LMFDB - Embedded newform 3762.2.b.a.989.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(989,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3762.989");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3762.b (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:	$30.0397212404$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			989.5
Character	$\chi$	$=$	3762.989
Dual form			3762.2.b.a.989.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.00309i q^{5} +3.31087i q^{7} -1.00000 q^{8} +3.00309i q^{10} +(3.24754 + 0.673411i) q^{11} -4.99159i q^{13} -3.31087i q^{14} +1.00000 q^{16} +5.87757 q^{17} -1.00000i q^{19} -3.00309i q^{20} +(-3.24754 - 0.673411i) q^{22} +0.771532i q^{23} -4.01857 q^{25} +4.99159i q^{26} +3.31087i q^{28} +8.92687 q^{29} -3.81106 q^{31} -1.00000 q^{32} -5.87757 q^{34} +9.94287 q^{35} -1.04032 q^{37} +1.00000i q^{38} +3.00309i q^{40} -5.68449 q^{41} +7.56719i q^{43} +(3.24754 + 0.673411i) q^{44} -0.771532i q^{46} -5.44408i q^{47} -3.96189 q^{49} +4.01857 q^{50} -4.99159i q^{52} -5.33171i q^{53} +(2.02232 - 9.75267i) q^{55} -3.31087i q^{56} -8.92687 q^{58} -0.367024i q^{59} -4.68980i q^{61} +3.81106 q^{62} +1.00000 q^{64} -14.9902 q^{65} +2.00040 q^{67} +5.87757 q^{68} -9.94287 q^{70} +13.5343i q^{71} +12.6777i q^{73} +1.04032 q^{74} -1.00000i q^{76} +(-2.22958 + 10.7522i) q^{77} -4.97264i q^{79} -3.00309i q^{80} +5.68449 q^{82} +10.4352 q^{83} -17.6509i q^{85} -7.56719i q^{86} +(-3.24754 - 0.673411i) q^{88} +11.6524i q^{89} +16.5265 q^{91} +0.771532i q^{92} +5.44408i q^{94} -3.00309 q^{95} -7.71791 q^{97} +3.96189 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 36 q^{2} + 36 q^{4} - 36 q^{8} + 4 q^{11} + 36 q^{16} - 16 q^{17} - 4 q^{22} - 28 q^{25} - 8 q^{31} - 36 q^{32} + 16 q^{34} - 16 q^{35} + 16 q^{37} - 24 q^{41} + 4 q^{44} - 68 q^{49} + 28 q^{50}+ \cdots + 68 q^{98}+O(q^{100}) $ 36 * q - 36 * q^2 + 36 * q^4 - 36 * q^8 + 4 * q^11 + 36 * q^16 - 16 * q^17 - 4 * q^22 - 28 * q^25 - 8 * q^31 - 36 * q^32 + 16 * q^34 - 16 * q^35 + 16 * q^37 - 24 * q^41 + 4 * q^44 - 68 * q^49 + 28 * q^50 + 28 * q^55 + 8 * q^62 + 36 * q^64 - 32 * q^65 + 16 * q^67 - 16 * q^68 + 16 * q^70 - 16 * q^74 + 32 * q^77 + 24 * q^82 + 56 * q^83 - 4 * q^88 - 48 * q^91 + 8 * q^97 + 68 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3762\mathbb{Z}\right)^\times$.

$n$	$343$	$2377$	$2927$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		−	3.00309i		−	1.34302i	−0.740993	−	0.671512i	$-0.765645\pi$
$5$		−	3.00309i		−	1.34302i	0.740993	−	0.671512i	$-0.234355\pi$
$6$		0			0
$7$			3.31087i			1.25139i	0.780067	+	0.625697i	$0.215185\pi$
$7$			3.31087i			1.25139i	−0.780067	+	0.625697i	$0.784815\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$			3.00309i			0.949662i
$11$	3.24754	+	0.673411i	0.979170	+	0.203041i
$11$	3.24754	+	0.673411i	0.979170	+	0.203041i
$12$		0			0
$13$		−	4.99159i		−	1.38442i	−0.721697	−	0.692209i	$-0.756638\pi$
$13$		−	4.99159i		−	1.38442i	0.721697	−	0.692209i	$-0.243362\pi$
$14$		−	3.31087i		−	0.884869i
$15$		0			0
$16$	1.00000			0.250000
$17$	5.87757			1.42552			0.712760	−	0.701408i	$-0.247445\pi$
$17$	5.87757			1.42552			0.712760	+	0.701408i	$0.247445\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		−	3.00309i		−	0.671512i
$21$		0			0
$22$	−3.24754	−	0.673411i	−0.692378	−	0.143572i
$23$			0.771532i			0.160875i	0.996760	+	0.0804377i	$0.0256318\pi$
$23$			0.771532i			0.160875i	−0.996760	+	0.0804377i	$0.974368\pi$
$24$		0			0
$25$	−4.01857			−0.803715
$26$			4.99159i			0.978931i
$27$		0			0
$28$			3.31087i			0.625697i
$29$	8.92687			1.65768			0.828839	−	0.559487i	$-0.189002\pi$
$29$	8.92687			1.65768			0.828839	+	0.559487i	$0.189002\pi$
$30$		0			0
$31$	−3.81106			−0.684487			−0.342243	−	0.939611i	$-0.611187\pi$
$31$	−3.81106			−0.684487			−0.342243	+	0.939611i	$0.611187\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$	−5.87757			−1.00799
$35$	9.94287			1.68065
$36$		0			0
$37$	−1.04032			−0.171027			−0.0855134	−	0.996337i	$-0.527253\pi$
$37$	−1.04032			−0.171027			−0.0855134	+	0.996337i	$0.527253\pi$
$38$			1.00000i			0.162221i
$39$		0			0
$40$			3.00309i			0.474831i
$41$	−5.68449			−0.887768			−0.443884	−	0.896084i	$-0.646400\pi$
$41$	−5.68449			−0.887768			−0.443884	+	0.896084i	$0.646400\pi$
$42$		0			0
$43$			7.56719i			1.15399i	0.816749	+	0.576993i	$0.195774\pi$
$43$			7.56719i			1.15399i	−0.816749	+	0.576993i	$0.804226\pi$
$44$	3.24754	+	0.673411i	0.489585	+	0.101520i
$45$		0			0
$46$		−	0.771532i		−	0.113756i
$47$		−	5.44408i		−	0.794100i	−0.917797	−	0.397050i	$-0.870034\pi$
$47$		−	5.44408i		−	0.794100i	0.917797	−	0.397050i	$-0.129966\pi$
$48$		0			0
$49$	−3.96189			−0.565985
$50$	4.01857			0.568312
$51$		0			0
$52$		−	4.99159i		−	0.692209i
$53$		−	5.33171i		−	0.732366i	−0.930543	−	0.366183i	$-0.880664\pi$
$53$		−	5.33171i		−	0.732366i	0.930543	−	0.366183i	$-0.119336\pi$
$54$		0			0
$55$	2.02232	−	9.75267i	0.272689	−	1.31505i
$56$		−	3.31087i		−	0.442434i
$57$		0			0
$58$	−8.92687			−1.17216
$59$		−	0.367024i		−	0.0477825i	−0.999715	−	0.0238912i	$-0.992394\pi$
$59$		−	0.367024i		−	0.0477825i	0.999715	−	0.0238912i	$-0.00760554\pi$
$60$		0			0
$61$		−	4.68980i		−	0.600468i	−0.953866	−	0.300234i	$-0.902935\pi$
$61$		−	4.68980i		−	0.600468i	0.953866	−	0.300234i	$-0.0970647\pi$
$62$	3.81106			0.484005
$63$		0			0
$64$	1.00000			0.125000
$65$	−14.9902			−1.85931
$66$		0			0
$67$	2.00040			0.244387			0.122194	−	0.992506i	$-0.461007\pi$
$67$	2.00040			0.244387			0.122194	+	0.992506i	$0.461007\pi$
$68$	5.87757			0.712760
$69$		0			0
$70$	−9.94287			−1.18840
$71$			13.5343i			1.60623i	0.595826	+	0.803114i	$0.296825\pi$
$71$			13.5343i			1.60623i	−0.595826	+	0.803114i	$0.703175\pi$
$72$		0			0
