LMFDB - Embedded newform 4012.2.b.b.237.36

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			237.36
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79340i q^{3} -1.19915i q^{5} +1.20458i q^{7} -0.216279 q^{9} +O(q^{10})$	$q+1.79340i q^{3} -1.19915i q^{5} +1.20458i q^{7} -0.216279 q^{9} +0.564620i q^{11} -6.78189 q^{13} +2.15056 q^{15} +(-2.99791 - 2.83064i) q^{17} -5.87932 q^{19} -2.16030 q^{21} +2.74169i q^{23} +3.56203 q^{25} +4.99232i q^{27} -8.94813i q^{29} +4.79978i q^{31} -1.01259 q^{33} +1.44448 q^{35} -5.32745i q^{37} -12.1626i q^{39} -6.25269i q^{41} +8.95816 q^{43} +0.259352i q^{45} +9.79572 q^{47} +5.54898 q^{49} +(5.07646 - 5.37645i) q^{51} +11.8332 q^{53} +0.677067 q^{55} -10.5440i q^{57} +1.00000 q^{59} +0.445356i q^{61} -0.260526i q^{63} +8.13254i q^{65} -13.0800 q^{67} -4.91694 q^{69} +9.22255i q^{71} -13.5216i q^{73} +6.38814i q^{75} -0.680133 q^{77} -15.6374i q^{79} -9.60206 q^{81} -6.77674 q^{83} +(-3.39437 + 3.59496i) q^{85} +16.0476 q^{87} -2.83173 q^{89} -8.16937i q^{91} -8.60792 q^{93} +7.05022i q^{95} -12.9863i q^{97} -0.122115i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			1.79340i			1.03542i	0.855556	+	0.517710i	$0.173215\pi$
$3$			1.79340i			1.03542i	−0.855556	+	0.517710i	$0.826785\pi$
$4$		0			0
$5$		−	1.19915i		−	0.536278i	−0.963380	−	0.268139i	$-0.913591\pi$
$5$		−	1.19915i		−	0.536278i	0.963380	−	0.268139i	$-0.0864087\pi$
$6$		0			0
$7$			1.20458i			0.455290i	0.973744	+	0.227645i	$0.0731026\pi$
$7$			1.20458i			0.455290i	−0.973744	+	0.227645i	$0.926897\pi$
$8$		0			0
$9$	−0.216279			−0.0720929
$10$		0			0
$11$			0.564620i			0.170239i	0.996371	+	0.0851197i	$0.0271273\pi$
$11$			0.564620i			0.170239i	−0.996371	+	0.0851197i	$0.972873\pi$
$12$		0			0
$13$	−6.78189			−1.88096			−0.940480	−	0.339850i	$-0.889624\pi$
$13$	−6.78189			−1.88096			−0.940480	+	0.339850i	$0.889624\pi$
$14$		0			0
$15$	2.15056			0.555273
$16$		0			0
$17$	−2.99791	−	2.83064i	−0.727101	−	0.686531i
$17$	−2.99791	−	2.83064i	−0.727101	−	0.686531i
$18$		0			0
$19$	−5.87932			−1.34881			−0.674405	−	0.738362i	$-0.735600\pi$
$19$	−5.87932			−1.34881			−0.674405	+	0.738362i	$0.735600\pi$
$20$		0			0
$21$	−2.16030			−0.471416
$22$		0			0
$23$			2.74169i			0.571682i	0.958277	+	0.285841i	$0.0922728\pi$
$23$			2.74169i			0.571682i	−0.958277	+	0.285841i	$0.907727\pi$
$24$		0			0
$25$	3.56203			0.712406
$26$		0			0
$27$			4.99232i			0.960773i
$28$		0			0
$29$		−	8.94813i		−	1.66163i	−0.556552	−	0.830813i	$-0.687876\pi$
$29$		−	8.94813i		−	1.66163i	0.556552	−	0.830813i	$-0.312124\pi$
$30$		0			0
$31$			4.79978i			0.862066i	0.902336	+	0.431033i	$0.141851\pi$
$31$			4.79978i			0.862066i	−0.902336	+	0.431033i	$0.858149\pi$
$32$		0			0
$33$	−1.01259			−0.176269
$34$		0			0
$35$	1.44448			0.244162
$36$		0			0
$37$		−	5.32745i		−	0.875828i	−0.899017	−	0.437914i	$-0.855717\pi$
$37$		−	5.32745i		−	0.875828i	0.899017	−	0.437914i	$-0.144283\pi$
$38$		0			0
$39$		−	12.1626i		−	1.94758i
$40$		0			0
$41$		−	6.25269i		−	0.976506i	−0.872702	−	0.488253i	$-0.837634\pi$
$41$		−	6.25269i		−	0.976506i	0.872702	−	0.488253i	$-0.162366\pi$
$42$		0			0
$43$	8.95816			1.36611			0.683053	−	0.730369i	$-0.260652\pi$
$43$	8.95816			1.36611			0.683053	+	0.730369i	$0.260652\pi$
$44$		0			0
$45$			0.259352i			0.0386619i
$46$		0			0
$47$	9.79572			1.42885			0.714426	−	0.699711i	$-0.246688\pi$
$47$	9.79572			1.42885			0.714426	+	0.699711i	$0.246688\pi$
$48$		0			0
$49$	5.54898			0.792711
$50$		0			0
$51$	5.07646	−	5.37645i	0.710847	−	0.752854i
$52$		0			0
$53$	11.8332			1.62541			0.812706	−	0.582674i	$-0.197994\pi$
$53$	11.8332			1.62541			0.812706	+	0.582674i	$0.197994\pi$
$54$		0			0
$55$	0.677067			0.0912956
$56$		0			0
$57$		−	10.5440i		−	1.39658i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$			0.445356i			0.0570220i	0.999593	+	0.0285110i	$0.00907656\pi$
$61$			0.445356i			0.0570220i	−0.999593	+	0.0285110i	$0.990923\pi$
$62$		0			0
$63$		−	0.260526i		−	0.0328232i
$64$		0			0
$65$			8.13254i			1.00872i
$66$		0			0
$67$	−13.0800			−1.59797			−0.798985	−	0.601350i	$-0.794630\pi$
$67$	−13.0800			−1.59797			−0.798985	+	0.601350i	$0.794630\pi$
$68$		0			0
$69$	−4.91694			−0.591930
$70$		0			0
$71$			9.22255i			1.09451i	0.836964	+	0.547257i	$0.184328\pi$
$71$			9.22255i			1.09451i	−0.836964	+	0.547257i	$0.815672\pi$
$72$		0			0
$73$		−	13.5216i		−	1.58258i	−0.611440	−	0.791291i	$-0.709409\pi$
$73$		−	13.5216i		−	1.58258i	0.611440	−	0.791291i	$-0.290591\pi$
$74$		0			0
$75$			6.38814i			0.737639i
$76$		0			0
$77$	−0.680133			−0.0775083
$78$		0			0
$79$		−	15.6374i		−	1.75934i	−0.475582	−	0.879671i	$-0.657763\pi$
$79$		−	15.6374i		−	1.75934i	0.475582	−	0.879671i	$-0.342237\pi$
$80$		0			0
$81$	−9.60206			−1.06690
$82$		0			0
$83$	−6.77674			−0.743844			−0.371922	−	0.928264i	$-0.621301\pi$
$83$	−6.77674			−0.743844			−0.371922	+	0.928264i	$0.621301\pi$
$84$		0			0
$85$	−3.39437	+	3.59496i	−0.368171	+	0.389928i
$86$		0			0
$87$	16.0476			1.72048
$88$		0			0
$89$	−2.83173			−0.300163			−0.150081	−	0.988674i	$-0.547954\pi$
$89$	−2.83173			−0.300163			−0.150081	+	0.988674i	$0.547954\pi$
