LMFDB - Embedded newform 624.4.n.d.623.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,4,Mod(623,624)] 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("624.623"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.n (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [56,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			623.20
Character	$\chi$	$=$	624.623
Dual form			624.4.n.d.623.18

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.39946 + 4.60897i) q^{3} +1.65893 q^{5} -4.60037 q^{7} +(-15.4851 - 22.1181i) q^{9} -54.6366i q^{11} +(25.2167 + 39.5110i) q^{13} +(-3.98055 + 7.64597i) q^{15} +43.8926i q^{17} +80.0323 q^{19} +(11.0384 - 21.2030i) q^{21} +22.8507 q^{23} -122.248 q^{25} +(139.098 - 18.2989i) q^{27} +223.650i q^{29} +24.7864 q^{31} +(251.818 + 131.099i) q^{33} -7.63171 q^{35} +318.937i q^{37} +(-242.611 + 21.4174i) q^{39} +253.460 q^{41} +21.9864i q^{43} +(-25.6888 - 36.6924i) q^{45} +78.3054i q^{47} -321.837 q^{49} +(-202.300 - 105.319i) q^{51} -302.530i q^{53} -90.6385i q^{55} +(-192.035 + 368.866i) q^{57} -480.544i q^{59} -332.203 q^{61} +(71.2375 + 101.752i) q^{63} +(41.8327 + 65.5461i) q^{65} -844.699 q^{67} +(-54.8293 + 105.318i) q^{69} +651.949i q^{71} +455.933i q^{73} +(293.330 - 563.437i) q^{75} +251.349i q^{77} -246.129i q^{79} +(-249.421 + 685.004i) q^{81} +1417.67i q^{83} +72.8149i q^{85} +(-1030.79 - 536.639i) q^{87} +891.167 q^{89} +(-116.006 - 181.765i) q^{91} +(-59.4742 + 114.240i) q^{93} +132.768 q^{95} +1596.67i q^{97} +(-1208.46 + 846.056i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 64 q^{9} + 8 q^{13} + 1208 q^{25} + 4680 q^{49} + 1616 q^{61} + 480 q^{69} + 1568 q^{81}+O(q^{100}) $ 56 * q - 64 * q^9 + 8 * q^13 + 1208 * q^25 + 4680 * q^49 + 1616 * q^61 + 480 * q^69 + 1568 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	−2.39946	+	4.60897i	−0.461777	+	0.886996i
$3$	−2.39946	+	4.60897i	−0.461777	+	0.886996i
$4$		0			0
$5$	1.65893			0.148379			0.0741897	−	0.997244i	$-0.476363\pi$
$5$	1.65893			0.148379			0.0741897	+	0.997244i	$0.476363\pi$
$6$		0			0
$7$	−4.60037			−0.248397			−0.124198	−	0.992257i	$-0.539636\pi$
$7$	−4.60037			−0.248397			−0.124198	+	0.992257i	$0.539636\pi$
$8$		0			0
$9$	−15.4851	−	22.1181i	−0.573524	−	0.819189i
$10$		0			0
$11$		−	54.6366i		−	1.49760i	−0.662798	−	0.748799i	$-0.730631\pi$
$11$		−	54.6366i		−	1.49760i	0.662798	−	0.748799i	$-0.269369\pi$
$12$		0			0
$13$	25.2167	+	39.5110i	0.537988	+	0.842953i
$13$	25.2167	+	39.5110i	0.537988	+	0.842953i
$14$		0			0
$15$	−3.98055	+	7.64597i	−0.0685182	+	0.131612i
$16$		0			0
$17$			43.8926i			0.626208i	0.949719	+	0.313104i	$0.101369\pi$
$17$			43.8926i			0.626208i	−0.949719	+	0.313104i	$0.898631\pi$
$18$		0			0
$19$	80.0323			0.966351			0.483176	−	0.875523i	$-0.339483\pi$
$19$	80.0323			0.966351			0.483176	+	0.875523i	$0.339483\pi$
$20$		0			0
$21$	11.0384	−	21.2030i	0.114704	−	0.220327i
$22$		0			0
$23$	22.8507			0.207161			0.103580	−	0.994621i	$-0.466970\pi$
$23$	22.8507			0.207161			0.103580	+	0.994621i	$0.466970\pi$
$24$		0			0
$25$	−122.248			−0.977984
$26$		0			0
$27$	139.098	−	18.2989i	0.991457	−	0.130431i
$28$		0			0
$29$			223.650i			1.43209i	0.698053	+	0.716047i	$0.254050\pi$
$29$			223.650i			1.43209i	−0.698053	+	0.716047i	$0.745950\pi$
$30$		0			0
$31$	24.7864			0.143606			0.0718029	−	0.997419i	$-0.477125\pi$
$31$	24.7864			0.143606			0.0718029	+	0.997419i	$0.477125\pi$
$32$		0			0
$33$	251.818	+	131.099i	1.32836	+	0.691556i
$34$		0			0
$35$	−7.63171			−0.0368570
$36$		0			0
$37$			318.937i			1.41711i	0.705658	+	0.708553i	$0.250652\pi$
$37$			318.937i			1.41711i	−0.705658	+	0.708553i	$0.749348\pi$
$38$		0			0
$39$	−242.611	+	21.4174i	−0.996126	+	0.0879368i
$40$		0			0
$41$	253.460			0.965459			0.482730	−	0.875769i	$-0.339645\pi$
$41$	253.460			0.965459			0.482730	+	0.875769i	$0.339645\pi$
$42$		0			0
$43$			21.9864i			0.0779742i	0.999240	+	0.0389871i	$0.0124131\pi$
$43$			21.9864i			0.0779742i	−0.999240	+	0.0389871i	$0.987587\pi$
$44$		0			0
$45$	−25.6888	−	36.6924i	−0.0850992	−	0.121551i
$46$		0			0
$47$			78.3054i			0.243022i	0.992590	+	0.121511i	$0.0387739\pi$
$47$			78.3054i			0.243022i	−0.992590	+	0.121511i	$0.961226\pi$
$48$		0			0
$49$	−321.837			−0.938299
$50$		0			0
$51$	−202.300	−	105.319i	−0.555444	−	0.289168i
$52$		0			0
$53$		−	302.530i		−	0.784069i	−0.919951	−	0.392034i	$-0.871771\pi$
$53$		−	302.530i		−	0.784069i	0.919951	−	0.392034i	$-0.128229\pi$
$54$		0			0
$55$		−	90.6385i		−	0.222213i
$56$		0			0
$57$	−192.035	+	368.866i	−0.446239	+	0.857150i
$58$		0			0
$59$		−	480.544i		−	1.06037i	−0.847883	−	0.530183i	$-0.822123\pi$
$59$		−	480.544i		−	1.06037i	0.847883	−	0.530183i	$-0.177877\pi$
$60$		0			0
$61$	−332.203			−0.697282			−0.348641	−	0.937256i	$-0.613357\pi$
$61$	−332.203			−0.697282			−0.348641	+	0.937256i	$0.613357\pi$
$62$		0			0
$63$	71.2375	+	101.752i	0.142462	+	0.203484i
$64$		0			0
$65$	41.8327	+	65.5461i	0.0798263	+	0.125077i
$66$		0			0
$67$	−844.699			−1.54025			−0.770123	−	0.637896i	$-0.779805\pi$
$67$	−844.699			−1.54025			−0.770123	+	0.637896i	$0.779805\pi$
$68$		0			0
$69$	−54.8293	+	105.318i	−0.0956620	+	0.183751i
$70$		0			0
$71$			651.949i			1.08975i	0.838518	+	0.544874i	$0.183423\pi$
$71$			651.949i			1.08975i	−0.838518	+	0.544874i	$0.816577\pi$
$72$		0			0
$73$			455.933i			0.731000i	0.930811	+	0.365500i	$0.119102\pi$
$73$			455.933i			0.731000i	−0.930811	+	0.365500i	$0.880898\pi$
$74$		0			0
$75$	293.330	−	563.437i	0.451610	−	0.867467i
$76$		0			0
$77$			251.349i			0.371998i
$78$		0			0
$79$		−	246.129i		−	0.350528i	−0.984521	−	0.175264i	$-0.943922\pi$
$79$		−	246.129i		−	0.350528i	0.984521	−	0.175264i	$-0.0560779\pi$
$80$		0			0
