LMFDB - Embedded newform 4012.2.b.a.237.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.4
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.37

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.95725i q^{3} +2.24490i q^{5} +1.49294i q^{7} -5.74532 q^{9} +O(q^{10})$	$q-2.95725i q^{3} +2.24490i q^{5} +1.49294i q^{7} -5.74532 q^{9} -3.01181i q^{11} -1.07234 q^{13} +6.63873 q^{15} +(3.78928 - 1.62523i) q^{17} -2.42497 q^{19} +4.41499 q^{21} +4.21990i q^{23} -0.0395857 q^{25} +8.11860i q^{27} +7.90298i q^{29} -1.58375i q^{31} -8.90666 q^{33} -3.35150 q^{35} +4.71406i q^{37} +3.17117i q^{39} +3.40074i q^{41} -11.0609 q^{43} -12.8977i q^{45} -3.19986 q^{47} +4.77114 q^{49} +(-4.80622 - 11.2058i) q^{51} +1.46065 q^{53} +6.76121 q^{55} +7.17124i q^{57} -1.00000 q^{59} +5.45542i q^{61} -8.57740i q^{63} -2.40729i q^{65} -4.21042 q^{67} +12.4793 q^{69} -0.188503i q^{71} +8.17855i q^{73} +0.117065i q^{75} +4.49644 q^{77} -4.28961i q^{79} +6.77276 q^{81} +6.93340 q^{83} +(3.64849 + 8.50656i) q^{85} +23.3711 q^{87} +1.22169 q^{89} -1.60093i q^{91} -4.68356 q^{93} -5.44382i q^{95} +0.466099i q^{97} +17.3038i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	2.95725i		−	1.70737i	−0.520791	−	0.853684i	$-0.674363\pi$
$3$		−	2.95725i		−	1.70737i	0.520791	−	0.853684i	$-0.325637\pi$
$4$		0			0
$5$			2.24490i			1.00395i	0.864882	+	0.501975i	$0.167393\pi$
$5$			2.24490i			1.00395i	−0.864882	+	0.501975i	$0.832607\pi$
$6$		0			0
$7$			1.49294i			0.564277i	0.959374	+	0.282138i	$0.0910438\pi$
$7$			1.49294i			0.564277i	−0.959374	+	0.282138i	$0.908956\pi$
$8$		0			0
$9$	−5.74532			−1.91511
$10$		0			0
$11$		−	3.01181i		−	0.908094i	−0.890978	−	0.454047i	$-0.849980\pi$
$11$		−	3.01181i		−	0.908094i	0.890978	−	0.454047i	$-0.150020\pi$
$12$		0			0
$13$	−1.07234			−0.297413			−0.148707	−	0.988881i	$-0.547511\pi$
$13$	−1.07234			−0.297413			−0.148707	+	0.988881i	$0.547511\pi$
$14$		0			0
$15$	6.63873			1.71411
$16$		0			0
$17$	3.78928	−	1.62523i	0.919035	−	0.394177i
$17$	3.78928	−	1.62523i	0.919035	−	0.394177i
$18$		0			0
$19$	−2.42497			−0.556327			−0.278163	−	0.960534i	$-0.589726\pi$
$19$	−2.42497			−0.556327			−0.278163	+	0.960534i	$0.589726\pi$
$20$		0			0
$21$	4.41499			0.963429
$22$		0			0
$23$			4.21990i			0.879910i	0.898020	+	0.439955i	$0.145006\pi$
$23$			4.21990i			0.879910i	−0.898020	+	0.439955i	$0.854994\pi$
$24$		0			0
$25$	−0.0395857			−0.00791715
$26$		0			0
$27$			8.11860i			1.56243i
$28$		0			0
$29$			7.90298i			1.46755i	0.679394	+	0.733773i	$0.262243\pi$
$29$			7.90298i			1.46755i	−0.679394	+	0.733773i	$0.737757\pi$
$30$		0			0
$31$		−	1.58375i		−	0.284451i	−0.989834	−	0.142225i	$-0.954574\pi$
$31$		−	1.58375i		−	0.284451i	0.989834	−	0.142225i	$-0.0454258\pi$
$32$		0			0
$33$	−8.90666			−1.55045
$34$		0			0
$35$	−3.35150			−0.566506
$36$		0			0
$37$			4.71406i			0.774986i	0.921873	+	0.387493i	$0.126659\pi$
$37$			4.71406i			0.774986i	−0.921873	+	0.387493i	$0.873341\pi$
$38$		0			0
$39$			3.17117i			0.507794i
$40$		0			0
$41$			3.40074i			0.531106i	0.964096	+	0.265553i	$0.0855545\pi$
$41$			3.40074i			0.531106i	−0.964096	+	0.265553i	$0.914445\pi$
$42$		0			0
$43$	−11.0609			−1.68676			−0.843382	−	0.537314i	$-0.819439\pi$
$43$	−11.0609			−1.68676			−0.843382	+	0.537314i	$0.819439\pi$
$44$		0			0
$45$		−	12.8977i		−	1.92267i
$46$		0			0
$47$	−3.19986			−0.466747			−0.233374	−	0.972387i	$-0.574977\pi$
$47$	−3.19986			−0.466747			−0.233374	+	0.972387i	$0.574977\pi$
$48$		0			0
$49$	4.77114			0.681591
$50$		0			0
$51$	−4.80622	−	11.2058i	−0.673005	−	1.56913i
$52$		0			0
$53$	1.46065			0.200636			0.100318	−	0.994955i	$-0.468014\pi$
$53$	1.46065			0.200636			0.100318	+	0.994955i	$0.468014\pi$
$54$		0			0
$55$	6.76121			0.911681
$56$		0			0
$57$			7.17124i			0.949854i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$			5.45542i			0.698496i	0.937030	+	0.349248i	$0.113563\pi$
$61$			5.45542i			0.698496i	−0.937030	+	0.349248i	$0.886437\pi$
$62$		0			0
$63$		−	8.57740i		−	1.08065i
$64$		0			0
$65$		−	2.40729i		−	0.298588i
$66$		0			0
$67$	−4.21042			−0.514384			−0.257192	−	0.966360i	$-0.582797\pi$
$67$	−4.21042			−0.514384			−0.257192	+	0.966360i	$0.582797\pi$
$68$		0			0
$69$	12.4793			1.50233
$70$		0			0
$71$		−	0.188503i		−	0.0223712i	−0.999937	−	0.0111856i	$-0.996439\pi$
$71$		−	0.188503i		−	0.0223712i	0.999937	−	0.0111856i	$-0.00356056\pi$
$72$		0			0
$73$			8.17855i			0.957227i	0.878026	+	0.478614i	$0.158861\pi$
$73$			8.17855i			0.957227i	−0.878026	+	0.478614i	$0.841139\pi$
$74$		0			0
$75$			0.117065i			0.0135175i
$76$		0			0
$77$	4.49644			0.512416
$78$		0			0
$79$		−	4.28961i		−	0.482619i	−0.970448	−	0.241310i	$-0.922423\pi$
$79$		−	4.28961i		−	0.482619i	0.970448	−	0.241310i	$-0.0775769\pi$
$80$		0			0
$81$	6.77276			0.752529
$82$		0			0
$83$	6.93340			0.761040			0.380520	−	0.924773i	$-0.375745\pi$
$83$	6.93340			0.761040			0.380520	+	0.924773i	$0.375745\pi$
$84$		0			0
$85$	3.64849	+	8.50656i	0.395734	+	0.922665i
$86$		0			0
$87$	23.3711			2.50564
$88$		0			0
$89$	1.22169			0.129499			0.0647493	−	0.997902i	$-0.479375\pi$