$90$		0			0
$91$		−	8.16937i		−	0.856383i
$92$		0			0
$93$	−8.60792			−0.892600
$94$		0			0
$95$			7.05022i			0.723337i
$96$		0			0
$97$		−	12.9863i		−	1.31855i	−0.751900	−	0.659277i	$-0.770862\pi$
$97$		−	12.9863i		−	1.31855i	0.751900	−	0.659277i	$-0.229138\pi$
$98$		0			0
$99$		−	0.122115i		−	0.0122731i
$100$		0			0
$101$	15.7177			1.56397			0.781983	−	0.623299i	$-0.214208\pi$
$101$	15.7177			1.56397			0.781983	+	0.623299i	$0.214208\pi$
$102$		0			0
$103$	−18.8995			−1.86223			−0.931114	−	0.364729i	$-0.881162\pi$
$103$	−18.8995			−1.86223			−0.931114	+	0.364729i	$0.881162\pi$
$104$		0			0
$105$			2.59053i			0.252810i
$106$		0			0
$107$		−	9.21894i		−	0.891228i	−0.895225	−	0.445614i	$-0.852985\pi$
$107$		−	9.21894i		−	0.891228i	0.895225	−	0.445614i	$-0.147015\pi$
$108$		0			0
$109$			8.69650i			0.832973i	0.909142	+	0.416487i	$0.136739\pi$
$109$			8.69650i			0.832973i	−0.909142	+	0.416487i	$0.863261\pi$
$110$		0			0
$111$	9.55425			0.906849
$112$		0			0
$113$		−	10.8045i		−	1.01640i	−0.861240	−	0.508199i	$-0.830311\pi$
$113$		−	10.8045i		−	1.01640i	0.861240	−	0.508199i	$-0.169689\pi$
$114$		0			0
$115$	3.28771			0.306580
$116$		0			0
$117$	1.46678			0.135604
$118$		0			0
$119$	3.40974	−	3.61124i	0.312571	−	0.331042i
$120$		0			0
$121$	10.6812			0.971019
$122$		0			0
$123$	11.2136			1.01109
$124$		0			0
$125$		−	10.2672i		−	0.918326i
$126$		0			0
$127$	−2.90644			−0.257904			−0.128952	−	0.991651i	$-0.541161\pi$
$127$	−2.90644			−0.257904			−0.128952	+	0.991651i	$0.541161\pi$
$128$		0			0
$129$			16.0656i			1.41449i
$130$		0			0
$131$			13.8548i			1.21050i	0.796035	+	0.605251i	$0.206927\pi$
$131$			13.8548i			1.21050i	−0.796035	+	0.605251i	$0.793073\pi$
$132$		0			0
$133$		−	7.08215i		−	0.614100i
$134$		0			0
$135$	5.98657			0.515242
$136$		0			0
$137$	5.94936			0.508288			0.254144	−	0.967166i	$-0.418206\pi$
$137$	5.94936			0.508288			0.254144	+	0.967166i	$0.418206\pi$
$138$		0			0
$139$		−	16.0659i		−	1.36269i	−0.731962	−	0.681346i	$-0.761395\pi$
$139$		−	16.0659i		−	1.36269i	0.731962	−	0.681346i	$-0.238605\pi$
$140$		0			0
$141$			17.5676i			1.47946i
$142$		0			0
$143$		−	3.82919i		−	0.320213i
$144$		0			0
$145$	−10.7302			−0.891094
$146$		0			0
$147$			9.95152i			0.820788i
$148$		0			0
$149$	−0.867969			−0.0711068			−0.0355534	−	0.999368i	$-0.511319\pi$
$149$	−0.867969			−0.0711068			−0.0355534	+	0.999368i	$0.511319\pi$
$150$		0			0
$151$	7.69235			0.625995			0.312997	−	0.949754i	$-0.398667\pi$
$151$	7.69235			0.625995			0.312997	+	0.949754i	$0.398667\pi$
$152$		0			0
$153$	0.648385	+	0.612207i	0.0524188	+	0.0494940i
$154$		0			0
$155$	5.75568			0.462307
$156$		0			0
$157$	−20.9266			−1.67012			−0.835062	−	0.550155i	$-0.814569\pi$
$157$	−20.9266			−1.67012			−0.835062	+	0.550155i	$0.814569\pi$
$158$		0			0
$159$			21.2216i			1.68298i
$160$		0			0
$161$	−3.30260			−0.260281
$162$		0			0
$163$		−	4.63463i		−	0.363012i	−0.983390	−	0.181506i	$-0.941903\pi$
$163$		−	4.63463i		−	0.363012i	0.983390	−	0.181506i	$-0.0580972\pi$
$164$		0			0
$165$			1.21425i			0.0945293i
$166$		0			0
$167$		−	15.9930i		−	1.23758i	−0.785557	−	0.618789i	$-0.787623\pi$
$167$		−	15.9930i		−	1.23758i	0.785557	−	0.618789i	$-0.212377\pi$
$168$		0			0
$169$	32.9941			2.53801
$170$		0			0
$171$	1.27157			0.0972396
$172$		0			0
$173$		−	7.78024i		−	0.591521i	−0.955262	−	0.295760i	$-0.904427\pi$
$173$		−	7.78024i		−	0.591521i	0.955262	−	0.295760i	$-0.0955730\pi$
$174$		0			0
$175$			4.29077i			0.324351i
$176$		0			0
$177$			1.79340i			0.134800i
$178$		0			0
$179$	−12.3247			−0.921192			−0.460596	−	0.887610i	$-0.652364\pi$
$179$	−12.3247			−0.921192			−0.460596	+	0.887610i	$0.652364\pi$
$180$		0			0
$181$			19.0861i			1.41866i	0.704879	+	0.709328i	$0.251001\pi$
$181$			19.0861i			1.41866i	−0.704879	+	0.709328i	$0.748999\pi$
$182$		0			0
$183$	−0.798701			−0.0590417
$184$		0			0
$185$	−6.38844			−0.469687
$186$		0			0
$187$	1.59823	−	1.69268i	0.116875	−	0.123781i
$188$		0			0
$189$	−6.01368			−0.437431
$190$		0			0
$191$	9.57292			0.692673			0.346336	−	0.938110i	$-0.387426\pi$
$191$	9.57292			0.692673			0.346336	+	0.938110i	$0.387426\pi$
$192$		0			0
$193$		−	3.91168i		−	0.281569i	−0.990040	−	0.140785i	$-0.955038\pi$
$193$		−	3.91168i		−	0.281569i	0.990040	−	0.140785i	$-0.0449625\pi$
$194$		0			0
$195$	−14.5849			−1.04445
$196$		0			0
$197$		−	4.48496i		−	0.319540i	−0.987154	−	0.159770i	$-0.948925\pi$
$197$		−	4.48496i		−	0.319540i	0.987154	−	0.159770i	$-0.0510753\pi$
$198$		0			0
$199$		−	15.1850i		−	1.07644i	−0.842805	−	0.538219i	$-0.819097\pi$
$199$		−	15.1850i		−	1.07644i	0.842805	−	0.538219i	$-0.180903\pi$
$200$		0			0
$201$		−	23.4576i		−	1.65457i
$202$		0			0
$203$	10.7788			0.756522
$204$		0			0
$205$	−7.49794			−0.523679
$206$		0			0
$207$		−	0.592969i		−	0.0412142i
$208$		0			0
$209$		−	3.31958i		−	0.229620i
$210$		0			0
$211$			8.61658i			0.593190i	0.955003	+	0.296595i	$0.0958511\pi$
$211$			8.61658i			0.593190i	−0.955003	+	0.296595i	$0.904149\pi$
$212$		0			0
$213$	−16.5397			−1.13328
$214$		0			0
$215$		−	10.7422i		−	0.732613i
$216$		0			0
