LMFDB - Embedded newform 4012.2.b.b.237.21

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:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.21
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.274529i q^{3} +3.55822i q^{5} +0.321791i q^{7} +2.92463 q^{9} +O(q^{10})$	$q-0.274529i q^{3} +3.55822i q^{5} +0.321791i q^{7} +2.92463 q^{9} -0.790275i q^{11} -4.51068 q^{13} +0.976836 q^{15} +(-0.792973 - 4.04613i) q^{17} -3.21569 q^{19} +0.0883410 q^{21} +3.60173i q^{23} -7.66094 q^{25} -1.62649i q^{27} -7.35707i q^{29} -10.7779i q^{31} -0.216954 q^{33} -1.14500 q^{35} -8.58392i q^{37} +1.23831i q^{39} -7.36832i q^{41} -7.18947 q^{43} +10.4065i q^{45} -2.61004 q^{47} +6.89645 q^{49} +(-1.11078 + 0.217694i) q^{51} -5.58406 q^{53} +2.81197 q^{55} +0.882801i q^{57} +1.00000 q^{59} +7.98834i q^{61} +0.941120i q^{63} -16.0500i q^{65} +12.9523 q^{67} +0.988781 q^{69} -10.4862i q^{71} +5.27767i q^{73} +2.10315i q^{75} +0.254303 q^{77} -1.75505i q^{79} +8.32738 q^{81} -12.3958 q^{83} +(14.3970 - 2.82157i) q^{85} -2.01973 q^{87} +3.48022 q^{89} -1.45149i q^{91} -2.95886 q^{93} -11.4421i q^{95} +16.6285i q^{97} -2.31126i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * 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
$3$		−	0.274529i		−	0.158500i	−0.996855	−	0.0792498i	$-0.974748\pi$
$3$		−	0.274529i		−	0.158500i	0.996855	−	0.0792498i	$-0.0252525\pi$
$4$		0			0
$5$			3.55822i			1.59128i	0.605767	+	0.795642i	$0.292867\pi$
$5$			3.55822i			1.59128i	−0.605767	+	0.795642i	$0.707133\pi$
$6$		0			0
$7$			0.321791i			0.121626i	0.998149	+	0.0608128i	$0.0193693\pi$
$7$			0.321791i			0.121626i	−0.998149	+	0.0608128i	$0.980631\pi$
$8$		0			0
$9$	2.92463			0.974878
$10$		0			0
$11$		−	0.790275i		−	0.238277i	−0.992878	−	0.119138i	$-0.961987\pi$
$11$		−	0.790275i		−	0.238277i	0.992878	−	0.119138i	$-0.0380132\pi$
$12$		0			0
$13$	−4.51068			−1.25104			−0.625518	−	0.780209i	$-0.715112\pi$
$13$	−4.51068			−1.25104			−0.625518	+	0.780209i	$0.715112\pi$
$14$		0			0
$15$	0.976836			0.252218
$16$		0			0
$17$	−0.792973	−	4.04613i	−0.192324	−	0.981331i
$17$	−0.792973	−	4.04613i	−0.192324	−	0.981331i
$18$		0			0
$19$	−3.21569			−0.737729			−0.368865	−	0.929483i	$-0.620253\pi$
$19$	−3.21569			−0.737729			−0.368865	+	0.929483i	$0.620253\pi$
$20$		0			0
$21$	0.0883410			0.0192776
$22$		0			0
$23$			3.60173i			0.751013i	0.926820	+	0.375507i	$0.122531\pi$
$23$			3.60173i			0.751013i	−0.926820	+	0.375507i	$0.877469\pi$
$24$		0			0
$25$	−7.66094			−1.53219
$26$		0			0
$27$		−	1.62649i		−	0.313017i
$28$		0			0
$29$		−	7.35707i		−	1.36617i	−0.730337	−	0.683087i	$-0.760637\pi$
$29$		−	7.35707i		−	1.36617i	0.730337	−	0.683087i	$-0.239363\pi$
$30$		0			0
$31$		−	10.7779i		−	1.93578i	−0.251384	−	0.967888i	$-0.580886\pi$
$31$		−	10.7779i		−	1.93578i	0.251384	−	0.967888i	$-0.419114\pi$
$32$		0			0
$33$	−0.216954			−0.0377668
$34$		0			0
$35$	−1.14500			−0.193541
$36$		0			0
$37$		−	8.58392i		−	1.41119i	−0.708616	−	0.705594i	$-0.750680\pi$
$37$		−	8.58392i		−	1.41119i	0.708616	−	0.705594i	$-0.249320\pi$
$38$		0			0
$39$			1.23831i			0.198289i
$40$		0			0
$41$		−	7.36832i		−	1.15074i	−0.817894	−	0.575369i	$-0.804858\pi$
$41$		−	7.36832i		−	1.15074i	0.817894	−	0.575369i	$-0.195142\pi$
$42$		0			0
$43$	−7.18947			−1.09638			−0.548192	−	0.836353i	$-0.684684\pi$
$43$	−7.18947			−1.09638			−0.548192	+	0.836353i	$0.684684\pi$
$44$		0			0
$45$			10.4065i			1.55131i
$46$		0			0
$47$	−2.61004			−0.380714			−0.190357	−	0.981715i	$-0.560965\pi$
$47$	−2.61004			−0.380714			−0.190357	+	0.981715i	$0.560965\pi$
$48$		0			0
$49$	6.89645			0.985207
$50$		0			0
$51$	−1.11078	+	0.217694i	−0.155541	+	0.0304833i
$52$		0			0
$53$	−5.58406			−0.767030			−0.383515	−	0.923535i	$-0.625287\pi$
$53$	−5.58406			−0.767030			−0.383515	+	0.923535i	$0.625287\pi$
$54$		0			0
$55$	2.81197			0.379166
$56$		0			0
$57$			0.882801i			0.116930i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$			7.98834i			1.02280i	0.859342	+	0.511401i	$0.170873\pi$
$61$			7.98834i			1.02280i	−0.859342	+	0.511401i	$0.829127\pi$
$62$		0			0
$63$			0.941120i			0.118570i
$64$		0			0
$65$		−	16.0500i		−	1.99076i
$66$		0			0
$67$	12.9523			1.58237			0.791187	−	0.611574i	$-0.209463\pi$
$67$	12.9523			1.58237			0.791187	+	0.611574i	$0.209463\pi$
$68$		0			0
$69$	0.988781			0.119035
$70$		0			0
$71$		−	10.4862i		−	1.24449i	−0.782824	−	0.622244i	$-0.786221\pi$
$71$		−	10.4862i		−	1.24449i	0.782824	−	0.622244i	$-0.213779\pi$
$72$		0			0
$73$			5.27767i			0.617704i	0.951110	+	0.308852i	$0.0999448\pi$
$73$			5.27767i			0.617704i	−0.951110	+	0.308852i	$0.900055\pi$
$74$		0			0
$75$			2.10315i			0.242851i
$76$		0			0
$77$	0.254303			0.0289805
$78$		0			0
$79$		−	1.75505i		−	0.197458i	−0.995114	−	0.0987292i	$-0.968522\pi$
$79$		−	1.75505i		−	0.197458i	0.995114	−	0.0987292i	$-0.0314778\pi$
$80$		0			0
$81$	8.32738			0.925265
$82$		0			0
$83$	−12.3958			−1.36062			−0.680308	−	0.732926i	$-0.738154\pi$
$83$	−12.3958			−1.36062			−0.680308	+	0.732926i	$0.738154\pi$
$84$		0			0
$85$	14.3970	−	2.82157i	1.56158	−	0.306043i
$86$		0			0
$87$	−2.01973			−0.216538
$88$		0			0