$73$			12.6777i			1.48382i	0.670502	+	0.741908i	$0.266079\pi$
$73$			12.6777i			1.48382i	−0.670502	+	0.741908i	$0.733921\pi$
$74$	1.04032			0.120934
$75$		0			0
$76$		−	1.00000i		−	0.114708i
$77$	−2.22958	+	10.7522i	−0.254084	+	1.22533i
$78$		0			0
$79$		−	4.97264i		−	0.559466i	−0.960078	−	0.279733i	$-0.909754\pi$
$79$		−	4.97264i		−	0.559466i	0.960078	−	0.279733i	$-0.0902460\pi$
$80$		−	3.00309i		−	0.335756i
$81$		0			0
$82$	5.68449			0.627747
$83$	10.4352			1.14541			0.572706	−	0.819761i	$-0.305894\pi$
$83$	10.4352			1.14541			0.572706	+	0.819761i	$0.305894\pi$
$84$		0			0
$85$		−	17.6509i		−	1.91451i
$86$		−	7.56719i		−	0.815991i
$87$		0			0
$88$	−3.24754	−	0.673411i	−0.346189	−	0.0717858i
$89$			11.6524i			1.23515i	0.786511	+	0.617577i	$0.211886\pi$
$89$			11.6524i			1.23515i	−0.786511	+	0.617577i	$0.788114\pi$
$90$		0			0
$91$	16.5265			1.73245
$92$			0.771532i			0.0804377i
$93$		0			0
$94$			5.44408i			0.561514i
$95$	−3.00309			−0.308111
$96$		0			0
$97$	−7.71791			−0.783635			−0.391818	−	0.920043i	$-0.628154\pi$
$97$	−7.71791			−0.783635			−0.391818	+	0.920043i	$0.628154\pi$
$98$	3.96189			0.400212
$99$		0			0
$100$	−4.01857			−0.401857
$101$	5.63251			0.560456			0.280228	−	0.959934i	$-0.409590\pi$
$101$	5.63251			0.560456			0.280228	+	0.959934i	$0.409590\pi$
$102$		0			0
$103$	2.65084			0.261195			0.130597	−	0.991435i	$-0.458310\pi$
$103$	2.65084			0.261195			0.130597	+	0.991435i	$0.458310\pi$
$104$			4.99159i			0.489465i
$105$		0			0
$106$			5.33171i			0.517861i
$107$	18.0820			1.74805			0.874026	−	0.485879i	$-0.161500\pi$
$107$	18.0820			1.74805			0.874026	+	0.485879i	$0.161500\pi$
$108$		0			0
$109$		−	17.4431i		−	1.67074i	−0.549686	−	0.835371i	$-0.685253\pi$
$109$		−	17.4431i		−	1.67074i	0.549686	−	0.835371i	$-0.314747\pi$
$110$	−2.02232	+	9.75267i	−0.192820	+	0.929881i
$111$		0			0
$112$			3.31087i			0.312848i
$113$		−	0.423629i		−	0.0398517i	−0.999801	−	0.0199258i	$-0.993657\pi$
$113$		−	0.423629i		−	0.0398517i	0.999801	−	0.0199258i	$-0.00634301\pi$
$114$		0			0
$115$	2.31698			0.216060
$116$	8.92687			0.828839
$117$		0			0
$118$			0.367024i			0.0337873i
$119$			19.4599i			1.78389i
$120$		0			0
$121$	10.0930	+	4.37386i	0.917549	+	0.397623i
$122$			4.68980i			0.424595i
$123$		0			0
$124$	−3.81106			−0.342243
$125$		−	2.94731i		−	0.263616i
$126$		0			0
$127$			2.22458i			0.197399i	0.995117	+	0.0986997i	$0.0314683\pi$
$127$			2.22458i			0.197399i	−0.995117	+	0.0986997i	$0.968532\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	14.9902			1.31473
$131$	13.4918			1.17879			0.589394	−	0.807846i	$-0.299366\pi$
$131$	13.4918			1.17879			0.589394	+	0.807846i	$0.299366\pi$
$132$		0			0
$133$	3.31087			0.287089
$134$	−2.00040			−0.172808
$135$		0			0
$136$	−5.87757			−0.503997
$137$		−	13.2515i		−	1.13215i	−0.824353	−	0.566076i	$-0.808461\pi$
$137$		−	13.2515i		−	1.13215i	0.824353	−	0.566076i	$-0.191539\pi$
$138$		0			0
$139$		−	3.71160i		−	0.314814i	−0.987534	−	0.157407i	$-0.949687\pi$
$139$		−	3.71160i		−	0.314814i	0.987534	−	0.157407i	$-0.0503134\pi$
$140$	9.94287			0.840326
$141$		0			0
$142$		−	13.5343i		−	1.13577i
$143$	3.36139	−	16.2104i	0.281093	−	1.35558i
$144$		0			0
$145$		−	26.8082i		−	2.22630i
$146$		−	12.6777i		−	1.04922i
$147$		0			0
$148$	−1.04032			−0.0855134
$149$	−0.782444			−0.0641003			−0.0320502	−	0.999486i	$-0.510204\pi$
$149$	−0.782444			−0.0641003			−0.0320502	+	0.999486i	$0.510204\pi$
$150$		0			0
$151$			4.68719i			0.381438i	0.981645	+	0.190719i	$0.0610820\pi$
$151$			4.68719i			0.381438i	−0.981645	+	0.190719i	$0.938918\pi$
$152$			1.00000i			0.0811107i
$153$		0			0
$154$	2.22958	−	10.7522i	0.179665	−	0.866437i
$155$			11.4450i			0.919283i
$156$		0			0
$157$	−19.0620			−1.52131			−0.760655	−	0.649157i	$-0.775122\pi$
$157$	−19.0620			−1.52131			−0.760655	+	0.649157i	$0.775122\pi$
$158$			4.97264i			0.395602i
$159$		0			0
$160$			3.00309i			0.237415i
$161$	−2.55444			−0.201318
$162$		0			0
$163$	7.03872			0.551315			0.275657	−	0.961256i	$-0.411104\pi$
$163$	7.03872			0.551315			0.275657	+	0.961256i	$0.411104\pi$
$164$	−5.68449			−0.443884
$165$		0			0
$166$	−10.4352			−0.809929
$167$	−18.1622			−1.40543			−0.702716	−	0.711470i	$-0.748030\pi$
$167$	−18.1622			−1.40543			−0.702716	+	0.711470i	$0.748030\pi$
$168$		0			0
$169$	−11.9159			−0.916611
$170$			17.6509i			1.35376i
$171$		0			0
$172$			7.56719i			0.576993i
$173$	−20.0969			−1.52794			−0.763970	−	0.645252i	$-0.776752\pi$
$173$	−20.0969			−1.52794			−0.763970	+	0.645252i	$0.776752\pi$
$174$		0			0
$175$		−	13.3050i		−	1.00576i
$176$	3.24754	+	0.673411i	0.244793	+	0.0507602i
$177$		0			0
$178$		−	11.6524i		−	0.873385i
$179$		−	18.4938i		−	1.38229i	−0.722715	−	0.691146i	$-0.757106\pi$
$179$		−	18.4938i		−	1.38229i	0.722715	−	0.691146i	$-0.242894\pi$
$180$		0			0
$181$	11.0037			0.817898			0.408949	−	0.912557i	$-0.365895\pi$
$181$	11.0037			0.817898			0.408949	+	0.912557i	$0.365895\pi$
$182$	−16.5265			−1.22503
$183$		0			0
$184$		−	0.771532i		−	0.0568781i
$185$			3.12416i			0.229693i
$186$		0			0
$187$	19.0876	+	3.95802i	1.39583	+	0.289439i
$188$		−	5.44408i		−	0.397050i
$189$		0			0
$190$	3.00309			0.217867
$191$		−	19.9599i		−	1.44425i	−0.691763	−	0.722125i	$-0.743166\pi$
$191$		−	19.9599i		−	1.44425i	0.691763	−	0.722125i	$-0.256834\pi$
$192$		0			0
$193$		−	3.64379i		−	0.262285i	−0.991364	−	0.131143i	$-0.958135\pi$
$193$		−	3.64379i		−	0.262285i	0.991364	−	0.131143i	$-0.0418646\pi$
$194$	7.71791			0.554114
$195$		0			0
$196$	−3.96189			−0.282992
$197$	24.2098			1.72488			0.862440	−	0.506160i	$-0.168935\pi$
$197$	24.2098			1.72488			0.862440	+	0.506160i	$0.168935\pi$
$198$		0			0
$199$	19.1882			1.36022			0.680108	−	0.733112i	$-0.261933\pi$
$199$	19.1882			1.36022			0.680108	+	0.733112i	$0.261933\pi$
