LMFDB - Embedded newform 1932.2.k.a.1609.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1609,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1932.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.k (of order $2$, degree $1$, 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:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.16
Character	$\chi$	$=$	1932.1609
Dual form			1932.2.k.a.1609.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.41621 q^{5} +(2.53101 - 0.770692i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.41621 q^{5} +(2.53101 - 0.770692i) q^{7} -1.00000 q^{9} +3.81567i q^{11} +6.45743i q^{13} +1.41621i q^{15} -3.92234 q^{17} -0.414030 q^{19} +(0.770692 + 2.53101i) q^{21} +(-4.75626 + 0.614837i) q^{23} -2.99436 q^{25} -1.00000i q^{27} +0.316832 q^{29} +0.348692i q^{31} -3.81567 q^{33} +(3.58444 - 1.09146i) q^{35} +0.720567i q^{37} -6.45743 q^{39} -1.82254i q^{41} +8.25316i q^{43} -1.41621 q^{45} -3.93692i q^{47} +(5.81207 - 3.90127i) q^{49} -3.92234i q^{51} +7.37616i q^{53} +5.40378i q^{55} -0.414030i q^{57} -5.43153i q^{59} +11.2869 q^{61} +(-2.53101 + 0.770692i) q^{63} +9.14504i q^{65} +10.0521i q^{67} +(-0.614837 - 4.75626i) q^{69} +10.9485 q^{71} -14.7376i q^{73} -2.99436i q^{75} +(2.94071 + 9.65752i) q^{77} +4.54134i q^{79} +1.00000 q^{81} -11.2420 q^{83} -5.55483 q^{85} +0.316832i q^{87} +6.10525 q^{89} +(4.97669 + 16.3438i) q^{91} -0.348692 q^{93} -0.586351 q^{95} +6.25498 q^{97} -3.81567i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{9}+O(q^{10}) $ 32 * q - 32 * q^9	$ 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) $ 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1932\mathbb{Z}\right)^\times$.

$n$	$829$	$925$	$967$	$1289$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.41621			0.633346			0.316673	−	0.948535i	$-0.397434\pi$
$5$	1.41621			0.633346			0.316673	+	0.948535i	$0.397434\pi$
$6$		0			0
$7$	2.53101	−	0.770692i	0.956633	−	0.291294i
$7$	2.53101	−	0.770692i	0.956633	−	0.291294i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			3.81567i			1.15047i	0.817989	+	0.575234i	$0.195089\pi$
$11$			3.81567i			1.15047i	−0.817989	+	0.575234i	$0.804911\pi$
$12$		0			0
$13$			6.45743i			1.79097i	0.445093	+	0.895484i	$0.353171\pi$
$13$			6.45743i			1.79097i	−0.445093	+	0.895484i	$0.646829\pi$
$14$		0			0
$15$			1.41621i			0.365663i
$16$		0			0
$17$	−3.92234			−0.951306			−0.475653	−	0.879633i	$-0.657788\pi$
$17$	−3.92234			−0.951306			−0.475653	+	0.879633i	$0.657788\pi$
$18$		0			0
$19$	−0.414030			−0.0949849			−0.0474925	−	0.998872i	$-0.515123\pi$
$19$	−0.414030			−0.0949849			−0.0474925	+	0.998872i	$0.515123\pi$
$20$		0			0
$21$	0.770692	+	2.53101i	0.168179	+	0.552313i
$22$		0			0
$23$	−4.75626	+	0.614837i	−0.991748	+	0.128202i
$23$	−4.75626	+	0.614837i	−0.991748	+	0.128202i
$24$		0			0
$25$	−2.99436			−0.598873
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	0.316832			0.0588342			0.0294171	−	0.999567i	$-0.490635\pi$
$29$	0.316832			0.0588342			0.0294171	+	0.999567i	$0.490635\pi$
$30$		0			0
$31$			0.348692i			0.0626269i	0.999510	+	0.0313134i	$0.00996901\pi$
$31$			0.348692i			0.0626269i	−0.999510	+	0.0313134i	$0.990031\pi$
$32$		0			0
$33$	−3.81567			−0.664223
$34$		0			0
$35$	3.58444	−	1.09146i	0.605880	−	0.184490i
$36$		0			0
$37$			0.720567i			0.118460i	0.998244	+	0.0592302i	$0.0188646\pi$
$37$			0.720567i			0.118460i	−0.998244	+	0.0592302i	$0.981135\pi$
$38$		0			0
$39$	−6.45743			−1.03402
$40$		0			0
$41$		−	1.82254i		−	0.284633i	−0.989821	−	0.142317i	$-0.954545\pi$
$41$		−	1.82254i		−	0.284633i	0.989821	−	0.142317i	$-0.0454551\pi$
$42$		0			0
$43$			8.25316i			1.25859i	0.777165	+	0.629297i	$0.216657\pi$
$43$			8.25316i			1.25859i	−0.777165	+	0.629297i	$0.783343\pi$
$44$		0			0
$45$	−1.41621			−0.211115
$46$		0			0
$47$		−	3.93692i		−	0.574258i	−0.957892	−	0.287129i	$-0.907299\pi$
$47$		−	3.93692i		−	0.574258i	0.957892	−	0.287129i	$-0.0927009\pi$
$48$		0			0
$49$	5.81207	−	3.90127i	0.830295	−	0.557324i
$50$		0			0
$51$		−	3.92234i		−	0.549237i
$52$		0			0
$53$			7.37616i			1.01319i	0.862183	+	0.506597i	$0.169097\pi$
$53$			7.37616i			1.01319i	−0.862183	+	0.506597i	$0.830903\pi$
$54$		0			0
$55$			5.40378i			0.728645i
$56$		0			0
$57$		−	0.414030i		−	0.0548396i
$58$		0			0
$59$		−	5.43153i		−	0.707125i	−0.935411	−	0.353562i	$-0.884970\pi$
$59$		−	5.43153i		−	0.707125i	0.935411	−	0.353562i	$-0.115030\pi$
$60$		0			0
$61$	11.2869			1.44514			0.722570	−	0.691297i	$-0.242961\pi$
$61$	11.2869			1.44514			0.722570	+	0.691297i	$0.242961\pi$
$62$		0			0
$63$	−2.53101	+	0.770692i	−0.318878	+	0.0970981i
$64$		0			0
$65$			9.14504i			1.13430i
$66$		0			0
$67$			10.0521i			1.22806i	0.789281	+	0.614032i	$0.210454\pi$
$67$			10.0521i			1.22806i	−0.789281	+	0.614032i	$0.789546\pi$
$68$		0			0
$69$	−0.614837	−	4.75626i	−0.0740177	−	0.572586i
$70$		0			0
$71$	10.9485			1.29934			0.649671	−	0.760215i	$-0.274907\pi$
$71$	10.9485			1.29934			0.649671	+	0.760215i	$0.274907\pi$
$72$		0			0
$73$		−	14.7376i		−	1.72491i	−0.506137	−	0.862453i	$-0.668927\pi$
$73$		−	14.7376i		−	1.72491i	0.506137	−	0.862453i	$-0.331073\pi$
$74$		0			0
$75$		−	2.99436i		−	0.345759i
$76$		0			0
$77$	2.94071	+	9.65752i	0.335125	+	1.10058i
$78$		0			0
$79$			4.54134i			0.510940i	0.966817	+	0.255470i	$0.0822302\pi$
$79$			4.54134i			0.510940i	−0.966817	+	0.255470i	$0.917770\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−11.2420			−1.23397			−0.616983	−	0.786976i	$-0.711645\pi$