$89$	3.48022			0.368903			0.184452	−	0.982842i	$-0.440949\pi$
$89$	3.48022			0.368903			0.184452	+	0.982842i	$0.440949\pi$
$90$		0			0
$91$		−	1.45149i		−	0.152158i
$92$		0			0
$93$	−2.95886			−0.306819
$94$		0			0
$95$		−	11.4421i		−	1.17394i
$96$		0			0
$97$			16.6285i			1.68837i	0.536056	+	0.844183i	$0.319914\pi$
$97$			16.6285i			1.68837i	−0.536056	+	0.844183i	$0.680086\pi$
$98$		0			0
$99$		−	2.31126i		−	0.232291i
$100$		0			0
$101$	3.23224			0.321620			0.160810	−	0.986985i	$-0.448589\pi$
$101$	3.23224			0.321620			0.160810	+	0.986985i	$0.448589\pi$
$102$		0			0
$103$	−8.44326			−0.831939			−0.415970	−	0.909379i	$-0.636558\pi$
$103$	−8.44326			−0.831939			−0.415970	+	0.909379i	$0.636558\pi$
$104$		0			0
$105$			0.314337i			0.0306761i
$106$		0			0
$107$			0.348368i			0.0336780i	0.999858	+	0.0168390i	$0.00536027\pi$
$107$			0.348368i			0.0336780i	−0.999858	+	0.0168390i	$0.994640\pi$
$108$		0			0
$109$		−	5.13969i		−	0.492293i	−0.969233	−	0.246146i	$-0.920836\pi$
$109$		−	5.13969i		−	0.492293i	0.969233	−	0.246146i	$-0.0791644\pi$
$110$		0			0
$111$	−2.35654			−0.223673
$112$		0			0
$113$			15.2868i			1.43806i	0.694977	+	0.719032i	$0.255415\pi$
$113$			15.2868i			1.43806i	−0.694977	+	0.719032i	$0.744585\pi$
$114$		0			0
$115$	−12.8158			−1.19508
$116$		0			0
$117$	−13.1921			−1.21961
$118$		0			0
$119$	1.30201	−	0.255171i	0.119355	−	0.0233915i
$120$		0			0
$121$	10.3755			0.943224
$122$		0			0
$123$	−2.02282			−0.182392
$124$		0			0
$125$		−	9.46820i		−	0.846862i
$126$		0			0
$127$	−0.653722			−0.0580084			−0.0290042	−	0.999579i	$-0.509234\pi$
$127$	−0.653722			−0.0580084			−0.0290042	+	0.999579i	$0.509234\pi$
$128$		0			0
$129$			1.97372i			0.173776i
$130$		0			0
$131$		−	19.2667i		−	1.68334i	−0.539993	−	0.841669i	$-0.681573\pi$
$131$		−	19.2667i		−	1.68334i	0.539993	−	0.841669i	$-0.318427\pi$
$132$		0			0
$133$		−	1.03478i		−	0.0897267i
$134$		0			0
$135$	5.78739			0.498100
$136$		0			0
$137$	4.92887			0.421101			0.210551	−	0.977583i	$-0.432474\pi$
$137$	4.92887			0.421101			0.210551	+	0.977583i	$0.432474\pi$
$138$		0			0
$139$			9.86388i			0.836644i	0.908299	+	0.418322i	$0.137382\pi$
$139$			9.86388i			0.836644i	−0.908299	+	0.418322i	$0.862618\pi$
$140$		0			0
$141$			0.716534i			0.0603430i
$142$		0			0
$143$			3.56467i			0.298093i
$144$		0			0
$145$	26.1781			2.17397
$146$		0			0
$147$		−	1.89328i		−	0.156155i
$148$		0			0
$149$	−21.7676			−1.78327			−0.891634	−	0.452757i	$-0.850440\pi$
$149$	−21.7676			−1.78327			−0.891634	+	0.452757i	$0.850440\pi$
$150$		0			0
$151$	0.217646			0.0177118			0.00885589	−	0.999961i	$-0.497181\pi$
$151$	0.217646			0.0177118			0.00885589	+	0.999961i	$0.497181\pi$
$152$		0			0
$153$	−2.31916	−	11.8335i	−0.187493	−	0.956678i
$154$		0			0
$155$	38.3503			3.08037
$156$		0			0
$157$	20.8393			1.66316			0.831580	−	0.555405i	$-0.187437\pi$
$157$	20.8393			1.66316			0.831580	+	0.555405i	$0.187437\pi$
$158$		0			0
$159$			1.53299i			0.121574i
$160$		0			0
$161$	−1.15900			−0.0913424
$162$		0			0
$163$		−	2.41772i		−	0.189370i	−0.995507	−	0.0946852i	$-0.969816\pi$
$163$		−	2.41772i		−	0.189370i	0.995507	−	0.0946852i	$-0.0301844\pi$
$164$		0			0
$165$		−	0.771969i		−	0.0600977i
$166$		0			0
$167$		−	17.0993i		−	1.32318i	−0.749865	−	0.661591i	$-0.769881\pi$
$167$		−	17.0993i		−	1.32318i	0.749865	−	0.661591i	$-0.230119\pi$
$168$		0			0
$169$	7.34621			0.565093
$170$		0			0
$171$	−9.40471			−0.719196
$172$		0			0
$173$			13.6742i			1.03963i	0.854280	+	0.519813i	$0.173998\pi$
$173$			13.6742i			1.03963i	−0.854280	+	0.519813i	$0.826002\pi$
$174$		0			0
$175$		−	2.46522i		−	0.186353i
$176$		0			0
$177$		−	0.274529i		−	0.0206349i
$178$		0			0
$179$	11.3501			0.848347			0.424174	−	0.905581i	$-0.360565\pi$
$179$	11.3501			0.848347			0.424174	+	0.905581i	$0.360565\pi$
$180$		0			0
$181$			12.3040i			0.914552i	0.889325	+	0.457276i	$0.151175\pi$
$181$			12.3040i			0.914552i	−0.889325	+	0.457276i	$0.848825\pi$
$182$		0			0
$183$	2.19303			0.162114
$184$		0			0
$185$	30.5435			2.24560
$186$		0			0
$187$	−3.19756	+	0.626666i	−0.233828	+	0.0458264i
$188$		0			0
$189$	0.523388			0.0380709
$190$		0			0
$191$	2.12027			0.153418			0.0767088	−	0.997054i	$-0.475559\pi$
$191$	2.12027			0.153418			0.0767088	+	0.997054i	$0.475559\pi$
$192$		0			0
$193$		−	20.3601i		−	1.46555i	−0.680471	−	0.732775i	$-0.738225\pi$
$193$		−	20.3601i		−	1.46555i	0.680471	−	0.732775i	$-0.261775\pi$
$194$		0			0
$195$	−4.40619			−0.315534
$196$		0			0
$197$		−	4.03733i		−	0.287648i	−0.989603	−	0.143824i	$-0.954060\pi$
$197$		−	4.03733i		−	0.287648i	0.989603	−	0.143824i	$-0.0459399\pi$
$198$		0			0
$199$		−	21.2652i		−	1.50745i	−0.657189	−	0.753726i	$-0.728255\pi$
$199$		−	21.2652i		−	1.50745i	0.657189	−	0.753726i	$-0.271745\pi$
$200$		0			0
$201$		−	3.55578i		−	0.250806i
$202$		0			0
$203$	2.36744			0.166162
$204$		0			0
$205$	26.2181			1.83115
$206$		0			0
$207$			10.5337i			0.732146i
$208$		0			0
$209$			2.54128i			0.175784i
$210$		0			0
$211$		−	12.0537i		−	0.829813i	−0.909864	−	0.414906i	$-0.863814\pi$
$211$		−	12.0537i		−	0.829813i	0.909864	−	0.414906i	$-0.136186\pi$
$212$		0			0
