i	0.999708	−	0.0241724i	$-0.00769507\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4366.44	+	7562.90i	0.187669	+	0.325052i
$816$		0			0
$817$	−10328.3	−	5963.06i	−0.442279	−	0.255350i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−29818.0	−	17215.4i	−1.26754	−	0.731817i	−0.293022	−	0.956106i	$-0.594661\pi$
$821$	−29818.0	−	17215.4i	−1.26754	−	0.731817i	−0.974523	+	0.224289i	$0.927994\pi$
$822$		0			0
$823$	−5944.70	−	10296.5i	−0.251785	−	0.436105i	0.712232	−	0.701944i	$-0.247684\pi$
$823$	−5944.70	−	10296.5i	−0.251785	−	0.436105i	−0.964017	+	0.265839i	$0.914351\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			22873.4i			0.961775i	0.876782	+	0.480887i	$0.159685\pi$
$827$			22873.4i			0.961775i	−0.876782	+	0.480887i	$0.840315\pi$
$828$		0			0
$829$	13966.3	−	8063.46i	0.585127	−	0.337823i	−0.178041	−	0.984023i	$-0.556976\pi$
$829$	13966.3	−	8063.46i	0.585127	−	0.337823i	0.763168	+	0.646200i	$0.223643\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−15242.2	+	30806.7i	−0.633986	+	1.28138i
$834$		0			0
$835$	1663.57	−	2881.39i	0.0689464	−	0.119419i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6878.32			0.283034			0.141517	−	0.989936i	$-0.454802\pi$
$839$	6878.32			0.283034			0.141517	+	0.989936i	$0.454802\pi$
$840$		0			0
$841$	−56808.4			−2.32926
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3244.05	−	5618.85i	0.132069	−	0.228751i
$846$		0			0
$847$	12086.5	+	19464.9i	0.490314	+	0.789637i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−38293.3	+	22108.6i	−1.54251	+	0.890569i
$852$		0			0
$853$		−	45697.2i		−	1.83428i	−0.398562	−	0.917141i	$-0.630491\pi$
$853$		−	45697.2i		−	1.83428i	0.398562	−	0.917141i	$-0.369509\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7365.24	−	12757.0i	−0.293573	−	0.508483i	0.681079	−	0.732210i	$-0.261511\pi$
$857$	−7365.24	−	12757.0i	−0.293573	−	0.508483i	−0.974652	+	0.223727i	$0.928178\pi$
$858$		0			0
$859$	2791.43	+	1611.64i	0.110876	+	0.0640143i	0.554413	−	0.832242i	$-0.312943\pi$
$859$	2791.43	+	1611.64i	0.110876	+	0.0640143i	−0.443536	+	0.896256i	$0.646276\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−29743.0	−	17172.1i	−1.17319	−	0.677342i	−0.218762	−	0.975778i	$-0.570202\pi$
$863$	−29743.0	−	17172.1i	−1.17319	−	0.677342i	−0.954430	+	0.298436i	$0.903535\pi$
$864$		0			0
$865$	734.102	+	1271.50i	0.0288557	+	0.0499796i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	9030.16i		−	0.352505i
$870$		0			0
$871$	34469.7	−	19901.1i	1.34094	−	0.774193i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5758.82	+	10757.1i	−0.222496	+	0.415608i
$876$		0			0
$877$	5875.03	−	10175.8i	0.226209	−	0.391806i	−0.730472	−	0.682942i	$-0.760700\pi$
$877$	5875.03	−	10175.8i	0.226209	−	0.391806i	0.956682	+	0.291136i	$0.0940333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2036.67			0.0778855			0.0389428	−	0.999241i	$-0.487601\pi$
$881$	2036.67			0.0778855			0.0389428	+	0.999241i	$0.487601\pi$
$882$		0			0
$883$	8621.63			0.328585			0.164293	−	0.986412i	$-0.447466\pi$
$883$	8621.63			0.328585			0.164293	+	0.986412i	$0.447466\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16347.4	+	28314.6i	−0.618820	+	1.07183i	0.370882	+	0.928680i	$0.379056\pi$