$200$	4.01857			0.284156
$201$		0			0
$202$	−5.63251			−0.396302
$203$			29.5557i			2.07441i
$204$		0			0
$205$			17.0711i			1.19229i
$206$	−2.65084			−0.184693
$207$		0			0
$208$		−	4.99159i		−	0.346104i
$209$	0.673411	−	3.24754i	0.0465808	−	0.224637i
$210$		0			0
$211$			6.95465i			0.478778i	0.970924	+	0.239389i	$0.0769471\pi$
$211$			6.95465i			0.478778i	−0.970924	+	0.239389i	$0.923053\pi$
$212$		−	5.33171i		−	0.366183i
$213$		0			0
$214$	−18.0820			−1.23606
$215$	22.7250			1.54983
$216$		0			0
$217$		−	12.6179i		−	0.856562i
$218$			17.4431i			1.18139i
$219$		0			0
$220$	2.02232	−	9.75267i	0.136344	−	0.657525i
$221$		−	29.3384i		−	1.97351i
$222$		0			0
$223$	−15.7239			−1.05295			−0.526476	−	0.850190i	$-0.676487\pi$
$223$	−15.7239			−1.05295			−0.526476	+	0.850190i	$0.676487\pi$
$224$		−	3.31087i		−	0.221217i
$225$		0			0
$226$			0.423629i			0.0281794i
$227$	−19.8004			−1.31420			−0.657101	−	0.753803i	$-0.728217\pi$
$227$	−19.8004			−1.31420			−0.657101	+	0.753803i	$0.728217\pi$
$228$		0			0
$229$	−7.86405			−0.519671			−0.259835	−	0.965653i	$-0.583668\pi$
$229$	−7.86405			−0.519671			−0.259835	+	0.965653i	$0.583668\pi$
$230$	−2.31698			−0.152777
$231$		0			0
$232$	−8.92687			−0.586078
$233$	−2.35291			−0.154144			−0.0770721	−	0.997026i	$-0.524557\pi$
$233$	−2.35291			−0.154144			−0.0770721	+	0.997026i	$0.524557\pi$
$234$		0			0
$235$	−16.3491			−1.06650
$236$		−	0.367024i		−	0.0238912i
$237$		0			0
$238$		−	19.4599i		−	1.26140i
$239$	−24.4821			−1.58362			−0.791808	−	0.610770i	$-0.790860\pi$
$239$	−24.4821			−1.58362			−0.791808	+	0.610770i	$0.790860\pi$
$240$		0			0
$241$		−	1.52554i		−	0.0982687i	−0.998792	−	0.0491344i	$-0.984354\pi$
$241$		−	1.52554i		−	0.0982687i	0.998792	−	0.0491344i	$-0.0156463\pi$
$242$	−10.0930	−	4.37386i	−0.648805	−	0.281162i
$243$		0			0
$244$		−	4.68980i		−	0.300234i
$245$			11.8979i			0.760131i
$246$		0			0
$247$	−4.99159			−0.317607
$248$	3.81106			0.242003
$249$		0			0
$250$			2.94731i			0.186404i
$251$		−	18.4091i		−	1.16197i	−0.813914	−	0.580985i	$-0.802667\pi$
$251$		−	18.4091i		−	1.16197i	0.813914	−	0.580985i	$-0.197333\pi$
$252$		0			0
$253$	−0.519558	+	2.50558i	−0.0326643	+	0.157524i
$254$		−	2.22458i		−	0.139582i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	21.5082i		−	1.34164i	−0.741618	−	0.670822i	$-0.765941\pi$
$257$		−	21.5082i		−	1.34164i	0.741618	−	0.670822i	$-0.234059\pi$
$258$		0			0
$259$		−	3.44435i		−	0.214022i
$260$	−14.9902			−0.929653
$261$		0			0
$262$	−13.4918			−0.833529
$263$	28.0416			1.72912			0.864559	−	0.502530i	$-0.167598\pi$
$263$	28.0416			1.72912			0.864559	+	0.502530i	$0.167598\pi$
$264$		0			0
$265$	−16.0116			−0.983586
$266$	−3.31087			−0.203003
$267$		0			0
$268$	2.00040			0.122194
$269$		−	28.3167i		−	1.72650i	−0.504779	−	0.863248i	$-0.668426\pi$
$269$		−	28.3167i		−	1.72650i	0.504779	−	0.863248i	$-0.331574\pi$
$270$		0			0
$271$			22.7758i			1.38353i	0.722123	+	0.691765i	$0.243166\pi$
$271$			22.7758i			1.38353i	−0.722123	+	0.691765i	$0.756834\pi$
$272$	5.87757			0.356380
$273$		0			0
$274$			13.2515i			0.800553i
$275$	−13.0505	−	2.70615i	−0.786974	−	0.163187i
$276$		0			0
$277$		−	8.06995i		−	0.484876i	−0.970167	−	0.242438i	$-0.922053\pi$
$277$		−	8.06995i		−	0.484876i	0.970167	−	0.242438i	$-0.0779472\pi$
$278$			3.71160i			0.222607i
$279$		0			0
$280$	−9.94287			−0.594200
$281$	20.3290			1.21272			0.606362	−	0.795188i	$-0.292628\pi$
$281$	20.3290			1.21272			0.606362	+	0.795188i	$0.292628\pi$
$282$		0			0
$283$		−	16.7483i		−	0.995580i	−0.867297	−	0.497790i	$-0.834145\pi$
$283$		−	16.7483i		−	0.995580i	0.867297	−	0.497790i	$-0.165855\pi$
$284$			13.5343i			0.803114i
$285$		0			0
$286$	−3.36139	+	16.2104i	−0.198763	+	0.958540i
$287$		−	18.8206i		−	1.11095i
$288$		0			0
$289$	17.5458			1.03211
$290$			26.8082i			1.57423i
$291$		0			0
$292$			12.6777i			0.741908i
$293$	24.3860			1.42464			0.712321	−	0.701854i	$-0.247644\pi$
$293$	24.3860			1.42464			0.712321	+	0.701854i	$0.247644\pi$
$294$		0			0
$295$	−1.10221			−0.0641730
$296$	1.04032			0.0604671
$297$		0			0
$298$	0.782444			0.0453258
$299$	3.85117			0.222719
$300$		0			0
$301$	−25.0540			−1.44409
$302$		−	4.68719i		−	0.269718i
$303$		0			0
$304$		−	1.00000i		−	0.0573539i
$305$	−14.0839			−0.806443
$306$		0			0
$307$			13.2730i			0.757533i	0.925492	+	0.378766i	$0.123652\pi$
$307$			13.2730i			0.757533i	−0.925492	+	0.378766i	$0.876348\pi$
$308$	−2.22958	+	10.7522i	−0.127042	+	0.612663i
$309$		0			0
$310$		−	11.4450i		−	0.650031i
$311$		−	9.95863i		−	0.564702i	−0.959311	−	0.282351i	$-0.908886\pi$
$311$		−	9.95863i		−	0.564702i	0.959311	−	0.282351i	$-0.0911143\pi$
$312$		0			0
$313$	−8.55364			−0.483480			−0.241740	−	0.970341i	$-0.577718\pi$
$313$	−8.55364			−0.483480			−0.241740	+	0.970341i	$0.577718\pi$
$314$	19.0620			1.07573
$315$		0			0
$316$		−	4.97264i		−	0.279733i
$317$		−	8.69979i		−	0.488629i	−0.969696	−	0.244315i	$-0.921437\pi$
$317$		−	8.69979i		−	0.488629i	0.969696	−	0.244315i	$-0.0785629\pi$
$318$		0			0
$319$	28.9904	+	6.01145i	1.62315	+	0.336577i
$320$		−	3.00309i		−	0.167878i
$321$		0			0
$322$	2.55444			0.142354
$323$		−	5.87757i		−	0.327037i
$324$		0			0
$325$			20.0591i			1.11268i
$326$	−7.03872			−0.389839
$327$		0			0
$328$	5.68449			0.313873
$329$	18.0247			0.993732
$330$		0			0
$331$	29.4762			1.62016			0.810079	−	0.586321i	$-0.199424\pi$
$331$	29.4762			1.62016			0.810079	+	0.586321i	$0.199424\pi$
$332$	10.4352			0.572706
$333$		0			0
$334$	18.1622			0.993790
$335$		−	6.00738i		−	0.328218i
$336$		0			0
$337$		−	9.17830i		−	0.499974i	−0.968249	−	0.249987i	$-0.919574\pi$
$337$		−	9.17830i		−	0.499974i	0.968249	−	0.249987i	$-0.0804263\pi$
$338$	11.9159			0.648142
$339$		0			0
$340$		−	17.6509i		−	0.957254i