$81$	−249.421	+	685.004i	−0.342141	+	0.939649i
$82$		0			0
$83$			1417.67i			1.87481i	0.348239	+	0.937406i	$0.386780\pi$
$83$			1417.67i			1.87481i	−0.348239	+	0.937406i	$0.613220\pi$
$84$		0			0
$85$			72.8149i			0.0929163i
$86$		0			0
$87$	−1030.79	−	536.639i	−1.27026	−	0.661308i
$88$		0			0
$89$	891.167			1.06139			0.530694	−	0.847563i	$-0.321931\pi$
$89$	891.167			1.06139			0.530694	+	0.847563i	$0.321931\pi$
$90$		0			0
$91$	−116.006	−	181.765i	−0.133634	−	0.209387i
$92$		0			0
$93$	−59.4742	+	114.240i	−0.0663138	+	0.127378i
$94$		0			0
$95$	132.768			0.143387
$96$		0			0
$97$			1596.67i			1.67131i	0.549252	+	0.835657i	$0.314913\pi$
$97$			1596.67i			1.67131i	−0.549252	+	0.835657i	$0.685087\pi$
$98$		0			0
$99$	−1208.46	+	846.056i	−1.22681	+	0.858908i
$100$		0			0
$101$		−	1177.81i		−	1.16036i	−0.814488	−	0.580180i	$-0.802982\pi$
$101$		−	1177.81i		−	1.16036i	0.814488	−	0.580180i	$-0.197018\pi$
$102$		0			0
$103$		−	300.997i		−	0.287943i	−0.989582	−	0.143971i	$-0.954013\pi$
$103$		−	300.997i		−	0.287943i	0.989582	−	0.143971i	$-0.0459873\pi$
$104$		0			0
$105$	18.3120	−	35.1743i	0.0170197	−	0.0326920i
$106$		0			0
$107$	−1054.22			−0.952475			−0.476238	−	0.879317i	$-0.658000\pi$
$107$	−1054.22			−0.952475			−0.476238	+	0.879317i	$0.658000\pi$
$108$		0			0
$109$			1013.62i			0.890704i	0.895356	+	0.445352i	$0.146922\pi$
$109$			1013.62i			0.890704i	−0.895356	+	0.445352i	$0.853078\pi$
$110$		0			0
$111$	−1469.97	−	765.277i	−1.25697	−	0.654387i
$112$		0			0
$113$			1948.14i			1.62182i	0.585174	+	0.810908i	$0.301026\pi$
$113$			1948.14i			1.62182i	−0.585174	+	0.810908i	$0.698974\pi$
$114$		0			0
$115$	37.9077			0.0307384
$116$		0			0
$117$	483.425	−	1169.58i	0.381989	−	0.924167i
$118$		0			0
$119$		−	201.923i		−	0.155548i
$120$		0			0
$121$	−1654.16			−1.24280
$122$		0			0
$123$	−608.169	+	1168.19i	−0.445827	+	0.856359i
$124$		0			0
$125$	−410.168			−0.293492
$126$		0			0
$127$			1299.81i			0.908183i	0.890955	+	0.454091i	$0.150036\pi$
$127$			1299.81i			0.908183i	−0.890955	+	0.454091i	$0.849964\pi$
$128$		0			0
$129$	−101.334	−	52.7555i	−0.0691628	−	0.0360067i
$130$		0			0
$131$	−1995.14			−1.33066			−0.665330	−	0.746549i	$-0.731709\pi$
$131$	−1995.14			−1.33066			−0.665330	+	0.746549i	$0.731709\pi$
$132$		0			0
$133$	−368.179			−0.240039
$134$		0			0
$135$	230.754	−	30.3567i	0.147112	−	0.0193532i
$136$		0			0
$137$	1263.42			0.787890			0.393945	−	0.919134i	$-0.371110\pi$
$137$	1263.42			0.787890			0.393945	+	0.919134i	$0.371110\pi$
$138$		0			0
$139$		−	441.052i		−	0.269133i	−0.990905	−	0.134567i	$-0.957036\pi$
$139$		−	441.052i		−	0.269133i	0.990905	−	0.134567i	$-0.0429643\pi$
$140$		0			0
$141$	−360.907	−	187.891i	−0.215559	−	0.112222i
$142$		0			0
$143$	2158.75	−	1377.75i	1.26240	−	0.805689i
$144$		0			0
$145$			371.020i			0.212493i
$146$		0			0
$147$	772.235	−	1483.33i	0.433285	−	0.832267i
$148$		0			0
$149$	−1851.20			−1.01783			−0.508914	−	0.860818i	$-0.669953\pi$
$149$	−1851.20			−1.01783			−0.508914	+	0.860818i	$0.669953\pi$
$150$		0			0
$151$	2670.01			1.43896			0.719478	−	0.694515i	$-0.244381\pi$
$151$	2670.01			1.43896			0.719478	+	0.694515i	$0.244381\pi$
$152$		0			0
$153$	970.822	−	679.684i	0.512982	−	0.359145i
$154$		0			0
$155$	41.1191			0.0213081
$156$		0			0
$157$	296.337			0.150638			0.0753192	−	0.997159i	$-0.476002\pi$
$157$	296.337			0.150638			0.0753192	+	0.997159i	$0.476002\pi$
$158$		0			0
$159$	1394.35	+	725.909i	0.695466	+	0.362065i
$160$		0			0
$161$	−105.122			−0.0514580
$162$		0			0
$163$	−41.4335			−0.0199100			−0.00995498	−	0.999950i	$-0.503169\pi$
$163$	−41.4335			−0.0199100			−0.00995498	+	0.999950i	$0.503169\pi$
$164$		0			0
$165$	417.750	+	217.484i	0.197102	+	0.102613i
$166$		0			0
$167$			130.547i			0.0604912i	0.999542	+	0.0302456i	$0.00962895\pi$
$167$			130.547i			0.0604912i	−0.999542	+	0.0302456i	$0.990371\pi$
$168$		0			0
$169$	−925.241	+	1992.67i	−0.421138	+	0.906996i
$170$		0			0
$171$	−1239.31	−	1770.16i	−0.554226	−	0.791624i
$172$		0			0
$173$			2113.53i			0.928834i	0.885617	+	0.464417i	$0.153736\pi$
$173$			2113.53i			0.928834i	−0.885617	+	0.464417i	$0.846264\pi$
$174$		0			0
$175$	562.386			0.242928
$176$		0			0
$177$	2214.81	+	1153.05i	0.940540	+	0.489652i
$178$		0			0
$179$	2146.63			0.896349			0.448175	−	0.893946i	$-0.352074\pi$
$179$	2146.63			0.896349			0.448175	+	0.893946i	$0.352074\pi$
$180$		0			0
$181$	−90.3221			−0.0370917			−0.0185458	−	0.999828i	$-0.505904\pi$
$181$	−90.3221			−0.0370917			−0.0185458	+	0.999828i	$0.505904\pi$
$182$		0			0
$183$	797.109	−	1531.11i	0.321989	−	0.618487i
$184$		0			0
$185$			529.095i			0.210269i
$186$		0			0
$187$	2398.15			0.937807
$188$		0			0
$189$	−639.901	+	84.1819i	−0.246275	+	0.0323986i
$190$		0			0
$191$	−2064.15			−0.781972			−0.390986	−	0.920397i	$-0.627866\pi$
$191$	−2064.15			−0.781972			−0.390986	+	0.920397i	$0.627866\pi$
$192$		0			0
$193$			862.814i			0.321796i	0.986971	+	0.160898i	$0.0514391\pi$
$193$			862.814i			0.321796i	−0.986971	+	0.160898i	$0.948561\pi$
$194$		0			0
$195$	−402.476	+	35.5301i	−0.147805	+	0.0130480i
$196$		0			0
$197$	3824.14			1.38304			0.691520	−	0.722357i	$-0.256941\pi$
$197$	3824.14			1.38304			0.691520	+	0.722357i	$0.256941\pi$
$198$		0			0
$199$			5263.10i			1.87483i	0.348213	+	0.937416i	$0.386789\pi$
$199$			5263.10i			1.87483i	−0.348213	+	0.937416i	$0.613211\pi$
$200$		0			0
$201$	2026.83	−	3893.19i	0.711250	−	1.36619i
$202$		0			0
$203$		−	1028.87i		−	0.355727i
$204$		0			0
$205$	420.473			0.143254
$206$		0			0
$207$	−353.846	−	505.413i	−0.118812	−	0.169704i
$208$		0			0
$209$		−	4372.70i		−	1.44721i
$210$		0			0
$211$			1902.37i			0.620685i	0.950625	+	0.310342i	$0.100444\pi$