$887$	−16347.4	+	28314.6i	−0.618820	+	1.07183i	−0.989701	+	0.143147i	$0.954278\pi$
$888$		0			0
$889$	−41.4509	+	77.4275i	−0.00156380	+	0.00292108i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−22857.5	+	13196.8i	−0.856549	+	0.494529i
$894$		0			0
$895$		−	9609.16i		−	0.358881i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−32428.9	−	56168.4i	−1.20307	−	2.08378i
$900$		0			0
$901$	−2230.64	−	1287.86i	−0.0824788	−	0.0476192i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3705.92	−	2139.61i	−0.136120	−	0.0785892i
$906$		0			0
$907$	−16338.3	−	28298.7i	−0.598129	−	1.03599i	−0.993097	−	0.117295i	$-0.962578\pi$
$907$	−16338.3	−	28298.7i	−0.598129	−	1.03599i	0.394968	−	0.918695i	$-0.370756\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	9801.48i		−	0.356463i	−0.983989	−	0.178231i	$-0.942962\pi$
$911$		−	9801.48i		−	0.356463i	0.983989	−	0.178231i	$-0.0570375\pi$
$912$		0			0
$913$	7026.04	−	4056.49i	0.254686	−	0.147043i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	11765.7	+	18948.4i	0.423705	+	0.682367i
$918$		0			0
$919$	−11747.6	+	20347.5i	−0.421674	+	0.730361i	−0.996103	−	0.0881935i	$-0.971891\pi$
$919$	−11747.6	+	20347.5i	−0.421674	+	0.730361i	0.574429	+	0.818554i	$0.305224\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	33982.8			1.21187
$924$		0			0
$925$	35472.5			1.26089
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17874.3	−	30959.1i	0.631255	−	1.09337i	−0.356041	−	0.934470i	$-0.615874\pi$
$929$	17874.3	−	30959.1i	0.631255	−	1.09337i	0.987296	−	0.158895i	$-0.0507930\pi$
$930$		0			0
$931$	−1515.24	−	23589.8i	−0.0533404	−	0.830425i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2283.00	+	1318.09i	−0.0798524	+	0.0461028i
$936$		0			0
$937$		−	34509.4i		−	1.20317i	−0.798808	−	0.601587i	$-0.794535\pi$
$937$		−	34509.4i		−	1.20317i	0.798808	−	0.601587i	$-0.205465\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12219.9	−	21165.5i	−0.423334	−	0.733236i	0.572929	−	0.819605i	$-0.305807\pi$
$941$	−12219.9	−	21165.5i	−0.423334	−	0.733236i	−0.996263	+	0.0863688i	$0.972474\pi$
$942$		0			0
$943$	−29510.7	−	17038.0i	−1.01909	−	0.588371i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2055.28	−	1186.61i	−0.0705253	−	0.0407178i	0.464323	−	0.885666i	$-0.346298\pi$
$947$	−2055.28	−	1186.61i	−0.0705253	−	0.0407178i	−0.534848	+	0.844948i	$0.679631\pi$
$948$		0			0
$949$	−24111.6	−	41762.5i	−0.824758	−	1.42852i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			18404.5i			0.625583i	0.949822	+	0.312791i	$0.101264\pi$
$953$			18404.5i			0.625583i	−0.949822	+	0.312791i	$0.898736\pi$
$954$		0			0
$955$	−5677.98	+	3278.18i	−0.192393	+	0.111078i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	60.6354	+	1889.94i	0.00204173	+	0.0636385i
$960$		0			0
$961$	11007.6	−	19065.7i	0.369494	−	0.639982i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1132.65			−0.0377836
$966$		0			0
$967$	−55423.0			−1.84311			−0.921553	−	0.388252i	$-0.873079\pi$
$967$	−55423.0			−1.84311			−0.921553	+	0.388252i	$0.873079\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3770.87	−	6531.34i	0.124627	−	0.215861i	−0.796960	−	0.604032i	$-0.793560\pi$