$217$	−5.78174			−0.392490
$218$		0			0
$219$	24.2496			1.63864
$220$		0			0
$221$	20.3315	+	19.1971i	1.36765	+	1.29134i
$222$		0			0
$223$	19.6359			1.31491			0.657457	−	0.753492i	$-0.271632\pi$
$223$	19.6359			1.31491			0.657457	+	0.753492i	$0.271632\pi$
$224$		0			0
$225$	−0.770391			−0.0513594
$226$		0			0
$227$		−	17.2664i		−	1.14601i	−0.819552	−	0.573005i	$-0.805778\pi$
$227$		−	17.2664i		−	1.14601i	0.819552	−	0.573005i	$-0.194222\pi$
$228$		0			0
$229$	−7.99676			−0.528441			−0.264220	−	0.964462i	$-0.585115\pi$
$229$	−7.99676			−0.528441			−0.264220	+	0.964462i	$0.585115\pi$
$230$		0			0
$231$		−	1.21975i		−	0.0802536i
$232$		0			0
$233$		−	24.3644i		−	1.59616i	−0.602550	−	0.798081i	$-0.705849\pi$
$233$		−	24.3644i		−	1.59616i	0.602550	−	0.798081i	$-0.294151\pi$
$234$		0			0
$235$		−	11.7466i		−	0.766262i
$236$		0			0
$237$	28.0441			1.82166
$238$		0			0
$239$	−4.97573			−0.321853			−0.160927	−	0.986966i	$-0.551448\pi$
$239$	−4.97573			−0.321853			−0.160927	+	0.986966i	$0.551448\pi$
$240$		0			0
$241$		−	10.0453i		−	0.647073i	−0.946216	−	0.323537i	$-0.895128\pi$
$241$		−	10.0453i		−	0.647073i	0.946216	−	0.323537i	$-0.104872\pi$
$242$		0			0
$243$		−	2.24336i		−	0.143911i
$244$		0			0
$245$		−	6.65408i		−	0.425113i
$246$		0			0
$247$	39.8730			2.53706
$248$		0			0
$249$		−	12.1534i		−	0.770190i
$250$		0			0
$251$	8.24542			0.520446			0.260223	−	0.965548i	$-0.416204\pi$
$251$	8.24542			0.520446			0.260223	+	0.965548i	$0.416204\pi$
$252$		0			0
$253$	−1.54801			−0.0973227
$254$		0			0
$255$	−6.44720	−	6.08746i	−0.403739	−	0.381212i
$256$		0			0
$257$	−17.0017			−1.06054			−0.530269	−	0.847829i	$-0.677909\pi$
$257$	−17.0017			−1.06054			−0.530269	+	0.847829i	$0.677909\pi$
$258$		0			0
$259$	6.41737			0.398756
$260$		0			0
$261$			1.93529i			0.119791i
$262$		0			0
$263$	−3.00933			−0.185563			−0.0927815	−	0.995686i	$-0.529576\pi$
$263$	−3.00933			−0.185563			−0.0927815	+	0.995686i	$0.529576\pi$
$264$		0			0
$265$		−	14.1898i		−	0.871673i
$266$		0			0
$267$		−	5.07842i		−	0.310794i
$268$		0			0
$269$			19.3968i			1.18264i	0.806435	+	0.591322i	$0.201394\pi$
$269$			19.3968i			1.18264i	−0.806435	+	0.591322i	$0.798606\pi$
$270$		0			0
$271$	1.69220			0.102794			0.0513970	−	0.998678i	$-0.483633\pi$
$271$	1.69220			0.102794			0.0513970	+	0.998678i	$0.483633\pi$
$272$		0			0
$273$	14.6509			0.886715
$274$		0			0
$275$			2.01119i			0.121279i
$276$		0			0
$277$			19.0746i			1.14608i	0.819527	+	0.573040i	$0.194236\pi$
$277$			19.0746i			1.14608i	−0.819527	+	0.573040i	$0.805764\pi$
$278$		0			0
$279$		−	1.03809i		−	0.0621489i
$280$		0			0
$281$	−21.6400			−1.29093			−0.645467	−	0.763788i	$-0.723337\pi$
$281$	−21.6400			−1.29093			−0.645467	+	0.763788i	$0.723337\pi$
$282$		0			0
$283$			27.6229i			1.64201i	0.570921	+	0.821005i	$0.306586\pi$
$283$			27.6229i			1.64201i	−0.570921	+	0.821005i	$0.693414\pi$
$284$		0			0
$285$	−12.6439			−0.748957
$286$		0			0
$287$	7.53190			0.444594
$288$		0			0
$289$	0.974975	+	16.9720i	0.0573515	+	0.998354i
$290$		0			0
$291$	23.2895			1.36526
$292$		0			0
$293$	−15.3433			−0.896367			−0.448183	−	0.893942i	$-0.647929\pi$
$293$	−15.3433			−0.896367			−0.448183	+	0.893942i	$0.647929\pi$
$294$		0			0
$295$		−	1.19915i		−	0.0698175i
$296$		0			0
$297$	−2.81876			−0.163561
$298$		0			0
$299$		−	18.5938i		−	1.07531i
$300$		0			0
$301$			10.7909i			0.621975i
$302$		0			0
$303$			28.1880i			1.61936i
$304$		0			0
$305$	0.534051			0.0305797
$306$		0			0
$307$	3.03723			0.173344			0.0866719	−	0.996237i	$-0.472377\pi$
$307$	3.03723			0.173344			0.0866719	+	0.996237i	$0.472377\pi$
$308$		0			0
$309$		−	33.8944i		−	1.92819i
$310$		0			0
$311$		−	7.05485i		−	0.400044i	−0.979791	−	0.200022i	$-0.935899\pi$
$311$		−	7.05485i		−	0.400044i	0.979791	−	0.200022i	$-0.0641013\pi$
$312$		0			0
$313$			6.10368i			0.345000i	0.985010	+	0.172500i	$0.0551845\pi$
$313$			6.10368i			0.345000i	−0.985010	+	0.172500i	$0.944816\pi$
$314$		0			0
$315$	−0.312411			−0.0176024
$316$		0			0
$317$			5.36501i			0.301329i	0.988585	+	0.150664i	$0.0481413\pi$
$317$			5.36501i			0.301329i	−0.988585	+	0.150664i	$0.951859\pi$
$318$		0			0
$319$	5.05229			0.282874
$320$		0			0
$321$	16.5332			0.922795
$322$		0			0
$323$	17.6257	+	16.6422i	0.980721	+	0.925999i
$324$		0			0
$325$	−24.1573			−1.34001
$326$		0			0
$327$	−15.5963			−0.862477
$328$		0			0
$329$			11.7998i			0.650543i
$330$		0			0
$331$	−7.13788			−0.392333			−0.196167	−	0.980571i	$-0.562849\pi$
$331$	−7.13788			−0.392333			−0.196167	+	0.980571i	$0.562849\pi$
$332$		0			0
$333$			1.15222i			0.0631410i
$334$		0			0
$335$			15.6849i			0.856957i
$336$		0			0
$337$			3.72226i			0.202765i	0.994848	+	0.101382i	$0.0323265\pi$
$337$			3.72226i			0.202765i	−0.994848	+	0.101382i	$0.967673\pi$
$338$		0			0
$339$	19.3767			1.05240
$340$		0			0
$341$	−2.71005			−0.146758
$342$		0			0
$343$			15.1163i			0.816204i
$344$		0			0
$345$			5.89617i			0.317439i
$346$		0			0
$347$			9.93694i			0.533443i	0.963774	+	0.266721i	$0.0859404\pi$
$347$			9.93694i			0.533443i	−0.963774	+	0.266721i	$0.914060\pi$
$348$		0			0