$83$	−11.2420			−1.23397			−0.616983	+	0.786976i	$0.711645\pi$
$84$		0			0
$85$	−5.55483			−0.602506
$86$		0			0
$87$			0.316832i			0.0339679i
$88$		0			0
$89$	6.10525			0.647155			0.323578	−	0.946202i	$-0.395114\pi$
$89$	6.10525			0.647155			0.323578	+	0.946202i	$0.395114\pi$
$90$		0			0
$91$	4.97669	+	16.3438i	0.521699	+	1.71330i
$92$		0			0
$93$	−0.348692			−0.0361576
$94$		0			0
$95$	−0.586351			−0.0601583
$96$		0			0
$97$	6.25498			0.635097			0.317549	−	0.948242i	$-0.397140\pi$
$97$	6.25498			0.635097			0.317549	+	0.948242i	$0.397140\pi$
$98$		0			0
$99$		−	3.81567i		−	0.383490i
$100$		0			0
$101$			8.90949i			0.886527i	0.896391	+	0.443264i	$0.146179\pi$
$101$			8.90949i			0.886527i	−0.896391	+	0.443264i	$0.853821\pi$
$102$		0			0
$103$	−13.4353			−1.32382			−0.661908	−	0.749585i	$-0.730253\pi$
$103$	−13.4353			−1.32382			−0.661908	+	0.749585i	$0.730253\pi$
$104$		0			0
$105$	1.09146	+	3.58444i	0.106515	+	0.349805i
$106$		0			0
$107$			0.502808i			0.0486083i	0.999705	+	0.0243042i	$0.00773702\pi$
$107$			0.502808i			0.0486083i	−0.999705	+	0.0243042i	$0.992263\pi$
$108$		0			0
$109$			3.88249i			0.371875i	0.982562	+	0.185938i	$0.0595323\pi$
$109$			3.88249i			0.371875i	−0.982562	+	0.185938i	$0.940468\pi$
$110$		0			0
$111$	−0.720567			−0.0683932
$112$		0			0
$113$		−	11.9492i		−	1.12409i	−0.827108	−	0.562043i	$-0.810016\pi$
$113$		−	11.9492i		−	1.12409i	0.827108	−	0.562043i	$-0.189984\pi$
$114$		0			0
$115$	−6.73583	+	0.870735i	−0.628120	+	0.0811965i
$116$		0			0
$117$		−	6.45743i		−	0.596989i
$118$		0			0
$119$	−9.92749	+	3.02292i	−0.910052	+	0.277110i
$120$		0			0
$121$	−3.55936			−0.323578
$122$		0			0
$123$	1.82254			0.164333
$124$		0			0
$125$	−11.3217			−1.01264
$126$		0			0
$127$	2.57622			0.228602			0.114301	−	0.993446i	$-0.463537\pi$
$127$	2.57622			0.228602			0.114301	+	0.993446i	$0.463537\pi$
$128$		0			0
$129$	−8.25316			−0.726650
$130$		0			0
$131$		−	2.76384i		−	0.241478i	−0.992684	−	0.120739i	$-0.961474\pi$
$131$		−	2.76384i		−	0.241478i	0.992684	−	0.120739i	$-0.0385264\pi$
$132$		0			0
$133$	−1.04792	+	0.319090i	−0.0908658	+	0.0276686i
$134$		0			0
$135$		−	1.41621i		−	0.121888i
$136$		0			0
$137$			0.0166810i			0.00142515i	1.00000		0.000712577i	$0.000226820\pi$
$137$			0.0166810i			0.00142515i	−1.00000		0.000712577i	$0.999773\pi$
$138$		0			0
$139$			9.74742i			0.826765i	0.910557	+	0.413383i	$0.135653\pi$
$139$			9.74742i			0.826765i	−0.910557	+	0.413383i	$0.864347\pi$
$140$		0			0
$141$	3.93692			0.331548
$142$		0			0
$143$	−24.6394			−2.06045
$144$		0			0
$145$	0.448699			0.0372624
$146$		0			0
$147$	3.90127	+	5.81207i	0.321771	+	0.479371i
$148$		0			0
$149$		−	2.75818i		−	0.225959i	−0.993597	−	0.112980i	$-0.963961\pi$
$149$		−	2.75818i		−	0.225959i	0.993597	−	0.112980i	$-0.0360395\pi$
$150$		0			0
$151$	−14.6689			−1.19373			−0.596867	−	0.802340i	$-0.703588\pi$
$151$	−14.6689			−1.19373			−0.596867	+	0.802340i	$0.703588\pi$
$152$		0			0
$153$	3.92234			0.317102
$154$		0			0
$155$			0.493819i			0.0396645i
$156$		0			0
$157$	21.3297			1.70230			0.851148	−	0.524926i	$-0.175907\pi$
$157$	21.3297			1.70230			0.851148	+	0.524926i	$0.175907\pi$
$158$		0			0
$159$	−7.37616			−0.584968
$160$		0			0
$161$	−11.5643	+	5.22177i	−0.911395	+	0.411533i
$162$		0			0
$163$	17.6971			1.38615			0.693073	−	0.720867i	$-0.256256\pi$
$163$	17.6971			1.38615			0.693073	+	0.720867i	$0.256256\pi$
$164$		0			0
$165$	−5.40378			−0.420683
$166$		0			0
$167$			3.34633i			0.258947i	0.991583	+	0.129473i	$0.0413287\pi$
$167$			3.34633i			0.258947i	−0.991583	+	0.129473i	$0.958671\pi$
$168$		0			0
$169$	−28.6984			−2.20757
$170$		0			0
$171$	0.414030			0.0316616
$172$		0			0
$173$		−	11.1048i		−	0.844286i	−0.906529	−	0.422143i	$-0.861278\pi$
$173$		−	11.1048i		−	0.844286i	0.906529	−	0.422143i	$-0.138722\pi$
$174$		0			0
$175$	−7.57878	+	2.30773i	−0.572902	+	0.174448i
$176$		0			0
$177$	5.43153			0.408259
$178$		0			0
$179$	−17.4752			−1.30616			−0.653078	−	0.757290i	$-0.726523\pi$
$179$	−17.4752			−1.30616			−0.653078	+	0.757290i	$0.726523\pi$
$180$		0			0
$181$	12.8079			0.952005			0.476002	−	0.879444i	$-0.342085\pi$
$181$	12.8079			0.952005			0.476002	+	0.879444i	$0.342085\pi$
$182$		0			0
$183$			11.2869i			0.834352i
$184$		0			0
$185$			1.02047i			0.0750265i
$186$		0			0
$187$		−	14.9664i		−	1.09445i
$188$		0			0
$189$	−0.770692	−	2.53101i	−0.0560596	−	0.184104i
$190$		0			0
$191$			24.8388i			1.79727i	0.438697	+	0.898635i	$0.355440\pi$
$191$			24.8388i			1.79727i	−0.438697	+	0.898635i	$0.644560\pi$
$192$		0			0
$193$	9.14938			0.658587			0.329294	−	0.944228i	$-0.393189\pi$
$193$	9.14938			0.658587			0.329294	+	0.944228i	$0.393189\pi$
$194$		0			0
$195$	−9.14504			−0.654890
$196$		0			0
$197$	−11.5979			−0.826317			−0.413158	−	0.910659i	$-0.635574\pi$
$197$	−11.5979			−0.826317			−0.413158	+	0.910659i	$0.635574\pi$
$198$		0			0
$199$	16.5243			1.17138			0.585690	−	0.810535i	$-0.300824\pi$
$199$	16.5243			1.17138			0.585690	+	0.810535i	$0.300824\pi$
$200$		0			0
$201$	−10.0521			−0.709023
$202$		0			0
$203$	0.801906	−	0.244180i	0.0562828	−	0.0171381i
$204$		0			0
$205$		−	2.58109i		−	0.180271i
$206$		0			0
$207$	4.75626	−	0.614837i	0.330583	−	0.0427341i
$208$		0			0
$209$		−	1.57980i		−	0.109277i
$210$		0			0
$211$	5.78310			0.398125			0.199063	−	0.979987i	$-0.436210\pi$