$89$	1.22169			0.129499			0.0647493	+	0.997902i	$0.479375\pi$
$90$		0			0
$91$		−	1.60093i		−	0.167823i
$92$		0			0
$93$	−4.68356			−0.485662
$94$		0			0
$95$		−	5.44382i		−	0.558524i
$96$		0			0
$97$			0.466099i			0.0473251i	0.999720	+	0.0236626i	$0.00753273\pi$
$97$			0.466099i			0.0473251i	−0.999720	+	0.0236626i	$0.992467\pi$
$98$		0			0
$99$			17.3038i			1.73910i
$100$		0			0
$101$	3.96095			0.394129			0.197065	−	0.980391i	$-0.436859\pi$
$101$	3.96095			0.394129			0.197065	+	0.980391i	$0.436859\pi$
$102$		0			0
$103$	6.67977			0.658177			0.329088	−	0.944299i	$-0.393259\pi$
$103$	6.67977			0.658177			0.329088	+	0.944299i	$0.393259\pi$
$104$		0			0
$105$			9.91121i			0.967235i
$106$		0			0
$107$		−	1.62040i		−	0.156650i	−0.996928	−	0.0783251i	$-0.975043\pi$
$107$		−	1.62040i		−	0.156650i	0.996928	−	0.0783251i	$-0.0249572\pi$
$108$		0			0
$109$			17.9721i			1.72142i	0.509099	+	0.860708i	$0.329979\pi$
$109$			17.9721i			1.72142i	−0.509099	+	0.860708i	$0.670021\pi$
$110$		0			0
$111$	13.9406			1.32319
$112$		0			0
$113$			6.76876i			0.636751i	0.947965	+	0.318376i	$0.103137\pi$
$113$			6.76876i			0.636751i	−0.947965	+	0.318376i	$0.896863\pi$
$114$		0			0
$115$	−9.47326			−0.883386
$116$		0			0
$117$	6.16093			0.569578
$118$		0			0
$119$	2.42637	+	5.65715i	0.222425	+	0.518590i
$120$		0			0
$121$	1.92903			0.175366
$122$		0			0
$123$	10.0568			0.906793
$124$		0			0
$125$			11.1356i			0.996002i
$126$		0			0
$127$	3.80040			0.337231			0.168615	−	0.985682i	$-0.446070\pi$
$127$	3.80040			0.337231			0.168615	+	0.985682i	$0.446070\pi$
$128$		0			0
$129$			32.7097i			2.87993i
$130$		0			0
$131$			9.85569i			0.861096i	0.902568	+	0.430548i	$0.141680\pi$
$131$			9.85569i			0.861096i	−0.902568	+	0.430548i	$0.858320\pi$
$132$		0			0
$133$		−	3.62033i		−	0.313922i
$134$		0			0
$135$	−18.2255			−1.56860
$136$		0			0
$137$	21.3663			1.82545			0.912724	−	0.408577i	$-0.133975\pi$
$137$	21.3663			1.82545			0.912724	+	0.408577i	$0.133975\pi$
$138$		0			0
$139$			5.33104i			0.452173i	0.974107	+	0.226086i	$0.0725931\pi$
$139$			5.33104i			0.452173i	−0.974107	+	0.226086i	$0.927407\pi$
$140$		0			0
$141$			9.46278i			0.796910i
$142$		0			0
$143$			3.22967i			0.270079i
$144$		0			0
$145$	−17.7414			−1.47334
$146$		0			0
$147$		−	14.1095i		−	1.16373i
$148$		0			0
$149$	−8.27383			−0.677819			−0.338909	−	0.940819i	$-0.610058\pi$
$149$	−8.27383			−0.677819			−0.338909	+	0.940819i	$0.610058\pi$
$150$		0			0
$151$	17.1854			1.39853			0.699265	−	0.714863i	$-0.253511\pi$
$151$	17.1854			1.39853			0.699265	+	0.714863i	$0.253511\pi$
$152$		0			0
$153$	−21.7706	+	9.33749i	−1.76005	+	0.754891i
$154$		0			0
$155$	3.55537			0.285575
$156$		0			0
$157$	19.0706			1.52200			0.761002	−	0.648750i	$-0.224708\pi$
$157$	19.0706			1.52200			0.761002	+	0.648750i	$0.224708\pi$
$158$		0			0
$159$		−	4.31952i		−	0.342560i
$160$		0			0
$161$	−6.30004			−0.496513
$162$		0			0
$163$			8.02061i			0.628222i	0.949386	+	0.314111i	$0.101706\pi$
$163$			8.02061i			0.628222i	−0.949386	+	0.314111i	$0.898294\pi$
$164$		0			0
$165$		−	19.9946i		−	1.55658i
$166$		0			0
$167$		−	3.17910i		−	0.246006i	−0.992406	−	0.123003i	$-0.960747\pi$
$167$		−	3.17910i		−	0.246006i	0.992406	−	0.123003i	$-0.0392525\pi$
$168$		0			0
$169$	−11.8501			−0.911546
$170$		0			0
$171$	13.9322			1.06543
$172$		0			0
$173$			7.55120i			0.574107i	0.957915	+	0.287053i	$0.0926757\pi$
$173$			7.55120i			0.574107i	−0.957915	+	0.287053i	$0.907324\pi$
$174$		0			0
$175$		−	0.0590990i		−	0.00446746i
$176$		0			0
$177$			2.95725i			0.222280i
$178$		0			0
$179$	18.1213			1.35445			0.677223	−	0.735778i	$-0.263183\pi$
$179$	18.1213			1.35445			0.677223	+	0.735778i	$0.263183\pi$
$180$		0			0
$181$			0.863183i			0.0641599i	0.999485	+	0.0320799i	$0.0102131\pi$
$181$			0.863183i			0.0641599i	−0.999485	+	0.0320799i	$0.989787\pi$
$182$		0			0
$183$	16.1330			1.19259
$184$		0			0
$185$	−10.5826			−0.778048
$186$		0			0
$187$	−4.89489	−	11.4126i	−0.357950	−	0.834569i
$188$		0			0
$189$	−12.1206			−0.881641
$190$		0			0
$191$	−14.8608			−1.07529			−0.537646	−	0.843171i	$-0.680686\pi$
$191$	−14.8608			−1.07529			−0.537646	+	0.843171i	$0.680686\pi$
$192$		0			0
$193$		−	25.0731i		−	1.80480i	−0.430894	−	0.902402i	$-0.641802\pi$
$193$		−	25.0731i		−	1.80480i	0.430894	−	0.902402i	$-0.358198\pi$
$194$		0			0
$195$	−7.11897			−0.509800
$196$		0			0
$197$		−	8.58909i		−	0.611948i	−0.952040	−	0.305974i	$-0.901018\pi$
$197$		−	8.58909i		−	0.611948i	0.952040	−	0.305974i	$-0.0989820\pi$
$198$		0			0
$199$		−	0.831894i		−	0.0589714i	−0.999565	−	0.0294857i	$-0.990613\pi$
$199$		−	0.831894i		−	0.0589714i	0.999565	−	0.0294857i	$-0.00938695\pi$
$200$		0			0
$201$			12.4513i			0.878244i
$202$		0			0
$203$	−11.7986			−0.828103
$204$		0			0
$205$	−7.63432			−0.533204
$206$		0			0
$207$		−	24.2447i		−	1.68512i
$208$		0			0
$209$			7.30354i			0.505197i
$210$		0			0
$211$			3.37534i			0.232368i	0.993228	+	0.116184i	$0.0370663\pi$
$211$			3.37534i			0.232368i	−0.993228	+	0.116184i	$0.962934\pi$
$212$		0			0
$213$	−0.557451			−0.0381959
$214$		0			0
$215$		−	24.8305i		−	1.69343i