$213$	−2.87878			−0.197251
$214$		0			0
$215$		−	25.5817i		−	1.74466i
$216$		0			0
$217$	3.46824			0.235440
$218$		0			0
$219$	1.44887			0.0979058
$220$		0			0
$221$	3.57684	+	18.2508i	0.240605	+	1.22768i
$222$		0			0
$223$	−7.14711			−0.478606			−0.239303	−	0.970945i	$-0.576919\pi$
$223$	−7.14711			−0.478606			−0.239303	+	0.970945i	$0.576919\pi$
$224$		0			0
$225$	−22.4054			−1.49370
$226$		0			0
$227$		−	4.98272i		−	0.330714i	−0.986234	−	0.165357i	$-0.947122\pi$
$227$		−	4.98272i		−	0.330714i	0.986234	−	0.165357i	$-0.0528777\pi$
$228$		0			0
$229$	−15.7120			−1.03828			−0.519138	−	0.854690i	$-0.673747\pi$
$229$	−15.7120			−1.03828			−0.519138	+	0.854690i	$0.673747\pi$
$230$		0			0
$231$		−	0.0698137i		−	0.00459340i
$232$		0			0
$233$		−	1.27726i		−	0.0836762i	−0.999124	−	0.0418381i	$-0.986679\pi$
$233$		−	1.27726i		−	0.0836762i	0.999124	−	0.0418381i	$-0.0133214\pi$
$234$		0			0
$235$		−	9.28712i		−	0.605825i
$236$		0			0
$237$	−0.481812			−0.0312971
$238$		0			0
$239$	−6.09195			−0.394055			−0.197028	−	0.980398i	$-0.563129\pi$
$239$	−6.09195			−0.394055			−0.197028	+	0.980398i	$0.563129\pi$
$240$		0			0
$241$		−	8.81366i		−	0.567738i	−0.958863	−	0.283869i	$-0.908382\pi$
$241$		−	8.81366i		−	0.567738i	0.958863	−	0.283869i	$-0.0916181\pi$
$242$		0			0
$243$		−	7.16557i		−	0.459671i
$244$		0			0
$245$			24.5391i			1.56775i
$246$		0			0
$247$	14.5049			0.922927
$248$		0			0
$249$			3.40301i			0.215657i
$250$		0			0
$251$	−16.3569			−1.03244			−0.516219	−	0.856456i	$-0.672661\pi$
$251$	−16.3569			−1.03244			−0.516219	+	0.856456i	$0.672661\pi$
$252$		0			0
$253$	2.84636			0.178949
$254$		0			0
$255$	−0.774604	−	3.95241i	−0.0485076	−	0.247509i
$256$		0			0
$257$	−19.1374			−1.19376			−0.596879	−	0.802331i	$-0.703593\pi$
$257$	−19.1374			−1.19376			−0.596879	+	0.802331i	$0.703593\pi$
$258$		0			0
$259$	2.76223			0.171637
$260$		0			0
$261$		−	21.5167i		−	1.33185i
$262$		0			0
$263$	−12.0064			−0.740348			−0.370174	−	0.928963i	$-0.620702\pi$
$263$	−12.0064			−0.740348			−0.370174	+	0.928963i	$0.620702\pi$
$264$		0			0
$265$		−	19.8693i		−	1.22056i
$266$		0			0
$267$		−	0.955423i		−	0.0584710i
$268$		0			0
$269$			19.2527i			1.17385i	0.809640	+	0.586927i	$0.199663\pi$
$269$			19.2527i			1.17385i	−0.809640	+	0.586927i	$0.800337\pi$
$270$		0			0
$271$	−27.9752			−1.69937			−0.849685	−	0.527291i	$-0.823208\pi$
$271$	−27.9752			−1.69937			−0.849685	+	0.527291i	$0.823208\pi$
$272$		0			0
$273$	−0.398478			−0.0241170
$274$		0			0
$275$			6.05424i			0.365085i
$276$		0			0
$277$			12.7629i			0.766851i	0.923572	+	0.383425i	$0.125256\pi$
$277$			12.7629i			0.766851i	−0.923572	+	0.383425i	$0.874744\pi$
$278$		0			0
$279$		−	31.5215i		−	1.88714i
$280$		0			0
$281$	26.9407			1.60715			0.803574	−	0.595205i	$-0.202929\pi$
$281$	26.9407			1.60715			0.803574	+	0.595205i	$0.202929\pi$
$282$		0			0
$283$		−	12.7663i		−	0.758879i	−0.925217	−	0.379439i	$-0.876117\pi$
$283$		−	12.7663i		−	0.758879i	0.925217	−	0.379439i	$-0.123883\pi$
$284$		0			0
$285$	−3.14120			−0.186069
$286$		0			0
$287$	2.37106			0.139959
$288$		0			0
$289$	−15.7424	+	6.41695i	−0.926023	+	0.377468i
$290$		0			0
$291$	4.56500			0.267605
$292$		0			0
$293$	0.211883			0.0123784			0.00618918	−	0.999981i	$-0.498030\pi$
$293$	0.211883			0.0123784			0.00618918	+	0.999981i	$0.498030\pi$
$294$		0			0
$295$			3.55822i			0.207168i
$296$		0			0
$297$	−1.28537			−0.0745847
$298$		0			0
$299$		−	16.2463i		−	0.939545i
$300$		0			0
$301$		−	2.31350i		−	0.133348i
$302$		0			0
$303$		−	0.887344i		−	0.0509766i
$304$		0			0
$305$	−28.4243			−1.62757
$306$		0			0
$307$	14.6564			0.836482			0.418241	−	0.908336i	$-0.362647\pi$
$307$	14.6564			0.836482			0.418241	+	0.908336i	$0.362647\pi$
$308$		0			0
$309$			2.31792i			0.131862i
$310$		0			0
$311$		−	4.65608i		−	0.264022i	−0.991248	−	0.132011i	$-0.957857\pi$
$311$		−	4.65608i		−	0.264022i	0.991248	−	0.132011i	$-0.0421434\pi$
$312$		0			0
$313$		−	6.83132i		−	0.386129i	−0.981186	−	0.193065i	$-0.938157\pi$
$313$		−	6.83132i		−	0.386129i	0.981186	−	0.193065i	$-0.0618427\pi$
$314$		0			0
$315$	−3.34871			−0.188679
$316$		0			0
$317$		−	12.2162i		−	0.686131i	−0.939311	−	0.343066i	$-0.888535\pi$
$317$		−	12.2162i		−	0.686131i	0.939311	−	0.343066i	$-0.111465\pi$
$318$		0			0
$319$	−5.81411			−0.325528
$320$		0			0
$321$	0.0956371			0.00533794
$322$		0			0
$323$	2.54995	+	13.0111i	0.141883	+	0.723957i
$324$		0			0
$325$	34.5560			1.91682
$326$		0			0
$327$	−1.41099			−0.0780282
$328$		0			0
$329$		−	0.839888i		−	0.0463046i
$330$		0			0
$331$	−12.8496			−0.706277			−0.353139	−	0.935571i	$-0.614886\pi$
$331$	−12.8496			−0.706277			−0.353139	+	0.935571i	$0.614886\pi$
$332$		0			0
$333$		−	25.1048i		−	1.37574i
$334$		0			0
$335$			46.0871i			2.51801i
$336$		0			0
$337$		−	31.5627i		−	1.71933i	−0.510856	−	0.859666i	$-0.670672\pi$
$337$		−	31.5627i		−	1.71933i	0.510856	−	0.859666i	$-0.329328\pi$
$338$		0			0
$339$	4.19668			0.227933
$340$		0			0
$341$	−8.51753			−0.461250
$342$		0			0
$343$			4.47175i			0.241452i
$344$		0			0
$345$			3.51830i			0.189419i
$346$		0			0
$347$			5.92731i			0.318195i	0.987263	+	0.159097i	$0.0508584\pi$