$211$	5.78310			0.398125			0.199063	+	0.979987i	$0.436210\pi$
$212$		0			0
$213$			10.9485i			0.750176i
$214$		0			0
$215$			11.6882i			0.797126i
$216$		0			0
$217$	0.268734	+	0.882544i	0.0182429	+	0.0599110i
$218$		0			0
$219$	14.7376			0.995875
$220$		0			0
$221$		−	25.3282i		−	1.70376i
$222$		0			0
$223$			18.2037i			1.21901i	0.792781	+	0.609506i	$0.208632\pi$
$223$			18.2037i			1.21901i	−0.792781	+	0.609506i	$0.791368\pi$
$224$		0			0
$225$	2.99436			0.199624
$226$		0			0
$227$	7.39741			0.490983			0.245492	−	0.969399i	$-0.421051\pi$
$227$	7.39741			0.490983			0.245492	+	0.969399i	$0.421051\pi$
$228$		0			0
$229$	13.8540			0.915501			0.457751	−	0.889081i	$-0.348655\pi$
$229$	13.8540			0.915501			0.457751	+	0.889081i	$0.348655\pi$
$230$		0			0
$231$	−9.65752	+	2.94071i	−0.635418	+	0.193485i
$232$		0			0
$233$	30.0698			1.96994			0.984968	−	0.172737i	$-0.0552611\pi$
$233$	30.0698			1.96994			0.984968	+	0.172737i	$0.0552611\pi$
$234$		0			0
$235$		−	5.57548i		−	0.363704i
$236$		0			0
$237$	−4.54134			−0.294991
$238$		0			0
$239$	−6.45467			−0.417518			−0.208759	−	0.977967i	$-0.566942\pi$
$239$	−6.45467			−0.417518			−0.208759	+	0.977967i	$0.566942\pi$
$240$		0			0
$241$	19.1500			1.23356			0.616779	−	0.787137i	$-0.288437\pi$
$241$	19.1500			1.23356			0.616779	+	0.787137i	$0.288437\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	8.23108	−	5.52499i	0.525864	−	0.352979i
$246$		0			0
$247$		−	2.67357i		−	0.170115i
$248$		0			0
$249$		−	11.2420i		−	0.712431i
$250$		0			0
$251$	20.3278			1.28308			0.641540	−	0.767090i	$-0.278296\pi$
$251$	20.3278			1.28308			0.641540	+	0.767090i	$0.278296\pi$
$252$		0			0
$253$	−2.34602	−	18.1483i	−0.147493	−	1.14098i
$254$		0			0
$255$		−	5.55483i		−	0.347857i
$256$		0			0
$257$		−	18.5661i		−	1.15812i	−0.815285	−	0.579060i	$-0.803420\pi$
$257$		−	18.5661i		−	1.15812i	0.815285	−	0.579060i	$-0.196580\pi$
$258$		0			0
$259$	0.555335	+	1.82376i	0.0345069	+	0.113323i
$260$		0			0
$261$	−0.316832			−0.0196114
$262$		0			0
$263$			24.8672i			1.53338i	0.642020	+	0.766688i	$0.278097\pi$
$263$			24.8672i			1.53338i	−0.642020	+	0.766688i	$0.721903\pi$
$264$		0			0
$265$			10.4462i			0.641702i
$266$		0			0
$267$			6.10525i			0.373635i
$268$		0			0
$269$			11.8372i			0.721726i	0.932619	+	0.360863i	$0.117518\pi$
$269$			11.8372i			0.721726i	−0.932619	+	0.360863i	$0.882482\pi$
$270$		0			0
$271$		−	16.9783i		−	1.03136i	−0.856782	−	0.515678i	$-0.827540\pi$
$271$		−	16.9783i		−	1.03136i	0.856782	−	0.515678i	$-0.172460\pi$
$272$		0			0
$273$	−16.3438	+	4.97669i	−0.989174	+	0.301203i
$274$		0			0
$275$		−	11.4255i		−	0.688984i
$276$		0			0
$277$	21.2648			1.27768			0.638838	−	0.769341i	$-0.279415\pi$
$277$	21.2648			1.27768			0.638838	+	0.769341i	$0.279415\pi$
$278$		0			0
$279$		−	0.348692i		−	0.0208756i
$280$		0			0
$281$		−	3.40757i		−	0.203278i	−0.994821	−	0.101639i	$-0.967591\pi$
$281$		−	3.40757i		−	0.203278i	0.994821	−	0.101639i	$-0.0324087\pi$
$282$		0			0
$283$	8.95787			0.532490			0.266245	−	0.963905i	$-0.414217\pi$
$283$	8.95787			0.532490			0.266245	+	0.963905i	$0.414217\pi$
$284$		0			0
$285$		−	0.586351i		−	0.0347324i
$286$		0			0
$287$	−1.40462	−	4.61288i	−0.0829121	−	0.272290i
$288$		0			0
$289$	−1.61527			−0.0950162
$290$		0			0
$291$			6.25498i			0.366674i
$292$		0			0
$293$	−25.8886			−1.51243			−0.756215	−	0.654324i	$-0.772953\pi$
$293$	−25.8886			−1.51243			−0.756215	+	0.654324i	$0.772953\pi$
$294$		0			0
$295$		−	7.69216i		−	0.447855i
$296$		0			0
$297$	3.81567			0.221408
$298$		0			0
$299$	−3.97027	−	30.7132i	−0.229606	−	1.77619i
$300$		0			0
$301$	6.36064	+	20.8889i	0.366621	+	1.20401i
$302$		0			0
$303$	−8.90949			−0.511837
$304$		0			0
$305$	15.9846			0.915274
$306$		0			0
$307$			10.4476i			0.596275i	0.954523	+	0.298138i	$0.0963654\pi$
$307$			10.4476i			0.596275i	−0.954523	+	0.298138i	$0.903635\pi$
$308$		0			0
$309$		−	13.4353i		−	0.764306i
$310$		0			0
$311$		−	32.2923i		−	1.83113i	−0.402169	−	0.915565i	$-0.631744\pi$
$311$		−	32.2923i		−	1.83113i	0.402169	−	0.915565i	$-0.368256\pi$
$312$		0			0
$313$	13.9154			0.786543			0.393271	−	0.919422i	$-0.371343\pi$
$313$	13.9154			0.786543			0.393271	+	0.919422i	$0.371343\pi$
$314$		0			0
$315$	−3.58444	+	1.09146i	−0.201960	+	0.0614967i
$316$		0			0
$317$	6.23595			0.350246			0.175123	−	0.984547i	$-0.443968\pi$
$317$	6.23595			0.350246			0.175123	+	0.984547i	$0.443968\pi$
$318$		0			0
$319$			1.20893i			0.0676869i
$320$		0			0
$321$	−0.502808			−0.0280640
$322$		0			0
$323$	1.62396			0.0903598
$324$		0			0
$325$		−	19.3359i		−	1.07256i
$326$		0			0
$327$	−3.88249			−0.214702
$328$		0			0
$329$	−3.03415	−	9.96439i	−0.167278	−	0.549355i
$330$		0			0
$331$	2.22381			0.122232			0.0611158	−	0.998131i	$-0.480534\pi$
$331$	2.22381			0.122232			0.0611158	+	0.998131i	$0.480534\pi$
$332$		0			0
$333$		−	0.720567i		−	0.0394868i
$334$		0			0
$335$			14.2359i			0.777790i
$336$		0			0
$337$		−	1.25545i		−	0.0683886i	−0.999415	−	0.0341943i	$-0.989113\pi$
$337$		−	1.25545i		−	0.0683886i	0.999415	−	0.0341943i	$-0.0108865\pi$
$338$		0			0
$339$	11.9492			0.648991
$340$		0			0
$341$	−1.33049			−0.0720503
$342$		0			0
$343$	11.7037	−	14.3535i	0.631943	−	0.775015i
$344$		0			0
$345$	−0.870735	−	6.73583i	−0.0468788	−	0.362645i
$346$		0			0
$347$	−28.4606			−1.52784			−0.763922	−	0.645309i	$-0.776729\pi$
$347$	−28.4606			−1.52784			−0.763922	+	0.645309i	$0.776729\pi$