$216$		0			0
$217$	2.36445			0.160509
$218$		0			0
$219$	24.1860			1.63434
$220$		0			0
$221$	−4.06338	+	1.74280i	−0.273333	+	0.117233i
$222$		0			0
$223$	2.97040			0.198913			0.0994565	−	0.995042i	$-0.468290\pi$
$223$	2.97040			0.198913			0.0994565	+	0.995042i	$0.468290\pi$
$224$		0			0
$225$	0.227433			0.0151622
$226$		0			0
$227$			21.1479i			1.40364i	0.712357	+	0.701818i	$0.247628\pi$
$227$			21.1479i			1.40364i	−0.712357	+	0.701818i	$0.752372\pi$
$228$		0			0
$229$	12.1085			0.800152			0.400076	−	0.916482i	$-0.368984\pi$
$229$	12.1085			0.800152			0.400076	+	0.916482i	$0.368984\pi$
$230$		0			0
$231$		−	13.2971i		−	0.874884i
$232$		0			0
$233$		−	2.46603i		−	0.161555i	−0.996732	−	0.0807775i	$-0.974260\pi$
$233$		−	2.46603i		−	0.161555i	0.996732	−	0.0807775i	$-0.0257403\pi$
$234$		0			0
$235$		−	7.18337i		−	0.468591i
$236$		0			0
$237$	−12.6855			−0.824009
$238$		0			0
$239$	9.55827			0.618273			0.309137	−	0.951018i	$-0.399960\pi$
$239$	9.55827			0.618273			0.309137	+	0.951018i	$0.399960\pi$
$240$		0			0
$241$			5.91528i			0.381036i	0.981684	+	0.190518i	$0.0610168\pi$
$241$			5.91528i			0.381036i	−0.981684	+	0.190518i	$0.938983\pi$
$242$		0			0
$243$			4.32707i			0.277582i
$244$		0			0
$245$			10.7107i			0.684284i
$246$		0			0
$247$	2.60039			0.165459
$248$		0			0
$249$		−	20.5038i		−	1.29938i
$250$		0			0
$251$	−8.64205			−0.545482			−0.272741	−	0.962088i	$-0.587930\pi$
$251$	−8.64205			−0.545482			−0.272741	+	0.962088i	$0.587930\pi$
$252$		0			0
$253$	12.7095			0.799041
$254$		0			0
$255$	25.1560	−	10.7895i	1.57533	−	0.675664i
$256$		0			0
$257$	8.22887			0.513303			0.256651	−	0.966504i	$-0.417381\pi$
$257$	8.22887			0.513303			0.256651	+	0.966504i	$0.417381\pi$
$258$		0			0
$259$	−7.03779			−0.437307
$260$		0			0
$261$		−	45.4052i		−	2.81051i
$262$		0			0
$263$	−7.59902			−0.468575			−0.234288	−	0.972167i	$-0.575276\pi$
$263$	−7.59902			−0.468575			−0.234288	+	0.972167i	$0.575276\pi$
$264$		0			0
$265$			3.27903i			0.201429i
$266$		0			0
$267$		−	3.61283i		−	0.221102i
$268$		0			0
$269$			14.5603i			0.887757i	0.896087	+	0.443879i	$0.146398\pi$
$269$			14.5603i			0.887757i	−0.896087	+	0.443879i	$0.853602\pi$
$270$		0			0
$271$	−4.49903			−0.273296			−0.136648	−	0.990620i	$-0.543633\pi$
$271$	−4.49903			−0.273296			−0.136648	+	0.990620i	$0.543633\pi$
$272$		0			0
$273$	−4.73436			−0.286536
$274$		0			0
$275$			0.119225i			0.00718951i
$276$		0			0
$277$			3.58861i			0.215618i	0.994172	+	0.107809i	$0.0343836\pi$
$277$			3.58861i			0.215618i	−0.994172	+	0.107809i	$0.965616\pi$
$278$		0			0
$279$			9.09918i			0.544754i
$280$		0			0
$281$	−28.3876			−1.69346			−0.846731	−	0.532021i	$-0.821433\pi$
$281$	−28.3876			−1.69346			−0.846731	+	0.532021i	$0.821433\pi$
$282$		0			0
$283$			8.39894i			0.499265i	0.968341	+	0.249633i	$0.0803099\pi$
$283$			8.39894i			0.499265i	−0.968341	+	0.249633i	$0.919690\pi$
$284$		0			0
$285$	−16.0987			−0.953607
$286$		0			0
$287$	−5.07708			−0.299691
$288$		0			0
$289$	11.7172	−	12.3169i	0.689249	−	0.724524i
$290$		0			0
$291$	1.37837			0.0808014
$292$		0			0
$293$	17.6639			1.03194			0.515969	−	0.856607i	$-0.327432\pi$
$293$	17.6639			1.03194			0.515969	+	0.856607i	$0.327432\pi$
$294$		0			0
$295$		−	2.24490i		−	0.130703i
$296$		0			0
$297$	24.4516			1.41883
$298$		0			0
$299$		−	4.52516i		−	0.261697i
$300$		0			0
$301$		−	16.5132i		−	0.951802i
$302$		0			0
$303$		−	11.7135i		−	0.672924i
$304$		0			0
$305$	−12.2469			−0.701255
$306$		0			0
$307$	−20.8768			−1.19150			−0.595752	−	0.803168i	$-0.703146\pi$
$307$	−20.8768			−1.19150			−0.595752	+	0.803168i	$0.703146\pi$
$308$		0			0
$309$		−	19.7537i		−	1.12375i
$310$		0			0
$311$		−	29.8750i		−	1.69405i	−0.531550	−	0.847027i	$-0.678390\pi$
$311$		−	29.8750i		−	1.69405i	0.531550	−	0.847027i	$-0.321610\pi$
$312$		0			0
$313$		−	25.0291i		−	1.41473i	−0.706849	−	0.707364i	$-0.749884\pi$
$313$		−	25.0291i		−	1.41473i	0.706849	−	0.707364i	$-0.250116\pi$
$314$		0			0
$315$	19.2554			1.08492
$316$		0			0
$317$			18.6032i			1.04486i	0.852682	+	0.522430i	$0.174974\pi$
$317$			18.6032i			1.04486i	−0.852682	+	0.522430i	$0.825026\pi$
$318$		0			0
$319$	23.8022			1.33267
$320$		0			0
$321$	−4.79193			−0.267460
$322$		0			0
$323$	−9.18889	+	3.94114i	−0.511283	+	0.219291i
$324$		0			0
$325$	0.0424493			0.00235466
$326$		0			0
$327$	53.1480			2.93909
$328$		0			0
$329$		−	4.77719i		−	0.263375i
$330$		0			0
$331$	20.8163			1.14417			0.572083	−	0.820196i	$-0.306135\pi$
$331$	20.8163			1.14417			0.572083	+	0.820196i	$0.306135\pi$
$332$		0			0
$333$		−	27.0838i		−	1.48418i
$334$		0			0
$335$		−	9.45197i		−	0.516417i
$336$		0			0
$337$			34.7211i			1.89138i	0.325073	+	0.945689i	$0.394611\pi$
$337$			34.7211i			1.89138i	−0.325073	+	0.945689i	$0.605389\pi$
$338$		0			0
$339$	20.0169			1.08717
$340$		0			0
$341$	−4.76996			−0.258308
$342$		0			0
$343$			17.5736i			0.948883i
$344$		0			0
$345$			28.0148i			1.50827i
$346$		0			0
$347$		−	1.48725i		−	0.0798396i	−0.999203	−	0.0399198i	$-0.987290\pi$
$347$		−	1.48725i		−	0.0798396i	0.999203	−	0.0399198i	$-0.0127102\pi$