$347$			5.92731i			0.318195i	−0.987263	+	0.159097i	$0.949142\pi$
$348$		0			0
$349$	−25.5288			−1.36653			−0.683264	−	0.730171i	$-0.739440\pi$
$349$	−25.5288			−1.36653			−0.683264	+	0.730171i	$0.739440\pi$
$350$		0			0
$351$			7.33655i			0.391596i
$352$		0			0
$353$	29.9796			1.59565			0.797827	−	0.602887i	$-0.205983\pi$
$353$	29.9796			1.59565			0.797827	+	0.602887i	$0.205983\pi$
$354$		0			0
$355$	37.3123			1.98033
$356$		0			0
$357$	−0.0700520	−	0.357440i	−0.00370755	−	0.0189177i
$358$		0			0
$359$	21.9472			1.15833			0.579165	−	0.815210i	$-0.303379\pi$
$359$	21.9472			1.15833			0.579165	+	0.815210i	$0.303379\pi$
$360$		0			0
$361$	−8.65935			−0.455755
$362$		0			0
$363$		−	2.84837i		−	0.149501i
$364$		0			0
$365$	−18.7791			−0.982943
$366$		0			0
$367$		−	9.57717i		−	0.499924i	−0.968256	−	0.249962i	$-0.919582\pi$
$367$		−	9.57717i		−	0.499924i	0.968256	−	0.249962i	$-0.0804181\pi$
$368$		0			0
$369$		−	21.5497i		−	1.12183i
$370$		0			0
$371$		−	1.79690i		−	0.0932904i
$372$		0			0
$373$	−15.2926			−0.791823			−0.395912	−	0.918289i	$-0.629571\pi$
$373$	−15.2926			−0.791823			−0.395912	+	0.918289i	$0.629571\pi$
$374$		0			0
$375$	−2.59930			−0.134227
$376$		0			0
$377$			33.1854i			1.70913i
$378$		0			0
$379$		−	17.7427i		−	0.911382i	−0.890138	−	0.455691i	$-0.849392\pi$
$379$		−	17.7427i		−	0.911382i	0.890138	−	0.455691i	$-0.150608\pi$
$380$		0			0
$381$			0.179466i			0.00919431i
$382$		0			0
$383$	−27.9274			−1.42702			−0.713511	−	0.700644i	$-0.752896\pi$
$383$	−27.9274			−1.42702			−0.713511	+	0.700644i	$0.752896\pi$
$384$		0			0
$385$			0.904867i			0.0461163i
$386$		0			0
$387$	−21.0266			−1.06884
$388$		0			0
$389$	−21.1403			−1.07186			−0.535928	−	0.844263i	$-0.680038\pi$
$389$	−21.1403			−1.07186			−0.535928	+	0.844263i	$0.680038\pi$
$390$		0			0
$391$	14.5731	−	2.85608i	0.736993	−	0.144438i
$392$		0			0
$393$	−5.28927			−0.266808
$394$		0			0
$395$	6.24485			0.314213
$396$		0			0
$397$			3.31179i			0.166214i	0.996541	+	0.0831070i	$0.0264843\pi$
$397$			3.31179i			0.166214i	−0.996541	+	0.0831070i	$0.973516\pi$
$398$		0			0
$399$	−0.284077			−0.0142216
$400$		0			0
$401$		−	15.0297i		−	0.750549i	−0.926914	−	0.375275i	$-0.877548\pi$
$401$		−	15.0297i		−	0.750549i	0.926914	−	0.375275i	$-0.122452\pi$
$402$		0			0
$403$			48.6158i			2.42173i
$404$		0			0
$405$			29.6307i			1.47236i
$406$		0			0
$407$	−6.78366			−0.336253
$408$		0			0
$409$	17.1284			0.846946			0.423473	−	0.905909i	$-0.360811\pi$
$409$	17.1284			0.846946			0.423473	+	0.905909i	$0.360811\pi$
$410$		0			0
$411$		−	1.35312i		−	0.0667444i
$412$		0			0
$413$			0.321791i			0.0158343i
$414$		0			0
$415$		−	44.1070i		−	2.16513i
$416$		0			0
$417$	2.70792			0.132608
$418$		0			0
$419$			11.2983i			0.551957i	0.961164	+	0.275978i	$0.0890018\pi$
$419$			11.2983i			0.551957i	−0.961164	+	0.275978i	$0.910998\pi$
$420$		0			0
$421$	8.24782			0.401974			0.200987	−	0.979594i	$-0.435585\pi$
$421$	8.24782			0.401974			0.200987	+	0.979594i	$0.435585\pi$
$422$		0			0
$423$	−7.63342			−0.371150
$424$		0			0
$425$	6.07492	+	30.9972i	0.294677	+	1.50358i
$426$		0			0
$427$	−2.57057			−0.124399
$428$		0			0
$429$	0.978607			0.0472476
$430$		0			0
$431$			21.2735i			1.02471i	0.858774	+	0.512355i	$0.171227\pi$
$431$			21.2735i			1.02471i	−0.858774	+	0.512355i	$0.828773\pi$
$432$		0			0
$433$	−17.4919			−0.840605			−0.420303	−	0.907384i	$-0.638076\pi$
$433$	−17.4919			−0.840605			−0.420303	+	0.907384i	$0.638076\pi$
$434$		0			0
$435$		−	7.18665i		−	0.344574i
$436$		0			0
$437$		−	11.5820i		−	0.554044i
$438$		0			0
$439$		−	5.40966i		−	0.258189i	−0.991632	−	0.129094i	$-0.958793\pi$
$439$		−	5.40966i		−	0.258189i	0.991632	−	0.129094i	$-0.0412071\pi$
$440$		0			0
$441$	20.1696			0.960457
$442$		0			0
$443$	−23.9527			−1.13803			−0.569014	−	0.822328i	$-0.692675\pi$
$443$	−23.9527			−1.13803			−0.569014	+	0.822328i	$0.692675\pi$
$444$		0			0
$445$			12.3834i			0.587030i
$446$		0			0
$447$			5.97583i			0.282647i
$448$		0			0
$449$		−	19.8950i		−	0.938902i	−0.882958	−	0.469451i	$-0.844452\pi$
$449$		−	19.8950i		−	0.938902i	0.882958	−	0.469451i	$-0.155548\pi$
$450$		0			0
$451$	−5.82300			−0.274194
$452$		0			0
$453$		−	0.0597502i		−	0.00280731i
$454$		0			0
$455$	5.16474			0.242127
$456$		0			0
$457$	5.32121			0.248916			0.124458	−	0.992225i	$-0.460281\pi$
$457$	5.32121			0.248916			0.124458	+	0.992225i	$0.460281\pi$
$458$		0			0
$459$	−6.58098	+	1.28976i	−0.307174	+	0.0602008i
$460$		0			0
$461$	−16.8450			−0.784549			−0.392275	−	0.919848i	$-0.628312\pi$
$461$	−16.8450			−0.784549			−0.392275	+	0.919848i	$0.628312\pi$
$462$		0			0
$463$	33.3140			1.54824			0.774118	−	0.633042i	$-0.218194\pi$
$463$	33.3140			1.54824			0.774118	+	0.633042i	$0.218194\pi$
$464$		0			0
$465$		−	10.5283i		−	0.488237i
$466$		0			0
$467$	−14.1098			−0.652923			−0.326461	−	0.945211i	$-0.605856\pi$
$467$	−14.1098			−0.652923			−0.326461	+	0.945211i	$0.605856\pi$
$468$		0			0
$469$			4.16793i			0.192457i
$470$		0			0
$471$		−	5.72101i		−	0.263610i
$472$		0			0
$473$			5.68165i			0.261243i
$474$		0			0
$475$	24.6352			1.13034