$211$			1902.37i			0.620685i	−0.950625	+	0.310342i	$0.899556\pi$
$212$		0			0
$213$	−3004.81	−	1564.33i	−0.966603	−	0.503221i
$214$		0			0
$215$			36.4739i			0.0115698i
$216$		0			0
$217$	−114.027			−0.0356712
$218$		0			0
$219$	−2101.38	−	1094.00i	−0.648394	−	0.337559i
$220$		0			0
$221$	−1734.24	+	1106.83i	−0.527863	+	0.336892i
$222$		0			0
$223$	5158.67			1.54910			0.774551	−	0.632511i	$-0.217976\pi$
$223$	5158.67			1.54910			0.774551	+	0.632511i	$0.217976\pi$
$224$		0			0
$225$	1893.03	+	2703.89i	0.560897	+	0.801153i
$226$		0			0
$227$		−	1869.22i		−	0.546541i	−0.961937	−	0.273270i	$-0.911895\pi$
$227$		−	1869.22i		−	0.546541i	0.961937	−	0.273270i	$-0.0881053\pi$
$228$		0			0
$229$		−	2540.62i		−	0.733139i	−0.930391	−	0.366570i	$-0.880532\pi$
$229$		−	2540.62i		−	0.733139i	0.930391	−	0.366570i	$-0.119468\pi$
$230$		0			0
$231$	−1158.46	−	603.103i	−0.329961	−	0.171780i
$232$		0			0
$233$		−	3006.01i		−	0.845196i	−0.906317	−	0.422598i	$-0.861118\pi$
$233$		−	3006.01i		−	0.845196i	0.906317	−	0.422598i	$-0.138882\pi$
$234$		0			0
$235$			129.903i			0.0360594i
$236$		0			0
$237$	1134.40	+	590.578i	0.310917	+	0.161866i
$238$		0			0
$239$		−	2829.85i		−	0.765889i	−0.923771	−	0.382945i	$-0.874910\pi$
$239$		−	2829.85i		−	0.765889i	0.923771	−	0.382945i	$-0.125090\pi$
$240$		0			0
$241$		−	2951.84i		−	0.788982i	−0.918900	−	0.394491i	$-0.870921\pi$
$241$		−	2951.84i		−	0.788982i	0.918900	−	0.394491i	$-0.129079\pi$
$242$		0			0
$243$	−2558.68	−	2793.21i	−0.675472	−	0.737386i
$244$		0			0
$245$	−533.905			−0.139224
$246$		0			0
$247$	2018.15	+	3162.16i	0.519885	+	0.814588i
$248$		0			0
$249$	−6533.99	−	3401.65i	−1.66295	−	0.865745i
$250$		0			0
$251$	1456.36			0.366234			0.183117	−	0.983091i	$-0.441381\pi$
$251$	1456.36			0.366234			0.183117	+	0.983091i	$0.441381\pi$
$252$		0			0
$253$		−	1248.48i		−	0.310243i
$254$		0			0
$255$	−335.602	−	174.717i	−0.0824164	−	0.0429066i
$256$		0			0
$257$			2029.62i			0.492623i	0.969191	+	0.246311i	$0.0792186\pi$
$257$			2029.62i			0.492623i	−0.969191	+	0.246311i	$0.920781\pi$
$258$		0			0
$259$		−	1467.23i		−	0.352004i
$260$		0			0
$261$	4946.71	−	3463.25i	1.17315	−	0.821340i
$262$		0			0
$263$	6532.66			1.53164			0.765819	−	0.643056i	$-0.222334\pi$
$263$	6532.66			1.53164			0.765819	+	0.643056i	$0.222334\pi$
$264$		0			0
$265$		−	501.876i		−	0.116340i
$266$		0			0
$267$	−2138.32	+	4107.36i	−0.490125	+	0.941447i
$268$		0			0
$269$		−	5078.45i		−	1.15107i	−0.817776	−	0.575536i	$-0.804793\pi$
$269$		−	5078.45i		−	1.15107i	0.817776	−	0.575536i	$-0.195207\pi$
$270$		0			0
$271$	5338.90			1.19673			0.598367	−	0.801222i	$-0.295816\pi$
$271$	5338.90			1.19673			0.598367	+	0.801222i	$0.295816\pi$
$272$		0			0
$273$	1116.10	−	98.5282i	0.247435	−	0.0218432i
$274$		0			0
$275$			6679.22i			1.46463i
$276$		0			0
$277$	752.446			0.163213			0.0816066	−	0.996665i	$-0.473995\pi$
$277$	752.446			0.163213			0.0816066	+	0.996665i	$0.473995\pi$
$278$		0			0
$279$	−383.822	−	548.229i	−0.0823613	−	0.117640i
$280$		0			0
$281$	−7177.82			−1.52382			−0.761909	−	0.647684i	$-0.775738\pi$
$281$	−7177.82			−1.52382			−0.761909	+	0.647684i	$0.775738\pi$
$282$		0			0
$283$			2174.61i			0.456775i	0.973570	+	0.228387i	$0.0733453\pi$
$283$			2174.61i			0.456775i	−0.973570	+	0.228387i	$0.926655\pi$
$284$		0			0
$285$	−318.573	+	611.924i	−0.0662127	+	0.127183i
$286$		0			0
$287$	−1166.01			−0.239817
$288$		0			0
$289$	2986.44			0.607864
$290$		0			0
$291$	−7359.00	−	3831.15i	−1.48245	−	0.771774i
$292$		0			0
$293$	−8725.85			−1.73983			−0.869914	−	0.493204i	$-0.835826\pi$
$293$	−8725.85			−1.73983			−0.869914	+	0.493204i	$0.835826\pi$
$294$		0			0
$295$		−	797.191i		−	0.157336i
$296$		0			0
$297$	−999.792	−	7599.83i	−0.195333	−	1.48480i
$298$		0			0
$299$	576.217	+	902.853i	0.111450	+	0.174627i
$300$		0			0
$301$		−	101.146i		−	0.0193685i
$302$		0			0
$303$	5428.48	+	2826.11i	1.02923	+	0.535827i
$304$		0			0
$305$	−551.102			−0.103462
$306$		0			0
$307$	−164.031			−0.0304943			−0.0152471	−	0.999884i	$-0.504853\pi$
$307$	−164.031			−0.0304943			−0.0152471	+	0.999884i	$0.504853\pi$
$308$		0			0
$309$	1387.29	+	722.232i	0.255404	+	0.132965i
$310$		0			0
$311$	−8812.84			−1.60685			−0.803425	−	0.595406i	$-0.796991\pi$
$311$	−8812.84			−1.60685			−0.803425	+	0.595406i	$0.796991\pi$
$312$		0			0
$313$	7228.47			1.30536			0.652679	−	0.757634i	$-0.273645\pi$
$313$	7228.47			1.30536			0.652679	+	0.757634i	$0.273645\pi$
$314$		0			0
$315$	118.178	+	168.799i	0.0211384	+	0.0301928i
$316$		0			0
$317$	9131.24			1.61786			0.808930	−	0.587906i	$-0.200047\pi$
$317$	9131.24			1.61786			0.808930	+	0.587906i	$0.200047\pi$
$318$		0			0
$319$	12219.5			2.14470
$320$		0			0
$321$	2529.55	−	4858.84i	0.439831	−	0.844842i
$322$		0			0
$323$			3512.83i			0.605137i
$324$		0			0
$325$	−3082.68	−	4830.14i	−0.526143	−	0.824394i
$326$		0			0
$327$	−4671.72	−	2432.13i	−0.790051	−	0.411307i
$328$		0			0
$329$		−	360.234i		−	0.0603658i
$330$		0			0
$331$	−3166.25			−0.525779			−0.262889	−	0.964826i	$-0.584675\pi$
$331$	−3166.25			−0.525779			−0.262889	+	0.964826i	$0.584675\pi$
$332$		0			0
$333$	7054.28	−	4938.78i	1.16088	−	0.812744i
$334$		0			0
$335$	−1401.30			−0.228541
$336$		0			0
$337$	−4294.44			−0.694163			−0.347081	−	0.937835i	$-0.612827\pi$
$337$	−4294.44			−0.694163			−0.347081	+	0.937835i	$0.612827\pi$
$338$		0			0
$339$	−8978.89	−	4674.48i	−1.43854	−	0.748918i
$340$		0			0
$341$		−	1354.25i		−	0.215064i
$342$		0			0
$343$	3058.50			0.481467
$344$		0			0
$345$	−90.9582	+	174.715i	−0.0141943	+	0.0272648i
$346$		0			0
$347$	−4226.06			−0.653794			−0.326897	−	0.945060i	$-0.606003\pi$
$347$	−4226.06			−0.653794			−0.326897	+	0.945060i	$0.606003\pi$