$348$		0			0
$349$			2.65625i			0.142186i	0.997470	+	0.0710928i	$0.0226487\pi$
$349$			2.65625i			0.142186i	−0.997470	+	0.0710928i	$0.977351\pi$
$350$		0			0
$351$	6.45743			0.344672
$352$		0			0
$353$		−	7.69144i		−	0.409374i	−0.978827	−	0.204687i	$-0.934382\pi$
$353$		−	7.69144i		−	0.409374i	0.978827	−	0.204687i	$-0.0656176\pi$
$354$		0			0
$355$	15.5053			0.822934
$356$		0			0
$357$	−3.02292	−	9.92749i	−0.159990	−	0.525418i
$358$		0			0
$359$		−	30.8134i		−	1.62627i	−0.582077	−	0.813134i	$-0.697760\pi$
$359$		−	30.8134i		−	1.62627i	0.582077	−	0.813134i	$-0.302240\pi$
$360$		0			0
$361$	−18.8286			−0.990978
$362$		0			0
$363$		−	3.55936i		−	0.186818i
$364$		0			0
$365$		−	20.8715i		−	1.09246i
$366$		0			0
$367$	−12.2628			−0.640111			−0.320056	−	0.947399i	$-0.603702\pi$
$367$	−12.2628			−0.640111			−0.320056	+	0.947399i	$0.603702\pi$
$368$		0			0
$369$			1.82254i			0.0948777i
$370$		0			0
$371$	5.68475	+	18.6692i	0.295138	+	0.969255i
$372$		0			0
$373$		−	5.25466i		−	0.272076i	−0.990704	−	0.136038i	$-0.956563\pi$
$373$		−	5.25466i		−	0.272076i	0.990704	−	0.136038i	$-0.0434369\pi$
$374$		0			0
$375$		−	11.3217i		−	0.584648i
$376$		0			0
$377$			2.04592i			0.105370i
$378$		0			0
$379$			27.3170i			1.40318i	0.712582	+	0.701589i	$0.247526\pi$
$379$			27.3170i			1.40318i	−0.712582	+	0.701589i	$0.752474\pi$
$380$		0			0
$381$			2.57622i			0.131984i
$382$		0			0
$383$	5.25226			0.268378			0.134189	−	0.990956i	$-0.457157\pi$
$383$	5.25226			0.268378			0.134189	+	0.990956i	$0.457157\pi$
$384$		0			0
$385$	4.16465	+	13.6770i	0.212250	+	0.697046i
$386$		0			0
$387$		−	8.25316i		−	0.419531i
$388$		0			0
$389$		−	32.6799i		−	1.65694i	−0.560036	−	0.828469i	$-0.689213\pi$
$389$		−	32.6799i		−	1.65694i	0.560036	−	0.828469i	$-0.310787\pi$
$390$		0			0
$391$	18.6556	−	2.41160i	0.943456	−	0.121960i
$392$		0			0
$393$	2.76384			0.139417
$394$		0			0
$395$			6.43146i			0.323602i
$396$		0			0
$397$		−	16.6473i		−	0.835502i	−0.908561	−	0.417751i	$-0.862818\pi$
$397$		−	16.6473i		−	0.835502i	0.908561	−	0.417751i	$-0.137182\pi$
$398$		0			0
$399$	−0.319090	−	1.04792i	−0.0159745	−	0.0524614i
$400$		0			0
$401$			10.6081i			0.529743i	0.964284	+	0.264871i	$0.0853295\pi$
$401$			10.6081i			0.529743i	−0.964284	+	0.264871i	$0.914670\pi$
$402$		0			0
$403$	−2.25165			−0.112163
$404$		0			0
$405$	1.41621			0.0703718
$406$		0			0
$407$	−2.74945			−0.136285
$408$		0			0
$409$			27.2050i			1.34520i	0.740006	+	0.672600i	$0.234823\pi$
$409$			27.2050i			1.34520i	−0.740006	+	0.672600i	$0.765177\pi$
$410$		0			0
$411$	−0.0166810			−0.000822814
$412$		0			0
$413$	−4.18604	−	13.7473i	−0.205982	−	0.676459i
$414$		0			0
$415$	−15.9209			−0.781528
$416$		0			0
$417$	−9.74742			−0.477333
$418$		0			0
$419$	22.3426			1.09151			0.545753	−	0.837946i	$-0.316244\pi$
$419$	22.3426			1.09151			0.545753	+	0.837946i	$0.316244\pi$
$420$		0			0
$421$			2.38171i			0.116078i	0.998314	+	0.0580388i	$0.0184847\pi$
$421$			2.38171i			0.116078i	−0.998314	+	0.0580388i	$0.981515\pi$
$422$		0			0
$423$			3.93692i			0.191419i
$424$		0			0
$425$	11.7449			0.569711
$426$		0			0
$427$	28.5673	−	8.69874i	1.38247	−	0.420961i
$428$		0			0
$429$		−	24.6394i		−	1.18960i
$430$		0			0
$431$			7.36006i			0.354521i	0.984164	+	0.177261i	$0.0567236\pi$
$431$			7.36006i			0.354521i	−0.984164	+	0.177261i	$0.943276\pi$
$432$		0			0
$433$	19.3355			0.929207			0.464604	−	0.885519i	$-0.346197\pi$
$433$	19.3355			0.929207			0.464604	+	0.885519i	$0.346197\pi$
$434$		0			0
$435$			0.448699i			0.0215135i
$436$		0			0
$437$	1.96923	−	0.254561i	0.0942011	−	0.0121773i
$438$		0			0
$439$			2.31909i			0.110684i	0.998467	+	0.0553420i	$0.0176249\pi$
$439$			2.31909i			0.110684i	−0.998467	+	0.0553420i	$0.982375\pi$
$440$		0			0
$441$	−5.81207	+	3.90127i	−0.276765	+	0.185775i
$442$		0			0
$443$	−27.4269			−1.30309			−0.651545	−	0.758610i	$-0.725879\pi$
$443$	−27.4269			−1.30309			−0.651545	+	0.758610i	$0.725879\pi$
$444$		0			0
$445$	8.64629			0.409873
$446$		0			0
$447$	2.75818			0.130458
$448$		0			0
$449$	9.19492			0.433935			0.216968	−	0.976179i	$-0.430383\pi$
$449$	9.19492			0.433935			0.216968	+	0.976179i	$0.430383\pi$
$450$		0			0
$451$	6.95422			0.327462
$452$		0			0
$453$		−	14.6689i		−	0.689203i
$454$		0			0
$455$	7.04802	+	23.1462i	0.330416	+	1.08511i
$456$		0			0
$457$		−	7.93551i		−	0.371207i	−0.982625	−	0.185604i	$-0.940576\pi$
$457$		−	7.93551i		−	0.371207i	0.982625	−	0.185604i	$-0.0594241\pi$
$458$		0			0
$459$			3.92234i			0.183079i
$460$		0			0
$461$			25.2115i			1.17422i	0.809508	+	0.587108i	$0.199734\pi$
$461$			25.2115i			1.17422i	−0.809508	+	0.587108i	$0.800266\pi$
$462$		0			0
$463$	−10.9267			−0.507807			−0.253904	−	0.967230i	$-0.581715\pi$
$463$	−10.9267			−0.507807			−0.253904	+	0.967230i	$0.581715\pi$
$464$		0			0
$465$	−0.493819			−0.0229003
$466$		0			0
$467$	−27.7340			−1.28337			−0.641687	−	0.766966i	$-0.721765\pi$
$467$	−27.7340			−1.28337			−0.641687	+	0.766966i	$0.721765\pi$
$468$		0			0
$469$	7.74711	+	25.4421i	0.357728	+	1.17481i
$470$		0			0
$471$			21.3297i			0.982821i
$472$		0			0
$473$	−31.4913			−1.44797
$474$		0			0
$475$	1.23976			0.0568839
$476$		0			0
$477$		−	7.37616i		−	0.337731i
$478$		0			0
$479$	6.79293			0.310377			0.155188	−	0.987885i	$-0.450402\pi$
$479$	6.79293			0.310377			0.155188	+	0.987885i	$0.450402\pi$
$480$		0			0
$481$	−4.65301			−0.212159
$482$		0			0