$348$		0			0
$349$	23.8478			1.27655			0.638273	−	0.769810i	$-0.279649\pi$
$349$	23.8478			1.27655			0.638273	+	0.769810i	$0.279649\pi$
$350$		0			0
$351$		−	8.70588i		−	0.464686i
$352$		0			0
$353$	−17.2421			−0.917703			−0.458852	−	0.888513i	$-0.651739\pi$
$353$	−17.2421			−0.917703			−0.458852	+	0.888513i	$0.651739\pi$
$354$		0			0
$355$	0.423171			0.0224596
$356$		0			0
$357$	16.7296	−	7.17538i	0.885424	−	0.379761i
$358$		0			0
$359$	13.2366			0.698599			0.349299	−	0.937011i	$-0.386420\pi$
$359$	13.2366			0.698599			0.349299	+	0.937011i	$0.386420\pi$
$360$		0			0
$361$	−13.1195			−0.690501
$362$		0			0
$363$		−	5.70461i		−	0.299414i
$364$		0			0
$365$	−18.3601			−0.961009
$366$		0			0
$367$			10.9894i			0.573642i	0.957984	+	0.286821i	$0.0925986\pi$
$367$			10.9894i			0.573642i	−0.957984	+	0.286821i	$0.907401\pi$
$368$		0			0
$369$		−	19.5383i		−	1.01712i
$370$		0			0
$371$			2.18066i			0.113214i
$372$		0			0
$373$	−20.9572			−1.08512			−0.542562	−	0.840016i	$-0.682546\pi$
$373$	−20.9572			−1.08512			−0.542562	+	0.840016i	$0.682546\pi$
$374$		0			0
$375$	32.9309			1.70054
$376$		0			0
$377$		−	8.47466i		−	0.436467i
$378$		0			0
$379$		−	5.87497i		−	0.301777i	−0.988551	−	0.150888i	$-0.951787\pi$
$379$		−	5.87497i		−	0.301777i	0.988551	−	0.150888i	$-0.0482134\pi$
$380$		0			0
$381$		−	11.2387i		−	0.575777i
$382$		0			0
$383$	−6.43631			−0.328880			−0.164440	−	0.986387i	$-0.552582\pi$
$383$	−6.43631			−0.328880			−0.164440	+	0.986387i	$0.552582\pi$
$384$		0			0
$385$			10.0941i			0.514441i
$386$		0			0
$387$	63.5482			3.23033
$388$		0			0
$389$	−15.2444			−0.772920			−0.386460	−	0.922306i	$-0.626302\pi$
$389$	−15.2444			−0.772920			−0.386460	+	0.922306i	$0.626302\pi$
$390$		0			0
$391$	6.85832	+	15.9904i	0.346840	+	0.808668i
$392$		0			0
$393$	29.1457			1.47021
$394$		0			0
$395$	9.62976			0.484526
$396$		0			0
$397$		−	8.02334i		−	0.402680i	−0.979521	−	0.201340i	$-0.935470\pi$
$397$		−	8.02334i		−	0.402680i	0.979521	−	0.201340i	$-0.0645296\pi$
$398$		0			0
$399$	−10.7062			−0.535981
$400$		0			0
$401$		−	4.16400i		−	0.207940i	−0.994580	−	0.103970i	$-0.966845\pi$
$401$		−	4.16400i		−	0.207940i	0.994580	−	0.103970i	$-0.0331547\pi$
$402$		0			0
$403$			1.69832i			0.0845994i
$404$		0			0
$405$			15.2042i			0.755502i
$406$		0			0
$407$	14.1978			0.703760
$408$		0			0
$409$	−22.9198			−1.13331			−0.566656	−	0.823954i	$-0.691763\pi$
$409$	−22.9198			−1.13331			−0.566656	+	0.823954i	$0.691763\pi$
$410$		0			0
$411$		−	63.1855i		−	3.11671i
$412$		0			0
$413$		−	1.49294i		−	0.0734626i
$414$		0			0
$415$			15.5648i			0.764047i
$416$		0			0
$417$	15.7652			0.772025
$418$		0			0
$419$			15.0445i			0.734970i	0.930030	+	0.367485i	$0.119781\pi$
$419$			15.0445i			0.734970i	−0.930030	+	0.367485i	$0.880219\pi$
$420$		0			0
$421$	−20.0170			−0.975571			−0.487785	−	0.872964i	$-0.662195\pi$
$421$	−20.0170			−0.975571			−0.487785	+	0.872964i	$0.662195\pi$
$422$		0			0
$423$	18.3842			0.893871
$424$		0			0
$425$	−0.150001	+	0.0643360i	−0.00727613	+	0.00312076i
$426$		0			0
$427$	−8.14460			−0.394145
$428$		0			0
$429$	9.55095			0.461124
$430$		0			0
$431$		−	21.7104i		−	1.04575i	−0.852408	−	0.522877i	$-0.824859\pi$
$431$		−	21.7104i		−	1.04575i	0.852408	−	0.522877i	$-0.175141\pi$
$432$		0			0
$433$	8.02892			0.385845			0.192923	−	0.981214i	$-0.438203\pi$
$433$	8.02892			0.385845			0.192923	+	0.981214i	$0.438203\pi$
$434$		0			0
$435$			52.4658i			2.51554i
$436$		0			0
$437$		−	10.2331i		−	0.489517i
$438$		0			0
$439$			21.1180i			1.00791i	0.863730	+	0.503954i	$0.168122\pi$
$439$			21.1180i			1.00791i	−0.863730	+	0.503954i	$0.831878\pi$
$440$		0			0
$441$	−27.4117			−1.30532
$442$		0			0
$443$	−25.3490			−1.20437			−0.602185	−	0.798357i	$-0.705703\pi$
$443$	−25.3490			−1.20437			−0.602185	+	0.798357i	$0.705703\pi$
$444$		0			0
$445$			2.74257i			0.130010i
$446$		0			0
$447$			24.4678i			1.15729i
$448$		0			0
$449$		−	7.29057i		−	0.344063i	−0.985091	−	0.172032i	$-0.944967\pi$
$449$		−	7.29057i		−	0.344063i	0.985091	−	0.172032i	$-0.0550331\pi$
$450$		0			0
$451$	10.2424			0.482294
$452$		0			0
$453$		−	50.8216i		−	2.38781i
$454$		0			0
$455$	3.59394			0.168486
$456$		0			0
$457$	22.9576			1.07391			0.536957	−	0.843610i	$-0.319574\pi$
$457$	22.9576			1.07391			0.536957	+	0.843610i	$0.319574\pi$
$458$		0			0
$459$	13.1946	+	30.7636i	0.615872	+	1.43592i
$460$		0			0
$461$	−20.9390			−0.975226			−0.487613	−	0.873060i	$-0.662132\pi$
$461$	−20.9390			−0.975226			−0.487613	+	0.873060i	$0.662132\pi$
$462$		0			0
$463$	6.07465			0.282313			0.141157	−	0.989987i	$-0.454918\pi$
$463$	6.07465			0.282313			0.141157	+	0.989987i	$0.454918\pi$
$464$		0			0
$465$		−	10.5141i		−	0.487581i
$466$		0			0
$467$	−7.01734			−0.324724			−0.162362	−	0.986731i	$-0.551911\pi$
$467$	−7.01734			−0.324724			−0.162362	+	0.986731i	$0.551911\pi$
$468$		0			0
$469$		−	6.28589i		−	0.290255i
$470$		0			0
$471$		−	56.3967i		−	2.59862i
$472$		0			0
$473$			33.3131i			1.53174i
$474$		0			0
$475$	0.0959943			0.00440452
$476$		0			0
$477$	−8.39193			−0.384240
$478$		0			0