$341$	−12.3766	−	2.56641i	−0.670229	−	0.138979i
$342$		0			0
$343$			10.0588i			0.543124i
$344$		−	7.56719i		−	0.407996i
$345$		0			0
$346$	20.0969			1.08042
$347$	−9.03702			−0.485133			−0.242566	−	0.970135i	$-0.577989\pi$
$347$	−9.03702			−0.485133			−0.242566	+	0.970135i	$0.577989\pi$
$348$		0			0
$349$		−	16.9667i		−	0.908205i	−0.890949	−	0.454103i	$-0.849960\pi$
$349$		−	16.9667i		−	0.908205i	0.890949	−	0.454103i	$-0.150040\pi$
$350$			13.3050i			0.711182i
$351$		0			0
$352$	−3.24754	−	0.673411i	−0.173094	−	0.0358929i
$353$			24.0307i			1.27902i	0.768781	+	0.639512i	$0.220864\pi$
$353$			24.0307i			1.27902i	−0.768781	+	0.639512i	$0.779136\pi$
$354$		0			0
$355$	40.6448			2.15720
$356$			11.6524i			0.617577i
$357$		0			0
$358$			18.4938i			0.977428i
$359$	30.9906			1.63562			0.817811	−	0.575486i	$-0.195187\pi$
$359$	30.9906			1.63562			0.817811	+	0.575486i	$0.195187\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	−11.0037			−0.578341
$363$		0			0
$364$	16.5265			0.866225
$365$	38.0724			1.99280
$366$		0			0
$367$	18.7089			0.976599			0.488300	−	0.872676i	$-0.337617\pi$
$367$	18.7089			0.976599			0.488300	+	0.872676i	$0.337617\pi$
$368$			0.771532i			0.0402189i
$369$		0			0
$370$		−	3.12416i		−	0.162418i
$371$	17.6526			0.916478
$372$		0			0
$373$			26.1185i			1.35236i	0.736735	+	0.676181i	$0.236366\pi$
$373$			26.1185i			1.35236i	−0.736735	+	0.676181i	$0.763634\pi$
$374$	−19.0876	−	3.95802i	−0.986998	−	0.204664i
$375$		0			0
$376$			5.44408i			0.280757i
$377$		−	44.5592i		−	2.29492i
$378$		0			0
$379$	1.81358			0.0931572			0.0465786	−	0.998915i	$-0.485168\pi$
$379$	1.81358			0.0931572			0.0465786	+	0.998915i	$0.485168\pi$
$380$	−3.00309			−0.154055
$381$		0			0
$382$			19.9599i			1.02124i
$383$			7.82110i			0.399639i	0.979833	+	0.199820i	$0.0640356\pi$
$383$			7.82110i			0.399639i	−0.979833	+	0.199820i	$0.935964\pi$
$384$		0			0
$385$	32.2899	+	6.69563i	1.64564	+	0.341241i
$386$			3.64379i			0.185464i
$387$		0			0
$388$	−7.71791			−0.391818
$389$		−	25.3857i		−	1.28711i	−0.765401	−	0.643554i	$-0.777459\pi$
$389$		−	25.3857i		−	1.28711i	0.765401	−	0.643554i	$-0.222541\pi$
$390$		0			0
$391$			4.53473i			0.229331i
$392$	3.96189			0.200106
$393$		0			0
$394$	−24.2098			−1.21967
$395$	−14.9333			−0.751377
$396$		0			0
$397$	−27.5933			−1.38487			−0.692434	−	0.721481i	$-0.743462\pi$
$397$	−27.5933			−1.38487			−0.692434	+	0.721481i	$0.743462\pi$
$398$	−19.1882			−0.961818
$399$		0			0
$400$	−4.01857			−0.200929
$401$			17.2571i			0.861776i	0.902405	+	0.430888i	$0.141800\pi$
$401$			17.2571i			0.861776i	−0.902405	+	0.430888i	$0.858200\pi$
$402$		0			0
$403$			19.0233i			0.947616i
$404$	5.63251			0.280228
$405$		0			0
$406$		−	29.5557i		−	1.46683i
$407$	−3.37847	−	0.700559i	−0.167464	−	0.0347254i
$408$		0			0
$409$		−	27.3940i		−	1.35454i	−0.735733	−	0.677272i	$-0.763162\pi$
$409$		−	27.3940i		−	1.35454i	0.735733	−	0.677272i	$-0.236838\pi$
$410$		−	17.0711i		−	0.843079i
$411$		0			0
$412$	2.65084			0.130597
$413$	1.21517			0.0597947
$414$		0			0
$415$		−	31.3379i		−	1.53832i
$416$			4.99159i			0.244733i
$417$		0			0
$418$	−0.673411	+	3.24754i	−0.0329376	+	0.158842i
$419$		−	10.3841i		−	0.507295i	−0.967297	−	0.253647i	$-0.918370\pi$
$419$		−	10.3841i		−	0.507295i	0.967297	−	0.253647i	$-0.0816303\pi$
$420$		0			0
$421$	−5.45710			−0.265963			−0.132981	−	0.991119i	$-0.542455\pi$
$421$	−5.45710			−0.265963			−0.132981	+	0.991119i	$0.542455\pi$
$422$		−	6.95465i		−	0.338547i
$423$		0			0
$424$			5.33171i			0.258931i
$425$	−23.6194			−1.14571
$426$		0			0
$427$	15.5273			0.751421
$428$	18.0820			0.874026
$429$		0			0
$430$	−22.7250			−1.09590
$431$	9.65282			0.464960			0.232480	−	0.972601i	$-0.425316\pi$
$431$	9.65282			0.464960			0.232480	+	0.972601i	$0.425316\pi$
$432$		0			0
$433$	−26.2314			−1.26060			−0.630301	−	0.776351i	$-0.717068\pi$
$433$	−26.2314			−1.26060			−0.630301	+	0.776351i	$0.717068\pi$
$434$			12.6179i			0.605681i
$435$		0			0
$436$		−	17.4431i		−	0.835371i
$437$	0.771532			0.0369074
$438$		0			0
$439$		−	3.73505i		−	0.178264i	−0.996020	−	0.0891320i	$-0.971591\pi$
$439$		−	3.73505i		−	0.178264i	0.996020	−	0.0891320i	$-0.0284093\pi$
$440$	−2.02232	+	9.75267i	−0.0964101	+	0.464940i
$441$		0			0
$442$			29.3384i			1.39549i
$443$			27.8080i			1.32120i	0.750739	+	0.660599i	$0.229698\pi$
$443$			27.8080i			1.32120i	−0.750739	+	0.660599i	$0.770302\pi$
$444$		0			0
$445$	34.9933			1.65884
$446$	15.7239			0.744550
$447$		0			0
$448$			3.31087i			0.156424i
$449$			34.6728i			1.63631i	0.574999	+	0.818154i	$0.305003\pi$
$449$			34.6728i			1.63631i	−0.574999	+	0.818154i	$0.694997\pi$
$450$		0			0
$451$	−18.4606	−	3.82799i	−0.869276	−	0.180253i
$452$		−	0.423629i		−	0.0199258i
$453$		0			0
$454$	19.8004			0.929280
$455$		−	49.6307i		−	2.32672i
$456$		0			0
$457$			16.9656i			0.793617i	0.917901	+	0.396808i	$0.129882\pi$
$457$			16.9656i			0.793617i	−0.917901	+	0.396808i	$0.870118\pi$
$458$	7.86405			0.367463
$459$		0			0
$460$	2.31698			0.108030
$461$	23.8211			1.10946			0.554730	−	0.832031i	$-0.312822\pi$
$461$	23.8211			1.10946			0.554730	+	0.832031i	$0.312822\pi$
$462$		0			0
$463$	20.5137			0.953354			0.476677	−	0.879079i	$-0.341841\pi$
$463$	20.5137			0.953354			0.476677	+	0.879079i	$0.341841\pi$
$464$	8.92687			0.414419
$465$		0			0
$466$	2.35291			0.108996
$467$			27.2191i			1.25955i	0.776778	+	0.629774i	$0.216853\pi$
$467$			27.2191i			1.25955i	−0.776778	+	0.629774i	$0.783147\pi$
$468$		0			0
$469$			6.62307i			0.305825i
$470$	16.3491			0.754127
$471$		0			0
$472$			0.367024i			0.0168937i
$473$	−5.09583	+	24.5748i	−0.234306	+	1.12995i
$474$		0			0
$475$			4.01857i			0.184385i
$476$			19.4599i			0.891943i
$477$		0			0
$478$	24.4821			1.11979
$479$	−1.54399			−0.0705467			−0.0352734	−	0.999378i	$-0.511230\pi$