$476$		0			0
$477$	−16.3313			−0.747760
$478$		0			0
$479$			16.4199i			0.750244i	0.926976	+	0.375122i	$0.122399\pi$
$479$			16.4199i			0.750244i	−0.926976	+	0.375122i	$0.877601\pi$
$480$		0			0
$481$			38.7193i			1.76545i
$482$		0			0
$483$			0.318181i			0.0144777i
$484$		0			0
$485$	−59.1678			−2.68667
$486$		0			0
$487$			36.1869i			1.63978i	0.572518	+	0.819892i	$0.305967\pi$
$487$			36.1869i			1.63978i	−0.572518	+	0.819892i	$0.694033\pi$
$488$		0			0
$489$	−0.663735			−0.0300151
$490$		0			0
$491$	−43.0710			−1.94377			−0.971884	−	0.235462i	$-0.924340\pi$
$491$	−43.0710			−1.94377			−0.971884	+	0.235462i	$0.924340\pi$
$492$		0			0
$493$	−29.7677	+	5.83396i	−1.34067	+	0.262748i
$494$		0			0
$495$	8.22399			0.369641
$496$		0			0
$497$	3.37437			0.151361
$498$		0			0
$499$		−	9.72241i		−	0.435235i	−0.976034	−	0.217617i	$-0.930172\pi$
$499$		−	9.72241i		−	0.435235i	0.976034	−	0.217617i	$-0.0698285\pi$
$500$		0			0
$501$	−4.69426			−0.209724
$502$		0			0
$503$		−	9.41513i		−	0.419800i	−0.977723	−	0.209900i	$-0.932686\pi$
$503$		−	9.41513i		−	0.419800i	0.977723	−	0.209900i	$-0.0673139\pi$
$504$		0			0
$505$			11.5010i			0.511789i
$506$		0			0
$507$		−	2.01675i		−	0.0895669i
$508$		0			0
$509$	−16.5250			−0.732460			−0.366230	−	0.930524i	$-0.619352\pi$
$509$	−16.5250			−0.732460			−0.366230	+	0.930524i	$0.619352\pi$
$510$		0			0
$511$	−1.69830			−0.0751286
$512$		0			0
$513$			5.23027i			0.230922i
$514$		0			0
$515$		−	30.0430i		−	1.32385i
$516$		0			0
$517$			2.06265i			0.0907153i
$518$		0			0
$519$	3.75396			0.164780
$520$		0			0
$521$		−	8.97192i		−	0.393067i	−0.980497	−	0.196533i	$-0.937032\pi$
$521$		−	8.97192i		−	0.393067i	0.980497	−	0.196533i	$-0.0629684\pi$
$522$		0			0
$523$	11.7801			0.515106			0.257553	−	0.966264i	$-0.417084\pi$
$523$	11.7801			0.515106			0.257553	+	0.966264i	$0.417084\pi$
$524$		0			0
$525$	−0.676775			−0.0295369
$526$		0			0
$527$	−43.6090	+	8.54661i	−1.89964	+	0.372296i
$528$		0			0
$529$	10.0275			0.435979
$530$		0			0
$531$	2.92463			0.126918
$532$		0			0
$533$			33.2361i			1.43962i
$534$		0			0
$535$	−1.23957			−0.0535912
$536$		0			0
$537$		−	3.11594i		−	0.134463i
$538$		0			0
$539$		−	5.45009i		−	0.234752i
$540$		0			0
$541$			5.33311i			0.229288i	0.993407	+	0.114644i	$0.0365728\pi$
$541$			5.33311i			0.229288i	−0.993407	+	0.114644i	$0.963427\pi$
$542$		0			0
$543$	3.37782			0.144956
$544$		0			0
$545$	18.2881			0.783378
$546$		0			0
$547$		−	14.3523i		−	0.613661i	−0.951764	−	0.306830i	$-0.900732\pi$
$547$		−	14.3523i		−	0.613661i	0.951764	−	0.306830i	$-0.0992685\pi$
$548$		0			0
$549$			23.3630i			0.997107i
$550$		0			0
$551$			23.6581i			1.00787i
$552$		0			0
$553$	0.564759			0.0240160
$554$		0			0
$555$		−	8.38509i		−	0.355927i
$556$		0			0
$557$	−22.6940			−0.961574			−0.480787	−	0.876838i	$-0.659649\pi$
$557$	−22.6940			−0.961574			−0.480787	+	0.876838i	$0.659649\pi$
$558$		0			0
$559$	32.4294			1.37162
$560$		0			0
$561$	0.172038	+	0.877823i	0.00726346	+	0.0370617i
$562$		0			0
$563$	32.6791			1.37726			0.688630	−	0.725113i	$-0.258213\pi$
$563$	32.6791			1.37726			0.688630	+	0.725113i	$0.258213\pi$
$564$		0			0
$565$	−54.3940			−2.28837
$566$		0			0
$567$			2.67968i			0.112536i
$568$		0			0
$569$	39.5002			1.65594			0.827968	−	0.560775i	$-0.189497\pi$
$569$	39.5002			1.65594			0.827968	+	0.560775i	$0.189497\pi$
$570$		0			0
$571$		−	21.3218i		−	0.892289i	−0.894961	−	0.446145i	$-0.852797\pi$
$571$		−	21.3218i		−	0.892289i	0.894961	−	0.446145i	$-0.147203\pi$
$572$		0			0
$573$		−	0.582077i		−	0.0243166i
$574$		0			0
$575$		−	27.5926i		−	1.15069i
$576$		0			0
$577$	28.5478			1.18846			0.594230	−	0.804295i	$-0.297457\pi$
$577$	28.5478			1.18846			0.594230	+	0.804295i	$0.297457\pi$
$578$		0			0
$579$	−5.58944			−0.232289
$580$		0			0
$581$		−	3.98886i		−	0.165486i
$582$		0			0
$583$			4.41294i			0.182765i
$584$		0			0
$585$		−	46.9403i		−	1.94074i
$586$		0			0
$587$	22.7008			0.936964			0.468482	−	0.883473i	$-0.344801\pi$
$587$	22.7008			0.936964			0.468482	+	0.883473i	$0.344801\pi$
$588$		0			0
$589$			34.6585i			1.42808i
$590$		0			0
$591$	−1.10837			−0.0455921
$592$		0			0
$593$	−0.710075			−0.0291593			−0.0145796	−	0.999894i	$-0.504641\pi$
$593$	−0.710075			−0.0291593			−0.0145796	+	0.999894i	$0.504641\pi$
$594$		0			0
$595$	0.907956	+	4.63284i	0.0372226	+	0.189928i
$596$		0			0
$597$	−5.83793			−0.238931
$598$		0			0
$599$	−37.7174			−1.54109			−0.770545	−	0.637385i	$-0.780016\pi$
$599$	−37.7174			−1.54109			−0.770545	+	0.637385i	$0.780016\pi$
$600$		0			0
$601$		−	6.75497i		−	0.275541i	−0.990464	−	0.137770i	$-0.956006\pi$
$601$		−	6.75497i		−	0.275541i	0.990464	−	0.137770i	$-0.0439936\pi$
$602$		0			0
$603$	37.8807			1.54262
$604$		0			0
$605$			36.9182i			1.50094i
$606$		0			0
$607$		−	2.84747i		−	0.115575i	−0.998329	−	0.0577875i	$-0.981595\pi$
$607$		−	2.84747i		−	0.115575i	0.998329	−	0.0577875i	$-0.0184046\pi$
$608$		0			0
$609$		−	0.649931i		−	0.0263365i
$610$		0			0
$611$	11.7731			0.476287
$612$		0			0
$613$	30.2798			1.22299			0.611494	−	0.791249i	$-0.290569\pi$
$613$	30.2798			1.22299			0.611494	+	0.791249i	$0.290569\pi$