$479$		−	25.1480i		−	1.14904i	−0.818490	−	0.574521i	$-0.805188\pi$
$479$		−	25.1480i		−	1.14904i	0.818490	−	0.574521i	$-0.194812\pi$
$480$		0			0
$481$		−	5.05506i		−	0.230491i
$482$		0			0
$483$			18.6308i			0.847731i
$484$		0			0
$485$	−1.04635			−0.0475121
$486$		0			0
$487$		−	13.3023i		−	0.602784i	−0.953500	−	0.301392i	$-0.902549\pi$
$487$		−	13.3023i		−	0.602784i	0.953500	−	0.301392i	$-0.0974513\pi$
$488$		0			0
$489$	23.7189			1.07261
$490$		0			0
$491$	10.9480			0.494078			0.247039	−	0.969005i	$-0.420542\pi$
$491$	10.9480			0.494078			0.247039	+	0.969005i	$0.420542\pi$
$492$		0			0
$493$	12.8442	+	29.9466i	0.578473	+	1.34873i
$494$		0			0
$495$	−38.8453			−1.74597
$496$		0			0
$497$	0.281423			0.0126236
$498$		0			0
$499$			14.4735i			0.647921i	0.946071	+	0.323960i	$0.105014\pi$
$499$			14.4735i			0.647921i	−0.946071	+	0.323960i	$0.894986\pi$
$500$		0			0
$501$	−9.40139			−0.420023
$502$		0			0
$503$		−	1.80908i		−	0.0806627i	−0.999186	−	0.0403314i	$-0.987159\pi$
$503$		−	1.80908i		−	0.0806627i	0.999186	−	0.0403314i	$-0.0128414\pi$
$504$		0			0
$505$			8.89195i			0.395686i
$506$		0			0
$507$			35.0437i			1.55634i
$508$		0			0
$509$	11.2422			0.498302			0.249151	−	0.968465i	$-0.419848\pi$
$509$	11.2422			0.498302			0.249151	+	0.968465i	$0.419848\pi$
$510$		0			0
$511$	−12.2101			−0.540141
$512$		0			0
$513$		−	19.6874i		−	0.869219i
$514$		0			0
$515$			14.9954i			0.660777i
$516$		0			0
$517$			9.63735i			0.423850i
$518$		0			0
$519$	22.3308			0.980212
$520$		0			0
$521$		−	13.1001i		−	0.573925i	−0.957942	−	0.286963i	$-0.907354\pi$
$521$		−	13.1001i		−	0.573925i	0.957942	−	0.286963i	$-0.0926455\pi$
$522$		0			0
$523$	−34.0152			−1.48738			−0.743690	−	0.668525i	$-0.766926\pi$
$523$	−34.0152			−1.48738			−0.743690	+	0.668525i	$0.766926\pi$
$524$		0			0
$525$	−0.174770			−0.00762761
$526$		0			0
$527$	−2.57397	−	6.00129i	−0.112124	−	0.261420i
$528$		0			0
$529$	5.19244			0.225758
$530$		0			0
$531$	5.74532			0.249326
$532$		0			0
$533$		−	3.64674i		−	0.157958i
$534$		0			0
$535$	3.63764			0.157269
$536$		0			0
$537$		−	53.5891i		−	2.31254i
$538$		0			0
$539$		−	14.3697i		−	0.618949i
$540$		0			0
$541$			18.3517i			0.789001i	0.918896	+	0.394501i	$0.129082\pi$
$541$			18.3517i			0.789001i	−0.918896	+	0.394501i	$0.870918\pi$
$542$		0			0
$543$	2.55265			0.109545
$544$		0			0
$545$	−40.3456			−1.72822
$546$		0			0
$547$		−	15.1212i		−	0.646536i	−0.946307	−	0.323268i	$-0.895218\pi$
$547$		−	15.1212i		−	0.646536i	0.946307	−	0.323268i	$-0.104782\pi$
$548$		0			0
$549$		−	31.3432i		−	1.33769i
$550$		0			0
$551$		−	19.1645i		−	0.816435i
$552$		0			0
$553$	6.40412			0.272331
$554$		0			0
$555$			31.2954i			1.32842i
$556$		0			0
$557$	−20.6857			−0.876481			−0.438240	−	0.898858i	$-0.644398\pi$
$557$	−20.6857			−0.876481			−0.438240	+	0.898858i	$0.644398\pi$
$558$		0			0
$559$	11.8610			0.501666
$560$		0			0
$561$	−33.7498	+	14.4754i	−1.42492	+	0.611152i
$562$		0			0
$563$	11.9714			0.504533			0.252267	−	0.967658i	$-0.418824\pi$
$563$	11.9714			0.504533			0.252267	+	0.967658i	$0.418824\pi$
$564$		0			0
$565$	−15.1952			−0.639267
$566$		0			0
$567$			10.1113i			0.424635i
$568$		0			0
$569$	−24.9699			−1.04679			−0.523396	−	0.852089i	$-0.675335\pi$
$569$	−24.9699			−1.04679			−0.523396	+	0.852089i	$0.675335\pi$
$570$		0			0
$571$			44.4325i			1.85944i	0.368265	+	0.929721i	$0.379952\pi$
$571$			44.4325i			1.85944i	−0.368265	+	0.929721i	$0.620048\pi$
$572$		0			0
$573$			43.9472i			1.83592i
$574$		0			0
$575$		−	0.167048i		−	0.00696638i
$576$		0			0
$577$	−29.2954			−1.21958			−0.609792	−	0.792561i	$-0.708747\pi$
$577$	−29.2954			−1.21958			−0.609792	+	0.792561i	$0.708747\pi$
$578$		0			0
$579$	−74.1475			−3.08147
$580$		0			0
$581$			10.3511i			0.429437i
$582$		0			0
$583$		−	4.39921i		−	0.182197i
$584$		0			0
$585$			13.8307i			0.571828i
$586$		0			0
$587$	−4.66193			−0.192418			−0.0962092	−	0.995361i	$-0.530672\pi$
$587$	−4.66193			−0.192418			−0.0962092	+	0.995361i	$0.530672\pi$
$588$		0			0
$589$			3.84056i			0.158248i
$590$		0			0
$591$	−25.4001			−1.04482
$592$		0			0
$593$	−44.9858			−1.84734			−0.923672	−	0.383185i	$-0.874827\pi$
$593$	−44.9858			−1.84734			−0.923672	+	0.383185i	$0.874827\pi$
$594$		0			0
$595$	−12.6997	+	5.44696i	−0.520639	+	0.223304i
$596$		0			0
$597$	−2.46012			−0.100686
$598$		0			0
$599$	−13.9851			−0.571414			−0.285707	−	0.958317i	$-0.592228\pi$
$599$	−13.9851			−0.571414			−0.285707	+	0.958317i	$0.592228\pi$
$600$		0			0
$601$		−	4.55998i		−	0.186006i	−0.995666	−	0.0930028i	$-0.970353\pi$
$601$		−	4.55998i		−	0.186006i	0.995666	−	0.0930028i	$-0.0296465\pi$
$602$		0			0
$603$	24.1902			0.985101
$604$		0			0
$605$			4.33047i			0.176059i
$606$		0			0
$607$			2.76623i			0.112278i	0.998423	+	0.0561388i	$0.0178789\pi$
$607$			2.76623i			0.112278i	−0.998423	+	0.0561388i	$0.982121\pi$
$608$		0			0
$609$			34.8915i			1.41388i
$610$		0			0
$611$	3.43133			0.138817
$612$		0			0
$613$	30.6406			1.23756			0.618782	−	0.785563i	$-0.287626\pi$
$613$	30.6406			1.23756			0.618782	+	0.785563i	$0.287626\pi$
$614$		0			0
$615$			22.5766i			0.910376i