$348$		0			0
$349$			1683.16i			0.258159i	0.991634	+	0.129080i	$0.0412023\pi$
$349$			1683.16i			0.258159i	−0.991634	+	0.129080i	$0.958798\pi$
$350$		0			0
$351$	4230.59	+	5034.45i	0.643339	+	0.765582i
$352$		0			0
$353$	−2459.28			−0.370805			−0.185402	−	0.982663i	$-0.559359\pi$
$353$	−2459.28			−0.370805			−0.185402	+	0.982663i	$0.559359\pi$
$354$		0			0
$355$			1081.54i			0.161696i
$356$		0			0
$357$	930.654	+	484.506i	0.137970	+	0.0718285i
$358$		0			0
$359$		−	5407.08i		−	0.794916i	−0.917620	−	0.397458i	$-0.869892\pi$
$359$		−	5407.08i		−	0.794916i	0.917620	−	0.397458i	$-0.130108\pi$
$360$		0			0
$361$	−453.826			−0.0661650
$362$		0			0
$363$	3969.11	−	7623.98i	0.573895	−	1.10236i
$364$		0			0
$365$			756.363i			0.108465i
$366$		0			0
$367$			12629.5i			1.79634i	0.439653	+	0.898168i	$0.355101\pi$
$367$			12629.5i			1.79634i	−0.439653	+	0.898168i	$0.644899\pi$
$368$		0			0
$369$	−3924.87	−	5606.06i	−0.553714	−	0.790894i
$370$		0			0
$371$			1391.75i			0.194760i
$372$		0			0
$373$	2468.27			0.342633			0.171316	−	0.985216i	$-0.445198\pi$
$373$	2468.27			0.342633			0.171316	+	0.985216i	$0.445198\pi$
$374$		0			0
$375$	984.183	−	1890.45i	0.135528	−	0.260326i
$376$		0			0
$377$	−8836.63	+	5639.70i	−1.20719	+	0.770448i
$378$		0			0
$379$	2729.43			0.369924			0.184962	−	0.982746i	$-0.440784\pi$
$379$	2729.43			0.369924			0.184962	+	0.982746i	$0.440784\pi$
$380$		0			0
$381$	−5990.77	−	3118.84i	−0.805555	−	0.419378i
$382$		0			0
$383$		−	11643.5i		−	1.55341i	−0.629865	−	0.776704i	$-0.716890\pi$
$383$		−	11643.5i		−	1.55341i	0.629865	−	0.776704i	$-0.283110\pi$
$384$		0			0
$385$			416.971i			0.0551969i
$386$		0			0
$387$	486.297	−	340.462i	0.0638756	−	0.0447201i
$388$		0			0
$389$			4944.04i			0.644404i	0.946671	+	0.322202i	$0.104423\pi$
$389$			4944.04i			0.644404i	−0.946671	+	0.322202i	$0.895577\pi$
$390$		0			0
$391$			1002.98i			0.129725i
$392$		0			0
$393$	4787.28	−	9195.55i	0.614469	−	1.18029i
$394$		0			0
$395$		−	408.312i		−	0.0520111i
$396$		0			0
$397$			8136.65i			1.02863i	0.857601	+	0.514316i	$0.171954\pi$
$397$			8136.65i			1.02863i	−0.857601	+	0.514316i	$0.828046\pi$
$398$		0			0
$399$	883.431	−	1696.92i	0.110844	−	0.212913i
$400$		0			0
$401$	7334.78			0.913420			0.456710	−	0.889616i	$-0.349028\pi$
$401$	7334.78			0.913420			0.456710	+	0.889616i	$0.349028\pi$
$402$		0			0
$403$	625.031	+	979.338i	0.0772581	+	0.121053i
$404$		0			0
$405$	−413.772	+	1136.38i	−0.0507667	+	0.139425i
$406$		0			0
$407$	17425.6			2.12225
$408$		0			0
$409$			3604.80i			0.435809i	0.975970	+	0.217904i	$0.0699221\pi$
$409$			3604.80i			0.435809i	−0.975970	+	0.217904i	$0.930078\pi$
$410$		0			0
$411$	−3031.52	+	5823.04i	−0.363829	+	0.698855i
$412$		0			0
$413$			2210.68i			0.263391i
$414$		0			0
$415$			2351.82i			0.278184i
$416$		0			0
$417$	2032.79	+	1058.29i	0.238720	+	0.124280i
$418$		0			0
$419$	14407.6			1.67986			0.839928	−	0.542698i	$-0.182597\pi$
$419$	14407.6			1.67986			0.839928	+	0.542698i	$0.182597\pi$
$420$		0			0
$421$			886.554i			0.102632i	0.998682	+	0.0513159i	$0.0163415\pi$
$421$			886.554i			0.102632i	−0.998682	+	0.0513159i	$0.983658\pi$
$422$		0			0
$423$	1731.97	−	1212.57i	0.199081	−	0.139379i
$424$		0			0
$425$		−	5365.78i		−	0.612421i
$426$		0			0
$427$	1528.26			0.173203
$428$		0			0
$429$	1170.18	+	13255.5i	0.131694	+	1.49180i
$430$		0			0
$431$			4428.41i			0.494916i	0.968899	+	0.247458i	$0.0795953\pi$
$431$			4428.41i			0.494916i	−0.968899	+	0.247458i	$0.920405\pi$
$432$		0			0
$433$	−9403.31			−1.04364			−0.521818	−	0.853057i	$-0.674746\pi$
$433$	−9403.31			−1.04364			−0.521818	+	0.853057i	$0.674746\pi$
$434$		0			0
$435$	−1710.02	−	890.249i	−0.188481	−	0.0981245i
$436$		0			0
$437$	1828.79			0.200190
$438$		0			0
$439$			3981.63i			0.432876i	0.976296	+	0.216438i	$0.0694440\pi$
$439$			3981.63i			0.432876i	−0.976296	+	0.216438i	$0.930556\pi$
$440$		0			0
$441$	4983.69	+	7118.41i	0.538137	+	0.768644i
$442$		0			0
$443$	−17073.6			−1.83113			−0.915564	−	0.402173i	$-0.868255\pi$
$443$	−17073.6			−1.83113			−0.915564	+	0.402173i	$0.868255\pi$
$444$		0			0
$445$	1478.39			0.157488
$446$		0			0
$447$	4441.89	−	8532.12i	0.470009	−	0.902809i
$448$		0			0
$449$	11870.4			1.24766			0.623830	−	0.781560i	$-0.285576\pi$
$449$	11870.4			1.24766			0.623830	+	0.781560i	$0.285576\pi$
$450$		0			0
$451$		−	13848.2i		−	1.44587i
$452$		0			0
$453$	−6406.60	+	12306.0i	−0.664477	+	1.27635i
$454$		0			0
$455$	−192.446	−	301.537i	−0.0198286	−	0.0310687i
$456$		0			0
$457$			4774.61i			0.488724i	0.969684	+	0.244362i	$0.0785785\pi$
$457$			4774.61i			0.488724i	−0.969684	+	0.244362i	$0.921421\pi$
$458$		0			0
$459$	803.188	+	6105.36i	0.0816767	+	0.620858i
$460$		0			0
$461$	−1697.50			−0.171498			−0.0857489	−	0.996317i	$-0.527328\pi$
$461$	−1697.50			−0.171498			−0.0857489	+	0.996317i	$0.527328\pi$
$462$		0			0
$463$	−1618.85			−0.162493			−0.0812464	−	0.996694i	$-0.525890\pi$
$463$	−1618.85			−0.162493			−0.0812464	+	0.996694i	$0.525890\pi$
$464$		0			0
$465$	−98.6637	+	189.516i	−0.00983961	+	0.0189002i
$466$		0			0
$467$	−3709.58			−0.367578			−0.183789	−	0.982966i	$-0.558836\pi$
$467$	−3709.58			−0.367578			−0.183789	+	0.982966i	$0.558836\pi$
$468$		0			0
$469$	3885.93			0.382592
$470$		0			0
$471$	−711.049	+	1365.81i	−0.0695614	+	0.133616i
$472$		0			0
$473$	1201.26			0.116774
$474$		0			0
$475$	−9783.79			−0.945076
$476$		0			0
$477$	−6691.38	+	4684.71i	−0.642300	+	0.449682i
$478$		0			0
$479$		−	4710.56i		−	0.449334i	−0.974436	−	0.224667i	$-0.927871\pi$
$479$		−	4710.56i		−	0.449334i	0.974436	−	0.224667i	$-0.0721294\pi$
$480$		0			0
$481$	−12601.5	+	8042.52i	−1.19455	+	0.762385i
$482$		0			0