$479$	−1.54399			−0.0705467			−0.0352734	+	0.999378i	$0.511230\pi$
$480$		0			0
$481$			5.19283i			0.236772i
$482$			1.52554i			0.0694865i
$483$		0			0
$484$	10.0930	+	4.37386i	0.458774	+	0.198812i
$485$			23.1776i			1.05244i
$486$		0			0
$487$	16.5914			0.751828			0.375914	−	0.926654i	$-0.377329\pi$
$487$	16.5914			0.751828			0.375914	+	0.926654i	$0.377329\pi$
$488$			4.68980i			0.212297i
$489$		0			0
$490$		−	11.8979i		−	0.537494i
$491$	−33.1297			−1.49512			−0.747562	−	0.664192i	$-0.768776\pi$
$491$	−33.1297			−1.49512			−0.747562	+	0.664192i	$0.768776\pi$
$492$		0			0
$493$	52.4683			2.36305
$494$	4.99159			0.224582
$495$		0			0
$496$	−3.81106			−0.171122
$497$	−44.8104			−2.01002
$498$		0			0
$499$	−27.7467			−1.24211			−0.621056	−	0.783766i	$-0.713296\pi$
$499$	−27.7467			−1.24211			−0.621056	+	0.783766i	$0.713296\pi$
$500$		−	2.94731i		−	0.131808i
$501$		0			0
$502$			18.4091i			0.821637i
$503$	−6.41063			−0.285836			−0.142918	−	0.989735i	$-0.545648\pi$
$503$	−6.41063			−0.285836			−0.142918	+	0.989735i	$0.545648\pi$
$504$		0			0
$505$		−	16.9150i		−	0.752706i
$506$	0.519558	−	2.50558i	0.0230972	−	0.111387i
$507$		0			0
$508$			2.22458i			0.0986997i
$509$		−	21.3460i		−	0.946147i	−0.881023	−	0.473073i	$-0.843145\pi$
$509$		−	21.3460i		−	0.946147i	0.881023	−	0.473073i	$-0.156855\pi$
$510$		0			0
$511$	−41.9744			−1.85684
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			21.5082i			0.948686i
$515$		−	7.96072i		−	0.350791i
$516$		0			0
$517$	3.66610	−	17.6799i	0.161235	−	0.777559i
$518$			3.44435i			0.151336i
$519$		0			0
$520$	14.9902			0.657364
$521$		−	30.2415i		−	1.32490i	−0.749104	−	0.662452i	$-0.769516\pi$
$521$		−	30.2415i		−	1.32490i	0.749104	−	0.662452i	$-0.230484\pi$
$522$		0			0
$523$		−	9.82644i		−	0.429680i	−0.976649	−	0.214840i	$-0.931077\pi$
$523$		−	9.82644i		−	0.429680i	0.976649	−	0.214840i	$-0.0689230\pi$
$524$	13.4918			0.589394
$525$		0			0
$526$	−28.0416			−1.22267
$527$	−22.3998			−0.975750
$528$		0			0
$529$	22.4047			0.974119
$530$	16.0116			0.695500
$531$		0			0
$532$	3.31087			0.143545
$533$			28.3746i			1.22904i
$534$		0			0
$535$		−	54.3019i		−	2.34768i
$536$	−2.00040			−0.0864040
$537$		0			0
$538$			28.3167i			1.22082i
$539$	−12.8664	−	2.66798i	−0.554195	−	0.114918i
$540$		0			0
$541$			42.3404i			1.82036i	0.414218	+	0.910178i	$0.364055\pi$
$541$			42.3404i			1.82036i	−0.414218	+	0.910178i	$0.635945\pi$
$542$		−	22.7758i		−	0.978303i
$543$		0			0
$544$	−5.87757			−0.251999
$545$	−52.3832			−2.24385
$546$		0			0
$547$			3.94544i			0.168695i	0.996436	+	0.0843474i	$0.0268805\pi$
$547$			3.94544i			0.168695i	−0.996436	+	0.0843474i	$0.973119\pi$
$548$		−	13.2515i		−	0.566076i
$549$		0			0
$550$	13.0505	+	2.70615i	0.556474	+	0.115391i
$551$		−	8.92687i		−	0.380297i
$552$		0			0
$553$	16.4638			0.700112
$554$			8.06995i			0.342859i
$555$		0			0
$556$		−	3.71160i		−	0.157407i
$557$	2.32476			0.0985034			0.0492517	−	0.998786i	$-0.484316\pi$
$557$	2.32476			0.0985034			0.0492517	+	0.998786i	$0.484316\pi$
$558$		0			0
$559$	37.7723			1.59760
$560$	9.94287			0.420163
$561$		0			0
$562$	−20.3290			−0.857526
$563$	−3.29794			−0.138991			−0.0694957	−	0.997582i	$-0.522139\pi$
$563$	−3.29794			−0.138991			−0.0694957	+	0.997582i	$0.522139\pi$
$564$		0			0
$565$	−1.27220			−0.0535218
$566$			16.7483i			0.703982i
$567$		0			0
$568$		−	13.5343i		−	0.567887i
$569$	6.61627			0.277368			0.138684	−	0.990337i	$-0.455713\pi$
$569$	6.61627			0.277368			0.138684	+	0.990337i	$0.455713\pi$
$570$		0			0
$571$			30.5375i			1.27796i	0.769225	+	0.638978i	$0.220642\pi$
$571$			30.5375i			1.27796i	−0.769225	+	0.638978i	$0.779358\pi$
$572$	3.36139	−	16.2104i	0.140547	−	0.677790i
$573$		0			0
$574$			18.8206i			0.785558i
$575$		−	3.10046i		−	0.129298i
$576$		0			0
$577$	−38.8350			−1.61672			−0.808360	−	0.588688i	$-0.799645\pi$
$577$	−38.8350			−1.61672			−0.808360	+	0.588688i	$0.799645\pi$
$578$	−17.5458			−0.729809
$579$		0			0
$580$		−	26.8082i		−	1.11315i
$581$			34.5496i			1.43336i
$582$		0			0
$583$	3.59043	−	17.3149i	0.148700	−	0.717111i
$584$		−	12.6777i		−	0.524608i
$585$		0			0
$586$	−24.3860			−1.00737
$587$			3.15215i			0.130103i	0.997882	+	0.0650515i	$0.0207212\pi$
$587$			3.15215i			0.130103i	−0.997882	+	0.0650515i	$0.979279\pi$
$588$		0			0
$589$			3.81106i			0.157032i
$590$	1.10221			0.0453772
$591$		0			0
$592$	−1.04032			−0.0427567
$593$	−9.24115			−0.379488			−0.189744	−	0.981834i	$-0.560766\pi$
$593$	−9.24115			−0.379488			−0.189744	+	0.981834i	$0.560766\pi$
$594$		0			0
$595$	58.4399			2.39580
$596$	−0.782444			−0.0320502
$597$		0			0
$598$	−3.85117			−0.157486
$599$		−	8.49532i		−	0.347109i	−0.984824	−	0.173555i	$-0.944475\pi$
$599$		−	8.49532i		−	0.347109i	0.984824	−	0.173555i	$-0.0555253\pi$
$600$		0			0
$601$		−	6.52893i		−	0.266321i	−0.991095	−	0.133160i	$-0.957487\pi$
$601$		−	6.52893i		−	0.266321i	0.991095	−	0.133160i	$-0.0425125\pi$
$602$	25.0540			1.02113
$603$		0			0
$604$			4.68719i			0.190719i
$605$	13.1351	−	30.3103i	0.534018	−	1.23229i
$606$		0			0
$607$		−	45.3963i		−	1.84258i	−0.388876	−	0.921290i	$-0.627137\pi$
$607$		−	45.3963i		−	1.84258i	0.388876	−	0.921290i	$-0.372863\pi$
$608$			1.00000i			0.0405554i
$609$		0			0
$610$	14.0839			0.570241
$611$	−27.1746			−1.09937
$612$		0			0
$613$			22.4162i			0.905382i	0.891667	+	0.452691i	$0.149536\pi$
$613$			22.4162i			0.905382i	−0.891667	+	0.452691i	$0.850464\pi$
$614$		−	13.2730i		−	0.535657i
$615$		0			0
$616$	2.22958	−	10.7522i	0.0898323	−	0.433218i
$617$		−	13.5245i		−	0.544476i	−0.962230	−	0.272238i	$-0.912236\pi$
$617$		−	13.5245i		−	0.544476i	0.962230	−	0.272238i	$-0.0877637\pi$
$618$		0			0
$619$	19.1199			0.768495			0.384248	−	0.923230i	$-0.374461\pi$