$616$		0			0
$617$			19.0529i			0.767040i	0.923533	+	0.383520i	$0.125288\pi$
$617$			19.0529i			0.767040i	−0.923533	+	0.383520i	$0.874712\pi$
$618$		0			0
$619$			30.4411i			1.22353i	0.791040	+	0.611765i	$0.209540\pi$
$619$			30.4411i			1.22353i	−0.791040	+	0.611765i	$0.790460\pi$
$620$		0			0
$621$	−34.2597			−1.37479
$622$		0			0
$623$			1.82390i			0.0730730i
$624$		0			0
$625$	−25.1964			−1.00785
$626$		0			0
$627$	21.5984			0.862557
$628$		0			0
$629$	7.66144	+	17.8629i	0.305482	+	0.712239i
$630$		0			0
$631$	−29.0363			−1.15592			−0.577959	−	0.816066i	$-0.696151\pi$
$631$	−29.0363			−1.15592			−0.577959	+	0.816066i	$0.696151\pi$
$632$		0			0
$633$	9.98173			0.396738
$634$		0			0
$635$			8.53151i			0.338563i
$636$		0			0
$637$	−5.11627			−0.202714
$638$		0			0
$639$			1.08301i			0.0428433i
$640$		0			0
$641$		−	27.4497i		−	1.08420i	−0.840314	−	0.542099i	$-0.817630\pi$
$641$		−	27.4497i		−	1.08420i	0.840314	−	0.542099i	$-0.182370\pi$
$642$		0			0
$643$			36.3969i			1.43536i	0.696376	+	0.717678i	$0.254795\pi$
$643$			36.3969i			1.43536i	−0.696376	+	0.717678i	$0.745205\pi$
$644$		0			0
$645$	−73.4301			−2.89131
$646$		0			0
$647$	−27.9147			−1.09744			−0.548720	−	0.836006i	$-0.684885\pi$
$647$	−27.9147			−1.09744			−0.548720	+	0.836006i	$0.684885\pi$
$648$		0			0
$649$			3.01181i			0.118224i
$650$		0			0
$651$		−	6.99225i		−	0.274048i
$652$		0			0
$653$			17.7873i			0.696069i	0.937482	+	0.348035i	$0.113151\pi$
$653$			17.7873i			0.696069i	−0.937482	+	0.348035i	$0.886849\pi$
$654$		0			0
$655$	−22.1251			−0.864498
$656$		0			0
$657$		−	46.9884i		−	1.83319i
$658$		0			0
$659$	−43.1382			−1.68043			−0.840214	−	0.542255i	$-0.817571\pi$
$659$	−43.1382			−1.68043			−0.840214	+	0.542255i	$0.817571\pi$
$660$		0			0
$661$	−14.4749			−0.563007			−0.281504	−	0.959560i	$-0.590833\pi$
$661$	−14.4749			−0.563007			−0.281504	+	0.959560i	$0.590833\pi$
$662$		0			0
$663$	5.15389	+	12.0164i	0.200161	+	0.466680i
$664$		0			0
$665$	8.12728			0.315163
$666$		0			0
$667$	−33.3498			−1.29131
$668$		0			0
$669$		−	8.78422i		−	0.339618i
$670$		0			0
$671$	16.4307			0.634299
$672$		0			0
$673$			12.3170i			0.474786i	0.971414	+	0.237393i	$0.0762929\pi$
$673$			12.3170i			0.474786i	−0.971414	+	0.237393i	$0.923707\pi$
$674$		0			0
$675$		−	0.321381i		−	0.0123700i
$676$		0			0
$677$		−	7.07339i		−	0.271852i	−0.990719	−	0.135926i	$-0.956599\pi$
$677$		−	7.07339i		−	0.271852i	0.990719	−	0.135926i	$-0.0434010\pi$
$678$		0			0
$679$	−0.695856			−0.0267045
$680$		0			0
$681$	62.5396			2.39652
$682$		0			0
$683$		−	6.66530i		−	0.255040i	−0.991836	−	0.127520i	$-0.959298\pi$
$683$		−	6.66530i		−	0.255040i	0.991836	−	0.127520i	$-0.0407018\pi$
$684$		0			0
$685$			47.9653i			1.83266i
$686$		0			0
$687$		−	35.8078i		−	1.36615i
$688$		0			0
$689$	−1.56632			−0.0596719
$690$		0			0
$691$		−	22.0599i		−	0.839197i	−0.907710	−	0.419598i	$-0.862171\pi$
$691$		−	22.0599i		−	0.839197i	0.907710	−	0.419598i	$-0.137829\pi$
$692$		0			0
$693$	−25.8335			−0.981332
$694$		0			0
$695$	−11.9677			−0.453959
$696$		0			0
$697$	5.52699	+	12.8863i	0.209350	+	0.488105i
$698$		0			0
$699$	−7.29267			−0.275834
$700$		0			0
$701$	21.0139			0.793683			0.396842	−	0.917887i	$-0.370106\pi$
$701$	21.0139			0.793683			0.396842	+	0.917887i	$0.370106\pi$
$702$		0			0
$703$		−	11.4315i		−	0.431146i
$704$		0			0
$705$	−21.2430			−0.800058
$706$		0			0
$707$			5.91345i			0.222398i
$708$		0			0
$709$			17.8355i			0.669825i	0.942249	+	0.334913i	$0.108707\pi$
$709$			17.8355i			0.669825i	−0.942249	+	0.334913i	$0.891293\pi$
$710$		0			0
$711$			24.6452i			0.924268i
$712$		0			0
$713$	6.68329			0.250291
$714$		0			0
$715$	−7.25030			−0.271146
$716$		0			0
$717$		−	28.2662i		−	1.05562i
$718$		0			0
$719$		−	48.0006i		−	1.79012i	−0.445946	−	0.895060i	$-0.647133\pi$
$719$		−	48.0006i		−	1.79012i	0.445946	−	0.895060i	$-0.352867\pi$
$720$		0			0
$721$			9.97247i			0.371394i
$722$		0			0
$723$	17.4929			0.650570
$724$		0			0
$725$		−	0.312845i		−	0.0116188i
$726$		0			0
$727$	3.14923			0.116798			0.0583992	−	0.998293i	$-0.481400\pi$
$727$	3.14923			0.116798			0.0583992	+	0.998293i	$0.481400\pi$
$728$		0			0
$729$	33.1145			1.22646
$730$		0			0
$731$	−41.9126	+	17.9765i	−1.55019	+	0.664883i
$732$		0			0
$733$	−7.58307			−0.280087			−0.140043	−	0.990145i	$-0.544724\pi$
$733$	−7.58307			−0.280087			−0.140043	+	0.990145i	$0.544724\pi$
$734$		0			0
$735$	31.6743			1.16833
$736$		0			0
$737$			12.6810i			0.467109i
$738$		0			0
$739$	33.6496			1.23782			0.618911	−	0.785461i	$-0.287574\pi$
$739$	33.6496			1.23782			0.618911	+	0.785461i	$0.287574\pi$
$740$		0			0
$741$		−	7.69000i		−	0.282499i
$742$		0			0
$743$		−	15.3296i		−	0.562390i	−0.959651	−	0.281195i	$-0.909269\pi$
$743$		−	15.3296i		−	0.562390i	0.959651	−	0.281195i	$-0.0907308\pi$
$744$		0			0
$745$		−	18.5739i		−	0.680496i
$746$		0			0
$747$	−39.8346			−1.45747
$748$		0			0
$749$	2.41916			0.0883941
$750$		0			0
$751$		−	34.2835i		−	1.25102i	−0.780215	−	0.625512i	$-0.784890\pi$
$751$		−	34.2835i		−	1.25102i	0.780215	−	0.625512i	$-0.215110\pi$
$752$		0			0