$483$	252.235	−	484.502i	0.0237621	−	0.0456431i
$484$		0			0
$485$			2648.77i			0.247989i
$486$		0			0
$487$	15573.1			1.44904			0.724521	−	0.689253i	$-0.242061\pi$
$487$	15573.1			1.44904			0.724521	+	0.689253i	$0.242061\pi$
$488$		0			0
$489$	99.4182	−	190.966i	0.00919396	−	0.0176601i
$490$		0			0
$491$	2749.87			0.252749			0.126375	−	0.991983i	$-0.459666\pi$
$491$	2749.87			0.252749			0.126375	+	0.991983i	$0.459666\pi$
$492$		0			0
$493$	−9816.57			−0.896788
$494$		0			0
$495$	−2004.75	+	1403.55i	−0.182034	+	0.127444i
$496$		0			0
$497$		−	2999.21i		−	0.270690i
$498$		0			0
$499$	−14090.3			−1.26406			−0.632031	−	0.774943i	$-0.717779\pi$
$499$	−14090.3			−1.26406			−0.632031	+	0.774943i	$0.717779\pi$
$500$		0			0
$501$	−601.687	−	313.243i	−0.0536555	−	0.0279335i
$502$		0			0
$503$	−7264.60			−0.643961			−0.321981	−	0.946746i	$-0.604349\pi$
$503$	−7264.60			−0.643961			−0.321981	+	0.946746i	$0.604349\pi$
$504$		0			0
$505$		−	1953.90i		−	0.172174i
$506$		0			0
$507$	−6964.07	−	9045.75i	−0.610030	−	0.792378i
$508$		0			0
$509$	16396.8			1.42785			0.713927	−	0.700221i	$-0.246915\pi$
$509$	16396.8			1.42785			0.713927	+	0.700221i	$0.246915\pi$
$510$		0			0
$511$		−	2097.46i		−	0.181578i
$512$		0			0
$513$	11132.3	−	1464.51i	0.958096	−	0.126042i
$514$		0			0
$515$		−	499.334i		−	0.0427248i
$516$		0			0
$517$	4278.34			0.363948
$518$		0			0
$519$	−9741.17	−	5071.33i	−0.823872	−	0.428914i
$520$		0			0
$521$		−	9914.89i		−	0.833741i	−0.908966	−	0.416870i	$-0.863127\pi$
$521$		−	9914.89i		−	0.833741i	0.908966	−	0.416870i	$-0.136873\pi$
$522$		0			0
$523$			9431.14i			0.788518i	0.918999	+	0.394259i	$0.128999\pi$
$523$			9431.14i			0.788518i	−0.918999	+	0.394259i	$0.871001\pi$
$524$		0			0
$525$	−1349.43	+	2592.02i	−0.112179	+	0.215476i
$526$		0			0
$527$			1087.94i			0.0899270i
$528$		0			0
$529$	−11644.8			−0.957085
$530$		0			0
$531$	−10628.7	+	7441.30i	−0.868639	+	0.608145i
$532$		0			0
$533$	6391.42	+	10014.5i	0.519405	+	0.813837i
$534$		0			0
$535$	−1748.87			−0.141328
$536$		0			0
$537$	−5150.76	+	9893.73i	−0.413913	+	0.795058i
$538$		0			0
$539$			17584.1i			1.40519i
$540$		0			0
$541$			16945.0i			1.34662i	0.739360	+	0.673310i	$0.235128\pi$
$541$			16945.0i			1.34662i	−0.739360	+	0.673310i	$0.764872\pi$
$542$		0			0
$543$	216.725	−	416.292i	0.0171281	−	0.0329002i
$544$		0			0
$545$			1681.52i			0.132162i
$546$		0			0
$547$		−	12763.9i		−	0.997707i	−0.866686	−	0.498854i	$-0.833754\pi$
$547$		−	12763.9i		−	0.997707i	0.866686	−	0.498854i	$-0.166246\pi$
$548$		0			0
$549$	5144.21	+	7347.70i	0.399908	+	0.571206i
$550$		0			0
$551$			17899.2i			1.38391i
$552$		0			0
$553$			1132.29i			0.0870700i
$554$		0			0
$555$	−2438.58	−	1269.54i	−0.186508	−	0.0970975i
$556$		0			0
$557$	−3718.90			−0.282899			−0.141450	−	0.989945i	$-0.545176\pi$
$557$	−3718.90			−0.282899			−0.141450	+	0.989945i	$0.545176\pi$
$558$		0			0
$559$	−868.704	+	554.423i	−0.0657286	+	0.0419492i
$560$		0			0
$561$	−5754.27	+	11053.0i	−0.433058	+	0.831831i
$562$		0			0
$563$	−6518.14			−0.487934			−0.243967	−	0.969784i	$-0.578449\pi$
$563$	−6518.14			−0.487934			−0.243967	+	0.969784i	$0.578449\pi$
$564$		0			0
$565$			3231.83i			0.240644i
$566$		0			0
$567$	1147.43	−	3151.27i	0.0849867	−	0.233406i
$568$		0			0
$569$			5245.52i			0.386474i	0.981152	+	0.193237i	$0.0618986\pi$
$569$			5245.52i			0.386474i	−0.981152	+	0.193237i	$0.938101\pi$
$570$		0			0
$571$		−	19960.8i		−	1.46293i	−0.681877	−	0.731467i	$-0.738836\pi$
$571$		−	19960.8i		−	1.46293i	0.681877	−	0.731467i	$-0.261164\pi$
$572$		0			0
$573$	4952.85	−	9513.60i	0.361097	−	0.693606i
$574$		0			0
$575$	−2793.45			−0.202600
$576$		0			0
$577$			20237.0i			1.46010i	0.683392	+	0.730051i	$0.260504\pi$
$577$			20237.0i			1.46010i	−0.683392	+	0.730051i	$0.739496\pi$
$578$		0			0
$579$	−3976.68	−	2070.29i	−0.285432	−	0.148598i
$580$		0			0
$581$		−	6521.81i		−	0.465697i
$582$		0			0
$583$	−16529.2			−1.17422
$584$		0			0
$585$	801.970	−	1940.25i	0.0566793	−	0.137127i
$586$		0			0
$587$		−	10732.0i		−	0.754609i	−0.926089	−	0.377305i	$-0.876851\pi$
$587$		−	10732.0i		−	0.754609i	0.926089	−	0.377305i	$-0.123149\pi$
$588$		0			0
$589$	1983.72			0.138774
$590$		0			0
$591$	−9175.90	+	17625.4i	−0.638656	+	1.22675i
$592$		0			0
$593$	−3971.72			−0.275040			−0.137520	−	0.990499i	$-0.543913\pi$
$593$	−3971.72			−0.275040			−0.137520	+	0.990499i	$0.543913\pi$
$594$		0			0
$595$		−	334.976i		−	0.0230801i
$596$		0			0
$597$	−24257.5	−	12628.6i	−1.66297	−	0.865754i
$598$		0			0
$599$	14579.7			0.994511			0.497256	−	0.867604i	$-0.334341\pi$
$599$	14579.7			0.994511			0.497256	+	0.867604i	$0.334341\pi$
$600$		0			0
$601$	17702.5			1.20150			0.600749	−	0.799438i	$-0.294869\pi$
$601$	17702.5			1.20150			0.600749	+	0.799438i	$0.294869\pi$
$602$		0			0
$603$	13080.3	+	18683.1i	0.883367	+	1.26175i
$604$		0			0
$605$	−2744.15			−0.184406
$606$		0			0
$607$		−	16473.2i		−	1.10153i	−0.834662	−	0.550763i	$-0.814337\pi$
$607$		−	16473.2i		−	1.10153i	0.834662	−	0.550763i	$-0.185663\pi$
$608$		0			0
$609$	4742.04	+	2468.74i	0.315529	+	0.164267i
$610$		0			0
$611$	−3093.92	+	1974.60i	−0.204856	+	0.130743i
$612$		0			0
$613$		−	13263.0i		−	0.873880i	−0.899491	−	0.436940i	$-0.856062\pi$
$613$		−	13263.0i		−	0.873880i	0.899491	−	0.436940i	$-0.143938\pi$
$614$		0			0
$615$	−1008.91	+	1937.95i	−0.0661516	+	0.127066i
$616$		0			0
$617$	−4653.47			−0.303633			−0.151816	−	0.988409i	$-0.548512\pi$
$617$	−4653.47			−0.303633			−0.151816	+	0.988409i	$0.548512\pi$
$618$		0			0
$619$	−2551.25			−0.165660			−0.0828298	−	0.996564i	$-0.526396\pi$
$619$	−2551.25			−0.165660			−0.0828298	+	0.996564i	$0.526396\pi$
$620$		0			0
$621$	3178.47	−	418.143i	0.205391	−	0.0270201i