$349$	16.0432			0.858775			0.429387	−	0.903120i	$-0.358729\pi$
$349$	16.0432			0.858775			0.429387	+	0.903120i	$0.358729\pi$
$350$		0			0
$351$		−	33.8574i		−	1.80717i
$352$		0			0
$353$	−0.866248			−0.0461057			−0.0230529	−	0.999734i	$-0.507339\pi$
$353$	−0.866248			−0.0461057			−0.0230529	+	0.999734i	$0.507339\pi$
$354$		0			0
$355$	11.0593			0.586964
$356$		0			0
$357$	6.47640	+	6.11503i	0.342767	+	0.323642i
$358$		0			0
$359$	29.4865			1.55624			0.778119	−	0.628118i	$-0.216174\pi$
$359$	29.4865			1.55624			0.778119	+	0.628118i	$0.216174\pi$
$360$		0			0
$361$	15.5665			0.819287
$362$		0			0
$363$			19.1557i			1.00541i
$364$		0			0
$365$	−16.2145			−0.848704
$366$		0			0
$367$		−	14.5988i		−	0.762051i	−0.924565	−	0.381025i	$-0.875571\pi$
$367$		−	14.5988i		−	0.762051i	0.924565	−	0.381025i	$-0.124429\pi$
$368$		0			0
$369$			1.35232i			0.0703992i
$370$		0			0
$371$			14.2541i			0.740034i
$372$		0			0
$373$	−18.7235			−0.969465			−0.484733	−	0.874662i	$-0.661083\pi$
$373$	−18.7235			−0.969465			−0.484733	+	0.874662i	$0.661083\pi$
$374$		0			0
$375$	18.4132			0.950852
$376$		0			0
$377$			60.6853i			3.12545i
$378$		0			0
$379$		−	21.8231i		−	1.12098i	−0.828163	−	0.560488i	$-0.810614\pi$
$379$		−	21.8231i		−	1.12098i	0.828163	−	0.560488i	$-0.189386\pi$
$380$		0			0
$381$		−	5.21240i		−	0.267039i
$382$		0			0
$383$	17.6275			0.900722			0.450361	−	0.892847i	$-0.351295\pi$
$383$	17.6275			0.900722			0.450361	+	0.892847i	$0.351295\pi$
$384$		0			0
$385$			0.815584i			0.0415660i
$386$		0			0
$387$	−1.93746			−0.0984866
$388$		0			0
$389$	−11.7294			−0.594702			−0.297351	−	0.954768i	$-0.596103\pi$
$389$	−11.7294			−0.594702			−0.297351	+	0.954768i	$0.596103\pi$
$390$		0			0
$391$	7.76073	−	8.21935i	0.392477	−	0.415670i
$392$		0			0
$393$	−24.8472			−1.25338
$394$		0			0
$395$	−18.7516			−0.943497
$396$		0			0
$397$		−	32.8927i		−	1.65084i	−0.564520	−	0.825420i	$-0.690939\pi$
$397$		−	32.8927i		−	1.65084i	0.564520	−	0.825420i	$-0.309061\pi$
$398$		0			0
$399$	12.7011			0.635851
$400$		0			0
$401$		−	9.79236i		−	0.489007i	−0.969648	−	0.244504i	$-0.921375\pi$
$401$		−	9.79236i		−	0.489007i	0.969648	−	0.244504i	$-0.0786250\pi$
$402$		0			0
$403$		−	32.5516i		−	1.62151i
$404$		0			0
$405$			11.5144i			0.572153i
$406$		0			0
$407$	3.00799			0.149100
$408$		0			0
$409$	−14.3786			−0.710977			−0.355488	−	0.934681i	$-0.615685\pi$
$409$	−14.3786			−0.710977			−0.355488	+	0.934681i	$0.615685\pi$
$410$		0			0
$411$			10.6696i			0.526292i
$412$		0			0
$413$			1.20458i			0.0592738i
$414$		0			0
$415$			8.12636i			0.398907i
$416$		0			0
$417$	28.8126			1.41096
$418$		0			0
$419$		−	1.17302i		−	0.0573057i	−0.999589	−	0.0286529i	$-0.990878\pi$
$419$		−	1.17302i		−	0.0573057i	0.999589	−	0.0286529i	$-0.00912174\pi$
$420$		0			0
$421$	3.06357			0.149309			0.0746546	−	0.997209i	$-0.476215\pi$
$421$	3.06357			0.149309			0.0746546	+	0.997209i	$0.476215\pi$
$422$		0			0
$423$	−2.11861			−0.103010
$424$		0			0
$425$	−10.6787	−	10.0828i	−0.517991	−	0.489088i
$426$		0			0
$427$	−0.536469			−0.0259616
$428$		0			0
$429$	6.86727			0.331555
$430$		0			0
$431$		−	5.03954i		−	0.242746i	−0.992607	−	0.121373i	$-0.961270\pi$
$431$		−	5.03954i		−	0.242746i	0.992607	−	0.121373i	$-0.0387297\pi$
$432$		0			0
$433$	−19.8369			−0.953300			−0.476650	−	0.879093i	$-0.658149\pi$
$433$	−19.8369			−0.953300			−0.476650	+	0.879093i	$0.658149\pi$
$434$		0			0
$435$		−	19.2435i		−	0.922656i
$436$		0			0
$437$		−	16.1193i		−	0.771089i
$438$		0			0
$439$			12.9200i			0.616636i	0.951283	+	0.308318i	$0.0997660\pi$
$439$			12.9200i			0.616636i	−0.951283	+	0.308318i	$0.900234\pi$
$440$		0			0
$441$	−1.20013			−0.0571488
$442$		0			0
$443$	9.80471			0.465836			0.232918	−	0.972496i	$-0.425173\pi$
$443$	9.80471			0.465836			0.232918	+	0.972496i	$0.425173\pi$
$444$		0			0
$445$			3.39568i			0.160971i
$446$		0			0
$447$		−	1.55661i		−	0.0736253i
$448$		0			0
$449$			41.2582i			1.94710i	0.228480	+	0.973549i	$0.426624\pi$
$449$			41.2582i			1.94710i	−0.228480	+	0.973549i	$0.573376\pi$
$450$		0			0
$451$	3.53039			0.166240
$452$		0			0
$453$			13.7955i			0.648167i
$454$		0			0
$455$	−9.79634			−0.459259
$456$		0			0
$457$	32.0203			1.49785			0.748923	−	0.662658i	$-0.230571\pi$
$457$	32.0203			1.49785			0.748923	+	0.662658i	$0.230571\pi$
$458$		0			0
$459$	14.1315	−	14.9666i	0.659600	−	0.698579i
$460$		0			0
$461$	−12.8193			−0.597056			−0.298528	−	0.954401i	$-0.596496\pi$
$461$	−12.8193			−0.597056			−0.298528	+	0.954401i	$0.596496\pi$
$462$		0			0
$463$	−9.65804			−0.448847			−0.224424	−	0.974492i	$-0.572050\pi$
$463$	−9.65804			−0.448847			−0.224424	+	0.974492i	$0.572050\pi$
$464$		0			0
$465$			10.3222i			0.478682i
$466$		0			0
$467$	−20.1862			−0.934106			−0.467053	−	0.884229i	$-0.654684\pi$
$467$	−20.1862			−0.934106			−0.467053	+	0.884229i	$0.654684\pi$
$468$		0			0
$469$		−	15.7559i		−	0.727541i
$470$		0			0
$471$		−	37.5297i		−	1.72928i
$472$		0			0
$473$			5.05796i			0.232565i
$474$		0			0
$475$	−20.9423			−0.960900
$476$		0			0
$477$	−2.55927			−0.117181
$478$		0			0
$479$			2.78186i			0.127106i	0.997978	+	0.0635532i	$0.0202432\pi$