$753$			25.5567i			0.931338i
$754$		0			0
$755$			38.5796i			1.40405i
$756$		0			0
$757$	−0.772990			−0.0280948			−0.0140474	−	0.999901i	$-0.504472\pi$
$757$	−0.772990			−0.0280948			−0.0140474	+	0.999901i	$0.504472\pi$
$758$		0			0
$759$		−	37.5852i		−	1.36426i
$760$		0			0
$761$	−18.9799			−0.688021			−0.344010	−	0.938966i	$-0.611786\pi$
$761$	−18.9799			−0.688021			−0.344010	+	0.938966i	$0.611786\pi$
$762$		0			0
$763$	−26.8312			−0.971356
$764$		0			0
$765$	−20.9617	−	48.8729i	−0.757873	−	1.76700i
$766$		0			0
$767$	1.07234			0.0387199
$768$		0			0
$769$	46.1565			1.66445			0.832224	−	0.554440i	$-0.187068\pi$
$769$	46.1565			1.66445			0.832224	+	0.554440i	$0.187068\pi$
$770$		0			0
$771$		−	24.3348i		−	0.876397i
$772$		0			0
$773$	6.76290			0.243244			0.121622	−	0.992576i	$-0.461190\pi$
$773$	6.76290			0.243244			0.121622	+	0.992576i	$0.461190\pi$
$774$		0			0
$775$			0.0626941i			0.00225204i
$776$		0			0
$777$			20.8125i			0.746644i
$778$		0			0
$779$		−	8.24669i		−	0.295468i
$780$		0			0
$781$	−0.567735			−0.0203152
$782$		0			0
$783$	−64.1611			−2.29293
$784$		0			0
$785$			42.8117i			1.52802i
$786$		0			0
$787$			29.0128i			1.03419i	0.855927	+	0.517097i	$0.172987\pi$
$787$			29.0128i			1.03419i	−0.855927	+	0.517097i	$0.827013\pi$
$788$		0			0
$789$			22.4722i			0.800031i
$790$		0			0
$791$	−10.1053			−0.359304
$792$		0			0
$793$		−	5.85006i		−	0.207742i
$794$		0			0
$795$	9.69690			0.343914
$796$		0			0
$797$	44.3010			1.56922			0.784611	−	0.619989i	$-0.212863\pi$
$797$	44.3010			1.56922			0.784611	+	0.619989i	$0.212863\pi$
$798$		0			0
$799$	−12.1251	+	5.20052i	−0.428957	+	0.183981i
$800$		0			0
$801$	−7.01898			−0.248004
$802$		0			0
$803$	24.6322			0.869252
$804$		0			0
$805$		−	14.1430i		−	0.498475i
$806$		0			0
$807$	43.0584			1.51573
$808$		0			0
$809$		−	20.2454i		−	0.711792i	−0.934526	−	0.355896i	$-0.884176\pi$
$809$		−	20.2454i		−	0.711792i	0.934526	−	0.355896i	$-0.115824\pi$
$810$		0			0
$811$			35.7471i			1.25525i	0.778516	+	0.627625i	$0.215973\pi$
$811$			35.7471i			1.25525i	−0.778516	+	0.627625i	$0.784027\pi$
$812$		0			0
$813$			13.3047i			0.466618i
$814$		0			0
$815$	−18.0055			−0.630704
$816$		0			0
$817$	26.8223			0.938392
$818$		0			0
$819$			9.19787i			0.321400i
$820$		0			0
$821$			12.4527i			0.434602i	0.976105	+	0.217301i	$0.0697252\pi$
$821$			12.4527i			0.434602i	−0.976105	+	0.217301i	$0.930275\pi$
$822$		0			0
$823$			41.2150i			1.43667i	0.695700	+	0.718333i	$0.255095\pi$
$823$			41.2150i			1.43667i	−0.695700	+	0.718333i	$0.744905\pi$
$824$		0			0
$825$	0.352577			0.0122751
$826$		0			0
$827$		−	55.5146i		−	1.93043i	−0.261452	−	0.965216i	$-0.584201\pi$
$827$		−	55.5146i		−	1.93043i	0.261452	−	0.965216i	$-0.415799\pi$
$828$		0			0
$829$	6.78773			0.235747			0.117874	−	0.993029i	$-0.462392\pi$
$829$	6.78773			0.235747			0.117874	+	0.993029i	$0.462392\pi$
$830$		0			0
$831$	10.6124			0.368140
$832$		0			0
$833$	18.0792	−	7.75421i	0.626406	−	0.268668i
$834$		0			0
$835$	7.13677			0.246978
$836$		0			0
$837$	12.8579			0.444433
$838$		0			0
$839$		−	28.1738i		−	0.972669i	−0.873773	−	0.486334i	$-0.838334\pi$
$839$		−	28.1738i		−	0.972669i	0.873773	−	0.486334i	$-0.161666\pi$
$840$		0			0
$841$	−33.4571			−1.15369
$842$		0			0
$843$			83.9492i			2.89136i
$844$		0			0
$845$		−	26.6023i		−	0.915147i
$846$		0			0
$847$			2.87991i			0.0989550i
$848$		0			0
$849$	24.8378			0.852430
$850$		0			0
$851$	−19.8929			−0.681918
$852$		0			0
$853$			10.7655i			0.368603i	0.982870	+	0.184301i	$0.0590023\pi$
$853$			10.7655i			0.368603i	−0.982870	+	0.184301i	$0.940998\pi$
$854$		0			0
$855$			31.2765i			1.06963i
$856$		0			0
$857$		−	8.88869i		−	0.303632i	−0.988409	−	0.151816i	$-0.951488\pi$
$857$		−	8.88869i		−	0.303632i	0.988409	−	0.151816i	$-0.0485121\pi$
$858$		0			0
$859$	−0.858330			−0.0292858			−0.0146429	−	0.999893i	$-0.504661\pi$
$859$	−0.858330			−0.0292858			−0.0146429	+	0.999893i	$0.504661\pi$
$860$		0			0
$861$			15.0142i			0.511683i
$862$		0			0
$863$	−5.08941			−0.173246			−0.0866228	−	0.996241i	$-0.527607\pi$
$863$	−5.08941			−0.173246			−0.0866228	+	0.996241i	$0.527607\pi$
$864$		0			0
$865$	−16.9517			−0.576375
$866$		0			0
$867$	−36.4242	−	34.6508i	−1.23703	−	1.17680i
$868$		0			0
$869$	−12.9195			−0.438264
$870$		0			0
$871$	4.51499			0.152985
$872$		0			0
$873$		−	2.67789i		−	0.0906327i
$874$		0			0
$875$	−16.6248			−0.562021
$876$		0			0
$877$		−	4.11026i		−	0.138794i	−0.997589	−	0.0693968i	$-0.977893\pi$
$877$		−	4.11026i		−	0.138794i	0.997589	−	0.0693968i	$-0.0221075\pi$
$878$		0			0
$879$		−	52.2367i		−	1.76190i
$880$		0			0
$881$		−	3.57714i		−	0.120517i	−0.998183	−	0.0602584i	$-0.980808\pi$
$881$		−	3.57714i		−	0.120517i	0.998183	−	0.0602584i	$-0.0191925\pi$
$882$		0			0
$883$	40.8213			1.37374			0.686872	−	0.726778i	$-0.258983\pi$
$883$	40.8213			1.37374			0.686872	+	0.726778i	$0.258983\pi$
$884$		0			0
$885$	−6.63873			−0.223159
$886$		0			0
$887$		−	4.41335i		−	0.148186i	−0.997251	−	0.0740929i	$-0.976394\pi$
$887$		−	4.41335i		−	0.148186i	0.997251	−	0.0740929i	$-0.0236061\pi$
$888$		0			0