$622$		0			0
$623$	−4099.70			−0.263645
$624$		0			0
$625$	14600.6			0.934435
$626$		0			0
$627$	20153.6	+	10492.1i	1.28367	+	0.668286i
$628$		0			0
$629$	−13999.0			−0.887402
$630$		0			0
$631$	−18981.5			−1.19753			−0.598764	−	0.800926i	$-0.704341\pi$
$631$	−18981.5			−1.19753			−0.598764	+	0.800926i	$0.704341\pi$
$632$		0			0
$633$	−8767.95	−	4564.67i	−0.550545	−	0.286618i
$634$		0			0
$635$			2156.29i			0.134756i
$636$		0			0
$637$	−8115.64	−	12716.1i	−0.504793	−	0.790942i
$638$		0			0
$639$	14419.9	−	10095.5i	0.892710	−	0.624997i
$640$		0			0
$641$			23963.8i			1.47662i	0.674460	+	0.738311i	$0.264376\pi$
$641$			23963.8i			1.47662i	−0.674460	+	0.738311i	$0.735624\pi$
$642$		0			0
$643$	−13446.0			−0.824661			−0.412330	−	0.911034i	$-0.635285\pi$
$643$	−13446.0			−0.824661			−0.412330	+	0.911034i	$0.635285\pi$
$644$		0			0
$645$	−168.107	−	87.5179i	−0.0102623	−	0.00534266i
$646$		0			0
$647$	−9699.46			−0.589374			−0.294687	−	0.955594i	$-0.595215\pi$
$647$	−9699.46			−0.589374			−0.294687	+	0.955594i	$0.595215\pi$
$648$		0			0
$649$	−26255.3			−1.58800
$650$		0			0
$651$	273.604	−	525.546i	0.0164721	−	0.0316402i
$652$		0			0
$653$		−	4389.49i		−	0.263054i	−0.991313	−	0.131527i	$-0.958012\pi$
$653$		−	4389.49i		−	0.263054i	0.991313	−	0.131527i	$-0.0419880\pi$
$654$		0			0
$655$	−3309.81			−0.197443
$656$		0			0
$657$	10084.4	−	7060.19i	0.598827	−	0.419246i
$658$		0			0
$659$	−21977.3			−1.29911			−0.649555	−	0.760314i	$-0.725045\pi$
$659$	−21977.3			−1.29911			−0.649555	+	0.760314i	$0.725045\pi$
$660$		0			0
$661$		−	11415.5i		−	0.671726i	−0.941911	−	0.335863i	$-0.890972\pi$
$661$		−	11415.5i		−	0.671726i	0.941911	−	0.335863i	$-0.109028\pi$
$662$		0			0
$663$	−940.068	−	10648.9i	−0.0550667	−	0.623782i
$664$		0			0
$665$	−610.784			−0.0356168
$666$		0			0
$667$			5110.54i			0.296673i
$668$		0			0
$669$	−12378.0	+	23776.1i	−0.715340	+	1.37405i
$670$		0			0
$671$			18150.5i			1.04425i
$672$		0			0
$673$	−1930.71			−0.110584			−0.0552922	−	0.998470i	$-0.517609\pi$
$673$	−1930.71			−0.110584			−0.0552922	+	0.998470i	$0.517609\pi$
$674$		0			0
$675$	−17004.4	+	2237.01i	−0.969629	+	0.127559i
$676$		0			0
$677$		−	19368.1i		−	1.09952i	−0.835322	−	0.549760i	$-0.814719\pi$
$677$		−	19368.1i		−	1.09952i	0.835322	−	0.549760i	$-0.185281\pi$
$678$		0			0
$679$		−	7345.28i		−	0.415149i
$680$		0			0
$681$	8615.19	+	4485.13i	0.484779	+	0.252380i
$682$		0			0
$683$		−	22753.2i		−	1.27471i	−0.770569	−	0.637356i	$-0.780028\pi$
$683$		−	22753.2i		−	1.27471i	0.770569	−	0.637356i	$-0.219972\pi$
$684$		0			0
$685$	2095.92			0.116907
$686$		0			0
$687$	11709.6	+	6096.13i	0.650292	+	0.338547i
$688$		0			0
$689$	11953.3	−	7628.78i	0.660933	−	0.421819i
$690$		0			0
$691$	−23400.1			−1.28825			−0.644126	−	0.764920i	$-0.722779\pi$
$691$	−23400.1			−1.28825			−0.644126	+	0.764920i	$0.722779\pi$
$692$		0			0
$693$	5559.36	−	3892.18i	0.304737	−	0.213350i
$694$		0			0
$695$		−	731.676i		−	0.0399339i
$696$		0			0
$697$			11125.0i			0.604578i
$698$		0			0
$699$	13854.6	+	7212.82i	0.749685	+	0.390292i
$700$		0			0
$701$			28147.9i			1.51659i	0.651909	+	0.758297i	$0.273968\pi$
$701$			28147.9i			1.51659i	−0.651909	+	0.758297i	$0.726032\pi$
$702$		0			0
$703$			25525.3i			1.36942i
$704$		0			0
$705$	−598.720	−	311.698i	−0.0319845	−	0.0166514i
$706$		0			0
$707$			5418.36i			0.288230i
$708$		0			0
$709$		−	18233.4i		−	0.965822i	−0.875669	−	0.482911i	$-0.839579\pi$
$709$		−	18233.4i		−	0.965822i	0.875669	−	0.482911i	$-0.160421\pi$
$710$		0			0
$711$	−5443.91	+	3811.35i	−0.287149	+	0.201036i
$712$		0			0
$713$	566.387			0.0297494
$714$		0			0
$715$	3581.22	−	2285.60i	0.187315	−	0.119548i
$716$		0			0
$717$	13042.7	+	6790.11i	0.679341	+	0.353670i
$718$		0			0
$719$	7021.51			0.364198			0.182099	−	0.983280i	$-0.441711\pi$
$719$	7021.51			0.364198			0.182099	+	0.983280i	$0.441711\pi$
$720$		0			0
$721$			1384.70i			0.0715241i
$722$		0			0
$723$	13604.9	+	7082.84i	0.699824	+	0.364334i
$724$		0			0
$725$		−	27340.7i		−	1.40056i
$726$		0			0
$727$		−	21025.0i		−	1.07259i	−0.844031	−	0.536295i	$-0.819824\pi$
$727$		−	21025.0i		−	1.07259i	0.844031	−	0.536295i	$-0.180176\pi$
$728$		0			0
$729$	19013.3	−	5090.67i	0.965976	−	0.258633i
$730$		0			0
$731$	−965.040			−0.0488280
$732$		0			0
$733$		−	25864.5i		−	1.30331i	−0.758516	−	0.651655i	$-0.774075\pi$
$733$		−	25864.5i		−	1.30331i	0.758516	−	0.651655i	$-0.225925\pi$
$734$		0			0
$735$	1281.09	−	2460.75i	0.0642906	−	0.123491i
$736$		0			0
$737$			46151.5i			2.30667i
$738$		0			0
$739$	279.361			0.0139059			0.00695296	−	0.999976i	$-0.497787\pi$
$739$	279.361			0.0139059			0.00695296	+	0.999976i	$0.497787\pi$
$740$		0			0
$741$	−19416.8	+	1714.09i	−0.962608	+	0.0849778i
$742$		0			0
$743$		−	7307.21i		−	0.360801i	−0.983593	−	0.180401i	$-0.942261\pi$
$743$		−	7307.21i		−	0.360801i	0.983593	−	0.180401i	$-0.0577394\pi$
$744$		0			0
$745$	−3071.02			−0.151025
$746$		0			0
$747$	31356.1	−	21952.8i	1.53582	−	1.07525i
$748$		0			0
$749$	4849.79			0.236592
$750$		0			0
$751$			1579.19i			0.0767318i	0.999264	+	0.0383659i	$0.0122152\pi$
$751$			1579.19i			0.0767318i	−0.999264	+	0.0383659i	$0.987785\pi$
$752$		0			0
$753$	−3494.49	+	6712.32i	−0.169118	+	0.324848i
$754$		0			0
$755$	4429.37			0.213512
$756$		0			0
$757$	28840.0			1.38469			0.692344	−	0.721567i	$-0.256578\pi$
$757$	28840.0			1.38469			0.692344	+	0.721567i	$0.256578\pi$
$758$		0			0
$759$	5754.22	+	2995.69i	0.275184	+	0.143263i
$760$		0			0
$761$	12157.2			0.579103			0.289551	−	0.957162i	$-0.406494\pi$
$761$	12157.2			0.579103			0.289551	+	0.957162i	$0.406494\pi$
$762$		0			0
$763$		−	4663.01i		−	0.221248i
$764$		0			0
$765$	1610.53	−	1127.55i	0.0761160	−	0.0532897i