$619$	19.1199			0.768495			0.384248	+	0.923230i	$0.374461\pi$
$620$			11.4450i			0.459641i
$621$		0			0
$622$			9.95863i			0.399305i
$623$	−38.5797			−1.54566
$624$		0			0
$625$	−28.9439			−1.15776
$626$	8.55364			0.341872
$627$		0			0
$628$	−19.0620			−0.760655
$629$	−6.11452			−0.243802
$630$		0			0
$631$	−9.61661			−0.382831			−0.191416	−	0.981509i	$-0.561308\pi$
$631$	−9.61661			−0.382831			−0.191416	+	0.981509i	$0.561308\pi$
$632$			4.97264i			0.197801i
$633$		0			0
$634$			8.69979i			0.345513i
$635$	6.68062			0.265112
$636$		0			0
$637$			19.7761i			0.783559i
$638$	−28.9904	−	6.01145i	−1.14774	−	0.237996i
$639$		0			0
$640$			3.00309i			0.118708i
$641$			5.17195i			0.204280i	0.994770	+	0.102140i	$0.0325689\pi$
$641$			5.17195i			0.204280i	−0.994770	+	0.102140i	$0.967431\pi$
$642$		0			0
$643$	−12.1805			−0.480350			−0.240175	−	0.970730i	$-0.577205\pi$
$643$	−12.1805			−0.480350			−0.240175	+	0.970730i	$0.577205\pi$
$644$	−2.55444			−0.100659
$645$		0			0
$646$			5.87757i			0.231250i
$647$		−	20.3812i		−	0.801268i	−0.916238	−	0.400634i	$-0.868790\pi$
$647$		−	20.3812i		−	0.801268i	0.916238	−	0.400634i	$-0.131210\pi$
$648$		0			0
$649$	0.247158	−	1.19193i	0.00970180	−	0.0467872i
$650$		−	20.0591i		−	0.786781i
$651$		0			0
$652$	7.03872			0.275657
$653$			2.93909i			0.115016i	0.998345	+	0.0575078i	$0.0183154\pi$
$653$			2.93909i			0.115016i	−0.998345	+	0.0575078i	$0.981685\pi$
$654$		0			0
$655$		−	40.5173i		−	1.58314i
$656$	−5.68449			−0.221942
$657$		0			0
$658$	−18.0247			−0.702674
$659$	41.3613			1.61121			0.805604	−	0.592454i	$-0.201841\pi$
$659$	41.3613			1.61121			0.805604	+	0.592454i	$0.201841\pi$
$660$		0			0
$661$	−3.58345			−0.139380			−0.0696900	−	0.997569i	$-0.522201\pi$
$661$	−3.58345			−0.139380			−0.0696900	+	0.997569i	$0.522201\pi$
$662$	−29.4762			−1.14562
$663$		0			0
$664$	−10.4352			−0.404964
$665$		−	9.94287i		−	0.385568i
$666$		0			0
$667$			6.88736i			0.266680i
$668$	−18.1622			−0.702716
$669$		0			0
$670$			6.00738i			0.232085i
$671$	3.15816	−	15.2303i	0.121920	−	0.587960i
$672$		0			0
$673$			6.69968i			0.258254i	0.991628	+	0.129127i	$0.0412174\pi$
$673$			6.69968i			0.258254i	−0.991628	+	0.129127i	$0.958783\pi$
$674$			9.17830i			0.353535i
$675$		0			0
$676$	−11.9159			−0.458306
$677$	3.34298			0.128481			0.0642405	−	0.997934i	$-0.479538\pi$
$677$	3.34298			0.128481			0.0642405	+	0.997934i	$0.479538\pi$
$678$		0			0
$679$		−	25.5530i		−	0.980636i
$680$			17.6509i			0.676881i
$681$		0			0
$682$	12.3766	+	2.56641i	0.473924	+	0.0982729i
$683$		−	30.3946i		−	1.16302i	−0.813540	−	0.581509i	$-0.802463\pi$
$683$		−	30.3946i		−	1.16302i	0.813540	−	0.581509i	$-0.197537\pi$
$684$		0			0
$685$	−39.7955			−1.52051
$686$		−	10.0588i		−	0.384047i
$687$		0			0
$688$			7.56719i			0.288497i
$689$	−26.6137			−1.01390
$690$		0			0
$691$	3.08430			0.117332			0.0586662	−	0.998278i	$-0.481315\pi$
$691$	3.08430			0.117332			0.0586662	+	0.998278i	$0.481315\pi$
$692$	−20.0969			−0.763970
$693$		0			0
$694$	9.03702			0.343041
$695$	−11.1463			−0.422803
$696$		0			0
$697$	−33.4110			−1.26553
$698$			16.9667i			0.642198i
$699$		0			0
$700$		−	13.3050i		−	0.502882i
$701$	22.3588			0.844479			0.422240	−	0.906484i	$-0.361244\pi$
$701$	22.3588			0.844479			0.422240	+	0.906484i	$0.361244\pi$
$702$		0			0
$703$			1.04032i			0.0392362i
$704$	3.24754	+	0.673411i	0.122396	+	0.0253801i
$705$		0			0
$706$		−	24.0307i		−	0.904407i
$707$			18.6485i			0.701350i
$708$		0			0
$709$	32.5620			1.22289			0.611445	−	0.791287i	$-0.290588\pi$
$709$	32.5620			1.22289			0.611445	+	0.791287i	$0.290588\pi$
$710$	−40.6448			−1.52537
$711$		0			0
$712$		−	11.6524i		−	0.436693i
$713$		−	2.94035i		−	0.110117i
$714$		0			0
$715$	−48.6813	−	10.0946i	−1.82058	−	0.377515i
$716$		−	18.4938i		−	0.691146i
$717$		0			0
$718$	−30.9906			−1.15656
$719$			37.8993i			1.41340i	0.707511	+	0.706702i	$0.249818\pi$
$719$			37.8993i			1.41340i	−0.707511	+	0.706702i	$0.750182\pi$
$720$		0			0
$721$			8.77659i			0.326857i
$722$	1.00000			0.0372161
$723$		0			0
$724$	11.0037			0.408949
$725$	−35.8733			−1.33230
$726$		0			0
$727$	−15.4961			−0.574720			−0.287360	−	0.957823i	$-0.592778\pi$
$727$	−15.4961			−0.574720			−0.287360	+	0.957823i	$0.592778\pi$
$728$	−16.5265			−0.612514
$729$		0			0
$730$	−38.0724			−1.40912
$731$			44.4767i			1.64503i
$732$		0			0
$733$			40.1017i			1.48119i	0.671952	+	0.740595i	$0.265456\pi$
$733$			40.1017i			1.48119i	−0.671952	+	0.740595i	$0.734544\pi$
$734$	−18.7089			−0.690560
$735$		0			0
$736$		−	0.771532i		−	0.0284390i
$737$	6.49637	+	1.34709i	0.239297	+	0.0496207i
$738$		0			0
$739$		−	28.1975i		−	1.03726i	−0.854998	−	0.518632i	$-0.826442\pi$
$739$		−	28.1975i		−	1.03726i	0.854998	−	0.518632i	$-0.173558\pi$
$740$			3.12416i			0.114847i
$741$		0			0
$742$	−17.6526			−0.648048
$743$	−46.4920			−1.70563			−0.852813	−	0.522216i	$-0.825105\pi$
$743$	−46.4920			−1.70563			−0.852813	+	0.522216i	$0.825105\pi$
$744$		0			0
$745$			2.34975i			0.0860883i
$746$		−	26.1185i		−	0.956265i
$747$		0			0
$748$	19.0876	+	3.95802i	0.697913	+	0.144719i
$749$			59.8672i			2.18750i
$750$		0			0
$751$	14.6027			0.532860			0.266430	−	0.963854i	$-0.414156\pi$
$751$	14.6027			0.532860			0.266430	+	0.963854i	$0.414156\pi$
$752$		−	5.44408i		−	0.198525i
$753$		0			0
$754$			44.5592i			1.62275i
$755$	14.0761			0.512281
$756$		0			0
$757$	−11.2801			−0.409980			−0.204990	−	0.978764i	$-0.565716\pi$
$757$	−11.2801			−0.409980			−0.204990	+	0.978764i	$0.565716\pi$
$758$	−1.81358			−0.0658721
$759$		0			0
$760$	3.00309			0.108934
$761$	36.1020			1.30869			0.654347	−	0.756194i	$-0.272943\pi$
$761$	36.1020			1.30869			0.654347	+	0.756194i	$0.272943\pi$
$762$		0			0
$763$	57.7518			2.09076
$764$		−	19.9599i		−	0.722125i