$889$			5.67375i			0.190291i
$890$		0			0
$891$		−	20.3982i		−	0.683367i
$892$		0			0
$893$	7.75957			0.259664
$894$		0			0
$895$			40.6805i			1.35980i
$896$		0			0
$897$	−13.3820			−0.446813
$898$		0			0
$899$	12.5164			0.417445
$900$		0			0
$901$	5.53482	−	2.37390i	0.184392	−	0.0790862i
$902$		0			0
$903$	−48.8335			−1.62508
$904$		0			0
$905$	−1.93776			−0.0644134
$906$		0			0
$907$		−	16.1543i		−	0.536394i	−0.963364	−	0.268197i	$-0.913572\pi$
$907$		−	16.1543i		−	0.536394i	0.963364	−	0.268197i	$-0.0864278\pi$
$908$		0			0
$909$	−22.7569			−0.754800
$910$		0			0
$911$			57.3386i			1.89971i	0.312687	+	0.949856i	$0.398771\pi$
$911$			57.3386i			1.89971i	−0.312687	+	0.949856i	$0.601229\pi$
$912$		0			0
$913$		−	20.8821i		−	0.691095i
$914$		0			0
$915$			36.2171i			1.19730i
$916$		0			0
$917$	−14.7139			−0.485897
$918$		0			0
$919$	−8.42540			−0.277928			−0.138964	−	0.990297i	$-0.544377\pi$
$919$	−8.42540			−0.277928			−0.138964	+	0.990297i	$0.544377\pi$
$920$		0			0
$921$			61.7380i			2.03434i
$922$		0			0
$923$			0.202139i			0.00665349i
$924$		0			0
$925$		−	0.186609i		−	0.00613568i
$926$		0			0
$927$	−38.3774			−1.26048
$928$		0			0
$929$			12.2878i			0.403149i	0.979473	+	0.201574i	$0.0646058\pi$
$929$			12.2878i			0.403149i	−0.979473	+	0.201574i	$0.935394\pi$
$930$		0			0
$931$	−11.5699			−0.379187
$932$		0			0
$933$	−88.3477			−2.89238
$934$		0			0
$935$	25.6201	−	10.9885i	0.837867	−	0.359364i
$936$		0			0
$937$	22.1490			0.723575			0.361788	−	0.932261i	$-0.382167\pi$
$937$	22.1490			0.723575			0.361788	+	0.932261i	$0.382167\pi$
$938$		0			0
$939$	−74.0173			−2.41546
$940$		0			0
$941$		−	10.6116i		−	0.345929i	−0.984928	−	0.172964i	$-0.944665\pi$
$941$		−	10.6116i		−	0.345929i	0.984928	−	0.172964i	$-0.0553345\pi$
$942$		0			0
$943$	−14.3508			−0.467325
$944$		0			0
$945$		−	27.2095i		−	0.885124i
$946$		0			0
$947$		−	0.530828i		−	0.0172496i	−0.999963	−	0.00862479i	$-0.997255\pi$
$947$		−	0.530828i		−	0.0172496i	0.999963	−	0.00862479i	$-0.00274539\pi$
$948$		0			0
$949$		−	8.77017i		−	0.284692i
$950$		0			0
$951$	55.0143			1.78396
$952$		0			0
$953$	−20.1191			−0.651720			−0.325860	−	0.945418i	$-0.605654\pi$
$953$	−20.1191			−0.651720			−0.325860	+	0.945418i	$0.605654\pi$
$954$		0			0
$955$		−	33.3611i		−	1.07954i
$956$		0			0
$957$		−	70.3892i		−	2.27536i
$958$		0			0
$959$			31.8986i			1.03006i
$960$		0			0
$961$	28.4917			0.919088
$962$		0			0
$963$			9.30973i			0.300002i
$964$		0			0
$965$	56.2868			1.81194
$966$		0			0
$967$	−17.3599			−0.558258			−0.279129	−	0.960254i	$-0.590046\pi$
$967$	−17.3599			−0.558258			−0.279129	+	0.960254i	$0.590046\pi$
$968$		0			0
$969$	11.6549	+	27.1738i	0.374411	+	0.872949i
$970$		0			0
$971$	−17.8149			−0.571708			−0.285854	−	0.958273i	$-0.592277\pi$
$971$	−17.8149			−0.571708			−0.285854	+	0.958273i	$0.592277\pi$
$972$		0			0
$973$	−7.95890			−0.255151
$974$		0			0
$975$		−	0.125533i		−	0.00402028i
$976$		0			0
$977$	3.70446			0.118516			0.0592580	−	0.998243i	$-0.481127\pi$
$977$	3.70446			0.118516			0.0592580	+	0.998243i	$0.481127\pi$
$978$		0			0
$979$		−	3.67948i		−	0.117597i
$980$		0			0
$981$		−	103.256i		−	3.29670i
$982$		0			0
$983$			51.0396i			1.62791i	0.580928	+	0.813955i	$0.302690\pi$
$983$			51.0396i			1.62791i	−0.580928	+	0.813955i	$0.697310\pi$
$984$		0			0
$985$	19.2817			0.614365
$986$		0			0
$987$	−14.1273			−0.449678
$988$		0			0
$989$		−	46.6757i		−	1.48420i
$990$		0			0
$991$			32.5736i			1.03473i	0.855763	+	0.517367i	$0.173088\pi$
$991$			32.5736i			1.03473i	−0.855763	+	0.517367i	$0.826912\pi$
$992$		0			0
$993$		−	61.5589i		−	1.95351i
$994$		0			0
$995$	1.86752			0.0592044
$996$		0			0
$997$			14.5336i			0.460283i	0.973157	+	0.230142i	$0.0739189\pi$
$997$			14.5336i			0.460283i	−0.973157	+	0.230142i	$0.926081\pi$
$998$		0			0
$999$	−38.2716			−1.21086

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.4	✓	40
17.16	even	2	inner	4012.2.b.a.237.37	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.4	✓	40	1.1	even	1	trivial
4012.2.b.a.237.37	yes	40	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.95725i q^{3} +2.24490i q^{5} +1.49294i q^{7} -5.74532 q^{9} +O(q^{10})\)	\(q-2.95725i q^{3} +2.24490i q^{5} +1.49294i q^{7} -5.74532 q^{9} -3.01181i q^{11} -1.07234 q^{13} +6.63873 q^{15} +(3.78928 - 1.62523i) q^{17} -2.42497 q^{19} +4.41499 q^{21} +4.21990i q^{23} -0.0395857 q^{25} +8.11860i q^{27} +7.90298i q^{29} -1.58375i q^{31} -8.90666 q^{33} -3.35150 q^{35} +4.71406i q^{37} +3.17117i q^{39} +3.40074i q^{41} -11.0609 q^{43} -12.8977i q^{45} -3.19986 q^{47} +4.77114 q^{49} +(-4.80622 - 11.2058i) q^{51} +1.46065 q^{53} +6.76121 q^{55} +7.17124i q^{57} -1.00000 q^{59} +5.45542i q^{61} -8.57740i q^{63} -2.40729i q^{65} -4.21042 q^{67} +12.4793 q^{69} -0.188503i q^{71} +8.17855i q^{73} +0.117065i q^{75} +4.49644 q^{77} -4.28961i q^{79} +6.77276 q^{81} +6.93340 q^{83} +(3.64849 + 8.50656i) q^{85} +23.3711 q^{87} +1.22169 q^{89} -1.60093i q^{91} -4.68356 q^{93} -5.44382i q^{95} +0.466099i q^{97} +17.3038i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more