$766$		0			0
$767$	18986.8	−	12117.7i	0.893838	−	0.570464i
$768$		0			0
$769$			12578.7i			0.589858i	0.955519	+	0.294929i	$0.0952960\pi$
$769$			12578.7i			0.589858i	−0.955519	+	0.294929i	$0.904704\pi$
$770$		0			0
$771$	−9354.44	−	4869.99i	−0.436955	−	0.227482i
$772$		0			0
$773$	5024.92			0.233808			0.116904	−	0.993143i	$-0.462703\pi$
$773$	5024.92			0.233808			0.116904	+	0.993143i	$0.462703\pi$
$774$		0			0
$775$	−3030.09			−0.140444
$776$		0			0
$777$	6762.41	+	3520.56i	0.312227	+	0.162548i
$778$		0			0
$779$	20285.0			0.932973
$780$		0			0
$781$	35620.3			1.63200
$782$		0			0
$783$	4092.55	+	31109.1i	0.186789	+	1.41986i
$784$		0			0
$785$	491.603			0.0223517
$786$		0			0
$787$	11668.3			0.528501			0.264250	−	0.964454i	$-0.414875\pi$
$787$	11668.3			0.528501			0.264250	+	0.964454i	$0.414875\pi$
$788$		0			0
$789$	−15674.9	+	30108.8i	−0.707276	+	1.35856i
$790$		0			0
$791$		−	8962.15i		−	0.402854i
$792$		0			0
$793$	−8377.05	−	13125.7i	−0.375129	−	0.587776i
$794$		0			0
$795$	2313.13	+	1204.23i	0.103193	+	0.0537230i
$796$		0			0
$797$		−	16725.9i		−	0.743365i	−0.928360	−	0.371683i	$-0.878781\pi$
$797$		−	16725.9i		−	0.743365i	0.928360	−	0.371683i	$-0.121219\pi$
$798$		0			0
$799$	−3437.03			−0.152182
$800$		0			0
$801$	−13799.9	−	19710.9i	−0.608731	−	0.869477i
$802$		0			0
$803$	24910.7			1.09474
$804$		0			0
$805$	−174.390			−0.00763531
$806$		0			0
$807$	23406.4	+	12185.6i	1.02100	+	0.531539i
$808$		0			0
$809$		−	32990.1i		−	1.43371i	−0.697222	−	0.716855i	$-0.745581\pi$
$809$		−	32990.1i		−	1.43371i	0.697222	−	0.716855i	$-0.254419\pi$
$810$		0			0
$811$	28635.4			1.23986			0.619929	−	0.784658i	$-0.287161\pi$
$811$	28635.4			1.23986			0.619929	+	0.784658i	$0.287161\pi$
$812$		0			0
$813$	−12810.5	+	24606.8i	−0.552625	+	1.06150i
$814$		0			0
$815$	−68.7354			−0.00295423
$816$		0			0
$817$			1759.62i			0.0753505i
$818$		0			0
$819$	−2223.94	+	5380.50i	−0.0948848	+	0.229560i
$820$		0			0
$821$	−41284.8			−1.75499			−0.877497	−	0.479582i	$-0.840788\pi$
$821$	−41284.8			−1.75499			−0.877497	+	0.479582i	$0.840788\pi$
$822$		0			0
$823$		−	4389.00i		−	0.185894i	−0.995671	−	0.0929471i	$-0.970371\pi$
$823$		−	4389.00i		−	0.185894i	0.995671	−	0.0929471i	$-0.0296288\pi$
$824$		0			0
$825$	−30784.3	−	16026.5i	−1.29912	−	0.676330i
$826$		0			0
$827$		−	17206.1i		−	0.723476i	−0.932280	−	0.361738i	$-0.882183\pi$
$827$		−	17206.1i		−	0.723476i	0.932280	−	0.361738i	$-0.117817\pi$
$828$		0			0
$829$	32350.9			1.35536			0.677679	−	0.735357i	$-0.262986\pi$
$829$	32350.9			1.35536			0.677679	+	0.735357i	$0.262986\pi$
$830$		0			0
$831$	−1805.47	+	3468.00i	−0.0753681	+	0.144770i
$832$		0			0
$833$		−	14126.3i		−	0.587570i
$834$		0			0
$835$			216.569i			0.00897566i
$836$		0			0
$837$	3447.74	−	453.565i	0.142379	−	0.0187306i
$838$		0			0
$839$			6096.97i			0.250883i	0.992101	+	0.125441i	$0.0400347\pi$
$839$			6096.97i			0.250883i	−0.992101	+	0.125441i	$0.959965\pi$
$840$		0			0
$841$	−25630.2			−1.05089
$842$		0			0
$843$	17222.9	−	33082.3i	0.703664	−	1.35162i
$844$		0			0
$845$	−1534.91	+	3305.71i	−0.0624883	+	0.134580i
$846$		0			0
$847$	7609.77			0.308707
$848$		0			0
$849$	−10022.7	−	5217.90i	−0.405157	−	0.210928i
$850$		0			0
$851$			7287.92i			0.293568i
$852$		0			0
$853$		−	40668.0i		−	1.63241i	−0.577764	−	0.816204i	$-0.696074\pi$
$853$		−	40668.0i		−	1.63241i	0.577764	−	0.816204i	$-0.303926\pi$
$854$		0			0
$855$	−2055.94	−	2936.58i	−0.0822357	−	0.117461i
$856$		0			0
$857$		−	538.343i		−	0.0214579i	−0.999942	−	0.0107290i	$-0.996585\pi$
$857$		−	538.343i		−	0.0214579i	0.999942	−	0.0107290i	$-0.00341520\pi$
$858$		0			0
$859$			34633.0i			1.37563i	0.725887	+	0.687814i	$0.241429\pi$
$859$			34633.0i			1.37563i	−0.725887	+	0.687814i	$0.758571\pi$
$860$		0			0
$861$	2797.80	−	5374.11i	0.110742	−	0.212717i
$862$		0			0
$863$		−	26459.3i		−	1.04367i	−0.853047	−	0.521833i	$-0.825248\pi$
$863$		−	26459.3i		−	1.04367i	0.853047	−	0.521833i	$-0.174752\pi$
$864$		0			0
$865$			3506.20i			0.137820i
$866$		0			0
$867$	−7165.85	+	13764.4i	−0.280698	+	0.539173i
$868$		0			0
$869$	−13447.7			−0.524950
$870$		0			0
$871$	−21300.5	−	33374.9i	−0.828633	−	1.29835i
$872$		0			0
$873$	35315.3	−	24724.7i	1.36912	−	0.958538i
$874$		0			0
$875$	1886.92			0.0729025
$876$		0			0
$877$		−	44087.8i		−	1.69754i	−0.528765	−	0.848768i	$-0.677345\pi$
$877$		−	44087.8i		−	1.69754i	0.528765	−	0.848768i	$-0.322655\pi$
$878$		0			0
$879$	20937.4	−	40217.1i	0.803412	−	1.54322i
$880$		0			0
$881$		−	25467.1i		−	0.973903i	−0.873429	−	0.486951i	$-0.838109\pi$
$881$		−	25467.1i		−	0.973903i	0.873429	−	0.486951i	$-0.161891\pi$
$882$		0			0
$883$		−	9351.33i		−	0.356396i	−0.983995	−	0.178198i	$-0.942973\pi$
$883$		−	9351.33i		−	0.356396i	0.983995	−	0.178198i	$-0.0570267\pi$
$884$		0			0
$885$	3674.23	+	1912.83i	0.139557	+	0.0726544i
$886$		0			0
$887$	−11869.9			−0.449328			−0.224664	−	0.974436i	$-0.572128\pi$
$887$	−11869.9			−0.449328			−0.224664	+	0.974436i	$0.572128\pi$
$888$		0			0
$889$		−	5979.60i		−	0.225590i
$890$		0			0
$891$	37426.3	+	13627.5i	1.40722	+	0.512389i
$892$		0			0
$893$			6266.96i			0.234844i
$894$		0			0
$895$	3561.11			0.133000
$896$		0			0
$897$	−5543.83	+	489.402i	−0.206358	+	0.0182170i
$898$		0			0
$899$			5543.48i			0.205657i
$900$		0			0
$901$	13278.8			0.490990
$902$		0			0
$903$	466.177	+	242.695i	0.0171798	+	0.00894395i
$904$		0			0
$905$	−149.838			−0.00550364
$906$		0			0
$907$			46364.2i			1.69735i	0.528913	+	0.848676i	$0.322600\pi$
$907$			46364.2i			1.69735i	−0.528913	+	0.848676i	$0.677400\pi$
$908$		0			0
$909$	−26050.9	+	18238.5i	−0.950554	+	0.665494i
$910$		0			0
$911$	−3851.00			−0.140054			−0.0700271	−	0.997545i	$-0.522309\pi$