$765$		0			0
$766$		−	7.82110i		−	0.282588i
$767$	−1.83203			−0.0661509
$768$		0			0
$769$			1.14170i			0.0411708i	0.999788	+	0.0205854i	$0.00655300\pi$
$769$			1.14170i			0.0411708i	−0.999788	+	0.0205854i	$0.993447\pi$
$770$	−32.2899	−	6.69563i	−1.16365	−	0.241294i
$771$		0			0
$772$		−	3.64379i		−	0.131143i
$773$		−	28.4950i		−	1.02489i	−0.858719	−	0.512447i	$-0.828739\pi$
$773$		−	28.4950i		−	1.02489i	0.858719	−	0.512447i	$-0.171261\pi$
$774$		0			0
$775$	15.3150			0.550132
$776$	7.71791			0.277057
$777$		0			0
$778$			25.3857i			0.910123i
$779$			5.68449i			0.203668i
$780$		0			0
$781$	−9.11415	+	43.9532i	−0.326130	+	1.57277i
$782$		−	4.53473i		−	0.162162i
$783$		0			0
$784$	−3.96189			−0.141496
$785$			57.2448i			2.04316i
$786$		0			0
$787$		−	32.2977i		−	1.15129i	−0.817700	−	0.575644i	$-0.804751\pi$
$787$		−	32.2977i		−	1.15129i	0.817700	−	0.575644i	$-0.195249\pi$
$788$	24.2098			0.862440
$789$		0			0
$790$	14.9333			0.531304
$791$	1.40258			0.0498701
$792$		0			0
$793$	−23.4096			−0.831298
$794$	27.5933			0.979250
$795$		0			0
$796$	19.1882			0.680108
$797$			30.7011i			1.08749i	0.839251	+	0.543745i	$0.182994\pi$
$797$			30.7011i			1.08749i	−0.839251	+	0.543745i	$0.817006\pi$
$798$		0			0
$799$		−	31.9979i		−	1.13201i
$800$	4.01857			0.142078
$801$		0			0
$802$		−	17.2571i		−	0.609368i
$803$	−8.53732	+	41.1714i	−0.301275	+	1.45291i
$804$		0			0
$805$			7.67124i			0.270376i
$806$		−	19.0233i		−	0.670065i
$807$		0			0
$808$	−5.63251			−0.198151
$809$	−42.2005			−1.48369			−0.741846	−	0.670570i	$-0.766050\pi$
$809$	−42.2005			−1.48369			−0.741846	+	0.670570i	$0.766050\pi$
$810$		0			0
$811$			37.5447i			1.31837i	0.751980	+	0.659186i	$0.229099\pi$
$811$			37.5447i			1.31837i	−0.751980	+	0.659186i	$0.770901\pi$
$812$			29.5557i			1.03720i
$813$		0			0
$814$	3.37847	+	0.700559i	0.118415	+	0.0245546i
$815$		−	21.1379i		−	0.740429i
$816$		0			0
$817$	7.56719			0.264743
$818$			27.3940i			0.957807i
$819$		0			0
$820$			17.0711i			0.596147i
$821$	−44.5145			−1.55357			−0.776783	−	0.629768i	$-0.783150\pi$
$821$	−44.5145			−1.55357			−0.776783	+	0.629768i	$0.783150\pi$
$822$		0			0
$823$	−43.4007			−1.51285			−0.756427	−	0.654078i	$-0.773057\pi$
$823$	−43.4007			−1.51285			−0.756427	+	0.654078i	$0.773057\pi$
$824$	−2.65084			−0.0923463
$825$		0			0
$826$	−1.21517			−0.0422812
$827$	−18.6485			−0.648471			−0.324235	−	0.945976i	$-0.605107\pi$
$827$	−18.6485			−0.648471			−0.324235	+	0.945976i	$0.605107\pi$
$828$		0			0
$829$	−26.2638			−0.912178			−0.456089	−	0.889934i	$-0.650750\pi$
$829$	−26.2638			−0.912178			−0.456089	+	0.889934i	$0.650750\pi$
$830$			31.3379i			1.08775i
$831$		0			0
$832$		−	4.99159i		−	0.173052i
$833$	−23.2863			−0.806822
$834$		0			0
$835$			54.5427i			1.88753i
$836$	0.673411	−	3.24754i	0.0232904	−	0.112319i
$837$		0			0
$838$			10.3841i			0.358712i
$839$			13.1011i			0.452300i	0.974093	+	0.226150i	$0.0726139\pi$
$839$			13.1011i			0.452300i	−0.974093	+	0.226150i	$0.927386\pi$
$840$		0			0
$841$	50.6890			1.74790
$842$	5.45710			0.188064
$843$		0			0
$844$			6.95465i			0.239389i
$845$			35.7847i			1.23103i
$846$		0			0
$847$	−14.4813	+	33.4168i	−0.497583	+	1.14821i
$848$		−	5.33171i		−	0.183092i
$849$		0			0
$850$	23.6194			0.810140
$851$		−	0.802636i		−	0.0275140i
$852$		0			0
$853$			12.0774i			0.413521i	0.978392	+	0.206760i	$0.0662921\pi$
$853$			12.0774i			0.413521i	−0.978392	+	0.206760i	$0.933708\pi$
$854$	−15.5273			−0.531335
$855$		0			0
$856$	−18.0820			−0.618030
$857$	9.45066			0.322828			0.161414	−	0.986887i	$-0.448394\pi$
$857$	9.45066			0.322828			0.161414	+	0.986887i	$0.448394\pi$
$858$		0			0
$859$	−6.50772			−0.222041			−0.111020	−	0.993818i	$-0.535412\pi$
$859$	−6.50772			−0.222041			−0.111020	+	0.993818i	$0.535412\pi$
$860$	22.7250			0.774916
$861$		0			0
$862$	−9.65282			−0.328776
$863$		−	28.5792i		−	0.972846i	−0.873723	−	0.486423i	$-0.838301\pi$
$863$		−	28.5792i		−	0.972846i	0.873723	−	0.486423i	$-0.161699\pi$
$864$		0			0
$865$			60.3529i			2.05206i
$866$	26.2314			0.891380
$867$		0			0
$868$		−	12.6179i		−	0.428281i
$869$	3.34863	−	16.1489i	0.113595	−	0.547813i
$870$		0			0
$871$		−	9.98516i		−	0.338334i
$872$			17.4431i			0.590697i
$873$		0			0
$874$	−0.771532			−0.0260974
$875$	9.75819			0.329887
$876$		0			0
$877$			46.3280i			1.56439i	0.623036	+	0.782193i	$0.285899\pi$
$877$			46.3280i			1.56439i	−0.623036	+	0.782193i	$0.714101\pi$
$878$			3.73505i			0.126052i
$879$		0			0
$880$	2.02232	−	9.75267i	0.0681722	−	0.328762i
$881$			30.4522i			1.02596i	0.858400	+	0.512980i	$0.171459\pi$
$881$			30.4522i			1.02596i	−0.858400	+	0.512980i	$0.828541\pi$
$882$		0			0
$883$	32.1999			1.08361			0.541807	−	0.840503i	$-0.317740\pi$
$883$	32.1999			1.08361			0.541807	+	0.840503i	$0.317740\pi$
$884$		−	29.3384i		−	0.986757i
$885$		0			0
$886$		−	27.8080i		−	0.934228i
$887$	−41.5220			−1.39417			−0.697086	−	0.716988i	$-0.745520\pi$
$887$	−41.5220			−1.39417			−0.697086	+	0.716988i	$0.745520\pi$
$888$		0			0
$889$	−7.36530			−0.247024
$890$	−34.9933			−1.17298
$891$		0			0
$892$	−15.7239			−0.526476
$893$	−5.44408			−0.182179
$894$		0			0
$895$	−55.5386			−1.85645
$896$		−	3.31087i		−	0.110609i
$897$		0			0
$898$		−	34.6728i		−	1.15705i
$899$	−34.0209			−1.13466
$900$		0			0
$901$		−	31.3375i		−	1.04400i
$902$	18.4606	+	3.82799i	0.614671	+	0.127458i
$903$		0			0
$904$			0.423629i			0.0140897i
$905$		−	33.0451i		−	1.09846i
$906$		0			0
$907$	1.06675			0.0354208			0.0177104	−	0.999843i	$-0.494362\pi$
$907$	1.06675			0.0354208			0.0177104	+	0.999843i	$0.494362\pi$
$908$	−19.8004			−0.657101
$909$		0			0
$910$			49.6307i			1.64524i
$911$			13.6459i			0.452110i	0.974115	+	0.226055i	$0.0725829\pi$
$911$			13.6459i			0.452110i	−0.974115	+	0.226055i	$0.927417\pi$