$614$		0			0
$615$		−	7.19764i		−	0.290237i
$616$		0			0
$617$			18.3190i			0.737495i	0.929530	+	0.368747i	$0.120213\pi$
$617$			18.3190i			0.737495i	−0.929530	+	0.368747i	$0.879787\pi$
$618$		0			0
$619$		−	25.8192i		−	1.03776i	−0.854847	−	0.518880i	$-0.826349\pi$
$619$		−	25.8192i		−	1.03776i	0.854847	−	0.518880i	$-0.173651\pi$
$620$		0			0
$621$	5.85816			0.235080
$622$		0			0
$623$			1.11990i			0.0448680i
$624$		0			0
$625$	−4.61472			−0.184589
$626$		0			0
$627$	0.697655			0.0278617
$628$		0			0
$629$	−34.7317	+	6.80682i	−1.38484	+	0.271406i
$630$		0			0
$631$	15.2284			0.606234			0.303117	−	0.952953i	$-0.401973\pi$
$631$	15.2284			0.606234			0.303117	+	0.952953i	$0.401973\pi$
$632$		0			0
$633$	−3.30910			−0.131525
$634$		0			0
$635$		−	2.32609i		−	0.0923079i
$636$		0			0
$637$	−31.1077			−1.23253
$638$		0			0
$639$		−	30.6684i		−	1.21322i
$640$		0			0
$641$		−	26.5774i		−	1.04974i	−0.851181	−	0.524872i	$-0.824113\pi$
$641$		−	26.5774i		−	1.04974i	0.851181	−	0.524872i	$-0.175887\pi$
$642$		0			0
$643$			21.7428i			0.857452i	0.903435	+	0.428726i	$0.141037\pi$
$643$			21.7428i			0.857452i	−0.903435	+	0.428726i	$0.858963\pi$
$644$		0			0
$645$	−7.02293			−0.276528
$646$		0			0
$647$	6.03816			0.237384			0.118692	−	0.992931i	$-0.462130\pi$
$647$	6.03816			0.237384			0.118692	+	0.992931i	$0.462130\pi$
$648$		0			0
$649$		−	0.790275i		−	0.0310210i
$650$		0			0
$651$		−	0.952134i		−	0.0373171i
$652$		0			0
$653$		−	3.80240i		−	0.148799i	−0.997229	−	0.0743997i	$-0.976296\pi$
$653$		−	3.80240i		−	0.148799i	0.997229	−	0.0743997i	$-0.0237041\pi$
$654$		0			0
$655$	68.5552			2.67867
$656$		0			0
$657$			15.4352i			0.602186i
$658$		0			0
$659$	48.9863			1.90824			0.954118	−	0.299429i	$-0.0967963\pi$
$659$	48.9863			1.90824			0.954118	+	0.299429i	$0.0967963\pi$
$660$		0			0
$661$	39.4094			1.53285			0.766424	−	0.642335i	$-0.222034\pi$
$661$	39.4094			1.53285			0.766424	+	0.642335i	$0.222034\pi$
$662$		0			0
$663$	5.01038	−	0.981949i	0.194587	−	0.0381357i
$664$		0			0
$665$	3.68197			0.142781
$666$		0			0
$667$	26.4982			1.02601
$668$		0			0
$669$			1.96209i			0.0758589i
$670$		0			0
$671$	6.31298			0.243710
$672$		0			0
$673$			17.6100i			0.678817i	0.940639	+	0.339408i	$0.110227\pi$
$673$			17.6100i			0.678817i	−0.940639	+	0.339408i	$0.889773\pi$
$674$		0			0
$675$			12.4604i			0.479601i
$676$		0			0
$677$		−	23.9415i		−	0.920146i	−0.887881	−	0.460073i	$-0.847823\pi$
$677$		−	23.9415i		−	0.920146i	0.887881	−	0.460073i	$-0.152177\pi$
$678$		0			0
$679$	−5.35089			−0.205348
$680$		0			0
$681$	−1.36790			−0.0524181
$682$		0			0
$683$			45.6195i			1.74558i	0.488095	+	0.872790i	$0.337692\pi$
$683$			45.6195i			1.74558i	−0.488095	+	0.872790i	$0.662308\pi$
$684$		0			0
$685$			17.5380i			0.670092i
$686$		0			0
$687$			4.31339i			0.164566i
$688$		0			0
$689$	25.1879			0.959583
$690$		0			0
$691$		−	35.6406i		−	1.35583i	−0.735139	−	0.677916i	$-0.762883\pi$
$691$		−	35.6406i		−	1.35583i	0.735139	−	0.677916i	$-0.237117\pi$
$692$		0			0
$693$	0.743744			0.0282525
$694$		0			0
$695$	−35.0979			−1.33134
$696$		0			0
$697$	−29.8132	+	5.84288i	−1.12926	+	0.221315i
$698$		0			0
$699$	−0.350646			−0.0132626
$700$		0			0
$701$	15.2467			0.575859			0.287930	−	0.957652i	$-0.407033\pi$
$701$	15.2467			0.575859			0.287930	+	0.957652i	$0.407033\pi$
$702$		0			0
$703$			27.6032i			1.04108i
$704$		0			0
$705$	−2.54959			−0.0960229
$706$		0			0
$707$			1.04010i			0.0391172i
$708$		0			0
$709$		−	23.0171i		−	0.864426i	−0.901772	−	0.432213i	$-0.857733\pi$
$709$		−	23.0171i		−	0.864426i	0.901772	−	0.432213i	$-0.142267\pi$
$710$		0			0
$711$		−	5.13288i		−	0.192498i
$712$		0			0
$713$	38.8193			1.45379
$714$		0			0
$715$	−12.6839			−0.474351
$716$		0			0
$717$			1.67242i			0.0624576i
$718$		0			0
$719$			4.70328i			0.175403i	0.996147	+	0.0877013i	$0.0279521\pi$
$719$			4.70328i			0.175403i	−0.996147	+	0.0877013i	$0.972048\pi$
$720$		0			0
$721$		−	2.71696i		−	0.101185i
$722$		0			0
$723$	−2.41961			−0.0899862
$724$		0			0
$725$			56.3621i			2.09323i
$726$		0			0
$727$	36.0475			1.33693			0.668464	−	0.743745i	$-0.266952\pi$
$727$	36.0475			1.33693			0.668464	+	0.743745i	$0.266952\pi$
$728$		0			0
$729$	23.0150			0.852407
$730$		0			0
$731$	5.70105	+	29.0895i	0.210861	+	1.07592i
$732$		0			0
$733$	10.7704			0.397815			0.198908	−	0.980018i	$-0.436261\pi$
$733$	10.7704			0.397815			0.198908	+	0.980018i	$0.436261\pi$
$734$		0			0
$735$	6.73670			0.248487
$736$		0			0
$737$		−	10.2359i		−	0.377043i
$738$		0			0
$739$	−23.3002			−0.857111			−0.428555	−	0.903515i	$-0.640977\pi$
$739$	−23.3002			−0.857111			−0.428555	+	0.903515i	$0.640977\pi$
$740$		0			0
$741$		−	3.98203i		−	0.146283i
$742$		0			0
$743$			38.1044i			1.39791i	0.715164	+	0.698957i	$0.246352\pi$
$743$			38.1044i			1.39791i	−0.715164	+	0.698957i	$0.753648\pi$
$744$		0			0
$745$		−	77.4538i		−	2.83769i
$746$		0			0
$747$	−36.2532			−1.32643
$748$		0			0
$749$	−0.112102			−0.00409610
$750$		0			0
$751$		−	32.2283i		−	1.17603i	−0.808851	−	0.588013i	$-0.799910\pi$
$751$		−	32.2283i		−	1.17603i	0.808851	−	0.588013i	$-0.200090\pi$
$752$		0			0