$483$	−5.22177	−	11.5643i	−0.237599	−	0.526194i
$484$		0			0
$485$	8.85834			0.402237
$486$		0			0
$487$	41.7910			1.89373			0.946866	−	0.321628i	$-0.104230\pi$
$487$	41.7910			1.89373			0.946866	+	0.321628i	$0.104230\pi$
$488$		0			0
$489$			17.6971i			0.800292i
$490$		0			0
$491$	30.3317			1.36885			0.684426	−	0.729082i	$-0.260053\pi$
$491$	30.3317			1.36885			0.684426	+	0.729082i	$0.260053\pi$
$492$		0			0
$493$	−1.24272			−0.0559693
$494$		0			0
$495$		−	5.40378i		−	0.242882i
$496$		0			0
$497$	27.7107	−	8.43789i	1.24299	−	0.378491i
$498$		0			0
$499$	−3.93356			−0.176091			−0.0880453	−	0.996116i	$-0.528062\pi$
$499$	−3.93356			−0.176091			−0.0880453	+	0.996116i	$0.528062\pi$
$500$		0			0
$501$	−3.34633			−0.149503
$502$		0			0
$503$	−12.9395			−0.576945			−0.288472	−	0.957488i	$-0.593147\pi$
$503$	−12.9395			−0.576945			−0.288472	+	0.957488i	$0.593147\pi$
$504$		0			0
$505$			12.6177i			0.561478i
$506$		0			0
$507$		−	28.6984i		−	1.27454i
$508$		0			0
$509$		−	23.5446i		−	1.04360i	−0.853068	−	0.521799i	$-0.825261\pi$
$509$		−	23.5446i		−	1.04360i	0.853068	−	0.521799i	$-0.174739\pi$
$510$		0			0
$511$	−11.3582	−	37.3011i	−0.502456	−	1.65010i
$512$		0			0
$513$			0.414030i			0.0182799i
$514$		0			0
$515$	−19.0271			−0.838434
$516$		0			0
$517$	15.0220			0.660666
$518$		0			0
$519$	11.1048			0.487449
$520$		0			0
$521$	−20.0306			−0.877555			−0.438778	−	0.898596i	$-0.644588\pi$
$521$	−20.0306			−0.877555			−0.438778	+	0.898596i	$0.644588\pi$
$522$		0			0
$523$	6.82214			0.298311			0.149156	−	0.988814i	$-0.452344\pi$
$523$	6.82214			0.298311			0.149156	+	0.988814i	$0.452344\pi$
$524$		0			0
$525$	−2.30773	−	7.57878i	−0.100718	−	0.330765i
$526$		0			0
$527$		−	1.36769i		−	0.0595773i
$528$		0			0
$529$	22.2440	−	5.84865i	0.967128	−	0.254289i
$530$		0			0
$531$			5.43153i			0.235708i
$532$		0			0
$533$	11.7689			0.509769
$534$		0			0
$535$			0.712080i			0.0307859i
$536$		0			0
$537$		−	17.4752i		−	0.754110i
$538$		0			0
$539$	14.8860	+	22.1769i	0.641184	+	0.955229i
$540$		0			0
$541$	−15.9600			−0.686174			−0.343087	−	0.939304i	$-0.611473\pi$
$541$	−15.9600			−0.686174			−0.343087	+	0.939304i	$0.611473\pi$
$542$		0			0
$543$			12.8079i			0.549640i
$544$		0			0
$545$			5.49841i			0.235526i
$546$		0			0
$547$	−19.8572			−0.849033			−0.424517	−	0.905420i	$-0.639556\pi$
$547$	−19.8572			−0.849033			−0.424517	+	0.905420i	$0.639556\pi$
$548$		0			0
$549$	−11.2869			−0.481714
$550$		0			0
$551$	−0.131178			−0.00558836
$552$		0			0
$553$	3.49997	+	11.4942i	0.148834	+	0.488782i
$554$		0			0
$555$	−1.02047			−0.0433166
$556$		0			0
$557$		−	29.8239i		−	1.26368i	−0.775100	−	0.631839i	$-0.782301\pi$
$557$		−	29.8239i		−	1.26368i	0.775100	−	0.631839i	$-0.217699\pi$
$558$		0			0
$559$	−53.2942			−2.25410
$560$		0			0
$561$	14.9664			0.631880
$562$		0			0
$563$	−27.5948			−1.16298			−0.581492	−	0.813552i	$-0.697531\pi$
$563$	−27.5948			−1.16298			−0.581492	+	0.813552i	$0.697531\pi$
$564$		0			0
$565$		−	16.9225i		−	0.711935i
$566$		0			0
$567$	2.53101	−	0.770692i	0.106293	−	0.0323660i
$568$		0			0
$569$		−	31.8982i		−	1.33724i	−0.743603	−	0.668622i	$-0.766884\pi$
$569$		−	31.8982i		−	1.33724i	0.743603	−	0.668622i	$-0.233116\pi$
$570$		0			0
$571$		−	1.17986i		−	0.0493757i	−0.999695	−	0.0246878i	$-0.992141\pi$
$571$		−	1.17986i		−	0.0493757i	0.999695	−	0.0246878i	$-0.00785918\pi$
$572$		0			0
$573$	−24.8388			−1.03765
$574$		0			0
$575$	14.2420	−	1.84105i	0.593931	−	0.0767769i
$576$		0			0
$577$			17.7282i			0.738033i	0.929423	+	0.369017i	$0.120305\pi$
$577$			17.7282i			0.738033i	−0.929423	+	0.369017i	$0.879695\pi$
$578$		0			0
$579$			9.14938i			0.380235i
$580$		0			0
$581$	−28.4536	+	8.66410i	−1.18045	+	0.359448i
$582$		0			0
$583$	−28.1450			−1.16565
$584$		0			0
$585$		−	9.14504i		−	0.378101i
$586$		0			0
$587$			14.3682i			0.593039i	0.955027	+	0.296520i	$0.0958260\pi$
$587$			14.3682i			0.593039i	−0.955027	+	0.296520i	$0.904174\pi$
$588$		0			0
$589$		−	0.144369i		−	0.00594861i
$590$		0			0
$591$		−	11.5979i		−	0.477074i
$592$		0			0
$593$		−	16.6218i		−	0.682574i	−0.939959	−	0.341287i	$-0.889137\pi$
$593$		−	16.6218i		−	0.682574i	0.939959	−	0.341287i	$-0.110863\pi$
$594$		0			0
$595$	−14.0594	+	4.28107i	−0.576378	+	0.175507i
$596$		0			0
$597$			16.5243i			0.676296i
$598$		0			0
$599$	41.9563			1.71429			0.857144	−	0.515077i	$-0.172237\pi$
$599$	41.9563			1.71429			0.857144	+	0.515077i	$0.172237\pi$
$600$		0			0
$601$		−	44.5323i		−	1.81651i	−0.418418	−	0.908255i	$-0.637415\pi$
$601$		−	44.5323i		−	1.81651i	0.418418	−	0.908255i	$-0.362585\pi$
$602$		0			0
$603$		−	10.0521i		−	0.409355i
$604$		0			0
$605$	−5.04079			−0.204937
$606$		0			0
$607$		−	37.6908i		−	1.52982i	−0.644135	−	0.764912i	$-0.722782\pi$
$607$		−	37.6908i		−	1.52982i	0.644135	−	0.764912i	$-0.277218\pi$
$608$		0			0
$609$	0.244180	+	0.801906i	0.00989467	+	0.0324949i
$610$		0			0
$611$	25.4224			1.02848
$612$		0			0
$613$		−	44.4554i		−	1.79554i	−0.440466	−	0.897769i	$-0.645187\pi$
$613$		−	44.4554i		−	1.79554i	0.440466	−	0.897769i	$-0.354813\pi$
$614$		0			0
$615$	2.58109			0.104080
$616$		0			0
$617$			42.6756i			1.71805i	0.511930	+	0.859027i	$0.328931\pi$
$617$			42.6756i			1.71805i	−0.511930	+	0.859027i	$0.671069\pi$
$618$		0			0
$619$	15.2695			0.613734			0.306867	−	0.951752i	$-0.400719\pi$
$619$	15.2695			0.613734			0.306867	+	0.951752i	$0.400719\pi$
$620$		0			0