$479$			2.78186i			0.127106i	−0.997978	+	0.0635532i	$0.979757\pi$
$480$		0			0
$481$			36.1302i			1.64740i
$482$		0			0
$483$		−	5.92287i		−	0.269500i
$484$		0			0
$485$	−15.5725			−0.707112
$486$		0			0
$487$		−	35.1232i		−	1.59158i	−0.605570	−	0.795792i	$-0.707055\pi$
$487$		−	35.1232i		−	1.59158i	0.605570	−	0.795792i	$-0.292945\pi$
$488$		0			0
$489$	8.31174			0.375870
$490$		0			0
$491$	−16.3223			−0.736613			−0.368307	−	0.929704i	$-0.620062\pi$
$491$	−16.3223			−0.736613			−0.368307	+	0.929704i	$0.620062\pi$
$492$		0			0
$493$	−25.3289	+	26.8257i	−1.14076	+	1.20817i
$494$		0			0
$495$	−0.146435			−0.00658177
$496$		0			0
$497$	−11.1093			−0.498322
$498$		0			0
$499$		−	4.76824i		−	0.213456i	−0.994288	−	0.106728i	$-0.965963\pi$
$499$		−	4.76824i		−	0.213456i	0.994288	−	0.106728i	$-0.0340373\pi$
$500$		0			0
$501$	28.6819			1.28141
$502$		0			0
$503$			27.8201i			1.24044i	0.784429	+	0.620218i	$0.212956\pi$
$503$			27.8201i			1.24044i	−0.784429	+	0.620218i	$0.787044\pi$
$504$		0			0
$505$		−	18.8479i		−	0.838721i
$506$		0			0
$507$			59.1716i			2.62790i
$508$		0			0
$509$	31.8898			1.41349			0.706745	−	0.707468i	$-0.250163\pi$
$509$	31.8898			1.41349			0.706745	+	0.707468i	$0.250163\pi$
$510$		0			0
$511$	16.2879			0.720534
$512$		0			0
$513$		−	29.3515i		−	1.29590i
$514$		0			0
$515$			22.6635i			0.998672i
$516$		0			0
$517$			5.53086i			0.243247i
$518$		0			0
$519$	13.9531			0.612472
$520$		0			0
$521$		−	19.5058i		−	0.854566i	−0.904118	−	0.427283i	$-0.859471\pi$
$521$		−	19.5058i		−	0.854566i	0.904118	−	0.427283i	$-0.140529\pi$
$522$		0			0
$523$	−25.7686			−1.12678			−0.563391	−	0.826191i	$-0.690503\pi$
$523$	−25.7686			−1.12678			−0.563391	+	0.826191i	$0.690503\pi$
$524$		0			0
$525$	−7.69505			−0.335840
$526$		0			0
$527$	13.5864	−	14.3893i	0.591835	−	0.626809i
$528$		0			0
$529$	15.4831			0.673180
$530$		0			0
$531$	−0.216279			−0.00938570
$532$		0			0
$533$			42.4051i			1.83677i
$534$		0			0
$535$	−11.0549			−0.477946
$536$		0			0
$537$		−	22.1031i		−	0.953820i
$538$		0			0
$539$			3.13306i			0.134951i
$540$		0			0
$541$			12.9792i			0.558017i	0.960289	+	0.279009i	$0.0900058\pi$
$541$			12.9792i			0.558017i	−0.960289	+	0.279009i	$0.909994\pi$
$542$		0			0
$543$	−34.2289			−1.46890
$544$		0			0
$545$	10.4284			0.446705
$546$		0			0
$547$		−	7.67782i		−	0.328280i	−0.986437	−	0.164140i	$-0.947515\pi$
$547$		−	7.67782i		−	0.328280i	0.986437	−	0.164140i	$-0.0524849\pi$
$548$		0			0
$549$		−	0.0963211i		−	0.00411088i
$550$		0			0
$551$			52.6090i			2.24122i
$552$		0			0
$553$	18.8365			0.801012
$554$		0			0
$555$		−	11.4570i		−	0.486323i
$556$		0			0
$557$	17.0374			0.721899			0.360949	−	0.932585i	$-0.382453\pi$
$557$	17.0374			0.721899			0.360949	+	0.932585i	$0.382453\pi$
$558$		0			0
$559$	−60.7533			−2.56959
$560$		0			0
$561$	3.03565	+	2.86627i	0.128165	+	0.121014i
$562$		0			0
$563$	−35.0979			−1.47920			−0.739601	−	0.673046i	$-0.764986\pi$
$563$	−35.0979			−1.47920			−0.739601	+	0.673046i	$0.764986\pi$
$564$		0			0
$565$	−12.9562			−0.545072
$566$		0			0
$567$		−	11.5665i		−	0.485747i
$568$		0			0
$569$	6.60353			0.276835			0.138417	−	0.990374i	$-0.455798\pi$
$569$	6.60353			0.276835			0.138417	+	0.990374i	$0.455798\pi$
$570$		0			0
$571$		−	26.0165i		−	1.08876i	−0.838840	−	0.544378i	$-0.816766\pi$
$571$		−	26.0165i		−	1.08876i	0.838840	−	0.544378i	$-0.183234\pi$
$572$		0			0
$573$			17.1681i			0.717206i
$574$		0			0
$575$			9.76597i			0.407269i
$576$		0			0
$577$	24.5374			1.02151			0.510753	−	0.859728i	$-0.329367\pi$
$577$	24.5374			1.02151			0.510753	+	0.859728i	$0.329367\pi$
$578$		0			0
$579$	7.01521			0.291542
$580$		0			0
$581$		−	8.16316i		−	0.338665i
$582$		0			0
$583$			6.68125i			0.276709i
$584$		0			0
$585$		−	1.75890i		−	0.0727214i
$586$		0			0
$587$	−15.7405			−0.649682			−0.324841	−	0.945769i	$-0.605311\pi$
$587$	−15.7405			−0.649682			−0.324841	+	0.945769i	$0.605311\pi$
$588$		0			0
$589$		−	28.2195i		−	1.16276i
$590$		0			0
$591$	8.04332			0.330858
$592$		0			0
$593$	−28.6638			−1.17708			−0.588540	−	0.808468i	$-0.700297\pi$
$593$	−28.6638			−1.17708			−0.588540	+	0.808468i	$0.700297\pi$
$594$		0			0
$595$	−4.33044	−	4.08881i	−0.177531	−	0.167625i
$596$		0			0
$597$	27.2328			1.11456
$598$		0			0
$599$	−35.9389			−1.46842			−0.734211	−	0.678921i	$-0.762448\pi$
$599$	−35.9389			−1.46842			−0.734211	+	0.678921i	$0.762448\pi$
$600$		0			0
$601$		−	45.3256i		−	1.84887i	−0.381342	−	0.924434i	$-0.624538\pi$
$601$		−	45.3256i		−	1.84887i	0.381342	−	0.924434i	$-0.375462\pi$
$602$		0			0
$603$	2.82892			0.115202
$604$		0			0
$605$		−	12.8084i		−	0.520736i
$606$		0			0
$607$			36.8702i			1.49651i	0.663409	+	0.748257i	$0.269109\pi$
$607$			36.8702i			1.49651i	−0.663409	+	0.748257i	$0.730891\pi$
$608$		0			0
$609$			19.3307i			0.783318i
$610$		0			0
$611$	−66.4335			−2.68761
$612$		0			0
$613$	3.24520			0.131073			0.0655363	−	0.997850i	$-0.479124\pi$
$613$	3.24520			0.131073			0.0655363	+	0.997850i	$0.479124\pi$
$614$		0			0
$615$		−	13.4468i		−	0.542227i
$616$		0			0
$617$			8.05121i			0.324129i	0.986780	+	0.162065i	$0.0518153\pi$