$753$			4.49045i			0.163641i
$754$		0			0
$755$			0.774432i			0.0281845i
$756$		0			0
$757$	−16.2790			−0.591672			−0.295836	−	0.955239i	$-0.595598\pi$
$757$	−16.2790			−0.591672			−0.295836	+	0.955239i	$0.595598\pi$
$758$		0			0
$759$		−	0.781408i		−	0.0283633i
$760$		0			0
$761$	−7.64226			−0.277032			−0.138516	−	0.990360i	$-0.544233\pi$
$761$	−7.64226			−0.277032			−0.138516	+	0.990360i	$0.544233\pi$
$762$		0			0
$763$	1.65390			0.0598753
$764$		0			0
$765$	42.1061	−	8.25207i	1.52235	−	0.298354i
$766$		0			0
$767$	−4.51068			−0.162871
$768$		0			0
$769$	−6.32640			−0.228136			−0.114068	−	0.993473i	$-0.536388\pi$
$769$	−6.32640			−0.228136			−0.114068	+	0.993473i	$0.536388\pi$
$770$		0			0
$771$			5.25378i			0.189210i
$772$		0			0
$773$	−12.1610			−0.437402			−0.218701	−	0.975792i	$-0.570182\pi$
$773$	−12.1610			−0.437402			−0.218701	+	0.975792i	$0.570182\pi$
$774$		0			0
$775$			82.5691i			2.96597i
$776$		0			0
$777$		−	0.758313i		−	0.0272043i
$778$		0			0
$779$			23.6942i			0.848934i
$780$		0			0
$781$	−8.28701			−0.296532
$782$		0			0
$783$	−11.9662			−0.427636
$784$		0			0
$785$			74.1509i			2.64656i
$786$		0			0
$787$			36.3524i			1.29582i	0.761715	+	0.647912i	$0.224358\pi$
$787$			36.3524i			1.29582i	−0.761715	+	0.647912i	$0.775642\pi$
$788$		0			0
$789$			3.29611i			0.117345i
$790$		0			0
$791$	−4.91917			−0.174905
$792$		0			0
$793$		−	36.0328i		−	1.27956i
$794$		0			0
$795$	−5.45471			−0.193459
$796$		0			0
$797$	32.6574			1.15678			0.578392	−	0.815759i	$-0.303680\pi$
$797$	32.6574			1.15678			0.578392	+	0.815759i	$0.303680\pi$
$798$		0			0
$799$	2.06969	+	10.5606i	0.0732205	+	0.373607i
$800$		0			0
$801$	10.1784			0.359635
$802$		0			0
$803$	4.17080			0.147185
$804$		0			0
$805$		−	4.12399i		−	0.145352i
$806$		0			0
$807$	5.28542			0.186055
$808$		0			0
$809$			40.3111i			1.41726i	0.705578	+	0.708632i	$0.250687\pi$
$809$			40.3111i			1.41726i	−0.705578	+	0.708632i	$0.749313\pi$
$810$		0			0
$811$			9.87855i			0.346883i	0.984844	+	0.173441i	$0.0554887\pi$
$811$			9.87855i			0.346883i	−0.984844	+	0.173441i	$0.944511\pi$
$812$		0			0
$813$			7.68000i			0.269349i
$814$		0			0
$815$	8.60278			0.301342
$816$		0			0
$817$	23.1191			0.808834
$818$		0			0
$819$		−	4.24509i		−	0.148335i
$820$		0			0
$821$			14.3857i			0.502066i	0.967979	+	0.251033i	$0.0807703\pi$
$821$			14.3857i			0.502066i	−0.967979	+	0.251033i	$0.919230\pi$
$822$		0			0
$823$		−	46.2245i		−	1.61128i	−0.592403	−	0.805642i	$-0.701821\pi$
$823$		−	46.2245i		−	1.61128i	0.592403	−	0.805642i	$-0.298179\pi$
$824$		0			0
$825$	1.66207			0.0578658
$826$		0			0
$827$		−	45.8632i		−	1.59482i	−0.603438	−	0.797410i	$-0.706203\pi$
$827$		−	45.8632i		−	1.59482i	0.603438	−	0.797410i	$-0.293797\pi$
$828$		0			0
$829$	−31.4416			−1.09201			−0.546007	−	0.837781i	$-0.683853\pi$
$829$	−31.4416			−1.09201			−0.546007	+	0.837781i	$0.683853\pi$
$830$		0			0
$831$	3.50380			0.121545
$832$		0			0
$833$	−5.46870	−	27.9040i	−0.189479	−	0.966815i
$834$		0			0
$835$	60.8431			2.10556
$836$		0			0
$837$	−17.5302			−0.605931
$838$		0			0
$839$		−	26.5384i		−	0.916208i	−0.888898	−	0.458104i	$-0.848529\pi$
$839$		−	26.5384i		−	0.916208i	0.888898	−	0.458104i	$-0.151471\pi$
$840$		0			0
$841$	−25.1265			−0.866431
$842$		0			0
$843$		−	7.39601i		−	0.254732i
$844$		0			0
$845$			26.1394i			0.899224i
$846$		0			0
$847$			3.33873i			0.114720i
$848$		0			0
$849$	−3.50473			−0.120282
$850$		0			0
$851$	30.9170			1.05982
$852$		0			0
$853$			1.46875i			0.0502890i	0.999684	+	0.0251445i	$0.00800459\pi$
$853$			1.46875i			0.0502890i	−0.999684	+	0.0251445i	$0.991995\pi$
$854$		0			0
$855$		−	33.4640i		−	1.14445i
$856$		0			0
$857$		−	11.7343i		−	0.400835i	−0.979711	−	0.200418i	$-0.935770\pi$
$857$		−	11.7343i		−	0.400835i	0.979711	−	0.200418i	$-0.0642299\pi$
$858$		0			0
$859$	−37.3648			−1.27487			−0.637436	−	0.770503i	$-0.720005\pi$
$859$	−37.3648			−1.27487			−0.637436	+	0.770503i	$0.720005\pi$
$860$		0			0
$861$		−	0.650925i		−	0.0221835i
$862$		0			0
$863$	21.2450			0.723189			0.361595	−	0.932335i	$-0.382232\pi$
$863$	21.2450			0.723189			0.361595	+	0.932335i	$0.382232\pi$
$864$		0			0
$865$	−48.6557			−1.65434
$866$		0			0
$867$	1.76164	+	4.32175i	0.0598284	+	0.146774i
$868$		0			0
$869$	−1.38697			−0.0470498
$870$		0			0
$871$	−58.4236			−1.97961
$872$		0			0
$873$			48.6322i			1.64595i
$874$		0			0
$875$	3.04678			0.103000
$876$		0			0
$877$			18.3508i			0.619663i	0.950791	+	0.309832i	$0.100273\pi$
$877$			18.3508i			0.619663i	−0.950791	+	0.309832i	$0.899727\pi$
$878$		0			0
$879$		−	0.0581682i		−	0.00196197i
$880$		0			0
$881$			0.293171i			0.00987719i	0.999988	+	0.00493860i	$0.00157201\pi$
$881$			0.293171i			0.00987719i	−0.999988	+	0.00493860i	$0.998428\pi$
$882$		0			0
$883$	37.3029			1.25534			0.627671	−	0.778478i	$-0.284008\pi$
$883$	37.3029			1.25534			0.627671	+	0.778478i	$0.284008\pi$
$884$		0			0
$885$	0.976836			0.0328360
$886$		0			0
$887$			16.6582i			0.559328i	0.960098	+	0.279664i	$0.0902230\pi$
$887$			16.6582i			0.559328i	−0.960098	+	0.279664i	$0.909777\pi$
$888$		0			0
$889$		−	0.210362i		−	0.00705530i
$890$		0			0