$912$		0			0
$913$	33.8887	+	7.02718i	1.12155	+	0.232566i
$914$		−	16.9656i		−	0.561172i
$915$		0			0
$916$	−7.86405			−0.259835
$917$			44.6698i			1.47513i
$918$		0			0
$919$			50.2666i			1.65814i	0.559144	+	0.829071i	$0.311130\pi$
$919$			50.2666i			1.65814i	−0.559144	+	0.829071i	$0.688870\pi$
$920$	−2.31698			−0.0763886
$921$		0			0
$922$	−23.8211			−0.784506
$923$	67.5577			2.22369
$924$		0			0
$925$	4.18058			0.137457
$926$	−20.5137			−0.674123
$927$		0			0
$928$	−8.92687			−0.293039
$929$			32.2263i			1.05731i	0.848837	+	0.528655i	$0.177303\pi$
$929$			32.2263i			1.05731i	−0.848837	+	0.528655i	$0.822697\pi$
$930$		0			0
$931$			3.96189i			0.129846i
$932$	−2.35291			−0.0770721
$933$		0			0
$934$		−	27.2191i		−	0.890635i
$935$	11.8863	−	57.3220i	0.388724	−	1.87463i
$936$		0			0
$937$			37.7134i			1.23204i	0.787729	+	0.616021i	$0.211257\pi$
$937$			37.7134i			1.23204i	−0.787729	+	0.616021i	$0.788743\pi$
$938$		−	6.62307i		−	0.216251i
$939$		0			0
$940$	−16.3491			−0.533248
$941$	−16.4780			−0.537167			−0.268584	−	0.963256i	$-0.586556\pi$
$941$	−16.4780			−0.537167			−0.268584	+	0.963256i	$0.586556\pi$
$942$		0			0
$943$		−	4.38576i		−	0.142820i
$944$		−	0.367024i		−	0.0119456i
$945$		0			0
$946$	5.09583	−	24.5748i	0.165680	−	0.798994i
$947$			22.4636i			0.729967i	0.931014	+	0.364984i	$0.118925\pi$
$947$			22.4636i			0.729967i	−0.931014	+	0.364984i	$0.881075\pi$
$948$		0			0
$949$	63.2820			2.05422
$950$		−	4.01857i		−	0.130380i
$951$		0			0
$952$		−	19.4599i		−	0.630699i
$953$	13.3899			0.433743			0.216871	−	0.976200i	$-0.430415\pi$
$953$	13.3899			0.433743			0.216871	+	0.976200i	$0.430415\pi$
$954$		0			0
$955$	−59.9415			−1.93966
$956$	−24.4821			−0.791808
$957$		0			0
$958$	1.54399			0.0498841
$959$	43.8741			1.41677
$960$		0			0
$961$	−16.4758			−0.531478
$962$		−	5.19283i		−	0.167423i
$963$		0			0
$964$		−	1.52554i		−	0.0491344i
$965$	−10.9426			−0.352256
$966$		0			0
$967$			43.6332i			1.40315i	0.712597	+	0.701574i	$0.247519\pi$
$967$			43.6332i			1.40315i	−0.712597	+	0.701574i	$0.752481\pi$
$968$	−10.0930	−	4.37386i	−0.324402	−	0.140581i
$969$		0			0
$970$		−	23.1776i		−	0.744189i
$971$			20.3377i			0.652669i	0.945254	+	0.326335i	$0.105814\pi$
$971$			20.3377i			0.652669i	−0.945254	+	0.326335i	$0.894186\pi$
$972$		0			0
$973$	12.2886			0.393956
$974$	−16.5914			−0.531623
$975$		0			0
$976$		−	4.68980i		−	0.150117i
$977$			11.0146i			0.352387i	0.984356	+	0.176194i	$0.0563785\pi$
$977$			11.0146i			0.352387i	−0.984356	+	0.176194i	$0.943621\pi$
$978$		0			0
$979$	−7.84686	+	37.8417i	−0.250787	+	1.20943i
$980$			11.8979i			0.380066i
$981$		0			0
$982$	33.1297			1.05721
$983$		−	12.2196i		−	0.389743i	−0.980829	−	0.194872i	$-0.937571\pi$
$983$		−	12.2196i		−	0.389743i	0.980829	−	0.194872i	$-0.0624290\pi$
$984$		0			0
$985$		−	72.7044i		−	2.31656i
$986$	−52.4683			−1.67093
$987$		0			0
$988$	−4.99159			−0.158804
$989$	−5.83833			−0.185648
$990$		0			0
$991$	15.9075			0.505319			0.252660	−	0.967555i	$-0.418695\pi$
$991$	15.9075			0.505319			0.252660	+	0.967555i	$0.418695\pi$
$992$	3.81106			0.121001
$993$		0			0
$994$	44.8104			1.42130
$995$		−	57.6240i		−	1.82680i
$996$		0			0
$997$		−	20.0207i		−	0.634061i	−0.948415	−	0.317030i	$-0.897314\pi$
$997$		−	20.0207i		−	0.634061i	0.948415	−	0.317030i	$-0.102686\pi$
$998$	27.7467			0.878305
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.b.a.989.5	✓	36
3.2	odd	2		3762.2.b.b.989.32	yes	36
11.10	odd	2		3762.2.b.b.989.5	yes	36
33.32	even	2	inner	3762.2.b.a.989.32	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3762.2.b.a.989.5	✓	36	1.1	even	1	trivial
3762.2.b.a.989.32	yes	36	33.32	even	2	inner
3762.2.b.b.989.5	yes	36	11.10	odd	2
3762.2.b.b.989.32	yes	36	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3762.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.00309i q^{5} +3.31087i q^{7} -1.00000 q^{8} +3.00309i q^{10} +(3.24754 + 0.673411i) q^{11} -4.99159i q^{13} -3.31087i q^{14} +1.00000 q^{16} +5.87757 q^{17} -1.00000i q^{19} -3.00309i q^{20} +(-3.24754 - 0.673411i) q^{22} +0.771532i q^{23} -4.01857 q^{25} +4.99159i q^{26} +3.31087i q^{28} +8.92687 q^{29} -3.81106 q^{31} -1.00000 q^{32} -5.87757 q^{34} +9.94287 q^{35} -1.04032 q^{37} +1.00000i q^{38} +3.00309i q^{40} -5.68449 q^{41} +7.56719i q^{43} +(3.24754 + 0.673411i) q^{44} -0.771532i q^{46} -5.44408i q^{47} -3.96189 q^{49} +4.01857 q^{50} -4.99159i q^{52} -5.33171i q^{53} +(2.02232 - 9.75267i) q^{55} -3.31087i q^{56} -8.92687 q^{58} -0.367024i q^{59} -4.68980i q^{61} +3.81106 q^{62} +1.00000 q^{64} -14.9902 q^{65} +2.00040 q^{67} +5.87757 q^{68} -9.94287 q^{70} +13.5343i q^{71} +12.6777i q^{73} +1.04032 q^{74} -1.00000i q^{76} +(-2.22958 + 10.7522i) q^{77} -4.97264i q^{79} -3.00309i q^{80} +5.68449 q^{82} +10.4352 q^{83} -17.6509i q^{85} -7.56719i q^{86} +(-3.24754 - 0.673411i) q^{88} +11.6524i q^{89} +16.5265 q^{91} +0.771532i q^{92} +5.44408i q^{94} -3.00309 q^{95} -7.71791 q^{97} +3.96189 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 36 q^{2} + 36 q^{4} - 36 q^{8} + 4 q^{11} + 36 q^{16} - 16 q^{17} - 4 q^{22} - 28 q^{25} - 8 q^{31} - 36 q^{32} + 16 q^{34} - 16 q^{35} + 16 q^{37} - 24 q^{41} + 4 q^{44} - 68 q^{49} + 28 q^{50}+ \cdots + 68 q^{98}+O(q^{100}) \) 36 * q - 36 * q^2 + 36 * q^4 - 36 * q^8 + 4 * q^11 + 36 * q^16 - 16 * q^17 - 4 * q^22 - 28 * q^25 - 8 * q^31 - 36 * q^32 + 16 * q^34 - 16 * q^35 + 16 * q^37 - 24 * q^41 + 4 * q^44 - 68 * q^49 + 28 * q^50 + 28 * q^55 + 8 * q^62 + 36 * q^64 - 32 * q^65 + 16 * q^67 - 16 * q^68 + 16 * q^70 - 16 * q^74 + 32 * q^77 + 24 * q^82 + 56 * q^83 - 4 * q^88 - 48 * q^91 + 8 * q^97 + 68 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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