$617$			8.05121i			0.324129i	−0.986780	+	0.162065i	$0.948185\pi$
$618$		0			0
$619$		−	41.3037i		−	1.66013i	−0.557663	−	0.830067i	$-0.688302\pi$
$619$		−	41.3037i		−	1.66013i	0.557663	−	0.830067i	$-0.311698\pi$
$620$		0			0
$621$	−13.6874			−0.549256
$622$		0			0
$623$		−	3.41106i		−	0.136661i
$624$		0			0
$625$	5.49819			0.219928
$626$		0			0
$627$	5.95334			0.237753
$628$		0			0
$629$	−15.0801	+	15.9712i	−0.601283	+	0.636815i
$630$		0			0
$631$	−5.75140			−0.228960			−0.114480	−	0.993426i	$-0.536520\pi$
$631$	−5.75140			−0.228960			−0.114480	+	0.993426i	$0.536520\pi$
$632$		0			0
$633$	−15.4530			−0.614200
$634$		0			0
$635$			3.48526i			0.138309i
$636$		0			0
$637$	−37.6326			−1.49106
$638$		0			0
$639$		−	1.99464i		−	0.0789068i
$640$		0			0
$641$			5.01188i			0.197957i	0.995090	+	0.0989787i	$0.0315576\pi$
$641$			5.01188i			0.197957i	−0.995090	+	0.0989787i	$0.968442\pi$
$642$		0			0
$643$		−	44.2604i		−	1.74546i	−0.488203	−	0.872730i	$-0.662347\pi$
$643$		−	44.2604i		−	1.74546i	0.488203	−	0.872730i	$-0.337653\pi$
$644$		0			0
$645$	19.2651			0.758562
$646$		0			0
$647$	1.68354			0.0661869			0.0330935	−	0.999452i	$-0.489464\pi$
$647$	1.68354			0.0661869			0.0330935	+	0.999452i	$0.489464\pi$
$648$		0			0
$649$			0.564620i			0.0221633i
$650$		0			0
$651$		−	10.3690i		−	0.406392i
$652$		0			0
$653$			10.1938i			0.398913i	0.979907	+	0.199456i	$0.0639176\pi$
$653$			10.1938i			0.398913i	−0.979907	+	0.199456i	$0.936082\pi$
$654$		0			0
$655$	16.6141			0.649166
$656$		0			0
$657$			2.92443i			0.114093i
$658$		0			0
$659$	8.86756			0.345431			0.172715	−	0.984972i	$-0.444746\pi$
$659$	8.86756			0.345431			0.172715	+	0.984972i	$0.444746\pi$
$660$		0			0
$661$	−3.88174			−0.150982			−0.0754910	−	0.997146i	$-0.524052\pi$
$661$	−3.88174			−0.150982			−0.0754910	+	0.997146i	$0.524052\pi$
$662$		0			0
$663$	−34.4280	+	36.4626i	−1.33707	+	1.41609i
$664$		0			0
$665$	−8.49259			−0.329328
$666$		0			0
$667$	24.5330			0.949921
$668$		0			0
$669$			35.2149i			1.36149i
$670$		0			0
$671$	−0.251457			−0.00970739
$672$		0			0
$673$		−	9.49461i		−	0.365990i	−0.983114	−	0.182995i	$-0.941421\pi$
$673$		−	9.49461i		−	0.365990i	0.983114	−	0.182995i	$-0.0585792\pi$
$674$		0			0
$675$			17.7828i			0.684460i
$676$		0			0
$677$		−	23.6839i		−	0.910246i	−0.890428	−	0.455123i	$-0.849595\pi$
$677$		−	23.6839i		−	0.910246i	0.890428	−	0.455123i	$-0.150405\pi$
$678$		0			0
$679$	15.6431			0.600325
$680$		0			0
$681$	30.9655			1.18660
$682$		0			0
$683$		−	25.4349i		−	0.973240i	−0.873614	−	0.486620i	$-0.838230\pi$
$683$		−	25.4349i		−	0.973240i	0.873614	−	0.486620i	$-0.161770\pi$
$684$		0			0
$685$		−	7.13421i		−	0.272584i
$686$		0			0
$687$		−	14.3414i		−	0.547158i
$688$		0			0
$689$	−80.2514			−3.05733
$690$		0			0
$691$			7.48165i			0.284616i	0.989822	+	0.142308i	$0.0454523\pi$
$691$			7.48165i			0.284616i	−0.989822	+	0.142308i	$0.954548\pi$
$692$		0			0
$693$	0.147098			0.00558780
$694$		0			0
$695$	−19.2655			−0.730782
$696$		0			0
$697$	−17.6991	+	18.7450i	−0.670401	+	0.710019i
$698$		0			0
$699$	43.6950			1.65270
$700$		0			0
$701$	−37.4544			−1.41463			−0.707316	−	0.706898i	$-0.750094\pi$
$701$	−37.4544			−1.41463			−0.707316	+	0.706898i	$0.750094\pi$
$702$		0			0
$703$			31.3218i			1.18133i
$704$		0			0
$705$	21.0663			0.793403
$706$		0			0
$707$			18.9333i			0.712059i
$708$		0			0
$709$			19.5713i			0.735016i	0.930020	+	0.367508i	$0.119789\pi$
$709$			19.5713i			0.735016i	−0.930020	+	0.367508i	$0.880211\pi$
$710$		0			0
$711$			3.38203i			0.126836i
$712$		0			0
$713$	−13.1595			−0.492827
$714$		0			0
$715$	−4.59179			−0.171723
$716$		0			0
$717$		−	8.92347i		−	0.333253i
$718$		0			0
$719$			1.37742i			0.0513689i	0.999670	+	0.0256845i	$0.00817652\pi$
$719$			1.37742i			0.0513689i	−0.999670	+	0.0256845i	$0.991823\pi$
$720$		0			0
$721$		−	22.7661i		−	0.847854i
$722$		0			0
$723$	18.0152			0.669992
$724$		0			0
$725$		−	31.8735i		−	1.18375i
$726$		0			0
$727$	−12.4574			−0.462018			−0.231009	−	0.972952i	$-0.574203\pi$
$727$	−12.4574			−0.462018			−0.231009	+	0.972952i	$0.574203\pi$
$728$		0			0
$729$	−24.7829			−0.917887
$730$		0			0
$731$	−26.8558	−	25.3573i	−0.993297	−	0.937874i
$732$		0			0
$733$	13.5840			0.501735			0.250868	−	0.968021i	$-0.419284\pi$
$733$	13.5840			0.501735			0.250868	+	0.968021i	$0.419284\pi$
$734$		0			0
$735$	11.9334			0.440171
$736$		0			0
$737$		−	7.38520i		−	0.272038i
$738$		0			0
$739$	37.9653			1.39658			0.698289	−	0.715816i	$-0.253945\pi$
$739$	37.9653			1.39658			0.698289	+	0.715816i	$0.253945\pi$
$740$		0			0
$741$			71.5081i			2.62692i
$742$		0			0
$743$		−	10.6464i		−	0.390579i	−0.980746	−	0.195290i	$-0.937435\pi$
$743$		−	10.6464i		−	0.390579i	0.980746	−	0.195290i	$-0.0625647\pi$
$744$		0			0
$745$			1.04083i			0.0381330i
$746$		0			0
$747$	1.46566			0.0536259
$748$		0			0
$749$	11.1050			0.405768
$750$		0			0
$751$		−	16.1764i		−	0.590287i	−0.955453	−	0.295143i	$-0.904633\pi$
$751$		−	16.1764i		−	0.590287i	0.955453	−	0.295143i	$-0.0953675\pi$
$752$		0			0
$753$			14.7873i			0.538880i
$754$		0			0