$621$	0.614837	+	4.75626i	0.0246726	+	0.190862i
$622$		0			0
$623$	15.4525	−	4.70527i	0.619090	−	0.188513i
$624$		0			0
$625$	−1.06197			−0.0424790
$626$		0			0
$627$	1.57980			0.0630912
$628$		0			0
$629$		−	2.82631i		−	0.112692i
$630$		0			0
$631$		−	14.0071i		−	0.557614i	−0.960347	−	0.278807i	$-0.910061\pi$
$631$		−	14.0071i		−	0.557614i	0.960347	−	0.278807i	$-0.0899390\pi$
$632$		0			0
$633$			5.78310i			0.229858i
$634$		0			0
$635$	3.64845			0.144784
$636$		0			0
$637$	25.1922	+	37.5310i	0.998149	+	1.48703i
$638$		0			0
$639$	−10.9485			−0.433114
$640$		0			0
$641$		−	1.11878i		−	0.0441890i	−0.999756	−	0.0220945i	$-0.992967\pi$
$641$		−	1.11878i		−	0.0441890i	0.999756	−	0.0220945i	$-0.00703347\pi$
$642$		0			0
$643$	−5.79011			−0.228340			−0.114170	−	0.993461i	$-0.536421\pi$
$643$	−5.79011			−0.228340			−0.114170	+	0.993461i	$0.536421\pi$
$644$		0			0
$645$	−11.6882			−0.460221
$646$		0			0
$647$		−	18.1981i		−	0.715442i	−0.933828	−	0.357721i	$-0.883554\pi$
$647$		−	18.1981i		−	0.715442i	0.933828	−	0.357721i	$-0.116446\pi$
$648$		0			0
$649$	20.7249			0.813525
$650$		0			0
$651$	−0.882544	+	0.268734i	−0.0345896	+	0.0105325i
$652$		0			0
$653$	−21.9672			−0.859642			−0.429821	−	0.902914i	$-0.641423\pi$
$653$	−21.9672			−0.859642			−0.429821	+	0.902914i	$0.641423\pi$
$654$		0			0
$655$		−	3.91416i		−	0.152939i
$656$		0			0
$657$			14.7376i			0.574969i
$658$		0			0
$659$		−	24.6452i		−	0.960041i	−0.877257	−	0.480020i	$-0.840629\pi$
$659$		−	24.6452i		−	0.960041i	0.877257	−	0.480020i	$-0.159371\pi$
$660$		0			0
$661$	−3.14516			−0.122333			−0.0611663	−	0.998128i	$-0.519482\pi$
$661$	−3.14516			−0.122333			−0.0611663	+	0.998128i	$0.519482\pi$
$662$		0			0
$663$	25.3282			0.983666
$664$		0			0
$665$	−1.48406	+	0.451896i	−0.0575495	+	0.0175238i
$666$		0			0
$667$	−1.50693	+	0.194800i	−0.0583487	+	0.00754268i
$668$		0			0
$669$	−18.2037			−0.703797
$670$		0			0
$671$			43.0672i			1.66259i
$672$		0			0
$673$	−0.0745018			−0.00287183			−0.00143592	−	0.999999i	$-0.500457\pi$
$673$	−0.0745018			−0.00287183			−0.00143592	+	0.999999i	$0.500457\pi$
$674$		0			0
$675$			2.99436i			0.115253i
$676$		0			0
$677$	24.4091			0.938116			0.469058	−	0.883167i	$-0.344593\pi$
$677$	24.4091			0.938116			0.469058	+	0.883167i	$0.344593\pi$
$678$		0			0
$679$	15.8315	−	4.82067i	0.607555	−	0.185000i
$680$		0			0
$681$			7.39741i			0.283469i
$682$		0			0
$683$	4.37727			0.167492			0.0837458	−	0.996487i	$-0.473312\pi$
$683$	4.37727			0.167492			0.0837458	+	0.996487i	$0.473312\pi$
$684$		0			0
$685$			0.0236237i			0.000902616i
$686$		0			0
$687$			13.8540i			0.528565i
$688$		0			0
$689$	−47.6310			−1.81460
$690$		0			0
$691$		−	18.7370i		−	0.712788i	−0.934336	−	0.356394i	$-0.884006\pi$
$691$		−	18.7370i		−	0.712788i	0.934336	−	0.356394i	$-0.115994\pi$
$692$		0			0
$693$	−2.94071	−	9.65752i	−0.111708	−	0.366859i
$694$		0			0
$695$			13.8043i			0.523629i
$696$		0			0
$697$			7.14862i			0.270773i
$698$		0			0
$699$			30.0698i			1.13734i
$700$		0			0
$701$			30.5807i			1.15502i	0.816385	+	0.577508i	$0.195975\pi$
$701$			30.5807i			1.15502i	−0.816385	+	0.577508i	$0.804025\pi$
$702$		0			0
$703$		−	0.298336i		−	0.0112520i
$704$		0			0
$705$	5.57548			0.209985
$706$		0			0
$707$	6.86647	+	22.5500i	0.258240	+	0.848081i
$708$		0			0
$709$		−	32.1030i		−	1.20565i	−0.797872	−	0.602826i	$-0.794041\pi$
$709$		−	32.1030i		−	1.20565i	0.797872	−	0.602826i	$-0.205959\pi$
$710$		0			0
$711$		−	4.54134i		−	0.170313i
$712$		0			0
$713$	−0.214389	−	1.65847i	−0.00802892	−	0.0621101i
$714$		0			0
$715$	−34.8945			−1.30498
$716$		0			0
$717$		−	6.45467i		−	0.241054i
$718$		0			0
$719$			32.8318i			1.22442i	0.790696	+	0.612209i	$0.209719\pi$
$719$			32.8318i			1.22442i	−0.790696	+	0.612209i	$0.790281\pi$
$720$		0			0
$721$	−34.0049	+	10.3545i	−1.26641	+	0.385620i
$722$		0			0
$723$			19.1500i			0.712195i
$724$		0			0
$725$	−0.948709			−0.0352342
$726$		0			0
$727$	−15.8639			−0.588361			−0.294180	−	0.955750i	$-0.595047\pi$
$727$	−15.8639			−0.588361			−0.294180	+	0.955750i	$0.595047\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	32.3717i		−	1.19731i
$732$		0			0
$733$	−24.3132			−0.898027			−0.449014	−	0.893525i	$-0.648224\pi$
$733$	−24.3132			−0.898027			−0.449014	+	0.893525i	$0.648224\pi$
$734$		0			0
$735$	5.52499	+	8.23108i	0.203792	+	0.303608i
$736$		0			0
$737$	−38.3557			−1.41285
$738$		0			0
$739$	−42.2514			−1.55424			−0.777122	−	0.629350i	$-0.783321\pi$
$739$	−42.2514			−1.55424			−0.777122	+	0.629350i	$0.783321\pi$
$740$		0			0
$741$	2.67357			0.0982160
$742$		0			0
$743$			43.6045i			1.59969i	0.600205	+	0.799846i	$0.295086\pi$
$743$			43.6045i			1.59969i	−0.600205	+	0.799846i	$0.704914\pi$
$744$		0			0
$745$		−	3.90616i		−	0.143110i
$746$		0			0
$747$	11.2420			0.411322
$748$		0			0
$749$	0.387511	+	1.27262i	0.0141593	+	0.0465004i
$750$		0			0
$751$			0.977002i			0.0356513i	0.999841	+	0.0178256i	$0.00567438\pi$
$751$			0.977002i			0.0356513i	−0.999841	+	0.0178256i	$0.994326\pi$
$752$		0			0
$753$			20.3278i			0.740786i
$754$		0			0
$755$	−20.7741			−0.756047
$756$		0			0
$757$		−	49.8339i		−	1.81125i	−0.424084	−	0.905623i	$-0.639404\pi$
$757$		−	49.8339i		−	1.81125i	0.424084	−	0.905623i	$-0.360596\pi$
$758$		0			0
$759$	18.1483	−	2.34602i	0.658742	−	0.0851550i
$760$		0			0
$761$		−	27.2166i		−	0.986600i	−0.869859	−	0.493300i	$-0.835790\pi$
$761$		−	27.2166i		−	0.986600i	0.869859	−	0.493300i	$-0.164210\pi$