$971$	3770.87	−	6531.34i	0.124627	−	0.215861i	0.921587	+	0.388172i	$0.126893\pi$
$972$		0			0
$973$	12706.0	+	6802.15i	0.418638	+	0.224118i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21592.6	+	12466.5i	−0.707071	+	0.408227i	−0.809975	−	0.586464i	$-0.800520\pi$
$977$	−21592.6	+	12466.5i	−0.707071	+	0.408227i	0.102905	+	0.994691i	$0.467186\pi$
$978$		0			0
$979$		−	8570.61i		−	0.279793i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18025.7	−	31221.4i	−0.584872	−	1.01303i	−0.994891	−	0.100951i	$-0.967811\pi$
$983$	−18025.7	−	31221.4i	−0.584872	−	1.01303i	0.410019	−	0.912077i	$-0.365522\pi$
$984$		0			0
$985$	−1564.26	−	903.127i	−0.0506006	−	0.0292142i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	21974.1	+	12686.8i	0.706508	+	0.407902i
$990$		0			0
$991$	15099.7	+	26153.4i	0.484013	+	0.838335i	0.999831	−	0.0183632i	$-0.00584552\pi$
$991$	15099.7	+	26153.4i	0.484013	+	0.838335i	−0.515819	+	0.856698i	$0.672512\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	5963.33i		−	0.190000i
$996$		0			0
$997$	−4898.38	+	2828.08i	−0.155600	+	0.0898357i	−0.575778	−	0.817606i	$-0.695301\pi$
$997$	−4898.38	+	2828.08i	−0.155600	+	0.0898357i	0.420178	+	0.907442i	$0.361968\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.bt.d.593.12		48
3.2	odd	2	inner	1008.4.bt.d.593.13		48
4.3	odd	2		504.4.bl.a.89.12	yes	48
7.3	odd	6	inner	1008.4.bt.d.17.13		48
12.11	even	2		504.4.bl.a.89.13	yes	48
21.17	even	6	inner	1008.4.bt.d.17.12		48
28.3	even	6		504.4.bl.a.17.13	yes	48
84.59	odd	6		504.4.bl.a.17.12	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.12	✓	48	84.59	odd	6
504.4.bl.a.17.13	yes	48	28.3	even	6
504.4.bl.a.89.12	yes	48	4.3	odd	2
504.4.bl.a.89.13	yes	48	12.11	even	2
1008.4.bt.d.17.12		48	21.17	even	6	inner
1008.4.bt.d.17.13		48	7.3	odd	6	inner
1008.4.bt.d.593.12		48	1.1	even	1	trivial
1008.4.bt.d.593.13		48	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.bt (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.35769 + 2.35159i) q^{5} +(8.74105 - 16.3277i) q^{7} +O(q^{10})\)	\(q+(-1.35769 + 2.35159i) q^{5} +(8.74105 - 16.3277i) q^{7} +(-8.39023 + 4.84410i) q^{11} -67.7228i q^{13} +(-50.1037 - 86.7822i) q^{17} +(59.6837 + 34.4584i) q^{19} +(-126.981 - 73.3122i) q^{23} +(58.8133 + 101.868i) q^{25} +284.951i q^{29} +(-197.116 + 113.805i) q^{31} +(26.5285 + 42.7234i) q^{35} +(150.784 - 261.166i) q^{37} +232.403 q^{41} -173.051 q^{43} +(-191.489 + 331.669i) q^{47} +(-190.188 - 285.443i) q^{49} +(22.2602 - 12.8520i) q^{53} -26.3072i q^{55} +(-371.973 - 644.275i) q^{59} +(-343.708 - 198.440i) q^{61} +(159.257 + 91.9468i) q^{65} +(293.861 + 508.981i) q^{67} +501.792i q^{71} +(616.668 - 356.033i) q^{73} +(5.75368 + 179.336i) q^{77} +(-466.039 + 807.203i) q^{79} -837.407 q^{83} +272.102 q^{85} +(-442.322 + 766.124i) q^{89} +(-1105.76 - 591.968i) q^{91} +(-162.064 + 93.5678i) q^{95} -1175.54i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{7}+O(q^{10}) \) 48 * q + 24 * q^7	\( 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) \) 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Introduction

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