$755$		−	9.22432i		−	0.335707i
$756$		0			0
$757$	50.8517			1.84824			0.924118	−	0.382108i	$-0.124802\pi$
$757$	50.8517			1.84824			0.924118	+	0.382108i	$0.124802\pi$
$758$		0			0
$759$		−	2.77620i		−	0.100770i
$760$		0			0
$761$	−0.0134063			−0.000485977			−0.000242989	−	1.00000i	$-0.500077\pi$
$761$	−0.0134063			−0.000485977			−0.000242989		1.00000i	$0.500077\pi$
$762$		0			0
$763$	−10.4757			−0.379245
$764$		0			0
$765$	0.734131	−	0.777514i	0.0265426	−	0.0281111i
$766$		0			0
$767$	−6.78189			−0.244880
$768$		0			0
$769$	29.0410			1.04725			0.523623	−	0.851950i	$-0.324580\pi$
$769$	29.0410			1.04725			0.523623	+	0.851950i	$0.324580\pi$
$770$		0			0
$771$		−	30.4909i		−	1.09810i
$772$		0			0
$773$	21.8675			0.786519			0.393260	−	0.919428i	$-0.371347\pi$
$773$	21.8675			0.786519			0.393260	+	0.919428i	$0.371347\pi$
$774$		0			0
$775$			17.0970i			0.614141i
$776$		0			0
$777$			11.5089i			0.412880i
$778$		0			0
$779$			36.7616i			1.31712i
$780$		0			0
$781$	−5.20723			−0.186329
$782$		0			0
$783$	44.6719			1.59645
$784$		0			0
$785$			25.0942i			0.895652i
$786$		0			0
$787$			26.1478i			0.932068i	0.884767	+	0.466034i	$0.154318\pi$
$787$			26.1478i			0.932068i	−0.884767	+	0.466034i	$0.845682\pi$
$788$		0			0
$789$		−	5.39692i		−	0.192136i
$790$		0			0
$791$	13.0149			0.462756
$792$		0			0
$793$		−	3.02036i		−	0.107256i
$794$		0			0
$795$	25.4480			0.902547
$796$		0			0
$797$	−34.3693			−1.21742			−0.608711	−	0.793392i	$-0.708313\pi$
$797$	−34.3693			−1.21742			−0.608711	+	0.793392i	$0.708313\pi$
$798$		0			0
$799$	−29.3667	−	27.7281i	−1.03892	−	0.980951i
$800$		0			0
$801$	0.612443			0.0216396
$802$		0			0
$803$	7.63456			0.269418
$804$		0			0
$805$			3.96032i			0.139583i
$806$		0			0
$807$	−34.7862			−1.22453
$808$		0			0
$809$		−	39.2875i		−	1.38128i	−0.723200	−	0.690638i	$-0.757330\pi$
$809$		−	39.2875i		−	1.38128i	0.723200	−	0.690638i	$-0.242670\pi$
$810$		0			0
$811$		−	23.7166i		−	0.832804i	−0.909181	−	0.416402i	$-0.863291\pi$
$811$		−	23.7166i		−	0.832804i	0.909181	−	0.416402i	$-0.136709\pi$
$812$		0			0
$813$			3.03479i			0.106435i
$814$		0			0
$815$	−5.55764			−0.194676
$816$		0			0
$817$	−52.6679			−1.84262
$818$		0			0
$819$			1.76686i			0.0617391i
$820$		0			0
$821$		−	5.08368i		−	0.177422i	−0.996057	−	0.0887109i	$-0.971725\pi$
$821$		−	5.08368i		−	0.177422i	0.996057	−	0.0887109i	$-0.0282747\pi$
$822$		0			0
$823$			11.1226i			0.387710i	0.981030	+	0.193855i	$0.0620991\pi$
$823$			11.1226i			0.387710i	−0.981030	+	0.193855i	$0.937901\pi$
$824$		0			0
$825$	−3.60687			−0.125575
$826$		0			0
$827$		−	51.5725i		−	1.79335i	−0.442687	−	0.896676i	$-0.645975\pi$
$827$		−	51.5725i		−	1.79335i	0.442687	−	0.896676i	$-0.354025\pi$
$828$		0			0
$829$	10.5832			0.367570			0.183785	−	0.982967i	$-0.441165\pi$
$829$	10.5832			0.367570			0.183785	+	0.982967i	$0.441165\pi$
$830$		0			0
$831$	−34.2083			−1.18667
$832$		0			0
$833$	−16.6353	−	15.7071i	−0.576381	−	0.544220i
$834$		0			0
$835$	−19.1781			−0.663686
$836$		0			0
$837$	−23.9621			−0.828250
$838$		0			0
$839$		−	5.85068i		−	0.201988i	−0.994887	−	0.100994i	$-0.967798\pi$
$839$		−	5.85068i		−	0.201988i	0.994887	−	0.100994i	$-0.0322022\pi$
$840$		0			0
$841$	−51.0690			−1.76100
$842$		0			0
$843$		−	38.8092i		−	1.33666i
$844$		0			0
$845$		−	39.5650i		−	1.36108i
$846$		0			0
$847$			12.8664i			0.442095i
$848$		0			0
$849$	−49.5388			−1.70017
$850$		0			0
$851$	14.6062			0.500695
$852$		0			0
$853$		−	27.1396i		−	0.929241i	−0.885510	−	0.464620i	$-0.846191\pi$
$853$		−	27.1396i		−	0.929241i	0.885510	−	0.464620i	$-0.153809\pi$
$854$		0			0
$855$		−	1.52481i		−	0.0521475i
$856$		0			0
$857$			37.4357i			1.27878i	0.768883	+	0.639389i	$0.220813\pi$
$857$			37.4357i			1.27878i	−0.768883	+	0.639389i	$0.779187\pi$
$858$		0			0
$859$	−24.7401			−0.844120			−0.422060	−	0.906568i	$-0.638693\pi$
$859$	−24.7401			−0.844120			−0.422060	+	0.906568i	$0.638693\pi$
$860$		0			0
$861$			13.5077i			0.460341i
$862$		0			0
$863$	−48.3614			−1.64624			−0.823120	−	0.567867i	$-0.807769\pi$
$863$	−48.3614			−1.64624			−0.823120	+	0.567867i	$0.807769\pi$
$864$		0			0
$865$	−9.32971			−0.317220
$866$		0			0
$867$	−30.4376	+	1.74852i	−1.03371	+	0.0593828i
$868$		0			0
$869$	8.82918			0.299509
$870$		0			0
$871$	88.7069			3.00572
$872$		0			0
$873$			2.80865i			0.0950585i
$874$		0			0
$875$	12.3677			0.418105
$876$		0			0
$877$			8.35495i			0.282127i	0.990001	+	0.141063i	$0.0450521\pi$
$877$			8.35495i			0.282127i	−0.990001	+	0.141063i	$0.954948\pi$
$878$		0			0
$879$		−	27.5167i		−	0.928115i
$880$		0			0
$881$			32.8888i			1.10805i	0.832499	+	0.554026i	$0.186909\pi$
$881$			32.8888i			1.10805i	−0.832499	+	0.554026i	$0.813091\pi$
$882$		0			0
$883$	−51.0878			−1.71924			−0.859620	−	0.510933i	$-0.829300\pi$
$883$	−51.0878			−1.71924			−0.859620	+	0.510933i	$0.829300\pi$
$884$		0			0
$885$	2.15056			0.0722904
$886$		0			0
$887$			52.4200i			1.76009i	0.474889	+	0.880046i	$0.342488\pi$
$887$			52.4200i			1.76009i	−0.474889	+	0.880046i	$0.657512\pi$
$888$		0			0
$889$		−	3.50105i		−	0.117421i
$890$		0			0
$891$		−	5.42152i		−	0.181628i