$762$		0			0
$763$	2.99221	+	9.82665i	0.108325	+	0.355749i
$764$		0			0
$765$	5.55483			0.200835
$766$		0			0
$767$	35.0737			1.26644
$768$		0			0
$769$	31.9223			1.15115			0.575573	−	0.817750i	$-0.304779\pi$
$769$	31.9223			1.15115			0.575573	+	0.817750i	$0.304779\pi$
$770$		0			0
$771$	18.5661			0.668641
$772$		0			0
$773$	27.2617			0.980536			0.490268	−	0.871572i	$-0.336899\pi$
$773$	27.2617			0.980536			0.490268	+	0.871572i	$0.336899\pi$
$774$		0			0
$775$		−	1.04411i		−	0.0375055i
$776$		0			0
$777$	−1.82376	+	0.555335i	−0.0654272	+	0.0199225i
$778$		0			0
$779$			0.754587i			0.0270359i
$780$		0			0
$781$			41.7757i			1.49485i
$782$		0			0
$783$		−	0.316832i		−	0.0113226i
$784$		0			0
$785$	30.2072			1.07814
$786$		0			0
$787$	−23.0958			−0.823277			−0.411638	−	0.911347i	$-0.635043\pi$
$787$	−23.0958			−0.823277			−0.411638	+	0.911347i	$0.635043\pi$
$788$		0			0
$789$	−24.8672			−0.885295
$790$		0			0
$791$	−9.20916	−	30.2436i	−0.327440	−	1.07534i
$792$		0			0
$793$			72.8844i			2.58820i
$794$		0			0
$795$	−10.4462			−0.370487
$796$		0			0
$797$	−3.80488			−0.134776			−0.0673879	−	0.997727i	$-0.521467\pi$
$797$	−3.80488			−0.134776			−0.0673879	+	0.997727i	$0.521467\pi$
$798$		0			0
$799$			15.4419i			0.546295i
$800$		0			0
$801$	−6.10525			−0.215718
$802$		0			0
$803$	56.2339			1.98445
$804$		0			0
$805$	−16.3774	+	7.39510i	−0.577228	+	0.260643i
$806$		0			0
$807$	−11.8372			−0.416688
$808$		0			0
$809$	9.70521			0.341217			0.170609	−	0.985339i	$-0.445427\pi$
$809$	9.70521			0.341217			0.170609	+	0.985339i	$0.445427\pi$
$810$		0			0
$811$			8.10574i			0.284631i	0.989821	+	0.142316i	$0.0454548\pi$
$811$			8.10574i			0.284631i	−0.989821	+	0.142316i	$0.954545\pi$
$812$		0			0
$813$	16.9783			0.595454
$814$		0			0
$815$	25.0628			0.877910
$816$		0			0
$817$		−	3.41705i		−	0.119548i
$818$		0			0
$819$	−4.97669	−	16.3438i	−0.173900	−	0.571100i
$820$		0			0
$821$	51.1589			1.78546			0.892729	−	0.450594i	$-0.148788\pi$
$821$	51.1589			1.78546			0.892729	+	0.450594i	$0.148788\pi$
$822$		0			0
$823$	37.6472			1.31230			0.656149	−	0.754631i	$-0.272184\pi$
$823$	37.6472			1.31230			0.656149	+	0.754631i	$0.272184\pi$
$824$		0			0
$825$	11.4255			0.397785
$826$		0			0
$827$		−	19.0680i		−	0.663059i	−0.943445	−	0.331530i	$-0.892435\pi$
$827$		−	19.0680i		−	0.663059i	0.943445	−	0.331530i	$-0.107565\pi$
$828$		0			0
$829$			1.40163i			0.0486805i	0.999704	+	0.0243403i	$0.00774851\pi$
$829$			1.40163i			0.0486805i	−0.999704	+	0.0243403i	$0.992251\pi$
$830$		0			0
$831$			21.2648i			0.737667i
$832$		0			0
$833$	−22.7969	+	15.3021i	−0.789865	+	0.530186i
$834$		0			0
$835$			4.73909i			0.164003i
$836$		0			0
$837$	0.348692			0.0120525
$838$		0			0
$839$	42.0355			1.45123			0.725614	−	0.688102i	$-0.241556\pi$
$839$	42.0355			1.45123			0.725614	+	0.688102i	$0.241556\pi$
$840$		0			0
$841$	−28.8996			−0.996539
$842$		0			0
$843$	3.40757			0.117363
$844$		0			0
$845$	−40.6428			−1.39815
$846$		0			0
$847$	−9.00880	+	2.74317i	−0.309546	+	0.0942566i
$848$		0			0
$849$			8.95787i			0.307433i
$850$		0			0
$851$	−0.443031	−	3.42720i	−0.0151869	−	0.117483i
$852$		0			0
$853$			25.0768i			0.858614i	0.903159	+	0.429307i	$0.141242\pi$
$853$			25.0768i			0.858614i	−0.903159	+	0.429307i	$0.858758\pi$
$854$		0			0
$855$	0.586351			0.0200528
$856$		0			0
$857$		−	0.825211i		−	0.0281887i	−0.999901	−	0.0140943i	$-0.995513\pi$
$857$		−	0.825211i		−	0.0281887i	0.999901	−	0.0140943i	$-0.00448652\pi$
$858$		0			0
$859$		−	27.4033i		−	0.934988i	−0.883996	−	0.467494i	$-0.845157\pi$
$859$		−	27.4033i		−	0.934988i	0.883996	−	0.467494i	$-0.154843\pi$
$860$		0			0
$861$	4.61288	−	1.40462i	0.157207	−	0.0478693i
$862$		0			0
$863$	−16.6654			−0.567295			−0.283648	−	0.958929i	$-0.591545\pi$
$863$	−16.6654			−0.567295			−0.283648	+	0.958929i	$0.591545\pi$
$864$		0			0
$865$		−	15.7267i		−	0.534725i
$866$		0			0
$867$		−	1.61527i		−	0.0548576i
$868$		0			0
$869$	−17.3283			−0.587821
$870$		0			0
$871$	−64.9110			−2.19942
$872$		0			0
$873$	−6.25498			−0.211699
$874$		0			0
$875$	−28.6553	+	8.72552i	−0.968725	+	0.294976i
$876$		0			0
$877$	11.1019			0.374883			0.187442	−	0.982276i	$-0.439980\pi$
$877$	11.1019			0.374883			0.187442	+	0.982276i	$0.439980\pi$
$878$		0			0
$879$		−	25.8886i		−	0.873201i
$880$		0			0
$881$	−36.1891			−1.21924			−0.609620	−	0.792694i	$-0.708678\pi$
$881$	−36.1891			−1.21924			−0.609620	+	0.792694i	$0.708678\pi$
$882$		0			0
$883$	−43.5853			−1.46676			−0.733380	−	0.679819i	$-0.762059\pi$
$883$	−43.5853			−1.46676			−0.733380	+	0.679819i	$0.762059\pi$
$884$		0			0
$885$	7.69216			0.258569
$886$		0			0
$887$		−	35.1239i		−	1.17934i	−0.807642	−	0.589672i	$-0.799257\pi$
$887$		−	35.1239i		−	1.17934i	0.807642	−	0.589672i	$-0.200743\pi$
$888$		0			0
$889$	6.52044	−	1.98547i	0.218689	−	0.0665905i
$890$		0			0
$891$			3.81567i			0.127830i
$892$		0			0
$893$			1.63000i			0.0545459i
$894$		0			0
$895$	−24.7484			−0.827249
$896$		0			0
$897$	30.7132	−	3.97027i	1.02548	−	0.132563i
$898$		0			0
$899$			0.110477i			0.00368460i
$900$		0			0
$901$		−	28.9318i		−	0.963857i
$902$		0			0
$903$	−20.8889	+	6.36064i	−0.695138	+	0.211669i
$904$		0			0
$905$	18.1386			0.602948
$906$		0			0
$907$			47.1105i			1.56428i	0.623103	+	0.782140i	$0.285872\pi$
$907$			47.1105i			1.56428i	−0.623103	+	0.782140i	$0.714128\pi$
$908$		0			0
$909$		−	8.90949i		−	0.295509i