$911$	−3851.00			−0.140054			−0.0700271	+	0.997545i	$0.522309\pi$
$912$		0			0
$913$	77456.7			2.80771
$914$		0			0
$915$	1322.35	−	2540.01i	0.0477766	−	0.0917707i
$916$		0			0
$917$	9178.41			0.330532
$918$		0			0
$919$		−	25293.7i		−	0.907903i	−0.891026	−	0.453951i	$-0.850014\pi$
$919$		−	25293.7i		−	0.907903i	0.891026	−	0.453951i	$-0.149986\pi$
$920$		0			0
$921$	393.586	−	756.013i	0.0140815	−	0.0270483i
$922$		0			0
$923$	−25759.2	+	16440.0i	−0.918607	+	0.586271i
$924$		0			0
$925$		−	38989.4i		−	1.38591i
$926$		0			0
$927$	−6657.48	+	4660.98i	−0.235880	+	0.165142i
$928$		0			0
$929$	32476.6			1.14696			0.573479	−	0.819220i	$-0.305593\pi$
$929$	32476.6			1.14696			0.573479	+	0.819220i	$0.305593\pi$
$930$		0			0
$931$	−25757.3			−0.906727
$932$		0			0
$933$	21146.1	−	40618.1i	0.742007	−	1.42527i
$934$		0			0
$935$	3978.36			0.139151
$936$		0			0
$937$	11069.0			0.385920			0.192960	−	0.981207i	$-0.438191\pi$
$937$	11069.0			0.385920			0.192960	+	0.981207i	$0.438191\pi$
$938$		0			0
$939$	−17344.4	+	33315.8i	−0.602785	+	1.15785i
$940$		0			0
$941$	−16698.2			−0.578476			−0.289238	−	0.957257i	$-0.593402\pi$
$941$	−16698.2			−0.578476			−0.289238	+	0.957257i	$0.593402\pi$
$942$		0			0
$943$	5791.73			0.200005
$944$		0			0
$945$	−1061.55	+	139.652i	−0.0365421	+	0.00480728i
$946$		0			0
$947$		−	2558.01i		−	0.0877762i	−0.999036	−	0.0438881i	$-0.986025\pi$
$947$		−	2558.01i		−	0.0877762i	0.999036	−	0.0438881i	$-0.0139745\pi$
$948$		0			0
$949$	−18014.4	+	11497.1i	−0.616198	+	0.393269i
$950$		0			0
$951$	−21910.1	+	42085.6i	−0.747090	+	1.43503i
$952$		0			0
$953$			3280.46i			0.111505i	0.998445	+	0.0557527i	$0.0177558\pi$
$953$			3280.46i			0.111505i	−0.998445	+	0.0557527i	$0.982244\pi$
$954$		0			0
$955$	−3424.29			−0.116029
$956$		0			0
$957$	−29320.2	+	56319.1i	−0.990373	+	1.90234i
$958$		0			0
$959$	−5812.18			−0.195709
$960$		0			0
$961$	−29176.6			−0.979377
$962$		0			0
$963$	16324.7	+	23317.2i	0.546267	+	0.780257i
$964$		0			0
$965$			1431.35i			0.0477480i
$966$		0			0
$967$	−49833.7			−1.65723			−0.828617	−	0.559817i	$-0.810872\pi$
$967$	−49833.7			−1.65723			−0.828617	+	0.559817i	$0.810872\pi$
$968$		0			0
$969$	−16190.5	−	8428.91i	−0.536754	−	0.279438i
$970$		0			0
$971$	39079.2			1.29157			0.645783	−	0.763521i	$-0.276531\pi$
$971$	39079.2			1.29157			0.645783	+	0.763521i	$0.276531\pi$
$972$		0			0
$973$			2029.00i			0.0668519i
$974$		0			0
$975$	29658.7	−	2618.24i	0.974195	−	0.0860007i
$976$		0			0
$977$	−30647.0			−1.00357			−0.501783	−	0.864994i	$-0.667322\pi$
$977$	−30647.0			−1.00357			−0.501783	+	0.864994i	$0.667322\pi$
$978$		0			0
$979$		−	48690.4i		−	1.58953i
$980$		0			0
$981$	22419.3	−	15696.0i	0.729655	−	0.510840i
$982$		0			0
$983$			49203.3i			1.59648i	0.602339	+	0.798240i	$0.294235\pi$
$983$			49203.3i			1.59648i	−0.602339	+	0.798240i	$0.705765\pi$
$984$		0			0
$985$	6344.00			0.205215
$986$		0			0
$987$	1660.31	+	864.368i	0.0535442	+	0.0278755i
$988$		0			0
$989$			502.403i			0.0161532i
$990$		0			0
$991$			7980.56i			0.255813i	0.991786	+	0.127907i	$0.0408258\pi$
$991$			7980.56i			0.255813i	−0.991786	+	0.127907i	$0.959174\pi$
$992$		0			0
$993$	7597.30	−	14593.1i	0.242792	−	0.466363i
$994$		0			0
$995$			8731.13i			0.278186i
$996$		0			0
$997$	−5464.05			−0.173569			−0.0867844	−	0.996227i	$-0.527659\pi$
$997$	−5464.05			−0.173569			−0.0867844	+	0.996227i	$0.527659\pi$
$998$		0			0
$999$	5836.20	+	44363.4i	0.184834	+	1.40500i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.n.d.623.20	yes	56
3.2	odd	2	inner	624.4.n.d.623.39	yes	56
4.3	odd	2	inner	624.4.n.d.623.38	yes	56
12.11	even	2	inner	624.4.n.d.623.17	✓	56
13.12	even	2	inner	624.4.n.d.623.19	yes	56
39.38	odd	2	inner	624.4.n.d.623.40	yes	56
52.51	odd	2	inner	624.4.n.d.623.37	yes	56
156.155	even	2	inner	624.4.n.d.623.18	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.n.d.623.17	✓	56	12.11	even	2	inner
624.4.n.d.623.18	yes	56	156.155	even	2	inner
624.4.n.d.623.19	yes	56	13.12	even	2	inner
624.4.n.d.623.20	yes	56	1.1	even	1	trivial
624.4.n.d.623.37	yes	56	52.51	odd	2	inner
624.4.n.d.623.38	yes	56	4.3	odd	2	inner
624.4.n.d.623.39	yes	56	3.2	odd	2	inner
624.4.n.d.623.40	yes	56	39.38	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.39946 + 4.60897i) q^{3} +1.65893 q^{5} -4.60037 q^{7} +(-15.4851 - 22.1181i) q^{9} -54.6366i q^{11} +(25.2167 + 39.5110i) q^{13} +(-3.98055 + 7.64597i) q^{15} +43.8926i q^{17} +80.0323 q^{19} +(11.0384 - 21.2030i) q^{21} +22.8507 q^{23} -122.248 q^{25} +(139.098 - 18.2989i) q^{27} +223.650i q^{29} +24.7864 q^{31} +(251.818 + 131.099i) q^{33} -7.63171 q^{35} +318.937i q^{37} +(-242.611 + 21.4174i) q^{39} +253.460 q^{41} +21.9864i q^{43} +(-25.6888 - 36.6924i) q^{45} +78.3054i q^{47} -321.837 q^{49} +(-202.300 - 105.319i) q^{51} -302.530i q^{53} -90.6385i q^{55} +(-192.035 + 368.866i) q^{57} -480.544i q^{59} -332.203 q^{61} +(71.2375 + 101.752i) q^{63} +(41.8327 + 65.5461i) q^{65} -844.699 q^{67} +(-54.8293 + 105.318i) q^{69} +651.949i q^{71} +455.933i q^{73} +(293.330 - 563.437i) q^{75} +251.349i q^{77} -246.129i q^{79} +(-249.421 + 685.004i) q^{81} +1417.67i q^{83} +72.8149i q^{85} +(-1030.79 - 536.639i) q^{87} +891.167 q^{89} +(-116.006 - 181.765i) q^{91} +(-59.4742 + 114.240i) q^{93} +132.768 q^{95} +1596.67i q^{97} +(-1208.46 + 846.056i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 64 q^{9} + 8 q^{13} + 1208 q^{25} + 4680 q^{49} + 1616 q^{61} + 480 q^{69} + 1568 q^{81}+O(q^{100}) \) 56 * q - 64 * q^9 + 8 * q^13 + 1208 * q^25 + 4680 * q^49 + 1616 * q^61 + 480 * q^69 + 1568 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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