$892$		0			0
$893$	−57.5922			−1.92725
$894$		0			0
$895$			14.7792i			0.494015i
$896$		0			0
$897$	33.3462			1.11340
$898$		0			0
$899$	42.9491			1.43243
$900$		0			0
$901$	−35.4748	−	33.4954i	−1.18184	−	1.11590i
$902$		0			0
$903$	−19.3523			−0.644005
$904$		0			0
$905$	22.8871			0.760794
$906$		0			0
$907$		−	5.76160i		−	0.191311i	−0.995415	−	0.0956555i	$-0.969505\pi$
$907$		−	5.76160i		−	0.191311i	0.995415	−	0.0956555i	$-0.0304947\pi$
$908$		0			0
$909$	−3.39940			−0.112751
$910$		0			0
$911$			49.7166i			1.64719i	0.567182	+	0.823593i	$0.308034\pi$
$911$			49.7166i			1.64719i	−0.567182	+	0.823593i	$0.691966\pi$
$912$		0			0
$913$		−	3.82628i		−	0.126631i
$914$		0			0
$915$			0.957766i			0.0316628i
$916$		0			0
$917$	−16.6893			−0.551130
$918$		0			0
$919$	10.4763			0.345581			0.172791	−	0.984959i	$-0.444722\pi$
$919$	10.4763			0.345581			0.172791	+	0.984959i	$0.444722\pi$
$920$		0			0
$921$			5.44696i			0.179483i
$922$		0			0
$923$		−	62.5463i		−	2.05874i
$924$		0			0
$925$		−	18.9765i		−	0.623945i
$926$		0			0
$927$	4.08757			0.134253
$928$		0			0
$929$			16.4773i			0.540602i	0.962776	+	0.270301i	$0.0871232\pi$
$929$			16.4773i			0.540602i	−0.962776	+	0.270301i	$0.912877\pi$
$930$		0			0
$931$	−32.6242			−1.06922
$932$		0			0
$933$	12.6522			0.414213
$934$		0			0
$935$	−2.02979	−	1.91653i	−0.0663811	−	0.0626773i
$936$		0			0
$937$	−16.0491			−0.524300			−0.262150	−	0.965027i	$-0.584431\pi$
$937$	−16.0491			−0.524300			−0.262150	+	0.965027i	$0.584431\pi$
$938$		0			0
$939$	−10.9463			−0.357220
$940$		0			0
$941$			41.1241i			1.34061i	0.742087	+	0.670304i	$0.233836\pi$
$941$			41.1241i			1.34061i	−0.742087	+	0.670304i	$0.766164\pi$
$942$		0			0
$943$	17.1429			0.558251
$944$		0			0
$945$			7.21133i			0.234584i
$946$		0			0
$947$		−	14.0892i		−	0.457839i	−0.973445	−	0.228919i	$-0.926481\pi$
$947$		−	14.0892i		−	0.457839i	0.973445	−	0.228919i	$-0.0735192\pi$
$948$		0			0
$949$			91.7020i			2.97677i
$950$		0			0
$951$	−9.62159			−0.312002
$952$		0			0
$953$	52.2989			1.69413			0.847064	−	0.531491i	$-0.178368\pi$
$953$	52.2989			1.69413			0.847064	+	0.531491i	$0.178368\pi$
$954$		0			0
$955$		−	11.4794i		−	0.371465i
$956$		0			0
$957$			9.06078i			0.292893i
$958$		0			0
$959$			7.16651i			0.231419i
$960$		0			0
$961$	7.96210			0.256842
$962$		0			0
$963$			1.99386i			0.0642513i
$964$		0			0
$965$	−4.69071			−0.150999
$966$		0			0
$967$	−33.3205			−1.07152			−0.535758	−	0.844372i	$-0.679974\pi$
$967$	−33.3205			−1.07152			−0.535758	+	0.844372i	$0.679974\pi$
$968$		0			0
$969$	−29.8462	+	31.6099i	−0.958797	+	1.01546i
$970$		0			0
$971$	22.5734			0.724415			0.362207	−	0.932098i	$-0.382023\pi$
$971$	22.5734			0.724415			0.362207	+	0.932098i	$0.382023\pi$
$972$		0			0
$973$	19.3527			0.620420
$974$		0			0
$975$		−	43.3237i		−	1.38747i
$976$		0			0
$977$	−26.8609			−0.859356			−0.429678	−	0.902982i	$-0.641373\pi$
$977$	−26.8609			−0.859356			−0.429678	+	0.902982i	$0.641373\pi$
$978$		0			0
$979$		−	1.59885i		−	0.0510995i
$980$		0			0
$981$		−	1.88087i		−	0.0600515i
$982$		0			0
$983$			1.28886i			0.0411082i	0.999789	+	0.0205541i	$0.00654303\pi$
$983$			1.28886i			0.0411082i	−0.999789	+	0.0205541i	$0.993457\pi$
$984$		0			0
$985$	−5.37816			−0.171362
$986$		0			0
$987$	−21.1617			−0.673584
$988$		0			0
$989$			24.5605i			0.780978i
$990$		0			0
$991$			38.1800i			1.21283i	0.795149	+	0.606414i	$0.207392\pi$
$991$			38.1800i			1.21283i	−0.795149	+	0.606414i	$0.792608\pi$
$992$		0			0
$993$		−	12.8011i		−	0.406229i
$994$		0			0
$995$	−18.2092			−0.577270
$996$		0			0
$997$			34.1554i			1.08171i	0.841115	+	0.540856i	$0.181900\pi$
$997$			34.1554i			1.08171i	−0.841115	+	0.540856i	$0.818100\pi$
$998$		0			0
$999$	26.5964			0.841472

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.36	yes	46
17.16	even	2	inner	4012.2.b.b.237.11	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.11	✓	46	17.16	even	2	inner
4012.2.b.b.237.36	yes	46	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.79340i q^{3} -1.19915i q^{5} +1.20458i q^{7} -0.216279 q^{9} +O(q^{10})\)	\(q+1.79340i q^{3} -1.19915i q^{5} +1.20458i q^{7} -0.216279 q^{9} +0.564620i q^{11} -6.78189 q^{13} +2.15056 q^{15} +(-2.99791 - 2.83064i) q^{17} -5.87932 q^{19} -2.16030 q^{21} +2.74169i q^{23} +3.56203 q^{25} +4.99232i q^{27} -8.94813i q^{29} +4.79978i q^{31} -1.01259 q^{33} +1.44448 q^{35} -5.32745i q^{37} -12.1626i q^{39} -6.25269i q^{41} +8.95816 q^{43} +0.259352i q^{45} +9.79572 q^{47} +5.54898 q^{49} +(5.07646 - 5.37645i) q^{51} +11.8332 q^{53} +0.677067 q^{55} -10.5440i q^{57} +1.00000 q^{59} +0.445356i q^{61} -0.260526i q^{63} +8.13254i q^{65} -13.0800 q^{67} -4.91694 q^{69} +9.22255i q^{71} -13.5216i q^{73} +6.38814i q^{75} -0.680133 q^{77} -15.6374i q^{79} -9.60206 q^{81} -6.77674 q^{83} +(-3.39437 + 3.59496i) q^{85} +16.0476 q^{87} -2.83173 q^{89} -8.16937i q^{91} -8.60792 q^{93} +7.05022i q^{95} -12.9863i q^{97} -0.122115i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 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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