$910$		0			0
$911$			45.0198i			1.49157i	0.666185	+	0.745786i	$0.267926\pi$
$911$			45.0198i			1.49157i	−0.666185	+	0.745786i	$0.732074\pi$
$912$		0			0
$913$		−	42.8957i		−	1.41964i
$914$		0			0
$915$			15.9846i			0.528434i
$916$		0			0
$917$	−2.13007	−	6.99532i	−0.0703411	−	0.231006i
$918$		0			0
$919$			32.1007i			1.05891i	0.848339	+	0.529453i	$0.177603\pi$
$919$			32.1007i			1.05891i	−0.848339	+	0.529453i	$0.822397\pi$
$920$		0			0
$921$	−10.4476			−0.344260
$922$		0			0
$923$			70.6989i			2.32708i
$924$		0			0
$925$		−	2.15764i		−	0.0709427i
$926$		0			0
$927$	13.4353			0.441272
$928$		0			0
$929$		−	15.0188i		−	0.492751i	−0.969174	−	0.246376i	$-0.920760\pi$
$929$		−	15.0188i		−	0.492751i	0.969174	−	0.246376i	$-0.0792397\pi$
$930$		0			0
$931$	−2.40637	+	1.61524i	−0.0788655	+	0.0529374i
$932$		0			0
$933$	32.2923			1.05720
$934$		0			0
$935$		−	21.1954i		−	0.693165i
$936$		0			0
$937$	−29.1357			−0.951821			−0.475910	−	0.879494i	$-0.657881\pi$
$937$	−29.1357			−0.951821			−0.475910	+	0.879494i	$0.657881\pi$
$938$		0			0
$939$			13.9154i			0.454111i
$940$		0			0
$941$	23.6796			0.771934			0.385967	−	0.922513i	$-0.373868\pi$
$941$	23.6796			0.771934			0.385967	+	0.922513i	$0.373868\pi$
$942$		0			0
$943$	1.12057	+	8.66848i	0.0364907	+	0.282284i
$944$		0			0
$945$	−1.09146	−	3.58444i	−0.0355051	−	0.116602i
$946$		0			0
$947$	−15.6928			−0.509948			−0.254974	−	0.966948i	$-0.582067\pi$
$947$	−15.6928			−0.509948			−0.254974	+	0.966948i	$0.582067\pi$
$948$		0			0
$949$	95.1670			3.08925
$950$		0			0
$951$			6.23595i			0.202215i
$952$		0			0
$953$			52.8288i			1.71129i	0.517562	+	0.855646i	$0.326840\pi$
$953$			52.8288i			1.71129i	−0.517562	+	0.855646i	$0.673160\pi$
$954$		0			0
$955$			35.1768i			1.13829i
$956$		0			0
$957$	−1.20893			−0.0390790
$958$		0			0
$959$	0.0128559	+	0.0422199i	0.000415140	+	0.00136335i
$960$		0			0
$961$	30.8784			0.996078
$962$		0			0
$963$		−	0.502808i		−	0.0162028i
$964$		0			0
$965$	12.9574			0.417114
$966$		0			0
$967$	−31.0844			−0.999607			−0.499804	−	0.866139i	$-0.666595\pi$
$967$	−31.0844			−0.999607			−0.499804	+	0.866139i	$0.666595\pi$
$968$		0			0
$969$			1.62396i			0.0521692i
$970$		0			0
$971$	−6.31232			−0.202572			−0.101286	−	0.994857i	$-0.532296\pi$
$971$	−6.31232			−0.202572			−0.101286	+	0.994857i	$0.532296\pi$
$972$		0			0
$973$	7.51226	+	24.6709i	0.240832	+	0.790911i
$974$		0			0
$975$	19.3359			0.619244
$976$		0			0
$977$			40.8629i			1.30732i	0.756788	+	0.653660i	$0.226767\pi$
$977$			40.8629i			1.30732i	−0.756788	+	0.653660i	$0.773233\pi$
$978$		0			0
$979$			23.2956i			0.744532i
$980$		0			0
$981$		−	3.88249i		−	0.123958i
$982$		0			0
$983$	35.5720			1.13457			0.567285	−	0.823522i	$-0.307994\pi$
$983$	35.5720			1.13457			0.567285	+	0.823522i	$0.307994\pi$
$984$		0			0
$985$	−16.4250			−0.523345
$986$		0			0
$987$	9.96439	−	3.03415i	0.317170	−	0.0965781i
$988$		0			0
$989$	−5.07435	−	39.2541i	−0.161355	−	1.24821i
$990$		0			0
$991$	3.81745			0.121265			0.0606327	−	0.998160i	$-0.480688\pi$
$991$	3.81745			0.121265			0.0606327	+	0.998160i	$0.480688\pi$
$992$		0			0
$993$			2.22381i			0.0705705i
$994$		0			0
$995$	23.4019			0.741889
$996$		0			0
$997$		−	11.0193i		−	0.348985i	−0.984659	−	0.174492i	$-0.944172\pi$
$997$		−	11.0193i		−	0.348985i	0.984659	−	0.174492i	$-0.0558284\pi$
$998$		0			0
$999$	0.720567			0.0227977

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.k.a.1609.16	yes	32
3.2	odd	2		5796.2.k.d.5473.11		32
7.6	odd	2	inner	1932.2.k.a.1609.13	✓	32
21.20	even	2		5796.2.k.d.5473.21		32
23.22	odd	2	inner	1932.2.k.a.1609.15	yes	32
69.68	even	2		5796.2.k.d.5473.22		32
161.160	even	2	inner	1932.2.k.a.1609.14	yes	32
483.482	odd	2		5796.2.k.d.5473.12		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.k.a.1609.13	✓	32	7.6	odd	2	inner
1932.2.k.a.1609.14	yes	32	161.160	even	2	inner
1932.2.k.a.1609.15	yes	32	23.22	odd	2	inner
1932.2.k.a.1609.16	yes	32	1.1	even	1	trivial
5796.2.k.d.5473.11		32	3.2	odd	2
5796.2.k.d.5473.12		32	483.482	odd	2
5796.2.k.d.5473.21		32	21.20	even	2
5796.2.k.d.5473.22		32	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.41621 q^{5} +(2.53101 - 0.770692i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.41621 q^{5} +(2.53101 - 0.770692i) q^{7} -1.00000 q^{9} +3.81567i q^{11} +6.45743i q^{13} +1.41621i q^{15} -3.92234 q^{17} -0.414030 q^{19} +(0.770692 + 2.53101i) q^{21} +(-4.75626 + 0.614837i) q^{23} -2.99436 q^{25} -1.00000i q^{27} +0.316832 q^{29} +0.348692i q^{31} -3.81567 q^{33} +(3.58444 - 1.09146i) q^{35} +0.720567i q^{37} -6.45743 q^{39} -1.82254i q^{41} +8.25316i q^{43} -1.41621 q^{45} -3.93692i q^{47} +(5.81207 - 3.90127i) q^{49} -3.92234i q^{51} +7.37616i q^{53} +5.40378i q^{55} -0.414030i q^{57} -5.43153i q^{59} +11.2869 q^{61} +(-2.53101 + 0.770692i) q^{63} +9.14504i q^{65} +10.0521i q^{67} +(-0.614837 - 4.75626i) q^{69} +10.9485 q^{71} -14.7376i q^{73} -2.99436i q^{75} +(2.94071 + 9.65752i) q^{77} +4.54134i q^{79} +1.00000 q^{81} -11.2420 q^{83} -5.55483 q^{85} +0.316832i q^{87} +6.10525 q^{89} +(4.97669 + 16.3438i) q^{91} -0.348692 q^{93} -0.586351 q^{95} +6.25498 q^{97} -3.81567i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9}+O(q^{10}) \) 32 * q - 32 * q^9	\( 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) \) 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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