$891$		−	6.58092i		−	0.220469i
$892$		0			0
$893$	8.39309			0.280864
$894$		0			0
$895$			40.3862i			1.34996i
$896$		0			0
$897$	−4.46007			−0.148917
$898$		0			0
$899$	−79.2941			−2.64461
$900$		0			0
$901$	4.42801	+	22.5939i	0.147518	+	0.752711i
$902$		0			0
$903$	−0.635125			−0.0211356
$904$		0			0
$905$	−43.7805			−1.45531
$906$		0			0
$907$			2.76660i			0.0918634i	0.998945	+	0.0459317i	$0.0146257\pi$
$907$			2.76660i			0.0918634i	−0.998945	+	0.0459317i	$0.985374\pi$
$908$		0			0
$909$	9.45311			0.313540
$910$		0			0
$911$		−	36.8402i		−	1.22057i	−0.792182	−	0.610285i	$-0.791055\pi$
$911$		−	36.8402i		−	1.22057i	0.792182	−	0.610285i	$-0.208945\pi$
$912$		0			0
$913$			9.79609i			0.324203i
$914$		0			0
$915$			7.80329i			0.257969i
$916$		0			0
$917$	6.19985			0.204737
$918$		0			0
$919$	2.41604			0.0796978			0.0398489	−	0.999206i	$-0.487312\pi$
$919$	2.41604			0.0796978			0.0398489	+	0.999206i	$0.487312\pi$
$920$		0			0
$921$		−	4.02360i		−	0.132582i
$922$		0			0
$923$			47.3000i			1.55690i
$924$		0			0
$925$			65.7609i			2.16221i
$926$		0			0
$927$	−24.6934			−0.811039
$928$		0			0
$929$			50.3555i			1.65211i	0.563588	+	0.826056i	$0.309420\pi$
$929$			50.3555i			1.65211i	−0.563588	+	0.826056i	$0.690580\pi$
$930$		0			0
$931$	−22.1768			−0.726816
$932$		0			0
$933$	−1.27823			−0.0418473
$934$		0			0
$935$	−2.22982	−	11.3776i	−0.0729228	−	0.372088i
$936$		0			0
$937$	−19.8173			−0.647402			−0.323701	−	0.946159i	$-0.604927\pi$
$937$	−19.8173			−0.647402			−0.323701	+	0.946159i	$0.604927\pi$
$938$		0			0
$939$	−1.87540			−0.0612013
$940$		0			0
$941$			33.2523i			1.08399i	0.840380	+	0.541997i	$0.182332\pi$
$941$			33.2523i			1.08399i	−0.840380	+	0.541997i	$0.817668\pi$
$942$		0			0
$943$	26.5387			0.864220
$944$		0			0
$945$			1.86233i			0.0605816i
$946$		0			0
$947$			5.11283i			0.166145i	0.996544	+	0.0830724i	$0.0264733\pi$
$947$			5.11283i			0.166145i	−0.996544	+	0.0830724i	$0.973527\pi$
$948$		0			0
$949$		−	23.8058i		−	0.772770i
$950$		0			0
$951$	−3.35371			−0.108752
$952$		0			0
$953$	−23.0256			−0.745873			−0.372936	−	0.927857i	$-0.621649\pi$
$953$	−23.0256			−0.745873			−0.372936	+	0.927857i	$0.621649\pi$
$954$		0			0
$955$			7.54440i			0.244131i
$956$		0			0
$957$			1.59614i			0.0515960i
$958$		0			0
$959$			1.58606i			0.0512167i
$960$		0			0
$961$	−85.1640			−2.74723
$962$		0			0
$963$			1.01885i			0.0328319i
$964$		0			0
$965$	72.4456			2.33211
$966$		0			0
$967$	−40.9600			−1.31718			−0.658592	−	0.752500i	$-0.728848\pi$
$967$	−40.9600			−1.31718			−0.658592	+	0.752500i	$0.728848\pi$
$968$		0			0
$969$	3.57193	−	0.700037i	0.114747	−	0.0224884i
$970$		0			0
$971$	−55.7898			−1.79038			−0.895190	−	0.445684i	$-0.852960\pi$
$971$	−55.7898			−1.79038			−0.895190	+	0.445684i	$0.852960\pi$
$972$		0			0
$973$	−3.17411			−0.101757
$974$		0			0
$975$		−	9.48664i		−	0.303816i
$976$		0			0
$977$	43.4019			1.38855			0.694275	−	0.719710i	$-0.255725\pi$
$977$	43.4019			1.38855			0.694275	+	0.719710i	$0.255725\pi$
$978$		0			0
$979$		−	2.75033i		−	0.0879010i
$980$		0			0
$981$		−	15.0317i		−	0.479925i
$982$		0			0
$983$		−	33.2906i		−	1.06180i	−0.847433	−	0.530902i	$-0.821853\pi$
$983$		−	33.2906i		−	1.06180i	0.847433	−	0.530902i	$-0.178147\pi$
$984$		0			0
$985$	14.3657			0.457730
$986$		0			0
$987$	−0.230574			−0.00733925
$988$		0			0
$989$		−	25.8945i		−	0.823398i
$990$		0			0
$991$			25.7140i			0.816832i	0.912796	+	0.408416i	$0.133919\pi$
$991$			25.7140i			0.816832i	−0.912796	+	0.408416i	$0.866081\pi$
$992$		0			0
$993$			3.52759i			0.111945i
$994$		0			0
$995$	75.6664			2.39879
$996$		0			0
$997$		−	11.8217i		−	0.374396i	−0.982322	−	0.187198i	$-0.940059\pi$
$997$		−	11.8217i		−	0.374396i	0.982322	−	0.187198i	$-0.0599406\pi$
$998$		0			0
$999$	−13.9616			−0.441726

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.21	✓	46
17.16	even	2	inner	4012.2.b.b.237.26	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.21	✓	46	1.1	even	1	trivial
4012.2.b.b.237.26	yes	46	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)

\(f(q)\)	\(=\)	\(q-0.274529i q^{3} +3.55822i q^{5} +0.321791i q^{7} +2.92463 q^{9} +O(q^{10})\)	\(q-0.274529i q^{3} +3.55822i q^{5} +0.321791i q^{7} +2.92463 q^{9} -0.790275i q^{11} -4.51068 q^{13} +0.976836 q^{15} +(-0.792973 - 4.04613i) q^{17} -3.21569 q^{19} +0.0883410 q^{21} +3.60173i q^{23} -7.66094 q^{25} -1.62649i q^{27} -7.35707i q^{29} -10.7779i q^{31} -0.216954 q^{33} -1.14500 q^{35} -8.58392i q^{37} +1.23831i q^{39} -7.36832i q^{41} -7.18947 q^{43} +10.4065i q^{45} -2.61004 q^{47} +6.89645 q^{49} +(-1.11078 + 0.217694i) q^{51} -5.58406 q^{53} +2.81197 q^{55} +0.882801i q^{57} +1.00000 q^{59} +7.98834i q^{61} +0.941120i q^{63} -16.0500i q^{65} +12.9523 q^{67} +0.988781 q^{69} -10.4862i q^{71} +5.27767i q^{73} +2.10315i q^{75} +0.254303 q^{77} -1.75505i q^{79} +8.32738 q^{81} -12.3958 q^{83} +(14.3970 - 2.82157i) q^{85} -2.01973 q^{87} +3.48022 q^{89} -1.45149i q^{91} -2.95886 q^{93} -11.4421i q^{95} +16.6285i q^{97} -2.31126i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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